aa r X i v : . [ h e p - ph ] D ec Bounding b → sµ + µ − tensor operators from B → K ∗ ( X s ) γ Namit Mahajan ∗ Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India
Tensor operators are often invoked as specific new physics operators beyond the standard modelin an effort to explain anomalies in rare B-decays and CP asymmetries. Specifically, b → sµ + µ − tensor operators are invoked in the study of semi-leptonic decays, both inclusive and exclusive. Inthis note we use the data on b → s radiative decay modes and CP asymmetries to tightly constrainthe tensor operators. It is found that constraints thus obtained are tighter than those from semi-leptonic modes. We also comment on b → ss ¯ s tensor operators that help in explaining the B → φK ∗ polarization puzzle, and b → sτ + τ − operators with tensor structure. PACS numbers: 13.25.Hw, 13.20.-v, 12.60.-i
Absence of flavour changing neutral currents at thetree level within the standard model (SM) makes themvery sensitive to quantum corrections due to heavy par-ticles in the loops. Rare decays, both inclusive and ex-clusive, of the type b → sγ , b → sg and b → sℓ + ℓ − are among the most promising channels in our searchfor possible new physics (NP) beyond SM. The studyof CP violation, and its origin, has been one of the mainaims of the B-factories. Thanks to excellent experimentalprecision reached at the B-factories, and also at CLEOand TeVatron, we now have accurate measurements ofbarnching ratios and CP asymmteries for many rare de-cay processes. LHCb has to be added to this list andhas already begun to yield competitive results even withsmall amount of data collected till now (see for example[1], [2], [3]). The situation is expected to improve drasti-cally over the next few years [4]. It is also not improbablethat the first glimpse of NP or the absence of it at theelectroweak scale will be from rare B-decays rather thanthe direct searches at LHC.Till date SM has turned out to be consistent with al-most all the available experimental data, though thereare some anomalies or unexplained features that seemto call for physics beyond SM (see for example [5]).Experimental observations and measurements have es-tablished the dominance of Cabibbo-Kobayashi-Maskawa(CKM) phase as the prominent source of CP violationas far as the low energy sector is concerned. Any at-tempt to infer hints of new physics (NP) need to ensurethat we have quantitatively exhausted all the possibil-ities within SM, including sub-leading effects and anyother neglected contributions based on some assump-tions. Semi-leptonic and radiative decays of B-mesonsoffer a unique opportunity to explore the possibility ofNP, including new sources of CP violation beyond theCKM phase. The inclusive decays, radiative and semilep-tonic, are theoretically more under control while the ex-clusive decays are relatively easier experimentally. Atpresent, the inclusive rate BR ( B → X s γ ) and the ex-clusive rate BR ( B → K ( ∗ ) ℓ + ℓ − ) and associated lepton ∗ Electronic address: [email protected] forward-backward asymmetry provide the most stringentconstraints on any new physics model, even in a quitemodel independent manner. Latest experimental resultsindicate good agreement with SM expectations for thesemodes but also leave a bit of a room for NP.When going beyond SM, new and heavier particles(more massive than the electroweak scale) can bring intotally new contributions not present in SM. An exam-ple could be left-right symmetric models which naturallylead to operators in the low energy theory that have dif-ferent chiral structure than SM (see for example [6]).Other popular examples include supersymmetric theo-ries which not only have different chiral structures forthe operators but also bring along completely new op-erators with naturally large coefficients or strengths [7].The aim of the current and future experiments is to accu-rately measure all possible observables and thereby endup tightly constraining the possible structures/operators,and if possible revealing the specific type of NP present.The effective Hamiltonian responsible for the semi-leptonic and radiative b → s transitions within SM isgiven by [8] H SMeff = − G F √ V ∗ tb V ts h X i =1 C i Q i + n C γ Q γ + C g Q g + C V Q V + C A Q A + C S Q S + C P Q P + X X C ′ X Q ′ X oi + V ∗ ub V us [ ... ] + H.C (1)where C ’s are the relevant Wilson coefficients while Q ’sare four fermion operators. Here, Q , are the current-current operators, while Q − and Q − are the QCDpenguin and electroweak (EW) penguin operators. Op-erators Q , Q , Q and Q have ( V − A ) ⊗ ( V + A )structure while all others have ( V − A ) ⊗ ( V − A ) struc-ture. Q γ and Q g are the electromagnetic and chromo-magnetic dipole operators while Q V and Q A are thevector and axial-vector semi-leptonic operators. Q S and Q P are the scalar and pseudoscalar semi-leptonic oper-ators. The primed operators can be obtained from theunprimed ones by making the replacement L ↔ R . Theterms proportional to V ∗ ub V us are usually neglected dueto the smallness of | V ub | . For what concerns us here, theexplicit form of some of the operators is: Q = (¯ c Lβ γ µ b Lα )(¯ s Lα γ µ c Lβ ) ,Q = (¯ c Lα γ µ b Lα )(¯ s Lβ γ µ c Lβ ) ,Q γ = e π m b (¯ s α σ µν Rb α ) F µν ,Q g = g s π m b (¯ s α σ µν R λ
Aαβ b β ) G Aµν ,Q V = e π m b (¯ s α γ µ Lb α )(¯ ℓγ µ ℓ ) ,Q A = e π m b (¯ s α γ µ Lb α )(¯ ℓγ µ γ ℓ ) ,Q S = e π m b (¯ s α Rb α )(¯ ℓℓ ) ,Q P = e π m b (¯ s α Rb α )(¯ ℓγ ℓ ) (2)The SM Wilson coefficients at scale µ = m b (approxi-mately) read: C ∼ − . , C ∼ . , C − ∼ O (10 − ) ,C , ∼ O (10 − ) , C ∼ − . α, C ∼ . αC γ ∼ − . , C g ∼ − . , C V ∼ . , C A ∼ − . V ub , which we neglect here.Due to the extreme smallness of C S,P ∼ m ℓ m b /m W within SM, the corresponding operators are usually ne-glected. The primed Wilson coefficients typically read C ′ X ∼ ( m s /m b ) C X , implying that they are expected tobe suppressed and hence neglected. Specifically, withinSM C ′ γ ∼ − . , C ′ g ∼ − . , C ′ V = 0 = C ′ A The scalar and pseudo-scalar operators can have en-hanced coefficients in many extensions of NP beyond SM,and can be cleanly probed in modes like B s → µ + µ − [9].In any extension of SM, either the Wislon coefficients ofthe operators already present get new non-negligible con-tributions or there are new operators induced in the lowenergy theory or both. In many analyses of semi-leptonicdecays, tensor operators are considered [10], [11]. Ten-sor operators are also invoked to explain the polarizationpuzzle in B → φK ∗ since they have the capability tosignificantly enhance the transverse polarization fractionand hence explain the experimental data [12]. We con-sider the following tensor operators with the scale of newphysics denoted by Λ Q TLL = (¯ s α σ µν Lb α )( ¯ f σ µν Lf ) ,Q TLL = (¯ s α σ µν Lb α )( ¯ f σ µν Rf ) ,Q TLL = (¯ s α σ µν Rb α )( ¯ f σ µν Lf ) ,Q TLL = (¯ s α σ µν Rb α )( ¯ f σ µν Rf ) (3) such that the additional terms in the effective Hamilto-nian read H NPeff = − X AB C TAB Q TAB = − G F √ V ∗ tb V ts X AB C TAB Q TAB (4)In the above equations, f refers to any light chargedfermion. We shall be specifically interested in f = µ .The Wilson coefficients are in general complex quan-tities but to simplify the analysis we assume them to bereal here. This then broadly refers to Minimal FlavourViolation scenario where it is assumed that CKM is theonly source of CP violation (see for example [13]). Thisshould only be taken as a simplifying assumption whentrying to make model independent statements. Else,we should write all possible relevant operators and al-low for complex coefficients. Then a detailed fit to thedata would yield the best fit values for the coefficients.This however requires a very large data set and a com-plicated analysis. In the meantime, one could just focuson a smaller sub-set of operators and try to constrainthem in a somewhat independent fashion. This is thetypical approach that is followed generally and we alsoadhere to that for the present study. Given the newtensor operators, the next task would be to study theireffect on various processes. The obvious ones f = µ are the semileptonic channels like B → K ( ∗ ) ( X s ) µ + µ − .By comparing the theoretical branching ratios and otherobservables like forward-backward, CP and any otherpossible asymmetries with the experimentally availabledata, constraints on the coefficients of the new opera-tors are obtained. For example, the rate of the inclusivesemileptonic process B → X s µ + µ − leads to a relation | C T | + 4 | C T E | < C T and C T E are the co-efficients of the following two operators that are usuallyconsidered in the literature: O T = (¯ sσ µν b )(¯ µσ µν µ ) O T E = i (¯ sσ µν b )(¯ µσ αβ µ ) ǫ αβµν (5)It is clear that the above two operators in Eq.(5)whenadded in suitable combinations are equivalent to the fourtensor operators listed in Eq.(3). There is however onesmall but potentially crucial difference. The Wilson coef-ficients of operators generated via suitable linear combi-nations of the operators O T and O T E are not all different,and therefore much more tightly constrained, althoughthere is a priori no reason for some of the coefficients tobe equal.We now study the effect of the tensor operators on b → sγ processes. It is clear that due to the dipole struc-ture of the operators involved, these operators will di-rectly contribute to b → sγ . Fig. 1 shows the Feynmandiagrams for operator insertions leading to new contri-butions to the process. For the present case, we studythe effects when f = µ . Thus only the left diagramcontributes (other possible diagrams where the photon isattached to the external quark lines are not shown). Toevaluate the effect of all the four tensor operators listedin Eq.(3), we consider the following general structure Q T = (¯ s α σ µν
12 (1 − aγ )2 b α )( ¯ f σ µν
12 (1 − a ′ γ ) f ) (6)with a, a ′ = ± f = u, d, s, c, e, mu, tau f = s FIG. 1: Feynman diagrams (drawn using the package Jaxo-Draw [14]) for generating b → sγ via the operator insertions(crossed circles denote the operator insertions). The righthand diagram gives the second insertion possible when thelight fermion f is the strange quark. On evaluating the diagram with the operator in Eq.(6)one finds that the chirality factors involved yield a non-zero contribution only if a = a ′ . This simply implies thatonly Q TLL and Q TRR contribute and can thus be bounded.The other two operators are totally unconstrained fromthe present analysis. This is precisely the potentially im-portant difference between the basis considered in Eq.(3)and the one usually employed in the study of semi-leptonic decays, ie the one in Eq.(5). On evaluatingthe loop diagram, one finds for both a = a ′ = +1 and a = a ′ = −
1, the same factor F Loop = 16 Q f m f m b ln m f µ R ! where Q f , m f and µ R are the light fermion charge,mass and renormalization scale respectively. We set µ R = m b = 4 . m b in the above expression has been introduced for conve-nience such that the new contribution finally takes thefamiliar form of Q ( ′ )7 γ . Denoting by ∆ C γ and ∆ C ′ γ thecontributions to respective Wilson coefficients due to thenew physics effects, one has∆ C ( ′ )7 γ = F Loop C TRR ( LL ) (7)For the case of muon, F µLoop ∼ .
5. For the case of strangequark, the F sLoop ∼ .
8. However, for the tau lepton orcharm quark, F c,τLoop ∼ O (10). For the present analysis,we set F µLoop = 2. It may be worthwhile to mention thatthe Wilson coefficients for the tensor operators at scales m b and m W are related by [15] C TAB ( m b ) = (cid:18) α s ( m W ) α s ( m b ) (cid:19) / (3 β s ) C TAB ( m W ) ∼ O (1) (8)where β s = 11 − N activef /
3. This then implies that as afirst approximation, the changes due to running may be
Observable HFAG average [5] BR ( B → K ∗ γ ) (42 . ± . × − BR ( B → X s γ ) (3 . ± . × − A CP ( b → sγ ) ( − . ± . S K ∗ γ − . ± . b → sγ system neglected. One therefore has the following at the scale m b : C γ → C γ + ∆ C γ ; C ′ γ → C ′ γ + ∆ C ′ γ (9)We note in passing that if the light fermion f is a quark,then the same set of operators will also contribute tothe chromomagnetic dipole operators ie to ∆ C ( ′ )8 g . Theanswers can be easily read off after making appropriatechanges in ∆ C ( ′ )7 γ . At the scale m b , the Wilson coefficientsmix and read [16] C γ ( m b ) ∼ − .
31 + 0 . C NP γ ( m W ) + 0 . C NP g ( m W ) C g ( m b ) ∼ − .
15 + 0 . C NP g ( m W ) (10)It is the extra contribution which is labeled ∆ C γ, g andwhen translating into constraints on the coefficients ofspecific tensor operators, these relations could be in-verted and the constraints read off.The branching ratio and time dependent CP asymme-try have been measured for B → K ∗ γ . Also available arethe very precise measurement of the inclusive branchingfraction BR ( B → X s γ ). The direct CP asymmetry inthe inclusive radiative mode can also be considered. Theexperimental situation is summarised in Table 1.We consider the exclusive mode first. The decay rate(or the branching fraction) reads [17] BR ( B → K ∗ γ ) = ( V ∗ tb V ts ) G F m b m B π (cid:18) − m K ∗ m B (cid:19) ×| T (0) | ( | C γ | + | C ′ γ | ) (11)where T is the form factor at q = 0, while the timedependent CP asymmetry reads [18] S K ∗ γ ≃ | C γ | + | C ′ γ | Im ( e − iφ d C γ C ′ γ ) (12)In the above equation, φ d describes the mixing in the B d sector, ie sin( φ d ) = S ψK S , time dependent mixinginduced CP asymmetry. We employ the experimentallymeasured value S ψK S = 0 . ± .
02 [5] in the numericalanalysis. We consider 1 σ and 2 σ ranges for the branch-ing ratio and mixing induced CP asymmetry respectively.The reason for choosing 2 σ range for S K ∗ γ is to includepossible error due to S ψK S , and we then employ the cen-tral value in the analysis. Fig. 2 shows the constraints in∆ C γ -∆ C ′ γ plane. The constraints on C TLL,RR are read-ily obtained from Eq.(7). From Fig. 2 it is clear thatdemanding that BR ( B → K ∗ γ ) and S K ∗ γ are withinthe experimental ranges yields very tight constraints onthe Wilson coefficients, more stringent than the maxi-mally allowed ones from the semi-leptonic processes. Asan example and a check, we looked at the representativevalues of C T and C T E employed in [11] and check whetherthey yield consistent values for both BR ( B → K ∗ γ )and S K ∗ γ . We find that for most of the representativepairs ( C T , C T E ) either or both the observables fail to fallwithin the experimentally allowed ranges. At this pointit is rather important to clearly mention that the valuesemployed in [11] are the ones that give maximal deviationfrom SM expectations for the observables studied. How-ever, smaller values are also consistent with their anal-ysis [19]. In no way this invalidates the analysis in [11]but the main point of this exercise was to illustrate thatonce the tensor operators are appropriately contracted inorder to obtain additional contributions to C γ and C ′ γ ,only a restricted region of parameter space survives. Thistherefore shows the power and importance of combiningthe constraints from a direct analysis like semi-leptonicmodes and indirect ones like radiative modes. c b a - - - D C Γ D C Γ ’ FIG. 2: Scatter plot of allowed region in ∆ C γ -∆ C ′ γ plane.Both the parameters are varied from -1 to 1. Fig. 2 has three regions which are allowed once con-straints from BR ( B → K ∗ γ ) and S K ∗ γ are included.Region ’c’ corresponds to small deviations from the SMvalues and hence is kind of expected. Bulk of the re-gion ’a’ corresponds to the case where the sign of C γ gets flipped and C ′ γ is not too large. This region naivelyspeaking is strongly disfavoured from B → X s ℓ + ℓ − rate.However, it is important to carefully check whether with C ′ γ also present this result still holds or not. Region ’b’is the most interesting region since both the coefficientsare non-negligible and of similar magnitude. A completeand consistent study will involve considering the effectof C ′ γ (and other relevant chirality flipped operators)on semi-leptonic modes in conjunction with the radiativemodes discussed here. This is beyond the scope of thepresent work and will be dealt elsewhere.From Eq.(7) it is clear that if f = τ , due to large F τLoop , the corresponding Wilson coefficients C T,τLL,RR willbe smaller. When translated in terms of C T , the valuesobtained are consistent with those obtained in [15].We now consider the tensor operators when f = s . Such operators are invoked in order to explain the po-larization puzzle in B → φK ∗ modes. Authors of [12]study a host of observables available in B → φK ∗ modesand obtain the following best fit values: C TLL ( f = s ) ∼ × − e iφ LL e iδ LL with φ LL = − . , δ LL = 1 . C TRR ( f = s ) ∼ . × − e iφ RR e iδ RR with φ RR =0 . , δ RR = 2 .
36 where φ ’s and δ ’s are the weak andstrong phases expressed in radians. We use these and findthat they yield consistent values for both BR ( B → K ∗ γ )and S K ∗ γ .We have explicitly checked that the inclusive b → sγ rate also yields similar constraints. Another powerfulobservable is the direct CP asymmetry in B → X s γ .Following [16], for a cut on photon energy, E γ > (1 − δ γ ) E maxγ with δ γ = 0 . A CP ( b → sγ ) ∼ . | C γ | + | C ′ γ | ) [1 . Im ( C C ∗ γ ) − . Im ( C g C ∗ γ + C g C ′ ∗ γ )+0 . Im ( C C ∗ g ) − . Im ( ǫ s C (13) × ( C ∗ γ − . C ∗ g + C ′ ∗ γ − . C ′ ∗ g ))]where ǫ s = V ∗ ub V us / ( V ∗ tb V ts ).We have checked that our results are consistent with A CP ( b → sγ ), and so are the values for Wilson coeffi-cients obtained by [12]. It is interesting to notice thatseemingly small Wilson coefficients as in [12] which asexpected would be consistent with the constraints fromradiative modes still are able to explain the polarizationpuzzle in B → φK ∗ . Very recently, a similar situationhas arisen in B s → K ∗ ¯ K ∗ where again the longitudinalfraction is found to be much lower than the expectations[2]. This is a b → s ¯ dd penguin dominated mode and isexpected to be an important channel in search of newphysics. It would be interesting to see if similar tensoroperators can explain the polarization puzzle and remainconsistent with the constraints from radiative modes. Weleave this for a separate study. It is also noteworthy thatto explain the polarization puzzle other type of opera-tors are also considered, eg right handed currents [20],(pseudo-)scalar operators [21]. We would like to empha-size that any operator of the form (¯ s Γ b )(¯ q Γ ′ q ), whereΓ , Γ ′ are Dirac structures would in principle generatenew contributions to dipole operators and one should ex-plictly check whether they pass the simple tests discussedabove.In this note we have studied the impact of tensor op-erators corresponding to physics beyond SM, that areinvoked in the study of semi-leptonic decays b → sℓ + ℓ − and b → s ¯ ss to explain polarization puzzle in B → φK ∗ modes, on radiative modes b → sγ and CP asymmetries.We have shown that two of the tensor operators with chi-rality LL and RR can be stringently constrained. Theother two operators with chiral structure LR and RL donot contribute to the radiative mode and therefore areleft unconstrained from the present analysis. We havealso eluded to a potential difference between the casewhen all the Wilson coefficients for these operators aretaken as free parameters and the case when due to spe-cific choice of the operators some of the coefficients areequal to each other. This according to us may be overrestrictive. We have found that the tensor operators endup generating new contributions to dipole operators withboth the chiralities. The Wilson coefficients have beenassumed to be real but extension to complex coefficientsis straightforward. It is known that complex coefficientsyield a far more richer phenomenology (see for example[22] for effects of complex coefficients on B → K ∗ ℓ + ℓ − )and in fact presence of new phases may fake the effectsnaively expected due to the presence of new class of op-erators. Also, if the operator under consideration has thelight fermion as a quark, then a similar contribution isgenerated for the chromomagnetic dipole operators andtherefore would contribute non-trivially to A CP ( b → sγ )and BR ( b → sg ). An enhanced b → sg rate may help in explaining some of the issues in rare B-decays like charmcounting, large B → Kη ′ rate and semi-leptonic branch-ing ratio (see for example [23]). Further, the presence ofchirality flipped dipole operator calls for a separate anal-ysis including semi-leptonic decays. We leave all these fora future study. In conclusion, we have shown that rathertight constraints can be obtained on the tensor operatorsby studying their impact on radiative modes. 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