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Nuclear polarization corrections for the S-levels of electronic and muonic deuterium
Physacs Leners B 319 (1993) 7-12 North-Holland
PHYSICS LETTERS B
Nuclear polarization corrections for the S-levels of electronic and muonic deuterium Y a n g Lu a n d R. R o s e n f e l d e r Paul Scherrer l~t~tatute. ('!1-5232 i "dhgen PSi. Switzerland
Received 23 September 1993. revised manuscript received 25 October 1993 Edator: C. Mahaux We calculate the second.order corrections to the atomic ener~. level shifts m ordinary and muomc deuterium due to virtual expiations of the deuteron which arc important for ongoing and planned precise expenmcnts in these systems. We use a method for light atoms m which the shift is expressed as integrals over the longitudinal and transverse inelastic structure functions of the nucleus and employ the structure functions arising from separable N N potentials. Special emphasis is put on gauge mvanance which requires a consistent inclusion of interaction currents and seagull terms. The effect of the D-wa~,c component of the deuteron is investigated for the leading longitudinal contnbuuon. We also estimate the shift for plonic deuterium.
I. Recently the isotope shift o f the 2 S - I S transition in electronic hydrogen and deuterium has been measured with a thirty-fold increase in accuracy compared to previous experiments [ I ] and prospects are good to increase the present accuracy of 37 ppb (or 22 kHz ) by another order of magnitude. At this level not only various Q E D corrections and the finite size o f t h e deuteron are important but also virtual excitations of the deuteron (the so-called nuclear polarization) cannot be neglected anymore. Using a rough square.well model of the deuteron, together with the dipole and the closure approximation, the IS shift due to virtual Coulomb excitation was estimated to be about - 1 9 kHz [2] ~,h,ch is just the present experimental uncertainty. In light and heav.~ muonic atoms the nuclear polarization shift generally limits the accuracy with which nuclear sizes can be extracted since a reliable calculation o f these corrections requires knowledge o f t h e whole nuclear spectrum [ 3 ]. The deuteron is unique among all nuclei in that this information is available: the quantum-mechanical two-body problem is solvable and realistic N:V potentials describe bound and scattering states rather well. Therefore, unlike the usual case, nuclear polarization corrections are calculable for the deuteron - presumably with an accuracy at the percent level.
It is the purpose o f the present note to evaluate this shift within a more realistic model for the deuteron, avoiding the use of the closure approximation, taking into account all multipole excitations and including also transverse excitations which have been neglected up to now. For the purpose o f a planned experiment at PSI with muonic hydrogen [4] which may be extended to deuterium [5] we also evaluate the corresponding shifts in the muonic case.
2. We will calculate the nuclear polarization shifts in the atomic S-states o f the lepton. These are more difficult to evaluate than the corresponding shifts in higher orbits where only the longest range multipole is of importance and therefore the only nuclear structure mformation needed is the electric dipole polarizabihty o f the nucleus. In contrast, many multipoles contribute to virtual excitations from atomic S-states since there is an overlap between leptonic and nuclear wavefunctions. In light nuclei, however, the relevant scales (Bohr radius vs. nuclear radius) are vastly different so that the S-wave lepton wavefunction can be considered as approximately constant over the nuclear volume ¢ , 0 ( x ) "" ¢ . , : the lepton then just acts as a static source with four-momentum k = ( m , 0 ) . We will evaluate the S-wave nuclear polarization
0370-2693/9315 06.00 (~) 1993-Elsevier Science Publishers B.V. All rights reserved SSDI 0370-2693(93 )El 381.7
Volume 319. number 1.2,3
PHYSICS LEVI FRS B
23 December 19q3
A, ,,~ = - 8-2R,01¢',,o (0).: /
dq
,,d
0
%
#
*.,
Fig I. Second-order contributmns to the nuclear polanzatton shift: (a) box graph. (b) crossed graph. (cl seagull graph.
shtfis along the lines of ref. [6], Le. not by a multipole decomposition but by integrating over the inelastic structure functions of the nucleus. The diagrams which contribute to nuclear polarization are shown in fig. 1. Note that we need the "seagulV contribution of fig. Ic for gauge invariance in a nonrelat~v~stlc system like the nucleus. Under the mentioned simplifications one obtains
/
._ _
A~,, ° _
(4n,~):m It)., ' I m
t.,,lq.k)D~Plq)D";(
= Zd((o'
+ 1-o-
+ Kz(q.w)St(q. to)] + R.s(q))f"(q).
(3)
Here R ~ is a correction factor for the variation of the leptonic wave function over the nucleus, q is the magnitude of the three-momentum transfer and ¢o the energy transfer to the nucleus. The kernels K~,.r(q,to) are gtven in the appendtx of ref. [6] for fully relativistic kinematics of the lepton. Rs(q) is the contribution from the internal seagull. F m a l b , f ( q ) = [ I + q:/{0.71GeV:)I : describes the electromagnetic formfactor of the nucleon. Actually eq. (3) not onl) holds in the Coulomb gaugc but is gauge mvariant providcd qUT,~ (q, - q ) = 0. This in turn requires current conservation and six'cial conditions for the seagull term [8,9] whtch will be discussed below.
d'tq
( 2n)4 q)Tp,(q,-q).
where tv~ (q. k ) is the leptonic tensor [6 ] , ,, = e-' = 1/137.036 the fine-structure constant and m is the lepton mass. Furthermore. i,)aVlq) denotes the photon propagator and T v , ( q . - q ) the forward virtual nuclear Compton amplitude. To be more precise, the latter is that part o f the full Compton amplitude in which the nucleus is m an excited mtermedmte state. It can be expressed m terms of its imaginar) part, i.e. by the inelastic longitudinal and transverse structure functions
St,r
x (]d¢o[l(L(q.to)SL(q,to) o
E.,,)le~.~lOt.t
WOe',".
(2)
~'1# (I
Here to' = to - q:/4M ts the internal excttation energ)', Eo < 0 the ground state energy, and Oi. t are the operators for longitudinal and transverse excitations respectively, in Coulomb gauge one obtains [6]
3. Unlike ref. [6] where a phenomenological model for the structure functions of *:C had to be used. the deuteron allows for a consistent calculatlvn of these quantittes after a nucleon-nucleon interaction has been chosen. For simplicity we take a separable potential of the form ( M is the nucleon mass) ).
f'tp,p') = - .-~ g(pJg(p').
(4)
This is not realistic in the modern sense lit lacks the one-plon exchange tail and all other comphcations of the .VN force) but it gives a fairly good description of the iow-eneriD' N N interaction which should be suffio e n t for the present accuracy of isotope shift experiments. Most important for the present purposes it allows for an analytic evaluation of the structure functions. For example, the longitudinal structure function is obtained as
SLlq,~o) = ~.M Im
+ n
/
d~pl~o(p -
(
!q)12(5 w ' + E o - - ~
(12(co'+l"o,q) ) ! + ;.C'(to' + Eo)
(5)
"
Obviousl) the first term is just the impulse approximation to the structure function whereas the last one de-
Volume 319. number 1.2,3
PHYSICS LETTERS B
scribes the final-state interaction. The functions C (E) and I(E,q) are gtven in the appendix of ref. [I0]. Due to the simple form o f e q . (4) the final-state interaction o n b acts in states with angular momentum zero which should be a good approximation for lowenergy processes. Note that the internal charge operator is p(q) = exp(lq.r[2) where the factor 1[2 arises from the transformat,on to internal coordinates. For the Yamaguchi choice [ i I ] I
g) (P) - p: + /I: "
~"[T.p(q)]
The linear terms due to the potential generate an interaction current A Jr (j,,) whereas the second-order terms give rise to the interaction seagull AB, j (y. z ) both of which can be obtained by the appropriate functional derivative ofeq. (9). With the separable potential (8) one arrives at the following expression for the matrix element of the interaction current Lo'.AJ(q)~'/ = - ~
-q. (.l"°""b(q) + J'"P'"'(q)),p) = O, (7}
(V, + V,,)
I
(61
all integrals can be performed analytically. Details will be given elsewhere. The transverse nuclear polarization shift receives different contnbut~ons: first, we have the standard contribut,on from the transverse (with respect to q) pan of the convect,on current d,o~,)(q ) which does not have a final-state interaction term s,nce all exerted states necessarily have at least angular momentum one. Second, the spin current J"P'"' (q) involves the magnettc moments and spins of neutron and proton and gives rise to spin-flip excitations. Since the procedure of calculating the transverse structure with these currents is similar to the longitudinal case the exphcit expressions v~ill not be given here. Note that the above charge and current operators obey current conservation ,n the form
23 December 1993
/ d.gU,-,, ½qlg(.'+ (t
-.I
,.'q),
(I0)
0
which satisfies
(II)
LdllV.p(ql] .- q . AJ (q )[p) = O.
so that all together our currents are conserved under the time-evolution of the full Hamihonian. Similarly one obtains an explicit expression for the matrix element of the interaction seagull which comes in addition to the kinematical (,nternal) seaguU 6,j/2M. It can be checked then that the gauge relation ~ ' . [ p ' ( q ) . A J A ( q ) ] L D ) = q(p'lAB~(q)[p)
(121
is fulfilled wh,ch is needed for full gauge invariance of the Compton ampl,tude [8,9]. The seagull contribution in eq. (3) now has the form
where T is the kinetic energy operator. 4. A separable potential of the type (41 is nonlocal and equ,valent to a momentum-dependent potentml 3. l'
_
M
f
~3,T.q
I
(8)
( i 31
where
d~rd3x.¢ (r + ~ x ) . ¢ ( r - ~x)
, exp(-i@- !x),r>(r e x p ( - i p . ~x).
.
~'(q) = ~.,
i,
du--u. [ h 2 ( 0 1 - h Z ( u q ) ] .
(14)
0
Mimmal couplingpr - p p - eA(rpl in the two-body Hamihonian lI = l" + I" then produces a power series in the electromagnetic coupling constant e
lI = Ill
,
,.,,
c jf d3vJv(y).-lu() ')
+ ~ e 2 [ dJyd~z..I,(y)..Ij(:.)B,O..z.) + .... J
(9)
h(.'4.'1
=
f
dJp~uolP)g(p -- ~tX).
(151
5. We now turn to the numerical results o f o u r calculation. We have evaluated the nuclear polarization shift consistently wtth the Yamaguchi separable form (6) using the value ,8 = 286 MeV. We have checked
PHYSICS LE'T']"r..RS B
Volume 319. number 1.2.3
numerically that our longitudinal structure function fulfills the non-energy-weighted sum rule
j doJSl(q,o~) = l - ko2(q)
(16)
0
to better than I p a n in i0 s. Here Fo(q) is the elastic formfactor calculated directly from the ground state wave function. The electric dipole polarizability (which is not the phi) relevant quantity for Swave shifts) was found to be 0.613 fm ~ compared to experimental values of 0.61-0.70 fm 3 [12]. The point rms-radius m this model is 1.92 fm compared to the experimental value 1.96 fm. This shows that the simple Yamaguchi parametrization describes the deuteron properties and therefore the low-energy triplet NN-mteract~on reasonably well. For the spinflip excitations one also needs the parameters in the singlet channel. Again follo~,ing Yamaguchi we assume ,8, = ,8, and determine the corresponding strength parameter from the singlet scattering length a . . . . 23.69 fro. For non-relativistic point hydrogen wave functions one has j4~,,o(0)l" = l / ( h a h n 3) where aB = I/D*rnrt-d) is the Bohr radius and m,,.e the reduced mass o f the lepton. Writing A¢,,0 = - ~
A?'.
(17)
the shift A?- is then independent of the atomtc state. We have evaluated the double integral in eq. (3) by Gauss-Legendre numerical integration with up to 3 × 72 points. Our results for the different contributions and for the total A?" are listed in table I. It should be emphasized that the integrand from the transverse convection current has a l/q-divergence for small q which, however, is exactly cancelled by the seagull due to the gauge condition for the two-photon operator. This can also be seen from eqs. ( 3 ) and ( 13 ): at low q the kernel Kr(q.to) behaves like - i/4mq(o [6] and Siegert's theorem tells us that
fd¢oIs~(0,~)_
l 2M +x
(18)
0
where ~¢ =_ K(0) is the dipole enhancement factor (K = 0.176 for the Yamaguchi potential). However, l0
23 I.X,-,ccmber 1993
Table I Contributions to the nuclear polarization shift AT (see eq. (17) ) for electronic tel and muonic (,u) deuterium in Coulomb gauge. The Yamaguchi S-wave separable potentaal has been used throughout. The different contnbuttons are labeled b) L: Iongttudinal. 7"~¢°~'~ + S: transverse convectlon current + seagull, rts~ttt): transverse spin current; A( T + S). mteractton transverse current + interaction seagull. Contnbutaon
e [kHz)
/J [MeVI
L
-18.31 -2.25 +0.33 -0.31 -20.54
-11.77 -0.06 +0.03 -0.02 - II .82
y.tco~,~ + .S" 7'it°'n~ A(7" + S) total
the resulting contribution to the energy shift is exactly opposite m s~gn to the q ~ 0-limit of the seagull contribution (I 3). We have checked numerically that our transverse structure function fulfills the sum rule (18) to sufficient accuracy. Consequently we onl) give the combined contribution of transverse convection and seagull excitations in table 1. It is seen that it ts bigger in electronic deuterium than in muonic deuterium because the electron velocity is higher in the first case. As the spin current contribution vanishes for q = 0 ~t can be given separately. However. numerically it turns out to be of no great importance. The same can be said of the interaction terms which nearly cancel the spin contribution. The smallness of the interaction terms is welcome since the nonlocality of the Yamaguchi separable potential is somehow artificial and pull partly simulates exchange current effects. It should be kept in mind that the individual contributions are gaugedependent and that only the total A?" is a meaningful physical quantity. The size of the transverse and seagull terms, however, indicates the errors one usually makes if only the longitudinal excitations are taken into account. As to the numerical accuracy, we have checked that the results in table 1 are accurate to one part in the last digtt. In order to estimate the model dependence of these results we also have calculated the dominating longitudinal contribution for the Tabakin separable potenttal [13] which describes both attraction at low energies and repulsion at higher energies in the S-wave phase shift. As the principal value integrals in the
Volume 319. number 1.2.3
PIIYSICS LETTERS B
Table 2 Longitudinal nuclear polarizatmn shift ..$7 for different separable N N-potentials. NN-potenttal
e [kHz]
g [MeVJ
Yamaguchl S-wa~,e [11] Tabakm S-wave [13] YamaguchiS+D-wave [14]
-18.31 -18.54 - 18.45
-11.77 -11.02 -11.86
structure functmn (5) had now to be calculated numerically the computing time for the nuclear polarizatmn shift increased considerably. Again the sum rule was checked and an electric dtpole polarizability of 0.623 fm 3 ~as obtained. The values for the shift given in table 2 are estimated to have an accuracy of about three parts m the last dtg~t. Despite the different functtonal parametrizatton of g (p) the result for A~" is very close to the Yamaguchi one which shows that only low-energy properties of the N N - m t e r a c t i o n are ~mportant for the nuclear polarization shift. Finally we also have investigated the influence of the D-state admixture in the deuteron by' using the Yamagucht st'parable potential with tensor force [ 14 ] g(p) = g)(p) + ~
1
T) ( p ) S r . ~ ) .
(19)
with Tr(p)
-
( p ; tp: + ;.,.): .
S p . ( p ) = 3 o r ' p op2, ' P
(20)
op • o,,.
(2 1)
The original parameter values of ref. [141 lead to an asymptotic D / S rano of 0.0285 whtch ts quite reasonable when compared with modern values [I 5J. The dipole polarizability is calculated to be 0.625 fm ~. Since the algebra including the tensor force is more involved the sum rule check (to one part m 10 ~) is nontrivial. To convert these numbers to the actual nuclear polarization shifts for the S-levels we need the finite size correction factors R,o. Using the approximate atomic wavefunctions of ref. [ 161 one obtains in first order in the ratio nuclear radius/Bohr radius R,o "" I
3.06 (r2"J" aB
(22)
23 December 1993
The numerical factor in this equation was determined by evaluating the rauo of various moments of the charge distribution with the Yamaguchi ground state wavefuncuon. Eq. (22) gives n.mdependent correction factors R'") = 0.9793 and ~ t , , = 0.99989. O f course, on the present level of accurac.~ one can practically neglect these correction factors. We estimate the accuracy' of our theoretical predictions in the following way.: the accuracy of the calculated longatudinal shift is taken as three times the model-dependence shown in table 2 and we assign a 20% error to the transverse current contribution and 50% one to the interaction pieces. Adding these errors linearly' we therefore arrive at the final result for the nuclear polarization shifts in electronic and muonic deutermm ,,o = ( - 20.5 -¢- 1.3) ~-~ kHz. qllm
i
A¢~0 = t- 11.6 ± 0.5) ~-~ MeV.
(23) (24)
If the future measurement of the 2P-2S transition in muonic deuterium reaches the planned accuracy of 0.05 MeV [ 5 ] the nuclear polarization shift in the 2Slevel will be an ~mportant ingredient for analysing the experiment. We also have esumated the nuclear polarization shift m piomc deuterium by' replacing the muon mass by the p~on mass. The longitudinal and the transverse convection current contribution of the present formalism should give a reasonable value also for a heavy spin zero panicle because to a good approximation it can bc treated nonrelatwistically with no difference between a Dirac and a K l e i n - G o r d o n description. In this way we obtain I
a,~'~:' ~_ -28 ~ MeV.
(25~
For n = I this is a factor of t ~ o smaller than the precision aimed at in an ongoing experiment at PSI to measure the strong mteractmn shifts in pionic hydrogen [17]. If the future isotope shift experiments in electronic deuterium actually reduce the experimental accuracy to about I kHz [ I ] it would be worlh~hile to repeat the present calculation with a modern NN-potentml hke the Paris potential [18]. Finally'. before discrepanctes between theor)' and experiment in the isotope II
Volume 319, number 1.2.3
I)HYSIC.'S I.E i TELLSB
shifts are taken serious one should include secondorder effects also in the anal.,,sis o f e l e c t r o n - d e u t e r o n scattering e x p e r i m e n t s ~ h l c h extract the r o o t - m e a n .square radius o f t h e deuteron. In the case o f J-'C similar diserepanc~es betg'een electron scattering data and m u o m c energ.~ shifts seem to disappear [191 when second-order effects are taken into account m the analysis o f both experiments. We thank Andreas Schreiber for helpful discussions and a critical reading o f the manuscript. We are grateful to Leo Ssmons and Pieter G o u d s m i t for p r o v i d i n g us with e x p e r i m e n t a l information.
References [ I ] F. ~hmld1-Kalcr. D. Lelbfncd. M. Wcltz and T 9,. llansch. Ph'.s. Rc~. Left. 70 11993p 2261. [2] K. Pachuckl. D. Lesbfned and T.W. Hansth. Ph~,s. Re'. ..',, 48 11993~ RI.
12
2"~December 1993
[31 R. Roscnfelder, in: Muon,c Atoms and Molecules. eds L.A. Schaller and C. Petitjean (Btrkauser. 1993) p. 95. [4] E Za'.attmt et al.. PSI proposal R-93-06.1 (1993). [5l L. S,mons. private communication. [6] R. Roscnfelder. Nucl. Phys. A 393 (1983) 301. [71H. Grotch and D.R. Yenme, Rev. Mod. Phys. 41 (1969) 350. [8] J.L. Friar and M Rosen. Ann. Phys. 87 (1974) 289. 191H. Arcnhosel. Z. Ph~,s. A 297 { 1980) 129. [101R. Rosenfclder. Ann. Phys. 128 (1980) 188. [11 ] Y. Yamaguchl, Phys. Re,,. 95 (1954) 1626. [12]J.L. Friar and S. Falheros, Ph~,s. Rev. C 29 ~19841 232. [13] F. Tabakm. Phys. Re,,. 174 (1968) 1208. 114] Y. Yamaguchl and Y. Yamaguchl, Phys. Rev. 95 I1954~ 1635. II5]T.E.O. Ericson and M. Rosa-Clot. Ann Rev. Nucl. Part. , ~ . 35 (1985) 271. [161J.L Frtar. Z. Ph~,s. A 292 (19791 I. [ 17 ] P.F.A. Goudsm,t. pr,'.ate commumcation. [18] M. Lacombc et al., Ph)s. Re',. C 21 (1980) 861. [ 19] E.A.JM. Ofl'erman et al.. Phys. Re'.. C 44 ( 1991 ) 1096.
PHYSICS LETTERS B
Nuclear polarization corrections for the S-levels of electronic and muonic deuterium Y a n g Lu a n d R. R o s e n f e l d e r Paul Scherrer l~t~tatute. ('!1-5232 i "dhgen PSi. Switzerland
Received 23 September 1993. revised manuscript received 25 October 1993 Edator: C. Mahaux We calculate the second.order corrections to the atomic ener~. level shifts m ordinary and muomc deuterium due to virtual expiations of the deuteron which arc important for ongoing and planned precise expenmcnts in these systems. We use a method for light atoms m which the shift is expressed as integrals over the longitudinal and transverse inelastic structure functions of the nucleus and employ the structure functions arising from separable N N potentials. Special emphasis is put on gauge mvanance which requires a consistent inclusion of interaction currents and seagull terms. The effect of the D-wa~,c component of the deuteron is investigated for the leading longitudinal contnbuuon. We also estimate the shift for plonic deuterium.
I. Recently the isotope shift o f the 2 S - I S transition in electronic hydrogen and deuterium has been measured with a thirty-fold increase in accuracy compared to previous experiments [ I ] and prospects are good to increase the present accuracy of 37 ppb (or 22 kHz ) by another order of magnitude. At this level not only various Q E D corrections and the finite size o f t h e deuteron are important but also virtual excitations of the deuteron (the so-called nuclear polarization) cannot be neglected anymore. Using a rough square.well model of the deuteron, together with the dipole and the closure approximation, the IS shift due to virtual Coulomb excitation was estimated to be about - 1 9 kHz [2] ~,h,ch is just the present experimental uncertainty. In light and heav.~ muonic atoms the nuclear polarization shift generally limits the accuracy with which nuclear sizes can be extracted since a reliable calculation o f these corrections requires knowledge o f t h e whole nuclear spectrum [ 3 ]. The deuteron is unique among all nuclei in that this information is available: the quantum-mechanical two-body problem is solvable and realistic N:V potentials describe bound and scattering states rather well. Therefore, unlike the usual case, nuclear polarization corrections are calculable for the deuteron - presumably with an accuracy at the percent level.
It is the purpose o f the present note to evaluate this shift within a more realistic model for the deuteron, avoiding the use of the closure approximation, taking into account all multipole excitations and including also transverse excitations which have been neglected up to now. For the purpose o f a planned experiment at PSI with muonic hydrogen [4] which may be extended to deuterium [5] we also evaluate the corresponding shifts in the muonic case.
2. We will calculate the nuclear polarization shifts in the atomic S-states o f the lepton. These are more difficult to evaluate than the corresponding shifts in higher orbits where only the longest range multipole is of importance and therefore the only nuclear structure mformation needed is the electric dipole polarizabihty o f the nucleus. In contrast, many multipoles contribute to virtual excitations from atomic S-states since there is an overlap between leptonic and nuclear wavefunctions. In light nuclei, however, the relevant scales (Bohr radius vs. nuclear radius) are vastly different so that the S-wave lepton wavefunction can be considered as approximately constant over the nuclear volume ¢ , 0 ( x ) "" ¢ . , : the lepton then just acts as a static source with four-momentum k = ( m , 0 ) . We will evaluate the S-wave nuclear polarization
0370-2693/9315 06.00 (~) 1993-Elsevier Science Publishers B.V. All rights reserved SSDI 0370-2693(93 )El 381.7
Volume 319. number 1.2,3
PHYSICS LEVI FRS B
23 December 19q3
A, ,,~ = - 8-2R,01¢',,o (0).: /
dq
,,d
0
%
#
*.,
Fig I. Second-order contributmns to the nuclear polanzatton shift: (a) box graph. (b) crossed graph. (cl seagull graph.
shtfis along the lines of ref. [6], Le. not by a multipole decomposition but by integrating over the inelastic structure functions of the nucleus. The diagrams which contribute to nuclear polarization are shown in fig. 1. Note that we need the "seagulV contribution of fig. Ic for gauge invariance in a nonrelat~v~stlc system like the nucleus. Under the mentioned simplifications one obtains
/
._ _
A~,, ° _
(4n,~):m It)., ' I m
t.,,lq.k)D~Plq)D";(
= Zd((o'
+ 1-o-
+ Kz(q.w)St(q. to)] + R.s(q))f"(q).
(3)
Here R ~ is a correction factor for the variation of the leptonic wave function over the nucleus, q is the magnitude of the three-momentum transfer and ¢o the energy transfer to the nucleus. The kernels K~,.r(q,to) are gtven in the appendtx of ref. [6] for fully relativistic kinematics of the lepton. Rs(q) is the contribution from the internal seagull. F m a l b , f ( q ) = [ I + q:/{0.71GeV:)I : describes the electromagnetic formfactor of the nucleon. Actually eq. (3) not onl) holds in the Coulomb gaugc but is gauge mvariant providcd qUT,~ (q, - q ) = 0. This in turn requires current conservation and six'cial conditions for the seagull term [8,9] whtch will be discussed below.
d'tq
( 2n)4 q)Tp,(q,-q).
where tv~ (q. k ) is the leptonic tensor [6 ] , ,, = e-' = 1/137.036 the fine-structure constant and m is the lepton mass. Furthermore. i,)aVlq) denotes the photon propagator and T v , ( q . - q ) the forward virtual nuclear Compton amplitude. To be more precise, the latter is that part o f the full Compton amplitude in which the nucleus is m an excited mtermedmte state. It can be expressed m terms of its imaginar) part, i.e. by the inelastic longitudinal and transverse structure functions
St,r
x (]d¢o[l(L(q.to)SL(q,to) o
E.,,)le~.~lOt.t
WOe',".
(2)
~'1# (I
Here to' = to - q:/4M ts the internal excttation energ)', Eo < 0 the ground state energy, and Oi. t are the operators for longitudinal and transverse excitations respectively, in Coulomb gauge one obtains [6]
3. Unlike ref. [6] where a phenomenological model for the structure functions of *:C had to be used. the deuteron allows for a consistent calculatlvn of these quantittes after a nucleon-nucleon interaction has been chosen. For simplicity we take a separable potential of the form ( M is the nucleon mass) ).
f'tp,p') = - .-~ g(pJg(p').
(4)
This is not realistic in the modern sense lit lacks the one-plon exchange tail and all other comphcations of the .VN force) but it gives a fairly good description of the iow-eneriD' N N interaction which should be suffio e n t for the present accuracy of isotope shift experiments. Most important for the present purposes it allows for an analytic evaluation of the structure functions. For example, the longitudinal structure function is obtained as
SLlq,~o) = ~.M Im
+ n
/
d~pl~o(p -
(
!q)12(5 w ' + E o - - ~
(12(co'+l"o,q) ) ! + ;.C'(to' + Eo)
(5)
"
Obviousl) the first term is just the impulse approximation to the structure function whereas the last one de-
Volume 319. number 1.2,3
PHYSICS LETTERS B
scribes the final-state interaction. The functions C (E) and I(E,q) are gtven in the appendix of ref. [I0]. Due to the simple form o f e q . (4) the final-state interaction o n b acts in states with angular momentum zero which should be a good approximation for lowenergy processes. Note that the internal charge operator is p(q) = exp(lq.r[2) where the factor 1[2 arises from the transformat,on to internal coordinates. For the Yamaguchi choice [ i I ] I
g) (P) - p: + /I: "
~"[T.p(q)]
The linear terms due to the potential generate an interaction current A Jr (j,,) whereas the second-order terms give rise to the interaction seagull AB, j (y. z ) both of which can be obtained by the appropriate functional derivative ofeq. (9). With the separable potential (8) one arrives at the following expression for the matrix element of the interaction current Lo'.AJ(q)~'/ = - ~
-q. (.l"°""b(q) + J'"P'"'(q)),p) = O, (7}
(V, + V,,)
I
(61
all integrals can be performed analytically. Details will be given elsewhere. The transverse nuclear polarization shift receives different contnbut~ons: first, we have the standard contribut,on from the transverse (with respect to q) pan of the convect,on current d,o~,)(q ) which does not have a final-state interaction term s,nce all exerted states necessarily have at least angular momentum one. Second, the spin current J"P'"' (q) involves the magnettc moments and spins of neutron and proton and gives rise to spin-flip excitations. Since the procedure of calculating the transverse structure with these currents is similar to the longitudinal case the exphcit expressions v~ill not be given here. Note that the above charge and current operators obey current conservation ,n the form
23 December 1993
/ d.gU,-,, ½qlg(.'+ (t
-.I
,.'q),
(I0)
0
which satisfies
(II)
LdllV.p(ql] .- q . AJ (q )[p) = O.
so that all together our currents are conserved under the time-evolution of the full Hamihonian. Similarly one obtains an explicit expression for the matrix element of the interaction seagull which comes in addition to the kinematical (,nternal) seaguU 6,j/2M. It can be checked then that the gauge relation ~ ' . [ p ' ( q ) . A J A ( q ) ] L D ) = q(p'lAB~(q)[p)
(121
is fulfilled wh,ch is needed for full gauge invariance of the Compton ampl,tude [8,9]. The seagull contribution in eq. (3) now has the form
where T is the kinetic energy operator. 4. A separable potential of the type (41 is nonlocal and equ,valent to a momentum-dependent potentml 3. l'
_
M
f
~3,T.q
I
(8)
( i 31
where
d~rd3x.¢ (r + ~ x ) . ¢ ( r - ~x)
, exp(-i@- !x),r>(r e x p ( - i p . ~x).
.
~'(q) = ~.,
i,
du--u. [ h 2 ( 0 1 - h Z ( u q ) ] .
(14)
0
Mimmal couplingpr - p p - eA(rpl in the two-body Hamihonian lI = l" + I" then produces a power series in the electromagnetic coupling constant e
lI = Ill
,
,.,,
c jf d3vJv(y).-lu() ')
+ ~ e 2 [ dJyd~z..I,(y)..Ij(:.)B,O..z.) + .... J
(9)
h(.'4.'1
=
f
dJp~uolP)g(p -- ~tX).
(151
5. We now turn to the numerical results o f o u r calculation. We have evaluated the nuclear polarization shift consistently wtth the Yamaguchi separable form (6) using the value ,8 = 286 MeV. We have checked
PHYSICS LE'T']"r..RS B
Volume 319. number 1.2.3
numerically that our longitudinal structure function fulfills the non-energy-weighted sum rule
j doJSl(q,o~) = l - ko2(q)
(16)
0
to better than I p a n in i0 s. Here Fo(q) is the elastic formfactor calculated directly from the ground state wave function. The electric dipole polarizability (which is not the phi) relevant quantity for Swave shifts) was found to be 0.613 fm ~ compared to experimental values of 0.61-0.70 fm 3 [12]. The point rms-radius m this model is 1.92 fm compared to the experimental value 1.96 fm. This shows that the simple Yamaguchi parametrization describes the deuteron properties and therefore the low-energy triplet NN-mteract~on reasonably well. For the spinflip excitations one also needs the parameters in the singlet channel. Again follo~,ing Yamaguchi we assume ,8, = ,8, and determine the corresponding strength parameter from the singlet scattering length a . . . . 23.69 fro. For non-relativistic point hydrogen wave functions one has j4~,,o(0)l" = l / ( h a h n 3) where aB = I/D*rnrt-d) is the Bohr radius and m,,.e the reduced mass o f the lepton. Writing A¢,,0 = - ~
A?'.
(17)
the shift A?- is then independent of the atomtc state. We have evaluated the double integral in eq. (3) by Gauss-Legendre numerical integration with up to 3 × 72 points. Our results for the different contributions and for the total A?" are listed in table I. It should be emphasized that the integrand from the transverse convection current has a l/q-divergence for small q which, however, is exactly cancelled by the seagull due to the gauge condition for the two-photon operator. This can also be seen from eqs. ( 3 ) and ( 13 ): at low q the kernel Kr(q.to) behaves like - i/4mq(o [6] and Siegert's theorem tells us that
fd¢oIs~(0,~)_
l 2M +x
(18)
0
where ~¢ =_ K(0) is the dipole enhancement factor (K = 0.176 for the Yamaguchi potential). However, l0
23 I.X,-,ccmber 1993
Table I Contributions to the nuclear polarization shift AT (see eq. (17) ) for electronic tel and muonic (,u) deuterium in Coulomb gauge. The Yamaguchi S-wave separable potentaal has been used throughout. The different contnbuttons are labeled b) L: Iongttudinal. 7"~¢°~'~ + S: transverse convectlon current + seagull, rts~ttt): transverse spin current; A( T + S). mteractton transverse current + interaction seagull. Contnbutaon
e [kHz)
/J [MeVI
L
-18.31 -2.25 +0.33 -0.31 -20.54
-11.77 -0.06 +0.03 -0.02 - II .82
y.tco~,~ + .S" 7'it°'n~ A(7" + S) total
the resulting contribution to the energy shift is exactly opposite m s~gn to the q ~ 0-limit of the seagull contribution (I 3). We have checked numerically that our transverse structure function fulfills the sum rule (18) to sufficient accuracy. Consequently we onl) give the combined contribution of transverse convection and seagull excitations in table 1. It is seen that it ts bigger in electronic deuterium than in muonic deuterium because the electron velocity is higher in the first case. As the spin current contribution vanishes for q = 0 ~t can be given separately. However. numerically it turns out to be of no great importance. The same can be said of the interaction terms which nearly cancel the spin contribution. The smallness of the interaction terms is welcome since the nonlocality of the Yamaguchi separable potential is somehow artificial and pull partly simulates exchange current effects. It should be kept in mind that the individual contributions are gaugedependent and that only the total A?" is a meaningful physical quantity. The size of the transverse and seagull terms, however, indicates the errors one usually makes if only the longitudinal excitations are taken into account. As to the numerical accuracy, we have checked that the results in table 1 are accurate to one part in the last digtt. In order to estimate the model dependence of these results we also have calculated the dominating longitudinal contribution for the Tabakin separable potenttal [13] which describes both attraction at low energies and repulsion at higher energies in the S-wave phase shift. As the principal value integrals in the
Volume 319. number 1.2.3
PIIYSICS LETTERS B
Table 2 Longitudinal nuclear polarizatmn shift ..$7 for different separable N N-potentials. NN-potenttal
e [kHz]
g [MeVJ
Yamaguchl S-wa~,e [11] Tabakm S-wave [13] YamaguchiS+D-wave [14]
-18.31 -18.54 - 18.45
-11.77 -11.02 -11.86
structure functmn (5) had now to be calculated numerically the computing time for the nuclear polarizatmn shift increased considerably. Again the sum rule was checked and an electric dtpole polarizability of 0.623 fm 3 ~as obtained. The values for the shift given in table 2 are estimated to have an accuracy of about three parts m the last dtg~t. Despite the different functtonal parametrizatton of g (p) the result for A~" is very close to the Yamaguchi one which shows that only low-energy properties of the N N - m t e r a c t i o n are ~mportant for the nuclear polarization shift. Finally we also have investigated the influence of the D-state admixture in the deuteron by' using the Yamagucht st'parable potential with tensor force [ 14 ] g(p) = g)(p) + ~
1
T) ( p ) S r . ~ ) .
(19)
with Tr(p)
-
( p ; tp: + ;.,.): .
S p . ( p ) = 3 o r ' p op2, ' P
(20)
op • o,,.
(2 1)
The original parameter values of ref. [141 lead to an asymptotic D / S rano of 0.0285 whtch ts quite reasonable when compared with modern values [I 5J. The dipole polarizability is calculated to be 0.625 fm ~. Since the algebra including the tensor force is more involved the sum rule check (to one part m 10 ~) is nontrivial. To convert these numbers to the actual nuclear polarization shifts for the S-levels we need the finite size correction factors R,o. Using the approximate atomic wavefunctions of ref. [ 161 one obtains in first order in the ratio nuclear radius/Bohr radius R,o "" I
3.06 (r2"J" aB
(22)
23 December 1993
The numerical factor in this equation was determined by evaluating the rauo of various moments of the charge distribution with the Yamaguchi ground state wavefuncuon. Eq. (22) gives n.mdependent correction factors R'") = 0.9793 and ~ t , , = 0.99989. O f course, on the present level of accurac.~ one can practically neglect these correction factors. We estimate the accuracy' of our theoretical predictions in the following way.: the accuracy of the calculated longatudinal shift is taken as three times the model-dependence shown in table 2 and we assign a 20% error to the transverse current contribution and 50% one to the interaction pieces. Adding these errors linearly' we therefore arrive at the final result for the nuclear polarization shifts in electronic and muonic deutermm ,,o = ( - 20.5 -¢- 1.3) ~-~ kHz. qllm
i
A¢~0 = t- 11.6 ± 0.5) ~-~ MeV.
(23) (24)
If the future measurement of the 2P-2S transition in muonic deuterium reaches the planned accuracy of 0.05 MeV [ 5 ] the nuclear polarization shift in the 2Slevel will be an ~mportant ingredient for analysing the experiment. We also have esumated the nuclear polarization shift m piomc deuterium by' replacing the muon mass by the p~on mass. The longitudinal and the transverse convection current contribution of the present formalism should give a reasonable value also for a heavy spin zero panicle because to a good approximation it can bc treated nonrelatwistically with no difference between a Dirac and a K l e i n - G o r d o n description. In this way we obtain I
a,~'~:' ~_ -28 ~ MeV.
(25~
For n = I this is a factor of t ~ o smaller than the precision aimed at in an ongoing experiment at PSI to measure the strong mteractmn shifts in pionic hydrogen [17]. If the future isotope shift experiments in electronic deuterium actually reduce the experimental accuracy to about I kHz [ I ] it would be worlh~hile to repeat the present calculation with a modern NN-potentml hke the Paris potential [18]. Finally'. before discrepanctes between theor)' and experiment in the isotope II
Volume 319, number 1.2.3
I)HYSIC.'S I.E i TELLSB
shifts are taken serious one should include secondorder effects also in the anal.,,sis o f e l e c t r o n - d e u t e r o n scattering e x p e r i m e n t s ~ h l c h extract the r o o t - m e a n .square radius o f t h e deuteron. In the case o f J-'C similar diserepanc~es betg'een electron scattering data and m u o m c energ.~ shifts seem to disappear [191 when second-order effects are taken into account m the analysis o f both experiments. We thank Andreas Schreiber for helpful discussions and a critical reading o f the manuscript. We are grateful to Leo Ssmons and Pieter G o u d s m i t for p r o v i d i n g us with e x p e r i m e n t a l information.
References [ I ] F. ~hmld1-Kalcr. D. Lelbfncd. M. Wcltz and T 9,. llansch. Ph'.s. Rc~. Left. 70 11993p 2261. [2] K. Pachuckl. D. Lesbfned and T.W. Hansth. Ph~,s. Re'. ..',, 48 11993~ RI.
12
2"~December 1993
[31 R. Roscnfelder, in: Muon,c Atoms and Molecules. eds L.A. Schaller and C. Petitjean (Btrkauser. 1993) p. 95. [4] E Za'.attmt et al.. PSI proposal R-93-06.1 (1993). [5l L. S,mons. private communication. [6] R. Roscnfelder. Nucl. Phys. A 393 (1983) 301. [71H. Grotch and D.R. Yenme, Rev. Mod. Phys. 41 (1969) 350. [8] J.L. Friar and M Rosen. Ann. Phys. 87 (1974) 289. 191H. Arcnhosel. Z. Ph~,s. A 297 { 1980) 129. [101R. Rosenfclder. Ann. Phys. 128 (1980) 188. [11 ] Y. Yamaguchl, Phys. Re,,. 95 (1954) 1626. [12]J.L. Friar and S. Falheros, Ph~,s. Rev. C 29 ~19841 232. [13] F. Tabakm. Phys. Re,,. 174 (1968) 1208. 114] Y. Yamaguchl and Y. Yamaguchl, Phys. Rev. 95 I1954~ 1635. II5]T.E.O. Ericson and M. Rosa-Clot. Ann Rev. Nucl. Part. , ~ . 35 (1985) 271. [161J.L Frtar. Z. Ph~,s. A 292 (19791 I. [ 17 ] P.F.A. Goudsm,t. pr,'.ate commumcation. [18] M. Lacombc et al., Ph)s. Re',. C 21 (1980) 861. [ 19] E.A.JM. Ofl'erman et al.. Phys. Re'.. C 44 ( 1991 ) 1096.