A generalized Crewther relation and the V scheme: analytic results in fourth-order perturbative QCD and QED

Abstract

Using the analytic $\overline{\mathrm{MS}}$ scheme, three-loop contribution to the perturbative Coulomb-like part of the static color potential of a heavy quark–antiquark system, we obtain an analytic expression for the fourth-order $\beta$-function in the gauge-invariant effective V scheme in the case of the generic simple gauge group. We also present the Adler function of electron–positron annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule in the V scheme up to $a^4_s$ terms. We demonstrate that at this level of the perturbation theory in this effective scheme, the generalized Crewther relation, which connects the flavor nonsinglet contributions to the Adler and Bjorken polarized sum rule functions, is satisfied. Starting from the $a^2_s$ order, it contains a conformal symmetry breaking term that factors into the conformal anomaly $\beta(a_s)/a_s$ and the polynomial in powers of $a_s$. We prove that this relation also holds in other gauge-invariant renormalization schemes. The obtained results allows revealing the difference between the V-scheme $\beta$-function in QED and the Gell-Mann–Low $\Psi$-function. This distinction arises due to the presence of the light-by-light type scattering corrections first appearing in the static potential at the three-loop level.

Appendix A

We consider the question related to the integral representation of multiple zeta values. In general, these functions are defined as

$$\zeta_{m_1,\dots,m_k}=\sum_{i_1=1}^\infty\sum_{i_2=1}^{i_1-1}\dotsb \sum_{i_k=1}^{i_{k-1}-1}\prod_{j=1}^k \frac{\operatorname{sign}(m_j)^{i_j}}{i_j^{|m_j|}}. $$

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They were studied in detail in a number of works on the subject (see, e.g., [89]–[91], [61]). We use the Hurwitz–Lerch zeta function $\Phi(z, s, q)$

$$\Phi(z,s,q)=\sum_{k=0}^\infty\frac{z^k}{(k+q)^s} $$

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and its integral representation

$$\Phi(z,s,q)=\frac{1}{\Gamma(s)}\int_0^1\frac{x^{q-1}(-\log x)^{s-1}}{1-zx}\,dx, $$

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which is valid for $\operatorname{Re}(q)>0$, $\operatorname{Re}(s)>0$ and $z\in[-1;1)$ or $\operatorname{Re}(s)>1$ and $z=1$.

Then, for the constant $\zeta_{-5,-1}$ with transcendence of weight 6, appearing in calculating the three-loop correction to the static potential [13], we can write [28]

$$\begin{aligned} \, \zeta_{-5,-1}&=\sum_{k=1}^\infty\frac{(-1)^k}{k^5} \sum_{i=1}^{k-1}\frac{(-1)^i}{i} =\frac{15}{16}\zeta_5\log 2-\sum_{k=1}^\infty\frac{\Phi(-1,1,k)}{k^5}= \nonumber \\ &=\frac{15}{16}\zeta_5\log 2 -\int_0^1\frac{dx}{x(x+1)}\sum_{k=1}^\infty\frac{x^k}{k^5} =\frac{15}{16}\zeta_5\log 2-\zeta_6+\int_0^1\,dx\frac{\mathrm{Li}_5(x)}{x+1}. \end{aligned}$$

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Therefore, the constant $s_6=\zeta_6+\zeta_{-5,-1}$ can be represented in the form [90]

$$s_6=\frac{15}{16}\zeta_5\log 2 +\int_0^1dx\,\frac{\mathrm{Li}_5(x)}{x+1}\approx 0.9874414. $$

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Similarly, we can obtain an integral representations for multiple zeta values with specific arguments arising in the intermediate calculations in [13]:

$$\zeta_{5,2}=\zeta_5\zeta_2-\zeta_7+\int_0^1dx\, \frac{{\mathrm{Li}_5}(x)\log x}{1-x}\approx 0.0385751,$$

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$$\zeta_{-5, 2}=-\frac{15}{16}\zeta_5\zeta_2+\frac{63}{64}\zeta_7 +\int_0^1dx\,\frac{{\mathrm{Li}_5}(-x)\log x}{1-x}\approx 0.0271089.$$

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For instance, the function $\zeta_{5,3}$ occurs in the computation of the $\overline{\mathrm{MS}}$-scheme $\beta$-function of the $O(N)$-symmetric $\phi^4$ theory in the six-loop approximation [92] (in the notations in that paper, $\zeta_{3,5}$):

$$\zeta_{5,3}=\zeta_3\zeta_5-\zeta_8 -\frac{1}{2}\int_0^1dx\,\frac{\mathrm{Li}_5(x)\log^2(x)}{1-x}\approx 0.0377077. $$

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Appendix B

It is interesting to note some common features of the CBK relation and the action sum rule [93]–[97] (in lattice QCD, it is also known as the Michael sum rule). Indeed, both of them contain a conformal anomaly term, reflecting the effect of conformal symmetry violation. However, the second relation can be directly used in the nonperturbative region as well.

We recall that the conformal anomaly in the trace of the energy–momentum tensor of a massless $SU(N_c)$ gauge theory in the Euclidean domain has the form [98]–[100]

$$T_{\mu\mu}(x)=\frac{\beta(a_s)}{2a_s}F^a_{\mu\nu}(x)F^a_{\mu\nu}(x) =2\frac{\beta(a_s)}{a_s}\mathcal L(x), $$

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where $\mathcal L(x)$ is the gluon gauge part of the Euclidean Lagrangian density of the $SU(N_c)$ theory, expressed trough the Euclidean chromoelectric and chromomagnetic fields

$$\mathcal L(x)=\frac{1}{4}F^a_{\mu\nu}(x)F^a_{\mu\nu}(x) =\frac{1}{2}(\vec E(x)^2+\vec B(x)^2). $$

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We note that owing to a change in the metric signature, the square of the Euclidean electric field has an opposite sing to its Minkowskian counterpart, while signs of the squares of the Euclidean and Minkowskian magnetic fields coincide. The action sum rule relates a certain combination of the static potential to the Euclidean chromoelectric and chromomagnetic condensates and the $\beta$-function [93]–[97],

$$\widetilde V(r)+r\,\frac{\partial\widetilde V(r)}{\partial r} =\frac{\beta(a_s)}{a_s}\biggl\langle\int d^3x\,(\vec E(x)^2+\vec B(x)^2)\biggr\rangle_r, $$

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where $\widetilde V(r)$ is the static potential in coordinate space including the confining and nonconfining components and $\langle\,{\cdot}\,\rangle_r$ is the vacuum expectation value in the presence of a static quark–antiquark pair spaced apart from each other at a distance $r$ excluding the analogous contribution without these field sources.

It would be interesting to study the possible relation of the action sum rule and the CBK relation based on the first principles of quantum field theory.