The topological charge distribution P (Q) is calculated for lattice CP N−1 models. In order to suppress lattice cut-off effects, we employ a fixed point (FP) action. Through transformation of P (Q), we calculate the free energy F (θ) as a function of the θ parameter. For N=4, scaling behavior is observed for P (Q) and F (θ), as well as the correlation lengths ξ(Q). For N=2, however, scaling behavior is not observed, as expected. For comparison, we also make a calculation for the CP 3 model with a standard action. We furthermore pay special attention to the behavior of P (Q) in order to investigate the dynamics of instantons. For this purpose, we carefully consider the behavior of γ eff , which is an effective power of P (Q) (∼ exp(−CQ γ eff )), and reflects the local behavior of P (Q) as a function of Q. We study γ eff for two cases, the dilute gas approximation based on the Poisson distribution of instantons and the Debye-Hückel approximation of instanton quarks. In both cases, we find behavior similar to that observed in numerical simulations. §1. IntroductionIt is interesting to study the phase structure of asymptotic free theories such as QCD and the CP N −1 model. Non-perturbative studies of the phase structure of such theories are necessary in order to understand why effects of the topological term (θ term) are suppressed in Nature. The θ term affects the dynamics at low energy and is expected to lead to rich phase structures. 1) Actually, in the Z(N) gauge model, it has been shown by use of free energy arguments that oblique confinement phases could emerge and that an interesting phase structure may be realized. 2) In this paper we are concerned with the dynamics of the θ vacuum of CP N −1 models with a topological term, which have several dynamical properties in common with QCD. We believe that study of the two-dimensional model will be useful in acquiring information about realistic physics.From the numerical point of view, the topological term introduces a complex Boltzmann weight in the Euclidean lattice path integral formalism. The complex nature of the weight prevents one from straightforwardly applying the standard algorithm used for Monte Carlo simulations. This problem can be circumvented * )
Topological charge distributions in the two-dimensional CP 2 model with the θ-term are calculated. In strong coupling regions, the topological charge distribution is approximately given by a Gaussian form as a function of the topological charge, and this behavior leads to a first order phase transition at θ = π. In weak coupling regions, this distribution exhibits non-Gaussian form, and the first order phase transition disappears. The free energy as a function of θ displays "flattening" behavior at θ = θ f < π when we calculate the free energy directly from the topological charge distribution. A possible origin of this flattening phenomenon is proposed. * )
It was often argued that the real part of the s-channel resonant amplitude cannot be known from information on s-channel resonances in the meaning of local or semilocal duality since they will lead to a null result by a cancellation in the integrand.!) We will show that a non-null result 1s obtained when the resonance correlation is correctly taken into account.The s-t dual amplitude fH corresponding to the H-type urbaryon rearrangement diagram2l is given by a sum of infinite schannel resonant amplitudes expressed by the Breit-Wigner form:with e=MF. The resonances do not exist randomly but occur at a constant impact parameter b0 =51GeV (generalized Ogawa's rule). This mechanism is called "resonance correlation", and the correlated resonances as a whole are called "effective resonance". BJ According to this rule j9 (M 2 , l)-iJ·D(l 2 -(b0MI2) 2 ), fHbecomes as follows after the sum over M 2 is carried out, and the sum over l is replaced by the integral over impact parameter b = l I ( vs I 2) :.u'J, (s, x) with e' =b0bF I vs, x=b 2 , x 0 =b0 2 and y = -t, where .J,{ denotes the helicity change.For the sake of simplicity we approximate iJ 4A (s, x) by a constant, then (2) becomes in the narrow width approximation fi{(s,y) =2ii}4,·H51J( ../x0 y)where HW is the first kind Hankel function. The Neuman function N 4 A behaves almost similarly to J4H 1 except in a very small y region. The form (4) just corresponds to the rotation phase exp ( -in:a(t)) in the Regge representation of fH, which means that direct channel resonances as a whole give the real to imaginary ratio in a manner consistent to the t-channel picture (duality). Now we will discuss the physical meaning of fH given by the resonance correlation. The real and imaginary parts of the Breit-Wigner form in the b representation (3) are J
Character expansion developed in real space renormalization group (RSRG) approach is applied to U(1) lattice gauge theory with θ-term in 2 dimensions. Topological charge distribution P (Q) is shown to be of Gaussian form at any β(inverse coupling constant). The partition function Z(θ) at large volume is shown to be given by the elliptic theta function. It provides the information of the zeros of partition function as an analytic function of ζ = e iθ (θ = theta parameter). These partition function zeros lead to the phase transition at θ = π. Analytical results will be compared with the MC simulation results. In MC simulation, we adopt (i)"set method" and (ii)"trial function method".
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