UM–P–92/42

Relativistic Corrections to the Neutron Electric Dipole Moment in Valence Quark Models

John P. Costella and Bruce H. J. McKellar

School of Physics, The University of Melbourne, Parkville 3052, Australia

and

Institute for Nuclear Theory, University of Washington, Seattle WA 98195, U. S. A.

Abstract

We show that, in valence quark models, the relativistic corrections to the SU(6) relation for the contribution of the electric dipole moments of the quarks to the electric dipole moment of the neutron can be expressed as a multiplicative correction factor. The correction factor is evaluated in light cone wavefunction models, in the bag model, and in relativistic mean-field models, and is found to lie between 1/3 and 1.

We also show that, in these models, there is a linear relation between the correction to the SU(6) value for g_{A} and that for the electric dipole moment.

I. INTRODUCTION

The neutron electric dipole moment (EDM), D_{n}, is non-zero only when symmetry under both parity and time-reversal are broken. That P and CP violation have been observed in K_{L} → 2π makes it plausible that the neutron EDM (i.e. a term D\overline{ψ}γ_{5}σ_{μν}ψ in the electromagnetic current) will be observed, although only an upper limit

|D_{n}| < 1.1 × 10^{−25} e cm  (1)

is currently available [1]. Theoretically, the value of D_{n} is determined by contributions from the electric dipole moments of the quarks (D_{q}), the colour electric dipole moments of the quarks and gluons, P and T violating q–q interactions and long distance effects [2]. In this paper we concentrate on the contribution from the EDMs of the valence quarks. D_{n} is generally calculated from the D_{q} using the non-relativistic SU(6) relation:

D_{n}^{(V)} = \frac{1}{3} (4D_{d} − D_{u})  (2)

Ellis [3] has emphasised the need for the examination of relativistic corrections to (2). We will refer to the SU(6) value of D_{n}^{(V)} as D_{0}, i.e.

D_{0} ≡ \frac{1}{3} (4D_{d} − D_{u})

and the relativistic value of D_{n}^{(V)} as D. The ratio D/D_{0} will be referred to as κ_{D}.

It is reasonable to expect that relativistic effects should be significant. Even one of the key papers establishing the non-relativistic constituent quark model [4] required a constituent quark mass of order 300 MeV and a radius of the hadron of order 0.6 fm ≈ 330 MeV^{−1}. Thus p/m ≈ 1, and relativistic corrections cannot be expected to be negligible.

We can get some idea of what to expect by examining the magnetic dipole moments, where relativistic corrections to the SU(6) relations are known to be important [5–9]. Indeed, in that case the relativistic corrections destroy the agreement between theory and experiment for the octet magnetic moments. We will give the results of an analogous calculation for the electic dipole moments in section II.

Other relativistic models of the valence quark wavefunctions of the hadron are the bag model [10], and the mean-field potential model [11]. The bag model, which does not respect translation invariance, has been criticised as a model of the magnetic dipole moments [12]. In the case of the electric dipole moment there is no contribution from the γ_{μ}, so that the corrections from the spurious centre of mass motion in bag models may not be so important. Results from these models are similar and are discussed in section III.

In all of these models we find the same relationship between the relativistic corrections to the EDM and the relativistic corrections to g_{A}. This relationship is discussed in section IV. Our analysis may be regarded as analogous to the relation between magnetic moments and the axial vector matrix elements discussed by Karl [9].

II. LIGHT CONE WAVEFUNCTION MODELS

In this section, we use the light-front formalism as developed by Berestetskii and Terent’ev [5], in a completely analogous way to their calculation of the nucleon magnetic moments [6]. (The correction to the three-body Melosh transformation [13] is not required for nucleon calculations.) We extend equation (23) of reference [6], which gives the hadron electromagnetic form factors in terms of the electromagnetic interactions of the quarks, to include the electric dipole moment terms:

{F_{1}^{Λ}δ_{λλ′} + iF_{2}^{Λ} (k ·ε ·σ_{λλ′}) + D^{Λ} (k ·σ_{λλ′})} δ_{ΛΛ′}
= 3 ∫ dΓψ_{Λ′λ′}^{∗} {F_{1}^{(c)} + iF_{2}^{(c)} (k ·ε ·σ^{(c)}) + D^{(c)} (k ·σ^{(c)})} e^{−ikρ^{(c)}}ψ_{Λλ}

Application of the Melosh transformation following the analysis of references [6] and [7] yields the result

κ_{D} ≡ \frac{D}{D_{0}} = 1 − ζ  (3)

where ζ is a complicated integral over the quark relative momenta, that depends on the chosen form of the quark wavefunction. The calculation of κ_{D} is almost identical to the calculation of the contribution of the quark anomalous magnetic moments to the anomalous magnetic moment of the nucleon, which is no surprise since

γ_{5}σ_{μν} = \frac{i}{2}ε_{μνρλ}σ^{ρλ}

One can evaluate the correction factor κ_{D} above by using some specific choice for the wavefunction; for example, with the gaussian form Φ ∼ exp(−M_{0}^{2}/α^{2}) (where M_{0} is the effective mass, which is a function of the momenta of the constituent quarks; see [5]), a constituent quark mass of 300 MeV, and a choice of the free parameter α that correctly reproduces the neutron magnetic moment [7], this model gives κ_{D} = 0.993, which is a very small variation from the non-relativistic value. For this form of wave function it is possible to demonstrate that κ_{D} is a monotonically decreasing function of α/m, which approaches the value κ_{D} = 1/2 as α/m → ∞ (the extreme relativistic limit).

It is instructive to apply the same technique to the axial vector charged current A_{μ}^{(+)}. It is well known [4] that the nonrelativistic quark model gives the SU(6) value g_{A(0)} = 5/3. The lightcone wavefunction methods applied to A_{μ}^{(+)} give

\frac{g_{A}}{g_{A(0)}} = 1 − 2ζ  (4)

where ζ is the same complicated integral as in (3). (It is amusing to note that the extreme relativistic limit discussed above gives g_{A} → 0 — we do not have any physical interpretation of this extreme relativistic quenching of g_{A}, which seems to be a peculiarity of the light cone wavefunction model.)

Eliminating ζ from equations (3) and (4) gives us

κ_{D} ≡ \frac{D}{D_{0}} = \frac{1}{2} \{1 + \frac{g_{A}}{g_{A(0)}} \}  (5)

One might then, as an improved estimate, use the experimental nucleon axial coupling constant [14]

g_{A} = 1.261 ± 0.004

in equation (5) to obtain

κ_{D} ≈ 0.878

which indicates that the relativistic corrections to the EDM of the hadrons are on the order of 10%.

III. BAG AND MEAN-FIELD POTENTIAL MODELS

For our purposes the bag model may be regarded as equivalent to a potential model, since the essential feature we require is the structure of the single particle relativistic wave function, viz.

ψ = ( \begin{matrix}f_{j}(r)\sqrt{\frac{j+m_{j}}{2j}} Y_{j−1/2,m_{j}−1/2} \\ −f_{j}(r)\sqrt{\frac{j−m_{j}}{2j}} Y_{j−1/2,m_{j}+1/2} \\ g_{j}(r)\sqrt{\frac{j+1−m_{j}}{2(j+1)}} Y_{j+1/2,m_{j}−1/2} \\ g_{j}(r)\sqrt{\frac{j+1+m_{j}}{2(j+1)}} Y_{j+1/2,m_{j}+1/2} \end{matrix} )

where j is the total angular momentum, m_{j} its projection, f_{j}(r) and g_{j}(r) the solutions of the (coupled) radial equations, and we are using the Dirac–Pauli representation of the gamma matrices.

That the bag model determines the radial wavefunctions f_{j}(r) and g_{j}(r) by imposing boundary conditions, and that mean-field potential models determine them from an assumed or calculated mean-field potential in the Dirac equation, is not important for our calculation.

Using the P and T violating electromagnetic current

J_{μ}^{P̸ T̸} = D_{q}\overline{ψ}γ_{5}σ_{μν}ψ

and the axial vector current

A_{μ}^{(+)} = g_{A}\overline{ψ}γ_{μ}γ_{5}ψ

one readily obtains

\frac{D}{D_{0}} = F_{j}^{2} + \frac{1}{3}G_{j}^{2}  (6)

and

\frac{g_{A}}{g_{A(0)}} = F_{j}^{2} − \frac{1}{3}G_{j}^{2}  (7)

where we have defined the quantities

F_{j}^{2} ≡ ∫_{0}^{∞} r^{2}dr |f_{j}(r)|^{2}

and

G_{j}^{2} ≡ ∫_{0}^{∞} r^{2}dr |g_{j}(r)|^{2}

In addition to (6) and (7), normalisation of the wavefunction requires

F_{j}^{2} + G_{j}^{2} = 1  (8)

Elimination of F_{j}^{2} and G_{j}^{2} from (6) and (7) using (8) gives the relation:

κ_{D} = \frac{1}{2} \{1 + \frac{g_{A}}{g_{A(0)}} \}

just as before.

In this case, using equations (6), (7) and (8), and the fact that F_{j}^{2} and G_{j}^{2} are by definition positive-indefinite, one can also see that the relativistic modification to the valence quark contribution to to g_{A} and D, regardless of the bag boundary conditions or potentials assumed, must be in the range

\frac{1}{3} ≤ \frac{D}{D_{0}} ≤ 1

and

−\frac{1}{3} ≤ \frac{g_{A}}{g_{A(0)}} ≤ 1

respectively.

IV. DISCUSSION

The most significant result of our calculations is that in all the models considered, we find that the relativistic wavefunction corrections to D and g_{A} are related by

κ_{D} ≡ \frac{D}{D_{0}} = \frac{1}{2} \{1 + \frac{g_{A}}{g_{A(0)}} \}  (9)

We emphasise that all of the models we have considered are independent quark models, which ignore interactions between the quarks, and which ignore QCD except through the “mean field” seen by the quarks. Many contributions to D (e.g. quark and gluon electric dipole moments, P and T violating quark–quark interactions, long distance effects), and to g_{A} (e.g. gluonic contributions, anomalous contributions, sea quark effects, long distance effects) have been omitted from our considerations, so that strictly speaking equation (9) applies only to the valence quark contributions to both D and g_{A}.

However, equation (9) does answer Ellis’s call for an estimate of the relativistic corrections to the valence quark contribution to D_{n}. It shows that the relativistic effects alter the valence quark contributions to D by about 10% — and in the most extreme and unrealistic models by no more than a factor of 3. Given that the level of uncertainty in the estimates of D_{q} can be as much as a factor of 10, as can the uncertainties in the other effects which can contribute to D [2], we must regard the relativistic corrections considered here as well controlled.

V. ACKNOWLEDGEMENTS

Helpful discussions with Prof. J. M. Namyslowski, Prof. L. Wilets, Dr. J. W. G. Wignall and Dr. S. Bass are gratefully acknowledged.

This work was supported in part by the Australian Research Council, an Australian Postgraduate Research Allowance and a Dixson Research Scholarship. We warmly thank the Institute for Nuclear Theory at the University of Washington for its hospitality and the United States Department of Energy Grant #DOE/ER40561 for partial support during the completion of this work.

REFERENCES

[1] I. S. Altarev et. al., Phys. Lett. B 276, 242 (1992); K. F. Smith et. al., Phys. Lett. B 234, 191 (1990).

[2] X.–G. He, B. H. J. McKellar and S. Pakvasa, Int. J. Mod. Phys. A 4, 5011 (1989).

[3] J. Ellis, Nucl. Inst. and Meth. A 284, 33 (1989).

[4] A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D 12, 147 (1975).

[5] V. B. Berestetskii and M. V. Terent’ev, Yad. Fiz. 24, 1044 (1976).

[6] V. B. Berestetskii and M. V. Terent’ev, Yad. Fiz. 25, 653 (1977).

[7] N. E. Tupper, B. H. J. McKellar and R. C. Warner, Aust. J. Phys. 41, 19 (1988).

[8] P. L. Chung and F. Coester, Phys. Rev. D 44, 229 (1991).

[9] G. Karl, Phys. Rev. D 45, 247 (1992).

[10] A. W. Thomas, Advances in nuclear physics, Vol. 13, eds E. W. Vogt and J. Negele (Plenum, New York), ch. 1.

[11] L. Wilets, Non Topological Solitons (World Scientific, Singapore).

[12] S. J. Brodsky and S. D. Drell, Phys. Rev. D 22, 2236 (1980).

[13] L. A. Kondratyuk and M. V. Terent’ev, Yad. Fiz. 31, 1087 (1980).

[14] Particle Data Group, Phys. Lett. B 239, VIII.8 (1990).