P, T Violating Moments

Introduction

1.2 Fundamental Symmetry Violation .1 Background.1Background

1.2.3 P, T Violating Moments

Since we know Nature can violate symmetries, and we have cosmological motivations for symmetry violation, we have reason to expect electromagnetism and/or the strong force have their discrete symmetries broken on a a subatomic scale. Generically, both forces can admit interactions and terms that violate 𝑃and 𝑇 symmetries, but so far no such interactions have been observed in Nature. In this section we provide a discussion of how such symmetry violation manifests in electromagnetic interactions.

𝑃 and 𝑇-violation can can result in symmetry violating electromagnetic mo- ments of fundamental particles. These moments can be derived generally, either from a standard multipole expansion of charge and current, or from a decomposition

8If fewer than 3 generations are invovled, the CKM phase is trivial can be transformed away by a unitary operation.

of the electromagnetic current operator, 𝑗_𝜇, into Lorentz invariant form factors. The latter approach is detailed in Refs. [35–38], and we outline it here schematically.

Consider the matrix element of 𝑗_𝜇connecting generic initial and final particle states with a given spin 𝑆 and momenta 𝑘 , 𝑘^′. The matrix element ⟨𝑘^′, 𝑆|𝑗_𝜇|𝑘 , 𝑆⟩ can be factored into its Lorentz invariant constituents, known as form factors, labeled as 𝐹

1(𝑞²), 𝐹

2(𝑞²), . . ., and parameterized in terms of the 4-momentum transfer 𝑞2= (𝑘^′−𝑘)². There are generically 6𝑆+1 form factors for a given spin𝑆.

For now, we consider the case of a fundamental spin-¹₂ particle, such as the electron, giving us four separate terms. In the non-relativistic rest frame, 𝑞2 → 0, and the form factors can be identified9with various properties of our spin-¹₂particle, some more familiar than others:

𝐹1(0) =𝑄 (charge) (1.1)

1 2𝑚

(𝐹

1(0) +𝐹

2(0)) =𝜇 (magnetic dipole moment) (1.2)

− 1 2𝑚

𝐹3(0) =𝑑 (electric dipole moment) (1.3) 1

𝑚2

𝐹4(0) =𝑎 (anapole moment). (1.4) The first two quantities, the charge𝑄and magnetic moment𝜇, are familiar properties of all subatomic particles, including electrons. The electric dipole moment has a clear classical analogue, and the anapole moment describes a “torodial” magnetic moment. We note all of these quantities are intrinsic. Of these quantities, only𝑄 is invariant under rotations. While the magnitude of the moments (including the anapole) are fixed, they are vector observables that must be oriented along the same axis as the spin: 𝜇® =𝜇𝑆®, 𝑑®=𝑑𝑆®, and𝑎®=𝑎𝑆®. We now give some reasons for such a constraint.

First, recall our spin-¹₂ particle transforms under rotations according to the Wigner D-matrices, D^(𝑆=12)

, and this was obtained by demanding our particle properties remain invariant under changes to our reference frame. Imagine we were to add another, second physical axis to describe our particle, describing 2𝑆^′+1 hypothetical orientations of some dipole moment 𝑑®. To maintain invari- ance under rotations, we demand this second axis also transform according to the D-matrices10, and our particle’s rotation properties are now given byD⁽¹2)⊕ D^(𝑆′) =

9For composite particles such as protons or neutrons, the form factors besides𝐹

1 are generally hard to compute.

10If we did not do this, then𝑑®would pick out an absolute direction in space, which is Lorentz violating.

D^{(|𝑆′−1}2|) ⊗ D^{(|𝑆′−1}2+1|) ⊗. . .D^(𝑆′+12)

. Our simple spin-¹₂ particle has turned into a coupled angular momentum problem consisting of(2(¹

2) +1) (2𝑆^′+1)orientations, contradicting our initial classification of the particle’s rotational symmetries. If we tried to use this two-axis state to describe a wavefunction of identical electrons in the same spatial state, we would conclude that we have (2𝑆^′+1) times morecon- figurations available than what we physically observe with electrons under the Pauli exclusion principle. Furthermore, we run into a deeper issue—by the spin-statistics theorem, if𝑆^′ is half-integer, then our combined two-axis particle is now a boson, and Pauli exclusion does not apply at all! In Nature, we observe spin-¹₂ electrons that only have 2-fold internal degrees of freedom, which is only consistent with the case that𝑑®∝ ®𝑆.

For the sake of argument, let us proceed as if 𝑑®can point at an arbitrary angle relative to𝑆®. If we try to measure𝑑®, we will run into problems. Since our particle has angular momentum, any components of𝑑®perpendicular to 𝑆®will be averaged away by the spin, leaving only the projection𝑑®· ®𝑆. Since𝑆®only has𝑆_𝑧defined due to the commutation relationships of angular momenta, the transverse components of𝑆® and𝑑®vanish, and we can only measure 𝑑_𝑧 ∝ 𝑆_𝑧. This argument can be generalized to angular momenta larger than𝑆 = ¹

2 via the Wigner-Eckart theorem, presented in Sec. 2.1.2. In the language of spherical tensor operators [39], all electromagnetic moments of a given rank are therefore proportional to the angular momentum tensor of the same rank, which classifies the rotational symmetries of our particle.

Now we move on to considering the symmetry behaviors of these moments interacting with external fields. One approach is to consider the coupling of 𝑗_𝜇 to the photon field in the QED Lagrangian [35]. Instead, we pursue the low energy equivalent, considering the Hamiltonian derived from the non-relativistic limit of the Lagrangian [19, 38]. If the Hamiltonian is left changed by our symmetry operation, then the symmetry is broken. First, we consider the charge𝑄. While𝑄

−→ −𝐶 𝑄, the Couloumb interaction scales as𝐻_𝑄 ∝𝑄 𝜙, where𝜙is the charge-dependent electric potential, and therefore 𝐻_𝑄

−→𝐶 𝐻_𝑄. Similarly, for the magnetic moment, we have 𝜇

−→ −𝐶 𝜇

−𝑇

→ 𝜇, recalling that 𝜇® ∝ 𝑄𝑆®. But also, the interaction Hamiltonian is 𝐻_𝜇 =− ®𝜇· ®𝐵, and magnetic fields are generated by currents which are𝐶- and𝑇-odd (see Table 1.1), so we have 𝐻_𝜇

−𝑇

→ 𝐻_𝜇, and a similar argument shows 𝐻_𝜇 is also 𝐶-symmetric.

The anapole moment 𝑎 has a non-relativistic interaction Hamiltonian given by [37, 38] 𝐻_𝑎 ∝ 𝑎𝑆®· (∇ × ®𝐵− ^{𝜕 𝐸}

𝜕 𝑡). Note the anapole only interacts with elec-

tromagnetic sources or sinks, which means it is only nonzero in matter. Since ∇ is 𝑃-odd and 𝜕 𝑡 is𝑇-odd, we see the term in the parentheses is𝐶-odd, 𝑃-odd and 𝑇-odd. The spin is𝑇-odd as well, so the resulting anapole Hamiltonian is𝐶-odd, 𝑃-odd, and𝑇-even, and satisfies𝐶 𝑃𝑇 invariance as expected. The𝐶-odd nature of the anapole means it cannot have long-distance effects [35]. Finally, we consider the electric dipole moment, given by𝐻_𝑑 =−𝑑𝑆®· ®𝐸. The spin𝑆is𝑇-odd, while the electric field, generated by charge distributions, is 𝑃-odd and𝐶-odd. By intuition (and by𝐶 𝑃𝑇)𝑑is𝐶-odd, and we therefore have that𝐻_𝑑is𝑃-odd,𝑇-odd,𝐶-even.

We emphasize that in the above discussion,𝑑refers to a permanent moment, and is not taken to be𝑃-odd, though often it is presented as such when hand-waving. The permanent dipole moment 𝑑®∝ ®𝑆 of a point particle is not the same as a composite dipole moment𝐷® ∝ ®𝑟that we will encounter in atoms and molecules, distinguished by capital𝐷. Since𝐷® is explicitly𝑃-odd and𝑇-even, its interaction with the𝑃-odd 𝐸field is𝑃-even,𝑇-even. Further, one can show that⟨𝐷⟩must vanish for states with well-defined parity [40, 41], and therefore there are no permanent dipole moments of atoms and molecules at zero field, which are good parity eigenstates. For an atom this makes sense given its spherical symmetry. For a simple diatomic molecule, the spherical symmetry is reduced to cylindrical symmetry, with 𝐷® pointing along the symmetry axis. However, the molecule eigenstate still has well-defined parity.

We can think of the molecule as constantly rotating, causing ⟨ ®𝐷⟩ = 0 in the lab frame. As we shall see later, only by applying an external field and breaking the 𝑃 symmetry of the molecule do we begin to an induce a dipole moment in the system.

The total dipole moment can then be decomposed into 𝑇-even contributions and 𝑇-odd contributions, and we search for the latter.

We finally return to the case of an arbitrary spin-𝑆 particle, with 6𝑆 +1 total invariant form factors. In Ref. [35], these form factors are tabulated according to their total interaction symmetry under𝐶,𝑃, and𝑇, and we reproduce their results in Table 1.2. For this thesis, we will only focus on the𝑃, 𝑇-odd moments. The𝐶-odd moments are discussed further in Refs. [42–44].

If we have a particle with spin 𝑆 ≥ 1, we can now support additional 𝑃, 𝑇- violating moments, which correspond to higher order multipoles. Of particular interest is the magnetic quadrupole moment (MQM)M, which is𝑃, 𝑇-odd and has never been observed, in contrast to the electric quadrupole moment (EQM)Q, which is𝑃, 𝑇-even, and a commonly observed property of nuclei. Similar to how𝑑 and𝜇 have analogous mathematical forms, so too can we draw comparisons between M

Table 1.2: Table denoting the𝐶 , 𝑃, 𝑇symmetry properties of the 6𝑆+1 electromag- netic moments obtained from the form factor decomposition of the electromagnetic current. When two values are given separated by the semi-colon, the left value refers to half-integer𝑆; the right refers to integer𝑆. Table reproduced from Ref. [35].

𝐶 𝑃 𝑇 Number of moments + + + 2𝑆+1

− − + 𝑆+ ¹

2;𝑆

− + − 𝑆− ¹

2;𝑆 + − − 2𝑆

andQ.

The EQM and MQM are both rank 2 moments, which means they are described by two spatial indices, and referred to as tensors. To obtain a rotationally invariant interaction Hamiltonian, these moments must be contracted with tensor quantities that also have two indices. The EQM naturally interacts with electric field gradients, 𝐻_Q ∝ Q𝑖 𝑗∇𝑖𝐸_𝑗, while the MQM interacts with magnetic field gradients, 𝐻_M ∝ M𝑖 𝑗∇𝑖𝐵_𝑗. As with the EDM, the MQM must point along the spin 𝑆, but since it is a tensor quantity, we must construct a rank 2 representation of𝑆. The common choice is the irreducible, traceless tensor, given by:

M𝑖 𝑗 =M 3

2𝑆(2𝑆−1)𝑇_{𝑖 𝑗}. (1.5) Here, we have defined the tensor 𝑇_{𝑖 𝑗} = {𝑆_𝑖, 𝑆_𝑗} − ²

3𝛿_{𝑖 𝑗}𝑆(𝑆 +1), where {𝐴, 𝐵} = 𝐴 𝐵+𝐵 𝐴is the anti-commutator, and𝛿_{𝑖 𝑗} is the Kronecker delta, with𝛿_{𝑖 𝑗} =1 if𝑖= 𝑗 and 0 otherwise. The 2𝑆− 1 factor in the denominator means the MQM is only well-defined for 𝑆 ≥ 1. The quantity M B M𝑧 𝑧 is the “magnitude” of the MQM, which can be seen by evaluating eq. 1.5 for a polarized spin, 𝑆_𝑧 = 𝑆. The M𝑖 𝑗

form is given in Cartesian coordinates. In the language of spherical tensor operators (discussed in Sec. 2.1.2 and Ch. 5 of Ref. [39]), we can write the MQM as:

𝑇²

𝑝(M) =

√ 6 𝑆(2𝑆−1)𝑇²

𝑝(𝑆, 𝑆) (1.6)

where𝑝is the index labeling the 5 components in the spherical basis, running from 𝑝 =−2,−1,0,1,2. Returning to the form of𝐻_M, we have an odd number of𝑇-odd quantities (2 spin components in M, and one from the magnetic field 𝐵), and the dependence on the gradient results in 𝑃-odd behavior as well, and therefore the MQM interaction is𝑃, 𝑇-odd,𝐶-even.

Classically, we can think of an MQM as two opposite current loops, with magnetic moments±𝑗 𝑎, separated by a distance𝑟, where 𝑗 is electric current and 𝑎 is area. Using atomic units, the MQM is given by M ∝ 𝜇_𝐵𝑎

0, or alternatively with nuclear units, 𝜇_𝑁fm. Clearly, the MQM is the mangnetic dipole analogue of the EDM, with units of dipole×distance instead of charge×distance. The SI units of the MQM are current×volume, or A m³. We note some papers write the MQM by factoring out the speed of light and setting𝑐 =1, resulting in the same units for MQMs and EQMs, charge×area.

Finally, for completeness, we provide the classical formula for the MQM. The classical form of the MQM multipole is given by [45, 46]:

←→ M = 1

2

∫ d𝑟®

®

𝑟( ®𝑟× ®𝐽) + ( ®𝐽× ®𝑟) ®𝑟

(1.7) where the notation

←→

M indicates we are dealing with a tensor quantity. Like the previous forms we provided of the MQM, this form is also traceless and symmetric.

Further, classical vector potential generated by

←→

M is given by:

𝐴®( ®𝑟) = 𝜇

0

4𝜋 ˆ 𝑟 ×←→

M ×𝑟ˆ

𝑟3 (1.8)

and the magnetic field can be obtained by the usual relation, 𝐵® = ∇ × ®𝐴. A generalization to higher order moments can be found in Jackson [47].

So far, our discussion has focused on permanent 𝑃, 𝑇 violating moments of fundamental particles. Early on, physicists realized that these permanent moments can also manifest in composite systems. The first example was the work of Ramsey and Purcell in 1957 [48], where they placed the first limits on the EDM of the neutron (𝑑_𝑛 < 10⁻²⁰ 𝑒 cm). We might naively expect a nonzero neutron EDM, given the neutron is made up of oppositely charged quarks. However, no EDM has been found so far, with current experimental bounds limiting the neutron EDM to 𝑑_𝑛 < 1.8×10⁻²⁶𝑒cm, and even more sensitive experiments currently underway [49].

The neutron EDM is explicitly a probe of𝑃, 𝑇 violation in the strong force, which manifests in the quantum chromodynamics (QCD) vacuum angle, parameterized by the𝜃 parameter. The experimental neutron EDM bound translates to a limit of 𝜃 ≲ 10⁻¹⁰ [7], though the exact value can vary in the literature as the calculation is challenging. The question of the small or zero value of𝜃 constitutes the strong- CP problem, a rich field of physics that has given rise to the theory of axion-like particles, a potential dark matter candidate. We do not discuss this further, but direct the reader to the excellent review in Ref. [50] for more information.

1.3 Atoms and Molecules