The complete computation of high-pT W and Z production in second-order QCD

Nuclear Physics B319 (1989) 37-71 North-Holland, Amsterdam

THE COMPLETE COMPUTATION OF HIGH-p w W AND Z P R O D U C T I O N IN S E C O N D - O R D E R Q C D Peter B. ARNOLD High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA

and Fermi National Accelerator Laboratory, P. 0. Box 500, Batavia, IL 60510, USA

M. HALL RENO Departmento de Fisica, Centro de lnvestigacion y Estudios Avanzados del LP.N., Apartado Postal 14-740, Mexico 07000, D.F.

and Fermi National Aecelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA

Received 28 November 1988

We present a full computation of do/dq2T for W, Z and virtual "y production to second order in QCD. This includes qg, gg, and singlet q7:1+ qq collisions in addition to the usual non-singlet q~ + qq. The primary motivation is the importance of quark-gluon collisions at Tevatron energies. Attention is restricted to high and moderate qv, avoiding the resummation necessary at small transverse momentum. We give analytic results for all constituent cross-sections d o / d q ~ d y and, for W and Z production, numerically convolute them with structure functions.

1. Introduction T h e transverse m o m e n t u m ( q v ) d i s t r i b u t i o n of single W p r o d u c t i o n provides a useful test of the s t a n d a r d m o d e l a n d a means of m e a s u r i n g as [1]. F u r t h e r m o r e , W p r o d u c t i o n is a b a c k g r o u n d to new physics. In this p a p e r , we e x a m i n e the qv d i s t r i b u t i o n s of virtual p h o t o n s a n d weak gauge bosons. W e restrict o u r a t t e n t i o n to large qa- a n d calculate to second o r d e r in Q C D . By including the full a~ c o n t r i b u tion, we r e d u c e the theoretical errors associated with the leading o r d e r ( O ( a s ) ) result. U s i n g these expressions together with p a r t o n d i s t r i b u t i o n functions evolved at t w o - l o o p order, we m a k e p r e d i c t i o n s for the qv d i s t r i b u t i o n s at the SPS a n d T e v a t r o n Colliders. T h e s e c o n d - o r d e r corrections have previously been calculated for the m o s t significant processes at C E R N energies: the non-singlet c o n t r i b u t i o n to qC1 annihilat i o n [2, 3]. A t T e v a t r o n energies, however, q u a r k - g l u o n scattering is equally im0550-3213/89/$03.500Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

38

P.B. Arnold, M. Hall Reno /

Wand Z production

portant*. This process has already been approached for the specific case of qg -o W + 2 jets [4]. The technique is to calculate the amplitude and then perform all of the phase-space integrals numerically. Unfortunately, such computations are valid only when the jets are well separated; singularities appear when the jets become collinear with each other or the beam axis. These singularities are related to the break-down of perturbation theory for the differential cross-section in those cases. For example, when the invariant mass s= of the two jets becomes small, powers of a s are countered by factors of large logarithms, In s2, of the small scale. For small, fixed s 2, the differential cross-section cannot then be calculated perturbatively. If we were really interested in the cross-section as a function of s=, we would have to perform some sort of resummation to proceed. The details of the small s 2 behavior are not necessary if we are interested in the inclusive cross-section for producing a W with any number of jets. Integrating over the phase space of the jets, no small scale remains to give large logarithms and so high-order contributions should be controlled. For two-jet production, this phasespace integral diverges as the jets become collinear. The divergence is cancelled by a similar divergence in the one-loop corrections to single-jet production. One can cancel the divergences only after integrating over the relative phase-space of the two jets. The complexity of the calculation lies in that this integral must therefore be performed analytically rather than numerically. When the W has small transverse momentum qT, there is again a small scale and large logarithms ln(q~-/Q2).Perturbation theory for do/dq~ breaks down and one must either perform a resummation or restrict attention to the total cross-section o by integrating analytically in qT- These techniques have been carried out to first order in QCD [5], and some of the second-order terms have been analyzed in the qT -~ 0 limit [6]. We have attempted neither method with our second-order results and so do not consider very small values of qT. Our purpose in this paper will be to extend the previous work on non-singlet qq + qq scattering to qg, gg, and singlet qq + qq scattering. This will encompass all of the second-order processes. We do not distinguish the polarizations of the W bosons. The first-order diagrams L for W production through qct annihilation are shown in fig. 1"*. Part of the second-order contribution comes from the interference of these diagrams with the one-loop corrections V of fig. 2. The rest comes from two-jet production accompanying the W as in the diagrams G of fig. 3. The diagrams for W production through qg scattering can be obtained from L, V, and G by crossing. Those for production through gg scattering can be similarly obtained from G alone. The diagrams F of fig. 4 give the remaining contributions to qCl

* F o r definiteness, we discuss W production. We loosely use the terms "final state p a r t o n " and "jet" interchangeably. * * The diagrams of figs. 1 to 5 are taken from ref. [2].

P.B. Arnold, M. Hall Reno /

Wand Z production

Illifill

39

tiiiill

L].

L2

Fig. 1. Leading diagrams for q~ ~ g3'*. The looped lines are gluons, the wavy, unlooped lines are photons or W's.

vl

v2

v3

v4

v~

_.iiiii.ii iii..iiii LI,...... i..L ILl........ tY v6

v~

v8

v9

V]o

vzz

Fig. 2. Next-to-leading diagrams for qCt ~ g~,*.

G1

......."

G2

G3

G4

~i IL~

~ i~2 ~

[

LU.*,***,* t, 3 G5

°'~4 G6

G7

G8

Fig. 3. Diagrams for qq --, gg3'*.

scattering. T h e d i a g r a m s H of fig. 5 are for qq scattering. H 5 through H 8 should be i n c l u d e d o n l y if either the initial two quarks or final two quarks are identical. A f t e r n o t a t i o n a l preliminaries in sect. 2, we outline the calculational p r o c e d u r e using d i m e n s i o n a l regularization in sect. 3. A discussion of how to convert from M S f a c t o r i z a t i o n to a n y other scheme is described in sect. 4. In sect. 5, we d e m o n s t r a t e t h a t we c a n sideskip most of the difficulties associated with Y5 in d i m e n s i o n a l

40

P.B. Arnold, M. Hall Reno /

vl

Wand Z production

v2

v3

v,

Z v, Fig. 4. Diagrams for qcl "-* qF::F/'*.

~

:

3

-~/3

~

3

4

-f~/~ 4

~

4

.L

3

~

4

Fig. 5. Diagrams for qq ~ qq-f*.

regularization. Finally, in sect. 6, we discuss numerical techniques and results for W and Z production at the SPS and Tevatron Colliders. The analytic parton level cross-sections for ~,* production with MS factorization appear in appendix A. Recipes for converting these formulae to W or Z production are shown in appendix B.

2. Notation The four-momenta of the colliding hadrons a r e P1 and P2 while those of the colliding partons are Pl and P2. q is the four-momenta of the produced W boson, Z boson, or virtual photon; qT and y are its transverse momentum and rapidity; and

P.B. Arnold, M. HallReno / Wand Z production

41

Q2 = q2. The hadronic and constituent Mandelstam variables are defined by:

S = ( P , + P2) 2,

T = ( P I - q ) 2,

U=(P2-q)2;

(2.1)

s=(pl +p2)2,

t=(pl-q)

u = ( p 2 - q ) 2.

(2.2)

2,

When two-jets accompany the W, their invariant mass s 2 is given by s 2 = s + t + u - Q2. s 2 sometimes appears in distributions as 1/(s2)A+ or defined by

(ln(s2)/s2))A ÷ which are

1

1

foAdse (s2)~-~-+f(s2) = foA dse--(f(Sz)s2

.

(2.3)

(Ins.)

.

-f(O)),

lns.

fo ds2\ s. a+f(Sz)=fods2-~2(f(sz)-f(O))"

(2.4)

(2.5)

A is generally used for the upper limit of s 2 integrations and appears explicitly in some formulas. The group structure of Q C D is handled generically with N c the dimension of a single fermion representation and the Casimir C v defined by F~TaTa= CF. For QCD, these are

Nc = 3

CF_4

(2.6)

T R is half of the number of fermion flavors and so for five flavors is T R = ~.

(2.7)

3. Procedure

We will now briefly review the procedure of the calculation to fix notation and indicate to the reader what is involved. Much of this discussion is taken from Ellis et al., and we refer the reader there or to several reviews for more detail [2, 7]. We want to calculate a quantity d6 that can be convoluted with structure functions to obtain the hadronic cross-section:

dq

dy = ,,, f dx dx L.(Xl,

~'J ~x 2, M ) d -da, ~ u (j .a ~ ( M 2 ) , xlP,, x2P2).

(3.1)

P.B. Arnold, M. Hall Reno /

42

Wand Zproduction

A and B indicate hadron flavors; i and j indicate parton flavors. There are two sources of terms subleading in as: those from subleading terms of d6 and those from subleading terms in the definition and evolution of the structure functions. Both will appear in the bare calculation of parton cross-sections and must be disentangled to yield d6. Step 1. Calculate the bare result s do/dtdu of the Feynman graphs. The loop integrals and the phase space integrals for the particles accompanying the W are done with dimensional regularization. We will denote the first- and second-order contributions by do <1) and do ~2). Step 2. Fix your conventions for defining the relationship between the renormalized and bare structure functions: OLS

(3.2)

In general, R has the form,

Ri,_j(z,M

1 F(l - e) (4rrp,21~ 2)=-TPi,_j(z)~(lz~) M2 ] +Ci,__j(z),

(3.3)

where the O(e) pieces are irrelevant and where the P are the Altarelli-Parisi functions:

Pq~_q(~) = c ;

I+Z 2 (1--Z)~

3 1 +-23(z-1) ,

Pg,__q(Z)=CF[I+(1--Z)Z]/Z, Pg'-g(Z)=2Nc [ ( 1 - z') ~

,

+-+z(1-z)-2z

]

Pq~_g(z)= ½(z2+ (1- z)2).

(3.4)

The distribution 1/(1 - z ) + is defined by

fol d z ( l f(z) -z)+

-= fo1dz

f(z)1 ---f(1) z

(3.5)

The definition of the non-singular part Ci~ j of eq. (3.3) is undetermined, however, and reflects a freedom of choice in the definition of the structure functions. For our analytic calculations, we have chosen the MS factorization

P.B. Arnold, M. Hall Reno / Wand Z production

43

scheme defined by C ~g i~j ---

Step 3.

0

(3.6)

"

Calculate the factorized cross-section d6. It is given at order a2 by:

S d~ij

s doij

dtdu

dtdu Ors ~k foldZiRk~i(Zl, M 2xsd°(1) 2~r ) - - - ~ pa_zlp 18 ( 2 1 ( s + t - Q 2 )

wu)

_ 2x) ~ s doi(~ as ~" f 1 dzzRk~-J(Z2'M 7 - - ) p2~z2P2 8(z2(s+u-Q2)+t) 2~r k ~0

(3.7) The z 1 and z 2 integrals are trivial. Step 4. The two-jet collinear divergences will be associated with the limit s 2 ---+ 0 and will appear as inverse powers such as Sz1-L Such terms will cause divergences when the x a and x 2 integrals are performed. These poles may be made manifest by using the identity:

1

1 ~(s2)[l_elnA + ~e21n2 A]

1

+

(S2)A+

-

(lns2]

$2

] A+ + o ( d ) .

(3.8) The poles will now all cancel explicitly between all collinear divergences, the virtual diagrams, and the factorization pieces. The results of this step are given in appendix A. Step 5. Now that the poles are gone, we have an expression that can be evaluated numerically. We are ready to convolve d6 with the structure functions to compute the hadronic cross-section do/dq 2 dy. For the sake of evaluating distributions such a s 1 / ( s 2 ) A +, it is convenient to rewrite the x 1 and x 2 integrals as [2, 8]

foldx~ fo~dX20(s2)~ f~v¢+e y ( x 1 S

+d xT1-

02) fo AIDS2

dx 2

02) fo

(3.9)

44

P.B. Arnold, M. Hall Reno / Wand Zproduction

where Q2),

(3.10)

A2= Tf~+eY(x2S + T-Q2) + Q2 + x2(U-Q2 ),

(3.11)

A 1 = U+ Xl(S + T-

Tf~+= ~ Q 2 s q~ + ~

(3.12)

.

4. Factorization schemes

For the analytic results displayed in appendix A, we used MS factorization. But one may make contact with any factorization scheme simply by adding the correction terms determined by eq. (3.7): s drij

dtdu

s d r i Ms dtdu 2,~ (s + t-- Q2) ck._, ~ + - t - 0 2, M2 - T i - -

%

.1

.

.

.(

--upl/.+,

e2~

M2 ) sdoi~)p~-

--t

2~r (s+u-Q2) Ck'-j s + u - Q 2'

~

,p#(s+, 02) (4.1)

In particular, for our numeric work we have used the structure functions of Diemoz, Ferroni, Longo and Martinelli [9]. They define the quark structure functions from deep inelastic scattering so that xF2(x,

Q2) = X[fq(X, Q2) +/q(x, Q2)].

(4.2)

This choice determines [10] Cq~q(Z) = CF[(I

+22)(ln(l-z) ) 1-z

j+

3 1 2 (l-z)+

-l -+lzn z2 1-z

+3 + 2 z - (9 + ~r 2) 8(1 - z)],

1--z

Cq,__g(z)=l[(z2+(l-z)2)ln(----~)+6z(1-z)].

(4.3)

P.B. Arnold. M. HallReno / Wand Zproduction

45

For the remaining coefficients, Diemoz et al. take the simplest choice which enforces momentum conservation: Cg~q(Z)

=

- Cq~_q(Z),

Cg~g(z)

=

-4TRCq~g(Z ) .

(4.4)

In the preceding, the subscript q represents a single flavor of quark and the coefficient functions are defined identically for anti-quarks. To integrate eq. (4.1) as in step 5, one may transform the distributions in z to the variable s 2 using the following identities:

z, =- - u / ( s + t - Q2),

(4.5) (4.6)

8 0 - zl) = ( , 2 - u) 8(s2), 1 (1-zl)+

l--Z1

- (s2-u)

+

1 - +6(s2)ln A ], (s2)a+

1

1

(4.7)

l n ( s 2 - u ) + ½ 8 ( s 2 ) ln2 A ] u

$2 ] A +

(4.8) The remaining relations come from taking z 1 ~ z 2 and u ~ t in the above.

5. Regularization of % The implementation of 75 in dimensional regularization is subtle and problematical. The technique we choose is completely straightforward to implement calculationally, but the underlying assumptions should be carefully laid out. We shall make a simple definition following Chanowitz et al. [11] in cases where we have traces involving an even number of 75's. In the cases we have traces involving an odd number, the extension of the Dirac algebra from 4 to d dimensions will be shown to be irrelevant, and we will simply evaluate the traces in 4 dimensions. From Chanowitz et al., we assume the following properties of 7s:

(1) (2)

Ys anticommutes with all y~ in d dimensions,

.y25 ~

1

(3) tr(3,s3,"y"3,°7 ") = 4ie ~°" + O ( d - 4) × ambiguity on the four-dimensional subspace/t, v, p, ~-= 0,1,2, 3,

46

P.B. Arnold, M. Hall Reno /

Wand Z production

i

V -A

V -A

2

Fig. 6. Diagram for IG212.

and we assume (4)

tr('y57"3'~3'°3' ") is anti-symmetric.

The anomaly is intimately related to the fact that the ambiguous term above cannot be explicitly defined. In our calculation, however, this term will never be relevant. Let us now consider squared amplitudes where each fermion loop connects to an even number of W's. For example, consider [G2[2 shown in fig. 6. Each W vertex has vectorial and axial contributions. We first argue that the VA terms of the squared amplitude do not contribute. This is because the Dirac traces will all have one factor of tr(~,sy~,~3'Py" ). Once we have performed the phase-space integrals over P3 and P4, only three vectors remain: Pl, P2, and q. But there is no non-zero contraction of three vectors with the anti-symmetric tr(ysyuy~yP3"). So the VA terms vanish once integrated over the phase space of the final partons. In the limit of massless fermions, the AA terms are identical to the VV terms because one 3'5 can be anticommuted to the other and removed with y~ = 1. We can therefore treat W's as photons if we use the effective coupling constant: g2 _

+

(5.a)

The only interferences that involve fermion loops connected to an odd number of W's and that do not vanish are 2Re(F 5 + F6)*(Fv + F8) and its brothers 2 Re(H 1 + Hz)*(H 3 + H4) and 2 Re(H 5 + H6)*(H 7 + H8). In neither case is there any divergence when we do the momentum integrals; the regularization does not matter and we can do the traces unambiguously in four dimensions. To show this, it is not enough to know that the resulting answer is in fact finite. A trace of order

P.B. Arnold, M. Hall Reno / i

47

Wand Z production 1 '4

(

;

)

;

2

)

2 (b)

(a)

Fig. 7. The two routings of qv for 2Re(F6* Fs).

4 - d could multiply a divergent integral that gave 1 / ( 4 - d). If we took the limit d ~ 4 before doing the integral, we would falsely conclude that the result was zero. We must verify that this does not occur. We use an argument by J.C. Collins [12] to show that the coefficient of possible divergences is exactly zero for any d near 4. Consider 2 Re(F6* Fs). There are two ways to route the large q x of the W as shown in fig. 7. Divergences can arise when the other lines have small transverse momentum and correspond to potential collinear singularities of the gluons. In the first routing, the left gluon can become collinear with P2 and P4- The only unsuppressed gluon polarization e - is then parallel to its m o m e n t u m k. In this limit, the gluon's upper vertex may be rewritten as e - . F = k . I ' / [ k [ . k . F is in a form where we can apply a Ward identity; it must cancel as shown in fig. 8 [13]. Therefore, our potential divergence must cancel against the similar one of 2Re(Fs*F8) shown in fig. 9. In a similar way, all divergences of 2 Re(F 5 + F 6 ) * ( F 7 + FS) must pair up and cancel. This argument does not rely on d being exactly 4 and is independent of the nature of the W vertex and the precise definition of 3'5To conclude this section, we note that it would be nice to check the entire calculation by repeating it with an explicit definition of 3'5 in d dimensions such as iY0Y1Y2Y3 [14]. Such a definition avoids the ambiguity of tr(3'53'~3'"3'°3'~ ) but no longer anticommutes with all of the 3'~. The calculation becomes more difficult computationally, and we have not attempted it.

K/~

+ /.L

= 0 /.L

Fig. 8. Application of the Ward identity.

48

P.B. Arnold, M. Hall Reno /

Wand Z production

1

(_

;

_)

(

-

-)

Fig. 9. The divergence of 2 Re(Fff Is) that cancels the first one of fig. 7.

6. Numerical techniques and results The d x ds 2 integrands of eq. (3.9) are in fact quite peaked at low values of x and s 2. To numerically integrate in a reasonable amount of time, it behooves us to transform to new variables that smooth out the integrand. The peak in s 2 occurs around s 2 - q2 and the peak in the outer x integration around ~ / S above its lower limit Xmi. = e -+Yr~+. The integrand is smoothed out by changing variables to ~ = l n ( ( x - Xmi,)+ e f ~ S - ) ,

(6.1)

~ = ln(s 2 + eq~-).

(6.2)

By trial and error, e = 0.1 gives good convergence. We will be working with five flavors and ignore the top quark. Correspondingly, we evolve a s ( M 2) using Nf = 5 in d°/s

d ( l n M 2)

/30

2

4~-as

~1

3

(6.3)

16~r 2a~'

where /30 = 11 - ( 2 / 3 ) N f ,

/3a = 102 - (38/3)N t .

(6.4)

F o r AQCD, calculated with five flavors, we will consider 100, 175, and 250 MeV,

which match to four-flavor values of 160, 260, and 360 MeV at the b threshold. When not otherwise indicated, we will in particular use the middle value of A (4 flavors) = 260 MeV. We have chosen the renormalization scale ~2 of the constituent cross-section to be the same as the factorization scale M 2 of the structure functions. We will examine M 2 at both of the physical scales of the problem: Q2 and q~-. The sensitivity to this choice will help give us an error estimate for the calculation.

P.B. Arnold, M. Hall Reno / 10 3

'

I

L

I

I

I 60

'

I

'

I

W a n d Z production '

I

'

I

'

I

49 i

I

L

I 180

t

i0 4

'>. i0

s

"xJ

i0 -~ ~J

10 7 b

i 0 -8

i0 -9

2O

I

40

t

I 80

I

I I I 100 120 qT [GeV]

I

i 140

,

1 160

I

200

Fig. 10. W (solid) and Z (dashed) production at ~ = 1.8 TeV. In each case, the theoretical error estimate is delineated by the two lines.

Finally, our particular choices of weak parameters are: M w = 81.8 GeV, sin 2 0w = 1 -

2 2 Mw/Mz,

M z = 92.6 GeV

(6.5) (6.6)

sin 0c = 0.219.

The results of all this machinery are shown in fig. 10 for W and Z production at the Tevatron. We have normalized the differential cross-sections with respect to the total cross-sections. The analytic form of the total cross-sections are known to first non-leading order [15], and we have convolved with the Diemoz et al. structure functions that we use for the rest of our computation*. For the factorization scale M 2 = Q2 and A(4 flavors) = 260 MeV, these cross-sections are: o(W+)=9.68nb,

o(Z)=6.05nb

for~=l.8TeV,

M 2=Q2,

(6.7)

o(W +)=2.80nb,

a(Z)=l.78nb

for~=630GeV,

M 2=Q2.

(6.8)

For contact with previous work, we show the predictions along with experimental data [16] for SPS energies in fig. 11. But the data is not yet very sensitive to the size * Consistent with our coupling constant definitions in appendix B, we have used weak coupling of a w = ~/2 G F M 2 / r r . Note that the exact choice of normalization cancels in the ratio ( d o / d q T ) / ( q T / 0 ) "

P.B. Arnold, M. Hall Reno /

50 10 3

'

[

10 -4 '>

'

[

'

I

'

@

Wand Z production I

'

r

o

UA1

t

i

'

[

10 -~

3 10 -6

i

b q:J 10 -v

10 -8

10 -°

10,0

~ 0

I ~0

,

I 40

,

I 60

i

I i 80 qr [GeV]

I 100

i

I 120

,

I 140

, 160

Fig. 11. P r e d i c t i o n s a n d e x p e r i m e n t for W p r o d u c t i o n at V~ = 630 G e V . T h e t w o lines d e l i n e a t e the e s t i m a t e d e r r o r o f the t h e o r e t i c a l c a l c u l a t i o n .

of the second-order contributions. Indeed, on the large logarithmic scale of these graphs, little difference between the first and second-order calculations could be seen, and so these graphs are not very informative about our calculation. We have instead tabulated our results in tables 1 and 2. Results are given for the choices M 2 = Q2 and M 2 = q~ of factorization scale. In the latter case, we have evaluated the total cross-section at M 2 =
o(Z)=6.13nb,

for ~- = 1.8 TeV,

M 2 = (20 GeV) 2 ,

(6 9) a(W +)=3.80nb,

o(Z)=2.56nb,

for v~ = 630 GeV,

M 2 = (10 GeV) 2.

(6.10) Our results are given for qx down to 20 GeV. This is roughly where the resummation becomes important as resummed first-order results become first-order perturbative results [17]. To show the second-order results visually, fig. 12 displays the K factor: of the total of first and second order contributions to the first alone. convenient to plot, the K factor is unphysical and sensitive to the choice

need for equal to the ratio Though of scale.

P.B. Arnold, M. Hall Reno /

51

W a n d Z production

TABLE 1 in GeV 2 as a function of qT in GeV for ~/~ = 1.8 TeV and A(4 flavors) = 260 MeV. Values for M 2 = Q2 a n d m 2 = q2 are given with the total cross-section o evaluated at M 2 = Q2 a n d M 2 = ( q 2 ) respectively (do/dqT)/(qTo)

W production M z = Q2

M 2 = q2

M s = Q2

20 40 60 80 100 120 140 160 180 200

4.85×10 4 5.02×10 5 1.00×10 s 2.62×10 6 8.10×I0 7 2.81×I0 7 1.06×10 v 4 . 3 0 × 1 0 -8 1.84 x 10 8 8.21×I0 9

5.70×10 4 5.56×10 s 1.06×10 5 2.67×10 6 7.98×10 7 2.70×10 7 1.00×10 7 4.01×10 8 1.69 × 10 -8 7.49×10 9

5.03×10 4 5.42×10 5 1.13×10 5 3.i0×10 6 9.99×10 7 3.60×10 7 1.41×10 7 5.92×10 8 2.61 × 10 -~ 1.21×10 8

Here, our convention m2=

is t o n o r m a l i z e

Q2. In this graph,

M 2 ___ Q 2 a n d results

vs.

contribution

one

can

M

more for

deeply the dependence a fixed

choice

of

between

4 5 5 6 7 7 7 ~ ~ ~

is c l e a r

are two scales here which are sometimes

the literature:

the scale where

the second-order

the choice

at

of scale

on the choice of scale by plotting

q-r- I t

of scale. There

of these scales disappear.

5.80×10 5.91×10 1.18×10 3.12×10 9.75×10 3.44×10 1.33×10 5.49×10 2.40 × 10 1.10×10

cross-section.

helps stabilize the first-order one, making

where the total turns over. Looking

M 2 = q2

with respect to the first-order cross-section

see the difference

M 2 = q2 for the full second-order

F i g . 13 e x p l o r e s the

Z production

qT

that

the

second-order

it l e s s s e n s i t i v e t o t h e c h o i c e attached

correction

favored

status in

is z e r o a n d

the scale

a t a s m a l l e r v a l u e o f q-r i n fig. 14, h o w e v e r , b o t h

Numerically,

the contribution

o f g g --+ q ~ W

is w h a t k e e p s

TABLE 2 Same as table 1 but for ~ - = 630 GeV W production

Z production

qT

M 2 = Q2

M 2 = q~

M 2 = Q2

20 40 60 80 IO0 120 140 160

2.65 x lO 4 1.64×10 5 1.93×10 6 2.91×10 7 4 . 8 8 × 1 0 .8 8.61×10 9 1.53×10 9 2 . 6 2 × i 0 lO

2.48 × 10 -4 1.36×10 5 1.51x10 6 2 . 1 5 × 1 0 -7 3.44×10 8 5 . 8 9 × 1 0 -9 1.01×10 9 1.67×10-1°

2.88 x 10 4 1.93×10 5 2 . 4 6 × 1 0 .6 4.02×10 7 7.39×10 8 1.43×10 8 2.77×10 9 5 . 2 0 × 1 0 lo

M 2 = q2 2.56 x 10 1.54×10 1.85×10 2.88×10 5.06×10 9.56×10 1.82×109 3.36×10

4 5 6 7 8 9 io

P.B. Arnold, M. Hall Reno /

52 2

'

'

'

I

'

'

'

'

Wand Z production

[

'

'

'

'

r

. . . .

1.8

1.6

1.4

1.2

............

1

"_'_-_-~ - -

M2=Q~ .8

................................. ............

WI2=qT 2 .6 i

i

i

I

i

l

i

i

50

P

i

~

~

100

i

l

r

I

i

J

150

200

qv [GeV] Fig. 12. d a ( M 2 = O 2 ) / d o ( 1 ) ( M 2 = Q2) and d o ( M 2 = q~)/dotl)(M 2 = Q2) where d o m e a n s d o / d ( q 2) a n d d o O) is the first order c o n t r i b u t i o n alone. Q u a n t i t i e s are for W + p r o d u c t i o n at V~ = 1.8 TeV. The dashed line is d o O ) ( M 2 = q2)/do(1)(M2 = Q2).

I0

3

~

Ist

t"-,

7

o

b

I

llJll

I

10

I

,

,

~Jlli

30

I

100

I

300

~

,

Ill

000

M [GeV] Fig. 13. The d e p e n d e n c e of d o / d ( q ~ ) on the factorization scale M. This is for W + p r o d u c t i o n at V/~ = 1.8 TeV a n d qv = 100 GeV. Separate lines are given for A(4 flavors) = 160, 260, 360 MeV. Both 1st and 1st + 2nd o r d e r results are shown.

P.B. Arnold, M. Hall Reno / '

'''"1

'

W and Z production

]

......

I

I

, , Illll

'

53 I

.....

I

I I Ill

3 >

b

1

A4-360

&=260 A4=160

0

I I rllll

I 10

30

M [GeV]

,

100

300

1000

Fig. 14. Same as fig. 13 b u t qT = 20 GeV.

the second-order correction positive here at small M. In our opinion, the most straightforward thing to do is focus on the physical scales q2 and QR and use the discrepancy between them to estimate the error. Note that figs. 13 and 14 also show the sensitivity of the results to AQCD. There are four sources of theoretical error: the choice of renormalization scale that minimizes higher-order corrections, the value of A QCD, the parametrization of the structure functions, and the growing need for resummation at small qT- We have addressed the first two by examining the variation when A(4 flavors) = 160, 260, and 360 and M 2 = Q2, q2. For the Tevatron, the variation at the low qT and high qT ends of our results is roughly _+10% relative to the mean of the m 2 = Qz and M 2 = q~ values listed in table 1. We have taken this error for the entire range, giving the band drawn in fig. 10. This is not an extremely conservative estimate as only some sources of error can be addressed. However, it would be surprising if the true error were more than double this estimate. For the SPS, we find a variation of roughly _+10% at the low qT end and _+35% at the high qT end. We have extrapolated for intermediate qT to get the band of fig. 11. It is worth noting that taking the ratio of d o / d q 2 with o improved the error for low qT and aggravated it for high qT: the variations in d o / d q 2 alone were roughly _+15% and _+20% respectively. For the Tevatron, taking the ratio changed the variation only slightly. Finally, to give the reader a feel for the numerical contributions of different processes, we show their relative contribution at M 2 = Q2 in figs. 15 and 16. For an independent study of the subject of this paper, we refer the reader to the work of Gonsalves, et al. [18].

54

P.B. A r n o ~ M . H a H R e n o / W a n d Z p r o d u c t i o n .6

I l i l

.4

A ~

I

. . . .

i

i

i

i

i

I

i

i

1

~

~

.3

b

B

"g"

C

.2

.

•

.................................. .1

~

i

i

i

0

I

i

i

i

i

50

I

J

J

~:'22 i

i

I

i

i

i

g i

I

150

100

qT [GeV]

I

200

Fig. 15. Relative contributions of different processes to W production at ~ - = 1.8 TeV using M 2 = O 2. First-order contributions are (A) qcl "" gW and (B) qg --+ qW. Second-order contributions are from (C) [qgl '" gW + ggWl + IF~ + F212, (D) qg --, qW + qgW and gig + CtW + gtgW, (E) gg --, qClW, (F) remaining qgt --+ qgtW , and (G) qq ---, qqW and C]gt --, giftw .

1 .8 .6 A , v

b

%" "O

F

.4 s

.2

-.2 ,

-.4

0

,

,

,

I

50

,

,

,

,

I

,

L

I00

,

,

I

,

150

qT [GeV] Fig. 16. Same as fig. 15 but at ~ = 630 GeV.

,

,

I

200

,

P.B. Arnold, M. Hall Reno /

W and Z production

55

We are greatly indebted to Keith Ellis for his advice and help in this project. Many analytic results for generic integrals were obtained from him and checked by us numerically. We are indebted to R.J. Gonsalves, J. Pawlowski and C.-F. Wai for sharing their results before publication and for pointing out a sign error in our original version of eq. (A.2). We would also like to thank John Collins for useful discussions and for the argument concerning 7s explained in the text. We thank J. Stirling for pointing out some typographic errors.

Appendix A

In this appendix, we display the analytical results for s d o / d t du for the various processes of interest. MS factorization has been used with dimensional regularization in 4 - 2 e dimensions. For notational simplicity, the results are given for massive photon production rather than directly for W and Z production, ef is the appropriate fermion charge of + 32 or - ½. ef2 is a slightly loose notation for the product of the two fermion charges relevant to the diagram at hand. With the exception of 2(F5 + F6)*(F 7 + 1=8), all of the diagrams calculated give identical results for axial and vectorial photons. For the exception, the different results are labeled by the superscripts (A) and (V). Not all interference terms are listed. Having performed the phase-space integral on the final particles accompanying the photon, there are some simple relations. Interchanging u *-* t in the results will take F> F3, Fs, F6 ~ F2, F4, F7, Fa

Ha, H2, H5, H6 ~ H3, H4, H7, Hs.

(A.1)

Recall that diagrams H 5 through H 8 should be included only if either the two initial quarks or the two final quarks are identical. For non-identical final quarks, some of the H's are related to the F's by: IHx + H212 = IH5 + H6I 2 = 11=5+ F6I 2 , 2(H{ v) + H(2v))*(H~v) + H(4v)) = - 2(Fs~V~+ F6(V))*Cv7(v) + F~V)),

2(H{A) + H~A))* (H~A) + H~4A)) = + 2(Fs(A) + F(A) )* (FT(A) + F(A) ) .

(A.2)

For identical final quarks, the H combinations will be smaller than the F combinations by the symmetry factor 1. Finally, we have the relation 2(F~V) + F2~V))*(F~V)+ F~V)) = 0. 2(F 1 + F2)*(F3 + F4) is zero for vectorial photons, W bosons, and Z bosons. The squared diagram has a loop of the final fermions connected to three gauge bosons: two gluons and the photon. Any vectorial contribution will vanish by Furry's

56

P.B. Arnold, M. Hall Reno /

W a n d Zproduction

theorem. For W's and Z's, the axial contribution is proportional to tr r = 0 and also vanishes. Strictly, the axial coupling of the Z only vanishes if weak doublets are complete, i.e. we are not between the t and b thresholds. The extra processes which contribute between such thresholds are analyzed in refs. [18, 19]. We find their contribution to the total cross-section is numerically small and less than 1% at Tevatron or SPS energies. We will now present the results associated with the various terms of the squared diagrams. Many of the results for the F and H diagrams have already been presented by Ellis et al. Unfortunately, though they give exactly the combinations relevant to photon production, their expressions sum too many terms to be applied to W production. (See appendix B.) We have recalculated these processes in the relevant combinations. We will make use of the following definitions:

•"~--2rraNcCFF(1---7) 5{" Kqq = N g '

--Q-T-]

~7-

'

)g Kqg- Nc(2NcCF

)g ) ,

K g g - ,2,T t i V c CF,2 ) , (A.4)

X2 = (u + t) 2 - 4s2Q 2,

[ (u t

T0(Q 2, u, t) = (1 - e) t +

(A.5)

+

ut

]

- 2e .

(A.6)

T0(Q 2, u, t) is proportional to the squared matrix element for the lowest order process for qCt --+ g3'*.

A.1. THE DIAGRAMS IF21Li + Yq,~l~l2 For q~ scattering, the result is taken directly from Ellis et al. [2]. The corresponding result for qg can be obtained by analytic continuation. The continuation is not completely trivial as one must take care with the branches of the logarithms that arise in the second-order cross-term 2 Re[(E/2=xLi)*02}l I Vi) ]. These logarithms yield terms like iTr when continued. If the result were calculated in euclidean space, where it is real, and continued to Minkowski space, unsquared i~r's would be dropped when we took the real part at the end. Keeping this in mind, it is possible to reconstruct the original and analytic euclidean expression. It is then safe to continue euclidean s to euclidean u and vice versa. Then we continue back to Minkowski space and finish by taking the real part.

~

V

I

I

J

÷

Jr

÷

~

Jr

I

J

~1 ~

I

~qw--m--.--I

~

P

÷

fd.~

÷

~

÷

c~ to

~1 ~

÷

÷

Jr

÷

t~

~1 ~

~

÷

i

4-

I

i

I

~1 ~

I

Oo

I

t~

~ I~

I

÷

I

÷

×

I

÷

t~

II

~

I

?

cJ

II

It

~J

~

o.~

~

P.B. Arnold, M. Hall Reno /

58

Rs,=Rl(s,t),

u Ru,= In ~-7

Wand Z production

Rs~=Rl(s,u ),

ln@22

'/7. 2

2 -

Rt~=R2(t,u),

(A.10)

Rz(t' u),

(A.11)

k..= G(s, u),

(A.12) '

R,,

= In

In ~-~

Rl(S, t)

= In

In

o2 q- ~-

(A.13)

RI(S , t),

+ ~ In2 02

51n2t-~--

} +Li2

-Li 2 ~

R2(t,u)=±ln2[ Q2--t 2 1 Q2

+ ½In2

+ Li 2

(A.14)

'

(Q2)

+Li2 ~

"

(A.15)

Li 2 is the dilogarithm function:

xdz Li2(x) = - fJ0 ~ - ln(1 - z).

(A.16)

The ultraviolet renormalization has been performed using MS subtraction.

A.2. T H E D I A G R A M S IE~=IG, I2 + IF1 + F2l 2 F O R qYq---, y*

For completeness, we reproduce this result from Ellis et al. [2]. These are the diagrams which, for qTq scattering, cancel the divergences of the previous one-loop processes. The contribution of IF1 + F2[ 2 may be isolated by considering only the

~"

~1~

~

I

~

i

+

~

~

+

~

~

+

+

, ~

+

+

+

i

~

+

i

~

+

r' ~ ' - ~

o:~

+

r~

i

,

+

4-

q-.

t o l b o

' b..,~ I i ~ '

4-

~1 ~

~:

o-, I ,~ .q-.

4-

i

+

4-

I

t._.. I

©

4-

+

i

i

5"

4-

..._,..,,

~1 ~

I K:b

+

r,,~

i

~

I

~

~

X

)J

S

~

~

+

~

4-

~1~

~

+

~

I

+

+

b3

+

~1~

I

0

5"

o

+

o

+

I

X

©

4-

I

I

~1 ~

7L

÷

i

i

t~

÷

t~

i

Jr

i

Jr

L

~J

I

-F

÷

I

Jr

÷

I

Jr

~

~

I

÷

J

J

÷

I

I

I

I

~1 ~

I

~

÷

÷

I

~ ÷

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I

H

#

+

I

i

t~

I

~1 ~

I

I

~1 ~

-F

i

I

~1 ~

I

~1 ~

t ~ i~

+

÷

I

I

I

÷

I

×

+

I

o~

I

~-_.m__-J

H

÷

H

cJ~

o

M

~

+

+

÷

~

÷

i

Jr

~

i

i

÷

~

÷

Jr

×

i

t

~

~

~

t

j

÷

i

~

~

~l

÷

~

I~

~

÷

Jr

'tA~

i

+

~J

I

t~

JF

i

Jr

fm~

Jr

t~

+

×

~

~

II

I

~

+

÷

~L ÷

I

t~

i

t~

÷

~ ÷

~

t~

t~

~w~

I

i

i

÷

I

~

t~

t~

I

÷

i~..

l

m

÷ r

I l~

t

i

÷

i

~

t~

I

I

÷

~

+

t

t

~

I

÷

~

L

i

m

~1

t~

Jr

÷

I

~

~

1

I

I

I

+

+

+

i

+

+

I

~

I

i

+

I

+

I

bJ

+

t~

+

~

I

+

t~

+

I

+

+

~

I

~,

+

+

~

~

+

~

+

+

nL

+

e,,

~

~

+

~

+

~

+

~

+

~

I

~

~

~

~

I

I

+

+

~

+

I

+

i

i

+

~ l ~~

~.

~+

I

~

~-.

I

"~ +

+

+

I

+ ~

I

~

+

+

=

.2. +

I

bJ

I

×

~

+

I

+

+

I

N

I

I

+

+

I

+

I

+

I

+

+

~

I

~1 ~

+

I

L

÷~.

+

~+

LL

+

"-~l ~

I

~

r~

÷

~

%

+

b~

I

÷

~

+ ~

I

+

t~

b~

I

I

+

I

+

J

r

I

+

t

I

_~_

i

I

+

~+

+

r~'~l

o

>

&

P.B, Arnold, M. Hall Reno /

64

+2CF ~

W a n d Z production

(,2-u)2

s2Q 8s 2

4(s 2 -

m

+Nc (s 2 - t) 2

1 1 (4ut+s2s+(4s_3s2)Q2)] , s(,+O~-,~) ~

u)

+

s(s 2 - t)

Nc

u t - s 2 Q 2 (s2_-u) 2 +

[3s2Qe(u_t)2(4(s+Q2) l )

x~(,+ o~_,~) ~

v

+(3s2+4Q2)(u-t)2-4s2Q2(s+Q 2)

+-, +ef- 7-2~{(u~t)}.

A.5. THE DIAGRAMS 2(F1 + F2)*(Fs + Fs)

2

sdo

__=e2Kqq

dtdu

1

°~s ( C F _ 7 N c ) _ _ I

s

2rr

ut

s 2 + 3uQ 2 × 2u

,

2

(

,~(s2-t)

[2s( s 2)

u+t s+ Q2-

s2t

)

2Q2-t+~2(s-Q2)

+~(u2+ut+st)

+

(s + u) 2 + - - + ,(,2-,) 2 s2 s2s 2

+

s2

V

(S 2 -- U) 2

4S2U 2

S2

S2~k2

1+12---~- ( u t - s 2 Q 2 ) + $2

1

+77

u2(s2 + Q2) _ 2tts2Q2

4-

~--(5tt + 17t)

+3stu + ( s / X 2)(u + t ) ( u ( u 2 - t 2) - s ( 3 t 2 - 2s2Q2))

2 In

+~(u+t)(uZ-t2)

s+

Q2

-s~-X

u2+2ss+u, in (o2 ---s2t,2)} s2

The coefficient of here.

]//S 2

.

st 2

is zero when s 2 --, 0; so 1/(s2)A+ and

1 / S 2 are

identical

P.B. Arnold, M. Hall Reno / W and Z production

65

A.6. T H E D I A G R A M S 2(F3 + F4)*(F5 + F6)

s do _ e 2 Kqq 1 a~ (C s 2~r ~ F 7Nc)

dtdu

48 ss2Q2(ut - s2Q2) 2 _ 2 [ Q 2 ( s 2 - u ) 2 + 3s(t-2Q2) 2] t ( s + Q 2 - s2) ~z t(s+@-,2) + V 1

1

+-(s2(s2+Q4)--l-7sQ2(sq-Q2)-s3-Q6)+

(Q2+s2-2s2u)

st

2 ( Q 2 _ 2s2Q2 s2(Q 2- u) + 2t - 5s - 8Q 2 + -- u + 2t + 2S2 + - +

st

+

(s 2 + t) 2 - 4Q2(s + t) S+ Q2- s 2

1[

2 4 ss2Q 2

q - ~ [ -- X4

u2

+ __(Q2_

--(u,-,2Q t

t

(s + Q2)2-s2 l n ( Q : ( s 2 - t ) 2 ) ;(;+ ~ ,-3 ,,2 2 (s-QZ(u Q2 2) + v s - ~i(, + ,2)

s) - u(3s + 3s 2 - Q2) _ s 2 ( t 2 _ 2s2t_ 6sQ2 + 2Q4)

t

+

S2 -- t

)

t "

(2Q2t + s + Q2 - s 2

s2(4s-s 2 - t)) + (s 2- u) 2 )]

+ 3(Sz- t)2 + 2sZ + 2 s t - 4 t 2 In (s2 -- t)2 + s2 In s2(2Q2

-2st(s+Q2-s2)

( s+Qz-s2+x s+

Q2

- s2-)t

- u) - Q2t t si J"

A.7. THE DIAGRAMS ]F3 + F4]2 We have corrected two typographic mistakes in the result appearing in Ellis et al. The published version contained a sign mistake on the 2t term of the second line

P.B. Arnold, M. Hall Reno /

66

W and Z production

and a mistake in the overall group factor of C F which should instead be ½.

-sdo -=y' dtdu

f

e 2 K q q ~..._2_ s X -s 2rr 2

Q2 -s2)

-2s+

(t-u)+4u-2t

+

l[u

+

3u~(u-,)[ 2,-.-,

l (s+Q2-s2+~)[ + ~-ln Q2 S +

+

S 2 --

In

Q2 S2

4s

+~s(2S+2s2+t-u)

1 Q2

S +

( 25u- 2+ 7 u + ~ S2

Q2 )k

S -t-

s)

l+-(3s+u-~s ;)]

S

S

3u - t - 7 - + - - - 2s -- S 2

S

+ U(2s2t-ut-2s2+4us2-3u2+2ss2-us

+V

s

In

s+

Q2

- s2-)t

~7~<_7~ (s-O~)+-V-(.-')(.+'-2s~)

X

z l + E e Kqq a~ f

s

{(u~t)}.

27r 2

A.8. THE DIAGRAMS IF5 + F612

sdo dt du

= e2Kqq

21[(s2-u)(s2e~s2

s 2~r 2

1 s

2 t2

2s2a - 2 s 2 + 1 t2 t

( ( 3 + In

M2QZ(s2.,~,~

- 1)2- 4 s2u _ 27~s [2_~ s2 -~-

t 2

t)z

+1)

2 - - + 1 t

))is(s~ ,)2 F(s~--~ s,+s22](s2) +

-

+ 2

In

[ s(u2 - t2) +--2u+3s] In ( s + Q022 - s 2 + X + ( . - , ) ( s + s ~ t)t3 tX t~2 S"}--$2--~ (? ) ,-u(s~-, ) ~ 9u+7s-5, + 4 s2 -1 +6 1 +-+ t2 st st t s u+ t _s } + t(s2-- t) (s2 t) ~ "

+

]

s+,~]

-

S -/-

- - 1 s

2(s+

e ~)

P.B. Arnold, M. Hall Reno / Wand Z production

67

A.9. THE DIAGRAMS 2(F~v~ + F6(v))*(F}v) + F~(v))

s do

e Kq

dtdu

s

+

u+t

s(s~ - ut)

(3s + 2Sz)(U + t)

-X

2 ut

~+ Q2 , 2 -

(s+s2-Q2)(u-t)

2ut

2 ut(Sz-U)(Sz-t)

2~r

1

×ln

+

a s2 1

2utX 2

s-Q 2 +2

-

s(u+t)(u-t)

s + Q 2 _ s2

ut(Q2

In

t)

(,2-- 7)(-;~-- 0

[s2(2Q2-u)-Q2t]

-~2~s-2 - t--)5

In

(2Q2-t)[t2+2(s~+s2-s2t)]

( Q2(s2-t)2)}

u~Q -2~ ~) (~ ~ F 2~ 2 )

--

2

2utX2

.)(Q2-

(s~ - @ ) 2 + (s~ - t) 2 + ,22 + ~

2

In

---st 2

2 1 +e~Kqq as s 2~r 2 {(u'~, t ) } . A.10. THE DIAGRAMS 2(F5(A) + F6(AJ)*(F}A) + F8(A))

sdo

2 1[ s / 2~r 2 ut

1 2s (u- 2s2)(u+ + t t(s2-u ) tX 2

d t du = e2 Kqq_ °ts

s

s + s 2 In (

+-~1

stu

3S + 2S 2 - - +

2 ( S - - U)

t

S+ Q 2 - s 2

2 t(s+@-s2)

t)

[ 12(Q 2 - t) + /

S(U2-- I 2 ) tX 2 u( u -- 2s2)

s+t-s

uZ + tZ + 4s( u + t ) +utt Q - - 2 s + e 2 Kqq ots2 1 s 2~r 2 { ( u o

s+ Qa- s2

t)}.

] (s+2 s2+)

1 In

s+

Q2

s2

-

X

ln u [ s 2 ( 2 Q Z - u ) - Q Z t ]

2

]( 21u s2 2 ),n stu

68

P.B. Arnold, M. Hall Reno /

A.11. THE DIAGRAMS 2(H1 + H2)*(H5 +

2

sdo d t du

Kqq a s ( C F e~

×

-

s

27r

1

Wand Z production

H6)

-4

-~Nc)--

-~-ln

t

ts22)2+S2nS2t22u)2,}

+

--

× (factor of 21 if identical final quarks }.

A.12. THE DIAGRAMS 2(H1 + H2)*(H7 + H8)

sdo

2 2 =ef2Kqq ots ( C F _ ½ N c ) -

dt du

s

2rr

stu

u2 + t 2 x

2s2Q 2 ut

× (factor of ½ if identical final quarks }.

Appendix B In this appendix, we show explicitly how to adapt the formula of appendix A to W and Z production. One must (1) include the flavor contractions and Cabbibo angles appropriate to each result; (2) use the appropriate weak coupling constants; and (3) sum over final flavors. The results are given below. One should use the formula of appendix A ignoring et2 and ~f e~ since charges and final flavor sums will be displayed explicitly below when relevant. For the proper overall normalization, a must be replaced by a w / 4 which we take in terms of G v as

a ~

a,v

v~GFM 2

4

4rr

(B.1)

Also, ignore any symmetry factors of ½ mentioned in the formula of appendix A as

P.B. Arnold, 114.Hall Reno /

Wand Zproduction

69

these are also displayed explicitly. ud---, W+:

Oij[(q~ ---, g'g*) + (qcl ---' ggY*) + 2TRIF1 + F212]

+

+ D,,(u'-" ,)]

+ Oij[2(F 1 + F2)*(F 5 + F6) + (u ~ t)], u u "-'+ W + :

(trD) 8ijlF3 + F412 + U,,IF5 + F612 + 8iY, i2(F3 + F4)*(F 5 + F6),

UU ~

[U.IFs + F6I 2 + Ujj(u ~

W+:

t)] + 8,jUii2(H , + Hz)*(H 7 + H8),

ud ~ W+:

UiilF5 + F6I 2 + ½Oi)2(H 1 + H2)*(H 5 + n 6 ) ,

ug ~ W+:

U,i [(qg ~ qy*) + (qg ~ qgy*)],

gg --+ W+:

(tr D)(gg ---, qCq'/*),

more:

u~d,d~fi,O~O

T,

and

UoD

in all the above

(B.2)

i and j are family indices of the colliding partons and are not to be implicitly summed above. The matrices O~j = IK~jl 2 ,

Dij = Re( K t K )ij,

Uij = Re( KK*)ij

(B.3)

contain the mixing angles. K is the KM matrix projected onto those particles for which we are above threshold. (In particular, KK t is not necessarily the unit matrix.) The symmetry which allows us to take u ~ d as directed is a combination of CP, isospin, and a 180 degree rotation of the plane of the W and the beam (which is equivalent to P in that plane). In our numerical work, we have included the flavors through b and ignored the t quark. We have also ignored terms of order (b content of proton) X (b mixing angles) and so have approximated COS 2 0 c

O=

sin2 0c

sin2 0c

COS 2 0 c

0

0

0 0 , 0

D--U=

1

0

0 0

1 0

0 0 0

(B.4)

P.B. Arnold, M. Hall Reno / Wand Z production

70

F o r Z production, the combinations are:

u d --~ Z :

~2d '

ufi ~ Z:

8ijgZ[(qFq---, 83'*) + (qYq--* gg3'*) + 2TRIF 1 + F212] +3ijg2u[2(F1 + F 2 + F 3 + F4)*(F 5 + F6) + (u ~-~ t)] +Sij(Nug 2 + NdgZ)lF3 + F412 +5~¢u+ ,

ud ~ Z:

~ud '

uu -+ Z:

1 g3ijg u2 [2(H1 + H2)*(H5 + H6 + H7 + H81 + (u ~_, t)] +~¢uu ,

ug~Z'

gu2 [ ( q g ~ q 3 ' * ) + ( q g ~ q g 3 ' * ) ] ,

gg --~ Z:

(Nug2u + Ndg2)(gg ~ q~3'*),

more:

u~dabove,

more:

u ~ ~ and d ~ d above,

(B.5)

where d~

= [ga2lF5 + F6I 2 + g2(u ~ t)] -+~'~v'"¢v)2/Ecv'~ 5b , s + F6~v')*(F~ v ) + F8~v') +"(A)~'(A)2[F(A)6a 6~ ~, 5 + F(A))*(Fv(a) + Vs(A))

(B.6)

N u and N d are the number of up and down flavors for which we are above threshold. The effective charges are

g,V) = ( M 2 / 4 M 2 ) l/2( r3 _ 4Q sin 2 0 w ) , g(A) = ( M 2 / 4 M 2 ) i / 2 r 3 '

g2=[g(V)]2+[g(a)12.

(B.7)

P.B. Arnold, M. Hall Reno /

Wand Z production

71

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[15]

[16] [17] [18] [19]

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