Parton distributions in radiative corrections to the cross section of electron-proton scattering

Abstract

The structure function approach and the parton picture, developed for the theoretical description of the deep inelastic electron-proton scattering, also proved to be very effective for calculation of radiative corrections in Quantum Electrodynamics. We use them to calculate radiative corrections to the cross section of electron-proton scattering due to electron-photon interaction, in the experimental setup with the recoil proton detection, proposed by A. A. Vorobyev to measure the proton radius. In the one-loop approximation, explicit expressions for these corrections are obtained for arbitrary momentum transfers. It is shown that, at momentum transfers small compared with the proton mass, various contributions to the corrections mutually cancel each other with power accuracy. In two loops, the corrections are obtained in the leading logarithmic approximation.

Appendices

Appendix A

The vacuum polarisation $\mathcal{P}(q^2)$ contains lepton (electron, muon, $\tau$-lepton ) and hadron contributions:

$$\begin{aligned} \mathcal{P}(q^2) = \mathcal{P}_e(q^2) + \mathcal{P}_\mu (q^2) +\mathcal{P}_\tau (q^2) + \mathcal{P}_h(q^2). \end{aligned}$$

(A.1)

One-loop lepton contribution $\mathcal {P}^{(1)}_{l}(q^2), \;\;( l=e, \mu , \tau )$ is well known (see, for example, [33]):

$$\begin{aligned}&\mathcal{P}^{(1)}_{l}(q^2) = \frac{\alpha }{\pi }\left( \frac{1}{3}\,\sqrt{1-\frac{4\,m_{l}^2}{q^2}}\,\left( 1+ \frac{2\,m_{l}^2}{q^2}\right) \, \right. \nonumber \\&\left. \times \ \ln \left( \frac{\sqrt{1-\frac{4m_{l}^2}{q^2}}+1}{\sqrt{1-\frac{4m_{l}^2}{q^2}}-1}\right) -\frac{4\,m_{l}^2}{3\,q^2} - \frac{5}{9}\right) . \end{aligned}$$

(A.2)

At $Q^2 =-q^2 \gg 4m_l^2$

$$\begin{aligned} \mathcal{P}^{(1)}_{l}(q^2) =\frac{\alpha }{3\pi }\left( \ln \left( \frac{Q^2}{m_l^2}\right) - \frac{5}{3}\right) ~, \end{aligned}$$

(A.3)

and at $Q^2 =-q^2 \ll 4m_l^2$

$$\begin{aligned} \mathcal{P}_{l}(q^2) = \frac{\alpha }{15\pi }\frac{Q^2}{m_l^2}~. \end{aligned}$$

(A.4)

The lepton contributions are known also in higher orders of perturbation theory (see, for example, [39, 40]). For us it is enough to know that the two-loop contribution contains only the first degree of $\ln \left( \frac{Q^2}{m_l^2}\right)$.

The hadron contribution $\mathcal{P}_h(q^2)$ is expressed in terms of the total cross section of one-photon electron-positron pair annihilation into hadrons

$$\begin{aligned} \mathcal{P}_h(q^2) = \frac{q^2}{4\pi ^2\alpha }\int _{4m^2_{\pi }}^\infty ds\frac{\sigma _{e^+e^ \rightarrow hadrons}(s)}{s-q^2}~. \end{aligned}$$

(A.5)

This contribution is small compared with $\alpha /\pi$ at $Q^2 < 4m^2_{\pi }$, becomes of order of $\alpha /\pi$ only at $Q^2\sim 4m^2_{\pi }$ and then grows logarithmically with $Q^2$. Recent review is given in [40].

Appendix B

To find the contribution $\delta ^e_{hard}$ of the one-photon emission with $\kappa '>\kappa _0$ to the radiative correction $\delta ^e$ at $Q^2\gg m^2$ it is convenient to introduce the intermediate scale $\kappa _1$ such that $Q^2\gg \kappa _1\gg m^2$. In the region $\kappa _1>\kappa '>\kappa _0$ one can put

$$\begin{aligned} W_g&=-\frac{\alpha }{2\pi }\Bigg [ \frac{Q^2}{\kappa '} \left( \ln \left( \frac{Q^4}{m^2(m^2+2\kappa ')}\right) -2 \right) \nonumber \\&\qquad +\frac{Q^2\kappa '}{(m^2+2\kappa ')^2}\Bigg ]~, \end{aligned}$$

(B.1)

$$\begin{aligned} W_l&=0~, \end{aligned}$$

(B.2)

so that

$$\begin{aligned}&F_1= -\frac{1}{2} W_g~,&\quad&F_2=2F_1~. \end{aligned}$$

(B.3)

Therefore in this region we have from Eqs. (22)–(24)

$$\begin{aligned} \frac{d\sigma ^{\gamma }}{dQ^2 dx } = - W_g\frac{d\sigma _{B}}{dQ^2}~. \end{aligned}$$

(B.4)

Using that in this region it is possible to put $\kappa ' = Q^2(1-x)/2$, it is easy to obtain the part $\delta ^{(1)}_{hard}$ of the correction $\delta ^e_{hard}$, defined by Eqs. (21) and (95), from this region:

$$\begin{aligned} \delta ^{(1)}_{hard}&= \frac{\alpha }{\pi }\int _{\kappa _0}^{\kappa _1}\frac{d\kappa '}{\kappa '} \Bigg [2 \Big (\ln \left( \frac{Q^2}{m^2}\right) -1 \Big )\nonumber \\&\quad - \ln \left( \frac{m^2+2\kappa '}{m^2}\right) + \frac{\kappa '^2}{(m^2+2\kappa ')^2} \Bigg ]\nonumber \\&= \frac{\alpha }{\pi }\Bigg [2 \ln \left( \frac{\kappa _1}{\kappa _0}\right) \Big (\ln \left( \frac{Q^2}{m^2}\right) -1 \Big ) \nonumber \\&\qquad -\frac{1}{2}\ln ^2\left( \frac{2\kappa _1}{m^2}\right) + \frac{1}{4}\Big (\ln \left( \frac{2\kappa _1}{m^2}\right) -1\Big )\nonumber \\&\qquad -\frac{\pi ^2}{6}\Bigg ]~, \end{aligned}$$

(B.5)

Using (67), we obtain

$$\begin{aligned}&\delta ^{e}_{vert}+ \delta _{soft}+\delta ^{(1)}_{hard}\nonumber \\&=\frac{\alpha }{\pi }\Bigg [2 \ln \left( \frac{2\kappa _1}{m^2}\right) \Big (\ln \left( \frac{Q^2}{m^2}\right) -1 \Big ) \nonumber \\&\quad -\frac{3}{2}\ln ^2\left( \frac{Q^2}{m^2}\right) + \frac{5}{2}\ln \left( \frac{Q^2}{m^2}\right) -\frac{1}{2}\ln ^2\left( \frac{2\kappa _1}{m^2}\right) \nonumber \\&\quad + \frac{1}{4} \ln \left( \frac{2\kappa _1}{m^2}\right) -\frac{\pi ^2}{6} -\frac{5}{4}\Bigg ]~. \end{aligned}$$

(B.6)

In the region $\kappa _{max}>\kappa '>\kappa _1$, i.e. $1-\frac{2\kappa _1}{Q^2}>x>x_{-}$ at $Q^2\gg m^2$ one can put

$$\begin{aligned} W_g&= -\frac{\alpha }{2\pi }\left[ \frac{1+x^2}{1-x}\ln \left( \frac{Q^2}{m^2x(1-x)}\right) \right. \nonumber \\&\qquad \left. +\frac{1-8x}{2(1-x)} \right] ~, \end{aligned}$$

(B.7)

and

$$\begin{aligned} W_l=\frac{\alpha }{2\pi }\frac{Q^2}{4x}~, \end{aligned}$$

(B.8)

so that

$$\begin{aligned} F_1 = \frac{1}{2}\left( \frac{4x^2\;W_l}{Q^2}-W_g\right) ~, \end{aligned}$$

(B.9)

$$\begin{aligned} F_2 =x\left( \frac{12x^2}{Q^2}\;W_l-W_g\right) = 2x F_1 +\frac{\alpha }{\pi }x^2~. \end{aligned}$$

(B.10)

As it is seen, the Callan-Gross relation [35] is violated in this region. It could be expected, since this relation is valid only in the collinear approximation.

Using Eqs. (22)–(24), we have for the part $\delta ^{(2)}_{hard}$ of the correction $\delta ^e_{hard}$ from this region:

$$\begin{aligned} \delta ^{(2)}_{hard}&= \int _{x_-}^{1-2\frac{\kappa _1}{Q^2}}dx \bigg [\frac{F_2(x, Q^2)}{x} \nonumber \\&-\frac{Q^2}{2(pl)}\frac{Q^2G_M^2 +4M^2G_E^2}{(4M^2+Q^2)R(1, Q^2)}\frac{F_2(x, Q^2)(1-x)}{x^2}\nonumber \\&+\frac{Q^2}{8(pl)^2}\frac{Q^2(Q^2+2M^2)G_M^2 -8M^4G_E^2}{(4M^2+Q^2)R(1, Q^2)}\nonumber \\&\quad \times \frac{F_2(x, Q^2)(1-x^2)}{x^3}\nonumber \\&-\frac{Q^2}{8(pl)^2}\frac{(Q^2G_M^2 -2M^2G_E^2)}{R(1, Q^2)}\nonumber \\&\quad \times \frac{F_2(x, Q^2)-2xF_1(x, Q^2)}{x^3}\bigg ] ~. \end{aligned}$$

(B.11)

Note that in the integral (B.11) the upper limit can be set equal to 1 in all terms except the first one. It gives

$$\begin{aligned} \delta ^{(2)}_{hard}&= \frac{\alpha }{2\pi }\Biggl [ \left( 3\ln \left( \frac{Q^2}{m^2}\right) -4\ln \left( \frac{2\kappa _1}{m^2}\right) \right. \nonumber \\&\ \left. -5 + x_- +\frac{x_-^2}{2}+2\ln (1-x_-) \right) \ln \left( \frac{Q^2}{m^2}\right) \nonumber \\&+\ln ^2\left( \frac{2\kappa _1}{m^2}\right) +\frac{7}{2}\ln \left( \frac{2\kappa _1}{m^2(1-x_-)}\right) \nonumber \\&-\ln ^2(1-x_-) + \frac{(1-x_-)}{2}(3+x_-)\ln (1-x_-) \nonumber \\&+ 2\text {Li}_2(1-x_-) -\frac{x_-}{2}(2+x_-)\ln x_- +\frac{5}{2} \nonumber \\&-\frac{x_-}{2}(3+2x_-)\nonumber \\&+\frac{Q^2}{4(pl)}\frac{Q^2G_M^2 +4M^2G_E^2}{(4M^2+Q^2)R(1, Q^2)}\,\phi _1(x_-) \nonumber \\&+\frac{Q^2}{16(pl)^2}\frac{Q^2(Q^2+2M^2)G_M^2 -8M^4G_E^2}{(4M^2+Q^2)R(1,Q^2)}\phi _2(x_-) \nonumber \\&+\frac{Q^2}{4(pl)^2}\frac{Q^2G_M^2 -2M^2G_E^2}{R(1, Q^2)}\,\phi _3(x_-) \Bigg ]~, \end{aligned}$$

(B.12)

with $\phi _{i}$ defined in Eqs. (69)–(72).

The intermediate parameter $\kappa _1$ disappears in the sum $\delta _{hard}^e = \delta ^{(2)}_{hard}+ \delta ^{(1)}_{hard}$:

$$\begin{aligned} \delta ^{e}_{hard}&= \frac{\alpha }{2\pi }\Bigg [ \ln \left( \frac{Q^2}{m^2}\right) \left( 3\ln \left( \frac{Q^2}{m^2}\right) \right. \nonumber \\&\qquad \left. -4\ln \left( \frac{2\kappa _0}{m^2}\right) -5\right) \nonumber \\&+4\ln \left( \frac{2\kappa _0}{m^2}\right) + 2 + \phi _0(x_-)\nonumber \\&+\frac{Q^2}{4(pl)}\frac{{Q^2}G_M^2 +4M^2G_E^2}{(4M^2+Q^2)R(1, Q^2)}\phi _1(x_-) \nonumber \\&+\frac{Q^2}{16(pl)^2}\frac{Q^2(Q^2+2M^2)G_M^2 -8M^4G_E^2}{(4M^2+Q^2)R(1, Q^2)} \phi _2(x_-) \nonumber \\&+\frac{Q^2}{4(pl)^2}\frac{Q^2G_M^2 -2M^2G_E^2}{R(1, Q^2)}\phi _3(x_-)\Bigg ]~. \end{aligned}$$

(B.13)

Appendix C

Straightforward integration gives

$$\begin{aligned}&\int ^{x_-}_0 dx\, V_2(x) = 4\text {Li}_2(x_-) \nonumber \\&\quad -4\ln (1-x_-)\ln \frac{(1-x_-)}{x_-} \nonumber \\&\quad -(\frac{4}{3}+4x_- +2x^2_-)\ln (1-x_-)\nonumber \\&\quad +3x_-(1+\frac{x_-}{2})\ln x_- -\frac{8}{3} x_- - \frac{7}{12} x^2_-~, \end{aligned}$$

(C.1)

$$\begin{aligned}&\int _{x_-}^1 dx\left( \frac{1}{x}-1\right) V_2(x) = 4\text {Li}_2(x_-)-\frac{2}{3} \pi ^2 \nonumber \\&\quad + \frac{1}{2}\ln ^2x_- +2(1-x_-^2)\ln (1-x_-) \nonumber \\&\quad -\left( \frac{5}{3}-\frac{3}{2} x_-^2\right) \ln x_-\nonumber \\&\quad +\frac{1}{12}(1-x_-)(31+7x_-) ~, \end{aligned}$$

(C.2)

$$\begin{aligned}&\int _{x_-}^1 dx\left( \frac{1}{x^2}-1\right) V_2(x) = 4\text {Li}_2(x_-)-\frac{2}{3} \pi ^2 \nonumber \\&\quad +\frac{1}{2}\ln ^2x_-\nonumber \\&\quad +2(1-x_-^2)\left( \frac{2}{x_-} +1\right) \ln (1-x_-) \nonumber \\&\quad - \left( \frac{1}{x_-}+\frac{5}{3} - 3 x_- -\frac{3}{2} x_-^2\right) \ln x_- \nonumber \\&\quad + (1-x_-)(\frac{2}{3x_-}+\frac{13}{4}+\frac{7}{12}x_-)~, \end{aligned}$$

(C.3)

$$\begin{aligned}&\int _{x_-}^1 dx\, S_2(x) = -\left( \frac{4}{3} +2x_- +x^2_-\right) \ln x_- \nonumber \\&\qquad - \frac{1}{9}(1-x_-)(22+13 x_- +4x_-^2)~, \end{aligned}$$

(C.4)

$$\begin{aligned}&\int _{x_-}^1 dx\, \frac{S_2(x)}{x} = - \ln ^2x_- - ({2}{x_-} +1)\ln x_- \nonumber \\&\qquad + \frac{(1-x_-)}{3}\left( \frac{4}{x_-}-11 -2x_-\right) ~, \end{aligned}$$

(C.5)

$$\begin{aligned}&\int _{x_-}^1 dx\, \frac{S_2(x)}{x^2} = - \ln ^2x_- + (\frac{2}{x_-} +1)\ln x_- \nonumber \\&\qquad + \frac{(1-x_-)}{3}\left( \frac{2}{x_-^2}+\frac{11}{x_-}-4\right) ~. \end{aligned}$$

(C.6)