1 Introduction

The ongoing measurements on the 125 GeV Higgs boson, h, at the Large Hadron Collider (LHC) have begun to probe directly its Yukawa interactions with fermions  [1,2,3,4,5,6]. In particular, for the branching fractions of the standard decay modes of h, the ATLAS and CMS experiments have so far come up with [1,2,3]

$$\begin{aligned}&\frac{\mathcal{B}\big (h\rightarrow b\bar{b}\big )}{\mathcal{B}\big (h\rightarrow b\bar{b}\big )_{\textsc {sm}}} \,=\, 0.70^{+0.29}_{-0.27}, \nonumber \\&\frac{\mathcal{B}(h\rightarrow \tau ^+\tau ^-)}{\mathcal{B}(h\rightarrow \tau ^+\tau ^-)_{\textsc {sm}}^{}} \,=\, 1.12^{+0.24}_{-0.22},\nonumber \\&\mathcal{B}(h\rightarrow e^+e^-) \,<\, 0.0019, \nonumber \\&\quad \mathcal{B}(h\rightarrow \mu ^+\mu ^-) \,<\, 0.0015, \end{aligned}$$
(1)

where the two upper limits 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\(^{-1}\) of Run-I data, CMS [4] has reported observing a slight excess of \(h\rightarrow \mu ^\pm \tau ^\mp \)  events with a significance of 2.4\(\sigma \), which if interpreted as a signal implies

$$\begin{aligned} \mathcal{B}(h\rightarrow \mu \tau )= & {} \mathcal{B}(h\rightarrow \mu ^-\tau ^+)+\mathcal{B}(h\rightarrow \mu ^+\tau ^-)\nonumber \\= & {} \big (0.84^{+0.39}_{-0.37}\big )\%, \end{aligned}$$
(2)

but as a statistical fluctuation translates into the bound [4]

$$\begin{aligned} \mathcal{B}(h\rightarrow \mu \tau ) \;<\; 1.51\% \quad \mathrm{at}~95\%~\mathrm{CL}. \end{aligned}$$
(3)

Its ATLAS counterpart has a lower central value and bigger error, \(\mathcal{B}(h\rightarrow \mu \tau )=(0.53\pm 0.51)\%\) corresponding to \(\mathcal{B}(h\rightarrow \mu \tau )<1.43\%\) at 95% CL [5]. Naively averaging the preceding CMS and ATLAS signal numbers, one would get \(\mathcal{B}(h\rightarrow \mu \tau )=(0.73\pm 0.31)\%\). More recently, upon analyzing their Run-II data sample corresponding to 2.3 fb\(^{-1}\), CMS has found no excess and given the bound \(\mathcal{B}(h \rightarrow \mu \tau )<1.20\%\) at 95% CL [9]. 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 [10] suggests lepton flavor violation, the SM contribution to lepton-flavor-violating Higgs decay via W-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 \(h\rightarrow \mu \tau \) 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 \(e\mu \) and \(e\tau \) channels to date have produced only the 95%-CL bounds  [6]

$$\begin{aligned} \mathcal{B}(h\rightarrow e\mu )< 0.036\%, \quad \mathcal{B}(h\rightarrow e\tau ) < 0.70\% \end{aligned}$$
(4)

from CMS and \(\mathcal{B}(h\rightarrow e\tau )<1.04\%\) from ATLAS [5].

In light of its low statistics, it is too soon to draw firm conclusions about the tantalizing tentative indication of \(h\rightarrow \mu \tau \) 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  [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57]. In this paper, we assume that \(\mathcal{B}(h\rightarrow \mu \tau )\sim 1\%\) 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 \(h\rightarrow \mu \tau \) 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 CP violation in the quark sector, the MFV hypothesis presupposes that Yukawa couplings are the only sources for the breaking of flavor and CP symmetries [58,59,60,61,62,63]. 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  [10], it is attractive to formulate leptonic MFV by incorporating new ingredients that can account for this fact [64]. 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 [64]. Less minimally, one could explicitly introduce right-handed neutrinos [64], or alternatively right-handed weak-SU(2)-triplet fermions [65], which transform nontrivially under an enlarged flavor group and play an essential role in the seesaw mechanism to endow light neutrinos with Majorana masses  [66,67,68,69,70,71,72,73,74,75]. One could also introduce instead a  weak-SU(2)-triplet of unflavored scalars [65, 76] which participate in the seesaw mechanism [77,78,79,80].Footnote 1 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 the lepton sectors.Footnote 2 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  [88] with SU(5) as the unifying gauge group  [91, 92].Footnote 3 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  \(h\rightarrow f\bar{f}'\). 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. [57] which shows that the CMS \(h\rightarrow \mu \tau \) signal interpretation can be explained under the MFV assumption provided that the right-handed neutrinos couple to the Higgs in some nontrivial way. In Sect. 3, we explore applying the MFV idea in the Georgi–Glashow SU(5) GUT [91], following the proposal of Ref. [88]. 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 \(\mathcal{B}(h\rightarrow \mu \tau )\sim 1\)%.  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 \(\mathcal{B}(h\rightarrow \mu \tau )\) 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 Sect. 4. An appendix contains some extra information.

2 Higgs fermionic decays with MFV

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

$$\begin{aligned} {\mathcal L}_\mathrm{m}^{}= & {} -(Y_u)_{kl}^{}\,\overline{Q}_{k,L\,}^{}U_{l,R\,}^{} \tilde{H} - (Y_d)_{kl}^{}\,\overline{Q}_{k,L\,}^{}D_{l,R\,}^{} H \nonumber \\&-\, (Y_\nu )_{kl}^{}\,\overline{L}_{k,L\,}^{}\nu _{l,R\,}^{}\tilde{H} - (Y_e)_{kl}^{}\,\overline{L}_{k,L\,}^{}E_{l,R\,}^{} H \nonumber \\&-\, \tfrac{1}{2} (M_\nu )_{kl}^{}\,\overline{(\nu _{k,R}){^\mathrm{c}}}\,\nu _{l,R}^{} \;+\; \mathrm{H.c.}, \end{aligned}$$
(5)

where summation over the family indices \(k,l=1,2,3\) is implicit, \(Y_{u,d,\nu ,e}\) denote 3 \(\times \) 3 matrices for the Yukawa couplings, \(Q_{k,L}\) \((L_{k,L})\) is a left-handed quark (lepton) doublet, \(U_{l,R}\) and \(D_{l,R\,}\) \(\bigl (\nu _{l,R}^{}\) and \(E_{l,R}\bigr )\) represent right-handed up- and down-type quarks (neutrinos and charged leptons), respectively, H stands for the Higgs doublet, \(\tilde{H}=i\tau _2^{}H^*\) with \(\tau _2^{}\) being the second Pauli matrix, \(M_\nu \) is a 3 \(\times \) 3 matrix for the Majorana masses of \(\nu _{l,R}\), and the superscript of \((\nu _{k,R})^\mathrm{c}\) refers to charge conjugation. We select the eigenvalues of \(M_\nu \) to be much greater than the elements of  \(v Y_\nu /\sqrt{2}\), so that the type-I seesaw mechanism becomes operational [66,67,68,69,70,71,72,73,74], leading to the light neutrinos’ mass matrix \(m_\nu ^{}=-(v^2/2)\, Y_\nu ^{}M_\nu ^{-1}Y_\nu ^{\textsc {t}}=U_{\textsc {pmns}\,}^{} \hat{m}_{\nu \,}^{}U_{\textsc {pmns}}^{\textsc {t}}\), which also involves the Higgs vacuum expectation value \(v\simeq 246\) GeV,  the Pontecorvo–Maki–Nakagawa–Sakata (PMNS [96, 97]) mixing matrix \(U_{\textsc {pmns}}\) for light neutrinos, and their eigenmasses \(m_{1,2,3}^{}\) in \(\hat{m}_\nu ^{}=\mathrm{diag}\bigl (m_1^{},m_2^{},m_3^{}\bigr )\). This suggests that [98]

$$\begin{aligned} Y_\nu ^{} = \frac{i\sqrt{2}}{v}\,U_{\textsc {pmns}}^{}\hat{m}^{1/2}_\nu OM_\nu ^{1/2}, \end{aligned}$$
(6)

where O is in general a complex orthogonal matrix, \(OO^{\textsc {t}}={1}\mathrm{l}\equiv \mathrm{diag}(1,1,1)\). 

Hereafter, we suppose that \(\nu _{k,R}\) are degenerate in mass, and so \(M_\nu ={\mathcal M}{1}\mathrm{l}\). The MFV hypothesis [63, 64] then implies that \({\mathcal L}_\mathrm{m}\) is formally invariant under the global flavor symmetry group \({\mathcal G}_\mathrm{f}=G_q\times G_\ell \),  where \(G_q^{}=\mathrm{SU}(3)_Q\times \mathrm{SU}(3)_U\times \mathrm{SU}(3)_D\) and \(G_\ell =\mathrm{SU}(3)_L\times \mathrm{O}(3)_\nu \times \mathrm{SU}(3)_E\). This entails that the above fermions are in the fundamental representations of their respective flavor groups,

$$\begin{aligned} Q_L^{}\rightarrow & {} V_Q^{}Q_L^{},\quad U_R^{} \rightarrow V_U^{}U_R^{},\quad D_R^{} \rightarrow V_D^{}D_R^{},\nonumber \\ L_L^{}\rightarrow & {} V_L^{}L_L^{}, \quad \nu _R^{} \rightarrow {\mathcal O}_\nu ^{}\nu _R^{}, \quad E_R^{} \rightarrow V_E^{}E_R^{}, \end{aligned}$$
(7)

where \(V_{Q,U,D,L,E}\in \mathrm{SU}(3)_{Q,U,D,L,E}\) are special unitary matrices and \({\mathcal O}_\nu \in \mathrm{O}(3)_\nu \) is an orthogonal real matrix [63, 64, 81]. Moreover, the Yukawa couplings transform under \({\mathcal G}_\mathrm{f}\) in the spurion sense according to

$$\begin{aligned} Y_u^{}\rightarrow & {} V_Q^{}Y_u^{}V^\dagger _U, \quad Y_d^{} \rightarrow V_Q^{}Y_d^{}V^\dagger _D,\nonumber \\ Y_\nu ^{}\rightarrow & {} V_L^{}Y_\nu ^{}{\mathcal O}_\nu ^{\textsc {t}},\quad Y_e^{} \rightarrow V_L^{}Y_e^{}V^\dagger _E. \end{aligned}$$
(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 \({\mathcal G}_\mathrm{f}\)-invariant and singlet under the SM gauge group [63, 64]. Of interest here are the combinations

$$\begin{aligned} \textsf {A}_q^{} = Y_u^{}Y_u^\dagger ,\quad \textsf {B}_q^{} = Y_d^{}Y_d^\dagger , \quad \textsf {A}_\ell ^{} = Y_\nu ^{}Y_\nu ^\dagger , \quad \textsf {B}_\ell ^{} = Y_e^{}Y_e^\dagger .\nonumber \\ \end{aligned}$$
(9)

Given that the largest eigenvalues of \(\textsf {A}_q\) and \(\textsf {B}_q\) are \(y_t^2=2m_t^2/v^2\sim 1\) and \(y_b^2=2m_b^2/v^2\sim 3\times 10^{-4}\), respectively, at the mass scale \(\mu \sim m_h^{}/2\), for our purposes we can devise objects containing up to two powers of \(\textsf {A}_q\) and drop contributions with \(\textsf {B}_q\), as higher powers of \(\textsf {A}_q\) can be connected to lower ones by means of the Cayley–Hamilton identity [99, 100]. As for \(\textsf {A}_\ell \), we assume that the right-handed neutrinos’ mass is big enough, \({\mathcal M}\sim 6\times 10^{14}\) GeV,  to make the maximum eigenvalue of \(\textsf {A}_\ell \) order 1, which fulfills the perturbativity condition [86, 87, 99, 100]. Hence, as in the quark sector, we will keep terms up to order \(\textsf {A}_\ell ^2\) and ignore those with \(\textsf {B}_\ell \), whose elements are at most \(y_\tau ^2=2m_\tau ^2/v^2\sim 10^{-4}\). Accordingly, the relevant spurion building blocks are

$$\begin{aligned} \Delta _q^{} = \zeta ^{}_1{1}\mathrm{l}+ \zeta ^{}_{2\,}\textsf {A}_q^{} + \zeta ^{}_{4\,}\textsf {A}_q^2,\quad \Delta _\ell ^{} = \xi ^{}_1{1}\mathrm{l}+ \xi ^{}_{2\,}\textsf {A}_\ell ^{} + \xi ^{}_{4\,}\textsf {A}_\ell ^2,\nonumber \\ \end{aligned}$$
(10)

where in our model-independent approach \(\zeta _{1,2,4}^{}\) and \(\xi _{1,2,4}^{}\) are free parameters expected to be at most of \({\mathcal O}(1)\) and with negligible imaginary components [86, 87, 99, 100], so that one can make the approximations \(\Delta _q^\dagger =\Delta _q^{}\) and \(\Delta _\ell ^\dagger =\Delta _\ell ^{}\). 

Thus, the desired \({\mathcal G}_\mathrm{f}\)-invariant effective operators that are SM gauge singlet and pertain to Higgs decays \(h\rightarrow f\bar{f}'\) into down-type fermions at tree level are given by [64]Footnote 4

$$\begin{aligned}&\mathcal{L}_{\textsc {mfv}}^{} = \frac{O_{RL}^{}}{\Lambda ^2} \,+\, \mathrm{H.c.}, \nonumber \\&O_{RL}^{} = (\mathcal{D}^\alpha H)^{\dagger \,}\overline{D}_R^{}Y_d^\dagger \Delta _q^{} \mathcal{D}_\alpha ^{}Q_L^{} + (\mathcal{D}^\alpha H)^{\dagger \,}\overline{E}_R^{}Y_e^\dagger \Delta _\ell ^{} \mathcal{D}_\alpha ^{}L_L^{},\nonumber \\ \end{aligned}$$
(11)

where the mass scale \(\Lambda \) characterizes the underlying heavy new physics and the covariant derivative \(\mathcal{D}^\alpha =\partial ^\alpha +(i g/2)\tau _a^{}W_a^\alpha +ig'Y'B^\alpha \) acts on \(H,Q_L,L_L\)  with hypercharges \(Y'=1/2,1/6,-1/2\), respectively, and involves the usual SU(2)\(_L\times \mathrm{U(1)}_Y\) gauge fields \(W_a^\alpha \)  and \(B^\alpha \), their coupling constants g and \(g'\), respectively, and Pauli matrices \(\tau _a^{}\), with \(a=1,2,3\) being summed over. There are other dimension-six MFV operators involving H and fermions, particularly

$$\begin{aligned}&i\bigl [H^{\dagger \,}\mathcal{D}_\alpha H-(\mathcal{D}_\alpha H)^\dagger H \bigr ] \overline{Q}_L^{}\gamma ^\alpha \Delta _{q1}^{}Q_L^{},\nonumber \\&g'\overline{D}_R^{}Y^\dagger _d\Delta _{q2\,}^{}\sigma _{\alpha \omega }^{} H^\dagger Q_L^{}B^{\alpha \omega },\nonumber \\&i\bigl [ H^\dagger \tau _{a\,}^{}\mathcal{D}_\alpha H - (\mathcal{D}_\alpha H)^\dagger \tau _a^{}H \bigr ] \overline{Q}_L^{}\gamma ^\alpha \Delta _{q3\,}^{}\tau _a^{}Q_L^{},\nonumber \\&g_{\,}\overline{D}_R^{}Y^\dagger _d\Delta _{q4\,}^{}\sigma _{\alpha \omega }^{} H^\dagger \tau _a^{}Q_L^{}W_a^{\alpha \omega } \end{aligned}$$
(12)

in the quark sector and

$$\begin{aligned}&i\bigl [H^{\dagger \,}\mathcal{D}_\alpha H-(\mathcal{D}_\alpha H)^\dagger H \bigr ] \overline{L}_L^{}\gamma ^\alpha \Delta _{\ell 1}^{}L_L^{}, \nonumber \\&g'\overline{E}_R^{}Y^\dagger _e\Delta _{\ell 2\,}^{}\sigma _{\alpha \omega }^{} H^\dagger L_L^{}B^{\alpha \omega },\nonumber \\&i\bigl [ H^\dagger \tau _{a\,}^{}\mathcal{D}_\alpha H - (\mathcal{D}_\alpha H)^\dagger \tau _a^{}H \bigr ] \overline{L}_L^{}\gamma ^\alpha \Delta _{\ell 3\,}^{}\tau _a^{}L_L^{},\nonumber \\&g_{\,}\overline{E}_R^{}Y_e^\dagger \Delta _{\ell 4\,}^{}\sigma _{\alpha \omega }^{} H^\dagger \tau _a^{}L_L^{}W_a^{\alpha \omega } \end{aligned}$$
(13)

in the lepton sector, where \(\Delta _{qn}\) and \(\Delta _{\ell n}\) are the same in form as \(\Delta _q\) and \(\Delta _\ell \), respectively, except they have their own coefficients \(\zeta _r^{}\) and \(\xi _r^{}\), but these operators do not induce \(h\rightarrow f\bar{f}'\) at tree level. In the literature the operators \(H^\dagger H_{\,}\overline{D}_R^{}Y_d^\dagger \Delta _q^{}H^\dagger Q_L^{}\) and \(H^\dagger H_{\,}\overline{E}_R^{}Y_e^\dagger \Delta _\ell ^{~}H^\dagger L_L^{}\)  are also often considered (e.g.,  [11, 12]), but they can be shown using the equations of motion for SM fields to be related to \(O_{RL}^{}\) and the other operators above [102].Footnote 5

It is worth remarking that there are relations among \(\Delta _q\) and \(\Delta _{qn}\) above (among their respective sets of coefficients \(\zeta _r^{}\)) 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 \(\Delta _\ell \),  \(\Delta _{\ell n}\), and the lepton operators in Eqs.  (11) and (13).Footnote 6 For these reasons, in our model-independent analysis on the contributions of \(O_{RL}^{}\) to \(h\rightarrow f\bar{f}'\) 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 \(h\rightarrow f\bar{f}'\). 

In view of \(O_{RL}^{}\) in Eq. (11) which is invariant under the flavor symmetry \(\mathcal{G}_\mathrm{f}\), it is convenient to rotate the fields and work in the basis where \(Y_{d,e}\) are diagonal,

$$\begin{aligned}&Y_d^{} \,=\, \mathrm{diag}\bigl (y_d^{},y_s^{},y_b^{}\bigr ),\quad Y_e^{} \,=\, \mathrm{diag}\bigl (y_e^{},y_\mu ^{},y_\tau ^{}\bigr ),\nonumber \\&\quad y_f^{} \,=\, \sqrt{2}\,m_f^{}/v, \end{aligned}$$
(14)

and \(U_k\), \(D_k\), \(\tilde{\nu }_{k,L}\), \(\nu _{k,R}\), and \(E_k\) refer to the mass eigenstates. Explicitly, \((U_1,U_2,U_3)=(u,c,t)\),  \((D_1,D_2,D_3)=(d,s,b)\), and \((E_1,E_2,E_3)=(e,\mu ,\tau )\). Accordingly,

$$\begin{aligned}&Q_{k,L}^{} = \left( \!\begin{array}{c} (V^\dagger _{\textsc {ckm}})_{kl\,}^{}U_{l,L}^{} \\ D_{k,L}^{} \end{array}\!\right) , \quad L_{k,L}^{} \,= \left( \!\begin{array}{c} (U_{\textsc {pmns}})_{kl}^{}\, \tilde{\nu }_{l,L}^{} \\ E_{k,L}^{} \end{array}\!\right) ,\nonumber \\&Y_u = V^\dagger _{\textsc {ckm}}\,\mathrm{diag}\bigl (y_u^{},y_c^{},y_t^{}\bigr ), \nonumber \\&\textsf {A}_q^{} = V^\dagger _{\textsc {ckm}}\, \mathrm{diag}\bigl (y_u^2,y_c^2,y_t^2\bigr )\, V_{\textsc {ckm}}^{} ,\nonumber \\&\textsf {A}_\ell ^{} = \frac{2\mathcal M}{v^2}\, U_{\textsc {pmns}\,}^{} \hat{m}^{1/2}_\nu O O^\dagger \hat{m}^{1/2}_\nu U_{\textsc {pmns}}^\dagger , \nonumber \\&\textsf {B}_q^{} = \mathrm{diag}\bigl (y_d^2,y_s^2,y_b^2\bigr ), \quad \textsf {B}_\ell ^{} \,=\, \mathrm{diag}\bigl (y_e^2,y_\mu ^2,y_\tau ^2\bigr ), \end{aligned}$$
(15)

where \(V_{\textsc {ckm}}\) is the Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix.

Now, we express the effective Lagrangian describing \(h\rightarrow f\bar{f}'\) as

$$\begin{aligned} \mathcal{L}_{h f\bar{f}'} = -\overline{f}\big (\mathcal{Y}_{f'f}^*P_L^{}+\mathcal{Y}_{ff'}^{}P_R^{} \big )f_{\,}'h, \end{aligned}$$
(16)

where \(\mathcal{Y}_{ff',f'f}\) are the Yukawa couplings, which are generally complex, and \(P_{L,R}=(1 \mp \gamma _5)/2\) are chirality projection operators.  This leads to the decay rate

$$\begin{aligned} \Gamma _{h\rightarrow f\bar{f}'}^{} \,=\, \frac{m_h^{}}{16\pi } \Big ( \big |\mathcal{Y}_{f'f}^{}\big |{^2} + \big |\mathcal{Y}_{ff'}^{}\big |{^2}\Bigr ), \end{aligned}$$
(17)

where the fermion masses have been neglected compared to \(m_h^{}\). Thus, from Eq. (11), which contributes to both flavor-conserving and -violating transitions, we find for \(h\rightarrow D_k^{}\bar{D}_l^{},E_k^-E_l^+\) 

$$\begin{aligned} \mathcal{Y}_{D_kD_l}^{}= & {} \mathcal{Y}_{D_kD_l}^{\textsc {sm}} \,-\, \frac{m_{D_l}^{}m_h^2}{2\Lambda ^2v}\,(\Delta _q)_{kl}^{},\end{aligned}$$
(18)
$$\begin{aligned} \mathcal{Y}_{E_kE_l}^{}= & {} \delta _{kl}^{}\,\mathcal{Y}_{E_kE_k}^{\textsc {sm}} \,-\, \frac{m_{E_l}^{}m_h^2}{2\Lambda ^2v}\,(\Delta _\ell )_{kl}^{}, \end{aligned}$$
(19)

where we have included the SM contributions, which are separated from the \(\Delta _{q,\ell }\) terms and can be flavor violating only in the quark case due to loop effects, and \(\mathcal{Y}_{ff}^{\textsc {sm}}=m_f^{}/v\) at tree level. Since approximately \(\Delta _{q,\ell }^{}=\Delta _{q,\ell }^\dagger \), it follows that in our MFV scenario \(|\mathcal{Y}_{ff'}|\gg |\mathcal{Y}_{f'f}|\) for \(ff'=ds,db,sb,e\mu ,e\tau ,\mu \tau \) and \(\mathcal{Y}_{ff}\) are real.

For \(\mathcal{Y}_{ds,db,sb}\), it is instructive to see how they compare to each other in the presence of \(\Delta _q\). In terms of the Wolfenstein parameters \((\lambda ,A,\rho ,\eta )\), the matrices \(\mathsf{A}_q\) and \(\mathsf{A}_q^2\) in \(\Delta _q\) are given by

$$\begin{aligned}&\mathsf{A}_q^{} \,\simeq \left( \begin{array}{ccc} \lambda ^6 A^2\big [(1-\rho )^2+\eta ^2\big ] &{} ~~ -\lambda ^5A^2(1-\rho +i\eta ) ~~ &{} \lambda ^3 A (1-\rho +i\eta ) \\ -\lambda ^5 A^2(1-\rho -i\eta ) &{} \lambda ^4 A^2 &{} -\lambda ^2 A \\ \lambda ^3 A (1-\rho -i\eta ) &{} -\lambda ^2 A &{} 1 \end{array}\right) \nonumber \\&\quad \simeq \, \mathsf{A}_q^2 \end{aligned}$$
(20)

to the lowest nonzero order in \(\lambda \simeq 0.23\) for each component, as \(y_u^2\ll y_c^2\sim 1.4\times 10^{-5}\sim 2\lambda ^8\)  and \(y_t^{}\sim 1\) at the renormalization scale \(\mu \sim m_h^{}/2\). If the \(\Delta _q\) part of \(\mathcal{Y}_{D_kD_l}\) for \(k\ne l\) is dominant, we then arrive at the ratio

$$\begin{aligned}&|\mathcal{Y}_{ds}| : |\mathcal{Y}_{db}| : |\mathcal{Y}_{sb}| \;\simeq \; \lambda ^3A|1-\rho +i\eta |m_s^{}:\lambda |1-\rho \nonumber \\&\quad +\,i\eta |m_b^{}:m_b^{} \;=\; 0.00016 : 0.21 : 1, \end{aligned}$$
(21)

the numbers having been calculated with the central values of the Wolfenstein parameters from Ref. [103]Footnote 7 as well as \(m_s^{}=57\) MeV  and \(m_b^{}=3.0\) GeV  at \(\mu \sim m_h^{}/2\). 

The SM coupling \(\mathcal{Y}_{D_kD_l}^{\textsc {sm}}\) with \(k\ne l\)  arises from one-loop diagrams with the W boson and up-type quarks in the loops. Numerically, we employ the formulas available from Ref. [104] to obtain \(\mathcal{Y}_{ds}^{\textsc {sm}}=(7.2+3.1i)\times 10^{-10}\), \(\mathcal{Y}_{db}^{\textsc {sm}}=-(9.2+3.8i)\times 10^{-7}\), \(\mathcal{Y}_{sb}^{\textsc {sm}}=(4.7-0.1i)\times 10^{-6}\), and relatively much smaller \(\big |\mathcal{Y}_{sd,bd,bs}^{\textsc {sm}}\big |\). 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 K–\(\bar{K}\), \(B_d\)–\(\bar{B}_d\), and \(B_s\)–\(\bar{B}_s\) oscillations, namely [101]

$$\begin{aligned} -5.9\times 10^{-10}< & {} \mathrm{Re}\big (\mathcal{Y}_{ds,\,sd}^2\big )<\, 5.6\times 10^{-10},\nonumber \\&\quad \big |\mathrm{Re}\big (\mathcal{Y}_{ds}^*\mathcal{Y}_{sd}^{}\big )\big | \,<\, 5.6\times 10^{-11}, \nonumber \\ -2.9\times 10^{-12}< & {} \mathrm{Im}\big (\mathcal{Y}_{ds,\,sd}^2\big )<\, 1.6\times 10^{-12}, \nonumber \\ -1.4\times 10^{-13}< & {} \mathrm{Im}\big (\mathcal{Y}_{ds}^*\mathcal{Y}_{sd}^{}\big ) \,<\, 2.8\times 10^{-13}, \nonumber \\ |\mathcal{Y}_{db,bd}^{}|^2< & {} 2.3\times 10^{-8},\quad |\mathcal{Y}_{db}^{~~}\mathcal{Y}_{bd}^{}\big | \,<\, 3.3\times 10^{-9}, \nonumber \\ |\mathcal{Y}_{sb,bs}^{}|^2< & {} 1.8\times 10^{-6},\quad |\mathcal{Y}_{sb}^{~~}\mathcal{Y}_{bs}^{}| \,<\, 2.5\times 10^{-7}.\nonumber \\ \end{aligned}$$
(22)

Hence there is ample room for new physics to saturate one or more of these limits. Before examining how the \(\mathcal{L}_{\textsc {mfv}}\) contributions may do so, we need to take into account also the \(h\rightarrow b\bar{b}\) measurement quoted in Eq. (1). Thus, based on the 90%-CL range of this number in view of its currently sizable error, we may impose

$$\begin{aligned} 0.4 \,<\, |\mathcal{Y}_{bb}^{}/\mathcal{Y}_{bb}^{\textsc {sm}}|^2 \,<\, 1.1, \end{aligned}$$
(23)

where \(\mathcal{Y}_{bb}^{\textsc {sm}}\simeq 0.0125\) from the central values of the SM Higgs total width \(\Gamma _h^{\textsc {sm}}=4.08\;\)MeV  and \(\mathcal{B}\big (h\rightarrow b\bar{b}\big ){}_{\textsc {sm}}^{}=0.575\) determined in Ref. [105] for \(m_h^{}=125.1\;\)GeV [10]. Upon applying the preceding constraints to Eq. (18), we learn that \(|\mathcal{Y}_{db}|^2<2.3\times 10^{-8}\) in Eq. (22) and the one in Eq. (23) are the most consequential and that the former can be saturated if at least both the \(\zeta _1^{}\) and the \(\zeta _2^{}\), or \(\zeta _4^{}\), terms in \(\Delta _q\) are nonzero. We illustrate this in Fig. 1 for \(\zeta _4^{}=0\), where the \(\zeta _2^{}/\Lambda ^2\) limits of the (blue) shaded areas are fixed by the just mentioned \(|\mathcal{Y}_{db}|\) bound and the \(\zeta _1^{}/\Lambda ^2\) values in these areas ensure that Eq. (23) is satisfied. Interchanging the roles of \(\zeta _2^{}\) and \(\zeta _4^{}\) would lead to an almost identical plot. If \(|\zeta _{1,2}^{}|\sim 1\), these results imply a fairly weak lower limit on the MFV scale \(\Lambda \) of around  50 GeV. 

Fig. 1
figure 1

Regions of \(\zeta _1^{}/\Lambda ^2\) and \(\zeta _2^{}/\Lambda ^2\) for \(\zeta _4^{}=0\) which fulfill the experimental constraints in Eqs. (22), (23). The \(\zeta _2^{}/\Lambda ^2\) range is determined by \(|\mathcal{Y}_{db}|^2<2.3\times 10^{-8}\) from Eq. (22)

Full size image

For the leptonic Yukawa couplings, \(\mathcal{Y}_{E_kE_l}\) in Eq. (19), the situation is different and not unique because the specific values and relative sizes of the elements of \(\mathsf{A}_\ell \) in \(\Delta _\ell \) can vary greatly [57]. In our MFV scenario with the type-I seesaw, this depends on the choices of the right-handed neutrinos’ mass \(\mathcal M\) and the orthogonal matrix O as well as on whether the light neutrinos’ mass spectrum \((m_1,m_2,m_3)\) has a normal hierarchy (NH) or an inverted one (IH).

For instance, if O is real, \(\mathsf{A}_\ell =\big (2\mathcal M/v^2\big )U_{\textsc {pmns}\,}^{} \hat{m}_{\nu \,}^{}U_{\textsc {pmns}}^\dagger \) from Eq. (15), and using the central values of neutrino mixing parameters from a recent fit to global neutrino data [106] we find in the NH case with \(m_1=0\)

$$\begin{aligned}&\mathsf{A}_\ell ^{} \,\simeq \frac{10^{-15}\mathcal {M}}{\mathrm GeV}\nonumber \\&\quad \times \left( \begin{array}{l@{\quad }l@{\quad }l} 0.12 &{} 0.19+0.12i &{} 0.01+0.14i \\ 0.19-0.12i &{} 0.82 &{} 0.7-0.02i \\ 0.01-0.14i &{} ~0.70+0.02i &{} 0.98 \end{array}\right) . \end{aligned}$$
(24)

Incorporating this and selecting \(\xi _4^{}=0\) in \(\Delta _\ell \) to be employed in Eq. (19), we then arrive at \(|\mathcal{Y}_{e\mu }|:|\mathcal{Y}_{e\tau }|:|\mathcal{Y}_{\mu \tau }|= |(\mathsf{A}_\ell )_{12}^{}|m_\mu ^{}:|(\mathsf{A}_\ell )_{13}^{}|m_\tau ^{}: |(\mathsf{A}_\ell )_{23}^{}|m_\tau ^{}\simeq 0.019:0.19:1\). Interchanging the roles of \(\xi _2^{}\) and \(\xi _4^{}\) would modify the ratio to 0.013 : 0.21 : 1. In the IH case with \(m_3=0\), the corresponding numbers are roughly about the same. These results for the Yukawas in the real-O case turn out to be incompatible with the following experimental constraints on the Yukawa couplings if we demand \(\mathcal{B}(h\rightarrow \mu \tau )\sim 1\)%  as CMS suggested, but with O being complex instead it is possible to satisfy all of these requirements [57].

For the first set of constraints, the direct-search limits in Eqs. (3) and (4) translate into [6]

$$\begin{aligned}&\sqrt{|\mathcal{Y}_{e\mu }^{}|^2+|\mathcal{Y}_{\mu e}^{}|^2} \,<\, 5.43\times 10^{-4},\nonumber \\&\sqrt{|\mathcal{Y}_{e\tau }^{}|^2+|\mathcal{Y}_{\tau e}^{}|^2} \,<\, 2.41\times 10^{-3}, \end{aligned}$$
(25)

and \(\sqrt{|\mathcal{Y}_{\mu \tau }|^2+|\mathcal{Y}_{\tau \mu }|^2}<3.6\times 10^{-3}\) under the no-signal assumption, while Eq. (2) for the \(h\rightarrow \mu \tau \) signal interpretation implies

$$\begin{aligned} 2.0\times 10^{-3} \,<\, \sqrt{\big |\mathcal{Y}_{\tau \mu }^{}\big |{^2} + \big |\mathcal{Y}_{\mu \tau }^{}\big |{^2}} \,<\, 3.3\times 10^{-3}. \end{aligned}$$
(26)

Additionally, the latest experimental bound \(\mathcal{B}(\mu \rightarrow e\gamma )<4.2\times 10^{-13}\) at 90% CL [107] on the loop-induced decay \(\mu \rightarrow e\gamma \) can offer a complementary, albeit indirect, restraint [11, 12, 101, 108, 109] on different couplings simultaneously [57]

$$\begin{aligned} \sqrt{\big |\big (\mathcal{Y}_{\mu \mu }^{}+r_\mu ^{}\big )\mathcal{Y}_{\mu e}^{} + 9.19\,\mathcal{Y}_{\mu \tau \,}^{}\mathcal{Y}_{\tau e}^{}\big |{^2} + \big |\big (\mathcal{Y}_{\mu \mu }^{}+r_\mu ^{}\big )\mathcal{Y}_{e\mu }^{} + 9.19\,\mathcal{Y}_{e\tau \,}^{}\mathcal{Y}_{\tau \mu }^{}\big |{^2}} \,<\, 4.4\times 10^{-7}, \end{aligned}$$
(27)

with \(r_\mu ^{}=0.29\) [101]. This could be stricter especially on \(\mathcal{Y}_{e\mu ,\mu e}\) 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 [101] from the data on \(\tau \rightarrow e\gamma ,\mu \gamma \) and leptonic anomalous magnetic and electric dipole moments are not competitive for our MFV cases. Finally, the \(h\rightarrow \mu ^+\mu ^-,\tau ^+\tau ^-\) measurements quoted in Eq. (1) are also relevant and may be translated into

$$\begin{aligned} \big |\mathcal{Y}_{\mu \mu }^{}/\mathcal{Y}_{\mu \mu }^{\textsc {sm}}\big |^2<\, 5,\quad 0.9 \,<\, \big |\mathcal{Y}_{\tau \tau }^{}/\mathcal{Y}_{\tau \tau }^{\textsc {sm}}\big |^2 \,<\, 1.3, \end{aligned}$$
(28)

where \(\mathcal{Y}_{\mu \mu }^{\textsc {sm}}\simeq 4.24\times 10^{-4}\)  and \(\mathcal{Y}_{\tau \tau }^{\textsc {sm}}\simeq 7.19\times 10^{-3}\)  from \(\mathcal{B}(h\rightarrow \mu ^+\mu ^-)_{\textsc {sm}}^{}=2.19\times 10^{-4}\) and \(\mathcal{B}(h\rightarrow \tau ^+\tau ^-)_{\textsc {sm}}^{}=6.30\)%  supplied by Ref. [105].

As pointed out in Ref. [57], the aforementioned leptonic MFV scenario with the O matrix in \({\textsf {A}}_\ell \) being real is unable to accommodate the preceding constraints, especially Eqs. (26) and  (27), even with the \(\xi _{1,2,4}^{}\) terms in \(\Delta _\ell \) contributing at the same time. Rather, it is necessary to adopt a less simple structure of \({\textsf {A}}_\ell \) with O being complex, which can supply extra free parameters to achieve the desired results, one of them being \(|\mathcal{Y}_{e\mu }/\mathcal{Y}_{\mu \tau }|\,\lesssim \,10^{-3}\). This possibility was already explored in Ref. [57] and therefore will not be analyzed further here.

3 Higgs fermionic decays in GUT with MFV

In the Georgi–Glashow grand unification based on the SU(5) gauge group [91]Footnote 8 the conjugate of the right-handed down-type quark, \((D_{k,R})^\mathrm{c}\), and the left-handed lepton doublet, \(L_{k,L}\), appear in the \(\varvec{\bar{5}}\) representations \(\psi _k^{}\), whereas the left-handed quark doublets, \(Q_{k,L}\), and the conjugates of the right-handed up-type quark and charged lepton, \((U_{k,R})^\mathrm{c}\) and \((E_{k,R})^\mathrm{c}\), belong to the 10 representations \(\chi _k^{}\). With three SU(5)-singlet right-handed neutrinos being included in the theory, the Lagrangian for fermion masses is [88, 92]

$$\begin{aligned} \mathcal{L}_\mathrm{m}^{\textsc {gut}}= & {} (\lambda _5)_{kl}^{}\, \psi _k^{\textsc {t}}\chi _{l\,}^{}\mathsf{H}_5^* + (\lambda _{10})_{kl}^{}\, \chi _k^{\textsc {t}}\chi _{l\,}^{}\mathsf{H}_5^{} \nonumber \\&+\, \frac{(\lambda _5')_{kl}^{}}{M_{\textsc {p}}}\,\psi _k^{\textsc {t}}\Sigma _{24}^{~~\;}\chi _{l\,}^{}\mathsf{H}_5^* \nonumber \\&+\, (\lambda _1)_{kl}^{}\,\nu _{k,R}^{\textsc {t}~~}\psi _{l\,}^{}\mathsf{H}_5^{} \,-\, \frac{(M_\nu )_{kl}^{}}{2}\, \nu _{k,R}^{\textsc {t}~~}\nu _{l,R}^{} \nonumber \\&+\; \mathrm{H.c.}, \end{aligned}$$
(29)

where SU(5) indices have been dropped, \(\mathsf{H}_5\) and \(\Sigma _{24}\) are Higgs fields in the 5 and 24 of SU(5), and compared to the GUT scale the Planck scale \(M_{\textsc {p}}\gg M_{\textsc {gut}}\). Since \(\mathcal{L}_\mathrm{m}^{\textsc {gut}}\) contains \(\mathcal{L}_\mathrm{m}\) for the SM plus 3 degenerate right-handed neutrinos, the Yukawa couplings in these Lagrangians satisfy the relations [88, 92]

$$\begin{aligned}&Y_u^\dagger \,\propto \, \lambda _{10}^{},\quad Y_d^\dagger \propto \, \lambda _5^{}+\epsilon \lambda _5',\quad Y_e^* \,\propto \, \lambda _5^{}-\tfrac{3}{2}\,\epsilon \lambda _5',\nonumber \\&Y_\nu ^\dagger =\, \lambda _1^{}, \end{aligned}$$
(30)

where \(\epsilon =M_{\textsc {gut}}/M_{\textsc {p}}\ll 1\). Evidently, in the absence of the dimension-five nonrenormalizable \(\lambda _5'\) term in \(\mathcal{L}_\mathrm{m}^{\textsc {gut}}\) the down-type Yukawas would be related by \(Y_d^{}\propto Y_e^{\textsc {t}}\) which is inconsistent with the experimental masses [92]. In this work, we do not include the corresponding term for the up-type quark sector, \((\lambda _{10}')_{kl}^{}\,\chi _k^{\textsc {t}}\Sigma _{24}^{~~\;} \chi _{l\,}^{}\mathsf{H}_5^{}/M_{\textsc {p}}^{}\)  [88], 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 \({\mathcal G}_\mathrm{f}^{\textsc {gut}}= \mathrm{SU}(3)_{\bar{5}}\times \mathrm{SU}(3)_{10}\times \mathrm{O}(3)_1\) the fermion fields and Yukawa spurions in \(\mathcal{L}_\mathrm{m}^{\textsc {gut}}\) transform as [88]

$$\begin{aligned} \psi\rightarrow & {} V_{\bar{5}\,}^{}\psi , \quad \chi \,\rightarrow \, V_{10\,}^{}\chi , \quad \nu _R^{} \,\rightarrow \, {\mathcal O}_1^{}\nu _R^{}, \nonumber \\ \lambda _5^{\scriptscriptstyle (\prime )}\rightarrow & {} V_{\bar{5}}^*\lambda _5^{\scriptscriptstyle (\prime )} V_{10}^\dagger ,\quad \lambda _{10}^{\scriptscriptstyle (\prime )} \,\rightarrow \, V_{10}^*\lambda _{10}^{\scriptscriptstyle (\prime )}V_{10}^\dagger ,\quad \lambda _1^{} \,\rightarrow \, {\mathcal O}_1^{}\lambda _1^{}V_{\bar{5}}^\dagger ,\nonumber \\ \end{aligned}$$
(31)

where we have assumed again that the right-handed neutrinos are degenerate, \(V_{\bar{5},10}\in \mathrm{SU(3)}_{\bar{5},10}\), and \({\mathcal O}_1\in \mathrm{O}(3)_1^{}\). It follows that the flavor transformation properties of the fermions and Yukawa coupling matrices in \(\mathcal{L}_\mathrm{m}\) are

$$\begin{aligned} Q_L^{}\rightarrow & {} V_{10}^{~~} Q_L^{},\quad U_R^{} \,\rightarrow \, V_{10}^* U_R^{},\quad D_R^{} \,\rightarrow \, V_{\bar{5}}^* D_R^{}, \nonumber \\ L_L^{}\rightarrow & {} V_{\bar{5}}^{~}L_L^{}, \quad E_R^{} \,\rightarrow \, V_{10}^* E_R^{},\quad Y_u^{} \rightarrow V_{10}^{~~} Y_u^{~} V_{10}^{\textsc {t}}, \nonumber \\ Y_d^{}\rightarrow & {} V_{10}^{~~} Y_d^{~} V_{\bar{5}}^{\textsc {t}}, \quad Y_e^{} \;\rightarrow \; V_{\bar{5}}^{~} Y_e^{~} V_{10}^{\textsc {t}},\quad Y_\nu ^{} \rightarrow V_{\bar{5}}^{~} Y_\nu ^{~} \mathcal{O}_1^{\textsc {t}}.\nonumber \\ \end{aligned}$$
(32)

As in the non-GUT scheme treated in the previous section, one can then put together the spurion building blocks \(\Delta _q^{}=\zeta _1^{}{1}\mathrm{l}+\zeta _{2\,}^{}\mathsf{A}_q^{}+\zeta _{4\,}^{}\mathsf{A}_q^2\) and \(\Delta _\ell ^{}=\xi _1^{}{1}\mathrm{l}+\xi _{2\,}^{}\mathsf{A}_\ell ^{}+\xi _{4\,}^{}\mathsf{A}_\ell ^2\), after dropping contributions involving products of down-type Yukawas, which have more suppressed elements.Footnote 9

In analogy to the non-GUT scenario, the effective operators of interest constructed out of the spurions and SM fields need to be invariant under both \({\mathcal G}_\mathrm{f}^{\textsc {gut}}\) and the SM gauge group. However, since \({\mathcal G}_\mathrm{f}^{\textsc {gut}}\) is significantly smaller than \({\mathcal G}_\mathrm{f}\), in the GUT MFV framework there are many more ways to arrange flavor-symmetry-breaking objects for the operators [88]. It is straightforward to see that those pertaining to Higgs decays into down-type fermions at tree level are given by

$$\begin{aligned} \mathcal{L}_{\textsc {mfv}}^{\textsc {gut}}= & {} \frac{1}{\Lambda ^2} (\mathcal{D}^\alpha H)^{\dagger \,}\overline{D}_R^{} \Big ( Y_d^\dagger \Delta _{q1}^{} + Y_e^*\Delta _{q2}^{} + \Delta _{\ell 3}^{\textsc {t}}Y_d^\dagger \nonumber \\&+\, \Delta _{\ell 4}^{\textsc {t}}Y_e^* + \Delta _{\ell 3}^{{\scriptscriptstyle \prime }\textsc {t}}Y_d^\dagger \Delta _{q1}^{\scriptscriptstyle \prime } + \Delta _{\ell 4}^{{\scriptscriptstyle \prime }\textsc {t}}Y_e^*\Delta _{q2}^{\scriptscriptstyle \prime } \Big ) \mathcal{D}_\alpha ^{}Q_L^{} \nonumber \\&+\, \frac{1}{\Lambda ^2} (\mathcal{D}^\alpha H)^{\dagger \,} \overline{E}_R^{} \Big ( Y_e^\dagger \Delta _{\ell 1}^{} + Y_d^*\Delta _{\ell 2}^{} + \Delta _{q3}^{\textsc {t}}Y_d^* \nonumber \\&+\,\Delta _{q4}^{\textsc {t}}Y_e^\dagger + \Delta _{q3}^{{\scriptscriptstyle \prime }\textsc {t}}Y_d^*\Delta _{\ell 2}^{\scriptscriptstyle \prime } + \Delta _{q4}^{{\scriptscriptstyle \prime }\textsc {t}}Y_e^\dagger \Delta _{\ell 1}^{\scriptscriptstyle \prime } \Big ) \mathcal{D}_\alpha ^{}L_L^{} \nonumber \\&+\, \mathrm{H.c.}, \end{aligned}$$
(33)

where \(\Delta _{qn}^{\scriptscriptstyle (\prime )}\) and \(\Delta _{\ell n}^{\scriptscriptstyle (\prime )}\) are the same in form as \(\Delta _q\) and \(\Delta _\ell \), respectively, but have their own coefficients \(\zeta _r^{\scriptscriptstyle (\prime )}\) and \(\xi _r^{\scriptscriptstyle (\prime )\,}\) \((r=1,2,4)\). We notice that, while the \(\Delta _{q1}\) and \(\Delta _{\ell 1}\) terms in \(\mathcal{L}_{\textsc {mfv}}^{\textsc {gut}}\) already occur in the non-GUT case, Eq. (11), the others are new here. In general, the different quark and lepton operators in Eq. (33) may be unrelated to each other, depending on the specifics of the underlying model, and so it is possible that only one or some of the terms in \(\mathcal{L}_{\textsc {mfv}}^{\textsc {gut}}\) dominate the nonstandard contribution to \(h\rightarrow f\bar{f}'\). Therefore, we will consider different possible scenarios below. As in the non-GUT framework of the last section, we will evaluate the contributions of \(\mathcal{L}_{\textsc {mfv}}^{\textsc {gut}}\) to Higgs decay model-independently and not deal with the constraints on the GUT-MFV counterparts of the operators in Eqs.  (12) and  (13), as the potential links among the \(\Delta \)s belonging to these various operators again depend on model details.

Working in the mass eigenstate basis, we derive from Eq. (33)

(34)

where now the column matrices \(D_{L,R}\) and \(E_{L,R}\) contain mass eigenstates, \(Y_{d,e}\) are diagonal and real as in Eq. (14), the formulas for \(\mathsf{A}_{q,\ell }\) in \(\Delta _{qn,\ell n}^{\scriptscriptstyle (\prime )}\), respectively, are those in Eq. (15), and

$$\begin{aligned} \textsf {C} \,=\, \mathcal{V}_{e_R}^{\textsc {t}}\mathcal{V}_{d_L}^{},\quad \textsf {G} \,=\, \mathcal{V}_{e_L}^{\textsc {t}}\mathcal{V}_{d_R}^{}, \end{aligned}$$
(35)

with \(\mathcal{V}_{d_L,d_R}\) and \(\mathcal{V}_{e_L,e_R}\) being the unitary matrices in the biunitary transformations that diagonalize \(Y_d\) and \(Y_e\), respectively. Since the elements of \(\mathcal{V}_{d_L,d_R}\) and \(\mathcal{V}_{e_L,e_R}\) are unknown, so are those of \(\textsf {C}\) and \(\textsf {G}\). Nevertheless, it has been pointed out in Ref. [88] that the two matrices have hierarchical textures. As indicated in Appendix, this implies that the limit \(\textsf {C}=\textsf {G}={1}\mathrm{l}\) is one possibility that may be entertained for order-of-magnitude considerations [88, 111]. It corresponds to neglecting the subdominant \(\lambda _5'\) contributions in Eq. (30). Due to the lack of additional information as regards \(\textsf {C}\) and \(\textsf {G}\), in the following we concentrate on this special scenario for simplicity, in which case the Yukawa couplings from Eq. (34) are

$$\begin{aligned} \mathcal{Y}_{D_kD_l}^{}= & {} \mathcal{Y}_{D_kD_{l}}^{\textsc {sm}} \,-\, \frac{m_h^2}{2\Lambda ^2v} \Big [ \big (\Delta _{q1}\big )_{kl\,}m_{D_l}^{} + \big (\Delta _{q2}\big )_{kl\,} m_{E_l}^{} \nonumber \\&+\, m_{D_k}^{} \big (\Delta _{\ell 3}\big )_{lk} + m_{E_k}^{} \big (\Delta _{\ell 4} \big )_{lk} \Big ] \nonumber \\&-\, \frac{m_h^2}{2\Lambda ^2v} \big ( \Delta _{q1}^{\scriptscriptstyle \prime }\hat{M}_d^{~} \Delta _{\ell 3}^{{\scriptscriptstyle \prime }\textsc {t}} + \Delta _{q2}^{\scriptscriptstyle \prime }\hat{M}_e^{~} \Delta _{\ell 4}^{{\scriptscriptstyle \prime }\textsc {t}} \big )_{kl}, \phantom {\int _{\int _|^|}} \nonumber \\ \mathcal{Y}_{E_kE_l}^{}= & {} \delta _{kl}^{}\,\mathcal{Y}_{E_kE_k}^{\textsc {sm}} \,-\, \frac{m_h^2}{2\Lambda ^2v} \Big [ \big (\Delta _{\ell 1}\big )_{kl\,} m_{E_l}^{} + \big (\Delta _{\ell 2}\big )_{kl\,} m_{D_l}^{} \nonumber \\&+\, m_{D_k}^{} \big (\Delta _{q3}\big )_{lk\,} + m_{E_k}^{} \big (\Delta _{q4}\big )_{lk} \Big ] \nonumber \\&-\, \frac{m_h^2}{2\Lambda ^2v} \big ( \Delta _{\ell 2}^{\scriptscriptstyle \prime }\hat{M}_d^{} \Delta _{q3}^{{\scriptscriptstyle \prime }\textsc {t}} + \Delta _{\ell 1}^{\scriptscriptstyle \prime }\hat{M}_e^{} \Delta _{q4}^{{\scriptscriptstyle \prime }\textsc {t}} \big )_{kl}, \end{aligned}$$
(36)

where \(\hat{M}_d^{}=Y_d^{~}v/\sqrt{2}=\mathrm{diag}(m_d,m_s,m_b)\) and \(\hat{M}_e^{}=Y_e^{~}v/\sqrt{2}=\mathrm{diag}(m_e,m_\mu ,m_\tau )\). 

To gain some insight into the potential impact of the new terms on these Yukawas, we can explore several different simple scenarios in which only one or more of the \(\Delta \)s are nonvanishing. If \(\Delta _{q1}\) and \(\Delta _{\ell 1}\) are the only ones present and independent of each other, their effects are the same as those of \(\Delta _q\) and \(\Delta _\ell \), respectively, investigated in the previous section and Ref. [57]. In the rest of this section, we look at other possible cases.

In the first one, we assume that \(\Delta _{\ell 2}\) is the only new source in Eq. (36). In view of the rough similarity between the \(\Delta _{\ell 1}\) and \(\Delta _{\ell 2}\) portions of \(\mathcal{Y}_{E_kE_l}\), due to \(m_\mu ^{}/m_s^{}\sim m_b^{}/m_\tau ^{}\sim 2\) at the renormalization scale \(\mu \sim m_h^{}/2\), we can infer that the situation in this case is not very different from its \(\Delta _\ell \) counterpart addressed briefly in the last section and treated more extensively in Ref. [57]. In other words, for the \(\Delta _{\ell 2}\) term alone to achieve \(\mathcal{B}(h\rightarrow \mu \tau )\sim 1\)%  and meet the other requirements described earlier simultaneously, the O matrix occurring in \(\textsf {A}_\ell \), as defined in  Eq. (15), must be complex in order to provide the extra free parameters needed to raise \(|\mathcal{Y}_{\mu \tau }|\) and reduce \(|\mathcal{Y}_{e\mu }|\) sufficiently. If \(\Delta _{\ell 1}\) is also nonvanishing and equals \(\Delta _{\ell 2}\), the picture is qualitatively unchanged. We have verified all this numerically.

Still another possibility with \(\Delta _{\ell n}\) is that all the \(\Delta _{q n}\) are absent and that \(\mathcal{Y}_{D_kD_l}\) and \(\mathcal{Y}_{E_kE_l}\) each have at least one \(\Delta _{\ell n}\). In this case, if, say, only \(\Delta _{\ell 1,\ell 3}\) are present and \(\Delta _{\ell 1}=\Delta _{\ell 3}\), we find that it is not possible to reach the desired \(|\mathcal{Y}_{\mu \tau }|>0.002\) and satisfy the constraints in the quark sector at the same time. The situation is not improved by keeping all the \(\Delta _{\ell n}\), while still taking them to be equal. However, if the \(\Delta _{\ell n}\) contributions to \(\mathcal{Y}_{D_kD_l}\) are weakened by an overall factor of 2 or more, at least part of the requisite range of \(|\mathcal{Y}_{\mu \tau }|\) can be attained.

Fig. 2
figure 2

Regions of \(\zeta _1^{}/\Lambda ^2\) and \(\zeta _2^{}/\Lambda ^2\) for \(\zeta _4^{}=0\) (cyan and dark blue) which satisfy the experimental constraints in Eqs. (25)–(28) if the \(\Delta _{q3}\) term is the only new-physics contribution in Eq. (36). For the orange and dark red regions, the roles of \(\zeta _2^{}\) and \(\zeta _4^{}\) are interchanged. The dark (blue and red) patches correspond to \(|\mathcal{Y}_{\tau \mu }|\simeq 0.0029\) and hence \(\mathcal{B}(h\rightarrow \mu \tau )\simeq 1\)%

Full size image
Table 1 Higgs–lepton Yukawa couplings if the \(\Delta _{q3}\) term with \(\zeta _4^{}=0\) is the only new-physics contribution in Eq. (36), and the resulting branching fractions of the \(\mu \rightarrow e\gamma \) decay and \(\mu \rightarrow e\) conversion in aluminum nuclei
Full size table

An interesting case is where \(\Delta _{q3}\) is nonvanishing and all of the other \(\Delta \)s in Eq. (36) are absent. This implies that the flavor changes depend entirely on the known CKM parameters and quark masses. Furthermore, \(|\mathcal{Y}_{\mu e,\tau e,\tau \mu }|\gg |\mathcal{Y}_{e\mu ,e\tau ,\mu \tau }|\), respectively, as can be deduced from Eq. (36). It turns out that the leptonic restrictions in Eqs. (25)–(28) can be satisfied together with only the \(\zeta _1^{}\) and \(\zeta _2^{}\), or \(\zeta _4^{}\), terms in \(\Delta _{q3}\) being present. We also find that the largest \(|\mathcal{Y}_{\tau \mu }|\) that can be attained is  \(\sim \) 0.0029.  We illustrate this in Fig. 2, where the cyan and dark blue (orange and dark red) areas correspond to only \(\zeta _{1,2}^{}\) \(\big (\zeta _{1,4}^{}\big )\) being nonzero. The widths of the two (colored) bands in this graph are controlled by the \(\mathcal{Y}_{\tau \tau }\) constraint, whereas the vertical and horizontal ranges are restrained by Eq. (26) as well as the \(\mathcal{Y}_{\mu \mu }\) constraint and Eq. (27). To show some more details of this case, we collect in Table  1 a few sample values of the Yukawa couplings in the allowed parameter space. Evidently, the predictions on \(\mathcal{Y}_{\mu \mu ,\tau \tau }\) can deviate markedly from their SM values and, therefore, will likely be confronted with more precise measurements of \(h\rightarrow \mu ^+\mu ^-,\tau ^+\tau ^-\) in the near future. As expected, the flavor-violating couplings obey the magnitude ratio \(|\mathcal{Y}_{\mu e}|:|\mathcal{Y}_{\tau e}|:|\mathcal{Y}_{\tau \mu }|\simeq |(\mathsf{A}_q)_{12}^{}|m_s^{}:|(\mathsf{A}_q)_{13}^{}|m_b^{}: |(\mathsf{A}_q)_{23}^{}|m_b^{}\simeq 0.00017:0.21:1\), compatible with Eq. (21). Also listed in the table are the branching fractions of the decay \(\mu \rightarrow e\gamma \) and \(\mu \rightarrow e\)  conversion in aluminum nuclei, computed with the formulas collected in Ref. [57] under the assumption that these transitions are induced by the Yukawas alone. The \(\mu \rightarrow e\gamma \) numbers are below the current experimental bound \(\mathcal{B}(\mu \rightarrow e\gamma )<4.2\times 10^{-13}\) [107], but not by very much. Hence they will probably be checked by the planned MEG II experiment with sensitivity anticipated to reach a  few times \(10^{-14}\) after 3 years of data taking [112]. Complementarily, the \(\mathcal{B}(\mu _{\,\!}\mathrm{Al}\rightarrow e_{\,\!}\mathrm{Al})\) results can be probed by the upcoming Mu2E and COMET searches, which utilize aluminum as the target material and are expected to have sensitivity levels under \(10^{-16}\) after several years of running [112].

In contrast to the preceding paragraph, if \(\Delta _{q4}\) instead of \(\Delta _{q3}\) is nonvanishing and the other \(\Delta \)s remain absent, the desired size of \(|\mathcal{Y}_{\tau \mu }|\) becomes unattainable, as it can be at most  \(\sim \) 0.001,  even with \(\zeta _{1,2,4}^{}\) being nonzero. If both \(\Delta _{q3,q4}\) are the only ones present and they are identical, we find \(|\mathcal{Y}_{\tau \mu }|\sim 0.0017\) to be the biggest achievable, somewhat below the lower limit in Eq. (26).

If instead \(\Delta _{q1}\) and \(\Delta _{q3}\) are the only ones nonvanishing and \(\Delta _{q1}=\Delta _{q3}\), the quark sector constraints in Eqs. (22) and (23) do not permit \(|\mathcal{Y}_{\tau \mu }|\) to exceed 0.00072, which is almost 3 times less than the required minimum in Eq. (26). This implies that, alternatively, if the \(\Delta _{q1}\) contribution to \(\mathcal{Y}_{D_kD_l}\) is decreased by an overall factor of 3 or more, at least part of the desired \(|\mathcal{Y}_{\tau \mu }|\) range can be reached and the other restrictions fulfilled.

Lastly, we look at the \(\Delta _{\ell 2}^{\scriptscriptstyle \prime }\hat{M}_d^{} \Delta _{q3}^{{\scriptscriptstyle \prime }\textsc {t}}\) and \(\Delta _{\ell 1}^{\scriptscriptstyle \prime }\hat{M}_e^{} \Delta _{q4}^{{\scriptscriptstyle \prime }\textsc {t}}\) parts in \(\mathcal{Y}_{E_kE_l}\). With \(\Delta _{q3}^{\scriptscriptstyle \prime }=\zeta _1^{\scriptscriptstyle \prime }{1}\mathrm{l}+ \zeta _2^{\scriptscriptstyle \prime }\mathsf{A}_q^{} + \zeta _4^{\scriptscriptstyle \prime }\mathsf{A}_q^2\) and \(\Delta _{\ell 2}^{\scriptscriptstyle \prime }=\xi _1^{\scriptscriptstyle \prime }{1}\mathrm{l}+ \xi _2^{\scriptscriptstyle \prime }\mathsf{A}_\ell ^{} + \xi _4^{\scriptscriptstyle \prime }\mathsf{A}_\ell ^2\), using in particular \(\mathsf{A}_q\) from Eq. (20) and \(\mathsf{A}_\ell \) from Eq. (24), we see that \(\Delta _{\ell 2}^{\scriptscriptstyle \prime }\hat{M}_d^{} \Delta _{q3}^{{\scriptscriptstyle \prime }\textsc {t}}\) has two more free parameters, \(\zeta _{2,4}^{\scriptscriptstyle \prime }\) \(\big (\xi _{^{\scriptstyle 2,4}}^{\scriptscriptstyle \prime }\big )\), compared to \(\Delta _{\ell 2}\hat{M}_d\) \(\big (\hat{M}_d\Delta _{^{\scriptstyle q3}}^{\textsc {t}}\big )\). It turns out, however, that the presence of additional parameters does not necessarily translate into more freedom for the \(\Delta _{\ell 2}^{\scriptscriptstyle \prime }\hat{M}_d^{} \Delta _{q3}^{{\scriptscriptstyle \prime }\textsc {t}}\) contributions due to the following reason. With \(\hat{M}_d\) being sandwiched between \(\Delta _{\ell 2}^{\scriptscriptstyle \prime }\) and \(\Delta _{q3}^{{\scriptscriptstyle \prime }\textsc {t}}\), in general \(\mathcal{Y}_{ff'}\) for \(f\ne f'\) can be comparable in size to \(\mathcal{Y}_{f'f}\) because they both have terms linear in \(m_b^{}\), as do \(\mathcal{Y}_{ee,\mu \mu }\), which is unlike the situation of the \(\mathcal{Y}_{E_kE_l}\) parts containing only one \(\Delta \). We find that, once the two extra free parameters are fixed to suppress the \(m_b^{}\) effects on \(\mu \rightarrow e\gamma \) as well as \(h\rightarrow \mu ^+\mu ^-\), the predictions for the various \(\mathcal{Y}_{E_kE_l}\) are not very different qualitatively from those in the \(\Delta _{\ell 2}\) \((\Delta _{q3})\) case examined earlier. Similarly, the implications of the contributions of \(\Delta _{\ell 1}^{\scriptscriptstyle \prime }\hat{M}_e^{} \Delta _{q4}^{{\scriptscriptstyle \prime }\textsc {t}}\) do not differ much from those of \(\Delta _{\ell 1}\hat{M}_e\) or \(\hat{M}_e\Delta _{q4}^{\textsc {t}}\) also discussed earlier.

The above simple scenarios have specific predictions for the flavor-conserving and -violating Yukawa couplings and hence are all potentially testable in upcoming measurements of \(h\rightarrow f\bar{f}'\)  and searches for flavor-violating charged-lepton transitions such as \(\mu \rightarrow e\gamma \). If the predictions disagree with the collected data, more complicated cases could be proposed in order to probe further the GUT MFV framework that we have investigated.

4 Conclusions

We have explored the flavor-changing decays of the Higgs boson into down-type fermions in the MFV framework based on the SM extended with the addition of right-handed neutrinos plus effective dimension-six operators and in its SU(5) GUT counterpart. As a consequence of the MFV hypothesis being applied in the latter framework, we are able to entertain the possibility that the recent tentative indication of \(h\rightarrow \mu \tau \) in the LHC data has some connection with potential new physics in the quark sector. Here the link is realized specifically by leptonic (quark) bilinears involving quark (leptonic) Yukawa combinations that control the leptonic (quark) flavor changes. We discuss different simple scenarios in this context and how they are subject to various experimental requirements. In one particular case, the leptonic Higgs couplings are determined mainly by the known CKM parameters and quark masses, and interestingly their current values allow the couplings to yield \(\mathcal{B}(h\rightarrow \mu \tau )\sim 1\)%  without being in conflict with other constraints. Forthcoming measurements of the Higgs fermionic decays and searches for flavor-violating charged-lepton decays will expectedly provide extra significant tests on the GUT MFV scenarios studied here.