Application of a Revised Moliere Theory to the Description of the Landau-Pomeranchuk Effect

Using the Coulomb corrections to some important parameters of a revised Moliere multiple scattering theory, we have obtained analytically and numerically the Coulomb corrections to the quantities of the Migdal theory of the Landau-Pomeranchuk (LPM) effect for sufficiently thick targets. We showed that the Coulomb correction to the spectral bremsstrahlung rate of this theory allows completely eliminating the discrepancy between the theory and experiment at leastfor high Z experimental targets.


Introduction
The theory of the multiple scattering of charged particles has been treated by several authors [1][2][3][4][5][6][7]. However, the most widely used at present is the multiple scattering theory of Molière [3] whose results are employed nowadays in most of the transport codes. It is of interest for numerous applications related to particle transport in matter; and it also presents the most used tool for taking into account the multiple scattering effects in experimental data processing. The DIRAC experiment [8] like many others [9] (the MuScat [10], MUCOOL [11] experiments, etc.) meets the problem of the excluding of multiple scattering effects in matter from obtained data. The standard theory of multiple scattering [8,9,10], proposed by Molière [3] and Fano [6], and some its modifications [10,12,13] are used for this aim.
As the Molière theory is currently used roughly for 10−300 GeV electron beams, the role of the high-energy corrections to the parameters of this theory becomes significant. Of especial importance is the Coulomb correction to the screening angular parameter, as this parameter also enters into other important quantities of the Molière theory.
Landau and Pomeranchuk were the first to show [14] that multiplicity of electron scattering processes on atomic nuclei in an amorphous medium results in the suppression of soft bremsstrahlung. The quantitative theory of this phenomenon was created by Migdal [15,16] 1 . Therefore, it received the name Landau-Pomeranchuk-Migdal (LPM) effect. The analogous effects are possible also at coherent radiation of relativistic electrons and positrons in a crystalline medium [18], in cosmic-ray physics [19] (e.g. in applications motivated by extremely high energy Ice-Cubes neutrino-induced showers with energies above 1 PeV [20]). Effects of this kind should manifest themselves in scattering of protons on the nuclei, what has recently been shown in Groning by the AGOR collaboration [21], at penetration of quarks and partons through the nuclear matter at the RHIC and LHC energies [22]. The QCD analogue of the LPM effect was examined in [23]; a possibility studying the LPM effect in oriented crystal at GeV energy was analyzed in [24]. Theoretically, an analogue of the LPM effect was considered for nucleon-nucleon collisions in the neutron stars and supernovae [25], and also in relativistic plasmas [26].
The results of a series of experiments at the SLAC [27,28,29] and CERN-SPS [30,31] accelerators on detection of the Landau-Pomeranchuk effect confirmed the basic qualitative conclusion that multiple scattering of ultrarelativistic charged particles in matter leads to suppression of their bremsstrahlung in the soft part of the spectrum. However, attempts to quantitatively describe the experimental data [27] faced an unexpected difficulty. For achieving satisfactory agreement of data with theory [15] the authors [27] had to multiply the results of their calculations in the Born approximation by a normalization factor R equal to 0.94 ± 0.01 ± 0.032, which had no reasonable explanation.
The alternate calculations [32,33] gave a similar result despite different computational basis [27]. The theoretical predictions are agreement with the spectrum of photon bremsstrahlung measured for 25 GeV electron beam and 0.7-6.0%L R 2 gold targets over the range 30< ω <500 MeV of the emitted photon frequency ω only within a normalization factor 0.94 [27] -0.93 [32]. The origin of the above small but significant disagreement between data and theory needs to be better understand [28].
In [33,34,35] the multiphoton effects was taken into account, and a comparison with SLAC E-146 data was carried out. Nevertheless, the problem of normalization remained and is still not clear. The other authors, except [32,35], do not discuss this normalization problem [29].
The aim of this work is to show that the discussed discrepancy can be explained at least for high Z targets if the corrections to the results of the Born approximation are appropriately considered on the basis of a revised version of the Molière multiple scattering theory [37,38].
The paper is organized as follows. In Section 2 we consider the basic formulae of the conventional [3] and a revised [37,38] small-angle Molière multiple scattering theory. We also calculate the Coulomb corrections to some important parameters of the Molière theory for the target materials used in [28]. Then, in Section 3 we present the results of the Migdal LPM effect theory for sufficiently tick targets and obtain the analytical and numerical results for the Coulomb corrections to the quantities of this theory in regimes of the large and the small LPM suppression based on the Coulomb corrections found in Section 2. Additionally, we obtain numerical results for Coulomb corrections to the asymptotes of the spectral radiation rate within the LPM theory analogue for a thin target [39,40]. Finally, in Section 4 we briefly sum up our results. Let w M (ϑ, L) be a spatial-angle particle distribution function in a homogenous medium, and ϑ is a twodimensional particle scattering angle in the plane orthogonal to the incident particle direction. For smallangle approximation ϑ 1 (sin ϑ ∼ ϑ), the above distribution function is the number of particles scattered in the angular interval dϑ after traveling through the target of thickness L. In the notation of Molière, it reads J 0 (ϑη) is the Bessel function, n 0 denotes the number density, σ 0 (θ) presents the Born differential scattering cross-section, and θ = ϑ − ϑ.
2 LR is the radiation length of the target material.
The function (1) satisfies the well-known Boltzmann transport equation, written here with the small angle approximation The Gaussian particle distribution function used in the Migdal LPM effect theory, which differs from (1), can be derived from the Boltzmann transport equation by the Fokker-Plank method [41].
One of the most important results of the Molière theory is that the scattering is described by a single parameter, the so-called screening angle (θ a or θ a ) where C E = 0.577 . . . is the Euler constant. More precisely, the angular distribution depends only on the logarithmic ratio of the characteristic angle θ c describing the foil thickness to the screening angle θ a , which characterizes the scattering atom.
In order to obtain a result valid for large angles, Molière defined a new parameter B by the transcendental equation The angular distribution function can then be written as The Molière expansion method is to consider the term y 2 ln(y 2 /4)/4B as a small parameter. Then, the angular distribution function is expanded in a power series in 1/B in which This method is valid for B ≥ 4.5 and ϑ 2 < 1. The first function w 0 (ϑ, L) has a simple analytical form where For small angles, i.e. ϑ/ϑ = ϑ/(θ c √ B) less than about 2, the Gaussian (12) is the dominant term. In this region, w 1 (ϑ, L) is in general less than w 0 (ϑ, L), so that the correction to the Gaussian is of order of 1/B, i.e. about 10%.
A good approximate representation of the distribution at any angle is with This approximation was applied by authors of [40] to the analysis of data [27,28] over the region ω < 30 MeV that will be shown in Section 3.
As show the classical works of Molière [3], the quantity (2) can be represented in the area of the important η values 0 ≤ η ≤ 1/θ c as where the screening angle θ a depends both on the screening properties of the atom and on the σ 0 (θ) approximation used for its calculation.
Using the Thomas-Fermi model of the atom and an interpolation scheme, Molière obtained θ a for the cases where σ 0 (θ) is calculated within the Born and quasiclassical approximations: Here, Z is the nuclear charge number of the target atom, α = 1/137 is the fine structure constant, and β = v/c is the velocity of a projectile in units of the velocity of light. The latter result (18) is only approximate (see critical remarks on its derivation in [41]). Below we will present exact analytical and numerical results for the screening angle and some other parameters of the Molière theory.

Molière
Very recently, it has been shown [38] that for any model of the atom the following rigorous relation determining the screening angular parameter θ a is valid: 19) or, equivalently, where ∆ CC is the Coulomb correction to the Born result, ψ is the logarithmic derivative of the gamma function Γ, and f (Zα/β) is an universal function of the Born parameter ξ = Zα/β, which is also known as the Bethe-Maximon function: .
To compare the approximate Molière result (18) with the exact one (20), we first present (18) in the form and also rewrite (20) as follows: Then we get: In order to obtain the relative difference between the approximate θ M a and exact θ a results for the screening angle we rewrite (22) and (23) in the following form and obtain for the ratio R CCM [θ a ] the expression We can also represent the relative difference (25) by the equation  (24), (26), and the ratio (28) in the range of nuclear charge 73 ≤ Z ≤ 92.  Figure 1).
From the Table 1 it is evident that the Coulomb correction δ CC [θ a ] has a large value, which ranges from around 30% for Z ∼ 70 up to 50% for Z ∼ 90. The relative difference between the approximate and exact results for this Coulomb correction varies from 17 up to 20% over the range 73 ≤ Z ≤ 92.
The relative difference δ CCM [θ a ] between the approximate θ M a and exact θ a results for the screening angle as well as R CCM [θ a ] = θ a /θ M a value does not vary significantly from one target material to another. Their sizes are 5.86 ± 0.22% for −δ CCM [θ a ] and 0.941 ± 0.002 for R CCM [θ a ] over the Z range studied. It is interesting that the latter value coincides with the normalization constant R = 0.94 ± 0.01 found in [27].
We show further that the above discrepancy between theory of the LPM effect and experiment [27,28,32] can be completely eliminated for heavy target elements on the basis of the Coulomb corrections to the screening angular parameter. For this purpose, we calculate also some additional Coulomb corrections to other important parameters of the Molière theory. Inserting (5) into (7) and differentiating the latter, we arrive at Accounting ϑ 2 = θ 2 c B (11), we get Finally, the relative Coulomb corrections can be represented as The Z dependence of the corrections (31), (32), and (34) is presented in Table 2 (see also Figure 1).  There exist two methods that allow to develop a rigorous quantitative theory of the Landau-Pomeranchuk effect. It is Migdal's method of kinetic equation [15,16] and the method of functional integration [32,33,35,36,42]. Neglecting numerically small quantum-mechanical corrections, we will adhere to version of the Landau-Pomeranchuk effect theory developed in [15].
Simple though quite cumbersome calculations yield the following formula for the electron spectral bremsstrahlung intensity averaged over various trajectories of electron motion in an amorphous medium (hereafter the units = c = 1, e 2 = 1/137 are used): where (dI/dω) 0 is the spectral bremsstrahlung rate without accounting for the multiple scattering effects in the radiation, and γ is the Lorentz factor of the scattered particle. The function Φ(s) accounts for the multiple scattering influence on the bremsstrahlung rate and reads It has simple asymptotes at the small and large values of the argument: For s 1, the suppression is large, and Φ(s) ≈ 6s. The intensity of radiation in this case is much less, than the corresponding result of Bethe and Heitler. If s ≥ 1, the function Φ(s) is close to a unit, and the following approximation is valid [18]: The formula (35) is obtained with the logarithmic accuracy. At s 1, (35) coincides within this accuracy with the Bethe-Heitler result If s 1, we have the large LPM suppression in comparison with (43). Let us notice that effect of a medium polarization is not considered here, i.e. it is assumed that the absolute permittivity of the medium ε(ω) = 1.

Applying the revised theory of Molière to the Migdal LPM effect theory
Now we obtain analytical and numerical results for the Coulomb corrections to the quantities of the Migdal LPM effect theory. In order to derive an analytical expression for the Coulomb correction to the Born spectral bremsstrahlung rate (dI/dω) 0 , we first write where In doing so, (44) becomes and the relative Coulomb correction reads Next, in order to obtain the relative Coulomb correction to the Migdal function Φ(s), we first derive corresponding correction to the parameter s 2 (39): This leads to the following relative Coulomb correction for s (41): For the asymptote Φ(s) = 6s (40), we get Then the total relative Coulomb correction to the spectral density of radiation in this asymptotic case becomes The regime of strong LPM suppression is not reached in the conditions of the experiment [27,28,29]. Therefore, we will carry out now calculation for the regime of small LPM suppression (42).
In order to obtain the relative correction δ CC [Φ(s)] in this regime, we first derive an expression for the Coulomb correction ∆ CC [Φ(s)] to the Migdal function Φ(s): This leads to the following relative Coulomb correction for Φ(s) (42): In Table 3 are listed the values of the relative Coulomb corrections to the quantities of (35) in the regime of small LPM suppression (42) for some separate s values from the range 1.0 ≤ s ≤ ∞ (e.g., for s = 1.1 and s = 1.5).  It will be seen from Table 4 that the Coulomb corrections δ CC [ dI/dω ] = −4.50 ± 0.05% (Z = 82) and δ CC [ dI/dω ] = −5.35 ± 0.06% (Z = 92) coincide within the experimental error with the sizes of the normalization correction −4.5 ± 0.2% for 2%L R lead target and −5.6 ± 0.3% for 3%L R uranium target, respectively (Table II in [28]).
We believe that this allows to understand an origin of the discussed in [27,28] normalization problem for high Z targets.

Application of Molière's theory to the description of the LPM effect analogue for a thin target
Experiment [27,28] caused considerable interest and stimulated development of various approaches to the study of the LPM effect, including an application of Molière's results to the description of an analogue of the LPM effect for a thin layer of matter [40] 5 .
In [40] it is shown that the region of the emitted photon frequencies naturally splits into two intervals, ω > ω c and ω < ω c , in first of which the LPM effect for sufficiently tick targets takes place, and in the second, there is its analogue for thin targets. The quantity ω c is defined here as ω c = 2γ 2 /L.
Application of the Molière multiple scattering theory to the analysis of experimental data [27,28] for a thin target in the second ω range is based on the use of the expression for the spatial-angle particle distribution function (1), which satisfies the standard Boltzmann transport equation for a thin homogenous foil and differs significantly from the Gaussian particle distribution of the Migdal LPM effect theory.
Besides, it determines an another expression for the spectral radiation rate in the context of the coherent radiation theory [40], which reads Here dI(ϑ) dω = 2e 2 π 2χ 2 + 1 χ χ 2 + 1 ln χ + χ 2 + 1 − 1 (57) with χ = γϑ/2. The latter expression is valid for consideration of the particle scattering in both amorphous and crystalline medium. The formula (57) has simple asymptotes at the small and large values of parameter χ: Replacing in this formula ϑ 2 by the average square value of the scattering angle ϑ 2 , we arrive at the following estimates for the average radiation spectral density value: In the experiment [27,28], the above frequency intervals correspond roughly to the following ω ranges: (ω > ω c ) ∼ (ω > 30 MeV) and (ω < ω c ) ∼ (ω < 30 MeV) for 25 GeV electron beam and 0.7 − 6.0%L R gold targets. Whereas in the first area the discrepancy between the LPM theory predictions and data is about 3.2 to 5% that requires the use of normalization factor 0.94 ± 0.01 ± 0.032, in the second area this discrepancy reaches ∼ 15%.
Using the second-order representation of the Molière distribution function (14), (15) for computing the spectral radiation rate (56) the authors of [40] were able to agree satisfactorily theory and 25 GeV and 0.7%L R data over the range ω < 30 MeV.
This result can be understood by considering the fact that the correction to the Gaussian first-order representation of the distribution function w M (ϑ) of order of 1/B B is about 12% for the used in calculations value B B = 8.46 [40].

Coulomb corrections in the coherent radiation theory for a thin target
Let us obtain the relative Coulomb correction to the average value of the spectral density of radiation for two limiting cases (59).
In the first case γ 2 ϑ 2 1, takin into account the equality (47), and (59), we get where B B ≈ 8.46 in the conditions of the discussed experiment [40].
In the second case γ 2 ϑ 2 1, we have For the latter quantity one can obtain The Coulomb correction becomes Taking into account (47), we arrive at a result: The numerical values of these corrections are presented below. The second asymptote is not reached [40] in the conditions of experiment [27,28]. Therefore we will also consider an another limiting case corresponding to these experimental conditions and taking into account the second term of the Molière distribution function expansion (9).
Inserting the second-order expression (14) for the distribution function into (56) and integrating its second term (15), we can arrive at the following expression for the electron radiation spectrum at µ 2 = γ 2 ϑ 2 1 [40]: In order to obtain the Coulomb correction to the Born spectral radiation rate from (66) Table 6. These corrections are not large. Their sizes are between two to four percent, i.e. of order of the systematic error in the experiment [27].  [39,40] and experiment [27,28] and coincides with the result of [33] obtained in the eikonal approximation (see Fig. 20a in [29] [28] in the regime of small LPM suppression.
5. Thus, we managed to show that the discussed discrepancy between theory and experiment [27,28] over the range 20 < ω < 500 MeV can be explained at least for heavy target elements on the basis of the obtained Coulomb corrections to the Born bremsstrahlung rate within the Migdal LPM effect theory.
7. The approach based on application of the improved Molière multiple scattering theory can be useful for the analysis of electromagnetic processes in strong crystalline fields at high energies, the description of cosmic-ray experiments, high-energy experiments with nuclear targets, etc.