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Ward-takahashi identity for vertex functions of sQED

Ward-Takahashi identity is an useful tool for calculating amplitude of scattering processes. In the high-order perturbative theory of sQED, propagator and vertex functions contain many high-order corrections. By using Ward-Takahashi identity, each vertex function is separated into two parts: ”longitudinal” and ”transverse ” part. The longitudinal part can be directly calculated from Ward-Takahashi identity. The transverse part depends on the expanding of specific orders of the theory. In this report, we present one method based on the Ward-Takahashi identity, to calculate this part of vertex functions at the one-loop order in arbitrary gauge and dimensions in sQED. | Communications in Physics, Vol. 23, No. 1 (2013), pp. 11-19 WARD-TAKAHASHI IDENTITY FOR VERTEX FUNCTIONS OF sQED HA THANH HUNG Department of Physics, Hanoi University of Education No. 2 LE THO HUE AND HOANG NGOC LONG Institute of Physics, VAST Abstract. Ward-Takahashi identity is an useful tool for calculating amplitude of scattering processes. In the high-order perturbative theory of sQED, propagator and vertex functions contain many high-order corrections. By using Ward-Takahashi identity, each vertex function is separated into two parts: ”longitudinal” and ”transverse ” part. The longitudinal part can be directly calculated from Ward-Takahashi identity. The transverse part depends on the expanding of specific orders of the theory. In this report, we present one method based on the Ward-Takahashi identity, to calculate this part of vertex functions at the one-loop order in arbitrary gauge and dimensions in sQED. I. INTRODUCTION We introduce a method in which Ward-Takahashi identity is used to decompose the vertex into to longitudinal part and transverse part. This form of vertex satisfies two conditions: (i) has no kinematics singularities in both two parts, (ii) the longitudinal part of a vertex has been fixed by scalar coefficient. II. PROPAGATORS AND VERTEX FUNCTIONS OF sQED IN BARE PERTURBATION. In the scalar Quantum Electrodynamics Dynamics (sQED), propagator and vertex functions in any gauge ξ are determined as follows [1]: µ ∆0µν = ν −gµν p2 +(1−ξ)pµ pν p4 ; arbitrary gauge ξ S 0 (p) = 1 p2 −m2 Fig. 1. Propagators of sQED in bare perturbative theory 12 HA THANH HUNG, LE THO HUE, AND HOANG NGOC LONG µ p k µ ν k p e2 Γ0µν = e2 gµν Γ0µ = (k + p)µ Fig. 2. Vertex functions of sQED in bare perturbative theory III. WARD-TAKAHASHI IDENTITY WITH 3-POINT VERTEX FUNCTION OF sQED . Propagators of scalar particles at one-loop order [1]: = −iΣ(p2 ) + = + Σ1 (p2 ) Σ2 (p2 ) Fig. 3. Propagator of complex scalar particle at one-loop. In .