Functional Integration and High Energy Scattering of Particles with Anomalous Magnetic Moments in Quantum Field Theory
aa r X i v : . [ h e p - t h ] D ec FUNCTIONAL INTEGRATION AND HIGH ENERGY SCATTERING OFPARTICLES WITH ANOMALOUS MAGNETIC MOMENTS IN QUANTUMFIELD THEORY
Nguyen Suan Han a,b, , Le Hai Yen a , Nguyen Nhu Xuan ca Department of Theoretical Physics,Vietnam National University, Hanoi, Vietnam. b The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. c Department of Theoretical Physics, Le Qui Don Technical University, Hanoi, Vietnam.
Abstract
The functional integration method is used for studying the scattering of a scalar pion onnucleon with the anomalous magnetic moment in the framework of nonrenomalizable quantumfield theory. In the asymptotic region s → ∞ , | t | ≪ s the representation of eikonal type forthe amplitude of pion-nucleon scattering is obtained. The anomalous magnetic moment leadsto additional terms in the amplitude which describe the spin flips in the scattering process. Itis shown that the renormalization problem does not arise in the asymptotic s → ∞ since theunrenomalized divergences disappear in this approximation. Coulomb interference is consideredas an application. Keywords:
Quantum scattering; anomalous magnetic moment. Email: [email protected]
Introduction
The eikonal approximation for the scattering amplitude of high-energy particles in quantumfield theory including quantum gravity has been investigated by many authors using various ap-proaches [1 − − µ →
0. It is pointed out that the eikonal approximationworks well in a wide energy range [21 − , , − πN elastic scattering amplitude expressed in the form of the functional integrals.To estimate the functional integrals we use the straight line path approximation, based on theidea of rectilinear paths of interacting particles of asymptotically high energies and small mo-mentum transfers. The third section is devoted to investigating the asymptotic behavior of thisamplitude in the limit of high energies s → ∞ , | t | ≪ s , and we obtained an eikonal or Glauberrepresentation of the scattering amplitude. As an application of the eikonal formula obtained infourth section, we consider the Coulomb interference in the scattering of charged hadrons. Here,we find a formula for the phase difference; this is a generalization of the Bether formula in theframework of relativistic quantum field theory. Finally, concluding remarks are presented.2 Construction of the two-particle scattering amplitude
We consider the scattering of a scalar particle (pion π ) on a Dirac particle with anomalous mag-netic moment (nucleon N ) at high energies and at fixed transfers in quantum field theory. Toconstruct the representation of the scattering in the framework of the functional approach wefirst find the two-particle Green’s function, once the Green’s function is obtained we consider themass respective to the external ends of the two particle lines.Using the method of variational derivatives we shall determine the two particles Green’sfunction G ( p , p | q , q ) by the following formula: G ( p , p | q , q ) = exp " i Z d kD µν ( k ) δ δA µ ( k ) δA ν ( − k ) G ( p , q | A ) G ( p , q | A ) .S ( A ) (cid:12)(cid:12)(cid:12) A =0 , (2 . S ( A ) is the vacuum expectation of the S matrix in the given external field A . Forsimplicity, we shall henceforth ignore vacuum polarization effects and also the contributions ofdiagrams containing closed nucleon loops; G ( p , q | A )-the Fourier of the Green’s function ( A. G ( p , q | A ) = i Z s dse i ( p − m ) s Z d xe i ( p − q ) x Z [ δ ν ] s exp[ ie Z s J µ A µ ] , (2 . R J A = R J µ ( z ) A µ ( z ), and J µ ( z ) is the current of the particle 1 definedby J µ ( z ) = 2 Z s dξν µ ( ξ ) δ ( z − x i + 2 Z ξ [ ν i ( η ) + p i ] dη ) . (2 . G ( p , q | A ) and the squaredGreen’s functions G ( p, q | A ) are identical [4], in eq. (2.1) we thus use the latter in eq. ( A. G ( p , q | A ) = i Z s e i ( p − m ) s ds Z d xe i ( p − q ) x T γ Z [ δ ν ] s exp { ie Z s J µ A µ ( x ) } , (2 . T γ is the symbol of ordering the γ µ matrices with respect to the ordering index ξ , and J µ ( z ) is the current of particle 2 defined by J µ ( z ) = 2 Z s dξ [ ν µ ( ξ ) + 12 σ µν ( ξ ) i∂ ν ] δ ( z − x i + 2 Z ξ [ ν i ( η ) + p ] dη ) . (2 . . Substituting (2 . , (2 .
4) into (2 .
1) and performing variational derivatives, for the two-particleGreen’s function we find the following expression: G ( p , p | q , q ) = i Y i =1 , Z ∞ ds i e i ( p i − m i ) s i Z [ δ ν i ] s i Z d x i e i ( p i − q i ) x i exp[ − ie Z D ( J + J ) ] ! , (2 . For simplicity, pion will be regarded as particle 1 and nucleon as particle 2.
J DJ = Z dz dz J µ ( z ) D µν ( z − z ) J ν ( z ) . Expanding expression (2 .
6) with respect to the coupling constant e and taking the functionalintegrals with respect to ν i ( η ), we obtain the well-known series of perturbation theory for thetwo-particle Green’s function. The term in exponent (2 .
6) we can rewrite in the following form: − ie Z D ( J + J ) = − ie Z DJ J − ie Z DJ − ie Z DJ , (2 . .
7) corresponds to the one-photon exchange between thetwo-particle and the remainder lead to radiative corrections to the lines of the two-particles.The scattering amplitude of two particles is expressed in the two particles Green’s functionby equation: i (2 π ) δ (4) ( p + p − q − q ) T ( p , p | q , q ) == 12 m u ( q ) " lim p i ,q i → m i ( p i − m i ) G ( p , p | q , q )( q i − m i ) u ( p ) , (2 . u ( q ) and u ( p ) on the mass shell satisfy the Dirac equation and the normalizationcondition u ( q ) u ( p ) = 2 m .The transition to the mass shell p i ; q i → m i ; calls for separating from formula (2 .
8) the poleterms ( p i − m i ) − and ( q i − m i ) − which cancel the factors ( p i − m i ) and ( q i − m i ). In per-turbation theory this compensation is obvious, since the Green’s function is sought by means ofmethods other than perturbation theory, the separation of the terms entails certain difficulties.We shall be interested in the structure of scattering amplitude as a whole, therefore the devel-opment of a correct procedure for going to the mass shell in the general case is very important.Many approximate methods that are reasonable from the physical point of view when used be-fore the transition on the mass shell , shift the positions of the pole of the Green’s function andrender the procedure of finding the scattering amplitude mathematically incorrect. In presentpaper we shall use a method for separating the poles of the Green’s functions that generalizes themethod introduced in Ref. [30] to finding the scattering amplitude in a model of scalar nucleoninteracting with scalar meson field, in which the contributions of closed nucleon loops are ignored.Substituting (2 .
6) into (2 .
8) we get(2 π ) δ ( p + p − q − q ) iT ( p , p | q , q )= 12 m u ( q ) " lim p i ,q i → m i Y i =1 , ( p i − m i )( q i − m i ) Z d x i e i ( p i − q i ) x i Z ∞ ds i Z ∞ dξe i ( p i − m i ) ! DJ J Z dλ exp[ − ie λ Z DJ J ] u ( p ) . (2 . .
9) we employ the operator of subtracting unity in the formula (2 .
9) from theexponent function containing the D-function in its argument in accordance with e − ie R DJ J − − ie Z dλDJ J e − iλ R DJ J . This corresponds to eliminating from the Green’s function the terms describing the propaga-tion of two noninteracting particles. Taking into account the identity: Y k =1 , Z ∞ ds k Z s k dξ k ... → Y k =1 , Z ∞ dξ k Z ∞ ξ ds k ... and making a change of the ordinary and the functional variables s i → s i + ξ i ; i = 1 , ,x i → x i − Z ξ i [ p + ν ( η )] dη,ν i ( η ) → ν i ( η − ξ ) − ( p − q ) θ ( η − s i ) , we transform eq. (2 .
9) into(2 π ) δ ( p + p − q − q ) iT ( p , p | q , q ) == 12 m u ( q ) " lim p i ,q i → m i Y i =1 , ( p i − m i )( q i − m i ) Z d x i e i ( p i − q i ) x i Z ∞ dξ i e i ( p i − m i ) ξ Z ∞ ds i e i ( q i − m i ) s i ! Z [ δ ν ] s ξ Z [ δ ν ] s ξ e DJ J Z dλ exp[ − ie λ Z DJ J ] u ( p ) . (2 . .
10) will be omitted. We now note that the integrals with respect to s i and ξ i give factors ( p i − m i ) − and ( q i − m i ) − ; i = 1 ,
2. Therefore, in eq. (2 .
10) we can go over themass shell with respect to the external lines of the particle using the relations [31] lim a,ε → ia Z ∞ e ias − ǫ f ( s ) ! = f ( ∞ ) , which holds for any finite function f ( s ). By means of the substitutions x = ( y + x ) / x = ( y − x ) / .
10) and performing the integration with respect to dy we can separateout the δ -function of the conservation of the four-momentum δ ( p + p − q − q ). As a result,the scattering amplitude takes the form 5 ( p , p | q , q ) = 12 m u ( q ) " e Y i =1 , Z [ δ ν i ] ∞−∞ Z d xe i ( p − q ) x × [ p + q + 2 ν (0)] D µν ( x )[ p + q + 2 ν (0)] ν Z dλ exp[ − ie λ Z DJ J ] u ( p ) , (2 . J µ ( k ; p , q | ν ) = 2 Z ∞−∞ dξ [ p θ ( ξ ) + q θ ( − ξ ) + ν ( η )] µ exp n ik [ p ξθ ( ξ ) + q ξθ ( − ξ ) + Z ξ ν ( η ) dη ] o ,J µ ( k ; p , q | ν ) = 2 Z ∞−∞ dξ n [ p θ ( ξ ) + q θ ( − ξ ) + ν ( η )] µ + 12 σ µν ( ξ ) i∂ ν o × exp n ik [ p ξθ ( ξ ) + q ξθ ( − ξ ) + Z ξ ν ( η ) dη ] o . Here, exp[ − ie λ R DJ J ] describes virtual-photon exchange among the scattering particles.The integration with respect to dλ ensures subtraction of the contribution of the freely propagat-ing particles from the matrix element. By going over to mass shell of external two particle Green’sfunction, we obtain an exact closed representation for the ”pion-nucleon” elastic scattering am-plitude, expressed in the form of the double functional integrals. We would like to emphasizethat eq. (2 .
11) can be applied for different ranges of energy.
The important point in our method is that the functional integrals with respect to δ ν are calcu-lated by the straight-line path approximation [2 , D -functions in (2 . .
11) in this approximation takes the form T ( p , p | q , q ) =12 m u ( q ) " e Z d xe i ( p − q ) x [ p + q ] D µν ( x )[ p + q ] ν Z dλ exp[ − ie λ Z DJ J ] u ( p ) . (3 . −→ p = −−→ p = −→ p and we direct the z-axis along the momentum −→ p : p = ( p , , , p = p z ); p = ( p , , , − p ) ,s = ( p + p ) = 4 p ; p = p = p , t = ( p − q ) = ( p − q ) ; (3 . db and db z in (3 .
1) we obtain for the scattering amplitude T ( s, t ) = − is u ( q )2 m Z d −→ b ⊥ e i ∆ −→ b ⊥ × T γ exp { ie Z ∞−∞ dτ Z ∞−∞ dτ J µ ( b p µ ) D µν ( b τ τ ) J ν ( b p , γ ( τ )) } − ! u ( p ) , (3 . b p µi = p µi / | p | , τ i = 2 | p | ξ i , ( i = 1 , b τ τ = −→ b ⊥ − p τ + p τ .Let us consider the asymptotic behavior of the elastic forward amplitude of the two-particles(3 .
1) in the region s → ∞ , | t | ≪ s . In this region, spinors u ( p ) and u ( p ), which are solutions ofthe Dirac equation [25] u ( p ) = ~σ~p | p | ! √ mψ p , ¯ u ( q ) = ¯ ψ q √ m (cid:18) , ~σ~p | p | (cid:19) , | ~p | ≈ | ~q | , (3 . ψ p and ψ q are ordinary two-component spinors.Using the expansion of J µ [ b p , γ ( τ )] with respect to the z component of the momentum andsubstituting (3 .
4) into (3 . T ( s, t ) = − is ¯ ψ q Z d~b ⊥ e i ∆ ~b ⊥ h e iχ ( b ) Γ ( b ) − i ψ p , (3 . χ ( b ) is the phase corresponding to the Coulomb interaction. This phase is determined by χ ( b ) = e (2 π ) Z d~k ⊥ e − i~k ⊥ ~b ⊥ µ + ~k ⊥ = e π K (cid:16) µ | (cid:12)(cid:12)(cid:12) ~b ⊥ (cid:12)(cid:12)(cid:12)(cid:17) , (3 . K (cid:16) µ (cid:12)(cid:12)(cid:12) ~b ⊥ (cid:12)(cid:12)(cid:12)(cid:17) - is the MacDonald function of zeroth order, and the expression Γ ( b ) is equalto Γ ( b ) = 12 (1 , − σ z ) T τ exp ( − iκ Z ∞−∞ dτ Z ∞−∞ dτ h ˆ p µ ~γ ⊥ ( τ ) × ~∂ ⊥ D cµρ ( b τ τ ) ˆ p ρ −− ˆ p µ (cid:20) γ z ( τ ) + γ ( τ ) p z p (cid:21) × (cid:2) ∂ z D cµ ( b τ τ ) − ∂ D cµz ( b τ τ ) (cid:3)) (cid:18) − σ z (cid:19) . (3 . h γ z + γ p z p i is actually withrespect to h γ z + γ p z p i = − m p , since (1 , − σ z ) h γ z + γ p z p i (1 , − σ z ) = 0. Therefore, the secondterm in the argument of the exponent in (3 .
7) can be ignored altogether. Thus, we haveΓ ( b ) = 12 (1 , σ z ) T τ exp (cid:20) − eκ Z ∞−∞ dτ Z ∞−∞ dτ ~γ ⊥ ( τ ) ~∂ ⊥ D ( b τ τ ) (cid:21) (cid:18) − σ z (cid:19) . (3 . h ~γ ⊥ ( τ ) ~∂ ⊥ D c ( b τ τ ) , ~γ ⊥ ( τ ′ ) ~∂ ⊥ D c (cid:16) b τ τ ′ (cid:17)i(cid:12)(cid:12)(cid:12) τ ′ = τ , (3 . ~γ ⊥ ( τ ) matrix in (3 .
8) does not depend on the ordering parameter τ and the T τ orderingexponential is equal to the ordinary exponential :Γ ( b ) = 12 (1 , σ z ) exp (cid:20) − eκ~γ ⊥ ~∂ ⊥ Z ∞−∞ dτ Z ∞−∞ dτ D ( b τ τ ) (cid:21) (cid:18) − σ z (cid:19) , σ z ) exp h − eκ π ~γ ⊥ ~∂ ⊥ K (cid:16) µ | ~b ⊥ (cid:17)i (cid:18) − σ z (cid:19) (3 . ~b ⊥ = ~ρ = ρ~n , ~n = ( cosφ, sinφ ), φ is the azimuthal angle inthe plane ( x, y ). Remembering further that[ ~n × ~σ ] z = − σ x sin ϕ + σ cos ϕ, [ ~n × ~σ ] z = 1 , (3 . ( b ) = exp { i [ ~n × ~σ ] z χ ( ~ρ ) } (3 . χ ( ρ ) is determined by χ ( ~ρ ) = eκ π ∂ ρ K ( µ | ρ | ) (3 . πN scattering amplitude T ( s, t ) = − is ¯ ψ q Z d~b ⊥ e i ∆ ~b ⊥ { exp [ iχ ( b ) + i ( ~n × ~σ ) z χ ( b )] − } ψ p . (3 . .
14) with respect to the angular variable [32], we obtain the amplitude T ( s, t ) = ¯ ψ q [ f ( s, ∆) + iσ y f ( s, ∆)] ψ p , (3 . f ( s, ∆) and f ( s, ∆) describe processes with and without spin flip, respectively, and theyare given by f ( s, ∆) = − π ? s Z ∞ ρdρJ (∆ ρ ) (cid:2) e iχ cos χ − (cid:3) , (3 . f ( s, ∆) = 4 πs Z ∞ ρdρJ (∆ ρ ) sin χ . (3 . . − (3 .
17) are finite, and therefore the renormaliza-tion problems does not arise in out approximation in the limit s → ∞ . Coulomb interference for particles with anomalous magnetic moment was considered for the firsttime in Ref. [39], in which the amplitude was actually only in the first Born approximation inthe Coulomb interaction. The relativistic eikonal approximation was used for the first time tocalculate Coulomb interference without allowance for spin [34]. It is interesting to use our resultsto consider Coulomb interference [33 −
39] in the scattering of the charges hadrons πN . The Scattering amplitude T ( s, t ) in c.m.s can be normalized by the expression σ tot = ImT ( s,t =0) s , dσd Ω = | T ( s,t ) | π s . χ em ( b ) → χ em ( b ) + χ h ( b ) T ( s, t ) = − is ¯ ψ q Z d~b ⊥ e i ∆ ~b ⊥ (exp [ iχ em ( b ) + iχ h ( b )] − ψ p , (4 . χ em ( b ) = χ ( b ) + i [ −→ n × −→ σ ] z χ ( b ), is eikonal phase that corresponds to the nuclear inter-action. For the following discussion, the Eq. (4 .
1) is rewritten in the form T ( s, t ) = T em ( s, t ) + T eh ( s, t ) , (4 . T em is the part of the scattering amplitude due to the electromagnetic interaction and de-termined by Eq. (3 .
14) or formulas (3 . − (3 . T eh ( s, t ) is the interference electromagnetichadron part of the scattering amplitude T eh ( s, t ) = e ϕ t T h ( s, t ) = − is ¯ ψ q Z d~b ⊥ e i ∆ ~b ⊥ (cid:16) e iχ h ( b ) − (cid:17) e iχ em ( b ) ψ p , (4 . φ t is the sum of the phase of the Coulomb and nuclear interaction, T h ( s, t ) is the purelynuclear amplitude obtained in the absence of an electromagnetic interaction. In the region ofhigh energies s → ∞ , | t | /s →
0, it is sufficient to retain only the terms linear in κ because κ issmall in the all the following calculations. Integrating in the expression (3 .
15) , we obtain T em ( s, t ) = 8 παs ∆ Γ(1 − iα )Γ(1 + iα ) exp ϕ em ¯ ψ q h − i κe σ y ∆ i ψ p , ϕ em = ie (cid:20) ln ∆ µ − γ (cid:21) , (4 . α = e / π , µ is the photon mass, and γ = 0 , ... is the Euler constant. Calculating T ch ( s, t ) we use the standard formulas T h ( s, t ) = ¯ ψ q f h ( s, t = 0) ψ p e R t , t = − ∆ (4 . f h ( s, t = 0) = s σ tot (cid:20) i + Ref h ( s, t = 0) Imf h ( s, t = 0) (cid:21) . (4 . . T emh ( s, t ) = T h ( s, t ) h eκ π σ y ∆ i exp ϕ t , ϕ t = − ie h ln ( Rµ ) + 2 γ i . (4 . T eh and T c ( s, t ) we find theexpression ϕ = ϕ t − ϕ c = − iα ln( R ∆) . (4 . psin θ ≃ pθ , p is the relativistic momentum in cms), thephase difference is equal to φ = 2 iαln Rpθ . This result is practically the same as Bethe’s [33]9
Conclusions
In the framework of the functional integration, a method is proposed for studying the scatteringof a scalar pion on nucleon with an anomalous magnetic moment in quantum field theory. Weobtained an eikonal representation of the scattering amplitude in the asymptotic region s →∞ , | t |≪ s . Allowance for the anomalous magnetic moment leads to the additional terms inthe amplitude that do not vanish as s → ∞ , and these describe spin flips of the particles inthe scattering process. It is shown that in the limit s → ∞ in the eikonal approximation therenormalization problem does not arise since the unrenomalized divergences disappear in thisapproximation. As an application of the eikonal formula obtained, we considered the Coulombinterference in the scattering of charged hadrons, and we found a formula for the phase difference,which generalizes the Bethe’s formula in the framework of relativistic quantum field theory. Acknowledgments.
We would like to express gratitude to Profs. B.M. Barbashov, A.V.Efremov,V.N. Pervushin for useful discussions. N.S.H. is also indebted to Profs. Randjbar-Daemi and Gi-anCarlo Ghirardi for support during my stay at the Abdus Salam ICTP in Trieste. This workwas supported in part by the Abdus Salam ICTP , JINR Dubna, TRIGA and Vietnam NationalUniversity under Contract QG.TD.10.02.
Appendix: The Green’s functions in the form of a functional in-tegral [40]
In this appendix we find the representation of the Green’s functions of the Klein-Gordon equa-tion and the Dirac equation for single particles in an external electromagnetic field A µ ( x ), ∂A µ ( x ) /∂x µ = 0 in the form of a functional integral. Let us consider the Klein-Gordon equationfor the Green’ function [( i∂ µ + eA µ ( x )) − m ] G ( x, y | A ) = − δ ( x − y ) . ( A. A.
1) in an operator form G ( x, y | A ) = i Z ∞ dξ exp (cid:26) i Z s ( i∂ µ ( ξ ) + eA µ ( x, ξ )) − im (cid:27) δ ( x − y ) , ( A. A. ∂ µ ( x, ξ ) and A µ ( x, ξ ) is considered as T ξ -exponent, where the ordering subscript ξ has meaning of propertime divided by mass m . All operators in ( A.
2) are assumed to be commuting functions thatdepend on the parameter ξ . The exponent in eq. ( A.
2) is quadratic in the differential operator ∂ µ .However, the transition from T ξ -exponent to an ordinary operator expression (”disentangling” thedifferentiation operators in the argument of the exponential function by terminology of Feynman Here we use all the notations presented in Ref. [4] ∂ µ ( x, ξ ) in eq. ( A.
2) by using the following formaltransformationexp (cid:26) i Z s dξ ( i∂ µ ( ξ )+ eA µ ( x, ξ )) (cid:27) = C Z δ ν exp (cid:26) − i Z s ν µ ( ξ ) dξ +2 i Z s (cid:20) i∂ µ ( ξ )+ eA µ ( x, ξ ) (cid:21)(cid:27) . ( A. A.
3) is taken in the space of 4-dimensionalfunction ν µ ( ξ ) with a Gaussian measure. The constant C µ is defined by the condition: C µ Z δ ν µ exp (cid:26) − i Z ν µ ( ξ ) dξ (cid:27) = 1 . ( A. A.
3) into ( A. (cid:20) i R s ν µ ( ξ ) ∂ µ ( ξ ) (cid:21) can be ”disentangled”and we can find a solution in the form of the functional integral: G ( x, y | A ) = i Z s dse − im s Z [ δ ν ] s exp (cid:20) ie Z s ν µ ( ξ ) A µ ( x − Z sξ ν ( η ) dη ) (cid:21) δ ( x − y − Z sξ ν ( η ) dη ) , ( A. δ ν ] s s = δ exp[ − i R s s ν µ ( η ) dη ]Π η d η R δ exp[ − i R s s ν µ ( η ) dη ]Π η d η , and [ δ ν ] s s is volume element of the functional space of the four-dimensional functions ν µ ( η )defined in the interval s ≤ η ≤ s .The expression for the Fourier transform of the Green’s function ( A.
5) takes the form. G ( p, q | A ) = Z d xd yG ( x, y | A ) = i Z s dξe i ( p − m ) s Z d xe i ( p − q ) x Z [ δ ν ] s exp[ ie Z s J µ A µ ] , ( A. R J A = R J µ ( z ) A µ ( z ), and J µ ( z ) is the current of the particle 1defined by J µ ( z ) = 2 Z s ν µ ( ξ ) δ ( z − x i + 2 p i ξ + 2 Z ξ ν i ( η ) dη ) . ( A. iγ µ ∂ µ − m + eγ µ A µ ( x )] G ( x, y | A ) = − δ ( x − y ) . ( A. G ( x, y | A )11 ( x, y | A ) = [ iγ µ ∂ µ + m + γ µ A µ ( x )] G ( x, y | A ) , ( A. G ( x, y | A ) satisfies[( i∂ µ + eA µ ( x )) − m + eσ µν ∂ µ A ν ( x )] G ( x, y | A ) = − δ ( x − y ) . ( A. A.
2) and ( A. σ µν related to spin ofparticle 2 G ( x, y | A ) = i Z s e − im s T γ Z [ δ ν ] s exp { ie Z s J µ A µ ( x ) } δ ( x − y − Z sξ ν ( η ) dη ) , ( A. T γ is the symbol of ordering the γ µ matrices with respect to the ordering index ξ , and J µ ( z ) is the current of the particle 2 defined by J µ ( z ) = 2 Z s [ ν µ ( ξ ) + 12 σ µν ( ξ ) i∂ ν ] δ ( z − x i + 2 p i ξ + 2 Z ξ ν i ( η ) dη ) . ( A. σ µν depends on ξ as an orderingindex, the solution of eq. (A.9) must contain γ ξ , therefore, T ξ remains in eq. (A.11). References