Transcendental structure of multiloop massless correlators and anomalous dimensions

We give a short account of recent advances in our understanding of the π- dependent terms in massless (Euclidean) 2-point functions as well as in generic anomalous dimensions (ADs) and β-functions. We extend the considerations of [1] by two more loops, that is for the case of 6- and 7-loop correlators and 7- and 8-loop renormalization group (RG) functions. Our predictions for the (π-dependent terms) of the 7-loop RG functions for the case of the O(n) 𝜙4 theory are in full agreement with the recent results from [2]. All available 7- and 8-loop results for QCD and the scalar O(n) ϕ4 theory obtained within the large Nf approach to the quantum field theory (see, e.g. [3]) are also in full agreement with our results.


Introduction
Since the seminal calculation of the Adler function at order α 3 s [4] it has been known that p-functions in QCD demonstrate striking regularities in terms proportional to π 2n (or, equivalently, even zetas 1 , with n being positive integer. Indeed, it was demonstrated in [4] for the first time a mysterious complete cancellation of all contributions proportional to ζ 4 ≡ π 4 90 (which generically appear in separate diagrams) while odd zetas terms (that is those proportional to ζ 3 and ζ 5 in the case under consideration) do survive and show up in the final result. Here by p-functions we understand (MS-renormalized) Euclidean Green functions 2 or 2-point correlators or even some combination thereof, expressible in terms of massless propagator-like Feynman integrals (to be named p-integrals below).
Since then it has been noted many times that all physical (that is scale-invariant) p-functions are indeed free from even zetas at order α 4 s (like corrections to the Bjorken (polarized) DIS sum rule) and some of them-like the Adler function-even at the next, in fact, 5-loop, α e α 4 s order [6]. On the other hand, the first appearance of ζ 4 in a one-scale physical quantity has been demonstrated in [7] for the case of the 5-loop scalar correlator.
It should be stressed that the limitation by QCD p-functions in the above discussion is essential. In general case scale-invariant p-functions do depend on even zetas already at 4 loops (see eq. (11.8) in [1]). 1 As is well known every even power 2n of π is uniquely related to the corresponding Euler ζ-function ζ2n ≡ i>0 1 i 2n , according to a rule ζ2n = r(n) π 2 n , with r(n) being a (known) rational number [5]. 2 Like quark-quark-qluon vertex in QCD with the external gluon line carrying no momentum.
To describe these regularities more precisely we need to introduce a few notations and conventions. Let F n (a, ℓ µ ) = 1 + 0≤j≤i 1≤i≤n g i,j (ℓ µ ) j a i (1.1) be a (renormalized) p-function in a one-charge theory with the coupling constant a 3 . Here Q is an (Euclidean) external momentum and ℓ µ = ln µ 2 Q 2 . The integer n stands for the (maximal) power of a appearing in the p-integrals contributing to F n . In the case of onecharge gauge theory and gauge non-invariant F we will always assume the case of the Landau gauge. In particularly all our generic considerations in this paper are relevant for QCD p-functions with a = αs(µ) 4 π . The F without n will stand as a shortcut for a formal series F ∞ . In terms of bare quantities 4 with the bare coupling constant, the corresponding renormalization constant (RC) and AD being The coefficients of the β-function β i are related to Z a in the standard way: A p-function F is called scale-independent if the corresponding AD γ ≡ 0. If γ = 0 then one can always construct a scale-invariant object from F and γ, namely: (1.7) Note that F si n+1 (a, ℓ µ ) starts from the first power of the coupling constant a and is formally composed from O(α n+1 s ) Feynman diagrams. In the same time is can be completely restored from F n and the (n + 1)-loop AD γ.
If not otherwise stated we will assume the so-called G-scheme for renormalization [8]. The scheme is natural for massless propagators. All ADs, β-functions and Z-factors are identical in MS-and G-schemes. For (finite) renormalized functions there exists a simple conversion rule. Namely, in order to switch from an G-renormalized quantity to the one in the MS-scheme one should make the following replacement in the former: ln µ 2 → ln µ 2 + 2 (µ is the renormalization scale, the limit of ǫ → 0 is understood).
An (incomplete) list of the currently known regularities 5 includes the following cases.
1. Scale-independent QCD p-functions F n and F si n with n ≤ 4 are free from π-dependent terms.
2. Scale-independent QCD p-functions F si 5 are free from π 6 and π 2 but do depend on π 4 .
3. The QCD β-function starts to depend on π at 5 loops only [13][14][15] via ζ 4 . In addition, there exits a remarkable identity [1] β ζ 4 5 = 4. If we change the MS-renormalization scheme as follows: with c 1 , c 2 and c 3 being any rational numbers, then all known QCD functionsF si 5 (ā, ℓ µ ) and the (5-loop) QCD β-functionβ(ā) both loose any dependence on π. This remarkable fact was discovered in [9]. 5. It should be also noted that no terms proportional to the first or second powers of π do ever appear in all known (not necessarily QCD!) p-functions and even in separate p-integrals at least at loop number L less or equal 5. This comes straightforwardly from the fact that the corresponding master p-integrals are free from such terms. The latter has been established by explicit analytic calculations for L = 2, 3 [8], L = 4 [16][17][18] and finally at L = 5 [19]. Note for the last case only a part of 5-loop master integrals was explicitly computed. However, there are generic mathematical arguments in favor of absence of contributions with weight one and two, that is π and π 2 in p-integrals at least with the proper choice of the basis set of transcendental generators [20,21]. By proper choice here we mean, essentially, a requirement that transcendental generators should be expressible in terms of rational combinations of finite p-integrals [22,23]. without use of π as a generator.
Our results below are in full agreement with these arguments. and RCs in dependence on the loop number L. The inverse power of ǫ stands for the maximal one in generic case; in particular cases it might be less.
It should be stressed that eventually every separate diagram contributing to F n and F si n+1 contains the following set of irrational numbers: ζ 3 , ζ 4 , ζ 5 , ζ 6 and ζ 7 for n = 4, ζ 3 , ζ 4 and ζ 5 for n = 3 as illustrated in Table 1. Thus, the regularities listed above are quite nontrivial and for sure can not be explained by pure coincidence.
In this paper 6 we first present a short discussion of recent advances in studying the structure of the π-dependent terms in massless (Euclidean) 2-point functions as well as in generic anomalous dimensions and β-functions. Then we extend the considerations of [1] by two more loops. Finally, we discuss remarkable connections between ǫ-expansion of 4-loop p-integrals and the D = 4 values of finite 5-, 6-, and 7-loop p-integrals.

Hatted representation: general formulation and its implications
The full understanding and a generic proof of points 1-5 above have been recently achieved in our work [1]. The main tool of the work was the so-called "hatted" representation of transcendental objects contributing to a given set of p-integrals.
Let us reformulate the main results of [1] in an abstract form. We will call the set of all L-loop p-integrals P L a π-safe one if the following is true.
(i) All p-integrals from the set can be expressed in terms of (M + 1) mutually independent (and ǫ-independent) transcendental generators This means that any p-integral F (ǫ) from P L can be uniquely 7 presented as follows where by F (ǫ) we understand the exact value of the p-integral F while the combination ǫ L F (ǫ, t 1 , t 2 , . . . , t M , π) should be a rational polynomial 8 in ǫ, t 1 . . . , t M , π. Every such 6 A preliminary version of the present work (not including the 8-loop case) was reported on the International Seminar "Loops and Legs in Quantum Field Theory" (LL2018) in St. Goar, Germany and published in [24]. 7 We assume that F (ǫ, t1, t2, . . . , π) does not contain terms proportional to ǫ n with n ≥ 1. 8 That is a polynomial having rational coefficients.
polynomial is a sum of monomials T α of the generic form with n ≤ L, n i and r α being some non-negative integers and rational numbers respectively. A monomial T α will be called π-dependent and denoted as T π,α if n M +1 > 0. Note that a generator t i with i ≤ M may still include explicitly the constant π in its definition, see below.
(ii) For every t i with i ≤ M let us define its hatted counterpart as follows: with {h iα } being rational polynomials in ǫ and T π,α are all π-dependent monomials as defined in (2.3). Then there should exist a choice of both a basis T and polynomials {h iα } such that for every L-loop p-integral F (ǫ, t i ) the following equality holds: We will call π-free any rational polynomial (with possibly ǫ-dependent coefficients) in As we will discuss below the sets P i with i = 3, 4, 5 are for sure π-safe (well, for L = 5 almost) while P 6 highly likely shares the property. For the case of P 7 the situation is more complicated (but still not hopeless!) as discussed in Section 5. In what follows we will always assume that every (renormalized) L-loop p-function as well as (L+1)-loop β-functions and anomalous dimensions are all expressed in terms of the generators t 1 , t 2 , . . . , t M +1 .
Moreover, for any polynomial P (t 1 , t 2 , . . . , π) we define its hatted version aŝ Let F L is a (renormalized, with ǫ set to zero) p-function, γ L and β L are the corresponding anomalous dimension and the β-function (all taken in the L-loop approximation). The following statements have been proved in [1] under the condition that the set P L is π-safe and that both the set T and the polynomials {h iα (ǫ)} are fixed.

No-π Theorem
(a) F L is π-free in any (massless) renormalization scheme for which corresponding βfunction and AD γ are both π-free at least at the level of L + 1 loops.
is π-free in any (massless) renormalization scheme provided the β-function is π-independent at least at the level of L + 1 loops.
3 π-structure of 3,4,5 and 6-loop p-integrals A hatted representation of p-integrals is known for loop numbers L = 3 [25], L =4 [16] and L = 5 [19]. In all three cases it was constructed by looking for such a basis T as well as polynomials h iα (ǫ) (see eq. (2.4)) that eq. (2.5) would be valid for sufficiently large subset of P L .
Let us consider the next-loop level, that is P 6 . In principle, the strategy requires the knowledge of all (or almost all) L-loop master integrals. On the other hand, if we assume the π-safeness of the set P 6 we could try to fix polynomials h iα (ǫ) by considering some limited subset of P 6 .
Actually, we do have at our disposal a subset of P 6 due to work [17] where all 4loop master integrals have been computed up to the transcendental weight 12 in their ǫ expansion. Every particular 4-loop p-integral divided by ǫ n can be considered as a (4 + n) loop p-integral. The collection of such (4+n)-loop p-integrals form a subset of P 4+n which we will refer to as P 4 /ǫ n . We have tried this subset for n = 1 and 2.
Here multiple zeta values are defined as [26] ζ n 1 ,n 2 := (3.9) Some comments on these eqs. are in order.
• The boxed entries form a set of π-independent (by definition!) generators for the cases of L = 3 (eq. . In what follows we will use for the boxed combinations in eqs. (3.1 -3.8) the notation • There is no terms proportional to single or second powers of π outside boxed combinations in relations (3.1-3.8). This fact directly leads to the absence of such terms in the (renormalized) 6-loop p-integrals and generic ADs and β-functions with the loop number not exceeding 7. Later we will see that the same is true for 7-loop p-integrals and 8-loop RG functions (assuming the conservative scenario as described in Section 5).
4 π-dependence of 7-loop β-functions and AD Using the approach of [1] and the hatted representation of the irrational generators (3.13) as described by eqs. (3.1)-(3.8) we can straightforwardly predict the π-dependent terms in the β-function and the anomalous dimensions in the case of any 1-charge minimally renormalized field model at the level of 7 loops. Our results read (the combination F tα 1 tα 2 ...tα n stands for the coefficient of the monomial (t α 1 t α 2 . . . t αn ) in F ; in addition, by F (1) we understand F with every generator t i from {t 1 , t 2 , . . . , t M +1 } set to zero).

Tests at 7 loops
With eqs. (4.1)-(4.38) we have been able to reproduce successfully all π-dependent constants appearing in the β-function and anomalous dimensions γ m and γ 2 of the O(n) ϕ 4 model which all are known at 7 loops from [2]. In addition, we have checked that the πdependent contributions to the terms of order n 6 f α 7 s in the QCD β-function as well as to the terms of order n 6 f α 7 s and of order n 5 f α 7 s contributing to the quark mass AD (all computed in [29][30][31]) within the framework of large N f [3,[32][33][34][35] approach are in full agreement with the constraints listed above.
Numerous successful tests at 4,5 and 6 loops have been presented in [1].

Hatted representation for 7-loop p-integrals and the π 12 subtlety
Motivated by the success of our derivation of the hatted representation for the 6-loop case we have decided to look on the next, 7-loop level. Within our approach this requires the knowledge of the ǫ-expansion of the 4-loop master integrals presented in [17] up to the transcendental weight 13. In principle, the methods employed by Lee and Smirnov are powerful enough to find such an expansion. One of the authors of [17] has provided us with ǫ-expansions for all 4-loop master p-integrals up to and including weight 13.
In fact, we have (well, almost) succeeded in constructing the hatted representation for the subset P 4 /ǫ 3 of P 7 . Our results are presented below 9 . The meaning of the question mark in front of ζ 12 in eqs. (5.4), (5.8) and (5.13) for hatted form of multiple zeta objects is as follows. Every integral from the set P 4 /ǫ 3 can either include at least one (or more) multiple zeta values from the collection ζ 5,3 , ζ 7,3 , ζ 5,3,3 , ζ 7,3,3 and ζ 6,4,1,1 or not.
Thus, the whole set P 4 /ǫ 3 can be represented as a union of two (non-intersecting!) subsets, namely, a simple one, S 4 /ǫ 3 , (that is without any dependence on multiple zeta values) and the rest N 4 /ǫ 3 .
The fact is that the hatted representation does exists for all p-integrals S 4 /ǫ 3 , while there is no way to replace the question marks in eqs above by some coefficients in order to meet eq. (2.5) for the p-integrals from N 4 /ǫ 3 . On the other hand, if we formally set to zero all terms proportional to ζ 12 in eqs. (5.1-5.13), then eq. (2.5) will be valid for the whole set P 4 /ǫ 3 "modulo" terms proportional to ζ 12 .
It is quite remarkable that the distinguished role of ζ 12 has been already established in [2] as a result of direct analytical calculations of quite complicated convergent 7-loop p-integrals.
Thus, we observe a nontrivial interplay between higher terms in the ǫ-expansion of 4-loop p-integrals and 7-loop finite p-integrals.
Certainly, the subset of the 7-loop p-integrals which has led to eqs. (5.1-5.13) is rather limited and our conclusions about π-structure of P 7 are not final. In principle, we can outline 3 possible scenarios.
Scenario 1 (pessimistic). There is no hatted representation for the set P 7 .
Scenario 2. (conservative) The master p-integrals from the difference P 7 \ P 4 /ǫ 3 can be presented in the hatted form modulo (explicitly) π-proportional terms with weight more or equal 12.  π-dependence of 8-loop β-functions and AD In this section we assume the conservative Scenario 2 and extend (following generic prescriptions elaborated in [1]) the predictions from Section 4 by one more loop for π-dependent terms with the transcendental weight not exceeding 11. The results read: 1 , (6.1) 1 γ ζ 7,3 7 6 , (6.10) 5 , (6.11)

Tests at 8 loops
Here we summarize all currently available evidence supporting assumptions (that is scenarios 2 and 3) leading to eqs. (6.1-6.21). First of all, we have checked that currently known 8-loop results for ADs and βfunctions are in full agreement to our predictions. Namely, we have successfully checked the following cases.
• Contributions of orders α 8 s N 7 f and α 8 s N 6 f to the QCD quark mass anomalous dimension [30,31].
• Contribution of order g 8 n 7 and g 8 n 6 to the β-function, the field anomalous dimension and to the mass anomalous dimension of the scalar O(n) φ 4 theory [35].

Conclusion
Using as input data essentially only deep ǫ expansions of the 4-loop master integrals [17] we have extended the hatted representation of 4-loop p-integrals of work [16] to the 5-, 6and 7-loop families of p-integrals. At 5-loop level we successfully reproduced the results of [19] which had been obtained by a direct calculation of a rather large subset of 5-loop master p-integrals. We have derived a set of generic model-independent predictions for π-dependent terms of RG-functions at 7 and 8 loops (at the latter case only for terms with weight less or equal to 11). All available 7-and 8-loop results are in agreement with our predictions.
Our results demonstrate a remarkable and somewhat mysterious (at least for us) connection between ǫ-expansions of the 4-loop p-integrals and D = 4 values of 5-, 6-and 7-loop finite p-integrals. Indeed, dealing only with 4-loop p-integrals we have been able to get some non-trivial information about 5-, 6-and 7-loop p-integrals. More precisely, we have found a set of proper transcendental generators which form a π-free basis for every known 5-,6, and 7-loop p-integrals provided that (i) the latter is expressible only in terms of multiple zeta values and (ii) all terms (if any) proportional to ζ 12 are discarded.
It would be interesting to see what new information can be extracted from expanding 4-loop master p-integrals to even higher orders in ǫ.