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


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


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


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


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


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


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


8/23

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


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


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


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


12/23


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


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


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


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


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


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


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


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


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


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


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

Documents

Noah Graham- An Introduction to Solitons and Oscillons