научная статья по теме STRONG VIOLATION AND THE NEUTRON ELECTRIC DIPOLE FORM FACTOR Физика

Текст научной статьи на тему «STRONG VIOLATION AND THE NEUTRON ELECTRIC DIPOLE FORM FACTOR»

ЯДЕРНАЯ ФИЗИКА, 2007, том 70, № 2, с. 377-385

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

STRONG CP VIOLATION AND THE NEUTRON ELECTRIC DIPOLE FORM FACTOR

© 2007 J. Kuckei1)*, C. Dib2)**, A. Faessler1) S. G. Kovalenko2)*****, V. E. Lyubovitskij1)******,

Received January 30, 2006

***, T. Gutsche1) K. Pumsa-ard1)

■k-k-k-k-k-k-k

We calculate the neutron electric dipole form factor induced by the CP-violating 0 term of QCD, within a perturbative chiral quark model which includes pion and kaon clouds. On this basis, we derive the neutron electric dipole moment and the electron—neutron Schiff moment. From the existing experimental upper limits on the neutron electric dipole moment we extract constraints on the 0 parameter and compare our results with other approaches.

PACS numbers: 12.39.Ki, 12.39.Fe, 11.30.Er, 11.30.Rd, 13.40.Em, 14.20.Dh

1. INTRODUCTION

The understanding of CP violation is one of the long-standing and challenging problems of particle physics. So far, the effects of CP violation have only been observed in the K and B hadron systems [1] and are in good agreement with the predictions of the Standard Model (SM) of electroweak interactions. Among other CP-odd observables, a great deal of effort has been directed to the study of electric dipole moments (EDMs) of leptons, neutrons, and neutral atoms. Both SM and various non-SM sources of CP violation have been considered (for recent reviews, see, e.g., [2—4]). These studies have been particularly stimulated by the expectation of great improvements (2 to 4 orders of magnitude) in the experimental sensitivities to EDMs in the next decade (for review, see [3]).

As it is well known, there are two sources of CP violation within the SM: the complex phase 5qkM of the Cabibbo—Kobayashi—Maskawa (CKM) quark mixing matrix in the weak interaction sector, and the 9 term in the strong interaction sector, which arises due to the nontrivial structure of the QCD vacuum [5—8]. The complex phase of the CKM matrix

''Institutfür Theoretische Physik, Universitat Tübingen, Germany.

2)Departamento de Fisica, Universidad Tecnica Federico Santa Maria, Valparaiso, Chile.

E-mail: kuckei@tphys.physik.uni-tuebingen.de

E-mail: cdib@fis.utfsm.cl

E-mail: amand.faessler@uni-tuebingen.de

E-mail: thomas.gutsche@uni-tuebingen. de

E-mail: Sergey.Kovalenko@usm.cl

E-mail: valeri.lyubovitskij@uni-tuebingen. de

E-mail: pumsa@tphys.physik.uni-tuebingen.de

provides a consistent explanation of the observed CP-odd effects in hadron decays, while it gives an imperceptible contribution to the EDMs, far below the sensitivity of present or foreseeable experiments. Indeed, the CKM prediction for the neutron EDM dn ranges from 10_31 to 10_33 e cm [9], while the present experimental upper limit [10] is

\dn\ < 0.63 x 10"25e cm.

(1)

On the other hand, CP violation induced by a 9 term leads to a sizable EDM for the neutron [11, 12] and may significantly contribute to atomic EDMs, while it is insignificant for CP violation in hadron decays. The non-observation of the neutron EDM and the atomic EDMs imposes a very strict upper bound on the value of 9, of the order of 10_10. This unnaturally small value of 9, which is otherwise not restricted by theory, is known as the strong CP problem. One elegant solution was proposed by Peccei and Quinn [13], which makes the 9 parameter to vanish dynamically. However, this mechanism also requires the appearance of a Goldstone boson, the axion, which remains to be discovered.

Other important contributions to the atomic and neutron EDMs may arise from possible physics beyond the SM. In particular, supersymmetric extensions of the SM offer additional mechanisms for CP violation [14—19] originating from complex phases in the soft SUSY-breaking terms and superpotential parameters (for a review, see [18]).

In the calculations of atomic and neutron EDMs one faces the problem of translating the effect of the CP violation introduced at the quark—gluon level to the processes at the hadronic or atomic level. This translation must resort to hadronic and

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nuclear models as well as to a careful treatment of the atomic electron wave functions in the case of atomic EDMs. Therefore, confidence of the estimates of the hadronic or atomic CP-violating observables depends on reliability of these models.

The problem of the neutron EDM has been studied within various theoretical approaches: current algebra and chiral perturbation theory (ChPT) [12, 20, 21], chiral quark models [22-24], lattice QCD [25, 26], QCD sum rules [27, 28], an approach based on solutions of Schwinger-Dyson and Bethe-Salpeter equations [29], etc.

In the present paper we apply to this problem the perturbative chiral quark model (PCQM) [30], that is a development of chiral quark models with a perturbative treatment of the pion cloud of nucleon [31 -33]. As shown in [30], the PCQM is successful in description of low-energy properties of light baryons such as the mass spectrum, the electromagnetic, axial, and strong form factors, including the quantities which receive a nontrivial contribution from the cloud of pseudoscalar mesons: meson-baryon sigma terms, strangeness content of the nucleon, etc. Compared to the models of [31-33], the PCQM contains several new features3): (i) the SU(3) extension of chiral symmetry in order to include the kaon- and eta-meson-cloud contributions; (ii) consistent formulation of perturbation theory both at the quark and baryon levels on the basis of the renormalization techniques and taking into account excited quark states in the meson-loop diagrams; (iii) incorporated constraints from the chiral symmetry (low-energy theorems); (iv) consistency with chiral perturbation theory for the case of the chiral expansion of the nucleon mass.

The purpose of the present work is to calculate within the PCQM not only the neutron EDM but also the neutron EDM form factor (EDFF) induced by the strong CP-violating 9 term. The neutron EDFF as function of the momentum transfer could be the next step in the experimental studies of the CP-odd structure of the neutron, after the measurement of its EDM. Also, it is known [34] that the atomic EDMs are sensitive to the nuclear Schiff moment, which depends on the neutron EDM square radius derived from the neutron EDFF.

This paper is organized as follows. In Section 2 we give a brief introduction to the PCQM. Section 3 deals with the calculation of the EDFF, the neutron EDM, and the electron-neutron Schiff moment induced by strong CP violation. Here, we extract the constraints for the CP-violating 9 parameter from the existing experimental data on the neutron EDM and compare our results to some other approaches.

Finally, we give a summary of our results and conclusions.

2. THE PERTURBATIVE CHIRAL QUARK MODEL

The basis of the PCQM [30] is an effective chiral Lagrangian describing the valence quarks of baryons as relativistic fermions moving in an external field (static potential) Veff (r) = S(r) + V(r) with r = |x|, which in the SU(3)-flavor version are supplemented by a cloud of Goldstone bosons (n,K,n). Treating Goldstone fields as small fluctuations around the three-quark core, the linearized effective Lagrangian is written as

Leff(x) = q(x)[i d - S(r) - y°V(r)]q(x)+ (2) 1 8

+ 1 E^(x)]2 + L?W (x) + LxSB(x).

i=1

Here we defined

F

Lf(1) (x) = -q(x)iY5^-L S (r)q(x). (3)

The additional term Lxsb in Eq. (2) contains the mass contributions both for quarks and mesons, which explicitly break chiral symmetry:

b

LxSB(x) = -q(x)Mq(x) - -tr[$2(x)M]. (4)

Here, $ = ^28=1 $i\i is the octet matrix of pseudoscalar mesons, F = 88 MeV is the pion decay constant in the chiral limit, M = diag{m, m, ms} is the mass matrix of current quarks (we restrict to the isospin symmetry limit mu = md = m) and B = = — (0\uu\0)/F2 is the quark condensate constant. We rely on the standard picture of chiral symmetry breaking, and for the masses of pseudoscalar mesons we use the leading term in their chiral expansion (i.e., linear in the current quark mass):

M2 = 2m B, MK = (m + ma)B,

(5)

2

M2 = -( m+ 2 m a )B. 3

3)For details, see [30].

In our analysis we use the following set of parameters:

m = 7 MeV, ms = 25m, (6)

B = M2+ /2m = 1.4 GeV.

The meson masses satisfy the Gell-Mann-Oakes-Renner and Gell-Mann-Okubo relations. In addition, the linearized effective Lagrangian fulfills PCAC. The properties of baryons, which are modeled as bound states of valence quarks surrounded by a meson cloud, are then derived using perturbation theory. At zeroth order, the unperturbed Lagrangian

simply describes a nucleon as three relativistic valence quarks which are confined by an effective one-body static potential Veff (r) in the Dirac equation. We denote the unperturbed three-quark ground state as |0O), with the normalization = 1. We expand

the quark field q in the basis of eigenstates generated by this potential as

q(x) baUa(x)exp(-iEat),

(7)

the ground state (for the excited quark states we proceed by analogy):

u0 (x) = N exp

x

2R2

1

ipax/R,

XsXfX* (9)

where the quark wave functions {ua} in orbits a are the solutions of the Dirac equation including the potential Veff(r). The expansion coefficients ba are the corresponding single-quark annihilation operators. All calculations are performed at an order of accuracy o(1/F2,m,ms). In the calculation of matrix elements, we project the quark diagrams on the respective baryon states. The baryon states are conventionally set up by the product of SU(6) spin-flavor and SU(3)c color wave functions, where the nonrelativistic single-quark spin wave function is replaced by the relativistic solution ua(x) of the Dirac equation.

In our description of baryons we use an effective potential Veff(r) = S(r) + j0V(r) which is given by a sum of a scalar potential S(r) providing confinement and the time component of a vector potential j0V(r). Obviously, other possible Lorenz structures (e.g., pseudoscalar or axial) are excluded by symm

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