Dr. Kenichi Konishi
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The '''Konishi anomaly''' is a set of exact quantum-mechanical relations | 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 traced back to the standard chiral U(1) anomalies and to supersymmetry. The Konishi anomaly 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 is broken. It also provides a strong consistency check for any dynamical calculation | 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 traced back to the standard chiral U(1) anomalies and to supersymmetry. The Konishi anomaly 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 is broken. It also provides a strong consistency check for any dynamical calculation | ||
| − | (e.g., instantons) or approximation scheme. | + | (e.g., instantons) or approximation scheme. |
=== Supersymmetric QCD and symmetry properties of vacua === | === Supersymmetric QCD and symmetry properties of vacua === | ||
Revision as of 20:25, 30 August 2008
Contents |
Konishi Anomaly
Definition
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 traced back to the standard chiral U(1) anomalies and to supersymmetry. The Konishi anomaly 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 is broken. It also provides a strong consistency check for any dynamical calculation (e.g., instantons) or approximation scheme.
Supersymmetric QCD and symmetry properties of vacua
In supersymmetric version of Quantum chromodynamics (QCD) -- SQCD below --, the matter chiral fields are left and right handed quark superfields \[ Q_i(x), {\tilde Q}^i(x), \quad i=1,2, \ldots, N_f, \], coupled to the gauge and gaugino fields represented by the superfield
\[
W^{\alpha}(x) = - \frac{1}{4} D^2 e^{-V} D^\alpha e^V
\]
in the standard manner. The appropriate color indices are suppressed above. The Konishi anomaly reads in this theory,
:<math Kanomaly>
- \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 \;.
</math>
where \[ D \] is some supersymmetric covariant derivative. By considering the vacuum expectation value of the both sides one gets
- <math kanomVev>
m_i \, \langle Q_i(x), {\tilde Q}^i(x) \rangle = \frac{g^2}{32 \pi} \langle \lambda^{\alpha}(x) \, \lambda_{\alpha}(x) \rangle \;.
</math>
for each squark, where the right hand side represents the gaugino condensation. On the left hand side, the commonly used notation is adopted for indicating the lowest (scalar) component of the quark superfields with the same symbols as that for the superfield. Explicit instanton calculation shows that the both sides of the equation is 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 vanishing quark masses, quite unlike the standard (i.e., non-supersymmetric) QCD.
The symmetry realization pattern in SQCD is indeed quite interesting and depends on the number of the flavors (types of quarks). For Nf less than Nc, equation (<ref> kanomVev </ref>) and instanton calculation tell us that there are no vacua at finite squark VEVs (called "run-away vacua"). It is not known whether such a theory finds a useful application in a physical theory. For Nf equal to Nc, there are finite vacua, among which is the vacuum with
\[
Q_i(x), {\tilde Q}^j(x) = \delta_i^j \, \Lambda^2 \;.
\]
in which the chiral symmetry 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 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(Nf)xSU(Nf)xU(1), for Nf \ge Nc+2, remained obscure, until the work by Seiberg (1994), who showed that the system in the conformal window, 3 Nc/2 \le Nf \le 3 Nc, flows into a non-trivial infrared-fixed point -- superconformal theory, and that the system exhibits the first example of non-Abelian electromagnetic duality.
General gauge theories
The Konishi anomaly for a generic gauge theory with gauge superfield W, and matter fields \[ \Phi_i \] reads
- <math KanomalyBis>
- \frac{1]{4} {\bar D}^2 \bar{\Phi}_i e^V \Phi_i = \frac{\delta}{\delta \Phi_i} {\cal W}(\Phi) \, \Phi_i + \frac{g^2}{32 \pi} \,C_2(\Phi) \, W^{\alpha}(x) \, W_{\alpha}(x) \;, \qquad {\rm no ~ sum ~ over} \,\, i,
</math>
where \[ {\cal W}(\Phi) \] 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
Supersymmetry breaking
Indeed, in a class of chiral gage theories (i.e., theories in which the left-handed and right-handed matters belong to distinct representations of the gauge group, unlike in QCD) the relations such as equation (<ref> KanomalyBis </ref>) and explicit instanton calculations are inconsistent if supersymmetry is assumed to be unbroken. In other words, in such theories, dynamical effects are
