Matter dependence of the four-loop QCD cusp anomalous dimension: from   small angles to all angles

Robin Br\"user; Andrey Grozin; Johannes M. Henn; Maximilian Stahlhofen

arXiv:1902.05076·hep-ph·June 26, 2019

Matter dependence of the four-loop QCD cusp anomalous dimension: from small angles to all angles

Robin Br\"user, Andrey Grozin, Johannes M. Henn, Maximilian Stahlhofen

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TL;DR

This paper calculates four-loop fermionic contributions to the QCD cusp anomalous dimension at small angles, tests a recent conjecture for angle dependence, and explores implications for large angles and the quark-antiquark potential.

Contribution

It provides the first four-loop fermionic contributions at small angles and tests the validity of a recent conjecture on angle dependence in QCD cusp anomalous dimension.

Findings

01

Conjecture fails for two of seven fermionic structures.

02

Results support the conjecture for remaining terms.

03

New analytic expressions for light-like cusp anomalous dimension.

Abstract

We compute the fermionic contributions to the cusp anomalous dimension in QCD at four loops as an expansion for small cusp angle. As a byproduct we also obtain the respective terms of the four-loop HQET wave function anomalous dimension. Our new results at small angles provide stringent tests of a recent conjecture for the exact angle dependence of the matter terms in the four-loop cusp anomalous dimension. We find that the conjecture does not hold for two of the seven fermionic color structures, but passes all tests for the remaining terms. This provides strong support for the validity of the corresponding conjectured expressions with full angle dependence. Taking the limit of large Minkowskian angle, we extract novel analytic results for certain terms of the light-like cusp anomalous dimension. They agree with the known numerical results. Finally, we study the anti-parallel lines…

Tables1

Table 1. Table 1: Four-loop contributions to Γ cusp ( ϕ ) subscript Γ cusp italic-ϕ \Gamma_{\text{cusp}}(\phi) and its limits in QCD as well as the HQET field anomalous dimension γ h subscript 𝛾 ℎ \gamma_{h} . The ∗ marks numerical results. The results [ × \times ] are obtained in the present paper.

color structure	sample diagram	$Γ_{cusp} (ϕ)$	$ϕ ≪ 1$	light-like	$γ_{h}$
${(T_{F} n_{f})}^{3} C_{R}$

Equations191

\displaystyle W=\frac{1}{N_{R}}\textrm{Tr}_{R}\big{\langle}0|\,P\,\exp{\left({ig\oint_{C}dx^{\mu}A_{\mu}(x)}\right)}|0\big{\rangle}=1+\mathcal{O}\!\left(g^{2}\right)\,.

\displaystyle W=\frac{1}{N_{R}}\textrm{Tr}_{R}\big{\langle}0|\,P\,\exp{\left({ig\oint_{C}dx^{\mu}A_{\mu}(x)}\right)}|0\big{\rangle}=1+\mathcal{O}\!\left(g^{2}\right)\,.

lo g W = lo g V (ϕ) - lo g V (0) = lo g Z + O (ϵ^{0}),

lo g W = lo g V (ϕ) - lo g V (0) = lo g Z + O (ϵ^{0}),

Γ_{cusp} (ϕ, α_{s})

Γ_{cusp} (ϕ, α_{s})

Γ_{cusp} (ϕ, α_{s})

Γ_{cusp} (ϕ, α_{s})

lo g Z =

lo g Z =

+ (\frac{α _{s}}{π})^{3} [- \frac{β _{0}^{2} Γ _{cusp}^{(1)}}{96 ϵ ^{3}} + \frac{β _{1} Γ _{cusp}^{(1)} + 4 β _{0} Γ _{cusp}^{(2)}}{96 ϵ ^{2}} - \frac{Γ _{cusp}^{(3)}}{6 ϵ}] + (\frac{α _{s}}{π})^{4} [+ \frac{β _{0}^{3} Γ _{cusp}^{(1)}}{512 ϵ ^{4}}

- \frac{β _{0} ( β _{1} Γ _{cusp}^{(1)} + 2 β _{0} Γ _{cusp}^{(2)} )}{256 ϵ ^{3}} + \frac{β _{2} Γ _{cusp}^{(1)} + 4 β _{1} Γ _{cusp}^{(2)} + 16 β _{0} Γ _{cusp}^{(3)}}{512 ϵ ^{2}} - \frac{Γ _{cusp}^{(4)}}{8 ϵ} + O (α_{s}^{5}) .

\frac{d lo g α _{s}}{d lo g μ} = - 2 ϵ + 2 β (α_{s}), β (α_{s}) = - k \geq 0 \sum (\frac{α _{s}}{4 π})^{k + 1} β_{k},

\frac{d lo g α _{s}}{d lo g μ} = - 2 ϵ + 2 β (α_{s}), β (α_{s}) = - k \geq 0 \sum (\frac{α _{s}}{4 π})^{k + 1} β_{k},

Γ_{cusp} (ϕ, α_{s}) = - i ϕ K (α_{s}) for ϕ \to i \infty .

Γ_{cusp} (ϕ, α_{s}) = - i ϕ K (α_{s}) for ϕ \to i \infty .

lo g (1 - \frac{d Σ _{h}}{d δ}) = - lo g Z_{h} + O (ϵ^{0}) .

lo g (1 - \frac{d Σ _{h}}{d δ}) = - lo g Z_{h} + O (ϵ^{0}) .

V (0) = 1 - \frac{d Σ _{h}}{d δ},

V (0) = 1 - \frac{d Σ _{h}}{d δ},

lo g V (0) = - lo g Z_{h} + O (ϵ^{0}) .

lo g V (0) = - lo g Z_{h} + O (ϵ^{0}) .

γ_{h} (α_{s}, ξ) = \frac{d lo g Z _{h}}{d lo g μ} .

γ_{h} (α_{s}, ξ) = \frac{d lo g Z _{h}}{d lo g μ} .

Γ_{cusp} =

Γ_{cusp} =

\displaystyle+(n_{f}T_{F})^{2}\Gamma_{\text{cusp}}^{f\!f\!R}+n_{f}T_{F}\left(C_{F}\Gamma_{\text{cusp}}^{f\!RF}+C_{A}\Gamma_{\text{cusp}}^{f\!RA}\right)\Big{]}+\mathcal{O}(\alpha_{s}^{4})\;.

\frac{d _{R} d _{R^{'}}}{N _{R}} \equiv \frac{d _{R}^{ab c d} d _{R^{'}}^{ab c d}}{N _{R}}, d_{R}^{ab c d} = STr [T_{R}^{a} T_{R}^{b} T_{R}^{c} T_{R}^{d}] .

\frac{d _{R} d _{R^{'}}}{N _{R}} \equiv \frac{d _{R}^{ab c d} d _{R^{'}}^{ab c d}}{N _{R}}, d_{R}^{ab c d} = STr [T_{R}^{a} T_{R}^{b} T_{R}^{c} T_{R}^{d}] .

Γ_{cusp}^{(4)} =

Γ_{cusp}^{(4)} =

+ n_{f} T_{F} (C_{R} C_{F}^{2} Γ_{cusp}^{f R F F} + C_{R} C_{F} C_{A} Γ_{cusp}^{f R F A} + C_{R} C_{A}^{2} Γ_{cusp}^{f R AA}) + n_{f} \frac{d _{R} d _{F}}{N _{R}} Γ_{cusp}^{f d R F} .

\frac{1}{N _{R}} Tr [T_{R}^{a} T_{R}^{b} T_{R}^{c} T_{R}^{d} T_{R}^{a} T_{R}^{b} T_{R}^{c} T_{R}^{d}] = \frac{1}{N _{R}} Tr [T_{R}^{a} T_{R}^{b} T_{R}^{c} T_{R}^{d}] Tr [T_{A}^{a} T_{A}^{b} T_{A}^{c} T_{A}^{d}] + \dots = \frac{d _{R} d _{A}}{N _{R}} + \dots,

\frac{1}{N _{R}} Tr [T_{R}^{a} T_{R}^{b} T_{R}^{c} T_{R}^{d} T_{R}^{a} T_{R}^{b} T_{R}^{c} T_{R}^{d}] = \frac{1}{N _{R}} Tr [T_{R}^{a} T_{R}^{b} T_{R}^{c} T_{R}^{d}] Tr [T_{A}^{a} T_{A}^{b} T_{A}^{c} T_{A}^{d}] + \dots = \frac{d _{R} d _{A}}{N _{R}} + \dots,

G (a_{1}, ..., a_{16}) = e^{4 ϵ γ_{E}} \int \frac{d ^{d} k _{1}}{i π ^{d /2}} \int \frac{d ^{d} k _{2}}{i π ^{d /2}} \int \frac{d ^{d} k _{3}}{i π ^{d /2}} \int \frac{d ^{d} k _{4}}{i π ^{d /2}} k = 1 \prod 16 D_{k}^{- a_{k}},

G (a_{1}, ..., a_{16}) = e^{4 ϵ γ_{E}} \int \frac{d ^{d} k _{1}}{i π ^{d /2}} \int \frac{d ^{d} k _{2}}{i π ^{d /2}} \int \frac{d ^{d} k _{3}}{i π ^{d /2}} \int \frac{d ^{d} k _{4}}{i π ^{d /2}} k = 1 \prod 16 D_{k}^{- a_{k}},

D_{1} = - 2 v \cdot k_{1} + 1,

D_{1} = - 2 v \cdot k_{1} + 1,

D_{4} = - 2 v \cdot k_{4} + 1,

D_{7} = - k_{3}^{2},

D_{10} = - (k_{1} - k_{3})^{2},

D_{13} = - (k_{2} - k_{4})^{2},

D_{16} = - 2 v \cdot (k_{1} - k_{3}) + 1,

topology 4: a_{15}, a_{16} = 0 and a_{10}, a_{11}, a_{12} \geq 0,

topology 4: a_{15}, a_{16} = 0 and a_{10}, a_{11}, a_{12} \geq 0,

topology 5: a_{1}, a_{3} = 0 and a_{4}, a_{5}, a_{10}, a_{11} \geq 0 .

γ_{h}^{f f R A} = (- \frac{35 ζ _{3}}{24} + \frac{π ^{4}}{120} - \frac{4157}{31104}) + ξ (- \frac{ζ _{3}}{24} + \frac{269}{7776}),

γ_{h}^{f f R A} = (- \frac{35 ζ _{3}}{24} + \frac{π ^{4}}{120} - \frac{4157}{31104}) + ξ (- \frac{ζ _{3}}{24} + \frac{269}{7776}),

γ_{h}^{f R F A} = (- \frac{15 ζ _{5}}{8} - \frac{105 ζ _{3}}{32} + \frac{23 π ^{4}}{960} + \frac{36503}{13824}) + ξ (- \frac{11 ζ _{3}}{32} - \frac{π ^{4}}{960} + \frac{767}{1536}),

γ_{h}^{f R AA} = (- \frac{2299 ζ _{5}}{1152} - \frac{3 ζ _{3}^{2}}{8} + \frac{37 π ^{2} ζ _{3}}{864} + \frac{1751 ζ _{3}}{256} - \frac{1529 π ^{4}}{69120} - \frac{π ^{2}}{48} + \frac{690965}{497664})

γ_{h}^{f R AA} = + ξ (- \frac{ζ _{5}}{576} + \frac{π ^{2} ζ _{3}}{432} + \frac{65 ζ _{3}}{192} - \frac{7 π ^{4}}{23040} + \frac{49729}{497664})

γ_{h}^{f R AA} = + ξ^{2} (- \frac{7 ζ _{3}}{768} + \frac{π ^{4}}{46080} - \frac{109}{9216}),

Γ_{cusp}^{f d R F} =

Γ_{cusp}^{f d R F} =

+ ϕ^{4} (\frac{71 ζ _{3}}{225} - \frac{16 π ^{2} ζ _{3}}{675} - \frac{4 ζ _{5}}{9} - \frac{23}{900} - \frac{157 π ^{2}}{8100} + \frac{49 π ^{4}}{8100})

+ ϕ^{6} (\frac{983 ζ _{3}}{33075} - \frac{32 π ^{2} ζ _{3}}{11025} - \frac{64 ζ _{5}}{1323} + \frac{797}{264600} - \frac{1333 π ^{2}}{595350} + \frac{421 π ^{4}}{595350}) + O (ϕ^{8}) .

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aainstitutetext: PRISMA Cluster of Excellence, Johannes Gutenberg University, 55128 Mainz, Germanybbinstitutetext: Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russiaccinstitutetext: Novosibirsk State University, Novosibirsk 630090, Russiaddinstitutetext: Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany

Matter dependence of the four-loop QCD cusp anomalous dimension: from small angles to all angles

Robin Brüser a,b,c

Andrey Grozin d

Johannes M. Henn a

Maximilian Stahlhofen

[email protected]

Abstract

We compute the fermionic contributions to the cusp anomalous dimension in QCD at four loops as an expansion for small cusp angle. As a byproduct we also obtain the respective terms of the four-loop HQET wave function anomalous dimension. Our new results at small angles provide stringent tests of a recent conjecture for the exact angle dependence of the matter terms in the four-loop cusp anomalous dimension. We find that the conjecture does not hold for two of the seven fermionic color structures, but passes all tests for the remaining terms. This provides strong support for the validity of the corresponding conjectured expressions with full angle dependence. Taking the limit of large Minkowskian angle, we extract novel analytic results for certain terms of the light-like cusp anomalous dimension. They agree with the known numerical results. Finally, we study the anti-parallel lines limit of the cusp anomalous dimension. In a conformal theory, the latter is proportional to the static quark-antiquark potential. We use the new four-loop results to determine parts of the conformal anomaly term.

††preprint:

MITP/18-125

MPP-2019-1

1 Introduction

The cusp anomalous dimension is a universal and ubiquitous quantity in QCD and the effective field theories describing its IR behavior as e.g. heavy quark effective theory (HQET) and soft collinear effective theory (SCET). It governs the IR singularity structure of QCD scattering amplitudes Ivanov:1985np ; Korchemsky:1985xu ; Korchemsky:1985ts ; Korchemsky:1985xj ; Korchemsky:1986fj . In the presence of massive partons the IR divergences are controlled by the angle dependent cusp anomalous dimension $\Gamma_{\textrm{cusp}}(\phi,\alpha_{s})$ . It can be determined from the UV divergences of a time-like Wilson loop with a cusp of (Euclidean) angle $\phi$ Korchemsky:1985xj . The light-like cusp anomalous dimension $K(\alpha_{s})$ relevant for scattering of massless partons emerges as the $\phi\to i\infty$ limit of $\Gamma_{\textrm{cusp}}(\phi,\alpha_{s})$ Korchemsky:1985xj ; Korchemsky:1991zp . It is the key ingredient to Sudakov resummation for scattering processes at high-energy colliders.

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