We study the classical chaos--order transition in the spatially homogenous
SU(2) Yang--Mills--Higgs system by using a quantal analog of Chirikov's
resonance overlap criterion. We obtain an analytical estimation of the range of
parameters for which there is chaos suppression.Comment: LaTex, 10 pages, to be published in Phys. Rev. 

Salasnich, Luca

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arXiv

Luca Salasnich

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Chaos suppression in the SU(2) Yang-Mills-Higgs system

We study the classical chaos--order transition in the spatially homogenous SU(2) Yang--Mills--Higgs system by using a quantal analog of Chirikov's resonance overlap criterion. We obtain an analytical estimation of the range of parameters for which there is chaos suppression

Salasnich, L

CERN Document Server

Chaos suppression in
the SU(2) Yang{Mills{Higgs system
Luca Salasnich1
Dipartimento di Fisica "G. Galilei" dell’Universita di Padova,
Via Marzolo 8, I 35131 Padova, Italy
Departamento de Fisica Atomica, Molecular y Nuclear
Facultad de Ciencias Fisicas, Universidad "Complutense" de Madrid,
Ciudad Universitaria, E 28040 Madrid, Spain
Abstract
We study the classical chaos{order transition in the spatially homogenous
SU(2) Yang{Mills{Higgs system by using a quantal analog of Chirikov’s res-
onance overlap criterion. We obtain an analytical estimation of the range of
parameters for which there is chaos suppression.
PACS: 11.15.-q; 05.45.+b; 03.65.Sq
1Electronic address: salasnich@padova.infn.it
1
In the last years there has been much interest in the chaotic behaviour
of classical eld theories. In particular have been studied Yang{Mills [1,2],
Yang{Mills{Higgs [3,4,5], Abelian{Higgs [6], and also Chern{Simons [7] sys-
tems.
Usually the order{chaos transition in these systems has been studied nu-
merically with Lyapunov exponents [8] and sections of Poincare [9]. Less
attention has been paid to analytical criteria. Some authors [4,6] have used
the curvature criterion of potential energy [10], but this criterion guarantees
only a local instability and can give incorrect results (for a fuller discussion
of this point see [11]).
In this work we study analytically the suppression of classical chaos in
the spatially homogenous SU(2) Yang{Mills{Higgs (YMH) system. We apply
a quantal analog [12] of the Chirikov resonance criterion [13] by using the
semiclassical quantization to calculate the critical value of the parameters
corresponding to the intersection of two neighboring quantal separatrices
[20].
Obviously, the constant eld approximation implies that our SU(2) YMH
system is a toy model for classical non{linear dynamics, with the attractive
feature that the model emerges from particle physics.
The lagrangian density for the SU(2) YMH system is given by [14]:
L = −
1
4
F aF
a +
1
2
(D)
+(D)− V () ; (1)
where:
F a = @A
a
 − @A
a
 + g
abcAbA
c
 ; (2)
(D) = @− igA
b
T
b ; (3)
2
with T b = b=2, b = 1; 2; 3, generators of the SU(2) algebra, and where the
potential of the scalar eld (the Higgs eld) is:
V () = 2jj2 + jj4 : (4)
We work in the (2+1){dimensional Minkowski space ( = 0; 1; 2) and choose
spatially homogenous Yang{Mills and the Higgs elds:
@iA
a
 = @i = 0; i = 1; 2 (5)
i.e. we consider the system in the region where space fluctuations of elds
are negligible compared to their time fluctuations.
In the gauge Aa0 = 0 and using the real triplet representation for the Higgs
eld we obtain:
L =
1
2
(
_~A1
2
+
_~A2
2
) +
_~
2
− g2[
1
2
~A21
~A22 −
1
2
( ~A1  ~A2)
2+
+( ~A21 + ~A
2
2)
~2 − ( ~A1  ~)
2 − ( ~A2  ~)
2]− V (~); (6)
where ~ = (1; 2; 3), ~A1 = (A11; A
2
1; A
3
1), and ~A2 = (A
1
2; A
2
2; A
3
2).
When 2 > 0 the potential V has a minimum in j~j = 0, but for 2 < 0
the minimum is:
j~0j = (
−2
4
)
1
2 = v
which is the non zero Higgs vacuum. This vacuum is degenerate and after
spontaneous symmetry breaking the physical vacuum can be chosen ~0 =
(0; 0; v). If A11 = q1, A
2
2 = q2, and the other components of the Yang{Mills
elds are zero, in the Higgs vacuum the hamiltonian of the system is:
H =
1
2
(p21 + p
2
2) + g
2v2(q21 + q
2
2) +
1
2
g2q21q
2
2 ; (7)
3
where p1 = _q1 and p2 = _q2. Obviously w2 = 2g2v2 is the mass term of the
Yang{Mills elds.
Classical chaos was demonstrated in a pure Yang{Mills system [1], i.e. in
a zero Higgs vacuum. Here we analyze the eect of a non zero Higgs vacuum
[3].
We introduce the action{angle variables by the canonical transformation:
qi = (
2Ii
!
)
1
2 cos i ; pi = (2Ii!)
1
2 sin i ; i = 1; 2: (8)
The hamiltonian becomes (see also [3]):
H = (I1 + I2)! +
1
v2
I1I2 cos
2 1 cos
2 2: (9)
By the new canonical transformation in slow and fast variables:
A1 = I1 + I2 ; A2 = I1 − I2 ;
1 = 1 + 2 ; 2 = 1 − 2 ; (10)
H can be written:
H = A1! +
1
4v2
(A21 −A
2
2) cos
2 (1 + 2) cos
2 (1 − 2): (11)
We now eliminate the dependence on the angles to order 1=v4 by resonant
canonical perturbation theory [15]. First we average on the fast variable 1.
This yields:
1
2
Z 2
0
d1 cos
2 (1 + 2) cos
2 (1 − 2) =
1
8
(2 + cos 42); (12)
and:
Hcl = A1! +
1
32v2
(A21 − A
2
2)(2 + cos 42): (13)
4
The dependence on 2 is now eliminated by a second canonical transforma-
tion. The Hamilton{Jacobi equation for the perturbation part is indeed:
[A21 − (
@S
@2
)2](2 + cos 42) = K; (14)
@S
@2
= 
vuutA21(2 + cos 42)−K
2 + cos 42
: (15)
and thus the Hamiltonian becomes:
H = B1 +
1
32v2
K(B1; B2); (16)
where:
B1 = A1; B2 =
1
2
I
d2
@S
@2
: (17)
It appears from the structure of this equation that the motion of our system
is similar that of a simple pendulum: for 0 < K < B21 rotational motion,
for K = B21 separatrix, and for B
2
1 < K < 3B
2
1 librational motion. On the
separatrix we have B21(2 + cos 42) = K, and:
B2 = 
2

Z b
a
dx
s
B21(2 + cos 4x)−K
2 + cos 4x
; (18)
where a = −
4
, b = 
4
for rotational motion, and a = −(K;B1), b =
+(K;B1) for librational motion, with:
(K;B1) = 
1
4
arccos(
K
B21
− 2): (19)
The appearance of a separatrix (which is not immediately obvious in the (p; q)
coordinates) accounts as is well known for the stochastic layers originating
near it [16]. This corresponds to local irregular behaviour of the quantum
5
spectrum; one of its manifestations is the local shrinking of the level spacing
and the tendency to avoided crossing [16,17].
The approximate hamiltonian (16) depends only on the actions so that a
semiclassical quantization formula can be obtained by the Bohr{Sommerfeld
quantization rules [15,18]. Set B1 = m1h and B2 = m2h, then, up to terms
of order h, the quantum spectrum is:
Em1;m2 = m1h! +
1
32v2
K(m1h;m2h); (20)
where K is implicitly dened by the relation:
m2h = 
2

Z b
a
dx
s
(m1h)2(2 + cos 4x)−K
2 + cos 4x
; (21)
with a = −
4
, b = 
4
for 0 < K < (m1h)2, and a = −(K;B1), b = +(K;B1)
for (m1h)2 < K < 3(m1h)2.
On the separatrix, where K = (m1h)2, m2 = m1, with:
 =
2

Z 
4
−
4
dx
s
1 + cos 4x
2 + cos 4x
: (22)
It is immediate to see that for m1 xed the function K, and hence the
semiclassical energy Em1;m2 , is a decreasing function of the secondary quan-
tum number m2, and we have a quantum multiplet [19].
We can calculate the value of the coupling constant 1=v2 corresponding to
the intersection of the separatrices of two neighboring quantum multiplets:
(m1 + 1)h!+
1
32v2
K[(m1 + 1)h; (m1 + 1)h] = m1h! +
1
32v2
K(m1h; m1h);
(23)
and so:
1
v2
=
−32h!
K[(m1 + 1)h; (m1 + 1)h]−K(m1h; m1h)
: (24)
6
In this way we have, in some sense, the quantal counterpart [12] of the method
of overlapping resonances developed by Chirikov [13]. The denominator can
be evaluated by the Taylor expansion and nally:
1
v2
=
24 −8!
@K
@B1
−  @K
@B2
35
B1=m1h;B2=m2h
: (25)
K is implicitly dened by the relation:
F [B1; B2; K(B1; B2)] = B2 −

2
Z 
4
−
4
dx
s
B21(2 + cos 4x)−K
2 + cos 4x
= 0; (26)
or:
F (B1; B2; K) = B2 − (B1; K) = 0: (27)
As a function of , 1=v2 can be written:
1
v2
= lim
K!B21
24 8! @@K
− @
@B1
35
B1=m1h
; (28)
where:
@
@K
= −
1

Z 
4
−
4
dx
1q
(2 + cos 4x)[B21(2 + cos 4x)−K]
@
@B1
=
2

Z 
4
−
4
dx
vuut B21(2 + cos 4x)
B21(2 + cos 4x)−K
: (29)
A similar procedure has been used for a more schematic model in [20]. The
result is:
1
v2
=
16!
m1h
; (30)
where m1h ’ E (the energy of the system) and ! = (2v2g2)
1
2 . Therefore
the chaos{order transition depends by the parameter  = v3g=E: if 0 <  <
p
2=32 a relevant region of the phase{space is chaotic, but if  >
p
2=32
7
the system becomes regular. This result shows that the value of the Higgs
eld in the vacuum v plays an important role: for large values makes the
system regular, in agreement with previous numerical calculations [3]. Also
the Yang{Mills coupling constant g has the same role. Instead if v and g are
xed there is an order{chaos transition increasing the energy E.
In conclusion, we have shown for the spatially homogenous SU(2) YMH
system that the quantum resonance criterion, which describes the onset of
widespread chaos associated to semiclassical crossing between separatices of
dierent quantum multiplets, gives an analytical estimation of the classical
chaos{order transition as a function of the higgs vacuum, the Yang{Mills
coupling constant and the energy of the system.
We observe that a classical chaos{order transition has been found numeri-
cally for the monopole solution [4] and the sphaleron solution [5] of the SU(2)
YMH theory. In the future, it will be of great importance to nd analytical
estimations of the onset of chaos also for these more realistic solutions.
*****
The author is greatly indebted to Profs. G. Benettin, S. Gra, V. R.
Manfredi and Dr. A. Vicini for many enlightening discussions. The au-
thor acknowledges Prof. J. M. G. Gomez for his kindly hospitality at the
Department of Atomic, Molecular and Nuclear Physics of "Complutense"
University, and the "Ing. Aldo Gini" Foundation for a partial support.
8
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9
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10


We study the classical chaos-order transition in the spatially homogeneous SU(2) Yang-Mills-Higgs system by using a quantal analogue of Chirikov's resonance overlap criterion. We obtain an analytical estimation of the range of parameters for which there is chaos suppression

Chaos Suppression in the SU(2) Yang--Mills--Higgs System

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