Baryonic R-parity violation and its running

Baryonic R-parity violation arises naturally once Minimal Flavor Violation (MFV) is imposed on the supersymmetric flavor sector at the low scale. At the same time, the yet unknown flavor dynamics behind MFV could take place at a very high scale. In this paper, we analyze the renormalization group (RG) evolution of this scenario. We find that low-scale MFV is systematically reinforced through the evolution, with the R-parity violating couplings exhibiting infrared fixed points. Intriguingly, we also find that if holomorphy is imposed on MFV at some scale, it is preserved by the RG evolution. Furthermore, low-scale holomorphy is a powerful infrared attractor for a large class of non-holomorphic scenarios. Therefore, supersymmetry with minimally flavor violating baryon number violation at the low scale, especially in the holomorphic case but not only, is viable and resilient under the RG evolution, and should constitute a leading contender for the physics beyond the Standard Model waiting to be discovered at the LHC.


Introduction
Supersymmetric particles have not yet shown up at the LHC. Though the current bounds on the masses of the supersymmetric particles depend on the assumed spectrum, they are now generally at or above the TeV scale. Such a large splitting in mass between Standard Model (SM) particles and their superpartners renders the model less radiatively stable, and requires delicate fine-tunings of its parameters to be viable. There is however one scenario in which these bounds are trivially relaxed. Most of the current searches assume that the minimal supersymmetric Standard Model (MSSM) incorporates R-parity as an exact global symmetry [1]. As is well-known, see e.g. [2] for a review, this forces superparticles to be present in pairs in all vertices, and thus renders the lightest superparticle (LSP) perfectly stable. Supersymmetric events are accompanied by a significant amount of missing energy carried away by the LSP. On the other hand, if R-parity is not exact, the LSP decays and these missing energy signatures are simply not there.
Removing R-parity may be fine for the LHC, but immediately allows for proton decay or neutron oscillations. The tight bounds on these observables imply that R-parity violating (RPV) couplings involving light fermion flavors must be really tiny. So, either the RPV couplings are globally suppressed, or they are highly hierarchical in flavor space. Remarkably, an adequate hierarchy is naturally obtained if the RPV couplings are aligned with the SM flavor couplings [3]. Indeed, under the assumption that there are no new flavor structures beyond the Yukawa couplings, lepton number violating couplings are forbidden while baryon number violating ones can be sizeable only when they involve the top flavor 1 . In practice, to export in a controlled way the SM flavor hierarchies onto the MSSM flavor couplings, we use the minimal flavor violation (MFV) approach [4,5], which is based on a well-defined symmetry principle [6].
In the present paper, our goal is to study the behavior of the minimally flavor violating RPV couplings under the renormalization group (RG) evolution. Indeed, if valid, the MFV hypothesis is likely to derive from a new flavor dynamics taking place at a very high scale (see e.g. Refs. [7][8][9][10][11]), and it is crucial to check whether it survives down to the low scale. We will see that this survival severely constrains the formulation of MFV, and in particular the viable flavor symmetry group. At the same time, once these constraints are in place, MFV is not only stable, it is even radiatively reinforced through the evolution.
Before being able to delve into the numerics of the evolution, we must set up the formalism. The first task is to construct a flavor-symmetric reparametrization of the RPV couplings in terms of the Yukawa couplings. At that stage, the RPV couplings need not satisfy MFV. Actually, this reparametrization provides a unique way to specify fully generic RPV couplings independently of the flavor basis chosen for the (s)quark fields, and thus extends to the RPV sector the procedure proposed in Refs. [12,13]. This is presented in Sections 2.1 and 2.2.
With this tool in hand, the second step is to impose MFV. In the R-parity conserving (RPC) sector, this is very easy: the reparametrization has to be natural, hence must involve at most O(1) coefficients [14]. In the RPV sector, however, we find that this O(1) criterion is neither stable nor welldefined. The reasons for this, and the conditions under which consistency is recovered, are detailed in Section 2.3.
Once these initial steps are completed, the numerical study of the RG evolution is undertaken in Section 3. Special emphasis is laid on the holomorphic implementation of MFV, as proposed in Ref. [15], for which we prove several unique features, most notably the RG invariance. A matrix identity based on Cayley-Hamilton identity is instrumental in this proof, as well as in the construction of the reparametrization. It is derived in Appendix A. Finally, the boundary conditions and mass spectrum of the CMSSM-like scenario used to illustrate the behavior of the RG evolution are collected in Appendix B.

From generic to MFV couplings
The MSSM flavor sector necessitates many parameters to be fully specified. Even restricted to (s)quarks, there are 73 real constants, to which 18 complex baryonic R-parity violating couplings must be added. Specifically, the supersymmetric parameters occur in the superpotential, where I, J, K = 1, 2, 3 denote flavor indices. Because of the understood contraction over colors, Y IJK udd = −Y IKJ udd , and R-parity violation is encoded into nine independent complex couplings [2]. In addition, there are several soft-breaking terms involving squark fields and with a priori non-trivial flavor structures, where m 2 Q , m 2 U , and m 2 D are hermitian squark mass terms, while the R-parity violating couplings again satisfy A IJK udd = −A IKJ udd because of the color contraction. At the same time, many of these parameters are constrained by the now precise data from flavor physics. Assuming SUSY particles are not far heavier than about one TeV, squark mixing cannot be large and R-parity violation must be limited. In an attempt at systematically embedding these constraints, the minimal flavor violation ansatz is particularly well suited [4,5]. As a starting point towards MFV, let us first construct an alternative parametrization of all these flavor couplings, using as a guiding principle only the U (3) 3 flavor symmetry [6] of the MSSM (s)quark kinetic terms.

Flavor-basis independence
When non-vanishing, all the flavor couplings break the U (3) 3 symmetry, which means that they vary when (s)quark undergo U (3) rotations in flavor space. Since quark mass terms originate from the Yukawa couplings Y u,d , this freedom is in general used to bring all but either the up-or the down-type left-handed quarks to their mass eigenstates. For example, when all quarks but the u L are mass eigenstates, the (s)quark fields are rotated to the basis where with m u,d the diagonal quark mass matrices, V CKM the CKM matrix, and v u,d the vacuum expecta- Obviously, performing the same unitary rotations on both quark and squark fields redefines the Y udd couplings as well as the soft-breaking terms. For example, if the singular value decompositions for the Yukawa couplings are denoted as Except if a flavor model is prescribed, V u,d R,L are unknown matrices so there is a numerical ambiguity in defining the whole flavor sector. More precisely, nonuniversal soft-breaking terms are unambiguously defined only once the Yukawa couplings in the same flavor basis are known. The situation is more critical for the RPV couplings since they are never universal. In other words, they always depend on the flavor basis. This issue was discussed for the R-parity conserving MSSM in Ref. [13]. Let us briefly review the strategy proposed there to circumvent the basis dependence. The trick is to write m 2 Q,U,D and A u,d directly in terms of the Yukawa couplings. To ensure their independence on the flavor basis, the Y u,d couplings are treated as spurions, and the RPC soft-breaking terms are written as manifestly U (3) 3 symmetric polynomial expansions in the spurions. Only then are we certain that performing U (3) 3 rotations of the (s)quark fields leaves invariant the expansion coefficients. Specifically, because both Y u and Y d transform non-trivially under U (3) Q , the most generic expansions are constructed by inserting in all possible ways SU (3) Q octets, i.e., by inserting and ⊕ denotes the presence of free expansion coefficients. These nine independent terms are sufficient to project an arbitrary complex matrix. They were obtained by first invoking the Cayley-Hamilton identity to remove linearly dependent terms (see Appendix A), and then select the nine terms involving the least number of Yukawa insertions [14]. In terms of these octets then, where both A 0 and m 0 set the soft-breaking scale. The crucial observation is that the real coefficients a q,u,d i and b q,u,d i and the complex coefficients c u,d i are independent of the flavor basis in which the (s)quark fields are defined. They thus permit to unambiguously parametrize the soft-breaking terms.
The goal in the next section is to construct the same kind of expansions for the RPV couplings. But before detailing this construction, let us recall the main properties or advantages of this procedure (for more details, see Ref. [13]): 1. As long as the coefficients are left free, any complex or hermitian matrix can be projected onto the basis (5). There are as many free coefficients as there are free parameters [16].
2. The MFV hypothesis is immediate to formulate: it requires all the coefficients to be of O(1) or smaller. By contrast, as discussed in Ref. [12], the coefficients are in general much larger than one whenever a new flavor structure not precisely aligned with the Yukawa couplings is present.
3. These expansions can be defined at any scale, so the RG evolution can be encoded into that of the expansion coefficients. In Refs. [14,17], the RPC coefficients were found to exhibit infrared "quasi"-fixed points. Interestingly, the RPV coefficients also show such a behavior, as will be discussed later on.
4. CP-violating sources are naturally separated into the new phases entering via the coefficients and those arising directly from the CKM phase present in the Yukawa couplings. 5. In practice, when none of the coefficients is large, many terms in these expansions can be dropped because X 2 u,d ≈ X u,d X u,d with the flavor trace X u,d 1. In addition, when tan β is not large, terms involving X d are negligible compared to those involving X u . In those cases, our procedure offers a simple phenomenological parametrization for a fully realistic flavor sector.
As advocated in Ref. [13], at least as long as no flavor model is introduced, the procedure of fixing flavor couplings through their flavor coefficients is far better than fixing them directly in some arbitrary basis, and should be implemented in the available numerical codes. In addition, experimental constraints can be translated as limits on the size of the various coefficients. Only in this case can one draw definitive basis-independent conclusions on the size of the new flavor couplings. In this respect, all the current flavor constraints, including EDM [12], flavor observables [18], or the extremely tight proton decay bounds [3], do allow for O(1) coefficients.

Generic RPV couplings
Let us now construct the expansions for the RPV couplings. Given that a generic Y udd introduces nine arbitrary complex parameters, the simplest polynomial expansions require nine independent terms. The strategy to chose them is to first consider possible contractions with epsilon tensors. This step was described in Ref. [3]. Here, we consider only the three simplest epsilon structures where either the epsilon tensor of SU where the two terms on the left-hand side enforce where O is an arbitrary complex matrix. The right-hand side retains a manifestly SU (3) Q invariant form since O transforms as an octet. Therefore, octets need to act on the Y u factor only, and the final set of nine terms can be chosen as (remember where λ q 1,...,9 are nine free complex parameters. A similar reduction can be done starting from Eq. (6b), leading to the alternative basis Finally, for the last structure, Eq. (6c), all octet insertions but those involving can be moved to the first index, and we remain with 12 possible terms. This time, there seems to be some latitude in the identification of the basis. For reasons that will be detailed below, the best choice is to keep two such X d insertions (which have to be antisymmetrized under J ↔ K): where the coefficients are ordered according to the number of Yukawa spurions. The RPV soft-breaking term A udd transforms exactly like Y udd under the SU (3) 3 symmetry, so admits the same expansions, up to a prefactor A 0 setting the soft-breaking scale, and of course a priori different coefficients. Therefore, and similarly for A U,D udd .

MFV limit for RPV couplings
At this stage, one may wonder why three different bases, Eqs. (9), (10), and (11), are constructed to parametrize Y udd . Indeed, any one of them is sufficient to project a completely arbitrary set of Y IJK udd couplings. Generalizing, it is clear that there is an infinity of equally valid bases of nine terms, at least from a mathematical point of view. Though this is indeed true when these bases are just meant to parametrize generic couplings, the situation changes when MFV is enforced. Indeed, we must make sure that the MFV limit is stable and well-defined; the second property quoted in Section 2.1. More precisely, if a flavor coupling is expressed as a combination of Yukawa spurions with the adequate symmetry properties and O(1) coefficients, then by definition it satisfies the MFV requirement. Thus, once projected on a specific choice of basis, it must give back O(1) coefficients only. This is trivial if that particular combination of Yukawa spurions is part of the basis, but not automatic otherwise, as we now explore.

Internal stability of the epsilon contractions
Within a given basis, i.e., for a given epsilon structure, the stability is ensured by the systematic use of the Cayley-Hamilton theorem. For example, if MFV holds, then O in Eq. (7) and (8) is at most of O(1), hence can be absorbed into the coefficients without upsetting their scaling [14]. Therefore, both the Y Q udd and Y D udd bases are internally consistent. On the other hand, the Y U udd basis must contain the λ u 3 and λ u 6 terms instead of for example 3 and λ u 6 were not part of the Y U udd basis, the other terms could not reproduce them with only O(1) coefficients because there is no matrix identity relating them. The converse holds though: the KN structures are so suppressed numerically that no large coefficients are generated when projected on the Y U udd basis of Eq. (11).

Incompatibility between epsilon contractions
The stability of MFV within a given basis can be ensured, but not that between the bases with different epsilon contractions. Consider for example the identity: It shows that projecting the λ d 1 structure of the Y D udd basis on the Y Q udd basis just produces the λ q 3 term, but that λ q . Thus, what is MFV for one basis is not necessarily MFV for another basis.
At this stage, there are two possible ways to restore a well-defined MFV principle. Either we combine terms from the three bases to construct a fully general one, or we constrain the possible U (1) breakings. For example, if only U (1) Q is broken, then the Y Q udd basis suffices. Indeed, once U (1) D and U (1) U are enforced, all the terms of the Y D udd and Y U udd bases are forbidden, since they involve an epsilon tensor acting in either SU (3) D or SU (3) U . This latter alternative will be followed here, because allowing for the simultaneous presence of different U (1)-breaking terms would cause also other difficulties, as detailed below.

Compatibility with the R-parity conserving MFV expansions
When constructing the expansions of the soft-breaking terms, Eq. (5), the invariance under U (3) 3 is enforced. In principle, if the invariance under SU (3) 3 is imposed instead, additional terms should occur in their expansions, like for example where ε LM N breaks U (1) Q and ε RJK breaks U (1) D , or where both epsilons break U (1) Q . In this latter case, having two epsilons acting in the same SU (3) actually preserves the corresponding U (1) symmetry, so this term must be redundant with those already present in Eq. (5). This can be checked explicitly by simplifying the epsilon contractions while maintaining the flavor symmetry manifest as On the contrary, the term of Eq. (14) does not match those already present in Eq. (5). Even worse, if projected onto the MFV basis of Eq. (5), it generates large non-MFV coefficients. So, if one insists on the pure SU (3) 3 invariance, with all the U (1) simultaneously broken, the usual MFV basis for the R-parity conserving soft-breaking terms has to be extended.
It should be stressed that this is not just a matter of principle. Through the RG evolution, the soft-breaking terms receive corrections from the RPV couplings. For example, the one-loop β function of m 2 D contains [19,20] (β m 2 Therefore, if Y udd or A udd contain epsilon tensors acting in different SU (3) spaces, terms similar to that in Eq. (14) will occur. In that case, MFV would only be maintained through the RG evolution provided additional terms are included in the expansions of Eq. (5). For the time being, we prefer not to follow that route. We thus stick to the terms in Eq. (5), but must allow for only a single flavored U (1) to be broken when constructing the expansions for the RPV couplings Y udd and A udd .

U(1) phases and Yukawa background values
At the beginning of the previous section, we stated that it is always possible to perform U (3) 3 rotations to reach a basis where, e.g.
However, only the invariance under SU (3) 3 was used in the construction of the Y udd expansion. Because of this mismatch, these expansions may not fully fulfill their role of rendering Y udd independent of the flavor basis: the unknown phases corresponding to the broken U (1) affect the coefficients. Let us be more precise. The singular value decompositions of the Yukawa couplings are where m u,d are diagonal and positive-definite, . Since two out of the three U (1)s of U (3) 3 have to remain exact, only the epsilon tensor of a single SU (3) can occur in the expansions of the RPV couplings. This constraint prevents the phases of the expansion coefficients from depending on the flavor basis. For example, if both Y Q udd and Y D udd are present, then the phases of the Y Q udd coefficients depend on arg(det(V u,d L )) and those of Y D udd on arg(det(V d R )), but both arg(det(V u,d L )) and arg(det(V d R )) cannot be set to zero in general. Therefore, for this and the other reasons discussed above, we will restrict our attention to scenarios where only a single U (1) is broken in the rest of the paper.

Renormalization group evolution
In the previous section, we have seen that simply asking for MFV to have a chance to remain valid through the running brings a strong restriction on its formulation. Only one U (1) can be broken at a time. Consequently, there are only three possible patterns of hierarchies for the RPV couplings when MFV is valid, and those depend only on tan β. For example, with tan β = 10, both Y udd and A udd /A 0 scale as in Table 1.
In the present section, we investigate in details the evolution of the coefficients. We start with the broken U (1) Q scenario, whose main interest is to cover the special case of holomorphic MFV [15]. As a result, we will see that this scenario has several unique properties, not shared by any other couplings under MFV. By contrast, the behavior of the broken U (1) D or U (1) U scenarios is more in line with that of the RPC soft-breaking terms [14,17]. This will be illustrated for the broken U (1) D case only. A detailed analysis of the U (1) U case is not very useful since it is similar. In addition, looking at Table 1, this scenario is much less interesting phenomenologically. First, the (s)top couplings are the largest when U (1) D is broken, but never exceed O(10 −13 ) for U (1) U . The same-sign top quark signals [21][22][23][24][25][26][27][28][29] at the LHC would thus essentially disappear, and be replaced by the more challenging two or three light-jet resonances. Second, the couplings involving the up quark are the largest when U (1) U is broken, hence the sparticles have to be heavier to pass the current bounds on the proton lifetime or neutron oscillation. Finally, note that in all three scenarios some RPV couplings are tiny. This can indirectly constrain the supersymmetric mass spectrum because the squark lifetimes have to be short enough to circumvent R-hadron signatures [15,25].
The RG evolution of the A udd couplings will also be discussed for the broken U (1) Q and broken U (1) D scenarios, though briefly. Indeed, the impact of A udd is very limited phenomenologically. Whenever an A udd coupling is large, the corresponding Y udd coupling is also large. So, if a squark can decay into two other squarks through A udd , it can also decay to the corresponding quarks through Y udd with a larger available phase-space. For this reason, except maybe for a slight reduction in the RPV branching ratios to quark final states, even a relatively large A udd coupling does not significantly affect the RPV signatures at the LHC.
Throughout this section, to illustrate the evolution of the RPV expansion coefficients in a realistic setting, we use the CMSSM-like parameter point described in Appendix B. We select the boundary conditions at the GUT scale so that, in the RPC case, the Higgs boson mass is close to 125 GeV. The impact of the RPV couplings on the particle spectrum is in general limited since most RPV couplings are very suppressed, hence will be neglected here.

RG invariance of MFV holomorphy
The holomorphic restriction of MFV proposed in Ref. [15] originates from the hypothesis that the flavor symmetry is dynamical at some scale M Flavor . There, the Yukawa spurions would either be true dynamical fields, or they would be directly related to those of this unknown flavor dynamics. At the same time, supersymmetry requires the superpotential to be holomorphic, so Y udd must be insensitive to Y † u and Y † d above the scale M Flavor . The most general flavor-symmetric expansion is then very simple, since there is only one way to write Y udd in terms of Y u and Y d : The holomorphic restriction thus respects MFV under the SU With only U (1) Q broken, it respects all the requirements discussed in the previous section and MFV is stable and well defined. However, the scale M Flavor at which holomorphy is imposed could be very high. Even if MFV is in itself stable, whether holomorphy is a reasonable approximation at the low scale is not obvious. Indeed, the RG equations of the Yukawa and Y udd couplings are coupled (we follow the notations of Ref. [19,20], but for a slight change of conventions in the indices): where t = log Q 2 . At one loop, γ U I U J , γ D I D J , and γ Q I Q J all involve "non-holomorphic" spurion insertions.
The consequence for the soft-breaking terms is well-known: even starting from universal squark masses m 2 Q = m 2 U = m 2 D = m 2 0 1 at the unification scale, the whole series of coefficients in Eq. (5) end up non-zero at the low scale. One would expect the same to happen for the Y udd coupling: the whole series of coefficients in Eq. (9) would appear at the low scale.
Interestingly, the holomorphy of Y udd holds at all scale because all these non-holomorphic effects precisely cancel out. This can be checked analytically: To reach the last line, we have used the matrix identity of Eq. (38a) in the form Therefore, the whole evolution of the holomorphic Y udd can be encoded into a single coefficient: The linear dependence of dλ/dt over λ ensures the RG invariance of λ = 0, when R-parity is unbroken. The beta function β λ involves only purely left-handed anomalous terms: its sole role is to compensate for the left-handed evolutions of the Yukawa couplings, since Y udd evolves according to right-handed anomalous terms only. This explains the mechanism behind the RG invariance 3 of the only that term both brings in just the required combination of righthanded quark anomalous dimensions, and at the same time leaves the rest as a pure flavor trace. No other structure could be RG invariant.
At the one-loop order, the beta function of the coefficient λ is [19,20] where g 1 , g 2 , and g 3 are the U (1) Y , SU (2) L , and SU (3) C gauge couplings (with the SU (5) normalization for the hypercharge). The leading order RG equation of λ can easily be solved. Indeed, the evolution of the Yukawa couplings depends quadratically on Y udd , whose maximal entry in the holomorphic case is about λ × 10 −4 when tan β ≈ 50. Except for very large non-MFV values of the coefficient, the impact of Y udd on Y u,d is completely negligible. So, the ratio between the coefficients at the GUT and SUSY scale is immediately found once the RPC evolution of the Yukawa and gauge couplings is known, Numerically, the right-hand side has only a very weak dependence on the rest of the MSSM parameters, essentially through threshold corrections. Though the sensitivity is a bit enhanced by the exponential, we find that with M SUSY ≈ 1 TeV, the ratio is quite stable, varying within 1/5 and 1/4. is turned on. Also, note that even though the leading coefficient is the largest, even a radiatively-induced A udd [M SUSY ] is not holomorphic at the low scale. In addition, these values are rather stable. If one starts with a non-vanishing A udd at the GUT scale, the RG evolution push the κ q i coefficients back to the same values as in Eq. (28). As shown in Fig. 1, this fixed-point

Holomorphy as an attractor
If Y udd is not holomorphic at some scale, it will remain so at all scales since the subleading expansion coefficients λ q i of Y Q udd are non-zero. Looking back at Eq. (9), it is clear that these coefficients do not multiply RG invariant structures. Rather, through the evolution, each of these coefficients contribute a priori to all the others.
What is remarkable is that the holomorphic scenario of Ref. [15] emerges as an infrared (IR) fixed point. Specifically, starting from some non-zero λ q i =1 at the GUT scale, they all evolve towards much reduced values at the low scale. For example, starting with λ q i [M GUT ] = (1, 1, 1, 1, 1, 1, 1, 1, 1) , we find that the leading coefficient is not affected by the others (we recover the same λ q 1 value as in the previous section), while all the others are suppressed by more than an order of magnitude: This convergence towards zero is effective even when the starting values λ q i =1 [M GUT ] are much larger than one, as illustrated in Fig. 2 for λ q 2 and λ q 3 . The scaling between the values at the GUT and SUSY scale is mostly linear, with for example . This observation has an important corollary: if any of the λ q i =1 is O(1) or larger at the low-scale, then they necessarily evolve towards non-MFV values at the GUT scale.
This behavior is similar to that of the coefficients of the RPC soft-breaking terms discussed in Refs. [14,17], but for two differences. First, it is much more pronounced in the present case. The IR i =1 are trivially independent of the SUSY parameters since they are simply zero. On the contrary, for the RPC soft-breaking terms, the IR values depend on the MSSM parameters (gluino mass, scalar masses, etc), hence were dubbed "quasi" fixed points in Ref. [14].
The presence of this unique and true fixed point is of immediate phenomenological relevance 4 . If MFV is active at some very high scale and if U (1) Q is broken, then to an excellent approximation, Y udd [M SUSY ] is holomorphic at the low scale since the subleading coefficients λ q i =1 [M SUSY ] are tiny. The non-holomorphic corrections to Y udd [M SUSY ], which are in any case rather suppressed since they involve more Yukawa couplings, are thus entirely negligible and the whole baryonic RPV sector can be parametrized by a single parameter.

Comparison with the broken U(1) D case
To illustrate how peculiar is the behavior of Y Q udd , let us perform the same analysis starting with Y D udd instead. To start with, let us evolve down the leading Y D udd structure, i.e., set At the low-scale, the whole series of nine coefficients is generated: These examples show that MFV is preserved through the running, but the subleading coefficients are not particularly reduced at the low scale. Compared to the broken U (1) Q scenario, the leading coefficient λ d 1 still evolves essentially independently of the others but the λ d i =1 do not converge towards zero. This can be seen in Fig. 3, where the evolutions of λ d 1 , λ d 2 , and λ d 3 are shown for various boundary conditions. Though a strong convergence of λ d 2 and λ d 3 towards their purely radiative values of Eq. (32) is apparent, these are not true fixed points. Indeed, being finite, they must necessarily depend on the specific MSSM scenario. In other words, for a different choice of MSSM parameters, λ d 2 and λ d 3 would run towards different values.
The existence of these IR fixed points implies that MFV at the low scale does not necessarily transcribe into MFV at the high scale. In view of Fig. 3 on the Yukawa couplings and on the soft-breaking terms is far from negligible. Given that the CMSSM parameters used throughout this work (see Appendix B) are quite fine-tuned to get a viable mass spectrum in the R-parity conserving case, especially a Higgs boson mass at around 125 GeV, the above numerical evaluations should be understood as illustrations for the behavior of the coefficients. udd and A D udd remain rather smooth, MFV is preserved down from the GUT scale, and quasi fixed points in the IR are apparent (the same could be said for Y U udd and A U udd in the broken U (1) U scenario). Actually, this is perfectly in line with the behaviors of the coefficients of the RPC sector [14,17]. Therefore, in view of the large Y D tds coupling, this scenario is worth investigating further, both experimentally and phenomenologically. In particular, a dedicated study of the impact of a Y D tds ∼ O(1) coupling on the supersymmetric spectrum, and thus indirectly on the Higgs boson mass, could prove very valuable, not least because it could in principle lessen the puzzling fine-tunings, and enlarge the viable MSSM parameter space.

Conclusions
In this paper, we have analyzed the behavior of the R-parity violating couplings under the renormalization group evolution. Particular emphasis is laid on the MFV restriction, since it permits to naturally pass all the bounds from proton decay or neutron oscillations even for relatively light superparticles [3]. To this end, the formulation of the MFV hypothesis in the RPV sector first had to be made more precise and robust. Specifically, our procedure and our main results are: 1. We have constructed a basis-independent parametrization for completely generic baryonic Rparity violating couplings, in a way similar to that proposed for the R-parity conserving softbreaking terms of the slepton sector in Ref. [12] and of the squark sector in Ref. [13]. It trades the 18 independent RPV couplings for 18 free expansion parameters, whose numerical values are independent of the flavor basis chosen for the (s)quark fields. As such, they thus fully encode the RPV sector. For example, experimental constraints translate into bounds on these coefficients, MFV is directly obtained by restricting these coefficients to O(1) values, they permit to set in an unambiguous way any boundary conditions for the RPV couplings, and they even suffice to describe the whole RG evolution of the RPV couplings.
2. We have shown that to impose MFV on the whole R-parity violating MSSM and at vastly different scales, it is necessary to restrict the flavor symmetry group. Out of the U (3) 3 symmetry of the (s)quark kinetic terms, only one U (1) can be broken at a time. Failure to do so generates ambiguities in the phases of the expansion coefficients, renders the O(1) naturality criterion for expansion coefficients ambiguous, and even invalidates the usual MFV expansions in the Rparity conserving sector. On the other hand, once properly set, the multi-scale MFV hypothesis is rather resilient. The RG evolution of the RPV expansion coefficients displays striking infrared fixed or quasi-fixed point behavior, ensuring that MFV at the low scale arises even from far from-MFV scenarios at the unification scale. The corollary also holds: if the expansion coefficients at the low scale are O(1) but far from their (quasi-) fixed points, then MFV is lost at the unification scale. In these respects, the RPV sector behaves very similarly to the R-parity conserving soft-breaking sector [14,17].
3. Finally, we have explored the RG behavior of the holomorphic MFV scenario. First, we proved analytically that the holomorphic restriction is RG invariant. This is far from trivial since the Yukawa spurions are never true dynamical fields, but has far-reaching consequences. In particular, it implies that holomorphy acts as a powerful infrared attractor for the RG evolution.
If present at the high scale, all the non-holomorphic corrections evolve towards zero at the low scale. Whether exact or approximate, low-scale holomorphy thus systematically emerges as the phenomenological paradigm once the broken flavor U (1) is that of the quark doublet. Intriguingly, this same flavored U (1) is the one already broken by the B +L anomaly of the weak interactions [31,32]. In other words, MFV, which must obviously hold in the SM, is compatible with the B + L anomaly only if that U (1) is broken [33]. Though the connection appears rather coincidental at present, it is thus tempting to conclude that low-scale holomorphy should hold, at least to a good approximation.
Imposing MFV on the R-parity violating couplings is not only a viable phenomenological approach, but it may also hint at some more fundamental aspects of the yet unknown origin to the flavor structures. For instance, the allowed flavor symmetry group and the RG properties of their MFV implementations may prove crucial in the search for a dynamical realization of MFV at some very high scale. Besides this model-building frontier, what is still lacking to fully validate this approach is, obviously, some experimental signals. We are thus eagerly waiting for the next round of searches at the LHC, which will hopefully usher us beyond the realm of the Standard Model.
the RPC case, running these values through SPheno 3.2.4 [34,35], the Higgs sector mass spectrum is m h 0 = 123 GeV together with m A 0 ≈ m H 0 ≈ m H ± = 2.0 TeV, while the SUSY spectrum is mg = 2.2 TeV , mχ± = (0.82, 1.5) TeV , mχ0 = (0.43, 0.82, 1.5, 1.5) TeV , At the low scale, the expansion coefficients for the RPC soft-breaking terms respect the MFV principle, with for example (see Eq. (5) This combined with the heavy sparticle masses ensure that all the flavor observables are in check in this scenario. Turning on the RPV couplings, the numerical evolutions are not computed with SPheno. Rather, we use custom Mathematica programs to solve the one-loop RG equations [19,20] of the RPV-MSSM between M SUSY and M GUT . We check that they agree at the percent level with SPheno in the CPand R-parity conserving case.
The RPV couplings Y udd and A udd are set at the GUT scale in a basis-independent way through their expansion coefficients. The multiscale boundary conditions To avoid spuriously large coefficients when the system of equations is solved exactly, both the evolution and the matching at M GUT have to be done using more than the default 16 digits of precision. Alternatively, if one sticks to the default precision, one should approximately solve the system of equations, under the constraint that the coefficients are the smallest as possible. We have not implemented that solution, because forcing Mathematica to work with somewhere between 25 to 30 digits does not prohibitively slow down the various routines.
We do not include the RPV threshold corrections, nor do we study the corrections to the Higgs mass brought in by Y udd , since our goal here is to illustrate the evolution of the RPV couplings. This is a very good approximation when the RPV couplings respect the MFV principle with either U (1) Q or U (1) U broken, since Y udd is then very suppressed, see Table 1. On the other hand, when U (1) D is broken, the Y tds coupling is large and can in principle affect significantly the spectrum. Though we neglect this effect here, it should be kept in mind.
Finally, let us stress that this benchmark is not tailored to induce an interesting phenomenology at the LHC. With its rather heavy spectrum, most signatures are suppressed, like for example in the same-sign top quark pair channel discussed e.g. in Refs. [21][22][23][24][25][26][27][28][29]. Indeed, though the neutralino LSP would decay exclusively to top quarks when either U (1) Q or U (1) D is broken, its pair production is not intense when all squark masses are above the TeV scale. To find more interesting benchmarks for the LHC is certainly interesting, but beyond our scope here.