Laser photons acquire circular polarization by interacting with a Dirac or Majorana neutrino beam

It is shown that for the reason of neutrinos being left-handed and their gauge-couplings being parity-violated, linearly polarized photons acquire their circular polarization by interacting with neutrinos. Calculating the ratio of linear and circular polarizations of laser photons interacting with either Dirac or Majorana neutrino beam, we obtain this ratio for the Dirac neutrino case, which is about twice less than the ratio for the Majorana neutrino case. Based on this ratio, we discuss the possibility of using advanced laser facilities and the T2K neutrino experiment to measure the circular polarization of laser beams interacting with neutrino beams in ground laboratories. This could be an additional and useful way to gain some insight into the physics of neutrinos, for instance their Dirac or Majorana nature.


I. INTRODUCTION
Since their appearance, neutrinos have always been extremely peculiar. Their charge neutrality, near masslessness, flavor mixing, oscillation, type of Dirac or Majorana, and in particular parityviolating gauge-coupling have been at the center of a conceptual elaboration and an intensive experimental analysis that have played a major role in donating to mankind the beauty of the standard model for particle physics. Several experiments studying solar, atmospheric and reactor neutrinos in past several years provide strong evidences supporting the existence of neutrino oscillations [1]. This implies that neutrinos are not exactly massless although they chirally couple to gauge bosons. Therefore they cannot be exactly two-component Weyl fermions. Instead, it is tempting to think of the nature of neutrinos, they might be fermions of Dirac type or Majorana type [2]. Dirac and Majorana neutrinos have different electromagnetic properties [3], which can be used to determine which type of neutrinos exists. The well-known example is the neutrino-less double beta decay [4], still to be experimentally verified. Some attention has been recently driven to study the neutrino interactions with laser beams; here we mention a few examples. The emission of νν pairs off electrons in a polarized ultra-intense electromagnetic (e.g., laser) wave field is analyzed in Ref. [5]. The electron-positron production rate has been calculated [6] using neutrinos in an intense laser field. In the frame of the standard model, by studying the elastic scattering of a muon neutrino on an electron in the presence of a linearly polarized laser field, multi-photon processes have been shown [7].
In this Letter, we quantitatively show and discuss linearly polarized photons acquire circular polarization by interacting with neutrinos, for the reason that neutrinos are left-handed and possess peculiar couplings to gauge bosons in a parity-violating manner. In particular we quantitatively calculate the circular polarization that a linearly polarized laser beam develops by interacting with a neutrino beam in ground laboratories. Moreover we show that using advanced laser facility one can possibly measure the circular polarization of laser photons interacting with a Dirac or Majorana neutrino beam produced by neutrino experiments, e.g., the Tokai-to-Kamioka (T2K) [8].
This could possibly provide an additional way to study some physics of neutrinos, for example, the nature of neutrinos, Dirac or Majorana type.
We recall that the similar photon-neutrino process was considered for the generation of circular polarizations of Cosmic Microwave Background (CMB) photons [9] interacting with cosmic background neutrinos (CNB), instead of photons [10] and electrons in the presence of magnetic field [11]. There the main calculation was done to obtain the power spectrum C V l of the circular polarization of CMB photons by the forward scattering between CMB photons and CNB neutrinos.

II. STOKES PARAMETERS
The polarization of laser beam is characterized by means of the Stokes parameters: the total intensity I, intensities of linear polarizations Q and U , as well as the intensity of circular polarizations V indicating the difference between left-and right-circular polarizations intensities.
The linear polarization can be represented by P L ≡ Q 2 + U 2 . An arbitrary polarized state of a photon (|k 0 | 2 = |k| 2 ), propagating in theẑ-direction, is given by where linear bases |ǫ 1 and |ǫ 2 indicate the polarization states in the x-and y-directions, and θ 1,2 are initial phases. Quantum-mechanical operators in this linear bases, corresponding to Stokes parameters, are given byÎ An ensemble of photons in a general mixed state is described by a normalized density matrix ρ ij ≡ ( |ǫ i ǫ j |/trρ), and the dimensionless expectational values for Stokes parameters are given by where "tr" indicates the trace in the space of polarization states. As shown below, in a quantum field theory, the density matrix describing Stokes parameters of polarized particle is represented in the phase space (x, k) in addition to the space of polarization states. Eqs. (3)(4)(5)(6) determine the relations between Stokes parameters and the 2 × 2 density matrix ρ of photon polarization states.

III. QUANTUM BOLTZMANN EQUATION FOR STOKES PARAMETERS
We express the laser field strength F µν = ∂ µ A ν − ∂ ν A µ , and free gauge field A µ in terms of plane wave solutions in the Coulomb gauge [12], where ǫ iµ (k) are the polarization four-vectors and the index i = 1, 2, representing two transverse polarizations of a free photon with four-momentum k and k 0 = |k|, k · ǫ i = 0 and ǫ i · ǫ j = −δ ij .
The creation operators a † i (k) and annihilation operators a i (k) satisfy the canonical commutation relation The density operator describing an ensemble of free photons in the space of energy-momentum and polarization state is given byρ where ρ ij (x, k) is the general density-matrix, analogous to Eqs. (3)(4)(5)(6), in the space of polarization states with the fixed energy-momentum "k" and space-time point "x". The number operator of photons D 0 ij (k) ≡ a † i (k)a j (k) and its expectational value with respect to the density-matrix (9) is defined by The time-evolution of the number operator D 0 ij (k) is governed by the Heisenberg equation where H I is the interacting Hamiltonian of photons with other particles in the standard model.
Calculating the expectational value of both sides of Eq. (11), one arrives at the following Quantum Boltzmann Equation (QBE) for the number operator of photons [13], where we only consider the interacting Hamiltonian H 0 I (t) of photons with neutrinos. The first term on the right-handed side is a forward scattering term, and the second one is a higher-order collision term, which will be neglected. In the following, we attempt to calculate the Stokes parameter V of Eq. (6) to show that a linearly polarized laser beam acquires the component of circular polarization while it interacts with a neutrino beam.

IV. PHOTON-NEUTRINO INTERACTION
First we consider Dirac neutrinos ψ ν l interacting with charged leptons ψ l and W ± µ gauge bosons in the standard model where we omit the unitary mixing matrix which is not relevant to the following calculations.
Because neutrino masses are approximately zero, compared with their energies (E ν l ≫ m ν l ), we adopt the two-component theory for zero-mass particles to describe neutrinos (see for example Ref. [12]) where neutrino energy q 0 = |q|, and helicity or chirality states U ± (q) = −V ± (q) are eigenstates of helicity operator Σ · q/|q| and chirality operator γ 5 , (+) representing the left-handed neutrino U + (q) and right-handed anti-neutrino V + (q); (−) representing the right-handed neutrino U − (q) and left-handed anti-neutrino V − (q). The annihilation and creation operators b r (d r ) and b † r (d † r ) for corresponding states satisfy following relations, For a given momentum q and chirality state (+), there exists only two independent states of lefthanded neutrino and right-handed anti-neutrino. The chiral projector (1 − γ 5 )/2 in the interacting Hamiltonian (13) implies that only left-handed neutrinos (+) participate the weak interaction. The relationsŪ r (q)γ µ U s (q) = 2q µ δ rs , are useful for the following calculations.
In the context of standard model SU L (2) × U Y (1) for the electro-weak interactions, two Feynman diagrams representing the leading-order contribution to the interaction between photons and neutrinos are shown in Fig. 1. The leading-order interacting Hamiltonian is given by where D αβ and S F are respectively W ± µ gauge-boson and charged-lepton propagators. In Eq. (17), our notation dq ≡ d 3 q/[(2π) 3 2q 0 ], the same for dp, dp ′ and dq ′ .
Then, using the commutators of Eqs. (8) and (15), as well as the following expectation values of operators [13], where ρ ss ′ (x, q) is the local matrix density and n ν (x, q) is the local spatial density of neutrinos in the momentum state q. Thus we define the neutrino distribution function n ν (x) and the average momentumq of neutrinos as For a neutrino beam we assume n ν (x, q) ∼ exp[−|q −q|/|q|], representing the most of neutrinos carry the momentum q ≈q. Then we arrive at Moreover, using dimensional regularization and Feynman parameterizations for four-momentum integration over l, we approximately obtain where neutrino and photon energies are much smaller than W ± µ -boson mass M W , i.e., E ν , E γ << M W .
The time evolution of Stokes parameter V of linearly polarized laser-photons is determined by Eqs. (12) and (25), where high-order terms suppressed at least by 1/M 4 W have been neglected. Using Eq. (16), we obtain where the Fermi and fine-structure constants are As follows, we will consider the neutrino beam is produced by high-energy muons (µ ± ) decay, the beam angle divergence θ div ∼ m µ ± /E µ ± ≪ 1, where E µ ± (m µ ± ) is the muon energy (mass). This implies that near to the source of high-energy muons (µ ± ) decay, the momenta of neutrinos are around the averaged oneq, given by Eq.(23). In this circumstance, we approximate Eq. (27) as In the laboratory frame, we set the laser-photon momentum k in theẑ-direction (the direction of incident laser beam), polarization vectors ǫ 1 (k) in thex-direction and ǫ 2 (k) in theŷ-direction. This sketch shows the relative angles θ and φ between the laser beam directionk and neutrino beam directionq.
whereq ≡q/|q| indicates the direction of neutrino beam, and n ν (x) ≈ const. along the beam direction. On the analogy of averaged momentumq of neutrino beam, k should be understood as an averaged momentum of photons in a laser beam in the directionk ≡ k/|k|, the Stokes parameters Q(x, k), U(x, k) and V (x, k) are the functions averaged over the momentum distribution of photons in a laser pulse. As shown in Fig. 2, we select thek along theẑ-direction, and θ and φ spherical angles ofq with respect to thek direction. Then Eq. (29) becomes where in the third line we set φ = 0 in the plane ofq and k. To discuss possible experimental relevance, we rewrite Eq. (30) as where the averaged energy-flux of neutrino beamF ν (x,q) ≡ c|q| n ν (x,q). Eq. (31) represents the ratio of circular and linear polarizations of laser beam interacting with the neutrino beam for a time interval ∆t.

VI. POLARIZED COMPTON SCATTERING
It is shown [13] that the photon circular-polarization is not generated by the forward Compton scattering of linearly polarized photons on unpolarized electrons. The reason is that the contributions from left-and right-handed polarized electrons to photon circular-polarization exactly cancel each other. In the following, we show the circular polarization generated by a polarized Compton scattering: a photon beam scattering on a polarized electron beam, whose the number-density of incident left-handed electrons is not equal to the right-handed one. For the sake of simplicity, we describe the polarized electron beam by the net number-density of left-handed electrons δn L,e , and consider relativistic electrons E e ≫ m e , E e ≫ E γ and q ≫ k ∼ p, where q, p and k are the momenta of incident electrons, incident and scattered photons respectively (see Fig. 3). Following the calculations of Ref. [13], we obtain the time evaluation of the Stokes parameter V for the circular polarization of scattered photons, which is proportional to δn L,e the averaged number-density of left-handed polarized electrons, herē q is the averaged momenta of incident polarized electrons andp ≡ p/|p|. This shows that linearly polarized photons scattering on polarized electrons acquire circular polarization, analogously with photon circular-polarization generated by linearly polarized photons scattering on left-handed neutrinos.

VII. MAJORANA NEUTRINOS
We turn to calculate the circular polarization of laser-photon beam due to its scattering with Majorana neutrinos [15], which are self-conjugated Dirac neutrinos (particle and anti-particle are identical), i.e., ψ M The interaction of Majorana neutrinos with charged particles via the W ± µ in the Standard Model SU L (2) × U Y (1) can be written as [15] where ψ c l is the conjugated lepton field. The first term is due to the left-handed Majorana neutrino, analogously to Eq. (13) for the left-handed Dirac neutrino, yielding Feynman diagrams in Fig. 1.
The second term comes from the conjugated left-handed Majorana neutrino ψ M ν (x) c , yielding "conjugated" Feynman diagrams, which are all the internal lines in Fig. 1 are replaced by their conjugated lines. The contribution from "conjugated" Feynman diagrams can be obtained by substitutions [15]: As a result, in Majorana neutrino case, the interacting Hamiltonian is This difference is due to the fact that in the standard model, see Eq. (13), there is only one species of the left-handed Dirac neutrino field, whereas the left-handed Majorana neutrino field has two species for it being a self-conjugated field, see Eq. (35). As a result, in the Majorana neutrino case, the interacting Hamiltonian H 0 I of Eq. (37) has an additional "conjugated" part, compared with the interacting Hamiltonian H 0 I of Eq. (17) in the Dirac neutrino case. These two parts in Eq. (37) have the same contribution to the V -parameter, which is equal to the contribution in the Dirac neutrino case. Thus, the V -contribution (44) from the photon and Majorana neutrino interaction is twice larger than the V -contribution (31) from photon and Dirac neutrino interaction.

VIII. NEUTRINO AND LASER-PHOTON BEAMS
Our result (31) shows that in order to produce a large intensity of photon circular-polarization for possible measurements, the intensities of neutrino and laser-photon beams should be large enough. In addition the interacting time ∆t of two beams, i.e., the spatial dimension ∆d of interacting spot of two beams (∆d ≈ c∆t) should not be too small. This depends on the sizes of two beams at interacting point, assuming sin 2 θ ∼ 1. Based on the Figs. 1 or 25 in Ref. [8] for the Tokaito-Kamioka (T2K) neutrino experiment (ND280), the maximal neutrino flux ∼ 10 13 cm −2 year −1 locating at energiesĒ ν ≈ |q| ∼ 0.5 GeV, we estimate the mean energy-flux of muon neutrino beam, F ν (x,q) ≈ |q| n ν (x,q)c ∼ 10 3 − 10 4 GeV/(cm 2 s).
The beam angle divergence θ div ∼ m µ /E µ ≈ 10 −2 , where the muon energy E µ ∼Ē ν . In addition, in the neutrino experiment (ND280), we estimate the spatial and temporal intervals ∆d and ∆t of interacting spot as ∆d ∼ 2R 0 + d · θ ∼ 100 cm and ∆t ≈ ∆d/c ∼ 10 −8 sec., where R 0 is the beam radius near to the source of muon neutrinos and d = 2.8 · 10 4 cm is the distance between the muon neutrino source and interacting spot with laser photons. Using optic laser beams, laser photon energy k 0 ∼ eV, we approximately obtain the ratio of Eq. (31) where the factor sin 2 θ is the order of unit. Eq. (46) represents the result for laser and neutrino beams (pulses) interacting once only. Suppose that by using mirrors or some laser facilities the laser beam can bent N -times in its traveling path, and all laser pulses are identical without interference, so that each laser pulse can interact with the neutrino beam N -times in its path, the ratio of Eq. (46) is approximately enhanced by a factor of N , since sin 2 θ = sin 2 (π − θ) for back and forth laser beams in opposite directions. Suppose that mirrors that trap laser beams can reflect 99.999% of laser light for a narrow range of wavelengths and angles. The absorption coefficient of the mirror is then C a ∼ 10 −5 , and the intensity Q of linearly polarized laser beam is reduced to Q → Q(1 − C a ) for each reflection, we approximately In general, we express our result as where k 0 is the averaged energy of laser photons, ∆t indicates time interaction, the averaged velocity and energy flux of the left-handed polarized electron beamv e = |q|/m e < 1 andF e ≡ |q|v e δn L,e .
In addition, Ref. [14] discussed the possibility of using intense muon neutrino beams, such as those available at proposed muon colliders, interacting with high powered lasers to probe the neutrino mass. The rate of photon-neutrino scattering (R γν ∼ 1/year or 3.1×10 −8 /s) was estimated by considering dramatically short pulsed lasers with energies of up to 1.6 × 10 7 J per pulse and very short pulse durations (∼ fs), which is near to the critical intensity ∼ 10 28−29 W/cm 2 for the production of electron-positron pairs. Even in this extremal powered lasers, the total probability for photon-neutrino scattering is too small to be observed. The main reason is that the interacting rate R γν is proportional to the photon-neutrino scattering cross-section σ γν ∝ α 2 (G F m ν ) 2 , which is very small.
To compare with the photon-neutrino scattering rate R γν obtained in Ref. [14], we estimate the rate of generating circular polarization of laser photons presented in this Letter. The quantity ∆V (k) of Eq. (31) represents the number of circularly polarized photons of energy |k| = k 0 ∼ eV per unit area (cm −2 ), unit time (s −1 ) in a laser pulse. Therefore the rate of generating circular polarization of laser photons can be estimated by where τ pulse is the time duration of a laser pulse, the effective area of photon-neutrino interaction is represented by the laser-beam size σ laser being smaller than the neutrino-beam size ∆d, and the laser repetition rate f pulse is the number of laser pulses per second. To have more efficiency, we assume that laser and neutrino beams are synchronized and the f pulse is equal to the repetition rate of neutrino beam f bunch , which is the number of neutrino bunches per second. In contrast with the rate of photon-neutrino scattering R γν ∼ σ γν ∝ α 2 (G F m ν ) 2 , the rate R V of Eq. (50) linearly depends on αG F via the ∆V (k) of Eq. (31). This implies that the rate R V of Eq. (50) should be much larger than the photon-neutrino scattering rate R γν considered in Ref. [14].
Combining Eqs. (48) and (50), we obtain where N γ is the number of photons in a laser pulse As a result, with a neutrino beamF ν ∼ 10 4 GeV cm −2 s −1 and a linearly polarized laser beam of energy k 0 ∼eV and powerP laser ≃ 10MW, the rate of generating circularly polarized photons R V ∼ 1/s (∼ 3 × 10 7 /year). This rate should be large enough for observations.

IX. CONCLUSION AND REMARK.
We show and discuss the reason why a linearly polarized photon acquires its circular polarization by interacting with a neutrino is due to the fact that the neutrino is left-handed and possesses chiral gauge-coupling to gauge bosons. Calculating the ratio of linear and circular polarizations of photons interacting with either Dirac or Majorana neutrinos, we obtain that this ratio in the