Six-loop beta functions in general scalar theory
PPrepared for submission to JHEP
Six-loop beta functions in general scalar theory
A. Bednyakov a,b, and A. Pikelner a a Joint Institute for Nuclear Research,Joliot-Curie, 6, Dubna 141980, Russia b P.N. Lebedev Physical Institute of the Russian Academy of Sciences,Leninskii pr., 5, Moscow 119991, Russia
E-mail: [email protected] , [email protected] Abstract:
We consider general renormalizable scalar field theory and derive six-loopbeta functions for all parameters in d = 4 dimensions within the MS-scheme. We do notexplicitly compute relevant loop integrals but utilize O ( n )-symmetric model counter-termsavailable in the literature. We consider dimensionless couplings and parameters with a massscale, ranging from the trilinear self-coupling to the vacuum energy. We use obtained resultsto extend renormalization-group equations for several vector, matrix, and tensor modelsto the six-loop order. Also, we apply our general expressions to derive new contributionsto beta functions and anomalous dimensions in the scalar sector of the Two-Higgs-DoubletModel. Corresponding author. a r X i v : . [ h e p - ph ] F e b ontents O ( n )-symmetric model 43.2 Matrix models 53.2.1 Real anti-symmetric field 63.2.2 O ( n ) × O ( m ) model 73.2.3 Complex anti-symmetric field 103.2.4 U ( n ) × U ( m ) model 113.2.5 Field in the adjoint representation of SU ( n ) 123.3 Higher rank O ( n ) × O ( n ) × O ( n ) tensor model 133.4 Two-Higgs Doublet Model 15 The renormalization group (RG) plays an essential role in high-energy physics and thetheory of critical phenomena. In particle physics, one can use RG to re-sum specificradiative corrections making theory predictions valid in a wide range of energy scales. Inthe study of critical phenomena, the RG approach allows one to study phase transitionsand predict critical exponents of the second-order transitions with high accuracy.A convenient tool to compute the RG functions that drive the dependence of modelparameters on the scale is to use a perturbative expansion of dimensional regularizedtheory [1] together with modified minimal MS subtraction of infinities. The latter appearin loop integrals and manifest themselves in d dimensions as poles in ε = (4 − d ) /
2. Onecancels the poles by a finite set of renormalization constants.There is significant progress in the calculation of beta functions and anomalous dimen-sions in the MS scheme. At the two-loop level, the RG functions are known in any generalrenormalizable quantum field theory (QFT) in d = 4 dimensions [2–6]. Despite severalcalculations of three-loop (and even four-loop) RG functions in particular particle-physicsmodels [7–16], general three-loop results are not yet available. Recently, an essential stephas been made in this direction [17–20]. 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– 1 –orresponding 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 functionsin a pure scalar model.Our paper does not follow this strategy and extends the results for general scalartheories up to six loops by more conventional technique, i.e., by computing contributionsfrom individual Feynman graphs. Such a leap in the loop level is due to the significantprogress in calculating critical exponents in scalar theories. Thanks to the authors ofref. [21], the required renormalization constants can be found given the diagram-by-diagramresults of the KR (cid:48) operation. Application of the latter to a Feynman graph produces thecorresponding counter-term in the MS scheme.We consider the following general renormalizable Lagrangian L = 12 ∂ µ φ a ∂ µ φ a − m ab φ a φ b − h abc φ a φ b φ c − λ abcd φ a φ b φ c φ d − t a φ a − Λ (1.1)for real scalar fields φ a . The mass parameters m ab , cubic h abc and quartic couplings λ abcd are symmetric in their indices. For completeness we also add the tadpole term proportionalto 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 allparameters of eq. (1.1). The RG function for a parameter A = { λ abcd , h abc , m ab , t a , Λ } isdefined as β A ≡ µ ∂A∂µ = (cid:88) l h l β ( l ) A , h = 116 π , (1.2)where β ( l ) A corresponds to the l -loop contribution. The field anomalous dimension is givenby φ a, = Z ab φ b ⇒ γ φab = Z − ac · µ ∂Z cb ∂µ = − µ ∂Z − ac ∂µ · Z cb (1.3)and is related to the field renormalization constant Z ab .The paper is organized as follows. Section 2 contains details of our calculation. Insection 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) modelspossessing different kinds of symmetries. Also, we extend known three-loop results forthe Two-Higgs-Doublet Model (2HDM) to six loops in section 3.4. Section 4 contains adiscussion of the results and conclusions. In appendix A we provide a derivation of the RGfunctions for dimensionful couplings in a general form. 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 ofref. [21] considered all the required six-loop graphs in the context of the O ( n )-symmetric– 2 –odel and made the corresponding counter terms available in a computer-readable form.One can adopt the latter for more complicated theories by changing model-dependentprefactors. In this way, six-loop renormalization-group functions for O ( n ) theory withcubic 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] modelfile. We use special mapping rules between its internal topology format and diagramtopologies, which are identified in ref. [21] and given in the Nickel index notation. Aftergenerating all needed two- and four-point functions with
DIANA and performing all neededindex contractions with
FORM [25], we substitute actual values for momentum integrals bycounter-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 upto the renaming of contracted indices. Also, since the corresponding numeric coefficientdepends only on the Feynman graph, we collect all the structures, which are different onlyby permutations of external indices abcd . As a consequence, we can cast our main resultfor β abcd into the form β abcd = (cid:88) l =1 h l n l (cid:88) i T ( l ) i,abcd C ( l ) i , (2.1)where n l = { , , , , , } is the number of unique tensor structures T ( l ) i,abcd at l loops.The coefficients C ( l ) i are pure numbers. To deduce the expressions for T ( l ) i,abcd , we made useof Nickel index notation [26] for graph representation of tensor contractions and utilizedthe GraphState package [27]. As an example, we give here one of the three-loop structures T (3)4 ,abcd ≡
14! [ λ ai i i λ bi i i λ ci i i λ di i i + perm . ] = , e123 | e23 | e3 | e | (2.2)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 bydifferent contractions between λ abcd indices and the corresponding coefficients. Given thesetables, we derive the beta functions for dimensionful parameters entering (1.1) employingthe 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 thevector φ a → φ a + x a . Contracting β abcd with one or more dummy fields x a , we can readilyobtain the expressions for β Λ , β a ≡ β t a , β ab ≡ β m ab , and β abc ≡ β h abc (see appendix A).Indeed, we consider β xxxx , β axxx , β abxx , together with β abcx , and identify λ abcx ≡ h abc , λ abxx ≡ m ab , λ axxx ≡ t a , λ xxxx ≡ We use compact notation β xxxx ≡ β abcd x a x b x c x d , etc. – 3 –nal answer (see ref. [6] for details). We can immediately identify corresponding tensorstructures in general expression for β abcd = = = = 0 , (2.3)where dotted lines represent dummy field x . We use tilde to denote the quantities withtadpole contribution removed, and write β Λ = 14! · ˜ β xxxx , β a = 13! · ˜ β axxx , β ab = 12 · ˜ β abxx , β abc = ˜ β abcx . (2.4)The tensor structures, including the corresponding graphs and coefficients for all theconsidered RG functions, can be found in the form of supplementary Mathematica files.
In this section we demonstrate the application of our general results to particular scalarmodels. It is worth mentioning that we heavily rely on
FORM [25] to deal with indexcontractions and algebraic simplifications in the case of matrix fields. O ( n ) -symmetric model Our first example is the well-known O ( n ) symmetric model, which has a long history in thestudy of critical phenomena (see ref. [21] and reference therein). The following EuclideanLagrangian describes the theory L = 12 (cid:126)φ ( − ∂ + m ) (cid:126)φ + λ
4! ( (cid:126)φ · (cid:126)φ ) + 12 g φφ d ab φ a φ b , (3.1)where (cid:126)φ = { φ a } , a = 1 , ..., n is a n -component scalar field. We also add a quadratic oper-ator involving traceless symmetric tensor d ab multiplied by a source g φφ . The anomalousdimension 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 γ φφ = − β g φφ + 2 γ φ (3.2)with β g φφ being the beta function of g φφ and γ φ corresponding to the anomalous dimensionof the field computed via eq. (1.3). This and other RG functions can be easily obtainedfrom our general result by means of substitutions λ abcd = λ δ ab δ cd + δ ac δ bd + δ ad δ bc ) , (3.3) m ab = m δ ab + g φφ d ab . (3.4)In our calculation we find perfect agreement with previous computations. Our newresult is related to the six-loop contribution to the beta function of the vacuum energy In ref. [30] the notation γ ˜ E = γ φφ is used. – 4 – 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 β v ≡ · π m β Λ = n n ( n + 2)96 g + n ( n + 2) g n + 8)(12 ζ − n ( n + 2) g (cid:104) ζ (3 n − n − ζ (4 n + 39 n + 146) − ζ (5 n + 22) − n + 13968 n + 64864 (cid:105) + n ( n + 2) g (cid:104) ζ (41 n − n − − ζ (70 n + 809 n + 2118)+ 96 ζ (45 n + 890 n + 19348 n + 67440) − ζ ( n + 294 n + 3088 n + 9496) − ζ (51 n + 700 n − n + 2024) − n − n − n − ζ (67 n + 1405 n + 4306) + 1152 ζ ζ (2 n + 145 n + 582) (cid:105) . (3.5)By simple rescaling λ → λ , one can easily get the six-loop contributions to theRG functions for the Standard Model Higgs potential parameters (including the vacuumenergy) from the results of O (4) theory. We consider matrix models with real and complex fields described by the following La-grangians L = 12 Tr (cid:2) φ ( − ∂ + τ ) φ T (cid:3) + λ
4! (Tr (cid:2) φφ T (cid:3) ) + λ
4! Tr (cid:2) φφ T φφ T (cid:3) (3.6)for real φ and L = Tr[ φ ( − ∂ + τ ) φ † ] + λ
4! (Tr[ φφ † ]) + λ
4! Tr[ φφ † φφ † ] (3.7)for complex φ . To deal with matrix models we make use of the following decomposition(see also ref. [34]) φ = N a (cid:88) a =1 χ a T a , (3.8)where χ a are real fields, and there are N a independent matrices T a , which encode all thedegrees of freedom present in φ . Substituting (3.8) into either (3.6) or (3.7), we can rewritethe Lagrangians in the form (1.1). One can see that we completely get rid of the initialmatrix 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 (cid:2) ( T a ) † ≡ ¯ T a (cid:3) Tr( T a T Tb ) + Tr( T b T Ta ) = 2 δ ab , for real φ, (3.9)Tr( T a ¯ T b ) + Tr( T b ¯ T a ) = δ ab , for complex φ. (3.10) In ref. [32] RG functions are defined as derivatives w.r.t ln µ = 2 ln µ . The factor in (3.5) is introducedfor convenience. – 5 –s a consequence, one can identify m ab = − τ δ ab , (3.11) λ abcd = λ (cid:104) T ab T cd + perm. (cid:105) + λ (cid:104) T abcd + perm. (cid:105) , (3.12)where T ab = Tr( T a T Tb ) ≡ T ab , T abcd = Tr( T a T Tb T c T Td ) ≡ T abcd , for real φ, (3.13) T ab = Tr( T a ¯ T b ) ≡ T ab , T abcd = Tr( T a ¯ T b T c ¯ T d ) ≡ T abcd , for complex φ. (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 weprovide some details of our calculations for the cases discussed in the literature. 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 canbe found in ref. [37].To use our general formulae, we utilize the decomposition (3.8) with N a = n ( n − and T a = t a corresponding to antisymmetric generators of SO ( n ). The latter satisfyTr( t a t b ) = T f δ ab , t aij t akl = T f δ il δ jk − δ ik δ jl ) . (3.15)To keep the standard normalization for the fields χ a , we use T f = 1 (see eq. (3.9)). Thenumber of terms in eq. (3.12) can be reduced (Tr t a t b . . . ≡ T ab... ) λ abcd = λ (cid:104) T ab T cd + T ac T bd + T ad T bc (cid:105) + λ (cid:104) T abcd + T abdc + T acbd (cid:105) , (3.16)where we used the cyclic symmetry of the trace operation and the fact that t Ta = − t a .By means of eq. (3.15) we write down the rules, which allow one to simplify the productsof traces involving t a with some of the indices contracted. Substituting (3.16) into thegeneral expression for β abcd , and performing the above-mentioned algebraic simplifications,we obtain β abcd of the form4! β abcd = f ( λ , λ , n ) (cid:104) T ab T cd + perm. (cid:105) + f ( λ , λ , n ) (cid:104) T abcd + perm. (cid:105) , (3.17)where f , ( λ , λ , n ) are some polynomials of their arguments. It is possible to extract thebeta functions for λ and λ from eq. (3.17) by applying suitable projectors. However, onecan also use the fact that by construction β abcd is symmetric in all the indices. Setting thelatter equal to each other in the end of calculation, we have β aaaa = f ( λ , λ , n )[ T aa ] + f ( λ , λ , n )[ T aaaa ] , no sum over a. (3.18)Comparing eqs. (3.18) and (3.16) with b = c = d = a , one can easily deduce that β λ = f ( λ , λ , n ) , β λ = f ( λ , λ , n ) . (3.19)– 6 –e utilize this approach to obtain relevant RG functions up to the six-loop level. Ourresults agree with that given in refs. [36, 37] . It is worth noting that for n = 2 and n = 3 the model is equivalent to one-component φ and the O (3)-vector theory consideredin sec. 3.1, respectively. Indeed, combining λ = λ + λ and computing β λ = β λ + β λ for n = 2 and n = 3 we get the expected results.Full six-loop beta functions and anomalous dimensions are available online as supple-mentary material. For convenience, we present here our expressions for the one-loop β (1) λ = λ n − n + 16) + λ λ n −
1) + λ , (3.20) β (1) λ = 4 λ λ + λ n − , (3.21) γ (1) τ ≡ − β (1) τ /τ = − λ n − n + 4) − λ n − , (3.22)and two-loop RG functions γ (2) φ = 1288 (cid:2) λ + λ (cid:3) ( n − n + 4) + λ λ
36 (2 n − , (3.23) β (2) λ = λ n − n − − (cid:20) λ λ λ (cid:21) (2 n − − λ λ
72 (5 n − n + 164) , (3.24) β (2) λ = λ λ
18 (5 n − n − − λ λ n −
1) + λ
24 ( n − n − , (3.25) γ (2) τ = 5144 (cid:2) λ + λ (cid:3) ( n − n + 4) + 5 λ λ
18 (2 n − . (3.26) O ( n ) × O ( m ) model Let us now consider a matrix model, which is invariant under O ( n ) × O ( m ) group. Itdescribes the critical thermodynamics of frustrated spin systems with noncollinear andnoncoplanar ordering (see, e.g., ref. [23] and references therein). In refs. [38] five-loopresults are presented in terms of u = λ + λ , and v = λ . Six-loop RG functions are alsoknown [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 onenon-zero element ( T a ) αi = (cid:112) T f · δ α, (( a −
1) div m )+1 δ i, (( a −
1) mod m )+1 , (3.27)where we introduce T f = 1 for convenience. As a consequence , we haveTr( T a T Tb ) = Tr( T Tb T a ) = T f δ ab , T aαi T aβj = T f δ αβ δ ij . (3.28) Note that in ref. [36] the notation λ i = g i is used and the RG functions are expanded in g i / (8 π ). Notice here that T a T Tb are n × n matrices, while T Ta T b have m × m dimension. – 7 –he quartic self-coupling is given by λ abcd = λ (cid:104) T ab T cd + T ac T bd + T ad T bc (cid:105) + λ (cid:104) T abcd + T abdc + T acbd + T acdb + T adbc + T adcb (cid:105) , (3.29)where to reduce the number of terms in LHS, we use the fact thatTr( A ) = Tr( A T ) , ( T a T Tb T c T Td ) T = T d T Tc T b T Ta so T abcd ≡ Tr( T a T Tb T c T Td ) = Tr( T d T Tc T b T Ta ) ≡ T dcba . (3.30)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 Tb .We use known results [23, 38] to cross-check our expressions, which at the one-looporder are given by β (1) λ = λ nm ) + 2 λ λ n + m ) + λ , (3.31) β (1) λ = 4 λ λ + λ m + n ) , (3.32) γ (1) τ ≡ − β (1) τ /τ = − λ nm ) − λ m + n ) , (3.33)while at two loops we have γ (2) φ = λ
36 (2 + mn ) + λ λ
18 (1 + m + n ) + λ
72 (3 + mn + m + n ) , (3.34) β (2) λ = − λ mn ) − λ λ m + n ) − λ λ
18 (87 + 5( mn + m + n )) − λ m + n ) , (3.35) β (2) λ = − λ λ mn ) − λ λ n + m ) + 29) − λ mn + 3( m + n )) , (3.36) γ (2) τ = 5 λ
18 (2 + mn ) + 5 λ λ m + n ) + 5 λ
36 (3 + mn + m + n ) . (3.37)In addition, we extend to the six-loop order the anomalous dimensions of quadratic oper-ators considered in refs. [39, 40]: Q (1) αiβj = φ αi φ βj − φ αj φ βi , (3.38) Q (2) αiβj = 12 ( φ αi φ βj + φ αj φ βi ) − n δ αβ φ δi φ δj − m δ ij φ αk φ βk + 1 nm δ αβ δ ij φ δk φ δk , (3.39) Q (3) ij = φ δi φ δj − m δ ij φ δk φ δk ≡ ˜ Q (3) αβij δ αβ , (3.40) Q (4) αβ = φ αk φ βk − n δ αβ φ δk φ δk ≡ ˜ Q (4) αβij δ ij (3.41)– 8 –nd belonging to different representations of O ( n ) × O ( m ). The operators can be treated inour approach in a similar fashion. We assume that the perturbations can be added to theLagrangian with the corresponding sources (“masses”) and rewritten in terms of χ -fieldsas, e.g., (cid:2) ˜ m ,αiβj (cid:3) (cid:104) Q (1) αiβj (cid:105) = (cid:104) ˜ m ,cd T cαi T dβj (cid:105) (cid:104) χ a χ b (cid:16) T aαi T bβj − T aαj T bβj (cid:17)(cid:105) ≡ m ,ab χ a χ b (3.42) m ,ab = 12 ˜ m ,cd (cid:104) T ac T bd − T adbc + ( a ↔ b ) (cid:105) . (3.43)Since the operators (3.41) do not mix under renormalization, we use the following substi-tutions Q (1) αiβj : m ab ⇒ ˜ m ,cd (cid:104) T ac T bd − T adbc + ( a ↔ b ) (cid:105) , (3.44) Q (2) αiβj : m ab ⇒ ˜ m ,cd (cid:20) T ac T bd + T adbc − n T acdb − m T abcd + ( a ↔ b ) (cid:21) + ˜ m ,cd nm T ab T cd , (3.45) Q (3) ij : m ab ⇒ ˜ m ,cd (cid:20) (cid:104) T acdb + ( a ↔ b ) (cid:105) − m T ab T cd (cid:21) , (3.46) Q (4) αβ : m ab ⇒ ˜ m ,cd (cid:20) (cid:104) T abcd + ( a ↔ b ) (cid:105) − n T ab T cd (cid:21) (3.47)and extract the beta functions ( β ˜ m i ) c,d ≡ − ˜ γ i · ˜ m i,cd of ˜ m i,cd , i = 1 , β ab . The RG functions for the operators Q ( i ) (3.41) are obtained by adding thecontribution from the field anomalous dimension γ χ = γ φ : γ Q i = ˜ γ i +2 γ φ . (3.48)A welcome check of the result is the fact that for v = 0 all γ Q i coincide. We also compareour expressions with that given in ref. [40] and find perfect agreement up to five loops .We present here our one-loop, γ (1) Q = − λ − λ , (3.49) γ (1) Q = − λ + λ , (3.50) γ (1) Q = − λ + (1 + n ) λ , (3.51) γ (1) Q = − λ + (1 + m ) λ , (3.52) Given eq. (3.30), one can prove that tensors multiplying ˜ m i,cd are symmetric in a ↔ b and c ↔ d . The results of ref. [40] are written in terms of ( u, v ) / (8 π ). – 9 –nd two-loop results γ (2) Q = λ mn ) + 2 λ λ m + n ) + λ
18 (1 − m − n ) , (3.53) γ (2) Q = λ mn ) + 2 λ λ m + n ) + λ
18 (9 + m + n ) , (3.54) γ (2) Q = λ mn ) + 2 λ λ m + 3 n ) + λ
18 (9 + mn + m + 3 n ) , (3.55) γ (2) Q = λ mn ) + 2 λ λ m + n ) + λ
18 (9 + mn + 3 m + n ) . (3.56)The six-loop expressions are available as supplementary files. 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 transitionsin quantum Fermi systems within the RG approach (see ref. [41]). We decompose the fieldvia (3.8) with N a = n ( n −
1) and antisymmetric T a = (cid:40) t a a = 1 , . . . , n ( n − / ,it a a = 1 + n ( n − / , . . . , n ( n − . The latter are written in terms of generators t a of SO ( n ). Given Tr( t a t b ) = T f δ ab , one canderive (cid:2) ( T a ) † ≡ ¯ T a (cid:3) Tr( T a ¯ T b ) + Tr( T b ¯ T a ) = − T f δ ab , T aij T akl = N a / (cid:88) b =1 (cid:16) t bij t bij + i t bij t bij (cid:17) = 0 ,T aij ¯ T akl = N a / (cid:88) b (cid:16) t bij ( − t bkl ) + it bij (+ it bkl ) (cid:17) = − T f ( δ il δ jk − δ ik δ jl ) . (3.57)One can see from eq. (3.10) that for T f = − / χ a are canonically normalized.The self-coupling (3.12) is given by λ abcd = λ (cid:104) T ab T cd + 11 permutations (cid:105) + λ (cid:104) T abcd + T abdc + T acbd T cadb + T bacd + T badc (cid:105) . (3.58)In writing the latter we take into account thatTr( A ) = Tr( A T ) , ( T a ¯ T b T c ¯ T d ) T = ( T d ) ∗ ( T c ) T ( T b ) ∗ ( T a ) T = ¯ T d T c ¯ T b T a , so, e.g., T abcd ≡ Tr( T a ¯ T b T c ¯ T d ) = Tr( ¯ T d T c ¯ T b T a ) = Tr( T a ¯ T d T c ¯ T b ) ≡ T adcb . – 10 –he expressions for the RG functions are available in literature up to the five-looplevel [42]. We extend these results up to six loops. The one-loop contributions read β (1) λ = λ
12 ( n − n + 8) + λ λ n −
1) + λ , (3.59) β (1) λ = λ λ + λ
12 (2 n − , (3.60) γ (1) τ ≡ − β (1) τ /τ = − λ
12 ( n − n + 2) − λ n − , (3.61)while two-loop corrections are γ (2) φ = λ
576 ( n − n −
2) + λ λ
144 ( n −
1) + λ n − n + 4) , (3.62) β (2) λ = λ
48 (3 n − n − − λ λ
36 ( n −
1) + λ λ
288 (15 n − n − − λ
24 ( n − , (3.63) β (2) λ = λ λ
144 (5 n − n −
82) + λ λ
36 (20 − n ) + λ
96 (7 n − n − , (3.64) γ (2) τ = 5 λ
288 ( n − n −
2) + 5 λ λ
72 ( n −
1) + 5 λ
576 ( n − n + 4) . (3.65)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 canbe used to study phase transitions in massless QCD and five-loop RG functions are availablein literature [43]. We compute the six-loop contributions by means of decomposition (3.8)with N a = 2 nm and T a being complex n × m matrices (c.f. eq. (3.27))( T a ) αi = (cid:112) T f (cid:40) δ α, (( a −
1) div m )+1 δ i, (( a −
1) mod m )+1 , a = 1 , . . . mniT a − mnαi , a = mn + 1 , . . . mn, (3.66)satisfying Tr( T a ¯ T b ) + Tr( T b ¯ T a ) = 2 T f δ ab , and T aαj T aβk = 0 , ¯ T ajα ¯ T akβ = 0 , T aαj ¯ T akβ = 2 T f δ αβ δ jk . (3.67)The calculations are carried out with T f = 1 / λ abcd = λ (cid:104) T ab T cd + 11 perms (cid:105) + λ (cid:104) T abcd + 11 perms (cid:105) , (3.68)where among all 24 permutations we exclude only those that correspond to the swappingbetween pairs of indices. In terms of g i = λ i / – 11 –ur calculation employs eq. (3.67) and renders at one loop β (1) λ = λ nm ) + λ λ n + m ) + λ , (3.69) β (1) λ = λ λ + λ n + m ) , (3.70) γ (1) τ ≡ − β (1) τ /τ = − λ nm ) − λ n + m ) , (3.71)and at two loops γ (2) φ = 1288 ( λ + λ )(1 + mn ) + λ λ
144 ( n + m ) , (3.72) β (2) λ = − λ
24 (7 + 3 mn ) − λ λ
36 ( m + n ) − λ λ
72 (41 + 5 mn ) − λ
12 ( m + n ) , (3.73) β (2) λ = − λ λ
72 (41 + 5 mn ) − λ λ
36 ( m + n ) − λ
24 (5 + mn ) , (3.74) γ (2) τ = 5144 ( λ + λ )(1 + mn ) + 5 λ λ
72 ( n + m ) . (3.75)The full results are available as supplementary files. It is worth noting that for m = n weget the four-loop results obtained in ref. [18] for the case of U ( n ) × U ( n ) model. SU ( n )In recent ref. [44] a model with φ being hermitian matrix field in the adjoint representationof 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]. Wegeneralize the Lagrangian of ref. [44] and include also a cubic term (see also refs. [34, 45,46]) together with the vacuum energy (we rescale f and λ for convenience) L = 12 Tr (cid:2) φ ( − ∂ + m ) φ (cid:3) + √ nf
3! Tr φ + λ (cid:0) Tr φ (cid:1) + nλ
4! Tr φ + Λ . (3.76)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( T a T b ) = T f δ ab , T aij T akl = T f (cid:18) δ il δ kl − n δ ij δ kl (cid:19) . (3.77)We utilize the normalization T f = 1 and substitute λ abcd = λ (cid:104) T ab T cd + T ac T bd + T ad T bc (cid:105) + nλ (cid:104) T abcd + T abdc + T acbd + T acdb + T adbc + T adcb (cid:105) (3.78) In ref. [18] the RG functions are written in terms of a u ≡ hnλ /
24 and a v ≡ hn λ / The term breaks Z symmetry φ → − φ imposed in ref. [44]. – 12 –ogether with h abc = √ nf (cid:104) T abc + T bac (cid:105) . (3.79)We obtain the RG functions up to the six-loop level, and at one loop we have β (1) λ = λ n ) + λ λ n −
6) + λ (3 + n ) , (3.80) β (1) λ = 4 λ λ + λ n − , (3.81) β (1) f = f (cid:2) λ + λ ( n − (cid:3) , (3.82) β (1) m = m (cid:2) λ ( n + 1) + λ (2 n − (cid:3) + f n − , (3.83) β (1)Λ = m n − . (3.84)The two-loop expressions are given by γ (2) φ = λ
36 (1 + n ) − λ λ
18 (3 − n ) + λ
72 (18 − n + n ) , (3.85) β (2) λ = − λ n ) + 22 λ λ − n ) − λ λ
18 (306 + 42 n + 5 n ) + λ n − n ) , (3.86) β (2) λ = λ λ n ) + 2 λ λ − n ) − λ − n + n ) , (3.87) β (2) f = f (cid:2) λ λ (105 − n ) − λ (35 + 3 n ) − λ (630 − n + 11 n ) (cid:3) (3.88) β (2) m = 5 m (cid:2) λ λ (3 − n ) − λ (1 + n ) − λ (18 − n + n ) (cid:3) + f (cid:2) λ (36 − n − n ) − λ (36 − n + 2 n ) (cid:3) , (3.89) β (2)Λ = − m f n − n − . (3.90)To compare our results with that of ref. [44], one has to take into account that thelatter correspond to f = 0 and are written in terms of g i / (8 π ) with g = ( nλ ) / g = λ /
6. We also use the expressions obtained by means of
ARGES [19] to cross-check γ φ and the beta functions for λ , λ , and m up to 4 loops. O ( n ) × O ( n ) × O ( n ) tensor model To give an example how to apply our general result to models with more complicated indexstructure, we consider evaluation of the beta functions in the model with O ( n ) × O ( n ) × O ( n )symmetry [48]. The model Lagrangian is L = 12 ∂ µ φ abc ∂ µ φ abc + λ T t φ φ φ φ + λ T p φ φ φ φ + λ T ds φ φ φ φ , (3.91)– 13 –here 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 toindicate contractions of triplets of indexes with φ abc . For convenience, we present the tensorstructures in the following pictorial form: T t = a b c a b c c b a c b a , T ds = a b c a b c c b a c b a , (3.92) T p = 13 a b c a b c c b a c b a + a b c a b c c b a c b a + a b c a b c c b a c b a (3.93)To map the model onto our general result, we associate open indices in general model (1.1)with multi index i k = { a k , b k , c k } , and rewrite the self-coupling (3.91) in the form: λ i i i i = λ T t( i i i i ) + λ T p( i i i i ) + λ T ds( i i i i ) (3.94)where ( i i i i ) denotes symmetrization. At one loop we get β (1)1 = 2 λ λ n ) + 4 λ λ + 4 λ , (3.95) β (1)2 = λ (2 + n ) + 4 λ λ n ) + 4 λ λ + λ n + n ) , (3.96) β (1)3 = 2 λ λ λ λ n + 2 λ λ n + n ) + λ n ) + λ n ) . (3.97)while two-loop contribution renders β (2)1 = λ
18 ( n − n − − λ λ n + 4 n + 13) − λ λ n − λ λ
54 ( n + 15 n + 93 n + 101) − λ λ λ n + 17 n + 17) − λ λ n + 82) − λ
81 (2 n + 13 n + 24) − λ λ , (3.98) β (2)2 = − λ n + n + 4) − λ λ
18 ( n + 12 n + 99 n + 98) − λ λ ( n + 2) − λ λ n + 18 n + 29) − λ λ λ n + 16) − λ
162 (5 n + 45 n + 243 n + 343) − λ λ n + 15 n + 29) − λ λ n + 82) , (3.99) β (2)3 = − λ n − λ λ n + n + 4) − λ λ
18 ( n + 3 n + 2) − λ λ n + 1)– 14 – λ λ λ n + 5 n + 17) − λ λ n − λ
81 ( n + 3 n + 5) − λ λ
54 (5 n + 15 n + 93 n + 97) − λ λ n + n + 1) − λ n + 14) . (3.100)Modulo rescaling λ i = 6 g i , the obtained expressions coincide with those given inref. [48]. Six-loop results can be found in the form of supplementary files. Motivated by three-loop calculation [13] in 2HDM model (see, e.g., refs. [49, 50] for review),we consider the following general renormalizable Higgs potential V = m Φ † Φ + m Φ † Φ − (cid:16) m Φ † Φ + h . c . (cid:17) + 12 λ (cid:16) Φ † Φ (cid:17) + 12 λ (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + (cid:20) λ (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + h . c . (cid:21) , (3.101)where Φ , are SU (2) doublets. The self-couplings λ − and the mass parameters m , m are real, while λ − , and m can be complex. Due to the freedom in redefinitionof Higgs-field basis, only 11 of 14 real parameters in eq. (3.101) are independent. Inref. [13] convenient variables [49, 50] and the so-called reparametrization invariants (see,e.g., ref. [51] for a comprehensive study) were used to compute the RG functions.In this work, we use another strategy and directly calculate the beta function of λ − together with the anomalous dimensions of m , m , and m from our general expressions.We enumerate all real components of two doublets Φ , and rewrite eq. (3.101) in the generalform (1.1) with indices a, b , etc. running from one to eight. We find full agreement withprevious results and extend the latter up to six loops. We have checked that our expressionsfor β λ ( β λ ) can be obtained from β λ ( β λ ) via the replacement λ ↔ λ and λ ↔ λ .One can use the same substitutions together with m ↔ m to get β m from β m . Wemake the six-loop results available as supplementary files. We considered the general renormalizable scalar QFT model and directly computed theRG functions for the quartic and cubic self-couplings, mass parameter, tadpole term, andvacuum energy. In deriving our results for dimensionless quantities, we used the expressionsfor the KR (cid:48) operation applied to individual Feynman integrals. The latter are publiclyavailable thanks to lengthy and non-trivial calculations of ref. [21]. To compute the RGfunctions of dimensionful parameters, we utilize the powerful dummy field technique.To validate our general results, we considered several scalar models discussed in thetheory of critical phenomena. We found perfect agreement with known results and extendthem by computing several missing six-loop contributions. Among the latter are the vac-uum energy beta function in the O ( n ) model, the anomalous dimensions of quadratic per-turbations in the O ( n ) × O ( m ) model, and the self-coupling beta functions for U ( n ) × U ( m ),– 15 –nd O ( n ) × O ( n ) × O ( n ) models and the model with the Higgs field in the adjoint represen-tation of the SU ( n ) group. Additionally, we extend the three-loop results for the generalTwo-Higgs-Doublet Model scalar sector to six loops.We believe that the obtained state-of-the-art RG functions are of immediate interestto the condensed-matter community. On the contrary, present six-loop results can hardlyfind their applications in phenomenological analyses of the Standard Model extensions inthe near future. However, it is convenient to estimate the influence of the high-order termson extended Higgs sector studies, which currently rely on the two- or three-loop RG. Publiccodes for RG analyses [20, 34, 52–54] can be equipped with our results to carry out thiskind of computations.We also note that the expression for vacuum energy beta function is relevant for ef-fective 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. Acknowledgments
We thank G.Kalagov, M.Kompaniets and N.Lebedev for fruitful discussions. We alsothank T. Steudtner for the correspondence regarding refs. [18, 19] and sharing his four-loop results. We are grateful to the Joint Institute for Nuclear Research for letting ususe their supercomputer“Govorun”. The work of A.B. is supported by the Grant of theRussian Federation Government, Agreement No. 14.W03.31.0026 from 15.02.2018. Thework of A.P. is supported by the Foundation for the Advancement of Theoretical Physicsand Mathematics “BASIS.”
A Deriving dimensionful couplings RG with Dummy field method
The author of ref. [18] introduced a convenient representation for four-loop quartic-couplingbeta function with all the self-couplings involving external indices explicitly factorized. Weadopt this ansatz to all loops β abcd = (cid:104) λ abcf γ φfd + λ abdf γ φfc + λ acdf γ φfb + λ bcdf γ φfa (cid:105) + (cid:2) λ abef λ cdgh (cid:13) ef | gh +5 perm. (cid:3) + (cid:2) λ abef λ cghi λ djkl (cid:52) ef | ghi | jkl +11 perm. (cid:3) + (cid:2) λ aefg λ bhij λ cklm λ dnop (cid:3) efg | hij | klm | nop + 23 perm. (cid:3) (A.1)and re-derive the RG functions for the dimensionful parameters entering the Lagrangian (1.1).In eq. (A.1) “perm.” denotes the terms, which can be obtained from the respective ex-pressions via non-equivalent permutations of external indices. It is convenient to represent– 16 – b cd ab cx ab xx ax xxλ abcd h abc m ab t a Table 1 . Representation of vertices corresponding to the parameters of the Lagrangian (1.1). Theindex “ x ” denotes the contraction with a dummy field. ef gh ef khg ilj
12 34 gf e op nmlkjih (cid:13) ef | gh (cid:52) ef | ghi | jkl (cid:3) efg | hij | klm | nop Table 2 . Graphical representation of the structures entering β abcd (A.1). The numbers encode thepositions of index groups (separated by vertical lines) in the corresponding expressions. eq. (A.1) in the following pictorial form: β abcd = cab dγ + 3 perm. + ba cd + 5 perm. + ba cd + 11 perm. + ba cd
12 34 + 23 perm. , (A.2)where external self-couplings are denoted by blue vertices (see table 1) and we use thenotation given in table 2 for the non-external parts of four-point functions. It is worthnoting that (cid:13) ab | cd , (cid:52) ab | cde | fgh , and (cid:3) 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 symmetriccouplings, and, thus, does not need to be explicitly symmetrized. Hoverer, we explicitlytake into account that an external index a, b, c , or d can be attached to any group via aquartic vertex. This corresponds the permutations indicated, e.g., in eq. (A.2). Due tothis, 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– 17 –he tadpole graphs discussed in sec. 2: β abc = xab cγ + 2 perm. + ba cx + ba cx + 2 perm. + ba cx + ba cx + 2 perm. + xa bc + 5 perm. + ba cx
12 34 + ba cx
23 41 + ba cx
34 12 + ba cx
41 23 + 5 perm. (A.3)Here the trilinear couplings correspond to red vertices (see table 1) and again we have toexplicitly take into account permutations of external indices. The analytic expression isgiven by β abc = (cid:104) h abf γ φfc + h acf γ φfb + h bcf γ φfa (cid:105) + (cid:2) λ abef h cgh (cid:0) (cid:13) ef | gh + (cid:13) gh | ef (cid:1) + 2 perm. (cid:3) + (cid:2) λ abef λ cghi h jkl (cid:0) (cid:52) ef | ghi | jkl + (cid:52) ef | jkl | ghi (cid:1) + 2 perm. (cid:3) + (cid:2) h aef λ bghi λ cjkl (cid:52) ef | ghi | jkl +5 perm. (cid:3) + (cid:2) λ aefg λ bhij λ cklm h nop (cid:0) (cid:3) efg | hij | klm | nop + (cid:3) nop | efg | hij | klm + (cid:3) klm | nop | efg | hij + (cid:3) hij | klm | nop | efg (cid:1) + 5 perm. (cid:3) (A.4)To obtain the beta function for mass parameter we contract eq. (A.3) with one more dummy– 18 –eld x c . Dividing the result by the factor of two, we get β ab = xax bγ + xbx aγ + ba xx + ba xx + ax bx + ax bx + ba xx + xx ab + xx ab + xa xb + xa xb + ( a ↔ b ) + ba xx
12 34 + xa bx
12 34 + xa xb
12 34 + ax bx
12 34 + ax xb
12 34 + xx ab
12 34 + ( a ↔ b ) , (A.5)where red dots denote mass parameter m ab insertions (c.f. table 1). The corresponding– 19 –nalytic expression is given by β ab = (cid:104) m af γ φfb + m bf γ φfa (cid:105) + (cid:2)(cid:0) λ abef m gh + h aef h bgh (cid:1) (cid:0) (cid:13) ef | gh + (cid:13) gh | ef (cid:1)(cid:3) + (cid:2) λ abef h ghi h jkl (cid:52) ef | ghi | jkl + m ef λ aghi λ bjkl (cid:0) (cid:52) ef | ghi | jkl + (cid:52) ef | jkl | ghi (cid:1)(cid:3)(cid:2) h aef h ghi λ bjkl (cid:0) (cid:52) ef | ghi | jkl + (cid:52) ef | jkl | ghi (cid:1) + ( a ↔ b ) (cid:3) + (cid:2) λ aefg λ bhij h klm h nop (cid:0) (cid:3) efg | hij | klm | nop + (cid:3) efg | klm | hij | nop + (cid:3) efg | klm | nop | hij + (cid:3) klm | efg | hij | nop + (cid:3) klm | efg | nop | hij + (cid:3) klm | nop | efg | hij (cid:1) + ( a ↔ b ) (cid:3) . (A.6)We proceed further and obtain the RG function for the tadpole term. Contractingeq. (A.5) with x b and dividing by the factor of 3, we get β a = xxx aγ + xa xx + xx xa + xa xx + xx ax + xx xa + xa xx
12 34 + ax xx
12 34 + xx ax
12 34 + xx xa
12 34 , (A.7)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 φfb + h aef m gh (cid:2) (cid:13) ef | gh + (cid:13) gh | ef (cid:3) + (cid:2) h aef h ghi h jkl (cid:52) ef | ghi | jkl + m ef λ aghi h jkl (cid:0) (cid:52) ef | ghi | jkl + (cid:52) ef | jkl | ghi (cid:1)(cid:3) + λ aefg h hij h klm h nop (cid:2) (cid:3) efg | hij | klm | nop + (cid:3) nop | efg | hij | klm + (cid:3) klm | nop | efg | hij + (cid:3) hij | klm | nop | efg (cid:3) . (A.8)One more contraction with the dummy field x a gives the beta function of the vacuumenergy: β Λ = xx xx + xx xx + xx xx
12 34 (A.9) We correct a couple of misprints in the corresponding expression in the published version of ref. [18]. – 20 –orresponding to β Λ = m ef m gh (cid:13) ef | gh + m ef h ghi h jkl (cid:52) ef | ghi | jkl + h efg h hij h klm h nop (cid:3) efg | hij | klm | nop . (A.10)Loop expansion of the structures (cid:13) ab | cd = (cid:88) l =1 (cid:13) ( l ) ab | cd , (A.11) (cid:52) ab | cde | f gh = (cid:88) l =1 (cid:52) ( l ) ab | cde | f gh, (A.12) (cid:3) abc | def | ghi | jkl = (cid:88) l =1 (cid:3) ( l ) abc | def | ghi | jkl. (A.13)can be found in a supplementary PDF file. References [1] G. ’t Hooft,
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