Interference between W-prime and W in single-top quark production processes

The paper is devoted to studies of prospects to search for hypothetical W-prime gauge bosons at hadron colliders via single-top quark production process. A special attention is paid to the interference between the Standard Model (SM) W and W-prime} boson contributions. A model independent analysis is performed for a wide interval of W-prime} masses potentially acceptable for a detection at the Tevatron and LHC. It is shown that the interference contribution to the cross section of the most promising s-channel single top production mode could be as large as 30 % for certain parameter points which is comparable to NLO effects computed in previous studies separately for the W-prime signal and the SM single-top background. The interference contribution affects particle distributions and has to be taken into account for more accurate W-prime signal and background simulation.


Introduction
The interactions of charged weak currents are realized in SM via exchange of charged massive gauge boson fields W + and W − . Although any additional charged massive bosons are not found yet experimentally their existence is predicted by various extensions of SM. The wide-common name for this vector boson field is W ′ . Such models like Non-Commuting Extended Technicolor [1], Composite [2,3] and Little higgs models [4,5,6], models of composite gauge bosons [7], Supersymmetric top-flavor models [8], Grand Unification [9] and Superstring theories [10,11,12] represent examples where extension of gauge group lead to appearing of W ′ . Physical properties and interaction parameters of W ′ can vary from model to model. For example the W ′ is presented in models of Universal extra-dimensions [13,14] as the lowest Kaluza-Klein mode of charged gauge boson W ± and has the same (V-A) chiral structure of interaction to fermion fields as the SM W ± . But in top-flavor models the W ′ boson interacts differently with fermions of the first two and third generations. It depends on magnitude of gauge coupling parameters g h and g l . If g h > g l the W ′ couples stronger to the third generation and weaker to the first two generations, and vice versa if g h < g l [15,16]. Frequently W ′ bosons are discussed in connection with so called Left-Right symmetric models [17,18,19,20,21,22,23]. The simplest extension of SM with Left-Right symmetry is based on U(1) × SU(2) L × SU(2) R gauge group. The left-handed fermions transform as doublets under SU(2) L and invariant under SU(2) R contrary to the right-handed ones which transform as doublets under SU(2) R and invariant under SU(2) L . Linear combinations of charged gauge fields produce massive eigenstates: W 1 = cos ζ W L +sin ζ W R , W 2 = − sin ζ W L +cos ζ W R , where W 1 is identified as observed W boson, and W 2 as a new W ′ boson, ζ is a mixing parameter between bosons for the right and left gauge groups. Parameter ζ is constrained to a very small value (ζ < 10 −3 ) to suppress (V+A) charged currents for light SM fermions in accord with experimental data [24,25]. In this case interactions of W ′ with fermions becomes almost purely right-handed. In Left-Right symmetric models the parity is broken spontaneously which leads to different masses for the SU(2) L and SU(2) R gauge bosons. There are two well-known variants of Left-Right models called manif est and pseudo − manif est for which the Cabibbo-Kobayashi-Maskawa matrices V L = V R and V L = V R * respectively.
Although the W ′ is not discovered yet there are experimental limits on its mass. Various models with W ′ contain many parameters, and indirect constraints of W ′ are highly model dependent. Indirect searches for W ′ being extracted from leptonic and semileptonic decays and also from cosmological and astrophysical data give very wide range for upper limits on W ′ mass varying from 549 GeV up to 23 TeV [26].
The direct limits on W ′ masses are based on hypothesis of purely right or lefthanded interacting W ′ with SM-like coupling constants. The limits from direct searches in leptonic decay modes of W ′ depend on a mass of a hypothetical righthanded neutrino. In case M W ′ < M ν R the decay mode W R → ν R , l is not kinematically allowed. The limits in this case have been extracted from light jet decay mode being M W ′ > 800 GeV [27]. In case M W ′ > M ν R the decay of W ′ to ν R and l is allowed, and the limit M W ′ > 786 GeV [28] has been obtained in this case from the leptonic decay modes combining both electron and muon channels.
One of the most promising and perspective way to search for W ′ is studying decays of W ′ to quarks of the third generation (W ′ → tb). This channel has relatively small QCD background comparing to light jet channel, and it is less model dependent. Searches in this channel at the Tevatron give a limit on W ′ mass M W ′ > 536 GeV at 95% CL in assumption M W ′ > M ν R . The assumption M W ′ < M ν R leads to slightly better limit M W ′ > 566 GeV [29] due to absence of the decay to ν R and correspondingly larger decay Br fraction to tb. The W ′ boson decaying to the top and bottom quarks contribute to the single-top production process. The single-top production being interesting itself for various aspects of the top quark physics gives perspective channel to search for W ′ [30]. In consequence that both W and W ′ contribute to single-top production process and have the couplings with the same fermion multiplets they should interfere to each other [31]. It should be noted that the interference becomes possible only for the left-handed interacting W ′ component because the SM W interacts only with left-handed electroweak currents being orthogonal to the right-handed interactions.
The aim of this paper is to study in more detail the interference phenomena between the SM W and the W ′ bosons in the single-top production process at the Tevatron and LHC energies.
Our paper is organized as follows. In the 2-nd section we present a simple analytical formula and a short analysis of the interference contribution. The 3-rd section describes results of numerical calculations. In the last section the summary and conclusion are given.

Interference between W ′ and W
To consider the W ′ production in a model independent way we write down the lowest dimension effective Lagrangian of W ′ interactions to quarks in most general form (possible higher dimension effective operators are not taken into account in our analysis): where a R q i q j , a L q i q j -left and right couplings of W ′ to quarks, g w = e/(s w ) is the SM weak coupling constant and V q i q j is the SM CKM matrix element. The notations are taken such that for so-called SM-like W ′ a L q i q j = 1 and a R q i q j = 0. As was mentioned a promising way to search for W ′ is the single-top quark production processes. There are three kinematically-different single-top production channels at hadron colliders, namely, s-channel, t-channel and associated W t channel. However for large W ′ mass region, which we are interested in, the W ′ contribution to the t-and associated W t channels becomes too small to be detectable, and the only s-channel where W ′ may contribute as a resonance remains important.
For the leading s-channel subprocess ud → tb the matrix element squared has the following form: where a L ud , a R ud -left and right couplings of W ′ to u, d quarks, and a L tb , a R tb -left and right couplings of W ′ to t, b quarks.
One can rewrite the formula in terms of Mandelstam variables using ( In these notations the formula reads as follows in complete agreement for the SM part (first term) with [32] and for W ′ part (last term) with the result from the paper [33]. The case of the SM-like W ′ corresponds to the couplings a L ud = a L tb = 1 and a R ud = a R tb = 0. One should stress that the interference (middle) term is proportional to the left couplings only because the SM W-boson has the only left (V − A) type of the interaction. The term containing the product of widths for any values ofŝ somewhere in the region between the W and W ′ boson masses, and it makes the interference term very small ifŝ is very close (equal) to one of the masses. However, in general, the interference term is not small being negative in the region of M 2 W <ŝ < M 2 W ′ and positive forŝ > M 2 W ′ (if the constants a L are positive). The interference term depends very weakly on the total width Γ W ′ of W ′ boson. As shown in the next section the interference contribution could be rather large and should be taken into account for a signal and background simulations in searches for W ′ if the W ′ has left-handed component in its interaction to fermions. Note that we assume here the couplings a L ud , a R ud , a L tb , a R tb to be real. Generalization of the formula to the complex couplings is straightforward and not given here.
After the integration over thet-variable the partonic cross section takes the form: The well-known SM cross section (first term) completely agrees with [34]. The QCD scale has been set to M W ′ . The couplings of W ′ to SM-fields have been implemented into CompHEP [35] which was used to compute the W ′ width, (see the Table 1), production cross sections, kinematical distributions and to generate unweighted events for different sets of W ′ masses. For the case of left-handed interacting W ′ we set a L ud = a L tb = 1, a R ud = a R tb = 0, for the right-handed case a L ud = a L tb = 0, a R ud = a R tb = 1 and for SM all the parameters are equal to zero a L ud = a L tb = a R ud = a R tb = 0. We simulate the process pp (pp) → W/W ′ → tb which includes 8 subprocesses with different parton combinations in the initial states.

Numerical illustrations
In the Tables 2, 3 the total cross section of the process pp (pp) → W/W ′ → tb, the contribution of the interference term (in %) and the contribution of the leading subprocess ud → W/W ′ → tb to the total cross section are listed as a function of W ′   TeV). Interference term is zero for the right interacting W ′ .
masses. The tables show that the contribution from subleading subprocesses is small for all values of W ′ mass. The cross sections for the purely left-handed interacting W ′ are smaller than for the right-handed one. This difference reflects the fact of negative contribution of the interference between the left-handed interacting W ′ and the SM W -boson which absent in the right-handed case. One can see that for the Tevatron ( Table 2) the interference term achieves a maximum about 33,4 % of the total cross section at M W ′ equal to 800 GeV, and about 14 % for the LHC (Table 3) at 1 TeV W ′ boson mass.
To illustrate the interference between the W ′ and the SM W bosons in more detail the Fig. 1 and Fig. 2 show the differential cross section for the s-channel single-top quark production as a function of the invariant mass of tb-system for three sets of W ′ masses 600 (upper plot), 800 (middle plot), and 1000 GeV (lower plot). For each value of W ′ masses the cases with purely left-handed and right-handed interactions of W ′ are plotted comparing to the SM process. All curves start from the reaction threshold at about M tb ≈ 180 GeV. In case of right-handed interactions of W ′ there is no interference, and the curve for the invariant mass distribution is the algebraic sum of two independent falling down SM W and the resonant W ′ distributions. The picture is significantly different in case of left-handed interacting W ′ where in addition to the resonance pike there is an area with a minimum due to destructive interference between the SM W and the W ′ boson contributions. This local minimum follows form the formula 4 and takes place at the value of tb invariant mass M tb =    We have simulated the complete chain of top quark decays taking into account all the spin correlations between top quark production and its subsequent decay. The separate study of angular correlations for these processes is given in the paper [36].

Conclusion
In this paper we focus our attention on the interference of SM W gauge boson with a hypothetical W ′ vector boson in single-top production process at hadron collider. Simple symbolic formula for matrix element squared of the leading subprocess shows explicitly in a model independent way a parameter dependence for general left-and right-handed couplings of W ′ boson to the SM fermions. As expected a maximal influence of the interference term takes place for the case of left-handed interactions of W ′ . Such a destructive interference may approach as large as 30% for the Tevatron leading to a local minimum in the invariant mass distribution at M tb = and being very small close to the resonance position in tb invariant mass. The NLO corrections have been computed separately for the SM single top s-channel process [34] (background for W ′ ) and for the W ′ contribution [33]. Such an approach works perfectly for the case of purely right-handed interacting W ′ . However a recipe how to proceed from the right-handed to a general coupling case by multiplying the answer by the coefficient a L ud 2 a L tb 2 + a R ud 2 a R tb 2 obviously does not work because the interference contribution and the additional term in W ′ contribution are proportional to different coupling combinations, the first to a L ud a L tb and the second a L ud 2 a R tb 2 +a R ud 2 a L tb 2 (see, Formulas 3,4). As follows from the Formulas 3, 4 in order to simulate the general coupling dependence one should perform the simulations for three cases, purely righthanded (a R tb = a R ud = 1, others equal to 0), purely left-handed (a L tb = a L ud = 1, others equal to 0), and mixed (for example, a R tb = 1, a L ud = 1, others equal to 0). Computation of the NLO corrections in general case including non-trivial interference remains to be done, however in a number of cases the NLO corrections can be extracted from the existing computations.
The interference contribution is important in general and has to be taken into account in searches for W ′ bosons performing MC event generation. Calculations described in this paper and correspondingly created Monte-Carlo event generator have been used in recent experimental searches for W ′ by the DØcollaboration [37]. The special study of angular correlations in the production and decay of W ′ is described in the separate paper [36].