The Quark Condensate in Multi-Flavour QCD - Planar Equivalence Confronting Lattice Simulations

Planar equivalence between the large $N$ limits of ${\cal N}=1$ Super Yang-Mills (SYM) theory and a variant of QCD with fermions in the antisymmetric representation is a powerful tool to obtain analytic non-perturbative results in QCD itself. In particular, it allows the quark condensate for $N=3$ QCD with quarks in the fundamental representation to be inferred from exact calculations of the gluino condensate in ${\cal N}=1$ SYM. In this paper, we review and refine our earlier predictions for the quark condensate in QCD with a general number $n_f$ of flavours and confront these with lattice results.

Abstract: Planar equivalence between the large N limits of N = 1 Super Yang-Mills (SYM) theory and a variant of QCD with fermions in the antisymmetric representation is a powerful tool to obtain analytic non-perturbative results in QCD itself. In particular, it allows the quark condensate for N = 3 QCD with quarks in the fundamental representation to be inferred from exact calculations of the gluino condensate in N = 1 SYM. In this paper, we review and refine our earlier predictions for the quark condensate in QCD with a general number n f of flavours and confront these with lattice results.

The QCD condensate and planar equivalence
One of the challenges in theoretical studies of QCD is to find analytic, non-perturbative methods for calculations of strong-coupling quantities such as the quark or gluon condensates. Such methods are an important complement to direct evaluations using lattice gauge theory and give extra physical insight into the underlying dynamical mechanisms. One proposal is to use the special properties of supersymmetric theories to perform exact non-perturbative calculations in N = 1 gauge theories, then relate these in suitable limits to infer results in QCD itself. This approach has been pioneered in refs. [1,2].
The key idea is to exploit a remarkable property of SU (N ) QCD with a single flavour of Dirac fermions in the antisymmetric representation, 1 QCD AS , viz. that as N is varied, it interpolates between three theories of special importance -pure Yang-Mills theory, QCD with one flavour of fundamental fermions and N = 1 super-Yang Mills theory.
Specifically, for N = 2, the antisymmetric representation becomes trivial and QCD AS becomes simply SU (2) Yang-Mills. For N = 3, the antisymmetric representation (which has dimension 1 2 N (N − 1)) coincides with the fundamental representation (dimension N ) and so QCD AS (N = 3) is identical to one-flavour SU (3) QCD. 2 In the large N limit, QCD AS (N → ∞) becomes equivalent to a theory with SU (N ) gauge group and a single real fermion in the adjoint representation (dimension N 2 − 1). Crucially, this theory is supersymmetric, viz. N = 1 super Yang Mills (SYM), and this is the key to being able to perform the exact non-perturbative calculations we exploit.
The relation of QCD AS at large N with N = 1 SYM has been extensively described in a series of earlier papers on "planar equivalence" [1][2][3][4]. It has been shown that in the 't Hooft large-N limit the two theories become equivalent in the common bosonic C-parity even sector. A necessary and sufficient condition for planar equivalence to hold is that charge conjugation symmetry is not broken spontaneously [5]. This was verified by a dedicated lattice simulation [6] (see also [7]) where it was shown that charge conjugation symmetry is broken if one dimension is compactified on a small-enough circle, but is restored at large (in particular infinite) compactification radius.
In this paper, we focus on a single issue -the prediction of the value of the quark condensate in QCD, its N and n f dependence, and its confrontation with lattice data. The gluino condensate [8,9] has been evaluated exactly in N = 1 SYM [10,11] and the idea here is to use QCD AS with varying N to infer the value of the quark condensate for Table 1. Renormalization group coefficients for general N and n f for the theories considered here. Gamma is the anomalous dimension for the condensate, i.e. the running mass anomalous dimension. In general, for a (complex) representation R with n f flavours, β 0 = 11 is the quadratic Casimir for the gauge group.
one-flavour N = 3 QCD by interpolating between N = 1 SYM at large N and pure Yang-Mills at N = 2, where of course the condensate disappears. For many flavours, we consider a generalisation to a theory, QCD AS−F , with one AS representation fermion and (n f − 1) fundamentals.
The calculation of the gluino condensate λλ ≡ λ aα λ a α in N = 1 SYM relies on the holomorphy of F -terms in supersymmetric theories to analytically continue a weakcoupling, semi-classical evaluation of the condensate in a deformed version of the theory to the strong-coupling regime of N = 1 SYM itself. Specifically, in ref. [10], additional matter fields with mass m are added allowing the condensate to be calculated from the one-instanton contribution in a weak-coupling regime at small non-zero m before taking m → ∞ to decouple the extra fields and recover the original theory. In ref. [11], N = 1 SYM itself is considered on a compactified space R 3 × S (with β the radius of the compactified dimension) and the condensate is evaluated initially in the limit of small β where the theory is weakly-coupled and the condensate is dominated by contributions from monopole configurations, both conventional BPS type and additional Kaluza-Klein monopoles. Both approaches agree and, quoting the result for the condensate for an SU (N ) gauge group in terms of the scale Λ M S appropriate to SYM (see below), we have Before proceeding, we need to carefully specify our conventions and the definitions of the key quantities used below. 3 First, for ease of reference, in Tables 1 and 2 we collect the Representation (R) Table 2. The Dynkin index T (R) and quadratic Casimir C 2 (R) for various representations of SU (N ). For a representation R of SU (N ) with generators t a they are defined as tr N and n f dependence of the main group theoretical parameters and the renormalization group coefficients for the theories considered here.
Our results are presented first in terms of renormalisation group invariant quantities, written in terms of the 't Hooft coupling. We define the RG invariant scale parameter and the RG invariant condensate for a Dirac fermion ψ as Expressed entirely in terms of these RG invariant quantities, the SYM gluino condensate is therefore noting that for N = 1 SYM both γ/β 0 and 3β 1 /β 2 0 are simply 1.
One flavour, QCD AS : To determine the condensate in one-flavour QCD, we start from the QCD AS theory, where planar equivalence has been firmly established. Our basic ansatz for the QCD AS condensate is where Ψ denotes a fermion in the AS representation of SU (N ) and the appropriate RG coefficients can be read off from Table 1. The content of (1.5) is that the most significant 1/N correction to the leading large N behaviour of Ψ Ψ c as determined by planar equivalence with the exact SYM result (1.4) is given by the relative (1 − 2/N ) factor. Assuming a smooth dependence of Ψ Ψ c on N in the QCD AS theory, this is the simplest interpolating factor between the large N SYM result and the vanishing of the condensate for N = 2, where the antisymmetric representation is trivial and QCD AS degenerates to pure SU (2) Yang-Mills. Notice that this factor is simply the ratio of the Dynkin indices for the AS and adjoint representations, a feature we may conjecture to be more generally valid. The remaining sub-leading corrections are encoded in the factor K AS = 1 + O(1/N ), which we initially assume to be relatively small.
Given the arbitrariness in the normalisation of the RG invariant condensates and scale parameters, it is natural to separate the dependence on the c,c factors explicitly on the rhs of (1.5). Notice 4 that in the ratio of ratios between (1.5) and (1.4) for the AS and adjoint represention condensates, both these factors are 1 + O(1/N ) since the RG factors in the exponents are both O(1/N ) (see Table 1), so could in principle be absorbed into the K AS factor. However, this would not be appropriate since they are clearly convention dependent whereas K AS should reflect the basic O(1/N ) physics of the theory.
Our prediction for the condensate can be expressed in several ways, which will be useful for comparing with lattice data. In particular, we may write where in the latter form, Ψ Ψ M S is µ-dependent and we need to find the 't Hooft coupling λ(µ) by inverting the relation (1.2) for Λ M S . Finally, as used in ref. [4], we could express the condensate entirely in terms of the 't Hooft coupling at scale µ, viz. we ultimately wish to discuss N = 3 QCD with quarks in the fundamental representation, we considered the hybrid theory QCD AS−F , viz. QCD with one AS and (n f −1) fundamental fermions (see footnote 2).
The demonstration of planar equivalence and matching of condensates with N = 1 SYM in this case involved comparison of Wilson loops and the construction from anomalous chiral Ward identities of a 'decoupling' current, which defines a sector in which the Goldstone bosons of spontaneously broken chiral symmetry do not affect the relevant correlation functions. These theoretical considerations are described at length in [4]. Here we just quote our conclusions for the RG-invariant condensates: for the AS fermion Ψ, while for the fundamental fermions q, Once again, the K factor for the AS representation is K AS = 1 + O(1/N ), with the n f dependence contained in the O(1/N ) terms. For the fundamental representation fermions, however, we do not necessarily need to impose this. All that is actually required is self-consistency for N = 3 when the two representations coincide, i.e. K F (1/3; n f ) = K AS (1/3; n f ).
These K factors encode the sub-dominant 1/N corrections, which we conjecture to be relatively small. Our initial condensate predictions for QCD are therefore based on taking the relevant K ≃ 1 and confronting these with lattice data. Further dynamical insight and assumptions may subsequently be used to refine the prediction. For example, in ref. [4] we used the argument that QCD with n f flavours, the K factors should go smoothly to zero as the conformal window is approached to estimate their flavour dependence, finding a rather mild dependence. Ideally, lattice simulations with sufficient precision to pin down the variation of the K factors for different numbers of flavours could ultimately give information on the behaviour of the condensate near the conformal window and the nature of the transition.

Numerical predictions and lattice data
We now specialise to QCD with N = 3 and n f fundamental flavours and present numerical predictions for the qq condensate based on the formulae above. These predictions will then be critically compared with available results from lattice gauge theory.
First we need to emphasise that the result of any calculation, analytic or lattice, is a dimensionless ratio, since the overall QCD scale is the free parameter of the theory. The cleanest way to present our results is therefore in terms of the ratio qq M S of the RG-invariant condensate to the QCD scale parameter in the M S scheme for the relevant number of flavours. For ease of comparison with the lattice, we adopt here the convention of ref. [15] for the RG-invariant condensate, viz. takec = β 0 /N . We denote this condensate by Σ RGI ≡ − qq c=β 0 /N . The results of the previous section imply: Our fundamental prediction (2.1) is shown in Figure 1, where we plot the ratio of the RG-invariant condensate to Λ M S for different numbers of flavours as a function of the K F parameter. As we see below, lattice data supports the view that K F is close to 1, so for orientation we list here our predictions taking K F = 1: It is also useful to express our results in terms of the M S condensate Σ M S = qq | µ=2 GeV . This requires the relation between Λ M S and the 't Hooft coupling λ(µ). In fact, for accuracy in the numerical predictions, we do this using the three-loop RG formula, rather than two-loop expression given above, viz.
This is shown, for N = 3, in Figure 2.
From the previous section, we have the following formula for the M S condensate in N = 3 QCD: This is plotted, taking K F = 1 and evaluating at the standard scale µ = 2 GeV, in Figure  3 for the ratio Σ   To confront these predictions with lattice results, we need to be careful about interpreting scale-dependent data in variants of 'real-world' QCD in which the number of flavours is varied. These are in principle distinct theories with their own independent free scale parameter Λ M S . Only for real-world QCD (which we consider as n f = 3 light flavours with quark masses taken into account) can predictions be unambiguously linked to experimental data, allowing results to be expressed in genuine MeV units. 5 This means that the only strictly meaningful comparisons to be made are between predictions of dimensionless ratios. For our purposes, this requires comparing our predictions to a lattice calculation that self-consistently determines the ratio of the condensate to Λ M S .
While, as we discuss in the appendix, there are several evaluations in the literature of the condensate for various n f , these are usually expressed in some definition of MeV units and are not linked to a self-consistent determination of Λ M S . This makes a precision confrontation of lattice data with our planar equivalence predictions difficult.
An exception is the recent work of Engel et al. [15] and the ALPHA lattice collaboration [16] in which they quote self-consistent evaluations of both the RG-invariant condensate Σ 1/3 RGI and the scale parameter Λ (2) M S for n f = 2. The condensate is determined by studying the rate of condensation of the low eigenvalues of the Dirac operator near the limit of vanishing quark mass. For the ratio, they quote 6 Comparing with eq.(2. 2) for n f = 2, this is in quite remarkable agreement with our K F = 1 prediction of 0.763.
To illustrate this further, in Figure 5 we restrict the plot of Σ 1/3 M S (see Figure 1) to n f = 2 and superimpose our prediction for the condensate as a function of K F with the one-sigma error band of the lattice result (2.5). The lattice constraint on K F is therefore K F (1/3; n f = 2) = 1.03 (16) (2.6) in excellent agreement with the planar equivalence prediction and our understanding that the corrections to K F ≃ 1 are relatively small. 5 In practice, a compromise is usually made whereby the MeV scale for QCD with n f = 3 is set by fixing some quantity which is considered to be only relatively weakly dependent on n f to its experimental, real-world QCD, value. This is of course potentially dangerous if we are to use lattice results to determine the n f -dependence of the condensate and constrain the KF parameter. 6 In ref. [15], the results are given in terms of an auxiliary scale F as M S /F = 3.6 (2) Setting MeV units by supplementing the theory with a quenched strange quark and fixing the scale through a fit to the physical decay constant FK, they quote The challenge to the lattice is therefore to extend determinations of the quark condensate in QCD to different numbers of flavours with the accuracy required to find a real discrimination amongst different n f . Comparison with the planar equivalence predictions may also be stringently tested by simulations for different numbers of colours, N = 3, or different fermion representations. For example, in ref. [17], a lattice study of the condensate was carried out in the quenched approximation with fermions in the AS, symmetric and adjoint representations of SU (N ) for various values of N . This broadly confirms the planar equivalence expectations and in particular the result (1.5) that to leading order, the ratio of condensates for different representations is given by the ratio of their Dynkin indices. In particular, we anticipate the following expression for the quark condensate in a theory with fermions in the symmetric (S) representation: