The massive 3-loop operator matrix elements with two

PoS(RADCOR2017)071

DESY 17-201, DO-TH 17/34

The massive 3-loop operator matrix elements withtwo masses and the generalized variable flavornumber scheme∗

J. Ablingera, J. Blümleinb, A. De Freitas†b, A. Goedickec‡, C. Schneidera,K. Schönwaldb and F. Wißbrocka,b

aResearch Institute for Symbolic Computation (RISC), Johannes Kepler University,Altenbergerstraße 69, A–4040, Linz, Austria.

bDeutsches Elektronen–Synchrotron, DESY, Platanenallee 6, D-15738 Zeuthen, Germany.cInstitut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT),76128 Karlsruhe, Germany.

We report on our latest results in the calculation of the two–mass contributions to 3–loop operatormatrix elements (OMEs). These OMEs are needed to compute the corresponding contributionsto the deep-inealstic scattering structure functions and to generalize the variable flavor numberscheme by including both charm and bottom quarks. We present the results for the non-singletand Agq,Q OMEs, and compare the size of their contribution relative to the single mass case.Results for the gluonic OME Agg,Q are given in the physical case, going beyond those presentedin a previous publication where scalar diagrams were computed. We also discuss our recentlypublished two–mass contribution to the pure singlet OME, and present an alternative method ofcalculating the corresponding diagrams.

RADCOR 2017 - 13th International Symposium on Radiative Corrections(Applications of Quantum Field Theory to Phenomenology),September 24 - 29, 2017St. Gilgen, Austria

∗This work was supported in part by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15) and the Euro-pean Commission through contract PITN-GA-2012-316704 (HIGGSTOOLS).

†Speaker.‡A. Hasselhuhn in previous publications.

c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/

mmizzaro

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PoS(RADCOR2017)071

The massive 3-loop OMEs with two masses and the generalized VFNS A. De Freitas

1. Introduction

Massive operator matrix elements (OMEs) constitute a key ingredient in the calculation of heavyquark corrections to deep–inelastic scattering (DIS) structure functions at large virtualities, as theyprovide a link between the corresponding massless Wilson coefficients and the massive ones. TheseOMEs are also needed to obtain the transition relations of the variable flavor number scheme(VFNS). Due to the accuracy of the currently available experimental data, the OMEs need to becalculated at three-loop order. Their knowledge is of importance for the precise measurement ofthe strong coupling constant αs(M2

Z) [1], of the parton distribution functions [2], and of the heavyquark masses mc and mb [3] from the world deep inelastic data.

In a series of publications, we have computed the single mass contributions at three loops to theOME Agq,Q [4], the T 2

F terms of the gluonic OME Agg,Q [5], the non-singlet contributions as well asall of the associated Wilson coefficients and structure functions [6–9], the pure singlet result [10],and the diagrams in the case of the ladder and V topologies of the OME AQg [11]. Furthermore,all diagrams contributing to AQg which result from master integrals obeying first order factorizingdifferential equations have been completed [12], as well as all color and ζ -value terms, which canbe determined using the method of arbitrarily large moments [13]. The logarithmic contributionsto all OMEs were given in [14]. Before a series of moments has been calculated for all massiveOMEs in Ref. [15].

At three loops, irreducible Feynman diagrams with two fermion loops of different masses ap-pear for the first time.1 The contributions from this type of diagrams cannot be ignored, since themass of the bottom quark is not considerably larger than the mass of the charm quark, which inparticular means that both quarks need to be decoupled simultaneously in the VFNS. The renor-malization of the OMEs in the 2-mass case has been performed in Ref. [17]. Here also the VNFShas been generalized to the 2-mass case.

In these proceedings, we report on our recent progress in the calculation of these two-massthree-loop contributions to the OMEs [17, 18]. In Section 2, we study the simplest of these OMEs,namely, A(3),NS

qq,Q and A(3)gq,Q (the tilde on top of the OMEs is used to denote the two–mass contri-

butions). In these cases, the dependence of the OMEs on the Mellin variable N and the massesfully factorizes. In Section 3, we show our recent results on the physical diagrams for A(3)

gg,Q, goingbeyond the results presented in [17], where a series of scalar diagrams were computed. In Sec-tion 4 we discuss the pure singlet case and show an alternative method of computing the Feynmandiagrams and the corresponding Feynman integrals to the one given in Ref. [18] and conclude inSection 5.

2. Operator matrix elements with a factorizing N and η dependence

The simplest OMEs containing irreducible two–mass contributions are the three–loop non–singletOME, A(3),NS

qq,Q , and the gluonic OME A(3)gq . In the case of these two OMEs, the dependence on the

ratio of the masses and the Mellin variable N factorizes completely, unlike the case in all otherOMEs, where these variables are intertwined in complicated functions, as we will see later.

1Reducible 2-mass contributions emerge already at NLO, cf. [16].

2

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All of the diagrams appearing in A(3),NSqq,Q and A(3)

gq contain two massive fermion bubbles, one ofwhich may be rendered effectively massless by using a Mellin–Barnes representation [19–23].

µ, a ν, b= g2

s TF4π(4π)−ε/2 (kµkν − k2gµν

)×∫ +i∞

−i∞

dσ

(m2

µ2

)σ (−k2)ε/2−σ Γ(σ − ε/2)Γ2(2−σ + ε/2)Γ(−σ)

Γ(4−2σ + ε).

(2.1)

This yields similar integrals as the ones appearing in the case where there is one massive and onemassless fermionic line [24,25]. One may now combine the denominators of the Feynman integralsusing Feynman parameters, integrate the momenta and perform the Feynman parameter integralsin terms of Euler Beta–functions. The OMEs will then be given by a linear combination of contourintegrals of the form

I ∝ Γ

[f1(ε,N), . . . , fi(ε,N)

fi+1(ε,N), . . . , fI(ε,N)

]×∫ +i∞

−i∞

dσ ησ

Γ

[g1(ε)+σ ,g2(ε)+σ ,g3(ε)+σ ,g4(ε)−σ ,g5(ε)−σ

g6(ε)+σ ,g7(ε)−σ

], (2.2)

where the f j and the g j are linear functions, N is the Mellin variable appearing in the operatorinsertion Feynman rules, and η is the ratio of the square of the masses

η =m2

2

m21, (2.3)

where we assume m1 > m2, i.e., η < 1.After closing the contour in (2.2) and collecting the residues, the integrals end up being ex-

pressed as a linear combination of generalized hypergeometric 4F3–functions [26]

I = ∑j

C j (ε,N) 4F3

[a j,1(ε),a j,2(ε),a j,3(ε),a j,4(ε)

b j,1(ε),b j,2(ε),b j,3(ε),η

]. (2.4)

Here and in the following finite and infinite sums occur which have to be summed. In the moreinvolved cases we thoroughly use difference field and ring theory algorithms [27–35] which areencoded in the package Sigma [36,37]. Since in (2.4) the parameters of the hypergeometric func-tions depend only on the dimensional regularization parameter ε , their corresponding expansionmay be performed with the code HypExp 2 [38]. The results of these expansions are then givenin terms of the following (poly)logarithmic functions [39–42],{

ln(η), ln(1−η), ln(

1−√η

1+√

η

), Li2 (

√η) , Li2 (η) , Li3 (

√η) , Li3 (η)

}. (2.5)

2.1 The flavor non–singlet contribution

The pole structure of the two–mass contribution to the non–singlet OME can be found from therenormalization procedure, which involves mass, coupling constant and operator renormalization,

3

PoS(RADCOR2017)071


as well as collinear factorization. After the subtraction of the single-mass terms, one obtains [17]

Â(3),NS

qq,Q = − 163ε3 γ

(0)qq β

20,Q−

4ε2

[23

β0,QγNS,(1)qq + γ

(0)qq β

20,Q (L1 +L2)

]− 2

ε

[β0,Qγ

NS,(1)qq (L2 +L1)

+γ(0)qq β

20,Q(L2

1 +L2L1 +L22)+4aNS,(2)

qq β0,Q−13

ˆγ(2),NSqq

]+ a(3),NS

qq,Q , (2.6)

where the γ(l)i j are the anomalous dimensions at l +1 loops, β0,Q =−4

3 TF , and

L1 = ln(

m21

µ2

), L2 = ln

(m2

2µ2

). (2.7)

The renormalized expression in the MS–scheme, treating the heavy quarks in the on-shell scheme,is given by

A(3),MS,NSqq,Q = γ

(0)qq β

20,Q

(23

L31 +

23

L32 +

12

L22L1 +

12

L21L2

)+β0,Qγ

NS,(1)qq

(L2

1 +L22)

+

{4aNS,(2)

qq β0,Q +12

β20,Qγ

(0)qq ζ2

}(L1 +L2)+8aNS,(2)

qq β0,Q + a(3),NSqq,Q . (2.8)

BothÂ(3),NS

qq,Q and A(3),MS,NSqq,Q vanish for N = 1 at all orders in ε due to fermion number conservation.

In Figure 1, we show a sample of the diagrams contributing to A(3),NSqq,Q . The remaining di-

agrams are related to these by the exchange m1 ↔ m2. In the case of the diagrams 2 and 3 inFigure 1, the pre–factors C j (ε,N) appearing in Eq. (2.4) will contain a sum arising from the vertexoperator Feynman rule (see Section 8.1 of Ref. [15]), which can be evaluated in terms of sin-gle harmonic sums using the Mathematica packages Sigma [36, 37], HarmonicSums [43–45],EvaluateMultiSums and SumProduction [46].

•••⊗

(1)

•••⊗

(2)

•••⊗

(3)

Figure 1: Diagrams for the two-mass contributions to A(3),NSqq,Q . The curly lines denote gluons, the dashed

arrow line represents the external massless quarks, while the thick solid arrow line represents a quark ofmass m1, and the thin arrow line a quark of mass m2. We assume m1 > m2. All diagrams have been drawnusing Axodraw [47].

For the constant part, a(3),NSqq,Q , which is the only term in Eq. (2.6) that is not determined by the

renormalization prescription, we obtain the following result

a(3),NSqq,Q = CFT 2

F

{(49

S1−3N2 +3N +2

9N(N +1)

)[−24

(L3

1 +L32 +(L1L2 +2ζ2 +5)(L1 +L2)

)

4

PoS(RADCOR2017)071


+η +1η3/2

(5η

2 +22η +5)(−1

4ln2(η) ln

(1+√

η

1−√η

)+2ln(η)Li2 (

√η)−4Li3 (

√η)

)+(1+√

η)2

2η3/2

(−10η

3/2 +5η2 +42η−10

√η +5

)[Li3 (η)− ln(η)Li2 (η)

]+

643

ζ3

+83

ln3(η)−16ln2(η) ln(1−η)+10η2−1

ηln(η)

]+

16(405η2−3238η +405

)729η

S1

+43

(3N4 +6N3 +47N2 +20N−12

3N2(N +1)2 − 403

S1 +8S2

)[43

ζ2 +(L1 +L2)2]

+89

(130N4 +84N3−62N2−16N +24

3N3(N +1)3 − 523

S1 +803

S2−16S3

)(L1 +L2)

+

[− R1

18N2(N +1)2η+

2(5η2 +2η +5

)9η

S1 +329

S2

]ln2(η)− 4R2

729N4(N +1)4η

+3712

81S2−

128081

S3 +25627

S4

}. (2.9)

Here S~a ≡ S~a(N) denote the (nested) harmonic sums [48]

Sb,~a(N) =N

∑k=1

(sign(b))k

k|b|S~a(k), S /0 = 1, b,ai ∈ Z\{0} . (2.10)

The Ri’s are polynomials in N and η , which we will not show here.This result is symmetric under the exchange of masses m1↔ m2, and agrees with the results

obtained in [17]2 for the individual fixed moments N = 2,4,6. Eq. (2.9) vanishes for N = 1as expected. A similar result is obtained in the case of the transversity contribution, where thechange γNS

qq → γNS,transqq [6, 51] and the corresponding one in the 2-loop OMEs must be performed

in Eq. (2.6), cf. [17].By a Mellin inversion of Eq. (2.9) we obtain the result in x-space [17]. In Figure 2, we plot the

ratio of the two–mass contribution to the complete T 2F term including the single mass results [6] as

a function of x and Q2, taking the masses of the heavy quarks to be those of the charm and bottomquarks in the on-shell scheme. The two–mass contributions become more important for largervalues of Q2, where the dependence on the ratio as a function of x becomes almost flat approachingvalues of O(0.4).

2.2 The gq-contribution

The genuine two-mass contributions to the OME A(3)gq,Q can be calculated in a similar way as ANS,(3)

qq,Q .A sample of the contributing diagrams is shown in Figure 3. There are three more diagrams relatedto these by the exchange m1↔ m2.

From the renormalization procedure, one obtains the following pole structure,

Â(3)

gq,Q = −16ε3 γ

(0)gq β

20,Q−

4ε2

[3γ

(0)gq β

20,Q (L2 +L1)+β0,Qγ

(1)gq

]+

1ε

[−6γ

(0)gq β

20,Q(L2

2 +L1L2 +L21)

2See also [49, 50].

5

PoS(RADCOR2017)071


0.001 0.01 0.1 1 x

-0.6

-0.4

-0.2

0.0

0.2

0.4

Aqq,QNS,H3L,2-mass �Aqq,Q

NS,TF2 H3L

Figure 2: The ratio of the genuine 2-mass contributions to A(3),NSqq,Q to the complete T 2

F -part of massive

3-loop OME A(3),NSqq,Q as a function of x and Q2, for mc = 1.59 GeV,mb = 4.78 GeV in the on-shell scheme.

Dash-dotted line: µ2 = 30 GeV2; Dotted line: µ2 = 50 GeV2; Dashed line: µ2 = 100 GeV2; Full line:µ2 = 1000 GeV2. The single mass contributions are given in Ref. [6]; from Ref. [17].

−3β0,Qγ(1)gq (L2 +L1)+

23

ˆγ(2)gq −12a(2)gq β0,Q

]+ a(3)gq,Q , (2.11)

and the renormalized operator matrix element in the MS–scheme reads

A(3),MSgq,Q = γ

(0)gq β

20,Q

(2L3

2 +2L31 +

32

L22L1 +

32

L21L2

)+

32

β0,Qγ(1)gq(L2

2 +L21)

+

{6a(2)gq β0,Q +

32

γ(0)gq β

20,Qζ2

}(L2 +L1)+12a(2)gq β0,Q + a(3)gq,Q .

(2.12)

•••••⊗

(1)

•••••⊗

(2)

•••••⊗

(3)

Figure 3: Diagrams for the two-mass contributions to A(3)gq,Q. The curly lines denote gluons, the dashed arrow

line represents the external massless quarks, while the thick solid arrow line represents a quark of mass m1,and the thin arrow line a quark of mass m2. We assume m1 > m2.

One obtains the following result for the constant part of the two–mass unrenormalized OME,

a(3)gq,Q = CFT 2F

1+(−1)N

2

{p(0)gq

[16(

L31 +L3

2 +

(L1L2 +2ζ2 +

263

)(L1 +L2)

)

6

PoS(RADCOR2017)071


− 43η3/2

((1+√

η)2R8Li3(−

√η)−

(1−√η

)2R9Li3(√

η))− 16

9ln3(η)

+

(2(1+√

η)2

3η3/2 R8Li2(−√

η)− 2(1−√η

)2

3η3/2 R9Li2(√

η)− 203η

(η

2−1))

ln(η)

+

((1+√

η)2

6η3/2 R8 ln(1+√

η)−(1−√η

)2

6η3/2 R9 ln(1−√η)− 163

S1

)ln2(η)

−6427

S31−

12827

S3−643

(ζ2 +

13

S2

)S1−

1289

ζ3

]− R10 ln2(η)

3η(N−1)N(N +1)2

+16[− 1(N +1)2 + p(0)gq

(83−S1

)]((L1 +L2)

2− 43(L1 +L2)S1

)+

[323

p(0)gq(S2−S2

1)− 64(8N +5)

9(N +1)3

](L1 +L2)−

64R11S1

27(N−1)N(N +1)3

+64(8N3 +13N2 +27N +16

)27(N−1)N(N +1)2

(S2

1 +S2 +3ζ2)− 8R12

243η(N−1)N(N +1)4

},

(2.13)

where p(0)gq is the color–stripped 1–loop anomalous dimension,

p(0)gq =2+N +N2

(N−1)N(N +1). (2.14)

Again, we will omit the explicit expressions of the polynomials Ri.The corresponding x-space expression is given in the appendix of Ref. [17]. Again, Eq. (2.13)

agrees with previously calculated fixed moments for N = 2,4,6. The behaviour of the ratio of thegenuine 2-mass contribution to the complete T 2

F 3-loop result for A(3)gq,Q at large values of Q2 is

similar to the one exhibited in the non-singlet case in Figure 2.

3. The gluonic operator matrix element A(3)gg

The three–loop contributions to Agg have no effect on the DIS structure functions at NNLO, sinceonly the corresponding two–loop contributions appear in the expressions for the massive Wilsoncoefficients (see Eq.s (2.20)–(2.24) of Ref. [17]). They are, however, needed in order to obtainthe gluonic transition relation of the VFNS at three loops. The calculation of A(3)

gg is considerablymore elaborate than those of the previous OMEs, and leads to new types of functions, namely,generalized sums in N-space and the generalized harmonic polylogarithms with square root lettersin the alphabet in x-space.

There is a total of 76 irreducible diagrams contributing to the two–mass part of A(3)gg , including

six diagrams with a ghost in the external lines. Only twelve of them, shown in Figure 4, are trulyindependent. All other diagrams are related to these either by symmetry, the exchange of massesor reversal of a fermion line.

In Ref. [17], we presented the results for the scalar versions of the first eight diagrams of Figure4. We have now completed the calculations of all but two of the diagrams in the physical case. Thedetails of how this calculation has been performed will be explained in a future publication. As apreview, we show here the result for one of the diagrams, namely diagram 6 of Figure 4. We get,

7

PoS(RADCOR2017)071


•••••⊗

(1)

•••••⊗

(2)

•••••⊗

(3)

•••••⊗

(4)

•••••⊗

(5)

•••••⊗

(6)

•••••⊗

(7)

•••••⊗

(8)

••••••⊗

(9)

••••••⊗

(10)

••••••⊗

(11)

•••••⊗

(12)

Figure 4: Independent diagrams for the two-mass contributions to A(3)gg . The curly lines denote gluons, the

dotted line represents a ghost, the thick solid arrow line represents a quark of mass m1, and the thin arrowline a quark of mass m2. We assume m1 > m2.

DB6 (N) = CAT 2

F1+(−1)N

2

{16(29N3 +41N2 +47N +47

)27(N−1)N(N +1)2ε3 +

1ε2

[8P5

81(N−1)2N2(N +1)3

−8(13N3 +21N2 +23N +27

)9(N−1)N(N +1)2 H0(η)− 8

(23N3 +19N2 +21N +13

)27(N−1)N(N +1)2 S1

]

+1ε

[4P6

81(N−1)3N3(N +1)4 +2(17N3−3N2−5N−21

)27(N−1)N(N +1)2 S2

+2(23N3 +43N2 +45N +61

)9(N−1)N(N +1)2 H2

0 (η)+2(29N3 +41N2 +47N +47

)27(N−1)N(N +1)2 S2

1

− 4P3

27(N−1)2N2(N +1)3 H0(η)+2(29N3 +41N2 +47N +47

)9(N−1)N(N +1)2 ζ2

+

(− 4P4

81(N−1)2N2(N +1)3 +4(7N3−N2−3N−7

)9(N−1)N(N +1)2 H0(η)

)S1

]

−53N3 +109N2 +111N +16327(N−1)N(N +1)2 H3

0 (η)− 23N3 +19N2 +21N +1381(N−1)N(N +1)2 S3

1

8

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+5N3 +11N2 +11N +17

9(N−1)N(N +1)2

(4H0(η)H0,1(η)−4H0,0,1(η)−2H2

0 (η)H1(η))

+4P8

45η(N−1)N(N +1)2

{H0(η)S1,1

(η

η−1,η−1

η,N)−S1,2

(η−1

η,

η

η−1,N)

+S1,2

(η

η−1,η−1

η,N)+S1,1,1

(η−1

η,1,

η

η−1,N)+S1,1,1

(η−1

η,

η

η−1,1,N

)}− 2P9

405(N−1)N(N +1)2ηS3 +

P14

3645(N−1)4N4(N +1)5(2N−5)(2N−3)(2N−1)η

+P15

90(N−1)N(N +1)η3/2

(H0(η)H0,−1

(√η)+H0(η)H0,1

(√η)− 1

4H0(η)2H1

(√η)

−2H0,0,−1(√

η)−2H0,0,1

(√η))

+P5

27(N−1)2N2(N +1)3 ζ2

+2−2N−3

(2NN

)P16

135(N−1)N(N +1)2(2N−5)(2N−3)(2N−1)η2

[N

∑i=1

22i(2ii

)i3−

N

∑i=1

22iS1 (i)(2ii

)i2

+H0(η)N

∑i=1

22i(2ii

)i2+

N

∑i=1

22i(η−1

)−iη i(2i

i

)i

S1,1

(η−1

η,1, i)

+12

H20 (η)

N

∑i=1

22i(η−1

)−iη i(2i

i

)i

−H0(η)N

∑i=1

22i(η−1

)−iη i(2i

i

)i

S1

(η−1

η, i)

−N

∑i=1

22i(η−1

)−iη i(2i

i

)i

S2

(η−1

η, i)]− 2(11N3 +47N2 +41N +89

)27(N−1)N(N +1)2 ζ3

+

(η

η−1

)N P11

540(N−1)2N2(N +1)2(2N−5)(2N−3)(2N−1)η

[S2

(η−1

η,N)

+H0(η)S1

(η−1

η,N)− 1

2H2

0 (η)−S1,1

(η−1

η,1,N

)]+

2−2N−4(2N

N

)P16

135(N−1)N(N +1)2(2N−5)(2N−3)(2N−1)η3/2

[8H0,0,−1

(√η)

+(H−1

(√η)+H1

(√η))

H20 (η)+8H0,0,1

(√η)−4H0(η)

(H0,−1

(√η)+H0,1

(√η))]

+

[−35N3 +63N2 +73N +81

27(N−1)N(N +1)2 S2 +−5N3 +23N2 +33N +41

9(N−1)N(N +1)2 H20 (η)

+2P1

27(N−1)2N2(N +1)3 H0(η)− 23N3 +19N2 +21N +139(N−1)N(N +1)2 ζ2

+P12

810(N−1)3N3(N +1)4(2N−5)(2N−3)(2N−1)η

]S1

+

[P5

81(N−1)2N2(N +1)3 −13N3 +21N2 +23N +27

9(N−1)N(N +1)2 H0(η)

]S2

1

+

[P2

81(N−1)2N2(N +1)3 +P7

45(N−1)N(N +1)2ηH0(η)

]S2

− 2P8

45(N−1)N(N +1)2η

[H2

0 (η)+2S1,1

(η−1

η,1,N

)]S1

(η

η−1,N)

9

PoS(RADCOR2017)071


+

[P13

810(N−1)3N3(N +1)4(2N−5)(2N−3)(2N−1)η

−13N3 +21N2 +23N +273(N−1)N(N +1)2 ζ2

]H0(η)+

[P10

540(N−1)2N2(N +1)3η

− P15

360(N−1)N(N +1)η3/2 H−1(√

η)]

H20 (η)

}, (3.1)

where the Pi’s are polynomials in N and η , and

Sb,~a(y,~x,N) =N

∑k=1

yk

kb S~a(~x,k), S /0 = 1, b,ai ∈ Z\{0}, y,xi ∈ C (3.2)

are generalized harmonic sums [45]. The Mellin inversion of this result can be performed using thepackage HarmonicSums. The results are given in terms of generalized iterated integrals definedby

G({ f1(τ), f2(τ), · · · , fn(τ)} ,z) =∫ z

0dτ1 f1(τ1)G({ f2(τ), · · · , fn(τ)} ,τ1) , (3.3)

with

G

({1τ,

1τ, · · · , 1

τ︸︷︷︸n times

},z

)≡ 1

n!lnn(z) . (3.4)

These will also appear in the case of the pure singlet OME discussed in the next section. Theletters in the alphabet of these iterated integrals, i.e., the functions fi(τ) in Eq. (3.3), may ingeneral depend on η .

4. The pure singlet OME

In the case of the pure singlet two–mass contribution, the generic pole structure is given by

Â(3),PS

Qq =163ε3 γ

(0)gq γ

(0)qg β0,Q +

1ε2

[4γ

(0)gq γ

(0)qg β0,Q (L1 +L2)+

23

γ(0)qg γ

(1)gq −

83

β0,QγPS,(1)qq

]+

1ε

[2γ

(0)gq γ

(0)qg β0,Q

(L2

1 +L1L2 +L22)+

{12

γ(0)qg γ

(1)gq −2β0,Qγ

PS,(1)qq

}(L2 +L1)

+23

ˆγ(2),PSqq −8a(2),PS

Qq β0,Q +2γ(0)qg a(2)gq

]+ a(3),PS

Qq . (4.1)

In the MS–scheme, one obtains the following renormalized expression

A(3),MS,PSQq = −1

2γ(0)gq γ

(0)qg β0,Q

(L2

2L1 +L21L2 +

43

L31 +

43

L32

)−{

14

γ(0)qg γ

(1)gq −β0,Qγ

PS,(1)qq

}(L2

2 +L21)

+

{4a(2),PS

Qq β0,Q− γ(0)qg a(2)gq −

12

β0,Qζ2γ(0)gq γ

(0)qg

}(L1 +L2)+8a(2),PS

Qq β0,Q−2γ(0)qg a(2)gq

+a(3),PSQq . (4.2)

There are sixteen diagrams contributing in this case, three of which are shown in Figure 5. Allothers are related to these by the exchange of masses and/or reversal of fermion lines.

10

PoS(RADCOR2017)071


•••••⊗

(1)

•••••⊗

(2)

•••••⊗

(3)

Figure 5: Diagrams for the two-mass contributions to A(3),PSQq . The curly lines denote gluons, the dashed

arrow line represents the external massless quarks, while the thick solid arrow line represents a quark ofmass m1, and the thin arrow line a quark of mass m2. We assume m1 > m2.

We use the same trick that we used before for ANSqq and Agq to decouple the mass coming from

the fermion loop without the operator insertion. For the fermion loop with the vertex operatorinsertion, we use

•••⊗µ, a ν, b

= 16δabTFg2s(∆.k)N−2

(4π)D/2 Γ(2−D/2)∫ 1

0dx xN(1− x)

(∆.k)∆µkν − k2∆µ∆ν

(m2− x(1− x)k2)2−D/2 .

(4.3)

One obtains the following result for diagram 3 in Figure 5,

D3(N) = −64CFT 2F(1+(−1)N)(m2

2µ2

) 32 ε (

1+ε

2

)Γ(N−1)

Γ(N +1+ ε/2)

× 12πi

∫ +i∞

−i∞dσ η

−σ Γ(−σ +N +1+ ε/2)Γ(−σ +2+ ε/2)Γ(−2σ +N +3+ ε)

Γ(−σ)Γ(−σ + ε)

×Γ

(σ − 3

2ε

)Γ(σ − ε/2)

Γ2(σ +2− ε)

Γ(2σ +4−2ε). (4.4)

The contour integral in Eq. (4.4) can be performed using the Mathematica package MB [52], to-gether with the extension MBresolve [53], which allows to resolve the singularity structure in ε

of this type of integrals by taking residues in σ . Once this is done, we can expand in ε . At O(ε0),we are left with a sum of terms containing the poles in ε of the integral and the original integralwith ε set to zero.

In the case where the operator insertion lies on a fermion line, we use the following expression,

••⊗

µ, a ν, b= 4δabTFg2

s(∆.k)N−2

(4π)D/2

∫ 1

0dx xN−2(1− x)

[

−2(x(1− x)(gµνk2−2kµkν)+m2gµν

) x2Γ(3−D/2)(∆.k)2

(m2− x(1− x)k2)3−D/2

+Γ(2−D/2)(2Nx+1−N)x(kµ∆ν + kν∆µ)(∆.k)(m2− x(1− x)k2)2−D/2

+Γ(2−D/2)((N−1)(1−2x)−Dx)xgµν(∆.k)2

(m2− x(1− x)k2)2−D/2

11

PoS(RADCOR2017)071


−Γ(1−D/2)N−11− x

(N(1− x)−1)∆µ∆ν

(m2− x(1− x)k2)1−D/2

], (4.5)

which allows us to write the result for diagram 1 in Figure 5 (as well as all related diagrams) interms of a linear combination of contour integrals similar to the one appearing in Eq. (4.4).

The Γ–functions in the integrand of Eq. (4.4) arise from the integration of Feynman parametersinto Euler Beta–functions. In Ref. [18], we left one of these Feynman parameters unintegrated,since it was already in the form of a Mellin transform. In the case of Eq. (4.4) this refers to theterm

Γ(−σ +N +1+ ε/2)Γ(−σ +2+ ε/2)Γ(−2σ +N +3+ ε)

=∫ 1

0dxx−σ+N+ε/2(1− x)−σ+1+ε/2. (4.6)

The exchange m1↔m2 implies also the change η→ 1/η , which means that in Ref. [18], for someof the diagrams the integrals depended on (ηx(1−x))−1, while for other diagrams, the dependencewas on η

x(1−x) . In the former case, we can close the contour to the left and take residues, since(ηx(1− x))−1 > 4 for all values of x ∈ (0,1). The latter case is a bit more complicated, since

η

x(1−x) will be smaller or bigger than 4, for η < 1, depending on the value of x, which means thatin some regions of x we had to close the contours to the left, while in the remaining regions wehad to close it to the right. On top of this, there are factors depending on N, such as the factorΓ(N−1)/Γ(N+1+ε/2) in (4.4), which after the ε expansion, lead to factors of the form 1/N and1/(N + 1). These factors were absorbed into the Mellin transform using integration by parts, andthe final result was then given in terms of integrals weighted by Heaviside θ–functions.

In these proceedings we show an alternative method of calculating the pure singlet diagrams.We work directly in the representation of the diagrams with the fully integrated Feynman parame-ters. Let us consider the contour integral in (4.4). After the ε expansion, one of the terms we get isgiven by

J =1

N +11

2πi

∫ −1/2+i∞

−1/2−i∞dσ η

−σ Γ(N +1−σ)Γ(−σ)2Γ(σ)2Γ(2−σ)Γ(2+σ)2

Γ(N +3−2σ)Γ(4+2σ). (4.7)

There is also a term with the same contour integral but a factor 1/N instead of 1/(N + 1) in thedecomposition of the diagram, and in the case of the diagrams where the operator insertion lies ona fermion line, also the factor 1/(N−1) appears in some cases.

Closing the contour to the left and summing residues, we obtain the following result

J = ηΓ(N +1)Γ(N +4)

(4S1(N +3)−2S1(N)−2ln(η)−3

)+ J′, (4.8)

where

J′ =∞

∑k=2

ηk

N +1B(k+N +1,k+2)

Γ(k)2Γ(2k−3)Γ(k−1)2Γ(k+1)2

{ln2(η)+2ln(η)

[S1(k+N)

−2S1(2k+N +2)−2S1(k−2)− 2k+S1(k+1)+2S1(2k−4)

]+

2k2

+

(S1(k+N)−2S1(2k+N +2)−2S1(k−2)− 2

k+S1(k+1)+2S1(2k−4)

)2

12

PoS(RADCOR2017)071


−S2(k+N)+4S2(2k+N +2)+2S2(k−2)−S2(k+1)−4S2(2k−4)

}. (4.9)

Unfortunately, this sum, as it stands, cannot be done using Sigma. Therefore, we proceed asfollows. First we reintroduce an integral in x using3

∫ 1

0dx xk+N(1− x)k+1 = B(k+N +1,k+2) (4.10)∫ 1

0dx xk+N(1− x)k+1 ln(x) = B(k+N +1,k+2)(S1(k+N)−S1(2k+N +2)) ,

(4.11)∫ 1

0dx xk+N(1− x)k+1 ln(1− x) = B(k+N +1,k+2)(S1(k+1)−S1(2k+N +2)) ,

(4.12)∫ 1

0dx xk+N(1− x)k+1 ln(x)2 = B(k+N +1,k+2)

[(S1(k+N)−S1(2k+N +2))2

−S2(k+N)+S2(2k+N +2)], (4.13)∫ 1

0dx xk+N(1− x)k+1 ln(1− x)2 = B(k+N +1,k+2)

[(S1(k+1)−S1(2k+N +2))2

−S2(k+1)+S2(2k+N +2)], (4.14)∫ 1

0dx xk+N(1− x)k+1 ln(x) ln(1− x) = B(k+N +1,k+2)

[−ζ (2)+S2(2k+N +2)

+(S1(k+1)−S1(2k+N +2))

×(S1(k+N)−S1(2k+N +2))], (4.15)

which leads to

J′ =∞

∑k=2

ηk

N +1

∫ 1

0dx (1− x)k+1xk+N Γ(k)2Γ(2k−3)

Γ(k−1)2Γ(k+1)2

{ln2(η)+2ln(η)

[−2S1(k−2)

−2k+2S1(2k−4)+ ln(1− x)+ ln(x)

]−4[

S1(k−2)+1k−S1(2k−4)

](ln(1− x)+ ln(x))

+4(

S1(k−2)+1k−S1(2k−4)

)2

+2ζ2 +2S2(k−2)+2k2 −4S2(2k−4)

+(ln(1− x)+ ln(x))2}. (4.16)

Now we do the binomial expansion of the term (1− x)k+1 and use

1N + l

∫ 1

0dx xN f (x) =

∫ 1

0dx xN+l−1

(∫ 1

0dy y−l f (y)−

∫ x

0dy y−l f (y)

), (4.17)

in order to absorb the 1N+l factor into the integrand. The integrals from 0 to x can be done using

∫ x

0dy yn ln(y) =

xn+1

n+1ln(x)− xn+1

(n+1)2 , (4.18)

3Transformations of this kind have been used in Ref. [54] before.

13

PoS(RADCOR2017)071


∫ x

0dy yn ln(1− y) = −S1(x,n)

n+1− xn+1

(n+1)2 +xn+1−1

n+1ln(1− x), (4.19)∫ x

0dy yn ln(y)2 =

2xn+1

(n+1)3 +xn+1

n+1ln2(x)− 2xn+1

(n+1)2 ln(x), (4.20)∫ x

0dy yn ln(1− y)2 =

ln2(x)n+1

[2

n+1S1(x,n+1)+2S1,1({1,x},n)−2x

+2ln(1− x)(S1(n+1)−S1(x,n+1)−1+ x

)+2x

−x(1− xn) ln2(1− x)+(1− x)(2ln(1− x)− ln2(1− x)

)], (4.21)∫ x

0dy yn ln(y) ln(1− y) =

1(n+1)2

[S1(x,n−1)− xn+1 ln(x)−

(xn+1−1

)ln(1− x)

]− xn ln(x)

n(n+1)+

1n+1

[S2(x,n−1)− ln(x)S1(x,n−1)−Li2(x)

+(xn+1−1

)ln(x) ln(1− x)

]+

(2n+1)xn

n2(n+1)2 +2xn+1

(n+1)3 . (4.22)

From this we get,

J′ =∫ 1

0dx xN+1

∞

∑k=2

k+1

∑i=0

(−1)iη

k(

k+1i

)Γ(k)2Γ(2k−3)

Γ(k−1)2Γ(k+1)2

{8(1− xi+k−1

)(i+ k−1)3

+1

(i+ k−1)2

[−4S1(x, i+ k−2)+4S1(i+ k−2)+

(1− xi+k−1

)(8S1(k−2)

+8k−8S1(2k−4)−4ln(η)

)+4xi+k−1 ln(x)+2

(2xi+k−1−1

)ln(1− x)

]+

1i+ k−1

{ln(η)

(2S1(x, i+ k−2)−2S1(i+ k−2)−2xi+k−1 ln(x)

)+ ln(x)

(2S1(x, i+ k−2)+4xi+k−1S1(k−2)+

4k

xi+k−1−4xi+k−1S1(2k−4))

+4(

S1(k−2)+1k−S1(2k−4)

)(S1(i+ k−2)−S1(x, i+ k−2)

)+2Li2(x)

−2S2(x, i+ k−2)−2S1,1({1,x}, i+ k−2)− π2

3xi+k−1− xi+k−1 ln2(x)

+ ln(1− x)(2S1(x, i+ k−2)−2S1(i+ k−1)

)+2S2(i+ k−2)+2S1,1(i+ k−2)

+(

1− xi+k−1)[

ln(η)

(−4S1(k−2)− 4

k+4S1(2k−4)+2ln(1− x)

)−4ln(1− x)

(S1(k−2)+

1k−S1(2k−4)

)+4(

S1(k−2)+1k−S1(2k−4)

)2

+2S2(k−2)+2k2 −4S2(2k−4)+ ln2(η)+ ln2(1− x)+2ln(x) ln(1− x)

]}}. (4.23)

Now we can do perform the double sum using Sigma and HarmonicSums. We finally obtain,

J′ =∫ 1

0dxxN+1

{− 1

24G2({

1τ,

√4τ−1

τ

},xη− x2

η

)− 1

12iG({√

4τ−1τ

,1τ,

1τ,

1τ

},x− x2

)

14

PoS(RADCOR2017)071


+1

288ln4((1− x)xη)+

12x2− x+6216x

ln3((1− x)xη)+

[ζ2

24+

66x2−15ηx−37x+24108x

−1136

ln(1−η)

]ln2((1− x)xη)+

[T2

54x+ ln(1− x)

(5η

36+

1136

ln(1−η)

)+ ln(x)

(1112

ln(1−η)− 7η

12

)− 11

9x ln(1−η)

]ln((1− x)xη)+

ζ2T3

27x+

ηT1

810−ζ2η

+

{1136

G({√

4τ−1√

4ητ−1τ

},x− x2

)− 20

9ηG({√

4τ−1√

4ητ−1},x− x2

)}[ζ2

+iG({

1τ,

√4τ−1

τ

},xη− x2

η

)]+

209

iηG({

1τ,

√4ητ−1

τ,√

4τ−1√

4ητ−1},x− x2

)+

112

iG({

1τ,

√4τ−1

τ

},xη− x2

η

)[ζ2 +G

({√4τ−1

τ,

√4ητ−1

τ

},x− x2

)]−11

36iG({

1τ,

√4ητ−1

τ,

√4τ−1

√4ητ−1

τ

},x− x2

)+

T3

27x

(1−4x(1− x)η

)3/2[−ζ2

+2i(

1+12

ln((1− x)xη)

)G({√

4τ−1τ

},xη− x2

η

)− iG

({1τ,

√4τ−1

τ

},xη− x2

η

)]+

112

iG({

1τ,

√4ητ−1

τ,

√4ητ−1

τ,

√4τ−1

τ

},x− x2

)+ ln2(x)

(7η

24− 11

24ln(1−η)

)+ ln2(1− x)

(−5η

72− 11

72ln(1−η)

)+ ln(1− x)

(1118

ln(1−η)− 127

η(8η−29))

+ ln(x)[− 1

54η(16η−169)+ ln(1− x)

(7η

12− 11

12ln(1−η)

)+

1118

ln(1−η)

]− i

12

(ln(η)+

173

)G({√

4τ−1τ

,1τ,

1τ

},x− x2

)+

(ln(x)

6− ln(η)

3

)Li3(η)+

Li4(η)

2

+

(112

ln((1− x)xη)− 12x2−13x+636x

)G({

1τ,

√4τ−1

τ,

√4τ−1

τ

},xη− x2

η

)+

(1+

12

ln((1− x)xη)

){−1

6iG({√

4τ−1τ

,

√4ητ−1

τ,

√4ητ−1

τ

},x− x2

)+

T4

36xG2({√

4τ−1τ

},xη− x2

η

)+

[409

iηG({√

4τ−1√

4ητ−1},x− x2

)−11

18iG({√

4τ−1√

4ητ−1τ

},x− x2

)]G({√

4τ−1τ

},xη− x2

η

)}−iη

(409+

209

ln((1− x)xη)

)G({√

4ητ−1τ

,√

4τ−1√

4ητ−1},x− x2

)+i(

1118

+1136

ln((1− x)xη)

)G({√

4ητ−1τ

,

√4τ−1

√4ητ−1

τ

},x− x2

)+

T4

36x

[iζ2G

({√4τ−1

τ

},xη− x2

η

)−G

({√4τ−1

τ,

1τ,

√4τ−1

τ

},xη− x2

η

)+ζ2 ln((1− x)xη)

]+Li2(x)

(13η

18− 11

18ln(1−η)+

Li2(η)

6

)− 1

6ln(1− x)Li3(η)

15

PoS(RADCOR2017)071


+G({√

4τ−1τ

,

√4ητ−1

τ

},x− x2

)(ζ2

12+

ln(η)

6+

112

ln(η) ln((1− x)xη)

)+

i12

G({

1τ,

√4τ−1

τ

},x− x2

)[12

ln2(η)+173

ln(η)+ζ2−13(5η−22)− 11

3ln(1−η)

+Li2(η)

]+

[124

ln2(1− x)+(

136

(12x−17)+112

ln(xη)

)ln(1− x)+

ln2(η)

8− 1

24ln2(xη)

+1

18x(1+

√η)(5η−5

√η−22)+

136

(12x+5) ln(x)+118

(6x−11) ln(η)

]Li2(η)

+√

η (5η−27){

ln(

1+√

η

1−√η

)[1

144ln2(1− x)+

(1

36(2x−1)+

172

ln(xη)

)ln(1− x)

+ln2(η)

48− 1

144ln2(xη)+

172

iG({

1τ,

√4τ−1

τ

},x− x2

)+

136

(2x−1) ln(x)+x

18ln(η)

+Li2(x)

36

]+

(−2x

9− 1

18ln(1− x)+

ln(x)18− ln(η)

9

)Li2 (√

η)+29

Li3 (√

η)− Li3(η)

36

− i18

G({√

4τ−1τ

},x− x2

)[12

ln(

1+√

η

1−√η

)(1+

12

ln((1− x)xη)

)−Li2 (

√η)

+Li2(η)

4

]+

(172

ln(1− x)− ln(x)72

+ln(η)

36

)Li2(η)

}+

[136

i ln3(η)− 1172

i ln2(η)

+1

36i(5η−22) ln(η)− 1

12G({

1τ,

√4τ−1

τ

},xη− x2

η

)ln(η)− 11iζ2

36− 1

27iη(8η−29)

− 112

iG({

1τ,

√4τ−1

τ,

√4τ−1

τ

},xη− x2

η

)+

i36

(5η−3ζ2−22+11ln(1−η)

−17ln(η)

)ln((1− x)x)+

1118

i ln(1−η)

(1+

ln(η)

2

)+

(5i36− 1

12i ln((1− x)xη)

)Li2(η)

+iLi3(η)

6

]G({√

4τ−1τ

},x− x2

)+

118

(11−12x)Li3(η)

}, (4.24)

where

T1 = 5(640x3−1011x2−171x+810

)+32ηx

(144x4−360x3 +320x2−120x+15

),(4.25)

T2 = 192η2x6−512η

2x5 +32η(18η +5)x4−η(416η +251)x3

+2(96η

2 +103η +30)

x2− (231η +32)x+12, (4.26)

T3 = 15x2−8x+3, (4.27)

T4 = 12x2−7x+6. (4.28)

All other integrals appearing in the calculation can be done in a similar way. This way of calculatingthe diagrams applies perfectly well in all of the cases where the operator insertion lies on theheaviest of the quarks, as long as we are assuming that η < 1. Taking into account that the diagramswhere the insertion lies on the lightest quark are related to the former diagrams by the changeη → 1/η , we can in principle obtain the results of the latter diagrams by analytic continuation to

16

PoS(RADCOR2017)071


the region where η > 1. This is, however, far less trivial compared with the Agg,Q case, since thesquare root letters in the alphabet of the iterated integrals introduce branch cuts that need to beanalyzed carefully. The complete result given in Ref. [18] has been obtained in a different way.

5. Conclusions

We have presented the two–mass contributions for a series of three–loop OMEs. The simplestones are A(3),NS

qq,Q and A(3)gq,Q, for which the N and η dependence factorizes. In the case of A(3)

gg,Q, wepresented results in the physical case, going beyond those given in [17], where a series of scalardiagrams were computed. In the pure singlet case, which was studied in [18], we presented a newway of computing the diagrams. This new way has the advantage of producing a result that is givenentirely by iterated integrals, without the need to introduce additional integrations on top of these,as we did in [18]. However, the results so far are only valid for half of the diagrams. In order tocover the other half, the analytic continuation η → 1/η still remain to be done.

With the full results for A(3)gg,Q within grasp, and having calculated all the other OMEs presented

here, the only remaining OME is A(3)Qg . Considering that even in the single mass case this OME

exhibits integrals with an elliptic behaviour [55,56], it doesn’t seem feasible that this OME will becomputed analytically any time soon. New methods will need to be applied in this case, which weplan to study at some point in the future.

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The massive 3-loop operator matrix elements with two