The $\gamma Z$-box correction and its impact on parity-violating deep-inelastic scattering

Technical Summary: The $\gamma Z$-box Correction and its Impact on Parity-Violating Deep-Inelastic Scattering

Problem Statement
Precision parity-violating (PV) measurements, such as those performed at Jefferson Lab (JLab), serve as sensitive probes of the electroweak interaction at energies well below the electroweak scale. The interpretation of these experiments relies on extracting effective electron-quark couplings, a process that necessitates the inclusion of electroweak radiative corrections. Among these, the $\gamma Z$-box correction is particularly critical.

Historically, the low-energy treatment of the $\gamma Z$-box has relied on a prescription introduced by Marciano and Sanda, where external momenta are set to zero and the associated infrared (IR) singularity is regulated by introducing an effective, finite quark mass. This mass is interpreted as the scale below which quarks behave as free particles. However, this approach suffers from a fundamental ambiguity: the value of the effective quark mass is not uniquely defined by the perturbative calculation. Different choices for this auxiliary scale have led to varying results for the $\gamma Z$-box contribution, introducing uncertainty into the extraction of low-energy electroweak parameters.

Methodology
The authors revisit the one-loop $\gamma Z$-box contribution to electron-quark scattering by performing a finite-mass calculation. Unlike conventional low-energy treatments, this approach retains finite electron ($m_e$) and quark ($m_q$) masses throughout the entire calculation, taking the low-energy limit only after evaluating the full amplitude.

Key methodological steps include:

1. Direct Amplitude Treatment: The calculation treats the IR behavior directly. At zero momentum transfer, the box contribution is finite, while the threshold limit develops a Coulomb singularity (Sommerfeld enhancement) dependent on particle velocity. This singularity is subtracted as it does not contribute to local low-energy couplings.
2. Mass Limits: The authors derive expressions for the box correction in two distinct limits:
   • Vanishing quark mass ($m_q \to 0$) with finite electron mass.
   • Vanishing electron mass ($m_e \to 0$) with finite quark mass.
     In both limits, large logarithmic terms ($\ln(M_Z^2/m^2)$) appear, but the scale of the logarithm and the accompanying constant terms depend on which fermion mass is retained.
3. Phenomenological Application: For the specific application to deep-inelastic scattering (DIS), the authors adopt the $m_q \to 0$ limit. In DIS, logarithmic quark-mass dependence can be absorbed into universal parton distribution functions (PDFs). However, the explicit quark-mass logarithm in the $\gamma Z$-box carries coefficients dependent on specific electroweak couplings and cannot be absorbed into universal PDFs. Setting $m_q = 0$ thus yields a perturbative contribution free from an arbitrary low-energy quark-mass scale, isolating the target-structure dependence for separate treatment.

Key Contributions and Results
The primary contribution of this work is the derivation of a well-defined perturbative $\gamma Z$-box contribution that eliminates the ambiguity associated with the effective quark-mass regulator.

• Analytic Expressions: The authors provide explicit formulas for the corrections to the effective couplings $C_{1q}$ and $C_{2q}$ in the $m_q \to 0$ limit (Eqs. 1 and 2). These expressions differ from previous low-energy analyses by determining the constant terms directly from the amplitude rather than relying on an auxiliary IR regulator.
• Reanalysis of JLab PVDIS Data: The authors reanalyzed the Jefferson Lab electron-deuteron measurement (originally reported in Ref. [5]) using updated Standard Model parameters and their new $\gamma Z$-box contribution.
  • The replacement of the conventional $\gamma Z$-box with the finite-mass result shifts all four extracted effective couplings ($C_{1u}, C_{1d}, C_{2u}, C_{2d}$) toward larger values.
  • The shifts are quantified as $\Delta C_{1u} = +0.0032$, $\Delta C_{1d} = +0.0036$, $\Delta C_{2u} = +0.0029$, and $\Delta C_{2d} = +0.0033$.
  • While small in absolute terms, these shifts are significant at the $10^{-3}$ level, which is relevant for the precision interpretation of PVDIS measurements, particularly for the $C_{2q}$ couplings to which PVDIS offers unique sensitivity.
  • Specifically, the combination $2C_{2u} - C_{2d}$ shifts from $-0.0973$ (using updated inputs but the old prescription) to $-0.0948$ (using the new prescription), a change of $+0.0025$.

Significance and Claims
The paper asserts that the agreement between previous extractions and Standard Model expectations does not validate the underlying box prescription, as the modification of this single contribution shifts the extracted couplings at a level comparable to the precision of the measurements.

The significance of the work is twofold:

1. Resolution of Ambiguity: It provides a consistent, well-defined perturbative starting point for the $\gamma Z$-box correction without introducing an arbitrary effective quark-mass scale.
2. Foundation for Non-Perturbative Separation: The calculation establishes a clear separation between the perturbative contribution (derived here) and the remaining non-perturbative hadronic dynamics. The authors note that a complete treatment of the $\gamma Z$-box for hadronic targets requires matching this perturbative result with process-dependent non-perturbative contributions.

The authors conclude that a consistent treatment of the box contribution is essential for deriving phenomenological constraints on electron-quark couplings from PVDIS and other precision parity-violation observables, such as atomic parity violation, where similar shifts (e.g., in the nuclear weak charge of $^{133}\text{Cs}$) have been shown to be comparable to current experimental uncertainties.