Russian Physics Journal, Vol. 56, No. 9, January, 2014 (Russian Original No. 9, September, 2013)
ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY COVARIANT EQUATIONS OF MOTION OF A SPIN 1/2 PARTICLE IN AN ELECTROMAGNETIC FIELD WITH ALLOWANCE FOR POLARIZABILITIES V. V. Andreev, O. M. Deryuzhkova, and N. V. Maksimenko
UDC 539.12
On the basis of a relativistic field-theoretic approach, equations for the interaction of an electromagnetic field with spin 1/2 hadrons have been constructed which take account of polarizabilities and anomalous magnetic moments. With the help of the Green’s function method of solving these equations the Compton scattering amplitude has been calculated with allowance for the polarizabilities of spin 1/2 particles. The magnetic and electric quasistatic polarizabilities have been computed for a spinor particle using the technique of calculating the matrix elements of Compton scattering. Keywords: polarizability, Lagrangian, Compton scattering, Green’s function. INTRODUCTION At the present time, many electrodynamic processes are known, on the basis of which it is possible to obtain experimental data on the polarizabilities of hadrons. In connection with this, the problem arises of achieving a selfconsistent covariant determination of the contribution of the polarizabilities to the amplitudes and cross sections of electrodynamic processes on hadrons [1, 2]. A problem of this sort can be solved within the framework of a fieldtheoretic covariant formalism of the interaction of the electromagnetic field with hadrons taking their polarizabilities into account. In [3–6] covariant methods were actively developed for constructing Lagrangians and the equations of the interaction of an electromagnetic field with hadrons in which the electromagnetic characteristics of these particles lay the foundation for the entire edifice. Recently, effective field Lagrangians have been used to describe the interaction of a low-energy electromagnetic field with nucleons on the basis of an expansion in powers of the inverse mass of the nucleon [7]. In practice, when using such Lagrangians to determine the amplitudes of specific electrodynamic processes, the physical interpretation of the constants arising in expansions of this sort plays an important role. Tied in with this, the determination of the signs with which the polarizabilities enter into the Lagrangian, with the interactions with a constant or variable electromagnetic field taken into account, acquires significance. In [8], on the basis of the correspondence principle between the classical and quantum theories, an effective covariant Lagrangian of the interaction of an electromagnetic field with spin 1/2 particles with allowance for their polarizabilities was presented within the framework of the field-theoretic approach. In the present paper, equations of the interaction of an electromagnetic field with spin 1/2 hadrons with polarizabilities and anomalous magnetic moments taken into account are derived within the framework of the fieldtheoretic approach on the basis of the effective Lagrangian presented in [8]. With the help of the Green’s function method of solving the electrodynamic equations [9–12] the amplitude of Compton scattering on spin 1/2 particles is obtained with their polarizabilities taken into account. Amplitude structures are identified that are analogous to polarizabilities, but arise as a consequence of electromagnetic interactions. The contributions of these structures to the polarizabilities of hadrons are analyzed.
F. Skorina Gomel’ State University, Gomel’, Belarus, e-mail:
[email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 86–91, September, 2013. Original article submitted May 13, 2013. 1064-8887/14/5609-1069 ©2014 Springer Science+Business Media New York
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1. COVARIANT EQUATIONS OF THE INTERACTION OF AN ELECTROMAGNETIC FIELD WITH A NUCLEON WITH POLARIZABILITIES TAKEN INTO ACCOUNT To determine the covariant equations of the interaction of an electromagnetic field with a nucleon with the contributions of the anomalous magnetic moment and the polarizabilities taken into account, we employ an effective Lagrangian of the form [8] ⎞ ⎞ 1 1 ⎛ ∧G 1 ⎛ ∧H L = − Fμν F μν + Ψ ⎜ i D − m ⎟ Ψ − Ψ ⎜ i D + m ⎟ Ψ . 4 2 ⎝ 2 ⎝ ⎠ ⎠
(1)
In the statement of the effective Lagrangian given by Eq. (1) we have introduced the following notation: ∧G ∧ G ieκ μν σ Fμν + ie A , D = ησν γ σ ∂ ν + 4m
(2)
∧H ∧ H ieκ μν σ Fμν − ie A , D = ∂ ν γ σ ησν − 4m
(3)
ησν = g σν +
2π ⎡αFσμ F μ ν + β Fσμ F μν ⎤ . ⎦ m⎣
(4)
1 Here Fμν = ∂ μ Aν − ∂ ν Aμ is the electromagnetic field tensor and Fσμ = εσμρν F ρν is its dual tensor, κ is the 2 anomalous magnetic moment, α and β are the electric polarizability and magnetic polarizability of the nucleon, and ∧
(
)
i μ ν γ γ − γ ν γ μ , and γ μ are four-dimensional matrices satisfying the commutation relations 2 γ μ γ ν + γ ν γ μ = 2g μν . If we substitute relations (2)–(4) into Eq. (1), then the effective Lagrangian can be represented as A = Aμ γ μ , σμν =
follows: ∧ 1 i I∧ eκ Ψσμν ΨFμν + K σν θσν , L = − Fμν F μν + Ψ ∂ Ψ − mΨΨ − eΨ A Ψ − 4 2 4m
(5)
I I G H 2π i ⎡αFσμ F μ ν + βFσμ F μν ⎤ , θσν = Ψγ σ ∂ ν Ψ , and ∂ ν = ∂ ν − ∂ ν . ⎣ ⎦ m 2 We single out in Lagrangian (5) the part associated with the polarizabilities of the nucleon:
where K σν =
L( α ,β) = −
2π ⎡αFμσ F μ ν + βFμσ F μν ⎤ θσν . ⎦ m⎣
(6)
1 Employing the relation Fμσ F μν = Fμσ F μν − δν σ Fμρ F μρ , Lagrangian (6) can be represented in the following form: 2 L( α ,β) = −
2π ⎡ β ( α + β ) Fμσ F μν θσν − θσσ Fμν F μν ⎤⎥ . m ⎢⎣ 2 ⎦
(7)
The form of Lagrangian (7) agrees with the effective Lagrangian of the same type in [13]. Expression (7) is a relativistic
(
)
field-theoretic generalization of the nonrelativistic relation H = − L( α,β ) = −2π αE 2 + βH 2 , which corresponds to the
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polarizabilities of the induced dipole moments in a constant electromagnetic field [14]. In the case of a variable electromagnetic field, the signs in front of the polarizabilities in the Lagrangian in the nonrelativistic approximation will change [15]; however, the structure of the tensor product in L( α ,β ) (7) does not change. In order to obtain the equations of the interaction of an electromagnetic field with a nucleon, we employ the effective Lagrangian (1) and the Lagrange–Euler field equations ⎛ ∂L ⎞ ∂L ⎛ ∂L ⎞ ∂L ⎛ ∂L ⎞ ∂L ∂μ ⎜ = 0, ∂ μ ⎜ = 0, ∂ μ ⎜ = 0. ⎟− ⎟− ⎟− ⎜ ∂ ( ∂ A ) ⎟ ∂Aν ⎜ ⎟ ⎜ ⎟ μ ν ⎠ ⎝ ⎝ ∂ ( ∂ μ Ψ ) ⎠ ∂Ψ ⎝ ∂ ( ∂ μ Ψ ) ⎠ ∂Ψ
As a result, we obtain ⎡ eκ ⎤ ∂ μ F μν = eΨγ ν Ψ − ∂ μ ⎢ Ψσμν Ψ + G μν ⎥ , ⎣ 2m ⎦
(8)
∧ ⎛ ∧G ⎞ i ν eκ μν σ σ ν σ Fμν Ψ , ⎜ i ∂− m ⎟ Ψ = e A Ψ − ⎡⎣ ∂ K σν γ Ψ + K σν γ ∂ Ψ ⎤⎦ + 2 4m ⎝ ⎠
(9)
(
)
∧ i ⎛ ∧H ⎞ eκ μν Ψ ⎜ i ∂+ m ⎟ = −Ψe A− ⎣⎡ ∂ ν Ψγ σ K σν + ∂ ν Ψ γ σ K σν ⎦⎤ − Ψ σ Fμν . 2 4m ⎝ ⎠
(
) (
)
(10)
If we restrict ourselves in Eqs. (9) and (10) to terms not higher than second order in the frequency of radiation, then it is possible to represent them in the form ⎛ ∧G ⎞ ⎛ ∧H ⎞ ⎜ i D − m ⎟ Ψ = 0, Ψ ⎜ i D + m ⎟ = 0, ⎝ ⎠ ⎝ ⎠ ∧G ∧H where D and D are defined in Eqs. (2) and (3). The antisymmetric tensor G μν in Eq. (8) has the form
G μν = −
(
∂L( α ,β ) 4π ⎡( α + β ) F μρ θ ρν − F ν ρ θ ρμ − βθρρ F μν ⎤ , = ⎦ ∂ ( ∂ μ Aν ) m ⎣
(
)
(11)
)
1 where θ ρν = θρν + θνρ . With the help of antisymmetric tensor (11), effective Lagrangian (7) can be represented as 2 follows: 1 L( α ,β) = − Fμν G μν . 4
If we restrict ourselves in Eqs. (9) and (10) to the contribution of the charge and the magnetic moment, then we obtain the well-known equations presented, for example, in [12]. 2. COVARIANT REPRESENTATION OF THE AMPLITUDE OF COMPTON SCATTERING ON A NUCLEON WITH THE CONTRIBUTION OF POLARIZABILITIES TAKEN INTO ACCOUNT
The Compton scattering amplitude, calculated with the contributions of the fine structure constant and the anomalous magnetic moment taken into account with second-order accuracy, is given in [10–12]. That having already
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been done, we will here determine the contribution of the electric and magnetic polarizabilities to the Compton scattering amplitude. Toward this end, we employ the Green’s function method [10–12]. We represent differential equation (9), in which we will take account of only the contribution of the polarizabilities, in integral form: Ψ ( x ) = Ψ ( 0 ) ( x ) + ∫ S F ( x − x′ ) ⋅V ( α ,β ) ( x′ ) dx′,
(
(12)
)
i where V ( α,β ) ( x′ ) = − ⎡⎣ ∂ ν K σν ( x′ ) γ σ Ψ ( x′ ) + K σν ( x′ ) γ σ ∂ ν Ψ ( x′ ) ⎤⎦ and the Green’s function S F ( x − x ′ ) satisfies the 2 ⎛ ∧ ⎞ equation ⎜ i ∂− m ⎟ S F ( x − x′ ) = δ ( x − x′ ) . We determine the matrix element S fi of scattering of a photon by a nucleon, ⎝ ⎠
following the approach in [10–12]. Toward this end, we convolve expression (12) with Ψ p2 ( r2 ) ( x ) in the limit t → +∞ and = ( −i ) Ψ p2 ( r2 ) ( x′ ) , where Ψ p2 ( r2 ) = make use of the relation ∫ Ψ p2 ( r2 ) ( x )S F ( x − x′ ) d 3 x t →+∞ ⋅e
ip2 x′
1
( 2π )
3/ 2
m ( r2 ) U ( p2 ) E2
. As a result, we obtain S fi = ( −i ) ∫ Ψ p2 ( r2 ) ( x′ ) ⋅V ( α ,β ) ( x′ ) d 4 x ′.
(13)
Employing boundary conditions in accordance with [10–12] together with crossed symmetry, we can represent expression (13) as
{
}
⎛ 1⎞ S fi = ⎜ − ⎟ ∫ Ψ p2 ( r2 ) ( x′ ) ∂ ν ⎣⎡ K σν 21 ( x′ ) γ σ Ψ p1 ( r1 ) ( x′ ) ⎦⎤ + K σν 21 ( x′ ) γ σ ∂ ν Ψ p1 ( r1 ) ( x′ ) d 4 x′. ⎝ 2⎠
(14)
Integrating by parts in expression (14) and employing the definition of the electromagnetic field tensor, we obtain S fi =
(
)
(
)
2πi ⎡ (2) μν (1) μν (2) μν 4 σ ρ ∫ (α + β) Fσμ F(1) + Fσμ F(2) θ ( 21)ν + β Fμν F(1) θ ( 21)ρ ⎦⎤d x′. m ⎣
(15)
If we take into account the wave functions of the nucleon and the photons in the initial and final states, expression (15) takes the form S fi =
imδ ( k1 + p1 − k2 − p2 )
( 2π )2 4ω1ω2 E1 E2
(16)
M,
where the amplitude М in expression (16) is represented as follows: M =
∧( λ2 ) ⎤ ⎧⎡ 2π ( r2 ) ⎪ ∧ U ( p2 ) ⎨ ⎢ k 2 eμ( λ2 ) − k2μ e ⎥ ⎡⎣ k1μ e( λ1 ) P − ( k1 P ) e( λ1 )μ ⎤⎦ + ⎡⎣ k2μ e( λ2 ) P − ( k2 P ) e( λ2 )μ ⎤⎦ m ⎥⎦ ⎪⎩ ⎢⎣
(
)
(
)
∧ ( λ1 ) ⎤ ⎫ ⎡∧ ⎪ λ ν λ μ × ⎢ k 1 eμ( λ1 ) − k1μ e ⎥ ⎬ ( α + β ) + mβ ⎡⎣ k2μ eν( λ2 ) − k2ν eμ( λ2 ) ⎤⎦ ⎡ k1μ e( 1 ) − k1ν e( 1 ) ⎤ U ( r1 ) ( p1 ) . ⎣ ⎦ ⎢⎣ ⎥⎪ ⎦⎭
Here eμ(λ1 ) and eμ(λ2 ) are the polarization vectors of the initial photon and the final photon, P =
(17)
1 ( p1 + p2 ) , k1 , p1 2
r r and k2 , p2 are the momenta of the initial and final photons and nucleons, and U ( 1 ) ( p1 ) and U ( 2 ) ( p2 ) are the
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bispinors of the initial and final nucleons. Let us now determine amplitude (17) in the rest frame of the target. We will restrict ourselves in М to terms not higher than second order in the radiation frequency. In this case we obtain [16]
) (
(
)
M = 4πω2 χ( r2 ) + ⎡ α e ( λ 2 ) e ( λ1 ) + β ⎡⎣ n2 e ( λ 2 ) ⎤⎦ ⎡⎣ n1 e (λ1 ) ⎤⎦ ⎤ χ( r1 ) , ⎣ ⎦ r + r where ω is the radiation frequency, n1 and n2 are unit vectors in the k1 and k2 directions, and χ( 2 ) and χ( 1 ) are
the spinors of the initial and final nucleons. If we take into account the contribution of the electric charge to the amplitude М along with the contribution of the polarizabilities α and β , we can represent М thus: ⎡⎛ e 2 ⎞ M = χ( r2 ) + ⎢⎜ − + 4πω2 α ⎟ e ( λ 2 ) e ( λ1 ) + 4πω2β ⎣⎡ n2 e ( λ 2 ) ⎦⎤ ⎣⎡ n1 e ( λ1 ) ⎦⎤ ⎠ ⎣⎝ m
(
)
(
⎤
)⎦⎥ χ(
r1 )
.
(18)
The differential cross section of Compton scattering, for example at the angle θ = 0D , calculated with the help of Eq. (18), has the form [16] 2
α d σ ⎛ αe ⎞ = ⎜ ⎟ − 2 e (α + β)ω 2 , dΩ ⎝ m ⎠ m
where α e =
e2 is the fine structure constant. 4π
3. QUASISTATIC POLARIZABILITIES OF SPIN 1/2 PARTICLES IN QED
An interesting peculiarity of particles not having structure due to strong interactions (polarizability, root-meansquare radius, etc.) is the presence of analogous characteristics manifested as a result of electromagnetic or weak interactions. Thus, it is well known that the anomalous magnetic moment of the electron is induced by higher orders of perturbation theory while for the proton it is there from the beginning. Let us find the quasistatic polarizabilities of the structureless fermions that arise in Compton scattering due to higher orders. It is well known that in general the forward ( θ = 0D ) and backward ( θ = π ) Compton scattering amplitudes T with accuracy to ω2 are written in the form λ′,σ′ λ ′,σ′ Tλ,σ (θ = 0) = 8πm f ω2 ( α E + βM ) δλ,λ′ δσ,σ′ , Tλ,σ (θ = π) = 8πm f ω2 ( α E − β M ) λδ−λ,λ′ δσ,−σ′ .
Here λ and λ′ are the helicities of the initial and final fermions, respectively, σ and σ′ are the helicities of the initial and final photons, respectively, and m f is the mass of the fermion. On the other hand, the possibility exists of calculating the matrix elements and, correspondingly, the Compton scattering amplitude within the framework of QED, including the order of perturbation theory in the constant α QED following the Born order (see, e.g., [17, 18]). In [19, 20] a technique was developed for calculating the polarizabilities of fermions within the framework of quantum field theories and models by comparing the corresponding matrix elements. The end result of this procedure in the given case is the relations α qE− s + βqM− s =
2 2 α QED 11 8α QED ⎛ 2ω ⎞ + ln ⎜ , ⎜ m ⎟⎟ 3πm3f 6 3πm3f ⎝ f ⎠
(19)
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α qE− s − βqM− s = −
2 2 αQED 59 4α QED ⎛ 2ω ⎞ ln ⎜ + ⎟, ⎝ λ ⎠ 3πm3f 6 3πm3f
(20)
where the parameter λ represents the infinitesimally small mass of the photon. Note that structures analogous to polarizabilities appear as a result of electromagnetic interactions; therefore, it makes sense to speak of the above quantities as quasi-polarizabilities of some kind which are corrections to the polarizabilities in the general case. For this reason, we have introduced the notation α qE− s and βqM− s for them. As follows from relations (19) and (20), quasistatic polarizabilities contain, in addition to their constant terms, non-analytic terms ~ ln ω , which diverge in the Thomson limit ( ω → 0 ). It is specifically the above-indicated property that has served as the reason why in [21, 22] structures (19) and (20) were called quasistatic polarizabilities. Equation (19) coincides with the expression obtained in [21, 22] while formula (20) was obtained here for the first time. And whereas the technique in [21] required significant efforts to calculate the cross sections and take the integrals in the Baldin sum rule [23], in the proposed technique the procedure reduces in fact to expanding the expressions for the matrix elements in the photon frequency ω . From relations (19) and (20) it is easy to find the electric ( α qE− s ) and magnetic ( βqM− s ) quasistatic polarizabilities and to estimate their contribution to the polarizabilities of the Dirac proton (a point fermion with zero anomalous magnetic moment). Setting m f = m p , and the parameter ω = 0.1m p , we find α qE− s + βqM− s ≈ −5.8 ⋅10−7 fm3 .
(21)
Comparing the obtained result with the experimental values [24] α (Ep ) + β(Ep ) = (13.8 ± 0.4 ) ⋅10−4 fm3 ,
(22)
it can be observed that the contribution of the given corrections is small and does not exceed even the experimental errors. Numerical estimates (21) and (22) agree with the estimates in [25].
CONCLUSIONS
Within the framework of the gauge-invariant approach we have obtained the equations of motion of a nucleon in an electromagnetic field in covariant form that takes account of its electric and magnetic polarizabilities. On the basis of a solution of the electrodynamic equations of motion of the nucleon obtained by the Green’s function method, we have shown that the developed covariant Lagrangian formalism for the interaction of low-energy photons with nucleons is in agreement with the low-energy theorem for Compton scattering. On the basis of the original technique, we have reproduced the well-known result for the combination of quasistatic polarizabilities α qE− s + βqM− s within the framework of QED and obtained a new expression for α qE− s − βqM− s . The unquestionable advantage of the technique for extracting polarizabilities, mentioned in Section 3, is its relative simplicity. The given approach opens up wide vistas for investigating the internal structure of nucleons and can be applied within the framework of various quantum field theories and models.
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