Top-quark decay into Upsilon-meson

The calculation of the partial width of the rare t-quark decay into Upsilon-meson, W-boson and b-quark (t ->\Upsilon W b) is presented. The branching ratio equals Br(t ->Upsilon W b) = 1.3 * 10^{-5} that make possible searches for this rare $t$-quark decay at LHC.


Introduction
In the SM the decay t → bW is by far the dominant one. The rates for other decay channels are predicted to be smaller by several orders of magnitude in the SM [1].
For example, for semi-exclusive t-quark decays the interaction of quarks among the tdecay products may lead to final states with one hadron (meson) recoiling against a jet. The decays of the top through an off-shell W with virtual mass M W * near to some resonance h, like π + , ρ + , K + , D + s , leads to the estimate as follows [1]: where the parameter f M is same as a well-known coupling f π . The typical values of the corresponding branching ratios are too small to be measured [1]: There are several two-body t-quark decay through flavour changing neutral currents: t → γ q, t → Z q, t → g q; q = u, c These processes in the SM can occur due to loop contribution only and are highly suppressed due to GIM mechanism. The estimated branching ratios are as follows [1]: In addition, it worth noting that almost all "interesting" t-quark rare decays have very small branching ratios and almost impossible to measured in experiment. Among rare top-quark decays one can single out the processes with the production of heavy quark QQ -pair (for example, bb) followed by the formation of a heavy M (QQ )meson. In this case, the description of such mesons production allows the use of the NRQCDmodel [2,3].
Note, that the top quarks production processes with subsequent t-quark decays into heavy quark QQ -pair is described within SM with high accuracy. Therefore, the search and study of such t-quark rare decays can allow, in particular, to find out in more detail which models formation of quarkonium (Color-Evaporation Model, the Color-Singlet Model or the Color-Octet Mechanism, see [4] for detailed discussion of various mechanisms) describe more correctly such processes.
In this article we calculate the t-quark decay widths into Υ-meson within NRQCD model [2,3]. As will be seen below, at least one decay channel has a "relatively" large branching fraction, providing an opportunity for experimental searches.

The effective bb Υ-vertex
Within NRQCD approach the integration on virtual momentum in the loop with two heavy quarks that entered into heavy M (QQ) meson (see fig. 1 Figure 1: bb Υ vertex within NRQCD approach effectively produces the following expression (see [2,3]) for details: where G = (m +p)/(m 2 − p 2 ) is fermion propagator, M = m Υ stands for Υ-meson mass, Ψ Υ is the Υ-meson wave function, {· · · } is other terms in the loop; ε µ is the polarization vector of the Υ-meson. The Υ-meson wave function at the origin of the R S (0) is related to the lepton decay width [2,3] as follows: Note, that in the final expression (3) the heavy b-quarks (entered in the heavy Υ-meson vertex) are considered to be on-shell with mass equals: Taking into account that (kε) = 0 we get the final expression for bbΥ vertex: In this section we present the evaluation of the two-body t-quark decay width within NRQCD Color Singlet model approach. This width was calculated previously in [5,6]. For the sake of completeness we repeat the evaluation of this quantity. The diagram describing this decay is shown in fig. 2. We set the mass of the light c-quark equals zero; m t is the mass of t-quark, M = m Υ is the Υ mass 1 .
The amplitude has the following form (see [2] for details): Then the decay width equals here Γ W is the total decay width of the W -boson. The resulted width (with m t = 172.5 GeV, |V bc | = 0.04 [8]) equals Γ(t → Υ c) = 6.35 × 10 −10 GeV (9) and is very similar to previous result [5]. At the same time this quantity is 2.5 smaller then result from [6]. This difference can be explained by the fact that authors used contributions of both color singlet and color octet to this decay channel (see [6] for details). For calculation of the branching ratio we use LO t-quark decay width value of and get the corresponding branching ratio for this decay channel

Top-quark decay t → Υ W b
It follows from previous section that two-body t-quark decay t → Υc is very small (see (11)). It is explained by very small value of |V bc | ≈ 0.04 and high virtuality of the W -boson ( p 2 (8)). To avoid such suppression factors we consider t-quark decay width additional bb-pair production in the final state: This decay process is described by 28 Feynman diagrams. We use the C++ version of the TopReX package [9] for calculation the decay width into this channel. The results equal However, the diagrams with bb pair production due to Higgs, Z-boson or γ exchange are highly suppressed (due to small couplings). As a result we have 4 diagrams, describing t → ΥW b decay channel. The diagrams with W -boson exchange (see fig. 3) are also highly suppressed (due to small couplings and high virtuality of intermediate u, c, t-quarks and W ). Therefore, the dominant contribution to the amplitude of t → ΥW b decay comes from two diagrams with gluon exchange (see fig. 4). Figure 4: The diagrams describing t → ΥW + b decay through gluon exchange.
The amplitude A has the form (the particle's momenta notations are shown in the fig. 4): The square of the full amplitude is rather cumbersome and we present it in the Appendix. As before we use the the C++ version of the TopReX package [9] for calculation of the decay width. In the table 1 we present the results for three Υ-meson states (Υ(1S), Υ(2S) and Υ(3S)).    As it seen the final W and b-quark are rather well separated, while b and Υ pair (the left curve, fig. 5) has an invariant mass very close to m b + m Υ . Therefore, one may expect that the Υ-meson will produce dominantly inside final b-jet. Now we present very rough estimates of the expected number of events for this rare tquark decay channel for LHC Run-2 option. We consider the process of tt-pair production with subsequent t-quark (ort-quark) decay int three Υ(nS) states t → Υ(nS)W b, n = 1, 23. The total tt cross section, extrapolated to the full phase space, is [10]: Then, for estimation the expected number of events we use the following options: -the LHC Run-2 integrated luminosity equals L tot = 100 fb −1 , -W + W − decay into lepton and quark pairs W + W − → e(µ) ν qq , -all three Υ states decay into charged leptons Υ(nS) → ee or µµ. As a result, at LHC Run-2 the expected number of events for tt-pair production with subsequent t → Υ W + b decay are as follows: The total number of events N = 80 is not very large. However, this number looks more or less suitable for the experimental study.

Conclusion
In this paper the calculation of the partial width for rare t-quark decay into Υ-meson (t → ΥW b) is presented. The decay width was evaluated within NRQCD-model. The calculated branching ratio equals Br(t → Υ(1S)W b) = 1.3×10 −5 that make possible searches for this rare t-quark decay at LHC.