[Paper Review] Leading and next to leading large $n_f$ terms in the cusp anomalous dimension and the quark-antiquark potential

Andrey Grozin|arXiv (Cornell University)|May 12, 2016

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper derives all-order resummations of leading and next-to-leading large-$n_f$ terms in the cusp anomalous dimension, heavy-quark field anomalous dimension, and quark-antiquark potential in QCD. Using large-$n_f$ QED as a calculable framework, it computes these quantities to next-to-leading $β_0$ order via exponentiated photon self-energies and light-by-light diagrams, yielding explicit formulas involving zeta values and $\beta_0$-expansions, with full agreement at two and three loops.

ABSTRACT

I discuss 3 related quantities: the cusp anomalous dimension, the HQET heavy-quark field anomalous dimension, and the quark-antiquark potential. Leading large $n_f$ terms can be calculated to all orders in $\\alpha_s$. Next to leading terms with the abelian color structure $C_F^2$ also can be found to all orders (but not non-abelian $C_F C_A$ terms). This talk is based on Appendices C and D in [arXiv:1510.07803].

Motivation & Objective

• To compute leading and next-to-leading large-$n_f$ contributions to the cusp anomalous dimension, HQET field anomalous dimension, and quark-antiquark potential in QCD.
• To extend all-order resummation techniques beyond leading-$n_f$ terms, focusing on abelian $C_F^2$ structures.
• To establish a systematic method using QED with $n_f$ massless leptons to compute these terms to next-to-leading $β_0$ order.
• To verify the results against known two- and three-loop calculations, confirming consistency in $C_F(T_F n_f)^{L-1}\alpha_s^L$ and $C_F^2(T_F n_f)^{L-2}\alpha_s^L$ terms.

Proposed method

• Use of large-$n_f$ QED with $C_F = T_F = 1$, $C_A = 0$, and $\beta_0 = -\frac{4}{3}n_f$ to model the abelian $C_F^2$-type large-$n_f$ contributions.
• Application of the $\beta_0$-expansion method, treating $1/\beta_0$ as a small parameter and computing to next-to-leading order.
• Exponentiation of the full photon propagator in coordinate and momentum space to resum leading-$n_f$ terms via $\log W = \text{exponent of full propagator}$.
• Incorporation of light-by-light diagrams at next-to-next-to-leading $β_0$ order to break the exponentiation, used to compute corrections.
• Derivation of the quark-antiquark potential via the vertex function $V(\omega,\omega;\varphi)$ and comparison with the cusp anomalous dimension via conformal symmetry.
• Use of conformal anomaly matching to relate $\delta\Gamma(\pi - \delta)$ and $\vec{q}^2 V(\vec{q})$, enabling cross-checking of results.

Experimental results

Research questions

• RQ1 Can all-order resummations of leading and next-to-leading large-$n_f$ terms in the cusp anomalous dimension be derived using QED analogies?
• RQ2 What is the structure of the $C_F^2$-type next-to-leading large-$n_f$ contributions in the cusp anomalous dimension and quark-antiquark potential?
• RQ3 How do light-by-light diagrams and conformal anomaly effects modify the exponentiation of the Wilson line and potential at next-to-leading $β_0$ order?
• RQ4 Can the $\beta_0$-expansion method be used to generate explicit formulas for $C_F^2(T_F n_f)^{L-2}\alpha_s^L$ terms to all orders in $\alpha_s$?
• RQ5 To what extent do the results reproduce known two- and three-loop results in QCD?

Key findings

• The leading large-$n_f$ terms in the cusp anomalous dimension and quark-antiquark potential are given by $C_F(T_F n_f)^{L-1}\alpha_s^L$ and $C_F(T_F n_f)^L\alpha_s^{L+1}$, respectively, and are resummed to all orders via exponentiated photon propagators.
• The next-to-leading large-$n_f$ terms with $C_F^2$ color structure are computed to all orders in $\alpha_s$ using a systematic algorithm based on $\beta_0$-expansion and QED, yielding explicit expressions in terms of zeta values.
• The two-loop quark-antiquark potential terms $C_F(T_F n_f)^2\alpha_s^3$ and $C_F^2 T_F n_f \alpha_s^3$ are reproduced exactly from the $\beta_0$-expansion method.
• The three-loop terms $C_F(T_F n_f)^3\alpha_s^4$ and $C_F^2(T_F n_f)^2\alpha_s^4$ are also correctly reproduced, confirming the method's consistency.
• The conformal anomaly relation $\Delta = \delta\Gamma(\pi - \delta) - \vec{q}^2 V(\vec{q})/4\pi$ is verified at next-to-leading $β_0$ order, with $\Delta$ computed up to $\mathcal{O}(\alpha_s^4)$ and matching known results.
• The $b^3/\beta_0^2$ term in the conformal anomaly vanishes, confirming consistency with the known $\mathcal{O}(\alpha_s^3)$ coefficient in (8), and the $\beta_0$-expansion yields a series in $b = \beta_0 \alpha_s / (4\pi)$ with coefficients involving $\zeta_3, \zeta_5, \zeta_7, \pi^4, \pi^6$, etc.