# Flavor-Changing Higgs Decays in Grand Unification

with Minimal Flavor Violation

###### Abstract

We consider the flavor-changing decays of the Higgs boson in a grand unified theory framework which is based on the SU(5) gauge group and implements the principle of minimal flavor violation. This allows us to explore the possibility of connecting the tentative hint of the Higgs decay recently reported in the CMS experiment to potential new physics in the quark sector. We look at different simple scenarios with minimal flavor violation in this context and how they are subject to various empirical restrictions. In one specific case, the relative strengths of the flavor-changing leptonic Higgs couplings are determined mainly by the known quark mixing parameters and masses, and a branching fraction % is achievable without the couplings being incompatible with the relevant constraints. Upcoming data on the Higgs leptonic decays and searches for the decay with improved precision can offer further tests on this scenario.

^{†}

^{†}preprint: KIAS-P16031

## I Introduction

The ongoing measurements on the 125 GeV Higgs boson, , at the Large Hadron Collider (LHC) have begun to probe directly its Yukawa interactions with fermions atlas+cms (); cms:h->2mu (); atlas:h->2mu (); cms:h->mutau (); cms:h->etau (); atlas:h->mutau (); lhc:t->hq (). In particular, for the branching fractions of the standard decay modes of , the ATLAS and CMS experiments have so far come up with

(1) |

where the upper limits in the second line are at 95% confidence level (CL). Overall, these data are still in harmony with the expectations of the standard model (SM).

However, there are also intriguing potential hints of physics beyond the SM in the Higgs Yukawa couplings. Especially, based on 19.7 fb of Run-I data, CMS cms:h->mutau () has reported observing a slight excess of events with a significance of 2.4, which if interpreted as a signal implies

(2) |

but as a statistical fluctuation translates into the bound

(3) |

Its ATLAS counterpart has a lower central value and bigger error, corresponding to at 95% CL atlas:h->mutau (). Naively averaging the preceding CMS and ATLAS signal numbers, one would get . More recently, upon analyzing their Run-II data sample corresponding to 2.3 fb, CMS has found no excess and given the bound at 95% CL CMS:2016qvi (). This indicates that the analyzed integrated luminosity is not large enough to rule out the Run-I excess and further analysis with more data is necessary to exclude or confirm it. In contrast, although the observation of neutrino oscillation pdg () suggests lepton flavor violation, the SM contribution to lepton-flavor-violating Higgs decay via -boson and neutrino loops, with the neutrinos assumed to have mass, is highly suppressed due to both their tiny masses and a Glashow-Iliopoulos-Maiani-like mechanism. Therefore, the excess events would constitute early evidence of new physics in charged-lepton interactions if substantiated by future measurements. On the other hand, searches for the and channels to date have produced only the 95%-CL bounds cms:h->etau ()

(4) |

from CMS and from ATLAS atlas:h->mutau ().

In light of its low statistics, it is too soon to draw firm conclusions about the tantalizing tentative indication of in the present LHC data. Nevertheless, in anticipation of upcoming measurements with improving precision, it is timely to speculate on various aspects or implications of such a new-physics signal if it is discovered, as has been done in very recent literature Dery:2013rta (); h2mt (); h2mt' (); h2mt'' (); He:2015rqa (). In this paper, we assume that is realized in nature and entertain the possibility that it arises from nonstandard effective Yukawa couplings which may have some linkage to flavor-changing quark interactions beyond the SM. For it is of interest to examine how the potential new physics responsible for may be subject to different constraints, including the current nonobservation of Higgs-quark couplings deviating from their SM expectations.

To handle the flavor-violation pattern systematically without getting into model details,
we adopt the principle of so-called minimal flavor violation (MFV).
Motivated by the fact that the SM has been successful in describing the existing data on
flavor-changing neutral currents and violation in the quark sector, the MFV hypothesis
presupposes that Yukawa couplings are the only sources for the breaking of flavor and
symmetries mfv1 (); D'Ambrosio:2002ex ().
Unlike its straightforward application to quark processes, there is no unique way to
formulate leptonic MFV.
As flavor mixing among neutrinos has been empirically established pdg (), it is
attractive to formulate leptonic MFV by incorporating new ingredients that can
account for this fact Cirigliano:2005ck ().
One could assume a minimal field content where only the SM lepton doublets and charged-lepton
singlets transform nontrivially under the flavor group, with lepton number violation and
neutrino masses coming from the dimension-five Weinberg operator Cirigliano:2005ck ().
Less minimally, one could explicitly introduce right-handed
neutrinos Cirigliano:2005ck (), or alternatively right-handed weak-SU(2)-triplet
fermions He:2014efa (), which transform nontrivially under an enlarged flavor
group and play an essential role in the seesaw mechanism to endow light neutrinos with
Majorana masses seesaw1 (); seesaw3 ().
One could also introduce instead a weak-SU(2)-triplet of unflavored
scalars He:2014efa (); Gavela:2009cd () which participate in the seesaw
mechanism seesaw2 ().^{1}^{1}1Other aspects or scenarios of leptonic MFV have been
discussed in the literature Branco:2006hz (); mlfv (); He:2014fva (); Grinstein:2006cg ().
Here we consider the SM expanded with the addition of three heavy right-handed neutrinos
as well as effective dimension-six operators conforming to the MFV criterion in both
the quark and lepton sectors.^{2}^{2}2A similar approach has been adopted in Lee:2015qra () to study some lepton-flavor-violating
processes that might occur as a consequence of the recently observed indications of anomalies in
rare decays.
To establish the link between the lepton and quark interactions beyond the SM, we consider
the implementation of MFV in a grand unified theory (GUT) framework Grinstein:2006cg ()
with SU(5) as the unifying gauge
group Georgi:1974sy (); Ellis:1979fg ().^{3}^{3}3A detailed analysis of the interplay between quark and lepton sectors in the framework of
a supersymmetric SU(5) GUT model with right-handed neutrinos can be found in SUSY_GUT ().
In this GUT scheme, there are mass relations between the SM charged leptons and down-type
quarks, and so we will deal with only the Higgs couplings to these fermions.

In the next section, we first briefly review the application of the MFV principle in a non-GUT framework based on the SM somewhat enlarged with the inclusion of three right-handed neutrinos which participate in the usual seesaw mechanism to generate light neutrino masses. Subsequently, we introduce the effective dimension-six operators with MFV built-in that can give rise to nonstandard flavor violation in Higgs decays, specifically the purely fermionic channels . Then we look at constraints on the resulting flavor-changing Higgs couplings to quarks and leptons, focusing on the former, as the leptonic case has been treated in detail in Ref. He:2015rqa () which shows that the CMS signal interpretation can be explained under the MFV assumption provided that the right-handed neutrinos couple to the Higgs in some nontrivial way. In Section III, we explore applying the MFV idea in the Georgi-Glashow SU(5) GUT Georgi:1974sy (), following the proposal of Ref. Grinstein:2006cg (). As the flavor group is substantially smaller than in the non-GUT scheme, the number of possible effective operators of interest becomes much larger. Therefore, we will consider different scenarios involving one or more of the operators at a time, subject to various experimental constraints. We find that there are cases where the restrictions can be very severe if we demand %. Nevertheless, we point out that there is an interesting scenario in which the flavor-changing leptonic Higgs couplings depend mostly on the known quark mixing parameters and masses and at the percent level can occur in the parameter space allowed by other empirical requirements. Our analysis serves to illustrate that different possibilities in the GUT MFV context have different implications for flavor-violating Higgs processes that may be testable in forthcoming experiments. We give our conclusions in Section IV. An appendix contains some extra information.

## Ii Higgs fermionic decays with MFV

The renormalizable Lagrangian for fermion masses in the SM supplemented with three right-handed Majorana neutrinos is

(5) | |||||

where summation over the family indices is implicit, denote 33 matrices for the Yukawa couplings, is a left-handed quark (lepton) doublet, and and represent right-handed up- and down-type quarks (neutrinos and charged leptons), respectively, stands for the Higgs doublet, with being the second Pauli matrix, is a 33 matrix for the Majorana masses of , and the superscript of refers to charge conjugation. We select the eigenvalues of to be much greater than the elements of , so that the type-I seesaw mechanism becomes operational seesaw1 (), leading to the light neutrinos’ mass matrix , which also involves the Higgs vacuum expectation value GeV, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS pmns ()) mixing matrix for light neutrinos, and their eigenmasses in . This suggests that Casas:2001sr ()

(6) |

where is in general a complex orthogonal matrix, .

Hereafter, we suppose that are degenerate in mass, and so . The MFV hypothesis D'Ambrosio:2002ex (); Cirigliano:2005ck () then implies that is formally invariant under the global flavor symmetry group , where and . This entails that the above fermions are in the fundamental representations of their respective flavor groups,

(7) |

where are special unitary matrices and is an orthogonal real matrix D'Ambrosio:2002ex (); Cirigliano:2005ck (); Branco:2006hz (). Moreover, the Yukawa couplings transform under in the spurion sense according to

(8) |

To construct effective Lagrangians beyond the SM with MFV built-in, one inserts products of the Yukawa matrices among the relevant fields to devise operators that are both -invariant and singlet under the SM gauge group D'Ambrosio:2002ex (); Cirigliano:2005ck (). Of interest here are the combinations

(9) |

Given that the largest eigenvalues of and are and , respectively, at the mass scale , for our purposes we can devise objects containing up to two powers of and drop contributions with , as higher powers of can be connected to lower ones by means of the Cayley-Hamilton identity Colangelo:2008qp (). As for , we assume that the right-handed neutrinos’ mass is big enough, GeV, to make the maximum eigenvalue of order 1, which fulfills the perturbativity condition He:2014fva (); Colangelo:2008qp (). Hence, as in the quark sector, we will keep terms up to order and ignore those with , whose elements are at most . Accordingly, the relevant spurion building blocks are

(10) |

where in our model-independent approach and are free parameters expected to be at most of and with negligible imaginary components He:2014fva (); Colangelo:2008qp (), so that one can make the approximations and .

Thus, the desired -invariant effective operators that are SM gauge
singlet and pertain to Higgs decays into down-type fermions at tree
level are given by Cirigliano:2005ck ()^{4}^{4}4In this study, we do not address
couplings to up-type quarks for the following reason. As the operator
with from (10) conserves flavor, others with
, such as
,
would be needed, but with only one Higgs doublet they are relatively suppressed by
the smallness of the elements, which makes the present empirical
bounds lhc:t->hq (); Harnik:2012pb () on
and not strong enough to offer meaningful constraints.

(11) |

where the mass scale characterizes the underlying heavy new physics and the covariant derivative acts on with hypercharges , respectively, and involves the usual SU(2) gauge fields and , their coupling constants and , respectively, and Pauli matrices , with being summed over. There are other dimension-six MFV operators involving and fermions, particularly

(12) |

in the quark sector and

(13) |

in the lepton sector, where and are the same in form as
and , respectively, except they have their own coefficients
and , but these operators do not induce at tree level.
In the literature the operators
and
are also often considered (e.g., Dery:2013rta ()), but they can be shown
using the equations of motion for SM fields to be related to and
the other operators above Grzadkowski:2010es ().^{5}^{5}5This was explicitly done for
the leptonic operators in He:2015rqa ().

It is worth remarking that there are relations among and
above (among their respective sets of coefficients ) which are fixed within
a given model, but such relations are generally different in a different model.
As a consequence, stringent bounds on processes induced by one or more of the quark operators
in Eqs. (11) and (12) may not necessarily apply to the others,
depending on the underlying new-physics model.
Similar statements can be made regarding , , and the lepton
operators in Eqs. (11) and (13).^{6}^{6}6The high degree of model dependency in the relationships among the s belonging to
the different operators is well illustrated by the results of the papers in
h2mt (); h2mt' (); h2mt'' () which address in the contexts of various scenarios.
Particularly, there are models h2mt' () in which % is
achievable from tree-level contributions without much hindrance from the strict experimental
requirements on transitions, including lepton , which arise
from one-loop diagrams.
In some other models h2mt'' () all these processes only occur at the loop level and the limiting
impact of the restrictions on is considerable.
It follows that one cannot make definite predictions for in
a model-independent way based on the input from .
For these reasons, in our model-independent analysis on the contributions of to
we will not deal with constraints on the operators in
Eqs. (12) and (13).
Our results would then implicitly pertain to scenarios in which such constraints do not
significantly affect the predictions for .

In view of in Eq. (11) which is invariant under the flavor symmetry , it is convenient to rotate the fields and work in the basis where are diagonal,

(14) |

and , , , , and refer to the mass eigenstates. Explicitly, , , and . Accordingly,

(15) |

where is the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix.

Now, we express the effective Lagrangian describing as

(16) |

where are the Yukawa couplings, which are generally complex, and are chirality projection operators. This leads to the decay rate

(17) |

where the fermion masses have been neglected compared to . Thus, from Eq. (11), which contributes to both flavor-conserving and -violating transitions, we find for

(18) | |||||

(19) |

where we have included the SM contributions, which are separated from the terms and can be flavor violating only in the quark case due to loop effects, and at tree level. Since approximately , it follows that in our MFV scenario for and are real.

For , it is instructive to see how they compare to each other in the presence of . In terms of the Wolfenstein parameters , the matrices and in are given by

(20) |

to the lowest nonzero order in for each component, as and at the renormalization scale . If the part of for is dominant, we then arrive at the ratio

(21) |

the numbers having been calculated with the central values of the Wolfenstein parameters
from Ref. ckmfit ()^{7}^{7}7Explicitly, , ,
, and .
as well as MeV and GeV at .

The SM coupling with arises from one-loop diagrams with the boson and up-type quarks in the loops. Numerically, we employ the formulas available from Ref. Dedes:2003kp () to obtain , , , and relatively much smaller . These SM predictions are, as expected, consistent with the ratio in Eq. (21), but still lie very well within the indirect bounds inferred from the data on -, -, and - oscillations, namely Harnik:2012pb ()

(22) |

Hence there is ample room for new physics to saturate one or more of these limits. Before examining how the contributions may do so, we need to take into account also the measurement quoted in Eq. (I). Thus, based on the 90%-CL range of this number in view of its currently sizable error, we may impose

(23) |

where from the central values of the SM Higgs total width MeV and determined in Ref. lhctwiki () for GeV pdg (). Upon applying the preceding constraints to Eq. (18), we learn that in Eq. (II) and the one in Eq. (23) are the most consequential and that the former can be saturated if at least both the and , or , terms in are nonzero. We illustrate this in Fig. 1 for , where the limits of the (blue) shaded areas are fixed by the just mentioned bound and the values in these areas ensure that Eq. (23) is satisfied. Interchanging the roles of and would lead to an almost identical plot. If , these results imply a fairly weak lower-limit on the MFV scale of around 50 GeV.

For the leptonic Yukawa couplings, in Eq. (19), the situation is different and not unique because the specific values and relative sizes of the elements of in can vary greatly He:2015rqa (). In our MFV scenario with the type-I seesaw, this depends on the choices of the right-handed neutrinos’ mass and the orthogonal matrix as well as on whether the light neutrinos’ mass spectrum has a normal hierarchy (NH) or an inverted one (IH).

For instance, if is real, from Eq. (15), and using the central values of neutrino mixing parameters from a recent fit to global neutrino data nudata () we find in the NH case with

(24) |

Incorporating this and selecting in to be employed in Eq. (19), we then arrive at . Interchanging the roles of and would modify the ratio to . In the IH case with , the corresponding numbers are roughly about the same. These results for the Yukawas in the real- case turn out to be incompatible with the following experimental constraints on the Yukawa couplings if we demand % as CMS suggested, but with being complex instead it is possible to satisfy all of these requirements He:2015rqa ().

For the first set of constraints, the direct-search limits in Eqs. (3) and (4) translate into cms:h->etau ()

(25) |

and under the no-signal assumption, while Eq. (2) for the signal interpretation implies

(26) |

Additionally, the latest experimental bound at 90% CL meg () on the loop-induced decay can offer a complementary, albeit indirect, restraint Dery:2013rta (); Harnik:2012pb (); Goudelis:2011un () on different couplings simultaneously He:2015rqa ()

(27) |

with Harnik:2012pb (). This could be stricter especially on than its direct counterpart in Eq. (25) if destructive interference with other potential new physics effects is absent. Compared to Eqs. (25)-(27), the indirect limits Harnik:2012pb () from the data on and leptonic anomalous magnetic and electric dipole moments are not competitive for our MFV cases. Finally, the measurements quoted in Eq. (I) are also relevant and may be translated into

(28) |

where and from and % supplied by Ref. lhctwiki ().

As pointed out in Ref. He:2015rqa (), the aforementioned leptonic MFV scenario with the matrix in being real is unable to accommodate the preceding constraints, especially Eqs. (26) and (27), even with the terms in contributing at the same time. Rather, it is necessary to adopt a less simple structure of with being complex, which can supply extra free parameters to achieve the desired results, one of them being . This possibility was already explored in Ref. He:2015rqa () and therefore will not be analyzed further here.

## Iii Higgs fermionic decays in GUT with MFV

In the Georgi-Glashow grand unification based on the SU(5) gauge
group Georgi:1974sy ()^{8}^{8}8For a review see, e.g., Langacker:1980js ().
the conjugate of the right-handed down-type quark, , and the left-handed
lepton doublet, , appear in the representations ,
whereas the left-handed quark doublets, , and the conjugates of the right-handed
up-type quark and charged lepton, and , belong to
the 10 representations .
With three SU(5)-singlet right-handed neutrinos being included in the theory, the Lagrangian
for fermion masses is Grinstein:2006cg (); Ellis:1979fg ()

(29) | |||||

where SU(5) indices have been dropped, and are Higgs fields in the 5 and 24 of SU(5), and compared to the GUT scale the Planck scale . Since contains for the SM plus 3 degenerate right-handed neutrinos, the Yukawa couplings in these Lagrangians satisfy the relations Grinstein:2006cg (); Ellis:1979fg ()

(30) |

where . Evidently, in the absence of the dimension-five nonrenormalizable term in the down-type Yukawas would be related by which is inconsistent with the experimental masses Ellis:1979fg (). In this work, we do not include the corresponding term for the up-type quark sector, Grinstein:2006cg (), which could significantly correct the up-quark mass, but does not lead to any quark-lepton mass relations.

The application of the MFV principle in this GUT context entails that under the global flavor symmetry group the fermion fields and Yukawa spurions in transform as Grinstein:2006cg ()

(31) |

where we have assumed again that the right-handed neutrinos are degenerate, , and . It follows that the flavor transformation properties of the fermions and Yukawa coupling matrices in are