Calculating anomalous dimensions in a φ3 field theory in 6 + ε dimensions using the methods of statistical mechanics

✍ Scribed by Gregory Dee; Siddhartha Sen

• Publisher: Elsevier Science
• Year: 1977
• Tongue: English
• Weight: 674 KB
• Volume: 107
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(77)90209-3

✦ Synopsis

The set of relations A, = 2 -2~6 -&z and ha = -& -9y4 -3~42 are derived for a @ theory in 6 + l dimensions, where AI , A, are eigenvalues which appear in statistical mechanics discussions of scale invariance and phase transitions, while ~4 and ~62 are the anomalous dimensions of the operators 4 and 4". The eigenvalues A,, As are calculated and the relations are used to determine ~6 and ~42 . Separate calculations of r+ and ~62 using field theory methods are also carried out. The statistical mechanics and field theory methods of calculating critical indices are briefly compared.

zation effects have to be taken into account in O(E) calculations, unlike the situation in a 4" field theory in (4 -E) dimensions. In Section 4 we derive formulas which relate the eigenvalues X1 , 2 A which we calculate in Section 3 to the anomalous dimensions yd and ~~2 . In Section 5 a quick sketch of the field theory method of calculating anomalous dimensions is included. These calculations have been done before, using slightly different techniques, by Mack [7] and by Woo and MacFarlane [8]. We include the material for the sake of completeness. Finally in Section 6 we summarize our results and draw a few conclusions.