赝$ε$展开与重整化耦合常数
Pseudo-$ε$ Expansion and Renormalized Coupling Constants at Criticality
摘要：
在准$ε$展开方法中，计算了确定非线性极化率$χ_{2k}$并进入标度状态方程的维度有效耦合常数$g_{2k}$的普适值.对于任意$n$，导出了$g_6$和$g_8$在临界状态下的伪$ε$展开式.还发现了状态方程中比值$R_6 = g_6/g_4^2$和$R_8 = g_8/g_4^3$的类似级数，并报道了由$β$函数的六环RG展开式降为Wilson固定点位置$g_{4}^*$的伪$ε$展开式.给出了$0 \le n \le 64$的数值结果，主要关注物理上重要的情形$n = 0, 1, 2, 3$ .四次和六次耦合的伪$ε$展开式具有快速减小的系数，因此Padé重算足以产生高精度的数值估计。此外，这些系列的直接求和与最佳截断给出的值的$g_4^*$和$R_6^*$几乎一样准确的Padé技术提供的。伪$ε$膨胀估计$g_8^*$和$R_8^*$被发现是远不如独立的重定方法上的低阶耦合。研究了伪$ε$展开方法在二维情况下的数值有效性.利用二维$λϕ^4$场论的$β$ -函数的五圈RG级数，对$g_4^*$进行了伪$ε$展开，得到了$n$的数值估计，其范围为0 ~ 64。讨论的方法给出了足够准确的值的$g_{4}^*$下降到$n = 2$，并导致公平的伊辛和聚合物（$n = 0$）模型的估计。
英文摘要：
Universal values of dimensional effective coupling constants $g_{2k}$ that determine nonlinear susceptibilities $χ_{2k}$ and enter the scaling equation of state are calculated for $n$-vector field theory within the pseudo-$ε$ expansion approach. Pseudo-$ε$ expansions for $g_6$ and $g_8$ at criticality are derived for arbitrary $n$. Analogous series for ratios $R_6 = g_6/g_4^2$ and $R_8 = g_8/g_4^3$ figuring in the equation of state are also found and the pseudo-$ε$ expansion for Wilson fixed point location $g_{4}^*$ descending from the six-loop RG expansion for $β$-function is reported. Numerical results are presented for $0 \le n \le 64$ with main attention paid to physically important cases $n = 0, 1, 2, 3$. Pseudo-$ε$ expansions for quartic and sextic couplings have rapidly diminishing coefficients, so Padé resummation turns out to be sufficient to yield high-precision numerical estimates. Moreover, direct summation of these series with optimal truncation gives the values of $g_4^*$ and $R_6^*$ almost as accurate as those provided by Padé technique. Pseudo-$ε$ expansion estimates for $g_8^*$ and $R_8^*$ are found to be much worse than that for the lower-order couplings independently on the resummation method employed. Numerical effectiveness of the pseudo-$ε$ expansion approach in two dimensions is also studied. Pseudo-$ε$ expansion for $g_4^*$ originating from the five-loop RG series for $β$-function of 2D $λϕ^4$ field theory is used to get numerical estimates for $n$ ranging from 0 to 64. The approach discussed gives accurate enough values of $g_{4}^*$ down to $n = 2$ and leads to fair estimates for Ising and polymer ($n = 0$) models.
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