

 11 Nov 94

BI-TP 94/51 UWITP-94/04

Probing the Higgs mechanism via fl fl ! W+W

\Gamma

A. Denner, R. Schustery Institut f"ur Theoretische Physik, Universit"at W"urzburg

Am Hubland, D-97074 W"urzburg, Germany

S. Dittmaierz Theoretische Physik, Universit"at Bielefeld Universit"atsstrasse, D-33501 Bielefeld, Germany

Abstract: We investigate the sensitivity of the reaction flfl ! W+W\Gamma to the Higgs sector based on the complete one-loop corrections in the minimal Standard Model and the gauged nonlinear oe-model. While this sensitivity is very strong for the suppressed cross-section of equally polarized photons and longitudinal W bosons, it is only marginal for the dominant mode of transverse polarizations. The corrections within the oe-model turn out to be UVfinite in accordance with the absence of log MH terms in the Standard Model with a heavy Higgs boson.

BI-TP 94/51 UWITP-94/04 November 1994

ySupported by the Deutsche Forschungsgemeinschaft. zSupported by the Bundesministerium f"ur Forschung und Technologie, Bonn, Germany.

1 Introduction

All present experimental results on electroweak physics confirm the conception that electromagnetic and weak interactions are unified in a SU(2) \Theta U(1) gauge theory. However, the underlying field theory cannot be of pure Yang-Mills type since the weak gauge bosons, the W\Sigma and Z boson, are empirically known to be massive. In the electroweak Standard Model (SM) this problem is solved by the well-known Higgs mechanism [1], i.e. by breaking the gauge symmetry spontaneously via a non-vanishing vacuum expectation value of an additional complex scalar SU(2) doublet. Whereas three of these four scalar fields are absorbed by the longitudinal degrees of freedom of the massive gauge bosons, a physical scalar field survives, the so-called Higgs boson. Of course, the Higgs mechanism cannot be conclusively confirmed before this particle is empirically detected. On the other hand, the Higgs-boson mass MH, which is a free parameter of the theory, enters all theoretical predictions within the SM at least via higher orders. Since the Higgs-mass dependence of low-energy observables turns out to be very mild, more precisely at most logarithmic at the one-loop level, only crude bounds on MH can be obtained from radiative corrections (RCs) to current precision measurements. Experimentally, the Higgs mass is only constrained by the lower bound MH ?, 60 GeV from LEP [2] but can well be in the TeV range.

The Higgs boson can be removed from the physical particle spectrum in two different ways. On the one hand, amplitudes can be calculated within the SM for finite MH, and subsequently asymptotically expanded for MH ! 1. Alternatively, the physical Higgs field can be eliminated by constraining the square of the Higgs-doublet field to be constant and equal to its (non-vanishing) vacuum expectation value. Then no physical Higgs particle exists from the beginning, but one is forced to introduce a non-linear representation of the Higgs sector leading to a non-renormalizable gauged non-linear oe-model (GNLSM). The relation between the heavy-Higgs limit and the GNLSM has been investigated for a SU(2) gauge theory and the SU(2) \Theta U(1) SM in Refs. [3] and [4,5], respectively. As expected, MH acts as an effective UV cut-off. The corresponding (logarithmic) one-loop divergences in the GNLSM can be identified with the log MH terms in the SM only up to finite constants, which have been calculated in Ref. [5]. Although the GNLSM is manifestly non-renormalizable, and its observables in general violate unitarity in the high-energy limit, an investigation of the GNLSM seems reasonable since it is equivalent to the SM in the unitary gauge with the physical Higgs field omitted. Consequently, by comparing theoretical predictions within the GNLSM and the SM for varying MH one may get insight into the influence of the mechanism of spontaneous symmetry breaking on specific observables. The discussion of these aspects for the cross-section of flfl ! W+W\Gamma represents the main issue of this paper.

The process flfl ! W+W\Gamma will be one of the most important reactions at future flfl colliders. In particular, the measurement of the corresponding cross-section yields direct information on possible anomalous flWW and flflWW couplings [6] widely independent of the couplings between Z and W\Sigma bosons. Moreover, a Higgs boson with a mass of several hundred GeV can be studied via the resonance contribution flfl ! H\Lambda ! W+W\Gamma , which is present owing to the flflH coupling induced at one-loop order. Since the structure of

1

this Higgs resonance has already been discussed in the literature [7,8], here we mainly concentrate on the case when the centre-of-mass energy is far below the Higgs mass MH.

We have calculated the full one-loop RCs to flfl ! W+W\Gamma including soft-photon bremsstrahlung both in the SM and GNLSM. A complete discussion of the SM RCs will be published elsewhere [9]; here we focus on the MH dependence of the SM corrections and their difference to the ones within the GNLSM. Despite of the non-renormalizability of the GNLSM, the corresponding one-loop RCs to flfl ! W+W\Gamma turn out to be ultraviolet finite. This fact is related to the absence of log MH terms in the SM corrections. The limit MH ! 1 indeed exists for the SM one-loop corrections, but for longitudinal polarized W bosons these one-loop corrected cross-sections violate unitarity for energies in the TeV range, as it is also the case in the GNLSM.

The paper is organized as follows: In Section 2 we discuss the MH dependence of the SM RCs and their difference to the ones within the GNLSM. The unitarity-violating effects for longitudinal W bosons are investigated in Section 3. Numerical results are presented in Section 4. Section 5 contains our conclusions.

2 Heavy-Higgs Standard Model versus gauged non-linear oe-model

The GNLSM is related (see e.g. Ref. [10]) to the SM in the unitary gauge without Higgs field by a Stueckelberg transformation [11]. Comparing the Lagrangians, one finds that the Feynman rules involving at most one unphysical scalar field are identical in the GNLSM and the SM with linearly realized Higgs sector. Vertices with at least two scalar fields are in general different. In particular, the WW'' and WWO/O/ couplings vanish in the GNLSM. By ' and O/ we denote the charged and neutral unphysical scalar fields, respectively. For the reaction flfl ! W+W\Gamma at one loop one simply has to omit all graphs that contain internal Higgs fields, or WW'' or WWO/O/ couplings in order to obtain the GNLSM results from the SM ones.

Obviously, the tree-level amplitudes agree in both models yielding

MBorn = 4ssff( 2M 2

W \Gamma t h2("

1 \Delta "2)(k1 \Delta "\Lambda +)(k2 \Delta "\Lambda \Gamma ) + 2("\Lambda + \Delta "\Lambda \Gamma )(k+ \Delta "1)(k\Gamma \Delta "2)

\Gamma 2("1 \Delta "\Lambda +)(k1 \Delta "\Lambda \Gamma )(k\Gamma \Delta "2) \Gamma 2("2 \Delta "\Lambda \Gamma )(k2 \Delta "\Lambda +)(k+ \Delta "1) + 2("1 \Delta "\Lambda \Gamma )(k1 \Delta "\Lambda +)(k\Gamma \Delta "2) + 2("2 \Delta "\Lambda +)(k2 \Delta "\Lambda \Gamma )(k+ \Delta "1) + s("1 \Delta "\Lambda +)("2 \Delta "\Lambda \Gamma )i \Gamma ("1 \Delta "2)("\Lambda + \Delta "\Lambda \Gamma )) + (`1' $ `2'; t ! u): (1) The momenta k and polarization vectors " of the incoming photons are labelled by `1',`2', the ones of the outgoing W\Sigma bosons by `\Sigma ', respectively; they are explicitly defined in the centre-of-mass (CM) system in Ref. [9]. The Mandelstam variables are given by

s = (k1 +k2)2 = 4E2; t = (k1 \Gamma k+)2 = M 2W \Gamma s2 (1\Gamma fi cos `); u = 2M 2W \Gamma s\Gamma t; (2)

2

with fi = q1 \Gamma M 2W=E2 denoting the velocity of the W bosons, and ` representing the scattering angle between photon `1' and W+.

For longitudinal W bosons the lowest-order matrix elements read explicitly

MBorn (*1 = *2; *\Sigma = 0) = 4ssff 2sM

2W

(M 2W \Gamma t)(M 2W \Gamma u) gjq

2jAEM2

W 4ssff

2sM 2W

ut ;

MBorn (*1 = \Gamma *2; *\Sigma = 0) = \Gamma 4ssff 2(s + 4M

2W)(ut \Gamma M 4W)

sfi2(M 2W \Gamma t)(M 2W \Gamma u) gjq

2jAEM2

W \Gamma 8ssff: (3)

Here and in the following we denote the Mandelstam variables s; t; u generically by q2. Note that in the high-energy limit jq2j AE M 2W the amplitude for equal photon helicities vanishes and that the other one contains no t- and u-channel pole in the leading term.

The calculation of the one-loop amplitude for flfl ! W+W\Gamma is simplified considerably by use of a non-linear gauge-fixing condition for the W-boson field, suggested in Ref. [12], rendering the fl'W coupling zero [9]. Using this gauge-fixing condition in the GNLSM as well, we have evaluated the difference of the one-loop matrix elements for flfl ! W+W\Gamma in the SM and the GNLSM

ffiMH = ffiMSM \Gamma ffiMGNLSM; (4) where ffiM always denotes one-loop contributions to the amplitude. In the limit of very large Higgs mass, M 2H AE jq2j; M 2W our result simplifies to

ffiMHfifififi

MH!1 =

ff2 3s2W(M 2W \Gamma t) (s("1 \Delta "

\Lambda +)("2 \Delta "\Lambda \Gamma ) + 2("1 \Delta "2)(k1 \Delta "\Lambda +)(k2 \Delta "\Lambda \Gamma )

\Gamma ("1 \Delta "\Lambda +)(k1 \Delta "\Lambda \Gamma )(k\Gamma \Delta "2) + ("1 \Delta "\Lambda \Gamma )(k1 \Delta "\Lambda +)(k\Gamma \Delta "2) \Gamma ("2 \Delta "\Lambda \Gamma )(k2 \Delta "\Lambda +)(k+ \Delta "1) + ("2 \Delta "\Lambda +)(k2 \Delta "\Lambda \Gamma )(k+ \Delta "1)) + (`1' $ `2'; t ! u); (5) whereas the exact analytical form of ffiMH for arbitrary Higgs mass is not very illuminating. In this context, we mention that we have derived (5) also using the effective Lagrangian for the difference of the SM limit MH ! 1 and the GNLSM given in Refs. [4,5].1

The logarithmic one-loop UV divergences occuring in the non-renormalizable GNLSM are directly related to the log MH terms in the SM with a heavy Higgs boson, i.e. MH can be regarded as an effective UV cut-off in this limit. This fact has already been pointed out in Ref. [4] and shown by explicit calculation in Ref. [5]. Thus, the absence of log MH terms in ffiMSM and the UV finiteness of ffiMGNLSM have the same root, however, the difference ffiMH is non-vanishing even for MH ! 1. Of course, all results derived in non-renormalizable models are not free from ambiguities or assumptions that fix these ambiguities so that such results have to be interpreted carefully. But the finiteness of ffiMGNLSM shows that the prediction for flfl ! W+W\Gamma within the GNLSM is independent

1More precisely, some missing counterterms involving ff11 had to be supplemented in the Feynman rules of Ref. [4], and the finite parts of the contributing ffi could be taken from Ref. [5].

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of any cut-off \Lambda whatever regularization procedure may be used. Note that such a \Lambda will play a role as "scale of new physics" if the GNLSM is embedded into a more complete field theory like the SM (where \Lambda , MH) or even beyond. Moreover, the difference ffiMH indicates to which extent the SM prediction might be modified by effects of new physics concerning the Higgs sector.

3 Production of longitudinal W bosons in the high-energy limit

Observables involving longitudinally polarized massive gauge bosons are most sensitive to deviations from the Yang-Mills interactions and the mechanism of spontaneous symmetry breaking of the underlying gauge theory for energies far above the scale of the gauge-boson masses. This is due to the well-known "gauge cancellations" which guarantee that the enhancement factor E=M of the longitudinal polarization vector (of a vector boson with energy E and mass M ) is cancelled between the individual contributions to the S-matrix elements. The equivalence theorem (ET) [13] states that in the SM the leading contribution to amplitudes involving external longitudinally polarized gauge bosons can be simply obtained by the replacement of this vector field by the corresponding unphysical scalar field, if all energy scales q2i are far above all masses mi, jq2i j AE m2i . Moreover, the ET can be generalized to the heavy-Higgs SM, jq2i j; M 2H AE m2i , and the GNLSM [14,15].

Applying the ET to flfl ! W+L W\Gamma L within the heavy-Higgs SM and the GNLSM, all one-loop RCs of the order M 2H=M 2W, q2=M 2W (q2 = s; t; u) can be obtained from Feynman diagrams of flfl ! '+'\Gamma involving only scalar inner particles, as can be deduced by power counting [15]. In the heavy-Higgs SM these diagrams are shown and calculated in Ref. [7] for equal photon helicities. The result for flfl ! W+L W\Gamma L with general photon helicities is given by

ffiMSM (*1 = *2; *\Sigma = 0) , \Gamma ff

2M 2H

2s2WM 2W (

M 2H s \Gamma M 2H + iMH\Gamma H + 2)

+ ff

2M 2H

2s2WM 2W (

M 2H

t

M 2H

t \Gamma 1! log 1 \Gamma

t M 2H ! +

M 2H

t + (t $ u)) ; (6)

ffiMSM (*1 = \Gamma *2; *\Sigma = 0) , \Gamma ff

2M 2H

s2WM 2W

\Theta ( M

2H(M 2H \Gamma t)

2ut "log

\Gamma s \Gamma iffl

M 2H ! log 1 \Gamma

t M 2H ! + Li2

t M 2H ! \Gamma

u

t log 1 \Gamma

t M 2H !#

\Gamma M

2H

2t +

1 8 +

M 2H(u \Gamma s \Gamma 2M 2H)

4st "Li2 1 +

s + iffl

M 2H ! \Gamma

ss2

6 # + (t $ u)) ; (7)

for s; \Gamma t; \Gamma u; M 2H AE M 2W; where (6) is in agreement with Ref. [7].2 The graphs which are relevant in the GNLSM are shown in Fig. 1. Note that the corresponding '''' coupling in the GNLSM implicitly

2The difference in the global sign is due to deviating phase conventions for the polarization vectors.

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jError: /typecheck in --put--
Operand stack:
--nostringval-- --nostringval-- --nostringval-- --nostringval-- 40.4972 467.291 --nostringval-- (j) 0 6.02973
Execution stack:
%interp_exit .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- false 1 %stopped_push 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- 1 1 0 --nostringval-- %for_pos_int_continue --nostringval-- --nostringval-- () --nostringval-- %string_continue --nostringval-- --nostringval-- --nostringval-- --nostringval--
Dictionary stack:
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Current allocation mode is local

