Chiral flavor violation from extended gauge mediation

Models of extended gauge mediation, in which large A-terms arise through direct messenger-MSSM superpotential couplings, are well-motivated by the discovery of the 125 GeV Higgs. However, since these models are not necessarily MFV, the flavor constraints could be stringent. In this paper, we perform the first detailed and quantitative study of the flavor violation in these models. To facilitate our study, we introduce a new tool called FormFlavor for computing precision flavor observables in the general MSSM. We validate FormFlavor and our qualitative understanding of the flavor violation in these models by comparing against analytical expressions. Despite being non-MFV, we show that these models are protected against the strongest constraints by a special flavor texture, which we dub chiral flavor violation (χFV). This results in only mild bounds from current experiments, and exciting prospects for experiments in the near future.


Introduction
The SUSY flavor problem is a serious challenge for models of weak-scale supersymmetry.
(For a review and original references, see [1,2].) Soft SUSY-breaking introduces many sources of flavor violation beyond the Standard Model Yukawas. Generic points of the MSSM parameter space are ruled out by myriad precision flavor constraints, such as neutral JHEP07(2015)040 specialize to a pair of representative type I Q-class and U -class models, the models I. 9 and I.13 from [20], where the messengers Φ D and Φ L have the SM gauge quantum numbers of the D and L matter fields. We have verified explicitly that the other type I squark models are very similar, both qualitatively and quantitatively. For reasons we will explain shortly, we expect the type II models are also qualitatively similar in their features. We will not consider the effect of turning on multiple EGMSB couplings, or couplings involving lepton flavor violation. Such couplings can be forbidden by appropriate choices of discrete symmetries. Finally, to focus exclusively on the SUSY flavor problem, we will not consider CP violation in this work, i.e., all the couplings κ i are taken to be real in the mass basis of the standard model particles. The (possibly stringent) constraints from CP violating observables, such as K and the neutron EDM, will be studied in a forthcoming publication [35]. The interactions in (1.2) result in flavor-violating contributions to the squark mass matrices Here each block (clockwise from upper left: LL, LR, RR, RL) is a 3 × 3 matrix. These should be added to the flavor-conserving GMSB contributions and the supersymmetric contributions, and together they contribute to precision flavor observables through a variety of one-loop diagrams involving the squarks and other superpartners. The calculation of these one-loop diagrams for general MSSM spectra is a tedious and laborious task, ideally suited for a computer program. There are several such programs that are publicly available; unfortunately, we found that they were all unsuitable for our purpose. These programs either assumed MFV, did not have a sufficiently broad list of flavor observables, were numerically unstable, or were found to have bugs, likely introduced in transcribing formulas by hand from the literature.
As a result, we found it necessary to develop a new tool called FormFlavor for the study of flavor physics. FormFlavor takes general MSSM spectra and computes the contributions to the various flavor observables shown in table 1. The novel aspect of FormFlavor is that the computation of one-loop Wilson coefficients in the MSSM is done completely from scratch, using the general-purpose packages FeynArts [36] and FormCalc [37]. 1 This avoids the problems associated with transcribing formulas from the literature, and it facilitates the inclusion of additional flavor observables in an automated and modular way. We intend to make FormFlavor publicly available; its usage and validation will be described in an upcoming publication [39].
Starting from the points of reduced tuning identified in [20], we will turn on κ 1,2 and use FormFlavor to investigate the constraints from precision flavor observables. Given that
In order to validate and interpret our numerical findings, we will compare them against analytic expressions for the flavor observables. Historically, the mass insertion approximation (MIA) has been utilized to interpret the influence of flavor-violating squark masses on JHEP07(2015)040 precision flavor observables (see e.g., [2] for a review of the MIA and original references). However, the utility of the MIA is limited when one or more of these mass insertions are O (1). Since that is precisely the interesting region of parameter space for our EGMSB models, the traditional MIA cannot be used here. Fortunately, in these EGMSB models there is another handle we can use to obtain an analytic understanding. From two powers of our anti-fundamental spurion, κ a , we can construct an adjoint+singlet of SU This matrix K governs all flavor violation from EGMSB. By exploiting the fact that it is only rank 1, together with the special properties of the soft masses for the type I models, we are able to obtain analytic formulas for the flavor observables that treat the flavor violation exactly. Then the only expansion that we do is in v/m SUSY , which is still an excellent approximation in these models, even when the flavor violation is O(1). With this technique, we are able to make precise estimates of the supersymmetric contributions to flavor observables, validate our numerical results in detail, and understand their qualitative features. Our paper is outlined as follows. In section 2, we describe the idea behind chiral flavor violation and how it applies to our models (1.1) in restricting the texture of the squark mass matrices. We then further restrict our attention to the type I squark models, and highlight some of their special features that will be useful in analyzing the flavor observables. In section 3, we turn to a detailed study of the flavor constraints on the EGMSB models (1.2). We illustrate these constraints in the κ 1 -κ 2 plane using a series of plots of flavor observables computed with FormFlavor. We also provide an analytic understanding of the features of these plots using the special properties of rank 1 χFV and the type I squark models. Finally in section 4, we conclude with a brief summary of our results, and a discussion of the promising future prospects for precision flavor tests of these EGMSB models. In the appendices, we detail our parameter deformation and various subtleties that arise there, provide a brief description of FormFlavor (postponing a more detailed manual and validation for an upcoming work [39]), compile the necessary expressions for flavor observables using a uniform notation, and present some other formulas used in this work.

Chiral flavor violation
In the absence of Yukawa couplings, the Standard Model flavor symmetry group is The assumption of MFV is that this flavor symmetry is broken only by the Yukawa couplings y u , y d and y , which transform as (3, The χFV ansatz results in a special texture of flavor violation in the MSSM, one which greatly suppresses the constraints from many flavor violating observables. In general, the 3 × 3 soft mass-squareds m 2 X transform as an adjoint+singlet of SU(3) X , while the A-terms transform in the same way as the Yukawas. Using spurions from only a single SU(3) X , one can obtain flavor violation only in m 2 X . To obtain flavor violation in any of the other soft terms, one must involve the Yukawa couplings. Thus these other soft masses inherit an MFV-like suppression.
The cases of interest for this paper are when X = Q or U . (We reserve a more general treatment for an upcoming publication [51].) Let us now examine these in more detail. To leading order in the Yukawa couplings, the symmetries of Q-class χFV imply that where Σ and Γ are built out of the SU(3) Q spurions. Substituting these into (1.3), we have under the third-generation dominant approximation (whereby y u and y d are nonzero only in the 33 component): We see that within the third-generation dominant approximation, no flavor violation appears in the RR block for either the up or down squark mass matrices. Also, the only flavor violation in the LR block involves the 3rd generation, and is v/m SUSY suppressed. These features greatly reduce the sensitivity of Q-class χFV to precision flavor constraints. As we will explain in more detail in section 3.1.1, what typically provide the most stringent flavor bounds on new physics (i.e., ∆m K and ∆m D ) involve the 1st and 2nd generations and involve simultaneous violation of flavor in both the left and right chiral sectors.
Meanwhile the U -class χFV models are even more insulated from constraints. Here, on symmetry grounds, and again to leading order in the Yukawas, we must have

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where now Σ and Γ are built out of the SU(3) U spurions. Again substituting these into (1.3) in the third-generation dominant approximation, we find: Thus there is no down-type flavor violation at all in the third-generation dominant approximation! As the majority of sensitive flavor probes involve the down sector, this eliminates almost all contributions to flavor observables. The only exception is ∆m D , which again has reduced sensitivity because there is no simultaneous left-right flavor violation in the 1st/2nd generations. (Down-type flavor observables can involve the up-squark RR block through chargino loops, but these must be mostly Higgsino-like, so they will be suppressed by Yukawa couplings.) Finally, we should comment on the role of the RG. The RGEs from the messenger scale to the weak scale add more terms, but the symmetry-based arguments given above -which were truncated at leading order in the Yukawas -clearly continue to hold with the inclusion of higher orders in the Yukawas. In particular, terms that are zero in the third-generation dominant limit remain zero under RG evolution. For this reason, we do not need to concern ourselves with the details of RG running to gain a qualitative understanding of the EGMSB flavor-violating contributions to our flavor observables.

χFV in type I squark models
The EGMSB models (1.1) are clearly examples of Q-class or U -class χFV. In this subsection, we will discuss some further features that are specific to the type I squark models that will help us analyze the Wilson coefficients for flavor observables in the next section.
We begin by quoting the explicit formulas for the soft masses for general type I squark models (see appendix C). For concreteness, we focus on Q-class models, where we have (in addition to the GMSB contributions),

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Here we are neglecting the (numerically irrelevant) down-Yukawa contributions; K is the rank one matrix of couplings defined in (1.4); d QH U = 2 is a multiplicity factor; and h(x) is an O (1) loop function (see (C.8) for the exact form). For our Q-class model of study (1.2), we have d Q = N = 6, d φ = 5, where N is the number of messengers. 2 We can see the χFV texture of (2.3) quite clearly in the formulas for δm 2 U and Aũ in (2.6). There are also additional features of these explicit formulas that go beyond the χFV ansatz. We notice that δm 2 Q , together with the quantities Γ and Σ introduced in (2.2), are all proportional to K: The simplicity of these relations is partly due to the rank 1 nature of the flavor violation. But in principle, on symmetry grounds alone, there could have been additional contributions to (2.7) proportional to the identity matrix and powers of the Yukawa couplings. These are absent due to the specific form of the type I squark couplings. This is the main reason we have focused on the type I models in this paper. The forms of the soft masses are more complicated in the type II models, and while we expect them to be similarly protected by their χFV flavor texture, understanding them at the level of analytical detail that we apply to the type I models is more difficult.
Using the relations (2.7), we now discuss the diagonalization of the squark mass matrices. Because the LR blocks are suppressed by v/m SUSY , the squark mass eigenvalues are given by those of the LL and RR blocks to a very good approximation. The RR blocks are already diagonal in the third-generation dominant approximation according to (2.3). The down-squark RR masses are just their minimal GMSB values m 2 0 , while the up-squark RR masses are given by m 2 0 , m 2 0 and m 2 0 +σy 2 t κ 2 3 ≡ m 2 RR . In practice the third eigenvalue is only a little offset from m 2 0 throughout the parameter space of our model, and this difference can be neglected.
The situation for the LL block is only a little more complicated. Since δm 2 Q is just proportional to K, the LL block of the squark mass matrices is diagonalized by the unitary transformation whereκ a = κ a /κ and a and b are any two orthonormal basis vectors for the orthogonal subspace κ ⊥ . The eigenvalues are m 2 0 , m 2 0 , and (Here we are ignoring the small differences between the minimal GMSB contributions to the Q, U , D soft masses coming from SU(2) × U(1).) In our EGMSB models, m 2 S corresponds roughly to the mass of the lightest squark, and moreover

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Thus the Wilson coefficients will generally be dominated by this lightest squark running in the loop, and m 2 S will play a central role in controlling the size of the flavor-violating effects in these models. In fact, because of (2.10), diagrams where the right-handed squarks propagate tend to be suppressed, and we will see that it suffices to focus on the LL flavor violation exclusively.

Flavor observables and constraints on EGMSB
In this section, we investigate the flavor constraints on our EGMSB models (1.2) in detail. For reference, the spectra corresponding to κ 1 = κ 2 = 0 are shown in figure 1. These are essentially the same points that were identified in [20] as being the least fine-tuned EGMSB models with m h = 125 GeV. Starting from these flavor-aligned points, we will perform a numerical scan in the flavor-violating parameter space (κ 1 /κ 3 , κ 2 /κ 3 ) of the models, using FormFlavor to compute the flavor observables and compare against their experimental values. Going through each observable X from table 1 in turn, we will exhibit plots of Here [X] TH denotes the SM prediction together with the EGMSB contribution, and [σ(X)] TH+EXP is the theoretical and experimental errors added in quadrature. A contour of 2, which is roughly the 95% CL exclusion limit, will be taken to indicate the point at which the EGMSB model is excluded by the given flavor observable. 3 Thus a contour of 2 does not represent a 95% confidence level exclusion for these observables. However, as it would require a substantial tuning for the standard model and the new physics contributions to cancel against one another, values larger than 1 are suspect. As we deform away from κ 1 = κ 2 = 0, there are numerous subtleties that must be taken into account regarding how the other parameters of the model are varied. These subtleties and specifics of the procedure are described in appendix A. In short, κ 1 and κ 2 are introduced in such a way that the superpartner mass eigenvalues and "net" A-terms are essentially held fixed.
In order to validate the numerical results from FormFlavor, we will compare them against analytical formulas for the flavor observables. This will also shed further qualitative insights on the role of χFV in weakening the flavor constraints. As discussed in the Introduction, the usual mass insertion approximation fails due to O (1) entries. Instead, we will use flavor symmetries and the special features of the type I models discussed above to characterize their exact κ dependence, to leading order in an expansion in v/m S . In general, the κ dependence arises through the lightest squark mass (2.9) and the unitary matrix U LL given in (2.  Mass / GeV but as discussed in the previous section, this dependence can be generally be neglected due to the heaviness of the right-handed squarks and the extra v/m S suppression.) The dependence on U LL is constrained by the flavor symmetries, and as explained in the previous paragraph, the squark mass eigenvalues are mostly held fixed in our parameter space. Thus it will be possible to fully characterize the features of the FormFlavor plots in terms of very simple functions of κ.

Meson mixing
We begin with the ∆F = 2 meson mixing observables. As we discuss in appendix D.1, for to a good approximation, while for X = B d and B q , to a good approximation. Here H eff is the ∆F = 2 effective Hamiltonian; its local shortdistance part (relevant for the MSSM contributions) is built out of the four-fermi operators, Here M, N = L, R label the chirality of the incoming quarks; and ab = 12, 12, 13, 23 for ∆m K , ∆m D , ∆m B d and ∆m Bs , respectively, while q is up-type for ∆m D and down-type for the rest.
The full result of FormFlavor is shown in figure 2. In this subsection, we will endeavor to understand its features analytically using the χFV ansatz and the special features of type I EGMSB identified in section 2.2.
The MSSM contributions to the ∆F = 2 observables are due to box diagrams involving the squarks and the gauginos. Here the great simplification of χFV is that any operator with an R index must transform non-trivially under SU(3) D or SU(3) U ; thus it is suppressed by χFV in our Q-class models. Furthermore, the O LL S and O LL T operators all involve an SU(2) L -breaking chirality flip, so they are dropped in the v = 0 approximation. Therefore, the only unsuppressed Wilson coefficient is C LL V . This also happens to be the only contribution to the one-loop SM Wilson coefficients, which proceeds through W exchange.
Since C LL V transforms in the square of the adjoint+singlet representation of SU(3) Q , with just left-handed squarks running in the loop, the only way it can depend on κ is, where f 1 is a dimensionless loop function depending on the LL squark mass eigenvalues. 4 Dimensional analysis fixes the dependence on the masses, and the rest of the dependence must be from the unitary matrix (2.8). Under this simplification, the meson mixing contributions are of the form (see appendix D.1 for more details and specific values of the parameters), JHEP07(2015)040 where f X is the decay constant for the meson; and B LL V,X is an O (1) hadronic parameter. This simple formula suffices to accurately describe the ∆F = 2 flavor observables.
Focusing on gluino boxes for simplicity, 5 we find from explicit computation that the leading Wilson coefficient is given by

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This clearly agrees with the general form (3.6). Here, x q = m 2 S /m 2 0 and x g = m 2 g /m 2 0 , and f ∆M,box g (x g , x q ) ≈ 0.05 is a loop function that we define in appendix E. As discussed earlier, since the relevant masses are roughly held fixed in our deformation, the loop function does not change across the parameter space. The QCD RG running from the SUSY scale to the meson scale is fairly mild for C LL V , yielding only a 20 − 30% suppression in the size of the Wilson coefficient [52].
The hadronic factors f 2 X B LL V,X are fairly similar across the mesons. We can define, So we expect deviations of the form, Noting thatκ 2 aκ 2 b is at most 1 4 , and comparing against table 1, we see that ∆m K and ∆m D should be most sensitive to EGMSB, while ∆m B d should be barely sensitive, and ∆m Bs completely insensitive. 6 These sensitivities are observed in the plots shown in figure 2. Moreover, theκ 2 For ∆m K and ∆m D , we can see that moving from κ 1 = κ 2 = κ 3 to the corners, constraints rise by the expected factor of ∼ 16/9. 7 These results from meson mixing may seem to conflict with the SUSY flavor problem, which suggests that, with O (1) flavor-violation, the SUSY scale needs to enter above ∼ 500 TeV due to constraints from ∆m K and ∆m D [49,50]. However, these constraints are driven by the C LR S Wilson coefficient, while χFV only generates the C LL V operator. In the MSSM, the contribution of the latter to ∆m K,D is suppressed by ∼ 10 −4 -10 −3 relative to the former. This is due to three separate effects that all work in the same direction. First, the hadronic matrix elements differ between these two operators, Next, the SUSY contributions to the Wilson coefficients are also quite different in size. From the MIA with O (1) mass-insertions, it is easy to see that C LR S /C LL V ∼ 30 [49]. Lastly, the QCD running from the SUSY scale to ∼ 2 GeV suppresses C LL V by 20-30%, but enhances C LR S by a factor of ∼ 3 [52]. All of these factors conspire to drop the scale of sensitivity to ∆m K and ∆m D to the TeV scale in χFV models. 6 As discussed in the Introduction, we are not considering CP violation in this work, in particular K .
Although it depends on the precise value of the CP violating phase, the expectation is that this will place a meaningful, tighter constraint on the parameter space when the phase is large. This and other CPV observables will be studied in an upcoming paper [35]. 7 Note that ∆mD is rotated by the Cabbibo angle θc relative to the other observables. This is a consequence of the the fact that, as discussed in in appendix A, our κ1, κ2, and κ3 directions are chosen to align with the low energy down, strange and bottom quark. For ∆mD, the dependence on κi proceeds through the LL sector of the up-squark mass matrix, where there is an additional rotation by VCKM .

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Unlike meson mixing in the previous subsection, the K ± → π ± νν observable (along with all other ∆F = 1 observables) enters as a matrix element squared, and thus interference with the standard model contribution can be important. The expression for this branching ratio is (see appendix D.2), where c + v 4 = 4.9 × 10 9 GeV 4 , C LL V,SM = (−1.21 + 0.39i) × 10 −10 GeV −2 , and all EGMSB effects are contained in C LL V and C RL V . These are the Wilson coefficients from the ∆F = 1 effective Hamiltonian built out of the four-fermi operators Here ab = 12 for K ± → π ± νν. In the MSSM, these Wilson coefficients arise through one-loop box and Z-penguin diagrams.
The full FormFlavor result is shown in figure 3. The general trend of K ± → π ± νν can again be understood through use of the features of rank 1 χFV discussed in section 2.2.
First of all, as for the ∆F = 2 observables, C RL V,ab must be zero in the third-generation dominant approximation -since it transforms in the adjoint+singlet of SU(3) D , using the available spurions one can only obtain something that is nonzero in the 33 component. Thus, we can focus on C LL V,ab . This transforms in the adjoint+singlet of SU(3) Q . Using flavor violation in the LL block only, the form of C LL V,ab is constrained by the symmetries to be: where . . . contains higher orders in v/m SUSY and other irrelevant terms, and f 2 is a dimensionless function of superpartner mass ratios. We note that (3.14) can come from box diagrams or Z-penguin diagrams, but in the latter case, the 1/m 2 Z from the Z propagator must be canceled out by two insertions of wino-Higgsino mixing. Insertions of LR mixing from the squark mass matrix would also cancel out the 1/m 2 Z , but the heavy right-handed squark masses suppress these contributions enough that they may be ignored. Now we will compare against explicit computations of the Wilson coefficients. We confirm that the C RL V coefficients are all negligible. For C LL V , we find that the chargino diagrams dominate, and they are given by x , x 2 , x µ ) ∼ 1.5 is a loop function of the sparticle mass-ratios x = m 2 /m 2 S , x 2 = M 2 2 /m 2 S , and x µ = µ 2 /m 2 S , defined in appendix E, and varies only mildly across the parameter space. This is fully consistent with the general form (3.14). This also explains JHEP07(2015)040 Figure 3. Contours of the difference in BR(K + → π + νν) between experiment and the SM + EGMSB predictions, given in units of the net uncertainty (added in quadrature). A contour of 2 corresponds roughly to a 95% exclusion. why the charginos dominate over the gluinos, since the gluinos can only give rise to Zpenguin diagrams that are nonzero by virtue of down sector LR squark-mixing insertions.
Substituting in numerically for the loop function, m S and α 2 , we find This simple function of κ, when added to the SM contribution and substituted into (3.12), reproduces well the features of figure 3. It grows in magnitude fastest along the lines κ 1 = ± κ 2 , asymptoting to the values ± 4 × 10 −11 GeV −2 . In the corners of the parameter space we obtain a deviation from the SM prediction of This is smaller than the current experimental+theoretical uncertainty shown in table 1. However, because the SM prediction for the BR is a little lower than the experimentally observed value, moving along the κ 1 = +κ 2 line slightly exacerbates the difference, while moving along the κ 1 = −κ 2 line slightly lessens it.

b → sγ and b → dγ
The expression for the b → sγ branching ratio (see appendix D.3) is,  where ab = 32 for b → sγ. These operators require SU(2) breaking, so their Wilson coefficients are zero in the v → 0 limit. Naively, this would mean that the result is negligible in our zeroth-order expansion in v/m SUSY . However, the SM Wilson coefficient is suppressed by m b /v, so here we consider one higher order in the v/m SUSY expansion in order to capture a numerically relevant result.
, it must be proportional to m d acting on the left (right). The latter is zero in the thirdgeneration dominant approximation, so it suffices to focus on C L A . With just left-handed squarks propagating in the loop, the Wilson coefficient must be given by: Here we assumed that m d is accompanied by a tan β enhancement, otherwise the entire effect is numerically negligible. Comparing with explicit calculation, we find again that charginos give the dominant contribution to the Wilson coefficient, through the quark-squark-Higgsino vertex. (Contributions of the form (3.20) can also arise through gluino and neutralino loops, but here the factor of m b tan β arises through the LR block of the down-squark mass matrix, so the diagrams are suppressed by heavy right-handed squarks propagating in the loop.) The dominant chargino diagram gives, where x µ = µ 2 /m 2 S , x 2 = M 2 2 /m 2 S , and f b→s/d γ,peng χ ± (x µ , x 2 ) ∼ 0.5 is a loop function defined in appendix E. This is fully in agreement with the general result (3.20). QCD running from the SUSY scale to the b pole [53] induces a mild 20% suppression to (3.21), and once again, we find this simple result is enough to account for the features of the FormFlavor plot in figure 4.

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Substituting in numerically for our parameter space (we take tan β = 10 as in [20]), we obtain: Clearly its effects are largest when κ 1 = 0, and along the κ 2 axis it has a maximum (minimum) at κ 2 /κ 3 = 1 (-1), while asymptoting back to zero as κ 2 /κ 3 → ±∞. Substituting this into (3.18) together with the SM Wilson coefficient, we find at these extrema a deviation from the SM prediction of Since the uncertainty on the measurement (combining the theoretical and experimental in quadrature) is 3.5 × 10 −5 , regions of exclusion are to be expected for b → sγ.
In practice, FormFlavor does find a region of exclusion in the bottom half of the parameter space, shown in figure 4. This exclusion is in part due to the current O (1σ) excess in the experimental measurement relative to the theoretical prediction, 8 which allows the ∼ 2σ change to constrain a large region of negative κ 2 in the plot. Uncertainty on this observable is comparable in size between theory and experiment, so improvements on either side could make this observable more constraining and in a direction that other observables currently have no sensitivity.
The observable b → dγ could also potentially place constraints. The EGMSB contribution to the Wilson coefficient is the same as in (3.21)-(3.22), but with κ 2 ↔ κ 1 . However, the SM contribution is of a different size, C L A,SM ∼ − (2.4 + 1.1i) × 10 −9 GeV −1 . Importantly, this is not much larger than the new physics contribution, so interference is very important. This yields an approximate deviation of, while the net uncertainty is 6.3 × 10 −6 . Again, a two sigma exclusion could be possible, but it would only be expected near κ 1 κ 3 = −1 due to the interference acting constructively. This is borne out by figure 4, which shows a maximum deviation of ≈ 2.3 times the net uncertainty. Because the measurement of b → dγ is fairly recent, a future improvement that pushes the experimental uncertainty to the 10-15% level would allow for b → dγ to place much tighter constraints on the region of constructive interference, i.e., κ 1 < 0.

B q → µ + µ −
As shown in figure 5, neither B s → µ + µ − nor B d → µ + µ − is at all close to constraining our EGMSB models. We include a brief discussion of these observables just for completeness sake. The branching ratio for B q → µ + µ − is [55,56] (see appendix D.4 for more detailed formulas):

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where X s = 5.36 × 10 7 and X d = 3.97 × 10 7 , and In the MSSM, the dominant contributions to B q → µ + µ − are the tan 3 β enhanced wave function correction diagrams with a heavy higgs propagator, see figure 6. Here gluinos, charginos and neutralinos can run in the loop, with both CP even and odd higgs states along the penguin line. For simplicity, we will quote the result only for gluinos; the answer for the others is very similar. As our higgs states are heavy, we use the relation With only left-handed squarks propagating in the loops, by symmetries the answer must be of the form: where a = 2 for B s → µµ and a = 1 for B d → µµ. Now the Yukawa coupling needed on symmetry grounds arises from the Higgs-quark-quark coupling. Explicit computation gives where x g = m 2 g /m 2 0 and x q = m 2 S /m 2 0 and the loop function (see appendix E) is f Bq→µ + µ − ,h-peng g (x q , x g ) ∼ 0.1. Since m A ∼ m SUSY , this leading contribution is of the form (3.27) as expected.
From the above expressions, we can translate to the phenomenologically useful parameters (3.29) These translate into maximum deviations from the SM branching fractions by These deviations are an order of magnitude smaller than the uncertainties on their respective measurements, and therefore neither B s → µ + µ − nor B d → µ + µ − place any meaningful constraints on our models. 9 JHEP07(2015)040

U-class models
As discussed in section 2.1, all U -class χFV models receive very few constraints from flavor observables. First, according to the χFV texture of U -class models (2.5), all flavorviolation is restricted to the up-squark sector only. As most potentially constraining flavor observables have external down-type quarks, chargino diagrams are required for sensitivity to flavor-violation. However, as is further shown in (2.5), there is no flavor-violation in the LL block; thus pure wino diagrams cannot contribute. Higgsino diagrams introduce Yukawa couplings which suppress contributions to down sector observables enough that none of these would be remotely constraining in the foreseeable future. We have verified all of these general results in the context of our type I EGMSB models. There can be constraints from the D-meson system. Contributions to ∆m D are as in the Q-class models, only now with the C RR V contribution dominating, so we expect very similar constraints from ∆m D . This is indeed shown in figure 7. Again, although the contour of 2 does not represent a 95% confidence level exclusion, values larger than 1 necessitate a cancellation between the standard model and the new physics contributions. It should be noted that a viable possibility is that the standard model contribution is in fact much smaller than the observed value, and the contour of one is actually where EGMSB entirely accounts for the ∆m D measurement. As with the Q-class models, the sensitivity vanishes as either κ 1 → 0 or κ 2 → 0, and increases most rapidly along the diagonals.

Summary and discussion
In this work, we performed a detailed investigation into the precision flavor constraints on extended GMSB models. These models, where the usual GMSB contributions are augmented by direct matter-messenger couplings, are well motivated in light of the recent JHEP07(2015)040 discovery of a Higgs boson near 125 GeV. However, since these models are not necessarily MFV, in their full, three-family generalizations they could be potentially dangerous from the point of view of flavor.
Our work required a computer program that could turn general MSSM spectra into precision flavor observables. We found that existing programs had various limitations -they either assumed MFV, were numerically unstable, or had incorrectly transcribed formulas from the literature. This motivated us to develop FormFlavor, a comprehensive package that computes flavor observables ab initio starting from the Feynman rules, and uses a modular framework that enables us to add new observables in a uniform and straightforward way.
Using FormFlavor, we studied the flavor constraints on the three-family generalizations of the EGMSB models of [20]. The results we encountered from this systematic study were interesting and unexpected. We found that despite the introduction of O (1) flavorviolating couplings, there are currently very few constraints on EGMSB models. To validate the numerical results of FormFlavor, we compared them in detail with analytic formulas for the Wilson coefficients derived using a combination of flavor symmetry arguments and direct calculation, and we found excellent agreement.
The mild flavor constraints in these models are illustrated in the summary plot of figure 8. U -class models only receive constraints from ∆m D . Q-class models are constrained by ∆m K and ∆m D in the corners of the plot, while the radiative b → sγ and b → dγ each exclude a single bubble near κ 2 = −κ 3 and κ 1 = −κ 3 , respectively. We note that the excluded region from b → sγ is largely due to the current ∼ 1σ discrepancy between the theoretical prediction and the measurement, which new theoretical work suggests will disappear [54].
The results from these models may seem at odds with the SUSY flavor problem. We have argued that the mildness of the flavor violation in these models originates from the fact that they obey the "chiral flavor violation" ansatz, whereby flavor is violated only by the Yukawas and spurions of a single SU(3) of the full SU(3) 5 SM flavor symmetry. We showed that χFV prevents many of the most problematic contributions to flavor observables from arising in the MSSM, and allows for O (1) flavor-violation in EGMSB models.

Future constraints
Although constraints are currently very mild, there is immense potential to further probe these models in the near future.
• For ∆m K , short-distance predictions exist with moderate uncertainty, i.e., (3.1 ± 1.2) × 10 −15 GeV [58], but the long-distance contributions are currently unknown. However, an accurate, full calculation on the lattice, including both long-and shortdistance contributions, may be coming in the near future, as promising preliminary work on the subject shows [59,60].
• Expected incremental improvements to the theoretical uncertainty of the standard model prediction could make ∆B d into a constraining observable soon. Estimates JHEP07(2015)040 suggest that both the bag parameter, B LL V,B d , and the relevant CKM elements could be calculated on the lattice to significantly improved levels by 2018 [61].
• Belle II [62] is expected to make significantly improved measurements to both b → sγ and b → dγ [63], which would allow for both of these observables to constrain more of the parameter space.
• NA62 [64] at CERN is projected to be able to measure BR(K ± → π ± νν) to a precision of about 10% [65] of the SM value. This is an improvement of more than an order of magnitude relative to the current measurement. In a few years, K + → π + νν will be one of the the most sensitive flavor observable to Q-class models. Perhaps more importantly, if NA62 were to measure a deviation from the standard model prediction, this model would provide a significant motivation to invest in an experiment like ORKA [66], that would be able to hone in on the parameter space.
Due to these potentially significant theoretical and experimental improvements, much of the parameter space in Q-class models could be probed in just a few years. The projected sensitivities are shown in figure 9. Although Q-class models have an exciting future in flavor, U -class models remain completely unconstrained. Short of lattice predictions for ∆m D , no observables in the current program are sensitive to these these models. Charm factories and precision top studies could someday explore this space in, for instance, c → uγ or t → c/uγ. However, for the moment, U -class models are resilient against flavor constraints.

Future directions
There are many avenues for future investigations. Here we list a few.
• As we alluded to earlier, the χFV ansatz is a general paradigm, and it provides a novel and realistic solution to the SUSY flavor problem. This texture, its possible origins, and consequences are worthy of further study [51].
• Another interesting question is that of CP violation in these models. The goal of this work was to focus on the SUSY flavor problem, and so the EGMSB couplings were intentionally assumed to be real. Allowing for arg κ i = 0 would give nontrivial contributions to CP-violating observables, some of which (such as K and the neutron EDM) are typically extremely constraining. An interesting question is to what extent the χFV texture protects EGMSB models from the SUSY CP problem before there is any conflict with data. We plan to study this in detail in an upcoming work [35].
• The possibility of heavily mixed squarks allows for very interesting collider signatures [68][69][70][71]. These EGMSB models provide a flavor-safe proof-of-concept motivation for experimental searches at ATLAS and CMS.
• While we made some effort to ensure that the flavor-violating A-terms do not destabilize the vacuum (see the discussion in appendix A), it would be interesting to study in more detail the vacuum stability of these flavor-violating EGMSB models, along the lines of [72].

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choose κ 1 , κ 2 and κ 3 to align with the low-energy down, strange and bottom quark, respectively. Thus in our choice of interaction basis, the down Yukawa is diagonal, but the up Yukawa is multiplied by V CKM . We ignore the small differences with our previous work where the alignment was with the top quark. The U -class models retain alignment with the up-type quarks.
• Turning on κ 1 and κ 2 while holding fixed the other parameters can significantly modify the spectrum, as shown in (2.6). At the least finely-tuned points, where a cancellation results in a stop lighter than the other squarks, this can either lead to stop tachyons, or it can lead to the stops being so heavy that the hypercharge tadpole contribution to the RG running quickly drives the sleptons tachyonic. To avoid these undesirable features, as we turn on κ 1,2 , we fix Λ, but vary M , so that the the oneloop contribution adapts in such a way that the lightest squark eigenvalue of the LL block in Q-class or RR block in U -class models is held fixed. This deformation enables us to prevent the interesting flavor-violation from vanishing and to maintain the squark masses at sensible values.
• Additionally, as can be seen from (2.6), if we were to fix κ 3 and turn on κ 1,2 , then the "net" A-term would increase. The "net" A-term in the type I models is aligned with the lightest eigenvalue direction of the rank 1 block, i.e., in Q-class models, The A-terms in all orthogonal squark directions vanish in the third-generation dominant limit. Increasing A/m S is potentially dangerous for vacuum stability [72]. In order to avoid these issues, we require that this "net" A-term remains constant. To achieve this we fix κ 3 κ everywhere in the parameter space so that as we increase κ 1 and κ 2 , κ 3 decreases to compensate.
• Finally, this deformation of κ 1,2,3 will also affect the Higgs mass. The one-loop corrections to the Higgs mass in the presence of general flavor violation have been computed in [73,74], and our preliminary studies of these suggest that m h may drift down by several GeV as we move out in κ 1,2 . However, the two-loop corrections are not yet known, and if the usual non-flavor-violating MSSM is any guide, these are likely to be important for an accurate determination of the Higgs mass. While it would be interesting to study this further, it is beyond the scope of the work. At the very least, one could imagine increasing the overall scale of the superpartners in order to compensate for any decrease in the Higgs mass. This would only serve to further alleviate the flavor constraints, so the qualitative value of χFV to these EGMSB models is unaffected.
Across a grid in (κ 1 , κ 2 ) constructed via this deformation of the benchmark point, we generate the soft spectrum at the messenger scale. All of the couplings and soft masses are then evolved down from the messenger scale to the SUSY scale using the fully general JHEP07(2015)040 3 × 3 MSSM β-functions. 11 At the SUSY scale, we apply the BMPZ QCD threshold corrections to the squarks and gluinos [75]. Finally, we use our new Mathematica package FormFlavor to compute the Wilson coefficients, RG evolve these coefficients down to the scale of interest (e.g., m b ) for each flavor observable, and compute the contributions to each flavor observable there. (The details of FormFlavor are briefly explained in appendix B and will be further fleshed out when the package is made publicly available [39].)

B Details of FormFlavor
FormFlavor is a general, modular flavor package written in Mathematica that takes general MSSM spectra as inputs and calculates a variety of flavor observables in situ starting from the Feynman rules. The package is flexible -new flavor observables can be straightforwardly added to it and, although currently implemented only for the MSSM, it can be extended to other models readily. Also, since processes are calculated in a uniform manner from first principles (rather than hardcoding formulas from the literature), the reliability of the code is greatly enhanced.
In FormFlavor, the process under consideration is specified in FeynArts [36] and all the relevant topologies for the various sparticle mediators are generated there. These topologies are then converted into their respective amplitudes via FormCalc [37]. The FormCalc output is analytically transformed into a particular Wilson operator basis for each process of interest, and the Wilson coefficients are extracted. Having performed this once for an observable the analytic expressions may be written to a process file and used in subsequent runs. The Wilson coefficients may then be piped into functions that calculate the flavor observables. 11 For simplicity, we do not run back and forth between the IR and UV to better specify the scale.

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The main components of FormFlavor are as follows: • Automated generation of one-loop diagrams for flavor processes (using FeynArts) • Automated calculation of Wilson coefficients from Feynman diagrams (using FormCalc) • Library of analytic one-loop integral functions • Routines for general MSSM spectrum input (SLHA2 [76]  This powerful flavor package will be released to the public and described in more detail in an upcoming work [39].

C Soft terms for type I squark models
In this appendix, we will specialize the fully general EGMSB formulas from [20] to the case of the type I squark models with general flavor-violating couplings κ i . These were quoted in (2.6) for the Q-class models, and we again focus on this case here. To translate to U -class models, one must take U ↔ Q, y u → y † u , Aũ → A † u and K → K T in all expressions.

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Before we begin, we must address a convention difference between this paper and that of [20] concerning the treatment of the right-handed quark chiral superfields U , D. In this paper, the Yukawas, squark mass matrices and A-terms are defined as: Thus, the Yukawas, A-terms, m 2 U and m 2 D are all transposed relative to those of [20]. Note that in these conventions,Ũ * ,D * are the scalar components of the chiral supermultipliets U , D.
The starting point of our derivation of (2.6) is eq. (2.19) in [20], which we repeat here for convenience: Here a, b, . . . run over all MSSM fields (including all gauge and flavor degrees of freedom). Meanwhile A, B, . . . run over all messenger fields similarly. In this appendix, we will use i, j, . . . to denote MSSM flavor indices. From (C.3), we immediately obtain the bilinear A-terms after summing over the N messenger multiplets and substituting λ aBC → κ i and d BC This becomes the trilinear A-term Aũ used in the paper via: Next let's consider the EGMSB contributions to the soft mass-squareds, starting with m 2 Q . From (C.3), we obtain: In the first line, we have introduced d cD B → 1 2 d φ , and we have used the fact that Q is the only MSSM field coupling to the messengers to set d cd a d BC c to zero. Additionally, as Q couples directly to the messengers, there is a one-loop term suppressed by Λ 2 M 2 [13],

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where h(x) is a loop function given by The EGMSB contribution to m 2 Hu is much simpler. Here only the third term of δm 2 ab in (C.3) contributes: where we have used d QU H = 3. Lastly, the EGMSB contribution to m 2 U also comes from just the third term of δm 2 ab : where d QH U = 2. Taking into account the need to transpose m 2 U to translate between the conventions of [20] and those of this paper, we obtain the correct result quoted in (2.6).

D Formulas for flavor observables
In this appendix, we collect formulas from the literature for the various flavor observables considered in this work. Along the way, we will streamline the different notations scattered throughout the literature into a uniform convention.
The uniform operator basis we will use was introduced in the text; we repeat it here for convenience. For dimension 5, we have: For dimension 6, we have: where M, N = L, R. P R = 1 2 (1 + γ 5 ), P L = 1 2 (1 − γ 5 ) are projection operators, σ µν = 1 2 [γ µ , γ ν ] and if f i carry color indices, they are contracted within a bilinear factor. The general effective Hamiltonian is then: where the sums runs over a complete basis of independent operators.

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Meson  Table 2. Properties of the meson mixing used in this work. f X and B LL V,X come from the FLAG review [77]. The other four non-perturbative B-parameters are taken from several sources: for ∆m K from [78] (at µ = 2 GeV), ∆m D from [79] (at µ = 3 GeV, rescaled for a common R D , and converted to our basis using B LL M X,12 and Γ X,12 are, respectively, the dispersive and absorptive parts of X|H eff |X . In terms of these quantities, the mass splitting is given by: For X = B d and B s where Γ X,12 M X,12 , this is well approximated by ∆m X ≈ 2|M X,12 | ≈ 2| X|H eff |X | (D.9) For X = K and D, where experimentally one finds that M X,12 /Γ X,12 is approximately real, and M X,12 and Γ X,12 are both predicted to be approximately real in the standard CKM convention (where the CPV phase is primarily in V td and V ub ), one has to a good approximation ∆m X ≈ 2 Re M X,12 ≈ 2 Re X|H eff |X (D.10) Finally, the short-distance part of the matrix elements in (D.9) and (D.10) (which is all that is relevant for the B q systems and for new physics) are given by

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where, (D. 12) and the B-parameters are non-perturbative corrections that have been computed on the lattice. In the Standard Model, only C LL V = 0. These values and the other parameters relevant for meson mixing are shown in table 2.
This corresponds to a dimension 6 effective Hamiltonian with f 1 = s, f 2 = d, f 3 = f 4 = ν . The branching ratios for K ± → π ± νν are given by [81], where v = 246 GeV and Here, the branching ratio has been included to remove dependence on the hadronic matrix element, and r K + = 0.901 contains isospin violating quark mass effects and electroweak corrections computed in [86]. In the standard model, the contributions from top loops are have no sensitivity to different generations, but the charm loop contributions do. In particular, the charm contributions are the same for e and µ, but differ for τ . Thus, the SM contribution can be expressed as, C LL V,SM, = α 2 πv 2 λ c X c + λ t X t (D. 15) where λ i = V * is V id , X t = 1.469 [41], X e c = X µ c = 1.055 × 10 −3 , and X τ c = 7.01 × 10 −4 [81]. In the absence of lepton flavor-violating new physics effects, the contributions from new physics are the same across lepton generations, i.e., C XY V,N P, = C XY V,N P, ≡ C XY V,N P . We can then give a simpler form to equation (D.13) BR(K ± → π ± νν) = c + v 4 C LL V + C RL Here, P c is the charm contribution averaged over the different neutrino flavors. Using P c simplifies the expression, but reduces the standard model charm contribution by about 3%. This represents only a 0.3% decrease in the overall SM contribution, which is completely negligible when compared to the theoretical uncertainty. Most importantly, the interference effects with new physics are properly captured under this simplification.

D.3 b → sγ and b → dγ
These observables correspond to dimension 5 effective Hamiltonians with f 1 = b and f 2 = s or d. The branching ratio is given by, where [87] c γ = 8π 2 2 6 π where the branching ratio to charm decays is used to remove sensitivity to the hadronic matrix element, and the Φ factor is introduced to account for the nontrivial phase space factor in the compared branching ratio (due mostly to the charm quark mass). Only C L A contains a standard model contribution, which is, where X SM = −0.3736. The observable b → dγ is defined completely analogously. The only difference is that the the standard model contribution differs. In particular, The branching ratio for B s,d → µ + µ − is [55,56]: where We compile some of the loop functions for our rank 1 χFV approximation below.
Starting with meson-mixing, we have, Here, x q = m 2 S /m 2 0 and x g = m 2 g /m 2 0 ; where m 0 and m S are the heaviest and lightest squark mass eigenvalues.
For K ± → π ± νν, there are two contributions, x 2 x 2 2 − x log x 2 Here x l = m 2 l /m 2 S , x 2 = M 2 2 /m 2 S and x µ = µ 2 /m 2 S . We have also dropped the neutralino boxes and Z-penguins that contribute at the O (10%) level or below.
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