Background field method in the larger N_{f} expansion of scalar QED

# Background field method in the larger Nf expansion of scalar QED

Zhi-Yuan Zheng CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences,
Beijing 100190, People’s Republic of China
School of Physical Sciences, University of Chinese Academy of Sciences,
No.19A Yuquan Road, Beijing 100049, China
Gai-Ge Deng Guangzhou University Library, Guangzhou University Guangzhou 510006, China Center for Astrophysics, Guangzhou University Guangzhou 510006, China
###### Abstract

Background field method, a method preserves gauge invariance explicitly, in this paper is used to calculate the beta function of scalar quantum electrodynamics in the larger- approximation. Our calculation carried out with an arbitrary gauge-fixing parameter has been done up to order , and the result of our calculation is independent of the gauge-fixing parameter . We also investigate the renormalon property of the beta function and that of the two point Green’s functions which are closely related to the beta function, finding 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.

background field method, beta function, renormalons, Borel transform

## I Introduction

The scale dependence of a renormalized running coupling is determined by its beta function which is a function of running coupling itself and is of fundamental importance. As is well-known, in the 1970s(Gross:1973id, ; Politzer:1973fx, ), it was the calculation of the beta function of QCD at one loop level that led to the discovery of asymptotic freedom in this theory which made theoretical physicists believe that this non-abelian gauge theory is the right theory for describing strong interactions. Since then, the passage of time has seen so many efforts been put into calculating the beta functions of various theories by various methods and techniques, with the calculations of the beta functions of QED (Kataev:2012rf, ; Baikov:2012zm, ; Baikov:2012rr, ) and that of QCD (Herzog:2017ohr, ; Luthe:2017ttc, ; Baikov:2016tgj, ) having been calculated to five loop order.

In some cases, the absence of a reasonable fixed points arising from the vanishing of the beta function of a theory calculated at present order, or, as we go to higher loop order, the seemingly increasing of the coefficients of the beta function calculated at present order, suggests that it’s meaningful to investigate the larger order behaviour of this theory. In this work, we shall, in the larger approximation, focus on the larger order behaviour of scalar quantum electrodynamics, which describes the dynamics of spinless charged fields interacting with photons. Our mainly focus is put on the evaluation of the beta function of scalar electrodynamics and the renormalon issues about the two point Green’s functions which in background field are closely related to the beta function.

As is well-known, the background field method, which preserves the gauge invariance of the effective action, is an efficient method to evaluate the beta function. In ordinary quantum gauge field theories, the classical gauge symmetries of a theory are broken by the introduction of a gauge-fixing term, which of course is gauge variant. The background field method presented in the literature (such as reference (Abbott:1980hw, ; Abbott:1981ke, ; Abbott:1983zw, )), which only fixes the gauge of the quantum field but not that of the background field, can be used to generate an effective action which is still gauge invariant with respect to background gauge transformations. And the preserved gauge invariance sets powerful constraints on the form of the effective action and leads to the simplification of calculating the renormalization factors. Aside form the obvious advantage of maintaining the gauge invariance explicitly, the background field method also can be used to calculate the scattering matrix(Abbott:1983zw, ).

An essential point, in the investigation of the large order behaviour of quantum field theories, is whether the results obtained usually by means of perturbation methods are convergent, and if it’s not the case, what can we do about it and what can we learn from it. The early investigation about this issue in quantum field theory can be traced back to the work of Dyson in (Dyson:1952tj, ) and others work in (Hurst:1952zk, ; Thirring:1953da, ) . In fact, our expression (usually expanded in powers of the running coupling) for a quantity obtained from using perturbation methods, is in general at best asymptotic rather than a convergent series(AD:2012kr, ).

The Borel transform, a mathematical technique, is extensively used to improve the convergence property of a series. To investigate the asymptotic behaviour of a expression expanded as a series, say , we can study its Borel transform , which by definition has a better convergence property than the original (Braaten:1998au, ). After the acquirement of , if there are no singularities in (a singularity in is called a renormalon (Braaten:1998au, )), we can recover . In the investigation of renormalon, the large approximation has been developed into a useful tool (Beneke:1998ui, ; Beneke:1994sw, ; Beneke:1994qe, ).

The remainder of this paper is organized as follows. In section II, we give a brief review of the background field method and derive the beta function in the background field method. In section III, we calculate the renormalization constant and show the equivalence between two approaches of background field method in calculating . In section IV, we investigate the renormalon issues of the two point Green’s functions closely related to the beta function in background field method and give a final expression (close form) for the beta function. Scheme dependence is also discussed in this section. In section V, some analytic and numerical results about the beta function are given. Finally, summary and conclusion are presented in section. VI.

## Ii A brief introduction of background field method

As is well known, in background field formalism the renormalization constant of the background gauge field (say ) and the coupling (say ) are related to each other through . Therefore, as has been done in the literature, to calculate the beta function of a gauge theory by means of background field method, we just need to calculate the two-point Green’s functions.

### ii.1 Background field method in Scalar QED

In this paper, we shall use the background field method to study scalar QED whose Lagrangian takes the form

 L=−14(Fμν)2+∣Dμϕ∣2−m2∣ϕ∣2, (1)

with

 Dμ=∂μ+ieAμ. (2)

In eq. (1), we have ignored the interaction terms between scalar fields . The Lagrangian of scalar QED shown in eq. (1), obviously, is invariant under a general gauge transformations of the form

 ϕ(x) →e−iα(x)ϕ, (3) Aμ(x) →Aμ(x)+1e∂μα(x). (4)

As is well-known, for practical calculation, we must choose a gauge-fixing term (or a gauge condition), which must be gauge variant under gauge transformation (3) and (4) and then breaks the gauge invariance. Various gauge-fixing terms have been used in the literature, two of them being the well-known Coulomb Gauge and Lorentz (or Landau) gauge.

In background field method, we can introduce a background field dependent gauge-fixing term to preserve the gauge invariance of the effective action. To illustrate this statement, first we note that to get the effective action (Weinberg:1996kr, ) we can replace the field in the conventional action with ( being the variable of integration in the functional integral, while being the background field), and then use the following formula to get the effective action

 exp[iΓ[ϕ0]]=∫1PI[∏rxdϕr(x)]exp{iI[ϕ+ϕ0]}, (5)

where the 1PI means that we include all diagrams, connected or not, each connected component being one-particle-irreducible. Therefore, it’s obvious that we can introduce a background field dependent gauge-fixing term which is gauge invariant under background gauge transformation to preserve the gauge invariance of the effective action.

In this work, we shall choose the following Lorentz covariant gauge-fixing term

 Lgf(Aμ,AμB)=−{∂μ(Aμ(x)−AμB(x))}22α0, (6)

where is the background field. Therefore, the gauge-fixing term and the “ordinary” Lagrangian obtained by adding the background fields to their corresponding quantum fields in the functional integral are

 Lgf =−(∂μAμ(x))22α, (7) L =−14(~Fμν)2+∣~Dμ~ϕ∣2−m2∣~ϕ∣2, (8)

where

 ~A =A+AB (9) ~ϕ =ϕ+ϕB (10) ~Dμ =∂μ+ie~Aμ, (11) ~Fμν =∂μ(Aν+AνB)−∂ν(Aμ+AμB). (12)

The gauge-fixing term shown in eq. (7), and the ordinary “ Lagrangian” shown in eq. (8) are all gauge invariant under the background field transformations

 AμB(x) →AμB(x)+1e∂μα(x), (13) Aμ(x) →Aμ(x), (14) ϕB(x) →e−iα(x)ϕB(x), (15) ϕ(x) →e−iα(x)ϕ(x). (16)

In this work, we have omitted the ghost part arising from the introduction of gauge-fixing term, because it does not couple to any fields and therefore has no effect.

Here a few remarks are in order here. To just preserve the gauge invariance of the effective action, any gauge-fixing term taking the following general form is admissible

 Lgf(Aμ,AμB)=−f(Aμ(x)−AμB(x))2α0 (17)

where is an arbitral Lorentz scalar function. For example, the following gauge-fixing term is possible

 Lgf(Aμ,AμB)=−(Aμ(x)−AμB(x))22α0, (18)

and the gauge-fixing term in the functional integral, accordingly, is

 Lgf(Aμ,AμB)=−(Aμ(x))22α0, (19)

which can be seen as a mass term for the photon field. However, this gauge-fixing term is difficult to use in our practical calculation since the form of the photon propagator. Therefore, for simplicity of calculation, in this paper we choose the gauge-fixing term shown in eq. (7).

The gauge invariance of the effective action follows from the gauge invariance of the gauge-fixing term shown in eq. (7) and that of the “Lagrangian” shown in eq. (8). And the following identity then can be proved

 Ze√ZA=1, (20)

where (to be defined later) is the renormalization constant for background photon field, the (to be defined later) is the renormalization constant for coupling. Here, following the treatment presented in (Weinberg:1996kr, ), we give a brief proof about identity .

The gauge invariance of the effective action guarantees that the divergence in the effective action, which is just a functional of background fields, takes the form

 Γ[AB,ψB]=∫dx−LA4(FμνB)2+Lϕ∣DμϕB∣2−Lmm2∣ϕB∣2. (21)

Adding this to the classical part obtained by setting all quantum parts in the ordinary “Lagrangian” shown in eq. (8) to zero, defining the renormalized quantities as

 ARμ(x) =√1+LAABμ(x), (22) ϕR(x) =√1+LϕϕB(x), (23) m2R =(1+Lm)m21+Lϕ, (24)

we get a renormalized effective action distinguished from the unrenormalized by superscript

 ΓR[AR,ψR]=∫dx−14(FRμν)2−m2R∣ϕR∣2+∣∂μϕR+ie0ABμϕR∣2. (25)

All the quantities, except the and , appearing in this renormalized effective action are renormalized quantities. So from the finiteness of the effective action, we conclude that must be finite and don’t renormalize, i.e. we can set

 Ze√ZA=1. (26)

Also along the treatment presented in(Abbott:1980hw, ; Abbott:1981ke, ), we can get the same identity.

### ii.2 Beta Function in background field formalism

In background field method, since the identity , the beta function is fully determined by the renormalization constant .

In dimensional regularization with dimensions, the bare and renormalized coupling are related by

 e0=μϵeZe, (27)

where, is the renormalization scale, the renormalization constant for the running coupling constant. Setting , basing on the independence of bare coupling on , we get

 βϵ(e)[2−e∂∂e]ZA=−2ϵeZA, (28)

where, , and , like any other renormalization constants in the minimal subtraction scheme, is usually written as

 ZA=1+∞∑i=1Z(i)Aϵi. (29)

The expression in the right hand side of eq. (28) has two terms without poles in dimensional regulator , one of them being of order , another of them being of order . This leads us to establish that must have a term proportional to (terms of order , , are excluded because of the absence of a corresponding part in the right hand side of eq. (28)). Taking this and the finiteness of the beta function into considerations, we can set

 βϵ(e)=−ϵe+β(e), (30)

where is the conventional beta function. Substituting eq. (30) in eq. (28),we have

 β(e)[2−e∂∂e]ZA=−ϵe2∂ZA∂e. (31)

The coefficients , according to this equation, are related to each other through

 β(e)[2−e∂∂e]Z(i)A=−e2∂∂eZ(i+1)A. (32)

The beta function as usual can be easily obtained, by setting , as

 β(e)=−12e2∂∂eZ(1)A. (33)

To conclude this subsection, we want to make the following remarks concerning the absence of pole term in . To illustrate this issue, it’s convenient to introduce a notation to represent the pole term coming from the contribution of the loop. Since our calculation is only up to order , in eq. (32) we can retain only the fist term (one loop beta function) of the beta function, that’s to say, in our approximation, we can write

 β1e3Nf[2−e∂∂e]Z(i,j)A=−e2∂∂eZ(i+1,j+1)A, (34)

where to be determined later is a constant of order . Since is proportional to , we immediately conclude that by setting . Then the identity can be got by successive setting .

## Iii Background field method calculation

In this section, for practical calculation, we define the renormalized quantities as follows

 AμB(x) =√ZAArμB(x), (35) Aμ(x) =√Z3Arμ(x), (36) e0 =Zee, (37) α0 =Zαα. (38)

In spinor QED, because of the Ward identity. In scalar QED this identity can be proved also through the use of the Ward identity (a brief proof is given in our appendix). Because of this identity, we, generally speaking, have two distinct way to carry out our calculation. In the first way, we cancel all the renormalization factors in the gauge-fixing term and get

 Lgf=−(∂μArμ(x))22α. (39)

In the second way, we split the gauge-fixing term as

 Lgf=−(∂μAμ(x))22α−(δZα)(∂μAμ(x))22α, (40)

with a symbol for . In what follows, borrowing the name from (Capper:1982tf, ), we shall call the first way “direct approach” and the second way “indirect approach”.

In practical calculation within background field method, as is well-known, we can avoid the introduction of renormalization procedures for some quantum fields which only appear in the internal lines. This can be applied to scalar field directly in this work. However, in “direct approach”, we can’t do this directly to the photon filed, since in this approach we can’t extract a factor from the gauge-fixing term shown in eq. (39) after its cancellation with to cancel that from vertexes. In this approach, we use this gauge-fixing term and a part to produce the covariant propagator

 DμνF=−i(gμνk2−(1−α)kμνk2)(k2)2, (41)

with being the momentum going through it, and the corresponding counterterms is generated from.

In “indirect approach”, the first term in eq. (40) dose not bring any problem to avoiding the renormalization procedure for the photon field because it contain an overall factor to cancel that from vertexes, while the second term contain an extra factor closely related to and brings problem. However, in our approximation (up to order ), we can avoid this renormalization procedure in the “indirect approach”, as long as the total effects of the second term of eq. (40) in our calculation vanishes, a matter to which we shall turn later.

### iii.1 Direct Approach Calculation

In this subsection, we shall calculate by means of the “direct approach” of the background field method; then in next subsection we shall prove the equivalence between this two approaches.

Firstly, the Feynman rules for propagators and vertexes shown in figure 1 and figure 2 are as follows

 F1.1 =ip2−m2, (42) F1.2 =−i(gμνp2+(α−1)pμpν)(p2)2, (43) F1.3 =−ie(pμ+kν), (44) F1.4 =2ie2gμν, (45) F2.1 =−ie(pμ+kν), (46) F2.2 =2ie2gμν, (47) F2.3 =2ie2gμν. (48)

Here, we have added a factor 2 in eq. (47), since we have two such kind of terms in our Lagrangian, while the factor 2 in eq. (45), and (48) come form the two choices of “contraction” we have.

Before the concrete calculation, we want to say that through out this paper we shall adopt the dimensional regularization in dimensions and the minimal subtraction like scheme (the calculation in this section is done in the minimal subtraction scheme). Having chosen the dimensional regularization (DR), we can ignore the third vertex shown in figure 2 since in this work this vertex is just used to produce “tadpole” diagrams whose contributions vanish in DR. Also, since in this work we are only concerned with the renormalization constant, we can set the mass to zero, as long as this does not introduce any IR problem.

Now, let’s begin our calculation with the evaluation of at one loop level. This can be done by calculating the first graph shown in figure 3 (since we have adopted dimensional regularization there is no contribution from the second graph in figure 3), whose contribution reads

 −i(gμνp2−pμpν)e2Nf48π2(4πμ2−p2)ϵD[ϵ], (49)

where is the renormalization scale, is the momentum flowing through this diagram, the number of scalar fields, and

 D(ϵ)=3Γ(1−ϵ)2Γ(ϵ)(3−2ϵ)Γ(2−2ϵ). (50)

Therefore, the corresponding at one loop level is

 Z13=−e2Nf48π2ϵ. (51)

Here and in what follows, we use a superscript in to indicate that this is the loop contribution to . Since the contribution of the one loop photon self energy graph (scalar bubble) shown in eq. (49) is transverse, we can establish that at one loop level the identity must hold—in fact this is enough for our approximation in this work. Obviously the value of equal to . This identity and the indicate that . Therefore, in our large approximation with calculation carried out only up to order (we assume that is of order ), we need not worry about the vertex corrections.

The calculations of higher order diagrams are a little more complicated since the appearance of overlapping divergence. The techniques involved in our calculation are the Gegenbauer polynomial technique (Chetyrkin:1980pr, ) and the integration by parts approach (Chetyrkin:1981qh, ; Grozin:2007zz, ; Smirnov:2006ry, ).

At two loop level, after dropping the tadpole diagrams, we have eight diagrams shown in figure 4 and figure 5 to consider. The total contributions of these diagrams to , without any dependence, read

 Z2A=−e4Nf128π4ϵ. (52)

A few remarks are in order here. To obtain the renormalization constants, we just need the divergence part in the results, but a detail calculation shows that the dependence are cancelled completely between two-loop diagrams (By this complete cancellation, we mean that not only the divergent part but also the remaining finite part are cancelled exactly and completely). This complete cancellation of dependence at two loop level will be used to prove the independence of later in the “indirect approach”.

Higher order contributions, in our large approximation up to order , come from diagrams generated by replacing the internal photon lines in figure 4 and figure 5 with “dressed” photon chains (photon propagator chain carrying some renormalized one-loop scalar bubbles) shown in figure 6; no other diagrams need be taken into consideration because of the suppress factor . Also, since the one loop photon self energy graph is transverse there are no dependence in these higher order diagram. The contributions of higher loop order diagrams to (here we give the results up to seven loop order, in later sections, we shall shown that the is completely determined by a simple function and give a closed form for at leading order) are as follows

 Z3A =−e6N2f18432π6ϵ2+49e6N2f221184π6ϵ, (53) Z4A =−e8N3f1769472π8ϵ3+49e8N3f21233664π8ϵ2−323e8N3f127401984π8ϵ, (54) Z5A =−e10N4f141557760π10ϵ4+49e10N4f1698693120π10ϵ3−323e10N4f10192158720π10ϵ2 −e10N4f(288ζ(3)−113)20384317440π10ϵ, (55)
 Z6A =−e12N5f10192158720π12ϵ5+49e12N5f122305904640π12ϵ4−323e12N5f733835427840π12ϵ3 −e12N5f(288ζ(3)−113)1467670855680π12ϵ2−e12N5f(−3920ζ(3)+65+16π4)4892236185600π12ϵ, (56) Z7A =−e14N6f684913065984π14ϵ6+7e14N6f1174136684544π14ϵ5−323e14N6f49313740750848π14ϵ4 −e14N6f(288ζ(3)−113)98627481501696π14ϵ3−e14N6f(−3920ζ(3)+65+16π4)328758271672320π14ϵ2 −e14N6f(5(5168ζ(3)+3456ζ(5)+87)−392π4)1972549630033920π14ϵ. (57)

Although, for the evaluation of the beta function it’s only necessary to write down the simple pole term in , we, here, write down all the pole terms in for checking the validity of our calculation. Seeing from the above results, we also find that there is no pole term () in , a mater which we have discussed in section II and will return to in section VI in a more directly way.

### iii.2 Indirect Approach Calculation

In this section, we shall prove the equivalence between the “direct approach” and the “indirect approach” of background field method in calculating .

In “indirect approach’, apart from the usual vertex, we have a new vertex (photon-photon vertex arising from the second term of the gauge fixing term shown in eq. (40)) shown in figure 7 to consider, whose Feynman rule is

 ~Dμν(k)=−i(1Zα−1)kμkνα, (58)

where, is the momentum going through this vertex, and the Lorentz indices.

As has been mentioned before, we first come to the issue about avoiding the introduction of renormalization procedure for gauge field. It’s obvious that if the effect of the new vertex shown in figure 7 is cancelled between diagrams, we can avoid the renormalization procedure for photon field by just dropping this term. In our larger approximation up to order , this issue, in fact, is closely related to the independencies.

The cancellation of dependencies at two loop level (here we just consider diagrams not carrying the new vertex) is the same as that in previous subsection with the only difference being that the coupling in the “direct approach” is the renormalized coupling while in “indirect approach” is the bare coupling (the two couplings attaching to the background external legs can be seen as the renormalized coupling because of ). From the complete cancellation of dependence in “direct approach” at two loop level, we can establish that the contributions from the longitudinal part of photon propagator are cancelled completely between two loop diagrams in “direct approach”. Also since in “indirect approach” the one loop photon self energy graph is obviously transverse, the contribution coming from diagram with any number of this one loop self energy graphs and any number of the new vertex with Feynman rule shown in eq. (58) vanishes, that’s is to say they can’t appear in a diagram simultaneously. Therefore the remaining dependencies come from the insertions of these new vertexes in the photon internal lines in the two loop diagrams. However, the effect of a insertion of a new vertex in the internal photon chain just result in an overall factor and maintain the longitudinal form since

 DμνF(k)~Dνρ(k)DρσF(k)∝~Dμσ(k). (59)

From this equation and the cancellation of dependencies at two loop level, we conclude that, in our approximation, the independencies and the vanishing of the total contribution of the new vertex are proved. Therefore, in “indirect approach”, we can avoid all renormalization procedures for quantum fields.

Now, we are only left with diagrams without the new vertex to consider. In what follows, we shall prove that a diagram (except the one loop diagram) interpreted in the ‘indirect approach‘ is a sum of an infinite number of diagrams interpreted in the “direct approach”. Of course, at one loop level there is no difference between these two kinds of approach in calculating .

In our large approximation up to order , two loop diagrams (except the one loop diagram), which contribute to in “direct approach” have been shown in figure 4 and figure 5; in “indirect approach”, two loop diagrams contributing to are, in shape, look like the diagrams in figure 4 and figure 5, with the only difference being that in “direct approach”, all the couplings in our calculation are the renormalized coupling, while in “indirect approach” all the couplings appearing in our calculation are the bare coupling (except the two attaching to the external legs). Another difference between these two approaches in higher loop order is that the “one loop photon self energy diagram (scalar bubble)” in “indirect approach” is unrenormalized while usually renormalized in “direct approach”. Therefore, in our approximation, to prove the equivalence between these two approach, we can focus on the equivalence of the “photon propagator chain” in these two approaches.

Our procedures for proving the equivalence between “indirect approach” and “direct approach” is similar to that used in a work of us twozzy2017 ()(in that work (still unpublished when we write this paper) we focus on the large order behaviour of QED).

In loop level within “indirect approach”, the most general “photon propagator chain” is of the form

 −i(gμνk2−kμkν)(−k2)2+nϵ(e0)2+2n(−B[ϵ])n, (60)

with

 B[ϵ]=Nf(4πμ2)ϵΓ(1−ϵ)2Γ(ϵ)16π2(3−2ϵ)Γ(2−2ϵ) (61)

where the appearance of the extra in the denominator of the “propagator”, the are a consequence of the insertion of unrenormalized scalar bubbles. For later convenience, we have put the two bare coupling arising from the two vertexes linked to “photon propagator chain” in eq. (60).

To prove the equivalence, we first express the bare couplings in eq. (60) in terms of the renormalized coupling by means of Taylor expansion

 (e0)2n+2=e2n+2(ZA)n+1=e2n+2(1+∞∑k=01(k+1)!k∏i=0(n+1+i)(−Z1A)k+1+O(1/Nf)), (62)

from which eq. (60) can be rewritten as

 −i(gμνk2−kμkν)(−k2)2+nϵ(−B[ϵ])ne2n+2(1+∞∑k=01(k+1)!k∏i=0(n+1+i)(−Z1A)k+1+O(1/Nf)). (63)

In “direct approach”, we encounter a set of diagrams each of which carries unrenormalized scalar bubbles and a certain number of counterterms. Let’s consider one of these diagrams, say containing counterterms; the “photon propagator chain”of this Feynman diagram is

 −igμνk2−kμkν(−k2)2+nϵ(e)2+2n(−B[ϵ])n(−Z13)k+1. (64)

Note that an interchange between a counterterms and a unrenormalized scalar bubbles does not bring any change in the expression for the “photon propagator chain”. Therefore we have equivalent diagrams in “direct approach”, and the number of diagram equivalent to diagram is

 Cnn+k+1=(n+k+1)!n!(k+1)!=1(k+1)!k∏i=0(n+1+i), (65)

this is just the coefficient of in eq. (63). Multiplying expression (65) with expression (64), taking the summation over , recalling , and then comparing the result of these two operation with eq. (62), we can conclude that the equivalence between the “direct approach” and “indirect approach” is proved.

## Iv Investigation of the renormalons

As is shown in previous section, to calculate the beta function , we, by means of background field method, just need to calculate the corresponding two point Green’s functions. In this section we shall investigate the renormalon property of this kind of two point Green’s functions (TPGFs) in the large approximation by two different way.

### iv.1 A brief review of Borel transform

In quantum field theory, as is well-known, to extend our calculation to include all Feynman diagrams is impossible and beyond our calculational powers. Most successful applications of quantum field theory are based on the use of perturbation methods used when the interactions between particles (elementary or composite particles) are weak. And our results obtained by means of perturbation methods are usually expressed as a series

 R[g]=∑rngn, (66)

An important issue in any series is whether the series are convergent or not. For example, in some cases the coefficient may grows as , which indicates that the convergence radio of is zero(Weinberg:1996kr, ).

There is a well-known mathematical technique called Borel transformation which can be used to improve the convergence property of a series. The Borel transform of , in this work, is defined as

 BR[t]=∑rnn!gn, (67)

which obviously have a better convergence property than the original series . After the acquirement of the , the recovering of is formally done through

 gR[g]=∫∞0e−tgBR[t]dt. (68)

However, if there are singularities in , we can’t guarantee the validity of this equation. A singularity in is called a ultraviolet or infrared renormalon (which name you call it depends on how this renormalon appears), and the renormalon may prevent us from using eq. (68) to recover . However, according to eq. (68), the singularity on the negative real axis does not pose any problem to prevent us from using eq. (68) to recover since for the recovering of we only need for real positive .

### iv.2 The Borel transform of two-point Green’s functions—LTR approach

As has been shown in section 3, in the “direct approach” of background field method, the higher order diagrams which contribute in our approximation are generated by inserting a certain number of renormalized one-loop photon self energy graphs (scalar bubbles) shown in figure 3 into the photon line of the two loop diagrams shown in figure 4 and figure 5.

The insertion of an unrenormalized scalar bubble into the photon line of diagrams shown in figure 4 and figure 5 just leads to a multiplicative factor

 g(4πμ2−k2)ϵ−F[ϵ]ϵ, (69)

where , is the momentum going through the scalar bubbles and is given by,

 F[ϵ]=3Γ(1−ϵ)2Γ(ϵ+1)(3−2ϵ)Γ(2−2ϵ), (70)

while an insertion of a counterterms does not bring any change except an divergent factor (here we want to emphasise that through out this section we shall proceed in the Landau gauge—this choice of gauge does not bring any essential changes in our investigation about the larger order behaviour of the two point Green’s function, because we have proved the independence before in section 3).

Firstly, let’s begin our investigation with a diagram containing solely unrenormalized scalar bubbles (Note that the total number of these diagrams is eight, since the number of the prototype two loop diagrams shown in figure 4 and figure 5 is eight). The total expression for these eight diagrams reads

 −i(gρνp2−pρpν)Πn(g), (71)

with

 Πn(g)=gn(n+2)ϵn+1π[ϵ,(n+2)ϵ], (72)

where is the external momentum, and as usual are the Lorentz indices, and

 π[ϵ,s]=(−F[ϵ])s/ϵ−2H[s,ϵ]. (73)

For later illustration, we find the following function is useful

 π′[ϵ,s]=(F[ϵ])s/ϵ−2H[s,ϵ] (74)

where is analytic in at and of the form

 H[u+2ϵ,ϵ]=(4πμ2−p2)(u+2ϵ)e4Nf(u+2ϵ){−Γ(1−ϵ)2Γ(1−u−2ϵ)Γ(1−u−ϵ)Γ(u+ϵ)× Γ(u+2ϵ)(u2(4ϵ+3)+u(12ϵ2+ϵ−6)+8(ϵ(ϵ2+ϵ−4)+2))128π4(2ϵ−3)Γ(u+1)Γ(−u−3ϵ+3)Γ(−u−2ϵ+3)Γ(u+ϵ+1)+ 2G(ϵ,u−1)+5G(ϵ,u)+2G(ϵ,u+1)256π4(2ϵ−3)} (75)

This expression can be further reduced by means of the integration by part technique to

 H[u+2ϵ,ϵ]=(4πμ2−p2)(u+2ϵ)e4Nf(u+2ϵ){−Γ(1−ϵ)2Γ(1−u−2ϵ)Γ(1−u−ϵ)Γ(u+2ϵ)× u3(4ϵ−7)+u2(ϵ(24ϵ−65)+38)+2u(ϵ−1)(ϵ(24ϵ−67)+37)+16(ϵ−1)2(2(ϵ−3)ϵ+3)128π4(2ϵ−3)(u+2ϵ−2)(u+2ϵ−1)Γ(u+1)Γ(−u−3ϵ+3)Γ(−u−2ϵ+3) −(u(u+3ϵ−2)+4(ϵ−1))G(ϵ,1+u)256π4(2ϵ−3)(u+2ϵ−2)(u+2ϵ−1)}, (76)

where is defined proportional to

 ∫ddl1ddl2(2π)2d1(l21)(l22)(l23)(l24)(l25)1+u (77)

where, , , , and . This scalar integral appears as a result of the overlapping divergence we encounter in calculating diagrams generated by inserting the scalar bubbles into the photon line of the first diagram shown in figure 4 and is convergent for . In next subsection, we shall give some details about this function. Here the most important property of this function, which we shall use is that there is no pole in when .

Firstly, we define the following series

 π[ϵ,s] =∞∑i=0πj[ϵ]sj,π′[ϵ,s]=∞∑i=0π′j[ϵ]sj (78) π0[ϵ] =∞∑j=0gjϵj,π′0[ϵ]=∞∑j=0g′jϵj (79)

The important point in these four definitions, as can been seen from eq. (76), is that there are no pole terms in these four expressions.

New diagrams which contribute in our approximation can be generated by replacing some or all of the unrenormalized scalar bubbles in those eight diagrams by the counterterms. Taking all these diagrams into consideration, we get the following result

 Πtn(g)=gn{n∑j=01ϵn+11n+2−jn!j!(n−j)!π[ϵ,(n+2−j)ϵ]}, (80)

where the combinatorial factor is come from the number of choices we have in replacing just scalar bubbles with the counterterms. We also can rewrite this equation in terms of , with the result being

 Πtn(g)=(−g)n{n∑j=01ϵn+1(−1)jn+2−jn!j!(n−j)!π′[ϵ,(n+2−j)ϵ]}, (81)

Substituting eq. (78) in eq. (81), we have

 Πtn(g)=(−g)n{n+1∑i=0π′i[ϵ]ϵn+1−in∑j=0(−1)jn!j!(n−j)!(n+2−j)i−1}. (82)

The following combinatoric identity shown in PalanquesMestre:1983zy (); Beneke:1992ch () reduces our sum over to sum over only two special case and

 n∑k=0Ckn(−1)k(n+2−k)i−1=0,(1<=i<=n). (83)

For the case , the sum over has been given in PalanquesMestre:1983zy () as

 n∑j=0Cjn(−1)j1n+2−j=(−1)n(n+2)(n+1) (84)

For the case , the sum over , we find, is

 n∑k=0Ckn(−1)k(n+2−k)n=n! (85)

Thus, eq. (82) can be rewritten as

 Πtn(g)=gn{1(n+1)(n+2)π′0[ϵ]ϵn+1+(−1)nn!π′n+1[ϵ]}, (86)

where the , according to eq. (73) is given by

 π′0[ϵ]=H[0,ϵ]F[ϵ]2=−e4Nf(3−2ϵ)2(ϵ−4)(2ϵ−1)Γ(2−2ϵ)1152π4Γ(1−ϵ)2Γ(3−ϵ)Γ(ϵ+1). (87)

Since the second term in the bracket of eq. (86) suffers from no pole in , this equation indicates that the renormalization constant is totally determined by function . This partially reflect an important aspect of the renormalizable theories that only one new divergence arises when we go to a new loop order.

Now we turn to the Borel transform of the two point Green’s function

 BΠ[t] =∞∑n=0Πtn(t)n!=∞∑n=0tn(n+2)!π′0[ϵ]ϵn+1+∞∑n=0π′n+1[ϵ](−t)n =∞∑n=0n+1∑i=0tn(n+2)!g′iϵn+1−i+∞∑n=0π′n+1[ϵ](−t)n. (88)

The sum over is truncated at n+1, since we are only interested in pole terms (especially simple pole term since only this kind of pole is related to beta function which is of physical importance) and finite terms.

Extracting the simple pole term in eq. (88), we have

 B1Π[t]=∞∑n=0tng′n(n+2)!1ϵ=1ϵt2∞∑n=0tn+2g′n(n+2)!=1ϵt2H[t], (89)

with

 d2dt2H[t]=∞∑n=0g′ntnn!=Bπ′0[t], (90)

Where is the Borel transform of . Therefore, taking , into consideration, we can write eq. (89) as

 B1Π[t]=1ϵt2∫t0dx∫x0dyBπ′0[y]. (91)

By the property of the Borel transform and the introduction of a new function , we can rewrite eq. (91) as

 B1Π[t]=1ϵt2Bπ2[t]. (92)

Having investigated the Borel transform of the simple pole term, we turn to the Borel transform of the beta function. Implicit in our discussion given above is that we write the renormalization constant in the form

 Z(1)A=∞∑i=0rngn. (93)

Therefore the beta function, according to eq. (33) is given by

 β(e)=−12e2