The Scalar Field Theory At Finite Temperature With RGOPT Method

The fundamental theory for the strong interaction is Quantum Chromodynamics.As an important part of the standard model,it is very successful in explaining high energy phenomena.Although great progress has achieved in nonperturbative calculation(such as QCD sum rules,lattice QCD),some difficulties are not solved yet.QCD at finite temperature is the fundamental theory in explaining thermal QCD phenomena.At finite temperature and density the phase diagram of QCD exhibits abundance,such as the confinement-deconfinement phase transition,the chiral phase transition and the color superconductivity.Since phase transitions are essentially nonperturbative phenomena,reliable nonperturbative methods are necessary.The lattice QCD simulation is regarded as the most reliable method in investigating QCD nonperturbative phenomena,but applying it reasonably to nonzero chemical potential is still a challenge.Therefore,effective theories(such as quark meson models,chiral models and NJL models)are very important in finite temperature regime,which set important base in technology and inspiration for further investigating realistic QCD.The calculation in finite temperature field theory is more involved than in zero temperature case.One of the important challenge in the field theory at finite temperature is how to approximate reasonably.The perturbation expansion at nonzero temperature depends not only on the interaction coupling constant but also on temperature,the series is expanded by the product of coupling constant and temperature which restricts the accessible range of perturbation.Even when perturbation can be used,the convergence is slower than in the zero temperature case.To obtain reliable result,the higher order calculations are necessary.Therefore,how to perform reasonable approximations is critical.The common used approximations at finite temperature include the mean field approximation,large N approximation and hard thermal loop expansion,they all face the problem of how to take the higher order effects into account in order to obtain more reliable result in these approximation.The renormalization group improved optimized perturbative theory(RGOPT)method used in this thesis is essentially a variation method and it is the improvement of optimized perturbation method.The fundamental idea of the optimized perturbative theory is by introducing variational parameter with mass dimension and artificial expansion,the new Lagrangian is the interpolation of the original one and a solvable theory.After expandingperturbatively and setting artificial expansion parameter to one,one demands that any physical quantity must be insensitive to the variational parameter so that in lower order calculation higher order effects can be included.In the same time,one demand the physics quantity satisfies the renormalization group equation,which on the one hand will enforce restricts on the interpolating Lagrangian,on the other hand will solved partly the renormalization scale dependence to obtain better convergence of the perturbative series.In this thesis we investigate the standard scalar field theory in both vanishing mass and nonzero mass cases by using RGOPT.We find that in nonzero mass case even at the lowest order,renormalization group invariance will lead to important consequence so that we have to take measure to reconcile RG invariance and OPT.The free energy is given in this thesis,they can be used conveniently to investigate other quantities such as pressure and entropy.