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تسجيل مستخدم جديدCritical, crossover, and correction-to-scaling exponents for isotropic Lifshitz points to order $boldsymbol{(8-d)^2}$
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A two-loop renormalization group analysis of the critical behaviour at an isotropic Lifshitz point is presented. Using dimensional regularization and minimal subtraction of poles, we obtain the expansions of the critical exponents $ u$ and $eta$, the crossover exponent $phi$, as well as the (related) wave-vector exponent $beta_q$, and the correction-to-scaling exponent $omega$ to second order in $epsilon_8=8-d$. These are compared with the authors recent $epsilon$-expansion results [{it Phys. Rev. B} {bf 62} (2000) 12338; {it Nucl. Phys. B} {bf 612} (2001) 340] for the general case of an $m$-axial Lifshitz point. It is shown that the expansions obtained here by a direct calculation for the isotropic ($m=d$) Lifshitz point all follow from the latter upon setting $m=8-epsilon_8$. This is so despite recent claims to the contrary by de Albuquerque and Leite [{it J. Phys. A} {bf 35} (2002) 1807].
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We comment on a recent letter by L. C. de Albuquerque and M. M. Leite (J. Phys. A: Math. Gen. 34 (2001) L327-L332), in which results to second order in $epsilon=4-d+frac{m}{2}$ were presented for the critical exponents $ u_{{mathrm{L}}2}$, $eta_{{m
athrm{L}}2}$ and $gamma_{{mathrm{L}}2}$ of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.
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