On the Origin of Two-Loop Neutrino Mass from SU(5) Grand Unification

In this work we propose a renormalizable model based on the $SU(5)$ gauge group where neutrino mass originates at the two-loop level without extending the fermionic content of the Standard Model (SM). Unlike the conventional $SU(5)$ models, in this proposed scenario, neutrino mass is intertwined with the charged fermion masses. In addition to correctly reproducing the SM charged fermion masses and mixings, neutrino mass is generated at the quantum level, hence naturally explains the smallness of neutrino masses. In this set-up, we provide examples of gauge coupling unification that simultaneously satisfy the proton decay constraints. This model has the potential to be tested experimentally by measuring the proton decay in the future experiments. Scalar leptoquarks that are naturally contained within this framework can accommodate the recent B-physics anomalies.


Introduction
Even though the Standard Model (SM) of particle physics is highly successful in describing the interactions of the fundamental particles, it has several drawbacks, such as not being able to explain the neutrino mass, existence of Dark Matter (DM) and the origin of matterantimatter asymmetry of the universe. Among them, one of the most important downside of the SM is that the neutrinos remain massless. However, experimentally neutrino oscillation has been observed, hence the neutrinos must have acquired mass in some unspecified mechanism yet to be discovered. Due to these short coming, the SM begs for extensions.
Grand Unified Theories (GUTs) [1][2][3] are the leading candidates beyond the Standard Model (BSM) since they are ultraviolet complete theories and come with many aesthetic features.
Among different possibilities, SU (5) GUT is the simplest choice, this is the only simple group that contains SM as a subgroup and has the same rank as the SM group. SU (5) GUT can incorporate gauge coupling unification, it relates quarks with leptons and quantization of the electric charge can also be understood.
In the first proposed SU (5) GUT by Georgi and Glashow [2], the three families of fermions of the SM belong to the 5 F i +10 F i (i = 1−3 is the generation index) representations of the SU (5). Quarks and leptons are unified in these representations as can be seen from their decompositions under the SM: where, i = (ν i e i ) T and q i = (u i d i ) T . Interestingly, these multiplets contain only the fermions that are present in the SM, no additional fermions need to be introduced to cancel the gauge anomalies.
To describe our universe, the SU (5) gauge symmetry needs to be broken to that of the SM at the high scale: SU (5) → SU (3) C × SU (2) L × U (1) Y that can be achieved by employing a Higgs field in the 24 H -dimensional 1 representation [2]. When this field acquires a vacuum expectation value (VEV) in the SM singlet direction, the GUT symmetry is spontaneously broken down to the SM. Then at the low energy scale, the SM gauge symmetry is spontaneously broken by the 5 H -dimensional representation: As a result, the SM Higgs contained in the 5 H -Higgs generates masses to all the charged fermions. However, this scenario predicts special mass relations m e = m d , m µ = m s [6] at the GUT scale that is ruled out by the experimental data. The shortcomings of the Georgi-Glashow model are: 1 Alternatively, instead of 24 H Higgs, SU (5) breaking to the SM group can also be achieved by using 75 H Higgs [4,5].
-it predicts wrong mass relations among the charged fermions.
-it fails to achieve gauge coupling unification.
In 2 Neutrino mass in renormalizable SU (5) GUTs The first shortcoming of the Georgi-Glashow model listed above can be fixed in two different ways: one approach to correct the bad mass relations is to add higher-dimensional operators [7], which we do not pursue. The alternative approach that we are interested in, is to work within the renormalizable framework that requires extension of the minimal Higgs sector.
As aforementioned, the SM fermions belong to the 5 F + 10 F -multiplets of SU (5): The Higgs fields that can generate masses for the charged fermions can be identified from the fermion bilinears [8]: where SU (5) group indices are explicitly shown and i, j = 1 − 3 are the generation indices.
From this Lagrangian, the down-type quark and the charged-lepton mass matrices are given by: T . This normalization follows the relation: 2v 2 5 + 12v 2 45 = v 2 , with v = 174 GeV. The above relations clearly violate the simple mass relations of the Georgi-Glashow model and, in the Yukawa sector, there are enough parameters to fit all the charged fermions masses and mixings. In Yukawa matrices, and the up-quark mass matrix is not related to the down-quark and chagred lepton mass matrices and is a symmetric complex matrix, M U = M T U . In this renormalizable model, one can also achieve gauge coupling unification and the model is safe from too rapid proton decay [9]. So the minimal model extended by 45 H Higgs can simultaneously solve the first two shortcomings of the Georgi-Glashow model listed above except the last one. This renormalizable model consists of fermion fields given in Eqs. (1.1)-(1.2) and scalar fields as given below in Eqs. (2.9)-(2.11), and for brevity we refer to this model as MRSU5 (minimal renormalizable SU (5) GUT) for the rest of the text.
. (2.11) Note however that in this minimal model the neutrinos are still massless just like the SM.
Extension must be made to make this set-up phenomenologically viable. Here we briefly review the different possibilities of generating non-zero neutrino mass 2 in the context of renormalizable SU (5) GUT.
• Tree-level: To incorporate neutrino mass, the simplest possibility is to add at least two righthanded Majorana neutrinos ν c (1,1,0) to MRSU5 model that are singlets of SU (5).
This possibility can give rise to neutrino masses by using the type-I seesaw mechanism [11][12][13][14]. Since this extension involves GUT group singlets, this approach may not be aesthetic and it is preferable to have multiplets that are non-singlets under the gauge group. Neutrino mass can be generated via type-II seesaw scenario [15][16][17][18]  to the MRSU5 [24], this scenario generates neutrino mass in a combination of type-III [25] and type-I seesaw mechanisms. This scenario makes use of the fermionic weak triplet lies in the adjoint representation, (1, 3, 0) ⊂ 24 F . These are the simple possibilities to incorporate neutrino mass at the tree-level by extending the MRSU5 model by one (type-II) or more (type-I and type-III) multiplets.
• Loop-level: Neutrino mass can also be generated at the quantum level within the SU (5) GUT framework and is a very interesting alternative possibility. The first radiative model 5 of neutrino mass generation for Majorana type particles was proposed by Zee [27] in the context of the SM gauge group by extending the SM by another Higgs doublet and a singly charged scalar singlet. In Zee model, neutrino mass is generated at the one loop level as shown in Fig. 1 (Feynman diagram on the left). The SU (5) 2 For a recent general review on neutrino mass generation mechanisms for Majorana type neutrinos see Ref. [10]. 3 Extension by 15 H Higgs was first considered within the non-renormalizable SU (5) context [19][20][21]. 4 Extension by 24 F fermions was first considered within the non-renormalizable SU (5) context [23]. 5 First radiative model was proposed for Dirac type neutrinos [26]. GUT embedding of Zee mechanism with the use of 10 H -dimensional Higgs was first proposed in Ref. [28], also pointed out in Ref. [29] and just recently studied in the renormalizable context in Ref. [30]. by 10 H Higgs and three generations of vector-like leptons 5 F + 5 F . In the models mentioned above, the particle running in the loop are colorless, however, neutrino mass can also be generated via Zee mechanism while colored particles run through the loop. For such leptoquark mechanism of neutrino masses at the one-loop level within the SU (5) GUT framework, see for example Ref. [33].

The proposed model
In the models discussed above, the neutrino mass can be generated at the tree-level via seesaw mechanism or at the one-loop level via Zee mechanism within the context of renormalizable SU (5) GUT. In this work, we for the first time construct a realistic SU (5) GUT, where the origin of the neutrino mass is realized at the two-loop level. In our construction, we restrict ourselves by demanding the following requirements: -model should be renormalizable.
-the only symmetry of the theory is SU (5) gauge symmetry.
-no additional fermions are added compared to the already existing ones in the SM.
-automatic vanishing of the neutrino mass at the tree-level and at the one-loop level.
-in our search, we restrict ourselves to the representations of dimension < 100.
along with the already existing Yukawa interactions given in Eq. (2.6) combinedly determine the neutrino mass. Hence, neutrino mass does not appear to be completely detached, rather gets intertwined with the charged fermion masses. Here the Yukawa coupling Y 5 is a symmetric 3 × 3 matrix in the generation space. Since neutrino mass appears at the twoloop level in this model, the neutrino masses are highly suppressed compared to the charged fermions, hence naturally explains the smallness of the neutrino masses. The decomposition of this Higgs fields under the SM is as follows: .
In the context of the SM gauge group, the possibility of generating neutrino mass via two-loop is well known [34]. The simplest possibility is to add a singly charged scalar and a doubly charged scalar both singlets under the SM group and is commonly known as Zee- Babu model [35] as shown in Fig. 1 (Feynman diagram on the right). Many variations of the Zee-Babu model are proposed in the literature by extending the SM particle content.
Note that in both the one-loop (Zee model) and the two-loop (Zee-Babu model) neutrino mass mechanism, at least two new multiplets need to be added to the theory to generate non-zero neutrino mass. In the original Zee model, in addition to a second SM like Higgs doublet, a singly charged scalar singlet needs to be added. In the original version of the Zee-Babu model, again two BSM multiplets, one singly charged singlet and one doubly charged singlet need to be introduced. Below, we show that to realize two-loop neutrino mass in the context of renormalizable SU (5) GUT, at least two new multiplets need to be added to the MRSU5.
Our framework incorporates Zee-Babu mechanism to explain the extremely small neutrino mass. The Yukawa coupling given in Eq. (3.12) has doubly charged scalar couplings to two charged leptons (suppressing the group indices): (3.14) Now to complete the loop-diagram, one must introduce at least one more Higgs multiplet, which is however not arbitrary but unambiguously determined by the group theory.
The simplest possibility is to add a 40 H -dimensional representation that has the following decomposition under the SM: ) ⊕ η 6 (8, 1, 1).

(3.15)
Note that 40 H Higgs has an iso-spin doublet η 1 = (η − 1 η −− 1 ) T with a hypercharge of Y = −3/2 that is necessary to close the loop-diagram. The SU (5) invariant scalar potential contains cubic terms relevant for neutrino mass generation that are of the form: are the singly charged scalars from the SM like doublets. And the relevant quartic terms in the potential to complete the loop-diagram are of the form: With the simultaneous presence of the Yukawa coupling Eq. Here we compute the neutrino mass matrix. First note that the breaking of the EW symmetry allows mixings of the particles carrying the same electric charge. Mixing among the singly charged fields are induced by the quartic terms of Eq. (3.17), whereas for the doubly charged particles are induced by the cubic terms of Eq. (3.28). After the breaking of the EW symmetry, the part of the scalar potential containing the relevant mixing terms are given by: Here the off-diagonal entries are the mixing terms as already mentioned above and the diagonal entries are the mass terms which for simplicity we do not write down explicitly, however can be computed straightforwardly from the full potential. In the next section, we will construct part of the scalar potential that is relevant for the study of the gauge coupling unification. For simplicity, treating all the parameters of the the scalar potential appearing in Eq. (3.18) to be real, the transformation between the weak basis and the mass basis for the CP-even neutral fields, the doubly charged scalars and the singly charged scalars can be written as: Here, the fields labeled with H (H + i , H ++ i ) represent the mass eigenstates and G + is the Goldstone boson. This leads to: Where we have defined, and we have made use of the notation: c ω = cos ω, s ω = sin ω, c ij = cos θ ij , s ij = sin θ ij .
Furthermore, we get: Then, the neutrino mass matrix is evaluated to be: To make life simple, we assume m + 1 = m + 2 and furthermore use the approximation m c,a,b, >> m k,l , which is valid since charged lepton masses are small compared to the BSM charged scalars running in through the loop. Then one finds [37], with, Here we note that the neutrino masses do not decouple from that of the charged fermion masses, rather gets entangled with them. To find the correlation, we express the Yukawa couplings Y 1 and Y 2 in terms of the down-quark and charged-lepton masses matrices from Eqs. (2.7)-(2.8) as: As a result, Y + 1,2 can also be expressed in terms of down-quark and charged-lepton masses matrices: Here we have gone to a basis where the up-quark and charged lepton mass matrices are diagonal. In this rotated basis, the Y 5 matrix takes the form: E c Y 5 E cT . To get to these relations, we used the following convention for diagonalization of the charged fermion masses: Hence the requirement of both L ⊃ Y 5 10 F 10 F 50 H and L ⊃ Y 6 5 F 5 F 10 H Yukawa couplings into the theory are required, where Y 5 is a symmetric 3 × 3 matrix where as Y 6 is antisymmetric 3×3 matrix in the flavor space. However, presence of the Yukawa coupling Y 6 and the allowed gauge invariant cubic term µ 5 H 5 H 10 * H in the scalar potential automatically leads to one-loop diagram via Zee-mechanism [28] (left diagram in Fig. 1). This is why, such an embedding which was realized in [38], cannot be a true two-loop neutrino mass model. Similar conclusion can be reached for the SU (5) model presented in [39], due to the presence of 10 H Higgs, in addition to their two-loop diagram, one-loop diagram of the Zee-type automatically appears. So the model presented in this work is unique in its features.

Scalar potential and the Higgs Bosons mass spectrum
As aforementioned, the minimal model consists of the Higgs set 24 H , 5 H , and 45 H . However, this model is still defective since neutrinos remain massless. In the previous section it is shown that to build a true two-loop neutrino mass model the minimal model needs to be extended by two more Higgs multiplet 40 H and 50 H . In this section, we compute the Higgs mass spectrum of the set 5 H + 24 H + 40 H + 45 H + 50 H after the GUT symmetry is broken spontaneously. This analysis is performed to find the Higgs mass relationships, which will be relevant for our study of the gauge coupling unification performed in the next section. The tensorial properties of all the Higgs multiplets in our framework are presented in table I. where, The SM singlet component, Φ 1 (1, 1, 0) of the adjoint Higgs acquires VEV, Φ H ≡ V GU T and breaks the GUT symmetry down to the SM group. The minimization condition demands: The multiplets (3, 2, − 5 6 ) and (3, 2, 5 6 ) from 24 H field correspond to the Goldstone bosons and hence eaten up by the massive gauge bosons. The masses of the other multiplets in 24 H are given by: The mass spectrum of the multiplets residing in 5 H Higgs by neglecting mixing with other fields are given by: Similarly, ignoring the mixing of the fields, the mass spectrum of the multiplets contained in the 45 H Higgs are given by: Mass spectrum of the multiplets of 40 H field: And finally the mass spectrum of the multiplets residing in 50 H field: These mass spectrum helps one to understand whether splitting among different multiplets originating from the same field is possible or not. Splitting among different multiplets for some of the fields is necessary to achieve unification to be discussed in the next section.
From the mass spectrum computed above, it can be realized that, due to enough number of parameters, there is no mass relationship among the multiplets of 40 H . This is also true for 5 H , 24 H and 45 H . However, which is not true for the multiplets contained in 50 H and from the above calculation we find: For the study of the gauge coupling unification, we impose the mass relations as derived above.
Till now, we have ignored the mixings among the multiplets having the same quantum number coming from different Higgs representations. For completeness here we take into account such mixings. Note that the relevant mixing terms are contained in the V mix term given in Eq. (4.48). Now taking these mixed terms into consideration, the mixing between the iso-spin doublets, (1, 2, 1 2 ) present in 5 H and 45 H representations are given by: with, with, Note that, due to the breaking of the GUT symmetry, all the multiplets acquire mass of the order of the GUT scale. However, to break the SM symmetry, the SM like Higgs doublet needs to be kept at the EW scale. This can be achieved by imposing the well known fine-tuning condition in the doublet mass matrix Eq. (4.77). This fine-tuning does not leave any color triplet Higgs light that can be seen from the corresponding mass matrix given in Eq. (4.79).

Gauge coupling unification and proton decay constraints
In this section we present few different scenarios where successful gauge coupling unification within our framework can be achieved which are also in agreement with the proton decay bounds. For the gauge couplings the renormalization group equations can be written as: In this way, the equations become: 2 r η 4 η 5 (6, 2, 1 6 ) − 14 15 r η 5 − 2 3 r η 5 η 6 (8, 1, 1)  The main experimental test of the existence of GUTs is via the detection of the proton decay yet to be observed. In GUT models in the non-supersymmetric framework, the leading contribution to the proton decay is due to the gauge mediated d = 6 operators. In SU (5) GUT, the gauge bosons that are responsible for the proton to decay are (3, 2, − 5 6 )+(3, 2, 5 6 ) ⊂ 24 G . The most stringent experimental bound on the proton lifetime comes from the gauge mediated proton decay mode: p → π 0 e + and the corresponding decay width is given by [42,43] : here, m p is the proton mass, the running factor of the relevant operators give A ≈ 1.8 and, The c-coefficients given in Eq. To suppress proton decay one would expect these fields to have mass of the order of the GUT scale. For our analysis we assume that these fields are sufficiently heavy so that the corresponding dangerous proton decay operators are suppressed.
For the purpose of comparison, at first we discuss the coupling unification scenario within the minimal renormalizable model. Note that to achieve high GUT scale value, one needs to keep scalars light that provide negative contribution to the B 12 . In the MRSU5 model, other than the SM like doublets, such negative contribution is provided by the Φ 2 , Σ 5 , Σ 7 multiplets, see table II. Among these three, Σ 5 mediate proton decay, on the other hand, Φ 2 and Σ 7 do not and can be very light. However, keeping Σ 5 at the GUT scale and the other two fields light fails unification test, so Σ 5 must be light as well within this scenario.
To avoid proton decay bounds, this multiplet needs to be heavier than about 10 10 GeV by assuming natural values of the Yukawa couplings [9], however, for smaller values of the Yukawa couplings, this multiplet can be kept at lower scale. In this minimal scenario, by fixing m Σ 5 = 10 10 GeV and m φ 1 = m Σ 1 = m Σ 7 = m Φ 2 = M Z we find, to achieve unification at the one-loop, one needs m Φ 3 = 7.28 × 10 5 GeV and the corresponding unification scale is 3.02 × 10 16 GeV which agrees with [9].

Case
Multiplets (GeV)  D, E) we demonstrate how gauge coupling unification can be restored in our model. For this analysis, we fix the masses of the two SM like iso-spin doublets as m φ 1 ,Σ 1 = v EW . We also fix m Σ 7 = 3.5 TeV, since from the collider bounds it is required that m Σ 7 > 3.1 TeV provided that the Yukawa couplings take natural values [47]. We also take m Φ 2,3 = 1 TeV and fix m Σ 5 = 10 10 GeV (except for case E). Furthermore, for the cases A, C, D: m χ 1 = 1 TeV, for case B: m η 1 = 1 TeV and for case E: m η 1 = m χ 1 = 1 TeV are assumed. With these assumptions and using the mass relations derived in Sec. 4, the results are presented in the table III with the corresponding unified value of the gauge coupling constant, the scale of the unification and the estimation of the associated proton lifetime for each scenario. It should be pointed out that the for case C, even though the proton life time is estimated to be somewhat below the current upper bound τ p (p → π 0 e + ) > 1.6 × 10 34 years, small threshold corrections near to the GUT scale can make this scenario viable. Even though these choices are not unique, but clearly demonstrate how successful gauge coupling unification consistent with proton decay bounds can be achieved within our set-up. Though no firm prediction can be made about the proton decay within this framework, but in all the provided examples here, the proton decay rate is very close to the current experimental bound and has the potential to be tested in near future.

B-physics anomalies
Recently, the B-physics anomalies have gained a lot of attention in the high energy physics community. In this section we briefly discuss how these anomalies can be accommodated in our set-up. Related to the neutral-current process b → s , the LHCb experiment has measured the lepton flavor universality (LFU) ratios R K ( * ) = B(B→K ( * ) µµ) B(B→K ( * ) ee) and the results [48,49] seem to be lower than the SM prediction with a significance of 2.5σ [50,51]. On the other hand, related to the charged-current process b → c ν, several experiments [52][53][54] have found that LFU ratios R D ( * ) = B(B→D ( * ) τ ντ ) B(B→D ( * ) ν ) are larger than the SM prediction, here = e, µ. The deviations of the measurements on R D ( * ) from the SM are within 2 − 3σ confidence level [55][56][57][58].
To be consistent with these experimental measurements, one or more BSM particles around the TeV scale needs to be introduced. In the literature there exist few different approaches to explain these deviations from the SM, among them leptoquarks (LQs) mechanism is one of most popular candidate. Presence of the LQs are well motivated since they arise naturally from unified theories, such as within our framework. Recently it is pointed out [59] that a TeV scale LQ with quantum number (3, 2, − 7 6 ) originating from 45 H and 50 H generates a combination of effective scalar and tensor operators at the low energies to explain R D ( * ) anomaly. Another LQ with quantum number (3, 3, − 1 3 ) residing at the sub-TeV scale originating from 45 H introduces an effective V − A operator at the low energy that can explain the R ( * ) anomaly. Simultaneous presence of the Yukawa couplings of 45 H and 50 H Higgs are required for explaining the two anomalies. Since our set-up has both the 45 H and 50 H representations, we expect that the proposed model in this work can simultaneously explain the neutrino mass generation and B-anomalies. A combined analysis of the charged fermion spectrum, neutrino oscillation data and B-physics anomalies needs to be performed to confirm this. Whereas establishing such a link among all these different phenomena in a unified framework is interesting, however this is beyond the scope of the this work. For detail on the SU (5) embedding to accommodate B-physics anomalies and the relevant low scale phenomenology we refer the readers to this work [59].

Conclusions
Grand unification based on the SU (5) gauge group is one of the leading candidates for the ultraviolet completion of the SM. The minimal SU (5) GUT has many attractive features, however fails to incorporate neutrino mass. In this work, we have proposed a renormalizable