Top Quark Rare Decays via Loop-Induced FCNC Interactions in Extended Mirror Fermion Model

Flavor changing neutral current (FCNC) interactions for a top quark $t$ decays into $Xq$ with $X$ represents a neutral gauge or Higgs boson, and $q$ a up- or charm-quark are highly suppressed in the Standard Model (SM) due to the Glashow-Iliopoulos-Miami mechanism. Whilst current limits on the branching ratios of these processes have been established at the order of $10^{-4}$ from the Large Hadron Collider experiments, SM predictions are at least nine orders of magnitude below. In this work, we study some of these FCNC processes in the context of an extended mirror fermion model, originally proposed to implement the electroweak scale seesaw mechanism for non-sterile right-handed neutrinos. We show that one can probe the process $t \to Zc$ for a wide range of parameter space with branching ratios varying from $10^{-6}$ to $10^{-8}$, comparable with various new physics models including the general two Higgs doublet model with or without flavor violations at tree level, minimal supersymmetric standard model with or without $R$-parity, and extra dimension model.

For the heavy top quark t, the story is quite different. Since there is no time for the heavy top quark to form bound states, we can discuss its free decay, like the dominant decay mode t → W + b at tree level or its rare FCNC decays. The SM branching ratios B(t → Xq) where X denotes one of the following neutral particle Z, γ, g or h in SM, and q denotes the light u or c quark, vary in the range 10 −17 − 10 −12 [2], which are unobservable at the present technology. However, in many models beyond the SM, branching ratios for some of these processes of order up to 10 −3 can be achieved.
Observations of these rare top quark FCNC decays at the Large Hadron Collider (LHC) with significant larger branching ratios than the SM predictions would then be clear signals, albeit indirect, of new physics.
Indeed LHC can be considered as a top quark factory, estimated to produce 10 8 tt pair with an integrated luminosity of 100 inverse femtobarn. For an updated review on top quark properties at the LHC, see [3]. The current limits for t → Zq  (2) and for t → γq, we have the limits from CMS [8] B(t → γu) ≤ 1.3 × 10 −4 , Projected limits for the above as well as other FCNC processes for the top quark are expected to be improved constantly in the future at the LHC. Thus searching for or discovery of any one of these FCNC rare top decays t → Xq at the LHC would be providing interesting constraints or discriminations of various new physics models in the future.
Over the years FCNC top quark decays had been studied intensively in the literature for many new physics models, like the minimal supersymmetric standard model (MSSM) with [9] or without [10,11] R-parity, flavor conserving [12] or flavor violating [13][14][15][16] two Higgs doublet model (2HDM), aligned two Higgs doublet model (A2HDM) [17], warped extra-dimensions [18,19], and effective Lagrangian framework [20], etc. Branching ratios for FCNC top quark decays in all these models are typically many orders of magnitude above the SM and some of them may lead to detectable signals at the LHC.
In this work, we compute the FCNC decays of t → V q (V = Z, γ; q = u, c) in an extension of mirror fermion model [21] originally proposed by one of us [22]. In contrast with various left-right symmetric models, the model in [22] did not include the gauge group SU (2) R while adding the mirror partners of the SM fermions. Despite having the same SM gauge group, the scalar sector must be enlarged. In additional to employ the bi-triplets in the Georgi-Machacek (GM) model [23,24] and a Higgs singlet to implement the electroweak scale seesaw mechanism for the non-sterile righthanded neutrino masses [22], one needs to add a mirror Higgs doublet [25] in the scalar sector so as to make consistency with the various signal strengths of the 125 GeV Higgs measured at the LHC. We will briefly review this class of mirror fermion model and its further extension with a horizontal A 4 symmetry in the following section.
We layout the paper as follows. In Section II, after giving a brief highlight on some of the salient features of the model, we present the relevant interaction Lagrangian.
Our calculation and analysis are presented in Section III and IV respectively. We finally summarize our results in Section V. Analytical expressions for the loop functions are collected in the Appendix.

A. A Lightning Review
As already eluded to in the Introduction, the mirror fermion model in [22] was devised to implement the so-called electroweak scale seesaw mechanism for the neutrino masses. We first list the particle content of the model in Table I for further discussions. One special feature of this mirror model is to treat the right-handed neutrino in each generation to be non-sterile by grouping it with a new heavy mirror right-handed charged lepton into a weak doublet l M Ri , regarded as the mirror of the SM doublet l Li with i labelling the generation. When the Higgs singlet φ 0S develops a small vacuum expectation value (VEV) of order 10 5 eV, through its Yukawa couplings between the SM lepton doublets and their mirror partners, it can provide a small Dirac mass term for the neutrinos. On the other hand, when the tripletχ field with hypercharge Y /2 = 1 in Table I develops a VEV of order v SM = 246 GeV, a Majorana mass term of electroweak scale can be generated through its Yukawa couplings among these new mirror lepton doublets. Details of this electroweak scale seesaw mechanism in the mirror fermion model can be found in [22].
Note that the other triplet ξ which has zero hypercharge is grouped withχ to form the bi-triplets in the GM model to maintain the custodial symmetry and therefore the ρ parameter equals unity at tree level. In [26], the potential dangerous contributions from the GM triplets to the S and T oblique parameters are shown to be partially cancelled by the opposite contributions from mirror fermions such that the model is still healthy against electroweak precision tests.
Mirrors of other SM fermions, both leptons and quarks, can be introduced in the same way as listed in Table I. Searches for these heavy mirror fermions at the LHC were presented in [27] and [28] for the mirror quarks and mirror leptons respectively.
In order to reproduce the signal strengths of h 125 → γγ and h 125 → Zγ for the 125 GeV Higgs observed at the LHC, a mirror Higgs doublet Φ M of the SM one Φ was introduced [25]. We note that mixing effects among the two doublets as well as with the triplet ξ must be taken into account in order to satisfy the LHC results. processes µ → eγ [30], µ − e conversion [31] and h 125 → τ µ [29], as well as for the electron electric dipole moment [32]. Here we will explore its implication in the rare FCNC top decays. Implications of the A 4 symmetry for the quark masses and mixings will be given in [33].

B. Interaction Lagrangian for Quarks and Their Mirrors
Here we will write down the interactions for the quarks and their mirrors that are relevant to the FCNC processes t → V q that we are studying. Since the result for t → gq can be easily obtained from that of t → γq, we will not present detailed formulas for the former process. As for t → hq, one must consider the mixing effects from the more complicated Higgs sector in the extended mirror model. We will leave it for future work.

Quark Yukawa Couplings with A 4 Symmetry
Recall that the tetrahedron symmetry group A 4 has four irreducible representations 1, 1 , 1 , and 3 with the following multiplication rule 3 × 3 = 3 1 (23, 31, 12) + 3 2 (32, 13, 21) 2 . Using the above A 4 multiplication rules one can construct new Yukawa couplings in the leptonic sector, which are both gauge invariant and A 4 symmetric, to implement small Dirac neutrino masses in electroweak seesaw and to discuss charged lepton mixings [21].
In the same vein, one can write down the following new Yukawa couplings for the quarks and their mirrors (both in the flavor basis with subscripts "0") with the scalar singlets, which are both gauge invariant and A 4 symmetric, where g Q,u,d

1S
and g Q,u,d

2S
are in general complex coupling constants. Implications of the above Yukawa interactions on the quark mixings will be presented in [33].
Next we move to the physical basis by making the following unitary transformations on the left-handed fields and similarly for the right-handed fields We can then recast the Yukawa interactions in the following form with are three field-dependent three by three matrices which can be decomposed in terms of the four scalar fields according to where and similar decompositions for M u S (φ) and M d S (φ) with M u,k and M d,k obtained by the substitutions of g Q iS → g u iS and g d iS respectively in Eq. (8). Introducing the following combinations of the coupling matrices M Q,k and M q,k (k = 0, 1, 2, 3) with the fermion mixing matrices we arrive at the final form of the Yukawa interactions We have combined the four scalars φ 0S and φ S into φ kS with k = 0, 1, 2, 3. For the FCNC rare top decays that we are studying, only V u,k (L,R) are relevant.

Neutral Currents
We also need the neutral current interactions for the SM Z boson and photon couple to the quarks and their mirrors.
The above neutral current interactions in SM and the new Yukawa couplings can induce FCNC decay t → V q at one-loop level as depicted by the three Feynman diagrams in Fig. 1.
The effective Lagrangian for t → Zq and t → γq can be expressed as where q = (u, c); Z µν = ∂ µ Z ν − ∂ ν Z µ and F µν = ∂ µ A ν − ∂ ν A µ ; and A L,R , A L,R and C L,R are dimensionless quantities.
In terms of the dimensionless mass ratios the partial decay rate for t → Zq is given by where λ(1, y, z) For a given model, the dimensionless quantities A L,R and C L,R can be determined.
In the mirror fermion model, these quantities are induced at one loop level, as depicted by the Feynman diagrams in Fig. 1. Their analytical expressions are given in the Appendix.
Similarly, for t → γq we have with . The expressions for A L and A R are also given in the Appendix.

IV. ANALYSIS
In our numerical analysis, we will make the following assumptions on the parameter space of the model.
(1) First, we will take all the unknown Yukawa couplings to be real and assume g q iS = g Q iS for q = (u, d) and i = 0, 1, 2. We will explore how our results depend on the couplings g Q iS . We note that it has been shown recently [34] in the extended mirror fermion model [25] the complex values of some of these Yukawa couplings, combined with the electroweak scale seesaw mechanism generating the minuscule neutrino masses, one can provide a solution to the strong CP problem without introducing axion.
(2) Since only the product V CKM = (V u L ) † V d L are known experimentally, we will study the following scenarios for illustrative purpose.
(3) For the three generation of mirror quark masses, we assume and vary the common mirror quark mass M from 150 to 800 GeV. We note that mirror fermions in this class of electroweak scale mirror fermion model are expected to have masses of electroweak scale to satisfy unitarity [27].
(4) For the scalars φ kS , their masses are necessarily small since they are link to the Dirac neutrino masses [21,22]. We set their masses m kS all equal 10 MeV.
Our numerical results are not sensitive to this choice as long as m kS m q M m .
(5) For the SM parameters, we use [35] m t = 173.21 GeV , m c = 1.275 GeV , m u = 2.3 MeV ,  In Fig. (2), we show the contour plots of log B(t → γu) (left panel) and log B(t → γc) (right panel) on the (log 10 (g Q 0S ), log 10 (g Q 1S )) plane in the case of Scenario 1 with g Q 2S set to be 10 −3 . Fig. (3) is similar as Fig. (2  Mirror quarks may be pair produced at the LHC [27]. Once produced the heavier mirror fermions may cascade into lighter ones by emitting an on-shell or off-shell SM W -boson, depending on the detail mass spectrum of the mirror fermions. The lightest mirror quark will then decay into its SM partner with any one of the scalar singlets φ kS via the new Yukawa interactions which are responsible to the FCNC decays of the top quark studied here. In the mirror lepton case, the corresponding Yukawa couplings g L iS are necessarily small since they are responsible for providing small Dirac masses to the neutrinos in the electroweak scale seesaw mechanism. Assuming the lightest mirror fermion is u M . In Fig. (6), we plot the contours of the decay length of u M in the (M, log 10 (g Q 0S )) plane. We take all the Yukawa couplings to be the same just for illustrations. One can see that for very small Yukawa couplings < 10 −6 , can the decay length reach a few mm for a displaced vertex. Search strategies for the mirror fermions would then be quite different from the usual cases, involving not merely the missing energies but displaced vertices as well [27]. Current experiments at the LHC have the capability to perform such kind of searches. Further studies of this issue are warranted.
Nevertheless, for the mirror quarks, there is no a priori reason that these new Yukawa couplings have to be very small except that there are stringent constraints from the mixings between SM fermions and their mirrors. The mixing angle is roughly of order g Q iS φ kS /M . For g Q iS ∼ 1, φ kS ∼ 1 MeV and M ∼ 500 GeV, this mixing angle is about 2 × 10 −6 . A full analysis taking into the account of the mixing effects is beyond the scope of this paper.
In Figs. (7) and (8) At 14 TeV the total cross section for tt production at the LHC is about 598 pb.
With a luminosity of 300 (1000) fb −1 , we thus expect 180 (6) events of t → Zc for a branching ratio of 10 −6 (10 −8 ) before any experimental cuts. For the other processes t → Zu and t → γq, their branching ratios are typically smaller by 1 − 2 orders of magnitude.
For the gluon mode t → gq its partial width is about 42 times larger than that of the photon mode t → γq. The LHC limits for the branching ratios of t → gu and t → gc are 2.0 × 10 −5 and 4.1 × 10 −4 respectively from CMS [37], and 4.0 × 10 −5 and 20 × 10 −5 respectively from ATLAS [38]. These branching ratios are extracted from the single top production via FCNC interactions from gluon plus up-or charmquark initial states. They are still 1 − 2 orders of magnitude above our theoretical predictions.
It is also interesting to consider FCNC processes involving both the heavy top quark and the 125 GeV Higgs, the two heaviest particles in the SM. One particular important process is t → hq, which LHC has obtained the following limits [39,40] In the mirror fermion model, the mirror Higgs as well as the GM triplets could be an imposter for the 125 GeV Higgs due to mixing effects, which must be taken into account. This work will be reported elsewhere [41].
To minimize cluttering in our expressions given below, we define where V u,k (L,R) are given by Eq. (9). The individual contributions from each diagrams can be computed using the automated tools FormCalc and LoopTools in FeynArts [42]. The results are listed as follows. and and where ∆ = 2/ − γ E + ln 4π with = 4 − d is the regulator in dimensional regularization and γ E being the Euler's constant, one can easily verify that the divergences in the three diagrams summed up to nil leading to finite results for C L and C R .
Only  Since the PV functions C 1 , C 2 , C 11 , C 12 and C 22 do not have ultraviolet divergences, A L and A R are finite, as one should expect for they are the coefficients of the nonrenormalizable magnetic and electric dipole operators.
Form Factors for t → γq and t → gq A L and A R can be obtained from the above A L and A R respectively by replacing The decay rate for t → gq can be obtained from that of t → γq simply by replacing the top quark electric charge 2 3 e by the strong coupling g s and multiply the final result by an overall color factor (N 2 C − 1)/2N C where N C is the number of color. Thus Γ(t → gq) Taking N C = 3, α s = 0.11 and α −1 em = 128, this ratio is about 42. Next-to-leading order QCD corrections to the processes t → γq, t → Zq and t → gq can be found