New Physics in $B_{d,s}$-$\Bbar_{d,s}$ mixings and $B_{d,s}\to\mu^+\mu^-$ decays

A new way of probing new physics in the $B$ meson system is provided. We define double ratios for the observables of $B_{d,s}$-$\Bbar_{d,s}$ mixings and $B_{d,s}\to\mu^+\mu^-$ decays, and find simple relations between the observables. By using the relations we predict the yet-to-be measured branching ratio of $B_d\to\mu^+\mu^-$ to be $(0.809\sim1.03)\times 10^{-10}$, up to the new physics models.

Recent discovery of a Higgs boson at the large hadron collider (LHC) opens a new era of high energy physics. It may take time to confirm whether the new particle is really the Higgs boson of the standard model (SM), but it looks more and more like the SM Higgs.
The discovery of the Higgs boson would mean a completion of the SM. On the other hand, we have many reasons to believe that there must be new physics (NP) beyond the SM.
Unfortunately, the LHC up to now has not reported any clues of NP. But it is too early to say that there is no NP at all. B d,s mesons are good test beds for NP. Especially, B d,s -B d,s mixings and B d,s → µ + µ − decays are loop induced phenomena in the SM and very sensitive to NP effects. Current status of experiments is well compatible with the SM predictions.
But there is still some room for NP, as discussed in [4,6,7]. In this paper, we provide a where c a are the new couplings and f i is some function of c a /c b . For another observable O j we can also define a similar quantity R ab j which would be ≃ f j (c a /c b , we can find out from the double ratio relations which NP is realized in B physics. In this paper we specifically consider flavor changing scalar (un)particles and vector boson (Z ′ ) scenarios. Actually it was already known that ∆M q and Br(B q → µ + µ − ) can be related to each other [8]. In our approach, R ab i are directly proportional to the new physics effects, so the resulting relations are solely those of new physics. The relations might be different for various models, which makes it easier to see which kind of new physics is realized.
The new physics couplings adopted in this analysis are summarized as follows [4,9]: where P L,R = (1 ∓ γ 5 )/2. In L U one can also include the right-handed couplings, but here (and in [9]) only the minimal extension of the SM are considered for simplicity.
First consider the B d,s -B d,s mixing. The mixing effect is parametrized as the following where and where the subscript V (S) stands for Z ′ (H) contributions. Explicitly [6,7], where The expectation values of the operators Q a i are For the case of ∆ bq R = 0, up to the leading order of ∆ bq L . Now we define a double ratio R Z ′ ∆M as where the result of Eq. (23) is applied. Similarly, for the scalar contribution (with ∆ bq R = 0), We assumed here that the light-quark dependence on P a i (µ H , B q ) is negligible [10], and thus In the scalar unparticle scenario [9], Here where M SM 12 is the SM contribution and with d U being the scaling dimension of scalar unparticle operator. The double ratio for scalar unparticle is where we put c bq U L ≡c bq U L · V * tb V tq . For realc bq U L , one has Ifc bq U L is purely imaginary, one gets a similar result. Now we move to B d,s → µ + µ − decays. The relevant effective Hamiltonian is given by where the operators O i are For B s decay it is convenient to define [11] Br where and the asymmetric parameter where R H(L) exp −Γ where The standard model contribution is with η Y = 1.012 and For Z ′ model, while other coefficients are vanishing. Using ∆ sb L,R (Z ′ ) = ∆ bs L,R (Z ′ ) * , one has Br(B s → µ + µ − ) Br(B s → µ + µ − ) SM − 1 ≃ y s 1 + y s cos(2θ Bs Y + θ Bs S ) − 1 and up to O(y s ∆ L,R ∆ A ). For ∆ R = 0 and ∆ bq L =∆ bq L V tq where∆ bq L is real, the double ratio remarkably reduces to (49) In this case the ratio R Z ′ ∆M = (∆ bs L /∆ bd L ) 2 , and thus one arrives at a very simple relation For neutral scalar H, the coefficients are One can define a double ratio R H µµ similar to Eq. (48). For simplicity we assume that ∆ R = 0 and ∆ bq L =∆ bq L V tq with real∆ bq L . Note that in this case For the case of ∆ µµ S (H) = 0, the double ratio reduces to be On the other hand if ∆ µµ P = 0, For scalar unparticles [12], and thus A ∆Γ = cos(2ϕ P − φ U s ). Here φ U s is the phase of ∆ U in Eq. (26). For realc bq U L , c ℓ U L , cos(2ϕ P − φ U s ) ≃ 1 up to O(c U L ) 4 , and the double ratio is where the result of Eq. (30) is used. Our results are summarized as follows: The reason why R H µµ ∼ R H ∆M is that in R H µµ , Br/Br SM − 1 is non-vanishing only at O(c 2 ), due to the fact that ∆ µµ P is pure imaginary [4]. Numerically, Eqs. (62)-(65) are where R ∆M = 0.712 is used. The above results can be used to predict the yet-to-be-measured branching ratio, Br(B d → µ + µ − ). Table I shows the predicted values of Br(B d → µ + µ − ).
Note that the values of Table I [13], the relative size of NP in ∆M d,s (= h d,s ) is currently 0.2 ∼ 0.3, and would be 0.1 in near future ("Stage I" where the LHCb will end). As for B d → µ + µ − , current upper bound is almost order of magnitude larger than the SM prediction. It is predicted in [14] that at 2σ, 0.3 × 10 −10 Br(B d → µ + µ − ) 1.8 × 10 −10 . If the measured branching ratio does not lie within this window, it would be a clear indication of NP. It is also found in [14] that although the measured value of Br(B s → µ + µ − ) provide constraints on NP, there are still sizable regions allowed for C S − C ′ S and C P − C ′ P parameter space.
Besides the current status of NP searches, we need NP for various reasons (dark matter for example). Although there have been no smoking-gun signals for NP up to now, we believe that the SM is not (and should not be) the full story of particle physics. In this context the double ratio analysis might be very promising with the coming flavor precision era, and can be also applied to the K meson systems.