Phase Diagram of the Abelian Higgs Model Using the Auxiliary Mass Method

Kenzo Ogure 0 0 Department of Physics, Kobe University , Kobe 657-8501, Japan We study a phase transition in the Abelian Higgs model at finite temperature using the auxiliary mass method. We vary the gauge coupling constant q andthe scalar self-coupling constant . We find that a first-order phase transition disappears at a tricritical point. It can be fit by the form q2 very well. The value of the parameter depends on the range of . Our results can be appliedto experiments on high-temperature superconductors and liquidcrystals. In particular, the tricritical point is c = 0.28 for superconductors, where q = 2e. - sition, but it is almost impossible to reach the narrow critical region of an ordinary superconductor. Fortunately, the critical region of a high-temperature superconductor is much broader than that of the ordinary one, due to its small coherent length and high critical temperature. 31) Though there have been some experiments with high-temperature superconductors, 32), 33) they have not yet been able to demonstrate the existence or location of a tricritical point. On the other hand, many experiments have been done using liquid crystals. 34) - 51) Most of them indicate that a tricritical point exists, although the result of Ref. 46) indicates that there is no tricritical point and that the phase transition is always of first order. It is suitable to investigate the phase transition using a finite temperature field theory, 52) which is based only on a statistical mechanism. However, its perturbation theory is often unreliable, due to an infrared divergence, even when using the method of ring resummation, which improves the convergence of the perturbation theory. The loop expansion parameter of the perturbation theory is not , but, rather, T at finite temperature T , due to the infrared effect even after the ring resumm mation. 2), 53), 54) Here, is a coupling constant, and m is a mass at finite temperature. For a second order or a weakly first order phase transition, the coherent length m1 becomes very large around the critical temperature. Since the loop expansion parameter mT can be large in such cases, the perturbation theory is unreliable even if is small. According to the perturbation theory, the phase transition is of first order for any parameter value, 54) - 56) but the results are unreliable for 1. We do not have conclusive results regarding the the properties of the phase transition, in spite of several attempts, indicating those using the CJT method 57) and the gapequation method. 58) In the present paper, we use a recently developed auxiliary mass method to reveal the properties of the phase transition. 11), 59) - 62) This method considerably improves the ordinary perturbation theory both qualitatively and quantitatively for scalar field theories. 61), 62) It is based on the following simple idea. We add an auxiliary large mass M ( T ) to a true mass in order to make the expansion parameter mT small. We can rely on an effective potential calculated with the perturbation theory, owing to the large auxiliary mass. We then calculate the effective potential at the true mass by integrating an evolution equation, which we give below. Finally, we obtain physical quantities from the effective potential: the order of the phase transition, the strength of the first order phase transition, the critical exponents of the second order phase transition, and so on. 2. Auxiliary mass method for the Abelian Higgs model We now apply the auxiliary mass method to the Abelian Higgs model explicitly. The Lagrangian density of this model is the following: 1 F F + |( iqA )|2 m2||2 3! ||4. L = 4 We take the true mass squared to be negative, m2 = 2, in order for the theory to exhibit spontaneous symmetry breaking (SSB) at zero temperature. With this notation, the G-L parameter is = 6q2 . We first take the mass squared of the Higgs boson to be a large positive value, m2 = M 2( T 2). This auxiliary mass for the scalar field is sufficient to improve the perturbation theory, since we have no 3or 4-point self-interactions of the gauge field in the Abelian Higgs model, in contrast to the non-Abelian Higgs model. We express the effective potential calculated using the Landau gauge in terms of = 2|| as V (m2 = M 2) = + J (m21) + J (m22) + 2J (m2A) + J (m2AL), dk k2 log 1 exp k2 + m2 3 This effective potential is reliable, due to the large auxiliary mass. The ring resummation is needed only for the gauge field, since the effect of the resummation on the scalar field is small in comparison with the auxiliary mass M . We next construct an evolution equation that describes the change of an effective potential as the mass varies. Since we introduced the auxiliary mass only for the scalar field, the evolution equation is the same as that of the O(N ) invariant scalar model for N = 2. The evolution equation, which relates the derivative of the effective potential with respect to the mass squared of the Higgs boson to the full propagators, has been obtained in Ref. 62) as +i+ Here, Sig = Sig(m2, 2, k02, k2, T ) and NG = NG(m2, 2, k02, k2, T ) are the full self-energies, which correspond to massive and massless modes in the broken phase. We then carry out the following replacement by ignoring the momentum dependence of the self-energies:) ) This is the leading order approximation of a systematic approximation. 63) The evolution equation here is obtainedby assuming that the effective action has the simple form = 0 d d3x ( c)2 + V (c2) . We fix the coefficient of the kinetic term to unity andignore terms which contain more than two derivatives. We also ignore the effect of the gauge field on the effective action. The gauge fieldtherefore contributes to results only through the initial effective potential in the present paper. m2 + 2 + NG(0, 0, , m2, T ) 1 V . 6 This replacement allows us to convert Eq. (2.4) into the partial differential equation Numerical results and discussion We solved the partial differential equation using the improved Crank-Nicholson method in Ref. 61). The mesh size is reduced sufficiently for the convergence of the results here. We give the results with initial mass, M 2 = T 2, because we need the auxiliary mass, whose temperature is of the same or larger order. We obtained almost the same results for M 2 = 4T 2 and M 2 = T 2/4. Below, we use mass units in which = 1. We show the effective potential at the critical temperature for various gauge coupling constants in Fig. 1. It is seen that the first order phase transition becomes weaker for smaller values of the gauge coupling constant and finally disappears. This indicates the existence of a tricritical point. The location of the tricritical point is clear in Fig. 2, which displays the ratio of the critical field expectation value to the critical temperature as a function of the gauge coupling constant. We observe that the results obtained with the auxiliary mass method and with th (...truncated)