New Results on the 3-Loop Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering

0.1 MB·English

by  J. Ablinger

DESY12-247,DO-TH12/39,SFB/CPP12-104,LPN12-141, arXiv:1212.5950[hep-ph]
3 New Results on the 3-Loop Heavy Flavor Wilson
1
0
Coefficients in Deep-Inelastic Scattering
2

n
a
J

2

]
h
JakobAblingera, JohannesBlümleinb,Abilio DeFreitas∗b, AlexanderHassel-
p
- huhna,b,SebastianKleinc,CarstenSchneidera,FabianWißbrockb
p
e a ResearchInstituteforSymbolicComputation(RISC)JohannesKeplerUniversity,
h
Altenbergerstraße69,A-4040Linz,Austria
[
b DeutschesElektronen-Synchrotron,DESY,Platanenalle6,D-15738Zeuthen,Germany.

2 cInstitutfürTheoretischePhysikE,RWTHAachenUniversity,D-52056Aachen,Germany.
v
0
5
9
We report on recent results obtained for the 3-loop heavy flavor Wilson coefficients in deep-
5
. inelasticscattering(DIS)atgeneralvaluesoftheMellinvariableN atlargerscalesofQ2. These
2
1 concerncontributionstothegluonicladder-topologies,thetransitionmatrixelementsinthevari-
2
able flavor scheme of O(n T2) and O(T2), and first results on higher 3-loop topologies. The
1 f F F
: knowledgeoftheheavyflavorWilsoncoefficientsat3-looporderisofimportancetoextractthe
v
i partondistributionfunctionsanda (M2)incompleteNNLOQCDanalysesoftheworldprecision
X s Z
dataonthestructurefunctionF (x,Q2).
r 2
a
36thInternationalConferenceonHighEnergyPhysics
4-11July2012
Melbourne,Australia
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
NewResultsonthe3-LoopHeavyFlavorWilsonCoefficients... AbilioDeFreitas
The massive Wilson coefficients in deep-inelastic scattering are known to be expressible in
the limit of high virtualities Q2 ≫m2 as convolutions between massive operator matrix elements
(OMEs) and massless Wilson coefficients [1]. Here m denotes the heavy quark mass. The gen-
eral structure of the Wilson coefficients to O(a 3) has been derived in [2]. These massive Wilson
s
coefficientsareinturnconvolutedwithpartondistributionfunctionstoobtaintheheavyflavorcon-
tributions to DIS structure functions at leading twist. They have been calculated for the twist-2
heavy flavorcontributions totheunpolarized structure functions atleading[3]andnext-to-leading
order [4] 1 for general values of Q2. Since the massless Wilson coefficients are known by now at
3-looporder[6],itremainstocomputetheOMEsanalyticallyatO(a 3),inordertoobtainthemas-
s
siveWilsoncoefficientsatNNLO.ThesecoefficientswillallowforaconsistentNNLOanalysisof
thedeep-inelastic worlddataatQ2>20GeV2,cf.[7].
∼
Intheseproceedings,wediscussrecentprogressobtainedinthisdirection. Ouraimistocalcu-
lateallcontributing OMEsforgeneralvaluesoftheMellinvariable N. Animportantpreviousstep
towards this goal was the computation of the moments of the massive OMEsfor N =2...10(14)
contributing in the fixedand variable 2 flavorschemes [2]. The3-loop heavy flavorcorrections to
F (x,Q2)intheasymptoticcasewerecalculated in[9]. Firstresults forgeneralvaluesofN forthe
L
colorfactorfactorsT2C werecalculatedin[10]fortwoheavyquarklinesofthesamemass. The
F A,F
case of two different quark masses was considered in [10,11] for fixed moments. Results for the
colorfactors n T2C forgeneral N wereobtainedin[12,13]andthecalculation of3-loopladder
f F A,F
topologies was performed in [14]. Two–loop results up to O(e ) were obtained in [15]. Here the
massiveOMEsarecomputedforon-shellexternalmasslesspartons. Thecaseofamassiveon-shell
external fermionlinewasstudied attwoloopsin[16]incase ofQED.
In the following we will describe the methods used to perform these computations. We gen-
erate the Feynman diagrams using QGRAF[17]. After the numerators of these diagrams are con-
tracted with appropriate projectors we end up with a large set of scalar integrals. Many of these
integrals arecalculated usingavarietyofapproaches, namely,
1. Modernsummationalgorithms, implemented inthe MathematicapackageSigma[18].
2. Themethodofhyperlogarithmsforconvergentintegrals, generalizingthemethoddeveloped
in[19]tooneadditional variable x.
3. Mellin-Barnes integralrepresentations [20].
4. Theuseofintegration byparts identities [21]toexpress allintegrals intermsofasmallset
ofmastersintegrals.
Wewillfocushereonthefirsttwomethodsandshowafewexamples. TheFeynmandiagrams
with operator insertions may be turned into nested sums [22]. These infinite and finite sums may
be solved using Sigma whenever they have a representation in terms of elements of difference-
andproductfields. Thisincludesdivergentdiagrams,sincethedifferentpolesandpowersine may
be separated. Letus consider the scalar integrals associated with the ladder diagrams like the one
shown in Fig. 1. In this diagram, the loop fermion is massive and the momentum of the external
1ForapreciseimplementationinMellinspacesee[5].
2Inusingvariableflavorschemesacorrectscalematchingisofimportance[8].
2
NewResultsonthe3-LoopHeavyFlavorWilsonCoefficients... AbilioDeFreitas
l
q q −l k
p → → p
q −p k−l k −p
l −p
Figure1: 3-loopladderdiagramcontainingacentrallocaloperatorinsertion.
gluons is p, with p2 =0. We consider the case where all powers of propagators are equal to one,
and in the numerator of the integral we only have the operator insertion (D ·l)N. The result after
Feynmanparameterization andcalculation oftheloop-momentum integralsturnsouttobe[14]
I = i(D .p)Na3sSe3Iˆ , (1)
1a (m2)2−32e 1a
whereSe isthespherical factor Se =exp e2(gE−ln(4p )) ,and
(cid:2) (cid:3)
Iˆ = −exp −3eg G (2−3e /2)(cid:213) 7 1dw q (1−w1−w2)w−1e/2w2−e/2(1−w1−w2)
1a 2 E i 2−3e/2
(cid:18) (cid:19) i=1Z0 1+w w3 +w w4
11−w3 21−w4
×we/2(1−w )−1+e/2we/2(1−w )−1+e/2(1−(cid:16)w w −w w −(1−w(cid:17)−w )w )N . (2)
3 3 4 4 5 1 6 2 1 2 7
Expanding the polynomial that appears raised to the Nthpower, one can see that the w - and
1
w -integrals, can be written in terms of an Appell hypergeomteric function. After an appropri-
2
ate analytic continuation, we end up with the following representation of the integral in terms of
multiplesums,
Iˆ = exp −23eg E G (2−3e /2) (cid:229)¥
1a
(N+1)(N+2)(N+3)
(cid:0) (cid:1) m,n=0(
N(cid:229) +2 N+3 (t−e /2)m(N+2+e /2)m+n(N+3−t−e /2)n
t (N+4−e )
t=1(cid:18) (cid:19) m+n
t,t−e /2,m+1+e /2,n+1+e /2,N+3−t,N+3−t−e /2
×G
N+4−e ,m+1,n+1,m+t+1+e /2,N+n−t+4+e /2
(cid:20) (cid:21)
−N(cid:229) +3s(cid:229)−1 s N+3 (−1)s(r−e /2)m(s−1+e /2)m+n(s−r−e /2)n
r s (s+1−e )
s=1r=1(cid:18) (cid:19)(cid:18) (cid:19) m+n
r,r−e /2,s−r,m+1+e /2,n+1+e /2,s−r−e /2
×G . (3)
m+1,n+1,m+r+1+e /2,s−r+n+1+e /2,s+1−e
)
(cid:20) (cid:21)
Wecannowexpandine ,andtheresultingmultiplesumscanthenbeperformedusing thepackage
Sigma. The result for this and other integrals can be written in terms of harmonic sums S [23]
~a
3
NewResultsonthe3-LoopHeavyFlavorWilsonCoefficients... AbilioDeFreitas
andtheirgeneralizations S (~x ;N)[24,25]3:
~a
(cid:229)N sign(b)k
S (N) = S (k), S (k)=1
b,~a k|b| ~a 0/
k=1
N h k
S (h ,~x ;N) = (cid:229) S (~x ;k), S =1, h ,x ∈R. (4)
b,~a kb ~a 0/
k=1
Omittingtheexplicitdependence oftheharmonicsumsonN,weobtain
4(N+1)S +4 2S 1 S4
Iˆ = − 1 z + 2,1,1 + −2(3N+5)S − 1
1a (N+1)2(N+2) 3 (N+2)(N+3) (N+1)(N+2)(N+3) 3,1 4
(cid:26)
4(N+1)S −4N 5N+6 2(3N+5)S2 9+4N
+ 1 S +2 (2N+3)S + S + 1 + S2
N+1 2,1 1 N+1 3 (N+1)(N+2) 4 2
(cid:20) (cid:21)
7N+11 5N 5 N 4(2N+3)
+ 2 + S − S2 S + S3+ S
(N+1)(N+2) N+1 1 2 1 2 N+1 1 (N+1)2(N+2) 1
(cid:20) (cid:21)
1 2N+3
− (2N+3)S +8 . (5)
2 4 (N+1)3(N+2)
(cid:27)
This result was checked using MATAD [27] for the fixed moments N =1...10. Other, more in-
volved, integrals calculated inasimilarwayweregiveninRef.[14].
Thesecondmethodwehaveusedtocomputetheintegrals isbasedonanalgorithm originally
proposed in [19]. It is applicable when the integral turns out to be finite, even in case for local
operator insertions for a fixed integer value of the Mellin variable N. We have generalized this
method to the case allowing for one non-vanishing fermion mass and local operator insertions
in order to find the general N-representations for convergent 3-loop topologies. We work in the
a -representation andobtainintegrals oftheform
I (N)= ··· da da da da da da da da T d 1−(cid:229) a . (6)
4 1 2 3 4 5 6 7 8 U2V2 i
Z Z i !
Thecorresponding graphpolynomials ofagraph Garegivenby
• U =(cid:229) (cid:213) a ,whereT denotesthespanning treesofG.
T l∈/T l
• V =(cid:229) a .
l∈massive l
• Dodgson polynomials [28] T arise from the operator insertions. Theform ofthese poly-
nomialswilldepend onthespecificoperatorinsertion weare considering.
The integrals given by (6) are projective integrals, where one a -parameter may be set to one
eliminatingthed –distribution. Theoperatorssitonon-shelldiagramswhichobeyspecificsymme-
tries. These are generally not obeyed by the operator insertion. TheFeynman parameter integrals
arenowperformed intermsofhyperlogarithms [19] L(−→w,z):C\S →C,where
• S ={s ,s ,...,s }aredistinctpointsinCwhichmaycontain integration variables.
0 1 N
3Cyclotomicandgeneralizedcyclotomicharmonicsumsandpolylogarithmsandtheirrelationshavebeentreated
in[26].
4
NewResultsonthe3-LoopHeavyFlavorWilsonCoefficients... AbilioDeFreitas
• →−w isawordoverthealphabet A={a ,a ,...,a }, where each letter a corresponds toa
0 1 N i
points .
i
−→
L(w,z)isuniquelydefinedbythefollowingproperties :
1
L({},z)=1, and L(0n,z,)= logn(z) for n≥1
n!
¶ L({a−→w},z)= 1 L(−→w,z) for z∈C\S
¶ z i z−s
i
−→ →−
If w isnotoftheform w=(0,0,···,0), thenlimL(w,z)=0. (7)
z→0
Forexample, L(a,z)=log(z−s )−log(s ).
i i i
Thehyperlogarithms satisfyshufflerelations, e.g.
L({a ,a },z)L({a },z)=L({a ,a ,a },z)+L({a ,a ,a },z)+L({a ,a ,a },z) . (8)
1 2 3 3 1 2 1 3 2 1 2 3
The points to which the indices a correspond may contain further integration variables. Using
i
theseproperties afterpartialfractioning andintegration byparts,onecanexpressanyprimitivefor
expressions consisting of rational and hyperlogarithmic functions in terms of different hyperloga-
rithmic functions. These primitives have to be evaluated at the respective integration limits. Due
to the operator-insertions leading to power-type functions, the integrals do not fit directly into the
frameworkofthealgorithmforgeneralvaluesof N. Inordertoobtainthecorresponding extension
agenerating function isconstructed bythemapping,
¥
p(a ,···,a )N → (cid:229) xkp(a ,···,a )k = 1 . (9)
1 n 1 n 1−x p(a ,···,a )
k=0 1 n
Figure2: A3-loopBenzdiagram.
PerformingtheFeynman-parameterintegrations thenleads toanexpressionwhichcontainshyper-
logarithmsL inthevariablex. Usingthismethod,thescalarintegralwithallpowersofpropagators
w
equal tooneassociated withthe diagram shown inFig.2, corresponding toaBenz-type topology,
yields
1
I(x) = z 2L({−1},x)−2(−1+2x)L({1},x)−4L({1,1},x)
3
(1+N)(2+N)x
(
h i
−3L({−1,0,0,1},x)+2L({−1,0,1,1},x)−2xL({0,0,1,1},x)+3xL({0,1,0,1},x)
−xL({0,1,1,1},x)+(−3+2x)L({1,0,0,1},x)+2xL({1,0,1,1},x)−L({1,0,1,1,1},x)
5
NewResultsonthe3-LoopHeavyFlavorWilsonCoefficients... AbilioDeFreitas
−(5x−1)L({1,1,0,1},x)+xL({1,1,1,1},x)−2L({1,0,0,1,1},x)+3L({1,0,1,0,1},x)
+2L({1,1,0,0,1},x)+2L({1,1,0,1,1},x)−5L({1,1,1,0,1},x)+L({1,1,1,1,1},x) .
)
(10)
Finally, theNthcoefficient ofthisexpression in xhas tobeextracted analytically inordertoundo
themapping(9). ThisisachievedusingtheGetMomentsoptionofthepackageHarmonicSums
[25]. One may also use guessing-methods to obtain the corresponding difference equation based
on a huge number of moments, cf. [29]. Fora more complicated graph with non-trivial argument
structure in xwewereabletoproduce ∼1500moments[30]. Oneobtains fromEq.(10)
1 648+1512N+1458N2+744N3+212N4+32N5+2N6
I(N) =
(N+1)(N+2)(N+3) (1+N)3(2+N)3(3+N)3
(
2 −1+(−1)N+N+(−1)NN N 1 1
− z −(−1)NS − S3+ S4− S
(1+N) 3 −3 6(1+N) 1 24 1 4 4
(cid:0) (cid:1)
7+22N+10N2 19 1+4N+2N2 9 (−9+4N)
− S − S2− S2+ S − S
2(1+N)2(2+N) 2 8 2 2(1+N)2(2+N) 1 4 2 3(1+N) 3
(cid:0) (cid:1)
(−1+6N) 54+207N+246N2+130N3+32N4+3N5
−2(−1)NS + S + S
−2,1 (1+N) 2,1 (1+N)3(2+N)2(3+N)2 1
(−2+7N) 13
+4z S − S S + S S −7S S −7S +10S . (11)
3 1 2 1 3 1 2,1 1 3,1 2,1,1
2(1+N) 3
)
Anotherexample, wherethistechnique hasbeenapplied, isshowninFig.3.
Figure3: Asecondexampleofa3-loopBenztopology.
Inthiscasetheresultis
1 2 1−13(−1)N+(−1)N23+N+N−7(−1)NN+3(−1)N21+NN
I(N) = z
3
(N+1)(N+2) (1+N)(2+N)
( (cid:0) (cid:1)
1 (−1)N (−1)N(3+2N) 5(−1)N 2(−1)N(3+N)
+ S + S3− S + S2+ S
(2+N) 3 2(2+N) 1 2(1+N)2(2+N) 2 2 2 (1+N)(2+N) 2,1
(−1)N(3+2N) (−1)N 3(−1)N(4+3N) 2
+ S2− S S2+ S +3(−1)NS + S
2(1+N)2(2+N) 1 2 2 1 (1+N)(2+N) 3 4 (2+N) −2,1
(−1)N(5+7N)
−12(−1)NS z + S S +3(−1)NS S +4(−1)NS S −4(−1)NS
1 3 1 2 1 3 2,1 1 3,1
2(1+N)(2+N)
6
NewResultsonthe3-LoopHeavyFlavorWilsonCoefficients... AbilioDeFreitas
4 (−1)N22+N−3(−2)NN+3(−1)N21+NN 1
− S ,1 −5(−1)NS
1,2 2,1,1
(1+N)(2+N) 2
(cid:0) (cid:1) (cid:18) (cid:19)
2 −(−1)N22+N−13(−2)NN+5(−1)N21+NN 1
+2(−1)Nz S (2)+ S ,1,1
3 1 1,1,1
(1+N)(2+N) 2
(cid:0) (cid:1) (cid:18) (cid:19)
1 1
−2(−1)NS 2, ,1 −(−1)NS 2, ,1,1 . (12)
1,1,2 1,1,1,1
2 2
)
(cid:18) (cid:19) (cid:18) (cid:19)
Notice the presence of generalized harmonic sums and highly divergent factors in the limit
N →¥ , such as 2N. It can be shown, however, that the complete expression is convergent in this
limit and possesses a well-defined asymptotic expansion for N →¥ . In general neither the repre-
sentationinindividualnestedsumsorbyiteratedintegralsshowsthisproperty,butacorresponding
combination oftermsdoes.
We calculated the contributions of O(n T2C ) to all massive OMEs completely [12,13].
f F A,F
Furthermore, first systematic results were obtained for the case of graphs containing two massive
fermionlineswithm =m . AtypicalgraphisshowninFig.4inthegluoniccase.
1 2
Figure4: A3-loopgraphcontainingtwomassivefermionlinesm =m andanoperatorinsertion.
1 2
Oneobtains
1 1 74N3−455N2+381N−210 1
I= + − S (N)
105e 2 e 44100(N−1)N(N+1) 210 1
" #
8903N3+39537N2−114440N+36576
+ S (N)
1
2822400(N+1)(2N−3)(2N−1)
P 1
+ 1 + S (N)2+S (N)+3z
148176000(N−1)2N2(N+1)2(2N−3)(2N−1) 840 1 2 2
(cid:16) (cid:17)
+2−2N−9(N−1)N(5N−6) 2N −7z −(cid:229)N 4j +(cid:229)N 4jS1(j) , (13)
3(2N−3)(2N−1) (cid:18)N (cid:19) 3 j=1 2jj j3 j=1 2jj j2 !
(cid:0) (cid:1) (cid:0) (cid:1)
P =1795487N8−7087789N7+10654130N6−5797102N5+6828839N4−16594069N3
1
+9651144N2+902160N−1058400. (14)
Integrals of this type usually contain finite binomial and inverse binomial sums, which even may
benested.
In conclusion, we have seen that the methods shown here allow us to obtain analytic expres-
sionsatgeneralvaluesofN for3–loopintegralscontributingtothemassiveOMEswhichcouldnot
be obtained byother methods before. Wecontinue working onthe setofintegrals that weneed in
ordertoobtainallnecessaryoperatormatrixelements. Inparticular,wearestudyingthepossibility
7
NewResultsonthe3-LoopHeavyFlavorWilsonCoefficients... AbilioDeFreitas
also to extend the method of hyperlogarithms. Application of these methods to the more compli-
cated case of non-planar integrals are underway. The package Sigma and related packages are
continuously beingupgraded tobeabletomeetthechallenges thatkeeparisinginthisendeavor.
Acknowledgment. WewouldliketothankF.Brownfordiscussions. TheFeynmandiagramshave
been drawn using Axodraw. This work has been supported in part by DFG Sonderforschungs-
bereich Transregio 9, Computergestützte Theoretische Teilchenphysik, by the Austrian Science
Fund (FWF) grant P203477-N18, by the EU Network LHCPHENOnetPITN-GA-2010-264564,
andERCStartingGrantPAGAPFP7-257638.
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