Neutrino decoherence in a fermion and scalar background

We consider the decoherence effects in the propagation of neutrinos in a background composed of a scalar particle and a fermion due to the non-forward neutrino scattering processes. Using a simple model for the coupling of the form f̄RνLφ we calculate the contribution to the imaginary part of the neutrino self-energy arising from the non-forward neutrino scattering processes in such backgrounds, from which the damping terms are determined. In the case we are considering, in which the initial neutrino state is depleted but does not actually disappear (the initial neutrino transitions into a neutrino of a different flavor but does not decay into a fφ pair, for example), we associate the damping terms with decoherence effects. For this purpose we give a precise prescription to identify the decoherence terms, as used in the context of the master or Linblad equation, in terms of the damping terms we have obtained from the calculation of the imaginary part of the neutrino self-energy from the non-forward neutrino scattering processes. The results can be directly useful in the context of Dark Matter-neutrino interaction models in which the scalar and/or fermion constitute the dark-matter, and can also serve to guide the generalizations to other models and/or situations in which the decoherence effects in the propagation of neutrinos originate from the non-forward scattering processes may be important. As a guide to estimating such decoherence effects, the contributions to the absorptive part of the self-energy and the corresponding damping terms are computed explicitly in the context of the model we consider, for several limiting cases of the momentum distribution functions of the background particles.


Introduction and Summary
Several extensions of the standard electroweak theory involve the coupling of neutrinos to scalar particles (φ) and fermions (f ) of the generic formf R ν L φ. Such couplings have been considered recently in the context of Dark Matter-neutrino interactions [1,2,3,4,5,6,7]. Those interactions can produce nonstandard contributions to the neutrino index of refraction and effective potential when the neutrino propagates in a background of those particles. In Ref. [8] we considered the real part of the self-energy of a neutrino that propagates in a medium consisting of fermions and scalars, with a coupling of that form. From the self-energy, the neutrino and antineutrino effective potential and dispersion relations were then determined.
In the presence of these interactions there can also be damping terms in the neutrino effective potential and index of refraction, as a consequence of processes such as ν + φ ↔ f and ν +f ↔φ, that may become possible depending on the kinematics conditions. In Ref. [9], we extended our previous work to calculate the imaginary part (or more precisely the absorptive part) of the neutrino self-energy, in a fermion and scalar background due to thef R ν L φ interaction. From the imaginary part of the self-energy the damping terms in the effective potential and dispersion relation were obtained.
Here we note that, in addition to the effects we have mentioned, the presence of those couplings in general can induce decoherence effects in the propagation of neutrinos due to the neutrino non-forward scattering process [10,11,12,13,14]. To be more precise, here we consider various neutrino flavors (ν La ) interacting with a scalar and fermion with a coupling of the form L int = a λ afR ν La φ + h.c. (1.1) In this case, the scattering processes of the form ν a + x → ν b + x, where x = f, φ, can induce decoherence effects in the propagation of neutrinos, independently of the possible damping effects already mentioned. From the calculational point of view, the first step in our strategy is to determine the contribution of such processes to the absorptive part of the self-energy, from which we obtain the corresponding contribution to the damping matrix Γ by the usual method. However, in the present case, in which the initial neutrino state is depleted but does not actually disappear (the initial neutrino transitions into a neutrino of a different flavor but does not decay into a f φ pair, for example), the effects of the non-forward scattering processes are more properly interpreted in terms of decoherence phenomena rather than damping. The second step in our strategy is to give a precise prescription to identify the decoherence terms, as used in the context of the master or Linblad equation, in terms of the damping matrix Γ that we obtain from the calculation of the imaginary part of the neutrino self-energy due to the non-forward neutrino scattering processes. In writing Eq. (1.1) we assume the presence of only one scalar and one fermion field. Despite this simplification our work illustrates some features that can serve as a guide when considering more general cases or situations not envisioned here. They can be applied, for example, in the context of models in which sterile (ν Ls ) neutrinos have secret gauge interactions of the formν Ls γ µ ν Ls A ′ µ [15], when a sterile neutrino propagates in a background of sterile neutrinos and A ′ bosons. They can also be applied in models in which sterile neutrinos interact with the active neutrinos via coupling of the form λ aν c Rs ν La φ [3,6]. The formulas we obtain for the damping and decoherence terms can be applied in the context of such models with minor modifications. As usual, the formulas involve integrals over the momentum distribution functions of the background particles. As a guide to estimating such decoherence effects, the contributions to the absorptive part of the self-energy and the corresponding damping terms are computed explicitly in the context of the model we consider, for several limiting cases of the momentum distribution functions of the background particles.
In Section 2 we review the method we used previously to determine the dispersion relation and damping term for a single neutrino generation propagating in an f φ background, and then extend it here to the case of several neutrino generations, in particular to determine the damping matrix from the calculation of the self-energy. In Section 3 we carry out the calculation of the absorptive part of the self-energy that arises from the non-forward neutrino scattering processes. For definiteness we consider the special situation in which there are no φ scalars in the background (the heavy φ limit ), so the background consists of the f fermions and the antiparticles only. The final result in that section is the formula for the damping matrix, expressed in terms of integrals over the background fermion distribution functions and the coupling constants g a defined in Eq. (1.1). In Section 4 we formulate the interpretation of the damping matrix so determined as a decoherence effect and its relation to the Linblad equation and the stochastic evolution of the state vector [16,17,18,19,20]. The result is a well-defined formula for the "jump" operators in that context. In Section 5 the integrals involved are evaluated explicitly for different conditions of the fermion background. Our conclusions and outlook are given in Section 6, and some details of the derivations are provided in the Appendix.
2 Self-energy and the damping matrix

Dispersion relation for a single neutrino generation
In order to set down our notation and conventions it is useful to first review briefly the case of only one neutrino generation coupled in Eq. (1.1), considered in Refs. [8,9]. We denote by u µ the velocity fourvector of the background medium and by k µ the momentum of the propagating neutrino. In the background Figure 1: One-loop diagram for the neutrino thermal self-energy matrix elements in an f φ background. medium's own rest frame, 1) and in this frame we also write k µ = (ω, κ) .
In this work we consider only one background medium, which can be taken to be at rest, and therefore we adopt Eqs. (2.1) and (2.2) throughout. The dispersion relation ω( κ) and the spinor of the propagating mode are determined by solving the equation where Σ ef f is the neutrino thermal self-energy. Σ ef f can be decomposed in the form where Σ r is the dispersive part and Σ i the absorptive part, with In the context of thermal field theory where Σ 11 is the 11 element of the thermal self-energy matrix. On the other hand, Σ i is conveniently obtained from the formula where Σ 12 (k) is the 12 element of the neutrino thermal self-energy matrix, while is the fermion distribution function, written in terms of a dummy variable z, and the variable x ν is given by To the lowest order, Σ 11 and Σ 12 are determined by evaluating the diagram shown in Fig. 1. The chirality of the neutrino interactions imply that [21] Σ = V µ (ω, κ)γ µ L , (2.11) and correspondingly We have indicated explicitly that, in general, both V µ r,i are functions of ω and κ. Ordinarily we will omit those arguments but we will restore them when needed.
The results obtained in Ref. [9] are summarized as follows. Writing the neutrino and antineutrino dispersion relations in the form On the other hand, for the imaginary part, where n µ is defined in Eq. (2.17). If the correction due to the n · ∂V r (ω, κ)/∂ω in the denominator can be neglected, the formulas in Eq. (2.18) reduce to In any case, Eqs. (2.15) and (2.18), allow us to obtain the neutrino and antineutrino dispersion relation and damping from the self-energy.

Several generations -equation for the flavor spinors
Our aim here is to extend the above considerations to the case of several neutrino generations. In this case V µ r,i , as well as Σ r,i , are matrices in flavor space. As already stated, in this work we assume that the distribution functions of the background particles are isotropic. In this case V µ is a function only of k µ and u µ and no other vectors. One traditional way to take this into account is to parameterize V µ in the form (2.20) For our present purposes we find more convenient to proceed as follows. The isotropy assumption is equivalent to assume that in that frame. For completeness we note that this can be written in a general way by introducing and In what follows we adopt the conventions defined by Eqs.
where ξ (λ) is a flavor spinor, representing the amplitude in flavor space. Eq. (2.26) implies that (2.29) Substituting Eq. (2.28) in Eq. (2.3) and using Eq. (2.29), we get the following equation for the flavor spinor where we have used the fact that and we have indicated explicitly that, in general, V (u) , V (t) are functions of ω and κ [22].
Eq. (2.30) is an implicit equation that in principle determines the dispersion relations for ω (λ) ( κ) for each mode. The next step is to linearize the equation, by substituting the zeroth order solution, ω = λκ in V (u) , V (t) . Thus, Eq. (2.30) becomes Eq. (2.32) determines the positive and negative frequency dispersion relations ω (±) ( κ). According to the decomposition in Eq. (2.13), we define The neutrino Hamiltonian is identified, as usual, by associating it with the positive frequency solution; that is we set For the antineutrino, we look at the equation for In this way, the equations are or explicitly, where n µ has been defined in Eq. (2.17). In summary, for either neutrinos or antineutrinos, we have the eigenvalue equation for ξ, with H r and Γ being Hermitian matrices in flavor space, calculated in terms of the vector V µ , . (2.40) In coordinate space, this translates to the evolution equation Figure 2: Two-loop diagram for the neutrino thermal self-energy matrix element Σ 12 in an f φ background. In principle we have to consider the various thermal vertices A = 1, 2 and B = 1, 2. However, in the heavy φ limit, only the diagonal components of the φ thermal propagator are non-zero and therefore only one diagram, with A = 1 and B = 2, must be considered. In the labels referring to the various neutrino families, we use the indices a, b running over the neutrino flavors and i, j, k running over the neutrino modes with a definite dispersion relation in the medium. For simplicity of notation, we have labeled k ′ = p − p ′ + k.

Non-forward scattering terms
In the case that several neutrino flavors couple to f φ as indicated in Eq. (1.1), the damping matrix Γ receives another contribution, from the diagrams depicted in Fig. 2. From a physical point of view these diagrams correspond to contributions to the damping matrix Γ due to the various neutrino non-forward scattering processes that can occur in the presence of the background particles f φ, schematically of the form ν a + x ↔ ν b + x, where x = f, φ (and similar ones with f and/or φ crossed). This contrasts with the processes involved in the diagrams of Fig. 1, which are associated with decay type process like ν a + φ ↔ f and related ones. To distinguish the two types of contributions to Γ, we denote by Γ (1,2) the contribution the one-loop diagram (Fig. 1) and the two-loop diagram (Fig. 2), respectively. Our main observation is that Γ (2) has a structure that lends itself to a formulation as decoherence terms that in turn allows us to go beyond the evolution equation Eq. (2.41) to consider its effects. However, before going in that route we calculate explicitly Γ (2) by direct evaluation of the diagrams in Fig. 2. 3.1 Calculation of Σ 12 from Fig. 2 From Fig. 2, taking into account the sign difference between type 1 and type 2 vertices, where Recall that (Σ i ) ba is given by Eq. (2.8) and then (Γ (2) ) ba is obtained from Eq. (2.40) with (V µ i ) ba identified according to Eq. (2.12).
We will assume that m φ is larger than both the background temperature and the incoming neutrino energy so that we can work in the heavy φ limit. In this case only the diagonal elements of the thermal φ propagator are non-zero, and therefore in the vertices in Fig. 2 only the case A = 1, B = 2 has to be considered. Then using ∆ 4 Tr RiS which corresponds to the collapsed diagram shown in Fig. 3. The components of the propagator matrices are given by with n F defined in Eq. (2.9) and θ is the step function. We have defined For our purposes, it will be more convenient to use the identity and express S We now consider the propagator to use for the internal neutrino line. In principle we would use the formulas appropriate for the propagating neutrino including the background effects, taking into account the relationship between the neutrino flavor states and the mode states with definite dispersion relation. However, we adopt the perturbative approach and neglect the effect of the non-zero neutrino masses and/or dispersion relations in the calculation of Σ 12 . In this case the neutrino propagator S (ν) AB (k ′ )) cd is diagonal in flavor space, with all the elements actually being the same since all the neutrinos have the same mass (zero) and the same chemical potential. Specifically, where and We now work the product of the fermion propagators in Eq. (3.3) as follows. Using the relation which follows from Eq. (3.2), the following identity is readily derived (see Appendix A), where where We now let k ′ be an arbitrary variable, but insert the factor δ (4) (k ′ + p ′ − p − k) and integrate over k ′ . Thus, The next step is to carry out the integrals over p 0 , p ′ 0 , k ′ 0 . We do the integral over k ′ 0 first. With the help of the delta function we obtain where To arrive at Eq. (3.19) we have changed variables κ → − κ in the term with the σ ν (−k ′ ) factor, and we have defined with E defined in Eq. (3.15). Using the relations  (3.20), and the process that contributes to the ν(k) damping via Eq. (3.26). To simplify the notation we are using the shorthands shown in Eq. (3.29) for the various distribution functions.
Next we carry out the integration over p 0 , p ′ 0 in a similar way. In analogy with Eq. (3.23), we will use and similarly for p ′ µ . In Eq. (3.26) we have introduced the factors E ν,λλ ′ and Eν ,λλ ′ (with λ, λ ′ being ±), which are defined as follows, and similarly for Eν ,λλ ′ . Using Eq. (3.24) and the corresponding formulas for n F (x ′ f ) in Eq. (3.20), the explicit formulas are given in Table 1. To simplify the notation in the formulas summarized in Table 1 we have introduce the shorthands The formulas for Eν ,λλ ′ are obtained from those for E ν,λλ ′ by making the replacement f ′ ν → (1 −f ′ ν ).
Using the relations (3.30) and the analogous relations for σ ν (k ′ ), makes it evident that the matrix element can be expressed as a sum of terms of the form involving the amplitudes for the processes as well as the processes obtained by crossing f (p), f (p ′ ), ν i (k ′ ). Each term in the of terms that appear within the bracket in Eq. (3.26) correspond to one such process, and its inverse. The factors of E ν,λλ ′ , Eν ,λλ ′ incorporate the statistical effects of the background. As is well known [23], for Fermi systems the inverse reactions are inhibited as a consequence of the Pauli blocking effect, and they contribute additively to the depletion of the state. The formulas given in Table 1 reflect the fact that the contributions from the direct and the inverse process are given by the sum of their rates instead of their difference as in the bosonic case. For example, E ν,++ can be rewritten in the form which is just the sum of the statistical factors for the direct and inverse process indicated in Eq. (3.33). In similar fashion it can be verified that the terms in Eq. (3.26) and the associated E ν,λλ ′ and Eν ,λλ ′ can be identified with the various processes as indicated in Table 1. Similar identifications can be made for the antineutrino matrix elementv For some conditions, some of these processes will be kinematically forbidden and will not contribute. We now assume that the conditions are such that, for ω > 0, the only processes that are kinematically accessible are the one shown above, and the following one, (3.36) These correspond to the the first and the fourth terms, respectively, in the list of terms that appear within the bracket in Eq. (3.26). Alternatively, for ω < 0, the only kinematically accessible processes arē which correspond to the fifth and eighth terms within the bracket in Eq. (3.26). In addition we will assume that there no neutrinos or antineutrinos in the background, therefore we set f ν and fν to zero. Then, where, as we have mentioned, if ω > 0 only the first two terms in the bracket contribute, while for ω < 0 only the last two contribute.

Formula for V µ i
We now express V µ i as follows. Using Eqs. (3.5) and (3.12), It is useful to note that The remark below Eq. (3.38) is equivalent to say that V Thus, finally, putting

Formula for Γ (2)
From Eq. (2.40), remembering Eq. (3.45), . (3.49) Denoting by Γ (ν) , Γ (ν) the matrix for neutrinos or antineutrinos, respectively, explicitly using Eq. (3.48), respectively, where, for x = f,f , we define We have used Eq. (3.46) and the analogous relation between k µ and n µ , and in the expression for I we use the fact that the delta function implies that p ′ · k = p · k ′ , while for we use p · k = p ′ · k ′ . Therefore, (3.54) From Eqs. (3.51) and (3.52) we then have 4 Non-forward scattering as a decoherence effect Our main observation here is that Γ (2) , given in Eq. (3.50) by direct evaluation of the diagrams in Fig. 2, has a structure that lends itself to a formulation as decoherence terms in the context of the Linblad equation, and the notion of the stochastic evolution of the state vector [16,17,18,19,20]). Thus we will assume that, for kinematic reasons, Γ (1) is zero and that Γ (2) is the only contribution to the damping matrix. The idea then is to assume that the evolution due to the damping effects described by Γ (2) is accompanied by a stochastic evolution that cannot be described by the coherent evolution of the state vector. Let us then consider the evolution of the state vector (using a generic notation) with In an interval dt the state vector would have evolved coherently to The norm of this vector is Thus, we interpret p as the probability that the system has decayed (1 − p is the survival probability) due to the coherent (but non-Hermitian) evolution. We now assume that this coherent evolution is accompanied by stochastic processes that cause the system to "jump" from the initial state to a set of possible states, thus causing the damping.
To define the construction, suppose specifically that Γ has the form In other words, suppose that we can find a set of matrices L i such that Γ can be written in this form. As we will verify later on, this is indeed the case for Γ (2) . Let us call Therefore, from Eq. (4.5), Now we want to say that the stochastic processes cause the state vector to "jump" to any of the (normalized) state vectors with a probability π i . Of course the condition is that which we satisfy by assuming that π i = p i dt . (4.11) The main assumption is that the evolution of the system, taking into account both the coherent and stochastic evolution, is described by the density matrix (in the sense that we can use it to calculate averages of quantum expectation values) which is the Linblad equation [24]. From Eq. (3.50), it is immediately evident that Γ (2) has the form given in Eq. (4.6). Indeed, to be more precise, only one such matrix L is needed, for ℓ = ν,ν, with the identification (4.15) In summary we assert that, in the situation that Γ (1) is zero (or negligible) so that the damping matrix is given by Γ (2) , determined from Fig. 2 and which has the form given in Eq. (3.50) (under the approximations and idealizations we have made), then its effects are more effectively taken into account in the context of the evolution equation for the flavor density matrix, in this case, for neutrinos or antineutrinos, with L (ℓ) identified in Eq. (4.15). In what follows we compute the integrals involved in the expressions for γ (ν,ν) explicitly for some idealized situations, which nevertheless should serve as starting point to consider more general and/or realistic cases.

Example of calculation of integrals
Eqs. (3.54) and (3.55) serve as the basis for the calculation of the matrix L using Eq. (4.15) in a number of useful cases. For illustrative purposes and a guide to applications to realistic and/or potentially important situations, here we evaluate explicitly the integrals involved for some specific simple cases of the background conditions.
We assume that f x ≪ 1 so that we can set (1 − f x (E p ′ )) → 1. Then where The evaluation of the integrals J 1,2 is straightforward, as shown in Appendix B. Here we quote the results for the particular cases of an ultrarelativistic or a non-relativistic fermion background. Although the idealizations and approximations we have made to arrive at these formulas may limit their applicability to realistic situations, the simplicity of these results can be used as a guide and benchmarks when considering specific applications of practical interest.

Ultrarelativistic background
Specifically we assume that As shown in Appendix B, in this case where p = | p|. Then from Eq. (5.1), remembering that ω κ = κ. For a completely degenerate where p F x is the Fermi momentum, (Fermi gas) .
The Fermi momentum is given in terms of the number density f x of the background fermions by p F x = (3π 2 n x ) Using the above results in Eq. (3.55) we can consider some specific example situations. For example, for a completely degenerate f gas (and nof particles), φ (Fermi f gas) , (5.8) or, for a completely degeneratef gas (and no f particles), while for a classical gas (equal number of f andf ) (5.10)

Nonrelativistic background
Here we assume that As shown in Appendix B, in this case where is the total number density of f orf . Thus, from Eq. (3.55), (5.14)

Generalizations
As already mentioned in Section 1, the method we have followed, and the formulas we have obtained for the jump operators, can be applied with minor modifications to other model interactions of potential interest. In particular, they can be applied to study the effects of the non-forward scatering of neutrinos when they propagate through a matter background due to the standard weak interactions of the neutrinos with the electrons and nucleons. For example, consider the contribution from the electron background. In the local limit of the weak interactions, the kinematics of the diagrams involved are similar to those of the heavy φ limit that we have assumed. The corresponding jump operators would be given by formulas of the same form as those in Eqs. (3.55) and (4.15), with obvious replacements. That is, m φ → m W , while the couplings g a would have the standard weak coupling g as a common factor times another factor that is the same for ν µ,τ but different for ν e due to the charged-current interaction of the ν e with the electrons. Similarly, the integrals would involve the background electron number density, but the specific kinematic factors involved must be determined by explicit calculation. The details of the calculation would be similar to those presented above.

Conclusions and outlook
In this work we have considered the damping effects in the propagation of neutrinos in a background composed of a scalar particle and a fermion with an interaction of the form given in Eq. (1.1), due to the non-forward neutrino scattering processes. Specifically, we calculated the contribution to the imaginary part of the neutrino thermal self-energy arising from the non-forward neutrino scattering processes in such backgrounds, from which the damping matrix is determined. Since in this case the initial neutrino state is depleted but does not actually disappear we have argued that the damping matrix should be associated with decoherence effects. Following this suggestion we have given a precise prescription to determine the decoherence terms, as used in the context of the master or Linblad equation, in terms of the damping terms we have obtained from the calculation of the non-forward neutrino scattering contribution to the imaginary part of the neutrino self-energy. The main result is a well-defined formula for the "jump" operators in that context, expressed in terms of integrals over the background fermion distribution functions and the couplings constants of the interaction of the neutrinos with the background particles in the model we consider. The results can be useful in the context of Dark Matter-neutrino interaction models in which the scalar and/or fermion constitute the dark-matter, and can also serve to guide the application to other models and/or situations that have been considered recently using the Linblad equation (e.g., [10]) in which the decoherence effects in the propagation of neutrinos may be important. For reference and guidance purposes we have evaluated the integrals involved explicitly for some conditions of the background. Despite those simplifications the results illustrate some features that can serve as a guide when considering more general cases or situations not envisioned here. The work of S. S. is partially supported by DGAPA-UNAM (Mexico) Project No. IN103019.
A Derivation of Eq. (3.14) Here we show the details leading to Eq. (3.14). We will use the fact that the x ′ s satisfy We then have Using e x = 1 n F (x) − 1 (A. 3) we now work the first factor, , (A. 4) and using this in Eq. (A.2) we get Eq. (3.14).
B Calculation of integrals J 1,2 in Eq. (5.2) Since J 1,2 are a scalar integrals, we choose to do the integration in the frame in which p µ = (m f , 0) (the lab frame). We label the quantities in that frame with an asterisk, k µ = (ω * k , k * ) and similarly for k ′µ , and therefore and where the requirement that −1 ≤ cos θ * k ′ ≤ 1 implies For J 2 we proceed similarly, with the replacement p · k ′ → p · k = m f ω * κ in the integrand, and thus, In order to use Eqs. (B.3) and (B.5) in Eq. (5.1), we express ω * ′ min and ω * ′ max in terms of E p and | p| by means of the relation with v p = | p|/E p . This allows the angular integration in Eq. (5.1) to be carried out in straightforward fashion, leaving only the integration over E p , which depends on the distribution function, to be performed. As usual we can consider special cases for illustrative purposes.

B.2 Nonrelativistic background
Here we assume that ω κ ≫ m f ≫ T .