Solving Single Mass Feynman Integrals at 2 and 3 Loops · The contact between DESY & RISC has been established after a plenary talk ofB. Buchbergerat ACAT 2005 at DESY (not on Gr

The first five years Mathematical Structures Physics Goals The Heavy Flavor Structure Functions: Some Examples

Solving Single Mass Feynman Integrals at 2 and 3Loops

Johannes Blumlein, DESY

5 Years of DESY-RISC Collaboration, May 2012


Contents

1 The first five years

2 Mathematical Structures

3 Physics Goals

4 The Heavy Flavor Structure Functions: Some Examples


The first five years

The contact between DESY & RISC has been established after a plenarytalk of B. Buchberger at ACAT 2005 at DESY (not on Grobner bases buton the Theorema project).

Bruno notified me C. Schneider’s work & many more activities on whichpotential cooperations might be possible.

In September 2005 I have been visiting RISC for an intense discussion day.

I had just started with the Summerstudent S. Klein something new(without knowing yet).

So, everything was very much on time.

Carsten could not be there, because being on honeymoon.

Several other visits followed and in 2006 Peter Paule and CarstenSchneider came to DESY for Seminars and to prepare our contract.



Contract signed at Linz in February 2007.



During the summer of 2005 I noticed that the heavy flavor 3-loopcorrections to the structure function FL in the asymptotic region were notcalculated, but were completely doable with pieces available, provided oneunderstood the associated mechanism.

This drove me into a new type of 3-loop calculations and S. Klein startedhis Diploma with me on a related matter at 2 loops.

Of course we were after the ultimate integrator from the very beginning indiscussions with C. Schneider, M. Kauers and P. Paule. Despite ofwonderful and extremely powerful algorithms around, 2- and 3-loopproblems were (and are) still hard.

In 2006 S. Klein, A. De Freitas, the late W.L. van Neerven and myself stillhad our own integration and summation techniques. It became clear that2-loops (if at all) might be solved, but nothing more.

Things became lengthly and tedious. I.e. re-write rules alone, althoughbeing clever in part, will not help.

Sigma came into operation.

In 2008 Sigma was used quite systematically for the 2-Loop O(ε)corrections, the first common large paper with RISC.



Can one unravel the general N expressions from a finite number ofmoments and how voluminous are contemporary problems ?

In 2008 M. Kauers, S. Klein, C. Schneider and myself worked this out forquantities up to the 3-loop massless Wilson coefficients.

Of course, the problem had to be solved practically.

Individual cases needed 5114 moments leading to difference equations oforder 35 and degree 1000, with rational coefficients of ∼ 13.000digits/13.000 digits filling 450 books after 3 weeks computational timeusing guessing. Sigma solved this after another week.

However, there are no algorithms to produce an input of that size in caseof new calculations at the moment.

A next important step consisted in computing a significant series of fixedN Mellin moments for the 3-loop corrections to F2 during 2008/09 withI. Bierenbaum and S. Klein and solve the renormalization problem.

These results (and technologies) also allow for comparisons on theongoing general N calculations.


Mathematical Structures

Which are the relevant mathematical structures ?

History until 1998: ζk , Sp,n(f (x)), [Lik(f (x))]

It took a while to understand that the emergence of f (x) should bethoroughly avoided, except sufficiently simple cases.

1998: usual harmonic sums Sa1,...,an (N) =∑N

k=1(sign(a1))k

k|a1|Sa2,...,an (k) and

the Harmonic Polylogarithms over {−1, 0, 1} turned out to be moresystematic; J. Vermaseren; J.B.; E. Remiddi & J. Vermaseren

∼ 2000: quasi-shuffle algebras M. Hoffman

2001: Generalized harmonic sums Moch, Uwer, Weinzierl based on a suggestion byGoncharov; numerator weights αi ∈ R.

> 2004: structural relations for harmonic sums (esp. differentiation)J. Ablinger, JB, C. Schneider

2008: Hyperlogarithms F. Brown

2009: MZV data mine JB, D. Broadhurst, J. Vermaseren


Mathematical Structures

2010: cyclotomic harmonic sums, special numbers and HPLsJ. Ablinger, JB, C. Schneider

Finite harmonic sums are worked out in explicit form to w=8.J. Ablinger, JB, C. Schneider

Generalized harmonic sums are algorithmically available, including those ofthe cyclotomic case. J. Ablinger, JB, C. Schneider

Package: HarmonicSums J. Ablinger


Emerging Functions

2 Loops: p+1Fq functions suffice as representations of the 2-pointFeynman integrals with operator insertion

3 Loops: Further hypergeometric functions emerge, e.g. the Appellfunctions.

As further sums over these structures have to be carried out one just cansay that general hypergeometric play a role.

These functions are introduced to allow for a well-defined expansion in thedimensional parameter ε = D − 4.

One may generally show that these structures for the general frame.JB 2009; JB, C. Schneider, F. Stan, 2010


Physics Goals: The strong coupling at highest precision

αs(M2Z ) from NNLO DIS(+) analyses

αs(M2Z )

BBG 0.1134 +0.0019−0.0021 valence analysis, NNLO

GRS 0.112 valence analysis, NNLOABKM 0.1135± 0.0014 HQ: FFNS Nf = 3JR 0.1124± 0.0020 dynamical approachJR 0.1158± 0.0035 standard fitMSTW 0.1171± 0.0014ABM 0.1147± 0.0012 FFNS, incl. combined H1/ZEUS dataABM11J 0.1134− 0.1149± 0.0012 Tevatron jets (NLO) incl.CTEQ 0.118± 0.005NNPDF 0.1174± 0.0006± 0.0001Gehrmann et al. 0.1153± 0.0017± 0.0023 e+e− thrustAbbate et al. 0.1135± 0.0011± 0.0006 e+e− thrust

BBG 0.1141 +0.0020−0.0022 valence analysis, N3LO

∆THαs = αs(N3LO)− αs(NNLO) + ∆HQ = +0.0009± 0.0006HQ


Physics Goals: The strong coupling at highest precision

αs(M2Z ) from further processes

αs(M2Z )

3 jet rate 0.1175± 0.0025 Dissertori et al. 2009Z -decay 0.1190± 0.0026 BCK 2008τ decays 0.1202± 0.0019 BCK 2008τ decays 0.1212± 0.0014 Pich 2010τ decays 0.1180± 0.0008 Beneke, Jamin 2008lattice 0.1183± 0.0008 HPQCD 2008Average 2011 0.1185± 0.0008 S. Bethke

Despite the statistical and systematic errors are getting smaller, there is no finalconsensus on the value of αs(M2

Z ) yet.


Physics Goals: Kinematic Quark and Gluon Distributions

µ=2 GeV, nf=4

0

2

4

6

8

10

10-5

10-4

10-3

10-2

xG

x0

1

2

3

0.05 0.1 0.15 0.2 0.25

xG

x

0

0.25

0.5

0.75

1

10-5

10-4

10-3

10-2

x(uS+d

S)/2

x0

0.05

0.1

0.15

0.2

0.05 0.1 0.15 0.2 0.25

x(uS+d

S)/2

x

0

0.2

0.4

0.6

0.8

0.2 0.4 0.6 x

x(uV

+uS)

x(dV

+dS)

-0.02

0

0.02

0.04

0.06

10-4

10-3

10-2

10-1

x(dS-u

S)

x

0

0.5

1

1.5

10-5

10-4

10-3

10-2

x

x(s+s-)/2

0

0.025

0.05

0.075

0.1

0.05 0.1 0.15 0.2 0.25 x

x(s+s-)/2

S. Alekhin, JB, S. Moch, 2012


Physics Goals: Higgs Search

200150100

1.0

0.9

0.8

0.7

0.6

HERAPDF 1.0(αS = 0.1176)

HERAPDF 1.0(αS = 0.1145)JR09VFNNLO

ABKM 09

MSTW 2008

NNLO at µR = µF = 12MH

σ(gg → H) [fb]

MH [GeV]

200180160140120100

2000

1800

1600

1400

1200

1000

800

600

400

200

0

The gg → H cross section as a function of MH when the four NNLO PDF sets, MSTW, ABKM, JR and HERAPDF, are used. In the

inserts, shown are the deviations with respect to the central MSTW value; Baglio, Djouadi, Godbole, 2011

The exclusion limits depend on the pdf’s and the value of αs(MZ ) used. In particular,the predictions vary by up to 40 % for Tevatron.


Physics Goals: Higgs Search

(GeV)Hm200 300 400 500 600 700 800

MS

TW

σ/σ

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

MSTW08_90CL

JR09

ABM11NNPDF21_as_0119

NNPDF21_as_0114

Ch. Anastasiou et al., 2012


Heavy Flavor Corrections: the Status.

Leading Order: Witten, 1976 Nucl.Phys.B; Babcock, Sivers, 1978 Phys.Rev.D; Shifman, Vainshtein, Zakharov, 1978

Nucl.Phys.B; Leveille, Weiler, 1979 Nucl.Phys.B; Gluck, Reya, 1979 Phys.Lett.B; Gluck, Hoffmann, Reya, 1982 Z.Phys.C.

Next-to-Leading Order: Laenen, van Neerven, Riemersma, Smith, 1993 Nucl. Phys. B

Next-to-Leading Order: Large Q2/m2Buza, Matiounine, Smith, Migneron, van Neerven, 1996

Nucl.Phys.B] IBP; Bierenbaum, JB, Klein, 2007 Nucl.Phys.B] via pFq’s, more compact results.

O(α2sε) contributions Bierenbaum, JB, Klein, Schneider 2008 Nucl.Phys.B, Bierenbaum, JB, Klein, 2009

Phys.Lett.B

3-Loops: Moments for F2: N = 2...10(14) Bierenbaum, JB, Klein 2009 Nucl.Phys.B

Moments for Transversity: N = 1...13 JB, Klein, Todtli 2009 Phys.Rev.D

3-Loops: General N: Ablinger, Blumlein, Klein, Schneider, Wißbrock 2011 Nucl.Phys.B contrib. ∝ nfto F2 (all N):

various other 3-loop papers underway


Heavy flavor contributions to F2

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 102

103

104

F2(x

,Q2)

10-3

10-2

10-1

1 10 102

103

104

F2

,c(x

,Q2)

LO charm contributions: PDFs from [Alekhin, Melnikov, Petriello, 2006.]

→ different scaling violations,→ massive contributions at lower values of x are of order 20%-35%.Hence for the prediction of cross sections at the LHC the precise knowledge of all

PDFs and the exact value of αs(M2Z ) is needed. αs(M2

Z ) = 0.1135± 0.0014


Representation for F2 at Q2 > 10m2

in the asymptotic region FL is known for general values of N to NNLO[Blumlein, De Freitas, van Neerven, Klein, 2006.]

F2 for NF massless and one heavy quark flavor:[Bierenbaum, Blumlein, Klein, 2009.]

FQQ(2,L)

(x, NF + 1, Q2,m2) =

NF∑k=1

e2k

{LNSq,(2,L)

x, NF + 1,Q2

m2,m2

µ2

⊗ [fk (x, µ2, NF ) + f

k(x, µ2

, NF )]

+1

NF

[LPSq,(2,L)

x, NF + 1,Q2

m2,m2

µ2

⊗ Σ(x, µ2, NF ) + LS

g,(2,L)

x, NF + 1,Q2

m2,m2

µ2

⊗ G(x, µ2, NF )

]}

+e2Q

[HPSq,(2,L)

x, NF + 1,Q2

m2,m2

µ2

⊗ Σ(x, µ2, NF ) + HS

g,(2,L)

x, NF + 1,Q2

m2,m2

µ2

⊗ G(x, µ2, NF )

]

⊗ denotes the Mellin convolution

[A ⊗ B](x) =

∫ 1

0

∫ 1

0dx1dx2 δ(x − x1x2)A(x1)B(x2) ,

The asymptotic representation for F2(x ,Q2) becomes effective at Q2 ≥ 10 ·m2


Heavy flavor Wilson Coefficients

In this limit the massive Wilson coefficients up to O(a3s ) read

LNSq,(2,L)(nf + 1) = a2s

[A

(2),NSqq,Q (nf + 1)δ2 + C

(2),NSq,(2,L)(nf )

]

+ a3s

[A

(3)qq,Q,NS(nf + 1)δ2 +A

(2),NSqq,Q (nf + 1)C

(1),NSq,(2,L)(nf + 1) + C

(3),NSq,(2,L)(nf )

]

LPSq,(2,L)(nf + 1) = a3s

[A

(3),PSqq,Q (nf + 1) δ2 +A

(2)gq,Q(nf ) nf C

(1)g,(2,L)(nf + 1) + nf

ˆC(3),PSq,(2,L)(nf )

]

LSg,(2,L)(nf + 1) = a2sA

(1)gg,Q(nf + 1)nf C

(1)g,(2,L)(nf + 1) + a3s

[A

(3)qg,Q(nf + 1) δ2

+A(1)gg,Q(nf + 1) nf C

(2)g,(2,L)(nf + 1) +A

(2)gg,Q(nf + 1) nf C

(1)g,(2,L)(nf + 1)

+ A(1)Qg(nf + 1) nf C

(2),PSq,(2,L)(nf + 1) + nf

ˆC(3)g,(2,L)(nf )

],

HPSq,(2,L)(nf + 1) = a2s

[A

(2),PSQq (nf + 1) δ2 + C

(2),PSq,(2,L)(nf + 1)

]+ a3s

[A

(3),PSQq (nf + 1) δ2

+ C(3),PSq,(2,L)(nf + 1) +A

(2)gq,Q(nf + 1) C

(1)g,(2,L)(nf + 1)

+A(2),PSQq (nf + 1) C

(1),NSq,(2,L)(nf + 1)

],

HSg,(2,L)(nf + 1) = as

[A

(1)Qg(nf + 1) δ2 + C

(1)g,(2,L)(nf + 1)

]+ a2s

[A

(2)Qg(nf + 1) δ2

+ A(1)Qg(nf + 1) C

(1),NSq,(2,L)(nf + 1) + A

(1)gg,Q(nf + 1) C

(1)g,(2,L)(nf + 1)

+ C(2)g,(2,L)(nf + 1)

]+ a3s

[A

(3)Qg(nf + 1) δ2 + A

(2)Qg(nf + 1) C

(1),NSq,(2,L)(nf + 1)

+ A(2)gg,Q(nf + 1) C

(1)g,(2,L)(nf + 1) + A

(1)Qg(nf + 1)

{C

(2),NSq,(2,L)(nf + 1)

+ C(2),PSq,(2,L)(nf + 1)

}+ A

(1)gg,Q(nf + 1) C

(2)g,(2,L)(nf + 1) + C

(3)g,(2,L)(nf + 1)

]


Fixed moments N = 2 ... 10 (12, 14) are known [Bierenbaum, Blumlein, Klein, 2009]

The renormalization of this problem has been worked out by [Bierenbaum, Blumlein, Klein,

2009]

Through the renormalization the general structure of the unrenormalized OME’sis known

Example:

A(3)Qg

=( m2

µ2

)3ε/2[γ

(0)qg

6ε3

((NF + 1)γ(0)

gq γ(0)qg + γ(0)

qq

[γ

(0)qq − 2γ(0)

gg − 6β0 − 8β0,Q

]+ 8β2

0

+28β0,Qβ0 + 24β20,Q + γ(0)

gg

[γ

(0)gg + 6β0 + 14β0,Q

])+

1

6ε2

(γ

(1)qg

[2γ(0)

qq − 2γ(0)gg

−8β0 − 10β0,Q

]+ γ(0)

qg

[γ

(1),PSqq {1 − 2NF } + γ(1),NS

qq + γ(1),NSqq + 2γ(1)

gg − γ(1)gg − 2β1

−2β1,Q

]+ 6δm

(−1)1

γ(0)qg

[γ

(0)gg − γ

(0)qq + 3β0 + 5β0,Q

])+

1

ε

( γ(2)qg

3− NF

ˆγ(2)qg

3

+γ(0)qg

[a

(2)gg,Q

− NF a(2),PSQq

]+ a

(2)Qg

[γ

(0)qq − γ

(0)gg − 4β0 − 4β0,Q

]+γ

(0)qg ζ2

16

[γ

(0)gg

{2γ(0)

qq

−γ(0)gg − 6β0 + 2β0,Q

}− (NF + 1)γ(0)

gq γ(0)qg + γ(0)

qq

{−γ(0)

qq + 6β0

}− 8β2

0

+4β0,Qβ0 + 24β20,Q

]+δm

(−1)1

2

[−2γ(1)

qg + 3δm(−1)1

γ(0)qg + 2δm

(0)1γ

(0)qg

]

+δm(0)1γ

(0)qg

[γ

(0)gg − γ

(0)qq + 2β0 + 4β0,Q

]− δm(−1)

2γ

(0)qg

)+ a

(3)Qg

]


LPS2,q

a(3),PSqq,Q = CFT

2FNF

{128

27

(N2 +N + 2

)2

(−1 +N)N2 (1 +N)2 (2 +N)S31 − 64

27

(266N4 + 181N5 + 269N3 + 230N2 + 74N6 + 16N7 + 44N − 24

)

N3 (−1 +N) (2 +N)2 (1 +N)3S21

+128

9

(N2 +N + 2

)2

(−1 +N)N2 (1 +N)2 (2 +N)S1S2 +

64

81

P1(N)

(−1 +N)N4 (1 +N)4 (2 +N)3+

32

3

(N2 +N + 2

)2

(−1 +N)N2 (1 +N)2 (2 +N)ζ2 S1

−64

27

(266N4 + 181N5 + 269N3 + 230N2 + 74N6 + 16N7 + 44N − 24

)

N3 (−1 +N) (2 +N)2(1 +N)

3 S2 +256

27

(N2 +N + 2

)2

(−1 +N)N2 (1 +N)2(2 +N)

S3

− 32

243

P2(N)

N5 (−1 +N) (2 +N)4(1 +N)

5 − 16

9

P3(N)

N3 (−1 +N) (2 +N)2(1 +N)

3 ζ2 +224

9

(N2 +N + 2

)2

(−1 +N)N2 (1 +N)2(2 +N)

ζ3

}

P1(N) = 10101N7 + 4737N8 + 14923N6 + 17085N5 + 14133N4 + 5944N3 + 144− 48N + 568N2 + 1352N9 + 181N10

P2(N) = (311482N10 + 105173N11 + 636490N9 + 966828N8 + 1126568N7 + 968818N6 + 550813N5 + 169250N4 − 864− 1008N

+12104N3 − 3408N2 + 21728N12 + 2074N13

P3(N) = 266N4 + 181N5 + 269N3 + 230N2 + 74N6 + 16N7 + 44N − 24

(complete OME)


LS2g

a(3),0qg,Q = NFT

2F

{CF

[N2 +N + 2

N(N + 1)(N + 2)

[−56

9S4 +

32

27S3S1 +

8

9S2S

21 +

4

9S22 +

4

27S41 +

256

9S1ζ3

]

−16(10N3 + 13N2 + 29N + 6)

81N2(1 +N)(2 +N)

[S31 + 3S2S1

]+

32(5N3 − 16N2 +N − 6)

81N2(1 +N)(2 +N)S3

+8(109N4 + 291N3 + 478N2 + 324N + 40)

27N2(1 +N)2(2 +N)S2

+8(215N4 + 481N3 + 930N2 + 748N + 120)

81N2(1 +N)2(2 +N)S21 − R4(N)

243N2(1 +N)3(2 +N)S1

− 64(N2 +N + 2)R5(N)

9(N − 1)N3(1 +N)3(2 +N)2ζ3 +

R6(N)

243(N − 1)N6(1 +N)6(2 +N)5

]

+CA

[N2 +N + 2

N(N + 1)(N + 2)

[−56

9S4 −

128

9S−4 +

160

27S3S1 −

4

9S22 +

8

9S2S

21

− 4

27S41 − 64

9S2,1S1 −

128

9S3,1 +

64

9S2,1,1 −

256

9ζ3S1

]

+32(5N4 + 20N3 + 41N2 + 49N + 20)

81N(1 +N)2(2 +N)2[S31 + 12S2,1 − 3S2S1

]

+64

81

(5N4 + 38N3 + 59N2 + 31N + 20)

N(1 +N)2(2 +N)2S3 +

128

27

(5N2 + 8N + 10)

N(1 +N)(2 +N)S−3

+512

9

(N2 +N + 1)(N2 +N + 2)

(N − 1)N2(1 +N)2(2 +N)2ζ3 −

16R7(N)

81N(1 +N)3(2 +N)3S2

−32(121N3 + 293N2 + 414N + 224)

81N(1 +N)2(2 +N)S−2 −

R8(N)

81N(1 +N)3(2 +N)3S21

+16R9(N)

243(N − 1)N2(1 +N)4(2 +N)4S1 +

8R10(N)

243(N − 1)N5(1 +N)5(2 +N)5

]}

(complete OME)


Six massive lines and vertex insertion

I4 =Q1(N)

2(1 +N)5(2 +N)5(3 +N)5+

Q2(N)

(1 +N)2(2 +N)2(3 +N)2ζ3 +

(−1)N (65 + 101N + 56N2 + 13N3 +N4)

2(1 +N)2(2 +N)2(3 +N)2S−3

+(−24− 5N + 2N2)

12(2 +N)2(3 +N)2S31 −

1

2(1 +N)(2 +N)(3 +N)S22 +

1

(2 +N)(3 +N)S21S2

+Q4(N)

4(1 +N)3(2 +N)2(3 +N)2S21 −

3

2S5 − Q5(N)

6(1 +N)2(2 +N)2(3 +N)2S3 − 2S−2,−3 − 2ζ3S−2 − S−2,1S−2

+(−1)N (65 + 101N + 56N2 + 13N3 +N4)

(1 +N)2(2 +N)2(3 +N)2S−2,1 +

(59 + 42N + 6N2)

2(1 +N)(2 +N)(3 +N)S4 +

(5 +N)

(1 +N)(3 +N)ζ3S1 (2)

− Q6(N)

4(1 +N)3(2 +N)2(3 +N)2S2 − ζ3S2 −

3

2S3S2 − 2S2,1S2 +

(99 + 225N + 190N2 + 65N3 + 7N4)

2(1 +N)2(2 +N)2(3 +N)S2,1

+Q3(N)

(1 +N)4(2 +N)4(3 +N)4S1 −

(11 + 5N)

(1 +N)(2 +N)(3 +N)ζ3S1 −

Q7(N)

4(1 +N)2(2 +N)2(3 +N)2S2S1 − S2,3

+(53 + 29N)

2(1 +N)(2 +N)(3 +N)S3S1 −

3(3 + 2N)

(1 +N)(2 +N)(3 +N)S1S2,1 +

(−79− 40N +N2)

2(1 +N)(2 +N)(3 +N)S3,1 − 3S4,1

+ S−2,1,−2 +2N+1 (−28− 25N − 4N2 +N3)

(1 +N)2(2 +N)(3 +N)2S1,2

(1

2, 1

)− (−7 + 2N2)

(1 +N)(2 +N)(3 +N)S2,1,1

+5S2,2,1 + 6S3,1,1 +2N (−28− 25N − 4N2 +N3)

(1 +N)2(2 +N)(3 +N)2S1,1,1

(1

2, 1, 1

)

− (5 +N)

(1 +N)(3 +N)S1,1,2

(2,

1

2, 1

)− (5 +N)

2(1 +N)(3 +N)S1,1,1,1

(2,

1

2, 1, 1

)

How does the 2N -behaviour cancel?


More Results :

APSqq,Q ,Aqg,Q complete

all O(nfT2FCA,F ) terms are available

Most ladder topologies completed

First Benz topologies calculable

Moments N = 2, 4, 6 for different mass 3-loop graphs

a large class of polarized moments and general N results calculated


Several Theses:

Diploma :

S. Klein

J. Ablinger

F. Wißbrock

PhD :

S. Klein

F. Stan

J. Ablinger & 2 in preparation


More on the current research in loop integrals and assoc.structures:

I. Bierenbaum: Mellin-Barnes techniques in physics

A. De Freitas: 2-loop integrals in case of external massive lines

A. Hasselhuhn: 3-loop integrals: ladder and gluonic contributions

M. Hoang: 3-loop corrections and resummantions

M. Hoffman: Quasi-shuffle algebras

T. Riemann: Mellin-Barnes techniques in physics

F. Wißbrock: The hyperlogarithmic method

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Solving Single Mass Feynman Integrals at 2 and 3 Loops · The contact between DESY & RISC has been established after a plenary talk ofB. Buchbergerat ACAT 2005 at DESY (not on Gr