Stability of an Oscillon in the SU(2) Gauged
Higgs Model
by
Ruza Markov
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
BACHELOR OF SCIENCE
,t, tilec
CHUSETTS
INS1
TiE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
0 F TECHNOLOGY
UN 0 7 2005
>F5Ae
2;j
Jun, 2005
L IBRARIES
®2005 Ruza Markov. All Rights Reserved.
The author hereby grants to MIT permission to reproduce and to
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
Author
............... .......................................
,
Department of Physics
May 6, 2005
Certified by ...
.............................
Edward H. Farhi
Professor, Department of Physics
Thesis Supervisor
Accepted
by.Acceptedl
by ..............
...
.......-...........................
David E. Pritchard
Professor, Department of Physics
Senior Thesis Coordinator
4 HlVES
2
Stability of an Oscillon in the SU(2) Gauged Higgs Model
by
Rtuza Markov
Submitted t the Department of Physi(Cs
(on
May 6. 005, i partial fulfillment (ofthe
re(quirments f)orte (legre of
BACHELOR OF SCIENCE
Abstract
Os(cillonsare localiz(edsolutions of nonlinear fiel(dtheories that oscillate without (lissiI)ati(on. W' have
1m1herically found a famlily()fvery long-lived ()s(cillons
in the sheri(cal
a-nsatz of the SL(2) gauged Higgs model -the standard mnoclelof the weak interactions without electronmagnetism and fermions. In this thesis, we study the stability of
these obje-cts. We dlothis b adding a massless mode to the model and coupling it to
the oscillating fields contained in the Higgs doublet. Such a mode is expected to provide a decaying mechanism for the oscillons. However, numerical investigation shows
that our oscillons do not decay if the massless mode is sufficiently weakly coupled
and suggests that our oscillons are stable long-lived solutions that could substantially
influence the dynamics of this theory.
Thesis Supervisor: Edward H. Farhi
Title: Professor, Department of Physics
3
4
Acknowledgments
I would like to thank the other Oscillators for their devoted work on this project.
They are: Eddie Farhi. Noah Grahamn, Vishesh Khemani and Ruben Rosales. The
first three chap)ters of this Thesis follow (closelythe work we i)blishe(l un(ldr the title
An Oscillon in the S(U(2) Gauged HigqgsModel.
5
6
Contents
1
Introduction
11
2 SU(2) Gauged Higgs Model
13
3 Numerical Setup and Results
17
4 Stability
25
5 Conclusions
33
I
8
List of Figures
3-1 Energy density as a function of position and time (left). Total energ-v
ain(leergy in the b)ox of size 80 (righlt).................
3-2
19
Evolution of g-ug-e invariant variables jO(r = 0, t)l, and JfO(r = 1(), )
cbtainecl with parameters d = -0.1 and d, = -3.2 ............
3-3
.
20
Io(r. t)I (top left). y(r. /)I (top right). .fot(r, t) (bottom left). and ~(r. t)
(bottomn right) as functions of time at dlifferent radii. for dt =-0.1
(d = -3.2
.
and
................................
1
3-4 The dlependence of the evolution on the initial configuration. We show
ceergv density as a fun(ction of position anl( time obtainled wNith(11=
---0.1 and six different values of the initial parameter d2 ........
22
3-.5 The (decay of the einvelope of 15) at the origin as a function of timne.
The field is displayed in cvan. The maximal points of the beats (blue)
are fitted with an exponential. The best fit. with \2
9.4, is shown
i red ............................................
23
4-1 Timie evolution of the new field, ), for (
as a function of r and t (left) and
= 0.06 and c2 = 0.1.
)
as a function of t for different
radii (right). The field is oscillating with the beat frequency and the
aml)litude of these oscillations is not in(creasing .7...........
2
4-2 The change of gauge invariant fields from the values they had before the
i.iew field was intro(lduced for c'l = 0.06 with (2 = 0.1 (left) al(
(right). We show, from the top down,
(2
=
(r, t.). \(r, t) . fo1 (r, t) and
,(r, t) as a functionoftimefor differentradii. . . . . . . . . . .
9
0.'
29
4-3 Influence of a change in coupling constant c.
The difference of each
of the gauge invariant fields obtained with c = 0.06 and c., = -0.2
an(d those ob)taine(l with c = 0.08 and c = -0.2, sowing
left.
(r. t)I top right, f(,(r, t) bottom left and
functions of time at different radii ...................
O(r. t)| top
(r. ) bottom right as
........
.
30
4-4 Influence of a change in coupling constant c,2. The difference of each
of the gauge invariant fields obtained with c = 0.06 and c, = 0.1 and
those obtained with ('l = 0.06 and c = -0.2. showing 6(r. t)l top
left. \(r, ) top right. ,fo1 (r, t1)bottom left ad
functions of time at different radii ....................
10
(r, t) bottoInI righlt as
31
Chapter
1
Introduction
Ill nonlinear classical field theories. one canllset
p a local disturbance that would.
generally. disperse away. But, sometimes, the disturbance can result in a localized
solution that oscillates without dissipation.
Such solutions are clled
scillonusor
breathers.
Often. oscillons are not infinitely long-lived and only live for extremely long times.
But. for p)hysi(csa)li(cations,
the distinc(tion between a
infinite ad
a very loIl(
lifetime is irrelevant. As long as the object's lifetime is significantly larger than the
nlatural tiniescales of the p)rob)lem.it can hlave significant eff(ects (o the (dynIlaIics ()f
the theory.
We have found an oscillon in the gauged SI((2) Higgs model which is the standard
mno(lelof the weak interac(tions without ele(ctromiagnetism
iandfermiions.
e worke(d
in the spherical ansatz and numerically evolved the classical equations of motion.
Even after verv long runs. ex(ee(ding times of 30.000 in natural units, we have never
seen this oscillon decay.
In this thesis, we stlldy the stability of the oscillon we found. This object owes its
extre'mIelvlog life to its oscillation fre(quenc(ywhi('chis lwer than anly natural imode
of the sstem.
In order to test its stability we introduce an additional effectively
imassless mode that provides a possible decay mechanism. We study the effects of
this massless rnlo(deoni the evolution of our oscillon.
11
12
Chapter 2
SU(2) Gauged Higgs Model
We consider lanSlU(2) gauge theory coupled to a doublet Higgs in 3 + 1 dinensions.
which is euivalent
to the weak interac(tions sector of the Standardt MIo(el(ignoring
fermnionsand electromagnetisml. The Lagrangian density for this theory is
1
£(x,.t)=
-I TrFFP',2 +2Tr (L)(L)t L) -
where F,,. = 0,4,-VA[,
A (Tr() (I) - V
2)2
- iqg[A4t,
A,] DD = (01,- i.qA4tL)..4, =
(2.1)
and
-4To/;')
indices run over one time and three spatial dimensions. WVehave defined the 2 x 2
nmatrix(1)representing
the Higos doublet
by
t)
)
2
; 1
(2.2)
,:2
We follow the conventions of [1], except we use the metric ds2 =
The spherical ansatz
is given b
Aj(x, t)
(I (x. t)
=
1
-ao(r.
2g
= -
-
t)o
- dx 2.
expressig the gaige field .4 ancidthe Higos fieli
D in terms of six real functions a((r, t). al(r. t). ( (r, t),
0o(x. t)
2
r, t),
(r. t) and v(r. ):
fT
(1 (r. t)co ki + (--
- [(r,t) + ,1(r.
t)c -
(o , - r. xr ) +
I.
., ( .
['
.
eI
.
J
Ok
-.
(2- 3)
g
13
where i is the unit three-vector in the radial direction and
are the Pauli matrices.
,/r and
For the fields of the full theory to be regular at the origin. o,o(. ar-y./r.
'
must vanish as r - 0. The theory reduced to this spherical ansatz has a residual (1)
gauge invariance consisting of gauge transformations of the form exp[i(r,
t)o
;/2]
with Q)(0. ) =0.
In the spherical ansatz we obtain the Lagrangian density
4 7£C(r,
oI)
[I~1
-g-K
.
D,,-(t1
r ftv +(DI')*D, +r (DIL
)*D
-
2- -
2) , r (|)2
2
2 2
o-- Re'(i\*0
.-( \-+-I)
.(0,
=- =(k.
2
fJ,2
(2.4)
where the incdicesnow run over t and r and
ia,..
0,,
DlA
,, =
Und(ler the re(lduced
~o,).
(1) g
\
+ Il
i(, -
(),,- D,,.O
(-
rauge
invariance.
the
=
+
.
a,,)().-
omplex scalar fields
and
_1),
(2.5)
have
charges of 1 and 1/2 respectivelyv.a, is the gauge field. j, is the field strenith. and
D[I is the (covarianlit
derivative. The indices are raise(l and lowered with the 1 + 1
dimensional
metric ds 2 = dt2 - dr 2 .
The equations of motion for the reduced theory are
"(,'(r-f.L)
= /;[D.\*\-\ * ] +.','
o'[6 .]
*2 [DO(,6o-*
-
i~~o:'
=
i
_'
,
D)'r2D,+ (\
[l)'
~
2A
9(,./.)
+ 1)+2 r
1 2]
K' \K-i) -tioK
1 ()
F _
O.
(2.6)
These equ(ationscan be obtained either by varying the Lagrangian density (2.4) or
b, imposilng the spherical ansatz on the equations of motion of the full theory.
Altlholgh the theory is described b six fields, (10. 1.
.
.
(, ianld, there are
only four indepelndent degrees of freedom consisting of the three lV-bosons and the
massive
iggs. The remniainingclegrees of freedom are gauge artifacts. WNemay fix
14
thle o-augeby a timne-dependent gauge transformation followed bv a tinie-indepenldent
gauge transformation which removes the remaining gauge freedom. We choose to first
set (10 (r. t) = 0 everywhere, an(l then apply a timle-in(el)end(elltgaugetransformation
to set
(r. =) 0)
0 initially. The choice of ao()=() gauge makes the covariant time
dlerivatives eqtual to the regular time derivatives.
As we chose a0o
() there are only five fields that evolve in time. To obtain time
evolution., for each of the four fields p a.
ct, and -t we must specify the profile at t = 0
as a fn(ctionll of r as well as the time (lerivativeof the p)rofile at t = 0 as a function
of r. The time derivative of l(r, t) at t = () is then determined by imposing Gauss's
= 0. In this a-lge. Gauss's Law
Law whichl is the first equation in (2.6) with index
is
i
2
t(r
,al(r. t))
=
'2-
[6(r.t)*0,6(r. t)
1
-
+- [\(r. t)*.(r. t)-
Ot@(r.t)*(r. t)
t(r. t)*\ (r.t)].
(2.7)
We evolve these five fields according to the Equations (2.6). Configurations obeying,
Gauss's Law at the initial time will obey it for all times. Thus we have fully specified
the initial value problem by providing initial value data for the four real degrees of
fr'eedom(ontailned(lin /,. v, . and .
15
16
Chapter 3
Numerical Setup and Results
III order
(
to
imulate the extremely Onglifetiie
((iuritelh
highly stalble numerical, techniques.
Laglrangin
inl E(qulation (2.4).
of os(illos.
we require
Ne cliscretize the systemn at the level of the
Xe (1OOS a fixedt spatial latti(e
sacing
Ar, pla(Cing
the scalar fields at the sites of the lattice and the gauge field al on the links. Thus each
of the five scala.r fields A(r. t) is replaced by a set of functions A\{t"'(t). defined at each
lattice )(int. Frther o. we rellace a first-or(ler lttice
lge-(ovriant
erivattive
by the discrete expression
, A{ +A}(t)ex
(xp [- go
D,.A(r, t)
(t)Ar]-
A
(t)
(3.1)
Ar
where g is the charge of the scalar field A and r = r/Ar
labels the lattice point
correspndlding to radilus r.
We vairv the spattially (liscretizedLatgranlgianto btain secoi(l-or(der acurate
lhit-
tice equat ions of motion. In the limit of continuum time evolution. this system has
.(:act(conservation
of eiierg- rln(lexact gauge iivarial(ce.
NVe
iiollitor these (qualtities
to detect numerical errors and determine whether our time steps are short enough.
However. they do not tell us whether our spatial grid is fine enough. because even
for
very coarse
,grid
or
systeim conserves eergy
Ind obeys Gauss's Law. In ll our
simulations, these invariants hold to aln accuracy of roughly one part in 106, and we
see no numnerical instabilities even after extremely long runs.
17
Since we chose to work in a ao(r. t) = () gauge. the covariant time derivative
coincides with the ordinary time derirvative. Thus, the discrete second-order time
(derivativ(' is given 1)xby
D2A(r. )
A'{j}(t+ At) - 2A{tI(t)
+ {}(t - At)
2
(3.2)
(At)
Our time evolution is simply inverting this second-order differential. which appears
in the e(lllations of mlotiorn,i order to solve for ,\A{t'}(t+ At). Th'lus.we compute ech
new timne step fronmthe previous two. This approach is stable for At < Ar so we
(hoose to work with
fixed( time step At = Ar/2.
In order to monitor the energy conservation and gauge invariance of the solution.
it is neessary to compute first-order til(e (derivatives.
e comu)ite the derivative at
time to by subtracting the result at t o - At front the result at to + zi and dividing
by 2At. rather than comparing adjacent time steps t and t + At. We chose this
(lefinlitionlof the first (derivative b)ecaluse.in the continuummlimit, it is accurate to the
second ordler in At unlike the one using adjacent time steps which is only accurate to
the first crder.
Since -we are considering classical dynamics, the theory is invariant under overall
rescalings of the Lagrangian densitv. As a result. the theory is completely specified
by the choice of the ratio of the Higgs mass
mw = g'l/2.
7'
2.Ato the IV-boson mass
For generic values of this ratio. we observe localized configurations
that oscillate lbut are spreading out and decaying. However. in a )articullar case
m.Hu= 2n-
these oscillations exhibit no observable decay in our simulation. This is
the only ]mnassratio at which we found stable oscillons in this theory. We chose to
work wNith q =
2 and A = 1 whilch give this milass ratio.
Also, we chose to work il
natural units with the vacuum expectation value of the 0 field
= 1.
The localized initial configuration we start from is given b
o(r.
-
0)
I+
and
(r t = )
-i (
+
'/)
(3.3)
here
d1. . and ' parameterize the initial configration.
e set initial time ewhere ctl, d.2. ad tc parameterize the initial configuration. We set initial time deriv18
atives eqlual to zero everywhere. Then, we evolve this configuration according to
the equations of motion and observe its evolution. To ensure that we are not seeing numllerical
trtifacts we monitor
auge-invarilant quantities. A (lear coi( e are the
energy and energy density. As described in [1], there are also four gauge-invarialnt
field 1O(r.t)lI(rnt)[. Jl(r,t), and (r.t) = arg[i(r.t)()(r,t)*)2],
that completely
determine the state of the sstem.
WVedisplay results for d =-0.1.
d2
=-3.2,
and
12 although small changes
! =
inl these p)aralleters give siuilar results. In the left panel of Figure 3-1 we show the
energy- density as a function of t and r. A localized time dependent object is clearly
visible. The right panel shows that the initial configuration sheds about a cquarter of
its (llenrya ui ickly
(
settles into the localize(l oscilloll which (h)es not decay visibly.
0
0.45
0.4
0.4
8000
80
0.35
0.3
-
75
total energy
energyin the boxof size80
0.25
- 16000
0.2
70
0.15
24000
0.1
65
0.05
32000
0
20
40
r
60
AN
vNv,.
80
U
----eUUu
---IUUUU
----dUUUu
t
Figure 3-1: Energy (denlsity as
function of position and time (left).
and(l energy in the box of size 80 (rilght).
Tlotal energy
In the upper left panel of Figure 3-2 we show the gauge invariant magnitude of (5
ait the riginl, 1(5(r
= 0. t) .
at
= ,
lThefield
is oscillating aout
1. By counting the number of oscillations. the frequency of the oscillation
is measured to be 0.2239. As the mass of the Higgs is
propagat;ng
the vacuum expectation value
miode ()f the Higgs field is
2/2r
9 the lowest frequency of any
= 0.22.51 which is above the oscillol
frequency. This is an essential property. Any oscillon owes it long life to a frequency
of oscillation which is belowvany natural mode of the system. Such an object cannot
19
decav into modes which canl carry energy away.
We also show that the gauge invariant quantity fol(r = 10,t) is oscillating with
a frequency of 0.1120 - exactly half of the frequency of oscillations of lo within the
accuracy of the measurement. The gauge boson mass is
/2 so the lowest frequency
of these )rop)agating modes is x/22/47 = 0.1125 which i also aove the oscillation
frequency. Similar analysis holds for the other two gauge invariant fields, \(r. t)] and
,(jr,t) where \(r, t)I oscillates with samnefrequency as [6(r.t)[while (r. t) oscillates
with same frelen(y
as jo (r, t).
In the lower panels of Figure 3-2 we plot the two fields over a. much longer time
so that their oscillations are not visil)le. However we (
see beats in these profiles.
The beat frequency of both fields is about 4. x 10-1.
I
1.15
1.1
1.05
7:_
0
C~
1
7,
Sigif c n
0.95
l
$o
0.9
0.85
0.8
I
I
i
6020
6000
6040
6060
6080
61(00
t
0.06
1.15
1.1
I.... -
0.04
VY
V
V
VI
A
IAAAAAAAAAAAAAA
-
_ 1.05
c5
0.l
1
_P
0.95
VVVVV V
0.85
0.8 I
0
i!
_
_
00
5000
100
10000
150I00
15000 20000
50
25000
0
-0.02
'IIIIII"""'"
0.9
0.02
-0.04
00
30000
-0.06
0
5000
10000
15000
t
20000
25000
30000
Figure 3-2: Evolution of gauge invariant variables 1,(r = O,t)l. and fol(r = 10.t)
obtained with parameters d =-0.1 and d.
2 =-3.2.
20
In Figure 3-3 we show the profiles of four gauge invariant fields. {kI(r.t) I[(r. t),
(r.t) = arg[iX(r,t)(o(r.,t)*)
fol (r. t), and
2],
as a function of time for four different
radii.
l
. -r
1
_
Q
- r=0I -
r=101--
r=20
r=301
r=40
1.5
1.4Ir
-
1.3
-e
1
,,2
C
1.1
.
.
T
v
I
0851
~,
0
5000
10000
15000
,",I
O9
20000
5000
t
0.10I1t_-
-
I
10000
15000
20000
t
d
1
I
i'
r
7
0.8i
C
0.6
_
_
_
0.4
0.2
--0
-
-0.2
-(
-0.4
-0.6
-01
-0,2 - r=OI0 --2
0
-
5000
t
r=101
r=20
10000
t
~tl
r=301 --- r=40I
15000
-1
20000
I
_iLIlk
0
-r=0l
--- -r=10
5000
-r=201
10000
r=30 - ---r=40
15000
20000
t
Figure 3-3: I( (r. t)l (top left). Ix(r,t)l (top right), f (r.t) (bottom left). and (r.,t)
(bottom right) as functions of time at different radii. for d =-0.1 and d2
-3.2.
Althoigh a particular initial configuration is displayve(ld,
the details of the starting
point are not essential. In the beginning. the initial configuration varies substantially.
It sheds energy andclfinally settles into the stable configuration. From the left panel
of Figure 3-1 one can see energy leaving the snmallregion aroun(d the origin dring
the shedding. The fact that our initial configuration spontaneously settles into an
oscillon culdcl mean that oscillons can form from generic initial conditions and are
21
not imp)robable solutions.
However, changing the initial conditions does affect the nature of the oscillation.
InIFigure 3-4 we show that the bl)eatfreq(uency of tile oscillon (dep)ends(onthe initial
configuration. There. parameter d2 is varied until the oscillon becomes unstable.
d2=2.6
d2=2.8
d2=3.2
100(:
1000
100(
200(
2000
200C
E 300(
3000
300C
400(
4000
400C
500(
5000
500C
600(
0
20
40
60
80
6000
0
40
20
d2=3.4
60
80
600C
---- 0
20
d2=3.5
40
60
80
60
80
d2=3.6
i
1000
2000
a)
E
100(
200(
i
3000
4000
300C
400C
I
5000
U
6000
e:~V
4+
OU
OU
500C
0
20
40
r
60
80
600(
0
20
40
r
Figure 3-4: The dependence of the evolution on the initial configuration. W-e show
energ-v density as a function of position and time obtained with d = -0.1 and six
different alues of the initial parameter d.
We never saw the death of the oscillon shown i Figures 3-1, 3-2 and 3-3. But a
carefull eve canl notice a slight decay in the maximal value that the beats of the fiel(ls
achieve. This effect is displayed in cyan in Figure 3-5 for 161at the origin. where the
vertical axis is stretched until this decay is clearly visible. This figure also shows that
there is a l)eat in te
TThe maximal
l)eats of this field.
points of the beats of [( (r. t) at the origin were read off and fitted
with a shifted exponential function of time. ac¢- ' + c where c = 1 is the acuum
22
1 1nn
1.
u
-L
I
2
data
\,
1.104
I
X 1 = 9.4
nirve
I-fittd
~........._.
,j
y=a*exp(-bt)+c
TI
a = 0.016 0.001
xl",
T
\x
%I,
1.102
|
T
'sb
2) x 10- 5
=(5 ±+
I
`-
1.1
'
C=
1.094 ± 0.002
-IT
1.098
i
no- _
1I U;U
50 0 0
I
.
.
.
10000
..
_._
15000
.
20000
t
25000
30000
_
35000
Figure 3-5: The decay of the envelope of 1]0 at the origin as function of time. The
field is displayed i cvan. The maximal points of the beats (blue) are fitted with an
exponential. he l)est fit. with \2= 9.4, is shown in red.
value. The fitted curve and its parameters are shown in Figure 3-5, as well. Although
the decay rate b = (5 ± 2) x 10-5, could not be determined very precisely due to
insufficient d(eta. the ('constant offset. c = 1.094±0.009 is determlilled with relative error
of less than 0.2%c. The fact that this offset does not equal the vacuum expectation
value ' =: 1 suggests that the os(cillol is not (decaying but only settling into a stable
configuration.
23
24
Chapter 4
Stability
In order to study the stability of the oscillon described in Chapter 3 we introduce
t Inatssless scalar field t) coupled to the Higgs doubl)let. This is a good (che(kof the
robustness of our object as an uncoupled oscillon owes its long life to the fact that its
freq(uen(cyof oscillation is lower than alny natural mode of the system, so there are no
modes that can carry its energy away. The addition of a massless field may provide
a decay nlechanismn for the oscillon.
The new Larangitm
we choose is
LC(x.t)
))'i/
= [-TrF"F,~,
--(A-
4
+ Tr (D"i' ) D)(
+
c1t)-( 2 2) (TrttT-
(4.1)
which reduces to the Latgrtangianin equation (2.1) when O(x, t) = 0 everywhere.
In the spherical ansatz this Lagrtangian is
£(r,t)
4
~~~
.22
~
~~~~~~~~~~~~~
(Dt\)*DI\
+[r
r-.l
, + (D~'\)*D y,-i+
r'2(Dl,/)*
(Db16)*Do+ g aOa'.ti)
~()l2 _ 2 (I 2 + 1)1,.32*2)
2r2
]22
,~~~~~~~~~~~~~~~~~~.
,.
'2
.(2
47,~
1i
g2-(A
(,I-)
-C2)
(
-
)
the indexes running
~ith
over t a1nd r and all the fields
(4.)
onlv on r ad(4lepending
t.
with the indexes running over t and r and all the fields depending only on r and t.
25
as define( i Chapter 2. The regularity of this theory at r =
a,(, ca. a, - oa/r, /r
and
must vanish as r
implies that. O90/'r,
0.
--
The equations of motion for this spherical Lagrangian are
2f.,.) = i[vy\*\ - \*),\]
O"(r
i\*=
--o
[DO'rDpL ( \|
oDI -- t 24A-
~
=.2
_'12
0'L( r°
2
=
-2r'2
-) (-
22
+ -+
1)
.,I
(\
11 _12
+ 2,0
22c.)
1(-1--
92)
2\
(-2tL)
c1\ - c1)
:d
I
[L2
r2[D,)(*6
- O*D]
(4.3)
g-u-
Again. we will fix the gaullgeby first setting a((r,t) = 0 everywhere, and then
a)pplving a time-ind(lel)enldent (auge tranlsfoirmation to set (l (r, t
To obtain
for each of the five real fields f, P.
v
time evolution,
0) = 0 initially.
.
a,nd 1) we must
specify the profile at t = 0 as a function of r as well as the time derivative of the
l)rofile at t =
as a function of r. Timetilme (lerivative of oI (r, t) at
determined by iposing
= 0 is then
Gauss's Law, the first eqluation in (4.3) with index
= 0,
which in this gauge read(s:
0,(r Otal(r.t)) =
2
[(r. t)*&to(r, t) - (to(r. t)*o(r, t)]
+ [dr' )*9f(
I)-d9t\(T t) \i(r t).
(44)
We evolve the five fields according to equations (4.3) and the configurations obeying Gauss's Law at the initial time then obey it for all timles. Thus we have fully
specified the iitial
value problem by providing initial value data for the five real
clegreesof freedom contained in tl, v. c,
. and 0. corresponding to the five degrees
of freed(lonmof this theory.
In order to see how the newlv introduced field
os(illonIlswe (choosethe samileinitial configuration fr
gave rise to an oscillon.
. =-3.2
influences the stability of our
t,
, a(, and
as the one that
We arbitrarily choose the oscillon with (1 = -().1 and
as the essential properties of the oscillons did not clependconilthe initial
26
configuration. Also. we choose a static initial configuration for the new field given by
0(r)
= O. 90(r)/Ut = 0. From the last equation in (4.3) we see that if cl 4 0 this
static Zero field configuIation
will evolve and move away fom zero.
We are interested in knowing if the new field 0 carries energy away from the
oscillon. Because the system now has a natural mode lower than the frequency of
oscillation of the oscillon we think that energy will be transferring from the oscillon
to the massless field. As a massless mode cannot keep the energy confined. we expect
to see the energy dissip)atilngout of the snmallregion near the origin.
for c1 = ().06 and c = 0.1. The initial static
Figure 4-1 shows the evolution of
zero field evolves and pulses with the frequency of the beats in the other fields. The
amplitude and freq(uency of these plses
is not (lhalging in te.
This shows that
althouoh there is some energy stored in the new field, contrary to our expectations
this ainolnt
is not increasing in time. The ener-v is not (contiimnousl transferring
friomi
the massive fields to the massless field. After the oscillon has stabilized, the energy
in the box of size 80 is 61.6 in this case while it was 61.8 before ) was introduced.
nntA
U.4
0.035
-r=O
r2
r20
-r10
r=01
r30 -r=40
0.035
0.03
501
I
0.03
0025
0.025
0.02
_ fooc
0.015
0.01
150
0.005
2000
0
10
20
30
40
50
60
70
80
1^
0.02
0.015
0 ._01K , .
",r
^.--
0005
.005os
°.°°--~v.~.
n
ann
U
UUU
/
i",
1nnn
UUUU
1nn
I uJU
.mnv
ZUUUU
t
t
0 as a
func(tion o)fr ad(l t (left) and 0)as a function of t for (lifferentrad(lii(right). The field
is oscillating with the beat frequency and the amplitude of these oscillations is not
Fignre 4-1: lime evolution
of the new field. 0, for
('
= 0.06 and c = 0.1.
increasing,.
In order to see how the
ew evolving massless mode effects the evolution of the
other fields we compare the gauge invariant variables without the new field (or, equiv27
alentlv, with cl = 0 and c = 0) and with
present. as illustrated in Figure 4-2. The
right column is the difference of the results obtained with cl = 0.06, C2 = 0.1 and
those with no
and c., =-0.2
while the left olumn comp)ares the results obtained with c' = 0.06
to those with no
. Row by row. from top to bottom, we show the
foi(r.t) and (r.t).
change of 6c(r,t)}. y(r.t),
bWe
see that the change in the fields
is quasi-periodic and does not grow in time. The 'period' of this (quasi-periodic evolution of fo(rt)
and
(rt)
is twice that for 16(rt)j and {\(r,t)j.
We also see that
the period' is longer for c.2 = -0.2 than c.2= 0.1.
The dependence on the coupling constant c1 is shown in Figure 4-3. where we
plot the difference of each of the four gauge invariant fields obtained with different
(couliling ( onstaIlt c'l and the salne
(c..
Spec('ifically. Nveused (c
= -0.2
and ci = 0.06
for one rn while c = 0.08 for the other, and plotted the difference between the gautge
invariant fields l)etweenlthese two runs. We see that the difference b)etweellthe fields
is, on average. not growing in timne.but is periodic and oscillates around zero. The
period of this oscillation for 1(5(rt)l and \(r, t) I is the same nd very nearly equals
half of that for fol (r. t) tand (r, t).
Figure 4-2 shows that the changes in the fields due to introducing d with the two
different values of c are very similar. Results obtained with c = 0.1 and c2 = -0.2
(both having (l = 0.06) are directly compared in Figure 4-4. where we plot the
difference of each of the four gauge invariant fields obtained with different coupling
constant
and c
(,.
e see that the amount by which
(r,) t) I and
\(r',
) (lifferfor c = 0.1
-0.2 is growing at first and then decreasing after a time of about 15000
suggesting that it actually oscillates with period of about 30000. On the other hand.
aTimountsi) . which / l(rt, ) and ,(r, t) differ is
rowing over the time of the run of
200)00.but the properties noticed above suggest that they are actually cquasi-periodic
with p)eriodof about 60000. A longer run is necessary to further test the (ldependence
on the coupling constant
.
28
c,=0.06and c2=-0.2
c=0.06 and c2=0.1
--r=0 I-
025
r=10 201
r
=
r=30 1
r=40 I
---
I
0.25
O
7
=r -r101I
,
r=30 1 ---
-=201
r=40
015
o0
-00.1
-3
-
02
o
-
L
0.1
0.05
I
a~
0
i
-0~c
-0.05
-C
-0 1
-0- -0 15
-0.
-c
-0.2
_n
10000
5000
20000
15000
0
5000
10000
15000
205300
t
0 25
---
.r=0
=10'
---
r=20
r=30
I
r=40
-
.02
-r=I
r=10I
20
-
=301- -- r=40
¢o 0.
0.1
.2
I
i
02
0
0
,-4
0
0.0
Z_
-
x,:
-0
-0 0
-=
x= -0.
-0
-0
-0 1
-
)
-u
-0.1
10000
5000
0
15000
20000
5000
0
10000
0.25
-
02
i
20000
15000
t
t
0.25
--- r=Oj--r=101
--- r=20j
r=301
----
4
j
015-
I
n,
r=40j!I
,
r=0 I
r=10 -----
r=201
r=30 I - -
r=40
i
-
01
-
0.I
-
z-
-0 05
__~
-00c
-0
-P
-0.1
-0,15-0.2
_no_
/
-- IJ
-
0
10000
5000
200()0
15000
0
5000
10000
t
1.5, -
,*v
I
=
I --- r=
--I
-
.....
F
r=101 I
-r=20
15000
20000
t
f~
~
=
15. .9
q
- -
-
-
--
,=30 1 - -=40
I rO-
-
r=10
r=201
r=30
-
,:o
r=40
1.
Z___p
0
o0.51
I11-P
-0.5
v
-0. 5-
-0
I
"-p
-1
_
o
5000
10000
15000
ll
20000
1
0
t
5000
10000
15000
20000
t
Figure 4-2: The change of gauge invariant fields from the values they had before the
new field was introduced for l = 0.06 with c = 0.1 (left) and c = .2 (right). We
(r, t) , /ol(r./) and ((r' t) as a func('tion of time
show, froim the top down, (r. t),
for different ra(lii.
29
0.25r
E--r=O
I-
r=10
-
r=20
r=30 -
r=40
_
--- r=0 -
r=10 ---
r=20
r=30
r=40
0.2-
A
0
9
0.1
0.
0
I?
.0I
0
ol
0.
0.0
; 0I-
'2'
x..
-0.
I
.-
-0.0
-0
-0.
-0.I
-0.250
0
5_n
V
5000
10000
t
15000
0
20000
-- r=101
0.2L E-r=o
-- P
10000
t
15000
20C
100
I.W
'I D-
U.Zb l -
10
9
011
5000
- -r=201
I
r=301
r=40
0.1
1:
(0
-P
I
0.150.1
0.05
5
10
I6
9
0II-
--9
-C,
-0.15-
-0.2 0
5000
10000
t
15000
200(
JO
t
Figllre 4-3: Influence of a chage in couplillng, constant (c1 .
the g(augeinvariant fields otained with c = 0.06 and(1c,with cl = 0.08 and c = -0.2, showing 1((r t)j top left,
bottom left and (r. t) bottom right as functions of time at
30
TIhe (lifferelnce of each of
-0.2 1and(
those obtailled
(r.t)l top right. f(rt)
different radii.
,:
0.2
I.J
u
0.25r
---
r--r=O
-
r=10 -
r=201
r=301
- r=40
l
i'
,5
I
e
r--O-
r=101-
r=20
.2,]:
r30IJ
--
r=-30
r40
r=401
I
15F
0.*
.1
0
1c0.0.-c
0.1
-0.
0
1
oC?
l~
11
)
-5
).r_
-C
-0
-0.1
uw
-0.
15F
i
-
-02
0
-0.
-0c
I
{,4_ S.[~ r)E
Zz
H)
0
5000
I
10000
15000
20000
~
5000
10000
t
15000
20000
C
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
r-r=0l I---r=101
-r=20 r
=30] -
1j I--
r=40
0.1
,1
,
r=301
r40
6
I
0. I
4
O.z
~,
15
C
r=101 -- r=20I
F~~~~~~~~~~~~~i
0.8
0.0
__O
r----
v~
0.
0
2
-0
-0.015'
v
-0..1 _
0
Filre
-.
4
D.1 _ _~~~~~~
-U.b F
-0.8
5000
10000
t
15000
4--4: Influenc(e of a chan-ge in
200DO
oupling
0
5000
onstant
c.
10000
t
15000
The differen(e
20000
f each of
the gwuge invariant fields o)taine(l with cl = 0.06 ad (' = 0.1 an(l those obtained
with c = 0.06 and c = -0.2. showing b(r.,t)j top left. , (r. t) I top right f(rt)
bottom left and ((r, t) bottom right as functions of time at different radii.
31
32
Chapter 5
Conclusions
Ill this Thesis we studied the stability of an oscillon in thle SU(2) gauged Higgs model
bv introclducinga mnasslessfield coupled to the oscillating Higgs doublet. Contrary to
our exp)ectations we found that
a
sufficiently weaklv coupled inassless field (does not
destabilize the oscillon. We saw that onlv a small amount of energy is transfered to
the new field and that this amount dloes not grow in time. We also saw that the new
field does not introdluce an increasing change in the gauge invarianIt fields describ)ilng
the state of the system. This suggests that our oscillon is a stable long-livecl solutions
which (ll sulbstalitiallyinfluence the dynamics of this theory.
The next natural question is: how does this finding depend on the strength of
the coupling? How strong must the coupling be for the oscillon to become unstable?
t\ (l,
did not pursue this issue as our numerical simulation
couplings
)becomesunstable for larger
the results stop obeying Gauss's Law and conserving energy. Further
investigation is needled to find the source of this numIerical instabl)ility. Then, the
robustness of the oscillon could be tested for a strongly coupled massless field.
33
34
Bibliography
[1] E. Farhi, K. Rajagopal, and R. Singleton, hep-phl/9503268. Phys. Rev. D52
(1995) 2394.
[2] R. Rajaraman. Solitons and Instantonls (North-Holland, Amstercldam.1982).
[3] C. N.ash and S. Sen. Topology and Geometry for Physicsts, (Academic Press,
London, 1983).