A New Spin on Neutrino Quantum Kinetics

Recent studies have demonstrated that in anisotropic environments a coherent spin-flip term arises in the Quantum Kinetic Equations (QKEs) which govern the evolution of neutrino flavor and spin in hot and dense media. This term can mediate neutrino-antineutrino transformation for Majorana neutrinos and active-sterile transformation for Dirac neutrinos. We discuss the physical origin of the coherent spin-flip term and provide explicit expressions for the QKEs in a two-flavor model with spherical geometry. In this context, we demonstrate that coherent neutrino spin transformation depends on the absolute neutrino mass and Majorana phases.

We present and discuss the quantum kinetic equations (QKEs) which govern neutrino flavor and spin evolution in hot, dense, and anisotropic media. A novel feature of these QKEs is the presence of a coherent spin-flip term which can mediate neutrino-antineutrino transformation for Majorana neutrinos and active-sterile transformation for Dirac neutrinos. We provide an alternative derivation of this term based on a standard effective Hamiltonian.
PACS numbers: 14.60.Pq, 97.60.Bw, 26.50.+x, 13.15.+g In this letter we present in compact and general form the quantum kinetic equations (QKEs) describing the evolution of neutrinos in hot and dense media, accounting for kinetic, flavor, and the often neglected spin degrees of freedom [1]. QKEs are the essential tool to obtain a complete description of neutrino transport in the early universe, core collapse supernovae, and compact object mergers, valid before, during, and after the neutrino decoupling epoch (region). A self-consistent treatment of neutrino transport is highly relevant because in such environments neutrinos carry a significant fraction of the energy and entropy, and through their flavor-and energydependent weak interactions play a key role in setting the neutron-to-proton ratio [2], a critical input for the nucleosynthesis process. In this letter we emphasize the physical content of the QKEs and discuss in simple terms the need to include spin degrees of freedom, the origin of the novel spin mixing term, and the connection with the more familiar spin-(flavor) oscillations induced by neutrino magnetic moments in a magnetic field.
Prior to Ref.
[1], state-of-the-art QKEs coupling flavor and kinetic degrees of freedom were developed for isotropic environments in Refs. [3][4][5][6][7][8]. Some of these form the basis for studies of neutrino mixing in the early universe, for example Refs. [9][10][11]. Within the relativistic many-body framework of Refs. [3,4] the "density matrix" is identified with the ensemble average of generalized number operators f αβ ( p) ∼ a α ( p) † a β ( p) , where a † α ( p) represents the creation operator of a left-handed massless neutrino of momentum p and flavor α. The QKEs are then derived treating masses and weak interactions as a perturbation. These QKEs encode the standard coherent evolution (with vacuum, medium, and self-interaction effective Hamiltonian) and display a non-abelian matrix structure for the inelastic collision term [3,4]. Even within the coherent regime (in which inelastic collisions are neglected), in environments with high neutrino density the physics can be very rich and non-trivial effects have been discovered, such as collective neutrino oscillations (see [12] and references therein).
In Ref.
[1] the QKEs describing the evolution of Majorana neutrinos were derived using field-theoretic meth-ods. These QKEs generalize the work of Ref. [3] in two respects: (i) They include spin degrees of freedom and allow for general anisotropy. As we will show later, this provides a potentially interesting "handle" to distinguish Dirac from Majorana neutrinos. (ii) They include effects up to second order in small ratios of scales characterizing the neutrino environments we are interested in. Specifically, we treat neutrino masses, masssplitting, and matter potentials induced by forward scattering, as well as external gradients as much smaller than the typical neutrino energy scale E, set by the temperature or chemical potential: namely The inelastic scattering can also be characterized by a potential Σ inelastic ∼ Σ 2 forward which we therefore power-count as Σ inelastic /E ∼ O( 2 ). This power-counting is tantamount to the statement that physical quantities vary slowly on the scale of the neutrino de Broglie wavelength.
In the most general terms a neutrino ensemble is described by the set of all 2n-field Green's functions, encoding n-particle correlations. These obey coupled integrodifferential equations, equivalent to the BBGKY equations [13]. As discussed in Refs. [1,3], for weakly interacting neutrinos (Σ/E ∼ O( , 2 )) the set of coupled equations can be truncated by using perturbation theory to express all higher order Green's functions in terms of the two-point functions. In this case the neutrino ensemble is characterized by the full set of oneparticle correlations. 2 One-particle states of massive Dirac neutrinos and antineutrinos are specified by the three-momentum p, the helicity h ∈ {L, R}, and the family label i (for eigenstates of mass m i ), with corresponding annihilation operators a i, p,h and b j, p,h satisfying the canonical anti-commutation relations {a i, p,h , a † j, p ,h } = (2π) 3 2 ω i ( p) δ hh δ ij δ (3) ( p − p ), etc., where ω i ( p) = p 2 + m 2 i . Then, the ensemble is specified by the ma- where . . . denotes the ensemble average and the normalization factor can be chosen as n ij = 2ω i ω j /(ω i + ω j ). 3 For inhomogeneous backgrounds, the density matrices depend also on the space-time label, denoted by x in what follows. Despite the intimidating index structure, the physical meaning of the generalized density matrices f ij hh ( p) and f ij hh ( p) is dictated by simple quantum mechanical considerations: the diagonal entries f ii hh ( p) represent the occupation numbers of neutrinos of mass m i , momentum p, and helicity h; the off diagonal elements f ij hh ( p) represent quantum coherence of states of same helicity and different mass (familiar in the context of neutrino oscillations); f ii hh ( p) represent coherence of states of different helicity and same mass, and finally f ij hh ( p) represent coherence between states of different helicity and mass.
In summary, ensembles of neutrinos and anti-neutrinos are described by the 2n f × 2n f matrices, where we have suppressed the generation indices (each block f hh is a square n f × n f matrix). For Dirac neutrinos, one needs both F andF , with f LL andf RR describing the occupation of active states. For Majorana neutrinos, one can choose the phases so that a i ( p, h) = b i ( p, h) and therefore f hh =f T hh (with transposition acting on flavor indices). Therefore in this case the dynamics is specified by f ≡ f LL ,f ≡f RR = f T RR , and φ = f LR and one needs evolution equations only for the matrix F [1]: Strictly speaking. the above discussion in terms of creation and annihilation operators makes sense only within the mass eigenstate basis [17]. One can still define "flavor basis" density matrices f αβ in terms of the mass-basis f ij as f αβ = U αi f ij U * βj , where U is the unitary transformation ν α = U αi ν i that puts the inverse neutrino propagator in diagonal form. While the QKEs can be written in any basis, we present results below in the "flavor" basis.
The description in terms of creation and annihilation operators presented so far has a simple counterpart in the QFT approach of Ref. [1]. In that language, the basic dynamical object is the Wigner Transform G ij (p, x) of the neutrino statistical two-point function G ij (x, r) ≡ 1/2 [ν i (x + r/2),ν j (x − r/2)] (i.e. Fourier transform w.r.t. the variable r). The Wigner Function G ij (p, x) has sixteen spinor components (scalar, pseudoscalar, vector, axial-vector, tensor). For ultra-relativistic neutrinos of momentum p, it is useful to express all Lorentz tensors in terms of a basis formed by two light-like four-vectors n µ (p) = (1,p) andn µ (p) = (1, −p) (n · n =n ·n = 0, n ·n = 2) and two transverse four vectors where P L,R ≡ (1 ∓ γ 5 )/2, x ± ≡ x 1 ± ix 2 , and the upper (lower) indices refer to Φ (Φ † ). These components can be collected in a 2n f × 2n f matrix In the free theory, the positive and negative frequency integrals ofF coincide with particle and antiparticle density matrices of Eq. (3) defined in terms of ensemble averages of creation and annihilation operators: In the interacting theory, we take the above equations as definitions of neutrino and antineutrino density matrices. After appropriate spinor projections and integration over positive and negative p 0 , the equations of motion for G ij (p, x) lead to QKEs for F ( p, x) andF ( p, x). A detailed derivation is given in Ref.
[1]. Keeping terms up to O( 2 ) in the power counting discussed in the introduction, the QKEs take the compact 2n f × 2n f form: The differential operator on the left-hand side generalizes the "Vlasov" term. The first term on the right-hand side controls coherent evolution due to mass and forward scattering and generalizes the standard MSW [18][19][20]. Finally, the second term encodes inelastic collisions and generalizes the standard Boltzmann collision term used in supernova neutrino analyses [21][22][23][24][25][26][27][28][29]. We now describe in detail each term. The Vlasov and commutator term depend only on the neutrino mass matrix m and the potential induced by forward scattering on matter and other neutrinos. In the field-theory approach, forward scattering is encoded in the one-loop diagrams of Fig. 1. The self-energy Σ(x) itself has non-trivial spinor structure. The components entering the QKEs are [1] which can be arranged in the 2n f × 2n f structure These potentials can be further projected along the basis vectors: with light-like component Σ κ ≡ n · Σ approximately along the neutrino trajectory; and space-like component Σ i ≡ x i · Σ, roughly transverse to the neutrino trajectory.
With this notation in hand, and further defining ∂ κ ≡ n · ∂, ∂ i ≡ x i · ∂, the generalized Vlasov operators are Using ω ± = | p| ± Σ κ for the neutrino (+) and antineutrino (−) energies, the differential operators D and D are simply The Hamiltonian-like operators controlling the coherent evolution are given by with where Σ ± L,R ≡ 1/2 (Σ 1 L,R ± iΣ 2 L,R ).H L,R can be obtained from H L,R by flipping the sign of the entire term multiplying 1/(2| p|).   2: Feynman graphs contributing to Π(p, x). Notation as in Fig. 1 term H m complete the set O( 2 ) terms, and can be as important as m 2 /| p| in supernova environments.
Finally, the collision terms in Eq. (10) are The 2n f × 2n f gain and loss potentials Π ± can be extracted from a calculation of the two-loop self-energies of Fig. 2, after appropriate spinor projections and integrations over positive and negative p 0 . The entries of this matrix can be expressed in terms of neutrino density matrices and distribution functions of the medium particles (electrons, etc.). The contribution to ν − ν scattering neglecting spin coherence is given in Ref.
[1], while a full analysis including scattering off neutrinos, electrons, and low-density nucleons will be presented elsewhere [30].
Here, we simply remark that the collision term has a nondiagonal matrix structure in both flavor [3,4] and spin space [1], which is often missed in heuristic treatments of the QKEs [31]. We now elaborate on a qualitatively new effect captured by the QKEs in Eq. (10), which is the possibility of coherent conversion between helicity states through the n f × n f off-diagonal block H m of the full 2n f × 2n f hamiltonian H. Note that Eq. (18) shows that H m is non-zero as long as: (i) The vector potentials Σ µ L,R have components transverse to the test neutrino momentum, which requires an anisotropic environment such as the supernova envelope; (ii) Σ + R,L contain an axial component such as in the Standard Model (SM), so that Eq. (18) does not trivially vanish.
An important feature of the new term is that it induces qualitatively different effects for Dirac and Majorana neutrinos. In the Dirac case, H m converts active left-handed neutrinos to sterile right-handed states. On the other hand, in the Majorana case H m enables conversion of neutrinos into antineutrinos.
Given the potentially high impact of the spin-flip term, here we sketch an alternative way to derive it, within a standard 2n f -level effective Hamiltonian approach to fermion mixing. The effective hamiltonian H can be identified by computing amplitudes between massless states labeled by momentum p, helicity h ∈ {L, R}, and flavor i ∈ {1, ..., n f }, namely The amplitudes are computed in perturbation theory, with interaction Lagrangian (we suppress flavor indices) where m is the mass matrix and Σ µ L,R are the vector potentials arising from interactions with the medium (Fig. 1). For example, up to O( 2 ) corrections proportional to G F mf LR , the neutrino contribution from the left-diagram of Fig. 1 is Matter fields, such as electrons, give similar contributions, with different overall coefficients and flavor structure. From Eq. (23) it is clear that in general the potential has space-like components. Even in the simplest "bulb" model [32] for supernovae one has Σ R (x) = 0, and neutrinos emerging from the neutrino sphere non-radially will see a non-zero transverse component from Σ R (x).
To first order in perturbation theory, consistent with Eqs. (16), (17), and (18), we find H LL = n(p) · Σ R , H RR = n(p) · Σ L , and H LR,RL = 0. This result encodes the known medium birefringence effect [33]: from parity-violating interactions (Σ L = Σ R ) left-handed and right-handed states of momentum p have different energies. Defining the axial potential as Σ µ A ≡ Σ µ L − Σ µ R , the energy splitting is proportional to Σ 0 A − Σ A ·p. To second order in perturbations, keeping both m 2 and m × Σ L,R terms, we again reproduce Eqs. (16), (17), and (18). In the one-flavor limit H LR takes the simple form where x + = x 1 + i x 2 ( x i are the space-like components of x µ i (p)), so that the mixing hamiltonian is proportional to the components of Σ A perpendicular to p.
In summary, SM interactions in a non-isotropic background induce a coupling of the neutrino axial current to an axial-vector potential Σ µ (22)). Σ 0 A induces the well known birefringence effect. The space-like potential Σ A has a twofold effect: (i) its component Σ A ·p parallel to the neutrino propagation gives an additional contribution to the energy splitting of L and R states; (ii) its component Σ A · x + (p) transverse to p induces mixing of the L and R states. These effects are flavor dependent, as Σ µ A carries flavor indices.
The Majorana case is obtained by replacing ν R → ν c L , and setting to zero the diagonal elements µ ii ν . Assuming typical magnetic fields in a supernova envelope (B ∼ 10 10−12 G) and Majorana transition magnetic moments a factor of 100 larger than the SM values, so that µ ij ν B(r) ∼ 10 −18 eV (50km/r) 2 , Refs [37,38] find that magnetic spin-flip transitions lead to significant effects on collective neutrino oscillations in supernovae. A naive estimate based on Σ ∼ √ 2G F (n ν − nν), with net neutrino density n ν − nν, suggests that H m ≥ 10 −18 eV at r ≤ 100 km (for m ν = 0.01 eV and | p| = 10 MeV), in the same ballpark as the magnetic term. While these estimates are rough since they ignore the flavor structure and the effects of geometry, combined with the results of Refs. [37,38], they nevertheless suggest potential implications for supernova neutrinos.
In this letter we have presented in compact form the QKEs which govern the evolution of the flavor and spin states of both Majorana and Dirac neutrinos propagating in hot and dense media. We also have discussed the physical origin and meaning of the various terms in the QKEs, and sketched a heuristic derivation of the spinflip term H m to elucidate and complement the formal derivation of this in Ref.
[1]. Novelly, our QKEs can treat environments which have anisotropy in either or both of the matter or neutrino fields. Anisotropy along these lines is the origin of H m -mediated spin-flip and, hence, neutrino-antineutrino inter-conversion. Compact object environments like those in core collapse supernovae or binary neutron star mergers can be significantly anisotropic where neutrino interactions are important for dynamics or nucleosynthesis. In turn, these interactions and their effects can be sensitive to the ratios of ν e 's andν e 's, and this highlights the importance of understanding and modeling neutrino spin-flip if we are to truly understand these astrophysical phenomena.
This work was supported in part by NSF grant PHY-1307372 at UCSD, by the LDRD Program at LANL, and by the University of California Office of the President and the UC HIPACC collaboration. We would also like to acknowledge support from the DOE/LANL Topical Collaboration. We thank J. Carlson, J. F. Cherry, and S. Reddy for discussions.