Four-loop cusp anomalous dimension in QED

The 4-loop CF3TFnl and 5-loop CF4TFnl terms in the HQET field anomalous dimension γh are calculated analytically (the 4-loop one agrees with the recent numerical result [1]). The 4-loop CF3TFnl and 5-loop CF4TFnl terms in the cusp anomalous dimension Γ(φ) are calculated analytically, exactly in φ (the φ → ∞ asymptotics of the 4-loop one agrees with the recent numerical result [2]). Combining these results with the recent 4-loop dF Fnl contributions to γh and to the small-φ expansion of Γ(φ) up to φ4 [3], we now have the complete analytical 4-loop result for the Bloch-Nordsieck field anomalous dimension in QED, and the small-φ expansion of the 4-loop QED cusp anomalous dimension up to φ4.


JHEP06(2018)073
1 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., [4][5][6]). 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 [7]: The HQET field anomalous dimension γ h is known up to three loops [8,9]. 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 [10] (n l is the number of light flavors), C 2 F (T F n l ) 2 [11,12], [13] 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 ; JHEP06(2018)073 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 (section 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 [11,14]. At [15], where Γ l is the light-like cusp anomalous dimension. Several color structures of the 4-loop Γ(ϕ) are also known: [11,12]. 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 [11,12] and the large-N c N 2 c n 2 l result [18]. Large-N c results for Γ l at n 1 l [2, 18] and n 0 l [2,20] 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 (section 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 [10], C 2 F (T F n l ) 2 [11,12], C 3 F T F n l (section 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 (section 4). Combining the 4-loop Γ(ϕ) full results for C F (T F n l ) 3 [16], C 2 F (T F n l ) 2 [11,12], C 3 F T F n l (section 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 (section 4).

HQET field anomalous dimension: the
This is a QED problem. Due to exponentiation [21,22], 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 (figure 1). In QED with n l = 0 there is only 1 single-web diagram: figure 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 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. 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 L (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 [23] The coordinate-space full photon propagator is the Fourier transform of (2.4): The sum of single-web diagrams (figure 1) in the Landau gauge, analytically continued to Euclidean t = −iτ , is (2.10) where the 1-loop HQET integral can be calculated in coordinate space (2.9), or as a Fourier transform of the momentumspace HQET propagator. Now we re-express log W via the renormalized α(µ) at µ 0 = 2e −γ /τ (it is sufficient to do this in the 1-loop term) and obtain Extracting log Z h and differentiating it in log µ, we obtain γ h . Restoring the color factors and adding the gauge dependent term, 2 we obtain In the arbitrary covariant gauge the extra term to be added to w1 (2.10) is Γ(−ε)e −γε a0A = Γ(−ε)e −γε a(µ0)α(µ0)/(4π), because in QED Zα = Z −1 A . Hence the extra term to be added to log Z h is purely 1-loop: −(a/ε)α/(4π). In QED d log (a(µ)α(µ)) /d log µ = −2ε exactly, and hence the extra term in γ h (2.13) is also purely 1-loop: 2aα/(4π).
We have reproduced the C 2 F T F n l term in the 3-loop anomalous dimension [8,9] by a simpler method. The coefficient of in perfect agreement with the numerical result 0.1894 ± 0.0030 (table III in [1]). where the sum runs over all single-web diagrams. Diagrams in which all photon vertices are to the left (or to the right) of the J vertex cancel in w i (ϕ) − w i (0). The remaining 2-leg webs are shown in figure 2. At 4 loops 4-leg webs appear; they have been calculated (at ϕ 1) in [3]. 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 (3.9) We obtain Similarly to (2.12), we substitute (3.10) into log W (ϕ) W (0) = log Z J + (finite) 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 Γ(ϕ) = 4(ϕ coth ϕ − 1)

JHEP06(2018)073
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