Five-loop additive renormalization in theφ4theory and amplitude functions of the minimally renormalized specific heat in three dimensions

Abstract

We present an analytic five-loop calculation for the additive renormalization constant A(u, ǫ) and the associated renormalization-group function B(u) of the specific heat of the O(n) symmetric φ 4 theory within the minimal subtraction scheme. We show that this calculation does not require new fiveloop integrations but can be performed on the basis of the previous five-loop calculation of the four-point vertex function combined with an appropriate identification of symmetry factors of vacuum diagrams. We also determine the amplitude function F + (u) of the specific heat in three dimensions for n = 1, 2, 3 above T c and F − (u) for n = 1 below T c up to five-loop order, without using the ǫ = 4 − d expansion. Accurate results are obtained from Borel resummations of B(u) for n = 1, 2, 3 and of the amplitude functions for n = 1. Previous conjectures regarding the smallness of the resummed higherorder contributions are confirmed. Combining our results for B(u) and F + (u) for n = 1, 2, 3 with those of a recent three-loop calculation of F − (u) for general n in d = 3 dimensions we calculate Borel resummed universal amplitude

Functional differential equations for the free energy and the effective energy in the broken-symmetry phase of φ4-theory and their recursive graphical solution

“…By doing so, we have obtained a result which is relevant for the calculation of universal amplitude ratios [13,14] by an additive renormalization of the vacuum energy above [15,16] and below the critical point. According to Eq.…”

“…A single dot represents the energy shift If the cubic and the quartic interactions K and L in (2.1) vanish, the functional integral in (2.6) is Gaussian and can be immediately calculated to obtain for the negative free energy 14) where the trace of the logarithm of the kernel is defined by the series [6, p. 16] Tr ln…”

Universal amplitude ratios from numerical studies of the three-dimensional O(2) model

“…In Table 6 we list the results for the Goldstone-boson-decay constant F at various J-values. We performed simulations at H = 0.0001 on lattices with linear extensions L = 8, 10,12,16,20,24,30,36,40,48 and 56. By construction the ǫ-expansion is only applicable in a range where m π L < ∼ 1.…”

Critical exponents and equation of state of three-dimensional spin models

“…Only the ratio A + /A − is measured experimentally to high precision [7]. We report a selection of theoretical determinations: IHT+ 4 He is our IHT computation, using as input for α the experimental value α = −0.01285(38) [19]; IHT ⋆ is a complete IHT computation, without experimental input [8]; FT is a g expansion in fixed dimension [16]; ε-exp is obtained by ε expansion [17]. The value of A + /A − is strongly correlated with the value of α, and all disagreement between IHT ⋆ and experiment can be reconduced to the discrepancy in α.…”

“…HT+LT is a combination of HT and lowtemperature expansion [12,13]; the other theoretical determinations are the same discussed for the critical exponents, and are taken from Refs. [2] (IHT), [14,15] (MC), and [16][17][18] (FT). For experimental data, see Refs.…”