Table Of ContentDifferential Reduction Techniques for the Evaluation
of Feynman Diagrams
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1
0
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S.A. Yost∗†
n
a Dept. ofPhysics,TheCitadel,171MoultrieSt.,Charleston,SC29409,USA
J E-mail: [email protected]
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1 V.V. Bytev
JointInstituteforNuclearResearch,141980Dubna(MoscowRegion),Russia
]
h E-mail: [email protected]
p
- M.Yu.Kalmykov‡
h
t II.InstitutfürTheoretischePhysik,UniversitätHamburg,LuruperChaussee149,22761
a
m Hamburg,Germany
[ JointInstituteforNuclearResearch,141980Dubna(MoscowRegion),Russia
E-mail: [email protected]
1
v BerndA. Kniehl‡
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4 II.InstitutfürTheoretischePhysik,UniversitätHamburg,LuruperChaussee149,22761
3 Hamburg,Germany
2
E-mail: [email protected]
.
1
0 B.F.L.Ward §
1 Dept. ofPhysics,BaylorUniversity,OneBearPlace,Waco,TX76798,USA
1
E-mail: BFL [email protected]
:
v
i
X Stablereductionmethodswillbeimportantintheevaluationofhigh-orderperturbativediagrams
r appearing in QCD and mixed QCD-electroweak radiative corrections at the LHC. Differential
a
reductiontechniquesare usefulforrelating hypergeometricfunctionswith shiftedvaluesof the
parameters. Wepresentapropositionrelatingthenumberofmasterintegralsintheexpansionof
aFeynmandiagramtothenumberofderivativesinadifferentialreduction.
35thInternationalConferenceofHighEnergyPhysics-ICHEP2010,
July22-28,2010
ParisFrance
∗Speaker.
†ThisresearchhasbeensupportedinpartbyUSDOEgrantDE-SC0000528andTheCitadelFoundation.
‡Thisresearchhasbeensupported inpartbyBMBFGrantNo.05HT6GUA,DFGGrantsNo.KN365/3–1 and
KN365/3–2,andHGFGrantNo.HA101.
§ThisresearchhasbeensupportedinpartbyUSDOEgrantDE-FG02-09ER41600.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
DifferentialReductionTechniquesfortheEvaluationofFeynmanDiagrams S.A.Yost
Techniques for the evaluation of physical cross sections have traditionally been based on the
direct evaluation of Feynman diagrams. For obtaining stable numerical results, a deeper under-
standing of the analytical structure of Feynman diagrams and Green functions is desirable. In the
framework of dimensional regularization,[1] any multi-loop Feynman diagram can be expanded
as a Laurent series in powers of the regularization parameter e . An understanding of the analytic
functions inthise -expansion givesusefulinformation abouttheFeynmandiagram.
Thehypergeometric function representation isobtained from amultiple Mellin-Barnes repre-
sentation[2]oftheFeynmanintegral, whichmaybewritten
P p G ((cid:229) m a z +c )
F (a ,b ,c,d ,g )= dz ...dz j=1 s=1 js s j za 1...za m ,
js kr i j Zg+iR 1 mP qk=1G ((cid:229) mr=1bkrzr+dk) 1 m
where a ,b ,c,d ∈R, a ∈C, and z are Mandelstam variables. This integral can be written
js kr i j k k
as a sum of multiple residues of the integrated expression, resulting in a linear combination of
Horn-type hypergeometric functions oftheform
¥ P K G ((cid:229) r m m +g )
H (~g ;~s ;~x)= (cid:229) j=1 a=1 ja a j xm1···xmr ,
r P L G ((cid:229) r n m +s ) 1 r
m1,m2,···,mr=0 k=1 b=1 kb b k !
with m ,n ∈ Q, g ,s ∈ C, and their derivatives with respect to the parameters s ,g . (See,
ab ab j k k j
for example, Ref. [3].) In general, the hypergeometric representation of a Feynman diagram is
not unique, since non-linear transformations of the arguments z are possible. This is equivalent
k
toquadratic, cubic, andmorecomplicated transformations oftheassociated hypergeometric func-
tions. Arecentexampleisconsidered inRef.[3,4].
It is useful to see what can be learned about Feynman diagrams from their representation in
terms of Horn-type hypergeometric functions. A series of publications [5] have presented and
explored thefollowingproposition:
(i) Eachterminthehypergeometric representation ofaFeynmandiagramshouldhavethesame
numberofderivatives, independent ofthetypeofhypergeometric functions appearing.
(ii) The number of master integrals required to represent a Feynman diagram should coincide
withthenumberofderivativesplusone.
Thesestatements arebothbasedonadifferential reductionalgorithm (Ref.[6]andreferences
therein) which constructs raising and lowering operators which shift the upper and lowerparame-
ters of the hypergeometric function by one unit. The reduction procedure can be used to express
thehypergeometric functions appearing intheexpression foraFeynmandiagram intermsofaset
ofbasisfunctions. Specifically, adifferential reduction algorithm canbeapplied toanyHorn-type
hypergeometric function, relating functions withshiftedarguments according to
L ¶ k
R(~z)H(~b +~m;~l +~n;~z)= (cid:229) P(~z) H(~b ;~l ;~z),
k ¶ z ···¶ z
k=0 k1 kr
where~m,~naresetofintegernumbersandR,P arepolynomial functions. (Theserelations arealso
k
usefulinconstructing thee -expansion ofHypergeometric Functions[7].)
Considerthestandardhypergeometric representation ofaFeynmandiagram,
k
F (n,~j;~z)= (cid:229) S (n,~j,~z)H (~b ;~l ;~x ) (1)
a a a a
a=1
2
DifferentialReductionTechniquesfortheEvaluationofFeynmanDiagrams S.A.Yost
where ~j is a list of the powers of the propagators in the Feynman diagram, n is the space-time
dimension, ~x are the arguments of the hypergeometric functions, which are related the kinematic
invariants of the Feynman diagram, {b ,l } are linear combinations of~j and n with polynomial
a a
coefficients, and S are rational functions of the variables~z with coefficients depending on n and
a
~j.1 The number of basis elements for the Horn hypergeometric functions H can be found by
a
constructing the Pfaff system of differential equations for this function. However, when some of
the parameters or differences between parameters of the functions H are integers, the number of
a
derivatives isreduced. Ourproposition (i)isthat,regardlessofthetypeoffunctionsinther.h.s. of
Eq.(1),thenumberofbasiselementsisthesame(uptoamoduleofrationalfunctions).
Being a sum of holonomic functions, F (~j;~z) is also holonomic. Thus, the number of basis
elements on the r.h.s. of Eq. (1) is equal to the number of master-integrals F (~z) that may be
k
derived from the l.h.s. by applying the integration-by-parts (IBP) technique [9], which may be
written symbolically as F (n,~j;~z) = (cid:229) h B (n,~j;z)F (n;z) . The number h of nontrivial master
k=1 k k
integrals following from IBPwhich arenot expressible intermsofgammafunctions isthen equal
tothenumberofbasiselementsLforeachtermofr.h.s. ofEq.(1).
Thegeneralized hypergeometric functions ofonevariable whichhavebeenanalyzed indetail
inRef.[5]areallinfullagreement withtheproposition.
References
[1] G.’tHooftandM.Veltman,Nucl.Phys.B44(1972)189.
[2] V.A.Smirnov,FeynmanIntegralCalculus,Springer,Berlin2006.
[3] V.DelDuca,C.Duhr,E.W.N.GloverandV.A.Smirnov,JHEP1001(2010)042
[arXiv:0905.0097].
[4] B.A.KniehlandO.V.Tarasov,Nucl.Phys.B833(2010)298[arXiv:1001.3848].
[5] V.V.Bytevetal.,DifferentialReductionAlgorithmsforHypergeometricFunctionsAppliedto
FeynmanDiagramCalculation,inproceedingsofInternationalLinearColliderWorkshop2008:
LCWS08andILC08,Chicago,2008[arXiv:0902.1352];V.V.Bytev,M.Yu.KalmykovandB.A.
Kniehl,Nucl.Phys.B836(2010)129[arXiv:0904.0214].
[6] M.Yu.Kalmykovetal.,FeynmanDiagrams,DifferentialReduction,andHypergeometricFunctions,
inproceedingsofXIIInt.WorkshoponAdvancedComputingandAnalysisTechniquesinPhysics
Research,Erice,2008PoS(ACAT08)125[arXiv:0901.4716].
[7] M.Yu.Kalmykov,JHEP04(2006)056[arXiv:hep-th/0602028];M.Yu.Kalmykov,B.F.L.
WardandS.A.Yost,JHEP02(2007)040[arXiv:hep-th/0612240];ibid.,JHEP10(2007)048
[arXiv:0707.3654];ibid.,JHEP11(2007)009[arXiv:0708.0803];M.Yu.Kalmykovand
B.A.Kniehl,Nucl.Phys.B809(2009)365[arXiv:0807.0567];ibid.,Phys.Part.Nucl.41(2010)
942[arXiv:1003.1965].
[8] L.U.AncaraniandG.Gasaneo,J.Phys.A42(2009)395208;ibid.,J.Phys.A43(2010)085210.
[9] F.V.Tkachov,Phys.Lett.B100(1981)65;K.G.Chetyrkin,F.V.Tkachov,Nucl.Phys.B192(1981)
159.
1In general, Eq. (1) may contain derivatives of Horn functions with respect to their parameters. [3] However,
thereissome evidence [8]that suchexpressions canagainbeexpressedintermsof linearcombination of Horn-type
hypergeometricfunctions.
3
