Vector-like quarks and New Physics in the flavour sector

We present a detailed analysis of recent flavour data in the framework of a simple extension of the Standard Model, where a Q = 2/3 vector-like isosinglet quark is added to the spectrum. Constraints from all the relevant quark flavour sectors are used. Important deviations from Standard Model expectations in different observables such as the semileptonic asymmetry in Bd decays, AdSL, the time-dependent CP asymmetry in Bs → J/ΨΦ, and rare decays such as K+ → π+ν̄ν can be obtained.


Introduction
We consider an extension of the Standard Model (SM) where one isosinglet vector-like quark T with charge Q = 2/3 is added to the spectrum [1,2,3]. After diagonalization of the up and down mass matrices, the 3 × 3 mixing matrix connecting standard quarks is no longer unitary, but a submatrix of a larger 4 × 4 unitary matrix U . The charged and neutral current interactions have the form where d ≡ (d, s, b), u ≡ (u, c, t, T ) and V is a 4 × 3 submatrix of the matrix U : The submatrix V (3×3) , i.e. the upper left 3 × 3 block within U , is not a unitary matrix, since V (3×3) V † (3×3) = 1 (3×3) . These deviations of unitarity of the "would-be standard" mixing matrix Summarizing, the addition of one isosinglet vector-like up quark provides: • A new mass eigenstate in the up sector. It can give new contributions to amplitudes involving virtual up quarks, as for example, in kaon or B meson mixings. • A mixing matrix V which is not 3×3 unitary anymore, allowing for deviations of the mixing elements V ij from SM values. • Modified couplings to the Z boson in the up sector, including tree level flavour changing and reduced flavour conserving couplings.
With these ingredients we can expect • modifications in the bd sector that can alleviate the existing tensions, • new contributions to the B 0 s -B 0 s (dispersive) mixing amplitude M (s) 12 that can significantly modify its phase, and thus the B s → J/ΨΦ time dependent CP asymmetry, • that the deviations from 3 × 3 unitarity can change the B d and B s (absorptive) mixing amplitudes Γ (q) 12 , and produce larger-than-standard values for the semileptonic asymmetries A q SL , • modifications in the rates of several rare decays.

Experimental constraints
To reflect the abundant experimental information that constrains modifications of the flavour sector such as the ones introduced in the present scenario, we have considered the following observables.
• Tree level observables, whose extraction from experiment is presumably unaffected by New Physics (NP) effects. These observables include moduli of the CKM elements in the first and second rows. For the third row the only relevant measurement is the one of the ratio of branching fractions . The physical phase γ, is also a tree level observable. The actual values are collected in table 1. Finally, the decay B + → τ + ν τ is also a tree level process which participates in the so called tensions in the bd sector 2 . Table 1. Tree level observables [4,6].
1 They are naturally suppressed by ratios m 2 /m 2 T , where m denotes generically the standard quark masses [1]. This natural suppression of FCNC is crucial in order to make the model plausible. 2 Notice however that, as it is helicity suppressed and proportional to |V ub | 2 , sizeable NP contributions may appear in different beyond SM scenarios, but not in our case. • Observables related to B 0 d -B 0 d and B 0 s -B 0 s mixings: we consider time-dependent CP asymmetries A J/ΨK S and A J/ΨΦ (the "golden" channel in each system), mass and width differences ∆M B d , ∆Γ d , and ∆M Bs , ∆Γ s , additional CP asymmetries involving different combinations of invariant phases, sin(2ᾱ), sin(2β +γ) (and cos(2β) which removes a discrete ambigity in fixing 2β = sin −1 (A J/ΨK S )), and, finally, semileptonic asymmetries A d SL , A s SL and A b SL . The actual values are collected in table 2. Table 2. B d and B s mixing-related observables [4,5,7].
• Electroweak precision observables, in particular the oblique parameters S and T (U too, but its role is negligible): ∆S = 0.02 ± 0.11 , ∆T = 0.05 ± 0.12 , with a correlation coefficient 0.879. • D 0 -D 0 mixing: we require that the Z-mediated, short distance, contribution to the D 0 -D 0 mixing amplitude does not give a larger than observed mixing parameter x D . Notice that, as long distance contributions to D 0 -D 0 are also present, we are not requiring that the short distance ones fully account for the observed x D = (0.8 ± 0.2) · 10 −2 .

Results
We summarize the results of a complete numeric analysis of the model [3] in tables 5 to 8, where experimental results (where available), SM expectations and the predictions of the model are displayed together as colored bars corresponding to 1, 2 and 3σ ranges. Table 5 illustrates how the mixing matrix can depart from the SM tight 3 × 3 unitary structure: invariant phases such as β and β s and mixing elements such as |V tb | and |V ub |, span much wider ranges than in the SM.  Table 6 illustrates how the model adequately reproduces constraints related to B d and B s mixings, together with K and ε / K . Notice in addition that the CP asymmetry in B s → J/ΨΦ, less constraining from the experimental point of view, can sizeable depart from SM expectations (and it does so at a level that LHCb will be sensitive to).
Finally,  Besides the information summarized in the previous tables, correlations among observables provide a huge playground to put the model to the test.
• Figure 1(a) illustrates how deviations from SM expectations are correlated for A d SL and Br(B → τ ν) (controlled by |V ub |); red ellipses correspond to SM 68%, 95% and 99% CL regions.
• Figure 1(b) shows the strong correlation among A s SL and A J/ΨΦ when non standard values for both are present. The red cross indicates the SM prediction. Notice that within the SM the allowed range of variation is too small to be resolved with the scales of the figure.
• Figure 1(c) shows how significant deviations from |V tb | can only be achieved for relatively light values of m T . • Figure 1(d) shows how both Br(K L → π 0 νν) and Br(K + → π + νν) can deviate from SM expectations (the red cross indicates the central value, the red bars over the axes stand for the 68%, 95% and 99% CL ranges), and do following a well defined pattern.

Conclusions
We have presented an overview of a detailed analysis of flavour data in the context of a simple extension of the Standard Model, that includes an additional Q = 2/3 vector-like isosinglet quark. Experimental constraints from all the relevant quark flavour sectors are imposed and yet important deviations from Standard Model expectations can be present. This has been illustrated with different individual observables and important correlations.