THES-TP 2000/07

hep-ph/0007110.

May 2000

The process at a Collider.^{†}^{†}†Partially supported by the European
Community grant ERBFMRX-CT96-0090.

G.J. Gounaris, P.I. Porfyriadis

Department of Theoretical Physics, Aristotle University of Thessaloniki,

Gr-54006, Thessaloniki, Greece.

Abstract

The helicity amplitudes for the process are studied to 1-loop order in the minimal SUSY (MSSM) model, where is the CP-odd Higgs particle. Simple exact analytic formulae are obtained, in terms of the and Passarino-Veltman functions; in spite of the fact that the loop-diagrams often involve different particles running along their sides. For a usual mSUGRA set of parameters, is expected. If SUSY is realized in Nature, these expressions should be useful for understanding the Higgs sector.

## 1 Introduction

If in the future
Linear Colliders (LC) [1],
the option to develop high energy
collisions will also be available, then many
new opportunities for new physics (NP) searches
should arise. Employing back-scattering of laser photons,
this option transforms
an^{1}^{1}1In this case it would be best to run LC
in its mode, [2, 3]. LC
to essentially a Collider
() with about of the
initial energy and a comparable luminosity
[2, 3].
The importance of stems from the
fact that the cross sections
for gauge boson and top production in collisions
at sufficiently high energies, are often considerably larger
than the corresponding quantities in the
case [4, 5].

To some extent, such an enhancement should arise for Higgs production also. For the neutral Higgs particles in particular, an may act as a Higgs factory which can be used to study their detail properties, including possible Higgs anomalous couplings [6, 7]. Since the anomalous gauge boson, top and Higgs couplings are interconnected and constitute an important possible source of new physics, an should be very helpful for its identification. In case the NP scale is very high, such forms of NP may be described by the complete list of operators involving gauge bosons and/or quarks of the third family presented in [8].

Alternatively, it may turn out that the NP scale is nearby, as it would be expected in the usual SUSY scenario [9]. In such a case many neutral spinless particles of Higgs and sneutrino type may exist, and an may be used for an s-channel production of the CP-even light and heavy neutral Higgs bosons and respectively, as well as the CP-odd . The study of the various branching ratios, and the polarization of the incoming photons, could then be very helpful to establish and disentangle the nature of these Higgs particles [11, 12].

Once any of these spinless bosons is discovered, its properties should be carefully looked at, in order to be sure that they fulfill the SUSY expectations. Motivated by this, we study in this paper the process in the context of a minimal SUSY model, where no new sources of CP violation, apart from those already known in the Standard Model (SM) Yukawa potential, are assumed to exist. Thus, the various new SUSY couplings are taken to be real, but no specific assumption on their relative magnitudes or signs is made [10]. As we will see below, in such a case, there are only two independent helicity amplitudes for , denoted below as and , where the indices describe the helicities of the incoming photons.

It is also interesting to study the phases of these amplitudes.
The motivation for this stems from the recent observation
in [13, 14], that at c.m energies
, out of the many
independent helicity amplitudes for the processes
,
only the two helicity conserving amplitudes
and are important, which moreover turn out
to be almost purely imaginary^{2}^{2}2For in SM the further
assumption is made that the standard Higgs particle
is light; e.g. below ..
The physical reason for this result is not very
clear [13, 14]. Therefore,
it seems worthwhile to investigate what happens
in other processes, like e.g. the
neutral Higgs boson production, which, as
the neutral gauge bosn production,
also vanish at tree order and they
first appear at the 1-loop level.

Below, in Section 2 we give an overall view of the helicity amplitudes in SUSY. The needed SUSY vertices appear in Appendix A, while the corresponding contributions to the amplitudes are given in Appendix B. The results are expressed in terms of and Passarino-Veltman functions only [15], using expressions analogous to those encountered in the calculation [14]. Finally in Section 3, we give our Conclusions.

Coming now to the related studies already existing in the
literature, we first remark that has
been studied in SM by Jikia [16]. In the non-linear
gauge defined in (A.1) and used here,
the only contributing
diagrams involve or top-loops, similar to those appearing
in Figs.1,3. We have repeated the
calculations of [16]
and agree with the results,
apart from the overall sign of the^{3}^{3}3For the gauge boson
polarization vectors, here and in [13, 14],
we use the same conventions as in [18].
The only difference is that we use the JW convention
[19], which introduces an additional minus
to the polarization vector of a longitudinal
”Number 2” Z and affects
. amplitude.
For the top contributions, our results are fully consistent with
those of [17].
The relevant amplitudes are presented
and compared to those of
at the end of Section 2.

In [20] a calculation of in a general SUSY model has been presented in terms of the general and of Passarino and Veltman functions. The production of two neutral Higgs pairs in SUSY models at hadronic Colliders has also been studied in [21]; where of course the complications from loops involving -bosons, or ”single” and ”mixed” charginos, are avoided. Finally the processes and have also appeared in a non-Supersymmetric gauge model involving a two Higgs doublet scalar sector [22, 23].

## 2 An overall view of the amplitudes.

The invariant helicity amplitudes for^{4}^{4}4We use the same
conventions as in [13, 14].
are denoted as , where
describe the helicities of the incoming photons, and
the kinematics are defined in Appendix B. Assuming that the SUSY
Higgs potential is CP-invariant we get (see (B.2))

(1) |

which implies that there are only two independent helicity amplitudes, and .

As in [13, 14], we employ the non-linear gauge of [24], which implies the gauge fixing and FP-ghost interactions of (A.1, A.2), leading to the conclusion that there are no , vertices. The diagrams contributing to -pair production are then given in Figs.1-4.

The contribution to the and amplitudes from the diagrams in Fig.1 consists of two types. The first is induced by the two diagrams in the first line in Fig.1 and describes the -pole contributions appearing in (B.19, B.20). The diagrams in the second to last line of Fig.1 involve loops in which and/or are running along their internal lines. These induce the second type of contributions contained in (B.22, B.23), and expressed in terms of the -functions explained in (B.8-B.14); as well as the functions , , defined in (B.15, B.16). The contributions (B.22, B.23) give the largest effect to the amplitudes, for the numerical applications considered below.

The chargino loop contribution is described by the diagrams in Fig.2. It consists also of an -pole contribution given in (B.24); the box contributions involving a ”single chargino”-loop giving (B.26, B.27); and the ”mixed chargino” contribution (B.28, B.29), arising when both charginos are running along the loop. Analytically, the later is the most complicated one. Nevertheless, it is simple enough to be possible to write it. Numerically, it has to be taken into account only when both charginos are relatively light.

The and quark contributions are described by the diagrams in Fig.3. They are given in (B.31) for the -pole contribution, and in (B.33, B.34) for the box diagrams.

As an example of a sfermion contribution, we only considered the one arising from the -loop, described by the diagrams in Fig.4. Their contributions are given by (B.35-B.38).

Table 1: mSUGRA parameters in Figs. 5-7. [25].

mSUGRA(1) | mSUGRA(2) | mSUGRA(3) (light stop) | |

3 | 30 | 3 | |

at the Unification scale | |||

100 | 160 | 100 | |

200 | 200 | 200 | |

0 | 600 | -715 | |

+ | + | + | |

at the Electroweak scale | |||

152 | 150 | 153 | |

316 | 263 | 435 | |

375 | 257 | 489 | |

97.7 | 108 | 101 | |

379 | 257 | 490 | |

383 | 269 | 495 | |

-373 | -258 | -500 | |

128 | 132 | 138 | |

346 | 295 | 454 | |

295.4 | 353 | 133 | |

494.2 | 469 | 491 |

For the numerical applications we use the three CP-invariant mSUGRA set of parameters introduced in [25, 10] and presented in Table 1. For the electromagnetic coupling we take . The results are shown in Figs.5-7.

The real and imaginary parts of the helicity amplitudes and are presented in Figs. 5-7 [26]. As indicated there, the most important contributions to the amplitudes arise from the -loop diagrams presented in the 2nd to last line of Fig.1 and appearing in (B.22, B.23). At sufficiently high energies, these contributions are mainly imaginary. But the predominance of the imaginary parts of the amplitudes is not so strong, as the one observed in the gauge boson production cases [13, 14].

As indicated in Figs. 5-7, the chargino contribution is generally quite important; while the -quark contribution is somewhat smaller; and the stop contribution is negligible for the above cases.

For the -quark contribution we also remark that in the mSUGRA(1) and mSUGRA(3) cases, where is small, the -contribution is negligible compared to the top one. On the contrary, for the mSUGRA(2) case of , the -quark contribution may be more important than the -quark one.

For comparison, we have also looked at the and amplitudes for in the Standard Model. The results for are given in Fig.8. For the we find that the -loop contributions is comparable the -one, and the amplitude is never particularly imaginary. It is only for , for which there is no Higgs-pole contribution; that at energies , the -loop is more important than the top-one, and the imaginary part of the amplitude becomes predominant.

The unpolarized cross section for the sets of parameters in Table 1, are given in Fig.9. It lies in the range of , which is similar but somewhat smaller, than the result expected for in SM for [16]. This result does not seem particularly sensitive to SUSY parameters like e.g. ; but mainly depends on the -mass. It should also be compared to the situation for a single or production studied in [27]. We also remark that a cross section at the -level may be observable, if a luminosity of e.g. is realized in TESLA [2, 3].

## 3 Conclusions

The Higgs sector, which is responsible for giving masses to almost all particles immediately after our Universe started, is definitely the most fascinating part of the present elementary particle theory. Motivated by this and assuming that the SUSY option is chosen by nature, we have studied here the process .

In the non-linear gauge used here, the types of contributing diagrams may be divided into two categories constructed on the basis of whether an s-channel neutral Higgs-pole is involved or not. Each category may then be further divided into three classes, on the basis of whether their loops involve the -pair, charginos or sfermions. General formulae have been presented which allow the description of the process in any SUSY model, minimal or non-minimal.

For the numerical applications we only considered three SUGRA examples presented in Table 1, leading to an heavier than . Excluding the forward and backward regions, the cross-section is found in the region.

At sufficiently high energies, both amplitudes and are found to be to largely imaginary; an effect reminiscent, but not so predominant, as the one noticed in neutral gauge boson production [13, 14]. On the contrary, nothing like this appears for the amplitude in , in the Standard Model. It seems that the predominance of the imaginary part of a loop amplitude at high energies, is somehow associated with the predominance of a -involving loop. The understanding of such properties may be useful for new physics searches; since e.g. for they determine the way the interference between the ”old” and possible forms of ”new” physics may appear [28].

Thus, after the discovery of and the study of the single production process [11], the study of the double production through , should certainly be useful for verifying the Higgs identification.

Appendix A: The MSSM vertices for .

In order to reduce the number of diagrams contributing to , we use the nonlinear gauge defined by the gauge fixing term

(A.1) |

which is free from and vertices [24]. The implied ghost-photon and ghost-scalar field interactions then are

(A.2) | |||||

The complete list of diagrams contributing to the process
in the present gauge, appear in
Figs.1-4. Below we give the interaction
Lagrangian describing the vertices for these sets of diagrams.

The diagrams in Fig.1 describe the loop contribution (together of course with the accompanying ghost and Goldstone ones). The relevant vertices involve, in addition to the gauge boson self-interactions present in SM, also the triple and quartic vertices [9]

(A.3) | |||||

On the basis of this we define the -couplings
^{5}^{5}5For the definition of the scalar sector mixing angles we
follow the standard notation of e.g. [10].

(A.4) |

and the -couplings

(A.5) |

which are used in Appendix B for expressing
the (Higgs-) loop
contribution of the diagrams in Fig.1,
as well as the s-channel -pole diagrams
contained in Figs.2-4.

The chargino loop contribution is described
by the diagrams in
Fig.2. To define them we first list
the chargino mass matrix term as^{6}^{6}6The
gaugino fields are defined so that
they satisfy . In such
a case there is no in front of
in (A.6).

(A.6) |

Assuming that in MSSM there no new sources of CP-violation, apart from those already known in the Yukawa part of SM; we take the quantities as real, but of arbitrary sign. is the usual charge conjugation matrix, and the index indicates transposition of the spinorial field. In terms of

(A.7) |

the physical chargino masses are expressed as

(A.8) |

The mixing-angles in the and sectors respectively, are defined so that they always lie in the second quarter

(A.9) |

They are written as

(A.10) |

We always describe the chargino field so that it absorbs a positive chargino particle; i.e. . Using this and the sign-quantities

(A.11) |

the neutral gauge boson-chargino couplings are written as

(A.12) | |||||

with

(A.13) | |||||

(A.14) |

(A.15) |

where the sign-factors
are given
in^{7}^{7}7 These expressions are equivalent to those given e.g.
in [29], where a more common definition
of the -angles is employed. (A.11).

The corresponding chargino-neutral Higgs vertices are

(A.16) | |||||

where

(A.17) | |||||

The appearance in
(A.15, A.17) of the
sign-factors defined in (A.11), guarantees
that the physical charginos always have positive masses;
irrespective of the signs
of the arbitrary real parameters and .
These signs are of course intimately related to the definition of
the chargino mixing angles employed in
(A.9, A.10).

We next turn to and quark loop contribution. The relevant diagrams for the -quark case are shown in Fig.3. The necessary vertices are determined by

(A.18) | |||||

where are the and quark charges. The implied quark couplings are

(A.19) |

and correspondingly fro the -couplings.

Finally, for the stop loop contribution, the relevant interaction Lagrangian is

(A.20) | |||||

where, as usual, .
The various neutral Higgs- couplings are determined
from the coefficients of the various terms
in (A.20). For two examples, we note^{8}^{8}8As usual,
in the definition of the
and
couplings from the last two terms in
(A.20), the relevant coefficient is multiplied by
a 2, due to the identity of the two -fields.

For determining the corresponding couplings to the physical we write

(A.21) |

where is fully determined by^{9}^{9}9 The quantities
, in (A.22)
are the usual soft SUSY breaking
parameters in which the small D-contributions have also been
included.

(A.22) |

while is also real. We observe from (A.22) that

since , by definition. We have checked that this stop-mixing-formalism is equivalent to the usual one found e.g. in [9, 10, 30, 14].

Appendix B: The MSSM contributions to .

The invariant helicity amplitudes for the process

(B.1) |

are denoted as^{10}^{10}10Their sign is related to the sign of
the