Six-loop beta functions in general scalar theory

We consider general renormalizable scalar field theory and derive six-loop beta functions for all parameters in d = 4 dimensions within the $\overline{MS}$-scheme. We do not explicitly compute relevant loop integrals but utilize $O(n)$-symmetric model counter-terms available in the literature. We consider dimensionless couplings and parameters with a mass scale, ranging from the trilinear self-coupling to the vacuum energy. We use obtained results to extend renormalization-group equations for several vector, matrix, and tensor models to the six-loop order. Also, we apply our general expressions to derive new contributions to beta functions and anomalous dimensions in the scalar sector of the Two-Higgs-Doublet Model.


Introduction
The renormalization group (RG) plays an essential role in high-energy physics and the theory of critical phenomena. In particle physics, one can use RG to re-sum specific radiative corrections making theory predictions valid in a wide range of energy scales. In the study of critical phenomena, the RG approach allows one to study phase transitions and predict critical exponents of the second-order transitions with high accuracy.
A convenient tool to compute the RG functions that drive the dependence of model parameters on the scale is to use a perturbative expansion of dimensional regularized theory [1] together with modified minimal MS subtraction of infinities. The latter appear in loop integrals and manifest themselves in d dimensions as poles in ε = (4 − d)/2. One cancels the poles by a finite set of renormalization constants.
There is significant progress in the calculation of beta functions and anomalous dimensions in the MS scheme. At the two-loop level, the RG functions are known in any general renormalizable quantum field theory (QFT) in d = 4 dimensions [2][3][4][5][6]. Despite several calculations of three-loop (and even four-loop) RG functions in particular particle-physics models [7][8][9][10][11][12][13][14][15][16], general three-loop results are not yet available. Recently, an essential step has been made in this direction [17][18][19][20]. The main idea is to enumerate all possible "tensor" structures that can appear in the RG functions at a certain loop level and compute the corresponding unknown coefficients by matching them to specific models. Not long ago, this approach allowed authors of the paper [18] to derive general three-loop RG functions in a pure scalar model.
Our paper does not follow this strategy and extends the results for general scalar theories up to six loops by more conventional technique, i.e., by computing contributions from individual Feynman graphs. Such a leap in the loop level is due to the significant progress in calculating critical exponents in scalar theories. Thanks to the authors of ref. [21], the required renormalization constants can be found given the diagram-by-diagram results of the KR operation. Application of the latter to a Feynman graph produces the corresponding counter-term in the MS scheme.
We consider the following general renormalizable Lagrangian for real scalar fields φ a . The mass parameters m 2 ab , cubic h abc and quartic couplings λ abcd are symmetric in their indices. For completeness we also add the tadpole term proportional to t a , and the vacuum energy term Λ.
Here we present the six-loop RG equations in the MS-scheme for the field φ a and all parameters of eq. (1.1). The RG function for a parameter A = {λ abcd , h abc , m 2 ab , t a , Λ} is defined as where β

(l)
A corresponds to the l-loop contribution. The field anomalous dimension is given by and is related to the field renormalization constant Z ab . The paper is organized as follows. Section 2 contains details of our calculation. In section 3 we apply our general results to the cases known in the literature. In particular, we consider vector (section 3.1), matrix (section 3.2), and tensor (section 3.3) models possessing different kinds of symmetries. Also, we extend known three-loop results for the Two-Higgs-Doublet Model (2HDM) to six loops in section 3.4. Section 4 contains a discussion of the results and conclusions. In appendix A we provide a derivation of the RG functions for dimensionful couplings in a general form.

Details of calculation
As the calculation method, we decided to use an approach similar to the one in ref. [21], based on the direct computation of the necessary counter-terms from individual diagrams. However, in our work, we avoid the calculation of any loop integrals. The authors of ref. [21] considered all the required six-loop graphs in the context of the O(n)-symmetric model and made the corresponding counter terms available in a computer-readable form. One can adopt the latter for more complicated theories by changing model-dependent prefactors. In this way, six-loop renormalization-group functions for O(n) theory with cubic anisotropy [22] and O(n) × O(m) symmetric model [23] were derived.
To perform calculations with general Lagrangian (1.1), we prepare a DIANA [24] model file. We use special mapping rules between its internal topology format and diagram topologies, which are identified in ref. [21] and given in the Nickel index notation. After generating all needed two-and four-point functions with DIANA and performing all needed index contractions with FORM [25], we substitute actual values for momentum integrals by counter-terms from the available tables [21]. It is trivial to extract the RG functions γ ab and β abcd ≡ β λ abcd from the first ε pole in the sum of counter-terms.
The obtained results involve a certain number of tensor structures, i.e., products of (up to 12) general couplings λ abcd with all but four (two) indices contracted in β abcd (γ ab ). We can simplify corresponding expressions by identifying tensor structures identical up to the renaming of contracted indices. Also, since the corresponding numeric coefficient depends only on the Feynman graph, we collect all the structures, which are different only by permutations of external indices abcd. As a consequence, we can cast our main result for β abcd into the form i,abcd , we made use of Nickel index notation [26] for graph representation of tensor contractions and utilized the GraphState package [27]. As an example, we give here one of the three-loop structures where we indicate the corresponding Nickel index and emphasize the normalization of T (l) i,abcd together with the fact that the latter are symmetric in abcd. We provide a table containing a minimal set of unique tensor structures formed by different contractions between λ abcd indices and the corresponding coefficients. Given these tables, we derive the beta functions for dimensionful parameters entering (1.1) employing the so-called dummy field method [6,28,29]. The core of the technique is to introduce "dummy" non-propagating field(s) x a , e.g., by shifting all (or just one) components of the vector φ a → φ a + x a . Contracting β abcd with one or more dummy fields x a , we can readily obtain the expressions for β Λ , β a ≡ β ta , β ab ≡ β m 2 ab , and β abc ≡ β h abc (see appendix A). Indeed, we consider 1 β xxxx , β axxx , β abxx , together with β abcx , and identify λ abcx ≡ h abc , λ abxx ≡ 2m 2 ab , λ axxx ≡ 3!t a , λ xxxx ≡ 4!Λ. The only subtlety here is that we have to remove contributions from external leg renormalization, leading to tadpole diagrams in the 1 We use compact notation βxxxx ≡ β abcd xax b xcx d , etc.
final answer (see ref. [6] for details). We can immediately identify corresponding tensor structures in general expression for β abcd where dotted lines represent dummy field x. We use tilde to denote the quantities with tadpole contribution removed, and write The tensor structures, including the corresponding graphs and coefficients for all the considered RG functions, can be found in the form of supplementary Mathematica files.

From general results to specific models
In this section we demonstrate the application of our general results to particular scalar models. It is worth mentioning that we heavily rely on FORM [25] to deal with index contractions and algebraic simplifications in the case of matrix fields.

Warming up with O(n)-symmetric model
Our first example is the well-known O(n) symmetric model, which has a long history in the study of critical phenomena (see ref. [21] and reference therein). The following Euclidean Lagrangian describes the theory where φ = {φ a }, a = 1, ..., n is a n-component scalar field. We also add a quadratic operator involving traceless symmetric tensor d ab multiplied by a source g φφ . The anomalous dimension 2 γ φφ of the corresponding operator is related to the so-called crossover exponent (see, e.g., refs. [30,31]) and can be found in our approach as with β g φφ being the beta function of g φφ and γ φ corresponding to the anomalous dimension of the field computed via eq. (1.3). This and other RG functions can be easily obtained from our general result by means of substitutions 3) In our calculation we find perfect agreement with previous computations. Our new result is related to the six-loop contribution to the beta function of the vacuum energy 2 In ref. [30] the notation γẼ = γ φφ is used. β Λ for g φφ = 0 (see refs. [32,33] for the five-loop expression). Using the notation g ≡ hλ (c.f. ref. [32]) we have 3 By simple rescaling λ → 3!λ, one can easily get the six-loop contributions to the RG functions for the Standard Model Higgs potential parameters (including the vacuum energy) from the results of O(4) theory.

Matrix models
We consider matrix models with real and complex fields described by the following Lagrangians for real φ and for complex φ. To deal with matrix models we make use of the following decomposition (see also ref. [34]) where χ a are real fields, and there are N a independent matrices T a , which encode all the degrees of freedom present in φ. Substituting (3.8) into either (3.6) or (3.7), we can rewrite the Lagrangians in the form (1.1). One can see that we completely get rid of the initial matrix indices of φ and replace them with a single one a = 1, . . . , N a . Given eqs. (3.6) and (3.7), for the fields χ a to be canonically normalized, we have to ensure that (T a ) † ≡T a As a consequence, one can identify (3.14) In eq. (3.12) all 24 permutations of the indices abcd are taken into account. Obviously, the number of terms can be reduced in specific models. In the following subsections we provide some details of our calculations for the cases discussed in the literature.

Real anti-symmetric field
The Lagrangian of the model is given by eq. (3.6) with φ being an antisymmetric n × n matrix, φ T = −φ. The model was considered in refs. [35,36] and the four-loop results can be found in ref. [37].
To use our general formulae, we utilize the decomposition (3.8) with N a = n(n−1) 2 and T a = t a corresponding to antisymmetric generators of SO(n). The latter satisfy To keep the standard normalization for the fields χ a , we use T f = 1 (see eq. (3.9)). The number of terms in eq. (3.12) can be reduced (Trt a t b . . . ≡ T ab... ) where we used the cyclic symmetry of the trace operation and the fact that t T a = −t a . By means of eq. (3.15) we write down the rules, which allow one to simplify the products of traces involving t a with some of the indices contracted. Substituting (3.16) into the general expression for β abcd , and performing the above-mentioned algebraic simplifications, we obtain β abcd of the form where f 1,2 (λ 1 , λ 2 , n) are some polynomials of their arguments. It is possible to extract the beta functions for λ 1 and λ 2 from eq. (3.17) by applying suitable projectors. However, one can also use the fact that by construction β abcd is symmetric in all the indices. Setting the latter equal to each other in the end of calculation, we have , no sum over a.
We utilize this approach to obtain relevant RG functions up to the six-loop level. Our results agree with that given in refs. [36,37] 4 . It is worth noting that for n = 2 and n = 3 the model is equivalent to one-component φ 4 and the O(3)-vector theory considered in sec. 3.1, respectively. Indeed, combining λ = λ 1 + 1 2 λ 2 and computing β λ = β λ 1 + 1 2 β λ 2 for n = 2 and n = 3 we get the expected results.
Full six-loop beta functions and anomalous dimensions are available online as supplementary material. For convenience, we present here our expressions for the one-loop and two-loop RG functions Let us now consider a matrix model, which is invariant under O(n) × O(m) group. It describes the critical thermodynamics of frustrated spin systems with noncollinear and noncoplanar ordering (see, e.g., ref. [23] and references therein). In refs. [38] five-loop results are presented in terms of u = λ 1 + λ 2 , and v = λ 2 . Six-loop RG functions are also known [23] in terms of g i = λ i . The Landau-Wilson Lagrangian can be written in the form (3.6) with φ = {φ αi } being n × m real matrix field, and α = 1, . . . , n, i = 1, . . . , m.
To compute relevant RG functions from our general result we interpret χ a in eq. (3.8) as N a = n · m matrix elements of φ, so that each of n × m real matrices T a has only one non-zero element where we introduce T f = 1 for convenience. As a consequence 5 , we have The quartic self-coupling is given by 29) where to reduce the number of terms in LHS, we use the fact that To extract the RG functions, we substitute (3.29) together with (3.11) into β abcd , β ab and γ ab and use the rules (3.28) to simplify the products of traces involving T a and T T b . We use known results [23,38] to cross-check our expressions, which at the one-loop order are given by In addition, we extend to the six-loop order the anomalous dimensions of quadratic operators considered in refs. [39,40]: αβij δ ij (3.41) and belonging to different representations of O(n)×O(m). The operators can be treated in our approach in a similar fashion. We assume that the perturbations can be added to the Lagrangian with the corresponding sources ("masses") and rewritten in terms of χ-fields as, e.g., Since the operators (3.41) do not mix under renormalization, we use the following substi- ij : m 2 ab ⇒m 2 3,cd and extract the beta functions (βm2 i ) c,d ≡ −γ i ·m 2 i,cd ofm 2 i,cd , i = 1, 4 from the corresponding terms in β ab . The RG functions for the operators Q (i) (3.41) are obtained by adding the contribution from the field anomalous dimension γ χ = γ φ : (3.48) A welcome check of the result is the fact that for v = 0 all γ Q i coincide. We also compare our expressions with that given in ref. [40] and find perfect agreement up to five loops 7 .

Complex anti-symmetric field
Let us now generalize the model discussed in sec. 3.2.1 and consider complex antisymmetric n×n matrices φ. The corresponding Lagrangian (3.7) can be used to study phase transitions in quantum Fermi systems within the RG approach (see ref. [41]). We decompose the field via (3.8) with N a = n(n − 1) and antisymmetric T a = t a a = 1, . . . , n(n − 1)/2, it a a = 1 + n(n − 1)/2, . . . , n(n − 1).
The latter are written in terms of generators t a of SO(n). Given Tr(t a t b ) = T f δ ab , one can derive (T a ) † ≡T a One can see from eq. (3.10) that for T f = −1/2 the fields χ a are canonically normalized. The self-coupling (3.12) is given by In writing the latter we take into account that so, e.g., The expressions for the RG functions are available in literature up to the five-loop level 8 [42]. We extend these results up to six loops. The one-loop contributions read β (1) All results at six loops are available online as supplementary material.

U (n) × U (m) model
Consider now eq. (3.7) with general complex n × m matrix field φ = {φ αi }. The model can be used to study phase transitions in massless QCD and five-loop RG functions are available in literature [43]. We compute the six-loop contributions by means of decomposition (3.8) with N a = 2nm and T a being complex n × m matrices (c.f. eq. The calculations are carried out with T f = 1/2 and the following representation of general self-coupling λ abcd = λ 1 12 T ab T cd + 11 perms + λ 2 12 T abcd + 11 perms , where among all 24 permutations we exclude only those that correspond to the swapping between pairs of indices. Our calculation employs eq. (3.67) and renders at one loop β (1) (3.71) and at two loops The full results are available as supplementary files. It is worth noting that for m = n we get the four-loop results obtained in ref. [18] for the case of U (n) × U (n) model. 9

Field in the adjoint representation of SU (n)
In recent ref. [44] a model with φ being hermitian matrix field in the adjoint representation of SU (n) is analyzed both with perturbative and non-perturbative methods. In addition, the model was also considered as an example of application of the ARGES code [19]. We generalize the Lagrangian of ref. [44] and include also a cubic term 10 (see also refs. [34,45,46]) together with the vacuum energy (we rescale f and λ 2 for convenience) Obviously, we can easily treat the model in our approach by means of the decomposition (3.8) with T a being SU (n) generators. The latter satisfy the well-known relations [47] Tr We utilize the normalization T f = 1 and substitute We obtain the RG functions up to the six-loop level, and at one loop we have β (1) The two-loop expressions are given by To compare our results with that of ref. [44], one has to take into account that the latter correspond to f = 0 and are written in terms of g i /(8π 2 ) with g 1 = (nλ 2 )/6 and g 2 = λ 1 /6. We also use the expressions obtained by means of ARGES [19] to cross-check γ φ and the beta functions for λ 1 , λ 2 , and m 2 up to 4 loops.

Higher rank O(n) × O(n) × O(n) tensor model
To give an example how to apply our general result to models with more complicated index structure, we consider evaluation of the beta functions in the model with O(n)×O(n)×O(n) symmetry [48]. The model Lagrangian is where we follow the naming scheme from ref. [48] for the interaction terms as "tetrahedral", "pillow" and "double-sum". Again, we use fields as tensor indices of the structures T i to indicate contractions of triplets of indexes with φ abc . For convenience, we present the tensor structures in the following pictorial form: (3.92) where (i 1 i 2 i 3 i 4 ) denotes symmetrization. At one loop we get

(3n + 14). (3.100)
Modulo rescaling λ i = 6g i , the obtained expressions coincide with those given in ref. [48]. Six-loop results can be found in the form of supplementary files.
In this work, we use another strategy and directly calculate the beta function of λ 1−7 together with the anomalous dimensions of m 2 11 , m 2 22 , and m 2 12 from our general expressions. We enumerate all real components of two doublets Φ 1,2 and rewrite eq. (3.101) in the general form (1.1) with indices a, b, etc. running from one to eight. We find full agreement with previous results and extend the latter up to six loops. We have checked that our expressions for β λ 2 (β λ 7 ) can be obtained from β λ 1 (β λ 6 ) via the replacement λ 1 ↔ λ 2 and λ 6 ↔ λ 7 . One can use the same substitutions together with m 2 11 ↔ m 2 22 to get β m 2 22 from β m 2 11 . We make the six-loop results available as supplementary files.

Conclusion
We considered the general renormalizable scalar QFT model and directly computed the RG functions for the quartic and cubic self-couplings, mass parameter, tadpole term, and vacuum energy. In deriving our results for dimensionless quantities, we used the expressions for the KR operation applied to individual Feynman integrals. The latter are publicly available thanks to lengthy and non-trivial calculations of ref. [21]. To compute the RG functions of dimensionful parameters, we utilize the powerful dummy field technique.
To validate our general results, we considered several scalar models discussed in the theory of critical phenomena. We found perfect agreement with known results and extend them by computing several missing six-loop contributions. Among the latter are the vacuum energy beta function in the O(n) model, the anomalous dimensions of quadratic perturbations in the O(n)×O(m) model, and the self-coupling beta functions for U (n)×U (m), and O(n) × O(n) × O(n) models and the model with the Higgs field in the adjoint representation of the SU (n) group. Additionally, we extend the three-loop results for the general Two-Higgs-Doublet Model scalar sector to six loops.
We believe that the obtained state-of-the-art RG functions are of immediate interest to the condensed-matter community. On the contrary, present six-loop results can hardly find their applications in phenomenological analyses of the Standard Model extensions in the near future. However, it is convenient to estimate the influence of the high-order terms on extended Higgs sector studies, which currently rely on the two-or three-loop RG. Public codes for RG analyses [20,34,[52][53][54] can be equipped with our results to carry out this kind of computations.
We also note that the expression for vacuum energy beta function is relevant for effective potential V eff (φ) RG improvement (see, e.g., ref. [55]). Moreover, in recent ref. [56], the vacuum energy function's role is emphasized in the effective field theory approach to V eff (φ) computation in models with many different scales.  Table 1. Representation of vertices corresponding to the parameters of the Lagrangian (1.1). The index "x" denotes the contraction with a dummy field.  eq. (A.1) in the following pictorial form: where external self-couplings are denoted by blue vertices (see table 1) and we use the notation given in table 2 for the non-external parts of four-point functions. It is worth noting that ab|cd , ab|cde|f gh , and abc|def |ghi|jkl do not need to be symmetric w.r.t. permutations of (group of) indices. Each group of indices is contracted with symmetric couplings, and, thus, does not need to be explicitly symmetrized. Hoverer, we explicitly take into account that an external index a, b, c, or d can be attached to any group via a quartic vertex. This corresponds the permutations indicated, e.g., in eq. (A.2). Due to this, we distinguish index groups and mark them by numbers (c.f., table 2).
Let us now contract the expression (A.2) with external dummy field x d and exclude the tadpole graphs discussed in sec. 2: Here the trilinear couplings correspond to red vertices (see table 1) and again we have to explicitly take into account permutations of external indices. The analytic expression is given by + λ abef λ cghi h jkl ef |ghi|jkl + ef |jkl|ghi + 2 perm. + h aef λ bghi λ cjkl ef |ghi|jkl +5 perm. + λ aef g λ bhij λ cklm h nop ef g|hij|klm|nop + nop|ef g|hij|klm + klm|nop|ef g|hij + hij|klm|nop|ef g + 5 perm. (A.4) To obtain the beta function for mass parameter we contract eq. (A.3) with one more dummy field x c . Dividing the result by the factor of two, we get where red dots denote mass parameter m 2 ab insertions (c.f. table 1). The corresponding analytic expression is given by 11 β ab = m 2 af γ φ f b + m 2 bf γ φ f a + λ abef m 2 gh + h aef h bgh ef |gh + gh|ef + λ abef h ghi h jkl ef |ghi|jkl +m 2 ef λ aghi λ bjkl ef |ghi|jkl + ef |jkl|ghi h aef h ghi λ bjkl ef |ghi|jkl + ef |jkl|ghi + (a ↔ b) + λ aef g λ bhij h klm h nop ef g|hij|klm|nop + ef g|klm|hij|nop + ef g|klm|nop|hij + klm|ef g|hij|nop + klm|ef g|nop|hij + klm|nop|ef g|hij + (a ↔ b) . (A.6) We proceed further and obtain the RG function for the tadpole term. Contracting eq. (A.5) with x b and dividing by the factor of 3, we get where the orange vertex corresponds the tadpole parameter t a of the Lagrangian (1.1). The analytic form of eq. (A.7) looks like β a = t f γ φ f b + h aef m 2 gh ef |gh + gh|ef + h aef h ghi h jkl ef |ghi|jkl +m 2 ef λ aghi h jkl ef |ghi|jkl + ef |jkl|ghi + λ aef g h hij h klm h nop ef g|hij|klm|nop + nop|ef g|hij|klm + klm|nop|ef g|hij + hij|klm|nop|ef g . (A.8) One more contraction with the dummy field x a gives the beta function of the vacuum energy:  11 We correct a couple of misprints in the corresponding expression in the published version of ref. [18]. can be found in a supplementary PDF file.