# Minimal flavour violation and SU(5)-unification

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## Abstract

Minimal flavour violation in its strong or weak versions, based on \(U(3)^3\) and \(U(2)^3\), respectively, allows suitable extensions of the standard model at the TeV scale to comply with current flavour constraints in the quark sector. Here we discuss considerations analogous to minimal flavour violation (MFV) in the context of *SU*(5)-unification, showing the new effects/constraints that arise both in the quark and in the lepton sector, where quantitative statements can be made controlled by the CKM matrix elements. The case of supersymmetry is examined in detail as a particularly motivated example. Third generation sleptons and neutralinos in the few hundred GeV range are shown to be compatible with current constraints.

### Keywords

Higgs Boson Yukawa Coupling Neutrino Masse Effective Field Theory Quark Sector## 1 Introduction

The discovery of the Higgs boson and the measurement of some of its couplings, together with the number of different flavour measurements performed in the last 15 years or so, have raised the tests of the standard model (SM) to a qualitatively higher level. On one side there is the reported evidence for a linear relation, \(m_i = \lambda _i v\), between the masses and the couplings to the Higgs boson of the SM particles (for the moment the heavier ones). On the other side several of the flavour-changing SM loops have been experimentally confirmed with strengths as expected in the Cabibbo–Kobayashi–Maskawa (CKM) picture of flavour physics. Altogether it is appropriate to say that the ensemble of these tests have turned the Yukawa couplings of the Higgs boson in the SM into an element of physical reality. At the same time this strikingly underlines what is perhaps the major weakness of the SM itself: its inability to predict any of these couplings. This is the current status of the flavour problem in the SM, which strongly motivates the efforts to increase the precision of the mentioned tests, now typically at the \(10\div 30\) % level.

When trying to go beyond the SM, the description of flavour faces a further problem of different nature. If new particles are expected at the TeV scale, the compliance with the flavour tests is highly non trivial. Attempts to achieve it rest on dynamical assumptions, on flavour symmetries or on a combination thereof. Examples of the first kind are gauge or anomaly mediation of supersymmetry breaking, whereas a combinations of symmetries (typically *U*(1) factors) and dynamics is invoked in models of alignment. Based on symmetries alone, minimal flavour violation (MFV) is the way to make new physics at the TeV scale compatible with flavour tests. By MFV phenomenologically defined we mean here that in an Effective Field Theory (EFT) approach the only relevant operators are the ones that correspond to the Flavour Changing Neutral Current effects occurring in the SM, weighted by a common scale and by the standard CKM factors up to (possibly flavour dependent) coefficients of order unity. As briefly recalled in the next section there exists a *strong* version of MFV [2, 3, 4] based on the \(U(3)^3\) flavour group (or equivalent) and a *weak* version, based on the \(U(2)^3\) flavour group [5, 6] (or equivalent).

MFV, as recalled above, refers to the quark sector. Is there something analogous that can be said on the lepton sector, always having in mind new physics not far from the TeV scale? When asking such question, what comes immediately to mind is the issue of neutrino masses, whose nature (Dirac or Majorana) and origin (at low or high energy among other issues) are unknown.^{1} This is a difficulty. Perhaps the very small neutrino masses do not influence at all the flavour structure of the charged lepton sector. If so, however, what is left that can possibly constrain it? In the SM, without neutrino masses, one remains with individual lepton number conservation. With extra particles at the TeV scale individual lepton number conservation is unlikely, but, leaving out neutrino masses, one seems to lose any way to argue further in a truly quantitative way.

In this paper we discuss to what extent *SU*(5) unification can avoid this impasse. By *SU*(5) unification we simply mean that there exist definite *SU*(5)-invariant Yukawa couplings that give rise, after symmetry breaking, to realistic quark masses and mixings as well as to the observed charged lepton masses. In the low energy theory this leads both to deviations from MFV in the quark sector and to a definite pattern of flavour violation in the charged lepton sector, always controlled by the CKM matrix elements. The compatibility of such patterns with current bounds will be discussed in general as well as, in particular, considering the possible existence of supersymmetric particles at the TeV scale.

The relation between *strong* MFV and *SU*(5)-unification has been first discussed in [1]. In this paper we analyse the relation of *SU*(5)-unification with both versions of MFV, *strong* or *weak*, as defined above, pointing out specific differences between the two cases.

## 2 Strong and weak minimal flavour violation

*V*is the CKM matrix. Here with \(Y_u\) and \(Y_d\) we denote the low energy Yukawa couplings with canonically normalised quark fields, which in general differ from the original symmetry-breaking parameters but have necessarily the same transformation properties under \(U(3)^3\) and can equally well be used to characterise the symmetry breaking in the EFT.

^{2}This suggests to consider \(U(2)^3\) rather than \(U(3)^3\) as the relevant symmetry with \(U(2)^3\) breaking described by small parameters. The only minimal set of spurions that can do this is

## 3 \(U(3)^2\) and *SU*(5)-unification

*SU*(5), the minimal set of Yukawa couplings for realistic charged fermion masses is

*SU*(5) as indicated, each with an \(SU(2)\times U(1)\) breaking component of similar size, and flavour indices are everywhere left understood. The inclusion of a coupling to \(H_{45}\) is necessary to account for the different \(\mu - s\) and \(e - d\) masses. A possibility to describe neutrino masses is to introduce a triplet of fermions,

*N*, not transforming under

*SU*(5) and include in \(\mathscr {L}_Y \) the further terms \(N Y_N \bar{F} H_5 + N M N\). We assume that the elements of \(Y_N\) are small enough not to influence the considerations developed in the following. This is certainly consistent, e.g., if any of the elements of the matrix

*M*is less than \(10^{11}\) GeV. As mentioned, we do this to limit the impact of our ignorance on the values of \(Y_N\) and

*M*separately.

*SU*(5) case is to consider the symmetry

*T*and \(\bar{F}\) as \(T = (3,1)\) and \(\bar{F} = (1,3)\) [1]. Furthermore one assumes that \(U(3)^2\) is only broken in the directions

*h*is the only light Higgs doublet and the multiplets

*Q*,

*u*,

*d*,

*e*, each with a flavour index and canonically normalised kinetic terms, have definite transformation properties under \(U(3)_T\times U(3)_{\bar{F}}\)

*V*is again the CKM matrix. On the contrary \(\lambda _e\) has the form

### 3.1 \(U(3)^2\) and lepton flavour violation

The presence of two spurions \(\lambda _d^T\) and \(\lambda _e\) with the same transformations properties under \(U(3)^2\), one of which dependent on unknown mixing matrices, is the source of potentially large deviations from MFV, particularly from chirality-breaking down-quark operators. If compared with the current bounds, an even stronger direct constraint arises from lepton flavour violation (LFV) and, more specifically, from the \(\mu \rightarrow e + \gamma \) transition.

## 4 \(U(2)^2\) and *SU*(5)-unification

*SU*(5)-invariant Yukawa Lagrangian

^{3}

Before proceeding, let us note that \(U(2)^2\), unlike \(U(3)^2\), makes natural room for the successful relations \(m_b \approx m_\tau \) and \(m_\mu \approx 3 m_s\), valid at unification. This only requires that \(\Delta _1\) be sufficiently smaller than \(\Delta _2\) not to undo the last relation arising from the coupling to \(H_{45}\). At the same time \(\Delta _1\) must be capable to give the proper relation between \(m_e\) and \(m_d\). We assume in the following that all the elements of \(\Delta _1\) are at most of the order needed to this purpose. This in turn implies that the relative alignment between the \(\lambda _d\) and \(\lambda _e^T\) matrices is, without any further assumption or tuning, of order \(m_d/m_s\) both on the left and on the right side.

### 4.1 \(U(2)^2\) and LFV

In analogy with the discussion in the \(U(3)^2\) case, the presence of two spurions with the same transformation properties in the down and charged lepton sectors is a source of potentially large flavour violations. In the \(U(2)^2\) case, however, there are two significant differences. As just said, in 1–2 flavour space \(\lambda _d\) and \(\lambda _e^T\) are misaligned only by relative rotations of the order of \(m_d/m_s\). Furthermore, due to the small \(U(2)^2\) breaking, the diagonalisation of both \(Y_d\) and \(Y_e\) in 2–3 flavour space is obtained by small rotations of the same order \(\epsilon \).

*A*,

*B*are the misalignment matrices between \(\lambda _e\) and \(\lambda _D^T\) in the 1-2 sector, of order \(m_d/m_s\). One gets therefore the bound

### 4.2 Electric dipole moments

^{4}

*U*(2) group factor or, differently stated, that the diagonalisation matrix \(U_u\) is present on both sides of \(Y_u^D\) in Eq. (31). Barring cancellations this gives the limits

Upper bounds on the coefficients of the operators discussed in the text, normalised to \(\Lambda = 3\) TeV

Observable | \(\mu \rightarrow e \gamma \) | e EDM | u EDM | d EDM | \(\epsilon '\) | \(A_{CP}^{\Delta C=1}\) |

Coefficient | \(|c^{\mu \rightarrow e\gamma }|\) | \( |\text {Im}(\tilde{c}_e^{EDM})| \) | \( |\text {Im}(\tilde{c}_u^{EDM})|\) | \( |\text {Im}(c_d^{EDM})|\) | \(|c^{\Delta S=1} \text {sin}\phi |\) | \(|c^{\Delta C=1}|\) |

Upper bound | \(5\times 10^{-4}\) | \(1.6 \times 10^{-5} \) | \(1.2\times 10^{-2}\) | \(5.6 \times 10^{-3}\) | \(6.5 \times 10^{-2}\) | 0.2 |

### 4.3 \(U(2)^2\) and quark flavour violation

## 5 \(U(2)^2\) in supersymmetry

The picture that emerges from the previous sections is that \(U(2)^2\) gives rise to several new effects than the ones normally considered in MFV, with the relevant feature, as in MFV, that their flavour structure, both in the quark and, more interestingly, in the charged lepton sector, is always controlled by the CKM matrix elements. It is this feature that allows to make quantitative considerations.^{5}

The model we consider is a generic supersymmetric *SU*(5)-theory with a Yukawa superpotential that gives rise to \(\mathscr {L}_Y^{U(2)}\) as in Eq. (24) and with soft supersymmetry-breaking terms generated by supergravity. In the flavour sector the entire theory is invariant under \(U(2)^2\) as in Eqs. (25) and (26). On this basis we shall consider the low energy theory in two different ways. We implement the general case as discussed above or we take universal *A*-terms at least when restricted to the 1-2 sector.

*A*-terms and in the squared masses for squarks and leptons. The Yukawa couplings \(Y_{u,d,e}\), with the usual meaning of the angle \(\beta \),

*A*-terms

*a*-factors are mass terms of similar order of magnitude, related to the low energy scale of effective supersymmetry breaking. Finally the squared masses have two different forms. Up to negligibly small terms quadratic in \(\lambda _{e, d, u}\), the squared masses for \(\tilde{L}\) and \(\tilde{d}_R\) are diagonal and degenerate in 1-2 sector, whereas the mass terms for \(\tilde{e}_R, \tilde{u}_R\) and \(\tilde{Q}\) have contributions controlled by the spurion \(\mathbf {V}\), i.e.and similar for \(M_{\tilde{Q}}^2, M_{\tilde{u}}^2\).

### 5.1 \(\mu \rightarrow e + \gamma \)

*A*-terms were universal, at least in the 1-2 sector, one would have no \(\lambda _d^T\) term in (46) and (47) and, in the basis under consideration, we would haveOn the contrary, in the general case, in which both \(\lambda _e\) and \(\lambda _d^T\) are present, it is the latter that dominates. Therefore, in this case we havewhere on both sides of the diagonal term there appear two unitary 1–2 transformations of order \(m_d/m_s\), representing precisely the misalignment of \(\lambda _d^T\) with \(\lambda _e\).

*A*-terms universal) the only flavour-changing matrix present in these interactions is \(U_e\). As such, a GIM-like cancellation takes place, controlled by the non-degeneracy of the charged sleptons, of relative order unity between the third and the first two generations and of relative order \(\epsilon ^2\) within the first two generations. As a consequence the \(\mu \rightarrow e \gamma \) amplitude receives contributions proportional to \(m_\mu U^*_e(\mu \tau )U_e(e \tau )\) (a 1–2/3 effect) or to \(m_\mu \epsilon ^2 U^*_e(\mu \mu )U_e(e\mu )\) (a 1–2 effect), both equal to \(m_\mu V_{ts}^*V_{td}\) up to a factor of order unity. In the case of a general \(A_e\)-term the presence of the misalignment matrices in Eq. (51) inhibits the GIM-like cancellation in the diagram of Fig. 1b, which then becomes the dominant contribution to the amplitude, proportional to \(m_d\) and mediated by exchanges of sleptons of the first two generations.

Representative values for the size of these effects, taken incoherently, and normalised to the current limit, \(BR(\mu \rightarrow e \gamma ) < 5.3\times 10^{-13}\) [13], are shown in Figs. 2 and 3 both for general and universal \(A_e\)-term. Consistently with this bound, the largest possible value of \(BR(\tau \rightarrow \mu \gamma )\) can be reached with a universal \(A_e\)-term, at \(10^{-9}\) level. One should remember the order one unknown factor in front of each of these amplitudes.

### 5.2 Electron EDM

## 6 Summary and conclusions

The effort to increase the precision of current flavour tests of the SM, now at the \(10\div 30\) % level, is a strongly motivated task of particle physics in itself. At the same time this effort could give indirect signals of the existence of new particle at the TeV scale, complementary to the potentiality of their direct search in high energy collisions. Although not exclusively, nevertheless a strong basis for this statement is the possibility that MFV be at work in some extension of the SM. Especially in its weak form, based on the \(U(2)^3\) flavour group, phenomenological MFV can explain the absence of new signals so far, while making plausible their emergence in foreseen flavour physics experiments.

While MFV has a predictive content in the quark sector, this is relatively less the case when one tries to extend it to the lepton sector, due to the uncertainties related to the description of neutrino masses. To overcome this problem here we have proposed a predictive scheme based on extending MFV considerations to *SU*(5)-unification. As far as the quark sector is concerned, weak MFV can be made consistent with *SU*(5)-unification without introducing new strong constraints, even though some interesting CP-violating effects appear both through \(\Delta S=1\) and \(\Delta C=1\) chirality-breaking operators. In the charged lepton sector, on the other hand, one predicts flavour violations with intensities also controlled, to a good approximation, by the CKM mixing angles. From a general EFT point of view Table 1 is an effective summary of our findings. As shown there, not unexpectedly, the current limits on \(\mu \rightarrow e \gamma \) as on the electron EDM represent strong constraints.

Although not exclusively, supersymmetry is the obvious arena where these considerations might be of relevance. For this reason we have considered their implementation in a realistic supersymmetric *SU*(5)-theory with soft supersymmetry-breaking terms generated by supergravity. Without any extra assumption \(\mu \rightarrow e \gamma \) and the electron EDM with maximal CP-violating phases require charged sleptons of the first two generations in the \(1\div 3\) TeV range, as shown in Figs. 2 and 4. Sleptons of the first two generations in the few hundred GeV range can be made compatible with the flavour scheme proposed here provided the *A*-terms are universal and the CP-violating phases contributing to the electron EDM are not maximal. Third generation sleptons and neutralinos in the few hundred GeV range are in any case consistent with present bounds. This is illustrated in Figs. 3 and 5. The weaker bound on the mass of the third generation leptons comes from the fact that in any event the communication between them and the first two lepton generations is controlled by the small CKM matrix elements.

## Footnotes

- 1.
- 2.
- 3.
Note a small abuse of notation: here and below the matrices \(\lambda _{u,d,e}\) act in the 1–2 flavour space, unlike the case of Sect. 3 where they act on the full 1–2–3 space.

- 4.
For brevity we do not discuss the chromo-magnetic dipole operators for the up and down quarks, but they lead to similar bounds on the corresponding coefficients. The contribution of the charm chromo-electric dipole, \(c^{CEDM}_c m_c/\Lambda ^2\), to the three gluon Weinberg CP-violating operator gives also a significant bound \(|\text {Im}(c^{CEDM}_c)|\lesssim 3\times 10^{-2} (\Lambda /\text {3 TeV})^2\) [19]. Note also that along this line by

*e*,*u*,*d*we mean specifically the first generation particles and not the flavour triplets as in Sect. 3. - 5.

## Notes

### Acknowledgments

R.B. thanks Dario Buttazzo and Filippo Sala for discussions at the early stage of this work and Gino Isidori and David Straub for useful exchanges. F.S. thanks Diptimoy Ghosh for useful discussions. This work is supported in part by the European Programme “Unification in the LHC Era”, contract PITN-GA-2009-237920 (UNILHC) and by MIUR under the contract 2010YJ2NYW-010.

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