Abstract
Full Text
Reports of the Academy of Sciences of the USSR
- Vol. 148, No. 1
PHYSICS
N. F. NELIPA
ON THE THEORY OF RADIATIVE PRODUCTION OF ELECTRON–POSITRON PAIRS ON A NUCLEUS
(Presented by Academician Ya. B. Zel’dovich on 10 VII 1962)
A theoretical analysis of the process of radiative pair production on nuclei:
\(\gamma+\text{nucleus}\to \text{recoil nucleus}+e^+ + e^-+\gamma'\) was carried out in several papers \((^{1-3})\). Various special cases were considered in them. The purpose 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 \(\sim 1/137\) of the pair-production cross section).
Let \(k_1(\omega_1,\mathbf{k}_1)\), \(k_2(\omega_2,\mathbf{k}_2)\), \(p_1(\varepsilon_1,\mathbf{p}_1)\), \(p_2(\varepsilon_2,\mathbf{p}_2)\), \(q(0,\mathbf{p}_1+\mathbf{p}_2+\mathbf{k}_2-\mathbf{k}_1)\) denote the energy–momentum vectors, respectively, of the incident and emitted photons, the positron, the electron, and the recoil nucleus (we neglect the kinetic energy of the recoil nucleus).
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 and positron 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 is written as follows*:
\[ d\sigma = -r_0^2 \frac{Z^2\alpha^2 |\mathbf{p}_1|\,|\mathbf{p}_2|\,|\mathbf{k}_2|\,d\Omega_{p_1}\,d\Omega_{p_2}\,d\Omega_{k_2}\,d\varepsilon_1\,d\varepsilon_2} {2(2\pi)^4\omega_1 q^4} \frac{1}{16}\operatorname{Sp}F . \tag{1} \]
Here
\[ \begin{aligned} \operatorname{Sp}F = \operatorname{Sp}\Bigg\{ &\frac{\gamma_\nu(\hat l_1+1)\gamma_0(\hat f_1+1)\gamma_\mu} {-p_2k_2\cdot p_1k_1} + \frac{\gamma_\nu(\hat l_1+1)\gamma_\mu(\hat f_2+1)\gamma_0} {-p_2k_2[p_1q-q^2/2]} + \frac{\gamma_0(\hat l_2+1)\gamma_\nu(\hat f_1+1)\gamma_\mu} {p_1k_1[p_2q-q^2/2]} \\ &+ \frac{\gamma_\mu(\hat l_3+1)\gamma_\nu(\hat f_2+1)\gamma_0} {p_2k_1[p_1q-q^2/2]} + \frac{\gamma_\mu(\hat l_3+1)\gamma_0(\hat f_3+1)\gamma_\nu} {-p_1k_2\cdot p_2k_1} + \frac{\gamma_0(\hat l_2+1)\gamma_\mu(\hat f_3+1)\gamma_\nu} {-p_1k_2[p_2q-q^2/2]} \Bigg\} (1-\hat p_1) \\ \times \Bigg\{ &\frac{\gamma_\mu(\hat f_1+1)\gamma_0(\hat l_1+1)\gamma_\nu} {-p_2k_2\cdot p_1k_1} + \frac{\gamma_0(\hat f_2+1)\gamma_\mu(\hat l_1+1)\gamma_\nu} {-p_2k_2[p_1q-q^2/2]} + \frac{\gamma_\mu(\hat f_1+1)\gamma_\nu(\hat l_2+1)\gamma_0} {p_1k_1[p_2q-q^2/2]} \\ &+ \frac{\gamma_0(\hat f_2+1)\gamma_\nu(\hat l_3+1)\gamma_\mu} {p_2k_1[p_1q-q^2/2]} + \frac{\gamma_\nu(\hat f_3+1)\gamma_0(\hat l_3+1)\gamma_\mu} {-p_1k_2\cdot p_2k_1} + \frac{\gamma_\nu(\hat f_3+1)\gamma_\mu(\hat l_2+1)\gamma_0} {-p_1k_2[p_2q-q^2/2]} \Bigg\} (\hat p_2+1); \end{aligned} \tag{2} \]
\[ l_1=p_2+k_2,\qquad l_2=p_2-q,\qquad l_3=p_2-k_1,\qquad f_1=-p_1+k_1,\qquad f_2=-p_1+q, \]
\[ f_3=-p_1-k_2,\qquad q=p_1+p_2+k_2-k_1,\qquad \text{the scalar product }\; ab=a_0b_0\left[1-\frac{|\mathbf a|\,|\mathbf b|}{a_0b_0}\cos(ab)\right], \qquad r_0=\alpha=e^2/4\pi . \]
* In what follows, Feynman notation is used throughout (see, for example, \((^4)\)); energies and momenta are expressed in units of the electron rest mass.
Generally speaking, after multiplication in (2) it is necessary to calculate 21 traces*:
\[ \operatorname{Sp} F= \frac{\operatorname{Sp}11}{\gamma_{11}}+ \frac{\operatorname{Sp}22}{\gamma_{22}}+\cdots+ \frac{\operatorname{Sp}66}{\gamma_{66}}+ 2\left[ \frac{\operatorname{Sp}12}{\gamma_{12}}+\cdots+ \frac{\operatorname{Sp}23}{\gamma_{23}}+\cdots+ \frac{\operatorname{Sp}34}{\gamma_{34}}+\cdots+ \frac{\operatorname{Sp}45}{\gamma_{45}}+\cdots+ \frac{\operatorname{Sp}57}{\gamma_{56}} \right]. \tag{3} \]
However, in fact one may restrict oneself to finding expressions only for 7 traces: 11, 12, 14, 15, 22, 23, 24, 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 \(\gamma_{13}=-p_{2}k_{2}(p_{1}k_{1})^{2}[p_{2}q-q^{2}/2]\), etc.
Table 1
| Connection of traces | Form of substitution | Connection of traces | Form of substitution |
|---|---|---|---|
| \(11\to55\) | \(l_{1}\to l_{3},\ f_{1}\to f_{3}\) | \(15\to26\) | \(l_{1}\to f_{2},\ f_{1}\to -p_{1},\ -p_{1}\to f_{3}\) \(f_{3}\to l_{2},\ l_{3}\to p_{2},\ p_{2}\to l_{1}\) |
| \(12\to13\) | \(l_{1}\leftarrow f_{1},\ -p_{1}\leftarrow p_{2},\ f_{2}\to l_{2}\) | \(15\to34\) | \(l_{1}\to p_{2},\ f_{1}\to l_{2},\ -p_{1}\to f_{1}\) \(f_{3}\to -p_{1},\ l_{3}\to f_{2},\ p_{2}\to l_{3}\) |
| \(12\to45\) | \(l_{1}\to l_{3},\ f_{1}\to f_{3}\) | \(22\to33\) | \(l_{1}\to f_{1},\ f_{2}\to l_{2},\ -p_{1}\rightleftarrows p_{2}\) |
| \(12\to56\) | \(f_{1}\to l_{3},\ l_{1}\to f_{3},\ -p_{1}\leftarrow p_{2},\ f_{2}\to l_{2}\) | \(22\to44\) | \(l_{1}\to l_{3}\) |
| \(14\to16\) | \(l_{1}\rightleftarrows f_{1},\ -p_{1}\rightleftarrows p_{2},\ f_{2}\to l_{2},\ l_{3}\to f_{3}\) | \(22\to66\) | \(l_{1}\to f_{3},\ f_{2}\to l_{2},\ -p_{1}\to p_{2}\) |
| \(14\to35\) | \(l_{1}\to f_{3},\ f_{1}\rightleftarrows l_{3},\ -p_{1}\leftarrow p_{2},\ f_{2}\to l_{2}\) | \(23\to46\) | \(l_{1}\to l_{3},\ f_{1}\to f_{3}\) |
| \(14\to25\) | \(l_{1}\leftarrow l_{3},\ f_{1}\to f_{3}\) | ||
| \(24\to36\) | \(l_{1}\to f_{3},\ f_{2}\to l_{2},\ l_{3}\to f_{1},\ -p_{1}\rightleftarrows p_{2}\) |
Calculations of the basic traces lead to the following result:
\[ \begin{aligned} \frac{1}{16}\operatorname{Sp}11={}& 2p_{2}l_{1}\{\overline{p_{1}l_{1}}(f_{1}^{2}-1) -2\overline{l_{1}f_{1}}(p_{1}f_{1}+2) -2(p_{1}f_{1}+1)\}\\ &+l_{1}^{2}\{2p_{1}f_{1}(\overline{p_{2}f_{1}}+2) +\overline{p_{1}p_{2}}(1-f_{1}^{2}) +4(\overline{p_{2}f_{1}}+f_{1}^{2}+1)\}\\ &-4f_{1}^{2}(p_{2}l_{1}+\overline{p_{1}l_{1}}-\tfrac14\,\overline{p_{1}p_{2}}-1) -2p_{1}l_{1}(\overline{p_{2}l_{1}}-4\overline{l_{1}f_{1}}-2)\\ &+4(4\overline{l_{1}f_{1}}-\overline{p_{2}f_{1}}+\overline{p_{1}l_{1}}-\tfrac14\,\overline{p_{1}p_{2}}+1); \end{aligned} \]
\[ \begin{aligned} \frac{1}{16}\operatorname{Sp}12={}& l_{1}^{2}\{f_{1}f_{2}\cdot p_{1}p_{2} +\overline{f_{2}p_{1}}(f_{1}p_{2}+2) -\overline{f_{1}p_{1}}\cdot \overline{f_{2}p_{2}} -\overline{f_{1}p_{2}} +2t_{1}(p_{20}+2f_{10})-2\}\\ &+2l_{1}p_{2}\{\overline{f_{1}l_{1}}(1-f_{2}p_{1}) -\overline{f_{1}f_{2}}\cdot p_{1}l_{1} +\overline{f_{1}p_{1}}\cdot \overline{f_{2}l_{1}} -p_{1}f_{2}-2t_{1}t_{2}+1\}\\ &+\overline{p_{1}f_{1}}(\overline{f_{2}p_{2}}-4\overline{f_{2}l_{1}}) +\overline{p_{1}f_{2}}(4\overline{f_{1}l_{1}}-\overline{f_{1}p_{2}}+2) +f_{1}f_{2}(4p_{1}l_{1}-p_{1}p_{2})\\ &+2\left\{\tfrac12\,\overline{f_{1}p_{2}}-2\overline{l_{1}f_{1}} +t_{1}(2t_{2}+2l_{10}-p_{20})-1\right\}; \end{aligned} \]
\[ \begin{aligned} \frac{1}{16}\operatorname{Sp}14={}& \alpha_{14}+\beta_{14} +\overline{p_{2}f_{2}}(l_{3}l_{1}+\overline{f_{1}l_{3}}+2l_{10}p_{10}) +\overline{p_{2}l_{3}}(l_{1}p_{1}+\overline{f_{1}p_{1}}-f_{2}f_{1}-\overline{f_{2}l_{1}})\\ &+\overline{f_{2}l_{3}}(\overline{p_{2}l_{1}}+f_{1}p_{2}) -\overline{p_{1}p_{2}}(l_{1}l_{3}+\overline{f_{1}l_{3}}) -p_{1}l_{3}(f_{1}p_{2}+\overline{l_{1}p_{2}}) -\overline{l_{1}f_{1}}(\overline{f_{2}p_{1}}-1)\\ &-l_{1}p_{1}\cdot \overline{f_{1}f_{2}} +\overline{l_{1}f_{2}}\cdot \overline{f_{1}p_{1}} +2l_{30}(p_{1}p_{2}f_{20}-p_{1}f_{2}p_{20})\\ &+2p_{20}(l_{1}f_{1}l_{30}-l_{1}l_{3}f_{10}-f_{1}l_{3}l_{10}-l_{30})\\ &+(f_{2}l_{1}-\overline{p_{1}f_{2}}-\overline{p_{1}l_{1}}+1)(p_{2}f_{1}+l_{3}f_{1})\\ &+(\overline{f_{1}l_{1}}+\overline{p_{1}l_{1}}+f_{1}p_{1}+1)(\overline{f_{2}p_{2}}+\overline{l_{3}f_{2}})\\ &+(f_{1}p_{1}-\overline{l_{1}f_{2}}-\overline{p_{1}f_{2}}+1)(\overline{p_{2}l_{1}}+\overline{l_{3}l_{1}})\\ &-(f_{2}l_{1}+\overline{f_{1}f_{2}}+\overline{f_{1}l_{1}}+1)(p_{1}p_{2}+l_{3}p_{1}) -\overline{f_{2}p_{1}}-2t_{1}t_{2}+1; \end{aligned} \]
* In the notation \(ik\), the first number corresponds to the ordinal number of the factor in the first curly bracket of expression (2), and the second number to that in the second curly bracket.
\[ \begin{aligned} \frac{1}{16}\operatorname{Sp}15={}& 2p_1p_2\bigl(l_1f_1\cdot \overline{f_3l_3}+l_1f_3\cdot \overline{f_1l_3}-\overline{l_1l_3}\cdot \overline{f_1f_3}\bigr) +p_1p_2\bigl(2\overline{l_1f_1}-\overline{l_1f_3}+2\bigr)-\\ &-f_1p_1(l_1f_3+\overline{l_1l_3}) +f_1l_3\bigl(l_1p_1+\overline{l_1f_3}+l_1p_2-p_1f_3-\overline{f_3p_2}-2\bigr)+\\ &+f_3p_2\bigl(\overline{l_1f_1}-l_1p_1-\overline{f_1p_1}+1\bigr) +\overline{f_1f_3}\,(p_1p_2+l_1l_3)+\\ &+\overline{f_1f_3}\,(l_1p_1+l_1p_2+l_3p_1+l_3p_2-2) -l_1f_1\bigl(p_1l_3+\overline{p_1f_3}+f_3l_3+l_3p_2-1\bigr)+\\ &+f_1p_2\bigl(\overline{l_1l_3}-p_1l_3+\overline{l_3f_3}-\overline{p_1f_3}+1\bigr)+\\ &+l_1f_3\bigl(l_3p_1-\overline{f_1p_2}+l_3p_2+2l_{30}f_{10}-2p_{10}p_{20}-2\bigr) -2l_1f_1(f_{30}+p_{20})l_{30}-\\ &-\overline{l_1l_3}\,(p_1f_3+\overline{f_3p_2}+2) +\overline{l_1p_2}\,(f_3p_1+\overline{f_3l_3}-p_1l_3+1) +l_1l_3(\overline{p_2p_1}+2p_{20}f_{10})+\\ &+\overline{l_3f_3}\,(2p_1p_2-\overline{f_1p_1}-l_1p_1+1) +\overline{f_1l_3}\cdot p_1p_2-(\overline{f_1p_1}-1)p_2l_3+\\ &+2f_{30}p_{10}l_1p_2-l_1p_1\cdot\overline{p_2l_3} -\overline{f_3p_1}-l_3p_1-l_1p_1-\overline{f_1p_1}+1; \end{aligned} \]
\[ \begin{aligned} \frac{1}{16}\operatorname{Sp}22={}& l_1^2\left[2f_2p_2(\overline{p_1f_2}-1)+f_2^2(4-p_1p_2)-4(2\overline{f_2p_1}-1)+\overline{p_1p_2}\right]+\\ &+2l_1p_2\left[\overline{p_1l_1}(f_2^2-1)+2p_1f_2(2-l_1f_2)-2(f_2^2+1-l_1f_2)\right]-\\ &-f_2^2\left[4(\overline{l_1p_1}-1)-\overline{p_1p_2}\right] -2\overline{f_2p_1}\left[f_2p_2+4(1-l_1f_2)\right]-p_1p_2+\\ &+2f_2p_2+4(\overline{l_1p_1}-2l_1f_2+1); \end{aligned} \]
\[ \begin{aligned} \frac{1}{16}\operatorname{Sp}23={}& -\alpha_{23}+2t_1\left[p_2l_2t_2-(f_1l_2+l_1l_2)p_{20}-(l_1p_2+f_1p_2)t_{20}+2t_3-t_2\right]+\\ &+4t_1t_3l_1f_1 +2t_3\left[(f_2f_1+l_1f_2)p_{10}+(p_1f_1+l_1p_1)f_{20}-p_1f_2t_2-t_2\right]+\\ &+\overline{p_1p_2}\cdot\overline{f_2l_2} -p_1l_2\cdot f_2p_2 -\overline{p_1t_2}\,(\overline{p_2l_2}+1) -l_1p_1\cdot f_2f_1-\overline{l_1t_1}\,(\overline{p_1f_2}-1)+\\ &+\overline{l_1f_2}\cdot p_1f_1 +p_2l_2(1+\overline{l_1f_1}) +l_1l_2\cdot f_1p_2 -\overline{l_1p_2}\cdot\overline{f_1l_2}+1; \end{aligned} \]
\[ \begin{aligned} \frac{1}{16}\operatorname{Sp}24={}& 2l_1l_3\left[2p_2f_2(\overline{p_1f_2}-1)-\overline{p_1p_2}(f_2^2-1)+\frac12\right]+\\ &+f_2^2\left(l_1p_2+l_1l_3+l_3p_2+\overline{p_1p_2}+\overline{l_1p_1}+\overline{l_3p_1}-2\right)-\\ &-2\overline{f_2p_1}\left(f_2l_3+f_2p_2+f_2l_1+l_3p_2+l_1l_3+l_1p_2-2\right)+\\ &+2(l_3f_2+f_2p_2+l_1f_2-1)-\overline{l_3p_1}+l_1p_2-\overline{p_1p_2}-\overline{l_1p_1}+l_3p_2, \end{aligned} \]
where
\[ \begin{gathered} t_1=f_{20}-p_{10},\qquad t_2=l_{10}+f_{10},\qquad t_3=l_{20}+p_{20};\\ \alpha_{14}=l_1\widetilde f_1\,[\widetilde p_1p_2\cdot f_2l_3+\widetilde p_2l_3\cdot\widetilde p_1f_2-\widetilde p_2f_2\cdot\widetilde p_1l_3]+\\ \quad +l_1\widetilde p_2\,[\widetilde f_1f_2\cdot\widetilde p_1l_3-\widetilde f_1p_1\cdot f_2l_3-\widetilde f_1l_3\cdot\widetilde p_1f_2]+\\ \quad +l_1p_1[\widetilde f_1\widetilde p_2\cdot f_2l_3+\widetilde f_1l_3\cdot\widetilde p_2f_2-\widetilde f_1f_2\cdot\widetilde p_2l_3] +l_1f_2[\widetilde f_1p_1\cdot\widetilde p_2l_3-\widetilde f_1p_2\cdot\widetilde p_1l_3-\widetilde f_1l_3\cdot\widetilde p_2p_1]+\\ \quad +l_1l_3[\widetilde f_1p_2\cdot p_1f_2+\widetilde f_1f_2\cdot\widetilde p_1p_2-\widetilde f_1p_1\cdot\widetilde p_2f_2], \end{gathered} \]
\(\beta_{14}\) is obtained from \(\alpha_{14}\) by the substitutions \(\widetilde p_2\to\widetilde f_2\), \(\widetilde p_1\to p_1\), \(f_2\to p_2\), and \(\alpha_{23}\) by the substitutions \(\widetilde f_1\to p_1\), \(\widetilde p_2\to\widetilde f_2\), \(\widetilde p_1\to\widetilde f_1\), \(f_2\to p_2\), \(l_3\to l_2\), with \(\widetilde{ab}=\widetilde a b=\overline{ab}\), \(\widetilde{\widetilde a b}=ab\).
Let us emphasize that, when obtaining the expressions for the other traces from the basic ones, the corresponding substitutions indicated in Table 1 must also be extended to the quantities \(f_{10}, f_{20}, f_{30}, l_{10}, l_{20}, l_{30}, p_{10}, p_{20}\); for example, in order to obtain 46 from 23, along with the substitution \(l_1\to l_3\), \(f_1\to f_3\), one must make the substitution \(l_{10}\to l_{30}\) and \(f_{10}\to f_{30}\) (in this case \(f_{10}=-p_{10}+k_{10}\), etc.).
To compute the value of the differential cross section, one must find the trace quantities 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 this work.
Lebedev Physical Institute
Academy of Sciences of the USSR
Received
15 VI 1962
References Cited
- M. A. Markov, Dokl. Akad. Nauk SSSR, 20, No. 2–3 (1938).
- B. de Tollis, G. Jona-Lasinio, R. S. Liotta, Nuovo Cim., 18, 545 (1960).
- N. F. Nelipa, Report of the Lebedev Physical Institute, Academy of Sciences of the USSR, 1959.
- N. N. Bogolyubov, D. V. Shirkov, Introduction to the Quantum Theory of Wave Fields, Moscow, 1957.