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Exact NLO strong interaction corrections to the Delta F=2 effective Hamiltonian in the MSSM

We compute the order alpha_s^3 (next-to-leading) corrections to the Wilson coefficients of the Delta F=2 effective Hamiltonian in the Minimal Supersymmetric Standard Model with completely arbitrary soft terms. These results are relevant for phenomenological studies of neutral meson mixings in the presence of large squark mass splittings, in particular in hierarchical scenarios of supersymmetry. These corrections achieve a considerable reduction of the uncertainty related to the matching scale ambiguity, and are therefore numerically relevant. We briefly analyze the effect of certain mass splittings on the size of the NLO corrections, and compare it with the case of almost degenerate squarks in the mass insertion approximation.


Introduction
There is an almost absolute consensus among the particle physics community that unknown physics must exist at energies of around the TeV. Indeed, the minimal Higgs mechanism, so far completely untested, is quite unlikely to be the true mechanism for electroweak symmetry breaking; and even if it were, an additional explanation should be provided for the stabilization of the electroweak scale. Another indication of new physics around these scales is given by, for example, the dark matter problem. Low-energy supersymmetry is a leading candidate to fill that gap since it provides natural and simple explanations to those conundrums.
However, over the last decade, two other things have become clear. First, that the structure of flavor symmetry breaking of the Standard Model is a very peculiar one, and second, that, quite unexpectedly, flavor physics experiments confirm it to great accuracy. Indeed, any new physics model one can think of leads naturally to violent flavor breaking, and the experimental limits on flavor-changing neutral currents (FCNC's) are among the most stringent constraints that any model must satisfy.
The Minimal Supersymmetric Standard Model (MSSM) is no exception: an enormous set of new flavor violating parameters arise from its supersymmetry-breaking sector, which in the quark sector can be understood as a general misalignment between quark and squark mass matrices. More specifically, a simultaneous flavor rotation of quark and squark fields diagonalizing the Yukawa matrices leads to the so called super-CKM basis, in which no tree-level FCNC couplings are present. But in this basis the squark mass matrix M 2 is not generally diagonal, and an extra rotation of the squark fields is necessary to diagonalize this matrix, introducing strong interaction flavor violating couplings. Therefore, SUSY contributions to flavor violating observables mediated by strong interactions will easily compete with the SM contributions, which are driven by weak interactions, and well tested experimentally. Flavor constraints on these flavor violating parameters have been studied extensively, and the conclusion to be taken from those studies is, basically, that it is quite difficult to reconcile the relatively low SUSY masses required by naturalness with a generic flavor structure of the soft breaking terms.
A solution to this problem is to assume that the structure of flavor violation of the new physics respects the hypothesis of Minimal Flavor Violation (MFV), by virtue of which the Yukawa couplings are the only source of flavor symmetry breaking. However, this is quite a pessimistic scenario for new physics searches in the quark sector, and it is useful for many purposes to go beyond it. Indeed, present indications of new physics in B d,s and K processes [1,2,3] seem to require a departure from MFV.
In the MSSM, a suitable approach for phenomenological studies that takes into consideration all these issues, is to assume that squark masses are very nearly degenerate. This case is parametrized by a squark mass matrix of the form (M 2 ) ij = M 2 s (1 + δ) ij , where M s is an "average" squark mass, and δ is a matrix with entries much smaller than one. These small parameters, or mass insertions, parametrize the departure from MFV, and constraints on their values derived from flavor observables provide valuable information for model building. They are usually treated as expansion parameters, which at leading order (linear and quadratic in δ for ∆F = 1 and ∆F = 2 processes respectively), define what is coventionally know as the Mass Insertion Approximation (MIA) [4]. Strong bounds have been derived for these quantities from FCNC processes [5,6,7,8] and vacuum stability requirements [9,10] as well as from charged-current processes [11].
However, it is also of interest to consider other scenarios with non-degenerate squark masses. For example, it has been argued that a "hierarchical" setup in which the first two generations of squarks are much heavier than the rest of the SUSY spectrum (lying near the electroweak scale), can satisfy naturalness criteria [12]. In this framework, correlation patterns between ∆F = 1 and ∆F = 2 observables can be quite different from the ones in the degenerate case. Indeed, as shown in Ref. [13], in these scenarios the bounds from B → X s γ can be partially evaded, allowing for a large B s mixing phase. This would be of great relevance if the experimental indication for such a large phase [14,15,2] is confirmed.
In order to perform phenomenological studies of these scenarios, corresponding calculations of flavor violating processes have to be performed. Here we focus on ∆F = 2 processes, that is, K −K, D −D and B d,s −B d,s mixing. These low energy observables are more conveniently computed in the framework of an effective theory in which heavy modes have been integrated out. The most general effective hamiltonian relevant for ∆F = 2 processes is given in Eq. (6), where the Wilson coefficients C i contain the information from heavy modes. Then the observables are expressed as functions of the Wilson coefficients and the matrix elements of the operators, and arise from the ∆F = 2 amplitude in the effective theory: where we have written the matrix elements of the operators in terms of tree level matrix elements O (0) . The matrix elements must be computed using some non-perturbative approach, for example in the lattice. The SUSY contributions are encoded inside the Wilson coefficients, which are evaluated by matching the full theory (MSSM) onto the effective theory at some matching scale µ. According to the renormalization group (RG) prescription for the resummation of large logarithms, the matching scale µ must be close to the SUSY scale, and the scale at which matrix elements are computed must be close to a relevant mass scale in the effective theory (for example m b in the case of B −B mixing), and the Wilson coefficients at the matching scale must be used as initial conditions for the RG evolution that provides the Wilson coefficients at the low scale. This evolution is governed in particular by the anomalous dimensions of the operators. Leading order (LO) strong interaction matching conditions in the MSSM have been computed in Refs. [16,17,18], and arise from the squark-gluino box diagrams shown in Appendix A.1. The corresponding next-to-leading order (NLO) corrections arise from the two loop diagrams shown in Appendices A.2 -A. 4, and have been computed in Ref. [19] within the Mass Insertion Approximation. The anomalous dimension matrix for the complete set of operators (Eq. (7)) has been computed at NLO in QCD in Refs. [20,21].
The purpose of this paper is to present the computation of the NLO matching conditions beyond the MIA, in the presence of arbitrary soft terms, and in particular for arbitrary squark mass splittings. The motivation for a NLO determination of the matching conditions is three-fold. First, LO matching conditions are both scale-and schemeindependent, so in order to get scheme independent results and NLO scale invariance it is necessary to go beyond the leading order. Second, the leading order corrections are proportional to α 2 s . Since at LO neither the renormalization scale or the scheme can be specified for the strong coupling, LO results show a particularly high uncertainty related to the scheme and scale ambiguities. Third, this uncertainty is particularly severe due to the large anomalous dimensions of the ∆F = 2 operators involved. These uncertainties are largely cured by the NLO corrections, from about 10-15% to a few percent, as shown explicitly in Ref. [19].
The ∆F = 2 amplitude in the MSSM up to NLO can be written as where F (0) i and F (1) i are the LO and NLO contributions, and we have factored out a common α 2 s . The matching of the full theory onto the effective theory is performed by imposing that the effective and MSSM amplitudes are equal at and below a matching scale µ. An order by order identification of Eqs. (1) and (2) leads to the following formula for the Wilson coefficients: Thus, the NLO matching calculation requires the computation of the matrix r and the functions F i . The matrix r is obtained from the renormalization of the operators of the effective theory, and its computation is described in Section 2. This matrix has been computed before in several renormalization schemes (see Refs. [20,21,19]), and we agree with their results. The functions F i are obtained computing the one-and two-loop diagrams in the MSSM (see Appendix A). The details of this computation are described in Section 3, and the functions F (1) i obtained here are the main new results of this paper. Both calculations, the effective and the full theory amplitudes, are carried out in the NDR scheme (dimensional regularization with anticommuting γ 5 ), with modified minimal subtraction (MS) of ultraviolet divergencies. Also, we choose massless external quarks with zero external momenta. This choice of external states introduces infrared (IR) divergencies from diagrams in which a gluon connects two external legs. We regularize these divergencies with an unphysical gluon mass λ. While both r and F (1) i depend on the external states (and are therefore IR divergent), this dependence cancels in the Wilson coefficients, as expected.
After describing the computation of the relevant amplitudes, we discuss briefly in Section 4 some issues related to the renormalization of ultraviolet divergencies and the renormalization scale dependence. In Section 5 we mention some of the checks that can be done to ensure the correctness of the results. In Section 6 we show how reduce the exact results to the MIA and to the MIA with non-degenerate squarks, which allows to compare our results with those in Ref. [19]. Finally, some results are presented in Section 7.
Before getting down to brass tacks, we would like to set some notation and specify some definitions related with rotation and mass matrices beyond leading order. At tree level, the super-CKM basis is defined by doing a joint rotation in flavor space of quark and squark fields such as to diagonalize the tree level Yukawa matrices. The resulting squark mass matrix is not diagonal and defines the tree level mass insertions. In this basis there are no tree level FCNC's. This matrix is diagonalized by an additional rotation of the squark fields,d where (q I L ,q I R ) denote the squark fields in the (tree level) super-CKM basis, andq is the mass eigenbasis. The rotation matrices Γ L and Γ R are 3 × 6 matrices, and the indices U, D will be omitted hereon, which raises no confusion. In the mass eigenbasis, tree level FCNC's appear; for example a flavor changingq i -q j -g vertex is generated with the following Feynman rule: where T a are the color matrices and P L,R are the chiral projectors. At NLO, a subtlety arises because a squark-gluino loop generates a finite flavorchanging self energy for the quark fields (see Fig. 2). At this point, one must specify what is meant by the super-CKM basis at NLO, since the definitions for the mass insertions depend on that choice. We believe that the most natural definition for the super-CKM basis is the one for which quark fields do not mix at one loop, and tree-level FCNC's are absent. In this case the rotation matrices, the mass insertions and the CKM matrix differ from the tree level ones. This criterium does not coincide with that in Ref. [11], but it has no effect when comparing our results with Ref. [19], since this is not an issue in the degenerate case. A comment in favor of the criterium adopted in Ref. [11] is that mass insertions are directly related with SUSY-breaking parameters in the lagrangian. A further discussion on this issue is provided in Section 4, and the formulae necessary to switch from one criterium to the other is provided in Appendix C.

Effective Hamiltonian for ∆F = processes at NLO
The most general effective Hamiltonian for ∆F = 2 processes up to operators of dimension six can be written as where C i are the Wilson coefficients and O i are the dimension six ∆F = 2 operators. In four dimensions there are eight independent operators of this type. Here we choose the following basis: where P L,R = (1∓γ 5 )/2 are the usual chiral projectors. The operatorsÕ 1,2,3 are obtained from O 1,2,3 by exchanging L ↔ R. To simplify the notation throughout the paper we focus on the case of B s −B s mixing; for the cases of K, D and B d mixing one should make obvious replacements of the quark fields. In order to define the effective Hamiltonian beyond leading order, one must specify a renormalization scheme. Here we choose to regularize ultraviolet divergencies in dimensional regularization, where d = 4 identities between operators do not hold. This means that one must complete the effective Hamiltonian with a set of evanescent operators that vanish in 4 dimensions but can give finite contributions beyond leading order if they are multiplied by a divergence. The choice of a set of evanescent operators is not unique, and different sets lead to different subtractions, so specifying this set is necessary to fix the renormalization scheme. Here we choose the set of evanescent operators given in Ref. [21]. By requiring that the matrix elements of evanescent operators vanish in four dimensions one can omit these operators altogether from the effective Hamiltonian once the renormalization has been performed. The calculation of the NLO effective Hamiltonian amounts basically to the NLO renormalization of the operators. The matrix elements of the bare operators and of the renormalized operators are related through the renormalization constants: where Z s,b are the quark wave function renormalization factors and Z ij is the renormalization matrix necessary to renormalize properly the operators in the effective theory. The effective theory amplitude can then be written as where we have written Z ′ ij = δ ij + δZ ′ ij and the second term contains the counterterms. In order to obtain the NLO renormalized operators O i ren in terms of tree level matrix elements O i (0) , one must compute the one loop gluonic corrections such as those shown in Fig. 1. For UV divergencies we use naive dimensional regularization with modified minimal substraction (MS-NDR). Moreover, we choose zero external momenta and set the quark masses to zero. This simplifies the computation but introduces IR divergencies from the soft gluon region. We regularize this divergencies using a gluon mass λ. The same IR divergencies should appear in the full theory, and cancel in the matching, providing a non-trivial check of the calculation.
All the one loop diagrams contributing to the NLO renormalized operators are proportional to the same loop integral, and the amplitude can be written as The computation then provides the coefficients A ij , which must be extracted up to O(ǫ): The sum over j runs over both physical and evanescent operators, the later giving finite contributions to A ǫ ij .
The counterterms can now be fixed according to the MS scheme: δZ ′ ij = (α s /16πǫ)A 0 ij , which provides the leading order anomalous dimension matrix of the operators (7) in QCD: where we have used the well known QCD quark wave function renormalization factor, Z q = 1 − (α s /4πǫ) C F . We obtain in agreement with, for example, refs. [26,19]. The missing 3×3 block corresponding to the operatorsÕ 1,2,3 has been omitted: they do not mix with the other operators and their anomalous dimensions are the same as for O 1,2,3 . Finally, the NLO amplitude in the effective theory is given by For the NLO matrix r we obtain The first term is the IR divergent piece that must cancel in the matching procedure. The second term is the NLO contribution, and it is scheme dependent: it is valid only in the MS-NDR scheme. However it can be easily translated to dimensional reduction (DRED) and Regularization-Independent (RI) schemes by using the formulae in Refs. [19,20]. In fact, as pointed out in Ref. [20], the matrix r can be thought of as defining the renormalization scheme, and Eq. (13) as a definition the renormalized operators. The same scheme must be used in the full theory calculation and in the evaluation of the matrix elements in order to obtain scheme-independent results for physical amplitudes.

Details of the calculation in the MSSM
The NLO ∆F = 2 amplitude in the MSSM is obtained by computing the one-and twoloop Feyman diagrams listed in Appendix A. The one-loop contributions are the box diagrams shown in appendix A.1; the computation of these graphs is standard and will not be reviewed here. Renormalization of the two-loop diagrams requires the inclusion of these boxes with vertex counterterms, as well as pentagons containing the self-energy counterterms. These one loop integrals must then be computed up to and including terms of O(ǫ), which provide the finite scale-and scheme-dependent contributions to the NLO amplitude. We will come back to this in section 4. The computation of the two-loop diagrams is done in four steps: 1. Partial fractioning of the denominators and tensor reduction to reduce the Feynman integrals to a set of scalar integrals with three scalar propagators.
2. Decomposition of the scalar integrals down to a set of one-and two-loop master integrals. These master integrals are known functions of masses and contain divergencies up to O(1/ǫ 2 ).

Fierz rearrangement and spinor transpositions in order to express the spinor structures in terms of tree level matrix elements of the operators.
Let us describe in some detail each of these steps.
1. Since we are choosing massless quarks with zero external momenta, there are only two independent momenta appearing in the Feynman integrals. One can then decompose the denominators such that each diagram can be expressed as a sum of terms of the following type where the f 's are some functions that depend in general on all the masses appearing in the Feynman diagram (including the fictitious gluon mass), and ∆q ≡ q 1 −q 2 . Also, Γ and Γ represent Dirac and color structures. The spinors s and b might be u or v spinors and they might appear transposed and in different order, according to the chosen reference order adopted to keep track of the relative signs of interfering Feynman graphs [22]. The tensor integrals are momentum-independent (again, because external momenta are zero), so they can be expressed in terms of scalar integrals multiplied by metric tensors. For example, In this way we can express each diagram as a sum of terms of the form At this point, care must be taken when manipulating the Dirac structures Γ andΓ after contraction with the metric tensors. In particular, in D = 4 − 2ǫ dimensions, structures of the type (γ µ γ ν γ λ · · ·) ⊗ (γ µ γ ν γ λ · · ·) cannot be reduced like in D = 4, and evanescent structures (of order O(ǫ)) must be introduced. According to the NDR prescription here adopted, however, one can freely anti-commute the γ 5 and use the usual anti-commutation relations for γ matrices.
2. The resulting scalar integrals can be further reduced to a set of one-and twoloop master integrals by the method of recurrence relations [23]. This reduction can be performed automatically using the Mathematica program TARCER [24]. In this way the scalar integrals can be expressed in terms of various one-loop tadpole integrals and a single two-loop master integral, .
The result for this master integral with arbitrary masses is given in ref. [25].

Once all the loop integrations have been performed, the resulting expression for the Feynman diagram consists of a sum of terms of the form
where A, B and C are some functions of gluino and squark masses. Since in this case one-loop corrections are finite, no 1/ǫ 2 divergencies can appear, so all 1/ǫ 2 terms should (and do) cancel directly. We also get automatic cancellation of 1/ǫ terms for the diagrams shown in appendix A.4, as it should be. The rest of the diagrams (those shown in appendix A.3) contain 1/ǫ divergencies that cancel against one-loop diagrams with vertex and self-energy counterterms. We will discuss the details of the renormalization in section 4.

4.
After all the divergencies have been removed, 4D Fierz identities can be used and transposition of spinors can be performed to put all spinor and Dirac structures in suitable form. These structures must then appear in the precise combinations that constitute the tree level matrix elements of the physical operators, as for example, This, however, does not occur for individual diagrams, but only for certain groups of diagrams (and of course for the amplitude as a whole). Therefore this step provides an interesting check of the calculation, in particular of the relative sign between the different diagrams. The NLO quantities F

Renormalization
In order to reduce the ultraviolet divergencies arising from the two-loop graphs, appropriate counterterms must be introduced. In particular, the squark-quark-gluino vertex, as well as the quark, squark and gluino wave functions and masses have to be renormalized. For reference, the relevant renormalization factors in the MS-NDR scheme up to O(α s ) are which are defined in the usual way. Also, a non-diagonal squark-mass counterterm is necessary, because at one loop divergent flavor-changing squark propagators are generated. There are two ways of dealing with this issue: 1) Renormalize the squark rotation matrices so that non-diagonal squark masses are zero at one loop. In this case all the finite pieces of the one loop corrections cancel with the renormalized non-diagonal mass insertions, which are not zero anymore but of order O(α s ). 2) Renormalize the nondiagonal squark-mass parameters minimally, and include the finite pieces of the loop corrections to the flavor-changing squark propagator. In this case the squark rotation matrices are defined such that renormalized mass insertions are zero at the matching scale.
We choose the second option, and renormalize the squark mass parameters according to m 2 bare A consequence of all this is that, even though we are working in the basis of diagonal squark masses, non-diagonal mass parameters do run with the scale and contribute to the renormalization group equation.
There is an additional issue related with the fact that NDR is a regularization scheme that breaks supersymmetry. In particular, the coupling g s appearing in the quark-gluongluon vertex and the couplingĝ s that appears in the quark-squark-gluino vertex, receive different radiative corrections in the NDR scheme. In the effective theory only g s appears, so when doing the matching it is convenient to have the MSSM amplitude expressed solely in terms of this coupling. Being this is an O(α s ) effect, one can set g s =ĝ s in the twoloop amplitude, but it gives a finite contribution of O(α 2 s ) from the LO amplitude. At the end, this effect is corrected for by performing in the one loop amplitude the following replacement (see [27,19] Finally, one must address the wave-function renormalization of the external states. There are two types of contributions: gluon corrections and squark-gluino corrections, both contributing a factor of −C F to Z q (see eq. (17)). In addition, squark-gluino corrections contribute finite pieces of two types:

Flavor diagonal corrections:
These give a finite contribution to the on-shell quark wave-function renormalization constant,

Flavor changing corrections:
The presence of quark flavor changing squark-gluino loops (see Fig. 2) can be handled in two different ways, in relation to two different definitions of the super-CKM basis: 1) Tree-level definition of the super-CKM basis: This is the usual definition, in which the tree level Yukawa matrices are diagonal. At one loop, quarks of different flavors mix through the radiative corrections in Fig. 2, and one must include these corrections in the external legs.
2)"On-shell" definition of the super-CKM basis: In this case the quark superfields are (finitely) renormalized with matrix-valued renormalization factors, that induce a perturbative rotation in flavor space. These counterterms are defined at each order in perturbation theory such as to render the quark self-energies diagonal at the given order, effectively canceling the flavor-changing self-energies of Fig. 2.
As mentioned in the introduction, the mass insertions (and the squark rotation matrices) that arise in each scheme represent different quantities. Here we assume an on-shell definition of the super-CKM basis, and omit the flavor-changing externalleg corrections in our computations. For completeness, we provide in Appendix C the explicit relationship between squark rotation matrices and mass insertions in both definitions. The full results for the NLO Wilson coefficients consistent with the usual tree-level definition of the super-CKM basis can be obtained by adding an O(α s ) correction to the rotation matrices Γ ij L,R in the LO results, as explained in detail in Appendix C.
Once all the renormalization factors have been defined, one can write down explicitly the renormalization group equation for the Wilson coefficients, The renormalization group functions are given by Therefore, with the notation of Eq. (2), the explicit form of the NLO RG-equation reads where appropriate sums over i, j, k, q, q ′ are understood.

Checks of the calculation
There are several checks that can be made of the NLO calculation: 1. Cancellation of UV divergencies: All the UV divergencies arising from the two-loop graphs (specifically those in appendix A.3) must be cancelled by one loop graphs with the insertions of the counterterms specified by the renormalization factors in eqs. (17) and (18). 2. Projection onto tree-level matrix elements: The NLO amplitude must be expressible in terms of tree-level matrix elements of the physical operators, that is, . As mentioned before, this does not happen for individual two-loop graphs, and provides a check of interference between diagrams.
3. Cancellation of IR divergences: Both the amplitude in the effective theory and the amplitude in the MSSM depend on log λ, where λ is the gluon mass introduced to regularize the IR divergencies. This dependence must cancel completely in the matching when combining both amplitudes; that is, F k r ki must be IRfinite. This is a non trivial check involving three completely independent pieces of the calculation: the matrix r from the renormalization of the effective operators, F 4. NLO renormalization scale independence: The RGE (eq. (23)) must be fullfilled.
There is a close relationship between cancellation of UV divergencies and fulfillment of the RGE, but it is nevertheless a convenient check.
We have verified that our results fulfill these requirements. Moreover, a final check consists in reducing the results to the degenerate MIA and comparing them with the results obtained in ref. [19]. In the next section we address the issue of how to reduce the exact results to the MIA and the NDMIA.
6 Reduction to MIA and NDMIA 6

.1 Reduction to the Mass Insertion Approximation
Any Wilson coefficient computed in this paper has the structure of a sum of terms of the following type, where A, B, C, . . . are either L or R, and a sum is understood running over the indices i, j, k, l =d L ,s L ,b L ,d R ,s R ,b R , and q, q ′ = d, s, b, (for f 6,2 also q = u, c, t and k = u L ,c L ,t L ,ũ R ,c R ,t R ). These terms can be translated into functions of the entries of the squark mass matrix in the super-CKM basis (M 2 ) using the following relations, In the MIA, the diagonal elements in M 2 are assumed to be equal to an "average" squark mass M 2 s , and the off-diagonal elements (called mass insertions and denoted by ∆ AB qq ′ ), are assumed to be much smaller than M 2 s . In this way, a power expansion on the dimensionless mass insertions δ AB qq ′ ≡ ∆ AB qq ′ /M 2 s ≪ 1 can be made, keeping in this case only the leading terms δ 2 . The mass eigenvalues are then M 2 s (1 + O(δ)), so the functions in Eq. (24) can be Taylor-expanded around the average squark mass: Using the relations (25) it is easy to see that, for example, and similar for the other terms.
We have checked that our results, when reduced to the MIA, reproduce exactly the results in Ref. [19].

Reduction to the Non-degenerate MIA
From the exact results obtained in this paper, one of the most straightforward generalizations of the MIA results that can be obtained is the case in which the diagonal elements of the squark mass matrix are non-degenerate. In this case, squark masses can be widely different while keeping mass insertions small. We call this the Non-degenerate Mass Insertion Approximation (NDMIA), in which diagonal masses are non-degenerate but mass insertions are kept only up to second order.
In order to reduce the terms in Eq. (24) to the NDMIA, we first expand the functions aroundm = 0 and apply the relations (25), and then we split the mass matrix in two pieces: M 2 = X + ∆, where X is diagonal and ∆ ≪ X. Then we can resum the series in X, and keep only terms quadratic in ∆. For example, with the function F given by .
From the NDMIA results one can also recover the Wilson coefficients in the MIA by taking the limit in which all Xq's are equal. In the non-degenerate case, the dimensionless mass insertions can be defined normalizing by any of the different squark mass parameters. It is customary to normalize by some "average" squark mass, although it makes no real difference as long as in the degenerate limit the definition coincides with the one in the MIA. We shall understand that they are normalized by the smaller diagonal entry of the squark mass matrix.  Figure 3: Relative importance of the full NLO result with respect to the LO, for C 2 and C 3 , as a function of x h ≡m 2 12 /m 2 g , wherem 12 is a common mass for first and second generation squarks. The dashed lines correspond to the degenerate (MIA) scenario. We have chosen mg = µ = 350 GeV, and 400 GeV for third generation squark masses.

Results
The full results for the LO and NLO Wilson coefficients in the MS-NDR scheme are presented in Appendix B. In this section we present some results for the NLO Wilson coefficients in a scenario with a hierarchy of masses, making a comparison with the MIA. In line with the rest of the paper, we keep focusing on the B s system for illustration.
The considered scenario is obtained from the mass insertion approximation with non-degenerate diagonal entries, taking a common mass for the first two generation squarks (m 12 ), different from a common mass for third generation squarks (m 3 ). This corresponds to the "hierarchical" scenario of Ref. [13], wherem 3 is assumed to be near the electroweak scale, andm 12 is allowed to be heavy, up to several TeV. We therefore denote x h ≡m 2 12 /m 2 g and x l ≡m 2 3 /m 2 g , for heavy and light respectively. Note thatm 12 andm 3 are related to the true masses by corrections of O(δ). In this case the mass insertions are normalized tom 3 , which corresponds to the MIA definition when x h → x l .
The plots in Fig. 3 illustrate the relative importance of the NLO corrections, as a function of the mass splitting between the light and heavy squarks. The NLO correction is typically a ∼ 10% effect in the degenerate case, but its importance increases with the mass splitting. For heavy squarks of about a TeV, the NLO contribution to C 3 can be up to a ∼ 25% correction. However it should be mentioned that this is true in the NDR scheme and could vary in other schemes. Also, this depends on the matching scale µ, but a high sensitivity to µ would be related to large scale ambiguity at LO that is efficiently reduced at NLO. This means that the NLO contributions are either numerically important, or they achieve a considerable reduction of theoretical errors, and both situations are not easily disentangled.
In order to analyze more closely the role of the mass splittings in the NLO corrections, we consider C 1 , C 4 and C 5 as a function x l , for different splittings between x l and x h . This is shown in Fig. 4, where the dashed lines correspond to x l = x h (that is, the degenerate case), and -departing smoothly from that limit-the solid lines show increasing values of x h /x l . In these plots we take δ L,R = δ R,L = 0 and mg = µ = 350 GeV. We as a function of x l ≡m 2 3 /m 2 g , wherem 3 is a common mass for third generation squarks. The plots are in units of (α 3 s /π)δ 2 LL and (α 3 s /π)δ LL δ RR for C 1 and C 4,5 respectively. The different lines correspond to different values of x h ≡m 2 12 /m 2 g , and range from the MIA case, see that increasing the heavy scale tends to reduce systematically the size of the NLO contribution, being largest in the degenerate case.
As mentioned before, this is scheme dependent (although the conclusion might be more general). In any case, the true impact of the NLO corrections can only be established by analyzing their effect on observables. A full phenomenological analysis of these corrections and their impact on the bounds on the mass insertions (beyond the mass insertion approximation) is worthwhile, and will be presented elsewhere.

Conclusions
In this paper we have presented the computation of the NLO strong interaction corrections to the Wilson coefficients relevant for ∆F = 2 processes in the MSSM, beyond the mass insertion approximation. The full results for the Wilson coefficients in the NDR scheme are given in Appendix B.
These results are relevant for two reasons. First, NLO corrections are necessary to cancel renormalization scheme and scale dependence from the renormalized operators. This has the effect of a considerable reduction in the theoretical error. Second, in order to study scenarios with significant mass splittings one must depart from the degenerate mass insertion approximation. We have shown some illustrative examples of the effect of mass splittings in the NLO Wilson coefficients as compared to the degenerate case.
A full phenomenological study of neutral meson mixing incorporating these new corrections will be presented in the future. Corresponding calculations for ∆F = 1 processes are underway, and when available will allow to perform a complete analysis of correlations between decay and mixing observables of neutral mesons at NLO in α s .

Acknowledgements
I would like to thank Luca Silvestrini, Enrico Franco and Miguel Nebot for useful discussions, as well as Ulrich Nierste, Andreas Crivellin and Diego Guadagnoli for correspondence. J.V. is associated to the Dipartimento di Fisica, Università di Roma 'La Sapienza'.

A List of diagrams
In this appendix we present the set of one-and two-loop feynman diagrams that contribute to the MSSM amplitude. We focus on the case of B s mixing; for other neutral meson systems the external quarks must be changed. Dashed lines and i, j, k, l denote squarks, solid arrowed lines and q denote quarks and solid lines without arrows are (majorana) gluinos. We show all the possible topologies with the different insertions of external quark flavors, but an additional multiplicity is present interchanging initial and final external quarks (whenever they correspond to different contractions). For example, at leading order there are two diagrams of each of the two topologies shown in Appendix A.1, with appropriate signs given by a reference order of external legs.

B Wilson coefficients
In this appendix we present the complete expressions for the Wilson coefficients at NLO in the MS-NDR scheme. We omit the results forC 1,2,3 , which are obtained from C 1,2,3 by exchanging L ↔ R in the squark rotation matrices. The coefficients are split into LO and NLO: and depend on the matching scale µ through α s , mg andm i , and explicitly through log(m 2 g /µ 2 ). The functions depend on the squark masses through the variables x i defined as x i ≡m 2 i /m 2 g . The LO Wilson coefficients are given by where a sum is understood over i, j =d L ,s L ,b L ,d R ,s R ,b R . The functions G, H are: The NLO coefficient functions are the main result of this paper. They are scheme dependent; we present the results in the MS-NDR scheme, but can be translated easily to other schemes using formulae analogous to that presented in Ref [19]. The function Li 2 (x) denotes the dilogarithm defined in the usual way: Appropriate sums over indices are understood, in particular q, q ′ = d, s, b and Q = u, d, s, c, b, t, and i, j, k run over left and right-handed squarks of u or d-type depending on the quark they appear with in the rotation matrix: for example a term containing Γ ku L contains a sum over k =ũ L ,c L ,t L ,ũ R ,c R ,t R . All dependence on u-type quarks and squarks come only from diagrams with gluino self-energies. The NLO Wilson coefficients read The coefficients g ijk , h ijk , m ijk , n ijkl and p ijkl are the following, Finally, the functions f ij appearing in the part with four squark rotation matrices are given by: i + x 4 i (215 + 178x j ) + 4x 3 i (40 + x j (−183 + 32x j )) +x 2 j (54 + x j (160 + (215 − 42x j )x j )) + 2x 2 i (27 + x j (−18 + x j (59 + 64x j ))) +2x i x j (16 + x j (−18 + x j (−366 + 89x j )))) log +x i x 2 j (5 + 9x j (2 + x j )) + x 2 i x j (5 + x j (−74 + 5x j ))) log x i log x j C Flavor changing quark self-energies: Relationship between the tree-level and on-shell definitions of the super-CKM basis In this appendix we elaborate on the relationship between the two different definitions of the super-CKM basis addressed in Section 4. The flavor changing self-energies of Fig. 2 can be written as 1 Σ q ′ q (p) = Σ RL q ′ q (p 2 )P L + Σ LR q ′ q (p 2 )P R + / p [ Σ LL q ′ q (p 2 )P L + Σ RR q ′ q (p 2 )P R ].
When inserted as external legs, the quark propagator will provide a chiral enhancement with the chirality-flipping part of the self-energy, and we will keep only this contribution. Moreover we will expand in the external momentum, which is justified by the fact that the SUSY masses are much higher than the external quark masses. Then, the self energies we are interested in are given by Σ RL,LR q ′ q (0) = 2α s 3π mg k g(m 2 k /m 2 g ) Γ kq ′ * R,L Γ kq L,R with g(x) = (1 − x + x log x)/(1 − x).
For the case of B s mixing, the corrected couplings of the external quarks are, where, in the approximation m b ≫ m s ≫ m d , and keeping only the chirally enhanced terms, and correspondingly for L ↔ R. At this point, we denote by Γ (0) the squark rotation matrices associated with the tree-level definition of the super-CKM basis, and Γ the rotation matrices in the on-shell case. At tree-level both coincide; at the one loop level, the relationship between both definitions is given by and correspondingly for L ↔ R. The full results for the NLO Wilson coefficients in terms of Γ (0) -corresponding to the tree-level definition of the super-CKM basis-can be obtained by making the substitutions of Eq. (40) in the LO expressions given in Eq. (29).
In the degenerate MIA case, these corrections cancel. Consider for example the coefficient C 1 in Eq. (29). Making the substitutions of Eq. (40) in the rotation matrices with index i, and making the reduction to the MIA (see Section 6.1) gives, at order α 3 s , two terms: and both cancel due to the relative signs in Eqs. (40). The same happens with the other two terms arising from the Γ jq L , and in the other Wilson coefficients. Therefore, the two bases coincide in the mass insertion approximation with degenerate squarks.
Finally, the relationship between the mass insertions defined in each basis can be deduced from Eqs. (40) and (25). As we have just seen, both are equivalent in the degenerate case. In the non-denerate case, one such relation is (at leading order in the mass insertion expansion), with the notation of Section 6.2. Analogue relations for other mass insertions can be found accordingly. From this last relation one can also see that the effect disappears in the degenerate case. The effect of the squark-gluino corrections to the external legs (or the difference between the two definitions of the super-CKM basis), turns out to be numerically important because of the chiral enhancement. While this article was under revision, a phenomenological study of these corrections has been performed in Ref. [29], with the conclusion that the bounds on some mass insertions (corresponding to the tree-level definition of the super-CKM basis) are modified considerably. These corrections have obviously no effect on the phenomenological bounds derived for the mass insertions associated to the on-shell definition of the super-CKM basis, and in this scheme the NLO corrections are well defined in the limit of vanishing quark masses. However, it is not a claim of the present paper that the on-shell scheme is in any way preferred to the tree-level one. In fact, the tree level scheme might be more convenient if one wishes to relate phenomenological bounds on these type of low energy processes to specific mechanisms of SUSY breaking. This issue has been already discussed in Ref. [29]. In those cases one can add the flavor-changing self-energies as given in that paper to our NLO results, or do the substitutions of Eqs. (40) or (43) directly.