]>NUPHB14561S05503213(19)30029X10.1016/j.nuclphysb.2019.02.001Quantum Field Theory and Statistical SystemsFig. 1RG flows of renormalized coupling constants. The left picture corresponds to n < nc, the right one – to n > nc. Symbols in boxes mark Gaussian, Ising, Heisenberg and cubic fixed points.Fig. 1Fig. 2Padé–Borel–Leroy estimates of nc based upon approximants [1/4], [2/3], [3/2], [4/1], [1/3], [2/2] and [3/1] as functions of the parameter b. The curves are depicted only within the intervals where corresponding Padé approximants are free from the “dangerous” (positive axis) poles.Fig. 2Fig. 3Dependence of the marginal spin dimensionality value on the order of RG approximation. The upper curve (“ε expansion”) represents the estimates obtained earlier from the fiveloop ε expansion for nc.Fig. 3Table 1Padé triangle for the ε expansion of nc. Here Padé estimate of kth order (lower line, RoC) is the number given by corresponding diagonal approximant [L/L] or by a half of the sum of the values given by approximants [L/L−1] and [L−1/L] when a diagonal approximant does not exist. Three estimates are absent because corresponding Padé approximants have poles close to the physical value ε = 1.Table 1M ∖ L012345
0424.5885−1.285815.5412−41.0807
12.66673.12832.79173.06842.5692
2–2.89302.95762.8828
31.9518–2.9138
4–2.7887
50.4549

RoC42.33333.12832.84242.95762.8983
Table 2Padé–Borel–Leroy estimates of nc obtained from ε expansion (25) under the optimal value of the shift parameter bopt = 1.845. The estimate of kth order (lower line, RoC) is the number given by corresponding diagonal approximant [L/L] or by a half of the sum of the values given by approximants [L/L−1] and [L−1/L] when a diagonal approximant does not exist. Two estimates are absent because corresponding Padé approximants turn out to be spoiled by dangerous poles.Table 2M ∖ L012345
0424.58848−1.2858415.5412−41.0807
12.759963.059882.870422.922832.91341
2–2.933942.911322.91499
32.577752.914192.91416
4–2.91416
52.39138

RoC42.38003.05992.90222.91132.9146
Table 3Padé triangle for the ε expansion of β. Five estimates are absent because corresponding Padé approximants have poles lying between ε = 0 and ε = 2.Table 3M ∖ L0123456
00.50.36110.37920.34210.43010.17791.059
10.39130.37710.36700.36820.36480.3740
20.3791–0.36810.36730.3674
30.35860.3715–0.3674
4–0.36740.3693
50.2983–
6–
Table 4Padé–Borel–Leroy estimates of β obtained from corresponding ε expansion under the optimal value of the shift parameter bopt = 3.460. Several boxes are empty because of dangerous poles spoiling corresponding Padé–Borel–Leroy approximants.Table 4M ∖ L0123456
00.50.36110.37920.34210.43010.17791.059
10.39520.37680.36740.36740.36640.3703
20.3808–0.36740.36740.3672
30.36530.3728––
4–0.36970.3690
50.34740.3691
6–
Table 5Padé triangle for the ε expansion of γ. Five estimates are absent because corresponding Padé approximants have poles close to the physical value ε = 1.Table 5M ∖ L0123456
011.22221.34891.31981.48851.04512.651
11.2857–1.32521.34461.36631.3925
21.42751.35431.39391.3733–
31.29051.38481.37701.3879
4–1.37541.3832
50.9069–
6–
Table 6Padé–Borel–Leroy estimates of γ obtained from sixloop ε expansion under the optimal value of the shift parameter bopt = 0.090. Empty boxes correspond to the approximants spoiled by dangerous poles.Table 6M ∖ L0123456
011.22221.34891.31981.48851.04512.651
1––1.32631.34381.37111.3825
2–––––
31.18931.36311.3604–
4–1.36051.3629
51.142–
6–
Table 7The values of critical exponents for the cubic class of universality obtained by means of Padé–Borel–Leroy resummation of the sixloop ε expansions. Corresponding Padé estimates and the differences between Padé–Borel–Leroy estimates and their Padé counterparts are also presented.Table 7n = 3αβγδην
PBL resum.−0.09(9)0.3684(13)1.368(12)4.733(4)0.036(3)0.700(8)
Pade resum.−0.11(6)0.368(3)1.379(8)4.772(17)0.038(4)0.703(5)
Difference0.02(11)0.0004(33)−0.011(44)−0.039(17)−0.002(5)0.003(9)
Table 8Sixloop estimates of critical exponents versus scaling relations.Table 8Scaling relation:1234
Deviation from zero−0.005(14)0.016(13)0.0121(36)0.007(45)
Table 9The values of correctiontoscaling exponents ω1 and ω2 for the cubic class of universality obtained by means of Padé–Borel–Leroy resummation of the sixloop ε expansions. Corresponding Padé estimates and the differences between Padé–Borel–Leroy estimates and their Padé counterparts are also presented.Table 9n = 3ω1ω2
PBL resum.0.799(4)0.005(5)
Pade resum.0.78(11)0.008(38)
Difference0.02(11)−0.003(38)
Table 10Marginal order parameter dimensionality nc given by the ε expansion technique, 3D RG approach, Monte Carlo simulations and the pseudoε expansion machinery. By the number of loops we mean the order of approximation.Table 10Number of loopsncPaperncPaperncPaper
ε expansion3D RGOthers
14[5]1974Monte Carlo
22.333[5]19742.0114[44]19833[23]1998
33.128[5]19743.003[45]1984
42.918[21]19972.9[17]1989
2.96(11)This work20192.89(2)[25]2000
52.958[20]1995
<3[22]19972.892.92[24]2000
2.855[21]1997
2.87(5)[26]2000
2.91(3)This work2019Pseudoε expansion
62.915(3)This work20192.89(4)[26]20002.86(1)[32]2016
2.862(5)[28]2000
Table 11Critical exponent values given by multiloop ε expansion calculations versus those resulting from 3D RG analysis. Error bar for four loop estimate of η is absent because it can not be evaluated within approach described in Sec. 4.1.Table 11Number of loopsηνPaperηνPaper
ε expansion3D RG
3–0.700[45]1984
40.0340.68(3)This work20190.03310.6944[17]1989
0.03320.6996[25]2000
50.0375(5)0.6997(24)[46]19980.025(10)0.671(5)[24]2000
0.0374(22)0.701(4)[26]2000
0.0353(21)0.686(13)This work2019
60.036(3)0.700(8)This work20190.0333(26)0.706(6)[26]2000
Table 12Comparison of critical exponents for cubic (this work) and Heisenberg ([35]) classes of universality for n = 3. The numbers with asterisk were obtained from sixloop ε expansion estimates for η and ν via scaling relations.Table 12n = 3αβγδην
Cubic−0.09(9)0.3684(13)1.368(12)4.733(4)0.036(3)0.700(8)
Heisenberg−0.118(6)*0.3663(12)*1.385(4)*4.781(3)*0.0378(5)0.7059(20)
Sixloop ε expansion study of threedimensional nvector model with cubic anisotropyL.Ts.Loran Ts.AdzhemyanElla V.IvanovaMikhail V.KompanietsAndreyKudlis⁎andrewkudlis@gmail.comAleksandr I.SokolovSt. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, RussiaSt. Petersburg State University7/9 Universitetskaya nab.St. Petersburg199034Russia⁎Corresponding author.Editor: Hubert SaleurAbstractThe sixloop expansions of the renormalizationgroup functions of φ4 nvector model with cubic anisotropy are calculated within the minimal subtraction (MS) scheme in 4−ε dimensions. The ε expansions for the cubic fixed point coordinates, critical exponents corresponding to the cubic universality class and marginal order parameter dimensionality nc separating different regimes of critical behavior are presented. Since the ε expansions are divergent numerical estimates of the quantities of interest are obtained employing proper resummation techniques. The numbers found are compared with their counterparts obtained earlier within various fieldtheoretical approaches and by lattice calculations. In particular, our analysis of nc strengthens the existing arguments in favor of stability of the cubic fixed point in the physical case n=3.1IntroductionAs is well known, the systems undergoing continuous phase transitions demonstrate the universal critical behavior. This leads to the concept of classes of universality introduced decades ago. They are determined by the general properties of the system such as spatial dimensionality, symmetry, and the number of order parameter components, thereby its microscopic nature does not play any role in the vicinity of phase transition temperature. There is a set of universal parameters such as critical exponents, critical amplitude ratios, etc. that characterize the critical behavior of the systems belonging to the same universality class.The analysis of critical phenomena in a broad variety of materials can be performed on the base of threedimensional O(n)symmetric φ4 field model. In case of onecomponent – scalar – order parameter (n=1) one deals with the Ising model describing phase transitions in uniaxial ferromagnets, simple fluids, binary mixtures, and many other systems. There is also a great numbers of substances with the vector ordering, e.g. easyplane ferromagnets, superconductors and superfluid helium4 (n=2), Heisenberg ferromagnets (n=3), quarkgluon plasma in some models of quantum chromodynamics (n=4), superfluid helium3 (n=18) and the neutron star matter (n=10). On the other hand, if we consider real materials with more or less complex structure, some anisotropy of the order parameter often exists. Perhaps, simplest example of such a material is a cubic ferromagnet.Initially, to describe its thermodynamics near Curie point the O(3)symmetric theory neglecting crystal anisotropy has been used. The detailed analysis performed later within the renormalizationgroup (RG) approach has shown, however, that for proper description of the critical behavior of real cubic crystals one should take into account the presence of the anisotropy, i.e. add to the Landau–Wilson Hamiltonian an extra term invariant with respect to the cubic group of transformations. It looks as g2∑α=1nφ4α, where φα is nvector ordering field and g2 – anisotropic coupling constant. This new quartic coupling, in particular, accounts for the fact that in real ferromagnets (n=3) the vector of magnetization “feels” the crystal anisotropy and can lie only along the axes or spatial diagonals of cubic unit cell in the ordered phase.This model with two coupling constants – g1 (isotropic) and g2 – was carefully examined since 1972 [1] by many researches. As was found, its RG equations describing evolution of quartic couplings under T→Tc possess four fixed points: Gaussian (0,0), Ising (0,gI⁎), Heisenberg (gH⁎,0) and cubic (g1⁎,g2⁎). One of the most important issues involved in the study is the determination of the stability of these fixed points or, in other words, what critical regime takes place in real ferromagnets. Analyzing the RG flows it was shown that the first two points are always unstable for arbitrary values of order parameter dimensionality n whereas the last two of them corresponding to the Heisenberg (isotropic) and cubic (anisotropic) modes of critical behavior compete with each other. Which regime turns out to be stable depends on n. For n<nc, where nc is some marginal value of spin dimensionality, the isotropic (Heisenberg) critical regime is stable while for n>nc the cubic critical behavior is realized. If initial (“bare”) values of coupling constants lie outside the regions of fixed points attraction critical fluctuations strongly modify the behavior of the system converting the secondorder phase transition into the firstorder one. Fig. 1 illustrates the situation.Thus, in the case n>nc the cubic quartic term is certainly relevant and has to be taken into account. This results in the emergence of new class of universality corresponding to the anisotropic – cubic – critical behavior. So, the value of nc becomes of prime physical importance since it determines the true regime of the critical behavior in real cubic ferromagnets and of some other systems of interest.Detailed study of the nvector cubic model including evaluation of critical exponents and nc was carried out by many groups [2–32] having used both fieldtheoretical methods and lattice calculations. Early numerical estimates of nc obtained in the lowerorder approximations within the ε expansion approach [2,4–6] and in the frame of 3D RG machinery [12,15,16] turned out to be in favor of the conclusion that nc>3, while lattice calculations implied nc is practically equal to 3 [13]. This made the study of the cubic class of universality less interesting from the physical point of view. Later, however, the higherorder analysis including resummation of RG perturbative series was performed and shown that numerical value of nc falls below 3 [17–22,24–26,28,32]. To date, the most advanced estimates of nc obtained within the ε expansion, 3D RG and pseudoε expansion approaches are nc=2.855,2.87 [21,26], nc=2.89,2.91 [24,26] and nc=2.86 [28,32], respectively.These numbers differ from each other appreciably what may be considered as a stimulus to find the value of nc with higher accuracy. On the other hand, recently the ε expansions of record length – sixloop – for O(n)symmetric φ4 field theory [33–35] were calculated. This paves the way to analysis of the critical behavior of the cubic model within the highestorder ε approximation including getting precise numerical estimates for critical exponents and nc. Such an analysis is the aim of this work.The paper is organized as follows. In Sec. 2 we write down the fluctuation Hamiltonian (Landau–Wilson action) of nvector cubic model and describe the renormalization procedure. In Sec. 3 the sixloop ε expansions for β functions, critical exponents and nc are calculated. The sixloop ε series for cubic fixed point coordinates and critical exponents are also presented here for the physically interesting case n=3. In Sec. 4 the ε expansions for “observables” – nc and critical exponents – are resummed and corresponding numerical estimates are found. In Sec. 5 the numbers obtained are discussed and compared with their counterparts given by alternative fieldtheoretical approaches and extracted from the lowerorder approximations. Sec. 6 contains the summary of main results and concluding remarks.2Model and renormalizationIn this work we address the fieldtheoretical RG approach in spatial dimensionality D=4−ε.11Original sixloop calculations [33–35] were performed in space dimension D=4−2ε which is more common for high energy physics. The critical behavior of the cubic model is governed by the wellknown Landau–Wilson action with two coupling constants(1)S=∫dDx{12[(∂φ0α)2+m02φ0α2]+14![g01Tαβγδ(1)+g02Tαβγδ(2)]φ0αφ0βφ0γφ0δ}, where φ0α is ncomponent bare field, g01 and g02 being the bare coupling constants. The tensor factors T(1) and T(2) entering the O(n)invariant and cubic terms respectively are as follows(2)Tαβγδ(1)=13(δαβδγδ+δαγδβδ+δαδδγβ),Tαβγδ(2)=δαβγδ,δα1…αn={1,α1=α2=…=αn0,otherwise. In particular,(3)Tαβγδ(1)Tαβγδ(1)=n(n+2)3,Tαβγδ(1)Tαβγδ(2)=n,Tαβγδ(2)Tαβγδ(2)=n. The action (1) is seen to be physical (positively defined) if g02>−g01 for g01>0 and g02>−ng01 for negative g01.The model is known to be multiplicatively renormalizable. The bare parameters g10, g20, m02, φ0 can be expressed via the renormalized ones g1,g2,m2,φ by means of the following relations(4)m02=m2Zm2,g01=g1μεZg1,g02=g2μεZg2,φ0=φZφ,Z1=Zφ2,Z2=Zm2Zφ2,Z3=Zg1Zφ4,Z4=Zg2Zφ4. Using these relations we arrive to the renormalized action(5)SR=∫dDx{12[Z1(∂φα)2+Z2m2φα2]+14![Z3g1μεTαβγδ(1)+Z4g2μεTαβγδ(2)]φαφβφγφδ}, where μ is an arbitrary mass scale introduced to make couplings g1 and g2 dimensionless. Renormalization constants are defined in a way enabling to absorb divergences from all Green functions, so that renormalized Green functions are free of divergences. Due to multiplicative renormalizability of the model it is enough to remove divergences in two and fourpoint oneparticle irreducible Green functions:(6)Γαβ(2)=Γ(2)δαβ,Γαβγδ(4)=Γ1(4)T(1)αβγδ+Γ2(4)Tαβγδ(2),(7)Γ1(4)=3(Tαβγδ(1)−Tαβγδ(2))n(n−1)Γαβγδ(4),Γ2(4)=(n+2)Tαβγδ(2)−3Tαβγδ(1)n(n−1)Γαβγδ(4).In this paper we employ the Minimal Subtraction (MS) scheme where renormalization constants acquire only pole contributions in ε and depend only on ε and coupling constants:(8)Zi(g1,g2,ε)=1+∑k=1∞Zi(k)(g1,g2)ε−k. Renormalization constants can be found from the requirement of the finiteness of renormalized two and fourpoint oneparticle irreducible Green functions. Another way to calculate renormalization constants is use of Bogolubov–Parasiuk R′ operation:(9)Zi=1+KR′Γ¯i, where R′ – incomplete Bogoludov–Parasiuk Roperation, K – projector of the singular part of the diagram and Γ¯i – normalized Green functions of the basic theory (see e.g. [36,37]) defined by the following relations:(10)Γ¯1=∂∂m2Γ(2)p=0,Γ¯2=12(∂∂p)2Γ(2)p=0Γ¯3=1g1μεΓ1(4)p=0,Γ¯4=1g2μεΓ2(4)p=0.One of the most important advantages of the Bogoludov–Parasiuk approach is that counterterms of the diagrams computed for O(1)symmetric (scalar) model can be easily generalized to any theory with nontrivial symmetry due to the factorization of the tensor structures (see e.g. [38–40]). To calculate tensor factors for particular diagrams of the cubic model (1) one should apply projectors (7) to it. Such an operation can be automated with FORM [41] and GraphState [42] while counterterm values can be taken from data obtained in the course of recent 6loop calculations for O(n)symmetric model [35].3Sixloop expansions for RG functions, cubic fixed point coordinates, critical exponents and ncThe RG functions, i.e. β functions and anomalous dimensions γφ, γm2 are related to renormalization constants Zi by the following relations:(11)βi(g1,g2,ε)=μ∂gi∂μg01,g02=−gi[ε−g1∂Zgi(1)∂g1−g2∂Zgi(1)∂g2],i=1,2,γj(g1,g2)=μ∂logZj∂μg01,g02=−g1∂Zj(1)∂g1−g2∂Zj(1)∂g2,j=φ,m2, where Zi(1) – coefficients at first pole in ε from (8).We calculated the RG functions as series in renormalized coupling constants up to sixloop order. They are found analytically and presented in Tables 1, 2, 3 and 4 of Supplementary materials (see Appendix A) in the form(12)βi=gi[−ε+∑l=16∑k=0lCβik,(l−k)g1kg2l−k],i=1,2,(13)γj=∑l=16∑k=0lCγjk,(l−k)g1kg2l−k,j=φ,m2. The critical regimes of the system are controlled by the fixed points (g1⁎,g2⁎) of RG equations that are zeroes of β functions:(14)β1(g1⁎,g2⁎,ε)=0,β2(g1⁎,g2⁎,ε)=0. As was already mentioned, for the model under consideration there are four fixed points: Gaussian (0,0), Ising (0,gI⁎), Heisenberg (gH⁎,0) and cubic (g1⁎,g2⁎). Since sixloop ε expansions analysis of Ising and Heisenberg models have been performed earlier [33–35] we concentrate on the cubic critical behavior. To calculate ε expansions for critical exponents we have to find those for coordinates of the cubic fixed point. Solving (14) by means of iterations in ε for the cubic fixed point we find:(15)g1⁎=εn+ε2(−10627n3+12527n2−1927n)+∑k=36Cg1(k)εk+O(ε7),g2⁎=ε(n−4)3n+ε2(42481n3−17827n2+3127n+1781)+∑k=36Cg2(k)εk+O(ε7), where higherorder coefficients Cg1(k), Cg2(k) are presented in Tables 5 and 6 of Supplementary materials (see Appendix A).To fully characterize the cubic class of universality, we need to calculate the critical exponents α, β, γ, η, ν and δ. They can be expressed via γm2⁎≡γm2(g1⁎,g2⁎) and γφ⁎≡γφ(g1⁎,g2⁎) in the following way:(16)α=2−D2+γm2⁎,β=D/2−1+γφ⁎2+γm2⁎,γ=2−2γφ⁎2+γm2⁎,η=2γφ⁎,ν=12+γm2⁎,δ=D+2−2γφ⁎D−2+2γφ⁎. The critical exponents are related to each other by wellknown scaling relations and only two of them may be referred to as independent.It is instructive to present ε expansions of cubic fixed point coordinates for physically important case n=3. They are as follows:(17)g1⁎=13ε+98729ε2+ε3[−28ζ(3)729−61975708588]+ε4[30308ζ(3)177147+2ζ(4)729+200ζ(5)2187−48973747344373768]++ε5[+54608659ζ(3)114791256+101851ζ(4)708588−325ζ(6)39366−1519ζ(7)6561−−5375ζ(3)259049−2305600930431338925209984]+ε6[24368284757ζ(3)27894275208+597666691ζ(4)1721868840−−1112573461ζ(5)645700815−7725253ζ(6)9565938+16586384ζ(7)7971615+176698ζ(8)13286025++2911136ζ(9)4782969−101024906ζ(3)2215233605+14080ζ(3)3531441−28412ζ(4)ζ(3)177147++115696ζ(5)ζ(3)177147+90592ζ(3,5)4428675−20057900878765108452942008704]+O(ε7),(18)g2⁎=−19ε+1182187ε2+ε3[4354392125764−260ζ(3)2187]++ε4[−231404ζ(3)531441−226ζ(4)2187+920ζ(5)2187+2579118431033121304]++ε5[−291502339ζ(3)344373768−692465ζ(4)2125764+760450ζ(5)531441+22925ζ(6)39366−−31115ζ(7)19683+52853ζ(3)2177147+10778617093314016775629952]+ε6[−547951382833ζ(3)418414128120−−631200319ζ(4)1033121304+1732037966ζ(5)645700815+17543357ζ(6)9565938−120541604ζ(7)23914845−−209656711ζ(8)39858075+86923264ζ(9)14348907+880268036ζ(3)2645700815+490496ζ(3)31594323++1185542ζ(4)ζ(3)2657205−708704ζ(5)ζ(3)1594323+12497456ζ(3,5)13286025++4366735502557371626794130130560]+O(ε7), where ζ(3,5) is double zeta value [35]:(19)ζ(3,5)=∑0<n<m1n3m5≃0.037707672985. To give an idea about the numerical structure of these expansions we present them also with the coefficients in decimals:(20)g1⁎=0.33333ε+0.13443ε2−0.13363ε3+0.16124ε4−0.43104ε5++1.3278ε6+O(ε7),g2⁎=−0.11111ε+0.053955ε2+0.061933ε3+0.050592ε4−0.18841ε5++0.95219ε6+O(ε7).The character of a fixed point and, in particular, its stability is determined by the eigenvalues ω1, ω2 of the matrix(21)Ω=(∂β1(g1,g2)∂g1∂β1(g1,g2)∂g2∂β2(g1,g2)∂g1∂β2(g1,g2)∂g2) taken at g1=g1⁎, g2=g2⁎. If both eigenvalues are positive the fixed point is stable and describes true critical behavior. At the same time, the roles of ω1 and ω2 in governing the cubic critical behavior are quite different. The eigenvalue ω1 determines the rate of flow to the cubic fixed point along the radial direction in the plane (g1,g2), while ω2 controls approaching this point normally to the radial ray. In particular, when n→nc the cubic fixed point tends to coincide with Heisenberg one and ω2 goes to zero. So, the dependence of ω2 on n and its numerical value at n=3 are essential in the problem we study. That is why here we write down the ε expansion for ω2 only. It reads:(22)ω2=εn−43n+ε2(n−1)(−848+660n+72n2−19n3)81n3(n+2)+∑k=36Cω2(k)εk+O(ε7), where coefficients Cω2(k), along with those for ω1, are presented in Tables 7 and 8 of Supplementary materials (see Appendix A).With ε expansion for ω2 in hand we can find ε series for the marginal dimensionality of the fluctuating field nc. It may be extracted from the equation(23)ω2(nc,ε)=0. Solving it by iterations in ε we obtain:(24)nc=4−2ε+ε2[5ζ(3)2−512]+ε3[15ζ(4)8+5ζ(3)8−25ζ(5)3−172]++ε4[93ζ(3)128+15ζ(4)32−3155ζ(5)1728−125ζ(6)12+11515ζ(7)384−−229ζ(3)2144−1384]+ε5[1709ζ(3)6912−2657ζ(3,5)160+279ζ(4)512+4879ζ(5)20736−−21175ζ(6)6912+182663ζ(7)41472+237079ζ(8)2560−2554607ζ(9)23328−21685ζ(3)23456−−1793ζ(3)3324−229ζ(4)ζ(3)96−3455ζ(5)ζ(3)216+9710368]+O(ε6) or, in decimals,(25)nc=4−2ε+2.588476ε2−5.874312ε3+16.82704ε4−56.62195ε5+O(ε6).Sixloop ε expansions for critical exponents η and ν corresponding to the cubic class of universality result directly from those for anomalous dimensions and scaling relations (16). In its turn, sixloop ε expansions for γφ and γm2 originate from RG series (13) and ε expansions for the cubic fixed point coordinates. Since ε expansions for the critical exponents under arbitrary n are extremely lengthy they are presented in Tables 9 and 10 of Supplementary materials (see Appendix A). Here we write down them only for physically interesting case n=3:(26)η=5243ε2+4433236196ε3+ε4[2102395229582512−85659049ζ(3)]+ε5[−211933ζ(3)19131876−−214ζ(4)19683+880ζ(5)19683+302817233223154201664]+ε6[−123938827ζ(3)55788550416−−211933ζ(4)25509168+80933ζ(5)3188646+1100ζ(6)19683−80458ζ(7)531441+169100ζ(3)214348907−−12007171241972301961339136]+O(ε7)==0.020576ε2+0.018768ε3−0.0082681ε4+0.022634ε5−−0.065781ε6+O(ε7),(27)ν−1=2−49ε−3832187ε2+ε3[4002187ζ(3)−1812292125764]++ε4[52279ζ(3)531441+100ζ(4)729−3760ζ(5)6561−457929312066242608]++ε5[6730303ζ(3)172186884+52279ζ(4)708588−357650ζ(5)1594323−4700ζ(6)6561+38710ζ(7)19683−−20032ζ(3)2177147+189983504952008387814976]+ε6[−12508116067ζ(3)167365651248+6730303ζ(4)229582512++137705935ζ(5)1549681956−1076375ζ(6)3188646+10154279ζ(7)19131876+94237301ζ(8)15943230−−101478944ζ(9)14348907−44681927ζ(3)2129140163−560896ζ(3)31594323−10016ζ(4)ζ(3)59049−−1565872ζ(5)ζ(3)1594323−2714888ζ(3,5)2657205+21979362510179650717652052224]+O(ε7)==2−0.44444ε−0.17513ε2+0.13460ε3−0.34969ε4++0.99461ε5−3.48637ε6+O(ε7).Of significant interest is also the critical exponent of susceptibility γ which is usually measured in experiments and extracted from lattice calculations. Coefficients of its ε expansion at the cubic fixed point under arbitrary n are presented in Table 11 of Supplementary materials (see Appendix A). For n=3 this expansion is as follows:(28)γ=1+ε29+ε22772187+ε3[−200ζ(3)2187+859311062882]++ε4[−87775ζ(3)1062882−50ζ(4)729+1880ζ(5)6561+23261567516560652]++ε5[−10826597ζ(3)172186884−87775ζ(4)1417176+346225ζ(5)1594323+2350ζ(6)6561−19355ζ(7)19683++10016ζ(3)2177147+2452679419125524238436]+ε6[−384088139ζ(3)83682825624−10826597ζ(4)229582512−+240030707ζ(5)3099363912+1913375ζ(6)6377292−23980511ζ(7)38263752−94237301ζ(8)31886460+50739472ζ(9)14348907+51810395ζ(3)2258280326+5008ζ(3)ζ(4)59049+782936ζ(3)ζ(5)1594323+280448ζ(3)31594323+1357444ζ(3,5)2657205−32326689118181339706506528]+O(ε7)==1+0.22222ε+0.12666ε2−0.029080ε3+0.16865ε4+−0.44336ε5+1.6059ε6+O(ε7).All calculated ε expansions are rather complicated and need to be checked up. We compared them with known fiveloop series [20] and found complete agreement. In the Ising (g1→0) and Heisenberg (g2→0) limits our ε expansions are found to reduce to their counterparts for O(n)symmetric model [35] under n=1 and arbitrary n respectively. Our ε expansions should also obey some exact relations appropriate to the cubic model with n=2. Such a system possesses a specific symmetry: if the field φα undergoes the transformation(29)φ1→φ1+φ22,φ2→φ1−φ22, the coupling constants are also transformed:(30)g1→g1+32g2,g2→−g2, but the structure of the action itself remains unchanged [1]. Since the RG functions are completely determined by the structure of the action, the RG equations should be invariant with respect to any transformation conserving this structure [10]. It means that under the transformation (30) the β functions should transform in an analogous way while all the observables including critical exponents should be invariant with respect to above replacement (see [10,29,43] for details and extra examples). The expansions (12) and (13) do satisfy these symmetry requirements. Moreover, transformation (30) converts the Ising fixed point into cubic one and vice versa making them dual under n=2. Sixloop ε expansions (15) reproduce this duality.4Resummation and numerical estimatesWith sixloop ε expansions in hand we can obtain advanced numerical estimates for all the quantities of interest. It is well known that ε expansions as other fieldtheoretical perturbative series are divergent and for getting proper numerical results some resummation procedures have to be applied. In this paper we address the methods of resummation based upon Padé approximants [L/M] which are the ratios of polynomials of orders L (numerator) and M (denominator) and Borel–Leroy transformation. The Padé–Borel–Leroy technique enables one to optimize the resummation procedure by tuning the shift parameter b and proved to yield accurate numerical estimates for basic models of phase transitions. Much simpler Padé technique that is certainly less powerful will be also used, mainly in order to clear up to what extent the numerical results depend on the resummation procedure. Note that both approaches do not require a knowledge of higherorder (Lipatov's) asymptotics of the ε expansions coefficients finding of which is a separate nontrivial problem.4.1Resummation strategy and error estimationApplication of Padé approximants and use of Padé–Borel–Leroy resummation technique are rather straightforward and were described in detail in a good number of papers and books. At the same time, the determination of the final estimate of the quantity to be found and evaluation of corresponding error bar (apparent accuracy) are somewhat ambiguous procedures. The point is that the choice of a subset of approximants which can be accepted as working and used to get the asymptotic or averaged estimate of a given order usually may be tuned within a very wide range what may lead to unreliable (unstable) results and overestimation of the accuracy.Here we suggest clear and consistent strategy for calculating estimates with Padé approximants and Padé–Borel–Leroy technique which is aimed to yield the stable results and reasonable error estimates from order to order. While finding numerical values of physical quantities with Padé approximants we use the following procedure. To estimate the value in kth order of perturbation theory we take into consideration approximants of k and k−1 orders (particular values of [L/M] depend on the observable). The reason of accounting for such a subset is to provide the results stable from order to order while keeping the contribution from kth order dominant. From this set of approximants we exclude “maximally offdiagonal” ones, in particular [0/M] and [L/0] as they are known to possess bad approximating properties. We exclude also approximants which have poles in the interval ε∈[0,2εphys] (in our case εphys=1). The reason for this is as follows: if there is a pole in ε∈[0,εphys] the approximant simply cannot be used to estimate the value at εphys=1, but even if the pole lying outside this area is still close to εphys=1 such an approximant cannot give reliable estimate as unphysical pole contribution dominates in this case. Particular choice of the upper bound (2εphys), namely multiplier 2 is based on our experience and tries to keep a balance between dropping out unsuitable approximants and keeping a total number of working approximants as large as possible.To estimate the error bar (apparent accuracy) we consider values given by different approximants as “independent measurements” of the quantity and use tdistribution tp,n with p=0.95 confidence level, i.e. estimates for the value itself and its error are calculated with the following formulas:(31)〈x〉=x1+…+xnn,Δx=t0.95,n(〈x〉−x1)2+…+(〈x〉−xn)2n(n−1).In the case of Padé–Borel–Leroy resummation the procedure is almost the same except the fact that we have an additional – tuning – parameter b. For each particular value of b we perform Borel–Leroy transformation of the original series, construct Padé approximants of k and k−1 orders for Borel–Leroy transform and drop out approximants [0/M], [L/0] and those spoiled by pole(s) on positive real axis. To find the optimal value of b we perform discrete scan over b∈[0,20] with Δb=0.01 and search for the value of b which minimizes the standard deviation. The final estimate and error bar are then calculated with (31) for this value of b.4.2Marginal field dimensionality ncLet us start from the estimation of the fluctuating field marginal dimensionality nc. As seen from (25) ε expansion for nc is alternating and its coefficients rapidly grow in modulo. The former property makes employing Padé approximants not meaningless. The results of Padé resummation of the series (25) under the physical value ε=1 are shown in Table 1. Applying the procedure described in section 4.1 to the data collected in Table 1 we obtain nc(4)=2.9±0.4, nc(5)=2.94±0.12 and nc(6)=2.89±0.14 as the fourloop, fiveloop and sixloop estimates respectively. These estimates are seen to converge to the value close to 2.9 but the rate of convergence and the accuracy are certainly very low.Since higherorder coefficients of the ε expansion for nc are big and rapidly grow use of Borel–Leroy transformation that factorially weakens such a growth should significantly accelerate the convergence and refine the estimate itself. This transformation looks as follows(32)f(x)=∑i=0∞cixi=∫0∞e−ttbF(xt)dt,F(y)=∑i=0∞ciΓ(i+b+1)yi. Padé–Borel–Leroy resummation procedure consists of transformation (32) and analytical continuation of the Borel transform F(y) by means of Padé approximants. It includes also the choice (tuning) of the shift parameter b enabling one to achieve the fastest convergence of the iteration scheme. The results of the Padé–Borel–Leroy resummation of the sixloop series for nc are presented in Fig. 2 and Table 2. The figure shows the behavior of relevant sixloop and fiveloop Padé–Borel–Leroy estimates as functions of the parameter b and illustrates, in particular, the emergence of the optimal value bopt. Note that the curves in Fig. 2 are drawn only within the regions where Padé approximants of the Borel–Leroy transform have no positive axis poles. Padé–Borel–Leroy estimates of various approximants obtained under the optimal value of b which was found to be bopt=1.845 are collected in Table 2.As is seen the application of Padé–Borel–Leroy machinery indeed makes the iteration faster convergent and corresponding estimates much less oscillating. Being processed according to our strategy (Section 4.1) the numbers presented in Table 2 give us nc(4)=2.96±0.11, nc(5)=2.91±0.03 and nc(6)=2.915±0.003 at the four, five and sixloop levels. The last, highestorder value(33)nc=nc(6)=2.915±0.003 we accept as a final result of our calculations.4.3Critical exponentsSince the coordinates of the fixed points depend on the normalization conditions adopted their numerical values being nonuniversal are not interesting from the physical point of view. That is why further we proceed directly to evaluation of critical exponents characterizing the cubic class of universality at n=3. Starting from the sixloop ε expansions for η and ν−1 and using wellknown scaling relation we obtain ε expansions for exponents α, β, γ, ν and δ. Then we perform Padé and Padé–Borel–Leroy resummation of all the series in hand. As the Padé–Borel–Leroy resummation procedure turns out to be most effective (regular and fast convergent) for β and γ we present here details of evaluation of these two exponents. Numerical values of β and γ obtained within Padé and Padé–Borel–Leroy resummation approaches are collected in Tables 3, 4, 5 and 6. Similar tables were calculated for the exponents α, δ, η and ν. All the final estimates and error bars obtained according to the resummation procedure suggested in Section 4.1 are presented in Table 7.What is the accuracy of numerical results just found? Some idea on the point may be obtained looking at the differences between the Padé–Borel–Leroy and Padé estimates presented in Table 7. However, much more definite conclusions concerning an actual accuracy of our calculations can be made on the base of the analysis to what extent the numbers obtained obey exact scaling relations between the critical exponents. One can combine critical exponents in different ways. We choose the next set of independent relations:(34)1)γν(2−η)−1=0,2)2βν(1+η)−1=0,3)5−ηδ(1+η)−1=0,4)β+α+γ2−1=0, that are “normalized to unity” to get the estimates of accuracy more uniform. Since the calculated values of critical exponents are approximate they can not meet the scaling relations precisely and emerging discrepancies may be considered as a measure of achieved accuracy. The discrepancies relevant to scaling relations (34) along with their error bars originating from the estimates of the critical exponents themselves (Table 7, upper line) are presented in Table 8. As is seen the deviations from exact scaling relations are small demonstrating the consistency of our approach and indicating that actual computational uncertainty of found numerical estimates is of order of 0.01.To finalize this section, in Table 9 we present, for completeness, the values of correctiontoscaling exponents ω1 and ω2 obtained by resummation of corresponding ε expansions. Despite the fact that zero lies inside the error bar for ω2 the median value of this exponent, being very small, turns out to be positive. Moreover, keeping in mind the results of independent evaluation of nc we may state that the value of ω2 given by sixloop ε expansion analysis is certainly positive. More accurate estimates for ω2 can be obtained within the higherorder (sevenloop, etc.) approximations or by means of more sophisticated resummation procedure such as Borel transformation combined with conformal mapping which will be a subject of a separate paper.5DiscussionIn this section we will compare our results with those obtained earlier within the lowerorder approximations and by alternative methods.The first quantity of interest is the marginal spin dimensionality for which we get the value nc=2.915(3). It is worthy to note that the ε expansion for this quantity has rapidly growing coefficients (see eq. (25)) what prevents Padé approximants from giving accurate enough numerical results while Padé–Borel–Leroy approach yields stable estimates with an accuracy increasing from order to order. The results of previous studies performed within the ε expansion approach and RG machinery in fixed dimensions (3D RG) as well as the numbers extracted from the Monte Carlo simulations and the sixloop pseudoε expansion are aggregated in the Table 10.In addition, the values of nc collected in Table 10 are depicted at Fig. 3 to visualize the trend these values demonstrate under increasing order of approximation. This trend enable us to conclude that nc is certainly less than 3 for the 3D cubic model that justifies the significance of studying the cubic class of universality.The other quantities of prime physical importance are critical exponents of the cubic universality class. We should stress that to get estimates for critical exponents we perform resummation of the series for each exponent separately and afterwards checked a validity of several scaling relations (34). Despite the fact that sometimes the relations are satisfied with inaccuracies exceeding corresponding error bar estimates, these deviations are not too large lying within 3σ interval. This may be considered as a proof of the consistency of the results obtained and a demonstration of the numerical power of the ε expansion approach.It is worthy to compare our estimates with their analogs given by the lowerorder approximations and with the results of multiloop 3D RG analysis. The data enabling one to do such a comparison are collected in Table 11. The numbers presented in both columns are seen to rapidly converge to the asymptotic values that differ from each other only tiny coinciding in fact within the declared error bars. It confirms the conclusion that the field theory is a powerful instrument enabling one to get precise numerical results provided the calculations are performed in high enough perturbative order. On the other hand, addressing the sixloop ε approximation shifts the estimates only slightly indicating that they should be very close to the exact values still unknown.Another point to be discussed is to what extent – quantitatively – the critical exponents of the cubic class of universality differ from those of the 3D Heisenberg model. Since for n=3 the cubic fixed point lies near the Heisenberg one corresponding differences are known to be rather small. In Table 12 we present the estimates of critical exponents for cubic and Heisenberg classes of universality obtained in the sixloop approximation. As expected, the differences between numerical values of critical exponents for these two classes are really small. So, it is hardly believed that measuring critical exponents in physical or computer experiments one can distinguish between cubic and Heisenberg critical behaviors.6ConclusionTo summarize, we performed sixloop RG analysis of the critical behavior of nvector φ4 model with cubic anisotropy in the framework of ε expansion approach employing the minimal subtraction scheme. We calculated ε expansions for marginal spin dimensionality nc and critical exponents α, β, γ, δ, η, ν, ω1, ω2 for the cubic class of universality. We resummed these diverging series with Padé approximants and using Padé–Borel–Leroy technique. Obtained numerical estimates for critical exponents turn out to be selfconsistent in the sense that they are in accord, within the computational uncertainties, with the scaling relations. Sixloop contributions are found to shift fiveloop estimates only slightly but they improve numerical results considerably diminishing their error bars. Our results confirm and strengthen the conclusion that cubic ferromagnets (D=3, n=3) belong to cubic class of universality and their critical behavior is described by critical exponents differing from those of 3D Heisenberg model. At the same time, the critical exponents of 3D cubic and Heisenberg models are numerically so close to each other that it makes their behaviors practically indistinguishable if one limits himself by measuring critical exponents only.AcknowledgementIt is a pleasure to thank Professor M. Hnatič and Professor M.Yu. Nalimov for fruitful discussions. E.I. and A.K. are especially grateful to the Professor Hnatič for support and hospitality during their stay in Slovakia. This work has been supported by Foundation for the Advancement of Theoretical Physics “BASIS” (grant 1812431).Appendix ASupplementary materialSupplementary material related to this article can be found online at https://doi.org/10.1016/j.nuclphysb.2019.02.001.In Supplementary materials we present expansions of RG functions and critical exponents for arbitrary n. In rg_expansion_coefficients.pdf we list coefficients Cki,j of the expansions of beta functions β1, β2 (12), anomalous dimensions γϕ, γm2 (13) and ε expansions of coordinates of the cubic fixed point (15), correctiontoscaling exponents ω1, ω2 (22) and critical exponents η (26), 1/ν (27) and γ (28) corresponding to cubic universality class.Additionally, for RG functions (β1(g1,g2), β2(g1,g2), γϕ(g1,g2), γm2(g1,g2)) we provide Mathematica file with their expansions (rg_expansion.m). For critical exponents we present Mathematica files for all nontrivial fixed points: cubic (cubic_crit_exp.m), Ising (ising_crit_exp.m) and Heisenberg (heisenberg_crit_exp.m). Each file contains ε expansion for exponents α, β, γ, δ, η, ν as well as for 1/ν, correctiontoscaling exponents ω1, ω2 and coordinates of fixed points g1⁎, g2⁎. In the file corresponding to cubic fixed point (cubic_crit_exp.m) we also present expansion for marginal spin dimensionality nc. Appendix ASupplementary materialThe following is the Supplementary material related to this article.MMC 1Here we list coefficients of the expansions of beta functions, anomalous dimensions and epsilon expansions of coordinates of the cubic fixed point, correctiontoscaling exponents and some critical exponents corresponding to cubic universality class.MMC 1MMC 2Here we present the symbolic 6loop expansions of the renormalization group functions.MMC 2MMC 3This file contains the epsilonexpansions of all critical exponents for cubic fixed point.MMC 3MMC 4This file contains the epsilonexpansions of all critical exponents for Heisenberg fixed point.MMC 4MMC 5This file contains the epsilonexpansions of all critical exponents for Ising fixed point. Ending in .m files are suitable for Mathematica.MMC 5References[1]K.G.WilsonM.E.FisherPhys. Rev. Lett.281972240[2]A.AharonyPhys. Rev. B819734270[3]D.WallaceJ. Phys. C619731390[4]I.J.KetleyD.J.WallaceJ. Phys. 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