The $\beta$-function for Yukawa theory at large $N_f$

We compute the $\beta$-function for a massless Yukawa theory in a closed form at the order $\mathcal{O}(1/N_f)$ in the spirit of the expansion in a large number of flavours $N_f$. We find an analytic expression with a finite radius of convergence, and the first singularity occurs at the coupling value $K=5$.


Introduction
The success of the Standard Model in describing the electroweak scale phenomena notwithstanding the apparent problems with the high-energy behaviour have lead to revival of interest in better understanding the UV properties of general gauge-Yukawa theories, see e.g. Refs [1][2][3]. In particular, gauge-Yukawa theories with a large number of fermion flavours, N f , provide interesting candidates within the asymptotic-safety framework as opposed to the traditional asymptotic-freedom paradigm [4,5].
The groundwork for these considerations was laid few decades ago with the computation of the leading large-N f behaviour of the gauge β-functions [6][7][8] for N f fermion charged under the gauge group; see also Refs [9,10]. The leading 1/N f contribution to the β-function is obtained by resumming the gauge self-energy diagrams with ever increasing chain of fermion bubbles constituting a power series in K = αN f /π. It was noticed that this series has a finite radius of convergence; in the case of U(1) gauge group K = 15/2. Furthermore, the leading 1/N f contribution to the U(1) β-function has a negative pole at K = 15/2, thereby suggesting that this behaviour could cure the Landau-pole behaviour of the SM U(1) coupling, see e.g. Refs [9,11,12].
Recently, a further step towards a more complete understanding of these models was achieved by working out the leading 1/N f contribution from the gauge sector to a Yukawa coupling [13]; an extension to semi-simple gauge groups was discussed in Ref. [14]. However, only a single fermion flavour was assumed to couple to the scalar, and the scalar self-energy remained uneffected by the N f fermion bubbles. Our work is the first step to bridge this remaining gap: we provide the leading 1/N f β-function for pure Yukawa theory, where N f flavours of fermions couple to the scalar field via Yukawa interaction. We leave the more detailed study within a general gauge-Yukawa framework for future work. Interestingly, the pure Yukawa model is closely related to the Gross-Neveu-Yukawa model, whose critical exponents have been recently computed up to 1/N 2 f [15,16]; see also the earlier studies on the Gross-Neveu model e.g. Refs [17,18].
The paper is organized as follows: In Sec. 2 we introduce the framework and notations and in Sec. 3 give the expressions for the renormalization constants. In Sec. 4 we perform the resummations of the bubble chains and give closed form expressions for the renormalization constants. In Sec. 5 we collect the results, and write down the final expression for the β-function, and in Sec. 6 we conclude. Explicit formulas for the loop integrals are given in Appendix A.

The framework and definitions
We consider the massless Yukawa theory for a real scalar field, φ, and a fermionic multiplet, ψ, consisting of N f flavours interacting through the usual Yukawa interaction: We define the rescaled coupling,  which is kept constant in the limit N f → ∞. The β-function of the rescaled coupling, K, can then be expanded in powers of 1/N f as 3) The purpose of this paper is to compute F 0 and F 1 (K). The former is entirely fixed at the one-loop level and can be derived just by rescaling the well-known result for the β-function at that order, while the evaluation of F 1 (K) requires the resummation of diagrams in Fig. 1 involving all-order fermion-bubble chains. The β-function can be obtained from where G 1 is defined by and Z S , Z F , and Z V are the renormalization constants for the scalar wave function, the fermion wave function, and the 1PI vertex, respectively. The scalar wave function renormalization constant is determined via where Π 0 (p 2 , K 0 , ) is the scalar self-energy divided by p 2 , where p is the external momentum. Here and in the following, divX denotes the poles of X in . The self-energy can be written as where Π (1) gives the one-loop result, and Π (n) the n-loop part containing n − 2 fermion bubbles in the chain, and summing over the topologies given in Fig. 1a. Other contributions are of higher order in 1/N f and are thus omitted. For the fermion self-energy and vertex renormalization constants, the lowest non-trivial contributions are already O(1/N f ), and we, therefore, have where Σ (n) is depicted in Fig. 1b with n − 1 fermion bubbles. Similarly, where V (n) again contains n − 1 fermion bubbles and is shown diagrammatically in Fig 1c. Finally, we briefly comment on the scalar three-point and four-point functions, assuming that they are generated via fermion loops: the former exactly vanishes for massless fermions, while the latter is found to be already O(1/N f ) at the lowest order. Therefore, they can be neglected for the purpose of our analysis.

Renormalization constants
In this section our goal is to extract the contributions to the renormalization constants that are O(1/N F ) and relevant for the computation of the β-function.
Our starting point for Z S is Eq. (2.6). Using the expansion of the scalar self-energy, Eq. (2.7), we obtain and substituting Eqs (2.8) and (2.10), the first term between brackets can be written as The Π (1) part corresponds to the one-loop diagram and is given by 2) in Appendix A.1, and we have introduced the notation Π F to indicate the finite part of Π (1) . Then, where the higher poles, i.e., higher than 1/ , arise from the product of two divergent parts and will be omitted because they play no role in what follows. Then, at the lowest order in 1/N f , Therefore, every time Z K K appears in the argument of Σ 0 and V 0 , it can be replaced by K 1 − K −1 ; the additional contributions are higher order in 1/N f . For Eq. (3.4), we arrive at Similarly, the second term of Eq. (3.1) reads Altogether, we can write Z S as where the explicit functional dependence on (p 2 , ) has been omitted to lighten the notation. Using the binomial expansion, and performing a shift in the summation, n → n − i, we find our final expression for Z S : (3.10) We notice that Eq. (3.10) differs essentially from its counterpart in the QED [7] because of the contribution from the fermion self-energy and the vertex, which exactly cancel in QED because of the Ward identity.
The expression for Z F can be derived from Eq. (2.8) in a similar manner: where we have again performed the same shift n → n − i in the last line. The derivation of Z V is completely analogous, and we can readily write the expression for Z V :

The vertex
By explicit computation, the n-loop contribution to V 0 is where G(n 1 , n 2 ) is defined in Eq. (A.2). We notice that, as in Ref. [7], Eq. (4.1) allows for the following expansion: and v j (p 2 , ) are regular in the limit → 0 for all j. In particular, v 0 ( ) is independent of p 2 and is explicitly given by Substituting Eqs (4.1) and (4.2) in Eq. (3.12), we find: Then, by using the result of Ref. [7], Eq. (4.5) gets simplified to Expanding and keeping only the 1/ pole of Eq. (4.7), we find the closed formula for Z V : (4.9)

The fermion self-energy
The n-loop contribution to Σ 0 is found to be where σ(n, , and σ j (p 2 , ) are regular for → 0. Again, σ 0 ( ) is independent of p 2 , and it is given by (4.13) Using the same procedure as in the previous section, we find that only σ 0 ( ) contributes to Z F . Keeping only the 1/ pole, the closed formula for Z F is (4.14)
In view of Eq. (3.10), we define

2Π
(1) where ξ(p 2 , , n) ≡ n Π with ξ j ( , p 2 ) regular for → 0 for all j. In particular, ξ 0 ( ) is independent of p 2 and is explicitly given by (4.20) Then, using the above definitions, Eq. (3.10) can be written as (4.21) Moreover, we find that and therefore the expression for Z S can be significantly simplified: where in the second line we extended the sum over j up to ∞ without affecting the result, since all the terms for j > n − 1 are finite. The function ξ(p 2 , , 1), corresponding to can be evaluated by taking in Eq. (4.18) the limit n → 1, although the latter is formally defined for n ≥ 2. We find the following expression: ≡ ξ( , 1). (4.25) Few comments are in order: Eq. (4.25) ensures that Z S is independent of the external momentum p 2 , as it should. This result comes from an exact cancellation among the different contributions of the scalar self-energy, the fermion self-energy, and the vertex in Eq. (4.18). In particular, we find that π(p 2 , , 1) = 2 3 and therefore which is equivalent to Eq. (4.25). Interestingly, Eq. (4.26) only holds for n = 1. All in all, the p 2 independence of Eq. (4.25) provides a non-trivial check for our computation.

The β-function
Using the results of the previous section together with Eq. (2.5), we can finally proceed to evaluating the β-function. First, we find that Now, it is straightforward to compute the β-function: Recalling Eq. (4.27) and using ξ 0 (0) = − 3 2 , Eq. (5.2) can be further simplified to Finally, by comparison with Eq. (2.3), we see that F 0 = 1 and where We plot the integrand, I 1 (t), in Fig. 2. We have checked that our β-function agrees at the leading order in N f up to four-loop level by comparing with the result of Ref. [19].
Finally, let us comment on the pole structure: the integrand, I 1 (t), has the first pole occuring at t = 3, which results in a logarithmic singularity for F 1 (K) around K = 3. Due to the sign of I 1 (t), we see that F 1 (K) approaches large negative values for K → 3 − . This suggests the existence of a UV fixed point at K UV 3 such that F 1 (K UV ) = −N f . Moreover, it can be shown that there is an infrared fixed point K IR , symmetric to K UV with respect to K = 3; see e.g. Ref. [9]. The same qualitative analysis applies to the singularity at K = 5.

Conclusions
We have computed the leading 1/N f contribution for the β-function in Yukawa theory with N f fermion flavours coupling to a real scalar. We obtained a closed form expression for the β-function up to order O(1/N f ). This expression has a finite radius of convergence, and the first singularity occurs at K = 3.
The present result adds an interesting ingredient to models with a large number of fermions, and makes a contribution to better understand the UV behaviour of gauge-Yukawa theories. Curiously, the location of the first singularity of the β-function is the same as for the non-abelian gauge theory at the order 1/N f [8,9], which might suggest an interplay of the gauge and Yukawa contributions. This study is, however, left for future work.