On the four-loop cusp anomalous dimension€¦ · On the four-loop cusp anomalous dimension Evaluating four-loop QCD form factors The photon-quark form factor, which is a building

On the four-loop cusp anomalous dimension

On the four-loop cusp anomalous

dimension

Johannes M. Henn

Mainz University

PRISMA Cluster of Excellence

Nordita, Aspects of Amplitudes, June 30, 2016


Based on collaboration with Alexander Smirnov, Vladimir

Smirnov and Matthias Steinhauser

J. Henn, A. Smirnov, V. Smirnov and M. Steinhauser,arXiv:1604.03126, JHEP 1605 (2016) 066


J. Henn, A. Smirnov, V. Smirnov and M. Steinhauser,arXiv:1604.03126, JHEP 1605 (2016) 066The first next-to-next-to-next-to-next-to-leading order (N4LO)contribution to a three-point function within QCD.



Evaluating four-loop QCD form factors




Massless planar four-loop vertex integrals





Perspectives



The photon-quark form factor, which is a building block forN4LO cross sections.



The photon-quark form factor, which is a building block forN4LO cross sections.It is a gauge-invariant part of virtual forth-order corrections forthe process e+e− → 2 jets, or for Drell-Yan production athadron colliders.



The photon-quark form factor, which is a building block forN4LO cross sections.It is a gauge-invariant part of virtual forth-order corrections forthe process e+e− → 2 jets, or for Drell-Yan production athadron colliders.Let Γµq be the photon-quark vertex function.The scalar form factor is

Fq(q2) = −

1

4(1 − ǫ)q2Tr

(

p2/ Γµqp1/ γµ)

,

where D = 4 − 2ǫ, q = p1 + p2 and p1 (p2) is the incoming(anti-)quark momentum.



The photon-quark form factor, which is a building block forN4LO cross sections.It is a gauge-invariant part of virtual forth-order corrections forthe process e+e− → 2 jets, or for Drell-Yan production athadron colliders.Let Γµq be the photon-quark vertex function.The scalar form factor is

Fq(q2) = −

1

4(1 − ǫ)q2Tr

(

p2/ Γµqp1/ γµ)

,

where D = 4 − 2ǫ, q = p1 + p2 and p1 (p2) is the incoming(anti-)quark momentum.The large-Nc asymptotics of Fq(q

2) → planar Feynmandiagrams.



Three-loop results[P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnovand M. Steinhauser’09,T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, andC. Studerus’10]



Three-loop results[P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnovand M. Steinhauser’09,T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, andC. Studerus’10]Analytic results for the three missing coefficients[R. N. Lee, A. V. Smirnov and V. A. Smirnov’10]



Three-loop results[P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnovand M. Steinhauser’09,T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, andC. Studerus’10]Analytic results for the three missing coefficients[R. N. Lee, A. V. Smirnov and V. A. Smirnov’10]Analytic results for the three-loop master integrals up toweight 8[R. N. Lee and V. A. Smirnov’10]motivated by a future four-loop calculation.



The fermionic corrections (∼ nf ) to Fq in the large-Nc limit,to the four-loop order.



Numerical four-loop calculations[R. H. Boels, B. A. Kniehl & G. Yang’16]



Numerical four-loop calculations[R. H. Boels, B. A. Kniehl & G. Yang’16]

Partial results for some individual integrals[A. von Manteuffel, E. Panzer & R. M. Schabinger’15]



We apply

qgraf for the generation of Feynman amplitudes;



We apply


q2e and exp for writing down form factors in terms ofFeynman integrals



We apply



FIRE and LiteRed for the IBP reduction to masterintegrals.



We apply




Calculations in generic ξ-gauge for checks.



We apply





Fq = 1 +∑

n≥1

(

α0s

4π

)n (µ2

−q2

)(nǫ)

F (n)q .



We apply





Fq = 1 +∑

n≥1

(

α0s

4π

)n (µ2

−q2

)(nǫ)

F (n)q .

Our result is the fermionic contribution to F(4)q in the large-Nc

limit.



F(4)q |large-Nc

=

1

ǫ7

[

1

12N3

c nf

]

+1

ǫ6

[

41

648N2

c n2f −

37

648N3

c nf

]

+1

ǫ5

[

1

54Ncn

3f +

277

972N2

c n2f

+

(

41π2

648−

6431

3888

)

N3c nf

]

+1

ǫ4

[

(

215ζ3

108−

72953

7776−

227π2

972

)

N3c nf

+11

54Ncn

3f +

(

5

24+

127π2

1944

)

N2c n

2f

]

+1

ǫ3

[

(

229ζ3

486−

630593

69984+

293π2

2916

)

N2c n

2f

+

(

2411ζ3

243−

1074359

69984−

2125π2

1296+

413π4

3888

)

N3c nf +

(

127

81+

5π2

162

)

Ncn3f

]

+O

(

1

ǫ2

)



Exponentiation of infrared (soft and collinear) divergences



Exponentiation of infrared (soft and collinear) divergences

log(Fq)|pole part|(αs4π)4 =

{

1

ǫ5

[

25

96β3

0CFγ0cusp

]

+1

ǫ4

[

CF

(

−13

96β2

0γ1cusp −

5

12β1β0γ

0cusp

)

−1

4β3

0γ0q

]

+1

ǫ3

[

CF

(

5

32β2γ

0cusp +

3

32β1γ

1cusp +

7

96β0γ

2cusp

)

+1

4β2

0γ1q +

1

2β1β0γ

0q

]

+1

ǫ2

[

−1

4β2γ

0q −

1

4β1γ

1q −

1

4β0γ

2q −

1

32CFγ

3cusp

]

+1

ǫ

[

γ3q

4

]}

,



The coefficients of the cusp and collinear anomalous dimensions

γx =∑

n≥0

(

αs(µ2)

4π

)n

γnx ,



The coefficients of the cusp and collinear anomalous dimensions

γx =∑

n≥0

(

αs(µ2)

4π

)n

γnx ,

with x ∈ {cusp, q}.

γ0cusp

= 4 ,

γ1cusp

=

(

−4π2

3+

268

9

)

Nc −40nf

9,

γ2cusp

=

(

44π4

45+

88ζ3

3−

536π2

27+

490

3

)

N2c

+

(

−64ζ3

3+

80π2

27−

1331

27

)

Ncnf −16n2

f

27,

γ3cusp

=

(

−32π4

135+

1280ζ3

27−

304π2

243+

3463

81

)

Ncn2f +

(

128π2ζ3

9+ 224ζ5

−44π4

27−

16252ζ3

27+

13346π2

243−

60391

81

)

N2c nf +

(

64ζ3

27−

32

81

)

n3f + . . .



γ0q = −

3Nc

2, γ1

q =

(

π2

6+

65

54

)

Ncnf +

(

7ζ3 −5π2

12−

2003

216

)

N2c ,

γ2q =

(

−π4

135−

290ζ3

27+

2243π2

972+

45095

5832

)

N2c nf +

(

−4ζ3

27−

5π2

27+

2417

1458

)

N

+ N3c

(

−68ζ5 −22π2ζ3

9−

11π4

54+

2107ζ3

18−

3985π2

1944−

204955

5832

)

,

γ3q = N3

c

[(

−680ζ2

3

9−

1567π6

20412+

83π2ζ3

9+

557ζ5

9+

3557π4

19440−

94807ζ3

972

+354343π2

17496+

145651

1728

)

nf

]

+

(

−8π4

1215−

356ζ3

243−

2π2

81+

18691

13122

)

Ncn3f

+

(

−2

3π2ζ3 +

166ζ5

9+

331π4

2430−

2131ζ3

243−

68201π2

17496−

82181

69984

)

N2c n

2f + . . .



We reproduce results up to three loops[A. Vogt’01; C.F. Berger’02; S. Moch, J.A.M. Vermaseren &A. Vogt’04,05; P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov,V.A. Smirnov & M. Steinhauser’09; T. Becher &M. Neubert’09; T. Gehrmann, E.W.N. Glover, T. Huber,N. Ikizlerli & C. Studerus’10]




The N3c n

3f term of γ3

cusp agrees with[M. Beneke & V.M. Braun’94]




The N3c n

3f term of γ3


Agreement of the n2f term with

[Davies, B. Ruijl, T.Ueda, J. Vermaseren & A. Vogt]talk at Loops and Legs 2016 by A. Vogt




The N3c n

3f term of γ3


Agreement of the n2f term with

[Davies, B. Ruijl, T.Ueda, J. Vermaseren & A. Vogt]talk at Loops and Legs 2016 by A. Vogt

All the other four-loop terms in γ3cusp and γ3

q , and for the finitepart, are new.



All planar four-loop on-shell form-factor integrals withp2

1 = p22 = 0, with q2 ≡ p2

3 = (p1 + p2)2

Fa1, . . . , a18 =

∫

. . .

∫

dDk1 . . . d

Dk4

(−(k1 + p1)2)a1(−(k2 + p1)2)a2(−(k3 + p1)2)a3

×1

(−(k4 + p1)2)a4(−(k1 − p2)2)a5(−(k2 − p2)2)a6(−(k3 − p2)2)a7

×1

(−(k4 − p2)2)a8(−k21 )

a9(−k22 )

a10(−k23 )

a11(−k24 )

a12

×1

(−(k1 − k2)2)−a13(−(k1 − k3)2)−a14(−(k1 − k4)2)−a15

×1

(−(k2 − k3)2)−a16(−(k2 − k4)2)−a17(−(k3 − k4)2)−a18

.



All planar four-loop on-shell form-factor integrals withp2

1 = p22 = 0, with q2 ≡ p2

3 = (p1 + p2)2

Fa1, . . . , a18 =

∫

. . .

∫

dDk1 . . . d

Dk4

(−(k1 + p1)2)a1(−(k2 + p1)2)a2(−(k3 + p1)2)a3

×1

(−(k4 + p1)2)a4(−(k1 − p2)2)a5(−(k2 − p2)2)a6(−(k3 − p2)2)a7

×1

(−(k4 − p2)2)a8(−k21 )

a9(−k22 )

a10(−k23 )

a11(−k24 )

a12

×1

(−(k1 − k2)2)−a13(−(k1 − k3)2)−a14(−(k1 − k4)2)−a15

×1

(−(k2 − k3)2)−a16(−(k2 − k4)2)−a17(−(k3 − k4)2)−a18

.

At most 12 indices can be positive.



FIRE → 99 master integrals.




[J. Henn, A.&V. Smirnov’13]: introduce an additional scale.p2

2 6= 0





2 6= 0

504 master integrals





2 6= 0


Use differential equations[A.V. Kotikov’91, Bern, Dixon & Kosower’94, E. Remiddi’97,T. Gehrmann & E. Remiddi’00, J. Henn’13,...]





2 6= 0







2 6= 0



[J. Henn’13]: use basis of integrals with constant leadingsingularities



Let f = (f1, . . . , fN) be primary master integrals (MI) for agiven family of dimensionally regularized (with D = 4 − 2ǫ)Feynman integrals.




Let x = (x1, . . . , xn) be kinematical variables and/or masses,or some new variables introduced to ‘get rid of square roots’.





DE:

∂i f (ǫ, x) = Ai(ǫ, x)f (ǫ, x) ,

where ∂i =∂∂xi

, and each Ai is an N × N matrix.





DE:

∂i f (ǫ, x) = Ai(ǫ, x)f (ǫ, x) ,

where ∂i =∂∂xi

, and each Ai is an N × N matrix.

[J. Henn’13]: turn to a new basis where DE take the form

∂i f (ǫ, x) = ǫAi(x)f (ǫ, x) .



In the case of two scales, i.e. with one variable in the DE, i.e.n = 1.

f ′(ǫ, x) = ǫ∑

k

ak

x − x (k)f (ǫ, x) .

where x (k) is the set of singular points of the DE and N × N

matrices ak are independent of x and ǫ.



In the case of two scales, i.e. with one variable in the DE, i.e.n = 1.

f ′(ǫ, x) = ǫ∑

k

ak

x − x (k)f (ǫ, x) .

where x (k) is the set of singular points of the DE and N × N

matrices ak are independent of x and ǫ.

For example, if xk = 0,−1, 1 then results are expressed interms of HPLs [E. Remiddi & J.A.M. Vermaseren]

H(a1, a2, . . . , an; x) =

∫ x

0

f (a1; t)H(a2, . . . , an; t)dt ,

where f (±1; t) = 1/(1 ∓ t), f (0; t) = 1/t



How to turn to a UT basis?




[J. Henn’13] Select basis integrals that have constantleading singularities [F. Cachazo’08] which aremultidimensional residues of the integrand. (Replacepropagators by delta functions).Based on experience in super Yang-Mills and a conjectureby [N. Arkani-Hamed et al.’12]

Transformations of the system of differential equationsbased on its singularities[Moser’59],[J. Henn’14],[R.N. Lee’14]











If you have almost reached the ε-form, make a small finalrotation of the current basis.See, e.g., [S. Caron-Huot, J. Henn,’14], [T. Gehrmann,A. von Manteuffel, L. Tancredi and E. Weihs’14]



We obtain differential equations with respect to x = p22/p

23

∂x f (x , ǫ) = ǫ

[

a

x+

b

1 − x

]

f (x , ǫ)

where a and b are x- and ǫ-independent 504 × 504 matrices.



We obtain differential equations with respect to x = p22/p

23

∂x f (x , ǫ) = ǫ

[

a

x+

b

1 − x

]

f (x , ǫ)

where a and b are x- and ǫ-independent 504 × 504 matrices.

Solving these equations in terms of HPL with letters 0 and 1.



Asymptotic behaviour at the points x = 0 and x = 1

f (x , ǫ)x→0=

[

1 +∑

k≥1

pk(ǫ)xk

]

x ǫaf0(ǫ) ,

f (x , ǫ)x→1=

[

1 +∑

k≥1

qk(ǫ)(1 − x)k

]

(1 − x)−ǫbf1(ǫ) ,




f (x , ǫ)x→0=

[

1 +∑

k≥1

pk(ǫ)xk

]

x ǫaf0(ǫ) ,

f (x , ǫ)x→1=

[

1 +∑

k≥1

qk(ǫ)(1 − x)k

]

(1 − x)−ǫbf1(ǫ) ,

Natural boundary conditions at the point x = 1:there is no singularity and it corresponds to propagator-typeintegrals which are known[P. Baikov and K. Chetyrkin’10; R. Lee and V. Smirnov’11]




f (x , ǫ)x→0=

[

1 +∑

k≥1

pk(ǫ)xk

]

x ǫaf0(ǫ) ,

f (x , ǫ)x→1=

[

1 +∑

k≥1

qk(ǫ)(1 − x)k

]

(1 − x)−ǫbf1(ǫ) ,

Natural boundary conditions at the point x = 1:there is no singularity and it corresponds to propagator-typeintegrals which are known[P. Baikov and K. Chetyrkin’10; R. Lee and V. Smirnov’11]

To obtain analytical results for the 99 one-scale MI, weperform (with the help of the HPL package [D. Maıtre’06])matching at the point x = 0.



Transporting boundary conditions at x = 1 to the point x = 0.



Transporting boundary conditions at x = 1 to the point x = 0.

0 1

p1 p2

p3

Re(x)

Im(x)

p1 p2

p3

x = 0

p2

p3

x = 1



In mathematics, the object transporting the boundaryinformation is called the Drinfeld associator.



In mathematics, the object transporting the boundaryinformation is called the Drinfeld associator.We construct the Drinfeld associator perturbatively in ǫ.




x ǫa =∑

x jεaj

Results for the 99 one-scale MI can be obtained from the‘naive’ part of the asymptotic expansion at x = 0, i.e. fromthe part corresponding to the zero eigenvalue of the matrix a,i.e. to j = 0.




x ǫa =∑

x jεaj


Two alternative descriptions of the asymptotic expansion:with DE and with with expansion by regions[M. Beneke and V. Smirnov’98]




x ǫa =∑

x jεaj



j = 0 corresponds to the hard-. . . -hard region while terms withj < 0 to other regions (soft, collinear, . . . ).




x ǫa =∑

x jεaj



j = 0 corresponds to the hard-. . . -hard region while terms withj < 0 to other regions (soft, collinear, . . . ). No positive j .



An example of our result

replacements

p1 p2k1 k2 k3

k4 + p1



An example of our result

replacements

p1 p2k1 k2 k3

k4 + p1

I12 =

∫

. . .

∫ 4∏

j=1

dDkj

(k24 )

2

k21k

22k

23 (k1 − k2)2(k2 − k3)2(k1 − k4)2

×1

(k2 − k4)2(k3 − k4)2(k1 + p1)2(k4 + p1)2(k4 − p2)2(k3 − p2)2



=1

576ε8+

1

216π2 1

ǫ6+

151

864ζ3

1

ǫ5+

173

10368π4 1

ǫ4+

[

505

1296π2ζ3 +

5503

1440ζ5

]

1

ǫ3+

+

[

6317

155520π6 +

9895

2592ζ23

]

1

ǫ2+

[

89593

77760π4ζ3 +

3419

270π2ζ5 −

169789

4032ζ7

]

1

ǫ

+

[

407

15s8a +

41820167

653184000π8 +

41719

972π2ζ2

3 −263897

2160ζ3ζ5

]

+O(ǫ) ,


Perspectives

General Nc . Non-planar diagrams


Perspectives


Other form-factors. More complicated integrals.


Perspectives


Other form-factors. More complicated integrals.

No conceptual problems. More powerful algorithms forIBP reduction. More powerful machines.

Documents

On the four-loop cusp anomalous dimension€¦ · On the four-loop cusp anomalous dimension Evaluating four-loop QCD form factors The photon-quark form factor, which is a building