11 Feb 2020 Improved Bethe-Heitler Formula

Home , Radiation length

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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

2 250 AMS-02 DAMPE CALET

) 200 Fermi-LAT -1 sec

-1 A sr 150 -2 m 2 B Gev ( - 100 +e + e F × 3 50 E

0 10 100 1000

Energy(Gev)

Figure 1: Cosmic electron-positron spectra multiplied by E3 as a function of energy. Data are taken from [2-5]. 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 ≃ 10−6g/cm2. This value is seven-orders of magnitude smaller than the usual standard value λ = 37 g/cm2.

Is it possible that there exists such a big diﬀerence in the radiation length? In order to explore the possibility of bremsstrahlung enhancement, 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

3 Figure 2: (a) The Rutherford cross section ∼ Z2α2/µ2; (b) The Bethe-Heitler cross 2 3 2 2 2 section ∼ Z α /me ln(me/µ ); (c) The anomalous bremsstrahlung cross section ∼ ∗2 3 2 2 2 ∗ Z α /µ ln(4Ei /µ ), Z is the aﬀective ionized charge number. parameter of the incident electron is large compared with 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 R2 ∼ 1/µ2, µ is the screening parameter (Fig. 2a). On the other hand, Bethe and Heitler predict a

2 strongly reduced bremsstrahlung cross section ∼ 1/me, 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

2 2 1/me replaces 1/µ 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 am-

4 plitudes 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 (Weizs¨acker-

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. Us- ing this method, we find that the parameter me 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

π sin θdθ 1 dσRuth ∼ 2 ∼ , (1.1) 2 θ µ 2 2 0 (sin + 2 ) µ Z 2 4Ei where θ is the scattering angle and Ei 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

π sin θdθ −q2 1 1 dσBress ∼ 2 ln ∼ ∼ , (1.2) 2 θ µ 2 2 2 2 θmin (sin + 2 ) B µ + me me Z 2 4Ei 2 2 2 2 where a weakly µ-dependent logarithm is neglected. Note that −q = me +4Ei sin (θ/2)

2 2 and B = me or µ . One can clearly see that the lower limit θmin of the integration is

2 determined by the condition −q ≥ B. Once there is a un-eliminated parameter me in

−q2/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 ≪ me.

A following interesting question is under what condition the parameter me can be

5 omitted in ln(−q2/B)? In this case, the contributions of the geometric cross section 1/µ2 will be restored. The straightforward answer is

−q2 |~q|2 ln → ln , (1.3) µ2 µ2

2 2 2 where ~q = 4Ei sin (θ/2) (at v → c). 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 Ei of electron, since they are far away from the source of the Coulomb center ﬁeld.

Besides, the integration on the space may go down to a lower limit |~q|2 = µ2 no blocking

2 from the cut-parameter me. There is a critical scale rc, when the impact parameter larger than rc (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 me and µ have diﬀerent physical meaning although they both have

2 2 the mass dimension in the natural units. Therefore, when me in ln(−q /µ ) is omitted

6 due to the recoil energy ν ≪ Ei, a more smaller parameter µ can be retained because µ is irrelevant to the energy Ei of the incident electron.

According to the above considerations, we derived a new bremsstrahlung formula, they are

3 2 2 3 2 I α 4Ei [1 + (1 − z) ](1 − z) 2α 4Ei dz dσ = 2 ln 2 dz → 2 ln 2 (1.4) me µ z me µ z in the normal media and

2α3 4E2 [1 + (1 − z)2](1 − z) 4α3 4E2 dz dσII = ln i dz → ln i (1.5) µ2 µ2 z µ2 µ2 z in the thin ionized gas, where z ≃ ω/Ei and taking the leading logarithmic (1/z) approximation. We will prove that dσI 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 the TOPT. Then we discuss the BH formula and compare these diﬀerent ver- sions for the bremsstrahlung formula at Sec. 3. The anomalous eﬀect in electron-positron pair creation is studied at Sec. 4. The improved cascade equation for electromagnetic shower is given at Sec. 5. The last section is a summary.

7 2 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]

meM0 2 4 4 dσ = |MpiPi→pf Pf k| (2π) δ (pi + Pi − pf − Pf − k) 2 2 2 (piPi) − meM0 q 3 3 3 2πd ~k med ~pf M0d P~f 3 3 3 P , (2.1) (2π) ω (2π) Ef (2π) Ef where the screening photon propagator in the matrix takes

ig µν . (2.2) q2 − µ2 + iǫ

Its 3-dimension component 1/(~q2 + µ2) 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 the TOPT, a covariant Feynman propagator in MpiPi→pf Pf k

4 iγ · l + me S = d l 2 2 , (2.3) Z l − me + iǫ may decompose into a forward and a backward components (Fig. 4)

ˆ 1 iγ · l + me SF = , forward (2.4) 2Eˆl ω + Ef − Eˆl and

1 iγ · ˆl + me SB = . backward (2.5) 2Eˆl −ω + Ef − Eˆl

8 Figure 3: Two elemental bremsstrahlung amplitudes.

Figure 4: The TOPT decomposition of Fig.3. The dashed lines indicate the time ordered of the process.

~ 2 2 ˆ ~ ˆ ~ Note that l(El, lT ,lL) is oﬀ-mass shell l =6 me, while l = (Eˆl, lT ,lL) or l = (−Eˆl, lT ,lL)

ˆ2 2 are on-mass shell, i.e., l = me.

The physical picture is frame-dependent in the 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 compli- cates the calculation with increasing the propagators. However, the backward component ~ will be suppressed at higher energy and small emitted angle. For example, we take ˆl along the z-direction, and define z as the momentum fraction of ˆl carried by the longitudinal momentum kL of photon,

9 k ω z = L ≃ , (2.6) ˆl Eˆl where we neglect the electron mass me at high energy and note that ˆl is on-mass shell.

At high energy and small emitted angle we denote

ˆ ˆ ˆ l =(Eˆl, lT , lL)=(Eˆl, 0, Eˆl), (2.7)

2 kT k =(ω,kT ,kL)= zEˆl + ,kT , zEˆl , (2.8) 2zEˆl ! and

2 kT pf =(Ef ,pf,T ,pf,L)= (1 − z)Eˆl + , −kT , (1 − z)Eˆl . (2.9) 2(1 − z)Eˆl !

If z =0,6 1 and Ei ≫ kT one can ﬁnd that

1 1 z(1 − z) SF ≃ ≃ 2 , (2.10) 2Eˆl ω + Ef − Eˆl kT which is much larger than

1 1 1 S ≃ ≃ . (2.11) B 2E ω − E + E 4zE2 ˆl f ˆl ˆl 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

10 Figure 5: Four TOPT diagrams after neglecting the contributions of the backward components at high energy and small scattering angle.

11 virtual photon cannot be ignored. Thus, the contributions of Figs. 5b and 5c are not negligible due to the coherence between Fig. 4a and 4c. The electron-atom scattering time is

1 τ ∼ , (2.12) ν

ν is the energy of the virtual photon. The radiation time is

1 Eˆlz(1 − z) T1,2 ∼ = 2 , (2.13) Ef + ω − Eˆl kT during this period the photon is emitted. Since Eˆl ∝ Ei and Eˆl ≫ kT , at high enough energy Ei and z =06 , 1 we always have

τ < T1,2 (2.14) 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

Fig. 4a) needs time T2 (or T1). Note that Fig. 4a,4c are time-ordered processes in the

TOPT description since the backward components are suppressed. This virtual photon 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 momentum ˆl, the process can further decompose into two sub-processes.

We discuss the process in Fig. 5a, Eq. (2.1) becomes

12 3 ~ 3 I meM0 2 M0d Pf med ~pf 4 4 dσ = |M ˆ | (2π) δ (p + P − p − P − k) a piPi→lPf 3 P 3 i i f f 2 2 2 (2π) E (2π) Ef (piPi) − meM0 f q 2 2 3~ 1 1 2 2πd k |Mˆ | l→pf k 3 2Eˆl ! Ef + ω − Eˆl ! (2π) ω

≡ dσˆadPa, (2.15) where

3 ~ 3 I meM0 2 M0d Pf med ~pf 4 4 dσˆ = |M ˆ | (2π) δ (p + P − p − P − k), a piPi→lPf 3 P 3 i i f f 2 2 2 (2π) E (2π) Ef (piPi) − meM0 f q (2.16) and

2 2 3~ 1 1 1 2 d k dP = |Mˆ | . (2.17) a 2 l→pf k 4π 2Eˆl ! Ef + ω − Eˆl ! ω

I We calculate dσˆa using

2 2 2 2 e (Ze) (4π) µ ν µ ν µν 2 |M ˆ | = l p + p l − g (l · pi − m ) piPi→lPf 4m2M 2(q2 − µ2)2 i i e e 0 h i µ ν µ ν µν 2 Pf Pi + Pi Pf − g (Pf · Pi − M0 ) , (2.18) h i ˆ where we use l to replace l in the matrix since El ≃ Eˆl for the small emitted angle. The result is

2 2 2 2 θ q 2 θ Z α 1 cos 2 − 2M 2 sin 2 dσˆI = 0 dΩ a 4E2 (sin2 θ + µ2/4E (E + ω))2 1+ 2Ei sin2 θ i 2 i f M0 2

2 θ q2 2 θ 2 2 cos − 2 sin Z α 1 2 2M0 2 = 2 dΩ 2 µ 2 θ 2 2 2 2Ei 2 θ 4Ei ((1 + ) sin + µ /4E ) 1+ sin 2EiM0 2 i M0 2 2 2 2 2 θ q 2 θ Z α 1 cos 2 − 2M 2 sin 2 ≃ 0 dΩ. (2.19) 4E2 (sin2 θ + µ2/4E2)2 1+ 2Ei sin2 θ i 2 i M0 2

13 Note that the q2-dependent term in Eq. (2,19) is absent when the target is a spin-0 particle, however, it does not change the following results.

A key point is that the TOPT allows us to separately calculate the splitting function

Pa and the sub-cross section dσˆ. For the sub-process e(ˆl) → e(pf )+ γ(k), we let z-axis ~ alon the direction of ˆl. Using Eqs. (2.7)-(2.9) we have

1 2Eˆlz(1 − z) = 2 , (2.20) Ef + ω − Eˆl kT

d3~k z(1 − z)dk2 ≃ T , (2.21) (2π)32ω 16π2z and

2 2 1+(1 − z) 2 |Mˆ | =2α k , (2.22) l→pf k z2(1 − z) T we obtain

α [1 + (1 − z)2](1 − z) dP = dzd ln k2 . (2.23) a 2π z T

2 ~ The calculation is accurate to ln kT since |kT |≪ Eˆl.

Combining equations (2.19) and (2.23), we have

2 2 2 2 θ q 2 θ 2 Z α 1 cos 2 − 2M 2 sin 2 α [1 + (1 − z) ](1 − z) dσI = 0 dΩdzd ln k2 a 4E2 (sin2 θ + µ2/4E2)2 1+ 2Ei sin2 θ 2π z T i 2 i M0 2

2 4 θ 2 θ 2Ei sin 2 2 2 1 − sin 2 + M 2 2Ei 2 θ 2 Z α 1 0 1+ sin 2 α [1 + (1 − z) ](1 − z) = M0 dΩdzd ln k2 . 4E2 (sin2 θ + µ2/4E2)2 1+ 2Ei sin2 θ 2π z T i 2 i M0 2 (2.24)

2 2 2 2 For a virtual mass −q , the integral in kT has an upper limit of order −q since kT origins from q2. Thus, we have

14 kT,max 2 2 dkT −q 2 = ln 2 , (2.25) ZkT,min kT µ

2 2 2 where we have −q >µ since kT is ordered and µ is a minimum quantity with the mass dimension. Thus, we can sum maximum contributions to the cross section.

The energy momentum conservation

2Ei(Ef + ω) 2 θ ν = Ei − Ef − ω = sin , (2.26) M0 2 and

−1 2Ei 2 θ Ef + ω = Ei 1+ sin . (2.27) M0 2!

Note that the 4-transfer momentum

θ θ q2 =2m2 − 4E (E + ω) sin2 ≃ 2m2 − 4E2 sin2 , (2.28) e i f 2 e i 2

2 where we take 1+2Ei/M0 sin (θ/2) ≃ 1 since the leading contributions are from θ → θmin.

We calculate the integrated bremsstrahlung cross section at a given initial energy

2 through the angle-integral. In general the term 2me in Eq. (2.28) can not be omitted

2 since the value of −q is not always large even at high energy. The lower limit θmin of integral is determined by

−2m2 +4E2 sin2 θ 4E2t′ 4E2 ln e i 2 = ln i = ln i + ln t′ ≥ 0, (2.29) µ2 µ2 µ2

2 ′ 2 2 where sin (θ/2) ≡ t and t ≡ t − me/(2Ei ).

Substituting Eqs. (2.25) and (2,28) into Eq. (2.24), we divide the integral into three parts.

15 Z2α2 4E2 π sinθdθ 1 α [1 + (1 − z)2](1 − z) dσI (1) = ln i 2π dz a 2 2 2 θ 2 2 2 2Ei 2 θ 4E µ θmin (sin + µ /4E ) 1+ sin 2π z i Z 2 i M0 2

πZ2α2 4E2 M π 2sin θ cos θ dθ 1 α [1 + (1 − z)2](1 − z) = ln i 0 2 2 dz 2 2 2 θ 2 2 2 M0 2 θ 2E µ 2Ei θmin (sin + µ /4E ) + sin 2π z i Z 2 i 2Ei 2 πZ2α2 4E2 M 1 2dt 1 α [1 + (1 − z)2](1 − z) = ln i 0 dz 2 2 2 2 2 M0 2E µ 2Ei ǫ (t + µ /4E ) + t 2π z i Z i 2Ei µ2 µ2 2 2 2 1+ 2 ǫ + 2 πZ α 4Ei M0 1 1 1 4Ei 1 1 4Ei = ln 2 2 + 2 ln − 2 − 2 ln 2 2 µ M0  µ M0 µ M0 µ M0 µ M0  2Ei µ Ei 2 − 1+ 2 − 2 1+ ǫ + 2 + 2 ǫ + 4E 2Ei 4E 2Ei 4E 2Ei 4E 2Ei 4E 2Ei i  i i i i    α [1 + (1 − z)2](1 − z) × dz 2π z 2 3 2 2 Z α 4Ei [1 + (1 − z) ](1 − z) ≃ 2 ln 2 dz, (2.30) me µ z

2 2 2 2 where t = sin (θ/2) and ǫ = tmin = me/(2Ei ). One can ﬁnd that me comes from the lower limit ǫ.

Z2α2 4E2 π sinθdθ sin2 θ α [1 + (1 − z)2](1 − z) dσI (2) = − ln i 2π 2 dz a 2 2 2 θ 2 2 2 2Ei 2 θ 4E µ θmin (sin + µ /4E ) 1+ sin 2π z i Z 2 i M0 2

πZ2α2 4E2 M π 2sin θ cos θ dθ sin2 θ α [1 + (1 − z)2](1 − z) = − ln i 0 2 2 2 dz 2 2 2 θ 2 2 2 M0 2 θ 2E µ 2Ei θmin (sin + µ /4E ) + sin 2π z i Z 2 i 2Ei 2 πZ2α2 4E2 M 1 2dt t α [1 + (1 − z)2](1 − z) = − ln i 0 dz 2 2 2 2 2 M0 2E µ 2Ei ǫ (t + µ /4E ) + t 2π z i Z i 2Ei 2 2 2 2 µ M0 µ µ M0 µ 2 2 2 2 1+ 2 2 ǫ + 2 πZ α 4Ei M0 1 4Ei 2Ei 4Ei 4Ei 2Ei 4Ei = − ln 2 2 + 2 ln − 2 − 2 ln 2 2 µ M0  µ M0 µ M0 µ M0 µ M0  2Ei µ Ei 2 − 1+ 2 − 2 1+ ǫ + 2 + 2 ǫ + 4E 2Ei 4E 2Ei 4E 2Ei 4E 2Ei 4E 2Ei i  i i i i    α [1 + (1 − z)2](1 − z) × dz. (2.31) 2π z

The contributions of this part can be neglected comparing with Eq. (2.30) at high energy.

Z2α2 4E2 2E2 π sinθdθ sin4 θ α [1 + (1 − z)2](1 − z) dσI (3) = ln i 2π i 2 dz a 2 2 2 2 θ 2 2 2 2Ei 2 θ 2 4E µ M θmin (sin + µ /4E ) (1 + sin ) 2π z i 0 Z 2 i M0 2

πZ2α2 4E2 E π 2sin θ cos θ dθ sin4 θ α [1 + (1 − z)2](1 − z) = ln i i 2 2 2 dz 2 2 2 θ 2 2 2 M0 2 θ 2 2E µ M0 θmin (sin + µ /4E ) ( + sin ) 2π z i Z 2 i 2Ei 2

16 πZ2α2 4E2 E 1 2dt t2 α [1 + (1 − z)2](1 − z) = ln i i dz 2 2 2 2 2 Ei 2 2E µ M0 ǫ (t + µ /4E ) ( + t) 2π z i Z i E0 4 2 4 2 µ M0 µ M0 2 2 2 4 2 4 2 πZ α 4Ei Ei 1 16Ei 4Ei 16Ei 4Ei = ln 2 2 + − 2 − 2 2  µ M0  µ M0 µ M0  2Ei µ M0 2 − 1+ 2 1+ ǫ + 2 ǫ + 4E 2Ei 4E 2Ei 4E 2Ei  i  i i     M0 +1 M0 + ǫ 2 2 2Ei 2 2Ei α [1 + (1 − z) ](1 − z) − 2 ln 2 − 2 ln 2 dz µ M0 3 µ µ M0 3 µ  2π z ( 4E2 − 2E ) 4E2 +1 ( 4E2 − 2E ) 4E2 + ǫ i i i i i i  2 3 2 2 2 4Z α Ei EiM0 4Ei [1 + (1 − z ) ](1 − z) ≃ 4 ln 2 ln 2 dz. (2.32) M0 me µ z

The contributions of this part can also be neglected comparing with Eq. (2.30) at Ei <

100 GeV . We will show that this restriction is unnecessary in thin ionization gas since me is replaced by µ in Eq. (2.30).

However, the second term is not so lucky. Through the numeric computations we ﬁnd

2 2 2 2 2 that the contribution of this term is almost ∼ βZ α /µ ln(4Ei /µ ) and β ∼ −0.5 is acceptable. Thus we have

2 3 2 2 I Z α 4Ei [1 + (1 − z) ](1 − z) dσa ≃ 2 ln 2 dz. (2.33) 2me µ z

Now we calculate the process in Fig.5d. Corresponding to Eq. (2.24) we have

3 ~ 3 I meM0 2 M0d Pf med ~pf 4 4 dσ = |Mˆ′ | (2π) δ (p + P − p − P − k) b l Pi→pf Pf 3 P 3 i i f f 2 2 2 (2π) E (2π) Ef (piPi) − meM0 f q 2 2 3~ 1 1 2 2πd k × |M ˆ′ | pi→l k 3 2Eˆl′ ! Eˆl′ + ω − Ei ! (2π) ω 2 2 2 2 θ q 2 θ Z α 1 1 − sin 2 − 2M 2 sin 2 = 0 2 2 θ 2 2 2(Ei−ω) 2 θ 4Ei (sin + µ /4Ef (Ei − ω)) 1+ sin 2 M0 2 α 1+(1 − z)2 −q2 × dΩdzd ln 2π z(1 − z) µ2

2 4 θ 2 θ 2Ef sin 2 2 2 2 1 − sin 2 + 2 Ef 2 θ Z α 1 M0 (1− sin ) ≃ µ 2 2 2 θ 2 2 2 2Ef 2 θ 4Ei (sin + µ /4E ) 1+ sin 2 f M0 2 α 1+(1 − z)2 −q2 × dΩdzd ln , (2.34) 2π z(1 − z) µ2

17 where we used

2 θ µ2 2 θ µ2 sin2 + = sin2 + 2 2Ef 2 θ 2 4Ef (Ei − ω)!  2 4E (1 − sin )−1) f M0 2   2 2 2 2 µ 2 θ µ 2 θ µ = (1 − ) sin + 2 ≃ sin + 2 . (2.35) 2Ef M0 2 4Ef ! 2 4Ef !

We denote

α dσ ≡ dσˆ PdzdΩ. (2.36) 2π

Using the exchange i ↔ f between Eqs. (2.24) and (2.34), we have

2 I I Ef I 2 σˆb =σ ˆa ≃ σˆa(1 − z) . (2.37) Ei On the other hand,

1+(1 − z)2 P = . (2.38) b z(1 − z)

2 I Moving factor (1 − z) fromσ ˆb to Pb, we have

I I I I dσa = dσb , σˆa =σ ˆb , Pa = Pb, (2.39)

Thus, we have

2 3 2 2 I I I Z α 4Ei [1 + (1 − z) ](1 − z) dσ = dσa + dσb = 2 ln 2 dz. (2.40) me µ z

We keep leading order term 1/z, i.e., the leading logarithmic (1/z) approximation

(LL(1/z)A) and obtain the integrated cross section

2 3 2 I 2Z α 4Ei Ei σ ≃ 2 ln 2 ln . (2.41) me µ ωmin

18 2 2 2 We ﬁnd that the parameter me enters 1/(µ + me) ≃ 1/me through the lower limit

2 of integration. The resulting integrated cross section is always suppressed by ∼ 1/me as similar to the BH formula (see the next section).

We consider a diﬀerent example: the electric ﬁeld of ionized atom can extend to a large space in the thin ionosphere, where the recoil of the target atom can be neglected

ν ≪ Ei if the impact parameter is large enough. Using δ(Ei − Ef − ω) the scattering potential becomes time-independent. Now Eq.(2.28) is replaced by

θ θ |~q|2 = |~p − ~p |2 =4|~p|2 sin2 ≃ 4E2 sin2 , (2.42) f i 2 i 2 where |p| ≃ Ei in the unit c = 1. Corresponding to Eq.(2.30) we have

II dσa (1) µ2 µ2 2 2 2 1+ 2 ǫ + 2 πZ α 4Ei M0 1 1 1 4Ei 1 1 4Ei = ln 2 2 + 2 ln − 2 − 2 ln 2 2 µ M0  µ M0 µ M0 µ M0 µ M0  2Ei µ Ei 2 − 1+ 2 − 2 1+ ǫ + 2 + 2 ǫ + 4E 2Ei 4E 2Ei 4E 2Ei 4E 2Ei 4E 2Ei i  i i i i   α [1 + (1 − z)2](1 − z)  × dz 2π z 2Z2α3 4E2 [1 + (1 − z)2](1 − z) ≃ ln i dz, (2.43) µ2 µ2 z

2 2 where ǫ = tmin = µ /(4Ei ), since the lower limit θmin of integral is determined by

4E2 sin2 θ ln i 2 ≥ 0 (2.44) µ2

Similar to Eqs. (2.40) and (2.41), we have

2Z2α3 4E2 [1 + (1 − z)2](1 − z) dσII = dσII + dσII = ln i dz, (2.45) a b µ2 µ2 z and

2 3 2 II 4Z α 4Ei Ei σ ≃ 2 ln 2 ln . (2.46) µ µ ωmin

19 2 2 2 2 The improved BH formula contains the factor ln 4Ei /µ rather than ln4Ei /me since we assume that all bremsstrahlung photons have a transverse cut-oﬀ of order µ and

µ ≪ me. A main distinguish of two bremsstrahlung formulas is a large diﬀerence in the

2 2 cross section, which originates from 1/µ and 1/me in front of the logarithmic factor.

Comparing with this eﬀect, the logarithmic diﬀerence between ln 1/µ and ln 1/me can be neglected.

The results show that the bremsstrahlung cross section is almost proportional to the geometric area of the atomic Coulomb ﬁeld ∼ R2, rather than a weaker ln R-dependence that the BH formula predicted.

The new bremsstrahlung formulas dσII 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. 3.

20 3 Comparing with the Bethe-Heitler formula

The diﬀerential cross section of the BH formula [1] for bremsstrahlung is

2 3 2 2 2 2 Z α |~pf | dω dΩcdΩk ~pf sin θf 2 2 ~pi sin θi 2 2 dσB−H = 2 4 2 (4Ei − ~q )+ 2 (4Ef − ~q ) 2π |~pi| ω |~q | " (Ef −|~pf | cos θf ) (Ei −|~pi| cos θi)

2 2 2 ~pi sin θi + ~pf sin θf |~pi|| ~pf | sin θi sin θf cos φ 2 2 2 +2ω − 2 (2Ei +2Ef − ~q ) , (Ei −|~pi| cos θi)(Ef −|~pf | cos θf ) (Ei −|~pi| cos θf )(Ef −|~pf | cos θf ) # (3.1)

~ ~ where θi, θf are the angles between k and ~pi, ~pf respectively, φ is the angle between (~pik) plane and (~pf~k) plane.

After integral over angles in Eq. (3.1) at E ≫ me (but still keeping me), the cross section can be simpliﬁed as

2 3 Z α dω 4 2 2 2 2EiEf 1 dσB−H ≃ 2 2 (Ei + Ef − EiEf ) log − . (3.2) me ω Ei 3 meω 2 Unfortunately, Eq.(3.2) does not present the screening eﬀect since it uses a pure

2 Coulomb potential. Bethe and Heitler consider that the term 2EiEf /meω in Eq.(3.2) is ∼ µ. Thus, they write

2 3 Z α dω 4 2 2 2 me 1 dσB−H = 2 2 (Ef + Ei − Ef Ei) log − , (3.3) me ω Ei 3 µ 2! or

2 3 Z α dω 4 2 2 2 −1/3 1 dσB−H = 2 2 (Ef + Ei − Ef Ei) log(137Z ) − , (3.4) me ω Ei 3 2 where the Thomas-Fermi model is used.

For comparison, we rewrite Eq.(3.3) as

α dσ =σ ˆ P dv, (3.5) B−H B−H 2π B−H 21 where

2 2 8πZ α me 1 σˆB−H = 2 log − , (3.6) me µ 2! and

dω 1 2 2 2 PB−H dz = 2 Ei + Ef − EiEf . (3.7) ω Ei 3

Using z = ω/Ei (note that ν = 0 in the BH formula) and dω/ω = dz/z, we have

1+(1 − z)2 2(1 − z) PB−H (z)= − . (3.8) " z 3z #

Thus, the BH formula (3.3) becomes

2 3 2 4Z α me 1 1+(1 − z) 2(1 − z) dσB−H ≃ 2 log − − dz. (3.9) me µ 2! " z 3z #

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]

4 2 2 θ 3 β sin 2 , NR dσSoft dσRuth 2α Ei = ln  (3.10) 2 dΩ dΩ π ωmin  |~q|  ln 2 − 1. ER me  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]

|~q| dk W ∼ . (3.11) Zµ k We insert Eq.(3.11) to Eq.(3.10) at the ER limit and get

22 2 2 2 2 2 ER Z α 2α Ei |~q| |~q| 1 |~q| dσSoft ≃ dΩ 2 ln ln 2 ln 2 − ln 2 . (3.12) 2 2 θ µ2 π ωmin me µ 2 µ Z 4E sin + 2 i 2 4E i

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 dσI the

2 integrated cross section will be suppressed by a factor 1/me as similar to dσB−H . For further comparison, we use the Coloumb potential without recoil to replace dσˆI in Eq.

(2.24) and take the LL(1/z) approximation, the result

2 3 π 2 2 3 2 I Z α sin θdθ −q dz 4Z α me dz dσ ∼ 2 ln ≃ ln . (3.13) 2 2 θ µ 2 2 2 2 Ei θmin (sin + 2 ) µ z me µ z Z 2 4Ei 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, dσII shows that the bremsstrahlung cross section has a big enhance at the thin ionized gas. We remember that the cross section dσII is valid at ν → 0, where the contributions of Figs. 4b,c should be included. However, the double logarithm in

Eq.(3.12) contributes the following factor

2 3 2 2 2 2 Z α me 2 4Ei 4Ei 4Ei dσ ∝ 2 ln 2 + ln 2 + ln 2 ln 2 + ...... , (3.14) me " µ me me µ #

Obviously, these corrections are negligible if comparing with dσII . Therefore, we suggest that dσII is an valuable bremsstrahlung formula in the thin ionized gas.

23 I According to the above discussions, we conclude that dσ (or dσB−H ) applies to bremsstrahlung in most media, but we should use dσII for the bremsstrahlung process in the thin ionized gas at high energy. We suggest to test our prediction in the ionosphere.

A straightforward way may help us to understand the anomaly bremsstrahlung. For the electron going through the bremsstrahlung, the probability for it to radiate a photon can be estimated by

α P rob. ∼ P (z), (3.15) p · k e→eγ where p is the electron momentum and k is the photon momentum and z = k0/(k0 + p0) is the energy fraction carried by the electron. Note that

|~k| p · k = |~p|2 − m2|~k|−|~p||~k| cos θ> m2 + ϑ(m4). (3.16) e 2|~p| e e q −2 Therefore the probability is proportional to me . However, when there is the coulomb potential, a single momentum kick due to the Coulomb potential is ~kc ∼ ϑ(1/R) ≪ me, which which will modify the denominator to

|~k| p · k → (p + k + k )2 − m2 ∼ m2 − 2~k · ~p − ~k2, (3.17) e e 2|~p| e c c

2 it is still of the order of me for the energy region where is not large enough. However for extremely large |~p|, there is enough room in the phase space for the two terms to cancel, which results in

~ 2 2 2 p · k ∼ −kc ∼ µ ≪ me, (3.18) and hence the probability may be enhanced to be proportional to 1/µ2.

24 4 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 TOPT describes pair creation in Fig. 6. The contributions of the interferant processes between

2 Figs. 1a and 1c can be neglected at the the thin ionosphere since it is also 1/me-suppressed.

The cross section of pair creation in Fig. 7a at the leading order approximation reads

[10]

2πM0 2 4 4 + dσγ→e = P |MkPi→p+Pf p− | (2π) δ (k + Pi − p+ − Pf − p−) Ei ω|v − c| 3 3 3 med ~p+ med ~p− M0d P~f × 3 3 3 P , (4.1) (2π) E+ (2π) E− (2π) Ef 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

P energy. Therefore, M0/Ei ≃ 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 frame- work,

2 2 3 1 1 2 med ~p− + dσγ→e = |Mk→ˆlp− | 2 2Eˆl ! Eˆl + E− − ω ! (2π) E−ω 3 ~ 3 2 M0d Pf med ~p+ 4 4 ×|Mˆ | (2π) δ (k + P − p − P − p ). (4.2) lPi→Pf p+ 3 P 3 i + f − (2π) Ef (2π) E+

ˆ ~ ˆ2 2 where l =(Eˆl, lT ,lL) is on-mass shell, i.e., l = me.

We take ~k along the z-direction, and deﬁne z as the momentum fraction of k = |~k| carried by the longitudinal momentum of electron,

25 Figure 6: The TOPT decomposition for pair production.

Figure 7: Four TOPT diagrams after neglecting the contributions of the backward components at high energy and small scattering angle.

26 pL E z = − ≃ − , (4.3) k ω where we neglect the electron mass me at high energy. At high energy and small emitted

l2 l2 T ˆ T angle we denote k = [ω, 0,ω], p− = [zω + 2zω ,lT ,zω] and l = [(1 − z)ω + 2(1−z)ω , −lT , (1 − z)ω]. Trough a simple calculation, one can get

α 2 dσ − ≡ dσdˆ Ω P dz ln~l γ→e 2π T

2 2M0 4 2 2 P α 2 = |Mˆ | d Pf δ(P − M )Θ(E )me|~p−|dE−dΩ Pdzd ln~l lPi→Pf p+ (2π)2 f 0 f 2π T 2 |Mˆ | 2meM0 lPi→Pf p+ α ~2 = 2 |~p−| dΩ Pdzd ln lT (2π) 2[M0 +(ω − E−) − (ω − E−) cos θ] 2π q2 2 2 2 θ 2 θ Z α 1 cos 2 − 2M 2 sin 2 α = 0 dΩ Pdzd ln~l2 2 2(ω−E−) 2 θ T 4ωme 2 θ µ2 1+ sin 2π sin + 2 M0 2 2 4E+ q2 2 2 2 θ 2 θ Z α 1 cos 2 − 2M 2 sin 2 = 0 dΩ 2 2 2(ω−E−) 2 θ 4ω 2 θ µ2 1+ sin sin + 2 M0 2 2 4E+ α 1 − z z ~2 × z + dzd ln lT . (4.4) 2π z 1 − z At the last step, we use

1 1 ≃ , (4.5) 2Eˆl 2(1 − z)ω

1 1 2z(1 − z)ω ≃ = , (4.6) l2 l2 2 Eˆ + E− − ω T T lT l (1 − z)ω + 2(1−z)ω + zω + 2zω − ω

2 2 1 − z z |Mk→ˆlp− | =2αlT + , (4.7) z 1 − z to calculate

2 2 3 α ~2 2π 1 1 2 med ~p− P dz ln lT = |Mk→ˆlp− | 3 2π ω 2Eˆl ! Eˆl + E− − ω ! (2π) E−

27 α me 1 − z z ~2 = z + dzd ln lT , (4.8) 2π ω z 1 − z where me/ω will move to dσˆ.

Now we integral over angle in Eq. (4.4) and result is

3 2 α 4ω 2 2 dσ + = ln (1 − z) (1 − z) + z dz. (4.9) γ→e 2µ2 µ2 h i Consideringγ → e+ = γ → e−, we have

3 2 α 4ω 2 2 dσ = dσ + + dσ − = ln (1 − z)[(1 − z) + z ]dz, (4.10) γ→e γ→e γ→e µ2 µ2 where e = e+ + e−.

28 5 The improved electromagnetic cascade equation

High-energy electrons traversing matter lose energy by radiation. The secondary photon products a pair of electron-positron, which can further radiate. At each step the number of particles increases while the average energy decreases, until the energy falls below the critical energy. This phenomenon is called the electromagnetic shower. The evolution of the energy spectra of electrons and photons in a shower is described by the cascade equation, which couples bremsstrahlung of electron and pair production of photon. The theoretical basis of the electromagnetic cascade equation is the BH formula for bremsstrahlung and pair creation. Therefore, we modify the cascade equation in this section.

We denote X and λ as the depth and the radiation length in unity g/cm2. The photon

ﬂux Φγ and electron/positron ﬂux Φe satisfy the coupled equations in the electromagnetic cascade process [14]

∞ 1 dΦγ (ω, t) dEi ω = Pe→γ Φe(Ei, t) − Φγ (ω, t) dzPγ→e(z), (5.1) dt Zω Ei Ei Z0 and

∞ 1 ∞ dΦe(Ef , t) dEi Ef dω Ef = Pe→e Φe(Ei, t)−Φe(Ef , t) dzPe→e(z)+ Pγ→e Φγ (ω, t). dt ZEf Ei Ei Z0 ZEf ω ω (5.2)

Where the cascade kernel Pe→γ(z)dtdz (or Pe→e(z)dtdz) is the probability for an electron/positron of energy Ei to radiate a photon of energy ω = zEi (or to an electron/positron of energy Ef = zEi) in traversing dt = dX/λ, while Pγ→e(z)dtdz is the probability for a photon of energy ω to radiate an electron/positron of energy Ef = zEi in traversing dt = dX/λ.

29 A following key step is to extract the cascade kernels from the bremsstrahlung- and

2 2 pair production-cross creation. The logarithmic ln 4Ei /µ 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. (2.45) and

(4.10) have similar form as that in the QCD equation except the diﬀerent normalization coeﬃcients and two factors Ef /Ei and E−/ω. The former is because of the reason that

Eqs. (5.1) and (5.2) are scaled by the radiation length λ, while the later two factors are arisen from the deﬁnition of the parton (electron) distribution. We imitate the QCD evolution equation and suggest to insert

E E 1= f i , (5.3) Ei Ef and

E ω 1= − , (5.4) ω E− into Eqs. (2.40) and (4.10), where Ef /Ei (or E−/ω) incorporates with the ﬂux Φe (or Φγ ), while Ei/Ef (or ω/E−) belongs to the cascade kernels. Using the normalized condition

[14]

1 dzzPe→e(z)=1, (5.5) Z0 we get

3 1+ z2 P = , (5.6) e→e 4 1 − z

3 1+(1 − z)2 P = , (5.7) e→γ 4 z 30 and

3 P = [z2 + (1 − z)2]. (5.8) γ→e 4 Our numeric solutions show that the above mentioned substitutions don’t inﬂuent the anomalous bremsstrahlung eﬀect.

The traditional cascade equation has a same form as Eqs. (5.1) and (5.2) but with the diﬀerent cascade kernels, they are [14]

1 − z 4 Pe→γ(z)= z + +2b , (5.9) z 3

z 4 Pe→e(z)=1 − z + +2b (5.10) 1 − z 3 and

2 1 4 1 P (z)= − b +( +2b)(z − )2. (5.11) γ→e 3 2 3 2 The parameter b = [18 ln(183/Z1/3)]−1 =0.0122 relates with the screening eﬀects.

We compare the cascade kernels of the traditional equation and our improved equation in Fig. 8. One can ﬁnd the diﬀerence between them is negligible. Note that the cascade equation is scaled by the radiation length λ. Therefore the integrated cross sections σa→b are not appear in the cascade equation. The solution Φ(E, t) should be replaced by

Φ(E,X) using X = tλ, where λ contains the anomalous eﬀect in bremsstrahlung and pair creation.

Note that the cascade kernel is normalized in Eq. (5.5), therefore, the anomalous bremsstrahlung eﬀect does not directly appear in the solutions of the cascade equation.

When we estimate the value of the radiation length λ as shown in Sec. 1, this eﬀect is shown by the cross section.

31 10

6 a b P 4

Pe e 2

Pg e

0 0.0 0.2 0.4 0.6 0.8 1.0 z

Figure 8: Comparison of the cascade kernels for a new cascade equation (the solid curves) and the BH-formula-based equation (the dashed curves)

6 Summary

The BH formula successfully describes bremsstrahlung of high energy electrons. The

2 integrated cross section of the BH formula is restricted in a region ∼ 1/me, 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.

ACKNOWLEDGMENTS I would like to thank an anonymous reviewer for his/her insightful comments on the manuscript. Especially, the statement in Eqs. (3.15)-(3.18)

32 is proposed by this reviewer. The work is supported by the National Natural Science of

China (No.11851303).

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