b\to sν\barν decay in the MSSM: Implication of b\to sγat large tan beta
aa r X i v : . [ h e p - ph ] J a n TU-796 b → sν ¯ ν decay in the MSSM: Implication of b → sγ atlarge tan β Youichi Yamada
Department of Physics, Tohoku University, Sendai 980-8578, Japan
Abstract
The decay b → sν ¯ ν is discussed in the minimal supersymmetric standard modelwith general flavor mixing for squarks, at large tan β . In this case, in addition to thechargino loop contributions which were analyzed in previous studies, tan β -enhancedcontributions from the gluino and charged Higgs boson loops might become sizablecompared with the standard model contribution, at least in principle. However,it is demonstrated that the experimental bounds on the new physics contributionsto the radiative decay b → sγ should strongly constrain these contributions to b → sν ¯ ν , especially on the gluino contribution. We also briefly comment on apossible constraint from the B s → µ + µ − decay. PACS: 13.20.He, 12.60.Jv 1
Introduction
Recently, there has been significant experimental improvements in the measurements offlavor-changing neutral current (FCNC) processes of B mesons at B factories and Teva-tron. For the b → s transition, experimental data for b → sγ and b → sl + l − ( l = e, µ )decays, B s − ¯ B s oscillation, and B s → µ + µ − decay have already started to constrainpossible contributions from new physics beyond the standard model.Here we focus our attention to one of the b → s processes, the decay into neutrinopairs [1, 2], b → sν ¯ ν. (1)It is known that the decays of the B mesons induced by the partonic process (1), especiallythe inclusive branching ratio BR( ¯ B → X s ν ¯ ν ), have small theoretical uncertainty due tothe absence of photonic penguin and strong suppression of light quark contributions. Onthe other hand, experimental search of the decay (1) is a hard task. At present, only theupper bounds are known for both inclusive [3] and exclusive [4] branching ratios, at 90%C.L., X ν Br( ¯ B → X s ν ¯ ν ) < . × − , X ν Br( B + → K + ν ¯ ν ) < . × − , X ν Br( ¯ B → K S ν ¯ ν ) < . × − , X ν Br( ¯ B → K ∗ ν ¯ ν ) < . × − , X ν Br( B + → K ∗ + ν ¯ ν ) < . × − , (2)which are still one order of magnitude larger than the standard model predictions for theinclusive [5] and exclusive [6] modes, X ν Br( ¯ B → X s ν ¯ ν ) SM = (3 . ± . × − , X ν Br( ¯ B → Kν ¯ ν ) SM = (3 . +1 . − . ) × − , X ν Br( ¯ B → K ∗ ν ¯ ν ) SM = (1 . +0 . − . ) × − . (3)Future upgrades of the B factories [7] will extend the search region for the exclusive decays.For example, Br( B + → K + ν ¯ ν ) around the level of the standard model prediction (3) isexpected to be measured at the precision of 20% with integrated luminosity 50–100 ab − .On the other hand, a future e + e − collider running on the Z -boson resonance (GIGA-Z)has a potential [8] to produce very large number of Z → b ¯ b events, and possibility togreatly improve previous studies of the inclusive modes [3] at the LEP I, to measure theinclusive branching ratio. 2n this paper, we consider the decay (1) in the framework of the minimal supersym-metric standard model (MSSM) [9] with general flavor mixing of squarks, and study thecontributions of new particles, namely the supersymmetric (SUSY) particles and Higgsbosons. In cases where the value of tan β , the ratio of the vacuum expectation valuesof two Higgs boson doublets in the MSSM, is not much larger than unity, it is shown[10, 11] that the chargino-squark loops give main part of the new physics contributionsto the decay, and may become sizable when large flavor mixing is present in the left-rightmixing part of the up-type squark mass matrix. Note that this is also the case for theSUSY contributions to the related decays K → πν ¯ ν [12, 13].At large tan β , say similar to or larger than m t /m b ∼
40, the MSSM loop contributionsother than charginos might become also important, at least in principle. For example,gluino-squark loop contributions are generated by tan β -enhanced large left-right mixingof down-type squarks. When, in addition, sizable mixing between down-type squarks inthe second and third generations are present, gluino contribution might become sizable.It is also possible that, as explained later, charged Higgs boson might give sizable loopcontributions due to the flavor-changing effective Higgs-quark couplings, generated by O (tan β ) SUSY loop corrections, as pointed out in Ref. [14] for the K → πν ¯ ν decays.However, parameters in the SUSY and Higgs sectors should receive stringent constraintsfrom existing measurements of the FCNC processes, which might suppress possible mag-nitudes of their contributions to b → sν ¯ ν . In this paper, we will present a rough estimateof the possible constraints from the decay b → sγ , by showing correlations between thenew physics contributions to the Wilson coefficients for b → sν ¯ ν and those for b → sγ ,for each SUSY/Higgs sector separately. We will also comment on the implication of the B s → µ + µ − decay to the Higgs boson contributions.The paper is organized as follows. In Sec. 2, we present basic formulas for the analysisof the b → sν ¯ ν decay in the MSSM. In Sec. 3, numerical results for the new physicscontributions in the MSSM to b → sν ¯ ν are presented as correlations with those to b → sγ for each new physics sector. An additional constraint from the B s → µ + µ − decay on theHiggs boson contributions is briefly commented in Sec. 4. Finally, conclusion is given inSec. 5. b → sν ¯ ν decay in the MSSM The b → sν ¯ ν decay is described by the effective Hamiltonian, in the notation of Ref. [6], H eff = − G F √ K ∗ ts K tb [ C ν O L + C ′ ν O R ] , (4)where K ij is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Here the relevant operatorsare O L = α π (¯ s L γ µ b L )(¯ ν L γ µ ν L ) , (5)3 R = α π (¯ s R γ µ b R )(¯ ν L γ µ ν L ) . (6)The inclusive branching ratio is then expressed in terms of the Wilson coefficients ( C ν , C ′ ν ) in Eq. (4) as [11] X ν Br( ¯ B → X s ν ¯ ν ) ∼ N ν α π Br( ¯ B → X c e ¯ ν e ) | K tb K ∗ ts | | K cb | ( | C ν | + | C ′ ν | ) , (7)up to the QCD corrections and O ( m c /m b ) corrections to the semileptonic decays ¯ B → X c e ¯ ν e . Interference between C ν and C ′ ν appears in the branching ratios of the exclusivemodes ¯ B → ( Kν ¯ ν, K ∗ ν ¯ ν, · · · ) [2, 6]. Note that, in the massless quark limit, ( C ν , C ′ ν ) areindependent of the renormalization scale in QCD.In the MSSM, the interaction (4) is generated by the Z -boson penguin and box dia-grams. The standard model particles only contribute to C ν , giving at the leading orderin QCD [15, 10, 1, 2, 6], C ν, SM = − θ W x x − h x + x − x −
2) log x i , (8)where x = m t /m W . Numerically, C ν, SM is about − . m t = 171 GeV.New particles in the MSSM, namely the SUSY particles and Higgs bosons, may con-tribute to both C ν and C ′ ν , C ν = C ν, SM + C ν (new) , C ′ ν = C ′ ν (new) ,C ( ′ ) ν (new) = C ( ′ ) ν, ˜ g + C ( ′ ) ν, ˜ χ ± + C ( ′ ) ν, ˜ χ + C ( ′ ) ν,H ± . (9) C ( ′ ) ν (new) consists of the contributions of the gluino ˜ g - down type squark loops, chargino˜ χ ± - up-type squark loops, neutralino ˜ χ - down-type squark loops, and charged Higgs bo-son H ± - top quark loops. Below we list the analytic forms of these one-loop contributionsfor each sector: C ν, ˜ g = − g s e K ∗ ts K tb (Γ † DL ) i (Γ DR ) ik (Γ † DR ) kj (Γ DL ) j C ( ˜ d i , ˜ d j , ˜ g ) , (10) C ′ ν, ˜ g = 4 g s e K ∗ ts K tb (Γ † DR ) i (Γ DL ) ik (Γ † DL ) kj (Γ DR ) j C ( ˜ d i , ˜ d j , ˜ g ) , (11) C ν, ˜ χ ± = a C ∗ ik a Cjl e K ∗ ts K tb h − δ kl (Γ UL ) iγ (Γ † UL ) γj C (˜ u i , ˜ u j , ˜ χ ± k )+ δ ij V ∗ k V l { C (˜ u i , ˜ χ ± k , ˜ χ ± l ) − } − δ ij U k U ∗ l m ˜ χ ± k m ˜ χ ± l C (˜ u i , ˜ χ ± k , ˜ χ ± l ) (cid:21) + a C ∗ ik a Cil m W e K ∗ ts K tb U k U ∗ l m ˜ χ ± k m ˜ χ ± l D (˜ u i , ˜ χ ± k , ˜ χ ± l , ˜ l − ) , (12) C ′ ν, ˜ χ ± = b C ∗ ik b Cjl e K ∗ ts K tb h δ kl (Γ UR ) iγ (Γ † UR ) γj C (˜ u i , ˜ u j , ˜ χ ± k )4 δ ij U k U ∗ l { C (˜ u i , ˜ χ ± k , ˜ χ ± l ) − } − δ ij V ∗ k V l m ˜ χ ± k m ˜ χ ± l C (˜ u i , ˜ χ ± k , ˜ χ ± l ) (cid:21) − b C ∗ ik b Cil m W e K ∗ ts K tb U k U ∗ l D (˜ u i , ˜ χ ± k , ˜ χ ± l , ˜ l − ) , (13) C ν, ˜ χ = a N ∗ ik a Njl e K ∗ ts K tb h − δ kl (Γ DR ) iγ (Γ † DR ) γj C ( ˜ d i , ˜ d j , ˜ χ k )+ δ ij ( N ∗ k N l − N ∗ k N l ) { C ( ˜ d i , ˜ χ k , ˜ χ l ) − } + 12 δ ij ( N k N ∗ l − N k N ∗ l ) m ˜ χ k m ˜ χ l C ( ˜ d i , ˜ χ k , ˜ χ l ) (cid:21) + a N ∗ ik a Nil m W e K ∗ ts K tb (cid:20) f N ∗ k f N l D ( ˜ d i , ˜ χ k , ˜ χ l , ˜ ν ) + 12 f N k f N ∗ l m ˜ χ k m ˜ χ l D ( ˜ d i , ˜ χ k , ˜ χ l , ˜ ν ) (cid:21) , (14) C ′ ν, ˜ χ = b N ∗ ik b Njl e K ∗ ts K tb h δ kl (Γ DL ) iγ (Γ † DL ) γj C ( ˜ d i , ˜ d j , ˜ χ k ) − δ ij ( N k N ∗ l − N k N ∗ l ) { C ( ˜ d i , ˜ χ k , ˜ χ l ) − }− δ ij ( N ∗ k N l − N ∗ k N l ) m ˜ χ k m ˜ χ l C ( ˜ d i , ˜ χ k , ˜ χ l ) (cid:21) − b N ∗ ik b Nil m W e K ∗ ts K tb (cid:20) f N k f N ∗ l D ( ˜ d i , ˜ χ k , ˜ χ l , ˜ ν ) + 12 f N ∗ k f N l m ˜ χ k m ˜ χ l D ( ˜ d i , ˜ χ k , ˜ χ l , ˜ ν ) (cid:21) , (15) C ν,H ± = h t cos β e x tH ( x tH − (1 − x tH + log x tH ) , (16) C ′ ν,H ± = − ( ˆ Y d ) α K ∗ tα K tβ ( ˆ Y d ) ∗ β sin β e K ∗ ts K tb x tH ( x tH − (1 − x tH + log x tH ) , (17)where x tH = m t /m H ± , h t = g m t / ( √ m W sin β ). We assume flavor degeneracy in thelepton and slepton sectors. The formulas (10–17) are derived from previous studies of the b → sν ¯ ν decays in the MSSM [10, 11], as well as related works on the K → πν ¯ ν decays[12, 13, 14]. C , ( a, b, c ) ≡ C , ( m a , m b , m c ) and D , ( a, b, c, d ) ≡ D , ( m a , m b , m c , m d )are the three-point functions for the Z -penguin diagrams and four-point functions for thebox diagrams, respectively [16], in the convention of Ref. [17]. Ultraviolet divergence of C cancels out in the formulas (10–15). We ignore the masses of ( u, d, c ) quarks, andinclude those of ( s, b ) only when they are multiplied by tan β . In this approximation, theneutral Higgs boson contributions to C ( ′ ) ν vanish.The couplings and mixing matrices in Eqs. (10–17) are given as follows: The squarkmixing matrices (Γ QL , Γ QR ) ( Q = U, D ) give relations between the mass eigenstates ˜ q i =(˜ u i , ˜ d i )( i = 1 −
6) to the gauge eigenstates in the “super-CKM” basis (˜ q Lα , ˜ q Rα )( α = 1 − q α = [ u α = ( u, c, t ) , d α = ( d, s, b )]5y SUSY transformation, as˜ q Lα = (Γ † QL ) αj ˜ q j , ˜ q Rα = (Γ † QR ) αj ˜ q j . (18)These matrices are determined to diagonalize the 6 × M q = M QLL ( M QRL ) † M QRL M QRR ! , ( M QLL ) αβ = ( m QLL ) αβ + ( m (0) Q ) † ( m (0) Q ) + δ αβ ( I q L − e q sin θ W ) m Z cos 2 β, ( M QRR ) αβ = ( m QRR ) αβ + ( m (0) Q )( m (0) Q ) † + δ αβ e q sin θ W m Z cos 2 β, ( M URL ) αβ = ( m URL ) αβ − m (0) U µ ∗ cot β, ( M DRL ) αβ = ( m DRL ) αβ − m (0) D µ ∗ tan β. (19)In Eq. (19), off-diagonal elements of the soft SUSY breaking mass matrices ( m QLL,RR,RL )induce flavor mixings which are not constrained by the CKM matrix in general, andmay cause potentially large FCNC. ( m (0) Q ) αβ are the “bare” mass matrices of the quarks.For the up-type squarks, it is just the running mass matrix ( m (0) U ) αβ = ( m U ) αβ =diag( m u , m c , m t ) ∼ diag(0 , , m t ) in the standard model. For the down-type squarks,in contrast, ( m (0) D ) αβ may substantially deviate from the standard model mass matrix( m D ) αβ = diag( m d , m s , m b ), as explained later. The quark-squark-chargino and quark-squark-neutralino couplings ( a Cikα , b
Cikα , a
Nikα , b
Nikα ) are then given in terms of the mixingmatrices for squarks (18), for charginos (
V, U ), and for neutralinos N [18], as a Cikα = g (Γ UL ) iβ V ∗ k K βα − h t (Γ UR ) i V ∗ k K tα ,b Cikα = − (Γ UL ) iβ U k K βγ ( ˆ Y d ) ∗ αγ ,a Nikα = √ − g N ∗ k + g Y N ∗ k )(Γ DL ) iα + ( ˆ Y d ) βα N ∗ k (Γ DR ) iβ ,b Nikα = √ g Y N k (Γ DR ) iα + ( ˆ Y d ) ∗ αβ N k (Γ DL ) iβ , (20)Finally, f N k ≡ N k − tan θ W N k in Eqs. (14, 15) denote the neutrino-sneutrino-neutralinocouplings.We need some explanation for ( ˆ Y d ) αβ , the bare Yukawa coupling matrix for down-typequarks. We start from the effective lagrangian for the couplings of d iR to the Higgs bosondoublets ( H D , H U ) in the MSSM, after integrating out the SUSY particles, L eff = − ( ˆ Y d ) ij ¯ d iR ( d jL H D − K ∗ kj u kL H − D ) − (∆ Y d ) ij ¯ d iR ( d jL H ∗ U + K ∗ kj u kL H − U ) + (h . c) . (21)The couplings (∆ Y d ) ij are forbidden at the tree-level by supersymmetry, but induced bySUSY particle loops with soft SUSY breaking. The running mass matrix in the standard6odel ( m D ) αβ = diag( m d , m s , m b ) is then given by( m D ) αβ = √ m W g cos β [ ˆ Y d + tan β ∆ Y d ] αβ , ≡ [ m (0) D + δm D ] αβ . (22)Although the loop-generated ∆ Y d is suppressed relative to the tree-level coupling ˆ Y d , itscontribution to m D , δm D , is enhanced by tan β , as seen in Eq. (22), and may becomenumerically comparable to the tree-level part m (0) D ∝ ˆ Y d at large tan β [19]. On the otherhand, the couplings of ( d iR , ˜ d iR ) to heavier Higgs bosons ( H , A , H ± ) and higgsinos ˜ H D are determined by ˆ Y d , as shown in Eqs. (17, 20), without tan β -enhanced contributionsfrom ∆ Y d . As a consequence, at large tan β , these couplings may significantly deviatefrom the tree-level values [20] given in terms of ( m D ) αβ and, since ∆ Y d is not flavordiagonal in general, include flavor-mixing parts not determined by the CKM matrix, evenin the super-CKM basis. The bare quark mass matrix m (0) D should be also used in themass matrix (19) of the down-type squarks, which also receives no contributions from∆ Y d . The correction (22) therefore affects the masses and mixing matrices (Γ DL , Γ DR )of the down-type squarks, generating additional flavor mixing for squarks. These tan β -enhanced corrections to the down-type quarks and squarks are often comparable to thetree-level contributions in the MSSM at large tan β , and should be included in realisticanalysis of processes involving these particles [20, 21].Now we turn to the behavior of the SUSY and Higgs contributions (10–17) to ( C ν , C ′ ν ).The main part of these contributions comes from the Z penguin diagrams through effective Z µ ¯ s L γ µ b L and Z µ ¯ s R γ µ b R vertices. Appearance of these vertices needs both the mixingbetween the second and third generations of quarks/squarks, and the SU(2) × U(1) gaugesymmetry breaking in the loops. For small or moderate value of tan β , the largest SU(2)breaking in the loops are provided by the top quark and squarks. As a consequence, C ν,H ± (16) and C ν, ˜ χ ± (12) are relevant. The former, however, is suppressed by 1 / tan β and only relevant for tan β ∼
1, which is disfavored by experimental lower limit on themass of the lightest Higgs boson. Therefore, only the latter, C ν, ˜ χ ± , is left as a potentiallyimportant SUSY contribution to b → sν ¯ ν . Previous studies have shown [11, 6, 5] that C ν, ˜ χ ± is enhanced by large M URL , especially by its flavor-mixing parts. Similar behavioris observed for the chargino contributions to the K → πν ¯ ν decays [13].At large tan β , however, other contributions to b → sν ¯ ν have the possibility to be-come sizable, by the following reasons: First, the SU(2)-breaking left-right mixing of thedown-type squarks ( M DRL ) increases as tan β and may enhance the gluino contribution.Second, off-diagonal parts of the effective Yukawa coupling ˆ Y d in Eq. (22) induce theflavor-changing couplings of the down-type quarks, which are enhanced by tan β and notnecessarily suppressed by the corresponding CKM matrix elements or quark masses. Es-pecially, the element ( ˆ Y d ) , induced by flavor mixing in M DRR , might give large Yukawacouplings of s R and enhance C ′ ν,H ± . This is similar to the case of K → πν ¯ ν at largetan β [14], where loop-induced couplings (( ˆ Y d ) , ( ˆ Y d ) ) give large effective ¯ s R d R Z cou-7ling. Therefore, the gluino (10, 11) and charged Higgs boson (17) contributions must beconsidered in the analysis of b → sν ¯ ν at large tan β . b → sν ¯ ν and cor-relation with b → sγ We present numerical results for the new physics contributions (10–17) to the b → sν ¯ ν de-cay in the MSSM. We concentrate on the cases with large tan β , which were not consideredin previous studies.In the estimation of possible magnitudes of the new physics contributions (10–17) to b → sν ¯ ν , we need to take into account the constraints on SUSY and Higgs parametersfrom other FCNC processes. In this section, we consider the implication of the constraintsfrom the radiative decay b → sγ . This constraint is expected to be crucial since theSU(2) × U(1) breaking and flavor mixing between quarks/squarks in the second and thirdgenerations, which are necessary to enhance the contributions to b → sν ¯ ν , may also givelarge contributions to b → sγ . Another reason to focus on b → sγ is the rather goodagreement between experimental data [22] and the standard model prediction [23] of theinclusive branching ratio Br( ¯ B → X s γ ). Indeed, the decay b → sγ in the MSSM havebeen shown [24, 10, 25, 26, 27, 28, 29, 21] to give strong constraints on the Higgs andSUSY parameters. It should also be noted that the SUSY contributions to b → sγ areenhanced by tan β [25, 26].Here we do not attempt precise calculation of the experimental constraints from b → sγ . Instead, we present a very rough estimation of the expected constraints in terms ofthe Wilson coefficients ( C , C ′ )( µ ) for b → sγ , defined as H eff = − G F √ K ∗ ts K tb ( C ( µ ) O ( µ ) + C ′ ( µ ) O ′ ( µ )) , O = e π m b ( µ )(¯ s L σ µν b R ) F µν , O ′ = e π m b ( µ )(¯ s R σ µν b L ) F µν . (23)Below we show the correlations between C ( ′ ) ν (new), Eqs. (10–17), and new physicscontributions to C ( ′ )7 , C ( ′ )7 (new), for each sector of new physics: namely, the gluino-squark,chargino-squark, and charged Higgs boson-top quark loop contributions, varying squarksmixing parameters which are relevant to b → sν ¯ ν . For simplicity, we assume the flavorstructures of the soft SUSY breaking terms in the squark mass matrices (19) as m QXX = M Q δ qXX ) δ qXX ) ( XX = LL, RR ) , (24)8 URL = m t A u ) ( A u ) , (25)Since CP violation is not essential for the analysis in this paper, all SUSY and Higgsparameters, including those in Eqs. (24, 25) are set to be real. We also set m DRL = 0in Eq. (19) since its contribution to M DRL is, when the vacuum stability bounds [30]is applied, O ( m b M ˜ Q ) and subdominant compared to the second term m (0) D µ ∗ tan β = O ( m b tan βM ˜ Q ). Note that the condition (24) for m QLL may be imposed only either˜ Q = ˜ U or ˜ Q = ˜ D , due to the SU(2) symmetry ( m ULL ) αβ = K αγ ( m DLL ) γδ K ∗ βδ .We calculate the new physics contributions to C ( ′ ) ν and C ( ′ )7 at the leading one-looporder (see Refs. [10, 25, 26, 27] for the formulas of C ( ′ )7 ), but improved by including thetan β -enhanced corrections to the quark/squark Yukawa couplings from Eq. (22) and, for C ( ′ )7 , also from the proper vertex corrections to the u iR couplings to ( H , A , H ± ) [28, 29],in the effective lagrangian formalism [28]. In these formulas, we use the running quarkmasses and α s at the renormalization scale µ = M ˜ Q , calculated from m t (pole) = 171GeV, m b ( m b ) = 4 . m s (2GeV) = 95 MeV, m q (others) = 0 and α s ( m Z ) = 0 . C ( ′ ) ν ( µ ) and C ( ′ )7 ( µ ) at the renormalization scale µ = M ˜ Q . For SUSY and Higgsparameters, we fix the following parameters: tan β = 50, M ˜ Q = 500 GeV, m ˜ g = 500 GeV, M = 300 GeV, M = 150 GeV, while varying other parameters. We also impose thebounds m ˜ χ ± >
100 GeV and m ˜ q >
250 GeV, suggested by experimental search limits forSUSY particles.For each sector of the new physics, rough estimates of the bounds on the contributionsto ( C ν , C ′ ν ) are obtained by requiring that the magnitudes of ( C , C ′ )(new) should besmaller than the standard model contribution C , SM ( µ ∼ m W ) ∼ − . The gluino-squark contributions C ( ′ ) ν, ˜ g are induced by the flavor and left-right mixing ofthe down-type squarks. In Fig. 1, the gluino contribution C ν, ˜ g is shown as a correlationwith C , ˜ g , for parameter scan over ( δ dLL ) = [ − . , . δ dRR ) = [ − . , . µ =[ − , A u ) and ( A u ) are set to 0. Correlation between C ′ ν, ˜ g and C ′ ,sg for the same parameters is obtained from Fig. 1 by changing the sign of the horizontalaxis. Large | C ν, ˜ g | is obtained for large negative µ and large ( δ LL,RR ) , which causelarge ˜ b R − ˜ s L mixing. It is seen that | C ν, ˜ g | can be larger than 1, which gives about30 % correction to the standard model prediction of the decay width (7). However, byrequiring | C , ˜ g | < | C , SM ( µ W ) | ∼ .
2, magnitudes of C ν, ˜ g are constrained to be muchsmaller than C ν, SM ∼ − .
8. Therefore, without very precise cancellation between newphysics contributions to b → sγ , gluino contributions to b → sν ¯ ν should be completelynegligible, even for tan β ≫
1, to satisfy the bound from b → sγ . These vertex corrections also appear in C ν,H ± . However, we ignored the corrections in Eq. (16), since C ν,H ± itself is strongly suppressed by 1 / tan β and numerically negligible. C ( ′ ) ν, ˜ χ , Eqs. (14, 15). Similarto the gluino contributions, C ( ′ ) ν, ˜ χ are induced by the ˜ b − ˜ s mixing in the loops. However,due to small couplings, these contributions are much smaller than the gluino contributions C ( ′ ) ν, ˜ g for most parameter regions and therefore not discussed here. -10-8-6-4-2 0 2 4 6 8 10-1.5 -1 -0.5 0 0.5 1 1.5 C ( g l u i no ) C ν (gluino) Figure 1:
Correlation between C ν, ˜ g and C , ˜ g . Parameters are tan β = 50, µ = [ − , δ dLL,RR ) = [ − . , . | C , ˜ g | < . The chargino-squark loop contributions C ν, ˜ χ ± , Eq. (12), have been studied in previousworks [11, 6] at small or moderate value of tan β . In these works, it has been shown thatthey might give sizable contributions, larger than the uncertainty of the standard modelpredictions (3), for large flavor-mixing element of m URL in Eq. (25), especially its (˜ t R ,˜ c L )-mixing element ( m URL ) ∼ ( A u ) m t .Figure 2 shows the correlation between C ν, ˜ χ ± and C ( ′ )7 , ˜ χ ± , for varying parameters over( A u ) = [ − , A u ) = [ − , δ uLL ) = [ − . , . µ = 500 GeV, m ˜ l ± L = 400 GeV, and ( δ uRR ) = 0. Forthese parameters, C ′ ν, ˜ χ ± is negligibly small ( < .
02) and not shown here. As is the case ofthe gluino contributions, SUSY parameters which give large C ν, ˜ χ ± tend to also give large10 ( ′ )7 , ˜ χ ± . The resulting constraint on C ν, ˜ χ ± gets tighter as tan β increases, since C ( ′ )7 , ˜ χ ± areenhanced by tan β while C ν, ˜ χ ± is not. Nevertheless, the correlation is not so strong as inthe gluino sector, as seen in Fig. 2. This is due to the different dependences of C ν, ˜ χ ± and C ( ′ )7 , ˜ χ ± on two A-term elements, ( A u ) and ( A u ) in Eq. (25). In fact, as seen in Fig. 2,we may have | C ν, ˜ χ ± | > | C ( ′ )7 , ˜ χ ± | < .
2. Even larger value of C ν, ˜ χ ± might bepossible by careful choice of the SUSY parameters. The resulting deviations of the decaywidths from the standard model predictions (3) could be proved at future B factories,if the theoretical uncertainties of the exclusive widths in Eq. (3), mainly coming fromthe meson form factors, are reduced. However, one must note that the large charginocontribution is realized by the fine tuning between SUSY parameters, especially ( A u ) and ( A u ) , to realize small C ( ′ )7 , ˜ χ ± . -3-2-1 0 1 2 3 4-1.5 -1 -0.5 0 0.5 1 1.5 C ( c ha r g i no ) C ν (chargino) |C’ |<0.2|C’ |>0.2 Figure 2:
Correlation between C ν, ˜ χ ± and C , ˜ χ ± for parameters ( A u ) = [ − , A u ) = [ − , δ uLL ) = [ − . , . | C ′ , ˜ χ ± | smaller (larger)than | C , SM | ∼ . As discussed in the previous section, only C ′ ν,H ± , Eq. (17), is relevant at large tan β . Thiscontribution comes from the H − ¯ s R t L coupling ∼ ( ˆ Y d ) α K ∗ tα ∼ ( ˆ Y d ) , which is generatedby the flavor mixing involving ˜ s R through the tan β -enhanced loop corrections (22). In11ig. 3, we show the correlations between C ′ ν,H ± and C ( ′ )7 ,H ± at µ = −
500 GeV, ( A u ) = 0GeV, ( A u ) = 0 GeV, ( δ dLL ) = [ − . , . δ dRR ) = [ − . , . m H ± = [400 , b → sγ , the main parts of C ( ′ )7 ,H ± are not enhanced by tan β . Moreover, the correlations between C ′ ν,H ± and C ,H ± isseverely affected by different parameter dependences of two generation-mixing H ± cou-plings: the effective ¯ s R t L H − coupling ∼ ( ˆ Y d ) in C ′ ν,H ± and C ′ ,H ± , and O (tan β ) propervertex corrections to the ¯ s L t R H − coupling [28, 29] in C ,H ± . As a result, similar to thecase of chargino contributions, there is possiblity to have sizable C ′ ν,H ± while keeping C ( ′ )7 ,H ± small. -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 -4 -2 0 2 4 ( C ’ , C )( H + ) C’ ν (H + ) C’ C Figure 3:
Correlation between C ν,H ± and C ( ′ )7 ,H ± at tan β = 50, ( δ dLL,RR ) = [ − . , . m H ± = [400 , B s → µ + µ − on the H ± contribution In addition to b → sγ , several other b → s FCNC processes have been measured in recentexperiments. Since most of these measurements show rather good consistency with thestandard model predictions, they should give additional constraints on the SUSY andHiggs parameters, and their contributions to b → sν ¯ ν . For example, measurements of the B s − ¯ B s oscillation [31] impose constraints on the ˜ b − ˜ s mixing, especially on ( δ dLL ) and12 δ dRR ) [32]. Here we just show a case of these constraints: implication of the upper limitof the branching ratio for B s → µ + µ − on the H ± contributions C ′ ν,H ± , at large tan β .As seen in Eq. (17), large value of C ′ ν,H ± is obtained for parameters which give largeeffective H − ¯ s R t L Yukawa coupling ∼ ( ˆ Y d ) . As discussed in Sect. 2, the parameter ( ˆ Y d ) also gives the flavor-changing ( H , A )¯ s R b L couplings of the heavier neutral Higgs bosons( H , A ). On the other hand, at large tan β , this coupling gives “tree-level” contributionsto the B s → µ + µ − decay by the Higgs penguin diagrams [33, 21], which are often muchlarger than the standard model contributions by orders of magnitude. Requiring thatthese Higgs penguin contributions do not saturate the experimental upper bound Br( B s → µ + µ − ) < − at 95% C.L. [34], and neglecting mass difference between ( H , A ), thecondition | ( ˆ Y d ) | + | ( ˆ Y d ) | < . β ( m A /
500 GeV) , (26)is imposed on the b − s mixing Yukawa couplings (( ˆ Y d ) , ( ˆ Y d ) ) at the renormalizationscale µ b ∼ m b . In the approximation of neglecting the QCD running between µ b and M ˜ Q ,and also the O (tan β )-enhanced correction to the ¯ t L b R H + coupling ∼ ( ˆ Y d ) , Eq. (26)implies the bound | C ′ ν,H ± | < .
15 for tan β = 50 and m A < C ν, SM . We expect that this strong constraint still holds when morerigorous estimation of B s → µ + µ − is adopted. We have studied the flavor-changing decay b → sν ¯ ν in the MSSM, at large tan β andwith general flavor mixing of squarks. This case is interesting since the gluino and H ± loops, which are negligible at moderate value of tan β and with minimal flavor viola-tion for squarks, are enhanced and might give contributions to this decay, comparable tothe standard model and chargino loop contributions. This is due to the tan β -enhancedSU(2) × U(1) gauge symmetry breaking and flavor mixing in the down-type squark sec-tor, and loop-generated effective flavor-changing couplings of the charged Higgs boson toquarks and squarks. However, the contributions to b → sν ¯ ν by new physics should beconstrained by experimental data for other b → s processes.In this paper, we have focused our attention to the constraints from the radiative decay b → sγ , since both of the b → sν ¯ ν and b → sγ decays are enhanced by the SU(2) × U(1)symmetry breaking and flavor mixing between the second and third generations of thequarks/squarks in the loops. As a very rough estimation of the constraints by b → sγ , wehave calculated the correlations between new physics contributions to the Wilson coeffi-cients C ( ′ ) ν and C ( ′ )7 for the b → sν ¯ ν and b → sγ decays, respectively, for each new physicssector: gluino-squark, chargino-squark, and charged Higgs-quark loops. Calculation hasbeen done at the leading order, but including tan β -enhanced corrections to the quarkYukawa couplings in the loops. It has been demonstrated that the requirement that thenew physics contributions C ( ′ )7 (new) for each sector are smaller than C , SM strongly con-13trains the new physics contributions C ( ′ ) ν (new). Especially, the gluino contributions C ( ′ ) ν, ˜ g are suppressed much below C ν, SM due to their strong correlation with C ( ′ )7 , ˜ g . In contrast,although the constraints by C ( ′ )7 are also tight for chargino and charged Higgs boson con-tributions, there still remains a possibility that their contributions to C ( ′ ) ν become sizable, O (10)%, while keeping contributions to C ( ′ )7 below C , SM .As an example of the constraints by other b → s processes, we have also consideredthe Higgs penguin contributions to the decay B s → µ + µ − , which might become muchlarger than the standard model contribution at large tan β . It has been shown that thepresent experimental upper bound of the decay ratio may impose strong constraints on C ′ ν,H ± , suppressing it much below C ν, SM .For more realistic analysis of b → sν ¯ ν in the MSSM and estimation of the new physicscontributions, we need to scan over wider parameter space, including correlations betweendifferent contributions to C ( ′ ) ν , main parts of the QCD corrections and hadronic effects,and constraints from other flavor-changing processes using more precise formulas of thenew physics contributions. We leave such studies for future works. Acknowledgements
The author thanks Francesca Borzumati for earlier collaboration. The work was sup-ported in part by the Grant-in-Aid for Scientific Research on Priority Areas from theMinistry of Education, Culture, Sports, Science and Technology of Japan, No. 16081202and 17340062.
References [1] G. Buchalla and A. J. Buras, Nucl. Phys. B , 285 (1993); G. Buchalla andA. J. Buras, Nucl. Phys. B , 225 (1993); M. Misiak and J. Urban, Phys. Lett. B , 161 (1999) [arXiv:hep-ph/9901278]; G. Buchalla and A. J. Buras, Nucl. Phys.B , 309 (1999) [arXiv:hep-ph/9901288].[2] P. Colangelo, F. De Fazio, P. Santorelli and E. Scrimieri, Phys. Lett. B , 339(1997) [arXiv:hep-ph/9610297]; D. Melikhov, N. Nikitin and S. Simula, Phys. Lett.B , 290 (1997) [arXiv:hep-ph/9704268]; D. Melikhov, N. Nikitin and S. Sim-ula, Phys. Rev. D , 6814 (1998) [arXiv:hep-ph/9711362]; D. Melikhov, N. Nikitinand S. Simula, Phys. Lett. B , 171 (1998) [arXiv:hep-ph/9803269]; T. M. Aliev,A. ¨Ozpineci and M. Savcı, Phys. Lett. B , 77 (2001) [arXiv:hep-ph/0101066].[3] R. Barate et al. [ALEPH Collaboration], Eur. Phys. J. C , 213 (2001)[arXiv:hep-ex/0010022].[4] K. F. Chen et al. [BELLE Collaboration], Phys. Rev. Lett. , 221802 (2007)[arXiv:0707.0138 [hep-ex]]. 145] C. Bobeth et al. , Nucl. Phys. B , 252 (2005) [arXiv:hep-ph/0505110].[6] G. Buchalla, G. Hiller and G. Isidori, Phys. Rev. D , 014015 (2001)[arXiv:hep-ph/0006136].[7] A. G. Akeroyd et al. [SuperKEKB Physics Working Group], arXiv:hep-ex/0406071;J. L. Hewett et al. , arXiv:hep-ph/0503261; M. Bona et al. , arXiv:0709.0451 [hep-ex].[8] A. Ali, D. Benson, I. I. Y. Bigi, R. Hawkings and T. Mannel, arXiv:hep-ph/0012218.[9] H. P. Nilles, Phys. Rep. , 1 (1984); H. E. Haber and G. L. Kane, Phys. Rep. ,75 (1985).[10] S. Bertolini, F. Borzumati, A. Masiero and G. Ridolfi, Nucl. Phys. B , 591 (1991).[11] Y. Grossman, Z. Ligeti and E. Nardi, Nucl. Phys. B , 369 (1996) [Erratum-ibid. B , 753 (1996)] [arXiv:hep-ph/9510378]; T. Goto, Y. Okada, Y. Shimizuand M. Tanaka, Phys. Rev. D , 4273 (1997) [Erratum-ibid. D , 019901(2002)] [arXiv:hep-ph/9609512]; A. J. Buras, P. Gambino, M. Gorbahn, S. J¨agerand L. Silvestrini, Nucl. Phys. B , 55 (2001) [arXiv:hep-ph/0007313]; C. Bo-beth, A. J. Buras, F. Kr¨uger and J. Urban, Nucl. Phys. B , 87 (2002)[arXiv:hep-ph/0112305].[12] S. Bertolini and A. Masiero, Phys. Lett. B , 343 (1986); G. F. Giudice, Z. Phys. C , 57 (1987); I. I. Y. Bigi and F. Gabbiani, Nucl. Phys. B , 3 (1991); G. Coutureand H. K¨onig, Z. Phys. C , 167 (1995) [arXiv:hep-ph/9503299].[13] Y. Nir and M. P. Worah, Phys. Lett. B , 319 (1998) [arXiv:hep-ph/9711215];A. J. Buras, A. Romanino and L. Silvestrini, Nucl. Phys. B , 3 (1998)[arXiv:hep-ph/9712398]; G. Colangelo and G. Isidori, J. High Energy Phys. ,009 (1998) [arXiv:hep-ph/9808487]. A. J. Buras, G. Colangelo, G. Isidori, A. Ro-manino and L. Silvestrini, Nucl. Phys. B , 3 (2000) [arXiv:hep-ph/9908371];A. J. Buras, T. Ewerth, S. J¨ager and J. Rosiek, Nucl. Phys. B , 103 (2005)[arXiv:hep-ph/0408142]; G. Isidori, F. Mescia, P. Paradisi, C. Smith and S. Trine, J.High Energy Phys. , 064 (2006) [arXiv:hep-ph/0604074].[14] G. Isidori and P. Paradisi, Phys. Rev. D , 055017 (2006) [arXiv:hep-ph/0601094].[15] T. Inami and C. S. Lim, Prog. Theor. Phys. , 297 (1981) [Erratum-ibid. , 1772(1981)].[16] G. Passarino and M. J. G. Veltman, Nucl. Phys. B , 151 (1979).[17] K. Hagiwara, S. Matsumoto, D. Haidt and C. S. Kim, Z. Phys. C , 559 (1994)[Erratum-ibid. C , 352 (1995)] [arXiv:hep-ph/9409380].1518] J. F. Gunion and H. E. Haber, Nucl. Phys. B , 1 (1986) [Erratum-ibid. B ,567 (1993)].[19] T. Banks, Nucl. Phys. B303 , 172 (1988); R. Hempfling, Phys. Rev. D , 6168(1994); L. J. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D , 7048 (1994)[arXiv:hep-ph/9306309]; M. Carena, M. Olechowski, S. Pokorski and C. E. M. Wag-ner, Nucl. Phys. B426 , 269 (1994) [arXiv:hep-ph/9402253]; T. Blaˇzek, S. Raby andS. Pokorski, Phys. Rev. D , 4151 (1995) [arXiv:hep-ph/9504364].[20] M. Carena, S. Mrenna and C. E. M. Wagner, Phys. Rev. D , 075010 (1999)[arXiv:hep-ph/9808312]; K. S. Babu and C. F. Kolda, Phys. Lett. B , 77(1999) [arXiv:hep-ph/9811308]; F. Borzumati, G. R. Farrar, N. Polonsky andS. Thomas, Nucl. Phys. B555 , 53 (1999) [arXiv:hep-ph/9902443]; H. Eberl, K. Hi-daka, S. Kraml, W. Majerotto and Y. Yamada, Phys. Rev. D , 055006 (2000)[arXiv:hep-ph/9912463]; M. Carena, D. Garcia, U. Nierste and C. E. M. Wagner,Nucl. Phys. B577 , 88 (2000) [arXiv:hep-ph/9912516]; H. E. Haber et al. , Phys.Rev. D , 055004 (2001) [arXiv:hep-ph/0007006]; M. J. Herrero, S. Pe˜naranda andD. Temes, Phys. Rev. D , 115003 (2001) [arXiv:hep-ph/0105097]; D. A. Demir,Phys. Lett. B , 193 (2003) [arXiv:hep-ph/0303249];[21] A. J. Buras, P. H. Chankowski, J. Rosiek and L. S lawianowska, Phys. Lett. B , 96 (2002) [arXiv:hep-ph/0207241]; A. J. Buras, P. H. Chankowski, J. Rosiekand L. S lawianowska, Nucl. Phys. B , 3 (2003) [arXiv:hep-ph/0210145];J. Foster, K. i. Okumura and L. Roszkowski, Phys. Lett. B , 102 (2005)[arXiv:hep-ph/0410323]; J. Foster, K. i. Okumura and L. Roszkowski, J. High En-ergy Phys. , 094 (2005) [arXiv:hep-ph/0506146]; G. Isidori and P. Paradisi,Phys. Lett. B , 499 (2006) [arXiv:hep-ph/0605012]; E. Lunghi, W. Porod andO. Vives, Phys. Rev. D , 075003 (2006) [arXiv:hep-ph/0605177].[22] E. Barberio et al. [Heavy Flavor Averaging Group (HFAG) Collaboration],arXiv:0704.3575 [hep-ex].[23] M. Misiak et al. , Phys. Rev. Lett. , 022002 (2007) [arXiv:hep-ph/0609232].[24] S. Bertolini, F. Borzumati and A. Masiero, Phys. Lett. B , 437 (1987).[25] N. Oshimo, Nucl. Phys. B404 , 20 (1993); M. A. D´ıaz, Phys. Lett. B ,278 (1993) [arXiv:hep-ph/9303280]; R. Barbieri and G. F. Giudice, Phys. Lett.B , 86 (1993) [arXiv:hep-ph/9303270]; Y. Okada, Phys. Lett. B , 119(1993) [arXiv:hep-ph/9307249]; R. Garisto and J. N. Ng, Phys. Lett. B ,372 (1993) [arXiv:hep-ph/9307301]; F. M. Borzumati, Z. Phys. C , 291 (1994)[arXiv:hep-ph/9310212]; S. Bertolini and F. Vissani, Z. Phys. C , 513 (1995)[arXiv:hep-ph/9403397]. 1626] F. M. Borzumati, M. Olechowski and S. Pokorski, Phys. Lett. B , 311 (1995)[arXiv:hep-ph/9412379]; R. Rattazzi and U. Sarid, Nucl. Phys. B501 , 297 (1997)[arXiv:hep-ph/9612464]; H. Baer, M. Brhlik, D. Casta˜no and X. Tata, Phys. Rev. D , 015007 (1998) [arXiv:hep-ph/9712305]; T. Blaˇzek and S. Raby, Phys. Rev. D ,095002 (1999) [arXiv:hep-ph/9712257].[27] F. Borzumati, C. Greub, T. Hurth and D. Wyler, Phys. Rev. D , 075005 (2000)[arXiv:hep-ph/9911245].[28] G. Degrassi, P. Gambino and G. F. Giudice, J. High Energy Phys. , 009 (2000)[arXiv:hep-ph/0009337]; M. Carena, D. Garcia, U. Nierste and C. E. M. Wag-ner, Phys. Lett. B , 141 (2001) [arXiv:hep-ph/0010003]; K. i. Okumuraand L. Roszkowski, Phys. Rev. Lett. , 161801 (2004) [arXiv:hep-ph/0208101];K. i. Okumura and L. Roszkowski, J. High Energy Phys. , 024 (2003)[arXiv:hep-ph/0308102].[29] F. Borzumati, C. Greub and Y. Yamada, Phys. Rev. D , 055005 (2004)[arXiv:hep-ph/0311151]; G. Degrassi, P. Gambino and P. Slavich, Phys. Lett. B ,335 (2006) [arXiv:hep-ph/0601135].[30] J. A. Casas and S. Dimopoulos, Phys. Lett. B , 107 (1996)[arXiv:hep-ph/9606237].[31] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. , 021802 (2006)[arXiv:hep-ex/0603029]; A. Abulencia et al. [CDF - Run II Collaboration], Phys.Rev. Lett. , 062003 (2006) [arXiv:hep-ex/0606027]; A. Abulencia et al. [CDF Col-laboration], Phys. Rev. Lett. , 242003 (2006) [arXiv:hep-ex/0609040].[32] M. Ciuchini and L. Silvestrini, Phys. Rev. Lett. , 021803 (2006)[arXiv:hep-ph/0603114]; M. Endo and S. Mishima, Phys. Lett. B , 205(2006) [arXiv:hep-ph/0603251]; J. Foster, K. i. Okumura and L. Roszkowski, Phys.Lett. B , 452 (2006) [arXiv:hep-ph/0604121]; S. Baek, J. High Energy Phys. , 077 (2006) [arXiv:hep-ph/0605182]; R. Arnowitt, B. Dutta, B. Hu and S. Oh,Phys. Lett. B , 305 (2006) [arXiv:hep-ph/0606130].[33] S. R. Choudhury and N. Gaur, Phys. Lett. B , 86 (1999) [arXiv:hep-ph/9810307];K. S. Babu and C. F. Kolda, Phys. Rev. Lett. , 228 (2000) [arXiv:hep-ph/9909476];C. S. Huang, W. Liao, Q. S. Yan and S. H. Zhu, Phys. Rev. D , 114021 (2001)[Erratum-ibid. D , 059902 (2001)] [arXiv:hep-ph/0006250]; P. H. Chankowskiand L. S lawianowska, Phys. Rev. D , 054012 (2001) [arXiv:hep-ph/0008046].C. Bobeth, T. Ewerth, F. Kr¨uger and J. Urban, Phys. Rev. D64