Background field method in the large $N_f$ expansion of scalar QED

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.

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-N f approximation. Our calculation carried out with an arbitrary gauge-fixing parameter α has been done up to order 1/N f , and the result of our calculation is independent of the gauge-fixing parameter α. 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.

I. INTRODUCTION
The scale dependence of a renormalized running coupling g(u) is determined by its beta function β[g(u)] which is a function of running coupling g(u) itself and is of fundamental importance. As is well-known, in the 1970s [1,2], 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 [3][4][5] and that of QCD [6][7][8] 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 N f 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 [9][10][11]), 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 [11].
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 [12] and others work in [13,14] . 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 [15].
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 R(α s ), we can study its Borel transform B R [t], which by definition has a better convergence property than the original R(α s ) [16]. After the acquirement of B R [t], if there are no singularities in B R [t] (a singularity in B R [t] is called a renormalon [16]), we can recover R(α s ). In the investigation of renormalon, the large N f approximation has been developed into a useful tool [17][18][19].
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 Z A and show the equivalence between two approaches of background field method in calculating Z A . 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 Z A ) and the coupling (say Z g ) are related to each other through Z g Z 1 2 A = 1. 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.

A. 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 with In eq. (1), we have ignored the interaction terms V (φ * φ) between scalar fields φ. The Lagrangian of scalar QED shown in eq. (1), obviously, is invariant under a general gauge transformations of the form 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 [20] we can replace the field φ in the conventional action with φ + φ 0 (φ being the variable of integration in the functional integral, while φ 0 being the background field), and then use the following formula to get the effective action where the 1PI means that we include all diagrams, connected or not, each connected component being one-particleirreducible. 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 where A B 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 The gauge-fixing term L gf shown in eq. (7), and the ordinary " Lagrangian" L shown in eq. (8) are all gauge invariant under the background field transformations 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 where f is an arbitral Lorentz scalar function. For example, the following gauge-fixing term is possible and the gauge-fixing term in the functional integral, accordingly, is 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 where Z A (to be defined later) is the renormalization constant for background photon field, the Z e (to be defined later) is the renormalization constant for coupling. Here, following the treatment presented in [20], we give a brief proof about identity Z e √ Z A = 1.
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 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 we get a renormalized effective action distinguished from the unrenormalized by superscript R All the quantities, except the e 0 and A B , appearing in this renormalized effective action are renormalized quantities. So from the finiteness of the effective action, we conclude that e 0 A B µ must be finite and don't renormalize, i.e. we can set Also along the treatment presented in [9,10], we can get the same identity.

B. Beta Function in background field formalism
In background field method, since the identity Z e √ Z A = 1, the beta function is fully determined by the renormalization constant Z A .
In dimensional regularization with 4 − 2 dimensions, the bare and renormalized coupling are related by where, µ is the renormalization scale, Z e the renormalization constant for the running coupling constant. Setting Z e = 1/ √ Z A , basing on the independence of bare coupling e 0 on µ, we get where, β (e) = µ de dµ , and Z A , like any other renormalization constants in the minimal subtraction scheme, is usually written as 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 0 . This leads us to establish that β (e) must have a term proportional to (terms of order 2 , 3 , . . . 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 β (e) into considerations, we can set where β(e) is the conventional beta function. Substituting eq. (30) in eq. (28),we have The coefficients Z (i) A , according to this equation, are related to each other through The beta function as usual can be easily obtained, by setting i = 0, as To conclude this subsection, we want to make the following remarks concerning the absence of pole term 1/ i in Z i A . To illustrate this issue, it's convenient to introduce a notation Z (i,j) A to represent the pole term 1/ i coming from the contribution of the jth loop. Since our calculation is only up to order 1/N f , 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 where β 1 to be determined later is a constant of order O(e 0 ). Since Z (1,1) A is proportional to e 2 , we immediately conclude that Z

III. BACKGROUND FIELD METHOD CALCULATION
In this section, for practical calculation, we define the renormalized quantities as follows In spinor QED, Z α = Z 3 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 In the second way, we split the gauge-fixing term as with δZ α a symbol for 1 Zα − 1. In what follows, borrowing the name from [21], 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 φ 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 Z 3 from the gauge-fixing term shown in eq. (39) after its cancellation with Z α to cancel that from vertexes. In this approach, we use this gauge-fixing term and a A 2 part − to produce the covariant propagator with k 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 Z 3 to cancel that from vertexes, while the second term contain an extra factor closely related to Z 3 and brings problem. However, in our approximation (up to order 1/N f ), 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.

A. Direct Approach Calculation
In this subsection, we shall calculate Z A 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 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 4 − 2 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 m to zero, as long as this does not introduce any IR problem. Now, let's begin our calculation with the evaluation of Z 3 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 where µ is the renormalization scale, p is the momentum flowing through this diagram, N f the number of scalar fields, and .
Therefore, the corresponding Z 3 at one loop level is Here and in what follows, we use a superscript i in Z i to indicate that this is the ith loop contribution to Z. 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 Z 3 = Z α must hold-in fact this is enough for our approximation in this work.
Obviously the value of Z 1 A equal to Z 1 3 . This identity and the Z e √ Z A = 1 indicate that Z 1 e Z 1 3 = 1. Therefore, in our large N f approximation with calculation carried out only up to order 1/N f (we assume that e 2 N f is of order O(1)), 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 [22] and the integration by parts approach [23][24][25]. At two loop level, after dropping the tadpole diagrams, we have eight diagrams shown in 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 α 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 α dependence at two loop level will be used to prove the α independence of Z A later in the "indirect approach". Higher order contributions, in our large N f approximation up to order 1/N f , 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 → FIG. 6: Photon chain consideration because of the suppress factor 1/N f . Also, since the one loop photon self energy graph is transverse there are no α dependence in these higher order diagram. The contributions of higher loop order diagrams to Z A (here we give the results up to seven loop order, in later sections, we shall shown that the Z A is completely determined by a simple function and give a closed form for Z A at leading order) are as follows Although, for the evaluation of the beta function it's only necessary to write down the simple pole term in Z A , we, here, write down all the pole terms in Z A for checking the validity of our calculation. Seeing from the above results, we also find that there is no pole term 1 i (i ≥ 2) in Z i A , a mater which we have discussed in section II and will return to in section VI in a more directly way.

B. 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 Z A .
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 where, k is the momentum going through this vertex, µ and ν 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 N f approximation up to order 1/N f , this issue, in fact, is closely related to the α independencies.
The cancellation of α 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 Z e √ Z A = 1). From the complete cancellation of α 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 α 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 From this equation and the cancellation of α dependencies at two loop level, we conclude that, in our approximation, the α 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 Z 1 A . In our large N f approximation up to order 1/N f , two loop diagrams (except the one loop diagram), which contribute to Z A in "direct approach" have been shown in figure 4 and figure 5; in "indirect approach", two loop diagrams contributing to Z A 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 two [26](in that work (still unpublished when we write this paper) we focus on the large order behaviour of QED).
In n + 2 loop level within "indirect approach", the most general "photon propagator chain" is of the form with where the appearance of the extra (−k 2 ) n in the denominator of the "propagator", the (−e 2 0 B[ ]) n are a consequence of the insertion of n 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 from which eq. (60) can be rewritten as In "direct approach", we encounter a set of diagrams each of which carries n unrenormalized scalar bubbles and a certain number of counterterms. Let's consider one of these diagrams, say D k+1 containing k + 1 counterterms; the "photon propagator chain"of this Feynman diagram is 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 D k+1 is this is just the coefficient of (−Z 1 A ) k+1 in eq. (63). Multiplying expression (65) with expression (64), taking the summation over k, recalling Z 1 A = Z 1 3 , 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 β(e), 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 N f approximation by two different way.

A. 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 An important issue in any series is whether the series are convergent or not. For example, in some cases the coefficient r n may grows as n!, which indicates that the convergence radio of R[g] is zero [20].
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 R[g], in this work, is defined as

B. 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 where g = e 2 N f 48π 2 , k is the momentum going through the scalar bubbles and F [ ] is given by, while an insertion of a counterterms does not bring any change except an divergent factor g/ (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 α independence before in section 3). Firstly, let's begin our investigation with a diagram containing solely n 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 with Π n (g) = g n (n + 2) n+1 π[ , (n + 2) ], where p is the external momentum, ρ and ν as usual are the Lorentz indices, and For later illustration, we find the following function is useful where H[s, ] is analytic in s at s = 0 and of the form This expression can be further reduced by means of the integration by part technique to where G[ , 1 + u] is defined proportional to where, l 3 = l 1 − p, l 4 = l 2 − p, l 5 = l 1 − l 2 , and d = 4 − 2 . 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 Re(u) < 1. 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 G[ , 1 + u] when u = n . Firstly, we define the following series 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 n unrenormalized scalar bubbles in those eight diagrams by the counterterms. Taking all these diagrams into consideration, we get the following result where the combinatorial factor n!/(j!(n − j)!) is come from the number of choices we have in replacing just j scalar bubbles with the counterterms. We also can rewrite this equation in terms of G [ , u], with the result being Substituting eq. (78) in eq. (81), we have The following combinatoric identity shown in [27,28] reduces our sum over i to sum over only two special case i = 0 and i = n + 1 For the case i = 0, the sum over j has been given in [27] as For the case i = n + 1, the sum over j, we find, is Thus, eq. (82) can be rewritten as Π t n (g) = g n { 1 (n + 1)(n + 2) where the π 0 [ ], according to eq. (73) is given by Since the second term in the bracket of eq. (86) suffers from no pole in , this equation indicates that the renormalization constant Z A is totally determined by function π 0 [ ]. 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 The sum over i 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 By the property of the Borel transform and the introduction of a new function π (2) [x] = x 2 π 0 [x], we can rewrite eq. (91) as 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 Z (1) A in the form Therefore the beta function, according to eq. (33) is given by where we have used e∂r n /∂e = 4r n (there is a global factor e 4 in all r n , as can be seen from eq. (75) and eq. (76)) and e∂g n /∂e = 2ng n . Eq. (94) suggests that the convergent property of the beta function is worse than that of the simple pole part of Z A . Form the definition of the Borel transform, we obtain the Borel transform of the beta function with Since the appearance of the differentiation, eq. (95) explicitly shows that the Borel transform of the beta function is more vulnerable to suffers from renormalons than the Borel transform of the simple pole part of Z A . Now, let's turn to the Borel transform of the finite term (renormalized Borel transform). The renormalized Borel transform in the minimal subtraction scheme is obtained by subtract only the pole terms in the original Borel transform. Thus from eq. (88), we get the renormalized Borel transform Here also by the introduction of a new function π (1) we can rewrite this equation as Since we shall take the limit = 0, the second terms in these two brackets cancel each other. And the first term in the second bracket, according to eq. (74) and eq. (78) is given by Therefore eq. (98) can be combine into a compatible form We shall derive it again in another way and investigate it more detail in next subsection.

C. The Borel transform of two-point Green's functions-RTL approach
In previous section, our investigation about the renormalons issues is vastly simplified by noting two combinatoric identities. In this subsection, we shall use a different approach inspired by that in [16] to reinvestigate the Borel transform issues of the two point Green's function more detailedly and further discuss the scheme dependence of the Borel transform in the context of this approach(Recently we have used this approach to study the large order behaviour of spinor QED in [26]). The essential point of this approach lies in the observation that in our approximation to investigate the renormalon issues of the two point Green's functions, we can first investigate the renormalon issues of the photon propagator chain.
Obviously an insertion of a renormalized scalar bubble into the photon line with momentum k leads to a multiplicative factor Π 0 (k 2 , g) where g = e 2 N f /48π 2 . And the Borel transform of the photon chain obviously is This can be rewritten as [16] B µν The result of the first step is still a very length expression of the form −i(g µν p 2 − p µ p ν )H[u + 2 ]/u + 2 where H[u + 2 , ] have been shown in eq. (75) and eq. (76). The overall factor −i(g µν p 2 − p µ p ν ) is the required polynomial in the external momentum p and shall be suppressed later. In the second step, we can make the following trick By definition, as is shown in previous section, the pole terms in the Borel transform of the two point Green's function take the form t i / j or u i / j . The maintenance of this form, as will be clear later, in this subsection is guaranteed by our trick on the second term of eq. (104). The first term of eq. (104) does not suffers from pole because of the analyticity of H[u + 2 , ] in u + 2 at u + 2 = 0 as can be seen from eq. (76). Therefore, we can take the limit → 0 within this term and then get For the resolution of the second term, we, by means of the α integral representation, can write the denominator as Substituting this identity in the second term of eq. (104), doing some elementary computation, we get An essential point in obtainment of this result we want to emphasize is that since we are concerned with the renormalization constant, the here take a positive real part. The first term of this expression does not have pole terms and can be written as As regarding to the second term of eq. (107), the exponential factor exp(−u/ ) can be expanded in powers of −u/ , and the P ( ) = H[0, ] /F [ ] 2 = π (1) ( ) given below can be expanded in power of These two expansion grantee that the pole terms take the form u i / j . After expanding this two components, we can write the second term of eq. (107) as the following series where P is the coefficient of the series P ( ) = P n n . Here, a few remarks are in order. Firstly, since the absence of pole term in eq. (105) and eq. (108), the pole terms of the Borel transform of TPGFs only come from eq. (110), that's is to say the renomalization constant Z A is fully determined by the function P ( ). Secondly, the function P ( ) is only a function of and have nothing to do with the external momentum p-in fact, this momentum independence of P ( ) reflects and grantees the momentum independence of the renormalization constant Z A .
Eq. (110) is a component part of the Borel transform of the TPGFs, according to which we can write the Borel transform of Z A in the minimal subtraction scheme as The second identity is guaranteed by P 0 = 0 which is obvious from eq. (109) and guarantees that there are no negative powers of t in our Borel transform.
Having got the Borel transform of Z A , we can recover Z A , or rather the coefficients of Z A (say r m ) in various ways, such as differentiating the Borel transform B Z A [t] with respect to t enough times and then setting t = 0, or multiplying m! back-all these two methods give the same result. Here, for clarity, we write Z A as then the Borel transform of it is Comparing eq. (113) with eq. (111), we get and then get Eq. (115) explicitly shows that there is no pole term 1/ i (i ≥ 2) in Z i A -for example, at three loop level,that's is to say m = 1, for Z 3 A to have pole term 1/ 3 , we must choose n = 0, but P n = 0; eq. (115) also shows the relationship between the pole terms in Z A explicitly.
Among all the pole terms in Z A , we are only interested in the simple pole part. Extracting the simple pole part in Z A , we get and then the beta function (here we don't include the one loop beta function) according to its definition is given by where P 1 = P (x)/x = π 0 (x). By introducing a new beta function β(g) = 2gβ(e)/e, we can also write The formal Borel transform of the beta function β(e), according to eq. (117) is given by As has been mentioned in [17], for the analyticity of subtraction function say S[u] (except at u = 0), there is a requirement that the renormalization group functions have convergent region, or at least they don't diverge as fast as factorials. Our results given above up to leading order show that the beta function does have a convergent region g < 5/2 which is obvious seen from eq. (109). Now, we turn to the renormalized Borel transform of TPGFs, which, as has been said before, in the minimal subtraction scheme is obtained by subtracting all the pole terms in the Borel transform. Therefore, the renormalized Borel transform consists of three parts of which two parts suffering from no pole have been shown in eq. (105) and eq. (108) and the third part of which according to eq. (110) is given by where B P [t] is the Borel transform of function P (x). Adding all three component of renormalized Borel transform, we have Before further study, we want to emphasize that the RTL approach used in this subsection and LTR approach used in previous subsection give the same resluts, which is obvious by comparing the results of this two subsection. As can be seen form eq. (121), the fist term of the renomalized Borel transform of TPGFs, except a exponential factor exp{−t( 8 3 − γ)}, is just the result of taking the limit = 0 in the original "expression" for our result of the fist step; the second terms can be seen as a "correction" to this operation.
The explicit form of the first term of the renomalized Borel transform, according to eq. (75) is where G[u] is the result of carefully setting = 0 in G[ , u] which has been expressed in [22] as a double sum; and according to that formula we find that To investigate the singularities of this function, we transform this into another form given in [28] which can be obtained by making the replacements m = k and n = l − k and using the identity The function G[u] given in [28] is of the form Eq. (125) define a function suffers from pole at u = −N with N being positive integers from the individual terms of the double sum. As has been mentioned before the convergent region of the G function is Re(u) < 2. In fact this function has a continuation with double pole at (postive or negative) integers (except u = 1). This analytic continuation is performed by the following symmetry between G Having got all the ingredient needed for our discussion about the renormalized Borel transform, we now turn to this investigation. The renormalized Borel transform shown in eq. (122) has an obvious singularity in t at t = 0 arising from the singularities of G[0] and G[−1] and the singulatity of the first term in the bracket of eq. (122), which has nothing to do with the large order behaviour-in fact near t = 0 D[t] is given by(in what follows we shall ignored the overall factor exp(t(γ − 8 3 ))( 4πµ 2 −p 2 ) t e 4 N f which has no singularities) Since we have subtracted the pole term, this singularities in the total rernormalized Borel transform is unwelcome. Noting that in the second term 1 t 2 B P [t] of expression (121), there is also a singularity in t at t = 0-near t = 0 this term behave as Therefore we conclude that in the total renormalized Borel transform there is no singularity in t at t = 0. After the cancelation of singularity at u = 0 with the 1 t 2 B P [t], D[t] still suffers from two kinds of singularities. The first kind of singularities having nothing to do with the singularity of the G function come from the singularities of the fist term in the bracket of eq. (122) which becomes singular when t = −1, −2. The second kind of singularities come from the singularities of G function which becomes singular when its argument t become an integer not equivalent to 1(Some properties of G function which we shall use in this work has been given in [28]).
When t is an integer and t = −1, −2, the singularities of D[t] come from the singularities of G function. And all these poles are in fact double pole since the pole in G function are double poles. When t = −1, −2, there are some cancelations of poles between the fist term and the second term in the bracket of eq. (122). Near t = −1, D[t] behaves as This singularity at t = −1 is required to disappear in spinor QED because of the absence of a gauge invariant operator of dimension two (in general the singularity at t = −n is accounted for by operator of dimension 2n [28]). Near t = −2, D[t] behaves as The singularities of D[t] at t = n,n = 1, 2, . . . are called ultraviolet renormalon since these singularities originate from high-momentum regions of integration in the loop integral. And these singularities destroy the Borel sumability of the series because they are on the positive real axis. Accordingly, the singularities at t = n,n = −1, −2, . . . are called infrared renormalon since they originate from low-momentum regions of integration in the loop integral.

D. Scheme dependence of the Borel transform
Our presentations given above is based on the adoption of the minimal subtraction scheme. In this subsection, we want to generalise our investigation to the minimal subtraction like scheme including the M S scheme and the minimal subtraction scheme. Fortunately our method prescribed above can be applied to the M S scheme with little changes.
The main difference between the M S scheme and the minimal subtraction scheme is that after the subtractions of the pole terms we can still subtract some finite terms in the M S scheme. And the renormalization factors, in the M S scheme, are usually of the following form where S usually is chosen to be of the form S = 1 + a + O( 2 ). In M S scheme the beta function is still determined by the simple pole of Z A , to be more prise by Z n,1 , and given by [29] β(e) = −e ∞ n=1 ng n Z n,1 .
Also, in M S scheme the finiteness of the beta function still allows us to determine the higher poles in Z A from the simple poles of it.
In M S scheme, the first change appears in the multiplicative factor Π 0 (k 2 , g) arising from the insertion of scalar bubbles into the photon propagator chain; with Since our insertion of renormalized scalar bubbles start with three loop level, the extra factor S 2 , which will be combined with an over all factor e 4 N f to generate (gS ) 2 = g 2 . Here a few remarks are in order. Since in our methods we start with the investigation of the Borel transform of the "photon propagator chain", this two extra coupling g 2 should not be seen as the "coupling" related to the Borel parameter t. That's is to say, if we write Π(g) = n Π n (g) = g 2 n r n g n, then the Borel transform of Π(g) is given by and the Borel transform of the renormalization constant Z A is given by where r n,p indicates the pole terms in r n which is universal in M S-like scheme. Having this in mind, we can establish that in M S-like schemes the beta functions determined by the simple pole of r n,b are the same beta function. Obviously, the renormalized Borel transform is scheme dependent. This scheme dependence is not surprising, since the finite results obtained by renormalization can be changed by varying the renormalization scheme and the Borel transform is defined as a Borel transform with respect to a renormalized coupling which is also scheme dependent (except we choose a physical coupling which is difficult to use in practical calculation).
In M S scheme the scheme dependent renormalized Borel transform changes into The first term of this expression obviously depend only on the part of S -this property of the Borel transform is in accordance with a general property of M S scheme that we have various choices for S , but only the part of S affected the renormalized Green's function [29]. Obviously, the second term is the same for all MS-like scheme-the S 2 in P ( ) dose not take participate in the expansion of P ( ) and in the end becomes 1 when we take the limit = 0. Eq. (139) also shows that the position of the renormalon of D[t] is not changed when we change our M S scheme.

V. THE VALUE OF THE BETA FUNCTION
The beta functions β(e) and β(g) not including one loop beta function are given by eq. (117) and eq. (118) respectively, where the coefficients P n can be got by expanding P (x) in powers of x. Doing this expansion work up to, for example, order O(x 8 ), we can get the beta function up to 9 loop level (β 1 (e,g) have nothing to do with scalar bubbles insertions and obviously is eg(g = e 2 N f /48π 2 ) for β(e) and 2g for β(g)).  Seeing from this result, we find that the beta function consists of the riemann zeta function ζ(s) defined as and some other rational number. This can be understand from the following relationship between the riemann zeta function and the Γ function and the following relationship between π 2n and ζ(2n) where B 2n is Bernoulli numbers which are a sequence of rational numbers. From formula (142) and eq. (109), we can establish that the Euler constant γ will not enter into our expression for the beta function. The result shown in eq. (140) is a little more complicated. In table. 1, we give some numerical values of the coefficients P n omitting the overall factore 4 N f /2304π 4 from which we can get the numerical results for the beta function through eq. (117) and eq. (118).
Also by using eq. (117) or eq. (118), we can get an "all-order" result (by this we mean that we include all contribution contributing at order 1/N f ) for any fixed running coupling. Although the beta function depends on N f , there are still something definite for the beta function β(e) or beta function β(g). For example, these two beta functions have only one zero at g = 0 and are always positive in the convergent region g < 5/2.

VI. SUMMARY AND CONCLUSION
In this paper, we have calculated the beta function in the context of background field method. Obviously, the similarity of Feynman rules shown in eq. (42-47), and the identities Z 1 = Z 2 (or Z e = Z − 1 2 3 ) , Z 3 = Z α which we shall prove in our Appendix, grantee the equivalence between the normal field approach and the "direct approach" of background field method in calculating the beta function.
In our work, we have made clear the role played by the gauge parameter by carrying out its renormalization in the "direct approach" and the "indirect approach" of the background field method; also, a closely related issue about the avoidance of the renormalization procedures for quantum fields have been discussed. The equivalence between these two approach has been proved in detail.
Furthermore, we have investigated the renormalon property of the two point Green functions which in background field method are closely related to the beta function. In this investigation, we find that the renormalization factor Z A and hence the beta function having a convergent region is solely determined by a very simple function. We also discuss the scheme dependence issues, finding or proving that the beta function is scheme independent in M S scheme (extending this study to other scheme is a further work), while the renormalized Borel transform suffers from renormalons is scheme dependent.