The Flavour of Natural SUSY

An inverted mass hierarchy in the squark sector, as in so-called"natural supersymmetry", requires non-universal boundary conditions at the mediation scale of supersymmetry breaking. We propose a formalism to define such boundary conditions in a basis-independent manner and apply it to generic scenarios where the third-generation squarks are light, while the first two generation squarks are heavy and near-degenerate. We show that not only is our formalism particularly well-suited to study such hierarchical squark mass patterns, but in addition the resulting soft terms at the TeV scale are manifestly compatible with the principle of minimal flavour violation, and thus automatically obey constraints from flavour physics.


Introduction
In supersymmetric extensions of the Standard Model (SM), any particles with sizeable couplings to the Higgs sector are expected to have masses not too far above the electroweak scale. This concerns in particular the squarks of the third generation, which should be lighter than about a TeV in order not to create a severe naturalness problem. By contrast, the squarks of the first two generations could well be much heavier. This possibility is particularly attractive because the bounds from supersymmetry (SUSY) searches at the LHC are strongest by far for the first two generation squarks, and because flavour constraints are also easier to satisfy when they are very heavy. The scenario of an inverted mass hierarchy in the squark sector, typically combined with a small higgsino mass parameter and a not too heavy gluino (see e.g. [1][2][3][4] and references therein), is commonly dubbed "natural" or "effective" SUSY, and is increasingly becoming the new paradigm of SUSY phenomenology.
In the Minimal Supersymmetric Standard Model (MSSM) with boundary conditions at the Grand Unification (GUT) scale, light stops and sbottoms with otherwise very heavy squarks are especially interesting because they can lead to radiatively induced large stop mixing [5][6][7]. The latter is needed in the MSSM to obtain a 126 GeV Higgs mass while keeping the stops reasonably light. More precisely, if the first two generation squarks have masses of the order of 10 TeV, and if supersymmetry breaking is mediated at a very high scale such as M GUT ≈ 10 16 GeV, then the stop masses at the low scale receive significant negative contributions from two-loop running (or possibly even from one-loop running if there is a non-vanishing hypercharge D-term). This allows to realize a sizeable ratio |A t /mt|, where A t is the stop trilinear parameter and mt is the average stop mass, leading to large one-loop corrections to the lightest Higgs mass. However, in precisely this situation where radiative corrections to the spectrum from the first two generations are important, they may also induce a significant misalignment between the squark and quark mass matrices. The resulting flavour-changing neutral currents (FCNCs) are tightly constrained by experiment. The effects of such a split squark spectrum on flavour observables have already been investigated in [8][9][10][11] (see also [12][13][14][15][16][17][18][19] for some recent discussions on FCNCs in selected models with light third generation squarks). Here, we propose to shed light on this issue using a different strategy.
Firstly, having assumed a very high mediation scale, hierarchical squark soft terms at the low scale have to be obtained from some non-universal boundary conditions through the renormalization group evolution. But even just prescribing such boundary conditions in a model-independent way is nontrivial, since they depend on the chosen flavour basis. Our first result is to propose a formalism to define general soft term boundary conditions in a basis-independent manner. Secondly, we apply this formalism to the cases where either a subset or all of the third-generation squarks are light, while the first two generation squarks are heavy and near-degenerate. It turns out that not only is our formalism particularly well-suited to study such squark mass patterns, but in addition the resulting TeV-scale soft terms are in many cases manifestly compatible with the minimal flavour violation principle (MFV) 1 , as defined in Ref. [20]. In addition, whenever a departure from MFV is observed, it can be quantified precisely. Clearly, realizing split squark scenarios in this way is of great advantage because it helps ensure that there will be no conflict with bounds on D −D and K −K mixing observables, which one might otherwise expect for generic hierarchical soft terms.
In Section 2, we briefly recall the essentials of the SUSY flavour problem, the concept of MFV, 1 In this respect, it should be stressed that our strategy to prove the compatibility of hierarchical squark mass patterns with FCNC constraints is different from that of Ref. [11], which relied on a modified MFV principle, based on a smaller flavour symmetry group. and present our procedure to define fully generic and non-universal boundary conditions for softbreaking terms. In Section 3 we use this scheme to parametrize the boundary conditions leading to third-generation squarks much lighter than the first two generations, and characterize their flavour properties. Section 4 contains our conclusions. In the appendix, we address some technical subtleties regarding the definition and running of the CKM matrix and show that our scheme allows to easily deal with, and correct for, CKM-induced uncertainties in the renormalization group (RG) running.

The SUSY flavour sector
The matter fields of the supersymmetric Standard Model transform under a global non-abelian flavour symmetry SU This symmetry is explicitly broken by the Yukawa superpotential 2 as well as by the soft mass matrices for the squarks and sleptons, and by the soft trilinear terms.
In the following, we concentrate on the quark-squark sector. Contrary to Y e which can always be diagonalized via a suitable SU(3) L × SU(3) E transformation, at most one of the matrices Y u and Y d can be chosen diagonal. After electroweak symmetry breaking, the Yukawa matrices are diagonalized by The misalignment of left-handed quarks is encoded in the CKM matrix, V CKM = V L † u V L d . Rotating quarks and squarks by the same unitary transformations defines the super-CKM basis, in which the squark mass matrices take the form In terms of the interaction-basis soft masses m 2 Q,U,D and trilinear terms T u,d , Note in particular that a different transformation on the left-squarks serves to define T u and T d .
The supersymmetric flavour problem arises because the 3 × 3 soft-term blocks in the mass matrices Eqs. (3) are a priori generic 3 × 3 matrices. So, rotating the squark fields into their mass eigenstate basis leads to flavour-violating gaugino-quark-squark vertices, which in turn induce new contributions to the flavour-changing neutral currents at the one-loop level. This problem can be ameliorated in three well-known ways: 1. Degeneracy: The sfermion mass matrices may be proportional to the unit matrix, up to small corrections, whereupon flavour changing processes are suppressed by the small mass differences between the squarks. The misalignment angles with respect to the super-CKM basis may nevertheless be large.

Alignment:
The sfermion masses and trilinears may be approximately diagonal in the super-CKM basis, while the mass differences between the squarks may still be large. In that case FCNCs are suppressed by small misalignment angles. Note however that exact alignment is impossible unless m 2 Q is proportional to the unit matrix, as is evident from Eqs. (3).
3. Decoupling: The sfermions may be heavy, such that dangerous sfermion loop effects are suppressed by large propagator denominators.
Under the hypothesis of MFV, the SU(3) 5 flavour symmetry of the MSSM is assumed to be broken only by the Yukawa matrices. These are the only allowed flavour symmetry-breaking spurions, but can be used to parametrize the squark soft-breaking terms [20] (see also, e.g., [22][23][24]). This leads to an efficient suppression of FCNCs through a combination of the first two effects, because the CKM mixing angles and the Yukawa couplings of all but the third generation are numerically small. Before moving on to more general settings, let us briefly recall how this proceeds in practice.

Squark soft-breaking terms under the MFV hypothesis
The MFV hypothesis is formulated in two steps (see, e.g., Refs. [25,26] for more details). First, let us define the matrices transforming as bifundamentals under an SU(3) Q rotation which sends Q → V Q Q: Given that m 2 Q also transforms as a bifundamental, (m 2 Q ) T → V * Q (m 2 Q ) T V T Q , the following expansion is compatible with the flavour symmetry: Likewise, given their transformation rules under G F , the right-handed squark mass matrices and the trilinear terms are expanded as where the a q,u,d i , b q,u,d i must be real because the mass terms are hermitian, but the c u,d i can be complex. The flavour basis is held fixed in each of the expansions in Eqs. (7)- (9), and is for convenience usually chosen as that where (transpositions originate from the SLHA2 conventions [21] adopted in Eq. (1)) so that all soft terms are kept in the super-CKM basis. The fact that two different bases are used is irrelevant, since the expansion coefficients do not depend on the flavour basis chosen to compute them.
Let us stress that, at this stage, we have only introduced a reparametrization of the squark soft terms. There is no loss of generality in these expansions since any 3 × 3 matrix can be projected on the above basis made of powers of A and B [27]. The three hermitian mass matrices plus the two complex trilinear matrices amount together to 63 real parameters, and we introduced the 27 real coefficients (a q,u,d 1,...,6 , b q,u,d 1,2,3 ) plus 18 complex coefficients (c u,d 1,...,9 ) to fully parametrize them. Thus, as long as all these coefficients are free, no approximation is being made. For instance, it is straightforward to recast constraints on mass insertions into constraints on linear combinations of the a q,u,d i , b q,u,d i , and c u,d i [25].
Although the above basis matrices are linearly independent, and therefore span the entire space of hermitian or complex 3 × 3 matrices, this basis is somewhat peculiar. Because of the large hierarchy in the Yukawa couplings, numerically one has B 2 ≈ tr(B)B and A 2 ≈ tr(A)A; that is, some of the basis matrices are nearly parallel in flavour space. In addition, the only non-diagonal structure provided by A and B is the very hierarchical CKM matrix. Therefore, numerically expanding a generic 3 × 3 matrix requires coefficients spanning several orders of magnitude, typically up to the order of m 2 t /m 2 u ∼ 10 10 . By contrast, the MFV hypothesis requires these coefficients to be at most of O(1) when m 0 and A 0 are of the order of the typical soft mass scale. This restricts the structure of the flavoured soft-breaking terms and in particular the absolute and relative size of their off-diagonal entries. Indeed, once the coefficients are required to be O(1), small misalignments among basis vectors can no longer be enhanced. The soft terms directly inherit the hierarchies of the Yukawa couplings, and flavour violation is entirely tuned by the misalignment between the Y u Y † u and Y d Y † d terms, i.e. by the CKM matrix. Note that for O(1) coefficients, since small misalignments among basis vectors can no longer be enhanced, the expansions can be restricted to terms linear in both A and B. Furthermore, whenever tan β is not large, ||A|| ||B||, and only a handful of terms are relevant.

Non-universal squark soft-breaking terms beyond MFV
In this work, we are interested in models where the soft terms are neither universal nor necessarily MFV-like at some very high mediation scale M GUT ≈ 10 16 GeV. Though physically sound in principle, such boundary conditions are difficult to set in practice because one must specify not only their numerical values, but also the basis for the squark fields in which they are given. Of course, if we had a flavour model in which both the Yukawa couplings and the flavoured softbreaking terms could be predicted in the same basis, the above issue would not arise. Lacking this, it is compulsory to deal with the basis dependence in some way to draw any conclusion about the flavour structure of the MSSM at the electroweak scale.
In Refs. [28,29], this problem was addressed phenomenologically by assuming that all the unknown mixing matrices V L,R u,d are CKM-like, i.e. highly hierarchical or even diagonal at some ). Given some numerical values for the soft terms at the GUT scale, this nevertheless leaves several bases (all of which are equally practical) in which to interpret them. For example, all bases where either Here, we propose a simple and well-defined procedure to fully circumvent the issue of basis dependence of non-universal boundary conditions for soft-breaking terms. Specifically, the boundary conditions should be defined by a set of coefficients a q,u,d i , b q,u,d i , and c u,d i , see Eqs. (7)- (9). As explained in the previous section, any possible boundary condition can be expressed in this way, and the coefficients are basis independent. Such a procedure has many other desirable features (below, we use the short-hand 1. The soft masses and trilinear terms at any scale Q admit expansions of the form (7)- (9), where both soft terms and Yukawa couplings are understood as those at the scale Q. Thus, the running of the soft masses and trilinear terms can be represented by that of the flavour coefficients. Their renormalization group equations (RGEs) were studied in Refs. [30,31]. Typically, not only are the evolutions of the coefficients from Q = M GUT down to the TeV scale smooth and bounded, but they even exhibit infrared "quasi"-fixed points, whose values mostly depend on the non-flavoured MSSM parameters.
2. The β-functions of the soft masses and trilinear terms are naturally compatible with the expansions (7)- (9), and the running of the various coefficients sum up different physical effects. For example, the leading coefficients a q,u,d  7. As analysed in the appendix, provided none of the leading coefficients are particularly large, the soft-term expansions are largely independent on the precise parametrization of the CKM matrix. In particular, the coefficients are similar using the full CKM matrix or its CP-conserving limit, no matter how this limit is taken. By contrast, off-diagonal entries of the soft terms can deviate by tens of percent depending on the chosen CKM matrix. This observation is useful in practice since it permits to compute the coefficients under some simplifying assumptions (CP-limit, no threshold corrections, and/or no experimental errors for the CKM parameters), and then to reconstruct with an excellent accuracy the physical soft terms and thereby reliably compute all the flavour observables.
8. Last but not least, it is easy and straightforward to parametrize boundary conditions where the third-generation squarks are split from the first two generations, since Y u Y † u and Y d Y † d do have precisely such a hierarchy. This possibility will be explored in detail in the next section.
To be complete, we should point out that there is one practical issue that needs to be kept in mind. Since the basis matrices span several orders of magnitude and are approximately linearly dependent, it is necessary to maintain a high level of accuracy in the numerical evaluations, otherwise instabilities can easily arise. This is especially true when computing the coefficients of highly suppressed terms such as a u,d 6 or b u,d 3 . For the same reason, a perfectly unitary representation of the CKM matrix must be used, otherwise spuriously large coefficients can arise.

Split squarks and MFV
The peculiar structure of the MSSM Yukawa couplings should have its origin in some unknown flavoured dynamics at some high scale M F . If supersymmetry breaking is mediated at a scale greater than M F , then one can reasonably expect that this flavour dynamics will also generate some non-trivial flavour structures for the soft mass terms and the trilinear couplings. In that sense, expressing the soft terms directly in terms of the Yukawa couplings through the expansions (7)-(9) can be regarded as an attempt at capturing the relationships between them. If this picture is correct, the expansion coefficients at the scale M F would not be random, but would derive from the flavour dynamics at that scale. It is thus quite possible that the various coefficients would actually follow a very definite pattern.
With the above idea in mind, our goal is to design flavour structures leading to spectra with light third-generation squarks at the low scale. There are many ways to achieve this. A first possibility is to impose where · denotes the trace in flavour space. When the free parameter α q is close to one, in the basis where Y u is diagonal, m 2 Q has its first two entries nearly degenerate and much larger than the third, which is precisely what we aim for. Note, however, that in this particular case the value of (m 2 Q ) 33 receives large negative loop corrections from (m 2 U ) 33 . In order to generate a realistic spectrum, the GUT-scale (m 2 Q ) 33 cannot not be chosen too small, and/or sizeable positive corrections from the gaugino masses are needed to overcome this effect. At the low scalẽ t L andb L then end up much lighter than all the other squarks.
It should be remarked that compared to the usual procedure of setting our procedure requires the same number of free parameters. But, at the same time, setting the initial conditions in our way is entirely independent of the flavour basis, while Eq. (15) in principle requires one to specify also the four mixing matrices V L,R u,d . In addition, the parameter α q could bear some physical meaning. First, because Y u Y † u −1 is factored out, its RG evolution is very flat over the whole range down to the electroweak scale. Typically α q changes by 20% during the evolution. (We explicitly show the evolution of α q for a different scenario in the following discussion, see Fig. 2.) Second, it is tempting to imagine that some unknown flavour dynamics sets α q to exactly one at the scale M F . However, since M F = M GUT , one would then have α q [M GUT ] close but not exactly equal to one. Thus, the only phenomenological constraint on this parameter is for it to evolve down to a value smaller than one at the low scale, so as to avoid inducing negative eigenvalues for the stop or sbottom squarks and the ensuing colour symmetry breaking.
A very interesting feature of the boundary condition Eqs. (14) is that even if left squark masses are highly hierarchical, it nevertheless respects the MFV principle since Y u Y † u ≈ y 2 t is of O(1) at all scales. So, once evolved to the low scale, we can immediately predict that these initial conditions should be compatible with flavour constraints.
Other scenarios can be constructed along the same lines. For instance, to also split thet R from the first-and second-generation squarks, one can further impose which is also compatible with the MFV principle when α u ≈ 1. As opposed to the above scenario, the condition that both m 2 U and m 2 Q be hierarchical is radiatively stable (provided that the other states which couple strongly to the stop sector, such as the up-type Higgs and the gauginos, are not too heavy). Together with a small µ parameter, this constitutes a way to realize "natural supersymmetry" within MFV. An example for the typical evolution of the leading expansion coefficients for such a natural SUSY-MFV scenario is given in Fig 1. The RG evolution and computation of the mass spectrum is done with SPheno [33, 34] with boundary conditions adapted according to Eqs. (7)-(9). The a 1 coefficients are not shown because they remain very close to unity, with deviations at the level of less than a percent. The evolution of the a q 3 and a u 2 coefficients is much steeper than that of the other a i . The reason for this is that a q 3 and a u 2 are dominated by the running of y t ; when the y t dependence is factored out, the evolution is very flat, see Fig. 2.
On the other hand, there is no way to split the right sbottom from the first two generations without moving away from MFV. Indeed, all the non-trivial terms in the expansion of m 2 D are sandwiched between Y † d and Y d , which are small when tan β is not very large. Specifically, the simplest way to lighten all third generation squarks is to impose with α q,u,d ≈ 1. Clearly, unless tan β is very large, m 2 D significantly deviates from the MFV assumption. One might worry that this setting conflicts with current flavour constraints, which  Fig. 1. The plot serves to confirm the flatness of the evolution of α. Moreover, it illustrates that the evolution of the flavour coefficients a q 3 and a u 2 is dominated by the RG evolution of y t , which is factored out here.
would thus disfavour lightb R squarks. However, this is not the case. First, note that a large at the low scale is harmless, since it does not contribute to the δ d RR mass insertions (this is evident in a basis where Y d is diagonal). The impact of a large a d 2 at the high scale is less obvious, since it can drive other coefficients towards large non-MFV values through the RGE evolution. However, as illustrated in Fig. 3, this effect turns out to be quite limited numerically. Though some coefficients are indeed initially driven towards large values, the quasi-fixed point behaviour of the RGE evolution then kicks in and brings them back to MFV-like values at the low scale (see e.g. the coefficient a d 4 in Fig. 3). So, even if the low-scale coefficients are not strictly compatible with the MFV principle, they are sufficiently close to MFV to pass all flavour constraints (we also checked this explicitly by direct computation of the flavour observables, using the SUSY FLAVOR 2.02 code [35]).
There is another scenario worth considering. Imagine that for some reasons, the shift from universality induced by the yet unknown flavour dynamics occurs only in the SU Plugging this structure in the soft-breaking expansion, they can be parametrized at the scale M GUT : for some O(1) coefficients a q,u,d i and c u,d i , which we set to one for simplicity. Note how the a u,d is sandwiched between Y † u,d and Y u,d . Again, this input respects the MFV requirement. The only difference with the first scenario is to impose inverted hierarchies in the trilinear terms at the unification scale. Such a pattern does not survive to the evolution however. Looking at the expansion of the trilinear terms, the leading c u,d 1 and subleading c u,d i =1 coefficients do not evolve at the same speed, especially when the former are driven by the gluino mass. So, the cancellation present at the unification scale does not happen at the low scale, and trilinear terms end up being quite similar to those obtained with the first scenario. In this respect, the difficulty mentioned there to obtain a viable spectrum applies here also; a dedicated numerical analysis would be needed to conclude on the valid parameter space of these scenarios.

Conclusions
Third-generation squarks below the TeV scale are an essential requirement for supersymmetry to be natural, while the squarks of the first two generations are likely much heavier. Therefore it is important to study the physics of non-universal squark masses, and of inverted squark mass hierarchies in particular. In phenomenological approaches which prescribe the soft terms at the TeV scale, such as the pMSSM, this is possible to a limited extent only, since effects arising from the renormalization group running from the mediation scale are not accounted for. In particular, these effects could lead to radiatively induced flavour-violating squark mass mixings. Given the tight experimental constraints from flavour observables, to fully grasp the implications of non-universal squark masses, one should be careful to account for such effects.
In this paper we have studied non-universal squark masses in the case that SUSY breaking is mediated at the GUT scale. We have shown how split squark mass matrices (and trilinears) can be conveniently and generally prescribed in a basis-independent way, and investigated their renormalization group evolution.
When requiring only the top squarks to be light, and the first two generations to be nearly mass degenerate, the most natural prescription automatically respects the principle of minimal flavour violation at the GUT scale. Since MFV is preserved during the RG evolution of the soft terms down to the TeV scale, bounds on FCNCs can be easily evaded.
For more general hierarchical soft terms at the GUT scale, the compatibility with flavour observables is not automatic, even though generic soft terms tend to be attracted towards MFV-like structures in the infrared [30,31]. We have confirmed this tendency for the particularly relevant case where all third-generation squarks, including the right-sbottom, are light compared to the squarks of the first two generations. While this scenario strongly violates the MFV hypothesis at the GUT scale, the soft terms become increasingly MFV-like during the running, and end up compatible with flavour constraints at the low scale.
Our analysis puts the increasingly popular framework of "natural SUSY" on a more solid footing, showing that it is actually possible to obtain a natural SUSY spectrum at the TeV scale from well-motivated GUT-scale boundary conditions without having to worry about RG-induced flavour violation. Furthermore, our formalism for defining non-universal soft terms in a basis independent way should be very useful for further studies of the supersymmetric flavour problem beyond minimal flavour violation. A full exploration, within our scheme, of the parameter space leading to natural SUSY is left for a subsequent work.
In the present section, our goal is to show that these issues can be circumvented if the squark soft mass terms and trilinear terms are defined through their expansion coefficients. Indeed, to a large extent, these do not depend on the precise value of the CKM matrix entries. So, once the expansion coefficients at the low scale have been computed under some approximation, it is a simple matter to reconstruct with an excellent accuracy the physical soft terms by plugging back the physical CKM matrix. Let us illustrate this procedure.

CP-conserving limit for the CKM matrix
As a first approximation, the MSSM evolution is often computed in the CP-conserving limit. To this end, the CP violating phase of the SM must somehow be disposed of. There is no unique way to achieve this, since there is no unique way to parametrize the CKM matrix itself, and no matter the chosen procedure, the modulus of at least one of the CKM entries is significantly affected.
Let us take m 2 Q as an example. If the true, complex CKM matrix is used, and using the same scenario as in Fig. 1, then The purely CP-violating coefficients b 1...3 are entirely induced through the RG running. Numerically, their contributions to Im(m 2 Q ) are extremely suppressed because they are tuned by the small Jarlskog invariant Im The bulk of Im(m 2 Q ) actually comes from Im(Y u Y † u ), see Eq. (19). Let us now compare this with the results in the CP-conserving limit. The most frequent CP-conserving prescription is to set δ 13 = 0 in the conventional CKM parametrization. This is the prescription adopted in the RGE codes SPheno [34] and SOFTSUSY [37]. 3 The only CKM entry significantly affected by this is V td , |V td | δ 13 =0 = 0.0058 vs. |V td | = 0.0085 .
As a consequence of Eq. (19), the (1, 2) and (1, 3) entries of m 2 Q are then significantly reduced, since they are induced by V * tb V td and V * ts V td respectively: 3 Reference [37] explicitly mentions that the CP-conserving limit may induce significant uncertainties.
If used to compute FCNC observables, this approximation is particularly dangerous for the b → d and s → d transitions. First, the SM and charged Higgs contributions to Z, γ penguins and boxes, dominated by the top quark contributions hence tuned by V * tb V td , are systematically underestimated. This could still be cured by plugging back the correct CKM matrix in the relevant vertices. This procedure fails, however, to cure the also underestimated gaugino-induced FCNC contributions tuned by (M LL d ) 13 and (M LL d ) 12 . On the other hand, it is easy to check that the expansion coefficients discussed above stay very close to the ones obtained in the CP-violating case. If we project m 2 Q [1 TeV] δ 13 =0 using the CP-conserving Yukawa matrices Y u,d [1 TeV] δ 13 =0 , we find This remains true for all the other soft terms: the shift in the coefficients is below the percent level for the first five coefficients, and of a few percent for the last four. Thanks to this stability, we can use the coefficients computed in the CP-conserving limit together with the true, CPviolating Yukawa couplings to reconstruct the true CP-violating soft-breaking terms with an excellent accuracy. To be precise, this means that if we compute where Since such small differences are irrelevant phenomenologically, and since the other soft-breaking terms are equally well reproduced, it is a simple matter to cure at the same time all the contributions to the FCNC from the artefacts of the CP-conserving limit. In practice, it is thus possible to perform the RGE study in the CP-conserving limit, use our prescription on the output file to reconstruct the full-fledged CP-violating flavour structures, and then pass it on to codes like SUSY FLAVOR [35] to compute reliably the supersymmetric contributions to the FCNC. This is what we actually did to check the compatibility of the scenarios described in the main text with current flavour constraints.
This procedure works no matter the CP conserving prescription. Let us compare, for instance, the δ 13 → 0 limit to the η → 0 limit. In the latter case, |V td | is reduced only by about 7%, while V ub is suppressed by nearly 60%, |V ub | η=0 = 0.00132 vs. |V ub | = 0.00349 .
which nearly matches the real part of the CP-violating result (but stays significantly off for the absolute parts). This can be understood from Eq. (19): the RGE corrections proportional to Y u Y † u depend, to an excellent approximation, only on the third row of the CKM matrix, which stays close to the true one. The η → 0 limit therefore mostly affects tree-level charged-current flavour-changing observables like B → τ ν, and this is easily cured by plugging back the true From them, the reconstructed soft termm 2 Q matches m 2 Q up to corrections of the order of 10 −7 × m 2 0 , which is again more than enough phenomenologically. It should be stressed here that our prescription works particularly well when the soft-breaking terms respect the MFV hypothesis, i.e., when none of the leading expansion coefficients are exceedingly large. In that case, their values are extremely resilient to changes in the CKM parameters, and the prescription reproduces the soft-breaking terms with an impressive accuracy. Beyond MFV, the coefficients in the CP-conserving and violating cases are not necessarily as close. For example, taking the scenario detailed in Fig. 3, we find that coefficients vary by up to about 20%. But, crucially, these variations affect mostly the subleading coefficients, whose phenomenological impact is very limited. As a result, the CP-conserving coefficients still permit to reconstruct the full CP-violating soft-breaking terms with an accuracy better than 1%. Thus, even though we have not tested extensively the range of validity of the prescription when moving out of the MFV framework, we expect it remains accurate for a broad range of flavour-compatible scenarios.

Threshold corrections and experimental errors on the CKM matrix
It is well known that the CKM matrix runs very slowly. So, for simplicity, when it is evolved using the MSSM beta functions already from the electroweak scale, it is not subsequently corrected for threshold effects. There is, however, a coincidental fact that tends to slightly enhance the error induced by this procedure: the SM and MSSM beta functions for the CKM parameters have opposite signs [38]. As shown in Fig. 4, neglecting the former, the CKM angles are underestimated at all scales. As a result, CKM-driven flavour mixing in the squark soft-breaking terms, i.e. those arising from both the RGE effects and the GUT-scale boundary conditions, are underestimated.
Numerically, the effect on the CKM parameters is small but not entirely negligible. Let us use their SM running between M Z and 1 TeV as a measure of their sensitivity to threshold corrections, see Fig. 4. The variations of the Wolfenstein parameters are all much smaller than their corresponding experimental errors, except for the A parameter [39], which increases by about 2% from M Z to 1 TeV. In view of this, we can estimate the impact of neglecting CKM threshold effects on the soft terms by decreasing the A parameter. As a rough estimate, we send the A parameter to the low end of its 2σ range, A = 0.823 +0.018 −0.042 . Still using the scenario of Fig. 1, this leads to which deviates by up to about 10% from the values in Eq. (20). This shows that even supposedly negligible shifts in the CKM parameters can build up sizeable effects in the soft-breaking terms. The expansion coefficients, on the other hand, are the same up to completely negligible shifts of the order of 10 −7 . In other words, these coefficients are essentially independent of the threshold corrections even though soft-breaking terms can deviate significantly. So, whenever the threshold effects for the CKM running are not fully taken care of, one can rely on the same strategy as for the CP-limit, i.e., compute the coefficients and then reconstruct accurately the soft-breaking terms by plugging in the physical CKM matrix.
As an interesting corollary, the stability of the coefficients offers a very simple procedure to estimate the impact of the CKM experimental errors on the soft-breaking terms. Only one run is needed with the central values of the CKM parameters to get the expansion coefficients, and once known, it suffices to vary the CKM matrix entering the Yukawa couplings used to reconstruct the soft-breaking terms at the low scale. Let us illustrate this procedure. First, we perform the RGE evolution starting with the electroweak-scale CKM matrix obtained by shifting all the Wolfenstein parameters to the extremes of their 2σ ranges [36]: We do not take into account the correlations between these parameters. The ranges of values for the soft-breaking term entries are then This represents sizeable shifts, up to 30% (40%) for the real (imaginary) parts. On the other hand, the expansion coefficients do not change significantly: the first five of each expansion being shifted by less than 10 −6 , while the last three of each expansion by less than 10 −4 . They are thus essentially constant over the experimental ranges for the CKM parameters. This confirms that once the experimental errors on the CKM matrix at a given scale are known, the full RGE analysis needs to be performed only once to derive those on the soft-breaking terms at that scale.
Since this rather indirect but nevertheless significant impact of the errors on the CKM matrix elements is in general neglected, this could greatly improve and simplify the study of their effect on the flavour constraints for a given scenario.