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Comparison of recent massive muon pair data with the asymptotically free parton model
Volume 71B, number 1
PIIYSICS LETTERS
7 November 1977
COMPARISON OF RECENT MASSIVE MUON PAIR DATA WITH THE ASYMPTOTICALLY
FREE PARTON MODEL*
J. KOGUT 1 and Junko SHIGEM1TSU
Laboratory of NuclearStudies, Cornell Uni~'ersity,Ithaca,N. Y. 14853, USA Received 2 August 1977 We present calculations of ~,W2 and of massive muon pair production cross sections in the kinematic ranges of recent experiments. These calculations test the asymptotically free parton model and excellent agreement with the data is found. Estimates of the transverse momentum (ql.)/w. expected in massive dimuon experiments are made. For s = 750 GeV2 and Mtm -~. 10 GeV, we predict
~---{FB(x't)= 4, Jx - ~
~A PBA(X/Y)FA(Y't) (1)
and have been studied in the literature [4,6]. In eq. (1) ~' Supported in part by the National Science Foundation. I Alfred P. Sloan Foundation Fellow.
as(t ) is the strong interaction momentum dependent fine structure constant, as(t) _ g2 (t) _ 12rr • ~ 25 In(Q2/A 2)
(2)
in QCD with four quark flavors [8]. All of this makes contact with experiment through the relation for deep inelastic electron scattering [6,8],
uW2(x, Q2)
= ~ i
e 2 F/(x, t)
(3)
and the c.m. cross section to produce muon pairs of invariant mass x/rQ- ' / i n proton-proton collisions [ 1 - 4 ] ,
3Q 4 • -~ E dQd~zdqzqz=O=4rr°t2 1 ~i.
e 2 6.(x,
t)b]~-(x, t)
(4)
where x 2 = Q2/s and x,/sis the invariant mass of the colliding protons. Eq. (4) is the generalization of the DrellYan formula [9] to asymptotically free field theories. It states that the dominant process creating a muon pair of mass-squared Q2 is the annihilation of partons of size scale Q - 1 [ 1]. It is the purpose of this note to present evaluations of eqs. (3) and (4) and observe that they explain the available data very well. In addition, estimates of the transverse momenta of the muon pair will be made assuming QCD. The results are encouraging but not very well tested at this time. These successes reinforce our hope that recently published estimates of W~ and Z 0 production in the asymptotically free parton model should be useful [5]. 165
Volume 71 B, number 1
PHYSICS LETTERS
7 November 1977
To calculate vW2(x, Q2) we must fit the functions a convenient value of t. This is done at Q ~ 3.5 GeV 2 in the SLAC kinematic range. Since tile gluon distributions enter eq. (1) we must also exercise some theoretical prejudice to begin. We choose distributions quite similar (but not identical) to those discussed by Barger and Phillips in ref. [10],
Next the momentum scale A 2 which enters the expression for % ( 0 must be set. We present curves using the values A 2 = 0.25 and 0.10 as suggested by other investigations [11]. The sensitivity to this choice can be read off the resulting curves. Now eq. (1) can be analyzed since the functions PBAhave been computed in ref. [7]. The resulting vW2(x, Q2) is shown in fig. 1 a-d, where the data points [12] have also been FuValence = X/x [0.594(1 - x2) 3 + 0.461(1 - x2) 5 reproduced. The agreement is good and could certainly be improved by experimenting with the initial gluon + 0.621(1 - x 2 ) 7] distributions, sea distributions and alternative ~caling (5) variables. The paucity of data points does not warrant FdValence = ~ [0.072(1 - x2) 3 + 0.206(1 - x 2 ) 5 such an investigation at this time, but we l'*ok forward to the accumulation of more data and more stringent +0.621(1 -- x2)71 , tests in the future. We now turn to massive muon pair production in = 0, b Glue = 3.276(1 - x ) 6, proton-proton collisions. This topic is the main con~uuea = 0.8(1 - X) 6 F Sea c ,d,s cern of our article. Data at s = 750 GeV 2 and M 2 - M /.L,u 2
F.(x, t) at
0.5 Q=~5.5GeV =
0.4
0.4
0.3
Oz~ IIGlfiff i
. . . . At.O.I
"~'~,~,
--At
"0'25
0.5 ~N
~N
' ~'"~,
0.2
0.2
0.1
0.1
(C)
i
0.2
0.4
x
0.6
0.8
1.0
°.4
o.
,.o
05 0 =~ 6 GoV= 04
'
~
. . . . A=*0"I
0P,
/I 1
.... A'=o.i
\x~,(
- - A'-0.25
--A',oz5 0.3
(D)
',
,
(d)
0.2
~' 0.2
XNO
OI
I
0.2
I ~XN XN N ~
0.4
0.15 x
OI
0.8
x 0.2
XXXx
04
0.6
0.8
IO
g
. . . . . . . . 2 ~ 2 Q2 Ftg. 1. Asymptotic freedom calculations of vW2 as a function of the Bjorken scahng vartable x for (a) Q ~ 3.5 GeV , (b) 2 2 2 2 2 2 6GeV ,(c) Q ~ l l G e V ,and(d) Q ~22GeV .TheparameterA sets the scale of the coupling constant.
166
Volume 71B, number 1
PHYSICS LETTERS
dimuon events. These studies [14] should be re-evaluated. 3. 11
ELob = 4 0 0 G e V
i0-~
.....
Ai,O.IGeV =
-
A==025
-
--.--
7 November 1977
GeV =
Naive p a r l o n model
I 0 -36
¥
E u o-
\
g2(Q2) (q2) = const.
i0-==
47r
• Q2 . ~(Q2/s, g2(Q2))
(6)
where the dimensionless function c-j is complicated but computable. Numerical studies of eq. (6) are in progress, but rough estimates of (q2) can be made quite simply. Since quark-antiquark annihilation creates the massive muon pair [2], 6
O
I0 Male (GeV)
12
14
16
Fig. 2. Asymptotic freedom calculation of massive muon pair production cross section in the c.m. for A2 = 0.25 and 0.1 GeV2. The dash-dotted curve follows from assuming no scale breaking, i.e., using the distribution functions of eq. (5) for all Mug. The data points are taken from ref. [ 13 ]. The theoretical curves are for an isoscalar target.
(q2) ~½ [~o2(x, Q2))q + @2(x, ' Q2))~. + (q~-)] (7) where xx' = Q2/s and (p2(x, Q2)) is the mean transverse momentum squared of quarks of size scale Q - 1 in the proton. But a simple calculation yields [I 5],
(8)
R : OL/O T ~ 4(p2(x, Q2))/Q2 = Q2 in the range 36 to 144 GeV 2 have been reported [13]. A comparison between eq. (4) using Fi(x, t) computed as discussed above and the experimental points is shown in fig. 2. We examine the curve in three M ranges: 1.6 < Muu < 9 GeV. The agreement between theory and experiment is excellent. Both the shape and normalization of the theory curves match the experimental data points with surprising accuracy. 2.9 < Muu < 11 GeV. Presumably the bumps here reflect the existence of a family of resonances containing new heavy quarks [13]. Our article only addresses the background (!) to this discovery. Previous comparisons between theory and experiment did not separate these enhancements from the nonresonant
so eq. (7) can be estimated [2], (q2) ~ ¼ Q2 [R(x, Q2) + R(x', Q2)].
(9)
R has been calculated for deep inelastic scattering off protons assuming asymptotic freedom [ 11 ]. A simple but approximate analytic formula for these numerical results is [4],
4n 2 16 R~ -25
1 ln(Q2/A2)
• (l - x ) .
-x)] (10)
167
Volume 71B, number 1
PI1YSICS LETTERS
Table 1 For s = 750 GeV we tabulate~ are obtained assuming an exponential fit exp(-b q±) is made to the data. The .scale parameter A2 = 0.1 GeV 2 .
(q~)ta~(GeV2/c2) b(GeV/c) -I (qk)au
Q(GeV) .
.
.
.
.
6 8 l0 12 14
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1.53 2.24 2.94 3.56 4.04
.
.
.
.
1.98 1.64 1.43 1.30 1.22
.
.
.
.
.
.
.
.
.
.
.
(GeV/c) .
.
.
.
.
1.01 1.22 1.40 1.54 1.64
7 November 1977
transverse m o m e n t u m distributions have been dropped, etc. However, a critical study of our estimate leads us to predict with considerable confidence that (q±) should exceed I. 1 GeV/c in this kinematic region. It is amusing to note that ~/±) ~ 1.5 GeV/c has been reported in previous massive m u o n pair experiments [16]. These results encourage us to a t t e m p t a serious numerical analysis of eq. (6). We also look forward to confronting the asymptotically free parton model with more precise data in the future.
References C o m b i n i n g eqs. (9) and (10) and recalling t h a t x ~ x ' for the experimental measurements of interest we have
(q2)uu
~ - - 8 . (1 - Q x ~ s ) 25
Q2 In(Q2/A2)
.
(11)
Given A 2 as determined from the fits to vW2, this equation provides a parameter-free estimate of (q2). It is recorded in table 1 for the values s and Q2 in the data of fig. 2. Since the experimentalists fit the q i distributions by e x ~ e n t i a l forms, e x r _ ~ l ), we also tabulate b = x/6/(q~) and (q t) = x/2(q~)/3. We note the following trends: I. (q±) increases more rapidly at Q ~ 6 GeV than at Q ~ 14 GeV. 2. (ql) is large - in excess of 1.4 GeV/c for Q between 10 and 14 GeV/c - and not related to the mean transverse m o m e n t a of hadrons produced in purely hadronic reactions. How seriously should table 1 be taken? We feel that the trends noted above should be observed in the data, b u t the details have been estimated only crudely, e.g. the quark's transverse m o m e n t u m due to the soft forces confining it in the p r o t o n has been neglected, possible differences between quark and antiquark
168
[1] J.B. Kogut, Phys. Lett. 65B (1976) 377. [2] J. Ilinchliffe and C.II. Llewellyn Smith, Phys. Lett. 66B (1977) 281. [3] D.E. Soper, Phys. Rev. Left. 38 (1977) 461. [4] II.D. Politzer, ltarvard preprint, May 1977. [5] J.B. Kogut and J. Shigemitsu, Cornell preprint CLNS-363, July 1977 (submitted to Nucl. Phys. B). [6] J. Kogut and L. Susskind, Phys. Rev. D9 (1974) 697, 3391. [7] G. Altarelli and G. Parisi, Ecole Normale Sup~rieure preprint, March 1977. [8] D.J. Gross and k'. Wilczek, Phys. Rev. D8 (1973) 3633; D9 (1974) 980; II. Georgi and H.D. Politzer, Phys. Rev. D9 (1974)416. [9] S.D. Drell and T.-M. Yan, Phys. Rev. Lett. 25 (1970) 316. [10] V. Barger and R.J.N. Phillips, Nucl. Phys. B73 (1974) 269. [ 11 ] A. DeRujula, H. Georgi and H.D. Politzer, Ann. Phys. (N.Y.) 103 (1977) 315. 112] It.L. Anderson et al., Phys. Rev. Lett. 38 (1977) 1450. [ 13] SLAC Summer School on Deep Inelastic Phenomena, July 1977. These data were presented by Dr. W.R. Innes and it improves the less accurate data appearing in S.W. llerb et at., Phys. Rev. Lett. 39 (1977) 252. [ 14] D. Antreasyan et at., Test of scaling in muon-pair production by hadrons, Univ. of Chicago preprint, June 1977. [15] R.P. Feynman, Photon hadron interactions (W.A. Benjamin, New York, 1972). [16] K.J. Anderson et at., Phys. Rev. Lett. 37 (1976) 799.
PIIYSICS LETTERS
7 November 1977
COMPARISON OF RECENT MASSIVE MUON PAIR DATA WITH THE ASYMPTOTICALLY
FREE PARTON MODEL*
J. KOGUT 1 and Junko SHIGEM1TSU
Laboratory of NuclearStudies, Cornell Uni~'ersity,Ithaca,N. Y. 14853, USA Received 2 August 1977 We present calculations of ~,W2 and of massive muon pair production cross sections in the kinematic ranges of recent experiments. These calculations test the asymptotically free parton model and excellent agreement with the data is found. Estimates of the transverse momentum (ql.)/w. expected in massive dimuon experiments are made. For s = 750 GeV2 and Mtm -~. 10 GeV, we predict
~---{FB(x't)= 4, Jx - ~
~A PBA(X/Y)FA(Y't) (1)
and have been studied in the literature [4,6]. In eq. (1) ~' Supported in part by the National Science Foundation. I Alfred P. Sloan Foundation Fellow.
as(t ) is the strong interaction momentum dependent fine structure constant, as(t) _ g2 (t) _ 12rr • ~ 25 In(Q2/A 2)
(2)
in QCD with four quark flavors [8]. All of this makes contact with experiment through the relation for deep inelastic electron scattering [6,8],
uW2(x, Q2)
= ~ i
e 2 F/(x, t)
(3)
and the c.m. cross section to produce muon pairs of invariant mass x/rQ- ' / i n proton-proton collisions [ 1 - 4 ] ,
3Q 4 • -~ E dQd~zdqzqz=O=4rr°t2 1 ~i.
e 2 6.(x,
t)b]~-(x, t)
(4)
where x 2 = Q2/s and x,/sis the invariant mass of the colliding protons. Eq. (4) is the generalization of the DrellYan formula [9] to asymptotically free field theories. It states that the dominant process creating a muon pair of mass-squared Q2 is the annihilation of partons of size scale Q - 1 [ 1]. It is the purpose of this note to present evaluations of eqs. (3) and (4) and observe that they explain the available data very well. In addition, estimates of the transverse momenta of the muon pair will be made assuming QCD. The results are encouraging but not very well tested at this time. These successes reinforce our hope that recently published estimates of W~ and Z 0 production in the asymptotically free parton model should be useful [5]. 165
Volume 71 B, number 1
PHYSICS LETTERS
7 November 1977
To calculate vW2(x, Q2) we must fit the functions a convenient value of t. This is done at Q ~ 3.5 GeV 2 in the SLAC kinematic range. Since tile gluon distributions enter eq. (1) we must also exercise some theoretical prejudice to begin. We choose distributions quite similar (but not identical) to those discussed by Barger and Phillips in ref. [10],
Next the momentum scale A 2 which enters the expression for % ( 0 must be set. We present curves using the values A 2 = 0.25 and 0.10 as suggested by other investigations [11]. The sensitivity to this choice can be read off the resulting curves. Now eq. (1) can be analyzed since the functions PBAhave been computed in ref. [7]. The resulting vW2(x, Q2) is shown in fig. 1 a-d, where the data points [12] have also been FuValence = X/x [0.594(1 - x2) 3 + 0.461(1 - x2) 5 reproduced. The agreement is good and could certainly be improved by experimenting with the initial gluon + 0.621(1 - x 2 ) 7] distributions, sea distributions and alternative ~caling (5) variables. The paucity of data points does not warrant FdValence = ~ [0.072(1 - x2) 3 + 0.206(1 - x 2 ) 5 such an investigation at this time, but we l'*ok forward to the accumulation of more data and more stringent +0.621(1 -- x2)71 , tests in the future. We now turn to massive muon pair production in = 0, b Glue = 3.276(1 - x ) 6, proton-proton collisions. This topic is the main con~uuea = 0.8(1 - X) 6 F Sea c ,d,s cern of our article. Data at s = 750 GeV 2 and M 2 - M /.L,u 2
F.(x, t) at
0.5 Q=~5.5GeV =
0.4
0.4
0.3
Oz~ IIGlfiff i
. . . . At.O.I
"~'~,~,
--At
"0'25
0.5 ~N
~N
' ~'"~,
0.2
0.2
0.1
0.1
(C)
i
0.2
0.4
x
0.6
0.8
1.0
°.4
o.
,.o
05 0 =~ 6 GoV= 04
'
~
. . . . A=*0"I
0P,
/I 1
.... A'=o.i
\x~,(
- - A'-0.25
--A',oz5 0.3
(D)
',
,
(d)
0.2
~' 0.2
XNO
OI
I
0.2
I ~XN XN N ~
0.4
0.15 x
OI
0.8
x 0.2
XXXx
04
0.6
0.8
IO
g
. . . . . . . . 2 ~ 2 Q2 Ftg. 1. Asymptotic freedom calculations of vW2 as a function of the Bjorken scahng vartable x for (a) Q ~ 3.5 GeV , (b) 2 2 2 2 2 2 6GeV ,(c) Q ~ l l G e V ,and(d) Q ~22GeV .TheparameterA sets the scale of the coupling constant.
166
Volume 71B, number 1
PHYSICS LETTERS
dimuon events. These studies [14] should be re-evaluated. 3. 11
ELob = 4 0 0 G e V
i0-~
.....
Ai,O.IGeV =
-
A==025
-
--.--
7 November 1977
GeV =
Naive p a r l o n model
I 0 -36
¥
E u o-
\
g2(Q2) (q2) = const.
i0-==
47r
• Q2 . ~(Q2/s, g2(Q2))
(6)
where the dimensionless function c-j is complicated but computable. Numerical studies of eq. (6) are in progress, but rough estimates of (q2) can be made quite simply. Since quark-antiquark annihilation creates the massive muon pair [2], 6
O
I0 Male (GeV)
12
14
16
Fig. 2. Asymptotic freedom calculation of massive muon pair production cross section in the c.m. for A2 = 0.25 and 0.1 GeV2. The dash-dotted curve follows from assuming no scale breaking, i.e., using the distribution functions of eq. (5) for all Mug. The data points are taken from ref. [ 13 ]. The theoretical curves are for an isoscalar target.
(q2) ~½ [~o2(x, Q2))q + @2(x, ' Q2))~. + (q~-)] (7) where xx' = Q2/s and (p2(x, Q2)) is the mean transverse momentum squared of quarks of size scale Q - 1 in the proton. But a simple calculation yields [I 5],
(8)
R : OL/O T ~ 4(p2(x, Q2))/Q2 = Q2 in the range 36 to 144 GeV 2 have been reported [13]. A comparison between eq. (4) using Fi(x, t) computed as discussed above and the experimental points is shown in fig. 2. We examine the curve in three M ranges: 1.6 < Muu < 9 GeV. The agreement between theory and experiment is excellent. Both the shape and normalization of the theory curves match the experimental data points with surprising accuracy. 2.9 < Muu < 11 GeV. Presumably the bumps here reflect the existence of a family of resonances containing new heavy quarks [13]. Our article only addresses the background (!) to this discovery. Previous comparisons between theory and experiment did not separate these enhancements from the nonresonant
so eq. (7) can be estimated [2], (q2) ~ ¼ Q2 [R(x, Q2) + R(x', Q2)].
(9)
R has been calculated for deep inelastic scattering off protons assuming asymptotic freedom [ 11 ]. A simple but approximate analytic formula for these numerical results is [4],
4n 2 16 R~ -25
1 ln(Q2/A2)
• (l - x ) .
-x)] (10)
167
Volume 71B, number 1
PI1YSICS LETTERS
Table 1 For s = 750 GeV we tabulate
(q~)ta~(GeV2/c2) b(GeV/c) -I (qk)au
Q(GeV) .
.
.
.
.
6 8 l0 12 14
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1.53 2.24 2.94 3.56 4.04
.
.
.
.
1.98 1.64 1.43 1.30 1.22
.
.
.
.
.
.
.
.
.
.
.
(GeV/c) .
.
.
.
.
1.01 1.22 1.40 1.54 1.64
7 November 1977
transverse m o m e n t u m distributions have been dropped, etc. However, a critical study of our estimate leads us to predict with considerable confidence that (q±) should exceed I. 1 GeV/c in this kinematic region. It is amusing to note that ~/±) ~ 1.5 GeV/c has been reported in previous massive m u o n pair experiments [16]. These results encourage us to a t t e m p t a serious numerical analysis of eq. (6). We also look forward to confronting the asymptotically free parton model with more precise data in the future.
References C o m b i n i n g eqs. (9) and (10) and recalling t h a t x ~ x ' for the experimental measurements of interest we have
(q2)uu
~ - - 8 . (1 - Q x ~ s ) 25
Q2 In(Q2/A2)
.
(11)
Given A 2 as determined from the fits to vW2, this equation provides a parameter-free estimate of (q2). It is recorded in table 1 for the values s and Q2 in the data of fig. 2. Since the experimentalists fit the q i distributions by e x ~ e n t i a l forms, e x r _ ~ l ), we also tabulate b = x/6/(q~) and (q t) = x/2(q~)/3. We note the following trends: I. (q±) increases more rapidly at Q ~ 6 GeV than at Q ~ 14 GeV. 2. (ql) is large - in excess of 1.4 GeV/c for Q between 10 and 14 GeV/c - and not related to the mean transverse m o m e n t a of hadrons produced in purely hadronic reactions. How seriously should table 1 be taken? We feel that the trends noted above should be observed in the data, b u t the details have been estimated only crudely, e.g. the quark's transverse m o m e n t u m due to the soft forces confining it in the p r o t o n has been neglected, possible differences between quark and antiquark
168
[1] J.B. Kogut, Phys. Lett. 65B (1976) 377. [2] J. Ilinchliffe and C.II. Llewellyn Smith, Phys. Lett. 66B (1977) 281. [3] D.E. Soper, Phys. Rev. Left. 38 (1977) 461. [4] II.D. Politzer, ltarvard preprint, May 1977. [5] J.B. Kogut and J. Shigemitsu, Cornell preprint CLNS-363, July 1977 (submitted to Nucl. Phys. B). [6] J. Kogut and L. Susskind, Phys. Rev. D9 (1974) 697, 3391. [7] G. Altarelli and G. Parisi, Ecole Normale Sup~rieure preprint, March 1977. [8] D.J. Gross and k'. Wilczek, Phys. Rev. D8 (1973) 3633; D9 (1974) 980; II. Georgi and H.D. Politzer, Phys. Rev. D9 (1974)416. [9] S.D. Drell and T.-M. Yan, Phys. Rev. Lett. 25 (1970) 316. [10] V. Barger and R.J.N. Phillips, Nucl. Phys. B73 (1974) 269. [ 11 ] A. DeRujula, H. Georgi and H.D. Politzer, Ann. Phys. (N.Y.) 103 (1977) 315. 112] It.L. Anderson et al., Phys. Rev. Lett. 38 (1977) 1450. [ 13] SLAC Summer School on Deep Inelastic Phenomena, July 1977. These data were presented by Dr. W.R. Innes and it improves the less accurate data appearing in S.W. llerb et at., Phys. Rev. Lett. 39 (1977) 252. [ 14] D. Antreasyan et at., Test of scaling in muon-pair production by hadrons, Univ. of Chicago preprint, June 1977. [15] R.P. Feynman, Photon hadron interactions (W.A. Benjamin, New York, 1972). [16] K.J. Anderson et at., Phys. Rev. Lett. 37 (1976) 799.