Nucleon spin-averaged forward virtual Compton tensor at large $Q^2$

The nucleon spin-averaged forward virtual Compton tensor determines important physical quantities such as electromagnetically induced mass differences of nucleons and two-photon exchange contributions in hydrogen spectroscopy. It depends on two kinematic variables: $\nu$, the virtual photon energy in the proton rest frame, and $Q^2$, the photon’s invariant four-momentum squared. Using the operator product expansion, we calculate the tensor’s large-$Q^2$ behavior for $\nu =0$, including for the first time the full spin-2 contribution and correcting a previous result in the literature for the spin-0 contribution. Implications for the proton radius puzzle are discussed.

Figure 1

Forward scattering of virtual photon of momentum $q^{\mu }$ on a hadron of momentum $k^{\mu }$.

Figure 2

Quark matching condition. The right-hand side denotes a sum over operators in Eq. (7).

Figure 3

Matching condition onto gluon operators. The three diagrams on the left-hand side denote the contributions $T_a^{\mu \nu }$, $T_b^{\mu \nu }$, and $T_c^{\mu \nu }$ in Eq. (10). The right-hand side denotes a sum over operators in Eq. (7). Permutations of external states are not displayed.

Figure 4

Spin-0 contribution to $Q^2{W}_1(0,Q^2)/2m_p^2$ as a function of $Q^2$. The final results are given with $n_f=4$ in the left-hand plot. For comparison, results with $n_f=3$ are in the right-hand plot. The black band with vertical stripes displays the result with perturbative uncertainty estimated by renormalization scale variation $Q/2<\mu <2Q$. The solid bands represent hadronic uncertainty from local matrix elements in Table 1. In the $n_f=4$ evaluation, dark and light solid bands are obtained using $f_c^{(0)}$ (pQCD) and $f_c^{(0)}$ (lattice), respectively. The bottom dashed blue line and top dash-dotted red line give the separate quark and gluon contributions to the central value.

Figure 5

Spin-2 contribution to $Q^2{W}_1(0,Q^2)/2m_p^2$ as a function of $Q^2$. The dark gray band represents hadronic uncertainty. The vertically hatched band represents perturbative uncertainty, estimated by varying renormalization scale $Q/2<\mu <2Q$. Upper dash-dotted and lower dashed blue lines represent the contributions of the up quark and down quark to the central value, and the lower red hatched band shows the gluon contribution and uncertainty.

Figure 6

Total $Q^2{W}_1(0,Q^2)/2m_p^2$ as a function of $Q^2$. The dashed red and blue lines denote central values for the spin-0 and spin-2 contributions. The vertical hatched black band is the total contribution, with perturbative and hadronic errors from each of the spin-0 and spin-2 contributions added in quadrature.

Figure 7

$W_1(0,Q^2)$ as a function of $Q^2$. Solid black curves on the left and right are from Eqs. (33) and (35), as explained in the text. Intermediate red curves are interpolations from the central curves and envelopes.

Figure 8

$W_1(0,Q^2)$, after subtracting the $Q^2=0$ limit and scaling by the kinematic factor $k(Q)$ in Eq. (38), as a function of $Q$. The contribution to the muonic hydrogen Lamb shift is proportional to the area under the curve, cf. Eq. (37).

Figure 9

Two-photon contribution to the Lamb shift in muonic hydrogen, adjusted to common proton elastic form factors (see the text). Black circles denote previous work, and the blue square denotes the present work. The red triangle is the summary of Ref. [4] used in the CREMA muonic hydrogen extraction of $r_E^p$.

Figure 10

Different calculations of $W_1(0,Q^2)-W_1(0,Q^2{)}^{\mathrm{SIFF}}$. The black solid line, black dashed line, and solid band, give the “third-order” (${\overline{T}}_1^{(3)}$), “fourth-order” (${\overline{T}}_1^{(3)}+{\overline{T}}_1^{(4)}$), and “fourth-order plus $\mathrm{\Delta }$” (${\overline{T}}_1^{(3)}+{\overline{T}}_1^{(4)}+{\overline{T}}_1^{(\mathrm{\Delta })}$) results of Ref. [3], using the parameters of Ref. [3] and their prescription for determining $\delta \beta$. The solid band adds in quadrature the errors from $c_1,c_2,c_3,\beta$, and $g_{\pi \mathrm{N}\mathrm{\Delta }}$. The dash-dotted black line gives the fourth-order plus $\mathrm{\Delta }$ contribution (${\overline{T}}_1^{(3)}+{\overline{T}}_1^{(4)}+{\overline{T}}_1^{(\mathrm{\Delta })}$), Taylor expanded through $\mathcal{O}(Q^4)$. The bottom blue solid line gives the “third-order” [$T_1^{(\mathrm{NB})}(0,Q^2)$] result of Ref. [38], using its parameters.

Figure 11

Same as Fig. 8, but with $W_1(0,Q^2)$ at low $Q^2$ taken as $W_1(0,Q^2{)}^{\mathrm{SIFF}}$ with dipole nucleon form factors and deviation from $W_1(0,Q^2{)}^{\mathrm{SIFF}}$ given by the band in Fig. 10.