Four-loop cusp anomalous dimension in QED

The 4-loop $C_F^3 T_F n_l$ and 5-loop $C_F^4 T_F n_l$ terms in the HQET field anomalous dimension $\gamma_h$ are calculated analytically (the 4-loop one agrees with the recent numerical result [arXiv:1801.08292]). The 4-loop $C_F^3 T_F n_l$ and 5-loop $C_F^4 T_F n_l$ terms in the cusp anomalous dimension $\Gamma(\varphi)$ are calculated analytically, exactly in $\varphi$ (the $\varphi\to\infty$ asymptotics of the 4-loop one agrees with the recent numerical result [arXiv:1707.08315]). Combining these results with the recent 4-loop $d_{FF} n_l$ contributions to $\gamma_h$ and to the small-$\varphi$ expansion of $\Gamma(\varphi)$ up to $\varphi^4$ [arXiv:1708.01221] (recently extended to $\varphi^6$ [arXiv:1807.05145]) we now have the complete analytical 4-loop result for the Bloch--Nordsieck field anomalous dimension in QED, and the small-$\varphi$ expansion of the 4-loop QED cusp anomalous dimension up to $\varphi^6$.


Introduction
QCD problems with a single heavy quark Q having momentum P = M Q v + p (where M Q is the on-shell mass and v is some vector with v 2 = 1) can be described by heavy quark effective theory (HQET) if characteristic heavy-quark residual momentum p, as well as characteristic gluon and light-quark momenta k i , are M Q (see, e.g., [5][6][7]). The heavy quark is described by the field where we use the MS scheme, and Z h is a minimal renormalization constant. We use the covariant gauge: −(∂ µ A µa 0 ) 2 /(2a 0 ) is added to the Lagrangian, the gauge-fixing parameter is renormalized by the same minimal constant as the gluon field: a 0 = Z A (α s (µ), a(µ))a(µ). The HQET heavy-quark field anomalous dimension is defined as γ h = d log Z h /d log µ. The h v0 coordinate-space propagator in the v rest frame has the form where W (t) is the straight Wilson line along v of length t. The heavy-quark field is QCD and HQET are related by the matching coefficient z [8]: The HQET field anomalous dimension γ h is known up to three loops [9,10]. In the first of these papers, it was obtained as a by-product of the three-loop calculation of the heavyquark field renormalization constant in the on-shell scheme Z os Q , from the requirement that the renormalized matching coefficient z(µ) (1.3) must be finite; in the second paper, it was confirmed by a direct HQET calculation. Several color structures of the 4-loop result are also known: C F (T F n l ) 3 [11] (n l is the number of light flavors), C 2 F (T F n l ) 2 [12,13], [14] using the finiteness of z(µ)), d F F n l [3]. Here C R (R = F , A) are the standard quadratic Casimirs: t a R t a R = C R 1 R (t a R are the generators in the representation R, 1 R is the corresponding unit matrix); Tr t a F t b F = T F δ ab ; R (the brackets mean symmetrization), and N c = Tr 1 F . The remaining terms are known numerically [1], from the numerical 4-loop Z os Q using the finiteness of z(µ) (1.3). Here I calculate the C L−1 F T F n l α L s terms up to L = 5 analytically (sect. 2); the L = 4 term agrees with the numerical result [1].
If the heavy-quark velocity is substantially changed (e. g., a weak decay into another heavy quark), we have HQET with 2 unrelated fields h v , h v . At the effective-theory level this is described by the current The minimal renormalization constant Z J is gauge invariant (unlike Z h ) because the current J 0 is color singlet. The anomalous dimension of this current, also known as the cusp anomalous dimension, is defined as The QCD cusp anomalous dimension Γ(ϕ) is known up to three loops [12,15]. At ϕ 1 it is a regular series in ϕ 2 . At ϕ 1 it is Γ l ϕ + O(ϕ 0 ) [16], where Γ l is the light-like cusp anomalous dimension. Several color structures of the 4-loop Γ(ϕ) are also known: [12,13]. The d F F n l term is known at ϕ 1 up to ϕ 4 [3] 1 . For the ϕ 1 asymptotics (i. e. Γ l ), both n 2 l terms are known from combining the C 2 F (T F n l ) 2 result [12,13] and the large-N c N 2 c n 2 l result [19]. Large-N c results for Γ l at n 1 l [2, 19] and n 0 l [2,21] are also known analytically. Contributions of individual color structures of Γ l at n 1,0 l are only known numerically [2]. Here I calculate the C L−1 F T F n l α L s terms up to L = 5 in Γ(ϕ) analytically, as exact functions of ϕ (sect. 3). In particular, I find their ϕ 1 asymptotics; the analytical L = 4 result agrees with the numerical one [2].
In QED without light lepton flavors (n l = 0), as explained below, both γ h and Γ(ϕ) are exactly given by the one-loop formulas. When n l = 0, higher corrections appear. Combining the 4-loop γ h results for C F (T F n l ) 3 [11], C 2 F (T F n l ) 2 [12,13], C 3 F T F n l (sect. 2) and d F F n l [3] structures, I obtain the complete analytical 4-loop result for the Bloch-Nordsieck field anomalous dimension γ h in QED (sect. 4). Combining the 4-loop Γ(ϕ) full results for C F (T F n l ) 3 [17], C 2 F (T F n l ) 2 [12,13], C 3 F T F n l (sect. 3) structures with the d F F n l term [3] (expansion up to ϕ 4 ), I obtain the expansion of the 4-loop QED Γ(ϕ) up to ϕ 4 (sect. 4).

HQET field anomalous dimension: the
This is a QED problem. Due to exponentiation [22,23], the coordinate-space propagator of the Bloch-Nordsieck field (i. e. the straight Wilson line W ) is where w i are single-web diagrams. Due to C parity conservation in QED, webs have even numbers of legs ( fig. 1). In QED with n l = 0 there is only 1 single-web diagram: fig. 1a with the free photon propagators. Therefore, log W is exactly 1-loop; the β function is 0, and hence γ h is also exactly 1-loop. At n l > 0 corrections to the photon propagator in fig. 1a appear. Webs with 4 legs ( fig. 1b) first appear at 4 loops; they have been calculated in [3]. All contributions to log W (2.1) are gauge invariant except the 1-loop one, because proper vertex functions with any numbers of photon legs are gauge invariant and transverse with respect to each photon leg due to the QED Ward identities. a b The full momentum-space photon propagator in the covariant gauge is where Π(k 2 ) is the photon self-energy: (e 2 0 has dimensionality m 2ε , so that the power of −k 2 is obvious; γ is the Euler constant). Only the 0-loop term in (2.2) is gauge dependent. Writing Π L asΠ L n l + (n >1 l terms), we obtain in the Landau gauge a 0 = 0 (2.4) The MS charge renormalization is (note that here we call the L-loop β function coefficient β L , not β L−1 as usually done; this makes subsequent formulas more logical). In QED log 1 − Π(k 2 ) = log Z α + (finite); writing β L =β L n l + (n >1 l terms), we see that 1/ε terms inΠ L are related toβ L : Here the β function coefficients are [24] and [25] The coordinate-space full photon propagator is the Fourier transform of (2.4): The sum of single-web diagrams ( fig. 1) in the Landau gauge, analytically continued to Euclidean t = −iτ , is
We have reproduced the C 2 F T F n l term in the 3-loop anomalous dimension [9,10] by a simpler method. The coefficient of in perfect agreement with the numerical result 0.1894 ± 0.0030 (Table III in [1]). Renormalization constants cannot depend on kinematics of Green functions we choose to calculate, and so we choose t = t to have a single-scale problem. We have The the L-loop n 1 l contribution is (L ≥ 2)

QCD cusp anomalous dimension
whereD µν L (x) is given by (2.9). We can write it, together with the 1-loop Landau-gauge contribution, in the form where The integrals I 1,2 (ϕ) are

5)
(3.9) We obtain and re-express it via α(µ 0 ) (it is sufficient to do this in the 1-loop term). Note thatV (ϕ) does not depend on L; as a result, termsβ L−1V (ϕ) cancel in log Z J (in contrast to the first line of (2.13) where they contributed because of the 1/L term in (2.11)). Differentiating log Z J we obtain Thus we have reproduced the 3-loop C 2 F T F n l term in [12,15]. The coefficient of 2T F n l C 3 F (α s /(4π)) 4 in the light-like cusp anomalous dimension Γ l is −4 80ζ 5 − 148 3 ζ 3 − 143 9 ≈ −31.055431 , in perfect agreement with the numerical result −31.00 ± 0.4 ( Table 2 in [2]). The C L−1 F T F n l terms in the quark-antiquark potential in Coulomb gauge are given by a single Coulomb-gluon propagator: where α s is taken at µ 2 = q 2 . The terms up to α 4 s agree with [26]. The cusp anomalous dimension at Euclidean angle ϕ E = π − δ, δ → 0, is related to the quark-antiquark potential [27] δ Γ(π − δ) δ→0 = q 2 V ( q ) 4π ; (3.13) this relation follows from conformal invariance, and in QCD it is broken by extra terms proportional to coefficients of the β function [12,15]. Comparing (3.11) with (3.12), we see that the relation (3.13) for the C L−1