Axial meson exchange and the relation of hydrogen hyperfine splitting to electron scattering

Volume 19, number 6

PHYSICS

AXIAL MESON HYPERFINE

EXCHANGE SPLITTING

LETTERS

1 December 1965

AND THE RELATION OF HYDROGEN TO ELECTRON SCATTERING *

S. D. DRELL and J. D. SULLIVAN Stanford

Lznear

Accelerator

Center,

Stanford

Unzverszty,

Stanford,

Calzfomza

Received 1 November 1965

In a recent note, Fenster et al.[l] suggested that the exchange of an axial vector meson (Jpc = l++) might remove much of the apparent discrepancy between theory and experiment for the ground state hyperfine splitting of hydrogen. The existence of such an electromagnetic Gamow-Teller type mteraction was postulated by Nambu and collaborators m earlier publications [2] in order to mamtain divergence free axial currents. Its strength was determined on the basis of a theoretical model and the observed r” - 2~ decay rate; its phase was undetermined by their arguments. We wish to show m this letter that such a suggestion is difficult to reconcile with existing data on the ratio of positron-proton to electron-proton elastic scattermg [3]. The diagram considered by Fenster et al. is shown m fig. 1, where the axial vector meson of mass mA is indicated by A. We denote its phenomenological couplmgs to protons and electrons by Band b respectively. The true coupling of A to electrons presumably occurs through a two photon intermediate state; however, it is not necessary to concern ourselves with the detailed mechanism here. The diagram of fig. 1 gives for the S-matrix element between e-p states **

S

e-p; e-p

g@‘-

= (2n)464Cpr+I’ -p-Z)

qCLqV/m2

A

q’-rni

where q = p’ - p. From eq. (1) in the non-relativistic

(1)

’

limit one infers an effective hyperfme Ha.miltonmn 6H

= - +rewp

6(re-

r’p)

(2)

mA m addition to the usual Fermi-Segre

term

,2

H= 12memp

gpae*up

6(re-rp)

(3)

3

where g = 5.58 is the proton gfactor. Hence, the exchange of an axial vector meson generates a fractional c Range in the triplet-singlet ground state hyperfine splitting u = VT- VS of $=

- (_!&)(‘2;mP).

(4)

A If the apparent discrepancy between theory and experiment [4] Vept -Vtl=ory = (45 Vexpt

l

17) x 10-6 a 4(1 * $) x 10-5

1s attributed entirely to the axial exchange mechanism the resulting couplmg is determined to be * Work supported by the U.S. Atomic Energy Commission. ** Couplmgs of the form E(P)q ‘~5 u(p) may be omltted smce the electron CL limit me - 0.

516

axial current

1s dwergenceless

m the

PHYSICS

Volume 19, number 6

1 December 1965

LETTERS

We should perhaps point out the relation of the axial vector exchange of the type considered here to the exhaustive discussion of the hydrogen hyperfine splitting presented in the latest analysis of Iddings [4]. He expressed the proton structure contributions to the hyperfine splitting in terms of a &spersion relation for the forward Compton scattering y + p - y + p of off-mass shell photons. The weight functions m this dispersion relation are measurable in e-p scattering. There remains, however, the possible presence of subtraction terms proportional to the photon mass. (On-the-mass-shell Compton scattering has a subtraction term uniquely fixed by the Thomson limit.) Axial vector exchange (with couplmg to the electron line via two photons) is precisely an off-the-mass-shell subtraction term of this type; it has no absorptive part in the e-p channel, and it vanishes as the photons go on the small shell (l+ - y + y is forbidden by angular momentum conservation and statistics for real photons). The S-matrix element eq. (1) also occurs in electron and positron-proton scattering in addition to the usual one photon exchange contribution and leads to a first order correction to the Rosenbluth cross section that is readily calculated to be [5]

where 1

(Y2 cos2 $e

M = 4E2 sin4 $6 [1+2E/mp

sina $01

I

I

I

I

I

I

I

06

06

l3-

q2 7a-4X0

(P)

EXPERIMENT

P

and GR and G are familiar electric and magnetic . form . fat7, ors normalized to GE(O) = 1, GM(O) = $gp = 2.79. Two features of eq. (7) are of particular experimental interest. The last term changes sign from + for positron-proton to - for electronproton scattering since it is an interference term between the odd charge conjugation amplitude for photon exchange and the even amplitude one for axial exchange (equivalent to two photons). Also the last term has a factor (E/mp) leading to a deviation from the Rosenbluth straight line [5].

l2-

E is the mcident electron energy,

t b’d 00 I

lO--

P

--q-

P

50’

130”

Ogo

p

T

I I-

($)THE~RY

P'

-7rA

9

Fig. 1. Axial vector exchange contributionto the electron-proton interaction.

1

OS%

I 02

I 04 )Q’( -)

(GeV/d

Fig. 2. The positron-proton to electron-proton differential cross section ratio versus momentumtransfer. Shownare the experimentalpointa of Browman et al. and the correapondlngpomta that wouldbe expected on the basis of axial vector exchangeusing the coupling of eq. (6) and a form factor varying as GE@) . The theoretical points are calculated for mA = 1.5 mp and are msensltive to this choice. 517

Volume 19, number 6

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PHYSICS

LETTERS _______

_ ______~~ _

1 I:ecember 1965 ____~ ._ ____._

If (Bb) has the mmus sign required to fit the hyperfme splitting we see lmrnec~~*sl~ly frl),il eq. (7) C 1 Using eqs. (6) and (7) and allowing Bb to have a iri~m far\:~r varying that (dddn)e+p/(dddsl)e-p like %(q2) we have calculated the ratios shown m fig. 2 at energies and angles ct)i‘r.:spor!‘*ulg !o the experimental p mts of Browman et al. [3]. It is clear from that figure that axial WCtor exchange sufficient to explain all or most of the hyperfine dscrepancy is difficult if not impon$rrh;c:to rl:concile with the observed ratio. The theoretical pomts are msensitive to the choice of the axldl meson mass for all rnA2> -2 as assumed here and their error bars reflect th? uncertainly m eq. (6). Only wjth an ad hoc assumption that the coupling Bb has a form factor that falls for large q2 < 0 much more ral!ldly than GE(q2) and GM(q2) = 2 78 X GE(q2) is it possible to avoid t.hs contradiction. AltcrnWvely one must turn to additional and compensating 2~ exchange contributions to e*p scattering wluch have not been mdicated by earlier studies [6] or one must look for an interpretation of the apparent hyperfmc sphttmg discrepancy m eq. (5) m terms of other inadequacies of the theoretical calculations of the 2y exchange contribution. Pseudoscalar exchange cannot provide such a compensation since its coupling via two photons to the eiectron line vamshes m the tigh energy limit me - 0. [6, footnote on page 361 It 1s also possible m principle to test for this axial vector exchange by means of electron-proton scattermg alone [6]. From eq. (7) we see that in the plot of (du/dO)/(du/&)M versus tan2 $% ai fixed q2 the explicit presence of E/mpmthe interference term causes a departure from straght lme behavlour. However, with the coupling parameter of eq. (6) It 1s found that experiments to better than a 1% accuracy would be required in order to detect the prehcted deviation from lmearlty with present accelerator (includmg SLAC) parameters. This does not seem to be feasible. References 1. S. Fenster, R. K(lberle and Y.Nambu, Physics Letters 19 (1965)513 Iddmgs [4] has also speculated that a l+ meson could play a role 111 the hyperfme discrepancy.

2. J.Cron 1and Y.Nambu, to be published; Y.Nambu and J.J.Sakural, Phys.Rev.Letters 11 (1963) 42; P.G.O.Freund

and Y.Nambu, to be published.

3. A.Browman, F.Lm and C.Schaerf, Phys.Rev.139 (1965) B1079. 4. W. E. Cleland, J. M. Bailey, M.Eckhause, V. W. Hughes, R. M. Mobley, H. Prepost and J. E . Rothberg, Phys. Rev. Letters 13 (1964) 202; S.B.Crampton, D.KIeppner andN.F.Ramsey, Phys.Rev.Letters 11 (1963) 338; C.K.Iddmgs, Phys.Rev. 138 (1965) B446, C.K.Iddmgs and P.M.Plaizman, Phys.Rev.113 (1959) 192; C.K.Iddings and P.M. Platzman, Phys.Rev. 115 (1959) 919. 5. These features of axial vector exchange have already been pomted out by D.Flamm and W.Kummer, Nuovo Cimento 28 (1963) 33. 6. S.D.Drell and S.Fubml, Phys.Rev.113 (1959) 741.

*****

FORM

FACTORS FOR WEAK DECIMET-OCTET

AND ELECTROMAGNETIC TRANSITIONS

R. OEHME The Ennco Ferma Institute for Nuclear Studzes and Department of Physzcs. The Unrverszty of Chzcago, Chzcago, Illznors Received 5 November 1965

The spurion scheme of broken U(6,6) - symmetry [l] implies that vertex functions are approximately covariant with respect to the collinear group U(6)q [2]. The consequences of this scheme for the electro* This work was supported m part by the U.S. Atomic Energy Commission. 518