Z. Phys. C - Particles and Fields 51, 107-112 Partkztes

andF �9 Springer-Verlag 1991

Cross sections for five jet production above the W + W- threshold in e + e - annihilation N. Brown

Rutherford Appleton Laboratory, Chilton, Didcot, Oxon O X l l 0QX, UK

Received 8 November 1990

Abstract. We describe the calculation of five jet pro- duction from the W + W - intermediate state in e+e - annihilation in lowest order QCD. Some results for integrated cross-sections are presented.

I Introduction

The production of jets in e+e - annihilation provides a crucial testing ground for the accepted theory of strong interactions; quantum chromodynamics (QCD). Below the W + W - threshold the dominant source of jet events is the 'point like' process via an intermediate photon or Z boson. As energies rise above this threshold, the decay of the W boson into qc7 pairs gives rise to an alternative source of jet events. To give an idea of the magnitude of these processes, the total cross section for W + W -

production at a centre of mass energy , f~ = 200 GeV is approximately 20pb. The branching fraction for a W to decay into hadrons is about 2/3 (assuming that the top quark is too heavy for the W to decay into). This gives a hadronic cross section of about 9 pb. This contrasts with a Born level cross section of about 18.2 pb for the process e +e- ~ q~ via a virtual Z or photon at the same energy. These use the following values for standard model parameters: Mz = 91.1 GeV, M w = 80.6 GeV, F z = 2.5GeV, Fw = 2.06GeV, c(= 1/128, es=0.11, sin2 0w = 0.23, as we shall throughout this study. All five flavours of light quarks are treated as massless, as is the electron.

The kinematics of these two processes, however, are quite different. To lowest order the W + W - process has four partons in the final state giving rise to a characteristic signal of four jets. The point like process will be dominantly two jet, the four jet production being suppressed by two powers of the strong coupling es. Thus for multijet configurations the W + W - process is expected to do- minate over the point like one, and this is borne out by theoretical studies [1].

At next order in Q C D it is possible for one of the quarks from the decay of the W to radiate a gluon, leading to a five parton final state. Again we expect this to

dominate over the point like production of five jets, as is demonstrated in [2]. In Fig. 1 we show the difference in cross sections for five jet rates, based on an invariant mass cut y. Here we have demanded that all five partons in the final state satisfy s~ = (pz + pj)Z > ys, i ~ j , where the Pl are the parton momenta. The importance of this process is that it will allow us to check that W hadronic decays are in agreement with QCD. Whilst it would be very surprising if this were not so, it is, nevertheless, of importance to test this. As was shown in [2] and can be seen from Fig. 1, the event rates will be sufficient to do this.

In this paper we detail the calculation of the five jet process via the W + W - intermediate state at lowest order in QCD perturbation theory. The lowest order calcula- tion for five jet production via the point like process was presented by Franck et al. [3]. Because of the very different kinematic and flavour structure of the two processes there will be virtually no interference between them. This is further underlined by the fact that for the point like process the qglqgl9 final state is somewhat suppressed relative to the qg1999 final state.

2 The calculation

There are twelve Feynman diagrams contributing to the process e+e - ~ qgtqgl9 via the W + W - intermediate state. Three of these are shown in Fig. 2, the remaining diagrams are obtained by attaching the gluon to different quark lines. Because of the colour structure, the diagrams involving W + ~ qflg, W - ~ qgl do not interfere with those involving W + --*qgl, W - ~ q g t g . We can therefore treat these two cases separately. We shall detail the calculation of the first case, merely giving the result for the second.

We use the spinor method of Kleiss and Stirling [4], where the basic quantities are the spinor products (see Appendix A)

s(pl,P2) = t~+(Pl)U (P2) = -- s ( p 2 , P l )

t ( p l , P 2 ) = u - ( p l ) u + ( p 2 ) = s ( p 2 , p l ) * , (1)

where �9 denotes complex conjugation, and the + / - denote helicity, with left handed helicity being conven- tionally chosen as negative. These spinor products are

108

o.

1

0"1

0'01 \, \

\ \ \ \

\ \

1,ool 1 I I I I ooq 0.0 o.os 0.04 0.05

Y

WW

OCD

1 1

Fig. l. Cross sections for five jet production as a function o f invariant mass cut y. The W+W contribution is shown with a full line, the point like contribution with a dashed line

just a complex function of their m o m e n t a arguments and are easily incorpora ted into a compute r program. Note that all other spinor products vanish for massless spinors. F r o m these spinor products we can easily obta in the vector inner p roduc t via

2pi"p2 = ]s(pl,P2)l 2. (2)

Note that for massless particles we can replace anti-particle spinors by particle spinors of opposi te helicity, and so work entirely in terms of u spinors.

An impor tan t p roper ty we shall need is the Chisholm identity

7u~,(p~)Tuu~(p2) = 2u~(pz)~(px ) + 2u_~(p~)~_,~(p2). (3)

Following Kleiss and Stirling we write the polar izat ion vector for a gluon of helicity 2 and m o m e n t u m k as

~z(k)y~'ua(p) e~(k) - (4)

Here p is any light-like vector with p.k # 0. A change in p consti tutes a gauge t ransformat ion and does not affect physical quantities. This polar izat ion vector satisfies the required properties, as is shown in [4]. It is easy to show using (3) that

Cz = (ua(p)O~(k) + u_ ~(k)O_ z (p) ) /x /wk , (5)

we shall also use the completeness relation for massless

q

e ~

ct

e ~ g

"4- e

e-

q

e

Fig. 2. Three of the Feynman diagrams contributing to the process e+e ~q(lq(lg via the W § W- intermediate state. The other diagrams are obtained by attaching the gluon to different quark lines

spinors to write

p = u + (p)a + (p) + u_ (p)a_ (p) (6)

for any light-like vector p. We shall use m o m e n t a assignments as follows

e+(p+)e- (p_)~ W+(ka)W-(k3),

W+(k2) ~ q(pa){l(P2)g(k), (7)

W-(k3) -* q(p3)(l(P4),

where we have k 2 = Pl + P2 + k and k 3 = P3 + P4. We write the ampl i tude for this process (ignoring an

overall factor of i) as

M~ = {(F~(p +,p_) + Z~ +,p_))T~aP(k2,k3)

+ NaP(p+,p_,k2,k3)}D~o(k, pl ,p2,P3,p4 ). (8)

First of all we explain this expression. The terms F~, Z~ arise from the s-channel pho ton and Z diagrams. They consti tute the factors f rom the initial e lec t ron-pos i t ron line, together with the pho ton or Z p ropaga to r factor and the coupling constants from each end of the p ropa- gator.

t s +, p _ ) = ~L(o + % u ~ ( p _ ) s

Z~(p +, p _ ) = a~(p + % ( g v - gA~5)u~(p- )

_ g 2

"( 2(s - M~z+-iMzFz) ) (9)

Here a denotes the helicity of the electron and positron, gA = - 1/2, gv = - 1/2 + 2 sinZ0w, are the vector and axial coupling strengths, g is the weak coupling constant and e is the electric coupling constant satisfying e = g sin 0w. Here we have used gauge invariance to eliminate terms from the numerator of the propagators which are not proportional to g,~.

N ~0 arises from the electron positron line from the neutrino t-channel exchange diagrams. We have

g~ ~ _ ( p + ) ~ q ~ - ~+Gu_(p_) N#P(P+,P-'kz 'k3) = ~ ~ ~ + ) 2

(10)

Because of the left handed couplings of the W, only negative helicity is allowed to flow along the electron- positron line in these diagrams.

The quantity T~PP(k2, k3) arises from the vector boson triple couplings, with the coupling constants removed, we have

r cr = g~a(2k p) + (k 3 - k2)ag I~p q- (-- 2kP3)g ~p. (11)

Here we have used the fact that terms proportional to k~z and k p will vanish when contracted with the final state W decays by gauge invariance.

The factor D~ ~ comes from the final state W decays, where 2 is the gluon helicity. We have

D~P( k, Pl, P2, P3, P4) 1 1 g2

= Dw(k 2) Dw(k~) 2 u-(p3)TPu-(P4)Q~(k'pI'P2)" (12)

Here Dw(q 2) is a W propagator factor

Dw(q2) = qZ _ M 2, + i M w F w , (13)

and Q~ arises from the decay W § ~qFtg. Again we have used gauge invariance to remove terms in the numerators of the W propagators which are not proportional to g,~. Using (4) we have

1 a p Q~(k, pl,P2) = ~p~.k (A u-(Pl)7 u-(pz)

+ BZO_(pl)TPu_(k) + CZfi_(k)7'u_(p2)), (14)

where

A z = u-(Pl)r O-(Pz)r 2p1 "k 2pz.k '

B ~ = u-(k)r 2pz. k '

C~ _ t~_ (p 1)r (k) 2pl"k (15)

Once again only negative helicity spinors contribute because of the left handed nature of the W boson. These expressions simplify somewhat if we choose the p = P2 in (4). Using the Chisholm identity (3) we get

109

A + = t(pDP2)s(k, pI) 2p l ' k '

B + = 1 ,

C- - t(pl' k)s(p2' k) 2pl "k '

A - _ t(pl, k)s(p2, k)

2p l"k

(16)

where B- = C + = 0. In these expressions we have sup- pressed colour indices and taken out the QCD vertex factor ig~T~, which is the strong coupling constant multiplied by an SU(3) colour matrix. In our final matrix element squared we must therefore multiply by 2 2 g~ NcCe where N c = 3 is the number of colours and C~ = 4/3 is the appropriate colour Casimir.

To obtain our final expression for the amplitude, we simply multiply out (8), using the Chisholm identity to reduce everything to spinor products. For example, the expression

a~(p + )Y'G(P - )a _ (P3)Gu - (P4) (17)

can be written as

2u-(p3){u~(p-)5,(p+) + u_~(p+)F~_~(p_)}u_(p4), (18)

and this equals

2t(p3,p_)s(p+,p4 ) a = +, 2t(p3,p+)s(p_,p4 ) a = - . (19)

Similar expressions are obtained for other terms in the expansion of (8). We shall give our results in terms of the quantity

J(a, b, c, d) = s(p., pb)t(pc, Pd)" (20)

Here a, b, c, d refer to momentum labels and will be taken from +, - , l, 2, 3, 4, 5, where we use P5 - k.

3 M a t r i x e l ements

Using the expressions above it is just an algebraic exercise to obtain expressions for the matrix elements..First of all we consider W + ~qqg , W--*qq . If the helicity of the incoming electron-positron line is a = + then only the photon and Z diagrams contribute and we have

( ~ (gv = gA)gZ "~ l g2 M~+ = 2 ( s - M2z + iMzrz) J ~ Dw(k22)Dw(k~)

�9 (H l -1- H 2 q- H3) , (21)

where

H 1 = 2(J(1, 4, 3, 1) + J(2, 4, 3, 2) + J(5, 4, 3, 5))

�9 (A~J( +,2,1, --) + B~J( +,5,1, --) + C~J( +,2, 5, -)),

H z = (d(+, 3, 3, - ) + J (+ ,4 ,4 , - ) - J ( + , 1, 1, - )

- d ( + , 2, 2, - ) - J ( + , 5, 5, - ) )

�9 (AaJ(4, 2, 1, 3)+ Ba(4, 5, 1, 3)+ Ca J(4, 2, 5, 3)),

H 3 = -2(AZ(J(3, 2, 1,3)+J(4,2, 1,4))

�9 B~(J(3, 5, 1, 3) + J(4, 5, 1, 4))

�9 C~(J(3 ,2 ,5 ,3)+J(4 ,2 ,5 ,4)) )J(+,4 ,3 , - ) . (22)

Here the quantities A~,B a, C A are given in (16).

110

When the incoming electron positron helicity is a = - , then all diagrams contribute and we obtain

MZ_=_fe_ 2 (gvq.-gA)g 2 " ] 1 g2

ks 2 ( s - ~222 + ~ z Fz ) J x/~z 'k Dw(k~) Dw(k2)

9 4 1 �9 ( O 4 --t- 0 5 -1- H 6 ) -1- Ow(k2)Ow(k~) x~2" k

1 (P~ + P2 + k - p+)2 Hv, (23)

where

H4 = 2(J(1,4, 3, 1) + J(2, 4, 3, 2) + J(5, 4, 3, 5))

�9 (AZJ(2,-, +, 1)+ BzJ (5 , - , +, 1)+ CzJ(2,- , +, 5)),

H5 =(J(3 , - , + , 3 ) + J ( 4 , - , + , 4 ) - J ( 1 , - , +, 1)

- J ( 2 , - , + , 2 ) - J ( 5 , - , +,5))

�9 (A~J(4,2,1,3)+BZ(4,5,1,3)+CZJ(4,2,5,3)), (24)

H 6 = - 2(AZ(J(3, 2, 1, 3) + J(4, 2, 1,4))

�9 BZ(J(3, 5, 1, 3)+ J(4, 5, 1,4))

�9 Cz(J(3, 2, 5, 3) + J(4, 2, 5, 4)))J(4, - , + , 3),

and

H 7 = AZJ(4, - , + , 1)(J(2, 1, 1,3) + J(2, 5, 5, 3)

- J ( 2 , +, + ,3))+BaJ(4, - , + , l)(J(5, l, 1,3)

+ J(5,2,2, 3 ) - J (5 , +, + , 3 ) ) + CzJ(4 , - , +,5)

�9 (J(2, 1, 1, 3)+J(2, 5,5, 3 ) - J (2 , +, +, 3)). (25)

After summing over spins and colour and averaging over initial spins, the transition probability for the W + ~ q~19, W- ---r qc~ process IMll a is then

2 1 2 2 + 2 +IM+ +lm-12) . IM~I =-~g~NcCr(IM+l +IM+-I z I 2 (26)

The calculation for the process where W + ---'qgl, W - ~qg19 is very similar. In fact for the photon and Z exchange diagrams the amplitudes are simply obtained from the expressions above by using the interchanges p~,--,p3 and pz~--~p,~ (thus here P4 is the vector used to define the gluon polarization, as opposed to P2 for W+~qglg, W - ~ q g t above), and by multiplying by an overall minus sign from the antisymmetry of the triple gauge vertices. If we refer to the amplitudes for this process as M~ ~, then since only the photon and Z diagrams contribute to the amplitude with a = + we have

M] + + (27) = - Mz [ploP3 p 2 ~ p 4 �9

The neutrino exchange diagrams are slightly different and we have

M,a = ( ~ ( g v + g A ) g Z ~

1 ffz I

" . / ~ ' k Dw(k~)Dw(k~) (n'4 + H5 + U'6) W

g4 1 1

+ Dw(k2)Dw(k2) x/-~4"k (Pl +P2+ _p+)Z H'7. (28)

Here we have

H'i= - H,lp,~p3 p 2 ~ p 4 i=4 ,5 ,6 , (29)

as before. For the neutrino diagrams H i is given by

H'7 = A'zJ(4, --, +, 1)(J(2, 1, 1, 3) -- J(2, +, +, 3))

+ B"~J(5, - , +, 1)(J(2, 1, 1, 3) - J(2, +, +, 3))

+ C'aJ(4, - , +, 1)(J(2, 1, 1, 5 ) - J (2 , +, +, 5)), (30)

where A'~,B '~, C '~ are obtained from AZ, B ~, C ~ by the usual interchange p~*-~p3,pz~-~p4 . After summing over colours and spins and averaging over initial spins we obtain the transition probability I MEI 2 for the W + --* q~, W- ~ qqo as

1M212 = 1 2 2 M,++ ~gsNcCAI I2+lM'-+l~+lM'+-IZ +lM'--I2). (31)

4 Cross sections

The cross section for e+e-~qqqqg via the W + W - intermediate state is given as

aww(5 - j e t )

1 1 - 2s (2n) 11 ~dSq~(P + + P - ~ P l '}- P2 +P3 +P4 + k)

�9 ([M112 + IM2I 2) VKM. (32)

Here d 5 q~is the measure for five body phase space without factors of 72, and VKM is a factor to take account of quark mixing in the W decays. Since we are not allowing the W to decay to the top quark, we have VKM = 4, reflecting the fact there are two W bosons decaying and two available generations of quarks to decay into. If the top quark were much lighter than the W, then we would have VKM = 6. If the top quark were just a little bit lighter than W, then the situation would be slightly more complicated since some decay channels might be allowed, e.g. W + ~tti, whereas W + ~ t b might be forbidden. In all these situations the total W width, F w should be modified accordingly.

It is convenient to integrate the two transition proba- bilities IMl l 2 and Im2l 2 separately�9 The transition probability is strongly peaked when k~ ~MZw and k~'~MEw. In the first case for W+~qgtg, W---*qgl we have k 2 = Pl + P2 + k, k 3 = P3 + P4, whereas for W + ~qq, W ~qc19 we have k 2 = p l + p 2 , k 3 = P 3 + P 4 + k . For IMxl 2 we can reflect this peaking of the transition probability by writing the phase space integral ydSq~ - (P+ +P---+Px +P2 +P3+P4 +k) as

fdk2dk2d2cl)(p + + p ---*kz + k3)d 3 ~(k2--*p~ + P2 + k)

"d2 q)(k3 ---~ P3 q- P4), (33)

with a related expression for the phase space integral for IM212�9 We then change integration variables

k 2 - M Z w = M w F w t a n O , i = 2,3. (34)

This gives us the jacobians

1 dk 2 - . - - 2 2 (MwFw + (k~ - ME)2)dOi, (35) M w F w

which is equivalent to

1 dkU - - - IDw(k~)12 dO, (36)

M w F w

and it can be seen that this change of variables removes the dependence on the W propagator factors in the transition probability, making the integrand more smoothly dependent on the integration variables. The integrations to obtain Fig. 1 were performed using the Monte Carlo routine VEGAS, with the integration error being less than 1~o.

To check the calculation we let the W width F w become small. In the narrow width approximation the five jet cross section factorizes and we have

aww(5- j e t ) r ~ o . . . . (~ F(W~qgl ) F(W--+q~lg) ~ rw j

(37)

The partial width F ( W ~ q?lg) is calculated by demanding that the three final state parton momenta involved in this decay satisfy (Pi+pj)2 > y s i ~ j for various y. It is important that this condition to define the three parton final state be Lorentz invariant. We put no constraints on the quark momenta in the decay W~q~l. We then verified that (37) does indeed hold for our 5-jet cross section.

5 Conclusions

We have shown how the process e+e---+qqqqg can be calculated using the spinor techniques of Kleiss and Stirling. The expressions obtained can be easily coded and depend on the complex functions s and t, the details of which are given in Appendix A. This once again demonstrates the usefulness of these calculational techniques in obtaining matrix elements for higher order tree level processes.

As can be seen from Fig. 1, this process dominates over the point like one at LEP200 energies, and with the canonical integrated luminosity of 500pb-1, we should see hundreds of these five jet events. These will allow us to test the QCD corrections to W-+qgl decay, just as in the past we have tested the QCD corrections to the q~ decay of virtual photons and Z bosons.

Finally, from the calculation detailed in this paper it is easy to obtain matrix elements for related processes such as e+e --+ # - ~qglg etc. In Appendix B we give some details on this.

Appendix A: Spinor products

As in [4] we need two 4-vectors ko, k 1 to define our spinor products�9 They must satisfy the following pro- perties:

ko .ko=0, k l . k l = - l , ko-k l=0 . (38)

Next, we define a basic negative helicity spinor u_(ko) with its complex phase defined by

u_ (ko)t~_ (k0) = o9_/~o, (39)

111

where e)+ = �89 _+ 75). The corresponding positive helicity spinor is defined by

u + (k o) = g 1 u_ (k o). (40)

From these we can construct spinors for any light-like massless momentum p by

uz(p) = pu_ x(ko)/ 2x/2~o. (41)

This is valid provided p. k does not vanish or become too small, an extremely unlikely occurrence when the mo- menta are generated by monte-carlo techniques. The two non-vanishing spinor products are

s(pl , p2) =- a+ ( p l ) u - (p2) = - s(p~, p 1),

t(pl,P2) - ff-(Pa)u+(p2) = s(p2,Pl)*. (42)

s(pl, P2) can be evaluated as

s(pl,p2) = {(Pl 'ko)(p2kO - (p~k O(p2ko) �9 # v p a . -leu~p~koklplpz}/x/(pl ko)(p2"ko), (43)

where we use the metric (+ , , , ) and/;o123 = + 1. For the purposes of calculation we make an explicit choice for k o and kl, taking p~= o x y z

(Pi, Pi, Pi, Pi ), kg = (1, 1, 0, 0), (44)

k~ = (0, 0, 1, 0),

giving us

+i z'rP~ " z FpO_p~ql/2

(45)

As was previously states the spinor product also gives us the vector product via

2pl"P2 = Is(pl,p2)l 2. (46)

Appendix B: Other processes

The calculation of the five parton matrix element via the W + W - intermediate can be easily adapted to give us the matrix elements for other possible decays of the W bosons. For example the process W + ~q~tg, W - ~ l -9g has a spin and colour summed and averaged transition probability IMi ]2 where

1 2 IMl-12 = Ncc IM11 �9 (47)

where we must also set VK~ = 2 to reflect that only one W decays into hadrons. Here P3 is the charged lepton momentum and P4 is that of the neutrino.

For W + ~ l + vt, W - ~ qflg the transition probability is the same as (47) except we replace M 1 with M 2. Here pl is the neutrino momentum and P4 is that of the charged lepton. Again VKM = 2.

Finally, the simpler lower order process where both W bosons decay into a fermion anti-fermion pair, has a spin and colour summed and averaged transition probability ]Mo] 2, where M o is obtained from M 1 by

112

mult iplying by x~22"k, setting B ~ = C ~ = A - ~ 0, A § = l. Then all we set the m o m e n t u m k = P5 equal to zero. For completeness we obtain

iMol2 x +12 =zRN(IM0 + l m o [ 2 ) . (48)

Here we have

M ; = 2(s ~ + ~ / z r z ) Dw(k~)Dw(k~) ' (49)

where

H + = 2 J ( + , 2 , 1, - ) ( J ( 1 , 4 , 3, 1) + J(2, 4, 3, 2))

+ J(4, 2, 1, 3 ) ( J (+ , 3, 3, - ) + J ( + , 4 , 4 , - )

- J ( + , 1, 1 , - ) - J ( + , 2 , 2 , - ) )

- 2J( + , 4, 3, - ) ( J ( 3 , 2, 1, 3) + J(4, 2, 1, 4). (50)

M o is given by

(~2 + 2 2 (_gv gA)g ~ ~ g- H -

M o = 2(s - M~ + iMzFz)JDw(kZ)Dw(k 2)

g 4 1

+ Dw(k22)Dw(k3) 2 (Pl + P2 -- p+)2 H ' - . (51)

Here we have

H - = 2J(2, - , + , 1)(J(1, 4, 3, 1) + J(2, 4, 3, 2))

+ J(4, 2, 1, 3)(J(3, - , + , 3) + J(4, - , + , 4)

- J (1 , - , + , 1) - J(2 , - , + , 1)) - 2J(4, - , + , 3)(J(3, 2, 1, 3) + J(4, 2, 1, 4)),

H ' - = J(4, - , + , 1)(J(2, 1, 1, 3) - J(2, +, +, 3)). (52)

The m o m e n t a assignments are

e+(p+)e-(p_)~ W~ +(kz)W-(k3),

W+(k2)~ f(pl) f(p2), W - (k3) ~ f'(P3)f'(P4). (53)

For both the W bosons decaying into quarks we have the Q C D factor RN = N 2 - 9 and VKM = 4. For one W decaying into quarks, the other into leptons, we have Rs = Nc = 3 and VKM = 2. For both W bosons decaying into leptons we have RN = 1 and VKM = 1.

References

1. A. B6hm (ed.): ECFA workshop on LEP200, vol. 1, CERN-87-08, ECFA 87/108 (1987)

2. N. Brown: Rutherford Report RAL-90-079 3. N.K. Faick, D. Graudenz, G. Kramer: Phys. Lett. B220 (1989)

299; Nucl. Phys. B328 (1989) 317 4. R. Kleiss, W.J. Stirling: Nucl. Phys. B262 (1985) 235