Seesaw Scale, Unification, and Proton Decay

We investigate a simple realistic grand unified theory based on the $SU(5)$ gauge symmetry which predicts an upper bound on the proton decay lifetime for the channels $p \to K^+ \bar{\nu}$ and $p \to \pi^+ \bar{\nu}$, i.e. $\tau (p \to K^+ \bar{\nu}) \lesssim 3.4 \times 10^{35}$ and $\tau (p \to \pi^+ \bar{\nu}) \lesssim 1.7 \times 10^{34}$ years, respectively. In this context, the neutrino masses are generated through the type I and type III seesaw mechanisms, and one predicts that the field responsible for type III seesaw must be light with a mass below 500 TeV. We discuss the testability of this theory at current and future proton decay experiments.


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fermions. The last possibility is the simplest, and in this case, the theory can be more predictive. We will discuss this possibility in detail.
• Neutrino Masses: As in the Standard Model, the neutrinos are massless in SU (5). One can generate neutrino masses through the different seesaw mechanisms: -Type I [5]: The masses can be generated by adding at least two copies of right-handed neutrinos, 1 i .
-Type II [6]: The masses can be generated by the addition of a new Higgs representation, 15 H .
In this case, the field important for the seesaw mechanism, ∆ ∼ (1, 3, 1), lives in the 15 H representation. See Ref. [7] for a simple model that implements the type II seesaw mechanism.
-Zee mechanism [14]: One can generate neutrino masses at the one-loop level by adding two new Higgs fields: a charged singlet in 10 H and a second Higgs doublet in 45 H . See Ref. [15] for the implementation of this mechanism in a simple renormalizable model.
Following the above discussion, one can think about different realistic extensions of the Georgi-Glashow model. In this article, we investigate a simple renormalizable extension of the Georgi-Glashow model that corrects the three major problems with the Georgi-Glashow model: neutrino masses, consistent charged fermion masses, and unification of gauge couplings. In this theory, one can achieve a consistent relation between the charged fermion masses by adding vector-like fermions in the 5 and5 representations. The neutrino masses are generated through the type I and type III seesaw mechanisms, but in this case, the new vector-like fermions also play a crucial role. In this context, we show that we can achieve the unification of the gauge couplings in agreement with the low energy constraints. We find that the field generating neutrino masses through the type III seesaw mechanism must be light, i.e. M ρ 3 < 500 TeV. We discuss the predictions for proton decay and show that the theory predicts an upper bound on the proton decay lifetime for the channels p → K +ν , i.e. τ (p → K +ν ) 3.4 × 10 35 years, and p → π +ν , i.e. τ (p → π +ν ) 1.7 × 10 34 years. We discuss the constraints on the spectrum of the theory and the generation of neutrino masses in detail. The model proposed in this article can be considered as one of the most appealing realistic extensions of the Georgi-Glashow model.
This article is organized as follows: In section II, we discuss the main features of our model: the unification constraints, the generation of neutrino and charged fermion masses, and the predictions for proton 3 decay. We summarize our main results in section III.

II. THEORETICAL FRAMEWORK
We focus on a simple renormalizable theory with the following properties: • Fermions: As in the original Georgi-Glashow model, we have the Standard Model fermions in thē 5 and 10 representations. We add the 24 representation to generate neutrino mass and, in the spirit of the Standard Model, three copies of vector-like fermions in5 and 5 representations to achieve a realistic relation between the charged fermion masses. We list the representations to set our notation: • Gauge Bosons: The Standard Model gauge bosons live in the adjoint representation, and we use the following notation: • Scalar Sector: We stick to the minimal Higgs sector of the Georgi-Glashow model: Now we describe the splitting between the fields in the new fermionic representations: • New Seesaw Fields: We write the following terms relevant for the mass of the 24: After the grand unified symmetry is broken, the masses for the fields in the 24 representation are In the same spirit as the splitting of the 24, we can definem 5 = M D /M L to write the mass of the down-type quarks as a function of the mass of the leptons.
• Yukawa couplings: The Yukawa terms relevant to understanding the generation of fermion masses are given by:

A. Gauge Unification Constraints
The pragmatic way to determine if one can achieve gauge coupling unification in agreement with the low energy constraints is to assume unification at the high scale and to constrain the full spectrum of the theory using the allowed freedom. The equations for the running of the gauge couplings are given by: and M I is the mass of any new particle living in the great desert. These equations can be rewritten in a more suitable form in terms of the differences of the coefficients and the low energy observables. Assuming unification, these equations can be reduced to: ln where These equations can be used to constrain the spectrum of the theory.   In Table I, we list the contributions to the B ij coefficients in the theory. The relevant equations for our analysis can be explicitly written as: where B SM 12 = 109/15, B SM 23 = 23/6 and r 5 = r L − r D . We assume that the colored triplet in the 5 H lives 6 at the high scale because it mediates proton decay. We note that r 5 is only a function of the mass splittinĝ m 5 , i.e. r 5 = lnm 5 /(ln M GU T − ln M Z ). Since unification is only sensitive to the splitting in the mass of the representations, we can eliminate the overall mass scales and write the above equations in a simple way:  In Fig.1, we show the parameter space allowed by the unification of gauge couplings in the plane Log 10 M GUT − Log 10m24 . We note that even though there is a large overall parameter space, the parameter space that gives unification compatible with proton decay is limited. Specifically, we find thatm 24 10 9 , m 5 10 −2 , and that Σ 8 and Σ 3 should live near the electroweak scale. We find the maximum possible GUT scale is 10 15.5 GeV. The bounds coming from proton decay experiments will be discussed in the next section.
We have shown the unification and proton decay constraints on the mass splitting for the fermionic fields living in the 24, 5 , and5 representations, but it is also useful to explicitly show the allowed masses for these fields. In Fig. 2, we show the allowed parameter space for the masses of the new fermions for the most optimistic case: We find that the seesaw field generating neutrino masses through the type III seesaw, ρ 3 , has a mass at the multi-TeV scale, with an upper bound of M ρ 3 ≤ 500 TeV. This is an interesting result which allows for the possibility to test the type III seesaw mechanism at current or future 7 colliders.  The most dramatic prediction of grand unified theories is the decay of the proton. In this theory, it is important to understand if we can satisfy the current proton decay bounds and if we can hope to test its predictions at current or future proton decay experiments. The relevant decay widths for the proton decay into charged leptons or neutrinos are given by: where where the parameter k 1 = g GU T / √ 2M GU T , and A encodes the information for the running of the operators.
The numerical values we use are A SR ≈ 1.5 and A QCD ≈ 1.2 [2]. The matrix elements present in the different decay channels can be computed using lattice QCD. We use the values reported in the recent lattice study [19]: π 0 (ud) L u L p = 0.134(5)(16) GeV 2 , π 0 (ud) R u L p = −0.131(4)(13) GeV 2 , The c-coefficients [20] in the above decay channels are given by: where α, β = 1, 2 and i = 1, 2, 3. The mixing matrices are defined as: where the matrices U, E, D and N define the diagonalization of the Yukawa couplings: In our theory, Y u = Y T u , and this allows us to make a clean prediction for the decay channels into antineutrinos. Therefore, the different proton decay channels can be written in a simple way: where V ij U D are the elements of the V CKM matrix. We note that one cannot predict the decay width for the channel p → e + π 0 since we do not known the mixing matrices V 2 and V 3 . However, the decay width for the proton decay channels into anti-neutrinos are predicted, and one can use them to define the lower bound on the GUT scale imposing the proton decay experimental bounds. In Fig. 4, we show the values for the unified gauge coupling, α GU T , in the most interesting scenario allowed by proton decay. Using these results, we can predict the proton decay lifetime for the most relevant proton decay channels. In Fig. 5, we show the predictions for the p → K +ν and p → π +ν channels. These results are striking because we can predict an upper bound on proton decay, i.e. τ (p → K +ν ) 3.4 × 10 35 years and τ (p → π +ν ) ≤ 1.7 × 10 34 years. In Fig. 6, we show the predictions for the p → e + π 0 channel, but since the theory does not predict the relevant mixing matrices, we cannot make a strong prediction. 3 . The red line shows the current proton decay lifetime, i.e. τ (p → π 0 e + ) > 1.6 × 10 34 years [22]. The orange dashed line shows the projected bound on proton decay lifetime from the Hyper-Kamiokande collaboration, i.e. τ (p → π 0 e + ) > 8 × 10 34 years [18].
We emphasize that in contrast to other GUT theories, where the correction of the fermion mass relations sacrifices the prediction Y u = Y T u , we find clean channels which do not depend on any unknown mixing matrix. This allows us to set an upper bound on the proton decay lifetime; thus, there is hope to test this theory in future proton decay experiments.

C. Neutrino Masses
It is important to show that one can generate at least two massive neutrinos in the context of this theory.
The relevant terms in the Lagrangian for our discussion are given by: The achievement of unification together with proton decay bounds imposes a well-defined hierarchy in the neutral fermionic sector; see Fig. 7. As we have shown, the minimum mass splitting in 24 required by proton decay bounds is nine orders of magnitude, which places ρ 3 and ρ 0 near the electroweak and GUT scale, respectively. As we show in Fig. 2, one has more freedom for the mass splitting in 5 and 5 representations, and the vector-like fermions could live anywhere in the great desert.
According to the established hierarchy, M ρ 0 M L M ρ 3 M ν , we first integrate out the heaviest neutral field, ρ 0 . This generates a contribution to the neutrino mass and some mixing terms between N and N c , all suppressed by M ρ 0 . At this point, the mass matrix for the N and N c fields is: where we keep only the lowest order terms. The N and N c fields can be written as a linear combination of the new fields: where the mixing angle θ is defined by the diagonalization of the mass matrix: Clearly, since M L M ρ 0 v 2 5 , the mixing angle is θ ∼ π/4, and the eigenvalues are ±M L . Note that we can always rotate the field N 2 → N 2 e i π 2 to define positive masses. According to the mass hierarchy in the neutral fermions, we can integrate N 1 and N 2 out, which leads to the following effective Lagrangian for the light degrees of freedom: where we do not include terms of order (M L M ρ 0 ) −1 and we neglect corrections to the ρ 3 mass suppressed by M L . We note that in the limit θ → π/4, there is no contribution to the light neutrino mass term from the vector-like leptons; however, we point out that they do contribute to the effective coupling between ν and ρ 3 . This will be a key point to predict a consistent neutrino mass spectrum.
Finally, by integrating out ρ 3 , the final seesaw takes place, and a one generates a new contribution to the light neutrinos suppressed by M ρ 3 . The effective mass term for the light neutrinos is given by: We summarize below the different contributions to the light neutrino mass term in a schematic way: where × symbolises a tree-level interaction between the fermions, ⊗ represents a mass term insertion, and 13 we define the effective vertex: We note that with only the contribution of the 24, the model would predict two massless neutrinos; thus, the presence of the 5 and5 is crucial to guarantee the consistency of the theory with experiment.

D. Charged Fermion Masses
As we discussed above, one of the main problems of the Georgi-Glashow model is that one predicts In this theory, we can achieve a consistent relation between the masses for charged leptons and down quarks. We can compute the masses using the following terms: We find the following mass matrix for the down-type quarks: where we have neglected the mixing proportional to d c ρ (3,2) and D c ρ (3,2) since they enter in the light neutrino mass matrix. The mass matrix for the charged leptons is given by: Clearly, there is enough freedom to have a consistent relation between the masses of the charged leptons and down quarks. We refer the reader to Ref. [23] for a detailed study on the role of 5 and5 representations in the achievement of realistic charged fermion masses at the low scale.

III. SUMMARY
We have investigated a simple, realistic grand unified theory based on SU (5) where one can generate fermion masses consistent with experiment and predict an upper bound on proton decay for the channels with antineutrinos: τ (p → K +ν ) 3.4 × 10 35 years and τ (p → π +ν ) 1.7 × 10 34 years. In this context, we can have a consistent relation between the charged lepton and down quark masses due to the presence of the new vector-like fermions. The neutrino masses are generated through the type I and type III seesaw mechanisms, and we find that the field responsible for the type III seesaw mechanism must be light, i.e.
M ρ 3 500 TeV. This theory can be considered as one of the appealing candidates for unification based on SU (5), as it can be tested in current or future proton decay experiments. The RGEs at two-loop level can be written as where α i = g 2 i /4π and the Y are the Yukawa couplings. The b i and b ij are given by Here we follow and use the notation of the Ref. [24].