Real-time dynamics of the $O(4)$ scalar theory within the fRG approach
Yang-yang Tan, Yongrui Chen, Wei‐jie Fu
Abstract
In this paper, the real-time dynamics of the O(4) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> scalar theory is studied within the functional renormalization group formulated on the Schwinger-Keldysh closed time path. The flow equations for the effective action and its n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> -point correlation functions are derived in terms of the "classical'' and "quantum’’ fields, and a concise diagrammatic representation is presented. An analytic expression for the flow of the four-point vertex is obtained. Spectral functions with different values of temperature and momentum are obtained. Moreover, we calculate the dynamical critical exponent for the phase transition near the critical temperature in the O(4) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> scalar theory in 3+1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> dimensions, and the value is found to be z\simeq 2.023 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>≃</mml:mo> <mml:mn>2.023</mml:mn> </mml:mrow> </mml:math> .