Generalǫ-expansionforscalar one-loopfeynmanintegrals

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1 Generalǫ-expansionforscalar one-loopfeynmanintegrals KhiemHongPHAN DeutschesElektronen-SynchrotronDESY,Platanenallee6, D 15738Zeuthen,Germany. Incollaborationwith JohannesBlümleinandTordRiemannDESY,Zeuthen 1/0

2 Outline 1 Introduction Recurrencerelationsforscalarone-loopn-point integrals 3 Analyticresultsforscalarone-looptwo-,three-, four-pointintegrals 4 ConclusionandOutlook /0

3 Introduction 1 FuturecollidersLHC,ILC 1 focus: tostudybrout-englert-higgsbosonproperties;topquark properties;vectorbosonproduction; aswellasdirectlysearchfornewphysicssignalssusy, ExtraDimension,etc. = Themeasurementsareperformedathighprecision. Suchhigherprecisionurgentlyrequiresthecalculationof one-loopmulti-legcorrections;two-loopandhigherloop correctionstoseveralprocessesofinterest. 1 PhysicsataHigh-LuminosityLHCwithATLAS:arXiv: ;CMS:arXiv: ;ILCTechnicalDesign Report:arXiv: /0

4 Introduction 1 Scalarone-loopintegralsingeneralspace-timedimension d =n+ǫatgeneralscales;massesandpropagatorindices areveryimportant: tobuildcountertermsforevaluatingtwo-loopcorrections. togaingoodnumericalstabilitycuregramdeterminant problematone-loopcorrections one-loopintegralswithd =n+ǫwithn =,4, must betakenintoaccount:davydychev,phys.lett.b ;j.fleischeretal.,phys. Rev.D ;J.M.Campbelletal.,Nucl.Phys.B ;etc. Scalarone-loopintegralshavebeenperformedbymany authors: Davydychevetal:Mellin-BarnesrepresentationJ.Math.Phys ,etc;GeometricalapproachesJ.Math.Phys ,etc. Tarasov et al:recurrencerelation for multi-loop integral in shifteddimensionnucl.phys.proc.suppl ;nucl.phys.b /0

5 Introduction AMBRE:automaticMellin-Barnesrepresentationformulti-loop integralsj.gluzaetal.,comput.phys.commun ;etc. Thesecalculationshavenotyetcompletedtotreatscalarand tensorone-loopintegralswithgeneral ǫ-expansionatgeneral scales,massesandpropagatorindices. Inthistalk:basedonthemethodTarasovetal.,Nucl.Phys.B Asystematicstudyofthescalarone-looptwo-,three-,and four-pointintegralsispresented,consideringallcasesofmass assignment;externalinvariantswithpropagatorindices ν =1. Wealsoperformextensivecomparisonwiththeliterature. Analyticresultsareexpandedinhigherpowerofǫ. ApackageinMathematicaandFortranwillbebuiltinthenear future. 5/0

6 Generalremarks Kinematicvariablesaredefinedas: Y 11 Y 1... Y 1n Y 1 Y... Y n λ ij k =.....,. Y 1n Y n... Y nn Y ij = p i p j +m i +m j, Õ Ô ½ Õ ÔÒ Õ Ô ¾ d I n d d k 1 =, iπ d/ D 1 D...D n D j = k p j m j ;d =4+ǫ. p 1 p n p 1 p n... p 1 p n p n 1 p n g ij k = n p 1 p n p p n... p p n p n 1 p n p 1 p n p n 1 p n... p n 1 p n p n 1 p n ;...k = λ ij...k g ij...k. 6/0

7 Recurrencerelationforscalarone-loopn-pointfunctions Õ Ô ½ Õ ÔÒ Õ Ô ¾ I d n = d d k iπ d/ 1 D 1 D...D n, D j = k p j m j. Recurrencerelation Tarasovetal.,Nucl.Phys.B I d n =b n d n k λ ij k d n+1 1 k=1 λ ij k r=0 k =, k I m n d =I d n 1, a n = Γa+n k Γa theboundaryterm:b n d I n d whend. r k r k I d+r n. 7/0

8 Scalarone-looptwo-pointfunctions Recurrencerelationfortwo-pointfunctions I d =b d k λ ij d 1 k=1 λ ij r=0 r 1 r k I d+r. k = λ iji d πγ d [ Γ 1 d = Γ d 1 r d ij + iλ ij m j 1 d r=0 k I d+r = 1 rγd Γ1 d Γ d +r m k d +r. d 1 r d r m j i λ ij 1 m j + r + jλ ij m i 1 d j λ ij r=0 1 m i d 1 r d r m r i. 8/0

9 Scalarone-looptwo-pointfunctions λ ij I d πγ d Γ 1 d = Γ d 1 r d i λ ij ij + iλ ij m j 1 d F 1 [ 1, d 1 ; m j d ; 1 m j + + jλ ij m i 1 d j λ ij 1 m i F 1 [ 1, d 1 ; m i d ; d 1 providedthatre >0, m j m <1and i <1. Note:forphysiccalresults,onemustbeperformedanalyticcontinuationfor F 1 nextslide!., TheGausshypergeometricseriesaregivenL.J.Slater,GeneralizedHypergeometric Functions.CambridgeUniversityPress,1966. F 1 [ a,b; c; z = a n b n z n, a n = Γa+n, c n Γa n=0 9/0

10 Scalarone-looptwo-pointfunctions 1 Theintegralrepresentationfor F 1 is F 1 [ a,b; c; z = Γc 1 duu b 1 1 u c b 1 1 zu a, ΓbΓc b 0 providedthat z <1andRec >Reb >0. Analyticcontinuation z >1 F 1 [ a,b; c; z = ΓcΓb a [ a,1+a c; Γc aγb z a F 1 1+a b; + ΓcΓa b ΓaΓc b z b F 1 [ b,1+b c; 1+b c; 1 z 1 z, 10/0

11 Scalarone-looptwo-pointfunctions Allspecialcasesaretreated: =0; =m i m j,m i =m j =0and g ij =0 comparisonwithdavydychevetal.,j.math.phys ;nucl.phys.b Take =0asanexample,wefirstapply F 1 [ a,b; c; F 1 [ a,b; c; [ ΓcΓc a b z = Γc aγc b F 1 a,b; a +b c+1; [ F 1 c a b ΓcΓa +b c +1 z ΓaΓb [ z = 1 z a a,c b; F 1 c; Wearriveatanotherrepresenation [ g ij I d Γ d = iλ ij F 1 m j 4 d = I d = Γ d d 3 1, 4 d ; z z 1 p ij +m i m j p ij. 3 ; 1 m j 1 z c a,c b; c a b+1; 1 z, +i j term. d m i +i j term. 11/0

12 Scalarone-loopthree-pointfunctions I d 3 = b 3 d 3 k λ ijk d k=1 λ ijk r=0 r 1 r k I d+r 3. k Where πγ d k I d+r 3 = +r Γ 1 d r [ Γ d 1 +r rd +r ij +Γ 1 d { +r j λ ij m i d 1 m i i λ ij 1 m j + [ d 1+r F +r, 1 ; 1 [ + iλ ij mj d 1+r F 1 1 m j j λ ij 1 m i m i d +r; d } +r, 1 ; mj. +r; d 1/0

13 Scalarone-loopthree-pointfunctions I d 3 = 8 πγ d gijk k d Θτ ijk +C 0 ijk +C0 ikj +C0 kji, Θτ ijk = θ g ijk θ p ij θ p jk θ p ki θk m i θk m j θk m k. θx = { 1 if x >0, 0 if x 0. Cijk 0 d π Γ d 1 1 r = ij k λ ijk Γ d Γ d 1 λ ij λ ijk Γ d 1 1 k λ ijk + Γ d λ ij λ ijk { i λ ij { j λ ij + iλ ij 1 m i 1 m j 1 m j + jλ ij 1 m i [ 1, }F d ; r ij 1 d 1 ; k m d 1 d i F1 ;1; 1 ;d ;m i, m i k m d 1 d j F1 ;1; 1 ;d ;m j, m } j, k providedthat m i <1, <1,andRe d >0. k 13/0

14 Scalarone-loopthree-pointfunctions 1 AseriesforAppellF 1 functionisl.j.slater,generalizedhypergeometricfunctions. CambridgeUniversityPress,1966. F 1 a;b,b ;c;x,y = m=0n=0 providedthat x <1and y <1. ThesingleintegralrepresentationforF 1 is a m+n b m b n x m y n, c m+n m!n! F 1 a;b,b ;c;x,y = Γc 1 = duu a 1 1 u c a 1 1 xu b 1 yu b, Γc aγa 0 providedthatrec >Rea >0and x <1, y <1. 3 AnalyticcontinuationforF 1 studiedindetailinp.o.m.olsson,j.math. Phys ;wecanalsoperformitsanalyticcontinuationby usingamellin-barnesrepresenation. 14/0

15 Scalarone-loopthree-pointfunctions Allspecialcasesaretreated: =0; =m ifori =1,,3;massless; k =0andg ijk =0 comparisonwithdavydychevetal.,phys.rev.d ;Nucl.Phys.BProc.Suppl Takemasslesscaseasanexample,weconfirmb 3 =0inthiscase. Takingm i 0,inpreviousresultoneconfirmsthat m i d 1 F 1 d ;1;1 ;d ;0,0 0. Wearriveat I d 3 = π Γ d 1 Γ d Γ 1 d 1 p j +p k p i [ 1, 1 F ; λp i,p j,p k 1 d 1 ; p j +p k p i { } + ijk jki. d p i 4 { } + ijk kij ItagreeswithDavydychevetal.,Phys.Rev.D /0

16 Scalarone-loopfour-pointfunctions I d 4 = 8π3 Γ d d Γ d 3 g ijkl r d 3 ijkl Θτ ijkl +D 0 ijkl +D0 lijk +D0 klij +D0 jkli, Θτ ijkl = θ g ijkl θ p ij θ p jk θ p kl θ p li θkl m i θkl m j θkl m k θkl m l D 0 ijkl = Γ d [ 8π d l λ d 3 g ijk r ijkl,1; r ijk ijk F 1 d Θτ ijk + λ ijkl 1; kl { π Γ Γ d d d 1 1 l λ ijkl k λ ijk r + ij Γ d 1 λ ijkl λ ijk λ ij iλ ij 1 m j 1 d 3 1 r F 1 ;1, 1 ij k ;d 1 ; kl, k Γ Γ d d 1 l λ ijkl k λ ijk j λ ij k Γ d λ ijkl λ ijk λ ij k m i m i m d 1 d 3 i FS,1,1;1,1, 1 ;d,d,d ; m i m i m i, kl m, i k m i } + i j term + {i,j,k k,i,j} + {i,j,k j,k,i}. 16/0

17 Scalarone-loopfour-pointfunctions 1 TheSaranfunctionS.Saran,Ganita ;ActaMath., : F S α 1,α,α,β 1,β,β 3,γ 1,γ 1,γ 1 ;x,y,z = α 1 r α m+n β 1 r β m β 3 n = x r y m z n. γ 1 r+m+n r!m!n! r,m,n=0 TheintegralrepresentationofF S is: F S α 1,α,α,β 1,β,β 3,γ 1,γ 1,γ 1 ;x,y,z = 1 Γγ 1 = dt tγ 1 α t α 1 1 Γα 1 Γγ 1 α x+tx β 1 F 1 α,β,β 3 ;γ 1 α 1 ;ty,tz, providedthat x <1, y <1, z <1andReγ α >0. 3 TheanalyticcontinuationforF S canbeperformedusinga Mellin-Barnesrepresenation. 17/0

18 Scalarone-loopfour-pointfunctions Allspecialcasesaretreated: =0; =m ifori =1,,3; massless;k =0;kl =0andg ijkl =0. Takemasslesscaseasanexample,weconfirmb 4 =0inthis case.takingm i 0,inthepreviousresultoneconfirmsthat d 3 m i d 1 F S,1,1;1,11 ;d,d,d ;0,0,0 = = m i d 1 Γ d 1 Γ d 3 Γ 3 dt t1 t d 5 = m i d Wearriveat π Γ d Γ d 1 gij l λ ijkl k λ ijk D 0 ijkl = rd 1 ij Γ d 1 1 rij k F 1 λ ijkl λ ijk d 3 ;1,1 ;d 1 ;, r ij kl k +{i,j,k j,k,i}+{i,j,k k,i,j}. λ ij 18/0

19 Conclusionandoutlook Conclusion Asystematicstudyofthescalarone-looptwo-,three-,and four-pointintegralswaspresentedwithconsideringall casesofmassassignmentsandexternalinvariants. Wehavecomparedourresultstotheliteratureindetail. Acompleteanalyticsolutionwaspresented. Thecalculationprovidesanimportantbuildingblockfor higherloopcalculation. Outlook Higherpowersofǫ-expansionsforscalarone-loop integralsarederived. ApackageinMathematicaandFortranwillbebuiltinthe nearfuture. 19/0

20 Thankyouverymuchforattention! 0/0