Self Energy Correction in Light Front QED And Coherent State Basis
SSelf Energy Correction in Light Front QED And Coherent StateBasis
Jai D. More ∗ and Anuradha Misra † Department of Physics,University of Mumbai,Santa Cruz(E), Mumbai, India-400098 (Dated: November 10, 2018)
Abstract
We discuss the calculation of fermion self energy correction in Light Front QED using a coherentstate basis. We show that if one uses coherent state basis instead of fock basis to calculate thetransition matrix elements, the true infrared divergences in electron mass renormalization δm getcanceled up to O ( e ) in Light Front gauge. We have also verified this cancellation in Feynmangauge up to O ( e ). PACS numbers: 11.10.Ef,12.20.Ds,12.38.Bx ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] S e p . INTRODUCTION LSZ formalism of quantum field theory is based on the assumption that at large times,the dynamics of incoming and outgoing particles in a scattering process is governed by thefree Hamiltonian, i.e. the asymptotic Hamiltonian H as is the same as free Hamiltonian: H as = lim | t |→∞ H = H (1)However, it was pointed out by Kulish and Faddeev [1] that this assumption does nothold for theories in which either • long range interactions like QED are present or • the incoming and outgoing states are bound states like in QCD.Kulish and Faddeev (KF) proposed the method of asymptotic dynamics and showed thatin QED, at large times, when one takes into account the long range interaction between theincoming and outgoing states, then H as = H + V as (2) V as was shown to be non-zero in QED and was used to construct the asymptotic M¨olleroperators Ω A ± = T exp (cid:20) − i (cid:90) ∓∞ V as ( t ) dt (cid:21) which leads to the coherent states | n : coh (cid:105) = Ω A ± | n (cid:105) , as the asymptotic states, where | n (cid:105) is the n particle Fock state. It was then shown thatthe transition matrix elements evaluated between these coherent states are infra-red (IR)divergence free.In this talk, I will discuss the issue of cancellation of IR divergences in the electron massrenormalization in light front QED. 2or the sake of completeness, we state the notation followed by us [2]. Our metric tensoris g µν = − − so that the four vector is defined as x µ = ( x + , x − , x ⊥ )where x + = ( x + x ) √ , x − = ( x − x ) √ , x ⊥ = ( x , x )Four momentum is p = ( p + , p − , p ⊥ ) and the mass shell condition is p − = p ⊥ + m p + In LFFT, there are two kinds of IR divergences1) Spurious IR divergences, which are divergences arising due to k + → P − = H ≡ H + V + V + V , H = (cid:90) d x ⊥ dx − { i ξγ − ↔ ∂ − ξ + 12 ( F ) − a + ∂ − ∂ k a k } V = e (cid:90) d x ⊥ dx − ¯ ξγ µ ξa µ V = − ie (cid:90) d x ⊥ dx − dy − (cid:15) ( x − − y − )( ¯ ξa k γ k )( x ) γ + ( a j γ j ξ )( y ) V = − e (cid:90) d x ⊥ dx − dy − ( ¯ ξγ + ξ )( x ) | x − − y − | ( ¯ ξγ + ξ )( y ) ξ ( x ) and a µ ( x ) can be expanded in terms of creation and annihilation operators as ξ ( x ) = (cid:90) d p ⊥ (2 π ) / (cid:90) dp + √ p + (cid:88) s = ± [ u ( p, s ) e − i ( p + x − − p ⊥ x ⊥ ) b ( p, s, x + )+ v ( p, s ) e i ( p + x − − p ⊥ x ⊥ ) d † ( p, s, x + )] ,a µ ( x ) = (cid:90) d q ⊥ (2 π ) / (cid:90) dq + √ q + (cid:88) λ =1 , (cid:15) λµ ( q )[ e − i ( q + x − − q ⊥ x ⊥ ) a ( q, λ, x + ) + e i ( q + x − − q ⊥ x ⊥ ) a † ( q, λ, x + )] , The creation and annihilation operator satisfy { b ( p, s ) , b † ( p (cid:48) , s (cid:48) ) } = δ ( p + − p (cid:48) + ) δ ( p ⊥ − p (cid:48)⊥ ) δ ss (cid:48) = { d ( p, s ) , d † ( p (cid:48) , s (cid:48) ) } , (3)[ a ( q, λ ) , a † ( q (cid:48) , λ (cid:48) )] = δ ( q + − q (cid:48) + ) δ ( q ⊥ − q (cid:48)⊥ ) δ λλ (cid:48) . (4)The light cone time dependence of the interaction Hamiltonian is given by H I ( x + ) = V ( x + ) + V ( x + ) + V ( x + )where V ( x + ) = e (cid:88) i =1 (cid:90) dν (1) i [ e − iν (1) i x + ˜ h (1) i ( ν (1) i ) + e iν (1) i x + ˜ h (1) † i ( ν (1) i )] (5)˜ h (1) i ( ν (1) i )’s are the three point QED interaction vertices and ν (1) i is the light-front energytransferred at the vertex ˜ h (1) i . dν (1) i is the integration measure. For example, (cid:90) dν (1)1 = 1(2 π ) / (cid:90) [ dp ][ dk ] (cid:112) p + (6)and ν (1)1 = p − − k − − ( p − k ) − . The expressions for V ( x + ) and V ( x + ) can be found in Ref. [7].Following the KF method, H as is evaluated by taking the limit x + → ∞ in exp [ − iν (1) i x + ],which contains the time dependence of this term in the interaction Hamiltonian H int . If4 (1) i → H int does not vanish in large x + limit. One can then use KF method to obtain the asymptotic Hamiltonian and to constructthe asymptotic M¨oller operator which leads to the coherent states.Here, I will illustrate the construction of asymptotic Hamiltonian using the 3-point vertexonly. As seen from the light-cone time dependence of V ( x + ), the non-zero contribution toasymptotic interaction Hamiltonian comes from the regions in which ν (1) i ’s vanish. It hasbeen shown [3] that out of the four light-cone energy differences, ν (1) i ’s in Eq. (5), two cannever be zero. A convenient way to define the asymptotic region is by requiring k ⊥ < k + ∆ p + k + < p + ∆ m . where ∆ = p + ∆ E . ∆ E is an energy cutoff which may be choosen to be the experimentalresolution. One can verify that in this region ν (1) i = p − − k − − ( p − k ) − →
0. Therefore, wecan define V as ( x + ) as V as ( x + ) = e (cid:88) i =1 , (cid:90) dν (1) i Θ ∆ ( k )[ e − iν (1) i x + ˜ h (1) i ( ν (1) i ) + e iν (1) i x + ˜ h (1) † i ( ν (1) i )] (7)where Θ ∆ ( k ) is given by Θ ∆ ( k ) = θ (cid:18) k + ∆ p + − k ⊥ (cid:19) θ (cid:18) p + ∆ m − k + (cid:19) Taking the limit, k + → k ⊥ →
0, in all slowly varying functions of k. Performing the x + integration and neglecting the 4-point instantaneous term, one obtains the asymptoticM¨oller operator Ω A ± which gives the asymptotic statesΩ A ± | n : p i (cid:105) = exp (cid:20) − e (cid:90) dp + d p ⊥ (cid:88) λ =1 , [ d k ][ f ( k, λ : p ) a † ( k, λ ) − f ∗ ( k, λ : p ) a ( k, λ )] (cid:21) ρ ( p ) | n : p i (cid:105) (8)where f ( k, λ : p ) = p µ (cid:15) µλ ( k ) p · k θ (cid:18) k + ∆ p + − k ⊥ (cid:19) θ (cid:18) p + ∆ m − k + (cid:19) , (9)Taking into account, the asymptotic limit of instantaneous interaction also, one obtains [7]Ω A ± | n : p i (cid:105) = exp (cid:20) − e (cid:90) dp + d p ⊥ (cid:88) λ =1 , [ d k ][ f ( k, λ : p ) a † ( k, λ ) − f ∗ ( k, λ : p ) a ( k, λ )]+ e (cid:90) dp + d p ⊥ (cid:88) λ ,λ [ d k ][ d k ][ g ( k , k , λ , λ : p ) a † ( k , λ ) a ( k , λ ) − g ( k , k , λ , λ : p ) a ( k , λ ) a † ( k , λ )] (cid:21) ρ ( p ) | n : p i (cid:105) (10)5here g ( k , k , λ , λ : p ) = − p + p · k − p · k + k · k δ ( k − k ) g ( k , k , λ , λ : p ) = 4 p + p · k − p · k − k · k δ ( k − k ) (11)We have used light-cone time ordered perturbation theory to calculate the transitionmatrix elements in both Fock basis and the coherent state basis. The transition matrix isgiven by the perturbative expansion T = V + V p − − H V + · · · Conventionally, electron mass shift δm is obtained by calculating the matrix element of thisseries, T pp , between the initial and final one electron electron Fock states | p, s (cid:105) : δm = p + (cid:88) s T pp (12)Expanding T pp in powers of e as T pp = T (1) + T (2) + · · · (13)one gets T ( n ) , the O ( e n ) contribution to fermion self energy correction. II. MASS RENORMALIZATION UP TO O ( e ) In O ( e ), fermion self energy correction is represented by the diagrams shown in Fig. 1.Fig. 1(b) is a tree level diagram and does not have any vanishing denominator. Neglectingthe contribution of Fig. 1(b) ( a )( p, s ) ( p, σ ) k ( p, s ) ( p, σ )( b ) k FIG. 1: Diagrams for O ( e ) self energy correction in fock basis corresponding to T in LF gauge T (1) pp ≡ T (1) ( p, p ) = (cid:104) p, s | V p − − H V | p, s (cid:105) (14)6ollowing the standard procedure in Fock basis, we obtain δm [7],( δm a ) IR = − e (2 π ) (cid:90) d k ⊥ (cid:90) dk +1 k +1 ( p · (cid:15) ( k )) ( p · k ) (15)We have performed this calculation in Feynman gauge also. QED Lagrangian in Feynmangauge with additional PV fields is given by [8]: L = (cid:88) i =0 ( − i (cid:20) − F µνi F i,µν + 12 µ i A µi A iµ −
12 ( ∂ µ A iµ ) (cid:21) + (cid:88) i =0 ( − i ¯ ψ i ( iγ µ ∂ µ − m i ) ψ i − e ¯ ψγ µ ψA µ . (16) ( a )( p, s ) ( p, s ) k ( p, s ) ( p, s )( b ) k FIG. 2: Diagrams for O ( e ) self energy correction in fock basis in Feynman gauge. In (a), wavy linecorresponds to physical photon ( i = 0) and in (b) curly line corresponds to PV photon ( i = 1 , Here, there are additional contributions to self energy correction due to diagram inFig. 2(b), in which the curly line denotes the massive PV field. Note that there are noinstantaneous diagrams in Feynman gauge as the non-local terms in the Hamiltonianget canceled by similar terms for PV fields. Moreover, for massive PV photons thereis no vanishing denominator. Thus, the diagram in Fig. 2(b) does not contribute to IRdivergences. To conclude, up to one loop order, there is only one diagram that can give IRdivergences in both LF and Feynman gauge. k ( p, s ) ( p, σ ) k ( p, s ) ( p, σ ) FIG. 3: Additional diagrams in coherent state basis for O ( e ) self energy correction T (cid:48) ( p, p ) = (cid:104) p, s : f ( p ) | V | p, s : f ( p ) (cid:105) (17)contains O ( e ) terms. In particular, in O ( e ), one can also get a contribution from diagramsin Fig. 3(a), which represents a situation where a soft photon accompanying the incomingparticle is absorbed and Fig. 3(b) which represents a situation where a soft photon is emitted.However, the two particle states are indistinguishable from the single particle state due tofinite experimental resolution. The contribution of these coherent state diagrams is foundto be T (cid:48) ( p, p ) = e (2 π ) (cid:90) d k ⊥ p + (cid:90) dk +1 k +1 × u ( p, s (cid:48) ) (cid:15) / λ ( k ) u ( p, s ) f ( k , λ : p ) (18)where the form of f ( k, λ : p ) in Eq. 9 ensures that the integrals are performed only over asmall region around k + = 0, k ⊥ = 0. Using Eq. (12) and simplifying further we obtain( δm ) (cid:48) = e (2 π ) (cid:90) d k ⊥ (cid:90) dk +1 k +1 ( p · (cid:15) ( k )) Θ ∆ ( k ) p · k (19)Adding the IR divergent contributions arising due to p.k → O ( e ). III. MASS RENORMALIZATION UP TO O ( e ) In O ( e ), the contribution to self energy correction in the Fock basis comes from diagramswhich contain • only 3-point vertices which is represented by Fig. 4. • both 3-point and 4-point vertices which is shown in Fig. 5. • only 4-point vertices which is represented by Fig 6.Transition matrix element for O ( e ) correction to self energy is given by T (2) = T + T + T + T + T ( p, σ )( p, s ) ( a ) k ( p, σ )( p, s ) ( b ) k k ( p, σ )( p, s ) ( c ) k k FIG. 4: Diagrams for O ( e ) self energy correction corresponding to T in fock basis in LF gauge k ( p,σ )( p,s ) ( a ) k ( p,σ )( p,s ) ( b ) k k ( p,σ )( p,s ) ( c ) k k ( p,σ )( p,s ) ( a ) k k ( p,σ )( p,s ) ( b ) k k k ( p,σ )( p,s ) ( a ) k k ( p,σ )( p,s ) ( a ) k ( p,σ )( p,s ) ( b ) k k ( p,σ )( p,s ) ( c ) k k FIG. 5: Diagrams for O ( e ) self energy correction in fock basis corresponding to T , T and T respectively. ( p,σ )( p,s ) ( a ) k ( p,σ )( p,s ) ( b ) k k ( p,σ )( p,s ) ( c ) k k FIG. 6: Diagrams for O ( e ) self energy correction in fock basis corresponding to T , where T = (cid:104) p, s | V p − − H V p − − H V p − − H V | p, s (cid:105) (20) T = (cid:104) p, s | V p − − H V p − − H V | p, s (cid:105) (21) T = (cid:104) p, s | V p − − H V p − − H V | p, s (cid:105) (22) T = (cid:104) p, s | V p − − H V p − − H V | p, s (cid:105) (23) T = (cid:104) p, s | V p − − H V | p, s (cid:105) (24)We have calculated these diagrams using light-cone time ordered perturbation theory andhave shown that there are IR divergences present in the Fock basis in Figs. 4 and 5.Transition matrix element for O ( e ) correction to self energy due to the additional contri-bution in coherent state basis is given by T (2) + T (cid:48) + T (cid:48) + T (cid:48) + T (cid:48) a )( p,s ) ( p,σ ) k k ( b ) k k ( p,s ) ( p,σ )( c ) k k ( p,s ) ( p,σ ) k k ( d )( p,s ) ( p,σ )( e )( p,s ) ( p,σ ) k k ( f )( p,s ) ( p,σ ) k k FIG. 7: Additional diagrams in coherent state basis for O ( e ) self energy correction correspondingto T . ( p,σ )( p,s ) ( a ) k k ( p,σ )( p,s ) ( b ) k k ( p,σ )( p,s ) ( c ) k k ( p,σ )( p,s ) ( d ) k k FIG. 8: Additional diagrams in coherent state basis for O ( e ) self energy correction correspondingto T . where T (cid:48) = (cid:104) p, s : f ( p ) | V p − − H V p − − H V | p, s : f ( p ) (cid:105) , (25) T (cid:48) = (cid:104) p, s : f ( p ) | V p − − H V | p, s : f ( p ) (cid:105) , (26) T (cid:48) = (cid:104) p, s : f ( p ) | V p − − H V | p, s : f ( p ) (cid:105) + (cid:104) p, s : f ( p ) | V p − − H V | p, s : f ( p ) (cid:105) (27) T (cid:48) = (cid:104) p, s : f ( p ) | V | p, s : f ( p ) (cid:105) (28)In Ref. [7], it was shown that in O ( e ), the IR divergences get canceled if coherent state basisis used in LF gauge. In Section 2, we have verified this cancellation in Feynman gauge alsoup to O ( e ) thereby establishing the usefulness of the coherent state basis in both LF and11eynman gauge. We have also verified that IR divergences cancel up to O ( e ) in Feynmangauge. Details can be found in Ref. [9]. ( p,σ )( p,s ) ( p,σ )( p,s )( a ) ( b ) k k k k k k k ( p,σ )( p,s ) k ( p,σ )( p,s )( c ) ( d ) ( p,σ )( p,s ) ( f ) k k ( p,σ )( p,s ) ( e ) k k k k ( p,σ )( p,s ) ( h ) k k ( p,σ )( p,s ) ( g ) ( p,σ )( p,s ) ( p,σ )( p,s )( i ) k k k k ( p,σ )( p,s ) k k ( p,σ )( p,s ) k k FIG. 9: Additional diagrams in coherent state basis for O ( e ) self energy correction correspondingto T . ( p,σ )( p,s ) ( a ) k k ( p,σ )( p,s ) ( b ) k k ( p,σ )( p,s ) ( c ) k k ( p,σ )( p,s ) ( d ) k k FIG. 10: Additional diagrams in coherent state basis for O ( e ) self energy correction correspondingto T . V. CONCLUSION
We have shown that the true IR divergences get canceled up to O ( e ) when coherent statebasis is used to calculate the transition matrix elements in lepton self energy calculation inlight-front QED in LF gauge as well as in Feynman gauge. The cancellation of IR divergencesbetween real and virtual processes is known to hold in equal time QED to all orders. Itwould be interesting to verify this all order cancellation in LFQED. The present work is aninitial step in this direction. It is well known that IR divergences do not cancel in QCDin higher orders. This is related to the fact that the asymptotic states are bound states.Connection between asymptotic dynamics and IR divergences can possibly be exploited toconstruct an artificial potential that may be used in bound state calculations in LFQCD. ACKNOWLEDGEMENTS
I would like to thank University of Delhi and organizers of LC2012 for travel support.Part of this work was done under a DST sponsored project No. SR/S2/HEP-17/2006. [1] P.P. Kulish and L.D. Faddeev, Theor. Math. Phys. , 745 (1970).[2] D. Mustaki, S. Pinsky, J. Shigemitsu and K.G. Wilson, Phys. Rev. D , 3411 (1991).[3] Anuradha Misra, Phys. Rev. D , 4088 (1994).[4] Anuradha Misra, Phys. Rev. D , 5874 (1996).[5] Anuradha Misra, Phys. Rev. D , 125017 (2000).[6] Anuradha Misra, Few-Body Systems 36, 201-204 (2005).[7] Jai D. More and Anuradha Misra, Phys. Rev. D , 065037 (2012).[8] S. S. Chabysheva and J. R. Hiller, Phys. Rev. D , 034001 (2011).[9] Jai D. More and Anuradha Misra, Phys. Rev. D , 085035 (2013)., 085035 (2013).