Noah Graham- An Introduction to Solitons and Oscillons
Transcript of Noah Graham- An Introduction to Solitons and Oscillons
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
1/23
An Introduction to Solitons and Oscillons
Noah GrahamMiddlebury College
July 23, 2007
1
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
2/23
Static solitons: The kink
Consider the 1 + 1 dimensional field theory given by
S[] =m2
d2x 1
2
()()
U() where U() = m
2
82
12
This is a double-well
potential with minima at
= 1. The (nonlinear) fieldequation is
(x, t) = (x, t) U((x, t))1 0 1
U()
We will look for a static solution that goes from = 1 atx =
to = +1 at x = +
.
2
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
3/23
Solving for the kink
We know how to solve (x) = U((x)) in a different context: if xwere t and (x) were x(t), this would be ordinary Newtonian
dynamics of a particle of unit mass in the potential U.
So the solution we are looking for
rolls from one maximum of U tothe other. Throughout this motion, its
(conserved) energy is equal to zero:
1
2(x)2 U() = 0.
So our kink (antikink) should have
kink(x) =
2U(kink)
kink(x) = tanhmx
2
1 0 1
U()
4 1 0 1 4
1
0
1
x
kink
3
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
4/23
Static localized solutions: Solitons
Many static solutions, like the kink and sine-Gordon soliton, are the
lowest energy configurations in a particular topological class, and
thus are automatically stable against deformations. Other solitons
are local minima of the energy without topological structure.
Other topological solitons include: the magnetic monopole in
SU(2) gauge theory with an adjoint Higgs [t Hooft, Polyakov] andmagnetic flux lines in superconductors [Abrikosov, Nielsen, Olesen].
Solitons typically carry such exotic charges and are of particular
interest in the early universe (also string theory, condensed matter).
But they dont appear in every theory. For example, a scalar theory
in more than one space dimension has no static solitons they
lower their energy by shrinking. [Derrick]
4
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
5/23
Time-dependent localized solutions: Oscillons/Breathers
More importantly, the Standard Model of particle physics has no
known stable, localized, static classical solutions. (It does have
instanton processes and an unstable sphaleron.)
Solutions that are time-dependent but still localized evade Derricks
theorem and can exist in a wider variety of field theory models.
If solitons or oscillons form from a thermal background, they can
provide a mechanism for sustained departures from equilibrium,
which can be of particular interest in the early universe, especially
baryogenesis.
5
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
6/23
An integrable system
The Sine-Gordon model
S[] =m2
d2x
1
2()() m2 (1 cos )
similarly has static (anti)soliton solutions: (x) = 4 arctan emx
This theory has an equivalent dual description in which thesolitons are fundamental (fermionic) particles. [Coleman]
It is also integrable, with an infinite set of conserved charges. Sowe can solve its dynamics analytically. [Dashen, Hasslacher, and Neveu]
For example, collide a soliton and antisoliton. They pass right
through each other with only a phase shift:
(x, t) = 4 arctan
sinh mut
u cosh mx
where u is the incident speed and = 1/
1 u2.6
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
7/23
An integrable system II
Letting u = i/ we obtain an exact breather:
(x, t) = 4 arctan
sin mt
cosh mx
where now = 1/1 +
2
Temporal frequency is = m < m.
Spatial width is 1/ = 1/(m).
Amplitude is controlled by . For small , we can construct anapproximate solution of this form for any potential based on
the leading nonlinear terms.
At large distances, the field is small and a linear analysis holds:
8e
|x
|sin t with 2 = m2
2.
7
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
8/23
Q-balls: Stability via conserved charge
Q-balls are time-dependent solutions requiring only a single global
charge, in a nonintegrable theory with no static solitons. [Coleman]
S[] =
d4x
12
()() 12
M2||2 + A||3 ||4The charge is
Q =1
2id3x t
t .
We fix the charge Q by a Lagrange multiplier and obtain the
Q-ball as a local minimum of the energy
E[
] = d
3
x1
2|t i|2 +
1
2||2 +
U(||
) +Q
where U(||) = 12
(M2 2)||2 A||3 + ||4
The Q-ball solution has simple time dependence: (x, t) = eit
(x).8
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
9/23
Q-balls: Stability via conserved charge II
We thus obtain the energy function
E[] =
d3x
1
22 + 1
2U()
+ Q
which is to be minimized over variations of and .
The equation for is
d2
dr2(r) + 2
rd
dr(r) = U()
which is again analogous to ordinary
Newtonian mechanics, but now with
time-dependent friction. 0 0.5 1 1.5 2 2.5 3 3.5 40. 5
0. 4
0. 3
0. 2
0. 1
0
0.1
0.2
0.3
U
()
A simple overshoot/undershoot analysis shows that for a given , a
solution exists with
0 as r
. Then minimize this energy
over to find the exact, periodic Q-ball solution.9
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
10/23
Kink breathers: Forever = a very long time
Suppose we consider breathers, like we saw in the sine-Gordonmodel, but now for the 4 theory in 1 + 1 dimensions. This modelhas static soliton solutions but does not have a useful conserved
charge (we always have = 1 at infinity), and is not integrable. Sothere are no simple expressions for exact breathers.
But for the right ranges of initial velocities, numerical simulationsshow breathers that live for an indefinitely long time.
[Campbell et. al.]
Breathers are stable to all orders in the multiple scale expansion.[Dashen, Hasslacher, and Neveu]
After much debate, the current consensus is that in thecontinuum, such configurations do decay, however, due toexponentially-suppressed non-perturbative effects. [Segur & Kruskal]
For physical applications this distinction is generally irrelevant.10
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
11/23
4 oscillons in three dimensions
What about a real scalar 4 theory in three dimensions? It is not
integrable, has no conserved charges, and no static solitons.
There are still
(approximate)oscillons!
They live a long time,
then suddenly decay.[Gleiser]
0 1000 2000 3000 4000 50000
20
40
60
80
t
totalenergy
and
energy
in
box
ofradius
10
0 1000 2000 3000 4000 50002
0
2
4
t
(0,
t)
11
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
12/23
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
13/23
Oscillon decay
Q: How does such a configuration decay?
A: By coupling to higher-frequency harmonics: 2, 3, etc.
If we cut off the high frequencies with a lattice such that
2 >
m2 + 4/(x)2, then no such harmonics would exist, and the
oscillon would be absolutely stable.
Even without this limitation, however, oscillons can live for an
unnaturally long time.
13
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
14/23
Almost the Standard Model
We would like to apply these ideas to the weak interactions in the
Standard Model. We begin by ignoring fermions and
electromagnetism. (Later we will restore electromagnetism.)
The Higgs is a scalar, SU(2) fundamental: =
12
.
The gauge field is a vector, SU(2) adjoint:
A = ( A0 A1 A2 A3 ) where A = Aa
a
2
The action is
S[, A] =
d4x
(D)D 1
2tr FF
v
2
2
2
with F = A A ig[A, A] and D = ( igA).14
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
15/23
Almost the Standard Model II
We have:
3 real massive vector bosons (W
and Z0, degenerate for us)
with mW =gv2 .
1 real massive scalar (Higgs boson) with mH = v
2.
This is a gauge theory, so any remaining degrees of freedom are
irrelevant gauge artifacts. There are no physical massless degrees
of freedom (because we have ignored electromagnetism).
To make the problem tractable, we will write an ansatz for our
field configurations that is as close to spherically symmetric as
possible: It is invariant under simultaneous rotations of real space
and isospin space. [Dashen, Hasslacher, and Neveu]15
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
16/23
Higgs and Gauge fields in the spherical ansatz
Write as a 2 2 matrix: =
2 11 2
, so that
0
1
= .
The ansatz is:
A(x, t) =a1(r, t)x( x)
2g+
(r, t)( x( x)) + (r, t)(x )2gr
A0(x, t) =
a0(r, t)
x
2g (x, t) =
((r, t) + i(r, t)
x)
g
Ansatz is preserved under U(1) gauge transformations:
A A ig [(r, t)] ( x) exp [i(r, t) x]
For regularity, , , , a1 r and r must vanish at r = 0.
16
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
17/23
Effective 1-d theory
Form reduced fields in 1 + 1 dimensions:
(r, t) = (r, t) + i(r, t) D = ( i2
a)
(r, t) = (r, t) + i((r, t) 1) D = ( ia)a = ( a0(r, t) a1(r, t) ) f = a awhere now , = 0, 1.
A U(1) gauge theory!
Gauge transformation:
a a i(r, t) ei(r,t)/2 ei(r,t)
has charge 1/2 and mass mH.
has charge 1 and mass mW.
17
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
18/23
Effective 1-d theory II
The Higgs-gauge action becomes
S[,,a] =4
g2 dt
0dr
(D)D + r2(D)D
14
r2ff 12r2
(||2 1)2 12
(||2 + 1)||2
Re(i2) g2
r2||2 g
2v2
2
2
Work in a0 = 0 gauge. Still have freedom to make
time-independent gauge transformations. Use this freedom to set
a1(r, t = 0) = 0. ta1(r, t = 0) is determined from the reduced
Gausss Law once the other fields are specified.
So a configuration is specified by initial values and first time
derivatives of the two complex quantities and . (Four degrees
of freedom, as expected.)18
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
19/23
Spherical ansatz results
We do find oscillons in the spherical ansatz, but only if mH = 2mW.
In a further reduction of the spherical ansatz, this ratio can beexplained using a small amplitude expansion.[Stowell, Farhi, Graham, Guth, Rosales]
The oscillon is stable for as long as we can run numerically, with aringing or beat pattern superimposed on the basic oscillations.
r
t
0 20 40 60 80
0
8000
16000
24000
32000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Energy density
0 10000 20000 3000060
65
70
75
80
85
t
total energy
energy in the box of size 80
19
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
20/23
Electromagnetism returns
Now restore electromagnetism.
Breaks isospin symmetry, so we wont stay in the sphericalansatz.
Do a full 3-d simulation starting from spherical ansatz initialconditions, with no rotational symmetry assumptions. (We
could also use an axially symmetric ansatz.)
Z0 is now split in mass from W.
Massless photon a danger to oscillon stability.
20
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
21/23
Electroweak results
Spherical ansatz oscillons are modified but remain stable formH = 2mW! (Not stable for mH = 2mZ.) [Graham]
Dangerous photon is disarmed because fields settle into anelectrically neutral configuration.
Beats decay more rapidly with photon coupling included.
Observed solution has energy
E 30 TeV and size
r0 0.05 fm.(1 mass unit 114 GeV.)
0 10,000 20,000 30,000 40,000 50,000
50
100
150
200
250
300
time
energyinb
oxofradius28
=1
=0.95
21
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
22/23
Conclusions
While solitons are easier to study, oscillons can appear in a wider
range of theories.
Conserved charges, integrability and existence of static solitons are
helpful for finding oscillons, but not necessary for oscillons to exist.
All oscillon solutions found numerically are attractors, or we neverwould have found them. In simple models, oscillons have been
shown to appear spontaneously from thermal initial conditions in an
expanding universe. [Farhi, Graham, Guth, Iqbal, Rosales, Stamatopoulos]
Even if oscillons are not perfectly stable, those that decay over
unnaturally long time scales can be equally interesting.
22
-
8/3/2019 Noah Graham- An Introduction to Solitons and Oscillons
23/23
Acknowledgements
This work was done in collaboration with: E. Farhi (MIT)
V. Khemani (industry)
R. Markov (Berkeley)
R. R. Rosales (MIT)
With support from: National Science Foundation
Research Corporation
Vermont-EPSCoRMiddlebury College
Current collaborators on this project: E. Farhi (MIT)
M. Gleiser (Dartmouth)A. Guth (MIT)
N. Iqbal (MIT)
R. R. Rosales (MIT)
J. Thorarinson (Dartmouth)23