Related to the di¬erence in the topological spaces of the abelian and non-

abelian Higgs ¬elds, signi¬cantly di¬erent phenomena occur in the spontaneous

symmetry breakdown. In the Georgi“Glashow model, the loss of rotational sym-

metry in color space is not complete. While the con¬guration (107) changes

under the (global) color rotations (108) and does therefore not re¬‚ect the invari-

ance of the Lagrangian or Hamiltonian of the system, it remains invariant under

rotations around the axis in the direction of the Higgs ¬eld ± ∼ φ0 . These trans-

formations form a subgroup of the group of rotations (108), it is the isotropy

group (little group, stability group) (for the de¬nition cf. (69)) of transformations

which leave φ0 invariant

Hφ0 = {h ∈ SU (2)|hφ0 = φ0 } . (109)

The space of the zeroes of V , i.e. the space of vectors φ of ¬xed length a, is S 2

which is a homogeneous space (cf. the discussion after (68)) with all elements

being generated by application of arbitrary transformations g ∈ G to a (¬xed)

φ0 . The space of zeroes of V and the coset space G/Hφ0 are mapped onto each

other by

Fφ0 : G/Hφ0 ’ {φ|V (φ) = 0} , Fφ0 (˜) = gφ0 = φ

g

with g denoting a representative of the coset g . This mapping is bijective. The

˜

space of zeroes is homogeneous and therefore all zeroes of V appear as an im-

age of some g ∈ G/Hφ0 . This mapping is injective since g1 φ0 = g2 φ0 implies

˜ ˜ ˜

’1

g1 φ0 g2 ∈ Hφ0 with g1,2 denoting representatives of the corresponding cosets

g1,2 and therefore the two group elements belong to the same equivalence class

˜

(cf. (56)) i.e. g1 = g2 . Thus, these two spaces are homeomorphic

˜ ˜

G/Hφ0 ∼ S 2 . (110)

It is instructive to compare the topological properties of the abelian and non-

abelian Higgs model.

Topological Concepts in Gauge Theories 45

• In the abelian Higgs model, the gauge group is

G = U (1)

and by the requirement of gauge invariance, the self-interaction is of the form

V (φ) = V (φ— φ).

The vanishing of V determines the modulus of φ and leaves the phase unde-

termined

V = 0 ’ |φ0 | = aeiβ .

After choosing the phase β, no residual symmetry is left, only multiplication

with 1 leaves φ0 invariant, i.e.

H = {e} , (111)

and thus

G/H = G ∼ S 1 . (112)

• In the non-abelian Higgs model, the gauge group is

G = SU (2),

and by the requirement of gauge invariance, the self-interaction is of the form

V (φ) = V (φ2 ) , φ2 = φa 2 .

a=1,3

The vanishing of V determines the modulus of φ and leaves the orientation

undetermined

ˆ

V = 0 ’ φ0 = aφ0 .

ˆ

After choosing the orientation φ0 , a residual symmetry persists, the invari-

ance of φ0 under (true) rotations around the φ0 axis and under multiplica-

tion with an element of the center of SU (2) (cf. (62))

H = U (1) — Z2 , (113)

and thus

G/H = SU (2)/ U (1) — Z2 ∼ S 2 . (114)

5.2 The Higgs Phase

To display the physical content of the Georgi“Glashow model we consider small

oscillations around the ground-state con¬gurations (107) “ the normal modes

of the classical system and the particles of the quantized system. The analysis

of the normal modes simpli¬es greatly if the gauge theory is represented in the

unitary gauge, the gauge which makes the particle content manifest. In this

gauge, components of the Higgs ¬eld rather than those of the gauge ¬eld (like

46 F. Lenz

the longitudinal gauge ¬eld in Coulomb gauge) are eliminated as redundant

variables. The Higgs ¬eld is used to de¬ne the coordinate system in internal

space

„a „3

φ(x) = φa (x) = ρ(x) . (115)

2 2

Since this gauge condition does not a¬ect the gauge ¬elds, the Yang“Mills part

of the Lagrangian (80) remains unchanged and the contribution of the Higgs

¬eld (82) simpli¬es

1 1

L = ’ F aµν Fµν + ‚µ ρ‚ µ ρ + g 2 ρ 2 A’ A+ µ ’ V (|ρ|) ,

a

(116)

µ

4 2

with the “charged” components of the gauge ¬elds de¬ned by

1

A± = √ (A1 “ iA2 ). (117)

µ µ µ

2

For small oscillations we expand the Higgs ¬eld ρ(x) around the value in the

zero-energy con¬guration (107)

|σ|

ρ(x) = a + σ(x), a. (118)

To leading order, the interaction with the Higgs ¬eld makes the charged compo-

nents (117) of the gauge ¬elds massive with the value of the mass given by the

value of ρ(x) in the zero-energy con¬guration

M 2 = g 2 a2 . (119)

The ¬‚uctuating Higgs ¬eld σ(x) acquires its mass through the self-interaction

m2 = Vρ=a = 2 a2 . (120)

σ

The neutral vector particles A3 , i.e. the color component of the gauge ¬eld along

µ

the Higgs ¬eld, remains massless. This is a consequence of the survival of the non-

trivial isotropy group Hφ0 ∼ U (1) (cf. (109)) in the symmetry breakdown of the

gauge group SU (2). By coupling to a second Higgs ¬eld, with expectation value

pointing in a color direction di¬erent from φ0 , a further symmetry breakdown

can be achieved which is complete up to the discrete Z2 symmetry (cf. (114)).

In such a system no massless vector particles can be present [8,40].

Super¬cially it may appear that the emergence of massive vector particles in

the Georgi“Glashow model happens almost with necessity. The subtleties of the

procedure are connected to the gauge choice (115). De¬nition of a coordinate

system in the internal color space via the Higgs ¬eld requires

φ = 0.

This requirement can be enforced by the choice of form (controlled by a) and

strength » of the Higgs potential V (104). Under appropriate circumstances,

quantum or thermal ¬‚uctuations will only rarely give rise to con¬gurations where

Topological Concepts in Gauge Theories 47

φ(x) vanishes at certain points and singular gauge ¬elds (monopoles) are present.

On the other hand, one expects at ¬xed a and » with increasing temperature the

occurrence of a phase transition to a gluon“Higgs ¬eld plasma. Similarly, at T =

0 a “quantum phase transition” (T = 0 phase transition induced by variation

of external parameters, cf. [41]) to a con¬nement phase is expected to happen

when decreasing a, » . In the unitary gauge, these phase transitions should be

accompanied by a condensation of singular ¬elds. When approaching either the

plasma or the con¬ned phase, the dominance of the equilibrium positions φ = 0

prohibits a proper de¬nition of a coordinate system in color space based on the

the color direction of the Higgs ¬eld.

The fate of the discrete Z2 symmetry is not understood in detail. As will

be seen, realization of the center symmetry indicates con¬nement. Thus, the Z2

factor should not be part of the isotropy group (113) in the Higgs phase. The

gauge choice (115) does not break this symmetry. Its breaking is a dynamical

property of the symmetry. It must occur spontaneously. This Z2 symmetry must

be restored in the quantum phase transition to the con¬nement phase and will

remain broken in the transition to the high temperature plasma phase.

5.3 Topological Excitations

As in the abelian Higgs model, the non-trivial topology (S 2 ) of the manifold of

vacuum ¬eld con¬gurations of the Georgi“Glashow model is the origin of the

topological excitations. We proceed as above and discuss ¬eld con¬gurations of

¬nite energy which di¬er in their topological properties from the ground-state

con¬gurations. As follows from the expression (106) for the energy density, ¬nite

energy can result only if asymptotically, |x| ’ ∞

φ(x) ’ aφ0 (x)

B’0

a

[Di φ((x))] = [‚i δ ac ’ g Ab (x)]φc (x) ’ 0 ,

abc

(121)

i

where φ0 (x) is a unit vector specifying the color direction of the Higgs ¬eld. The

last equation correlates asymptotically the gauge and the Higgs ¬eld. In terms

of the scalar ¬eld, the asymptotic gauge ¬eld is given by

1 1

Aa ’ φ ‚i φc + φa Ai ,

abc b

(122)

i

ga2 a

where A denotes the component of the gauge ¬eld along the Higgs ¬eld. It is

arbitrary since (121) determines only the components perpendicular to φ. The

asymptotic ¬eld strength associated with this gauge ¬eld (cf. (78)) has only a

color component parallel to the Higgs ¬eld “ the “neutral direction” (cf. the

de¬nition of the charged gauge ¬elds in (117)) and we can write

1 a ij 1

φ ‚ φ ‚ φ + ‚ i Aj ’ ‚ j Ai .

F aij = with F ij = abc a i b j c

φF , (123)

ga3

a

48 F. Lenz

One easily veri¬es that the Maxwell equations

‚i F ij = 0 (124)

are satis¬ed. These results con¬rm the interpretation of Fµν as a legitimate

¬eld strength related to the unbroken U (1) part of the gauge symmetry. As

the magnetic ¬‚ux in the abelian Higgs model, the magnetic charge in the non-

abelian Higgs model is quantized. Integrating over the asymptotic surface S 2

which encloses the system and using the integral form of the degree (49) of the

map de¬ned by the scalar ¬eld (cf. [42]) yields

1 4πN

B · dσ = ’ φ ‚ φ ‚ φ dσ i = ’

ijk abc a j b k c

m= . (125)

2ga3 g

S2 S2

No contribution to the magnetic charge arises from ∇ — A when integrated over

a surface without boundary. The existence of a winding number associated with

the Higgs ¬eld is a direct consequence of the topological properties discussed

above. The Higgs ¬eld φ maps the asymptotic S 2 onto the space of zeroes of V

which topologically is S 2 and has been shown (110) to be homeomorph to the

coset space G/Hφ0 . Thus, asymptotically, the map

S 2 ’ S 2 ∼ G/Hφ0

φ: (126)

is characterized by the homotopy group π2 (G/Hφ0 ) ∼ Z. Our discussion provides

an illustration of the general relation (61)

π2 (SU (2)/U (1) — Z2 ) = π1 (U (1)) ∼ Z .

The non-triviality of the homotopy group guarantees the stability of topological

excitations of ¬nite energy.

An important example is the spherically symmetric hedgehog con¬guration

xa

φa (r) r’∞ φa (r) = a ·

’’

0

r

which on the asymptotic sphere covers the space of zeroes of V exactly once.

Therefore, it describes a monopole with the asymptotic ¬eld strength (apart

from the A contribution) given, according to (123), by

xk r

B=’

ij ijk

F = , . (127)

gr3 g r3

Monopole Solutions. The asymptotics of Higgs and gauge ¬elds suggest the

following spherically symmetric Ansatz for monopole solutions

xa xj

[1 ’ K(agr)]

φa = a Aa = aij

H(agr) , (128)

i

gr2

r

with the boundary conditions at in¬nity

H(r) r’∞ 1, K(r) r’∞ 0 .

’’ ’’

Topological Concepts in Gauge Theories 49

As in the abelian Higgs model, topology forces the Higgs ¬eld to have a zero.

Since the winding of the Higgs ¬eld φ cannot be removed by continuous defor-