Solutions to large B and L breaking in the Randall-Sundrum model

The stability of proton and neutrino masses are discussed in the Randall-Sundrum model. We show that relevant operators should be suppressed, if the hierarchical Yukawa matrices are explained only by configurations of wavefunctions for fermions and the Higgs field along the extra dimension. We assume a $Z_N$ discrete gauge symmetry to suppress those operators. In the Dirac neutrino case, there is an infinite number of symmetries which may forbid the dangerous operators. In the Majorana neutrino case, the discrete gauge symmetries should originate from $U(1)_X$ gauge symmetries which are broken on the Planck brane. We also comment on the $n-\bar{n}$ oscillation as a phenomenon which can distinguish those discrete gauge symmetries.


Introduction
Models with low scale quantum gravity [1,2] have been intensely studied, since they can account for the hierarchy between the electroweak and the fundamental scales. However, in the low cut-off theories, higher dimensional operators induce in general fast proton decays or too large neutrino masses, since these operators are suppressed only by the low cut-off scale ∼ 1 TeV 1 . It has been argued that these problems can be solved by some symmetries [4] or field configurations along the extra dimensions [5].
The solution by field configurations along the extra dimensions is interesting, since it also explains the hierarchical Yukawa matrices without introducing extra spectator fields [5,6,7]. In fact, there are some attempts in higher dimensional models to explain the hierarchical Yukawa matrices [8,9,10] and sufficiently long proton lifetime [5] by field configurations along the extra dimensions. However, these benefits do not go together in general, since the observed fermion masses require sufficient overlaps of wave functions for fermions, while the long proton lifetime requires small overlaps. For example, the proton lifetime is too short in the Randall-Sundrum (RS) model, if the observed fermion masses are explained by particular field configurations [9].
The purpose of this paper is to study these problems in the RS model. In the next section, we analyze the proton lifetime and neutrino masses. We show that relevant operators must be suppressed substantially by hand, if the hierarchical Yukawa matrices are explained only by field configurations. In section 3, we look for symmetries which may suppress those operators naturally. Here, we focus on discrete gauge symmetries [11] 2 , since global symmetries may be explicitly broken by the topological effects of gravity [14], and there is no continuous gauge symmetry which supresses the dangerous operators. We introduce a Z N discrete gauge symmetry to supress those operators. In the Dirac neutrino case, there is an infinite number of symmetries which may forbid the dangerous operators. In the Majorana neutrino case, discrete gauge symmetries cannot forbid the dangerous operators. However, if the discrete gauge symmetries originate from U(1) X gauge symmetries, and they are broken on the Planck brane, then the dangerous operators can be suppressed. Finally, we comment on n −n oscillation as a phenomenon which distinguishes between those discrete gauge symmetries. 1 For other constraints (from flavor changing neutral currents, precision electroweak measurements, etc.), see [3] and references therein.
2 Discrete gauge symmeties in extra dimension models have been studied in [12]. However, their model is different from ours in some ways. For example, right-handed neutrinos do not acquire large Majorana masses in their model. In our model, they acquire large Majorana masses and the usual seesaw mechanism [13] is realized, as we will show in section 3.2.
In some extra dimension models, the configurations of fermions and the Higgs field along the extra dimensions are employed to explain the proton stability, neutrino masses, and the hierarchical Yukawa matrices. However, in the RS model, the field configurations by themselves can not explain the above issues simultaneously, as we will show in this section.
First, we summarize our setup. The metric of the RS model is where σ = k|y|, and k ∼ M P = 2.4 × 10 18 GeV is the AdS curvature. The fifth dimension y is compactified on an orbifold S 1 /Z 2 . Two 3-branes reside at the fixed points y = 0, y = πR, The operators which generate 4D Majorana neutrino masses are where the upper suffix c denotes the charge conjugation. The 4D Majorana neutrino masses (m ν ) ij are given by where v ≡ e −πkR H ∼ 100 GeV is the VEV of the Higgs field, and T (c) is (see Appendix A for details).
The 5D Yukawa interactions which generate 4D Dirac mass terms are where (y e ) ij ∼ O(1) are dimensionless 5D Yukawa couplings. The 4D Dirac mass matrix (m e ) ij is given by Using m τ ∼ (m e ) 33 ∼ 1 GeV, we have where we have used (y e ) 33 ∼ 1 and T (c) < 1 for c ∼ 1. This neutrino mass is well above the limit m ν < 0.68 eV obtained from the WMAP observations [15] and the implications of where U L , U R , D L , D R , E L and E R are 3 × 3 unitary matrices. The eigenvalues of fermion mass matrices are where dimensionless 5D Yukawa couplings y u , y d and y e are ∼ O(1). where (see Appendix A for details). Taking the mixing into account, the suppression scales of the Using Eq. (11), we see that these summations are approximately given by 3 1 Thus we have Since y u , y d and y e are ∼ O(1) in Eq.(10), we have The decay rates of p → π 0 e + induced by the dimension 6 operators O 1 =û R1dR1ûL1êL1 and where W (k − q ′ ) is the form factor of the p → π matrix element and k, k ′ and q are four momenta of proton, positron and pion, respectively. The momentum dependence of W is weak and W ≃ −0.15 GeV 2 [17], so that Thus the suppression scales obtained in Eq. (16) are too small to explain the observed proton lifetime.
We have seen that operators concerning with the Majorana neutrino masses and the proton decay should be suppressed by small factors, or forbidden by some symmetries. We consider that the former solution contradicts with the philosophy of the RS model, that is, solving the hierarchy problem without fine tunings. In the next section, we look for the symmetries which may suppress these operators.

Discrete gauge symmetry
In this section, we look for the symmetries which may suppress the dangerous operators. We concentrate on gauge symmetries, since any global symmetries may be explicitly broken by the topological effects of gravity [14]. In addition, we consider discrete gauge symmetries, since there is no continuous anomaly-free symmetry except for U(1) B−L , which can not supress the dangerous operators for the proton decay. We introduce only one discrete gauge symmetry: the gauge symmetry of the action is assumed to be SU (

Z N symmetry for the Dirac neutrino case
Here we discuss Z N discrete gauge symmetries in the Dirac neutrino case. Those Z N symmetries must respect the Yukawa terms. The Z N charge of each field is constrained as Table 1. We can q L u R d R l L e R ν R H Z N m m m p p p 0 Table 1: Z N charges consistent with the Yukawa interactions. Here we set the charge of the Higgs field to 0 by using a gauge rotation of U(1) Y .
set the Z N charge of the Higgs field to 0 without loss of generality by using a gauge rotation of The anomaly cancellation conditions which include Z N are where r i are integers and η = 1, 0 for N = even, odd.  [18,19]. There is an infinite number of Z N (p, m) discrete gauge symmetries which satisfy these constraints.
Now we examine which operators should be suppressed by symmetries. First, we consider operators with dimension n ≥ 12. The proton lifetime derived from a dimension n operator is roughly estimated as where M n is the suppression scale of the proton decay operator. For n ≥ 12, τ p is longer than the experimental bound τ p > 1.

Z N symmetry for the Majorana neutrino case
Let us consider the case where the seesaw mechanism [13] induces the light Majorana neutrino masses. Naively, this seems to be impossible, since the charges of Mν R ν R are the same as those of HHl L l L which induce too large neutrino masses. However, this is not the case, when the discrete gauge symmetries originate from U(1) X gauge symmetries.
Let us consider that the scalar field Φ which breaks the U(1) X symmetry lives on the Planck brane 6 , and ν R acquire the Majorana masses through the coupling with Φ: where M ij are given by Assuming Φ ∼ M P , M ij take values between M T and M P for 0 < c νi , c νj < 1.
The operators HHl L l L also couple with Φ, since their U(1) X charges are the same as those of ν R ν R . Thus the dangerous Majorana neutrino mass terms appear only through combinations with Φ: which should vanish, since H do not overlap with Φ.
Thus, the operators HHl L l L induced directly from the Φ condensation are negligible. We now consider Yukawa interaction terms H †l L ν R and the Majorana mass terms Mν R ν R to estimate neutrino masses. A model with these mass terms was suggested in [21], and the effective light neutrino masses are approximately given by 5 The symmetries in this series are independent of each other for the following reason. Equivalent discrete gauge symmetries are related through the charge conjugation or U (1) Y gauge rotation. Under the convention of Table 1, any symmetry equivalent to Z N (p, m) takes the form Z nN (np, n(m+ kN 3 )). Thus equivalent symmetries have a common value of N/p. 6 A model in which the lepton number symmetry is broken on the Planck brane for the Dirac neutrino case is discussed in [20].
which take values between ∼ 1 GeV and ∼ 10 −33 eV for 0 < c li , c νj < 1. Thus observations concerning with neutrino masses are easily explained.
Let us count the Z N discrete gauge symmetries which satisfy the anomaly cancellation conditions and respect the Majorana mass terms 7 . The anomaly cancellation conditions are the same as those of the last subsection. Since the Majorana mass terms Mν R ν R are induced from the interaction terms of Φ and ν R , there arises another condition for p and N, where r 5 is an integer. Thus the discrete gauge symmetries are There are other discrete symmetries which satisfy these constraints. However, they are embedded in the above symmetries, or U(1) Y -gauge equivalents of those [23].
All of these symmetries allow some of the dangerous operators, whose Z N charges are 2p, 4p, 6p, 3m ± p, 3m ± 3p and 6m. However, depending on the U(1) X charges of fermions, these operators can be suppressed. To see this, we analyze general properties of higher dimensional operators which include Φ.
Consider an operator which consists of 2n fermions and a scalar field Φ where Ψ i are 5D Dirac fields and ψ i are their zero modes. Then the effective suppression scale M 2n is given by where we used Φ ∼ M P and Eq. (12). Thus any higher dimensional operator with nonzero U(1) X charge is Planck suppressed in 4D effective theory.
For all discrete symmetries in Eq.(27), we can set the U(1) X charges of fermions so that the dangerous operators have to couple with Φ. For example, consider the case where the U(1) X charges of Φ, q L and l L are 1, 3 and 2, respectively. The U(1) X symmetry is broken to Z 1 (0, 0) in low energy. In this case, all the dangerous operators have nonzero U(1) X charges, and become Planck suppressed operators.
We comment on n −n oscillation as a phenomenon which would distinguish between the above discrete gauge symmetries. The n −n oscillation is induced by the dimension 9 operator In the case of Z 2 (1, 1), Z 9 (0, 2) and Z 18 (9, 1) symmetries, this operator is always Planck suppressed, since u R and d R have nonzero U(1) X charges. In the case of completely broken U(1) X , the U(1) X charge of (u R d R d R ) 2 can be set to zero. Then the suppression scale is determined by the configurations of quarks, and could be tuned to the current lower bound, which is evaluated to be 10 5 GeV [24]. Thus the n −n oscillation could be observed in future experiments 8 , if the discrete gauge symmetry is Z 1 (0, 0).

Conclusion
It has been argued that short proton lifetime and too large neutrino masses are most likely induced in the RS model, if the hierarchical Yukawa matrices are explained only by the field configurations along the extra dimension. We have confirmed that the above unwanted phenomenon are inevitable, and hence searched for the discrete gauge symmetries which may forbid the dangerous operators in the Dirac neutrino case and the Majorana neutrino case. We have found that there is an infinite number of such symmetries for the Dirac neutrino case. For the Majorana neutrino case, the discrete gauge symmetries should originate from U(1) X gauge symmetries, and they should be broken on the Planck brane. Furthermore, if the U(1) X symmetry is completely broken, the n −n oscillation could be observed in future experiments.

A Profile of 5D fermion
Here we summarize the convention for fermions in the bulk [26]. The kinetic and mass terms of a 5D Dirac field Ψ are where e A M is the vielbein, and Ω M is the spin connection. The 5D Dirac mass m D is odd, and parametrized as m D = cσ ′ . The 5D Dirac field Ψ is decomposed to where ψ and we have T (c) and P (c) which we have used in this paper are defined as