A few thoughts on $\theta$ and the electric dipole moments

I highlight a few thoughts on the contribution to the dipole moments from the so-called $\theta$ parameter. The dipole moments are known can be generated by $\theta$. In fact, the renowned strong $\cal{CP}$ problem was formulated as a result of non-observation of the dipole moments. What is less known is that there is another parameter of the theory, the $\theta_{QED}$ which becomes also a physical and observable parameter of the system when some conditions are met. This claim should be contrasted with conventional (and very naive) viewpoint that the $\theta_{\rm QED}$ is unphysical and unobservable. A specific manifestation of this phenomenon is the so-called Witten effect when the magnetic monopole becomes the dyon with induced electric charge $e'=-e \frac{\theta_{QED}}{2\pi}$. We argued that the similar arguments suggest that the electric magnetic dipole moment $\mu$ of any microscopical configuration in the background of $\theta_{QED}$ generates the electric dipole moment $\langle d_{\rm ind} \rangle $ proportional to $\theta_{QED}$, i.e. $\langle d_{\rm ind}\rangle= - \frac{\theta_{\rm QED} \cdot \alpha}{\pi} \mu$. We also argue that many $\cal{CP}$ correlations such as $ \langle \vec{B}_{\rm ext} \cdot\vec{E}\rangle = -\frac{\alpha\theta_{\rm QED}}{\pi}\vec{B}^2_{\rm ext}$ will be generated in the background of an external magnetic field $\vec{B}_{\rm ext} $ as a result of the same physics.


I. INTRODUCTION AND MOTIVATION
The leitmotiv of the present work is related to the fundamental parameter θ in Quantum Chromodynamics (QCD), as well as the axion field related to this parameter.The θ parameter was originally introduced in the 70s.Although the θ term can be represented as a total derivative and does not change the equation of motion, it is known that this parameter is a fundamental physical parameter of the system on the non-perturbative level.It is known that the θ = 0 introduces P and CP violation in QCD, which is most well captured by the renowned strong CP problem.
In particular, what is the most important element for the present notes is that the θ parameter generates the neutron (and proton) dipole moment which is known to be very small, d n 10 −26 e • cm, see e.g.review in Physics Today [1].It can be translated to the upper limit for θ 10 −10 .The strong CP problem is formulated as follows: why parameter θ is so small in strongly coupled gauge theory?The proton electric dipole moment d p , similar to the neutron dipole moment d n will be also generated as a result of non-vanishing θ.In particular, a future measurement of the d p on the level d p 10 −29 e • cm will be translated to much better upper limit for θ 10 −13 .
On the other hand, one may also discuss a similar theta term in QED.It is normally assumed that the θ QED parameter in the abelian Maxwell Electrodynamics is unphysical and can be always removed from the system.
The arguments are based on the observation that the θ QED term does not change the equation of motion, which is also correct for non-abelian QCD.However, in contrast with QCD when π 3 [SU (3)] = Z, the topological mapping for the abelian gauge group π 3 [U (1)] = 0 is trivial.This justifies the widely accepted view that θ QED does not modify the equation of motions (which is correct) and does not affect any physical observables and can be safely removed from the theory (which is incorrect as we argue below).We emphasize here that the claim is not that θ QED vanishes.Instead, the (naive) claim is that the physics cannot depend on θ QED irrespective to its value.While these arguments are indeed correct for a trivial vacuum background when the theory is defined on an infinitely large 3+1 dimensional Minkowski space-time, it has been known for quite sometime that the θ QED parameter is in fact a physical parameter of the system when the theory is formulated on a non-simply connected, compact manifold with non-trivial π 1 [U (1)] = Z, when the gauge cannot be uniquely fixed, see the original references [15,16] and review [17].Such a construction can be achieved, for example, by putting a system into a back-ground of the magnetic field or defining a system on a compact manifold with non-trivial topology.In what follows we treat θ QED as a new fundamental (unknown) parameter of the theory.
Roughly speaking, the phenomena, in all respects, are very similar to the Aharonov-Bohm and Aharonov Casher effects when the system is highly sensitive to pure gauge (but topologically nontrivial) configurations.
In such circumstances the system cannot be fully described by a single ground state 1 .Instead, there are multiple degenerate states which are classified by a topological index.The physics related to pure gauge configurations describing the topological sectors is highly nontrivial.In particular, the gauge cannot be fixed and defined uniquely in such systems.This is precisely a deep reason why θ QED parameter enters the physical observables in the axion Maxwell electrodynamics in full agreement with very generic arguments [15][16][17].Precisely these contributions lead to the explicit θ QED -dependent effects, which cannot be formulated in terms of conventional propagating degrees of freedom (propagating photons with two physical polarizations).
The possible physical effects from θ QED have also been discussed previously [19,20] in the spirit of the present notes.We refer to our paper [21] with explicit and detail computations of different observable effects (such as induced dipole moment, induced current on a ring, generating the potential difference on the plates, etc) when the system is defined on a nontrivial manifold, or placed in the background of the magnetic field.
It is important to emphasize that some effects can be proportional to θ QED , as opposed to θQED as commonly assumed or discussed for perturbative computations.Precisely this feature has the important applications when some observables are proportional to the static time-independent θ QED , and, in general, do not vanish even when θQED ≡ 0, see below. 1 We refer to [18] with physical explanation (in contrast with very mathematical papers mentioned above) of why the gauge cannot be uniquely fixed in such circumstances.In paper [18] the socalled "modular operator" has been introduced into the theory.The exp(iθ) parameter in QCD is the eigenvalue of the large gauge transformation opeartor, while exp(iθ QED ) is the eigenvalue of the modular operator from [18].This analogy explicitly shows why θ QED becomes a physically observable parameter in some circumstances.

II. AXION θ FIELD AND VARIETY OF TOPOLOGICAL PHENOMENA
Our starting point is the demonstration that the θ QED indeed does not enter the equations of motion.As a direct consequence of this observation, the corresponding Feynman diagrams at any perturbation order will produce vanishing result for any physical observable at constant θ QED .Indeed, which shows that θQED and not θ QED itself enters the equations of motion.In our analysis we ignored spatial derivatives ∂ i θ QED as they are small for non-relativistic axions.This anomalous current (1) points along magnetic field in contrast with ordinary E&M , where the current is always orthogonal to B. Most of the recent proposals [9][10][11][12][13][14] to detect the dark matter axions are precisely based on this extra current (1) when θ is identified with propagating axion field oscillating with frequency m a .
We would like to make a few comments on the unusual features of this current.First of all, the generation of the very same non-dissipating current (1) in the presence of θ has been very active area of research in recent years.However, it is with drastically different scale of order Λ QCD instead of m a .The main driving force for this activity stems from the ongoing experimental results at RHIC (relativistic heavy ion collider) and the LHC (Large Hadron Collider), which can be interpreted as the observation of such anomalous current (1).
The basic idea for such an interpretation can explained as follows.It has been suggested by [22,23] that the socalled θ ind -domain can be formed in heavy ion collisions as a result of some non-equilibrium dynamics.This induced θ ind plays the same role as fundamental θ and leads to a number of P and CP odd effects, such as chiral magnetic effect, chiral vortical effect, and charge separation effect, to name just a few.This field of research initiated in [24] became a hot topic in recent years as a result of many interesting theoretical and experimental advances, see recent review papers [25,26] on the subject.
In particular, the charge separation effect mentioned above can be viewed as a result of generating of the in-duced electric field in the background of the external magnetic field B ext and θ QED = 0.This induced electric field E ind separates the electric charges, which represents the charge separation effect.Then formula (2) essentially implies that the electric field locally emerges in every location where magnetic field is present in the background of the θ QED = 0.
The effect of separation of charges can be interpreted as a generation of the electric dipole moment in such unusual background.Indeed, for a table-top type experiments it has been argued in [21] that in the presence of the θ QED the electric and magnetic dipole moments of a topologically nontrivial configuration (such as a ring or torus) are intimately related: which obviously resembles the Witten's effect [27] when the magnetic monopole becomes the dion with electric charge e ′ = −(eθ QED /2π).
To support this interpretation we represent the magnetic dipole moment m ind as a superposition of two magnetic charges g and −g at distance L 3 apart, where L 3 can be viewed as the size of the compact manifold in construction [21] along the third direction2 .As the magnetic charge g is quantized, g = 2π e , formula (3) can be rewritten as This configuration becomes an electric dipole moment d ind with the electric charges e ′ = −(eθ QED /2π) which precisely coincides with the Witten's expression for e ′ = −(eθ QED /2π) in terms of the θ QED according to [27].This construction is justified as long as magnetic monopole size is much smaller than the size of the entire configuration L 3 such that the topological sectors from monopole and anti-monopole do not overlap and cannot untwist themselves.The orientation of the axis L 3 also plays a role as it defines the L 1 L 2 plane with non-trivial mapping determined by π 1 [U (1)] = Z, see below.If our arguments on justification of this formula are correct it can be applied to all fundamental particles including electrons, neutrons, and protons because the typical scale L 3 ∼ m −1 e ∼ 10 −11 cm, while magnetic monopole itself be assumed to be much smaller in size.In this case the expression (3) derived in terms of the path integral in [21] assumes the form where µ is the magnetic moment of any configuration, including the elementary particles: µ e , µ p , µ n .As emphasized in [21,28] the corresponding expression can be represented in terms of the boundary terms, which normally emerge for all topological effects.
The observed upper limit for d e < 10 −29 e • cm implies that θ QED < 10 −16 .We do not have a good explanation of why this parameter is so small.This question is not addressed in the present work.It is very possible that a different axion field must be introduced into the theory which drives θ QED to zero, similar to conventional axion resolution of the strong CP problem [2][3][4][5][6][7][8].
The equation similar to (5), relating the electric and magnetic dipole moments of the elementary particles was also derived in [29,30] where it has been argued that for time-dependent axion background the electric dipole moment of the electron d e will be generated 3 , and it must be proportional to the magnetic moment of the electron µ e and the axion field θ(t).The absolute value for the axion field θ 0 ≈ 3.7 • 10 −19 was fixed by assuming the axions saturate the dark matter density today.While the relation (5) and the one derived in [29,30] look identically the same (in the static limit m a → 0 and proper normalization) the starting points are dramatically different: we begin with canonically defined fundamental unknown constant θ QED = 0 while computations of [29,30] are based on assumption of time dependent axion fluctuating field saturating the DM density today, which obviously implies a different normalization for θ.Still, both expressions identically coincide in the static m a → 0 limit.
The identical expressions with precisely the same coefficients (for time dependent [29,30] and time independent (5) formulae) in static limit m a → 0 relating the electric dipole and magnetic dipole moments strongly suggest that the time dependent expression [29,30] can be smoothly extrapolated to (5) with constant θ QED .This limiting procedure can be viewed as a slow adiabatic process when θ ∝ m a → 0 and the θ becomes the timeindependent parameter, θ → θ QED when the same normalization is implemented 4 .
We want to present one more argument suggesting that the constant θ QED may produce physical effects including the generating of the electric dipole moment.Indeed the S θ term in QED in the background of the uniform static magnetic field along z direction can be rewritten as follows where 2πκ The expression on the right hand side is still a total divergence, and does not change the equation of motion.
In fact, the expression in the brackets is identically the same as the θ term in 2d Schwinger model, where it is known to be a physical parameter of the system as a result of nontrivial mapping π 1 [U (1)] = Z, see e.g.[34] for a short overview of the θ term in 2d Schwinger model in the given context 5 .
The expression (6) shows once again that θ QED parameter in 4d Maxwell theory becomes the physical parameter of the system in the background of the magnetic field 6 .In such circumstances the electric field will be 4 A different approach on computation of the time dependent dipole moment due to the fluctuating θ parameter was developed recently in [33].The corresponding expression given in [33] approaches a finite non-vanishing constant value if one takes the consecutive limits t → ∞ and after that the static limit ma → 0 by representing e/(2m) = µ in terms of the magnetic moment of a fermion.In this form it strongly resembles the expression derived in [29,30]. 5In this exactly solvable 2d Schwinger model one can explicitly see why the gauge cannot be uniquely fixed, and, as the consequence of this ambiguity, the θ becomes observable parameter of the system.The same 2d Schwinger model also teaches us how this physics can be formulated in terms of the so-called Kogut-Susskind ghost [35] which is the direct analog of the Veneziano ghost in 4d QCD. 6The parameter κ which classifies our states is arbitrary real number.It measures the magnetic physical flux, which not necessary assumes the integer values.
induced along the magnetic field in the region of space where the magnetic field is present according to (2).This relation explains why the electric dipole moment of any configuration becomes related to the magnetic dipole moment of the same configuration as equation ( 5) states.
The topological arguments for special case (6) when the external magnetic field is present in the system suggest that the corresponding configurations cannot "unwind" as the uniform static magnetic field B z enforces the system to become effectively two-dimensional, when the θ QED parameter is obviously a physical parameter, similar to analogous analysis in the well-known 2d Schwinger model, see footnote 5.
The practical implication of this claim is that there are some θ QED -dependent contributions to the dipole moments of the particles.While the θ QED does not produce any physically measurable effects for QED with trivial topology, or in vacuum, we expect that in many cases as discussed in [21] and in present work the physics becomes sensitive to the θ QED which is normally "undetectable" in a typical scattering experiment based on perturbative analysis of QED.We want to list below several CP odd correlations which will be generated in the presence of θ QED , and which could be experimentally studied by a variety of instruments.
The generation of the induced electric field (2) unambiguously implies that the following CP odd correlation will be generated Another CP odd correlation which can be also studied is as follows: where one should average over entire ensemble of particles with magnetic moments µ i , which are present in the region of a non-vanishing magnetic field B ext .The induced electric field (2) will coherently accelerate the charged particles along B ext direction such that particles will assume on average non-vanishing momentum p i along B ext .As a result of this coherent behaviour the following CP odd correlation for entire ensemble of particles is expected to occur One should add that the dual picture when the external magnetic field B ext is replaced by external electric field E ext also holds.For example, instead of (2) the magnetic field will be induced in the presence of the strong external electric field E ext , as e.g. in the proposal [36] to measure the proton EDM when the E ext is directed along the radial component, such that the correlation similar to (7) will be also generated

III. CONCLUSION AND FUTURE DIRECTIONS
The topic of the present notes on the dipole moments of the particles and antiparticles in the presence of the θ QED is largely motivated by the recent experimental advances in the field, see [36,37].There are many other CP odd phenomena which accompany the generation of the dipole moments.All the relations discussed in the present notes, including (5) or (7) are topological in nature and related to impossibility to uniquely describe the gauge fields over entire system, as overviewed in the Introduction.
Essentially the main claim is that the θ QED should be treated as a new fundamental parameter of the theory when the system is formulated on a topologically nontrivial manifold, and in particular, in the background of a magnetic field which enforces a non-trivial topology, as argued in this work.I believe that the very non-trivial relations such as (5) or (7) which apparently emerge in the system at nonvanishing θ and θ QED is just the tip of the iceberg of much deeper physics rooted to the topological features of the gauge theories.
In particular, the θ dependent portion of the vacuum energy could be the source of the Dark Energy today (at θ = 0) in the de Sitter expanding space as argued in [38,39].Furthermore, these highly non-trivial topological phenomena in strongly coupled gauge theories can be tested in the QED tabletop experiments where the very same gauge configurations which lead to the relation similar to (5) or (7) may generate an additional Casimir Forces, as well as many other effects as discussed in [28,34,40].What is even more important is that many of these effects in axion electrodynamics can be in principle measured, see [41][42][43][44][45] with specific suggestions and proposals.I finish on this optimistic note.

ACKNOWLEDGEMENTS
These notes appeared as a result of discussions with Dima Budker and Yannis Semertzidis during the conference "Axions across boundaries between Particle Physics, Astrophysics, Cosmology and forefront Detection Technologies" which took place at the Galileo Galilei Institute in Florence, June 2023.I am thankful to them for their insisting to write some notes on the dipole moments of the particles and their relations to the fundamental parameters of the theory, the θ and the θ QED .I am also thankful to participants of the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP) workshop on "quantum sensors and new physics" for their questions during my presentation.Specifically, I am thankful to Yevgeny Stadnik for the long discussions on topics related to refs [29-33].These notes had been largely completed during the MIAPbP workshop in August 2023.Therefore, I am thankful to the MIAPbP for the organization of this workshop.The MIAPbP is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-EXC-2094 -390783311.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada.