Flavor changing nucleon decay

Recent discovery of neutrino large mixings implies the large mixings in the diagonalizing matrices of $\bf\bar 5$ fields in $SU(5)$ grand unified theory (GUT), while the diagonalizing matrices of $\bf 10$ fields of $SU(5)$ are expected to have small mixings like Cabibbo-Kobayashi-Maskawa matrix. We calculate the predictions of flavor changing nucleon decays (FCND) in $SU(5)$, $SO(10)$, and $E_6$ GUT models which have the above features for mixings. We found that FCND can be the main decay mode especially for GUT with higher rank unification group. Recent report for $P\rightarrow \pi^0\mu^+$ process implies $E_6$ or $SO(10)$ GUT if the signal is from the nucleon decay.


I. INTRODUCTION
One of the most exciting discoveries in elementary particle physics among the latest 20 years is neutrino oscillation [1,2], which leads to massive neutrinos and large neutrino mixing angles. Interestingly, this discovery gives an evidence of grand unified theory (GUT) [3], in which unification of forces and unification of quarks and leptons are realized. This evidence for unification of quarks and leptons makes the idea of GUT quite promising, because for unification of forces we have already known an experimental evidence that three gauge couplings meet at a scale, the GUT scale Λ G [4], especially in supersymmetric (SUSY) GUT [5]. Moreover, the large neutrino mixing angles imply not only large mixing angles of doublet lepton l but also those of right-handed down quark d c R in SU (5) GUT because5 field of SU (5) contains l and d c R . This suggests an interesting possibility that the flavor changing processes are seen in nucleon decay which is the most important prediction of GUT 1 . In this paper, we study the flavor changing nucleon decay and propose that the flavor changing nucleon decay can be a key observation for GUT.

II. QUALITATIVE EVIDENCE FOR SU (5) UNIFICATION
First, we explain the qualitative evidence for the unification of matters. In SU (5) GUT, Yukawa interactions with Higgs fields 5 H and5 H are given by L Y = (Y u ) ij 10 i 10 j 5 H + (Y d,e ) ij 10 i5j5H where 10 fields contain doublet quark q, right-handed up quark u c R , and right-handed charged lepton e c R , and the last term can be obtained from (Y νD ) ij5i 1 j 5 H + (M νR ) ij 1 i 1 j by integrating the right-handed neutrino fields 1 i . Here i = 1, 2, 3 is the index for generation, and Y u , Y d,e , Y νD , and M νR are Yukawa matrix of up type quarks, that of down type quarks and charged leptons, Dirac neutrino Yukawa matrix, and right-handed neutrino mass matrix, respectively. These unified structures for Yukawa interactions are corresponding to the classification of the hierarchies of the observed quark and lepton masses that up type quark masses have the strongest hierarchy, neutrino masses have the weakest, and down type quark and charged lepton masses have middle hierarchies if neutrino mass hierarchy is normal (not inverted). Moreover, if we assume that 10 fields induce stronger hierarchy in Yukawa couplings than5 fields, these various hierarchies for quark and lepton masses can be explained. Furthermore, this assumption explains that quark mixings are smaller than lepton mixings at the same time if we use a reasonable expectation that the stronger hierarchy leads to smaller mixings. This brilliant chemistry between the Yukawa structure in SU (5) GUT and the observed hierarchies of quark and lepton masses and mixings is quite non-trivial, and therefore it can be regarded as an experimental signature for unification of quarks and leptons in SU (5) GUT. [7][8][9] is more attractive because the assumption in the SU (5) GUT can be derived, and as a result, various Yukawa matrices can be derived from one basic Yukawa hierarchy [9]. The fundamental representation in E 6 is divided into SO(10)(SU (5)) representations as 27 = 16(10 +5 + 1) + 10(5 +5 ′ ) + 1(1).
This 27 includes one generation quarks and leptons in addition to one pair of vector-like fields 5 +5 and a singlet.
If we introduce three 27 i (i = 1, 2, 3) for three generation quarks and leptons, we have six5 fields. Three of six 5 fields become superheavy after developing the vacuum expectation values (VEVs) of 27 H and 27 C through the Yukawa interactions where the VEV of 27 H breaks E 6 into SO(10) and the VEV of 27 C breaks SO(10) into SU (5). Once we fix Y H , Y C , 27 H , and 27 C , 3 × 6 mass matrix of three 5s and six5s is determined, and therefore, three massless modes5 0 i are fixed. Here we assume that these Yukawa couplings Y H and Y C have strong hierarchy corresponding to the hierarchy of 10 of SU (5). Typically, we take where a unit of hierarchy λ ∼ 0.22 is taken to be around the Cabibbo mixing to obtain Cabibbo-Kobayashi-Maskawa (CKM) matrix [10]. The O(1) coefficients are omitted usually in this paper. Then, twō 5 fields from 27 3 become superheavy unless 27 H ≪ 27 C because they have larger Yukawa couplings and therefore have larger mass parameters. The three massless modes5 0 i come from the first two generation fields 27 1 and 27 2 which have smaller Yukawa couplings. As a result, three5 0 i , whose main modes typically become (5 1 ,5 ′ 1 ,5 2 ), induce milder Yukawa hierarchy than 10 i fields, that is nothing but what we assume in the SU (5) GUT to obtain realistic hierarchies of quark and lepton masses and mixings. Note that5 0 2 ∼5 ′ 1 + λ ∆5 3 has Yukawa couplings through the mixing with5 3 when the Higgs 5 H and5 H are included in 10 H of SO(10) in 27 H . Then we can obtain realistic Yukawa hierarchies as when ∆ ∼ 2.5. The right-handed neutrino masses are obtained from where X, Y =H,C, Λ is the cutoff scale, after develop- Here we take Y XY ∼ Y H ∼ Y C . All quark and lepton mass matrices can be diagonalized by unitary matrices for 10 fields and5 fields and we can obtain realistic CKM matrix V CKM ∼ V 10 and the Maki-Nakagawa-Sakata (MNS) matrix [11] V MNS ∼ V5, when ∆ ∼ 2.5. Note that the important prediction (V MNS ) 13 ∼ (V CKM ) 12 , which was confirmed by recent neutrino experiments as (V MNS ) 13 ∼ 0.15 [2], is caused bȳ 5 0 3 ∼5 2 . Therefore, to obtain the realistic hierarchies of quark and lepton masses and mixings, it is essential that the5 ′ 1 , which comes from 10 of SO(10), becomes the second generation5 field5 0 2 . That structure is important to study of the prediction of the nucleon decay in the next section.
Note that the relation5 0 2 ∼5 ′ + λ ∆5 3 can be realized even in SO(10) unification, if 10 of SO(10), which pro-vides5 ′ , is introduced as a matter field [12]. Therefore, we have three GUT models which satisfy the Yukawa hierarchy hypothesis, "10 fields induce stronger hierarchy in Yukawa couplings than5 fields". Their unification groups are SU (5), SO(10), and E 6 . Next, we study how to identify these unification group by observing various partial nucleon decay widths.

IV. NUCLEON DECAY
In this paper, we concentrate on the nucleon decay via dimension 6 operators [13], because the nucleon decay via dimension 5 operators [14] is strongly dependent on the explicit model of GUT Higgs sector which is expected to have big modification to solve the most difficult problem called the doublet-triplet splitting problem [15] and because it is strongly suppressed in natural GUT in which the difficult problem is solved with natural assumption [12,16,17].
The dimension 6 effective operators which induce nucleon decay in E 6 GUT are produced via mediation by SU (5) superheavy gauge boson X, SO(10) superheavy gauge boson X ′ , and E 6 superheavy gauge boson X ′′ as [19] where g G is the unified gauge coupling and the superheavy gauge boson masses M X , M X ′ , and M X ′′ are dependent on the VEVs of the GUT Higgs which break E 6 into the SM gauge group. Here, large character denotes 5 ′ field which comes from 10 of SO(10). In the SO(10) GUT, we just take M X ′′ → ∞, and in SU (5) GUT, we take M X ′ , M X ′′ → ∞ and neglect the interactions which include the large character fields. Note that the nucleon decay via dimension 6 operators depends on Yukawa couplings, although this is via gauge interactions. The situation is similar to the weak interaction. The weak interaction is also the gauge interaction, but we have CKM mixings which are determined by Yukawa couplings. For the nucleon decay, the nucleon decay via dimension 6 operators depends on the diagonalizing matrices for Yukawa matrices. However, we have already understood the mixings in GUT as the qualitative evidence for the SU (5) GUT. Especially for the diagonalizing matrices, V 10 and V5 are fixed as CKM matrix and MNS matrix, respectively, except O(1) coefficients. Therefore, these ambiguities are almost fixed by our understanding of Yukawa structures. Therefore, we can compare the predictions of nucleon decays in SU (5), SO(10) and E 6 GUTs.
Important observation to find useful nucleon decay modes for identification of unification group is that all four fermions in the first term in Eq. (8) come from 10 of SU (5) fields, and in the other terms two of four fermions come from5 fields. Since X ′ and X ′′ gauge interactions induce only the effective interactions with5 fields, we should look for the nucleon decay modes in which the operators with5 fields are significant to identify the unification group.
Since all operators with5 fields include a lepton doublet while 10 field includes no neutrino, the modes with neutrino can be important to identify the unification group. The decay mode N → π 0ν 2 has been studied in the literature for the identification [18,19]. Especially in Ref [19], we have shown that two ratios R 1 ≡ Γ(N → π 0ν )/Γ(P → π 0 e + ) and R 2 ≡ Γ(P → K 0 µ + )/Γ(P → π 0 e + ) are useful to identify three unification group as in Fig. 1, where we have 10 5 model points for each unification group SU (5)(black points), SO(10)(red points), and E 6 (blue points) and the magnitudes of the O(1) coefficients of diagonalizing matrices are determined randomly between 0.5 and 2. We adopt superheavy gauge boson masses M X = M X ′ = √ 2M X ′′ as in the previous paper [19]. 3 In the calculations in this paper, we use the hadron matrix elements calculated by lattice [20], and the renormalization factors of the minimal SUSY SU (5) GUT as A R = 3.6 for the operators which include a right-handed charged lepton e c R and A R = 3.4 for the operators which include the doublet leptons l as the reference values [21]. The ratio R 2 is sensitive to flavor structure of the second generation, and very useful to identify SO(10) and E 6 unification group. Interestingly, 2 In the decay modes which include neutrino, we sum up over the flavor of neutrino because the nucleon decay detectors do not distinguish neutrino types. 3 In this paper, we have not fixed V u c R = 1 (and V d c R = 1 for SU (5)), which are adopted in Ref. [19]. Theoretically we can fix those diagonalizing matrices without loss of generality. If we have not imposed any constraints to the other diagonalizing matrices, it would not produce any changes in the results. However, in our analysis, we constrained the O(1) coefficients of the other diagonalizing matrices, and therefore, the results depends on whether these conditions are imposed or not. We think that the results without these conditions become similar to the results with these conditions with wider allowed range for the O(1) coefficients. Therefore, distributions of model points have become wider in this paper than in the previous one.

FIG. 1:
The distribution of 10 5 model points for SU (5)(black), SO(10)(red), and E6(blue) GUTs with horizontal axis R1 = Γ(N → π 0ν )/Γ(P → π 0 e + ) and vertical axis R2 = Γ(P → K 0 µ + )/Γ(P → π 0 e + ). The superheavy gauge boson masses are taken to be MX = M X ′ = √ 2M X ′′ . R 1 can be larger than one especially for higher rank unification group like E 6 . Of course the results are strongly dependent on the mass spectrum of superheavy gauge bosons. If M X ′′ ≫ M X ′ = M X , the E 6 model points shrink to SO(10) model points, and when M X ′ becomes much larger than M X , the SO(10) model points shrink to the SU (5) model points. However, we can say that if R 1 > 0.5, SU (5) is implausible and if R 1 > 1, E 6 is preferable. Unfortunately, the detection efficiency for the mode N → π 0ν is not so high as P → π 0 e + mode [22,23] , and therefore, it requires extremely more powerful experiments to observe the mode N → π 0ν even if R 1 > 1.
In this paper, we propose novel modes which may be useful for the identification of unification group. Essential point is that5 fields have large mixings in diagonalizing matrices while 10 fields have small mixings. And therefore, flavor changing nucleon decay, for example, P → π 0 µ + or P → K 0 e + , becomes more important for higher rank unification group. In Figs. 2 and 3, we have calculated the two ratios R 3 ≡ Γ(P → π 0 µ + )/Γ(P → π 0 e + ) and R 4 ≡ Γ(P → K 0 e + )/Γ(P → π 0 e + ) with horizontal axis R 1 in 10 5 model points of SU (5)(black), SO(10)(red), and E 6 (blue) GUTs with the superheavy gauge boson masses M X = M X ′ = √ 2M X ′′ . Interestingly, the SU (5) model points are clearly separated from SO(10) and E 6 model points in Fig. 2, while Fig. 1 has no such separation. One more interesting point is that there are a lot of model points with R 3 > 1. Since the detection efficiency of the P → π 0 µ + is as large as that of P → π 0 e + [22], the flavor changing nucleon decay mode P → π 0 µ + can be found earlier than P → π 0 e + if R 3 > 1. On the contrary, R 4 is comparatively smaller, mainly because the mode Γ(P → K 0 e + ) has the phase space suppression and smaller hadron matrix elements. Note that there is a tendency to obtain larger R 1 for larger R 3 .
Although it may not be so clear in these figures, GUT
with larger rank unification group predicts larger FCND. Actually, it is seen in concrete numbers of model points with R 3 > 1 (17% in E 6 , 0.7% in SO(10) and 0.5% in SU (5)). It must be useful to stress the advantage of the neutrino modes like N → π 0ν for identification of unification group, although such modes have disadvantage for the detection. The most important feature for Γ(N → π 0ν ) is that the value becomes larger for GUT with larger rank unification group, especially when 10 fields have small mixings. Actually, when V 10 = 1, we can show that Γ(N → π 0ν ) Γ SU (5)  Horizontal axis is R3 = Γ(P → π 0 µ + )/Γ(P → π 0 e + ) and vertical axis is partial lifetime for P → π 0 e + and P → π 0 µ + , which are proportional to (MX /gG) 4 . The superheavy gauge boson masses are taken to be summed the flavor of neutrinos, that is important in this calculation. Obviously R 1 becomes larger for larger unification group. This feature is quite important to identify the unification group.

V. DISCUSSION AND SUMMARY
Recently, two events have been found in the signal region for the process P → π 0 µ + [24], though these are still consistent with the background expected to be 0.9 event mainly from atmospheric neutrino events. If the signature for the flavor changing nucleon decay P → π 0 µ + has been found in SuperKamiokande, higher rank unification group like SO(10) or E 6 is preferable when the mixings of 10 fields are small. The predicted partial lifetime for M X /g G = 1 × 10 16 GeV is presented in Fig.  4. Obviously, for larger R 3 , longer partial lifetime for P → π 0 e + and shorter partial lifetime for P → π 0 µ + are obtained. In SU (5), both partial lifetimes become longer than in SO(10) and E 6 . If the signature is from the real nucleon decay process, it is obvious that the usual MSSM predicted value M X /g G ∼ 3 × 10 16 is too large to explain the events even if the ambiguities in Hadron matrix elements [20] are taken into account. Therefore, to explain the signal, larger unification gauge coupling g X (it requires extra vector-like fields in addition to the MSSM fields.), and/or smaller superheavy gauge boson mass M X are required. Note that both features are predicted in the natural GUT [12,16,17], in which the nucleon decay via dimension 6 operators is enhanced while that via dimension 5 is suppressed.
Which mode will be found next? We expect that Γ(N → π 0ν ) can be larger than Γ(P → π 0 e + ), since R 3 is positively correlated with R 1 as in Fig. 2. However, since the detection efficiency for the mode N → π 0ν is much smaller than that for P → π 0 e + , we can predict that next mode should be P → π 0 e + . Of course, the other modes, N → π 0ν and P → K 0 e + , are expected to be found in future experiments like HyperKamiokande [25]. The observation of these modes is quite important and gives us critical hints for studying GUT models.
In this paper, we have emphasized the importance of flavor changing nucleon decay, whose observation may identify the unification group. Especially, the mode P → π 0 µ + is important because the detection efficiency is as large as the usual mode P → π 0 e + . The partial lifetime of P → π 0 µ + can be shorter than that of P → π 0 e + especially in E 6 GUT. Of course, our results are strongly dependent on our important assumptions for diagonalizing matrices of quarks and leptons, V 10 ∼ V CKM and V5 ∼ V MN S , and for the unification group which are restricted to SU (5), SO (10), and E 6 . Therefore, our results are not directly applied to the models which do not satisfy our assumptions like the GUT models in Refs. [6].
Although most of model points predict longer partial lifetime of P → π 0 µ + than that of P → π 0 e + , it is important to pay attention to the mode P → π 0 µ + even if the present signal for P → π 0 µ + is from the back ground processes.