Estimates for W±, Z0, and μ-pair production in the asymptotically-free parton model

Nuclear Physics B129 (1977) 461-482 © North-Holland Publishing Company

E S T I M A T E S F O R W-+, Z 0, AND p-PAIR PRODUCTION IN THE ASYMPTOTICALLY-FREE PARTON MODEL * J. K O G U T ** and J u n k o S H I G E M I T S U Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14853 Received 13 July 1977

Curves are presented for massive t~-pair, W-+ and Z 0 production from proton-proton collisions at center-of-mass energies between 100 and 600 GeV assuming m w ~ 60 GeV and m z ~ 77.5 GeV. The calculations are done using a generalization of the Drell-Yan formula to asymptotically free quantum chromodynamics. The cross sections are large and interesting, and in many cases the asymptotic freedom estimates exceed the naive quark parton model calculations.

1. I n t r o d u c t i o n One o f the main purposes o f the n e x t generation o f p r o t o n accelerators already proposed or under c o n s t r u c t i o n is to find the intermediary o f the weak force. According to Weinberg and Salam [1] the mass o f the W e should be a b o u t 60 GeV and that o f the Z ° should be about 80 GeV if we accept the experimental value [2] of the Weinberg angle, sin 2 0 w ~ 0.4. These masses lie well within the capabilities o f the new machines. Estimates of the p r o d u c t i o n cross sections for W ~ and Z ° have been m a d e by m a n y theorists ***. Most calculations e m p l o y the generally successful quark parton m o d e l t (QPM) to estimate the rate for hadron + hadron -+ W e + anything using the m e c h a n i s m first suggested by Drell and Yan [5]. In the QPM the m o s t efficient way to produce a high mass V+/l- pair, W +- or Z °, is to have a quark from one hadron annihilate i n c o h e r e n t l y with an anti-quark from the second hadron (fig. 1). Given the quark distribution functions as measured from electron and n e u t r i n o scattering, one can then calculate these Drell-Yan processes straightforwardly and find that the p r o d u c t i o n cross sections are large and interesting.

* ** *** ?

Supported in part by the National Science Foundation. Alfred P. Sloan Foundation Fellow. Work very similar to ours is being pursued by Hinchliffe and Llewellyn Smith [3]. A few of the huge assortment of articles on the subject include those of ref. [4 ] which we have used in some detail. 461

462

J. Kogut, J. Shigemitsu / Asymptotically-free parton model

Fig. 1. Quark annihilation into a massive muon pair.

The issue that will be considered here is the effect of asymptotic freedom on these naive predictions. Recent deep inelastic muon scattering experiments [6] show the patterns of scale breaking predicted by Quantum Chromodynamics (QCD) [7]. In terms of quark distribution functions, one finds that as Q2 (minus the fourmomentum squared of the virtual photon) increases, the probability to find quarks of large longitudinal fraction inside the target decreases while the probability for small longitudinal fraction increases. To make a massive state through the Drell-Yan q~ annihilation process requires quarks and anti-quarks in the hadrons with substantial longitudinal fractions. This raises the possibility that W± production might be badly suppressed in QCD. Happily, we find that this is not the case. For x/~ >~ 500 GeV (s is the square of the center-of-mass energy of the reaction) the W +-production cross section is larger than in the naive quark patton model. Roughly speaking, the reason for this is the enhancement of antiquark distributions as Q2 grows and the fact that when s is very large the longitudinal fractions of the contributing quarks can be quite small. Of course, nearer threshold the cross sections are suppressed somewhat (see below). This paper is organized into several short sections. In sect. 2 we briefly discuss the generalization of the naive QPM description of Drell-Yan processes to QCD [8]. In sect. 3 the calculation of Q2-dependent quark distribution functions is sketched and quantitative results are presented. In sect. 4 the production cross sections for massive/~+bt-, W± and Z ° are estimated. In addition, the single lepton spectra for hadron + hadron -~ W+-(Z°) + anything -~ ~± + anything are presented. In many cases the muon signal is very dear; the rates to produce muons with transverse momenta on the order of 30 GeV/c are very accessible.

2. Annihilation of asymptotically free partons The naive QPM rests on the assumption that quarks are free at short distances [9]. However, even in asymptotically free field theories this assumption is too strong * ; the virtual processes in which a quark splits into a quark and a gluon or a gluon splits into two gluons or a quark antiquark pair are important on all length scales. These fluctuations can be incorporated into a parton description by introducing Q2-dependent distribution functions [ 10] ; let Fi(x, t) dx/x be the number of partons of type i with longitudinal fraction between x and x + dx which are resolved * The physics underlying the formal results of ref. [7] is discussed in parton language in ref. [10].

J. Kogut, J. Shigemitsu ~Asymptotically-free parton model

463

by a virtual photon of four-momentum squared q2 - _Q2 = A2et: Renormalization group improved perturbation theory allows one to calculate Fi(x, t + dt) in terms of Fi(x, t) from the basic vertices of the theory. This will be discussed in detail in the next section. The functions Fi(x, t) are measured in deep inelastic scattering experiments. For example, as discussed in ref. [ 10],

vW2(x, Q2)= ~ i

ei2Fi(x, t ) .

(2.1)

For this class of experiments this patton interpretation is equivalent to the use of the Callan-Symanzik equation and the operator product expansion. However, since the parton interpretation is more general, it can be applied to a wider class of phenomena. Of interest here are the reactions, proton + proton-+~+g - + anything

(2.2a)

W-++ anything

(2.2b)

Z ° + anything.

(2.2c)

On the basis of physical arguments and perturbation theory it has been suggested [8] that the cross section for reaction (2.2a) is d°

dQ 2

l

3Q 4 3

i

where Q2 is the invariant mass-squared of the muon pair and t = ln(Q2/A2). The extra factor ½ in eq. (2.3) arises from the three colors of the annihilating quarks. Recall that in the naive patton model eq. (2.3) applies with Fi(x, t) replaced by Q2-independent distribution functions. The additional Q2 dependence in eq. (2.3) arises from the fluctuations described above which effectively clothe the constituents. What is the theoretical status of eq. (2.3)? In particular, if vW 2 is given by eq. (2.1), under what circumstances does eq. (2.3) follow? Halliday [ 11] has studied this problem in perturbation theory to all orders (leading logarithm approximation) in the Abelian gluon model and has verified the connection. Physical arguments have also been presented [8] and low-order calculations for QCD [12] * also support his result, but more powerful analyses of this problem are necessary. One must also inquire into the size of the correction terms to eq. (2.3). Recall that in the naive QPM bremsstrahlung production (fig. 2) of massive muon pairs is suppressed by a power of Q2 order by order in perturbation theory. However, a survey of graphs (fig. 2, for example) indicates that for asymptotically free partons the sup* This analysis considers O(g2) corrections to the naive Drell-Yan picture.

464

J. Kogut, J. Shigemitsu ~Asymptotically-free parton model

\

F

Fig. 2. Bremsstrahlung of a massivemuon pair. The quanta exchanged between the quarks is a gluon.

pression is only logarithmic in Q2 (by powers ofg2(Q 2) ~ (ln Q2)-l, the running coupling constant of QCD), so it may be, therefore, that while eq. (2.3) is indicative of the general character of the physical cross section, it is not reliable in quantitative detail except at enormous values of Q2. In fact, by the same argument, if strong interactions were described by a fixed point field theory, eq. (2.3) would not be correct order by order in perturbation theory. These observations illustrate why this class of reactions is difficult to deal with using only general theoretical tools: the dominant class of graphs contributing to massive muon production is model dependent. As is common theoretical practice, we will apply eq. (2.3) and accept the fact that it is at best a leading logarithm (lowest order renormalization group) approximation. An important characteristic of eq. (2.3) which distinguishes asymptotic freedom from the naive QPM is the transverse momentum q± distribution of the center-ofmass of the muon pair. On very simple grounds one expects that [8,3]

(q~)/Q2 = const [g2(Q2)/4rr2 ] . ~ (Q2/s),

(2.4)

where ~depends on the wave functions of the colliding protons. In the simplest form of the QPM (q~) is a constant on the order of several hundred MeV. Present experiments [13] hint at the correctness ofeq. (2.4)but they are not decisive at this time. Of course, eq. (2.4) implies that the We and Z ° produced via the Drell-Yan mechanism will have appreciable transverse momenta. We will have to deal with this fact in some of our numerical estimates below.

3. Calculating Fi(x, t) To evaluate the cross sections of interest we must predict the quark and gluon structure functions for huge values of Q2. This will be done using parton model language. The basic concepts for this approach have been laid down in references [10 and 14]. Recently Altarelli and Parisi [15] have done detailed calculations in this language and have rederived results obtained earlier with the Callan-Symanzik equation and the operator product expansion. As usual, the advantage of the parton approach is its simplicity and general character.

J. Kogut, J. Shigemitsu /Asymptotically-free parton model

~a)

465

(b)

(e) (d) Fig. 3. Basic (order g) vertices of QCD, corresponding to eqs. (3. la)-(3, ld). To begin, one introduces a function [15] P ~ ( z , t)which is the probability density to find a parton of type B at momentum scale t + dt inside a parton of type A that was resolved at m o m e n t u m scale t (t -= In Q2/A2). z is the fraction of A's longitudinal momentum carried by B. The functions.PBA are easily computed from the basic vertices of QCD using renormalization group improved perturbation theory. In QCD there are the following possibilities: "gluon in a quark":

Pgq(Z, t) = ~a(t)

Pgq(Z) dt

"quark in a gluon":

~(t) Pqg(Z, t) = ~ Pqg(2) d t ,

"quark in a quark":

a(t) , Pqq(Z, t) = ~ - Pqq(Z) dt,

z 4:1 ,

(3.1c)

"gluon in a gluon":

~(t) , Pgg(Z, t) = ~ Pgg(Z) dt,

z 4:1 ,

(3.1d)

(3.1a)

(3.1b)

where a(t) = g2(t)/4n and g(t) is the running coupling constant. The expressions (3.1 a)-(3.1 d) correspond to the figs. 3a d respectively. In addition, the trivial graphs of fig. 4 contribute to Pqq and Pgg, so for all z, ~ oe(t) Pqq(Z, t) = 6(z - 1) + 4r~-rr Pqq(Z) d t ,

(a)

(3.2a)

(b)

Fig. 4. Contributions to: (a) Pqq, and (b) Pgg.

466

J. Kogut, J. Shigemitsu /Asymptotically-free parton model

egg(Z, g)= ~ ( z - 1)-I-o((t) egg(Z)dt,

(3.2b)

4rr

where

Pqq(Z) = Pqq(Z) + CqS(Z - 1),

(3.3a)

Pgg(Z) = P~g(Z) + CgS(z - 1).

(3.3b)

The constants Cq and Cg can be determined from the condition that the total longitudinal fraction of the constituents sum up to the longitudinal fraction of the parent.

1 f z[Pqq(Z, t)+ Pgq(Z, t)] dz : 1, 0

(3.4)

1

f z [.Pgg(Z, t) + 2fPqg(Z, t)] dz : 1,

0 where f = number of quark flavors. A final constraint expresses conservation of fermion number, 1

(3.5)

f Pqq(Z, t) dz = 1. 0

The functions PBA govern the Q2 dependence of the parton distribution functions FA(X, t) dx/x used in sect. 2. Since PBA is a conditional probability which represents the splittings of partons of scale t into those of scale t + dt we have

x

dFq(x' t) = -~n

x

x

}

ypqq(~) dtFq(y, t) dy +ypqg(y) dtFg(y, t) dy , x

Y

Y

or dt

47r x ~ -

Fq(y, t) + eqg

Pqq

t) .

(3.6)

= FqCv, t) .

(3.7)

Fg02,

Similarly for the distribution of gluons, dFg(x, at

t) _ a(t) ~ xdy 4n x ~

Pgg

x

Fg(y, t) + Pgq

x

To use this scheme in practice we need the functions PBA and the quark distributions Fi(Y, t) at some t and at all 0 < y < 1. The conditional probabilities can be computed from the graphs of fig. 3. This is best done using infinite momentum

J. Kogut,J. Shigemitsu/Asymptotieally-freepartonmodel

467

frame perturbation theory and has been carried out in ref. [ 15],

Pqq(Z)=-~{38(lnz)+2(1

z)+4Ii~zl

, +}

Pqg(Z) = z 2 + (1 - z) 2 , 8 {l+(1-z) Pgq(Z) = -~ z

(3.8)

2} " '

Pgg(Z)=g-~8(lnz)+12{(1-z)(z+l)+ I~-z~ } where the distribution [z/(1 - z)]+ is defined by 1

1 dy (x/y) [ N ( y ) - N ( x ) ]

f dy .7 X(y) = l n ( 1 - x)X(x)+ f --fy It_lx/yx/y_]+ x

~x/-~

x

and its presence is a reflection of the spin-1 character of the gluons in QCD. We have also obtained eq. (3.8) from the Callan-Symanzik equation and the operator product expansion just to check the physical picture in quantitative detail. Next we need all the distributions at one value of t. This step involves some theoretical guesswork for several reasons. First, the gluon distributions are not measured directly and, second, the momentum scale in the definition of t must be set. Within the context of QCD with 4 flavors and 3 colors, many authors have used the SLAC electroproduction and the scanty neutrino data available to determine the quark distribution functions. Sum rules and theoretical models have also led to estimates of Fg. We will use the fits of Barger et al. [16] which applies at Q2 ~ 3.5 GeV 2, in the SLAC kinematic range. The mass scale in t, A 2, has been chosen to be A 2 ~ 0.25 GeV 2 which is within the ranges preferred by other authors [17]. We have checked that with these choices the most recent muon scattering data [6], which show the scale breaking patterns of QCD, is reproduced by eqs. (3.6) and (3.7). This agreement is not very sensitive to the precise value of A2: other choices such as A 2 ~ 0.1 GeV 2 or A 2 ~ 0.5 GeV 2 also pass this test. The distribution functions of Barger et al. are (we have modified their "sea" distribution function slightly)

Fi(y, t)

FuValence(x)

=

V'x(0.594(1 - x2) a + 0.461(1 - x2) s + 0.621(1

-

x2)7)

,

F~alence(x) = X/x(0.072(1 - x2) a + 0.206(1 - x2) s + 0.621(1 - x2)7), Fsea(x)

=

0.609(1 - x) 6 .

(3.9)

Fsea(x) includes contributions from the total SU(3) symmetric sea. We assume no

468

0.8 0.7 0.6

/i

\

//

x

\

~5

~'-x \\ \ \

Fu V(x) 0.4

,'

~\\

0.3

[ 0.2

',,,02~3.5G,v2 ,,,

\

Xo 2-~o ,,,

'\

02 ~4OOOX

/

\X

~x\

"'\ ~

\\\\

",

0.1 "

,

,

,

0.1

0.2

03

" ~ ," ~ ' ' "

,

,

Q4

0.5

0.6

0.7

0.8

0.9

1.0

x

Fig. 5. Distribution functions FVp(x) for valence up-quarks. The dotted curve shows the input distribution function of eq. (3.9). (Q2n - 3.5 GeV2). The dashed and solid curves correspond to Q2 ~ 70 GeV 2 and Q2 ~ 4000 GeV~respectively. 0.8

0.7

0.6

0.5

FdV(x) 0.4

0.3

~.

" " ' . . Q2- 35 GeV2

0.2

~;

Q2~4000~

" X

-.

O.I ,

,

0.1

0.2

,~ - - ~ - ~ - - 0.5

0.4

0.5

,

0.6

0.7

x

Fig. 6. Same as fig. 5 for valence down-quarks.

0.8

0.9

i 1.0

J. Kogut, J. Shigemitsu /Asymptotically-free parton model 0.8

u,d,s

Sea i 0 2 ~ 3 . 5 Q2 ~ 4 0 0 0 .

469

GeV 2 . . . . . . GeV 2

~

Charmed Seo IQ2-,,4000. GeV 2 . . . . . . . .

0.7

0.6

0.5 se0

F(x} 0.4

0.3

0.2

0.1

\

..

0.1

0.2

0.5

0.4

0.5

0.6

0.7

0.8

0.9

1.0

x

Fig. 7. Sea distribution functions for the u, d, s sea at Q2 _ 3.5 GeV 2 and Q2 _ 4000 GeV 2 (dotted and solid curves) and for the charmed sea at Q2 _ 4000 GeV 2 (dash-dotted curve).

1.0 .9 .8 z .7 0 I--

~

.6 I

GLUONS

.5 =E

2.4

QUARKS

~o.3

=E

.2 ANTIQUARKS I 5.

I

IO.

I

50.

I

IOO.

I 500.

i

IOOO.

500(

Q2 (GeV z)

Fig. 8. Fraction of proton's m o m e n t u m carried by: gluons (dotted curve), quarks (solid curve), and antiquarks (dashed curve) versus Q2.

470

J. Kogut, J. Shigemitsu / Asymptotically-free parton model

charmed quarks at Qg ~ 3.5 GeV 2. These will be pair-created from gluons in accordance with eq. (3.6). For the gluon distribution we take Fg(x) = 3.465(1 - x) 6 .

(3.10)

In figs. 5 - 7 we plot these distribution functions at higher values of Q2. Finally, in fig. 6 we plot the fraction of the proton's longitudinal momentum carried by gluons, quarks, and anti-quarks versus Q2. The fact that the anti-quark distributions become more substantial at large Q2 is crucial for the annihilation reactions of interest here: we shall see that this growth will more than compensate for the decline of the quark distributions at large x and there will be accessible kinematic regions where it is more likely to produce W's given asymptotically free partons than naive point-like ones.

4. Annihilation cross-section c:lculations Now we turn to the cross sections. The processes are shown in fig. 9. The kinematics for the observed particles is: Pl, P2 = four-momenta of colliding hadrons; S= (Pl +P2) 2 ;

p = four-momentum of detected lepton; Q = four-momentum of virtual 7, We or Z ° ;

(4.1)

Ui = --(Pi -- P)2/S" Hats decorate the definitions of patton kinematics:

kl, k 2 : four-momenta of annihilating partons, ki = (k O, k 3, ki2) = (k O, xi p3, ki3_) ,

I

I 1" ( /"

A+-,Z

Pl

,y

2

Fig. 9. Drell-Yan picture of W +-, Z O, 7 production in a pp collision.

J. Kogut, J. Shigemitsu / Asymptotically 4~'eepatton model

4 71

=(k 1+k2)2=Q2 ,

hi = - ( k i - p)2/~ .

(4.2)

The connection between parton and hadron kinematic variables simplifies in that region where xip 3 ~ [kiz], [I 1 ~-~IAI/X 2 ,

II 2 ~ ' H 2 / X 1 ,

(4.3)

S'~X1X2S.

(In our discussion of transverse momentum smearing below we will see that

xip~ >>Ikiil is indeed true for the bulk of the cross sections.) The general structure for the processes proton + proton ~

do _1 dQ2

~

fdXl

3 { i, 7}

Xl

+ anything

dX2 Fi(Xl, t)b)~x2, t ) 6 ( X l X 2 S _ Q 2 ) b i ] ( Q 2 ) X2

'

(4.4) '

where oi, fis the relevant parton cross section, the 1 comes from matching colors of the annihilating partons and the sum over i and j matches the parton types appropriately. A clear signal for the existence of the We and the Z ° will be charged leptons at huge transverse momenta. So, we are interested in the differential cross sections proton + proton -+ W+- + anything, +v

proton + proton ~ Z ° + anything. ~ / . t + + ktThe parton subprocesses are shown in fig. 10. First consider the production and decay of the W÷. For the diagram of fig. 10a in the Weinberg-Salam model,

d Oi3 E d3 p

dOi'7 "trsdf/ldU2

0~2

t i 2 ~ ( ~ l + /~/2

1)

4 sin40w 1~- m 2 + imwI'w] 2 '

(4.5)

where l"w is the full width of the W. In the quark model I'w is roughly eight times the width for the decay W -+/lu, Pw = 8 I ' w - ~ v

2 c~ 3 sin20w r n w '

(4.6)

472

J.

Kogut, J. Shigemitsu/Asymptotically-freeparton model

q'~

k

~i

"/k'

,

///'~±

"%

i

/ M'+

T

",~,-

(a)

(b)

~

'

/

\/~-

lel Fig. 10. Quark amplitudes for the processes qq- ~ We -> g-+vand qq ~ 3, or Z0 -->#+g-.

and we shall accept the experimental value sinZ0w = 0.4 for which mw = 60 GeV. The differential cross section for proton + proton -+ W+ + anything -+/~+ + anything is, therefore, 1--u 1

do" _ 1

Ed3p

-3

~ f (i,J}

d.,v - Fi(uz/y, t) Ff(Ul/(1 - y), t) y(1 - - y )

~2

X

(1 - y)2

4 sin40w

Ip~/y(1

y) - m~v + imwI'wl 2 + (ul +~ b/2).

(4.7)

It is convenient to plot the differential cross section for 90 ° scattering in the c.m. where u 1 = u2 = P.L/V'-S. Then eq. (4.7) becomes

1-p~/~

Ed;p--

90 ° in c.m.

~2 × s4i n 4 0 ~ "

3 (i, 7}

,/s

y(1 -- y )

/\X/sO ~- y ) '

(1 _ y ) 2 Ip~/y(1 - y ) - m~v + imwPwJ z '

(4.8)

where the summation runs over the pairs (i, j } = {u, d } and (c, g}. In writing down eq. (4.7) and (4.8) a serious idealization has been made:we have neglected the transverse momenta of the annihilating quarks. As mentioned in sect. 2 and as emphasized in refs. [3,8],

(k/21) = const, gZ(Q2_____))Q2 4n2

(4.9)

J. Kogut, J. Shigemitsu /Asymptotically-free parton model

473

which is not generally negligible here. In fact to calculate the lepton spectra with precision one would have to use the full apparatus of ref. [8] and calculate the patton distribution functions in transverse momenta as well as longitudinal fraction. The transverse momenta of the partons, eq. (4.9), will smear the momentum spectra given by eqs. (4.7) and (4.8). We will take this effect into account in a phenomenological fashion which will be discussed in detail below. Suffice it to say, that for energies well above the W boson threshold, the unsmeared cross sections are good guides to the more realistic calculations. Hence we shall first present simple calculations which neglect the transverse momenta of the partons and at the end of this section estimate the effects of the transverse momenta of the W-*, Z ° and 7. Next consider the Z ° and 3' contributions to the/x + spectra. The graphs of figs. 10b and 10c interfere, so the parton differential cross section reads E

d~ d3p

d~ - - - 2a26(£/1 + £/2 - 1) Gi(£/1, £/2), rrgd/t 1d/t2

(4.10)

where

Gi(£/1, £/2) =s~ (1

2£/1£/2) +

[(a 2 + b2)(A 2 + B2)(1 - 2£/1£/2) + 4abAiBi(£/1 - £/2)1 m2) 2 + m z2 P z2]

cos40w sin40w[(S

2ei(s - m2)[aAi(1 - 2£/1/t2) + bBi(£/l £/2)] sin20w cos20wg[(s - m~) 2 + m z2 F z2] -

(4.11)

where

A

A

UIU 2 Fz-

mz = 77.5 G e V ,

,

2wnz

3 sin20wCOS20w

a = sin 2 0 w - 1 ,

(1 b

2 sin20w + 8 s i n 4 0 w ) ,

--

1 4

'

and for i = u or c,

ei = _23,

Ai

=

1 _ 2 s i n 2 0 w + -4,

Bi

=1

Bi

-

,

while for i = d or s,

ei

-

1

3,

Ai = 1 sin20w _ 1 ,

1

4 "

Then, the differential hadronic cross section is

E~

d° I90° in c.m. -- - -4 a3 2 i=1

j-Pz/X/Sy(1 p±/x/s

_y---------~Fi

,t

J. Kogut, J. Shigemitsu ~Asymptotically-free patton model

474

>

t N

-

o

¢_ .................

..? ~>

¢,j

i ......

/

~

~'O

d

~

~

/

I

.....

~o

'

'

'

o

I ........

o

t~

Izmo~ m -!.o6 I ~ - ] lop

-8

:zk ÷t __

--

,

_

% ÷

I ~ / ~ /

I-

/

J

J

o

b

'o

,o ( zAeO/z'"0~ IN0 u ! o 0 6 d ~ p ~ I

'_o

~Z~Cr 4°

N

I 0 "3a

10.3?

I0

ilk

i'y

I

20

,'"

P. (GeV/¢)

"""

~:,

~,\,,,//

\ ///'~

I

50

t',/

t,."

t

\ "\ 40

I

",,I'

I~

A~

,,",

,,) ! w~'--,.F* ~'

Jl

~'-S= 300 GeV

\

Fig. 11 (continued).

I

.i

b

.E

>.

to'4O

I0

,d~'

Id3?I

I 20

I 30 P, (GeV/c)

W*--,, u.+ u

[

IX,Z °-,. F÷,u.-

.... I 4,.0

4

##1!

~-S = 400 GeV

"4,

<.,.

id 4°

I

I0

I

20

,

P. (GaV/c)

30

W+----,. F ÷ v

i 4.0

= 500 GeV

\ I

I I0

,o~O

-37 I0 r

I 20

P, (GaV/c)

I 30

W+--,,.F+ v

40

L

I

7, Z°-~/~÷/~-

=600 GeV

x/s = 100 GeV to x/s = 600 GeV ((a) to (f)).

Fig. 11. E da/d3p at 90 ° in the pp c.m. system versus p j_ for a p+ from W+ decay or ~,, Z 0 decay. Dotted curves were obtained using input distribution functions of eq. (3.9) (no scale breaking). Solid curves show the results of using Q2 Cependent distribution functions, x/s values range from

b~

i d ~e

t(~ 37

?

477

J. Kogut, J. S h i g e m # s u ~ A s y m p t o t i c a l l y - f r e e parton m o d e l

Fi(x/~(Plty),t) Gi(hl= l - y , ft2=y).

(4.12)

Given the quark distributions computed in sect. 3 one can evaluate these cross sections numerically for various values of s. The results are shown in figs. 11 a - f . On these graphs we also include the calculations using the Barger et al. [ 16] distributions at all t, i.e. assuming exact Bjorken scaling. This allows us to see the effects of asymptotic freedom very simply. Note that the asymptotic freedom curves are smaller near threshold but exceed the naive QPMpredictions at ~/s ~ 400 500 GeV. The physical origin of this favorable effect was explained in the Introduction. Of course, the precise shape and normalization of these curves are somewhat uncertain (since, for example, the input gluon distributions are not measured directly), but their gross features are probably reliable. Next, consider massive muon pair production (through 3' or Z °) and the cross section do/dmu+u_. The graphs of figs. 10b, c contribute to the parton cross section 4rc~2 {e~4 °i(o2) = T Q2 +

_

(a2+b2)(A2+B2) c°s40wsin4Ow[(Q2 m2) 2 + m 2 p ~ ]

2ei(Q2 m 2) aA i

}

2 sin 2 0 WCOS2~-W~ [(-Q2 ~ m 2 ) 2 + m 2z Pz]

"

(4.13)

and give a hadron cross section, do _ 4 ~ . dQ 3Q •

~ T

x

Fi(x,t)F~(r,t)oi(Q2) '

(4.14)

where r = Q2/s. Eq. (4.14) is plotted in figs. 1 2 a - c for various values o f s and shown on the same figures are the naive QPM expectations. In all the figures there is a huge resonant peak atop an appreciable/~+~- continuum. Finally, we must estimate the effect of the transverse momentum of the produced W+- on the spectra of its decay products. To proceed we will combine our theoretical expectation eq. (2.4) with data on the mean transverse momenta of massive/~+/~- pairs. Using the notation of eq. (2.4) recent muon pair experiments [13] suggest a q± distribution which widens as Qz grows. If one attempts a simple exponential fit e - b q ± , then the slope parameter b is b -1 ~ 0 . 2 + 0.08 x / ~

(4.15)

with considerable uncertainty in the coefficients due to the large experimental errors. Then the mean transverse momentum of a massive muon pair is

/o; q±e-bq±dq±=~,2

(q±)o2 = f q~ e-Oq±dq± o

(4.16)

k

10.39

io -3°

i0 -~

i0 - ~

i

60

i

50

...

/

i 80

M,u.+p. - (GoV)

70

I

,

/

90

X

'

I Z°---~y+ V-

I I I I i

II

i , I00

.,/s = zoo GeV

I II0

Fig. 12.

,

60

II

/',

50

70

80 90 i M.+/~- (GeV)

;llt

I00

I IO

I'

//li

' ,o_~

~I~=

~'~

II

= 4 0 0 GeV

~ Io.36 ,,~

I0"a

,o~

.,¢=.

J. Kogut,J. Shigemitsu~Asymptotically-freepartonmodel

479

I d 34 ,J"S=600

GeV

Z°.-~/j.+~

id 35

-

>. (,9

~, i~ 36 bzL

16~r

\

I

I

I

50

60 '

I

I

1

90

IOO

I10

I

70 80 M/z+/~.- (GeV)

Fig. 12. do/dm+~t_near the Z0 resonance with scale breaking (solid curve) and without scale breaking (dotted curve) for (a) x/s = 200 GeV, (b) x/s = 400 GeV and (c) x/s = 600 GeV.

which has the value 10 GeV for Q2 in the mw ~ 60 GeV region. This fit probably overestimates the transverse m o m e n t a in the mw region, but it is satisfactory for our purposes here since we want to argue that transverse momentum smearing will not badly obscure the W and Z signals plotted in figs. 1 l a - f . (The reader should be aware o f other interpretations of this data [18].) Since the mean transverse momenta of the annihilating quarks (roughly, 5 - 7 GeV/c) is considerably smaller than 1 mw = 30 GeV, it is an adequate first approximation to take account of the transverse momentum smearing by replacing eq. (4.8) by

d3--'-fil9oOinc.m. 4 s i n 4 0 w 3

X F/(x/~(Pll_y),y ) f

{i,]}pllx/s N(b) e-bq±(1 q±)2

F. L y(1-y) -

y(1-y) Fi

,t

-- y)2 d2q± (4.17)

2

+ mwF w

J. Kogut, J. Shigemitsu /Asymptotically-free patton model

480 where

N(b) e -bqs is t h e n o r m a l i z e d s p e c t r u m o f t h e W +, max

q± 27rN(b)

/

e-bq±q±dq± = 1 ,

(4.18a)

o

a n d in t h e W p r o p a g a t o r we h a v e

g _ (p± _ ½ q±)2

y(] -y)

(4.18b)

,

as is easily verified. So, in this r o u g h a p p r o x i m a t i o n , we h a v e left the p a r t o n kinem a t i c s in the d i s t r i b u t i o n s F i u n c h a n g e d b u t h a v e a c c o u n t e d for the a p p r e c i a b l e

t°-371 •~ =400 GeV W+~p.+z/ 1(538

%%

~\~,

io"39

/~ r, z*-~m ~-

10-40

m

i

i

|

I0

20

30

40

!

P.t (GeV/c) Fig. 13. Effect of q.L smearing at ~/s = 400 GeV on the p± distribution of leptonic decay products. (Q = momentum of W+, % Z°.) Solid curves are the same as in fig. 1 ld. In the dotted (dashed) curves we have taken into account the nonzero transverse momentum of the prodhced W+(',(, Z O) in a phenomenological way (see text).

J. Kogut, J. Shigemitsu /Asymptotically.free parton model

481

transverse momentum of the W+ in the parton annihilation kinematics (i.e. the propagator) and the integration over the e x p ( - b q ± ) distribution. We have evaluated eq. (4.17) numerically and show the results in fig. 13 for x/~ = 400 GeV. Several observations are in order: (i) The sharp, unsmeared peaks for W+ -->~t+u and Z ° -->~t+#- are lost, but the lepton spectra are still huge and now extend considerably beyond p± of 30 GeV/c. (ii) The single lepton spectrum from the Z ° is partially obscured by the W ~ / J signal due to smearing. This suggests that measurements of dcr/dmu+ u . will reveal the Z ° more easily. In conclusion, although transverse momentum smearing is appreciable the presence of W+ -+/~+u in the data will be clear.

5. Conclusions We have calculated the cross sections for massive muon pairs, W-+ and Z ° production in proton-proton collisions and have seen that for x/s sufficiently large these quantities are larger in asymptotically free QCD than in the naive QPM. These calculations were based on a simple extension of the Drell-Yan annihilation process to interacting field theories and, although several analyses support eq. (2.3), a more thorough justification is desirable. Throughout this article we have used the Weinberg-Salam model as a guide for m w and mz. But Bjorken [19] has argued that these estimates apply to a wide range of theories, so our curves may be more generally useful. The reader should be aware of the work by Hinchliffe and C.H. Llewellyn Smith [3] which is very similar to ours and a recent thorough set of calculations in the naive QPM by R.F. Peierls, T.L. Trueman and Ling-Lie Wang [4]. The authors thank M. Peskin and S. Rudaz for discussions and help with the derivation of eq. (3.8) from the operator product formalism.

References [1] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, Elementary particle theory, ed. N. Svartholm (Almquist and Wiksell, Stockholm, 1968) p. 367. [2] F.J. Hasert et al., Aachen-Brussels-CERN-Ecole Polytechnique-Milan-Orsay-London Collaboration, Phys. Lett. 46B (1973) 138. [3] I. Hinchliffe and C.H. Llewellyn Smith, Phys. Lett. 66B (1977) 281. [4] S.M. Berman, J.D. Bjorken and J. Kogut, Phys. Rev. D4 (1971) 3388; R.L Jaffe and J.R. Primack, Nucl. Phys. B61 (1973) 317; R.F. Peierls, T.L. Trueman and L.-L. Wang, Brookhaven preprint (March 1977). [5] S.D. Drell and T.-M. Yan, Phys. Rev. Lett. 25 (1970) 316.

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J. Kogut, J. Shigemitsu /Asymptotically-free parton model

[6] H.L. Anderson et al., Phys. Rev. Lett. 38 (1977) 1450. [7] D.J. Gross and F. Wilczek, Phys. Rev. D8 (1973) 3633; D9 (1974) 980; H. Georgi and H.D. Politzer, Phys. Rev. D9 (1974) 416. [8] J.B. Kogut, Phys. Lett. 65B (1976) 377. [9] R.P. Feynman, Photon hadron interactions (Benjamin, New York, 1972). [10] J. Kogut and L. Susskind, Phys. Rev. D9 (1974) 697, 3391. [11] I.G. Halliday, Nucl. Phys. B103 (1976) 343. [12] H.D. Politzer, Harvard preprint (May 1977). [13] K.J. Anderson et al., Phys. Rev. Lett. 37 (1976) 799. [ 14] G. Parisi, Proc. XI Rencontre de Moriond on Weak Interactions, ed. Tran Thanh Van (1976). [15] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298; K.J. Kim and K. Schilcher, Universit~t Mainz preprint (May 1977). [16] V. Barger and R.J.N. Phillips, Nucl. Phys. B73 (1974) 269. [17] A. De R~jula, H. Georgi and H.D. Politzer, Ann. of Phys. 103 (1977) 315. [18] J. Gunion, Univ. of California, Davis, preprint (1976); M. Duong-van, K.V. Vasavada and R. Blankenbecler, SLAC-PUB-1882, (Feb. 1977). [19] J.D. Bjorken, Phys. Rev. D15 (1977) 1330.