Renormalization group dependence of the QCD coupling

The general relation between the standard expansion coefficients and the beta function for the QCD coupling is exactly derived in a mathematically strict way. It is accordingly found that an infinite number of logarithmic terms are lost in the standard expansion with a finite order, and these lost terms can be given in a closed form. Numerical calculations, by a new matching-invariant coupling with the corresponding beta function to four-loop level, show that the new expansion converges much faster.

The general relation between the standard expansion coefficients and the beta function for the QCD coupling is exactly derived in a mathematically strict way. It is accordingly found that an infinite number of logarithmic terms are lost in the standard expansion with a finite order, and these lost terms can be given in a closed form. Numerical calculations, by a new matching-invariant coupling with the corresponding beta function to four-loop level, show that the new expansion converges much faster.
In practical applications, it is convenient to have an explicit expression of the α as a function of the renormalization point u. The standard approach is to expand it to a series of L = 1/ ln(u 2 /Λ 2 ), where Λ is the QCD scale parameter. However, how the expansion coefficients are connected to the beta function is not generally known, though one can find the relation to order 3 in Ref. [8], and to order 4 in Ref. [9]. In this letter, the general relation between the expansion coefficients and the beta * Email: gxpeng@ihep.ac.cn, gxpeng@lns.mit.edu function are provided. It is accordingly found that an infinite number of terms like ln j L are lost in the standard expansion with a finite order, and these lost terms can be given in a closed form. Numerical calculations, by a new matching-invariant coupling with the corresponding beta function to four-loop level, show that the new expansion, with the lost terms included, converges much faster.
To solve Eq. (1), let's definé With the series expression for β(α) in Eq. (1), one can easily get an explicit expressioń which indicates thatβ(α) is analytic at α = 0, and can thus be expanded to a Taylor series aś The expansion coefficientsβ k can be obtained from the normal mathematical formulá Another easy way to obtain these coefficients is to use the recursive relatioń with the obvious initial conditionβ 0 = β 2 /β 1 − β 1 /β 0 . Let L = 1/ ln(u/Λ), where Λ is a dimensional parameter, then Eq. (1) becomes −L 2 dα dL = β(α), or − dL where C ′ is the constant of integration. Let α = L β0 Y (L), then where L * ≡ (β 1 /β 2 0 )L, C ≡ (β 2 0 /β 1 )C ′ − ln β 0 , and The graved beta functionβ k can be easily obtained from the expression for the acute beta functionβ k in Eq. (6) or (7), and here are the results: l,s =β ls+1+1 +β 2βls+1 , B (2) l,s =β ls+2+2 , andβ i = β i /β 1 (i = 0, 1, 2, . . .). Now suppose we have a solution of the form where the square cup operator, , has been defined in the appendix. Substituting these expressions into Eq. (9), we can obtain all f i,j by comparing the corresponding coefficients of u * i ln j u. For (i, j) = (0, 0), (1, 0), and (1, 1), we have f 0,0 = 1, Please note, there are only terms of f i ′ <i,j ′ <j on the right hand side of this equation. Therefore, it is a recursive relation. Here are the solution to order 5: f 4,2 = 6C 2 + 13C + 6β 1 + 9/2, (25) · · · These correspond to the standard form in Ref. [8] at order 3, and agrees to that in Ref. [9] at order 4, i.e., To give a general representation for the expansion coefficients f i,j , introduce a set of new functionsβ i by the recursive relation where K ≡ (−1) k−1 (1+1/k). From the initial conditions β 0 = 1 andβ 1 = C, one can easily get allβ i from Eq. (27). For C = 0, for example, we havē Please note, even when one sets all β i>2 to zero, noβ k>2 will be zero. This is one of the most important reason for us to know the general relation between f i,j and the beta function.
It is found that f i,j satisfies where k 0 Φ l are the Taylor coefficients of ln k (1 + x), i.e., ln (32) One can naturally consider to prove Eq. (31) by mathematical induction. We have numerically checked it to order 100. Now we directly used it to give an expression for f i,j , i.e., where P (i) j,k can be obtained by repeatedly using Eq. (31). For k = i − j and k = i − j − 1, it is easy to get where Q is a function of three non-negative integers and defined by For a given i, one can regard P (i) = P (i) j,k as a matrix of order i + 1. The specially simple elements are: P In the traditional minimum subtraction scheme (MS), the strong coupling α(u) as a function of the renormalization point u is not continuous at the quark masses. Let's derive a matching-invariant coupling by absorbing loop effects into the MS definition and give the corresponding beta function to four-loop level.
Suppose the new coupling α ′ is connected to the original coupling α by Then, using the matching conditionα = ∞ j=0 C j α j+1 with the matching coefficients [9,10] C 0 = 1, C 1 = 0, C 2 = 11/72, where an overhead check means decreasing N f by one flavor to the corresponding (N f − 1)-flavor effective theory. Accordingly, comparing the coefficients of α in the equalityα ′ = α ′ yields Assume

Substitution into Eq. (42) then gives
whose solution is a 0 = a 0,0 , a 1 = a 1,0 , a 2 = a 2,0 + 11 72 To definitely fix the new coupling, one needs to choose a i,0 . The simplest choice would be a i,0 = δ i,0 . With this convention, one has where a 3,1 = 7037/1536 + 82043ζ 3 /27648 ≈ 8.148377983. Then the new matching-invariant coupling is The renormalization equation for α ′ is The primed beta function β ′ i can be obtained as such. Operating with u d du on both sides of Eq. (38), applying Eqs. (49) and (1), and then comparing coefficients will give i k=0 β ′ k k+2 0 a i−k − (k + 1)a k β i−k = 0, namely, β ′ i are given by the recursive relation On application of Eqs. (46) and (47), one immediately has the following explicit expressions for the new beta function: · · · with β It should be mentioned that a different expression for β ′ 2 was previously given in Ref. [11]. The difference is caused by the fact that a wrong value for C 2 was quoted there [12].
As an application of the general relation between f i,j and the beta function, one can develop another expansion which converges much faster. For this one can observe, more carefully, the standard expansion Representing the terms in this expansion with the corresponding coefficients f i,j , all the terms can be arranged in a matrix as The standard expansion corresponds to summing the terms row by row. When one takes the expansion to a finite order, i.e., replacing the ∞ in Eq. (55) with a positive integer N , as has been done in the usual way, then all the terms like f j,j (N < j < ∞) on the diagonal and f j+1,j on the next to diagonal are missed, although these terms are all known and have nothing to do with beta functions. Generally, the terms f j+k,j for 0 < j < ∞ on the kth next to diagonal involves only β 0≤l≤k−2 . But all the terms f j+k,j with j > N are lost, though no such terms are zero even when one sets all β i>2 to zero.
To include the contribution from the terms just mentioned, we can consider to sum over diagonals, which can be achieved by taking i = j + k in Eq. (55), i.e., where the expressions for f j+k,j can be obtained from Eq. (33): From these expressions, we can give compact form to I k : where x ≡ 1 + (β 1 /β 2 0 ) ln ln(u/Λ)/ ln(u/Λ), and X ≡ L * /(β 1 x) = 1/[β 2 0 ln(u/Λ) + β 1 ln ln(u/Λ)].  In Eq. (57), there are an infinite number of logarithmic terms like ln j x which are included in I k , even when one takes the expansion to a finite order, say α = N k=0 I k . The first several I k s have been worked out to a closed form in Eqs. (60)-(62). The procedure is, to some extent, similar to that in Ref. [13] for the matching function. Here we can, in fact, give the closed form for I k to arbitrary order: In Eqs. (63) and (55), there are two arbitrary constants: Λ and C. Because the renormalization group equation (49) or (1) is of the first order, only one of them is independent. So we can arbitrarily take one of them, while the other is determined by giving an initial condition. It nearly becomes standard, nowadays, to take C = 0 [14], which makes expressions somewhat simpler, and α(m Z ) = 0.1187/π, where M Z = 91.1876 GeV is the mass of Z bosons. Setting C = 0 requires distinct Λ for different effective flavor regimes, and we use Λ 6 , Λ 5 , Λ 4 , and Λ 3 for u > m t , m b < u < m t , m c < u < m b , and m s < u < m c , respectively, where the relevant quark masses are taken, in the present calculations, to be m t = 175 GeV, m b = 4.2 GeV, m c = 1.2 GeV, and m s = 100 MeV.
In Fig. 1, the coupling is shown as a function of the renormalization point, calculated from Eq. (63) with the infinity replaced by N , and the order N from 1 to 4. The same calculation has also been performed from the conventional expansion in Eq. (55). The relative difference between the results from Eq. (63) and Eq. (55) is shown in Fig. 2. It can be seen that, with decreasing u, the difference becomes more and more significant.  To compare the convergence speed of Eqs. (63) and (55), all the Λ i (i = 3-6) are listed in Tab. I. There are two columns corresponding to each Λ i , the left column is for Eq. (63) while the right column is for Eq. (55). It is obvious that the new expansion (63) converges much faster than the original expansion (55). Even at the leading-order (N = 1), the corresponding Λ i for Eq. (63) has nearly approached to its value at order 4. So in practical applications, it should be very accurate to calculate the coupling simply by α = β 0 β 2 0 ln(u/Λ) + β 1 ln ln(u/Λ) .
In summary, the general relation between the standard expansion coefficients and the beta function is carefully derived. A matching-invariant coupling is given with the corresponding beta function to four-loop level. A new expansion for the coupling is then deduced, which is in principle more accurate than the conventional expansion due to the inclusion of an infinite number of logarithmic terms in a closed form.