Unitarity-Cuts, Stokes' Theorem and Berry's Phase

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by  Pierpaolo Mastrolia

Unitarity-Cuts, Stokes’ Theorem and Berry’s Phase
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0 Pierpaolo Mastrolia∗†
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CentroStudieRicerche“E.Fermi”,PiazzadelViminale1,I-00184,Roma,Italy
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DipartimentodiFisica,UniversitàdiSalerno,ViaPontedonMelillo,I-84084Fisciano,Italy
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J TheoryDepartment,CERN,CH-1211Geneva23,Switzerland

8 E-mail: [email protected]
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h Two-particle unitarity-cuts of scattering amplitudes can be efficiently computed by applying
p Stokes’ Theorem, in the fashion of the Generalised Cauchy Theorem. Consequently, the Op-
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p tical Theorem can be related to the Berry Phase, showing how the imaginary part of arbitrary
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one-loopFeynmanamplitudescanbeinterpretedasthefluxofacomplex2-form.
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RADCOR2009-9thInternationalSymposiumonRadiativeCorrections(ApplicationsofQuantumField
TheorytoPhenomenology),
October25-302009
Ascona,Switzerland
∗Speaker.
†InmemoryofmyfriendThomas(Binoth),whoreactedwithhischaracteristicenthusiasmwhenIfirstshewhim
theresultsnowcollectedinthismanuscripts. IacknowledgeTaniaRobensforclarifyingdiscussions. Myparticipation
totheconferencewassupportedbyHepTools.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Unitarity-Cuts,Stokes’TheoremandBerry’sPhase PierpaoloMastrolia
1. Introduction
Unitarityandgeometricphasesaretwoubiquitouspropertiesofphysicalsystems. TheBerry
phase is the phase acquired by a system when it is subjected to a cyclic evolution, resulting only
fromthegeometricalpropertiesofthepathtraversedintheparameterspacebecauseofanholonomy
[1, 2]. Unitarity represents the probability conservation in particle scattering processes described
by the unitary scattering operator, S. The relation, S=1+i T, between the S-operator and the
transition operator, T, leads to the Optical Theorem, −i(T −T†)=T†T . The matrix elements
of this equation between initial and final states are expressed, in perturbation theory, in terms of
Feynman diagrams. The evaluation of the right hand side requires the insertion of a complete
set of intermediate states. Therefore, since −i(T −T†) = 2 ImT, the Optical Theorem yields
the computation of the imaginary part of Feynman integrals from a sum of contributions from all
possibleintermediatestates.
The Cutkosky-Veltman rules, implementing the unitarity conditions, allow the calculation of
thediscontinuityacrossabranchcutofanarbitraryFeynmanamplitude, whichcorrespondstoits
imaginarypart[3]. Accordingly,theimaginarypartofagivenFeynmanintegralcanbecomputed
byevaluatingthephase-spaceintegralobtainedbycuttingtwointernalparticles,whichamountsto
applyingtheon-shellconditionsandreplacingtheirpropagatorsbythecorrespondingδ-function,
(p2−m2+i0)−1→(2πi)δ(+)(p2−m2).Inlaterstudiestheproblemoffindingthediscontinuityof
a Feynman integral associated to a singularity was addressed in the language of homology theory
anddifferentialforms[4].
More recently multi-particle cuts have been combined with the use of complex momenta for
on-shellinternalparticlesintoveryefficienttechniques,by-nowknownasunitarity-basedmethods,
tocomputescatteringamplitudesforarbitraryprocesses. Thesemethodsexploittwogeneralprop-
ertiesofscatteringamplitudeslikeanalyticity,grantingthatamplitudesaredeterminedbytheirown
singularity-structure, and unitarity, granting that the residues at the singular points factorize into
productsofsimpleramplitudes. Unitarityandanalyticitybecometoolsforthequantitativedetermi-
natonofone-loopamplitudes[5]whenmergedwiththeexistenceofanunderlyingrepresentation
of amplitudes as a combination of basic scalar one-loop functions [6]. These functions, known as
MasterIntegrals(MI’s),aren-pointone-loopintegrals,I (1≤n≤4),withtrivialnumerator,equal
n
to1,characterisedbyexternalmomentaandinternalmassespresentinthedenominator. Ingeneral,
thefulfillmentofmultiple-cutconditionsrequiresloopmomentawithcomplexcomponents. Since
the loop momentum has four components, the effect of the cut-conditions is to fix some of them
accordingtothenumberofthecuts. Anyquadruple-cut[11]fixestheloop-momentumcompletly,
yielding the algebraic determination of the coefficients of I ,(n ≥ 4); the coefficient of 3-point
n
functions, I , are extracted from triple-cut [12, 13, 14, 15, 16, 17]; the evaluation of double-cut
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[18, 19, 20, 14, 21, 16, 22, 23, 17] is necessary for extracting the coefficient of 2-point functions,
I ;andfinally,inprocessesinvolvingmassiveparticles,thecoefficientsof1-pointfunctions,I ,are
2 1
detectedbysingle-cut[16,24,25]. Incaseswherefewerthanfourdenominatorsarecut, theloop
momentumisnotfrozen: thefree-componentsareleftoverasphase-spaceintegrationvariables.
In[26,27],Iintroducedanovelefficientmethodfortheanalyticevaluationofthecoefficients
ofone-loop2-pointfunctionsviadouble-cuts. Spun-offfromthespinor-integrationtechnique[18,
19, 20], that method is an application of Stokes’ Theorem. Due to a special decomposition of the
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Unitarity-Cuts,Stokes’TheoremandBerry’sPhase PierpaoloMastrolia
loop-momentum,thedouble-cutphase-spaceintegraliswrittenasparametricintegrationofrational
function in two complex-conjugated variables. By applying Stokes’ Theorem, the integration is
carried on in two simple steps: an indefinite integration in one variable, followed by Cauchy’s
ResidueTheoremintheconjugatedone.
The coefficients of the 2-point scalar functions, being proportional to the rational term of
the double-cut, can be directly extracted from the indefinite integration by Hermite Polynomial
Reduction.
2. Double-Cut
Thetwo-particleLorentzinvariantphase-space(LIPS)intheK2-channelisdefinedas,
(cid:90) (cid:90)
d4Φ= d4(cid:96) δ(+)((cid:96)2−m2)δ(+)(((cid:96) −K)2−m2), (2.1)
1 1 1 1 2
where Kµ is the total momentum across the cut. We introduce a suitable parametrization for (cid:96)µ
1
[26, 20], in terms of four massless momenta, which is a solution of the two on-shell conditions,
(cid:96)2=m2 and((cid:96) −K)2=m2,
1 1 1 2
1−2ρ(cid:16) (cid:17)
(cid:96)µ = pµ+zz¯qµ+zεµ+z¯εµ +ρKµ , (2.2)
1 1+zz¯ + −
where p andq aretwomasslessmomentawiththerequirements,
µ µ
p +q =K , p2=q2=0, 2 p·q=2 p·K =2q·K ≡K2 ; (2.3)
µ µ µ
thevectorsεµ andεµ areorthogonaltoboth pµ andqµ,withthefollowingproperties1,
+ −
ε2 =ε2 =0=ε ·p=ε ·q, 2ε ·ε =−K2 . (2.4)
+ − ± ± + −
√
The pseudo-threshold ρ = (K2+m2−m2− λ)/(2K2) , with λ = (K2)2+(m2)2+(m2)2−
1 2 1 2
2K2m2−2K2m2−2m2m2 , depends only on the kinematics. The complex conjugated variables
1 2 1 2
z and z¯ parametrize the degrees of freedom left over by the cut-conditions. Because of (2.2), the
LIPSin(2.1)reducestotheremarkableexpression,
(cid:90) (cid:90)(cid:90) dz∧dz¯
d4Φ=(1−2ρ) . (2.5)
(1+zz¯)2
Thedouble-cutofagenericn-pointamplitudeintheK2-channelisdefinedas
(cid:90)
∆≡ d4ΦAtree((cid:96) )Atree((cid:96) ), (2.6)
L 1 R 1
whereAtreearethetree-levelamplitudessittingatthetwosidesofthecut,seeFig.1. Byusing(2.5)
L,R
fortheLIPS,and(2.2)fortheloop-momentum(cid:96)µ,onehas,
1
(cid:90)(cid:90) Atree(z,z¯)Atree(z,z¯)
∆=(1−2ρ) dz∧dz¯ L R , (2.7)
(1+zz¯)2
1In terms of spinor variables that are associated to massless momenta, we can define pµ =(1/2)(cid:104)p|γµ|p] and
qµ =(1/2)(cid:104)q|γµ|q],henceεµ =(1/2)(cid:104)q|γµ|p]andεµ =(1/2)(cid:104)p|γµ|q].
+ −
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Unitarity-Cuts,Stokes’TheoremandBerry’sPhase PierpaoloMastrolia
Figure1: Double-cutofone-loopamplitudeintheK2-channel.
wherethetree-amplitudesAtreeandAtreearerationalinzandz¯. Sinceρ isindependentofzandz¯,its
L R
presenceintheintegrandisunderstood. ByapplyingaspecialversionofthesocalledGeneralised
Cauchy Formula also known as the Cauchy-Pompeiu Formula [28], one can write the two-fold
integration in z- and z¯-variables appearing in Eq.(2.7) simply as a convolution of an unbounded
z¯-integralandacontourz-integral2 [26],
(cid:73) (cid:90) Atree(z,z¯)Atree(z,z¯)
∆=(1−2ρ) dz dz¯ L R , (2.8)
(1+zz¯)2
where the product AtreeAtree is a rational function of z and z¯, and the integration contour has to
L R
bechosenasenclosingallthecomplexz-poles. TheequivalenceofEq.(2.7)andEq.(2.8)isdueto
Stokes’Theorem[26]. Accordingly,thedouble-cut∆in(2.7)isthefluxofa2-form,corresponding
toanintegraloverthecomplextangentbundleoftheRiemannsphere3,wherethecurvature2-form
isdefinedasΩ=1/(1+|z|2)2 dz∧dz¯.
2.1 Coefficientofthe2-pointfunction
The formula in Eq.(2.8) can be intrgrated straightforwardly in two steps. To begin with the
integration,wefindaprimitivewithrespecttoz¯,sayF,bykeepingzasindependentvariable,
(cid:90) Atree(z,z¯)Atree(z,z¯)
F(z,z¯)= dz¯ L R , (2.9)
(1+zz¯)2
sothat∆becomes,
(cid:73)
∆=(1−2ρ) dzF(z,z¯). (2.10)
SinceF istheprimitiveofarationalfunction,itsgeneralformcanonlycontaintwotypesofterms:
arationaltermandalogarithimcone,
F(z,z¯)=Frat(z,z¯)+Flog(z,z¯). (2.11)
Thecoefficientofa2-pointfunctionintheK2-channelwillappearin∆rat,namelytheresultofthe
Residue Theorem in z applied only to Frat. The choice of p and q specified in Eqs.(2.3) grants
thatthereexistsapoleatz=0associatedtothe2-pointfunctionintheK2-channel,I (K2);while
2
2Therolesofzandz¯canbeequivalentlyexchanged.
3In[26]ithasbeenshownthatthedouble-cutofthescalar2-pointfunction,∆I =(cid:82)d4Φamountstotheintegral
2
(cid:82)(cid:82)Ω=−2πi.ThisresultcorrespondstotheintegrationofthefirstChernclass,(i/π)(cid:82)(cid:82)Ω=2.
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Unitarity-Cuts,Stokes’TheoremandBerry’sPhase PierpaoloMastrolia
thereductionofhigher-pointfunctionsthathaveI (K2)assubdiagramcangeneratepolesatfinite
2
z-values. TheResidueTheoremhastobeappliedbyreadingalltheresiduesinz,andsubstituting
thecorresponding complex-conjugatevalues wherez¯appears. Since itcanbe shown[26]that the
double-cut of the 2-point scalar function in the K2-channel amounts to ∆I =−2πi (1−2ρ), the
2
coefficientofthe2-pointfunctioncanbefinallyextractedfromtheratio,
∆rat (cid:16) (cid:17)
c = =− Res Frat(z,z¯)+Res Frat(z,z¯) , (2.12)
2 z=0 z(cid:54)=0
∆I
2
whichwilldependonρ.
Recently, the method just described has been succesfully applied to the completion of the
analyticcalculationoftheone-loopvirtualcorrectionstoH+2jetsviagluonfusion[29,30].4
3. OpticalTheoremandBerry’sPhase
In [27] the following observation was made. In the double-cut integral (2.7), we did not
make any assumptions on the tree-level amplitudes sewn along the cut, thus providing a general
framework to the integration method developed in [26]. If we now choose Atree =A∗,tree, that is
L m→2
theconjugatescatteringamplitudeofaprocessm→2,andAtree=Atree ,thatistheamplitudeofa
R n→2
processn→2,then∆reads,
(cid:90) (cid:104) (cid:105) (cid:110) (cid:111)
∆= d4ΦA∗,tree Atree =−i Aone−loop−A∗,one−loop =2Im Aone−loop , (3.1)
m→2 n→2 n→m m→n n→m
whichisthedefinitionofthetwo-particlediscontinuityoftheone-loopamplitudeAone−loop across
n→m
the branch cut in the K2-channel, corresponding to the field-theoretic version of the Optical The-
orem for one-loop Feynman amplitudes. On the other side, because of Stokes’ Theorem in (2.7,
2.8),onehas,
(cid:90)(cid:90) A∗,tree Atree (cid:73) (cid:90) A∗,tree Atree
∆=(1−2ρ) dz∧dz¯ m→2 n→2 =(1−2ρ) dz dz¯ m→2 n→2 , (3.2)
(1+zz¯)2 (1+zz¯)2
whichprovidesageometricalinterpretationoftheimaginarypartofone-loopscatteringamplitudes,
as a flux of a complex 2-form through a surface bounded by the contour of the z-integral (the
contourshouldencloseallthepolesinzexposedintheintegrandaftertheintegrationinz¯[26]).
Given the equivalence of (3.1) and (3.2), a correspondence between the imaginary part of
scatteringamplitudesandtheanholonomyofBerry’sphasedoesemerge,sincethelatterisindeed
definedasthefluxofa2-forminpresenceofcurvedspace[1,2]. Inthiscontext,onecouldestablish
aparalleldescriptionbetweentheAharonov-Böhm(AB)effect[31]andthedouble-cutofone-loop
Feynmanintegrals. Accordingly,letusfollowtheevolutioninFig.1fromthelefttotheright. The
twoparticlesproducedintheA -scattering,goingaroundtheloopandinitiatingtheA -process,at
L R
theA -interactionpointwouldexperienceaphase-shiftduetothenon-trivialgeometryineffective
R
momentumspaceinducedbytheon-shellconditions. AsintheAB-effect,theanholonomyphase-
shift is a consequence of Stokes’ Theorem, and here it corresponds to the imaginary part of the
one-loopFeynmanamplitude.
4SeealsoS.Badger,andC.Williamsintheseproceedings.
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Unitarity-Cuts,Stokes’TheoremandBerry’sPhase PierpaoloMastrolia
References
[1] M.V.Berry,Proc.Roy.Soc.Lond.A392(1984)45.
[2] A.D.ShapereandF.Wilczek(ed.),Adv.Ser.Math.Phys.5(1989)1.
[3] L.D.Landau,Nucl.Phys.13,181(1959).R.E.Cutkosky,J.Math.Phys.1,429(1960).R.J.Eden,
P.V.Landshoff,D.I.Olive,J.C.Polkinghorne,TheAnalyticSMatrix,CambridgeUniversityPress,
1966.M.J.G.Veltman,Physica29(1963)186.E.Remiddi,Helv.Phys.Acta54(1982)364.
[4] R.C.Hwa,andV.L.Teplitz,MathematicalPhysicsMonographs,W.A.BenjaminInc.,1966.
[5] Z.Bern,L.J.Dixon,D.C.DunbarandD.A.Kosower,Nucl.Phys.B425(1994)217;Nucl.Phys.B
435(1995)59.
[6] G.PassarinoandM.J.G.Veltman,Nucl.Phys.B160(1979)151.
[7] G.Ossola,C.G.PapadopoulosandR.Pittau,Nucl.Phys.B763,147(2007);JHEP0707,085(2007).
[8] R.K.Ellis,W.T.GieleandZ.Kunszt,JHEP0803(2008)003.
[9] W.T.Giele,Z.KunsztandK.Melnikov,JHEP0804(2008)049.
[10] C.F.Bergeretal.,Phys.Rev.D78(2008)036003.
[11] R.Britto,F.CachazoandB.Feng,Nucl.Phys.B725,275(2005).
[12] P.Mastrolia,Phys.Lett.B644,272(2007).
[13] T.Binoth,G.Heinrich,T.GehrmannandP.Mastrolia,Phys.Lett.B649,422(2007).
[14] D.Forde,Phys.Rev.D75,125019(2007).
[15] N.E.J.Bjerrum-Bohr,D.C.DunbarandW.B.Perkins,JHEP0804,038(2008).
[16] W.B.Kilgore,arXiv:0711.5015[hep-ph].
[17] S.D.Badger,JHEP0901(2009)049.
[18] R.Britto,E.Buchbinder,F.CachazoandB.Feng,Phys.Rev.D72,065012(2005).
[19] R.Britto,B.FengandP.Mastrolia,Phys.Rev.D73,105004(2006).
[20] C.Anastasiou,R.Britto,B.Feng,Z.KunsztandP.Mastrolia,Phys.Lett.B645,213(2007);JHEP
0703,111(2007).
[21] R.BrittoandB.Feng,Phys.Rev.D75,105006(2007);JHEP0802,095(2008).
[22] R.Britto,B.FengandP.Mastrolia,Phys.Rev.D78(2008)025031.
[23] R.Britto,B.FengandG.Yang,JHEP0809(2008)089.
[24] E.W.N.GloverandC.Williams,JHEP0812(2008)067.
[25] R.BrittoandB.Feng,Phys.Lett.B681,376(2009).
[26] P.Mastrolia,Phys.Lett.B678(2009)246.
[27] P.Mastrolia,toappearinLett.Math.Phys.,arXiv:0906.3789.
[28] M.J.Ablowitz,andA.S.Fokas,ComplexVariables,CambridgeTextsinAppliedMathematics,2003,
2ndEdition.
[29] S.Badger,E.W.NigelGlover,P.MastroliaandC.Williams,toappearinJHEP,arXiv:0909.4475.
[30] S.Badger,J.M.Campbell,R.K.EllisandC.Williams,JHEP0912(2009)035.
[31] Y.AharonovandD.Bohm,Phys.Rev.115(1959)485.
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