#  Taming the pinch singularities in the two-loop neutrino self-energy in a
medium

José F<EMAIL_ADDRESS>
Laboratory of Theoretical Physics, Department of Physics
University of Puerto Rico, Río Piedras, Puerto Rico 00936
Sarira<EMAIL_ADDRESS>
Instituto de Ciencias Nucleares
Universidad Nacional Autónoma de Mexico
Circuito Exterior, C. U.
A. Postal 70-543, 04510 Mexico DF, Mexico

###### Abstract

We consider the calculation of the thermal self-energy of a neutrino that
propagates in a medium composed of fermions and scalars interacting via a
Yukawa-type coupling, in the case that the neutrino energy is much larger than
the fermion and scalar masses, as well as the temperature and chemical
potentials of the background. In this kinematic regime the one-loop
contribution to the imaginary part of the self-energy is negligible. We
consider the two-loop contribution and we encounter the so-called pinch
singularities which are known to arise in higher loop self-energy calculations
in Thermal Field Theory. With a judicious use of the properties and
parametrizations of the thermal propagators the singularities are treated
effectively and actually disappear. From the imaginary part of the self-
energy, we obtain a precise formula for the damping matrix expressed in terms
of integrals over the background particle distributions. The formulas predict
a specific dependence of the damping terms on the neutrino energy, depending
on the background conditions. For guidance in estimating the effects in
specific contexts, we compute the damping terms for several limiting cases of
the momentum distribution functions of the background particles. We discuss
briefly the connection between the results of our calculations for the damping
matrix and the decoherence effects described in terms of the Lindblad
equation.

##  1 Introduction and Summary

In several models and extensions of the standard electro-weak theory the
neutrinos interact with scalar particles ($\phi$) and fermions ($f$) via a
coupling of the form

$L_{int}=\sum_{a}g_{a}\bar{f}_{R}\nu_{La}\phi+h.c\,.$ (1.1)

For definiteness we are assuming the presence of only one $f$ and $\phi$,
while the indices $a,b,c,...$ label the neutrino flavors. Those interactions
produce nonstandard contributions to the neutrino index of refraction and
effective potential when the neutrino propagates in a background of those
particles. Couplings of this form have been considered recently in the context
of Dark Matter-neutrino interactions[1, 2, 3, 4, 5, 6, 7, 8, 9]. Similar
effects occur due to neutrino-neutrino-scalar interactions of the form
$\bar{\nu}^{c}_{Rb}\nu_{La}\phi$ when a neutrino propagates in a neutrino
background. This can occur in the environment of a supernova, where the
neutrino-neutrino interactions lead to the collective neutrino oscillations
and related phenomena (see for example Refs. [10] and [11] and the works cited
therein), and it can also occur in the hot plasma of the Early Universe before
the neutrinos decouple[12, 13].

In previous works we have presented various calculations related to the
propagation of neutrinos in that kind of background[14, 15, 16]. In Ref. [14]
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 the form
given in Eq. (1.1). We calculated the real part (or more precisely the
dispersive part) of the neutrino thermal self-energy, denoted by $\Sigma_{r}$,
from which the dispersion relation and effective potential are determined.
Those interactions can also induce processes such as $\nu+\phi\leftrightarrow
f$ and $\nu+\bar{f}\leftrightarrow\bar{\phi}$, depending on the kinematic
conditions, that produce damping terms in the neutrino dispersion relation and
index of refraction. Thus in Ref. [15], we continued our work to calculate the
imaginary part (or more precisely the absorptive part) of the neutrino thermal
self-energy, denoted by $\Sigma_{i}$, in a scalar and fermion background due
to the $\bar{f}_{R}\nu_{L}\phi$ interaction. From $\Sigma_{i}$ the
corresponding contribution to the damping matrix $\Gamma$ in the dispersion
relation were obtained. The calculations in Ref. [15] were based on the one-
loop diagram for the neutrino self-energy.

In Ref. [16] we noted that those couplings can induce decoherence effects, of
the form discussed in recent works [17, 18, 19, 20, 21], due to the neutrino
non-forward scattering process $\nu_{a}+x\rightarrow\nu_{b}+x$, where
$x=f,\phi$. As observed in Ref. [16], the contribution to $\Gamma$ due to
these processes can be determined from the two-loop calculation of
$\Sigma_{i}$. Thus, in that reference we performed the two-loop calculation of
$\Sigma_{i}$ and $\Gamma$ or the case in which the background contains only
the fermions $f$, assuming that the $\phi$ particle is heavy enough and the
conditions are such that there are no $\phi$ particles in the background.
Under those conditions, the two-loop contribution to $\Gamma$ is the relevant
one since the two-body processes that contribute in one-loop are kinematically
forbidden.

The present work is a continuation of that previous work. Here we consider the
situation in which both $f$ and $\phi$ may be present in the background. We
are particularly interested in the kinematic regime

$\kappa>m_{\phi},m_{f},T\,,$ (1.2)

where $\kappa$ is the neutrino momentum and $T$ the background temperature.
That is, both $f$ and $\phi$ are relatively light compared to the neutrino
energy. We refer to this as the _light background_. It is the kinematic regime
that is relevant in the context of the possible existence of light scalars as
dark-matter and the effects they may have on neutrino experiments, that has
been explored in the recent literature[22, 23]. The results can be useful also
for the studies of the environmental decoherence effects in long baseline
neutrino oscillation experiments that have been carried out recently[24, 25,
26, 27]. Again, in this kinematic regime the two-body processes that
contribute in one-loop are inhibited and the two-loop contribution is the
relevant one.

Apart from the relevance for the applications already mentioned, from a
calculational point of view the present calculation has a technical merit.
There is one important technical issue that shows up in the kinematic regime
we are considering in the present case and those we considered previously. The
two-loop diagrams for the self-energy, from which the damping matrix is
determined, suffer from the so-called _pinch-singularities_[28]. These arise
from the fact that in the present case some of the diagrams contain a product
of two thermal propagators with the same momentum. Since the thermal
propagators involve the on-shell delta functions, such products are ill-
defined. As we show, by a judicious use of the properties and parametrizations
of the thermal propagators, the expressions for the diagrams can be rearranged
such that the pinch singularities are absent in the final expressions,
allowing a straightforward evaluation of the self-energy and whence the
damping terms. While the conventional wisdom is that indeed such singularities
actually disappear, our calculations provide an explicit proof of that fact in
a concrete and non-trivial example that can be generalized to other
calculations.

The final results are well defined formulas for the damping terms in the
neutrino dispersion relation (or effective potential) in terms of the model
parameters (i.e., couplings $g_{a}$, masses $m_{f,\phi}$) and the enviromental
parameters (e.g. temperature). In practical applications, the possible values
of all the parameters involved vary significantly depending on the context,
i.e., astrophysical, cosmological or neutrino oscillations. For example, in
the context of a supernova (such as SN1987A) the neutrino interactions with
$\phi$ as a cold dark matter candidate can have effects on the observed
neutrino flux for $m_{\phi},T\sim\mbox{a few}$ MeV[1]. But these, and similar
considerations in other contexts, depend on the particle physics model as
well. Thus, for example, while we concentrate here on the calculation
involving the $L_{int}$ interaction term, in isolation from the Standard Model
interactions, the two-loop diagrams in some models may involve the standard
particles and/or other non-standard gauge boson interactions as well.
Nevertheless, subject to the limitation of the _light background_ condition
stated above, the particular results we obtain for the damping terms can be
used in the context of many such models and conditions, and in fact the method
to treat the pinch singularities is applicable to those more general cases as
well.

In summary, our plan is as follows. In Section 2 we summarize the framework in
which we carry out the calculations. There we explain that, while the
effective potential is determined from the one-loop diagram for the self-
energy, in the kinematic regime we consider [Eq. (1.2)] the damping is
determined form the two-loop diagrams. In Section 3 we calculate the
dispersive part of the self-energy and determine the effective potential. In
Section 4 we consider the calculation of the two-loop contribution to the
absorptive part of the self-energy, from which the damping matrix is
determined. There we indicate the problem of the pinch singularities, and
present our treatment to resolve it. The net result, summarized in Section
4.4, is the set of formulas for the two-loop contributions to the absorptive
part of the self-energy, free from the singularities. In Section 5 we evaluate
explicitly the corresponding expressions for the damping matrix in the case of
a scalar background under various conditions, and indicate the path to
generalize such calculations to consider more complicated backgrounds. There
we also discuss briefly the connection between the damping matrix thus
determined and the decoherence described in terms of the Lindblad equation.
Finally Section 6 has our conclusions.

##  2 Preliminaries - effective potential and the damping matrix

To be self-contained we summarize the following material borrowing from Ref.
[16]. We denote by $k^{\mu}$ the momentum four-vector of the propagating
neutrino, and as usual we denote by $u^{\mu}$ the velocity four-vector of the
background medium. In the background medium’s own rest frame, it takes the
form

$u^{\mu}=(1,\vec{0})\,,$ (2.1)

and in this frame we write

$k^{\mu}=(\omega,\vec{\kappa})\,.$ (2.2)

Since we are considering only one background medium, it can be taken to be at
rest and therefore we adopt Eqs. (2.1) and (2.2) throughout.

Let us consider first the case of one neutrino propagating in the medium. The
dispersion relation and the spinor of the propagating mode are determined by
solving the equation

$\left(k\hbox to0.0pt{\hss/}-\Sigma_{eff}\right)\psi_{L}(k)=0\,,$ (2.3)

where $\Sigma_{eff}$ is the neutrino thermal self-energy. It can be decomposed
in the form

$\Sigma_{eff}=\Sigma_{r}+i\Sigma_{i}\,,$ (2.4)

where $\Sigma_{r,i}$ are the dispersive and absorptive parts,

$\displaystyle\Sigma_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\Sigma_{eff}+\bar{\Sigma}_{eff}\right)\,,$
$\displaystyle\Sigma_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2i}\left(\Sigma_{eff}-\bar{\Sigma}_{eff}\right)\,,$
(2.5)

respectively, with

$\bar{\Sigma}_{eff}=\gamma^{0}\Sigma^{\dagger}_{eff}\gamma^{0}\,.$ (2.6)

In the context of thermal field theory $\Sigma_{r}$ is given in terms of the
$11$ element of the thermal self-energy matrix by

$\Sigma_{r}=\Sigma_{11r}\equiv\frac{1}{2}(\Sigma_{11}+\overline{\Sigma}_{11})\,.$
(2.7)

On the other hand, $\Sigma_{i}$ is more conveniently determined in terms of
the $12$ element of the neutrino thermal self-energy matrix by the formula

$\Sigma_{i}=\frac{\Sigma_{12}}{2in_{F}(x_{\nu})}\,.$ (2.8)

Here

$n_{F}(z)=\frac{1}{e^{z}+1}\,,$ (2.9)

is the fermion distribution function, written in terms of a dummy variable
$z$, and the variable $x_{\nu}(k)$ is given by

$x_{\nu}(k)=\beta k\cdot u-\alpha_{\nu}\,,$ (2.10)

where $T=1/\beta$ is the temperature and $\alpha_{\nu}$ is the neutrino
chemical potential.

The chirality of the neutrino interactions imply that333In a strict sense this
is correct in the massless neutrino limit, which is valid in practice in the
approximation that the neutrino mass is neglected in the calculation of the
relevant diagrams for the self-energy.

$\Sigma_{eff}=V^{\mu}\gamma_{\mu}L\,.$ (2.11)

Here and below we use the notation $L$ and $R$ for the left and right chiral
projection matrices $L,R=\frac{1}{2}(1\mp\gamma_{5})$, respectively.
Corresponding to the decomposition in Eq. (2.4) we also write

$V^{\mu}=V^{\mu}_{r}+iV^{\mu}_{i}\,.$ (2.12)

and

$\Sigma_{r,i}=V^{\mu}_{r,i}\gamma_{\mu}L\,.$ (2.13)

In general $V^{\mu}_{r,i}$ are functions of $\omega$ and $\vec{\kappa}$. We
omit those arguments ordinarily but we will restore them when needed.

Writing the neutrino and antineutrino dispersion relations in the form

$\omega^{(\nu,\bar{\nu})}(\kappa)=\omega^{(\nu,\bar{\nu})}_{r}(\kappa)-\frac{i\gamma^{(\nu,\bar{\nu})}(\kappa)}{2}\,,$
(2.14)

the solution of Eq. (2.3) gives

$\omega^{(\nu,\bar{\nu})}_{r}=\kappa+V^{(\nu,\bar{\nu})}_{eff}$ (2.15)

where $V^{(\nu,\bar{\nu})}_{eff}$ are the effective potentials

$\displaystyle V^{(\nu)}_{eff}$ $\displaystyle=$ $\displaystyle n\cdot
V_{r}(\kappa,\vec{\kappa})=V^{0}_{r}(\kappa,\vec{\kappa})-\hat{\kappa}\cdot\vec{V}_{r}(\kappa,\vec{\kappa})\,,$
$\displaystyle V^{(\bar{\nu})}_{eff}$ $\displaystyle=$ $\displaystyle-n\cdot
V_{r}(-\kappa,-\vec{\kappa})=-V^{0}_{r}(-\kappa,-\vec{\kappa})+\hat{\kappa}\cdot\vec{V}_{r}(-\kappa,-\vec{\kappa})\,,$
(2.16)

with

$n^{\mu}=(1,\hat{\kappa})\,.$ (2.17)

On the other hand, for the imaginary part,

$\displaystyle-\frac{\gamma^{(\nu)}(\vec{\kappa})}{2}$ $\displaystyle=$
$\displaystyle\frac{n\cdot
V_{i}(\kappa,\vec{\kappa})}{1-n\cdot\left.\frac{\partial
V_{r}(\omega,\vec{\kappa})}{\partial\omega}\right|_{\omega=\kappa}}\,,$
$\displaystyle-\frac{\gamma^{(\bar{\nu})}(\vec{\kappa})}{2}$ $\displaystyle=$
$\displaystyle\frac{n\cdot
V_{i}(-\kappa,-\vec{\kappa})}{1-n\cdot\left.\frac{\partial
V_{r}(\omega,-\vec{\kappa})}{\partial\omega}\right|_{\omega=-\kappa}}\,,$
(2.18)

where $n^{\mu}$ is defined in Eq. (2.17). We will retain only the dominant
contribution to $n\cdot V_{i}$ in the numerator, which in our case is the two-
loop term as we argue below. Then to leading order the formulas in Eq. (2)
reduce to

$\displaystyle-\frac{\gamma^{(\nu)}(\vec{\kappa})}{2}$ $\displaystyle=$
$\displaystyle n\cdot V_{i}(\kappa,\vec{\kappa})\,,$
$\displaystyle-\frac{\gamma^{(\bar{\nu})}(\vec{\kappa})}{2}$ $\displaystyle=$
$\displaystyle n\cdot V_{i}(-\kappa,-\vec{\kappa})\,.$ (2.19)

neglecting the correction due to the $n\cdot\partial
V_{r}(\omega,\vec{\kappa})/\partial\omega$ term in the denominator.

In the case of various neutrino flavors, the vector $V^{\mu}$ in Eq. (2.11) is
a matrix in the neutrino flavor space. As shown in Ref. [16], the
generalization of the above discussion is that the dispersion relations of the
propagating modes are determined by solving the following eigenvalue equation
in flavor-space,

$\left(H_{r}-i\frac{\Gamma}{2}\right)\xi=\omega\xi\,,$ (2.20)

where $H_{r}$ and $\Gamma$ are Hermitian matrices in flavor space given by

$\displaystyle H_{r}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\kappa+n\cdot
V_{r}(\kappa,\vec{\kappa})&(\nu)\\\ \kappa-n\cdot
V^{\ast}_{r}(-\kappa,-\vec{\kappa})&(\bar{\nu})\,,\end{array}\right.$ (2.23)
$\displaystyle-\frac{1}{2}\Gamma$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}n\cdot
V_{i}(\kappa,\vec{\kappa})&(\nu)\\\ n\cdot
V^{\ast}_{i}(-\kappa,-\vec{\kappa})&(\bar{\nu})\,.\end{array}\right.$ (2.26)

In coordinate space, this translates to the evolution equation

$i\partial_{t}\xi(t)=\left(H_{r}-i\frac{\Gamma}{2}\right)\xi(t)\,.$ (2.27)

We refer to $\Gamma$ as the damping matrix, and to its elements as the damping
terms. Our purpose in this work is to determine the contribution to $H_{r}$,
and specially $\Gamma$, due to the presence of the _light background_.

The lowest order diagram is shown in Fig. 1. From that diagram we obtain
$\Sigma_{11}$ and $\Sigma_{12}$, and whence the dispersive and absorptive
parts $\Sigma_{r,i}$ by means of Eqs. (2.7) and (2.8). The corresponding one-
loop contribution to $V_{r}$ and $\Gamma$ are then obtained from Eq. (2.23).

Figure 1:  One-loop diagram for the neutrino self-energy matrix
$(\Sigma_{AB})_{ba}$ in a background of fermions $f$ and scalars $\phi$. $a,b$
are neutrino flavor indices while $A$ and $B$ label the thermal vertices, that
can take the values 1 or 2.

However, as in the cases discussed in Refs. [16] and [29], the one-loop
contribution to $\Sigma_{12}$ is negligible in this case also. The reason is
that such contributions arise from the two-body neutrino processes such as
$\nu+\phi\leftrightarrow f$, which are inhibited by the kinematics in the
regime we are considering [i.e, Eq. (1.2)].

To be more specific, the one-loop damping term is due to real processes like

$\displaystyle(A)$ $\displaystyle\nu+\bar{f}$
$\displaystyle\leftrightarrow\bar{\phi}\quad\mbox{(if $m_{\phi}>m_{f}$)}$
$\displaystyle(B)$ $\displaystyle\nu+\phi$ $\displaystyle\leftrightarrow
f\quad\mbox{(if $m_{f}>m_{\phi}$)}\,.$ (2.28)

The calculation of the one-loop damping terms for all such conditions was
carried out in Ref. [15]. Let us consider (A). As shown in that reference, the
damping is maximum for values of the neutrino momentum

$\displaystyle\kappa$ $\displaystyle\sim$
$\displaystyle\frac{m^{2}_{\phi}}{T}\quad\mbox{(if $T\gg m_{f}$)}$
$\displaystyle\kappa$ $\displaystyle\sim$
$\displaystyle\frac{m^{2}_{\phi}}{m_{f}}\quad\mbox{(if $T\ll m_{f}$)}\,.$
(2.29)

Outside of those ranges the damping becomes exponentially small. Analogous
considerations apply to case (B) as well. As a result, in the kinematic regime
we are considering, the damping matrix is determined by the two-loop diagrams
for $\Sigma_{12}$ shown in Fig. 2.

Figure 2:  Two-loop diagram for the 12 element of the neutrino self-energy
matrix in a background of fermions $f$ and scalars $\phi$, obtained by
inserting additional propagators in the one-loop diagram in Fig. 1. The
meaning of the labels $A,B$ are the same as in Fig. 1 while $a,b,c,d$ are
neutrino flavor indices. To simplify the notation we are setting
$k^{\prime}=k+p-p^{\prime}$. Diagram C does not exist if $\phi$ is a complex
scalar field. For a real scalar diagram C must be included.

In summary, the neutrino and antineutrino effective potential is given by Eq.
(2.16), and the damping by Eq. (2.19), where $V^{\mu}_{r}$ is determined from
the calculation of $\Sigma_{11}$ using the one-loop diagram in Fig. 1, while
$V^{\mu}_{i}$ is determined from $\Sigma_{12}$ calculated from the two-loop
diagrams in Fig. 2. As already stated, in writing Eq. (2.19) we are neglecting
the correction due to the $n\cdot\partial
V_{r}(\omega,\vec{\kappa})/\partial\omega$ in the overall denominator in Eq
(2.18), which corresponds to keep the leading order of the dominant term. As
we have emphasized, the expressions corresponding to the diagrams in Fig. 2
suffer from the pinch singularities. After handling the singularities,
$\Sigma_{12}$ is expressed in terms of integrals over the background particles
distribution functions that can be evaluated in principle once the background
conditions are specified. Correspondingly, the final formula for the damping
matrix that we determine by means of Eq. (2.23) is expressed in terms of
integrals over the background particle distribution functions that we will
evaluate explicitly for some illustrative cases.

##  3 Effective potential

###  3.1 Dispersive part $\Sigma_{r}$

The contribution of the diagram in Fig. 1 to the 11 component of the neutrino
thermal self-energy matrix is given by

$-i(\Sigma_{11})_{ba}=\int\,\frac{d^{4}p}{(2\pi)^{4}}i\Delta^{(\phi)}_{11}(p-k)(ig^{\ast}_{b}R)iS^{(f)}_{11}(p)(ig_{a}L)\,.$
(3.1)

We write the 11 components of the $f$ and $\phi$ thermal propagators in the
form

$\displaystyle iS^{(f)}_{11}(p)$ $\displaystyle=$ $\displaystyle(p\hbox
to0.0pt{\hss/}+m_{f})\left[\frac{i}{p^{2}-m^{2}+i\epsilon}-2\pi\delta(p^{2}-m^{2}_{f})\eta_{f}(p)\right]\,,$
$\displaystyle i\Delta^{(\phi)}_{11}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{p^{2}-m^{2}_{\phi}+i\epsilon}+2\pi\delta(p^{2}-m^{2}_{\phi})\eta_{\phi}(p)\,,$
(3.2)

where

$\displaystyle\eta_{f}(p)$ $\displaystyle=$ $\displaystyle
n_{F}(x_{f}(p))\theta(p\cdot u)+n_{F}(-x_{f}(p))\theta(-p\cdot u)\,,$
$\displaystyle\eta_{\phi}(p)$ $\displaystyle=$ $\displaystyle
n_{B}(x_{f}(p))\theta(p\cdot u)+n_{F}(-x_{f}(p))\theta(-p\cdot u)\,.$ (3.3)

with $\theta$ being the unit step function. Here $n_{F}$ is the fermion
momentum distribution function defined in Eq. (2.9), and $n_{B}$ is the
corresponding one for bosons,

$n_{B}(z)=\frac{1}{e^{z}-1}\,.$ (3.4)

The variables $x_{f,\phi}$ are given by

$\displaystyle x_{f}(p)=\beta p\cdot u-\alpha_{f}\,,$ $\displaystyle
x_{\phi}(p)=\beta p\cdot u-\alpha_{\phi}\,,$ (3.5)

where $\alpha_{f,\phi}$ are the chemical potentials. Discarding the pure
vacuum contribution in Eq. (3.1) we then have

$\Sigma_{r}=\Sigma^{(f)}_{r}+\Sigma^{(\phi)}_{r}\,,$ (3.6)

where

$\displaystyle\left(\Sigma^{(f)}_{r}\right)_{ba}$ $\displaystyle=$
$\displaystyle-g^{\ast}_{b}g_{a}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{p\hbox
to0.0pt{\hss/}L}{(p-k)^{2}-m^{2}_{\phi}}\delta(p^{2}-m^{2}_{f})\eta_{f}(p)\,,$
(3.7) $\displaystyle\left(\Sigma^{(\phi)}_{r}\right)_{ba}$ $\displaystyle=$
$\displaystyle g^{\ast}_{b}g_{a}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{(p\hbox
to0.0pt{\hss/}+k\hbox
to0.0pt{\hss/})L}{(p+k)^{2}-m^{2}_{f}}\delta(p^{2}-m^{2}_{\phi})\eta_{\phi}(p)\,.$
(3.8)

###  3.2 Effective potential in the _light background_

For completeness and to make the present work self-contained, here we consider
specifically the case of the light background in the sense of Eq. (1.2). In
correspondence to Eq. (2.13), we write the $\Sigma^{(x)}_{r}$ ($x=f,\phi$) in
the form

$\Sigma^{(x)}_{r}=V^{(x)\mu}_{r}\gamma_{\mu}L\,,$ (3.9)

and therefore

$V^{\mu}_{r}=V^{(f)\mu}_{r}+V^{(\phi)\mu}_{r}\,.$ (3.10)

From Eqs. (3.7) and (3.8) we then have

$\displaystyle\left(V^{(f)\mu}_{r}(\omega,\kappa)\right)_{ba}$
$\displaystyle=$
$\displaystyle-(g^{\ast}_{b}g_{a})\int\frac{d^{4}p}{(2\pi)^{3}}\frac{p_{\mu}}{(p-k)^{2}-m^{2}_{\phi}}\delta(p^{2}-m^{2}_{f})\eta_{f}(p)\,,$
(3.11) $\displaystyle\left(V^{(\phi)\mu}_{r}(\omega,\kappa)\right)_{ba}$
$\displaystyle=$
$\displaystyle(g^{\ast}_{b}g_{a})\int\frac{d^{4}p}{(2\pi)^{3}}\frac{(p^{\mu}+k^{\mu})}{(p+k)^{2}-m^{2}_{f}}\delta(p^{2}-m^{2}_{\phi})\eta_{\phi}(p)\,.$
(3.12)

Under the conditions that we are considering [i.e., Eq. (1.2)], we can make
the replacement

$\displaystyle\frac{1}{(p+k)^{2}-m^{2}_{f}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{2p\cdot k+k^{2}}\,,$
$\displaystyle\frac{1}{(p-k)^{2}-m^{2}_{\phi}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{-1}{2p\cdot k-k^{2}}\,,$ (3.13)

in Eqs. (3.11) and (3.12). Furthermore, since the effective potential is
defined by setting $\omega=\kappa$ [i.e., Eq. (2.23)], we can set

$k^{\mu}=\kappa n^{\mu}\,,$ (3.14)

where $n^{\mu}$ is defined in Eq. (2.17). Thus,

$\displaystyle\left(V^{(f)\mu}_{r}(\kappa,\kappa)\right)_{ba}$
$\displaystyle=$
$\displaystyle(g^{\ast}_{b}g_{a})\int\frac{d^{4}p}{(2\pi)^{3}}\frac{p_{\mu}}{2p\cdot
k}\delta(p^{2}-m^{2}_{f})\eta_{f}(p)\,,$ (3.15)
$\displaystyle\left(V^{(\phi)\mu}_{r}(\kappa,\kappa)\right)_{ba}$
$\displaystyle=$
$\displaystyle(g^{\ast}_{b}g_{a})\int\frac{d^{4}p}{(2\pi)^{3}}\frac{(p^{\mu}+\kappa
n^{\mu})}{2p\cdot k}\delta(p^{2}-m^{2}_{\phi})\eta_{\phi}(p)\,.$ (3.16)

Carrying out the integral over $p^{0}$ with the help of the delta function, we
then have

$\displaystyle\left(V^{(f)\mu}_{r}(\kappa,\vec{\kappa})\right)_{ba}$
$\displaystyle=$
$\displaystyle\frac{g^{\ast}_{b}g_{a}}{2\kappa}\int\frac{d^{3}p}{(2\pi)^{3}2E_{f}}\frac{p^{\mu}}{p\cdot
n}\left[f_{f}(E_{f})+f_{\bar{f}}(E_{f})\right]\,,$
$\displaystyle\left(V^{(\phi)\mu}_{r}(\kappa,\vec{\kappa})\right)_{ba}$
$\displaystyle=$
$\displaystyle\frac{g^{\ast}_{b}g_{a}}{2\kappa}\int\frac{d^{3}p}{(2\pi)^{3}2E_{\phi}}\frac{1}{p\cdot
n}\left\\{p^{\mu}\left[f_{\phi}(E_{\phi})+f_{\bar{\phi}}(E_{\phi})\right]\right.$
(3.17) $\displaystyle\mbox{}+\left.\kappa
n^{\mu}\left[f_{\phi}(E_{\phi})-f_{\bar{\phi}}(E_{\phi})\right]\right\\}\,,$

and

$V^{(x)\mu}_{r}(-\kappa,-\vec{\kappa})=\left.V^{(x)\mu}_{r}(\kappa,\vec{\kappa})\right|_{n^{\mu}\rightarrow-n^{\mu}}\,.$
(3.18)

In Eq. (3.2) we have introduced the $f$ and $\phi$ momentum distribution
functions (in the rest frame of the medium)

$\displaystyle f_{f,\bar{f}}(E_{f})$ $\displaystyle=$
$\displaystyle\frac{1}{e^{\beta E_{f}\mp\alpha_{f}}+1}\,,$ $\displaystyle
f_{\phi,\bar{\phi}}(E_{\phi})$ $\displaystyle=$
$\displaystyle\frac{1}{e^{\beta E_{\phi}\mp\alpha_{\phi}}-1}\,,$ (3.19)

and it is understood that in each integral,

$p^{\mu}=(E_{x},\vec{p})\qquad(x=f,\phi)\,,$ (3.20)

with

$E_{x}=\sqrt{{\vec{p}}{\,{}^{2}}+m^{2}_{x}}\,.$ (3.21)

Thus remembering that $n^{2}=0$,

$\left(n\cdot V^{(x)}(\kappa,\vec{\kappa})\right)_{ba}=-\left(n\cdot
V^{(x)}(-\kappa,-\vec{\kappa})\right)_{ba}=\frac{g^{\ast}_{b}g_{a}}{2\kappa}\int\frac{d^{3}p}{(2\pi)^{3}2E_{x}}\left[f_{x}(E_{x})+f_{\bar{x}}(E_{x})\right]\,.$
(3.22)

From Eq. (2), we then have

$\left(V^{(\nu)}_{eff}(\vec{\kappa})\right)_{ba}=\left(V^{(\bar{\nu})}_{eff}(\vec{\kappa})\right)_{ba}=(V_{f})_{ba}+(V_{\phi})_{ba}\,,$
(3.23)

where ($x=f,\phi$)

$(V_{x})_{ba}=\frac{g^{\ast}_{b}g_{a}}{2\kappa}J_{x}\,,$ (3.24)

and we have defined

$J_{x}\equiv\int\frac{d^{3}p}{(2\pi)^{3}2E_{x}}\left[f_{x}(E_{x})+f_{\bar{x}}(E_{x})\right]\,.$
(3.25)

Eqs. (3.23)-(3.25) reveal a number of differences in contrast with the
Wolfenstein term that gives standard matter contribution to the effective
potential[30]. The effective potential in this case is momentum-dependent,
proportional to $1/\kappa$, has the same sign for neutrinos and antineutrinos,
and does not vanish in a particle-antiparticle symmetric background. In fact,
as is well known, for practical purposes the parameters
$g^{\ast}_{b}g_{a}J_{f}$ and $g^{\ast}_{b}g_{a}J_{\phi}$ act as contributions
to the vacuum squared mass matrix, with the same value (and sign) for
neutrinos and antineutrinos (see e.g., Ref. [31]).

It is a simple matter to evaluate $J_{x}$ for different conditions of the
background. For example, and for reference purposes, in the non-relativistic
(NR) limit, or in the ultra-relativistic (UR) limit and zero chemical
potential,

$J_{\phi}=\left\\{\begin{array}[]{ll}\frac{1}{2m_{\phi}}(n_{\phi}+n_{\bar{\phi}})&\mbox{(NR)}\\\\[12.0pt]
\frac{T^{2}}{12}&\mbox{(UR)}\,,\end{array}\right.$ (3.26)

and similarly for $J_{f}$. In Eq. (3.26), $n_{\phi,\bar{\phi}}$ are the total
number densities of $\phi$ and $\bar{\phi}$, respectively, i.e.,

$n_{\phi,\bar{\phi}}=\int\frac{d^{3}p}{(2\pi)^{3}}f_{\phi,\bar{\phi}}(E_{\phi})\,.$
(3.27)

In Eq. (3.26) we have assumed that $\phi$ is complex. For a real $\phi$, in
the NR limit

$J_{\phi}=\frac{n_{\phi}}{m_{\phi}}\qquad\mbox{(NR)}\,.$ (3.28)

In the UR limit the formula in Eq. (3.26) holds in this case as well.

##  4 Two-loop diagrams - absence of pinch singularities

We reiterate that we assume $\kappa$ to be high enough so that the conditions
such as those given in Eq. (2) (or the analogous ones in case (B)) are
satisfied and therefore the one-loop contribution to the damping matrix,
arising from the two-body processes shown in Eq. (2) is negligible. As we have
already mentioned, in that case the damping terms arise from the two-loop
diagrams for $\Sigma_{12}$ shown in Fig. 2, which are obtained by inserting
additional propagators in the one-loop diagram in Fig. 1. The corresponding
expressions for their contributions to $\Sigma_{12}$ are

$\displaystyle-i\left(\Sigma^{(A)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{A,B}\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{12}(p^{\prime})(i\eta_{2}g_{a}L)i\Delta^{(\phi)}_{2B}(p^{\prime}-k)i\Delta^{(\phi)}_{A1}(p^{\prime}-k)$
(4.1)
$\displaystyle\mbox{}\times(-1)\mbox{Tr}\left\\{(i\eta_{A}g_{c}L)\left(iS^{(\nu)}_{AB}(k^{\prime})\right)_{cd}(i\eta_{B}g^{\ast}_{d}R)iS^{(f)}_{BA}(p)\right\\}\,,$
$\displaystyle-i\left(\Sigma^{(B)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{A,B}\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{21}(p)i\Delta^{(\phi)}_{AB}(p^{\prime})(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{1A}(k+p)(i\eta_{A}g_{c}L)$
(4.2)
$\displaystyle\mbox{}\times\left(iS^{(\nu)}_{AB}(k^{\prime})\right)_{cd}(i\eta_{B}g^{\ast}_{d}R)iS^{(f)}_{B2}(k+p)(i\eta_{2}g_{a}L)\,,$
$\displaystyle-i\left(\Sigma^{(C)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{A,B}\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{2A}(p)i\Delta^{(\phi)}_{1B}(p^{\prime})(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{1A}(k-p^{\prime})(i\eta_{A}g_{c}L)$
(4.3)
$\displaystyle\mbox{}\times\left(iS^{(\nu)}_{AB}(k^{\prime})\right)_{cd}(i\eta_{B}g^{\ast}_{d}R)iS^{(f)}_{B2}(k+p)(i\eta_{2}g_{a}L)\,,$

where

$k^{\prime}=k+p-p^{\prime}\,.$ (4.4)

As indicated in Figs. 1 and 2, the subscripts $A,B$ label the internal thermal
vertices, and each of them can take the values 1 or 2. Correspondingly, the
factors $\eta_{A,B}$ take into account the sign of the coupling associated
with each vertex type, $\eta_{1}=1,\eta_{2}=-1$.

Apart from the fact that there are many diagrams because we have to sum over
all the values of the internal thermal vertices $A,B$ in each one, if we
attempt to evaluate the expressions for each literally, we encounter the pinch
singularity problems in diagrams A and B. Take for example diagram A. Since
the $\phi$ propagators have the same momentum, there will be diagrams in which
two delta functions (coming from the thermal part of the propagators) with the
same argument, appear. Something similar happens with the $f$ propagators also
in diagram B, while diagram C does not have this problem. Therefore, before we
can proceed we must prove that those _pinch_ singularities actually disappear.
This is what we do next. The end result is a simplified expression for each
contribution $\Sigma^{(A,B,C)}_{12}$ that we then evaluate explicitly (in the
kinematic regime that we are considering). For convenience and future
reference, the relevant formulas are summarized in Section 4.4.

###  4.1 Diagram A

We write the contribution from diagram A in Fig. 2, given in Eq. (4.1), in the
form

$\displaystyle-i\left(\Sigma^{(A)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{A,B}\int\frac{d^{4}p^{\prime}}{(2\pi)^{4}}(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{12}(p^{\prime})(i\eta_{2}g_{a}L)$
(4.5) $\displaystyle\mbox{}\times
i\Delta^{(\phi)}_{2B}(p^{\prime}-k)\left(-i\pi^{(\phi)}_{BA}(p^{\prime}-k)\right)i\Delta^{(\phi)}_{A1}(p^{\prime}-k)\,,$

where

$-i\pi^{(\phi)}_{BA}(p^{\prime}-k)=(-1)\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\mbox{Tr}\left\\{(i\eta_{A}g_{c}L)\left(iS^{(\nu)}_{AB}(k^{\prime})\right)_{cd}(i\eta_{B}g^{\ast}_{d}R)iS^{(f)}_{BA}(p)\right\\}\,.$
(4.6)

The main point is that in this form we can manipulate the integration in Eq.
(4.5), and in particular the pinch singularity, using the symmetry properties
and the parametrization of the scalar propagator as well as the scalar self-
energy. Thus we study (and manipulate)

$X_{21}(q)\equiv\sum_{A,B}\Delta^{(\phi)}_{2B}(q)\,\pi^{(\phi)}_{BA}(q)\,\Delta^{(\phi)}_{A1}(q)\,,$
(4.7)

which we write in matrix notation,

$X_{21}=\left(\tilde{\Delta}^{(\phi)}\tilde{\pi}^{(\phi)}\tilde{\Delta}^{(\phi)}\right)_{21}\,.$
(4.8)

$\tilde{\Delta}^{(\phi)}$ and $\tilde{\pi}^{(\phi)}$ are the scalar thermal
propagator and self-energy matrices. We omit the momentum argument $q$ except
when required.

We will use the parametrizations given in Eqs (2.9) and (2.34) of Ref. [32],
namely

$\displaystyle\tilde{\Delta}^{(\phi)}$ $\displaystyle=$ $\displaystyle
U_{\phi}\left(\begin{array}[]{cc}\Delta^{(\phi)}_{F}&0\\\
0&-\Delta^{(\phi)\ast}_{F}\end{array}\right)U_{\phi}\,,$ (4.11)
$\displaystyle\tilde{\pi}^{(\phi)}$ $\displaystyle=$ $\displaystyle
U^{-1}_{\phi}\left(\begin{array}[]{cc}\pi^{(\phi)}&0\\\
0&-\pi^{(\phi)\ast}\end{array}\right)U^{-1}_{\phi}\,,$ (4.14)

where $\Delta^{(\phi)}_{F}$ is the (vacuum) Feynman propagator. The matrix
$U_{\phi}$ is given by

$U_{\phi}=\frac{1}{\sqrt{1+\eta_{\phi}(q)}}\left(\begin{array}[]{cc}1+\eta_{\phi}(q)&\eta_{\phi}(q)+\theta(-q\cdot
u)\\\ \eta_{\phi}(q)+\theta(q\cdot u)&1+\eta_{\phi}(q)\end{array}\right)\,,$
(4.15)

where $\eta_{\phi}$ is defined in Eq. (3.1). Therefore, we have

$\tilde{X}\equiv\tilde{\Delta}^{(\phi)}\tilde{\pi}^{(\phi)}\tilde{\Delta}^{(\phi)}=U_{\phi}\left(\begin{array}[]{cc}\Delta^{(\phi)}_{F}\pi^{(\phi)}\Delta^{(\phi)}_{F}&0\\\
0&-\Delta^{(\phi)\ast}_{F}\pi^{(\phi)\ast}\Delta^{(\phi)\ast}_{F}\end{array}\right)U_{\phi}\,.$
(4.16)

The key point is that the problematic terms
$\Delta^{(\phi)}_{F}\Delta^{(\phi)\ast}_{F}$ that give rise to the pinch
singularity are actually absent. The quantity that enters in the expression
for $\Sigma^{(A)}_{12}$ is

$\displaystyle X_{21}$ $\displaystyle=$ $\displaystyle
U_{\phi\,11}U_{\phi\,21}\left[\Delta^{(\phi)}_{F}\pi^{(\phi)}\Delta^{(\phi)}_{F}-\Delta^{(\phi)\ast}_{F}\pi^{(\phi)\ast}\Delta^{(\phi)\ast}_{F}\right]$
(4.17) $\displaystyle=$ $\displaystyle(\eta_{\phi}(q)+\theta(q\cdot
u))\left[\Delta^{(\phi)}_{F}\pi^{(\phi)}\Delta^{(\phi)}_{F}-\Delta^{(\phi)\ast}_{F}\pi^{(\phi)\ast}\Delta^{(\phi)\ast}_{F}\right]\,.$

Finally, it should be remembered that $\pi$ can be calculated by using the
formulas

$\displaystyle\mbox{Re}\,\pi^{(\phi)}$ $\displaystyle=$
$\displaystyle\mbox{Re}\,\pi^{(\phi)}_{11}\,,$
$\displaystyle\mbox{Im}\,\pi^{(\phi)}$ $\displaystyle=$
$\displaystyle\frac{i\pi^{(\phi)}_{12}}{2[\eta_{\phi}(q)+\theta(-q\cdot
u)]}=\frac{i\pi^{(\phi)}_{21}}{2[\eta_{\phi}(q)+\theta(q\cdot u)]}\,,$ (4.18)

where the $\pi^{(\phi)}_{AB}$ are given by the expressions in Eq. (4.6).

Thus in summary the procedure we need to follow is the following:

  1. 1.

Compute $\mbox{Re}\,\pi^{(\phi)}_{11}(p^{\prime}-k)$ and
$\pi^{(\phi)}_{12}(p^{\prime}-k)$ using Eq. (4.6), and from this determine
$\pi$ using Eq. (4.1).

  2. 2.

Determine the quantity

$X_{21}(q)=(\eta_{\phi}+\theta(q\cdot
u))\left[\Delta^{(\phi)}_{F}\pi^{(\phi)}\Delta^{(\phi)}_{F}-\Delta^{(\phi)\ast}_{F}\pi^{(\phi)\ast}\Delta^{(\phi)\ast}_{F}\right]\,,$
(4.19)

where $q=p^{\prime}-k$.

  3. 3.

Compute $\Sigma^{(A)}_{12}$ from

$\displaystyle-i\left(\Sigma^{(A)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}p^{\prime}}{(2\pi)^{4}}(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{12}(p^{\prime})(i\eta_{2}g_{a}L)iX_{21}(p^{\prime}-k)\,.$
(4.20)

At this point, by means of Eq. (4.20), these instructions provide a complete
and consistent procedure for calculating the contribution to the neutrino
self-energy from diagram A, which is free from the pinch singularities.
However, for our purposes, we can simplify the explicit computation as
follows.

  4. 4.

As we argue below, in the kinematic regime we are considering, the
$\mbox{Re}\,\pi^{(\phi)}$ term in Eq. (4.19) contributes negligibly, so that
we can take $X_{21}$ to be

$X_{21}=-\frac{1}{2}\left[\Delta^{(\phi)}_{F}\pi^{(\phi)}_{21}\Delta^{(\phi)}_{F}+\Delta^{(\phi)\ast}_{F}\pi_{21}\Delta^{(\phi)\ast}_{F}\right]\,.$
(4.21)

The argument concerning the contribution from $\mbox{Re}\,\pi^{(\phi)}$ is the
following. The term $(\Delta^{(\phi)}_{F})^{2}-(\Delta^{(\phi)\ast}_{F})^{2}$
contributes only when the scalar is on-shell, that is
$(p^{\prime}-k)^{2}\approx m^{2}_{\phi}$. Since the external neutrino momentum
$k$ as well as the fermion momentum $p^{\prime}$ are on shell in Eq. (4.20),
the $\mbox{Re}\,\pi^{(\phi)}$ term corresponds to a contribution to
$\Sigma^{(A)}_{12}$ arising from a process involving two-body subproceses
$\nu\phi\leftrightarrow f$. Under the kinematic conditions we are considering,
such sub-processes are suppressed by the same (two-body) kinematics that
suppress the one-loop contributions to $\Sigma^{(\nu)}_{12}$ (and the _raison-
de-entre_ for considering the two-loop contributions) and therefore those
terms can be neglected.

  5. 5.

More specifically, in our case in which we consider the high $k$ limit, we
take

$\Delta^{(\phi)}_{F}(p^{\prime}-k)\rightarrow\frac{-1}{2p^{\prime}\cdot k}\,,$
(4.22)

which we write in the form

$\Delta^{(\phi)}_{F}(p^{\prime}-k)\rightarrow\Delta_{0}(p^{\prime},-k)\,,$
(4.23)

where we define

$\Delta_{0}(\ell_{1},\ell_{2})\equiv\frac{1}{2\ell_{1}\cdot\ell_{2}}\,.$
(4.24)

This gives

$\displaystyle-i\left(\Sigma^{(A)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}p^{\prime}}{(2\pi)^{4}}(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{12}(p^{\prime})(i\eta_{2}g_{a}L)$
(4.25)
$\displaystyle\mbox{}\times\left(-i\Delta_{0}(p^{\prime},-k)\right)\left(-i\pi^{(\phi)}_{21}(p^{\prime}-k)\right)i\Delta_{0}(p^{\prime},-k)\,.$

This is just the expression that we would obtain from Eq. (4.5) by considering
only the term with $A=1,B=2$, and retain only the high $k$ limit of the vacuum
part of the scalar propagator.

Thus explicitly,

$\displaystyle-i\left(\Sigma^{(A)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{c,d}\int\frac{d^{4}p^{\prime}}{(2\pi)^{4}}\frac{d^{4}p}{(2\pi)^{4}}(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{12}(p^{\prime})(i\eta_{2}g_{a}L)(-i\Delta_{0}(p^{\prime},-k))i\Delta_{0}(p^{\prime},-k)$
(4.26)
$\displaystyle\mbox{}\times(-1)\left\\{\mbox{Tr}(i\eta_{1}g_{c}L)\left(iS^{(\nu)}_{12}(k^{\prime})\right)_{cd}(i\eta_{2}g^{\ast}_{d}R)iS^{(f)}_{21}(p)\right\\}\,.$

###  4.2 Diagram B

We now consider the contribution from diagram B in Fig. 2, given in Eq. (4.2).
Here the pinch singularity involves the fermion propagator. We write it in the
form

$\displaystyle-i\left(\Sigma^{(B)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{A,B}\int\frac{d^{4}p}{(2\pi)^{4}}i\Delta^{(\phi)}_{21}(p)(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{1A}(k+p)$
(4.27)
$\displaystyle\mbox{}\times\left(-i\Sigma^{(f)}_{AB}(k+p)\right)iS^{(f)}_{B2}(k+p)(i\eta_{2}g_{a}L)\,,$

where

$-i\Sigma^{(f)}_{AB}(k+p)=\sum_{c,d}\int\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{AB}(p^{\prime})(i\eta_{A}g_{c}L)\left(iS^{(\nu)}_{AB}(k^{\prime})\right)_{cd}(i\eta_{B}g^{\ast}_{d}R)\,.$
(4.28)

Therefore, in analogy with the previous case, here we denote by $q=k+p$ the
momentum of the virtual $f$, and consider the quantity

$Y_{12}(q)\equiv\sum_{A,B}S^{(f)}_{1A}\Sigma^{(f)}_{AB}S^{(f)}_{B2}=\left(\tilde{S}^{(f)}\tilde{\Sigma}^{(f)}\tilde{S}^{(f)}\right)_{12}\,,$
(4.29)

using the parametrization of the fermion propagator,

$\displaystyle\tilde{S}^{(f)}$ $\displaystyle=$ $\displaystyle
U_{f}\left(\begin{array}[]{cc}S^{(f)}_{F}&0\\\
0&-\bar{S}^{(f)}_{F}\end{array}\right)U_{f}\,,$ (4.32)
$\displaystyle\tilde{\Sigma}^{(f)}$ $\displaystyle=$ $\displaystyle
U^{-1}_{f}\left(\begin{array}[]{cc}\Sigma^{(f)}&0\\\
0&-\bar{\Sigma}^{(f)}\end{array}\right)U^{-1}_{f}\,,$ (4.35)

where $S^{(f)}_{F}$ is the vacuum Feynman propagator. The matrix $U_{f}$ is
given by taking the expression given in Eq. (4.15) and replacing
$\eta_{\phi}\rightarrow-\eta_{f}$, with $\eta_{f}$ defined in Eq. (3.1).

Therefore in correspondence with the scalar case here we have,

$\tilde{Y}\equiv\tilde{S}^{(f)}\tilde{\Sigma}^{(f)}\tilde{S}^{(f)}=U_{f}\left(\begin{array}[]{cc}S^{(f)}_{F}\Sigma^{(f)}S^{(f)}_{F}&0\\\
0&-\bar{S}^{(f)}_{F}\bar{\Sigma}^{(f)}\bar{S}^{(f)}_{F}\end{array}\right)U_{f}\,.$
(4.36)

Thus as in the scalar case, the problematic terms
$S^{(f)}_{F}\bar{S}^{(f)}_{F}$ that give rise to the pinch singularity are
actually absent. The quantity that enters in the expression for
$\Sigma^{(B)}_{12}$ is

$\displaystyle Y_{12}$ $\displaystyle=$ $\displaystyle
U_{f\,11}U_{f\,12}\left[S^{(f)}_{F}\Sigma^{(f)}S^{(f)}_{F}-\bar{S}^{(f)}_{F}\bar{\Sigma}^{(f)}\bar{S}^{(f)}_{F}\right]$
(4.37) $\displaystyle=$ $\displaystyle-(\eta_{f}(q)-\theta(-q\cdot
u))\left[S^{(f)}_{F}\Sigma^{(f)}S^{(f)}_{F}-\bar{S}^{(f)}_{F}\bar{\Sigma}^{(f)}\bar{S}^{(f)}_{F}\right]\,.$

Finally, it should be remembered that $\Sigma$ can be calculated by using the
formulas

$\displaystyle\mbox{Re}\,\Sigma^{(f)}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(\Sigma^{(f)}+\bar{\Sigma}^{(f)}\right)=\mbox{Re}\,\Sigma^{(f)}_{11}\,,$
$\displaystyle\mbox{Im}\,\Sigma^{(f)}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2i}\left(\Sigma^{(f)}-\bar{\Sigma}^{(f)}\right)=\frac{\Sigma^{(f)}_{12}}{2i[\eta_{f}(q)-\theta(-q\cdot
u)]}=\frac{\Sigma^{(f)}_{21}}{2i[\eta_{f}(q)-\theta(q\cdot u)]}\,,$ (4.38)

where the $\Sigma^{(f)}_{AB}$ are given by the expressions in Eq. (4.28).

Thus in summary the procedure we need to follow is the following:

  1. 1.

Compute $\mbox{Re}\,\Sigma^{(f)}_{11}(k+p)$ and $\Sigma^{(f)}_{12}(k+p)$ using
Eq. (4.28), and from this determine $\Sigma^{(f)}(k+p)$ using Eq. (4.2).

  2. 2.

Determine the quantity

$Y_{12}(q)=-(\eta_{f}(q)-\theta(-q\cdot
u))\left[S^{(f)}_{F}\Sigma^{(f)}S^{(f)}_{F}-\bar{S}^{(f)}_{F}\bar{\Sigma}^{(f)}\bar{S}^{(f)}_{F}\right]\,.$
(4.39)

  3. 3.

Compute $\Sigma^{(B)}_{12}(k)$ from

$\displaystyle-i\left(\Sigma^{(B)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}i\Delta^{(\phi)}_{21}(p)(i\eta_{1}g^{\ast}_{b}R)iY_{12}(p+k)(i\eta_{2}g_{a}L)\,.$
(4.40)

This is the result for diagram B, analogous to Eq. (4.20) for diagram A. It is
free from the pinch singularities, and while it provides a consistent starting
point, we can again simplify the explicit computation by proceeding as we did
for diagram A.

  4. 4.

As we argue below, in the kinematic regime we are considering, the
$Re\,\Sigma^{(f)}$ term in Eq. (4.39) contributes negligibly, so that we can
take $Y_{12}$ to be

$Y_{12}=-\frac{1}{2}\left[S^{(f)}_{F}\Sigma^{(f)}_{12}S^{(f)}_{F}+\bar{S}^{(f)}_{F}\Sigma^{(f)}_{12}\bar{S}^{(f)}_{F}\right]\,.$
(4.41)

The argument concerning the contribution from $\mbox{Re}\,\Sigma^{(f)}$ is
similar to the scalar case in the discussion of Diagram (A). Schematically,
the term $S^{(f)\,2}_{F}-\bar{S}^{(f)\,2}_{F}$ contributes only when the $f$
is on-shell, that is $(k+p)^{2}\approx m^{2}_{f}$. Since the external neutrino
momentum $k$ as well as the scalar momentum $p$ are on shell in Eq. (4.40),
the $\mbox{Re}\,\Sigma^{(f)}$ term corresponds to a contribution to
$\Sigma^{(B)}_{12}$ arising from a process involving two-body subprocesses
$\nu\phi\leftrightarrow f$. Again, under the kinematic conditions we are
considering, such sub-processes are suppressed by the same (two-body)
kinematics that suppress the one-loop contributions to $\Sigma^{(\nu)}_{12}$
(and the _raison-de-entre_ for considering the two-loop contributions) and
therefore those terms can be neglected.

  5. 5.

More specifically, in our case in which we consider the high $k$ limit, we
take

$S^{(f)}_{F}(k+p)\rightarrow S^{(f)}_{0}(p,k)\equiv\frac{k\hbox
to0.0pt{\hss/}+p\hbox to0.0pt{\hss/}+m_{f}}{2p\cdot k}\,,$ (4.42)

which gives

$\displaystyle-i\left(\Sigma^{(B)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}i\Delta^{(\phi)}_{21}(p)(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{0}(p,k)$
(4.43)
$\displaystyle\mbox{}\times\left(-i\Sigma^{(f)}_{12}(p+k)\right)\left(-iS^{(f)}_{0}(p,k)\right)(i\eta_{2}g_{a}L)\,.$

This is just the expression that we would obtain from Eq. (4.27) by
considering only the term with $A=1,B=2$, and retain only the high $k$ limit
of the vacuum part of the $f$ propagator.

Thus, explicitly,

$\displaystyle-i\left(\Sigma^{(B)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{21}(p)i\Delta^{(\phi)}_{12}(p^{\prime})(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{0}(p,k)$
$\displaystyle\mbox{}\times(i\eta_{1}g_{c}L)\left(iS^{(\nu)}_{12}(k^{\prime})\right)_{cd}(i\eta_{2}g^{\ast}_{d}R)\left(-iS^{(f)}_{0}(p,k)\right)(i\eta_{2}g_{a}L)\,.$

###  4.3 Diagram C

The contribution to the self-energy is

$\displaystyle-i\left(\Sigma^{(C)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{A,B}\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{2A}(p)i\Delta^{(\phi)}_{1B}(p^{\prime})(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{1A}(k-p^{\prime})(i\eta_{A}g_{c}L)$
(4.45)
$\displaystyle\mbox{}\times\left(iS^{(\nu)}_{AB}(k^{\prime})\right)_{cd}(i\eta_{B}g^{\ast}_{d}R)iS^{(f)}_{B2}(k+p)(i\eta_{2}g_{a}L)\,,$

Considering the sum over $A,B$, there are four terms, corresponding to the
combinations $AB=12,21,11,22$. Consider the third one, that is $A=B=1$. The
integrand contains the factor $S^{(f)}_{12}(k+p)\Delta^{(\phi)}_{21}(p)$, and
as a consequence of the delta functions involved, the momentum integration
will be suppressed by the same two-body kinematics we have already alluded to.
Similar arguments apply to the term $A=B=2$, which contains the factor
$S^{(f)}_{12}(k-p^{\prime})\Delta^{(\phi)}_{12}(p^{\prime})$.

On the other hand, the terms corresponding to $AB=12,21$ are not suppressed in
this way. We therefore write

$\left(\Sigma^{(C)}_{12}\right)_{ba}=\left(\Sigma^{(CI)}_{12}\right)_{ba}+\left(\Sigma^{(CII)}_{12}\right)_{ba}\,,$
(4.46)

where

$\displaystyle-i\left(\Sigma^{(CI)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{21}(p)i\Delta^{(\phi)}_{12}(p^{\prime})(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{11}(k-p^{\prime})(i\eta_{1}g_{c}L)$
(4.47)
$\displaystyle\mbox{}\times\left(iS^{(\nu)}_{12}(k^{\prime})\right)_{cd}(i\eta_{2}g^{\ast}_{d}R)iS^{(f)}_{22}(k+p)(i\eta_{2}g_{a}L)\,,$

and

$\displaystyle-i\left(\Sigma^{(CII)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{22}(p)i\Delta^{(\phi)}_{11}(p^{\prime})(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{12}(k-p^{\prime})(i\eta_{2}g_{c}L)$
(4.48)
$\displaystyle\mbox{}\times\left(iS^{(\nu)}_{21}(k^{\prime})\right)_{cd}(i\eta_{1}g^{\ast}_{d}R)iS^{(f)}_{12}(k+p)(i\eta_{2}g_{a}L)\,.$

For the purpose of carrying out the momentum integrations it is more
convenient to relabel the momentum variables in the expression for
$\left(\Sigma^{(CII)}_{12}\right)_{ba}$ according to the diagram shown in Fig.
3, which corresponds to

$\displaystyle-i\left(\Sigma^{(CII)}_{12}(k\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{22}(p-k)i\Delta^{(\phi)}_{11}(k-p^{\prime})(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{12}(p^{\prime})(i\eta_{2}g_{c}L)$
(4.49)
$\displaystyle\mbox{}\times\left(iS^{(\nu)}_{21}(k^{\prime\prime})\right)_{cd}(i\eta_{1}g^{\ast}_{d}R)iS^{(f)}_{12}(p)(i\eta_{2}g_{a}L)\,.$

Figure 3:  Equivalent form of diagram C in Fig. 2, with a relabeling of the
internal momentum variables. Here we have defined
$k^{\prime\prime}=p+p^{\prime}-k$.

Invoking once again the two-body kinematics argument, we see that in Eq.
(4.47) the contribution from the thermal part of the diagonal $f$ propagators
is suppressed, as well as the contribution from the thermal part of the
diagonal $\phi$ propagators in Eq. (4.49). Thus, approximating the vacuum part
of the diagonal propagators by their high $k$ limit,

$\displaystyle-i\left(\Sigma^{(CI)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{21}(p)i\Delta^{(\phi)}_{12}(p^{\prime})(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{0}(-p^{\prime},k)(i\eta_{1}g_{c}L)$
(4.50)
$\displaystyle\mbox{}\times\left(iS^{(\nu)}_{12}(k^{\prime})\right)_{cd}(i\eta_{2}g^{\ast}_{d}R)\left(-iS^{(f)}_{0}(p,k)\right)(i\eta_{2}g_{a}L)\,,$
$\displaystyle-i\left(\Sigma^{(CII)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle\sum_{c,d}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}\left(-i\Delta^{(\phi)}_{0}(p,-k)\right)i\Delta^{(\phi)}_{0}(p^{\prime},-k)(i\eta_{1}g^{\ast}_{b}R)iS^{(f)}_{12}(p^{\prime})(i\eta_{2}g_{c}L)$
(4.51)
$\displaystyle\mbox{}\times\left(iS^{(\nu)}_{21}(k^{\prime\prime})\right)_{cd}(i\eta_{1}g^{\ast}_{d}R)iS^{(f)}_{12}(p)(i\eta_{2}g_{a}L)\,.$

###  4.4 Summary

In summary, we are left with four contributions,

$\left(\Sigma_{12}\right)_{ba}=\left(\Sigma^{(A)}_{12}\right)_{ba}+\left(\Sigma^{(B)}_{12}\right)_{ba}+\left(\Sigma^{(CI)}_{12}\right)_{ba}+\left(\Sigma^{(CII)}_{12}\right)_{ba}\,,$
(4.52)

given by Eqs. (4.26), (4.2), (4.50) and (4.51). In those formulas,
$\Delta^{(\phi)}_{0}$ and $S^{(f)}_{0}$ are the free fermion and scalar
propagators, respectively, in the high energy limit, defined in Eqs. (4.24)
and (4.42). From the expressions in those equations it follows that only B and
CI contribute in a pure $\phi$ background (no $f$ in the background) while in
a pure $f$ background (no $\phi$ in the background) only A and CII contribute.
In either case, the diagrams CI and CII exist only if $\phi$ is a real scalar.
For a complex $\phi$ only diagrams A and B exist. In the next section we
consider precisely the former case, namely a pure $\phi$ background. We
determine the damping terms according to the scheme explained in Section 2,
and evaluate explicitly the integrals involved for some illustrative
background conditions.

Before moving ahead, it is worth to emphasize the following. Only diagrams A
and B suffer from the pinch singularities. Eqs. (4.26) and (4.2) give us
convenient starting points to compute the contribution from diagrams A and B,
respectively, within the high-momentum approximation that we restrict
ourselves here. However, the pinch singularities are already _tamed_ in Eqs.
(4.20) and (4.40), respectively, and those expressions can be used to compute
the corresponding contributions in other situations of interest.

##  5 A pure $\phi$ background

For definiteness, in the remainder of this work we consider only a $\phi$
background with no fermions $f$. We will calculate $\Sigma_{i}$ for this case
next. From the result we will determine the damping matrix in Section 5.2.

###  5.1 Expression for $\Sigma_{i}$

The starting point is the expression for each of the diagrams B and CI we have
obtained in Eqs. (4.2) and (4.50), explicitly

$\displaystyle-i\left(\Sigma^{(B)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle(2K_{ba})\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{21}(p)i\Delta^{(\phi)}_{12}(p^{\prime})$
$\displaystyle\mbox{}\times
RiS^{(f)}_{0}(p,k)LiS^{(\nu)}_{12}(k^{\prime})R\left(-iS^{(f)}_{0}(p,k)\right)L\,,$
$\displaystyle-i\left(\Sigma^{(CI)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle(2K_{ba})\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}i\Delta^{(\phi)}_{21}(p)i\Delta^{(\phi)}_{12}(p^{\prime})$
(5.1) $\displaystyle\mbox{}\times
RiS^{(f)}_{0}(-p^{\prime},k)LiS^{(\nu)}_{12}(k^{\prime})R\left(-iS^{(f)}_{0}(p,k)\right)L\,,$

where

$K_{ba}=\frac{1}{2}g^{\ast}_{b}g_{a}\left(\sum_{c}|g_{c}|^{2}\right)\,.$ (5.2)

We have used the fact that the neutrino propagator is diagonal in flavor
space, and we are defining $K_{ba}$ with the factor of $1/2$ for later
convenience when we identify $\Sigma_{i}$. It it understood that for a complex
$\phi$ only $\Sigma^{(B)}_{12}$ contributes, while if $\phi$ is real then
$\Sigma^{(CI)}_{12}$ also contributes but in that case the chemical potential
$\alpha_{\phi}=0$ in the final evaluation of the integrals involving the
distribution functions.

We write the formulas for the propagators as follows. For the fermion $f$,
remembering Eq. (4.42), we have

$RS^{(f)}_{0}(p,k)L=S_{f}(p,k)L\,,$ (5.3)

with

$S_{f}(p,k)\equiv\frac{p\hbox to0.0pt{\hss/}+k\hbox to0.0pt{\hss/}}{2p\cdot
k}\,.$ (5.4)

For the neutrino and the scalar propagators,

$\displaystyle S^{(\nu)}_{12}(k^{\prime})$ $\displaystyle=$ $\displaystyle
2\pi i\delta(k^{\prime\,2})n_{F}(x^{\prime}_{\nu})\epsilon(k^{\prime}\cdot
u)Ls_{\nu}(k^{\prime})\,,$ $\displaystyle\Delta^{(\phi)}_{21}(p)$
$\displaystyle=$ $\displaystyle-2\pi
i\delta(p^{2}-m^{2}_{\phi})e^{x_{\phi}}n_{B}(x_{\phi})\epsilon(p\cdot u)\,,$
$\displaystyle\Delta^{(\phi)}_{12}(p^{\prime})$ $\displaystyle=$
$\displaystyle-2\pi
i\delta(p^{\prime\,2}-m^{2}_{\phi})n_{B}(x^{\prime}_{\phi})\epsilon(p^{\prime}\cdot
u)\,,$ (5.5)

where

$s_{\nu}(k^{\prime})=k\hbox to0.0pt{\hss/}^{\prime}\,,$ (5.6)

$\epsilon(z)=\theta(z)-\theta(-z)$ with $\theta(z)$ being the unit step
function, the fermion and boson distribution functions $n_{F,B}$ are defined
in Eqs. (2.9) and (3.4), and

$\displaystyle x^{\prime}_{\nu}$ $\displaystyle=$ $\displaystyle\beta
k^{\prime}\cdot u-\alpha_{\nu}\,,$ $\displaystyle x_{\phi}$ $\displaystyle=$
$\displaystyle\beta p\cdot u-\alpha_{\phi}\,,$ $\displaystyle
x^{\prime}_{\phi}$ $\displaystyle=$ $\displaystyle\beta p^{\prime}\cdot
u-\alpha_{\phi}\,.$ (5.7)

It is useful to remember that the relation $k^{\prime}=k+p-p^{\prime}$
(actually the definition of $k^{\prime}$), implies the following relation

$x_{\nu}+x_{\phi}=x^{\prime}_{\nu}+x^{\prime}_{\phi}\,.$ (5.8)

Substituting Eq. (5.1) in Eq. (5.1), we then have

$\displaystyle-i\left(\Sigma^{(B)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle-(2K_{ba})\int\frac{d^{4}p}{(2\pi)^{3}}\frac{d^{4}p^{\prime}}{(2\pi)^{3}}\frac{d^{4}k^{\prime}}{(2\pi)^{3}}(2\pi)^{4}\delta^{(4)}(k^{\prime}+p^{\prime}-k-p)$
$\displaystyle\mbox{}\times{}\delta(p^{2}-m^{2}_{\phi})\delta(p^{\prime\,2}-m^{2}_{\phi})\delta(k^{\prime\,2})\epsilon(p\cdot
u)\epsilon(p^{\prime}\cdot u)\epsilon(k^{\prime}\cdot u)$
$\displaystyle\mbox{}\times\left[S_{f}(p,k)s_{\nu}(k^{\prime})S_{f}(p,k)L\right]E^{\prime}\,,$
$\displaystyle-i\left(\Sigma^{(CI)}_{12}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle-(2K_{ba})\int\frac{d^{4}p}{(2\pi)^{3}}\frac{d^{4}p^{\prime}}{(2\pi)^{3}}\frac{d^{4}k^{\prime}}{(2\pi)^{3}}(2\pi)^{4}\delta^{(4)}(k^{\prime}+p^{\prime}-k-p)$
(5.9)
$\displaystyle\mbox{}\times\delta(p^{2}-m^{2}_{\phi})\delta(p^{\prime\,2}-m^{2}_{\phi})\delta(k^{\prime\,2})\epsilon(p\cdot
u)\epsilon(p^{\prime}\cdot u)\epsilon(k^{\prime}\cdot u)$
$\displaystyle\mbox{}\times\left[S_{f}(-p^{\prime},k)s_{\nu}(k^{\prime})S_{f}(p,k)L\right]E^{\prime}\,,$

where

$E^{\prime}\equiv
e^{x_{\phi}}n_{B}(x_{\phi})n_{B}(x^{\prime}_{\phi})n_{F}(x^{\prime}_{\nu})\,.$
(5.10)

In writing Eq. (5.1) we have taken $k^{\prime}$ to be an arbitrary variable
but inserted a factor of $\delta^{(4)}(k+p-k^{\prime}-p^{\prime})$ and
integrate over $k^{\prime}$.

Letting $X=B,CI$, the corresponding contributions to the absorptive part of
the self-energy, identified by

$\left(\Sigma_{i}\right)_{ba}=\frac{\Sigma_{12}}{2in_{F}(x_{\nu})}\,,$ (5.11)

are then given by

$\displaystyle\left(\Sigma^{(X)}_{i}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle-
K_{ba}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{d^{4}p^{\prime}}{(2\pi)^{3}}\frac{d^{4}k^{\prime}}{(2\pi)^{3}}(2\pi)^{4}\delta^{(4)}(k+p-k^{\prime}-p^{\prime})$
(5.12)
$\displaystyle\mbox{}\times\delta(p^{2}-m^{2}_{\phi})\delta(p^{\prime\,2}-m^{2}_{\phi})\delta(k^{\prime\,2})\epsilon(p\cdot
u)\epsilon(p^{\prime}\cdot u)\epsilon(k^{\prime}\cdot u)M^{(X)}_{\nu}E\,,$

where we have used the identity

$\frac{1}{n_{F}(x_{\nu})}E^{\prime}\equiv
E=n_{B}(x_{\phi})\left(1+n_{B}(x^{\prime}_{\phi})\right)-n_{F}(x^{\prime}_{\nu})\left(n_{B}(x_{\phi})-n_{B}(x^{\prime}_{\phi})\right)\,,$
(5.13)

and we have defined

$\displaystyle M^{(B)}_{\nu}$ $\displaystyle=$ $\displaystyle
S_{f}(p,k)s_{\nu}(k^{\prime})S_{f}(p,k)L\,,$ $\displaystyle M^{(CI)}_{\nu}$
$\displaystyle=$ $\displaystyle
S_{f}(-p^{\prime},k)s_{\nu}(k^{\prime})S_{f}(p,k)L\,.$ (5.14)

Next we carry out the integrals over $p^{0},p^{\prime\,0},k^{\prime\,0}$.
Starting with $k^{\prime\,0}$,

$\displaystyle\left(\Sigma^{(X)}_{i}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle-
K_{ba}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{d^{4}p^{\prime}}{(2\pi)^{3}}\frac{d^{3}\kappa^{\prime}}{(2\pi)^{3}2\omega_{\kappa^{\prime}}}\delta(p^{2}-m^{2}_{\phi})\delta(p^{\prime\,2}-m^{2}_{\phi})\epsilon(p\cdot
u)\epsilon(p^{\prime}\cdot u)$ (5.15)
$\displaystyle\mbox{}\times(2\pi)^{4}\left\\{\delta^{(4)}(k+p-k^{\prime}-p^{\prime})M^{(X)}_{\nu}E_{\nu}-\delta^{(4)}(k+p+k^{\prime}-p^{\prime})M^{(X)}_{\bar{\nu}}E_{\bar{\nu}}\right\\}\,,$

where

$\displaystyle E_{\nu}$ $\displaystyle=$ $\displaystyle
n_{B}(x_{\phi})\left(1+n_{B}(x^{\prime}_{\phi})\right)-f_{\nu}(\omega_{\kappa^{\prime}})\left(n_{B}(x_{\phi})-n_{B}(x^{\prime}_{\phi})\right)$
$\displaystyle E_{\bar{\nu}}$ $\displaystyle=$ $\displaystyle
n_{B}(x^{\prime}_{\phi})\left(1+n_{B}(x_{\phi})\right)+f_{\bar{\nu}}(\omega_{\kappa^{\prime}})\left(n_{B}(x_{\phi})-n_{B}(x^{\prime}_{\phi})\right)\,,$
(5.16)

and

$M^{(X)}_{\bar{\nu}}=\left.M^{(X)}_{\nu}\right|_{k^{\prime}\rightarrow-k^{\prime}}\,,$
(5.17)

understanding that from now on

$k^{\prime}=(\omega_{\kappa^{\prime}},\vec{\kappa}^{\prime})\,,$ (5.18)

with $\omega_{\kappa^{\prime}}=|\vec{\kappa}^{\prime}|\equiv\kappa^{\prime}$.
To arrive at Eq. (5.15) we have also made the change of variable
$\vec{\kappa}^{\prime}\rightarrow-\vec{\kappa}^{\prime}$ in the second term.
Proceeding in a similar way with the integrals over $p^{0}$ and
$p^{\prime\,0}$,

$\displaystyle\left(\Sigma^{(X)}_{i}(k)\right)_{ba}$ $\displaystyle=$
$\displaystyle-
K_{ba}\int\frac{d^{3}p}{(2\pi)^{3}2E_{p}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}2E_{p^{\prime}}}\frac{d^{3}\kappa^{\prime}}{(2\pi)^{3}2\omega_{\kappa^{\prime}}}$
(5.19)
$\displaystyle\mbox{}\times\sum_{\lambda,\lambda^{\prime}}(2\pi)^{4}\left\\{\delta^{(4)}(k+\lambda
p-k^{\prime}-\lambda^{\prime}p^{\prime})M^{(X)}_{\nu,\lambda\lambda^{\prime}}E_{\nu,\lambda\lambda^{\prime}}\right.$
$\displaystyle\mbox{}-\left.\delta^{(4)}(k+\lambda
p+k^{\prime}-\lambda^{\prime}p^{\prime})M^{(X)}_{\bar{\nu},\lambda\lambda^{\prime}}E_{\bar{\nu},\lambda\lambda^{\prime}}\right\\}\,,$

where

$\displaystyle M^{(B)}_{\nu,\lambda\lambda^{\prime}}$ $\displaystyle=$
$\displaystyle S_{f}(\lambda p,k)s_{\nu}(k^{\prime})S_{f}(\lambda p,k)L\,,$
$\displaystyle M^{(CI)}_{\nu,\lambda\lambda^{\prime}}$ $\displaystyle=$
$\displaystyle
S_{f}(-\lambda^{\prime}p^{\prime},k)s_{\nu}(k^{\prime})S_{f}(\lambda p,k)L\,,$
$\displaystyle M^{(X)}_{\bar{\nu},\lambda\lambda^{\prime}}$ $\displaystyle=$
$\displaystyle\left.M^{(X)}_{\nu,\lambda\lambda^{\prime}}\right|_{k^{\prime}\rightarrow-k^{\prime}}\,.$
(5.20)

We have defined

$\displaystyle
E_{\nu,\lambda\lambda^{\prime}}=\left.E_{\nu}\right|_{p\rightarrow\lambda
p,\,p^{\prime}\rightarrow\lambda^{\prime}p^{\prime}}$ (5.21)

and similarly for $E_{\bar{\nu},\lambda\lambda^{\prime}}$, and from now on $p$
and $p^{\prime}$ are on-shell. The formulas are given explicitly in Table 1.

$E_{\nu,++}=f(1+f^{\prime})-f^{\prime}_{\nu}(f-f^{\prime})$ | $\nu_{a,b}(k)+\phi(p)\leftrightarrow\nu_{i}(k^{\prime})+\phi(p^{\prime})$
---|---
$E_{\nu,-+}=-(1+\bar{f})(1+f^{\prime})+f^{\prime}_{\nu}(1+\bar{f}+f^{\prime})$ | $\nu_{a,b}(k)\leftrightarrow\nu_{i}(k^{\prime})+\bar{\phi}(p)+\phi(p^{\prime})$
$E_{\nu,+-}=-f\bar{f}^{\prime}-f^{\prime}_{\nu}(1+f+\bar{f}^{\prime})$ | $\nu_{a,b}(k)+\phi(p)+\bar{\phi}(p^{\prime})\leftrightarrow\nu_{i}(k^{\prime})$
$E_{\nu,--}=(1+\bar{f})\bar{f}^{\prime}+f^{\prime}_{\nu}(\bar{f}-\bar{f}^{\prime})$ | $\nu_{a,b}(k)+\bar{\phi}(p^{\prime})\leftrightarrow\nu_{i}(k^{\prime})+\bar{\phi}(p)$
$E_{\bar{\nu},++}=(1+f)f^{\prime}+\bar{f}^{\prime}_{\nu}(f-f^{\prime})$ | $\nu_{a,b}(k)+\bar{\nu}_{i}(\bar{k}^{\prime})+\phi(p)\leftrightarrow\phi(p^{\prime})$
$E_{\bar{\nu},-+}=-\bar{f}f^{\prime}-\bar{f}^{\prime}_{\nu}(1+\bar{f}+f^{\prime})$ | $\nu_{a,b}(k)+\bar{\nu}_{i}(\bar{k}^{\prime})\leftrightarrow\bar{\phi}(p)+\phi(p^{\prime})$
$E_{\bar{\nu},+-}=-(1+f)(1+\bar{f}^{\prime})+\bar{f}^{\prime}_{\nu}(1+f+\bar{f}^{\prime})$ | $\nu_{a,b}(k)+\bar{\nu}_{i}(\bar{k}^{\prime})+\phi(p)+\bar{\phi}(p^{\prime})\leftrightarrow 0$
$E_{\bar{\nu},--}=\bar{f}(1+\bar{f}^{\prime})-\bar{f}^{\prime}_{\nu}(\bar{f}-\bar{f}^{\prime})$ | $\nu_{a,b}(k)+\bar{\nu}_{i}(\bar{k}^{\prime})+\bar{\phi}(p^{\prime})\leftrightarrow\bar{\phi}(p)$
Table 1: Correspondence between the $E_{\nu,\lambda\lambda^{\prime}}$ and
$E_{\bar{\nu},\lambda\lambda^{\prime}}$ factors defined in Eq. (5.21), and the
process that contributes to the $\nu(k)$ damping via Eq. (5.19). To simplify
the notation we are using the shorthands shown in Eq. (5.1) for the various
distribution functions.

To simplify the notation in the formulas summarized in Table 1 we have
introduce the shorthands

$\displaystyle f=f_{\phi}(E_{p}),\quad
f^{\prime}=f_{\phi}(E_{p^{\prime}}),\quad
f^{\prime}_{\nu}={f_{\nu}(\omega_{\kappa^{\prime}})}$
$\displaystyle\bar{f}=f_{\bar{\phi}}(E_{p}),\quad\bar{f}^{\prime}=f_{\bar{\phi}}(E_{p^{\prime}}),\quad\bar{f}^{\prime}_{\nu}=f_{\bar{\nu}}(\omega_{\kappa^{\prime}})\,.$
(5.22)

The formulas for $E_{\bar{\nu},\lambda\lambda^{\prime}}$ are obtained from
those for $E_{\nu,\lambda\lambda^{\prime}}$ by making the replacement
$f^{\prime}_{\nu}\rightarrow(1-\bar{f}^{\prime}_{\nu})$. Each of the terms in
Eq. (5.19) represents a contribution to $\Sigma_{i}$ from a specific physical
process, as indicated in Table 1.

###  5.2 Damping matrix

We write Eq. (5.1) in the form

$M^{(X)}_{\nu}=\frac{N^{(X)}L}{D^{(X)}}\,,\\\ $

where

$\displaystyle D^{(B)}$ $\displaystyle=$ $\displaystyle(2p\cdot k)^{2}\,,$
$\displaystyle D^{(CI)}$ $\displaystyle=$ $\displaystyle(-2p^{\prime}\cdot
k)(2p\cdot k)\,,$ (5.23)

and

$\displaystyle N^{(B)}$ $\displaystyle=$ $\displaystyle(k\hbox
to0.0pt{\hss/}+p\hbox to0.0pt{\hss/})k\hbox to0.0pt{\hss/}^{\prime}(k\hbox
to0.0pt{\hss/}+p\hbox to0.0pt{\hss/})\,,$ $\displaystyle N^{(CI)}$
$\displaystyle=$ $\displaystyle(k\hbox to0.0pt{\hss/}-p\hbox
to0.0pt{\hss/}^{\prime})k\hbox to0.0pt{\hss/}^{\prime}(k\hbox
to0.0pt{\hss/}+p\hbox to0.0pt{\hss/})\,.$ (5.24)

The expressions for the $N^{(X)}$ are reduced by using the identity

$\gamma_{\lambda}\gamma_{\mu}\gamma_{\nu}=C_{\lambda\mu\nu\rho}\gamma^{\rho}+i\epsilon_{\lambda\mu\nu\rho}\gamma^{\rho}\gamma^{5}\,,$
(5.25)

where

$C_{\lambda\mu\nu\rho}=g_{\lambda\mu}g_{\nu\rho}-g_{\lambda\nu}g_{\mu\rho}+g_{\lambda\rho}g_{\mu\nu}\,.$
(5.26)

After the integrations over $k^{\prime},p,p^{\prime}$ the only vectors
remaining are $k$ and $u$, and then the terms with the antisymmetric tensor
vanish. Thus we can replace in the integrand

$N^{(X)}\rightarrow\ell^{(X)\rho}\gamma_{\rho}\,,$ (5.27)

where

$\displaystyle\ell^{(X)}_{\rho}$ $\displaystyle=$ $\displaystyle
C_{\lambda\mu\nu\rho}(k+p)^{\lambda}k^{\prime\,\mu}(k+p)^{\nu}\,,$
$\displaystyle\ell^{(X)}_{\rho}$ $\displaystyle=$ $\displaystyle
C_{\lambda\mu\nu\rho}(k-p^{\prime})^{\lambda}k^{\prime\,\mu}(k+p)^{\nu}\,.$
(5.28)

Then, corresponding to each diagram $X=B,CI$ we have

$\displaystyle\left(V^{(X)\mu}_{i}(\omega,\vec{\kappa})\right)_{ba}$
$\displaystyle=$ $\displaystyle-
K_{ba}\int\frac{d^{3}p}{(2\pi)^{3}2E_{p}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}2E_{p^{\prime}}}\frac{d^{3}\kappa^{\prime}}{(2\pi)^{3}2\omega_{\kappa^{\prime}}}$
(5.29)
$\displaystyle\mbox{}\times\sum_{\lambda,\lambda^{\prime}}\left(\frac{\ell^{(X)\mu}_{\lambda\lambda^{\prime}}}{D^{(X)}_{\lambda\lambda^{\prime}}}\right)(2\pi)^{4}\left\\{\delta^{(4)}(k+\lambda
p-k^{\prime}-\lambda^{\prime}p^{\prime})E_{\nu,\lambda\lambda^{\prime}}\right.$
$\displaystyle\mbox{}+\left.\delta^{(4)}(k+\lambda
p+k^{\prime}-\lambda^{\prime}p^{\prime})E_{\bar{\nu}\lambda\lambda^{\prime}}\right\\}\,,$

where

$\displaystyle\ell^{(X)\mu}_{\lambda\lambda^{\prime}}$ $\displaystyle\equiv$
$\displaystyle\left.\ell^{(X)\mu}\right|_{p\rightarrow\lambda
p,p^{\prime}\rightarrow\lambda^{\prime}p^{\prime}},$ $\displaystyle
D^{(X)}_{\lambda\lambda^{\prime}}$ $\displaystyle\equiv$
$\displaystyle\left.D^{(X)}\right|_{p\rightarrow\lambda
p,p^{\prime}\rightarrow\lambda^{\prime}p^{\prime}}\,.$ (5.30)

Since the formulas for $\Gamma$ are given in terms of $n\cdot
V_{i}(\kappa,\vec{\kappa})$ (for neutrinos), or $n\cdot
V_{i}(-\kappa,-\vec{\kappa})$ (for antineutrinos), we consider the evaluation
of $\Sigma_{i}$ for $k^{\mu}=\omega n^{\mu}$ (which in particular implies
$k^{2}=0$), and in the end put $\omega=\pm\kappa$ (for neutrinos or
antineutrinos, respectively). From now on we thus set $k^{2}=0$.

We evaluate $n\cdot\ell$ and $n\cdot\ell^{\prime}$, putting $k^{\mu}=\omega
n^{\mu}$ as we already stated. Then doing the algebra, remembering to set
$n^{2}=0$,

$\displaystyle n\cdot\ell^{(B)}$ $\displaystyle=$ $\displaystyle 2(n\cdot
p)(k^{\prime}\cdot p)\,,$ $\displaystyle n\cdot\ell^{(CI)}$ $\displaystyle=$
$\displaystyle-(n\cdot p)(k^{\prime}\cdot p^{\prime})-(n\cdot
p^{\prime})(k^{\prime}\cdot p)\,,$ (5.31)

and therefore

$\displaystyle\frac{n\cdot\ell^{(B)}}{D^{(B)}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\omega}\frac{k^{\prime}\cdot p}{k\cdot p}\,,$
$\displaystyle\frac{n\cdot\ell^{(CI)}}{D^{(CI)}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\omega}\frac{k^{\prime}\cdot p^{\prime}}{k\cdot
p^{\prime}}+\frac{1}{4\omega}\frac{k^{\prime}\cdot p}{k\cdot p}\,.$ (5.32)

Up to this moment we have only used straightforward algebra to arrive here
from Eq. (5.2). We now invoke the high energy limit we are considering. The
momentum delta functions set $k^{\prime}=k+p-p^{\prime}$. Therefore, to
leading order in $k$, we put $k^{\prime}\rightarrow k$ in the above and for
either diagram we have

$\frac{n\cdot\ell^{(X)}}{D^{(X)}}=\frac{1}{2\omega}\,.$ (5.33)

For the antineutrino part, the delta function gives
$k^{\prime}=-k-p+p^{\prime}$, therefore the replacement is
$k^{\prime}\rightarrow-k$. Putting all this together we then have, from Eq.
(5.29),

$\displaystyle\left(n\cdot V^{(X)}_{i}(\omega,\omega\hat{\kappa})\right)_{ba}$
$\displaystyle=$
$\displaystyle-\frac{K_{ba}}{2\omega}\int\frac{d^{3}p}{(2\pi)^{3}2E_{p}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}2E_{p^{\prime}}}\frac{d^{3}\kappa^{\prime}}{(2\pi)^{3}2\omega_{\kappa^{\prime}}}$
(5.34)
$\displaystyle\mbox{}\times\sum_{\lambda,\lambda^{\prime}}(2\pi)^{4}\left\\{\delta^{(4)}(k+\lambda
p-k^{\prime}-\lambda^{\prime}p^{\prime})E_{\nu,\lambda\lambda^{\prime}}\right.$
$\displaystyle\mbox{}-\left.\delta^{(4)}(k+\lambda
p+k^{\prime}-\lambda^{\prime}p^{\prime})E_{\bar{\nu}\lambda\lambda^{\prime}}\right\\}\,.$

The damping matrix is given by

$-\frac{1}{2}\Gamma^{(2)}=\left\\{\begin{array}[]{ll}n\cdot
V^{(B)}_{i}(\kappa,\vec{\kappa})&(\nu)\\\ n\cdot
V^{(B)\ast}_{i}(-\kappa,-\vec{\kappa})&(\bar{\nu})\,,\end{array}\right.$
(5.35)

or

$-\frac{1}{2}\Gamma^{(2)}=\left\\{\begin{array}[]{ll}n\cdot
V^{(B)}_{i}(\kappa,\vec{\kappa})+n\cdot
V^{(CI)}_{i}(\kappa,\vec{\kappa})&(\nu)\\\ n\cdot
V^{(B)\ast}_{i}(-\kappa,-\vec{\kappa})+n\cdot
V^{(CI)\ast}_{i}(-\kappa,-\vec{\kappa})&(\bar{\nu})\,,\end{array}\right.$
(5.36)

for a complex or real $\phi$, respectively. The final expressions for both
diagram contributions $n\cdot V^{(X)}_{i}$, given in Eq. (5.34), are formally
the same. But it must be understood that for a real $\phi$ the distribution
functions of the $\phi$ have $\alpha_{\phi}=0$, or equivalently
$f_{\bar{\phi}}=f_{\phi}$.

Not all the terms in Eq. (5.34) contribute, depending on whether $\omega$ is
positive or negative. Equivalently, the corresponding processes are inhibited
by the kinematics. In addition we will assume that there are no neutrinos in
the background. The result is that for the neutrinos ($\omega$ positive) only
the terms $E_{\nu,++},E_{\nu,--}$ contribute, while for the antineutrinos
($\omega$ negative) only $E_{\bar{\nu},++},E_{\bar{\nu},--}$ contribute.

Denoting by $\Gamma^{(\nu)}$ and $\Gamma^{(\bar{\nu})}$ the matrices for
neutrinos and antineutrinos, respectively, for the case of a complex $\phi$ we
then have from Eq. (5.34),

$\displaystyle\frac{1}{2}\Gamma^{(\nu)}_{ba}$ $\displaystyle=$
$\displaystyle\frac{K_{ba}\gamma^{(\nu)}}{2\kappa}\,,$
$\displaystyle\frac{1}{2}\Gamma^{(\bar{\nu})}_{ba}$ $\displaystyle=$
$\displaystyle\frac{K^{\ast}_{ba}\gamma^{(\bar{\nu})}}{2\kappa}\,,$ (5.37)

where

$\displaystyle\gamma^{(\nu)}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}2E_{p}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}2E_{p^{\prime}}}\frac{d^{3}\kappa^{\prime}}{(2\pi)^{3}2\omega_{\kappa^{\prime}}}$
$\displaystyle\mbox{}(2\pi)^{4}\left\\{\delta^{(4)}(k+p-k^{\prime}-p^{\prime})E_{\nu,++}+\delta^{(4)}(k+p^{\prime}-k^{\prime}-p)E_{\nu,--}\right\\}\,,$
$\displaystyle\gamma^{(\bar{\nu})}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}2E_{p}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}2E_{p^{\prime}}}\frac{d^{3}\kappa^{\prime}}{(2\pi)^{3}2\omega_{\kappa^{\prime}}}$
(5.38)
$\displaystyle\mbox{}(2\pi)^{4}\left\\{\delta^{(4)}(k+p^{\prime}-k^{\prime}-p)E_{\bar{\nu},++}+\delta^{(4)}(k+p-k^{\prime}-p^{\prime})E_{\bar{\nu},--}\right\\}\,,$

with

$\displaystyle E_{\nu++}$ $\displaystyle=$ $\displaystyle
f_{\phi}(E_{p})(1+f_{\phi}(E_{p^{\prime}}))\,,$ $\displaystyle E_{\nu--}$
$\displaystyle=$ $\displaystyle
f_{\bar{\phi}}(E_{p^{\prime}})(1+f_{\bar{\phi}}(E_{p}))\,,$ $\displaystyle
E_{\bar{\nu}++}$ $\displaystyle=$ $\displaystyle
f_{\phi}(E_{p^{\prime}})(1+f_{\phi}(E_{p}))\,,$ $\displaystyle
E_{\bar{\nu}--}$ $\displaystyle=$ $\displaystyle
f_{\bar{\phi}}(E_{p})(1+f_{\bar{\phi}}(E_{p^{\prime}}))\,.$ (5.39)

Relabeling the $p,p^{\prime}$ integration variables in some terms, we can see
that $\gamma^{(\bar{\nu})}=\gamma^{(\nu)}$. Therefore, in explicit form, Eq.
(5.2) becomes

$\displaystyle\frac{1}{2}\Gamma^{(\nu)}_{ba}$ $\displaystyle=$
$\displaystyle\frac{K_{ba}\gamma^{(\phi)}}{2\kappa}\,,$
$\displaystyle\frac{1}{2}\Gamma^{(\bar{\nu})}_{ba}$ $\displaystyle=$
$\displaystyle\frac{K^{\ast}_{ba}\gamma^{(\phi)}}{2\kappa}\,,$ (5.40)

where

$\displaystyle\gamma_{\phi}$ $\displaystyle\equiv$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}2E_{p}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}2E_{p^{\prime}}}\frac{d^{3}\kappa^{\prime}}{(2\pi)^{3}2\omega_{\kappa^{\prime}}}$
(5.41)
$\displaystyle\mbox{}(2\pi)^{4}\delta^{(4)}(k+p-k^{\prime}-p^{\prime})\left\\{f_{\phi}(E_{p})(1+f_{\phi}(E_{p^{\prime}}))+f_{\bar{\phi}}(E_{p})(1+f_{\bar{\phi}}(E_{p^{\prime}}))\right\\}\,.$

As already stated, Eq. (5.2) holds for a complex $\phi$. For a real $\phi$ the
formula is the same but with the replacement $2\kappa\rightarrow\kappa$, and
putting $f_{\bar{\phi}}=f_{\phi}$ in Eq. (5.41).

###  5.3 Example evaluation of integrals

In the dilute gas approximation (i.e., neglecting the terms with the product
of the distribution function),

$\gamma_{\phi}=\int\frac{d^{3}p}{(2\pi)^{3}2E_{p}}(f_{\phi}(E_{p})+f_{\bar{\phi}}(E_{p}))J\,,$
(5.42)

where

$J=\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}2E_{p^{\prime}}}\frac{d^{3}\kappa^{\prime}}{(2\pi)^{3}2\omega_{\kappa^{\prime}}}(2\pi)^{4}\delta^{(4)}(k+p-k^{\prime}-p^{\prime})\,.$
(5.43)

It is straightforward to evaluate the $J$ integral. Let us define

$q=k+p\,,$ (5.44)

and

$s=q^{2}\,.$ (5.45)

We then obtain

$J=\frac{1}{4\pi}\left(\frac{p^{\prime\,\ast}}{\sqrt{s}}\right)\theta(\sqrt{s}-E^{\ast}_{p^{\prime}})\theta(E^{\ast}_{p^{\prime}}-m_{\phi})\,,$
(5.46)

where

$E^{\ast}_{p^{\prime}}=\frac{s+m^{2}_{\phi}}{2\sqrt{s}}\,,$ (5.47)

and

$p^{\prime\,\ast}=\sqrt{E^{\ast\,2}_{p^{\prime}}-m^{2}_{\phi}}=\frac{s-m^{2}_{\phi}}{2\sqrt{s}}\,.$
(5.48)

From the definition in Eq. (5.45),

$\displaystyle s$ $\displaystyle=$ $\displaystyle m^{2}_{\phi}+2k\cdot p$
(5.49) $\displaystyle=$ $\displaystyle m^{2}_{\phi}+2\kappa
E_{p}(1-v_{p}\cos\theta_{p})\,,$

where

$\cos\theta_{p}=\hat{\kappa}\cdot\hat{p}\,,$ (5.50)

and

$v_{p}=\frac{|\vec{p}|}{E_{p}}\,.$ (5.51)

Thus for any value of $\vec{p}$, we have $s>m^{2}_{\phi}$, and this implies
that the two step functions in Eq. (5.46) are automatically satisfied.
Therefore, we can take $J$ to be simply

$J=\frac{1}{8\pi}\left(\frac{s-m^{2}_{\phi}}{s}\right)=\frac{1}{8\pi}\left(1-\frac{m^{2}_{\phi}}{s}\right)=\frac{1}{8\pi}\left(1-\frac{m^{2}_{\phi}}{m^{2}_{\phi}+\kappa
E_{p}(1-v_{p}\cos\theta_{p})}\right)$ (5.52)

In principle we can use this to do the remaining integral over $\vec{p}$ to
evaluate $\gamma_{\phi}$ for different background distribution functions.
However, since we are interested in the high $\kappa$ limit, we retain just
the leading term

$J=\frac{1}{8\pi}\,,$ (5.53)

which in turn gives

$\gamma_{\phi}=\frac{1}{8\pi}J_{\phi}\,.$ (5.54)

Thus from Eq. (5.2), for a complex $\phi$,

$\frac{1}{2}\Gamma^{(\nu)}_{ba}=\frac{K_{ba}J_{\phi}}{16\pi\kappa}\,,$ (5.55)

where $J_{\phi}$ is defined in Eq. (3.25). For a real $\phi$

$\frac{1}{2}\Gamma^{(\nu)}_{ba}=\frac{K_{ba}J_{\phi}}{8\pi\kappa}\,,$ (5.56)

and $J_{\phi}$ is evaluated putting $f_{\bar{\phi}}=f_{\phi}$. In either case
the formula for $\Gamma^{(\nu)}_{ba}$ is obtained by replacing
$K_{ba}\rightarrow K^{\ast}_{ba}$. Thus, for example, using Eq. (3.26), Eq.
(5.55) yields

$\frac{1}{2}\Gamma^{(\nu)}_{ba}=K_{ba}\left\\{\begin{array}[]{ll}\frac{n_{\phi}+n_{\bar{\phi}}}{32\pi\kappa
m_{\phi}}&\mbox{(NR)}\\\\[12.0pt]
\frac{T^{2}}{192\pi\kappa}&\mbox{(UR)}\,.\end{array}\right.$ (5.57)

These are valid for a complex $\phi$. For a real $\phi$, the corresponding
formulas are

$\frac{1}{2}\Gamma^{(\nu)}_{ba}=K_{ba}\left\\{\begin{array}[]{ll}\frac{n_{\phi}}{8\pi\kappa
m_{\phi}}&\mbox{(NR)}\\\\[12.0pt]
\frac{T^{2}}{96\pi\kappa}&\mbox{(UR)}\,.\end{array}\right.$ (5.58)

For the antineutrinos, the damping matrix is given by the same formulas, but
replacing $K_{ba}\rightarrow K^{\ast}_{ba}$.

Comparing Eq. (5.55) (or Eq. (5.56)) with Eq. (3.24) we see that the ratio of
the imaginary part (damping) to the real part of the effective potential is
$\sim g^{2}/16\pi$. This contrasts with the result in the case of a normal
matter background. In that case the same ratio is further suppressed by the
mass factor $\kappa m_{e}/m^{2}_{W}$ (for $\kappa>m_{e}$) or
$\kappa^{2}/m^{2}_{W}$ (for $m_{e}>\kappa$)[29]. Therefore, in situations
where the effective potential due to a light scalar background may be
relevant, the damping effects may be important since they are not suppressed
by the mass factors. On the other hand, the relative importance of such
damping effects may be negligible if all $g_{a}$ couplings are too small.

###  5.4 Discussion

We have obtained Eqs. (5.57) and (5.58), or their more general versions given
in Eqs. (5.55) and (5.56), by purposely considering a background with only
$\phi$ particles and no fermions $f$. However the inclusion of the fermion
contribution can be carried out straightforwardly in analogous fashion. It
involves calculating in a similar way the contributions denoted by
$\Sigma^{(A)}_{12}$ and $\Sigma^{(CII)}_{12}$ in Eq. (4.52).

From a physical point of view, the damping matrix $\Gamma$ induces decoherence
effects in the propagation of neutrinos. As emphasized in our previous work
Ref. [16], and illustrated again here, the contribution to $\Gamma$ from the
neutrino non-forward scattering process $\nu_{a}+x\rightarrow\nu_{b}+x$, where
$x=f,\phi$, can be determined from the two-loop calculation of $\Sigma_{i}$.
However, since in this case the initial neutrino state is depleted but the
neutrino does not actually disappear (the initial neutrino transitions into a
neutrino of a different flavor but does not decay into a $f\phi$ pair, for
example), we have argued that the effects of the non-forward scattering
processes are more appropriately interpreted in terms of decoherence phenomena
rather than damping. Specifically, the damping matrix should be associated
with decoherence effects in terms of the Lindblad equation and the notion of
the stochastic evolution of the state vector[33, 34, 35, 36, 37]. The idea is
to assume that the evolution due to the damping effects described by $\Gamma$
is accompanied by a stochastic evolution that cannot be described by the
coherent evolution of the state vector. As discussed in detail in Ref. [16],
the result of this idea is that the evolution of the system is described by
the density matrix $\rho$ (in the sense that we can use it to calculate
averages of quantum expectation values) that satisfies the Lindblad equation,

$\partial_{t}\rho=-i[H_{r},\rho]+\sum_{n}\left\\{L_{n}\rho
L^{\dagger}_{n}-\frac{1}{2}L^{\dagger}_{n}L_{n}\rho-\frac{1}{2}\rho
L^{\dagger}_{n}L_{n}\right\\}\,,$ (5.59)

where the $L_{n}$ matrices, representing the _jump operators_ , are related to
$\Gamma$ by

$\Gamma=\sum_{n}L^{\dagger}_{n}L_{n}\,.$ (5.60)

We refer to the terms involving the jump operators in the right-hand-side of
Eq. (5.59) as the decoherence terms.

The damping matrix that we have determined from the two-loop self-energy
calculation can be expressed in this form. For example, consider a real $\phi$
background. Eq. (5.56) can be written as

$\Gamma^{(\ell)}_{ba}=(L^{(\ell)\dagger}_{\phi}L^{(\ell)}_{\phi})_{ba}=\sum_{c}(L^{(\ell)}_{\phi})^{\ast}_{cb}(L^{(\ell)}_{\phi})_{ca}\qquad(\ell=\nu,\bar{\nu})\,,$
(5.61)

with

$\displaystyle(L^{(\nu)}_{\phi})_{ca}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{J_{\phi}}{8\pi\kappa}}\,g_{c}g_{a}\,,$
$\displaystyle(L^{(\bar{\nu})}_{\phi})_{ca}$ $\displaystyle=$
$\displaystyle(L^{(\nu)}_{\phi})^{\ast}_{ca}\,.$ (5.62)

The $L$ matrices are expressed in terms of integrals over the background
particles distribution functions. Going a step further, consider for
illustrative purposes the NR limit. Then using Eq. (3.28),

$\displaystyle(L^{(\nu)}_{\phi})_{ca}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{n_{\phi}}{8\pi\kappa m_{\phi}}}\,g_{c}g_{a}\,,$
$\displaystyle(L^{(\bar{\nu})}_{\phi})_{ca}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{n_{\phi}}{8\pi\kappa
m_{\phi}}}\,g^{\ast}_{c}g^{\ast}_{a}\,.$ (5.63)

It is straightforward to consider the addition of fermions $f$ in the
background, or in fact more complicated superpositions of different background
species. The evaluation of $\Sigma^{(A,CII)}_{12}$ would result in a matrix
$L_{f}$ contributing in Eq. (5.60), so that

$\Gamma^{(\ell)}=L^{(\ell)\dagger}_{\phi}L^{(\ell)}_{\phi}+L^{(\ell)\dagger}_{f}L^{(\ell)}_{f}\,.$
(5.64)

The matrix $L^{(\ell)}_{f}$ would be given in terms of the fermion
distribution by formulas analogous to Eq. (5.4). In general these formulas
predict, for example, a specific dependence of the decoherence terms on the
neutrino energy, depending on the background conditions. This complements the
studies of the decoherence effects that are based on general considerations at
a phenomenological level without a calculation of the decoherence terms. We do
not pursue this any further here, but our results and calculations show the
path for further applications along these lines.

##  6 Conclusions and outlook

In this work we have been concerned with the calculation of the damping terms
that result from non-forward scattering processes when neutrinos propagate in
a background of fermions ($f$) and scalars ($\phi$) interacting via a Yukawa-
type interaction. We determine the contribution of those processes to the
damping matrix $\Gamma$ from the two-loop calculation of the imaginary part of
the thermal neutrino self-energy using the methods of thermal field theory
(TFT). In the context of TFT the two-loop self-energy diagrams suffer from the
so-called _pinch singularities_ , which appear because the expressions contain
two thermal propagators with the same momentum argument. A significant effort
in this work was to show how those singularities are effectively handled by a
judicious use of the properties and parametrizations of the thermal
propagators. The final result of that exercise is a set of formulas for the
two-loop contribution to the imaginary part of the self-energy from which the
damping matrix is determined. The formulas are well-defined integrals over the
background particle momentum distribution functions, which can be evaluated
straightforwardly for different background conditions. For concreteness, we
considered in detail a pure $\phi$ background, with no fermions $f$. We
obtained the corresponding formulas for the damping terms, and evaluated them
in some specific limits of the $\phi$ distribution functions.

As a guide to applications, we discussed briefly in Section 5.4 the connection
between $\Gamma$ and the decoherence described in terms of the Lindblad
equation. There we showed the explicit formulas obtained for the _jump
operators_ that appear in the Lindblad equation using the results of the
calculation of $\Gamma$. We indicated how this approach can be extended to
consider more general backgrounds that include the fermions $f$ or other
particles.

The results we have presented extend our previous work and can be used to
study the decoherence effects in a variety of physical contexts and
environments. As a by-product, we have presented a detailed calculation of the
imaginary part of the two-loop neutrino thermal self-energy, controlling the
pinch singularities, using and illustrating a method that can be applied to
perform similar calculations consistently in other situations of interest.

The work of S. S. is partially supported by DGAPA-UNAM (Mexico) Project No.
IN103019.

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