###### Abstract

The scalar and vector leptoquark pair production cross sections in hadronic collisions are calculated. In a model independent analysis we consider the most general and conserving couplings of gluons to both scalar and vector leptoquarks described by an effective low–energy Lagangian which obeys invariance. Analytrical expressions are derived for the differential and integral scattering cross sections including the case of anomalous vector leptoquark couplings, and , to the gluon field. Numerical predictions are given for the kinematic range of the TEVATRON and LHC. The pair production cross sections are also calculated for the resolved photon contributions to at HERA and LEP LHC, and for the process at possible future linear colliders and colliders. Estimates of the search potential for scalar and vector leptoquarks at present and future high energy colliders are given.

DESY 96–174

October 1996

Leptoquark Pair Production in Hadronic

Interactions

Johannes Blümlein, Edward Boos, and Alexander Kryukov

DESY –Zeuthen,

Platanenallee 6, D–15735 Zeuthen, Germany

Institute of Nuclear Physics, Moscow State University,

RU–119899 Moscow, Russia

## 1 Introduction

In many extensions of the Standard Model new bosons are predicted which carry both lepton and baryon number [1]. There is indeed a close relation between these quantum numbers in the Standard Model since the triangle anomalies are cancelled by the requirement

(1) |

for each fermion family, which renders the theory renormalizable. Here , and denote the electromagnetic, left-, and right-handed neutral current charges, respectively. Leptoquark states emerge naturally as the gauge bosons in Grand Unified Theories [2]. In these scenarios their couplings are not baryon number conserving and their masses are situated in the range . On the other hand, leptoquarks with B- and L-conserving couplings may exist in the mass range accessible at high energy colliders. These states will be considered in the present paper. They may be either sought through virtual effects in low energy processes, from which already severe constraints on their fermionic couplings were derived [3], or searched for at high energy colliders as at LEP [4], HERA [5], and TEVATRON [6]. These searches constrained further the allowed mass and coupling ranges being limited by other experiments [7] previously. At present the most stringent constraints come from the TEVATRON and exclude leptoquarks associated with the first and second family with masses in the range [8]. For the third generation scalar leptoquarks currently the mass range is excluded.

In the case of single leptoquark production the scattering cross sections
are always proportional to one
of the fermionic couplings [9],
, which are
constrained to be rather small from low energy
processes up to masses of
^{1}^{1}1For even higher
masses the fermion couplings of leptoquarks can still be of the
order . Single production at proton colliders
has been studied
in [10] recently.. Thus the search limits always
lead to combined bounds on the couplings and
the leptoquark masses.
For leptoquark pair production, however, the small size
of the fermionic
couplings does not severely constrain
the scattering cross sections due to the finite bosonic contributions.
The strength of the leptoquark couplings to the gauge bosons,
, and , is determined by the coupling constant
of the respective gauge field up to eventual anomalous contributions
in the case of vector leptoquarks,
which are described by the parameters and ,
with
, and . As will be shown for the hadronic
contributions to the different
scattering cross sections, a combination of
anomalous couplings exists
for which they become minimal. Therefore one may constrain the
allowed mass ranges for leptoquarks directly. As will be shown,
the corresponding minimizing
couplings are in general not those of the Yang-Mills type
or the minimal vector boson couplings.

In the present paper the cross sections for scalar and vector leptoquark pair production in hadronic interactions are calculated. They apply for and scattering at the TEVATRON and LHC, respectively, but also for hadronic interactions at , , and colliders. In the latter cases they emerge as the resolved photon contributions which accompany the respective direct photon processes [11]–[13].

The paper is organized as follows.
In section 2 the basic notations are introduced. Section 3 contains the
derivation of the partonic scattering cross sections for scalar
and vector leptoquark pair production including the anomalous
couplings and . Here it is also shown how the
scattering cross sections for the scalar and the vector case, in
the absence of anomalous couplings, can be obtained using factorization
relations of the amplitude. The hadronic production cross sections are
derived in section 4. For , , and scattering
the direct photon contributions
calculated previously [11]–[13]
are added to obtain a complete description. The numerical results
are presented in section 5 ^{2}^{2}2The code for
the different
processes can be obtained on request from .
A detailed description of the code LQPAIR 1.0 is given
in [14].
and section 6 contains the conclusions. The Feynman rules used in the
present calculation are summarized in Appendix A. Appendix B contains
the coefficients which
describe the differential and integrated pair
production cross sections for vector leptoquarks in the presence of
anomalous couplings.

## 2 Basic Notations

Following earlier investigations [11]–[13],[15]
we consider the class
of leptoquarks introduced in [16]. The fermionic couplings
of these states are dimensionless, baryon and lepton number conserving,
family–diagonal, and invariant.
These
leptoquarks are color triplets.
As outlined in ref. [15], these conditions do widely induce also
the couplings to the gauge bosons of the Standard Model.
For the scalar states their
bosonic couplings are determined completely. In the case of
vector leptoquarks, the
Yang–Mills type couplings may be supplemented by anomalous couplings
which are specified by two parameters and .
The couplings and ,
corresponding to different gauge fields,
are not generally related.
The hadronic processes depend on
the parameters
and only. Since
most of the fermionic couplings
of the
leptoquarks are bounded to be very small in the mass range up to
[3],
we will neglect their contribution
in the following and consider pair production through
bosonic couplings only^{3}^{3}3
The hadronic pair production cross sections
for leptoquarks of different flavor or
is possible through quark or quark–antiquark scattering
and depends on the fermionic couplings as ..

The effective Lagrangian describing the interaction of the scalar and vector leptoquarks with gluons is given by

(2) |

where

(3) |

(4) |

Here, denotes the strong coupling constant, are the generators of , and are the leptoquark masses, and and are the anomalous couplings. The field strength tensors of the gluon and vector leptoquark fields are

(5) |

with the covariant derivative given by

(6) |

The parameters and are assumed to be real. They are related to the anomalous ’magnetic’ moment and ’electric’ quadrupole moment of the leptoquarks in the color field

(7) |

Since we wish to keep the analysis as model independent as possible we assume that these quantities are independent. At present there are no direct bounds on the parameters and . Below we will consider the range of . The hadronic production cross sections are found to vary significantly for parameters in this range. As will be shown, searches at the TEVATRON will be able to constrain this range further. The above choice covers both the cases of Yang–Mills type couplings, , and the minimal vector couplings, .

The Feynman rules relevant for the processes studied in the present paper are summarized in Appendix A.

## 3 Partonic Cross Sections

Before we study the pair production cross sections for leptoquarks at different colliders we present the partonic cross sections. The diagrams of the contributing subprocesses and are shown in figure 1 and 2. Let us first consider the case of vanishing anomalous couplings, i.e. the simplified situation in which the production cross sections depend on the gauge coupling only.

In non-Abelian gauge theories the amplitudes for a series of scattering processes can be factorized into a group and a Lorentz part [17]. As will be shown this applies to and for the special case of vanishing anomalous vector couplings . This representation also yields a particularly simple result for the differential cross sections in comparison with expressions obtained otherwise [18, 19]. The former case has been dealt with in ref. [17] using this method, however, the scattering cross section obtained disagrees with other results [18]. Therefore we will recalculate the cross sections for both cases showing the factorization of the group factor and the Lorentz part in detail before deriving the more general result for finite anomalous couplings and .

### 3.1 Factorization Relations

We use a physical gauge for the gluon fields. The gluon polarization vectors obey and . The matrix elements for the above processes can be written as

(8) |

with the channel index, , and . The numerators in the matrix element have been written in terms of the respective group factors and the Lorentz parts . The following relations are valid.

(9) | |||||

(10) | |||||

(11) |

Since , , and , eq. (10) is the commutation relation of the generators of . The Lorentz terms consist out of the and channel terms supplemented by the parts of the sea–gull diagram (see figure 1) corresponding to the group factor and , respectively. The contributions to the Lorentz parts and are given in table 1 for the scalar and vector case. One finds that

(12) |

with the gluon polarization vectors. These relations result into

(13) |

has thus the form of an Abelian amplitude. Since the leptoquarks described by (2) are color triplets or antitriplets one obtains

(14) |

Scalar Field | Vector Field, | |
---|---|---|

Table 1: contributions to the Lorentz part of
the scattering amplitudes
. The polarization vectors of
the vector field are denoted by .

The Lorentz parts of the amplitude yield

(15) |

and

(16) |

Comparing (13) with other results obtained in
earlier calculations, we agree with the scattering cross sections
for scalar pair production derived in [18]
but disagree with those found
in [17, 20, 21]^{4}^{4}4
We checked that the difference in
in ref. [20] could be explained by leaving out the ghost term
in the Feynman gauge.
The result obtained in
ref. [22] agrees with that given in [20] and
was used to derive numerical
results. Later the same authors revised this expression,
see [23],
and agree with eq. (18).
Since ref. [11] in [24] refers to two different
expressions for it
remains
unclear on which relation the numerical calculation presented
was based..
Our result for pair production of vector color
triplets agrees with that
given in [19, 21]^{5}^{5}5A numerical illustration of this
relation has been given in [25] for the gluonic contributions
recently..

### 3.2 Scalar Leptoquarks

The differential and integral pair production cross sections for and scattering are

(17) | |||||

(18) |

and

(19) |

(20) |

with ,
, the cms
energy and the leptoquark scattering angle in the parton--parton
cms. All quark flavors have been dealt with
as massless^{6}^{6}6This is a sufficient approximation in the mass
and energy range
and considered in
the present paper..
Eqs. (17, 18) result from (13) directly.
Eqs. (19,20) are known for a long time,
cf. ref. [26].

### 3.3 Vector Leptoquarks

While for the case of vanishing anomalous couplings,
, the pair production cross section
follows from eq. (13) it cannot be
derived by the technique discussed in section 3.1
for finite anomalous couplings because
eq. (12) does not hold. For the
general case we have performed the calculation
of by
CompHEP [27] in the Feynman gauge^{7}^{7}7The
Feynman rules given in appendix A have been implemented into
CompHEP as a new model..
We also checked gauge invariance explicitly
using FORM [28] in the gauge
with a free gauge parameter. For these calculations,
the diagrams in figure 1 must be supplemented
by an initial state ghost contribution.

The differential and integral pair production cross sections for scattering are

(21) |

with

(22) |

(23) |

(24) |

The functions and are obtained after a lengthy calculation and are given in appendix B. Similar to the case of [11] and [13] scattering, the contributions linear in either or do not contain unitarity violating pieces .

For scattering the cross section reads

(25) |

with

(26) |

The integrated cross section is

(27) |

where

(28) |

The functions
and are
listed in appendix B. For
eqs. (25,27) agree
with those found in [21]
specifying the color factor^{8}^{8}8Note that this result disagrees
with eq. (4) in [25] by a factor of ..
Relations for
the special cases
[29] and [15] have been obtained for other reactions
previously.

Contrary to the case of gluo-production (23), terms are contained even in the contribution because we did not impose a relation between the fermionic and bosonic couplings of the leptoquarks for Yang-Mills type leptoquark-gauge boson couplings. To restore unitarity, graphs with lepton exchange would have to be added for the process . The value required for the adjusted fermionic couplings, however, would be too large to be consistent with the limits derived in [3]. The Lagrangian (2) is assumed to parametrize leptoquark interactions for not too large energies, i.e. in the threshold range. It has to be supplemented by further terms restoring the correct high energy behaviour for . These terms are model dependent and are related to the specific scenario leading to leptoquarks in the mass range of to . Approaching high energies, symmetry breaking scales are passed and the respective Higgs terms contribute.

## 4 Production Cross Sections

Subsequently we will calculate the differential and integral hadronic production cross sections for leptoquark pair production at different colliders. In the present paper we will apply the collinear parton model to describe the initial state of the hard scattering process . The densities are the parton densities in the case of the or collisions. For the resolved photon contributions in collisions one of the densities is the probability for finding a quark, antiquark, or gluon in the electron in a neutral current process. These distributions are described by the convolution

(29) |

where is the photon density in an electron and denotes the density of parton in the photon. The convolution of the densities is given by

(30) |

In the same way the parton densities for photoproduction of leptoquark pairs at colliders are described. For colliders, at which the photon beams are prepared by laser back–scattering, the distribution is given by the Compton spectrum , eq. (59).

In , , and scattering, in addition to the resolved photon subprocesses, direct contributions due to and fusion are present which have been studied in refs. [11] and [12, 13], respectively.

In terms of the generalized partonic distributions the differential cross section reads:

(31) |

Here, denotes the rapidity of one of the leptoquarks,

(32) |

with and the leptoquark energy and longitudinal momentum. is the transverse momentum in the laboratory frame, , and

(33) |

The Mandelstam variables and in the cms are

(34) | |||||

(35) |

The differential cross sections in the partonic sub-systems have been given in eqs. (17,19,21,25) in section 3. The single differential distributions and are derived from eq. (31). Eq. (33) constrains the rapidity and ranges to

(36) | |||

(37) |

where .

Finally the integral cross sections are

(38) |

To be specific, we list the relations for the different contributions to the integrated cross sections for , and scattering explicitly below.

### 4.1 and scattering

Here the total cross section consists of contributions from quark–antiquark annihilation and gluon–gluon fusion

(39) |

where

(40) | |||||

(41) |

, and denote the quark, antiquark and gluon distributions of the proton (antiproton), and is the factorization scale.

### 4.2 scattering

In scattering, the two contributions to the production cross section are the direct process [11] and the resolved photon process. Due to the photon-leptoquark coupling, the direct contribution, , contains a factor while the resolved one, , does not depend on the leptoquark charge. We use the Weizsäcker–Williams approximation (WWA) to describe the photon spectrum both in the case of the direct and resolved photon contributions. This approximation is known to hold at an accuracy of 10 to 15 %.

The total cross section is

(42) |

with (cf. [11])

(43) |

where , , , , , with , and the four momenta of the proton, the incoming and outgoing electron. The boundaries are given by

(44) |

where , and and are the electron and proton mass, respectively. denotes the Weizsäcker-Williams distribution, see e.g. [30] :

(45) |

To parametrize the scales we choose the kinematic limits

(46) |

The cross sections in the subsystem are

(47) |

where