Radiative Corrections in Møller Scattering for PRad Experiment at Thomas Jefferson National Accelerator Facility (TJNAF)

Abstract

By applying several methods for removing infrared divergences, electromagnetic radiative corrections in low-energy Møller scattering are calculated under conditions of the PRad experiment at Thomas Jefferson National Accelerator Facility (TJNAF).

Appendices

Appendix

BREMSSTRAHLUNG PHASE SPACE

We now depict in the center-of-mass frame the final-particle vectors (see Fig. 5), employing the auxiliary vector ${\mathbf{p}_{5}}=-{\mathbf{p}}$. For the energy of the final electron with 4-momentum $p_{3}$, we obtain

$${p_{3}}_{0}=\frac{\sqrt{s^{3}}+2p_{0}^{2}\sqrt{s}-3p_{0}s+Ap_{0}s_{q}}{2s-4p_{0}\sqrt{s}+2p_{0}^{2}(1-A^{2})},$$

(A.1)

where

$$s_{q}=\sqrt{s(\sqrt{s}-2p_{0})^{2}+4m^{2}\left[p_{0}^{2}(A^{2}-1)+2p_{0}\sqrt{s}-s\right]}$$

and

$$A=\cos(\widehat{{\mathbf{p}_{5}},{\mathbf{p}_{3}}})$$

(A.2)

$${}=\sin\theta_{3}\sin\theta_{5}\cos\varphi_{5}+\cos\theta_{3}\cos\theta_{5}$$

$${}=-\sin\theta_{3}\sin\theta_{p}\cos\varphi_{p}-\cos\theta_{3}\cos\theta_{p}.$$

In the ultrarelativistic approximation, the expression for the energy assumes the following compact form:

$${p_{3}}_{0}=\frac{\sqrt{s}}{2}\frac{\sqrt{s}-2p_{0}}{\sqrt{s}-p_{0}(1+A)}.$$

Taking into account the foregoing, we obtain the phase space in the form

$$d\Phi_{3}=\frac{|{\mathbf{p}_{3}}|}{4{p_{4}}_{0}{\mathcal{F}}}d\cos\theta_{3}d\varphi_{3}\frac{d^{3}{\mathbf{p}}}{2{p}_{0}}.$$

(A.3)

Integration with respect to the azimuthal angle $\varphi_{3}$ yields $2\pi$ because of symmetry with respect to the rotation of the system about the beam axis, and we also have

$${\mathcal{F}}=1+\frac{{p_{3}}_{0}(1-A|{\mathbf{p}}|/|{\mathbf{p}_{3}}|)}{\sqrt{{p_{3}}_{0}^{2}-2A|{\mathbf{p}}||{\mathbf{p}_{3}}|+|{\mathbf{p}}|^{2}}}.$$

(A.4)

Further, we proceed to perform integration with respect ${\mathbf{p}}$; that is,

$$d^{3}{\mathbf{p}}=|{\mathbf{p}}|^{2}d|{\mathbf{p}}|d\cos\theta_{p}d\varphi_{p},\quad\theta_{p}=\pi-\theta_{5},$$

$$\varphi_{p}=\pi+\varphi_{5}.$$

Upon employing the vector ${\mathbf{p}_{5}}$, the final-particle vectors assume the following form in the center-of-mass frame:

$${\mathbf{p}_{3}}=(|{\mathbf{p}_{3}}|\sin\theta_{3},0,|{\mathbf{p}_{3}}|\cos\theta_{3}),$$

(A.5)

$${\mathbf{p}_{5}}=(|{\mathbf{p}}|\sin\theta_{5}\cos\varphi_{5},|{\mathbf{p}}|\sin\theta_{5}\sin\varphi_{5},|{\mathbf{p}}|\cos\theta_{5}),$$

$${\mathbf{p}_{4}}={\mathbf{p}_{5}}-{\mathbf{p}_{3}}.$$

Clearly, the energy ${p_{4}}_{0}=\sqrt{m^{2}+|{\mathbf{p}_{4}}|^{2}}$ can be calculated on the basis of the set of Eqs. (A.5).

We will now express all radiative invariants in terms of the photon energy, azimuthal angle, and polar angle as [28]

$$z_{1}=2p_{0}{p_{1}}_{0}-2|{\mathbf{p}}||{\mathbf{p}_{1}}|\cos\theta_{p},$$

(A.6)

$$v_{1}=2p_{0}{p_{2}}_{0}+2|{\mathbf{p}}||{\mathbf{p}_{2}}|\cos\theta_{p},$$

$$z=2p_{0}{p_{3}}_{0}+2|{\mathbf{p}}||{\mathbf{p}_{3}}|A,$$

$$v=2p_{0}(\sqrt{s}-{p_{3}}_{0})-2|{\mathbf{p}}||{\mathbf{p}_{3}}|A.$$

We note that, in all of the formulas, $p_{0}$ and $|{\mathbf{p}}|$ are different: in these quantities, we retain the photon mass $\lambda$ (that is, $p_{0}^{2}-|{\mathbf{p}}|^{2}=\lambda^{2}\rightarrow 0$), which we use in the following as an infinitesimal parameter to regularize infrared divergences.