THE ASTROPHYSICAL JOURNAL, 482:372–376, 1997 June 10
© 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.

Magnetic Photon Splitting: Computations of Proper-Time Rates and Spectra

MATTHEW G. BARING 1 AND ALICE K. HARDING

Laboratory for High Energy Astrophysics, Code 661, NASA Goddard Space Flight Center, Greenbelt, MD 20771; baring@lheavx.gsfc.nasa.gov, harding@twinkie.gsfc.nasa.gov

Received 1996 September 6; accepted 1997 January 11


ABSTRACT

     The splitting of photons γ → γγ in the presence of an intense magnetic field has recently found astrophysical applications in polar cap models of γ-ray pulsars and in “magnetar” (i.e., neutron stars with extremely high fields) scenarios for soft gamma repeaters. Numerical computation of the polarization-dependent rates of this third-order QED process for arbitrary field strengths and energies below pair creation threshold is difficult; thus, early analyses focused on analytic developments and simpler asymptotic forms. The recent astrophysical interest spurred the use of the S-matrix approach by Mentzel, Berg, and Wunner to determine splitting rates. In this paper, we present numerical computations of a full proper-time expression for the rate of splitting that was obtained by Stoneham and is exact up to the pair creation threshold. While the numerical results derived here are in accord with the earlier asymptotic forms that are due to Adler, our computed rates still differ by as much as a factor of 3 from the S-matrix reevaluation of Wilke and Wunner, reflecting the extreme difficulty of generating accurate S-matrix numerics for fields below about 4.4 × 1013 G. We find that our proper-time rates appear to be very accurate and exceed Adler's asymptotic specializations significantly only for photon energies just below pair threshold and for supercritical fields, but always by less than a factor of ∼2.6. We also provide a useful analytic series expansion for the scattering amplitude valid at low energies.

Subject headings: gamma rays: bursts—gamma rays: theory—pulsars: general—radiation mechanisms: nonthermal—stars: magnetic fields—stars: neutron

FOOTNOTES

     1 Compton Fellow, Universities Space Research Association.

§1. INTRODUCTION

     The exotic quantum electrodynamical (QED) process of the splitting of a photon into two photons (γ → γγ) in the presence of a strong magnetic field has recently been of interest in the study of γ-ray pulsars and soft gamma repeaters (SGRs). Many radio pulsars are known to have spin-down fields in excess of 1012 G, with about 12 exceeding 1013 G, including the γ-ray pulsar PSR 1509-58. This source is unusual among the seven known γ-ray pulsars in that it has a hard power-law spectrum that turns over sharply around 1 MeV (Ulmer et al. 1993; Hermsen et al. 1997), whereas other pulsars (e.g., the Crab; see Nolan et al. 1993) for which the fields are nearer 1012 G have emission extending beyond 1 GeV. A question then naturally arises: does the high field of PSR 1509-58 cause its spectrum to differ so much from that of the other γ-ray pulsars? It appears that magnetic photon splitting, in concert with the more familiar process of single-photon pair creation, may provide the answer (Harding, Baring, & Gonthier 1996, 1997) in the context of polar cap models, inhibiting emission above a few MeV for a reasonable range of cap sizes. Photon splitting may also be quite important in the most recently discovered (Ramanamurthy et al. 1996) γ-ray pulsar, PSR 0656+14, which has a spin-down field of 9.3 × 1012 G, between that of the Crab and PSR 1509-58.

     Also of interest to high-energy astrophysicists are soft gamma repeaters (SGRs), the transient sources that are observed to have sporadic periods of high γ-ray activity. There are three known repeating sources, all with subsecond durations and soft spectra (kT ∼ 30 keV; for SGR 1900+14, see Kouvelioutou et al. 1993). The identification of radio (SGR 1806-20: Kulkarni & Frail 1993) and X-ray (e.g., 1979 March 5 repeater; Rothschild et al. 1994) counterparts to SGRs spawned much interest in these objects. It has been suggested that SGRs are magnetars, i.e., neutron stars with extremely high fields, in excess of the quantum critical field strength B$\mathstrut{_{{\rm cr}}}$=m$\mathstrut{^{2}_{e}}$ c$\mathstrut{^{3}}$eℏ=4.413×10$\mathstrut{^{13}}$ G (when the cyclotron and electron rest-mass energies are comparable); the spin-down estimate for the 1979 March 5 repeater is B ∼ 6 × 1014 G (Duncan & Thompson 1992), assuming its age is that of N49, its associated supernova remnant. Such fields far exceed those found in most radio pulsars and can perhaps be generated by enhancement that is due to dynamo action (Duncan & Thompson 1992). Baring (1995) proposed the potential importance of photon splitting for softening emission spectra in SGRs via a splitting cascade to reproduce the observed emission spectra, yielding an estimate of B ≳ 2 × 1014 G for SGR 1806-20. The observed stability of the spectra from outburst to outburst argues in favor of an equatorial emission region (Baring & Harding 1995; Harding & Baring 1996). Splitting can also be quite influential in SGR spectral models in the absence of cascade formation (e.g., Thompson & Duncan 1995).

     To date, astrophysical models that invoke splitting in the environments of γ-ray pulsars (Harding, Baring, & Gonthier 1996, 1997; Chang, Chen, & Ho 1996) and SGRs (e.g., Baring 1995; Baring & Harding 1995; Thompson & Duncan 1995; Harding & Baring 1996; Chang et al. 1996) use relatively simple approximations to the splitting rates that were derived by Adler et al. (1970, see also Adler 1971; Bialynicka-Birula & Bialynicki-Birula 1970), using effective Lagrangian techniques, which are valid for low photon energies ϵmec2 and fields satisfying ϵB ≪ Bcr. Restriction to this asymptotic regime has until recently been necessitated by the total lack of more general evaluations of splitting rates that are sufficiently amenable for use in astrophysical models: formal expressions for the third-order QED process of γ → γγ are very complicated. This has been a real deficiency since the ϵB ∼ Bcr regime can easily be realized in both SGRs and pulsars. This paper addresses this limitation by presenting numerical determinations of splitting rates for arbitrary field strengths and photon energies below the pair creation threshold of 2mec2, as obtained from the general proper-time QED formulation (a technique due to Schwinger) of splitting presented by Stoneham (1979).

     In this paper, our numerical computation of Stoneham's splitting rates focuses on the polarization mode ⊥ → ‖‖ that is permitted in the limit of weak dispersion by energy-momentum kinematic selection rules (see Adler 1971); ⊥ → ⊥⊥ and ‖ → ⊥‖ are the other two modes permitted by QED in the nondispersive limit, and results for them are similar to those presented here (Baring, Harding, & Weise 1997; hereafter BHW97). Here ⊥ and ‖ denote the orientation (perpendicular and parallel, respectively) of the photon electric vector relative to the plane containing its momentum and the field line. Our computed rates vary continuously from the low-energy limit and reproduce widely accepted analytic asymptotic forms at both low and highly supercritical field strengths. The numerical computations presented here therefore appear extremely reliable for use in astrophysical applications and concur with the recent numerical work of Baier, Milshtein, & Shaisultanov (1996), which resulted from their alternative Schwinger-type analysis of photon splitting. Note, however, that our rates differ dramatically from the numerical evaluation of the S-matrix determination of the splitting rate by Mentzel, Berg, & Wunner (1994; see also Wunner, Sang, & Berg 1995) that is orders of magnitude greater than the earlier proper-time determinations, and differs by as much as a factor of ∼3 from the recently updated S-matrix numerics of Wilke & Wunner (1996; see also the erratum to Wunner, Sang, & Berg 1995), which still show an inability to produce accurate results at subcritical field strengths, the domain applicable to pulsar models. This discrepancy underlies the inherent difficulty in computing S-matrix rates for relatively low fields and contrasts the reliability of the proper-time computations presented in this paper and also by Baier et al. (1996).

§2. PHOTON SPLITTING RATES AND SPECTRA

     Magnetic splitting γ → γγ is a relatively recent prediction of QED. A decade of controversy followed the earliest calculations before Bialynicka-Birula & Bialynicki-Birula (1970), Adler et al. (1970), and Adler (1971) performed, via an effective Langrangian technique, the first correct evaluations of its rate, including asymptotic forms in the limit of photon energies well below pair creation threshold, which varied as B6 when B ≪ Bcr. Their rate determinations neglected dispersion in the birefringent, magnetized vacuum, although the effects of such dispersion on the possibility of energy and momentum nonconservation and associated polarization selection rules was discussed in detail by Adler (1971). The early controversy was fueled by the inherent difficulties in calculating the rates of this third-order process by standard QED techniques, and the physics of photon splitting at high magnetic field strengths is still the subject of debate.

     The most general calculation of photon splitting rates was performed by Stoneham (1979), who used Schwinger's proper-time method to determine formal expressions for the splitting rate for ϵ below pair creation threshold and arbitrary field strengths. This analysis fully included dispersive effects that are due to magnetized vacuum in formal expressions for the rates, leaving the rates for the different polarization modes of splitting as unmanageable nine-dimensional integrals. Fortunately, in the limit of zero vacuum dispersion, Stoneham's formulae specialize to amenable triple integrals that involve only moderately complicated combinations of elementary functions (see Adler 1971, Papanyan & Ritus 1972 and Baier, Mil'shtein, & Shaisultanov 1986 for alternative presentations, and the recent compact results of Baier, Milshtein, & Shaisultanov 1996 [hereafter BMS96] and Adler & Schubert 1996). We have analytically developed these triple integral expressions (see BHW97) to facilitate our numerical purposes, eliminating leading-order terms (linear in photon energies) that cancel exactly, leaving scattering amplitudes that are cubic in photon energies. Such cancellations plague numerical evaluations of splitting rates and can be a principal source of error. Hence their elimination via an algebraic development is highly desirable. Readers interested in the details of our analytic reduction of Stoneham's (1979) expressions can refer to the more extensive presentation in BHW97, where the full expressions that are computed in this paper are exhibited.

     We note that while such analytic developments are expedient and very useful, it is possible to proceed by numerically subtracting off cancellations in a well-chosen manner. This is the approach discussed in Adler & Schubert (1996; see their eq. [17]), who have developed codes to compute the integrands (i.e., double integrals) of the triple integrals composing the scattering amplitude. Their codes, which are publicly available on the World Wide Web (the address is listed in Adler & Schubert 1996), demonstrate numerically the equivalence of the integrands that emerge from the work of Adler 1971, Stoneham (1979), BMS96, and also Adler & Schubert's presentation. Unlike the results presented in this paper and the work of BMS96, Adler & Schubert (1996) do not compute the full triple integral for the amplitude, a computationally demanding extra step that is necessary to obtain results of use for astrophysical applications. The numerical algorithm used to obtain the results of this paper involves a mixture of Simpson's rule and Gauss-Laguerre quadrature to obtain accurate differential and total splitting rates.

     For quite general photon energies and magnetic field strengths, the differential attenuation coefficient (i.e., the rate divided by c) of the photon splitting mode ⊥ → ‖‖ can be written in the form (BHW97)



for an initial photon of energy ϵ and produced photons of energies ϵ1 and ϵ2 = ϵ - ϵ1. Here ℳ denotes the scattering amplitude, scaled according to Stoneham's notation; other polarization modes assume a similar form. Hereafter, dimensionless units will be used, with B being expressed in units of Bcr and photon energies ϵ (initial) and ϵ1 and ϵ - ϵ1 (final) being scaled by mec2. In equation (1), αf is the fine structure constant, = 3.86 × 10-11 cm is the electron Compton wavelength over 2π, and &thetas; is the angle the photon momenta make with the field lines; equation (1) is strictly valid only in the zero-dispersion limit, for which the initial and final photon momenta are collinear. The triple integral expression we obtained for the scattering amplitude ℳ, somewhat lengthy for presentation here, has an integrand consisting purely of terms that are cubic combinations of ϵ - ϵ1 and ϵ1, multiplied by an exponential that is quadratic in these energies, all divided by the common factor ϵϵ1(ϵ - ϵ1).

     When the incident photon energy is low, namely ϵ ≪ 1 (i.e., 511 keV), these energy dependences in the integrand cancel exactly, the exponential loses its energy dependence, and dramatic analytic simplification is possible. The scattering amplitude then becomes dependent only on the field strength:



for arbitrary B. The form of ℳ$\mathstrut{_{1}}$&parl0;B&parr0; is just that given in Adler (1971) and equation (41) of Stoneham (1979), and is that used in the recent astrophysical models of SGRs (e.g., Baring & Harding 1995) and γ-ray pulsars (e.g., Harding, Baring, & Gonthier 1996, 1997). In the limit of B ≪ 1, ℳ$\mathstrut{_{1}}$&parl0;B&parr0;≈26/315, while in the limit of B ≫ 1, ℳ$\mathstrut{_{1}}$&parl0;B&parr0;≈1/&parl0;6B$\mathstrut{^{3}}$&parr0;. For arbitrary B and energies well below pair creation threshold, a series expansion of the scaled scattering amplitude ℳ$\mathstrut{_{{\bot}{\rightarrow}{\vert}{\vert}}}$ in terms of photon energy was obtained by BHW97:



where the produced photons assume dimensionless energies ϵ1 and ϵ2 = ϵ - ϵ1, and



and



The expansion in equation (3) is a result that is quite useful in pulsar models and can be reliably used for ϵ ≲ 0.5. For B ≲ 0.3, this series reduces to



     Recently, Baier, Milshtein, & Shaisultanov (1996) have derived the high-field limit (i.e., B ≫ 1) for splitting at arbitrary energies below pair creation threshold (ϵ ≤ 2); in the notation used here, their result becomes (for &thetas; = 90°)



for ϵ2 = ϵ - ϵ1. We have reproduced this formula identically from Stoneham's (1979) formulae and also our algebraic reduction of his expressions (BHW97). For photon energies far below pair creation threshold, equation (7) approaches 1/(6B3), the high-field limit of ℳ$\mathstrut{_{1}}$&parl0;B&parr0;, and the series expansion in energy maps over to the high-field limit of equation (3), for which ℳ$\mathstrut{_{11}}$∼-ℳ$\mathstrut{_{12}}$∼1/&parl0;30B$\mathstrut{^{3}}$&parr0;. The inverse cubic dependence of this result on the field implies an attenuation coefficient that is virtually independent of B in highly supercritical regimes.

     Total splitting attenuation coefficients are obtained by integrating equation (1) over produced energies 0 ≤ ϵ1 ≤ ϵ. For ϵ ≪ 1 and B ≪ 1, ℳ$\mathstrut{_{{\bot}{\rightarrow}{\vert}{\vert}}}$→ℳ$\mathstrut{_{1}}$&parl0;0&parr0;=26/315 and this integration is trivial, leading to a total attenuation coefficient of



In the limit of highly supercritical fields (B ≫ 1), no analytic evaluation of the total splitting rate (proportional to ϵ$\mathstrut{^{2}_{1}}$ϵ$\mathstrut{^{2}_{2}}$ times the square of the formula in eq. [7]) is apparent, as extensive development leaves some intractable terms. However, the approximation



is accurate to better than 1.5% at all energies below pair creation threshold and suffices for numerical purposes. The asymptotic formulae represented by equations (6) and (7) and also the approximate result in equation (9) provide strong checks on the accuracy of the numerical computations presented here.

     The computed attenuation coefficients for the splitting mode ⊥ → ‖ ‖ obtained by integrating equation (1) over produced photon energies ϵ1 are displayed in Figure 1a, for different field strengths, together with the magnetic pair creation rates for the ⊥ polarization (discussed in Daugherty & Harding 1983; Harding, Baring, & Gonthier 1997). Two characteristic properties of QED radiation processes in strong fields are immediately apparent. First, photon splitting, like pair creation, is a strongly increasing function of both the magnetic field strength (when B ≲ 1) and also of the energy of the incident (absorbed) photon. Second, at energies near pair threshold, as a third-order QED process, photon splitting has attenuation coefficients several orders of magnitude smaller than pair creation, a first-order process. Another striking feature of Figure 1a is how closely the coefficients resemble a ϵ5 power law, even for highly supercritical fields; equation (9) clearly reveals that the high B asymptotic limit (Fig. 1a, dashed curve) has at most modest (i.e., less than a factor of ∼2.6) deviations above the ϵ5 dependence near pair threshold. Further, Adler's (1971) simplest analytic form (∝ϵ5B6) is quite accurate below pair creation threshold for B ≲ 0.2. Deviations from ϵ5 behavior are illustrated in Figure 1b, where the ratio T$\mathstrut{^{{\rm tot}}_{{\bot}{\rightarrow}{\vert}{\vert}}}$/T$\mathstrut{_{0}}$ is plotted as a function of incident photon energy. For the different field strengths, quasi-horizontal portions of the curves indicate ϵ5 dependence; clearly from the figure, this simple asymptotic form is obeyed over a wide range of phase space. Figure 1b also demonstrates the impressive agreement between the numerical results obtained here and those of BMS96 and Adler (1971). The intersections of the various curves with the ordinate axis roughly defines corresponding values of (315/26) ℳ$\mathstrut{_{1}}$&parl0;B&parr0;.

Fig. 1

      Figure 2 exhibits differential photon production spectra, i.e., ℳ$\mathstrut{^{2}_{{\bot}{\rightarrow}{\vert}{\vert}}}$, normalized to unity. A remarkable property emerges: except for very near pair threshold, or for highly supercritical fields, the spectra closely approximate the low-energy asymptotic limit of ϵ$\mathstrut{^{2}_{1}}$&parl0;ϵ-ϵ$\mathstrut{_{1}}$&parr0;$\mathstrut{^{2}}$/30. Magnetic broadening of the spectrum (i.e., with increasing B), a feature of other strong field QED radiation processes, does however become very significant near pair creation threshold. These numerical results are also in accord with the recent alternative presentation of BMS96.

Fig. 2

§3. DISCUSSION

     Motivation for this numerical computation of proper-time results has been enhanced by a new result on the rates of photon splitting: Mentzel, Berg, & Wunner (1994, hereafter MBW94) presented an S-matrix calculation of the polarized splitting rates. While their formal development is equivalent (Weise, Baring, & Melrose 1997) to an earlier S-matrix formulation of splitting in Melrose & Parle (1983a, 1983b), their presentation of numerical results (see also Wunner, Sang, & Berg 1995, where the implications of their calculations were discussed in astrophysical contexts) was in violent conflict with the splitting results obtained by Adler et al. (1970), Adler (1971), Bialynicka-Birula & Bialynicki-Birula (1970), Papanyan & Ritus (1972), Stoneham (1979) and a number of other more recent works. The numerical rates of Mentzel, Berg, & Wunner (1994) for B = 0.1 and B = 1 are larger by orders of magnitude than our computations (and also those of BMS96). Their S-matrix rates exhibit a weak energy dependence well below pair threshold, and also exceed the pair creation rate at threshold (ϵ = 2) for B = 0.1. Wilke & Wunner (1996) have very recently retracted the earlier S-matrix rates (see also the recent erratum for Wunner, Sang, & Berg 1995), confirming a coding error in the numerics of MBW94. Their numerical reevaluation, a sample of which is depicted in Figure 1a, is much closer to the proper-time results but still differs significantly (by as much as a factor of 3.3 at ϵ = 0.1, B = 1, and 45% at ϵ = 1.9, B = 1 for the illustrated cases) from our determinations, unless the field is highly supercritical. The S-matrix and proper-time techniques should produce equivalent numerical results, and indeed they have done so demonstrably in the case of pair production (see Daugherty & Harding 1983; Tsai & Erber 1974). The proper-time analysis, although nontrivial, is computationally much more amenable than the S-matrix approach and has been reproduced in the limit of B ≪ 1 by numerous authors. Clearly, the work of Wilke & Wunner (1996), which does not exhibit any results below about 0.6Bcr, still demonstrates an inability to produce suitably accurate results at critical and subcritical field strengths. This point is not made clear in the recent erratum to Wunner, Sang, & Berg (1995). These inaccuracies can dramatically impact astrophysical applications to pulsar models (e.g., Harding, Baring, & Gonthier 1997), where factors of 3 become very significant. Such discrepancies underline the inherent difficulty in computing S-matrix rates for relatively low fields, where many Landau level quantum numbers for the three intermediate states must be summed over (e.g., see Weise et al. 1997) and contrasts the reliability of the proper-time computations presented in this paper, and by BMS96, that emanate from relatively simple integrals.

     In conclusion, this paper presents the major features of our numerical computations of proper-time expressions for photon splitting attenuation coefficients and spectra. Our numerical results are quite consistent with several analytic limits obtained by various groups, including the very recent approach of Adler & Schubert (1996), comparable to the recent numerical computations of BMS96, and well-behaved in the light of physical intuition related to strong field QED processes; they still differ significantly from and represent a very useful improvement over the S-matrix reevaluation of Wilke & Wunner (1996), which becomes problematic at field strengths below ∼4 × 1013 G. We believe that our proper-time computations can be reliably and efficiently used in astrophysical models.

ACKNOWLEDGMENTS

     We thank Stephen Adler and George Pavlov for reading the paper and for comments helpful to the improvement of the manuscript. This work was supported through Compton Gamma-Ray Observatory Guest Investigator Phase 5 and NASA Astrophysics Theory Program grants. M. G. B. thanks the Institute for Theoretical Physics at the University of California, Santa Barbara for support (under NSF grant PHY 94-07194) during part of the period in which work for this paper was completed. We thank Ramin Sina for making his pair creation code available to us for use in the preparation of Figure 1a.

REFERENCES

FIGURES


Full image (24kb) Full image (20kb) | Discussion in text
     FIG. 1.—(a) The computed photon splitting attenuation coefficients for polarization mode ⊥ → ‖‖, as functions of the incident photon energy ϵ, for field strengths B = 0.1, 1, 10 (solid lines below 2mec2, in ascending order), and the asymptotic rate resulting from the insertion of eq. (7) in eq. (1), labeled as B ≫ 1 (dashed curve). Photons are assumed to propagate orthogonally to the field lines (&thetas; = 90°). For comparison, the pair creation rates (Daugherty & Harding 1983) for ⊥ photons at two different field strengths are depicted above 2mec2. The filled circles denote the S-matrix determination of Wilke & Wunner (1996) for B = 1. (b) The ratio of the total attenuation coefficient T$\mathstrut{^{{\rm tot}}_{{\bot}{\rightarrow}{\vert}{\vert}}}$ to the low-energy, low-magnetic field asymptotic limit T0 (see eq. [8]) as a function of energy for different field strengths B, as labeled. The open squares represent ratios determined from the computations of Adler (1971), with the two points near ϵ = 0.1 being the ϵ = 0 results from Adler's Fig. 8. The filled triangles are derived from the numerical evaluations in Fig. 2 of BMS96. Impressive agreement between our results and those of Adler (1971) and BMS96 is evident.

Full image (21kb) | Discussion in text
     FIG. 2.—The computed photon splitting differential rates, normalized to unity and expressed as functions of the energy ϵ1 of one of the produced photons, for polarization mode ⊥ → ‖ ‖. The top three curves are for field strengths B = 0.1, B = 100, and B = 103 for initial photon energies ϵ = 1 (i.e., 511 keV), with the broad distribution corresponding to ϵ = 2 and B = 103. The B = 0.1 curves closely resemble the asymptotic low-energy form of ϵ$\mathstrut{^{2}_{1}}$&parl0;ϵ-ϵ$\mathstrut{_{1}}$&parr0;$\mathstrut{^{2}}$/30, which can be deduced from eq. (1), while the B = 103 distributions closely resemble the shape of the square of the high field limit in eq. (7).