Solution to the perturbative infrared catastrophe of hot gauge theories.

The free energy of a nonabelian gauge theory at high temperature T can be calculated to order g using resummed perturbation theory, but the method breaks down at order g. A new method is developed for calculating the free energy to arbitrarily high accuracy in the high temperature limit. It involves the construction of a sequence of two e ective eld theories by rst integrating out the momentum scale T and then integrating out the momentum scale gT . The free energy decomposes into the sum of three terms, corresponding to the momentum scales T , gT , and gT . The rst term can be calculated as a perturbation series in g(T ), where g(T ) is the running coupling constant. The second term in the free energy can be calculated as a perturbation series in g(T ), beginning at order g. The third term can also be expressed as a series in g(T ) beginning at order g, with coe cients that can be calculated using lattice simulations of 3-dimensional QCD. Leading logarithms of T=(gT ) and of gT=(gT ) can be summed up using renormalization group equations. The perturbative infrared catastrophe of a nonabelian gauge theory at high temperature is one of the most important unsolved problems in thermal eld theory. The problem, which was rst identi ed by Linde in 1979 [1, 2], is that perturbative calculations of the free energy of a nonabelian gauge theory at high temperature T with weak coupling g break down at order g. In contrast, there seems to be no obstacle to calculating the free energy of an abelian gauge theory like QED to arbitrarily high order in the coupling constant e. This problem casts a shadow over all applications of QCD perturbation theory to describe a quark-gluon plasma at high temperature, because the catastrophe a icts any observable at su ciently high order in g. The catastrophe also arises in applications of the electroweak gauge theory and grand-uni ed theories to describe phase transitions in the early universe, since they also involve nonabelian gauge theories at high temperature. The catastrophe can be avoided for static observables in hot QCD by calculating them directly using lattice simulations, but the computational resources that are required increase rapidly with T . Moreover, there are many problems, such as the calculation of dynamical observables or the calculation of static observables at nonzero baryon density, which are more easily addressed by perturbative methods than by lattice simulations. For these reasons, it is important to solve the problem of the perturbative infrared catastrophe. In this Letter, we solve the problem by constructing a sequence of two e ective eld theories. The rst e ective theory, which is obtained by integrating out the momentum scale T , is equivalent to thermal QCD at length scales of order 1=(gT ) or larger. The second e ective theory, which is obtained by integrating out the scale gT from the rst e ective theory, is equivalent to thermal QCD at length scales of order 1=(gT ) or larger. If T is su ciently large, the parameters of the two e ective theories can be calculated as perturbation series in the running coupling constant g(T ). The second e ective theory is inherently nonperturbative, so that the e ects from the scale gT must be calculated by lattice simulations. To explain the origin of the infrared catastrophe, we rst consider the case of the abelian