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Supersymmetric large tan(beta) corrections to DeltaM_(d,s) and B_(d,s) ->mu+mu- revisited

We point out that in the minimal supersymmetric standard model terms from the mixing of Higgs and Goldstone bosons which are connected to the renormalization of tan(beta) via Slavnov-Taylor identities give rise to corrections that do not vanish in the limit where the supersymmetric particles are much heavier than the Higgs bosons. These additional contributions have important phenomenological implications as they can lead to potentially large supersymmetric effects in DeltaM_d and to a significant increase of DeltaM_s relative to the standard model prediction for a light pseudoscalar Higgs A0. We calculate all the missing one-loop pieces and combine them with the known effective non-holomorphic terms to obtain improved predictions for the B_(d,s)-anti-B_(d,s) mass differences DeltaM_(d,s) and the branching ratios of B_(d,s) ->mu+mu- in the large tan(beta) regime of the minimal supersymmetric standard model with minimal flavor violation.


I. INTRODUCTION
In minimal supersymmetric (SUSY) extensions of the standard model (SM) soft SUSY breaking terms are introduced that explicitly violate the underlying symmetry without spoiling the cancellation of quadratically divergent radiative corrections to the Higgs and other scalar masses. These soft terms must have positive mass dimension and the scale M SUSY associated with them should be below a few TeV to naturally maintain the hierarchy between the electroweak scale v and the Planck or any other very large energy scale. While theoretically little is known definitely about the origin and mechanism of the SUSY breaking itself, the soft mass terms will be measured and constrained as superpartners are detected. If SUSY is the solution to the hierarchy problem, then the Tevatron may, and the LHC will likely, find direct evidence for it. Meanwhile, the soft terms are also indirectly constrained by low-energy observables such as ∆M d,s , B(B → X s γ), B(B d,s → µ + µ − ), B(B + → τ + ν τ ), and (g − 2) µ . From these measurements one can learn about the structure of SUSY breaking.
For large sparticle masses, i.e., v/M SUSY → 0, the effects of SUSY degrees of freedom can be absorbed into the coupling constants of local operators in an effective theory that arises after decoupling the heavy particles. The corresponding low-energy theory is a two-Higgs-doublet model (THDM) of type II. In order for the THDM to be well-defined beyond tree-level, the effective couplings need to be calculated in the limit of unbroken SU (2) × U (1) symmetry. At the matching scale, some of the corrections to the effective couplings do not vanish for M SUSY → ∞ if the Higgsino parameter µ is assumed to be of comparable size, i.e., µ = O(M SUSY ). In addition, some of the corrections can be enhanced by the ratio tan β = v u /v d of the vacuum expectation values (VEVs) v d,u of the two Higgs doublets H d,u that separately give masses to the down-and up-type fermions. As a result, they can be sizable, of order α s tan β ≃ 1 for values of tan β ≫ 1, and need to be resummed if applicable.
In the minimal supersymmetric SM (MSSM) with minimal-flavor-violation (MFV), four different types of large tan β contributions that affect the interactions between Higgs bosons and SM fermions have been identified: (i) corrections to the vertices between a Higgs boson and down-type fermions [1], which are interpreted as corrections to the Yukawa couplings y d and resummed [2], (ii) similar corrections to the vertices between a Higgs boson and up-type fermions [3] which are not resummed, (iii) corrections to the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [4], and (iv) flavorchanging neutral Higgs vertex corrections [5], which do not appear at tree-level.
The purpose of this article is it to point out that there are additional terms from mixing of Higgs and Goldstone bosons [6] which are connected to the renormalization of tan β. These new contributions have important phenomenological implications as they can lead to potentially large SUSY effects in ∆M d and to a significant increase of ∆M s relative to the SM prediction in a certain region of the allowed parameter space. Both findings go against common lore [7,8,9,10,11,12], but they are an unavoidable consequence of the analysis presented here.
This article is organized as follows. In the next section we derive the tan β enhanced corrections to the effective Higgs interactions with quarks of the third generation, including all terms that arise from the one-loop mixing between eigenstates of the two Higgs doublets. Analytic formulas for the neutral Higgs contributions to ∆M d,s and B(B d,s → µ + µ − ) are presented in Sec. III. Sec. IV contains a numerical analysis of ∆M d,s in the MFV MSSM with large tan β taking into account all relevant constraints from flavor and collider physics. Concluding remarks are given in Sec. V.

II. EFFECTIVE THEORY
Beyond leading order, the physical Higgs fields A 0 and H ± can mix with the longitudinal components of the FIG. 1: Diagrams in the MSSM that lead to non-decoupling corrections to the G 0 A 0 mixing self-energy. See text for details.
gauge bosons fields Z 0 and W ± through loop corrections. This mixing has to be removed for on-shell momenta by suitable rotations of the fields. Due to the connection between the longitudinal gauge bosons and the Goldstone bosons G 0 and G ± , this procedure also implies field renormalization for the mixing between Higgs and Goldstone bosons. The relation between the terms with gauge and Goldstone bosons is given by Slavnov-Taylor identities (STIs), derived from the invariance of two-point functions under Becchi-Rouet-Stora-Tyutin transformations. In our case the relevant STIs are given to one-loop order by where and hatted quantities denote amputated and renormalized mixing self-energies. The ellipses stand for terms that vanish in the limit k 2 → M 2 S which are irrelevant for the further discussion.
In the renormalized mixing self-energies, besides the field renormalization counterterms δZ S ′ S , the contributions from the tadpole renormalization, δt d,u , and from the renormalization of the VEVs, δZ d,u , need to be included. The tadpole counterterms are fixed by the requirement that the properly minimized scalar potential should have no finite tadpole terms,t d,u = t d,u + δt d,u = 0. The renormalization of the VEVs,v d,u = (1 + δZ d,u )v d,u , translates into the renormalization of the gauge boson masses and tan β, δ tan β = (δZ u − δZ d ) tan β. Here quantities without hat denote unrenormalized contributions.
Decomposing the V S mixing self-energy viaΣ V S µ (k) = k µΣ V S (k 2 ) and using the above conventions the renormalized self-energies can be written aŝ where again terms that vanish on-shell have been omitted. Furthermore the abbreviations s β = sin β, c β = cos β, t β = tan β, etc. have been used and the tadpoles of the H d,u fields expressed in terms of those of the Higgs mass eigenstates h 0 and H 0 . Requiring now that the mixing between longitudinal gauge and Higgs bosons vanishes when the Higgs boson momentum is on-shell,Σ V S (M 2 S ) = 0, one obtains for the field renormalization counterterms The presence of δt β in the above equalities illustrates that beyond tree-level a definition of tan β is needed, which in turn controls the extent of mixing between Higgs and Goldstone boson fields. Several prescriptions for the renormalization of tan β have been studied in the literature, but from a careful analysis [13] it was found that in order to avoid large higher order corrections, the best scheme is to use DR renormalization for tan β, which is manifestly processindependent to all orders and gauge-independent at the one-loop level within the class of R ξ gauges. If only the leading contributions for large tan β are retained, the counterterm δ tan β is thus identical to zero.
In the limit tan β ≫ 1, the leading contribution to the field renormalization counterterms involving SUSY loops are of zeroth order in tan β. The Feynman diagrams that lead to the non-decoupling corrections to the G 0 A 0 mixing self-energy are shown in Fig. 1. Calculating these, the corresponding G ± H ∓ diagrams, and the tadpole counterterms in the limit of unbroken SU (2) × U (1) symmetry we find and The variables xf = m 2 fL /m 2 fR and x i = M 2 i /µ 2 denote the ratios of the left-and right-handed sfermion masses and the soft SUSY breaking masses M 1,2 and the Higgsino parameter µ squared. We assume CP conservation, so all soft SUSY breaking terms are real.
For sizable values of the tri-linear soft SUSY breaking couplings A t,b the correction ǫ GP is dominated by the stop and sbottom contributions, which are proportional to the square of the Yukawa couplings y t,b . The stau and bino corrections are numerically insignificant. Assuming the values of |µ|, |A t |, mt L , and mt R to be degenerate and keeping only the stop contribution one finds ǫ GP ≃ −sign(µA t )/(32π 2 ) ≃ −3 × 10 −3 sign(µA t ), while |ǫ GP | can reach ≃ 10 −2 for tan β ≫ 1 and natural choices of sfermion masses and soft SUSY breaking parameters in the TeV range.
The corrections (4) as such are not tan β enhanced. However, since they describe the mixing between eigenstates of the two Higgs doublets, they replace tan β suppressed Higgs-fermion couplings with the non-suppressed Goldstone-fermion couplings. The tan β suppression at tree-level is therefore effectively lifted at the one-loop level. Obviously, this can only occur at the next-toleading order (NLO) and there are no resummable enhanced tan β corrections beyond that order.
The term ǫ GP can also be derived from the STIs of the CP -even neutral Higgs sector. In this case one demands un-mixing of the on-shell Higgs bosons at loop level, and needs to take into account the renormalization of other parameters of the Higgs potential in the broken phase, such as the on-shell renormalization of the gauge boson masses. We find As a result, the mixing of h 0 and H 0 receives particularly large contributions from ǫ GP if the difference between the masses M h 0 and M H 0 is small. The term ǫ GP essentially describes the mixing between the two Higgs doublets at one loop. Therefore the resulting contributions are universal for the different CP and charge scalar eigenstates, up to the different mixing in the CP -even neutral sector, which is described here through the Higgs masses in Eq. (7).
The diagonalization of the Higgs mass matrix necessarily induces diagonal and off-diagonal couplings of the neutral and charged scalar to the quark fields. In the basis of neutral and charged Higgs mass eigenstates the large tan β corrections to the effective Higgs interactions with quarks of the third generation can be cast into the following form Here G F denotes the Fermi constant, V eff ij are the physical CKM matrix elements, and m f are running MS masses evaluated at a scale of order m t , which are connected to the Yukawa couplings through The subscripts L and R indicate the chirality of the quark fields involved in the interaction.
As the field renormalization counterterms δZ S ′ S in Eqs. (4) and (7) cancel the momentum-independent part of the mixing self-energies Σ S ′ S (k 2 ), dimension four operators related to the mixing of the scalar fields {S, removed from the effective theory. Alternatively, if the vanishing of the S ′ S mixing in the full theory is not enforced by suitable renormalization conditions, then diagrams with insertions of dimension four operators that mix the scalar fields S ′ and S will contribute on the effective side. If implemented correctly, the two strate-gies lead naturally to identical results for physical observables.
The unique role of neutral Higgs double penguin (DP) contributions to ∆M d,s and B(B d,s → µ + µ − ) has been extensively discussed in the literature [7,8,9,10,12,14]. In the following, we extend these analyses by incorporating the effects due to the new term ǫ GP . This allows us to obtain improved predictions for ∆M d,s and B(B d,s → µ + µ − ) in the large tan β regime of the MFV MSSM based on the SU (2) × U (1) symmetry limit.
Predicting the B d,s -B d,s mass differences involves integrating out heavy degrees of freedom at a scale of order m t by matching on to the effective Hamiltonian In the MFV MSSM with large tan β the numerically dominant contributions to ∆M d,s are induced by the two effective operators Combining the flavor-changing neutral Higgs couplings of Eq. (8) we find that the initial conditions of the corresponding Wilson coefficients are given by where and Here we have used the approximations sin β ≃ 1 and cos β ≃ 0 valid for tan β ≫ 1. The same relations are employed in the following whenever it is justified. The first line of Eq. (18) where the factors P LR 2 = 2.56 and P SLL 1 = −1.06 condense renormalization-group-improved NLO QCD corrections [16] and the relevant matrix elements [17]. In our numerical analysis we employ the unquenched staggered three flavor results f B d = 216(22)MeV [18] and f Bs = 260(29)MeV [19] for the B d,s -meson decay constants obtained by the HPQCD Collaboration, while we take |V eff * tb V eff td | = 86(14) × 10 −4 , |V eff * tb V eff ts | = 41.3(7) × 10 −3 [20], M B d = 5.2793GeV, and M Bs = 5.3696GeV [21].
In the case of the rare decays B d,s → µ + µ − , the effective Hamiltonian that arises after removing all heavy particles as active degrees of freedom is given by where the electromagnetic coupling α em and the weak mixing angle s W are naturally evaluated at the electroweak scale [22].
In the large tan β regime of the MFV MSSM only two effective operators can have a sizable impact on B d,s → µ + µ − , namely The same flavor-changing Higgs vertices that generate the dominant contribution to ∆M d,s induce enhanced tan β corrections to the Wilson coefficients of the semileptonic operators Q S and Q P . The effective couplings of Eq. (8) lead to the following matching conditions where m µ = 105.66 MeV [21]. While C P remains unchanged with respect to the analytic expression first presented in [7], C S picks up an additional contribution due to ǫ GP given by the second term in the second line of Eq. (23).
The DP contribution to the branching ratios of B d,s → µ + µ − can be expressed to a very good approximation in terms of the initial conditions C S and C P as where the B d,s -meson lifetimes are taken to be τ B d = 1.527ps and τ Bs = 1.454ps [23].

IV. NUMERICAL ANALYSIS
We are now in the position to analyze the impact of the correction ǫ GP on the prediction of ∆M d,s in the MFV MSSM with large tan β, taking into account the constraints from the low-energy observables ∆M K , |ǫ K |, B(B → X s γ), B(B → X s l + l − ), B(B d,s → µ + µ − ), and B(B + → τ + ν τ ), as well as the limit on the lightest Higgs boson mass M h 0 . In the calculation of the flavor physics observables all relevant contributions stemming from A 0 , h 0 , H 0 , H ± , andχ ± exchange are taken into account. More precisely, in the case of ∆M K and |ǫ K | we rely on the formulas given in [10], while our calculation of B(B → X s γ) includes all tan β enhanced charged Higgs and chargino corrections [3] as well as the DP contribution [9]. In the case of ∆M s we combine the neutral Higgs effects with the tan β resummed terms from charged Higgs box diagrams [7,8], while for B(B + → τ + ν τ ) we employ the formula first derived in [24]. In all cases we supplement the existing expressions with the corrections stemming from the new term ǫ GP and use the complete formulas Eqs. (9) to (13) forǫ 3 , ǫ 0 , ǫ Y , ǫ ′ 0 , and ǫ ′ Y . Another important difference with respect to the preceding analyses [7,8,9,10,12] is the fact that we do not evaluate the mixing angle α and the masses M h 0 and M H 0 appearing in Eqs. (18) and (23) at tree-level, but include the dominant one-loop corrections [15], which are essential to obtain M h 0 > M Z . While the inclusion of these higher order corrections turns out to have a minor impact on ∆M s and B d,s → µ + µ − , we find that they have a profound effect on ∆M d , as they invalidate the common assumption [7,8,9,10,12] that C SLL 1 gives only a negligible contribution to the B d -B d mass difference. 2 In our numerical analysis we focus on scenarios with heavy sparticles and the mass scale of the Higgs sector close to the electroweak scale. We allow the parameters to float freely in the following ranges: 10 ≤ tan β ≤ 60, 100 GeV ≤ M A 0 ≤ 500 GeV, 1 TeV ≤m i ≤ 2 TeV 2 In [25] it has been claimed that in ∆Ms the contribution due to C SLL 1 may amount to 80% (45%) of C LR 2 for M A 0 = 150 GeV (200 GeV). We are unable to reproduce these results.
wherem i = mt L , mt R , mb R , mτ L , mτ R , M 1 , M 2 , Mg, and 1 TeV ≤ |M i | ≤ 2 TeV forM i = µ, A t , A b , A τ . The SM parameters are fixed to α s = α MS s (m t ) = 0.109, m t = 165 GeV, m b = 3 GeV, m s = 0.06 GeV, m d = 0.003 GeV, m τ = 1.78 GeV, α em = α MS em (M Z ) = 1/127.9, and s 2 W = sin 2θ W = 0.231 [21]. In order to find the boundaries of the allowed parameter space we perform an adaptive scan of the 14 SUSY variables employing the method advocated in [26]. The correctness of the obtained results has been independently verified by a scanning procedure based on random walk techniques.
To simplify our numerical analysis we set all CKM factors and non-perturbative parameters to their central values and combine experimental and theoretical uncertainties into bounds corresponding to 95% confidence levels (CLs) by adding theory errors linearly. Severe constraints on the SUSY parameter space follow from B(B → X s γ), B(B s → µ + µ − ), B(B + → τ + ν τ ), and M h 0 . In the case ofB → X s γ the most recent SM calculations [27] are used and B(B → X s γ) is required to lie in the interval 2.7 ≤ B(B → X s γ) × 10 4 ≤ 4.4. Since the SM prediction of B(B → X s γ) is now lower than the experimental world average [23] by about 1.4 σ, a cancellation between the constructive charged Higgs corrections and the chargino contribution is easier to achieve than in the past, where the theoretical result used to be above the experimental one. As far as B s → µ + µ − is concerned all parameter points are required to satisfy B(B s → µ + µ − ) < 1.0 × 10 −7 [28], while, in view of the sizable experimental [29] and theoretical uncertainties, we use 0.2 ≤ B(B + → τ + ν τ ) × 10 4 ≤ 2.5 in the case of B + → τ + ν τ . Because the interference between SM and charged Higgs corrections is necessarily destructive [30], B + → τ + ν τ may become the most stringent constraint in the near future, in particular if improved measurements of B(B + → τ + ν τ ) will not differ much from the SM expectation. Finding B(B + → τ + ν τ ) close to its SM prediction would also have a important effect on our numerical analysis since light values of M A 0 would be disfavored in such a case. Concerning the lightest neutral Higgs boson we ensure M h 0 > 114.4 GeV [31].
The constraints from K-and the remaining B-physics observables are much less restrictive. ∆M K and |ǫ K | are allowed to differ from their experimental values [21] by ±50% and ±40%, while we reject parameter points that reverse the sign of the amplitude A(b → sγ) with respect to the SM, as they correspond to B(B → X s l + l − ) values higher than the measurements [32] by around 3 σ [33]. In the case of B d → µ + µ − we require B(B d → µ + µ − ) < 3.0 × 10 −8 [28]. Notice that, we do not take into account the experimental constraint from (g − 2) µ , because in our scenario the slepton sector parameters are uncorrelated with the ones of the squark sector, so that (g − 2) µ does not lead to any restriction.
The MFV MSSM prediction for the mass differences ∆M d,s as a function of M A 0 can be seen in Fig. 2. The blue (dark gray) areas correspond to the full re-sults obtained from Eqs. (17) to (20), while the orange (medium gray) and yellow-green (light gray) regions are obtained after removing successively the contributions due to ǫ GP and the one-loop corrections to α, M h 0 , and  [36] and the CKM factors and B d,s -meson masses given earlier by adding all errors in quadrature. For a critical discussion of hadronic uncertainties in ∆M SM d,s we refer to [20]. From Fig. 2 it is evident that, whereas one-loop corrections to α, M h 0 , and M H 0 have only a minor impact on ∆M s , which slowly loses importance with increasing M A 0 , they are essential to obtain a correct prediction in the case of ∆M d . While already this result is interesting by itself, truly spectacular effects arise after the inclusion of the new term ǫ GP . Now large negative and positive corrections to ∆M d (∆M s ) of up to −0.14 ps −1 and 0.05 ps −1 (−5.6 ps −1 and 1.5 ps −1 ) are possible in the mass window 130 GeV M A 0 160 GeV without violating any existing constraint from flavor and collider physics. These large corrections typically arise for 15 tan β 30 and µ, A t 1.5 GeV. Their size is (slightly) less pronounced for larger (smaller) values of mt L and mt R (Mg), while they are highly uncorrelated with the remaining SUSY parameters. Therefore they do not correspond to exceptional points in the large tan β and small M A 0 region of the MSSM parameter space.
As pointed out above, the small M A 0 region may be severly constrained by a more precise determination of B(B + → τ + ν τ ). The large effects shown in Fig. 2 occur only for B(B + → τ + ν τ ) 10 −4 . If future measurements should find a value in the ballpark of 1.5 × 10 − 4 with a small error of 30% or less, large corrections to ∆M d,s would be excluded within the MSSM with MFV.
It is also compelling to analyze the impact of neutral Higgs DP contributions in the double ratio R DP sd = (∆M DP s /∆M SM s )/(∆M DP d /∆M SM d ), which has smaller hadronic uncertainties than the ratios ∆M DP d,s /∆M SM d,s themselves. The size of the possible departures of R DP sd from the SM value 1 should be compared to the total uncertainty of the double ratio in Fig. 3, where the blue (dark gray) area represents the full result derived from Eqs. (17) to (20). The 68% CL and central value of the tree-level determination R exp sd = 1.04(54) is indicated by the light gray band and the dotted line. The quoted range of R exp sd is obtained from Eq. (25) using λ = 0.2257 (21), R b = 0.42(4), γ = 67(31) • [37] and the remaining input specified above by adding all uncertainties in quadrature. Because of the poor knowledge of γ from B → D ( * ) K ( * ) the double ratio R exp sd is only weakly constrained at the moment.
Two properties of R DP sd deserve a special mention. Our numerical analysis reveals that i) effects due to ǫ GP and the one-loop corrections to α, M h 0 , and M H 0 cancel almost entirely in the double ratio R DP sd since their size is strongly correlated between ∆M DP d and ∆M DP s and ii) the MFV MSSM with large tan β predicts R DP sd ≤ 1 as demonstrated in Fig. 3. The impact of a future precision measurement of γ by the LHCb experiment [38] is also illustrated in this figure. Assuming γ = 67(5) • but ξ unaltered leads to R exp sd = 1.04 (11). The corresponding 68% CL and central value is indicated by the dark gray band and the dotted line in Fig. 3. It can be seen that even with this improvement in precision, R exp sd offers only limited potential for exclusion of new physics through a deviation from its SM value. A similar conclusion has been drawn in [20].

V. CONCLUSIONS
To conclude, we have pointed out that in the MSSM terms from the mixing between eigenstates of the two Higgs doublets give rise to tan β enhanced corrections that do not vanish in the limit where the SUSY particles are much heavier than the Higgs bosons. We have calculate all one-loop corrections of this type by matching the full MSSM on to an effective two-Higgs-doublet model of type II. After combining the missing effective couplings with all known non-holomorphic terms we obtain improved predictions for the B d,s -B d,s mass differences ∆M d,s and the branching ratios of B d,s → µ + µ − in the large tan β regime of the MSSM with minimal-flavorviolation. Our numerical analysis shows that these universal contributions have striking phenomenological implications as they can lead to large SUSY effects in ∆M d and to a significant enhancement of ∆M s relative to the SM prediction for a light pseudoscalar Higgs boson A 0 with mass in the range between 130 GeV and 160 GeV.