Distinguishing between SU(5) and flipped SU(5)

We study in detail the d=6 operators for proton decay in the two possible matter unification scenarios based on SU(5) gauge symmetry. We investigate the way to distinguish between these two scenarios. The dependence of the branching ratios for the two body decays on the fermion mixing is presented in both cases. We point out the possibility to make a clear test of flipped SU(5) through the decay channel p \to \pi^+ \bar{\nu}, and the ratio \tau(p \to K^0 e^+_{\alpha}) / \tau(p \to \pi^0 e^+_{\alpha}).

into account the general dependence on fermion mixing (for early analyses see [22,23,24]). With this work we seek to remedy that. Namely we investigate all d = 6 proton decay operators in two different GUT models based on SU (5). We then confront the signatures of the two unifying schemes pointing out the way to distinguish between them. We also point out the way to test flipped SU (5).
The manuscript is organized as follows. In Section II we briefly review the key properties of both SU (5) and flipped SU (5) unified theory. Section III is devoted to the general discussion of d = 6 operators in both scenarios. In Section IV we specify all the branching ratios for the independent channels for proton decay. That section contains the main results of our work. Finally, we conclude in the last section. The Appendix contains useful decay rate formulas used throughout the manuscript. (5) The smallest special unitary group that contains the Standard Model (SM) gauge group is SU (5). The SU (5) grand unified theory [25,26] is an anomaly free theory, where we have partial matter unification for each family in three representation: 10, 5 and 1. The singlet is identify with the right handed neutrino. In the SM language we have: The electric charge is a generator of conventional SU (5). However, it is possible to embed the electric charge in such a manner that it is a linear combination of the generators operating in both SU (5) and an extra U (1), and still reproduce the SM charge assignment. This is exactly what is done in a flipped SU (5) [22,27,28,29]. The matter now unifies in a different manner, which can be obtained from the SU (5) assignment by a flip: d C ↔ u C , e C ↔ ν C , u ↔ d and ν ↔ e. In the case of flipped SU (5) the gauge bosons responsible for proton decay are: (X ′ , Y ′ ) = (3, 2, −1/3).

II. MATTER UNIFICATION BASED ON SU
The electric charge of Y ′ is −2/3, while X ′ has the same charge as Y . Since the gauge sector and the matter unification differ from SU (5) case, the proton decay predictions are also different [22].
Flipped SU (5) is well motivated from string theory scenarios, since we do not need large representations to achieve the GUT symmetry breaking [29]. Another nice feature of flipped SU (5) is that the dangerous d = 5 operators are suppressed due to an extremely economical missing partner mechanism. This allows us to concentrate our attention to the gauge d = 6 contributions.
We next analyze the possibility to test two realistic grand unified theories: the SU (5) and flipped SU (5) theory. We make an analysis of the operators in each theory, and study the physical parameters entering in the predictions for proton decay. We do not commit to any particular model for fermion masses, in order to be sure that we can test the grand unification idea.

III. D=6 OPERATORS
In the Georgi-Glashow SU (5) matter unification case, the gauge d = 6 operators contributing to the decay of the proton are [2,3,4]: On the other hand, flipped SU (5) matter unification yields: In the above expressions k 1 = g 5 M −1 (X,Y ) , and In these theories the diagonalization of the Yukawa matrices is given by the following bi-unitary transformations: Using the operators listed in Eqs. (1), the effective operators for each decay channel in the SU (5) case upon Fierz transformation take the following form in the physical basis [21]: where In the case of flipped SU (5) (see Eqs. (2)) the effective operators are where We use the subscripts SU (5) and SU (5) ′ to distinguish the two scenarios. The mixing matrices where K 1 and K 2 are diagonal matrices containing three and two phases, respectively. The leptonic mixing Notice that in general to predict the lifetime of the proton in SU (5), due to the presence of d = 6 operators, we have to know k 1 , V 1b 1 , V 2 , V 3 , while in flipped SU (5) we have to know k 2 , V 1b 1 , V 3 , V 4 and U EN . In addition we have to know three diagonal matrices containing CP violating phases, K 1 , K 2 and K 3 , in the case that the neutrino is Majorana. In the Dirac case there is an extra matrix with two more phases.
From the above equations, we see that there are no decays into ν C in SU (5), and in flipped SU (5) into e C , since these are singlets in the corresponding scenarios.
IV. FLIPPED SU (5) VERSUS SU (5) There are only seven independent relations for all coefficients of the gauge d=6 operators contributing to nucleon decay [21]. Therefore, if we want to test a grand unified theory, the number of physical quantities entering in the proton decay amplitude must be less than that. This is important to know in order to see if it is possible to test a GUT scenario.
Since we cannot distinguish between the neutrino flavors in the proton decay experiments, in order to compute the branching ratios into antineutrinos we have to sum over all of them. Using the expressions in the Appendix, and the following relations: we can write down the ratios between the lifetimes in both theories for the decays into antineutrinos.
They are: where The same procedure can be done for the decays into charged leptons: Eqs. (11), (13), and (14) are the most general equations that we could write in the two scenarios and will help in future to distinguish between them if proton decay is found. In other words, for a given model of fermion masses, using the above equations we could see the difference in the predictions for proton decay. Unfortunately, as one can appreciate, the branching ratios depend on too many unknown factors, including the new CP violating phases. (These, in principle, could be defined in a particular model for CP violation.) Therefore it is impossible to test those scenarios in general through the decay of the proton unless we known the flavor structure of the SM fermions.
Since we cannot make clear predictions in the most general case, let us consider special cases in these two matter unification scenarios based on SU (5) and compare them.
where K u is a diagonal matrix containing three CP violating phases. Therefore we get: In this case, as has been shown [21], the clean channels, i.e., the channels that we have to look to test this scenario, are: where Notice that we have two expressions for k 1 , which are independent of the unknown mixing matrices and the CP violating phases. Therefore it is possible to test SU (5) grand unified theory with symmetric up Yukawa matrices through these two channels [21]. Notice that these results are valid for any unified model based on SU (5) with Y U = Y T U . For example, this includes the case of minimal SUSY SU (5) with two extra Higgses in the fundamental and antifundamental representations. The case of modified missing doublet SUSY SU (5) model [30,31] is also included in our analysis.
B. Renormalizable flipped SU (5) In renormalizable flipped SU (5) we have Y D = Y T D , so D C = DK d , where K d is a diagonal matrix containing three CP violating phases. In this case the coefficients entering in the proton decay predictions are: Using these equations we get the following relations: where: We can say that the renormalizable flipped SU (5) can be verified by looking at the channel p → π +ν , and using the correlation stemming from Eq. (19c). This is a nontrivial result and can help us to test this scenario, if proton decay is found in the next generation of experiments. It is one of the main results of this work. If this channel is measured, we can know the predictions for decays into charged leptons using Eq. (19b) for a given model for fermion masses. Therefore it is possible to differentiate between different fermion mass models.
Note the difference between Eqs. (16b) and (19a); there appears a suppression factor for the channel p → π + ν in the case of SU (5).
Since the nucleon decays into K mesons are absent in the case of flipped SU (5), that is an independent way to distinguish this model from SU (5), where these channels are always present.

V. CONCLUSIONS
We have investigated in model independent way the predictions coming from the gauge d = 6 operators in the two possible matter unification scenarios based on SU (5) gauge symmetry. We write down the most general ratios between the lifetimes in SU (5) and flipped SU (5) theory for each channel, providing the way to distinguish between them. We find that in general it is very difficult to test flipped SU (5). However, in the case of renormalizable flipped SU (5) model, the decay channel p → π + ν, which is a clean channel, and the ratio τ (p → K 0 e + α )/τ (p → π 0 e + α ) could be used to test this theory. If the decay of the proton is found in future, our results will be useful to analyze the predictions in these theories.

APPENDIX
Using the chiral Lagrangian techniques (see reference [32]), the decay rate of the different channels due to the presence of the gauge d = 6 operators are given by: In the above equations m B is an average Baryon mass satisfying m B ≈ m Σ ≈ m Λ , D, F and α are the parameters of the chiral lagrangian, and all other notation follows [32]. Here all coefficients of four-fermion operators are evaluated at M Z scale. A L takes into account renormalization from