Fermion mixings vs d=6 proton decay

It is well known, although sometimes ignored, that not only the d=5 but also d=6 proton decay depends on fermion mixings. In general we study carefully the dependence of d=6 decay on fermion mixings using the effective operator approach. We find that without specifying a theory it is impossible to make clear predictions. Even in a given model, it is often not possible to determine all the physical parameters. We point out that it is possible to make a clear test of any grand unified theory with symmetric Yukawa couplings. We discuss in some detail realistic theories based on SU(5) and SO(10) gauge symmetry.


I. INTRODUCTION
The decay of the proton is the most dramatic prediction coming from matter unification. Since the paper by Pati and Salam in 1973 [1], proton decay has been the most important constraint for grand unified theories [2][3][4][5][6][7][8][9][10][11][12][13]. There are different operators contributing to the nucleon decay in GUTs, in supersymmetric scenarios the d = 5 contributions are the most important, but quite model dependent. They depend on the whole SUSY spectrum, on the structure of the Higgs sector and on fermion masses. In recent years these contributions have been under discussion, in order to understand if the minimal supersymmetric SU(5) [2,3] is ruled out [14,15]. There are several solutions to this very important issue in the context of the minimal SUSY SU (5) [16,17].
The d = 6 contributions for proton decay in general are the second more important, but they are less model dependent. From the non-diagonal part of the gauge field we get the gauge contributions, which basically depend only on fermion masses. The remaining d = 6 operators coming from the Higgs sector are less important and they are quite model dependent, since we can have different structures in the Higgs sector. There are several models where due to a specific structure of the Higgs sector, the d = 5 operators contributing to the decay of the proton are not present [18].
In general we study in detail the gauge d = 6 contributions. Assuming that in the future the decay of the proton will be measured, we analyze all possible information that we could get from these experiments. Using this information we will study the possibility to test the realistic grand unified theories based on the SU(5) and SO(10) gauge groups. Our analysis is valid in supersymmetric and non-supersymmetric GUT scenarios.

II. D=6 OPERATORS
Using the properties of the Standard Model fields we can write down the possible d = 6 operators contributing to the decay of the proton, which are SU (3) [5][6][7]: In the above expressions k 1 = g GU T M −1 (X,Y ) , and , the electric charge of Y ′ is −2/3, while X ′ has the same charge as Y . In SO(10) theories all these superheavy fields are present. Notice that all these operators conserve B − L, i.e. the proton always decays into an antilepton. A second selection rule ∆S/∆B = −1, 0 is satisfied for those operators [19].
Using the operators listed above, we can write the effective operators for each decay channel in the physical basis: where: 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 due to the presence of d = 6 operators we have to know k 1 , k 2 , V 1b 1 , V 2 , V 3 , V 4 and U EN . In addition we have three diagonal matrices containing CP violation 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.

III. TWO BODIES DECAY CHANNELS OF THE NUCLEON
As we know the gauge d = 6 operators conserve B − L, therefore the nucleon decays into a meson and an antilepton. Let us analyze all different channels. Assuming that in the proton decay experiments [20] one can not distinguish the flavour of the neutrino and the chirality of charged leptons in the exit channel, and using the chiral Lagrangian techniques (see reference [21]), 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 [21]. Here all coefficients of four-fermion operators are evaluated at M Z scale. A L takes into account renormalization from M Z to 1 GeV. ν i = ν e , ν µ , ν τ and e β = e, µ.
Let us analyze all different channels. When the nucleon decays into a strange meson plus an antineutrino the amplitudes (eqs. 13 and 18) of these channels are proportional to a linear combination of the coefficients c(ν i , s, d C ) and c(ν i , d, s C ) . In the case of the nucleon decays into a light unflavored meson plus an antineutrino, the amplitudes (eqs. 14, 19 and . Looking at the channels with a charged antilepton, we see that the amplitudes (eqs. 15, 17 and 21) of the channels with a light meson are proportional to a linear combination of the coefficients c(e α , d C ) and c(e C α , d), while in the case that we have a strange meson they are proportional to a linear combination of c(e α , s C ) and c(e C α , s) (eq. 16). If the neutrinos are Dirac-like we have extra channels to the decay of the nucleon, where we have the decays into ν C i and a meson. The amplitudes in this case are proportional to c(ν C i , d, d C ) , c(ν C i , s, d C ) and c(ν C i , d, s C ) respectively. Notice that from the radiative decays [22] we get the same information as in the case of the decays into a charged antilepton. Note that from the equations 13-21 we can get only seven relations for all coefficients of the gauge d=6 operators contributing to nucleon decay. 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 seven. This is an important result which helps us to know when it is possible to test a GUT scenario. However, as we will see in the next section looking only at the antineutrino channels we can get interesting predictions.

IV. TESTING GUT MODELS
Let us analyze the possibility to test the realistic grand unified models, the SU(5) and SO(10) theories respectively. Let us make an analysis of the operators in each theory, and study the physical parameters entering in the predictions for proton decay. Here we do not assume any particular model for fermion masses, in order to be sure that we can test the grand unification idea.
In these models the diagonalization of the Yukawa matrices is given by: • a GUT based on SU (5) Let us start with the simplest grand unified theories, which are based on the gauge group SU (5). In these theories the unification of quark and leptons is realized in two irreducible representations, 10  (eq. 2) contributing to the decay of the proton. Using the equations 9-12, and taking k 2 ≡ 0 the coefficients for the proton decay predictions are given by: We see from these expressions that in order to make predictions in any theory based on the SU(5) gauge group using proton decay, we have to know k 1 , V 1i 1 and the matrices V 2 , and V 3 . Note that in a SU(5) theory there are not decays into a ν C , even if the neutrino is a Dirac-like particle (see eq. 28), it could be a possibility to distinguish a SU(5) theory of the rest of GUTs. In order to compute the decay rate into antineutrinos we must use the following relation: Using this expression we can see that the antineutrino channel depends on the matrices V 1 and K 1 . Since we have only three independent equations (eqs. 13, 14 and 18) for these channels, it is clear that we can not test a GUT theory based on SU (5). Notice that from the channels with charged leptons is even more difficult to get some information, due to the presence of the matrices V 2 , V 3 and the elements V 1i 1 . In the naive case without all CP violation sources beyond V CKM we could get the information about V 1 from the nucleon decays into antineutrinos.
Let us analyze a particular case, the unrealistic minimal SU(5) model, where Y U = Y T U and Y D = Y T E (see reference [23]), in this case we have the following relations: c(e α , d C β ) unreal−min Notice that in this naive GUT model, all the channels are determined by V CKM . Unfortunately it is a prediction that we lost in the case of realistic versions of SU (5). However, if this modification of the theory does not change the relation Y U = Y T U , we could test a SU(5) theory from the nucleon decays into an antineutrino (see equation 29).
• a GUT model with symmetric Yukawa couplings.
There are many examples of grand unified theories with symmetric Yukawa couplings. This is the case of SO(10) [4] theories with two Higgses 10 H and 126 H , including the minimal supersymmetric SO(10) model [24,25].
In reference [26] has been investigated the dependence of the d = 6 gauge contributions on fermion mixings. They consider two different cases, the naive minimal SO (10), where all fermion masses arise from Yukawa couplings to 10 H , and the case where we have the Higgses 10 H and 126 H . Assuming only two generations, and neglecting the possible mixings which appear when the neutrino mass matrix is diagonalized, they showed approximately that the predictions for the decay channels p → π +ν and p → K 0 l + do not change in the different models for fermion masses. At the same time, it has been showed that the predictions for the decays p → K 0 e + , and p → µ + π are quite different in these two scenarios for fermion masses.
In this section we will analyze the properties of all decays in those theories, using the fact that the Yukawa matrices are symmetric. We will take into account the mixings of the third generation and all possible CP violation effects.
In theories with symmetric Yukawa couplings we get the following relations for the mixing matrices, U C = UK u , D C = DK d and E C = EK e , where K u , K d and K e are diagonal matrices containing three CP violating phases. In those cases Using these relations the coefficients in equations 9-12 are given by: Notice all overall phases in the different coefficients. In order to compute the decay rate into an antineutrino we need the following expression: Using the above expression, and equation 13 we find that it is possible to determine the factor k 1 = g GU T /M (X,Y ) : where: Notice that we have an expression for k 1 , which is independent of the unknown mixing matrices and the CP violating phases. In other words, we find that the amplitude of the decay p → K +ν is independent of all unknown mixings and CP violating phases, this only depends on the factor k 1 . Therefore it is a possibility to test any grand unified theory with symmetric Yukawa matrices through this channel.
Once we know k 1 , and using the expression 14 we can find the factor k 2 , solving the following equation: with: Using the condition Q 2 > 1, we get the following relation: It is a clear prediction of a GUT model with symmetric Yukawa couplings.
Using the expressions 13,14,18,19, and 20 we can get the following relations: with: Notice that using the expressions for k 1 and k 2 (eqs. 38 and 43), and the relation between the different decay rates of the neutron and the proton into an antineutrino (eqs. 45-48), we can conclude that it is possible to make a clear test of a grand unified theory with symmetric Yukawa couplings.
As we say before, there are realistic SO(10) theories with symmetric Yukawa couplings. In a SO(10) theory all fermions of a family live in the 16 F spinor representation [4]. In this case the coefficients for the gauge d=6 operators are given by the equations 9-12.
We have already studied the case of GUT models with symmetric Yukawa couplings, where we pointed out the possibility to make a consistent check of these theories. In order to predict the decay rates into charged antileptons in this case, we have to know the matrices K 2 and V DE (see eqs. 33 and 34). In those SO(10) theories there is a specific expression for the matrix V DE : To compute the amplitude for proton decay into charged antileptons we need the following expression: Therefore the amplitude of the channels with charged antileptons always depend on the matrices K 2 and V DE . Therefore it is not possible to make a clear test of the theory through those channels, they are useful to distinguish between different models for fermion masses with symmetric Yukawa matrices. Notice that in reference [26] has been showed that the predictions for the decay channel p → l + K 0 are the same in different models for fermion masses, however as we can appreciate from equation 51 it is not true in the general case when we consider all generations and the extra CP violating phases.

V. CONCLUSIONS
We have studied in detail the predictions coming from the gauge d = 6 operators, the less model dependent contributions for proton decay. Analyzing the different decay channels, we find that there are only seven independent equations for the coefficients involved in the two bodies decay channels for proton decay. In general we could say that the number of physical parameters involved in those predictions must be less than seven. We have pointed out that it is possible to make a clear test of any grand unified theory with symmetric Yukawa couplings through the decay of the nucleon, since in these cases the decay rates of the nucleon into an antineutrino are independent of the mixings matrices and the new sources of CP violation beyond V CKM and V l , they depend only on the factors k 1 and k 2 . The relations between the decays of the proton and the neutron into an antineutrino have been found. Notice that it is the case of realistic grand unified theories based on the SO(10) gauge group. The predictions for the decay channels with charged leptons are not the same in different models for fermion masses with symmetric Yukawa couplings, therefore they could be useful to distinguish between different models. Our results are valid in supersymmetric and non-supersymmetric scenarios.

ACKNOWLEDGMENTS
We would like to thank Goran Senjanović and Borut Bajc for invaluable discussions and comments. We would like to thank Ariel Garcia, Lotfi Boubekeur and Brett Viren for comments. We would like to thank the High Energy Section of the ICTP for hospitality and support.