The scalar glueball operator, the a-theorem, and the onset of conformality

We show that the anomalous dimension $\gamma_G$ of the scalar glueball operator contains information on the mechanism that leads to the onset of conformality at the lower edge of the conformal window in a non-Abelian gauge theory. In particular, it distinguishes whether the merging of an UV and an IR fixed point -- the simplest mechanism associated to a conformal phase transition and preconformal scaling -- does or does not occur. At the same time, we shed light on new analogies between QCD and its supersymmetric version. In SQCD, we derive an exact relation between $\gamma_G$ and the mass anomalous dimension $\gamma_m$, and we prove that the SQCD exact beta function is incompatible with merging as a consequence of the $a$-theorem; we also derive the general conditions that the latter imposes on the existence of fixed points, and prove the absence of an UV fixed point at nonzero coupling above the conformal window of SQCD. Perhaps not surprisingly, we then show that an exact relation between $\gamma_G$ and $\gamma_m$, fully analogous to SQCD, holds for the massless Veneziano limit of large-N QCD. We argue, based on the latter relation, the $a$-theorem, perturbation theory and physical arguments, that the incompatibility with merging may extend to QCD.


Introduction
In Quantum Chromodynamics (QCD) with a few massless fermions conformal symmetry is lost in a highly non trivial way. One single breaking phenomenon manifests itself in two forms: asymptotic freedom and confinement 1 -one cannot exist without the other. The recently proposed solution [1] for the scalar glueball two-point function in the 't Hooft limit of large-N QCD entails this property.
A new family of gauge theories is known to arise from QCD, for a sufficiently large number N f of massless fermions in the fundamental representation, as well as in other representations of the gauge group. Such family is called the conformal window, ranging from N c f where the zerotemperature theory deconfines and chiral symmetry is restored, to N U V f where ultraviolet freedom is lost. Theories for N c f < N f < N U V f have a non-trivial, i.e., interacting infrared fixed point (IRFP) where they are conformal. A conformal window also arises in supersymmetric versions of non-Abelian gauge theories [2]. Above the conformal window N f > N U V f , infrared freedom leads to the possibility of realising theories with an additional non-trivial ultraviolet fixed point (UVFP) [3]. For these reasons, all theories with N f > N c f may lead to new paradigms for particle dynamics beyond the standard model.
Just below the conformal window N f N c f , the phenomenologically interesting possibility of a preconformal behaviour characterised by a walking, i.e., slow-running 2 gauge coupling has been proposed [4,5]. Theories with a preconformal behaviour would not differ from QCD as far as their fixed point structure is concerned, i.e., they must be confining and asymptotically free 3 . The preconformal behaviour is directly related to the nature of the mechanism that opens the conformal window at N c f , and it should in some way affect the evolution from the UV to the IR of observables. It has been shown that a conformal phase transition [6][7][8] -equivalently a Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two-dimensional spin systems [9][10][11] -leads to the walking phenomenon for N f N c f , and the associated preconformal behaviour of physical observables known as Miransky scaling or BKT scaling [6][7][8][9][10][11]. Interestingly, it was observed in [12] that a gauge coupling beta function leading to the loss of conformality at N c f via the merging of a pair of (UV and IR) fixed points is a simple way to realise this preconformal scaling. An interesting and still open question is if a conformal phase transition with its preconformal scaling can be realised without the need for a UV-IR fixed point merging mechanism at N c f . Ultimately, is a conformal phase transition realised at all in QCD? A phase transition of another nature at N c f , for example a first order one [13], is allowed and it would not lead to precursor effects.
In this letter we show that the anomalous dimension of the scalar glueball operator is a powerful probe of the mechanism for the loss of conformality at N c f ; its N f dependence along the IRFP line inside the conformal win-2 At least on a finite energy range [µ IR , µ U V ]. 3 In other words, no phase transition is expected to occur between QCD and preconformal theories with N f N c f at zero temperature.
dow reveals if a UV-IR fixed point merging does or does not occur. The following section 2 provides the anomalous dimension of the scalar glueball operator in terms of the beta function of the gauge coupling at a fixed point; this exact formula is valid for a non-Abelian gauge theory with fermions in any representation. In section 3, by means of this simple formula, we derive the scalar glueball anomalous dimension in perturbation theory, along the IRFP line for N f > N c f fermions in the fundamental representation, we compare with large-N predictions and discuss analogies and differences between QCD and supersymmetric QCD. We then show in section 4 how the prediction of a UV-IR fixed-point merging mechanism differs from the previous results. Inspired by the recent bounds on the location of the lower edge of the conformal window of QCD [14], we can now critically compare the implications of perturbative arguments with those of other proposed scenarios for the conformal window of QCD. We discuss this and prospects in the concluding section.

The scalar glueball operator and its anomalous dimension
It is well known that the anomalous dimension of the scalar glueball operator Tr(G 2 ) ≡ G a µν G aµν is constrained by the trace anomaly, i.e., the nonzero contribution to the trace of the energy-momentum tensor. The trace anomaly of QCD that enters the matrix elements of renormalised gauge invariant operators is 4 with the beta function The dimension of a quantum operator O is dictated by the scaling equation The nonrenormalisation of T µ µ implies that it scales classically, i.e., d T µ µ = 4 in four dimensions, and the scaling equation (3) applied to equation (1) gives for with β (α) the derivative of the beta function with respect to α. At a fixed point, β(α * ) = 0 and the anomalous dimension is a physical property of the system, renormalisation scheme independent.

Perturbative Results
It is instructive to determine γ G in perturbation theory, where the beta function is known to a given loop order inside the conformal window. Later we compare this result with the Veneziano limit of large-N QCD (N → ∞, N f /N = const) [18]. The perturbative beta function of equation (2) can be expressed as a series The quantities a and α a ≡ can interchangeably be used, times appropriate group invariants, as the expansion parameter and l denotes the number of loops involved in the calculation of b l and b l = b l /(4π) l . From now on we use the coefficientsb l , also used in [19,20] for the numerical analysis, while b l are used in [21][22][23]. The coefficientsb 1 andb 2 are universal [24][25][26][27] and given byb here written in terms of the quadratic Casimir invariants C f ≡ C 2 (R) and C A ≡ C 2 (G), for, respectively, the representation R to which the N f fermions belong and the adjoint representation. The quantity T f ≡ T (R) is the trace invariant for the representation R.
Coefficients of higher order are scheme-dependent [28,29] and have been calculated up to four-loop order in the M S scheme [21][22][23]. In Table 1 we list a subset of their values, for the SU (N = 3) theory with 6 ≤ N f ≤ 12 Dirac fermions in the fundamental representation: β (α) with respect to α can also be expressed in terms of  Table 2: Infrared zeros α IR,n at n-loop order, n = 2, 3, 4, for the SU(N = 3) beta function with N f = 6, . . . , 12 Dirac fermions in the fundamental representation. At four loops, the negative zero is not listed here, α IR,4 is the positive one closest to the origin, and a third zero α UV,4 occurs farther from the origin (possibly an artefact of the perturbative expansion).
the coefficientsb l The value α IR of the IRFP coupling is one root of the equation β(α) = 0 5 . At two loops α IR, At four loops one has a cubic equation with three zeros, one of which is negative [30]. The numerical results for the case of interest are summarised in Table 2, and agree with those reported in [19,20].
In all cases, the derivative β (α IR ) is positive and increases along the IRFP line for decreasing N f . The disappearance of the zero occurs for N f > 8 at two loops, while it shifts to lower N f at three and four loops, suggesting a lower endpoint of the conformal window in the range 7 < N f < 8 at four loops -provided the zero can be taken as sufficient condition. Note that at two loops the disappearance of the zero is determined by the change of sign ofb 2 , implying that the fixed point disappears at infinite coupling α IR,2 → ∞. This behaviour, however, is likely to be an artefact of the truncated perturbative expansion; the same singularity occurs in β (α IR,2 ).
It is most interesting to compare these results with the implications of the exact beta function derived in [18] for the large-N massless limit of QCD with N f fundamental fermions in the Veneziano limit (N f , N → ∞, N f /N =const), which manifests salient analogies and differences with the exact beta function of SQCD [31][32][33]. From inspection of the beta function [18] 5 Note that the zero of the beta function is in general a necessary, but not sufficient condition for the existence of a stable IRFP. 6 Equation (10) is written in terms of the canonical coupling gc = √ N g in [18].
with the anomalous dimension factor and the fermion mass anomalous dimension both of O(N g 2 ), and the comparison with the NSVZ beta function of SQCD [31][32][33], one concludes that the absence of supersymmetry generates the new anomalous dimension contribution in Z for QCD. The fate of a zero of the beta function, possibly associated to a stable fixed point, will depend on the numerator and denominator of equation (10): the pole generates a cusp in the RG flow of g, at N g 2 = 4π 2 , unless the numerator has a zero before the pole is hit. And a zero associated to an IRFP inside the conformal window must be shown to be renormalisation scheme independent. Remarkably, this has been shown to be true for the beta function in equation (10) at the lower edge of the conformal window, located at N f /N = 5/2 where the stability of the glueball kinetic term is lost [18].
A fundamental difference between SQCD and QCD is the presence in SQCD of a phase just below the lower edge, for N + 2 ≤ N f ≤ 3N/2, where the only description that makes physical sense is in terms of the dual variables that describe a free non-Abelian magnetic phase [2]; the original electric theory is infinitely coupled. The fact that the IR fixed point disappears at infinity g IR → ∞ at the lower edge of the conformal window, or equivalently the presence of the free magnetic phase, can coexist with the occurrence of a cusp in the RG flow of the gauge coupling. This occurs in SQCD when γ m (g 2 = 8π 2 /N ) ≥ 1 − 3N/N f [2], for which the pole of the beta function is hit before the zero of the numerator.
The absence of the same phase in QCD calls instead for a differentiable flow, thus without cusps, across and below the lower edge of the conformal window. It is rewarding that the beta function in [18] can realise this property. It also supports that the lower edge singularity forb 2 = 0 of the two-loop beta function arises as an artefact of truncated perturbation theory, and that the IR fixed point does not disappear at infinity at the lower edge, nor it merges with a UV fixed point. In this context, the N f = 0 case is instructive. In [34] a renormalisation scheme for the large-N Yang-Mills exact beta function has been constructed, where the canonical coupling is shown to coincide with the physical effective charge g phys entering the static inter-quark potential with nonzero string tension σ. The effective charge g phys in the Coulomb potential is observed to saturate to a constant at large distances in lattice SU (3) Yang-Mills [35] in agreement with the effective bosonic string theory prediction [36], and it saturates according to the large-N beta function of [34]. In other words: the beta function of g phys develops a zero, while conformal symmetry remains broken due to the linear confining contribution to the potential (non-zero string tension) dominating the large distance behaviour. Provided a RG transformation between the canonical coupling and the effective charge exists, we should not expect a qualitatively different behaviour for N f = 0. Coming back to γ G , since the large-N beta function in [18] reproduces the two-loop one up to O(1/N 2 ) contributions, the predicted γ G in this case should also reproduce the two-loop result up to O(1/N 2 ); on the other hand, its complete expression can be expected to remove the twoloop singularity forb 2 = 0 at the lower edge of the conformal window. In summary, perturbation theory as well as the large-N solution predict an increasing magnitude of the anomalous dimension |γ G | = |β (α * |) for N f N c f .

Merging of UV and IR fixed points.
We show that the behaviour discussed above is opposite to the one implied by a UV-IR fixed point merging mechanism at N c f . In this scenario we can assume without loss of generality that close to N c f , where the merging would occur, we have [12]: For completeness, we recall the properties of this beta function first discussed in [12]. It has zeros at α ± = α c ± N f − N c f . As shown in Fig. 1 (top), α ± are distinct and real for N f > N c f , they coincide α + = α − = α c for N f = N c f (the UV-IR fixed point merging) and become complex for N f < N c f , thus leading to the disappearance of the conformal window. Below the conformal window, for N f sufficiently close to N c f , the gauge coupling increases towards the IR from an initial value α U V at some UV scale µ U V , as qualitatively shown in Fig. 1  (bottom). In the region where the beta function is small, the coupling will "walk" until it blows up at some IR scale µ IR . The latter is obtained by integrating the beta function on [µ U V , µ IR ], it defines the longest correlation length Here, we first observe that this beta function develops a local maximum at N f = N c f , i.e., β (α c ) = 0, with decreasing |β | along the IRFP line for decreasing N f N c f . In other words, oppositely to what predicted by perturbation theory, the magnitude of the scalar glueball anomalous dimension γ G is predicted to decrease for decreasing N f inside the conformal window, and vanishes at its lower edge. This may pose some problems. It is rather counterintuitive to have the anomalous dimension of the scalar glueball operator, a probe of confinement, that diminishes while the theory is approaching the confined phase and the screening of the gauge force diminishes. On the contrary, perturbation theory predicts what we do expect. Secondly, one has to guarantee that the beta function of equation (14) smoothly metamorphoses into the perturbative QCD beta function for any N f < N c f and α α c ; another non-trivial test to be passed is that the correct UV behaviour of correlation functions dictated by asymptotic freedom must be reproduced. Finally, of interest beyond QCD remains the question if, and if so how, Miransky/BKT scaling can occur without a beta function with a UV-IR fixed point merging.

Prospects
We have shown that the scalar glueball operator, a probe of confinement, carries information on the nature of the lower edge of the conformal window and the zero temperature phase that precedes it. In particular, perturbation theory and the large-N limit of QCD predict that the magnitude of its anomalous dimension increases inside the conformal window along the IRFP line for N f N c f , while it decreases in the presence of a UV-IR fixed point merging mechanism and vanishes at the lower edge; we observe that it is more difficult to reconcile the latter behaviour, which is associated to Miransky/BKT scaling, with the fact that theories do confine below the conformal window.
Conversely, the trend of the anomalous dimension γ G predicted by perturbation theory, as well as by the large-N limit of QCD, is rather intuitive. These results together with the predicted location of the lower edge of the conformal window of QCD, see Table 2 and [18], offer a picture quite consistent with the recently determined bound on the lower edge 6 < N c f < 8 [14] based on a lattice study. This analysis also suggests that a nonperturbative determination of γ G for varying N f inside the conformal window would finally establish if a UV-IR fixed point merging is realised or not in QCD. It would also provide a measure of how close the complete theory is to the predictions of perturbation theory and its large-N limit. This measure can be achieved on the lattice through the study of the two-point correlation function of the scalar glueball operator; such studies, however, are a notoriously difficult challenge [37]. We also recognise that the Wilson flow proposed in [38,39] can be a useful tool in this context, in order to discriminate between a conformal and a confining behaviour in the theory formulated on a lattice [40]. Inside the conformal window, and in the deconfined hightemperature phase of QCD, the AdS/CFT correspondence [41] as well as conformal bootstrap can also offer valuable insights. Finally, an extension of this study to theories with different gauge groups and/or fermionic matter in higher-dimensional representations would also be instructive and it has relevance for phenomenology beyond the standard model.