Flavor Changing Neutral Currents in the Asymmetric Left-Right Gauge Model

In the $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{(B-L)/2}$ extension of the standard model, a minimal (but asymmetric) scalar sector consists of one $SU(2)_R \times U(1)_{(B-L)/2}$ doublet and one $SU(2)_L \times SU(2)_R$ bidoublet. Previous and recent studies have shown that this choice is useful for understanding neutrino mass as well as dark matter. The constraints from flavor changing neutral currents mediated by the scalar sector are discussed in the context of the latest experimental data.

neutral scalars is derived. It is shown that under a simple assumption, all such effects depend only on two scalar masses which are almost degenerate in addition to an unknown unitary 3 × 3 matrix V R which is the right-handed analog of the well-known CKM matrix V CKM for left-handed quarks. In Sec. 4 the experimental data on the K −K, B d −B d , and B s −B s mass differences, as well as the recent data on B s → µ + µ − , are compared against their SM predictions to constrain the two scalar masses assuming that (A) V R = V CKM and (B) V R = 1. In Sec. 5 there are some concluding remarks.
The most general Higgs potential consisting of Φ R , η, andη is given by [2] where all parameters have been chosen real for simplicity. Let φ 0 R = v R and η 0 1,2 = v 1,2 , then the minimum of V has a solution where v 2 v 1 , i.e. with In the limit v 2 = 0, the physical Higgs bosons are φ ± 2 and h I = √ 2Im(φ 0 2 ) with masses squared and three linear combinations of with the 3 × 3 mass-squared matrix Since v 1 /v R is known to be small, h 1,2,R are approximately mass eignestates, with h 1 almost equal to the observed 125 GeV scalar boson at the Large Hadron Collider (LHC). Note also that h 2 is almost degenerate with h I in mass. We can make this even more precise by having small λ 4 and f 1,3 .
There are two charged gauge bosons W ± L and W ± R in the 2 × 2 mass-squared matrix given by With our assumption that v 2 v 1 , W L − W R mixing is negligible. The present LHC bound on the W R mass is 3.7 TeV [7].
There are three neutral gauge bosons, i.e. W 3L from SU (2) L , W 3R from SU (2) R , and B from U (1) (B−L)/2 , with couplings g L , g R , and g B respectively. Let them be rotated to the following three orthonormal states: where 1 The photon A is massless and decouples from Z and Z , the latter two forming a masssquared matrix given by .
The neutral-current gauge interactions are given by The present LHC bound on the Z mass is 4.1 TeV [8]. The Z − Z mixing is given by which is then less than 3.6 × 10 −4 for g R = g L and within precision measurement bounds.

Yukawa Sector and the FCNC Structure
The fermion content is well-known, i.e.
with the electric charge given by Q = I 3L + I 3R + (B − L)/2. Now the Yukawa couplings between the quarks and the neutral members of the scalar bidoublets are In the limit v 2 = 0, both up and down quark masses come from only v 1 . Hence where U L,R and D L,R are unitary matrices, with being the known quark mixing matrix for left-handed charged currents and the corresponding unknown one for their right-handed counterpart.
Whereas Z and Z couple diagonally to all quarks, nondiagonal terms appear in the scalar Yukawa couplings. Using Eqs. (18), (19) and (20), the FCNC structure is then completely determined, i.e.
for the up quarks, and for the down quarks. Hence h 1 behaves as the SM Higgs boson, and at tree-level, all FCNC effects come from h 2 and h I . We may thus use present data to constrain these two masses.
Note that all FCNC effects are suppressed by quark masses, so we have an understanding of why they are particularly small in light meson systems.
The analog of Eq. (18) for leptons is Hence If neutrinos are Dirac fermions, then good approximation. The analog of Eq. (22) for charged leptons is then In the following we consider the contributions of Eqs. (21), (22), and (25) to a number of processes sensitive to them in two scenarios: We compare the most recent experimental data with theoretical SM calculations to obtain constraints coming from the mass differences ∆M K , ∆M B d , ∆M Bs of the neutral meson systems of with an upper limitB (B d → µ + µ − ) LHCb < 3.4 × 10 −10 at 95% confidence-level. These values are in agreement with the next-to-leading-order (NLO) electroweak (EW) as well as NNLO QCD predictions [10,11]: Nevertheless, new physics (NP) contributions are possible within the error bars. In addition, the K-K and B q -B q mixings, which interfere to obtain time-averaged decay widths [12,13,14], may also provide possible signals of NP.

∆M B q and ∆M K
In the SM, other than long-distance contributions [17], B q −B q and K −K mixings occur mainly via the well-known box diagrams with the exchange of W ± bosons and the (u, c, t) quarks. In the asymmetric left-right model, the new scalars h 2 and h I have additional tree-level contributions. We consider the usual operator analysis with Wilson coefficients obtained from the renormalization group (RG). The mass difference between the two mass eigenstates of a neutral meson system (see [19,21] for details) may be obtained from the ∆F = 2 effective Hamiltonian [22,23,24] where the operators relevant to the SM and the new scalar contributions are [11] O for the B d −B d and B s −B s systems. In the case of K −K, we just change b to s and q to d in the above. P R and P L are right-and left-handed projection operators (1 ± γ 5 )/2, respectively. α and β are color indices. We follow the details in [22] with recent updates [11,25] for B q as well as [17] for K. After ignoring terms that are suppressed by light quark masses, we obtain with λ x ≡ V xs V * xd . The Inami-Lim function S 0 (x i , x j ) with x q ≡ (m q (m q )/m W ) 2 describes the electroweak corrections in one loop [26]. The factors η i are perturbative QCD corrections at NLO [22], as well as [27]( [23]) for the new B q (K) terms. Since the QCD corrections generate nondiagonal entries, the color mixed operators should be considered as well at low scale [28] (see also [15,29,30]).
Noting that O 2,3 = Õ 2,3 in QCD, we consider the relevant operators for B q −B q mixing in terms of their bag parameters [11,31], and and Bq include all nonperturbative effects. The lattice calculation has been done in [11] for B q with in the scheme of [29], as well as [16] for K. The renormalization group evolution effects are considered in [23,27].
In the asymmetric left-right model, the tree-level h 2 and h I contributions to the Wilson coefficients at the new physics scale µ NP are , and the matrix V d comes from the second term of Eq. (22).
The B q mass difference is thus given by where η 4 3.90, η 22 2.25 and η 32 −0.12, [27,32]. Similarly, the K 0 mass difference is where P 1,2 are given in [17,23] and a recently updated lattice simulation [16]. Hence Note that φ M s may deviate [14] from the SM value, i.e. φ M s = φ SM s + φ N P s . A nonzero φ N P s would contribute to the CP violation effect in the B s → (J/ψ)φ decay (see [33] and the recent review [21]

B s → µ + µ −
The scalars h 2 and h I contribute not only to the mass difference of B s , but also to the decay of B s → µ + µ − at tree level. The SM contribution is dominated by the operator O SM 10 , so we ignore other possible SM operators [24,25]. The effective Hamiltonian is given by [10,37] where α em is the fine structure constant, and s 2 W ≡ sin 2 θ W with θ W the weak mixing angle.
The operators are defined as Including the b quark mass m b makes those operators as well as their Wilson coefficients to be renormalization-group invariant [25]. For the NLO SM contribution, we use a numerical value approximated by [25] C where m p t is the t quark pole mass. The contributions of NLO EW and NNLO QCD have been computed by [38,39,40,41,42]. The non-SM Wilson coefficients are given by tree-level h 2 or h I exchange, i.e.
, and the matrix V p comes from the second term of Eq. (25).
The form factors are From the above, the branching fraction of B s → µ + µ − is then [12] B where m Bs , τ Bs and f Bs denote the mass, lifetime and decay constant of the B s meson, respectively. The amplitudes P and S are defined as [14] P ≡ C SM 10 + To compare against experimental data, the time-integrated branching fraction is discussed extensively in [12,13,14,43], i.e.
where y s = ∆Γ s /2Γ s (Γ s being the average B s decay width) and [33] with

Numerical Analysis
We now discuss the experimental constraints on the two scalar masses m 2 and m I . We allow for the theoretical uncertainties in computing ∆M K , ∆M Bq and B s → µ + µ − which arise mainly from the decay constant f Bq (and the bag parametersB (i) q ) and the combination of CKM matrix elements |V * ts V tb | (i.e. |V cb | as well as |V ub |, from the unitarity of V CKM ) [11].
We note that there is a long-standing discrepancy between the determinations of V ub from inclusive and exclusive B decays. We adopt the recent averaged CKM matrix elements by the CKMfitter group [35], and use running quark masses [44]. Our input parameters are given in Table 1, and the scales used are {µ K , µ b , µ NP } = {2, 3, 1000} GeV.  Table XIII of [11] in the scheme of [29]  Eqs. (39) and (40). The B s → µ + µ − contribution comes from Eqs. (47) and (48 restricts it to only a thin line in the region of heavier masses, i.e. m 2 m I . Their overlap shows a strong constraint indicated by an arrow (cyan) in Fig. 1. If the ∆M K constraint is included, then this tiny allowed region is ruled out if only the short-distance (SD) contribution is considered. Adding the long-distance (LD) contributions from π and η exchange [46,47] ∆m with a consistent overlap with the data may be obtained. Although the LD contributions are still not well understood, with somewhat large uncertainties [17], these terms shift the SM contribution and allow Scenario A to survive. In summary, the above constraints with LD physics allow the masses to lie within the region 20.0 TeV ≤ m 2 m I ≤ 22.8 TeV.
In Scenario B, the asymmetric mixing matrix elements e.g. Hence lighter m 2 , m I masses from ∆M Bq are not ruled out in the (blue, green) area of Fig. 1 where |λ 3 | = 4π has been used. The two branches (purple) represent the model restrictions on (m 2 , m I ) depending on the sign of λ 3 . If a value of |λ 3 | less than 4π is used, then the region between these two branches will be filled in. Since our model contribution to B s → µ + µ − is proportional to v 2 which is always assumed to be small so far, there is no constraint from it unless v 2 is sizeable. For |λ 3 | = 4π, if we also assume v 2 < 0.5v 1 , then within 1σ of the B s → µ + µ − experimental rate, the allowed region cuts off for small (m 2 , m I ), as shown (purple) in Fig. 1. The allowed region with λ 3 = 4π in Scenario B is indicated by an arrow (cyan) in the subgraph, i.e. 1.80 ≤ m I ≤ 2.45 TeV. For λ 3 < 4π, a thin region opens up above the purple line. As for ∆M K in Scenario B, this result is not affected whether LD contributions are included or not.
From Eq. (21), we see that D 0 −D 0 mixing is suppressed by down-quark masses in the asymmetric left-right model. It does not provide a tighter constraint [32,48,49].

Concluding Remarks
We have studied the possible contributions of the heavy scalars h 2 and h I in the asymmetric left-right model to B q −B q mixings as well as B s → µ + µ − . We find that improvements of the fit to experimental data within 1σ are possible, as shown in Fig. 1. In the scenario with the right-handed charged-current mixing matrix V R equal to V CKM , we predict m 2 m I to be between 20.0 and 22.8 TeV. If V R = 1, then m I 1.80 to 2.45 TeV, and m 2 2.00 to 2.60 TeV for λ 3 = 4π and small v 2 .
If the doublet Φ R is replaced with the triplet (ξ ++ R , ξ + R , ξ 0 R ), the FCNC analysis remains the same. What will change is that ν R will acquire a large Majorana mass and the usual neutrinos will get seesaw Majorana masses. A doubly-charged physical scalar ξ ±± R will also appear and decays to e ± e ± . In addition, there are more candidates for predestined dark matter [3], i.e. scalar SU (2) L triplet, fermion singlet, fermion bidoublet, fermion SU (2) L triplet, and fermion SU (2) R triplet.