Anomalous Dimensions of Quark Masses in the Three-Loop Approximation

Abstract

The results of calculation of the three-loop radiative correction to the renormalization constant of fermion masses for non-abelian gauge theory interacting with fermions are presented. Dimensional regularization and the t’Hooft-Veltman minimal subtraction scheme are used. The method of calculation is described in detail. The renormalization group function ${{\gamma }_{m}}$ determining the behavior of the effective mass of fermions is presented. The anomalous dimensions of fermions for QED and QCD up to three loops are given. All calculations were performed on a computer with the help of the SCHOOONSCHIP system for analytical manipulations.

APPENDIX

The three-loop 1PI diagrams of the fermion propagator are given in Fig. 1 and Fig. 2.

Fig. 1.

Fig. 2.

The contributions to $Z_{2}^{{ - 1}}$ from different sets of the diagrams ($- KR{\kern 1pt} '$):

Figure 1:

$$\begin{gathered} - \,\,\frac{{{{C}_{F}}}}{{{{\varepsilon }^{3}}}}\left[ { - \frac{7}{4}{{C}^{2}} - \frac{4}{3}C{{C}_{F}} + \frac{1}{3}Ctf} \right] \\ - \,\,\frac{{{{C}_{F}}}}{{{{\varepsilon }^{2}}}}\left[ {\frac{{219}}{{24}}{{C}^{2}} + \frac{{35}}{6}C{{C}_{F}} - \frac{4}{3}{{C}_{F}} - \frac{{19}}{6}Ctf - \frac{4}{3}{{C}_{F}}tf} \right] \\ - \,\,\frac{{{{C}_{F}}}}{\varepsilon }\left[ { - \frac{{233}}{{12}}{{C}^{2}} + \frac{{17}}{2}C{{C}_{F}} - \frac{7}{{12}}C_{F}^{2} + \frac{{43}}{6}Ctf} \right. \\ \left. { - \,\,\frac{7}{3}{{C}_{F}}tf} \right] - \frac{{{{C}_{F}}}}{\varepsilon }C\zeta (3)\left( {\frac{5}{2}C - 2{{C}_{F}}} \right). \\ \end{gathered} $$

Figure 2a:

$$\frac{{C_{F}^{2}}}{{{{\varepsilon }^{2}}}}\left( {\frac{5}{{12}}C - \frac{1}{3}tf} \right) + \frac{{C_{F}^{2}}}{\varepsilon }\left( {\frac{3}{8}C - \frac{1}{6}tf} \right).$$

Figure 2b:

$$\begin{gathered} - \,\,\frac{{{{C}_{F}}}}{{{{\varepsilon }^{2}}}}\left[ {\frac{{175}}{{72}}{{C}^{2}} - \frac{{55}}{{18}}Ctf + \frac{8}{9}{{t}^{2}}{{f}^{2}}} \right] \\ - \,\,\frac{{{{C}_{F}}}}{\varepsilon }\left[ { - \frac{{2171}}{{432}}{{C}^{2}} + \frac{{527}}{{108}}Ctf + 2{{C}_{F}}tf - \frac{{20}}{{27}}{{t}^{2}}{{f}^{2}}} \right]. \\ \end{gathered} $$

Figure 2c:

$$\begin{gathered} - \,\,\frac{{C_{F}^{2}}}{{{{\varepsilon }^{3}}}}\left[ {\frac{1}{3}C - \frac{1}{6}{{C}_{F}}} \right] + \frac{{C_{F}^{2}}}{{{{\varepsilon }^{2}}}}\left[ {3C - \frac{7}{{12}}{{C}_{F}} - \frac{2}{3}tf} \right] \\ - \,\,\frac{{C_{F}^{2}}}{\varepsilon }\left[ {\frac{{91}}{{24}}C - 2\zeta (3)C + \frac{1}{{12}}{{C}_{F}} - \frac{5}{6}tf} \right]. \\ \end{gathered} $$

Figure 2d:   $- \tfrac{2}{{{{\varepsilon }^{3}}}}C_{F}^{2}C$.

The contributions to ${{Z}_{{\overline \psi \psi }}}$ from the diagrams with the insertion $- {\kern 1pt} \otimes {\kern 1pt} -$ were calculated simultaneously with the diagrams shown in Figs. 1 and 2. In fact, for the fermion propagator instead of ${{\hat {p}} \mathord{\left/ {\vphantom {{\hat {p}} {{{p}^{2}}}}} \right. \kern-0em} {{{p}^{2}}}}$ we used $(\hat {p} + {{m)} \mathord{\left/ {\vphantom {{m)} {{{p}^{2}}}}} \right. \kern-0em} {{{p}^{2}}}}$, and after multiplication of all propagators we kept only the part without the mass term and the part linear in $m$. The coefficient in front of $m$ determines the contribution of the diagrams with the mass insertion. For example, from the diagram

we get the following diagrams with the insertion:

The contributions to ${{Z}_{{\bar {\psi }\psi }}}$ from different sets of diagrams ($- KR{\kern 1pt} '$)

Figure 1:

$$\begin{gathered} - \,\,\frac{{{{C}_{F}}}}{{{{\varepsilon }^{3}}}}\left[ {\frac{{31}}{3}{{C}^{2}} + \frac{{50}}{3}C{{C}_{F}} + 8C_{F}^{2} - 4Ctf - \frac{8}{3}{{C}_{F}}tf} \right] \\ - \,\,\frac{{{{C}_{F}}}}{{{{\varepsilon }^{2}}}}\left[ { - \frac{{181}}{6}{{C}^{2}} - \frac{{80}}{3}{{C}_{F}}C + 8C_{F}^{2} + \frac{{34}}{3}Ctf + \frac{8}{3}{{C}_{F}}tf} \right] \\ - \,\,\frac{{{{C}_{F}}}}{\varepsilon }\left[ {\frac{{181}}{4}{{C}^{2}} - \frac{{145}}{3}C{{C}_{F}} + 17C_{F}^{2} - 13Ctf + \frac{{20}}{3}{{C}_{F}}tf} \right] \\ - \,\,\frac{{{{C}_{F}}}}{\varepsilon }\zeta (3)\left[ { - \frac{3}{2}{{C}^{2}} + 20{{C}_{F}}C - 16C_{F}^{2} - 8Ctf} \right]. \\ \end{gathered} $$

Figure 2a:

$$\begin{gathered} - \,\,\frac{{C_{F}^{2}}}{{{{\varepsilon }^{3}}}}\left[ {\frac{{10}}{3} - \frac{8}{3}tf} \right] - \frac{{C_{F}^{2}}}{{{{\varepsilon }^{2}}}}\left[ { - \frac{{17}}{3}C + 4tf} \right] \\ - \,\,\frac{{C_{F}^{2}}}{\varepsilon }\left[ { - C + \frac{4}{3}tf} \right]. \\ \end{gathered} $$

Figure 2b:

$$\begin{gathered} - \,\,\frac{{{{C}_{F}}}}{{{{\varepsilon }^{3}}}}\left[ {\frac{{175}}{{36}}{{C}^{2}} - \frac{{55}}{9}Ctf + \frac{{16}}{9}{{t}^{2}}{{f}^{2}}} \right] \\ - \,\,\frac{{{{C}_{F}}}}{{{{\varepsilon }^{2}}}}\left[ { - \frac{{337}}{{27}}{{C}^{2}} + \frac{{346}}{{27}}Ctf + 4{{C}_{F}}tf - \frac{{64}}{{27}}{{t}^{2}}{{f}^{2}}} \right] \\ - \,\,\frac{{{{C}_{F}}}}{\varepsilon }\left[ {\frac{{18685}}{{1296}}{{C}^{2}} - \frac{{1915}}{{324}}Ctf - 17{{C}_{F}}tf - \frac{{80}}{{81}}{{t}^{2}}{{f}^{2}}} \right] \\ - \,\,\frac{{{{C}_{F}}}}{\varepsilon }\zeta (3)\left[ { - {{C}^{2}} - 8Ctf + 16C\,{{~}_{F}}tf} \right]. \\ \end{gathered} $$

Figure 2c:

$$\begin{gathered} - \frac{{C_{F}^{2}}}{{{{\varepsilon }^{3}}}}\left[ {6C + \frac{8}{3}{{C}_{F}} - \frac{8}{3}tf} \right] - \frac{{C_{F}^{2}}}{{{{\varepsilon }^{2}}}}\left[ { - 17C - 10{{C}_{F}} + 4tf} \right] \\ - \,\,\frac{{C_{F}^{2}}}{\varepsilon }\left[ {\frac{{80}}{3}C + 5{{C}_{F}} - \frac{{16}}{3}tf - 16\zeta (3)C + 16\zeta (3){{C}_{F}}} \right]. \\ \end{gathered} $$

Figure 2d: 0.