PHYSICAL REVIEW C 71, 045503 (2005)

Vacuum polarization radiative correction to parity violating electron scattering on heavy nuclei

A. I. Milstein∗1Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia

O. P. Sushkov†2School of Physics, University of New South Wales, Sydney 2052, Australia

(Received 14 September 2004; published 25 April 2005)

The effect of vacuum polarization on the parity violating asymmetry in elastic electron-nucleus scattering isconsidered. Calculations are performed in the high-energy approximation, with an exact treatment of the electricfield of the nucleus. It is shown that the radiative correction to the parity violating asymmetry is logarithmicallyenhanced and the value of the correction is about −1%.

DOI: 10.1103/PhysRevC.71.045503 PACS number(s): 24.80.+y, 25.30.Bf, 21.10.Gv

Experimental investigations of the proton ρp(r) and neutronρn(r) densities in heavy nuclei are important in the testing ofnuclear models. These densities are also important in manyapplications, such as precise calculations of atomic paritynonconservation. Although the charge distribution (protondensity) is known fairly well, mainly from data on elasticelectron scattering, the determination of the neutron densitywith good accuracy is a complicated problem [1]. It wassuggested by Donnelly et al. [2] that parity violation (PV) inelectron-nucleus scattering be used to determine the neutrondistribution. The idea behind this suggestion is that themediator of the weak interaction, the Z boson, interactsmainly with neutrons. The parity violating asymmetry isdefined as

APV = dσ+/d� − dσ−/d�

dσ+/d� + dσ−/d�, (1)

where dσ+/d� and dσ−/d� are cross sections for the scatter-ing of right-handed and left-handed electrons, respectively.The asymmetry can be measured experimentally, and todetermine the neutron distribution one needs to calculate thecorresponding asymmetry. The scattering potential is of theform

V (r) = VC(r) + γ5 A(r), A(r) = GF

2√

2�W (r). (2)

Here VC(r) is the Coulomb potential of the nucleus, GF =1.16639 × 10−5 GeV−2 is the Fermi constant, γ5 is theDirac matrix, and �W (r) is the density of the weak charge,given by

�W (r) = −N�n(r) + (1 − 4 sin2 θW )Z�p(r), (3)

where θW is the Weinberg angle (sin2 θW ≈ 0.23), N is thenumber of neutrons, and Z is the number of protons. Thedensities are normalized such that

∫d r�p, n(r) = 1. In this

work we consider high-energy scattering, so the electron masscan be neglected compared to its energy. In this limit the cross

∗Electronic address: A.I.Milstein@inp.nsk.su†Electronic address: sushkov@phys.unsw.edu.au

sections dσ+/d� and dσ−/d� correspond to scattering fromthe potentials VC(r) + A(r) and VC(r) − A(r), respectively.The PV asymmetry is due to interference between the Coulombamplitude and the amplitude corresponding to the axialpotential A(r). Since neutrons dominate in the weak-chargedensity �W (r) (3), the asymmetry APV is sensitive to theshape of �n(r). Though the value of APV is small and itsmeasurement is complex, there is a proposal [3] to performsuch an experiment.

Detailed theoretical investigation of the asymmetry APV

has been performed by Horowitz [4], Vretenar et al. [5], andHorowitz et al. [6]. It has been shown that APV is sensitiveto the small difference between �p(r) and �n(r). It has alsobeen shown that the Coulomb distortions of the electron-wavefunctions significantly modify the asymmetry compared tothat obtained in the plane-wave approximation. In this caseradiative corrections can be enhanced. These corrections havenot been considered before.

It is known that the electron self-energy and vertexcorrections related to radiation of virtual and real photons arelarge, owing to double logarithms [7–9]. The vertex correctionto the Born scattering cross section is [10]

�σ = exp

(−2α

πln

−q2

m2ln

E

�

)− 1, (4)

where q is the momentum transfer, E and m are the electronenergy and mass, respectively, and � is the resolutionwith respect to bremsstrahlung. For the conditions of theproposal [3] (E = 850 MeV, q = 90 MeV,� = 4 MeV) thecorrection is −23%. However, the double-logarithm correctionis mainly due to soft photons and therefore cancels out in theparity violating asymmetry (1). The single-logarithm vertexcorrection, in our opinion, does influence the asymmetry.Calculation of the vertex correction is a difficult problem thatlies outside of the scope of the present work. In this articlewe calculate the vacuum polarization radiative correction toAPV. We find that, similar to radiative corrections to atomicparity violation [11,12], the correction we calculate here islogarithmically enhanced. The logarithm is ln(λC/r0) , wherer0 ≈ 1.5Z1/3 fm is the nuclear radius and λC is the electronCompton wavelength.

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A. I. MILSTEIN AND O. P. SUSHKOV PHYSICAL REVIEW C 71, 045503 (2005)

Let us first calculate the asymmetry (1) without anyradiative corrections. Such a calculation has been performedbefore in Refs. [4–6] using a summation over partial waves.Here we use a different method: the high-energy small-angleapproximation. The scattering amplitude in this approximationreads

f = − ip

2π

∫dρ exp[−iqρ]{exp[−iχ (ρ)] − 1},

(5)

χ (ρ) =∫ ∞

−∞dz V (r), r =

√z2 + ρ2,

where q = p2 − p1 is the momentum transfer, the z-axisis directed along the vector p = ( p2 + p1)/2, and ρ is atwo-dimensional vector orthogonal to p. Here p1 is the initialmomentum of the electron and p2 is the final momentum.The validity of Eq. (5) has been discussed in numerous works(see the book in Ref. [13] and references therein). At first,this formula was derived using wave functions in the eikonalapproximation. Then, Olsen et al. [14] demonstrated that therange of validity is wider: The semiclassical approximation issufficient to justify Eq. (5) even if the eikonal approximationis not valid. Corrections to Eq. (5) are considered in Ref. [15];see also Ref. [16]. For high-energy small-angle scatteringthe semiclassical approximation, and hence formula (5), iswell justified and we will use it. From Eqs. (1) and (5),we obtain

APV = 2Re[f ∗

CfW ]

|fC |2 ,

fC = − ip

2π

∫dρ exp[−iqρ]{exp[−iχC(ρ)] − 1},

fW = − p GF

4√

2π

∫dρ exp[−iqρ − iχC(ρ)] (6)

×∫ ∞

−∞dz �W (r),

χC(ρ) =∫ ∞

−∞dz VC(r).

To simplify further numerical integrations, it is convenient toexpress χC(ρ) in terms of �p,

χC(ρ) = 2Zα �(ρ),

�(ρ) = ln(ρ/L) + 4π

∫ ∞

ρ

drr�p(r) (7)

×[r ln

(r +

√r2 − ρ2

ρ

)−

√r2 − ρ2

].

Here L � r0 is an arbitrary constant; the cross sections andthe asymmetry are independent of the constant. The function�(ρ) has the following properties:

�(ρ) ={

4π∫ ∞

0 drr2�p(r) ln(2r/L) − 1, at ρ → 0,

ln(ρ/L) , at ρ � r0.(8)

Taking in Eq. (6) the integral over the angle of the vector ρ

and using Eq. (7), we obtain

fC = 2Zαp

q2

{4π

∫ ∞

0dρρ J0(qρ) exp[−2iZα�(ρ)]

×∫ ∞

ρ

drr�p(r)√r2 − ρ2

− 2iZα

∫ ∞

0

dρ

ρJ0(qρ)

× exp[−2iZα�(ρ)] �2(ρ)

},

(9)

fW = −p GF√2

∫ ∞

0dρρJ0(qρ) exp[−2iZα�(ρ)]

×∫ ∞

ρ

dr r �W (r)√r2 − ρ2

,

where

�(ρ) = ρ∂�(ρ)

∂ρ= 1 − 4π

∫ ∞

ρ

drr�p(r)√

r2 − ρ2 ,

and J0(x) is the Bessel function. Amplitudes in the form (9)are very convenient for numerical integration because of thefast convergence of the integrals. In the Born approximation(plane-wave approximation) the amplitudes and the asymme-try are

f BC = 2Zαp

q2Fp(q), f B

W = −p GF FW (q)

4π√

2,

ABPV = − q2GF FW (q)

4π√

2ZαFp(q)= q2GF

4π√

2α(10)

×[NFn(q)

ZFp(q)+ 4 sin2 θW − 1

],

where Fp(q), Fn(q), and FW (q) are Fourier transforms (formfactors) of corresponding densities. If the neutron and protondistributions coincide, �n(r) = �p(r), then the Born approxi-mation asymmetry (10) is given by

A0 = GF q2

4π√

2α

[N

Z+ 4 sin2 θW − 1

]. (11)

We will use this asymmetry as a reference point. Figure 1shows the dependence of the ratio APV/A0 on the momentumtransfer q for Pb (Z = 82) at �n(r) = �p(r) (solid curve) andat �n(r) = 0.953�p(0.95r) (dashed curve). For �p(r) we usethe three-parameter Fermi charge density [4].

�p(r) = �01 + w(r/r0)2

1 + exp[(r − r0)/a],

r0 = 6.4 fm, a = 0.54 fm, (12)

w = 0.32, �0 = 6.95 × 10−4 fm−3.

The plots in Fig. 1 are in excellent agreement with the results ofthe partial wave analysis [4]. The figure clearly demonstratesa high sensitivity to the difference between �n(r) and �p(r)as well as the importance of the Coulomb distortion. Thedistortion is especially important at small q. At q � 1/r0 the

045503-2

VACUUM POLARIZATION RADIATIVE CORRECTION TO . . . PHYSICAL REVIEW C 71, 045503 (2005)

0 0. 5 1 1.5 2-1

-0.5

0

0. 5

1

q(fm−1)

A PV/A

0

FIG. 1. Dependence of the ratio APV/A0 for Pb (Z = 82) onthe momentum transfer q. The solid line corresponds to �n(r) =�p(r) and the dashed line corresponds to �n(r) = λ3�p(λr) with λ =0.95. The reference Born approximation asymmetry A0 is given byEq. (11).

Coulomb and the weak amplitudes are of the form

fC = 2Zαp

q2

(qL

2

)2iZα�(1 − iZα)

�(1 + iZα),

(13)

fW = −p GF√2

∫ ∞

0dρρ exp[−2iZα�(ρ)]

∫ ∞

ρ

drr�W (r)√r2 − ρ2

.

So at q → 0 the weak amplitude fW is independent of q

whereas the phase of the Coulomb amplitude fC stronglydepends on q. This explains the change of sign of APV inFig. 1 at small q.

Let us proceed now to the vacuum polarization radiativecorrection. We use the Uehling potential [17] to describethe effect of vacuum polarization. It is known that there isa Wichmann-Kroll correction to the Uehling potential [18].However, this correction is very small, even at Zα ∼ 1 (seethe review paper [19]), and therefore we neglect it. The chargedensity �vp(r) induced by polarization of the electron-positronvacuum, expressed in units of the elementary charge |e|, is ofthe form (see, e.g., Ref. [19])

�vp(r) = Zα

3π

∫ ∞

1dτ

√τ − 1

τ

(τ + 1/2)

τ 2

×{�p(r) − m2τ

π

∫d R

exp(−2m√

τ |r − R|)|r − R| �p(R)

}.

(14)

This charge density corresponds to the Uehling potential[17]. For small distances, r � λC , expression (14) can betransformed to

�vp(r) → 2Zα

3π

{[ln(λC/r) − C − 5/6)] �p(r)

+ 1

r

∫ r

0dR �p(R) − 1

2

∫ ∞

0dR

�p(R)

r + R

0 2 4 6 8 10 12-0. 2

0.0

0.2

0.4

0.6

0.8

1.0

r(fm)

103 �(f

m−

3 )

FIG. 2. The solid line is the vacuum polarization charge density�vp(r) for Pb in units 10−3|e|/fm3. The dashed line is the protondensity �p(r) in units 10−3 fm−3. The proton density is normalized tounity.

− 1

2

∫ 2r

0dR

�p(R) − �p(r)

|R − r|− 1

2

∫ ∞

2r

dR�p(R)

R − r

}, (15)

where C ≈ 0.577 is the Euler constant. Note that Eq. (15) is notsingular at r = 0, as all the divergent terms cancel out. Outsidethe nucleus, at r � r0, Eq. (14) is equivalent to the well-knownformula for the induced charge density of a pointlike nucleus

�vp(r) = −2Zα m2

3π2r

∫ ∞

1dx

√x2 − 1

×(

1 + 1

2x2

)exp(−2mrx). (16)

At λC � r � r0 it gives

�vp(r) = − Zα

6π2r3. (17)

A direct numerical integration of (14) is straightforward.In Fig. 2 the solid line shows the vacuum polarizationcharge density �vp(r) for Pb. For comparison the dashed linepresents the proton density (12), which is normalized to unity,

0 0.5 1 1.5 2

0

-0.5

-1

-1.5

-2

q(fm−1)

∆(%

)

FIG. 3. The vacuum polarization radiative correction to the parityviolation asymmetry (%) versus the momentum transfer.

045503-3

A. I. MILSTEIN AND O. P. SUSHKOV PHYSICAL REVIEW C 71, 045503 (2005)

∫d r�p(r) = 1. We see that in the vicinity of the nucleus the

vacuum polarization charge density is practically proportionalto the proton density. With logarithmic accuracy this factimmediately follows from Eq. (15):

�vp(r) ∼ 2Zα

3πln

(λC

r0

)�p(r). (18)

It is interesting that inside a sphere of radius 8 fm the vacuumpolarization charge is rather large and equal to 0.47|e|. Soone can say that the electric charge of the Pb nucleus is82.47. Certainly the excess charge (0.47) is compensated bya negative charge of −0.47 distributed between r0 and λC

[see Eq. (17)]. Using Eqs. (15), (16), and (9), one can easilycalculate the asymmetry Atot that accounts for the effect ofvacuum polarization. One should replace �p(r) in (9) by�p(r) + 1

Z�vp(r). Note that it is not necessary to account

for the effect of vacuum polarization on �W (r) because thecorresponding correction contains the additional suppressionfactor 1 − 4 sin2 θW in the electron-Z0 vector vertex in theelectron loop. With this account of the radiative correction,the parity violating asymmetry is reduced:

APV → APV(1 + �). (19)

The value of the vacuum polarization radiative correction �

versus the momentum transfer q is plotted in Fig. 3. In the

interesting range of momenta, q = 0.2 − 2 fm−1, the radiativecorrection is about −1%. Note that, except for the very smallWichmann-Kroll correction that has been neglected, the resultshown in Fig. 3 is exact in Zα. At Zα � 1 an estimatefor the polarization operator correction to the PV asymmetrycan be easily obtained using the Born approximation. In thisapproximation only the denominator Fp(q) in Eq. (10) isinfluenced by the correction. Therefore the radiative correctiondue to (18) is

�(Born) ∼ − 2α

3πln

(λC

r0

)≈ −0.7%. (20)

This estimate agrees surprisingly well with the exact result forPb presented in Fig. 3.

We have calculated the effect of vacuum polarization onthe parity violating asymmetry in elastic electron-nucleusscattering. The vacuum polarization radiative correction islogarithmically enhanced and the value of the correction isabout −1%.

A.I.M. gratefully acknowledges the School of Physics atthe University of New South Wales for warm hospitality andfinancial support during a visit. The work was also supportedby RFBR Grant No. 03-02-16510.

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