Comparison with experimental data of different theoretical approaches to high-energy electron bremsstrahlung including quantum coherence effects

The basic expressions for the differential nuclear bremsstrahlung cross section at high electron energy, as derived under different theoretical approaches and approximations to quantum coherence effects, are compared. The Baier-Katkov treatment is reformulated to allow introduction of the same value of the radiation length in all calculations. A dedicated Monte Carlo code is employed for obtaining photon energy spectra in the framework of the Baier-Katkov approach taking into account multiphoton emission, attenuation by pair production, and pile-up with photons from the background. The results of Monte Carlo simulations for both the Migdal and Baier-Katkov descriptions are compared to all available data that show the Landau-Pomeranchuk-Migdal suppression. The issue of the sensitivity of the experiments to the difference of the two approaches is investigated.

Figure 1

Comparison of the best available bremsstrahlung cross sections (SM-LO continuous line), obtained with the Furry-Sommerfeld-Maue wave functions at the leading order, with the BH high-energy approximation (dashed line) given in Eq. (2). The power spectrum $kd\sigma /dk$ is actually shown to cancel the BH divergence at low values of $k$. The SM-LO results take into account the screening correction with the Olsen-Maximon-Wergeland additivity rule and the atomic form factors by Hubbell et al. [49, 50].

Figure 2

Ratio of the main term of the BK radiation intensity to the Migdal one; see Eq. (22). To compare both approaches on the same footing, the Coulomb correction is omitted from the BK expression and $X_0^{\mathrm{c}}$ is used for $X_0$ in both Eq. (10) and Eq. (12).

Figure 3

Relative difference of the radiation length, $X_0$, obtained from the BK expression, Eq. (28), using for $L_1$ the value of Eq. (20) from the value tabulated in Ref. [38] and used in the present work by a redefinition of $L_1$, as explained in the text.

Figure 4

Comparison of the predicted power spectra for a 287-GeV electron beam impinging on a $128\text{−}\mu \mathrm{m}$-thick iridium target. Different curves are calculated according to the BH, Migdal, BD, and BK approaches as specified in the legend. For the latter, the main term as well as the correction term are reported separately. Polarization effects are completely negligible for the photon energy range considered.

Figure 5

Comparison of the predicted power spectra for a 287-GeV electron beam impinging on a 3% $X_0$ aluminum target. Different curves are calculated according to the BH, Migdal, and BK approaches both including and excluding dielectric suppression as specified in the legend. For the latter, the main term as well as the correction term are reported separately.

Figure 6

Comparison of the BK approach (continuous line) [see Eqs. (21) and (25)] with the CERN LPM data for a 287-GeV electron beam impinging on a $128\text{−}\mu \mathrm{m}$-thick iridium target (solid triangles). The result of a Monte Carlo simulation, based on the same theory, and with all other interaction processes disabled, is also shown (solid circles).

Figure 7

Difference of the approximated analytic multiphoton correction factor $f$ [see Eq. (30)] with respect to the exact Monte Carlo calculations [see Eq. (31)] for a 287-GeV electron beam impinging on a $128\text{−}\mu \mathrm{m}$-thick iridium target. The three approximate expressions developed by Baier and Katkov are considered: Eq. (32), (33), and (36) (solid circles, squares, and triangles, respectively).

Figure 8

Difference $(s-r)/s$ of a simulation $s$ with respect to a reference one $r$ for two cases. In the first (solid circles), $s$ is the complete simulation with all processes and $r$ the simulation with only bremsstrahlung always considering 25-GeV electrons impinging on a 3% $X_0$ aluminum target. In the second (solid squares), $s$ is the simulation with bremsstrahlung and pair production enabled and $r$ the simulation with only bremsstrahlung.

Figure 9

Comparison of a full Monte Carlo simulation, employing the BK approach and taking into account all secondary processes, with the CERN LPM data for a 287-GeV electron beam impinging on a $128\text{−}\mu \mathrm{m}$-thick iridium target. The target alone (solid circles) and the target and background with background subtracted (solid squares) are shown.

Figure 10

Difference between the simulated first photon spectrum and the BK cross section for an electron impinging on iridium with an energy $E=25\mathrm{GeV}$. The dielectric suppression is taken into account.

Figure 11

Comparison of the simulations based on the Migdal (solid circles) and BK approaches (solid squares) with the SLAC E-146 data at 8 GeV (solid triangles).

Figure 12

Comparison of the simulations based on the Migdal (solid circles) and BK approaches (solid squares) with the SLAC E-146 data at 25 GeV (solid triangles).

Figure 13

Values of ${\chi }^2/n_{\mathrm{DF}}$ for the comparison of the simulations based on the Migdal (solid symbols) and BK approaches (open symbols) with the SLAC E-146 data at 8 (circles) and 25 GeV (squares) as a function of the atomic number of the target $Z$. The bars represent the standard deviation of the reduced ${\chi }^2$.

Figure 14

Comparison of the simulations based on the Migdal (solid circles) and BK approaches (solid squares) with the CERN LPM data at 149 GeV (solid triangles).

Figure 15

Comparison of the simulations based on the Migdal (solid circles) and BK approaches (solid squares) with the CERN LPM data at 207 GeV (solid triangles).

Figure 16

Comparison of the simulations based on the Migdal (solid circles) and BK approaches (solid squares) with the CERN LPM data at 287 GeV (solid triangles).

Figure 17

Values of ${\chi }^2/n_{\mathrm{DF}}$ for the comparison of the simulations based on the Migdal (solid symbols) and BK approaches (open symbols) with the CERN LPM data at 149 (circles), 207 (squares), and 287 GeV (triangles) as a function of the atomic number of the target $Z$. The bars have the same meaning as in Fig. 13.

Figure 18

Comparison of the simulations based on the Migdal (solid circles) and BK approaches (solid squares) with the CERN LOW-$Z$ data at 178 GeV (solid triangles).

Figure 19

Values of the $D_1({\nu }_0)$ and $D_2({\nu }_0)$ functions defined by Baier and Katkov obtained with the numerical procedure described in the text.

Figure 20

Plot of the rational function approximations given by Eqs. (a1) and (a2) to $D_1$ and $D_2$ with the numerical values of the parameters reported in Table 4. Note that two different parametrizations are used below and above ${\nu }_0=0.1$ and they join smoothly both in value and slope.

Figure 21

Plot of the relative deviation of the rational function approximations given by Eqs. (a1) and (a2) to the true values of $D_1$ and $D_2$.

Figure 22

Value of the photon energy $k=k_{\mathrm{d}}$ at which ${\rho }_{\mathrm{c}}=1$ (no dielectric suppression) and ${\tilde{\rho }}_{\mathrm{c}}=1$ (dielectric suppression) as a function of the impinging electron energy $E$ for three materials. Without dielectric suppression, there is always a value of $k=k_{\mathrm{d}}$ for every value of $E$ (dashed line). With dielectric suppression, there is no value of $k=k_{\mathrm{d}}$ for $E$ below a minimum value $E_{\mathrm{d}}$. For $E$ above $E_{\mathrm{d}}$, there are two values of $k=k_{\mathrm{d}}^{\mathrm{l}}$ and $k=k_{\mathrm{d}}^{\mathrm{u}}$ for one value of $E$: therefore a lower (dotted line) and an upper (solid line) branch are present. The two branches join for $E=E_{\mathrm{d}}$.

Figure 23

Minimum value of the impinging electron energy $E=E_{\mathrm{d}}$ for which the discontinuity in the first derivative of the cross section is present, when dielectric suppression is taken into account, as a function of the atomic number of the target element. Both the Migdal (solid circles) and BK approaches (solid squares) are considered.

Figure 24

Total bremsstrahlung cross section for the radiation of a photon with energy $k>\mathrm{TCUT}=10\mathrm{keV}$ by an electron with energy $E$ impinging on the indicated material. The vertical scale on the left is used for copper and iridium and the one on the right for carbon. Three approaches are considered: the BK one without dielectric suppression (dashed lines), the BK one with dielectric suppression (solid lines), and, finally, the one by Migdal with dielectric suppression (dotted lines).

Figure 25

Difference between the exact value of the maximum $q_{\max }(E)$ of the function $q$ [see Eq. (d3)] and the result obtained by the sixth-order polynomials in $E$ fitted to a set of precalculated values. Both the case without (dashed lines) and with (solid lines) dielectric suppression are considered.