$\mathbf{γZ}$ Box at Low Energy

TL;DR

This work evaluates the one‑loop γZ box correction to low‑energy electron–quark scattering with nonzero fermion masses, translating the result into shifts of the parity‑violating four‑fermion couplings $C_{iq}$ in the low‑energy effective Lagrangian. By performing a full mass‑dependent calculation and reducing the amplitudes to standard scalar integrals, the authors carefully treat threshold behavior and infrared singularities, introducing a Sommerfeld enhancement term that is subtracted to define finite $C_{iq}^{ m eff}$. A central result is that taking the quark mass to zero at the end yields finite, regulator‑independent low‑energy couplings, allowing a clean separation into a perturbative, massless‑quark piece and a nonperturbative hadronic remainder encoded in form factors $F_{ m γZ}^{ep}(s,t)$. The forward‑limit results and mass‑dependent threshold expressions illuminate the role of hadronic structure in determining the proton weak charge, with implications for polarized electron scattering and atomic parity violation, and point to future work using dispersion relations or lattice QCD to supplement the perturbative results and to address Coulomb corrections.

Abstract

We calculate the 1-loop $γZ$ box-graph correction to electron-quark scattering at low energy and low momentum transfer. Both electron and quark masses are kept non-zero. From our result, we extract coupling constants for the low-energy effective Lagrangian with parity-violating 4-fermion interaction terms. We study the zero-mass limits and show that a non-zero electron mass is sufficient to obtain finite, well-defined couplings which are insensitive to a hadronic mass cutoff. We finally discuss the impact of our results on the determination of the weak charge of the proton from polarized electron-proton scattering.

Figures (6)

• Figure 1: The four $\gamma Z$-box graphs.
• Figure 2: The box-graph corrections $\delta_{box}\hat{C}_{iu}(s, t=0)$ as a function of the Mandelstam variable $s$, evaluated at fixed $t = 0$ for a quark mass of $M = 0.3$ GeV, see Eqs. (\ref{['eq:deltaboxC0st']} - \ref{['eq:deltaboxC3st']}) in the appendix. For $\delta_{box}\hat{C}_{0u}(s,t=0)$ we inserted a zoom-in at threshold to show that the function is in fact smooth. The vertical dash-dotted lines indicate the position of the threshold, $s_0 = (m+M)^2$.
• Figure 3: The subtracted box-graph corrections $\delta_{box}C_{iu}^{\mathrm{eff}}$ as a function of the Mandelstam variable $s$, evaluated at fixed $t = 0$ for a quark mass of $M = 0.3$ GeV. The orange dashed lines show the value of $\delta_{box}C_{iu}$ at threshold ($s\xrightarrow[]{} (M+m)^2$) as given in Eqs. (\ref{['C0thresholdgeneric']} - \ref{['C3thresholdgeneric']}). The vertical dash-dotted lines indicate the position of the threshold.
• Figure 4: The subtracted box-graph corrections $\delta_{box}C_{id}^{\mathrm{eff}}$ as in Fig. \ref{['fig:CiuSE']} but now for down quarks as a function of the Mandelstam variable $s$, evaluated at fixed $t = 0$ for a quark mass of $M = 0.3$ GeV. The orange dashed lines show the value of $\delta_{box}C_{iu}$ at threshold ($s\xrightarrow[]{} (M+m)^2$) as given in Eqs. (\ref{['C0thresholdgeneric']} - \ref{['C3thresholdgeneric']}). The vertical dash-dotted lines indicate the position of the threshold.
• Figure 5: The box-graph corrections $\delta_{box}C_{i u}^{\mathrm{eff}}$ as a function of the Mandelstam variable $s$ at fixed $t=0$ for different effective quark masses $M$: blue for $0.1~\mathrm{GeV}$, orange for $0.2~\mathrm{GeV}$, green for $0.3~\mathrm{GeV}$, yellow for $0.4~\mathrm{GeV}$, and purple for $0.5~\mathrm{GeV}$.