Abstract
In the SU (6) × SU (2) R string-inspired model, we evolve the couplings and the masses down from the string scale M S using the renormalization group equations and minimize the effective potential. This model has the flavor symmetry including the binary dihedral groupD 4 . We show that the scalar mass squared of the gauge non-singlet matter field possibly goes negative slightly below the string scale. As a consequence, the precocious radiative breaking of the gauge symmetry down to the standard model gauge group can occur. In the present model, the large Yukawa coupling which plays an important role in the symmetry breaking is identical with the colored Higgs coupling related to the longevity of the proton.
Introduction
In the minimal supersymmetric standard model(MSSM), it is well-known that the spontaneous breaking of the gauge symmetry SU(2) L × U(1) Y → U(1) em is caused around the electroweak scale by the radiative effect due to the large top Yukawa coupling. [1] On the other hand, in many supersymmetric GUT models, it is assumed that by taking the wine-bottle type of the Higgs potential by hand, the spontaneous breaking of a large gauge symmetry such as SU (5) or SO(10) takes place via Higgs mechanism at high energies around O(10 16 GeV). In order to clarify whether or not the spontaneous breaking of the large gauge symmetry occurs at such a large energy scale, we need to address the underlying string theory which yields the GUT-type models. The radiative breaking of the large gauge symmetry occurs if the mass squared of a gauge non-singlet scalar field goes negative precociously as one evolves down from the string scale. Then it is of importance to study whether or not the radiative effect due to the large Yukawa couplings resulting from the underlying theory causes the scalar mass squared to be driven negative at a large energy scale.
In the extra U(1) 2 string-inspired model it has been already found that the radiative effect due to the large Yukawa couplings possibly breaks down one of the extra U(1) gauge symmetries around O(10 15 GeV). [2] In this paper we consider the SU(6) × SU(2) R string-inspired model, which contains many phenomenologically attractive features. [3, 4, 5, 6, 7, 8, 9] In this model we evolve couplings and masses down from the string scale M S using the renormalization group(RG) equations and minimize the effective potential. The purpose of this paper is to explore whether the gauge symmetry breaking occurs or not at very large energy scale. Studying the RG evolution from the string scale M S , we show that the scalar mass squared of the gauge non-singlet matter field possibly goes negative slightly below the string scale. This implies that the precocious breaking of the gauge symmetry SU(6) × SU(2) R can occur due to the radiative effect. In this model the large Yukawa coupling which plays an important role in the symmetry breaking is identical with the colored Higgs coupling related to the longevity of the proton. This symmetry breaking triggers off the subsequent symmetry breaking. [10] Thus we obtain the sequential symmetry breaking
where SU(4) P S and G SM represent the Pati-Salam SU(4) [11] and the standard model gauge group, respectively.
In the framework of the string theory we are prohibited from adding extra matter fields by hand. In the effective theory from string, the matter contents and the Lagrangian are strongly constrained due to the topological and the symmetrical structure of the compact space. This situation is in sharp contrast to the conventional GUT-type models. For instance, in the perturbative heterotic string we have no adjoint or higher representation matter(Higgs) fields. Also, in the context of the brane picture, matter fields belong to the bi-fundamental or the anti-symmetric representations under the gauge group such as SU(M) × SU(N). In the present model, under SU(6) × SU(2) R , gauge non-singlet matter fields consist of (15, 1), (6 * , 2) and their conjugates. Within the rigid framework we have to find out the path from the string scale physics to the low-energy physics. ¿From this point of view we study the RG evolution of couplings and masses from the string scale and explore the hierarchical path of the gauge symmetry breaking.
To the SU(6) × SU(2) R string-inspired model we introduce the flavor symmetry
The cyclic group Z M and the binary dihedral groupD 4 have R symmetries, while Z N has not. Introduction of the binary dihedral groupD 4 is motivated by the phenomenological observation that the R-handed Majorana neutrino mass for the third generation has nearly the geometrically averaged magnitude of M S and M Z . Further, the binary dihedral flavor symmetryD 4 is an extention of the Rparity. In Ref. [9] , solving the anomaly-free conditions under many phenomenological constraints coming from the particle spectra, we found a large mixing angle(LMA)-MSW solution with (M, N) = (19, 18) , in which appropriate flavor charges are assigned to the matter fields. In Refs. [8, 9] we have assumed that the scalar mass squared of the gauge non-singlet field goes negative slightly below M S . The results are in good agreement with the experimental observations about fermion masses and mixings and also about hierarchical energy scales including the GUT scale, the µ scale and the Majorana mass scale of the R-handed neutrinos. Then we carry out the present analysis of the RG evolution of the scalar masses squared on the basis of the SU(6) × SU(2) R model with the flavor symmetry
This paper is organized as follows. In section 2, after explaining main features of the SU(6)×SU(2) R string-inspired model with the flavor symmetry Z 19 ×Z 18 ×D 4 , we exhibit the superpotential. We point out that if the soft scalar mass squared is driven negative, the spontaneous breaking of the gauge symmetry SU(6) × SU(2) R down to G SM occurs in two steps sequentially. In section 3 we study the RG evolutions of couplings and masses down from M S . It is found that the scalar mass squared of the gauge non-singlet matter field possibly goes negative slightly below the string scale. The final section is devoted to summary and discussion.
SU (6) × SU (2) R Model and the Scalarpotential
The SU(6) × SU(2) R string-inspired model considered here is studied in detail in Refs. [3, 4, 5, 6, 7, 8, 9] . To begin with, we review the main features of the model.
(i). The gauge group G = SU(6) × SU(2) R can be obtained from E 6 through the Z 2 flux breaking on a multiply-connected manifolds K. [12, 13, 14] To be more specific, the nontrivial holonomy U d on K is of the form
where I 3R represents the third direction of the SU(2) R . The symmetry breaking of G down to G SM can take place via the Higgs mechanism without matter fields of adjoint or higher representations. SU(6) × SU(2) R is the largest one of such gauge groups. [15] (ii). Matter consists of the chiral superfields of three families and the one vector-like multiplet, i.e.,
in terms of E 6 . The superfields Φ in 27 of E 6 are decomposed into the irreducible
where g and g c and H u and H d represent the colored Higgs and the doublet Higgs superfields, respectively, N c is the right-handed neutrino superfield, and S is an SO(10) singlet. It should be noted that the doublet Higgs and the color-triplet Higgs fields belong to the different irreducible representations of G 
as shown in Eq.(3). As a consequence, the triplet-doublet splitting problem is solved naturally. [3] (iii). As the flavor symmetry, we introduce the Z 19 × Z 18 and theD 4 symmetries and regard Z 19 and Z 18 as the R and the non-R symmetries, respectively. Since the numbers 19 and 18 are relatively prime, we can combine these symmetries as
Solving the anomaly-free conditions under many phenomenological constraints coming from the particle spectra, we obtain a LMA-MSW solution with the Z 342 charges of matter superfields as shown in Table 1 . [9] In this solution we assign the Grassmann number θ, which has the charge (−1, 0) under Z 19 × Z 18 , the charge 18 under Z 342 . The assignment of "D 4 charges" to matter superfields is given in Table 2 , where σ i (i = 1, 2, 3) represent the Pauli matrices and
The σ 3 transformation yields the R-parity. Namely, the R-parities of the superfields Φ i (i = 1, 2, 3) for three generations are all odd, while those of the Φ 0 and Φ are even.
(iv). There are two types of gauge invariant trilinear combinations 
The flavor symmetry requires that in the superpotential these trilinear combinations are multiplied by some powers of φ 0 φ or ψ 0 ψ. Concretely, the superpotential terms are of the forms
The exponents are determined by the constraints coming from the flavor symmetry.
To be more specific, we have 
The coefficients z 0 , z, h 0 , h, z ij , h ij and m ij are O(1) constants. The relation ζ 0 = 0 means that only the superfield φ 0 takes part in the renormalizable interaction with the large Yukawa coupling at M S . The powers of φ 0 φ can be replaced all or in part with those of ψ 0 ψ subject to the flavor symmetry. When φ 0 and φ develop the non-zero vacuum expectation values(VEV's), the above non-renormalizable terms become the effective Yukawa couplings with hierarchical patterns. [16] In addition, the superpotential contains the other types of the non-renormalizable term
with λ i = O(1). The flavor symmetry Z 342 ×D 4 yields n = 81 and m = 4. In the present model we study the minimum point of the scalarpotential. We will assume that the supersymmetry is broken at the string scale due to the hidden sector dynamics and that the supersymmetry breaking is communicated gravitationally to the observable sector via the soft supersymmetry breaking terms. As mentioned above, there exists a large Yukawa coupling at the string scale only for φ 0 . Then the radiative corrections of the soft scalar masses squared due to the Yukawa coupling are sizable only for φ 0 . On the other hand, the R-parity odd superfields φ i and ψ i (i = 1, 2, 3) which appear in pair in the superpotential term, hardly receive the radiative corrections in the region around M S . The soft scalar masses squared of the R-parity odd superfields remain positive in the wide energy region. Therefore, the F-flat conditions for φ i and ψ i (i = 1, 2, 3) require
Thus in order to minimize the scalarpotential, it is sufficient for us to confine ourselves to the R-parity even sector. In the R-parity even sector we have the superpotential W 1 . The scalarpotential is given by
where V soft represents the soft supersymmetry breaking terms
The one-loop correction ∆V 1−loop is of the form[17]
where bosons(fermions) contribute with positive(negative) sign in the supertrace and the mass M has to be considered to be a function of φ 0 , φ, ψ 0 and ψ. In the above equations, for simplicity we denote the scalar components of the superfields by the same letters as the superfields. The soft scalar masses squaredm are assumed to take a universal positive value at the string scale. We evolve down from the string scale the scalar masses squared by using the RG equations. Since the one-loop correction Eq. (14) is linear in ln Q 2 , it is possible to find a value of Q such that this correction is quite small in the minimum of the potential. Therefore, it is sufficient to treat the minimization by simply using the RG-improved tree-level potential. Ifm is driven negative slightly below the string scale, the gauge symmtry could be spontaneously broken irrespective ofm 
Thus, ifm 
in a feasible parameter region of the coefficients λ i , wherem φ = |m |. Since we obtain | φ 0 | > | ψ 0 |, the gauge symmetry breaking occurs in two steps as
When the gauge symmetry SU(6) × SU(2) R is broken down to SU(4) P S × SU(2) L × SU(2) R , the field φ 0 (15, 1) is decomposed as
Needless to say, the field φ 0 (1, 1, 1) develops the non-zero VEV | φ 0 |. In addition, the field ψ 0 (6 * , 2) is decomposed as
A question arises as to which field of ψ 0 (4 * , 1, 2) and ψ 0 (1, 2, 2) develops the non-zero VEV | ψ 0 |. As seen in Eq. (7), the field ψ 0 (1, 2, 2) has the coupling φ 0 (1, 1, 1)ψ 0 (1, 2, 2) 2 , which is the third term of Eq. (8) . Below the scale | φ 0 |, this term induces th µ term. The F-flat condition for ψ 0 (1, 2, 2) requires
at a large energy scale. Consequently, the non-zero VEV | ψ 0 | is attributed to ψ 0 (4 * , 1, 2). Thus we obtain G SM below the scale | ψ 0 |.
We now calculate the energy scale of the gauge symmetry breaking. Since we have n = 81 and m = 4, the VEV's | φ 0 | and | ψ 0 | are smaller than M S but not far from M S . By taking M S ∼ 5 × 10 17 GeV [18] andm φ ∼ 10 3 GeV, we obtain
Therefore, for the present model to be consistent, it is necessary thatm 2 φ0 +m 2 φ is deriven negative slightly below M S . In the next section, evolving couplings and masses down from the string scale using the RG equations, we show thatm 
The RG evolutions of scalar masses
The one-loop RG equation for the SU(6) gauge coupling g 6 is given by
where the variable t is defined as ln(Q/M S ) and b 6 = 3. Similarly, we have the one-loop RG equation for the SU(6) gaugino mass M 6
These equations are easily solved as
where the variable u is defined as
The constants g 0 and M 1/2 represent the values of the gauge coupling and the gaugino mass at the string scale M S , respectively. At the string scale the renormalizable term of the superpotential is of the simple form
This term induces the soft breaking A term
Here we assume that the Yukawa coupling z 0 and the soft breaking parameter A 0 are real. In this case the RG equations for z 0 , A 0 ,m 
Concretely, the RG evolutions are expressed as
. In these expressions we define three dimensionless parameters
and three functions
Since we are interested in the precocious breaking of the gauge symmetry, the RG evolution of the couplings and the masses is carried out in the energy region ranging from M S to M S /5. Since 4π/g 2 0 takes a value around 15 in the present model [3] , we have g 0 ≃ 0.9 and then the region considered here of the variable u becomes 1.0 ∼ 0.95. We now proceed to accomplish the numerical study as to whether or notm 
Summary and discussion
In the SU(6)×SU(2) R string-inspired model with the flavor symmetry Z 19 ×Z 18 ×D 4 , we evolve couplings and masses down from the string scale M S using the RG equations. In the feasible parameter region of a Yukawa coupling and the soft supersymmetry breaking masses, the scalar mass squared of the gauge non-singlet matter field possibly goes negative slightly below the string scale. This implies that the precocious radiative breaking of the gauge symmetry SU(6) × SU(2) R can occur due to the radiative effect. This symmetry breaking triggers off the subsequent symmetry SUGRA GUT model, as to which some difficulties have been pointed out concerning the proton lifetime, [20] our result is consistent with the present experimental data [21] . The longevity of the proton is in connection with the precocious gauge symmetry breaking through the common large Yukawa coupling.

{'paper_id': '6219366', 'title': 'A ug 2 00 2 hep-ph / 0209004 AUE-02-02 / KGKU-02-02 Precocious Gauge Symmetry Breaking in SU ( 6 ) × SU ( 2 ) R Model', 's2_url': 'https://api.semanticscholar.org/CorpusID:6219366', 'source': 'S2ORC'}