Critical Look at β -Function Singularities at Large N

We propose a self-consistency equation for the β functions for theories with a large number of flavors, N , that exploits all the available information in the Wilson-Fisher critical exponent, ω , truncated at a fixed order in 1 =N . We show that singularities appearing in critical exponents do not necessarily imply singularities in the β function. We apply our method to (non-)Abelian gauge theory, where ω features a negative singularity. The singularities in the β function and in the fermion mass anomalous dimension are simultaneously removed providing no hint for a UV fixed point in the large- N limit.

We propose a self-consistency equation for the β functions for theories with a large number of flavors, N, that exploits all the available information in the Wilson-Fisher critical exponent, ω, truncated at a fixed order in 1=N. We show that singularities appearing in critical exponents do not necessarily imply singularities in the β function. We apply our method to (non-)Abelian gauge theory, where ω features a negative singularity. The singularities in the β function and in the fermion mass anomalous dimension are simultaneously removed providing no hint for a UV fixed point in the large-N limit. DOI: 10.1103/PhysRevLett.123.131602 Introduction.-There are indications that perturbative series in quantum field theory are, in general, asymptotic series with zero radius of convergence. In theories with a large number of flavorlike degrees of freedom, N, a reorganization of the perturbative expansion in powers of 1=N is convenient. It can be shown that at fixed order in 1=N expansion, the number of diagrams contributing grows only polynomially rather than factorially: convergent series are obtained that can be summed up within their radius of convergence.
There is a vast literature on resummed results corresponding to the first few orders in 1=N expansion, mainly for RG functions obtained via direct diagram resummation or critical-point methods, see, e.g., Refs. .
Since the perturbative series at fixed order in 1=N are convergent, singularities in the (generically complex) coupling are expected. The appearance of such singularities on the real-coupling axis seems to be true for all the d ¼ 4 theories analyzed so far, thereby having a dramatic effect on RG flows. In particular, the appearance of singularities in the coefficients of the 1=N expansion for gauge and Yukawa β functions have inspired speculations of a possible UV fixed point [23][24][25][26][27][28][29].
More generally, the UV fate of gauge theories for which asymptotic freedom is lost has broad theoretical interest, and this is in fact the case of matter-dominated theories. There, a nontrivial zero of the β function can be envisaged if the large-N resummation produces a contribution to β functions such that lim g→r β 1=N ðgÞ ¼ −∞, where r is the radius of convergence of the 1=N series. Near the singularity, the Oð1=NÞ contribution exceeds the leadingorder result, and it is clear that a zero must emerge. Unfortunately, close to the radius of convergence the perturbative expansion in 1=N is broken, and higher order cannot be neglected. Further shadow on the existence of the fixed point as a consistent conformal field theory is cast by studying anomalous dimensions of other operators in the vicinity of the β-function singularity: in the case of large-N QED truncated at Oð1=NÞ, the anomalous dimension of the fermion mass diverges [1,2], and it was recently pointed out that in the large-N QCD the anomalous dimension of the glueball operator breaks the unitarity bound [30]. Recently, the first lattice simulations to investigate the existence of possible fixed points appeared [31]. Even though these studies are not yet conclusive, no support for the fixed point is found.
In this Letter we provide quantitative evidence that these singularities are an artifact of the fixed-order large-N expansion of the β function. This follows from the observation that a fixed-order truncation in 1=N in the critical exponents is not equivalent to the same-order truncation in β functions, see also Ref. [32]. Instead, a fixed-order critical exponent induces higher-order terms in 1=N. These do not significantly affect the result far from the singular point, but are relevant near the singularity signaling a breakdown of the 1=N expansion. In particular, close to the radius of convergence these contributions diverge with alternating signs. Remarkably, such contributions can be resummed, and the final result is free of singularities. This conclusion is essential for the studies speculating on the UV fixed point, since they fully rely on the existence of a singularity in the β function.
We demonstrate this method concretely for fourdimensional gauge β function and Gross-Neveu (GN) model in two dimensions. Generically, we find that the fixed-order singularities are removed and the appearance of a fixed point is not supported within the large-N framework.
β function from the critical exponents.-Following Ref. [32], we review the general form for the β function in the large-N limit written in terms of the critical exponent, ω. This critical exponent gives the slope of the β function at the Wilson-Fisher (WF) fixed point, βðg c Þ ¼ 0, where d is the dimension of spacetime (in the literature, this equation is often found as ω ¼ −β 0 =2. We omit this factor for notational convenience). The large-N expansion of the β function can be incorporated by using the following ansatz: where d c is the critical dimension of the coupling g (in QED, for example, g ¼ α=π, and should not be confused with the simple gauge coupling), b and c are modeldependent one-loop coefficients, and the functions F n satisfying F n ð0Þ ¼ 0 are all-order in x ≡ gN.
Using the ansatz one can relate the coupling value at the WF fixed point, g c , to the spacetime dimension, d, and, consecutively, find the slope of the β function, β 0 ðg c Þ ¼ ωðdÞ.
In Ref. [32], we noticed that the critical exponent ω ð1Þ contributes to the β function also beyond Oð1=NÞ. The same holds for each ω ðjÞ : it contributes to all F n with n ≥ j. In the following, we denote the contribution of ω ð1Þ ; …; ω ðjÞ to F n , n ≥ j, by F ðjÞ n . It is worth to stress that these contributions are necessary in order to obtain the correct perturbative result from the critical point formalism. It is tempting to assume that these originate from a specific class of nested diagrams.
Since ω ð1Þ , or equivalently F 1 , is known, all the F ð1Þ n can be computed. These induced coefficients are found in closed form as where the c ðkÞ m are defined iteratively: It follows from Eq. (4) that if F 1 ðxÞ features a negative singularity at a given x, this results into a sequence of singularities of alternating signs in F in Fig. 1. This means that the negative pole in the β function driven by F 1 is not guaranteed to persist when all the F ð1Þ n are taken into account. In the next section, we show that all the F ð1Þ n 's can be actually resummed, and the final result features no singularity.
Self-consistency equation.-A direct resummation of the F ðjÞ n terms is not straightforward, and we therefore employ a different approach. Denoting the relation β 0 ðg c Þ ¼ ωðdÞ is rewritten as where the dimension and the critical coupling are related via Equation (8) would provide an exact solution if ω were known to all orders. However, in practice this is not the case, but rather we have access to the contributions induced by ω ð1Þ ; …; ω ðjÞ only. Nonetheless, a consistent solution to Eq. (8) incorporating all known coefficients can be achieved by truncating the critical exponent to which corresponds to truncating F n to F ðjÞ n in F ðx; NÞ, Eq. (7). The resulting function is denoted by F ðjÞ ðx; NÞ.  Let us now concentrate on the simplest case j ¼ 1, where the truncation leads to the following differential equation for F ð1Þ : where we have used Eq. (9). If the critical exponent as a function of the space-time dimension is known, this is a nonlinear first-order differential equation for F ð1Þ . Traditionally, this has been solved order by order in the 1=N expansion. Indeed, neglecting the backreaction of F ð1Þ on the right-hand side of Eq. (11) gives the standard solution F ð1Þ ðx; ∞Þ ≡ F 1 ðxÞ. The advantage now is that we can solve Eq. (11) as it is and only afterwards take the large-N limit. This is equivalent to resumming all the F ð1Þ n 's, given explicitly in Eq. (4), that we know to be important near the singularity.
Where the 1=N expansion is under control, the one-loop term in the β function, g 2 bN, dominates and, in particular, no zero can emerge for a large enough N. However, there exist examples in which the critical exponent, ω ð1Þ , features a singularity for some real value of d, potentially affecting the previous conclusion. For instance, in QED the first singularity of ω ð1Þ QED occurs at d ¼ −1 translating to the Oð1=NÞ singularity of the β function at x ¼ 7.5.
Let us first consider a model where b and the singularity in ω are of same sign: the higher-order terms would just enhance the singularity and lead to a Landau pole as is the case of super-QED at Oð1=NÞ [33] and in OðNÞ model at On the contrary, if the singularity and b are of opposite sign, as in QED and QCD, Eq. (11) yields a smooth solution which, close to the would-be singularity at x ¼ x s , approaches a scaling solution of the form: where a is typically Oð1Þ and implicitly defined by This indicates that the alternating singularities in the F ð1Þ n can be resumed to yield a finite contribution. By using Eq. (12) and recalling that x ¼ gN, the β function is found to be Given that a and b need to have the same sign due to the boundary condition F ð0; NÞ ¼ 0 and ωðd c Þ ¼ 0, a zero cannot emerge neither for g ≳ g s , nor for g < g s , where the one-loop coefficient dominates.
When the Oð1=N 2 Þ term, ω ð2Þ , is included in the analysis, there are two possibilites: (1) the closest singularity at x ¼ x ð2Þ s is positive, and (2) the closest singularity at x ¼ x ð2Þ s is negative. In the first case, the β function clearly grows faster than before close to x ð2Þ s , and no zero can appear. If the new singularity is closer, this rather implies a Landau pole. As for the regular points before the first singularity, the contribution of ω ð2Þ is negligible for a large enough N. An example of this behavior is given by the OðNÞ model [34].
In the second case, the same reasoning for resumming the alternating singularities applies and gives the asymptotic scaling in Eq. (14) with a modified coefficient a, valid for g ≳ minðg s ; g ð2Þ s Þ. The same procedure generalizes to any finite order ω ðjÞ .
To summarize, the singularities appearing in fixed-order critical exponents do not necessarily imply singularities in the β function. In particular no hint for a UV zero is found in the large-N limit, as its existence relied entirely on the presence of a singularity.
Finally, we emphasize that the resummation we have employed is relevant also beyond the case when the β function features singularities on the positive real axis. In the following we will show that the wild oscillations in the β function of the Gross-Neveu (GN) model-which naïvely would lead to infinitely many alternating IR and UV zeroes-can be resummed in the same way.
As explicit examples we consider two classes of models: four-dimensional gauge theories and the GN model in two dimensions. For the latter, the critical exponent is known up to Oð1=N 2 Þ, allowing us to study the effect of higher-order corrections.
QED and QCD.-The critical exponent for a general gauge β function is known up to Oð1=NÞ and is given in d ¼ 2μ by [11] ω ð1Þ ð2μÞ ¼ η ð1Þ ð2μÞ where T F and C F are the index and quadratic Casimir of the fermion representation, respectively, C A is the Casimir of the adjoint representation, and η ð1Þ reads For the Abelian case, the first singularity occurs at μ ¼ −1=2, while the non-Abelian system has a singularity already at μ ¼ 1. 123, 131602 (2019) 131602-3

PHYSICAL REVIEW LETTERS
We compute the β function by solving Eq. (11) numerically for a benchmark value N ¼ 100. In the notation of Eq. (11), QED corresponds to The scaling solutions are given by a QED ≈ 4.995, a QCD ≈ 1.985. In Fig. 2 we show the numerical solution to Eq. (11) for QCD with N ¼ 100; for QED the plot looks qualitatively the same. As expected from the general analysis above, the singularities and the putative UV fixed points at x ¼ 3 for QCD and x ¼ 7.5 for QED have both disappeared.
In the QED case, the fermion mass anomalous dimension has a singularity at the same coupling value as the first singularity of ω ð1Þ , x ¼ 7.5. A fixed point in this coupling region would have the operatorψψ violating the unitarity bound. Similarly to the critical exponent, ω, we truncate the fermion mass anomalous dimension to Oð1=NÞ: where the Oð1=NÞ result is given by γ ð1Þ m ð2μÞ ¼ −2η ð1Þ ð2μÞ=ðμ − 2Þ [10]. Evaluating Eq. (18) with the solution for F ð1Þ , we obtain γ m in the same truncation as the β function. We find that the singularity in γ m is also removed, and the anomalous dimension reaches a constant value above x ¼ 7.5 given bỹ For N ¼ 100, we findγ m ≈ −0.14. Gross-Neveu model.-The critical exponent, λðdÞ ¼ β 0 ðg c Þ, for the GN model is currently known up to Oð1=N 2 Þ [18]. The Oð1=NÞ coefficient is explicitly given by while the expression for λ 2 ðdÞ can be explicitly found in Ref. [18].
In the notations of Eq. (2), the GN model is characterized by d c ¼ 2, b ¼ −1, and c ¼ 2. We solve again Eq. (11) numerically for the benchmark value N ¼ 100 both using only the Oð1=NÞ and Oð1=N 2 Þ critical exponent, λ. We show the resulting β functions in Fig. 3 along with the β function computed directly up to Oð1=N 2 Þ using Eq. (2). The scaling solution using only λ ð1Þ is given by a ð1Þ ≈ −8.6, while including λ ð2Þ modifies this to a ð2Þ ≈ −6.3. The result shows no hint for an IR fixed point in agreement with previous studies [35,36].
Conclusions.-We have shown that singularities in a fixed-order large-N critical exponent do not necessarily imply singularities in the β function. This is due to the fact that a fixed-order critical exponent generates contributions to every subsequent order in 1=N in the β function. We proposed a self-consistency equation to properly include these contributions.
In the case of negative singularities that have inspired speculations of UV fixed points, it turns out that the same singularity appears with alternating signs at higher-order terms, and resumming these contributions yields an asymptotic linear growth of the β function rather than a UV zero.
As concrete examples we showed this scaling behavior in the case of QED, QCD, and the GN model. For QED and QCD, the singularities are removed and in the GN model the wild oscillations tamed. For QED, this procedure simultaneously cures the singularity of the fermion mass anomalous dimension.  We stress that the linear scaling is very sensitive to the higher-order corrections in 1=N to the critical exponent that could potentially turn it into a Landau-pole behavior. Nonetheless, the emergence of a fixed point remains incompatible within any finite set of higher-order corrections. Our result invalidates fixed points based on the singularities in the large-N β function. A hypothetical fixed point could thus be supported only through a nonperturbative computation beyond the 1=N expansion.
We thank John Gracey for valuable comments and are grateful for the numerous fruitful discussions during the MASS conference. The CP 3 -Origins centre is partially funded by the Danish National Research Foundation, Grant No. DNRF:90.