On the Moduli Space of Noncommutative Multi-solitons at Finite θ

We study the finite θ correction to the metric of the moduli space of noncommutative multi-solitons in scalar field theory in (2+1) dimensions. By solving the equation of motion up to order O(θ−2) explicitly, we show that the multi-soliton solution must have the same center for a generic potential term. We examine the condition that the multi-centered configurations are allowed. Under this condition, we calculate the finite θ correction to the metric of the moduli space of multi-solitons and argue the possibility of the non right-angle scattering of two solitons. We also obtain the potential between two solitons. Solitons in field theories on noncommutative spacetime are useful for studying nonperturbative effects. These play also an important role in string theory and condensed matter physics(see [1] for reviews). In particular, the noncommutative soliton in 2 + 1 dimensional scalar field theory [2] provides an interesting and nontrivial example since it does not exist in scalar field theory on commutative spacetime. Various aspects of noncommutative scalar solitons have been studied [3, 4, 5, 6, 7, 8, 9]. In ref. [3], the multi-soliton solutions and their moduli space are studied in the limit of large noncommutative parameter θ. The geodesic in the moduli space of two solitons describes the scattering of the soliton in the adiabatic approximation[10]. In refs. [4, 5, 6], it is shown that the right-angle scattering of solitons occurs at the head-on collision. The scattering is also studied for various noncommutative solitons [11]. It is an interesting problem whether noncommutative solitons in scalar field theory exist at finite θ. In this case, we need to consider the attractive or repulsive force between solitons. Recently, Durhuus and Jonsson pointed out that there are no multi-soliton solutions which interpolate smoothly between n overlapping solitons and n solitons with an infinite separation at the lowest order perturbation in θ−1[7]. Therefore multi-solitons at finite θ are in general unstable and decay into infinitely separate or overlapping solitons. This heavily depends on the shape of potential term. In fact, for the potential V (φ) with 1/V ′′(0)+1/V ′′(λ) = 0, where φ = 0 and λ are two critical points of V (φ), energies for the above two configurations agree with each other. Hence the moduli space approximation looks still good for such a potential. In this paper, we study the θ correction to the noncommutative scalar soliton. We solve the static equation of motion explicitly in scalar field theory up to the order O(θ−2). We examine some consistency conditions which appears in the process of solving the equation of motion. From these conditions, we show that noncommutative multi-solitons must have the same center for the potential with 1/V ′′(0) + 1/V ′′(λ) 6= 0. On the other hand, multi-centered configurations are allowed for the potential with 1/V ′′(0) + 1/V ′′(λ) = 0. Using the perturbed solutions, we may calculate the finite θ correction to the metric of the moduli space for the multi-solitons. We argue the possibility of the non right-angle scattering for scattering of two solitons for finite θ. This would provide interesting physics in contrast with the right-angle scattering in the large θ limit. We also study the force