The 2H Electric Dipole Moment in a Separable Potential Approach

Measurement of the electric dipole moment (EDM) of 2H or of 3He may well come prior to the coveted measurement of the neutron EDM. Exact model calculations for the deuteron are feasible, and we explore here the model dependence of such deuteron EDM calculations. We investigate in a separable potential approach the relationship of the full model calculation to the plane wave approximation, correct an error in an early potential model result, and examine the tensor force aspects of the model results as well as the effect of the short range repulsion found in the realistic, contemporary potential model calculations of Liu and Timmermans. We conclude that, because one-pion exchange dominates the EDM calculation, separable potential model calculations should provide an adequate picture of the 2H EDM until better than 10% measurements are achieved.


Introduction
With the discovery of parity (P) violation as suggested by Lee and Yang [1], Landau [2] recognized that charge conjugation along with parity (CP) invariance would imply that the electric dipole moment of the neutron should be zero.The Standard Model predicts values of EDMs which are smaller than are currently experimentally detectable.Thus, an unambiguous observation of a non zero EDM would imply an undiscovered source of CP violation [3,4].The implied new physics could arise in the strong interaction sector (e.g., the θ term), or the weak interaction sector (e.g., second order W boson exchange).
Both PT violating interactions as well as P conserving but T violating interactions may give rise to an EDM [3].However, one-pion exchange contributes only to the former.We focus our attention here on the effects of the violation of PT invariance in the nuclear potential, because a method has been proposed to directly measure the EDM of a charged ion in a storage ring [5].
This prospect makes the deuteron attractive for an EDM investigation.As one will see, the contributions of the proton and neutron EDMs tend to cancel, so that the PT violating interaction should become a significant contributor to the 2 H EDM. Because the deuteron is well understood, reliable calculations are possible.

Analysis
The total one-body contribution to the deuteron EDM of the neutron and proton is the sum of their individual EDMs: a e-mail: bfgibson@lanl.gov The neutron and proton EDMs can arise from a variety of sources.Liu and Timmermans [6] estimate: which has been expressed in terms of the Ḡ(i) X , the product of the strong coupling constant g XNN and the associated PT violating meson-nucleon coupling constant ḡ(i) X .We note that the theoretical uncertainty in the estimate for d (1)  D is sizable.
However, Liu and Timmermans also estimated the twobody contribution from the PT violating NN interaction to the deuteron EDM to be Therefore, for the deuteron it is clear that the nuclear physics component of d (2)  D dominates.Even an uncertainty of 40% in d (1)  D makes but a minor contribution.For this reason we investigate the model aspects of the d (2)  D dominant term in the 2 H EDM in some detail.
To aid in our discussion that follows, we separate the two-body contribution to the deuteron EDM d (2)  D into a plane-wave term d PW and a final-state re-scattering contribution d MS : We note that d PW and d MS are proportional to the coupling constant combination g πNN ḡπNN /16π.
EPJ Web of Conferences tial of Mongan [8].He reported a value of -0.91 e fm for a physical pion mass of the exchanged meson.Khriplovich and Korkin [9] later estimated d (2) D using a zero-range approximation in the chiral limit and reported a value of -0.96 e fm.
Using the contemporary potential models Av 18 , Reid93, and Nijm II [10], Liu and Timmermans [6] estimated the value for d (2)  D to be -0.73 ± 0.01 e fm within the range of uncertainly defined by the three potential models.
The implications of these results are several.
-If the zero range calculation is correct, then the results of Liu and Timmermans suggest that the final-state interaction reduces d (2)  D by less than 25%.-Such a reduction is considerably greater than the reduction reported by Avishai.In fact, we find the Avishai result to be in error by a factor of 2; we believe that the corrected result should be -0.48 e fm.-The significant difference between the results of Avishai and Liu and Timmermans suggest that the EDM is sensitive to the potential model fit to the NN scattering data.
Thus, we are led to explore the sensitivity of the 2 H EDM to the potential model representation of the NN scattering data.That includes the softness of the rank-one Mongan potentials compared with the contemporary potential models emplyed by Liu and Timmermans and the sensitivity to the short range repulsion in the deuteron wave function.

Original Mongan 3 P 1 Results
To explore the accuracy of the results for the deuteron EDM, we begin that process by using a set of Mongan's [8] rankone 3 P 1 potentials.The separable potential form factors are tabulated in Table 1 along with the range parameters β and the potential strengths λ.In Table 2 we quote the measure of the quality of the 1969 potential fits to the data as provided by Mongan, the sum of the squares of the residuals: The fits are semi-quatitative at best, which is indicative of the lower precision of the experimental data in 1969.For the deuteron in this study we have used the rankone Yamaguchi-Yamaguchi [11] 4% and 7% (P D ) potentials plus the Unitary Pole Approximation (UPA) for the Reid soft core (RSC) potential [12] generated in 1973 [13].
To compare with Avishai, we look at d (2)  D as a function of the mass of the exchanged meson for the P D = 4% YY deuteron in Table 3. Avishai did not specify the Mongan potential he used, but it is clear that our results are not consistent with his.The difference is of the order of a factor of 2. He used an overall strength coefficient of A = g πNN ḡπNN /8π , whereas we find the denominator should be 16π.Thus, we believe the Avishai result to be in error by a factor of 2. Therefore, we suggest that the correct result for Avishai should be -0.46 e fm.
Table 3.The dependence of the deuteron EDM d (2)  D on the mass of the exchanged particle m in the potential V. We compare the results for three Mongan potentials with the results of Avishai [7].In each case a Yamaguchi-Yamaguchi potential with a 4% Dstate probability was used.To compare with Khriplovich and Korkin, we look at the results in Table 4.The nominal agreement between that of Khriplovich and Korkin and the RSC (UPA) is likely fortuitous.Nevertheless, given that the result of Khriplovich and Korkin was obtained in the chiral limit of a zero range approximation, the overall agreement is reasonable.Moreover, short range repulsion would seem to account for only a 10% reduction in d PW .

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Table 4. Variation in the 2 H EDM with the strength of the tensor force and the role of short range repulsion.For the 3 P 1 potential the Mongan Case I (1968) was used.The 1969 Mongan potentials were fit to early phase shift data.In contrast Liu and Timmermans used potentials that reproduce the 1993 Nijmegen phase shift analysis [14].
To explore the sensitivity to the 3 P 1 phase shifts, we have refit the Mongan potentials to the 1993 Nijmegen phase shifts.The new parameters are summarized in Table 5 along with The fit to the phase shifts is illustrated by quoting values of δ at 1 and 100 MeV.We compare in Table 6 and Table 7 results for the original Mongan potentials with results for the contemporary Mongan potentials fit to the 1993 phase shifts, for the YY deuteron models with 4% and 7% D-state probabilities.Table 6.The deuteron EDM values for the two YY rank-one deuteron 3 S 1 -3 D 1 potentials and four different rank-one 3 P 1 potentials.The 3 P 1 potentials are the original Mongan rank-one potentials.
3 S 1 -3 D 1 YY 4% YY 7% The d MS values for the contemporary Mongan potentials exhibited in Table 7 are significantly smaller, bringing d (2)  D into closer agreement with the result of Liu and Timmermans.The variation with P D is comparable to that shown for the d PW values.
Table 7.The deuteron EDM values for the two YY rank-one deuteron 3 S 1 -3 D 1 potentials and four different rank-one 3 P 1 potentials.The 3 P 1 potentials are of the Mongan form with parameters adjusted to fit the 1993 Nijmegen phase shifts.In Table 8 and Table 9 we compare results for the original Mongan potentials with results for the contemporary Mongan potentials fit to the 1993 phase shifts, for the RSC (UPA) deuteron model.Again, the results in Table 9 for the contemporary Mongan potentials are in much better agreement with that of Liu and Timmermans.In summary, the differences due to the improved fit to the NN scattering data are significant.The effect of the 3 P 1 interaction is much reduced; d MS is smaller for the contemporary Mongan potentials fit to the 1993 phase shift analysis.This suggests that the potentials fit to the more recent scattering data are weaker, affecting the EDM calculation less.Moreover, for the RSC deuteron calculations shown in Table 8, the short range repulsion in the deuteron 04016-p.3EPJ Web of Conferences has removed most of the 3 P 1 model dependence.Furthermore, the separable potential model result agrees with that of Liu and Timmermans d (2)  D = -0.73 ± 0.01 e fm within about 10%.

Conclusions
We find Avishai's original calculation to be in error by a factor of 2; we believe that the corrected result should be -0.46 e fm.This would make his estimate for the 2 H EDM more consistent with the plane wave result of Khriplovick and Korkin as well as the realistic potential model calculations of Liu and Timmermans.
The EDM calculation is sensitive to the quality of the fit to the 3 P 1 NN scattering data.The resulting d (2)  D values, based upon the contemporary Mongan potentials fit to the Nijmegen 1993 phase shifts, differ strikingly from the results based upon the original Mongan potentials, especially for the YY models of the deuteron.
Fitting to the Nijmegen 1993 phase shifts reduces the contribution of d MS considerably, suggesting that the contemporary Mongan 3 P 1 potentials are significantly weaker.Thus, it may be possible to treat the 3 P 1 interaction as a perturbation when calculating the 3 He EDM.
The d MS term is smaller for the RSC (UPA) calculation than for the YY calculations, which leads to d (2)  D being closer to the Liu andTimmerman result for the realistic, contemporary potentials.Moreover, the dependence of the RSC (UPA) result on a particular 3 P 1 potential is reduced to the point that d (2)  D is almost independent of the 3 P 1 contemporary Mongan potential model used.
Finally, until the precision of 2 H EDM measurements is considerably enhanced (i.e., reaches a level of better than 10%), it would appear that a separable potential approach to modeling the EDM should be more than adequate.

Table 1 .
The parameters of the rank one Mongan [8] 3 P 1 potentials.The strength of the potentials is related to Mongan's strength C R by λ = C R c .For Case III (1969) the function Q 1 (z) is the Legendre function of the second kind.

Table 5 .
Parameters for the contemporary Mongan potentials that give a best fit to the 1993 Nijmegen np phase shifts.Also included is the χ 2 = 11 i=1 |δ Th i − δ exp i | 2 and shown are the fits to the phase shifts δ at 1 and 100 MeV.

Table 8 .
The deuteron EDM values for the RSC (UPA) rank-one deuteron 3 S 1 -3 D 1 potential and four different rank-one 3 P 1 potentials.The 3 P 1 potentials are the original Mongan rank-one potentials.

Table 9 .
The deuteron EDM values for the RSC (UPA) rank-one deuteron 3 S 1 -3 D 1 potential and four different rank-one 3 P 1 potentials.The 3 P 1 potentials are of the Mongan form with parameters adjusted to fit the 1993 Nijmegen phase shifts.