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A dissertation, dated September 2003 and originally written and defended in Spanish, filed in the particle physics folder as a posted arXiv copy. It studies dynamical chiral symmetry breaking in QED3 with Schwinger-Dyson equations, covering Ward-Green-Takahashi identities, Landau-Khalatnikov-Fradkin transformations, gauge dependence of the mass and condensate, and a perturbation-guided construction of the vertex for massive fermions.

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arXiv:hep-th/0404138 v1 20 Apr 2004Gauge Invariance and Construction of the Fermion-Boson Vertex in QED3 Alfredo Raya Monta˜ no September, 2003 2 Preface The study of phisics at the level of the fundamental constitu ents of the universe has represented a challenge for the human mind. The clasiﬁca tion of such objects by their properties is very helpful for understanding the in teractions among them. One of these properties, their mass, remains as one of t he most intriguing, since, up to date, there is no theory that can explain its orig in. The manner in which we understand the interactions at the fundamental l evel is by the exchange of mediators in the Standar Model (SM) of Particle P hysics. All the processes are then described in terms of Feynman diagrams, a nd in order to compare with the experimental results, one must calculate t he amplitudes for a given event, adding the contributions of all the possible w ays in which the event can take place. It is true that this model satisfactori ly reproduces the dynamics of the particles through the symmetries of the math ematical objects that describe such particles, but it assumes that they are ma ssless. On the other hand, in the diﬀerent high energy laboratories around the wo rld, masses for the fundamentel constituents of the universe have been measure d, in some cases very accurately. This reﬂects the fact of the breaking of the symmetry of the SM. In order to conciliate this discrepancy between theory a nd experiment, the quest for a self consistent theoretical model and its experi mental corroboration that explains the origin of the masses of the fundamental blo cks of the universe has begun. The most famous as well as popular of these models is the so-ca lled Higgs mechanism, which explains that the mass of the particles eme rges by their in- teraction with the Higgs boson, a phenomenom that involves t he Spontaneous Breaking of the Chiral Symmetry in the SM. The search for this boson in particle accelerators in the US and in Europe has impulsed the technol ogical develop- ment, due to the technical diﬃculties and low budgets to prod uce and detect it; and the theoretical advance, which oﬀers simpler channels t owards its discovery. At the same time, it has promoted the increase of the number of students and scientists dedicated to this branch of physics. In spite of t he eﬁciency on which the SM describes the phenomenology of the fundamental parti cles, there are still some problems. The Higgs boson has not been discovered experimentally yet. The SM as- sumes that this boson is fundamental, but it does not exist a f undamental scalar in nature, at least, it has not been observed yet. On the other hand, when we try to extend the SM to include some other forces by construct ing a Grand 3 4 Uniﬁcation Theory, we face the hierarchy problem, and in ord er to solve it, one must perform an unnatural ﬁne tunning of the Higgs parameter s. Supersymme- try or SUSY takes care of this problem at the expense of doubli ng the spectrum of particles. Fundamental scalars are still there and SUSY k eeps on eluding us. As an alternative, models have been propossed in which the ex sistence of a scalar particle is no longer necessary for the symmetry bre aking, since it is given in the presence of condensates. Some of these models ar e Technicolor and its Extended versions, Top Condensate and Top Color. Pre dictions from these models are obtained by non perturbative calculations , that even in the simplest cases are very hard to perform and they are based in s everly incorrect assumptions. For example, in the Top Condensate Models in th e context of the Schwinger-Dyson Equations (SDE)-which we shortly will talk about-, the QDC corrections are only calculated in Landau gauge. If one r epeats the same excercise in other gauges, one ﬁnds that the top quark mass de pends upon the gauge, which is physically senseless. Trying to solve this s ituation in complicated theories like QCD is a formidable problem. In this thesis we d iscuss similar topics in a simple model, QED in 2+1 dimensions. We decided to study the Dynamical Breaking of Chiral Symmetr y in the context of Schwinger-Dyson equations. These equations are an inﬁnite set of integral relations among the Green’s functions of a Quantum Field Theory, and they provide the analitic structure that such functions mus t have. Their imple- mentation is extended to the High Energy Physics as well as Nu clear Physics, therefore, the solutions for these equations are of interes t for a wide sector of physicists around the world. Solutions to the SDE are obtained by truncating the inﬁnite t ower of integral equations, and the main task for the solution of these equati ons should be to incorporate all of the gauge identities of the theory in orde r to calculate gauge parameter independent physical observables. To pursue thi s objective, analy- tical studies, up to where the complexity of the theory allow s, are performed. Such studies are either based on the use of ans¨ atze for the Gr eens function or in their spectral representation. There are also succesful attempts to describe the dynamics of the fundamental particles by discretising t he space-time, i. e., assumming a lattice where the dynamics takes place. Then , the diﬀerent processes are calculated with the unavoidable use of comput ers, which the more powerful, the more useful in this context. Even though there have been attempts to solve the DSE in QCD an d in QED in a four dimensional space-time, such studies face a diﬃ culty inheret to four-dimensional theories: Ultraviolet divergences. Add ed to this, the initial assumptions to truncate the inﬁnite tower of SDE have not all owed to clearly identify the sources of the gauge dependence of physical obs ervables, nor the role which play the gauge invariance constraints in the restaura tion of the gauge in- dependence of these quantities, like the Ward Identities fo r QED of the Slavnov- Taylor Identities for QCD, or some others, like the Landau-K halatnikov-Fradkin transformations in QED, which describe the precise manner i n which the Green’s function vary under a gauge transformation. Therefore, we decided to focus or research in Quantum Electr odynamics on 5 a plane, or QED3, where, besides of enjoying of the beneﬁts wh ich theories of phenomena occurring on planar surfaces oﬀer, we can ident ify exactly the role of the initial assuptions made to solve the SDE and the ga uge invariance constraints, since QED3 lacks of ultraviolet divergences. The special features of the electromagnetic dynamics on a pl ane are stu- died in Chapter 1, while the derivation of the SDE as well as th e Ward-Green- Takahashi Identities (WGTI) and the Landau-Khalatnikov-F radkin (LKF) trans- formations is carried out in Chapter 2. For a better understa nding of the phe- nomenom of the Dynamical Generation of Fermion Masses, in Ch apter 3 we solve the SDE for the Fermion Propagator with the only assupt ion that the fermions interact among them in the simplest known way, that is, assuming that the vertex of the intaraction is only the bare one. This a llows us to identify the sources of the gauge dependence for the two relevant phys ical observables, the Euclidean mass and the Chiral Condensate. This scenario also allows us to study the role of the WGTI in the restauration of the gauge ind ependence of the above mentioned physical observables. The conclusion t hat the imposition of the WGTI is a necesary, but not suﬃcient condition to guara ntee the gauge independence of the physical observables is translated as t he necessity for the incorporation of other gauge invariance constraints. The L KF transformations are the next ingredient to consider, but their implementati on is not as simple as the WGTI. In Chapter 4 we describe the manner to implement t hese trans- formations as a requirement that the solutions for the SDE mu st fulﬁll. Being then only left with the initial assumption about the vertex, in Chapter 5 we advocated our attention to remove it by constructing the ver tex for the electro- magnetic interaction making use of Perturbation Theory as a guide. The main reason for this is that in Perturbation Theory, gauge identi ties (WGTI and LKF transformations) and the gauge independence of physical ob servables are satis- ﬁed order by order. Then if our initial assumption for the fer mion-boson vertex reduces to its Feynman expansion in the weak coupling regime , we stand our best chance to obtain the correct gauge behavior in the nonpe rturbative regime. Our construction is the most ambitious of its kind, because, besides considering the gauge invariance constraints of QED3, it deals with mass ive fermions in the interaction. Our vertex provides the ﬁrst insight of the nonperturbative interaction, because when we write an eﬀective Ward identit y for the transverse part of the vertex (i.e., a relation between the transverse v ertex and the fermion propagator), we are left with an interaction explicitly ind ependend of the elec- tromagnetic coupling. Consequences of this construction a s well as the possible paths for the implementation of this vertex in similar studi es are discussed in Chapter 6, where besides we oﬀer our conclusions. We keep the hope that the technique we developed can be applied in future to alternati ve models to the SM which help us to discover the origin of masses for the funda mental particles, where more reliable calculations are required with improve d vertex ans¨ atze. 6 From this work there have been published the following paper s : 1.Constructing the fermion-boson vertex in three-dimension al QED. A. Bashir y A. Raya . Phys. Rev. D64105001 (2001). 2.Gauge dependence of mass and the condensate in chirally asym metric phase of QED3. A. Bashir, A. Huet y A. Raya . Phys. Rev. D66025029 (2002). 3.Landau-Khalatnikov-Fradkin transformation and the fermi on propagator in Quantum Electrodynamics .A.Bashir y A. Raya . Phys. Rev. D66 105005 (2002). This dissertation was originally written and defended in sp anish. Contents 1 Quantum Electrodynamics on a Plane 9 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Dirac Equation and the Lagrangian . . . . . . . . . . . . . . . . . 13 1.3.1 2×2 Representation . . . . . . . . . . . . . . . . . . . . . 13 1.3.2 4×4 Representation . . . . . . . . . . . . . . . . . . . . . 16 1.4 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Schwinger-Dyson Equations and Gauge Invariance Constrai nts 23 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Electromagnetic Action . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Schwinger-Dyson Equations (SDE) . . . . . . . . . . . . . . . . . 25 2.3.1 SDE for the Photon Propagator . . . . . . . . . . . . . . 26 2.3.2 SDE for the Fermion Propagator . . . . . . . . . . . . . . 30 2.3.3 SDE for the Fermion-Boson Vertex . . . . . . . . . . . . . 31 2.3.4 Solving the SDE . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 Ward-Green-Takahashi Identity . . . . . . . . . . . . . . . . . . 35 2.5 Landau-Khalatnikov-Fradkin Transformations . . . . . . . . . . . 37 3 Gauge Dependence of Physical Observables 43 3.1 The Fermion Propagator . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Eﬀect of the Wavefunction Renormalization . . . . . . . . . . . . 45 3.3 Eﬀect of the Ward-Green-Takahashi Identity . . . . . . . . . . . 48 3.4 Dimensional Regularization Method . . . . . . . . . . . . . . . . 5 1 3.5 Towards the Full Vertex . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.1 Curtis-Penington Vertex . . . . . . . . . . . . . . . . . . . 56 3.5.2 Burden-Roberts Vertex . . . . . . . . . . . . . . . . . . . 57 3.5.3 Dong-Munczek-Roberts Vertex . . . . . . . . . . . . . . . 58 3.5.4 Burden-Tjiang Vertex . . . . . . . . . . . . . . . . . . . . 59 3.5.5 Bashir-Pennington Vertex . . . . . . . . . . . . . . . . . . 60 7 8 CONTENTS 4 Landau-Khalatnikov-Fradkin Transformations and the Fer mion Propagator 65 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Fermion Propagator and the LKF Transformation . . . . . . . . 67 4.3 Three Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . 68 4.4 Four Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4.1 Case α= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4.2 Case m>>p . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.3 Case of Weak Coupling . . . . . . . . . . . . . . . . . . . 73 5 Constructing the Vertex 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 Longitudinal and Transverse Vertex to One-Loop . . . . . . . . . 78 5.2.1 The Fermion Propagator . . . . . . . . . . . . . . . . . . . 78 5.2.2 Longitudinal Vertex to One Loop . . . . . . . . . . . . . . 79 5.2.3 Full Vertex to One Loop . . . . . . . . . . . . . . . . . . . 79 5.2.4 Transverse Vertex to One loop . . . . . . . . . . . . . . . 102 5.3 Non-perturbative Form of the Vertex (Eﬀective Transver se WTI) 107 5.3.1 On the Gauge Parameter Dependence of the Vertex . . . 107 5.3.2 Non-perturbative Vertex (Eﬀective Transverse WTI) . . . 109 5.4 Comments on the d−dimensional Case . . . . . . . . . . . . . . . 111 5.4.1 Fermion Propagator . . . . . . . . . . . . . . . . . . . . . 111 5.4.2 The Transverse Vertex . . . . . . . . . . . . . . . . . . . . 111 5.5 Comments on the Two Loop Case . . . . . . . . . . . . . . . . . 112 5.5.1 Two-Loops Fermion Propagator . . . . . . . . . . . . . . 114 5.5.2 Two-Loops Vertex . . . . . . . . . . . . . . . . . . . . . . 115 6 Discussion and Conclusions 117 Chapter 1 Quantum Electrodynamics on a Plane 1.1 Introduction Even though we live in a four-dimensional space-time –three of its dimensions being spatial and one temporal–, many theories are also inte resting when they are formulated in other dimensions. Phenomena like the Dyna mical Breaking of Chiral Symmetry in Quantum Field Theories are more easily understood when such theories restric their dynamics to less than four d imensions. Three- dimensional models are very useful to understand phenomena that happen on planar surfaces, where precisely one counts with two spatia l dimensions and one temporal. They are also helpful to study the high tempera ture behavior of four-dimensional theories, since these theories have as in ﬁnite-temperature limit their three-dimensional counterparts [1, 2]. In such cases , a three-dimensional theory is interesting not only as an academic model, but beca use they have a direct physical interpretation. QED3 is one of such models. This is the subject of our interest in this thesis and we start from an introducti on to it in this chapter. In QED3, the phenomenom of Dynamical Mass Generation has bee n studied as an alternative mechanism to that of Higgs in order to expla in how funda- mental particles acquire such property (an excelent review can be found in [3]). Although the Higgs mechanism oﬀers a solution to this questi on, it is partial in the sense that it relates particles to their masses, but it sa ys nothing about the value of these masses. Besides, if the Higgs boson is detecte d in experiments at LEP and/or Tevatron, it can take a while before determinin g whether it is fundamental or not. The theory we study has a coupling e2with dimensions of mass, which allows the dynamically generated mass to be related to this couplin g, and it is no longer necessary to introduce by hand a cut-oﬀ mass scale for this model, since e2deﬁnes this scale. Even more, this theory is superrenormali zable, and lacks 9 10 Quantum Electrodynamics on a Plane of ultraviolet divergences, one of the most serious obstacl es to study physical phenomena in four-dimensional theories. The order of diver gence of QED3 is [4] : w= 3−f−1 2b−1 2n, (1.1) wherefis the number of external fermions in a diagram, bthe number of external photons and nthe perturbative order of the diagram. It can be shown that the Green’s functions become ﬁnite in the ultraviolet r egion. However, it does not mean that the theory is free from divergences. In o ur case, the problem is that there are infrared divergent integrals, at l east in the massless theory. That is why one is forced to start from certain assump tions that remove these divergences to solve the Schwinger-Dyson Equations ( SDE). Therefore, we have a very interesting model, from which we ca n learn pretty much about the Dynamical Generation of Masses and the analyt ic structure of the Fermion Propagator in general [5], but which is mathemat ically more easily treatable than in four dimensions. Results we obtained can s erve as a guide to more complicated theories, like Grand Uniﬁcation Theories . This theory, besides, posseses a direct physical relevance . QED3 with a dinamically generated mass has been propossed as a model for two-dimensional superconductivity, specially for the discovery of superco nducting quasiplanar oxides at high temperature, like La2CuO 4yYBa 2Cu6[6]. There are other models that describe bidemensional superconductivity at h igh temperature, like anyon models, but they are problematic, because they lead to parity violation, which is not observed in nature. There are also lattice studies on the structure of the Fermio n and Photon Propagators which allow us to understand the Dynamical Gene ration of Masses. In this kind of studies, the search for the relevant critical exponents is more easily carried out that in analytical studies in the continu um. Also, there are studies of QED3 at ﬁnite temperature, due to the potentially predictive power of this model to explain early universe phenomena, and it is a lso considered as the ﬁrst step towards the comprehension of the hadronic stru cture, since in its unquenched version, i. e., considering vacuum polarizatio n eﬀects, the theory exhibits conﬁnement [7]. This can be seen in a heuristical ma nner by considering the classical potential1 V(x)≡/integraldisplay∞ −∞dx0/integraldisplayd3q (2π)3ei(q·x+q0x0)e2∆T(q) =/integraldisplayd2q (2π)2eiq·xe2∆T(q), (1.2) where ∆T(q) =1 q2[1 + Π(q)](1.3) 1Although the arguments of Green’s functions are the momenta squared, in order to sim- plify the notation we will consider that G=G(p). Quantum Electrodynamics on a Plane 11 is related to the transverse part of the photon propagator as : ∆T µν(q) =/parenleftbigg −gµν+qµqν q2/parenrightbigg∆T(q) q2. (1.4) Setting Π(q) = 0, one obtains V(x) =e2 2πlne2x, (1.5) which is a logarithmically conﬁnement potential. e2is obviously the electro- magnetic coupling. In this chapter we will show the dynamics of the electromagne tically interac- ting objects on a plane. We start by explicitly showing the th ree-dimensional equivalent to the Lorentz Group. Next we study the Dirac equa tion explicitly, using 2×2 and 4×4 representations for the Dirac matrices. Then, we study the chiral symmetry, that we will be considering throughout the next chapters, and the discrete symmetries of QED3. In the end, we shall discuss why we prefer to study QED3 in its representation 4 ×4 in this thesis. 1.2 Lorentz Group Lorentz Group can be represented by 3 ×3 matrices, and there are 9 parameters to classify them. However, the relation Λµ αgµνΛν β=gαβ (1.6) ﬁxes 6 of these parameters, leaving only three, correspondi ng to two boosts, each along the corresponding axis, and one rotation. Explicit representation for the boosts are : B1(γ) = γ γv 0 γv γ 0 0 0 1 , (1.7) B2(γ) = γ0γv 0 1 0 γv0γ . Obviouslyγ=√ 1−v2. To obtain the inﬁnitesimal generators, we perform the parametrization γ= coshψ. (1.8) Then, B1(ψ) = coshψsinhψ0 sinhψcoshψ0 0 0 1 , (1.9) B2(ψ) = coshψ0 sinhψ 0 1 0 sinhψ0 coshψ . 12 Quantum Electrodynamics on a Plane These inﬁnitesimal generators satisfy : Ki=−idBi dψ/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=0, (1.10) which explicitly correspond to the matrices K1= 0−i0 −i0 0 0 0 0 , (1.11) K2= 0 0−i 0 0 0 −i0 0 . A rotation on the plane is described by the matrix R(θ) = 1 0 0 0 cosθsinθ 0−sinθcosθ . (1.12) This matrix has the inﬁnitesimal generator J=−idR dθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle θ=0= 0 0 0 0 0−i 0i0 . (1.13) Now, these three generators are in one-to-one corresponden ce with the parame- ters of the Lorentz Group. They obey the following commutati on relations : [K1,K2] =iJ [J,K1] =−iK2 [J,K2] = +iK1. (1.14) As it is well known (see, for instance, [8]), pure Lorentz tra nsformations do not for a group. Let us observe that Jis a Hermitian matrix, but neither K1nor K2are. However, with the deﬁnitions A1=iK1, A 2=−iK2A3=J, (1.15) relations (1.14) can be re-written as : [Ai,Aj] =iǫijkAk, , (1.16) where ǫijk=  +1 if (ijk) is an even permutation of (123) −1 if (ijk) is an odd permutation of (123) 0 else. Therefore, we conclude that the Lorentz Group on a plane is is omorphic to the SU(2) group. Owing to the reduced number of degrees of freedom o n a plane, Dirac equation can also be studied in its matrix representat ion of dimensions less than 4. We discuss this matter in the next section. Quantum Electrodynamics on a Plane 13 1.3 Dirac Equation and the Lagrangian In order to solve the Dirac Equation, let us recall that Paul A drien Maurice’s idea (see, for instance [9]) was precisely to write down a Ham iltonian with linear dependence on ∂0, and consequently on ∂i, in such a way that its most general form was Hψ= (α·P+βm)ψ, (1.17) with the requirement that it must fulﬁll the relativistic re lation H2ψ= (P2+m2)ψ. (1.18) Dirac found that αandβare no numbers, but matrices with the following characteristics : •α1,α2yα3commute with each other. •α2 1=α2 2=α2 3=β2= 1. Dirac equation can be written then as : (i∝ne}ationslash∂−m)ψ= 0, (1.19) with∝ne}ationslash∂=γµ∂µ, where γµ≡(β,βα). (1.20) These matrices satisfy the Cliﬀord’s algebra : {γµ,γν}= 2gµν. (1.21) Once the representation for the γmatrices has been chosen, one can proceed to solve the Dirac equation. 1.3.1 2×2Representation In QED3, only three γmatrices are necessary to write down the Dirac Equation. The problem is that these matrices should be trated in a speci al manner when the number of dimensions of the space-time is other than four . In the three- dimensional case, a basis for the Cliﬀord’s algebra is given by the monimials [10] : 1, γ0, γ1, γ2, γ0γ1, γ0γ2, γ1γ2, γ0γ1γ2. (1.22) If Γirepresents any element of this set of monomials, it follows t hat the matrix γ=γ0γ1γ2satisﬁes [γ,Γi] = 0, (1.23) Then, by Schur’s lemma, γ=cI , (1.24) wherecis a constant. We also have that (γ)2=−1, (1.25) 14 Quantum Electrodynamics on a Plane therefore, c=±i. (1.26) We can see that we have two inequivalent representations for the Cliﬀord’s algebra, depending upon the choice of the two signs for γ. In the case of 2 ×2 matrices, Pauli’s σmatrices represent the Dirac matrices. We start with the metric gµν= 1 0 0 0−1 0 0 0−1  (1.27) and proceed to identify [4] : γ0=σ3, γ1=iσ1, γ2=iσ2. (1.28) The (anti-)commutation relations among these matrices are then : {γµ,γν}= 2gµν, (1.29) γµγν=gµν−iǫµναγα. (1.30) These relations imply γµγµ= 3, (1.31) γµ∝ne}ationslashpγµ=−∝ne}ationslashp, (1.32) ǫµναǫβγα=δβ µδγ ν−δγ µδβ ν, (1.33) ǫαβγǫαβγ = 3, (1.34) and we have the traceology : Tr[γµ] = 0, (1.35) Tr[γµγν] = 2gµν, (1.36) Tr[γµγνγρ] =−2iǫµνρ, (1.37) Tr[γµγνγργσ] = 2 (gµνgρσ−gµρgνσ+gµσgνρ). (1.38) Let us notice that the Lorentz indices run from 0 to 2. In our ca se γ0=β=σ3, γ1=βα1=iσ1, γ2=βα2=iσ2. (1.39) To solve the equation, let us proceed in the standard way, tha t is, let us consider ﬁrst the particle at rest. The corresponding equation reads [iγ0∂0−m]ψ(t) = 0,/bracketleftbigg i/parenleftbigg1 0 0−1/parenrightbigg ∂0−m/parenleftbigg1 0 0 1/parenrightbigg/bracketrightbigg/parenleftbiggψA(t) ψB(t)/parenrightbigg = 0, or,/parenleftigg ∂ψA(t) ∂t −∂ψB(t) ∂t/parenrightigg =−im/parenleftbigg ψA(t) ψB(t)/parenrightbigg . (1.40) Quantum Electrodynamics on a Plane 15 So, we ﬁnd the solutions : ψA(t) =e−imtψA(0), ψB(t) =eimtψB(0), (1.41) such that the wavefunction can be written as : ψ(t) =/parenleftbigge−imtψA(0) eimtψB(0)/parenrightbigg , (1.42) with ψ(0) =/parenleftbiggψA(0) ψB(0)/parenrightbigg . (1.43) If we look for independent solutions of the form ψ1(0) =/parenleftbigg1 0/parenrightbigg , ψ 2(0) =/parenleftbigg0 1/parenrightbigg , (1.44) we ﬁnd that the solutions for a particle at rest are : ψ1(t) =/parenleftbigg 1 0/parenrightbigg e−imt, E > 0, (1.45) ψ2(t) =/parenleftbigg0 1/parenrightbigg eimt, E < 0. For a moving particle, we look for solutions of the form ψ(x) =e−ix·pu(p). (1.46) Let us observe that ∂µψ(x) =∂µ[e−ix·p]u(p) =−ipµe−ix·pu(p) =−ipµψ(x), (1.47) therefore, we can write the Dirac equation as follows : (γµpµ−m)u(p) = 0. (1.48) Since γµpµ=γ0p0−/vector γ·/vector p =/parenleftbiggE−(p2+ip1) p2−ip1−E/parenrightbigg , (1.49) in matrix form we have /parenleftbigg E−m−(p2+ip1) p2−ip1−E−m/parenrightbigg/parenleftbigg uA(p) uB(p)/parenrightbigg = 0, (1.50) which leads us to the following system of equations for the sp inor componets : (E−m)uA(p)−(p2+ip1)uB(p) = 0, (p2−ip1)uA(p)−(E+m)uB(p) = 0. (1.51) 16 Quantum Electrodynamics on a Plane To solve this system, we choose uA(p) = 1, such that uB(p) =/bracketleftbiggp2−ip1 E+m/bracketrightbigg , (1.52) ⇒ψ1(x) =/parenleftigg 1 p2−ip1 E+m/parenrightigg e−ip·x. This is the positive energy solution. If now we choose uB(p) = 1, uA(p) =/bracketleftbiggp2+ip1 E−m/bracketrightbigg , (1.53) ⇒ψ2(x) =/parenleftigg p2+ip1 E−m 1/parenrightigg e−ip·x, which is the negative energy counterpart. 1.3.2 4×4Representation In this case, we choose the Weyl or chiral representation for theγmatrices [9] : γ0=/parenleftbiggσ30 0−σ3/parenrightbigg , γ1=/parenleftbiggiσ10 0−iσ1/parenrightbigg , γ2=/parenleftbiggiσ20 0−iσ2/parenrightbigg . (1.54) To solve the Dirac equation, once again we start with the equa tion for the particle at rest, which reads /parenleftbigg/parenleftbiggiσ30 0−iσ3/parenrightbigg ∂0−m/parenleftbigg1 0 0 1/parenrightbigg/parenrightbigg/parenleftbiggψA(t) ψB(t)/parenrightbigg = 0,(1.55) /parenleftbiggiσ3∂ ∂t−mc 0 0−iσ3∂ ∂t−mc/parenrightbigg/parenleftbiggψA(t) ψB(t)/parenrightbigg = 0, and that leads su to the following system of equations for the spinors : iσ3∂ψA ∂t=mψA(t) (1.56) iσ3∂ψB ∂t=−mψB(t). If we decompose the spinors into their components in the form : ψA(t) =/parenleftbiggψA1(t) ψA2(t)/parenrightbigg , ψB(t) =/parenleftbiggψB1(t) ψB2(t)/parenrightbigg , (1.57) we obtain now the following system of equations : i∂ψA1(t) ∂t=−mψA1(t), (1.58) Quantum Electrodynamics on a Plane 17 i∂ψA2(t) ∂t=mψA2(t), i∂ψB1(t) ∂t=mψB1(t), i∂ψB2(t) ∂t=−mψB2(t), whose solutions are ψA1(t) = e−imtψA1(0), (1.59) ψA2(t) = eimtψA2(0), ψB1(t) = eimtψB1(0), ψB2(t) = e−imtψB2(0). Then, the wavefunction ψ(t) can be written as ψ(t) = e−imtψA1(0) eimtψA2(0) eimtψB1(0) e−imtψB2(0) . (1.60) We can, as we did before, choose independent solutions such t hat the particle at rest can be described as ψ1(t) = 1 0 0 0 e−imt, ψ 2(t) = 0 1 0 0 eimt, (1.61) ψ3(t) = 0 0 1 0 eimt, ψ 4(t) = 0 0 0 1 e−imt. When the particle is moving, the Dirac equation is expressed as : /parenleftbiggσ3−iσ1p1−iσ2p2−m 0 0−σ3+iσ1p1+iσ2p2−m/parenrightbigg/parenleftbigguA(p) uB(p)/parenrightbigg = 0, (1.62) from where we obtain the system of equations [σ3−iσ1p1−iσ2p2−m]uA(p) = 0, (1.63) [−σ3+iσ1p1+iσ2p2−m]uB(p) = 0. Once more, if we decompose the spinors into their components uA(p) =/parenleftbigguA1(p) uA2(p)/parenrightbigg , uB(p) =/parenleftbigguB1(p) uB2(p)/parenrightbigg , (1.64) 18 Quantum Electrodynamics on a Plane we arrive to the system of equations : (E−m)uA1(p)−(p2+ip1)uA2(p) = 0, (1.65) (p2−ip1)uA1(p)−(E+m)uA2(p) = 0, −(E+m)uB1(p)−(p2+ip1)uB2(p) = 0, −(p2−ip1)uB1(p)−(E−m)uB2(p) = 0. To solve such system, we take advantage of the consequences o f choosing a particular normalization for the spinors components, as di splayed below : •uA1(p) = 1⇒uA2(p) =p2−ip1 E+m, •uA2(p) = 1⇒uA1(p) =p2+ip1 E−m, •uB1(p) = 1⇒uB2(p) =p2−ip1 E−m, •uB2(p) = 1⇒uB1(p) =p2+ip1 E+m. Therefore, we have the independent solutions u1(p) = 1 p2−ip1 E+m 0 0 , u 2(p) = p2−ip1 E+m 1 0 0 , (1.66) u3(p) = 0 0 1 p2−ip1 E−m , u 4(p) = 0 0 p2+ip1 E+m 1 . Since the solutions are decoupled, we can merge the correspo nding solutions for positive and negative energy as uP(p) = 1 p2−ip1 E+m p2+ip E+m 1 (1.67) uN(p) = p2+ip1 E−m 1 1 p2−ip E−m , where we understand uP(p) =/parenleftbiggu1 P u2 P/parenrightbigg , uN(p) =/parenleftbiggu1 N u2 N/parenrightbigg . (1.68) In this way we represent the solutions to the Dirac equation. Quantum Electrodynamics on a Plane 19 1.4 Discrete Symmetries We are now in position to discuss the discrete symmetries of D irac equation. For that purpose, we will follow the work of [13], where we use only the 2×2 representation for the Dirac matrices. We should ﬁrst intro duce two Dirac ﬁelds ψandφwhich satisfy, respectively, (iγα∂α−γ∂2−m)ψ(x) = 0, (1.69) (−iγα∂α+γ∂2−m)φ(x) = 0, where the index αruns from 0 to 1 and γ=γ0γ1γ2. In order to study the charge conjugation operation C, we must write the Dirac equation when the ﬁeldψinteracts with an external magnetic ﬁeld : {iγα∂α−γ∂2−e(γαAα+iγA2)−m}ψ(x) = 0. (1.70) It follows that, under charge conjugation, {iγα∂α−γ∂2+e(γαAα+iγA2)−m}ψC(x) = 0. (1.71) On the other hand, we have {iγα∂α−γ∂2+e(γαAα+iγA2)−m}C¯ψT(x) = 0, (1.72) whereCin the charge conjugation matrix and Tdenotes the matrix transpose operation. Comparing these equation, we obtain that ψC(x) =C¯ψT(x), (1.73) which is the usual charge conjugation operation. ParityPin three dimensions corresponds to the inversion of one axis , sayx, because the inversion of both the axis can be obtained from a r otation ofπof the plane. Dirac equation modiﬁed by a Parity transformatio n then reads : (iγ0∂0−iγ1∂1−γ∂2−m)ψP(t,−x,y) = 0. (1.74) For the ﬁeld φ, we rewrite the corresponding Dirac equation without perfo rming the Parity operation as follows : (iγ0∂0+iγ1∂1+γ∂2−m)φ(t,x,y) = 0. (1.75) Multiplying by 1 = γ0γ0those terms with γ1andγ, our equation then reads : (iγ0∂0+iγ1γ0γ0∂1+γγ0γ0∂2−m)φ(t,x,y) = 0, (1.76) (iγ0∂0−iγ0γ1γ0∂1−γ0γγ0∂2−m)φ(t,x,y) = 0, (iγ0γ0∂0−iγ0γ1∂1−γ0γ∂2−mγ0)γ0φ(t,x,y) = 0, (iγ0∂0−iγ1∂1−γ∂2−m)γ0φ(t,x,y) = 0, (1.77) 20 Quantum Electrodynamics on a Plane such that, under a comparison, ψP(t,−x,y) =γ0φ(t,x,y). (1.78) It is convenient at this stage to take a closed look at the mass terms of QED3. If we write the four-dimensional spinors as ψ=/parenleftbiggψ1 ψ2/parenrightbigg , (1.79) a Parity transformation acts on them in the following way : ψ1→σ1ψ2, ψ 2→σ1ψ1. (1.80) Obviously, the ordinary mass term is invariant under this tr ansformation, m¯ψψ=m(ψ† 1ψ2+ψ† 2ψ1), (1.81) since Pauli matrices are unitary and hermitian. The other ma ss term ˜m¯ψ1 2[γ3,γ5]ψ= ˜m(ψ† 1σ3ψ1+ψ† 2σ3ψ2), (1.82) is invariant under chiral transformations (1.89), but not u nder Parity, (1.78). For the Time Reversal τ, we will need to relate somehow the ﬁelds ψandφ. Dirac equation under a Time Revarsal operation for the ﬁeld ψreads : (−iγ0∂0+iγ1∂1−γ∂2−m)ψτ(−t,/vector x) = 0. (1.83) Using the same reasoning as before, the ﬁeld φsatisﬁes : (iγ0∂0+iγ1∂1−γ∂2−m)C¯φT(t,/vector x) = 0,(1.84) (−iγγγ0∂0−iγγγ1∂1−γ∂2−m)C¯φT(t,/vector x) = 0, (iγγ0γ∂0+iγγ1γ∂1−γ∂2−m)C¯φT(t,/vector x) = 0, (iγγ0∂0+iγγ1∂1−γγ∂2+mγ)γC¯φT(t,/vector x) = 0, (−iγ0∂0−iγ1∂1+γ∂2−m)γC¯φT(t,/vector x) = 0, (−iγ0∂0−iγ1γ0γ0∂1+γγ0γ0∂2−m)γC¯φT(t,/vector x) = 0, (−iγ0∂0+iγ0γ1γ0∂1−γ0γγ0∂2−m)γC¯φT(t,/vector x) = 0, (−iγ0γ0∂0+iγ0γ1∂1−γ0γ∂2−mγ0)γ0γC¯φT(t,/vector x) = 0, (−iγ0∂0+iγ1∂1−γ∂2−m)γ0γC¯φT(t,/vector x) = 0, from where we conclude that ψτ(−t,/vector x) =γ0γC¯φT(t,/vector x). (1.85) We see that if we require the massive spin-1/2 ﬁelds to be inva riant under C, Pandτrespectively, we need to introduce the ﬁeld φwhich satisﬁes Dirac equation with the second representation for γ. It is worth to mention that if Quantum Electrodynamics on a Plane 21 originally the lagrangian does not include such ﬁeld, as in o ur case, the usual mass term breaks Pandτ, but retains CPτ. Finally, if we consider the transformation φ→φ′=iγφ, (1.86) we observe that φ′satisﬁes the same equation as φ, except for the sign for the mass term. The boost along the yaxis and the rotation around it for the ﬁeld φcorrespond to the same transformations for the ﬁeld ψ, but in the opposite direction, due to the sign of γ. On the other hand, φ′behaves as ψunder Lorentz transformations, although with the opposite sign f or the mass term. This corresponds to the fact that both the representations o f the Cliﬀord’s algebra are the same (under transformations of γ) as the representation of the Lorentz Group. Therefore, we can conclude that if we require invariance under C,Pandτ, there must exist two ﬁelds whith the opposite sign for their mass terms, and under C,Pandτ, such ﬁeld are interchanged. The invariant lagrangian under discrete transformations i s given by L=i¯ψ(γα∂α+iγ∂2)ψ+m¯ψψ+i¯φ′(γα∂α+iγ∂2)φ′−m¯φ′φ′. (1.87) 1.5 Chiral Symmetry As we mentioned before, we can choose the 2 ×2 representation for the γµ matrices as the Pauli matrices, and we then use two-dimensio nal spinors. How- ever, there is no other 2 ×2 matrix which anticommutes with the σmatrices, therefore, the massive theory does not posses a greater symm etry than the massles one [11]. This is an obstacle to deﬁne chirality. Tha t is why we use four-dimensional spinors and also those γµmatrices from the four-dimesnional space-time. The massless theory in this case is invariat und er two chiral-like tranformations : ψ→eiαγ3ψ, (1.88) ψ→eiαγ5ψ. (1.89) and, therefore, the lagrangian is invariant under a global U(2) symmetry with the generators 1,γ3,γ5,/bracketleftbig γ3, γ5/bracketrightbig . (1.90) This symmetry, however, is broken with a mass term of the form m¯ψψ. Also, a dynamically generated mass will break this symmetry, as in four-dimensions. In QED3 there is another posible mass term which is invariant under Chiral transformations, but not under Parity (as we saw in the last s ection). Such term has the form 1 2˜m¯ψ[γ3, γ5]ψ. 22 Quantum Electrodynamics on a Plane We are only considering the typical mass term, since it has be en shown, by analyzing the eﬀective potential, that solutions to the SDE with this mass term are energetically preferred [11, 12]. Another reason to do s o is the conservation of parity in QED3. As a result, the Lagrangian that we shall be considering in this thesis is the one we are familiar with in QED4, i.e., L=¯ψ(iγµ∂µ−m)ψ−1 4FµνFµν−1 2ξ(∂µAµ)2. (1.91) where the notations carry the usual meaning. These are the main features of Quantum Electrodynamics on a p lane. To study the phenomenon of Dynamical Mass Generation, the one w e are concern- ing with, we must ﬁrstly study the gauge structure of QED3, in particular we must know the Schwinger-Dyson equations in this context, an d also we must explore two of the consequences of gauge covariance of the th eory: the Ward- Green-Takahashi Identities, which relate Green’s functio ns among them, and the Landau-Khalatnikov-Fradkin transformations of these fun ctions, which realize their gauge behavior under a variation of gauge. As the Lagra ngian does not change its form, the derivation of the Schwinger-Dyson equa tions etc. does not really diﬀer from the one in 3 spatial dimensions. We take up t hese derivations in the next chapter. Chapter 2 Schwinger-Dyson Equations and Gauge Invariance Constraints 2.1 Introduction As in any other theory, the electromagnetic dynamics can be o btained, in arbi- trary dimensions, from the lagrangian and its correspondin g action S=/integraldisplay ddxL(φ(x),∂µφ(x)). (2.1) The equation of motion for the ﬁeld φis obtained after impossing the staticity condition on the action δS= 0. (2.2) From this condition, we obtain the Euler-Lagrange equation s for this ﬁeld ∂µδL δ(∂µφ)−δL δφ= 0. (2.3) In this chapter we start from the QED action. Functional deri vatives of this ac- tion lead us to the Schwinger-Dyson Equations (SDE) for the G reen’s functions. We also repeat the derivation of the Ward-Green-Takahashi I dentity (WGTI) and the Landau-Khalatnikov-Fradkin (LKF) transformation s, two of the gauge identities of QED necessary in order to ensure that solution s to the SDE in the study of the Dynamical Generation of Masses with a Non Pertur bative Vertex reproduce gauge parameter independent physical observabl es, as they must be. 23 24 Schwinger-Dyson Equations and Gauge Invariance Constrain ts 2.2 Electromagnetic Action Electromagnetic quantization can be obtained either from t he canonical formu- lation or by means of the Feynman’s Path Integral. We prefer t he later scheme, since the integral formulation allows us to neatly obtain th e SDE. Richard’s idea was that in order to know the transition amplitude betwe en two quantum states, we must sum over all possible histories in which such transition can take place, i. e., U(tf,xf←ti,xi) =/integraldisplay DxDpeiS, (2.4) whereDxandDpare the measures. We saw before that the electromagnetic dynamics is obtained from Dirac equation (1.19) (iγµ∂µ−m)ψ= 0. This equation can be obtained from the free lagrangian densi ty Lfree=¯ψ(i∂µγµ−m)ψ. (2.5) We must consider the interaction between fermions and the el ectromagnetic ﬁeld, given by the term eγµAµ. To make sure that the lagrangian density being invariant under global gauge transformations, we must repl ace the ordinary derivatives by covarian derivatives by using the minimal su bstitution principle : ∂µ→Dµ=∂µ−ieAµ. (2.6) In this way, the lagrangian density is expressed as : Lint=¯ψ(iγµDµ−m)ψ. (2.7) We still need to consider the interaction of the magnetic ﬁel d with itself. This is obtained from the term −1 4FµνFµν. (2.8) Gathering terms, QED action is S[¯ψ,ψ,Aµ] =/integraldisplay ddx N/summationdisplay f=1¯ψf(iγµDf µ−mf)ψf−1 4FµνFµν , (2.9) wherefis a ﬂavor label and Nis the number of diﬀerent fermion ﬂavors. Although we are considering bare quantities, the results we obtain are also valid for renormalizad quantities, provided this process is perf ormed properly. As for the Green’s functions, its bareness will be shown explicitl y in order to distinguish them from the corresponding complete functions. Let us turn our atention to the gererating functional Z[¯η,η,Jµ] =/integraldisplay dµ(¯ψ,ψ,A )e/parenleftbig iS[¯ψ,ψ,A µ]+i/integraltext ddx/bracketleftbig/summationtext f(¯ψfηf+¯ηfψf)+AµJµ/bracketrightbig/parenrightbig ,(2.10) Schwinger-Dyson Equations and Gauge Invariance Constrain ts 25 where ¯ηf,ηfandJµare the sources for the fermions, antifermions and photons, respectively, and where it is deﬁned, as in [3] dµ(¯ψ,ψ,A ) = ΠfD¯ψfDψfΠµDAµ. (2.11) To complete the operational deﬁnition of QED, let us note tha t the action is invariant under the abelian local transformations ψ(x)→ψλ(x) = e−ieλ(x)ψ, ¯ψ(x)→¯ψλ(x) = eieλ(x)¯ψ (2.12) Aµ(x)→Aλ µ(x) =Aµ(x)−∂µλ(x) whereλ(x) is an arbitrary scalar function and, for the time being, we h ave supressed ﬂavor labels. Under such circumstances, the gene rating functional is senseless, since for every single ﬁelds conﬁguration {¯ψ(x),ψ(x),Aµ(x)}, due to the gauge inavriance, there exists an inﬁnite number of rela ted conﬁgurations {¯ψλ(x),ψλ(x),Aλ µ(x)}, which have the same action S[¯ψ,ψ,Aµ] =S[¯ψλ,ψλ,Aλ µ]. (2.13) The Grassman integration over ¯ψandψyield the same result, independently ofλ(x), since the corresponding Jacobian is unity. Then, there is a divergence in the functional integration over the ﬁeld Aµ. The correct deﬁnition of the measure must ensure that the integration over tha gauge ﬁeld is extended only to inequivalent conﬁgurations under gauge transformation s. This problem can be solved by introducing the Fadeev-Popov d eterminant. The net eﬀect of this procedure in QED is simply to introduce a gauge ﬁxing term in the action. A commonly used choice for this term is S[¯ψ,ψ,Aµ]→Sξ[¯ψ,ψ,Aµ] =S[¯ψ,ψ,Aµ]−1 2ξ/integraldisplay ddx(∂µAµ)2, (2.14) whereξis the gauge ﬁxing parameter. 2.3 Schwinger-Dyson Equations (SDE) It is known, from some time ago, that from the ﬁeld equations f or a Quantum Field Theory, it can be derived a system of coupled integral e quations which relates the Green’s functions of such theory among them. Thi s inﬁnite tower of equations is known as the Schwinger-Dyson Equations [14] . We will use the integral functionals formulation to derive the SDE followi ng the works on [3] and [15]. We start from eq. (2.14) and the gererating functional (2.10 ). Let us take into account that the fermionic ﬁelds {¯ψ,ψ}and their sources {¯η,η}are ele- ments if the Grassman algebra, that is, all of these ﬁeld anti commute among 26 Schwinger-Dyson Equations and Gauge Invariance Constrain ts themselves; and that Aµand its source Jµarec-numbers. We also take the standard notations and conventions, where ∝ne}ationslashA=Aµγµ=gµνAµγν,{γµ,γν}= 2gµν,etc. Let us note that for an electron, the physical charge must be ephys=−e, where, by deﬁnition, e=|e|is the magnitude of the charge of the electron. 2.3.1 SDE for the Photon Propagator Let us consider the generating functional (2.10). The gener ating functional for connected Green’s functions G[¯η,η,Jµ] is given by Z[¯η,η,Jµ] =eG[¯η,η,J µ]. (2.15) To obtain the SDE corresponding to the Photon Propagator we s imply use the fact that the functional integral of a total functional deri vative vanishes with the apropriate boundary conditions. For example, 0 =/integraldisplay dµ(¯ψ,ψ,Jµ)δ δAµ(x)e/braceleftbig i/parenleftbig Sξ[¯ψ,ψ,A µ]+/integraltext ddx[¯ψfηf+¯ηfψf+AµJµ]/parenrightbig/bracerightbig =/integraldisplay dµ(¯ψ,ψ,Jµ)/braceleftbiggδSξ δAµ(x)+Jµ(x)/bracerightbigg e/braceleftbig i/parenleftbig Sξ[¯ψ,ψ,A µ]+/integraltext ddx[¯ψfηf+¯ηfψf+AµJµ]/parenrightbig/bracerightbig (2.16) =/braceleftbiggδSξ δAµ(x)/bracketleftbigg −δ iδη,δ iδ¯η,δ iδJµ/bracketrightbigg +Jµ(x)/bracerightbigg Z[¯η,η,Jµ]. Diﬀerentiating the action (2.14), we immediately obtain δSξ δAµ(x)=/bracketleftbigg ∂ρ∂ρgµν−/parenleftbigg 1−1 ξ/parenrightbigg ∂µ∂ν/bracketrightbigg Aν+/summationdisplay fef¯ψfγµψf, (2.17) from where it follows that, after we divide by Z, we can write eq. (2.17) as : /bracketleftbigg ∂ρ∂ρgµν−/parenleftbigg 1−1 ξ/parenrightbigg ∂µ∂ν/bracketrightbiggδG iδJν(x)+ /summationdisplay fef/parenleftbiggδG δηf(x)γµδG δ¯ηf(x)+δ δηf(x)/bracketleftbigg γµδG δ¯ηf(x)/bracketrightbigg/parenrightbigg =−Jµ(x).(2.18) This equation represents a compact form of the non perturbat ive equivalent to the Maxwell’s equations. This is useful for us to obtain an ex pression for the photon vacuum polarization. Now we can take the Legendre tra nsformation and introduce the generating functional for one-particle irre ducible (1PI) Green’s functions, Γ[ ¯ψ,ψ,Aµ] : G[¯η,η,Aµ]≡iΓ[¯ψ,ψ,Aµ] +i/integraldisplay ddx/bracketleftbig¯ψfηf+ ¯ηfψf+AµJµ/bracketrightbig . (2.19) Schwinger-Dyson Equations and Gauge Invariance Constrain ts 27 From the Grassman integration, it follows that Z[¯ψ,ψ,Aµ] and consequently G[¯η,η,Jµ] depend only on even powers of ¯ ηandη, which in turn implies that setting ¯η=η= 0 after taking the derivative of G(orZ), we will have nonvan- ishing results only for the same number of derivatives with r espect to ¯ηandη. Similarily, in the absence of derivatives with respect to th e fermionic ﬁelds, it can be seen that only an even number of derivatives of ZandGwith respect to Jµsurvive when we set J= 0. From the last expresion, we have Aµ(x) =δG iδJµ(x), ψf(x) =δG iδ¯ηf(x),¯ψf(x) =−δG iδηf(x),(2.20) Jµ(x) =−δΓ δAµ(x), ηf(x) =−δΓ δ¯ψf(x),¯ηf(x) =δΓ δψf(x). From here we obtain expressions for ¯ψ,ψandAµin terms of ¯ η,ηandJµand viceversa; for example, ¯ψf α(x) =¯ψf α[¯η,η,Jµ] =iδG[¯η,η,Jµ] δηf α, with the spinorial indices explicitly shown. It is now easy t o see that, setting J= 0 after we diﬀerentiate Γ, we will have nonvanishing result s only when we have the same number of derivatives of ¯ψandψ, in analogy with the case of G. Making use of the expresions (2.21), let us take a look at the f ollowing term : i/integraldisplay ddzδ2G δηf α(x)δ¯ηhγ(z)δ2Γ δψhγ(z)δ¯ψg β(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle η=¯η=ψ=¯ψ=0 =/integraldisplay ddzδψh γ(z) δηf α(x)δηg β(y) δψhγ(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle η=¯η=ψ=¯ψ=0 =δηg β(y) δηf α(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0=δαβδfgδd(x−y).(2.21) Therefore, when the fermionic sources (¯ η,η) are null, we can write eq. (2.18) as : δΓ δAµ(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0=/bracketleftbigg ∂ρ∂ρgµν−/parenleftbigg 1−1 ξ/parenrightbigg ∂µ∂ν/bracketrightbigg Aν(x) −i/summationdisplay fefTr[γµSf F(x,x,[Aµ])], (2.22) where we have identiﬁed the term Sf F(x,y,[Aµ]) =iδG δηf(y)δ¯ηf(x)=−iδG δ¯η(x)δηf(y)(2.23) as the Fermion Propagator of ﬂavor fin an extermal magnetic ﬁeld Aµ. One of the consequences of eq. (2.21) is that the inverse of this G reen’s function is 28 Schwinger-Dyson Equations and Gauge Invariance Constrain ts given by : Sf F(x,y,[Aµ])−1=δ2Γ δψf(x)δ¯ψf(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0. (2.24) Obviously, the complete Green’s function for the fermion SF(x,y) is obtained after setting Aµ= 0 in eq. (2.24). To obtain the corresponding SDE for the photon polarization tensor, we only need to act with δ/δAν(y) on eq. (2.22) and set Jµ(x) = 0. Let us see : δ2Γ δAµ(x)δAν(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle Aµ=ψ=¯ψ=0 =/bracketleftbigg ∂ρ∂ρgµν−/parenleftbigg 1−1 ξ/parenrightbigg ∂µ∂ν/bracketrightbigg δd(x−y) −i/summationdisplay fefTr γµδ δAν(y)/parenleftigg δ2Γ δψf(x)δ¯ψf(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0/parenrightigg−1 .(2.25) The right hand side of this equation can be interpreted in a be tter way by observing that : δ δAν(y)/parenleftigg δ2Γ δψf(x)δ¯ψf(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0/parenrightigg−1 =−/integraldisplay dduddw/parenleftigg δ2Γ δψf(x)δ¯ψf(w)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0/parenrightigg−1 ×δ δAν(y)δ2Γ δψf(u)δ¯ψf(w)/parenleftigg δ2Γ δψf(w)δ¯ψf(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0/parenrightigg−1 ,(2.26) an analogous result to d dx[A(x)A−1(x) =I] = 0 =dA(x) dxA−1(x) +A(x)dA−1(x) dx ⇒dA−1(x) dx=−A−1(x)dA(x) dxA−1(x),(2.27) which hold for ﬁnite-dimensional matrices. Equation (2.26 ) involves the fermion- boson Vertex of [1-PI] efΓf µ(x;y,z) =δ δAµ(x)δ2Γ δψf(x)δ¯ψf(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle 0=Aµ=ψ=¯ψ, (2.28) which should not be confused with the generating functional Γ. Similarily to the fermionic case, the second derivative of Γ with respect t oAµgenerates the inverse of the photon propagator (∆−1)µν(x,y) (left hand side of eq. (2.25)). Schwinger-Dyson Equations and Gauge Invariance Constrain ts 29 Therefore, from eqs. (2.22), (2.25) and (2.26), we obtain th e SDE for the inverse of the photon propagator : (∆−1)µν(x,y) =δ2Γ δAµ(x)δAν(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle Aµ=ψ=¯ψ=0 =/bracketleftbigg ∂ρ∂ρgµν−/parenleftbigg 1−1 ξ/parenrightbigg ∂µ∂ν/bracketrightbigg δd(x−y) + Πµν(x,y),(2.29) where we have identiﬁed the photon polarization tensor Π µν: Πµν(x,y) =i/summationdisplay f(ef)2/integraldisplay ddz1ddz2Tr[γµSf F(x,z1)Γf ν(y;z1,z2)Sf F(z2,x)]. (2.30) Making use of the traslational invariance, we can write the p hoton propagator in momentum space ∆µν(q) =−gµν+ (qµqν/(q2+iǫ)) q2+iǫ1 1 + Π(q)−ξqµqν (q2+iǫ)2, (2.31) where, as usual, we deﬁne the scalar polarization Π( q) as : Πµν(q)≡(−gµνq2+qµqν)Π(q). Let us note that Π( q) is independent of the gauge parametre ξin QED, as a result of current conservation. At the lowest order in Pertu rbation Theory, we have that Π( q) = 0, as well as Γf ν(y;z1,z2) =γνδd(y−z1)δd(y−z2) y (i∝ne}ationslash∂−mf)Sf F(x,y) =δd(x−y).(2.32) Once we factorized ef, there is no explicit ﬂavor dependence for the proper ver- tex. We have seen that from the second derivative of the gener ating functional Γ[¯ψ,ψ,Aµ] we obtain the photon and fermion propagators, and from the t hird one, the proper vertex of the fermion-boson interaction. Ge nerally, higher or- der derivatives of Γ[ ¯ψ,ψ,Aµ] yield the corresponding proper Green’s functions, where the number and type of derivatives yield the number and type of legs in the proper Green’s functions. The SDE for the Photon Propaga tor is shown in Diagram (1)1. -1 -1= - Diagram (1) : SDE for the Photon Propagator. 1Diagrams were generated with AXODRAW [16] 30 Schwinger-Dyson Equations and Gauge Invariance Constrain ts The representation of the SDE in momentum space is immediate ly obtained by taking the Fourier transform of the expression in coordinat e space, or, more easily, using the usual Feynman rules for the diagrams based in the lowest order perturbative contribution to the non perturbative quantit ies. For example, for the photon polarization tensor we obtain iΠµν(q) = (−1)/summationdisplay f(ef)2/integraldisplayddk (2π)d Tr[(iγµ)(iSf F(k))(iΓf ν(k,k+q))(iSf F(k+q))], (2.33) where the factor ( −1) arises from the fermion loop as usual. 2.3.2 SDE for the Fermion Propagator Following a similar procedure, we can derive the integral eq uation for the Fermion Propagator starting from 0 =/integraldisplay dµ(¯ψ,ψ,A )δ δ¯ψ(x)e/braceleftbig i/parenleftbig Sξ[¯ψ,ψ,A µ]+/integraltext ddx[¯ψfηf+¯ηfψf+AµJµ]/parenrightbig/bracerightbig (2.34) =/braceleftbiggδSξ δ¯ψ(x)/bracketleftbigg −δ iδη,δ iδ¯η,δ iδJ/bracketrightbigg +ηf(x)/bracerightbigg Z[¯η,η,Jµ] =/bracketleftbigg ηf(x) +/parenleftbigg i∝ne}ationslash∂−mf+efγµδ iδJµ(x)/parenrightbiggδ iδ¯η(x)/bracketrightbigg Z[¯η,η,Jµ].(2.35) The last line of this expresion in the functional non perturb ative equivalent to the Dirac equation. As before, we act with δ/δηf(y) on this expresion to obtain δd(x−y)Z[¯η,η,Jµ]/vextendsingle/vextendsingle η=¯η=0 −/parenleftbigg i∝ne}ationslash∂−mf+efγµδ iδJµ(x)/parenrightbigg Z[¯η,η,Jµ]/vextendsingle/vextendsingle/vextendsingle/vextendsingle η=¯η=0Sf F(x,y; [Aµ]) = 0,(2.36) with obvious notation. Now, using eqs. (2.15) and (2.21), we can rewrite : δd(x−y) −/parenleftbigg i∝ne}ationslash∂−mf+ef∝ne}ationslashA(x; [J]) +efγµδ iδJµ(x)/parenrightbigg Sf F(x,y; [Aµ]) = 0,(2.37) which deﬁnes the non perturbative connected two-points Gre en’s function. The electromagnetic potential vanishes in the absence of an external source, that is,Aµ(x; [J= 0]) = 0, in such a way that it is only written to exhibit the content of the remaining functional derivation for eq. (2.3 7), which can be done exploiting the identity (2.26) : δ iδJµ(x)Sf F(x,y; [Aµ]) Schwinger-Dyson Equations and Gauge Invariance Constrain ts 31 =/integraldisplay ddzδAν(z) iδJµ(x)δ δAν(z)/parenleftigg δ2Γ δψf(x)δ¯ψf(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0/parenrightigg−1 =−ef/integraldisplay ddzdduddwδAν(z) iδJµ(x)Sf F(x,u)Γν(u,w;z)Sf F(w,y) =−ef/integraldisplay ddzdduddwi∆µν(x,z)Sf F(x,u)Γν(u,w;z)Sf F(w,y),(2.38) where in the last line we take J= 0. From here, in the absence of external sources, eq. (2.37) is equivalent to the expression : δd(x−y)−(i∝ne}ationslash∂−mf)Sf F(x,y) =−i(ef)2/integraldisplay ddzdduddw∆µν(x,z)γµSf F(x,u)Γν(u,w;z)Sf F(w,y).(2.39) It is usual to write the SDE corresponding to the inverse Ferm ion Propaga- tor. Therefore, multiplying by Sf−1 F(y,y′), integrating with respect to yand relabelingy′=y: Sf−1 F(x,y)−(i∝ne}ationslash∂−mf)δd(x−y) =−i(ef)2/integraldisplay ddzddu∆µν(x,z)γµSf F(x,u)Γν(u,y;z).(2.40) The photon porpagator couples eqs. (2.40) and (2.30). In thi s way, one observes that the equations for the two-points functions couple to ea ch other, and both depend on the three-points Green’s function Γfµ. This is the ﬁrst indication of a general rule which says that the SDE for an n-points function is coupled to others of the same order or lower orders, and to functions of o rder (n+ 1). Diagram (2) shows the SDE corresponding to the Fermion Propa gator -1 -1 = - Diagram (2) : SDE for the Fermion Propagator. 2.3.3 SDE for the Fermion-Boson Vertex The corresponding equation for the three-point vertex can b e obtained in a similar fashion. For completeness, we present it in momentu m space, where it is written more concisely : iΓf µ(p′,p) =iγµ+/summationdisplay g/integraldisplayddl (2π)d(iSg F(p′+l)) ×(iΓg µ(p′+l,p+l))(iSg F(p+l))Kgf(p+l,p′+l,l).(2.41) Kis the fermion-antifermion scattering kernel. The diagram atic representation of this equation is shown in Diagram (3) : 32 Schwinger-Dyson Equations and Gauge Invariance Constrain ts = - Diagram (3) : SDE for the Vertex. Clearly, Γ µcouples to the two-point function for the fermion SF, and to the fermion-antifermion scattering amplitude M, a four-point function, which again illustrates the general rule. To solve the SDE for the Fermion Propagator, the simplest app roximation for the Vertex that has been used is the so-called rainbow or l adder approxima- tion, which consists in approximate Mby iterating the lowest order perturba- tive contribution to the kernel K, along with the substitution of the fermionic propagators by their bare couterparts, S0f F(p) = 1/[∝ne}ationslashp−mf]. This and other approximation will be discussed below, avoiding ﬂavor labe ls. 2.3.4 Solving the SDE Quenched Approximation In massless QED3 in the quenched approximation [7, 17, 18], w hich corresponds to neglect fermion-loop contributions to the vacuum polari zation, that is, to take Π(q) = 0 (2.42) in the Photon Propagator, we face infrared divergences with the ordinary Per- turbation Theory. A commonly used remedy for this situation is to soften the infrared behavior of the Photon Propagator by including fer mion-loop contri- butions to the vacuum polarization. At the lowest order for a fermion of mass m, this contribution to the Polarization Scalar is : Π(k) =α k2/bracketleftbigg 2m+k2−m2 karcsin/parenleftbiggk√ k2+ 4m2/parenrightbigg/bracketrightbigg . (2.43) In a theory with Nmassless fermions, the Polarization Scalar is then : Π(k) =˜α k, (2.44) where ˜α=Ne2 8, Schwinger-Dyson Equations and Gauge Invariance Constrain ts 33 such that the photon propagator behaves as 1 /qforq2→0, that is, the infrared divergence has been softened without altering the ultravio let properties of the propagator. The quenched approximation in the SDE for the Fermion Propag ator corres- ponds to Diagram (4) : -1 -1 = - Diagram (4) : SDE for the Fermion Propagator (quenched appro ximation) and at the one-loop level in the ordinary Perturbation Theor y there is no dis- tiction between the quenched and the unquenched approximat ions. Rainbow or Ladder Approximation In the study of the Dynamical Generation of Masses, one commo nly used ap- proximation is to set [3] Γµ(k,p) =γµ, (2.45) which is known as the rainbow or ladder approximation. If thi s approximation is added to the quenched one, the SDE for the Fermion Propagat or decouples for the corresponding equations for the Photon Propagator a nd for the Vertex. The following Diagram : -1 -1 = - Diagram (5) : SDE for the Fermion Propagator (quenched appro ximation and bare vertex) describes this conjunction of approximations. Beyond the Rainbow Approximation One of the problems with the rainbow approximation is the vio lation of gauge co- variance, particularly of the Ward-Green-Takahashi Ident ity. The correct form for the fermion-boson vertex is crucial to restore the gauge covariance of the SDE and should be such that the above mentioned identity is fu lﬁlled, among other requirements. Ball and Chiu [19] have studied the stru cture of this vertex and have propossed their now famous ansatz, which we will dis cus afterwards. The restoration of the gauge covariance for the physical obs ervables is one of the main motivations for the construction of the fermion-bo son vertex, which we will carry out in Chapter5. 34 Schwinger-Dyson Equations and Gauge Invariance Constrain ts 1/NExpanssion Ordinary Perturbation Theory in terms of the coupling seems to break down due to the infrared divergences of the Green’s functions. These divergences can be avoided making use of another expanssion parameter. We can t akeNfermion ﬂavors and expand in 1 /N[5]. It has been shown that massless QED3 is ﬁnite order by order in this approximation, and besides, the infra red behavior of the Photon Propagator softenes, as we pointed out before. There fore, we have a theory with Nmassless fermions and we take the large- Nlimit, keeping Ne2 ﬁxed in the weak coupling regime. If we write e2=8 N, (2.46) the new perturbative expanssion for the Photon Propagator i s shown in Dia- gram (6) : -1 -1= +N + O(1/N) Diagram (6) : SDE for the Fermion Propagator ( 1/Nexpanssion). and after its evaluation, it yield a Polarization Scalar for the Photon Π(q) =q, (2.47) which leads to a 1 /qbehavior for the Photon Propagator in the infrared domain. The SDE for the Fermion Propagator in this scheme is given by SF(p)−1=∝ne}ationslashp−8i N/integraldisplayd3k (2π)3γµSF(p)Γν(p,k)∆µν(p−k). (2.48) There exists a controversy on whether in the study of this equ ation, see for example Pennington et. al. [20] and Atkinson et. al. [21], DCSB takes place for arbitrary number of ﬂavours or there exists a critical nu mber of such ﬂavours separating the chirally symmetric and asymmetric phases of unquenched QED. However, we do not take up these matters in this thesis. Gauge Technique There exist a scheme to solve the SDE which diﬀers substantia lly in the method with the previously mentioned studies : The Gauge Technique [25, 26, 27, 28, 29, 30, 31, 32, 33]. This scheme, based in Minkowski space, as sumes that the elements of the SDE (Propagators and Vertices) have a spectr al representation, in term of which the SDE are reformulated and directly solved . For exam- ple, it assumes that there exists a spectral function ρψsuch that the Fermion Schwinger-Dyson Equations and Gauge Invariance Constrain ts 35 Propagator can be written as : SF(p) =/integraldisplay∞ −∞dωρψ(ω) ∝ne}ationslashp−ω, (2.49) and the fermion-boson Vertex has a similar form. In fact, an a nsatz in this scheme is SF(p)ΓGT µ(p,q)SF(q) =/integraldisplay∞ −∞dωρψ(ω)1 ∝ne}ationslashp−ωγµ1 ∝ne}ationslashq−ω, , (2.50) Inserting these expressions into the SDE, we obtain a linear equation for the spectral density. This is an important feature of the Gauge T echnique : It reduces the SDE to linear equations. Another advantage is th at the Ward- Green-Takahashi identity is automatically taken into acco unt. 2.4 Ward-Green-Takahashi Identity One of the consequences of gauge covariance is that Green’s f unctions obey certain identites which relate one of these functions to the others. Thes are called Ward-Geen-Takahashi identities (WGTI), [34, 35, 36 ], and they come out from the Becci-Rouet-Stora-Tyutin (BRST) symmetry. Th ey play a crucial role in the proof for the renormalizability of the theory. On e of them, simply known as the WGTI, relates the [1-PI] Vertex to the propagato rs, and it has been widely implemented in SDE studies based either on the Ga uge Technique, for intance,[25, 26, 27, 28, 29, 30, 32, 33], and on making an a nsatz for the fermion-boson vertex, [19, 37, 38, 39, 40, 41, 42, 43]. We derive this identity following the textbook [8], startin g from the gene- rating functional (2.10) with the action (2.14), which incl udes the gauge ﬁxing term. Let us recall that without such term (and the source ter ms), the la- grangian corresponding to this action is gauge invariant. T his makesZto be inﬁnite and spoils the search for the photon propagator. In o rder to ﬁnd a ﬁnite propagator, we are forced to introduce a gauge ﬁxing term (an d a ghost term, which in the abelian case, we can absorbed into the normaliza tion). This means that the lagrangian related to the action (2.14) is no longer gauge invariat. The physical consequences of the theory, expressed in terms of G reen’s functions, should not depend upon the gauge, in such a way that Zmust be gauge inva- riant. This is a nontrivial requirement, and leads us to a diﬀ erential equation forZ, which we will ﬁnd below. Let us take the transformations (2.13) to be inﬁnitesimal, t hat is, Aµ→Aµ+∂µλ(x) ψ→ψ−ieλ(x)ψ ¯ψ→¯ψ+ieλ(x)¯ψ. (2.51) Under these transformations, neither the gauge ﬁxing term, nor the source terms are gauge invariant, in such a way that the integrand of Zacquires a factor e/braceleftbig i/integraltext dx[−1 ξ(∂µAµ)∂ρ∂ρλ+Jµ∂µλ+ieλ(¯ηψ−¯ψη)]/bracerightbig , (2.52) 36 Schwinger-Dyson Equations and Gauge Invariance Constrain ts which, being λinﬁnitesimal, can be rewritten as 1 +i/integraldisplay dx/bracketleftbigg −1 ξ(∂µAµ)∂ρ∂ρ+∂µJµλ−ie(¯ηψ−¯ψη)/bracketrightbigg λ, (2.53) where we have integrated by parts to remove the derivative op erator from λ. Gauge invariance of Zimplies that the operator (2.53), when acting on Z, is merely the identity. Since λis an arbitrary function, this implies that /bracketleftbigg −1 ξ∂ρ∂ρ(∂µAµ) +∂µJµλ−ie(¯ηψ−¯ψη)/bracketrightbigg Z= 0. (2.54) Substituting the ﬁelds by derivatives with respect to their sources, ψ→1 iδ δ¯η,¯ψ→1 iδ δη, Aµ→1 iδ δJµ, (2.55) we ﬁnd the following functional diﬀerential equation for Z: /bracketleftbiggi ξ∂ρ∂ρ∂µδ δJµ−∂µJµ−e/parenleftbigg ¯ηδ δ¯η−ηδ δη/parenrightbigg/bracketrightbigg Z[¯η,η,Jµ] = 0. (2.56) Taking the transformation (2.15), the last expression can b e written as an equa- tion forG: i ξ∂ρ∂ρ∂µδG δJµ−∂µJµ−e/parenleftbigg ¯ηδG δ¯η−ηδG δη/parenrightbigg = 0, (2.57) whereG=G[¯η,η,Jµ]. Finally, let us turn this expression into an equation for the vertex function Γ, given by the transformation (2.19). M aking use of the identities (2.21), eq. (2.57) becomes : −1 ξ∂ρ∂ρ∂µAµ(x) +∂µδΓ δAµ(x)−ieψδΓ δψ(x)+ie¯ψδΓ δ¯ψ(x)= 0. (2.58) Now, taking the functional derivative with respect to ¯ψ(x1) andψ(y1), and setting ¯ψ=ψ=A= 0, the ﬁrst term vanishes, and therefore, −∂µ xδ3Γ δ¯ψ(x1)δψ(y1)δAµ(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯ψ=ψ=A=0=ieδ(x−x1)δ2Γ δ¯ψ(x1)δψ(y1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0 −ieδ(x−y1)δ2Γ δ¯ψ(x1)δψ(y1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0.(2.59) The left hand side of this equation is the derivative of the [1 -PI] fermion-boson vertex (2.28), and the next two terms are the inverses of the e xact Fermion Propagators (2.23). The content of eq. (2.59) becomes clear if we expres it in momentum space. For such purpose, we deﬁne the proper vert ex function Γµ(k,p,q) as /integraldisplay dxdx 1dy1ei(qx1−ky1−px)δ3Γ δ¯ψ(x1)δψ(y1)δAµ(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯ψ=ψ=A=0 =ie(2π)4δ(q−k−p)Γµ(k,p,q).(2.60) Schwinger-Dyson Equations and Gauge Invariance Constrain ts 37 On the other hand, we deﬁne the Fermion Propagator in momentu m space as : /integraldisplay dx1dy1ei(qx1−ky1)δ2Γ δ¯ψ(x1)δψ(y1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ψ=¯ψ=0= (2π)4δ(q−p)iS−1 F(p).(2.61) Therefore, multiplying eq. (2.59) by ei(qx1−ky1−px)and integrating over x,x1 andy1, we have qµΓµ(k,p,q) =S−1 F(k)−S−1 F(p). (2.62) or, in the limit k→p, ∂S−1 F ∂pµ= Γµ(p,p). (2.63) WGTI is one of the requirements for the restoration of the gau ge covariance of the physical observables that have been employed in SDE stud ies. Its imple- mentation is pretty much simple, and it has been widely used. There exist also a WGTI for the Photon Propagator, which is gi ven by the expression : qµΠµν(q) = 0. (2.64) This expression is useful, because it allows us to deﬁne the P olarization Scalar in the traditional way. The fact that Πµνis transverse, leads us to the masslessness for the photon. We no longer take into account this identity. 2.5 Landau-Khalatnikov-Fradkin Transformations In a gauge ﬁeld theory, Green’s functions transfrom in a spec iﬁc manner un- der a variation of gauge. In Quantum Electrodynamics, and in honor to Lev Davidovich and his collegues who ﬁrs obatained them, these t ransformations carry the name of Landau-Khalatnikov-Fradkin (LKF) transf ormations, [44, 45, 46]. These were also derived by Johnson and Zumino throug h functional methods,[47, 48]2. LKF transformations are non perturbative in nature, and therefore, they have the potential to play an important role to address the problems of gauge invariance which plague the strong coupli ng SDE studies. In general, the rules governning these transformation are f ar from simple. The fact that they better describe their essence in coordinate s pace, make them even more complex. As a result, these transformations have playe d a less signiﬁcant and practical role than desired in SDE studies. We display its derivation below, following the work of Zumin o [48]. We start by noticing that in Landau gauge, the Photon Propagator can b e written as ∆µ,ν(x; 0) =/bracketleftbigg gµν−∂µ∂ν ∂2/bracketrightbigg ∆c(x), (2.65) 2Fukuda, Kubo and Yokoyama have looked for a possible formali sm where renormalization constants of the wave function are in fact gauge invariant [4 9] 38 Schwinger-Dyson Equations and Gauge Invariance Constrain ts where ∆c(x) is the so-called Feynman function ∆c(x) =−δ(x) ∂2. (2.66) In an arbitrary covariant gauge, the Photon Propagator can b e parametrized by an arbitrary function ∆ din the form : ∆µν(x; ∆d) = ∆µν(x; 0) +∂µ∂ν∆d(x). (2.67) Let us recall now that the generating functional is expresse d as Z[¯η,η,Jµ] =∝an}bracketle{t0|Tei/integraltext dx(¯ηψ+¯ψη+AµJµ)|0∝an}bracketri}ht. (2.68) Just to obtain the LKF transformations, for the moment we are not assuming that ∂µJµ= 0. (2.69) Now, under a gauge transformation, Zλ[¯η,η,Jµ] =Z0/bracketleftbig ¯ηeieλ,ηe−ieλ,Jµ/bracketrightbigeie/integraltext dxJ µ∂µλ, (2.70) which can be written in a diﬀerential form with the expressio n iδZ δλ=/parenleftbigg ∂µJµ+eηδ δη−e¯ηδ δ¯η/parenrightbigg Z . (2.71) With these deﬁnitions, it is easy to verify that the generati ng functional satisﬁes the following set of diﬀerential equations : /braceleftbigg ∂σ/parenleftbigg ∂µδ iδJσ−∂σδ iδJµ/parenrightbigg +(δσ µ+aµ∂σ)/parenleftbigg ieδ iδηγσδ iδ¯η−Jσ/parenrightbigg/bracerightbigg Z= 0 (2.72) /braceleftbigg/bracketleftbigg γµ/parenleftbigg ieδ iδJµ/parenrightbigg +m/bracketrightbiggδ iδ¯η−η/bracerightbigg Z= 0 (2.73) /braceleftbigg −δ iδη/bracketleftbigg −γµ/parenleftbigg ∂µ+ieδ iδJµ/parenrightbigg +m/bracketrightbigg −¯η/bracerightbigg Z= 0 (2.74) /parenleftbigg aµδ iδJµ+λ/parenrightbigg Z= 0.(2.75) The vector operator aµis introduced ﬁrstly for the sake of consistency of the notation. It satisﬁes ∂µaµ=−1, (2.76) and a convenient choice for it deﬁnes the diﬀerent gauges as w ell. For instance, aµ=∂µ(−∂2−iǫ)−1(2.77) deﬁnes the Landau gauge, and for a Lorentz frame, characteri zed by a time-like unitary vector nµ, aµ=∂µ+nµ(n·∂) ∂2+ (n·∂)2(2.78) Schwinger-Dyson Equations and Gauge Invariance Constrain ts 39 corresponds to the Coulomb gauge. Turning our attention bac k to the gene- rating functional, if an object F, constructed as a functional derivative of the generating functional, is gauge invariant in the sense that it does not change with the choice of λ, then we have that δF δλ= 0. (2.79) In particular, if we take the Fermion Propagator as SF(x,y) =1 iZδ2Z δη(y)δ¯η(x), (2.80) we obtain from eq. (2.70) or (2.71), after setting η= ¯η= 0, Zλ[0,0,Jµ]SFλ(x,y) =e/bracketleftbig ie(λ(x)−λ(y))+i/integraltext ∂µJµλ/bracketrightbig Z0[0,0,Jµ]SF0(x,y),(2.81) or, iδ δλ(z)[Z[0,0,Jµ]SF(x,y)] = [∂µJµ(z)−eδ(x−z) +eδ(y−z)]Z[0,0,Jµ]SF(x,y).(2.82) To obtain a convenient expression for the generating functi onal which allows us to deduce the LKF transformations, let us consider ﬁrst the F ermion Propagator in an external magnetic ﬁeld Bµgiven by ˜SF[x,y;Bµ]≡˜SF(B) =δ(x−y) γµ(∂µ−ieBµ) +m. (2.83) We write then the vaccum Polarization Scalar, in obvious not ation, as : Π(B) =e−Tr(ln˜SF(B)˜S−1 F(0)). (2.84) By direct veriﬁcation we have the identity /parenleftbiggδ δBµ−eδ iδηγµδ iδ¯η/parenrightbigg [ei¯η˜SF(B)ηΠ(B)] = 0. (2.85) The generating functional can then be written in the followi ng way : Z[¯η,η,Jµ] = ei¯η˜SF/bracketleftbig δ iδJµ/bracketrightbig ηΠ/bracketleftbiggδ iδJµ/bracketrightbigg e[i 2(Jµ+aµ∂ρJρ)∆c(Jµ+aµ∂ρJρ)−i∂ρJρλ−i 2∂ρJρ∆d∂σJσ].(2.86) This expression satisﬁes eqs. (2.72) to (2.74) automatical ly, and eq.(2.75) is satisﬁed when we take ∆ d= 0. We can verify directly this sentence, except in 40 Schwinger-Dyson Equations and Gauge Invariance Constrain ts the case of eq. (2.72), where we should observe ﬁrstly that, o n acting with the operator ∂σ/parenleftbigg ∂µδ iδJσ−∂σδ iδJµ/parenrightbigg in the last exponential of (2.86), we obtain a factor Jµ+aµ∂ρJρ. (2.87) Those factors with λand ∆ddo not contibute. We must now move this factor to the left of the terms of (2.86) which contain derivatives w ith respect to Jµ. The net eﬀect of this operation is to replace Jµ→Jµ−ieδ iδηγµδ iδ¯η, (2.88) by virtue of (2.85). This veriﬁes eq. (2.72). We also take adv antage of Π/bracketleftbiggδ iδJµ/bracketrightbigg ∂ρJρ=∂ρJρΠ/bracketleftbiggδ iδJµ/bracketrightbigg (2.89) and ˜SF/bracketleftbiggδ iδJµ;x,y/bracketrightbigg ∂ρJρ(z) = [∂ρJρ(z)−eδ(x−z) +eδ(y−z)]˜SF/bracketleftbiggδ iδJµ;x,y/bracketrightbigg ,(2.90) to ﬁnd the relation Z[0,0,Jµ]SF[x,y;Jµ] =˜SF/bracketleftbiggδ iδJµ;x,y/bracketrightbigg Z[0,0,Jµ], (2.91) which involves the propagator (2.80). This expression is ve ry useful to obtain in a direct way the LKF transformation for the Fermion Propag ator. We sa- tart from considering an inﬁnitesimal transformation of th e function ∆ d. Such variation induces a transfrormation on the generating func tional given by δZ=i 2/integraldisplay/integraldisplayδ δλ(δ∆d)δ δλZ . (2.92) Now, setting η= ¯η= 0, the induced change in the generating functional without fermionic sources is δZ[0,0,Jµ] =−i 2/integraldisplay/integraldisplay ∂µJµ(δ∆d)∂ρJρZ[0,0,Jµ], (2.93) or, in ﬁnite form Z′[0,0,Jµ] =e−i 2/integraltext/integraltext ∂µJµ(δ∆d)∂ρJρZ[0,0,Jµ]. (2.94) Schwinger-Dyson Equations and Gauge Invariance Constrain ts 41 Therefore, for the Fermion Propagator we have δ(Z[0,0,Jµ]SF) =i 2/integraldisplay/integraldisplayδ δλ(δ∆d)δ δλ(Z[0,0,Jµ]SF), (2.95) which, along with eq. (2.82), implies S′ F(x,y;Jµ) = eie2[δ∆D(x−y)−δ∆d(0)]+ie/integraltext [δ∆d(x−z)−δ∆d(y−z)]∂ρJρ(z)dz SF(x,y;Jµ). (2.96) ForJ= 0 we ﬁnd S′ F(x,y; 0) =eie2[δ∆d(x−y)−δ∆d(0)]SF(x,y; 0) (2.97) Then, for a ﬁnite change of ∆ d, the transformation law for the Fermion Propa- gator reads : SF(x;ξ) =SF(x; 0)e−i[∆d(0)−∆d(x)]. (2.98) For the Vertex, we have Bµ(z;x,y|∆) =Bµ(z;x,y|0)e−i[∆d(0)−∆d(x−y)] +SF(x−y; 0)e−i[∆d(0)−∆d(x−y)]∂ ∂zµ[∆d(x−z)−∆d(z−y)],(2.99) whereBµis the non-amputated vertex, deﬁned in momentum space in ter ms of the amputated vertex Γ µas : Bµ(k,p) =SF(k)Γν(k,p)SF(p)∆µν(q). (2.100) The transformation rule for the partially amputated Vertex Λµ(k,p) =SF(k)Γµ(k,p)SF(p) (2.101) follows from eqs. (2.67), (2.98) and (2.99), and it is simply given by : Λµ(z;x,y|∆d) = Λµ(z;x,y|0)e−i[∆d(0)−∆d(x−y)]. (2.102) In the usual covariant way for gauge ﬁxing, the Photon Propag ator takes the form : ∆µν(q,ξ) =1 q2[1 + Π(q)]/parenleftbigg gµν−qµqν q2/parenrightbigg +ξqµqν q4, (2.103) which is obtained by taking ∆ din eq. (2.67) as : ∆d(x) =−iξe2µ4−d/integraldisplayddq (2π)de−iq·x q4, (2.104) wheree2is the dimensionless electromagnetic coupling, and µis the ’t Hooft mass scale in Dimensional Regularization. 42 Schwinger-Dyson Equations and Gauge Invariance Constrain ts These LKF transformations are ruled by very complex laws. Be ing written in coordinate space adds to their complexity, and they have b een less used in the context of SDE, as compared to the WGTI, particularly in t he study of the phenomenon of Dynamical Mass Generation. To have a better understanding of the role that either the WGT I and the LKF transformations play in the restoration of gauge indepe ndence for the phy- sical observables, it is necessary to ﬁrst know the phenomen on we are dealing with, and those assumptions that simplify the most the SDE, t o our knowledge, to make use of the bare vertex. This is the scenario we will dev elop in the next chapter. Chapter 3 Gauge Dependence of Physical Observables In a gauge theory, at the level of physical observables, gaug e symmetry reﬂects as the fact that they be independent of the gauge parameter. Per turbation theory respects these requirements and besides the Ward-Green-Ta kahashi identities (WGTI) and the Landau-Khalatnikov-Fradkin (LKF) transfor mations remain valid at every level of approximation. However, this has not been achieved in general in the non perturbative study of gauge ﬁeld theories through Schwinger- Dyson equations (SDE) carried out so far although signiﬁcan t progress has been made. The gauge technique of Salam, Delbourgo and later collabora- tors, [25, 26, 27, 28, 29, 30, 31], was developed to incorpora te the constraint imposed by WGTI. However, as pointed out in [33], gauge techn ique can be- come completely reliable only after incorporating transve rse Green functions with correct analytic and gauge-covariance properties. An other method widely used to explore the non-perturbative structure of the SDE is to make an ansatz for t he full fermion-boson vertex and then study the gauge de pendence of the physical observables related to the phenomenon of Dynamica l Chiral Symme- try Breaking (DCSB). This method has been quite popular in fo ur dimensional Quantum Electrodynamics (QED). For example, the vertex ansatz proposed by Curtis and Pennington, [37], has been extensively used to st udy the gauge de- pendence of the fermion propagator and the dynamical genera tion of fermion mass in Quenched QED, e.g., [38, 39, 40, 41]. Later on, in the w ork of Bashir and Pennington, [42, 43], an improved vertex which achieves complete gauge independence of the critical coupling above which mass is dy namically gener- ated is proposed. These methods use the cut-oﬀ regularizati on to study the gauge dependence of the physical observables. As the cut-oﬀ method in general does not respect gauge symmetry, a criticism of these works h as been raised recently, [50, 51, 52]. They suggest dimensional regulariz ation scheme to study t he chirally asymmetric phase of QED so that the possible gau ge dependence coming from the inappropriate regulator could be ﬁltered ou t. 43 44 Gauge Dependence of Physical Observables Three dimensional Quantum Electrodynamics (QED3) provide s us with a neat laboratory to study DCSB as it is ultraviolet well-beha ved and hence the source of gauge non-invariance ﬁnds its roots only in the sim plifying assumptions employed and notin the choice of the regulator. Burden and Roberts, [7], studied the gauge dependence of the chiral condensate in que nched QED3 and proposed a vertex which appreciably reduces this gauge depe ndence in the range 0−1 of the covariant gauge parameter ξ. Unfortunately, the choice of their vertex does not transform correctly under the operation of c harge conjugation. Moreover, the selected range of values for ξis very narrow, close to the vicinity of the Landau gauge. In this chapter, we undertake the calcul ation of the Euclidean mass of the fermion (referred to as massfrom now onwards) and the condensate for a wide range of values of ξin the bare vertex approximation in the followig schemes : •SettingF(p) = 1. •Including the equation for F(p), and •Making a partial use of the WGTI. We have not achieved a complete gauge independence neither f or the mass nor for the chiral condensate, which is a sing for the necesit y of the construction and use of the full vertex. 3.1 The Fermion Propagator In quenched QED3, the SDE for the fermion propagator in the Mi nkowski space can be written as : S−1 F(p) =S0−1 F(p)−ie2/integraldisplayd3k (2π)3Γν(k,p)SF(k)γµ∆0 µν(q),(3.1) whereq=k−p,eis the electromagnetic coupling, Γν(k,p) is the full fermion- photon vertex, S0 F(p) and ∆0 µν(q) are the bare fermion and photon propagators deﬁned as S0 F(p) = 1/∝ne}ationslashp, ∆0 µν(q) =−gµν q2+ (1−ξ)qµqν q4, (3.2) andSF(p) is the full fermion propagator, which we prefer to write in t he fol- lowing most general form : SF(p) =F(p) ∝ne}ationslashp−M(p). (3.3) F(p) is referred to as the wavefunction renormalization and M(p) as the mass function and ξis the usual covariant gauge parameter. Eq. (3.1) is a matrix equation. It consists of two independen t equations, which can be decoupled by taking its trace after multiplying it with 1 and∝ne}ationslashp, Gauge Dependence of Physical Observables 45 respectively. Making use of Eqs. (3.2,3.3) and replacing th e full vertex by its bare counterpart, these equations can be written as : 1 F(p)= 1 +α 2π2p2/integraldisplay d3kF(k) k2+M2(k)1 q4 /bracketleftbig −2(k·p)2+ (2−ξ)(k2+p2)k·p−2(1−ξ)k2p2/bracketrightbig , M(p) F(p)=α(2 +ξ) 2π2/integraldisplay d3kF(k)M(k) k2+M2(k)1 q2, (3.4) whereα=e2/(4π) as usual. Carrying out angular integration after the Wick rotation to the Euclidean space, the above equations acquir e the form : 1 F(p)= 1−αξ πp2/integraldisplay∞ 0dkk2F(k) k2+M2(k)/bracketleftbigg 1−k2+p2 2kpln/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+p k−p/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg ,(3.5) M(p) F(p)=α(ξ+ 2) πp/integraldisplay∞ 0dkkF(k)M(k) k2+M2(k)ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+p k−p/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (3.6) A trivial solution to Eq. (3.6) is M(p) = 0, which corresponds to the usual perturbative solution. We are interested in a non-trivial s olution by solving Eqs. (3.5) and (3.6) simultaneously. Such a solution for M(p) is related to the massmand the chiral condensate ∝an}bracketle{t¯ψψ∝an}bracketri}ht. Assuming a simple analytic con- tinuation from Minkowski to the Euclidean space, neglectin g the rotation of the integration contour, we deﬁne m=M(m). It is true that this is not the physical mass for the fermion, and we do not expect it to be exa ctly gauge in- variant. However, since m∼M(0), we can consider it as an eﬀective mass. At most we can expect this Euclidean mass to be approximately ga uge invariant, in the sense that it is close to the physical mass [40]. On the o ther hand, in reference [7], Burden and Roberts demonstrated that the sta ndard deﬁnition of the fermion condensate (Eq. (3.9) in [7]) in terms of an integ ral over the mass function is in excellent numerical agreement with the predi ction of the operator product expansion [53] which allows us to write ∝an}bracketle{t¯ψψ∝an}bracketri}ht= 4p2M(p)/(2 +ξ) (in units ofe4) in the limit when p2→∞. Such an expansion is valid only for values of the gauge parameter in the range [0 ,1]. As expected, we ﬁnd that in this limit,M(p) falls as 1/p2so that the condensate does not depend upon the momentum variable p. We shall study the gauge dependence of the mass and the condensate in the next section. 3.2 Eﬀect of the Wavefunction Renormalization In studying DCSB, it has been a common practice to make the app roximation F(p) = 1 so that we only have to solve Eq. (3.6). The justiﬁcation f or this approximation stems from the fact that perturbatively F(p) = 1 +O(αξ/π). If αis small and we are suﬃciently close to the Landau gauge, one w ould naturally expect that F(p)≈1. Although, it has been quite customary to employ this approximation, there exist several works which include bot h the equations. We 46 Gauge Dependence of Physical Observables 00.10.20.30.40.5 0.001 0.01 0.1 1 10 100 1000M(p) pMASS FUNCTION IN VARIOUS GAUGES (BARE VERTEX WITH F=1 AND WITHOUT WGTI)ξ=0 ξ=1 ξ=2 ξ=3 ξ=4 ξ=5 Figure 3.1: Mass Function M(p) in theF(p) = 1 approxmation. study the eﬀects of neglecting the wavefunction renormaliz ation quantitatively. Fig. (3.1) depicts the mass function M(p) forF(p) = 1 in various gauges. As expected, the mass function is roughly a constant for low val ues ofpand falls as 1/p2for large values of p. The integration region chosen is from 10−3to 103 and we select 26 points per decade. The mass probes low moment um region of this graph, whereas, the condensate is extracted from its as ymptotic behaviour. Obviou sly, the mass seems to vary in more or less equally spac ed steps with the variation of the gauge parameter. In order to obtain a quanti tative value of the mass, we select neighbouring points paandpb(pa>pb), such thatM(pa)<pa andM(pb)>pb. We then approximate the mass by the following relation : m=M(pb)−M(pa) pb−pa(m−pa) +M(pa). (3.7) As for the condensate, the ﬁgure does not distinguish betwee n the results for various gauges. Therefore, we have to look at the numbers exp licitly. Table (1) shows the value of the condensate for ξranging from 1−5. Momentum p is displayed in units of e2and the condensate in units of 10−3e4. The point p= 1000 was chosen to calculate the condensate. This number se ems suﬃciently large as the 1 /p2behaviour seems to set in much earlier ( p≈300), as noted also in [7]. In Figs. (3.2) and (3.3) we display the gauge dependen ce of the chiral condensate and the mass for F(p) = 1 in a wide range of values of the gauge parameter. The condensate varies heavily with the change of gauge, roughly twice per unit change in the value of ξ. Gauge dependence of the mass is not too diﬀerent either. Gauge Dependence of Physical Observables 47 -505101520253035 0 1 2 3 4 5Condensate ξCONDENSATE IN VARIOUS GAUGES (BARE VERTEX WITH F=1 AND WITHOUT WGTI) Figure 3.2: Condensate <¯ψψ> in theF(p) = 1 approxmation. Repeating the exercise by taking both the equations, namely Eqs. (3.5) and (3.6), into account, we see similar qualitative behaviour o f the mass function. It is roughly a constant for low values of pand falls as 1 /p2for large values of p, Fig. (3.4). The large pbehaviour is also evident from the entries in Table (2). As for the wavefunction renormalization, it also is constant f or small values of p. As pbecomes large it goes to 1, Fig. (3.5). In Table (2) we also giv e a comparison with the work of Burden and Roberts, [7]. As mentioned earlie r, they restrict themselves to the close vicinity of the Landau gauge, where o ur results are in excellent agreement. We investigate the gauge dependence o f the condensate as well as the mass far beyond the Landau gauge. A graphical desc ription can be found in Figs. (3.6) and (3.7). The following points are impo rtant to note : •The wavefunction renormalization plays an extremely impor tant role in restoring the gauge invariance of the chiral condensate as w ell as the mass of the fermion. Although the qualitative behaviour of the ma ss function in various regimes of momenta remains largely unchanged, wh ether or not we employ the approximation F(p) = 1, quantitative dependence of the physical observables mentioned above on the covariant gaug e parameter ξ reduces a great deal by including the wavefunction renormal ization. •As we move away from the Landau gauge towards large positive v alues ofξ, the gauge dependence of the condensate as well as the mass kee ps dimini- 48 Gauge Dependence of Physical Observables 00.10.20.30.40.50.6 0 1 2 3 4 5Mass ξMASS IN VARIOUS GAUGES (BARE VERTEX WITH F=1 AND WITHOUT WGTI) Figure 3.3: Mass in the F(p) = 1 approxmation. shing, without resorting to any sophisticated ansatze for the fermion-boson interaction. 3.3 Eﬀect of the Ward-Green-Takahashi Iden- tity The bare photon propagator which appears in Eq. (3.1) can be s plit up in longitudinal and transverse parts as follows : ∆0 µν(q) = ∆0T µν(q)−ξqµqν q4, (3.8) where ∆0T µν(q) =−gµν/q2+qµqν/q4. Employing this decomposition, we can rewrite Eq. (3.1) as : S−1 F(p) =S0−1 F(p)−ie2/integraldisplayd3k (2π)3Γν(k,p)SF(k)γµ∆0T µν(q) +ie2ξ/integraldisplayd3k (2π)3Γν(k,p)SF(k)γµqµqν q4. (3.9) It is well known that the use of the WGTI, in the equivalent of t he last term of Eq. (3.9) in QED4, ﬁlters out a spurious term which is an art ifact of using the gauge dependent cut-oﬀ regulator. Therefore, one is nat urally motivated to use this decomposition in dimensions other than four. Now em ploying the bare vertex ansatz Γµ(k,p) =γµ, multiplying Eq. (3.9) by 1 and ∝ne}ationslashprespectively and Gauge Dependence of Physical Observables 49 00.020.040.060.080.10.120.14 0.001 0.01 0.1 1 10 100 1000M(p) pMASS FUNCTION IN VARIOUS GAUGES (BARE VERTEX WITHOUT WGTI)ξ=0 ξ=1 ξ=2 ξ=3 ξ=4 ξ=5 Figure 3.4: Mass Function incliding the equation for the Wav efunction Renor- malization. Wick-rotating to the Euclidean space, we obtain the followi ng equations : 1 F(p)= 1 +α 2π2p2/integraldisplay d3kF(k) k2+M2(k)1 q4 /bracketleftbigg 2(q·p)(q·k)+ξ F(p)[p2(q·k)+M(k)M(p)(q·p)]/bracketrightbigg ,(3.10) M(p) F(p)=α 2π2/integraldisplay d3kF(k) k2+M2(k)1 q2 /bracketleftbigg 2M(k)−ξ q21 F(p)[M(k)(p·q)−M(p)(k·q)]/bracketrightbigg .(3.11) On carrying out angular integration, 1 F(p)= 1 +αξ πp2/integraldisplay∞ 0dkk2F(k)/F(p) k2+M2(k)/bracketleftigg p2 k2−p2+p 2kln/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+p k−p/vextendsingle/vextendsingle/vextendsingle/vextendsingle +M(k)M(p)/braceleftbigg1 k2−p2−1 2kpln/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+p k−p/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightbigg/bracketrightigg , (3.12) M(p) F(p)=α π/integraldisplay∞ 0dkk2F(k) k2+M2(k)/bracketleftigg 2M(k) kpln/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+p k−p/vextendsingle/vextendsingle/vextendsingle/vextendsingle −ξ F(p)/braceleftbiggM(k)−M(p) k2−p2−M(k) +M(p) 2kpln/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+p k−p/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightbigg/bracketrightigg .(3.13) As the terms of the type 1 /(k2−p2) are harder to deal with numerically, we 50 Gauge Dependence of Physical Observables 0.20.40.60.811.21.4 0.001 0.01 0.1 1 10 100 1000F(p) pF(p) IN VARIOUS GAUGES (BARE VERTEX WITHOUT WGTI)ξ=0 ξ=1 ξ=2 ξ=3 ξ=4 ξ=5 Figure 3.5: Wavefunction Renormalization. use the approximation F(p) = 1 to analyze the eﬀect of the WGTI. Under this simpliﬁcation, we only have to solve M(p) =α π/integraldisplay∞ 0dkk2 k2+M2(k)/bracketleftigg 2M(k) kpln/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+p k−p/vextendsingle/vextendsingle/vextendsingle/vextendsingle −ξ/braceleftbiggM(k)−M(p) k2−p2−M(k) +M(p) 2kpln/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+p k−p/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightbigg/bracketrightigg .(3.14) The mass function obtained on solving Eq. (3.14) is depicted in Fig. (3.8), which, along with Table (3), reveals that its qualitative behaviou r remains unchanged both for small and large values of p. In Figs. (3.9) and (3.10), we compare the gauge dependence of the condensate and the mass with and with out the usage of the WGTI. We ﬁnd that the gauge dependence of these quantit ies seems to increase by incorporating the said identity. Similar behav iour was observed in QED4 by Gusynin et. al. [50] in studying the gauge dependence of the critical coupling above which mass is generated. They carried out a nu merical analysis of the criticism raised by Dong et. al. [18] on the work of Atki nson et. al. [54] who did not employ the WGTI as suggested in Eq. (3.9), res ulting in the appearance of a spurious cut-oﬀ dependent term1. Gusynin et. al. found that if one employs the WGTI, the critical coupling is more steeply g auge dependent. We ﬁnd similar behaviour for the mass and the condensate in QE D3 in the approximation F(p) = 1 in this section. 1To trace back the origin of this error, consult the footnote o n page 7680 of the referenc e [42] Gauge Dependence of Physical Observables 51 -505101520253035 0 1 2 3 4 5Condensate ξCONDENSATE IN VARIOUS GAUGES (BARE VERTEX WITHOUT WGTI)F=1 including F Figure 3.6: Condensate: Comparison between the cases with a nd without the usage ofF(p) = 1. 3.4 Dimensional Regularization Method In this section we compare our numerical results with those o btained by em- ploying the dimensional regularization scheme, [50, 51, 52 ]. For simplicity, we restrict ourselves only to the Landau gauge without incorpo rating the WGTI. In this case, the equation for the mass function acquires the following form in Euclidean space in arbitrary dimensions : M(p) = 4πα(d−1)/integraldisplayddk (2π)dM(k) k2+M2(k)1 q2, (3.15) whereαis a dimensionful coupling except in four dimensions. We deﬁ ned= 4−2ǫand α=αdµ2ǫ, (3.16) αdbeing dimensionless. We now use the volume element ddk=kd−1dkdΩd, wheredΩdis thed-dimensional solid angle deﬁned as dΩd=d−1/productdisplay l=1sind−1−lθldθl. The angleθd−1varies from 0 to 2 π, whereas all other angles vary from 0 to π. Choosing θ1to be the angle between kandp, we can easily carry out the remaining angular integrations to arrive at : M(p) =2(d−1)α (4π)d−1 2Γ(d−1 2)/integraldisplay∞ 0dk2kd−2M(p) k2+M2(p)/integraldisplayπ 0dθ1sind−2θ1 q2.(3.17) 52 Gauge Dependence of Physical Observables 00.10.20.30.40.50.6 0 1 2 3 4 5Mass ξMASS IN VARIOUS GAUGES (BARE VERTEX WITHOUT WGTI)F=1 including F Figure 3.7: Mass: Comparison between the cases with and with out the usage ofF(p) = 1. Using the standard formula, [55], /integraldisplayπ 0dxsin2σ−1x [1 + 2acosx+a2]λ=B(σ,1/2)2F1(λ,λ−σ+ 1/2,µ+ 1/2;a2) |a|<1, integration over θ1yields : M(p) =(3−2ǫ)α (4π)1−ǫΓ(2−ǫ)/integraldisplay∞ 0dk2(k2)1−ǫM(k) k2+M2(k)/bracketleftbigg1 k22F1/parenleftbigg 1,ǫ; 2−ǫ;p2 k2/parenrightbigg θ(k2−p2)+1 p22F1/parenleftbigg 1,ǫ; 2−ǫ;k2 p2/parenrightbigg θ(k2−p2)/bracketrightbigg . (3.18) This equation was studied in detail in [50] in four dimension s, takingǫto be a small positive number. The factor ( k2)−ǫin the numerator regulates the otherwise divergent behaviour of the integrand for large mo menta. As noted in [50], the hypergeometric function does not play any role in r egularization and hence can simply be replaced by F(1,0; 2,z) = 1. In case of three dimensions, the hypergeometric function develops a pole for k2=p2, as is obvious from the following identity : 2F1(1,ǫ; 2−ǫ; 1) =1−ǫ 1−2ǫ. (3.19) As was pointed out earlier, such terms are hard to deal with nu merically. Due to increasing computational time and memory, we go only u p toǫ= 0.48, starting from ǫ= 0.4. To obtain satisfying results, we need to use increasingly more points per decade as we approach closer to ǫ= 0.5. For instance, we use Gauge Dependence of Physical Observables 53 00.10.20.30.40.50.60.7 0.001 0.01 0.1 1 10 100 1000M(p) pMASS FUNCTION IN VARIOUS GAUGES (BARE VERTEX WITH F=1 AND WITH WGTI)ξ=0 ξ=1 ξ=2 ξ=3 ξ=4 ξ=5 Figure 3.8: Mass Function incliding the eﬀect of the WGTI for F(p) = 1. 100 points per decade for ǫ= 0.46. The problems of ever increasing computa- tional time and memory limited us to use 140 points per decade for the case ofǫ= 0.48. Despite this large number, we belive that the correspond ing result falls short of the desired accuracy. As a result, there is a sl ight rise at the end of the ﬂat region of the mass function, and the ﬁnal descent be gins rather late, Fig. (3.11). The problematic pole for ǫ= 0.5 in Eq. (3.19) corresponds to the relatively well-controlled singularity in the following e xpression 2F1/parenleftbigg 1,1 2;3 2;z2/parenrightbigg =1 2zln1 +z 1−z(3.20) forz→1. A comparison between the mass function obtained from tech niques based upon the dimensional regularization scheme and the on e computed in Section 3.3 is also depicted in Fig (3.11). Taking the numeri cal limitation for ǫ= 0.48 into account, we note that as ǫapproaches the value of 0 .5, we get closer and closer to the result obtained previously, where we work i n 3-dimensions to start with. 3.5 Towards the Full Vertex WGTI with the bare vertex is valid only if bare fermion propag ators are in- volved. When we use full propagators, physical observables calculated from the corresponding SDE exhibit gauge dependence. Temptatio n emerges then to move towards the full vertex, and the ﬁrst requirement we mus t think of is that it should satisfy the WGTI. Hence we face the construction of the Ball-Chiu vertex detailed displayed below. The full vertex satisﬁes W GTI qµΓµ(k,p) =S−1 F(k)−S−1 F(p). (3.21) 54 Gauge Dependence of Physical Observables 010203040506070 0 1 2 3 4 5Condensate ξCONDENSATE IN VARIOUS GAUGES (BARE VERTEX WITH AND WITHOUT WGTI FOR F=1)IWGT No IWGT Figure 3.9: Eﬀect of the WGTI on the gauge dependence of the co ndensate for F(p) = 1. This relation allows us to decompose the complete vertex int o its longitudinal Γµ L(k,p) and transverse Γµ T(k,p) parts : Γµ(k,p) = Γµ L(k,p) + Γµ T(k,p), (3.22) where the transverse part satisﬁes qµΓµ T(k,p) = 0 y Γµ T(p,p) = 0, (3.23) and therefore, remains undetermined by the WGTI. Following the work of Ball and Chiu, we deﬁne the longitudinal part of the vertex in term s of the Fermion Propagator. In the limit k→p, WGTI is written as : Γµ(p,p) =∂ ∂pµS−1 F(p). (3.24) Substituting the expresion for the full Fermion Propagator , we observe that Γµ(p,p) =∂ ∂pµS−1 F(p) =∂ ∂pµ∝ne}ationslashp−M(p) F(p) =γµ F(p)+ 2pµ∝ne}ationslashp∂ ∂p2−2pµ∂ ∂p2M(p) F(p). (3.25) After we make the substitutions 1 F(p)→1 2/bracketleftbigg1 F(k)+1 F(p)/bracketrightbigg , Gauge Dependence of Physical Observables 55 00.10.20.30.40.50.6 0 1 2 3 4 5Mass ξMASS IN VARIOUS GAUGES (BARE VERTEX WITH AND WITHOUT WGTI FOR F=1)IWGT No WGTI Figure 3.10: Eﬀect of the WGTI on the gauge dependence of the m ass for F(p) = 1. pµ→1 2(kµ+pµ), ∝ne}ationslashp→1 2(∝ne}ationslashk+∝ne}ationslashp), ∂ ∂p21 F(p)→1 k2−p2/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg , ∂ ∂p2M(p) F(p)→1 k2−p2/bracketleftbiggM(k) F(k)−M(p) F(p)/bracketrightbigg , (3.26) we can deﬁne the longitudanal or Ball-Chiu (BC) vertex as : Γµ BC=γµ 2/bracketleftbigg1 F(k)+1 F(p)/bracketrightbigg +1 2(∝ne}ationslashk+∝ne}ationslashp)(k+p)µ (k2−p2)/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg +(k+p)µ (k2−p2)/bracketleftbiggM(k) F(k)−M(p) F(p)/bracketrightbigg . (3.27) Without loss of generality, the transverse vertex can be exp ressed as : Γµ T(k,p) =8/summationdisplay i=1τi(k2,p2,q2)Tµ i(k,p), (3.28) with the apropriate {Tµ}basis. Functions τ1,τ4,τ5andτ7are proportional to the massm, and therefore in massless studies they do not appear. Durin g the last few years, a programme has been started towards the cons truction of vertex ans¨ atze which impose constraints on the transverse part of the vertex. Some of the most famous attempts are discussed below. 56 Gauge Dependence of Physical Observables 00.020.040.060.080.10.120.14 0.001 0.01 0.1 1 10 100 1000M(p) pMASS FUNCTION WITH D. R. (BARE VERTEX IN LANDAU GAUGE)ε=0.40 ε=0.42 ε=0.44 ε=0.46 ε=0.48 d=3 without D. R. Figure 3.11: Mass Function for various values of ǫ. Notice that ǫ= 0.5 corres- ponds to 3 dimensions. The result for ǫ= 0.48 could not be achived with the desired accuracy due to the computing time and memory increa se. 3.5.1 Curtis-Penington Vertex In four dimensions, the question of why cannot we set all of th eτ’s as zero was answered by Curtis and Pennington, arguing to Perturbation Theory and the multiplicative renormalizability properties of the Fermi on Propagator. Their vertex (CP) Γµ CP= Γµ BC+1 2/bracketleftbigg1 F(k)−1 F(p)/bracketrightbiggγµ(k2−p2)−(k+p)µ∝ne}ationslashq d(k,p), (3.29) where d(k,p) =(k2−p2)2+ [M2(k) +M2(p)]2 k2+p2, (3.30) comes out from the following assumptions : •Transverse vertex must agree with perturbative results at t he one-loop level in the relevant kinematic regime k>>p . •Fermion Propagator obtained from this vertex should be mult iplicatively renormalizable. •This Propagator should transform correctly under its LKF tr ansformation. •Vertex should have correct charge conjugation properties. •Transverse vertex should vanish in Landau gauge. •Transverse Vertex should not depend upon the angle between f ermion momenta. Gauge Dependence of Physical Observables 57 •The only contributing coeﬃcient is τ6. •Transverse vertex as well as the longitudinal one are writen without ex- plicit dependence on the gauge parameter ξ2. Transverse vertex is written in such a way that, in the massle ss case, their τ6=1 2k2+p2 (k2−p2)2/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg (3.31) is antisymmetric under the interchange of kandp. This vertex, however, does not lead to a completelly gauge in dependent critical coupling. In the massless case, it exhibits a kinem atic singularity when k2→p2. In comparison with Perturbation Theory at one-loop, CP ver tex agrees with perturbative results only in the kinematic regi me above mentioned. Multiplicative renormalizability for the Fermion Propaga tor is achieved only in the leading term of the power law for the wavefunction reno rmalization. Finally, the LKF transformation of this propagator is valid only up to the leading logarithmic term. 3.5.2 Burden-Roberts Vertex Burden and Roberts have parametrized a slight modiﬁcation t o the BC vertex in the following way [7] : Γµ BR=/bracketleftbigg a1 F(k)+ (1−a)1 F(p)/bracketrightbigg γµ +(k+p)µ((1−a)∝ne}ationslashk−a∝ne}ationslashp) k2−p2/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg −(k+p)µ k2−p2/bracketleftbiggM(k) F(k)−M(p) F(p)/bracketrightbigg , (3.32) where a=1 2+δ. (3.33) With the assumptions that the vertex should satisfy the WGTI and that it should be free from kinematical singularities to all orders in Perturbation The- ory, their parametrization allows to optimize δsuch that the most gauge inde- pendence of the chiral condensate is achieved. They report δ= 0.03 as the best value. However, we observe that the BR vertex can be written a s : Γµ BR= Γµ BC+δ/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg [k2−p2]/braceleftbig γµ(p2−k2) + (k+p)µ∝ne}ationslashq/bracerightbig = Γµ BC+δτ6Tµ 6, (3.34) 2This, as we will show, is not possible. 58 Gauge Dependence of Physical Observables with τ6= (k2−p2)/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg . (3.35) Charge conjugation symmetry for the vertex requires τ6to be antisymmetric under the interchage of kandp, but the BR vertex is symmetric. Besides, for δ= 0.03 this vertex does not agree with perturbative results, and therefore we neglect it as a good choice. 3.5.3 Dong-Munczek-Roberts Vertex Dong et. al. [18] have proposed a vertex (DMR) that satisﬁes WGTI and ensures gauge covariance of the Fermion Propagator in the ma ssless case by means of the so-called Transversality Condition : /integraldisplayddk (2π)d∆T µν(p−k)γµSF(k)Γν(k,p) = 0. (3.36) DMR vertex is constructed under the assumption that the coeﬃ cionts of the transverse part are independent of the angle between fermio n momenta. In it, the following constraints are set on the transverse coeﬃcie nts : fi= 0 for i∝ne}ationslash= 3,8 f3=1 2/parenleftbiggd 2−1/parenrightbigg f8 f8=1 d 2−1(d−1)I3 I1−I3, (3.37) dis the number of space-time dimensions and I1=k2p2I1 I2=1 2/parenleftbig (k2+p2)I1−1/parenrightbig I3=1 2[k2+p2]I2, (3.38) where they deﬁne In=/integraldisplay dΩd1 (k−p)2n, (3.39) being the solid angle /integraldisplay dΩd≡1 N/bracketleftbigg/integraldisplayπ 0dθ2sind−2θ2/integraldisplayπ 0dθ3sind−3θ3.../integraldisplay2π 0dθd−1/bracketrightbigg ,(3.40) and N=2πd/2 Γ(d/2). (3.41) Gauge Dependence of Physical Observables 59 Functionsfiare related to the τiin the form : τi=1 k2−p2/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg fi. (3.42) In this construction it is shown that the eﬀect of including o nlyτ6as in the CP vertex can also be reproduced by considering τ3andτ8. This vertex, however, exhibit logarithmic kinematic singularities. It has the co rrect charge conjugation properties and leads to a multiplicatively renormalizable Fermion Propagator. 3.5.4 Burden-Tjiang Vertex Using a similar reasoning, Burden and Tjiang [56] have decon structed a one- parameter ans¨ atze family for massless QED3. Constraints o n the transverse coeﬃcients in the BT vertex are the following : ¯f=−2(1 +β)I(k,p) J(k,p) f3=−βI(k,p) J(k,p) f6= 0, (3.43) where I(k,p) =(k2+p2)2 16kpln/parenleftbigg(k+p)2 (k−p)2/parenrightbigg −1 4(k2+p2) J(k,p) =(k2−p2)2 16kpln/parenleftbigg(k+p)2 (k−p)2/parenrightbigg −1 4(k2+p2), (3.44) They have used the relation (3.42) and they deﬁne ¯τ=τ8+ (k2+p2)τ2. (3.45) Choosingβ= 1 leads to the DMR vertex. This deconstruction is based upon the assumptions that the transverse vertex vanishes in Land au gauge and takes τ6= 0. Transversality Condition is the strongest argument to e nsure the gauge covariance of the Fermion Propagator. A possible drawback m entioned by the authors is that for any value of β, fork2=p2butkµ∝ne}ationslash=pµ, the BT vertex exhibits a logarithmic singularity. A criticism to ths vertex has been exposed by Bashir et. al. [57]. They as- sure thatτ6cannot be taken as zero and that the transverse coeﬃcients sh ould depend upon the angle between fermion momenta. One importan t observation is that the parameter β, introduced explicitly gauge parameter independent by the authors, should have such dependence as pointed out by th e corresponding perturbative calculation. Finally, in this work they point out that the Transver- sality Condition is not valid beyond the one-loop level. 60 Gauge Dependence of Physical Observables 3.5.5 Bashir-Pennington Vertex With the assumptions that the transverse vertex vanishes in Landau gauge and thet it has no dependence upon the angle between fermion mome nta, Bashir and Pennington [42]. have constructed a vertex that satisﬁe s : •WGTI, •that the Fermion Propagator is multiplicatively renormali zable, •that with the correct choice of the functions that deﬁne this vertex, it agrees with Perturbation Theory in the weak coupling regime is war- rantied, •oﬀers a strictly gauge independent critical coupling. BP Vertex imposes integral and diferential contraints on th e functions that deﬁne its transverse part. For the massless case, Kızılers¨ u, Bashir and Penning- ton [58] removed the initial assumptions, including all the corrections in the exponent of the power law for the wavefunction renormalizat ion. This is done by introducing new functions and constraints on them. Continuing with the programme, we should start an ambitious quest towards the construction of a vertex for the fermion-boson interact ion in massive QED3, taking into account all the nice features of the above mentio ned vertices, but getting rid of their drawbacks. It is clear that the inclusio n of the WGTI is common in all the vertex ans¨ atze. This is due to the fact that its implementation is straghtforward. Demanding correct gauge behavior of Gre en’s functions is a nontrivial requirement. We will show thos in the next chapte r, where we study the LKF transformation for the Fermion Propagator at the tre e level. Gauge Dependence of Physical Observables 61 Appendix: Tables Below we display the Tables that contain the numeric results of our calculation : ξ p4 2+ξp2M(p) 0.0 1000 2.31109 642.233 2.31117 316.228 2.31119 0.5 1000 3.61103 642.233 3.61119 316.228 3.61124 1.0 1000 5.19982 642.233 5.20009 316.228 5.20017 1.2 1000 5.91622 642.233 5.91654 316.228 5.91664 1.5 1000 7.07745 642.233 7.07787 316.228 7.07800 2.0 1000 9.24390 642.233 9.24452 316.228 9.24473ξ p4 2+ξp2M(p) 2.5 1000 11.6992 642.233 11.7001 316.228 11.7003 3.0 1000 14.4432 642.233 14.4444 316.228 14.4448 3.5 1000 17.4761 642.233 17.4777 316.228 17.4782 4.0 1000 20.7977 642.233 20.7998 316.228 20.8005 4.5 1000 24.4081 642.233 24.4108 316.228 24.4117 5.0 1000 28.3073 642.233 28.3106 316.228 28.3117 TABLE 1. Condensate in Various Gauges for F(p) = 1 62 Gauge Dependence of Physical Observables ξ p4 2+ξp2M(p)4 2+ξp2M(p) BHR BR 0.0 1000 2.31109 2.316 642.233 2.31117 316.228 2.31119 0.5 1000 1.77309 1.775 642.233 1.77313 316.228 1.77306 1.0 1000 1.44791 1.447 642.233 1.44793 316.228 1.44780 1.2 1000 1.35288 1.352 642.233 1.35288 316.228 1.35274 1.5 1000 1.23591 642.233 1.23591 316.228 1.23574 2.0 1000 1.09014 642.233 1.09012 316.228 1.08992 Gauge Dependence of Physical Observables 63 ξ p4 2+ξp2M(p)4 2+ξp2M(p) BHR BR 2.5 1000 0.98597 642.233 0.98594 316.228 0.98571 3.0 1000 0.90941 642.233 0.90938 316.228 0.90912 3.5 1000 0.85201 642.233 0.85197 316.228 0.85169 4.0 1000 0.80839 642.233 0.80833 316.228 0.80803 4.5 1000 0.77497 642.233 0.77491 316.228 0.77458 5.0 1000 0.74933 642.233 0.74927 316.228 0.74891 TABLE 2. Condensate In Various Gauges Including F(p) 64 Gauge Dependence of Physical Observables ξ p4 2+ξp2M(p) 0.0 1000 2.31109 642.233 2.31117 316.228 2.31119 0.5 1000 4.30196 642.233 4.30221 316.228 4.30248 1.0 1000 6.93473 642.233 6.93529 316.228 6.93609 1.2 1000 8.16826 642.233 8.16900 316.228 8.17011 1.5 1000 10.2122 642.233 10.2133 316.228 10.2150 2.0 1000 14.1356 642.233 14.1375 316.228 14.1406ξ p4 2+ξp2M(p) 2.5 1000 18.7057 642.233 18.7085 316.228 18.7136 3.0 1000 23.9226 642.233 23.9268 316.228 23.9345 3.5 1000 29.7865 642.233 29.7924 316.228 29.8037 4.0 1000 36.2975 642.233 36.3056 316.228 36.3212 4.5 1000 43.4556 642.233 43.4663 316.228 43.4872 5.0 1000 51.2608 642.233 51.2746 316.228 51.3020 TABLE 3. Condensate in Various Gauges for F(p) = 1 using WGTI Chapter 4 Landau-Khalatnikov- Fradkin Transformations and the Fermion Propagator 4.1 Introduction Looking for the gauge independence of the physical observab les, we found that the Ward-Green-Takahashi Identity (WGTI) is a necessary co ndition for it, but not suﬃcient. We see, therefore, the need for the incorporat ion of other gauge invariance constraints for this purpose. We incorporate th e Landa-Khalatnikov- Fradnin (LKF) transformations, [44, 45, 46, 47, 48], which t ell us the manner in which Green’s functions change under a variation of gauge . Rules governing these transformations are far from simple. In counterpart, WGTI [34, 35, 36], are more simple, and therefore, they have been widely implem ented. We can make larger this set of identities by transforming also the g auge parameter ξ[59, 60], and arrive to the Nielsen Identities (NI). One adva ntage of these identities over the conventional Ward identities is that ∂/∂ξ becomes part of the new relations involving Green’s functions. This fact wa s exploited in [61, 62] to prove the gauge independenc e of some quantities relat ed to two-point Green’s functions at the one-loop level and to all orders in P erturbation Theory, respectively. Since it is a diﬃcult task to establish tha gau ge independence of the physical observables in the Schwinger-Dyson Equations (SDE) study, NI can play a signiﬁcant role in addressing this issue, along with t he WGTI and the LKF transformations. However, in this chapter, we have focu sed only in the later. The LKF transformation for the three-point vertex is compli cated and ham- pers direct extraction of analytical restrictions on its st ructure. Burden and 65 66Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator Roberts, [7], carried out a numerical analysis to compare th e self-consistency of various ansatze for the vertex, [19, 37, 63], by means of its LKF transformati on. In addition to these numerical constraints, indirect analy tical insight can be obtained on the nonperturbative structure of the vertex by d emanding correct gauge covariance properties of the fermion propagator. In t he context of gauge technique, examples are [64, 65, 66]. Concerning the works b ased upon choosing a vertex ansatze , references [18, 37, 42, 43, 56, 58] employ this idea1. However, all the work in the later category has been carried out for mas sless QED3 and QED4. The masslessness of the fermions implies that the ferm ion propagator can be written only in terms of one function, the so called wav efunction renor- malization, F(p). In order to apply the LKF transform, one needs to know a Green function at least in one particular gauge. This is a fo rmidable task. However, one can rely on approximations based on perturbati on theory. It is customary to take F(p) = 1 in the Landau gauge, an approximation justiﬁed by one loop calculation of the massless fermion propagator i n arbitrary dimen- sions, see for example, [67]. The LKF transformation then im plies a power law forF(p) in QED4 and a simple trigonometric function in QED3. To impr ove upon these results, one can take two paths: (i) incorporate t he information contained in higher orders of perturbation theory and (ii) s tudy the massive theory. As pointed out in [58], in QED4, the power law structu re of the wave- function renormalization remains intact by increasing ord er of approximation in perturbation theory although the exponent of course gets contribution from next to leading logarithms and so on. In [58], constraint was obtained on the 3-point vertex by considering a power law where the exponent of this power law was not restricted only to the one loop fermion propagator. I n QED3, the two loop fermion propagator was evaluated in [22, 57, 68], where it was explicitly shown that the the approximation F(p) = 1 is only valid upto one loop, thus violating the transversality condition advocated in [56]. The result found there was used in [23] to ﬁnd the improved LKF transform. In the present Chapter, we calculate the LKF transformed fer mion propa- gator in massive QED3 and QED42. We start with the simplest input which corresponds to the lowest order of perturbation theory, i.e .,S(p) = 1/i∝ne}ationslashp−min the Landau gauge. On LKF transforming, we ﬁnd the fermion pro pagator in an arbitrary covariant gauge. In the case of QED3, we obtain the result in terms of basic functions of momenta. In QED4, the ﬁnal expression i s in the form of hypergeometric functions. Coupling αenters as parameter of this transcenden- tal function. A comparison with perturbation theory needs t he expansion of the hypergeometric function in terms of its parameters. We use t he technique deve- loped by Moch et. al. , [69], for the said expansion. We compare our results with the one loop expansion of the fermion propagator in QED4 and Q ED3, [70, 71], and ﬁnd perfect agreement upto terms independent of the gaug e parameter at one loop, a diﬀerence permitted by the structure of the LKF tr ansformations. We believe that the incorporation of LKF transformations, a long with WGTI, 1A criticism of the vertex construction in [56] was raised in [ 57]. 2In the context of gauge technique, gauge covariance of the sp ectral functions in QED was studied in [64, 65, 66]. Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator 67 in the SDE can play a key role in addressing the problems of gau ge invariance. For example, in the study of the SDE of the fermion propagator , only those assumptions should be permissible which keep intact the cor rect behaviour of the Green functions under the LKF transformations, in addit ion to ensuring that the WGTI is satisﬁed. It makes it vital to explore how two and three point Green functions transform in a gauge covariant fashion. In t his Chapter, we consider only a two point function, namely, the fermion prop agator. 4.2 Fermion Propagator and the LKF Transfor- mation We start by expanding out the fermion propagator, in momentu m and coordi- nate spaces respectively, in its most general form as follow s : SF(p;ξ) =A(p;ξ) +iB(p;ξ) ∝ne}ationslashp≡F(p;ξ) i∝ne}ationslashp−M(p;ξ), (4.1) SF(x;ξ) =∝ne}ationslashxX(x;ξ) +Y(x;ξ), (4.2) where we explicitly write the gauge in which we specify the Fe rmion Propagator. Motivated from the lowest order perturbation theory, we tak e F(p; 0) = 1 and M(p; 0) =m. (4.3) Perturbation theory also reveals that this result continue s to hold true to one loop order for the wavefunction renormalization. Eqs. (4.1 ,4.2) are related to each other through the following Fourier transforms : SF(p;ξ) =/integraldisplay ddxeip·xSF(x;ξ) (4.4) SF(x;ξ) =/integraldisplayddp (2π)de−ip·xSF(p;ξ), (4.5) wheredis the dimension of space-time. Let us recall that the LKF tra nsforma- tion relating the coordinate space fermion propagator in La ndau gauge to the one in an arbitrary covariant gauge reads (2.98) : SF(x;ξ) =SF(x; 0)e−i[∆d(0)−∆d(x)], and, also that (2.104) ∆d(x) =−iξe2µ4−d/integraldisplayddp (2π)de−ip·x p4. Takingψto be the angle between xandp, we can write ddp=dppd−1sind−2ψdψΩd−2, 68Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator where Ωd−2= 2π(d−1)/2/Γ ((d−1)/2). Hence ∆d(x) =−iξe2µ4−df(d)/integraldisplay∞ 0dppd−5/integraldisplayπ 0dψsind−2ψe−ipxcosψ, (4.6) wheref(d) = Ωd−2/(2π)d. Performing angular and radial integrations, we arrive at the following equation ∆d(x) =−iξe2 16(π)d/2(µx)4−dΓ/parenleftbiggd 2−2/parenrightbigg . (4.7) With these tools at hand, the procedure now is as follows : •Start with the lowest order fermion propagator and Fourier t ransform it to coordinate space. •Apply the LKF transformation law. •Fourier transform the result back to momentum space. 4.3 Three Dimensional Case Employing Eqs. (4.1,4.2,4.3,4.5), the lowest order three d imensional fermion propagator in Landau gauge in the position space is given by X(x; 0) =−e−mx(1 +mx) 4πx3, (4.8) Y(x; 0) =−me−mx 4πx. Once in the coordinate space, we can apply the LKF transforma tion law using expression (4.7) explicitly in three dimensions : ∆3(x) =−iαξx 2, (4.9) whereα=e2/4π. The fermion propagator in an arbitrary gauge is then SF(x;ξ) =SF(x; 0)e−(αξ/2)x. (4.10) For Fourier transforming back to momentum space, we use A(p;ξ) =−F(p;ξ)M(p;ξ) p2+M2(p;ξ)=/integraldisplay d3xeip·xY(x;ξ) (4.11) iB(p;ξ) =−ip2F(p;ξ) p2+M2(p;ξ)=/integraldisplay d3xp·xeip·xX(x;ξ). Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator 69 Performing the angular integration, we get A(p;ξ) =−m p/integraldisplay∞ 0dxsinpxe−(m+αξ/2)x, (4.12) B(p;ξ) =1 p/integraldisplay∞ 0dx x2(1 +mx) [pxcospx−sinpx]e−(m+αξ/2)x,(4.13) and the radial integration then yields A(p;ξ) =−4m 4p2+ (2m+αξ)2(4.14) B(p;ξ) =−4p2+αξ(2m+αξ) 4p2+ (2m+αξ)2+αξ 2parctan[2p/(2m+αξ)].(4.15) We can now arrive at the following expressions for the wavefu nction renormali- zation and the mass function, respectively : F(p;ξ) =−αξ 2parctan[2p/(2m+αξ)] +2p(4p2+α2ξ2) φ(p;ξ) −αξ(4p2+αξ(2m+αξ))arctan[2p/(2m+αξ)] φ(p;ξ),(4.16) M(p;ξ) =8p3m φ(p;ξ), (4.17) where φ(p;ξ) = 2p(4p2+αξ(2m+αξ))−αξ(4p2+(2m+αξ)2)arctan[2p/(2m+αξ)]. (4.18) In the massless limit, we immediately recuperate the well-k nown results : Fnm(p;ξ) = 1−αξ 2parctan2p αξ, (4.19) Mnm(p;ξ) = 0. In the weak coupling, we can expand out Eqs.(4.16,4.17) in po wers ofα. To O(α), we ﬁnd F(p;ξ) = 1 +αξ 2p3/bracketleftbig (m2−p2) arctan [p/m]−mp/bracketrightbig , (4.20) M(p;ξ) =m/bracketleftbigg 1 +αξ 2p3/braceleftbig (m2+p2) arctan [p/m]−mp/bracerightbig/bracketrightbigg .(4.21) Let us compare these results with the ones obtained in [71] : F1−loop(p;ξ) = 1 +αξ 2p3/bracketleftbig (m2−p2) arctan [p/m]−mp/bracketrightbig , (4.22) M1−loop(p;ξ) =m/bracketleftbigg 1+α 2p3/braceleftbig [ξ(m2+p2)+4p2]arctan[p/m]−ξmp/bracerightbig/bracketrightbigg .(4.23) 70Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator We of course only expect the results to be in agreement upto a t erm propor- tional toα, as allowed by the structure of the LKF transformations. The re is no such term in Eq. (4.22). Therefore, the agreement is exact . Eq. (4.21) and Eq. (4.23) become identical only after we subract out the non -vanishing term in the Landau gauge from Eq. (4.23) to write out the subtracted mass function at one loop as : MS 1−loop(p;ξ) =m/bracketleftbigg 1 +αξ 2p3/braceleftbig (m2+p2)arctan[p/m]−mp/bracerightbig/bracketrightbigg .(4.24) One can numerically check that without the above mentioned s ubtraction, Eqs. (4.21,4.23) approach the same value only in the large momentum regime. 4.4 Four Dimensional Case Employing Eqs. (4.1,4.2,4.3,4.5), the lowest order four di mensional fermion pro- pagator in coordinate space is given by X(x; 0) =−m2 4π2x2K2(mx), (4.25) Y(x; 0) =−m2 4π2xK1(mx), (4.26) whereK1andK2are Bessel functions of the second kind. In order to apply the LKF transformation in four dimensions, we expand Eq. (4.7) a roundd= 4−ǫ and use the following identities Γ/parenleftig −ǫ 2/parenrightig =−2 ǫ−γ+O(ǫ), xǫ= 1 +ǫlnx+O(ǫ2), to obtain ∆4(x) =iξe2 16π2−ǫ/2/bracketleftbigg2 ǫ+γ+ 2 lnµx+O(ǫ)/bracketrightbigg . (4.27) Note that we cannot write a similar expression for ∆ 4(0) because of the presence of the term proportional to ln x. Therefore, we introduce a cut-oﬀ scale xmin. Now ∆4(xmin)−∆4(x) =−iln/parenleftbiggx2 x2 min/parenrightbiggν , (4.28) whereν=αξ/4π. Hence SF(x;ξ) =SF(x; 0)/parenleftbiggx2 x2 min/parenrightbigg−ν . (4.29) For Fourier transforming back to momentum space we use the fo llowing expres- sions A(p;ξ) =−F(p;ξ)M(p;ξ) p2+M2(p;ξ)=/integraldisplay d4xeip·xY(x;ξ) (4.30) Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator 71 iB(p;ξ) =−ip2F(p;ξ) p2+M2(p;ξ)=/integraldisplay d4xp·xeip·xX(x;ξ). On carrying out angular integration, we obtain : A(p;ξ) =−m2 px2ν min/integraldisplay∞ 0dxx−2ν+1K1(mx)J1(px), (4.31) B(p;ξ) =−m2/integraldisplay∞ 0dxx−2ν+1K2(mx)J2(px). (4.32) The radial integration then yields : A(p;ξ) =−1 m/parenleftbiggm2 Λ2/parenrightbiggν Γ(1−ν)Γ(2−ν)2F1/parenleftbigg 1−ν,2−ν; 2;−p2 m2/parenrightbigg ,(4.33) B(p;ξ) =−p2 2m2/parenleftbiggm2 Λ2/parenrightbiggν Γ(1−ν)Γ(3−ν)2F1/parenleftbigg 1−ν,3−ν; 3;−p2 m2/parenrightbigg ,(4.34) where we have identiﬁed 2 /xmin→Λ. The above equations imply F(p;ξ) =Γ(1−ν) 2m2Γ(3−ν)2F1(1−ν,3−ν; 3;−p2/m2)/parenleftbiggm2 Λ2/parenrightbiggν /bracketleftigg 4m2Γ2(2−ν)2F2 1/parenleftbigg 1−ν,2−ν; 2;−p2 m2/parenrightbigg +p2Γ2(3−ν)2F2 1/parenleftbigg 1−ν,3−ν; 3;−p2 m2/parenrightbigg/bracketrightigg ,(4.35) M(p;ξ) =2m2F1/parenleftbig 1−ν,2−ν; 2;−p2/m2/parenrightbig (2−ν)2F1(1−ν,3−ν; 3;−p2/m2). (4.36) Eqs. (4.35,4.36) constitute the LKF transformation of Eqs. (4.3). We shall now see that although Eqs. (4.3) correspond to the lowest order p ropagator, their LKF transformation, Eqs. (4.35,4.36), is nonperturbative in nature and contains information of higher orders. 4.4.1 Case α= 0 Let us switch oﬀ the coupling and put α= 0 which implies ν= 0. Now using the identity 2F1(1,2; 2;−p2/m2) = 2F1(1,3; 3;−p2/m2) = (1 +p2/m2)−1,(4.37) it is easy to see that F(p;ξ) = 1 andM(p;ξ) =m, (4.38) which coincides with the lowest order perturbative result a s expected. 72Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator 4.4.2 Case m >> p In the limit m >> p , the hypergeometric functions in Eqs. (4.35,4.36) can be easily expanded in powers of p2/m2, using the identity 2F1/parenleftbigg α,β;γ;−p2 m2/parenrightbigg = 1−αβ γp2 m2+O/parenleftbiggp2 m2/parenrightbigg2 . (4.39) Retaining onlyO(p2/m2) terms, we arrive at : F(p;ξ) =Γ(1−ν)Γ(2−ν) (1−ν/2)/parenleftbiggm2 Λ2/parenrightbiggν /bracketleftigg 1+2ν 3/parenleftbigg 1−5ν 8/parenrightbiggp2 m2+O/parenleftbiggp2 m2/parenrightbigg2/bracketrightigg , M(p;ξ) =m (1−ν/2)/bracketleftigg 1 +ν 6(1−ν)p2 m2+O/parenleftbiggp2 m2/parenrightbigg2/bracketrightigg .(4.40) Now carrying out an expansion in αand substituting ν=αξ/4π, we get the followingO(α) expressions : F(p;ξ) = 1 +αξ 4π/bracketleftbigg 2γ−1 2+2p2 3m2+ lnm2 Λ2/bracketrightbigg , (4.41) M(p;ξ) =m/braceleftbigg 1 +αξ 8π/bracketleftbigg 1 +p2 3m2/bracketrightbigg/bracerightbigg . (4.42) Let us now compare these expressions against the one-loop pe rturbative evalu- ation of the massive fermion propagator, see e.g., [70] : F1−loop(p;ξ) = 1−αξ 4π/bracketleftbigg Cµǫ+/parenleftbigg 1−m2 p2/parenrightbigg (1−L)/bracketrightbigg , (4.43) M1−loop(p;ξ) =m+αm π/bracketleftbigg/parenleftbigg 1+ξ 4/parenrightbigg +3 4(Cµǫ−L)+ξ 4m2 p2(1−L)/bracketrightbigg ,(4.44) where L=/parenleftbigg 1 +m2 p2/parenrightbigg ln/parenleftbigg 1 +p2 m2/parenrightbigg , C=−2 ǫ−γ−lnπ−ln/parenleftbiggm2 µ2/parenrightbigg . Knowing the fermion propagator even in one particular gauge is a prohibitively diﬃcult task. Therefore, Eqs. (4.3) have to be viewed only as an approximation. For the wavefunction renormalization F(p; 0), this approximation is valid upto one loop order, whereas, for the mass function, it is true onl y to the lowest order. Therefore we cannot expect the LKF transform of Eqs. ( 4.3) to yield Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator 73 correctly each term in the perturbative expansion of the fer mion propagator. However, it should correctly reproduce all those terms at ev ery order of expan- sion which vanish in the Landau gauge at O(α) and beyond. Therefore, we expect Eq. (4.43) to be exactly reproduced and Eq. (4.44) to b e reproduced upto the terms which vanish in the Landau gauge at O(α). After subtracting these terms, the resulting subtracted mass function is : MS 1−loop(p;ξ) =m+αξm 4π/bracketleftbigg 1 +m2 p2(1−L)/bracketrightbigg . (4.45) In the limit m→∞, the wavefunction renormalization acquires the form F(p;ξ)1−loop= 1 +αξ 4π/bracketleftbigg −Cµǫ−1 2+2p2 3m2/bracketrightbigg , (4.46) while the subtracted mass function is MS 1−loop(p;ξ) =m/braceleftbigg 1 +αξ 8π/bracketleftbigg 1 +p2 3m2/bracketrightbigg/bracerightbigg . (4.47) The last two expressions are in perfect agreement with Eqs. ( 4.41,4.42) after we make the identiﬁcation : −Cµǫ→2γ+ lnm2 Λ2. (4.48) 4.4.3 Case of Weak Coupling The casem >> p is relatively easier to handle as we merely have to expand 2F1(β,γ;δ;x) in powers of xand retain only the leading terms. If we want to obtain a series in powers of the coupling alone, we need the ex pansion of the hypergeometric functions in terms of its parameters βandγ. We follow the technique developed in [69]. One of the mathematical object s we shall use for such an expansion are the Z-sums deﬁned as : Z(n;m1,...,mk;x1,...,xk) =/summationdisplay n≥i1>i2>...>i k>0xi1 1 im1 1...xik k imk k. (4.49) Forx1=...=xk= 1 the deﬁnition reduces to the Euler-Zagier sums, [72, 73] : Z(n;m1,...,mk; 1,...,1) =Zm1,...,m k(n). (4.50) Euler-Zagier sums can be used in the expansion of Gamma funct ions. For positive integers nwe have [69]: Γ(n+ǫ) = Γ(1 +ǫ)Γ(n)/bracketleftbig 1 +ǫZ1(n−1) +...+ǫn−1Z11...1(n−1)/bracketrightbig .(4.51) The ﬁrst sum Z1(n−1), e.g., is just the ( n−1)-th harmonic number, Hn−1, of order 1 : Z1(n−1) =n−1/summationdisplay i=11 i≡Hn−1. (4.52) 74Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator With these deﬁnitions in hand, we proceed to expand a hyperge ometric function, 2F1(1 +ε,2 +ε; 2;x), as an example, assuming |x|<1 : 2F1(1 +ε,2 +ε; 2;x) = 1 +Γ(2) Γ(1 +ε)Γ(2 +ε) ∞/summationdisplay n=1Γ(1 +ε+n)Γ(2 +ε+n) Γ(2 +n)xn n!(4.53) = 1 +1 (1 +ε)Γ2(1 +ε) ∞/summationdisplay n=1(1 +ε+n)(ε+n)2Γ2(ǫ+n) Γ(2 +n)xn n!.(4.54) Employing Eq. (4.51), we can expand the last expression in po wers ofεto any desired order of approximation. We shall be interested only in terms uptoO(α). 2F1(1 +ε,2 +ε; 2;x) = 1 +∞/summationdisplay n=1xn−ε∞/summationdisplay n=1xn+ε∞/summationdisplay n=12 + 3n n(n+ 1)xn +2ε∞/summationdisplay n=1Hn−1xn. (4.55) Performing the summations, we obtain 2F1(1 +ε,2 +ε; 2;x) =1 1−x/bracketleftbigg 1−ε/braceleftbigg 1 +1 +x xln (1−x)/bracerightbigg/bracketrightbigg .(4.56) Similarly, 2F1(1 +ε,3 +ε; 3;x) =1 1−x −ε/braceleftbigg1 x+3 21 1−x+/parenleftbigg1 +x x2+2 1−x/parenrightbigg ln (1−x)/bracerightbigg .(4.57) Substituting back into Eqs. (4.35,4.36) and identifying ε=−ν, we obtain F(p;ξ) = 1−αξ 4π/bracketleftbigg −2γ−lnm2 Λ2+/parenleftbigg 1−m2 p2/parenrightbigg (1−L)/bracketrightbigg ,(4.58) M(p;ξ) =m+αξm 4π/bracketleftbigg 1 +m2 p2(1−L)/bracketrightbigg , which matches exactly onto the one loop result of Eqs. (4.43, 4.45) after the same identiﬁcation as before, i.e., (4.48). Therefore, we h ave seen that the LKF transformation of the bare propagator contains important i nformation of higher orders in perturbation theory. LKF transformations, being non perturbative in nature, tel l us the non per- turbative way in which the Fermion Propagator [74] and the Ve rtex transform Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator 75 under a gauge variation. Most of the works on SDE violate thes e transfor- mations, with the exception of few of them which include the c orresponding transformation to the Fermion Propagator at one-loop level . Since in Pertur- bation Theory all of the gauge invariance constraints are va lid order by order, we can exploit this fact to construct a vertex ansatz such tha t in its pertur- bative expansion, it automatically satisﬁes, along with it s associated Fermion Propagator, their corresponding LKF transformation, as we will do in the next Chapter. Apendix Most of the integrals involved in this Chapter are listed bel ow for a quick refe- rence [55, 75] : /integraldisplayπ 0dψsind−2ψcosψe−ipxcosψ=−i√π/parenleftigpx 2/parenrightig1−d 2Γ/parenleftbiggd−1 2/parenrightbigg Jd 2(px),(4.59) /integraldisplay∞ 0xd/2−1Jd/2(ax) =Γ(d/2) 21−d/2ad/2. (4.60) For the three dimensional case, the needed integrals are : /integraldisplayπ 0dθsinθe−ipxcosθ=2 sinpx px, (4.61) /integraldisplayπ 0dθcosθsinθe−ipxcosθ= 2i/bracketleftbiggcospx px−sinpx (px)2/bracketrightbigg ,(4.62) /integraldisplay∞ 0dpp3 (p2+m2)/bracketleftbiggcospx px−sinpx (px)2/bracketrightbigg =−π 2(1 +mx) x2e−mx,(4.63) /integraldisplay∞ 0dppsinpx (p2+m2)=π 2e−mx, (4.64) 1 p/integraldisplay∞ 0dx x2e−ax[pxcospx−sinpx] =−1 +a parctanp a, (4.65) 1 p/integraldisplay∞ 0dx xe−ax[pxcospx−sinpx] =a a2+p2−1 parctanp a,(4.66) /integraldisplay∞ 0dxsinpxe−(m+αξ/2)x=p (m+αξ/2)2+p2.(4.67) For the four dimensional case, we used the following integra ls in particular : /integraldisplayπ 0dθsin2θe−ipxcosθ=π pxJ1(px), (4.68) 76Landau-Khalatnikov-Fradkin Transformations and the Ferm ion Propagator /integraldisplay∞ 0dppν+1Jν(px) (p2+m2)µ+1=mν−µxµ 2µΓ(µ+ 1)Kν−µ(mx), (4.69) /integraldisplay∞ 0dxx−λKµ(ax)Jν(bx) =aλ−ν−1bν 2λ+1Γ(1+ν)Γ/parenleftbiggν−λ+µ+1 2/parenrightbigg Γ/parenleftbiggν−λ−µ+1 2/parenrightbigg (4.70) × 2F1/parenleftbiggν−λ+µ+1 2,ν−λ−µ+1 2;ν+1;−b2 a2/parenrightbigg . Some of the series used in our calculation are as follows : ∞/summationdisplay n=1Hn−1xn=−xln(1−x) 1−x, (4.71) ∞/summationdisplay n=1n+ 1 n(n+ 2)xn=−2 +x 4x−(1 +x2)ln(1−x) 2x2,(4.72) ∞/summationdisplay n=11 (n+ 1)(n+ 2)xn=2−x 2x+(1−x)ln(1−x) x2. (4.73) Chapter 5 Constructing the Vertex 5.1 Introduction In studies on the Dynamical Breaking of Chiral Symmetry (DBC S) in QED3 in the quenched and unquenched approximations [7, 11, 12, 17 , 18, 50, 56, 76, 77, 78, 79, 80, 81, 82], we look for gauge independent physica l observables. For that purpose, we have seen previously the necessity of in cluding the Ward- Green-Takahashi Identity (WGTI) as well as the Landau-Khal atnikov-Fradkin (LKF) transformations. We know that, in Perturbation Theor y, these gauge invariance constraints are satisﬁed order by order, thus, t he vertex should be modiﬁed in every order of approximation. This fact has been e xploited by, for example, [22, 23, 57, 68]. In this Chapter we carry out the con structionn of the non perturbative Vertex based on its perturbative counterp art, following the work [83]. Making use of the WGTI, that relates the Vertex to the Fermion Propagator, one part of the Vertex, called longitudinal, can be expresse d in terms of the said propagator [19]. We perform the evaluation of the Fermion Pr opagator at the one-loop level and hence we determine the longitudinal vert ex at the same order. We also calculate the Full vertex at one-loop, and a mere subs traction of the longitudinal part yields the transverse part, which is not ﬁ xed by the WGTI. According to the choice of Ball and Chiu, later modiﬁed by Kız ılers¨ u et. al. [70], the transverse vertex can be expressed in terms of 8 ind ependent spin structures. Vertex should be free from kinematical singula rities. Ball and Chiu choose the basis in such a way that the coeﬃcient of every elem ent of this basis is independently free from kinematical singularities in Fe ynman gauge. It was later shown by Kızılers¨ u et. al. [70] that a similar calculation to the one of Ball and Chiu in an arbitrary covariant gauge does not have the sam e nice feature. Consequently, they proposed a modiﬁed basis whose coeﬃcien ts are free from kinematical singularities in an arbitrary covariant gauge . The calculation in the present Chapter conﬁrms that all the vectors of the modiﬁed b asis also retain this feature for massive QED3. The ﬁnal result for the transv erse vertex is 77 78 Constructing the Vertex written in terms of basic functions of the momenta in a form su itable for its extension to the non-perturbative domain, following the id eas of Curtis and Pennington [37]. Using perturbative constraints as a guide, we carry out a con struction of the non-perturbative vertex, which has no explicit dependence on the coupling α. This vertex has an explicit dependence on the gauge paramete rξ. For practical purposes of the numerical study of DCSB, we also construct an eﬀective vertex which shifts the angular dependence from the unknown Fermio n Propagator functions to the known basic functions, without changing it s perturbative pro- perties at the one-loop level. We believe that this vertex sh ould lead to a more realistic study of the dynamically generated masses throug h the corresponding SDEs. 5.2 Longitudinal and Transverse Vertex to One- Loop 5.2.1 The Fermion Propagator One-loop Fermion Propagator can be obtained by evaluating t he graph in Dia- gram (7). -1 p-1 p kq = - Diagram 7 : One-loop Correction to the Fermion Propagator. This graph corresponds to the following equation : iSF(p)−1=iS0−1 F(p) +e2/integraldisplayd3k (2π)3γµS0 F(k)γν∆0 µν(q),(5.1) whereq=k−p.The bare Fermion and Photon Propagators are, respectively : S0 F(p) =1 ∝ne}ationslashp−m, ∆0 µν(q) =−/bracketleftbig q2gµν+ (ξ−1)qµqν/bracketrightbig /q4, (5.2) wheremis the bare mass of the fermion and SF(p) represents the full propaga- tor, deﬁned in eq. (3.3). Taking the trace of Eq. (5.1), having multiplied it with ∝ne}ationslashpand with 1 re- spectively, one can obtain two independent equations. On si mplifying, these equations can be written as : 1 F(p)= 1 +i4παξ p2/integraldisplayd3k (2π)31 q4(k2−m2)/bracketleftbig (k2+p2)k·p−2k2p2/bracketrightbig (5.3) Constructing the Vertex 79 M(p) F(p)=m−i4πα(ξ+ 2)/integraldisplayd3k (2π)3m q2(k2−m2), (5.4) On Wick rotating to the Euclidean space and carrying out angu lar and radial integrations, we arrive at : 1 F(p)= 1−αξ 2p2/bracketleftbig m−(m2+p2)I(p)/bracketrightbig , M(p) F(p)=m[1 +α(ξ+ 2)I(p)], (5.5) where we have used the simplifying notation I(p2) = (1//radicalbig −p2)arctan/radicalbig −p2/m2. Equations (3.3) and (5.5) form the complete Fermion Propaga tor at one loop. 5.2.2 Longitudinal Vertex to One Loop We take the longitudinal part of the Vertex as the one that sat isﬁes the WGTI, i. e., the Ball-Chiu Vertex, eq. (3.27). On substituting eq. (5.5) in the said expression, we obtain : Γµ L(= Γµ BC) =/bracketleftbigg 1 +αξ 4σ1/bracketrightbigg γµ+αξ 4σ2[kµ∝ne}ationslashk+pµ∝ne}ationslashp+kµ∝ne}ationslashp+pµ∝ne}ationslashk] +α(ξ+ 2)σ3[kµ+pµ], (5.6) where σ1=m2+k2 k2I(k) +m2+p2 p2I(p)−mk2+p2 k2p2, σ2=1 k2−p2/bracketleftbiggm2+k2 k2I(k)−m2+p2 p2I(p) +mk2−p2 k2p2/bracketrightbigg , σ3=m[I(k)−I(p)]. (5.7) Eqs. (5.6) and (5.7) give the longitudinal part of the fermio n-photon vertex to one loop for the massive QED3. 5.2.3 Full Vertex to One Loop Full Vertex can be obtained from Diagram (8) : qk pqk pqk-w p-ww = - Diagram 8 : One-loop Correction to the Vertex. 80 Constructing the Vertex and can be expressed as : Γµ(k,p) =γµ+ Λµ. (5.8) Using the Feynman rules, ΛµtoO(α) is simply given by : −ieΛµ=/integraldisplay Md3w (2π)3(−ieγα)iS0 F(p−w)(−ieγµ)iS0 F(k−w)(−ieγβ)i∆0 αβ(w), (5.9) where the loop integration is to be performed in Minkowski sp ace (in Euclidean space, these deﬁnitions are modiﬁed by the correspondig fac tors ofi). Λµcan be expressed as : Λµ=−iα 2π2/braceleftigg /bracketleftbig γα∝ne}ationslashpγµ∝ne}ationslashkγα+m(4kµ+ 4pµ−∝ne}ationslashpγµ−γµ∝ne}ationslashk)−m2γµ/bracketrightbig J(0) −[γα∝ne}ationslashpγµγνγα+γαγνγµ∝ne}ationslashkγα+ 6mgµν]J(1) ν+γαγνγµγλγαJ(2) νλ +(ξ−1)/bracketleftigg γµK(0)−[γν∝ne}ationslashpγµ+γµ∝ne}ationslashkγν+ 2mgµν]J(1) ν (5.10) +/bracketleftbig γν∝ne}ationslashpγµ∝ne}ationslashkγλ+m(γν∝ne}ationslashpγµγλ+γνγµ∝ne}ationslashkγλ) +m2γνγµγλ/bracketrightbig I(2) νλ/bracketrightigg/bracerightigg , where the integrals K(0),J(0),J(1) µ,J(2) µν,I(0),I(1) µandI(2) µνare : K(0)=/integraldisplay Md3w1 [(p−w)2−m2] [(k−w)2−m2] J(0)=/integraldisplay Md3w1 w2[(p−w)2−m2] [(k−w)2−m2] J(1) µ=/integraldisplay Md3wwµ w2[(p−w)2−m2] [(k−w)2−m2] J(2) µν=/integraldisplay Md3wwµwν w2[(p−w)2−m2] [(k−w)2−m2] I(0)=/integraldisplay Md3w1 w4[(p−w)2−m2] [(k−w)2−m2] I(1) µ=/integraldisplay Md3wwµ w4[(p−w)2−m2] [(k−w)2−m2] I(2) µν=/integraldisplay Md3wwµwν w4[(p−w)2−m2] [(k−w)2−m2]. (5.11) We evaluate these integrals following the techniques devel oped in [19, 57, 68, 70]. The results are tabulated below, employing the notatio n ∆2= (k·p)2−k2p2 andX0= (2/iπ2)X(0)forX=I,J,K . Constructing the Vertex 81 The Scalar Integral J(0) In arbitrary dimensions, J(0)=/integraldisplayddw [(k−w)2−m2][(p−w2)−m2]w2. (5.12) Employing Feynman parametrization, J(0)= Γ(3)/integraldisplay ddw/integraldisplay1 0dα1/integraldisplay1 0dα2/integraldisplay1 0dα3 ×δ(αs−1) [α1[(k−w)2−m2] +α2[(p−w)2−m2] +α3w2]3,(5.13) with αs=α1+α2+α3. (5.14) We deﬁneDas D=α1[(k−w)2−m2] +α2[(p−w)2−m2] +α3w2 =αsw2−2(α1k·w+α2p·w) +α1(k2−m2) +α2(p2−m2) =αs/bracketleftigg w2−2(α1k+α2p)·w αs+(α1k+α2p)2 α2s −(α1k+α2p)2 α2s+α1(k2−m2) +α2(p2−m2) αs/bracketrightigg =αs/bracketleftigg/parenleftbigg w−α1k+α2p αs/parenrightbigg2 + 1 α2s/braceleftigg αs[α1(k2−m2) +α2(p2−m2)]−(α1k+α2p)2/bracerightigg/bracketrightigg .(5.15) Let w−α1k+α2p αs→w. (5.16) Then, J(0)= 2/integraldisplay1 0dα1/integraldisplay1 0dα2/integraldisplay1 0dα3δ(αs−1) α3s(5.17) ×/integraldisplayddw /bracketleftbig w2+α−2s{αs[α1(k2−m2)+α2(p2−m2)]−(α1k+α2p)2}/bracketrightbig3. Making use of the formulas /integraldisplayddw wn= 0 /integraldisplayddw (w2−s)n= (−1)niπd 2Γ/parenleftbig n−d 2/parenrightbig Γ(n)sd 2−n, (5.18) 82 Constructing the Vertex we can write J(0)=−iπd 2Γ/parenleftbigg 3−d 2/parenrightbigg/integraldisplay1 0dα1/integraldisplay1 0dα2/integraldisplay1 0dα3 (5.19) ×δ(αs−1) αd−3s{−αs[α1(k2−m2) +α2(p2−m2)] + (α1k+α2p)2}3−d 2. At this point, we remind the Cheng-Wo Theorem [84] : Cheng-Wu Theorem 1 If I=/integraldisplay1 0n/productdisplay i=1dαiδ/parenleftigg 1−n/summationdisplay i=1αi/parenrightigg F(α), (5.20) then we can write I=/integraldisplay∞ 0′/productdisplay dαi/integraldisplay1 0′′/productdisplay dαiδ/parenleftigg 1−′′/summationdisplay αi/parenrightigg F(α), (5.21) where the set of αhas been split into two nonempty sets {α′}={α1,...,αk},{α′′}={αk+1,...,αn}. (5.22) Making use of this theorem, J(0)=−iπd 2Γ/parenleftbigg 3−d 2/parenrightbigg/integraldisplay∞ 0dα3/integraldisplay∞ 0dα2/integraldisplay1 0dα1 (5.23) ×δ(α1−1) αd−3s{−αs[α1(k2−m2) +α2(p2−m2)] + (α1k+α2p)2}3−d 2. After a trivial integration over α1, we obtain D= (1 +α2+α3)d−3/bracketleftbig m2(1 +α2+α3)(1 +α2)−k2(1 +α2+α3−1) −p2(1 +α2+α3)α2+α2 2p2+ 2α2k·p]3−d 2. (5.24) Using−2k·p=q2−k2−p2, D= (1+α2+α3)d−3[m2(1+α2+α3)(1+α2)−α3k2−α2α3p2−α2q2]3−d 2.(5.25) Therefore, J(0)=−iπd 2Γ/parenleftbigg 3−d 2/parenrightbigg/integraldisplay∞ 0dα3/integraldisplay∞ 0dα2 (1 +α2+α3)d−3 ×1 [m2(1 +α2+α3)(1 +α2)−α3k2−α2α3p2−α2q2]3−d 2.(5.26) Constructing the Vertex 83 In our three-dimensional case J(0)=−iπ3 21 2π1 2/integraldisplay∞ 0dα3/integraldisplay∞ 0dα2 ×1 [m2(1 +α2+α3)(1 +α2)−α3k2−α2α3p2−α2q2]3 2 =iπ2 2J0, (5.27) where J0=−/integraldisplay∞ 0dα3/integraldisplay∞ 0dα2 ×1 [m2(1 +α2+α3)(1 +α2)−α3k2−α2α3p2−α2q2]3 2.(5.28) Being simpler the integration over α3ﬁrst, we take D=m2(1 +α2+α3)(1 +α2)−α3k2−α2α3p2−α2q2 =α3[m2(1 +α2)−k2−α2p2] +m2(1 +α2)2−α2q2 =α3[α2(m2−p2) + (m2−k2)] +m2(1 +α2)2−α2q2.(5.29) Therefore, J0=−/integraldisplay∞ 0dα2/integraldisplay∞ 0dα3 [α3[α2(m2−p2)+(m2−k2)]+m2(1+α2)2−α2q2]3 2 = 2/integraldisplay∞ 0dα21 [α2(m2−p2) + (m2−k2)] ×1 {α3[α2(m2−p2) + (m2−k2)] +m2(1 +α2)2−α2q2}1 2/vextendsingle/vextendsingle/vextendsingle/vextendsingleα3=∞ α3=0 =−2/integraldisplay∞ 0dα21 [α2(m2−p2) + (m2−k2)]1 [m2(1 +α2)2−α2q2]1 2 =−2 (m2−p2)/integraldisplay∞ 0dα21/bracketleftig α2+m2−k2 m2−p2/bracketrightig1 [m2(1 +α2)2−α2q2]1 2.(5.30) Let α2+m2−k2 m2−p2=z⇒dα2=dz . (5.31) Integration limits are transformed in the following way : α2= 0⇒z=m2−k2 m2−p2α2→∞⇒z→∞. (5.32) Now we rewrite the integrand as : m2(1 +α2)2−α2q2=m2/parenleftbigg 1 +z−m2−k2 m2−p2/parenrightbigg2 −/bracketleftbigg z−m2−k2 m2−p2/bracketrightbigg q2 84 Constructing the Vertex =m2+/bracketleftbigg z2+(k2−p2)2 (m2−p2)2+ 2zk2−p2 m2−p2/bracketrightbigg −zq2+q2m2−k2 m2−p2 =m2z2+z/bracketleftbigg2m2(k2−p2)−q2(m2−p2) m2−p2/bracketrightbigg +m2(k2−p2)2+q2(m2−k2)(m2−p2) (m2−p2)2 =cz2+bz+a≡R, (5.33) where c=m2, b=2m2(k2−p2)−q2(m2−p2) m2−p2, a=m2(k2−p2)2+q2(m2−k2)(m2−p2) (m2−p2)2=χ (m2−p2)2.(5.34) Consequently, J0=−2 m2−p2/integraldisplay∞ m2−k2 m2−p2dz z1√ R=2 (m2−p2)1√−aarctan2a+bz 2√−a√ R/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞ m2−k2 m2−p2. (5.35) We consider 1. Evaluation at inﬁnity lim z→∞2a+bz 2√−a√ R=b 2√−a√c =m2(k2−p2)[2m2−k2−p2] +χ 2m√−χ(m2−k2).(5.36) 2. Evaluation at the lower limit 2a+bz 2√−a√ R/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=m2−k2 m2−p2=−χ+m2(k2−p2)(2m2−k2−p2) 2m√−χ(m2−p2),(5.37) and then, J0=−2√−χ/braceleftigg arctan m2(k2−p2)(2m2−k2−p2) +χ 2m√−χ(m2−k2)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright a  −arctan −χ+m2(k2−p2)(2m2−k2−p2) 2m√−χ(m2−p2)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright b /bracerightigg . (5.38) Constructing the Vertex 85 Since we diﬁned I(y) =1/radicalbig −y2arctan/radicalbigg −y2 m2, (5.39) we can write I(η2 iχ) =1/radicalbig −η2 iχarctan/radicalbigg −η2 iχ m2=1 ηi√−χarctanηi m√−χ.(5.40) Rewriting then eq. (5.38) with the identiﬁcations a=m2(k2−p2)(2m2−k2−p2) +χ 2m√−χ(m2−k2) =√−χ 2m/bracketleftbigg −/braceleftbiggm2(k2−p2)(2m2−k2−p2) +χ 2χ(m2−k2)/bracerightbigg/bracketrightbigg ≡√−χ 2mη1, (5.41) where η1=−/braceleftbiggm2(k2−p2)(2m2−k2−p2) +χ 2χ(m2−k2)/bracerightbigg . (5.42) In a similar way b=−χ+m2(k2−p2)(2m2−k2−p2) 2m√−χ(m2−p2) =√−χ 2m/bracketleftbiggχ−m2(k2−p2)(2m2−k2−p2) χ(m2−p2)/bracketrightbigg ≡√−χ 2mη2, (5.43) where η2=/bracketleftbiggχ−m2(k2−p2)(2m2−k2−p2) χ(m2−p2)/bracketrightbigg . (5.44) Finally J0=/bracketleftbigg −η1I/parenleftbiggη1√χ 2/parenrightbigg +η2I/parenleftbiggη2√χ 2/parenrightbigg/bracketrightbigg . (5.45) The Scalar Integral K(0) We have, K(0)=/integraldisplayd3w [(p−w)2−m2][(k−w)2−m2]. (5.46) Employing Feynamn parametrization : 1 ab=/integraldisplay1 0dz [az+b(1−z)]2. (5.47) 86 Constructing the Vertex We take a= [(p−w)2−m2], b= [(k−w)2−m2]. (5.48) We deﬁne now D=z[(p−w)2−m2] + (1−z)[(k−w)2−m2] =z(p2−2p·w) + (1−z)(k2−2k·w) +w2−m2. (5.49) We perform now the following change of variable w→w′=w−k(1−z). (5.50) Then, we obtain D=zp2−2zp·w′−2p·kz(1−z) +k2z(1−z) +w′2−m2. (5.51) A second change of variable w′→w=w′−zp, (5.52) implies D=w2+ (k−p)2z(1−z)−m2 =w2+q2z(1−z)−m2. (5.53) Therefore, K(0)=/integraldisplay1 0dz/integraldisplay d3w1 [w2+q2z(1−z)−m2]2. (5.54) Using eqs. (5.18), we arrive at : K(0)=/integraldisplay1 0dz(−1)2iπ3 2Γ/parenleftbig1 2/parenrightbig Γ(2)[m2−q2z(1−z)]−1 2 =iπ2/integraldisplay1 0dz[m2−q2z(1−z)]−1 2 =2iπ2 /radicalbig −q2arctan/radicalbigg −q2 4m2 =iπ2I/parenleftigq 2/parenrightig , (5.55) which is our ﬁnal expression. Constructing the Vertex 87 The Tensor Integral J(1) µ We have J(1) µ=/integraldisplay d3wwµ w2[(p−w)2−m2] [(k−w)2−m2]. (5.56) We write this integral in its most general form as : J(1) µ=iπ2 2{kµJA(k,p) +pµJB(k,p)}. (5.57) Contracting with kµandpµ, we obtain the following system of equations : kµJ(1) µ=iπ2 2[k2JA+ (k·p)JB] pµJ(1) µ=iπ2 2[(k·p)JA+p2JB]. (5.58) On the other hand, pµJ(1) µ=(p2−m2) 2/integraldisplay d3w1 w2[(p−w)2−m2][(k−w)2−m2] −1 2/integraldisplay d3w1 [(k−w)2−m2]w2 +1 2/integraldisplay d3w1 [(p−w)2−m2][(k−w)2−m2] =(p2−m2) 2J(0)+1 2K(0)−1 2I(0,1,1). (5.59) I(0,1,1) will be diﬁned as a Master Integral when we calculate I(0), and we use the relation : p·w=1 2(p2+w2−(p−w)2−m2+m2). (5.60) In a similar way, we also have that : kµJ(1) µ=(k2−m2) 2J(0)+1 2K(0)−1 2I(1,0,1). (5.61) We can now solve the system of equations (5.58). On solving, w e ﬁnd : JA(k,p) =−2 ∆2/braceleftigg /bracketleftbig p2(k2−k·p)−m2(p2−k·p)/bracketrightbigJ0 4+k·pI(k) −p2I(p) +1 2(p2−k·p)I(q/2)/bracerightigg , JB(k,p) =JA(p,k). (5.62) Equations (5.57) and (5.62) form the ﬁnal answer. 88 Constructing the Vertex The Tensor Integral J(2) µν We have J(2) µν=/integraldisplay d3wwµwν w2[(p−w)2−m2] [(k−w)2−m2]. (5.63) We write this integral in its most general form as : J(2) µν=iπ3 2/braceleftbigggµν 3K0+/parenleftbigg kµkν−gµνk2 3/parenrightbigg JC+/parenleftbigg pµkν+kµpν−gµν2(k·p) 3/parenrightbigg JD +/parenleftbigg pµpν−gµνp2 3/parenrightbigg JE/bracerightbigg . (5.64) Contracting with pµ, pµJ(2) µν=iπ3 2/braceleftbiggpν 3K0+/parenleftbigg (k·p)kν−pνk2 3/parenrightbigg JC (5.65) +/parenleftbigg p2kν+(k·p)pν−pν2(k·p) 3/parenrightbigg JD+/parenleftbigg p2pν−pνp2 3/parenrightbigg JE/bracerightbigg . Performing the remaining contractions we obtain : pνpµJ(2) µν=iπ3 2/braceleftbiggp2 3K0+/parenleftbigg (k·p)2−k2p2 3JC/parenrightbigg +4 3p2(k·p)JD+2 3p4JE/bracerightbigg . (5.66) Other possible contractions are : kνpµJ(2) µν=iπ3 2/braceleftbigg(k·p) 3K0+2 3k2(k·p)JC+/parenleftbigg k2p2+(k·p)2 3/parenrightbigg JD +2 3p2(k·p)JE/bracerightbigg (5.67) and kµJ(2) µν=iπ3 2/braceleftbiggkν 3K0+2 3k2kνJC+/parenleftbigg (k·p)kν+k2pν−2 3kν(k·p)/parenrightbigg JD +/parenleftbigg (k·p)pν−kνp2 3/parenrightbigg JE/bracerightbigg , (5.68) wich, after a second contraction, yield : kνkµJ(2) µν=iπ3 2/braceleftbiggk2 3K0+2 3k4JC+4 3k2(k·p)JD+/parenleftbigg (k·p)2−k2p2 3/parenrightbigg JE/bracerightbigg (5.69) and pνkµJ(2) µν=iπ3 2/braceleftbigg(k·p) 3K0+2 3k2(k·p)JC+/parenleftbigg k2p2+1 3(k·p)2/parenrightbigg JD +2 3p2(k·p)JE/bracerightbigg . (5.70) Constructing the Vertex 89 On the other hand, pµJ(2) µν=/integraldisplay d3wp·wwν w2[(p−w)2−m2][(k−w)2−m2] =(p2−m2) 2/integraldisplay d3wwν w2[(p−w)2−m2][(k−w)2−m2] +/integraldisplay d3wwν [(p−w)2−m2][(k−w)2−m2] −1 2/integraldisplay d3wwν w2[(k−w)2−m2] =(p2−m2) 2J(1) ν+1 2Aν(k,p)−1 2Eν(k). (5.71) Similarly kµJ(2) µν=(k2−m2) 2J(1) ν+1 2Aν(k,p)−1 2Eν(p). (5.72) The new integrals needed are Eν(p) andAν(k,p). Before proceding with the calculation of J(2) µν, let us pay attention to them. First, Eν(p) =/integraldisplay d3wwν w2[(p−w)2−m2]. (5.73) Using Feynam Parametrization, Eν(p) =/integraldisplay1 0dx/integraldisplay d3wwν [x[(p−w)2−m2] + (1−x)w2]2.(5.74) Let D=x[(p−w)2−m2] + (1−x)w2 =w2+xp2−2xp·w−xm2. (5.75) We take now w=w′+xp, (5.76) such that D=w′2+p2x(1−x)−m2x, (5.77) and, consequently, Eν(p) =/integraldisplay1 0dx/integraldisplay d3w′w′ ν+pνx [w′2+p2x(1−x)−m2x]2 =pν/integraldisplay1 0dxx(−1)2iπ3 2Γ/parenleftbig 2−3 2/parenrightbig Γ(2)[−p2x(1−x) +m2x]3 2−2 =iπ2pν/integraldisplay1 0dxx1 2[−p2(1−x) +m2]−1 2 =iπ2pν p2/bracketleftigg m−(m2−p2)/radicalbig −p2arctan/radicalbigg −p2 m2/bracketrightigg k↔p=Eν(k). (5.78) 90 Constructing the Vertex The other integral we need to calculate is Aν(k,p) =/integraldisplay d3wwν [(p−w)2−m2][(k−w)2−m2]. (5.79) After Feynman parametrization, we write : Aν(k,p) =/integraldisplay1 0dz/integraldisplay d3wwν {z[(p−w)2−m2] + (1−z)[(k−w)2−m2]}2. (5.80) We deﬁne D=z[(p−w)2−m2] + (1−z)[(k−w)2−m2]. (5.81) After the change of variable w′=w−zp−(1−z)k, (5.82) we have D=w′2+q2z(1−z)−m2, (5.83) and, therefore, Aν(k,p) =/integraldisplay1 0dz/integraldisplay d3ww′ ν+pνz+kν(1−z) [w′2+q2z(1−z)−m2]2. (5.84) On simplifying, Aν(k,p) =/integraldisplay1 0dziπ2pνz+kν(1−z)/radicalbig m2−q2z(1−z) =iπ2(k+p)ν/integraldisplay1 0dzz/radicalbig m2−q2z(1−z) =iπ2(k+p)ν/radicalbig −q2arctan/radicalbigg −q2 4m2≡1 2K(0)(k+p)ν.(5.85) Let us turn back to the claculation of J(2) µν. Second contraction yields : pνpµJ(2) µν=(p2−m2) 2iπ3 2[(k·p)JA+p2JB]+1 2pνAν−pν 2Eν(k) kνpµJ(2) µν=(p2−m2) 2iπ3 2[k2JA+(k·p)JB]+1 2kνAν−kν 2Eν(k) kνkµJ(2) µν=(k2−m2) 2iπ3 2[k2JA+(k·p)JB]+1 2kνAν−kν 2Eν(p) pνkµJ(2) µν=(k2−m2) 2iπ3 2[(k·p)JA+p2JB]+1 2pνAν−pν 2Eν(p).(5.86) We arrive then to the following system of equations : /parenleftbigg (k·p)2−k2p2 3/parenrightbigg JC+4 3p2(k·p)JD+2 3p4JE=a Constructing the Vertex 91 2 3k4JC+4 3k2(k·p)JD+/parenleftbigg (k·p)2−k2p2 3/parenrightbigg JE=b 2 3k2(k·p)JC+/parenleftbigg k2p2+(k·p)2 3/parenrightbigg JD+2 3p2(k·p)JE=c,(5.87) where a=2 iπ3pµpνJ(2) µν−p2 3K0 b=2 iπ3kµkνJ(2) µν−k2 3K0 c=1 iπ3[kνpµJ(2) µν+pνkµJ(2) µν−(k·p) 3K0]. (5.88) Solutions to this system are : JC(k,p) =1 ∆2/braceleftigg /bracketleftbig p2(k·p−2k2)−m2(k·p−2p2)/bracketrightbigJA 2−p2(p2−m2)JB 2 +k·p k2(m2−k2)I(k) +1 2(k·p+p2)I(q/2)−mk·p k2/bracerightigg , JD(k,p) =1 2∆2/braceleftigg /bracketleftbig k2(3k·p−p2)−m2(3k·p−k2)/bracketrightbigJA 2 +/bracketleftbig p2(3k·p−k2)−m2(3k·p−p2)/bracketrightbigJB 2 −(m2−k2)I(k)−(m2−p2)I(p)−1 2(k+p)2I(q/2)+2m/bracerightigg , JE(k,p) =JC(p,k), (5.89) which completes or calculation of J(2) µν. The Scalar Integral I(0) We have I(0)=/integraldisplay d3w1 w4[(p−w)2−m2][(k−w)2−m2]. (5.90) In general, let us deﬁne the family of integrals : I(ν1,ν2,ν3) =/integraldisplay ddw1 [(p−w)2−m2]ν1[(k−w)2−m2]ν2[w2]ν3.(5.91) We use the following Integration by Parts (IBP) identity : /integraldisplay ddw∂ ∂wµ/bracketleftbigg(qi−w)µ [(p−w)2−m2]ν1[(k−w)2−m2]ν2[w2]ν3/bracketrightbigg = 0. (5.92) 92 Constructing the Vertex After the diﬀerentiation we obtain : /integraldisplay ddw(qi−m)/bracketleftigg 2ν1(p−w)µ Aν1+1Bν2Cν3+2ν2(k−w)µ Aν1Bν2+1Cν3−2ν2wµ Aν1Bν2Cν3+1/bracketrightigg +/integraldisplay ddw(−d) Aν1Bν2Cν3= 0, (5.93) since ∂ ∂wµwµ=gµ µ=d. (5.94) We have deﬁned A= (p−w)2−m2, B = (k−w)2−m2, C =w2. (5.95) Therefore, dI(ν1,ν2,ν3) =/integraldisplay ddw(qi−w) /bracketleftigg 2ν1(p−w)µ Aν1+1Bν2Cν3+2ν2(k−w)µ Aν1Bν2+1Cν3−2ν2wµ Aν1Bν2Cν3+1/bracketrightigg .(5.96) Assuming that qiis deﬁned such that q1=p,q2=kandq3= 0. Then, for i= 0 : dI(ν1,ν2,ν3) =/integraldisplay ddw(p−w) /bracketleftigg 2ν1(p−w)µ Aν1+1Bν2Cν3+2ν2(k−w)µ Aν1Bν2+1Cν3−2ν2wµ Aν1Bν2Cν3+1/bracketrightigg .(5.97) Let us observe the following products : 1. 2(p−w)·(k−w) = (p−w)2+ (k−w)2−(p−k)2−m2−m2+ 2m2 =A+B−[(p−k)2−2m2]. (5.98) 2. 2(p−w)·(−w) = (p−w)2−m2+w2−p2+m2=A+C−p2+m2.(5.99) 3. (p−w)2= (p−w)2−m2+m2=A+m2. (5.100) Constructing the Vertex 93 Substituting these products into eq. (5.97), we obtain : dI(ν1,ν2,ν3) =/integraldisplay ddw/bracketleftigg 2ν1[A+m2] Aν1+1Bν2Cν3+ν2[A+B−[(p−k)2−2m2]] Aν1Bν2+1Cν3 +ν3[A+C−(p2−m2)] Aν1Bν2Cν3+1/bracketrightigg =/integraldisplay ddw/bracketleftigg 2ν1+ν2+ν3 Aν1Bν2Cν3+ν2 Aν1−1Bν2+1Cν3+2ν1m2 Aν1+1Bν2Cν3 −ν2[(p−k)2−2m2] Aν1Bν2+1Cν3+ν3 Aν1−1Bν2Cν3+1−ν3(p2−m2) Aν1Bν2Cν3+1/bracketrightigg = (2ν1+ν2+ν3)I(ν1,ν2,ν3) +ν2I(ν1−1,ν2+2,ν3) +2m2ν1I(ν1+1,ν2,ν3)−ν2[(p−k)2−2m2]I(ν1,ν2+1,ν3) +ν3I(ν1−1,ν2,ν3+1)−ν3(p2−m2)I(ν1,ν2,ν3+ 1).(5.101) Rearranging the terms in such a way that those integrals with ν1+ν2+ν3 as the sum of their arguments are on the right hand side, while those with ν1+ν2+ν3+ 1 on the left hand side, we get : −2ν1m2I(ν1+1,ν2,ν3) +ν2[(p−k)2−2m2]I(ν1,ν2+1,ν3) +ν3(p2−m2)I(ν1,ν2,ν3+ 1) = (2ν1+ν2+ν3−d)I(ν1,ν2,ν3) +ν2I(ν1−1,ν2+ 1,ν3) +ν3I(ν1−1,ν2,ν3+ 1).(5.102) Fori= 2 we have : dI(ν1,ν2,ν3) =/integraldisplay ddw(k−w)µ /bracketleftigg 2ν1(p−w)µ Aν1+1Bν2Cν3+2ν2(k−w)µ Aν1Bν2+1Cν3−2ν3wµ Aν1Bν2Cν3+1/bracketrightigg = (ν1+ 2ν2+ν3)I(ν1,ν2,ν3) +ν1I(ν1+ 1,ν2−1,ν3) +ν3I(ν1,ν2−1,ν3+ 1) −[(p−k)2−2m2]ν1I(ν1+ 1,ν2,ν3) +2ν2m2I(ν1,ν2+ 1,ν3) −ν3(k2−m2)I(ν1,ν2,ν3+ 1) (5.103) or ν1[(p−k)2−2m2]I(ν1+ 1,ν2,ν3)−2ν2m2I(ν1,ν2+ 1,ν3) +ν3(k2−m2)I(ν1,ν2,ν3+ 1) = (ν2+ 2ν2+ν3−d)I(ν1,ν2,ν3) +ν1I(ν1+ 1,ν2−1,ν3) +ν3I(ν1,ν2,ν3+ 1), (5.104) and fori= 3 : dI(ν1,ν2,ν3) =/integraldisplay ddw(−wµ) 94 Constructing the Vertex /bracketleftigg 2ν1(p−w)µ Aν1+1Bν2Cν3+2ν2(k−w)µ Aν1Bν2+1Cν3−2ν3wµ Aν1Bν2Cν3+1/bracketrightigg = (ν1+ν2+ 2ν3)I(ν1,ν2,ν3) +ν1I(ν1+ 1,ν2,ν3−1) +ν2I(ν1+ν2+ 1,ν3−1) −ν1(p2−m2)I(ν1+ 1,ν2,ν3) −ν2(k2−m2)I(ν1,ν2+ 1,ν3) (5.105) or ν1(p2−m2)I(ν1+ 1,ν2,ν3) +ν2(k2−m2)I(ν1,ν2+ 1,ν3) = (ν1+ν2+ 2ν3−d)I(ν1,ν2,ν3) +ν1I(ν1+ 1,ν2,ν3−1) +ν2I(ν1,ν2+ 1,ν3−1). (5.106) Therefore, we need to solve the system of equations : −2ν1m2I(ν1+ 1,ν2,ν3) +ν2(q2−2m2)I(ν1+ν2+ 1,ν3) +ν3(p2−m2)I(ν1,ν2,ν3+ 1) =a ν1(q2−2m2)I(ν1+ 1,ν2,ν3)−2ν2m2I(ν1,ν2+ 1,ν3) +ν3(k2−m2)I(ν1,ν2,ν3+ 1) =b ν1(p2−m2)I(ν1+ 1,ν2,ν3) +ν2(k2−m2)I(ν1,ν2+ 1,ν3) +0I(ν1,ν2,ν3+1) =c,(5.107) where a= (2ν1+ν2+ν3−d)I(ν1+ν2+ν3) +ν2I(ν1−1,ν2+ 1,ν3) +ν3I(ν1−1,ν2,ν3+ 1) b= (ν1+ 2ν2+ν3−d)I(ν1+ν2+ν3) +ν1I(ν1+ 1,ν2−1,ν3) +ν3I(ν1,ν2−1,ν3+ 1) c= (ν1+ν2+ 2ν3−d)I(ν1+ν2+ν3) +ν1I(ν1+ 1,ν2,ν3−1) +ν2I(ν1,ν2+ 1,ν3−1). (5.108) In obvious notation :  −2m2ν1ν2(q2−2m2)ν3(p2−m2) ν1(q2−2m2)−2m2ν2ν3(k2−m2) ν1(p2−m2)ν2(k2−m2) 0  I1 I2 I3 = a b c , (5.109) which formally we can write as : SI=A. (5.110) Then, |S|= 2ν1ν2ν3[m2(k2−m2)2+m2(p2−m2)2+ (k2−m2)(p2−m2)(q2−2m2)], (5.111) Constructing the Vertex 95 and besides |S|I3=ν1ν2[(q2−2m2)(k2−m2) + 2m2(p2−m2)]a +ν1ν2[(q2−2m2)(p2−m2) + 2m2(k2−m2)]b +ν1ν2[4m4−(q2−2m2)2]c. (5.112) Forν1=ν2=ν3= 1, a=I(1,1,1) +I(0,2,1) +I(0,1,2) b=I(1,1,1) +I(2,0,1) +I(1,0,2) c=I(1,1,1) +I(2,1,0) +I(1,2,0) (5.113) withI(0)=I(1,1,2) andJ(0)=I(1,1,1). So, we need to calculate the integrals I(0,2,1)k↔p=I(2,0,1),I(0,1,2)k↔p=I(1,0,2) andI(2,1,0)k↔p=I(1,2,0). For that purpose, we deﬁne the integral Ipn(k,p,m 1,m2) =/integraldisplay ddw1 [(k−w)2−m2 1]p[(p−w)2−m2 2]n. (5.114) Using Feynam parametrization, we know that 1 AnBp=Γ(n+p) Γ(n)Γ(p)/integraldisplay1 0dxxn−1(1−x)p−1 1 [xA+ (1−x)B]n+p.(5.115) Let A= [(p−w)2−m2], B= [(k−w)2−m2], (5.116) and D=xA+ (1−x)B =x[p2+w2−2p·w−m2 2] + (1−x)[k2+w2−2k·w−m2 1] =w2−2w·[px+k(1−x)]+p2x+k2(1−x)−m2 2x−m2 1(1−x).(5.117) We make the change of variable w′=w−[px+k(1−x)], (5.118) such that w2−2w·[px+k(1−x)] =w′2−[px+k(1−x)]2. (5.119) In this way, Dis rewritten as : D=w′2−[px+k(1−x)]2+p2x+k2(1−x)−m2 2x−m2 1(1−x) =w′2+p2x(1−x) +k2x(1−x)−2k·px(1−x)−m2 2x−m2 1(1−x) =w′2+q2x(1−x)−m2 2x−m2 1(1−x). (5.120) 96 Constructing the Vertex Therefore, Ipn(k,p,m 1,m2) =Γ(n+p) Γ(n)Γ(p)/integraldisplay1 0dxxn−1(1−x)p−1/integraldisplay ddw′1 Dn+p =Γ(n+p) Γ(n)Γ(p)/integraldisplay1 0dxxn−1(1−x)p−1(−1)n+piπd/2Γ(n+p−d/2) Γ(n+p)sd 2−n−p =i(−1)n+pπd/2Γ(n+p−d/2) Γ(n)Γ(p)/integraldisplay1 0dxxn−1(1−x)p−1sd 2−n−p,(5.121) with s=−q2x(1−x) +m2 2x+m2 1(1−x). (5.122) For the integrals of our interest, we help ourselves with I11(k,p,m 1,m2) =/integraldisplayddw [(p−w)2−m2 2][(k−w)2−m2 1] =iπd 2Γ/parenleftbigg 2−d 2/parenrightbigg/integraldisplay1 0dxsd 2−2 =iπ2/integraldisplay1 0dxs−1 2, (5.123) where, in the last line, we set d= 3. So I(2,1,0) =1 2m1∂ ∂m1I11(k,p,m 1,m2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle m1=m2=m. (5.124) Now, ∂ ∂m1s−1 2=−m1(1−x)s−3 2, (5.125) and therefore, I(2,1,0) =−iπ2 2/integraldisplay1 0dx(1−x) [−q2x(1−x) +m2]3 2 =−iπ2 m[4m2−q2] k↔p=I(1,2,0). (5.126) On the other hand I(2,0,1) =1 2m1∂ ∂m1I11(k,p,m 1,m2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle k=m1=0,m2=m =−iπ2 2/integraldisplay1 0dxx [−p2x(1−x) +m2x]3 2 =−iπ2 m(m2−p2) k↔p=I(0,2,1). (5.127) Constructing the Vertex 97 To solveI(1,0,2), we take the integral Iν=/integraldisplay ddwwν w4[(p−w)2−m2]. (5.128) Then, pνIν=1 2(p2−m2)/integraldisplay ddw1 w4[(p−w)2−m2] +1 2/integraldisplay ddw1 w2[(p−w)2−m2]−1 2/integraldisplayddw w4 =1 2(p2−m2)I(1,0,2) +1 2I(1,0,1), (5.129) since tha last term vanishes in the dimensional regularizat ion scheme. It is easy to see that I(1,0,1) =I11(k,p,m 1,m2)/vextendsingle/vextendsingle/vextendsingle k=m1=0,m2=m =iπ2/integraldisplay1 0[−p2x(1−x) +m2x]−1 2 =2iπ2 /radicalbig −p2arctan/radicalbigg −p2 m2 k↔p=I(0,1,1). (5.130) To calculate explicitly Iν, we take in eq. (5.115) A= [(p−w)2−m2], B =w2, n= 1, p= 2. (5.131) So, in three dimensions, Iν=Γ(3) Γ(1)Γ(2)/integraldisplay1 0dx(1−x)/integraldisplay d3wwν 1 [Ax+B(1−x)]3.(5.132) Let w′=w−px, (5.133) Then, D=Ax+B(1−x) =w2+p2x−2p·wx−xm2 =w′2+p2x−p2x2−m2x =w′2+p2x(1−x)−m2x, (5.134) and therefore Iν= 2/integraldisplay1 0dx(1−x)/integraldisplay (d3w)pνx [w2+p2x(1−x)−m2x]3. (5.135) 98 Constructing the Vertex Using now (5.18), Iν= 2pν/integraldisplay1 0dxx(1−x)(−1)3iπ3 2Γ/parenleftbig3 2/parenrightbig Γ(3)[−p2x(1−x) +m2x]3 2 =−iπ2 2pν/integraldisplay1 0dxx−1 2(1−x)[−p2(1−x) +m2]−3 2 =−iπ2 2pν/bracketleftbigg/integraldisplay1 0dx/braceleftig x−1 2−x1 2/bracerightig [−p2(1−x) +m2]−3 2/bracketrightbigg =−iπ2 2[−p2]−3 2pν/bracketleftigg −2m2 m2−p2/radicalbigg −p2 m2+ 2 arctan/radicalbigg −p2 m2/bracketrightigg .(5.136) So that I(1,0,2) =2 p2−m2/bracketleftbigg pνIν−1 2I(1,0,1)/bracketrightbigg =2m (m2−p2)2iπ2 k↔p=I(0,1,2). (5.137) Finally, substiututing into (5.112) I(0)=1 χ/braceleftig q2(m2+k·p)J(0)+iπ2mL/bracerightig , (5.138) where L=q2(k2−m2)−(k2−p2)(k2+m2) (k2−m2)2 +q2(p2−m2) + (k2−p2)(p2+m2) (p2−m2)2. (5.139) This is our ﬁnal expression. The Tensor Integral I(1) µ We have I(1) µ=/integraldisplay Md3wwµ w4[(p−w)2−m2] [(k−w)2−m2].(5.140) We write this integral in its most general form as : I(1) µ=iπ2 2[kµIA(k,p) +pµIB(k,p)]. (5.141) Contracting with kµandpµwe obtain the following system of equations : kµI(1) µ=iπ2 2[k2IA+ (k·p)IB] pµI(1) µ=iπ2 2[(k·p)IA+p2IB]. (5.142) Constructing the Vertex 99 On the other hand pµI(1) µ=(p2−m2) 2/integraldisplay d3w1 w4[(p−w)2−m2][(k−w)2−m2] −1 2/integraldisplay d3w1 [(k−w)2−m2]w4 +1 2/integraldisplay d3w1 w2[(p−w)2−m2][(k−w)2−m2] =(p2−m2) 2I(0)+1 2J(0)−1 2I(0,1,2), (5.143) where we have used : p·w=1 2(p2+w2−(p−w)2−m2+m2). (5.144) We also have that : kµI(1) µ=(k2−m2) 2I(0)+1 2J(0)−1 2I(1,0,2). (5.145) We solve the system of equations (5.142), to ﬁnd : IA(k,p) =2 ∆2/braceleftigg /bracketleftbig k·p(p2−m2)−p2(k2−m2)/bracketrightbigI0 4+p·qJ0 4 +mp2 (m2−p2)2−mk·p (m2−k2)2/bracerightigg , IB(k,p) =IA(p,k). (5.146) Equations (5.141) and (5.146) form the complete solution. The Tensor Integral I(2) µν We have, I(2) µν=/integraldisplay Md3wwµwν w4[(p−w)2−m2] [(k−w)2−m2].(5.147) We express this integral in its most general form as : I(2) µν=iπ3 2/braceleftbigggµν 3J0+/parenleftbigg kµkν−gµνk2 3/parenrightbigg IC+/parenleftbigg pµkν+kµpν−gµν2(k·p) 3/parenrightbigg ID +/parenleftbigg pµpν−gµνp2 3/parenrightbigg IE/bracerightbigg . (5.148) Contracting with pµwe obtain : pµI(2) µν=iπ3 2/braceleftbiggpνJ0 3+/parenleftbigg (k·p)kν−pνk2 3/parenrightbigg IC +/parenleftbigg p2kν+(k·p)pν−pν2(k·p) 3/parenrightbigg ID+/parenleftbigg p2pν−pνp2 3/parenrightbigg IE/bracerightbigg .(5.149) 100 Constructing the Vertex Performing the remaining contractions, pνpµI(2) µν=iπ3 2/braceleftbiggp2 3J0+/parenleftbigg (k·p)2−k2p2 3/parenrightbigg IC+4 3p2(k·p)ID+2 3p4IE/bracerightbigg (5.150) and kνpµI(2) µν=iπ3 2/braceleftbigg(k·p) 3J0+/parenleftbigg2 3k2(k·p)/parenrightbigg IC+/parenleftbigg k2p2+(k·p)2 3/parenrightbigg ID +2 3p2(k·p)IE/bracerightbigg . (5.151) In the same way, kµI(2) µν=iπ3 2/braceleftbiggkν 3J0+2 3k2kνIC+/parenleftbigg (k·p)kν+k2pν−2 3kν(k·p)/parenrightbigg ID +/parenleftbigg (k·p)pν−kνp2 3/parenrightbigg IE/bracerightbigg , (5.152) from where, after a second contraction, we obtain : kνkµI(2) µν=iπ3 2/braceleftbiggk2 3J0+2 3k4IC+4 3k2(k·p)ID+/parenleftbigg (k·p)2−k2p2 3/parenrightbigg IE/bracerightbigg (5.153) and pνkµI(2) µν=pπ3 2/braceleftbigg(k·p) 3J0+2 3k2(k·p)IC+/parenleftbigg k2p2+1 3(k·p)2/parenrightbigg ID +2 3p2(k·p)IE/bracerightbigg . (5.154) On the other hand, pµI(2) µν=/integraldisplay d3wp·wwν w4[(p−w)2−m2][(k−w)2−m2] =(p2−m2) 2/integraldisplay d3wwν w4[(p−w)2−m2][(k−w)2−m2] +/integraldisplay d3wwν w2[(p−w)2−m2][(k−w)2−m2] −1 2/integraldisplay d3wwν w4[(k−w)w−m2] =(p2−m2) 2I(1) ν+1 2J(1) ν−1 2Eν(k). (5.155) Similarly, kµI(2) µν=(k2−m2) 2I(1) ν+1 2J(1) ν−1 2Eν(p). (5.156) Constructing the Vertex 101 From the second contraction we obtain : pνpµI(2) µν=(p2−m2) 2iπ3 2[(k·p)IA+p2IB]+1 2pνJ(1) ν−pν 2Eν(k) kνpµI(2) µν=(p2−m2) 2iπ3 2[k2IA+(k·p)IB]+1 2kνJ(1) ν−kν 2Eν(k) kνkµI(2) µν=(k2−m2) 2iπ3 2[k2IA+(k·p)IB]+1 2kνJ(1) ν−kν 2Eν(p) pνkµI(2) µν=(k2−m2) 2iπ3 2[(k·p)IA+p2IB]+1 2pνJ(1) ν−pν 2Eν(p).(5.157) In this way, we arrive at the following system of equations : /parenleftbigg (k·p)2−k2p2 3/parenrightbigg IC+4 3p2(k·p)ID+2 3p4IE=a 2 3k4IC+4 3k2(k·p)ID+/parenleftbigg (k·p)2−k2p2 3/parenrightbigg IE=b 2 3k2(k·p)IC+/parenleftbigg k2p2+(k·p)2 3/parenrightbigg ID+2 3p2(k·p)IE=c,(5.158) where a=2 iπ3pµpνI(2) µν−p2 3J0 b=2 iπ3kµkνI(2) µν−k2 3J0 c=1 iπ3/bracketleftbigg kνpµI(2) µν+pνkµI(2) µν−(k·p) 3J0/bracketrightbigg . (5.159) Solution to the system are : IC(k,p) =1 ∆2/braceleftigg p2J0+/bracketleftbig p2(k·p−2k2)−m2(k·p−2p2)/bracketrightbigIA 2 −p2(p2−m2)IB 2+ (k·p−2p2)JA 2−p2JB 2−k·p k2I(k) +mk·p k2(m2−k2)/bracerightigg , ID(k,p) =1 2∆2/braceleftigg −2k·pJ0+/bracketleftbig k2(3k·p−p2)−m2(3k·p−k2)/bracketrightbigIA 2 +/bracketleftbig p2(3k·p−k2)−m2(3k·p−p2)/bracketrightbigIB 2+ (3k·p−k2)JA 2 +(3k·p−p2)JB 2+I(k) +I(p)−m m2−k2−m m2−p2/bracerightigg , IE(k,p) =IC(p,k). (5.160) This completes the calculation of the vertex to O(α). 102 Constructing the Vertex 5.2.4 Transverse Vertex to One loop We can subtract from the full vertex, Eq. (5.11), the longitu dinal vertex, Eqs. (5.6) and (5.7), and obtain the transverse vertex to O(α). Let us recall that the trans- verse vertex Γµ T(k,p) can be written in terms of 8 basis vectors as follows : Γµ T(k,p) =8/summationdisplay i=1τi(k2,p2,q2)Tµ i(k,p), with the basis : Tµ 1= [pµ(k·q)−kµ(p·q)] Tµ 2= [pµ(k·q)−kµ(p·q)] (∝ne}ationslashk+∝ne}ationslashp) Tµ 3=q2γµ−qµ∝ne}ationslashq Tµ 4=q2[γµ(∝ne}ationslashk+∝ne}ationslashp)−kµ−pµ]−2(k−p)µkλpνσλν Tµ 5=qνσνµ Tµ 6=−γµ(k2−p2) + (k+p)µ∝ne}ationslashq Tµ 7=−1 2(k2−p2)[γµ(∝ne}ationslashk+∝ne}ationslashp)−kµ−pµ] + (k+p)µkλpνσλν Tµ 8=−γµkνpλσνλ+kµ∝ne}ationslashp−pµ∝ne}ationslashk, with σµν=1 2[γµ,γν]. (5.161) After a lengthy but straightforward algebra, the coeﬃcient sτican be identiﬁed. We prefer to write these out in the following form : τi(k,p) =αgi 5/summationdisplay jaij(k,p)I(lj) +ai6(k,p) k2p2 i= 1,···8,(5.162) wherel2 1=η2 1χ/4,l2 2=η2 2χ/4,l2 3=k2,l2 4=p2andl2 5=q2/4. Functions η1,η2 andχhave been deﬁned above. Similarly, the factors giare−g1=m∆2g2= 2m∆2g3= 2∆2g4=g5= 2m∆2g6= ∆2g7=mg8=m/4∆2. The coeﬃcients aijin the one loop perturbative expansion of the τi, Eq. (5.162), are tabulated below : a11(k,p) =−(ξ+ 2)η1(m2+k·p) a12(k,p) =a11(p,k) a13(k,p) = 4(ξ+ 2)(k2+k·p) (k2−p2) a14(k,p) =a13(p,k) a15(k,p) =−2(ξ+ 2) a16(k,p) = 0 a21(k,p) =−η1/braceleftigg/bracketleftbigg −q2 2m4+/braceleftbig (k·p)2−(k2+p2)(k·p) +k2p2/bracerightbig m2 Constructing the Vertex 103 −q2 4/braceleftbig (k·p)2+k2p2/bracerightbig/bracketrightbigg +(ξ−1) 2χ/bracketleftig −q4m8−q2/braceleftbig (k·p)2+ 2(k2+p2)k·p−5k2p2/bracerightbig m6 +3 2q2(k2+p2)∆2m4 +/braceleftig 2(k4+p4+k2p2)(k·p)3−7k2p2(k2+p2)(k·p)2 +10k4p4k·p−k4p4(k2+p2)/bracerightig m2 +1 2k2p2q2/braceleftbig (k2+p2)(k·p)2−4k2p2k·p+k2p2(k2+p2)/bracerightbig/bracketrightig/bracerightigg a22(k,p) =a21(p,k) a23(k,p) =1 (k2−p2)/bracketleftigg ξ/braceleftbig (k·p)3+k2(k·p)2−3k2p2k·p +2k4k·p+k4p2−2k6/bracerightbig m2/k2 +(k·p)3+ (2k2−p2)(k·p)2+k2p2k·p−2k4k·p−k2p4 +(ξ−1)/braceleftbig (k·p)3+p2(k·p)2−3k2p2k·p +2k4k·p+k2p4−2k4p2/bracerightbig/bracketrightigg a24(k,p) =a23(p,k) a25(k,p) =q2(m2+k·p) +(ξ−1)(q2m2+ (k·p)2−(k2+p2)(k·p) +k2p2) a26(k,p) =m∆2/braceleftigg k·p+(ξ−1) χ/bracketleftig q2k·pm4+ 2(k2+p2)∆2m2 −k2p2/braceleftbig 2(k·p)2+ (k2+p2)k·p−4k2p2/bracerightbig/bracketrightig/bracerightigg a31(k,p) =−η1 2/braceleftigg/bracketleftigg /braceleftbig −2(k·p)2+k4+p4/bracerightbig m4 +2/braceleftbig (k2+p2)(k·p)2+ (k2−p2)2k·p−k2p2(k2+p2)/bracerightbig m2 +1 2/braceleftbig −4(k·p)4+ (k2+p2)2(k·p)2+k2p2(k2−p2)2/bracerightbig/bracketrightigg +(ξ−1) χ/bracketleftigg q2/braceleftbig −2(k·p)2+k4+p4/bracerightbig m8 +2/braceleftig (k2+p2)[−2(k·p)3+ (k2+p2)(k·p)2+ (k4+p4)k·p] 104 Constructing the Vertex −k2p2(3k4+ 3p4−2k2p2)/bracerightig m6 −3 2q2∆2(k2−p2)2m4−2/braceleftig (k2+p2)(k4+p4−4k2p2)(k·p)3 −k2p2(k4+p4−6k2p2)(k·p)2 +2k4p4(k2+p2)k·p−k4p4(k2+p2)2/bracerightig m2 −1 2k2p2q2/braceleftig (k4+p4−6k2p2)(k·p)2+k2p2(k2+p2)2/bracerightig/bracketrightigg/bracerightigg a32(k,p) =a31(p,k) a33(k,p) =ξ/braceleftbig (k·p)3−k2(k·p)2−3k2p2k·p+2k4k·p−k4p2+ 2k6/bracerightbig m2/k2 +(ξ−2)/braceleftbig (k·p)3−(2k2−p2)(k·p)2 +k2p2k·p−2k4k·p+k2p4/bracerightbig a34(k,p) =a33(k,p) a35(k,p) =−(k4+p4−2(k·p)2)/bracketleftbig ξm2+ (ξ−2)k·p/bracketrightbig a36(k,p) =−m∆2/braceleftigg k·p(k2+p2)+2k2p2 +(ξ−1) χ/bracketleftig q2/braceleftbig (k2+p2)k·p+2k2p2/bracerightbig m4 +2(k2+p2)2∆2m2−k2p2(k+p)2/braceleftbig (k2+p2)k·p−2k2p2/bracerightbig/bracketrightig/bracerightigg a41(k,p) =−η1(ξ−1)(k2−p2) 2χ/bracketleftigg −q4m6+3q2/braceleftbig −(k2+p2)k·p+2k2p2/bracerightbig m4 +/braceleftig (k·p)2[4(k·p)2−3k4−3p4−26k2p2] +k2p2[24(k2+p2)k·p−3k4−3p4−14k2p2)]/bracerightigm2 2 +q2 2/braceleftig (k2+p2)(k·p)3+ 2k2p2(k·p)2 −3k2p2(k2+p2)k·p+ 2k4p4/bracerightig/bracketrightigg a42(k,p) =−a41(p,k) a43(k,p) =(ξ−1) k2/bracketleftbig (k2+k·p)(k·p)2+k2(2k2−3p2)k·p+k4(p2−2k2)/bracketrightbig a44(k,p) =−a43(p,k) a45(k,p) = (ξ−1)(k2−p2)q2 a46(k,p) =m(ξ−1)(k2−p2)∆2 χ/bracketleftbig q2k·pm2+ 2(k2+p2)(k·p)2 Constructing the Vertex 105 −2k2p2k·p−k2p2(k2+p2)/bracketrightbig a51(k,p) =−η1/braceleftigg ∆2+(ξ−1) 4χ/bracketleftigg −2q4m6 +6q2/braceleftbig 2k2p2−(k2+p2)k·p/bracerightbig m4−6k2p2q4m2 −q2/braceleftbig (k2−p2)2(k·p)2+2k2p2(k2+p2)k·p−k2p2(k2+p2)2/bracerightbig/bracketrightigg/bracerightigg a52(k,p) =a51(p,k) a53(k,p) =(ξ−1) k2/bracketleftbig (k·p)2+ 2k2k·p−k2(2k2+p2)/bracketrightbig a54(k,p) =a53(p,k) a55(k,p) = (ξ−1)q2 a56(k,p) =−m(ξ−1)∆2 χ/bracketleftbig q2(k2+p2)m2+2(k4+p4)k·p−2k2p2(k2+p2)/bracketrightbig a61(k,p) =−η1(k2−p2) 2/bracketleftigg q2m4−2/braceleftbig (k·p)2−(k2+p2)k·p+k2p2/bracerightbig m2 +q2 2/braceleftbig (k·p)2+k2p2/bracerightbig +q2(ξ−1) χ/bracketleftigg q2m8+ 2/braceleftbig (k·p)2+ (k2+p2)k·p−3k2p2/bracerightbig m6 −3 2q2∆2m4−2/braceleftbig (k2+p2)(k·p)3−k2p2(k·p)2−k4p4/bracerightbig m2 −1 2k2p2q2/braceleftbig (k·p)2+k2p2/bracerightbig/bracketrightigg/bracketrightigg a62(k,p) =−a61(p,k) a63(k,p) =−/bracketleftigg ξ/braceleftbig (k2+k·p)(k·p)2+k2(2k2−3p2)k·p −k4(2k2−p2)/bracerightbig m2/k2 +(ξ−2)/braceleftbig (2k2−p2+k·p)(k·p)2−k2(2k2−p2)k·p−k2p4/bracerightbig/bracketrightigg a64(k,p) =−a63(p,k) a65(k,p) =−q2(k2−p2)/bracketleftbig ξm2−(ξ−2)k·p/bracketrightbig a66(k,p) =−m(k2−p2)∆2/bracketleftigg k·p+(ξ−1) χ/parenleftbig q2k·pm4+ 2(k2+p2)∆2m2 −k2p2q2k·p/parenrightbig/bracketrightigg 106 Constructing the Vertex a71(k,p) =−η1(ξ−1) 4χ/bracketleftigg −2q6m6−6q4/braceleftbig (k2+p2)(k·p)−k2p2/bracerightbig m4 −3q2/braceleftbig ((k·p)2+k2p2)(k4+p4+6k2p2)−8k2p2(k2+p2)k·p/bracerightbig m2 +q2/braceleftig (k2−p2)2(k·p)3+ 4k2p2(k2+p2)(k·p)2 −k2p2(3k4+ 3p4+ 10k2p2)k·p+ 4k4p4(k2+p2)/bracerightig/bracketrightigg a72(k,p) =a71(p,k) a73(k,p) = (ξ−1)(k2−k·p) k2/parenleftbig (k·p)2+ 4k2k·p−2k4−3k2p2/parenrightbig a74(k,p) =a73(p,k) a75(k,p) = (ξ−1)q4 a76(k,p) =m(ξ−1)∆2 χ/bracketleftigg q2/braceleftigg (k2+p2)k·p−2k2p2/bracerightigg m2 +2(k4+p4)(k·p)2−4k2p2(k2+p2)k·p−k2p2(k4+p4−6k2p2)/bracketrightigg a81(k,p) =−η1(ξ+ 2) 2q2(m2+k·p) a82(k,p) =a81(p,k) a83(k,p) = 2(ξ+ 2)k·q a84(k,p) =a83(p,k) a85(k,p) =−(ξ+ 2)q2 a86(k,p) = 0. (5.163) An important point to note is that these coeﬃcients do not con tain any trigonometric function, as it has been extracted out for rai sing theτito a non- perturbative status. The τihave the required symmetry under the exchange of vectorskandp. All theτiare symmetric except τ4andτ6which are antisym- metric. Note that the form in which we write the transverse ve rtex makes it clear that each term in all the τiis either proportional to αI(l) orα/(k2p2). We shall see that this form provides us with a natural scheme to a rrive at its simple non-perturbative extension. A few comments in comparison with the work by Davydychev et. al. [67], are as follows: (i) None of the τiwe have calculated has kinematic singularity whenk2→p2. This clearly suggests that the choice of the basis {Tµ i}suggested by Kızılers¨ u et. al. is preferred over the one of Ball and Chiu (in QED3 as well) used by Davydychev et. al. [67]. In particular our τ4andτ7are independent of kinematic singularities. (ii) In three dimensions, thei r factorization of the common constant factor in Eq. (E.1) is singular. However, as the divergences completely cancel out, we ﬁnd our expressions more suitable for writing the transverse vertex in three dimensions. (iii) With the way we expressJ0, all the Constructing the Vertex 107 τiare written in terms of basic functions of kandpand a single trigonometric function of the form I(l). This form plays a key role in enabling us to make an easy transition to the possible non-perturbative str uct ure of the vertex, as explained in the next section. Moreover, with the given form ofJ0, a direct comparison can be made with the massless case. 5.3 Non-perturbative Form of the Vertex (Ef- fective Transverse WTI) 5.3.1 On the Gauge Parameter Dependence of the Vertex Let us ﬁrst look at the τiin the simpliﬁed massless case, with the notation k=√ −k2,p=/radicalbig −p2andq=/radicalbig −q2[57, 68], τ2=απ 41 kp(k+p)(k+p+q)2/bracketleftbigg 1 + (ξ−1)2k+ 2p+q q/bracketrightbigg ,(5.164) τ3=απ 81 kpq(k+p+q)2/bracketleftbig 4kp+ 3kq+ 3pq+ 2q2 + (ξ−1)(2k2+ 2p2+kq+pq)/bracketrightbig , (5.165) τ6=απ(2−ξ) 8k−p kp(k+p+q)2, (5.166) τ8=απ(2 +ξ) 21 kp(k+p+q). (5.167) It is interesting to note that the existence of the factor k−p kp=−/parenleftbigg1 k−1 p/parenrightbigg in eq. (5.166) puts τ6on a diﬀerent footing as compared to the rest of the τi. The reason is that in the massless limit, the Fermion Propaga tor is simply 1 F(p)= 1 +παξ 41 p, implying 1 F(k)−1 F(p)∝/bracketleftbigg1 k−1 p/bracketrightbigg . Therefore, the relation of τ6with the Fermion Propagator of the type [1 /F(k2)− 1/F(p2)] seems to arise rather naturally : τ6=−1 2ξ2−ξ (k+p+q)2/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg , (5.168) 108 Constructing the Vertex as noticed ﬁrst by Curtis and Pennington [37] (note however t hat their coeﬃcient is not the same). In the rest of the τi, the factor 1 /k−1/pdoes not arise. However, one could introduce it by hand to arrive at the follo wing expressions : τ2=−1 ξ1 (k2−p2)(k+p+q)2/parenleftbigg 1 + (ξ−1)2k+ 2p+q q/parenrightbigg ×/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg (5.169) τ3=−1 2ξ/bracketleftbig 4kp+ 3kq+ 3pq+ 2q2+ (ξ−1)(2k2+ 2p2+kq+pq)/bracketrightbig q(k−p)(k+p+q)2 ×/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg (5.170) τ8=−2(2 +ξ) ξ1 (k−p)(k+p+q)×/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg . (5.171) Equations. (5.168-5.171) represent a non-perturbative ve rtex which is in agree- ment with its complete one-loop expansion. This vertex has b een constructed in accordance with the form advocated, e.g., in [7, 37, 42]. T here are a couple of important points which need to be discussed here : •There is an explicit dependence on the gauge parameter, ξ. A widespread belief has been that the gauge dependence of the vertex shoul d solely arise through functions F(k2) andF(p2), and there should be no explicit appearance of the gauge parameter ξ. Such a belief has been expressed (or is reﬂected) in various works to date, e.g., [7, 18, 37, 42 , 56]. Here we show that at least in massless QED3, such a construction is no t possible. •One of the main reasons that the transverse vertex was believ ed to be proportional to the factor /bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg was the assumption that the transverse vertex vanishes in th e Landau gauge, based upon perturbative results. A complete one-loo p calcula- tion reveals that the transverse vertex does not vanish in th e Landau gauge. Moreover, an explicit presence of the gauge paramete r in the non- perturbative form of the vertex tells us that the presence of the factor [1/F(k)−1/F(p)] is no longer a guarantee that the transverse vertex vanishes in the Landau gauge. We now show that the explicit dependence of the vertex on the g auge pa- rameterξis unavoidable in massless QED3. We notice that at the one loo p level, each of the τican be written in the following form : τi(k,p,q) =αai(k,p,q) +αξ bi(k,p,q). Constructing the Vertex 109 On the other hand, Eq. (5.5) yields the following form for F: 1 F(p)= 1 +αξci(p). If we want to write the non-perturbative form of the τiin terms of 1 /F(p) and 1/F(k) alone and we do not expect explicit presence of α, the only way to get rid ofξdependence is to have b2Tµ 2+b3Tµ 3+b6Tµ 6+b8Tµ 8= 0. It is not possible as Tµ iform a linearly independent set of basis vectors. There- fore, any construction of the 3-point vertex will surely hav e an explicit depen- dence on the gauge parameter. Owing to these reasons, we real ize that to demand the transverse vertex to be proportional to [1 /F(k)−1/F(p)] is artiﬁ- cial (apart from τ6) and is not required. Therefore, we do not pursue this line of action anymore. In the next section, we move on to construc t the vertex for the massive case inspired from our perturbative results. 5.3.2 Non-perturbative Vertex (Eﬀective Transverse WTI) As pointed out in the previous section, each term in all the τiis either propor- tional to the trigonometric function αI(l) orα/(k2p2). On the other hand, the perturbative expressions for M(p) andF(p), Eqs. (5.5), permit us to write : 1 F(k)−1 F(p)=α k2p2ξ 2/bracketleftbig k2A(p)−p2A(k)/bracketrightbig , (5.172) where A(p) =/braceleftbig m−(m2+p2)I(p)/bracerightbig (5.173) and ξ 2(2+ξ)l2I(l)/bracketleftbiggM(l) F(l)−m/bracketrightbigg −/bracketleftbigg 1−1 F(l)/bracketrightbigg =ξ(m2+l2) 2l2αI(l).(5.174) In the massless limit, Eq. (5.172) simply reduces to 1 F(k)−1 F(p)=απξ 4/bracketleftbigg1 k−1 p/bracketrightbigg in the Euclidean space, as expected. It was in fact an analogo us massless ex- pression in the limit when k>>p that inspired Curtis and Pennington, [37], to propose their famous vertex in QED4. Here, we are extending t he reasoning to all the momentum regimes in the massive QED3. Fortunate simu ltaneouss oc- currence of the factor α/(k2p2) in all the 8 Eqs. (5.162) and Eq. (5.172), and the presence of the same trigonometric factor I(l) in the expressions for the vertex as 110 Constructing the Vertex well as the propagator, one naturally arrives at the followi ng non-perturbative form ofτi: τi=gi/braceleftigg5/summationdisplay j=1/parenleftigg 2aij(k,p)l2 j ξ(m2+l2 j)/bracketleftigg ξ 2(ξ+2)l2 jI(lj)/parenleftbiggM(lj) F(lj)−m/parenrightbigg −/parenleftbigg 1−1 F(lj)/parenrightbigg/bracketrightigg/parenrightigg +2ai6(k,p) ξ[k2A(p)−p2A(k)]/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg/bracerightigg . (5.175) By construction, in the weak coupling regime, this non-pert urbative form of the transverse vertex reduces to its corresponding Feynman expansion at the one loop level in an arbitrary covariant gauge and in all mome ntum regimes. Note that we have managed to write the transverse vertex sole ly as a func- tion of the fermion propagator. Therefore, eﬀectively, we h ave a WTI for this part of the vertex. We would like to emphasize that this is not a unique non- perturbative construction. However, it is probably the mos t natural and the simplest. A two loop calculation similar to the one presente d in this chapter, and the LKF transformation for the vertex should serve as tes ts of Eq. (5.175) or guides for improvement towards the hunt for the exact non-pe rturbative vertex. On practical side, the use of our perturbation theory motiva ted vertex in studies addressing important issues such as Dynamical Mass Generat ion for fundamen- tal fermions should lead to more reliable results, attempti ng to preserve key fe atures of gauge ﬁeld theories, e.g., gauge independence of p hysical observables. A computational diﬃculty to use the above vertex in such calc ulations could arise as the unknown functions FandMdepend on the angle between kand p. This would make it impossible to carry out angular integrat ion analytically in the SDE for the Fermion Propagator. This problem can be cir cumvented by deﬁning an eﬀective vertex which shifts the angular depen dence from the unknown functions FandMto the known basic functions of kandp. This can be done by re-writing the perturbative results, Eq. (5.162) , as follows : τi(k,p) =αgi/bracketleftbigg bi1(k,p)I(k) +bi2(k,p)I(p) +ai6(k,p) k2p2/bracketrightbigg ,(5.176) where bi1(k,p) =ai1(k,p)I(l1) I(l3)+ai3(k,p) +1 2ai5(k,p)I(l5) I(l3),(5.177) bi2(k,p) =ai2(k,p)I(l2) I(l4)+ai4(k,p) +1 2ai5(k,p)I(l2 5) I(l4).(5.178) This form can now be raised to a non-perturbative level exact ly as before, with the only diﬀerence that the functions FandMare independent of the angle between the momenta kandp: τi=gi/braceleftigg2/summationdisplay j=1/parenleftigg 2bij(k,p)κ2 j ξ(m2+κ2 j)/bracketleftigg ξ 2(ξ+ 2)κ2 jI(κj)/parenleftbiggM(κj) F(κj)−m/parenrightbigg Constructing the Vertex 111 −/parenleftbigg 1−1 F(κj)/parenrightbigg/bracketrightbigg/parenrightbigg +2ai6(k,p) ξ[k2A(p)−p2A(k)]/bracketleftbigg1 F(k)−1 F(p)/bracketrightbigg/bracerightigg ,(5.179) whereκ1=kandκ2=p. 5.4 Comments on the d−dimensional Case Following Davydychev et. al. , [67], two of the Master Integrals needed to cal- culate are : I(ν1,ν2,ν3)≡/integraldisplayddw [(p−w)2−m2]ν1[(k−w)2−m2]ν2[w2]ν3,(5.180) I(ν1,ν2,ν3)≡/integraldisplayddw [(p−w)2]ν1[(k−w)2]ν2[w2−m2]ν3. (5.181) In terms of these integrals, one can write out the Fermion Pro pagator and fermion-boson vertex in the context of QED in arbitrary dime nsions. 5.4.1 Fermion Propagator Fermion Propagator in the d-dimensional case at the one-loop order can be written as : 1 F(p)=e2 i(2π)d(d−2)ξ 2p2/bracketleftbig (p2+m2)I(0,1,1)−I(0,0,1)/bracketrightbig , M(p) F(p)=e2 i(2π)dm(d−1−ξ)I(0,1,1). (5.182) The integralI(0,1,1) is thed-dimensional counterpart of our I(y) function in the corresponding equation in three dimensions. 5.4.2 The Transverse Vertex The corresponding d-dimensional τ’s of the transverse vertex can be expressed as, [67] : τi=e2 i(2π)d/braceleftigg ti,0I(1,1,1) +ti,1[−(k·q)I(0,1,1) + (p·q)I(1,0,1) +q2I(1,1,0)] +ti,2(I(0,1,1) +I(1,0,1)−2I(1,1,0)) +ti,3/parenleftbigg I(0,1,1) +I(1,0,1)−2I(0,0,1) m2/parenrightbigg +ti,4(I(0,1,1) +I(1,0,1)) +ti,5I(0,1,1)−I(1,0,1) k2−p2/bracerightigg ,(5.183) 112 Constructing the Vertex where theti,j,j= 1...5 are basic functions of k,pandq, listed in [67]. A straightforward procedure to obtain a non-perturbative ve rtex in this case is a natural thing to ask for. However, a diﬁculty arises since th e functionI(1,1,1) in general cannot be rewritten as a linear combination of the integrals which appear in the Fermion Propagator in an obvious way. Note that in threee dimensions I(1,1,1) has the same structure as the integrals which deﬁne the Fermion Propagator, though with more complicated argument s. Therefore, we have not yet been able to write a simple non-perturbative ext ension of the transverse vertex in arbitrary dimensions. 5.5 Comments on the Two Loop Case As we have mentioned earlier in this chapter, the three point vertex must reduce to its all order Feynman expansion in the weak coupling regim e. What we have achieved in our work is the realization of this statement at t he one loop level. To be able to do so beyond this order, we must know the perturbati ve vertex at the higher loops to have a guide for the construction of its non-p erturbative coun- terpart. A good amount of progress has already been made in th e calculation of necessary integrals involved. Most of these attempts sur ge from the precision analysis in particle physics phenomenology at the next-to- next-to-leading order calculations in perturbation theory. However, the scatter ing (see for example [85]) or decay processes (see for example [86]) do not in gene ral require the calculation of vertex-type integrals with oﬀ-shell fermio ns and bosons. A com- plete two-loop calculation of the transverse vertex in arbi trary covariant gauge in arbitrary dimensions, to the best of out knowledge, still d oes not exist for oﬀ-shell particles in the legs. Using Dimensional regularization with d= 4−ǫ, the integrals appearing in the calculation of two-loop corrections take the generic fo rm [87] : I(p1,...,pn) =/integraldisplayddw1 (2π)d/integraldisplayddw2 (2π)d1 Pm1 1...Pmt tSn1 1...Snqq, (5.184) wherePiare the propagators, depending on w1,w2and the external momenta p1...pnwhileSiare scalar products of a loop momentum with an external momentum or of the two loop momenta. The integral is speciﬁed by the powers mi, (mi≥1) of all propagators ( P1,...,Pt) and by the section ( S1,...,Sq) of scalar products and their powers ( n1...nq), (ni≥0). Progress in the calculation of two-loop corrections to four -point amplitudes was based on an eﬃcient procedure to reduce the large number o f diﬀerent scalar integrals to a very limited number of so-called Maste r Integrals. The scalar integrals are related among each other by various ide ntities. One class of identities are the so-called integration by parts (IBP) ide ntities : /integraldisplayddw1 (2π)d/integraldisplayddw2 (2π)d∂ ∂wµ 2V(w,...) = 0, (5.185) Constructing the Vertex 113 whereVis any combination of propagators, scalar products and loop momen- tum vectors. Vcan be a vector or tensor of any rank. IBP identities are a generalization of (5.92) and they follow from the fact that t he integral over the total derivative with respect to any loop momentum vanishes . Another class of identities is obtained from the fact that al l integrals under consideration are Lorentz scalars, which are invariant und er Lorentz transfor- mations of the external momenta [88]. These Lorentz invaria nce (LI) identities are obtained from :/parenleftbigg pν 1∂ ∂pµ 1−pµ 1∂ ∂pν 1+...+pν n∂ ∂pµ n−pµ n∂ ∂pνn/parenrightbigg I(p1,...,pn) = 0.(5.186) Master Integrals can be analytically expressed in series of the Dimensional Regulator parameter ǫusing Feynman or Schwinger parametrizations. However, sometimes using these parametric representations does not leave an integral that can be solved easily. Diﬀerent techniques arose due to this s ituation, among others : •The Mellin-Barnes method is based on the representation for a sum to some power, as a contour integral over a complex variable and the integra- tion is then performed on straight contour lines parallel to the imaginary axis. After closing the contour, the result is the sum of all e nclosed residues that might be expressed as a hypergeometic series. For the tw o-loop boxes we have [89]. •The Negative Dimensions technique consists in rewriting th e integral over the parameters by introducing new ones through a multinomia l expansion. Many conditions have to be satisﬁed among the parameters, wh ich leads to the restriction that dmust be a negative integer. Some results are[90]. •Double-Integral representation, that consist on splittin g the loop space into the parallel and perpendicular subspaces to the extern al momentum. Then, using cylindrical instead of spherical coordinates, one can calculate the real and the imaginary parts of the integrals separately , performing contour integrals. For some results, see [91]. •Strategy of Expansion by regions [92], in which, instead of t he integration over the whole space of loop momenta, the integration is perf ormed only over some speciﬁc regions. The strategy consists in conside ring various regions of the loop momenta and expand, in every region, the i ntegrand in Taylor series with respect to the parameters that are cons idered small in the given region; integrate the integrand expanded in eve ry region and; put to zero any scaleless integral. •Dispersion relations are used to calculate self-energies f or the small exter- nal momentum behavior [93]. •Numerical strategies are used, where any loop integral is st ripped analy- tically of its IR singularities so that the ﬁnite integrals c an be performed numerically. [94]. 114 Constructing the Vertex •The analytic evaluation of MI can also be carried out without explicit integration over loop momenta by deriving diﬀerential equa tions for MI in internal propagator masses or external momenta. The equati ons can be solved with appropriate boundary conditions [95]. 5.5.1 Two-Loops Fermion Propagator The two-loops contributions to the fermion propagator are d epicted in the fol- lowing Diagram : Diagram 9 : Two-Loop Corrections to the Fermion Propagator. Using the Double-Integral representation method, the Ferm ion Propagator has been calculated in [91] for arbitrary masses. Scalar self-e nergies have been cal- culated using dispersion relations [93]. Using the Pinch Te chnique [96], oﬀ-shell self-energies have been calculated, in terms of Harmonic Po lylogarithms. Also, an asymptotic expansion has been used to calculate the on-sh ell master dia- gram for the two-loop propagator[97]. This expansion is tak en in the limit of the external momenta, and therefore the fermion masses, ten d to inﬁnity. On- shell self-energies master integrals with one mass have bee n calculated in [98]. In the light-cone gauge, the fermion propagator has been cal culated in [99]. Following the algorithm developed in [100], these correcti ons have been calcu- lated in an arbitrary covariant gauge for oﬀ-shell massive f ermions in arbitrary dimensions[101]. Constructing the Vertex 115 5.5.2 Two-Loops Vertex The two-loops corrections for the vertex are shown in Diagra m (10) : (a) (b) (c) (d) (e) Diagram 10 : Two-Loop Corrections to the Vertex. Some partial calculations to the full vertex using the doubl e integral repre- sentation, planar vertex in four dimensions has been presen ted in [91]. In Ref. [102] a recent on-shell calculation for the vertex has been d one at arbitrary momentum transfer in arbitrary dimensions. The planar vert ex ((a) in Figure 5) with essential on shell singularities has been calculate d in arbitrary dimen- sions [103] applying the strategy of expansion by regions. T he calculation of a the vertex function at zero momentum transfere and with spac e-like external momentum [104]. Vertex with two legs on-shell in arbitrary d imensions has been calculated in [105] using the Negative Dimensions tech nique. Numerically speaking, the massless vertex has been calculated in the Lan dau gauge [106]. For non-planar vertex diagrams ((b) in the corresponding ﬁg ure), a numerical method is propossed in [107]. In the Sudakov limit, the Verte x has been cal- culated using the Mellin-Barnes method in arbitrary dimens ions [108]. A full oﬀ shell calculation of the two-loops vertex has not been car ried out yet. A programme towards the construction of the full two-loop ver tex has to take into account the transverse part of it, so that one can check of mod ify the corres- 116 Constructing the Vertex ponding basis{Tµ i}. With this basis one can look for a nonperturbative vertex in arbitrary dimensions, or, at least, in our three dimensio nal case. Once we are able to construct a three-point vertex which agre es with its perturbative expansion in the weak coupling regime to all or ders, we shall rest assured that all the gauge identies are satisﬁed at every ord er of perturbation theory. We can then hope that the physical observables assoc iated with non- perturbative physics shall automatically be gauge indepen dent. Chapter 6 Discussion and Conclusions In this thesis we have focused our atention in incorporating gauge identities (Ward-Green-Takahashi identities and Landau-Khalatniko v-Fradkin transfor- mations) to the non perturbative study of the Schwinger-Dys on Equations in QED3. We know that Perturbation Theory is the only scheme whe re in each order of approximation, such identities are satisﬁed, and a lso it veriﬁed that, indeed, physical observable quantities are independent of the gauge parameter. However, in a non perturbative analysis of the Schwinger-Dy son Equations it has been impossible the conjunction of these two facts. A trace back for the origin of the gauge dependence points out to the fol- lowing problems : 1. Violaton of the Ward-Green-Takahashi Identity. 2. Incorrect gauge behavior of the Green’s functions. 3. Use of wrong technical tools, like gauge dependent regula rization schemes. In order to avoid that the technical details overlap the pure theoretical issues of the study, we choose QED3, because in a theory that lacks of ultraviolet divergences. Therefore, the source for the gauge dependenc e should be only due to the ﬁrst two points above mentioned. In Chapter 3 we considered the simplest scenario, and we calc ulated the mass and the chiral condensate by solving the Schwinger-Dys on Equation for the Fermion Propagator. We explicitly observed that these q uantities depend upon the gauge parameter. Since an incorrect regularizatio n cannot induce such dependence, the only room for improvement is in the three-po int vertex ansatz. An ansatz should be such that : •It satisﬁes the Ward-Green-Takahashi Identity, •The Fermion Propagator obtained from this ansatz and the ver tex itself should satisfy their corresponding Landau-Khalatnikov-F radkin transfor- mation. 117 118 Discussion and Conclusions The incorporation of the Ward-Green-Takahashi Identity is straightforward. However, as we did in Chapter 4, even in the simplest case, the Landau- Khalatnikov-Fradkin transformation for the Fermion Propa gator leads to a gauge behavior extremely complex for this Green’s function , and in spite of this, we were able to extact valuable information about its s tructure beyond the tree level. The search for a non perturbative vertex which le ads to this gauge bahavior for the Fermion Propagator is not at all a trivial ta sk. It is here that Perturbation Theory plays its role. If we can put forward to a non perturba- tive ansatz which reduces to its Feynman expansion to all ord ers in the weak coupling regime, we can hope that it automatically incorpor ates the Landau- Khalatnikov-Fradkin transformations not only for the Ferm ion Propagator, but for the Vertex itself. Therefore, in Chapter 5 we calculated the Fermion-Boson Ver tex at one-loop level outside of the quiral phase of QED3, previously studie d. We decompose this vertex into its longitudinal and transverse parts. The former guarantees the validity of the Ward-Green-Takahashi Identity, since i t is related to the Fermion Propagator calculated at the same order of approxim ation, while the later ensures that the Landau-Khalatnikov-Fradkin transf ormations are indeed satisﬁed. Taking advantage of the lack of additional constr aints, we exploited the perturbative form of the Fermion Propagator in such a way that we could write an eﬀective Ward-Green-Takahashi Identity for the tr ansverse vertex. This identity allows us to have the ﬁrst insight on the non perturb ative structure of the Fermion-Boson interaction, since the transverse verte x, related in this way to the Fermion Propagator does not depend explicitly on the e lectromagnetic coupling. We write, afterwards, the transverse vertex in a c onvenient form that will allow, to corroborate its non perturbative structure i n a future work at the two-loop level. The systematic construction of the three point vertex guara ntees that : •The Ward-Green-Takahash identity, which relates the Fermi on Propagator to the Fermion-Boson vertex, is satisﬁed non-perturbative ly. •Both the vertex and the resulting Fermion Propagator satisf y Landau- Khaltnikov-Fradkin transformations to O(α) andO(α2) respectively. •The vertex does not contain any kinematic singularities whe nk2→p2. •Most importantly, the vertex reduces to its correct perturb ative expansion atO(α) in the weak coupling regime in an arbitrary covariant gauge and for all momentum regimes. •The vertex has correct symmetry under the parity, charge con gugation and time reversal operations. This will allow that the technique as well as the reasoning us ed in our cons- truction could be applied in more complicated theories like QCD in a suitable and realistic way. 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