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Perturbative QCD and lepton charge symmetry at Z0 energies
Volume 108B, number 6
PHYSICS LETTERS
4 February 1982
PERTURBATIVE QCD AND LEPTON CHARGE SYMMETRY AT Z ° ENERGIES ~ David J.E. CALLAWAY High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Recewed 5 October 1981
Although the Dreil-Yan mechanism for lepton parr production is very successful qualitatively, large perturbative QCD corrections inhibit a quantitative comparison with experunent at energies presently accessible. It is shown that an appropriately defined charge asymmetry in the context of the parton model has small (5-15%) orderas(Q 2) perturbative corrections, despite large (50-250%) corrections to the cross section itself for typical experimental parameters. Calculations to order as(Q 2) of the asymmetry in lepton pair production from 15p collisions at x/rs= 540 GeV are presented. The charge asymmetry is large - czc ,~ 70-80 percent for a dflepton mass x / ~ ~ 60 GeV, where the asymmetry is maxmaized. The structure of the asymmetry near the Z ° resonance is also displayed.
The Drell-Yan mechanism [ 1], illustrated in fig. 1, is remarkably successful in explaining the qualitative features of lepton pair production in h a d r o n - h a d r o n collisions [2]. Corrections to this simple parton model picture occur from diagrams such as those displayed in fig. 2. These perturbative QCD corrections to the differential cross section can be evaluated systematically in a power series in C~s(Q2), the running strong coupling constant. For energies presently accessible experimentally such corrections are numerically large [ 3 - 8 ] , raising questions about the consisWork performed under the auspices of the United States Department of Energy.
q
(0)
i
.i
7::[
q
i
(b)
Fig. 1. Diagrams showing quarks and antiquaxks annihilating to produce lepton paixs: (a) the "DreU-Yan process",which proceeds by way of an intermediate photon; Co) the photon is replaced by a Z 0 (the weak neutral boson).
tency of the perturbative [ormalism for calculating the cross section. An interesting question is whether a measurable quantity exists in lepton pair production whose perturbative QCD corrections at present energies are smaI1. One such candidate is known as the charge asymmetry. This asymmetry has been discussed at length in the context of the parton model [9,10]. It arises as follows. The differential cross section for lepton pair production contains terms which are symmetric under the interchange of the lepton pair four-momenta ("charge-symmetric") and terms which are antisymmetric under this interchange ("charge-antisymmetric"). The ratio o f the charge-antisymmetric to the corresponding charge-symmetric cross section is detimed to be the charge asymmetry. The most significant contribution to this asymmetry involves the production o f lepton pairs via the axial vector portion of the neutral weak (Z 0) current. Other contributions to the asymmetry are mentioned below. Perturbative QCD corrections to the charge asymmetry seem to be much smaller than the corrections to the electroweak cross section. For antiproton-proton collisions at typical energies (x/~ -= 2 0 - 5 4 0 GeV), first order perturbative QCD corrections to the electroweak cross section are approximately 5 0 - 2 5 0 % of the parton model value, while the corresponding 421
Volume 108B, number 6
PHYSICS LETTERS
4 February 1982
]~' =~ (v~ + a~vs)~'~, lq'U= ~l(Vq + Aq "Y5),),uq, (o)
q
(1)
respectively. In the Weinberg-Salam model [ 11 ] (augmented by the GIM mechanism [12]) the various coupling constants are, in the lowest order, V~ = e'(1 - 4 sin20w),
A~ = - e '
Vq = e'(2T 3 - 4(eq/e) sin20w),
(2a,b) Aq =
e'2T3,(2c,d)
where e,=
e 4 sin 0 w cos 0 w '
(b) q
G
y~, Z 0
(e)
and e is the magnitude of the charge of the electron (e 2 = 47rol), eq is the electric charge of the quark q (q = u, d, s, etc.), T 3 is the third component of the weak isospin of the quark q (+~ for u quarks and - ~1 for d quarks etc.) and 0 w is the Weinberg mixing angle. The mass of the exchanged Z 0 boson in, e.g., fig. lb is given by = 3'I2
q Fig. 2. Diagrams illustrating some of the order a s perturba-
tire QCD corrections to the diagrams fig. 1 : (a) Quark-antiquark pair annihilate, producing a lepton pair (via a virtual photon or Z°) and a gluon. Crossed diagram not shown. (b) Initial gluon and quark scatter, radiating a virtual photon or Z°, which then couples to a lepton pair. Crossed diagram not shown. (c) Virtual gluon diagram. The interference of this diagram with those in fig. 1 gives an order as correction to the cross section derived from the diagrams fig. 1 alone. Self-energy corrections not shown.
e ~ (89 GeV) 2 4V~-GF sin20w COS20w
422
(4)
for sin20w = 0.225, the value used in this paper, and the Fermi weak coupling G F ~-. 1.16632 × 10 -5 GeV-2. It is useful to define the quantities
Nq(Q2)-(Q2-M2+ 2 eV---q-qe ~ / R (Q2), \ Q2 q ~/
(5a)
Dq(Q2) =- l + (2 (Q2 _ M 2 ! Vq v~ Q2 eq e~ (V2
corrections to the charge asymmetry are roughly 5 15% of the parton model result. Perturbative QCD corrections to the charge asymmetry are larger in p r o t o n - p r o t o n collisions, where initial gluon effects are important as well [6]. Details of the ~p calculation are now given. The neutral weak lepton and quark currents are given by
(3)
+
R (Q2) =
2 2 2 + Aq) (V~ +A~) ]R (Q2) e~ e-~ / ' q
(Q2)2 (Q2 - M ~z) 2 + M 2z r 2"
(58)
(5c)
Here e~ = - e is the leptonic charge and F is the width of the Z °, taken to be F ~ 2.5 GeV. The results given here do not depend sensitively on this value. Note that only the ratios of weak to electric charges appear in eqs. (5).
Volume 108B, number 6
PHYSICS LETTERS
The largest contributions to the ffp electroweak lepton pair production cross section come from diagrams involving the annihilation of an initial quark and antiquark, as displayed in figs. (2a) and (2c). Initial gluon diagrams such as the one shown in fig. (2b) gcve rise to much smaller effects in this case [6]. Thus each lepton pair is assumed to be produced by the annihilation of a quark of momentum p with an antiquark of momentum p', with
pU = X lP~l ,
p'U = x2t~2 ,
P1 = -gP0,
P2 = gP0,
(6a) s = 4P 2 .
(6b)
Here, p~z and P~ are the four-momenta of the incoming proton and antiproton. The hadrons and quarks are assumed to be massless and on-mass-shell. In addition, the following variables are defined:
0. =
+ 1;,
l.--
- I;
cos 02=--1"~/111,
cos ~ = I.L°Qj_/(IIj_IIQ.LI)
Ij'I=Q_L" ~"=-0,
x F--Qz/PO,
cos X = cos 02/(1 + x2/4r) 1/2 , 2 + 4r)1/2 -- XF], Xl-- ~a [(XF
d4o
[dQ2dxFd2~2~
X F,q(eq/e)Aq Nq(Q2)q(2l, Q2)t~(22, Q2)
(9)
Zq (eq/e) 2 Dq (Q2)q(xl, Q2)~(22 ' Q2) which is independent of¢Q. To a good approximation [10], the ffp charge asymmetry is given by the flu annihilation contribution alone, 1 Nu(Q 2) Au A~ 2 c o s x ac(PP) ~ 2rraDu(Q2 ) (en/e)(e2/e) 1 + cos2x '
(10)
which is independent of the parton distributions. Note that as 0 2 / M 2 ~ O,Nu(a2)/Du(a 2) ~ - 0 2 / M 2 and the charge asymmetry eq. (10) reduces to a result derived elsewhere [6,10]. Also notice that Nu(Q2 ) vanishes when 0 2 = M2/(1 + 2 Vu V2/e~) ~-.M 2 ,
d2~2~- sin 0~d02dq~ , X2-- ~1 [(x2+ 4r)l/2+XF]
(7)
x [
_ 1 2 cosx A2 ac(lSP) 21ra 1 + cos2x (e2/e)
(11)
7"=Q2/s,
where lu and l~- are the four-momenta of the positive and negative leptons, and the angles are defined in the hadronic cm system. The notation q (x 1, Q2) and q(x2, Q2) is also used to represent the (nonscaling) distributions of quarks and antiquarks of flavor q (q = u, d, s, etc.). The initial transverse momentum of the quarks and antiquarks is neglected. The integrated charge asymmetry is defined by [6]
- [ d4o txe - [dQ2__~Fd2 a 2
4 February 1982
/ [cos 02 ~--cos 02] t
+ [cos 02 *'--cos
}1 .
(8)
The integral d2Qj_ over the transverse momentum of the lepton pair has been performed separately for each term in the numerator and denominator of eq. (8). Both the numerator and denominator of this ratio are calculated by summing the appropriate quarkantiquark annihilation contributions. The parton model charge asymmetry is given by
and so also does the approximate expression for the charge asymmetry eq. (10). It is a general feature of weak asymmetries near a resonance that zeroes and sign changes in the asymmetries occur in the resonance region [13]. The details of the order as calculation of the integrated charge asymmetry are given elsewhere [6]. It should be noted, however, that the results given in ref. [6] are valid in the limit Q2/M2 ~ O. Here the asymmetry is evaluated in the Z 0 region as well, so that the full expressions eqs. (9) are retained. Thus in the present calculation the effects arising from the squared neutral current diagram fig. lb are included. In the order a s calculation [6] of the integrated charge asymmetry the various divergences are handled by dimensional regularization. When the integration over Q± is performed, the infrared divergences from the diagram fig. 2a (and the corresponding crossed diagram) cancel with those associated with the interference of the virtual diagram fig. 2c and the lowestorder diagrams figs. 1. The remaining (collinear) divergences are absorbed into the parton distributions in the usual fashion. In fig. 3 the integrated charge asymmetry calculated in the parton model (i.e. eq. (9)) is compared with the asymmetry calculated up to order a s . The 423
Volume108B,number6
PHYSICSLETTERS
08
O6o4
~P
"
/
Qe 0 2 0
-OZ -04_06 -
0.8 0
~ X
parton ~
parton ~x\ "~°(as) '~J I
I
20
40
~ 60
[
t
80
I00
120
Fig. 3. The charge asymmetry for lepton pair production from antiproton-proton collisionsat~s= 540 GeV is plotted as a function of the dilepton mass x/Q2 . Other parameter values are given m the text. The broken line disphys the asymmetry calculated in the parton model. The solid line is a plot of the asymmetry with order as(Q 2) corrections included. asymmetry is plotted versus the mass of the lepton pair, x/c~-, for x/~-'= 540 GeV, x~ = 0, and cos 0 = cos ¢~ = 1. The (nonscaling) Q -dependent parton distributions are taken from ref. [14], and the running coupling constant is taken to be 127r °¢s(Q2) = (33 - 2Nf) In (Q2/A2) '
(12)
with Nf = 5 and A = 0.3 GeV. For the range of Q2 considered here (10 GeV ~
4 February1982
Since the structure of these r n moments for large n gives the behavior of the cross section for large r, it follows that in the limit r ~ 1 the order as(Q2 ) charge asymmetry approaches its parton model value. Other large numerical corrections (such as the 7r2 from the analytic continuation of the q~17vertex function from spacelike to timelike values of Q2) occur in like multiplicative fashion in the charge symmetric and antisymmetric contributions to the cross section [6]. These large corrections therefore tend to cancel when the ratio comprising the charge asymmetry is taken. The remaining terms (which are different in the two cases) are negligible numerically. Note that if the large corrections can be summed (exponentiated) to all orders [8], the resulting large factor may drop out when the ratio comprising the charge asymmetry is taken. Contributions to the charge asymmetry may also arise from higher-order electromagnetic effects [9]. For the large dilepton masses of interest here (x/Q 2 10-100 GeV) such effects are presumably negligible, however. In addition, since such effects do not require the exchange of a Z 0 they may be isolated by considering the dependence of the asymmetry on dilepton mass near the Z 0 resonance. As the asymmetry arising from the axial part of the neutral weak current may be determined in this fashion, measurement of the charge asymmetry may also be useful for demonstrating the indefinite parity of the Z 0. In conclusion it has been shown that a suitably defined charge asymmetry in the parton model has unusually small perturbative QCD corrections to order as(Q2 ). In antiproton-proton collisions at energies of x/~-= 540 GeV this asymmetry is large enough (a c 70-80%) for dilepton masses ofVCQ2 ~- 60 GeV to permit its direct measurement. Such a measurement of this charge asymmetry may provide a quantitative test of perturbative QCD in lepton pair production. It is a pleasure to thank E.L. Berger, S.D. Ellis, E.M. Henley, J. Rutherfoord, and R.W. Williams for many stimulating discussions. This work was initiated at the University of Washington.
References [1] S.D. Dreli and T.-M. Yan, Phys. Rev. Lett. 25 (1970) 316,902; Ann. Phys. (N.Y.) 66 (1971) 578.
Volume 108B, number 6
PHYSICS LETTERS
[2] T.-M. Yan, Cornell preprmt CLNS-80/41 (1981), unpublished; E.L. Berger, Proc. Orbis Scientae, Coral Gables, January 1979, eds. A. Perlmutter, F. K_rausz and L.F. Scott (Plenum Press, New York, 1979); SLAC-PUB-2314 (1979); G. Matthiae, CERN preprint CERN-EP/80-183 (1980), to be published in the Rivista del Nuovo Cimento; R. Stroyonowskl, Lectures at the SLAC Summer Institute, 1979; SLAC report SLAC-PUB-2402 (1979), unpublished. [3] G. Altarelli, R.K. Ellis and G. Martinelli, Nucl. Phys. B143 (1978) 52; B146 (1978) 544 Erratum; B157 (1979) 461 ; B160 (1979) 301. [4] F. Halzen and D. Scott, Phys. Rev. D19 (1979) 216; J. Kubar-Andr6 and F.E. Palge, Phys. Rev. D19 (1979) 221; K. Harada, T. Kaneko and D. Sakai, Nucl. Phys. B155 (1979) 169; B165 (1980) 545 Erratum; K. Harada, Phys. Rev. D22 (1980) 663; B. Humpert and W.L. van Neerven, Phys. Lett. 84B (1979) 327; 89B (1979) 69; J. Abad and B. Humpert, Phys. Rev. D19 (1978) 221. [5] P. Aurenehe and J. Lindfors, Nucl. Phys. B185 (1981) 274; B175 (1981) 301; M. Hayashi, S. Homma and K. Katsuura, J. Phys. Soc. Japan 5, (1981) 1427.
4 February 1982
[6] D.J.E. CaUaway, University of Washington preprint RL0-1388-869 (1981), submitted to Annals of Physics. [7] R.K. Ellis, G. Martinelli and R. Petronzio, Phys. Lett. 104B (1981) 45. [8] G. Parisi, Phys. Lett. 90B (1979) 295; G. Curci and M. Greco, Phys. Lett. 92B (1980) 175. [9] R.W. Brown, V.K. Cung, K.O. Mikaellan and E.A. Paschos, Phys. Lett. 43B (1973) 403; R.W. Brown, K.O. Mikaelian and M.K. GaiUard, Nucl. Phys. B75 (1974) 112; K.O. Mikaelian, Phys. Lett. 55B (1975) 219; R. Budny, Phys. Lett. 45B (1973) 340. [10] D.J.E. Callaway, SJ). Ellis, E.M. Henley and W.-Y. P. Hwang, Nucl. Phys. B171 (1980) 59. [11] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Elementary particle theory: relativistic groups and analyticity (Nobel Symposium No. 8), ed. N. Svartholm (Almqvist and WikseU, Stockholm, 1968), p. 367. [12] S.L. Glashow, J. Iliopoulos and G. Maiani, Phys. Rev. D2 (1970) 1285. [13] D.J.E. Callaway, Phys. Rev. D23 (1981) 1547. [14] A.J. Buras and K.J.F. Gaemers, Nucl. Phys. B132(1978) 249. [15 ] B. Humpert and W.L. van Neerven, Phys. Lett. 93B (1980) 456.
425
PHYSICS LETTERS
4 February 1982
PERTURBATIVE QCD AND LEPTON CHARGE SYMMETRY AT Z ° ENERGIES ~ David J.E. CALLAWAY High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Recewed 5 October 1981
Although the Dreil-Yan mechanism for lepton parr production is very successful qualitatively, large perturbative QCD corrections inhibit a quantitative comparison with experunent at energies presently accessible. It is shown that an appropriately defined charge asymmetry in the context of the parton model has small (5-15%) orderas(Q 2) perturbative corrections, despite large (50-250%) corrections to the cross section itself for typical experimental parameters. Calculations to order as(Q 2) of the asymmetry in lepton pair production from 15p collisions at x/rs= 540 GeV are presented. The charge asymmetry is large - czc ,~ 70-80 percent for a dflepton mass x / ~ ~ 60 GeV, where the asymmetry is maxmaized. The structure of the asymmetry near the Z ° resonance is also displayed.
The Drell-Yan mechanism [ 1], illustrated in fig. 1, is remarkably successful in explaining the qualitative features of lepton pair production in h a d r o n - h a d r o n collisions [2]. Corrections to this simple parton model picture occur from diagrams such as those displayed in fig. 2. These perturbative QCD corrections to the differential cross section can be evaluated systematically in a power series in C~s(Q2), the running strong coupling constant. For energies presently accessible experimentally such corrections are numerically large [ 3 - 8 ] , raising questions about the consisWork performed under the auspices of the United States Department of Energy.
q
(0)
i
.i
7::[
q
i
(b)
Fig. 1. Diagrams showing quarks and antiquaxks annihilating to produce lepton paixs: (a) the "DreU-Yan process",which proceeds by way of an intermediate photon; Co) the photon is replaced by a Z 0 (the weak neutral boson).
tency of the perturbative [ormalism for calculating the cross section. An interesting question is whether a measurable quantity exists in lepton pair production whose perturbative QCD corrections at present energies are smaI1. One such candidate is known as the charge asymmetry. This asymmetry has been discussed at length in the context of the parton model [9,10]. It arises as follows. The differential cross section for lepton pair production contains terms which are symmetric under the interchange of the lepton pair four-momenta ("charge-symmetric") and terms which are antisymmetric under this interchange ("charge-antisymmetric"). The ratio o f the charge-antisymmetric to the corresponding charge-symmetric cross section is detimed to be the charge asymmetry. The most significant contribution to this asymmetry involves the production o f lepton pairs via the axial vector portion of the neutral weak (Z 0) current. Other contributions to the asymmetry are mentioned below. Perturbative QCD corrections to the charge asymmetry seem to be much smaller than the corrections to the electroweak cross section. For antiproton-proton collisions at typical energies (x/~ -= 2 0 - 5 4 0 GeV), first order perturbative QCD corrections to the electroweak cross section are approximately 5 0 - 2 5 0 % of the parton model value, while the corresponding 421
Volume 108B, number 6
PHYSICS LETTERS
4 February 1982
]~' =~ (v~ + a~vs)~'~, lq'U= ~l(Vq + Aq "Y5),),uq, (o)
q
(1)
respectively. In the Weinberg-Salam model [ 11 ] (augmented by the GIM mechanism [12]) the various coupling constants are, in the lowest order, V~ = e'(1 - 4 sin20w),
A~ = - e '
Vq = e'(2T 3 - 4(eq/e) sin20w),
(2a,b) Aq =
e'2T3,(2c,d)
where e,=
e 4 sin 0 w cos 0 w '
(b) q
G
y~, Z 0
(e)
and e is the magnitude of the charge of the electron (e 2 = 47rol), eq is the electric charge of the quark q (q = u, d, s, etc.), T 3 is the third component of the weak isospin of the quark q (+~ for u quarks and - ~1 for d quarks etc.) and 0 w is the Weinberg mixing angle. The mass of the exchanged Z 0 boson in, e.g., fig. lb is given by = 3'I2
q Fig. 2. Diagrams illustrating some of the order a s perturba-
tire QCD corrections to the diagrams fig. 1 : (a) Quark-antiquark pair annihilate, producing a lepton pair (via a virtual photon or Z°) and a gluon. Crossed diagram not shown. (b) Initial gluon and quark scatter, radiating a virtual photon or Z°, which then couples to a lepton pair. Crossed diagram not shown. (c) Virtual gluon diagram. The interference of this diagram with those in fig. 1 gives an order as correction to the cross section derived from the diagrams fig. 1 alone. Self-energy corrections not shown.
e ~ (89 GeV) 2 4V~-GF sin20w COS20w
422
(4)
for sin20w = 0.225, the value used in this paper, and the Fermi weak coupling G F ~-. 1.16632 × 10 -5 GeV-2. It is useful to define the quantities
Nq(Q2)-(Q2-M2+ 2 eV---q-qe ~ / R (Q2), \ Q2 q ~/
(5a)
Dq(Q2) =- l + (2 (Q2 _ M 2 ! Vq v~ Q2 eq e~ (V2
corrections to the charge asymmetry are roughly 5 15% of the parton model result. Perturbative QCD corrections to the charge asymmetry are larger in p r o t o n - p r o t o n collisions, where initial gluon effects are important as well [6]. Details of the ~p calculation are now given. The neutral weak lepton and quark currents are given by
(3)
+
R (Q2) =
2 2 2 + Aq) (V~ +A~) ]R (Q2) e~ e-~ / ' q
(Q2)2 (Q2 - M ~z) 2 + M 2z r 2"
(58)
(5c)
Here e~ = - e is the leptonic charge and F is the width of the Z °, taken to be F ~ 2.5 GeV. The results given here do not depend sensitively on this value. Note that only the ratios of weak to electric charges appear in eqs. (5).
Volume 108B, number 6
PHYSICS LETTERS
The largest contributions to the ffp electroweak lepton pair production cross section come from diagrams involving the annihilation of an initial quark and antiquark, as displayed in figs. (2a) and (2c). Initial gluon diagrams such as the one shown in fig. (2b) gcve rise to much smaller effects in this case [6]. Thus each lepton pair is assumed to be produced by the annihilation of a quark of momentum p with an antiquark of momentum p', with
pU = X lP~l ,
p'U = x2t~2 ,
P1 = -gP0,
P2 = gP0,
(6a) s = 4P 2 .
(6b)
Here, p~z and P~ are the four-momenta of the incoming proton and antiproton. The hadrons and quarks are assumed to be massless and on-mass-shell. In addition, the following variables are defined:
0. =
+ 1;,
l.--
- I;
cos 02=--1"~/111,
cos ~ = I.L°Qj_/(IIj_IIQ.LI)
Ij'I=Q_L" ~"=-0,
x F--Qz/PO,
cos X = cos 02/(1 + x2/4r) 1/2 , 2 + 4r)1/2 -- XF], Xl-- ~a [(XF
d4o
[dQ2dxFd2~2~
X F,q(eq/e)Aq Nq(Q2)q(2l, Q2)t~(22, Q2)
(9)
Zq (eq/e) 2 Dq (Q2)q(xl, Q2)~(22 ' Q2) which is independent of¢Q. To a good approximation [10], the ffp charge asymmetry is given by the flu annihilation contribution alone, 1 Nu(Q 2) Au A~ 2 c o s x ac(PP) ~ 2rraDu(Q2 ) (en/e)(e2/e) 1 + cos2x '
(10)
which is independent of the parton distributions. Note that as 0 2 / M 2 ~ O,Nu(a2)/Du(a 2) ~ - 0 2 / M 2 and the charge asymmetry eq. (10) reduces to a result derived elsewhere [6,10]. Also notice that Nu(Q2 ) vanishes when 0 2 = M2/(1 + 2 Vu V2/e~) ~-.M 2 ,
d2~2~- sin 0~d02dq~ , X2-- ~1 [(x2+ 4r)l/2+XF]
(7)
x [
_ 1 2 cosx A2 ac(lSP) 21ra 1 + cos2x (e2/e)
(11)
7"=Q2/s,
where lu and l~- are the four-momenta of the positive and negative leptons, and the angles are defined in the hadronic cm system. The notation q (x 1, Q2) and q(x2, Q2) is also used to represent the (nonscaling) distributions of quarks and antiquarks of flavor q (q = u, d, s, etc.). The initial transverse momentum of the quarks and antiquarks is neglected. The integrated charge asymmetry is defined by [6]
- [ d4o txe - [dQ2__~Fd2 a 2
4 February 1982
/ [cos 02 ~--cos 02] t
+ [cos 02 *'--cos
}1 .
(8)
The integral d2Qj_ over the transverse momentum of the lepton pair has been performed separately for each term in the numerator and denominator of eq. (8). Both the numerator and denominator of this ratio are calculated by summing the appropriate quarkantiquark annihilation contributions. The parton model charge asymmetry is given by
and so also does the approximate expression for the charge asymmetry eq. (10). It is a general feature of weak asymmetries near a resonance that zeroes and sign changes in the asymmetries occur in the resonance region [13]. The details of the order as calculation of the integrated charge asymmetry are given elsewhere [6]. It should be noted, however, that the results given in ref. [6] are valid in the limit Q2/M2 ~ O. Here the asymmetry is evaluated in the Z 0 region as well, so that the full expressions eqs. (9) are retained. Thus in the present calculation the effects arising from the squared neutral current diagram fig. lb are included. In the order a s calculation [6] of the integrated charge asymmetry the various divergences are handled by dimensional regularization. When the integration over Q± is performed, the infrared divergences from the diagram fig. 2a (and the corresponding crossed diagram) cancel with those associated with the interference of the virtual diagram fig. 2c and the lowestorder diagrams figs. 1. The remaining (collinear) divergences are absorbed into the parton distributions in the usual fashion. In fig. 3 the integrated charge asymmetry calculated in the parton model (i.e. eq. (9)) is compared with the asymmetry calculated up to order a s . The 423
Volume108B,number6
PHYSICSLETTERS
08
O6o4
~P
"
/
Qe 0 2 0
-OZ -04_06 -
0.8 0
~ X
parton ~
parton ~x\ "~°(as) '~J I
I
20
40
~ 60
[
t
80
I00
120
Fig. 3. The charge asymmetry for lepton pair production from antiproton-proton collisionsat~s= 540 GeV is plotted as a function of the dilepton mass x/Q2 . Other parameter values are given m the text. The broken line disphys the asymmetry calculated in the parton model. The solid line is a plot of the asymmetry with order as(Q 2) corrections included. asymmetry is plotted versus the mass of the lepton pair, x/c~-, for x/~-'= 540 GeV, x~ = 0, and cos 0 = cos ¢~ = 1. The (nonscaling) Q -dependent parton distributions are taken from ref. [14], and the running coupling constant is taken to be 127r °¢s(Q2) = (33 - 2Nf) In (Q2/A2) '
(12)
with Nf = 5 and A = 0.3 GeV. For the range of Q2 considered here (10 GeV ~
4 February1982
Since the structure of these r n moments for large n gives the behavior of the cross section for large r, it follows that in the limit r ~ 1 the order as(Q2 ) charge asymmetry approaches its parton model value. Other large numerical corrections (such as the 7r2 from the analytic continuation of the q~17vertex function from spacelike to timelike values of Q2) occur in like multiplicative fashion in the charge symmetric and antisymmetric contributions to the cross section [6]. These large corrections therefore tend to cancel when the ratio comprising the charge asymmetry is taken. The remaining terms (which are different in the two cases) are negligible numerically. Note that if the large corrections can be summed (exponentiated) to all orders [8], the resulting large factor may drop out when the ratio comprising the charge asymmetry is taken. Contributions to the charge asymmetry may also arise from higher-order electromagnetic effects [9]. For the large dilepton masses of interest here (x/Q 2 10-100 GeV) such effects are presumably negligible, however. In addition, since such effects do not require the exchange of a Z 0 they may be isolated by considering the dependence of the asymmetry on dilepton mass near the Z 0 resonance. As the asymmetry arising from the axial part of the neutral weak current may be determined in this fashion, measurement of the charge asymmetry may also be useful for demonstrating the indefinite parity of the Z 0. In conclusion it has been shown that a suitably defined charge asymmetry in the parton model has unusually small perturbative QCD corrections to order as(Q2 ). In antiproton-proton collisions at energies of x/~-= 540 GeV this asymmetry is large enough (a c 70-80%) for dilepton masses ofVCQ2 ~- 60 GeV to permit its direct measurement. Such a measurement of this charge asymmetry may provide a quantitative test of perturbative QCD in lepton pair production. It is a pleasure to thank E.L. Berger, S.D. Ellis, E.M. Henley, J. Rutherfoord, and R.W. Williams for many stimulating discussions. This work was initiated at the University of Washington.
References [1] S.D. Dreli and T.-M. Yan, Phys. Rev. Lett. 25 (1970) 316,902; Ann. Phys. (N.Y.) 66 (1971) 578.
Volume 108B, number 6
PHYSICS LETTERS
[2] T.-M. Yan, Cornell preprmt CLNS-80/41 (1981), unpublished; E.L. Berger, Proc. Orbis Scientae, Coral Gables, January 1979, eds. A. Perlmutter, F. K_rausz and L.F. Scott (Plenum Press, New York, 1979); SLAC-PUB-2314 (1979); G. Matthiae, CERN preprint CERN-EP/80-183 (1980), to be published in the Rivista del Nuovo Cimento; R. Stroyonowskl, Lectures at the SLAC Summer Institute, 1979; SLAC report SLAC-PUB-2402 (1979), unpublished. [3] G. Altarelli, R.K. Ellis and G. Martinelli, Nucl. Phys. B143 (1978) 52; B146 (1978) 544 Erratum; B157 (1979) 461 ; B160 (1979) 301. [4] F. Halzen and D. Scott, Phys. Rev. D19 (1979) 216; J. Kubar-Andr6 and F.E. Palge, Phys. Rev. D19 (1979) 221; K. Harada, T. Kaneko and D. Sakai, Nucl. Phys. B155 (1979) 169; B165 (1980) 545 Erratum; K. Harada, Phys. Rev. D22 (1980) 663; B. Humpert and W.L. van Neerven, Phys. Lett. 84B (1979) 327; 89B (1979) 69; J. Abad and B. Humpert, Phys. Rev. D19 (1978) 221. [5] P. Aurenehe and J. Lindfors, Nucl. Phys. B185 (1981) 274; B175 (1981) 301; M. Hayashi, S. Homma and K. Katsuura, J. Phys. Soc. Japan 5, (1981) 1427.
4 February 1982
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