Neutrino-impact ionization of atoms in searches for neutrino magnetic moment

Konstantin A. Kouzakov*

Department of Nuclear Physics and Quantum Theory of Collisions, Faculty of Physics, Moscow State University,Moscow 119991, Russia

Alexander I. Studenikin†

Department of Theoretical Physics, Faculty of Physics, Moscow State University, Moscow 119991, Russia

Mikhail B. Voloshin‡

William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USAand Institute of Theoretical and Experimental Physics, Moscow, 117218, Russia

(Received 28 January 2011; published 1 June 2011)

The ionization of atomic electrons by scattering of neutrinos is revisited. This process is the one studied

in the experimental searches for a neutrino magnetic moment using germanium detectors. Current

experiments are sensitive to the ionization energy comparable with the atomic energies, and the effects

of the electron binding should be taken into account. We find that the so-called stepping approximation to

the neutrino-impact ionization is in fact exact in the semiclassical limit and also that the deviations from

this approximation are very small already for the lowest bound Coulomb states. We also consider the

effects of electron-electron correlations and argue that the resulting corrections to the ionization of

independent electrons are quite small. In particular, we estimate that in germanium these are at a 1% level

at the energy transfer down to a fraction of keV. Exact sum rules are also presented as well as analytical

results for a few lowest hydrogenlike states.

DOI: 10.1103/PhysRevD.83.113001 PACS numbers: 13.15.+g, 14.60.St

I. INTRODUCTION

The neutrino magnetic moments (NMM) expected in thestandard model are very small and proportional to theneutrino masses [1]: �� � 3� 10�19�Bðm�=1 eVÞ with�B ¼ e=2m being the electron Bohr magneton, and m isthe electron mass. Thus any larger value of �� can ariseonly from physics beyond the standard model (a recentreview of this subject can be found in Ref. [2]). Currentdirect experimental searches [3–5] for a magnetic momentof the electron (anti)neutrinos from reactors have loweredthe upper limit on �� down to �� < 3:2� 10�11�B [5].These ultralow background experiments use germaniumcrystal detectors exposed to the neutrino flux from a reactorand search for scattering events by measuring the energy Tdeposited by the neutrino scattering in the detector. Thesensitivity of such a search to NMM crucially depends onlowering the threshold for the energy transfer T, due tothe enhancement of the magnetic scattering relative to thestandard electroweak one at low T. Namely, the differentialcross section d�=dT is given by the incoherent sum of themagnetic and the standard cross section, and for the scat-tering on free electrons the NMM contribution is given bythe formula [6,7]

d�ð�ÞdT

¼ 4���2�

�1

T� 1

E�

�¼ �

�2

m2

���

�B

�2�1

T� 1

E�

�;

(1)

where E� is the energy of the incident neutrino anddisplays a 1=T enhancement at low energy transfer.The standard electroweak contribution is constant in Tat E� � T:

d�EW

dT¼ G2

Fm

2�ð1þ 4sin2�W þ 8sin4�WÞ

�1þO

�T

E�

��

� 10�47 cm2=keV: (2)

In what follows we refer to these two types of contributionto the scattering as, respectively, the magnetic and theweak.The current experiments have reached threshold values

of T as low as a few keV and are likely to further improvethe sensitivity to low energy deposition in the detector. Atlow energies however one can expect a modification of thefree-electron formulas (1) and (2) due to the binding ofelectrons in the germanium atoms, where, e.g., the energyof the K� line, 9.89 keV, indicates that at least some ofthe atomic binding energies are comparable to the alreadyrelevant experiment values of T. Thus a proper treatment ofthe atomic effects in neutrino scattering is necessary andimportant for the analysis of the current and even more ofthe future data with a still lower threshold. Furthermore,there is no known means of independently calibrating

*kouzakov@srd.sinp.msu.ru†studenik@srd.sinp.msu.ru‡voloshin@umn.edu

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experimentally the response of atomic systems, such as thegermanium, to the scattering due to the interactions rele-vant for the neutrino experiments. Therefore one has to relyon a pure theoretical analysis in interpreting the neutrinodata. For the first time this problem was addressed inRef. [8], where a 2–3 times enhancement of the electro-weak cross section in the case of ionization from a 1sstate of a hydrogenlike atom with nuclear charge Zhad been numerically determined at neutrino energiesE� � �Zmc2. Subsequent numerical calculations withinthe Hartree-Fock-Dirac method for ionization from innershells of various atoms showed much lower enhancement(� 5–10%) of the electroweak contribution [9–12]. Theinterest in the role of atomic effects was renewed in severalrecent papers, which however are written with a ‘‘trial anderror’’ approach. The early claim [13] of a significantenhancement of the NMM contribution by the atomiceffects was later disproved [14,15] and it was argued [14]that the modification of Eqs. (1) and (2) by the atomicbinding effects is insignificant down to very low values ofT. It was subsequently pointed out [16] that the analysis ofRef. [14] is generally invalidated in multielectron systems,including atoms with Z > 1. Furthermore, the analysis ofRef. [14] is also generally invalidated by singularitiesof the relevant correlation function in the complex planeof momentum transfer, so that the claimed behavior of thecross section at low T applies only in the semiclassicallimit, although, as will be shown here, it gives a very goodapproximation to the actual behavior for an electron boundby a Coulomb potential.

In this paper we revisit the subject of neutrino scatteringon atoms at low energy transfer. We aim at describing thisprocess at T in the range of a few keVand lower, so that themotion of the electrons is considered as strictly nonrela-tivistic. Also in this range the energy of the dominant partof the incident neutrinos from the reactor is much largerthan T and we thus neglect any terms whose relative valueis proportional to T=E� [[in particular, in this range one canneglect the 1=E� term in Eq. (1) in comparison with 1=T].Furthermore, any recoil of the germanium atom as a wholeresults in an energy transfer less than 2E2

�=MGe, which atthe typical reactor neutrino energy is well below the con-sidered keV range of the energy transfer. Thus we formallyset the mass of the atomic nucleus to infinity and neglectany recoil by the atom as a whole. In particular, under theseconditions the interaction of the neutrino with the nucleuscan be entirely neglected, and only the scattering on theatomic electrons is to be considered.

We find that in the scattering on realistic atoms, such asgermanium, the so-called stepping approximation workswith very good accuracy. The stepping approach, intro-duced in Ref. [11] from an interpretation of numerical data,treats the process as scattering on individual independentelectrons occupying atomic orbitals and suggests thatthe cross section follows the free-electron behavior in

Eqs. (1) and (2) down to T equal to the ionization thresholdfor the orbital, and that below that energy the electron onthe corresponding orbital is ‘‘deactivated,’’ thus producinga sharp ‘‘step’’ in the dependence of the cross section on T.In the present paper, we consider general relations for thediscussed scattering on atomic systems in Sec. II andpresent in Appendix A sum rules for the theoretical objectsinvolved in the calculations.1 In Sec. III we prove that forthe scattering on individual electrons the stepping approxi-mation becomes exact in the semiclassical limit, so that itsapplicability is improved with the principal number n ofthe atomic orbital. We also find by an explicit calculation(Appendix B) for a hydrogenlike ground state, i.e., atn ¼ 1, that the deviation from the stepping behavior isless than 5% at the worst point, where the energy transferT is exactly at the threshold. The accuracy of the approachbased on considering the scattering on individual electronsis limited by the existence of the electron-electron corre-lations in the process. We consider the correction intro-duced by these correlations in Sec. IVand, in Sec. V, applythe derived formula to an estimate of the effect for germa-nium, using the Thomas-Fermi model. We find that thecorrelation correction grows at smaller T but is still small,of the order of a few percent, for T in the range of a fewhundred eV. We thus argue that the stepping approachdescribes the scattering cross section with a sufficient forpractical purposes accuracy, and that it can be applied tothe analysis of the present and future data of searches forNMM with germanium detectors down to the values of theenergy deposition T � 0:3 keV.

II. GENERAL FORMULAS FOR NEUTRINOSCATTERING ON ATOMIC ELECTRONS

In this section we briefly recapitulate the generalexpressions and introduce the relevant atomic objects forthe neutrino scattering on atomic electrons. We start withthe magnetic process and then also apply a similar treat-ment to the standard weak part of the cross section.The kinematics of the scattering of a neutrino on atomic

electrons is generally characterized by the componentsof the four-momentum transfer, the energy transfer T,and the spatial momentum transfer ~q, from the neutrinoto the electrons with two rotationally invariant variablesbeing T and q2 ¼ ~q2. At small T the electrons can betreated nonrelativistically both in the initial and in the finalstate, so that the process is that of scattering of an NMM

in the electromagnetic field A ¼ ðA0; ~AÞ of the electrons:

A0ð ~qÞ ¼ffiffiffiffiffiffiffiffiffiffi4��

p�ð ~qÞ= ~q2, ~Að ~qÞ ¼ ffiffiffiffiffiffiffiffiffiffi

4��p

~jð ~qÞ= ~q2, where

�ð ~qÞ and ~jð ~qÞ are the Fourier transforms of the electronnumber density and current density operators, respectively,

1The sum rules of Appendix A correct the omissions made inRef. [14].

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�ð ~qÞ ¼ XZa¼1

expði ~q � ~raÞ; (3)

~jð ~qÞ ¼ � i

2m

XZa¼1

�expði ~q � ~raÞ @

@~raþ @

@~raexpði ~q � ~raÞ

�;

(4)

and the sums run over the positions ~ra of all the Z electronsin the atom.

In this limit the expression for the double differentialcross section is given by [16]

d2�ð�ÞdTdq2

¼ 4���2

�

q2

��1� T2

q2

�SðT; q2Þ

þ�1� q2

4E2�

�RðT; q2Þ

�; (5)

where SðT; q2Þ, also known as the dynamical structurefactor [17], and RðT; q2Þ are

SðT; q2Þ ¼ Xn

�ðT � En þ E0Þjhnj�ð ~qÞj0ij2; (6)

RðT; q2Þ ¼ Xn

�ðT � En þ E0Þjhnjj?ð ~qÞj0ij2; (7)

with j? being the ~j component perpendicular to ~q andparallel to the scattering plane, which is formed bythe incident and final neutrino momenta. The sums inEqs. (6) and (7) run over all the states jni with energiesEn of the electron system, with j0i being the initial state.

Clearly, the factors SðT; q2Þ and RðT; q2Þ are relatedto, respectively, the density-density and current-currentGreen’s functions

FðT; q2Þ ¼ Xn

jhnj�ð ~qÞj0ij2T � En þ E0 � i

¼�0

���������ð� ~qÞ 1

T �H þ E0 � i�ð ~qÞ

��������0�; (8)

LðT; q2Þ ¼ Xn

jhnjj?ð ~qÞj0ij2T � En þ E0 � i

¼�0

��������j?ð� ~qÞ 1

T �H þ E0 � ij?ð ~qÞ

��������0

�; (9)

as

SðT; q2Þ ¼ 1

�ImFðT; q2Þ; (10)

RðT; q2Þ ¼ 1

�ImLðT; q2Þ; (11)

with H being the Hamiltonian for the system of electrons.For small values of q, in particular, such that q� T,only the lowest-order nonzero terms of the expansion of

Eqs. (10) and (11) in powers of q2 are of relevance (the so-called dipole approximation). In this case, one has [16]

RðT; q2Þ ¼ T2

q2SðT; q2Þ: (12)

Taking into account Eq. (12), the experimentally mea-sured singe-differential inclusive cross section is, to a goodapproximation, given by (see e.g. Refs. [14,16])

d�ð�ÞdT

¼ 4���2�

Z 4E2�

T2SðT; q2Þdq

2

q2: (13)

The standard electroweak contribution to the cross sec-tion can be similarly expressed in terms of the same factorSðT; q2Þ [14] asd�EW

dT¼ G2

F

4�ð1þ 4sin2�W þ 8sin4�WÞ

Z 4E2�

T2SðT; q2Þdq2;

(14)

where the factor SðT; q2Þ is integrated over q2 with a unitweight, rather than q�2 as in Eq. (13).The kinematical limits for q2 in an actual neutrino

scattering are explicitly indicated in Eqs. (13) and (14).At large E�, typical for the reactor neutrinos, the upperlimit can in fact be extended to infinity, since in thediscussed here nonrelativistic limit the range of momenta�E� is indistinguishable from infinity. The lower limit canbe shifted to q2 ¼ 0, since the contribution of the regionof q2 < T2 can be expressed in terms of the photoelectriccross section [14] and is negligibly small (at the level ofbelow 1% in the considered range of T). For this reason wehenceforth discuss the momentum-transfer integrals inEqs. (13) and (14) running from q2 ¼ 0 to q2 ¼ 1:

I1ðTÞ ¼Z 1

0SðT;q2Þdq

2

q2; and I2ðTÞ ¼

Z 1

0SðT;q2Þdq2:

(15)

For a free electron, which is initially at rest, the density-density correlator is the free particle Green’s function

FðFEÞðT; q2Þ ¼�T � q2

2m� i

��1; (16)

so that the dynamical structure factor is given bySðFEÞðT; q2Þ ¼ �ðT � q2=2mÞ, and the discussed here inte-

grals are in the free-electron limit as follows:

IðFEÞ1 ¼Z 1

0SðFEÞðT; q2Þ dq

2

q2¼ 1

T;

IðFEÞ2 ¼Z 1

0SðFEÞðT; q2Þdq2 ¼ 2m:

(17)

Clearly, these expressions, when used in Eqs. (13) and (14),result in the free-electron cross section in Eqs. (1) and (2).

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III. SCATTERING ON ONE BOUND ELECTRON

The binding effects generally deform the density-densityGreen’s function, so that both the integrals (15) are some-what modified. Namely, the binding effects spread the free-electron � peak in the dynamical structure function atq2 ¼ 2mT and also shift it by the scale of characteristicelectron momenta in the bound state. However, it turnsout that the free-electron expressions are quite robust inthe sense that in realistic systems the modifications ofthe integrals, relative to their free-electron limit, are quitesmall. As a formal statement, we will show in Appendix Athat when the function FðT; q2Þ is analytically continued inthe complex plane of q2 the free-electron expressions arevalid for the integrals over q2 extending from �1 to þ1,and in the case of the integral, similar to I2 i.e. with theweight q0, this property also holds for scattering on multi-electron atomic systems, while for that with the weight q�2

it generally holds only for the scattering on one electron, oron independent electrons. Clearly, the latter integrals overthe full axis of q2 differ from those of physical interest inEq. (15) by the contribution of negative q2, which althoughnumerically small even at low T still makes the scatteringon bound electrons different from that on free electrons.

In this section we consider the scattering on just oneelectron. The Hamiltonian for the electron has the formH ¼ p2=2mþ VðrÞ, and the density-density Green’s func-tion from Eq. (8) can be written as

FðT; q2Þ ¼ h0je�i ~q� ~r½T �Hð ~p; ~rÞ þ E0��1ei ~q� ~rj0i¼ h0j½T �Hð ~pþ ~q; ~rÞ þ E0��1j0i

¼�0

���������T � ~q2

2m� ~p � ~q

m�Hð ~p; ~rÞ þ E0

��1��������0

�;

(18)

where the infinitesimal shift T ! T � i is implied.Clearly, a nontrivial behavior of the latter expression in

Eq. (18) is generated by the presence of the operatorð ~p � ~qÞ in the denominator, and the fact that it does notcommute with the Hamiltonian H. Thus, an analyticalcalculation of the Green’s function as well as the dynami-cal structure factor is feasible in only a few specific prob-lems. In Appendix B we present such a calculation forionization from the 1s, 2s, and 2p hydrogenlike states. Inparticular, we find analytically that the deviation of thediscussed integrals (15) from their free values are verysmall: the largest deviation is exactly at the ionizationthreshold, where, for instance, each of the 1s integrals isequal to the free-electron value multiplied by the factorð1� 7e�4=3Þ � 0:957.2 Our findings thus substantiate the

results of the theoretical analysis carried out in Ref. [16],where the 1s case was examined numerically.The problem of calculating the integrals (15) however

can be solved in the semiclassical limit, where one canneglect the noncommutativity of the momentum ~pwith theHamiltonian, and rather treat this operator as a numbervector. Taking also into account that ðH � E0Þj0i ¼ 0, onecan then readily average the latter expression in Eq. (18)over the directions of ~q and find the formula for thedynamical structure factor:

SðT; q2Þ ¼ m

2pq

��

�T � q2

2mþ pq

m

�� �

�T � q2

2m� pq

m

��;

(19)

where p ¼ j ~pj and � is the standard Heaviside step func-tion. The expression in Eq. (19) is nonzero only in therange of q satisfying the condition �pq=m< T �q2=2m< pq=m, i.e. between the (positive) roots ofthe binomials in the arguments of the step functions:

qmin ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mT þ p2

p � p and qmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mT þ p2

p þ p.One can notice that the previously mentioned ‘‘spreadand shift’’ of the peak in the dynamical structure functionin this limit corresponds to a flat pedestal between qmin andqmax. The calculation of the integrals (15) with the expres-sion (19) is straightforward and yields the free-electronexpressions (17) for the discussed here integrals in thesemiclassical (WKB) limit3:

IðWKBÞ1 ¼ 1

T; IðWKBÞ

2 ¼ 2m: (20)

The difference from the pure free-electron case however isin the range of the energy transfer T. Namely, the expres-sions (20) are applicable in this case only above the ion-ization threshold, i.e. at T � jE0j. Below the threshold theelectron becomes ‘‘inactive.’’It is instructive to point out that the validity of the result

in Eq. (20) is based on the semiclassical approximation andis not directly related to the value of the energy T. Inparticular, for a Coulomb interaction the WKB approxi-mation is applicable at energy near the threshold [18]. ForT exactly at the threshold, T ¼ �E0, the criterion forapplicability of the semiclassical approach in terms of

the force F ¼ j ~Fj ¼ j½ ~p;H�j acting on the electron andthe momentum p of the electron is that [18] the ratio of thecharacteristic values mF=p3 is small. For the excitation ofa state with the principal number n this ratio behavesparametrically as 1=n.4Thus the applicability of a semi-classical treatment of the ionization near the thresholdimproves for initial states with large n. As previously

2It can also be noted that both integrals are modified in exactlythe same proportion, so that their ratio is not affected at any T:I2ðTÞ=I1ðTÞ ¼ 2mT. We find, however, that this exact propor-tionality is specific for the ionization from the ground state in theCoulomb potential.

3The appearance of the free-electron expressions here is notsurprising, since Eq. (19) can also be viewed as the one forscattering on an electron boosted to the momentum p.

4Indeed one has jFj ¼ �=r2 �m2�3n4 and p�m�=n, sothat mjFj=p3 � 1=n.

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mentioned, the modification of the integrals (15) by thebinding is already less than 5% for n ¼ 1, so that we fullyexpect this deviation to be smaller for the higher states, andeven smaller at larger values of T above the threshold dueto the approach to the free-electron behavior at T � E0.

We believe that the latter conclusion explains the so-called stepping behavior observed empirically [11] in theresults of numerical calculations. Namely, the calculatedcross section d�=dT for ionization of an electron from anatomic orbital follows the free-electron dependence on Tall the way down to the threshold for the correspondingorbital with a very small, at most a few percent, deviation.This observation led the authors of Ref. [11] to suggest thestepping approximation for the ratio of the atomic crosssection (per target electron) to the free-electron one:

fðTÞ d�=dT

ðd�=dTÞFE ¼ 1

Z

Xi

ni�ðT � jEijÞ; (21)

where the sum runs over the atomic orbitals with thebinding energies Ei and the filling numbers ni. Clearly,the factor fðTÞ simply counts the fraction of ‘‘active’’electrons at the energy T, i.e., those for which the ioniza-tion is kinematically possible. For this reason we refer tofðtÞ as an activation factor. We conclude here that thestepping approximation is indeed justified with a highaccuracy in the approximation of the scattering on inde-pendent electrons, i.e., if one neglects the two-electroncorrelations induced by the interference of terms in theoperator �ð ~qÞ in Eq. (3) corresponding to different elec-trons. In the next section we estimate the effect of suchan interference and find that the resulting correctionsare small, at least in atoms with large Z, such as thegermanium.

IV. TWO-ELECTRON CORRELATION

In this section we discuss the correction arising from acorrelation between two electrons. We consider the energyT and hence the relevant momentum transfer q as large incomparison with the atomic scale. In this way we estimatethe relevant parameter for the significance of the correla-tion effect.

We start by considering an isolated system of two elec-trons interacting among themselves through the Coulombpotential VðrÞ. The Hamiltonian for this system thus hasthe form

H ¼ P2

4mþ p2

mþ VðrÞ; (22)

where ~P ¼ �i@=@ ~R and ~p ¼ �i@=@~r are, as usual, themomenta conjugate to, respectively, the center of mass

coordinate ~R and the relative coordinate ~r.The spatial part of the wave function of the system

factorizes into the product ð ~RÞc ð ~rÞ, with ð ~RÞ andc ð~rÞ being, respectively, the center of mass and the relative

motion position-space wave functions, while the spin partwill be considered later. We consider here the system atrest, i.e., ðRÞ ¼ const, since a boost to a momentum Pdoes not change the cross section.The density-density Green’s function then takes the

form

FðT;q2Þ ¼Xn

h0jei ~q� ~r=2þ e�i ~q� ~r=2jnihnjei ~q� ~r=2þ e�i ~q� ~r=2j0iT� q2

4m�EnþE0

;

(23)

where the states j0i, jni and the energies E0, En refer tothe relative motion in the system with j0i standing for theinitial state. Clearly, it is implied in Eq. (23) that thecorresponding matrix elements for the (trivial) dynamicsof the system as a whole are already taken, which results inreplacing in the energy denominator the excitation energyT by its value corrected for the recoil of the system as awhole: T ! T � q2=4m.The cross terms between expði ~q � ~r=2Þ and

expð�i ~q � ~r=2Þ in the expression (23) result in the previ-ously discussed one-particle Green’s function

F1ðT; q2Þ ¼ 2

�0

���������T � q2

2m� ~p � ~q

m�H þ E0

��1��������0

�;

(24)

where the overall factor of 2 arises from the two identical(after averaging over the direction of ~q) cross terms, andphysically is corresponding to the presence of two particlesin the system.The contribution discussed here of the two-electron

correlation arises from the diagonal terms, whose contri-bution is given by

FcðT; q2Þ ¼ 2�Xn

h0jei ~q� ~r=2jnihnjei ~q�~r=2j0iT � q2

4m � En þ E0

; (25)

where, again, the two terms give the same contributionafter the averaging over the direction of ~q, which is ac-counted for by the factor of 2 in the latter expression. Thefactor � is the symmetry factor for the spin part of the two-electron system: � ¼ �1 for the spin-singlet state of thepair and � ¼ þ1 for the spin-triplet. The appearance ofthis factor can be explained as follows. The discussedcorrelation arises from the situation where an excitationof one electron by the operator � produces the same spatialwave function as an excitation of another one. In order forthe wave functions to be identical the spin variables of thetwo electrons should also be switched, in which the opera-tion results in the factor �. Clearly, no such factor arises inthe one-particle term (24) since the spin of both electronssimply ‘‘goes through.’’ It can also be mentioned that,naturally, the symmetry of the spatial wave function c ð ~rÞis opposite to �.One can notice that unlike in the one-particle contribu-

tion [Eq. (24)], where the momentum ~q flows in and out of

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the system, the correlation contribution in Eq. (25) corre-sponds to the net momentum ~q flowing into the system.Clearly, for noninteracting particles such contributionwould vanish and the whole correlation effect arises onlydue to the interaction between the electrons, in which theinteraction absorbs the momentum transfer. The term (25)can be graphically represented as shown in Fig. 1, wherethe system lines correspond to the propagation of thesystem in the potential V with the outer legs correspondingto the wave functions of the initial state in the momentumspace hc ð ~p0Þj and jc ð ~pÞi and the line between the action ofthe operators expði ~q � ~r=2Þ corresponding to the Green’sfunction. One can write in terms of these objects theexpression for FcðT; q2Þ as

FcðT; q2Þ ¼ 2Z d3p0

ð2�Þ3Z d3p

ð2�Þ3 cð ~p0Þ

�G

�E; ~p0 � ~q

2; ~pþ ~q

2

�c ð ~pÞ; (26)

where GðE; ~k0; ~kÞ is the Green’s function ðE�HÞ�1 in themomentum representation at the energy E ¼ T þ E0 �q2=4m. We shall consider separately the effect of theinteraction in the wave functions and in the Green’s func-tion. For the zeroth order Green’s function

G0ðE; ~k0; ~kÞ ¼ ð2�Þ3�ð3Þð ~k0 � ~kÞE� k2

m

(27)

and the exact wave functions one finds

Fð0Þc ðT; q2Þ ¼ 2�

Z d3p

ð2�Þ3c ð ~pþ ~qÞc ð ~pÞ

T � q2

2m � ð ~p� ~qÞm � p2

m þ E0

:

(28)

Let us consider now q as a large parameter in compari-son with the characteristic momenta p0 in c ð ~pÞ, beyondwhich the wave function falls off. At such values of q theproduct c ð ~pþ ~qÞc ð ~pÞ carries a suppression in only oneof the factors in two regions of ~p: one where p� p0 andthe other where j ~pþ ~qj � p0. Clearly, by shifting theintegration variable ~pþ ~q ! ~p one can readily see thatthe contribution of the latter region is the same as of thefirst one, so that one evaluates the integral in Eq. (28) byconsidering only the contribution of the region p � q andtaking it with a factor of 2. Then the leading at large qexpression for the function in Eq. (28) takes the form

Fð0Þc ðT; q2Þ ¼ 4�

c ð ~qÞT � q2

2m

Z d3p

ð2�Þ3 c ð ~pÞ ¼ 4�c ð ~qÞc ð0ÞT � q2

2m

:

(29)

The appearance of the wave function at the origin ~r ¼ 0,c ð0Þ, in this expression implies that at large q the consid-ered contribution to the correlation correction arises onlyfor an S-wave relative motion within the electron pair,which thus has to be a spin singlet, and therefore � ¼ �1.In fact, for a S-wave motion the momentum-space wave

function c ð ~qÞ can also be expressed at large q in terms ofthe position-space wave function at the origin c ð0Þ.Indeed, in the S-wave case the wave function is a functionof r: c ðrÞ. At small r the Coulomb repulsion between theelectrons dominates over all other interactions and theSchrodinger equation for c ðrÞ reads

� 1

mc 00 � 2

mrc 0ðrÞ þ �

rc ðrÞ ¼ Ec ðrÞ: (30)

By requiring the two singular at r ! 0 as 1=r terms tomatch in this expression, one finds that the derivative ofc ðrÞ over r at the origin is expressed in terms of c ð0Þ:

c 0ð0Þ ¼ m�

2c ð0Þ: (31)

A finite derivative over r implies that the gradient~rc ðrÞ ¼ ~rc 0ðrÞ=r is singular at r ¼ 0, so that the asymp-totic at large j ~qj behavior of the momentum-space wavefunction is proportional to 1=j ~qj4 with the coefficient de-termined by c 0ð0Þ, which in turn is determined, accordingto Eq. (31), by c ð0Þ:

c ð ~qÞjj ~qj!1 ¼ � 4�m�c ð0Þj ~qj4 : (32)

Using this relation in Eq. (29) one finds in the large q2 limit

Fð0Þc ðT; q2Þ ¼ 16��

mjc ð0Þj2q4

1

T � q2

2m

: (33)

The latter expression is manifestly proportional to thefirst power of the interaction between the electrons.Therefore a similar contribution can arise from the firstorder in the expansion of the Green’s function in theinteraction potential V. In this order one finds for thediscussed correlation part of FðT; q2Þ:Fð1Þc ðT;q2Þ

¼ 2�Z d3p0

ð2�Þ3Z d3p

ð2�Þ3

� c ð ~p0ÞVð� ~qþ ~p0 � ~pÞc ð ~pÞ½T� q2

2mþð ~p0� ~qÞm � p02

m þE0�½T� q2

2m�ð ~p� ~qÞm � p2

m þE0�;

(34)

where Vð ~kÞ is the Fourier transform of the potential, so thatfor the Coulomb repulsion between the electrons

FIG. 1. Graphical representation of the two-electron correla-tion. The external legs correspond to the momentum-spacewave function and the propagator is the Green’s function inthe potential V.

KOUZAKOV, STUDENIKIN, AND VOLOSHIN PHYSICAL REVIEW D 83, 113001 (2011)

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Vð ~kÞ ¼ 4��

k2: (35)

Considering as before the limit of large q and thusneglecting p and p0 in comparison with q, one readilyfinds that the result is again proportional to jc ð0Þj2,so that the effect remains only in the S wave (and hence� ¼ �1):

Fð1Þc ðT; q2ÞjmT;q2�p2;p02 ¼ �8��

jc ð0Þj2q2ðT � q2

2mÞ2: (36)

Collecting Eqs. (33) and (36) together one finds theestimate of the two-electron correlation part of FðT; q2Þin the limit of large T and q2:

FcðT; q2Þ ¼ 8��jc ð0Þj2�

2m

q4ðT � q2

2m� 1

q2ðT � q2

2mÞ2�:

(37)

The corresponding two-electron correlation correction tothe integrals for the neutrino scattering cross section is thencalculated by shifting T ! T � i and considering onlythe contribution of the singularity at q2 ¼ 2mT:

1

�

ZImFcðT; q2Þdq

2

q2¼ �4��

jc ð0Þj2mT3

;

1

�

ZImFcðT; q2Þdq2 ¼ 0:

(38)

Notice that the two-electron contribution to the integralrelevant for the standard electroweak scattering vanishes inthe discussed approximation due to a cancellation of thetwo terms in Eq. (37).

The above calculation shows, as expected, that at large qthe two-electron correlation arises only when the electronsare separated by a short distance. For this reason one canrelax the assumption we made in the beginning of thissection that the system of two electrons is in a free motion.Indeed the same result would apply in the situation, wherethe pair as a whole moves in a potential that is sufficientlysmooth so that the ‘‘tidal force’’ interaction with the rest ofthe atomic system does not overcome the Coulomb singu-larity of the repulsion between the electrons at distances oforder 1=q.

V. SCATTERING ON ATOMIC ELECTRONSIN GERMANIUM

In considering the neutrino scattering on actual atomsone needs to evaluate the dependence of the number ofactive electrons on T and generally also evaluate the effectof the two-electron correlations. The energies of the innerK, L, and M orbitals in the germanium atom are wellknown (see, e.g., Ref. [12], and references therein) andprovide the necessary data for a description of the neutrinoscattering by the stepping formula (21) down to the values

of the energy transfer T in the range of the binding of theMelectrons, i.e., at T > jEMj � 0:18 keV. The correspond-ing steps in the activation factor are shown in Fig. 2. It canbe mentioned that if one applies formulas of Appendix B tothe onset of the K shell step, i.e., just above 10.9 keV, thedifference in the plot step function would be practicallyinvisible in the scale of Fig. 2.Our goal in this section is to estimate the effect of the

two-electron correlations in the scattering on germanium.We shall estimate this effect by considering the atomicnumber Z as a large parameter and using the Thomas-Fermi model, which, in spite of its known shortcomings,appears to be appropriate for evaluating average bulkproperties of atomic electrons at large Z, such as in theproblem at hand.In the Thomas-Fermi model (see, e.g., Ref. [18]) the

atomic electrons are described as a degenerate free-electron gas in a master potential ðrÞ filling the momen-tum space up to the zero Fermi energy, i.e., up to themomentum p0ðrÞ such that p2

0=2m� e ¼ 0. The elec-

tron density nðrÞ ¼ p30=ð3�2Þ then determines the potential

ðrÞ from the usual Poisson’s equation. In the discussedpicture at an energy transfer T the ionization is possibleonly for the electrons whose energies in the potentialare above �T, i.e., with momenta above pTðrÞ withp2T=2m� e ¼ �T. The electrons with lower energy

are inactive. Calculating the density of the inactive elec-trons as p3

T=ð3�2Þ and subtracting their total number fromZ, one readily arrives at the formula for the activationfactor, i.e., the effective fraction of the active electronsZeff=Z as a function of T:

fðTÞ ¼ ZeffðTÞZ

¼ 1�Z x0ðTÞ

0

��ðxÞx

� T

T0

�3=2

x2dx; (39)

10−1

100

101

102

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

T (keV)

f

FIG. 2. The activation factor f for germanium in the steppingapproximation with the actual energies of the orbitals (solid line)and its interpolation in the Thomas-Fermi model (dashed line).

NEUTRINO-IMPACT IONIZATION OF ATOMS IN . . . PHYSICAL REVIEW D 83, 113001 (2011)

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where �ðxÞ is the Thomas-Fermi function, well known and

tabulated, of the scaling variable x ¼ 2ð4=3�Þ2=3m�Z1=3,the energy scale T0 is given by

T0 ¼ 2

�4

3�

�2=3

m�2Z4=3 � 30:8Z4=3 eV; (40)

and, finally, x0ðTÞ is the point where the integrand becomeszero, i.e., corresponding to the radius beyond which all theelectrons are active at the given energy T. The energy scaleT0 in germanium (Z ¼ 32) evaluates to T0 � 3:1 keV. TheThomas-Fermi activation factor for germanium calculatedfrom Eq. (39) is shown by the dashed line in the plot ofFig. 2. One can see that in the energy range it reasonablyapproximates the stepping behavior of the atomic orbitals.The discussed statistical model is known to approximatethe average bulk properties of the atomic electrons with a

relative accuracy OðZ�2=3Þ and as long as the essentialdistances r satisfy the condition Z�1 � m�r � 1, inwhich the condition in terms of the scaling variable x reads

as Z�2=3 � x � Z1=3. In terms of Eq. (39) for the numberof active electrons, the lower bound on the applicability of

the model is formally broken at T � Z2=3T0, i.e., at theenergy scale of the inner atomic shells. However, the effectof the deactivation of the inner electrons is small, of orderZ�1 in comparison with the total number Z of the electrons.On the other hand, at low T, including the most interestingregion of T � T0, the integral in Eq. (39) is determined bythe range of x of the order of 1, where the model treatmentis reasonably justified.

In order to apply the same model for an estimate ofthe correlation effect we replace in the estimated correla-tion contribution to the magnetic neutrino scattering inEq. (38) the factor jc ð0Þj2 by the total density of theelectrons that an active electron ‘‘sees’’ at its location inthe atom. Then the resulting correction to the integral I1 foran atom can be written in terms of the density naðrÞ of theactive electrons and the total density nðrÞ of the electrons inthe atom:

I1c ¼ � ��

2mT3

ZnaðrÞnðrÞd3r: (41)

It should be pointed out that the numerical coefficient inthis expression contains a factor of 1=8 as compared toEq. (38). This is because of a factor of 1=4 correspondingto the statistical weight of the spin-singlet state and anextra 1=2 compensating for the double counting of elec-trons in the pairs.

One can write the correction described by Eq. (41) interms of a correction to the activation factor in the Thomas-Fermi model as

fcðTÞ TI1cZ

¼ ��T1

T

�2Z 1

x0ðTÞ�3ðxÞdx

xþ

Z x0ðTÞ

0�3=2ðxÞ

���3=2ðxÞ �

��ðxÞ � T

T0

x

�3=2

�dx

x

; (42)

where the correlation energy scale T1 is given by

T1 ¼ffiffiffi2

p3�

m�2Z � 4:1Z eV (43)

and evaluates to about 131 eV in germanium. The plot ofthe estimated correlation correction in germanium isshown in Fig. 3. One can readily see that this correctionis below 1.5% at T � 0:3 keV and rapidly decreases athigher energy transfer. Clearly, this estimate refers only tothe magnetic part of the scattering, while for the weak partwe find no correlation effect in the considered order due tothe cancellation found in Eq. (38). We thus conclude that inthe range of values of T above a few hundred eV thecorrelation effect can be safely neglected for both contri-butions to the neutrino scattering on germanium.

VI. SUMMARY

We have considered the scattering of neutrinos on elec-trons bound in atoms. Our main finding is that the differ-ential over the energy transfer cross section given by thefree-electron formulas (1) and (2) and the stepping behav-ior of the activation factor given by Eq. (21) provides avery accurate description of the neutrino-impact ionizationof a complex atom, such as germanium, down to quite lowenergy transfer. The deviation from this approximation dueto the onset of the ionization near the threshold is less than5% (of the height of the step) for the K electrons, if oneapplies the analytical behavior of this onset that we find forthe ground state of a hydrogenlike ion. We also find that the

10−1

100

101

−0.025

−0.02

−0.015

−0.01

−0.005

0

T (keV)

fc

FIG. 3. The correlation correction fc to the activation factor ffor germanium in the Thomas-Fermi model.

KOUZAKOV, STUDENIKIN, AND VOLOSHIN PHYSICAL REVIEW D 83, 113001 (2011)

113001-8

free-electron expressions for the cross section are notaffected by the atomic binding effects in the semiclassicallimit and for independent electrons. For this reason weexpect that the deviation of the actual onset from a stepfunction at the threshold for ionization of higher atomicorbitals is even smaller than for the ground state, since themotion in the higher states is closer to the semiclassicallimit. Thus, our analytical results explain the numericallydetermined behaviors of the electroweak and magneticcontributions to the neutrino-impact ionization of variousatomic targets within the mean-field model [9–12].

The approximation of independent electrons lacks anaccount for the electron-electron correlations arisingfrom the Coulomb repulsion between the electrons in theatom. And to our knowledge, the present study gives thefirst theoretical consideration of the influence of suchcorrelations, which are beyond the mean-field model, onneutrino-impact ionization processes. We estimate thiseffect in the large Z limit using the Thomas-Fermi modeland show that it is small in germanium when the values ofthe energy transfer are above 0.2–0.3 keV. We thus arguethat for practical applications, i.e., for the analysis of dataof the searches for NMM, one can safely apply the free-electron formulas and the stepping approximation at theenergy transfer down to this range.

ACKNOWLEDGMENTS

We thank A. S. Starostin and Yu.V. Popov for useful andstimulating discussions. The work of K.A.K. (in part) andA. I. S. is supported by RFBR Grant No. 11-02-01509-a.K. A.K. also acknowledges partial support from RFBRGrant No. 11-01-00523-a. The work of M.B.V. is sup-ported in part by DOE Grant No. DE-FG02-94ER40823.

APPENDIX A: SUM RULES

We consider here the general sum rules for the dynami-cal structure factor SðT; q2Þ, which stem from the analy-ticity of the density-density Green’s function FðT; q2Þ at afixed T and complex q2 and also from its asymptoticbehavior at large jq2j. At a nonzero T the dynamicalstructure function, defined by Eq. (6), vanishes at q2 ¼ 0,due to the orthogonality of the excited states jni and theinitial state j0i in Eq. (6) since �ð0Þ reduces to a unitoperator. For this reason the function FðT; q2Þ is real atq2 ¼ 0 and thus satisfies in the complex plane the condi-tion FðT; zÞ ¼ FðT; zÞ. At a nonzero real q2 the imagi-nary part of this function is not vanishing for both positiveand negative q2, so that it has cuts along the real axisextending from zero to both infinities.5 On the other

hand, the asymptotic at large jq2j behavior of the Green’sfunction FðT; q2Þ is determined by the free-electronformula (16), since at jq2j ! 1 any interaction termscan be neglected. For a scattering on an atom with Zelectrons one finds

FðT; q2Þjjq2j!1 ! � 2mZ

q2: (A1)

This behavior enables one to write a dispersion relation forthe Green’s function with no subtractions:

FðT; q2Þ ¼ 1

�

Z 1

�1ImFðT;Q2ÞQ2 � q2 � i

dQ2: (A2)

By comparing the dispersion relation at q2 ! 1 with theasymptotic behavior in Eq. (A1) one readily finds the sumrule for an integral similar to I2, but extended to includealso the negative q2:Z 1

�1SðT; q2Þdq2 ¼ 2mZ; (A3)

where the dynamical structure function at negative q2 isdefined by the analytical continuation and Eq. (10), ratherthan by Eq. (6).In order to derive from Eq. (A2) a relation for an integral

similar to I1 it is necessary to consider the Green’s functionnear the origin, i.e., at q2 ! 0. In multielectron systemsthe behavior in this region is generally complicated by thetwo-electron correlations. For this reason we limit theconsideration here to the system with just one electron,Z ¼ 1. In such a system one has �ð ~qÞ ! 1 at q ! 0, so thatthe Green’s function in Eq. (8) is contributed by only theinitial state j0i:

FðT; 0Þ ¼ 1

T: (A4)

By comparing this formula with Eq. (A2) at Q2 ! 0 oneimmediately finds the sum rule

Z 1

�1SðT; q2Þ dq

2

q2¼ 1

T: (A5)

It should be pointed out that unlike the sum rule (A3) thislatter relation is generally invalidated in the multielectronsystem by the correlation effects. In fact, an indication ofsuch a difference in the behavior of the two integrals can beseen in Eq. (38), where the correlation effect vanishes forthe integral I2, but not for I1.The sum rule (A5) can also be derived from the

latter expression in Eq. (18). Indeed, one can rewrite theformula as

FðT; q2Þ ¼ 1

T � q2

2m

þ 1

T � q2

2m

��0

��������1

T � q2

2m � ð ~p� ~qÞm �H þ E0

ð ~p � ~qÞm

��������0

�

(A6)

5It is not clear what physical meaning can be ascribed in thisproblem to negative q2. However, a formal analytical continu-ation to negative q2 exists and results in a cut along the negativereal axis. It is the omission of this cut that resulted in a somewhatincorrect treatment of the problem in Ref. [14].

NEUTRINO-IMPACT IONIZATION OF ATOMS IN . . . PHYSICAL REVIEW D 83, 113001 (2011)

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and consider the expansion of the last term in powers ofð ~p � ~qÞ. Only the even terms in this expansion are non-vanishing, since the odd terms give zero due to the parity.One can readily see that in each term in the expansion thepole in q2 is of a higher order than the power of q2 in thenumerator, so that the imaginary part of each term integra-tes to zero in the integral as in Eq. (A5), while the term ofthe zeroth order in ð ~p � ~qÞ in Eq. (A6) gives the sum rule(A5). It is again important here that the integration runsover all values of q2, i.e., from �1 to þ1, since only inthis case all the poles of the terms in the expansion arewithin the integration range. Any restriction of the range ofintegration over q2 may leave some poles out so that thevanishing of the contribution of all higher terms in theexpansion is generally not guaranteed.

APPENDIX B: MOMENTUM-TRANSFERINTEGRALS FOR HYDROGENLIKE STATES

Consider the situation when the initial electron occupiesthe discrete nl orbital in a Coulomb potential Vð~rÞ ¼��Z=r. The dynamical structure factor for this hydrogen-like system is given by

SðnlÞðT; q2Þ

¼ mk

ð2�Þ31

2lþ 1

Xlml¼�l

Zd�kjh’�

~kj�ð ~qÞj’nlml

ij2; (B1)

where ’nlmlis the bound-state wave function, ’�

~kis the

outgoing Coulomb wave for the ejected electron with

momentum ~k, and k ¼ j ~kj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mT � p2

n

p, with pn ¼

�Zm=n being the electron momentum in the nth Bohrorbit. The closed-form expressions for the bound-freetransition matrix elements in Eq. (B1) can be found, forinstance, in Ref. [19]. In principle, they allow for perform-ing angular integrations in Eq. (B1) analytically. This task,however, turns out to be formidable for large valuesof n. Therefore, below we restrict our consideration tothe n ¼ 1, 2 states only, which nevertheless is enough fordemonstrating the validity of the semiclassical approachdeveloped in Sec. III.

Using results of Ref. [20], we can present the function(B1) when n ¼ 1, 2 as

SðnlÞðT; q2Þ ¼ 28mp6n

3½1� expð�2��Þ�� q2fnlðq2Þ

½ðq2 � k2 þ p2nÞ2 þ 4p2

nk2�2nþ1

� exp

��2� arctan

�2pnk

q2 � k2 þ p2n

��; (B2)

where the branch of the arctangent function should be usedthat lies between 0 and �, � ¼ �Zm=k is the Sommerfeldparameter, and

f1sðq2Þ ¼ 3q2 þ k2 þ p21; (B3)

f2sðq2Þ ¼ 8

�3q10 � ð32p2

2 þ 11k2Þq8

þ ð82p42 þ 72p2

2k2 þ 14k2Þq6

þ ð20p62 � 62p4

2k2 � 20p2

2k4 � 6k6Þq4

þ ðp22 þ k2Þ

�47

5p62 �

47

5p42k

2 � 7p22k

4 � k6�q2

þ ð4p22 þ k2Þðp2

2 þ k2Þ4�; (B4)

f2pðq2Þ ¼ 2p22

�36q8 � 48ðp2

2 þ k2Þq6

þ ð152p42 � 48p2

2k2 � 8k4Þq4

þ ðp22 þ k2Þ

�1712

15p42 þ

1568

15p22k

2 þ 16k4�q2

þ�44

3p22 þ 4k2

�ðp2

2 þ k2Þ3�: (B5)

Insertion of Eq. (B2) into the integrals (15) and integrationover q2, using the change of variable

2pnk

q2 � k2 þ p2n

¼ tanx

and the standard integrals involving the products of theexponential function and the powers of sine and cosinefunctions, yields

Ið1sÞ1 ðTÞ ¼ Ið1sÞ2 ðTÞ2mT

¼ T�1

1� expð� 2�ffiffiffiffiffiffiffiffiy1�1

p Þ1� exp

�� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

y1 � 1p

�

� exp

� �2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy1 � 1

p arctan

�y1 � 2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy1 � 1

p��

��1� 4

y1þ 16

3y21

�; (B6)

Ið2sÞ1 ðTÞ ¼ T�1

1� expð� 4�ffiffiffiffiffiffiffiffiy2�1

p Þ1� exp

�� 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

y2 � 1p

�

� exp

� �4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 � 1

p arctan

�y2 � 2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 � 1

p��

��1� 8

y2þ 80

3y22� 448

15y32þ 1792

15y42

�; (B7)

KOUZAKOV, STUDENIKIN, AND VOLOSHIN PHYSICAL REVIEW D 83, 113001 (2011)

113001-10

Ið2sÞ2 ðTÞ ¼ 2m

1� expð� 4�ffiffiffiffiffiffiffiffiy2�1

p Þ1� exp

�� 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

y2 � 1p

�

� exp

� �4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 � 1

p arctan

�y2 � 2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 � 1

p��

��1� 8

y2þ 80

3y22� 448

15y32þ 1024

15y42

�; (B8)

Ið2pÞ1 ðTÞ ¼ T�1

1� expð� 4�ffiffiffiffiffiffiffiffiy2�1

p Þ1� exp

�� 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

y2 � 1p

�

� exp

� �4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 � 1

p arctan

�y2 � 2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 � 1

p��

��1� 8

y2þ 80

3y22� 704

15y32þ 3328

45y42

�; (B9)

Ið2pÞ2 ðTÞ ¼ 2m

1� expð� 4�ffiffiffiffiffiffiffiffiy2�1

p Þ1� exp

�� 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

y2 � 1p

�

� exp

� �4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 � 1

p arctan

�y2 � 2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 � 1

p��

��1� 8

y2þ 80

3y22� 704

15y32þ 512

15y42

�; (B10)

where yn ¼ 2mT=p2n T=jEnj. The largest deviations of

these integrals from the free-electron analogs (17) occur atthe ionization threshold T ¼ jEnj. The corresponding rela-tive values in this specific case are

Ið1sÞ1

IðFEÞ1

¼ Ið1sÞ2

IðFEÞ2

¼ 1� 7

3e�4 ¼ 0:957 263 509 3;

Ið2sÞ1

IðFEÞ1

¼ 1� 1639

15e�8 ¼ 0:963 345 116 8;

Ið2sÞ2

IðFEÞ2

¼ 1� 871

15e�8 ¼ 0:980 520 803 4;

Ið2pÞ1

IðFEÞ1

¼ 1� 2101

45e�8 ¼ 0:984 337 622 6;

Ið2pÞ2

IðFEÞ2

¼ 1� 103

15e�8 ¼ 0:997 696 490 0:

The above results indicate a clear tendency: the larger n and

l, the closer IðnlÞ1 and IðnlÞ2 are to the free-electron values. The

departure from the free-electron behavior does not exceedseveral percent at most. These observations provide a solidbase for the semiclassical approach of Sec. III.

[1] K. Fujikawa and R. Shrock, Phys. Rev. Lett. 45, 963(1980).

[2] C. Giunti and A. Studenikin, Phys. At. Nucl. 72, 2089(2009).

[3] H. T. Wong et al. (TEXONO Collaboration), Phys. Rev. D75, 012001 (2007).

[4] A. G. Beda et al., Phys. At. Nucl. 70, 1873 (2009).[5] A. G. Beda et al., Phys. Part. Nucl. Lett. 7, 406 (2010);

arXiv:1005.2736.[6] G. V. Domogatskii and D.K. Nadezhin, Yad. Fiz. 12, 1233

(1970) [Sov. J. Nucl. Phys. 12, 678 (1971)].[7] P. Vogel and J. Engel, Phys. Rev. D 39, 3378 (1989).[8] Yu. V. Gaponov, Yu. L. Dobrynin, and V. I. Tikhonov, Yad.

Fiz. 22, 328 (1975) [Sov. J. Nucl. Phys. 22, 170 (1975)].[9] V. Yu. Dobretsov, A. B. Dobrotsvetov, and S.A. Fayans,

Yad. Fiz. 55, 2126 (1992) [Sov. J. Nucl. Phys. 55, 1180(1992)].

[10] S. A. Fayans, V. Yu. Dobretsov, and A. B. Dobrotsvetov,Phys. Lett. B 291, 1 (1992).

[11] V. I. Kopeikin, L. A. Mikaelyan, V.V. Sinev, and S. A.Fayans, Yad. Fiz. 60, 2032 (1997) [Phys. At. Nucl. 60,1859 (1997)].

[12] S. A. Fayans, L. A. Mikaelyan, and V.V. Sinev, Yad. Fiz.64, 1551 (2000) [Phys. At. Nucl. 64, 1475 (2001)]; arXiv:hep-ph/0004158.

[13] H. T. Wong, H. B. Li, and S. T. Lin, Phys. Rev. Lett. 105,061801 (2010).

[14] M. B. Voloshin, Phys. Rev. Lett. 105, 201801 (2010).[15] H. T. Wong, H. B. Li, and S. T. Lin, arXiv:hep-ph/

1001.2074v3, (November 2010).[16] K. A. Kouzakov and A. I. Studenikin, Phys. Lett. B 696,

252 (2011).[17] L. Van Hove, Phys. Rev. 95, 249 (1954).[18] L. D. Landau and E.M. Lifshits, Quantum Mechanics

(Non-relativistic Theory) (Pergamon, Oxford, 1977),3rd ed.

[19] Dz. Belkic, J. Phys. B 14, 1907 (1981).[20] A. R. Holt, J. Phys. B 2, 1209 (1969).

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