Proton Lifetime Upper Bound in Non-SUSY SU(5) GUT

In preparation for upcoming nucleon decay searches at Hyper-Kamiokande, it is important to derive a theoretical upper bound on the proton lifetime in a general class of grand unified theory (GUT) models. In this paper, we make an attempt along this direction for non-SUSY SU(5) models, under the mild restrictions that only one or two SM-decomposed multiplets are singularly light, and that the SU(5) gauge theory is asymptotically free and thus there are no too large representations in the model. We derive criteria for SM-decomposed multiplets that potentially enhance the proton lifetime when they are singularly light. We perform a numerical analysis on the proton lifetime and show that some choices of singularly light multiplets can provide a testable upper bound on the proton lifetime.

However, if a model contains extra scalar fields other than 5 and 24 and if the scalar potential is tuned in such a way that one or two SM-decomposed multiplets are singularly light compared to the other multiplets in the same SU (5) representations, the light multiplets modify the renormalization group equations (RGEs) and possibly enhance M X above the current experimental bound 2 . (To have a singularly light SM-decomposed multiplet(s), a fine-tuning of the scalar potential is mandatory. In this paper, we perform a phenomenological study and do not discuss the origin of this fine-tuning.) The next questions is, then, which choice of the singularly light SM-decomposed multiplets leads to an arbitrarily enhanced proton lifetime, and which choice leads to a proton lifetime bounded from above, and can further give a testable (i.e., within the scope of HK) proton lifetime upper bound. In this paper, we answer to this question by performing a systematic survey on a broad range of non-SUSY SU(5) GUT models. We only mildly restrict our study to the cases satisfying two conditions below: • There are only one or two singularly light SM-decomposed multiplets, and the rest of the SM-decomposed multiplets are mass-degenerate with the GUT gauge boson.
• The SU(5) gauge theory is asymptotically free even if all the scalar particles participate in the renormalization group evolutions. Thus, SU (5) representations with a large Dynkin index are not considered.
We will demonstrate in the main body of the paper that it is indeed possible to put a testable upper bound on the proton lifetime for some choices of singularly light SM-decomposed multiplets.
This paper is organized as follows: In Section 2, we revisit the 1-loop gauge coupling unification conditions in non-SUSY SU(5) GUT, and survey SM-decomposed multiplets that potentially enhance the proton lifetime when they are singularly light. Section 3 presents our main result; we display the proton lifetime for all patterns of singularly light SM-decomposed multiplets, and study in which cases the proton lifetime is bounded from above. Section 4 summarizes the paper.
where M eff , l eff A , l eff B are defined from the mass M r i and l r i A , l r i B of the two light SM-decomposed multiplets labelled by r 1 , r 2 as with M M i (i = 1, 2) being the common mass of the other SM-decomposed multiplets in the same SU(5) representation as the light multiplet r i . After eliminating M M , one obtains an analogous formula as Eq. (7).
It is interesting to compare the above conditions with the SUSY case. The GUT gauge boson mass in SUSY SU (5) reads The ratio M X /M Σ is proportional to 1/λ, where λ is the self-coupling of the GUT Higgs field and is not bounded from below theoretically. One finds that the proton lifetime in dimension-six processes becomes 2 4/3 ≃ 2.5 times larger when M X /M Σ increases by twice (the self-coupling of the GUT Higgs field decreases by half). Besides, both the dimension-five and six proton decay amplitudes are suppressed if a SM-decomposed multiplet whose l A and l B are both positive is singularly light [10]. In non-SUSY case, on the other hand, the GUT gauge boson mass is much insensitive to the threshold corrections of scalar multiplets near the GUT scale. The proton lifetime becomes merely 50% larger even if the mass ratio M X /M Σ is 10 times larger, which is a better situation for putting an upper bound on the proton lifetime.
The current bound on the dimension-six proton decay p → π 0 e + corresponds to M X > ∼ 6 × 10 15 GeV for the unified gauge coupling α U ≃ 1/35. Eqs. (5), (6) or Eqs. (8), (9) tell us that to satisfy the above bound on M X while having M H C in a reasonable range below the Planck scale, we need a singularly light multiplet with l r A < 0 and l r B > 0 and large −l r A and l r B , or two multiplets with l eff A < 0 and l eff B > 0 and large −l eff A and l eff B . If there is only one singularly light multiplet, one finds from Table 2 two candidates for it, 3 (6, 3, 1/3) in 50, (13) or (8,3,0) in 75, 3 It is understood that complex-conjugate fields are included implicitly.
However, Eq.(7) tells us that for small −l r B /l r A , the mass of the colored Higgs boson M H C is considerably reduced when M X is enhanced. On the other hand, M H C must be larger than roughly 10 10 GeV to avoid a dangerous dimension-six proton decay via the colored Higgs boson exchange, although the precise bound depends on the suppression from the Yukawa couplings.
From Eq.(7), we find that M X > ∼ 6 × 10 15 GeV and M H C 10 10 GeV are simultaneously achieved for −l r B /l r A > ∼ 1.1, which is not satisfied by either candidate 4 . It follows that we need (at least) two singularly light multiplets, with l eff A < 0 and l eff B > 0, −l eff A and l eff B being large, and −l eff B /l eff A > ∼ 1.1. There are two possible scenarios below: 1. (6, 3, 1/3) or (8, 3, 0) is light, and another 'assisting' multiplet whose l A is positive and whose −l B is not large (positive l B is favored) makes −l eff B /l eff A larger. The candidates of the assisting multiplet are 2. A multiplet with l A > 0 and l B > 0, such as (6, 2, −1/6) and (8, 2, 1/2), is light, and another assisting multiplet with l A < 0 and l B > 0 allows l eff B , l eff A to satisfy the conditions. Excluding the pairs in the scenario 1, we find the candidates of the assisting multiplet to be 4 If we drop the restriction that the SU(5) gauge theory is asymptotically free and instead adopt the criterion that the SU(5) gauge coupling remains perturbative up to about 10 18 GeV scale, we can have a larger SU(5) multiplet. It is then possible to construct a viable GUT model with only one singularly light SM-decomposed multiplet. Specifically, we are allowed to introduce a 126 ′ (5000), 175 ′ (1200), 175 ′′ (0300) or 280(1110) multiplet without conflicting the criterion that the SU(5) gauge coupling remains perturbative up to ∼ 10 18 GeV, and if (10, 3, 0) in 126 ′ (5000), 175 ′ (1200) or (15, 3, 1/3) in 175 ′′ (0300), 280(1110) is singularly light, the model satisfies −l r B /l r A > ∼ 1.1, l r A < 0, l r B > 0, and the condition that −l r A and l r B be large.
We note that (3, 3, −1/3) is contained in 45 and can cause proton decay. This multiplet should be heavier than roughly 10 10 GeV, depending on the Yukawa couplings of 45 representation.
In the next section, we will calculate the proton lifetime along the two scenarios above.

Numerical Results for Proton Lifetime
We calculate the proton lifetime for the dimension-six process in non-SUSY non-minimal SU (5) GUT using two-loop RGEs. We exclusively consider those GUT models that satisfy two restrictions below: • There are only one or two singularly light SM-decomposed multiplets, and the rest of the SM-decomposed multiplets are mass-degenerate with the GUT gauge boson. We have revealed in the previous section that the case with only one singularly light multiplet is not viable. Hence, we concentrate on the case with two singularly light multiplets.
• The SU(5) gauge theory is asymptotically free even if all the scalar particles participate in the renormalization group evolutions. Thus, SU (5) representations with a large Dynkin index are not considered.
Two comments are in order: • Because of the big exponent 42 for M X in Eq. (4), the detailed scalar mass spectrum does not significantly change the proton lifetime. Therefore, it is justifiable to ignore how the scalar mass spectrum is derived from a concrete scalar potential and approximate that all the multiplets other than the singularly light ones have the same mass as the GUT gauge boson.
• From the restriction that the SU(5) gauge theory is asymptotically free, the total of l i (found in Tables 1,2) must be less than 80, if we break SU(5) by a real 24 scalar. Accordingly, we do not employ 70 ′ or any representation with 100 or higher dimensions. Also, since the sum of l i of 50 and 70 is more than 80, we do not employ 50 and 70 simultaneously.
In the calculation of the proton lifetime, we use the proton decay hadronic matrix element α H = −0.014 GeV 3 at 2 GeV [12], and chiral Lagrangian parameters F = 0.463, D = 0.804.  The unlabelled blue lines correspond to cases with a different second lightest multiplet, and it is evident that all these cases are already excluded by SK.
Consequently, the proton lifetime is one digit smaller than the one-loop results.
We fix the colored Higgs mass M H C at 10 10 GeV when solving the unification conditions, which gives a larger value for the proton lifetime than the cases when M H C is larger. In fact, the proton lifetime is not sensitive to the colored Higgs mass, and if we take it to be 10 16 GeV, the proton lifetime becomes 1/2-1/3 of the values in the plots.
We show the proton lifetime as a function of the mass of the lightest SM-decomposed multiplet. In each figure, the horizontal solid line is the current experimental bound on p → π 0 e + partial lifetime [8] τ p > 1.6 × 10 34 years, and the horizontal dashed line is the 3σ discovery potential at HK with a 10 year exposure of 1-tank, 6.3 × 10 34 years [1]. The mass of the second lightest SM-decomposed multiplet also changes along each slope, and of course it is always larger than the mass of the leading lightest one.    . The red lines show the current bound at SK (solid), and the 3σ discovery potential at HK with a 10 year exposure of 1-tank (dashed). Each blue line with a label shows the proton partial lifetime when the leading lightest multiplet is as indicated by the label. The unlabelled blue lines correspond to cases with a different second lightest multiplet, and it is evident that all these cases are already excluded by SK. Interestingly, if the leading lightest multiplet is (8, 1, 1) or (6, 2, 5/6), the model can evade the current bound and predicts a proton lifetime within the coverage of HK.   . The red lines show the current bound at SK (solid), and the 3σ discovery potential at HK with a 10 year exposure of 1-tank (dashed). Each blue line with a label shows the proton partial lifetime when the leading and second lightest multiplets are as indicated by the label. The unlabelled blue lines correspond to cases with a different second lightest multiplet, and it is evident that all these cases are already excluded by SK.
In Fig.1, we plot the proton partial lifetime for p → π 0 e + process, τ p , when (6, 3, 1/3) is the leading lightest SM-decomposed multiplet (except for the doublet and triplet in 5). We consider all possibilities for the second lightest multiplet. However, since (6, 3, 1/3) is contained in 50, we do not employ a multiplet in 70 as the second lightest one, to maintain asymptotic freedom of the SU(5) gauge theory as mentioned in the first part of this section. The point at which all the lines gather corresponds to the case when the second lightest multiplet is actually mass-degenerate with the GUT gauge boson, namely, only (6, 3, 1/3) is singularly light. It is clear that this case is excluded by SK. All lines are below the current experimental bound except when the second lightest multiplet is (10, 1, −1) coming from 35. When the second lightest multiplet is (10, 1, −1), the proton lifetime can be far above the HK search range, which signals that it is impossible to put a phenomenologically meaningful bound on the proton lifetime. It is interesting to note that there is a theoretical motivation to employ 45 representation to build SU(5) GUT models, since models with 24, 45 and 5 scalars can accommodate realistic renormalizable Yukawa couplings. In the model that exclusively contains 24, 45 and 5 scalars, the only case that is not excluded by SK is the one where (8, 2, 1/2) and (3, 3, −1/3) are the singularly light multiplets, and the proton lifetime is given by the line of '(8, 2, 1/2) + (3, 3, −1/3)' in Fig. 5.
The representation 40 is not motivated in SU(5) GUT, but it is contained in 210 and 144 that break SO(10) symmetry, and so 40 can be motivated if we consider larger GUTs. As far as we know, there is no motivation to employ 35 and 70 representations, but we note that the combination (8, 3, 0)+(15, 1, −1/3) provides the largest proton lifetime under our criteria that the SU(5) gauge theory must be asymptotic free. In this case, the GUT gauge boson mass can be as large as the Planck scale.

Summary
We have surveyed all non-SUSY SU(5) models under the restrictions that (i) only one or two SM-decomposed multiplets (except for the electroweak and colored Higgses) are singularly light and that (ii) the SU(5) gauge theory is asymptotically free, namely, SU(5) representations with a large Dynkin index do not enter, and have attempted to derive a phenomenologically meaningful upper bound on the proton lifetime for various choices of the singularly light multiplets.
We have formulated criteria for singularly light multiplets that enhance the proton lifetime.
When two multiplets are light, the criteria are summarized as, (1) l eff It has been shown that when only one SM-decomposed multiplet is singularly light, the proton lifetime is always below the current SK bound.
When two multiplets are singularly light, the SK bound is evaded in a few cases, and we have successfully derived a testable upper bound in some of them. The results are summarized as follows: When the leading lightest multiplet is (6, 3, 1/3), all cases are excluded by SK, except for one case where the second lightest multiplet (10, 1, −1), in which case no meaningful bound on the proton lifetime is obtained. When the leading lightest multiplet is (8, 3, 0), six cases possibly evade the SK bound (if α   It is straightforward to restrict our study to theoretically well-motivated models, such as the model containing only 24+45+5 scalars, which can give realistic renormalizable Yukawa couplings.

Appendix: Gauge coupling unification
The gauge coupling unification conditions in SUSY SU(5) and SO(10) GUTs [16] are written as 6