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Title:
Scalar 1-loop Feynman integrals as meromorphic functions in
space-time dimension d
Author:
Khiem Hong Phan, Tord Riemann
Citation style:
Phan Khiem Hong, Riemann Tord.(2019). Scalar 1-loop
Feynman integrals as meromorphic functions in space-time dimension d.
"Physics Letters B" (Vol. 791, (2019), s. 257-264), doi
Contents lists available atScienceDirect
Physics
Letters
B
www.elsevier.com/locate/physletb
Scalar
1-loop
Feynman
integrals
as
meromorphic
functions
in
space-time
dimension d
Khiem
Hong Phan
a,
Tord Riemann
b,
c,
∗
aUniversityofScience,VietnamNationalUniversity,HoChiMinhCity,Vietnam bDESYDeutschesElektronen-Synchrotron,15738Zeuthen,Germany cInstituteofPhysics,UniversityofSilesia,40-007Katowice,Polanda
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Articlehistory:
Received27January2019
Receivedinrevisedform21February2019 Accepted26February2019
Availableonline28February2019 Editor:A.Ringwald
Keywords:
Massiveone-loopFeynmanintegrals Generalizedhypergeometricfunctions Tensorintegralreduction
The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimension d has been solved for the basis of scalar one-to four-point functions with indices one. In 2003 the solution of differenceequations in the space-time dimension allowed to determine the necessary classes of special functions: self-energies need ordinary logarithms and Gauss hypergeometric functions 2F1, vertices need additionally Kampé de Fériet-Appellfunctions F1,andboxintegralsalsoLauricella-Saranfunctions FS.Inthisstudy,alternative
recursiveMellin-Barnesrepresentationsareusedfortherepresentationofn-pointfunctionsintermsof
(n−1)-pointfunctions.TheapproachenabledthefirstderivationofexplicitsolutionsfortheFeynman integrals at arbitrary kinematics. In this article, we scetch our new representations for the general massive vertex and boxFeynman integralsand deriveanumerical approachfor the necessaryAppell functionsF1andSaranfunctions FS atarbitrarykinematicalarguments.
©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introduction
Wearestudyingscalarone-loopFeynmanintegrals,
Jn
(
d)
=
ddk iπ
d/2 1 Dν1 1 D ν2 2· · ·
D νn n,
(1)with inverse propagators Di
= (
k+
qi)
2−
m2i+
iε
. We assumeν
i=
1 as well as momentum conservation and all externalmo-menta to be incoming,
ne=1pe=
0. The qi are loop momentashiftsandwillbe expressed forapplicationsby the external mo-mentape.Dimensionsd
=
4+
2m−
2withm
≥
0 areofphysicalinterest because tensor one-loop Feynman integralsof rank r in
4
−
2dimensionsmaybeexpressed by scalarintegralstakenin higher dimensions up to d
=
4+
2r−
2[1]. Higher indices
ν
iwillalso appearin thereductions, butmay beeliminated by in-tegrationbypartsidentities,sothatacompletereductionbasisof higher-dimensionalscalarone- tofourpointintegralswithindizes one maybe derived. One-loopintegrals withvariableindices are alsoneededinthecontext oftheloop-by-loopMellin-Barnes
ap-*
Corresponding author at: DESY Deutsches Elektronen-Synchrotron, 15738 Zeuthen,Germany.E-mailaddress:[email protected](T. Riemann).
proachtomulti-loopintegralsoftheMathematicapackageAMBRE [3–6].
Thefirst termsofthe
-expansion ofone- tofour-point scalar functionsford
=
4−
2, untilincluding theconstant term, were given by G. t’Hooft and M. Veltman in 1978 [7]. A systematic numericaltreatmentofthenext termsoforder
-termswas per-formed in 1992 [8], and a systematic numerical approach was workedout in2001 [9]. It hasbeen shownin 2003[10,11] that representationsingeneraldimensiond,includingd
=
4−
2,will rely on certain multiple hypergeometric functions of the type
2F1
,
F1,
FS.Though,theexplicitsolutionsforarbitrarykinematicscouldnotbefound.
A scetch of the Feynman integrals at arbitrary kinematics in termsof2F1
,
F1,
FS andtheirexplicitnumericaldeterminationarethesubjectofthisletter.Thedependenceontheexternalmomenta
pe willbecontainedexclusivelyinthefunctionsRn:
Rn
≡
R12...n= −
λ
nGn
−
i
ε
.
(2)The Rn carry the causal regulator
−
iε
. The Cayley matrixλ
12...nwasintroducedin[12].ItiscomposedofthevariablesYi j,andits
determinant
λ
n is:https://doi.org/10.1016/j.physletb.2019.02.044
0370-2693/©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
258 K.H. Phan, T. Riemann / Physics Letters B 791 (2019) 257–264
λ
n≡
det(λ
12...n)
=
Y11 Y12. . .
Y1n Y12 Y22. . .
Y2n..
.
..
.
. .
.
..
.
Y1n Y2n. . .
Ynn,
(3) with Yi j=
Yji=
m2i+
m2j− (
qi−
qj)
2.
(4)Further,weusethe
(
n−
1)
×(
n−
1)
dimensionalGramdeterminantGn, Gn
≡ −
2ndet(
G12···n),
(5) and det(
G12···n)
=
(6)(
q1−
qn)
2. . .
(
q1−
qn)(
qn−1−
qn)
(
q1−
qn)(
q2−
qn)
. . .
(
q2−
qn)(
qn−1−
qn)
..
.
. .
.
..
.
(
q1−
qn)(
qn−1−
qn)
. . .
(
qn−1−
qn)
2.
Weusethespecialassignmentfortadpoles:
G1
= −
2.
(7)Bothdeterminants
λ
n andGn areindependentofacommonshiftoftheinternalmomentaqi.Further,weintroducethenotion R
(
i)
,R
(
i)
≡
r(
i)
−
iε
≡ −
det(λ
i)/
G1−
iε
=
m2i−
iε
,
(8)anduse,whereveritisuniquefromthecontext,
R1
≡
R(
i).
(9)We derived in [13] a newansatz, a recursionrelation forthe Feynmanintegralsdefinedin(1),
Jn
(
d)
=
−
1 2π
i c0+i∞ c0−i∞ ds(
−
s)(
d−n+1 2+
s)
2(
d−n2+1)
× (
s+
1)
Rn−s−1 n k=1∂
kRnk−Jn(
d+
2s),
(10)and its solution by a sequence of Mellin-Barnesrepresentations. Weusetherepresentation
∂
kRnfortheco-factoroftheCayleyma-trix,alsocalledsignedminorsine.g.[12]:
∂
kRn=
∂
Rn∂
mk2=
0 k n.
(11)Theoperatork−reducesann-pointFeynmanintegral Jn
(
d)
to(
n−
1
)
-pointintegrals Jn−1(
d)
byshrinkingthekthpropagator,1/
Dk:k− Jn
(
d)
=
ddk iπ
d/2 1 n j=k,j=1Dj.
(12)The recurrence relation (10) is the master integral for one-loop
n-pointfunctionsinspace-timedimensiond,representingthemby
n integralsover
(
n−
1)
-pointfunctionswithashifted,continuous dimensiond+
2s.Therecurrencestarts atn=
2 withthetadpoleJ1
(
d)
intheintegrand: J1(
d;
m2i)
=
ddk iπ
d/2 1 k2−
m2 i+
iε
= −
(
1−
d/
2)
(
m2i−
iε
)
1−d/2≡ −
(
1−
d/
2)
R11−d/2.
(13)Eqn. (10) contains for n
=
2 the termds
(
R1R2
)
s, multiplied by
-matrices with arguments depending on s, and is formally a Mellin-Barnes integral. Our representation is an alternative to Eq. (19) of[11].There, an infinitesum overa discretedimensional parameters wasderived inorderto representann-point function
Jn
(
d)
byintegrals Jn−1(
d+
2s)
.Thefurtherevaluationswilldepend,concerningthekinematics, exclusively on the R1
,
R2, etc. introduced in (2). Although, therewillarise exceptionalcaseswhenthespecificchoice ofthe exter-nal scalars
(
peipej)
or of internal mass squares m2
i will lead to
vanishing ordivergent determinants
λ
n orGn.In such cases,onehastogobacktointermediatedefinitionsandlookforspecific so-lutions.1 Seealsotheremarksin[14].
2. Massivevertexandboxfunctions
Representationsofthemassiveself-energy,vertexandbox inte-gralscanbederivediterativelyfrom(10) byclosingtheintegration contoursoftheMellin-Barnesintegralse.g.totherightandtaking the two seriesofresidues ofthe corresponding
-functions with arguments
(
−
s+ · · · )
. One Cauchy sumconstitutes the analogue of the so-calledboundaryorb-terms of [11], the other one has a genuined-dependence.BothsumstogetherrepresenttheFeynman integrals. Inour approach, closedanalytical expressions could be determinedforarbitrarykinematics.Thegeneralmassivevertexandboxintegrals J3
(
d),
J4(
d)
havefirst been presented at the conference “Loops and Legs 2018 (LL2018)”.Thevertexis
J3
(
d)
=
J123+
J231+
J312,
(14)withshortnotationsR3
=
R123,
R2=
R12etc.,and:J123
=
2−
d 2∂
3R3 R3∂
2R2 R2 R2 2√
1−
R1/
R2 (15)−
R d 2−2 2√
π
2d 2
−
1d 2
−
1 22F1 d
−
2 2,
1;
d−
1 2;
R2 R3+
R d 2−2 3 2F1 1,
1;
3 2;
R2 R3+
2−
d 2∂
3R3 R3∂
2R2 R2 R1 4√
1−
R1/
R2+
2R1 d 2−2 d−
2 F1 d−
2 2;
1,
1 2;
d 2;
R1 R3,
R1 R2−
R d 2−2 3 F1 1;
1,
1 2;
2;
R1 R3,
R1 R2+ (
R1(
1)
↔
R1(
2)).
1 Acompleteanalysisoftheexceptionalkinematicalcaseshasbeenperformedby
Weusetheabbreviation(11).Ford
→
4,boththesumsof expres-sions with2F1 and F1 in square brackets in(15) approach zero,thuscompensatingthepolefactor
(
2−
d/
2)
inthislimit.The J3staysfiniteatd
=
4,asitshouldbeforanymassive3-point func-tion.Andtheexpansionfor J123to ordern needs, inthiscase,
theevaluationofthecomponentstoorder
(
n+
1)
. Thecorrespondingmassivefour-pointfunctionis:J4
(
d)
=
J1234+
J2341+
J3412+
J4123,
(16)withR4
=
R1234,
R3=
R123,
R2=
R12etc.,and:J1234
=
2−
d 2∂
4R4 R4 b123 2−
R d 2−2 3 2F1 d−
3 2,
1;
d−
2 2;
R2 R3+
R d 2−2 4√
π
d−2 2
d−3 2
2F1 1 2
,
1;
1;
R2 R3+
d−2 2
d−3 2
√
π
4∂
3R3 R3∂
2R2√
1−
R1/
R2×
2F1 1 2,
1;
1;
R2 R3+
R d 2−2 2 d−
3F1 d−
3 2;
1,
1 2;
d−
1 2;
R2 R4,
R2 R3−
R d 2−2 4 F1 1 2;
1,
1 2;
3 2;
R2 R4,
R2 R3+
R1 8d−2 2
d−3 2
∂
3R3 R3∂
2R2 R2 1 1−
R1/
R3 1 1−
R1/
R2−
R d 2−2 1d−3 2
d 2
×
FS d−
3 2,
1,
1;
1,
1,
1 2;
d 2,
d 2,
d 2;
R1 R4,
· · ·
R1 R1−
R3,
R1 R1−
R2+
R d 2−2 4√
π
×
FS(
1 2,
1,
1;
1,
1,
1 2;
2,
2,
2,
R1 R4,
R1 R1−
R3,
R1 R1−
R2)
+ (
R1(
1)
↔
R1(
2))
+ (
2,
3,
1)
+ (
3,
1,
2),
(17) wherethefunctionb123isindependentofd,b123
=
1 2∂
3R3 R3∂
2R2 R2 R2 1−
R1 R2 2F1 1,
1;
3 2;
R2 R3−
1 2 R1 1−
R1 R2 F1 1;
1,
1 2;
2;
R1 R3,
R1 R2+ (
1↔
2).
(18)Here, it is R1
=
R1(
1)
and(11) definesderivatives like∂
2r2.Thetermb123,when multipliedwith
(
−
d−24)
Rd 2−2
3 ,equals the term
of J123 in (15) with d-independent F1 and FS. It replaces the
so-calledb3-termofthevertexintegralin[11] forarbitrary
kine-matics,whilethed-dimensionalpartsof J1234 agree.
For d
→
4, all the expressions in square brackets in (17) ap-proachzero,thuscompensatingthepoleof(
2−
d/
2)
inthislimit. As a result, the J4 stays finite at d=
4, as it should be for anymassive4-pointfunction. Andthe
expansion for J1234 toorder
n needs, in this case, the evalution of the components to order
(
n+
1)
.Thederivationsof J123and J1234weredoneunderthe
assump-tionthatthekinematicalargumentsx
,
y,
z ofthe2F1,
F1, FS fulfill|
x|,
|
y|,
|
z|
<
1. Nevertheless, the above formulae are validat ar-bitrary kinematicalarguments, for massive vertices at e(
d)
>
2 andforboxintegrals ate(
d)
>
3.In AppendixAto AppendixC wewillshowhowtocalculatethevarious F1 andFS forarbitrarycomplexarguments;for2F1 weassumethatsuchcalculationsare
well-known.
3. Numericalresults
The scalar one-loop basis consistsof one- to four-point func-tions.Ourtwo-pointfunction J2
(
d)
isincompleteagreementwith[11], whilefor J3
(
d)
and J4(
d)
our resultsare novel.Concerningnumericalresultsforthe3-pointfunctionswereferto several ta-bles in [13]. The kinematics was chosen such that the results of [11] couldbecompared.2 Anothernumericalcomparison,forabox integral J4
(
d)
withvanishingGramdeterminant,maybefoundin[15].
In Table 1 we show few examples of four-point functions in comparisonto other packages. Wedid not aimatmaximal accu-racyandclaimessentiallysixtoeightsafedigits(absolutevalues). Further, one propagator is massive and d
=
4 or d=
5, and we can also allow for complex masses at the internal lines. A sam-ple-expansionisreproduced forthegeneralizedhypergeometric function F1inTableB.2.
For thesafe numerical calculation ofmassive vertices J3 and
massive box integrals J4 we collect stablenumerical
representa-tions for the generalizedhypergeometric functions F1 and FS in
theAppendices.
4. Discussion
Themassiveone-loopFeynmanintegralshavebeenrepresented asmeromorphicfunctionsofspace-timed intermsofgeneralized hypergeometricfunctions. Manydetailsleft outherewillbe pub-lishedelsewhere.TheFeynmanintegralscanbecalculated numer-icallyatarbitrarykinematicsandarbitrarydimensiond,including potential pole locationsat d
=
4+
2m. For phenomenological or multi-loop applications,itis wishfulto havethe poleexpansions in closed analytical form. Their derivation is subjectof a subse-quentstudy.The newrecursion (10) hasa unique feature. Itallows to de-rive n-dimensional Mellin-Barnes integrals for n-point Feynman integrals.Generally,n-dimensionalintegralsareobtainedbysector decompositionmethods,whileintheMellin-Barnesapproach,asit isadvocatedinnumericalloopcalculations,thenumberof dimen-sion grows faster. Within the MBsuite, AMBRE generates for the mostgeneral massiven-point one-loop function an Nn
=
12n(
n+
1
)
-dimensional MB-integral; according to the number of entries2 WewouldliketothankOlegTarasovforahelpfuldiscussionconcerningthis
260 K.H. Phan, T. Riemann / Physics Letters B 791 (2019) 257–264
Table 1
Comparison of the box integral J4 defined in (17) with
theLoopToolsfunctionD0(p2
1,p22,p23,p24,(p1+p2)2,(p2+
p3)2,m2
1,m22,m23,m24)[16,17] atm22=m23=m24=0.Further
numerical referencesarethepackagesK.H.P_D0(PHK, un-published)andMBOneLoop[15].Externalinvariants:(p2
1= ±1,p2 2= ±5,p23= ±2,p42= ±7,s= ±20,t= ±1). (p2 1,p22,p23,p24,s,t) 4-point integral (−, −, −, −, −, −) d=4, m2 1=100 J4 0.00917867 LoopTools 0.0091786707 MBOneLoop 0.0091786707 (+, +, +, +, +, +) d=4, m2 1=100 J4 −0.0115927−0.00040603 i LoopTools −0.0115917−0.00040602 i MBOneLoop −0.0115917369−0.0004060243 i (−, −, −, −, −, −) d=5, m2 1=100 J4 0.00926895 K.H.P_D0 0.00926888 MBOneLoop 0.0092689488 (+, +, +, +, +, +) d=5, m21=100 J4 −0.00272889+0.0126488 i K.H.P_D0 (–) MBOneLoop −0.0027284242+0.0126488134 i (−, −, −, −, −, −) d=5, m2 1=100−10 i J4 0.00920065+0.000782308 i K.H.P_D0 0.0092006 +0.000782301 i MBOneLoop 0.0092006481+0.0007823090 i (+, +, +, +, +, +) d=5, m2 1=100−10 i J4 −0.00398725+0.012067 i K.H.P_D0 −0.00398723+0.012069 i MBOneLoop −0.0039867702+0.0120670388 i
Yi j inthesecondSymanzikpolynomial,F
(
x)
=
21xiYi jxj−
iε
.Foravertexorbox,N3
=
6,
N4=
10.Inthepresentapproach,itisonlyN3
=
3,
N 4=
4.Evidently,areplacementoftheoriginal kinemati-calinvariantsm2i,
(
pe,ipe,j)
orYi j bythealternatives Rn= −λ
n/
Gnis an essential building block and it might well be possible to find similar lower-dimensional MB-representations also formore involvedmulti-loopintegrals.
Basic numericalfeatures of the newn-dimensional MB-repre-sentation (10) have beenstudied in[15] in comparisonwith[2], withthepackage MBOneLoop,includingcasesofsmallor vanish-ingGramdeterminant.
Itisinterestingtocompareourresultsfor J3
(
d)
and J4(
d)
withthe earlierstudy [11]. The d-dependent part of J3
(
d)
aswell asmuch of the d-dependent part of J4
(
d)
agree with our results.Further, the expressions for the b-terms in [11] differ from our
d-independentparts,althoughincertainkinematicalregions they doagree numericallyfor J3
(
d)
.We find noagreement for J4(
d)
,duetothevariouscontributingb-terms.
Acknowledgements
T.R.wouldliketo thankJ. Fleischer,J. Gluza,M. Kalmykovand O. Tarasov for helpful discussions and J. Usovitsch for assistance in numerical comparisons. P.H.K.’s work is funded by the Viet-namNationalFoundationforScienceandTechnologyDevelopment (NAFOSTED) under grant number 103.01-2016.33. He would like to thank J. Blümlein and DESY for the opportunity to work in 2015and 2016as a guestscientist atZeuthen. The work ofT.R. issupportedinpartbya2015AlexandervonHumboldtHonorary Research Scholarshipof the Foundation for Polish Sciences (FNP) andbythePolish NationalScienceCentre(NCN)undertheGrant Agreement2017/25/B/ST2/01987.
Appendix A. TheAppellfunction F1andLauricella-Saran
function FS
Numerical calculations ofspecific Gauss hypergeometric func-tions 2F1, Appell functions F1 (Eqn. (1) of [18]), and
Lauricella-Saran functions FS (Eqn. (2.9) of [19]) are neededfor the scalar
one-loopFeynmanintegrals:
2F1
(
a,
b;
c;
x)
=
∞ k=0(
a)
k(
b)
k k! (
c)
k xk,
(A.1) F1(
a;
b,
b;
c;
y,
z)
=
∞ m,n=0(
a)
m+n(
b)
m(
b)
n m!
n! (
c)
m+n y mzn,
(A.2) FS(
a1,
a2,
a2;
b1,
b2,
b3;
c,
c,
c;
x,
y,
z)
(A.3)=
∞ m,n,p=0(
a1)
m(
a2)
n+p(
b1)
m(
b2)
n(
b3)
p m!
n!
p! (
c)
m+n+p xmynzp.
The
(
a)
k is the Pochhammer symbol. The series converge for|
x|,
|
y|,
|
z|
<
1, but the functions are needed for arbitrary ar-guments. All the 2F1,
F1,
FS are finite and have no pole termsin
. Practically all aspects of 2F1 are well-known and
imple-mented incomputer algebra systems, in Mathematicaasbuilt-in symbol
Hypergeometric2F1[a,b,c,z]
. There is no pub-lic FS-package, while the Appell function F1(
a;
b1,
b2;
c;
x,
y)
[18] is implemented in Mathematica as built-in symbol
Ap-pellF1[a,b1,b2,c,x,y]
and in few other public packages. Alltheimplementationshavesystematiclimitations.One approachtothenumerics of F1 and FS maybe basedon
Mellin-Barnesrepresentations.FortheGaussfunction2F1 andthe
Appellfunction F1,Mellin-Barnesrepresentationsare known.See
Eqn. (1.6.1.6)in[20], 2F1
(
a,
b;
c;
z)
=
1 2π
i(
c)
(
a)(
b)
(A.4)×
+i∞−i∞ ds
(
−
z)
s(
a+
s)(
b+
s)(
−
s)
(
c+
s)
,
andEqn. (10)in[18],whichisatwo-dimensionalMB-integral:
F1
(
a;
b,
b;
c;
x,
y)
=
1 2π
i(
c)
(
a)(
b)
(A.5)×
+i∞−i∞ dt
(
−
y)
t2F1(
a+
t,
b;
c+
t,
x)
×
(
a+
t)(
b+
t)(
−
t)
(
c+
t)
.
FortheLauricella-SaranfunctionFS wederivedthefollowing,new,
three-dimensionalMB-integral: FS
(
a1,
a2,
a2;
b1,
b2,
b3;
c,
c,
c;
x,
y,
z)
(A.6)=
1 2π
i(
c)
(
a1)(
b1)
+i∞−i∞ dt F1
(
a2;
b2,
b3;
c+
t;
y,
z)
× (−
x)
t(
a1+
t)(
b1+
t)(
−
t)
(
c+
t)
.
A generalnumerical evaluationof theserepresentations deserves some sophistication. Letusmentionthe simpleone-loop massive
QEDvertexforwhichnotrivialMB methodexistswhenthe kine-maticsisMinkowskian,aproblemdiscussede.g.in[21] andsolved in[4].It wasdemonstratedin[22] thatMBOneLoop,aforkofthe packageMBnumerics [14,15,23,24] maybe usedto solve (A.4) to (A.6) atarbitrarykinematicswithhighprecision.
Onemight alsotryto approach thegeneralized hypergeomet-ricfunctionsusingPochhammer’sdoubleloopcontours[25,26],or studythedefiningdifferentialequations[27–29],etc.Afterseveral trials, we decided to base our numerics on the integral repre-sentations of F1 proposed in [30] and FS proposed in [19];see
AppendixBandAppendixC.
Appendix B.TheAppellfunctionF1
Aone-dimensionalintegralrepresentationforF1 [30] isquoted
inEqn. (9)of[18]: F1
(
a;
b,
b;
c;
x,
y)
=
(
c)
(
a)(
c−
a)
(B.1)×
1 0 du u a−1(
1−
u)
c−a−1(
1−
xu)
b(
1−
yu)
b.
Weneedthreespecificcases,takenatd
≥
4.Namelyforvertices: F1v(
d)
≡
F1 d−
2 2;
1,
1 2;
d 2;
xc,
yc (B.2)=
1 2(
d−
2)
1 0 du ud2−2(
1−
xcu)
√
1−
ycu.
Integrabilityisviolatedatu
=
0 ifnote(
d)
>
2.Similarly,forbox integrals: Fb1(
d)
≡
F1 d−
3 2;
1,
1 2;
d−
1 2;
xc,
yc (B.3)=
1 2(
d−
3)
1 0 du ud/2−5/2(
1−
xcu)
√
1−
ycu=
F1v(
d−
1).
Integrabilityis violated atu
=
0 if not e(
d)
>
3. Finally forthe definitionoftheboxSaranfunction FS (C.1):F1S
(
yc,
zc)
≡
F1(
1;
1,
1 2;
3 2;
yc,
zc)
(B.4)=
1 2 1 0 du√
1−
u(
1−
ycu)
√
1−
zcu.
Thesingularityatu
=
1 isintegrable.Numericalchecks maybe performed usingtransformations of
F1 functionswithdifferentvaluesofx
,
y [31],e.g.F1
(
a;
b,
b;
c;
x,
y)
= (
1−
x)
−b(
1−
y)
−b×
F1 c−
a;
b,
b;
c;
x x−
1;
y y−
1.
(B.5)However,therelationsdonot allowtotransformtherealpartsof bothx
,
y tovaluessmallerthanone.AppendixB.1. Specificvaluesof2F1andF1atd
=
4Thevertexfunction(15) contains2F1 andF1 withspecific
val-uesatd
=
4: 2F1 1,
1;
3 2;
xc=
ArcSin(
√
x c)
√
1−
xc√
xc (B.6) and F1 1;
1,
1 2;
2;
xc,
yc=
2 ArcTanh √ xc√1−yc √ xc−yc√
x c√
xc−
yc−
2 ArcTanh√√xc xc−yc√
x c√
xc−
yc.
(B.7)Using logarithms only, ArcSin
(
z)
= −
iln(
iz+
√
1−
z2)
andArcTanh
(
z)
=
12[
ln(
1+
z)
−
ln(
1−
z)
]
. Eqn. (B.7) is only valid if(
xc−
yc)
hasawell-defined imaginarypart.For xc=
x−
iε
x andyc
=
y−
iε
y thisis not necessarilythe caseifε
x andε
y arein-dependent andbothinfinitesimal. So(B.7) hastobe usedwitha grainofcare.
Theboxfunction(17) containsadditional2F1 andF1with
spe-cificvaluesatd
=
4: 2F1 1 2,
1;
1;
xc=
√
1 1−
xc (B.8) and F1 1 2;
1,
1 2;
3 2;
xc,
yc=
√
1 1−
yc 2 F1(
1 2;
1,
3 2;
xc−
yc 1−
yc)
=
ArcTanhxc−yc 1−yc
√
xc−
yc.
(B.9)Eqn. (B.9) is only validif
(
xc−
yc)
hasa well-defined imaginarypart. Finally, we like to mention that we have no analogue to (B.8) and(B.9) forFS atd
=
4,namelyFS(
12,
1,
1;
1,
1,
12;
2,
2,
2;
xc,
yc
,
zc)
.The Appell function FS
1
=
F1(
1;
1,
12;
32
;
yc,
zc)
used in thein-tegrand of the definition of the Saran function (C.1) can also be simplified: F1
(
1;
1,
1 2;
3 2;
yc,
zc)
=
1 1−
zc 2 F1 1,
1;
3 2;
yc−
zc 1−
zc=
ArcSin yc−zc 1−zc√
(
yc−
zc)(
1−
zc)
.
(B.10)Both representations in(B.10) are onlyvalid whenthe imaginary partofthedifference
(
yc−
zc)
iswell-defined.Forthe Feynman integrals studied here, we have totake into account that xc
,
yc and zc may have, in general, uncorrelatedin-finitesimal imaginary parts, and so their difference may be not
well-defined. Letusremind that xc
=
R1/
R4, and yc=
R1/(
R1−
R3
)
,andzc=
R1/(
R1−
R2)
.Here,alltheRnhave,accordingto(2),identicalimaginaryparts
−
iε
.Thisleadstodifferentinfinitesimal imaginaryparts−
ε
x,
−
ε
y,
−
ε
z,withpotentiallydifferentsigns.So,one hasbasically two equivalent options. Either one treats
ε
x,
ε
yand
ε
z as independent quantities and avoids the appearance oftermslike
(
xc−
yc)
and(
yc−
zc)
.Oroneusestheexactknowledgeoftheimaginarypartsofthe Rn fromtheir definitionsandarrives
262 K.H. Phan, T. Riemann / Physics Letters B 791 (2019) 257–264
AppendixB.2. NumericalcalculationofFv
1
(
d)
Forxc
=
x−
i X andyc=
y−
iY ,Eqn. (B.2) maybeusedfornu-mericsif
(
X,
Y)
≥
const. >
0 orif(
x,
y)
<
1.The remaining cases(
X= −
ε
x,
Y= −
ε
y)
→ +
0 deserve a closer inspection. Theyap-pearfromFeynman integrals.We exemplify herethe first one of thetwomore involvedcases:1
<
x<
y and1<
y<
x and intro-duceanauxiliarysplitparameterum
=
1 2 1 y+
1 x with 0<
1 y<
um<
1 x<
1.
(B.11) Intheintegrandof F1 therewillacut beginatu=
1y anda poleariseatu
=
1x forinfinitesimalε
x,
ε
y.Asplitoftheintegralatum,1
0 du=
um 0 du+
1 um du≡
iL+
iR,
(B.12)will leadto a separation ofthe singularities.In both integralsat the right handside, the integrand is regular with one exclusion.
Wediscussnow severalopportunitiesofcalculations, all ofthem withanaccuracyofsixtoeightsafedigitsorbetter.
Ourmost carefulapproach persued the following ansatz with additionalsplittings: F1v
(
d)
=
IA+
I0+
IC+
ID+
IB+
IE=
lim R→+0 1 y−R 0+
1 y+R 1 y−R+
um 1 y+R+
1 x−R um+
1 x+R 1 x−R+
1 1 x+R(B.13)
Afterperforming thelimit R
→
0 whereverpossible,theintegralsA and B will give real contributions, and the others are purely imaginary: F1v
(
d)
= [
e F1v(
d)
] +
i[
mF1v(
d)
]
=
A+
sign(
ε
x)
sign(
ε
y)
B+
i sign(
ε
y) (
−
C+
D+
E)
.
(B.14) Itis I0=
0,
(B.15) A=
d−
2 2 1 y 0 du ud/2−2(
1−
xu)
√
1−
yu,
(B.16) B=
d−
2 2π
1 x y x−
1 xd/2−2,
(B.17) C=
d−
2 2 um 1 y du ud/2−2(
1−
xu)
√
yu−
1,
(B.18) D=
d−
2 2 1 x um du 1−
xu⎛
⎜
⎝
u d/2−2√
yu−
1−
x−d/2+2 y x−
1⎞
⎟
⎠
+
d−
2 2 1 y x−
1 xd/2−2ln
(
R)
−
ln(
1 2x−
1 2 y)
,
(B.19) E=
d−
2 2 1 1 x du 1−
xu⎛
⎜
⎝
u d/2−2√
yu−
1−
x−d/2+2 y x−
1⎞
⎟
⎠
+
d−
2 2 1 y x−
1 xd/2−2−
ln(
R)
+
ln(
1−
1 x)
.
(B.20) Theremaining R-dependencesin(B.19) and(B.20) dropoutinthe sumof D andE.Alternatively, witha subtractionineach ofthetwo partial in-tegrals in (B.12), one may regularize the integrand of Fv
1
(
d)
as follows: iL=
um 0 du gx(
u)
−
gx1 y
√
1−
yu+
i ana L,
(B.21) iR=
1 um du gy(
u)
−
gy 1 x 1−
xu+
i ana R,
(B.22) with ianaL= −
2 gx1 y
yc
!
1−
ycum−
1 (B.23)→ −
2 gx1 y
yc
−
1+
i sign(
ε
y)
!
yum−
1,
ianaR= −
gy 1 x xc ln 1−
xc 1−
xcum (B.24)→ −
gy 1 x x ln x−
1 1−
xum+
iπ
sign(
ε
x)
.
Finally, a simplestapproach will also do a reasonable numer-ics:Performmeanvalueintegrals,likee.g.thebuilt-infunctionof Mathematica: F1v
(
d)
=
lim →+0⎡
⎢
⎢
⎣
⎛
⎜
⎜
⎝
1 y− 0+
um 1 y+⎞
⎟
⎟
⎠ +
⎛
⎜
⎜
⎝
1 x− um+
1 1 x−⎞
⎟
⎟
⎠
⎤
⎥
⎥
⎦ .
(B.25)Ofcourse,acalculationwith,say,morethansixtoeightsafedigits, willdeserveanexplicitcontrolofthealgorithmicdetails.
Numerical examples for F1v
(
d)
are collected in Tables B.1 and (B.2).AppendixB.3. NumericalcalculationoftheboxAppellfunctionFb1
(
d)
Forthecalculation offour-pointFeynman integrals,one needs
Fb
1
(
d)
asintroducedin(B.3), bothford=
4 andford=
4+
2m−
2
ε
.Thebox F1-functionisrelatedtothevertexfunction F1v(
d)
by(B.3).Consequently,thenumericsoftheforegoingsubsectionsmay betakenover.
Appendix C. TheLauricella-SaranfunctionFs
Forthecalculationofthe4-pointFeynmanintegrals,oneneeds the Lauricella-Saran function FS [19]. Sarandefines FS as
three-fold sum(A.3),seeEqn. (2.9) in[19]. Hederivesa3-foldintegral representation in Eqn. (2.15) anda 2-fold integral in Eqn. (2.16). Wewillusethefollowingquiteusefulrepresentation,derivedatp. 304of[19]:
Table B.1
TheAppellfunctionF1ofthemassivevertexintegralsasdefinedin(B.2).Asaproofofprinciple,onlytheconstanttermofthe
expansionind=4−2εisshown,F1(1;1,12;2;x,y).Uppervalues:thiscalculation,(Appendix B.2),lowervalues:(B.7).
x−iεx y−iεy F1(1;1,12;2;x,y) +11.1−10−12i +12.1−10−12i −0.1750442480735 −0.0542281294732 i −0.17504424807351877884498289912 −0.054228129473304027882097641167 i +11.1−10−12i +12.1+10−12i +1.7108545293244 +0.0542281294732 i +1.71085452932433557134838204175 +0.05422812947148217381589270924 i +11.1+10−12i +12.1−10−12i +1.7108545304114 −0.0542281294732 i +1.71085452932433557134838204175 −0.05422812947148217381589270924 i +11.1+10−12i +12.1+10−12i −0.1750442480735 +0.0542281294733 i −0.17504424807351877884498289912 +0.054228129473304027882097641167 i +12.1−10−15i +11 .1−10−15i −0 .1700827166484 −0.0518684846037 i +12.1−10−10i +11.1−10−15i −0.17008271664800058101165749279 −0.05186848460465674976556525621 i +12.1−10−15i +11.1+10−15i −0.1700827166484 −1.7544202909955 i −0.17008271664844025647268817399 −1.75442029099557688735842562038 i +12.1+10−15i +11 .1−10−15i −0 .1700827166484 +1.7544202909955 i −0.17008271664844025647268817399 +1.75442029099557688735842562038 i +12.1+10−15i +11 .1+10−15i −0 .1700827166484 +0.0518684846037 i +12.1−10−10i +11 .1−10−15i −0 .17008271664800058101165749279 +0.05186848460465674976556525621 i +11.1−10−15i −12 .1 −0.0533705146518 −0.1957692111557 i −0.05337051465189944473349401152 −0.195769211155733985388920833693 i +11.1+10−15i −12 .1 −0.0533705146518 +0.1957692111557 i −0.05337051465189944473349401152 +0.195769211155733985388920833693 i −11.1 +12.1−10−12i +0 .1060864084662 −0.1447440700082i +0.10608640847651064287133527599 −0.144744070021333407167349619088 i −11.1 +12.1+10−12i +0 .1060864084662 +0.1447440700082i +0.10608640847651064287133527599 +0.144744070021333407167349619088 i −12.1 −11.1 +0.122456767687224028 +0.12245676768722402506513395161 Table B.2
TheAppellfunctionF1(1−;1,1
2;2−;xc,yc)asdefined in(B.2),neededford=4−2atxc=11.1−10−12i,yc= 12.1−10−12i. F1(1−;1,1 2;2−;xc,yc) +(−0.1750442480735 −0.05422812947328 i) +(−0.0086188585913 −0.39051763820462 i) +(+0.37518853545319 −0.34047477405516 i)2 +(+0.49765461883470 −0.00717399489427 i)3 +(+0.32835724868237 +0.23005850008124 i)4 +(+0.11199125312340 +0.25409725390712 i)5 +(−0.00954795237038 +0.17050760870656 i)6 +(−0.04217861994524 +0.08576862780838 i)7 FS
(
a1,
a2,
a2;
b1,
b2,
b3;
c,
c,
c,
x,
y,
z)
(C.1)=
(
c)
(
a1)(
c−
a1)
1 0 dtt c−a1−1(
1−
t)
a1−1(
1−
x+
tx)
b1 F1(
a2;
b2,
b3;
c−
a1;
t y,
tz).
Inourcase,thisbecomesFbS
(
d)
=
FS d−
3 2,
1,
1;
1,
1,
1 2;
d 2,
d 2,
d 2,
xc,
yc,
zc=
(
d2)
(
d−23)(
32)
(C.2)×
1 0 dt√
t(
1−
t)
d−25(
1−
xc+
xct)
F1(
1;
1,
1 2;
3 2;
yct,
zct)
Eqn.(C.2) isvalidif
e(
d)
>
3.Withagrainofcareonemayoften use(B.10) for F1S.BecausetheF1 underthet-integralisfiniteandsmooth,wehavetoconcentrateonlyontheterm1
/(
1−
xc+
xct)
,whichdevelopsapoleintheintegrationregionattx
= (
1−
x)/
x if e(
xc)
=
x>
1 andifm
(
xc)
= −
ε
x isinfinitesimal.AppendixC.1. Case(i)FbS
(
d)
atx≤
1Forx
=
1,theintegral(C.2) isnotwell-defined.Ifx<
1,adirect, stablenumericalintegrationofFS istrivialonceF1 isknown.AppendixC.2. Case(ii)Fb
S
(
d)
atx>
1Ifx
>
1,one hastoapply aregularization procedureto FbS(
d)
, asitisdescribedin(Appendix B.2),andwillgetastableresultforFS.Thecalculationofthe F1intheintegrandin(C.2) isdiscussed
inAppendixB.1.
Onenowhastostudythesingularitystructureofthet-integral
asafunctionofxc withregular F1S.Introduce
FbS
(
d)
=
1 0 dtgS(
t)
−
gS(
tx)
1−
x+
xt+
gS(
tx)
I reg S(
xc),
(C.3) with gS(
t)
=
√
t(
1−
t)
(d−5)/2F1S(
yct,
zct)
(C.4) and tx=
1−
1 x.
(C.5)Thefirst integralin(C.3) isnumericallystable, andwhatremains istocalculateanalyticallytheintegral
IregS
(
xc)
= +
1 xc 1 0 dt t−
txc=
1 xc ln 1−
1 txc.
(C.6)Forinfinitesimal
ε
x,wegetIregS
(
xc)
→
1
264 K.H. Phan, T. Riemann / Physics Letters B 791 (2019) 257–264
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