Novikov-Shifman-Vainshtein-Zakharov beta function

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Curator: Mikhail Shifman

The Novikov-Shifman-Vainshtein-Zakharov (NSVZ) beta function describes the exact evolution law of the gauge coupling \(\alpha\) in supersymmetric Yang-Mills theories in four dimensions, valid to all orders in the coupling constant. For \({\mathcal N} = 1,\,2,\,4\) theories without matter it gives the exact all-order beta function in terms of a few integer numbers which have a geometrical meaning. In supersymmetric Yang--Mills theories with matter the NSVZ formula relates the gauge coupling beta function to anomalous dimensions of the matter fields. Among many uses of the NSVZ beta function is determination of the edges of a conformal window within Seiberg's duality.


The NSVZ formula

The exact relation between the \(\beta\) function and the anomalous dimensions \(\gamma_i\) is (Novikov et al. 1983 and 1986, Shifman et al. 1986) \[\tag{1} \beta (\alpha) = -\frac{\alpha^2}{2\pi}\left[3\,T_G -\sum_i T(R_i)(1- \gamma_i ) \right]\left(1-\frac{T_G\,\alpha}{2\pi} \right)^{-1} \, , \]

where \(T(R_i)\) is the Dynkin index in the representation \(R_i\ ,\) \[ \mbox{Tr}\, (T^a T^b ) = T(R_i)\, \delta^{ab}\, , \] and \(T^a\) stands for the generator of the gauge group \(G\ ;\) the latter can be arbitrary. Moreover, \(T_G =T({\rm adjoint})\ ,\) and the \(\beta\) function is defined as \[ \mu\,\frac{\partial\alpha}{\partial \mu}\equiv \beta (\alpha) =-\frac{\beta_0}{2\pi } \alpha^2 -\frac{\beta_1}{4\pi^2 } \alpha^3 +...,\qquad \alpha = \frac{g^2}{4\pi} \] while the anomalous dimensions of the matter fields are \[\tag{2} \gamma = -d \ln Z (\mu ) /d\ln\mu \,. \]

\(T_G\) is also known as the dual Coxeter number. The sum in Eq. (1) runs over all matter supermultiplets which interact with the given gauge field. The NSVZ formula is valid for arbitrary (super)Yukawa interactions of the matter fields. The (super)Yukawa interactions show up only through the anomalous dimensions \(\gamma_i\ .\)

History and theoretical basis

In 1981 it was observed (Vainshtein et al. 1982) in the context of (nonsupersymmetric) Yang-Mills theory that the running of the gauge coupling \(\alpha\ ,\) as it emerges in the instanton measure, has a remarkable interpretation. The first coefficient \(\beta_0\) of the Gell-Mann-Low function can be represented as (for the SU(\(N\)) gauge group) \[ b = \frac{11}{3} N = \left( 4-\frac{1}{3}\right) N\,. \] Here \(4N\) represents an antiscreening contribution, which in perturbation theory (in the physical Coulomb gauge) is associated with the Coulomb gluon exchange and has no imaginary part, while \(-N/3\) is the normal screening contribution, the imaginary part of which is determined by unitarity (Khriplovich 1970 and Appelquist et al. 1977). Within instanton calculus, the term \(4N\) is entirely due to the zero modes. It has a geometrical meaning, and its calculation is trivial. The part which is relatively hard to obtain, \(-N/3\ ,\) comes from the nonzero modes. D'Adda and Di Vecchia established (D'Adda and Di Vecchia 1978) that the nonzero modes in supersymmetric theories cancel in the instanton measure at one loop. Then it was immediately realized (Novikov et al. 1983) that the cancellation would persist to all orders, and the \(\beta\) function would be exactly calculable. Thus, the first derivation of (1) is from the exact expression for the superinstanton measure which is, in turn, a part of superinstanton calculus (Novikov et al. 1985).

In \({\mathcal N}=1\) super-Yang-Mills theory the \(\beta\) function turns out to be a geometrical progression, \[\tag{3} \beta (\alpha) = -\frac{3T_G\alpha^2}{2\pi}\left( 1-\frac{T_G\alpha}{2\pi}\right)^{-1}\, . \]

An alternative way of derivation of Eq. (3) is from the exact expression for the gluino condensate (Shifman and Vainshtein 1988, Morozov et al. 1988). Being an observable quantity, it is renormalization-group invariant.

In fact, for \({\mathcal N} = 1,\,2,\,4\) theories without matter the exact expression for the superinstanton measure implies \[\tag{4} \beta (\alpha) = - \left(n_b-\frac{n_f}{2}\right)\, \frac{ \alpha^2}{2\pi}\left[ 1-\frac{\left(n_b-n_f\right)\,\alpha}{4\pi}\right]^{-1}\, , \]

where \(n_b\) and \(n_f\) count the gluon and gluino zero modes, respectively. For \({\mathcal N}= 2\ ,\) one gets \(n_b = n_f = 4T_G\ ,\) implying that the \(\beta\) function is one-loop. For \({\mathcal N}= 4\) the \(\beta\) function vanishes since \(n_f = 2n_b\ .\) Thus, in such theories the NSVZ formula (4) makes explicit the fact that all coefficients of the \(\beta\) function have a geometric interpretation - they count the number of the instanton zero modes which, in turn, is related to the number of nontrivially realized symmetries. Even more pronounced is the geometric nature of the coefficients in the two-dimensional Kähler sigma models, for obvious reasons: these models are geometrical themselves. The supersymmetric Kähler sigma models have extended supersymmetry, \({\mathcal N}= 2\ .\) Therefore, the \(\beta\) function is purely one-loop. The instanton calculation of the first coefficient for all nonexceptional compact homogeneous symmetric Kähler manifolds was carried out in (Novikov et al. 1984, Morozov et al. 1984).

In theories with matter, apart from the gluon and gluino zero modes, one has to deal with the zero modes of the matter fermions in the instanton background. While the gluon/gluino \(Z\) factors are related to the gauge coupling constant \(g^2\) itself, this is not the case for the \(Z\) factors of the matter fermions. The occurrence of the additional \(Z\) factors brings new ingredients into the analysis, the anomalous dimensions of the matter fields \(\gamma_i\ .\) Therefore, in theories with matter the exact instanton measure implies the exact relation between \(\beta\) and \(\gamma_i\) quoted in (1). It was shown (Hori 1999) that the running gauge coupling one obtains within \(D\)-brane engineering is compatible with Eq. (1).

The NSVZ formula was obtained in a number of alternative ways: from analysis of perturbation theory (Novikov et al. 1986 and Shifman et al. 1986) and, later, from consistency of the anomalies in supersymmetric gauge theories (Shifman and Vainshtein 1986). The both derivations are based on the fact that the inverse coupling constant \(g^{-2}\) gets complexified in supersymmetric gauge theories, \[ \frac{1}{g^2}\longrightarrow \frac{1}{g^2}- i\,\frac{\theta}{8\pi^2}\, , \] and the action is holomorphic with respect to the complexified \(g^{-2}\ .\) The latter assertion refers to the Wilsonean action, while the canonic action has a holomorphic anomaly (Dixon et al. 1991, Shifman and Vainshtein 1991). Equations (1), (3) present the \(\beta\) functions for the canonic coupling; the holomorphic anomaly is responsible for the denominator in these expressions. The distinction between the gauge couplings in the Wilsonean and canonic actions was first realized in (Shifman and Vainshtein 1986). For a more recent relevant discussion see Arkani-Hamed and Murayama (2000).

The original derivation of the NSVZ $\beta$ function from the Feynman graph analysis did not use any specific regularization although it was assumed, of course, that a regularization maintaining supersymmetry exists. Recently explicit calculations were performed in generic ${\cal N}=1$ supersymmetric Yang-Mills theories based on the higher covariant derivative regularization (Pimenov, Shevtsova, and Stepanyantz 2009) which exhibited the overall structure of the NSVZ $\beta$ function in a clear-cut manner (Stepanyantz 2011).


Straightforward extensions of the methods developed in connection with the NSVZ \(\beta\) function yield a number of exacts results, for instance, renormalization of the soft supersymmetry breaking parameters (Hisano and Shifman 1997) in all orders in the gauge coupling constant, the determination of the conserved \(R\) current for the theories lying in the Seiberg conformal window (Kogan et al. 1996), exact calculation of the conformal central charges (Anselmi et al. 1998), domain wall central charges (Chibisov and Shifman 1997), and so on.

Among other applications of the NSVZ \(\beta\) function one should mention determination of an infrared conformal window within Seiberg's duality (Seiberg 1994 and 1995) and the search for finite (superconformal) gauge theories in the class of \({\mathcal N}=1\) (Parkes and West 1984, Jones and Mezincescu 1984, Hamidi et al. 1984, Leigh and Strassler 1995, Lucchesi and Zoupanos 1997, Hanany et al. 1998, Benvenuti and Hanany 2006). In the former case the edges of the cormal window are obtained from the requirement of vanishing of the numerator in (1). In the latter case, the general idea is to arrange the matter sector in such a way that the conditions \(3\,T_G - \sum_i T(R_i)=0\) and \(\gamma_i=0\) are met simultaneously.

It is worth noting that a simplified version of Eq. (1), \[\tag{5} \beta (\alpha ) = \frac{\alpha^2}{\pi} \left[1-\gamma (\alpha )\right]\,, \]

applicable in supersymmetric QED, can be extracted from a complicated construction suggested by Clark, Piguet and Sibold in the late 1970s (Clark et al. 1978, 1980 and 1982). For a derivation of (5) along the NSVZ lines see (Shifman et al. 1986).


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Further reading

  • M. A. Shifman and A. I. Vainshtein, Instantons versus supersymmetry: Fifteen years later, arXiv:hep-th/9902018, published in M. Shifman, ITEP Lectures on Particle Physics and Field Theory, (World Scientific, Singapore, 1999), Vol. 2, p. 485.
  • M. A. Shifman, Exact results in gauge theories: Putting supersymmetry to work, Int. J. Mod. Phys. A 14, 5017 (1999) [arXiv:hep-th/9906049].

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