A note on the comparison between ABEL, CAIN and Guinea-Pig.

Results with parameters of "published paper"

We compared numbers of pairs by ABEL, CAIN and Guinea-Pig(GP). Parameters of linear collider are the same as those of published paper Particle Accelerators, 1993, Vol.41, pp29-39), and the major ones are expressed below;

Ecm = 1000GeV,  N=2.019 x 10^{10}/bunch,
sigma_y / sigma_x = 3.077nm / 372nm
beta_x/beta_y = 0.12356mm / 24.62mm
and headon collision.

For total numbers of generated pairs ( 5MeV < E_e < 500GeV), ABEL has about 20% less pairs than CAIN and GP, while CAIN agrees very well with GP(pair_q2=0, i.e. Q^2=m^2). There are three processes of LL (Landau-Lifshitz), BH (Bethe-Heitler) and BW (Breit-Wheeler) in incoherent pair creation. Among them, BH of ABEL was apparently less than those of CAIN and GP.

The three programs have two effects in processes with virtual photons; that is, (1) beam size effect and (2) external field effect. In order to study this discrepancy, pairs were generated without the two effects. For LL and BW, ABEL's numbers were 1.8 and 1.5 times higher than those of CAIN/GP, respectively, while BH pairs were same for the three. Since the absolute number of BW is two order of magnitudes less than LL and BH, the difference in BW was not seen in total. Next, the beam size effect was studied, where only the beam size effect was taken account of. The suppression factors in LL(BH) were 0.33 (0.37), 0.52 (0.47) and 0.51 (0.46) for ABEL, CAIN and GP, respectively. So the ABEL's factor was 0.63 (0.78) of those of CAIN and GP. Additional effects due to the external field were very similar for them, whose suppression factors in LL (BH) were 0.80 (0.99), 0.75 (1.0), 0.77 (1.0) for ABEL, CAIN and GP, respectively. With all effects, the LL of ABEL has 1.8 (generation) x 0.63 (beam size) x 1.05 (external field) = 1.19 and the BH of ABEL has 1.0 (generation) x 0.78(beam size) x 0.99 (external field) =0.77 relative to CAIN and GP. Since the generated number of BH pairs is dominant, the total generated number of ABEL was 20% less than those of CAIN and GP as mentioned before.

What was wrong in ABEL calculations ?

There are two problems; that is, (a) total cross sections of LL and BW and (b) the beam size effect. In ABEL there are patch factors corresponding to minimum angles of pair particles because of the ultra-relativistic approximation (m_e << E_e) employed in it. The patch factors have been determined by comparing "known" energy distributions in LL and BH. The "known" energy distributions are used in the original ABEL. They are expressed by,

   ds    56*(a*re)**2
  --- = -------------*2*log(1/x)*log(s/m**2*x)        (1)     for LL
   dx    9*pi * x

, (m/E1 << x << 1) , where , x=(E_e of final e+ or e-)/E1 , s=4*E1*E2 (E1,E2=beam energy (E0) ) (dependence on E2 ignored. replaced by E0), a=fine str.const., re=classical electron radius, m=rest mass, and see a reference paper of E.A.Kraev et al, Sov.J.Nucl.Phys.23(1976)85 , and

   ds    16*a*re**2                    s*x*(1-x)
  --- = -----------*(x**2-x+ 3/4)*(log----------  - 1/2 )   (2) for BH
   dx       3                          2*m**2

,(m/Ephoton < x < 1), where x=(E of final e+ or e-)/Ephoton , s=4*E1*Ephoton (E1=initial electron energy). Then, the patch factors were 0.70, 0.92 and 0.50 for LL, BH and BW, respectively. Actually, an integration of Eq. (1) is 1.55 times that of CAIN/GP for (0.005 < E_e<500GeV) while an integration of Eq.(2) is the same as them. Therefore, ABEL over-estimated the LL cross section by 1.55 at least.

Patch factors

We noticed that there is another method to determine the patch factors. A simple calculation shows that our approximated formulae in the published paper become exact ones if cosine is replaced with beta times cosine in everywhere, where beta is P_e/E_e. Therefore, the patch factors can be determined by equating a calculation of approximated formulae (the patch factors) with that of the "exact" formulae (beta x cosine). The resultant patch factors are 0.90, 0.92 and 0.92 for LL, BH and BW, respectively. For these calculations, we used BASES program of numerical integrations.

With the updated patch factors, ABEL generated almost the same numbers of BH and BW as CAIN/GP, while the number of LL was 1.18 times of them. The difference of 18% is considered to be due to further approximations employed in ABEL. This amount is within an accuracy of real photon approximation such as Q^2 scale ambiguity. A small correction in the LL cross section was applied too in order to correct the energy distribution at high energy. Now, we understand the 1.8 factor in the LL-calculation by 1.55 (energy function) x 1.18 (ABEL/BASES). The 1.5 factor in the BW was reduced to unity by a change of the patch factor from 0.5 to 0.92.

Beam size effect

In ABEL, the suppression factor due to the beam size effect is calculated by a ratio, R= n n' / n n, where n and n' are local and non-local beam-intensity, respectively. The non-local intensity is one at a point separated by an impact parameter and a sum of two impact parameters for BH and LL, respectively. The impact parameter corresponds to transverse displacement, therefore a vector, of virtual photon. Previously the direction of the impact parameter was allowed in all angular region, i.e. 2 pi, although its origin must be in the beam transverse profile, and probabilities for cases of R>1 are always set to be 1. In updated ABEL, the angular region was restricted in such way, and the cases of R>1 are properly taken account of by increasing the probabilities. The suppression factors become to be 0.46 and 0.44 for LL and BH, respectively, which should be compared with 0.52 and 0.47 of CAIN/GP.

Results of updated ABEL

With the above updates on the patch factors, the beam size effect, ABEL's number of pairs is almost same as those of CAIN and GP. For practical usages, the three programs give consistent results on background calculations (hit density etc.) based on JIM simulations even for the previous ABEL.