More on Meta-Stable Brane Configuration by Quartic Superpotential for Fundamentals

For the case where the gauge theory superpotential has a quartic term as well as the mass term for quarks, the nonsupersymmetric meta-stable brane configuration was found recently. By adding the orientifold 6-planes and the extra fundamental flavors to this brane configuration, we describe the meta-stable nonsupersymmetric vacua of the gauge theory with antisymmetric flavor as well as fundamental flavors in type IIA string theory.


Introduction
It is known that the dynamical supersymmetry breaking in meta-stable vacua [1,2] occurs in the standard N = 1 SQCD with massive fundamental flavors. The extra mass term for quarks in the superpotential implies that some of the F-term equations cannot be satisfied and then the supersymmetry is broken. The corresponding meta-stable brane realizations of type IIA string theory have been found in [3,4,5]. Very recently Giveon and Kutasov [6,7] have found the type IIA nonsupersymmetric meta-stable brane configuration where an additional quartic term for quarks in the superpotential is present. Geometrically, this extra deformation corresponds to the rotation of D6-branes along the (45)-(89) directions while keeping the other branes described in [3,4,5] unchanged. Classically there exist only supersymmetric ground states. By adding the orientifold 6-plane to this brane configuration [6], the meta-stable nonsupersymmetric vacua of the supersymmetric unitary gauge theory with symmetric flavor plus fundamental flavors is found [8].
Let us add an orientifold 6-plane and extra eight half D6-branes, located at the NS5'brane, into the brane configuration of [6] together with an extra NS5-brane and the mirrors for both D4-branes and rotated D6-branes. According to the observation of [9,10,11], this "fork" brane configuration contains the NS5'-brane embedded in an O6-plane at x 7 = 0. This NS5'-brane divides the O6-plane into two separated regions corresponding to positive x 7 and negative x 7 . Then RR charge of the O6-plane jumps from −4 to +4. Furthermore, eight semi-infinite D6-branes are present in the positive x 7 region. This is necessary for the vanishing of the six dimensional anomaly. Then the type IIA brane configuration consists of two NS5-branes, one NS5'-brane, D4-branes, rotated D6-branes, an O6-plane and eight half D6-branes. We'll see how the corresponding supersymmetric gauge theory, which is a standard N = 1 SQCD with massive flavors together with the extra matters, occurs in the context of dynamical supersymmetry breaking in meta-stable vacua.
In this paper, we study N = 1 SU(N c ) gauge theory with an antisymmetric flavor A, a conjugate symmetric flavor S, N f fundamental flavors Q and Q and eight fundamental flavorŝ Q in the context of dynamical supersymmetric breaking vacua. Now we deform this theory by adding both the mass term and the quartic term for quarks Q, Q in the fundamental representation of the gauge group [6]. Then we turn to the dual magnetic gauge theory [12]. The dual magnetic theory giving rise to the meta-stable vacua is described by N = 1 SU(2N f − N c + 4) gauge theory with dual matter contents. The difference between the brane configuration of [12] and the brane configuration of this paper is that the D6-branes are rotated in the (45)-(89) directions. By analyzing the magnetic superpotential, along the line of [6,7], we present the behaviors of gauge theory description and string theory description for the meta-stable vacua. In section 2, the type IIA brane configuration corresponding to the electric theory based on the N = 1 SU(N c ) gauge theory with above matter contents is given. In section 3, we construct the Seiberg dual magnetic theory which is N = 1 SU(2N f − N c + 4) gauge theory with corresponding dual matters. The rotation of D6-branes is encoded in the mass term for the meson field in the superpotential. In section 4, the nonsupersymmetric meta-stable minimum is found and the corresponding intersecting brane configuration of type IIA string theory is presented. In section 5, we comment on the future works.
Then the origin of the coordinates (x 6 , v, w) is located at the intersection of x 6 coordinate and O6-plane. The left NS5 L -brane is located at the left hand side of O6-plane while the right NS5 R -brane is located at the right hand side of O6-plane. The N c color D4-branes are suspended between NS5 L -brane and NS5 R -brane. Moreover the N f D6-branes are located between the NS5 L -brane and the middle NS5'-brane and its mirrors N f D6-branes are located between the middle NS5'-brane and the NS5 R -brane. The antisymmetric and conjugate symmetric flavors A and S are 4-4 strings stretching between D4-branes located at the left hand side of O6-plane and those at the right hand side of O6-plane, N f fundamental flavors Q and Q are strings stretching between N f D6-branes and N c color D4-branes and eight fundamental flavorsQ are strings stretching between eight half D6-branes which are on top of O6 − -plane and N c color D4-branes.
Let us deform this theory which has vanishing superpotential by adding both the mass term and the quartic term for N f fundamental quarks. The former can be achieved by "translating" the D6-branes along ±v direction leading to their coordinates v = ±v D6 [14] while the latter can be obtained by "rotating" the D6-branes [6] by an angle θ in (w, v)-plane. We denote them by D6 θ -branes which are at angle θ with undeformed unrotated D6-branes(0123789).
Then their mirrors N f D6-branes are rotated by an angle −θ in (w, v)-plane according to O6plane action and we denote them also by D6 −θ -branes 1 . Then, in the electric gauge theory, the deformed superpotential is given by where Λ is related to the scales of the electric and magnetic theories and ±v D6 is the v coordinate of D6 ∓θ -branes. Due to the last term, the flavor symmetry SU( Here the adjoint mass µ ≡ tan( π 2 − ω) is related to a rotation angle ω of NS5 L,R -branes in (w, v)-plane. In the limit of µ → ∞(or no rotations of NS5-branes ω → 0), the terms of 1 µ in (2.1) vanish. Let us summarize the N = 1 supersymmetric electric brane configuration with nonvanishing superpotential (2.1) in type IIA string theory as follows and draw it in Figure 1: • Two NS5-branes in (012345) directions with w = 0 By moving the D6 ±θ -branes from Figure 1 into the outside of NS5 L,R -branes, there exist N f flavor D4-branes connecting D6 ±θ -branes and the NS5 L,R -branes, and the gauge singlet field N appears. At energies much below the mass of N, the two brane descriptions coincide with each other. One can think of this new brane configuration as integrating the field N in from Figure 1 and the superpotential of this electric theory contains the interaction term between N with electric quarks, quadratic term and linear term for N [6]. The classical supersymmetric vacua of this brane configuration are characterized by the parameter k where k = 0, 1, · · · , N c and unbroken gauge symmetry in the k-th configuration is SU(N c − k). That is, the k D4-branes among N f D4-branes (stretched between NS5 R -brane and D6 −θ -branes) are reconnecting with those number of D4-branes stretched between the middle NS5'-brane and NS5 R -brane. Then those resulting k D4-branes are moving to ±v direction and the remaining (N c − k) D4-branes are stretching between the middle NS5'-brane and NS5 Rbrane and (N f − k) D4-branes are stretched between the NS5 R -brane and D6 −θ -branes(and their mirrors).  6 , v, w) is located at the intersection of x 6 and O6-plane. It is evident that the two deformations are characterized by both translation and rotation for D6-branes. As in [12], a combination of a middle NS5'-brane, O6 + -plane, O6 − -plane and eight half D6-branes is represented by NS5 ′ /O6/D6.

The N = 1 supersymmetric magnetic brane configuration
The magnetic theory is obtained by interchanging the D6 ±θ -branes and NS5 L,R -branes while the linking number is preserved. After one moves the left D6 θ -branes to the right all the way(and their mirrors, right D6 −θ -branes to the left) past the middle NS5'-brane and the right NS5 R -brane, the linking number counting [12] implies that one should add N f D4branes, corresponding to the meson M(≡ Q Q), to the left side of all the right N f D6 θbranes(and their mirrors). Note that when a D6-brane crosses the middle NS5'-brane, due to the parallelness of these, there was no creation of D4-branes. That is, when the D6 ±θ -branes approach the middle NS5'-brane, one should take θ = 0 limit(making D6-branes to be parallel to the middle NS5'-brane) and then after they cross the middle NS5'-brane, they return to the original positions given by D6 ±θ -branes as follows: Next, let us move the left NS5 L -brane to the right all the way past O6-plane (and its mirror, right NS5 R -brane to the left), and then the linking number counting [12] leads to the fact that the dual number of colors was (2N f − N c + 4). There was a creation of D4-branes when the NS5-brane crosses an O6-plane because they are not parallel to each other. From this, the constant term 4 in the dual color above arises, compared with the case of [15]. Now we draw the magnetic brane configuration in Figure 2 where some of the flavor D4-branes are recombined with those of (2N f − N c + 4) color D4-branes and those combined resulting D4-branes are moved into ±v direction. One takes k D4-branes from N f flavor D4-branes and reconnect them to those from (2N f − N c + 4) color D4-branes in Figure 2 such that the resulting branes are connecting from the D6 θ -branes to the NS5 L -brane directly. Their coordinates between D6 θ -branes and k D4-branes will be v = −v D6 in order to minimize the energy. This Figure 2 also can be obtained from the magnetic brane configuration of [6] by adding O6-planes and eight half D6-branes with the right presence of mirrors under the O6-plane action. Then the low energy dynamics is described by the dual magnetic theory with gauge group SU(2N f −N c +4) and this theory is higgsed down to SU(2N f −N c +4−k) in the k-th vacuum by nonzero vacuum expectation value for dual quarks which is determined later. The matters are N f flavors of fundamentals q, q, an antisymmetric flavor a, a conjugate symmetric flavor s, eight fundamentalsq, gauge singlet M which is magnetic dual of the electric meson field Q Q and other gauge singlet M that isQ Q. Then the superpotential including the interaction between the meson field M and dual matters with µ → ∞(ω → 0) limit is described by where the second and third terms originate from the two deformations (2.1), and the fourth term also comes from electric theory. The θ-dependent coefficient function, α, in front of quadratic term of the meson field also occurs in the geometric brane interpretation for the different supersymmetric gauge theory [16]. The α = 0 limit reduces to the theory given by [12]. The parameters α and m are the same as before in (2.1) 2 .
2 For arbitrary rotation angles of D6 ±θ -branes and N S5 ±ω -branes, there exist also other meson fields containing A or S:M 1 ≡ Q SA Q, P ≡ Q SQ and P ≡ QA Q. They couple to the dual quarks and other flavors in the superpotential via M 1 q q + P q sq + P qa q. As emphasized in [12], the geometric constraint (3.1) at the intersection between D6-branes and the middle NS5'-brane removes the presence of these gauge singlets, M 1 , P and P . That is, when D6 ±θ -branes are crossing the middle NS5'-brane, they do not produce any D4-branes because they are parallel at the origin x 6 = 0. This implies there is no M 1 term in the magnetic superpotential. In general, P and P -terms arise when D6 θ -branes intersect with its mirrors D6 −θ -branes around the origin x 6 = 0. But they are also parallel to each other due to (3.1). This leads to the fact that there are no P or P -terms in the magnetic superpotential. Therefore, we are left with (3.2).
k's eigenvalues 0 and (N f − k)'s eigenvalues m α : where k = 1, 2, · · · , N f and 1 N f −k is the (N f −k)×(N f −k) identity matrix [7]. The expectation value of M is represented by the fundamental string between the flavor brane displaced by the w direction and the color brane from Figure  Another deformation arises when we rotate the NS5 L,R -branes by an angle ±θ ′ in the (v, w)-plane. This rotation provides an adjoint field of SU(2N f − N c + 4) and couples to the magnetic dual matters. Integrating this adjoint field out leads to the fact that there exists a further contribution to the quartic superpotential for q and q. In particular, when the rotated NS5 ±θ ′ -branes are parallel to rotated D6 ±θ -branes, the coupling in front of M 2 in the magnetic superpotential vanishes.

Nonsupersymmetric meta-stable brane configuration
The theory has many nonsupersymmetric meta-stable ground states besides the supersymmetric ones we discussed in previous section. For the IR free region [12], the magnetic theory is the effective low energy description of the asymptotically free electric gauge theory. When we rescale the meson field as M = hΛΦ, then the Kahler potential for Φ is canonical and the magnetic quarks are canonical near the origin of field space. The higher order corrections of Kahler potential are negligible when the expectation values of the fields q, q, a, s and Φ are smaller than the scale of magnetic theory. Then the magnetic superpotential can be written in terms of Φ(or M) From this, one can read off the following quantities The classical supersymmetric vacua given by (3.4) and ( Figure 3. Both Φ n and ϕ ββ ϕ are n × n matrices. The supersymmetric ground state corresponds to the vacuum expectation vaules given by hΦ n = µ 2 µ φ 1 n , ϕ β = 0 = γ ϕ. Now the full one loop potential for Φ n ,φ ≡ ϕ β,ˆ ϕ ≡ γ ϕ [12] takes the form V |h| 2 = |Φ nφ | 2 + |Φ nˆ ϕ| 2 + |φˆ ϕ − µ 2 1 n + hµ φ Φ n | 2 + b|hµ| 2 tr Φ † n Φ n , where b is given by b = (ln 4−1) 8π 2 (2N f − N c + 4). Differentiating this potential with respect to Φ n and puttingφ = 0 =ˆ ϕ, one obtains for real µ and we assume here that µ φ << µ << Λ m . The vacuum energy V is given by V ≃ n|hµ 2 | 2 . Expanding around this solution, one obtains the eigenvalues for mass matrix forφ andˆ ϕ will be Then for | µ φ µ | 2 > |bh| 2 1−b|h| 2 ≃ |bh| 2 in order to avoid tachyons the vacuum (4.1) is locally stable. One can move n D4-branes, from (N f − k) D4-branes stretched between the NS5'-brane and the D6 θ -branes at w = v D6 cot θ, to the local minimum of the potential and the end points of these n D4-branes are at a nonzero w as in Figure 3. The remaining (N f − k − n) flavor D4-branes between the D6 θ -branes and the NS5'-brane are related to the corresponding eigenvaules of hΦ, i.e., µ 2 µ φ 1 N f −k−n . The coordinate of an intersection point between the (N f − k − n) D4-branes and the NS5'-brane is given by (v, w) = (0, v D6 cot θ). Finally, the remnant n "curved" flavor D4-branes between the D6 θ -branes and the NS5'-brane are related to the corresponding eigenvaules of hΦ n by (4.1). Note that since µ φ , the n D4-branes are nearer to the w = 0 located at the NS5 L -brane.
As explained in [6], this local stable vacuum decays to the supersymmetric ground states. The end points of n "curved" flavor D4-branes on the NS5'-brane approach those of the (2N f − N c + 4 − k) color D4-branes and two types of branes reconnect each other. For n ≤ (2N f − N c + 4 − k), the final brane configuration is nothing but the supersymmetric vacuum of Figure 2 with the replacement k → (k + n). When n > (2N ] flavor D4-branes remain. On the other hand, the n D4-branes can move to larger w and return to the Figure 2. Also some of the D4-branes approach the intersection point between D6 θ -branes and the NS5'-brane while the remaining D4-branes move to the one between D6 θ -branes and the NS5 L -brane. When D6 ±θ -branes are moved to the place between the NS5'-brane and the NS5 L,Rbranes, one gets the Figure 4 where the previous (N f − k) D6-branes in Figure 3 that were not connected to the NS5 L,R -branes, through the flavor D4-branes, are now connecting to the NS5 L,R -branes by the same number of D4-branes while the previous k D6-branes in Figure   3 that were connected to the NS5 L,R -branes, through the flavor D4-branes, are now not connecting to the NS5 L,R -branes. The former corresponds to a creation of D4-branes and the latter corresponds to an annihilation of D4-branes due to the Hanany-Witten transition [17,14]. In the remaining part, we focus on the gravitational potential of the NS5 L -brane. Let us remind that the branes are placed as follows: where we assume that D6 θ -branes are located at the nonzero y ≡ x 6 and the NS5 ′ -brane is located at y = 0(the origin). The dependence of the distance between the D6 θ -branes and NS5 ′ brane along the NS5 L -brane on w can be represented by Then the partial differentiation of ∆x with respect to w leads to an extra contribution for the computation of ∂ w θ i,w where we define θ i,w as with y m that is smallest value of y along the D4-brane. Then the energy density of the D4-brane was computed in [18,19] and is given by |w| 2 sin 2θ 1,w + (y 2 + |w| 2 ) sin 2θ 2,w (4.4) corresponding to (3.7) of [6] where τ 4 is a tension of D4-brane in flat spacetime.
Furthermore, the energy E(w) is zero from (4.4). This corresponds to the global minimal energy. For the parallel D6-branes and NS5'-brane(i.e., tan θ = 0), then the only stationary point is w = 0 [12]. If w = 0, then sin θ 1,w = 0 = sin θ 2,w from the first term of (4.8) but these are not physical solutions.
(4.9) Therefore, the brane configuration of Figure 4 has a local minimum where the end of D4brane are located at w given by (4.9). When the y goes to zero(θ 1,w = θ 2,w ≡ θ w ), one can approximate (4.9) and one gets tan θ w ≃ w tan θ 2ℓ s .
The gauge theory result is valid only when θ and v D6 ℓs are much smaller than g s while the classical brane construction with (4.9) is valid for any angle and the length parameters are of order ℓ s or larger.

Conclusions and outlook
In this paper, by adding the orientifold 6-planes and the extra fundamental flavors to the brane configuration [6], we have described the meta-stable nonsupersymmetric vacua of the gauge theory with antisymmetric flavor as well as fundamental flavors, through the Figures 3  and 4, from type IIA string theory.
It would be interesting to deform the theories given in [20,19,21,22,23] where one of the gauge group factor has the same matter contents as the one of the present paper and see how the meta-stable ground states appear. Along the lines of [18,19,24,21,23], when the D6 θ -branes are replaced by a single NS5 θ -brane, it would be interesting to see how the present deformation arises in these theories. It is also possible to deform the symplectic or orthogonal gauge group theory with massive flavors [25] by adding an orientifold 4-plane. Similar application to the product gauge group case [26,27,28] is also possible to study. It is an open problem to see how the type IIB description [29] is related to the present work. To construct a direct gauge mediation [30,31] for the present work is also possible open problem.