Nuclear structure corrections to deuterium hyperfine splitting

11 January

1996

PHYSICS

LETTERS

B

Physics Letters B 366 (1996) 13-17

Nuclear structure corrections to deuterium hyperfine splitting I.B. Khriplovich ’ , A.I. Milstein *, S.S. Petrosyan 3 Budker Institute

Received 6 September

of Nuclear

Physics, 630090

1995; revised manuscript Editor: C. Mahaux

Novosibirsk,

Russia

received 20 October

1995

Abstract The low-energy theorem for the forward Compton scattering is generalized to the case of an arbitrary target spin. The generalization is used to calculate the corresponding contribution to the deuterium hyperfine structure. The nuclear-structure corrections are quite essential in this case due to the deuteron large size. Within their estimated accuracy, corrections of this type calculated here remove the discrepancy between the theoretical and experimental values of the deuterium hypetfine splitting. 3 1.30.G~; 3 I .3O.Jv Low-energy theorem; Deuteron;

PACS: 21.45.+v; Keywords:

Hypefine

structure

1. The hyperfine (hf) splitting in deuterium ground state has been measured with high accuracy. The most precise experimental result for it was obtained with an atomic deuterium maser and constitutes [ 1]

v,,, = 327 384.352 522 2( 17) kHz.

(1)

Meanwhile the theoretical value of the hf splitting, extracted from Refs. [ 2,3] and including higher order pure QED corrections without proton structure and recoil effects, equals VQED =

327 339.27( 7) kHz.

(2)

In the present article we demonstrate that the discrepancy vex, -

VQED = 45

’ E-mail address: z E-mail address: 3 E-mail address:

kHz

(3)

[email protected]. [email protected]. Novosibirsk University.

0370-2693/96/$12.00 0 19% Elsevier Science B.V. All rights reserved SSDI 0370-2693(95)01354-7

can be removed by taking into account the effects due to the finite size of deuteron and its virtual excitations. Such effects are obviously much larger in deuterium than in hydrogen. One nuclear-structure contribution to the deuterium hf splitting was pointed out long ago [ 41 by some intuitive arguments, and then discussed in more detail in Refs. [ 5-71. Here we treat the deuteron structure effects in a systematic way. Not only the old result is reproduced, but new corrections are obtained, among them that generated by the deuteron electric and magnetic form-factors. To calculate some contributions to the deuterium hf structure we generalize the low-energy theorem for the Compton scattering to an arbitrary target spin. 2. According to the well-known low-energy theorem for the Compton scattering on a hadron [ 8,9], its amplitude is described by the pole relativistic Feynman diagrams, with the vertices including anomalous magnetic moments. We are interested here not in the

I.B. Khriplovich et al./Physics Letters B 366 (1996) 13-17

14

spin-independent Thomson amplitude which is of zeroth order in the photon frequency w, but in the next, spin-dependent term of the w expansion. This contribution as well can be easily obtained directly in the nonrelativistic approximation [ lo] . In this approximation the electromagnetic vertex can be immediately written for an arbitrary spin s:

v,= (&)2T

(g-s)

(s.[Ex

M2 = M2,,&e,

(-4;)

(g-

;)

xi(s.[e’xe]).

2 MI

=

Ml,,,,,ehe,, =

(7)

It produces one more piece in the scattering amplitude:

= (5-J2w

Here Z is the hadron charge, A is its mass as measured in the units of mp. The hadron g-factor is related as follows to its magnetic moment p measured in the nuclear magnetons e/2m,: g = p/s. In the forward scattering case, when the hadron is at rest (p = 0) and both photons have physical, transverse polarizations ( (ke) = (ke’) = O), this vertex reduces to the pure spin interaction. The nonrelativistic pole scattering amplitude generated by this interaction is

A]).

(8)

Taken together, expressions (5) and (8) generate the forward scattering amplitude:

(9) This is the generalization of the low-energy theorem we were looking for. In the particular case of a proton ( s = l/2, Z = A = 1) this formula reduces to the result of Refs. [ 8,9].

ug*i(s.[e’xe]).

However, this expression is incomplete. Indeed, being applied to a proton, it does not reproduce the wellknown result [8,9] according to which the forward scattering amplitude is proportional to (g - 2)2. The explanation was pointed out in Ref. [ 101: the nonrelativistic pole amplitude should be supplemented by a contact term generated by the spin-orbit interaction, which restores the agreement with the classical result

[WI. This contact term can be easily derived for an arbitrary spin (as well as the nonrelativistic pole contribution (5)) in the following way. The interaction Hamiltonian generating the equation of motion for spin in an external electric field E can be written to lowest nonvanishing order in U/C as (see book [ II], 841) (s.[Exu]).

After substituting

3. Being dependent on nuclear spin, the low-energy amplitude obtained contributes to the atomic hyperfine structure. However, to apply it to this problem, the amplitude should be modified. Indeed, both photons exchanged between the nucleus and electron are offmass-shell. So, here w # Ikl. Then, virtual photons have extra polarizations. We will use the gauge A0 = 0 where the photon propagator is Di,(u,

4rn k) = &-k2’

Do0 = Do,,, = 0.

into the Hamiltonian (6), we arrive at the following contact interaction:

c 10)

Now, first of all, the magnetic moment contribution Mlmn to the pole diagram changes to 2

g2iE,,,,,kkk(k.s)

A.

(11)

Second, the convection current, which is proportional to hk for a nucleus at rest, is operative now and induces the following nuclear-spin-dependent contribution to the forward scattering amplitude:

,,_P-ZeA f%

&,,+,+!;

M 3nvl= -

e

2

( > Gl

z

2xg

(12)

15

I.B. Khriplooich et al./Physics Letters B 366 11996) 13-17

We are ready now to write down the electronnucleus nuclear-spin-dependent scattering amplitude generated by the two-photon exchange and elastic hadronic channel: Tel = hai x bhn,

s

d4k

dimdjn yi(I-

k2 - 21k

mk4

+

M2mn +

R + m,)yj

(13)

M3mn).

Here 1, = (m,, O,O, 0) is the electron momentum. ‘Ihe structure ri( I- i+ rn,)rj reduces to -i o EijP/ where CTis the electron spin. We will calculate this Feynman integral with logarithmic accuracy. The final result for the relative magnitude of the discussed correction to the hf structure is &4g$12$

Ael = $2

. >

P

(14) At s = l/2, A - Z = 1 it agrees with the corresponding results [ 12,I3 ] for muonium (where the effective cut-off A is provided by the muon mass) and hydrogen (where the integral is cut off at the typical hadronic scale mP) . In the case of deuterium (s = 1, g = pd = 0.857, A - 2, Z = I), we are mainly interested in, the integration over the momentum transfer k is cut off at the inverse deuteron size K = 45.7 MeV. In this way we obtain the following result for the relative correction in deuterium: .

In the conclusion of this section let us mention strong numerical cancellation between A,1 and Ai”, their contributions to the hf structure being -3.2 kHz and 2.9 kHz respectively. 4. The low-energy Compton amplitude discussed above is just the linear term of the full amplitude expansion in o (and in jk12/w for virtual photons). One may expect, however, that for the deuteron with its small binding energy this approximation is insufficient even in the atomic problem considered here. And indeed, we will see below that the deuteron virtual excitations are far from being inessential to our problem, they strongly dominate the effect discussed. Since the contribution of large momentum transfers k > K has been calculated already (see formula ( 16)) we confine now to the region k < K. All calculations below are performed in the zero-range approximation (zra) for the deuteron which allows to obtain all the results in a closed analytical form. Let us start with the contribution of excitations induced by the spin interaction only. The corresponding scattering amplitude is 2

X

_ <

<

Ol[k x Sl,ln ><

Ol[k x

nl[k

x St],10

>

w-E,-1

S+l,ln>< nl[k

x S],lO

>

w+E,+I

(15)

(17) At larger momentum transfers, k > K, the amplitude of the Compton scattering on a deuteron is just the coherent sum of those amplitudes on free proton and neutron. This contribution to the hf structure can be also easily obtained with the above formulae. Since both nucleons are in the triplet state, one can substitute s/2 both for sp and for s,,. With the logarithmic integral cut off here at the usual hadronic scale mp = 770 MeV, we get in this way

(16) Here pp - 2.79 and /.L,,= -1.91 are the proton and neutron magnetic moments.

Here I = tc2/mP is the deuteron binding energy, E,, = is the energy of the intermediate state In > (all intermediate states belong to the continuous spectrum), and

p2/mp

S = ppoPeikr12 + pn@ne-ikr/2, where u,,(,,) is the proton (neutron) spin operator. When calculating this contribution, we will retain only terms logarithmic in the parameter E = Z/K = K/m,, << I. The log is generated by the integration over k, and to obtain it it is sufficient to put the exponents in S equal to unity. Then the operator S can induce only Ml transitions. In the zra the deuteron ground state is pure 3S, from which Ml transition is

16

I.B. Khriplouich et al./ Physics Letters B 366 (1996) 13-17

possible also to S-states only. But due to the orthogonality of the radial wave functions of different triplet states, the intermediate states confine to ‘SO. Since the total spin operator s = ( l/2) (a,, + a,) does not induce triplet-singlet transitions, the operator S reduces here to S --+ (Pup-Pun) ; (a, - 0,). In our problem of magnetic hf structure we need the antisymmetric part of tensor ( 17) which is linear in the deuteron spin s. It looks now as follows:

ground state. But in the zra all the states with 1 # 0 are free ones. It means that here we can use as the intermediate states just plane waves, eigenstates of the momentum operator. We will confine here to terms singular in the parameter c = K/m, < 1: 1/E and log E. The calculations are still rather tedious. They give the following result for this relative correction to the hf structure: A? _% I”

3cu m, --log5~~-~~

& -& 2K 7

-

=

K

mP

Iud

’ (22)

One of the terms in this correction,

X-

s

dp

1<’ So, p 13S, > I2

(27r)3 C-02 -

(p2

+ Kym;

me pu, 2K Pd

-a----,

’

(18)

where < I &, p 13Si > is the overlap integral of the ground state zra wave function (19) with the singlet one of the momentum p. This contribution to the electron-deuteron scattering amplitude is easily calculated with the logarithmic accuracy. Indeed, to this accuracy the energy denominator in formula ( 18) can be simplified to 1 02 - K"/rni ’ Then the integration over p reduces to the completeness relation. The resulting relative correction to the deuterium hf structure is

(20)

was obtained and discussed in Refs. [ 4-71. However, another term am,* 2K

pd’

is larger numerically. 5. In the case of hydrogen the corrections due to the finite distribution of the charge and magnetic moment were considered many years ago [ 141. For deuterium they should be obviously larger. In the zra the problem has here a closed analytical solution. Let us start with the second-order amplitude of the electron-deuteron scattering as induced by the deuteron charge and magnetic moment. The nucleus will be treated in the static limit. However, its finite charge and magnetic moment distributions will be taken into account by introducing the corresponding form-factors, Fct, and F,,,. This amplitude is v = -(47&5!P

Let us consider at last the inelastic contribution induced by the combined action of the convection and spin currents. Since the convection current is spinindependent, all the intermediate states are triplet ones as well as the ground state. Therefore, here operator S simplifies to S --+ s ( /Lpeifr/2 + /&&?+/2).

(21)

According to the common selection rules, neither 3Sr can be excited by the convection current from the

x-

J dq t.

Ls

x q

1

(2T)”

. Y0(f+4+me)y

Fcdq2)Fm(q2) q4

-r(i+Q+m,)y0

(23)

(l+q)2-mz

Here again I, = (m,,O,O,O), and qti = (0,q). This expression can be conveniently transformed to v

=

-8mea /- 4 2 Fcdq2> Fm(q2) To, ?T

(24)

I.B. Khriplovich et al./ Physics Letters B 366 (1996) 13-17

where To is the momentum-independent magnetic Born amplitude. The effect we are interested in, vanishes of course if unity is substituted for both form-factors. Therefore, the corresponding relative correction to the Born amplitude To and to the hf splitting is in fact (25) In the zra both deuteron form-factors have simple form: F&2)

=

Fm(q2)

==< 0~e’Q”2~0

>p

$

arctg&. (26)

Substituting it into formula (25), we get the following explicit expression for the correction discussed: *f

--a$l+210g2).

(27)

6. Our total result for the nuclear-structure corrections to the deuterium hf structure, comprising all the contributions, (15), (16), (20), (22), (27), is

pd

3n me mp (pp -cLR)2 + - - log 7 8~ mp Pd 3a

m,

=

mP

(28) Numerically this correction to the hf splitting in deuterium constitutes AV = 43 kHz.

Taking into account the zra used for the deuteron description and the neglect of contributions regular in K, we estimate the accuracy of the found number (29) as 25%. At least, there is no contradiction between our result and the lacking 45 kHz (see (3) ) . Clearly, the nuclear effects discussed are responsible for the bulk of the difference between the pure QED calculations and the experimental value of the deuterium hf splitting. The calculation of this hf correction based on more elaborate models of deuteron structure would serve as a sensitive check of these models. A more detailed paper on the subject is submitted to Zh. Eksp. Teor. Fiz. [ JETP]. That paper contains also a closed analytical result for the deuteron polarizability contribution to the Lamb shift. We are grateful to MI. Eides, H. Grotch and M.I. Strikman for useful discussions. This investigation was financially supported by the Program ‘Universities of Russia”, Grant No. 94-6.7-2053. References 111D.J. Wmeland and N.F. Ramsey, Phys. Rev. A 5 ( 1972) 821. 121G.T. Bodwin and D.R. Yennie, Phys. Rev. D 37 ( 1988) 498.

e-;(1+2log2)}

---

17

(29)

It is strongly dominated by the terms N 1/K (the first line in formula (28) ) which contribute 52 kHz to this number.

[31 N.F. Ramsey, in: Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, 1990). 141 A. Bohr, Phys. Rev. 73 ( 1948) 1109. [51 F.E. Low, Phys. Rev. 77 ( 1950) 361. [61 EE. Low and E.E. Salpeter, Phys. Rev. 83 ( I95 1) 478. [71 D.A. Greenberg and H.M. Foley, Phys. Rev. 120 ( 1960) 1684. isI EE. Low, Phys. Rev. 96 ( 1954) 1428. [91 M. Gell-Mann and M.L. Goldberger, Phys. Rev. 96 ( 1954) 1433. [101 A.1. Milstein, Siberian Journal of F%ysics ( 1995) No I, 43. [Ill V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Quantum Electrodynamics ( Pergamon Press, Oxford, 1982). ii21 R. Amowitt, whys. Rev. 92 ( 1953) 1002. [I31 H. Crotch and D.R. Yennie, Rev. Mod. Phys. 41 ( 1969) 350. iI41 A.C. Zemach, Phys. Rev. 104 ( 1956) 1771.