General scalar renormalisation group equations at three-loop order

For arbitrary scalar QFTs in four dimensions, renormalisation group equations of quartic and cubic interactions, mass terms, as well as field anomalous dimensions are computed at three-loop order in the $\overline{\text{MS}}$ scheme. Utilising pre-existing literature expressions for a specific model, loop integrals are avoided and templates for general theories are obtained. As an example, the results are applied to compute all renormalisation group equations in $U(n) \times U(n)$ scalar theories up to three-loop order.


Introduction
The renormalisation group (RG) is a key instrument to connect and extrapolate physics to different scales as well as to study critical phenomena. Hence, the computation of renormalisation group equations (RGEs) is a most crucial issue in these kind of works, requiring or at least desiring high accuracies. In spite of the advent of non-perturbative methods, e.g. Wilsonian RG [1][2][3][4] or the gradient flow [5,6], perturbative RGEs have stood the test of time, due to being systematic and extensible expansions that are reliable in the weak coupling regime.
Foundations for advancing the general framework have been laid in [18], obtaining Weyl consistency conditions on four-loop gauge and three-loop Yukawa β functions from the two-loop scalar quartic one. However, no progress towards general three-loop quartic β functions has been made, not even for pure scalar contributions, in spite of the multiloop success of O(n) models.
In this work, we break the stagnation of the fully general framework by extracting three-loop β functions and anomalous dimensions for an arbitrary scalar sector in the MS scheme. In Sec. 2 we will briefly review formalisms and introduce notations. Sec. 3 will detail our approach to extract RGEs, and we present general results in Sec. 4. We will apply these expressions to compute RGEs for a scalar U (n) × U (n) matrix model, which has been of special interest e.g. due to the walking regime [48][49][50][51][52] between its two complex fixed points [53], as well as in models with weakly coupled asymptotic safety in pertubatively exact settings [54][55][56][57][58] and extensions of the SM [59][60][61].
During the preparation of this manuscript, the work [62] has appeared with a similar scope of obtaining three-loop RGEs, but using a different method and renormalisation scheme.

General Framework
Any perturbatively renormalisable QFT in four dimension can be embedded in the template Lagrangian [8][9][10] which is formulated in terms of fermionic Weyl components ψ i , real scalar fields φ a and a generic gauge sector, including gauge-fixing and ghost terms L gf + L gh . Both scalar and fermionic indices a, b, ... and i, j, ... run over all field species and components, generation indices as well as gauge and flavour representations. All Yukawa couplings, scalar quartic and cubic interactions as well as fermion and scalar masses can hence be embedded into the respective tensor structures Y a ij , λ abcd , h abc , m ij and m 2 ab . In particular, these quantities are chosen to be symmetric in all their scalar or fermionic indices, e.g. λ abcd = λ cabd .
Using a dimensional regularisation in d = 4 − 2ǫ and the modified minimal subtraction scheme (MS) [7], a multiplicative renormalisation procedure is established in the bare action (2.1) via Here, µ labels the renormalisation scale. Similar substitutions for fermion and gauge fields apply, introducing field strength renormalisation factors Z and their corresponding counterterms δZ. In the same manner, g i and δg i are place holders for all couplings and their respective counter-terms. The numbers ρ i are determined by keeping the corresponding interaction operator d-dimensional after inserting the canonical dimensionality of the fields, e.g. ρ = 2 for quartics and ρ = 1 for gauge and Yukawa interactions. In minimal subtraction schemes, the counter-terms are independent of µ and can be expanded via The overall scale independence of bare couplings and fields thus leads to a relation of the leading poles in the counter-terms to β functions and field anomalous dimensions γ φ [63,64], which describe the renormalisation group flow of the renormalised couplings with respect to the scale µ. Using perturbation theory, these quantities can be obtained in a loop expansion Figure 1. Non-vanishing three-loop diagrams with quartic interactions contributing to the wave function renormalisation.
from the counter-terms of the same order. As those counter-terms of coupling constants are determined by the simple poles of the external leg contributions and proper vertex counter-terms, the n-loop order for each β-function contains two types of diagrams. There are tree-level couplings contracted with n-loop anomalous dimensions, as well as n-loop proper vertex corrections, but not a mix of those diagrams. For instance, In comination with the general ansatz (2.1), all momentum integrals and spinor summations can be resolved, and the RGEs (2.4) can be expressed in terms of contracted generalised couplings. This provides a template to conveniently obtain RGEs for any renormalisable QFT by using the embedding into (2.1), without the need of loop calculations. That however comes at the cost of having to conduct an involved computation for the general theory (2.1) once.

Three-loop scalar diagrams
In this section, we extend the state-of-the-art template RGEs by extracting the pure scalar part of the field anomalous dimensions and quartic β-functions. To this end, we will assume the QFT given by the Lagrangian General β functions for scalar masses and cubic interactions will be computed in the next section. For all these RGEs, gauge and fermionic interactions are additive features and will not invalidate the results, such that they can be computed separately. Following (2.5) and (2.6), we will list all scalar three-loop diagrams relevant for the computation of γ φ,3ℓ and β λ,3ℓ . In particular, the fact that momentum integrals of the shape = 0 (3.2) vanish in the MS scheme reduces the number of diagrams for the leg correction to 2, see Fig. 1, as well as 8 vertex contributions, see Fig. 2. The next step would be to compute the momentum integrals for these graphs as well as lower order diagrams with counter-term insertions. However, we will employ a different T (1) T (2) T (3) T (4) T (5) T (6) T (7) T (8) Figure 2. Non-vanishing three-loop diagrams with quartic interactions contributing to the proper vertex renormalisation.
Here we point out that the 10 open parameters κ i and τ j may be fixed by comparing the ansatz (3.3) against literature results that are available for specific models. Hence, this approach allows to obtain the most general RGEs by extrapolation of pre-existing results, without the need for explicit loop computations. This is possible because the number of unknown prefactors is relatively small.
Suitable three-loop computations are available for the SM Higgs sector [36][37][38][39], scalar O(n) theories [23][24][25][26][27][28][29][30][31][32], and the scalar potential of the Two-Higgs-Doublet-Model (THDM) [47]. The results for the SM Higgs [36][37][38][39] do not provide enough data to fix all coefficients. However, neglecting all other interactions, the complex doublet can actually be described by a O(4) real scalar. The general case of theories with n scalars and a O(n) symmetry is given by the Lagrangian (3.5) The respective three-loop RGEs [23] can be matched against the ansatz (3.3), where each power of n in the RGEs β λ,3ℓ /λ 4 , γ m 2 ,3ℓ /λ 3 and γ φ,3ℓ /λ 3 gives a seperate condition on the κ i and τ j . Unfortunately, the data extracted by this ansatz is insufficient for resolving all τ i . Alternatively, we will select a subset of the THDM, featuring two vector-like, complex doublet scalars Φ 1,2 . The quartic potential is protected by an U (2) Φ 1 × U (2) Φ 2 symmetry, where each scalar has its own subgroup. This allows for the permutation Φ 1 ↔ Φ 2 and hence λ 1 ↔ λ 2 . Translated into our own notation, the three-loop β functions computed in [47] β 3ℓ allow to extract the complete set of coefficients from (3.3) which yields the definite solution (3), featuring the zeta function ζ(3) ≈ 1.202. We find that the contractions K 2 , T 3 and T 7 do not give corrections to the RGEs. In fact, momentum integrals vanish for certain choices of external momenta in those three diagrams. This eliminates their contributions in the MS scheme, which is in accord with the calculation conducted in [23], where K 2 , T 3 and T 7 have been neglected. Curiously, in O(n) models like (3.5), the fact that τ 3 = 0 is the reason why leading-n contributions n 3 λ 4 are absent in β 3ℓ λ , see [23]. This procedure is extensible to higher loop orders. However, at four loops only O(n) results as in (3.5) are available for matching [23][24][25]. This model has already proven insufficient for the three-loop β functions, and higher orders will contain even more open parameters. Regarding the four-loop field anomalous dimension γ φ,4ℓ , [23] suggests that only four contractions remain after filtering out vanishing loop diagrams. However, due to redundancies in the matching conditions, the results in [23][24][25] are insufficient to extract all open parameters in γ φ,4ℓ .
Furthermore, completely new results can also be obtained using these expressions. This is demonstrated by computing RG equations for a U (n) × U (n) scalar theory, with the complex matrix field φ ab . The bare action is given by (4.5) In the large-n limit, two-loop results have been presented in [56], and can be extended for finite-n using [8][9][10][11][12]18]. For convenience, we introduce the 't Hooft couplings of the quartic sector α u = n u/(4π) 2 and Using (4.1), the field anomalous dimension reads up to three-loop order (4.7) From (4.4), the mass anomalous dimension γ m 2 = m −2 β m 2 (4.8) is obtained. Finally, (4.2) yields β functions for the 't Hooft couplings α u,v (4.10) We are looking forward to independent computational verifications of the U (n) × U (n) results (4.7)-(4.10) as well as the general RGEs (4.1)-(4.4). In summary, we have shown how fully general RGEs can be extracted from expressions obtained in much easier models. This represents an opportunity to break involved higherloop computations apart into smaller problems. In doing so, existing results can be reused. We have explored this technique for purely scalar RGEs as far as possible, and obtained novel three-loop results. Extracting general four-loop RGEs requires more input data. In principle, the idea can be used to extract other terms and/or different RGEs, possibly feeding off the data [35][36][37][38][39][40]43], and potentially supplemented Weyl consistency conditions [18]. Explorations into this direction are left for future work.