The contribution of the gluon-gluon subprocess to the Drell-Yan K-factor

Nuclear Physics B (Proc. Suppl.) 23B (1991) 3-8 North-Holland

3

THE CONTRIBUTION OF THE GLUON-GLUON SUBPROCESS TO THE DRELL-YAN K-FACTOR T. MATSUURA II

Institut

f u r Theoretische Physik,

U n i v e r s i t ~ t Hamburg

R. HAMBERG and W.L. VAN NEERVEN

Instituut-Lorentz,

U n i v e r s i t y of Leiden

ABSTRACT "

In t h i s t a l k we present the c a l c u l a t i o n of the gluon-gluon c o n t r i b u t i o n to the D r e l l - Y a n K - f a c t o r . The size of the cross section f o r g+g ÷ V+'X' i s compared w i t h those coming from the t h r e e o t h e r subprocesses, i . e . q+q + V + ' X ' , q+g + V+'X' and q+q ÷ V+'X' Furthermore, the dependence of the K - f a c t o r on the f a c t o r i z a t i o n and r e n o r m a l i z a t i o n scale i s analyzed f o r the c u r r e n t and f u t u r e colliders. This work ( I ) forms a p a r t of the ongoing

HI+H2 * V+'X' L~+

program to c a l c u l a t e the D r e l l - Y a n (DY) K - f a c t o r up to o r d e r 2 . Higher order c o r r e c t i o n s t o s K - f a c t o r s corresponding to s e m i l e p t o n i c pro-

where

cesses need to be c a l c u l a t e d because of the

decays i n t o a lepton p a i r

f o l l o w i n g reasons.

V

(l) ~2

i s one of the vector bosons of the

standard model

(W, Z or y ) , which subsequently

X denotes any f i n a l

I . The higher order c o r r e c t i o n s might be n o t i c e a b l e in view of the b e t t e r s t a t i s t i c s

(~,e2).

The symbol

hadronic s t a t e which mo-

menta are c o m p l e t e l y i n t e g r a t e d over.

The

c o l o u r averaged cross section is given by

a v a i l a b l e in recent and f u t u r e experiments. (E.g. UA2 experiment in r e f . 2. The o r d e r quite large.

~s

do =T.~v(Q2,M~)W(~,Q2) dQ2

2.)

(2)

c o r r e c t i o n s turn out to be

In the case of DY they are of the

order of 70% at f i x e d t a r g e t energies and of order 40% at l a r g e hadron c o l l i d e r Therefore we expect t h a t the o r d e r ion w i l l

• = Qz/S .

energies. 2

s

correct-

be a p p r e c i a b l e , too.

3. Higher order c o r r e c t e d K - f a c t o r s provide

Here

~V denotes the point like DY cross section.

Further

S

represents the C.M. energy of the

incoming hadrons

H~, H2,

the d i - l e p t o n p a i r mass.

and

Q2

stands f o r

From the DY mechanism

i t f o l l o w s t h a t the hadronic s t r u c t u r e f u n c t i o n W(~,Q 2)

us w i t h a t e s t of resummation techniques l i k e

can be w r i t t e n as

W(~,Q 2 ) =

the e x p o n e n t i a t i o n of s o f t gluon terms.

~ S dx, S dx, f dx 6 ( = - x , x 2 x ) i,jo o o

4. The p e r t u r b a t i o n s e r i e s of the K - f a c t o r serves as a t e s t f o r the p r e d i c t i v e power of im-

f~'(x

proved p e r t u r b a t i o n theory l i k e the P r i n c i p l e of Minimal S e n s i t i v i t y

( 3 ) ( p . M . S . ) or Fastest

Asymptotic Convergence ( 4 ) ( F . A . C . ) . Massive v e c t o r boson p r o d u c t i o n or d i - l e p t o n p a i r p r o d u c t i o n i s given by the r e a c t i o n 0920-5632/91/$03.50 Q 1991

Elsevier Science Publishers B.V.

where

fHk 1

distribution

,H2)f~2(x2,~2)Aij(x,Q2,~,M2),

(3)

denotes the parton (gluon, quark) f u n c t i o n of the hadron

depends on the mass f a c t o r i z a t i o n

Hk

scale

The QCD c o r r e c t i o n term denoted by a i j All rights reserved.

which N~. depends

4

T. Matsuura et a l . / T h e Drell-Yan K-factor

in a d d i t i o n to the mass f a c t o r i z a t i o n scale also on the r e n o r m a l i z a t i o n scale c o r r e c t i o n term

Aij

Mz.

N2

W via mass fac-

M.

In the subsequent part of t h i s t a l k we w i l l

The DY

can be i n f e r r e d from the

DY parton structure f u n c t i o n

scale

only show r e s u l t s f o r the t o t a l

i n c l u s i v e cross

section

torization.

° t o t ( S ) = S dQ2 BW(Q2,M~)W(~,Q2)

Wij(z,Q 2,M 2,~) =

where

(6)

BW(Q2,M~) denotes the B r e i t Wigner form.

In the discussion of the r e s u l t s we w i l l t r y to Z dx~ o~dx2 dx ~ ( z - x x2x) k,~ o o i

answer the f o l l o w i n g questions: a. How large is the

rki x~,pz,~)r..(x2,p 2 e j ,C)aks(X,Q2,p2,M2)

(4) where Wij

and r

(5)

d i f f e r e n t subprocesses the

as(M) which has been determined in the

0(~ s)

one?

0(~)

qq, qg, qq and gg to

part of the DY K-factor?

c. How does the cross section depend on the various parton distribution functions like

represents the s p l i t t i n g function. Like

DFLM(5), DO1(7), EHLQ(8) etc.?

i t depends on the renormalized coupling constant

c o n t r i b u t i o n to the

b. What is the r e l a t i v e c o n t r i b u t i o n of the f o u r

corresponds to the parton subprocess

i+j ÷ V+ 'X'

0(~)

K - f a c t o r compared with the

d. How does the cross section depend on the

MS scheme. The collinear divergences showing

d i f f e r e n t choices made f o r the f a c t o r i z a t i o n

up in

scale

W and r

have been regularized by n-

dimensional regularization (E~ n-4).

The

N and the r e n o r m a l i z a t i o n scale

M?

In t a b l e 1 we have presented the t o t a l cross

s p l i t t i n g functions as well as the correction

section f o r Z-production f o r three d i f f e r e n t

term are mass factorization dependent. In this

parametrizations i . e . DO1, EHLQ and DFLM4. The

talk all our results w i l l be presented in the

energies under c o n s i d e r a t i o n are V~S = 0.63;

DIS scheme. Up to order function

W (4)

2

S

the DY structure

receives contributions from

the following parton subprocesses i..e. q(q)g,

qq and gg

corrections.

qq,

including their radiative

Until this moment (1) we have not

computed the hard gluon 0(~s) qq and the

0(~s)

q(q)g

contribution to

subprocesses. Since

1.8; 16 and 40 TeV.

From t h i s t a b l e we i n f e r

t h a t at increasing energy the K - f a c t o r gets smaller (see also f i g . to the the

O(as)

0(~)

I).

This is wholly due

contribution.

On the other hand

contribution slowly increases and

becomes even larger than the at SSC energies.

O(es)

correction

This effect can be attributed

the DY correction term has been calculated in

to the

the DIS scheme, we need the patton distribution

negative over the whole energy range. Since

O(es) qg subprocess, which is always

H

functions

f~k(x) (3) in the same scheme in

vector boson production at large energies is

order to determinethe hadronic structure function

dominated by the small

W(T,Q2).

t h i s e f f e c t is not unexpected in view of the

For that purpose we have chosen the

DFLM (set 4) structure functions stated otherwise.

(5)

unless

For the running coupling

x~,x 2

region in (3)

steep r i s e of the gluon d i s t r i b u t i o n at small x values.

function

However one has to bear in

constant we adopt the expression in eq. lO of

mind t h a t the e x t r a p o l a t i o n of the e x i s t i n g

ref. 6 which is corrected up to two loops with

parametrizations of the gluon s t r u c t u r e function

the heavy flavour thresholds included (A=O.2GeV).

to small

For conveniencethe mass factorization scale

holds f o r the other parton distribution functions.

x

values cannot be t r u s t e d .

is chosen to be equal to the renormalization

Moreover we do not know the

0(~)

The same

contribution

T. Matsuura et al./ The Drell-Yan K-factor

5

Table I : The t o t a l cross section for Z-production in three approximations at SppS, Tevatron, LHC and SSC. The value between brackets is the sum of =the contributions from A(I),H and Aqg (I) " . 2 qQ Mz 91GeV, sln eW=0.227. Z production rate (nb) DO1

EHLQ

DFLM4

V~S=0.63 TeV Born O(a s)

1.37 1.78(-0.08)

2

0(~ s)

1.37

1.36

1.79(-0.08)

1.95

1.79(-0.07]

1.95

1.96

VS=I.8 TeV Born

4.71

0(~ s) 0(~)

5.85(-0.53) 6.45

4.55

4.93

Born

40.8

32.4

0(~ s) 0(~%)

46.3(-9.0) 52.3

37.0(-6.9) 41.8

5.64(-0.52) 6.22

6.16(-0.52] 6.78

V~S=I6. TeV L

I

Table 3: BoZ and BaW++W- for the Tevatron. We have used B(Z~e+e-) = 3.35 10 -2 and B(W÷eu) = 0.109 (see r e f . ( 1 ) ). MZ= 91GeV, MW= 80 GeV, sinZe W=0.227, sinZe C=0.05.

39.9

I 46.7(-7.4 52.5

¢S = 1.8 TeV

I

~ = 4 0 . TeV Born 0(~ s) 0(~)

I00.

71.1

I08.(-28.) 124.

76.1

79.0(-17.) 90.6

86.2(-17.) 98.3

Doi

I E,LQ [6FET4-]

CDF (91

B(Z* e+e-)'oZ (nb) Born

0.158

0.152

0.165

O(a s)

0.196

0.189

0.206

0(~)

0.216

0.208

0.227

0.197±0.012±0.032

B(W÷e~)'°W++W- (nb)

Table 2: Boz and BoW++W- f o r the SppS. We have used B(Z*e+e -) =3.35 10-2 and B(W+eu) : 0.109 (see r e f . ( I ) ) . MZ =91 GeV, MW=80 GeV, sin~e W=0.227, sinZec= 0.05. W-S = 0.63 TeV B(Z+ e+e-)'a z (pb) Born

45.8

45.9

45 7

0(~ s)

59.8

59.9

60 0

O(a2) s

65.4

65.5

65 6

70.4 ±5.5 ±4.0

B'W+e~"°W+{~ + W- (pb) Born 0(~ s) 0(~)

511. 666. 731 .

453 590 649

479. 628. 690.

660 m 15 ± 37

;ion; il;; ili; ii 4

2.06±O.04±0.34

T. Matsuura et al./The Drell-Yan K-factor

6

F i n a l l y we would l i k e to i n v e s t i g a t e the

of the qg subprocess to the K - f a c t o r y e t . I t is not impossible t h a t the l a t t e r can compensate

scale dependence of

the e f f e c t of the

l a t t e r should be scale independent since i t is a

0(~ s)

part.

Since the

experimental data are always taken at

x>O.Ol

W(~,Q2).

physical q u a n t i t y .

In p r i n c i p l e the

However, since the leading

the various parametrizations f o r the parton

and the next to leading logs in

distribution

c a l c u l a t e d up to a f i n i t e

functions show a d i f f e r e n t

behaviour at very small

x

values.

be c l e a r l y seen in t a b l e I .

At~

x

This can

will

= 0.63 TeV

become scale dependent.

choice of the r i g h t scale.

are only a s, W(~,Q2)

There are many

discussions in the l i t e r a t u r e

the three d i f f e r e n t parametrizations predict the

Aij

order in

concerning the

Some groups p r e f e r

same value f o r the t o t a l cross section whereas

PMS (3) whereas other people advocate FAC (4)

at l a r g e r energies (LHC and SSC) they lead to

Other groups vary the scale between some canon-

mutual d e v i a t i o n s of about 30%.

i c a l values in order to show the t h e o r e t i c a l

In f i g .

2 we have shown the c o n t r i b u t i o n s of

the various subprocesses to the K - f a c t o r f o r production. q~

Z-

From t h i s f i g u r e we i n f e r t h a t the

subprocess dominates the K - f a c t o r followed

by the

qg

reaction.

part of the

of the q q

qq process.

The dominance

process can be a t t r i b u t e d to the

expressions e x h i b i t a very small v a r i a t i o n under a wide range of scale choices.

A qq is c a l c u l a t e d in the DIS scheme. On

order c o n t r i b u t i o n (Born) is independent of as(M).

To s i m p l i f y t h i s discussion we choose

s o f t and v i r t u a l gluon terms appearing in

N=M.

be found in r e f . I .

the I% l e v e l .

gg

subprocess never exceeds

The same holds f o r the singlet iS)

Independent v a r i a t i o n s of

the r a t i o

In f i g .

N and

R = o(~,M)/c(M V, MV)I~=M

production at

~

= 0.63 TeV.

and the

if

unexpected that the

gg

sight i t

is

subprocess is so small

for

Z-

Here we obtain

an improvement in the dependence of

At f i r s t

M can

3 we have p l o t t e d

p a r t of the q~ r e a c t i o n (gluon exchange graphs) qq subprocess.

Semi-

to s a t i s f y t h i s requirement since the lowest

provided i t

the other hand the

The best s o l u t i o n

to the problem is to show t h a t the r e s u l t i n g

l e p t o n i c processes l i k e DY are a good candidate

In the case of SSC

energies the l a t t e r becomes even l a r g e r than the 0(~)

e r r o r of t h e i r p r e d i c t i o n .

R on

higher order c o r r e c t i o n s are included. How-

ever at higher energies (see r e f . I ) the corrected

t r i b u t i o n f u n c i t o n at small

corrected one. On the contrary i f we go to LHC or

x.

In the case of

R is not b e t t e r than the

0(~)

in view of the steep r i s e of the gluon dis-

0(~ s)

heavy f l a v o u r production (6) t h i s property is

SSC energies the

one of the main reasons f o r the importance of

l a r g e r d e v i a t i o n from 1 than the 0(~ s) corrected

this reaction.

However the cross section is not

R.

O(e~) c o r r e c t i o n s lead to a

Probably t h i s is due to the missing

function

c o n t r i b u t i o n from the

only but also by the c o r r e c t i o n term

aij

be a t t r i b u t e d to the wrong scale dependence of

(3).

qg

O(e~)

determined by the parton d i s t r i b u t i o n

subprocess or i t can

One can show (see r e f . I) t h a t at large c o l l i d e r

the e x i s t i n g parton d i s t r i b u t i o n

energies the integrand in (3) is dominated at

too small

small x~,x 2 and

been observed f o r

of

nij(x)

x=l.

Therefore the behaviour

at x=l is very important. A thorough

analysis ( I ) reveals t h a t steeply r i s e at and

aqq(X)

x+l

Aq~(X) and ~qg(X)

whereas

Agg(X), Aq~,S(X)

vanish in t h a t l i m i t .

This ex-

x values.

functions at

The same features as have

Z - p r o d u c t i o n are also d i s -

covered f o r W-production (see r e f . I ) .

There-

f o r e there is no need to discuss t h i s r e a c t i o n separately.

In t a b l e s 2 and 3 we have compared

our p r e d i c t i o n s f o r the process

p l a i n s why the l a s t three reactions are sup-

pp+W +ev~

pressed with respect to the other two.

the

p p ÷ Z ÷ e~

and

with the most recent r e s u l t s from

UA2 ( I ) and

CDF(9) groups. From t a b l e 2

T. Matsuura et a l . / T h e Drell-Yan K-factor

7

2.0

2 1.0 0.8

--'------~-- . . . . . . . . . . . . . . .

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

~o.6 tp

0.4 1.2 R -

0.1

.--1.o

o.5

,

,

,

, , , , , ,

I

5.o

i

i

lo.o

~/S(TeV)

,o.o

1,1 \\ \

Fig. I . The order

i

s

\ \\\

Z-production at a pp c o l l i d e r denoted by

\

corrected K - f a c t o r f o r 1.0

Ki .

I:KI;

2 : K2 . ~s1 c o n t r i b u t i o n to the K-

The order

f a c t o r f o r Z-production at a pp c o l l i d e r

0,9

denoted by K( i ) 3:K(1); 4 : K (2).

0.8

I

I

t

J [

I

1

i

]

3 ......................................

l

i

I t I

1

I

I

__-~- iZ.-- ~ -~ . ~ _ ~ - - =..~..-:.........

I

I

I

I

0.5

1.0

1.5

2.0

Fig. 3. Dependence of the mass f a c t o r i z a t i o n scale

P

represented by the q u a n t i t y

R= ~(N,N)/a(Mv,M V) /

/ /

4

°.- ---''°'" ....

~=

...;j

6

J5 ~

..'JS'~

z.. / .

ld 4

/

",'i

0.5

,

I I [

I

I

1.0

I

I

5.0

j

VS(TeV)

I

ill

I

I

I_

10.0

Fig. 2. The various c o n t r i b u t i o n s to the Kf a c t o r f o r Z-production at a p~ c o l l i d e r : ; K~2)'' 4 : K(2). 5

qq

6

qq,S

50.0

f o r Z-production at

0.63 TeV (SppS),

in u n i t s of

I

__

p is expressed

MV.

1 :Born;

2 : 0 ( ~ s)

3 :O(a~)

corrected.

corrected;

8

T. Matsuura et a l . / T h e Drell-Yan K-factor

one concludes that the 0(~ ) and O(a~) pres s dictions are in agreement with the UA2 data whereas the Born approximation is not.

In the

case of the CDF data one cannot distinguish between the corrected and uncorrected cross sections.

The UA2 data clearly indicate that

QCD corrections are necessary to bring the DY cross section into agreement with experiment. However at this moment they do not allow us to

discriminate between the

O(a s)

and O(a~)

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25th Savoie-