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The Neutron Electric Dipole Moment

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The Neutron Electric Dipole Moment H-P--S<J|o2_ UH-511-666-89 /03f UM-P-89/2 Mtj \OZ-1989/01 Jan. 1989 THE NEUTRON ELECTRIC DIPOLE MOMENT Xiao-Gang He and Bruce H. J. McKellar Research Center for High Energy Physics School of Physics, University of Melbourne Parkville, Vic. 3052, Australia and Sandip Pakvasa Department of Physics and Astronomy University of Hawaii at Manoa Honolulu. HI 96822, USA o2-^ - Abstract We have made a systematic study of the electric dipole moment (EDM) of neutron Dn in various models of CP violation. We find that (i) in the stundard KM model with 3 families the neutron EDM is in the range 1.4x10-33 < |Dn| < 1.6X10'31 e.cm, (ii) the two Higgs doublet model has approximately the same value of Dn as the standard model, 25 (iii) Dn in the Weinberg model is predicted to satisfy |Dn| > 10 ecm, (iv) in a class of left-right symmetric models Dn is of the order of 10-26±1 e.cm, 22 v) in supersymmetric models, Dn is of order 10~ $ ecm with <)> being the possible phase difference of the phases of gluino mass and the gluino-quark-squark mixing matrix, vi) the strong CP parameter 6 is found to be 9 < 10^, using the 25 present experimental limit that |Dn| < 2.6 x 10 ecm with 90% confidence. 2tt- 3 1 ) INTRODUCTION The electric dipole moment (EDM) of neutron (Dn) has been of interest to physicists for a long time. In 1950, Purcell and Ramsey!1! first considered the probelm of the existence of the neutron EDM. However at the time parity-P was assumed to be an exact conservation law, and in order to have a nonzero Dn, Purcell and Ramsey had to construct a rather unconventional P even EDM for 2 neutron. In 1957 Landaul ! observed that a non vanishing Dn was a signal of P and T (time reversal) non-conservation!3!. It was at about this time that P invariance was found not to be an exact conservation law of nature. The EDM of neutron violates both P and T. Even afier P invariance was found to be violated by the weak interactions, the prejudice was that T would be conserved, and if the CPT theorem is vald it must preserve the product of P and C(charge conjugation). When CP violation was observed in Kaon system!4! in 1964, the subject of the neutron EDM become of considerable theoretical and experimental interest. A recent review has been provided by Shabalinl5!. The current experimental situation is that two groups have reported values of -(1.4 ±0.6 ) x 1025 ecml6! and -(1.1 ± 0.7)x1025 ecml7! which correspond to an upper bound of |DnU 2.6 x10"25 e.cm at the 90% Confidence Level(C.L). There are many theoretical models which attempt to account for the observed CP violation in the K meson system, and in general they give different predictions for the neutron EDMf5!. This has provided an impetus for the experimenters •• particularly as some of the theories give values which are tantalizingly close to the present experimental upper bound. M This fact means that it is important to be satisfied that the calculation of Dn is reliable, and that the effect in different theories is calculated consistently, so that comparisons may be made in a consistent way. Surprisingly, a systematic, consistent study of the value of the neutron EDM in the different theories of CP violation has not been performed. It is our purpose in this paper to provide such a study, by integrating the various methods that have been used to calculate Dn, applying them to all of the models, and establishing the regions in which each method is valid. The most commonly used technique is the valance quark model in which the EDM of each valence quark is calculated and summed to obtain the neutron EDM. However in some models,most notably the standard model, the va'ence quark contribution is suppressed (in the standard model the quarks acquire a non-vanishing EDM only at the three loop order!8*9!). In this case both quark level exchange diagrams and hadronic level bop diagrams must also be considered, and the hadronic level loops turn out to give the dominant contribution! 10J, which is perhaps not surprising since it is known that the hadronic loops involving pions are logarithmically singular in the soft pion limit!11!. In some models, due to different origin of CP violation, quark EDM occurs at the one loop level and the calculations for the neutron EDM are different from the standard model. In this paper we consider all possible contributions to Dn in a systematic way, for the standard Kobayashi-Maskawa model, the standard model with two Higgs doublets, the Weinberg Higgs model , the left-right symmetric model and the supersymmetric standard model. We also for completeness briefly discuss the neutron EDM from the strong CP violation characterised by the 9 parameter. • 5 In section 2 we discuss possible contributions to the neutron EDM in a general, model independent way and set out the formalism for later use. These techniques are then applied in sections 3 to 8 to the Standard K-M model, the two Higgs doublet model, the Weinberg model, the left-right symmetric model and the supersymmetric standard model respectively. For completeness we a'so briefly discuss the neutron EDM in the strong CP violation model in section 8. Section 9 is devoted to a discussion of our results and in it we summarize the conclusions. 2) METHODS FOR CALCULATING ELECTRIC DIPOLE MOMENTS There are different model dependent contributions to the neutron EDM which we describe in this section. We also collect some general results in a form which can be readily particularized to specific models in the ensuing sections. 2.1) The Valence Quark Contribution This is conceptually the simplest contribution to the neutron EDM, in that one simply computes the quark EDM and sums to obtain Dn. If the EDM of the u and d quarks are Du and D^, the neutron EDM is then given by ( D ^= j(4Dd - Du) (2.1.1) The calculation of the individual quark moments is of course model dependent and is deferred to the subsequent sections. 6 2.2) Quark CDM contributions to the EDMl1?) A P and T violating coupling cf the gluon to a quark gives rise to a colour dipole moment (CDM) of the quark. We write the coupling in the form Glff (221) Hg = fls^q v%r" " • a where gs is the stroi.g QCD coupling constant, X are the SU(3) v Gell-Mann matrices, G^ a is the gluon field strength and fq is the quark CDM. This P and T violating interaction induces a change in the neutron wave function which results in a neutron EDM. To estimate this contribution to the neutron EDM we use the non- relativistic quark, model and treat Hg perturbatively. Non- relativistically, Hg becomes X Hg - -gsIfqok(-VG2) I. (2.2.2) a where G« is the time component of the gluon field potential, and the summation is over the spin. From equation (2.2.2) or equation (2.2.3) the P and T violating nature of Hg is immediately apparent. Now we introduce the unperturbed QCD Hamiltonian, Ho, in this approximation : D X a k k n (2.2.3) k 2m 2 ) and we may write Hg = i£ fg O (2.2.4) k Pk' "<> k To the lowest order in perturoation theory, the modified neutron wave function is |mxm| Hg |n> |n> = |n> + X m En ' Em (2.2.5) (1 + i£ fQ ak pk )|n> (c) Then the neutron EDM generated by the COM, D^ is given by (c) Vk - <ft|lQi'.||fl> i <n| XQiri. Z fq °k Pk |n> (2.2.6) i k = -<n| £ Qkfq Ok ln> k where Q\ is quark charge operator. We then immediately see that • 8 of = 5Te£fd+f<u> (2-2.7) 2.3) Hadronic loop contributions to the neutron EDM. it was shown by Barton and Whitel,3l that mesonic loops at the hadron level can lead to contributions to the EDM proportional to In(mM) which are thus singular in the soft meson limit. Two classes of diagrams, illustrated in Figures 2.1 and 2.2, may contribute. In figure 2.1 CP violation occurs at first order in the weak interactions, and in the diagrams of Figure 2.2 CP violation occurs only at second order in the weak interaction. Writing the strong interaction effective lagrangian as U - -V2gBrY5B'M + H.C, (2.3.1) and weak BB'M vertices as l_w - V2fe-'*B~B'M + H.C, (2.3.2) we evaluate these loop diagrams, obtaining tf - 0«» + D<'» ,,3.3, where 9 D'n.. 1 - "fV mBsin<-*)GB(mM> i2.3.4) An m _ and (I) eqf kg 2 (2.3.5) where mo, mn and mM are the masses of the baryon B, neutron n and meson M respectively. We have split D^ into two parts, the first of which arises from the sum of the graphs with the photon coupled to the meson and to the baryon through Dirac y coupling, and the second part arises from the graphs with the photon coupled to the baryon through Pauli CLV coupling. The anomalous magnetic moment coupling is written as K,, and we have assumed for simplicity that the Dirac and Pauli form factors are constant off the Baryon mass-shelU10!. The following functions appear on performing the integrals: 1 fl+v>l s"v i GB(mM) (2.3.6) 2 2 2 ?/ m v/ mM/mn, m )/ m , (2.3.7) ( B n -^•"•^^•^•^W^W^*""*^* 10 ( 1 , , y*- T f(s.v) tan i *- S+V ~2~ (2.3.8) and 3 3s-v-(s-v)2 1+v FB(m2) - f - (s-v) + ^ -W-p (s-v)[5s-v-(s-v)2]-4s f(s,v).
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