High-energy photoproduction cross section close to the end of spectrum
We consider the cross section of electron-positron pair production by a high-energy photon in a strong Coulomb field close to the end of electron or positron spectrum. We show that the cross section essentially differs from the result obtained in the Born approximation as well as form the result which takes into account the Coulomb corrections under assumption that both electron and positron are ultrarelativistic. The cross section of bremsstrahlung in a strong Coulomb field by a high-energy electron is also obtained in the region where the final electron is not ultrarelativistic.
The production of an electron-positron () pair by a photon in a strong atomic field has been investigated already since many years both theoretically and experimentally because of the importance of this process for various applications, see Refs. HGO1980 (); Hubbell2000 (). The cross section of this process in the Born approximation is known for arbitrary energy of the incoming photon, Refs. BH1934 (); Racah1934 () (we set throughout the paper). The effect of screening in this approximation can be easily taken into account using the atomic form factor JLS1950 (). For heavy atoms, it is however necessary to take into account the Coulomb corrections. These corrections are higher-order terms of the perturbation theory with respect to the parameter , where is the atomic charge number and is the fine-structure constant, with being the absolute value of the electron charge. The formal expression of the Coulomb corrections, exact in and , was derived in Ref. Overbo1968 (). This expression has a very complicated form which leads to difficulties in numerical computations. The difficulties grow as increases, so that numerical results have been so far obtained only for MeV, Ref. SudSharma2006 ().
In the high-energy region ( is the electron mass), considerations become greatly simplified. As a result, a simple form of the Coulomb corrections was obtained in Refs. BM1954 (); DBM1954 () in the leading approximation with respect to . However, this result has good accuracy only at energies MeV. The theoretical description of the Coulomb corrections for the total cross section at intermediate photon energies ( MeV) was based during a long time on the “bridging” expression derived in Overbo1977 (). This expression is actually an extrapolation of the results obtained at MeV. Results for the spectrum of one of the created particles at intermediate were practically absent. Recently, an important step was made in Ref. LMS2004 () where the first corrections of the order of to the spectrum as well as to the total cross section of photoproduction in a strong atomic field were derived. The correction to spectrum was obtained in the region where both produced particles are relativistic. It turns out that this correction is antisymmetric with respect to replacement , where and are the energy of the positron and the electron, respectively. Since the correction to the total cross section resulted to be very large, it is not related to the central region, where the created electron and positron are ultrarelativistic, and it comes from the region close to the end of spectrum where or . In Ref. LMS2004 (), the correction to the total cross section was obtained with the use of the dispersion relation for the forward Delbrück scattering amplitude but not by the direct integration of the spectrum. Note that the account for the correction to the total cross section leads to good description of available experimental data at intermediate photon energies, Refs. LMS2003 ().
In the present paper, we calculate the electron (positron) spectrum in the process of photoproduction in a strong Coulomb field in the case () and . We show that the Coulomb corrections drastically differ form that obtained in the region where and . In an analogous way, we have also derived the spectrum of bremsstrahlung in a strong Coulomb field in the region where the radiated photon has the energy close to that of the initial electron.
Ii General discussion
Following the usual Feynman rules (see, e. g., Ref. BLP1980 ()), the photoproduction cross section, at leading order in the interaction between the photon field and the electron-positron field, averaged over the polarization of the incoming photon and summed up over polarizations of electron and positron has the form
where is the four-momentum of the photon, and are the four-momenta of the electron and the positron, respectively, and and are their polarization indexes. Also, and are the corresponding positive-energy and negative-energy wave functions in a strong Coulomb field and are the Dirac matrices. In the following, we will calculate the spectrum, i.e., the cross section integrated over the angles of the vectors and . Due to rotational symmetry, this quantity is independent of the direction of the photon momentum . Therefore, we can average it over this direction, i.e., integrate both sides of Eq. (II) over . This results in the replacement
where . We can also use the known relations MS1983 ()
where and is the discontinuity on the cut of the electron Green’s function in the Coulomb field, to obtain
Since the representation (II) of the quantity is independent of the basis employed, we can also write in the form
and represent the spectrum as follows
where , . Here, is the positive-energy wave function with total angular momentum , parity , and projection of the total angular momentum along some quantization axis. The explicit form of this function is presented in Appendix A.
Below, we assume that . If the momentum of the electron is , then the formation length of the process is of the order of the Compton wavelength and the positron is ultrarelativistic with a typical angular momentum . This circumstance allows us to use the quasiclassical Green’s function obtained in Refs. MS1983a (); MS1983 () starting from a convenient integral representation derived in Ref. MS1982 () of the exact Green’s function of the Dirac equation in a Coulomb field. For the reader’s convenience, we present the formula for the discontinuity of this Green’s function in Appendix A, Eqs. (A). Moreover, it can be seen that if , then the main contribution to the integral in Eq. (6) is given by the region where the angles between vectors and are not small (or close to ). This is the region which also gives the main contribution to the bound-free photoproduction cross section at , see Ref. MS1993 (). In this region the expression in Eq. (A) becomes essentially simpler, see Eq. (A). By substituting Eq. (A) in Eq. (6) and by keeping the terms which do not contain highly oscillating functions, we arrive at
where (the notation is explained in Eq. (A)). By employing the explicit form (A) of the electron wave function, we finally arrive at the following expression of the cross section of pair production in the case of slow electron
where , , , and are the Legendre polynomials. The cross section on the other end of the spectrum, i.e. in the case of slow positron with momentum , is given by Eq. (II) with the replacement , , and . With the same procedure, one can also derive the spectrum of bremsstrahlung by a high-energy electron for the case where the final electron with momentum and energy is slow (). It turns out that this spectrum is given by the same formula (II) with the obvious substitutions and . Note that the correction to the bremsstrahlung spectrum of the order of in the case of initial and final electrons both ultrarelativistic was obtained recently in Ref. LMS2004 () from the corresponding results for pair production, and in Ref. LMSSch2005 () directly from the matrix element of bremsstrahlung.
It can be shown that three integrations in Eq. (II) can be performed analytically and the spectrum becomes
It is convenient to multiply the integrand in this formula by unity written in the form
to change the order of integration over the variables and , and to take the integral over (by exploiting the -function). After that, we pass from the variables and to the variables and such that and . As a result we obtain
This formula is still not convenient for numerical calculations because of the strong oscillations of the integrand in the vicinity of the point . In order to overcome this difficulty, we write
Then, by using the properties of the integrand , we make the following transformation
In the first step we changed the integration order of the variables and and then we performed the change of variable . In the second step, we changed the contour of integration, by exploiting the fact that the contribution along the straight path from the point to the point vanishes due to the exponential function (see Eq. (II) and also Fig. 1). This form of integral is appropriate for numerical calculations. Note that the integral (II) has zero imaginary part, so that it is not necessary to take real part of it afterwards.
The integration over the variable can be taken by using the relation Grad ()
The remaining integrations over the variables and can be performed numerically by employing the transformation (II). The largest contribution to the sum over is given by the term with . The contribution of the term with is essentially smaller, while that with is less than one percent even for large . In our numerical calculations here we have included terms with , and . The results for in units of at zero electron velocity is shown in Fig. 2 as a solid curve. At we obtain in agreement with previous results. In Ref. Deck1969 () by Deck et al., the following formula for at zero electron velocity was suggested
This formula is shown in Fig. 2 as a dashed curve and it is clear from the figure that Eq. (16) is applicable only at small values of . Also, note that the cross section in the Born approximation vanishes in the limit , since at it scales as . In Ref. SudSharma2006 () the results for was obtained at MeV and (which corresponds to ). These results are shown in Fig. 2 as a dotted curve starting from . One sees an excellent agreement of our results with those of Ref. SudSharma2006 (). At there is disagreement because at small and the contribution of the Born term is also important. Finally, we observe that, as expected, the spectrum of positron at small positron velocity tends to zero because in this case the wave functions are exponentially small (see Appendix A).
Iv Cross section at non-zero electron velocity
In order to obtain the spectrum for non-zero electron velocity, we substitute the explicit form of wave function (A) in Eq. (II) and use the relation for the confluent hypergeometric function. We come to the following expression for the cross section at
where is the hypergeometric function. After that we take numerically the integrals over the variables and .
The cross section in Eq. (IV) can be represented as
Unfortunately, the convergence of the series (19) is not fast, and it is necessary to take into account terms with rather large , especially at . In order to overcome this problem, we make the following approximation
where is a large integer and is given by the expression for with the replacement . The convergence of the series is much faster than the convergence of , and it is not necessary to take very large value of (we have seen that by choosing , an accuracy of about is reached), while the series can be summed analytically. Since the summation is not straightforward, we report some steps in the next paragraph.
iv.1 Calculation of
It is convenient to perform calculation of starting from Eq. (4). After the replacement , the expression for the discontinuity of the electron Green’s function is given by Eq. (A). By using this expression and by passing to the same variables as in the derivation of Eq. (II), we can again employ the convenient transformation (II). In this way, the integral over the variable can be easily performed and one obtains
Since the contour of integration over passes in the positive direction around the point , we can make a shift and take the integral over in using the relation Eerdelyi ()
Then we take the integral over with the help of Eq. (IV). Finally, we have
V Results and discussion
The cross section of the pair production process in the Born approximation is well known (see, for example, Ref. BLP1980 ()). In our limit ( and ), it has the form
In particular, at it is , while at the same cross section has the asymptotics
Now, the leading Coulomb correction to at reads DBM1954 ()
where , is the Euler constant. The correction (26) is independent of and it is the same for electron and positron (i. e., it is an even function of ). The next-to-leading correction was calculated recently in Ref. LMS2004 () and at it has the form
This correction has opposite sign for electron and positron since is an odd function of and it increases the cross section for slow electron while it decreases it for slow positron. In Fig. 3 and Fig. 4 we show our results for the Coulomb corrections to the spectrum for slow electron and positron, respectively at different values of (continuous curves). These results are compared with the asymptotic expressions (dashed curves) and (dotted curves). On the one hand, one can see that for each our results tend at large energies to the constant value . On the other hand, the next-to-leading correction essentially improves the agreement between exact results and asymptotic ones both for slow electron and positron.
In Ref. LMS2004 () the next-to leading correction to the total cross section, , was also obtained. It reads
where the function is related to the total cross section of the bound-free photoproduction,
The function is of the order of unity for all values of , see Ref. LMS2004 (). Since the Coulomb correction has opposite sign for electron and positron in the region where both particles are relativistic, the correction is determined by the region where momentum of electron or positron is of the order of . The correction can be obtained from our results using the relation
We conclude this section by discussing how it is possible to apply our results for photoproduction on heavy atoms, i.e., how the effects of screening can be accounted for. As it was pointed out above, the main contribution to the high-energy photoproduction cross section close to the end of spectrum of electron (or positron) comes from distances , where is the typical screening radius. Therefore, one can account for the screening by employing the prescription formulated in Refs. Pratt1971 (); Pratt1972 (); Overbo1979 (). Namely, the spectrum of slow electron in the screened Coulomb field can be obtain from that in unscreened field by means of the shift and values of the energy shift for various atoms are presented in Refs. Pratt1971 (); Pratt1972 (); Overbo1979 (). Analogously, for slow positron the corresponding shift is . In all cases . Thus, the effect of screening in our problem is important only for very small electron or positron velocities.
We have calculated exactly in the parameter the cross section of photoproduction in a Coulomb field at and (slow electron) or (slow positron). In the wide region, our results differ essentially from those obtained in the Born approximation as well as from the results which take into account the Coulomb corrections obtained at and . Therefore, the Coulomb correction to the spectrum can be approximated by its high-energy asymptotics only at rather large (). We have found that the cross section of bremsstrahlung in a strong Coulomb field by a high-energy electron in the region where a final electron has the energy coincides with the cross section of photoproduction at (slow electron). Finally, we have seen that the effect of screening for photoproduction close to the end of electron (positron) spectrum is important only for very small velocity of electron (positron).
We would like to thank R. N. Lee and V. M. Strakhovenko for valuable discussions. A. I. M. gratefully acknowledges the Max-Planck-Institute for Nuclear Physics for warm hospitality and financial support during his visit. The work was supported in part by RFBR under Grant No. 09-02-00024.
Appendix A Wave functions and Green’s function
The positive-energy wave function with total angular momentum , parity (), and projection of the total angular momentum on some quantization axis has the form BLP1980 ()