Abstract
Fixed points for scalar theories in 4 − ε, 6 − ε and 3 − ε dimensions are discussed. It is shown how a large range of known fixed points for the four dimensional case can be obtained by using a general framework with two couplings. The original maximal symmetry, O(N), is broken to various subgroups, both discrete and continuous. A similar discussion is applied to the six dimensional case. Perturbative applications of the a-theorem are used to help classify potential fixed points. At lowest order in the ε-expansion it is shown that at fixed points there is a lower bound for a which is saturated at bifurcation points.
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References
K.G. Wilson and M.E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972) 240 [INSPIRE].
A.B. Zamolodchikov, Renormalization group and perturbation theory near fixed points in two-dimensional field theory, Sov. J. Nucl. Phys. 46 (1987) 1090 [Yad. Fiz. 46 (1987) 1819] [INSPIRE].
A.W.W. Ludwig and J.L. Cardy, Perturbative evaluation of the conformal anomaly at new critical points with applications to random systems, Nucl. Phys. B 285 (1987) 687 [INSPIRE].
M. Lassig, Geometry of the renormalization group with an application in two-dimensions, Nucl. Phys. B 334 (1990) 652 [INSPIRE].
M. Lassig, Multiple crossover phenomena and scale hopping in two-dimensions, Nucl. Phys. B 380 (1992) 601 [hep-th/9112032] [INSPIRE].
M.R. Gaberdiel, A. Konechny and C. Schmidt-Colinet, Conformal perturbation theory beyond the leading order, J. Phys. A 42 (2009) 105402 [arXiv:0811.3149] [INSPIRE].
R. Poghossian, Two dimensional renormalization group flows in next to leading order, JHEP 01 (2014) 167 [arXiv:1303.3015] [INSPIRE].
C. Ahn and M. Stanishkov, On the renormalization group flow in two dimensional superconformal models, Nucl. Phys. B 885 (2014) 713 [arXiv:1404.7628] [INSPIRE].
A. Pelissetto and E. Vicari, Critical phenomena and renormalization group theory, Phys. Rept. 368 (2002) 549 [cond-mat/0012164] [INSPIRE].
L. Michel, J.-C. Toledano and P. Toledano,Landau free energies for n = 4 and the subgroups of o(4), in Symmetries and broken symmetries in condensed matter physics, N. Boccara ed., John Wiley & Sons Ltd, U.S.A., (1981), pg. 261.
J.-C. Toledano, L. Michel, P. Toledano and E. Brezin, Renormalization-group study of the fixed points and of their stability for phase transitions with four-component order parameters, Phys. Rev. B31 (1985) 7171.
D.M. Hatch, H.T. Stokes, J.S. Kim and J.W. Felix, Selection of stable fixed points by the Toledano-Michel symmetry criterion: six-component example, Phys. Rev. B 32 (1985) 7624.
J.S. Kim, D.M. Hatch and H.T. Stokes, Classification of continuous phase transitions and stable phases. I. Six-dimensional order parameters, Phys. Rev. B 33 (1986) 1774.
D.M. Hatch, J.S. Kim, H.T. Stokes and J.W. Felix, Renormalization-group classification of continuous structural phase transitions induced by six-component order parameters, Phys. Rev. B 33 (1986) 6196.
L. Michel, Renormalization-group fixed points of general n-vector models, Phys. Rev. B 29 (1984) 2777 [INSPIRE].
E. Vicari and J. Zinn-Justin, Fixed point stability and decay of correlations, New J. Phys. 8 (2006) 321 [cond-mat/0611353] [INSPIRE].
D.J. Wallace and R.K.P. Zia, Gradient properties of the renormalization group equations in multicomponent systems, Annals Phys. 92 (1975) 142 [INSPIRE].
J.A. Gracey, I. Jack and C. Poole, The a-function in six dimensions, JHEP 01 (2016) 174 [arXiv:1507.02174] [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
A. Stergiou, D. Stone and L.G. Vitale, Constraints on perturbative RG flows in six dimensions, JHEP 08 (2016) 010 [arXiv:1604.01782] [INSPIRE].
S. Gukov, RG flows and bifurcations, Nucl. Phys. B 919 (2017) 583 [arXiv:1608.06638] [INSPIRE].
L. Fei, S. Giombi and I.R. Klebanov, Critical O(N) models in 6 − ε dimensions, Phys. Rev. D 90 (2014) 025018 [arXiv:1404.1094] [INSPIRE].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Three loop analysis of the critical O(N) models in 6 − ε dimensions, Phys. Rev. D 91 (2015) 045011 [arXiv:1411.1099] [INSPIRE].
O.F. de Alcantara Bonfim, J.E. Kirkham and A.J. McKane, Critical exponents to order ϵ 3 for ϕ 3 models of critical phenomena in six ϵ-dimensions, J. Phys. A 13 (1980) L247 [Erratum ibid. A 13 (1980) 3785] [INSPIRE].
B. Grinstein, A. Stergiou, D. Stone and M. Zhong, Two-loop renormalization of multiflavor ϕ 3 theory in six dimensions and the trace anomaly, Phys. Rev. D 92 (2015) 045013 [arXiv:1504.05959] [INSPIRE].
R.K.P. Zia and D.J. Wallace, On the uniqueness of ϕ 4 interactions in two and three-component spin systems, J. Phys. A 8 (1975) 1089 [INSPIRE].
I. Jack and H. Osborn, Analogs for the c-theorem for four-dimensional renormalizable field theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
I.F. Herbut and L. Janssen, Critical O(2) and O(3) ϕ 4 theories near six dimensions, Phys. Rev. D 93 (2016) 085005 [arXiv:1510.05691] [INSPIRE].
J.A. Gracey and R.M. Simms, Six dimensional Landau-Ginzburg-Wilson theory, Phys. Rev. D 95 (2017) 025029 [arXiv:1701.03618] [INSPIRE].
A. Pelissetto, P. Rossi and E. Vicari, Large N critical behavior of O(n) × O(m) spin models, Nucl. Phys. B 607 (2001) 605 [hep-th/0104024] [INSPIRE].
J.A. Gracey, Chiral exponents in O(N) × O(m) spin models at O(1/N 2), Phys. Rev. B 66 (2002) 134402 [cond-mat/0208309] [INSPIRE].
J.A. Gracey, Critical exponent omega at O(1/N) in O(N) × O(m) spin models, Nucl. Phys. B 644 (2002) 433 [hep-th/0209053] [INSPIRE].
E. Brézin, J.C. Le Guillou and J. Zinn-Justin, Discussion of critical phenomena in multicomponent systems, Phys. Rev. B 10 (1974) 892 [INSPIRE].
D.J. Wallace and R.K.P. Zia, Harmonic perturbations of generalized Heisenberg spin systems, J. Phys. C 8 (1975) 839.
A. Aharony and M.E. Fisher, Critical behavior of magnets with dipolar interactions. I. Renormalization group near four dimensions, Phys. Rev. B 8 (1973) 3323.
A. Aharony, Critical behavior of anisotropic cubic systems, Phys. Rev. B 8 (1973) 4270 [INSPIRE].
D.J. Wallace, Critical behaviour of anisotropic cubic systems, J. Phys. C 6 (1973) 1390.
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Generalized F-theorem and the ϵ expansion, JHEP 12 (2015) 155 [arXiv:1507.01960] [INSPIRE].
R.K.P. Zia and D.J. Wallace, Critical behavior of the continuous N component Potts model, J. Phys. A 8 (1975) 1495 [INSPIRE].
N.V. Antonov, M.V. Kompaniets and N.M. Lebedev, Critical behaviour of the O(n)-ϕ 4 model with an antisymmetric tensor order parameter, J. Phys. A 46 (2013) 405002 [arXiv:1307.1991] [INSPIRE].
N.V. Antonov, M.V. Kompaniets and N.M. Lebedev, Critical behavior of the O(n) ϕ 4 model with an antisymmetric tensor order parameter: three-loop approximation, Theor. Math. Phys. 190 (2017) 204 [Teor. Mat. Fiz. 190 (2017) 239] [INSPIRE].
H. Kawamura, Generalized chiral universality, J. Phys. Soc. Jpn. 59 (1990) 2305.
S.A. Antonenko, A.I. Sokolov and K. Vaernshev, Chiral transitions in three-dimensional magnets and higher order ϵ expansions, Phys. Lett. A 208 (1995) 161.
D. Mukamel and S. Krinsky, ϵ-expansion analysis of some physically realizable n ≥ 4 vector models, J. Phys. C 8 (1975) L496.
N.A. Shpot, Critical behavior of the mn component field model in three-dimensions, Phys. Lett. A 133 (1988) 125.
N.A. Shpot, Critical behavior of the mn component field model in three-dimensions. 2: three loop results, Phys. Lett. A 142 (1989) 474 [INSPIRE].
A.I. Mudrov and K.B. Varnashev, Critical thermodynamics of three-dimensional M N component field model with cubic anisotropy from higher loop ϵ-expansion, J. Phys. A 34 (2001) L347 [cond-mat/0108298] [INSPIRE].
M. Stephen and J. McCauley Jr., Feynman graph expansion for tricritical exponents, Phys. Lett. A 44 (1973) 89.
A.L. Lewis and F.W. Adams, Tricritical behavior in two dimensions. 2. Universal quantities from the ϵ-expansion, Phys. Rev. B 18 (1978) 5099 [INSPIRE].
P. Basu and C. Krishnan, ϵ-expansions near three dimensions from conformal field theory, JHEP 11 (2015) 040 [arXiv:1506.06616] [INSPIRE].
K. Nii, Classical equation of motion and anomalous dimensions at leading order, JHEP 07 (2016) 107 [arXiv:1605.08868] [INSPIRE].
J. O’Dwyer and H. Osborn, ϵ-expansion for multicritical fixed points and exact renormalisation group equations, Annals Phys. 323 (2008) 1859 [arXiv:0708.2697] [INSPIRE].
R.D. Pisarski, Fixed point structure of ϕ 6 in three-dimensions at large N, Phys. Rev. Lett. 48 (1982) 574 [INSPIRE].
J. Hager and L. Schäfer, Θ-point behavior of diluted polymer solutions: can one observe the universal logarithmic corrections predicted by field theory?, Phys. Rev. E 60 (1999) 2071.
J.S. Hager, Six-loop renormalization group functions of O(n)-symmetric ϕ 6 -theory and ϵ-expansions of tricritical exponents up to ϵ 3, J. Phys. A 35 (2002) 2703 [INSPIRE].
P.K. Townsend, Consistency of the 1/N expansion for three-dimensional ϕ 6 theory, Nucl. Phys. B 118 (1977) 199 [INSPIRE].
T. Appelquist and U.W. Heinz, Vacuum stability in three-dimensional O(N) theories, Phys. Rev. D 25 (1982) 2620 [INSPIRE].
W.A. Bardeen, M. Moshe and M. Bander, Spontaneous breaking of scale invariance and the ultraviolet fixed point in O(n) symmetric (ϕ 63 ) theory, Phys. Rev. Lett. 52 (1984) 1188 [INSPIRE].
F. David, D.A. Kessler and H. Neuberger, The Bardeen-Moshe-Bander fixed point and the ultraviolet triviality of ϕ 6 in three-dimensions, Phys. Rev. Lett. 53 (1984) 2071 [INSPIRE].
H. Omid, G.W. Semenoff and L.C.R. Wijewardhana, Light dilaton in the large N tricritical O(N) model, Phys. Rev. D 94 (2016) 125017 [arXiv:1605.00750] [INSPIRE].
T. Appelquist and U.W. Heinz, Three-dimensional O(N) theories at large distances, Phys. Rev. D 24 (1981) 2169 [INSPIRE].
R.D. Pisarski, On the fixed points of ϕ 6 in three-dimensions and ϕ 4 in four-dimensions, Phys. Rev. D 28 (1983) 1554 [INSPIRE].
S. Yabunaka and B. Delamotte, Surprises in O(N) models: nonperturbative fixed points, large N limits and multicriticality, Phys. Rev. Lett. 119 (2017) 191602 [arXiv:1707.04383] [INSPIRE].
S. Giombi, I.R. Klebanov and G. Tarnopolsky, Bosonic tensor models at large N and small ϵ, Phys. Rev. D 96 (2017) 106014 [arXiv:1707.03866] [INSPIRE].
I. Jack, D.R.T. Jones and C. Poole, Gradient flows in three dimensions, JHEP 09 (2015) 061 [arXiv:1505.05400] [INSPIRE].
I. Jack and C. Poole, α-function in three dimensions: beyond the leading order, Phys. Rev. D 95 (2017) 025010 [arXiv:1607.00236] [INSPIRE].
D. Mukamel and S. Krinsky, Physical realizations of n ≥ 4-component vector models. II. ϵ-expansion analysis of the critical behavior, Phys. Rev. B 13 (1976) 5078.
E.J. Blagoeva et al., Fluctuation-induced first-order transitions in unconventional superconductors, Phys. Rev. B 42 (1990) 6124.
A.I. Mudrov and K.B. Varnashev, Three-loop renormalization-group analysis of a complex model with stable fixed point: critical exponents up to ϵ 3 and ϵ 4, Phys. Rev. B 57 (1998) 3562.
A.I. Mudrov and K.B. Varnashev, Stability of the three-dimensional fixed point in a model with three coupling constants from the ϵ expansion: three-loop results, Phys. Rev. B 57 (1998) 5704.
A. Cayley, On contour and slope lines, Phil. Mag. 18 (1859) 264.
J.C. Maxwell, On hills and dales, Phil. Mag. 40 (1870) 421.
D. Mukamel, Physical realizations of n ≥ 4 vector models, Phys. Rev. Lett. 34 (1975) 481 [INSPIRE].
L. Michel, The symmetry and renormalization group fixed points of quartic hamiltonians, in Symmetries in Particle Physics, Proceedings of a symposium celebrating Feza Gursey’s sixtieth birthday, B. Bars, A. Chodos and C.-H. Tze eds., Plenum Press, U.S.A., (1984), pg. 63 [INSPIRE].
G. Grinstein and D. Mukamel, Stable fixed points in models with many coupling constants, J. Phys. A 15 (1982) 233.
V. Bashmakov, M. Bertolini and H. Raj, Broken current anomalous dimensions, conformal manifolds and renormalization group flows, Phys. Rev. D 95 (2017) 066011 [arXiv:1609.09820] [INSPIRE].
O.F. de Alcantara Bonfim, J.E. Kirkham and A.J. McKane, Critical exponents for the percolation problem and the Yang-Lee edge singularity, J. Phys. A 14 (1981) 2391 [INSPIRE].
Y. Pang, J. Rong and N. Su, ϕ 3 theory with F 4 flavor symmetry in 6 − 2ϵ dimensions: 3-loop renormalization and conformal bootstrap, JHEP 12 (2016) 057 [arXiv:1609.03007] [INSPIRE].
P. Cvitanović, Group theory: birdtracks, Lie’s, and exceptional groups, Princeton University Press, Princeton U.S.A., (2008).
P. Dey, A. Kaviraj and A. Sinha, Mellin space bootstrap for global symmetry, JHEP 07 (2017) 019 [arXiv:1612.05032] [INSPIRE].
A. Petkou, Conserved currents, consistency relations and operator product expansions in the conformally invariant O(N) vector model, Annals Phys. 249 (1996) 180 [hep-th/9410093] [INSPIRE].
A.C. Petkou, C T and C J up to next-to-leading order in 1/N in the conformally invariant 0(N) vector model for 2 < d < 4, Phys. Lett. B 359 (1995) 101 [hep-th/9506116] [INSPIRE].
K. Diab, L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, On C J and C T in the Gross-Neveu and O(N) models, J. Phys. A 49 (2016) 405402 [arXiv:1601.07198] [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal bootstrap in Mellin space, Phys. Rev. Lett. 118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP 05 (2017) 027 [arXiv:1611.08407] [INSPIRE].
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Osborn, H., Stergiou, A. Seeking fixed points in multiple coupling scalar theories in the ε expansion. J. High Energ. Phys. 2018, 51 (2018). https://doi.org/10.1007/JHEP05(2018)051
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DOI: https://doi.org/10.1007/JHEP05(2018)051