TThe B d → K ∗ µ + µ − decay U. Egede a T. Hurth b J. Matias c M. Ramon c W.Reece da Imperial College London, London SW7 2AZ,United Kingdom b Institute for Physics, JohannesGutenberg-University, D-55099 Mainz, Germany c Universitat Aut`onoma de Barcelona, 08193Bellaterra, Barcelona, Spain d CERN, Dept. of Physics, CH-1211 Geneva 23,Switzerland
In this paper the potential for the discovery of newphysics in the exclusive decay B d → K ∗ µ + µ − is dis-cussed. Attention is paid to constructing observableswhich are protected from uncertainties in QCD formfactors and at the same time observe the symmetriesof the angular distribution. We discuss the sensitivityto new physics in the observables including the effectof CP -violating phases.
1. Introduction
MZ-TH/10-41With the LHC b experiment coming online,there is the prospects of performing precisionphysics in the B d → K ∗ µ + µ − channel withina few years. This means particular attention hasto be given to how the predictions from the phe-nomenology and the experimental measurementsare compared.First published results from BELLE [1] and B A B AR [2] based on O (100) decays alreadydemonstrate their feasibility.In [3], it was proposed to construct observ-ables that maximise the sensitivity to contribu-tions driven by the electro-magnetic dipole oper-ator O (cid:48) , while, at the same time, minimising thedependence on the poorly known soft form fac-tors. A (2)T is highly sensitive to new right-handed currents driven by the operator O (cid:48) [4], to which A FB is blind.Looking for the complete set of angular observ-ables sensitive to right-handed currents, one isguided to the construction of A (3)T and A (4)T [5] and A (5)T [6]. The observables A ( i )T (with i = 2 , , , K ∗ spin amplitudes as the fundamen-tal building block. This provides more freedomto disentangle the information on specific Wilsoncoefficients than just restricting oneself to use thecoefficients of the angular distribution as it wasrecently done in [7]. For instance, A (2)T , being di-rectly proportional to C (cid:48) enhances its sensitivityto the type of NP entering this coefficient. More-over using selected ratios of the coefficients of thedistribution, like in A ( i )T , the sensitivity to softform factors is completely canceled out at LO.1 a r X i v : . [ h e p - ph ] N ov
2. Differential decay distribution
The decay B d → K ∗ µ + µ − , with K ∗ → K − π + on the mass shell, is completely describedby four independent kinematic variables, thelepton-pair invariant mass squared, q , and thethree angles θ l , θ K , φ . Summing over the spinsof the final state particles, the differential decay distribution of B d → K ∗ (cid:96) + (cid:96) − can be written as d Γ dq d cos θ l d cos θ K dφ = 932 π J ( q , θ l , θ K , φ ) , (1)The dependence on the three angles can be mademore explicit: J ( q , θ l , θ K , φ )= J s sin θ K + J c cos θ K + ( J s sin θ K + J c cos θ K ) cos 2 θ l + J sin θ K sin θ l cos 2 φ + J sin 2 θ K sin 2 θ l cos φ + J sin 2 θ K sin θ l cos φ + ( J s sin θ K + J c cos θ K ) cos θ l + J sin 2 θ K sin θ l sin φ + J sin 2 θ K sin 2 θ l sin φ + J sin θ K sin θ l sin 2 φ . (2)The J i depend on products of the six complex K ∗ spin amplitudes, A L,R (cid:107) , A L,R ⊥ and A L,R in thecase of the SM with massless leptons. The L and R indicate a left and right handed currentrespectively. Each of these is a function of q .The amplitudes are just linear combinations ofthe well-known helicity amplitudes describing the B → Kπ transition: A L.R ⊥ , (cid:107) = ( H L.R +1 ∓ H L.R − ) / √ , A L,R = H L.R . (3)The J i will be bi-linear functions of the spin am-plitudes such as J s = 34 (cid:104) | A L ⊥ | + | A L (cid:107) | + | A R ⊥ | + | A R (cid:107) | (cid:105) , (4)with the expression for the eleven other J i termsgiven in [6].The amplitudes themselves can beparametrised in terms of the seven B → K ∗ form factors by means of a narrow-width ap-proximation. They also depend on the short-distance Wilson coefficients C i corresponding tothe various operators of the effective electroweakHamiltonian. The precise definitions of the formfactors and of the effective operators are givenin [5]. Assuming only the three most important SM operators for this decay mode, namely O , O , and O , and the chirally flipped ones, beingnumerically relevant, we have as an example A L,R ⊥ = N √ λ / (cid:20) V ( q ) m B + m K ∗ (cid:26) ( C (eff)9 + C (cid:48) (eff)9 ) ∓ ( C (eff)10 + C (cid:48) (eff)10 ) (cid:27) ++ 2 m b q ( C (eff)7 + C (cid:48) (eff)7 ) T ( q ) (cid:21) (5)where the C i denote the corresponding Wilson co-efficients, N is a normalisation and λ = m B + m K ∗ + q − m B m K ∗ + m K ∗ q + m B q ) . (6)There are similar expressions for the other spinamplitudes [5].When going from the six complex spin ampli-tude to the expression of the angular distribu-tion (2) with 12 J i terms, one would naturallyassume there is no loss of information. However,it turns out that there are a number of relationsbetween the J i terms; this in turns means thatthere are continuous transformations of the spinamplitudes that will result in the identical angu-lar distribution. The full derivation of these sym-metries can be found in [6], while here we justgive the result.In total four of the J i terms can be writtenas a function of the eight remaining J i . Thus,the differential distribution is invariant under thefollowing four independent symmetry transforma-tions of the amplitudes n (cid:48) i = (cid:20) e iφ L e − iφ R (cid:21) (cid:20) cos θ − sin θ sin θ cos θ (cid:21)(cid:20) cosh i ˜ θ − sinh i ˜ θ − sinh i ˜ θ cosh i ˜ θ (cid:21) n i , (7)where φ L , φ R , θ and ˜ θ can be varied indepen-dently and n i is defined as n = ( A L (cid:107) , A R (cid:107) ∗ ) ,n = ( A L ⊥ , − A R ⊥∗ ) , (8) n = ( A L , A R ∗ ) . Normally, there is the freedom to pick a sin-gle global phase, but as L and R amplitudes donot interfere here, two phases can be chosen arbi-trarily as reflected in the first transformation ma-trix. The interpretation of the third and fourthsymmetry is that they transform a helicity +1final state with a left handed current into a he-licity −
3. Comparing theory and experiment
As can be seen in the previous section it is pos-sible to express the full angular dependence interms of the effective Wilson coefficients. Fromthis it would seem that it is trivially possible toextract full knowledge of the Wilson coefficientsfrom a fit to experimental data on the angular dis-tribution. Unfortunately there are several prob-lems related to such an approach that will be de-scribed in turn.As we are dealing with an exclusive decay, wehave to rely on the calculation of form factors.These can come from either light cone sum rulecalculations in the low q region, or from lattice QCD calculations in the high q region but arein both cases subject to significant uncertainty.In the low q region below 6 GeV /c the use ofsoft collinear effective theory (SCET), allows forthe a reduction in the number of form factorsfrom seven to two, which we will subsequentlytake advantage of. From below we are limited to q > /c due to a logarithmic divergencein the SCET approach.Even after the form factors have been consid-ered, each spin amplitude has an uncertainty dueto Λ QCD /m b corrections. The level of these arenot known but dimensional considerations leadsone to expect them at the 10% level or below. Inour analysis we have made the effect explicit byillustrating the effect of Λ QCD /m b corrections atthe 5% and 10% level.The effect of charm loops will be importanteven outside the narrow resonance regions of the B d → J/ψ K ∗ and B d → ψ (2 S ) K ∗ due to theeffect of virtual cc pairs [8]. The effect is small for q < /c which is the region we consider.When comparing theory to experimental data,two approaches are in general taken. One canstart at the theory end with a physics model;from that we calculate the Wilson coefficients andsubsequently the spin amplitudes. The last stepwill lead to a loss of accuracy due to the formfactor uncertainties and the unknown Λ QCD /m b corrections. Finally we calculate the angular co-efficients J i which can be compared directly to anangular fit of experimental data.The other approach is to start from the exper-imental determination of the angular coefficientsand from that determine the spin amplitudes. Aswe have seen in the previous section this processis not well defined due to symmetries in the angu-lar distribution. Ignoring this point one can fromthe spin amplitudes get the Wilson coefficients(again suffering from form factor and Λ QCD /m b uncertainties) and go onto a physics model.As illustrated in Fig. 1 the experimental resultsand the theory can be compared at several dif-ferent levels. However, the uncertainties intro-duced in each direction means this is not opti-mal. We suggest instead to create a set of observ-ables which can be derived from both the theoryside and the experimental results as illustrated inFigure 1. Traditionally experiment and theoryare compared at some point along the routeof transforming one to the other. For B d → K ∗ µ + µ − this has the disadvantage of having alarge theoretical uncertainty from form factor and Λ QCD /m b uncertainties.Fig. 2. In this way the majority of the uncertaintydue to the soft form factors can be eliminated.
4. Constructing observables
When constructing observables as outlined inthe previous section, several constraints has tobe considered. • They should have sensitivity to a range ofnew physics models. In the work we presenthere, we have concentrated on sensitivity to models with right handed currents, but otherchoices are equally possible. • In the low q region, where there are onlytwo form factors, observables should be con-structed where their value cancel out to lead-ing order. • The effect of Λ QCD /m b corrections should bedemonstrated to be small in comparison toexpected differences between the SM and dif-ferent NP models. • The observable should be invariant underthe symmetries of the differential distribu-tion described above. Otherwise it would notbe well defined from an experimental pointof view. • The experimental resolution from a data setobtainable with LHC b or a super- B factoryshould be good enough to offer distinctionbetween models.Based on these constraints, a range of CP -conserving observables have been designed, des-ignated A ( i )T , with i = 2 , , , A (2)T , first proposedin [3], and defined as A (2)T = | A ⊥ | − | A (cid:107) | | A ⊥ | + | A (cid:107) | , (9)where | A i | = | A Li | + | A Ri | . It has a simpleform, free from form factor dependencies, in theheavy-quark ( m B → ∞ ) and large K ∗ energy( E K ∗ → ∞ ) limits: A (2)T = 2 (cid:104) Re (cid:16) C (cid:48) (eff)10 C (eff)10 ∗ (cid:17) + F Re (cid:16) C (cid:48) (eff)7 C (eff)7 ∗ (cid:17) + F Re (cid:16) C (cid:48) (eff)7 C (eff)9 ∗ (cid:17)(cid:105) |C (eff)10 | + |C (cid:48) (eff)10 | + F ( |C (eff)7 | + |C (cid:48) (eff)7 | ) + |C (eff)9 | + 2 F Re (cid:16) C (eff)7 C (eff)9 ∗ (cid:17) , (10)where F ≡ m b m B /q . The sensitivity tothe primed Wilson coefficients corresponding toright handed currents can clearly be seen from the equation and is illustrated in Fig. 3 where aNP contribution to C (cid:48) (eff)10 is considered. It canalso be seen how the theoretical errors are veryFigure 2. Observables can be constructed whichfrom the theoretical side offers cancellation ofform factors, and which from the experimentalside are well defined and have good experimentalsensitivity.small. In Fig. 4 the expected experimental res-olution with data from the LHC b experiment isillustrated.As another example, A (5)T = (cid:12)(cid:12) A L ⊥ A R (cid:107) ∗ + A R ⊥∗ A L (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) A L ⊥ (cid:12)(cid:12) + (cid:12)(cid:12) A R ⊥ (cid:12)(cid:12) + (cid:12)(cid:12) A L (cid:107) (cid:12)(cid:12) + (cid:12)(cid:12) A R (cid:107) (cid:12)(cid:12) . (11)as defined in [6], has a very different behaviourwith respect to NP contributions, and a compar-ison of experimental measurements of A (2)T and A (5)T , will be able to provide details of the un-derlying theory. The sensitivity to NP of A (5)T isillustrated in Fig. 5.
5. Conclusion
We have presented how the decay B d → K ∗ (cid:96) + (cid:96) − can provide detailed knowledge of NPeffects in the flavour sector. We developed amethod for constructing observables with specificsensitivity to some types of NP while, at the sametime, keeping theoretical errors from form factorsunder control. We demonstrate the possible im-pact of the unknown Λ QCD /m b corrections on theNP sensitivity of the observables. Experimen-tal sensitivity to the observables was evaluatedfor datasets corresponding to 10 fb − of data atLHC b . A phenomenological analysis of the A ( i )T observables reveals a good sensitivity to NP in-cluding the effects from CP -violating phases. SM C - - I GeV M A T H L Figure 3. A (2)T in the SM (green) and with NP in C (cid:48) (eff)10 = 3 e i π (blue); this value is allowed by themodel independent analysis of [9]. The inner linecorresponds to the central value of each curve.The dark orange bands surrounding it are theNLO results including all uncertainties (exceptfor Λ QCD /m b ). Internal light green/blue bands(barely visible) include the estimated Λ QCD /m b uncertainty at a ±
5% level and the external darkgreen/blue bands correspond to a ±
10% correc-tion for each spin amplitude. - - I GeV M A T H L Figure 4. For A (2)T we illustrate the expected sta-tistical experimental errors. The inner and outerbands correspond to 1 σ and 2 σ statistical errorswith a yield corresponding to a 10 fb − data setfrom LHC b . SMadb I GeV M A T H L Figure 5. A (5)T in the SM and with NP in boththe C (eff)7 and C (cid:48) (eff)7 Wilson coefficients. The cyanline ( a ) corresponds to ( C NP , C (cid:48) ) = (0 . e − i π ,0 . e iπ ), the brown line ( b ) to (0 . e i π , 0 . e i π )and the pink line ( d ) to (0 . e − i π , 0). The bandssymbolise the theoretical uncertainty as describedin Fig. 3. REFERENCES BELLE
Collaboration, J. T. Wei et. al. , Measurement of the Differential BranchingFraction and Forward-Backward Asymmetryfor B → K ( ∗ ) (cid:96) + (cid:96) − , Phys. Rev. Lett. (2009) 171801, [ arXiv:0904.0770 ].2.
BABAR
Collaboration, B. Aubert et. al. , Measurements of branching fractions, rateasymmetries, and angular distributions in the rare decays B → K(cid:96) + (cid:96) − and B → K ∗ (cid:96) + (cid:96) − , Phys. Rev.
D73 (2006)092001, [ hep-ex/0604007 ].3. F. Kruger and J. Matias,
Probing newphysics via the transverse amplitudes of B → K ∗ ( → K − π + ) (cid:96) + (cid:96) − at large recoil , Phys. Rev.
D71 (2005) 094009,[ hep-ph/0502060 ].4. E. Lunghi and J. Matias,
Huge right-handedcurrent effects in B → K ∗ ( → Kπ ) (cid:96) + (cid:96) − insupersymmetry , JHEP (2007) 058,[ hep-ph/0612166 ].5. U. Egede, T. Hurth, J. Matias, M. Ramon,and W. Reece, New observables in the decaymode B d → K ∗ (cid:96) + (cid:96) − , JHEP (2008) 032,[ arXiv:0807.2589 ].6. U. Egede, T. Hurth, J. Matias, M. Ramon,and W. Reece, New physics reach of thedecay mode ¯ B → ¯ K ∗ (cid:96) + (cid:96) − , JHEP (2010)001, [ ].7. W. Altmannshofer et. al. , Symmetries andAsymmetries of B → K ∗ µ + µ − Decays in theStandard Model and Beyond , JHEP (2009) 019, [ arXiv:0811.1214 ].8. A. Khodjamirian, T. Mannel, A. Pivovarov,and Y.-M. Wang, Charm-loop effect in B → K ( ∗ ) (cid:96) + (cid:96) − and B → K ∗ γ , .9. C. Bobeth, G. Hiller, and G. Piranishvili, CP Asymmetries in ¯ B → ¯ K ∗ ( → ¯ Kπ )¯ (cid:96)(cid:96) andUntagged ¯ B s , B s → φ ( → K + K − )¯ (cid:96)(cid:96) Decaysat NLO , JHEP (2008) 106,[ arXiv:0805.2525arXiv:0805.2525