Minimal $SU(3)\times SU(3)$ symmetry breaking patterns

We study the vacua of an $SU(3)\times SU(3)$-symmetric model with a bifundamental scalar. Structures of this type appear in various gauge theories such as the Renormalizable Coloron Model, which is an extension of QCD, or the Trinification extension of the electroweak group. In other contexts, such as chiral symmetry, $SU(3)\times SU(3)$ is a global symmetry. As opposed to more general $SU(N)\times SU(N)$ symmetric models, the $N=3$ case is special due to the presence of a trilinear scalar term in the potential. We find that the most general tree-level potential has only three types of minima: one that preserves the diagonal $SU(3)$ subgroup, one that is $SU(2)\times SU(2)\times U(1)$ symmetric, and a trivial one where the full symmetry remains unbroken. The phase diagram is complicated, with some regions where there is a unique minimum, and other regions where two minima coexist.


Introduction
that two inequalities involving the two quartic couplings are necessary and sufficient for that.
In Section 2 we present the renormalizable potential and the parameter space. In Sections 3-5 we identify all possible local minima. The conditions for having a potential bounded from below are derived in Section 6. We analyze the phase diagram of this theory, including all global minima, in Section 7. Section 8 includes our conclusions.

SU (3) × SU (3) with a scalar bifundamental
Consider an SU (3) 1 × SU (3) 2 symmetry with a scalar Σ transforming in the (3,3) representation. Thus, Σ is a 3 × 3 matrix with complex entries. The renormalizable potential of Σ is given by The dimensionless couplings λ and κ are real numbers. The mass-squared parameter, m 2 Σ , may be positive or negative. The phase rotation freedom of Σ allows us without loss of generality to choose the coefficient of the trilinear term (a mass parameter) to be real and satisfy µ Σ ≥ 0 . (2. 2) The potential V (Σ) has an accidental Z 3 symmetry. If µ Σ = 0, then the Z 3 symmetry is enhanced to a global U (1) Σ symmetry, with Σ carrying nonzero global charge. We also note that when both µ Σ = 0 and κ = 0 the potential has an enhanced SO (18) symmetry. Even though the scalar Σ has 18 degrees of freedom, upon an SU (3) 1 × SU (3) 2 transformation the most general form of its VEV is a diagonal 3 × 3 matrix. Furthermore, the diagonal SU (3) 1 × SU (3) 2 transformations, associated with the T 3 and T 8 generators, can be used to get rid of two phases. Thus, the most general VEV of Σ has four real parameters: Σ = diag(s 1 , s 2 , s 3 ) e iα/3 , with s i ≥ 0, i = 1, 2, 3 , and − π < α ≤ π . (2.3) The 1/3 in the complex phase of the VEV is due to the Z 3 symmetry. We seek the values of s i and α that correspond to local minima of the potential.
To identify the extrema of the V (Σ) potential, we need to find s i , i = 1, 2, 3 and α that satisfy the extremization (or more precisely stationarity) conditions, which are given by 1 2 two analogous equations for ∂V /∂s 2 and ∂V /∂s 3 (the i = 1, 2, 3 indices are cyclical), and finally ∂V ∂α This set of cubic equations in s i appears difficult to solve analytically; however, the first three equations can be replaced by a set of quadratic and linear equations as follows: To find the solutions to the set of equations (2.5) and (2.6) we will consider a few separate cases. A solution to the extremization conditions represents a local minimum if and only if the second derivative matrix has only positive eigenvalues. Denoting that matrix by where we defined ∆ ≡ λ s 2 Let us first apply these minimization conditions to the extrema located at the trivial solution to Eq. (2.6), s 1 = s 2 = s 3 = 0, for any α. Three of the eigenvalues of ∂ 2 V /(2∂s i ∂s j ) are equal to −m 2 Σ , while the fourth one is zero (representing a flat direction along α). Thus, there is a minimum with V (Σ) = 0 at s 1 = s 2 = s 3 = 0 provided m 2 Σ < 0.

SU (3)-symmetric vacuum
We now search for minima that have s 1 = s 2 = s 3 > 0, so that the VEV preserves an SU (3) symmetry, which is the diagonal subgroup of the SU (3) 1 × SU (3) 2 symmetry. The three equations (2.6) are then replaced by a single quadratic equation: The extremization condition (2.5) becomes sin α = 0. The phase α is further constrained by requiring stability of the potential. The second-derivative matrix shown in Eq. (2.7) has an eigenvalue equal to the 44 entry, namely s 3 1 cos α /µ Σ . Imposing that this is positive implies α = 0.
For the range of parameters where there are two solutions to the extremization conditions: Given that s i > 0, the above solution with positive sign is valid only when 3λ + κ > 0 , (3.4) while the solution with negative sign requires m 2 Σ < 0. We need to determine the regions of parameter space where these extrema satisfy the minimization conditions along the s i directions with i = 1, 2, 3. The 3 × 3 upper-left block of the second-derivative matrix shown in Eq. (2.7) may be written as follows: (3.5) where the elements of the 2 × 2 upper-left block of M 2 are given by 6) and the 33 entry of M 2 is The eigenvalues of M 2 are the squared-masses of the radial modes. SU (3) invariance implies that two eigenvalues are equal, M 2 2 = M 2 3 , because they are the squared-masses of different components (associated with the T 3 and T 8 generators) of an SU (3)-octet scalar. The third eigenvalue represents the squared-mass of an SU (3)-singlet scalar, and is given by where the + or − sign corresponds to the sign chosen for the extremum (3.3). This condition can never be satisfied by the negative solution (since m 2 Σ < 0 in that case), which thus is at most a saddle point.
The minimization condition (3.9) is automatically satisfied by the positive solution [given the constraint (3.4) in that case], so only M 2 3 > 0 remains to be imposed: For m 2 Σ > 0 we find that the positive solution from (3.3) represents a local minimum if and only if either κ ≥ 0 or else For m 2 Σ ≤ 0 the positive solution is a local minimum when −3λ < κ and λ < 0 , or − 3 2 λ < κ and λ > 0 , To derive the above conditions we used the constraints (3.2) and (3.4). The value of the potential at the SU (3)-symmetric vacuum is given by We will discuss the conditions for a global minimum in Section 7. Among the 18 degrees of freedom in Σ, there are 8 exactly massless Nambu-Goldstone Bosons (NGB's). The remaining 10 degrees of freedom are massive and can be decomposed into 8 + 1 + 1 under the unbroken SU (3) vacuum symmetry [8].
We now seek minima with two of the s i vanishing, so that the VEV preserves an SU (2) × SU (2) × U (1) symmetry. It is sufficient to set s 1 > 0 and s 2 = s 3 = 0, as this is equivalent up to SU (3) 1 × SU (3) 2 transformations to the cases s 1 = s 2 = 0 or s 1 = s 3 = 0. Another transformation, along the diagonal generators, can be used in this case to eliminate the phase α from the VEV (2.3). The extremization conditions (2.6) take a simple form, For (λ + κ)m 2 Σ > 0 the extremum is at Using the same rotation on the second derivative matrix ∂ 2 V /(2∂s i ∂s j ) as in Eq. (3.5), we find the eigenvalues The minimization condition M 2 1 > 0 is satisfied provided m 2 Σ > 0, which implies κ > −λ. As m Σ is real and its sign is irrelevant, we choose m Σ > 0. Given that M 2 Thus, an SU (2) × SU (2) × U (1)-symmetric local minimum exists at The value of the potential at this minimum is The degrees of freedom in the Σ field are grouped into 9 massless NGB's and 9 massive scalars. The latter can be decomposed into a complex scalar transforming as (2, 2, 0) under the unbroken SU (2) × SU (2) × U (1) vacuum symmetry, and a real singlet scalar.

Absence of less symmetric vacua
Let us now seek extrema with two of the s i equal but nonzero, so that the remaining symmetry of the Σ VEV is the diagonal It is sufficient to consider the case The solution s 1 = 0 to the first equation implies cos α = 0, due to the last two equations above. At this extremum, the second-derivative matrix [see Eq. (2.7)] is block diagonal, with one of the 2 × 2 blocks having the determinant equal to −s 4 2 < 0. Thus, at least one of the eigenvalues is negative so that the extremum at s 1 = 0 is only a saddle point.
The other solution to the first equation (5.2), sin α = 0, leads to more complications. One of the eigenvalues of the second-derivative matrix is given by its 44 entry, and is positive only for cos α = 1. Imposing this condition as well as the positivity condition (5.1), we find that the extremization conditions (5.2) have a solution, only for To see if the extremum (5.3) may be a minimum, we use the mass-squared matrix M 2 of Eq. (3.5), which in this case has the following nonzero elements: The determinant of M 2 is given by −(2λ+κ)s 2 2 M 4 3 , so a necessary minimization condition is 2λ + κ < 0 , (5.6) which in conjunction with (5.4) implies λ + κ < 0 and m 2 Σ < 0. Another necessary minimization condition is M 2 11 + M 2 22 > 0, which leads to The remaining minimization condition is M 2 3 > 0, implying

Asymptotic behavior
A necessary condition for the existence of a global minimum is that there are no runaway directions at large field values. In other words, V (Σ) must have a lower limit as s i → ∞.
At large field values, where the µ Σ and m Σ terms can be neglected, the potential (2.1) has the following asymptotic form: Hence, in the case where s 1 = s 2 = s 3 → ∞, the condition that V (Σ) is bounded from below is 3λ + κ > 0 (this was also derived in [9]). We point out that a separate condition for V (Σ) to be bounded from below is obtained in the case where s i → ∞ for a single value of i: λ + κ > 0. These two conditions can be combined as follows: which is a necessary condition to have V (Σ) bounded from below. We now prove that (6.2) is also a sufficient condition to have a bounded potential. For λ ≥ 0, the condition becomes κ > −λ so that For λ < 0, condition (6.2) becomes κ > −3λ > 0, which implies Therefore, (6.2) is the sufficient and necessary condition to have V (Σ) bounded from below.

Global minimum
As established in Sections 2-5, the renormalizable potential for a single bifundamental scalar allows only three possible vacua: Let us analyze which of these local minima represents a global minimum of the potential.
To this end we need to impose first the condition that V (Σ) is bounded from below, namely (6.2). In this case the regions of parameter space where the SU (3)-symmetric and SU (2) × SU (2) × U (1)-symmetric vacua exist, namely (3.13) and (4.4), are simpler. Three regions of parameter space have a single vacuum: where again we chose m Σ > 0 when m 2 Σ > 0. In the other regions there is competitions between two vacua. Studying the sign of the potential at the SU (3)-symmetric minimum, V  For the remaining region of parameter space, there is competition between the SU (3) and SU (2)×SU (2)×U (1) local minima. We need to compare the values of the potential at these minima, which are given in Eqs. (3.13) and (4.6). The SU (3) minimum is deeper, V One can check that the function defined above, ξ(λ, κ), is real and positive in this region In the gray-shaded region at κ/λ < −1 the potential is not bounded from below. of parameter space. As a result, we find the following possible vacua: SU (2)×SU (2)×U (1) local min.
The phase diagram of this model, based on Eqs. (7.2), (7.3) and (7.6), is shown in Figure 1 in the λ −1/2 µ Σ /m Σ versus κ/λ plane, for m Σ > 0 and λ > 0. Note that for λ > 0 the lower limit κ/λ > −1 is required in order to have the potential bounded from below, while there is no upper limit on κ/λ at tree level.
The region where the global minimum is SU (2) × SU (2) × U (1)-symmetric lies below the solid blue line in Figure 1, which is given by the function ξ(λ, κ)/ √ λ [see Eq. (7.5)]. In the region above or to the right of that line, the global minimum is SU (3)-symmetric.  A change of parameters that crosses the boundary between these two regions represents a first-order phase transition: both local minima exist for parameter points between the blue dashed line and the red dotted line of Figure 1. In between these two minima there is a shallow saddle point, of coordinates given in (5.3), which is SU (2) × U (1)-symmetric. In Figure 2 we show the potential for a point (µ Σ /m Σ = 0.2, κ = −0.21, λ = 1) from the ξ(λ, κ)/ √ λ curve, where the SU (3)-symmetric vacuum and the SU (2) × SU (2) × U (1)symmetric vacuum have the same depth and are global minima. The shallowness of the potential around both minima is related to the smallness of |µ Σ /m Σ | and |κ/λ|. The mass of the "angular mode" is parametrically smaller than the "radial mode".
The region where m Σ > 0 and λ < 0 has only the SU (3)-symmetric vacuum. In the phase diagram for m 2 Σ < 0, shown in Figure 3, there is competition between the SU (3) × SU (3) vacuum and the SU (3) vacuum, as described by the inequalities (7.2) and (7.3). On the boundary between the two regions defined in (7.3), given by the solid blue line in Figure 3, the two minima are degenerate. The saddle point that separates these two global minima corresponds to the negative-sign solution of Eq. (3.3).
In Figure 4, we show the potential for a point with m 2 Σ < 0, located on the boundary at µ Σ /|m Σ | = 1.8, κ = 0.12, λ = 0.2, where the depth of the potential is same at the two minima (V = 0), and at the saddle point it is given by V = 0.52 m 4 Σ . In the gray-shaded region at 3λ + κ < 0 the potential is not bounded from below.
Note that the inequalities (7.2), (7.3) and (7.6) do not explicitly refer to the cases where some parameters vanish. The reason for that is that the analysis in those cases becomes sensitive to loop corrections. 2 For example, λ = 0 at tree level makes the vertical axis ill defined in Figure 1, but 1-loop corrections would generate a nonzero λ. Likewise, m Σ = 0 is not stable against loops. By contrast, the µ Σ = 0 limit is protected by a global U (1) Σ symmetry, as discussed in Section 2.
We emphasize that although the global minimum of the potential will eventually be the vacuum, the universe might be stuck for a while in the shallower local minimum. Thus, a local minimum may be a viable vacuum provided that it is longer-lived than the age of the universe, and that the thermal history allows the universe to settle in it.

Conclusions
We have analyzed the vacuum structure of an SU (3) × SU (3)-symmetric renormalizable theory with a bifundamental scalar field. The parameter space is 4-dimensional, with two quartic couplings and two mass parameters. One of the latter, which is the coefficient of a cubic term in the potential, is not present in SU (N ) × SU (N )-symmetric theories for N = 3.
There are three possible types of vacua, with different symmetry properties: SU (3), SU (2) × SU (2) × U (1) and SU (3) × SU (3). Depending on which of these is a global minimum, and whether there are also some local minima, the parameter space is divided in 7 regions. These are described by Eqs. (7.2), (7.3) and (7.6). Remarkably, the phase diagram of the theory can be fully displayed in two-dimensional plots, namely Figures 1  and 3.
The cubic term in the potential, of coefficient µ Σ > 0, plays an important role in the selection of the possible vacua. Even when the bifundamental scalar has a positive squared-mass, i.e., m 2 Σ < 0 in the notation of Eq. (2.1), a nontrivial VEV is developed for µ Σ above a coupling-dependent value (see Figure 3), breaking the symmetry down to SU (3). For a negative squared-mass (or equivalently m Σ > 0), as µ Σ increases, the region with an SU (3)-symmetric vacuum is enlarged, while the region with an SU (2) × SU (2) × U (1)-symmetric vacuum is reduced (see Figure 1).
The vacuum structure of this theory is useful for various model building applications, including in the contexts of the ReCoM [8,9] and Trinification [15][16][17], or chiral symmetry breaking in strongly-coupled gauge theories [21]. In addition, it opens new possibilities for nonstandard cosmology, such as color-breaking in the early universe followed by color restoration at a lower temperature [22]. In particular, the presence of two minima of different symmetry properties, which for a range of parameters are nearly degenerate and separated by a shallow saddle point (see Figure 2) may lead to exotic cosmological or astrophysical phenomena.