Predictions for b ->ssdbar, ddsbar decays in the SM and with new physics

The b ->ssdbar and b ->ddsbar decays are highly suppressed in the SM, and are thus good probes of new physics (NP) effects. We discuss in detail the structure of the relevant SM effective Hamiltonian pointing out the presence of nonlocal contributions which can be about \lambda^{-4} (m_c^2/m_t^2) ~ 30% of the local operators (\lambda = 0.21 is the Cabibbo angle). The matrix elements of the local operators are computed with little hadronic uncertainty by relating them through flavor SU(3) to the observed \Delta S = 0 decays. We identify a general NP mechanism which can lead to the branching fractions of the b\to ss\bar d modes at or just below the present experimental bounds, while satisfying the bounds from K-Kbar and B_{(s)}-Bbar_{(s)} mixing. It involves the exchange of a NP field carrying a conserved charge, broken only by its flavor couplings. The size of branching fractions within MFV, NMFV and general flavor violating NP are also predicted. We show that in the future energy scales higher than 10^3 TeV could be probed without hadronic uncertainties even for b->s and b->d transitions, if enough statistics becomes available.


I. INTRODUCTION
The decays b → ssd and b → dds are highly suppressed in the SM: they are both loop and CKM suppressed (by six powers of small CKM elements V ts and/or V td ). As such they can be used for searches of New Physics (NP) signals [1,2,3,4,5,6,7,8]. The types of NP that would generate b → ssd and b → dds transitions will commonly also give contributions We address the second question first. For simplicity let us assume that NP contributions can be matched onto the SM operator basis, so that H ∆S = C sd . Using |C i 1 | = 1/(Λ i ) 2 one finds [9] K −K mixing : Λ sd > 1.0 · 10 3 TeV, with Im(C sd 1 ) additionally constrained from ε K . The above bounds should be compared with the following prediction for the b → ssd transition in the presence of NP with scale Λ b→ssd (see section V for derivation) B(B 0 →K 0 * K 0 * ) = 0.3 × 10 −6 10 TeV Λ b→ssd while the SM prediction for this branching ratio is of O(10 −15 ). Let us take as an estimate Λ b→ssd ∼ √ Λ bs Λ sd , a relation that holds in a wide set of NP models including the Minimal Flavor Violation (MFV) and Next-to-Minimal Flavor Violation (NMFV) frameworks. With enough statistics the bound on Λ bs can then be pushed up to 10 3 TeV and higher without running into SM background. The b → ssd decay modes could thus be used to constrain the NP flavor structure for b → s transitions as precisely as it is possible for s → d transitions from kaon physics. However, the statistics needed is very large. For instance, even to probe this type of flavor violating NP beyond the mixing bounds, the LHCb and Belle II luminosities will not be enough. In this scenario the K −K and B d −B d mixing bounds translate to B(b → dds) < ∼ 10 −13 and the bounds from K −K and B s −B s mixing translate to B(b → ssd) < ∼ 10 −11 . Does this mean that any NP discoveries using b → dds and b → ssd transitions are excluded at Belle II and LHCb? Certainly not. It is possible to have significant effects in b → dds and b → ssd while obeying the bounds from the meson mixing, if (i) the exchanged particle (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 cancellations in K −K mixing). Consider the NP Lagrangian of a generic form L flavor = g b→s (sΓb)X + g s→b (bΓs)X + g d→s (sΓd)X + g s→d (dΓs)X + h.c., (3) and assume that X carries a conserved quantum number broken only by the above terms. We also assume for simplicity that the field X couples to a fixed Dirac structure Γ. Integrating out the field X produces flavor-changing operators L eff = 1 M 2 X g d→s g * s→d (sΓd)(sΓd) + g b→s g * s→b (sΓb)(sΓb) + g b→s g * s→d (sΓb)(sΓd) + g d→s g * s→b (sΓb)(sΓd) , with the terms in the first line contributing to K −K mixing and B s −B s mixing, and in the second line to b → ssd decays (we also introducedΓ = γ 0 Γ † γ 0 ). It is now possible to set contributions to meson mixing to zero, while keeping b → ssd unbounded. This happens for instance, 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 .
In this way all the present experimental bounds can be satisfied, while branching ratios for b → ssd and b → dds induced decays are O(10 −6 ) (see section V for details).
The important ingredient in the above argument was that X carried a conserved quantum number, so that there were no terms in L eff of the form g 2 b→s (sΓb)(sΓb) + g 2 d→s (sΓd)(sΓd) + g * 2 s→b (sΓb)(sΓb) + g * 2 s→d (sΓd)(sΓd) . . . , 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, and the hierarchy in (5) is not possible (similarly K −K mixing bounds g d→s and g s→d to both be small). An explicit example of a NP scenario where only terms of the form (4) are generated is R−parity violating MSSM [4]. The R-parity violating term in the superpotential, W = λ ′ ijk L i Q jdk , leads toν iqLj d kR flavor violating coupling. Sneutrino exchange generates operators of the form (4), while operators of the form (6) are not generated, since the sneutrino 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 give dominant 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 analysis completely general using effective field theory. We will improve on the existing SM predictions, and also give predictions for general NP contributions. The most general local NP hamiltonian for b → ssd transition is [2] where c j are dimensionless Wilson coefficients, Λ NP the NP scale, and the operators are This happens in a large class of NP models, including the two-Higgs doublet model with small tan β, and the MSSM with conserved R parity [4]. The effects of the operators with non-standard chirality can be estimated using factorization.
The outline of the paper is as follows. In Section II we review the structure of the out that in addition to the local operators, the effective Hamiltonian contains also nonlocal operators which have not been included in the previous literature. In Section III we derive the flavor SU(3) relations for the matrix elements of the Q 1 operator. The resulting numerical predictions for b → ssd, dds decays in the SM are given in Section IV. NP predictions in the case of Q 1 operator dominance are discussed in Section V, while in Section VI the modifications needed for a general chiral structure are given. Three appendices contain further technical details.
In the SM the b → ssd, dds 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 → ssd, while the results for b → dds can be obtained through a replacement s ↔ d. The effective weak Hamiltonian for b → ssd is obtained in analogy to the one for K 0 −K 0 mixing [11,12,13,14], but with 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 is however suppressed by m 2 b /m 2 W , and is thus numerically negligible. Finally, in applications to K 0 −K 0 mixing the charm quark can be integrated out of the theory, while this cannot be 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 → ssd decays contains both local ∆S = 2 terms as well as nonlocal terms arising from T-products of ∆S = 1 effective weak Hamiltonians The local part is where the CKM structures are defined as The scaling of the three contributions in the local Hamiltonian (10) in terms of Cabibbo angle λ = 0.22 and quark masses is then: ∼ λ 7 x t , ∼ λ 3 x c and ∼ λ 7 x c (for b → dds all three terms are suppressed by another factor of λ). The third term in (10) can thus easily be neglected. Note also that there is no λ d c λ b c term. The resulting absence of large log 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 explained with the tree operators Q qq The insertions of tree operators with u and c quarks will generate contributions with CKM structure λ d c λ b c , that are not present in the local ∆S = 2 Hamiltonian (10), From dimensional analysis, the size of this contribution is roughly which is comparable to (10) and needs to be kept. Another set of contributions of comparable size coming from double ∆S = 1 weak Hamiltonian insertions has CKM structure λ d c λ b t . The nonlocal contributions proportional to λ d t λ b t , on the other hand, are power suppressed, scaling as m 2 c , compared to the corresponding ones in (10), which scale as m 2 t . These contributions can be safely neglected.
The appearance of nonlocal contributions is similar to the situation for K 0 −K 0 mixing, where the effective Hamiltonian below the charm scale contains the T-product of two ∆S = 1 operators mediating s → duū transitions, in addition to the local operator (sd The only difference is that in exclusive b → ssd decays the charm quark can not be integrated out because of the large momenta of the light mesons in the final state. The dominant nonlocal operators have CKM structure λ d c λ b t Eq. (B8) and λ d c λ b c Eq. (12). These operators contribute to the physical decay amplitude through rescattering effects with DD, Dπ,Dπ, · · · intermediate states. Their matrix elements are suppressed relative to those of the top box contribution ∼ C tt by λ −4 (m 2 c /m 2 t ) ≃ 30%, which suggests that the approximation of neglecting m 2 c suppressed (but CKM enhanced) nonlocal terms may be a reasonable first attempt.
We leave a complete calculation of the nonlocal contributions for the future and present only a partial evaluation of b → ssd branching ratios by relating the matrix elements of the local contributions (10) to the already measured charmless two body decays using flavor SU (3). We note that the nonlocal contributions were estimated in Ref. [15] using a hadronic saturation 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 the effective Hamiltonian (10) as The and the dimensionless coefficients κ i depend only on the CKM factors and calculable hard QCD coefficients. We have and similarly for κ dds . Numerically, the coefficients are (at µ = m b = 4.2 GeV, with CKM elements from [16]) The SU(3) symmetry relations derived below require also the C 1 + C 2 combination of Wilson coefficients, evaluated at the same scale µ = m b . At leading log order this is given by

III. SU(3) PREDICTIONS
We next show how two body B decay widths for b → ssd and b → dds transitions can be where the subscripts denote the isospin. They belong to the same SU(3) multiplet as the 15 in the decomposition of the b → duū tree operators [17] These operators contribute to ∆S = 0 decays such as B → ππ. The explicit expressions for 15 operators in (20) are We list the b → dds, ssd exclusive decays in Table I for B → P P and in Table II  These two reduced matrix elements also appear in the predictions for measured ∆S = 0 decays mediated by the operators in Eq. (20). This means that the B → P P matrix elements of the operators O ssd and O dds can be expressed in terms of ∆S = 0 decay amplitudes such as A(B 0 → π + π − ) and others. A similar analysis applies to B → P V decays, where there are four independent reduced matrix elements of the 15 operators: . These can again be expressed in terms of physical 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 of SU(3) decompositions quite intuitive. The two independent reduced matrix elements of the 15 operator are given in terms of graphical amplitudes [17,19] as This gives two relations between the graphical amplitudes T (tree), C (color-suppressed tree), 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 → ssd transitions (the decay amplitudes for B → P P modes in terms of these are collected in Table I). Equivalent relations apply between ∆S = 0 and b → dds decay amplitudes.
The most useful for our purposes is the relation (23). This gives the following prediction for the exclusive b → ssd decays and similarly for the b → dds decay Neglecting the 1/m b suppressed amplitudes e, a one also has 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 As a consequence the b → ssd and b → dds B → V V decays are longitudinally polarized in the same way as the B + → ρ + ρ 0 decay. (here the spectator participates in the weak interaction) [20,21].
We have T P,V + C P,V ∝ 10|15|3 ± 27|15|3 . The analogs of the relation (23) are then where the graphical amplitudes on the right-hand side are for ∆S = 0 decays. The expansion of the corresponding decay amplitudes in terms of graphical amplitudes can be found in Refs. [20,21]. Combining them with expansions in Table II gives the SU(3) relations for the t i + c i exclusive b → ssd decay amplitudes (for ∆S = 0 amplitude we only denote the final state) The B s decay amplitudes containing t i +c i are given in terms of the above b → ssd amplitudes where the 1/m b suppressed pure annihilation and exchange decay amplitudes are The relations for the b → dds transitions are derived in an analogous way, giving for the and The 1/m b suppressed pure annihilation and exchange amplitudes are The remaining B s mode is given by  Table IV. We only quote 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 → ssd decays The flavor ofK 0 * is tagged using the decay K 0( * ) → K + π − . The same decays withK 0 instead ofK 0 * , on the other hand, cannot be used to probe b → ssd transitions. The K 0 mixes withK 0 so that mass eigenstates K S,L are observed in the experiment. The "wrong kaon" decays listed above are thus only a subleading contribution in the SM rate. For easier comparison with previous calculations in the literature we will still quote results forB 0 →K 0K 0 , . . . , "branching ratios", knowing that these are unobservable in practice. Similar comments apply to b → dds transitions, where NP effects can be probed inB 0 → π 0 K 0 * , ρ 0 K 0 * , B − → π − K 0 * , ρ − K 0 * andB 0 s → K 0 * K 0 * decays, again using flavor tagged K 0 * decays.
We derive next numerical predictions for the branching fractions of the exclusive b → ssd, dds modes. The branching fraction of a given mode B q → M 1 M 2 is given by To predict b → ssd, dds decay amplitudes, A(B q → M 1 M 2 ), we use the SU(3) relations derived in Sec. III which relate them to the amplitudes of the already measured B + → π + π 0 , ρ + ρ 0 , and B → ρπ decays. The results are collected in Tables III and IV. As mentioned, we do not present results for the branching ratios of the 1/m b suppressed annihilation modes.
In the calculation of B → P V branching ratios we neglect the contributions of the small penguin dominated B →K * K, K * K decays in the SU(3) relations (with experimental upper bounds supporting this approximation). Furthermore, the application of the SU(3) relations requires that we know also the relative phases of the B → ρπ amplitudes. These phases are small, and can be neglected to a good approximation. This can be verified using the isospin pentagon relation Neglecting the relative phases, and using data from 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 where c i are coefficients specific to each final state calculated using the SU(3) relations and measured ∆S = 0 branching fractions. In the predictions we used the branching fractions for the ∆S = 0 modes listed in Table IV. We use τ (B + )/τ (B 0 ) = 1.071 ± 0.009 and τ (B 0 s )/τ (B 0 ) = 0.965 ± 0.017 [22]. Both Belle [23] and BABAR [24] collaborations presented the results of a search for these modes and report the 90% C.L. upper bounds (BABAR bounds are in square brackets) The quasi two-body decay B + →K 0 * π + is part of the B + → K − π + π + three body decay, The bounds on three body decays thus imply bound on two-body decays. These are 8 Mode  orders of magnitude or more above the estimates for the SM signal, but the situation could improve at a future super-B factory [25] or at LHCb. Note that B 0 → K 0 K + π − is observed in K S K + π − final states which also receives contributions from b → d penguin decay B 0 → K 0 K * 0 and from annihilation decay B 0 → K + K − . It thus cannot be used as a null probe of NP.

V. b → ssd AND b → dds TRANSITIONS IN THE PRESENCE OF NP
Next we consider the b → ssd and b → dds decays in the presence of generic NP. The most general local NP hamiltonian mediating the b → ssd and b → dds transitions was given in Eq. (7). In this section we will assume that NP matches onto the local operator . This is true for a large class of NP models, 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. 1  (14) as where Λ 0 = 2 1/4 /(2 G F |V ub V ud |) = 2.98 TeV and Q 1 is defined in (8) (the flavor dependence of Q 1 is not shown). The NP Hamiltonian for b → ssd is and similarly for b → dds decays.
From K −K and B s −B s mixing we have the bounds, Eq. (1),  Table VI and 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 present experimental bounds on b → ssd. As an illustration let us take g s→d = 0 and all the other couplings to be equal to 1. In this case the K −K mixing bound in (54) is trivially satisfied, Finally, we mention that, if b → ssd or b → dds modes are observed in the near future, this would imply nontrivial exclusions on the parameter space of the models. In particular models with g s→b ∼ g s→d and/or g b→s ∼ g d→s would be excluded as discussed in Appendix C.

B. NP with MFV and NMFV structures
Both MFV [28] and NMFV [29] fall in the class of new physics models where the b → ssd suppression scale Λ ssd is the geometric average of the NP scales in K −K and B s −B s mixing (1), Λ ssd ∼ √ Λ sd Λ bs > ∼ 173 TeV. In this paper we will restrict ourselves to MFV with small tan 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 and the b → ssd local operator is all of which depend only on one unknown parameter, the MFV scale Λ MFV . From a global fit the UTfit collaboration finds Λ MFV > 5.5 TeV [9]. We have also defined the suppression scales Λ sd , Λ bs , Λ ssd 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 as In NMFV the operators in (57) Table VI. For b → ssd decays the NP and SM contributions are roughly of the same size, while for b → dds the NP induced branching ratios are more than two orders of magnitude larger than the SM ones. This means that with enough statistics one could probe flavor violation without theoretical uncertainty to scales Λ ∼ 10 3 TeV both in 3 → 2 and 3 → 1 transitions and not just in 2 → 1 transitions as is possible now from K −K mixing. Of course, the statistics needed to achieve such an ambitious 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-standard chiralities Q 2−5 ,Q 1−5 . It is convenient to normalize the matrix elements of these operators to the ones of the SM operator Q 1 and similarly forQ 1−5 , where the ratio is denoted asr j . To obtain predictions for a b → ssd decay branching ratio due to a particular NP chiral structure, one only needs to multiply the results in Table VI with appropriate r 2 j orr 2 j . Using parity one can relate r j andr j , since P † Q j P =Q j . For B → P P (B → V P ) decays one then hasr 1 = ∓1, andr j = ∓r j , j = 2, . . . , 5.
For B → V V decays it is convenient to define ratios r λ,j ,r λ,j for final states with definite helicites, |V 1,λ V 2,λ , where λ = 0, ±. We then haver 1,± =r 1,0 = −1 and r j,± = −r j,∓ ,r j,0 = −r j,0 , j = 2, . . . , 5 We only need to compute the ratios r j , j = 2, . . . , 5. The ratiosr j are then already given by the above relations. To compute r j we use naive factorization [30], which suffices for the accuracy required here. Strictly speaking, naive factorization is not valid at leading order in the heavy quark expansion, but corresponds to assuming dominance of the softoverlap contributions in the complete SCET factorization formula [31], and keeping only terms of leading order in α s (m b ). In the QCDF approach, this corresponds to neglecting hard 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 1,2,4 , but not for Q 3,5 . To see this, one can rewrite Q 3 as a sum of color singlet and color octet terms using the color Fierz identity, and analogously for Q 5 . The matrix element of the color-singlet operator scales as N 1/2 c , while that of the color-octet scales as N −1/2 c . The two term in the above decomposition thus contribute 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 can be expressed in terms of r 2,4 . One has r 3,5 = 3r 2,4 , and The ratio r 2 is common to all the P V modes which depend only on the graphical amplitudes t P + c P (for which the spectator quark ends up in the pseudoscalar meson) and is given by . (63) Using f K * = 218 MeV, f ⊥ K * = 175 MeV and the form factors from Ref. [35] we find r 2 (K + K * 0 ) = 0.28 and r 2 (π +K * 0 ) = 0.27.
For the V V modes we quote only the ratios corresponding to longitudinally polarized vector mesons, which dominate the total rate. We find r 4 = −1/[2(N c + 1)] and .

(64)
Here V 1 denotes the neutral K * meson (K * 0 for b → ssd transitions, andK * 0 for the b → dds transitions), if V 1 , V 2 are different vector mesons. Numerically we find where we used the B → V form factors from Ref. [36].

VII. CONCLUSIONS
The exclusive rare B decays b → ssd and b → dds analyzed in this paper appear in the SM only at second order in the weak interactions and have thus very small branching fractions, but in NP models they can be greatly enhanced. We construct the complete effective Hamiltonian contributing to these modes in the SM, and point out the presence of nonlocal contributions, not included in previous work, which can contribute about 30% of the local term.
We show that the hadronic matrix elements of the local operators contributing to these exclusive decays in the SM can be determined using SU (3)  CKM matrix, the effective Hamiltonian is given by The top term in the Hamiltonian is a local operator where C(m 2 t /m 2 W , µ/m W ) is a Wilson coefficient. The box diagram with internal quarks q 1 , q 2 = u, c, on the other hand, is matched onto an effective Hamiltonian containing both local and nonlocal terms [12] The effective Hamiltonian H b (q 1 , q 2 ) mediates b → sq 1q2 transitions, and is given by can be obtained in the mass insertion approximation. The W ± coupling W + µ (ū L γ µ d L ) conserves chirality, which implies that only m 2 1 , m 2 2 terms are allowed, but not m 1 m 2 , which would require one mass insertion on each propagating line. The m 2 b term arises from two mass insertions on the incoming b quark line. This term is not present in K 0 −K 0 mixing.
On the other hand, in a theory with chiral-odd quark couplings, such as e.g. the charged Under renormalization, the local operator with Wilson coefficient A(µ/m W ) renormalizes multiplicatively, while the nonlocal operators mix into the local operators with coefficients Making use of the unitarity of the CKM matrix, it is possible to eliminate λ b u , λ d u as 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 proportional to the Wilson coefficient B(µ/m W ) is equal to 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 d c λ b c local term. The latter property does not hold in the presence of chiral-odd quark couplings, as for example in the 2HDM as discussed above.

APPENDIX B: ∆S = 2 WILSON COEFFICIENTS
In this appendix we show the translation of results obtained forK 0 − K 0 mixing to the case of b → ssd decays (the results for b → dds decays are equivalent). The results for K 0 − K 0 mixing were derived in [13] in the leading-log approximation, and in [14] in the next-to-leading log approximation.
We start with the Wilson coefficient C tt , which is obtained by matching the u, c, t loops at the weak scale onto the local operator (sb) V −A (sd) V −A . Below this scale, QCD radiative corrections introduce a correction η 2 (µ), so that at NLO The box function S 0 (x t ) with x t = m 2 t /M 2 W is the same as obtained in the one-loop matching at the m W scale forK 0 − K 0 mixing (external b quark leg can be considered as massless for the purpose of this calculation). It is given by [11] with the numerical value given form t (m t ) = 160.9 GeV. The QCD correction η 2 (µ) is obtained by solving the renormalization group equation At one-loop order, the anomalous dimension is γ + = α s /π, which gives using α S (m Z ) = 0.118 so that The coefficient D(µ) parameterizes the b quark mass effects, and is introduced by mixing from the nonlocal operators into the local operator This mixing has not been computed yet. We will neglect this contribution since it is suppressed by the small t λ d t nonlocal contributions due to insertions of two four-quark operators are power suppressed and can be neglected as discussed in appendix A. This is no longer true for top-charm contributions, where both local and nonlocal contributions are power suppressed by m 2 i /m 2 W , and mix under renormalization. We use the derivation of [14], which we adapt to the b → ssd process at hand. The local part of theK 0 −K 0 mixing weak Hamiltonian for µ above the charm quark mass (i.e. before charm quark is integrated out) is given by [14] The corresponding local part of the b → ssd effective Hamiltonian on the other hand is The RG evolution calculation for b → ssd process is the same as forK 0 − K 0 mixing, except that the total contribution is split into two because of two different CKM element structures in (B7). As shown in Appendix A, these structures have identical coefficients in the SM The same equality can be seen also in the anomalous dimension matrices for the running of these coefficients. Consider the nonlocal contribution to b → ssd with insertions of the tree operators T {Q 1,2 Q 1,2 }, which is given by When computing the mixing into the local operatorQ 7 , the terms in the first and the second brackets give the same contributions, since the quark masses are not relevant for the calculation of the anomalous dimensions (it does not matter whether c quark or u quark runs in the lower leg of the loop in Fig 1). This shows that the RG running for C ct , C tc is the same. Furthermore, this running is the same as that ofC 7 in K 0 −K 0 mixing. This can be seen by comparing (B8) with the nonlocal operator contributing toK 0 − K 0 mixing i,j=1,2 The two operators are identical, provided that one sets b → d in (B8). The same correspondence between K 0 −K 0 mixing and b → ssd applies also for the nonlocal contributions involving 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 7 (µ)x c π/α s , whereC 7 (µ) is obtained from RG evolution in the same way as forK 0 − K 0 mixing. A very compact form of RG equations was presented in [14] µ d dµ D =γ T · D, Here C is a vector of C i , i = 1, . . . 6, C ± = C 1 ± C 2 , 2 andC 7 was split toC 7 = C 7+ + C 7− , where the distribution between C 7+ and C 7− is arbitrary. At LO we have for the matching at weak scale D T (µ W ) = (1, 0, 0, 0, 0, 0, 0, 0), so that the nonzero value ofC 7 (µ) comes entirely from the running, from mixing with C 1 . At µ b the solution of RG running at LO is withγ = α S 4π γ (0) and V a matrix that diagonalizes the LO anomalous dimension matrix, γ 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 → ssd branching ratios can be large, if NP effects are due to exchange of particle(s) with conserved charge. The resulting effective weak Hamiltonian, Eq. (4), depends on four couplings, g s→d , g d→s , g b→s , g s→b and an overall mass scale M X , that in this appendix we set to M X = 10 TeV (this then fixes the overall normalization of g i ). In order to have large b → ssd branching ratios and simultaneously avoid bounds from K −K mixing and B s −B s mixing a hierarchy between couplings is required. Another way of looking at this is that, if a large b → ssd decay branching ratio (we will quantify what "large" means below) is found by Belle II and/or LHCb this would imply that a region of parameter space with g s→b ∼ g s→d and/or g b→s ∼ g d→s would be excluded. We show this below.
The experimental constraints from K −K mixing and B s −B s mixing give the following upper bounds (fixing M X = 10 TeV and using bounds from Eq. (1)) ε sd ≡ |g d→s g * s→d | ≤ We also define the following two ratios of coupling constants R = g s→b g s→d ,R = g b→s g d→s .
We now show that a measured lower bound on the b → ssd branching fraction excludes values of R,R that are close to 1. For definiteness, we assume that the NP field X couples to the quarks with the Dirac structure Γ = P R , as in RPV SUSY. Similar bounds can be derived for any other Dirac structure Γ.
The amplitude for theB → f transition mediated by the operator (sb)(sd), Eq. (4), is A(B → f ) = 1 M 2 X f |g d→s g * s→b Q 4 + g b→s g * s→dQ 4 |B = r 4 M 2 X f |Q 1 |B (g d→s g * s→b ∓ g b→s g * s→d ), where 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 1 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 meson mixing, Eq. (C1). The absolute value of the l.h.s on the other hand is assumed to be bounded from below from the measurement of b → ssd branching ratio, cf. Eq. (C3). We then have B 2 < |g b→s g * s→d ∓ g d→s g * s→b | 2 ≤ ε 2 sd |R| 2 + ε 2 bs 1 |R| 2 + 2ε sd ε bs . (C5) If B ≥ 2 √ ε sd ε bs , then the above inequality rules out a range of values for |R|, The same bound with ε sd ↔ ε bs holds also for |R|. The requirement B ≥ 2 √ ε sd ε bs corresponds to the requirement that B(B → f ) > 4B(B → f ) NMFV , with the NMFV predictions for branching ratios given in Table VI.