Introduction to Sphalerons - High Energy Physics at The University ...

Introduction to Sphalerons
Hridesh S. Kedia
Department of Physics
University of Chicago
Abstract
Sphalerons are defined to be static but unstable finite-energy solutions of the classical
field equations. This is a brief review of sphalerons and their significance in the Standard
Model. The sphaleron solution for the SU(2) Yang–Mills–Higgs theory as derived in [2] is
presented. The sphaleron is shown to have fractional charge and its role in baryon number
nonconservation is discussed. This work borrows heavily from [2].
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I. INTRODUCTION
The word ‘sphaleron’ is of Greek origin and means “ready to fall”. Sphalerons
are static unstable solutions of the classical field equations with finite total energy
of the fields. A sphaleron, being static and localized in space, is particlelike but
is unstable. The term ‘sphaleron’ was coined in [2] and used to describe static,
unstable solutions to the SU(2) Yang–Mills–Higgs theory.
Unlike solitons, sphalerons do not correspond to particle excitations in the
quantum theory since they are unstable solutions to the classical field equations.
Sphalerons separate topologically distinct vacuum states, ie vacuum state in one
gauge and vacuum state in another topologically inequivalent gauge.
Unfortunately all sphaleron solutions that I have managed to find in the literature
are complicated and so I will present only one of the many sphaleron solutions, which
is the sphaleron in the Yang–Mills–Higgs theory.
II. SPHALERONS IN THE YANG–MILLS–HIGGS THEORY
The sphaleron solution for the Yang–Mills–Higgs theory is a static solution to
the classical field equations, which is a saddle point of the energy functional and
therefore unstable.
Morse theory [8] tells us that there is a connection between the topology of a
smooth manifold and the stationary points of an arbitrary smooth function defined
on it. A standard example is a two-torus standing on end, with the height above
some reference plane the function defined on it (see Fig.. 1). The topology of the
torus requires that there must be atleast two saddle points (only one of which is
shown in the figure as the sphaleron S).
Sphaleron S
Vacuum
FIG. 1: Sphaleron S on top of a noncontractible loop in configuration space. The field
energy is zero for the vacuum and positive for the sphaleron S.
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[1] shows that in function space of all static fields which are solutions to the clas-
sical field equations for the Yang–Mills–Higgs theory, there exists a noncontractible
loop which begins and ends at the vacuum field configuration. They go on to sug-
gest that a saddle-point solution exists for the Yang–Mills–Higgs theory, by a bold
extension of the minimax idea by [6].
In this article, we consider a simplified version of the electroweak Standard Model
[7] without the hypercharge U(1) gauge field. This means, basically, that we set the
weak mixing angle θw ≡ arctan(g ′ /g) to zero, where g ′ and g are the coupling
constants of the U(1) and SU(2) gauge groups, respectively. Also, we take only one
family of quarks and leptons instead of the three known experimentally.
The SU(2) Yang–Mills gauge field is denoted by Aµ(x) ≡ A a µ(x) τ a /(2i), where
the τ a are the three Pauli matrices acting on weak isospin space and the component
fields A a µ(x) are real. (Repeated indices are summed over, unless stated otherwise.)
The complex Higgs field transforms as an isodoublet under the SU(2) gauge group
and is given by Φ(x) = (Φ1(x), Φ2(x)) t , where the suffix t stands for transpose.
The classical action of the gauge and Higgs fields reads
∫
ΓYMH = d 4 x
{
1
2 tr FµνF µν + (DµΦ) † (D µ (
Φ) − λ Φ † Φ − 1
2 v2
) }
2
, (2.1)
R 4
where Fµν ≡ ∂µAν − ∂νAµ + g[Aµ, Aν] is the SU(2) Yang–Mills field strength
and DµΦ ≡ (∂µ + gAµ)Φ the covariant derivative of the Higgs field. The theory
has Yang–Mills coupling constant g and quartic Higgs coupling constant λ, but
the classical dynamics depends only on the ratio λ/g 2 . The parameter v has the
dimension of mass and sets the scale of the Higgs expectation value. The three W
vector bosons then have equal mass, MW = g v. The single Higgs scalar boson has
a mass MH = 2 √ 2λ v .
The action (2.1) is invariant under a local gauge transformation
Φ ′ (x) = Λ(x) Φ(x) , gA ′ µ(x) = Λ(x) ( ) −1
gAµ(x) + ∂µ Λ(x) , (2.2)
for an arbitrary gauge function Λ(x) ∈ SU(2). In addition, there are certain global
SU(2) and U(1) symmetry transformations which operate solely on the Higgs field.
The field equations are as follows:
where
( ) a 1
DjFij = −
2 ig
[
Φ † σ a DiΦ − (DiΦ) † σ a ]
Φ
( ) a
DjFij = ∂jF a
ij + gϵ abc W b j F c
ij
3
(2.3)

and
DiDiΦ = 2λ ( Φ † Φ − 1
2 v2) Φ (2.4)
In order to find a sphaleron, we look for a solution that resembles the vacuum
configuration as r → ∞ and use a rotated version of the ansatz used in [5]. Doing
as stated above, we find that a solution to these SU(2) equations is possible with
fields of the form
and
W a
i σ a dx i = − 2ι
g f(gvr)dU ∞ (U ∞ ) −1
Φ = v √ h(gvr)U
2 ∞
( )
0
1
where
U ∞ = 1
r
(
)
z x + iy
−x + iy z
(2.5)
(2.6)
(2.7)
It is convenient now to introduce the dimensionless radial distance ξ = gvr. The
radial functions f and h which appear in the solutions to the field equations, are
functions of ξ.
For the fields 2.5, 2.6, the energy functional becomes
E = 4πv
g
∫ ∞
0
[
(
df
4
dξ
) 2
The field equations reduce to
+ 8
ξ2 [f(1 − f)]2 + 1
2 ξ2
(
dh
dξ
) 2
+ [h(1 − f)] 2 + 1
(
λ
4 g2 )
ξ 2 (h 2 − 1) 2
]
dξ (2.8)
ξ 2 d2f ξ2
= 2f(1 − f)(1 − 2f) −
dξ2 4 h2 (1 − f) (2.9)
d
(
2 dh
)
ξ = 2h(1 − f)
dξ dξ
2 + λ
g2 ξ2 (h 2 − 1)h (2.10)
The boundary conditions on the functions f and h are the following. Near
ξ = 0, f = αξ2 and h = βξ. As ξ → ∞, f = 1 − γexp(− 1ξ)
and h =
2
1 − (δ/ξ)exp(− √ 2λ/g2ξ). α, β, γ, and δ are constants of order unity which can
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only be determined by finding the complete solutions which are given as Ansatz in
[2].
The solution of 2.9 and 2.10 minimizes the energy functional 2.8, but the solution
is unstable with respect to fluctuations outside the class allowed by 2.5 and 2.6, which
is why it is a saddle-point.
III. FRACTIONAL CHARGE OF THE SPHALERON
For each family of quarks and leptons, which we assume to be massless, there is
a contribution to the divergence of the baryon current of the form
∂µj µ 1
B =
32π2 ϵµνρσ ( 1
2 g2F a µνF a ρσ + 1
2 g′2fµνfρσ) (3.1)
and equally for the divergence of the lepton current. For simplicity, we will only
consider the lightest family which consists of the u and d quarks, the electron and
the electron-type neutrino.
A consequence of 3.1 is that if the gauge and Higgs fields are time dependent, then
the baryonic charge is time dependent too. If the fields traverse a noncontractible
loop in configuration space, starting and finishing at the vacuum, with time as
the parameter, then the baryonic charge changes by an integer. This integer is
the Pontryagin index of the time-dependent gauge field, which equals the winding
number of the loop in configuration space.
Since the sphaleron represents a field configuration at the top of the energy barrier
between the vacuum in one gauge and the vacuum in a topologically inequivalent
gauge, one expects, if things are symmetrical, that the sphaleron has baryonic charge
1.
This is the case as the calculation in [2] shows.
2
The fact that the sphaleron has baryonic charge 1
2
ties in nicely with the physical
interpretation that the sphaleron separates topologically distinct vacua. Going by
the two-torus analogy, in configuration space, of all the homotopically equivalent
noncontractible loops that connect two topologically distinct vacua, the infimum of
the maximal energy configurations on all such loops corresponds to the sphaleron.
And the two topologically distinct vacua have baryon charges 0 and 1, with the
sphaleron having baryon charge 1.
The energy of the sphaleron is the barrier energy
2
separating the two vacua.
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IV. FINAL REMARKS
Sphaleron configurations arise because of the non-trivial topology of the config-
uration space of the classical field theory under consideration.
Sphaleron configurations might play an important role in explaining baryon num-
ber nonconservation in systems at high temperature such as the early universe.
Sphalerons can be used to explain baryon-number violating processes even when a
perturbative treatment may not be valid.
Thus the study of sphalerons is extremely important to electroweak baryogenesis.
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