Critical behavior of the 2d scalar theory: resumming the N8LO perturbative mass gap

Abstract

We apply the optimized perturbation theory (OPT) to resum the perturbative series describing the mass gap of the bidimensional ϕ4 theory in the ℤ2 symmetric phase. Already at NLO (one loop) the method is capable of generating a quite reasonable non-perturbative result for the critical coupling. At order-g7 we obtain gc = 2.779(25) which compares very well with the state of the art N8LO result, gc = 2.807(34). As a novelty we investigate the supercritical region showing that it contains some useful complimentary information that can be used in extrapolations to arbitrarily high orders.

The breakdown of resummed perturbation theory at high energies

“…It would be interesting to reconstruct this yet unknown function, for instance, in a 1/n-expansion (see, e.g., [52,53,63]). For a thorough investigation, more powerful resummation techniques may be needed (for some recent examples in ϕ 4 theory see, e.g., [74][75][76][77][78][79][80]).…”

Quantum phase transitions in a bidimensional O(N) × ℤ2 scalar field model

“…Following most applications (e.g., refs. [3,6,9]), we will evaluate the pole mass which means that all self-energies are to be evaluated on mass-shell (p 2 = −m 2 in Euclidean space). In this case, a semiclassical expansion in loops, to two-loop order yields…”

“…In order to get some numerical results, let us follow other applications [3,6,9] by simply setting µ = m. In this case, when in the subcritical region, where λ < λ c and χ = 0, the pole mass square is given by eq. (2.3), while terms proportional to L m that appear in there vanish.…”

“…(2.14) JHEP08(2022)028 Since M 2 = 0 at g = g c , one easily finds g c = 8 √ 6 19.59, which agrees with the two-loop result found in refs. [3,9] (once the different parametrizations represented by a 4! factor in the definition of the λφ 4 interaction have been adjusted).…”

“…For this reason, it has been extensively considered in the literature (for a far from complete list of references see, e.g., refs. [1][2][3][4][5][6][7][8][9]). The φ 4 2 model describes a simple non-integrable super-renormalizable theory which can display rich phase transition patterns.…”