Predictions for b -> ssdbar, ddsbar decays in the SM and with new physics
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Predictions for b → ss ¯ d and b → dd ¯ s decays in the SM and withnew physics Dan Pirjol and Jure Zupan National Institute for Physics and Nuclear Engineering,Department of Particle Physics, 077125 Bucharest, Romania Theory Division, Department of Physics,CERN, CH-1211 Geneva 23, Switzerland ∗ (Dated: November 15, 2018) Abstract
The b → ss ¯ d and b → dd ¯ s decays are highly suppressed in the SM, and are thus good probes ofnew physics (NP) effects. We discuss in detail the structure of the relevant SM effective Hamiltonianpointing out the presence of nonlocal contributions which can be about λ − ( m c /m t ) ∼
30% of thelocal operators ( λ = 0 .
21 is the Cabibbo angle). The matrix elements of the local operators arecomputed with little hadronic uncertainty by relating them through flavor SU(3) to the observed∆ S = 0 decays. We identify a general NP mechanism which can lead to the branching fractions ofthe b → ss ¯ d modes at or just below the present experimental bounds, while satisfying the boundsfrom K − ¯ K and B ( s ) − ¯ B ( s ) mixing. It involves the exchange of a NP field carrying a conservedcharge, broken only by its flavor couplings. The size of branching fractions within MFV, NMFVand general flavor violating NP are also predicted. We show that in the future energy scales higherthan 10 TeV could be probed without hadronic uncertainties even for b → s and b → d transitions,if enough statistics becomes available. ∗ On leave of absence from Faculty of mathematics and physics, University of Ljubljana, Jadranska 19, 1000Ljubljana, Slovenia, and Josef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia . INTRODUCTION The decays b → ss ¯ d and b → dd ¯ s are highly suppressed in the SM: they are both loopand CKM suppressed (by six powers of small CKM elements V ts and/or V td ). As such theycan be used for searches of New Physics (NP) signals [1, 2, 3, 4, 5, 6, 7, 8]. The types of NPthat would generate b → ss ¯ d and b → dd ¯ s transitions will commonly also give contributionsto K − ¯ K , B − ¯ B and B s − ¯ B s mixing. Since no clear deviations from the SM predictions areseen in the meson mixing, is it possible to have deviations in b → ss ¯ d and b → dd ¯ s transitionobservable at Belle II and at LHCb? A related question is: with improved statistics, can theexperiments using b → ss ¯ d and b → dd ¯ s decays push the bounds on flavor violation scalebeyond what can be achieved from the mixing observables?We address the second question first. For simplicity let us assume that NP contributionscan be matched onto the SM operator basis, so that H ∆ S = C sd ( ¯ d L γ µ s L )( ¯ d L γ µ s L ), H ∆ B = C bs (¯ s L γ µ b L )(¯ s L γ µ b L ) and H b → ss ¯ d = C b → ss ¯ d (¯ s L γ µ b L )(¯ s L γ µ d L ). Using | C i | = 1 / (Λ i ) one finds[9] K − ¯ K mixing : Λ sd > . · TeV ,B d − ¯ B d mixing : Λ bd >
210 TeV ,B s − ¯ B s mixing : Λ bs >
30 TeV , (1)with Im( C sd ) additionally constrained from ε K . The above bounds should be compared withthe following prediction for the b → ss ¯ d transition in the presence of NP with scale Λ b → ss ¯ d (see section V for derivation) B ( ¯ B → ¯ K ∗ ¯ K ∗ ) = 0 . × − (cid:16)
10 TeVΛ b → ss ¯ d (cid:17) , (2)while the SM prediction for this branching ratio is of O (10 − ). Let us take as an estimateΛ b → ss ¯ d ∼ √ Λ bs Λ sd , a relation that holds in a wide set of NP models including the MinimalFlavor Violation (MFV) and Next-to-Minimal Flavor Violation (NMFV) frameworks. Withenough statistics the bound on Λ bs can then be pushed up to 10 TeV and higher withoutrunning into SM background. The b → ss ¯ d decay modes could thus be used to constrain theNP flavor structure for b → s transitions as precisely as it is possible for s → d transitionsfrom kaon physics. However, the statistics needed is very large. For instance, even toprobe this type of flavor violating NP beyond the mixing bounds, the LHCb and Belle II2uminosities will not be enough. In this scenario the K − ¯ K and B d − ¯ B d mixing boundstranslate to B ( b → dd ¯ s ) < ∼ − and the bounds from K − ¯ K and B s − ¯ B s mixing translateto B ( b → ss ¯ d ) < ∼ − .Does this mean that any NP discoveries using b → dd ¯ s and b → ss ¯ d transitions areexcluded at Belle II and LHCb? Certainly not. It is possible to have significant effects in b → dd ¯ s and b → ss ¯ d while obeying the bounds from the meson mixing, if (i) the exchangedparticle (or a set of particles) X carries an approximately conserved global charge and, if(ii) additionaly there is some hierarchy in the couplings (or alternatively some cancellationsin K − ¯ K mixing). Consider the NP Lagrangian of a generic form L flavor = g b → s (¯ s Γ b ) X + g s → b (¯ b Γ s ) X + g d → s (¯ s Γ d ) X + g s → d ( ¯ d Γ s ) X + h.c. , (3)and assume that X carries a conserved quantum number broken only by the above terms. Wealso assume for simplicity that the field X couples to a fixed Dirac structure Γ. Integratingout the field X produces flavor-changing operators L eff = 1 M X (cid:2) g d → s g ∗ s → d (¯ s Γ d )(¯ s ¯Γ d ) + g b → s g ∗ s → b (¯ s Γ b )(¯ s ¯Γ b )+ g b → s g ∗ s → d (¯ s Γ b )(¯ s ¯Γ d ) + g d → s g ∗ s → b (¯ s ¯Γ b )(¯ s Γ d ) (cid:3) , (4)with the terms in the first line contributing to K − ¯ K mixing and B s − ¯ B s mixing, and inthe second line to b → ss ¯ d decays (we also introduced ¯Γ = γ Γ † γ ). It is now possible to setcontributions to meson mixing to zero, while keeping b → ss ¯ d unbounded. This happens forinstance, if g b → s ≪ g s → b , g s → d ≪ g d → s , or g b → s ≫ g s → b , g s → d ≫ g d → s . (5)In this way all the present experimental bounds can be satisfied, while branching ratios for b → ss ¯ d and b → dd ¯ s induced decays are O (10 − ) (see section V for details).The important ingredient in the above argument was that X carried a conserved quantumnumber, so that there were no terms in L eff of the form g b → s (¯ s Γ b )(¯ s Γ b ) + g d → s (¯ s Γ d )(¯ s Γ d ) + g ∗ s → b (¯ s ¯Γ b )(¯ s ¯Γ b ) + g ∗ s → d (¯ s ¯Γ d )(¯ s ¯Γ d ) . . . , (6)These would be generated for X = X † , which is impossible, if X carries a conserved charge.If terms (6) are present, then B s − ¯ B s mixing forces both g b → s and g s → b to be small, andthe hierarchy in (5) is not possible (similarly K − ¯ K mixing bounds g d → s and g s → d to both3e small). An explicit example of a NP scenario where only terms of the form (4) are gen-erated is R − parity violating MSSM [4]. The R -parity violating term in the superpotential, W = λ ′ ijk L i Q j ¯ d k , leads to ˜ ν i ¯ q Lj d kR flavor violating coupling. Sneutrino exchange gener-ates operators of the form (4), while operators of the form (6) are not generated, since thesneutrino carries lepton charge broken only by R -parity violating terms.A hierarchy of couplings in (5) is also present in (N)MFV models, if left-right terms givedominant contributions [10]. Both terms in (4) and (6) are generated, on the other hand,for FCNCs induced by Z ′ exchange, since Z ′ does not carry any conserved charge.In this paper we will not confine ourselves to a particular model but keep the analysiscompletely general using effective field theory. We will improve on the existing SM predic-tions, and also give predictions for general NP contributions. The most general local NPhamiltonian for b → ss ¯ d transition is [2] H NP = 1Λ (cid:16) X j =1 c j Q j + X j =1 ˜ c j ˜ Q j (cid:17) , (7)where c j are dimensionless Wilson coefficients, Λ NP the NP scale, and the operators are Q =(¯ s L γ µ b L )(¯ s L γ µ d L ) ,Q =(¯ s R b L )(¯ s R d L ) , Q = (¯ s αR b βL )(¯ s βR d αL ) ,Q =(¯ s R b L )(¯ s L d R ) , Q = (¯ s αR b βL )(¯ s βL d αR ) . (8)The ˜ Q j operators are obtained from Q j by L ↔ R exchange. In SM only Q is present. The b → dd ¯ s effective Hamiltonian is obtained by exchanging s ↔ d in the above equations, whilethe K − ¯ K and B s − ¯ B s mixing Hamiltonians follow from b → d and d → s replacements.The predictions following from the NP hamiltonian (7) require calculating the QCDmatrix elements of the four-quark operators. In this paper we show that the Q matrixelements can be related by SU(3) flavor symmetry to linear combinations of observable∆ S = 0 decay amplitudes. This gives clean predictions for the branching fractions of theexclusive b → ss ¯ d and b → dd ¯ s modes in the SM and the NP models where Q dominate.This happens in a large class of NP models, including the two-Higgs doublet model withsmall tan β , and the MSSM with conserved R parity [4]. The effects of the operators withnon-standard chirality can be estimated using factorization.The outline of the paper is as follows. In Section II we review the structure of theeffective Hamiltonian mediating the b → ss ¯ d, dd ¯ s decays in the Standard Model. We point4 ,cW tts sdW b ss d bs sdu,cu,c+bs dsWs sdWb u,cb FIG. 1: Matching the box diagrams contributing to b → ss ¯ d decays onto an effective theory with m W ≥ µ ≥ m b . The top quark box diagram (above) is matched onto a local four-quark operator,while the box diagrams with u, c internal quarks (below) are matched onto local and nonlocaloperators. The mixed top-charm and top-up box diagrams are power suppressed by m u,c /m W anddo not contribute at leading order, see Appendix A. out that in addition to the local operators, the effective Hamiltonian contains also nonlocaloperators which have not been included in the previous literature. In Section III we derive theflavor SU(3) relations for the matrix elements of the Q operator. The resulting numericalpredictions for b → ss ¯ d, dd ¯ s decays in the SM are given in Section IV. NP predictionsin the case of Q operator dominance are discussed in Section V, while in Section VI themodifications needed for a general chiral structure are given. Three appendices containfurther technical details. II. SM EFFECTIVE HAMILTONIAN FOR b → ss ¯ d AND b → dd ¯ s DECAYS
In the SM the b → ss ¯ d, dd ¯ s decays are mediated by the box diagram with internal u, c, t quarks, Fig. 1. For notational simplicity let us focus on the case of b → ss ¯ d , while the resultsfor b → dd ¯ s can be obtained through a replacement s ↔ d . The effective weak Hamiltonianfor b → ss ¯ d is obtained in analogy to the one for K − ¯ K mixing [11, 12, 13, 14], butwith several important differences. First, the CKM structure is more involved. Second,the presence of the massive b quark in the initial state introduces a correction, which ishowever suppressed by m b /m W , and is thus numerically negligible. Finally, in applications5o K − ¯ K mixing the charm quark can be integrated out of the theory, while this cannotbe done for exclusive B decays, where there is no clear separation between the charm mass m c and the energy scales relevant in nonleptonic exclusive B decays into two pseudoscalars.At scales m b ≤ µ ≤ m W , the effective weak Hamiltonian mediating b → ss ¯ d decayscontains both local ∆ S = 2 terms as well as nonlocal terms arising from T-products of∆ S = 1 effective weak Hamiltonians H ss ¯ d = H ∆ S =2 + R d d x T (cid:8) H ∆ S =1 d ( x ) , H ∆ S =1 b (0) (cid:9) . (9)The local part is H ∆ S =2 = G F m W π (cid:0) λ dt λ bt C tt + λ dc λ bt C ct + λ dt λ bc C tc (cid:1)(cid:2) (¯ sd ) V − A (¯ sb ) V − A (cid:3) , (10)where the CKM structures are defined as λ q ′ q = V qq ′ V ∗ qs . The Wilson coefficient coming fromthe top box loop is C tt ∼ O ( x t ) and from the top-charm box loop C ct = C tc ∼ O ( x c ), where x i = m i /m W . The scaling of the three contributions in the local Hamiltonian (10) in termsof Cabibbo angle λ = 0 .
22 and quark masses is then: ∼ λ x t , ∼ λ x c and ∼ λ x c (for b → dd ¯ s all three terms are suppressed by another factor of λ ). The third term in (10) canthus easily be neglected. Note also that there is no λ dc λ bc term. The resulting absence of largelog x c from the charm box contribution is sometimes called the super-hard GIM mechanism[14], and follows from the chiral structure of the weak interaction in the SM, as explainedin Appendix A. The precise values of the Wilson coefficients in (10) can be read off fromthe expressions for K − ¯ K mixing [11], where the RG running is performed only down toscale µ ∼ m b . For ¯ m t ( ¯ m t ) = 160 . m c ( ¯ m c ) = 1 .
27 GeV, α s ( m Z ) = 0 .
118 they are at µ = m b = 4 . C tt ( m b ) = 1 . C tc ( m b ) = 3 . x c = 9 . · − at leading order (LO)(see appendix B for the derivation).The nonlocal contributions in b → ss ¯ d transition, Eq. (9), come from insertions of ∆ S = 1effective weak Hamiltonians H ∆ S =1 b and H ∆ S =1 d (see also Fig. 1). The ∆ S = 1 effective weakHamiltonian H ∆ S =1 b is the same weak Hamiltonian relevant for hadronic B decays H ∆ S =1 b = G F √ (cid:16) X q,q ′ = u,c V ∗ qs V q ′ b X i =1 , C i Q qq ′ i,b − V ∗ ts V tb X j =3 C j Q bj (cid:17) , (11)with the tree operators Q qq ′ ,b = (¯ qb ) V − A (¯ sq ′ ) V − A , Q qq ′ ,b = (¯ q β b α ) V − A (¯ s α q ′ β ) V − A , and penguinoperators Q b , = (¯ sb ) V − A (¯ qq ) V ∓ A , Q b , = (¯ s α b β ) V − A (¯ q β q α ) V ∓ A , where the color indices α, β Q b − a sum over q = { u, d, s, c, b } is implied. The weak Hamiltonian H ∆ S =1 d follows from (11) by making the replacement b → d .Using CKM unitarity we can rewrite the CKM factors as V ∗ us V ub = − V ∗ ts V tb − V ∗ cs V cb .The insertions of tree operators with u and c quarks will generate contributions with CKMstructure λ dc λ bc , that are not present in the local ∆ S = 2 Hamiltonian (10), H cc = G F λ dc λ bc Z d d x X i,j =1 , C i C j T (cid:8) Q cci,d ( x ) Q ccj,b (0) + Q uui,d ( x ) Q uuj,b (0) −− Q cui,d ( x ) Q ucj,b (0) − Q uci,d ( x ) Q cuj,b (0) (cid:1)(cid:9) . (12)From dimensional analysis, the size of this contribution is roughly H cc ∼ G F π m c λ dc λ bc (¯ sd ) V − A (¯ sb ) V − A , (13)which is comparable to (10) and needs to be kept. Another set of contributions of comparablesize coming from double ∆ S = 1 weak Hamiltonian insertions has CKM structure λ dc λ bt . Thenonlocal contributions proportional to λ dt λ bt , on the other hand, are power suppressed, scalingas m c , compared to the corresponding ones in (10), which scale as m t . These contributionscan be safely neglected.The appearance of nonlocal contributions is similar to the situation for K − ¯ K mixing,where the effective Hamiltonian below the charm scale contains the T-product of two ∆ S = 1operators mediating s → du ¯ u transitions, in addition to the local operator (¯ sd ) V − A (¯ sd ) V − A .The only difference is that in exclusive b → ss ¯ d decays the charm quark can not be integratedout because of the large momenta of the light mesons in the final state.The dominant nonlocal operators have CKM structure λ dc λ bt Eq. (B8) and λ dc λ bc Eq. (12).These operators contribute to the physical decay amplitude through rescattering effectswith D ¯ D, Dπ, ¯ Dπ, · · · intermediate states. Their matrix elements are suppressed relativeto those of the top box contribution ∼ C tt by λ − ( m c /m t ) ≃ m c suppressed (but CKM enhanced) nonlocal terms may be areasonable first attempt.We leave a complete calculation of the nonlocal contributions for the future and presentonly a partial evaluation of b → ss ¯ d branching ratios by relating the matrix elements ofthe local contributions (10) to the already measured charmless two body decays using flavor7U(3). We note that the nonlocal contributions were estimated in Ref. [15] using a hadronicsaturation model, and were found to be suppressed relative to the local contributions.For the purpose of the SU(3) relations to be discussed below, it is useful to rewrite theeffective Hamiltonian (10) as H i = G F √ V ub V ∗ ud κ i O i , i = ss ¯ d, dd ¯ s. (14)The operators O i are O ss ¯ d = (¯ sb ) V − A (¯ sd ) V − A , O dd ¯ s = ( ¯ db ) V − A ( ¯ ds ) V − A , (15)and the dimensionless coefficients κ i depend only on the CKM factors and calculable hardQCD coefficients. We have κ ss ¯ d = √ G F m W (4 π ) V tb V ∗ ts V ub V ∗ ud ( V td V ∗ ts C tt + V cd V ∗ cs C ct ) , (16)and similarly for κ dd ¯ s . Numerically, the coefficients are (at µ = m b = 4 . κ ss ¯ d = (6 . · − ) e i ◦ , κ dd ¯ s = (1 . · − ) e − i ◦ . (17)The SU(3) symmetry relations derived below require also the C + C combination ofWilson coefficients, evaluated at the same scale µ = m b . At leading log order this is givenby ( C + C )( m b ) = (cid:18) α s ( M W ) α s ( m b ) (cid:19) / = η ( m b ) = 0 . . (18) III. SU(3) PREDICTIONS
We next show how two body B decay widths for b → ss ¯ d and b → dd ¯ s transitions can becalculated using flavor SU(3). As a first approximation we neglect the nonlocal charm-topcontributions, as justified in the previous section. Then the processes b → ss ¯ d and b → dd ¯ s are mediated in the SM only by the local operators O i in Eq. (15). Under flavor SU(3) theseoperators transform as O ss ¯ d = / , O dd ¯ s = , (19)8here the subscripts denote the isospin. They belong to the same SU(3) multiplet as the b → du ¯ u tree operators [17] C (¯ ub ) V − A ( ¯ du ) V − A + C ( ¯ db ) V − A (¯ uu ) V − A =12 ( C + C )( − √ / − √ / + 1 √ ( s )1 / ) + 12 ( C − C )( / − ( a )1 / ) . (20)These operators contribute to ∆ S = 0 decays such as B → ππ . The explicit expressions for operators in (20) are / = − √ bu )(¯ ud ) + (¯ bd )(¯ uu )] + 12 r
32 [(¯ bs )(¯ sd ) + (¯ bd )(¯ ss )] − √ bd )( ¯ dd ) , (21) / = − √ bu )(¯ ud ) + (¯ bd )(¯ uu )] + 1 √ bd )( ¯ dd ) . (22)We list the b → dd ¯ s, ss ¯ d exclusive decays in Table I for B → P P and in Table II for B → P V . The
P P final states transform under SU(3) as , , , the operators O ss ¯ d and O dd ¯ s are in , and thus there are only two reduced matrix elements, h | | i , h | | i .These two reduced matrix elements also appear in the predictions for measured ∆ S = 0decays mediated by the operators in Eq. (20). This means that the B → P P matrixelements of the operators O ss ¯ d and O dd ¯ s can be expressed in terms of ∆ S = 0 decayamplitudes such as A ( B → π + π − ) and others. A similar analysis applies to B → P V decays, where there are four independent reduced matrix elements of the operators: h S | | i , h A | | i , h | | i , h | | i . These can again be expressed in terms of phys-ical B → P V ∆ S = 0 amplitudes. We now derive these relations separately for the B → P P and B → P V final states. A. B → P P decays
We use the formalism of the graphical amplitudes [18], which makes the derivation ofSU(3) decompositions quite intuitive. The two independent reduced matrix elements of the operator are given in terms of graphical amplitudes [17, 19] as − √ h | | i = − C + C ( T + C ) = − κ ss ¯ d ( t + c ) , (23) h | | i = − C + C (cid:16)
15 ( T + C ) + A + E (cid:17) = − κ ss ¯ d (cid:16)
15 ( t + c ) + a + e (cid:17) . (24)9 ransition Mode Amplitude b → ss ¯ d B + → K + K t + cB → K K t + cB s → K π √ ( a + e ) B s → K + π − − ( a + e ) B s → K η q ( t + c + a + e ) b → dd ¯ s B + → ¯ K π + t + cB → ¯ K π √ ( t + c + a + e ) B → K − π + − ( a + e ) B s → ¯ K ¯ K t + c TABLE I: B → P P exclusive decays mediated by the b → ss ¯ d and b → dd ¯ s transitions. This gives two relations between the graphical amplitudes T (tree), C (color-suppressedtree), A (annihilation), E (exchange) in the ∆ S = 0 modes (the expression for B → P P decays can be found in [18]) and the corresponding graphical amplitudes t, c, a, e in b → ss ¯ d transitions (the decay amplitudes for B → P P modes in terms of these are collected inTable I). Equivalent relations apply between ∆ S = 0 and b → dd ¯ s decay amplitudes.The most useful for our purposes is the relation (23). This gives the following predictionfor the exclusive b → ss ¯ d decays A ( B + → K + K ) = A ( B → K K ) = κ ss ¯ d C + C √ A ( B + → π + π ) , (25)and similarly for the b → dd ¯ s decay A ( B + → ¯ K π + ) = A ( B s → ¯ K ¯ K ) = κ dd ¯ s C + C √ A ( B + → π + π ) . (26)Neglecting the 1 /m b suppressed amplitudes e, a one also has √ A ( B s → K η ) = κ ss ¯ d κ dd ¯ s A ( B → ¯ K π ) ≃ κ ss ¯ d C + C A ( B + → π + π ) . (27)The remaining amplitudes in Table I are proportional to e, a . They are 1 /m b suppressed,therefore we do not consider them further.The same SU(3) relations hold also for the decays into two vector mesons, B → V λ V λ ,separately for each helicity amplitude λ = 0 , ± . For example, the analog of Eq. (25) is A ( B + → K ∗ + λ K ∗ λ ) = A ( B → K ∗ λ K ∗ λ ) = κ ss ¯ d C + C √ A ( B + → ρ + λ ρ λ ) . (28)10 ransition Mode Amplitude b → ss ¯ d B + → K ∗ + K t V + c V B + → K + K ∗ t P + c P B → K ∗ K t P + t V + c P + c V B s → K ∗ π √ ( a P + e P ) B s → K ρ √ ( a V + e V ) B s → K ∗ η q ( t P + c P + a V + e V ) B s → K φ q ( t V + c V + a P + e P ) B s → K ∗ + π − − ( a P + e P ) B s → K + ρ − − ( a V + e V ) b → dd ¯ s B + → ¯ K ∗ π + t P + c P B + → ¯ K ρ + t V + c V B → ¯ K ∗ π √ ( t P + c P + a V + e V ) B → ¯ K ρ √ ( t V + c V + a P + e P ) B → K ∗− π + − ( a V + e V ) B → K − ρ + − ( a P + e P ) B s → ¯ K ∗ ¯ K t V + t P + c V + c P TABLE II: B → P V exclusive decays mediated by the b → ss ¯ d and b → dd ¯ s transitions. As a consequence the b → ss ¯ d and b → dd ¯ s B → V V decays are longitudinally polarized inthe same way as the B + → ρ + ρ decay. B. B → P V decays
Table II lists the decomposition of B → P V decays in terms of graphical amplitudes. Thesubscripts
P, V on t, c identify the final state meson that contains the spectator quark, whilethe subscripts on a, e denote the final state meson containing the q quark from ¯ b → ¯ q ¯ q q (here the spectator participates in the weak interaction) [20, 21].We have T P,V + C P,V ∝ h | | i ± h | | i . The analogs of the relation (23) are then t P + c P = κ ss ¯ d C + C ( T P + C P ) , t V + c V = κ ss ¯ d C + C ( T V + C V ) , (29)11here the graphical amplitudes on the right-hand side are for ∆ S = 0 decays. The expansionof the corresponding decay amplitudes in terms of graphical amplitudes can be found inRefs. [20, 21]. Combining them with expansions in Table II gives the SU(3) relations for the t i + c i exclusive b → ss ¯ d decay amplitudes (for ∆ S = 0 amplitude we only denote the finalstate) A ( B + → K ∗ + K ) = κ ss ¯ d C + C h − (cid:0) A ρ + π − − A ρ − π + (cid:1) − √ A ρ π + + (cid:0) A K ∗ ¯ K − A ¯ K ∗ K (cid:17) − (cid:0) A K ∗− K + − A K ∗ + K − (cid:1) + (cid:0) A ¯ K ∗ K + − A K ∗ + ¯ K (cid:1)(cid:3) , (30) A ( B + → K + K ∗ ) = κ ss ¯ d C + C h(cid:0) A ρ + π − − A ρ − π + (cid:1) − √ A ρ + π − (cid:0) A K ∗ ¯ K − A ¯ K ∗ K (cid:1) + (cid:0) A K ∗− K + − A K ∗ + K − (cid:1) − (cid:0) A ¯ K ∗ K + − A K ∗ + ¯ K (cid:1)i , (31) A ( B → K ∗ K ) = − κ ss ¯ d C + C √ (cid:0) A ρ + π + A ρ π + (cid:1) . (32)The B s decay amplitudes containing t i + c i are given in terms of the above b → ss ¯ d amplitudes A ( B s → K ∗ η ) = r (cid:2) A ( B + → K + ¯ K ∗ ) − A ( B s → K + ρ − ) (cid:3) , (33) A ( B s → K φ ) = r (cid:2) A ( B + → K ∗ + K ) − A ( B s → K ∗ + π − ) (cid:3) , (34)where the 1 /m b suppressed pure annihilation and exchange decay amplitudes are A ( B s → K ∗ + π − ) = −√ A ( B s → K ∗ π ) = − κ ss ¯ d C + C [ A ¯ K ∗ K + − A K ∗− K + − A ¯ K ∗ K ] , (35) A ( B s → K + ρ − ) = −√ A ( B s → K ρ ) = − κ ssd C + C [ A K ∗ + ¯ K − A K ∗ + K − − A K ∗ ¯ K ] . (36)The relations for the b → dd ¯ s transitions are derived in an analogous way, giving for the t i + c i amplitudes A ( B + → ¯ K ∗ π + ) = κ dd ¯ s C + C h(cid:0) A ρ + π − − A ρ − π + ) − √ A ρ + π − (cid:0) A K ∗ ¯ K − A ¯ K ∗ K (cid:1) + (cid:0) A K ∗− K + − A K ∗ + K − (cid:1) − (cid:0) A ¯ K ∗ K + − A K ∗ + ¯ K (cid:1)i , (37) A ( B + → ¯ K ρ + ) = κ dd ¯ s C + C h − (cid:0) A ρ + π − − A ρ − π + (cid:1) − √ A ρ π + + (cid:0) A K ∗ ¯ K − A ¯ K ∗ K (cid:1) − (cid:0) A K ∗− K + − A K ∗ + K − (cid:1) + (cid:0) A ¯ K ∗ K + − A K ∗ + ¯ K (cid:1)i , (38)and √ A ( B → ¯ K ∗ π ) = A ( B + → ¯ K ∗ π + ) − A ( B → K ∗− π + ) , (39) √ A ( B → ¯ K ρ ) = A ( B + → ¯ K ρ + ) − A ( B → K − ρ + ) . (40)12he 1 /m b suppressed pure annihilation and exchange amplitudes are A ( B → K ∗− π + ) = κ dd ¯ s C + C (cid:2) − A K ∗ + ¯ K + A K ∗ + K − + A K ∗ ¯ K (cid:3) , (41) A ( B → K − ρ + ) = κ dd ¯ s C + C (cid:2) − A ¯ K ∗ K + + A K ∗− K + + A ¯ K ∗ K (cid:3) . (42)The remaining B s mode is given by A ( B s → ¯ K ∗ ¯ K ) = − κ dd ¯ s C + C √ A ρ + π + A ρ π + ] . (43)These SU(3) relations will be used in the next Section to predict branching fractions ofexclusive b → ss ¯ d and b → dd ¯ s decays in the SM. The measured branching fractions of∆ S = 0 modes are then the inputs in the predictions and are collected in Table IV. We onlyquote results for those decays that are not 1 /m b suppressed. IV. SM PREDICTIONS FROM THE SU(3) RELATIONS
Experimentally one will be able to search for NP effects in the following b → ss ¯ d decays¯ B → ¯ K ∗ ¯ K ∗ , B − → K − ¯ K ∗ , ¯ B s → φ ¯ K ∗ . The flavor of ¯ K ∗ is tagged using the decay K ∗ ) → K + π − . The same decays with ¯ K instead of ¯ K ∗ , on the other hand, cannot beused to probe b → ss ¯ d transitions. The K mixes with ¯ K so that mass eigenstates K S,L are observed in the experiment. The ”wrong kaon” decays listed above are thus only asubleading contribution in the SM rate. For easier comparison with previous calculations inthe literature we will still quote results for ¯ B → ¯ K ¯ K , . . . , ”branching ratios”, knowingthat these are unobservable in practice. Similar comments apply to b → dd ¯ s transitions,where NP effects can be probed in ¯ B → π K ∗ , ρ K ∗ , B − → π − K ∗ , ρ − K ∗ and ¯ B s → K ∗ K ∗ decays, again using flavor tagged K ∗ decays.We derive next numerical predictions for the branching fractions of the exclusive b → ss ¯ d, dd ¯ s modes. The branching fraction of a given mode B q → M M is given by B ( B q → M M ) = τ B q | A ( B q → M M ) | | ~p | πm B q (44)To predict b → ss ¯ d, dd ¯ s decay amplitudes, A ( B q → M M ), we use the SU(3) rela-tions derived in Sec. III which relate them to the amplitudes of the already measured B + → π + π , ρ + ρ , and B → ρπ decays. The results are collected in Tables III and IV.13s mentioned, we do not present results for the branching ratios of the 1 /m b suppressedannihilation modes.In the calculation of B → P V branching ratios we neglect the contributions of the smallpenguin dominated B → ¯ K ∗ K, K ∗ ¯ K decays in the SU(3) relations (with experimental upperbounds supporting this approximation). Furthermore, the application of the SU(3) relationsrequires that we know also the relative phases of the B → ρπ amplitudes. These phases aresmall, and can be neglected to a good approximation. This can be verified using the isospinpentagon relation A ( ρ + π ) + A ( ρ π + ) = 1 √ A ( ρ + π − ) + A ( ρ − π + )) + √ A ( ρ π ) . (45)Neglecting the relative phases, and using data from Table IV, the left-hand side of thisequality is 6 . ± .
29 (in units of √B · ), which compares well with the right-hand sideof 6 . ± .
34. This justifies the assumption made of neglecting the relative phases of the B → ρπ amplitudes.To factor out the dependence on CKM elements, we also quote the predictions for B → P P, P V, V V modes in a common form as B ( B → X i ) = | κ ssd | ( C + C ) c i , (46)where c i are coefficients specific to each final state calculated using the SU(3) relations andmeasured ∆ S = 0 branching fractions. In the predictions we used the branching fractionsfor the ∆ S = 0 modes listed in Table IV. We use τ ( B + ) /τ ( B ) = 1 . ± .
009 and τ ( B s ) /τ ( B ) = 0 . ± .
017 [22].Both Belle [23] and BABAR [24] collaborations presented the results of a search for thesemodes and report the 90% C.L. upper bounds (BABAR bounds are in square brackets) b → dd ¯ s : B ( B + → K − π + π + ) < . . × − ; BELLE [BABAR] , (47) b → ss ¯ d : B ( B + → K + K + π − ) < . . × − ; BELLE [BABAR] , (48) B ( B → K K + π − ) < × − ; BELLE . (49)The quasi two–body decay B + → ¯ K ∗ π + is part of the B + → K − π + π + three body decay, B + → K + K ∗ is part of B + → K + K + π − , while B → K K ∗ is part of B → K K + π − .The bounds on three body decays thus imply bound on two-body decays. These are 814 ode c i [ × − ] B SM Literature B + → K + K . ± . . ± . · − . × − B → K K . ± . . ± . · − − B + → K ∗ + K . ± . . ± . · − . × − B + → K + K ∗ . ± . . ± . · − . × − B → K ∗ K . ± . . ± . · − − B + → K ∗ + K ∗ . ± . . ± . · − . × − B → K ∗ K ∗ . ± . . ± . · − − TABLE III: SU(3) predictions for the branching fractions of the b → ss ¯ d modes in the SM. Thelast column shows the predictions from a previous calculation [3].Mode c i [ × − ] B SM B + → ¯ K π + . ± . ± × − B s → ¯ K ¯ K . ± . ± × − B + → ¯ K ∗ π + . ± . ± × − B + → ¯ K ρ + . ± . ± × − B → ¯ K ∗ π . ± . ± × − B → ¯ K ρ . ± . ± × − B s → ¯ K ∗ ¯ K . ± . ± × − B + → ¯ K ∗ ρ + . ± . ± × − B s → ¯ K ∗ ¯ K ∗ . ± . ± × − TABLE IV: SU(3) predictions for the branching fractions of the b → dd ¯ s modes in the SM. orders of magnitude or more above the estimates for the SM signal, but the situation couldimprove at a future super-B factory [25] or at LHCb. Note that B → K K + π − is observedin K S K + π − final states which also receives contributions from b → d penguin decay B → ¯ K K ∗ and from annihilation decay B → K + K − . It thus cannot be used as a null probeof NP. 15 ode B ( × − ) Mode B ( × − ) B + → π π + . +0 . − . B → ρ ± π ∓ . ± . ρ π + . +1 . − . ρ + π − . ± . a ρ + π . +1 . − . ρ − π + . ± . b ρ + ρ . ± . ρ π . ± . Branching ratios for B → ππ, ρπ, ρρ decays, from Ref. [22] apart from: a ) the average of15 . ± . . ± . b ) the average of 7 . ± . . ± . V. b → ss ¯ d AND b → dd ¯ s TRANSITIONS IN THE PRESENCE OF NP
Next we consider the b → ss ¯ d and b → dd ¯ s decays in the presence of generic NP. Themost general local NP hamiltonian mediating the b → ss ¯ d and b → dd ¯ s transitions wasgiven in Eq. (7). In this section we will assume that NP matches onto the local operator Q in Eq. (7) with SM chirality ( V − A ) × ( V − A ). This is true for a large class of NPmodels, such as the two-Higgs doublet model with small tan β , or the constrained MSSM[4]. Effects of NP that matches to other chiral structures will be given in the next section. To simplify the notation we focus on b → ss ¯ d transitions — the expressions for b → dd ¯ s canbe obtained through a simple s ↔ d exchange — but show numerical results for both typesof decays.We consider three representative cases of NP: i) the exchange of NP fields that carrya conserved charge, where large effects are possible as explained in the Introduction, ii)minimally flavor violating (MFV) new physics with small tan β [28], NMFV [29] and iii)general flavor violation with a ∼ TeV scale suppression. For this analysis it is useful torewrite the b → ss ¯ d SM effective Hamiltonian (14) as H ss ¯ d = G F √ V ub V ∗ ud κ ss ¯ d O ss ¯ d = 1Λ e − iγ κ ss ¯ d Q , (50)where Λ = 2 / / (2 p G F | V ub V ud | ) = 2 .
98 TeV and Q is defined in (8) (the flavor dependenceof Q is not shown). The NP Hamiltonian for b → ss ¯ d is H NP ss ¯ d = c Λ η Q , (51) There is also the possibility that NP contributes through the ∆ S = 1 hamiltonians appearing in thenonlocal term in Eq. (9), for instance through a (¯ sb )(¯ cc ) term. We do not pursue this possibility further. c contains possible extra flavor hierarchy in the new physicstransitions, and Λ NP is the scale of NP. The hard QCD correction to the c coefficientdescribing the RG running from the weak scale m W to m b has been explicitly factored out, η ( m b ) = 0 .
85. It is now very easy to obtain the branching ratio in the presence of NP fromthe SM predictions, B NP ( B → f ) = (cid:12)(cid:12)(cid:12) Λ κ ss ¯ d c η Λ (cid:12)(cid:12)(cid:12) B SM ( B → f ) , (52)and similarly for b → dd ¯ s decays. A. NP with conserved charge
As discussed in the introduction it is possible to have large NP effects in b → ss ¯ d decays, ifthe transition is mediated by NP fields that carry a total conserved charge, and if in additionthere exists a hierarchy in the couplings. In this case we have for the Wilson coefficient inthe NP Hamiltonian (51) (cf. Eq. (4)) c Λ = 1 M X (cid:0) g b → s g ∗ s → d + g d → s g ∗ s → b (cid:1) . (53)From K − ¯ K and B s − ¯ B s mixing we have the bounds, Eq. (1), | g d → s g ∗ s → d | / M X < TeV , | g b → s g ∗ s → b | / M X <
130 TeV . (54)These bounds are trivially satisfied, if for instance g s → d = g b → s = 0, since then no mixingcontributions are induced. The b → ss ¯ d transitions, on the other hand, can still be large,if g d → s and g s → b are nonzero. Taking g d → s = g s → b = 1, the BABAR experimental boundon B ( B + → K + K ∗ ), Eq. (48), gives M X > . M X > . b → dd ¯ s from the BABAR bound on B + → ¯ K ∗ π + branching ratio. The resultingpredictions for b → ss ¯ d and b → dd ¯ s branching ratios are of O (10 − ) as shown in Table VIand may well be probed at Belle II and LHCb.A more generic situation may be that only one of the g i couplings is accidentally small.Unlike in the previous example, we choose M X such that we do not saturate the presentexperimental bounds on b → ss ¯ d . As an illustration let us take g s → d = 0 and all the othercouplings to be equal to 1. In this case the K − ¯ K mixing bound in (54) is trivially satisfied,17 ode ( ss ¯ d ) B X [ × ( . M X ) ] B NMFV [ × (
173 TeVΛ ssd ) ] B gen . B SM B + → K + K ∗ . × − . × − . × − (0 . ± . × − B + → K ∗ + K ∗ . × − . × − . × − (3 . ± . × − B → K ∗ K ∗ . × − . × − . × − (2 . ± . × − Mode ( dd ¯ s ) B X [ × ( . M X ) ] B NMFV [ × (
458 TeVΛ dds ) ] B gen . B SM B + → ¯ K ∗ π + . × − . × − . × − (3 . ± . × − B + → ¯ K ∗ ρ + . × − . × − . × − (14 . ± . × − B s → ¯ K ∗ ¯ K ∗ . × − . × − . × − (13 ± × − TABLE VI: Predictions for the branching fractions of the b → ss ¯ d, dd ¯ s modes in the presence of NPcarrying conserved charge ( B X ), with NMFV flavor structure ( B NMFV ), and general flavor violationat high scale ( B gen . ), in all cases assuming dominance of the SM operator (¯ sd ) V − A (¯ sb ) V − A (seealso text for details). The branching fractions B gen include (maximally constructive) interferencewith the SM amplitude, and were obtained using Λ ssd,dds = 10 TeV. Only modes which do notcontain K S,L are shown. while B s − ¯ B s mixing implies that M X >
30 TeV. The b → ss ¯ d branching ratios are B ( (cid:8) B + → K + K ∗ , B + → K ∗ + K ∗ , B → K ∗ K ∗ (cid:9) ) = (cid:8) . , . , . (cid:9) × − (cid:16) M X (cid:17) . (55)For b → dd ¯ s transitions the same choice for the values of coupling, g s → d = 0 and all theother g i = 1, sets M X >
210 TeV due to the present absence of NP effects in B d − ¯ B d mixing.This gives B ( (cid:8) B + → ¯ K ∗ π + , B + → ¯ K ∗ ρ + , B s → ¯ K ∗ ¯ K ∗ (cid:9) ) = (cid:8) . , . , . (cid:9) × − (cid:16) M X (cid:17) . (56)Finally, we mention that, if b → ss ¯ d or b → dd ¯ s modes are observed in the near future,this would imply nontrivial exclusions on the parameter space of the models. In particularmodels with g s → b ∼ g s → d and/or g b → s ∼ g d → s would be excluded as discussed in AppendixC. 18 . NP with MFV and NMFV structures Both MFV [28] and NMFV [29] fall in the class of new physics models where the b → ss ¯ d suppression scale Λ ss ¯ d is the geometric average of the NP scales in K − ¯ K and B s − ¯ B s mixing(1), Λ ss ¯ d ∼ √ Λ sd Λ bs > ∼
173 TeV. In this paper we will restrict ourselves to MFV with smalltan β , where the ∆ F = 2 processes are mediated by a single operator with ( V − A ) × ( V − A )structure [28]. This implies that the K − ¯ K and B s − ¯ B s mixing operators are V ts V ∗ td Λ (¯ s L γ µ d L ) ≡ sd (¯ s L γ µ d L ) , V tb V ∗ ts Λ (¯ b L γ µ s L ) ≡ bs (¯ b L γ µ s L ) , (57)and the b → ss ¯ d local operator is1Λ V tb V ∗ ts V td V ∗ ts (¯ b L γ µ s L )( ¯ d L γ µ s L ) ≡ ss ¯ d (¯ b L γ µ s L )( ¯ d L γ µ s L ) , (58)all of which depend only on one unknown parameter, the MFV scale Λ MFV . From a globalfit the UTfit collaboration finds Λ
MFV > . sd , Λ bs , Λ ss ¯ d that include the hierarchy of the NP induced flavor changing couplings,which in the MFV case are just the appropriate CKM matrix elements. They are related asstated above Λ ss ¯ d = √ Λ sd Λ bs .In NMFV the operators in (57) and (58) are still parameterically suppressed by the CKMmatrix elements, but the strict correlation between the Wilson coefficients is lost – they aremultiplied by O (1) complex coefficients. We then have approximately Λ ss ¯ d ∼ √ Λ sd Λ bs .Using the bounds from (1), Λ ss ¯ d > ∼
173 TeV, which gives b → ss ¯ d branching ratios of O (10 − ), Table VI. Similarly we have Λ dd ¯ s ∼ √ Λ sd Λ bd > ∼
458 TeV, giving b → dd ¯ s branchingratios of O (10 − ). In MFV the predicted branching ratios are much smaller, of the orderof the SM branching ratios in Table VI. The reason for the difference between NMFV andMFV is that in MFV NP the Wilson coefficient generating K − ¯ K mixing carries a weakphase (the same phase as it does in the SM), while in NMFV the NP contribution can bereal. C. General flavor violation with a high scale
As a final example we consider the case where NP is at the mass scale probed by K − ¯ K mixing, Λ ∼ TeV, and assume that flavor violating couplings are all of O (1). The19esulting branching fractions for b → ss ¯ d and b → dd ¯ s decays, assuming positive interferencebetween SM and NP contributions, are collected in Table VI. For b → ss ¯ d decays the NP andSM contributions are roughly of the same size, while for b → dd ¯ s the NP induced branchingratios are more than two orders of magnitude larger than the SM ones. This means thatwith enough statistics one could probe flavor violation without theoretical uncertainty toscales Λ ∼ TeV both in 3 → → → K − ¯ K mixing. Of course, the statistics needed to achieve such anambitious goal is well beyond the reach of present and planned flavor factories. VI. NP LEADING TO NON-SM CHIRALITIES
We now turn to the description of effects induced by the local operators with non-standardchiralities Q − , ˜ Q − . It is convenient to normalize the matrix elements of these operatorsto the ones of the SM operator Q r j ( B → M M ) ≡ h M M | Q j | B ih M M | Q | B i , (59)and similarly for ˜ Q − , where the ratio is denoted as ˜ r j . To obtain predictions for a b → ss ¯ d decay branching ratio due to a particular NP chiral structure, one only needs to multiplythe results in Table VI with appropriate r j or ˜ r j .Using parity one can relate r j and ˜ r j , since P † Q j P = ˜ Q j . For B → P P ( B → V P )decays one then has ˜ r = ∓ , and ˜ r j = ∓ r j , j = 2 , . . . , . (60)For B → V V decays it is convenient to define ratios r λ,j , ˜ r λ,j for final states with definitehelicites, | V ,λ V ,λ i , where λ = 0 , ± . We then have ˜ r , ± = ˜ r , = − r j, ± = − r j, ∓ , ˜ r j, = − r j, , j = 2 , . . . , r j , j = 2 , . . . ,
5. The ratios ˜ r j are then alreadygiven by the above relations. To compute r j we use naive factorization [30], which sufficesfor the accuracy required here. Strictly speaking, naive factorization is not valid at leadingorder in the heavy quark expansion, but corresponds to assuming dominance of the soft-overlap contributions in the complete SCET factorization formula [31], and keeping only20erms of leading order in α s ( m b ). In the QCDF approach, this corresponds to neglectinghard spectator scattering contributions [33, 34]. If needed, these assumptions can be relaxed.Naive factorization, or the vacuum insertion approximation, is also justified in the 1 /N c expansion for the matrix elements of the operators Q , , , but not for Q , . To see this, onecan rewrite Q as a sum of color singlet and color octet terms using the color Fierz identity, Q = (¯ s αR b βL )(¯ s βR d αL ) = 1 N c (¯ s R b L )(¯ s R d L ) + 2(¯ s R t a b L )(¯ s R t a d L ) , (62)and analogously for Q . The matrix element of the color-singlet operator scales as N / c ,while that of the color-octet scales as N − / c . The two term in the above decomposition thuscontribute at the same order in 1 /N c expansion, and both should in principle be kept.For the experimentally interesting B → P V and B → V V decay modes all the ratios canbe expressed in terms of r , . One has r , = 3 r , , and r = − / [2( N c + 1)].The ratio r is common to all the P V modes which depend only on the graphical am-plitudes t P + c P (for which the spectator quark ends up in the pseudoscalar meson) and isgiven by r = 18( N c + 1) f ⊥ V f T ( m V ) f V f + ( m V ) 2( m B − m P − m V ) m V ( m B + m P ) . (63)Using f K ∗ = 218 MeV, f ⊥ K ∗ = 175 MeV and the form factors from Ref. [35] we find r ( K + K ∗ ) = 0 .
28 and r ( π + ¯ K ∗ ) = 0 . V V modes we quote only the ratios corresponding to longitudinally polarizedvector mesons, which dominate the total rate. We find r k = − / [2( N c + 1)] and r k = − N c + 1) 3 f ⊥ V f V m V m B ~p T ( m V ) − ( m B − m V )( m B − m V − m V ) T ( m V )( m B + m V )( m B − m V − m V ) A ( m V ) − m B ~p m B + m V A ( m V ) . (64)Here V denotes the neutral K ∗ meson ( K ∗ for b → ss ¯ d transitions, and ¯ K ∗ for the b → dd ¯ s transitions), if V , V are different vector mesons. Numerically we find r k ( B + → K ∗ + K ∗ , K ∗ K ∗ ) = 0 . , (65) r k ( B + → ¯ K ∗ ρ + ) = 10 − , r k ( B s → ¯ K ∗ ¯ K ∗ ) = 0 . , where we used the B → V form factors from Ref. [36].21 II. CONCLUSIONS
The exclusive rare B decays b → ss ¯ d and b → dd ¯ s analyzed in this paper appear inthe SM only at second order in the weak interactions and have thus very small branchingfractions, but in NP models they can be greatly enhanced. We construct the completeeffective Hamiltonian contributing to these modes in the SM, and point out the presence ofnonlocal contributions, not included in previous work, which can contribute about 30% ofthe local term.We show that the hadronic matrix elements of the local operators contributing to theseexclusive decays in the SM can be determined using SU(3) flavor symmetry in terms ofmeasured ∆ S = 0 decay amplitudes. Detailed numerical predictions are given for all B → P P, V P, V V modes of experimental interest, both in the SM and for several examples ofNP models: NP with conserved global charge, (N)MFV models and general flavor violatingmodels.A general NP mechanism was identified which can enhance the branching fractions ofthese modes, while obeying existing constraints on NP in ∆ S = 2 mixing processes. Thismechanism represents a generalization of the sneutrino exchange in R parity violating SUSY.Any observation of such a decay mode gives a constraint on the ratio of flavor couplingsto the NP, and can exclude regions in the parameter space of the NP theory for branchingfractions observable at LHC-b and super-B factories. Acknowledgments
We thank B. Golob, S. Fajfer, S. Jaeger, and A. Weiler for comments and discussions.D.P. thanks the CERN Theory Division for hospitality during the completion of this work.
APPENDIX A: THE STRUCTURE OF THE EFFECTIVE HAMILTONIAN
Consider for definiteness the b → ss ¯ d transitions. At scales M W > µ > m b , thesetransitions are described by an effective Hamiltonian with propagating u, c quarks. Writingexplicitly the quarks propagating in the box diagram, and not assuming the unitarity of the22KM matrix, the effective Hamiltonian is given by H ss ¯ d = λ dt λ bt H ( t, t ) + X q ,q = u,c λ dq λ bq H ( q , q ) , (A1)with λ q ′ q = V qq ′ V ∗ qs (so that for instance λ dt = V td V ∗ ts . The top term in the Hamiltonian is alocal operator H ( t, t ) = G F m W C ( m t /m W , µ/m W )[(¯ sd ) V − A (¯ sb ) V − A ] , (A2)where C ( m t /m W , µ/m W ) is a Wilson coefficient. The box diagram with internal quarks q , q = u, c , on the other hand, is matched onto an effective Hamiltonian containing bothlocal and nonlocal terms [12] H ( q , q ) = G F (cid:26) A (cid:16) µm W (cid:17) m W [(¯ sd ) V − A (¯ sb ) V − A ] + B (cid:16) µm W (cid:17) ( m + m )[(¯ sd ) V − A (¯ sb ) V − A ]+ D (cid:16) µm W (cid:17) m b [(¯ sd ) V − A (¯ sb ) V − A ] (cid:27) + Z d d x T (cid:8) H d ( q , q )( x ) , H b ( q , q )(0) (cid:9) . (A3)The effective Hamiltonian H b ( q , q ) mediates b → sq ¯ q transitions, and is given by H b ( q , q ) = G F √ (cid:16) X i =1 , C i Q q q i,b + δ q q X j =3 C j Q bj (cid:17) . (A4) H d ( q , q ) mediates d → s ¯ q q and is given by a similar expression, with the replacement b → d . Note that there is no top-charm contribution at leading order in the m i /m W expansion. The top-charm box is matched in the effective theory onto six-quark operatorsof the form (¯ cb )(¯ sc )(¯ sd ), which are power suppressed by 1 /m W relative to the 4-quarkoperators shown. Such terms appear only after using the unitarity of the CKM matrix.The dependence on the light quark masses m , in the effective theory expression Eq. (A3)can be obtained in the mass insertion approximation. The W ± coupling W + µ (¯ u L γ µ d L ) con-serves chirality, which implies that only m , m terms are allowed, but not m m , whichwould require one mass insertion on each propagating line. The m b term arises from twomass insertions on the incoming b quark line. This term is not present in K − ¯ K mixing.On the other hand, in a theory with chiral-odd quark couplings, such as e.g. the chargedHiggs couplings H + (¯ u L d R ) in the 2HDM, another term can appear in Eq. (A3), proportionalto m m . Chirality prevents also the appearance of terms of the form m b m , m b m .Under renormalization, the local operator with Wilson coefficient A ( µ/m W ) renormalizesmultiplicatively, while the nonlocal operators mix into the local operators with coefficients B ( µ/m W ) , D ( µ/m W ). 23aking use of the unitarity of the CKM matrix, it is possible to eliminate λ bu , λ du as λ iu = − λ ic − λ it , i = b, d . This reproduces the effective Hamiltonian quoted in text Eq. (10).The terms proportional to A, C and D are combined into C tt , while the B term in Eq. (A4)reproduces the C tc and C ct coefficients. The total contribution of the local terms proportionalto the Wilson coefficient B ( µ/m W ) is equal to λ du λ bu · λ dc λ bu · m c + λ du λ bc · m c + λ dc λ bc · m c (A5)= − λ dc ( λ bc + λ bt ) · m c − λ bc ( λ dc + λ dt ) · m c + λ dc λ bc · m c = − ( λ dc λ bt + λ dc λ bt ) m c . This proves the two properties of the local effective Hamiltonian H ∆ S =2 stated in the text:i) the equality C tc = C ct , and ii) the absence of a λ dc λ bc local term. The latter propertydoes not hold in the presence of chiral-odd quark couplings, as for example in the 2HDM asdiscussed above. APPENDIX B: ∆ S = 2 WILSON COEFFICIENTS
In this appendix we show the translation of results obtained for ¯ K − K mixing to thecase of b → ss ¯ d decays (the results for b → dd ¯ s decays are equivalent). The results for¯ K − K mixing were derived in [13] in the leading-log approximation, and in [14] in thenext-to-leading log approximation.We start with the Wilson coefficient C tt , which is obtained by matching the u, c, t loopsat the weak scale onto the local operator (¯ sb ) V − A (¯ sd ) V − A . Below this scale, QCD radiativecorrections introduce a correction η ( µ ), so that at NLO C tt ( µ ) = η ( µ ) S ( x t ) + 18 π m b m W D ( µ/m W ) . (B1)The box function S ( x t ) with x t = m t /M W is the same as obtained in the one-loop matchingat the m W scale for ¯ K − K mixing (external b quark leg can be considered as massless forthe purpose of this calculation). It is given by [11] S ( x t ) = 4 x t − x t + x t − x t ) − x t log x t − x t ) = 2 . . (B2)with the numerical value given for ¯ m t ( ¯ m t ) = 160 . η ( µ ) isobtained by solving the renormalization group equation µ dη ( µ ) dµ = γ + η ( µ ) (B3)24t one-loop order, the anomalous dimension is γ + = α s /π , which gives using α S ( m Z ) = 0 . n f =5MS = 226 MeV) η ( µ b ) = (cid:18) α s ( M W ) α s ( m b ) (cid:19) / = 0 . , m b = 4 . , (B4)so that C tt ( µ b ) = 1 . . (B5)The coefficient D ( µ ) parameterizes the b quark mass effects, and is introduced by mixingfrom the nonlocal operators into the local operator m b (¯ sb ) V − A (¯ sd ) V − A . This mixing has notbeen computed yet. We will neglect this contribution since it is suppressed by the smallratio m b /m W ∼ . λ bt λ dt nonlocal contributions due to insertions of two four-quark operators are powersuppressed and can be neglected as discussed in appendix A. This is no longer true fortop-charm contributions, where both local and nonlocal contributions are power suppressedby m i /m W , and mix under renormalization.We use the derivation of [14], which we adapt to the b → ss ¯ d process at hand. The localpart of the ¯ K − K mixing weak Hamiltonian for µ above the charm quark mass (i.e. beforecharm quark is integrated out) is given by [14] H ¯ K − K eff = G F λ dc λ dt ˜ C ˜ Q , ˜ Q = m c g [(¯ sd ) V − A (¯ sd ) V − A ] . (B6)The corresponding local part of the b → ss ¯ d effective Hamiltonian on the other hand is G F m W π (cid:0) λ dc λ bt C ct + λ dt λ bc C tc (cid:1)(cid:2) (¯ sd ) V − A (¯ sb ) V − A (cid:3) , (B7)The RG evolution calculation for b → ss ¯ d process is the same as for ¯ K − K mixing, exceptthat the total contribution is split into two because of two different CKM element structuresin (B7). As shown in Appendix A, these structures have identical coefficients in the SM C ct = C tc .The same equality can be seen also in the anomalous dimension matrices for the runningof these coefficients. Consider the nonlocal contribution to b → ss ¯ d with insertions of thetree operators T { Q , Q , } , which is given by X i,j =1 , C i C j n λ dc λ bt (cid:0) Q uui,d Q uuj,b − Q cui,d Q ucj,b (cid:1) + λ dt λ bc (cid:0) Q uui,d Q uuj,b − Q uci,d Q cuj,b (cid:1)o . (B8)25hen computing the mixing into the local operator ˜ Q , the terms in the first and thesecond brackets give the same contributions, since the quark masses are not relevant for thecalculation of the anomalous dimensions (it does not matter whether c quark or u quarkruns in the lower leg of the loop in Fig 1). This shows that the RG running for C ct , C tc isthe same. Furthermore, this running is the same as that of ˜ C in K − ¯ K mixing. This canbe seen by comparing (B8) with the nonlocal operator contributing to ¯ K − K mixing X i,j =1 , C i C j λ dc λ dt (cid:0) Q uui,d Q uuj,d − Q cui,d Q ucj,d − Q cui,d Q ucj,d (cid:1) , (B9)The two operators are identical, provided that one sets b → d in (B8). The same corre-spondence between K − ¯ K mixing and b → ss ¯ d applies also for the nonlocal contributionsinvolving penguin operators.In conclusion, comparing the Eqs. (B6) and (B7) we find that for µ > m c , we have C ct ( µ ) = C tc ( µ ) = ˜ C ( µ ) x c π/α s , (B10)where ˜ C ( µ ) is obtained from RG evolution in the same way as for ¯ K − K mixing. A verycompact form of RG equations was presented in [14] µ ddµ ~D = ˆ γ T · ~D, (B11)with ˆ γ the 8 × ~D T = ( ~C T , C /C + , C − /C − ) . (B12)Here ~C is a vector of C i , i = 1 , . . . C ± = C ± C , and ˜ C was split to ˜ C = C + C − ,where the distribution between C and C − is arbitrary. At LO we have for the matching atweak scale ~D T ( µ W ) = (1 , , , , , , , C ( µ ) comes entirelyfrom the running, from mixing with C . At µ b the solution of RG running at LO is ~D ( µ ) = V (cid:16)h α S ( m W ) α S ( µ ) i ~γ (0) / β (cid:17) D V − , (B13)with ˆ γ = α S π γ (0) and V a matrix that diagonalizes the LO anomalous dimension matrix, γ (0) D = V − γ (0) T V . This gives ˜ C ( m b ) = 0 . , m b = 4 . , (B14) Here we caution about the definition of C , , which differs from the one in [11]. We use the definition,where C ( µ W ) ∼ C ( µ W ) ∼ C tc ( m b ) = 3 . x c = 9 . · − , (B15)where in the last equality we used m c = 1 .
27 GeV.
APPENDIX C: BOUNDS ON THE FLAVOR-CHANGING COUPLINGS
We have showed in the introduction that b → ss ¯ d branching ratios can be large, if NPeffects are due to exchange of particle(s) with conserved charge. The resulting effectiveweak Hamiltonian, Eq. (4), depends on four couplings, g s → d , g d → s , g b → s , g s → b and an overallmass scale M X , that in this appendix we set to M X = 10 TeV (this then fixes the overallnormalization of g i ). In order to have large b → ss ¯ d branching ratios and simultaneouslyavoid bounds from K − ¯ K mixing and B s − ¯ B s mixing a hierarchy between couplings isrequired. Another way of looking at this is that, if a large b → ss ¯ d decay branching ratio(we will quantify what ”large” means below) is found by Belle II and/or LHCb this wouldimply that a region of parameter space with g s → b ∼ g s → d and/or g b → s ∼ g d → s would beexcluded. We show this below.The experimental constraints from K − ¯ K mixing and B s − ¯ B s mixing give the followingupper bounds (fixing M X = 10 TeV and using bounds from Eq. (1)) ε sd ≡ | g d → s g ∗ s → d | ≤ M X Λ sd = 10 − , ε bs ≡ | g b → s g ∗ s → b | ≤ M X Λ bs = 0 . . (C1)We also define the following two ratios of coupling constants R = g s → b g s → d , ¯ R = g b → s g d → s . (C2)We now show that a measured lower bound on the b → ss ¯ d branching fraction excludesvalues of R, ¯ R that are close to 1. For definiteness, we assume that the NP field X couplesto the quarks with the Dirac structure Γ = P R , as in RPV SUSY. Similar bounds can bederived for any other Dirac structure Γ.The amplitude for the ¯ B → f transition mediated by the operator (¯ sb )(¯ sd ), Eq. (4), is A ( ¯ B → f ) = 1 M X h f | g d → s g ∗ s → b Q + g b → s g ∗ s → d ˜ Q | ¯ B i = r M X h f | Q | ¯ B i ( g d → s g ∗ s → b ∓ g b → s g ∗ s → d ) , (C3)27here the upper (lower) sign is for a P P ( P V ) final state. The combination of couplings g i can be written in terms of the ratios R, ¯ R defined in (C2) g b → s g ∗ s → d ∓ g d → s g ∗ s → b = g b → s g ∗ s → b R ∗ ∓ g d → s g ∗ s → d R ∗ = ( g d → s g ∗ s → d ) ¯ R ∓ ( g b → s g ∗ s → b ) 1¯ R . (C4)The products of coefficients on the r.h.s are now exactly the ones bounded from the mesonmixing, Eq. (C1). The absolute value of the l.h.s on the other hand is assumed to bebounded from below from the measurement of b → ss ¯ d branching ratio, cf. Eq. (C3). Wethen have B < | g b → s g ∗ s → d ∓ g d → s g ∗ s → b | ≤ ε sd | R | + ε bs | R | + 2 ε sd ε bs . (C5)If B ≥ √ ε sd ε bs , then the above inequality rules out a range of values for | R | ,12 ε sd [ B − ε sd ε bs − p B − ε sd ε bs ] ≤ | R | ≤ ε sd [ B − ε sd ε bs + p B − ε sd ε bs ] . (C6)The same bound with ε sd ↔ ε bs holds also for | ¯ R | . The requirement B ≥ √ ε sd ε bs corre-sponds to the requirement that B ( B → f ) > B ( B → f ) NMFV , with the NMFV predictionsfor branching ratios given in Table VI. [1] K. Huitu, D. X. Zhang, C. D. Lu and P. Singer, Phys. Rev. Lett. , 4313 (1998)[arXiv:hep-ph/9809566].[2] Y. Grossman, M. Neubert and A. L. Kagan, JHEP , 029 (1999) [arXiv:hep-ph/9909297].[3] S. Fajfer and P. Singer, Phys. Rev. D , 117702 (2000) [arXiv:hep-ph/0007132].[4] S. Fajfer, J. F. Kamenik and N. Kosnik, Phys. Rev. D , 034027 (2006)[arXiv:hep-ph/0605260].[5] S. Fajfer and P. Singer, Phys. Rev. D , 017301 (2002) [arXiv:hep-ph/0110233].[6] H. Y. Cai and D. X. Zhang, Commun. Theor. Phys. , 887 (2005) [arXiv:hep-ph/0410144].[7] S. Fajfer, J. F. Kamenik and P. Singer, Phys. Rev. D , 074022 (2004)[arXiv:hep-ph/0407223].[8] X. H. Wu and D. X. Zhang, Phys. Lett. B , 95 (2004) [arXiv:hep-ph/0312177].[9] M. Bona et al. [UTfit Collaboration], JHEP , 049 (2008) [arXiv:0707.0636 [hep-ph]].[10] A. L. Kagan, G. Perez, T. Volansky and J. Zupan, arXiv:0903.1794 [hep-ph].[11] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. , 1125 (1996)[arXiv:hep-ph/9512380].
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