Effect of Bare Mass on the Hosotani Mechanism

It is pointed out that the existence of bare mass terms for matter fields changes gauge symmetry patterns through the Hosotani mechanism. As a demonstration, we study an SU(2) gauge model with massive adjoint fermions defined on $M^4\otimes S^1$. It turns out that the vacuum structure changes at certain critical values of $mL$, where $m~(L)$ stands for the bare mass (the circumference of $S^1$). The gauge symmetry breaking patterns are different from models with massless adjoint fermions. We also consider a supersymmmetric SU(2) gauge model with adjoint hypermultiplets, in which the supersymmetry is broken by bare mass terms for the gaugino and squark fields instead of the Scherk-Schwarz mechanism.


Introduction
The Hosotani mechanism is one of the most important dynamical phenomena when one considers physics with compactified extra dimensions [1] [2]. The component gauge field 1 n 5 −3 + 4N f (1 + zn + (zn) 2 3 )e −zn × 2(1 + cos(2nθ)), where we have used Eq. (2). In order to see how the massive fermion affects the vacuum structure of the model, let us first consider asymptotic behaviors of the effective potential with respect to z. If z is large enough, the fermion contribution to the effective potential is suppressed due to the Boltzmann like factor e −zn in Eq. (4). This is consistent with the observation stated in the introduction that the Hosotani mechanism is governed infrared physics.
Then, the dominant contribution to the potential comes from the gauge sector in the model. Therefore, the vacuum configuration is given, in this limit, by θ = 0 (mod π) [2].
The SU(2) gauge symmetry is not broken in the limit 1 . On the other hand, if we take the massless limit, z → 0, the effective potential, as we have mentioned above, becomes identical to the one for the case of N f massless adjoint fermion. It has been known that the vacuum configuration for this case is given by θ = π/2 [5]. The SU(2) gauge symmetry is spontaneously broken down to U(1) in this limit. The above observations strongly suggest that there must exist certain critical values of z, at which the gauge symmetry breaking patterns change.
In order to confirm the existence of the expected critical values of z, let us study the stability of the configuration θ = 0 (π/2) with respect to z, which corresponds to the vacuum configuration in limit of z → ∞ (0). And we set N f = 1 for simplicity. By simple numerical calculations, we find that the sign of the second derivative of the effective potential at θ = π/2 (0) changes at z ≃ 1.54135 (1.13501) and becomes negative (positive) for larger values of it. Both configurations are stable for 1.13501 < ∼ z < ∼ 1.54135. If we compare the potential energy for the two configurations in the narrow region, we These observations mean that the gauge symmetry breaking patterns change at z * .
As far as our numerical analyses are concerned, there appears two degenerate minima at z * ≃ 1.40087 and is finite height of potential barrier between the configuration θ = 0 and θ = π/2. If z becomes smaller than z * , the minimum of the effective potential locates at θ = π/2, while if z becomes larger than z * , then, the vacuum configuration is given by θ = 0 (mod π). Hence, we conclude that 2 gauge symmetry breaking pattern = We have also computed the critical values of z in case N f = 3, 6, 10. They are given by z * ≃ 3.55154, 4.61892, 5.3521, respectively. If we take N f = 100, the critical value is about z * ≃ 8.378.
Let us comment on the adjoint Higgs scalar, which is originally the component gauge field for the S 1 direction. The mass term for the scalar, which is massless at the tree level, is generated through the quantum correction in the extra dimensions [1]. The mass term is given by estimating the second derivative of the effective potential (4) at the absolute minimum. Our numerical analyses tell us that the Higgs scalar is always massive for each gauge symmetry breaking pattern in Eq.(5).

Supersymmetric gauge model
In this section let us study the N = 1 supersymmetric SU(2) gauge model with N f adjoint hypermultiplets defined on M 4 ⊗ S 1 . The bare mass terms, which are gauge invariant, are introduced in such a way that they break the supersymmetry. Here we are interested in how the gauge symmetry breaking patterns are modified by such the supersymmetry breaking terms and comparing results with those by the model with the Scherk-Schwarz mechanism of supersymmetry breaking.
In the model we have a five-dimensional vectormultiplet V = (Aμ, Σ, λ D ), where Aμ is the five-dimensional gauge potential and λ D (Σ) stands for a Dirac spinor 3 (a real scalar). Here we call λ D gaugino. We also have N f adjoint hypermultiplets H = (ψ D , φ i ), where ψ D is a Dirac spinor and φ i(=1,2) is a complex scalar called squark. In case of the Scherk-Schwarz mechanism of supersymmetry breaking, the gaugino and squark masses are shifted by an unique nontrivial phase associated with the SU(2) R symmetry, so that they have different mass terms from their superpartners in four dimensions. The supersymmetry is broken by the unique phase. Here instead of resorting to the Scherk-Schwarz mechanism, we add the gauge invariant bare mass for the gaugino (m g ) and squarks (m s ), as one of the examples, to break the supersymmetry.
Following the standard procedure to calculate the effective potential for the constant background gauge field gL A y = diag(θ 1 , θ 2 ) = diag(θ, −θ), we obtain that 4 where we have defined z g ≡ m g L, z s ≡ m s L. The first and second lines in Eq. The effective potential is recast, by using Eq. (2), as where we have defined The function F (n, z i ) satisfies 0 ≤ F (n, z i ) ≤ 1, where the first (second) equality holds when z i → 0 (∞).
Let us first study the case of z g = z s (≡ z c ). The effective potential (7) becomes It is easy to see that the potential vanishes for N f = 1. This is because one can still have N = 1 supersymmetry by recombining the two massive fields into the same multiplet, so that there is one massless and one massive multiplet. And each multiplet is supersymmetric under the N = 1 supersymmetry. As a result, the total action is invariant under the supersymmetry.
The nonvanishing potential is given for N f ≥ 2. The supersymmetry is broken by an unique parameter z c in this case. Taking 0 ≤ F (n, z) ≤ 1 into account, the minimum of the potential is always located at θ = π/2, independent of the values of z c ( = 0). Thus, the SU(2) gauge symmetry is broken to U(1). This is the same result as the one obtained by the Scherk-Schwarz mechanism, in which the unique nontrivial phase associated with SU(2) R symmetry breaks the supersymmetry, and the vacuum configuration is given by θ = π/2 [6] 5 . Let us note that the adjoint Higgs scalar in this case is always massive except for z c = 0, where the potential vanishes due to the original N = 1 supersymmetry in five dimensions.
Let us next consider the case z g = z s with N f = 1. The sign of a function C ≡ F (n, z s ) − F (n, z g ) is important to determine the minimum of the effective potential (7). For z s < (>)z g , the sign of C is negative (positive), so that the configuration θ = 0, (π/2) is realized as the vacuum configuration. Therefore, depending on the relative magnitude between z g and z s , the vacuum configuration is different and accordingly, the gauge symmetry breaking patterns are different. If z g = z s , the effective potential vanishes due to the survived supersymmetry explained above. The adjoint Higgs scalar is always massive in this case.
Let us finally study the case N f = 2 with z g = z s . In order to demonstrate the possible effects of the bare mass on the gauge symmetry breaking patterns, we take z g to be 0.1, 1.0, 10 as an example. For each value of z g we study the behavior of the effective potential with respect to z s and find the minimum of the potential. We examine the stability of the configuration θ = 0 (mod π) and θ = π/2 with respect to z s for the given values of z g by studying the second derivative of the effective potential.
In case z g = 0.1, simple numerical calculations show that the configuration θ = 0 becomes unstable for z s > ∼ 0.0672937 ≡ z s1 , on the other hand θ = π/2 becomes stable for z s > ∼ 0.0706947 ≡ z s2 . The configuration that minimizes the effective potential in the narrow region between z s1 and z s2 is still given by the configuration which breaks the SU(2) gauge symmetry to U(1), though it is not θ = π/2. We confirm that by numerical 5 The effective potential for the case of the Scherk-Schwarz mechanism is given by 1 n 5 [1 − cos(nβ)] (2 + 2 cos(2nθ)) .
The Hosotani mechanism depends only on matter contents, so that we can quote the results obtained in [6].
analyses, the behavior of θ that minimizes the potential in the narrow region is that the θ increases gradually from zero at z s1 and approaches to π/2 at z s2 . Thus, we have shown that the bare mass terms for the gaugino and squark can affect the gauge symmetry breaking patterns through the Hosotani mechanism and the phase transition occurs at z s = z s1 for z g = 0.1. Hence, we obtain that 6 gauge symmetry breaking pattern = It should be noted that the very small values of θ, which is usually of order O(1), is possible in this case. This may affect mass spectrum in four dimensions. We will discuss this point in the last section.
We repeat the same analyses as above for the case z g = 1.0 (10) with N f = 2. The configuration θ = 0 becomes unstable for z s > ∼ 0.618288 (2.03287) ≡ z s1 , while the configuration θ = π/2 becomes stable for z s > ∼ 0.691531 (2.47766) ≡ z s2 . The qualitative behavior of θ that minimizes the effective potential in the narrow region is the same as that in the case z g = 0.1.
The adjoint Higgs scalar in this case can be massless unlike the previous cases. The second derivative of the effective potential evaluated at θ = 0 (π/2) vanishes for z s1 = 0.0672937 (0.0706947). Hence, the massless state of the Higgs scalar is possible for the fine tuned values of z s . In the other cases z g = 1.0, 10, we also have massless state of the adjoint Higgs scalar at the values of z s , where the second derivative of the potential evaluated at θ = 0, π/2 vanishes.
We have seen that the gauge symmetry breaking patterns change due to the existence of the bare mass terms for the gaugino and squark in the model. The SU(2) gauge symmetry is not broken for z s < z s1 , on the other hand, SU(2) is broken to U(1) for z s > z s1 for fixed values of z g in our examples.
If we add a bare mass term for the Dirac spinor ψ D in the hypermultiplet instead of the squark φ i , the structure of the effective potential is different from Eq.(6). It is easy to see that the SU(2) gauge symmetry is never broken for any nonzero values of the bare masses.

Conclusions and discussions
We have demonstrated that the existence of the bare mass affects the gauge symmetry breaking patterns through the Hosotani mechanism. We have explicitly shown that in the nonsupersymmetric SU(2) gauge model with the massive adjoint fermions defined on symmetry is unbroken (broken). The phase transition is the first order. The asymptotic behavior of the effective potential with respect to z also suggests the existence of the critical values of z ≡ mL: If the adjoint fermion is heavy enough, corresponding to z → ∞, it decouples from the effective potential and the gauge sector of the model dominates the potential. Hence, the SU(2) gauge symmetry is not broken through the Hosotani mechanism. On the other hand, if we take the massless limit of the fermion, that is, z → 0, the vacuum configuration breaks the SU(2) gauge symmetry to U(1) [5].
We have also studied the supersymmetric gauge model defined on M 4 ⊗S 1 . Instead of the Scherk-Schwarz mechanism of supersymmetry breaking, we have introduced the bare mass terms for the gaugino in the vectormultiplet and the squark in the hypermultiplet to break the supersymmetry. When the number of the hypermultiplet N f is equal to one, the critical point is given by z g = z s , where the potential vanishes due to the N = 1 supersymmetry. The SU(2) gauge symmetry is broken to U(1) for z s > z g , while the gauge symmetry is not broken for z s < z g . If N f ≥ 2 and z g = z s , then, the SU (2) gauge symmetry is always broken to U(1) as long as z c (≡ z g = z s ) = 0. In this case, the supersymmetry is broken by an unique bare mass z c . And the result is the same as the one obtained by the Scherk-Schwarz mechanism of supersymmetry breaking, in which the supersymmetry breaking parameter is also an unique and the gauge symmetry is always broken to U(1). In these cases the adjoint Higgs scalar cannot be massless except that the models have the accidental N = 1 supersymmetry.
We have considered the case z g = z s for N f = 2. We have shown the possible effect of the bare masses on the gauge symmetry breaking patterns through the Hosotani mechanism. By choosing the certain values of z g , we have investigated the configuration that minimizes the effective potential according to the change of the values of z s . And we have found the critical values of z s1 , above (below) which the gauge symmetry is broken (unbroken). The phase transition in the supersymmetric model is the second order unlike the case of the nonsupersymmetric model. We have also found that the massless state of the adjoint Higgs scalar appears for the fine tuned values of z s in this case.
There are many issues that are not discussed in this paper. Let us comment on a few of them. In the supersymmetric gauge model discussed in the section 3, it is important to determine the behavior of the order parameter θ with respect to z s precisely in the narrow region between z s1 and z s2 . As mentioned in the section, it is possible that the magnitude of the order parameter θ can be very small for (fine tuned) values of z s . Then, if particle does not have a bare mass term, the mass square of n = 0 mode in the Kaluza-Klein modes behaves like (θ/L) 2 , so that the order of the mass is highly reduced compared with the compactification scale 1/L at the tree level. Therefore, we expect the light particle in four dimensions through the Hosotani mechanism 7 .
It may be interesting to study the case of massive fundamental fermion instead of the adjoint one. It has been known that the SU(N) gauge symmetry is not broken for (supersymmetric) gauge model (with the Scherk-Schwarz mechanism) with massless fundamental fermion (matter) [7][6]. If we take the limit of the heavy bare mass, the fundamental fermion decouples from the effective potential and the potential is dominated by the gauge sector alone. Then, there are N physically equivalent vacua. On the other hand, in the massless limit, we expect that there is a single SU(N) symmetric vacuum for N = even and a doubly degenerate SU(N) vacuum for N = odd. Hence, we expect from these observations that there exist critical values of mL, at which a sort of phase transition, in which the number of the vacuum changes, occurs.
It may be interesting to consider higher rank gauge group and study the massive particle effect on the gauge symmetry breaking patterns. In particular, if we introduce the hierarchy among the bare masses, as we have done in the supersymmetric case, it may be expected to occur rich gauge symmetry breaking patterns. And it is also interesting to study the mass spectrum in four dimensions, taking the smallness of θ into account, as discussed above.
One can also expect the same phenomena in other extra dimensions such as the orbifold S 1 /Z 2 for example. According to the lessons obtained in this paper, the gauge symmetry breaking patterns change even in the case of the orbifold if particles possess bare mass terms. It is expected that degeneracy of equivalent classes of boundary conditions, which has been discovered and discussed recently in [8], may be lifted due to the effect of the bare mass. These problems are under investigation and will reported elsewhere.