- Email: [email protected]
The self-coupling of weak lepton currents in high energy neutrino and muon reactions
ANNALS
OF PHYSICS:
75, 491-502 (1973)
The Self-Coupling of Weak Lepton in High Energy Neutrino and Muon KAZUO
Currents Reactions*
FUJIKAWA
The Enrico Fermi Institute, UniLlersity of Chicago, Chicago, Illinois 60637 Received January 24, 1972
The transverse momentum distribution of the muons in the reaction Y + (z) + Y + p + p + (z) is discussed. The lower end of the energy spectrum of the muons in the previous study is also modified by using an improved interpolation formula.
In the previous study [I, 21 we discussed in detail the energy spectrum of the muons at fixed angle and also angular distributions in the process (see Fig. 1). v + (2) -
v + p + F; + (4.
(1)
We also discussed the angular distribution of the muons at fixed energy and energy spectra. Marked assymmetry between p- and TV+was found, and its origin was discussed.
FIG. 1. The process Y + (z) -+ v -I- p f P i- (Z).
However, we did not discuss the transverse momentum distribution of the mrlons in the process. In the present note we discuss this transverse momentum distribution based on the calculation reported in the previous study [l, 21. This problem has been discussed by Lovseth and Radomski [3]. It is straightforward to convert * Supported in part by the U. S. Atomic Energy Commission Contract No. AT (ll-l)-264.
491 Capyrieht All rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
492
FUJIKAWA I
I
I
I
I
I
-
Fe56 Target
I I I I 0 0.1 0.2 0.3 0.4 0.5
0.75
Pl (BeV) FIGURE I I.5 F
2
:
!
-7
Fes6 Torget E, ~40 BeV
!
I
$JJ] I -1.LLL.----.. 0.75
0.1 0.2 0.3 0.4 0.5
1.0
1.2
Pl (BeVl FIGURE -T
~~-~
3
,~ ~i-T-.--
T--
0.6b” 0.5r
0.4-
--I
Fe56 Target E,= 40 BeV El”? =I6 BeV ,/--.,y-~-
0.75 Pl (EeV) FIGURE
4
r -
HIGH ENERGY NEUTRINO AND MUON REACTIONS
0.1 0.2
0.3 0.4 0.5
0.75
1.0
493
1.2
PL (BeV)
FIGS. 2-5. Transverse momentum distribution d%/dEdp,at various energies of the outgoing muons off an 56Fe target for E, = 40 BeV. Figs. (2) and (4) correspond to Figs. (10) and (11) in Ref. [2]. All these graphs represent single-particle inclusive spectra. o,, = 3.03 x 1O-44 cme and z =
26.
the angular distribution atJixed energy in Refs. [l, 21into the transverse momentum distribution at fixed energy by using the relation d20/dE dp, = (sin 19/lp ( cos @(d20/dE d cos 0)
(2)
where E, p and cos 19denote the energy, momentum and scattering angle of the muon or the antimuon in the lab. system, respectively. Note that the conversion from the left hand side to the right hand side in Eq. (2) is not so easy due to the singular Jacobian at 8 = 0 [4]. We converted Figs. 10, 11, 13 and 14 in Ref. [2] into the transverse momentum distribution [S]. Those resulting distributions are shown in Figs. 2-7. Figures 3 and 5 were newly added to cover the entire range of the muon energy. We also add Figs. 8 and 9. See also the energy spectrum in Figs. 18 and 19. Our graphs always correspond to singleparticle inclusive spectra. In hadron physics it is customary to plot the distribution d2a/dE dpL2 = *(i/l p I2COS o)(d’a/dE d COS 0).
(3)
In this casethe Jacobian is regular around 0 = 0. We show the result in Figs. 10-13 for an 56Fetarget and for E, = 10 and 40 BeV. It seemsto be that the transverse momentum is not so small, as pointed out by Lovseth and Radomski [3]. This is also qualitativly expected from the momentum transfer from the leptonic system to the nuclear target. The momentum transfer is distributed in the region. t
= 1O-2N 10-3(BeV)2.
(4)
494
FUJIKAWA
0.1 0.2
0.3
6
FIGURE
1.5 1.5 -
Proton
,C-. I/
I
Target
_
E, q 4OBeV $==I6 BeV
t
l.O-
0.75
0.4 0.5 P* (BeV)
’
\
I
1
0.25 0.5 0.75
\
\
I
1.0
1.5
2.0
1 2.5
Pl (BeV)
FIGS. 6 and 7. Transverse momentum distribution d%/dE.dp, off a proton target at E, = 10 BeV. These graphs correspond to Figs. (13) and (14) in Ref. [2]. q, = 3.03 x 1O-44cm2.
See Fig. 19 in Ref. [2], and also Table I in Ref. [3]. The value oft in Eq. (4) is about the same as the pion mass squared, mrr2 = 2 x 1O-2 (BeV)2. It is generally believed [6] that the finite transverse momentum in the hadronic process is based on the finite (i.e., small) momentum transfer between the particles in the multiperipheral chain. If one takes the pion mass squared as a typical value of the momentum transfer in the hadron process, Eq. (4) suggests that the transverse momentum distribution in the process (1) may not be so distinct from that in hadron processes. We also show the transverse momentum distribution off a proton target in Figs. 14 and 15, and off a 20*Pb target in Figs. 16 and 17 for E, = 40 BeV. All these calculations are based on the results reported in Ref. [2] [5]. These figures,
HIGH
ENERGY
NEUTRINO
I
i
AND
I
I
MUON
495
REACTIONS I
I
Fe5’j Target E, =I0 BeV Ep,+=2BeV
0.1 0.2 0.3 0.4 0.5
0.75
PL (BeV) FIGURE
8
i r, < b 2 “, w
I
\
I
O;b 0.1 0.2 0.3 0.4 0.5 Pi (BeV)
1 c.75----hi
FIGS. 8 and 9. Transverse momentum distribution deo/dEifpo, x 1O-44 cm%.
off
an 6sFe target at Ev = 10 BeV.
o. = 3.03
compared with Figs. 2-5, clearly indicate the tendency we just expect. Note that the momentum transfer squared for a proton target is distributed in the region t = 10-l N 10” (BeV)2
(5)
Figures 14 and 15 suggest that the incoherent process off a proton target may be rather distinct from the semileptonic process at high energies. However, some model considerations [7] also suggest that the transverse momentum of the
lO-2
0
I
I
I
,
I
I
,
0.1 0.2 0.3 0.4 0.5 0.6 Pi (BeV)’ FIGURE 10
Positive Muon
5 t? -Y %7 b-e, “, w -tY ‘0° 0I.5
10-l 0
0.1 0.2 0.3 0.4 0.5 P: (BeV?
10 and 11. Transverse momentum distribution 10BeV. o, = 3.03 x 10-4Pcm8. z = 26. FIGS.
d20/dEdp,2 off an 68Fe target for I?, =
IO2 50
IO’ 5 b" -T "2 2;
IO0 0.5
10-l I 0
I 0.1
I \I 0.2 0.3
I 0.4
?\Fiziz 0.5 0.6
0.7
P: (BeV)* FIGURE 12 IO2 50
IO’
0.5
10-l
0.2 0.3 0.4 0.5 0.6 0.7 P: (BeV12 FIGS. 12 and 13. Transverse momentum distribution d20/dEdpLs off an 68Fe target for E, = 4oBev. ql = 3.03 x 1O-*4 cm2. 0
0.1
498
FUJIKAWA
secondary particles in the semileptonic process may depend on the “mass” of the current. Therefore, it is hard to say something definite about the transverse momentum of the hadrons in the semileptonic process [8].
0 -J-.1.
L.
.mL-.-i~-.
-.~.~
0.1 0.2 0.3 0.4 0.5
I
0.75
1.0
PL (BeV)
FIGURE 14
0.1 0.2 0.3 0.4 0.5
0.75
1.0
Pl (BeV)
FIGS. 14 and 15. Transverse momentum distribution 40 BeV. o. = 3.03 x 10-44cm2.
d%/dEdp,
off a proton target at Ey =
If the Adler process [9] dominates the energetic n+ production and the “shadow” effect proposed by Bell [lo] works in full, the semileptonic process would be significantly modified. It is hoped that the detailed experimental study of the semileptonic process at high energies will provide us more useful information how to separate the purely leptonic mun production from the background.
HIGH
ENERGY
NEUTRINO
AND
0.1 0.2 0.3 0.4 0.5 PA (Be’/) 40
MUON
0.75
FIGS. 16 and 17. Transverse momentum distribution BeV. (I~ = 3.03 x 1O-p4 cm*. .z = 82.
REACTIONS
1.0 d%/dEdp,
499
1.2
off a zosPb target at E, =
Next we would like to correct the lower end of the energy spectra in the previous study [2]. This spectrum was calculated from d2a/d cos 0 dE at fixed cos 8 based on an interpolation formula. The conversion of d‘b/d cos 8 dE at fixed cos 0 into d2a/d cos 8 dE at fixed E was not expected to be so accurate at the lower end of the spectrum [l I]. The point is that the integration mesh in the two-dimensional phase space (cos 8, E) was not uniformly taken to achieve the maximum accuracy of the numerical integration. Therefore the conversion from one distribution to the other is not trivial. However, following the question raised by Llewelyn Smith [12], we found that the total cross section for an 56Fe target at E, = 40 BeV obtained from f
g
dE
(6)
500
FUJIKAWA
grossly exceeds that obtained from
s
&dcosfl.
(7)
Equation (7) is more or less “exact” in our calculation [I, 21; the values of the total cross section in Refs. [l, 21 are based on Eq. (7). Therefore the lower-end peak in the energy spectrum, in particular, in Fig. 16-b in Ref. [2] is bot reliable; this discrepancy goes far beyond what we expected [I 11. We show the energy spectra based on an improved interpolation formula in Figs. 18 and 19. These
Fe56Target E,= IO BeV 1.0 cs
0
2.5
5.0 Er- (BeV)
7.5
IO
FIG. 18. Energy spectrum do/dE off an 6SFe target at Ev = 10 BeV. This diagram corresponds to Fig. 15 in Ref. [2]. o. = 3.03 x 10-4pcm2. .z = 26.
FIG. 19. Energy spectrum da/dE off an 66Fe target at E, to Fig. 16 in Ref. [2]. q, = 3.03 x lo-“cm2.
= 40
BeV. This diagram corresponds
HIGH
ENERGY
NEUTRINO
AND
MUON
501
REACTIONS
Target BeV
1~ -
20
30
-
E (BeV)
FIG. 20.
Energy spectrum da/dE off a proton target at Ev = 40 BeV. o,, = 3.03 x 1O-44 cma
Pbzo8 Target E, =40 BeV
t‘ CJ N 1. tui 317
1.0 .L ~ P.5 i
-
- Pi 1 7
r-
\\ 'A-
f' 0
/
-- ----_ IO
I------
20
30
--.
4’0-
_’;
Ep (BeV)
FIG. 21. Energy spectrum du/dE off a zOsPbtarget at E, = 40 BeV. o,, = 3.03 x IO-44 cd.
energy spectra differ from the previous values only at below 4 % of the lower end of the spectra. See Figs. 15 and 16 in Ref. [2]. Figure 17 in Ref. [2] remains unchanged; this spectrum was calculated by using the exact matrix element. For the sake of completeness we also add the energy spectra off a proton target and also off a 208Pb target at E, = 40 BeV in Figs. 20 and 21. These graphs show that the muons tend to accumulate near the origin when a larger momentum transfer to the nuclear (or nucleon) system is allowed. There is also a typographical mistake in the table for total cross sections in Ref. [2]. The total cross section for a zosPb target at E, = 40 BeV should read a(-) respectively.
= 7.27z2,
o(+) = 7.16~~
and
u(a) = 9.6z2,
502
FUJIKAWA
Note added in proof. The following authors also studied a problem similar to ours: A. V. BERKOV, L. M. VORONICA, AND E. P. SHABALIN, Soviet J. of Nuclear Phys. 14 (1972), 233; A. E. ASRATYAN, A. M. TIKHOMROV, AND E. P. SHABALIN, Soviet J. of Nuclear Phys. 14 (1972), 694; K. Korxn ET AL., Prog. Th. Phys. 46 (1971), 1150, and Kyoto Univ. report, RIFP 128 (1971). I am grateful to Dr. Llewellyn Smith for calling those works to my attention.
ACKNOWLEDGMENTS I am grateful to Professors S. B. Treiman and A. Mann for discussions.
REFERENCES
1. 2. 3. 4. 5.
6. 7. 8.
9. 10. 11.
12.
K. F~~KAWA, Thesis, Princeton University, May 1970, unpublished. K. FUJXAWA, Ann. Phys., N.Y. 68 (1971), 102. J. Lovsnrrr AND M. RADOMSKI, Phys. Rev. D 3 (1971), 2686. Because of this vanishing Jacobian at 0 = 0, the sharp peak of the angular distribution at 0 = 0 is rather misleading. When one calculates the transverse momentum distribution this sharp peak disappears due to the Jacobian. The computer output cards in the previous study which bear the distribution d%/dEiicos 6 for tied cos 0 and varying E were used. We converted this distribution into d%/dEdms 6 for tied E and varying cos 0 by an interpolation formula. Then we used Eqs. (2) and (3) to get transverse momentum distributions. See, for example, K. Wnso~, Cornell Preprint, CLNS - 131, November 1970. J. D. BJORKEN, SLAC-PUB-901 (1971). K. FUIIKAWA, Nuovo Cimento, 12A (1972), 117. The transverse momentum of p+ in the purely leptonic process could be distinct from that in the semileptonic process at extremely high energies because of different particle multiplicity. It is, however, dangerous to discuss this problem further without a detailed knowledge about the transverse momentum in the semileptonic process at high energies. If one knows the incident energy of the neutrino, one can form a sort of Dal&z plot for the muon pair in the final state. In this case it will be slightly easier to distinguish the purely leptonic process from the semileptonic process. S. ADLER, Phys. Rev. 135 (1964), B936. J. S. BELL, Lecture Note for the 1 lth Scottish Universities Summer School -in Physics, Edinburgh, 1970. We expected that the lower end of the energy spectrum may not be so accurate. See p. 51 of Ref. [I] and also Footnote 20 in Ref. 121. The consistency check of the interpolation formula was made at Ey = 10 BeV, based on Figs. 15 and 17 in Ref. [2]. Unfortunately these graphs showed a rather reasonable behavior as one can see by comparing Figs. 18 - 21 in the present note with those figures in Ref. 121. C. H. LLJZWELYNSMITH, Physics Reports 3C (1972), 264.
OF PHYSICS:
75, 491-502 (1973)
The Self-Coupling of Weak Lepton in High Energy Neutrino and Muon KAZUO
Currents Reactions*
FUJIKAWA
The Enrico Fermi Institute, UniLlersity of Chicago, Chicago, Illinois 60637 Received January 24, 1972
The transverse momentum distribution of the muons in the reaction Y + (z) + Y + p + p + (z) is discussed. The lower end of the energy spectrum of the muons in the previous study is also modified by using an improved interpolation formula.
In the previous study [I, 21 we discussed in detail the energy spectrum of the muons at fixed angle and also angular distributions in the process (see Fig. 1). v + (2) -
v + p + F; + (4.
(1)
We also discussed the angular distribution of the muons at fixed energy and energy spectra. Marked assymmetry between p- and TV+was found, and its origin was discussed.
FIG. 1. The process Y + (z) -+ v -I- p f P i- (Z).
However, we did not discuss the transverse momentum distribution of the mrlons in the process. In the present note we discuss this transverse momentum distribution based on the calculation reported in the previous study [l, 21. This problem has been discussed by Lovseth and Radomski [3]. It is straightforward to convert * Supported in part by the U. S. Atomic Energy Commission Contract No. AT (ll-l)-264.
491 Capyrieht All rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
492
FUJIKAWA I
I
I
I
I
I
-
Fe56 Target
I I I I 0 0.1 0.2 0.3 0.4 0.5
0.75
Pl (BeV) FIGURE I I.5 F
2
:
!
-7
Fes6 Torget E, ~40 BeV
!
I
$JJ] I -1.LLL.----.. 0.75
0.1 0.2 0.3 0.4 0.5
1.0
1.2
Pl (BeVl FIGURE -T
~~-~
3
,~ ~i-T-.--
T--
0.6b” 0.5r
0.4-
--I
Fe56 Target E,= 40 BeV El”? =I6 BeV ,/--.,y-~-
0.75 Pl (EeV) FIGURE
4
r -
HIGH ENERGY NEUTRINO AND MUON REACTIONS
0.1 0.2
0.3 0.4 0.5
0.75
1.0
493
1.2
PL (BeV)
FIGS. 2-5. Transverse momentum distribution d%/dEdp,at various energies of the outgoing muons off an 56Fe target for E, = 40 BeV. Figs. (2) and (4) correspond to Figs. (10) and (11) in Ref. [2]. All these graphs represent single-particle inclusive spectra. o,, = 3.03 x 1O-44 cme and z =
26.
the angular distribution atJixed energy in Refs. [l, 21into the transverse momentum distribution at fixed energy by using the relation d20/dE dp, = (sin 19/lp ( cos @(d20/dE d cos 0)
(2)
where E, p and cos 19denote the energy, momentum and scattering angle of the muon or the antimuon in the lab. system, respectively. Note that the conversion from the left hand side to the right hand side in Eq. (2) is not so easy due to the singular Jacobian at 8 = 0 [4]. We converted Figs. 10, 11, 13 and 14 in Ref. [2] into the transverse momentum distribution [S]. Those resulting distributions are shown in Figs. 2-7. Figures 3 and 5 were newly added to cover the entire range of the muon energy. We also add Figs. 8 and 9. See also the energy spectrum in Figs. 18 and 19. Our graphs always correspond to singleparticle inclusive spectra. In hadron physics it is customary to plot the distribution d2a/dE dpL2 = *(i/l p I2COS o)(d’a/dE d COS 0).
(3)
In this casethe Jacobian is regular around 0 = 0. We show the result in Figs. 10-13 for an 56Fetarget and for E, = 10 and 40 BeV. It seemsto be that the transverse momentum is not so small, as pointed out by Lovseth and Radomski [3]. This is also qualitativly expected from the momentum transfer from the leptonic system to the nuclear target. The momentum transfer is distributed in the region. t
= 1O-2N 10-3(BeV)2.
(4)
494
FUJIKAWA
0.1 0.2
0.3
6
FIGURE
1.5 1.5 -
Proton
,C-. I/
I
Target
_
E, q 4OBeV $==I6 BeV
t
l.O-
0.75
0.4 0.5 P* (BeV)
’
\
I
1
0.25 0.5 0.75
\
\
I
1.0
1.5
2.0
1 2.5
Pl (BeV)
FIGS. 6 and 7. Transverse momentum distribution d%/dE.dp, off a proton target at E, = 10 BeV. These graphs correspond to Figs. (13) and (14) in Ref. [2]. q, = 3.03 x 1O-44cm2.
See Fig. 19 in Ref. [2], and also Table I in Ref. [3]. The value oft in Eq. (4) is about the same as the pion mass squared, mrr2 = 2 x 1O-2 (BeV)2. It is generally believed [6] that the finite transverse momentum in the hadronic process is based on the finite (i.e., small) momentum transfer between the particles in the multiperipheral chain. If one takes the pion mass squared as a typical value of the momentum transfer in the hadron process, Eq. (4) suggests that the transverse momentum distribution in the process (1) may not be so distinct from that in hadron processes. We also show the transverse momentum distribution off a proton target in Figs. 14 and 15, and off a 20*Pb target in Figs. 16 and 17 for E, = 40 BeV. All these calculations are based on the results reported in Ref. [2] [5]. These figures,
HIGH
ENERGY
NEUTRINO
I
i
AND
I
I
MUON
495
REACTIONS I
I
Fe5’j Target E, =I0 BeV Ep,+=2BeV
0.1 0.2 0.3 0.4 0.5
0.75
PL (BeV) FIGURE
8
i r, < b 2 “, w
I
\
I
O;b 0.1 0.2 0.3 0.4 0.5 Pi (BeV)
1 c.75----hi
FIGS. 8 and 9. Transverse momentum distribution deo/dEifpo, x 1O-44 cm%.
off
an 6sFe target at Ev = 10 BeV.
o. = 3.03
compared with Figs. 2-5, clearly indicate the tendency we just expect. Note that the momentum transfer squared for a proton target is distributed in the region t = 10-l N 10” (BeV)2
(5)
Figures 14 and 15 suggest that the incoherent process off a proton target may be rather distinct from the semileptonic process at high energies. However, some model considerations [7] also suggest that the transverse momentum of the
lO-2
0
I
I
I
,
I
I
,
0.1 0.2 0.3 0.4 0.5 0.6 Pi (BeV)’ FIGURE 10
Positive Muon
5 t? -Y %7 b-e, “, w -tY ‘0° 0I.5
10-l 0
0.1 0.2 0.3 0.4 0.5 P: (BeV?
10 and 11. Transverse momentum distribution 10BeV. o, = 3.03 x 10-4Pcm8. z = 26. FIGS.
d20/dEdp,2 off an 68Fe target for I?, =
IO2 50
IO’ 5 b" -T "2 2;
IO0 0.5
10-l I 0
I 0.1
I \I 0.2 0.3
I 0.4
?\Fiziz 0.5 0.6
0.7
P: (BeV)* FIGURE 12 IO2 50
IO’
0.5
10-l
0.2 0.3 0.4 0.5 0.6 0.7 P: (BeV12 FIGS. 12 and 13. Transverse momentum distribution d20/dEdpLs off an 68Fe target for E, = 4oBev. ql = 3.03 x 1O-*4 cm2. 0
0.1
498
FUJIKAWA
secondary particles in the semileptonic process may depend on the “mass” of the current. Therefore, it is hard to say something definite about the transverse momentum of the hadrons in the semileptonic process [8].
0 -J-.1.
L.
.mL-.-i~-.
-.~.~
0.1 0.2 0.3 0.4 0.5
I
0.75
1.0
PL (BeV)
FIGURE 14
0.1 0.2 0.3 0.4 0.5
0.75
1.0
Pl (BeV)
FIGS. 14 and 15. Transverse momentum distribution 40 BeV. o. = 3.03 x 10-44cm2.
d%/dEdp,
off a proton target at Ey =
If the Adler process [9] dominates the energetic n+ production and the “shadow” effect proposed by Bell [lo] works in full, the semileptonic process would be significantly modified. It is hoped that the detailed experimental study of the semileptonic process at high energies will provide us more useful information how to separate the purely leptonic mun production from the background.
HIGH
ENERGY
NEUTRINO
AND
0.1 0.2 0.3 0.4 0.5 PA (Be’/) 40
MUON
0.75
FIGS. 16 and 17. Transverse momentum distribution BeV. (I~ = 3.03 x 1O-p4 cm*. .z = 82.
REACTIONS
1.0 d%/dEdp,
499
1.2
off a zosPb target at E, =
Next we would like to correct the lower end of the energy spectra in the previous study [2]. This spectrum was calculated from d2a/d cos 0 dE at fixed cos 8 based on an interpolation formula. The conversion of d‘b/d cos 8 dE at fixed cos 0 into d2a/d cos 8 dE at fixed E was not expected to be so accurate at the lower end of the spectrum [l I]. The point is that the integration mesh in the two-dimensional phase space (cos 8, E) was not uniformly taken to achieve the maximum accuracy of the numerical integration. Therefore the conversion from one distribution to the other is not trivial. However, following the question raised by Llewelyn Smith [12], we found that the total cross section for an 56Fe target at E, = 40 BeV obtained from f
g
dE
(6)
500
FUJIKAWA
grossly exceeds that obtained from
s
&dcosfl.
(7)
Equation (7) is more or less “exact” in our calculation [I, 21; the values of the total cross section in Refs. [l, 21 are based on Eq. (7). Therefore the lower-end peak in the energy spectrum, in particular, in Fig. 16-b in Ref. [2] is bot reliable; this discrepancy goes far beyond what we expected [I 11. We show the energy spectra based on an improved interpolation formula in Figs. 18 and 19. These
Fe56Target E,= IO BeV 1.0 cs
0
2.5
5.0 Er- (BeV)
7.5
IO
FIG. 18. Energy spectrum do/dE off an 6SFe target at Ev = 10 BeV. This diagram corresponds to Fig. 15 in Ref. [2]. o. = 3.03 x 10-4pcm2. .z = 26.
FIG. 19. Energy spectrum da/dE off an 66Fe target at E, to Fig. 16 in Ref. [2]. q, = 3.03 x lo-“cm2.
= 40
BeV. This diagram corresponds
HIGH
ENERGY
NEUTRINO
AND
MUON
501
REACTIONS
Target BeV
1~ -
20
30
-
E (BeV)
FIG. 20.
Energy spectrum da/dE off a proton target at Ev = 40 BeV. o,, = 3.03 x 1O-44 cma
Pbzo8 Target E, =40 BeV
t‘ CJ N 1. tui 317
1.0 .L ~ P.5 i
-
- Pi 1 7
r-
\\ 'A-
f' 0
/
-- ----_ IO
I------
20
30
--.
4’0-
_’;
Ep (BeV)
FIG. 21. Energy spectrum du/dE off a zOsPbtarget at E, = 40 BeV. o,, = 3.03 x IO-44 cd.
energy spectra differ from the previous values only at below 4 % of the lower end of the spectra. See Figs. 15 and 16 in Ref. [2]. Figure 17 in Ref. [2] remains unchanged; this spectrum was calculated by using the exact matrix element. For the sake of completeness we also add the energy spectra off a proton target and also off a 208Pb target at E, = 40 BeV in Figs. 20 and 21. These graphs show that the muons tend to accumulate near the origin when a larger momentum transfer to the nuclear (or nucleon) system is allowed. There is also a typographical mistake in the table for total cross sections in Ref. [2]. The total cross section for a zosPb target at E, = 40 BeV should read a(-) respectively.
= 7.27z2,
o(+) = 7.16~~
and
u(a) = 9.6z2,
502
FUJIKAWA
Note added in proof. The following authors also studied a problem similar to ours: A. V. BERKOV, L. M. VORONICA, AND E. P. SHABALIN, Soviet J. of Nuclear Phys. 14 (1972), 233; A. E. ASRATYAN, A. M. TIKHOMROV, AND E. P. SHABALIN, Soviet J. of Nuclear Phys. 14 (1972), 694; K. Korxn ET AL., Prog. Th. Phys. 46 (1971), 1150, and Kyoto Univ. report, RIFP 128 (1971). I am grateful to Dr. Llewellyn Smith for calling those works to my attention.
ACKNOWLEDGMENTS I am grateful to Professors S. B. Treiman and A. Mann for discussions.
REFERENCES
1. 2. 3. 4. 5.
6. 7. 8.
9. 10. 11.
12.
K. F~~KAWA, Thesis, Princeton University, May 1970, unpublished. K. FUJXAWA, Ann. Phys., N.Y. 68 (1971), 102. J. Lovsnrrr AND M. RADOMSKI, Phys. Rev. D 3 (1971), 2686. Because of this vanishing Jacobian at 0 = 0, the sharp peak of the angular distribution at 0 = 0 is rather misleading. When one calculates the transverse momentum distribution this sharp peak disappears due to the Jacobian. The computer output cards in the previous study which bear the distribution d%/dEiicos 6 for tied cos 0 and varying E were used. We converted this distribution into d%/dEdms 6 for tied E and varying cos 0 by an interpolation formula. Then we used Eqs. (2) and (3) to get transverse momentum distributions. See, for example, K. Wnso~, Cornell Preprint, CLNS - 131, November 1970. J. D. BJORKEN, SLAC-PUB-901 (1971). K. FUIIKAWA, Nuovo Cimento, 12A (1972), 117. The transverse momentum of p+ in the purely leptonic process could be distinct from that in the semileptonic process at extremely high energies because of different particle multiplicity. It is, however, dangerous to discuss this problem further without a detailed knowledge about the transverse momentum in the semileptonic process at high energies. If one knows the incident energy of the neutrino, one can form a sort of Dal&z plot for the muon pair in the final state. In this case it will be slightly easier to distinguish the purely leptonic process from the semileptonic process. S. ADLER, Phys. Rev. 135 (1964), B936. J. S. BELL, Lecture Note for the 1 lth Scottish Universities Summer School -in Physics, Edinburgh, 1970. We expected that the lower end of the energy spectrum may not be so accurate. See p. 51 of Ref. [I] and also Footnote 20 in Ref. 121. The consistency check of the interpolation formula was made at Ey = 10 BeV, based on Figs. 15 and 17 in Ref. [2]. Unfortunately these graphs showed a rather reasonable behavior as one can see by comparing Figs. 18 - 21 in the present note with those figures in Ref. 121. C. H. LLJZWELYNSMITH, Physics Reports 3C (1972), 264.