#### Simulations of Cold Electroweak Baryogenesis: hypercharge U(1) and the creation of helical magnetic fields

Received: May
Simulations of Cold Electroweak Baryogenesis: hypercharge U(1) and the creation of helical magnetic elds
Nottingham 0
U.K. 0
Zong-Gang Mou 0 1
Paul M. Sa n 0 2
Anders Tranberg 0 1
CP-violation. We 0
Open Access 0
c The Authors. 0
Theory, Nonperturbative E ects
0 4036 Stavanger , Norway
1 Faculty of Science and Technology, University of Stavanger
2 School of Physics and Astronomy, University of Nottingham
for the rst time in the Bosonic sector the full electroweak gauge group SU(2) U(1) and nd that the maximum generated baryon asymmetry is reduced by a factor of three relative to the SU(2)-only model of [1], but that the quench time dependence is very similar. In addition, we compute the magnitude of the helical magnetic elds, and nd that it is proportional to the strength of CP-violation and dependent on quench time, but is not proportional to the magnitude of the baryon asymmetry as proposed in [2, 3]. Astrophysical signatures of primordial magnetic helicity can therefore not in general be used as evidence that electroweak baryogenesis has taken place.
magnetic; elds; Cosmology of Theories beyond the SM; CP violation; Lattice Quantum Field
Contents
1 Introduction 2 3 3.1
Conclusion
A Lattice implementation
B Initial condition
C CP-violation,
Introduction
Variance of observables
CP-odd: the baryon asymmetry and helicity
Short and long time evolution
Hypercharge impact on asymmetry
Helical magnetic eld and baryogenesis
Sphaleron unravelling
The SU(2)
U(1)-Higgs model with CP-violation
quench time [1, 8].
rst-principles
considered hypercharge and the generation of magnetic
elds [12, 13], or have included
percharge on the
tion value, and
dependence of the baryon asymmetry itself.
The SU(2) U(1)-Higgs model with CP-violation
SU(2) and U(1) gauge elds, with the classical action
S =
dt d3x
+ e2 (t) y + ( y )2 +
y Tr W
The eld strength tensors are W
for SU(2) and B
for U(1), with the dual de ned by
e2 (t) =
2t= q)
The covariant derivative D is given by
= 12
around T = 1 GeV.
number, de ned as an integral over time
or equivalently in a spatial representation
Ncs;2(t)
Ncs;2(0) =
dt d3x Tr W
Ncs;2(t) =
WiaWjak
hNB(t)
NB(0)i = 3h[Ncs;2(t)
Ncs;2(0)]i:
(g0)2 Z
hypercharge
Chern-Simons number
Ncs;1(t) =
d3x ijkBiBjk;
2 =
These are naturally generalised to
outside of the unitary gauge, where we have de ned
= W 3 cos + B sin ;
= W 3 sin
B cos :
= naW a cos + B sin ;
= naW a sin
B cos ;
Nw =
dx3 ijkTr[U y@iU U y@j U U y@kU ];
U(x) =
hNB(t)
NB(0)i = 3h[(Ncs;2
Ncs;1)(t)
(Ncs;2
Ncs;1)(0)]i;
sition involving true fermion production always has
Ncs;1 = 0.
elds, respectively,
EW , EB, E .
na =
' =
to 'y@ '
a manifestly gauge covariant way as
'y(D ') sin
= @ (naW a)
@ (naW a)
value for the mixing angle , the eld A is the photon
eld throughout, but only at the
end of the transition is it massless [19].
We will include in our list of observables the
component, being composed of part of EB and part of EW .
! (0; 0; 1) everywhere,
mH = 2 2 = p
2 v = 125 GeV;
mW =
mZ =
= 0;
gv = 77:5 GeV;
= 88:4 GeV;
with our choices of parameters1
v = 246 GeV ; sin2
= 0:231 ;
e =
g =
= 0:63 ;
g0 =
= 0:35 :
In the A
1 Z
Nh =
d3x ijkAiFjk:
while our CP-even (P-even, C-even) observables are
2, all the energy components EB,
The gauge
impact on the nal results.
appendix A.
ip occurs. The
the evolution from the CP-conjugated initial data.
and energy is transferred to the gauge
elds as particles are created. At rst, only the
modes k
ensemble composed of such pairs.
a certain fraction ip with
1 or even in rare cases
2. Averaging over these instances of 0; 1; 2 gives our ensemble averages. We nd, that when CP-violation is not present, the vanishing of the CP-odd observables is not a result of the cancelling of positive and negative ips. No ips occur at all.
Energy
written as
E =
dt d3x y (x; t);
temperature is also smaller by a factor Tslowquench / (Einitial
E)1=4 ' 0:7 Tfastquench.
motion only, and not the gauge elds themselves.
Variance of observables
CP-odd quantities
w2 = hNw2 i
h2 = hNh2i
Chern-Simons number have been rescaled to t inside the plot.
cs;2 =
cs;1 =
w =
h =
the volume), describing the
(mH t)1:4 and
number as
hNcs;2i(t) =
cs;2(t)
Te (t)
cs;2(t)dt;
their di usion rate, and bias coe cient.
P violating term in the action is
dt cs;2(t)Ncs;
cs;2(t) =
up the di usion rate, and simply nd
hNcs;2i(t) =
CP-odd: the baryon asymmetry and helicity
Short and long time evolution
two quench-times in the gure4
0:294 exp( 0:003 mH t);
0:273 exp( 0:002 mH t):
minimum occurs at 0:71mH q + 10.
Ncs;2 ! Nw also produces good ts.
From this
mH t ' 300{500. Moreover, we see from
gure 7 that the winding number stays put after
{ 10 {
cp = 6:83.
400, and infer the
nal asymmetry from the value of Nw. In this
gure, we show the
accumulated errors from the time integration.
Hypercharge impact on asymmetry
metry itself.
{ 12 {
taking cp = 6:83.
strongly correlated with the
nal asymmetry (this was also observed in [8]). But we see
very little impact of including the hypercharge U(1).
Noting from
gure 10 (left)
Nw(t ! 1) = 1:39
the CP-symmetric ensemble.
{ 13 {
with mH q = 16.
Helical magnetic
eld and baryogenesis
hNcs;1i ' 1:56
that the dependence on cp is close to linear, and we nd
hNhi '
1:66 cp;
We show in
extrapolated to an asymptotic value as for Ncs;2 in
gure 7. We see that for both
observ{ 15 {
the magnetic eld, also generating helicity.
In [2], the prediction is that hNhi '
300hNcs;2i for the decay of Sphalerons. By order
Sphaleron unravelling
elds, with Nh
the equations of motion.
7In the plot we multiply Ncs by 3 to match the convention used in [3].
{ 17 {
Chern-Simons number is one, the helicity
uctuates around
Conclusion
timescale of
Higgs eld at minima of its mean value.
violation and quench time we discovered that, for a
xed quench time, the Chern-Simons
of the Higgs eld,
be used as a proxy for baryogenesis.
{ 18 {
there as well. This is currently under investigation [27].
Acknowledgments
Lattice implementation
L = X
Tr U0i
1 X Tr h(Di )y(Di )
Tr Uij
where cQ = 2 t
Q and
L2 is the CP-violating term,
momentum _ = [ (x + 0)
(x) =
U (x) (x + )Z (x)
and the gauge covariant derivative
U (x)
Z (x) =
V (x)
V (x)
gdx W a(x) ; V (x)
g0dx B (x) ;
of (A.1).
Eia(x) =
the Higgs eld:
electric elds on the lattice as,
Tr hi aUi(x+0)Uiy(x)
gdtdxi
; Ei(x) =
Vi(x+0)Viy(x)
Vi(x)Viy(x+0)
ig0dtdxi
(x+0) 2 (x)+ (x 0)
Tr[ y(x) (x)]
(x) + X 1 @ L2 k ;
where k is one 2
2 matrix in (1; i 1; i 2; i 3), so that
= P
k , and
(x + 0) = (x) + dt _ (x):
Vi(x + 0) = 4
dtdxiEi(x)
i dtdxiEi(x)5 Vi(x):
[Vij(x) Vji(x)+Vji(x j) Vij(x j)] ji ;
Tr hi aUi(x)Uj(x + i)Uiy(x + j)Ujy(x)
Tr hi aUjy(x
j)Ui(x
j)Uj(x + i j)Uiy(x)
For the U(1) gauge eld, we nd:
Ei(x) Ei(x 0)
For the SU(2) gauge elds:
Eia(x)
= X
ji =
J i;a =
Ui(x + 0) =
dtdxiEia(x)
+ X i a g dtdxiEia(x) Ui(x):
Tr h y(x)Ui(x) (x + i)Zi(x)i 3i ;
Tr h y(x)i aUi(x) (x + i)Zi(x)i :
{ 20 {
X Ei(x)
Ei(x i)
X Eia(x)
= 0
= a +
Eia(x) =
1 X Tr hi aUiy(x)i bUi(x)i Eib(x);
0 = g0
1 = g
2 = g
3 = g
the Higgs charge densities,
(x) = 0;
= 0; 1; 2; 3:
invariant combination,
A =
2 sin 1
dx g 2
U (x)
(x + )
Z (x)
dx g0 cos
Im [V (x)] :
Initial condition
Modes and initial charge. The Higgs eld
panded on the lattice as
and its canonical momenta _ are
ex(x) = p
(x) = p
p2!p
np + ; h p p yi = 2;
h p p yi = 2;
are complex
following three criteria:
(1) q1 < (N1
(2) q1 = (N1
(3) q1 = (N1
q1)%N1;
q1)%N1; q2 < (N2
q2)%N2;
q1)%N1; q2 = (N2
q2)%N2; q3 < (N3
q3)%N3;
whereas a \corner" mode obeys
Therefore, the elds can also be expanded into:
q1 = (N1
q1)%N1; q2 = (N2
q2)%N2; q3 = (N3
q3)%N3:
(x) = p
(x) = p
with real numbers ap , bp , cp , dp satisfying
p2!p
eipxr !p hcp +idp i r
ap + ibp
+h:c: + p
We only initialise unstable modes, whose momenta ful l
2 X
cos(pidxi)
H =
minimise H to zero,
; for x = ap ; bp ; cp ; dp
= 0; 1; 2; 3:
Under the parity inversion,
(x) !
Ui(x) ! U y( x
i
We assume the origin point is located at
x = (N1
s1; N2
on the lattice. So for
s2; N3
s3). The
charge conjugation operation is
(x) !
Ui(x) ! Ui (x);
Vi(x) ! Vi (x);
explicitly C-even ensemble of C-conjugate pairs.
CP-violation,
The CP-violating term is
On the lattice, we adopt
L2 =
4 2d4x
Tr [I01I23 + I02I31 + I03I12] ;
+ U y(x
+ U y(x
1 hU (x)U (x + )U y(x + )U y(x)
+ U (x)U y(x
+ )U y(x
)U (x
)U (x
)U (x
)U (x +
)U y(x)
)U (x
Here we list its derivatives with respect to di erent elds,
@@WLia2 =
3 CPgdxi
32 2m3W d4x
(K1[jk]
K1[kj]
+ Tr hi aUi(x)Uj(x + i) 0k(x + i + j)Uiy(x + j)Ujy(x)i
+ Tr hi aUi(x)Uj(x + i)Uiy(x + j) 0k(x + j)Ujy(x)i
+ Tr hi aUi(x)Uj(x + i)Uiy(x + j)Ujy(x) 0k(x)i
j)Uj(x
Tr hi aUi(x)Ujy(x + i j)Uiy(x
Tr hi aUi(x)Ujy(x + i j)Uiy(x
j)Uj(x
j)Uj(x
j)Uj(x
+ Tr hi aUi(x)Uiy(x + 0) jk(x + 0)
+ Tr hi aUi(x)Uiy(x + 0) jk(x)i
Tr hi aUi(x) jk(x + i)Uiy(x
Tr hi aUi(x) jk(x + i 0)Uiy(x
Tr hi aUi(x)Uiy(x
Tr hi aUi(x)Uiy(x
0) jk(x
(x) = j (x)j2I (x):
WL0a2 =
3 CPgdt
32 2m2W d4x
(K3[ijk] + K3[jki] + K3[kij]);
+ Tr hi aUi(x + 0) jk(x + i + 0)Uiy(x)i
+ Tr hi aUi(x + 0) jk(x + i)Uiy(x)i
+ Tr hi aUi(x + 0)Uiy(x) jk(x)i
1 @ L2 k =
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