1 Introduction

The Neutron Electric Dipole Moment and Probe of PeV Scale Physics

Amin Aboubrahim, Tarek Ibrahim***Email: tibrahim@zewailcity.edu.eg and Pran NathEmal: nath@neu.edu

University of Science and Technology, Zewail City of Science and Technology,
6th of October City, Giza 12588, Egypt333Permanent address: Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt
Department of Physics, Faculty of Science, Beirut Arab University, Beirut 11-5020, Lebanon444Email: amin.b@bau.edu.lb
Department of Physics, Northeastern University, Boston, MA 02115-5000, USA

Abstract

The experimental limit on the neutron electric dipole moment is used as a possible probe of new physics beyond the standard model. Within MSSM we use the current experimental limit on the neutron EDM and possible future improvement as a probe of high scale SUSY. Quantitative analyses show that scalar masses as large as a PeV and larger could be probed in improved experiment far above the scales accessible at future colliders. We also discuss the neutron EDM as a probe of new physics models beyond MSSM. Specifically we consider an MSSM extension with a particle content including a vectorlike multiplet. Such an extension brings in new sources of charge conjugation and parity (CP) violation beyond those in MSSM. These CP phases contribute to the EDM of the quarks and to the neutron EDM. These contributions are analyzed in this work where we include the supersymmetric loop diagrams involving the neutralinos, charginos, the gluino, squark and mirror squark exchange diagrams at the one loop level. We also take into account the contributions from the , , quark and mirror quark exchanges arising from the mixings of the vectorlike generation with the three generations. It is shown that the experimental limit on the neutron EDM can be used to probe such new physics models. In the future one expects the neutron EDM to improve an order of magnitude or more allowing one to extend the probe of high scale SUSY and of new physics models. For the MSSM the probe of high scales could go up to and beyond PeV scale masses.

Keywords:  Electric Dipole Moment, Neutron, vector multiplets
PACS numbers: 13.40Em, 12.60.-i, 14.60.Fg

1 Introduction

CP violation provides a window to new physics [For the early history of CP violation and for reviews see e.g.,[1, 2, 3, 4]]. One of the important manifestations of CP violation are that such violations generate electric dipole moment (EDM) for elementary particle, i.e., for the quarks and leptons. As is well known the EDM of elementary particles in the standard model are very small. For example, for the electron the EDM is estimated to be cm. The electroweak sector of the Standard Model gives an EDM for the neutron of size cm. These sizes are too small to be observed in any foreseeable experiment. The QCD sector of the standard model also produces a non-vanishing EDM for the neutron which is of size cm and satisfaction of Eq. (1) requires to be of size or smaller where is QCD phase which enters the QCD Lagrangian as . We assume the absence of such a term by a symmetry such as the Peccei-Quinn symmetry. In supersymmetric models there are a variety of new sources of CP violation and typically these new sources of CP violation lead to EDM of the elementary particles in excess of the observed limits. This phenomenon is often referred to as the SUSY EDM problem. Several solutions to this problem have been suggested in the past such as small CP phases  [5], mass suppression  [6] and the cancellation mechanism [7, 8] where various diagrams contributing to the EDMs cancel to bring the predicted EDM below the experimental value (for an alternate possibility see [9]). The recent data from the LHC indicates the Higgs mass to be GeV which requires a large loop correction to lift the tree level mass to the desired experimental value. The sizable loop correction points to a high SUSY scale and specifically large scalar masses. In view of this one could turn the indication of a large SUSY scale as a possible resolution of the EDM problem of supersymmetric models. In fact it has been suggested recently [10, 11, 12, 13, 14], that one can go further and utilize the current and future improved data on the EDM limits to probe mass scales far beyond those that may be accessible at colliders. We also note in passing that a large SUSY scale also helps suppress flavor changing neutral currents (FCNC) in supersymmetric models and helps stabilize the proton against rapid decay from baryon and lepton number violating dimension five operators in grand unified theories.

In this work we will focus on the neutron electric dipole moment of the light quarks which in turn generate an EDM of the neutron as a probe of high scale physics. The current experimental limit on the EDM of the neutron is [15]

(1)

Higher sensitivity is expected from experiments in the future [16]. In our analysis here we consider the neutron EDM as a probe of high scalar masses within the minimal supersymmetric standard model (MSSM) as well as consider the neutron EDM as a probe of an extension of MSSM with a vectorlike generation which brings in new sources of CP violation. A vectorlike generation is anomaly free. Further, a variety of grand unified models, string and D brane models contain vectorlike generations [17]. Vectorlike generations have been considered by several authors since their discovery would constitute new physics (see, e.g., [18, 19, 20, 21, 22, 23, 24, 25, 26, 13, 27, 28, 29, 30]).

Quark dipole moment have been examined in MSSM in previous works and a complete analysis at the one loop level is given in [7]. Here we compute the EDM of the neutron within an extended MSSM where the particle content contains in addition a vectorlike multiplet. The outline of the rest of the paper is as follows: In section 2 we give the relevant formulae for the extension of MSSM with a vectorlike generation. In section 3 we give the interactions of W and Z vector bosons with the quarks and mirror quarks of the extended model. Interactions of the gluino with quarks, squarks, mirror quarks and mirror squarks are given in section 4. Interactions of the charginos and neutralinos with quarks, squarks, mirror quarks and mirror squarks are given in section 5. An analysis of the electric dipole moment operator involving loop contributions from W and Z exchange, gluino exchange, and chargino and neutralino exchanges is given in section 6. A numerical estimate of the EDM of the neutron arising from these loop contributions is given in section 7. Here we also discuss the neutron EDM as a probe of PeV scale physics. Conclusions are given in section 8. In section 9 we give details of how the scalar mass square matrices are constructed in the extended model with a vectorlike generation.

2 Extension of MSSM with a Vector Multiplet

In this section we give details of the extension of MSSM to include a vectorlike generation. A vectorlike multiplet consists of an ordinary fourth generation of leptons, quarks and their mirrors. A vectorlike generation is anomaly free and thus its inclusion respects the good properties of a gauge theory. Vectorlike multiplets arise in a variety of unified models some of which could be low-lying. They have been used recently in a variety of analyses. In the analysis below we will assume an extended MSSM with just one vector multiplet. Before proceeding further we define the notation and give a very brief description of the extended model and a more detailed description can be found in the previous works mentioned above. Thus the extended MSSM contains a vectorlike multiplet. To fix notation the three generations of quarks are denoted by

(2)

where the properties under are also exhibited. The last entry in the braces such as is the value of the hypercharge defined so that . These leptons have interactions. We can now add a vectorlike multiplet where we have a fourth family of leptons with interactions whose transformations can be gotten from Eq. (2) by letting run from 1 to 4. A vectorlike quark multiplet also has mirrors and so we consider these mirror quarks which have interactions. The quantum numbers of the mirrors are given by

(3)

Interesting new physics arises when we allow mixings of the vectorlike generation with the three ordinary generations. Here we focus on the mixing of the mirrors in the vectorlike generation with the three generations. Thus the superpotential of the model allowing for the mixings among the three ordinary generations and the vectorlike generation is given by

(4)

where is the complex Higgs mixing parameter so that . The mass terms for the ups, mirror ups, downs and mirror downs arise from the term

(5)

where and stand for generic two-component fermion and scalar fields. After spontaneous breaking of the electroweak symmetry, ( and ), we have the following set of mass terms written in the four-component spinor notation so that

(6)

where the basis vectors in which the mass matrix is written is given by

(7)

and the mass matrix is given by

(8)

We define the matrix element of the mass matrix as so that,

(9)

The mass matrix is not hermitian and thus one needs bi-unitary transformations to diagonalize it. We define the bi-unitary transformation so that

(10)

Under the bi-unitary transformations the basis vectors transform so that

(11)

A similar analysis goes to the down mass matrix where

(12)

In general can be complex and we define their phases so that

(13)

We introduce now the mass parameter which defines the mass term in the (22) element of the mass matrix of Eq. (12) so that

(14)

Next we consider the mixing of the down squarks and the charged mirror sdowns. The mass squared matrix of the sdown - mirror sdown comes from three sources: the F term, the D term of the potential and the soft SUSY breaking terms. Using the superpotential of the mass terms arising from it after the breaking of the electroweak symmetry are given by the Lagrangian

(15)

where is deduced from , and while the is given by

(16)

For we assume the following form

(17)

Here , etc are the soft masses and , etc are the trilinear couplings. The trilinear couplings are complex and we define their phases so that

(18)

From these terms we construct the scalar mass squared matrices.

3 Interaction with W and Z vector bosons

Figure 1: and exchange contributions to the EDM of the up quark. Similar exchange contributions exist for the EDM of the down quark where and are interchanged and is replaced by in the diagrams above.
(19)

where

(20)
(21)

For the Z boson exchange the interactions that enter with the up type quarks are given by

(22)

where

(23)

and

(24)

where

(25)
(26)

For the Z boson exchange the interactions that enter with the down type quarks are given by

(27)

where

(28)

and

(29)

where

(30)
(31)

4 Interaction with gluinos

(32)

where

(33)

and

(34)

where is the phase of the gluino mass.

5 Interactions with charginos and neutralinos

Figure 2: Supersymmetric loop contributions to the EDM of the up-quark. Left panel: Loop diagram involving the neutralinos and up-squarks. Middle left panel: Gluino and up-squark loop contribution. Middle right panel: Loop contribution with chargino and d-squark exchange with the photon emission from the d-squark line. Right panel: Loop contribution with chargino and d-squark exchange with the photon emission from the chargino line. Similar loop contributions exist for the EDM of the down quark, where and are interchanged, and are interchanged and is replaced by in the diagrams above.

In this section we discuss the interactions in the mass diagonal basis involving squarks, charginos and quarks. Thus we have

(35)

such that,

(36)
(37)

and

(38)

such that,

(39)
(40)

with

(41)
(42)

and

(43)

We now discuss the interactions in the mass diagonal basis involving up quarks, up squarks and neutralinos. Thus we have,

(44)

such that

(45)
(46)

where

(47)
(48)

and

(49)
(50)

and where