Pendulum Leptogenesis
Neil D. Barrie
Kavli IPMU (WPI)

September 29, 2018

K. Bamba, NDB, A. Sugamoto, T. Takeuchi and K. Yamashita, arxiv:1610.03268(MPLA), and arxiv:1805.04826 (PLB).

N. D. Barrie (Kavli IPMU (WPI))

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Matter-Antimatter Asymmetry

The asymmetry is described quantitatively by,
η=nb − nb̄' 8.5 × 10−11s

The Sakharov Conditions
1

Baryon number violation

2

C and CP violation

3

Period of non-equilibrium

N. D. Barrie (Kavli IPMU (WPI))

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Ratchet Mechanism

Inspired by molecular motors in biological systems, and their ability togenerate directed motion.
Consider an inflaton and complex scalar carrying L charge during
reheating.
A derivative coupling between a complex scalar and inflaton.
Directed motion in the complex scalar phase gives a non-zero L
number density.

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The ModelInterplay between the inflaton and complex scalar during reheating,S=√dx 4 −g [g µν ∂µ φ∂ν φ∗ − V0 (φ, φ∗ ) ←→ i
1+ g µν ∂µ Φ∂ν Φ − U(Φ) + g µν φ∗ ∂µ φ ∂ν Φ ,2Λ

where V0 (φ, φ∗ ) = λφ∗ φ(φ − φ∗ )(φ∗ − φ) + ...

Satisfying the Sakharov Conditions
1

The complex scalar potential.

2

Derivative coupling interaction.

3

Reheating epoch.

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U(1)L Global Symmetry
Identify the U(1) symmetry with L charge, where φ has charge 2.
L preserving interactions, Lint = gL φ∗ ν̄Rc νR + yH H L̄νR + h.c.
Taking φ in polar form, φ = √12 φr e iθ ,V (φr , θ) = λφ4r sin2 θ + · · ·andnL = j

0= −2φ2rΦ̇θ̇ −Λ!

.

A non-zero nL requires driven motion in θ̇ .

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September 29, 2018

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N. D. Barrie (Kavli IPMU (WPI))

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Behaviour of the Inflaton
Assuming Starobinsky inflation (U(Φ) ≈ 12 µ2 Φ2 ) during reheating,Φ̈ +2
λφ4r
Φ̇ + Γ Φ̇ + µ2 Φ +sin(2θ) = 0 .
tΛNeed the inflaton (Torque) to be unaffected by θ, (µ2 Φ  λφ4r /Λ),Φ̈ +2+ Γ Φ̇ + µ2 Φ = 0 .
tWhen Γ  µ, the approximate solution to this equation isΦ(t)≈ΦiN. D. Barrie (Kavli IPMU (WPI))

tit

 



cos µ(t − ti ) .

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Behaviour of θUtilising the inflaton EoM, and neglecting Φ decay term,θ̈ + (Γθ + 3H) θ̇ + p sin(2θ) = −q(t) cos µ(t − ti ) ,where
p = λφ2r ,q(t) =µ2 ΦiΛtit .

This is analogous to a forced pendulum where
LHS represents acceleration, damping, and gravitation,
RHS is a sinusoidal driving torque with amplitude q and frequency µ.

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Possible Cases: p  q(t)Consider a large torque (p  q(t)),θ̈ + 3H θ̇ =
1 d  2 t θ̇ = −q(t) cos µ(t − ti ) ,2t dt

which gives,θ̇(t) =Φ̇Λ⇒

j0 = 0L violation has vanished, θ̇ oscillates around zero with Φ̇ .
Analogous to an effectively massless pendulum.

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Possible Cases: p  q(t)Consider a small torque (p  q(t)),θ̈ + 3H θ̇ + p sin(2θ) = 0
Friction term damps θ̇ until θ settles into a minima ⇒ no persistent
non-zero θ̇.
C and CP breaking term ignored.
Analogous to a very massive pendulum, unaffected by input torque.

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September 29, 201810 / 16

Sweet Spot Condition and Driven Motion
We need p ' q(t) ⇒ L, C and CP violating terms all contribute.
SSC: λφ2r 'µ 2 Φ i Hd
Λ Hi

Equivalent to Fd ' mgl ,Torque is time dependent so can only satisfy for a finite time.
Thus we want to consider,θ̈ + (Γθ + 3Hd )θ̇ + p sin(2θ) = −q(td ) cos µ(t − ti ) .

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Phase Locked States
Reparameterise the EoM,Θ̈ +q(td )1Θ̇ + sin Θ =cos(ωτ ) ,Q
pSolutions increasing monotonously in time with small modulations.
Known as “phase-locked states” in the study of forced pendulum.
General phase-locked state solution,
Θ(τ ) = Θ0 + nωτ −∞X

αm sin(mωτ − φm ) .

m=1

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N. D. Barrie (Kavli IPMU (WPI))

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Estimation of n and Parameter Constraints
Consistent with our initial assumptions,n'2λφ2r.
µ2

where n/2 = number of rotations of θ per oscillation of Φ.
After the SSC is violated no simultaneous violation of C, CP and L ,Can approximate the amplitude of oscillations as, sin 2θ ≈ HHd .
Parameter constraints (e.g. Φ is unaffected by θ),
1015 GeV > φr > µ ,1 > λ > 5 × 10−4 ,
310 > n > 1 .
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The Generated Asymmetry and Neutrino Masses
Assuming no additional entropy production,nLη reh =
≈ 0.04n ×sµφ2r
3Treh!

adarh

3.

Generated baryon asymmetry,ηTrh=.
ηobs
2λ · 108 GeV
L preserving interaction, generates a Majorana mass,
1014 GeV > mνR > 1011 GeV .
Via the seesaw mechanism,mν = yH2 vh2 /2mνR .
N. D. Barrie (Kavli IPMU (WPI))

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September 29, 201815 / 16

Conclusion and Future WorkInterplay between inflaton and scalar lepton during reheating,
Driven motion can be modelled as a forced pendulum,
Presence of phase-locked states,
Asymmetry linearly dependent on the reheating temperature.
Seesaw mechanism generates active neutrino masses.

Future work
Investigation of efficiency required.
Other cosmological implications
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