Improved Bethe-Heitler formula

The bremsstrahlung cross section of electron in the atomic electric field is re-derived using the time ordered perturbative theory. The results are compared with the Bethe-Heitler formula. We indicate that both the TOPT-description and a soft version for the bremsstrahlung process predict a strong screening parameter-dependent cross section, which is missed by previous bremsstrahlung theory.


Introduction
When electrons scatter off electric field of proton or nucleus, they can emit real photons. This is bremsstrahlung (braking radiation). Bethe and Heitler first gave a quantum description of the bremsstrahlung emission with the Coulomb potential of an infinite heavy atom [1]. The Bethe-Heitler (BH) formula is an elementary and important equation in quantum electromagnetic dynamics (QED) and astrophysics.
Recently, a puzzled difference of the energy spectra of electrons and positrons at GeV-TeV energy band in cosmic rays raises our doubts to the validity of the BH formula at high energy. These energy spectra have been measured at the atmosphere top by Alpha Magnetic Spectrometer (AMS) [2], Fermi Large Area Telescope (Fermi-LAT) [3], DArk Matter Particle Explorer (DAMPE) [4] and Calorimetric Electron Telescope (CALET) [5].
The discovery of the excess (or break) of the spectra in the GeV-TeV band causes big interest because it may be related to the new physical signals including dark matter.
However, a puzzled question is why the data of AMS and CALET are noticeably lower than that of DAMPE and Fermi-LAT at the measured energy band? This uncertainty in the key range makes us unable to understand the meaning of the signal correctly. The above measurements have been accumulated and improved over many years. Besides, Fermi-LAT, DAMPE and CALET use the similar calorimeter, while AMS employs a completely different kind of magnetic spectrometer. Therefore, the above difference seems not to be caused by the systematic or measurement errors.
We noticed that both AMS and CALET set on the international space station at ∼ 400 km, while Fermi-LAT and DAMPE are orbiting the Earth at 500 ∼ 560 km altitude. A naive suggestion is that the primary signals of electron-positron fluxes are weakened by the electromagnetic shower caused by the extremely thin atmosphere during its transmission from 500 km to 400 km. For this sake, we use the electromagnetic cascade  Figure 1: Cosmic electron-positron spectra multiplied by E 3 as a function of energy. Data are taken from [2][3][4][5]. A and B indicate the spectra at height ∼ 500 and ∼ 400 km. The curve A is an input of the electromagnetical cascade equation, and the curve B is the result of the cascade through ∼ 0.1λ. The corresponding bremsstrahlung cross section is seven-orders of magnitude larger than the prediction of the traditional theory. The difference between the curves A and B is explained as an anomalous bremsstrahlung effect in this work. equation to estimate the value of the corresponding radiation length λ, which may lead to the difference between the spectra as shown in Fig. 1. We find that λ ≃ 10 −6 g/cm 2 . This value is seven-orders of magnitude smaller than the usual standard value λ = 37 g/cm 2 .
Is it possible that there exists such a big difference in the radiation length? We review the derivation of the BH formula. The bremsstrahlung event contains the scattering of the incident electron on the nuclear electric field for the conservation of energy-momentum.
It is well known that the total cross section of the Rutherford cross section in a pure Coulomb field either in the classical or quantum theory is infinite, which origins from the following fact: the long-range 1/r potential has a significant contribution to the total cross section. If the impact parameter of the incident electron is large compared with Figure 2: (a) The Rutherford cross section ∼ Z 2 α 2 /µ 2 ; (b) The Bethe-Heitler cross section ∼ Z 2 α 3 /m 2 e ln(m 2 e /µ 2 ); (c) The anomalous bremsstrahlung cross section ∼ Z * 2 α 3 /µ 2 ln(4E 2 i /µ 2 ), Z * is the affective ionized charge number.
the atomic radius, the Coulomb field of the positive nuclear charge is completely screened by the electrons of a neutral atom. Therefore, the Coulomb cross section is limited in an atomic scale R 2 ∼ 1/µ 2 , µ is the screening parameter (Fig. 2a). On the other hand, Bethe and Heitler predict a strongly reduced bremsstrahlung cross section ∼ 1/m 2 e , which is much smaller than the geometric scattering cross section (Fig. 2b).
In order to find the reason of the difference presented in Fig. 2, we expose how a factor 1/m 2 e replaces 1/µ 2 in the BH formula. Unfortunately, the complex correlations between the scattering and radiation processes hindered Bethe and Heitler to obtain an analytical solution for the integrated cross section if the screening parameter was considered at the beginning derivation in the S-matrix. In deed, the parameter µ is introduced by a model at the last step in their derivation. Therefore, we can't track the whereabouts of the screening parameter in such way.
We re-derive the bremsstrahlung formula with the screening potential at high energy.
We find that if considering energy transfer due to the recoil effect, the interference amplitudes can be removed if the energy of electron is high enough. Thus we can further decompose the process, where the time ordered perturbative theory (TOPT) [6,7] is used to separate the scattering and radiation processes at the equivalent photon (Weizsäcker-Williams) approximation [8,9]. The above simplified method allows us to track how the screening parameter enters the final cross section from an original S-matrix element. Using this method, we find that the parameter m e in the radiation part enter into the denominator of the scattering part through a simple mathematical formula.
Let us use a simple example to illustrate our discovery. A typical integrated Rutherford cross section contains where θ is the scattering angle and E i the initial energy of electron. The result is proportional to the geometric area of an atomic electric field. Although scattering away from the target is weak, the cumulative contributions of scattering in a broad space lead to the divergence of the total cross section at µ → 0. On the other hand, a radiation factor combines with the scattering matrix element in bremsstrahlung and the integrated cross section becomes where a weakly µ-dependent logarithm is neglected. Note that −q 2 = m 2 e + 4E 2 i sin 2 (θ/2) and B = m 2 e or µ 2 . One can clearly see that the lower limit θ min of the integration is determined by the condition −q 2 ≥ B. Once there is a un-eliminated parameter m e in −q 2 /B, it will enter into the denominator after integrating the scattering angle. The result indicates that the bremsstrahlung cross section has a strong suppression since µ ≪ m e .
A following interesting question is under what condition the parameter m e can be omitted in ln(−q 2 /B)? In this case, the contributions of the geometric cross section 1/µ 2 will be restored. The straightforward answer is It means a no-recoil scattering. In this case, the bremsstrahlung cross section restores its geometric size and we call this as the anomalous bremsstrahlung effect. Theoretically, an infinite heavy atom can completely absorb the recoil effect as the Rutherford scattering. However, a target atom bound in the normal mater can not avoid the obviously recoil corrections due to the strong collisions. Therefore, the suppression in the bremsstrahlung cross section is a general phenomenon.
However, there is an exception as we have mentioned at the beginning of this work.
In the complete ionosphere about 400 ∼ 500 km height, the oxygen atoms are not only completely ionized, but its density is extremely thin. On average, there is only one atom per 1/1000000000 cubic centimeter. This is a big space with macroscopic scale ∼ 10 −3 cm. The nuclear Coulomb potential may spread into such a broader space, where the bremsstrahlung events may neglect the recoil energy comparing with a larger initial energy E i of electron, since they are far away from the source of the Coulomb center field (see Eq. (2.35)). Besides, the integration on the space may go down to a lower limit | q| 2 = µ 2 no blocking from the cut-parameter m 2 e (see Eq. (2.39)). There is a critical scale r c , when the impact parameter larger than r c (Figure 2c), where the bremsstrahlung cross section will restore the big geometric cross section. Thus, one can get a large enough increment of the cross section to explain the result in Fig. 1 since the accumulation of a large amount of soft photon radiation in a broad space.
We emphasize that m e and µ have different physical meaning although they both have the mass dimension in the natural units. Therefore, when m e in ln(−q 2 /µ 2 ) is omitted due to the recoil energy ν ≪ E i , a more smaller parameter µ can be retained because µ is irrelevant to the energy E i of the incident electron.
According to the above considerations, we derived a new bremsstrahlung formula, they in the normal media and in the thin ionized gas, where z ≃ ω/E i and taking the leading logarithmic (1/z) approximation. We will prove that dσ I is compatible with the BH formula.
Our discussion about the bremsstrahlung process also applies to pair production of electron-positron. We derived the improved formula. For testing the above anomalous effect, a modified cascade equation for the electromagnetic shower is given.
The paper is organized as follows. In Sec. 2 we detail the derivation of the bremsstrahlung formula using TOPT. Then we discuss the BH formula and a soft photon version for bremsstrahlung at Sec. 3. In Sec. 4 we compare these different versions for the bremsstrahlung formula. The anomalous effect in electron-positron pair creation is studied at Sec. 5. The improved cascade equation for electromagnetic shower is given at Sec. 6. The last section is a summary.

The bremsstrahlung cross section with screening potential
The BH formula assumes that the target atom is infinitely heavy. We consider a more general case in the following derivation: electron scattering off a finite heavy atom. The differential cross section of the bremsstrahlung emission ( Fig. 3) in covariant perturbation theory at the leading order approximation is [10] dσ = m e M 0 where the screening photon propagator in the matrix takes Its 3-dimension component 1/( q 2 + µ 2 ) corresponding to a potential ∼ e −rµ /r, i.e. the Coulomb potential vanishes at r > 1/µ ≡ R. R is the atom radius for a neutral atom.
According to TOPT, a covariant Feynman propagator in may decompose into a forward and a backward components (Fig. 4) and  The physical picture is frame-dependent in TOPT and it is not a relativistically invariant. The same physical process has different appearances in different frames, simple in some but complicated in others [11]. It seems that the TOPT decomposition complicates the calculation with increasing the propagators. However, the backward component will be suppressed at higher energy and small emitted angle. For example, we take l along the z-direction, and define z as the momentum fraction ofl carried by the longitudinal where we neglect the electron mass m e at high energy and note thatl is on-mass shell.
At high energy and small emitted angle we denotê and which is much larger than (2.11) Therefore, the contributions of the backward propagator are negligible at high energy.
This not only reduces the number of diagrams, but also allows us to factorize the complex Feynman graph due to the on-mass shell of the forward propagator. This is the theoretical base of the equivalent photon approximation.
We take the laboratory frame, where the target atom is at rest, but the incident electron has a high energy. This is an infinite momentum frame for the electron. In the above mentioned laboratory frame, both the longitudinal momentum and energy of the ν is the energy of the virtual photon. The radiation time is during this period the photon is emitted. Since El ∝ E i and El ≫ k T , at high enough energy E i and z = 0, 1 we always have for a not too large value of ν.
The virtual photon γ * with a short life τ triggers the event Fig. 4c (or the event can't trigger the following event 4a if it has triggered the event 4c. It also can't trigger the event 4c before it triggers the event 4a. It implies that the processes Fig. 4a,4c are incoherent. Therefore, the contributions of the interferant processes in Fig. 5b,5c can be neglected in our following discussions. After removing these coherent diagrams, using the on-mass shell of the momentuml, the process can further decompose into two sub-processes. We discuss the process in Fig. 5a, Eq. (2.1) becomes where and dP a = 1 4π 2 1 2El (2.17) We calculate dσ I a using where we use l to replacel in the matrix since E l ≃ El for the small emitted angle. The result is Note that the q 2 -dependent term in Eq. (2,19) is absent when the target is a spin-0 particle, however, it does not change the following results.
On the other hand, the calculation is accurate to ln k 2 T since | k T | ≪ El. We obtain In the calculation, we turn the z-axis direction from p i to l .
Combining equations (2.19) and (2.20), we have For a virtual mass −q 2 , the integral in k 2 T has an upper limit of order −q 2 since k 2 T origins from q 2 . Thus, we have The energy momentum conservation and (2.24) Note that the 4-transfer momentum where we take 1+2E i /M 0 sin 2 (θ/2) ≃ 1 since the leading contributions are from θ → θ min .
We calculate the integrated bremsstrahlung cross section at a given initial energy through the angle-integral. In general the term 2m 2 e in Eq. (2.25) can not be omitted since the value of −q 2 is not always large even at high energy. The lower limit θ min of integral is determined by where sin 2 (θ/2) ≡ t and t ′ ≡ t − m 2 e /(2E 2 i ).
The first term in Eq. (2.26) can be integrated and it contributes Z 2 α 2 /µ 2 ln(4E 2 i /µ 2 ) at the leading order approximation. However, the second term is not so lucky. Through the numeric computations we find that the contribution of this term is almost ∼ βZ 2 α 2 /µ 2 ln(4E 2 i /µ 2 ) and β ∼ −0.5 is acceptable. Thus we have Now we calculate the process in Fig.5d. Corresponding to Eq. (2.21) we have where After integral, we haveσ On the other hand, where |p| ≃ E i in the unit c = 1. Corresponding to Eq.(2.21) we have where Z = 1 if only one electron per atom is ionized, the Mott scattering formula with β = v/c ≃ 1 and c = 1 is used.
The results show that the bremsstrahlung cross section is almost proportional to the geometric area of the atomic Coulomb field ∼ R 2 , rather than a weaker ln R-dependence that the BH formula predicted.
The new bremsstrahlung formulas dσ II should include the contributions of Fig. 3b,c since ν ∼ 0 . However, we will prove that these corrections are negligible at high energy at Sec. 4.

Comparing with the Bethe-Heitler formula
The differential cross section of the BH formula [1] for bremsstrahlung is where θ i , θ f are the angles between k and p i , p f respectively, φ is the angle between ( p i k) plane and ( p f k) plane.
After integral over angles in Eq. (3.1) at E ≫ m e (but still keeping m e ), the cross section can be simplified as Unfortunately, Eq. is ∼ µ. Thus, they write where the Thomas-Fermi model is used.
For comparison, we rewrite Eq.(3.3) as and Using z = ω/E i (note that ν = 0 in the BH formula) and dω/ω = dz/z, we have Thus, the BH formula (3.8) becomes It is well known that at limit ω → 0 any process leading to photon emission can be factorized [13]. A corresponding factorized differential cross section at this approximation is [6,10] This equation describes the cross section for a single photon radiation at the elastic limit.
However, the integrated cross section integrates over all possible phase space, it does not only includes the contributions of elastic scattering (|k| = 0), but also inelastic scattering (|k| ∼ 0). The probability of such soft photons is proportional to [12] W ∼ | q| µ dk k . (3.11) We insert Eq.(3.11) to Eq.(3.10) at the ER limit and get dσ ER Sof t ≃ dΩ The first term in the absolute value symbol is known as the Sudakov double logarithm [12].
We consider bremsstrahlung of high energy electrons in the normal medium, where the nuclear Coulomb field is restricted inside the atomic scale. The larger the electron energy, the larger the energy transfer due to a stronger Coulomb field. In this case, the recoil of the target atom can not be neglected even in solid since the bound target can't completely absorb a strong recoil at such high energy (≫ 1 GeV ). According to dσ I the integrated cross section will be suppressed by a factor 1/m 2 e as similar to dσ B−H . For further comparison, we use the Coloumb potential without recoil to replace dσ I in Eq.
(2.21) and take the LL(1/v) approximation, the result is consistent with the BH formula (3.9) at the same approximation. There is a difference in the coefficients since we neglect the contributions of Figs. 5b and 5d.
On the other hand, dσ II shows that the bremsstrahlung cross section has a big enhance at the thin ionized gas. We remember that the cross section dσ II is valid at ν Obviously, these corrections are negligible if comparing with dσ II . Therefore, we suggest that dσ II is an valuable bremsstrahlung formula in the thin ionized gas.
According to the above discussions, we conclude that dσ I (or dσ B−H ) applies to bremsstrahlung in most media, but we should use dσ II for the bremsstrahlung process in the thin ionized gas at high energy. We suggest to test our prediction in the ionosphere.

The anomalous effect in electron-positron pair creation
Our discussion on bremsstrahlung also applies to electron-positron pair creation. We consider a high energy photon traversing the atomic Coulomb field. This photon has a certain probability of transforming itself into a pair of electron-positron. The time ordered perturbation theory (TOPT) describes pair creation in Fig. 6. The contributions of the interferant processes between Figs. 1a and 1c can be neglected at the the thin ionosphere since it is also 1/m 2 e -suppressed.
The cross section of pair creation in Fig. 7a at the leading order approximation reads [10] dσ γ→e where the screening photon propagator in the matrix takes Eq. (2.2). We take the laboratory frame, where the target atom is at rest, but the incident photon has a high energy. Therefore, M 0 /E P i ≃ 1 and note that |v − c| ≃ c = 1 in the nature unit.
The above cross section can be written as the factorization form in the TOPT framework, We take k along the z-direction, and define z as the momentum fraction of k = | k| carried by the longitudinal momentum of electron, Figure 6: The TOPT decomposition for pair production.
where we neglect the electron mass m e at high energy. At high energy and small emitted angle we denote k = (ω, 0, ω), (4.4) and where Z = 1 is only one electron per atom is ionized. At the last step, we have used and We take the laboratory frame. Similar to Eq. (2.35) we have if z = 0, 1. Now we integral over angle in Eq. (4.7) and result is Note that for a high energy event, for say, ω > 1 GeV , the small recoil energy ν comparing with ω may neglected. Thus, a factor incorporates with z[(1 − z)/z + z/(1 − z)] in Eq.(4.11) and it results in (1 − z) 3 + z 2 (1 − z).
Consideringγ → e + = γ → e − , we have where e = e + + e − . We denote X and λ as the depth and the radiation length in unity g/cm 2 . The photon flux Φ γ and electron/positron flux Φ e satisfy the coupled equations in the electromagnetic cascade process [14] dΦ

The improved electromagnetic cascade equation
and Where the cascade kernel P e→γ (z)dtdz (or P e→e (z)dtdz) is the probability for an electron/positron of energy E i to radiate a photon of energy ω = zE i (or to an electron/positron of energy E f = zE i ) in traversing dt = dX/λ, while P γ→e (z)dtdz is the probability for a photon of energy ω to radiate an electron/positron of energy E f = zE i in traversing dt = dX/λ.
A following key step is to extract the cascade kernels from the bremsstrahlung-and pair production-cross creation. The logarithmic ln 4E 2 i /mu 2 changes slowly with energy, it can be regarded a constant. The cascade kernels are irrelevant to the energy and they are functions of z in a so-called approximation A [14]. Interestingly, comparing with the QCD evolution equation [15], we find that the corresponding kernels in Eqs. (2.38) and

Summary
The BH formula successfully describes bremsstrahlung of high energy electrons. The integrated cross section of the BH formula is restricted in a region ∼ 1/m 2 e , which is much smaller than the geometric section of the target. Recently, the measured energy spectra of electrons-positrons at the GeV-TeV energy band in cosmic rays show two different sets.
According to the traditional bremsstrahlung theory, the above difference can't be caused by the the electromagnetic shower at the top of atmosphere, since the small integrated cross section implies that the energy loss of the shower at the thin ionosphere is negligible.
We find that an anomalous effect in bremsstrahlung and pair creation arises an unexpected big increment at the atmosphere top, which is missed by previous theory. This anomalous effect is caused by the accumulation of a large amount of soft photon radiation.
We derive the relating formula including an improved electromagnetic cascade equation.
These results may use to explain the above confusion in the electron-positron spectra.