K z1 z1 z2

{ K(z,2,%,) +PhilLucht | | { a 1 TheKinematic k-functiom andRelatedshine.” ~ e i— , , '- | .Contents: . |- 1.Introduction |- - 2.Geometric Interpretation ' 2.1 Elliptic Case ! 2.2 Hyperbolic Case ! 2.3 Terminology - 3.Some j-k formulas. , . 3.1 the j= integral representation.3.2someTieesels involving J=E :3.2.1Thecase{3}*1/(g-a) 3.2.2 The case f(z) »P(x) . . 3.3diagonalising elastic unitarity thehardway 3.4asumrepresentation forre --4.Some J+k' formulas. i4.1 the [4 integral representations 4.2someintegrals involvin, ay _ 4.2.1 Thecase¥(2) U/(z-a) _ e 4.3relation of4.2.1 to8fahous integral -4.4.theQ9(z)separation integral - i j —-- . i - 1 - -+ - 1 {- : e { i , __.Introduction The"k-function", wee - @ ~ ~~REWe)=ered Rayeel wen eee ee! —- ae -- e isone-ofasmallsetoffunctions whichappear quite-commonly inparticle kinematics calculations. Another ofthese unsung.heroes. ofthekinematic -high wireisthemorecommonplace “triangle function" ,_ . ~Aus ~Geghes® Ixy CeeAge>oO socalled-because thearea-of-a trianglepFsides-a,b, andc-isequaltosae .Thereseems.to benosuchglamorous interpretation.fork(x1¥42))80-we--justcallit- “thek-function."_ rs - --- Considered asafunction ofoneofitsvariables, thek-function isjuet— anon-descript quadratic form, hardly worthy ofconsideration. Nevertheless, its persistence inappearing 9211overtheplacer hasmotivated theauthortocollectseveral formulas involving thek-functior{ intooneplace. 7 ‘Noneof’thé‘fovmilas” presented hereisew.Ihfact,mostofthemare given in-an-appendix of-a1962paper ofdeAtfaro; Regge andRossetti [|}. @-Some-aleo-appear-in-the brief-appendix of!reference- (7Z.).--Gn-the otherhand,.onedoes_not_ find.theseresults. insuch.clomprehensive worksas-Bateman (2)and . Gradshteyn andRyzhik(¥¢). ae Lee @.GeonetricInterpretation, The——isofcoursesymmetricinitsthree arguments, whicharetakenherefoberegl,butwechoose oneargument 2toclassify“things"below. Oneisalwaysinterested inwher’éthek-funétion vanishes,and where itispositive ornegative. This information isbest summarized graphically. R@aAWD =eSeabee Qeea.-| 2.1theelliptic case.Forfixed2,/2f<1,theboundary k(z,2f1,22) =0appearsasanellipse inthefi-z2plane Thisellipse isrotated 45°relative totheaxes andisinscribed inthe “unit! square" whose edge isactually two unite.THNOGUODGKE isnegativdinsidetheellipse,positiveoutside.Inthefigure(|)wehavedrawnthisellipse forO<¢2<1. Obviously ittilts ‘the other way'if -1¢ 20. When 2-0 we"have &unit circle. _o r — ue =] —— . ' . a ;Le ’ ‘ t eee te -- +) ax) . -- ;|Cy|_ -- _ = - - - eee - =] - - a -FaSnob2,\Ay1y) -eaee2.2thehyperbolic case.Nowtheboundary} k(z,21122) =Oisahyperbola inthe _.21-2planeasshownpce kisnegative insidethehyperbola, positive outside. 7"The sketch isshown forz>1. For2<~1thehyperbola tilts theother way. ~ ata=wax 1+ €4the hyperbola isvery thin andnearly touches the unit box, afdas2goest01-€,ittransforms intstHin“ellipse “insidetheboxcyaabove. e- 1 Akey - pote HAV A ~oeoe -Bo WX) (=) _a a 4—a a BZ | WN/i .-ohy --Fay- \ . i = eee , H ’ . i 2 ,i ___Bebiterminologye Letusdefinesomepepterminology todescribe the_ a___locationofapoint(21122)inthe24-22Plane.tewillsaythat(21,22)_ s is"inside" ifboth (@\ <\and\2el<l) .Tf(@d>t and\2%>1, then 7 “(24122)is“outside”. Otherwise,Gaps"mixed."Theseregionsareshown below.Te /7 a ooopdsite|Ag)” east a 77; -a “==gieag—be ose -- - ce wee eee eT a oe ~3.SomeS=Kformless ~~~ f-- - ~ waeee eh _--31he\-K.Integral. Representation, —Theususlelastic unitanity’ integral~-hastheform92 oe -aeae an —-a@\= Sartore) 2... Cb). ewhere, 7 -.B=A(a) =ae"4JORG BY)es" ByAntroduing 1=Sae!5(@!-?') {negets, - - -so ne --——-}t- ari_0)=See"Sae!be“p-LaeS28). C2) - Weidentify this Zast~integral-as ajacobian, but“evalllite itdirectly: ° acd) ak vim -J=3G7SavBoB. spans we . FSAad Se-ae”—AMAG-29 ws¥)_ _-- _. ~ x<ase-8}|Sheek -_— SOBIGEES SISA OP TID ST . ; iwhere cos@*2 =BAL |, 4encesnoiisthat, © A=e)O-et)|See a oClerF=i) =.SCkese)) cee —\eencea")) sad[ =onKesia a , M, ’ Thus, wearriveatthefollowing"teers,fepresentation”TorJ-K*:- e@x 1 ek(22,2) 77Sy oe ~ rT See =NagS(etees -SOR(GRDeas).(3) A-K@ 2) ° ' - we kee _- a -. This derivation isvalid for(21122) either inside or-outside, where thesquare- rootisreal.For-(21s29) mixed,both sidesvanish; theleftsidedueto.ack), . mitheright sidebecause thesquare robt_goes imaginary andthedelta functionisnot"hit."Thus,theintegral represchtation Sooveisvalidforallreal 2421;29 +Twootherequivalent integret representations areobtained by permuting thez'sintheintegral. This issobecause theintegral isactually symmetric inthez's,astheleftside“Shows, together withthefactthatthe ~therepresentation isvalidfordll2,21pnd-22TFadaonegoudsokwk CA)teodesduneWtapleaigigReeAhbasquatraction Shamaltside. .3.2Soneintegralsinvolving(Xe,vebanabeinterested incertainintegrals_oftheform, 1~ L- eee 7 .: Qy 4)SCkKRaAR) 77 e Ter) =Yaesta SSK Ta) . Le __ .i~KEMyee) According toour discussion ofk,the seghents ofthe 2integration. which actually contribute to-the-c-integretien areee,shownbélow,(nowinthez-2,planey):> { aad Quo,-- - 4 = a a Gite) wsidw (Quon Bz,20) i ze . 7 7 :—(22)adsida (Serww®2d) @ za|. : foot woe eee ee - - 4 - a ; <7"Ayexgliniedpeat> as.setpe—----- e Reaej= Goad@ed 22 Lee Be=am,=Joranar .- “BeyE. FoOeeoGute)anikedsQoaafeqpale i1YDNomarTesouar ikWornueaupporting) asdrmonts okoll,dyMerofrcannywehowe,lywrsanican ofSBYweC4WV- v— - T@ey =hagh@) oj. Cs)whoB=2a,area -~.. 3.2.Aweas&@=CeoS)YihNaWyoCEYondea Lr - .e 2abssx Naeboo ~ ODeanquackeysaadusk-|77 - :5aeSCkeaz) .|_ar , (6) -2FSNSk@a Re).-—iAtk(aato. -- YMepoletea Snaggane dyLhowShawliqakin eortour Bact) ManROB.) peensqactinns ondwaegat,-- ! - i Ak@as) =\G@-eve@dy : To \o-a|e™ a-aie* : sO Teak , Lo. @ Wrign _ _. _ — . Bopading onubullasraSaaeyughalenar ajustVandCracorto: e=geee SJukdhTRER) | 25s7 _./hss,cowalonatheechoesree a. S+ etatro . i Pan . é -Ss,wowBACSy,wetrone |ao=eee- ——Nesdaak. woiho - -eo x ads eckedis) Sapa) 7 Tae)NaeheFees) 34(2)a \o fisCF). Wemrowlairaath QasmualVhgends ~ ACz=Renney+=‘viescoum8) @rdtamedio Nevoay,Sndibovbany? compan Ondy. Tad =TREMA _@) oCk(ea2) fi . (10)FYEHOmaXG- (ae)Uasidn Saat@Ceara *_gayrrien 2,22>) Cu :38)VERA ” wee ekViogyeratuying)Boskiealyybuhardwou, ; - an ae a. LS ooo —— - 4 : “ a "Wale . - . @o=Qasl eye?) =\ae’bopyeee(e)-LOCKER) sandySRAtnethise!andsymeoe ApgduiSd&@)dolooksided’and‘WLA-- CpSune, “yond: - ' .[Sactesace)|=%n[haeadewer)|«[Sadmade]. BMKeatonsoprasmmetion SxSE.|TheBhsogarndein C0Scamlas.Uartedd layoathFarey)see dovaMrsidar,ondbymaineMeepewgicevacr ohNe~Soyuodar_poteprewiinde, [--oeee Be) R= WEB . __ i --- @Queue droidfe] 4 . BA rn =x}7,Un)anhiayhe),~“AcmA~K(@68,88) =|feve“ Equoein, ORYeemedi oySho=)SBBRSIawd R(fa2ar) <0.CQe200zpan! OnJpagee Ly), — po . 4 x . { ’ | @5VTEWeedPeesansasudedtowy.Taurdagaol sepacmstalioss *kamadaqnroo2)“toa.Sollows:_ —SagS(e-am- TENET oh)= Gre) (8)-= \ VR@B20. asddSe-aacAGEVEN dab)= (2-2). Cw)y |:.(e182) .eGkeaa) LoLacsUe-%) Cis). +k(2,%,22) -es eee @yas Fagar TGV deg. Under coer oN(ante)WMxad. “tT me deAsduss anh Url Sung s . aa -* eGkeda _pty. yy Ta).Yeete)*YsKGERND=2A\r8*) WecomarendPinueesStee|3anhYHStinwing QuanombwWouuy, ushaspesiunn Leas uaMw(2,2)plane, Qdew,waasrataoniterofferoxarples) 0k @aayQeudt([8ee oo Sia SG} TIE pay-TeasaseFewep=ERS) : . |‘ i ¢: Wi ZeBee NERVE) Lah, a @5.25Vere 8)=205) a . 1 -~ Quydtackcoetaben cing? - dk TLL9gEy -- - -Nateaex Ace -nal~JaF=be- - : | . H owYd. Mok — ---a)20 i4)TORSaeSee -Le9g,aces dk@emy |-(6) FRB-2Mkeaes Arann [akaTeena |’ , Wh Ay Ifweassume thatadz_, thenk(a,z1+22)>0 and,choose thebranch {kK>@andthe principal branch ofthelogarithm. Forz.4a<.24 we.have, k@O andthe.answer canbe shuffled toread, { - - - - i ant <a yoo (BREA VIOeS —Re— e. SK@RRS T+ A J=R@RR) ’ - —pe te a - - Fora} 2,theresult. (/6)mstbetcotrected” byadding aterm.+aw/JK i “depending onwhether the integration contour!runs above orbelow the pole. Inthis- — casetheresult (/4)stands tiveasalprincipal partintegral. Ofcoursé with theright interpretation ofthebranches of{heLogarithm, equation ( )isreally i . velid inellcases. | , .H 43Qipao &C1Db2Loos sistent, - ~ \menever single variable dispersion relation arecombined withelastic uniterity, = - i thefollowing special case ofequation (1])tends toarise:( footnote: for example,inMandlestam's analysisofthedoabrespectralboundary.SeeCollinsand 4 Squirespage35.Notethesoxmxxex ‘yspersphical errorsintheirdefinition of-eo tunctiong|SeeSoeFrddlengatWatsry page605,oqumbins(2IF),(22)}.‘ondoe ear -- .- +Akon Ts Swtv a ~“Benmeth) 23 80re Bec . - - 4 ee eee ee - wet Foo=aa+JOS Ce_eosig! - ---- -os " . i, { |"rissahaqeadhreamleesemdeded onBenes) : 22) ‘i |e or\ a |4#9RSE By oe =.ax aOM.4 -.gSFane,-8(ROT =3%S Lws>ate Saha eosd|”. ‘ -- -Soe--gy-boo -=ary“CRE RR8)4HAE) -- -=42m,ou[ce -.De(aazdLeoB~:a Blean)=debo) una(eaye). =Br1(2,3,% (7) QeunlegredTecowNarronniecaeaforwardiB Nharend laotnd 1-. a re oe a . - cdo oOo =Yes3aSay8-2)\ \- | =X\ot\al |USCKG 22) Cie)| 35SarteN=KG,B29 . ewsaareelarretAframninCS}.Goufe |JroweCoytye wrsn So '; ©aee@ea, Rae QueoCkee) |eS2-=SKeah) YStNeayrEaee) ee ee . 1 ; sent aacThepoint,tonotice hereisthatontheright_side,\ he2~dependence isburied - ® -away-aspart-ofthedefinition. ofthe-boustiary: ofintegration in-thez'-2"plane——(seeellipse-tr-figure i})ywhereasjen-theleft-side thezdepéridérite is~ exteemelysimple.ne i~7 a . : ~~od _a~ wiTheOp—separation integral.Weotekaresult,analogousto(),00ttseens reasonable tostart directly with(1(0.).(footnote: Abarbanel andSaunders used this methody-although they have certdin-endpoint-errors )sFor(z',z") inside theunitbéxwehave, - -—| -— - | - . Suse) Tg ; °, Scklez, = . ”TC) een 7TRuleA") ‘i Y - Aus,Soa!_JSdatdobotleider,odsasSYBERG Be Looe {_ __ . .MNawwmomer Fomuse £5,BELPua)==Qua).dogk - - a ee wee{ a - °et ed fa!&agi OGKE Le)BDQm)y= ti\aeke 3S$\4,- SSSAMAA) =Gaho 3faEaSaauey Brom CII. DacasWeldate,beeones — - S eckt eaten -= Nahe \Fe ESPQRAG NSE Tewaw) ;. . ‘VeoOnRelagsckurs comWedeuraNomen and = -- i - : - -~ o-24) 7apy (2)dzQQ) EO@vore BS SR@8,2) yeas*) Wess 2= 2a+AGATE QashavecedusOe -o i ee ee - - 1 | : | OLD NoTes i 1 Meiyekd tathwb, 18_ . - wee —-.! en wee e@Fail,Sedeot_ _R(asb,e)__a. asta aol) -.ae - -2ne =La> ak\G-a.) ae _G2=betSOE —| we\SeMaweleaden .eye be-\Seiemifx) - (ye: a[Terns|.(Drewsesau&Asal,so20.) - ati@ - oa; ee -.aw“\ Sane~at)=No S(K-k) Gone a =oyx. wwe Ea(AF*). Oa,4per eG). oG-y= S(SE+) ©EULuy) =of@-a) , eee | _. @Reais aGENE>Onaw. |« !- se | -hoo _ ©Ca-ar)| Ve eee Seeye)=Jerrad ©Qdots, Rye ,|©ae 5 e.SseSCa-berak) =HYBeBCA+R)=DeOCH) bys:e(-¥\=e(“8a) =ofarse=oGaslbe-dJ) =oGata) YUVAL A, wwder aS(40,) Sseesi_——— eGara-) : : -Beke1 -Lote , @Vow,wel dakR=G-ax)(a-a-}. -eeeQwa, k eGs7 /.C) eG)=@(arax) +(ata) “ a @>a,).or. (aca). feefoe : PEBCave+ tone) =sore NES C Ney) See 7Sule@GJereceennedDekEe do.Oeeroggene i TS!alma catelW>I,Lelrt:CwSEK)>ind!obte!Dre,RateadeiYatnbetnaflScene'Cowbeide”, - &gvedaseaske<o.WrowWy|. aeo eg°= ®Sages¥nunsiteata\,edtMowYONDER&amagyamdLHS Nwormabue™ ,RUSdoarwa. @So,swhiedunt . =.HAWaste, \eldt, Ma _ m. lcs, (bel, Va. - @Wye ounkeke EV=dd. . :~ $ abe—Ways =S(aras) 7 Qe ee(aba~KETdag)eee ° YeasWoveeSea)=eceres | { pagetitles -- Aras\n5, e 0.Recapofprinciple resultsofthese25poges. .1.Origin ofthefamous kfunction. { 7~ 2.Geometric analysis oftheiefunction, f OO 1 ~ 3.dnintegral representation for]wk Q so ‘heConversion ofabove fohypérbolic fufdtional form. ~ aa 5.Exploration ‘ofalternative hyperbolic forms for 3above. ——-- ~~ ~-- --6r-First-attempt toprove Abarbanel- and.SauhdersWk repforI-(feilure). - | - 7.Evaluations fortheabove. 1 : : 8.More onthe above attempt. L Le . __9+Exercise ontheatbchxintegral. ae ae 10.First sign oferror inASequation (A.7). 11.The sheet structure of1-27. . ~12.Commentsontheintegral repforFk’.{ ~13.Atheorem using>k'. ~ nos - - Ih.Where does the a’+-bcos(x) “integral hate validity? -- oe —45.Atheorem relating -K™tok. ' —— ~ge- 16+Second attempt 0.prove.AS equation (A-?). Failure. . .e 17.Integral formsfortheIfunction ofStanley. _. 18.Third attempt atequation (A.7). Partifl success! 19.First attempt at“symmetry theorem" tofinish (A.7) proof. Errors made here. "20. Example shows uperror in19above} 7 —— ‘21. KCounterproof ofequation (i.7). Itcannot betrue!22."Second attempt’ etsymmetry thesren. sucess. -- - 23.Example verifies symmetry theorem. | - : 2h.Finally, thecorrect formof(A.&7) isfound, conditions for-velidity stated. 25.Improved version oftheHKintegral rep]validforall21and22.sThys,AeI. -| i i 1 ' , | @‘ =--- - 4 ~-- ~ - |- I | August4,1975 Recapofprinciple results sofar. i1.Our-firstsuccesswasacleargeousttteunderstandingof.thek(x,y,z)function. @Seepage 2forellipse, hyperbolas, fegions where k.GT.0, andsoon. 2.Next, weobtained anintegral represgntation for@CK)/J-K ,valid forallreal values ofthe arguments. Ifeny argupent leaves "the unit box", both sidés of thisrelationvanish,fordifferent iSeepage1andpage12.Result:oCkERR) g i SS OE S(e-2e, + Oancee -Jaana” Yas (-22. tJORRERY ws) Ofcoursethishassymmetric variatibna, oryoucanrunto24, 3.Onpage13weprovedalittletheorerinvolvinganintegralover(*k.ItwastSaeSCHL $3:eee ~& Saka) oO!ewixod -Also, somevariations shown onpage13. 4.Next,weconcoctedanintegralrepressntation forOGO/WKSeepages3445andAmproved version onpage25valid forlall 21and22,Regults sor@etd .aGagsle-aacscD bb) 5.Onpage15weshowedaconnection betweena-kanda+k.Resultwast ey; 1 \)deSkea) 1[T/ARGARS | -Qe BK iw =; Be RR TELava, 6.Thenwelooked atthesameintegral with#k.Seepages 17,18. Result wast aes)deoGkeaed) i!ea, p)omte JFKRAD ( \-$— =SaQtn dytatecee) =}2g, [ete +3K) = 3~BohRAEEGnoe) iG|S KCAR)SeA&= wreweEEuntenIl<iSaye-ay(RQ-22) a >Js 2 { =What NieYeauvens ganaunSunnyBOVETUREEK,YA) Sang g =i axwSGana)Se =DrsdiassnevasodotnFru. 7.Finally, onpages 22,23,2h wediscussed acertain symmetry property ofthe e integral repin6:forI,andwederivedtheresultonpage605ofGW: ! eS i & N Berl, Wish =TsUm\SSaw. SeRRae wun beaasiaaae} ob Cereus) 8+Thus,theAbarbanelandSaunderssufi(A.7)isincorrect,since...(seep24). 2 |. 6©Fosenur, omarder pages 2)-GA0.oadeoridunfesdoshinumsdenity ators:~oe ' ag ehoe i a : -Grama, wssrcountor: a2)=Ya be")(2) wwwomen. 2226] =ee!ares cos¢" ©wrod 4-\da,S(@-2")| - teSaedre) | . _Webuaobs2oud2”owwadeCay.Te,2!iaexpbrarkty cy‘ ’-a=Staun,BADR xYAR’SereSez’) @NoseaBowdole Wiisyrcdany: | a ,{ ba4 bone TeaeB)=SagSade!Sede!)8.22? iaTatcoe) _9,2) ° “ 4 QueLedoSamy: dadeae =datdg! Qs2!heyaiehivisl—momadlrallydo.cdrodu'.. So: ar| a TE DadSeaeTEEIHEcond’ =at° ae. \ ;:@te,waowromonlayBeMWUKug?, walawbaqationaanaateto \ : . , 14 oan SP. geo. . _x=Daa"SCa-2a-ieJet<8)ReachVaskMieMing Wihe"jurtea0 atr | (BMS \a oFasBV =2SSrrat MAKER) Sag’s(43), ©Veo. -, (8)o(ar-8). . —ecosgf= (a,— 22) , -+ Yee S}-2" Le Qua: si=\eekf sGM -Greedd Qnty?) =\-t-2 sth~2-3 yeCRRae =—k(2, B22) . weeeIS ‘-[OREOICs0ee Qua:sind=Jkane) Jroguanhex! r)‘AESTered i ©WoskteDeeomanGak,Leos <1?Gaand {sind<i.Sawaw(eos\*<1.AeCasQy<\.. de, \-s@F <1 esudyo. So e(\-ltead l= Qoanrsh.Qa SOCK2,23). » Wrs= WOCkKeaxd) . NKRED e amd1 \ ates=\ee (2)e+ 2OC-kCe,2,80) N “st N~K@,2,2) . 1 X © keaa =eet maes i : . ®DWaskwe(henwtoredWeout,R=OWwW2deonds? FroPa WUE Trower weFras Mieioacme Qhtv |Le?) oytaBuelwhat CAz2)~C-\eo Qedivouminadk la: Toyae =(=1)\Que: eist =alps, who=Suyguterta. alayearine. @Wosighs seem:aeaume (1S,doaciberDeWigar.fbhope deaued“ikWevaserdaakcudeNheunkequen: a)uwBal)ke@-wtS> Be. B)0,Wms Beh,Bee, akc hhadope da/de? Yo, - CranQEDHKnmwstlorwsgulasd + . \ : OmadeieearaS-1 |Lat2ez2l~eral=O e.2' 1(22-2 Ok(aad=(2)2L5oleae adOk B=@r),alsGeld so%aoe@) -QueweSpas.peonin . : eoENQuepoke 8 ‘ Seren GS) . / 1% SVamdB<0,afar te {INR _.Aristooanoan,~-Yeadon uscrleadone —| ; : | -We: Gada Co)>k=21do.Tua, kooueOewakeorien ontGODdhOreWaudonlaimedaside,Sha. @ies, Wee\ booe,Orel | 4 \Saitee PeaseC=Kae\seS@asoCk) VS ie -S Lr \ i. 7 aaa - weSore22WALalenlesLorgreRVoce Py _. zee —Ute 2.x0 a) 2=a *\Nee-4rer)/2 =afe+yen ia _Capettcies ontBear(@sda)|bok+slaE>way, zl. Yagavw, Wom@2)= (2, ‘=o odslope o.eeae Gad Gp),b~OaudSlopex0 hun BHA, B=4~ _— 2 Qeeatarso A U-eletae 0. S2@Yate at) 2eh=HyS|aese \.rVWareWokf2([>I. ae aiokco. 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