Critical Exponents without the Epsilon Expansion

We argue that the sharp-cuto Wilson renormalization group provides a powerful tool for the analysis of second-order and weakly rst-order phase transitions. In particular, in a computation no harder than the calculation of the 1-loop e ective potential, we show that the Wilson RG yields the xed point couplings and critical exponents of 3-dimensional O(N) scalar eld theory, with results close to those obtained in high-order "-expansion and large-N calculations. We discuss the prospects for an even more precise computation. E-mail address: [email protected] Weakly rst-order and second-order phase transitions have been important for some time in cosmology. Particular attention has been focused on the electroweak phase transition (EWPT), in an e ort to determine whether it is rst-order and thereby capable of generating the observed asymmetry between matter and antimatter in the universe [1]. However, progress has been complicated by infrared problems that invalidate the usual loop expansion [2]. In statistical mechanics, the usual remedy is to use the "-expansion [3, 4], and this has been applied to the EWPT [5]. However, it has been argued that the "-expansion may be unreliable for the EWPT [6], since there may be a xed point in three dimensions that is not visible in an expansion around four dimensions. In this letter we will show how a second-order phase transition can be analyzed by the \Wilson" or \exact" renormalization group (RG) [7, 8] directly in three dimensions, without recourse to the "-expansion. It appears straightforward to extend this method to rst-order phase transitions [14], which would enable us to determine the order and the dynamics of the EWPT and other cosmological phase transitions. There has recently been a resurgence of interest in the Wilson RG ([9]{[14]). An important contribution was made by Tetradis and Wetterich [15], who used the smooth-cuto Wilson RG to calculate critical exponents of a second-order phase transition, and found impressive agreement with the traditional methods mentioned above. Unfortunately, the use of the smooth cuto greatly complicates the computation, and numerical methods are required to determine the di erential ow equations, which are then solved numerically to obtain the exponents. We will nd, in contrast, that the sharp-cuto Wilson RG yields the xed point couplings and critical exponents in a simple and almost completely analytic form, requiring no more e ort than the calculation of the one-loop e ective potential. Even with a radically truncated action, the resultant values are within about 10% of those found by the traditional methods. In the appendix we discuss the problems that arise with both smooth and sharp cuto s in a more precise calculation. The Wilson RG follows the ow of the e ective action (free energy) as degrees of freedom are integrated out in successively shrinking momentum shells. This corresponds to calculating the e ective action with an infra-red (IR) cuto for the loop corrections, and sending ! 0. The e ective action then ows with , its couplings obeying di erential equations that we will study below. For high temperature systems, we are interested in the ow for the 3 dimensional theory, and if there is a second-order phase transition we expect to nd a xed point that is IR-attractive in all coupling directions except one. This corresponds to the fact that one linear combination of the UV couplings (i.e. the temperature) must be ne-tuned to a critical value if the theory is to ow into