Compton scattering in strong magnetic fields: Spin-dependent influences at the cyclotron resonance

# Compton scattering in strong magnetic fields: Spin-dependent influences at the cyclotron resonance

Peter L. Gonthier Hope College,
Department of Physics,
27 Graves Place
Holland, Michigan 49423, USA
Matthew G. Baring Department of Physics and Astronomy, MS-108,
Rice University,
P.O. Box 1892,
Houston, Texas 77251-1892, USA
Matthew T. Eiles Hope College,
Department of Physics,
27 Graves Place,
Holland, Michigan 49423, USA
Zorawar Wadiasingh Department of Physics and Astronomy, MS-108,
Rice University,
P.O. Box 1892,
Houston, Texas 77251-1892, USA
Caitlin A. Taylor Hope College,
Department of Physics,
27 Graves Place,
Holland, Michigan 49423, USA
Catherine J. Fitch Grinnell College,
Department of Physics,
1115 Eight Avenue,
Grinnell, Iowa 50112, USA
August 2, 2019
###### Abstract

The quantum electrodynamical (QED) process of Compton scattering in strong magnetic fields is commonly invoked in atmospheric and inner magnetospheric models of x-ray and soft gamma-ray emission in high-field pulsars and magnetars. A major influence of the field is to introduce resonances at the cyclotron frequency and its harmonics, where the incoming photon accesses thresholds for the creation of virtual electrons or positrons in intermediate states with excited Landau levels. At these resonances, the effective cross section typically exceeds the classical Thomson value by over 2 orders of magnitude. Near and above the quantum critical magnetic field of 44.13 TeraGauss, relativistic corrections must be incorporated when computing this cross section. This profound enhancement underpins the anticipation that resonant Compton scattering is a very efficient process in the environs of highly magnetized neutron stars. This paper presents formalism for the QED magnetic Compton differential cross section valid for both subcritical and supercritical fields, yet restricted to scattered photons that are below pair creation threshold.Calculations are developed for the particular case of photons initially propagating along the field, and in the limit of zero vacuum dispersion, mathematically simple specializations that are germane to interactions involving relativistic electrons frequently found in neutron star magnetospheres. This exposition of relativistic, quantum, magnetic Compton cross sections treats electron spin dependence fully, since this is a critical feature for describing the finite decay lifetimes of the intermediate states. Such lifetimes are introduced to truncate the resonant cyclotronic divergences via standard Lorentz profiles. The formalism employs both the traditional Johnson and Lippmann (JL) wave functions and the Sokolov and Ternov (ST) electron eigenfunctions of the magnetic Dirac equation. The ST states are formally correct for self-consistently treating spin-dependent effects that are so important in the resonances. It is found that the values of the polarization-dependent differential cross section depend significantly on the choice of ST or JL eigenstates when in the fundamental resonance, but not outside of it, a characteristic that is naturally expected. Relatively compact analytic forms for the cross sections are presented that will prove useful for astrophysical modelers.

non-thermal radiation mechanisms and magnetic fields and neutron stars and pulsars and X-rays
###### pacs:
12.20.Ds, 95.30.Cq, 95.85.Nv, 95.85.Pw, 97.60.Gb, 97.60.Jd, 98.70.Rz
preprint: APS/123-QED

Accepted for publication (August 2014) in Physical Review D.

## I Introduction

The physics of Compton scattering in strong magnetic fields has been studied fairly extensively over the last four decades, motivated at first by the discovery of cyclotron lines in accreting x-ray binary pulsars (see Truemper78 () for Her X-1, Wheaton79 () for 4U 0115+634, Makishima90 () for X0331+53, and Grove95 () for A0535+26), a genre of neutron stars. More recently, constraints on stellar magnetic dipole moments obtained from pulse timing observations have led to the identification of the exotic and highly magnetized class of neutron stars now known as magnetars [e.g., see VG97 () for the Anomalous X-ray Pulsar (AXP) 1E 1841-045, Kouv98 () for Soft Gamma-Ray Repeater (SGR) 1806-20, Wilson99 () and references therein for AXP 4U 0142+61, and Kouv99 () for SGR 1900+14] — this topical development has promoted a resurgence in the interest of this intriguing physical process. In classical electrodynamics, Thomson scattering in an external field evinces a pronounced resonance for incoming photons at the cyclotron frequency in the electron rest frame (ERF), within the confines of Larmor radiation formalism CLR71 (); GS73 (); BM79 (). This feature also appears in quantum formulations of magnetic Thomson scattering CLR71 (); deRHDM74 (); Herold79 () appropriate for fields Gauss. For neutron star applications it is often necessary to employ forms for the magnetic Compton scattering cross section that are computed in the relativistic domain. This is dictated by the atmospheric or magnetospheric fields of such compact objects possessing strengths either approaching, or exceeding (in the case of magnetars) the quantum critical field Gauss, i.e., that for which the electron cyclotron and electron rest mass energies are equal. Such results from quantum electrodynamics (QED) have been offered in various papers deRHDM74 (); Herold79 (); DH86 (); BAM86 () at various levels of analytic and numerical development. In particular, Refs. DH86 (); BAM86 () highlight the essential contributions provided by relativistic quantum mechanics, namely, the appearance of multiple resonances at various “harmonics” of the cyclotron fundamental and strong Klein-Nishina reductions that are coincident with electron recoil when the incident photon has an energy exceeding around in the electron’s initial rest frame. For photons incident at nonzero angles to the magnetic field, the harmonic resonances are not equally spaced in frequency DH86 (), and they correspond to kinematic arrangements that permit excitation of the intermediate virtual electron to various Landau levels — the discrete eigenvalues of energy transverse to the field.

Extant QED calculations of the magnetic Compton process in the literature DH86 (); BAM86 () emphasize frequency domains either away from the resonances, or in the wings of the resonances, and presuming infinitely long-lived intermediate states, and therefore possess divergent resonances at the cyclotron harmonics. This suffices for several astrophysical applications, for example, the consideration of Compton scattering contributions to opacity in forging atmospheric or photospheric structure in magnetars Ozel01 (); HoLai03 (); SPW09 (). However, for other applications that sample the resonances preferentially, such as the resonant Compton upscattering models of magnetar spectra and associated electron cooling in BH07 (); BWG11 (); Belob13 (), a refined treatment of the cross section in the resonances is necessary. The divergences appear in resonant denominators that emerge from Fourier transforms of the spatial and temporal complex exponentials in the wave functions: these denominators capture the essence of precise energy conservation at the peak of the resonance. Since the intermediate state is not infinitely long-lived, its energy specification is not exact, and consequently the divergences are unphysical, and must be suitably truncated. The appropriate approach is to introduce a finite lifetime or decay width to the virtual electrons for cyclotronic transitions to lower excited Landau levels, most commonly to the ground state. This introduces a Breit-Wigner prescription and forms a Lorentz profile in energy to express the finiteness of the cross section through any resonance WS80 (); BAM86 (); HD91 (); Graziani93 (); GHS95 (). Historically, when this approach has been adopted, spin-averaged widths (i.e., inverse decay times) for the virtual electrons have been inserted into the scattering formalism. While expedient, this is not precise in that a self-consistent treatment of the widths does not amount to a linear characterization of the overall spin dependence. In other words, averaging the spins in forming does not correctly account for the coupling of the spin dependence of the temporal decay of the intermediate electron with the spin dependence of the spatial portion of its wave functions, i.e., the spinors. Rectifying this oversight in prior work is a principal objective of this paper.

Another technical issue with calculating QED interactions in strong magnetic fields is the choice of the eigenstate solutions to the magnetic Dirac equation. Historically, several choices of wave functions have been employed in determinations of the Compton scattering cross section and cyclotron decay rates. The two most widely used wave functions are those of Johnson and Lippmann (JL) JL49 () and Sokolov and Ternov (ST) ST68 (). The JL wave functions are derived in Cartesian coordinates and are eigenstates of the kinetic momentum operator . The ST wave functions, specifically their “transverse polarization” states, are derived in cylindrical coordinates and are eigenfunctions of the magnetic moment (or spin) operator (with ) in Cartesian coordinates within the confines of the Landau gauge .

Given the different spin dependence of the ST and JL eigenstates, one must use caution in making the appropriate choice when treating spin-dependent processes. Herold, Ruder and Wunner HRW82 () and Melrose and Parle MP83a () have noted that the ST eigenstates have desirable properties that the JL states do not possess, such as being eigenfunctions of the Hamiltonian including radiation corrections, having symmetry between positron and electron states, and diagonalization of the self-energy shift operator. Graziani Graziani93 () and Baring, Gonthier and Harding BGH05 () noted that spin states in the ST formalism for cyclotron transitions are preserved under Lorentz boosts along B, a convenient property. In contrast, the JL wave functions mix the spin states under such a Lorentz transformation, and therefore are not appropriate for a spin-dependent formulation of the cyclotron process. In Graziani93 () it was observed that the ST wave functions are the physically correct choices for spin-dependent treatments and for incorporating widths in the scattering cross section. Although the spin-averaged ST and JL cyclotron decay rates are equal, their spin-dependent decay rates are not, except in the special case in which the initial component of momentum of the electron parallel to the magnetic field vanishes.

The differential cross sections for both ST and JL formalisms of magnetic Compton scattering are developed in parallel in this paper, for all initial and final configurations of photon polarization. They apply for kinematic domains below pair creation threshold. These are implemented for both spin-average cyclotronic decay rates and spin-dependent widths for the intermediate state, which are employed in a Breit-Wigner prescription to render the cyclotron resonances finite. This element of the analysis mirrors closely that in HD91 (). The ST formulation uses the general formalism presented by Sina96 (), while the JL case is an adaptation of the work of DH86 () and Getal00 (). The developments are specialized early on to the particular case of photons propagating along B in the ERF. This is actually quite an important case in astrophysical settings, since it corresponds to interactions where relativistic electrons are speeding along magnetic field lines above the stellar surface. In such cases, Lorentz boosting parallel to B collimates the interacting photon angles in the ERF almost along the local field line. The analysis spans a wide range of field strengths, focusing particularly on the regime around Gauss Gauss, and thereby magnetic domains pertinent to millisecond pulsars, young radio and gamma-ray pulsars, and magnetars. In particular, focus on the resonance regime is germane to Compton upscattering models BH07 (); FT07 (); ZTNR11 (); Belob13 () of the hard x-ray tails observed in quiescent emission above 10 keV from several magnetars kuip04 (); mereg05 (); goetz06 (); hartog08 (). The inverse Compton process, where ultrarelativistic electrons scatter seed x-ray photons from the surface of a neutron star, has gained popularity as the preferred mechanism for generating these powerful pulsed signals, primarily due to the efficiency of scattering in the cyclotron resonances for strong magnetic fields ZTNR11 (); Belob13 (); BH07 (); BWG11 (); NTZ08 (). The cross section calculations presented in this paper will also be pertinent to future computations of Compton opacity in dynamic plasma outflows that are postulated dt92 (); td96 () to be responsible for hard x-ray flaring activity seen in magnetars (e.g., see Lin11 () for SGR J0501+4516 and Lin12 () for AXP 1E 1841-045 and SGR J1550-5418).

After developing the general formalism for the scattering differential cross section in Sec. II, the exposition narrows the focus to the incoming photons beamed along the local field direction in the ERF, denoted by . This culminates in the relatively compact formulas for the polarization-dependent cross sections in Eq. (39) for ST, JL, and spin-averaged analyses. The specialization simplifies the mathematical development dramatically, since the associated Laguerre functions that appear as the transverse dependence (with respect to B) of the eigenfunctions of the magnetic Dirac equation reduce to comparatively simple exponentials. Furthermore, only the single resonance at the cyclotron fundamental appears. Various elements of the numerical character of the differential and total cross sections are presented in Sec. III. Outside the resonance, spin influences are purely linear in their contributions, and so both ST and JL formulations collapse to the spin-averaged case, as expected. In the resonance, appreciable differences between the ST, JL, and spin-averaged formulations arise, at the level of around 50% when Gauss, and rising to a factor of 3 for one polarization scattering mode in fields Gauss. The origin of this difference is in the fact that the coupling between the intermediate electron’s momentum parallel to the field and its spin in defining its decay width is dependent on the choice of eigenfunction solutions of the magnetic Dirac equation. It is in this resonant regime that the physically self-consistent, spin-dependent Sokolov and Ternov cross sections presented in this paper provide an important new contribution to the physics of magnetic Compton scattering.

An interesting anomaly emerges from spin-dependent influences at very low initial photon frequencies , i.e., well below the cyclotron fundamental, and is highlighted in Sec. III.3. In the domain , the finite lifetime of the intermediate state is inferior to the inverse frequency, and so the cross section saturates at a small constant value, times the Thomson value. This dominates the usual low-frequency behavior for the case CLR71 (); Herold79 (). In Sec. III.4, to facilitate broader utility of the new ST results offered here for use in astrophysical applications, compact analytic approximate expressions for the differential cross section are derived in Eq. (81), together with Eqs. (III.5), (83) and (III.5). These integrate to yield the approximate total cross sections in Eqs. (91) and (92), fairly simple results whose integrals can be efficiently computed when employing a convenient series expansion in terms of Legendre polynomials. These approximate resonant cross section results include both polarization-dependent and polarization-averaged forms, and are accurate to better than the 0.1% level. The net product is a suite of Compton scattering physics developments that can be easily deployed in neutron star radiation models.

The paper concludes with a discussion of the issue of photon dispersion in the magnetized vacuum. It is indicated that for astrophysically interesting field strengths, i.e., those below around Gauss, vacuum dispersion is generally small: the refractive indices of the birefringent modes deviate from unity by a few percent, at most, and generally much less. Accordingly, the influence of such dispersion on kinematic quantities pertaining to the photons is neglected in the scattering developments offered here.

## Ii Development of the Cross Section

The Compton cross section can be developed along the lines of the work of Daugherty & Harding DH86 () and more recent work of Sina Sina96 (). In this Section, we begin with more general elements of the formalism, and then specialize to our specific developments that focus on incident photons propagating along the magnetic field.

### ii.1 General formalism

The magnetic Compton scattering cross section can be expressed as integrations of the square of the -matrix element over the pertinent phase space factors for produced electrons and photons. The protocols for its development are standard in quantum electrodynamics, and, for unmagnetized systems, can be found in works by Jauch and Rohrlich JR80 () (see Secs. 8-6 and 11-1, therein, for nonmagnetic Compton formalism). When , the formulation is modified somewhat to take into account the quantization of momenta perpendicular to B (assumed to be in the direction throughout), and, to a large extent, parallel to the two-photon magnetic pair annihilation exposition in DB80 () [see Eq. (21) therein] because of crossing-symmetry relations. The total cross section for magnetic Compton scattering can be written by adapting Eq. (11-3) of JR80 (),

 σ=∫L3|Sfi|2(vrelT(L3d3kf((2πλto0.0pt−−)3(L3d3pf((2πλto0.0pt−−)3(→λto0.0pt−−2∫L3(1−βicosθi(|Sfi|2(λto0.0pt−−2cT(L3d3kf((2πλto0.0pt−−)3(Ldpf(2πλto0.0pt−−(BLdaf(2πλto0.0pt−−( (1)

using standard notation, where is the Compton wavelength of the electron over . In terms of the formation of the -matrix element, the time denotes the duration of the temporal integral, and the spatial integrations are over a cube of side length . Hereafter, the magnetic field strength will be expressed in units of the quantum critical value (Schwinger limit) Gauss, the field at which the electron cyclotron energy equals its rest mass energy. Also, throughout this paper, all photon and electron energies and momenta will be rendered dimensionless via scalings by and , respectively. The phase space correspondence for the scattered electrons, due to the quantization of their transverse energy levels, is routinely established: see Appendix E of Sina96 () or Sec. 4(c) of MP83b ().

The incoming electron speed is , parallel to the magnetic field, and the initial photon makes an angle with respect to the magnetic field direction, so that is the relative speed of the colliding photons and electrons. Eventually, the main focus of this paper will specialize to the particular case where the initial electron is in the ground state (lowest Landau level), and also possesses zero parallel momentum, , so that then . Following common practice, this will be referred to as the ERF, where it is understood that this applies to the initial electron throughout. One can always consider scattering in such a frame by performing a Lorentz boost parallel to B to eliminate any component of momentum of the initial electron parallel to the field.

Various definitions germane to Eq. (1) and kinematic identities are now outlined. The two Feynman diagrams for the scattering are depicted in Fig. 1. In general, the incoming electron and photon four-momenta are and , respectively, and represents the spatial location (dimensionless, i.e., in units of ) of the guiding center of the incoming electron. The corresponding quantities for the outgoing electron are , , and . The energies of the incoming and outgoing electrons in the quantizing field are generally given by

 Ej=√1+2jB+p2j,Eℓ=√1+2ℓB+p2ℓ (2)

for Landau level quantum numbers and , and dimensionless momenta and parallel to the field, respectively. The quantization of leptonic momenta transverse to the field implies that the correspondences and for their four-momenta are implicit in the symbolic depiction of Fig. 1.

For most of the paper, considerations are restricted to the ERF where the momentum component of the incoming electron parallel to B is set to zero. Differential cross sections for nonzero initial electron momenta along the field can be quickly recovered via Lorentz transformation of the forms presented in this paper; the total cross section is an invariant under such boosts. In this ERF specialization, one has and . The energy of the intermediate state assumes a similar form and is denoted by . The kinematic relations between the four-momenta of the incoming and outgoing species can be expressed via [e.g., see Eq. (15) of Getal00 ()]

 ωf=2(ωi−ℓB)r(1+√1−2(ωi−ℓB)r2sin2θf(,r=1(1+ωi(1−cosθf)( (3)

for the photon, which initially assumes an angle relative to B, and is scattered to an angle relative to the field direction. A simple rearrangement of this kinematic relation yields the following convenient form:

 (ωf)2sin2θf−2ωiωf(1−cosθf)+2(ωi−ℓB−ωf)=0. (4)

The specialization to cases is made throughout this paper, following Getal00 (); its astrophysical relevance is discussed soon below. Relaxation of the approximation will be developed in future work. The final electron’s parallel momentum and energy are given by

 pℓ=ωi−ωfcosθf,Eℓ=1+ωi−ωf≡√1+2ℓB+(ωi−ωfcosθf)2. (5)

The equivalence of the forms for can be derived from Eq. (3). In the appendixes, the quantities and are labeled by and for the special case of that will be generally adopted here. Note that more general kinematic identities for Eqs. (3) and (5), applicable for arbitrary incoming electron and photon momenta, can be found in DH86 ().

To facilitate the formation of the different cross section in terms of the angles of the outgoing photon, the identification is forged in Eq. (1), where . The -matrix element receives two contributions,

 Sfi=S(1)fi+S(2)fi, (6)

one for each of the two Feynman diagrams (e.g., see Sec. 8-2 of JR80 ())

 S(1)fi = −4πiα\sevenrm f∫d4x′∫d4xψ†f(x′)γμAμf(x′)GF(x′,x)γνAνi(x)ψi(x) (7) S(2)fi = −4πiα\sevenrm f∫d4x′∫d4xψ†f(x′)γμAμi(x′)GF(x′,x)γνAνf(x)ψi(x),

labeled (1) and (2) in Fig. 1, respectively. Here, and are the initial and final electron wave functions (see Appendix A for more details), and they are solutions of the magnetic Dirac equation:

 γμ(ℏc∂μ−ieAμ)ψ+mec2ψ=0, (8)

for Dirac gamma matrices and . The electron propagator or Green’s function, , satisfies the inhomogeneous counterpart Dirac equation, where the right-hand side of Eq. (8) is replaced by ; it is detailed at greater length just below. In Eq. (7), and are the initial and final photon vector potential functions, and they assume the generic form

 Aμ(x)=1(√2ω(L/λto0.0pt−−)3(ϵμexp{ikνxν}∝ei(k⋅x−ωt) (9)

for photon polarization vector and four-momentum (wave vector) . These are identical to their field-free forms: see Sec. 7-7 of BD64 () or Sec. 4-4 of Sakurai67 () for -matrix construction of unmagnetized Compton scattering in QED. Observe that due to the crossing symmetry, the photons are interchanged between contributions from the first and second Feynman diagrams.

The seminal papers of DH86 () and BAM86 () both originally derived QED formulations for Compton scattering in strong magnetic fields using JL JL49 () particle basis states for QED solutions to the Dirac wave equation. Later Sina96 () employed ST ST68 () “transverse polarization” basis states as solutions of the magnetic Dirac equation, incorporating spin-dependent widths at the cyclotron resonance. All of these works computed differential cross sections for scattering in the ERF and encompassed arbitrary angles of photon incidence relative to the magnetic field direction. In this study, we follow closely the development of Sina Sina96 (), specializing to the particular case of photon incidence angles along B, i.e., . This special case is an astrophysically important one for neutron star magnetospheres in that it applies to scatterings of x rays by ultrarelativistic electrons, when the laboratory angle of incidence of the incoming photons is Lorentz contracted to in the ERF. In particular, it is germane to Compton upscattering models of energetic x-ray production in magnetars BH07 (); BWG11 (). The specialization was explored by Getal00 () in JL scattering formalism, where it was highlighted that the single final Landau ground state of accounts for the entire cross section up to the cyclotron resonance at , above which transverse quantum numbers (i.e. excitations) begin to contribute. These connections provide ample motivation for restricting this work, our incipient study of spin-dependent resonant scattering, to ground-state–ground-state transitions.

In this presentation, the spin-dependent resonant width is included in a similar fashion to that in BAM86 (); HD91 (); GHS95 (): it represents the decay lifetime of the intermediate state, and therefore appears as an imaginary contribution to the energy of this virtual state. This modification therefore appears in the complex exponentials for the time dependence, and, after integration, yields complex corrections to the resonant denominators. Eventually, after squaring of the -matrix elements, its inclusion generates a truncation of all cyclotronic resonances via Lorentz profiles of width that depends on the spin of the intermediate electron or positron. This is an important inclusion in Compton scattering formalism that is required for precise computations of resonant upscattering spectra and associated electron cooling rates in models of x-ray and gamma-ray emission from neutron star magnetospheres. Following the work of GHS95 (), we can describe the bound state electron propagator [see Eq. (15) of DB80 (), which extends Eq. (6.48) of BD64 () to accommodate the quantization associated with the external magnetic field], including the appropriate widths in the expression

 GF(x′,x)=(L(2πλto0.0pt−−()2B∫dan∫dpn∞∑n=0{−iθ(t′−t)Δ+(x′,x)+iθ(t−t′)Δ−(x′,x)}, (10)

for

 Δ+(x′,x) = ∑s=±usn(x′)u†sn(x)exp[−i(En−iΓs/2)(t′−t)] (11) Δ−(x′,x) = ∑s=±vsn(x′)v†sn(x)exp[+i(En−iΓs/2)(t′−t)].

This form for the Green’s function can be applied to any choice of electronic wave functions that satisfies the magnetic Dirac equation [see Eq. [6.39] of BD64 ()]; in the absence of decay of the intermediate state, , and this reduces to the result in DB80 (). The in Eq. (10) is the unit step function implemented in the standard expansions of the Green’s functions, which is zero for negative arguments and unity for positive ones. The and contributions correspond to the positive and negative frequency portions of the Fourier transform BD64 (). The and constitute the spatial parts of the electron and positron wave functions, respectively. The quantities and denote the coordinate of the orbit center and longitudinal momentum component, respectively, of the intermediate state.

It is of crucial importance to understand that there is a coupling between the wave functions and and the excited state decay width that is spin dependent: one should not sum these spin dependences separately when computing the electron propagator. Outside cyclotronic resonances, the impact of this coupling virtually disappears as one can then set in Eq. (11). More particularly, for the scattering problem, the electron propagator captures motion parallel to B via the kinematics of Compton scattering. It is the presence of this parallel momentum of the intermediate state coupled intimately with spin [deducible from Eqs. (29) and (32) and supporting text below] that breaks the degeneracy between JL and ST formulations and renders the cross section in the resonance dependent on the choice of basis states, but only when decay widths are incorporated.

Before proceeding, some remarks about gauge choices are in order. As in Refs. Herold79 (); DB80 (), we use the standard Landau gauge to represent the field , where , [contrasting Johnson and Lippmann JL49 () who adopted ]. This freedom exploits the fact that the total cross section is independent of the choice of gauge for specifying the electron wave functions. Changing gauge introduces a complex exponential factor with the gauge modification as its argument. In other words, the contact transformation

 ψ(x,t)→ψ′(x,t)=ψ(x,t)exp{ie(ℏc(Λ(x)} (12)

yields as a solution of the transformed Dirac equation if is a solution of Eq. (8) for the original gauge. This phase change property is well known. We restrict considerations to spatial gauge transformations here, assuming time independence of the external field. Under this contact transformation, it is easily seen that the Green’s function defined by Eqs. (10) and (11) transforms according to

 GF(x′,x)→GF(x′,x)exp{ie[Λ(x′)−Λ(x)]}, (13)

since the integrations over and do not impact the spatial factors involving the s. Observe that here we have reverted to our natural unit convention . In contrast, the wave-function products in the -matrix expressions in Eq. (7) transform via an exponential factor that is precisely the complex conjugate of the one in Eq. (13). The quantized fields for the external photon lines only couple to the ambient magnetic field through vacuum dispersion, and, therefore, gauge-invariant absorptive processes (discussed in Sec. IV below), and so are not influenced by such a gauge transformation. Accordingly, the -matrices and the scattering differential cross section are independent of the choice of gauge.

Inserting the Green’s function into Eq. (7) for the first diagram, one obtains

 S(1)fi=−(2π)2α\sevenrm f(√ωiωf((λto0.0pt−−(L()3∫d3x′∫d3xe−ikf⋅x′eiki⋅x(L(2πλto0.0pt−−()2×B∫dan∫dpn{ϖ+(x′,x)−ϖ−(x′,x)}, (14)

where

 ϖ+(x′,x) = ∞∑n=0∑s=±[u†(t)ℓ(x′)Mfu(s)n(x′)][u†(s)n(x)Miu(r)j(x)] (15) ×∫∞−∞dtei(En−Γs/2−ωi−1)t∫∞tdt′ei(Eℓ+ωf−En+Γs/2)t′, ϖ−(x′,x) = ∞∑n=0∑s=±[u†tℓ(x′)Mfv(s)n(x′)][v†(s)n(x)Miu(r)j(x)] ×∫∞−∞dt′ei(Eℓ+ωf+En−Γs/2)t′∫∞t′dtei(−En+Γs/2−ωi−1)t.

The contribution from the second Feynman diagram can be similarly transcribed. The matrices express the polarization states in terms of gamma matrices, via the relations for the incoming photon and for the scattered photon, adopting the definitions in Mel13 () for the two orthogonal polarization vectors discussed in Appendix A. In this paper, we adopt the standard convention for the labeling of the photon linear polarizations: refers to the state with the photon’s electric field vector parallel to the plane containing the magnetic field and the photon’s momentum vector, while denotes the photon’s electric field vector being normal to this plane.

This polarization convention is appropriate for domains where one can neglect the dispersion of light propagation in either plasma, or the birefringent vacuum that is polarized by a large-scale electromagnetic field. Such a convention is commonplace in treatments of QED processes in strong magnetic fields, but it is not absolutely accurate in that the refractive index is not precisely unity. It then becomes an approximation, , to the true eigenmodes of propagation that are eigenvalues of the polarization tensor, which satisfy . Precise treatment of photon eigenmodes [see Eq. (46) of Adler71 () or Eqs. (11) and (12) of Shabad75 ()] in the magnetized vacuum generally would entail the addition of substantial or prohibitive mathematical complexity to scattering cross sections, and also rates for other processes, such as pair creation and cyclotron transitions. Fortunately, such dispersive modifications are generally a small influence for astrophysically interesting field strengths, even for magnetars. The character of vacuum dispersion and the small magnitude of its impact for the scattering problem is discussed at length in Sec. IV. There it becomes evident that the nondispersive approximation is appropriate for fields when the scattered photon perpendicular energy is below pair creation threshold, .

The development of the -matrix element in Eq. (7) mirrors that leading to Eqs. (3) and (4) in DH86 (). The temporal integrations are simply evaluated, leading to the appearance of the energy conservation function in Eq. (16) and the resonant denominators in Eq. (17) below. These steps culminate in the expression

 Sfi=−iα\sevenrm f(√ωiωf(λto0.0pt−−(L(δ(1+ωi−Eℓ−ωf)∞∑n=0∑s=±B∫dan∫dpn[T(1)n+T(2)n] (16)

after simple integration over the temporal dimensions. Here the sum over the index captures the Landau level quantum numbers of the intermediate state, and the sum over the index accounts for the different spins of this state. The spatial integrals are encapsulated in the terms

 T(1)n=S(1)u(1+ωi−En+iΓs/2(+S(1)v(1+ωi+En−iΓs/2( (17)

where

 S(1)u = [∫d3xe−ikf⋅xu†(t)ℓ(x)Mfu(s)n(x)][∫d3xeiki⋅xu†(s)n(x)Miu(r)j(x)] (18) S(1)v = [∫d3xe−ikf⋅xu†(t)ℓ(x)Mfv(s)n(x)][∫d3xeiki⋅xv†(s)n(x)Miu(r)j(x)]

and

 T(2)n=S(2)u(1−ωf−En+iΓs/2(+S(2)v(1−ωf+En−iΓs/2( (19)

where

 S(2)u = [∫d3xeiki⋅xu†(t)ℓ(x)Miu(s)n(x)][∫d3xe−ikf⋅xu†(s)n(x)Mfu(r)j(x)] (20) S(2)v = [∫d3xeiki⋅xu†(t)ℓ(x)Miv(s)n(x)][∫d3xe−ikf⋅xv†(s)n(x)Mfu(r)j(x)].

For the numerators , the index identifies the corresponding Feynman diagram in Fig. 1, and the subscripts mark the contributions from electron () and positron () propagators. Here, the and in Eqs. (18) and (20) represent the electron and positron spinor wave functions in the Landau state with energy quantum number . The indices and therein refer to the spin of the initial and final electron states, while the index refers to the spin of the intermediate lepton. The integrals appearing in the products within the numerators , termed vertex functions by MP83a (); MP83b (), are further developed in Appendix B, leading to the explicit appearance of functions familiar in -matrix calculations of QED processes in external magnetic fields ST68 (); HRW82 (); MP83a (); BGH05 (), including, specifically, expositions on Compton scattering Herold79 (); DH86 (); BAM86 (); GHS95 (); Getal00 (). These functions include exponentials and associated Laguerre functions in the photon variables that control the ensuing mathematical character of the cross section. Observe that the crossing symmetry relations

 ωi↔−ωf,ki↔−kf,ϵi↔ϵf (21)

identify the substitutions required to form the terms from , and vice versa.

It is immediately apparent from Eqs. (17) and (19) that the incorporation of widths that are dependent on the spin of the intermediate state imposes spin dependence on both the numerator and the denominator, which must first be developed separately and then summed. This is formally the correct protocol, and as we shall demonstrate, it leads to dependence of the resonant cross section on the spin of the excited virtual electron. If on the other hand, one were to implement the spin-averaged widths, then the terms are added within the denominator, thereby leading to significant simplification of the terms. This is the historically conventional approach that is employed for magnetic Compton scattering calculations away from the cyclotron resonances (i.e., when can be presumed), but is imprecise in such resonances. This is the crux of the offering here, providing the mandate for our refinements of the magnetic Compton cross section in the cyclotron resonances.

Using standard squaring techniques, the norm of the -matrix element can be expressed in the form

 ∣∣Sfi∣∣2 = × δ(1+ωi−Eℓ−ωf)exp(−k2⊥i+k2⊥f(2B()1(Ef(∣∣ ∣∣∞∑n=0[F(1)n,seiΦ+F(2)n,se−iΦ]∣∣ ∣∣2

with the fine-structure constant, and the time and length are those of the spacetime box for the perturbation calculation. Here standard terms and the phase factor emerges from the integrals of the products of the matrix elements in the numerators of the terms, which are similar to the ones in DH86 (); BAM86 (). The specific form for the phase factor is provided in Eq. (119) in Appendix B. Again, the labels for the terms correspond to the associated Feynman diagrams. The delta functions in express four-momentum conservation and emerge naturally from the Fourier transform manipulations of the incoming and outgoing plane-wave portions of the wave functions for the photons and electrons. The parameters and constitute the -coordinate orbit center of the incoming and outgoing electrons, respectively, and disappear from the -matrix after integration over and .

Observe that the exponential portion of the cross section that depends on the photon momenta perpendicular to the field, and , is explicitly isolated in this construction. The initial electron has a parallel momentum , a quantity that does not appear explicitly in the second function in Eq. (II.1) that describes momentum conservation parallel to the magnetic field. In contrast, (the parallel momentum of the scattered electron) appears in this function. This form for the square of the -matrix is just that in Eq. (6) of DH86 (), but specialized to the ERF case where the initial electron is in the ground state and possesses a zero component of momentum along B. Due to the azimuthal symmetry of the scattering, without loss of generality one can orient the coordinate system so that the initial photon momentum is along the axis by selecting such that and . On the other hand, is nonzero in general, and has both and components.

Inserting Eq. (II.1) into Eq. (1), the differential cross section in the rest frame of the electron can be readily obtained:

 dσ(dΩf(=3σ\sixrm T(8π(ωf(ωiK(e−(ω2isin2θi+ω2fsin2θf)/2B∣∣ ∣∣∞∑n=0∑s=±[F(1)n,seiΦ+F(2)n,se−iΦ]∣∣ ∣∣2 (23)

for general photon incidence angles , where

 K=Ef−pfcosθf=1+ωi−ωf−(ωicosθi−ωfcosθf)cosθf. (24)

Here is the Thomson cross section, with being the Compton wavelength of the electron. Whenever the initial and final electrons are in the ground state and the initial photon is parallel to the field such that , then Eq. (119) implies that . This expression for the cross section is of a form similar to that in Eq. (11) of DH86 (), with the denominator term later corrected HD91 (). Upon integration the resulting matrix elements represented by the integrals in Eqs. (18) and (20) are contained in the terms within the summations, which depend on the Landau level and spin quantum numbers of the intermediate states. At this juncture, a choice of electron basis states is required in order to evaluate the terms. The papers by DH86 (); HD91 () used JL spin states to evaluate the matrix elements following the work of DB80 (). On the other hand, Sina Sina96 () performed the requisite spatial integrals preserving the electron wave-function coefficients in general form, thereby allowing for the expedient development in either JL or ST basis states. However, Sina chose to focus on the resulting cross section within the context of the ST spin states, the preferred protocol. We take advantage of the manipulations in Sina96 () in order to develop the differential cross section for both basis states in parallel.

For the remainder of the paper, the focus is on the development of Compton scattering in strong fields in the specific case where the laboratory incident photon angles are parallel () to the external field in the electron rest frame, as was previously performed by Gonthier et al. Getal00 (). In that study, the role of the resonance was only considered in limited fashion, resulting in analytic descriptions of the cross section below and above the resonance. The bare resonance is divergent because an infinite lifetime for the intermediate state is thereby presumed. However, introducing a finite lifetime associated with the propagators truncates the resonance according to the prescription in Eqs. (17) and (19), with the width of the resonance being necessarily dependent on the spin of the intermediate state. Spin-dependent widths were incorporated into the differential cross section in the work of HD91 (), developing terms dependent on the spin of the intermediate state using ST eigenfunctions for determining the widths, but employing a JL formulation for the wave functions of the incoming and outgoing electrons. In our previous study BGH05 (), we showed the inherent difficulties with the JL basis states in that the spin states are not preserved under a Lorentz transformation along B. In contrast, we demonstrated that the ST electron wave functions, being eigenfunctions of the magnetic moment operator, behave correctly by preserving the spin states under Lorentz transformation, and form the appropriate set of states to describe the spin dependence of the resonance width. Accordingly, a greater emphasis is placed on the ST formulation below, presenting results that have not appeared before in the literature on magnetic Compton scattering.

### ii.2 Compton scattering for photons incident along the magnetic field

Motivated by the important astrophysical application of inverse Compton scattering in neutron stars, the focus narrows now to cases. In addition, our ensuing analysis will concentrate on the development of the main contribution to the resonant scattering, which is the final state, i.e., ground-state–ground-state transitions. Excited final electron states only become accessible when the incident photon energy exceeds the threshold DH86 (); Getal00 (). The majority of the cross section is dominated by the contribution, even somewhat above , as can been seen in Fig. 4 of Getal00 (). Moreover, BWG11 () highlights how cooling of relativistic electrons in neutron stars is dominated by interactions near the fundamental resonance. Accordingly, domains where contributions dominate are of the greatest interest to astrophysical applications. We note also that the nonresonant JL formalism for cases has already been presented in Getal00 (), and that away from resonances, the ST formulation will generate identical results.

For , only the first excited intermediate state contributes, collapsing the sum over in Eq. (23) to one term. The spin dependence of the resonance width (rate) is strongly dependent on the strength of the magnetic field (see Fig. 1 in our previous study BGH05 ()). With the restriction, the differential cross section possesses a simple dependence on , namely just complex phase factors from the factors in Eq. (23). Since then the terms do not depend on , the integration over the final is almost trivial, with cross terms proportional to that integrate to zero over the interval [see Eq. (4) in BAM86 () for the analogous inference for JL scattering formalism]. Performing this first then leads to a sum of squares of complex moduli of the matrix elements for the corresponding Feynman diagrams and appearing in the resulting form for the differential cross section:

 dσ(dcosθf(=3σ\sixrm T(4(ω2fe−ω2fsin2θf/2B(ωi(2ωi−ωf−ζ)(∑s=±[∣∣F(1)n=1,s∣∣2+∣∣F(2)n=1,s∣∣2] (25)

where

 ζ=ωiωf(1−cosθf). (26)

Observe that in developing Eq. (25), the identity derived from the scattering kinematics [See Eq. (4) using ] has been employed in the factor out in front, with given by Eq. (24). As indicated in Appendix B, the specialization restricts the sum over , selecting only the level of the intermediate state as contributing to the cross section. This is the leading order contribution, with a Kronecker delta evaluation of the Laguerre functions, , forcing the restriction. This simplifies the cross section dramatically, as in Herold79 (); Getal00 (). The matrix element terms comprise standard “energy-conservation” denominators, and numerators with terms, as listed in Eqs. (18) and (20), are given by

 F(m)n=1,s=S(m)u(ωm−Em+iΓs/2(+S(m)v(ωm+Em−iΓs/2(,m=1,2. (27)

Note that these terms defining the in Eq. (27) differ from the calligraphic used to define the terms in Eqs. (17) and (19) in that the calligraphic terms contain factors and functions arising out of the spatial integrals, while these terms here are only the products of the wave-function coefficients and the associated functions discussed in Appendix B. The two and terms here correspond to contributions from the electron and positron spinors of the intermediate state, with positive and negative energies, respectively. Observe that the numerators depend on , the spin quantum number of the virtual pairs. We have introduced some kinematic variables that depend on the Feynman diagram number , namely, total “incoming” energies

 ωm=1=1+ωi,ωm=2=1−ωf, (28)

and energies of the intermediate electron/positron state

 Em=1=√ω2i+ϵ2⊥,Em=2=√ω2fcos2θf+ϵ2⊥ (29)

for

 ϵ⊥=√1+2B (30)

as the threshold energy of the first Landau level. Using the identity ,

 pm=1=ωi,pm=2=−ωfcosθf (31)

define the components of momenta parallel to B that correspond to the .

The spin-dependent widths of the cyclotron resonance truncate the divergences at that would appear in Eq. (27) without their inclusion. In practice, because of the kinematics of scattering, only the diagram elicits such a divergence, as is evident from the inequality that is simply deduced from Eqs. (28) and (29). Here is the spin-averaged width in the frame of reference where the electron possesses no component of momentum along B. It is independent of the eigenfunction solutions of the Dirac equation, and its analytic form is given in Eqs. (43) and (44) below. The spin-correction factor does depend on the basis states being employed to describe the virtual particle; using the forms for found in BGH05 () [Eq. (1) therein for the ST case, and Eq. (53) for the JL states], one has

 ξST±=1∓1(ϵ⊥(,ξJL±=1∓Em+ϵ2⊥(ϵ2⊥(Em+1)( (32)

for the ST and JL basis states. Note that the Lorentz boost that would transform the intermediate electron from the zero parallel momentum () frame to the frame is employed to cast Eq. (53) of BGH05 () into the JL version for the spin-correction factor in Eq. (32). In the Lorentz profiles that will emerge when the squares of the s are taken, it will become apparent that the cross section at the peak of the truncated cyclotron resonance will scale as , so that the relative strengths of the resonant interaction for the two eigenfunction choices will scale with , and thereby be substantially spin-dependent when is not too much greater than unity.

The modulus squared of the matrix element terms can be evaluated in a similar manner as in HD91 (), keeping only terms that are of the highest order in , a small quantity. Specifically, combining the terms in Eq. (27) leads to the forms for that can routinely be determined from Eq. (27). Eliminating the term proportional to in the numerators and setting in the denominators permits the squares of the terms to be cast in a compact form, with numerators that employ the functions

 N(m)+ = ωm(S+u+S+v)+Em(S+u−S+v) (33) N(m)− = ωm(S−u+S−v)+Em(S−u−S−v),

where is the Feynman diagram number. The error incurred with this approximation is of the order of in the cyclotron resonance, which is always small (e.g., see Fig. 1 of BGH05 () for values of , or Fig. 3 of HL06 ()), being less than around for all field strengths, where is the fine-structure constant. With these terms defined, the spin-dependent factors can be isolated and the cross section in Eq. (25) can be expressed as

 dσ(dcosθf(=3σ\sixrm T(4(ω2fe−ω2fsin2θf/2B(ωi(2ωi−