

 18 Jul 1995

DESY 95-135 ISSN 0418-9833 July 1995

HIGGS BOSON PRODUCTION AND WEAK BOSON STRUCTURE\Lambda

Wojciech SLOMI 'NSKI and Jerzy SZWED

Institute of Computer Science, Jagellonian University,

Reymonta 4, 30-059 Krak'ow, Poland

Abstract The influence of the QCD structure of the weak bosons on the Higgs boson production in e-p scattering is studied. The energy and Higgs boson mass dependence of the cross-section, following from the new contributions, is calculated.

\Lambda Work supported by the Polish State Committee for Scientific Research (grant No. 2 PO3B 081 09) and the Volkswagen-Stiftung.

1 Introduction In our recent papers [1, 2] we have introduced the basic properties of the W and Z boson structure functions. In analogy to the photon case [3] it has been shown there, how the quark and gluon content of the intermediate bosons appears due to the QCD cascade. The corresponding evolution equations have been solved in the asymptotic regime with the solutions developing logarithmic Q2 growth of the quark and gluon densities. The form of weak couplings forces these densities to depend strongly on spin and flavour.

It is extremely interesting where the QCD structure of weak intermediate bosons may be observed experimentally. In the systematic investigation of possible processes we start with the Higgs boson production. The question put forward in this case is to what extent the inclusion of the 'resolved' W and Z can modify the predictions known before. We concentrate on the Higgs boson production in e-p scattering, quoting the e+e\Gamma scattering, where one should not expect large contributions, for completeness.

The paper is organized as follows. In Section 2 we recall the formalism introduced before to study the QCD structure of gauge bosons and quote the main results concerning W and Z, compared to the photon case. Section 3 presents the calculation of the cross-section for the Higgs boson production in e-p scattering at selected energies. The results are compared with the dominant production channel [4, 5] (W -W fusion) and earlier calculation of the 'resolved' photon contribution [6]. Summary, comments and conclusions are given in Section 4.

2 QCD structure of W and Z bosons In the standard model weak intermediate bosons are elementary (point-like) particles. Nevertheless, when observed by a very high Q2 particle, they can reveal QCD structure by collinear quark-gluon Bremsstrahlung. In this sense they become "composite".

The evolution equation for unpolarized parton A density, f BA (x; t), inside a composite weak intermediate boson B reads

df BA (x; t)

dt = XB PAB(x; t) \Omega f

B B (x; t) ; (1)

where t = ln(Q2=Q20) and the scale Q20, which depends on the particular process, will be discussed later. PAB(x; t) are splitting functions and the convolution is defined as

(P \Omega f )(x) j Z dx1 dx2 P (x1) f (x2) ffi(x \Gamma x1x2) : (2) For weak bosons the indices A; B go over quarks, antiquarks, gluons and point-like fl, W and Z.

In the following we will consider the leading-log QCD and 1-st order electroweak case. Denoting weak intermediate bosons by upper case letters and QCD partons by lower case ones, we have

fAB(x; t) = ffiABffi(1 \Gamma x) ; (3)P

iB(x; t) j ff

em

2ss PiB(x) ; (4)

Pik(x; t) j ffs(t)2ss Pik(x) : (5)

1

Substituting this into Eq.(1) we arrive at the following non-homogeneous evolution equations for the QCD content of a weak intermediate boson:

df Bi (x; t)

dt =

ffem

2ss P

Bi (x) + ffs(t)

2ss Xk Pik(x) \Omega f

Bk (x; t) (6)

.

In the lowest ffs order the splitting functions of the longitudinal W and Z bosons vanish and the transverse ones read

Pq\Sigma Z?(x) = P_q\Sigma Z?(x) = zq\Sigma s(x) ; (7) Pd\Gamma W \Gamma

? (x) = P_u+W

\Gamma ? (x) =

1

2 sin2 `W s(x) ; (8)

with s(x) = [x2 + (1 \Gamma x)2]=2.

The QCD splitting functions are taken in the standard form [7]. In the leading-log approximation with ffs(t)

ffs(t) = 2ssbt ; (9) (b = 11=2 \Gamma nf =3 for nf flavours) the t dependence of the equations (6) can be factorized out

f Bk (x; t) ' ffem2ss ~f Bk (x)t (10) leaving integral equations for the x dependence

~f Bi (x) = P Bi (x) + 1b X

k=q;_q;G

Pik(x) \Omega ~f Bk (x) ; (11)

Numerical solutions to the above equations can be found in Ref. [2]. They show in general that, apart from the t-dependent factor which at sub-asymptotic momentum transfers might be different (see discussion in the next Section), the quark and gluon structure of the weak bosons is reacher than that of the photon.

In the above review we have summed over the parton and weak boson polarizations (leaving out the longitudinal bosons which do not contribute in leading-log approximation). The same considerations can be repeated keeping the parton and boson polarizations fixed [8]. In fact, due to the particular form of weak couplings, we shall use in the following calculations partonic densities of given polarization inside polarized W and Z bosons.

3 Higgs production in e-p scattering The dominant mechanism of the Higgs boson production in electron-proton scattering is the W -W and Z-Z fusion (see Fig. 1a) [4, 5]. The new diagrams, which involve gauge boson structure, are shown in Fig. 1b-c. The case with the 'resolved' photon has been already studied some time ago [6]. Here we include in the calculation the W and Z bosons.

The approximations made in the following considerations require a word of warning. First, we will be using the equivalent boson approximation, known since long for the photon [9], and introduced in Ref. [10] for the W and Z. In the case of massive weak bosons, out

2

H a) p

e

g, Z, W g, Z, W

H b) p

G G

e

, Z, Wg

H

c) p

q, q q, q

e

g, Z, W

Figure 1: Diagrams contributing to the Higgs boson production in e-p scattering: a) dominant W -W fusion; b) gluonic part of the electroweak boson structure; c) quark part of the electroweak boson structure.

of the three possible polarization states only the transverse degrees of freedom develop the logarithmic factor, characteristic of photon emission. The density of transversely polarized gauge bosons inside unpolarized electron

f eB(x) / 1x ln Q

2 max + M

2B

Q2min + M 2B ! (12)

with x -- the boson momentum fraction, MB -- the gauge boson mass and Q2 -- the negative momentum squared of the emitted gauge boson. One sees that whereas in the photon case the logarithm is scaled by the electron mass, coming from Q2min, in the weak sector it is the mass M 2B which sets up the scale (explicit forms used in the considered processes are given in the next Section). Consequently, due to the large weak boson mass, the above logarithmic factor is responsible at presently available energies for the fact that there is more 'equivalent' photons in the electron than W 's and Z's. The accuracy of the

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equivalent boson approximation has been tested in the processes where exact calculation is also possible [5]. For example the ratio of the approximate to exact results in the Higgs production in e+e\Gamma scattering varies between a factor of two and few percent in the energy range between 500 GeV and 50 TeV and approaches 1 with increasing Higgs boson mass. Taking into account the above remarks one should treat the numerical results at energies corresponding to the existing accelerators only as estimates.

Another problem is the Z-fl interference. In general the neutral current exchange contains a coherent mixture of the Z and the photon. However in the probabilistic approach used here these possible interference terms are neglected.

The next approximation concerns the quark masses. In the QCD evolution equation all masses are neglected, they are used solely as thresholds, opening new flavour evolution when increasing Q2. A more delicate treatment requires the b quark threshold in the W structure evolution. In principle it appears already above the b quark mass due to the b_u production. However only after the channel b_t opens (pQ2 ? 180 GeV), the Kobayashi-Maskawa matrix elements squared add up to 1 and do not suppress the b quark production. In our calculation we neglect this suppression which potentially exists in the intermediate Higgs boson mass range.

Finally the scale Q20 of the QCD evolution requires some attention. In the leading-log approximation, which we are using, its value for Q2 AE Q20 is formally irrelevant (changing Q0 gives next-to-leading-log corrections). At finite Q2 the choice depends on the physical situation in the process. In e-p scattering we are interested in the electroweak bosons, 'as few off shell as possible' which practically means the mass negative and close to zero. Therefore, as concerns the masses, the difference between photons and weak bosons vanishes and in all cases we should take Q20 = \Lambda 2QCD.

The contribution to the total cross-section for the process e\Gamma p ! *=e\Gamma HX coming from the 'resolved' bosons (Fig. 1 b,c) reads

oeB = X

i;j;ff;fi Z

1 o/

dx

x f

eB

fi (x) Z

1

o/=x

dy

y f

Bfi iff (y)f Pjff o/xy ! ^oeij (13)

where o/ = M 2H=s with MH being the Higgs boson mass and s -- the total c.m. energy squared. The function f eBfi is the boson B (of polarization fi) density inside electron, f Bfiiff is the quark/antiquark/gluon (of polarization ff) density inside the boson B (of polarization fi) and f Pjff -- the antiquark/quark/gluon (of polarization ff) density inside the proton. The sum extends over partons (i; j = q; _q; G), their polarization ff and the boson polarization fi. The polarization-independent partonic cross-section

^oeGG = ss144p2 ` ffsss '

2 jN j2 GFM 2H

s (14) for gluon-gluon (Fig. 1b) and

^oeq_q = p2ss3 GFm

2q

s (15) for quark-antiquark annihilation (Fig.1c). Here GF is the Fermi coupling, mq -- the quark mass and N -- a function of the quark and Higgs boson masses [11].

The equivalent boson densities depend on the exchanged boson and its polarization:

f eW\Sigma (x) = ff4ss iw\Sigma + w\Upsilon (1 \Gamma x)2j 1x ln xs + M

2W

M 2W ! ; (16)

4

ZW

flfl+W+Z

ps = 314 GeV

MH [GeV] oe[pb]

20018016014012010080

10

\Gamma 5

10

\Gamma 6

10

\Gamma 7

10

\Gamma 8

ZW

flfl+W+Z

ps = 1:6 TeV

MH [GeV] oe[pb]

20018016014012010080

10

\Gamma 3

10

\Gamma 4

10

\Gamma 5

Figure 2: Contribution from the 'resolved' electroweak bosons to the cross-section for the Higgs boson production in e-p scattering at: a) ps = 314 GeV and b) ps = 1:6 TeV as function of the Higgs boson mass.

f eZ\Sigma (x) = ff4ss iz\Sigma + z\Upsilon (1 \Gamma x)2j 1x ln xs + M

2Z

M 2Z ! ; (17)

f efl\Sigma (x) = ff4ss i1 + (1 \Gamma x)2j 1x ln (1 \Gamma x)sxm

e ! ; (18)

where

w+ = 12 sin2 `

W ; w

\Gamma = 0; z+ = tan2 `W `1 \Gamma 12 sin2 `

W '

2

; z\Gamma = tan2 `W (19)

and me is the electron mass.

The parton densities f Bfiiff fulfill several relations [8], for example in the case of W \Gamma :

f W

\Gamma d

+ = f

W+ _u\Gamma ; (20)

f W

\Gamma d

\Gamma = f

W+ _u+ ; (21)

f W

+d

+ = f

W+

_d+ = f W

\Gamma _

d\Gamma = f

W\Gamma _u\Gamma = f W

\Gamma u

\Gamma = f

W+u

+ ; (22)

f W

+d

\Gamma = f

W+

_d+ = f W

\Gamma _

d+ = f

W\Gamma _u+ = f W

\Gamma u

+ = f

W+u

\Gamma ; (23)

f W

+G

\Gamma = f

W\Gamma G+ ; (24)

f W

+G

+ = f

W\Gamma G\Gamma : (25)

In the examples shown below we use their explicit asymptotic form following from the numerical solutions of the Eqs.(11). Only in the case of the photon we are able to see whether the asymptotic solutions lead to different results than the more realistic parametrisations

5

gluonsquarks quarks + gluons

p

s = 314 GeV

MH [GeV] oe[pb]

20018016014012010080

10

\Gamma 5

10

\Gamma 6

10

\Gamma 7

10

\Gamma 8

10

\Gamma 9

Figure 3: The 'resolved' W contribution to the cross-section for the Higgs boson production in e-p scattering at ps = 314 GeV as function of the Higgs mass. The gluonic and quark parts plotted separately.

of its structure. We have checked that the parametrisation of Ref.[12] (LAC3) gives the cross-section for the e-p Higgs boson production up to 30% larger in the considered energy and Higgs boson mass range.

The parton densities inside the proton f Pi (x) are taken from the parametrisation of Ref. [13] (MRS3).

The results for scattering energies ps = 314 and 1600 GeV are shown in Figs. 2 and 3. One sees that the 'resolved' W contribution is about a factor of 4 smaller than that of the 'resolved' fl at lower energies ps = 314 GeV and approaches one half at ps = 1:6 TeV (Fig.2). The Z contribution is the smallest for all considered energies and Higgs boson masses. In all cases the quark-antiquark diagram (Fig. 1c) is larger than the gluonic one (Fig. 1b), mainly due to presence of the b quark (Fig 3.). One should keep in mind that the dominant term in the Higgs production cross-section, the W -W fusion (Fig. 1a) [4, 5] exceeds the 'resolved' boson contribution by at least an order of magnitude. The conclusion is rather obvious: the QCD structure of the gauge bosons, used in e-p scattering, does not help in the hunt for the Higgs boson.

4 Summary In the paper we have considered the influence of the QCD structure of the electroweak gauge bosons on the e-p production of the Higgs boson. We have found that using the asymptotic

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form of the 'resolved' W and Z, the contribution to the cross-section of the W structure is of the same order as that of the photon, the Z term being slightly smaller. In general however the considered new diagrams cannot compete with the dominant channel i.e. the W -W fusion.

We have also checked the Higgs boson production via 'resolved' bosons in e+e+ scattering. As expected, the cross-section is suppressed additionally, as compared to the W -W and Z-Z fusion, by a factor of ff2 which multiplies the parton densities and consequently is negligible.

One general lesson follows from the above studies. The W structure function contributes to the considered processes approximately with the same strength as the structure of the photon. This means that one should look for its appearance in the reactions where the 'resolved' photon is known to dominate.

5 Acknowledgments This work has been performed during our visit to DESY, Hamburg. We would like to thank the DESY Theory Group for hospitality and the Volkswagen Foundation for financial support.

References

[1] W. Slomi'nski and J. Szwed, Phys. Lett. B 323 (1994) 427.

W. Slomi'nski and J. Szwed, in High Energy Spin Physics, Proc. of X Int. Symp. on high Energy Spin Physics, Nagoya, Japan, 1992; ed. T. Hasegawa et al. (Universal Academy, Tokyo, 1993);

[2] W. Slomi'nski and J. Szwed, Phys. Rev. D52 (1995). [3] E. Witten, Nucl. Phys. B120 , 189 (1977); C.H. Llewellyn Smith, Phys. Lett. 79B, 83

(1978); R.J. DeWitt et al., Phys. Rev. D19, 2046 (1979); T.F. Walsh and P. Zerwas, Phys. Lett. 36 B (1973) 195; R.L. Kingsley, Nucl. Phys. B 60 (1973) 45.

[4] Z. Hioki et al., Progr. Theor. Phys 69 (1983) 1484; D.A. Dicus and S.D. Willenbrock,

Phys. Rev. D 32 (1985) 1642.

[5] G. Altarelli, B. Mele and F. Pitolli, Nucl. Phys. B 287 (1987) 205. [6] W. Slomi'nski and J. Szwed, Acta Physica Polonica B 22 (1991) 859. [7] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298. [8] W. Slomi'nski and J. Szwed, unpublished. [9] C. Weizs"acker and E.J. Williams, Z. Phys. 88 , 612 1934). [10] G.L. Kane, W.W. Repko and W.B. Rolnick, Phys. Lett., 148B , 367 (1984). [11] R.N. Cahn and S. Dawson, Phys. Lett. 136 B (1984) 196. [12] H. Abramowicz, K. Charchula and A. Levy, Phys. Lett. B 269 (1991) 458. [13] A.D. Martin, W.J. Stirling and R.G. Roberts, Phys. Rev. D 47 867.

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