Production of heavy vector particles in interactions of leptons with the massless gauge bosons
aa r X i v : . [ h e p - ph ] D ec Production of heavy vector particles in interactions ofleptons with the massless gauge bosons
I. Alikhanov ∗ Institute for Nuclear Research of the Russian Academy of Sciences, 60-th OctoberAnniversary pr. 7a, Moscow 117312, Russia
Abstract
The cross section for vector leptoquark production in electron–gluon colli-sions is calculated analytically using the Lagrangian with the minimal cou-plings between the leptoquarks and the gauge fields of the standard model. Itis found that the cross section significantly exceeds the corresponding quan-tity previously presented in the literature. The cross section of exclusive W boson production in neutrino–photon scattering emerges as a by-productof this letter. The obtained results can be used for studies at ep colliders. Key words: leptoquark, electron, gluon, neutrino, photon
PACS:
1. Introduction
The cross section being the most important quantity for description andinterpretation of physics underlying interactions of particles connects the-ory and experiment. Accurate knowledge on it is therefore very importantfor deeper understanding already well established theories like the standard ∗ Email address: [email protected]
Preprint submitted to arXiv.org November 27, 2018 odel as well as for search of new physics.The effective Lagrangian of the leptoquark model proposed by Buchm¨uller,R¨uckl and Wyler in 1986 [1] obeys the symmetries of the standard modelgauge groups SU (3) C × SU (2) L × U (1) Y . Therefore one may expect thatdynamics of some processes in these models can formally coincide so thatthe calculated cross sections turn out also to be the same up to constantfactors associated with different couplings. This letter shows that exclusiveproduction of W bosons in neutrino–photon scattering and production ofvector leptoquarks in electron–gluon collisions represent such a situation inthe leading order of perturbation theory provided the leptoquarks minimallycouple to the gauge fields of the standard model. The cross sections of bothreactions are calculated analytically. A significant difference between theresult of this letter and previous analysis of the leptoquark production isfound.
2. Comparison of the models
Following [2] let us also postulate the Lagrangian with minimal couplingsbetween vector leptoquarks and the neutral gauge bosons of the standardmodel: L γ,Z, g = X V (cid:20) − G † µν G µν + M V V µ † V µ (1) − i X j = γ,Z, g g j V † µ V ν F µνj , where G µν = D µ V ν − D ν V µ is the field strength tensor of vector lepto-quarks, F µνj are the field strength tensors of the neutral gauge fields of the2tandard model, g j are the coupling constants for the interactions betweenthese gauge fields and the leptoquarks. The covariant derivative D µ is definedby D µ = ∂ µ − ieQA µ − ieQ Z Z µ − ig s λ a A aµ . (2)Here A µ , Z µ and A aµ are the photon, Z boson and gluon fields, respec-tively; e is the elementary electric charge, Q denotes the electric charge of aleptoquark, g s is the strong coupling constant, Q Z = ( T − Q sin θ W ) / sin θ W cos θ W ( T 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 verticesare similar to the
ZW W γW W ones [2]. The couplings of vector leptoquarksto the gluon fields have also analogy with the self-interaction of the gluons.For example, in Fig. 1 the explicit form of the Feynman rules for the γW W and g
V V vertices are given. One can see that they differ only by constantfactors.Let us consider two exclusive reactions. The first one is the W productionin neutrino–photon scattering allowed by the standard model [3]: ν l + γ → l + W, (3)where l =e , µ, τ .The second one is production of vector leptoquarks ( V ) in interactionsof left/right polarized electrons with gluons appearing in the Buchm¨uller–R¨uckl–Wyler leptoquark model [4]: 3 L/R + g → q + V. (4)The reactions (3) and (4) are closely related to each other from a formalpoint of view. To illustrate this, it is convenient to represent the consideredinteractions in the form of the Feynman diagrams so that the amplitudes con-tributing 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 Feyn-man rules for the vertices V eq respectively coincide in their structure withthose of the standard model for the vertices
W νl up to constant factors (seeFig. 4). As to the familiar rules corresponding to the γll and g qq vertices,they also differ from each other only by the coupling constants and the colorcoefficients. This means that in the limit of massless initial state leptonsthe leading order cross sections of the reactions (3) and (4) will have exactlythe same dependence on the Mandelstam variables and on the masses of thefinal state particles differing from each other only by a constant as well. Suchrelations between processes are well known in quantum field theory. Mostsimple QCD diagrams are exactly analogous to QED diagrams, and the QCDcross section is obtainable by appropriate replacements of the coupling con-stants in the QED one corresponding to the replacement of the photon by agluon [5].Suppose that σ ( s, M W , m l ) and σ ( s, M V , m q ) are the leading order crosssections of the reactions (3) and (4), where s is the center-of-mass (cm)energy squared, M i and m j are the masses of the final state particles. Then,in accordance with the above discussion, they must satisfy the following4ondition: σ ( s, M, m ) σ ( s, M, m ) = const. . (5)Note that both cross sections in (5) are taken with the same values of themasses. Therefore, it is enough to know one of the cross sections to find theother.
3. Calculation of the cross section
The cross section of the reaction (4) is calculated using the diagrams fromFig. 3 with the corresponding Feynman rules given in Figs. 1(b) and 4(b).The result reads σ ( s, M V , m q ) = λ α s F ( s, M V , m q ) , (6)where λ is the coupling constant corresponding to the V eq vertex, α s = g s / π , F ( s, m , m ) = 18 m n ps / (cid:2) s + 16 m − m (cid:0) s + m (cid:1) − m (cid:3) +2 2 m ( s − m s + 2 m ) + m ( s − ( s − m ) ( m − m )) − m s × log " ( p + m ) / + pm − m ( s − m ) − m ( s + 4 m s + 6 m ) − m ( s − m ) + 2 m s × log " ( p + m ) / + pm . (7)5ere p = p ( s − ( m + m ) ) ( s − ( m − m ) ) / √ s is the cm momen-tum of any of the final state particles.
4. 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 ( s, M W , m l ) is now reduced to justperforming the following obvious replacements of the coupling constants andthe masses of the final state particles in (6): λ → g √ , α s → α, M V → M W , m q → m l , (8)where g is the coupling of the weak charged current (related to the Fermicoupling constant G F by G F = √ g / M W , α is the fine structure con-stant. Note that the coefficient of α s is the color factor equal to 1 / σ ( s, M W , m l ) = g αF ( s, M W , m l ) , (9)The cross section for the reaction (3) in the leading order has also beenindependently calculated in [3], however in such a way that the masses ofthe final state leptons were neglected (let us denote this cross section by σ c ( s, M W , m l )). This means that (9) obtained in this letter and the resultof [3] must asymptotically coincide satisfying the following condition:6 c ( s, M W , m l ) σ ( s, M W , m l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ s ≫ M W + m l = 1 . (10)It should be emphasized that (10) is a criterion to verify the validity ofthe cross sections obtained in the present letter.Figure 5 shows that (10) is fulfilled as it must be in the case of cor-rect calculations. Moreover, there is a perfect coincidence between (9) and σ c ( s, M W , m l ) over the entire range of the energy for the case of the elec-tron in the final state owing to the vanishing electron mass in comparisonwith M W .
5. Comparison with previous calculations
The cross section for the reaction (4) in the leading order has also beencalculated in [4] whose result (let us denote it by σ c ( s, M V , m q )) significantlydiffers from (6). This fact is illustrated in Fig. 6, where dependences of bothcross 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 . σ c ( s, M V , m q ) to σ ( s, M V , m q ) shown in Fig. 7. One can see that there may be cases inwhich the cross section (6) exceeds the corresponding quantity from [4] byabout a factor of two. This is because different choices of the Lagrangianresponsible for interaction of the leptoquarks with the gluon fields whichlead to different Feynman rules for the g V V vertex [6]. In the present letterthe minimal couplings of the leptoquarks to the gauge fields of the standardmodel are assumed while in [4] the Lagrangian with anomalous interactionterms is used [7, 8]. 7 . Production of vector leptoquarks in electron–nucleon collisions
A standard convolution of the cross section (6) with the gluon distri-bution in the nucleon gives the cross section of inclusive vector leptoquarkproduction measurable in electron–proton collisions: σ e L p → V X ( s, M V , m q ) = Z x dx g( x, s ) σ ( xs, M V , m q ) , (11)where g( x, s ) is the gluon distribution function, x = ( M V + m q ) /s .Figure 8 shows the cross sections for the production of the ( eb ) and ( et )type vector leptoquarks evaluated using (11) with the gluon distributionfunction adopted from CTEQ5 [9].
7. Conclusions
The cross section for vector leptoquark production in electron–gluon colli-sions is calculated analytically using the SU (3) C × SU (2) L × U (1) Y -symmetricLagrangian with the minimal couplings between the leptoquarks and thegauge fields of the standard model. It is shown that from a formal point ofview this process is similar to exclusive production of W bosons in neutrino–photon scattering allowed by the standard model whose cross section is alsofound. The cross section of the leptoquark production obtained in this lettersignificantly exceeds the corresponding result of previous calculations pre-sented in the literature. The cross sections of inclusive production of the ( eb )and ( et ) vector leptoquarks in electron–nucleon interactions observable at ep colliders are evaluated. The presented analysis is applicable to studies of theproduction of vector leptoquarks of the other types as well.8 cknowledgements I thank P.M. Zerwas for useful comments on production of leptoquarks inseveral processes. I am also thankful to M. Spira for drawing my attentionto their results on production of leptoquarks at ep colliders as well as forproviding useful information. This work was supported in part by the RussianFoundation for Basic Research (grant 11-02-12043), by the Program for BasicResearch of the Presidium of the Russian Academy of Sciences ”NeutrinoPhysics and Neutrino Astrophysics” and by the Federal Target Program ofthe Ministry of Education and Science of Russian Federation ”Research andDevelopment in Top Priority Spheres of Russian Scientific and TechnologicalComplex for 2007-2013” (contract No. 16.518.11.7072). References [1] W. Buchm¨uller, R. R¨uckl, D. Wyler, Phys. Lett. B 191 (1987) 442;Erratum ibid. B 448 (1999) 320.[2] J.E. Cieza Montalvo, O.J.P. Eboli, Phys. Rev. D 47 (1993) 837.[3] D. Seckel, Phys. Rev. Lett. 80 (1998) 900.[4] A. Djouadi, T. K¨ohler, M. Spira, J. Tutas, Z. Phys. C 46 (1990) 679.[5] M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field The-ory, Addison-Wesley, Reading, Mass., 1995.[6] M. Spira, private communication.[7] J. Bl¨umlein, E. Boos, Nucl. Phys. B Proc. Suppl. 37 (1994) 181.98] T.M. Aliev, E. Iltan, N.K. Pak, Phys. Rev. D 54 (1996) 4263.[9] H.L. Lai, et al. , Eur. Phys. J. C 12 (2000) 375.10 igure CaptionsFig. 1:
The Feynman rules: (a) for the γW W vertex; (b) for the g
V V vertex.
Fig. 2:
Tree level Feynman diagrams describing the process ν l + γ → l + W . Fig. 3:
Tree level Feynman diagrams describing the process e +g → q + V . Fig. 4:
The Feynman rules: (a) for the
W νl vertex; (b) for the
V eq vertex.
Fig. 5:
Dependence of the ratio of the cross section σ c ( s, M W , m l ) takenfrom [3] to that calculated in the present letter on the cm energy in theresonance region. Fig. 6:
Dependence of the cross sections of production of vector lep-toquarks of mass 1000 GeV on the cm energy. Right panel: the ( eb ) typeleptoquark. Left panel: the ( et ) type leptoquark. The solid curves representthe results of this letter, the dashed curves are the calculations of [4]. Thecouplings λ = 1 and α s = 0 . Fig. 7:
Dependence of the ratio of the cross section σ c ( s, M V , m q ) takenfrom [4] to that calculated in the present letter on the cm energy for pro-duction of the ( et ) type vector leptoquark of mass 500 GeV (dash-dotted),700 GeV (dashed) and 1000 GeV (solid). The dotted line corresponds to σ c /σ = 1. 11 ig. 8: The cross section of the reaction e L p → V X as a functionof the leptoquark mass at √ s = 1800 GeV. Dashed curve: the ( eb ) typeleptoquark. Solid curve: the ( et ) type leptoquark. The coupling λ is dividedout, α s = 0 . igure 1:Figure 2: igure 3:Figure 4: e Γ ® eW Ν Μ Γ ® Μ W Ν Τ Γ ® Τ W
85 90 95 100 s @ GeV D Σ Σ Figure 5: s @ GeV D Σ H pb L s @ GeV D Σ H pb L M V = 1000 GeVm q = 172.9 GeV M V = 1000 GeVm q = 4.5 GeV Figure 6: q =
500 GeV700 GeV1000 GeV s @ GeV D Σ Σ Figure 7: , s =1800 GeVm q =4.5 GeVm q =172.9 GeV