Type I superconductivity upon monopole condensation in Seiberg–Witten theory

doi:10.1016/S0550-3213(01)00394-7

Nuclear Physics B

Volume 614, Issues 1–2, 29 October 2001, Pages 3-25

https://doi.org/10.1016/S0550-3213(01)00394-7 Get rights and content

Abstract

We study the confinement scenario in $N =2$ supersymmetric SU(2) gauge theory near the monopole point upon breaking of $N =2$ supersymmetry by the adjoint matter mass term. We confirm claims made previously that the Abrikosov–Nielsen–Olesen string near the monopole point fails to be a BPS state once next-to-leading corrections in the adjoint mass parameter taken into account. Our results shows that type I superconductivity arises upon monopole condensation. This conclusion allows us to make qualitative predictions on the structure of the hadron mass spectrum near the monopole point.

Introduction

According to Mandelstam and 't Hooft ideas [1] confinement of charges appears as a dual Meissner effect upon condensation of monopoles. Once monopoles condense the electric flux is confined in the (dual) Abrikosov–Nielsen–Olesen (ANO) vortex [2] connecting the heavy trial charge and anticharge. The vortex has a constant energy per unit length (the string tension T). This ensures that the confining potential between the heavy charge and anticharge increases linearly with their separation. However, since dynamics of monopoles is hard to control in non-supersymmetric gauge theories, this picture of confinement remained an unjustified qualitative scheme for many years.

The breakthrough in this direction was made by Seiberg and Witten in [3], [4]. Constructing the exact solution of the $N =2$ supersymmetric gauge theory they showed that the condensation of monopoles really occurs near the monopole point on the moduli space of the theory once $N =2$ supersymmetry is broken down to $N =1$ by the mass term of the adjoint matter [3].

More specifically, they considered the $N =2$ Yang–Mills theory with SU(2) as a gauge group [3]. The gauge symmetry is broken down to U(1) by the vev 〈ϕ〉=aτ₃/2 of the adjoint scalar field ϕ. The complex parameter a is the modulus on Coulomb branch of the theory. Near the monopole singularity on the Coulomb branch (the point where monopoles become massless) the effective low energy theory is the dual $N =2$ QED. This means that the theory has light monopole hypermultiplet coupled to dual photon multiplet in the same way as ordinary charges are coupled to the ordinary photon.

The $N =2$ supersymmetry is broken down to $N =1$ by adding a mass term μTrΦ² for the adjoint matter (Φ is the adjoint chiral superfield; its scalar component ϕ develops the vev discussed above). After the breaking the whole Coulomb branch shrinks to two points at which either the monopole or dyon become massless [3] (we give a brief review of the phenomenon in the next section). Consider, say, the monopole point. At this point the monopole condensate develops, its magnitude is proportional to the small parameter μ. The monopole condensation ensures U(1) confinement of trial heavy charges.

Since the work of Seiberg and Witten [3] it was quite important to understand to what extent this U(1) confinement of electric charges is similar to confinement of color we expect (but cannot control) in QCD. Moreover, we expect the QCD-like confinement also in $N =1$ supersymmetric QCD which can be obtained as a large μ limit of the theory under consideration. Unfortunately, we have no control on the dynamics of the theory in this limit (with exception of values of various chiral condensates which are known exactly at any μ [5], [6], [7]).

One distinction noticed by Douglas and Shenker [8] appears in SU(N_c) theories at N_c⩾3. Since SU(N_c) gauge group is broken down to U(1)^N_c−1 by the vevs of adjoint scalars there are N_c−1 winding numbers, one per each U(1) factor. Let us remind that the winding number n=0,±1,±2,… counts the flux of ‘magnetic’ field in the ANO vortex (it is an element of π₁(U(1))=Z). Numerous winding numbers lead to existence of too many hadronic states in the spectrum [8] (see also [9] for the brane interpretation of this result). Namely, the number of distinguishable families of $q ̄ q$ meson states produced by p-string/(p−1)-antistring pairs (i.e., objects of the type [n_i]=[0,…,0,−1,1,0,…,0]) is the integer part of (N_c+1)/2.

Another distinction, visible already in the SU(2) theory [10], is related with higher winding numbers, n>1. This also produces an extra multiplicity in the hadron spectrum if strings with higher winding numbers exist. In QCD or in the large μ limit of the present theory we expect classification of states under the center of the gauge group, Z₂ for SU(2), rather than Z.

Consider as an example the ANO vortex with two units of the flux, n=2. This string connects two quarks with two antiquarks producing an “exotic” state in the spectrum of the theory at small μ. Note, that the string with n=2, in principle, can be torn up by W boson pair creation. It does not happen while energy is less than 2m_W∼Λ which is a large quantity for μ⪡Λ (here Λ is the scale parameter of SU(2) theory). We do not expect such exotic states to appear in QCD.

The discussion above of “exotic” states in the hadron spectrum [10] is based on the purely topological reasoning. Here we consider the dynamical side of the problem. In fact, strings with multiple n are stable or unstable depending on the type of the superconductor. Namely, in the type I superconductor the tension T_n of the vortex with winding number n is less than nT₁ which is sum of tensions of n separated vortices with the unit flux. Therefore vortices with n>1 are stable and we really have a tower of “exotic” states in the spectrum.

On the contrary, in the type II superconductor vortices with n₀>1 are unstable against decay into n vortices with the unit winding number. Therefore, in this case “exotic” states are unstable and actually in the “real world” at strong coupling might not be observable at all.

The purpose of this paper is to find out the type of superconductivity at the monopole point in the SU(2) Seiberg–Witten theory perturbed by a small adjoint mass μ. The result will allow us to make a prediction about the presence of an “exotic” state in the hadron spectrum of the theory associated with multiple winding numbers of ANO vortices.

Note, that in this paper we consider the SU(2) gauge theory and discuss higher winding numbers |n|>1 in the single U(1) group. In Refs. [8], [9] the SU(N_c) gauge group is considered and p-strings with winding numbers |n_p|=1 in pth U(1) factor are studied. The string tensions are shown to be proportional to sin(πp/N_c) which is interpreted as a type I behavior of the p-string (see also the second paper in Ref. [11] for a study of a more general deformation of $N =2$ theory).

It was already noticed [9] that the mass term for the adjoint matter acts as a generalized Fayet–Iliopoulos [12] term to leading order in μ (see the next section for a brief review). To this order, although μ≠0, the extended $N =2$ supersymmetry is preserved in the effective low energy description near the monopole point. Then the superconductivity at the monopole point looks as being on the border between types I and II and the ANO string looks like BPS-saturated [13]. However, in [9] it was conjectured the BPS condition is spoiled by higher orders in the breaking parameter μ/Λ. In [11] authors came to the same conclusion.

In this paper we start by studying the effective theory in the vicinity of the monopole point. In leading order which accounts for linear in deviation terms we explicitly demonstrate that $N =2$ supersymmetry is preserved. This preservation was firstly derived in Ref. [9] within the brane construction. Our consideration shows that together with the preservation of $N =2$ supersymmetry a certain U(1) flavor symmetry is also preserved in the same order. In terms of the microscopic $N =1$ theory it is the R symmetry broken only by quantum anomalies. In the effective theory an anomalous nature of this U(1) shows up only in ‘nonperturbative’ corrections of order $μ/Λ$ .

Next we study these corrections. Taken them into account we show that $N =2$ SUSY is broken to $N =1$ and superconductivity at the monopole point turns out to be of type I. This result shows that the hadron spectrum of the theory looks very different from what we expect in QCD. As we explained above this means that the tower of “exotic” states with multiple fluxes is present in the hadron spectrum.

The paper is organized as follows. In Section 2 we review the confinement scenario near the monopole point. In Section 3 we consider the leading order perturbation in μ and show that $N =2$ supersymmetry remains unbroken to this order. In 5 The mass term perturbation, 6 Large we consider next-to-leading order corrections in μ/Λ. Section 7 contains our conclusions.

Section snippets

Monopole condensation

In this section we present a brief review on the monopole condensation in the broken $N =2$ gauge theory.

In the $N =1$ superfield notations the Seiberg–Witten solution leads to the following effective Lagrangian for the dual U(1) gauge field $V_{D} (x,θ, θ ̄)$ and its $N =2$ partner A_D(x,θ), $L_{eff} = 1 8π Im ∫ d^{2} θ τ_{D} (A_{D})W^{2}_{D} − 1 4π Im ∫ d^{2} θ d^{2} θ ̄ A ̄_{D} A(A_{D}).$ The solution is parametrized by vevs a(u)=〈A〉, a_D(u)=〈A_D〉 as functions of the modulus u=〈TrΦ²〉 where Φ is the the superfield describing the adjoint matter in the original

Fayet–Iliopoulos term in $N =2$ QED

In this section we consider the $N =2$ QED theory given by Eq. (4) and perturbed by the superpotential (9) with μ_D=0 and nonvanishing η. As it was mentioned above we neglect by “nonperturbative” corrections (a_D/Λ)^k and choose for simplicity $τ ̃_{D} (0)$ to be pure imaginary (i.e., taking the effective θ angle to be zero), $τ ̃_{D} (0)= 4πi g^{2} .$ The theory becomes the $N =2$ QED where the η perturbation appears as the $N =2$ generalization of the Fayet–Iliopoulos term. We show that although the η perturbation does

The BPS solution for string

In this section we consider the ANO string in QED with FI terms. We generalize to the $N =2$ supersymmetry (in 3+1D) the results of Ref. [16] obtained in the $N =1$ case (see also [15]).

Let us start from reminding basic facts about the ANO string in framework of the non-supersymmetric Abelian Higgs model. The model contains one complex scalar field ϕ interacting with electromagnetic field, $S_{AH} =∫ d^{4} x − 1 4g^{2} F^{2}_{μν} +| D_{μ} ϕ|^{2} − λ 2 |ϕ|^{2} −ξ^{2} .$ The field ϕ develops vev, ϕ²=ξ, the U(1) gauge group is broken. Photon

The mass term perturbation

In this section in addition to the linear in A_D term in $Δ W$ , see Eq. (9), we also switch on the mass term for A_D (the term μ_DA_D²/2 in Eq. (9)) in our low energy QED (4). It shifts the mass of the fermion field in the A_D supermultiplet, and also changes the scalar potential to the form $U(q,a)= g^{2} 2 |q|^{2} −| q ̃ |^{2}^{2} +2g^{2} q ̃ q+ i 2 η+ μ_{D} 2 a^{2} +2|a|^{2} |q|^{2} +| q ̃ |^{2} .$ In the covariant form (15) an introduction of μ_D can be viewed as an addition to the constant parameters ξ^f_g terms Δξ^f_g which are linear in the fields a and ā

Large μ_D limit

In this section we use the possibility to consider η and μ_D as an independent parameters ignoring the relations (10), of the Seiberg–Witten theory. Namely, we consider the limit of large μ_D, μ_D²⪢η. Note, that we still keep η⪡Λ² and μ_D⪡Λ in order to use the weak coupling QED description.

If the field A is heavy we can integrate it out. From (5) and (9) we get the effective superpotential depending on monopole fields $Q, Q ̃$ only (see [7], [20] where the similar integration is done for η=0) $W_{μ_{D}→∞} =− i μ_{D}$

Conclusions

In this paper we studied ANO vortices near the monopole point in the Seiberg–Witten theory perturbed by the mass of the adjoint matter. This perturbation reduces to FI F-term as well as to the mass term of the A_D field within the QED effective description. We showed explicitly that FI term does not break $N =2$ supersymmetry and the ANO string is BPS-saturated if the mass parameter μ_D=0. Then we break $N =2$ SUSY by nonzero μ_D and show that the superconductivity at the monopole point is of the type I.

Acknowledgements

Authors are grateful to Gregory Gabadadze, Amihay Hanany, Mikhail Shifman and Matthew Strassler for helpful discussions. We are thankful to the Institute for Theoretical Physics at Santa Barbara, where this work was initiated, for the support during the SUSY99 program provided by the NSF grant PHY 94-07194. A.Y. would like to thank the Theoretical Physics Institute at the University of Minnesota for hospitality and support. The work of A.V. is supported in part by DOE under the grant

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Nuclear Physics B

Type I superconductivity upon monopole condensation in Seiberg–Witten theory

Abstract

Introduction

Section snippets

Monopole condensation

Fayet–Iliopoulos term in N=2 QED

The BPS solution for string

The mass term perturbation

Large μD limit

Conclusions

Acknowledgements

Nucl. Phys. B

Phys. Lett. B

Nucl. Phys. B

Phys. Lett. B

Nucl. Phys. B

Nucl. Phys. B

Nucl. Phys. B

Nucl. Phys. B

Prog. Theor. Phys. Suppl.

Phys. Lett. B

Phys. Rev. D

Phys. Lett. B

Nucl. Phys. B

Phys. Lett. B

Phys. Rev. D

Fayet–Iliopoulos term in $N =2$ QED

Large μ_D limit