aa r X i v : . [ nu c l - t h ] S e p The Self-energy of Nucleon for the Anomalous Magnetic MomentSusumu Kinpara
National Institute of Radiological SciencesChiba 263-8555, Japan
Abstract
The anomalous part of the magnetic moment of nucleon is stud-ied. The electromagnetic vertex of nucleon is calculated using thepseudovector coupling pion-nucleon interaction. The self-energy ofnucleon suggested in the previous study is applied to the internal line.
The interaction between two nucleons and their kinematic behavior are in-teresting and it is described by the quantum field theoretical method like themeson-exchange model. Regardless of the strength of the coupling parame-ters it is possible to resolve the properties of the two-body system by usingthe non-perturbative approach for the Bethe-Salpeter equation. Besides themany-body system such as the finite nuclei also for the two-nucleon systemthe pion-exchange interaction is to be essential to interpret the experimentaldata. Particularly the pseudovector coupling is not made clear about therole on the phenomena and the relation between the nuclear force and thequantum corrections at the present time.Because of the derivative on the field of pion the vertex part containsthe variable on the four-momentum transfer and then which makes difficultto obtain the convergent result unlike the pseudoscalar coupling. While thedivergences in the perturbative expansion are not removed within the usualprocedure of the counter terms the generalized relation remains and which isformed by the arbitrary number of the Heisenberg operators. For the vertexfunction it is found to generate the non-perturbative terms in addition to theperturbative part. Then for example the self-energy of the nucleon propaga-tor results in the finite quantity which is applicable to the calculation of thehigher-order corrections.One of the interesting subjects is the investigation of the electromag-netic properties of nucleon. The form factor is associated with the magneticmoment of nucleon which is a basic property of fermion and there are ex-perimental data [1] to be explained by the method of the field theory. The1nomalous part is larger than the Dirac part based on the point-like struc-ture of nucleon. Thus the degrees of freedom of pion have an effect on theelectromagnetic form factor [2]. The lowest-order pion process is corrected bythe nucleon propagator with the self-energy under the pseudovector couplingpion-nucleon interaction suggested in our previous study [3].
The charged pion interacts with photons as the quantum system underthe lagrangian density given by L π − γ ( x ) = − F µν ( x ) F µν ( x )+( ∂ µ + ieA µ ( x )) φ ∗ ( x )( ∂ µ − ieA µ ( x )) φ ( x ) − m π φ ∗ ( x ) φ ( x ) , (1)in which φ ( x ), φ ∗ ( x ) and A µ ( x ) are the complex scalar field, the conjugatefield and the electromagnetic field respectively. The strength of the electro-magnetic field F µν is defined by F µν ≡ ∂ µ A ν − ∂ ν A µ . The operator φ ( x ) func-tions to destruct a negatively charged pion and create a positively chargedpion (the electric charge e >
0) and m π is the pion mass.The interaction between A µ ( x ) and φ ( x ) is obtained by the prescription ofthe minimal coupling ∂ µ → ∂ µ − ieA µ ( x ) which preserves the lagrangian in-variant under the local gauge transformation that is A µ ( x ) → A µ ( x )+ ∂ µ α ( x )and φ ( x ) → exp[ ieα ( x )] φ ( x ). To proceed calculation the Lorentz condition ∂ µ A µ ( x ) = 0 is imposed by determining the α ( x ) suitably and then the equa-tion of motion for A µ ( x ) is as follows ∂ µ F µν ( x ) = ∂ µ ∂ µ A ν ( x ) = J ν ( x ) , (2) J µ ( x ) ≡ e [ ~φ ( x ) × ∂ µ ~φ ( x )] − e A µ ( x )( φ ( x ) + φ ( x )) . (3)Here the third component of the conserved isovector current J µ ( x ) is ex-pressed in terms of the real isovector fields ~φ ( x ) = ( φ ( x ) , φ ( x ) , φ ( x ))instead of the fields φ ( x ) and φ ∗ ( x ) by using the relation between them2 ( x ) = ( φ ( x ) + i φ ( x )) / √
2. The subscript 3 in the first term tells that theneutral pion φ field is independent of J µ ( x ). The second term in Eq. (3)is not applied to the actual calculation below. The vertex diagram needsanother one of the photon propagators and so the numerical result would besuppressed by the factor ( ∼ e ).When the electromagnetic interaction for proton is taken into account theelectromagnetic current in terms of the nucleon fields ψ ( x ) and ¯ ψ ( x ) J µem ( x ) ≡ e ¯ ψ ( x ) γ µ τ + ψ ( x ) ( τ ± ≡ ± τ J µ ( x ) so that the source term in Eq. (2) isreplaced as J µ ( x ) → J µ ( x ) + J µem ( x ). Here τ is the third component of theisospin matrix.The photon-nucleon-nucleon three-point vertex part is used to investigatethe electromagnetic properties of nucleon. The vertex function Γ is connectedwith the nucleon propagator G and the expectation value about the currents J ( z ) and J em ( z ) by the following relation e Z d x ′ d y ′ G ( x − x ′ ) ∂ z · Γ( x ′ y ′ ; z ) G ( y ′ − y )= e i { δ ( z − x ) − δ ( z − y ) } τ + G ( x − y ) −h T[ ∂ z · ( J ( z ) + J em ( z )) ψ ( x ) ¯ ψ ( y )] i . (5)The second term in the right-hand side is left to proceed calculation of theanomalous part which is to be dropped by virtue of the current conservation.Eq. (5) is converted to the analogous one in momentum space. The relationbetween the vertex function Γ( p, q ) and the nucleon propagator G ( p ) is given( p − q ) · Γ( p, q ) = τ + ( G ( p ) − − G ( q ) − ) (6)as a function of the outgoing momentum p and the incoming momentum q .The vertex function Γ( p, q ) is divided into three partsΓ( p, q ) = Γ ( p, q ) + Γ π ( p, q ) + Γ em ( p, q ) , (7)in which Γ π ( p, q ) and Γ em ( p, q ) are the counterparts of the terms of the cur-rents J ( z ) and J em ( z ) in Eq. (5) accordingly. Γ ( p, q ) satisfies Eq. (6). Bythe conservation ( p − q ) · ¯ u ( p )Γ ( p, q ) u ( q ) = 0 using the Dirac spinors u ( q )and ¯ u ( p ) the component ∼ ( p − q ) is excluded from Γ ( p, q ).3 .2 The anomalous magnetic moment by the pion current For the calculation of the anomalous magnetic moment the pion currentcontribution Γ µπ ( p, q ) is the main part among the two components. Theisospin dependence is ∼ τ and it is roughly the same value of the observedratio between proton and neutron. By using the perturbative expansion thevertex function Γ µπ ( p, q ) isΓ µπ ( p, q ) = 2 τ ( f π m π ) Z d k (2 π ) ( − k + p + q ) µ × γ γ · ( k − p ) iG ( k ) γ γ · ( q − k ) i ∆ ( k − p ) i ∆ ( q − k ) (8)in the lowest-order approximation. Here G ( k ) = 1 / ( γ · k − M + iǫ ) and∆ ( k ) = 1 / ( k − m π + iǫ ) are the free propagators of nucleon with the mass M and pion.It is noted that the pseudovector coupling interaction is connected to thatof the pseudoscalar coupling and in the present case Γ µπ ( p, q ) is transferredto the pseudoscalar one by the equivalence relation ( f π m π ) = ( g π M ) betweenthese coupling constants and the replacement γ · ( k − p )2 M → γ · ( q − k )2 M →
1. Thenthe relation is such that the two interactions are agree with each other sup-posing the internal nucleon is restricted to the on-shell state ( k = M ).To perform the k -integral in Eq. (8) the Feynman formula is used and thevariables are converted from k to k ′ as k ′ = k − px − qy . The denominatorof the integrand becomes the simple form ( k ′ − M ρ ) where ρ ≡ ( x + y − + m π M ( x + y ) − Q M xy (9)and the four-momentum transfer Q ≡ p − q . Then it is appropriate to usethe method of the dimensional regularization integral. Because of the sharppeak at k ′ ∼ M ρ the quantity of the higher-order corrections would betreated approximately provided the form of it is analytic around the region.In the case of the pseudovector coupling the estimate of the integrationfor k is Γ π ∼ k and then the convergence of the k -integration is not clearalso about the ∼ ( p + q ) dependent term which is indispensable to cal-culate the anomalous part of the magnetic moment. The nucleon current¯ u ( p )Γ( p, q ) u ( q ) is expressed in the general form as¯ u ( p )Γ µ ( p, q ) u ( q ) = ¯ u ( p )[ γ µ F ( Q ) + iσ µν Q ν M F ( Q ) ] u ( q ) (10)4y means of the form factors F i ( Q ) ( i = 1 ,
2) as a function of Q . Herethe relation of the Gordon decomposition is useful and which is given by theform as ¯ u ( p )[2 M γ µ − ( p µ + q µ ) − iσ µν Q ν ] u ( q ) = 0.Our interest is F ( Q ) where the k -integral is done by the dimensionalregularization method using two parameters x and y to treat the denominatorof the integrand in Eq. (8) and it is F ( Q ) = τ ( 2 M f π πm π ) { ( 2 ǫ − γ − log M πµ ) I [ f ] + X i =2 , , , I [ f i ] } , (11) I [ f ] ≡ Z dx Z − x dy f ( x, y ) , (12) f ( x, y ) ≡ − x + y − x + y − − ( x + y )( x + y − , (13) f ( x, y ) ≡ − f ( x, y ) log ρ, (14) f ( x, y ) ≡ ( x + y − x + y − , (15) f ( x, y ) ≡ { ( x + y − − Q M ( x + y − x + y − xy } /ρ, (16) f ( x, y ) ≡ x + y − . (17)The infinitesimal quantity ǫ is stemmed from the shift of the space-time di-mension as 4 → − ǫ and γ is the Euler’s constant ( γ = 0 . · · · ). Theparameter µ is introduced to set the mass dimension of the interacting la-grangian density to 4 − ǫ . Since I [ f ] = 0 the parameters associated withthe integral on k are not significant to proceed the calculation. Then theresult of the magnetic moment is free from the divergence contrary to theestimate mentioned above. According to the dimensional regularization theadditional term is required to calculate the constant term ( ∼ ǫ ) correctlyand in the present case it is given by I [ f ]. It arises from the gamma matrix γ which is commutable with the extra pieces of the gamma matrix otherthan γ i ( i = 0 , , ,
3) due to the shift of the dimension.5 .3 The anomalous magnetic moment by the vertex correction
For the electromagnetic form factor of nucleon the vertex correction bythe pion propagator is important and as will be seen in the lowest-ordercalculation the magnitude of the numerical value is as large as that of the pioncurrent apart from the isospin matrix. In fact the observed ∼ τ dependenceof the anomalous part does not necessarily imply that the process is minortaking into account the non-perturbative effect of the self-energy of nucleon.The vertex function of the electromagnetic current Γ µem ( p, q ) isΓ µem ( p, q ) = − (1 + τ − ) ( f π m π ) Z d k (2 π ) i ∆ ( k ) × γ γ · k iG ( p − k ) γ µ iG ( − k + q ) γ γ · k (18)in the lowest-order approximation. As well as the form factor of the pioncurrent the equivalence relation exists if the two internal nucleon propagatorsare at the on-shell states (( p − k ) = M and ( q − k ) = M ).The k -integral is done same as the Γ µπ ( p, q ) case using ρ ′ given by ρ ′ ≡ ( x + y ) + m π M (1 − x − y ) − Q M xy (19)for the denominator ( k ′ − M ρ ′ ) . When Q = 0 both ρ ′ and ρ are positivedefinite and it suffices to do the integrals for the analytic functions. Theyare transferred to each other by the interchange x + y ↔ − x − y . The formfactor of the nucleon current corrected by the lowest-order pion process is F ( Q ) = (1 + τ − ) ( 2 M f π πm π ) { ( 2 ǫ − γ − log M πµ ) I [ f ] + X i =7 , , I [ f i ] } , (20) f ( x, y ) ≡ x + y ) − , (21) f ( x, y ) ≡ − f ( x, y ) log ρ ′ , (22) f ( x, y ) ≡ − x − y, (23) f ( x, y ) ≡ { − ( x + y ) − Q M (2 − x − y ) xy } /ρ ′ . (24)In Eq. (18) two additional terms on the commutability of γ cansel with eachother and do not contribute to the coefficient of the term ∼ p + q . So thereis not the effect on the magnetic moment. The divergent term vanishes by I [ f ] = 0 also in the part of the nucleon current.6 The numerical results and the effect of the self-energy
The numerical calculation is done by using the set of the parameters M = 939 MeV, m π = 139 . f π = 1 .
0. At Q = 0 the values of theintegrals I [ f i ] ( i = 2 , ,
4) are I [ f ] = 0 . I [ f ] = 0 . I [ f ] = 0 .
079 and I [ f ] = − /
3. Adding them the result is F (0) = a τ with a = 0 .
831 whichconstructs the electromagnetic pion current contribution for the anomalousmagnetic moment in the lowest-order. The part of the lowest-order vertexcorrection for the nucleon current is determined similarly by the integrals I [ f i ] ( i = 7 , ,
9) with the values of I [ f ] = − . I [ f ] = − / I [ f ] = − . F (0) = b (1 + τ − ) with b = − . Q = 0 is expressed as F exp (0) = κ p τ + + κ n τ − = z τ + + a ′ ( z ) τ + b ′ ( z ) (1 + τ − ) (25) a ′ ( z ) ≡ − z + 2 κ p − κ n b ′ ( z ) ≡ − z + κ p + κ n z is ascribed to the self-energy Σ( p )of the nucleon given by a series of γ · p − M . The experimental values ofthe proton and the neutron anomalous magnetic moments are κ p = 1 .
79 and κ n = − .
91 in units of the nuclear magneton ( µ N ) respectively [1]. The coef-ficients a ′ ( z ) and b ′ ( z ) are linearly dependent on z and the convergent resultsof the perturbative expansion may give the information on the magnitude ofthe self-energy. It is obtained by means of the non-perturbative part of thevertex function.In Fig. 1 the linear functions a ′ ( z ) and b ′ ( z ) on z are drawn along withthe results of the calculation of the coefficients a and b as the horizontallines. When z = 0 the a and b are different largely from the experimentallydetermined values a ′ (0) and b ′ (0) for lack of the self-energy in Eq. (6). Thissituation is changed by moving z to z = 2 at which the a/a ′ (2) and b/b ′ (2) arecomparable such as a/a ′ (2) ∼ b/b ′ (2) ∼ . z continues to increase. It indicates that the present calculation would be7mproved by the higher-order corrections such as to reduce the value of thecoupling constant f π in a and b ( ∼ f π ) effectively.Another kind of the higher-order correction is the inclusion of the self-energy for the nucleon propagator in the off-shell state. When there is notthe pion interaction the photon-nucleon-nucleon three-point vertex functionis in the perturbative expansion prescribed by the Feynman rule. The magni-tude of the fine structure constant as the parameter for the expansion makesus understand the iterative character of the method. Taking into account thepseudovector coupling interaction of pion this situation is changed becausethe parameter has the dimension of [MeV] − which implies the expansionseries are associated with divergences worse than that in the quantum elec-trodynamics.The non-perturbative treatment of the pion-nucleon-nucleon three-pointvertex function provides the non-perturbative term which is excluded fromthe rule of the perturbative expansion. Consequently the proper vertex func-tion is modified in comparison to the one without the term. As has beenseen in our previous study the effects on the propagators are not negligible.The self-energy of the nucleon propagator becomesΣ( k ) = M c M + m π − Mc − c = MM + m π c + O ( c ) (28)by the lowest-order approximation for the perturbative part of the vertexfunction [3] and which is expanded in the series of c ≡ γ · k − M .We do not use Eq. (28) directly for the calculation of the higher-ordercorrections of the anomalous magnetic moment. Alternatively the off-shellstate of the propagator is considered to have the effective mass M ′ definedby M ′ ≡ M + h Σ( k ) i , in which the bracket means Σ( k ) is replaced by anapproximate value independent of k . The approximation of the internalnucleon state achieves to include the effect of the higher-order correctionpreserving the simple form of the free propagator.The four-momentum k of the internal off-shell nucleon is the variable ofthe integral and permitted to take any values. Determining the value of theself-energy h Σ( k ) i we replace as γ · k → t and k → t in Eq. (28) using aparameter t with the dimension of mass. At the region 0 ≤ t ≤ M suitablefor the present study the effective mass M ′ is larger than M .There exists a way to determine the specific value of M ′ . The inverse ofthe exact propagator G − ( k ) has the zero at the parameter t = M + Σ( k )where γ · k → M + Σ is done in Σ( k ). Then the self-consistent equation on8 is derived by means of the form of Σ in Eq. (28). The positive solutionis Σ = M ( − q
17 + 8 m π /M ) / ∼ . M ∼
530 MeV roughly as large asthe σ meson mass. Consequently the coefficients on the pion current a andthe nucleon current b are a = 0 .
758 and b = − .
077 respectively and theratios become a/a ′ (2) ∼ b/b ′ (2) ∼ .
5. Improvement is observed however itis not considerable. While it is not possible to explain the data satisfactorilythe approximation is related to the structure of G ( k ). The magnitude of the anomalous part is larger than the Dirac part inthe magnetic moment of nucleon. The dependence on the isospin makesus convinced that the virtual processes of the charged pions actually takeplace and which is observed by the electromagnetic field. The pseudovectorcoupling interaction is useful to describe it quantitatively. The result is freefrom divergences within the lowest-order approximation. Inclusion of theself-energy of nucleon is necessary and its effect has been investigated inthe present study. The approximate propagator with the effective mass istractable since it is connected to the lowest-order calculation. The change ofthe nucleon mass may reflect the mixing of the continuum state beyond theone-particle state in the exact propagator.
References [1] Particle Data Group, Phys. Rev.
D66 (2002), p757 and p765.[2] G. F. Chew, R. Karplus, S. Gasiorowicz and F. Zachariasen, Phys. Rev. (1958), 265.[3] S. Kinpara, arXiv:nucl-th/1712.00930. a ′ ( z ) and b ′ ( z ) are shown as the linear function of z . The results of the calculations a and bb