Production of heavy vector particles in interactions of leptons with the massless gauge bosons

The cross section for vector leptoquark production in electron--gluon collisions is calculated analytically using the Lagrangian with the minimal couplings between the leptoquarks and the gauge fields of the standard model. It is found that the cross section significantly exceeds the corresponding quantity previously presented in the literature. The cross section of exclusive $W$ boson production in neutrino--photon scattering emerges as a by-product of this letter. The obtained results can be used for studies at $ep$ colliders.


Introduction
The cross section being the most important quantity for description and interpretation of physics underlying interactions of particles connects theory and experiment. Accurate knowledge on it is therefore very important for deeper understanding already well established theories like the standard model as well as for search of new physics.
The effective Lagrangian of the leptoquark model proposed by Buchmüller, Rückl and Wyler in 1986 [1] obeys the symmetries of the standard model gauge groups SU(3) C × SU(2) L × U(1) Y . Therefore one may expect that dynamics of some processes in these models can formally coincide so that the calculated cross sections turn out also to be the same up to constant factors associated with different couplings. This letter shows that exclusive production of W bosons in neutrino-photon scattering and production of vector leptoquarks in electron-gluon collisions represent such a situation in the leading order of perturbation theory provided the leptoquarks minimally couple to the gauge fields of the standard model. The cross sections of both reactions are calculated analytically. A significant difference between the result of this letter and previous analysis of the leptoquark production is found.

Comparison of the models
Following [2] let us also postulate the Lagrangian with minimal couplings between vector leptoquarks and the neutral gauge bosons of the standard model: Here A µ , Z µ and A a µ are the photon, Z boson and gluon fields, respectively; e is the elementary electric charge, Q denotes the electric charge of a leptoquark, g s is the strong coupling constant, Q Z = (T 3 −Q sin 2 θ W )/ sin θ W cos θ W (T 3 is the third component of the weak isospin, θ W is the Weinberg angle), λ a are the Gell-Mann matrices.
From (2) it follows that the Feynman rules for the ZV V and γV V vertices are similar to the ZW W γW W ones [2]. The couplings of vector leptoquarks to the gluon fields have also analogy with the self-interaction of the gluons. Let us consider two exclusive reactions. The first one is the W production in neutrino-photon scattering allowed by the standard model [3]: where l =e, µ, τ .
The second one is production of vector leptoquarks (V ) in interactions of left/right polarized electrons with gluons appearing in the Buchmüller- Rückl-Wyler leptoquark model [4]: The reactions (3) and (4) are closely related to each other from a formal point of view. To illustrate this, it is convenient to represent the considered interactions in the form of the Feynman diagrams so that the amplitudes contributing to (3) and (4) will look as shown in Fig. 2 and Fig. 3, respectively [3,4].
The leptoquark Lagrangian has such symmetry properties that the Feynman rules for the vertices V eq respectively coincide in their structure with those of the standard model for the vertices W νl up to constant factors (see in accordance with the above discussion, they must satisfy the following condition: Note that both cross sections in (5) are taken with the same values of the masses. Therefore, it is enough to know one of the cross sections to find the other.

Calculation of the cross section
The cross section of the reaction (4) is calculated using the diagrams from The result reads where λ is the coupling constant corresponding to the V eq vertex, α s = Here p = (s − (m 1 + m 2 ) 2 ) (s − (m 1 − m 2 ) 2 )/2 √ s is the cm momentum of any of the final state particles.

Verification of the validity of the cross section
The similarity between the cross sections of the reactions (3) and (4) discussed in Section 2 allows to verify the validity of the cross section (6).
Actually, the problem of calculation of σ 1 (s, M W , m l ) is now reduced to just performing the following obvious replacements of the coupling constants and the masses of the final state particles in (6): where g is the coupling of the weak charged current (related to the Fermi coupling constant G F by G F = √ 2g 2 /8M 2 W , α is the fine structure constant. Note that the coefficient of α s is the color factor equal to 1/2 for the process (4).

So that one finds
The cross section for the reaction (3) in the leading order has also been independently calculated in [3], however in such a way that the masses of the final state leptons were neglected (let us denote this cross section by σ c 1 (s, M W , m l )). This means that (9) obtained in this letter and the result of [3] must asymptotically coincide satisfying the following condition: It should be emphasized that (10) is a criterion to verify the validity of the cross sections obtained in the present letter.

Comparison with previous calculations
The cross section for the reaction (4) in the leading order has also been calculated in [4] whose result (let us denote it by σ c 2 (s, M V , m q )) significantly differs from (6). This fact is illustrated in Fig. 6, where dependences of both cross sections on the cm energy are shown for production of the (eb) and (et) type vector leptoquarks of mass 1000 GeV at λ = 1 and α s = 0.118.
The difference is more brightly reflected by the ratio of σ c 2 (s, M V , m q ) to σ 2 (s, M V , m q ) shown in Fig. 7. One can see that there may be cases in which the cross section (6) exceeds the corresponding quantity from [4] by about a factor of two. This is because different choices of the Lagrangian responsible for interaction of the leptoquarks with the gluon fields which lead to different Feynman rules for the gV V vertex [6]. In the present letter the minimal couplings of the leptoquarks to the gauge fields of the standard model are assumed while in [4] the Lagrangian with anomalous interaction terms is used [7,8].

Production of vector leptoquarks in electron-nucleon collisions
A standard convolution of the cross section (6) with the gluon distribution in the nucleon gives the cross section of inclusive vector leptoquark production measurable in electron-proton collisions: where g(x, s) is the gluon distribution function, x 0 = (M V + m q ) 2 /s. Figure 8 shows the cross sections for the production of the (eb) and (et) type vector leptoquarks evaluated using (11) with the gluon distribution function adopted from CTEQ5 [9].

Conclusions
The cross section for vector leptoquark production in electron-gluon collisions is calculated analytically using the SU ( [3] to that calculated in the present letter on the cm energy in the resonance region.