Tension between scalar/pseudoscalar new physics contribution to B_s --> mu+ mu- and B --> K mu+ mu-
aa r X i v : . [ h e p - ph ] A ug Tension between scalar/pseudoscalar new physics contributionto B s → µ + µ − and B → Kµ + µ − Ashutosh Kumar Alok, Amol Dighe, and S. Uma Sankar Tata Institute of Fundamental Research,Homi Bhabha Road, Mumbai 400005, India Indian Institute of Technology Bombay, Mumbai-400076, India
New physics in the form of scalar/pseudoscalar operators cannot lower the semileptonicbranching ratio B ( B → Kµ + µ − ) below its standard model value. In addition, we show thatthe upper bound on the leptonic branching ratio B ( B s → µ + µ − ) sets a strong constraint onthe maximum value of B ( B → Kµ + µ − ) in models with multiple Higgs doublets: with thecurrent bound, B ( B → Kµ + µ − ) cannot exceed the standard model prediction by more than2.5%. The conclusions hold true even if the new physics couplings are complex. Howeverthese constraints can be used to restrict new physics couplings only if the theoretical andexperimental errors in B ( B → Kµ + µ − ) are reduced to a few per cent. The constraintsbecome relaxed in a general class of models with scalar/pesudoscalar operators. PACS numbers: 13.20.He, 12.60.-i
One of the major aims of the large hadron collider (LHC), about to start operating soon, is tolook for Higgs particles within and beyond the standard model (SM). Even a direct observation ofa Higgs particle will not suffice to tell us whether it is the SM Higgs or not. An understanding ofpossible scalar/pseudoscalar new physics (SPNP) interactions through indirect means is thereforeextremely crucial.The flavor changing neutral interaction b → sµ + µ − serves as an important probe to test higherorder corrections to the SM as well as to constrain many new physics models. This four-fermioninteraction is responsible for the purely leptonic decay B s → µ + µ − , for the semileptonic decays B → ( K, K ∗ ) µ + µ − and also for the radiative leptonic decay B s → µ + µ − γ . The semileptonic decayshave been experimentally observed at BaBar and Belle [1, 2, 3, 4]. The pseudoscalar semileptonicdecay has the branching ratio B ( B → Kµ + µ − ) = (4 . +0 . − . ) × − , (1)which has been obtained with ∼
350 fb − of data. These values are consistent with the SM predic-tions [6, 7, 8, 9], and the experimental errors are expected to reduce to ∼
2% at the forthcomingSuper-B factories [10]. At the moment there is about 20% uncertainty in these SM predictionsdue to the error in the quark mixing matrix element V ts and the uncertainties related to stronginteractions. Improvements in the lattice calculations and the measurement of V ts are likely tobring this error down to a few per cent within the next decade.The purely leptonic decay B s → µ + µ − is highly suppressed in the SM, the prediction for itsbranching ratio being (3 . ± . × − [11]. The uncertainty in the SM prediction is mainlydue to the uncertainty in the decay constant f B s and V ts . This decay is yet to be observed inexperiments. Recently the upper bound on its branching ratio has been improved to [12] B ( B s → µ + µ − ) < . × − (95% C . L . ) , (2)which is still more than an order of magnitude away from its SM prediction. The decay B s → µ + µ − will be one of the important rare B decay channels to be studied at the LHC and we expect thatthe sensitivity of about 10 − can be reached in a few years [13].In the context of these decays, one needs to focus only on new physics from scalar/pseudoscalarinteractions, since (i) new physics in the form of vector/axial-vector operators is highly constrainedby the data on B → ( K, K ∗ ) µ + µ − as shown in [14], and (ii) new physics in the form of tensorand magnetic dipole operators does not contribute to B ( B s → µ + µ − ). A measured value of B ( B s → µ + µ − ) & − indicates that the new physics must be in the form of scalar/pseudoscalaroperators.We take the effective Lagrangian for the four-fermion transition b → sµ + µ − to be [6] L ( b → sµ + µ − ) = L SM + L SP , (3)where L SM = αG F √ π V tb V ⋆ts (cid:26) C eff9 (¯ sγ µ P L b ) ¯ µγ µ µ + C (¯ sγ µ P L b ) ¯ µγ µ γ µ + 2 C eff7 q m b (¯ siσ µν q ν P R b ) ¯ µγ µ µ (cid:27) , (4) L SP = αG F √ π V tb V ⋆ts (cid:26) ˜ R S (¯ s P R b ) ¯ µ µ + ˜ R P (¯ s P R b ) ¯ µγ µ (cid:27) . (5)Here P L,R = (1 ∓ γ ) / q is the sum of the µ + and µ − momenta. ˜ R S and ˜ R P are the scalarand pseudoscalar new physics couplings respectively, which in general can be complex. We usethe notation ˜ R S ≡ R S e iδ S , ˜ R P ≡ R P e iδ P . Here the phases are restricted to be 0 ≤ ( δ S , δ P ) < π ,whereas R S and R P can take positive as well as negative values. Within SM, the Wilson coefficientsin Eq. (4) have the following values [6]: C eff7 = − . , C eff9 = +4 .
138 + Y ( q ) , C = − . , (6)where the function Y ( q ) is given in [15]. These coefficients have an uncertainty of about 5%,which arises mainly due to their scale dependence.In Eq. (5), we have taken only P R in the quark bilinear, while the most general Lagrangian musthave a linear combination of P L and P R . Here we start by considering the simpler case becauseSPNP operators mostly arise due to multiple Higgs doublets. In such models, the coefficient of P R in the Lagrangian is much larger than that of P L [6]. In two Higgs doublet model, for instance,the coefficient of P L is smaller by a factor of m s /m b [16]. We shall examine the consequences ofconsidering the most general quark bilinear in the latter part of this Letter.In the following, we consider the interrelations between the contributions of L SP to the branchingratios of the decays B s → µ + µ − and B → Kµ + µ − . The effect of SPNP couplings on additionalobservables related to these decays, viz. forward-backward asymmetry in the semileptonic decayand the polarization asymmetry in the leptonic decay, has been studied in [5]. The contributionof L SP to B → K ∗ µ + µ − is so small [6] that no worthwhile correlation can be established betweenit and other decays. Also, L SP does not contribute to the radiative leptonic decay B s → µ + µ − γ [17, 18].We first consider the contribution of L SP to the decay rate of B s → µ + µ − . The branching ratiois given by B SP ( B s → µ + µ − ) = G F α m B s τ B s π | V tb V ∗ ts | f B s ( R S + R P ) . (7)Taking f B s = (0 . ± . B SP ( B s → µ + µ − ) = (1 . ± . × − ( R S + R P ) . (8)Note that the present experimental upper limit on B ( B s → µ + µ − ) is an order of magnitudelarger than the SM prediction. In the following, we will assume that the SPNP will provide anorder of magnitude increase of B ( B s → µ + µ − ). In such a situation, the SM amplitude can beneglected in the calculation of the branching ratio. Equating the expression in Eq. (8) to thepresent 95% C.L. upper limit in Eq. (2), we get the inequality( R S + R P ) ≤ . , (9)where we have taken the 2 σ lower bound for the coefficient in Eq. (8). Thus, the allowed regionin the R S – R P parameter space is the interior of a “leptonic” circle of radius r ℓ ≈ .
84 centered atthe origin, as indicated in both the panels of Fig. 1. As the upper bound on B ( B s → µ + µ − ) goesdown, the radius of the circle will shrink. R P R S R P R S FIG. 1: The allowed ranges of R S and R P , when the new physics couplings are real. In both figures,the dark grey circles centered at origin represent the regions allowed by the current 2 σ upper bound on B ( B s → µ + µ − ). The light grey annulus in each figure represents the parameter space allowed by B ( B → Kµ + µ − ) at 2 σ . The width of the annulus corresponds to the sum of the theoretical and experimentalerrors, both of which are taken to be 2%. In the left panel, we take B ( B → Kµ + µ − ) = (5 . ± . · − .The overlap between the allowed regions is represented by the black crescent. In the right panel we take B ( B → Kµ + µ − ) = (6 . ± . · − , where the allowed parameter spaces do not overlap. We now turn to the semileptonic decay B → Kµ + µ − . The measured branching ratio is consis-tent with the SM prediction, though there is a 25% error in the measurement and about 20% errorin the theoretical prediction due to uncertainties in V ts , form factors and Wilson coefficients (whichin turn depend on V ts ). With the addition of the SPNP contribution, the theoretical predictionfor the net branching ratio becomes [6] B ( B → Kµ + µ − ) = (cid:2) .
25 + 0 . R S + R P ) − . R P cos δ P (cid:3) (1 ± . × − , (10)In Eq. (10), the first term is purely due to the SM, the second term is purely due to SPNP andthe third term is due to the interference of the two. The theoretical errors arise from one tensorand two vector form factors in the SM, and a scalar form factor in SPNP (which is related toone of the SM vector form factors). We have made the simplifying assumption that the fractionaluncertainties in all the form factors are the same.Eq. (10) can be rewritten as B ( B → Kµ + µ − ) = (1 + ǫ ) B SM , (11)where ǫ is the fractional change in the branching ratio due to SPNP. The maximum negative valuethat ǫ can take is − . B ( B → Kµ + µ − ) by more than 0.5% below its standard model value. Indeed, if the theoreticaland experimental errors in this quantity were improved to 5%, with the central values unchanged,the discrepancy cannot be accounted for by SPNP at 2 σ .Let us first consider the case where the new couplings R S and R P are real, which is typical forthe class of models where the only charge-parity violation comes from the CKM matrix elements.Using Eqs. (1) and (10), we get R S + ( R P − . = B exp (0 . ± . × − − . , (12)where B exp is the measured value of B ( B s → Kµ + µ − ). The region in the R S – R P plane allowedby the measurement of B ( B s → Kµ + µ − ) is then an “semileptonic” annulus centered at (0 , . B exp is below the SM prediction by more than 0.5%. Then the radius of the circlebecomes imaginary, which implies that the discrepancy of the measurement with the SM cannotbe explained by SPNP.To illustrate the tension between the quantities B ( B s → µ + µ − ) and B ( B → Kµ + µ − ), weconsider the scenario where the errors in both B SM and B exp have been reduced to 2%, whilekeeping the upper limit on B ( B s → µ + µ − ) at its current value. The allowed R S – R P parameterspace is shown in Fig. 1. If the lower limit on B exp is small enough, the semileptonic annulus willoverlap with leptonic circle, as shown in the left panel. However, if the lower limit on B exp is largerthan a critical value (determined by the bound on the leptonic branching ratio), then there is noregion of overlap as shown in the right panel. In such a situation, the difference between B exp and B SM cannot be accounted for by SPNP because of the constraint coming from the leptonic mode.We represent the radius of the leptonic circle by r ℓ and the inner (outer) radius of the semilep-tonic annulus by r in ( r out ). There is tension between the two measurements if r in − r ℓ > . , (13)in which case the regions allowed by the two branching ratios do not overlap. Given the currentvalue of r ℓ = 0 .
84, we require 0 < r in < . σ lower limiton B exp should be between 4 . × − and 5 . × − . (We have added the theoretical andexperimental errors in quadrature.) If the upper bound on B ( B s → µ + µ − ) is improved by a factorof 5, the 2 σ range for the lower limit on B exp would be (4 . − . × − . For the tension to bemanifest in future experiments, the reduction of errors in B exp and B SM is the most crucial.When ˜ R S and ˜ R P are complex, the constraint Eq. (12) becomes R S + ( R P − .
36 cos δ P ) = B exp (0 . ± . × − − .
17 + (0 .
36 cos δ P ) . (14)For nonzero δ P , the center of the semileptonic annulus shifts along the R P axis, while the radiusof the annuli are almost unchanged. If the allowed regions do not overlap for δ P = 0 (as illustratedin the right panel of Fig. 1), then they will not overlap for any value of δ P . Hence the tensionbetween B ( B s → µ + µ − ) and B ( B → Kµ + µ − ) persists, and gives rise to the same constraints onthe semileptonic branching ratio even if the SPNP couplings are complex.In writing the effective SPNP Lagrangian in Eq. (5), we considered only the quark bilinear¯ sP R b . Lorentz Invariance of the Lagrangian also allows the bilinear ¯ sP L b in general. We can takethis generalization into account by replacing ¯ sP R b by ¯ s ( αP L + P R ) b , where α is the strength of the¯ sP L b bilinear relative to that of ¯ sP R b . With this modification, B ( B → Kµ + µ − ) is driven by thesum of the two quark bilinears with different chiralities, whereas B ( B s → µ + µ − ) depends on theirdifference [20]. The expressions for the branching ratios of the two processes considered here are: B ( B s → µ + µ − ) = (1 − α ) ( R S + R P ) (1 . ± . × − , (15) B ( B → Kµ + µ − ) = (cid:2) .
25 + 0 .
18 (1 + α ) ( R S + R P ) − .
13 (1 + α ) R P (cid:3) (1 ± . × − . (16)Here we have taken R S , R P and α to be real for simplicity. For α = 0, Eqs. (15) and (16) reduce toEqs. (8) and (10) respectively. For the special case α = 1, the new physics has no contribution to B s → µ + µ − because the quark bilinear is pure scalar and the corresponding pseudoscalar mesonto vacuum transition matrix element is zero. In such cases, B ( B s → µ + µ − ) is entirely due to theSM, and provides no constraints on B ( B → Kµ + µ − ).In Fig. 2, we show ǫ max , the maximum fractional deviation of B ( B → Kµ + µ − ) from its SMprediction as defined in Eq. (11), as a function of the 2 σ upper bound on B ( B s → µ + µ − ). Theminimum allowed value of ǫ is almost independent of the value of α and the leptonic upper bound,and is approximately − . α = 0, andthe maximum value of ǫ is restricted to +0 . α , this severe constraint is relaxed. For example, forthe models with α ≈ .
5, the value of ǫ may be as large as +0 .
7, as can be seen in the figure. Ingeneral for positive α values, ǫ max increases with α for α < .
0, and decreases thereafter. When α <
0, Eq. (15) indicates that the constraints on R S and R P should become more restrictive. Asa result, ǫ is constrained to be even smaller. From the figure, ǫ max for negative α are seen to bevery close to zero, and the corresponding ǫ max curves are almost overlapping. This implies that fornegative α , any significant deviation of B ( B → Kµ + µ − ) from SM is impossible with SPNP. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 ε m a x B(B s --> µ + µ - ) max X 10 α =2.5 α =1.5 α =0.5 α =0 α =-0.5 α =-1.5 α =-2.5 FIG. 2: ǫ max as a function of the 2 σ upper bound on B ( B s → µ + µ − ) for different values of α . We havetaken the theoretical error on B ( B s → µ + µ − ) to be 20%; decreasing it would further constrain ǫ max . For the measurements of B ( B s → µ + µ − ) and B ( B → Kµ + µ − ) to be compatible with SPNP,the lower bound on B ( B → Kµ + µ − ) should be less than (1 + ǫ max ) B SM . Thus, the upper boundon B ( B s → µ + µ − ) and the lower bound on B ( B → Kµ + µ − ) allow us to constrain the value of α in a class of models that involve new physics scalar/pseudoscalar couplings.In this letter, we have parameterized scalar/pseudoscalar new physics in terms of the effectiveoperators given in Eq. (5). In general, the introduction of new scalar/pseudoscalar fields into amodel leads to not only new effective operators but also modification of the coefficients of the SMoperators, e.g. the Wilson coefficients C , C and C shown in Eq. (4). However, it has beenshown that these modifications due to new scalar/pseudoscalar fields are very small [16, 21]. Wehave computed these changes in the two Higgs doublet model and found them to be at most 1%.Thus, our assumption of retaining the SM values for the Wilson coefficients, even in the presenceof new scalar/pseudoscalar fields, is valid.In summary, we have shown that in a class of models with new scalar/pseudoscalar operators,which includes models with multiple Higgs doublets, the SPNP couplings are strongly constrainedby the upper bound on B ( B s → µ + µ − ), and in turn restrict the allowed values of B ( B → Kµ + µ − )to within a narrow range around its SM prediction. Future precise measurements of these twobranching ratios have the potential not only to give an evidence for new physics, but also to revealthe nature of its Lorentz structure. However in order to achieve this, the theoretical as well asexperimental errors on B ( B → Kµ + µ − ) need to be reduced to a few per cent. Acknowledgments
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