Probing New Physics with the B_s to μ+ μ- Time-Dependent Rate
Andrzej J. Buras, Robert Fleischer, Jennifer Girrbach, Robert Knegjens
FFLAVOUR(267104)-ERC-35Nikhef-2013-007
Probing New Physics with the B s → µ + µ − Time-Dependent Rate
Andrzej J. Buras a,b , Robert Fleischer c,d ,Jennifer Girrbach a,b and Robert Knegjens ca TUM Institute for Advanced Study, Lichtenbergstr. 2a, D-85748 Garching, Germany b Physik Department, Technische Universit¨at M¨unchen, James-Franck-Straße,D-85748 Garching, Germany c Nikhef, Science Park 105, NL-1098 XG Amsterdam, Netherlands d Department of Physics and Astronomy, Vrije Universiteit Amsterdam, NL-1081 HVAmsterdam, Netherlands
Abstract
The B s → µ + µ − decay plays an outstanding role in tests of the Standard Modeland physics beyond it. The LHCb collaboration has recently reported the firstevidence for this decay at the 3 . σ level, with a branching ratio in the ballpark ofthe Standard Model prediction. Thanks to the recently established sizable decaywidth difference of the B s system, another observable, A µµ ∆Γ , is available, whichcan be extracted from the time-dependent untagged B s → µ + µ − rate. If tagginginformation is available, a CP-violating asymmetry, S µµ , can also be determined.These two observables exhibit sensitivity to New Physics that is complementaryto the branching ratio. We define and analyse scenarios in which these quantitiesallow us to discriminate between model-independent effective operators and theirCP-violating phases. In this context we classify a selection of popular New Physicsmodels into the considered scenarios. Furthermore, we consider specific modelswith tree-level FCNCs mediated by a heavy neutral gauge boson, pseudoscalaror scalar, finding striking differences in the predictions of these scenarios for theobservables considered and the correlations among them. We update the StandardModel prediction for the time-integrated branching ratio taking the subtle decaywidth difference effects into account. We find (3 . ± . × − , and discuss theerror budget. March 2013 a r X i v : . [ h e p - ph ] J un Introduction
The rare decay B s → µ + µ − is very strongly suppressed within the Standard Model (SM).As this need not be the case for SM extensions, which could dramatically enhance it,it has already for decades played an important role in constraining such extensions andgiving hope for seeing a clear signal of New Physics (NP) [1]. Using the parametricformula for its branching ratio from Ref. [2], and updating the values for the B s -decayconstant F B s [3] and the B s -lifetime τ B s [4], we find BR( B s → µ + µ − ) SM = (3 . ± . × − . (1)Thus, within the SM, only about one in every 300 million B s mesons is predicted todecay to a pair of muons.Concerning the measurement of this branching ratio, a complication arises due to thepresence of B s – ¯ B s oscillations [5]. In particular, LHCb has recently established a sizablevalue of the decay width difference ∆Γ s between the B s mass eigenstates [6]: y s ≡ Γ ( s )L − Γ ( s )H Γ ( s )L + Γ ( s )H = ∆Γ s s = 0 . ± . , (2)where Γ s = τ − B s denotes the average B s decay width. This quantity enters the time-integrated decay rate, which is at the origin of the measurement of the “experimental”branching ratio BR( B s → µ + µ − ). It is related to the “theoretical” branching ratio,referring to y s = 0, as follows [5]:BR( B s → µ + µ − ) = (cid:20) − y s A µµ ∆Γ y s (cid:21) BR( B s → µ + µ − ) , (3)where the observable A µµ ∆Γ equals +1 in the SM. Using the numerical value for the SMbranching ratio in (1) and the experimental value for y s in (2) givesBR( B s → µ + µ − ) SM = (3 . ± . × − , (4)which is the reference value for comparing the time-integrated experimental branchingratio with the SM.Over the last decade we have seen the upper bounds for the B s → µ + µ − branchingratio continuously move down thanks to the CDF and D0 collaborations at the Tevatronand the ATLAS, CMS and LHCb experiments at the LHC (for a review, see Ref. [7]). InNovember 2012, the LHCb collaboration reported the first evidence for the B s → µ + µ − decay at the 3 . σ level, with the following branching ratio [8]:BR( B s → µ + µ − ) LHCb = (3 . +1 . − . ) × − ∈ [1 . , . × − (95% C . L . ) . (5)The agreement with (4) is remarkable, although the rather large experimental error stillallows for sizable NP contributions. It is already obvious at this stage, however, thatthis will be a challenging endeavour. This should be compared with BR( B s → µ + µ − ) SM = (3 . ± . × − in Ref. [2], implying thatthe central value remains practically unchanged but the error has decreased significantly. We discussthe reason for this change in Subsection 2.3.
1s emphasized in Ref. [5], and discussed and illustrated in detail below, the observable A µµ ∆Γ entering (3) is sensitive to NP contributions. It is thus a complementary observableto the B s → µ + µ − branching ratio, offering independent information on the short-distance structure of this decay. The observable A µµ ∆Γ can be extracted from the untaggeddata sample, for which no distinction is made between initially present B s or ¯ B s mesons,once enough decay events are available for the decay-time information to be accuratelytaken into account [5, 9]. The conversion factor in (3) would then be determined fromthe data, thereby also allowing the extraction of the theoretical B s → µ + µ − branchingratio.If tagging information is included, requiring even more events to compensate theefficiency of distinguishing between initially present B s or ¯ B s mesons, a CP-violating,time-dependent rate asymmetry can be measured. As we assume that the muon helicitywill not be measured, this rate asymmetry is governed by another, third observable, S µµ , which vanishes in the SM but is very sensitive to new CP-violating effects entering B s → µ + µ − in extensions of the SM [5]. Analyses of CP-violating effects in B s ( d ) → (cid:96) + (cid:96) − decays, which neglected ∆Γ s effects, were performed for various models of NP inRefs. [10–13]. An analysis of Z (cid:48) models that including ∆Γ s effects was performed inRef. [14].The observables A µµ ∆Γ and S µµ both depend on the CP-violating B s – ¯ B s mixing phase φ s = φ SM s + φ NP s = − λ η + φ NP s , (6)where the numerical value of the SM piece, involving the Wolfenstein parameters λ and η of the CKM matrix, is given by − (2 . ± . ◦ . The LHCb analysis of CP-violationin the B s → J/ψφ decay currently gives the most precise experimental determination ofthis phase [6]: φ s = − (0 . ± . ◦ . (7)Hadronic uncertainties from doubly Cabibbo-suppressed penguin effects have been ne-glected in this measurement; these corrections have to be controlled once the experimen-tal precision improves further [16].As the measurements of the observables A µµ ∆Γ and S µµ refer to the era of the LHCbupgrade, we can assume that φ s will be known precisely once data for these observablesbecome available. The goal of the present paper is to investigate and illustrate how theexperimental knowledge of the trio from B s → µ + µ − ,BR( B s → µ + µ − ) , A µµ ∆Γ , S µµ , (8)will shed light on the possible presence of NP in this decay, which cannot be obtainedon the basis of the information on BR( B s → µ + µ − ) alone.Our paper is organized as follows: in Section 2 we recall the definitions of the ob-servables in (8) and discuss their various properties. In Section 3 we introduce variousscenarios for NP, classifying them in terms of four general parameters which can in prin-ciple be calculated in any fundamental model and are directly related to the physics of We avoid averages for φ s that include decays such as B s → J/ψf (980) because of the unsettledhadronic structure of the f (980) state [15]. s → µ + µ − . In this context we classify a selection of popular NP models into the con-sidered scenarios. In Section 4 we consider three classes of specific NP models. The firstone with tree level neutral gauge boson contributions to FCNC processes that could rep-resent Z (cid:48) models or models with flavour violating Z couplings. The second one in whichthe role of gauge bosons is taken over by scalar or pseudoscalar tree-level exchanges. Fi-nally, we present a third model in which both a scalar and a pseudoscalar with the samemass couple equally to quarks and leptons. We demonstrate how the measurements ofthe observables in (8) can distinguish between these three classes of models. In Section 5we summarize the highlights of our paper. B s → µ + µ − The general model-independent low-energy effective Hamiltonian for a B s → (cid:96) + (cid:96) − decayis [17, 18] H eff = − G F α √ π (cid:40) V ∗ ts V tb ,S,P (cid:88) i ( C i O i + C (cid:48) i O (cid:48) i ) + h . c (cid:41) , (9)where the operators are O = (¯ sγ µ P L b )(¯ lγ µ γ l ) , O (cid:48) = (¯ sγ µ P R b )(¯ lγ µ γ l ) , O S = m b (¯ sP R b )(¯ ll ) , O (cid:48) S = m b (¯ sP L b )(¯ ll ) , O P = m b (¯ sP R b )(¯ lγ l ) , O (cid:48) P = m b (¯ sP L b )(¯ lγ l ) (10)and α = e / π is the QED fine structure constant. The observables that we will calculatebelow can each be expressed in terms of the following combinations of Wilson Coefficients: P ≡ C − C (cid:48) C SM10 + m B s m µ (cid:18) m b m b + m s (cid:19) (cid:18) C P − C (cid:48) P C SM10 (cid:19) ≡ | P | e iϕ P ,S ≡ (cid:115) − m µ m B s m B s m µ (cid:18) m b m b + m s (cid:19) (cid:18) C S − C (cid:48) S C SM10 (cid:19) ≡ | S | e iϕ S . (11)In the SM C = C SM10 and C (cid:48) , C ( (cid:48) ) S and C ( (cid:48) ) P are all negligibly small, so that P SM = 1and S SM = 0. The Wilson coefficient C is given in the SM as follows C SM10 = − η Y sin − θ W Y ( x t ) = − . , (12)where Y ( x t ) is a one-loop function with x t = m t /M W [19], and the coefficient η Y is aQCD factor that for m t = m t ( m t ) is close to unity: η Y = 1 .
012 [20, 21]. The scalarWilson coefficients are thereby related to the parameter S by the numerical factor m b ( C S − C (cid:48) S ) = − . × S. (13)Note that while C and C (cid:48) are dimensionless, the coefficients C ( (cid:48) ) S and C ( (cid:48) ) P have dimen-sion GeV − . 3 .2 Time-Dependent Rates The time-dependent rate for a B s meson decaying to two muons with a specific helicity λ = L, R is given byΓ( B s ( t ) → µ + λ µ − λ ) = G F M W sin θ W π (cid:12)(cid:12) C SM10 V ts V ∗ tb (cid:12)(cid:12) F B s m B s m µ (cid:115) − m µ m B s × (cid:0) | P | + | S | (cid:1) × (cid:26) C λµµ cos(∆ M s t ) + S µµ cos(∆ M s t )+ cosh (cid:18) y s tτ B s (cid:19) + A µµ ∆Γ sinh (cid:18) y s tτ B s (cid:19) (cid:27) × e − t/τ Bs , (14)where τ B s ≡ / (Γ H + Γ L ) is the B s mean lifetime and y s is defined in (2).The time-dependent rate for a ¯ B s meson is obtained from the above expression byreplacing C λµµ → −C λµµ and S µµ → −S µµ . The time-dependent observables for both ratescan be expressed in terms of the parameters defined in (11) as [5, 22] C λµµ = − η λ (cid:20) | P S | cos( ϕ P − ϕ S ) | P | + | S | (cid:21) , (15) S µµ = | P | sin(2 ϕ P − φ NP s ) − | S | sin(2 ϕ S − φ NP s ) | P | + | S | , (16) A µµ ∆Γ = | P | cos(2 ϕ P − φ NP s ) − | S | cos(2 ϕ S − φ NP s ) | P | + | S | . (17)The phase φ NP s represents the CP-violating NP contributions to B s – ¯ B s mixing. It in-fluences the mixing-induced CP asymmetry in the B s ( ¯ B s ) → ψφ decays [23], with thelatter given by S ψφ = − sin φ s = sin(2 | β s | − φ NP s ) , V ts = −| V ts | e − iβ s (18)with β s (cid:39) − ◦ . In Ref. [1] and the papers reviewed there φ NP s = 2 ϕ B s .Only the observable C λµµ is dependent on the helicity of the final state i.e. it dependson the parameter η λ ≡ { +1: L ; − R } . The presence of the observable A µ + µ − ∆Γ is aconsequence of the sizable B s decay width difference ∆Γ s .In practice the muon helicities λ are very challenging to measure. If no attempt ismade to disentangle them, then we measure their sum:Γ( B s ( t ) → µ + µ − ) ≡ (cid:88) λ = L,R Γ( B s ( t ) → µ + λ µ − λ ) , Γ( ¯ B s ( t ) → µ + µ − ) ≡ (cid:88) λ = L,R
Γ( ¯ B s ( t ) → µ + λ µ − λ ) . (19)Observe from equations (14) and (15) that C λµµ , which was dependent on the muonhelicity, cancels in both sums [5]. 4he B s → µ + µ − helicity-summed time-dependent untagged rate is then given by (cid:104) Γ( B s ( t ) → µ + µ − ) (cid:105) ≡ Γ( B s ( t ) → µ + µ − ) + Γ( ¯ B s ( t ) → µ + µ − )= G F M W sin θ W π (cid:12)(cid:12) C SM10 V ts V ∗ tb (cid:12)(cid:12) F B s m B s m µ (cid:115) − m µ m B s × (cid:0) | P | + | S | (cid:1) × e − t/τ Bs [cosh ( y s t/τ B s ) + A µµ ∆Γ sinh ( y s t/τ B s )] . (20)Similarly, the helicity-summed time-dependent tagged rate asymmetry isΓ( B s ( t ) → µ + µ − ) − Γ( ¯ B s ( t ) → µ + µ − )Γ( B s ( t ) → µ + µ − ) + Γ( ¯ B s ( t ) → µ + µ − ) = S µµ sin(∆ M s t )cosh( y s t/τ B s ) + A µµ ∆Γ sinh( y s t/τ B s ) . (21)It is important to clarify that although there is no explicit term for direct CP violationin the rate asymmetry, this does not mean that the absolute values squared of S µµ and A µµ ∆Γ necessarily sum to one. These two observables also have an implicit dependence on C λµµ , the rate asymmetry for B s and ¯ B s decays to the specific helicity muon final states.This gives the relation |S µµ | + |A µµ ∆Γ | = 1 − |C λµµ | = 1 − (cid:20) | P S | cos( ϕ P − ϕ S ) | P | + | S | (cid:21) . (22)Thus if there are no new CP-violating phases in the mixing or decay amplitudes, ϕ P = ϕ S = φ NP s = 0 such that S µµ = 0, A µµ ∆Γ does not have to take its SM value of 1. Thepresence of a non-negligible scalar operator O ( (cid:48) ) S , so that | S | (cid:54) = 0, is sufficient to ensurethat A µµ ∆Γ (cid:54) = 1, as can also be seen from (17).In contrast to the branching ratio, the dependence on F B s cancels in both A µµ ∆Γ and S µµ , and these observables are also not affected by CKM uncertainties. Consequently,they are theoretically clean. Moreover, these observables are also not affected by theratio f d /f s of fragmentation functions, which are the major limitation of the precisionof the B s → µ + µ − branching ratio measurement at hadron colliders [24]. As A µµ ∆Γ does not rely on flavour tagging, which is difficult for a rare decay, it will be easier todetermine. Given enough statistics, a full fit to the time-dependent untagged rate willgive A µµ ∆Γ . With limited statistics, an effective lifetime measurement may be easier, whichcorresponds to fitting a single exponential to this rate. For a maximal likelihood fit, the B s → µ + µ − effective lifetime is equal to the time expectation value of (20) [9]: τ µµ ≡ (cid:82) ∞ t (cid:104) Γ( B s ( t ) → µ + µ − ) (cid:105) dt (cid:82) ∞ (cid:104) Γ( B s ( t ) → µ + µ − ) (cid:105) dt . (23)The untagged observable is then given by A µµ ∆Γ = 1 y s (cid:20) (1 − y s ) τ µµ − (1 + y s ) τ B s τ B s − (1 − y s ) τ µµ (cid:21) . (24) In principle, corrections arise from loop topologies with internal charm- and up-quark exchanges.However, these are strongly suppressed by the CKM ratio | V ∗ us V ub /V ∗ ts V tb | ∼ .
02, and are even furthersuppressed dynamically for B s → µ + µ − . These effects do hence not play any role for these observablesfrom the practical point of view. .3 The Branching Ratio A B s → µ + µ − branching ratio measurement amounts to counting all events over all(accessible) time, and is thus defined as the time integral of the untagged rate given in(20) [5, 9, 23]: BR( B s → µ + µ − ) ≡ (cid:90) ∞ (cid:104) Γ( B s ( t ) → µ + µ − ) (cid:105) dt. (25)LHCb has recently presented the first measurement of the B s → µ + µ − time-integrated rate [8] that we have given in (5). In contrast, the SM prediction for the B s → µ + µ − branching ratio in (1) is computed theoretically for one instant in time, namely at t = 0i.e. it neglects the effects of B s – ¯ B s mixing. Specifically, it is given byBR( B s → µ + µ − ) SM = τ B s (cid:104) Γ( B s ( t ) → µ + µ − ) (cid:105) (cid:12)(cid:12)(cid:12) t =0 , P =1 , S =0 = τ B s G F M W sin θ W π (cid:12)(cid:12) C SM10 V ts V ∗ tb (cid:12)(cid:12) F B s m B s m µ (cid:115) − m µ m B s ; (26)an updated numerical estimate is given in (1).It is now straightforward to derive the expressionBR( B s → µ + µ − )BR( B s → µ + µ − ) SM = | P | + | S | . (27)However, as not the theoretical but the experimental branching ratio is measured, it isuseful to introduce the following ratio [5]: R ≡ BR( B s → µ + µ − )BR( B s → µ + µ − ) SM = (cid:20) A µµ ∆Γ y s y s (cid:21) × ( | P | + | S | )= (cid:20) y s cos(2 ϕ P − φ NP s )1 + y s (cid:21) | P | + (cid:20) − y s cos(2 ϕ S − φ NP s )1 + y s (cid:21) | S | , (28)where the sizable decay width difference ∆Γ s enters. The parameter R is related to R defined in Ref. [5] by R = (1 − y s ) R . Combining the theoretical SM prediction in (1)with the experimental result in (5) gives R = 0 . +0 . − . ∈ [0 . , .
80] (95% C . L) . (29)This range should be compared with the SM value R SM = 1.Finally we would like to explain the origin of the reduced error in (1). To this endwe return to the basic parametric formula (18) for the theoretical branching ratio inRef. [2]. It turns out that the changes of the input parameters over the last six monthshave practically no impact on the central value obtained there. Indeed updating thecentral values of F B s and τ B s , we can cast this formula into the following expression:BR( B s → µ + µ − ) SM = 3 . × − (cid:18) M t . (cid:19) . (cid:18) F B s
225 MeV (cid:19) (cid:18) τ B s . (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) V ∗ tb V ts . (cid:12)(cid:12)(cid:12)(cid:12) . (30)6 t Τ B s F B s (cid:200) V tb (cid:42) V ts (cid:200) (cid:37) (cid:37) (cid:37) (cid:37) M t Τ B s (cid:68) M s B (cid:96) B s (cid:37) (cid:37) (cid:37) (cid:37) Figure 1: Error budgets for the two branching ratio calculations of B s → µ + µ − in theStandard Model given in (30) (left) and (33) (right).The most recent world averages for F B s [3] and τ B s [4] are F B s = (225 ±
3) MeV , τ B s = 1 . F B s = (227 ±
8) MeV and τ B s = 1 . τ B s is an experimental improvement, confirmation of the impressiveaccuracy on F B s is eagerly awaited. In Ref. [2] a more conservative approach has beenused, but here we follow Ref. [3], updating also τ B s . With unchanged input on M t and V ts with respect to Ref. [2] we arrive at (1) and consequently, after including the correctionfrom ∆Γ s , at (4).Now as stressed and analysed in [2, 25] additional modifications could come fromcomplete NLO electroweak corrections, which have just been completed (M. Gorbahn,private communication) and affect the overall factor in (30) by roughly 3%. The leftoveruncertainties due to unknown NNLO corrections are therefore fully negligible. Takingat face value the present error on F B s , the current error budget for the branching ratiois as follows: M t : 1 . , F B s : 2 . , τ B s : 0 . , | V ∗ tb V ts | : 4% , (32)It is also depicted in the left panel of Figure 1. Evidently, after completion of NLOelectroweak effects and improved values of F B s , the error on | V ∗ tb V ts | is now the largestuncertainty but this assumes that the error on F B s is indeed as small as obtained inRef. [3].While the small error on F B s is expected to be consolidated soon, the decrease of theerror in | V ts | appears to be much harder. In this context it should be recalled that thebranching ratio in question can also be calculated by using the mass difference ∆ M s [26].The updated parametric formula (13) of the latter paper readsBR( B s → µ + µ − ) SM = 3 . × − (cid:18) M t . (cid:19) . (cid:18) τ B s . (cid:19) (cid:18) . B B s (cid:19) (cid:18) ∆ M s . / ps (cid:19) . (33)7e note that among the uncertainties in (32), the largest two are absent and the uncer-tainty due to M t is reduced to 0 . M s is negligible [4]. Theerror budget for this expression reads M t : 0 . , τ B s : 0 . , ∆ M s : 0 . , ˆ B B s : 4 . , (34)where we used ˆ B B s = 1 . M s and the SM reproduces itsexperimental value. In view of this tacit assumption in (33), we prefer to use (1) as thepresent best estimate of the theoretical branching ratio. On the other hand, by using(33) we find, after the inclusion of ∆Γ s the correction,BR( B s → µ + µ − ) SM = (3 . ± . × − , (∆ M s ) (35)which agrees very well with (4). This updates the estimate (3 . ± . × − in Ref. [28],where ∆Γ s effects where not included and other input, in particular the value of τ B s , wasdifferent than now. Experiments have started honing in on the B s → µ + µ − time-integrated rate, or branchingratio, for which the observable R parameterises possible NP contributions. Next inline is a time-dependent analysis, first without tagging, giving A µµ ∆Γ , and then withtagging, giving S µµ . The end result will be three experimental observables, which, ifthere are scalar operators contributing to the decay mode, can each contain independentinformation (see the discussion around (22)).With the phase φ NP s already significantly constrained by the current data (7), thesethree observables depend on four unknowns: | P | , ϕ P , | S | , ϕ S . (36)Therefore we cannot in general solve for all of these model-independent NP parametersby considering the decay B s → µ + µ − alone. One solution is to invoke other b → sµ + µ − transitions like the decays B → Kµ + µ − and B → K ∗ µ + µ − . In particular, asemphasized in Ref. [29], observables in B → Kµ + µ − are sensitive to C S,P + C (cid:48) S,P , ratherthen differences of these coefficients, thereby allowing additional complementary testsand in principle the determination of all Wilson coefficients. But present form factoruncertainties in these decays do not yet provide significant new constraints on scalaroperators relatively to the ones obtained from B s → µ + µ − . In any case such analysiswould be beyond the scope of the present paper.In the spirit of the analysis of Z (cid:48) contributions to FCNC processes in Ref. [14], wherevarious scenarios for Z (cid:48) couplings to quarks have been considered, and an analogous8nalysis for tree-level scalar and pseudoscalar contributions to FCNC processes [30], wewill consider various scenarios for S and P that will allow us to reduce the number of freeNP parameters and eventually, with the help of future data, uniquely determine them.Our scenarios are motivated by generic features of NP models and, as we will see below,they result in a distinct phenomenology for the observables R , A µµ ∆Γ and S µµ . In thepresent section our analysis is dominantly phenomenological, although we discuss themotivation behind each scenario and the characteristic features of its phenomenology.Moreover we indicate what kind of fundamental physics could be at the basis of eachscenario considered and we survey specific models of NP and categorise them into thescenarios that we will list now.The five scenarios to be considered are as follows:(A) S = 0(B) P = 1(C) P ± S = 1(D) ϕ P , ϕ S ∈ { , π } (E) P = 0.The scenarios are intended to be limiting cases, i.e. although we are not aware of amodel that predicts P = 0, P ≈ S = 0 This scenario is realised if C S − C (cid:48) S = 0, leaving C ( (cid:48) )10 and C ( (cid:48) ) P free to take non-SMvalues as well as CP-violating phases. Thus models with only new gauge bosons orpseudoscalars naturally fall into this category and consequently, as we will see below,this scenario includes a number of popular BSM models. Also models with scalars canqualify, provided the scalars couple left-right symmetrically to quarks so that C S = C (cid:48) S .In this scenario the rate asymmetry between B s and ¯ B s decays to the individualmuon helicities vanishes: C λµµ = 0. Therefore the two time-dependent observables do notcarry independent information, being bound by the constraint |S µµ | + |A µµ ∆Γ | = 1 . (37)Specifically, A µµ ∆Γ = cos(2 ϕ P − φ NP s ) , S µµ = sin(2 ϕ P − φ NP s ) , (38)while the branching ratio observable is given by R = | P | (cid:20) y s cos(2 ϕ P − φ NP s )1 + y s (cid:21) . (39)9ormulae (38) and (39) are the basic expressions for this scenario. The three observ-ables in (38) and (39) are given in terms of two unknowns: | P | and ϕ P . As we assumeknowledge of φ NP s , A µµ ∆Γ and S µµ will allow an unambiguous extraction of the phase 2 ϕ P ,which in turn, with the help of R , will provide the value of | P | .The P parameter can also conveniently be expressed as P = 1 + ˜ P with˜ P = | ˜ P | e i ˜ ϕ P ≡ δC − C (cid:48) C SM10 + m B s m µ (cid:18) m b m b + m s (cid:19) (cid:18) C P − C (cid:48) P C SM10 (cid:19) . (40)where δC ≡ C − C SM10 . (41)In this notation all NP effects are contained in the parameter ˜ P . In the left panel ofFigure 2 we show the correlations between R and A µµ ∆Γ in Scenario A using this notation.We have varied ˜ P ∈ [0 , ϕ P . As will be discussed in detail in Section 4, the requirement for new gaugebosons or pseudoscalars to satisfy the B s mixing constraints implies that ˜ ϕ P ∼ π/ ϕ P ∼ , π , respectively.Note that in the case of no new phases, ϕ P ∈ { , π } , and φ NP s = 0, A µµ ∆Γ = 1 , S µµ = 0 , R = | P | . (42)While the first two results coincide with the SM, NP effects can still arise in R . In the CMFV scenario it is assumed that new low-energy effective operators beyondthose present in the SM are very strongly suppressed and that flavour violation andCP-violation are governed by the CKM matrix [31, 32]. Thus all the Wilson coefficientsaside from C are zero, and C is real. This translates into Scenario A , with the addedrestrictions that ϕ P = φ NP s = 0. Consequently the formula (42) applies and NP entersonly through the ratio R . Littlest Higgs Model with T-Parity (LHT)
Similar to CMFV, only SM operators are relevant in this framework but due to thepresence of new phases in the interactions of SM quarks with mirror quarks, CP asym-metries can differ from the SM ones. Therefore general formulae (38) and (39) applyhere. Typically BR( B s → µ + µ − ) is predicted to be larger than its SM value but it canonly be enhanced by 30% at most [33]. A significant part of this enhancement comesfrom the T-even sector that corresponds to the CMFV part of this model, while | S ψφ | and |S µµ | , governed by new phases in the mirror quark sector, are at most 0 . (cid:48) Models and RSc
As demonstrated in Ref. [14], larger effects than in LHT can be found in Z (cid:48) modelswith tree-level FCNC couplings. If new heavy neutral gauge bosons dominate NP con-tributions to FCNCs, S = 0 in these models, placing them automatically in ScenarioA . In contrast to CMFV and LHT, the presence of new operators implies a rather richphenomenology. Yet, in the absence of S the time-dependent observables are only sen-sitive to new CP violating phases and are not independent of one another. In Ref. [14]correlations between the S µµ observable and ∆ F = 2 observables have been found fordifferent scenarios for Z (cid:48) couplings with the size of effects that could be measured in thefuture provided the masses of these new gauge bosons do not exceed 2-3 TeV. We willreturn to this scenario in Section 4 showing results complementary to the ones presentedin Ref. [14].Smaller, but still measurable, effects have been found in 331 models in which newCP phases are present but no new operators [34]. Here NP effects in B s → µ + µ − arecomparable to the ones in the LHT model provided the mass of the new neutral gaugeboson does not exceed 2 TeV.Finally we mention the Randall–Sundrum model with custodial protection in whichNP contributions to B s → µ + µ − are governed by right-handed flavour-violating Z cou-plings to quarks but the resulting branching ratio is SM-like, with departures from SMprediction at most of order 15% [35]. Larger effects are found if the custodial protectionis absent and then left-handed couplings dominate [36]. Recently detailed analyses of Z couplings in similar scenarios related to partial compositeness have been presented inRefs. [37–39]. While having different goals than in Ref. [14], they also demonstrate thepower of B s → µ + µ − in distinguishing between various NP scenarios. Four Generation Models
In spite of the fact that the existence of a fourth generation seems to be very unlikely inview of the LHC data, in particular Higgs branching ratios, we just mention that it alsobelongs to
Scenario A . NP effects in B s → µ + µ − can still be sizable in these models.See Ref. [40] and references therein. Pseudoscalar Dominance
Also a model with NP dominated by tree-level FCNC contributions of a pseudoscalarbelongs to this class. It has been analysed recently in Ref. [30] and we will presentcomplementary implications of this model particularly suited to our paper in Section 4. P = 1 The simplest realisation of this scenario is C = C SM10 and C (cid:48) = C ( (cid:48) ) P = 0. However,pseudoscalars that couple left-right symmetrically to quarks, so that C P = C (cid:48) P , or aconspiracy of the form C − C (cid:48) = C SM10 are also allowed. The point is that in this11 . . . . . . . . ¯ R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . − . − . − . . . . . . . A ∆ Γ ( B s → µ + µ − ) SM | ˜ P | = | ˜ P | = . | ˜ P | = . ˜ ϕ P = π / ˜ ϕ P = π / ˜ ϕ P = π / P = 1 + | ˜ P | e i ˜ ϕ P , S = 0 | ˜ P | ≤ , ˜ ϕ P ∈ [ π , π ] | ˜ P | ≤ , ˜ ϕ P ∈ [0 , π ] ∪ [ π , π ]¯ R = 0 . +0 . − . . . . . . . . ¯ R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . − . − . − . . . . . . . A ∆ Γ ( B s → µ + µ − ) SM | S | = 1 | S | = 0 . | S | = 0 . ϕ S = π/ ϕ S = π / ϕ S = | S | , ϕ S free; P = 1¯ R = 0 . +0 . − . Excluded at 95% C . L . Figure 2: The correlation between the R and A µµ ∆Γ observables in Scenario A (left panel)and
Scenario B (right panel). In
Scenario A we have set P = 1 + ˜ P and S = 0 with ˜ P free to vary. In Scenario B P = 1 and S is free to vary.scenario only scalar operators O ( (cid:48) ) S drive new physics effects in B s → µ + µ − . In this sensethis case is complementary to Scenario A .As there are scalar operators present, there is a rate asymmetry in the B s and ¯ B s decays to the individual muon helicities. Therefore the two time-dependent observablesdo carry independent information. In this scenario the observables are given by A µµ ∆Γ = cos φ NP s − | S | cos(2 ϕ S − φ NP s )1 + | S | , S µµ = − sin φ NP s − | S | sin(2 ϕ S − φ NP s )1 + | S | ,R = 1 + y s cos φ NP s y s + | S | (cid:20) − y s cos(2 ϕ S − φ NP s )1 + y s (cid:21) . (43)Again, with precise value of φ NP s to be determined first, these three observables arein principle sufficient to determine the two NP unknowns, 2 ϕ S and | S | . Consequentlythe untagged observables R and A µµ ∆Γ are already sufficient to determine 2 ϕ S and | S | .Moreover, if all three observables are considered, correlations between them will resultthat depend on the precise value of φ NP s [30].In the right panel of Figure 2 we show the correlation between R and A µµ ∆Γ for differentvalues of S [5]. An interesting feature is that for no CP violating phase, ϕ S = { , π } , anincrease of | S | pushes A µµ ∆Γ →
0. But within current experimental bounds we have theprediction that A µµ ∆Γ cannot take a negative value. Moreover in this scenario | S | ≤ . .3.2 Examples of ModelsScalar Dominance A model with NP dominated by tree-level FCNC contributions of a scalar belongs tothis class. It has been analysed recently in Ref. [30] and we will present complementaryimplications of this model particularly suited to our paper in Section 4. P ± S = 1 The meaning of this scenario is clearer if we let P = 1 + ˜ P , with ˜ P defined in (40).Then the condition P ± S = 1 is equivalent to ˜ P = ∓ S i.e. in this scenario NP effectsto S and P are on the same footing. If we neglect contributions to C ( (cid:48) )10 and m µ withrespect to m B s , this scenario is realised if C ( (cid:48) ) S = ± C ( (cid:48) ) P .Letting ˜ P = − κS for κ = ±
1, the time dependent observables are A µµ ∆Γ = cos φ NP s − κ | S | cos( ϕ S − φ NP s )1 − κ | S | cos ϕ S + 2 | S | , S µµ = − sin φ NP s − κ | S | sin( ϕ S − φ NP s )1 − κ | S | cos ϕ S + 2 | S | , (44)which are in general independent. The branching ratio observable is R = 1 − κ | S | cos ϕ S + 2 | S | + y s [cos φ NP s − κ | S | cos( ϕ S − φ NP s )]1 + y s . (45)In the presence of a precise value of φ NP s , these three observables are sufficient to deter-mine the two NP unknowns ϕ S and | S | . Moreover, correlations between the involvedobservables characteristic for this scenario and additional tests are possible.The observable R is minimised by S crit = κ (1 + y s ) / φ NP s = 0, giving the lowerbound R ≥ − y s . (46)This lower bound, without the y s and phase considerations, was first observed in Ref. [41].A branching ratio measurement below this bound would thereby rule out this scenario.If we assume the new physics phase φ NP s in B s mixing is known, then the purelyuntagged observables A µµ ∆Γ and R can solve for S and ϕ S . Setting φ NP s = 0 for simplicity,we have the expressions | S | = | P − | = (cid:115) R (1 + y s )(1 − A µµ ∆Γ )2(1 + y s A µµ ∆Γ ) , cos ϕ S = − κ cos( ˜ ϕ P ) = (cid:115) (1 + y s A µµ ∆Γ )2 R (1 + y s )(1 − A µµ ∆Γ ) (cid:20) − R (1 + y s ) A µµ ∆Γ y s A µµ ∆Γ (cid:21) . (47)13 . . . . . . . . . ¯ R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . − . − . − . . . . . . . A ∆ Γ ( B s → µ + µ − ) S = − . S = 1 . S = 0 . S = 0 . S = 0 . S = 1 ϕ S = π/ ϕ S = π/ ϕ S = 0 SM S = 1 − P Excluded at 95% C . L . − − − −
45 0 45 90 135 180 ϕ S [deg] . . . . . . . . . . . . . . . . . | S | = | − P | A µ + µ − ∆Γ = 0 A µ + µ − ∆Γ = − . A µ + µ − ∆Γ = 0 . SM ¯ R = 0 . +0 . − . Excluded at 95% C . L . A µ + µ − ∆Γ ≤ Figure 3: Scenario C: P ± S = 1. Left panel: the correlation between the R and A µµ ∆Γ observables. Right panel: correlation between the | S | = | P − | and ϕ S = ˜ ϕ P + (1 + κ ) π/ R and A µµ ∆Γ in the limit φ NP s = 0. Observe the lower bound on R specified in (46). If, furthermore, ˜ ϕ P = ϕ S = { , π } we observe that A µµ ∆Γ can help to resolve the two possible solutions for S comingfrom a branching ratio measurement R .In the right panel of Figure 3 we show the correlation between ϕ S and | S | . Observethat the current measurement of R still allows a large range for both NP parameters.If A µµ ∆Γ were measured with a negative sign it would indicate large contributions fromNP. Moreover in this case the A µµ ∆Γ sharply cuts the R contour, so that a measurementof A µµ ∆Γ would distinguish between the magnitude and the phase of S up to the twofoldambiguity in ϕ S . A 2HDM in the decoupling regime, such that M H (cid:39) M A (cid:39) M H ± (cid:29) M h [42], has thegeneric feature that C S = − C P , C (cid:48) S = C (cid:48) P . (48)If the couplings of the heavy Higgs bosons are not left-right symmetric, so that either C S,P or C (cid:48) S,P are dominant , this corresponds to Scenario C . Thus the branching ratio hasa lower bound and a significant scalar NP contribution is indicated by negative values of A µµ ∆Γ . A precise measurement of the untagged observable A µµ ∆Γ can distinguish the phaseand magnitude of the NP Wilson coefficients. We will analyse a similar scenario in moredetail in Section 4.The above is true also for the MSSM, provided that NP contributions to vector-axialoperators, C (cid:48) , are negligible. The MSSM has the added advantage that large tan β In MFV this is the case. Namely C (cid:48) S,P /C S,P ∼ m s /m b . H orpseudoscalar A may be considerably lighter than the other. If this solo particle cangenerate the required FCNC, then we are in Scenario B or Scenario A respectively. ϕ P , ϕ S ∈ { , π } In this scenario we assume no CP violating phases in the B s → µ + µ − decay mode: ϕ P , ϕ S ∈ { , π } [5]. This is equivalent to all of the Wilson coefficients taking real values.Clearly this constraint can also be applied to the other scenarios discussed in this section,but this scenario is distinct in that S and P are allowed to remain arbitrary real values.Yet, in the presence of a non-vanishing NP phase φ NP s , the CP-asymmetry S µµ could benon-vanishing.The resulting time dependent observables in this scenario are A µµ ∆Γ = cos φ NP s (cid:20) | P | − | S | | P | + | S | (cid:21) , S µµ = − sin φ NP s (cid:20) | P | − | S | | P | + | S | (cid:21) , (49)and the branching ratio observable is given by R = | P | (cid:20) y s cos φ NP s y s (cid:21) + | S | (cid:20) − y s cos φ NP s y s (cid:21) . (50)Importantly, whereas the branching ratio observable R gives their squared sum, the A µµ ∆Γ is sensitive to the difference. With known φ NP s these three observables are sufficientto determine the two NP unknowns | P | and | S | . As sin φ NP s is already known to besmall, S µµ is also small in this scenario. Consequently A µµ ∆Γ and R will be the relevantobservables in this determination. With cos φ NP s very close to unity one finds then | P | = (1 + y s ) R (cid:20) A µµ ∆Γ y s A µµ ∆Γ (cid:21) , | S | = (1 + y s ) R (cid:20) − A µµ ∆Γ y s A µµ ∆Γ (cid:21) . (51)Finally a measurement of S µµ incompatible with the known value of φ NP s would excludethis scenario and indicate new CP violating phases in the decay.In Figure 4 we illustrate how measurements of R and A µµ ∆Γ can be used to pinpointthe parameters | S | and | P | (we have taken φ NP s = 0). Models with Minimal Flavour Violation (MFV), but without flavour blind phases, asformulated as an effective field theory in Ref. [46], belong naturally to this class. MFVprotects against any additional flavour structure or CP violation beyond what is already15 . . . . . . . . . | P | . . . . . . . . . . | S | A ∆ Γ = − . A ∆ Γ = − . A ∆ Γ = + . A ∆ Γ = + . A ∆ Γ = + . SM ¯ R = 0 . +0 . − . Illustration for A ∆Γ ( ϕ P,S = 0 , π ) Figure 4: Scenario D: ϕ P , ϕ S ∈ { , π } . The correlation between the | P | and | S | param-eters for varying values of A µµ ∆Γ . Also shown is the current measurement of R .present in the CKM matrix, while still allowing for additional, higher-dimensional, op-erators [46]. MFV therefore falls into Scenario D , with the added restriction that also φ NP s is zero. Thus in models with MFV, as seen in (51), the time-dependent untaggedobservable A µµ ∆Γ together with the branching ratio observable R are sufficient to disen-tangle the scalar contribution S from P . A measurement of S µµ (cid:54) = 0 would falsify MFV.Typical examples in this class are MSSM with MFV and 2HDM with MFV.An exception are models with MFV and flavour-blind phases, like the 2HDM withsuch phases, also known as 2HDM MFV [47]. In this case model specific details are nec-essary in order for the time-dependent observables to distinguish between the operatorsand phases. P = 0 In this scenario C ( (cid:48) ) P , C (cid:48) or δC destructively interfere with C SM to drive P to zero.Then non-zero values of the B s → µ + µ − observables will be driven purely by the opera-tors O ( (cid:48) ) S .This scenario is similar to Scenario A , in that there is no rate asymmetry betweenthe individual helicity decay modes. Thus the time-dependent observables are not inde-pendent: A µµ ∆Γ = − cos(2 ϕ S − φ NP s ) , S µµ = − sin(2 ϕ S − φ NP s ) . (52)The key difference, however, is that now only a scalar and not a gauge boson or pseu-doscalar is at work. Moreover, in the absence of new CP-violating phases A µµ ∆Γ = − Sce-nario A as seen in (42). This is also a limiting case of
Scenario D . The branching ratioobservable is given by R = | S | (cid:20) − y s cos(2 ϕ S − φ NP s )1 + y s (cid:21) . (53)16odel Scenario | P | ϕ P | S | ϕ S φ NP s CMFV A | P | | P | | S | Z (cid:48) A | P | ϕ P φ NP s | ∓ S | arg(1 ∓ S ) | S | ϕ S φ NP s | P | ϕ P φ NP s | S | ϕ S φ NP s Table 1: General structure of basic variables in different NP models. The last three casesapply also to the MSSM.We do not know any specific model that would naturally be placed in this scenario butwe will investigate in Section 4 whether requiring tree-level exchanges of a pseudoscalarto cancel SM contribution is still consistent with the data.
In Table 1 we collect the properties of the selected models discussed above with respectto the basic phenomenological parameters listed in (36) and the class they belong to. Wealso indicate whether the phase φ NP s can be non-zero in these models. In all cases | P | isgenerally different from zero as it contains the SM contributions. In order to distinguishbetween different models in each row of this table a more detailed analysis has to beperformed taking all existing constraints into account. However, already identifyingwhich of these four rows has been chosen by nature would be a tremendous step forward. B s Mixing
As the first class of specific models we consider Z (cid:48) models in which NP contributionsto FCNC observables are dominated by tree-level Z (cid:48) exchanges. A detailed analysis ofthese models has recently been presented in Ref. [14]. Also there the three observablesin (8) have been considered but the emphasis has been put on the correlations of themwith ∆ F = 2 observables, in particular S ψφ . Here we will complement this study bycomputing the correlations among R , A µµ ∆Γ , and S µµ , while taking the constraints from∆ F = 2 observables obtained in Ref. [14] into account.We define the flavour-violating couplings of Z (cid:48) to quarks as follows L FCNC ( Z (cid:48) ) = (cid:2) ∆ sbL ( Z (cid:48) )(¯ sγ µ P L b ) + ∆ sbR ( Z (cid:48) )(¯ sγ µ P R b ) (cid:3) Z (cid:48) µ , (54)where ∆ sbL,R ( Z (cid:48) ) are generally complex. 17e also define the Z (cid:48) couplings to muons L (cid:96) ¯ (cid:96) ( Z (cid:48) ) = (cid:2) ∆ (cid:96)(cid:96)L ( Z (cid:48) )(¯ (cid:96)γ µ P L (cid:96) ) + ∆ (cid:96)(cid:96)R ( Z (cid:48) )(¯ (cid:96)γ µ P R (cid:96) ) (cid:3) Z (cid:48) µ (55)and introduce ∆ µ ¯ µA ( Z (cid:48) ) = ∆ µ ¯ µR ( Z (cid:48) ) − ∆ µ ¯ µL ( Z (cid:48) ) . (56)Then the non-vanishing Wilson coefficients contributing to B s → µ + µ − are given asfollows: sin θ W C = − η Y Y ( x t ) − g M Z (cid:48) ∆ sbL ( Z (cid:48) )∆ µ ¯ µA ( Z (cid:48) ) V ∗ ts V tb , (57)sin θ W C (cid:48) = − g M Z (cid:48) ∆ sbR ( Z (cid:48) )∆ µ ¯ µA ( Z (cid:48) ) V ∗ ts V tb , (58)where g = 4 G F √ α π sin θ W . (59)As only the coefficients C and C (cid:48) are non-vanishing this NP scenario is governedby the formulae (38) and (39). Indeed this scenario is an example of Scenario A in which,in addition to S = 0, also the pseudoscalar contributions vanish. Yet, as P can differfrom unity and have a nontrivial phase, a rich phenomenology is found [14]. It is not our goal to present a full-fledged numerical analysis of all correlations includingpresent theoretical, parametric and experimental uncertainties as this would only washout the effects we want to emphasize. Therefore we simply choose the three parametersentering our formulae, F B s , τ ( B s ) and | V ts | to be in the ballpark of their present centralvalues: F B s = 225 . , τ ( B s ) = 1 .
503 ps , | V ts | = 0 . . (60)Other relevant input can be found in the Tables of Ref. [30].The main theoretical uncertainties in our analysis are due to the constraints on thecouplings ∆ sbL,R ( Z ) and ∆ sbL,R ( H ) coming from the experimental values of ∆ M s and S ψφ .Indeed the hadronic matrix elements of new operators are still subject to significantuncertainties. We will not recall the relevant formulae as they can be found in Ref. [14].We will use the full machinery presented in that paper, setting the relevant parametersat their central values and requiring ∆ M s and S ψφ to be in the ranges16 . / ps ≤ ∆ M s ≤ . / ps , − . ≤ S ψφ ≤ . . (61)Concerning the first range, it effectively takes the hadronic uncertainties into account.The second range corresponds to the 2 σ range for φ s in (7).As far as the direct lower bound on M Z (cid:48) from collider experiments is concerned, themost stringent bounds are provided by CMS experiment [48] but these constraints aremainly sensitive to the couplings of the Z (cid:48) to the light quarks which do not play any role18n our analysis. Moreover, the collider bounds on M Z (cid:48) are generally model dependent.While for the so-called sequential Z (cid:48) the lower bound for M Z (cid:48) is in the ballpark of 2 . M Z (cid:48) = 1 TeV. With the help of the formulaein Ref. [14] it is possible to estimate approximately, how our results would change for1 TeV ≤ M Z (cid:48) ≤ bsL (cid:54) = 0 and ∆ bsR = 0,2. Right-handed Scheme (RHS) with complex ∆ bsR (cid:54) = 0 and ∆ bsL = 0,3. Left-Right symmetric Scheme (LRS) with complex ∆ bsL = ∆ bsR (cid:54) = 0,4. Left-Right asymmetric Scheme (ALRS) with complex ∆ bsL = − ∆ bsR (cid:54) = 0.Note that the ordering in flavour indices in the couplings in these schemes is governedby the operator structure in B s – ¯ B s mixing [14, 30] and differs from the one in (54) and(66). In this context one should recall that∆ sbL,R ( Z (cid:48) ) = [∆ bsL,R ( Z (cid:48) )] ∗ , ∆ sbL,R ( H ) = [∆ bsR,L ( H )] ∗ , (62)where H stands for either scalar or pseudoscalar.The ranges for B s mixing given in (61) result in two allowed regions for the magnitudesand phases of the quark couplings ∆ sbL,R depending on the scheme chosen above. Theseregions in parameter space are dubbed oases . The oases for each case have a two folddegeneracy in the complex phase of the coupling. Where it is relevant we will distinguishbetween these two different oases using the colours blue and red.In order to perform the present analysis we assign ∆ µ ¯ µA ( Z (cid:48) ) = 0 .
5, as was donein Ref. [14]. In Section 4.2.3 and beyond, where we compare Z (cid:48) exchange with various(pseudo)scalar exchanges, this coupling will be allowed to vary. The sign of this couplingis crucial for the identification of various enhancements and suppressions with respectto SM branching ratio and CP asymmetries and impacts the search for successful oasesin the space of parameters that has been performed in Ref. [14]. If the sign of the Z (cid:48) coupling to muons will be identified in the future to be different from the one assumedhere, it will be straightforward, in combination with the discussion in Ref. [14], to findout how our results will be modified. In Figure 5 we show the correlation between S µµ and R for LHS (left) and RHS (right). Corresponding correlations between A µµ ∆Γ and R and between A µµ ∆Γ and S µµ are given in Figure 6 for LHS only. The two colours correspondto two oases in the values of the coupling ∆ L,R ( Z (cid:48) ) that are consistent with ∆ M s and S ψφ constraints.We observe in analogy with findings of Ref. [14] that the correlations in the LHSand RHS schemes have the same shape except the oases and consequently the colours inFigure 5 have to be interchanged. We conclude therefore that on the basis of the threeobservables considered by us it is not possible to distinguish between LHS and RHSschemes because in the RHS scheme one can simply interchange the oases to obtain the19 . . . . R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . . . . S µµ Z in LHS scenario with ˜∆ µµA ( Z ) =0.5 ¯ R = 0 . +0 . − . . . . . R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . . . . S µµ Z in RHS scenario with ˜∆ µµA ( Z ) =0.5 ¯ R = 0 . +0 . − . Figure 5: S µµ versus R for LHS (left) and RHS (right), assuming M Z (cid:48) = 1 TeV and∆ µµA ( Z (cid:48) ) = 0 .
5. Gray region: exp 1 σ range for R . The 2 σ CL combined fit region forthe Wilson coefficients C ( (cid:48) )10 come from a general b → sl + l − analysis given in Ref. [49]. . . . . R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . . . . A µµ ∆ Γ Z in LHS scenario with ˜∆ µµA ( Z ) =0.5 ¯ R = 0 . +0 . − . − . − . . . . S µµ − . − . . . . A µµ ∆ Γ Z in LHS scenario with ˜∆ µµA ( Z ) =0.5 Figure 6: A µµ ∆Γ versus R (left) and A µµ ∆Γ versus S µµ (right) for LHS, assuming M Z (cid:48) = 1 TeVand ∆ µµA ( Z (cid:48) ) = 0 .
5. Gray region in left panel: exp 1 σ range for R . The 2 σ CL combinedfit region for the Wilson coefficient C comes from a general b → sl + l − analysis givenin Ref. [49].same physical results as in LHS scheme. Consequently if one day we will have precisemeasurements of A µµ ∆Γ , S µ + µ − and BR( B s → µ + µ − ) we will still not be able to distinguishfor instance whether we deal with LHS scheme in the blue oasis or RHS scheme in thered oasis.As pointed out in Ref. [14], in order to make this distinction one has to considersimultaneously B → K ∗ µ + µ − , B → Kµ + µ − and b → sν ¯ ν transitions, which is beyondthe scope of our paper. However, we do include regions corresponding to the 2 σ CLcombined fits of Ref. [49] for the Wilson coefficients C and C (cid:48) , which result from thesetransitions, in Figures 5 and 6 where relevant. The combination of our oases and theseadditional constraints gives us valuable information. The allowed values for the threeobservables considered are, in this NP scenario,0 . ≤ A µµ ∆Γ ≤ . , . ≤ |S µ + µ − | ≤ . , . ≤ R ≤ (cid:26) . . . (63)20oreover, the smallest values of A µµ ∆Γ and largest values of |S µ + µ − | are obtained for small-est values of R . The non-zero values of S µ + µ − originate in Z (cid:48) models from requiring that∆ M s is suppressed with respect to its SM value in order to achieve a better agreementwith data. As we will see below for models with scalar or pseudoscalar exchanges, thisrequirement can also be satisfied for a vanishing S µ + µ − .If both LH and RH currents are present in NP contributions, but we impose a symme-try between LH and RH quark couplings, then NP contributions to B s → µ + µ − vanish.We thus find that A µµ ∆Γ = cos( φ NP s ) , S µµ = − sin( φ NP s ) . (64)The branching ratio observable is given by R = (cid:20) y s cos( φ NP s )1 + y s (cid:21) , (65)which can also be obtained from (38) and (39) by setting P = 1. In view of the smallnessof φ NP s the results for the three observables are very close to the SM values. Still thisexample shows that even if no departures from SM expectation will be found in B s → µ + µ − this does not necessarily mean that there is no NP around as with left-rightsymmetric couplings this physics cannot be seen in this decay except for small effectsfrom the B s mixing phase. This physics could then be seen in B → K ∗ µ + µ − and B → Kµ + µ − and b → sν ¯ ν transitions as demonstrated in Ref. [14].In the ALRS scheme NP contributions to B s → µ + µ − enter again with full power.Therefore the three observables in (8) offer as in the LHS and RHS schemes good test ofNP. In fact, as found in Ref. [14], after the ∆ B = 2 constraints are taken into account thepattern of NP contributions is similar to LHS scheme except that the effects are smallerbecause the relevant couplings have to be smaller in the presence of LR operators in∆ B = 2 in order to agree with the data on ∆ M s . Therefore we will not show the plotscorresponding to Figures 5 and 6. We will next consider tree-level pseudoscalar or scalar exchanges that one encountersin various models either at the fundamental level or in an effective theory. We willdenote by H any spin 0 particle, and will refer specifically to a scalar or pseudoscalaras H or A , respectively. It could in principle be the SM Higgs boson, but as therecent analysis in Ref. [30] shows, once the constraints from ∆ F = 2 processes are takeninto account, NP effects in B s → µ + µ − through a tree-level SM Higgs exchange are atmost 8% of the usual SM contribution and hardly measurable. The SM Higgs couplingto muons is simply too small. Therefore, what we have in mind here is a new heavyscalar or pseudoscalar boson encountered in 2HDM or supersymmetric models. Yet,in this subsection we will make the working assumption that either a neutral scalaror pseudoscalar tree-level exchange dominates NP contributions. A general analysis ofFCNC processes within such scenarios has been recently presented in Ref. [30]. Also21he observables in (8) have been analysed there, but with the emphasis put on theircorrelations with ∆ F = 2 observables, in particular S ψφ . Here we will complementthis study by computing the correlations among R , A µµ ∆Γ , and S µµ , while taking theconstraints from ∆ F = 2 observables computed in Ref. [30] into account.We define the flavour violating couplings of H as follows L FCNC ( H ) = (cid:2) ∆ sbL ( H )(¯ sP L b ) + ∆ sbR ( H )(¯ sP R b ) (cid:3) H (66)where ∆ sbL,R ( H ) are generally complex. Muon couplings ∆ µ ¯ µL,R ( H ) are defined in a similarway. Note that through (62) in LHS and RHS schemes only ∆ sbR ( H ) and ∆ sbL ( H ) arenon-vanishing, respectively.Then the relevant non-vanishing Wilson coefficients are given as follows m b ( M H ) sin θ W C S = 1 g M H ∆ sbR ( H )∆ µ ¯ µS ( H ) V ∗ ts V tb , (67) m b ( M H ) sin θ W C (cid:48) S = 1 g M H ∆ sbL ( H )∆ µ ¯ µS ( H ) V ∗ ts V tb , (68) m b ( M H ) sin θ W C P = 1 g M H ∆ sbR ( H )∆ µ ¯ µP ( H ) V ∗ ts V tb , (69) m b ( M H ) sin θ W C (cid:48) P = 1 g M H ∆ sbL ( H )∆ µ ¯ µP ( H ) V ∗ ts V tb , (70)where we have introduced ∆ µ ¯ µS ( H ) = ∆ µ ¯ µR ( H ) + ∆ µ ¯ µL ( H ) , ∆ µ ¯ µP ( H ) = ∆ µ ¯ µR ( H ) − ∆ µ ¯ µL ( H ) . (71)Note that m b has to be evaluated at µ = M H .From the hermiticity of the relevant Hamiltonian one can show that ∆ µ ¯ µS ( H ) is realand ∆ µ ¯ µP ( H ) purely imaginary. For convenience we define∆ µ ¯ µP ( H ) ≡ i ˜∆ µ ¯ µP ( H ) , (72)so that ˜∆ µ ¯ µP ( H ) is real.Already at this stage it is instructive to see how different scenarios introduced inSection 3 are realized in this case. Scenario A:
In this scenario S = 0. This can be realized simplest by setting ∆ µ ¯ µS ( H ) = 0, which,through (71), implies that ∆ µ ¯ µP ( H ) must be non-vanishing if the quark couplings differfrom zero. The second possibility are left-right symmetric coupling to quarks but thiswould automatically imply also the vanishing of pseudoscalar couplings giving P = 1.22 cenario B: We have just seen how this scenario can be obtained as a limiting case of scenario A . Inorder to have non-vanishing S in this case this is realized by setting ∆ µ ¯ µP ( H ) = 0, whichthrough (71) implies that ∆ µ ¯ µS ( H ) must be non-vanishing if the quark couplings differfrom zero. Scenario C:
To achieve this scenario the scalar coefficients should be equal up to a sign to thepseudoscalar ones. This requires the exchanged spin-0 particle to be a mixed scalar–pseudoscalar state, which is beyond the scope of the present analysis. We will insteadrealise
Scenario C in Section 4.3 by considering the presence of both a scalar and apseudoscalar with equal masses and equal couplings to quarks.
Scenario D:
Because a single scalar or pseudoscalar allows only S or P to deviate from its SM value,respectively, the intended usage case of this scenario, namely arbitrary but real valued S and P , cannot be realised. Scenario E:
In this concrete model in which there are no NP contributions to C ( (cid:48) )10 the vanishing of P implies: m B s m µ (cid:18) m b m b + m s (cid:19) (cid:18) C (cid:48) P − C P C SM10 (cid:19) = 1 . (73)We will investigate whether this condition is consistent with existing constraints whenthe relevant Wilson coefficients are given as in (69) and (70). Analogous to the case of tree-level Z (cid:48) exchanges we will use the results of the ∆ F = 2analysis in Ref. [30] to constrain the quark-scalar couplings in the schemes LHS, RHS,LRS and ALRS by imposing the conditions in (61). The next step is to set valuesfor the scalar and pseudoscalar muon couplings. For a single scalar particle H , theparameter | S | driving NP ( Scenario B ) is directly proportional to the muon coupling | ∆ µµS ( H ) | . However, for a single pseudoscalar particle A , the muon coupling ∆ µµP ( A )is not directly proportional to P , and the resulting NP observables thereby have a moreinvolved dependence on it. In Figure 7 we show the dependence of the observables R (left panel) and A µµ ∆Γ (right panel) with respect to muon coupling ˜∆ µµP ( A ) defined in (72)satisfying the B s mixing constraints for the LHS case. We observe that the parameterspace of the NP physics observables is very dependent on whether we pick a large orsmall coupling, and that a fixed coupling cannot do it justice. We further observe thatthe oases become indistinguishable if the sign of the coupling is not fixed.23 . − . − . − .
01 0 .
00 0 .
01 0 .
02 0 .
03 0 . µµP . . . . ¯ R ≡ B R ( B s → µ + µ − ) / B R S M ( B s → µ + µ − ) LHS with A − . − . − . − .
01 0 .
00 0 .
01 0 .
02 0 .
03 0 . µµP − . − . . . . A µµ ∆ Γ LHS with A Figure 7: The dependence of the observables R (left) and A µµ ∆Γ (right) on the pseudoscalarlepton coupling ˜∆ µ ¯ µP ( H ) satisfying the B s mixing constraints in the LHS case. For apseudoscalar with mass M A = 1 TeV.In order to compare the oases behaviour of the scalar and pseudoscalar we begin byfixing the muon couplings to∆ µ ¯ µS ( H ) = 0 . , ∆ µ ¯ µP ( A ) = i . , (74)and ∆ µ ¯ µP ( H ) = ∆ µ ¯ µS ( A ) = 0.As demonstrated in Ref. [30] these values are consistent with the allowed range for B ( B s → µ + µ − ) when the constraints on the quark couplings from B s − ¯ B s are takeninto account and M = 1 TeV. All other input parameters are as in Ref. [30]. The reasonfor choosing the scalar couplings to be larger than the pseudoscalar ones is that they aremore weakly constrained than the latter because the scalar contributions do not interferewith SM contributions. The constraints from b → s(cid:96) + (cid:96) − transitions do not have anyimpact in the (pseudo) scalar case as shown in Ref. [30].In Figure 8 we show the correlations of S µµ versus R satisfying B s mixing constraintsfor a single tree-level scalar (left) and pseudoscalar (right) exchange in the LHS scheme.For the scalar case the blue and red oases overlap. The red oases in the pseudoscalarcase corresponds to R <
R > Z (cid:48) exchange.As we stated earlier, fixing the pseudoscalar muon couplings to one value does notreveal the full structure of the NP parameter space. We therefore now consider the muoncouplings varied over the following range: | ∆ µ ¯ µS ( H ) | , | ∆ µ ¯ µP ( A ) | ∈ [0 . , . . (75)From here on we will ignore the sign of the lepton couplings, but again note that thisdegeneracy can be resolved in the pseudoscalar case if the blue or red oasis from the B s mixing constraints can be singled out. We thus also stop distinguishing between the twooases. 24 . . . . R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . . . . S µµ H in LHS scenario with ∆ µµS =0.04 ¯ R = 0 . +0 . − . . . . . R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . . . . S µµ A in LHS scenario with ˜∆ µµP =0.02 ¯ R = 0 . +0 . − . Figure 8: S µµ versus R for for LHS scheme with a scalar (left) and pseudoscalar (right)for M H = M A = 1 TeV. Gray region: exp 1 σ range for R .In Figure 9, R is plotted against A µµ ∆Γ (left panel) and S µµ (right panel) with theregions allowed by the B s mixing constraints overlayed for the various specific tree-level models discussed in this section. The scalar and pseudoscalar muon couplingshave been varied as just discussed, and also the Z (cid:48) muon couplings have been varied:∆ µµA ( Z (cid:48) ) ∈ [0 . , . Z (cid:48) modelthe allowed region has also been constrained by a 2 σ CL combined fit of the Wilsoncoefficient C from b → sl + l − transitions [49]. Focusing for the moment on modelswith single scalar ( H ) or pseudoscalar ( A ) exchanges, the following observations canbe made: • The branching ratio observable R can in the scalar case only be enhanced as thereis no interference with the SM contribution. On the other hand, in the pseudoscalarcase it can be suppressed or enhanced depending on which oasis of parameters ischosen. • The values of A µµ ∆Γ are positive for both H and A and for R within one σ exper-imental value close to unity. • S µµ can reach ± .
50 in both cases.We do not show the corresponding results in the RHS scheme as, similarly to thegauge boson case, the correlations in question have identical structure with the followingdifference between scalar and pseudoscalar cases originating in the absence and presenceof correlation with SM contributions, respectively: • In the scalar case the two correlations in question are invariant under the changeof LHS to RHS. • In the pseudoscalar case the structure of two correlations remain but going fromLHS to RHS the colours have to be interchanged as was the case for gauge bosons.25 . . . . R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . . . . A µµ ∆ Γ SM H (LHS) A (LHS) Z (LHS) H + A (MFV)¯ R = 0 . +0 . − . . . . . R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . . . . S µµ SM H (LHS) A (LHS) Z (LHS) H + A (MFV)¯ R = 0 . +0 . − . Figure 9: Overlay of the correlations for R versus A µµ ∆Γ (left) and S µµ (right) forthe various specific models considered. The lepton couplings are varied in the ranges | ∆ µµS,P ( H ) | ∈ [0 . , . µµA ( Z (cid:48) ) ∈ [0 . , . . ≤ A µµ ∆Γ ≤ . , |S µµ | ≤ . . (76)Finally we investigated whether the relation (73), representing Scenario E is stillconsistent with all available constraints. This is not the case if we take the pseudoscalarlepton coupling chosen in (74) and a mass for the pseudoscalar of 1 TeV. For the LHS andRHS schemes a lepton coupling of ∆ µ ¯ µP ( H ) ≈ ± i .
06 is needed to satisfy the relation. Ifa pseudoscalar does manage to make P vanish, then a scalar particle is needed to satisfythe lower bound on R . Such a model, with both a pseudoscalar and scalar particlepresent, is discussed in Section 4.3. Z (cid:48) Scenario
While the discussion presented above shows that the contributions of scalars and pseu-doscalars can be distinguished through the observables considered, more spectaculardifferences occur when one includes the Z (cid:48) scenario in this discussion. Indeed the cor-relation between S µµ and R in the left panel of Figure 5 has a very different structurefrom the case of pseudoscalar or scalar exchanges shown in Figure 8.In the right panel of Figure 9 an overlay of these regions is shown for LHS schemes,with the lepton couplings varied as given in (75). Similarly, in the left panel of Figure 9we show the correlation between A µµ ∆Γ and R , where strong contrasts between the allowedregions also emerge. The difference between the Z (cid:48) and pseudoscalar exchange is strikingbecause, unlike for a scalar, both particles generate Scenario A .The difference between the A -scenario and Z (cid:48) -scenario in question can be tracedback to the difference between the phase of the NP correction to ˜ P , which was definedin (40). As the phase δ in the quark coupling ∆ bsL from the analysis of B s -mixing in26ef. [30] is the same in both scenarios the difference enters through the muon couplings,which are imaginary in the case of the A -scenario but real in the case of Z (cid:48) . This iswhy the structure of correlations in both scenarios is so different. Taking also the signdifference between Z (cid:48) and pseudoscalar contributions to the b → sµ + µ − amplitude intoaccount, we find that P ( Z (cid:48) ) = 1 + r Z (cid:48) e iδ Z (cid:48) , P ( A ) = 1 + r A e iδ A (77)with r Z (cid:48) ≈ r A , δ Z (cid:48) = δ − β s , δ A = δ Z (cid:48) − π . (78)It turns out that the B s mixing constraints force the phase δ to be in the ballpark of90 ◦ and 270 ◦ for the blue and red oasis, respectively [14, 30]. This implies in the caseof the Z (cid:48) scenario, as seen in Figure 5, positive and negative value of S µµ for the blueand red oasis, respectively. Simultaneously R , where NP is governed by cos δ Z (cid:48) , can beenhanced or suppressed in each oasis. On the other hand, (78) implies that the phase δ A is in the ballpark of 0 ◦ and 180 for the blue and red oasis, respectively. Thereforethe asymmetry S µµ can vanish in both oases, while this was not possible in the Z (cid:48) case.As NP in R is governed by cos δ A , this enhances and suppresses R for blue and redoasis, respectively as clearly seen in Figure 8. In particular, R differs from its SM value,while this is not the case in the Z (cid:48) scenario. Finally, let us note that with larger valuesof muon couplings NP effects in R , A µµ ∆Γ and S µµ can be larger than shown in Figs. 5and 6 (see, for example, Figure 10).What is particularly interesting is that these differences are directly related to thedifference in the fundamental properties of the particles involved: their spin and CP-parity. As far as the last property is concerned, also differences between the implicationsof the pseudoscalar and scalar exchanges have been identified as discussed in detail above.They are related to the fact that the scalar contribution, being CP even, cannot interferewith the SM contribution. In this model we assume the presence of a scalar H and pseudoscalar A with equal (ornearly degenerate) mass M H . This is, for example, effectively realised in 2HDMs in adecoupling regime, where H and A are much heavier than the SM Higgs h and almostdegenerate in mass [42]. We will show that under specific assumptions this setup canreproduce Scenarios C , D or E .The couplings of the scalar and pseudoscalar to quarks are given in general by thefollowing flavour-violating Lagrangian: L FCNC ( H , A ) = (cid:2) ∆ sbL ( H )(¯ sP L b ) + ∆ sbR ( H )(¯ sP R b ) (cid:3) H + (cid:2) ∆ sbL ( A )(¯ sP L b ) + ∆ sbR ( A )(¯ sP R b ) (cid:3) A . (79)We will assume that the scalar and pseudoscalar couple with equal strength to quarks: L (cid:51) ¯ D L ˜∆ D R ( H + iA ) + h . c , (80)27here D = ( d, s, b ) and ˜∆ is a matrix in flavour space. Then∆ sbR ( H ) = ˜∆ sb , ∆ sbL ( H ) = (cid:104) ˜∆ bs (cid:105) ∗ , ∆ sbR ( A ) = i ˜∆ sb , ∆ sbL ( A ) = − i (cid:104) ˜∆ bs (cid:105) ∗ . (81)where in general ˜∆ sb , ˜∆ bs ∈ C . Scenario C:
To reproduce this scenario we set the pseudoscalar and scalar masses to be exactly equal: M H = M A = M H . Further relating the lepton couplings by a single real parameter˜∆ µ ¯ µ : ∆ µ ¯ µ ( H ) = ˜∆ µ ¯ µ , ∆ µ ¯ µ ( A ) = i ˜∆ µ ¯ µ (82)and inserting the lepton and quark couplings into formulae (67)–(70) we find: C S = − C P = 1 g SM M H m b sin θ W ˜∆ sb ˜∆ µµ V ∗ ts V tb (83) C (cid:48) S = C (cid:48) P = 1 g SM M H m b sin θ W (cid:104) ˜∆ bs (cid:105) ∗ ˜∆ µµ V ∗ ts V tb . (84)This simple model satisfies the relations in (48) and thereby belongs to Scenario C .These relations are in fact valid for all the quark coupling schemes: LHS, RHS, LRS andALRS. Yet the physics implications depend on the scheme considered: • In LHS and RHS schemes NP contributions to B s – ¯ B s mixing from scalar andpseudoscalar with the same mass cancel each other so that there is no constraintfrom B s – ¯ B s mixing. Thus NP effects in B s → µ + µ − can only be constrained bythe decay itself or other b → s(cid:96) + (cid:96) − transitions. • In LRS and ALRS schemes non-vanishing contributions from LR operators to B s –¯ B s mixing are present. Moreover we find C S = C (cid:48) S , C P = − C (cid:48) P (LRS) , (85) C S = − C (cid:48) S , C P = C (cid:48) P (ALRS) . (86)Therefore in the LRS case only pseudoscalar contributes to B s → µ + µ − ( ScenarioA ), while in the ALRS case only scalar contributes (
Scenario B ).We conclude therefore that in order to have an example of
Scenario C that differsfrom
Scenario A and B and moreover in which NP contributions to B s – ¯ B s mixing arepresent, we need both L and R couplings which are not equal to each other or do notdiffer only by a sign.An option to reproduce Scenario C with non-trivial constraints from mixing is givenby Minimal Flavour Violation (MFV). In the MFV formalism ˜∆ is constructed out of the28purion matrices Y U and Y D [46]. In principle the following constructions can contributeto the b → s FCNCs at leading order in the off-diagonal structure: Y U Y † U Y D , Y D Y † D Y U Y † U Y D , Y U Y † U Y D Y † D Y D . (87)However, the last two will in general receive dynamical (loop) suppressions. Thus, forsimplicity, we assume the first construction to be dominant. In the notation of Ref. [47],where MFV is discussed in the context of a general 2HDM with flavour blind phases(2HDM MFV ), this is equivalent to assuming | a | (cid:29) | a | , | a | . As a result we find˜∆ sb = (cid:15) y b y t V ∗ ts V tb , (cid:104) ˜∆ bs (cid:105) ∗ = (cid:15) ∗ y s y t V ∗ ts V tb = m s m b (cid:15) ∗ (cid:15) ˜∆ sb . (88)Thus under the above assumptions all of the quark couplings in (79) can be expressedin terms of a single NP parameter (cid:15) .The parameter (cid:15) is real in pure MFV but may be complex in 2HDM MFV [47]. Insertingrelation (88) into (84) we find C (cid:48) S = m s m b (cid:15) ∗ (cid:15) C S C (cid:48) P = − m s m b (cid:15) ∗ (cid:15) C P , (89)and observe a m s /m b suppression of the primed operators. In pure MFV, where (cid:15) is real,the parameters C ( (cid:48) ) S,P are also all real.
Scenario D: In Scenario D the parameters P and S are arbitrary but do not carry new CP violatingphases. The pure MFV model with a scalar and pseudoscalar that we just discussedis therefore a natural candidate. However, because this model was defined to satisfy Scenario C , as it stands we have P ± S = 1. If we continue to insist that the scalar andpseudoscalar should couple with equal strengths and phases to quarks as in (80), thenthere are two choices for making P and S arbitrary.One choice is to allow different couplings to leptons for the scalar and pseudoscalar i.e. | ∆ µ ¯ µ ( H ) | (cid:54) = | ∆ µ ¯ µ ( A ) | . In this case the constraints from B s mixing (discussed below)do not change, and only the current bounds on R must be satisfied.Alternatively, a non-trivial difference between the scalar mass M H and the pseu-doscalar mass M A can be introduced. In this case the lepton couplings can remainequal as defined in (82). The catch, however, is that now the LL and (to a much lesserextent in MFV) RR contributions to B s mixing no longer vanish. Thus the allowed massdifference, and thereby the arbitrariness of P and S is constrained by mixing. Scenario E:
This scenario requires that P = 0 and therefore that S alone generates a value of R largeenough to meet the current experimental bounds. As we are dealing with two spin-0particles, the pseudoscalar in the present model must satisfy the relation given in (73). It should be emphasized that in general this is not the case for 2HDM
MFV [47, 50]. See additionalcomments below.
29y definition
Scenario E allows S to have a new CP violating phase ϕ S . However,the relation in (73) requires that arg( C P − C (cid:48) P ) = 0. Therefore, because we required thatthe scalar and pseudoscalar should couple to quarks with equal strengths and phases (see(80)), it follows that ϕ S = 0. The Scenario E realisable in this model is therefore justa specific case of
Scenario D , where P and S are real and arbitrary, with the additionthat P is tuned to vanish.In the next section we will address whether, given that S and P are made to vary dueto a scalar–pseudoscalar mass difference and P is tuned to zero, the range of S allowedby mixing can satisfy the experimental bounds on R . Our numerical analysis for this model will focus on the above mentioned assumptionsthat produce
Scenario C . Specifically, we begin by assuming an exactly degenerate scalarmass M H , equal scalar and pseudoscalar lepton couplings and MFV. At the end of thissection we also briefly address the consequences of a scalar–pseudoscalar mass difference,which could produce Scenarios D and E .By imposing MFV on the flavour matrix ˜∆ introduced in (80), it follows that theanalogues of ˜∆ sb in the B d and K systems are related to it by˜∆ db = − V ∗ td V ∗ ts ˜∆ sb , ˜∆ ds = − m s m b V ∗ td V ∗ tb (cid:104) ˜∆ sb (cid:105) ∗ . (90)Therefore the value taken by ˜∆ sb should in principle not only satisfy the experimental B s mixing constraints, but also those of the B d and K systems. In practice, however, NPcontributions in this model to B d mixing are suppressed by a factor of m d /m s relativeto B s mixing and thereby very small. As a result, this model with MFV cannot relievethe current tensions in B d mixing between theory and experiment [51,52]. Contributionsto neutral Kaon mixing are totally negligible. We therefore proceed to only considerconstraints from B s mixing.The only contribution that survives in B s mixing is the LR one and this introducesthe following shift in the SM box function [30] S ( B s ) = S ( x t ) + [∆ S ( B s )] LR , (91)where [∆ S ( B s )] LR = 2 r LR [ ˜∆ sb ] ∗ ˜∆ bs M H ( V ts V ∗ tb ) = 2 r LR (cid:18) m s m b (cid:19) | (cid:15) | y b y t M H . (92)with r LR = − × TeV [30].The following observations should be made: • In spite of possible new flavour blind phases in the MFV scenario, these phases donot show up in B s -mixing, so that the CP asymmetry S ψφ remains at its SM value,still consistent with experiment. In Ref. [47] the ∆ B = 2 operator for a 2HDMwith MFV is also found to leave flavour-blind phases unconstrained in the limit30 . . . . R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . . . . A µµ ∆ Γ SM H (LHS) A (LHS) Z (LHS) H + A (MFV)¯ R = 0 . +0 . − . . . . . R ≡ BR( B s → µ + µ − ) / BR SM ( B s → µ + µ − ) − . − . . . . S µµ SM H (LHS) A (LHS) Z (LHS)¯ R = 0 . +0 . − . Figure 10: Overlay of the correlations for R versus A µµ ∆Γ (left) and S µµ (right) forthe various specific models considered. The lepton couplings are varied in the ranges | ∆ µµS,P ( H ) | ∈ [0 . , . µµA ( Z (cid:48) ) ∈ [0 . , . | a | (cid:29) | a | , | a | . In general this is not the case for 2HDM MFV and, as analysed inRefs. [47,50], S ψφ can receive NP contributions. Note that these phases also appearin the flavour-conserving Yukawa couplings, which contribute to the electric dipolemoments of various atoms and hadrons by the exchange of Higgs fields. But asshown for 2HDM MFV in Ref. [50] the present upper bounds on EDMs do not yethave any impact on the observables considered here . • On the contrary, ∆ M s receives a small (suppressed by m s /m b ) negative contributionwhich is good as the SM value is roughly 10% above its experimental value [1]. Thissuppression due to LR operators within a MFV framework was first pointed outfor the MSSM with MFV in Ref. [45]. • The fact that the flavour-blind phases are unconstrained through B s -mixing allowsus to obtain significant effects from them in B s → µ + µ − observables as we will seesoon.For both MFV and MFV we find the range: | ˜∆ sb | ∈ [0 . , . , (93)for M H = 1 TeV, which, as seen in (88), is consistent with the tacit assumption that (cid:15) should be small.To proceed with numerics for the B s → µ + µ − observables we must set the couplingof H and A to muons. In the context of Scenario C this means setting ˜∆ µµ as definedin (82). In order to compare with the single tree-level scalar and pseudoscalar modelsdiscussed in the previous section we begin by varying the coupling between [0 . , . We thank Minoru Nagai for enlighting comments on these issues.
00 1000 1500 2000 2500 M H [GeV] − . − . . . . A µµ ∆ Γ MFV with H+A and ˜∆ µµ =-0.05
500 1000 1500 2000 2500 M H [GeV] − . − . . . . A µµ ∆ Γ MFVbar with H+A and ˜∆ µµ =-0.05 Figure 11: The allowed region of A µµ ∆Γ versus the heavy scalar mass M H in MFV (leftpanel) and MFV (right panel). The allowed region satisfies the B s mixing constraintsand falls with the 2 σ C.L region of R : R ∈ [0 . , . A µµ ∆Γ plotted versus R for MFV with M H = 1 TeV.The allowed region from B s mixing constraints shown in this plot should be comparedwith the theoretical situation sketched for Scenario C in the left panel of Figure 3.By inspection of the theoretical plot one observes that the pure MFV model (with noflavour blind phases) corresponds to the outer border of the MFV region shown. It isinteresting to observe that in both models a negative A µµ ∆Γ is possible within the B s constraints mixing, in contrast to the tree-level models considered above with a single(pseudo)scalar or gauge boson. Because the flavour-blind phase in MFV is completelyunconstrained, almost the entire experimentally allowed region is left unconstrained by B s mixing in this model.In the right panel of Figure 9 we similarly show S µµ versus R in the MFV modelfor M H = 1 TeV. In the pure MFV model S µµ = 0 and therefore these plots are notinteresting.In a 2HDM with large tan β , which can generate a decoupled heavy scalar and pseu-doscalar as discussed here, the muon coupling is given by˜∆ µµ = − (cid:32) √ m µ v tan β (cid:33) = − . (cid:20) tan β (cid:21) , (94)which demonstrates that the (pseudo)scalar muon couplings can be larger than what wehave assumed so far. In Figure 10 we repeat the plots we have shown in Figure 9, butnow with the muon couplings varied over much larger ranges: | ∆ µ ¯ µS,P ( H , A ) | ∈ [0 . , . , | ∆ µ ¯ µA ( Z (cid:48) ) | ∈ [0 . , . . (95)This range of couplings gives a better impression of the full allowed parameter space, atthe cost of hiding some of the characteristic differences between the considered models.We do not again show the large allowed region of H + A model with MFV, but we doshow it with pure MFV in the A µµ ∆Γ versus R case (left panel).32odel M H [GeV] S R
MFV 900 – 1095 1.20 – 0.81 1.22 – 0.58LHS, RHS 1030 – 1310 0.94 – 0.56 0.75 – 0.26Table 2: Ranges allowed by B s mixing for the Scalar+Pseudoscalar model with non-degenerate masses. The model has been tuned so that P = 0, and thereby correspondsto Scenario E . The pseudoscalar mass is M A = 1 TeV.In Figure 11 we show the allowed range of A µµ ∆Γ with respect to the heavy scalar mass M H in the MFV (left panel) and MFV (right panel). In these plots we have fixed themuon couplings to ˜∆ µ ¯ µ = − .
03. The allowed range takes the B s mixing constraintsinto account and falls within the 2 σ C.L of R as defined in (29). We observe thatfor M H ≤ .
75 TeV negative values of A µµ ∆Γ are predicted in this scenario, while for M H ≥ . M H (cid:54) = M A . (96)We consequently shift our focus from Scenario C to Scenario E . Note, however, thata small mass difference is still consistent with a 2HDM in a decoupling regime, andwill approximately reproduce
Scenario C . In the presence of a non-zero mass differencethe following contribution to the SM box function in MFV or a LHS model no longervanishes: [∆ S ( B s )] LL = (cid:32) [ ˜∆ sb ] ∗ V ts V ∗ tb (cid:33) (cid:20) r LL ( M H ) M H − r LL ( M A ) M A (cid:21) (97)and an analogous expression for a RHS model after the replacement [ ˜∆ sb ] ∗ → ˜∆ bs and L with R with r RR = r LL = 50 TeV . In MFV we therefore find:[∆ S ( B s )] LL = (cid:18) m b m s (cid:19) ([∆ S ( B s )] RR ) ∗ = ( (cid:15) ∗ ) y b y t (cid:20) r LL ( M H ) M H − r LL ( M A ) M A (cid:21) . (98)We observe that: • The LL contribution, while suppressed relatively to the LR contribution throughsmaller hadronic matrix element ( | r LR | ≈ | r LL | ) and the splitting between scalarand pseudoscalar masses, does not suffer from m s /m b suppression. Thus whetherthe SLL contribution or LR dominates depends sensitively on the size of the masssplitting in question. The SRR contribution is totally negligible. • The SLL contribution now contains in principle a new flavour-blind phase in (cid:15) allowing for new CP-violating effects in B s mixing. • The formulae (92) and (98) are also valid in a non-MFV framework in which newflavour and CP-violating phases are present in (cid:15) .33or our numerics we fix the pseudoscalar mass to M A = 1 TeV and will continueto assume a universal lepton coupling as given in (94). The quark coupling required toset P = 0 has strength | ˜∆ sb | = 0 . sb ) = φ SM s / − ◦ . Thisthen poses the question of what the allowed range for the scalar mass M H is that iscompatible with the B s mixing constraints, and whether the resulting S satisfies theexperimental bounds on R .Because the scalar and pseudoscalar couple in the same way to quarks (see (80)), P = 0 implies that there are no new CP violating phases in the decay. The NP mixingphase φ NPs can still contribute (unless we assume MFV), and we find for the time-dependent observables: A µµ ∆Γ = − cos( φ NP s ) , S µµ = sin( φ NP s ) . (99)This change in sign for these observables with respect to the SM is a smoking gun signalof scalars dominating in the B s → µ + µ − decay. In Table 2 we summarise our results forthe allowed ranges of the scalar mass M H , parameter S and observable R for both MFVand a LHS/RHS quark model. Both models are seen to satisfy the current experimentalrange for R given in (29). We have performed the first detailed phenomenological analysis of the time-dependentrate for the B s → µ + µ − decay following the formalism developed in Ref. [5]. Our anal-ysis demonstrates that decay-time studies of B s → µ + µ − , which offer the observables A µµ ∆Γ and S µµ in addition to the branching ratio, allow for various NP scenarios to bedisentangled. Specifically, the presence of new scalar, pseudoscalar or gauge boson par-ticles can potentially be identified, which is not possible on the basis of the branchingratio alone.We have proposed a classification of various NP scenarios in terms of two complexvariables S and P that fully describe the three observables involved, and can be ex-pressed in terms of the fundamental parameters of a given model. The experimentaldetermination of S and P , accompanied by plausible model specific assumptions, willallow us to probe NP in this theoretically clean decay. We have illustrated this by placingseveral popular extensions of the SM into phenomenological scenarios introduced by us(see Table 1).We have further presented numerical analyses for the observables in question in mod-els for tree-level contributions to B s → µ + µ − mediated by heavy gauge bosons, scalarsand pseudoscalars. The plots in Figures 9 and 10 illustrate our general findings. Ourmain messages from these analyses are as follows. • The phenomenology of a tree-level Z (cid:48) exchange with respect to the studied ob-servables is very different in structure to that of spin-0 particles, in particularpseudoscalars. This is shown in Figure 9 (see also Figure 5 versus Figure 8). • In turn, the phenomenology of a scalar is more restricted than that of a pseu-doscalar. For instance, suppression of R with respect to its SM value would exclude34 NP scenario with only a single scalar, whereas such a suppression is possible fora single pseudoscalar. • For models with a single new particle with a mass of 1 TeV – specifically a gaugeboson, scalar or pseudoscalar – negative values of A µµ ∆Γ require large couplings tomuons and a significant deviation of the B s mixing phase φ s from its SM value. • On the contrary, a negative value of A µµ ∆Γ can naturally be explained in models withboth a scalar and pseudoscalar and a common mass M H ≤ . φ s from its SM value is notrequired. Furthermore, in these models R has a strict lower bound. • For a Z (cid:48) scenario the required suppression of ∆ M s , due to current tensions withexperiment, implies that S µµ (cid:54) = 0. • For a single pseudoscalar or scalar scenario, the required suppression of ∆ M s im-plies a departure from the SM value of BR( B s → µ + µ − ).The numerous plots and examples presented by us provide a roadmap for futureexperimental results of this outstanding rare B decay. Acknowledgements
We thank Christine Davies, Fulvia De Fazio, Martin Gorbahn, Minoru Nagai, DavidStraub and Robert Ziegler for discussions. RK would like to thank Andrzej Buras and hisgroup for hosting him at the IAS/TU in Munich. This research was financially supportedby the ERC Advanced Grant project “FLAVOUR” (267104) and the Foundation forFundamental Research on Matter (FOM).
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