CP violation and circular polarisation in neutrino radiative decay

The radiative decay of neutral fermions has been studied for decades but CP violation induced within such a paradigm has evaded attention. CP violation in these processes can produce an asymmetry between circularly polarised directions of the radiated photons and produces an important source of net circular polarisation in particle and astroparticle physics observables. The results presented in this work outlines the general connection between CP violation and circular polarisation for both Dirac and Majorana fermions and can be used for any class of models that produce such radiative decays. The total CP violation is calculated based on a widely studied Yukawa interaction considered in both active and sterile neutrino radiative decay scenarios as well as searches for dark matter via direct detection and collider signatures. Finally, the phenomenological implications of the formalism on keV sterile neutrino decay, leptogenesis-induced right-handed neutrino radiative decay and IceCube-driven heavy dark matter decay are discussed.


Introduction
For decades, studies of neutrinos have deepened our understanding of nature [1]. Although their very small but non-zero masses (for at least two of their generations) and lepton flavour mixing have been observed and verified by neutrino oscillation experiments, some fundamental questions about neutrinos such as their electromagnetic properties, CP violation, whether they are Dirac or Majorana fermions and if they have additional species existing in nature remain unknown.
The studies of neutrino radiative decays dates back fourty years [2][3][4] and beyond. Assuming neutrinos are electrically neutral fermions (Dirac or Majorana), their electromagnetic dipole moments (EDMs) can be generated at various loop levels and neutrino radiative decays ν i → ν f +γ are induced by off-diagonal parts of the EDMs [5][6][7][8][9][10]. Charged current interaction contributions in the Standard Model (SM) have previously been calculated at one-loop level in [5][6][7][8][9] and later studied in detail in [11,12]. However, these contributions are tiny due to the large mass hierarchy between the active neutrinos and the W boson as there is currently no positive experimental indication in favour of their JHEP04(2020)178 existence. Neutrino electromagnetic interactions therefore provide a tantalising probe for new physics (NP) beyond the SM (see [13] for a comprehensive review).
If more massive neutrinos exist, then these heavy neutrinos may decay to the lighter active neutrinos radiatively. These heavier neutrinos will consequently have a larger decay width due to the existence of such decay channels. Various hypothetical heavier neutrinos have been historically introduced, motivated by a combination of theoretical and phenomenological reasons. Some of the most famous ones are those introduced in the type-I seesaw mechanism [14][15][16][17][18][19], which was proposed in order to address the origin of sub-eV left-handed neutrino masses. Phenomenological motivations have suggested keV sterile neutrinos as dark matter (DM) candidates to explain the detection of a 3.5 keV X-ray line in [20,21] (for some representative reviews, see [22][23][24]). Very heavy DM was also proposed [25,26] in order to explain the IceCube data [27,28]. Radiative decays of such heavy particles may be more significant than those of active neutrinos due to their very large relative mass. Hence, radiative decay is typically a major channel of importance in detecting possible keV sterile neutrino DM.
CP violation may exist in various processes involving neutrinos. At low energy, neutrino oscillations provide the best way to clarify its existence in the neutrino sector. Combined analysis of current accelerator neutrino oscillation data [29] supports large CP violation in the appearance channel of neutrino oscillations [30,31]. The next-generation neutrino oscillation experiments DUNE and T2HK are projected to observe CP violation in the near future [32][33][34]. At high energy, the most well-studied process involving CP violation is the very heavy right-handed neutrino decaying into SM leptons and the Higgs boson. This effect is the source of the so-called thermal leptogenesis phenomenon, which can explain the observed matter-antimatter asymmetry in our Universe [35]. On the other hand, if these heavy neutrinos have lighter masses, specifically around the GeV scale, CP violation may appear in right-handed neutrino oscillations, which provides an alternative mechanism for leptogenesis [36] (See [37,38] for some reviews).
In this work we study CP violation in radiative decays of both Dirac and Majorana neutrinos. Whilst neutrino radiative decays have been extensively studied for some mass regions of neutrinos, CP violation in these processes has not been studied for a more general spectrum of mass scales with very few exceptions e.g. [39]. Recently, it was suggested in [40] that a net circular polarisation, specifically an asymmetry between two circularly polarised photons γ + and γ − , can be generated if CP is violated in neutrino radiative decays. Therefore, the circular polarisation of photons provides a potentially crucial probe to prove the existence of CP violation in the neutrino and DM sectors.
This work builds a formulation to describe both CP violation in neutrino radiative decays and also the resulting asymmetry between the produced photons γ + and γ − . In section 2, we outline the most general formalism of CP violation and circular polarisation in terms of form factors where the result is independent of the neutrino model or mass scale. In section 3, we discuss CP violation based on a simplified neutrino model. We begin this section with a discussion about the size of CP asymmetry for the SM contribution and then consider how CP violation can be enhanced via new interactions. A comprehensive analytical calculation of CP asymmetry based on Yukawa type NP interactions is then performed JHEP04(2020)178 in section 3.2, this type of simple interaction has a wide ensemble of phenomenological applications which is shown in section 4. Finally, we summarise our results in section 5.

The framework
In this section we shall set up the framework for computation of CP violation in neutrino radiative decays and the general connection with circular polarisation generated by such processes. Discussion in this section is fully independent of neutrino interactions and thus is applicable to any other electrically neutral fermion with mass at any scale.
Discussions in section 2.1 and 2.2 assume neutrinos are Dirac fermions. The extension to Majorana neutrinos will be given in section 2.3.

Matrix element for polarised particles
Assuming fermions are Dirac particles, the amplitude for the process ν i → ν f +γ ± is given by Here, u(p i ) and u(p f ) are spinors for the initial ν i and final ν f state neutrinos respectively. By momentum conservation, the photon momentum is q = p i − p f . The spinors include the spin polarisation of the fermions, this will be discussed in more detail in the next subsection in a specified inertial reference frame. The transition form factor is then parametrised as per [8][9][10]41] We will not consider electrically charged neutrinos, namely we require that f Q = 0. The modification to the result in the case of non-zero f Q will be mentioned at the end of this section. By requiring the photon to be on-shell q 2 = 0 and choosing the Lorenz gauge q · ε p = 0, the anapole does not contribute. In this case, only the electromagnetic dipole moment contributes to the neutrino radiative decay. We then rewrite the form factor as where f L,R fi = −f M fi ±if E fi and the chiral projection operators are defined as P L,R = 1 2 (1∓γ 5 ). The decay widths for ν i → ν f + γ ± are then given by The amplitudes M(ν i → ν f + γ ± ) are directly correlated with the coefficients which are derived in detail in appendix A. The sum of the decay widths for ν i → ν f + γ + and ν i → ν f + γ − yields the total radiative decay width Γ(ν i → ν f + γ).

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Again, if we only consider radiative decay for an electrically neutral antineutrino, the amplitudes of radiative decayν i →ν f + γ ± are then given by where v(p i ) and v(p f ) are antineutrino spinors. The decay width forν i →ν f ,s + γ l is By parametrising the form factor in a similar form to before, we havē Therefore, the amplitudes can be written in a similar fashion following eq. (2.5), i.e. by replacing f L fi and f R fi byf L if andf R if respectively (see the proof in appendix A). These formulae can be further simplified with the help of the CP T theorem, which is satisfied in all Lorentz invariant local quantum field theories with a Hermitian Hamiltonian. Due to CP T invariance,ν i →ν f + γ ∓ and ν f + γ ± → ν i have the same amplitude, and thusf M, is satisfied [13], leading tō Hence, amplitudes M(ν i →ν f + γ + ) can be simplified to Physical neutrinos and antineutrinos are related by a CP transformation which interchanges particles with antiparticles and replaces momentum by its parity conjugatẽ p = (p 0 , − p). The CP transformation reverses the momentum but preserves angular momentum. As a consequence, the polarisation is reversed. Performing a CP transformation for ν i (p i ) → ν f (p f ) + γ ± (q) gives rise to antineutrino channels with reversed 3D momentum and reversed photon polarisations in the final statesν i (p i ) →ν f (p f ) + γ ∓ (q). Since the amplitude is parity-invariant, the amplitude of the process is equivalent tō ν i (p i ) →ν f (p f ) + γ ∓ (q). Therefore, the radiative decay of antineutrinos can be represented as a CP conjugate of the decay of neutrinos (2.11) [13,42]. And eventually, we arrive at the identity (2.12) However, a CP violating source in the interaction may contribute at loop level and break this equality.

Correlation between CP asymmetry and circular polarisation
We define the CP asymmetry between the radiative decay ν i → ν f + γ + and its CP conjugate processν i →ν f + γ − as The CP asymmetry between ν i → ν f + γ − and its CP conjugate processν i →ν f + γ + , ∆ CP,− , is defined by exchanging + and − signs. The photon polarisation independent CP asymmetry is obtained by summing ∆ CP,+ and ∆ CP,− together which yields (2.14) It is also convenient to define the asymmetry between the radiated photons γ + and γ − as Given equal numbers for initial neutrinos and antineutrinos, ∆ +− represents the fraction where N γ + and N γ − are the number of polarised photons γ + and γ − produced by the radiative decays respectively. It is this source that generates circular polarisation for the radiated photons giving rise to a non-zero Stokes parameter V . Therefore, a non-zero ∆ +− is a source of circular polarisation for the photon produced by the radiative decay. Since the phase spaces are the same for neutrino and antineutrino channels, these formulae can be simplified to as well as (2.17) The total CP asymmetry and the asymmetry between γ + and γ − follows simple relations with ∆ CP,+ and ∆ CP,− as ∆ CP = ∆ CP,+ + ∆ CP,− , Therefore, we arrive at an important result that the generation of circular polarisation is essentially dependent upon CP asymmetry between neutrino radiative decay and its CP conjugate process. Note that we have not included any details related to the Lagrangian or JHEP04(2020)178 interactions yet. Given any neutral fermion, its radiative decay can always be parametrised by the electromagnetic dipole moments with coefficients f L fi and f R fi (as well asf L if andf R if for its antiparticle), we then arrive at the correlations between CP violation and circular polarisation in eq. (2.18) with their definitions in eqs. (2.16) and (2.17). Another source of asymmetry between polarised photons is the existence of an initial number asymmetry between neutrinos and antineutrinos [40]. There may be some other CP violating sources in particle physics which can induce this condition [43]. On the other hand, this kind of asymmetry is more likely to be generated in extreme astrophysical environments. For example, in supernovae explosions, the asymmetry between sterile neutrinos and antineutrinos may be generated because of the different matter effects during neutrino and antineutrino propagation [44,45]. In the rest of this paper, we will only consider circular polarisation directly produced by the CP violating decays between neutrinos and antineutrinos. Now we may turn our attention to obtaining non-zero CP violation for the radiative decay. For ν i → ν f + γ + and ν i → ν f + γ − , we parametrise the effective coefficients f L fi and f R fi , these should be obtained from the relevant loop calculations in the form without loss of generality. Here, we have used l to classify the different categories of loop contributions. For each loop category l, C l factorises out all coefficients of operators contributing to the diagram. K L l and K R l represents the pure loop kinematics after coefficients are extracted out. As a consequence, corresponding to the effective parameters forν i →ν f + γ ± , can always be represented in the form 1 The CP asymmetries with respect to the photon polarisations can then be simplified to Therefore, a non-zero CP asymmetry is determined by non-vanishing Im(C l C * l ) and nonvanishing Im(K L l K L * l ) (or Im(K R l K R * l )) from loops l and l . While the imaginary part of Im(C l C * l ) is straightforwardly obtained from the relevant terms in the Lagrangian, the main task is to compute the imaginary parts of K L l K L * l and 1 To clarify how this parametrisation is valid, we write out the subscripts explicitly, One can simplify f L if and f R if in the following steps. 1) The coefficient (C l ) if must be the complex conjugate of (C l ) fi since both processes are CP conjugates of one another. 2) (K L l ) if and (K R l ) if , as pure kinetic terms, must satisfy T parity, namely they must be invariant under the interchange of the initial and final state neutrinos ν i ↔ ν f , the chiralities must also be interchanged L ↔ R, namely, JHEP04(2020)178 In order to achieve non-zero values of these imaginary parts, one may apply the optical theorem which can be expressed as where the sum runs over all possible sets c of final-state particles [46]. Fixing a = ν i and b = ν f + γ, c has to include an odd number of fermions plus arbitrary bosons. All particles heavier than ν i cannot be included in c since this would violate energy-momentum conservation. In the next section, we will explicitly show how to derive a non-zero analytical result for Im(K R l K R * l ) based on a simplified NP model where Im(K L l K L * l ) is negligibly small.

CP violation in Majorana neutrino radiative decay
The above discussion is only limited to Dirac neutrinos. However, neutrinos may also be Majorana particles i.e. where the neutrino is identical to the antineutrino but with potentially different kinematics. In this case, both the neutrino and antineutrino modes must be considered together. The amplitude is then given by . Taking the explicit formulas for the amplitudes given in eq. (2.5) and (2.10), we obtain results with definite spins in the initial and final states as For Majorana fermions, the CP violation is identical to that obtained from P violation alone i.e. the CP asymmetry is essentially the same as the asymmetry between the two polarised photons ∆ M The CP asymmetry without considering the polarisation of the radiated photon is zero, namely, ∆ M CP = ∆ M CP,+ + ∆ M CP,− = 0. With the help of eq. (2.23), we can express ∆ M +− in the form of electromagnetc dipole parameters as (2.25) We will not discuss the Majorana case further here since the asymmetries are similarly straightforward to obtain once coefficients of the transition dipole moment are ascertained. At the end of this section, we comment on CP violation in electrically charged neutrino decay. In this scenario, the magnitudes of the neutrino and antineutrino decay modes are modified to

Calculating CP violation in radiative decay
Having provided a very general discussion on CP violation and circular polarisation for neutrino radiative decay in a mass scale and model independent way in the previous section, in the following sections, we will concentrate on a simplified example where a sterile neutrino radiatively decays ν s → ν i + γ and show how to obtain the exact form of the CP asymmetry and circular polarisation for the radiated photon. In this example, the initial and final state neutrinos are specified as ν i = ν s and ν f = ν i respectively. In this simplified case, we consider only one sterile neutrino generation and the three active neutrino generations with both ν s and ν i (for i = 1, 2, 3) being mass eigenstates. Extensions to multiple sterile neutrino generations are straightforward, and thus, will not be discussed here.
We will apply the above formulation in the following way. First, we estimate the size of CP violation from the SM contribution alone i.e. via the charged current interaction mediated by the W boson. Then, we consider the enhancement of CP violation by including NP Yukawa interactions for sterile neutrinos. Such Yukawa interactions have a wide array of applications with theoretical and phenomenological utility which we will outline in the following section. Finally, we list the simplified analytical result for CP violation and circular polarisation generated from the decay at the end of this section.

The Standard Model contribution
It is well known that the radiative decay can happen via one-loop corrections induced by SM weak interactions with SM particles (specifically with charged lepton α for α = e, µ, τ and the W boson) in the loop. The crucial operator is the charged-current interaction is where g is the EW gauge coupling constant and U αm represent the lepton flavour mixing.
Here we have m = i, s (where i = 1, 2, 3) representing the active light neutrino mass eigenstate ν i and the sterile neutrino mass eigenstate ν s . The one-loop Feynman diagrams for the radiative decay via the SM charged current interaction are shown in figure 1. 2 In the limit m 2 where m α and m W are the charged lepton and W boson masses respectively, we have the result for Γ µ fi given as where F α is a function obtained from the loop integrals and the Fermi constant is defined If m i is much smaller than the charged lepton masses, we arrive at the classic result [6,8] which is insensitive to neutrino masses. A more general neutrino mass-dependent result for F α with m i , m f up to the W boson mass has been given in [11,12]. In general, for m i < m W , F α is always positive, this is consistent with the optical theorem. From the above formulae, we obtain results for f L fi and f R fi given as factorising the SM contribution into a coefficient part and a purely kinetic part yields and with flavour index α = e, µ, τ . Since F α is real, both Im(K L α K L * β ) and Im(K R α K R * β ) vanish for any flacours α, β = e, µ, τ . In addition, by interchanging i ↔ s we notice that the one-loop SM contribution exactly satisfies f L fi =f R if and f R fi =f L if . Therefore, there is no CP violation coming from these diagrams.
For a sterile neutrino with mass smaller than the W boson mass, we comment that a non-zero CP violation can in principle be obtained after considering higher-loop SM contributions. We analyse this by applying the optical theorem once again. In order to JHEP04(2020)178 Namely, if the sterile neutrino is heavier than the W boson, an imaginary part can be obtained directly from the SM one-loop diagram, we will discuss this case in some of the following sections.

Enhancement by new physics
In order to enhance the CP violation in the radiative decay of the sterile neutrino, we include NP contributions. We being by introducing two new particles, one fermion ψ and one scalar φ with opposite electric charges Q and −Q respectively. Their couplings with neutrinos and the sterile neutrino are described by the following Yukawa interaction where λ m , with m = i, s (for i = 1, 2, 3), are complex coefficients to ν i and ν s , which are the active and sterile neutrino mass eigenstates respectively. Here, we only included one generation of φ and ψ respectively. The extension to more generations is straightforward and will be mentioned as necessary. Neither ψ or φ are supposed to be a specific DM candidate in this work and they can annihilate with their antiparticles due to their opposite electric charges. The full amplitude including the NP contribution for ν s → ν i + γ can then be written From eq. (3.10), we have the necessary expressions to compute the CP violation and asymmetry between the radiated photons γ + and γ − . As an example, we take ∆ CP,− to demonstrate an explicit calculation. The definition of ∆ CP,− has been given in eq.
With the help of the parametrisation in eq. (3.10) and assuming |K R l | = |K L l | for any loop l, we obtain ).
(3.11) For the two NP diagrams shown in figure 2, where a photon is radiated via the interaction between scalars φ and fermions ψ respectively, the amplitudes can be explicitly written as The coefficients C l NP (for l NP = 1, 2) are then simply obtained from inspection to be In this case, Im(C 1 C * 2 ) = 0 and the second part of eq. (3.11) vanishes. On the other hand the imaginary part is given by (3.14) We now turn to the loop contributions. Im(K R 1 K R * 2 ) does not need to be calculated since Im(C 1 C * 2 ) vanishes explicitly. Hence, the remaining term to be computed is Im(K R α K R * l NP ). Furthermore, since the SM contributions are always real, Im(K R α K R * l NP ) = −K R α Im(K R l NP ).

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In order to obtain CP violation between the radiative decay ν s → ν i + γ − and its CP conjugate channelν s →ν i +γ + for a Dirac-type sterile neutrino, a non-vanishing imaginary part Im(K R l NP ) is required, this can be summarised Following a similar approach to determine CP violation between ν s → ν i + γ + and its CP conjugate processν s →ν i + γ − , we obtain Due to the optical theorem, non-zero Im(K L l NP ) and Im(K R l NP ) can only be achieved if the sterile neutrino mass is larger than the sum of the charged scalar and the charged fermion masses, m s > m φ + m ψ . In the remainder of this section, our aim will be to compute these quantities.
Here, the loop integrals for the relevant diagrams shown in figure 2 will be calculated. Starting from the general form of the amplitude for sterile neutrino radiative decay ν s → ν i + γ ± given in eq. (2.1), we extract the purely kinetic terms K L l NP and K R l NP for l NP = 1, 2 as 4 The above results are obtained without any approximations. In order to derive further simplified analytical formulae, we consider the large mass hierarchy between ν s and ν i where m i m s , and may therefore take the limit m i → 0. In this case, K L l NP = 0 and after integrating over Feynman parameters z and x, K R l NP can be written as 4 Here, K L l NP and K R l NP represent (K L l NP )is and (K R l NP )is, respectively. Exchanging i with s, we obtain (K L l NP )si = (K R l NP )is and (K R l NP )si = 0, this is compatible with our previous statement that

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where ∆ φψ (y) = y m 2 K R l NP may have both real parts and imaginary parts. The real part Re(K R l NP ) is directly obtained by replacing ∆ φψ and ∆ ψφ with there absolute values, therefore simple analytical expressions for Re(K R l NP ) are difficult to obtain. However, in the hierarchical case m s m φ , m ψ approximate analytical expressions can be derived by expanding in powers of m 2 φ /m 2 s and m 2 ψ /m 2 s . Specifically, the leading-order results are given by Since we are chiefly interested in the CP violating component, we will focus on how to obtain and simplify the imaginary parts of K R l NP . Since m 2 φ , m 2 ψ ≥ 0, the imaginary and thus CP violating component in eq. (3.19) factorises when the argument of the logarithm is negative, by inspection we can see this occurs when Solutions at the boundaries of the CP violation conditions ∆ φψ (y) = 0 and ∆ ψφ (y) = 0 are y 1,2 (m φ , m ψ ) and y 1,2 (m ψ , m φ ) respectively. Therefore the conditions in eq. (3.22) in terms of y are fulfilled when y 1 (m φ , m ψ ) ≤ y ≤ y 2 (m φ , m ψ ) and y 1 (m ψ , m φ ) ≤ y ≤ y 2 (m ψ , m φ ) for the two diagrams respectively, where and µ 2 is defined as It should be noted that in both cases 0 < y 1 < y 2 < 1 is necessarily satisfied. Hence, the imaginary component of eq. (3.19) can now be written according to the complex logarithm definition as (3.25)

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Finally, integrating over the final Feynman parameter y leads to The requirement m s > m φ + m ψ leads to a positive µ 2 . In the mass-degenerate limit m s = m φ + m ψ , µ 2 = 0 and after some simplifications, it can be shown for this case that Im(K R 1 ) = Im(K R 2 ) = 0. In the massless limit m φ , m ψ → 0, these imaginary parts are approximately given by Im(K R 1 ) → −1/(16πm s ) and Im(K R 2 ) → 0. Since we need to compute ∆ CP,− to calculate CP violation, we apply eq. (3.15), which in this example can be written explicitly as is obtained by multiplying by a factor m 2 i /m 2 s which is strongly suppressed by the light active neutrino mass.
Here, we have defined I φψ , an order one normalised parameter which is defined via s I φψ and explicitly given by (3.28) See appendix B for more details regarding the calculation of the imaginary part of the loop diagrams.
In this example, we may safely ignore the and ∆ CP,+ is negligibly small. This result works for the Dirac neutrino case. In the Majorana neutrino case, from eq. (2.25), it is straightforward to apply a similar procedure and obtain

JHEP04(2020)178 4 Phenomenological applications of the formulation
We are now ready to discuss possible phenomenological implications of this suggested sterile neutrino model which has CP violation generated at one-loop level for radiative decays. The formulation based on the simplified example above has a wide array of possible applications. One direct application is the study of CP violation in keV neutrino DM radiative decay. We can also apply it to the general type-I seesaw mechanism where righthanded neutrinos are much heavier than the electroweak scale in order to recover light active neutrino masses. It is also of interest to consider its application for heavy neutrino DM motivated by the IceCube data.

keV sterile neutrino dark matter
The keV-scale sterile neutrino has been discussed extensively as a DM candidate (for example models, see [48][49][50][51][52]). Following the discussion in section 3.1, it is clear that the SM contribution at one-loop level cannot generate CP violation in keV neutrino radiative decay and a non-zero CP asymmetry can only be obtained at four-loop level. Therefore, we consider Yukawa interactions as shown in eq. (3.8).
We give a brief discussion on constraints to the sterile neutrino ν s and the new charged particles φ and ψ. Since ν s is assumed to be a DM candidate, the decay channel ν s → φψ introduced by the new interaction with φ and ψ must be controlled. The width of this channel is around where c ν = 1 for a Dirac neutrino and c ν = 2 for a Majorana neutrino. We require the width to be at least as small as the decay width of the SM Γ SM . We approximate Γ SM to the width of the dominant channels ν s → ν i ν jνi for any active neutrinos ν i and ν j [6,48,53] namely By introducing a parameter η representing the ratio of the two decay widths η = Γ NP /Γ SM , we can express |λ s | by η as |λ s | ≈ namely, an extremely small value for λ s is required. 5 The charged particles φ and ψ as in our previous formulation are assumed to be lighter than the sterile neutrino. Thus, they have to be at most millicharged to avoid significant modification to the precisely measured QED interactions at low energy. The Lamb shift imposes an upper bound for the millicharge Q 10 −4 e [54], which is valid for a scalar or fermion with a mass less than 1 keV.
Considering these bounds, we can roughly estimate the size of CP violation of ν s radiative decay. We also recall that the SM decay channel dominates the DM radiative decay while η < 1.

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In this case, we can approximate both f R fi and f L if in the denominator by f R fi,SM and it then follows that Therefore, we obtain the analytical result of the CP asymmmetry as where we have made the approximations F α ≈ 3/2 since m α m W , and denoted the phase of λ i (U † U ) is λ * s as δ is . In the limits m s m φ , m ψ , we have I φψ ≈ 1, and thus arrive at ∆ CP,− ∼ 10 −1 √ ηQ|λ i |, which is small due to the suppression by the millicharge Q. , and thus the enhancement by an order of magnitude is easily obtained from I φψ . By assuming a typical value of the millicharge Q ∼ 10 −4 e, the coupling λ i ∼ 10 −1 and η ∼ 1, we arrive at ∆ CP,− ∼ 10 −5 .
Other enhancements could be realised by considering the hierarchical mixing of the sterile neutrino with different active neutrinos.

Seesaw mechanism and leptogenesis
Our discussion thus far can also be generalised to the case of very heavy neutrinos. Heavy neutrinos with masses much higher than the EW scale are introduced in the seesaw mechanism to explain the tiny observed active neutrino masses. The heavy neutrinos are assumed to be Majorana particles in the mechanism. These particles, as originally proposed in [35], provide a class of scenarios where matter-antimatter asymmetry of the Universe is generated by the decays of heavy neutrinos by a process termed leptogenesis. Yukawa interactions involving heavy neutrinos provide the necessary source of CP violation between the decay N I → L α H and its CP conjugate N I →L α H † in leptogenesis. We address the fact that these interactions can also generate CP violation between the radiative decay N I → N J γ + and its CP conjugate process N I → N J γ − . 6 The CP asymmetry can be simply estimated with the help of the analytical result obtained in the last subsection. In order to achieve this, we first present the Yukawa interactions in the form Since we consider right-handed neutrinos to be much heavier than the W boson mass, the Goldstone-boson equivalence theorem can be applied. The main contributions to N I → N J γ are those loops involving charged leptons α and the Goldstone boson H + . Therefore, we can simply apply the formulation in section 3.2 by replacing masses m ψ and m φ with m α and m W respectively. Here, it is necessary to keep the charged JHEP04(2020)178 lepton masses as we will see later that it is essential to generate CP asymmetry. In this case, f L fi and f R fi are approximatively given by f L JI ≈ α C α K L α and f R JI ≈ α C α K R α with C α and K L,R α given by with K R 1,α and K R 2,α given by K R 1 and K R 2 in eq. (3.17) with masses m s , m i , m φ , m ψ replaced by M I , M J , m α and m W respectively. Assuming right handed neutrino masses M I M J , we can safely ignore the K L α contribution and arrive at the approximation of CP asymmetry shown in eq. (3.30).
The CP violation requires both non-zero values for Im(C α C * β ) and Im(K R α K R * β ). The former term given by Im(C α C * β ) = e 2 Im(λ αI λ * αJ λ βJ λ * βI ) is usually non-zero based on the complex Yukawa couplings which are necessary for leptogenesis. For the latter term, without considering the difference between charged lepton masses K R α = K R β and Im(K R α K R * β ) = 0 holds explicitly. Taking charged lepton masses into account and considering the hierarchy m α m W M I , we obtain the leading contribution (cf. eq. (3.19) and eq. (3.26)) Eventually, we arrive at the CP asymmetry as where for charged leptons, only the dominant τ mass has been considered. This formula takes a similar structure as the CP asymmetry of the N → L τ H decay in thermal leptogenesis (see e.g., in [38]), namely, the coefficient combination, Im(λ τ I [λ † λ] IJ λ * τ J ). The difference is that, while the asymmetry in thermal leptogenesis is suppressed by a loop factor, 7 the asymmetry here is not, but rather strongly suppressed by the mass hierarchy m 2 τ /M 2 I . Furthermore, we comment that the CP violation for heavy neutrino radiative decays are very hard to observe since the only way to access this quantity is to measure the circular polarisation of photons radiated from the decay. This is not possible to measure currently due to the very small size of ∆ +− . What presents an even larger challenge is that these processes happen in the very early stages of the evolution of the Universe. Thus, even if there is a large fraction of polarised photons produced, the asymmetry will be washed out by ubiquitous Compton scattering processes [55].
A possible way to enhance the CP asymmetry may be by considering a low-energy seesaw mechanism. For example, in the GeV sterile neutrino seesaw, there is no severe mass suppression between right-handed neutrino masses and the τ lepton mass to significantly reduce the CP asymmetry. Neutrinos at such a scale can explain baryon asymmetry 7 The leading order contribution of the N → Lτ H decay is at tree level and the CP violation appears at one-loop level.

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based on a different leptogenesis mechanism, specifically the Akhmedov-Rubakov-Smirnov mechanism [36]. Another advantage is that these neutrinos can be tested at the SHiP experiment [56]. The disadvantage is that since the neutrino mass is lower than the W boson mass, CP violation of the radiative decay cannot be generated at one-loop, but rather at two-loop level. Thus, a more complicated calculation is required for this case.

Heavy dark matter and IceCube
Very heavy neutrinos could also be DM candidates. In fact, a heavy neutrino DM N DM with mass around 10 2 TeV-PeV scale as a DM candidate [25,26] is motivated by the high energy neutrino component in excess of the well-known atmospheric events [27,57] by the IceCube experiment (see [58,59] for recent progresses and [60,61] for analysis combining with other experimental data). Examples of typical heavy neutrino DM models explaining these observations have been shown in [62][63][64][65][66]. At low energy, they may induce very weak effective Yukawa interactions between the DM neutrino with other fermions.
Since radiative decay of a DM candidate can proceed very slowly until the present day, the washout by Compton scattering in the early stage of Universe can be avoided. Given a sufficiently small Yukawa coupling λ α-DMLαH N DM , 8 we may easily estimate the size of CP asymmetry in the DM radiative decay. The tree-level decay to νZ is induced and is one of the main decay channels being tested at IceCube. On the other hand, this coupling also induces the radiative decay N DM → νγ which may result in CP violation. The CP violation, as discussed in the last subsection, would be suppressed by the ratio m 2 τ /M 2 DM 10 −6 .

Conclusion
In this work, we built a general framework for CP violation in neutrino radiative decays.
CP violation in such processes produces an asymmetry between the circularly polarised radiated photons and provides an important source of net circular polarisation that can be observed in particle and astroparticle physics experiments. The formulation between CP violation in neutrino radiative decays and the neutrino electromagnetic dipole moment at the form factor level is developed for both Dirac and Majorana neutrinos. We observed the model-independent connection between the decays and photon circular polarisation produced by these processes and concluded that CP violation directly determines the circular polarisation. Specifically in the Majorana neutrino case, the CP asymmetry is identical to the asymmetry of photon polarisations up to an overall sign difference. The contribution of a non-zero electric charge to neutrino decays is also discussed for completeness.
We then discussed how to generate non-vanishing CP violation through a generic new physics Yukawa interaction extension consisting of electrically charged scalar and fermion 8 This Yukawa coupling may be effectively induced. For example, in the Higgs induced RHiNo DM model [62,66,67], it is the dimension-five operator 1 ΛN c I NDMH † H with the thermal effect enhancing the mixing between DM with source neutrino NI which eventually enhances the DM production. This operator, together with the Yukawa coupling eq. (4.5) induces a very weak Yukawa coupling with coefficient λα-DM ∼ yαI v H M I ΛM DM in the limit MDM MI where vH is the Higgs VEV.

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states. Without introducing any source of electric charge for the neutrinos, these particles can decay only via the electromagnetic transition dipole moment. The explicit analytical result of CP violation for this model was derived and presented. This fundamental result is applicable when determining circular polarisation for both Dirac and Majorana fermions and can be exported for use in any models that generate radiative decays of this type. Finally, we included some brief discussion pertaining to the phenomenological implications of neutrinos at various mass scales. Firstly, the fomalism was applied to keV sterile neutrinos which are popular DM candidates and found CP violation and circular polarisation of the resulting radiated X-ray. We also considered the implications for much heavier sterile neutrinos of scale 1TeV which are required for the seesaw mechanism and leptogenesis. We argue that the CP source in the Yukawa coupling, which is essential for leptogenesis, can trigger CP violation for heavy neutrino radiative decays. The case of weakly interacting sterile neutrinos at a mass comparable to the electroweak scale is also interesting as it could produce exotic collider signatures as well as circular polarisation. We plan to compute the CP violation from such a process in future work. We also discussed the circular polarisation of γ-rays released from the radiative decay of the PeV scale dark matter motivated by IceCube data, however the size of this effect is too small to observe at current experimental sensitivities. where ξ S and η S are two-component spinors normalised to unity. Here, we include the polarisation index S for two independent spinors.

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To simplify the derivation, we prefer to work in the rest frame. Frame-independent results can be obtained straightforwardly from this case. In the rest frame, the initial sterile neutrino ν i is at rest p µ i = (m i , 0, 0, 0) T , and the photon is released in the +z direction with momentum q µ = (q, 0, 0, q) T . Conservation of momentum requires p µ f = (E f , 0, 0, −q) T with q = (m 2 i − m 2 f )/(2m i ) and E f = (m 2 i + m 2 f )/(2m i ). In this frame, S denotes spin along the +z direction i.e. S z , which takes values ± 1 2 . This geometry is shown in figure 3. The angular momentum along the z direction is conserved S z (ν i ) = S z (ν f )+S z (γ). For a fermion, S z = ±1/2 and for a massless photon, S z = ±1. Given the initial state ν i with spin S z (ν i ) = +1/2(−1/2), the only solution for spins in final states is S z (ν f ) = −1/2(+1/2) and S z (γ) = +1(−1). In other words, the released photon is the right-handed γ + (lefthanded γ − ).
For the photon moving in the +z direction, the polarisation vectors are as defined in [46] correspond to spin S z = +1 and −1, respectively. 9 9 Here we apply the convention in the textbook [46]. The definition of + in this convention has a sign difference from the one shown in [40]. Using the convention in [40] leads to a sign difference for iM(ν i,+

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In this frame, for the neutrino ν f moving in the −z direction, the spinors u S (p) and v S (p) with spin ± 1 2 are simplified to In the massless case, u + 1 2 and u − 1 2 are purely left-and right-handed respectively (because we have assumed ν f is moving in the −z direction). Spinors for initial neutrino ν i and antineutrinoν i are given by The amplitudes with definite spins in the initial and final states are then given by Here, ν i,+ We can generalise the result in eq. (A.7) to any inertial reference frame via spatial rotations and Lorentz boosts. These transformations change spins for fermions but leave photon polarisation invariant. Eventually, we obtain the Lorentz-invariant amplitudes M(ν i → ν f + γ ± ) and M(ν i →ν f + γ ± ) taking the same result as eq. (A.7) in any reference frame. Using the CP T -invariance property, namely,f R,L if = −f R,L if , we eventually arrive at eqs. (2.5) and (2.10). These are the most general results independent of either particle model or reference frame.

B Derivation of imaginary parts of the loop integrals
The two NP contributions to the sterile neutrino radiative decay given by the new proposed interactions are shown in figure 2. In order to compute their respective matrix elements,

(B.7)
Finally, applying the Ward identity q µ M µ = 0 and ignoring terms proportional to γ µ , since these are simply vertex corrections to the overall electric charge, 10 we only need to consider the tensor-like terms within Γ µ is to determine the form factor resulting from these diagrams. These are given by From these last expressions, we can identify the factors K L 1,2 and K R 1,2 given in eq. (3.17) and then integrate over the remaining Feynman parameters x and y as shown in eq. (3.19).
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 10 Notice that when both contributions are added the divergent terms cancel out.