FCNC B s and Λ b transitions: Standard Model versus a single Universal Extra Dimension scenario
aa r X i v : . [ h e p - ph ] F e b FCNC B s and Λ b transitions:Standard Model versus a single Universal Extra Dimension scenario P. Colangelo a , F. De Fazio a , R. Ferrandes a,b , T.N. Pham c a Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy b Dipartimento di Fisica, Universit´a di Bari, Italy c Centre de Physique Th´eorique,´Ecole Polytechnique, CNRS, 91128 Palaiseau, France
We study the FCNC B s → φγ, φν ¯ ν and Λ b → Λ γ, Λ ν ¯ ν transitions in the Standard Model and in ascenario with a single Universal Extra Dimension. In particular, we focus on the present knowledgeof the hadronic uncertainties and on possible improvements. We discuss how the measurements ofthese modes can be used to constrain the new parameter involved in the extra dimensional scenario,the radius R of the extra dimension, completing the information available from B-factories. Therates of these b → s induced decays are within the reach of new experiments, such as LHCb. PACS numbers: 12.60.-i, 13.25.Hw
I. INTRODUCTION
The heavy flavour Physics programmes at the hadronfacilities, the Tevatron at Fermilab and the Large HadronCollider (LHC) at CERN, include the analysis of heavyparticles, in particular B s and Λ b , which cannot be pro-duced at the e + e − factories operating at the peak ofΥ(4 S ) [1]. These programmes involve as an importanttopic the study of processes induced by Flavour Chang-ing Neutral Current (FCNC) b → s transitions, sincethey provide us with tests of the Standard Model (SM)and constraints for New Physics (NP) scenarios. Theobservation of B s − B s oscillations at the Tevatron [2]represents a great success in this Physics programme.Considering the experimental situation, data are avail-able at present for several b → s FCNC B meson de-cays. Together with the inclusive radiative B → X s γ branching ratio, the rates of a few exclusive radiativemodes, both for charged B : B ± → K ∗± γ , K (1270) ± γ and K ∗ (1430) ± γ , both for neutral B : B → K ∗ γ and K ∗ (1430) γ have been measured [3]. Moreover, thebranching fractions of B ± → K ( ∗ ) ± ℓ + ℓ − and B → K ( ∗ )0 ℓ + ℓ − , with ℓ = e, µ have been determined by theBelle and BaBar collaborations, which have also providedus with preliminary measurements of the lepton forward-backward asymmetry [4, 5] together with the K ∗ longi-tudinal helicity fraction in B → K ∗ ℓ + ℓ − [6]. Further-more, upper bounds for B ( B → K ( ∗ ) ν ¯ ν ) have been es-tablished [3]. Even not considering non leptonic b → s penguin induced B decays, the interpretation of which isnot straightforward, this wealth of measurements has al-ready severely constrained the parameter space of variousnon standard scenarios. Other tight information couldbe obtained by more precise data in the modes already observed, as well as by the measurement of the branch-ing fractions and of the spectra of B → K ( ∗ ) ν ¯ ν , and of B → K ( ∗ ) τ + τ − in which the τ polarization asymmetriesare sensitive to Physics beyond SM.Other processes induced by b → s transition involvethe hadrons B s , B c and Λ b , the decay modes of whichare harder to be experimentally studied, e.g. due totheir smaller production rate in b quark hadronizationwith respect to B mesons. However, at the hadron col-liders, in particular at LHC, the number of produced par-ticles is so large that even these processes are expectedto be observable, so that their study can contribute toour understanding of Physics of rare transitions withinand beyond the Standard Model description of elemen-tary interactions.In this paper we study a few b → s FCNC induced B s and Λ b decays, in particular B s → φγ, φν ¯ ν andΛ b → Λ γ, Λ ν ¯ ν , in SM and in a New Physics scenariowhere a single universal extra dimension is considered,the Appelquist, Cheng and Dobrescu (ACD) model [7].Models with extra dimensions have been proposed as vi-able candidates to solve some problems affecting SM [8],and within this class of NP models the ACD scenariowith a single extra dimension is worth investigating dueto its appealing features. Here, we do not discuss in de-tails the various aspects of the model, which have beenworked out in [7] and are summarized in [9]-[11]. Weonly recall that the model consists in a minimal exten-sion of SM in 4 + 1 dimensions, with the extra dimen-sion compactified to the orbifold S /Z . The fifth co-ordinate y runs from 0 to 2 πR , with y = 0, y = πR fixed points of the orbifold, and R the radius of the orb-ifold which represents a new physical parameter. All thefields are allowed to propagate in all dimensions, there-fore the model belongs to the class of universal extra di-mension scenarios. The four-dimensional description in-cludes SM particles, corresponding to the zero modes offields propagating in the compactified extra dimension,together with towers of Kaluza-Klein (KK) excitationscorresponding to the higher modes. Such fields are im-posed to be even under parity transformation in the fifthcoordinate P : y → − y . Fields which are odd under P propagate in the extra dimension without zero modesand correspond to particles with no SM partners.In addition to 5-d bulk terms, the Lagrangian of theACD model may also include boundary terms which rep-resent additional parameters of the theory and get renor-malized by bulk interactions, although being volume sup-pressed. A simplifying assumption is that such boundaryterms vanish at the cut-off scale, so that a minimal Uni-versal Extra Dimension model can be defined in whichthe only new parameter with respect to the StandardModel is the radius R of the extra dimension, an impor-tant feature as far as the phenomenological investigationof the model is concerned [12].The masses of KK particles depend on the radius R ofthe extra dimension. For example, the masses of the KKbosonic modes are given by [7], [9]-[11]: m n = m + n R n = 1 , , . . . (1.1)For small values of R these particles, being more andmore massive, decouple from the low energy regime.Thus, within this model candidates for dark matter areavailable, the Kaluza-Klein (KK) excitations of the pho-ton or of the neutrinos with KK number n = 1 [13, 14].This is related to another property of the ACD model,the conservation of the KK parity ( − j , with j the KKnumber [15]. KK parity conservation implies the absenceof tree level contributions of Kaluza Klein states to pro-cesses taking place at low energy, µ ≪ /R , a featurewhich permits to establish a bound: 1 /R ≥ − /R ≥
600 GeV assuming for the Higgs a mass of 115GeV, a constraint on the extra dimension radius whichcan be relaxed as low as 1 /R ≥
300 GeV with increasingHiggs mass [17]. Other bounds can be derived invok-ing cosmological arguments; in Ref.[18] it was found thatthe region of parameters preferred by cosmological con-straints in a single UED scenario corresponds to a Higgsmass between 185 and 245 GeV, to a lightest KK parti-cle between 810 and 1400 GeV and to maximal spittingbetween the first KK modes of 320 GeV; such constraintscome from the diffuse photon spectrum and from the nullsearches of exotic stable charged particles, assuming thatthe first excitation of the graviton is the lightest KK par-ticle. These bounds, however, are different if the lightestKK particle is not the first excitation of the graviton.The fact that KK excitations can influence processesoccurring at loop level suggests that FCNC transitions are particularly suitable for constraining the extra dimen-sion model, providing us with many observables sensitiveto the compactification radius R . For this reason, in[9, 10] the effective Hamiltonian governing b → s transi-tions was derived, and inclusive B → X s γ , B → X s ℓ + ℓ − , B → X s ν ¯ ν transitions, together with the B s ( d ) mix-ing, were studied. In particular, it was found that B ( B → X s γ ) allowed to constrain 1 /R ≥
250 GeV, abound updated by a more recent analysis based on theNNLO value of the SM Wilson coefficient c (defined inthe next section) [19] and on new experimental data to1 /R ≥
600 GeV at 95% CL, or to 1 /R ≥
330 GeV at 99%CL [20]. Concerning the exclusive modes B → K ∗ γ , B → K ( ∗ ) ℓ + ℓ − , B → K ( ∗ ) ν ¯ ν and B → K ( ∗ ) τ + τ − , itwas argued that the uncertainty related to the hadronicmatrix elements does not obscure the sensitivity to thecompactification parameter R , and that current data, inparticular the decay rates of B → K ∗ γ and B → K ∗ ℓ + ℓ − ( ℓ = e, µ ) can provide the bound 1 /R ≥ −
400 GeV[11, 21].In case of B s , the mode B s → µ + µ − was recognizedas the process receiving the largest enhancement with re-spect to the SM prediction; however, since the predictedbranching fraction is O (10 − ), the measurement is verychallenging [9]. A modest enhancement was found in B s → γγ [22], in B s → φℓ + ℓ − and in B s → ℓ + ℓ − γ [23].In the case of Λ b , for 1 /R ≃
300 GeV a sizeable effectof the extra dimension was found in Λ b → Λ ℓ + ℓ − [24],analogously to what obtained in B → K ( ∗ ) ℓ + ℓ − .For the modes B s → φν ¯ ν , Λ b → Λ γ and Λ b → Λ ν ¯ ν analyzed in this paper, no data are available at present,while for B s → φγ a first measurement of the branch-ing fraction has been recently carried out. Our aim isto work out a set of predictions for various observablesin the Standard Model, making use also of informationobtained at the B factories. Moreover, we consider theseprocesses in the extra dimension scenario studying thedependence on R which could provide us with furtherways to constrain such a parameter. To be conservative,we consider the range of 1 /R starting from 1 /R ≥ b → sγ and b → sν ¯ ν decays in SM and in theACD model. The case of B s is considered in Section III,while the Λ b transitions are the subject of Section IV.The last Section is devoted to the conclusions. II. b → sγ AND b → sν ¯ ν EFFECTIVEHAMILTONIANS
In the Standard Model the b → sγ and b → sν ¯ ν tran-sitions are described by the effective ∆ B = −
1, ∆ S = 1Hamiltonians H b → sγ = 4 G F √ V tb V ∗ ts c eff ( µ ) O ( µ ) (2.1)and H b → sν ¯ ν = G F √ α ( M W )2 π sin ( θ W ) V tb V ∗ ts η X X ( x t ) O L = c L O L (2.2)involving the operators O = e π (cid:2) m b (¯ s L σ µν b R ) + m s (¯ s R σ µν b L ) (cid:3) F µν (2.3)and O L = ¯ sγ µ (1 − γ ) b ¯ νγ µ (1 − γ ) ν , (2.4)respectively. Eq.(2.1) describes magnetic penguin dia-grams, while (2.2) is obtained from Z penguin and boxdiagrams, with the dominant contribution correspondingto a top quark intermediate state. G F is the Fermi con-stant and V ij are elements of the CKM mixing matrix;moreover, b R,L = 1 ± γ b , α is the electromagnetic con-stant, θ W the Weinberg angle and F µν denotes the elec-tromagnetic field strength tensor. The function X ( x t )( x t = m t M W , with m t the top quark mass) has been com-puted in [25] and [26]; the QCD factor η X is close to one,so that we can put it to unity [27, 28].In eq. (2.1) we use an effective coefficient c eff whichturns out to be scheme independent and takes into ac-count the mixing between the operators O with O un-der renormalization group evolution [29].In the ACD model no operators other than those in(2.1) and (2.2) contribute to b → sγ and b → sν ¯ ν transi-tions: the model belongs to the class of Minimal FlavourViolating models, where effects beyond SM are only en-coded in the Wilson coefficients of the effective Hamil-tonian [9, 10]. KK excitations only modify c and c L inducing a dependence on the compactification radius R .For large values of 1 /R , due to decoupling of massiveKK states, the Wilson coefficients reproduce the Stan-dard Model values, so that the SM phenomenology isrecovered. As a general expression, the Wilson coeffi-cients are represented by functions F ( x t , /R ) generaliz-ing their SM analogues F ( x t ): F ( x t , /R ) = F ( x t ) + ∞ X n =1 F n ( x t , x n ) , (2.5)with x n = m n M W and m n = nR . A remarkable result isthat the sum over the KK contributions in (2.5) is finiteat the leading order (LO) in all cases as a consequence ofa generalized GIM mechanism [9, 10]. When R → F ( x t , /R ) → F ( x t ) in that limit. For 1 /R of the order of a few hundreds of GeV the co-efficients differ from their Standard Model value; in par-ticular c is suppressed, as one can infer considering theexpressions collected in [9]-[11] together with the function X ( x t , /R ). Therefore, the predicted widths and spectraare modified with respect to SM. In case of exclusive de-cays, it is important to study if this effect is obscured bythe hadronic uncertainties, a discussion that we presentin the following two Sections for B s and Λ b , respectively. III. B s → φγ AND B s → φν ¯ ν DECAYS
The description of the decay modes B s → φγ and B s → φν ¯ ν involves the hadronic matrix elements of theoperators appearing in the effective Hamiltonians (2.1)-(2.2). In case of B s → φγ the matrix element of O canbe parameterized in terms of three form factors: < φ ( p ′ , ǫ ) | ¯ sσ µν q ν (1 + γ )2 b | B s ( p ) > = iǫ µναβ ǫ ∗ ν p α p ′ β T ( q ) ++ h ǫ ∗ µ ( M B s − M φ ) − ( ǫ ∗ · q )( p + p ′ ) µ i T ( q )+ ( ǫ ∗ · q ) " q µ − q M B s − M φ ( p + p ′ ) µ T ( q ) , (3.1)where q = p − p ′ is the momentum of the photon and ǫ the φ meson polarization vector. At zero value of q the condition T (0) = T (0) holds, so that the B s → φγ decay amplitude involves a single hadronic parameter, T (0). On the other hand, the matrix element of O L canbe parameterized as follows: < φ ( p ′ , ǫ ) | ¯ sγ µ (1 − γ ) b | B s ( p ) > = ǫ µναβ ǫ ∗ ν p α p ′ β V ( q ) M B s + M φ − i h ǫ ∗ µ ( M B s + M φ ) A ( q ) − ( ǫ ∗ · q )( p + p ′ ) µ A ( q ) M B s + M φ − ( ǫ ∗ · q ) 2 M φ q (cid:0) A ( q ) − A ( q ) (cid:1) q µ i (3.2)with a relation holding among the form factors A , A and A : A ( q ) = M B s + M φ M φ A ( q ) − M B s − M φ M φ A ( q ) (3.3)together with A (0) = A (0).The form factors represent a source of uncertaintyin predicting the B s decay rates we are considering.The other parameters are fixed to m b = 4 . ± . M B s = 5 . ± . τ B s = (1 . ± . × − s, V tb = 0 .
999 and V ts = 0 . ± . A. B s → φγ From the effective Hamiltonian (2.1), together with(2.3) and the matrix element (3.1), it is straightforwardto calculate the expression of the B s → φγ decay rate:Γ( B s → φγ ) = α (0) G F π | V tb V ∗ ts | ( m b + m s ) | c eff | [ T (0)] × M B s − M φ M B s ! . (3.4)This expression is useful to relate the branching fraction B ( B s → φγ ) to the measured value of B ( B d → K ∗ γ ),since B ( B s → φγ ) = T B s → φ (0) T B d → K ∗ (0) ! (cid:18) M B d M B s (cid:19) × M B s − M φ M B d − M K ∗ ! τ B s τ B d B ( B d → K ∗ γ ) (3.5)where we have indicated which hadronic matrix elementsthe form factors refer to. Eq.(3.5) shows that, in additionto measured quantities, the crucial quantity to predict B ( B s → φγ ) is the SU (3) F breaking parameter r definedby T B s → φ (0) T B d → K ∗ (0) = 1 + r , (3.6)as shown in Fig. 1 for positive values of r , using B ( B d → K ∗ γ ) = (4 . ± . × − and τ B d = (1 . ± . × − s [3], and combining in quadrature the uncertain-ties of the various quantites in (3.5). Detailed analysesof the range of values within which r can vary are notavailable, yet. Using r = 0 . ± .
006 estimated byLight Cone Sum Rules (LCSR) [30] we obtain the rangebounded by the dashed vertical lines in Fig. 1, whichallows us to predict: B ( B s → φγ ) = (4 . ± . × − . (3.7)Notice that the chosen value of r is smaller than ananalogous quantity parameterizing the ratio of leptonicconstants f B s /f B d , estimated as: r = 0 . ± .
03 [31].A compatible value of B ( B s → φγ ) is obtained in SMusing the form factor T (0) = (17 . ± . × − deter-mined by LCSR [30], although with a larger uncertainty: B ( B s → φγ ) = (4 . ± . × − . (3.8)These results must be compared to the measurement: B ( B s → φγ ) exp = (5 . +1 . − . ( stat ) +1 . − . ( syst )) × − (3.9)recently carried out by Belle Collaboration in a run atthe Υ(5 S ) peak [32]; this experimental result is affectedby a statistical and systematical uncertainty which areexpected to be reduced in the near future. r BR H B s ® Φ Γ L ´ r BR H B s ® Φ Γ L ´ FIG. 1: B ( B s → φγ ) as a function of the SU (3) F breakingparameter r in the ratio of B s → φ vs B d → K ∗ form factors T (0) (3.6). The dashed vertical lines show the range of r obtained by LCSR. In the single extra dimension scenario the modifica-tion of the Wilson coefficient c eff , corresponding to thevariation of the compactification radius R , changes theprediction for B ( B s → φγ ), as shown in Fig. 2 where weplot the branching ratio versus 1 /R . The width of theband reflects the uncertainty on the form factor quotedabove, as well as on the parameters m b and V ts . As for c eff , it is affected by the uncertainty due to the higherorder corrections in the ACD model. The inclusion ofnext-to-leading order QCD corrections would require thecontribution of two-loop diagrams involving KK gluoncorrections which at present is not known. One couldestimate the size of this correction by considering the ef-fect of the variation of the matching scale, as done in[20], where it was found that changing this scale between80 and 320 GeV the value of B ( B → X s γ ) is affected byan uncertainty not exceeding | +8 − % for 1 /R in the rangebetween 200 and 1500 GeV. For the exclusive modes con-sidered here, such an uncertainty, expected to be similar,must be combined in quadrature with the other errors,in particular with the hadronic uncertainty which is atpresent the largest one.We observe that for low values of 1 /R the branchingratio is reduced with respect to the SM expectation: at,e.g., 1 /R = 300 GeV B ( B s → φγ ) is smaller by 35%as a consequence of the lower value of c eff in the ACDmodel. This effect was already noticed in the analysisof B → X s γ and B → K ∗ γ . For higher values of 1 /R the lowering of the branching fraction is obscured by thehadronic uncertainty due to the form factor. B. B s → φν ¯ ν Analogously to the case of B → K ∗ ν ¯ ν [33], for thisdecay mode it is convenient to separately consider the
200 400 600 800 10000123456 (cid:144) R H GeV L BR H B s ® Φ Γ L ´ FIG. 2: B ( B s → φγ ) vs the inverse radius 1 /R of the compact-ified extra dimension in the ACD model. The Belle measure-ment is: B ( B s → φγ ) exp = (5 . +1 . − . ( stat ) +1 . − . ( syst )) × − ). missing energy distributions for longitudinally (L) andtransversely ( ± ) polarized φ mesons: d Γ L dx = 3 | c L | π | ~p ′ | M φ (cid:2) ( M B s + M φ )( M B s E ′ − M φ ) A ( q ) − M B s M B s + M φ | ~p ′ | A ( q ) (cid:3) , (3.10)and d Γ ± dx = 3 | ~p ′ | q π | c L | (cid:12)(cid:12) M B s | ~p ′ | M B s + M φ V ( q ) ∓ ( M B s + M φ ) A ( q ) (cid:12)(cid:12) . (3.11)In (3.10)-(3.11) x = E miss /M B s , with E miss the energyof the neutrino pair (missing energy); q is the momentumtransferred to the neutrino pair, ~p ′ and E ′ the φ three-momentum and energy in the B s meson rest frame, andthe sum over the three neutrino species has been carriedout.The missing energy distributions for polarized and un-polarized φ are depicted in Fig. 3 for 1 /R = 500 GeVand in the Standard Model. They are obtained using B s → φ form factors determined by LCSR [30]: A ( q ) = A (0)1 − q m A A ( q ) = r A − q m A + r A (cid:18) − q m A (cid:19) (3.12) V ( q ) = r V − q m R + r V − q m V where A (0) = 0 . ± .
030 and m A = 36 .
54 GeV ; r A = − . r A = 0 . A (0) = r A + r A = 0 . ± x d BR L , T H B s ® Φ ΝΝ L (cid:144) dx ´ x d BR H B s ® Φ ΝΝ L (cid:144) dx ´ FIG. 3: Missing energy distribution in B s → φν ¯ ν (upperpanel) for longitudinally (left curves) and transversally polar-ized φ meson (right curves). The sum over the three neutrinospecies is understood. The continuous lines bound the regioncorresponding to SM, the dashed lines the region correspond-ing to 1 /R = 500 GeV. In the lower panel the missing energydistribution for unpolarized φ is depicted (same notations). . m A = 48 .
94 GeV ; r V = 1 . r V = − . V (0) = r V + r V = 0 . ± . m R = 5 .
32 GeV, m V =39 .
52 GeV . The effect of the extra dimension consistsin a systematic increase of the various distributions inthe full range of missing energy; however, the hadronicuncertainty needs to be substantially reduced in order toclearly disentangle deviations from the Standard Modelpredictions which correspond to this value of 1 /R .In the SM the branching ratio is predicted: B ( B s → φν ¯ ν ) = (1 . ± . × − (3.13)therefore this mode is within the reach of future experi-ments, at least as far as the number of produced eventsis concerned, although the observation of a final state in-volving a neutrino-antineutrino pair is a challenging task,as observed also in [34]. The dependence of B ( B s → φν ¯ ν )on 1 /R is depicted in Fig. 4, where it is shown that thebranching fraction increases for low values of 1 /R : forexample, at 1 /R = 300 GeV there is a 23% enhancementwith respect to the SM expectation. For larger valuesof 1 /R the Standard Model prediction is recovered, thedependence on 1 /R being obscured by form factor uncer-tainties.
200 400 600 800 100001234 (cid:144) R H GeV L BR H B s ® Φ ΝΝ L ´ FIG. 4: B ( B s → φν ¯ ν ) vs the compactification parameter 1 /R in the ACD model. IV. Λ b → Λ γ AND Λ b → Λ ν ¯ ν DECAYS
In case of Λ b → Λ transitions the hadronic matrixelements of the operators O and O L in eqs. (2.1) and(2.2) involve a larger number of form factors. As a matterof fact, the various matrix elements can be written asfollows: < Λ( p ′ , s ′ ) | ¯ siσ µν q ν b | Λ b ( p, s ) > = ¯ u Λ (cid:2) f T ( q ) γ µ + if T ( q ) σ µν q ν + f T ( q ) q µ (cid:3) u Λ b (4.1) < Λ( p ′ , s ′ ) | ¯ siσ µν q ν γ b | Λ b ( p, s ) > = ¯ u Λ (cid:2) g T ( q ) γ µ γ + ig T ( q ) σ µν q ν γ + g T ( q ) q µ γ (cid:3) u Λ b (4.2) < Λ( p ′ , s ′ ) | ¯ sγ µ b | Λ b ( p, s ) > = ¯ u Λ (cid:2) f ( q ) γ µ + if ( q ) σ µν q ν + f ( q ) q µ (cid:3) u Λ b (4.3) < Λ( p ′ , s ′ ) | ¯ sγ µ γ b | Λ b ( p, s ) > = ¯ u Λ (cid:2) g ( q ) γ µ γ + ig ( q ) σ µν q ν γ + g ( q ) q µ γ (cid:3) u Λ b , (4.4)with u Λ and u Λ b the Λ and Λ b spinors.At present, a determination of all the form factors in(4.1)-(4.4) is not available. However, it is possible toinvoke heavy quark symmetries for the hadronic matrixelements involving an initial spin= heavy baryon com-prising a single heavy quark Q and a final light baryon;the heavy quark symmetries reduce to two the numberof independent form factors. As a matter of fact, in the infinite heavy quark limit m Q → ∞ and for a genericDirac matrix Γ one can write [35]: < Λ( p ′ , s ′ ) | ¯ s Γ b | Λ b ( p, s ) > = ¯ u Λ ( p ′ , s ′ ) (cid:8) F ( p ′ · v )+ v F ( p ′ · v ) (cid:9) Γ u Λ b ( v, s )(4.5)where v = pM Λ b is the Λ b four-velocity. The form fac-tors F and F depend on p ′ · v = M b + M − q M Λ b (forconvenience we instead consider them as functions of q through this relation). The expression (4.5) for a genericmatrix element not only shows that the number of inde-pendent form factors is reduced to two, but also that suchform factors are universal, since the same functions F and F describe both Λ b , both Λ c decays into Λ, envisag-ing the possibility of relating these two kind of processesif finite quark mass effects, in particular in the charmcase, are small. The relations between the form factorsin (4.1)-(4.4) and the universal functions in (4.5): f = g = f T = g T = F + M Λ M Λ b F f = g = f = g = F M Λ b f T = g T = q F M Λ b (4.6) f T = − ( M Λ b − M Λ ) F M Λ b g T = ( M Λ b + M Λ ) F M Λ b are strictly valid at momentum transfer close to the max-imum value q ≃ q max = ( M Λ b − M Λ ) . However, weextend their validity to the whole phase space, an as-sumption which introduces a model dependence in thepredictions.A determination of F and F has been obtained bythree-point QCD sum rules in the m Q → ∞ limit [36]. Inthe following we use the expressions for the functions F , F obtained by updating some of the parameters used in[36]. In particular, we use the PDG value of the Λ b mass: M Λ b = 5 . ± .
009 GeV [3]. Moreover, we fix the massdifference ∆ Λ b = M Λ b − m b , together with a constant f Λ b parameterizing a vacuum-current Λ b matrix element, tothe values computed in the infinite heavy quark masslimit in [37]: ∆ Λ b = 0 . ± . f Λ b = (2 . ± . × − GeV . Using these inputs, the obtained formfactors F , can be parameterized by the expressions: F , ( q ) = F , (0)1 + a , q + b , q (4.7)with F (0) = 0 . ± . a = − . − , b = − . × − GeV − , and F (0) = − . ± . a = − .
069 GeV − , b = 1 . × − GeV − .A few remarks are in order. First, the q (or p ′ · v )dependence of the two form factors turns out to be dif-ferent, at odds with what has been assumed in variousanalyses where the same q dependence for F and F is argued [38]. Second, while F and F are monotonicin q , their dependence is different from the simple ormultiple pole behaviour assumed in other analyses, withpole mass fixed by vector meson dominance (VMD) ar-guments. Finally, F is different from zero. This is no-ticeable, even though in various decay rates the termsinvolving F appear together with the suppressing fac-tor 1 /M Λ b . An analysis of the helicity amplitudes inΛ c → Λ ℓν carried out by the CLEO collaboration mea-suring various angular distributions in this decay processdemonstrated that F is not vanishing [39]. The result,based on the assumption that the heavy quark limit canbe applied in the charm case and on the hypothesis theboth F and F have the same dipolar q dependence, is: F ( q ) F ( q ) = − . ± . ± .
04 if the pole mass is fixed to M pole = M D ∗ s , or F ( q ) F ( q ) = − . ± . ± .
04 in corre-spondence to a fitted pole mass M pole = 2 . ± . ± . c → Λ ℓν transitions to determine the uni-versal form factors F and F , with the aim of employingthem to describe several Λ b transitions, was proposed in[40]. However, in such a proposal the q dependence ofthe two form factors must be assumed. If F , have botha dipolar q dependence, using the ratio F F determinedby the CLEO collaboration and the experimental valueof B (Λ c → Λ ℓν ), one gets: F ( q max ) = 1 . ± .
2. Fol-lowing the strategy proposed in [40] it is necessary toextrapolate F to the full kinematical range allowed inΛ b → Λ transitions by the assumed q dependence, andapply the result to Λ b → Λ γ and Λ b → Λ ν ¯ ν . We refrainfrom applying such a procedure, since the momentumrange where the extrapolation is required is very wideand, more importantly, the assumption that finite charmmass effects are negligible in case of Λ c should be justi-fied.It is worth mentioning that in the m b → ∞ limit andin the large recoil regime for the light hadron ( q ≃ E = v · p ′ → ∞ ), the relation F (0) F (0) = − M Λ E (4.8)can be derived using the Large Energy Effective The-ory/SCET framework at the leading order in the 1 m b and1 E expansion [41]. Direct calculations of F , in this limithave not been done, yet; the model (4.7) gives a ratio ofform factors compatible with (4.8).Admittedly, the knowledge of Λ b form factors deservesa substantial improvement; in the meanwhile, we use inour analysis the form factors in (4.7) stressing that theuncertainties we attach to the various predictions only take into account the errors of the parameters of the ex-pressions used for F and F . A. Λ b → Λ γ The expression of the Λ b → Λ γ decay rate:Γ(Λ b → Λ γ ) = α (0) G F | V tb V ∗ ts | m b π | c eff | M b × − M M b ! (cid:18) F (0) + M Λ M Λ b F (0) (cid:19) (4.9)is useful to relate also this mode to the observed B d → K ∗ γ transition: B (Λ b → Λ γ ) = F (0) T B d → K ∗ (0) ! (cid:18) M Λ M Λ b F (0) F (0) (cid:19) × (cid:18) M B d M Λ b (cid:19) M b − M M B d − M K ∗ ! τ Λ b τ B d B ( B d → K ∗ γ ) . (4.10)Using τ Λ b = (1 . ± . × − s [3], together withthe ratio of form factors F (0) F (0) = − . ± .
06 and F (0) T B d → K ∗ (0) = 1 . ± . T in [30], we predict: B (Λ b → Λ γ ) =(3 . ± . × − . The same result with a larger un-certainty comes from using T determined in [42]. Using(4.9) and the form factors (4.7), we get: B (Λ b → Λ γ ) = (3 . ± . × − . (4.11)The result suggests that this process is within the reachof LHC experiments. As for the effect of the extra dimen-sion in modifying the decay rate, in Fig. 5 we show how B (Λ b → Λ γ ) depends on 1 /R . Analogously to B → K ∗ γ and B s → φγ transitions, the branching fraction is sup-pressed for low values of 1 /R , being 30% smaller for1 /R = 300 GeV. B. Λ b → Λ ν ¯ ν Even for this mode it is useful to consider the missingenergy distribution in the variable x = E miss /M Λ b with E miss the energy of the neutrino - antineutrino pair: d Γ dx = | c L | | ~p ′ | π nh(cid:0) M b − M (cid:1) + q (cid:0) M b + M − q (cid:1)i × h F + M Λ M Λ b F i
200 400 600 800 1000012345 (cid:144) R H GeV L BR H L b ® L Γ L ´ FIG. 5: B (Λ b → Λ γ ) vs /R . The uncertainty shown by thedark band is mainly due to the errors on the form factors ofthe model (4.7). + h q (cid:0) M b − M (cid:1) − q (cid:0) M b + M + q (cid:1)i(cid:16) F M Λ b (cid:17) +6 M Λ q (cid:0) M b − M + q (cid:1)h F + M Λ M Λ b F i F M Λ b o . (4.12)In Fig. 6 we plot such a distribution in the SM andfor 1 /R = 500 GeV, showing the differences correspond-ing to such a value of the compactification radius. Thebranching ratio, which in SM is expected to be [43]: B (Λ b → Λ ν ¯ ν ) = (6 . ± . × − (4.13)depends on 1 /R as shown in Fig. 7. When 1 /R decreasesthe branching ratio increases of about 20% for 1 /R ≃ x d BR H L b ® L ΝΝ L (cid:144) dx ´ FIG. 6: Missing energy distribution for Λ b → Λ ν ¯ ν in theStandard Model (continuous lines) and for 1 /R = 500 GeV(dashed lines).
200 400 600 800 100005101520 (cid:144) R H GeV L BR H L b ® L ΝΝ L ´ FIG. 7: B (Λ b → Λ ν ¯ ν ) versus 1 /R . V. CONCLUSIONS
We have studied how a single universal extra dimen-sion could have an impact on several loop induced B s andΛ b decays. The analysis of processes involving these twoparticles will be among the main topics in the investiga-tions at the hadron colliders, especially at LHC. Since afew of the modes we have considered are difficult to recon-struct in a ”hostile” environment represented by hadroncollisions, we believe that the predictions we have workedout are useful to elaborate the measurement strategies.From the theoretical point of view, we have found thathadronic uncertainties due to the form factors in exclu-sive decays are not large in case of B s , where SU (3) F symmetry is also useful to exploit other measurementscarried out at the B factories. As for Λ b , the situation ismore uncertain. Calculations made in the infinite heavyquark limit should be corroborated by the analysis offinite heavy quark mass effects, and the various depen-dences on the momentum transfer should be confirmed.Such analyses deserve a dedicated effort. Acknowledgments
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