ABSTRACT (original article): We study the Bunch-Davies (BD) and Unruh-de Witt (UdW) de Sitter S-matrices in the presence of spin-1/2 fermions. Building on recent work, this enables us to correlate the de Sitter S-matrix with cosmological correlators. We consider a finite-time version of the UdW S-matrix to study
O(H)
corrections to some typical particle physics processes such as beta decay. Owing to the lack of time-reversal symmetry in the expanding Poincaré patch, we find signatures of intrinsic T-violation in polarized beta decay. The observable we study begins at
O(H)
. The possibility of T-violation was examined theoretically in the 1950s by Jackson, Treiman, and Wyld in flat space and has been probed more recently in the emiT experiment, with the purpose of examining fundamental T-violation coming from additional interactions in the Lagrangian. Our analysis places a lower bound on the intrinsic T-violation in the expanding Poincaré patch. At
O(H)
, we find both energy conserving and energy non-conserving contributions. Surprisingly, the energy-violating piece, in principle, can give large T-violation at fine-tuned values of the kinematical variables. CITATION (original article): Aman Goyal, Aneek Jana, Swapnanil Mandal and Aninda Sinha, Fermionic S-matrix and cosmological correlators: T-violation at O(H), JCAP11(2025)014. http://doi.org/10.1088/1475-7516/2025/11/014
Scattering amplitudes are interesting observables in quantum field theories (QFTs) which allow us to understand various aspects of the concerned QFT. In this work our aim is to compute scattering amplitudes involving fermionic particles in de Sitter spacetime, with Hubble constant
H
. However the propagators and mode functions of particles in de Sitter involve Hankel functions, and the time-integrals for computing the amplitude is hard to evaluate explicitly. Therefore, we do an
O(H)
expansion whose zeroth order term is just the flat-space answer and the
O(H)
part is the first correction that arises due to exponentially expanding background spacetime, that is de Sitter spacetime, in the small Hubble parameter limit, relevant for our present day universe. The interesting observation is that it is possible to construct combinations which reflect only the
O(H)
corrections thus opening up the possibility of obtaining the value of
H
from particle physics experiments. One such observable is the spin-asymmetry in beta decay in de Sitter spacetime. The goal of the notebook is to show how approximations to the mode functions mentioned in the paper can be used to calculate the
O(H)
correction to scattering in de Sitter spacetime.

Fermionic Scattering in de Sitter spacetime

Definitions

Basic Definitions

Identity matrices with dimensions 4 and 2
Id=IdentityMatrix[4];​​Id2=IdentityMatrix[2];
The following is 1 + 3 dimensional Minkowski metric in most negative signature
η={{1,0,0,0},{0,-1,0,0},{0,0,-1,0},{0,0,0,-1}};
Representation of flat space Dirac gamma matrices used in the paper, see Appendix A .2
γ0={{1,0,0,0},{0,1,0,0},{0,0,-1,0},{0,0,0,-1}};​​γ1={{0,0,0,1},{0,0,1,0},{0,-1,0,0},{-1,0,0,0}};​​γ2={{0,0,0,-I},{0,0,I,0},{0,I,0,0},{-I,0,0,0}};​​γ3={{0,0,1,0},{0,0,0,-1},{-1,0,0,0},{0,1,0,0}};
Dirac gamma matrices written as 4 - vector
γ={γ0,γ1,γ2,γ3};
The chirality matrix
γ5=Iγ0.γ1.γ2.γ3//Simplify;
3 - momentum vectors to be used later
p1vec={p1x,p1y,p1z};​​p2vec={p2x,p2y,p2z};​​p3vec={p3x,p3y,p3z};​​p4vec={p4x,p4y,p4z};
energies, from relativistic dispersion
ωp1=Sqrt[m1^2+p1x^2+p1y^2+p1z^2];​​ωp2=Sqrt[m2^2+p2x^2+p2y^2+p2z^2];​​ωp3=Sqrt[m3^2+p3x^2+p3y^2+p3z^2];​​ωp4=Sqrt[m4^2+p3x^2+p3y^2+p3z^2];
γ
μ
μ
p
objects
p1s=ωp1γ0-γ1p1x-γ2p1y-γ3p1z;​​p2s=ωp2γ0-γ1p2x-γ2p2y-γ3p2z;​​p3s=ωp3γ0-γ1p3x-γ2p3y-γ3p3z;​​p4s=ωp4γ0-γ1p4x-γ2p4y-γ3p4z;
σ . p object where p is the 3 - vector
sigmaDotp[pvec_]:=pvec[[1]]PauliMatrix[1]+pvec[[2]]PauliMatrix[2]+pvec[[3]]PauliMatrix[3];

First order expansions

In[]:=
$Assumptions={m>=0,k>=0,t∈Reals,H>=0};
The following is the O(H) correction to the scalar mode function as in Eq .2 .68 of the paper. The t - independent O(H) is irrelevant overall phase which may appear different than the one mentioned in the paper.
f1[t_,m_,k_]:=HI
1
8
9
m
Log
m
k
+Sqrt
m^2
k^2
+1+
k^2
2(k^2+m^2)
t+I
k^2
2Sqrt[k^2+m^2]
t^2
The following are the objects in Eq .2 .78 of the paper needed to construct the O(H) expansion of the spinor mode function in Eq .2 .80
OH=-
Hm-5m+6
2
k
+
2
m

24
3/2
(
2
k
+
2
m
)
+
1
2
tHm
2
k
+
2
m
-m
2
k
+
2
m
+
H
2
k
t^2
2
2
k
+
2
m
;​​OpH=+
Hm5m+6
2
k
+
2
m

24
3/2
(
2
k
+
2
m
)
-
1
2
tHm
2
k
+
2
m
+m
2
k
+
2
m
+
H
2
k
2
2
k
+
2
m
t^2;
Evaluating eq. 2.74 (we have done the substitution (t’ -> -t) as explained in the eq. (4.15)
The following objects are needed to construct the O(H) correction to the of fermion projectors u u_bar, v v_bar in de Sitter, see Eq .2 .78 of the paper .
d1=OH+Conjugate[OH/.{t->-t}]//FullSimplify​​d2=OH+Conjugate[OpH/.{t->-t}]//FullSimplify​​d3=OpH+Conjugate[OH/.{t->-t}]//FullSimplify​​d4=OpH+Conjugate[OpH/.{t->-t}]//FullSimplify
Out[]=
0
Out[]=
Hm-+2
2
k
+
2
m
t
2(
2
k
+
2
m
)
Out[]=
Hm-2
2
k
+
2
m
t
2(
2
k
+
2
m
)
Out[]=
0
From above we find that the diagonal elements of
r
α
u,v
is zero
Now we have various parts of Eq. 2.82 for the fermion mode-projectors
u
u
and
v
v
in de Sitter.
The following objects are zeroth order part of fermion mode projectors u u_bar, v v_bar, see Eq .2 .83
PrU[m_,pvec_,proj_]:=((Sqrt[m^2+pvec.pvec]+m)^2KroneckerProduct[{{1,0},{0,0}},proj]-(Sqrt[m^2+pvec.pvec]+m)KroneckerProduct[{{0,1},{0,0}},proj.sigmaDotp[pvec]]+(Sqrt[m^2+pvec.pvec]+m)KroneckerProduct[{{0,0},{1,0}},sigmaDotp[pvec].proj]-KroneckerProduct[{{0,0},{0,1}},sigmaDotp[pvec].proj.sigmaDotp[pvec]])/(Sqrt[m^2+pvec.pvec]+m)//FullSimplify;​​PrV[m_,pvec_,proj_]:=(KroneckerProduct[{{1,0},{0,0}},sigmaDotp[pvec].(Id2-proj).sigmaDotp[pvec]]-(Sqrt[m^2+pvec.pvec]+m)KroneckerProduct[{{0,1},{0,0}},sigmaDotp[pvec].(Id2-proj)]+(Sqrt[m^2+pvec.pvec]+m)KroneckerProduct[{{0,0},{1,0}},(Id2-proj).sigmaDotp[pvec]]-(Sqrt[m^2+pvec.pvec]+m)^2KroneckerProduct[{{0,0},{0,1}},(Id2-proj)])/(Sqrt[m^2+pvec.pvec]+m)//FullSimplify;
The following objects are O (H) part of fermion mode projectors u u_bar, v v_bar in de Sitter, see Eq .2 .84
αrU[m_,pvec_,proj_,t_]:=KroneckerProduct[{{0,1},{0,0}},proj.sigmaDotp[pvec]]
Hm(-2Sqrt[m^2+pvec.pvec]t)
2(m^2+pvec.pvec)
+KroneckerProduct[{{0,0},{1,0}},sigmaDotp[pvec].proj]
Hm(-2Sqrt[m^2+pvec.pvec]t)
2(m^2+pvec.pvec)
;​​αrV[m_,pvec_,proj_,t_]:=KroneckerProduct[{{0,1},{0,0}},sigmaDotp[pvec].(Id2-proj)]
Hm(+2Sqrt[m^2+pvec.pvec]t)
2(m^2+pvec.pvec)
+KroneckerProduct[{{0,0},{1,0}},(Id2-proj).sigmaDotp[pvec]]
Hm(+2Sqrt[m^2+pvec.pvec]t)
2(m^2+pvec.pvec)
;

Useful Equations

Defining the spin projectors (Eq. 2.87) to be used in above expressions as inputs.
In[]:=
projUp:={{Cos[θ/2]^2,Sin[θ/2]Cos[ϕ/2]Exp[-Iϕ]},{Sin[θ/2]Cos[ϕ/2]Exp[-Iϕ],Sin[θ/2]^2}};​​projDown:=Id2-projUp;
Table[Tr[γ[[a]].(Id-γ5).(PrU[m2,p2vec,projUp]).γ[[b]].(Id-γ5).(αrU[m1,p1vec,Id2,t])],{a,1,4},{b,1,4}]//Simplify​​Table[Tr[γ[[a]].(Id-γ5).(PrU[m2,p2vec,projUp]).γ[[b]].(Id-γ5).(αrV[m1,p1vec,Id2,t])],{a,1,4},{b,1,4}]//Simplify​​Table[Tr[γ[[a]].(Id-γ5).(PrV[m2,p2vec,projUp]).γ[[b]].(Id-γ5).(αrU[m1,p1vec,Id2,t])],{a,1,4},{b,1,4}]//Simplify​​Table[Tr[γ[[a]].(Id-γ5).(PrV[m2,p2vec,projUp]).γ[[b]].(Id-γ5).(αrV[m1,p1vec,Id2,t])],{a,1,4},{b,1,4}]//Simplify
Out[]=
{{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
Out[]=
{{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
Out[]=
{{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
Out[]=
{{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
The above is a very useful result as it says that if there is an unpolarized spinor then it gives a zero contribution to the |M|^2
Defining projectors along z - direction
In[]:=
projUpZ=projUp/.{θ->0}​​projDownZ=Id2-projUpZ
Out[]=
{{1,0},{0,0}}
Out[]=
{{0,0},{0,1}}

Pion Decay

In[]:=
p1vec4={ωp1,p1vec}//Flatten;
Defining, Eq. 4.12 in the paper
In[]:=
ε1:={Iωp1f1[t,m1,Sqrt[p1vec.p1vec]]+D[f1[t,m1,Sqrt[p1vec.p1vec]],t]-3H/2+Iωp1Ht,Ip1vecf1[t,m1,Sqrt[p1vec.p1vec]]}//Flatten;

Unpolarized Decay

We expand the |M|^2 (Eq. 4.9) up to first order in H. Zeroth order term is given by (We are factoring out
2
λ
-iΔω(t-t')
e
/(
∏
i
(2
ω
i
)
) )
​
In[]:=
Sum[η[[a,b]]η[[c,d]]Tr[γ[[a]].(Id-γ5).(p3s+Idm3).γ[[c]].(Id-γ5).(p2s-Idm2)](p1vec4[[b]]p1vec4[[d]]),{a,1,4},{b,1,4},{c,1,4},{d,1,4}]//Simplify
Out[]=
8
2
m1
p2xp3x+p2yp3y+p2zp3z+
2
m2
+
2
p2x
+
2
p2y
+
2
p2z
2
m3
+
2
p3x
+
2
p3y
+
2
p3z
+2p1yp1z(p2zp3y+p2yp3z)-p1y
2
m1
+
2
p1x
+
2
p1y
+
2
p1z

2
m2
+
2
p2x
+
2
p2y
+
2
p2z
p3y+p2y
2
m3
+
2
p3x
+
2
p3y
+
2
p3z
-p1z
2
m1
+
2
p1x
+
2
p1y
+
2
p1z

2
m2
+
2
p2x
+
2
p2y
+
2
p2z
p3z+p2z
2
m3
+
2
p3x
+
2
p3y
+
2
p3z
+
2
p1x
p2xp3x+
2
m2
+
2
p2x
+
2
p2y
+
2
p2z
2
m3
+
2
p3x
+
2
p3y
+
2
p3z
+
2
p1y
p2yp3y+
2
m2
+
2
p2x
+
2
p2y
+
2
p2z
2
m3
+
2
p3x
+
2
p3y
+
2
p3z
+
2
p1z
p2zp3z+
2
m2
+
2
p2x
+
2
p2y
+
2
p2z
2
m3
+
2
p3x
+
2
p3y
+
2
p3z
+p1xp1y(p2yp3x+p2xp3y)+p1z(p2zp3x+p2xp3z)-
2
m1
+
2
p1x
+
2
p1y
+
2
p1z

2
m2
+
2
p2x
+
2
p2y
+
2
p2z
p3x+p2x
2
m3
+
2
p3x
+
2
p3y
+
2
p3z

As the pion decay is 3 body process with 3-momentum conservation, the above volume element is zero as the decay is happening in a plane. Therefore, there are no O(H) corrections in this case.

Polarized decay

Here we are considering the case when we can distinguish between the two polarizations of charged lepton in the final state. We can consider positive z-axis as polarization direction without loss of generality. Zeroth order term Eq 4.14 is given by
Contribution from neutrino is zero as it is unpolarized. Contribution from charged lepton is given by, Eq. 4.15
There above term is equal to (4.17)(we have used 3-momentum conservation). The contribution from pion is given by, Eq. 4.16
Here again the term with momentum volume element goes to zero but we still have corrections due to pion. Using 3-momentum conservation we find this is equal to Eq.(4.17-4.18)

4-Fermi Interaction

Unpolarized Decay

We expand the Eq. 4.50 upto first order in H. Zeroth order term is given by
Because all the spinors are unpolarized it means that there are no contributions from them. Therefore correction is zero in this case as well Therefore the matrix element is simply eq 4.51

Polarized Decay

Here we are considering the case when we can distinguish between the two polarizations of incoming neutron. Again, We can consider positive z-axis as polarization direction without loss of generality. Zeroth order term now is given by
contributions to the correction will only be due to incoming neutron which is given by

Estimating the D

Following is the other guess
For guess 1
For guess 2
For guess 1
For guess 2
For guess 1
For guess 2

For guess 1

For guess 2

CITE THIS NOTEBOOK

Fermionic S-matrix and cosmological correlators: T-violation at O(H)​
by Aman Goyal, Aneek Jana, Swapnanil Mandal & Aninda Sinha​
Wolfram Community, STAFF PICKS, November 12, 2025
​https://community.wolfram.com/groups/-/m/t/3573982