with the changes just mentioned. We generate the independent loop and phase-
space momenta of the integration sampling points in the CM frame and compute
all the 4-momentum dot-products in the CM frame.

For our numerical study, we choose the benchmark point B (BP-B) mass
parameters (see p. 11) that has $M_{2}>M_{1}+2M_{U}$ allowing the loop to be
cut, giving a strong phase. Again, for the same reasons explained in the decay
case, only the ${\cal A}_{01}^{\prime(1)}$ leads to a baryon asymmetry. The
tree level scattering in SC-2 is exactly as before in Sec. 4.1, except now the
parameters are as in BP-B. We write the $\Delta\hat{\sigma}_{01}\,v$ for the
multiple operator scattering contribution for SC-2 in analogy with Eq. (43),
but now with the matrix element being ${\rm Im}(\hat{\cal
A}^{\prime(\sigma)}_{01})$, which is obtained from the decay contribution in
Eq. (31) by using the crossing relation of Eq. (38). We thermally average this
and compute for SC-2 the thermally averaged cross section difference
$\left<\Delta\hat{\sigma}_{01}^{n}\,v\right>$ between the process and
conjugate process analogous to Eq. (51), and, the baryon asymmetry from
scattering $\hat{\cal A}_{B}^{\prime(\sigma)}$ analogous to Eq. (52). We
present the results next.

For illustration, we give here from the multiple operator contribution, for
BP-B, some sample numerical values for the thermally averaged scattering cross
section difference between the process and conjugate process and the baryon
asymmetry. With the UV state propagator correction included with
$M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$ and $\hat{\Gamma}_{\xi}\\!\\!=\\!\\!1/2$,
the tree-level thermal average integrals $\left<{{{\cal
I}_{0}^{n}}}_{\lambda\lambda^{\prime}}^{(\sigma)}\right>$, for
$x\\!\\!=\\!\\!1/2$, for (SC-1, SC-2) respectively are: $\left<{{{\cal
I}_{0}^{1}}}_{LL}^{(\sigma)}\right>\\!=\\!(2.45,15.7)$, $\left<{{{\cal
I}_{0}^{1}}}_{LR}^{(\sigma)}\right>\\!=\\!(-1.1\times 10^{-3},0.027)$,
$\left<{{{\cal I}_{0}^{2}}}_{LL}^{(\sigma)}\right>\\!=\\!(2.45,15.7)$,
$\left<{{{\cal I}_{0}^{2}}}_{LR}^{(\sigma)}\right>\\!=\\!(-9\times
10^{-4},0.023)$. For the choice $g_{L}\\!\\!=\\!\\!(3/4)\,e^{-\pi/2}$,
$g_{R}\\!=\\!1/4$, $\tilde{g}\\!\\!=\\!\\!1$, for example, we find at
$x\\!\\!=\\!\\!1/2$, the LO thermally averaged cross-section for (SC-1,SC-2)
$n\\!\\!=\\!\\!1$ is
$\left<\sigma_{0}v\right>\\!=\\!(0.16,1)M_{\chi}^{2}/\Lambda^{4}$, and for
$n\\!\\!=\\!\\!2$ is
$\left<\sigma_{0}v\right>\\!=\\!(0.23,1.45)M_{\chi}^{2}/\Lambda^{4}$. For SC-2
$n\\!\\!=\\!\\!2$, $x\\!\\!=\\!\\!1/2$, the difference in the interference
term in the cross section
$\left<\Delta\hat{\sigma}_{01}\,v\right>\\!=\\!-5.2\times
10^{-4}M_{\chi}^{4}/\Lambda^{6}$, and the scattering baryon asymmetry is
$\hat{\cal A}_{B}^{\prime(\sigma)}\\!=\\!3.6\times
10^{-4}M_{\chi}^{2}/\Lambda^{2}$. We thus can write for
$M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$, for SC-2 $n\\!\\!=\\!\\!2$,
$\left<\sigma_{0}v\right>\\!\sim\\!1.45\times 10^{-2}\,g^{2}/\Lambda^{2}$,
$\left<\Delta\hat{\sigma}_{01}\,v\right>\\!\sim\\!-5.2\times
10^{-8}g^{4}/\Lambda^{2}$ and $\hat{\cal
A}_{B}^{\prime(\sigma)}\\!\sim\\!3.6\times 10^{-6}\,g^{2}$, including the
couplings scaling.

From the above $\left<\sigma_{0}v\right>$ and
$\left<\Delta\hat{\sigma}_{01}\,v\right>$ numbers for BP-B,
$x\\!\\!=\\!\\!1/2$, $g_{L}\\!\\!=\\!\\!(3/4)\,e^{-i\pi/2}$,
$g_{R}\\!\\!=\\!\\!1/4$, $\tilde{g}\\!\\!=\\!\\!1$, with the UV propagator
correction factor present, we extract the thermal averages of the LO cross
section and tree-loop interference functions parametrized in Eqs. (31) and
(32) of Ref. [2], and obtain for SC-2 $n\\!\\!=\\!\\!2$,
$\left<(1/(2\hat{E}_{n}\,2\hat{E}_{i}))\,\hat{f}_{00}^{(\sigma)}\right>\\!\approx\\!12.3$,
and
$\left<(1/(2\hat{E}_{n}\,2\hat{E}_{i}))\,\hat{f}_{01}^{(\sigma)}\right>\\!\approx\\!-0.1$,
for the multiple operator scattering contributions.

In Fig. 18 we show for BP-B, SC-2 $n\\!\\!=\\!\\!2$, $x\\!\\!=\\!\\!1/2$, the
dimensionless LO thermally averaged cross section
${\left<\hat{\sigma}_{0}\,v\right>}$ (in units of $M_{\chi}^{2}/\Lambda^{4}$),
the (negative of the) tree-loop interference cross section difference
$-\left<\Delta\hat{\sigma}_{01}v\right>$ (in units of
$M_{\chi}^{4}/\Lambda^{6}$), and the baryon asymmetry from scattering
$\hat{\cal A}_{B}^{\prime(\sigma)}$ (in units of $M_{\chi}^{2}/\Lambda^{2}$).
The UV propagator corrections are included with
$M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$ and $\hat{\Gamma}_{\xi}\\!\\!=\\!\\!1/2$.

Figure 18: For BP-B, SC-2 $n\\!\\!=\\!\\!2$, $x\\!\\!=\\!\\!1/2$, the
$\left<\sigma_{0}v\right>$ (in units of $M_{\chi}^{2}/\Lambda^{4}$) (left-
panel), $-\left<\Delta\hat{\sigma}_{01}v\right>$ (in units of
$M_{\chi}^{4}/\Lambda^{6}$) (middle-panel), and $\hat{\cal
A}_{B}^{\prime(\sigma)}$ (in units of $M_{\chi}^{2}/\Lambda^{2}$) (right-
panel) from the multiple-operator contributions, with the UV propagator
corrections included with $M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$ and
$\hat{\Gamma}_{\xi}\\!\\!=\\!\\!1/2$.

In Fig. 19 we show for BP-B the same quantities now as a function of $x$, for
$g_{L}\\!\\!=\\!\\!(3/4)e^{-i\pi/2}$, $g_{R}\\!\\!=\\!\\!1/4$,
$\tilde{g}\\!\\!=\\!\\!1$ for different choices of
$(M_{\chi}/\Lambda,\hat{\Gamma}_{\xi})$.

Figure 19:  For BP-B, the $\left<\sigma_{0}v\right>$ (in units of
$M_{\chi}^{2}/\Lambda^{4}$) (left), $-\left<\Delta\hat{\sigma}_{01}v\right>$
(in units of $M_{\chi}^{4}/\Lambda^{6}$) (middle), and $\hat{\cal
A}_{B}^{\prime(\sigma)}$ (in units of $M_{\chi}^{2}/\Lambda^{2}$) (right),
from the multiple operator SC-2 $n\\!\\!=\\!\\!2$ contributions for
$g_{L}=(3/4)e^{-i\pi/2}$, $g_{R}=1/4$, with UV propagator correction included
taking $(M_{\chi}/\Lambda,\hat{\Gamma}_{\xi})$ to be $(1/10,1/2)$ (solid),
$(1/10,1/10)$ (dot-dashed), $(1/100,5)$ (dotted), and, without UV propagator
correction (dashed).

We see that for $(M_{\chi}/\Lambda,\hat{\Gamma}_{\xi})\\!=\\!(1/100,5)$ (so
that $\tilde{\Gamma}_{\xi}\\!=\\!1/2$ as before), we find a similar shape as
for the $(1/10,1/2)$ case (with a factor of $1/10$ scaling of $x$), and the
size of $\left<\sigma_{0}v\right>$ and
$-\left<\Delta\hat{\sigma}_{01}v\right>$ are both higher by a factor of about
100, leaving $\hat{\cal A}_{B}^{(\sigma)}$ to be about the same. We can also
see that for $(M_{\chi}/\Lambda,\hat{\Gamma}_{\xi})\\!=\\!(1/10,1/10)$, we
find that the behavior is largely as for the $(1/10,1/2)$ case except for the
$\left<\sigma_{0}v\right>$ and $-\left<\Delta\hat{\sigma}_{01}v\right>$ both
being higher by a factor of about 5, again leaving $\hat{\cal
A}_{B}^{(\sigma)}$ to be about the same. These comparisons are without
including the powers of $M_{\chi}/\Lambda$ scaling factors that have been
pulled out as indicated above.

Denoting the multiple operator scattering cross section difference as
$\left<\Delta\hat{\sigma}^{\prime}_{01}\,v\right>$, we can compare, for
example for the $(M_{\chi}/\Lambda,\hat{\Gamma}_{\xi})\\!\\!=\\!\\!(1/10,1/2)$
case here with the single operator contribution
$\left<\Delta\hat{\sigma}_{01}\,v\right>$ in the previous section. Including
the $M_{\chi}/\Lambda$ scaling factors but suppressing the coupling
dependence, we find,
$\left<\Delta\hat{\sigma}^{\prime}_{01}\,v\right>/\left<\Delta\hat{\sigma}_{01}\,v\right>\\!\sim\\!(2\pi^{2})(\Lambda/M_{\chi})^{2}\left<\hat{f}_{01}^{{}^{\prime}(\sigma)}\right>/\left<\hat{f}_{01}^{(\sigma)}\right>\sim
10^{3}$. Similarly, comparing the multiple operator to single operator
scattering baryon asymmetry, for $M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$, we find
$\hat{\cal A}_{B}^{\prime(\sigma)}/\hat{\cal
A}_{B}^{(\sigma)}\\!\sim\\!2\pi^{2}(\Lambda/M_{\chi})^{2}\left<\hat{f}_{01}^{{}^{\prime}(\sigma)}\right>/\left<\hat{f}_{01}^{(\sigma)}\right>\cdot\left<\hat{f}_{00}^{(\sigma)}\right>/\left<\hat{f}_{00}^{{}^{\prime}(\sigma)}\right>\\!\sim\\!10^{4}$.

With the UV propagator correction included, we see from Fig. 19 that the
$\left<\Delta\hat{\sigma}_{01}\,v\right>$ is substantial only in the domain
$x\in(0.1,10)$ for $M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$. This domain shifts to
smaller $x$ as $M_{\chi}/\Lambda$ decreases. These are in contrast to the case
without the UV propagator correction (see the dashed curve), for which
$\left<\Delta\hat{\sigma}_{01}\,v\right>$ keeps increasing as $x$ decreases.
However, this growth in the $x\ll 1$ region is not to be trusted since we see
the UV propagator corrections modifies the behavior as the typical momenta
become bigger than $\Lambda$, bringing in sensitivity to the UV physics. This
is particularly true here for SC-2 as the UV correction is resonant since it
is in the s-channel. The scattering baryon asymmetry with and without the UV
correction are in good agreement for $x\gtrsim 1$, but differ for $x\lesssim
1$ where the latter keeps growing but the former turns around.

Thus, if the first operator in Eq. (2) is also present in the effective theory
in addition to the operator in Eq. (1), and if the kinematics allows (i.e. if
$M_{2}\\!>\\!M_{1}\\!+\\!2M_{U}$), the multiple operator scattering
contribution obtained from crossing Fig. 11 (left) could be dominant as it is
enhanced by a loop and $(\Lambda/M_{\chi})^{2}$ factors compared to the single
operator contribution. If any of these conditions is not met, the single
operator contribution of Sec. 4.1 involving only the operator in Eq. (1) would
remain as the only contribution to the scattering asymmetry.

##  5 Conclusions

In this work, we compute the baryon asymmetry generated in the decay and
scattering processes involving the electromagnetically charge-neutral fermion
$\chi$ and quark-like fermions ${U},{D}$, in the effective theory developed in
Ref. [2]. In the theory, the Dirac fermion $\chi$ with Dirac mass $M_{\chi}$
is split into a pair of Majorana fermions ${\cal X}_{n}$ due to the presence
of a baryon number violating $\chi$ Majorana mass. The $\chi,{U},{D}$
interaction is of the vector-vector (VV) form
$(1/\Lambda^{2})\,(\bar{\chi}\Gamma^{\mu}{U})\,(\bar{{D}^{c}}\Gamma_{\mu}{D})$.
Being a dimension-6 operator, it is suppressed by the cutoff scale $\Lambda$
as shown.

For two representative choices of parameters, benchmark points BP-A and BP-B,
we compute the unitary matrix that diagonalizes the $\chi$ mass matrix and
determine the VV effective vertex for the mass eigenstates ${\cal X}_{n}$.
Both the benchmark points we choose are in the pseudo-Dirac limit in which the
mass splitting between the ${\cal X}_{n}$ masses, namely, $M_{2}-M_{1}\ll
M_{\chi}$. We show the dependence of the ${\cal X}_{n}$ mass eigenvalues
$M_{n}$ on the Majorana mass and phase.

We identify loop amplitudes for the ${\cal X}_{n}$ decay and scattering
processes such that the interference term between them and the tree-level
amplitude is different for the process and its conjugate process, leading to a
baryon asymmetry. The phases in the Majorana mass, and in the VV interaction
coupling act like weak phases (opposite for process and conjugate process),
and the intermediate state ${D},{U}$ in a loop going on-shell produces a phase
of $\pi/2$ that acts like a strong phase (same for process and conjugate
process). These phases lead to $C$ and $CP$ violation, and along with the
baryon number violation due to the Majorana mass, produces a baryon asymmetry.

We refer to the loop processes that contain only the above VV effective
operator as the single operator contributions. We show that if the
$(1/\Lambda^{2})\,(\bar{\cal X}\Gamma^{\mu}{U})\,(\bar{U}\Gamma_{\mu}{\cal
X})$ effective operator is also present in the effective theory, and if the
kinematics allows it, additional loop contributions can be identified that
contain both of these effective vertices, that also produces a baryon
asymmetry. We refer to such contributions as multiple operator contributions.
In BP-A the baryon asymmetry is from the single operator contributions, while
in BP-B it is due to the multiple operator contributions.

We write down the Feynman diagrams for decay and scattering process, and in
the mass basis, compute the matrix elements, integrate over the loop momenta,
fold in the phase space, and compute the resulting baryon asymmetry. The
integrals we encounter are multi-scale and multi-dimensional, and we expend
considerable effort to make them computationally tractable. We present our
baryon asymmetry results in a general manner without picking a specific scale
of new physics, i.e. $\Lambda$ or $M_{\chi}$, but give scaling factors
involving the $\Lambda$ and $r_{M\Lambda}\\!\\!=\\!\\!M_{\chi}/\Lambda$ that
can be used to compute the actual numbers in any specific scenario with its
preference for the scale.

We compute numerically the decay rate asymmetry between ${\cal
X}_{n}\rightarrow{U}{D}{D}$ and ${\cal
X}_{n}\rightarrow\bar{U}\bar{D}\bar{D}$, for the benchmark points and for
sample coupling choices. For BP-A, we find the tree-level decay width
$\Gamma_{{\cal X}}\sim
10^{-6}\,\tilde{g}^{2}g_{L,R}^{2}\,M_{\chi}^{5}/\Lambda^{4}$, and from the
single operator contributions, we find the decay rate difference
$\Delta\hat{\Gamma}_{01}\sim-10^{-12}\,\tilde{g}^{4}g_{L,R}^{4}\,M_{\chi}^{9}/\Lambda^{8}$,
which yields the baryon asymmetry ${\cal A}_{B}\sim
10^{-6}\,\tilde{g}^{2}g_{L,R}^{2}\,M_{\chi}^{4}/\Lambda^{4}$. For BP-B, we
find $\Gamma_{{\cal X}}\sim
10^{-5}\,\tilde{g}^{2}g_{L,R}^{2}\,M_{\chi}^{5}/\Lambda^{4}$, and from the
multiple operator contributions, we find
$\Delta\hat{\Gamma}_{01}\sim-10^{-13}\,\tilde{g}^{2}g_{L,R}^{4}\,M_{\chi}^{7}/\Lambda^{6}$,
and ${\cal A}_{B}^{\prime}\sim
10^{-8}\,g_{L,R}^{2}\,M_{\chi}^{2}/\Lambda^{2}$. We show the dependence of the
baryon asymmetry on the theory parameters in Figs. 7, 8, and 13.

We use crossing relations on the decay matrix elements computed above to
obtain the scattering matrix elements. We then compute the thermally averaged
scattering cross section difference between the process ${\cal
X}_{n}\bar{Q}\rightarrow QQ$ and its conjugate process ${\cal
X}_{n}Q\rightarrow\bar{Q}\bar{Q}$ (with $Q=\\{{U},{D}\\}$) as a function of
$x=M_{\chi}/T$, with $T$ being the temperature of the bath in which the ${\cal
X}_{n}$ are interacting. In the thermal spectrum, there is a probability for
the ${\cal X}$ to have $p_{\chi}>\Lambda$, and therefore UV physics effects
can be sampled. We capture this effect by having a propagating UV state $\xi$
(as in Ref. [2]) resolve the VV effective interaction, and compute the baryon
asymmetry after including this correction. We find that the scattering cross
section asymmetry $\left<\Delta\sigma_{01}v\right>$ is generated predominantly
in the $0.01\lesssim x\lesssim 10$ range (cf. Figs. 17 and 19), indicating
that our effective theory computation still gives a fairly good estimate of
the size of the asymmetry generated in our framework. This also indicates to
us that for a truly accurate scattering cross section result, one better work
with a specific UV completion.

From our numerical computation, for BP-A, for $x=1/2$, from the single
operator contribution, we find the thermally averaged LO cross section
$\left<\sigma_{0}v\right>\sim 10\,g_{L,R}^{2}M_{\chi}^{2}/\Lambda^{4}$, and
from the single operator contributions, we find the thermally averaged cross
section difference between the process and conjugate process
$\left<\Delta\hat{\sigma}_{01}\,v\right>\sim-4\times
10^{-5}\,g_{L,R}^{4}M_{\chi}^{6}/\Lambda^{8}$, which yields the cross section
asymmetry $\hat{\cal A}_{B}^{(\sigma)}\sim 6\times
10^{-6}\,g_{L,R}^{2}M_{\chi}^{4}/\Lambda^{4}$. For BP-B, $x=1/2$, from the
multiple operator contribution, we find $\left<\sigma_{0}v\right>\sim
1.5g_{L,R}^{2}M_{\chi}^{2}/\Lambda^{4}$,
$\left<\Delta\hat{\sigma}_{01}\,v\right>\sim-5\times
10^{-4}\,g_{L,R}^{4}M_{\chi}^{4}/\Lambda^{6}$, and, $\hat{\cal
A}_{B}^{(\sigma)}\sim 4\times 10^{-4}\,g_{L,R}^{2}M_{\chi}^{2}/\Lambda^{2}$.
We vary the effective theory parameters about the benchmark points and show
their effects on the baryon asymmetry generated.

In summary, our finding is that the size of the baryon asymmetry generated in
the ${\cal X}_{n}$ decay and scattering processes that we identify looks very
interesting for explaining the observed BAU. Our results here would go in as
the collision term inputs into the Boltzmann equations (cf. Ref. [2]) that
govern the number densities of the baryons and antibaryons in the early
Universe. To determine the preferred mass scale $M_{\chi}$ that leads to the
observed BAU, we numerically solve the Boltzmann equations in a follow-up work
[3]. This would indicate the $M_{\chi}$ scale required to produce the observed
BAU. Knowing the scale would help us determine the prospects of observing such
new physics at accelerators such as the large hadron collider (LHC), or in
precision experiments such as the neutron-antineutron oscillation experiments
(see Ref. [2] and references therein, and Ref. [13]).

##  Appendix A Computation of the Matrix Element

Here we provide various technical details of computing the matrix elements. We
fall back on some first principles calculations to handle carefully the two
identical ${D}$ in the interaction, and the Majorana nature of the ${\cal
X}_{n}$. We start by discussing the normalization of the ${D}{D}{U}$ final
state. Following this, we compute the matrix element, which we split into two
fermion-line parts, the ${\cal X}{U}$ line part, and another ${D}{D}$ line
part in the final state and loops, and discuss them separately. We then put
them together to compute the tree-loop interference term in the mod-squared of
the matrix element, needed for the baryon asymmetry computation. We give the
details of how we renormalize (subtract) the $\bar{\cal X}_{m}{\cal X}_{n}$
2-point function part of diagram B, to be consistent with working in the mass
basis.

###  A.1 The ${D}{D}{U}$ state

In computing the matrix element, we project onto a color-singlet combination
of the 3-body ${U}{D}{D}$ final state as

$\left.|as{{\bf q}_{1}},bt{{\bf q}_{2}},cr{{\bf
q}_{U}}\right>=\frac{\epsilon^{abc}}{\sqrt{6}}\frac{1}{\sqrt{2}}\,\sqrt{2E_{\bf
q_{U}}}\sqrt{2E_{\bf q_{2}}}\sqrt{2E_{\bf q_{1}}}\,{\hat{a}_{{\bf
q}_{U}}}^{cr\,\dagger}{a_{{\bf q}_{2}}^{bt}}^{\dagger}{a_{{\bf
q}_{1}}^{as}}^{\dagger}\left.|0\right>\ ,$ (54)

where $a,b,c$ denote the color indices, $s,t,r$ the spin indices, and,
$q_{1},q_{2},q_{U}$ the momenta of the ${D},{D},{U}$ respectively, we
specialize to the $\epsilon^{abc}$ here for having a color singlet final state
with the $1/(\sqrt{2}\sqrt{6})$ providing the correct normalization, and the
fermionic creation operators anti-commute.

We can write the color-singlet state as a wave packet by introducing into Eq.
(54) a weight $K^{str}$ for the spin helicities, and $\phi({{\bf q}_{U}},{{\bf
q}_{1}},{{\bf q}_{2}})$ for the momenta. We pay particular attention to the
two identical particles, namely ${D}{D}$ (i.e. $\widebar{{D}^{c}}{D}$) in the
final-state, but distinguished by spin or color labels. Focusing on this
sector for now, some example combinations of the two spins are:
$K^{st}=\delta^{st}$ is a symmetric same spin combination;
$K^{st}=1,\forall\\{s,t\\}$, democratically combines all spins;
$K^{st}=\epsilon^{st}$ is an antisymmetric different spin combination. As
required for two fermions, we must get a minus sign under the interchange of
the two ${D}$ particles, i.e. under $as{{\bf q}_{1}}\leftrightarrow bt{{\bf
q}_{2}}$, which requires that $K^{st}\phi({\bf q_{1}},{\bf
q_{2}})\epsilon^{ab}$ combined must yield this minus sign. Since the last
(i.e. the color part) is already antisymmetric and will yield a minus sign
under interchange, the $K^{st}\phi({{\bf q}_{1}},{{\bf q}_{2}})$ part must
yield a plus sign. We note in particular that the simple state obtained by
taking $K^{st}=1,\forall\\{s,t\\}$, and $\phi({{\bf q}_{1}},{{\bf q}_{2}})=1$
as in Eq. (54) is a fine ${D}{D}$ state with the correct minus sign being
obtained on interchange of the two particles simply on account of the
anticommutation property of the $a^{\dagger}$.

###  A.2 The ${D}{D}$ final state part $(M^{12}_{0})$

Here we describe the computation of $(M^{12}_{0})$, the portion of the matrix
element involving the ${D}$ final state, encountered in Eq. (7). We pay
careful attention to the presence of more than one identical ${D}$ which lead
to multiple contractions in the matrix elements.

We start by being general in the effective interaction and include scalar-
scalar (SS) and vector-vector (VV) Lorentz structures as discussed in Ref.
[2], where for the VV interaction of Eq. 1 we have
$\widetilde{\Gamma}=\widetilde{G}=\tilde{g}\gamma^{\mu}$ and
$\Gamma=G^{n}_{V\mu}=\gamma_{\mu}\left(\hat{g}_{L_{n}}P_{L}+\hat{g}_{R_{n}}P_{R}\right)$,
while for the SS interaction we have a similar coupling structure but without
the $\gamma_{\mu}$. Here we explain details involving the contraction of the
field bilinear $\overline{{D}^{c}}\,\widetilde{\Gamma}{D}_{b}$ appearing in
the interaction, with the two identical particle ${D}{D}$ (final) state. We
use this to compute the $|M_{12}|^{2}$ appearing in the decay width
$\Gamma({\cal X}_{n}\rightarrow\widebar{{D}^{c}}{D}{U})$, discussed in Sec. 3,
where two identical ${D}$ particles appear in the final state since
$\widebar{{D}^{c}}{D}\sim{D}{D}$.

We evaluate the contraction of the field bilinear
$\overline{{D}^{c}}\,\widetilde{\Gamma}{D}_{b}$ with the 2-${D}$ state as in
Eq. (54), for now taking the simple weighting $K^{st}=1,\forall\\{s,t\\}$, and
$\phi({{\bf q}_{1}},{{\bf q}_{2}})=1$. Since the two ${D}$ are in the final
state, we evaluate the contraction with the hermitian conjugate of
$\overline{{D}^{c}}\,\widetilde{\Gamma}{D}$, namely
$\overline{{D}}\,\widebar{\widetilde{\Gamma}}{D}^{c}$, with
$\widebar{\widetilde{\Gamma}}\equiv\gamma^{0}\widetilde{\Gamma}^{\dagger}\gamma^{0}$.
Thus, one of the things we need to evaluate the decay amplitude $\Gamma({\cal
X}_{n}\rightarrow\widebar{{D}^{c}}{D}{U})$ is the matrix element
$M_{12}^{st}\equiv\left<btq_{2},asq_{1}|\overline{{D}}_{a^{\prime}}\,\widebar{\widetilde{\Gamma}}{D}_{b^{\prime}}^{c}|0\right>/\sqrt{2}$.
Now,
$\sum_{a^{\prime},b^{\prime}}\overline{{D}}_{a^{\prime}}(x)\,\widebar{\widetilde{\Gamma}}{D}_{b^{\prime}}^{c}(x)\epsilon^{a^{\prime}b^{\prime}}\supset\int\sum_{a^{\prime},b^{\prime},s^{\prime},t^{\prime}}[dq^{\prime}_{1}][dq^{\prime}_{2}]{a_{{\bf
q}_{1}^{\prime}}^{a^{\prime}s^{\prime}}}^{\dagger}{a_{{\bf
q}_{2}^{\prime}}^{b^{\prime}t^{\prime}}}^{\dagger}\epsilon^{a^{\prime}b^{\prime}}\bar{u}^{s^{\prime}}_{q_{1}^{\prime}}\widebar{\widetilde{\Gamma}}v^{t^{\prime}}_{q_{2}^{\prime}}e^{i(q^{\prime}_{1}+q^{\prime}_{2})x}+...$.
We thus have the factor $\left<0|{a_{\bf q_{1}}^{as}}{a_{\bf
q_{2}}^{bt}}{a_{\bf q_{1}^{\prime}}^{a^{\prime}s^{\prime}}}^{\dagger}{a_{\bf
q_{2}^{\prime}}^{b^{\prime}t^{\prime}}}^{\dagger}|0\right>$ in $M_{12}^{st}$,
which yields two contractions (due to the two identical ${D}$), to give

$-\delta({{\bf q}^{\prime}_{1}-{\bf q}_{1}})\delta({{\bf q}^{\prime}_{2}-{\bf
q}_{2}})\delta^{a^{\prime}a}\delta^{s^{\prime}s}\delta^{b^{\prime}b}\delta^{t^{\prime}t}+\delta({{\bf
q}^{\prime}_{1}-{\bf q}_{2}})\delta({{\bf q}^{\prime}_{2}-{\bf
q}_{1}})\delta^{a^{\prime}b}\delta^{s^{\prime}t}\delta^{b^{\prime}a}\delta^{t^{\prime}s}$.

The two contractions are shown in Fig. 20.

Figure 20: The left and right diagrams show the two contractions of the
$\overline{D}\,\bar{\widetilde{\Gamma}}\,D^{c}$ part of the interaction with
the ${D}{D}$ final state. The momentum flow, spin, and color indices are shown
as $(q,s,a)$ respectively.

Using the antisymmetry in the color indices, which flips the sign on the
second contraction (thus removing the relative minus sign), we obtain666We do
not include the factor of $(i\epsilon^{abc}/\Lambda^{2})$ from the interaction
vertex here but take it into account elsewhere. Also, we include elsewhere the
factor $e^{i(q_{1}+q_{2})x}/\sqrt{6}$. We choose to include only the
$1/\sqrt{2}$ factor from the final state normalization here.

$M_{12}^{st}=-\frac{1}{\sqrt{2}}(\bar{u}^{s}_{q_{1}}\widebar{\widetilde{\Gamma}}v^{t}_{q_{2}}+\bar{u}^{t}_{q_{2}}\widebar{\widetilde{\Gamma}}v^{s}_{q_{1}})\
.$ (55)

Taking the mod-square of this and summing over $s,t$, we have
$|M_{12}|^{2}=\sum_{s,t}|M_{12}^{st}|^{2}$, which is,

$|M_{12}|^{2}=\frac{1}{2}\left\\{{\rm Tr}[\widetilde{\Gamma}(\hbox
to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{1}+M)(\widebar{\widetilde{\Gamma}}-\widetilde{\Gamma}^{c})(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{2}-M)]+(q_{1}\leftrightarrow
q_{2})\right\\}\ ,$ (56)

where $\Gamma^{c}\equiv C\Gamma^{*}C$, and we have used the spinor identity

$\bar{v}_{q_{1}}^{s}\Gamma
u_{q_{2}}^{t}=\bar{v}_{q_{2}}^{t}(-\bar{\Gamma}^{c})u_{q_{1}}^{s}\ ,$ (57)

and similarly for $u,v$ flipped. We note that $\widebar{\bar{\Gamma}}=\Gamma$
and $\left(\widebar{\bar{\Gamma}}\right)^{c}=\Gamma^{c}$. In particular, these
properties apply to $\widetilde{\Gamma}$ also.

We first consider the $SS$ interaction with
$\widetilde{\Gamma}=\tilde{g}_{L}P_{L}+\tilde{g}_{R}P_{R}$. For this, we find
$\widebar{\widetilde{\Gamma}}=\widetilde{\Gamma}^{c}$, and Eq. (56) then
yields $|M_{12}^{SS}|^{2}=0$. In addition to the discussion in Ref. [2]
showing that the $SS$ interaction cannot be written down, if one were to
persist, we have now shown that it would not cause this decay with the two
identical ${D}$ in the final state. It should be remembered that this result
has been obtained for $K^{st}=1,\forall\\{s,t\\}$, and $\phi({{\bf
q}_{1}},{{\bf q}_{2}})=1$.

Continuing with the $SS$ interaction, we may still ask if contracting with a
2-${D}$ state with some other $K^{st}$ and $\phi({{\bf q}_{1}},{{\bf q}_{2}})$
may be nonzero. To address this question, we now generalize the above result.
Due to the presence of the $K^{st}$, writing the spinor products of Eq. (55)
summed over the spins as a ${\rm Tr}(...)$ will not work so straightforwardly.
To facilitate this, we introduce the corresponding helicity projectors and
then sum unrestricted over the spins, leading to ${\rm Tr}(...)$ but now with
the projectors. The helicity projectors for $u$-spinors is $\hat{P}_{s}({\bf
q})\equiv(1/2)(1\pm\hat{q}^{i}\Sigma^{i})$ for $s=\\{1,2\\}$, and for
$v$-spinors they are $\hat{P}^{\prime}_{s}({\bf
q})\equiv(1/2)(1\mp\hat{q}^{i}\Sigma^{i})$ (i.e.
$\hat{P}^{\prime}_{1,2}=\hat{P}_{2,1}$), with
$\Sigma^{i}=\begin{pmatrix}\sigma^{i}&0\\\ 0&\sigma^{i}\end{pmatrix}$, and
$\hat{q}^{i}$ is the unit 3-vector along ${\bf q}$. These projectors satisfy
$\hat{P}_{s}({\bf q})\hat{P}_{t}({\bf q})=\delta^{st}\hat{P}_{s}({\bf q})$.
Introducing these projectors and denoting $\hat{P}_{s}({\bf q}_{i})\equiv
P_{si}$ and summing unrestricted over $s^{\prime},t^{\prime}$ allows us to
generalize Eq. (55) as

$|M_{12}^{st}|^{2}=\sum_{s^{\prime},t^{\prime}}|K^{st}|^{2}|\phi({{\bf
q}_{1}},{{\bf
q}_{2}})|^{2}\frac{1}{2}|(\bar{u}^{s^{\prime}}_{q_{1}}P_{s1}\widebar{\widetilde{\Gamma}}P^{\prime}_{t2}v^{t^{\prime}}_{q_{2}}+\bar{u}^{t^{\prime}}_{q_{2}}P_{t2}\widebar{\widetilde{\Gamma}}P^{\prime}_{s1}v^{s^{\prime}}_{q_{1}})|^{2}\
.$ (58)

Since now the sum over $s^{\prime},t^{\prime}$ is unrestricted, we can write
this as a ${\rm Tr}(...)$. Using the notation $P_{s1}\Gamma
P^{\prime}_{t2}\equiv\Gamma_{s1t^{\prime}2}$ we have

$\displaystyle|M_{12}^{st}|^{2}=|K^{st}|^{2}|\phi({{\bf q}_{1}},{{\bf
q}_{2}})|^{2}\frac{1}{2}\left\\{{\rm
Tr}[\widetilde{\Gamma}_{t^{\prime}2s1}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{1}+M)(\widebar{\widetilde{\Gamma}}-\widetilde{\Gamma}^{c})_{s1t^{\prime}2}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{2}-M)]\right.$
$\displaystyle\left.+{\rm Tr}[\widetilde{\Gamma}_{s^{\prime}1t2}(\hbox
to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{2}+M)(\widebar{\widetilde{\Gamma}}-\widetilde{\Gamma}^{c})_{t2s^{\prime}1}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{1}-M)]\right\\}\ ,$ (59)

where we have used the identity
$(\overline{\Gamma_{t2s^{\prime}1}})^{c}=(\bar{\Gamma}^{c})_{s1t^{\prime}2}$.
Eq. (59) is the generalization we seek. For the $SS$ interaction, we have
$\widetilde{\Gamma}^{c}=\widebar{\widetilde{\Gamma}}$, and from Eq. (59) we
see that $|M_{12}^{st}|^{2}=0$ even for a general $K^{st}$. Again, even if we
persisted and considered the $SS$ interaction, it would not induce a decay
with ${D}^{s}{D}^{t}$ in the final state for any orientations of the spins.

We consider next the $VV$ interaction for $K^{st}=1,\forall\\{s,t\\}$, and
$\phi({{\bf q}_{1}},{{\bf q}_{2}})=1$. For the $VV$ interaction we have
$\widetilde{\Gamma}\equiv\widetilde{G}_{V}^{\mu}=\tilde{g}\gamma^{\mu}$, and
we must have $\tilde{g}_{L}=\tilde{g}_{R}\equiv\tilde{g}$ as we argued in Ref.
[2]. For this we have
$\widebar{\widetilde{\Gamma}}=-\widetilde{\Gamma}^{c}=\bar{\widetilde{G}}_{V}^{\mu}={\tilde{g}}^{*}\gamma^{\mu}$,
and from Eq. (55), we have

$(M^{12})^{st}_{\mu}=-\frac{1}{\sqrt{2}}(\bar{u}^{s}_{q_{1}}{\widebar{\widetilde{G}}_{V}}_{\mu}v^{t}_{q_{2}}+\bar{u}^{t}_{q_{2}}{\widebar{\widetilde{G}}_{V}}_{\mu}v^{s}_{q_{1}})\
.$ (60)

Similarly, for the conjugate process, we find

$(M^{c\,12})^{st}_{\mu}=-\frac{1}{\sqrt{2}}(\bar{u}^{s}_{q_{1}}(\widetilde{G}_{V})_{\mu}v^{t}_{q_{2}}+\bar{u}^{t}_{q_{2}}(\widetilde{G}_{V})_{\mu}v^{s}_{q_{1}})\
.$ (61)

The mod-squared of this (for different Lorentz indices $\mu,\tau$) summed over
$s,t$, is
$|{M_{12}}|^{2}_{\mu\tau}=\sum_{s,t}{M_{12}^{st}}_{\tau}^{\dagger}{M_{12}^{st}}_{\mu}$,
which, from Eq. (56), is

$\displaystyle|M^{12}|^{2}_{\mu\tau}$ $\displaystyle=$
$\displaystyle\left\\{{\rm Tr}[(\widetilde{G}_{V})_{\tau}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{1}+M){\widebar{\widetilde{G}}_{V}}_{\mu}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{2}-M)]+(q_{1}\leftrightarrow
q_{2})\right\\}\ ,$ (62) $\displaystyle=$
$\displaystyle|\tilde{g}|^{2}\left\\{{\rm Tr}[\gamma_{\tau}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{1}+M)\gamma_{\mu}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{2}-M)]+(q_{1}\leftrightarrow
q_{2})\right\\}$ $\displaystyle=$ $\displaystyle 2|\tilde{g}|^{2}{\rm
Tr}[\gamma_{\tau}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{1}+M)\gamma_{\mu}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{2}-M)]\ ,$ (63)

where the last line is due to the fact that the $(q_{1}\leftrightarrow q_{2})$
term gives the same contribution since the trace of an odd number of
$\gamma$-matrices is zero. We show at the end of Appendix A.3 that this can be
thought of equivalently as the cut of a loop diagram, and introduce the
notation $|M^{12}|^{2}=L_{12{}}$ to capture this equivalence. For the
conjugate process, we similarly have

$\displaystyle|M^{c\,12}|^{2}_{\mu\tau}$ $\displaystyle=$
$\displaystyle\left\\{{\rm Tr}[{\widebar{\widetilde{G}}_{V}}_{\tau}(\hbox
to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{1}+M)(\widetilde{G}_{V})_{\mu}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{2}-M)]+(q_{1}\leftrightarrow
q_{2})\right\\}\ ,$ (64) $\displaystyle=$
$\displaystyle|M^{12}|^{2}_{\mu\tau}\ .$

###  A.3 The ${D}{D}$ loop function in diagram A1

Here we compute the ${D}{D}$ Loop Function $L_{34}^{\sigma\nu}$ appearing in
diagram A1 in the loop contribution to the decay amplitude for ${\cal
X}_{n}\rightarrow\widebar{{D}^{c}}{D}{U}$, discussed in Sec. 3, and shown in
Fig. 5. We handle carefully the two identical ${D}$ particles appearing in the
loop.

For the general Lorentz structure of the interaction alluded to in Appendix
A.2, the loop function in coordinate space is

$L_{34}\propto\left<0|{\cal T}\
(\bar{D}_{a}\bar{\widetilde{\Gamma}}D^{c}_{b}\epsilon^{abc})_{y}(\bar{D}^{c}_{b^{\prime}}\widetilde{\Gamma}D_{a^{\prime}}\epsilon^{a^{\prime}b^{\prime}c^{\prime}})_{w}|0\right>,$

where ${\cal T}$ is the time-ordering symbol, and the two operator bilinears
are evaluated at different space-time points $y,w$. The $D,D^{c}$ fields when
contracted yield two propagators $\Delta_{yw}$ connecting the points. From the
usual mode expansion for the fields, in momentum space, we find the term

$L_{34}\propto\left<0|{b_{\bf q_{3}}^{as}}{b_{\bf q_{4}}^{bt}}{b_{\bf
q_{4}^{\prime}}^{b^{\prime}t^{\prime}}}^{\dagger}{b_{\bf
q_{3}^{\prime}}^{a^{\prime}s^{\prime}}}^{\dagger}|0\right>(\bar{v}_{q_{3}}^{s}\bar{\widetilde{\Gamma}}u_{q_{4}}^{t})(\bar{u}_{q^{\prime}_{4}}^{t^{\prime}}\widetilde{\Gamma}v_{q^{\prime}_{3}}^{s^{\prime}})\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}\
.$

From the first factor it is evident that two contractions are possible, which
is due to the two ${D}$ particles being identical. The two contractions yield

$\delta({{\bf q}_{4}-{\bf q}^{\prime}_{4}})\delta({{\bf q}_{3}-{\bf
q}^{\prime}_{3}})\delta^{aa^{\prime}}\delta^{ss^{\prime}}\delta^{bb^{\prime}}\delta^{tt^{\prime}}-\delta({{\bf
q}_{3}-{\bf q}^{\prime}_{4}})\delta({{\bf q}_{4}-{\bf
q}^{\prime}_{3}})\delta^{ab^{\prime}}\delta^{st^{\prime}}\delta^{ba^{\prime}}\delta^{ts^{\prime}}\
,$

and the relative minus sign in the second term is from two $b$’s being
(anti)commuted before contracting. These contractions are diagrammatically
shown in Fig. 21.

Figure 21: The first (left) and second (right) contraction for the loop
${D}{D}$ loop, showing the routing of momentum, the spin index, and the color
index, as $(q,s,a)$ respectively.

Enforcing the $\delta$ functions and summing over $s,t$, we get777The
integration over the loop momenta $q_{3},q^{\prime}_{U}$ is implied but is not
explicitly written down in our discussion here. We have also not included here
the $i/\Lambda^{2}$ factor, nor the denominators of the two fermion
propagators, nor a factor of $\epsilon^{abc}\epsilon^{abc^{\prime}}$. All of
these will be included elsewhere.

$\displaystyle L_{34}$ $\displaystyle=$
$\displaystyle\sum_{s,t}(\bar{u}_{q_{4}}^{t}\widetilde{\Gamma}v_{q_{3}}^{s}\bar{v}_{q_{3}}^{s}\bar{\widetilde{\Gamma}}u_{q_{4}}^{t}+\bar{u}_{q_{3}}^{s}\widetilde{\Gamma}v_{q_{4}}^{t}\bar{v}_{q_{3}}^{s}\bar{\widetilde{\Gamma}}u_{q_{4}}^{t})$
(65) $\displaystyle=$ $\displaystyle{\rm Tr}[\widetilde{\Gamma}(\hbox
to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{3}-M)(\bar{\widetilde{\Gamma}}-\widetilde{\Gamma}^{c})(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{4}+M)]\,$ (66)

where we have used the spinor identity of Eq. (57) and note that an extra
minus sign has flipped back the sign of the second term in Eq. (65) due to
interchanging the color indices $ab$.

In particular, for the $VV$ interaction of Eq. (1), we get

$\displaystyle(L_{34})^{\sigma\nu}$ $\displaystyle=$ $\displaystyle 2\,{\rm
Tr}[(\widetilde{G}_{V})^{\nu}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{3}-M){\widebar{\widetilde{G}}_{V}}^{\sigma}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{4}+M)]=2|\tilde{g}|^{2}\,{\rm
Tr}[\gamma^{\nu}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{3}-M)\gamma^{\sigma}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{4}+M)]\ .$ (67)

We note that the RHS is in fact symmetric under $(q_{3}\leftrightarrow q_{4})$
interchange since the only term which breaks this symmetry, namely the term
linear in $M_{D}$, is zero since the trace of an odd number of
$\gamma$-matrices is zero. Similarly, for the conjugate process we find

$\displaystyle(L^{c}_{34})^{\sigma\nu}$ $\displaystyle=$ $\displaystyle
2\,{\rm Tr}[{\widebar{\widetilde{G}}_{V}}^{\nu}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{3}-M)(\widetilde{G}_{V})^{\sigma}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{4}+M)]=(L_{34})^{\sigma\nu}\
.$ (68)

We note the similarity of the above $(L_{34})$ with the $|M^{12}|^{2}$ in Eq.
(63). Comparing the direction of momenta and making the translation
$q_{3}\rightarrow-q_{1}$, $q_{4}\rightarrow-q_{2}$, and
$({\widebar{\widetilde{G}}_{V}})\leftrightarrow(\widetilde{G}_{V})$, we obtain
$L_{12{}}$ and see that the mapping is exact. We thus can think of the
${D}{D}$ final state equivalently as a cut of the loop process
$|M^{12}|^{2}\equiv L_{12{}}$.

###  A.4 The tree-loop interference terms

We obtain here the contractions leading to the tree-loop interference terms
for the decay rate of ${\cal
X}_{n}\rightarrow{D}_{a}\overline{{D}^{c}}_{b}{U}_{c}$ depicted in Fig. 6. For
this purpose we write the mode expansion of the fermion fields, in the
notation of Ref. [9], as

$\psi(x)=\int d^{3}p\frac{1}{\sqrt{2E_{\bf p}}}\left(u_{p}a_{\bf
p}e^{-ipx}+v_{p}b_{\bf p}^{\dagger}e^{ipx}\right)\ ,\quad{\cal X}(x)=\int
d^{3}p\frac{1}{\sqrt{2E_{\bf p}}}\left(u_{p}c_{\bf p}e^{-ipx}+v_{p}c_{\bf
p}^{\dagger}e^{ipx}\right)\ ,$ (69)

where $\psi=(D,U)$ are Dirac fields, while ${\cal X}$ is a Majorana field. The
charge-conjugate field $\psi^{c}(x)=C\psi^{*}(x)$ has $a_{\bf
p}\leftrightarrow b_{\bf p}$, while ${\cal X}$ is self-conjugate, i.e. ${\cal
X}^{c}(x)={\cal X}(x)$. We use the VV interaction of Eq. (1) to obtain the
tree-loop interference term.

We start by deriving an expression for the $\chi-U$ fermion line in the tree-
loop interference term in coordinate space, which is obtained from

${{\cal M}_{01}^{\chi U\,\\!}}\sim\left<0|\right.c_{n}{\cal
T}\left((\widebar{U^{c}}{G_{V}^{n}}^{\tau}{\cal
X}_{n})\,(\bar{U}{\phantom{\,}\bar{G}_{V}^{m}\phantom{\,\\!}}^{\sigma}{\cal
X}_{m})\,(\widebar{{\cal
X}_{m}}{\phantom{\,}\bar{G}_{\Lambda}^{m}\phantom{\,\\!}}^{\mu}U^{c})\,(\widebar{U^{c}}{G_{\Lambda}^{n}}^{\nu}{\cal
X}_{n})\right)c_{n}^{\dagger}\left.|0\right>$ (70)

where ${\cal T}$ is the time-ordering symbol, and each $(...)$ is at a
different space-time point that is integrated over. Plugging in the mode
expansion of Eq. (69) into this gives us the momentum space diagram shown in
Fig. 22.

Figure 22: From this primitive diagram, the four different tree-loop
interference processes A, B, C, D can be generated by contracting the $q_{i}$
with the various $q^{\prime}_{j}$ as explained in the text. The direction of
the $q^{\prime}_{i}$ momentum is such as to be in line with the momentum
$q_{j}$ with which it is contracted.

This gives the expression for ${{\cal M}_{01}^{\chi U\,\\!}}$ in momentum
space given in Eq. (20). For the conjugate process, all fermion-line arrows
are flipped in the diagram, and we have

${{\cal M}_{01}^{c\,\chi U\,\\!}}\sim\left<0|\right.c_{n}{\cal
T}\left((\widebar{{\cal
X}_{n}}{\phantom{\,}\bar{G}_{\Lambda}^{n}\phantom{\,\\!}}^{\tau}U^{c})\,(\widebar{U^{c}}{G_{\Lambda}^{m}}^{\sigma}{\cal
X}_{m})\,(\widebar{{\cal
X}_{m}}{G_{V}^{m}}^{\mu}U)\,(\bar{U}{\phantom{\,}\bar{G}_{V}^{n}\phantom{\,\\!}}^{\nu}{\cal
X}_{n})\right)c_{n}^{\dagger}\left.|0\right>\ ,$ (71)

which yields a similar expression to that shown in Eq. (20) for ${{\cal
M}_{01}^{\chi U\,\\!}}$ but with the change
$G_{V}\leftrightarrow\bar{G}_{\Lambda}$.

We obtain next the expression for the ${D}$-loop in the tree-loop interference
term. What we need for this is

$\left<0|\right.a_{1}a_{2}{\cal
T}\left((\bar{{D}}\bar{\widetilde{G}}_{V}^{\sigma}{D}^{c})(\bar{{D}}\bar{\widetilde{G}}_{V}^{\mu}{D}^{c})\right)b_{4}^{\dagger}b_{3}^{\dagger}\left.|0\right>$
(72)

where $a_{i}$ is short for $a_{{\bf q}_{i}}$ etc. for the ${D}$ particle, and
where, again, each $(\bar{D}\cdot D^{c})$ field bilinear is at a different
space-time point that is integrated over. We have the particular choice of the
final $\left<0|\right.a_{2}a_{1}$ and the initial
$b_{4}^{\dagger}b_{3}^{\dagger}\left.|0\right>$ since our focus here is on the
contribution for the process (and not the conjugate process) that is relevant
for generating the baryon asymmetry, which are themselves to be obtained via
the VV interactions, i.e. from

$-\left<0|\right.{\cal
T}\left((\widebar{{D}^{c}_{2}}\widetilde{G}_{V}^{\tau}{D}_{1})(\bar{{D}}_{3^{\prime}}\bar{\widetilde{G}}_{V}^{\sigma}{D}^{c}_{4^{\prime}})(\bar{{D}}_{1^{\prime}}\bar{\widetilde{G}}_{V}^{\mu}{D}^{c}_{2^{\prime}})(\widebar{{D}^{c}_{4}}\widetilde{G}_{V}^{\nu}{D}_{3})\right)\left.|0\right>$
(73)

where we have shown this in momentum space with $D_{i}\equiv D_{q_{i}}$. This
is shown diagrammatically in Fig. 22. The overall minus sign is due to the
labeling of the momenta as in the figure.

Putting together the color factors associated with these vertices we have

$C_{01}^{(i)}=(\epsilon^{\alpha_{4}\alpha_{3}c^{\prime}}\epsilon^{\alpha_{2}^{\prime}\alpha_{1}^{\prime}c^{\prime}}\epsilon^{\alpha_{4}^{\prime}\alpha_{3}^{\prime}c}\epsilon^{\alpha_{2}\alpha_{1}c})/(\sqrt{6})^{2}\
,$ (74)

where the color index $\alpha_{i}$ is associated with $q_{i}$ etc., and the
$1/\sqrt{6}$ factor is from the color-part of the normalization of the
${D}{D}{U}$ final state (see Eq. (54)).

From the two bilinears
$(\bar{{D}}_{3^{\prime}}\bar{\widetilde{G}}_{V}^{\sigma}{D}^{c}_{4^{\prime}})(\bar{{D}}_{1^{\prime}}\bar{\widetilde{G}}_{V}^{\mu}{D}^{c}_{2^{\prime}})$,
we need two $a^{\dagger}$’s and two $b$’s to give us a nonzero contraction.
There are four such possibilities, namely,

$-\left<0|\right.a_{2}a_{1}\left(b_{3^{\prime}}b_{4^{\prime}}a^{\dagger}_{1^{\prime}}a^{\dagger}_{2^{\prime}}+a^{\dagger}_{3^{\prime}}a^{\dagger}_{4^{\prime}}b_{1^{\prime}}b_{2^{\prime}}+a^{\dagger}_{3^{\prime}}b_{4^{\prime}}a^{\dagger}_{1^{\prime}}b_{2^{\prime}}+b_{3^{\prime}}a^{\dagger}_{4^{\prime}}b_{1^{\prime}}a^{\dagger}_{2^{\prime}}\right)b_{4}^{\dagger}b_{3}^{\dagger}\left.|0\right>\
,$

where the momenta associated with $a^{\dagger},b^{\dagger}$ flow outward while
the momenta associated with the $a,b$ flow inward. These amount to attaching
the different $q_{i}$ to the various $q^{\prime}_{j}$ in Fig. 22. The
direction of the $q^{\prime}_{i}$ momentum is such as to be in line with the
momentum $q_{j}$ with which it is contracted. These four possibilities with
the adjacent $a$’s and $b$’s contracted (i.e. the momenta on them identified,
and so too with the associated color indices) correspond to the processes A,
B, C, D respectively shown diagrammatically in Fig. 6 where we have not shown
the diagram corresponding to the fourth term above (i.e. process D) which is
just a flip of process C about the horizontal fermion-line, i.e. process D
corresponds to interchanging $q_{1^{\prime}}\leftrightarrow q_{2^{\prime}}$
and $q_{3^{\prime}}\leftrightarrow q_{4^{\prime}}$ in process C. Any time
there is a cross of the contracted lines, we get a $(-1)$ factor associated
with an anticommutation of these fermionic operators. All such $(-1)$ factors
are multiplied to give us an overall sign of that term. Note that contracting
non-adjacent operators, in effect, gives a negative sign if it corresponds to
doing an odd number of anticommutations. There are totally four possibilities
within each term including the adjacent and non-adjacent contractions, which
can be viewed as twists of the various legs in the diagrams. Each contraction
also has the associated color factor. Including the spinor factors, we have
more fully,

$\displaystyle-\left<0|\right.a_{2}a_{1}\,\bar{v}_{2}\widetilde{G}_{V}^{\tau}u_{1}\left(b_{3^{\prime}}b_{4^{\prime}}a^{\dagger}_{1^{\prime}}a^{\dagger}_{2^{\prime}}\
\bar{v}_{3^{\prime}}\bar{\widetilde{G}}_{V}^{\sigma}{u}_{4^{\prime}}\
\bar{u}_{1^{\prime}}\bar{\widetilde{G}}_{V}^{\mu}{v}_{2^{\prime}}+a^{\dagger}_{3^{\prime}}a^{\dagger}_{4^{\prime}}b_{1^{\prime}}b_{2^{\prime}}\
\bar{u}_{3^{\prime}}\bar{\widetilde{G}}_{V}^{\sigma}{v}_{4^{\prime}}\
\bar{v}_{1^{\prime}}\bar{\widetilde{G}}_{V}^{\mu}{u}_{2^{\prime}}+\right.$
$\displaystyle\left.a^{\dagger}_{3^{\prime}}b_{4^{\prime}}a^{\dagger}_{1^{\prime}}b_{2^{\prime}}\
\bar{u}_{3^{\prime}}\bar{\widetilde{G}}_{V}^{\sigma}{u}_{4^{\prime}}\
\bar{u}_{1^{\prime}}\bar{\widetilde{G}}_{V}^{\mu}{u}_{2^{\prime}}+b_{3^{\prime}}a^{\dagger}_{4^{\prime}}b_{1^{\prime}}a^{\dagger}_{2^{\prime}}\
\bar{v}_{3^{\prime}}\bar{\widetilde{G}}_{V}^{\sigma}{v}_{4^{\prime}}\
\bar{v}_{1^{\prime}}\bar{\widetilde{G}}_{V}^{\mu}{v}_{2^{\prime}}\right)b_{4}^{\dagger}b_{3}^{\dagger}\,\bar{u}_{4}\widetilde{G}_{V}^{\nu}v_{3}\left.|0\right>\
.$ (75)

The initial-spin averaged total decay rate has all of the color and spin
indices summed over. It turns out that in all cases the overall sign
associated with twists within each process is undone by the color index
structure. Thus all four twists come with the same overall sign after
including color also, although the trace structure could be different.

In the way of an example, consider the first term, and contract the adjacent
indices associated with the $a,a^{\dagger}$ and $b,b^{\dagger}$, i.e. we have
$1\\!\\!=\\!\\!1^{\prime},2\\!\\!=\\!\\!2^{\prime},3\\!\\!=\\!\\!3^{\prime},4\\!\\!=\\!\\!4^{\prime}$,
and since no fermion contraction lines cross, the overall sign is +1. The
spinor factors with these contractions are
$-\bar{v}_{2}\widetilde{G}_{V}^{\tau}u_{1}\
\bar{v}_{3}\bar{\widetilde{G}}_{V}^{\sigma}{u}_{4}\
\bar{u}_{1}\bar{\widetilde{G}}_{V}^{\mu}{v}_{2}\
\bar{u}_{4}\widetilde{G}_{V}^{\nu}v_{3}$, which upon summing over all spins
yields $-{\rm Tr}[\widetilde{G}_{V}^{\tau}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{1}+M_{D})\bar{\widetilde{G}}_{V}^{\mu}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{2}-M_{D})]\,{\rm
Tr}[\bar{\widetilde{G}}_{V}^{\sigma}(\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/_{4}+M_{D})\widetilde{G}_{V}^{\nu}(\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/_{3}-M_{D})]$, as given in Eq.
(23). From Eq. (74), the associated color factor with this is
$C_{01}^{A}=\epsilon^{\alpha_{4}\alpha_{3}c^{\prime}}\epsilon^{\alpha_{2}\alpha_{1}c^{\prime}}\epsilon^{\alpha_{4}\alpha_{3}c}\epsilon^{\alpha_{2}\alpha_{1}c}/(\sqrt{6})^{2}$
with all these indices summed over, which yields
$C_{01}^{A}=2\delta^{c^{\prime}c}2\delta^{c^{\prime}c}/(\sqrt{6})^{2}=12/(\sqrt{6})^{2}=2$
as noted below Eq. (23). It turns out that all four twists yields the same
overall sign (after including the signs due to the anticommutations and the
color factor) and the same Tr[…] expression, explaining the factor of 4 for
the $L_{D}^{A}$ contribution in Eq. (23). While writing the spinor products as
${\rm Tr}[...]$ we sometimes make use of the spinor identity in Eq. (57).

A similar computation for all the four processes and twists therein gives the
expressions in Eq. (23), with the color-factors shown below that equation.
This explicitly tells us also that the expressions for processes C and D are
identical.

Turning now to the ${D}$-loop expressions for the conjugate process, we notice
that the changes are only of the form $u_{p}\leftrightarrow v_{p}$, which
could potentially change the sign of the $M_{D}$ terms in the ${\rm Tr}[...]$
expressions, and are therefore reflected only in terms with odd powers of
$M_{D}$. But such terms have the trace of an odd number of $\gamma$ matrices,
which is zero. Therefore we conclude that the ${D}$-loop expressions for the
conjugate process are the same as for the process, i.e.
$L^{c\,(i)}_{D}=L_{D}^{(i)}$ given in Eq. (23).

###  A.5 The renormalized (subtracted) $\bar{\cal X}_{m}{\cal X}_{n}$ 2-point
function

Here we pay attention to the renormalization conditions on the $\bar{\cal
X}_{m}{\cal X}_{n}$ 2-point function, and derive the subtraction term needed
for consistency with being in the mass basis. This is needed to correctly
incorporate the B1 diagram in Fig. 5, or equivalently the B diagram
contribution in Fig. 6 with $p^{\prime}_{n}=p_{n}$, since it involves this
2-point function. Due to the Majorana nature of the ${\cal X}_{n}$, the full
2-point function includes two different loop contributions, namely, one from
fields propagating in the loop, and another with their conjugate fields, which
we show diagrammatically in Fig. 23. The full 2-point function is the sum of
these, i.e. $\Sigma_{nm}(p_{n})=\Omega_{nm}(p_{n})+\Omega^{c}_{nm}(p_{n})$.

$\begin{array}[]{c}\raisebox{0.0pt}{\includegraphics[width=143.09538pt]{Omg.png}}\hskip
28.45274pt\raisebox{0.0pt}{\includegraphics[width=143.09538pt]{Omgc.png}}\end{array}$
$\hskip 14.22636pt\Omega_{nm}(p_{n})\hskip 128.0374pt\Omega^{c}_{nm}(p_{n})$

Figure 23:  The two different loop contributions to the $\bar{\cal X}_{m}{\cal
X}_{n}$ 2-point function $\Sigma_{nm}(p_{n})$.

We can further split up the $\Omega_{mn}$ as
$\Omega_{mn}=J^{\mu\nu}(M_{U})L_{\mu\nu}(M_{D})$, where the $J$ is the ${\cal
X}{U}$ fermion line, and the $L$ is the ${D}$ loop, and similarly
$\Omega^{c}_{mn}=J^{c\mu\nu}(M_{U})L^{c}_{\mu\nu}(M_{D})$.

We impose a renormalization condition such that the loop contribution to the
2-point function is canceled by a counterterm at the on-shell subtraction
points $p_{n}=p_{n_{0}}$ (for $n=1,2$), with
$p_{1_{0}}^{2},p_{2_{0}}^{2}=M_{1,2}^{2}$, leaving $M_{n}=M_{1,2}$ uncorrected
and as the renormalized masses, consistent with being in the mass-basis. We
take the counterterm Lagrangian as ${\cal L}^{\rm CT}\supset-\bar{\cal
X}_{m}\,(\delta\Sigma_{nm}/2)\,{{\cal X}_{n}}+{\rm h.c.}$, which we can split
into individual counterterms and write
$\delta\Sigma_{nm}=\delta\Omega_{nm}+\delta\Omega^{c}_{nm}$, with each piece
cancelling the corresponding $\Omega_{nm}(p_{n})$ contribution above, at the
subtraction point $p_{n_{0}}$. If $n=m$, we have $\Sigma_{nn}(p_{n_{0}})$ for
the same particle with both legs on-shell, and we can take
$\delta\Omega_{nn}=-\Omega_{nn}(p_{n_{0}})$ and
$\delta\Omega^{c}_{nn}=-\Omega^{c}_{nn}(p_{n_{0}})$, which implies
$\delta\Sigma_{nn}=-\Sigma_{nn}(p_{n_{0}})$. With this subtraction, there is
no contribution from the B diagram for $nm=\\{11,22\\}$ as they are entirely
canceled by the counter-terms. What remains then is the $n\\!\neq\\!m$ case,
with the ${\cal X}_{n}$-leg on-shell but the ${\cal X}_{m}$-leg necessarily
off-shell. Due to the Majorana nature of ${\cal X}_{n}$, we can show generally
that $\bar{\cal X}_{m}\Sigma_{nm}{\cal X}_{n}=\bar{\cal
X}_{n}(C\gamma^{0}\Sigma_{nm}^{T}\gamma^{0}C){\cal X}_{m}$ (no sum on $n,m$),
and therefore we must have the relation
$\Sigma_{mn}=C\gamma^{0}\Sigma_{nm}^{T}\gamma^{0}C$. In our theory with the VV
interaction, we find that
$\Omega_{mn}=C\gamma^{0}\Omega_{nm}^{cT}\gamma^{0}C$, and the above relation
is satisfied in an interesting way. This relation now has an important
consequence for us, namely, that for Majorana fermions, $\delta\Sigma_{nm}$
and $\delta\Sigma_{mn}$ cannot be independently chosen to simultaneously
cancel both $\Sigma_{21}(p_{2_{0}})$ and $\Sigma_{12}(p_{1_{0}})$. We can at
best cancel one of them, say $\Sigma_{12}(p_{1_{0}})$ by the counterterm
choice $\delta\Sigma_{12}=-\Sigma_{12}(p_{1_{0}})$, suggesting
$\delta\Omega_{12}=-\Omega_{12}(p_{1_{0}})$ and
$\delta\Omega^{c}_{12}=-\Omega^{c}_{12}(p_{1_{0}})$. Owing to the above
relation, this then fixes $\delta\Sigma_{21}$ also, which individually fixes
$\delta\Omega_{21}=-C\gamma^{0}{\Omega_{12}^{c}}^{\\!\\!T}(p_{1_{0}})\gamma^{0}C=-\Omega_{21}(p_{1_{0}})$
and
$\delta\Omega^{c}_{21}=-C\gamma^{0}{\Omega}^{T}_{12}(p_{1_{0}})\gamma^{0}C=-\Omega^{c}_{21}(p_{1_{0}})$.
Now with the counterterm subtraction taken into account, the contribution from
the B1 diagram is thus present only for $nm=21$, and is proportional to
$\Omega^{c}_{21}(p_{2})-\Omega^{c}_{21}(p_{1_{0}})$, not just to
$\Omega^{c}_{21}(p_{2})$ as one may have naively written down ignoring the
renormalization conditions. Similarly, the diagram with fields conjugated is
proportional to $\Omega_{21}(p_{2})-\Omega_{21}(p_{1_{0}})$. Cuts in
$\Omega_{nm}$ and $\Omega^{c}_{nm}$ leads to a discontinuity in the imaginary
piece. Although one may in principle choose the subtractions for the real and
imaginary pieces differently, we choose them similarly as above.

As an aside, we contrast this with the case of two Dirac fermions $\Psi_{n}$,
with each containing two (different) Weyl fields $\psi_{n\alpha}$ and
$\psi^{c}_{n\alpha}$. The 2-point functions $\bar{\Psi}_{n}\Psi_{m}$ now have
bilinears of the form $\psi^{c}_{n}\psi_{m}+h.c.$ . In the Dirac case also, an
identical argument can be used to cancel the 2-point functions
$\Sigma_{11},\Sigma_{22}$ at the on-shell subtraction scales
$p_{1_{0}},p_{2_{0}}$ and also $\Sigma_{12}$ at $p_{1_{0}}$. Now, the
$\psi^{c}_{1}\psi_{2}$ and $\psi^{c}_{2}\psi_{1}$ bilinears are independent,
allowing an independent counterterm to cancel the $\Sigma_{21}$ at $p_{2_{0}}$
also. Thus, in the Dirac case, for neither $\Psi_{1,2}$ do contributions of
the type of diagram B1 contribute since all the $\Sigma_{mn}$ can be canceled
at the respective $p_{n_{0}}$. This is in contrast to the Majorana case where
$\Sigma_{12}$ and $\Sigma_{21}$ are not independent and therefore cannot both
be canceled.

In our numerical analysis, we define dimensionless loop function that are
functions of the scaled (hatted) masses and momenta defined in Sec. 3.2,
namely $\hat{p}_{i}\equiv p_{i}/M_{n}$ and $\hat{M}_{i}\equiv M_{i}/M_{n}$. We
write from our earlier decomposition,
$\Omega_{nm}=(M_{n}^{5}/\Lambda^{4})\hat{J}_{nm}^{\mu\nu}(\hat{M}_{U})\hat{L}_{nm\mu\nu}(\hat{M}_{D})$,
where the $1/\Lambda^{4}$ follows from the 2 VV interactions, implying the
$M_{n}^{5}$ scaling for the $\Omega_{nm}$ since it has mass dimension +1. We
similarly define
$\Omega_{nm}^{c}=(M_{n}^{5}/\Lambda^{4})\hat{J}_{nm}^{c\mu\nu}\hat{L}^{c}_{\mu\nu}$.

From the above considerations, the renormalized 2-point function we use for
$nm=21$ is

$\displaystyle\Omega_{21}^{(ren)}(p_{2})$ $\displaystyle=$
$\displaystyle(1/\Lambda^{4})\,\left[M_{2}^{5}\hat{\Omega}_{21}(p_{2}/M_{2},M_{U}/M_{2},M_{D}/M_{2})-M_{1}^{5}\hat{\Omega}_{21}(p_{1_{0}}/M_{1},M_{U}/M_{1},M_{D}/M_{1})\right]\
,$ (76) $\displaystyle=$
$\displaystyle(1/\Lambda^{4})\,\left[M_{2}^{5}\hat{\Omega}_{21_{r}}-M_{1}^{5}\hat{\Omega}_{21_{s}}\right]=(M_{2}^{5}/\Lambda^{4})\,\left[\hat{J}_{21_{r}}\hat{L}_{21_{r}}-(M_{1}/M_{2})^{5}\hat{J}_{21_{s}}\hat{L}_{21_{s}}\right]\
,$

where $p_{1_{0}}$ is the subtraction scale that we take to be such that
$p_{1_{0}}^{2}=M_{1}^{2}$, and we have defined

$\displaystyle\hat{\Omega}_{21_{r}}$ $\displaystyle\equiv$
$\displaystyle\hat{\Omega}_{21}(p_{2}/M_{2},M_{U}/M_{2},M_{D}/M_{2})=\hat{J}_{21}^{\mu\nu}(p_{2}/M_{2},M_{U}/M_{2})\hat{L}_{21\,\mu\nu}(p_{2}/M_{2},M_{D}/M_{2})=\hat{J}_{21_{r}}\hat{L}_{21_{r}}\
,$ $\displaystyle\hat{\Omega}_{21_{s}}$ $\displaystyle\equiv$
$\displaystyle\hat{\Omega}_{21}(p_{1_{0}}/M_{1},M_{U}/M_{1},M_{D}/M_{1})=\hat{J}_{21}^{\mu\nu}(p_{1_{0}}/M_{1},M_{U}/M_{1})\hat{L}_{21\,\mu\nu}(p_{1_{0}},M_{D}/M_{1})=\hat{J}_{21_{s}}\hat{L}_{21_{s}}\
,$

where the $\hat{J}$ is due to the ${\cal X}$ fermion line, and the $\hat{L}$
is due to the ${D}$-loop. We similarly define the subtracted
$\Omega_{21}^{c\,(ren)}(p_{2})$. We recall that we have
$\Omega_{12}(p_{2})=C\gamma^{0}\Omega_{21}^{cT}(p_{2})\gamma^{0}C$ as already
pointed out above.

With this formalism in place, we are in a position to compute diagram B of
Fig. 6, which we should include only for $nm\\!=\\!21$ as explained above.
This is of the form
$(...)\Omega^{(ren)}_{12}\otimes\Omega^{c\,(ren)}_{21}(...)$, i.e.
$(\Omega_{12_{r}}-\Omega_{12_{s}})\otimes(\Omega^{c}_{21_{r}}-\Omega^{c}_{21_{s}})$,
and can be written as

$(...)[\hat{J}^{34}_{12_{r}}\hat{L}^{34}_{12_{r}}-(M_{1}/M_{2})^{5}\hat{J}^{34}_{12_{s}}\hat{L}^{34}_{12_{s}}]\,\otimes[\hat{J}^{12}_{21_{r}}\hat{L}^{12}_{21_{r}}-(M_{1}/M_{2})^{5}\hat{J}^{12}_{21_{s}}\hat{L}^{12}_{21_{s}}](...)\
,$

where the $34$ and $12$ superscripts denote the dependence on the momenta in
the left part and right part respectively of diagram B. Multiplying this out
we get four terms, each of which we decompose into the trace similar to
$\hat{\cal M}_{k\lambda}$ in Eq. (25) from the $\hat{J}$ part, times the
$D$-loop $\hat{L}^{12}\hat{L}^{34}$ from the $\hat{L}$ part. These four terms
can be written as $\sum_{a,b}\hat{\cal
M}^{ab}_{k\lambda}(\hat{M}_{U_{a}},\hat{M}_{U_{b}})\hat{L}_{a}^{12}(\hat{M}_{D_{a}})\hat{L}_{b}^{34}(\hat{M}_{D_{b}})$,
where $a,b=\\{r,s\\}$ with $r$ denoting that the evaluation is at $p_{2}$,
while $s$ the subtraction contribution evaluated at $p_{1_{0}}$. We generalize
Eq. (25) for diagram B to include the subtraction by adding the
$a,b=\\{r,s\\}$ indices and writing

$\displaystyle\hat{{\cal M}}^{ab}_{0}$ $\displaystyle\equiv$
$\displaystyle{\rm Tr}[\gamma^{\sigma}\hat{}\hbox to0.0pt{\hbox
to5.00002pt{\hfil$p$\hfil}\hss}/^{\prime}_{n}\gamma^{\mu}(\hat{\hbox
to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/}^{\prime}_{U}+\hat{M}_{U_{a}})\gamma^{\nu}\hat{}\hbox
to0.0pt{\hbox to5.00002pt{\hfil$p$\hfil}\hss}/_{n}\gamma^{\tau}(\hat{\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/}_{U}+\hat{M}_{U_{b}})]\ ,$
$\displaystyle\hat{{\cal M}}^{ab}_{1\lambda}$ $\displaystyle\equiv$
$\displaystyle\hat{M}_{n}\hat{M}_{m}{\rm
Tr}[\gamma^{\sigma}\gamma^{\mu}\hat{\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/}^{\prime}_{U}\gamma^{\nu}\gamma^{\tau}\hat{\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/}_{U}P_{\lambda}]\
,\quad\hat{{\cal
M}}^{ab}_{2\lambda}\equiv\hat{M}_{n}\hat{M}_{m}\hat{M}_{U_{a}}\hat{M}_{U_{b}}\,{\rm
Tr}[\gamma^{\sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\tau}P_{\lambda}]\ ,$
$\displaystyle\hat{{\cal M}}^{ab}_{3\lambda}$ $\displaystyle\equiv$
$\displaystyle M_{m}{\rm
Tr}[\gamma^{\sigma}\gamma^{\mu}(M_{U_{a}}\gamma^{\nu}\hat{}\hbox to0.0pt{\hbox
to5.00002pt{\hfil$p$\hfil}\hss}/_{n}\gamma^{\tau}\hat{\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/}_{U}+M_{U_{b}}\hat{\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/}^{\prime}_{U}\gamma^{\nu}\hat{}\hbox
to0.0pt{\hbox to5.00002pt{\hfil$p$\hfil}\hss}/_{n}\gamma^{\tau})P_{\lambda}]\
,$ $\displaystyle\hat{{\cal M}}^{ab}_{4\lambda}$ $\displaystyle\equiv$
$\displaystyle M_{n}{\rm Tr}[\gamma^{\sigma}\hat{}\hbox to0.0pt{\hbox
to5.00002pt{\hfil$p$\hfil}\hss}/^{\prime}_{n}\gamma^{\mu}(M_{U_{a}}\hat{\hbox
to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/}^{\prime}_{U}\gamma^{\nu}\gamma^{\tau}P_{\lambda}+M_{U_{b}}\gamma^{\nu}\gamma^{\tau}P_{\lambda}\hat{\hbox
to0.0pt{\hbox to5.00002pt{\hfil$q$\hfil}\hss}/}_{U})]\ ,$ (77)

with $\hat{M}_{U_{r}},\hat{M}_{U_{s}}=M_{U}/M_{2},M_{U}/M_{1}$, and
$\hat{M}_{{D}_{r}},\hat{M}_{{D}_{s}}=M_{D}/M_{2},M_{D}/M_{1}$. We generalize
the D-loop factors also to include the subtraction with the factors
$\hat{L}_{a}^{12}\hat{L}_{b}^{34}$ as mentioned above. Clearly the original
term $\hat{\cal M}_{k\lambda}\hat{L}_{D}$ of Eq. (27) in this new language is
$\hat{\cal M}_{k\lambda}^{rr}\hat{L}_{r}^{12}\hat{L}_{r}^{34}$, and we
generalize this to include the subtractions for diagram B by replacing in Eq.
(27) as

$\left[\hat{\cal
M}_{k\lambda}\hat{L}_{D}\right]\xrightarrow[(i)=(B)]{}\left[\hat{\cal
M}_{k\lambda}^{rr}\hat{L}_{r}^{12}\hat{L}_{r}^{34}-(M_{1}/M_{2})^{5}\hat{\cal
M}_{k\lambda}^{rs}\hat{L}_{r}^{12}\hat{L}_{s}^{34}-(M_{1}/M_{2})^{5}\hat{\cal
M}_{k\lambda}^{sr}\hat{L}_{s}^{12}\hat{L}_{r}^{34}+(M_{1}/M_{2})^{10}\hat{\cal
M}_{k\lambda}^{ss}\hat{L}_{s}^{12}\hat{L}_{s}^{34}\right]\ ,$ (78)

with the diagram B contribution present only for $nm=21$, and include it with
the other contributions in Sec. 3.2.

##  Appendix B The Phase-space and Loop Integration Variables

Here we arrive at our choice of independent variables for the 3-body phase-
space momenta and the loop momenta in the decay process ${\cal
X}_{n}(p_{n})\rightarrow{D}(q_{1}){D}(q_{2}){U}(q_{U})$, and write down the
phase-space and loop-integration measures. Our notation is that $q_{i}$ is a
4-vector, while ${\bf q}_{i}$ is the corresponding 3-vector with 3-magnitude
(radial component) ${q_{i}}_{r}=|{\bf q}_{i}|$. An on-shell particle has
$E_{i}=\sqrt{{q_{i}}^{2}_{r}+M_{i}^{2}}$. All final state particles are on-
shell; the ${D},{U}$ particles in loops are also on-shell given our focus on
computing the baryon asymmetry as explained in the text.

###  B.1 The 3-body phase-space variables and measure (for ${\cal X}$ decay)

We start by writing the 3-body phase-space integral of the matrix element as a
cut of the corresponding loop diagram as

$\int d\Pi_{3}\,|{\cal
A}|^{2}=\int\frac{1}{2}\frac{d^{4}q_{U}}{(2\pi)^{4}}\frac{d^{4}q_{1}}{(2\pi)^{4}}\frac{d^{4}q_{2}}{(2\pi)^{4}}(2\pi)^{3}\delta^{\rm
cut}_{12U}\,(2\pi)^{4}\delta^{4}(p_{n}-(q_{1}+q_{2}+q_{U}))|{\cal A}|^{2}\
,\quad$ (79)

where we put the three propagators on-shell by replacing them with 888 The $i$
is omitted in the expressions for $d\Pi_{3}$ here but is included elsewhere in
the text.  $(-2\pi i)^{3}\delta^{\rm cut}_{12U}$ where $\delta^{\rm
cut}_{12U}\equiv\delta(q_{U}^{2}-M_{U}^{2})\delta(q_{1}^{2}-M_{1}^{2})\delta(q_{2}^{2}-M_{2}^{2})$,
where for the decay channel, $M_{1}=M_{2}=M_{D}$. We include a factor of $1/2$
to account for the two identical ${D}$ in the final state. We deal with the
kinematical aspects of the phase-space here.

At the outset, we choose to enforce the overall energy-momentum conserving
$\delta$-function in Eq. (79) by performing the $d^{4}q_{2}$ integral, which
sets $q_{2}=p_{n}-(q_{1}+q_{U})$, and giving the form of Eq. (10). This leaves
8 integration variables, with the measure

$d^{4}q_{U}d^{4}q_{1}=dq_{U}^{0}dc_{U}d\phi_{U}d{q}_{U_{r}}{q}_{U_{r}}^{2}dq_{1}^{0}dc_{1}d\phi_{1}d{q_{1}}_{r}{q_{1}}_{r}^{2}$,

where $c_{i}\equiv\cos{\theta_{q_{i}}}$ and $\phi_{i}\equiv\phi_{q_{i}}$. We
have
$\delta(q_{i}^{2}-M_{i}^{2})=\delta({q_{i}^{0}}^{2}-E_{i}^{2})=\delta({q_{i}^{0}}-E_{i})/|2E_{i}|$
where $E_{i}^{2}={q_{i}}_{r}^{2}+M_{i}^{2}$ for each of the $q_{U},q_{1}$,
picking the positive root for a physical on-shell particle.

The Lorentz invariance of each factor allows us to take the $d^{4}q_{U}$ and
$d^{4}q_{1}$ in different frames. A convenient frame to consider is what we
call as the *-frame in which ${\bf p}_{n}^{*}={\bf q}_{U}^{*}$. In this frame,
3-momentum conservation implies ${\bf q}_{2}^{*}=-{\bf q}_{1}^{*}$, i.e. these
are back-to-back. The advantage of this frame is that the end-points of
${q_{1}}_{r}^{\\!*}$ are independent of the angular variables.

For a given 4-vector in the lab-frame, to obtain the corresponding 4-vector in
the *-frame, we make a Lorentz transformation as follows. We first form the
4-vector $k_{B}\equiv(p_{n}-q_{U})=(E_{B},{\bf k}_{B})$. The boost rapidity
$\eta$ that takes $k_{B}$ to $k_{B}^{*}\equiv(M_{B},{\bf 0})$, i.e. to the
$k_{B}$ rest-frame, is given by $t_{\eta}\equiv\tanh\eta={k_{B}}_{r}/E_{B}$.
The required Lorentz transformation is a boost with this rapidity in the
direction of the unit-vector $\\!\\!\hat{\,\,\bf k}_{B}={\bf
k}_{B}/{k_{B}}_{r}$. More explicitly, consider any lab-frame 4-vector
$p=(E_{p},{\bf p})$ for which we seek the corresponding *-frame 4-vector
$p^{*}=(E_{p}^{*},{\bf p}^{*})$. We first decompose as ${\bf p}={\bf
p}_{\parallel}+{\bf p}_{\perp}$, where ${\bf p}_{\parallel}=({\bf
p}\\!\cdot\\!\\!\\!\hat{\,\,\bf k}_{B})\,\\!\\!\hat{\,\,\bf k}_{B}$ and
${p_{\parallel}}_{r}=|{\bf p}_{\parallel}|$, and, ${\bf p}_{\perp}={\bf
p}-{\bf p}_{\parallel}$. Then, ${\bf p}^{*}={\bf p}_{\parallel}^{*}+{\bf
p}_{\perp}^{*}$ with ${\bf
p}_{\parallel}^{*}={p_{\parallel}}_{r}^{\\!*}\,\\!\\!\hat{\,\,\bf k}_{B}$, and

$\begin{pmatrix}E_{p}^{*}\\\ {p_{\parallel}}_{r}^{\\!*}\end{pmatrix}={\cal
B}_{\eta}\begin{pmatrix}E_{p}\\\ {p_{\parallel}}_{r}\end{pmatrix}{\rm where}\
{\cal B}_{\eta}\equiv\begin{pmatrix}c_{\eta}&-s_{\eta}\\\
-s_{\eta}&c_{\eta}\end{pmatrix}\ ;\quad{\bf p}_{\perp}^{*}={\bf p}_{\perp}\ ,$

with $s_{\eta}=\sinh\eta=t_{\eta}/\sqrt{1-t_{\eta}^{2}}$ and
$c_{\eta}=\cosh\eta=1/\sqrt{1-t_{\eta}^{2}}$. The inverse transformation that
takes a *-frame 4-vector to the lab-frame is obtained the same way, but with
the rapidity $-\eta$.

Using the Lorentz invariance of each measure as mentioned above, we take the
$d^{4}q_{U}\delta(q_{U}^{2}-M_{U}^{2})$ in the lab-frame (i.e. the ${\cal X}$
rest-frame for decays), and the
$d^{4}q^{*}_{1}\delta({q^{*}_{1}}^{2}-M_{1}^{2})$ and the remaining
$\delta({q^{*}_{2}}^{2}-M_{2}^{2})$ in the *-frame. The $dq_{U}^{0}$ and
$d{q^{*}_{1}}^{0}$ integrals can be performed enforcing the first two
$\delta$-functions, leaving

$d\Pi_{3}\,=\frac{1}{(2\pi)^{5}}\frac{1}{2}dc_{U}d\phi_{U}d{q}_{U_{r}}\frac{{q}_{U_{r}}^{2}}{2E_{U}}d\phi^{*}_{1}dc^{*}_{1}d{q_{1}^{*}}_{\\!r}\frac{{q_{1}^{*}}_{\\!r}^{2}}{2E^{*}_{1}}\delta({q^{*}_{2}}^{2}-M_{2}^{2})\
.$ (80)

We write the remaining $\delta-$function as
$\delta({q^{*}_{2}}^{2}-M_{2}^{2})\equiv\delta(f({q_{1}^{*}}_{\\!r}))$,
viewing the argument now as a function of ${q_{1}^{*}}_{\\!r}$, i.e.
$f({q_{1}^{*}}_{\\!r})={q^{*}_{2}}^{2}-M_{2}^{2}={E_{2}^{*}}^{2}-{q_{2}^{*}}_{\\!r}^{2}-M_{2}^{2}$,
with $E_{2}^{*}=E_{n}^{*}-E_{U}^{*}-E_{1}^{*}({q_{1}^{*}}_{\\!r})$ and
${q_{2}^{*}}_{\\!r}={q_{1}^{*}}_{\\!r}$, owing to ${\bf p}^{*}_{n}={\bf
q}^{*}_{U}$. We have
$\delta(f({q_{1}^{*}}_{\\!r}))=\delta({q_{1}^{*}}_{\\!r}-{{q_{1}^{*}}_{\\!r}}_{z})/|df/d{q_{1}^{*}}_{\\!r}|$
where ${{q_{1}^{*}}_{\\!r}}_{z}$ is the zero of $f$, and the denominator is
evaluated at the zero. We find that
${{q_{1}^{*}}_{\\!r}}_{z}=(M_{B}/2)\\{[1+(M_{1}^{2}-M_{2}^{2})/M_{B}^{2}]^{2}-4M_{1}^{2}/M_{B}^{2}\\}^{1/2}$,
$|df/d{q_{1}^{*}}_{\\!r}|=2({q_{1}^{*}}_{\\!r}/E_{1}^{*})(E_{n}^{*}-E_{U}^{*})$,
and $M_{B}^{2}=(E_{n}^{*}-E_{U}^{*})^{2}=(p_{n}-q_{U})^{2}$ being a Lorentz
invariant quantity can be computed in any frame. We can then perform the
$d{q_{1}^{*}}_{\\!r}$ integral enforcing the last $\delta-$function which sets
${{q_{1}^{*}}_{\\!r}}={{q_{1}^{*}}_{\\!r}}_{z}$, and we denote henceforth the
zero simply as ${q_{1}^{*}}_{\\!r}$. We finally have

$d\Pi_{3}=\frac{1}{(2\pi)^{5}}\frac{1}{2}dc_{U}d\phi_{U}d{q}_{U_{r}}d\phi^{*}_{1}dc^{*}_{1}\frac{{q}_{U_{r}}^{2}}{2E_{U}}\frac{{q_{1}^{*}}_{\\!r}}{4(E_{n}^{*}-E_{U}^{*})}\
,$ (81)

in terms of the five independent 3-body phase-space variables
$c_{U},\phi_{U},{q}_{U_{r}},c^{*}_{1},\phi^{*}_{1}$.

For the decay process, in the ${\cal X}$ rest-frame, the matrix element will
be independent of $\phi_{U}$ and the $d\phi_{U}$ integral just yields $2\pi$,
and we can include this factor and conveniently set $\phi_{U}=0$. The limits
of integration for the independent variables are straightforward, except for
the ${q}_{U_{r}}$, which is bounded above. Clearly, the minimum value of
${q}_{U_{r}}$ is $0$, i.e. ${q}_{U_{r}}^{(min)}=0$. We can find the maximum
value ${q}_{U_{r}}^{(max)}$ straightforwardly in the restframe of ${\cal X}$
i.e. with ${p_{n}}_{r}=0$, as follows. From the Lorentz invariance of
$M_{B}^{2}$, we can also write
$M_{B}^{2}=(p^{*}_{n}-q^{*}_{U})^{2}=(q_{1}^{*}+q_{2}^{*})^{2}=(E_{1}^{*}+E_{2}^{*})^{2}$
where we have used 4-momentum conservation and the fact that ${\bf
q}^{*}_{1}+{\bf q}^{*}_{2}=0$. We also have
$M_{B}^{2}=(E_{n}^{*}-E_{U}^{*})^{2}=(p_{n}-q_{U})^{2}=M_{n}^{2}+M_{U}^{2}-2(E_{n}E_{U}-{p_{n}}_{r}{q}_{U_{r}}c_{nU})$
with $c_{nU}={\bf p}_{n}\\!\cdot\\!{\bf q}_{U}/({p_{n}}_{r}{q}_{U_{r}})$, and
for ${p_{n}}_{r}=0$, we see that ${q}_{U_{r}}^{(max)}$ is attained when
$M_{B}$ takes its minimum value, which is for
${q_{1}^{*}}_{\\!r}={q_{2}^{*}}_{\\!r}=0$, and is
$M_{B}^{(min)}=(M_{1}+M_{2})$. From this we find
${q}_{U_{r}}^{(max)}=(M_{n}/2)\\{[1+(M_{U}^{2}-(M_{1}+M_{2})^{2})/M_{n}^{2}]^{2}-4M_{U}^{2}/M_{n}^{2}\\}^{1/2}$.
Thus, the ${q}_{U_{r}}$ domain of integration is
${q}_{U_{r}}\in(0,{q}_{U_{r}}^{(max)})$, while the other integrations are over
the usual full range.

###  B.2 The loop variables and measure (for ${\cal X}$ decay)

This analysis closely parallels that of the 3-body phase-space variables
discussed in Sec. B.1.

####  B.2.1 Two-loop integral measure

We have the two-loop integral measure (such as for Fig. 6) with three cut
propagators given by999 The $i$ is included elsewhere in the text.

$d\Pi^{\prime}_{3}=\frac{1}{2}\frac{d^{4}q^{\prime}_{U}}{(2\pi)^{4}}\frac{d^{4}q_{3}}{(2\pi)^{4}}\frac{d^{4}q_{4}}{(2\pi)^{4}}(2\pi)^{3}\delta^{\rm
cut}_{34U}\,(2\pi)^{4}\delta^{4}(p_{n}-(q_{3}+q_{4}+q^{\prime}_{U}))\ ,$ (82)

with $\delta^{\rm
cut}_{34U}=\delta({q^{\prime}_{U}}^{2}-M_{U}^{2})\delta(q_{3}^{2}-M_{3}^{2})\delta(q_{4}^{2}-M_{4}^{2})$
with $M_{3}=M_{4}=M_{D}$. We have shown also the overall 4-momentum conserving
$\delta-$function, which we enforce at the outset by performing the
$d^{4}q_{4}$ integral, which sets $q_{4}=p_{n}-q_{3}-q^{\prime}_{U}$, and
yields
$d\Pi^{\prime}_{3}=(1/2)\,(1/(2\pi)^{5})d^{4}q_{3}\,d^{4}q^{\prime}_{U}\,\delta^{\rm
cut}_{34U}$.

We follow a procedure identical to the 3-body phase-space discussion in Sec.
B.1, but with the following changes: $q_{U}\rightarrow q^{\prime}_{U}$,
$q_{1}\rightarrow q_{3}$, $q_{2}\rightarrow q_{4}$, $M_{1}\rightarrow M_{3}$,
$M_{2}\rightarrow M_{4}$, forming
$k^{\prime}_{B}\equiv(p_{n}-q^{\prime}_{U})=(E^{\prime}_{B},{\bf
k}^{\prime}_{B})$, and analogous to the *-frame earlier as the rest-frame of
$k_{B}$, we now construct the ${\oasterisk}$-frame as the rest-frame of the
$k^{\prime}_{B}$, i.e. ${k^{\prime}_{B}}^{\oasterisk}=(M^{\prime}_{B},{\bf
0})$, with ${M^{\prime}_{B}}^{2}={k^{\prime}_{B}}^{2}$. In the
${\oasterisk}$-frame the ${\bf q}_{3}^{\oasterisk}=-{\bf q}_{4}^{\oasterisk}$,
i.e. these are back-to-back. The boost rapidity that takes us from the lab-
frame to the ${\oasterisk}$-frame is now
$t^{\prime}_{\eta}\equiv\tanh\eta^{\prime}={k^{\prime}_{B}}_{r}/E^{\prime}_{B}$,
and the corresponding $p^{\oasterisk}=(E_{p}^{\oasterisk},{\bf
p}^{\oasterisk})$ with ${\bf p}^{\oasterisk}={\bf
p}_{\parallel}^{\oasterisk}+{\bf p}_{\perp}^{\oasterisk}$ is constructed in an
analogous way. We use the Lorentz invariance of each factor in the integration
measure and take $q^{\prime}_{U}$ in the lab-frame, and $q_{3}^{\oasterisk}$
in the ${\oasterisk}-$frame. We finally write the measure analogously as

$d\Pi^{\prime}_{3}=\frac{1}{(2\pi)^{5}}\frac{1}{2}dc^{\prime}_{U}d\phi^{\prime}_{U}d{q^{\prime}_{U}}_{\\!r}dc^{\oasterisk}_{3}d\phi^{\oasterisk}_{3}\frac{{q^{\prime}_{U}}_{\\!r}^{2}}{2E^{\prime}_{U}}\frac{{q_{3}}_{r}^{\\!\\!{\oasterisk}}}{4({E_{n}}^{\\!\\!{\oasterisk}}-{E^{\prime}_{U}}^{\\!\\!{\oasterisk}})}\
,$ (83)

in terms of the five independent loop variables
$c^{\prime}_{U},\phi^{\prime}_{U},{q^{\prime}_{U}}_{\\!r},c^{\oasterisk}_{3},\phi^{\oasterisk}_{3}$
for integration over the on-shell loop momenta. The domain of integration for
these variables are the usual full range, except for
${q^{\prime}_{U}}_{\\!r}\in(0,{q^{\prime}_{U}}_{\\!r}^{(max)})$ with
${q^{\prime}_{U}}_{\\!r}^{(max)}={q}_{U_{r}}^{(max)}$, as before, in the
${\cal X}$ rest-frame.

For each numerical integration sampling point, using the Lorentz
transformations given, we transform the $q_{1}^{*}$ and the
$q_{3}^{\oasterisk}$ to the lab-frame and compute all 4-momentum dot-products
using the lab-frame momenta
$\\{p_{n},q_{U},q_{1},q_{2},q^{\prime}_{U},q_{3},q_{4}\\}$. We also compute
the
$\epsilon^{\mu\nu\sigma\tau}{p_{i}}_{\mu}{p_{j}}_{\nu}{p_{k}}_{\sigma}{p_{l}}_{\tau}$
with the $p_{i,j,k,l}$ being any of these lab-frame momenta. Using the so
computed (Lorentz invariant) dot-products, we evaluate the matrix element,
fold in the $d\Pi_{3}\,d\Pi^{\prime}_{3}$ measures and obtain the required
integrals numerically.

####  B.2.2 One-loop integral measure

Here we write down the one-loop integral measure applicable to the diagram
such as the ${\cal A}_{01}^{\prime(1)}$ depicted in Fig. 11 (left). Appealing
to the Lorentz invariance of the loop measure, we write this in the $*$-frame
as

$d\Pi_{2}^{\prime}=\frac{1}{(2\pi)^{2}}dc_{3}^{*}d\phi_{3}^{*}\frac{{q_{3}^{*}}_{r}}{4(E_{n}^{*}-E_{U}^{*})}\
,$ (84)

where
${q_{3}^{*}}_{r}=(M_{B}/2)\\{[1-(M_{3}^{2}+M_{4}^{2})/M_{B}^{2}]^{2}-4M_{3}^{2}M_{4}^{2}/M_{B}^{4}\\}^{1/2}$
with $M_{3,4}=q_{3,4}^{2}$.

We point out a subtlety in the maximum value of ${q}_{U_{r}}$ in Fig. 11.
Since the left-cut also has to be enforced, 4-momentum conservation at the
left-most vertex puts an additional constraint, namely, we must now also have
${q}_{U_{r}}\leq{{\tilde{q}}_{U_{r}}}^{(max)}$ where
${{\tilde{q}}_{U_{r}}}^{(max)}=(M_{n}/2)\\{[1+(M_{U}^{2}-(M_{m}+M_{U})^{2})/M_{n}^{2}]^{2}-4M_{U}^{2}/M_{n}^{2}\\}^{1/2}$,
in addition to ${q}_{U_{r}}\leq{q}_{U_{r}}^{(max)}$ as earlier for the right-
cut. We must thus take the stricter upper-bound of these, i.e. we have the
${q}_{U_{r}}$ domain of integration for ${\cal A}_{01}^{\prime(1)}$ now given
as ${q}_{U_{r}}\in\left(0,{\rm
min}\\{{q}_{U_{r}}^{(max)},{{\tilde{q}}_{U_{r}}}^{(max)}\\}\right)$, while the
other integrations are over the usual full range.

###  B.3 The 2-body phase-space variables and measure (for ${\cal X}$
scattering)

Our choice of the final-state phase-space variables has similarities with the
considerations of Sec. B.1, but now for scattering it is a 2-body phase-space,
unlike being 3-body phase-space earlier. One change for scattering is that
instead of the final state momentum $q_{U}$ we now have the initial state
momentum $k_{i}$. Another change is that in decay, we could analyze the
process in the ${\cal X}$ rest-frame without any loss of generality, but in
scattering, we have to generalize this to have arbitrary ${p_{n}}_{r}$, i.e.
we no longer take the lab-frame to be the ${\cal X}$ rest-frame here. We make
these generalizations in our implementation of $d\Pi_{2}$ for our scattering
processes SC-1 and SC-2 discussed in Sec. 4.

Similar to the procedure of Sec. B.2, we now take the incoming momenta ${\bf
p}_{n}$ in terms of the independent $({p_{n}}_{r},c_{n},\phi_{n})$, and ${\bf
k}_{i}$ in terms of the independent variables $({k_{i}}_{r},c_{i},\phi_{i})$.
For unpolarized ${\cal X}$ scattering the matrix element will have no
$\phi_{n}$ dependence and we can integrate it trivially to get a factor of
$2\pi$, and conveniently set $\phi_{n}=0$.

The Lorentz invariance of $d\Pi_{2}$ allows us to compute it in any frame, and
we choose to compute it in the center-of-momentum (CM) frame, which is the
rest-frame of $k^{\prime}_{A}=p_{n}+k_{i}$.101010 This has parallels to the
*-frame in Sec. B.1, which was the rest-frame of $k_{B}$. We now write

$d\Pi_{2}=\frac{1}{(2\pi)^{2}}dc_{1}^{CM}d\phi^{CM}_{1}\frac{{q_{1}}_{r}^{CM}}{4M_{A}}\,\left[\frac{1}{2}\right]_{SC-2}\
,$ (85)

where $\theta_{1}\equiv\theta_{q_{1}}$, $\phi_{1}\equiv\phi_{q_{1}}$ with
$c_{1}\equiv\cos{\theta_{1}}$, $M_{A}^{2}={k^{\prime}_{A}}^{2}$. We have
$M_{A}=E_{CM}=E^{CM}_{n}+E_{i}^{CM}$, and
${q_{1}}_{r}^{CM}=(M_{A}/2)\,\\{[1+(M_{2}^{2}-M_{1}^{2})/M_{A}^{2}]^{2}-4M_{2}^{2}/M_{A}^{2}\\}^{1/2}$,
and since $M_{A}$ is frame-independent, we also have
$M_{A}^{2}=M_{n}^{2}+M_{i}^{2}+2(E_{n}E_{i}-{p_{n}}_{r}{k_{i}}_{r}c_{ni})$,
and without loss of generality we choose ${\bf p}_{n}$ to be along the z-axis,
and therefore, $c_{ni}=c_{i}$. We include an extra factor of $1/2$ only for
SC-2 due to two identical ${D}$ in the final state.

###  B.4 The loop variables and measure (for ${\cal X}$ scattering)

Our implementation of the loop integration measure in scattering closely
parallels the decay case in Sec. B.2, and we discuss here how we adapt this
for scattering. As mentioned in Sec. B.3 we generalize to have the scattering
${p_{n}}_{r}\neq 0$ here too.

####  B.4.1 Two-loop measure

We construct the ${\oasterisk}$ frame exactly as discussed before in Sec. B.2.
But using the Lorentz invariance of the measure, we choose to now take as the
independent variable ${{\bf q}^{\prime}_{U}}$ variables in the ${\oasterisk}$
frame (instead of the lab frame earlier). As before we take ${\bf q}_{3}$
independent variables also in the ${\oasterisk}$ frame. We can then write the
loop integration measure for the scattering channels as

$d\tilde{\Pi}^{\prime}_{3}=\frac{1}{(2\pi)^{5}}\frac{1}{2}d{c^{\prime}_{U}}^{\\!\\!{\oasterisk}}d{\phi^{\prime}_{U}}^{\\!\\!{\oasterisk}}d{{{q^{\prime}}_{\\!\\!U}}_{\\!r}^{\\!\\!{\oasterisk}}}dc^{\oasterisk}_{3}d\phi^{\oasterisk}_{3}\frac{{{{q^{\prime}}_{\\!\\!U}}_{\\!r}^{\\!\\!{\oasterisk}}}^{2}}{2{E^{\prime}_{U}}^{\\!\\!{\oasterisk}}}\frac{{q_{3}}_{r}^{\\!\\!{\oasterisk}}}{4({E_{n}}^{\\!\\!{\oasterisk}}-{E^{\prime}_{U}}^{\\!\\!{\oasterisk}})}\
,$ (86)

The domain of integration for these variables are the usual full range, except
for
${{{q^{\prime}}_{\\!\\!U}}_{\\!r}^{\\!\\!{\oasterisk}}}\in(0,{{{q^{\prime}}_{\\!\\!U}}_{\\!r}^{\\!\\!{\oasterisk}}}^{(max)})$.
Now, ${{{q^{\prime}}_{\\!\\!U}}_{\\!r}^{\\!\\!{\oasterisk}}}^{(max)}$ is
determined as follows. We first compute the Lorentz invariant
${M_{B}^{\prime}}^{2}=(p_{n}-q_{U}^{\prime})^{2}=({E_{n}}^{\\!\\!{\oasterisk}}-{E^{\prime}_{U}}^{\\!\\!{\oasterisk}})^{2}=M_{n}^{2}+M_{U}^{2}+2({{{q^{\prime}}_{\\!\\!U}}_{\\!r}^{\\!\\!{\oasterisk}}}^{2}-{E_{n}}^{\\!\\!{\oasterisk}}{E^{\prime}_{U}}^{\\!\\!{\oasterisk}})$.
But we also have from 4-momentum conservation
$M_{B}^{\prime}={E_{3}}^{\\!\\!{\oasterisk}}+{E_{4}}^{\\!\\!{\oasterisk}}$. We
realize that ${{{q^{\prime}}_{\\!\\!U}}_{\\!r}^{\\!\\!{\oasterisk}}}$ attains
its maximum value when ${q_{3}}_{r}^{\\!\\!{\oasterisk}}=0$, i.e. when
$M_{B}^{\prime}=2M_{D}$, and we obtain
${{{q^{\prime}}_{\\!\\!U}}_{\\!r}^{\\!\\!{\oasterisk}}}^{(max)}=(M_{n}^{2}/(4M_{D}))\\{[1+(M_{U}^{2}-4M_{D}^{2})/M_{n}^{2}]^{2}-4M_{U}^{2}/M_{n}^{2}\\}^{1/2}$.

Given the above independent momenta in the ${\oasterisk}$ frame, we transform
all the loop momenta to the lab-frame using the boost transformation that we
have already given in Sec. B.2. With all the momenta now available in the lab-
frame, we compute the 4-momentum dot-products, and the matrix element in terms
of those. We then fold in the loop integration measure discussed here and
obtain the required integrals numerically.

####  B.4.2 One-loop measure

For the multiple operator contribution in SC-2, we have
$p_{n}+k_{i}=q_{3}+q_{4}$ as can be seen from the crossed Fig. 10. Thus, it
would be most convenient to pick the independent loop momenta in the CM frame
as the ${\bf q}_{3}$ and ${\bf q}_{4}$ are back-to-back in this frame. We
follow our analysis in Sec. B.3 and write analogous to Eq. (85) the one-loop
measure for this case as

$d\Pi^{\prime}_{2}=\frac{1}{(2\pi)^{2}}dc_{3}^{CM}d\phi^{CM}_{3}\frac{{q_{3}}_{r}^{CM}}{4M_{A}}\
,$ (87)

where $q_{3}=q_{m}$ is the ${\cal X}_{m}$ momentum.

##  Appendix C Thermal Averaging Procedure

Our interest ultimately is in the baryon asymmetry generated due to ${\cal
X}_{n}$ decay and scattering taking place in a thermal bath of the hot early
Universe at temperature $T$. We assume that the ${\cal X}_{n}$ is not too far
out of equilibrium and take its distribution function to be the thermal
distribution. We compute the baryon asymmetry in terms of the thermal averages
of appropriate quantities, where we define the thermal average
$\left<\Theta\right>$ of any quantity $\Theta(\\{p_{i}\\},\\{q_{j}\\})$ with
initial state momenta $\\{p_{i}\\}$ and final state momenta $\\{q_{j}\\}$, as

$\left<\Theta\right>(\\{q_{j}\\})=\int\Pi_{i}\left(\frac{d^{3}p_{i}}{(2\pi)^{3}}\frac{f^{(0)}(E_{i})}{n^{(0)}_{i}}\right)\,\Theta(\\{p_{i}\\},~{}\\{q_{j}\\})\
,$ (88)

with the equilibrium Boltzmann distribution function $f(E)=e^{(\mu-E)/T}\equiv
e^{\mu/T}f^{(0)}(E)$ with $f^{(0)}(E)=e^{-E/T}$, for $E=\sqrt{{\bf
p}^{2}+M^{2}}$. For a species with $g$ degrees of freedom, we have the number
density $n=g\int d^{3}p/(2\pi)^{3}f(E)$, and we can write $n=n^{(0)}e^{\mu/T}$
with $n^{(0)}$ being the number density with $\mu=0$, being given by
$n^{(0)}=g\int d^{3}p/(2\pi)^{3}f^{(0)}(E)$, which we find to be
$n^{(0)}=gM^{2}T/(2\pi^{2})\,K_{2}(M/T)$ where $K_{2}$ is the modified Bessel
function of the second kind. We note the expansions $K_{2}(x)=2/x^{2}+{\cal
O}(1)$ for $x\ll 1$, and $K_{2}(x)=\sqrt{\pi/(2x)}e^{-x}+{\cal O}(1/x)$ for
$x\gg 1$, which for $x\\!\\!=\\!\\!M/T$ give the relativistic and non-
relativistic limits respectively as usual (see for example Ref.[8]).

We focus on thermal averages in scattering processes here since it is more
involved given the two initial state particles. We take the initial state
kinematical variables in the lab-frame, which in our early Universe setting
can be identified with the standard cosmological frame. For the spin averaged
scattering matrix element we expect a trivial dependence on
$(c_{n},\phi_{n},\phi_{i})$, and perform those integrations, leaving
$d^{3}p_{n}d^{3}k_{i}=8\pi^{2}d{p_{n}}_{r}{p_{n}}_{r}^{\\!\\!2}dc_{i}d{k_{i}}_{r}{k_{i}}_{r}^{2}$
over their usual domains ${p_{n}}_{r},{k_{i}}_{r}\in(0,\infty);\
c_{i}\in(-1,1)$.

We define $x\equiv M_{\chi}/T$, and the scaled variables $\hat{M}=M/M_{\chi}$,
$\hat{p}_{r}=p_{r}/M_{\chi}$, $\hat{E}=E/M_{\chi}$. In Fig. (24) (left) we
show the distribution function normalized as a probability distribution
function, namely,

$f_{P}^{(0)}(\hat{p}_{r})=\frac{1}{n^{(0)}}\frac{(4\pi)}{(2\pi)^{3}}\hat{p}_{r}^{2}f^{(0)}(\hat{p}_{r})=\frac{2\pi^{2}\hat{p}_{r}^{2}}{\hat{M}^{2}K_{2}(\hat{M}x)}\,xe^{-\hat{E}x}\
,$ (89)

normalized as $\int_{0}^{\infty}d\hat{p}_{r}f^{(0)}_{P}(\hat{p}_{r})=1$.

Figure 24: The probability distribution function $f_{P}^{(0)}$ vs.
$\hat{p}_{r}$ (left), and $(1/x)f_{P}^{(0)}$ vs. $\hat{\hat{p}}_{r}$ (right).
The right plot is for $x=\\{1/10,1/3,1,3,10,100\\}$ (solid to dotted).

We can equivalently write the energy probability distribution function

$f_{P}^{(0)}(\hat{E})=\frac{1}{n^{(0)}}\frac{(4\pi)}{(2\pi)^{3}}\hat{E}\sqrt{\hat{E}^{2}-\hat{M}^{2}}f^{(0)}(\hat{E})=\frac{2\pi^{2}\hat{E}\sqrt{\hat{E}^{2}-\hat{M}^{2}}}{\hat{M}^{2}K_{2}(\hat{M}x)}\,xe^{-\hat{E}x}\
,$ (90)

which is normalized as
$\int_{\hat{M}}^{\infty}d\hat{E}f^{(0)}_{P}(\hat{E})=1$. In Fig. (24) (right)
we show the probability distribution function $(1/x)f_{P}^{(0)}$ as a function
of $\hat{\hat{p}}_{r}\equiv x\hat{p}_{r}=p_{r}/T$, where we have
$\int_{0}^{\infty}d\,\hat{\hat{p}}_{r}(1/x)f^{(0)}_{P}(\hat{\hat{p}}_{r})=1$.

##  Appendix D UV Completion Corrections

As the momentum transfer in the effective VV interaction of Eq.(1) increases
to approach $\Lambda$, the point-like contact interaction will start to become
resolved by propagating states in the UV completion. This issue is
encountered, for instance, in the scattering channel when the Mandelstam
$s,t,u$ approaches $\Lambda^{2}$. For instance, in computing the thermal
average cross section in Eq. (50), momenta $p\gtrsim\Lambda$ are sampled from
the thermal spectrum. Another aspect is the fact that we rely on loop
corrections to obtain a CP violating effect leading to a baryon asymmetry, and
loop momenta being unrestricted, also resolve the effective vertex. Due to
these reasons, in a strict sense, an effective theory approach is not fully
accurate and one will have to work with a specific UV completion.

This issue is not so pressing for the ${\cal X}$ decay channel as the momenta
are limited by $M_{\chi}$ and the UV completion contributions appear only for
momenta $\gtrsim\Lambda$. This is true for loop momenta as well since we place
cuts on the intermediate state propagators to obtain a baryon asymmetry. In
any case we study the effects of UV physics on the decay processes as well to
confirm this expectation.

Even for scattering processes, our effective theory analysis should be
adequate for obtaining a reasonable estimate of the baryon asymmetry generated
for the following reasons: (a) the probability of sampling momenta
$p\sim\Lambda$ in the thermal spectrum for $T\sim M_{\chi}$ is not very large
for $M_{\chi}/\Lambda\ll 1$, and one can get a fairly good estimate using an
effective theory; (b) the baryon number violation in our setup is due to the
${\cal X}$, and any contributions from propagating UV physics states excited
in the large momentum regions in the loops should not nullify the effects we
have identified; (c) we take differences of loop amplitudes between a process
and conjugate process to obtain the baryon asymmetry, which should have
dominant contributions from the ${\cal X}$ sector. Thus, although the results
in the effective theory may not be fully accurate, it should capture the
dominant contributions. A big advantage is that it captures the behavior of a
whole class of UV completions, which is the spirit of our effective theory
analysis. We quantify next how good an approximation the effective theory
computation of the thermal average cross section is by comparing it to the
result with corrections from an example UV completion included.

We compute the ${\left<\sigma\,v\right>}$ in the effective theory and compare
it with that obtained by including corrections from the state that resolves
the effective vertex in the UV completion as the momentum transfer approaches
$\Lambda$. To do this, for illustration, we pick the UV completion A discussed
in Ref. [2], in which the vertex is resolved as the heavy state $\xi$ with
mass $M_{\xi}$ and decay width $\Gamma_{\xi}$ begins to propagate for
$p\gtrsim M_{\xi}$. The $\xi$ propagator is
$1/(p^{2}-M_{\xi}^{2}+iM_{\xi}\Gamma_{\xi})$. We focus on the scattering
channels SC-1 and SC-2 of Sec. 4. SC-1 involves $\xi$ exchange in the
t-channel, while SC-2 is in the s-channel. Deviations from a contact VV
interaction due to the propagating $\xi$ in $\sigma v$ can be captured by the
replacement at each vertex
$(1/\Lambda^{2})\rightarrow(1/\Lambda^{2})f_{\xi}(p)$, taking
$\Lambda=M_{\xi}$, where the correction factor due to a propagating $\xi$ is
$f_{\xi}(p)=1/(1-\tilde{p}^{2}-i\tilde{\Gamma}_{\xi})$ with
$\tilde{p}=p/\Lambda$ and $\tilde{\Gamma}_{\xi}=\Gamma_{\xi}/\Lambda$. We have
$f_{\xi}(p)\rightarrow 1$ as $\Lambda\rightarrow\infty$, in other words, the
effective theory is recovered in the limit of $r_{M\Lambda}\equiv
M_{\chi}/\Lambda\ll 1$.

For the s-channel scattering process (i.e. for SC-2), we subtract the on-shell
$\xi$ (resonance) contribution as this is included in the $\xi$ decay
contribution and should not also be included in the scattering channel (for a
detailed discussion of this aspect, see Ref. [12]). We compare two alternate
schemes for the subtraction: (a) use the principal value (PV) of the $\xi$
propagator, and, (b) cut-out the contribution under a total window width of
$5\tilde{\Gamma}_{\xi}$ centered at $\tilde{M}_{\xi}\equiv M_{\xi}/\Lambda=1$.
We implement (a) by defining the (finite) PV part as
$1/(p^{2}-M^{2}+i\epsilon)={\rm PV}(1/(p^{2}-M^{2}))-\pi i\delta(p^{2}-M^{2})$
isolating the singular (on-shell) part as the second term. We use the
representation $\delta(p^{2}-M^{2})=\lim_{\epsilon\rightarrow
0}[(\epsilon/\pi)/((p^{2}-M^{2})^{2}+\epsilon^{2})]$, and connect with a
physical interpretation by taking $\epsilon\equiv M_{\xi}\Gamma_{\xi}$. We
thus obtain
$f^{\rm(PV)}_{\xi}(p)=1/(1-\tilde{p}^{2}-i\tilde{\Gamma}_{\xi})-i\tilde{\Gamma}_{\xi}/((1-\tilde{p}^{2})^{2}+\tilde{\Gamma}_{\xi}^{2})$.
We see that our choice of the delta function ensures that the PV is real. We
view $f^{\rm(PV)}_{\xi}(p)$ as a (quadratic) function of the 3-momentum radial
component $p_{r}$, and since $p_{r}\geq 0$ we pick up only the positive pole
contribution.

We thus obtain UV propagator correction factors in the LO scattering
amplitudes given by $f_{\xi\,(t)}=f_{\xi}(p_{n}-q_{2})$ (for SC-1), and,
$f_{\xi\,(s)}=f_{\xi}(p_{n}+k_{i})$ (for SC-2). In the LO cross sections the
correction factors are
$f_{\xi\,(t)}^{(00)}=|f_{\xi}(p_{n}-q_{2})|^{2}=1/((1-\tilde{t})^{2}+\tilde{\Gamma}_{\xi}^{2})$
(for SC-1), and $f_{\xi\,(s)}^{(00)}=|f_{\xi}(p_{n}+k_{i})|^{2}={\rm
PV}(1/((1-\tilde{s})^{2}))$ (for SC-2), where
$(\tilde{s},\tilde{t})=(s,t)/\Lambda^{2}$ for the Mandelstam variables
$s=(p_{n}+k_{i})^{2}$ and $t=(p_{n}-q_{2})^{2}$, and
$\tilde{\Gamma}_{\xi}=\Gamma_{\xi}/\Lambda$ with $\Gamma_{\xi}$ being the
$\xi$ decay width, which we also parametrize as the dimensionless
$\hat{\Gamma}_{\xi}\equiv\Gamma_{\xi}/M_{\chi}$.

In Fig. 25 we show the LO $f_{\xi\,(s,t)}$ and $f_{\xi\,(s,t)}^{(00)}$ for the
t-channel SC-1, and the s-channel SC-2, in the CM frame for
$r_{M\Lambda}=M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$. With our parametrization of
the $\xi$ width,
$\hat{\Gamma}_{\xi}=\Gamma_{\xi}/M_{\chi}=r_{M\Lambda}\tilde{\Gamma}_{\xi}$,
we present our results for sample choices of $\hat{\Gamma}_{\xi}$. For SC-2 we
show the PV for $\hat{\Gamma}_{\xi}\\!\\!=\\!\\!1/2,1/10$, which correspond to
$\tilde{\Gamma}_{\xi}=1/20,1/100$ respectively, for
$M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$. The resonant enhancement is clearly seen,
with the peak being larger for a smaller value of $\hat{\Gamma}_{\xi}$. For
illustration, we have taken the mass parameters as in BP-A for $n=2$, and set
the CM frame phase space angular variables to $c_{1}^{CM}=1/2$, and
$\phi_{1}^{CM}=\pi/6$.

Figure 25:  The CM frame LO UV propagator correction $f_{\xi}$ for BP-A
parameters, $M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$, for SC-1 (top left) and SC-2
(top right), and $f_{\xi\,(s,t)}^{(00)}$ (bottom panel) for SC-1 (solid), and
SC-2 with $\hat{\Gamma}_{\xi}\\!\\!=\\!\\!1/2$ (dashed) and
$\hat{\Gamma}_{\xi}=1/10$ (dotted); the bottom right plot zooming in on the
resonance region.

We include the $f_{\xi}$ factor and obtain the ${\left<\sigma\,v\right>}$ by
numerical integration as detailed in Sec. 4. For illustration we evaluate the
${\left<\sigma\,v\right>}$ for $x=1/3$ and
$\tilde{\Gamma}_{\xi}=\Gamma_{\xi}/\Lambda=1/10$. Compared to the effective
theory value, we find the following ${\left<\sigma\,v\right>}$ numbers for
scheme (a) (with scheme (b) shown in parenthesis):
for $r_{M\Lambda}=1/10$, SC-1 is 66% (70%) lower, while the SC-2 is 200%
higher (33% lower);
for $r_{M\Lambda}=1/25$, SC-1 is 33% (33%) lower, while SC-2 is 600% (30%)
higher;
for $r_{M\Lambda}=1/50$, SC-1 is 10% (13%) lower, while SC-2 is 65% (45%)
higher;
for $r_{M\Lambda}=1/100$, SC-1 is 3% (4%) lower, while SC-2 is 10% (10%)
higher.
From this we learn that the relative difference in the thermal average cross
section between the effective theory and the UV completion A is at most an
${\cal O}(1)$ factor for $M_{\chi}/\Lambda<1/10$, and the agreement is better
than to 10% (10%) for $r_{M\Lambda}<1/100$.

For the tree-loop interference contribution of Fig. 6, we include the UV state
$\xi$ propagator correction by multiplying the $f_{\xi}(p)$ for each vertex,
which yields
$f^{(01)}_{\xi}=f_{\xi}(p_{n}-q^{\prime}_{U})\,f_{\xi}(q^{\prime}_{U}-p_{n}^{\prime})\,f_{\xi}(p^{\prime}_{n}-q_{U})\,f_{\xi}(q_{U}-p_{n})$,
which is equivalently,
$f^{(01)}_{\xi}=f_{\xi}(p_{n}-q^{\prime}_{U})\,f_{\xi}(q^{\prime}_{U}-p_{n}^{\prime})\,f^{*}_{\xi}(q_{U}-p^{\prime}_{n})\,f^{*}_{\xi}(p_{n}-q_{U})$,
and we take the PV part of each factor. For the UV correction factor for the
scattering channels we apply the crossing relations in Eq. (38) on this.

In Fig. 26 we show the $\hat{p}_{n_{r}}$ dependence of the CM frame UV
propagator correction factor $f_{\xi}^{(01)}$ for SC-1 and SC-2 for the tree-
loop interference contribution in diagrams A,B,C of (the crossed) Fig. 6 for
$M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$, where for SC-2 we show the PV for
$\hat{\Gamma}_{\xi}\\!\\!=\\!\\!1/2,1/10$ and the resonant enhancement is
clearly seen. The different curves for diagram $C$ is for the different
subdiagrams $Cij$ discussed below Eq. (23).

Figure 26:  The $\hat{p}_{n_{r}}$ dependence of the CM frame UV propagator
correction factor $f_{\xi}^{(01)}$ for the tree-loop interference contribution
in diagram A (left), diagram B (middle) and diagram C (right), for BP-A
parameters, and, $M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$, for SC-1 (solid), and
SC-2 with $\hat{\Gamma}_{\xi}\\!\\!=\\!\\!1/2$ (dashed) and
$\hat{\Gamma}_{\xi}=1/10$ (dotted).

For illustration we have set the CM frame phase space angular variables to
$c_{1}^{CM}=1/2$, and $\phi_{1}^{CM}=\pi/6$, and the loop variables to
$c^{\prime}_{U}=1/2$, $\phi^{\prime}_{U}=\pi/6$,
${q^{\prime}_{U}}_{\\!r}=0.15$, $c^{\oasterisk}_{3}=1/2$,
$\phi^{\oasterisk}_{3}=\pi/6$.

For the mixed-operator tree-loop interference correction factor in Fig. 11, we
have the UV propagator correction factors product
$f^{\prime(01)}_{\xi}=f_{\xi}(q_{3}+q_{U})\,f_{\xi}(q_{1}+q_{2})\,f^{*}_{\xi}(q_{1}+q_{2})$.
Again, for the scattering channels we apply the crossing relations on this. In
Fig. 27 we show the $\hat{p}_{n_{r}}$ dependence of the CM frame UV propagator
correction factor $f_{\xi}^{{}^{\prime}(01)}$ for SC-2 in the multiple-
operator tree-loop interference contribution of the (crossed) Fig. 11 diagram
for $M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$, for
$\hat{\Gamma}_{\xi}\\!\\!=\\!\\!1/2,1/10$. For this illustrative plot, we set
the phase space momentum angular variables at $c_{1}^{CM}=1/2$,
$\phi_{1}^{CM}=\pi/6$, and the loop momentum angular variables at $c_{m}=1/2$,
$\phi_{m}=\pi/6$.

Figure 27:  The $\hat{p}_{n_{r}}$ dependence of the CM frame UV propagator
correction factor $f_{\xi}^{{}^{\prime}(01)}$ in the multiple-operator tree-
loop interference contribution for SC-2 with BP-B parameters, and,
$M_{\chi}/\Lambda\\!\\!=\\!\\!1/10$, $\hat{\Gamma}_{\xi}\\!\\!=\\!\\!1/2$
(dashed) and $\hat{\Gamma}_{\xi}=1/10$ (dotted).

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