N. F. NELIPA

ON THE THEORY OF THE DOUBLE COMPTON EFFECT ON AN ELECTRON

(Presented by Academician Ya. B. Zel’dovich on 10 VII 1962)

The theoretical analysis of the double Compton effect has been carried out in a number of papers \({}^{1-7}\). In them various special cases were considered. The aim of the present article is to find a general expression for the differential cross section of the indicated process in the first nonvanishing approximation of perturbation theory (in order of magnitude it is equal to \(\sim \frac{1}{137}\) of the cross section of the single Compton effect).

Let us denote by \(k_1(\omega_1,\mathbf{k}_1)\), \(k_2(\omega_2,\mathbf{k}_2)\), \(k_3(\omega_3,\mathbf{k}_3)\), \(p_1(\varepsilon_1,\mathbf{p}_1)\), \(p_2(\varepsilon_2,\mathbf{p}_2)\) the energy–momentum vectors, respectively, of the incident and two emitted photons, of the initial electron, and of the recoil electron.

In the first nonvanishing approximation of perturbation theory, the contribution to the matrix element will be given by six diagrams, which are obtained by all possible permutations of the photon lines (with fixed electron lines).

After summation over the spins of the final particles and averaging over the spins of the initial particles, the expression for the differential cross section of the process under consideration will have the form*

\[ d\sigma = r_0^2 \frac{\alpha \omega_2 \omega_3 d\Omega_{k_2} d\Omega_{k_3} d\omega_3} {(2\pi)^2 16\omega_1 \varepsilon_2[1-v_2\cos(p_2 k_2)]} \frac{1}{16}\operatorname{Sp}F . \tag{1} \]

Here:

\[ r_0=\alpha=\frac{l^2}{4\pi}; \qquad v_2=\frac{|\mathbf{P}_2|}{\varepsilon_2}, \]

\[ \begin{aligned} \operatorname{Sp}F =&\operatorname{Sp}\Biggl\{ \Biggl[ \frac{\gamma_\mu(\hat g_0+1)\gamma_i(\hat f_1+1)\gamma_\nu}{p_2k_2\cdot p_1k_1} + \frac{\gamma_\mu(\hat g_0+1)\gamma_\nu(\hat f_2+1)\gamma_i}{-p_2k_2\cdot p_1k_3} \\ &\quad+ \frac{\gamma_i(\hat g_2+1)\gamma_\mu(\hat f_1+1)\gamma_\nu}{p_2k_3\cdot p_1k_1} + \frac{\gamma_\nu(\hat g_3+1)\gamma_\mu(\hat f_2+1)\gamma_i}{p_2k_1\cdot p_1k_3} \\ &\quad+ \frac{\gamma_\nu(\hat g_3+1)\gamma_i(\hat f_3+1)\gamma_\mu}{p_2k_1\cdot p_1k_2} + \frac{\gamma_i(\hat g_2+1)\gamma_\nu(\hat f_3+1)\gamma_\mu}{-p_2k_3\cdot p_1k_2} \Biggr] (\hat p_1+1) \\ &\quad\times \Biggl[ \frac{\gamma_\nu(\hat f_1+1)\gamma_i(\hat g_0+1)\gamma_\mu}{p_2k_2\cdot p_1k_1} + \frac{\gamma_i(\hat f_2+1)\gamma_\nu(\hat g_0+1)\gamma_\mu}{-p_2k_2\cdot p_1k_3} \\ &\quad+ \frac{\gamma_\nu(\hat f_1+1)\gamma_\mu(\hat g_2+1)\gamma_i}{p_2k_3\cdot p_1k_1} + \frac{\gamma_i(\hat f_2+1)\gamma_\mu(\hat g_3+1)\gamma_\nu}{p_2k_1\cdot p_1k_3} \\ &\quad+ \frac{\gamma_\mu(\hat f_3+1)\gamma_i(\hat g_3+1)\gamma_\nu}{p_2k_1\cdot p_1k_2} + \frac{\gamma_\mu(\hat f_3+1)\gamma_\nu(\hat g_2+1)\gamma_i}{-p_2k_3\cdot p_1k_2} \Biggr] (\hat p_2+1) \Biggr\}; \tag{2} \end{aligned} \]

* In what follows, Feynman’s notation is used throughout (see, for example, \({}^{8}\)). Energies and momenta are expressed in units of the electron rest mass.

\[ g_0=p_2+k_2,\qquad g_2=p_2+k_3,\qquad g_3=p_2-k_1,\qquad f_1=p_1+k_1, \]
\[ f_2=p_1-k_3,\qquad f_3=p_1-k_2, \]

the scalar product
\[ ab=a_0b_0\left[1-\frac{|a|\,|b|}{a_0b_0}\cos(ab)\right]. \]

Generally speaking, after multiplication in (2) one must compute 21 traces*:

\[ \operatorname{Sp}F=\frac{\operatorname{Sp}11}{\gamma_{11}}+ \frac{\operatorname{Sp}22}{\gamma_{22}}+\ldots+ \frac{\operatorname{Sp}66}{\gamma_{66}}+ \]
\[ +2\left[ \frac{\operatorname{Sp}12}{\gamma_{12}}+\ldots+ \frac{\operatorname{Sp}23}{\gamma_{23}}+\ldots+ \frac{\operatorname{Sp}34}{\gamma_{34}}+\ldots+ \frac{\operatorname{Sp}56}{\gamma_{56}} \right]. \tag{3} \]

However, in fact it is possible to restrict oneself to finding expressions only for 4 traces: 11, 12, 15, 14, since from them the expressions for the remaining traces are obtained by simple substitutions (see Table 1). The explicit form of the denominators \(\gamma_{ik}\) is easily obtained from (2), for example, \(\gamma_{12}=-(p_2k_2)^2(p_1k_1)(p_1k_3)\), etc.

Table 1

Computations of the basic traces lead to the following result:

\[ {}^{1}\!/_{16}\operatorname{Sp}11 =2\{g_0^2[2p_1f_1(f_1p_2-4)-p_1p_2(f_1^2-1)-4p_2f_1+8(1+f_1^2)]+ \]
\[ +2p_2g_0[(f_1^2-1)p_1g_0-2p_1f_1(f_1g_0-2)+4(g_0f_1-1)]+ \]
\[ +f_1^2[p_1p_2-4(p_1g_0+2p_2g_0-2)]+2p_1f_1(4f_1g_0-f_1p_2-4)+ \]
\[ +4(f_1p_2+g_0p_1-4g_0f_1+2)-p_1p_2\}; \]

\[ {}^{1}\!/_{16}\operatorname{Sp}12 =2\{2f_1f_2[2g_0p_1(p_2g_0-2)+p_1p_2(1-g_0^2)+1]- \]
\[ -2g_0p_2(g_0f_2+g_0f_1+g_0p_1+f_1f_2+f_2p_1+f_1p_1-2)+ \]
\[ +g_0^2[p_1p_2+f_1p_2+f_2p_2+2(f_1f_2+f_2p_1+f_2p_1)-4]+ \]
\[ +4(f_1g_0+p_1g_0+f_2g_0)-f_1p_2-p_1p_2-f_2p_2+2(p_1f_2+p_1f_1-2)\}; \]

\[ {}^{1}\!/_{16}\operatorname{Sp}14 =2\{-p_1p_2(g_0g_3\cdot f_1f_2-g_0f_1\cdot g_3f_2+g_0f_2\cdot g_3f_1)+ \]
\[ +f_2p_1(g_0g_3\cdot f_1p_2-g_0f_1\cdot g_3p_2+g_0p_2\cdot g_3f_1)+ \]
\[ +f_2p_2(g_0f_1\cdot g_3p_1-g_0g_3\cdot f_1p_1-g_0p_1\cdot g_3f_1)+ \]
\[ +f_1p_1(g_0f_2\cdot g_3p_2-g_0p_2\cdot g_3f_2)+f_1p_2(g_0p_1\cdot g_3f_2-g_0f_2\cdot g_3p_1)+ \]
\[ +f_1f_2(g_0p_2\cdot g_3p_1-g_3p_2\cdot g_0p_1)+g_0g_3(2p_2f_2-f_1p_2-p_1p_2+f_1p_1-1)+ \]
\[ +g_0p_2(-g_3f_1-g_3p_1+f_1p_1-1)+g_0p_1(g_3p_2-g_3f_1-f_1p_2+2f_1f_2-1)+ \]

* In the notation \(ik\), the first number corresponds to the ordinal number of the factor in the first square bracket of expression (2), and the second number—to that in the second square bracket.

\[ \begin{aligned} &+2g_0 f_2\left(g_3 f_1+g_3 p_1+f_1 p_2+p_1 p_2-\frac12\right) +g_0 f_1\left(g_3 p_2-g_3 p_1-p_1 p_2-1\right)+{}\\ &\quad +g_3 f_2\left(f_1 p_1-f_1 p_2-p_1 p_2-1\right) +g_3 p_2\left(f_1 f_2+f_2 p_1-1\right)+{}\\ &\quad +g_3 p_1\left(2+2f_1 p_2-f_1 f_2-f_2 p_2\right) -f_1 f_2\left(p_1 p_2+1\right)+{}\\ &\quad +f_1 p_1\left(f_2 p_2-1\right) -f_1 p_2\left(f_2 p_1-2\right) -f_2 p_1\left(1-f_1 g_3\right)-{}\\ &\quad -f_2 f_2\left(f_1 g_3+1\right) +2\left(g_3 f_1+p_1 p_2+2\right)\}; \end{aligned} \]

\[ \begin{aligned} \frac1{16}\operatorname{Sp}15 =2\,[&-4f_1 g_3(1+p_1p_2\cdot g_0f_3) +f_1g_0(2p_1p_2-f_3p_2-p_1g_3-1)+{}\\ &+p_1g_0(g_3f_1-g_3p_2+2f_3p_2-g_3f_3+2) +f_1p_1(2g_0f_3-f_3p_2-g_0g_3-1)+{}\\ &+f_1f_3(p_1p_2-g_3p_2-g_0p_2+2g_0g_3+2) +f_1p_2(2p_1g_3-g_3f_3+f_3g_0+2)+{}\\ &+f_1g_3(f_3p_2-2g_0p_2+2f_3p_1) +p_1g_3(g_0f_3+2-g_0p_2)-{}\\ &-f_3p_1(f_1p_2+g_0g_3+1) +2g_0f_3(g_3p_2-2) +p_1p_2(g_0g_3+2f_3g_3-4)+{}\\ &+2(g_0g_3+f_3p_2+1)-g_0p_2-f_3g_3-g_3p_2]. \end{aligned} \]

To compute the value of the differential cross section, one must find the values of the traces, using the last expressions and Table 1, substitute the results obtained into (3), and the latter into (1).

In conclusion I express my gratitude to M. A. Markov for his interest in the present work.

P. N. Lebedev Physical Institute
Academy of Sciences of the USSR

Received
15 VI 1962

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