Reduced uncertainty of the axial $\gamma Z$-box correction to the proton's weak charge

TL;DR

This paper provides an updated calculation of the $ ext{γ}Z$-box correction to the proton's weak charge, reducing uncertainty and improving the precision of low-energy weak mixing angle measurements.

Contribution

The authors present a refined analysis of the $ ext{γ}Z$-box correction, incorporating new data and symmetry considerations to halve the uncertainty in the axial contribution.

Findings

Confirmed previous results for the axial contribution to the $ ext{γ}Z$-box correction.

Reduced the uncertainty in the axial contribution by a factor of 2.

Estimated the additional contribution from parity-violating photon-hadron interactions and its uncertainty.

Abstract

We present the fully up-to-date calculation of the $\gamma Z$-box correction which needs to be taken into account to determine the weak mixing angle at low energies from parity-violating electron proton scattering. We make use of neutrino and antineutrino inclusive scattering data to predict the parity-violating structure function $F_3^{\gamma Z}$ by isospin symmetry. Our new analysis confirms previous results for the axial contribution to the $\gamma Z$-box graph, and reduces the uncertainty by a factor of~2. In addition, we note that the presence of parity-violating photon-hadron interactions induces an additional contribution via $F_3^{\gamma \gamma}$. Using experimental and theoretical constraints on the nucleon anapole moment we are able to estimate the uncertainty associated with this contribution. We point out that future measurements are expected to significantly reduce this latter uncertainty.

Tables

Table 1. Table 1: Individual contributions to □ γ ​ Z A ​ ( E ) superscript subscript □ 𝛾 𝑍 𝐴 𝐸 \Box_{\gamma Z}^{A}(E) and associated uncertainties for the P2 and QWeak experiments in units of 10 − 4 superscript 10 4 10^{-4} . The star at the resonance contribution indicates that it is 100% anticorrelated with the Regge contribution, see the discussion of the uncertainties in the text.

Contribution	$E=155$ MeV	$E=1.165$ GeV
elastic	4.7(3)	0.56(6)
DIS	30.4(1)	30.4(1)
$N\pi$	0.6(2)	0.7(2)
resonanes	2.3(7)*	​1.0(3)*
Regge	6.6(2.1)	7.0(2.2)
Total	44.6(2.1)	39.7(2.2)

Equations

$$Q_W^p=-\underset{|t|\to 0}{\lim }{\frac{4\sqrt{2}\pi \alpha }{G_F|t|}\frac{d{\sigma }_{+}-d{\sigma }_{-}}{d{\sigma }_{+}+d{\sigma }_{-}}|}_{E=0}.$$

$$\frac{\Delta s_W^2}{s_W^2}\approx 0.08\frac{\Delta Q_W^p}{Q_W^p},$$

$$Q_W^p$$

$${\square }_{\gamma Z}(E)={\square }_{\gamma Z}^V(E)+{\square }_{\gamma Z}^A(E),$$

$${\square }_{\gamma Z}^A(E)$$

$$f(r,t^{\prime })=\frac{1}{t^{\prime 2}}\mathrm{Re}\left[\ln \left(1-\frac{t^{\prime 2}}{r^2}\right)+2t^{\prime }{\tanh }^{-1}\left(\frac{t^{\prime }}{r}\right)\right]$$

$$\frac{1}{4\pi }\int d^{4}x{e}^{iq\cdot x}⟨p(p)|[J_{em}^{\mu }(x),(J_Z^{\nu }(0){)}_A]|p(p)⟩$$

$$J_{em}^{\mu }=\sum _q{e}_q\bar{q}{\gamma }^{\mu }q,$$

$$(J_Z^{\mu }{)}_V=\sum _q{g}_V^q\bar{q}{\gamma }^{\mu }q,$$

$${\square }_{\gamma Z}^A(0)=\frac{3}{2\pi }{\int }_0^{\infty }\frac{d{Q}^2}{Q^2}\frac{v_e(Q^2)\alpha (Q^2)}{1+Q^2/M_Z^2}{M}_3^{\gamma Z}(1,Q^2),$$

$$M_3^{\gamma Z}(1,Q^2)=\frac{4}{3}{\int }_0^{1}dx\frac{2r-1}{r^2}{F}_3^{\gamma Z}(x,Q^2)$$

$$F_3^{\gamma Z}=F_{3,\mathrm{el}}^{\gamma Z}+F_{3,\mathrm{inel}}^{\gamma Z},$$

$$F_{3,\mathrm{inel}}^{\gamma Z}=\{\begin{array}{ll}{F}_{3,\pi N}^{\gamma Z}+F_{3,\text{res}}^{\gamma Z}+F_{3,\mathbb{R}}^{\gamma Z}& (Q^2<2{\text{ GeV}}^2)\\ F_{3,\mathrm{DIS}}^{\gamma Z}& (Q^2>2{\text{ GeV}}^2)\end{array}$$

$${\square }_{\gamma Z}^A(E)=\sum _i{\square }_{\gamma Z}^{A,i}(E),$$

$$F_{3,\mathrm{el}}^{\gamma Z}(x,Q^2)=-G_M^p(Q^2)G_A(Q^2)\delta (1-x).$$

$${\int }_0^{1}dx{F}_{3,\mathrm{DIS}}^{\gamma Z}(x,Q^2)=$$

$$\frac{5}{3}\left[1-\frac{{\alpha }_s}{\pi }-c_2\frac{{\alpha }_s^2}{{\pi }^2}-c_3\frac{{\alpha }_s^3}{{\pi }^3}-c_4\frac{{\alpha }_s^4}{{\pi }^4}\right]-$$

$$c_2$$

$$c_3$$

$$c_4$$

$$s_3$$

$$s_4$$

$$J_{\text{em}}^{\mu }=J_{\text{em}}^{\mu (0)}+J_{\text{em}}^{\mu (1)}.$$

$$F_3^{\gamma Z}=\frac{1}{2}{F}_3^{\nu p+\bar{\nu }p}+F_{3(0)}^{\gamma Z}.$$

$$F_{3,\mathbb{R}(0)}^{\gamma Z}=\frac{f_{th}}{18}{C}_{WW}\frac{m_{\omega }^2}{m_{\omega }^2+Q^2}\frac{m_{a_1}^2}{m_{a_1}^2+Q^2}{\left(\frac{\nu }{{\nu }_0}\right)}^{{\alpha }_0}$$

$$F_{3,\mathbb{R}(1)}^{\gamma Z}=\frac{f_{th}}{2}{C}_{WW}\frac{m_{\rho }^2}{m_{\rho }^2+Q^2}\frac{m_{a_1}^2}{m_{a_1}^2+Q^2}{\left(\frac{\nu }{{\nu }_0}\right)}^{{\alpha }_0}$$

$$=\frac{1}{2}{F}_{3,\mathbb{R}}^{\nu p+\bar{\nu }p},$$

$$f_{th}=\Theta \left(W^2-W_{th}^2\right)\left(1-e^{(W_{th}^2-W^2)/{\Lambda }_{th}^2}\right),$$

$${\mathcal{L}}_{a_1\gamma }=\frac{e}{2}{g}_{a_1\gamma }{F}_{\mu \nu }{a}_1^{\mu \nu },$$

$$⟨N|(J_Z^{\mu }{)}_A|N⟩=g_A{\bar{u}}_N{\gamma }^{\mu }{\gamma }_5{\tau }_3{u}_N,$$

$$F_{3,\mathrm{el}}^{\gamma \gamma }=\frac{\delta g_A^{ep}}{g_A}{F}_{3,\mathrm{el}}^{\gamma Z}=-(0.20\pm 0.34)F_{3,\mathrm{el}}^{\gamma Z},$$

$$F_{3,\mathbb{R}}^{\gamma \gamma }=\frac{\delta g_A^{ep}}{g_A}{F}_{3,\mathbb{R}}^{\gamma Z},$$

$${\square }_{\gamma \gamma }^{PV}$$

$${\square }_{\gamma Z}^A=(44.6\pm 2.1)\times 10^{-4},$$

Full text

I Introduction

The precision measurement of $s_{W}^{2}\equiv\sin^{2}\theta_{W}$, where $\theta_{W}$ is the Standard Model (SM) weak mixing angle, in parity-violating (PV) electron scattering serves as a powerful tool to test the SM and search for physics beyond it. Since the energy dependence of $s_{W}^{2}$ is very precisely predicted within the SM Erler and Ferro-Hernández (2018), any significant deviation from it would be an indication of Beyond Standard Model (BSM) physics. Polarized elastic $ep$ scattering in the limit of vanishing beam energy $E$ and momentum transfer $t$, probes the so-called weak charge of the proton,

[TABLE]

Here, $\alpha$ is the fine structure constant and $G_{F}$ the Fermi constant. $d\sigma_{\pm}$ are the differential cross sections for scattering with right-handed and left-handed polarized electrons, respectively. At tree-level this quantity is given by $1-4s_{W}^{2}$ which is accidentally suppressed. This effectively leads to an enhancement in the sensitivity to $s_{W}^{2}$,

[TABLE]

which also implies an enhanced sensitivity to BSM effects that may enter the running of $s_{W}^{2}$. As a concrete example, the upcoming P2 experiment at the Mainz Energy-Recovering Superconducting Accelerator (MESA), that plans for a measurement of the proton weak charge to 1.4% with a beam energy of $E=155$ MeV will lead to a determination of $s_{W}^{2}$ to 0.15% precision Becker et al. (2018).

Due to its accidental suppression at tree level, one needs to carefully account for all the SM higher-order effects to $Q_{W}^{p}$ in order to properly translate the high-precision experimental measurements into a determination of the weak mixing angle and constraints on BSM parameters. Including one-loop electroweak radiative corrections, the proton weak charge reads Erler et al. (2003),

[TABLE]

Among them, the quantity $\Box_{\gamma Z}$ which denotes the contribution from the $\gamma Z$ box diagram (see Fig. 1) contains sensitivity to physics at the hadronic scale where perturbative calculations are unreliable, inducing a large theoretical uncertainty. It is in general a function of $E$, and can be split into two terms,

[TABLE]

where the superscript $V(A)$ denotes the contribution from the vector (axial-vector) weak neutral current on the hadron side. The axial box is non-zero at $E=0$, but suppressed by the small electron weak vector coupling $v_{e}=-(1-4s_{W}^{2})$. On the other hand, the vector box is not suppressed by any small coefficient, but is exactly zero at $E=0$.

Earlier studies of $\Box_{\gamma Z}$ Marciano and Sirlin (1983, 1984); Bardin et al. (2001) assume small $E$ and hence are sensitive only to the axial box. However, a re-analysis in 2009 based on a dispersion relation Gorchtein and Horowitz (2009) revealed a steep energy-dependence in $\Box_{\gamma Z}^{V}(E)$ which was not previously accounted for. This finding stimulated a large number of follow-up studies on the vector $\gamma Z$ box Sibirtsev et al. (2010); Rislow and Carlson (2011); Gorchtein et al. (2011); Hall et al. (2013); Gorchtein and Zhang (2015); Hall et al. (2016); Gorchtein et al. (2016) as the latter was found to play a dominant role in the extraction of the proton weak charge in the Qweak experiment which took data at $E=1.165$ GeV Androić et al. (2018). These studies consistently concluded that the theoretical uncertainty in $\Box_{\gamma Z}^{V}(E)$ increases with $E$, which also motivated future measurements of $Q_{W}^{p}$ (specifically, the P2 experiment) to be performed with lower beam energy.

Unlike its vector counterpart, there are much fewer follow-up theoretical studies of $\Box_{\gamma Z}^{A}(E)$ Blunden et al. (2011, 2012); Rislow and Carlson (2013); Gorchtein and Zhang (2015), despite its becoming increasingly important at small $E$. In particular, no attempt was made so far to relate the contributions from multi-hadron intermediate states at small momentum transfers below about 1 GeV2 to experimental data, and all existing studies of these contributions are based on ad hoc models where a systematic uncertainty analysis is difficult or impossible.

In this paper we present an updated study of the axial $\gamma Z$ box by adopting a Regge parameterization of the multi-hadron contributions with parameters fitted to inclusive $\nu p/\bar{\nu}p$ scattering data. This technique was recently introduced in the treatment of the axial $\gamma W$ box diagram in neutron and nuclear $\beta$-decay Seng et al. (2018, 2019). We will show that by employing the neutrino data one is able to further reduce the uncertainty in $\Box_{\gamma Z}^{A}$ quoted in Ref. Blunden et al. (2011) by a factor of 2. Within the same theoretical framework, we further discuss a contribution which arises from the parity-odd component in the $\gamma\gamma$ box. It possesses the same theoretical structure as $\Box_{\gamma Z}^{A}$ and is therefore inseparable from the latter. Using the current constraints on the nucleon anapole moment, we provide an estimate on the additional theory uncertainty induced by this term.

II Dispersive representation of $\Box_{\gamma Z}^{A}$

We start from the dispersive representation of $\Box_{\gamma Z}^{A}$ derived in Ref. Blunden et al. (2011) for the case of electron scattering in the forward direction,

[TABLE]

where

[TABLE]

with $x=Q^{2}/(W^{2}-M^{2}+Q^{2})$, $r=1+\sqrt{1+4M^{2}x^{2}/Q^{2}}$, $t^{\prime}=4MEx/Q^{2}$, and $M$ the nucleon mass. We will take into account that in the $\overline{\mathrm{MS}}$ scheme both $s_{W}^{2}$ and $\alpha$ are running, i.e., scale dependent parameters. We calculate them as described in Refs. Erler and Ferro-Hernández (2018); Erler (1999), respectively. The spin-independent, parity-odd structure function $F_{3}^{\gamma Z}$ is defined through the hadronic tensor,

[TABLE]

where $p$ is the 4-momentum of the incoming proton, $q$ the 4-momentum transfer with $q^{2}=-Q^{2}$, and $\varepsilon^{0123}=-1$. The electromagnetic and weak neutral currents read,

[TABLE]

with $e_{u}=2/3$, $e_{d}=e_{s}=-1/3$, $g_{V}^{q}=2(I_{3,L}^{q}-2e_{q}s_{W}^{2})$, and $g_{A}^{q}=-2I_{3,L}^{q}$. In particular, when $E=0$ one finds the simplification Seng et al. (2019),

[TABLE]

where

[TABLE]

is the first Nachtmann moment of $F_{3}^{\gamma Z}$ Nachtmann (1973, 1974).

It is important to identify the dominant contributions to the structure function $F_{3}^{\gamma Z}$ at different $Q^{2}$ and $W^{2}=(p+q)^{2}$, as we summarize in Fig. 2. One can write,

[TABLE]

separating the elastic (sometimes called Born) contribution that represents an isolated peak at $W^{2}=M^{2}$, and all the inelastic contributions that start to emerge above the single-pion production threshold $W^{2}=(M+m_{\pi})^{2}$. The latter can be further subdivided in $Q^{2}$:

[TABLE]

First, above $Q^{2}\simeq 2$ GeV2, in the deep-inelastic scattering (DIS) regime, the structure functions are well-described by the parton model including corrections from perturbative quantum chromodynamics (pQCD). This is corroborated by the observation that the pQCD-corrected Gross-Llewellyn-Smith (GLS) sum rule Gross and Llewellyn Smith (1969) is well satisfied at $Q^{2}>2$ GeV2 in $\nu p/\bar{\nu}p$ scattering experiments Kataev and Sidorov (1994); Kim et al. (1998).

In contrast, at lower $Q^{2}$ the effective degrees of freedom are hadrons. It was pointed out recently that nonperturbative contributions can in principle be obtained from lattice QCD Seng and Meissner (2019), but before such calculations are carried out and the respective uncertainties are well understood, we have to rely on a model constrained by the experimental input.

The lowest inelastic hadronic state, the $N\pi$ continuum, starts contributing above its production threshold. In the range $1.5\leavevmode\nobreak\ \mathrm{GeV}^{2}<W^{2}<5\leavevmode\nobreak\ \mathrm{GeV}^{2}$ one observes nucleon resonances. Above the two-pion threshold at $(M+2m_{\pi})^{2}$ multi-hadron states emerge which become dominant at large $W^{2}$. The Regge exchange picture provides an economical description of these contributions which we smoothly continue down to the two-pion threshold, as described in the following. We furthermore use the Vector Meson Dominance (VMD) picture to extend the Regge description to moderate values of $Q^{2}$. We label this contribution by the subscript “$\mathbb{R}$”. Correspondingly, we obtain the $\gamma Z$-box correction as a sum of five contributions,

[TABLE]

with $i=$ el, $\pi N$, res, $\mathbb{R}$, and DIS. Each contribution is computed by inserting the corresponding $F_{3,i}^{\gamma Z}$ into Eq. (5) and integrating over its support in $W$ and $Q^{2}$.

The elastic contribution is given in terms of the proton magnetic Sachs form factor $G_{M}^{p}$ and the axial form factor $G_{A}$,

[TABLE]

In the numerical evaluation of this part we adopt recent data on these form factors as given in Refs. Ye et al. (2018); Bhattacharya et al. (2011).

The DIS region contribution to the box diagram can be represented as an infinite sum of odd Mellin moments of $F_{3}^{\gamma Z}$ upon expanding the function $f(r,t^{\prime})$ in Eqs. (5,6) in powers of $x^{2}M^{2}/Q^{2}$ Blunden et al. (2011). None of the even Mellin moments of $F_{3}^{\gamma Z}$ appear due to the symmetry of the integrands in $x$. The result is almost $E$-independent. Furthermore, the size of contributions from the third and higher moments to $\Box_{\gamma Z}^{A}$ is about $10^{-5}$. It is two orders of magnitude below that from the first Mellin moment and has little impact on the final result. The first Mellin moment is independent of the details of the parton distribution functions and its pQCD correction has previously been considered up to $\mathcal{O}(\alpha_{s})$. For completeness, we include here the full $\mathcal{O}(\alpha_{s}^{4})$ expression Baikov et al. (2010a, b),

[TABLE]

The first term on the right hand side Baikov et al. (2010a) represents the iso-singlet piece satisfying the polarized Bjorken sum rule Bjorken (1966). It receives only contributions from non-singlet (connected) diagrams. The second term Baikov et al. (2010b) contributes only to the iso-triplet piece by singlet (disconnected) diagrams and enters the GLS Gross and Llewellyn Smith (1969) sum rule. The coefficients in numerical form are given by Baikov et al. (2010a, b),

[TABLE]

In the numerical evaluation, we use the running strong coupling constant $\alpha_{s}$ provided by the Mathematica code RunDec Chetyrkin et al. (2000) with $\alpha_{s}(M_{Z})=0.118$, and values for the number of flavors, $n_{f}$, changing by one unit at quark mass thresholds as implemented in RunDec. Integrating this expression with $Q^{2}_{\text{min}}=2$ GeV2 in Eq. (5) results in a contribution of $30.4\times 10^{-4}$ to $\Box_{\gamma Z}^{A}$. We note that our $\mathcal{O}(\alpha_{s}^{4})$ result differs from the $\mathcal{O}(\alpha_{s})$ result by only about 2%. We have also checked that using $Q^{2}_{\text{min}}=$1 GeV2 and the $\mathcal{O}(\alpha_{s})$ expression we obtain $32.9\times 10^{-4}$ in good agreement with Ref. Blunden et al. (2011).

Higher twist (HT) effects represent a source of the possible uncertainty in the lower $Q^{2}$ part of the DIS region, 2 GeV${}^{2}<Q^{2}<5$ GeV2. Since we have experimental data to compare with the pQCD prediction, we assume that the uncertainty due to HT contributions should not exceed that of the data, which is $\sim$5%. This uncertainty would be a systematical one, and we will use as the estimate 5% of the integral of the DIS contribution over this $Q^{2}$-range. Numerically, it amounts to 1.3$\times 10^{-5}$ which we include in the total error.

The $N\pi$ contribution is calculated within the framework of chiral perturbation theory, but with pointlike electroweak vertex couplings replaced by the Dirac and axial nucleon form factors, which suppresses their high-$Q^{2}$ contribution. Details of this calculation are described in Ref. Seng et al. (2019). We assign a generous 30% uncertainty due to higher chiral orders. This uncertainty, however, does not affect the total error budget.

As for the resonances, our treatment is the same as in Ref. Blunden et al. (2011). The only numerically relevant contribution is due to the $\Delta$-resonance, which we calculate using the parameterization in Ref. Lalakulich and Paschos (2005). As there are no assigned uncertainties for the values of the parameters, it is not possible to provide an error estimate for the $\Delta$-contribution to $\Box_{\gamma Z}^{A}$. This contribution is small, at the level of the overall uncertainty, and even a conservative 30% uncertainty hardly modifies the total error. Moreover, we note here that the resonance and multi-hadron contributions are anti-correlated, as it is their sum that is constrained by measured experimental cross sections. A significant increase in one will have to be compensated for by a reduction in the other, so that the overall effect is smaller than if this anti-correlation was naïvely neglected. We therefore opt to exclude the resonance uncertainty from the total error.

The treatment of multi-hadron intermediate states requires more care. Following Refs. Seng et al. (2018, 2019), we establish the correspondence between $F_{3}^{\gamma Z}$ and its charged current counterpart $F_{3}^{\nu p+\bar{\nu}p}$ by isospin symmetry. The electromagnetic current can be decomposed into iso-singlet and iso-triplet components,

[TABLE]

Likewise, we can also separate the structure function $F_{3}^{\gamma Z}$ into iso-singlet and iso-triplet parts, $F_{3}^{\gamma Z}=F_{3(0)}^{\gamma Z}+F_{3(1)}^{\gamma Z}$, where the axial part of the $Z$ current is purely iso-vector. Upon neglecting strange quarks, we find through an isospin rotation that the $(I=1)\otimes(I=1)$ isospin channel, $F_{3(1)}^{\gamma Z}$, equals half of the structure function $F_{3}^{\nu p+\bar{\nu}p}$ that accounts for the difference of the cross sections for inclusive $\nu p$ and $\bar{\nu}p$ scattering Onengut et al. (2006). We thus obtain the model-independent expression,

[TABLE]

There are two advantages in doing so, as (a) data exist for $F_{3}^{\nu p+\bar{\nu}p}$ at moderate $Q^{2}$ where a first-principle theory description is not available, and (b) $J_{\text{em}}^{\mu}$ is predominantly iso-vector, so that the contribution from $F_{3(0)}^{\gamma Z}$ is small. Therefore, utilizing neutrino data, one obtains a model-independent determination of the dominant iso-triplet piece, limiting the model dependence to the small iso-singlet piece. Apart from the use of isospin symmetry, our treatment differs from that in Ref. Blunden et al. (2011) in that we account for continuous backgrounds already starting at the two-pion threshold $W^{2}=(M+2m_{\pi})^{2}$.

Our parameterization is based on the leading Regge trajectory exchange picture introduced in Refs. Seng et al. (2018, 2019) where the reader can find more details. The respective diagrams are shown in Fig. 3. In this picture and making use of the nearly exact degeneracy between the $\omega$ and $\rho$ mesons, the iso-singlet and iso-vector pieces are related to each other and to their charged current counterpart in a simple way,

[TABLE]

with $\alpha_{0}\approx 0.477$ the intercept of the $\omega/\rho$ trajectory, $\nu=Q^{2}/2Mx$, and $\nu_{0}=1$ GeV a typical hadronic scale. The threshold function that suppresses the small-$W^{2}$ contribution was chosen in the form,

[TABLE]

where $\Lambda_{th}=1$ GeV, and $W_{th}^{2}=(M+2m_{\pi})^{2}$ is the two-pion threshold. As in Ref. Seng et al. (2018), we assume that it is sufficient to take $C_{WW}=C_{WW}(Q^{2})$ as linear, $C_{WW}=A_{WW}(1+B_{WW}Q^{2})$. The parameter values $A_{WW}=5.2\pm 1.5$ and $B_{WW}=1.08^{+0.48}_{-0.28}$ GeV*-2* Seng et al. (2018) have been obtained from a fit of the first Nachtmann moment of $F_{3}^{\nu p+\bar{\nu}p}$ using data from CCFR Kataev and Sidorov (1994); Kim et al. (1998), BEBC/Gargamelle Bolognese et al. (1983) and WA25 Allasia et al. (1985), after subtracting the elastic, $N\pi$, and resonance pieces. Using Eqs. (19) and (20) in Eq. (5) gives the Regge contribution to $\Box_{\gamma Z}^{A}$.

Our results for the $\Box_{\gamma Z}^{A}$ are shown as function of energy in Fig. 4. The upper panel displays all non-DIS contributions separately, while the lower panel shows their sum, the DIS contribution, the total, as well as a comparison with the previous evaluation of Ref. Blunden et al. (2011). A significant overlap of the uncertainty bands indicates an excellent agreement between the two calculations. In our analysis the uncertainty is reduced by a factor of almost 2.

We visualize the source of this uncertainty reduction in Fig. 5 where we display our Regge contribution to the first Nachtmann moment $M_{3}^{(1)}(Q^{2})/Q^{2}$ in comparison with the result of the approach of Ref. Blunden et al. (2011). That Ref. assumed no uncertainty beyond $Q^{2}=1$ GeV2 where the dashed red curve is matched to DIS. Instead, we match the Regge parametrization to DIS at $Q^{2}=2$ GeV2. The total uncertainty of the $\Box_{\gamma Z}^{A}$ associated with each approach is obtained by integrating the respective uncertainty band in the displayed range. We see that the use of data to determine the uncertainty of the dispersive calculation allows one to halve the associated theoretical error. The observed close agreement between the solid blue and dashed red curves in the range 1 GeV${}^{2}<Q^{2}<2$ GeV2 demonstrates that there is almost no sensitivity to the exact matching point in this range, and the respective error is contained in the shaded blue band.

In Tab. 1 we list the individual contributions for the kinematics of the two relevant experiments, P2 at MESA with a beam energy of $E=155$ MeV, and QWeak at JLAB with $E=1.165$ GeV. Our updated analysis provides predictions for the axial $\gamma Z$ box with an uncertainty at the level of 5%. For completeness, we quote here the previous evaluation by Ref. Blunden et al. (2011): $\Box_{\gamma Z}^{A}(E=0)=44(4)\times 10^{-4}$ and $\Box_{\gamma Z}^{A}(E=1.165\,{\rm GeV})=37(4)\times 10^{-4}$.

III Parity-violating photon-hadron interaction

In the previous sections we computed the hadronic structure-dependent one-loop corrections due to the exchange of a $\gamma$ and a $Z$ boson between the electron and the proton. This calculation requires information on the PV structure function $F_{3}^{\gamma Z}$ which we obtained by relating it to its charged current partner $F_{3}^{WW}$ by isospin symmetry. Since isospin breaking effects can be expected to be small, the uncertainty in this calculation is dominantly experimental. Together with $\Box_{\gamma Z}^{V}$ Gorchtein and Horowitz (2009); Sibirtsev et al. (2010); Rislow and Carlson (2011); Gorchtein et al. (2011); Hall et al. (2013); Gorchtein and Zhang (2015); Hall et al. (2016); Gorchtein et al. (2016) and other one-loop corrections Erler and Ferro-Hernández (2018) this completes the one-loop analysis for the parity-violating part of the cross section.

To go beyond this result, one will need to perform challenging calculations of two-loop effects. The situation is expected to be particularly complicated when addressing non-perturbative contributions. It is not evident how a complete two-loop calculation at the hadronic level will be viable since it will involve the analysis of time-ordered products of three currents. While these hadronic effects are not enhanced by large electroweak logarithms, they may still be of importance at the $10^{-3}-10^{-4}$ level which is the goal of the present analysis. Since it may not be possible to directly calculate these corrections, we aim at estimating the uncertainty which they may induce, together with the shift in the central value of $Q_{W}^{p}$.

Of particular interest is a subclass of two-loop hadronic corrections associated with the exchange of two bosons between the electron and proton, and another boson exchanged within the hadronic state. Since the loop integration is dominated by momenta of a typical hadronic scale, $\ell\lesssim\Lambda_{h}\sim 1$ GeV, one can conclude that the exchange of two heavy bosons, $W$ or $Z$, will lead to corrections of ${\cal O}(\alpha G_{F}\Lambda_{h}^{2})$ which are negligible. Furthermore, purely electromagnetic effects, i.e., diagrams with the exchange of photons only, cannot lead to a PV signature. Thus, only diagrams with at least one $Z$ boson need to be considered. Among these, the only significantly new contribution arises from the exchange of two photons between the electron and the hadronic system, with parity violation within the latter. Such PV effects may arise due to mixing of hadronic states of equal spin and opposite parity. The parity-odd effect in $2\gamma$-exchange thus induces a parity-odd structure function $F_{3}^{\gamma\gamma}$ in the $\gamma\gamma$ box. Its effect should be added to $\Box_{\gamma Z}^{A}$ as it is indistinguishable experimentally.

In order to estimate this effect we use the picture in which gauge bosons mix with vector and axial-vector mesons. The photon mixes mostly with the vector mesons $\rho$, $\omega$, and $\phi$. Hadronic PV interactions can induce $a_{1}$–$\rho$ mixing, effectively leading to $a_{1}$–$\gamma$ mixing (left panel of Fig. 6). Similarly, PV state mixing in the iso-scalar channel, such as $h_{1}$–$\omega$, gives rise to $h_{1}$–$\gamma$ mixing. But due to iso-vector dominance its numerical impact should be marginal (at the 10% level) and we disregard it. The Lagrangian describing the PV $a_{1}$–$\gamma$ interaction can be written as

[TABLE]

where $F^{\mu\nu}$ and $a_{1}^{\mu\nu}$ are the field strength tensors of the photon and the $a_{1}$, respectively, and $g_{a_{1}\gamma}$ is the PV coupling constant.

To study the elastic component of $F_{3}^{\gamma\gamma}$, we first recall that (neglecting strangeness and the iso-scalar axial coupling) the nucleon matrix element of the axial weak neutral current takes the form,

[TABLE]

where $g_{A}=-1.27641(56)$ has been measured precisely in neutron beta decay Märkisch et al. (2019); Brown et al. (2018). The inclusion of the so-called nucleon anapole moment effectively shifts the apparent value of the nucleon axial charge seen in PV electron scattering with respect to that in reactions involving charged current interactions, $g_{A}\rightarrow g_{A}+\delta g_{A}^{ep}$. Calculations based on hadronic parity violation in the framework of $SU(3)$ chiral perturbation theory Kaplan and Savage (1993), yield the result $\delta g_{A}^{ep}=0.26(43)$ Zhu et al. (2000). On the other hand, a global fit to PV electron scattering data not using theory constraints returns an even higher value, $\delta g_{A}^{ep}=0.66(63)$ Liu et al. (2007); González-Jiménez et al. (2014). These results are consistent with each other, as well as with zero, but have large uncertainties impacting the error of the parity-violating asymmetry. In the picture where the anapole moment arises due to the effective $a_{1}$–$\gamma$ mixing, as depicted in the left diagram of Fig. 6, the mixing strength $g_{a_{1}\gamma}$ is directly related to the value of $\delta g_{A}^{ep}$. The elastic contribution to $F_{3}^{\gamma\gamma}$ is then obtained as Gorchtein and Spiesberger (2016),

[TABLE]

where the numerical estimate is based on the theory result Zhu et al. (2000).

Here we note an analogous relation for the correction to the Regge contribution discussed in the previous section,

[TABLE]

which can be shown straightforwardly. This corresponds to a parity-odd effect which can be visualized by the right diagram of Fig. 6, where a photon mixes with an $a_{1}$ through Eq. (22), and then interacts with the nucleon through the exchange of a $\rho/\omega$ trajectory. Combining these two effects, we find an effective shift of $\Box_{\gamma Z}^{A}$ at P2 energies given by,

[TABLE]

One should keep in mind that further hadronic contributions not related to the anapole moment are possible, but their impact on the central value and the respective uncertainty is well below the target accuracy Gorchtein and Spiesberger (2016).

IV Summary and conclusions

We provided a thorough update of the $\Box_{\gamma Z}^{A}$ correction to PV electron-proton scattering. This calculation entails gathering all available information on the interference PV structure function $F_{3}^{\gamma Z}$ and its first Nachtmann moment over the full range of $Q^{2}$. The limiting cases of low and high $Q^{2}$ are governed by the elastic and DIS contributions, respectively. While these two contributions are known with good precision, the interpolation between them requires modelling inclusive hadronic contributions in the non-perturbative regime. This interpolation is the source of the uncertainty of the calculation. Luckily, the contributions from the intermediate $Q^{2}$ range are rather small, making the resulting model dependence not critical. In the past, the interpolation was performed by an essentially ad hoc procedure. In this work we invoke the isospin symmetry that is known to hold to a good extent, and relate the neutral current interference structure function $F_{3}^{\gamma Z}$ to its charged current counterpart $F_{3}^{WW}$ for which experimental data from inclusive neutrino and antineutrino scattering are available. Even though these data are not very precise, this procedure allowed us to better constrain the interpolation between the low and high $Q^{2}$ regimes, leading to a factor 2 reduction in the resulting uncertainty. In particular, for the P2 beam energy of $E=155$ MeV we obtain from Table 1,

[TABLE]

which is in good agreement with Ref. Blunden et al. (2011) but with the uncertainty reduced by a factor of about two. Including our result for $\gamma\gamma$ exchange with hadronic PV results in the total correction,

[TABLE]

This uncertainty is significantly larger than the one in Eq. (27). We note, however, that the P2 experiment Becker et al. (2018) will in any case aim at reducing the uncertainty of the proton’s anapole moment $\delta g_{A}^{ep}$ by a factor of four via a dedicated backward angle measurement. The relevant uncertainty will then be the one in Eq. (27), while the small remaining one due to the induced PV photon-hadron interaction is 100% correlated with, and needs to be added linearly to, the tree-level $g_{A}$ effect in the PV asymmetry.

Our calculation of the hadronic structure-dependent one-loop corrections due to the exchange of a $\gamma$ and a $Z$ boson between the electron and the proton has to be combined with other already known one-loop corrections Erler and Ferro-Hernández (2018); Gorchtein and Horowitz (2009); Sibirtsev et al. (2010); Rislow and Carlson (2011); Gorchtein et al. (2011); Hall et al. (2013); Gorchtein and Zhang (2015); Hall et al. (2016); Gorchtein et al. (2016). This completes the one-loop analysis for the parity-violating part of the cross section.

Until a complete two-loop calculation is performed, one should try to identify the leading two-loop effects. We have done this for hadronic PV effects entering through $\gamma\gamma$ exchange. Our result shows that two-loop contributions to the PV asymmetry in electron proton scattering are numerically relevant. It seems less important now to work on further improvements of the one-loop results; instead theoretical efforts should shift to the calculation of two-loop effects.

Acknowledgements: This work was supported by the German-Mexican research collaboration Grant No. 278017 (CONACyT) and No. SP 778/4-1 (DFG). CYS is supported in part by the DFG (Grant No. TRR110) and the NSFC (Grant No. 11621131001) through the funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD”, and also by the Alexander von Humboldt Foundation through a Humboldt Research Fellowship. MG acknowledges support by Helmholtz Institute Mainz. JE acknowledges support by PASPA (DGAPA–UNAM) and CONACyT Project No. 252167–F, and is grateful for hospitality and support by the excellence cluster PRISMA+ at JGU Mainz, as well as the Helmholtz-Institute Mainz.