DESY 13152
CERNPHTH/2013291
WITSCTP124
Light thirdgeneration squarks from flavour gauge messengers
Felix Brümmer, Moritz McGarrie and Andreas Weiler
SISSA/ISAS, I34136 Trieste, Italy
Deutsches ElektronenSynchrotron DESY, D22603 Hamburg, Germany
National Institute for Theoretical Physics, School of Physics, and Centre for Theoretical Physics, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa
CERN, Theory Division, Geneva, Switzerland
1 Introduction
In gaugemediated supersymmetry breaking, the SUSYbreaking hidden sector is charged under the gauge interactions of the supersymmetric Standard Model, and soft terms are induced by gauge boson, gaugino, and hiddensector loops. This mediation mechanism is attractive because it is predictive and well controlled: The soft terms for the visible sector depend on just a few parameters, and the underlying theory can be a fourdimensional, renormalizable (but typically strongly coupled) quantum field theory.
The automatic flavour universality of gaugemediated soft terms is a major advantage of gauge mediation, since it explains the absence of disastrous squark and sleptoninduced flavour changing neutral currents. It is becoming less attractive in the light of the results from the first LHC run, which point towards first and secondgeneration squarks heavier than 0.8 – 1.8 TeV [1, 2] for decoupled to equal mass gluinos, respectively. The constraints on thirdgeneration squarks are much weaker by comparison, for example 300 GeV stops are still allowed for LSP masses above 120 GeV [3, 4]. Moreover, light stops are often argued to be preferred by naturalness. A factor of two or more between the squark masses of the first and third generation is clearly at odds with flavour universality, even when taking into account the mass splittings that are induced by renormalization group running from the mediation scale to low energies. Additionally, it has been shown that the radiatively induced splittings do not ameliorate the finetuning problem [5].
Recently several models have been proposed which allow for flavour nonuniversal soft masses while retaining most of the predictivity of pure gauge mediation. In [6, 7, 8, 9, 10, 11], messenger fields were allowed to couple to, and mix with, the visible sector matter and Higgs fields in the superpotential. This may give additional nonuniversal contributions to the scalar soft masses. If the mattermessenger couplings are controlled by suitable flavour symmetries, FCNCs can still be suppressed sufficiently. When the first and second generation squarks are split due to the alignment of quark and squark mass matrices [11], this results in significantly weaker limits from direct LHC searches [12]. In [13, 14], an subgroup of the spurious flavour symmetry of the quark sector was gauged and taken to be higgsed by the Yukawa couplings. Its contributions to gauge mediation for the various squark masses then depends on the corresponding higgsing scales. For a suitably chosen scale of SUSY breaking mediation, large first and secondgeneration squark masses can be induced while keeping the third generation light. Similiar models based on abelian flavour symmetries were proposed earlier in [15, 16].
In the present paper we investigate an alternative possibility to obtain nonuniversal squark masses from a gauged flavour symmetry. In our model, supersymmetry breaking and flavour breaking are not disconnected, but are triggered by the same vacuum expectation values. This induces treelevel SUSY breaking masses for the broken gauginos, which in turn generate flavour nonuniversal soft masses through loops. Such “gauge messenger models”, where massive gauge multiplets couple directly to SUSY breaking, have been considered previously, mainly in the context of GUT breaking (see e.g. [17, 18, 19, 20, 21] for early work, and more recently [22, 23, 24, 25, 26]). To our knowledge the present model is the first which investigates the effects of gauge messengers for a spontaneously broken gauged flavour symmetry, or in fact for any extension of the Standard Model gauge group by a simple factor.
The dominant contribution of gauge messengers to the soft term spectrum is a tachyonic scalar soft mass squared which is generated at one loop. In a model which also contains ordinary chiral messenger fields charged under the SM gauge group, the oneloop tachyon can compete with the usual positive twoloop scalar masses, provided that the gauge coupling is somewhat smaller than the Standard Model gauge couplings. Since the supersymmetry breaking VEV is aligned with the top and bottom Yukawa couplings in flavour space, the negative contribution to the thirdgeneration squark masses is naturally much larger than the contributions to the first two generation squark masses. This leads to light stop and sbottom squarks.
This paper is organized as follows: In the next section we precisely define the class of models we are investigating, and present the leadingorder effect of flavour gauge messenger fields on the soft terms. In Section 3 we discuss the resulting superpartner mass spectra. We illustrate the effect of flavour gauge messengers using a number of parameter points in the MSSM and in the NMSSM. Section 4 is concerned with explicit example models for flavour and SUSY breaking: We show that the alignment between flavour symmetry breaking and SUSY breaking, which is a crucial ingredient in our models, can be realized in a simple model. Using this flavourbreaking pattern to generate realistic Yukawa textures, we can then compute the resulting contributions to flavourchanging neutral currents. We summarize our findings and conclude in Section 5.
2 Flavour gauge messengers in gauge mediation
The matter superfields of the supersymmetric Standard Model transform under an nonabelian flavour symmetry when the Yukawa couplings are switched off. Our main interest is an subgroup under which the quark superfields , and each transform as a .^{1}^{1}1Other flavour groups and representations, such as with , and , might also be of interest. We restrict ourselves to the quark sector here, although our construction could easily be extended to a model of lepton flavour, e.g. in order to embed it into a GUT model. The class of models we are considering is characterized by three essential features:

is a gauge symmetry,

it is spontaneously broken to at a scale , where the thirdgeneration Yukawa couplings are generated (while is broken completely at some lower scale, thus generating the remaining Yukawa couplings),

some of the vacuum expectation values which break also break supersymmetry.
Gauged quark flavour symmetries have been considered in supersymmetric model building for a long time (see e.g. [27, 13, 28, 31, 29, 34, 35, 14, 32, 33, 30, 37, 36] for some recent work). Among them is distinguished by being anomaly free with respect to the Standard Model gauge groups, so no new chiral matter with Standard Model charges needs to be added to promote it to a gauge symmetry. The idea of an approximate flavour symmetry acting on the first two generations also has a long history [38, 39, 40, 41, 42]. What is new here is mainly the third point: The same dynamics that leads to breaking may also be responsible for supersymmetry breaking. Later we will construct an explicit model where this mechanism is realized. For now we focus on the consequences for the squark soft term spectrum.
When a gauge symmetry such as is higgsed, the gauge fields and gauginos associated to the broken gauge generators become massive. If the breaking is nonsupersymmetric, in the sense that some charged fields acquire term vacuum expectation values, this will lead to treelevel SUSY breaking mass splittings between the broken gauge fields and gauginos. Thus they become messenger fields for gaugemediated supersymmetry breaking, inducing soft masses for the fields that are charged under through loops. (When allowing for nonzero terms, they can even induce soft masses at the tree level [43], but here we will only consider models in which the terms vanish.)
Gauge messengers for some general gauge group broken to were studied in great detail in [24]. This analysis was conducted using a formalism similar to general gauge mediation [44], which relies only on the assumption that the theory should be perturbative in the gauge coupling . The SUSYbreaking hidden sector itself, on the other hand, may be strongly coupled as one might expect for a realistic model of dynamical SUSY breaking. In [24] it was established that the leadingorder effect in on the visiblesector soft terms is a oneloop scalar soft mass
(1) 
Here are the generators of in the representation under which transforms, and is the piece, taken in a limit where becomes small but the gauge boson mass is kept constant, in the supertraced gauge supermultiplet propagators
(2) 
where
(3) 
The precise form of is modeldependent, and incalculable if the hidden sector is strongly coupled, but not essential for our purposes. However, it is important to note that the integral in Eq. (1) is typically negative. This has been shown to hold quite generally under certain weak assumptions [24], but is easiest to see explicitly when SUSY breaking is small, i.e. when the SUSYbreaking mass splittings within the gauge supermultiplet are much smaller than the gauge boson and gaugino masses themselves
Let us consider the small SUSY breaking case, where is broken to by the lowest and highest (i.e. term) component VEVs of some chiral superfields, and the SUSY breaking scale is suppressed compared to the mediation scale as set by the supersymmetric VEVs. Then the massive vector superfields can be integrated out supersymmetrically, and the leading effects of SUSY breaking mediation can be computed using the oneloop effective Kähler potential [45] (see also [46])
(4) 
Here the mass matrix for a massive vector field is given by
(5) 
where runs over all charged chiral superfields , and the are the generators of the corresponding representation, with . Splitting the into visible chiral superfields (which do not acquire vacuum expectation values) and hidden fields (which may acquire vacuum expectation values in both their lowest and components), Eq. (4) is seen to contain a term
(6) 
where
(7) 
The component of will contribute to the scalar soft masses at one loop,
(8) 
As already emphasized, these contributions are generally tachyonic, and nonvanishing if there are several with nonvanishing VEVs. In the case that all VEVs commute, this is seen by expanding the logarithm in Eq. (7) to obtain [24]
(9) 
where the inner products are defined in terms of the highest and lowestcomponent VEVs and by
(10) 
We are interested in the case where is a quark flavour symmetry with gauge coupling , and is the subgroup preserved by switching on only the top Yukawa coupling. The simplest way to break with realistic Yukawa matrices is to use two spurions , in the of (see e.g. [13]). The quark superpotential is
(11) 
with and . The simplest way to break with an term is to use a spurion in the . If in a basis where is diagonal and , then preserves . Eqns. (7) and (8) yield
(12) 
for any of the visiblesector fields transforming as under , up to corrections suppressed by small Yukawa couplings and CKM angles.
This model lacks an explanation for the flavour hierarchies, as well as a dynamical mechanism to align the SUSYbreaking term with the third generation in flavour space. Our main example will therefore use a different set of spurions, namely, , with untilded fields transforming as and tilded ones as . The dominant VEVs are
(13) 
and the top Yukawa coupling is generated by the operator
(14) 
The remaining Yukawa couplings are induced by subdominant supersymmetric VEVs for and , as we will explain in detail in Section 4. In that section we will also offer a dynamical explanation for the alignment of and . For this model, we find from Eqns. (7) and (8), again up to small corrections,
(15) 
The relative mass splittings in Eq. (15) have a simple grouptheoretic origin [24]. Since in this model all spurions transform in the same representation (up to conjugation) and all VEVs are aligned, in Eq. (7) is universal for all broken generators and can be chosen as , or more generally in Eq. (1). Then Eq. (1) becomes
(16) 
with the difference between the quadratic Casimirs of the representation and the representation of . For , , and the gauge coupling, we have
(17) 
Therefore, the oneloop contribution to the squark masssquared matrices can be written as
(18) 
where is some modeldependent characteristic mass scale; in the small SUSY breaking case, and we recover Eq. (15).
Eq. (18) must be interpreted with some care. First, it holds only at the scale of breaking, and second, it holds only in a particular flavour basis. Rotating to the superCKM basis will induce corrections, including small offdiagonal squark masses, which depend on the details of flavour symmetry breaking.
There are other soft terms induced by gauge messenger fields, but these will generically appear only at higher order in perturbation theory. For instance, gauge messengers induce oneloop terms, but is evidently subdominant with respect to the oneloop soft mass . There are also additional twoloop contributions to the scalar soft masses, and MSSM gaugino masses generated at threeloop order. For the rest of this paper, we will neglect these higherorder effects,^{2}^{2}2They may be relevant in models where the oneloop soft mass squared of Eq. (16) is suppressed for some reason. This is the case when the VEV of the scalar superpartner of the Goldstino is the only [46] or more generally the dominant [24] source of breaking. We will not consider such models here. and retain only the oneloop soft mass of Eq. (18). Indeed we will eventually take the gauge coupling to be very small, few , in order to obtain a realistic phenomenology, so higher loop orders can be safely neglected.
3 Soft terms and lowenergy spectrum
Clearly, the soft parameters induced by gauge messengers alone cannot account for a realistic superpartner mass spectrum: The squarks are tachyonic, and gaugino, slepton, and Higgs masses are tiny because they are induced only at higher loop order. We therefore need to consider more general models of gauge mediation where there are also contributions to the soft masses from hiddensector states charged under . The simplest such models are models with weakly coupled chiral messenger superfields, such as minimal gauge mediation. For concreteness, let us therefore assume that the matter and gaugino soft masses are as predicted by minimal gauge mediation (see [47] for a review) at the messenger scale , i.e. given in terms of an messenger index and the scale (taken to satisfy ) as
(19) 
To these we add the gauge messenger contributions to the squark masses of Eq. (18)
(20) 
We emphasize however that our mechanism as such does not rely on minimal gauge mediation: Similar conclusions will be reached whenever one assumes that the squark masses are flavourblind (as they generally are in conventional gauge mediation without gauged flavour symmetries) except for the gauge messenger contributions of Eq. (20). In particular, Eqs. (19) could be replaced by the soft masses obtained from any model of general gauge mediation. Moreover, the mediation scales for the chiral and gauge messengers could in general be distinct.
Assuming that is comparable with , the effect on the spectrum will mostly depend on the size of the extra gauge coupling . If is of the order of the Standard Model gauge couplings or larger, the tachyonic oneloop squark masses of Eq. (20) will be dominant over the positive twoloop squark masses of Eqs. (19), leading to an unrealistic spectrum. On the other hand, if is too small, there will be no noticeable effect coming from the gauge messengers at all. The most interesting parameter region is the one where the stop and sbottom masses from Eqs. (20) and (19) are of similar magnitude. This is typically the case for few , whereupon the stop and sbottom squarks become light, while the first and second generation squarks are less affected.
A wellknown benefit of large stop masses is of course that they allow one to accommodate a 125 GeV Higgs boson within the MSSM. This is because the lightest Higgs mass receives loop corrections proportional to . Another potentially large correction comes from the stop trilinear parameter . However, it is well known to be difficult to obtain a 125 GeV Higgs within pure gauge mediation, because is predicted to be negligibly small at the mediation scale. Lifting the lightest Higgs mass with only the radiatively induced then requires extremely heavy . These observations would thus seem to disfavour our gauge messenger model in connection with the MSSM.
It is important to note that is in fact not the case, since these arguments rest on rather too strong assumptions about SUSY breaking mediation. Within potentially realistic scenarios, our gauge messenger contribution to the stop mass may indeed make it easier to obtain a 125 GeV Higgs without having to resort to extreme parameter values. The crucial point here has actually been known for some time, although it is often ignored (as evidenced by the fact that phenomenological studies of “GMSB” benchmark scenarios are still being conducted): Pure gauge mediation has a problem [48]; a mechanism which solves this problem will generically give additional contributions to the Higgs soft masses and trilinear terms on top of the purely gaugemediated ones. Here by pure gauge mediation we mean any model in which the visible and hidden sector are coupled only by Standard Model gauge interactions. Then the higgsino mass parameter vanishes, as does the Higgs mass mixing parameter at the messenger scale.^{3}^{3}3There is a way to avoid this conclusion if one assumes that the origin of is unrelated to supersymmetry breaking, that it happens to be of the order of the soft mass scale by accident, and that at lower scales is induced radiatively. We will not consider this possibility as it leaves an unnatural coincidence of scales unexplained. To obtain realistic and terms, additional interactions between the Higgs sector and the SUSYbreaking hidden sector are needed, but these will affect also , and the trilinear terms in a modeldependent manner. For phenomenological studies of gauge mediation, it is therefore preferable to either rely on an explicit model which realizes this (and which ideally should allow one to calculate the resulting soft terms), or to leave all Higgs sector soft terms as free parameters.
It is highly nontrivial to build a calculable model which solves the  problem in gauge mediation, and the Higgs sector is not actually the focus of our study. We therefore choose to treat , , , , and as independent parameters, with the understanding that they could emerge from a variation of any of the more complete models on the market (see e.g. [49, 50]). By contrast, the soft terms in the matter and gaugino sectors are taken as predicted by minimal gauge mediation with additional gauge messengers.
To match to the Standard Model at low energies, the model parameters must be chosen such that both the electroweak scale and the lightest Higgs mass GeV are reproduced properly. In addition, the soft terms should be chosen such as not to be in conflict with LEP and LHC search bounds. This places severe constraints on the spectrum, in particular on the masses of the first two generation squarks and of the gluino, all of which should be significantly above a TeV.
Naturalness arguments, on the other hand, favour stop and gluino masses which are as low as possible. In the MSSM, the most natural remaining parameter region is characterized by subTeV stop squarks, with the Higgs mass accounted for by a maximal contribution from stop mixing. This in turn requires (where ). As we have argued above, a realistic gaugemediated model supplemented with additional Higgshidden sector interactions may well allow for large terms. Usually, however, it does not allow for reasonably light stops while at the same time evading the LHC bounds on the first generation squarks and gluinos. This is where the gauge messenger contributions can play a crucial role.
Fig. 2 shows the effect on the squark sector mass parameters in the MSSM, and the consequences for the lightest Higgs mass, as the gauge messenger contributions are switched on. The Higgs sector parameters were chosen to allow for maximal stop mixing when the gauge messenger contribution to the stop mass is sizeable. They are listed under “MSSMI” in Table 1. The resulting Higgs mass can be compatible with the LHC discovery when taking theory uncertainties into account. Of course maximal stop mixing is also possible with no gauge messenger contributions at all, but this would require either extremely large terms (of the order of TeV for the parameter point we are showing) or dangerously small firstgeneration squark and gluino masses (since they are tied to the stop masses in gaugemediated models without gauge messengers).
MSSMI  MSSMII  NMSSM  

GeV  GeV  GeV  
GeV  GeV  GeV  
GeV  
GeV  GeV  GeV  
GeV  GeV  GeV  
GeV  
0 
In the left panel of Fig. 3 we show the RG evolution of the stop and Higgs sector soft masses from the mediation scale to the TeV scale, for the same parameter point but keeping fixed. Note that the lighter stop soft mass, roughly given by , is negative at high energies (this is also the case for ; all other squark masses are positive at all scales). When running down towards the electroweak scale, it is driven positive by the gluino mass. Tachyonic boundary conditions for the stops have previously been employed to improve the finetuning in the MSSM [55], in particular also in the context of gauge mediation [56] and an based gauge messenger model [22].
For a generic direction in the space of the MSSM scalar fields, a negative running soft mass at high renormalization scales is no cause for concern (here 1 TeV denotes the soft mass scale). At first glance it would seem to induce a VEV of the order , where is some combination of MSSM gauge couplings. However, implies that the running treelevel potential at the scale is a poor approximation to the full effective potential, since the higher loop corrections would involve large logarithms. Instead one should use the treelevel potential at the scale , but there all squark masses are positive, so the additional vacuum is in fact spurious.
A potentially problematic case are the flat directions along which the quartic coupling vanishes, such that a large field expectation value could easily develop [57, 58]. If a mass along these directions becomes negative at large , the potential would be unbounded from below. In the presence of suitable higherdimensional operators all flat directions are lifted [61], and the runaway is stabilized, but additional vacua will appear in which electric charge and/or colour are broken. For the above model the most dangerous flat direction is the one associated with the operator , because it involves the two tachyonic fields and and only one positivemass field . We have checked that the mass along this direction remains positive at all scales up to , for all values of that yield a tachyonfree spectrum at the electroweak scale.
A somewhat more extreme case is shown in the right panel of Fig. 3, corresponding to the parameters listed under “MSSMII” in Table 1. This point serves to show that maximal stop mixing can even be purely radiatively induced in our model, although this comes at the price of a high mediation scale, a rather large (around TeV) gluino mass, and squarks which become tachyonic starting from around only GeV. Radiative effects, in particular due to the gluino mass, eventually drive the squark masses positive and the term large. Similar soft mass patterns have been discussed in [56]. For this model, the potential is indeed unbounded from below, which signals the appearance of additional charge and colourbreaking vacua. These can be problematic in two ways: Firstly, the universe could prefer to settle in them, rather than in the electroweak vacuum, during the early cosmological evolution. Secondly, even if our vacuum is the preferred one, one still needs to ensure that it does not decay on cosmological timescales. A detailed investigation of the constraints on negative squark masses from cosmology is beyond the scope of this paper, but would certainly be interesting to conduct (see also [58, 59, 60]).
Flavour gauge messengers may also be included in extensions of the MSSM where there is no need to rely on large corrections to the Higgs mass from the stop sector. For example, in the NMSSM, a SMlike Higgs with the proper mass can be obtained even with low stop masses and mixings, because there is an additional contribution to the Higgs quartic coupling coming from a superpotential term with a gauge singlet. Fig. 4 shows the squark masses in a random scan over the parameter space of the NMSSM Higgs sector (see also Table 1). For obvious reasons, the dependence of the squark masses on is similar as in the MSSM (Fig. 2); the difference between these plots is, however, that all of the points shown in Fig. 4 are compatible with a lightest Higgs mass of GeV.
Our examples show that it is possible to obtain a gaugemediated soft term spectrum with light thirdgeneration squarks from flavour gauge messengers, in a variety of scenarios. If hints of supersymmetry were to surface in stop or sbottom searches, this would be a natural way to explain the lightness of the third generation within gauge mediation.
Light stop squarks are often argued to alleviate the supersymmetric little hierarchy problem. This is because the Higgs potential is very sensitive to the stop masses, so if the stop masses are much larger than the electroweak scale, accidental cancellations are required in order to obtain the proper Fermi scale. Conversely, in a model with relatively light stops the electroweak scale can be naturally of the right order. Our mechanism provides an example of how this argument may fail (but fail in interesting way): While we can easily obtain subTeV stops, by playing off the positive contribution to the soft mass from standard gauge mediation against the negative contribution from flavour gauge messengers, these two contributions are individually large and independent. The usual measure of finetuning is the sensitivity of the electroweak scale with respect to variations of the independent fundamental model parameters. In our case, the electroweak scale depends sensitively on both (large) contributions to the stop mass, regardless of whether or not their sum is small, so by this standard we do not gain much in terms of finetuning from having light stops.^{4}^{4}4Any discussion on the subject of naturalness and finetuning, however, relies on assumptions about the UV completion. In [55, 22] the authors argue that light stops, or tachyonic highscale boundary conditions for the stop masses, could even be beneficial for naturalness. Their only benefit regarding naturalness is that, within the MSSM, less extreme values for the terms are needed to lift the Higgs mass.
4 An explicit model
4.1 Supersymmetry breaking and flavour symmetry breaking
The mechanism we have proposed relies on the alignment of supersymmetry breaking and breaking in flavour space. To show that this can be easily realized, let us construct a simple O’Raifeartaigh model as an example. The superpotential is
(21) 
where and are chiral superfields transforming as under , and transform as , and is a singlet. There is a symmetry under which , and carry charge . For later reference we note that there is also a non symmetry acting on , , and , with a subgroup which will be of interest for us. All fields except are odd under this .
We choose the parameters , , and to be real and positive, and such that and . Then the term potential is minimized at
(22) 
with a flat direction at tree level. Supersymmetry is broken because
(23) 
The oneloop effective potential will stabilize the remaining treelevel flat directions, with the VEV at or close to zero if the gauge coupling is small [65, 66].
With these VEVs, the term potential vanishes for . For there will be a nonvanishing term induced by and . Explicitly, in a gauge where ,
(24) 
In the absence of other fields taking VEVs, this term will push the vacuum away from the term pseudomoduli space of Eqs. (22). It is then easy to see that also in the new vacuum the term will be nonzero, which could induce dangerous VEVs for the squarks. We therefore assume that the overall term vanishes^{5}^{5}5This is a notable difference to models of treelevel gauge mediation [43], where a gauge symmetry is also broken by an term, but the ensuing term plays a crucial role for generating soft masses. due to another hidden sector field taking a VEV in the direction. For instance, if , an additional field in the will take a VEV , cancelling the term.
The terms of and break the flavour symmetry, and they are dynamically aligned in flavour space with the VEVs of and by the equations of motion. The gaugemediated soft terms are calculated as outlined in Section 2 (see also Appendix A). The result is Eq. (18) with
(25) 
Here we have neglected the , , and VEVs, as they will be small for .
In more general models, especially in strongly coupled ones, the small SUSY breaking limit need not be realized, and the flavourbreaking terms may be of the same order as the largest VEVs. This case of a single scale for SUSY breaking and gauge symmetry breaking is investigated in [23] for a symmetry instead of , and also in general in [24]. While the conclusions of Section 3 would remain unaffected, the relation between the VEVs and the scale would become more complicated than Eq. (25) (which holds only at the leading order in , or equivalently in ). For our purposes it is sufficient to consider the simpler case of Eq. (25).
In order to break completely, and to thereby generate realistic Yukawa matrices, additional fields charged under should take VEVs which are not aligned with . The simplest possibility is to add another pair of chiral superfields , in the whose VEVs are generated independently of SUSY breaking (and parametrically smaller than ). Then the flavourbreaking term of remains aligned with . Superpotential couplings or would spoil this alignment, but they are forbidden if and are even under the symmetry. Furthermore, the condition that no term should arise from the and VEVs fixes and to be equal up to a phase. Eq. (18) will receive small corrections from breaking; the dominant contribution is calculated in Appendix A.
To also obtain additional soft masses from minimal gauge mediation, the simplest possibility is to add flavoursinglet messenger fields , which transform as under and which couple to as
(26) 
where is an explicit messenger mass.^{6}^{6}6Note that the symmetry breaking superpotential of Eq. (26) will destabilize the SUSYbreaking vacuum. This is a common problem when trying to extend O’Raifeartaigh models into full models of minimal gauge mediation. If the explicit breaking is small, the SUSYbreaking minimum may persist as a metastable state, and the model may still be realistic. However, ultimately our SUSY breaking model should be regarded as a stepping stone towards a full model and is meant to merely illustrate the dynamical alignment of the term with the flavourbreaking VEV. Additionally, a number of Standard Model singlets charged under should be added to cancel the anomaly, and given a mass by coupling them to the breaking VEVs. Finally, there should also be heavy fields charged under that are integrated out at a somewhat higher scale, thereby generating the operators which ultimately induce the Yukawa couplings (see next section). We do not specify all these additional states because they will not have any significant effect on the visible sector – they affect the scalar soft masses only at the twoloop level in . For completeness, the field content as far as we have specified it is listed in Table 2.
field  

We do not advocate this model as a fully realistic hidden sector (for instance, in a full model one would expect all scales to be generated dynamically). However, it does exhibit all the characteristics which we used in Sect. 3, and these might well be found also in a more complete dynamical model of SUSY and flavour breaking:

There is a stepwise breaking of the flavour symmetry, , and , . , with the two steps triggered by the VEVs of

The SUSYbreaking fields , take part in the first breaking step. Their lowest components do not develop a significant VEV, but their terms are aligned with the VEVs of and by the equations of motion.

The terms of and induce SUSYbreaking gaugino masses for the broken flavour gauge bosons, which become gauge messengers.

Additional chiral messengers with Standard Model gauge charges also contribute to the visiblesector soft masses.
4.2 Yukawa and CKM hierarchies
The gauge messenger contribution to the squark soft masses will generally induce flavour changing neutral currents, which are strongly constrained by experiment. To calculate the effects of flavour violation, we need to specify the breaking pattern in more detail.
For the sake of concreteness, we will study a simple nonabelian FroggattNielsenlike flavour model as an example.^{7}^{7}7For other models also based on an horizontal symmetry, see e.g. [67, 68, 69]. As in the previous section, we introduce breaking fields , in the and , in the , which are treated as background fields from now on. We ignore , , , and ; direct superpotential couplings between them and the visible sector can be forbidden e.g. by symmetry.
In addition to the symmetry of the previous section under which and are odd and all other fields even, we impose a symmetry under which only and are charged with charges and respectively. We assume that all these fields develop vacuum expectation values satisfying
(27) 
where is an constant. In other words, the VEVs of and are aligned in flavour space, and the VEVs of and differ only by a phase; as we have argued in the previous section, this can easily be realized dynamically.
A further crucial assumption is that and are of the order of some cutoff scale , while and are suppressed with respect to by a factor . Without loss of generality, we can then choose a basis where
(28) 
where , , and . This shows explicitly that is broken to at a scale , and is subsequently completely broken at a lower scale .
Since the cutoff is of the order of the messenger scale, a potentially relevant source for soft masses are quartic terms coupling the hidden sector to the visible sector in the Kähler potential, such as . These terms will in fact be induced at the twoloop level by the usual gauge mediation diagrams. They are subdominant with respect to the oneloop gauge messenger contribution to the soft masses, provided that they are not already generated at one loop or at the tree level. This should be ensured by appropriate symmetries of the UV completion, analogous to messenger parity in ordinary gauge mediation.
The top Yukawa coupling is generated by the superpotential operator
(29) 
after replacing with its VEV. Further contributions to the uptype quark Yukawa matrix, suppressed by powers of , come from the superpotential operators
(30) 
Here we have dropped the indices in favour of the shorthand notation and . These operators give the leadingorder contributions to the matrix elements of . The contributions from all other operators allowed by and are of higher order in (except that any of the can be multiplied with a function of which is , but which can be absorbed in the couplings). The resulting Yukawa matrix is of the form
(31) 
For the down quark sector we can write down the equivalent terms with couplings . A realistic Yukawa hierarchy requires to be accidentally somewhat small, . The Yukawa matrix becomes
(32) 
The resulting Yukawa couplings are
(33) 
and the CKM matrix is
(34) 
The exact values for of the CKM angles and Yukawa couplings can be written, in an expansion in , as functions of , , , , and of the couplings and . For this roughly reproduces the observed flavour hierarchy, although some observables such as are slightly too suppressed, which needs to be compensated by the unknown coefficients.^{8}^{8}8A similar pattern was advocated e.g. in [70, 71, 72, 73]. We have checked that it is nevertheless possible to fit all quark masses and mixings with coefficients (see Appendix B for details). If the scale , associated with the uptype quarks, is taken different from the scale , at which the operators of Eqns. (29) and (30) for the downtype quarks are generated, one could improve the fit even further by having two distinct expansion parameters and [67], but we will not do so here.
In the flavour basis of Eqns. (31) and (32) the gauge messenger contributions to the squark soft masses Eq. (20) are diagonal but nonuniversal. Therefore, in the superCKM basis where the Yukawa matrices are diagonal, offdiagonal entries in the squark mass matrices will appear, inducing potentially dangerous FCNCs. In the next section we will investigate the constraints and possible observable consequences following from this.
4.3 Flavour violation
So far we have ignored the subleading offdiagonal squark masses which are also generated by gauge messengers. As we have already stated, Eq. (20) holds only in the flavour basis of Eq. (31). Rotating to the superCKM basis (in which we denote the soft matrices by ) induces
(35) 
To leading order, the offdiagonal terms in can be expressed in terms of CKM matrix entries. This is because is given by
(36) 
and