On a realization of $\{\beta\}$-expansion in QCD

We suggest a simple algebraic approach to fix the elements of the $\{ \beta \}$-expansion for renormalization group invariant quantities, which uses additional degrees of freedom. The approach is discussed in detail for N$^2$LO calculations in QCD with the MSSM gluino -- an additional degree of freedom. We derive the formulae of the $\{ \beta \}$-expansion for the nonsinglet Adler $D$-function and Bjorken polarized sum rules in the actual N$^3$LO within this quantum field theory scheme with the MSSM gluino and the scheme with the second additional degree of freedom. We discuss the properties of the $\{ \beta \}$-expansion for higher orders considering the N$^4$LO as an example.


Introduction
The knowledge of the detailed structure of QCD perturbation expansions is rather important for a variety of tasks of which the renormalization group optimization of the series is the best known. The detailed structure looks as a double series (or a matrix representation) rather than a usual series [1,2] even in the case of expansion of 'physical' quantities (having no anomalous dimension). We shall explore this structure for the QCD renormalization group invariant (RGI) one-scale dependent quantities as well as elaborate an algebraic approach to fix their elements within the {β}-expansion [1]. The knowledge of these expansion elements makes it possible to put the task of perturbation series optimization and helps to relate different quantities. In this sense our paper continues the investigations of the perturbation expansions in [1][2][3][4]. There, we discussed in detail the optimization of the perturbation expansion in N 2 LO in [3] beyond the Brodsky-Lepage-Mackanzie approach (BLM) [5]. Moreover, in [2,3], based on the {β}-expansion in this order, we related the pair of different RGI quantities. Now let us introduce the appropriate physically important quantities whose perturbation expansions are the most advanced. We JHEP04(2017)169 take as patterns of the RGI quantities the phenomenological important Bjorken polarized sum rules S Bjp (Q 2 , µ 2 ), S Bjp (Q 2 ) = g A 6 C Bjp NS Q 2 /µ 2 , a s (µ 2 ) + i q i C Bjp S Q 2 /µ 2 , a s (µ 2 ) , (1.1) and the Adler function D(Q 2 , µ 2 ), 2) where q i is the electric charge of the quark, g A -nucleon axial charge, d R -the dimension of the quark color representation. Perturbation expression for the nonsinglet (NS) coefficient functions of both the quantities at the renormalization scale µ 2 = Q 2 can be written down as D NS a s (µ 2 ) = 1 + n≥1 a n s (µ 2 ) d n , C Bjp NS a s (µ 2 ) = 1 + a s = α s /(4π), they are calculable in the MS -scheme and were obtained in order of O(a 4 s ) in [6]. We use here only the NS parts of these quantities, D = D NS , C Bjp = C Bjp NS , omitting the corresponding notation further in the text. The perturbation coefficients d n (c n ) in eqs. (1.3) are the combinations of only the color coefficients. Now, recall the structure of these perturbation coefficients. The {β}-expansion representation introduced in [1] prescribes to decompose d n or/and c n or any other of RGI quantity in the following way: where β i are the coefficients of the QCD β-function  [1][2][3] (see the beginning of section 4 for details). This kind of the expansion is the essential part of the procedures for the optimization of perturbation series, e.g., the decomposition (1.4b) was the starting point of the well-known BLM prescription [5] in NLO. Indeed, in the BLM the contribution β 0 d 2 [1] is transferred to the new normalization scale µ ′ of the coupling constant, , that often improves the expansion for many cases (e.g., d 2 [0] ≪ d 2 for the Adler function). For further high order development [1] the principle of maximum conformality (PMC) was proposed [7,8], which demands all of the β-terms in the decompositions eq. (1.4c), (1.4d), . . . to be accumulated into the new scale; therefore, the nth term of expansion turns into a n s d n PMC −→ a ′ n s d n [0] and one should know these proper elements d n [0]. This PMC approach does not mandatory lead to the improvement of expansion, the latter depends on distribution and signs of different (β) contributions for each perturbation order. Different optimization conditions require the knowledge of different β-terms in the decompositions of eq. (1.4), which were discussed in detail in sections 5 and 6 in [3].
At NLO of QCD the decomposition in (1.4b) looks evident because the term proportional to 4 3 T R n f unambiguously marks the contribution of the term proportional to [3] and the result in eq. (A.1b) in appendix A. How to fix the elements of the decomposition in higher orders? The consideration of color coefficients content of the d n is not enough for this, so one need to find additional conditions. To solve the problem, we introduce additional degrees of freedom (d.o.f.), new fields that interact following the universal gauge principle and enter only in intrinsic loops. Using the fermions in the adjoint representation (MSSM light gluino) as an additional d.o.f., we formulate a simple algebraic scheme to obtain the elements of the {β}-expansion and demonstrate the results in N 2 LO, eq. (1.4c), in the following sections 2.1 and 2.2. Moreover, based on the Crewther relation [9] we derive the relation between C and D in section 2.3. This algebraic scheme is well algorithmized and appropriate to apply to high loop results. It is applied to the N 3 LO expansion in eq. (1.4d) in section 3, which fixes completely the elements of expansion. The required expressions for d 4 , c 4 with the additional d.o.f. are expected to be calculated in future. In section 4, we discuss the general structure and properties of {β}-expansion for higher orders considering the N 4 LO as an example. The algebraic scheme fixes the expansion elements for this case too. Our main results are presented in Conclusion.
2 Algebraic approach for the {β}-expansion in N 2 LO

A simple illustration
Let us consider the task to fix the decomposition elements in eq. (1.4) algebraically, taking eq. (1.4c) as an example. The "renormalon" term d 3 [2] (or d n [n − 1] for any n) at the maximum power of β 0 can be identified by the maximum power of 4 3 T R n f (here it is proportional to C F 4 3 T R n f 2 ), or even calculated independently, see [10]. The corresponding JHEP04(2017)169 residual in the r.h.s. of eq. (1.4c) contains 5 Casimir coefficients where we put variable x = 4 3 T R n f . Taking eq. (2.1) at any three different values of x, (x 1 , x 2 , x 3 ) = X and compiling the coupled system of linear equations we can obtain the unique solution of this system under the evident condition that the corresponding determinant ∆ 3 , is not zero. The opposite condition ∆ 3 = 0 unambiguously means that the functions β 0 (x), this is just the case of QCD with only quark degrees of freedom (see the explicit expressions in eq. (B.1)). Due to this reason one cannot untangle contributions from β 0 and β 1 in N 2 LO without an additional constraint (see the discussion in [3]). In the case of an additional degree of freedom that contributes to both sides of eq. (2.1), i.e., to the coefficient d 3 and to β 0 , β 1 , one can obtain the unique solution. The goal of this note is to elaborate an algebraic scheme to obtain the decompositions in eqs. (1.4) using additional d.o.f. like ng -the number of MSSM gluino (we use y = 4 3 C A 2 ng) and, may be, other fields that interact following the universal gauge principle and appear only in intrinsic loops. The net effect of this field will be parameterized by means of the parameter z. Further, we shall suggest that the coefficients of perturbation expansion, like d n (c n ) in the l.h.s. of (1.4), as well as the coefficients of the β-function in the r.h.s. of (1.4) are calculated within the MS scheme and are known functions on the arguments x, y, z. To be more exact we consider the Adler function D(x, y) [11] as well as the β-coefficients β 0 (x, y), β 1 (x, y) presented in appendix B as functions on both the quark (x) and the MSSM gluinos (y) d.o.f. In this notation The results for the decomposition presented below are valid also for the coefficient function C Bjp (x, y) up to the replacement of the notation and for any RGI one-scale quantities.

The formalism of decomposition for D-function
To simplify the system of equations (SE) based on eq. (2.3) (the extended by the y d.o.f. eq. (2.1)), we take for the components of X: x 0 , x 1 , (x 01 , y 01 ), the special values -the roots of equations

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For this X 3 the SE 3 looks like = 0 that follows from eq. (2.2) or, can be obtained from the SE 3 (2.5) directly. Therefore the unique solution of the SE 3 is These values were obtained first in [1] using another trick; here they are presented explicitly in eq. (A.1) in appendix A.

How to obtain the {β}-expansion for C Bjp from one for D
To relate the already known structure of d 3 (the solutions in (2.6)) to the corresponding {β}-expansion of c 3 , we use the generalized Crewther relation (CR) [2,10], where K(a s ) = n=1 a n−1 s K n is a polynomial in a s . In the case of the β-function having identically zero coefficients β i = 0, the generalized CR (2.7) returns to its initial form [9] with only 1l in its r.h.s. that expresses the unbroken conformal symmetry. The later condition relates the d n [0], c n [0] elements in every order (see definition (4.1) and eq. (4.2) in [3]) The explicit closed solution of the relation (2. elements. The general relation (2.9) can be also treated as a prediction for C Bjp by means of D that is based on the {β}-expansion and CR. In the third order of a s the knowledge of the element c 3  i.e. to disentangle the contributions from c 3 [1]β 0 and c 3 [0, 1]β 1 , which were discussed in detail in section IVB in [3].
From another side one can use in the r.h.s. of eq. (2.7) the second term proportional to β(a s ) that expresses the conformal symmetry-breaking. This leads to the series of relations [2,3] for the elements of the different orders n at β n−1 , e.g.,  [3]. As a byproduct of the procedure we predicted the c 3 of C Bjp , see eq. (4.11) in [3], if the {β}-expansion for d 3 of D is already known and vice versa, C Bjp ⇆ D. This demonstrates that the elements of {β}-expansion provide appropriate "bricks" for complete determination of RGI quantities.

The {β}-expansion for Bjorken polarized SR and D-function in N LO
In the 5 loop case, d 4 (x) was first obtained in [12] as the polynomial with numerical coefficients, then all the color coefficients in decomposition for d 4 (x) and c 4 (x) were presented in [6]. Following the {β}-expansion we propose for these coefficients the decomposition,  [3] can be directly identified. The ten Casimirs (here A are distributed among all of the d 4 [ · ] elements, while the abelian elements of the box subgraphs with four gluon legs, related to color coefficients n f d abcd . These terms do not contribute to the renormalization of the charge 1 a s , see also the discussion of the subject in [13]. Although the corresponding 5-loop diagrams contain these one-loop boxes, further contraction of the subgraphs (see the discussion in [4]) do not contribute to β 0 . Due to this reason , y), where the (x, y)dependent part δd(x, y) is well recognized, while δd 4 (x, 0) δc 4 (x, 0) is already known from the result in [6] (see eq. (A.7) in appendix A). Therefore, the n f (ng)-dependence becomes partly separated from the charge renormalization for the first time in N 3 LO.

(3.4)
This value ∆ 6 (X 6 ) = 0, the solution of this SE 6 exists and unique, and can be obtained like the solution of eq. (2.6) for the N 2 LO in section 2.2. Therefore, to derive the {β}-expansion for d 4 , it is enough to obtain one at an additional single d.o.f. y, d 4 → d 4 (x, y) together with the coefficients β 0,1,2 (x, y) (see appendix B).
We present the solutions of SE 6 for a number of elements in the explicit form, taking the notation for the arguments (x ij , y ij ) and the function Y 4 for shortness, Just these elements will be used for the relation with similar elements in C Bjp . Of course, one can take another set X ′ 6 and construct the corresponding SE ′ 6 . In any case the solution for the elements d 4 [ · ] should be the same. The usage of the roots in eqs. (3.2) to construct X 6 leads to the simplification of the final SE.

Relations between the elements of D and C Bjp
Suppose that the coefficient functions for the Adler D-function d 4 (x, y) and the Bjorken SR c 4 (x, y) are known. Then, based on two terms in the r.h.s. of the Crewther relation, eq. (2.7), one can obtain for the sum of these functions d 4 (x, y)+c 4 (x, y) and their elements (see [2]) a series of the relations. In part, one can obtain from (2.8) the sum of "zero" elements, where D (3.9a) (3.9f) The r.h.s. s of (3.9a), (3.9c), (3.9e) are presented by mean of the already known results for d 3 and c 3 -(3.9b), (3.9d), see [3] and appendix A here. The last eq. (3.9e), suggested in [2], has already been verified and is put here for illustration and comparison with the two previous equations. Let us conclude, Let us mention thereupon an alternative approach to fix d 4 [ · ], c 4 [ · ] without additional d.o.f., which was suggested in [4] and was inspired by the structure of the r.h.s. of CR (2.7). The idea is based on the specific proposition that the perturbation series for D and C Bjp can be expanded in powers β(a s )/a s n similar to that had been proposed for the "conformal symmetry braking term" β(a s )K(a s ) in the r.h.s. of CR, see the presentation in eq. (6) in [2]. The results for the elements obtained within this approach differ from ours.

What can we get at 3 degrees of freedom x, y, z
Let us imagine that we have an additional third "intrinsic" d.o.f. that manifests itself as the parameter z. In this case d n = d n (x, y, z), β i = β i (x, y, z); therefore, one can use the points in (x, y, z) space to construct the set X:   The coefficient d 5 (x, y) is formed by the variety of 6-loop diagrams that get contributions from the intrinsic box-and pentagon-subgraphs with gluon legs that introduce into d 5 a specific (n f , ng)-dependence that does not relate to the charge renormalization. Indeed, the new color coefficients y) together with the contributions from the boxgraphs, which was already mentioned in section 3. The contributions from the latter boxgraphs, ngd abcd . All these contributions, proportional to n f , ng, are well recognized and can be accumulated in the specific term δd 5 (x, y), like it was done for δd 4 (x, y) in section 3. The element β 4 0 d 5 [4] should also be well recognized; therefore, one has 11 = 12 − 1 unknown elements d 5 [ · ]. By analogy with the previous lower orders procedure one can compile SE 11 based on eq. (4.1) with the rearranged l.h.s.
take the equation with the arguments at 11 points (X 11 ) on the plane (x, y). The SE 11 constructed in this way has unique solution with respect to the d 5 [ · ] elements under the condition the corresponding determinant of the system ∆ 11 (X 11 ) = 0. Let us take for these 11 components of X 11 the roots of the equations

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This choice of X 11 simplifies the set of equations, as we made sure in the previous cases of the constructing sets X 3,6 in eqs. (2.4), (3.2), respectively. The determinant corresponding to SE 11 ∆ 11 (X 11 ) = 0 but looks too cumbersome to show it here. It is clear that our algebraic scheme works further with the increasing perturbation order.

Conclusion
In this paper we have considered the important task of obtaining the elements of the detailed structure of the QCD perturbation expansion -the {β}-expansion for the renormalization group invariant quantities. The explicit knowledge of the elements of this expansion (i) gives a possibility to perform various kinds of optimization of the perturbation series; (ii) taken together with the Crewther relation it allows one to establish nontrivial relations between different physical quantities. We suggest an algebraic approach to fix the elements of the {β}-expansion for these quantities using additional degrees of freedom, and demonstrate that for the resolution of the detailed structure it is enough to use a single additional degree of freedom to the quark one. This approach is discussed in detail for N 2 LO calculations of the nonsinglet Adler Dfunction and for the Bjorken polarized sum rules C Bjp within QCD with the ng of MSSM gluinos -the additional degree of freedom. We derive the explicit formulae for the elements of the {β}-expansion for these quantities, named d n [ · ] and c n [ · ] respectively, see eq. (1.4d) in the actual case of N 3 LO within the aforementioned quantum field theory scheme. This {β}-expansion together with the explicit expressions for the elements d 4 [ · ] (c 4 [ · ]) can be considered as a prediction for any additional degrees of freedom that can be taken into consideration. Indeed, these degrees of freedom enter into either the well-known coefficients of the β-function, β i , or the well-recognized terms of the structure.
Another kind of predictions is provided by the relation between the elements d n [ · ] and c n [ · ] in virtue of the Crewther relation. We constructed the fixation procedure also for the case of two additional degrees of freedom. Finally, we discussed the structure and properties of {β}-expansion for higher orders considering the N 4 LO with the ng of MSSM gluinos as an example, where the expansion elements can be also fixed following to our algebraic procedure.
The next natural step in the development of this investigation would be the calculation of D or C Bjp with the additional degrees of freedom in N 3 LO. These results allow one to obtain relations between their elements that leads to the new predictions for one of them. Moreover, this provides the basis of optimization of the approximation for the physically important R e + e − →h (s)-ratio or for the Bjorken polarized sum rules.

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From the results in [6] it follows that

B The β-function coefficients
The required β-function coefficients with the Minimal Supersymmetric Model (MSSM) light gluinos ng [14], and the number n f of quark flavors, calculated in the MS scheme are where we have introduced appropriate rescaled variables x = 4 3 T R n f and y = 4 3 C A 2 ng after the first equality to simplify the expressions. The N 3 LO coefficient β 3 (n f , ng) has been