Demonstration of Confinement and Chiral Symmetry Breaking in $SO(N_c)$ Gauge Theories

We demonstrate that $SO(N_{c})$ gauge theories with matter fields in the vector representation confine due to monopole condensation and break the $SU(N_{F})$ chiral symmetry to $SO(N_{F})$ via the quark bilinear. Our results are obtained by perturbing the ${\cal N}=1$ supersymmetric theory with anomaly-mediated supersymmetry breaking.


INTRODUCTION
Ever since quarks were proposed as fundamental constituents of the proton, neutron, and numerous hadrons by Gell-mann and Ne'eman [1,2], it has been a mystery why they cannot be observed directly in experiments. At the same time, protons and neutrons bind in atomic nuclei due to the exchange of light pions predicted by Yukawa [3]. The binding of nuclei, and correspondingly the entire world of chemistry, hinges on pions being much lighter than protons, despite the fact that they are made of the same quarks. The first mystery was "explained" by postulating confinement of quarks by condensation of magnetic monopoles via the dual Meißner effect proposed by Mandelstam [4] and 't Hooft [5]. The second mystery was "explained" by postulating chiral symmetry breaking whose the Nambu-Goldstone bosons are the light pions proposed by Nambu and Jona-Lasinio [6,7]. In either case, however, it has been a challenge to derive these properties from the fundamental theory of strong interactions, Quantum ChromoDynamics (QCD).
It has been proposed recently [8] that one can study the dynamics of gauge theories using the supersymmetric version of the theory perturbed in a specific way called anomaly-mediated supersymmetry breaking (AMSB) [9,10] (see also [11,12] for earlier work containing some important aspects of AMSB). For other analyses of non-supersymmetric gauge theories via controlled supersymmetry breaking, see, for example, [13][14][15][16][17][18][19][20][21][22][23][24][25][26], as well as the more recent [27]. When AMSB was applied to SU (N c ) QCD, it was possible to derive chiral symmetry breaking for 1 < N F ≤ 3 2 N c , while the theory flows to a conformal fixed point for 3 2 N c < N F ≤ 3N c . Yet the SU (N c ) theory does not confine in the presence of quarks in the fundamental representation because any color charges can be screened.
The SO(N c ) theory with fermions in the vector representation is interesting because it does truly confine, since the spinor representation transforming under the Z 2 center cannot be screened. Therefore, we can hope to see the interplay between the condensation of monopoles on one hand, and fermion bilinears on the other hand. It turns out that we should focus on N F ≤ N c − 2 where we can demonstrate monopole condensation.
In this Letter, we sketch the essence of the analysis, while details are presented in a forthcoming companion paper [28], that will also contain a discussion of the cases where N F > N c − 2.

ANOMALY MEDIATION
Anomaly mediation of supersymmetry breaking (AMSB) is parameterized by a single number m that explicitly breaks supersymmetry in two different ways. One is the tree-level contribution based on the superpotential Note that Eq. (1) also breaks the U (1) R symmetry explicitly. When the superpotential does not include dimensionful parameters, this expression identically vanishes. In this case, there are the loop-level supersymmetry breaking effects from the superconformal anomaly [29]. In this Letter, we do not need the loop-level effects that can be neglected in the presence of the tree-level effects (1). The loop-level effects will be discussed in the companion paper [28] for special cases when they are needed.
We consider an SO(N c ) gauge theory with N F = N c −2 flavors Q i . In the supersymmetric limit, the theory is in an abelian Coulomb phase [30]. The D-flat directions are parameterized by the diagonal entries of the mesons M ij ≡ Q i Q j . As M ij are neutral under U (1) R , no superpotential can be generated, and there is a quantum moduli space. At a generic point M ij on the meson moduli space, the gauge symmetry is higgsed to a U (1), and so the theory only has a Coulomb branch. The effective gauge coupling τ = θ 2π + i8π g 2 of the theory is given on the Coulomb branch as a function of the SU (N F ) invariant arXiv:2106.10288v2 [hep-th] 9 Sep 2021 U ≡ detM only. There are singularities at the two points Around the singular point U = U 1 , the relevant light degrees of freedom are the monopoles E ± with magnetic charges ±1, which transform under the UV global wheref (t) = t + · · · is a holomorphic function in the neighborhood of t = 0. In practice, only the leading order inf matters for the stabilization of the minimum. Using canonically normalized fields we have whereŨ = detM andM = M/Λ,Ẽ ± = E ± / √ Λ are the canonically normalized meson and monopoles, respectively. Exactly atŨ =Ũ 1 ≡ 16Λ N F , 't Hooft anomaly matching is saturated byẼ ± ,M ij , and the photinos W α ∼ W α Q Nc−2 , whose charges are given in Table I. It is easy to verify that the U (1) R gravity 2 , U (1) 3 R , and U (1) R SO(N F ) 2 anomalies all match. Therefore, we know the degrees of freedom in the IR and their Kähler potentials are regular at this singularity. AMSB generates a tree-level contribution to the scalar potential from (1), producing the global minimum atŨ = U 1 . In particular, the scalar potential alongM ij =M δ ij is given locally as Note the (kN F ) −1 factor in the second line, which comes from the Kähler term kN FM †M forM , where k is an unknown O(1) normalization factor. The tree-level AMSB contribution is given by (1), i.e., This potential has a minimum at SinceM ij =M δ ij in this minimum, the global symmetry is broken spontaneously to SO(N f ), while U (1) R is explicitly broken by AMSB, and there are no 't Hooft anomalies to match. The most remarkable feature of the minimum (6) is the condensation of monopolesẼ ± , which gives an area law to non-trivial Wilson loop operators, indicating confinement [5,31,32]. This phenomenon is well known in the context of the breaking of N = 2 Seiberg-Witten theory to N = 1 by introducing a tree level superpotential for the matter field [33]. In [19,20], monopole condenstaion was shown in a non-supersymmetric theory by introducing soft SUSY breaking on top of the superpotential term for the Seiberg-Witten model. Here, monopole condensation and SUSY breaking emerge together as a result of AMSB. Furthermore, since the global SU (N F ) symmetry is broken to SO(N F ), this is an example of confinement with chiral symmetry breaking in a non-supersymmetric theory.
In the large m limit where all scalar superpartners decouple, we can connect the chiral symmetry breaking observed here to the familiar one due to fermion bilinears. To see this, note that in the large m limit the fermion bilinears are identified with the F -component of the meson chiral superfield: In other words, our analysis demonstrates the condensation of fermion bilinears in a non-supersymmetric theory, in addition to the monopole condensate. Around the singular point U = 0 the relevant light degrees of freedom are the dyons q ± i with magnetic charge ±1, which transform under the UV global symmetry These have a dynamically generated superpotential about U = 0 of where µ is an effective mass scale, t = U Λ 4−2Nc , and f (t) is a holomorphic function in the neighborhood of t = 0, normalized so that f (0) = 1. However, the scale µ can be absorbed into the normalization of the meson field M = M/µ and the theory at this point has no dimensionful parameters. Therefore the AMSB is loop-suppressed, and hence so is the vacuum energy. Consequently, the local AMSB minimum near this singularity is not the global minimum.

MONOPOLE CONDENSATION FOR NF < Nc − 2 VIA MASS DEFORMATIONS
In the above discussion of the theory with N F = N c −2, we explicitly saw monopole condensation in the nonsupersymmetric vacuum of the theory. Here we wish to study the cases with fewer flavors, by treating the latter as the N F = N c − 2 deformed by mass terms µ, with µ Λ. In this way, we will be able to interpret the theories with fewer flavors as also corresponding to monopole condensation all the way down to the pure SO(N c ) Yang-Mills case. On the other hand, we can also study the same theory with the Affleck-Dine-Seiberg (ADS) superpotential perturbed by AMSB. They must agree if we believe in the holomorphy argument that µ, m, and Λ can be varied without a phase transition.
We begin by considering the N F = N c − 2 theory in the supersymmetric limit, with just one mass term for the last flavor, The equation of motion forM N F N F gives whereM is the matrix of the remaining mesons.
On the other hand, the extra flavor can be integrated out first to give the ADS superpotential where there is another branch on which the superpotential vanishes; we have checked that this branch does not produce the global minimum when turning on AMSB. The SUSY theory runs away and does not have a ground state. Turning on AMSB stabilizes the runaway behavior of the superpotential at a large amplitude where the Kähler potential is canonical for ϕ Λ with M ij = ϕ 2 δ ij . The tree-level AMSB is which together with the scalar potential derived from the superpotential (11) gives a minimum Fig. 1 we show the minimum of the mass-deformed theory (9) in the presence of AMSB. As can be seen in the plot, the VEV of the first N c − 3 flavors interpolates between the minimum (6) for µ = 0, and the ADS+AMSB minimum (13) with N F = N c − 3 and Λ → Λ in the large µ limit. We can see that the monopole condensate persists in the large µ limit.
To correctly reproduce the ADS+AMSB minimum, we had to interpolate the Kähler potential between the neighborhood of detM ∼Ũ 1 , where it is canonical inM , to large detM , where the Kähler potential is canonical in ϕ ∼ M Λ. More specifically, we used the following interpolating Kähler potential in the numerical study: Interestingly, for µ < m, the UV theory itself is unstable, and has a runaway atẼ +Ẽ− = 0 andM i → ∞. This is a feature of the mass term in (9) in the presence of AMSB, and is unrelated to the dynamics of the gauge theory. Since this does not affect our analysis, we follow the local minimum which is continuously connected to the global minimum for µ > m. This accounts for the small "U-turn" of the curves in Fig. 1 between the red points (µ = 0) and the blue points (µ = m). Note that our argument regarding monopole condensation in the large µ limit is completely free of this subtlety. We explicitly checked that the same conclusions hold when integrating out more than one flavor, such that 0 ≤ N F ≤ N c − 2. Similarly to the N F = N c − 3 case, for N F = N c − 4 we have another branch with vanishing superpotential, which does not produce the global minimum in the presence of AMSB. For all N F in the range 0 ≤ N F ≤ N c − 2, we find that monopole condensation persists in the AMSB global minimum. Since in the µ → ∞ limit, all of the extra flavors effectively decouple, this is a demonstration of monopole condensation for the entire range 0 ≤ N F ≤ N c − 2.

LARGER NF
For N c − 2 < N F ≤ 3 2 (N c − 2), the SUSY limit has the IR description in terms of the free magnetic IR free run-away  1: Location of the minimum in the theory with AMSB and NF = Nc − 2, deformed by a mass term 1 2 µM N F N F for the last flavor. E is the VEV of the condensed monopoles, while Mi is the common VEV of the first Nc − 3 flavors. The different curves are labeled by the value of m/Λ. The curves start at the Nc − 2 minimum (6) for µ = 0. The VEV of Mi initially decreases by a very small amount but then increases as µ crosses m. As µ/Λ → ∞, the minimum goes over to the one given by eq. (13) with N F = Nc − 3, while the VEV for the monopoles E persists. The dashed line for µ = 50Λ is the relation (10). Note that the endpoints are a tiny bit off the dashed line as a result of the interpolating Kähler potential (14). We have chosen Nc = 13 for this plot. SO(N F − N c + 4) theory with magnetic quarks and mesons. With AMSB, the global minimum is obtained when the meson matrix has full rank and the magnetic quarks are integrated out, similar to the case of SU (N c ) QCD [8]. The low-energy limit is a pure SO(N F −N c +4) SUSY Yang-Mills (SYM) with the gaugino condensate, which is known to confine. With AMSB, the fermion biliniear also acquires a VEV, breaking the SU (N F ) global symmetry to SO(N F ).
On the other hand for 3 2 N c < N F ≤ 3N c , the SUSY theory flows to conformal fixed point. AMSB effects disappear by a power law towards the fixed point and the theory recovers supersymmetry. All these phases are summarized in Table II, which are discussed in much more detail in the companion paper [28]. CONCLUSIONS We studied the dynamics of the SO(N c ) gauge theory with fermions in the vector representation using its supersymmetric version perturbed by anomaly mediated supersymmetry breaking. We obtained the exact global minimum and demonstrated that the magnetic monopole and fermion bilinear condense for N F ≤ N c − 2, leading to both confinement and chiral symmetry breaking. We see no indication that there are phase transitions when varying m, µ, and Λ, consistent with the holomorphy argument. This encouraging observation supports the case that our results may persist in the non-supersymmetric limit m Λ. To the best of our knowledge, this is the first analytic demonstration of both confinement and chiral symmetry breaking in non-supersymmetric gauge theories.