Proc Natl Conf Theor Phys 36 (2011), pp 14-22 WARD-TAKAHASHI IDENTITY FOR VERTEX FUNCTIONS OF SQED H T HUNG, L T HUE, H N LONG Institute of Physics, VAST, P O Box 429, Bo Ho, Hanoi 10000, Vietnam Abstract Ward-Takahashi identity is an useful tool for calculating amplitude of scattering processes In the high-order perturbative theory of sQED, propagators and vertex functions include many high-order corrections By using Ward-Takahashi identity, each vertex function is separated into two parts: “longitudinal” and “transverse” The longitudinal part can be directly calculated from Ward-Takahashi identity The transverse part depends on the expanding of specific orders of the theory This paper will present one method based on the Ward-Takahashi identity, to calculate parts of vertex functions at the one-loop order in arbitrary gauge and dimensions in sQED I INTRODUCTION We introduce a method which use Ward-Takahashi identity to decompose the vertex into 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 fixed scalar coefficient that II PROPAGATORS AND VERTEX FUNCTIONS OF SQED IN BARE PERTURBATION In the scalar Quantum Electrodynamics Dynamics (sQED), propagators and vertex function in any gauge ξ are determined as follow: µ ∆0µν = ν −gµν p2 +(1−ξ)pµ pν p4 ; arbitrary gauge ξ S (p) = p2 −m2 Fig Propagators of sQED in bare perturbative theory WARD-TAKAHASHI IDENTITY FOR VERTEX FUNCTIONS OF SQED µ p k Γ0µ = (k + p)µ µ ν k p 15 e2 Γ0µν = e2 gµν Fig Vertex functions of sQED in bare perturbative theory = −iΣ(p2 ) + = + Σ1 (p2 ) Σ2 (p2 ) Fig Propagator of complex scalar particle at one-loop III WARD-TAKAHASHI IDENTITY WITH 3-POINT VERTEX FUNCTION OF SQED Propagators of scalar particles at one-loop order : In regular dimension the second diagram (tadpole) vanishes The one-loop propagator is given by: −e2 m2 l 2(m2 + p2 ) p2 ( ) Γ(1 − l){1 − F (2 − l, 1; l; ) m2 4π m2 m2 p2 (m2 − p2 )2 ) F (3 − l, 2; l; )} + (1 − ξ) m4 m2 S −1 = (1) The 3-point function of sQED at one-loop: µ µ µ = k p µ + k p µ + k p + k p k p In term of mathematical language, we have Γµ (k, p) = (k + p)µ + Γµ1 (k, p) + Γµ2 (p) + Γµ2 (k) In which: (2) 16 H T HUNG, L T HUE, H N LONG µ Γµ1 (k, p) = µ Γµ2 (p) = q p k For vertex functions: Γ1µ = µ q k Γµ2 (k) = q p k p −ie2 {4(kp)(k + p)µ J + [−8(kp)gµν − 2(k + p)µ (k + p)ν ]Jν1 + 4(k + p)ν Jµν (2π)2l + (k + p)µ K − 2Kµ1 + (ξ − 1)[(k + p)µ K + 4(k + p)µ P α K β Iαβ − 8pα K β Iµαβ − 2(k + p)µ (k + p)α Jα1 + 4(k + p)α Jµα − 2Kµ1 ]} (3) and e2 p2 pµ m2 π l−2 {[3 + ]Q (p) − Γ(1 − l)(m2 )l−1 (2π)2l p2 p2 p2 − m [Q1 (p) + (p2 − m2 )Q3 (p)]} + (ξ − 1) p2 Γµ2 (p) = (4) Ward-Takahashi identity for the 3-point vertex function: qµ Γµ (k, p) = S −1 (k) − S −1 (p) (5) in higher correlative orders we introduce: Γµ (k, p) = ΓµL (k, p) + ΓµT (k, p) (6) Longitudinal component and the transverse component is ΓµL (k, p) = S −1 (k) − S −1 (p) (k + p)µ ; ΓµT (k, p) = τ (k , p2 , q )T µ (k, p) k − p2 (7) Where T µ (k, p) = pqk µ − kqpµ = [q µ (k − p2 ) − (k + p)µ q ] (8) The condition of ΓµT (k, p) is: qµ ΓµT (k, p) = 0; ΓµT (p, p) = The function τ (k , p2 , q ) is reduced as follow: (9) WARD-TAKAHASHI IDENTITY FOR VERTEX FUNCTIONS OF SQED τ (k , p2 , q ) = e2 π {(k − 2m2 + p2 − 4kp)[−K0 + (m2 + kp)J0 ] 2(2π)d ∆2 2Q1 (p) 2 + [p (p − 3kp) + k (kp − 3p2 ) − 2m2 (p2 + kp)] k − p2 2Q1 (k) 2 − [k (k − 3kp) + p2 (kp − 3k ) − 2m2 (k + kp)] k − p2 +(ξ − 1)(m2 − k )(m2 − p2 )[J0 − (kp + m2 )I0 − 2Q3 (p) 2Q3 (k) (kp + p2 ) − (kp + k )]} k − p2 k − p2 17 (10) In which: ∆2 = (kp)2 − k p2 = (kq)2 − k q It is convenient to present τ (k , p2 , q ) = τ (k , p2 , q ) (11) in terms of propagators of scalar particle: [S −1 (k, ξ = 1) − S −1 (p, ξ = 1)] {(k − 2m2 + p2 − 4kp) 4∆2 [(m2 + k )Q1 (k) − (m2 + p2 )Q1 (p)] 2Q1 (p) 2 ×[−K0 + (m2 + kp)J0 ] [p (p − 3kp) + k (kp − 3p2 ) k − p2 2Q1 (k) 2 [k (k − 3kp) + p2 (kp − 3k ) − 2m2 (k + kp)]} −2m2 (p2 + kp)] − k − p2 [S −1 (k, ξ − 1) − S −1 (p, ξ − 1)] + (m2 − k )(m2 − p2 ) 2∆ [(m2 − k )Q3 (k) − (m2 − p2 )Q3 (p)] 2Q3 (k) 2Q3 (p) (kp + p2 ) − (kp + k )} (12) ×{J0 − (kp + m2 )I0 − 2 k −p k − p2 We define q µ = (k − p)µ ; P µ = (k + p)µ W-T identity for three-point function can be written in the form of qµ Γν − qν Γµ = (qµ Pν − qν Pµ ) S −1 (k) − S −1 (p) q + τ (k , p2 , q ) k − p2 Pµ Γν − Pν Γµ = (Pµ qν − Pν qµ ) k − p2 τ (k , p2 , q ) (13) (14) IV WARD-TAKAHASHI IDENTITY WITH 4-POINT VERTEX FUNCTION OF SQED Ward-Takahashi identity of 4-point function relates with 3-point vertex function by: k µ Γνµ (p , k ; p, k) = Γν (p + k, p) − Γν (p , p − k) k µ Γνµ (p , k ; p, k) = Γν (p , p + k ) − Γν (p − k , p) (15) 18 H T HUNG, L T HUE, H N LONG Following Eq (IV) we can determine the longitudinal component of 4-point vertex function based on 3-point vertex functions We denote: Qµ = kµ (p + p )k − kk (p + p )µ ; Rµ = kµ k k − k kν Qν = kν (p + p )k − kk (p + p )ν ; Rν = kν k k − k kν (16) Then 4-point vertex function is written by: Γµν T = ΓL µν + Γµν = Agµν + B11 (kk gµν − kν kµ ) + B12 Qν kµ + B13 Rν kµ + B21 kν Qµ + B22 Qν Qµ + B23 Rν Qµ + B31 kν Rµ + B32 Qν Rµ + B33 Rν Rµ (17) And now we can determine longitudinal and transverse component of 4-point vertex function as follow: ΓTµν ΓL µν = Agµν + B12 Qν kµ + B13 Rν kµ + B21 kν Qµ + B31 kν Rµ (18) = B11 (kk gµν − kν kµ ) + B22 Qν Qµ + B23 Rν Qµ + B32 Qν Rµ + B33 Rν Rµ (19) Factors of longitudinal component of 4-point vertex fuction is: S −1 (p + k) − S −1 (p) + S −1 (p − k) − S −1 (p ) kk S −1 (p + k) − S −1 (p) S −1 (p ) − S −1 (p − k) = − { − kk (p + k)2 − p2 p − (p − k)2 A = − B21 − k ΓT (p + k, p) − ΓT (p , p − k) } B12 = − B31 B13 S −1 (p − k ) − S −1 (p ) S −1 (p) − S −1 (p + k ) { − kk (p − k )2 − p p2 − (p + k )2 − k ΓT (p − k , p ) − ΓT (p, p + k ) } = − [(p + k)2 − p2 ]ΓT (p + k, p) + [(p − k)2 − p ]ΓT (p , p − k) (kk )2 = − [(p − k )2 − p ]ΓT (p − k , p ) + [(p + k )2 − p2 ]ΓT (p, p − +k ) (kk )2 (20) In which ΓT is the transverse component of 3-point vertex function and it is determined as follow: Γµ (p + k, p) = (2p + k)µ S −1 (p + k) − S −1 (p) + 2(kµ pk − k pµ )ΓT (p + k, p) (p + k)2 − p2 (21) Factors B11 , B22 , B23 , B32 and B33 of transverse component of 4-point vertex function will be calculate according to perturbative -orders of the theory We will introduce technique to calculate the transverse component of 4-point vertex function at one-loop The corresponding Feynman diagrams for this function are: WARD-TAKAHASHI IDENTITY FOR VERTEX FUNCTIONS OF SQED 19 For sake of simplicity, we denote q = k + p and q = k + p, then results of Feynman diagrams at one-loop are given by Γµν D1 (q) = − Γµν D4 (p, q) = ie2 (2π)D −2 Γµν D8 (p, q) = − (22) (p + q)µ q ν (p + q)µ ν K(q) + K (q) + (1 − ξ)(p + q)µ Lν (q) q − m2 q − m2 (23) ie2 µν ˜ g K(p − p ) − 2(p + p )µ Iµ (p, p ) + 4pp I(p, p ) (2π)D ˜ − p ) − 2(p + p )µ Iµ (p.p ) + 4p µ pν Jµν (p, p ) (ξ − 1)K(p + Γµν D9 (p, q) = − 2ie2 {−g µν K(q) + (1 − ξ)Lµν (q)} (2π)D (24) ie2 {2(p + q)µ pν I(p, q) − (p + q)µ I ν (p, q) − 4pν I µ (p + q) + 2I µν (p, q) (2π)D − (1 − ξ)[(p2 − m2 )(p + q)µ ]J ν (p, q) − 2(p2 − m2 )J µν (p, q) + 2Lµν (q) − (p + q)µ Lν (q)} (25) Γµν D15 (p, q, p ) = ie2 −2p ν K(p ) + K ν (p ) + (1 − ξ)(p − m2 )Lν (p ) (2π)D Γµν D19 (p, q, p ) = − ie2 4m2 µ ν (p + q) (p + q) + 2(2π)D q − m2 (q − m2 )2 T (1 − ξ) − L(q) 2 (q − m ) q − m2 Γµν D20 (p, q, p ) = − + Γµν D23 (p, q, p ) = − (26) K(q) (27) ie2 (p + q )µ (p + q )ν −4q K(q ) − T + 4qα K α (q ) 2(2π)D (q − m2 )2 (1 − ξ) T − 4qα K α (q ) + 4qα qβ Lαβ (q ) (28) ie2 4p q m2 − p µ ν (p + q) (p + q) + − I(p , q) 2(2π)D q − m2 q − m2 ˜ K(p − q) K(p ) K(q) 4p q m2 − p µ − 2(p + q) + + ( + − 1) q − m2 q − m2 q − m2 q − m2 q − m2 K ν (p ) K ν (q) (p + q)ν ˜ K(p − q) + + 2(q − m2 ) q − m2 q − m2 p − m2 − (1 − ξ) (p + q)µ (p + q)ν [(p − m2 )J(p , q) − L(q) − L(p )] q − m2 p − m2 ν − 2(p + q)µ [(p − m2 )J ν (p , q) − Lν (q) − L (p )] q − m2 × I ν (p , q) − (29) 20 H T HUNG, L T HUE, H N LONG ie2 (k + 2p)µ (k + 6p + p )ν [(−2m2 + (p − p)2 − 2pp )U (p.p q) 2(2π)D ˜ − p, p − p) − I(p, q) − I(p , q)] − 2(k + p + p ν [(−2m2 + (p − p)2 + I(q Γµν D27 (p, q, p ) = − 2pp )U µ (p, p , q)]) + I˜µ (q − p, p − p) (30) The remain Feynman diagrams is determined according to: µν µν µν µν µν µν µν Γµν D2 = ΓD1 (q ); ΓD5 = ΓD4 (p, q ); ΓD6 = ΓD4 (p , q ); ΓD7 = ΓD4 (p , q) (31) µν µν µν µν µν µν µν Γµν D10 = ΓD9 (p, q ); ΓD11 = ΓD9 (p , q ); ΓD12 = ΓD9 (p , q); ΓD16 = ΓD15 (p, q , p ) (32) µν µν µν µν µν µν µν Γµν D17 = ΓD15 (p , q , p); ΓD18 = ΓD15 (p , q, p); ΓD24 = ΓD23 (p, q , p ); ΓD25 = ΓD23 (p , q , p)(33) µν µν µν Γµν D26 = ΓD23 (p , q, p); ΓD28 = ΓD27 (k, p, q , p ) (34) µν µν µν µν Diagrams Γµν D3 ; ΓD13 ; ΓD14 ; ΓD21 vΓD22 vanish, so they not contribute And the calculating follow us to arrange terms of total 4-point vertex function in term of: Γµν = C0 gµν + C1 kµ kν + C2 kµ pν + C3 pµ kν + C4 kµ pν + C5 pµ kν +C6 pµ pν + C7 pµ pν + C8 pµ pν + C9 pµ pν , (35) in which factors Ci can be computed according to one-loop diagrams Now factors of transverse component of 4-point vertex function can be determined based on C i as follow: k2 k (kp + p2 − pp ) kp k (−p + pp + kp ) C1 + C2 − C3 + C4 kk (kk ) kk (kk )2 kp(−p + pp + kp ) kp kp (kp + p2 − pp ) + C − C + C7 (kk )2 kk (kk )2 kp (−p + pp + kp ) kp(kp + p2 − pp ) + C + C9 (kk )2 (kk )2 C6 + C + C + C (kp )(−2kp + kp ) + (kp)2 + k (k + 2kp − 2kp ) C6 − C + C − C (kp )(−2kp + kp ) + (kp)2 + k (k + 2kp − 2kp ) 2C2 + 2C4 − C6 + C7 + C8 − C9 (kp )(−2kp + kp ) + (kp)2 + k (k + 2kp − 2kp ) 2C2 − 2C4 + C6 + C7 − C8 − C9 − (kp )(−2kp + kp ) + (kp)2 + k (k + 2kp − 2kp ) B11 = − B22 = B23 = B32 = B33 = (36) WARD-TAKAHASHI IDENTITY FOR VERTEX FUNCTIONS OF SQED 21 V CONCLUSION Using Ward Takahashi Identity to present and and 4-point vertex functions as the sum of two parts First, Longitudinal part of 3-point vertex can be written in terms of complete scalar propagator while for 4-point vertex, this part is presented in terms of scalar propagators and transverse part of 3-point vertex function Second, transverse parts are not presented in term of fix components which depend on specific orders of the perturbative theory The thirst, this method can used to derived transverse part of 3-point and 4-point vertex functions at higher order of the perturbative theory REFERENCES [1] [2] [3] [4] A Bashir, Y Concha-Sanchez, R Delbourgo, Phys Rev D 76 (2007) 068009 A Bashir, Y Concha-Sanchez, R Delbourgo, Phys Rev D 80 (2009) 045007 H N Long, Basic of Particle Physics, 2006 Statistic Press P V Dong, H N Long, The Economical SU (3) × SU (3) × U (1) Gauge Model–Series of Monographs Basic Reseach, 2008 Vietnam Academy of Science and Technology Received 30-08-2011 22 H T HUNG, L T HUE, H N LONG Fig giando ... B32 = B33 = (36) WARD-TAKAHASHI IDENTITY FOR VERTEX FUNCTIONS OF SQED 21 V CONCLUSION Using Ward Takahashi Identity to present and and 4-point vertex functions as the sum of two parts First, Longitudinal... p2 τ (k , p2 , q ) (13) (14) IV WARD-TAKAHASHI IDENTITY WITH 4-POINT VERTEX FUNCTION OF SQED Ward-Takahashi identity of 4-point function relates with 3-point vertex function by: k µ Γνµ (p , k... component of 4-point vertex function at one-loop The corresponding Feynman diagrams for this function are: WARD-TAKAHASHI IDENTITY FOR VERTEX FUNCTIONS OF SQED 19 For sake of simplicity, we denote q =