# Background field method in the large \(N_f\) expansion of scalar QED

## Abstract

Using the background field method, we, in the large \(N_f\) approximation, calculate the beta function of scalar quantum electrodynamics at the first nontrivial order in \(1/N_f\) by two different ways. In the first way, we get the result by summing all the graphs contributing directly. In the second way, we begin with the Borel transform of the related two point Green’s function. The main results are that the beta function is fully determined by a simple function and can be expressed as an analytic expression with a finite radius of convergence, and the scheme-dependent renormalized Borel transform of the two point Green’s function suffers from renormalons.

## 1 Introduction

In quantum field theory, the beta functions which determine the flows of the coupling constants are of fundamental importance. As is well-known, in the 1970s [1, 2], it was the calculation of the beta function of a non-Abelian gauge theory (QCD) that led to the discovery of asymptotic freedom in this theory, which made theoretical physicists believe that this theory is the right theory for describing strong interactions. Since then, we have seen lots of efforts been put into calculating the beta functions of various theories, with the calculations of QED [3, 4, 5, 6] and QCD [7, 8, 9] having been calculated to five-loop order. In calculating the beta functions of gauge theories, the background field method which preserves the classical gauge invariance is an efficient method. In this method, we just need to calculate the related two point Green’s functions for the background gauge fields [10, 11, 12].

Generally apart from the first few coefficients of the beta functions, we know little about them. Therefore it’s meaningful to study the large order behaviour of quantum field theory under some approximation [13, 14, 15, 16]. An essential point, in the investigation of the large order behaviour of field theories, is whether the results obtained are convergent. The early investigations about this can be traced back to the works in Refs. [17, 18, 19]. In fact, our expressions obtained by perturbation methods, are generally at best asymptotic rather than convergent series [20]. The Borel transform, a mathematical technique, can be used to improve the convergence property of a series. To study the asymptotic behaviour of a series we can study its Borel transform which by definition has better convergence properties than the original series. After the acquirement of the Borel transform, if there are no singularities (renormalons), we can recover the original series [21, 22, 23, 24].

As is well-known, the beta functions provide us with useful information about the asymptotic behaviour of field theories, such as the asymptotic freedom in QCD (for a sufficiently large number of flavour we will lose this property). As regards the SM *U*(1) gauge theory, its one-loop beta function suggests that it may suffer from a Landau pole which can be avoided if there is a nontrivial UV fixed point arising from the zero of the beta function. However, according to a lattice result given in Ref. [25], there is no nontrivial fixed point in a *U*(1) gauge theory for \(N_f=4\) [15]. As has been shown in the literature [13, 14, 15, 16], the large \(N_f\) models can provide other possibilities; in Ref. [13], the beta function of spinor QED has been calculated at the leading order in \(1/N_f\), and the result suggests that there might be nontrivial (UV and IR) fixed points. As regards the scalar QED, the positive three-loop beta function shown in Ref. [26] suggests that the running coupling increases monotonically towards the ultraviolet and thus will suffer from a Landau pole. Hence, analogous to spinor QED and QCD, it is meaningful to study the large \(N_f\) behaviour of scalar QED – in this paper we shall calculate its beta function at the leading order in \(1/N_f\) and discuss whether there are some possibilities to find some fixed points to avoid the Landau pole just mentioned before.

The remainder of this paper is organized as follows. In Sect. 2, we give a brief introduction to the background field method, and in Sect. 3 derive the beta function in the background field method. In Sect. 4, we show the equivalence between two approaches of the background field method. In Sect. 5, we study the Borel transform of the two-point Green’s function and derive an analytic expression for the beta function. Scheme dependence issues are discussed in Sect. 6. In Sect. 7, some numerical results about the beta function are given. Discussions and conclusions are presented in Sect. 8.

## 2 A brief introduction to the background field method

*r*in a quantity means that this quantity is a bare/renormalized quantity). This Lagrangian is invariant under transformations

## 3 Beta function in the background field method

Through out this paper we will use the dimensional regularization (DR) procedure in \(4-2\varepsilon \) dimensions and choose the minimal subtraction (like) scheme – issues about the scheme dependence will be discussed in Sect. 6.

*j*-loop diagrams to \(Z_A\). This equation will act as a strong check on our calculations.

## 4 Application of the background field method

*AA*with the Feynman rule

### 4.1 Direct Approach

*m*to zero. The diagram we should calculate is the unrenormalized diagram shown in Fig. 1 where the scalar loop represents the total contribution of the \(N_f\) charged spinless fields, with the result being

*k*is the external momentum and

Higher order contributions come from diagrams generated by inserting some renormalized scalar bubbles shown in Fig. 1 into the internal photon lines of diagrams in Fig. 3; all other diagrams are suppressed by a factor of \(1/N_f\). Since the one-loop scalar bubble is transverse there is no \(\alpha \)-dependence in these higher order diagrams.

### 4.2 Indirect Approach

In the “indirect approach”, apart from the usual vertices, we have a new vertex *AA* to consider. Since the one-loop scalar bubble is transverse, the diagram carrying a photon chain having both insertions of this vertex and those of the scalar bubble will not contribute.

*AA*. Note that because

*AA*.

Now, we are only left with diagrams without the insertions of the new vertex *AA* to consider. The diagrams we should consider in the “indirect approach”, in shape, look like the corresponding diagrams in the “direct approach”, the main difference being that the one-loop scalar bubbles and the couplings (except the two couplings attaching to the two external background legs) in the “indirect approach” are unrenormalized (since in this approach we don’t introduce a renormalization procedure for the photon field), while usually renormalized in the “direct approach”. Since the one-loop scalar bubble is still transverse and there is no \(\alpha \)-dependence at the two-loop level (this follows from that in the two-loop level there is no contribution from the longitudinal part of the photon propagator), in the “indirect approach” there is no \(\alpha \)-dependence in our calculations. In what follows, we shall, in the Landau gauge, prove that a diagram (except the one-loop diagram) in the “indirect approach” is equivalent to a sum of an infinite number of diagrams in the “direct approach”. To prove this, we can focus on the equivalence of the photon chain between the diagrams considered [34].

*n*unrenormalized scalar bubbles and a certain number of counterterms. The photon chain of a Feynman diagram of this type with \(k+1\) counterterms is

*k*, recalling \(Z_A^1=Z_3^1\), we can establish that the equivalence between the “direct approach” and the “indirect approach” is proven.

## 5 The beta function and the Borel transform

In this section, we shall investigate the Borel transform of the two point Green’s function and derive an analytic expression for the beta function by two different approaches, which here we call LTR approach and RTL approach respectively. Having shown the equivalence between the “direct approach” and the “indirect approach” and the \(\alpha \)-independence of our calculations, in what follows we shall use the “direct approach” and proceed in the Landau gauge.

Before the concrete discussion, we want to say that since the similarities of Feynman rules and the identity \(Z_e=1/\sqrt{Z_3}\) (or \(Z_1=Z_2\)) proved in Appendix A, the calculation of the two point Green’s function in the normal field method is equivalent to that in the “direct approach”. This also can be understood from the property of the background field method. According to the presentation of Ref. [11], the effective action we get by using the background field method and the gauge-fixing term \(\mathscr {L}_{gf}(A,A_b)\) given in Eq. (6) is equal to the conventional effective action calculated with the gauge-fixing term \(\mathscr {L}_{gf}(A-A_b,A_b)\) and evaluated at \(A=A_b\). Since the calculation of the effective action just involves 1PI diagrams, we can neglect the terms in \(\mathscr {L}_{gf}(A-A_b,A_b)\) which have only zero or one quantum photon field, that’s is to say, we can calculate the conventional effective action with the usual gauge-fixing term \(\mathscr {L}_{gf}(A,0)\). Therefore in what follows, our investigation about the Borel transform of the two point Green’s function in the context of the “direct approach” of the background field method can be applied to the normal field method of scalar QED.

### 5.1 A brief introduction to Borel transform

*n*!, which indicates that this series has zero radius of convergence [27].

*R*[

*g*], in this work, is defined by

*R*[

*g*]. After the acquirement of \(B_R[t]\), the recovering of

*R*[

*g*] is formally done through

*R*[

*g*].

### 5.2 LTR approach

*k*gives a multiplicative factor

*n*unrenormalized scalar bubbles. The total expression for these diagrams is

*p*is the external momentum and

*s*at \(s=0\) and takes the form

*s*and \(\varepsilon \):

*n*unrenormalized scalar bubbles in those diagrams by their counterterms. Taking all these diagrams into consideration, denoting the result by \(\varPi _n^t(p,g)\), we have

*j*scalar bubbles with their counterterms.

*i*is truncated at \(n+1\) because we are only interested in the pole terms and the finite terms. Using the following combinatorial identity[13, 33]

*P*(

*x*). When we take the limit \(\varepsilon \rightarrow 0\), the second terms in these two brackets cancel each other, and the first term in the second bracket, according to Eqs. (37) and (42) is

### 5.3 RTL approach

In previous subsection, the derivation of \(Z_A\) and investigation of the Borel transform of the two point Green’s functions are simplified by using the combinatorial identity shown in Eq. (46). In this subsection, we will use a different approach inspired by the approach presented in Ref. [21] to derive \(Z_A\) and discuss the Borel transform of the two point Green’s function again (recently we have used this approach to study the large order behaviour of spinor QED in Ref. [34]). The essential point of this approach lies in the observation that in our approximation, to calculate the Borel transform of the two point Green’s function, we can first calculate the Borel transform of the photon chain.

*k*gives a multiplicative factor

*t*is the Borel parameter. This can be rewritten as [21]

*p*and will be omitted later. In the second step, we can make the following trick

*p*, which guarantees the momentum independence of the renormalization constant \(Z_A\).

*t*enough times and then setting \(t=0\), or multiplying

*m*! back – all these methods give the same result:

### 5.4 The beta function

*S*[

*u*], there is a requirement that the renormalization group functions have convergent regions, or at least they don’t diverge as fast as factorials. Our result given above in Eq. (71) shows that up to the leading order in \(1/N_f\) the beta function does have a convergent region \(g<5/2\), which can be seen from the explicit expression for the integrand; here we call it

*K*(

*x*) and depict its figure in Fig. 4. In this convergent region \(g<5/2\), \(\beta (e)\) is always positive. When

*g*approaches 5 / 2, the beta function encounters the first logarithmic singularity.

### 5.5 Renormalons

*D*[

*t*], according to Eq. (38), can be written as

*G*(

*u*) is given by

*G*(

*u*)

*G*(

*u*) is convergent for Re \(u<2\) and suffers from double poles at non-positive integers from individual terms of the sum; the analytic continuation of

*G*(

*u*) to the entire complex plane is performed under the symmetry \(G(1+u)=G(1-u)\) [33].

*D*[

*t*] suffers from a singularity at \(t=0\) arising from the singularities of

*G*(

*x*) at \(x=0,-1\) and the singularity of

*S*(

*x*) at \(x=0\). Near \(t=0\),

*D*[

*t*] behaves as

*D*[

*t*] should be cancelled. Note that in the first term of Eq. (68), there is also a singularity at \(t=0\) – near \(t=0\), this term, \(B_P[t]/t^2\), behaves as

*t*at \(t=0\).

Apart from the singularity at \(t=0\), *D*[*t*] still suffers from two kinds of singularities. The first kind of singularities come from the singularities of the function *S*(*t*) which becomes singular when \(t=-1,-2\). The second kind of singularities come from the singularities of the *G*-function which becomes singular when its argument becomes an integer not equivalent to 1.

*K*(

*t*) of

*D*[

*t*], then, near \(t=-1\),

*D*[

*t*] behaves as

*n*[33]). Near \(t=-2\),

*D*[

*t*] behaves as

*t*is an integer not equivalent to \(-1\) or \(-2\), the singularities of

*D*[

*t*] only come from the singularities of the

*G*-function. Since the singularities in the

*G*function are double poles, all these singularities are double poles.

Among the singularities of *D*[*t*], the singularities at \(t=n\), \(n=1,2,\ldots \) are called ultraviolet renormalons since they originate from high-momentum regions of integration in the loop integrals. These singularities destroy Borel summability of the series because they are on the positive real axis. The singularities at \(t=n\), \(n=-1,-2,\ldots \) are called infrared renormalons since they originate from low-momentum regions of integration in the loop integrals.

## 6 Scheme dependence

Our presentations given above is based on the adoption of the minimal subtraction scheme. In this section, we want to generalise our study to arbitrary minimal subtraction-like (MS-like) schemes such as \(\overline{MS}\).

*t*. By this we mean that if we write

*MS*-like schemes (here we should point out that the coefficients \(P_n\) have a factor \(e^4\) and can provide the required \(g^2\)). Having this and Eq. (86) in mind, we can establish that in all the

*MS*-like schemes the beta functions are the same beta function.

Numerical results for \(P_n\) (with the factor\(e^4 N_f/2304\pi ^4\) being ommited). The first three results are exact values

n | 1 | 2 | 3 | 4 | 5 |

\(P_n\) | 36 | \(-\,147\) | 161.5 | 58.2981 | \(-\,231.639\) |

n | 6 | 7 | 8 | 9 | 10 |

\(P_n\) | 140.374 | 17.8971 | \(-\,55.242\) | 20.0692 | 2.81043 |

n | 11 | 12 | 13 | 14 | 15 |

\(P_n\) | \( -\,4.01282 \) | 0.874324 | 0.172297 | \(-\,0.119144\) | \(1.47442\times 10^{-2}\) |

n | 16 | 17 | 18 | 19 | 20 |

\(P_n\) | \(4.4666\times 10^{-3}\) | \(-\,1.7065\times 10^{-3}\) | \(9.7507\times 10^{-5}\) | \(5.70104\times 10^{-5}\) | \(-\,1.30831\times 10^{-5}\) |

*t*approaches 0, the renormalized Borel transform doesn’t suffer from a singularity. When we change our scheme by changing \(S_{\varepsilon }\), the only change is the change in the argument of the exponential function in the second term of Eq. (95), which depends only on

*a*, the \(\varepsilon \) part of \(S_{\varepsilon }\). This property is in accordance with a general property of the \(\overline{MS}\) scheme that we have various choices for \(S_{\varepsilon }\), but only the \(\varepsilon \) part of \(S_{\varepsilon }\) affect the renormalized Green’s function [36]. Also since the locations of the renormalons are determined by \(H[p,0,-t]/t\), when we change our \(\overline{MS}\) scheme we don’t change the locations of the renormalons.

## 7 The numerical and analytic values of the beta function

*P*(

*x*) in powers of

*x*. Here, we give the first terms of \(\beta (e)\) by doing this expansion

Note that all the \(\varGamma \) functions in \(P(\varepsilon )\) are of the form \(\varGamma (a+b\varepsilon )\) (*a* and *b* being integers), and therefore can be changed into the standard form \(\varGamma (a+b\varepsilon )=C[a,\varepsilon ]\varGamma (1+b\varepsilon )\) where \(C[a,\varepsilon ]\) is a polynomial in *a* and \(\varepsilon \). Doing these standard transformations for all the \(\varGamma \) functions in \(P(\varepsilon )\), using formula (97), we can establish that the Euler constant \(\gamma \) doesn’t enter into our expression for the beta function.

## 8 Discussion and conclusion

In this paper, we in the large \(N_f\) approximation have calculated the beta function of scalar QED at the first nontrivial order in \(1/N_f\) by two different ways. We have derived an analytical expression with a finite radius of convergence for the beta function. In the convergent region \(g<5/2\), the beta function is always positive which indicates that in this region there are no nontrivial fixed points arising from the zeroes of the beta function. Scheme dependence issues also have been discussed. We have shown that the beta function is scheme-independent in *MS*-like schemes, while the renormalized Borel transform suffering from ultraviolet renormalons at \(t= n\) and infrared renormalons at \(t=-n\) (\(n=1,2\ldots \ldots \)), is scheme dependent. Furthermore, we have made clear the role played by the gauge parameter by carrying out its renormalization in both approaches (the “direct” approach and the “indirect approach”) of the background field method, and the equivalence between these two approaches has been proven.

The RTL approach we used in Sect. 5.3 can be generalized to other theories, such as Yukawa theory and Yang-Mills theory. Its generalization to Yukawa theory at the leading order in \(N_f\) is straightforward. When we extend this to Yang-Mills theory some new features appear. In Yang-Mills theory, we encounter three-gauge-boson vertices, and the vertex graph with a fermion loop and three external gauge fields doesn’t vanish. Therefore even at the leading order in \(1/N_f\) we have to deal with diagrams with two bubble chain insertions. In this case we can use the fact that the Borel transform of a product of series is a convolution, that is to say, at this order and even higher order we can replace each bubble chain in the diagram considered with its Borel transform, do the usual loop integrals and in the end do the convolution integral (more details can be found in Ref. [22]).

Finally, let’s turn to the singularity structurer of Eq. (71) which has implications for the existence of nontrivial fixed points in the beta function and makes a contribution to better understand the asymptotic behaviour of scalar QED. Obviously the integrand *K*(*x*) of expression in Eq. (71) suffers from poles at \(x=5/2+n\) \((n=0,1\ldots )\). The appearance of these poles is due to the singularity of \(\varGamma (4-2x)\) at \(x=5/2+n\) \((n=0,1\ldots )\) and therefore leads to logarithmic singularities of the beta function at \(g=5/2+n\) \((n=0,1\ldots )\), which usually can be dealt with by Cauchy principal value prescription [15, 16]. Assuming this prescription, it can be shown that there are a UV fixed point at \(g\lesssim 7/2\) and a symmetric IR fixed point at \(g\gtrsim 7/2\) (with the explicit value depending on the value of \(N_f\)). The same quantitative analysis can be extended to other poles at \(g=7/2+n\) \((n=1,2\ldots )\).

## Notes

### Acknowledgements

One of us (Z. Y. Zheng) is deeply indebted to Prof. Y. Q. Chen for numerous discussions and helpful suggestions. The work of Z. Y. Zheng is Supported by Key Research Program of Frontier Sciences, CAS, Grant No. QYZDY-SSW-SYS006. The work of G. G. Deng is partially supported by the National Natural Science Foundation of China (11733001, U1531245, 11590782).

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