Konishi anomaly

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Kenichi Konishi (2009), Scholarpedia, 4(3):7121. doi:10.4249/scholarpedia.7121 revision #91409 [link to/cite this article]
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Curator: Kenichi Konishi

The Konishi anomaly is a set of exact quantum-mechanical relations involving various composite operators in supersymmetric gauge theories. They contain anomalous terms as compared to what is expected from the classical field equation of motion. This anomaly can be understood as the combined effects of the standard \( U(1) \) chiral anomalies and of supersymmetry. The Konishi anomaly (in Konishi, 1984, see also Clark, et. al. 1979, and Gates, et. al. 1983) yields powerful constraints on the possible dynamical properties of the system, such as the way the global symmetry is realized at low energies, or whether or not supersymmetry itself is spontaneously broken. It also provides a strong consistency check for any dynamical calculation (e.g., instanton calculation) or approximation scheme.

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Konishi Anomaly

Supersymmetric QCD and symmetry properties of vacua

In the supersymmetric version of Quantum chromodynamics (QCD) with \(N_c\) colors (i.e. \(SU(N_c)\) gauge group), called Super Quantum chromodynamics (SQCD) below, the matter fields are contained in quark and antiquark chiral superfields \( Q_i(x), {\tilde Q}^i(x), \quad i=1,2, \ldots, N_f \ .\) These are coupled to the gauge and gaugino fields contained in the superfield \[ W^{\alpha}(x) = - \frac{1}{4} {\bar D}^2 e^{-V} D^{\alpha} e^V, \] in the standard manner, where \(D, \bar{D} \) are the supersymmetric covariant derivatives. The appropriate color indices are suppressed above. The Konishi anomaly reads in this theory, \[\tag{1} - \frac{1}{4} {\bar D}^2 {\bar Q}_i e^V Q_i + m_i \, Q_i(x) \, {\tilde Q}^i(x) + \frac{g^2}{32 \pi} W^{\alpha}(x) \, W_{\alpha}(x) =0 \;, \]

where repeated \( \alpha\) indices are summed, but no sum over repeated \(i\) indices is assumed. By considering the vacuum expectation value (VEV) of both sides one gets \[\tag{2} m_i \, \langle \, Q_i(x){\tilde Q}^i(x) \, \rangle = \frac{g^2}{32 \pi} \langle \lambda^{\alpha}(x) \, \lambda_{\alpha}(x) \rangle \; \]

for each squark, where the right hand side represents the gaugino condensate. On the left hand side, the commonly used notation is adopted for indicating the lowest (scalar) component of the quark superfields (squarks) with the same symbols as used for the superfield. Explicit instanton calculation shows that both sides of the equation are non-vanishing in general. Thus the squark condensates which determine the symmetry breaking pattern is determined by the quark mass ratios, even in the limit of vanishingly small quark masses, quite unlike the standard (i.e., non-supersymmetric) QCD.

The symmetry realization pattern in massless SQCD is indeed quite interesting and depends on the number of the flavors (types of quarks). For \( N_f \) less than \( N_c \ ,\) equation (2) and instanton calculation, together, tell us that there are no vacua at finite squark VEVs (this phenomenon is known as the "run-away vacua"). It is not known whether such a theory finds a useful application in a physical theory. For \( N_f\) equal to \( N_c\ ,\) there are finite-VEV vacua, among which is the vacuum with \[ \langle Q_i(x) {\tilde Q}^j(x) \rangle = \delta_i^j \, \Lambda^2 \] in which the chiral symmetry \( SU(N_f) \times SU(N_f) \times U(1)\) is broken to \( SU(N_f) \times U(1)\ .\) Theories with larger number of flavors do not generate instanton-induced potentials: the vacuum degeneracy of the theory remains intact, and in particular for each \(N_f>N_c\) there is a vacuum in which chiral symmetry of the underlying theory is unbroken.

Nevertheless, the question of how the system realizes dynamically the full chiral symmetry of the underlying theory, \(SU(N_f)\times SU(N_{f})\times U(1)\ ,\) for \(N_{f} \ge N_{c}+2, \) remained obscure, until the pioneering work by Seiberg in (Seiberg, 1995). He showed that, at the origin of the space of vacua (called the vacuum moduli space) and in the cases \( 3 N_{c}/2 \le N_{f} \le 3 N_{c}, \) the system flows into a non-trivial infrared-fixed point (superconformal) theory, and that the system exhibits an example of interacting non-Abelian electromagnetic duality.

General gauge theories

The Konishi anomaly for a generic gauge theory with gauge superfield \( W \) and the matter fields \( \Phi_i \) reads \[\tag{3} - \frac{1}{4} {\bar D}^2 \, \bar{\Phi}_i \, e^V\, \Phi_i = \frac{\delta { P}}{\delta \Phi_i} \, \Phi_i + \frac{g^2}{32 \pi} \,C_2(\Phi) \, W^{\alpha}(x) \, W_{\alpha}(x) \;, \qquad {\rm no ~ sum ~ over} \,\, i, \]

where \( { P}(\Phi) \) is the superpotential and \( \,C_2(\Phi) \) is the Dynkin index for the representation according to which the field \( \Phi \) transforms. The formula holds for each matter field, and by considering the vacuum expectation we find \[\tag{KanomalyBis:label exists!} \langle \frac{\delta { P}}{\delta \Phi_i} \, \Phi_i \rangle + \frac{g^2}{32 \pi} \,C_2(\Phi) \, \langle W^{\alpha}(x) \, W_{\alpha}(x) \rangle =0 \;, \]

for each field, meaning that there are many nontrivial relations among the chiral condensates.

The Konishi anomaly, (1) or (3), plays a key role in the determination of the exact beta function in supersymmetric QCD as in (Novikov, et. al., 1986), in the determination of the exact anomalous dimensions of various composite operators in suypersymmetric theories, for instance, see (Leigh and Strassler, 1995). These results are then used in the study of quantum superconformal theories, in particular, in the study of \( N=4 \) supersymmetric gauge theories.

Supersymmetry breaking

Indeed, in some class of chiral gauge theories (i.e., theories in which the left-handed and right-handed matter fields transform according to different representations of the gauge group, unlike in QCD), the relations such as the ones above (3) and explicit instanton calculations are shown to be consistent with each other, only if supersymmetry is assumed to be spontaneously broken. The simplest examples of such systems are found in chiral \(SU(2 N +1)\) gauge theories (see Meurice, et. al. 1984), but it is easy to construct other examples. As supersymmetry, if relevant in our physical world, must be spontaneously broken -- no supersymmetric partner of the electron or of the proton (degenerate in mass with these) are known -- this types of models may become an important part of our understanding of physical world.

Supergravity and gravitational anomaly

The Konishi anomaly can be derived also for supergravity models. In these theories the basic degrees of freedom contains graviton, gravitino and chiral superfields \( (\phi, \chi) \) (ignoring here the possible gauge fields and gauge fermions). It reads \[ {\bar \delta} \, ( {\bar \chi} \phi ) = \frac{\kappa^{2}}{384 \pi^{2}} \psi_{ab} \psi_{ab} + \ldots, \] where \( \psi_{ab}= e_{a}^{\mu} e_{b}^{\nu} (D_{\mu} \psi_{\nu}- D_{\nu} \psi_{\mu} )\) is the gravitino field. The one-loop calculation around the gravitational instanton (in this case, the Eguchi-Hanson instanton) shows that the gravitino condensate forms, showing that supersymmetry is indeed spontaneously broken by the quantum gravitational effects in any such theory. See for instance Konishi, et.al. (1988).

Generalized Konishi anomaly

More recently, a generalized form of the Konishi anomaly relations has been derived and used to solve the \( {N}=1 \) supersymmetric gauge theories with a generic superpotential of the form \[ { P}(\Phi, Q, {\tilde Q}) = W (\Phi) + {\tilde Q} \, m (\Phi) \, Q, \] where \(\Phi\) is a scalar multiplet in the adjoint representation of the gauge group, \( Q, {\tilde Q} \) represent the quark multiplets, and the mass matrix \( m(\Phi) \) can be a nontrivial function of \( \Phi\ .\) In a remarkable series of papers, Cachazo, Seiberg, Witten and others have shown how the detailed dynamical information on the vacuum (chiral condensates, symmetry breaking pattern, etc.) can be determined from these generalized Konishi anomaly relations. Indeed, the whole set of chiral condensates (vacuum expectation values of chiral composite operators) are encoded in the resolvent sets, \[ M= {\tilde Q} \, \frac{ 1 }{ z - \Phi} \, Q; \qquad R(z)= - \frac{ 1}{ 32 \pi^{2}}\,{\rm Tr}\, \frac{W_{\alpha}\, W^{\alpha} }{ z - \Phi}. \] where \( z \) is a complex parameter. The coefficients of the various inverse powers of \( z \) give the desired condensates. The solution of the anomaly equations for \( R(z) \) is (as proven by Cachazo, et.al. 2002) \[ 2 \, R(z) = W^[[:Template:\prime]](z) - \sqrt{ W^[[:Template:\prime]](z)^{2} + f(z) } \] where \( f(z) \) is directly related to the gaugino condensates in the strong \( \prod U(N_{i}) \) sectors. Once \( R(z) \) is known, all other condensates are determined by the relations, \[ \left[W^{\prime}(z)\, R(z)\right]_-=R(z)^2; \ :\]

\( 

  \left[(M(z) \, m(z))_i^j\right]_- = R(z)\, \delta_i^j \, ; \qquad \left[(m(z)\, M(z))_i^j\right]_- = R(z)\, \delta_i^j.

\) 

where the notation \( \left[O(z)\right]_- \) stands for keeping only the negative powers in the Laurent expansion of \( O(z)\ .\)

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