Marathe K. Topics in Physical Mathematics

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258 8 Yang–Mills–Higgs Fields

Equation (8.36)isalsoreferredtoasthemonopole equation.TheBianchi

identities imply that each solution (ω, φ) of the Bogomolnyi equations (8.36)

is a solution of the second order Yang–Mills–Higgs equations. In fact, one

can show that such solutions of the Yang–Mills–Higgs equations are global

minima on each connected component of the Yang–Mills–Higgs monopole

conﬁguration space C

deﬁned by

= {(ω,φ) ∈C| A

(ω,φ) < ∞, lim

y→∞

sup{

φ(x)|||x|≥y} =0},

where

φ(x)=|1 −|φ(x)||. Locally, the Bogomolnyi equations are obtained

by applying the reduction procedure to the instanton or anti-instanton equa-

tions on R

. No such ﬁrst order equations corresponding to the Yang–Mills–

Higgs equations with self-interaction potential are known. In particular, a

class of solutions of the Bogomolnyi equations on R

has been studied ex-

tensively. They are the most extensively studied special class of the Yang–

Mills monopole solutions. If the gauge group is a compact, simple Lie group,

then every solution (ω, φ) of the Bogomolnyi equations, satisfying certain

asymptotic conditions, deﬁnes a gauge-invariant set of integers. These inte-

gers are topological invariants corresponding to elements of the second homo-

topy group π

(G/J), where J is a certain subgroup of G obtained by ﬁxing

the boundary conditions. In the simplest example, when the gauge group

G = SU(2) and J = U(1), we have π

(G/J)

∼

)

∼

Z and there is only

one integer N (ω, φ) ∈ Z that classiﬁes the monopole solutions. It is called

the monopole number or the topological charge and is deﬁned by

N(ω, φ)=

4π



φ ∧F, (8.37)

wherewehavewrittenF for F

. It can be shown that, with suitable decay

of |φ| in R

, N(ω, φ) is an integer and we have

N(ω, φ)=

4π



φ ∧F

= lim

r→∞

4π



φ, F 

=deg{φ/|φ| : S

→ SU(2)},

where  ,  is the inner product in the Lie algebra. In fact, Groisser [170,

169] has proved that in classical SU(2) Yang–Mills–Higgs theories on R

with a Higgs ﬁeld in the adjoint representation, an integer-valued monopole

number is canonically deﬁned for any ﬁnite action smooth conﬁguration and

that the monopole conﬁguration space essentially has the homotopy type of

Maps(S

) (regarded as maps from the sphere at inﬁnity to the sphere in

8.6 Spontaneous Symmetry Breaking 259

the Lie algebra su(2)) with inﬁnitely many path components, labeled by the

monopole number.

The Yang–Mills equations and the Yang–Mills–Higgs equations share sev-

eral common features. As we noted above, locally, the Yang–Mills–Higgs

equations are obtained by a dimensional reduction of the pure Yang–Mills

equations. Both equations have solutions, which are classiﬁed by topological

invariants. For example, the G-instanton solutions over S

are classiﬁed by

(G). For simple Lie groups G this classiﬁcation goes by the integer deﬁned

by the Pontryagin index or the instanton number, whereas the monopole so-

lutions over R

are classiﬁed by π

(G/J). The ﬁrst order instanton equations

correspond to the ﬁrst order Bogomolnyi equations and both have solution

spaces that are parametrized by manifolds with singularities or moduli spaces.

However, there are important global diﬀerences in the solutions of the two

systems that arise due to diﬀerent boundary conditions. For example, no

translation-invariant non-trivial connection over R

can extend to S

.Ex-

tending the analytical foundations laid in [344, 386, 387], Taubes proved the

following theorem:

Theorem 8.9 There exists a solution to the SU(2) Yang–Mills–Higgs equa-

tions that is not a solution to the Bogomolnyi equations.

The corresponding problem regarding the relation of the solutions of the

full Yang–Mills equations and those of the instanton equations has been

solved in [350], where the existence of non-dual solutions to pure Yang–Mills

equations over S

is also established. Other non-dual solutions have been

obtained in [332]. The basic references for material in this section are Atiyah

and Hitchin [18] and Jaﬀe and Taubes [207]. For further developments see

[169,194].

8.6 Spontaneous Symmetry Breaking

We shall consider the spontaneous symmetry breaking in the context of a La-

grangian formulation of ﬁeld theories. We therefore begin by discussing some

examples of standard Lagrangian for matter ﬁelds, gauge ﬁelds, and their

interactions. An example of a Lagrangian for spin 1/2 fermions interacting

with a scalar ﬁeld φ

and a pseudo-scalar ﬁeld φ

is given by

L = −

ψ(D ψ + mψ)+y

ψψ + iy

ψγ

ψ, (8.38)

where ψ is the 4-component spinor ﬁeld of the fermion and

ψ is its conjugate

spinor ﬁeld and D = γ

∂

is the usual Dirac operator. Thus, the ﬁrst term in

the Lagrangian corresponds to the Dirac Lagrangian. The second and third

terms, representing couplings of scalar and pseudo-scalar ﬁelds to fermions

are called Yukawa couplings. The constants y

are called the Yukawa

coupling constants. They are introduced to give mass to the fermions. The

260 8 Yang–Mills–Higgs Fields

ﬁeld equations for the Lagrangian (8.38)aregivenby

(D + m)ψ = y

ψ + iy

ψ (8.39)

and the conjugate of equation (8.39). The Lagrangian for a spin 1/2fermion

interacting with an Abelian gauge ﬁeld (e.g., electromagnetic ﬁeld) is given

L = −

ψ(D

ψ + mψ) −

μν

, (8.40)

where D

is the Dirac operator coupled to the gauge potential ω and F

the corresponding gauge ﬁeld. A local expression for D

is obtained by the

minimal coupling to the local gauge potential A by

= γ

∇

≡ γ

(∂

− iqA

). (8.41)

We recall that the principle of minimal coupling to a gauge potential signiﬁes

the replacement of the ordinary derivative ∂

in the Lagrangian by the gauge-

covariant derivative ∇

. In the physics literature it is customary to denote

∂

by /∂ for any vector ∂

(∂

is regarded as a vector operator). The ﬁeld

equations corresponding to (8.40)aregivenby

∇

μν

+ ∇

νλ

+ ∇

λμ

=0, (8.42)

∂

μν

= J

:= iq

ψγ

ψ, (8.43)

(γ

∂

+ m)ψ = iqγ

ψ (8.44)

and the conjugate of equation (8.44). We note that (8.42) is often expressed

in the equivalent forms

F =0 or ∇

∗ F =0,

where δ

is the formal adjoint of ∇

.Equation(8.43) implies the conservation

of current ∂

= 0. The Lagrangian for spin 1/2 fermions interacting with

a non-Abelian gauge ﬁeld (e.g., an SU(N)-gauge ﬁeld) has the same form as

in equation (8.40) but with the operator D

deﬁned by

= γ

∇

≡ γ

(∂

− gT

), (8.45)

where {T

, 1 ≤ a ≤ N

− 1} is a basis for the Lie algebra su(N). The

expression for the current is given by

)

:= g

ψγ

ψ.

The law of conservation of current takes the form

∇

=0or∂

− f

=0,

8.6 Spontaneous Symmetry Breaking 261

where f

are the structure constants of the Lie algebra of su(N).

We now discuss the idea of spontaneous symmetry breaking in classical

ﬁeld theories by considering the Higgs ﬁeld ψ ∈ Γ (H

) with self-interaction

and then in interaction with a gauge ﬁeld. We note that most of the ob-

served symmetries in nature are only approximate. From classical spectral

theories as well as their quantum counterparts dealing with particle spectra,

we know that broken symmetries remove spectral degeneracies. Thus, for ex-

ample, one consequence of spontaneous symmetry breaking of isospin is the

observed mass diﬀerence between the proton and neutron masses (proton

and neutron are postulated to form an isospin doublet with isospin symmet-

ric Lagrangian). In general, if we have a ﬁeld theory with a Lagrangian that

admits an exact symmetry group K but gives rise to a ground state that is

not invariant under K, then we say that we have spontaneous symmetry

breaking. The ground state of a classical ﬁeld theory represents the vacuum

state (i.e., the state with no particles) in the corresponding quantum ﬁeld

theory. We now illustrate these ideas by considering an example of a classical

Higgs ﬁeld with self-interaction potential.

Let H

be a complex line bundle. Then ψ ∈ Γ (H

) is called a complex

scalar ﬁeld. We shall be concerned only with a local section in some co-

ordinate neighborhood with local coordinates (x

). Let us suppose that the

dynamics of this ﬁeld is determined by the Lagrangian L deﬁned by

L :=

(∂

ψ)(∂

ψ) − V (ψ), (8.46)

where

ψ = ψ

−iψ

is the complex conjugate of ψ.ThepotentialV is usually

taken to be independent of the derivatives of the ﬁeld. Let us consider the

following form for the potential:

V (ψ)=m

|ψ|

. (8.47)

The Lagrangian L admits U(1) as the global symmetry group with action

given by

ψ → e

iα

ψ, e

iα

∈ U (1).

We note that if we regard ψ as a real doublet, this action is just a rotation

of the vector ψ by the angle α and hence this Lagrangian is often called

rotationally symmetric. A classical state in which the energy attains its

absolute minimum value is called the ground state of the system. The

corresponding quantum state of lowest energy is called the vacuum state.

For the potential given by equation (8.47) the lowest energy state corresponds

to the ﬁeld conﬁguration deﬁned by ψ = 0. By considering small oscillations

around the corresponding vacuum state and examining the quadratic terms

in the Lagrangian we conclude that the ﬁeld ψ represents a pair of massive

bosons each with mass m. Let us now modify the potential V deﬁned by

equation (8.47) so that the ground state will be non-zero by changing the

262 8 Yang–Mills–Higgs Fields

sign of the ﬁrst term; i.e., we assume that

V (ψ)=−m

|ψ|

. (8.48)

As in the previous case the new Lagrangian admits U (1) as the global sym-

metry group. By rewriting the potential V of equation (8.48)intheform

V (ψ)=

(|ψ|

− a

)

−

, (8.49)

it is easy to see that the absolute minimum of the energy is (−m

/2). The

manifold of states of lowest energy is the circle ψ

+ ψ

= a

. A vacuum state

ψ corresponding to ψ has expectation value given by

|

ψ|

= a

Thus, we have a degenerate vacuum. Choosing a vacuum state breaks the

U(1) symmetry. The particle masses corresponding to the given system are

determined by considering the spectra of small oscillations about the vacuum

state. We shall carry out this analysis for the classical ﬁeld by choosing the

ground state ψ

:= a and expressing the Lagrangian in terms of the shifted

ﬁeld φ := ψ − ψ

. A simple calculation shows that the only term quadratic

in the ﬁeld variable φ is

quad

=2m

. (8.50)

The absence of a quadratic term in φ

in (8.50) and hence in the Lagrangian of

the shifted ﬁeld is interpreted in physics as corresponding to a massless boson

ﬁeld φ

. The ﬁeld φ

corresponds to a massive boson ﬁeld of mass

√

2m.The

massless boson arising in this calculation is called the Nambu–Goldstone

boson. The appearance of one or more such bosons is in fact a feature of a

large class of ﬁeld theories, with Lagrangians that admit a global continuous

symmetry group K, but which have ground states invariant under a proper

subgroup J ⊂ K. The assertion of such a statement is called Goldstone’s

theorem in the physics literature. The example that we have given above is

a special case of a version of Goldstone’s theorem that we now discuss.

Theorem 8.10 Let L denote a Lagrangian that is a function of the ﬁeld φ

and its derivatives. Suppose that L admits a k-dimensional Lie group K as

a global symmetry group. Let J (dim J = j) be the isotropy subgroup of K

ataﬁxedgroundstateφ

. Then by considering small oscillations about the

ﬁxed ground state one obtains a particle spectrum containing k − j massless

Nambu–Goldstone bosons.

However, there is no experimental evidence for the existence of such mass-

less bosons with the exception of the photon. Therefore, Lagrangian ﬁeld

theories that predict the existence of these massless bosons were considered

unsatisfactory. We note that Goldstone type theorems do not apply to gauge

8.7 Electroweak Theory 263

ﬁeld theories. In fact, as we observed earlier, gauge ﬁeld theories predict the

existence of massless gauge mesons, which are also not found in nature. We

now consider the phenomenon of spontaneous symmetry breaking for the

coupled system of gauge and matter ﬁelds in interaction. It is quite remark-

able that in such a theory the massless Goldstone bosons do not appear and

the massless gauge mesons become massive. This phenomenon is generally

referred to as the Higgs mechanism,orHiggs phenomenon, although

several scientists arrived at the same conclusion in considering such coupled

systems (see, for example, Coleman [80]). A geometric formulation of spon-

taneous symmetry breaking and the Higgs mechanism is given in [148].

8.7 Electroweak Theory

We now discuss spontaneous symmetry breaking in the case of a Higgs ﬁeld

with self-interaction minimally coupled to a gauge ﬁeld and explain the corre-

sponding Higgs phenomena. For the Higgs ﬁeld we choose an isodoublet ﬁeld

ψ (i.e., ψ ∈ Γ (H

), where ρ is the fundamental 2-dimensional complex repre-

sentation of the isospin group SU(2)) in interaction with an (SU(2) ×U(1))-

gauge ﬁeld. This choice illustrates all the important features of the general

case and also serves as a realistic model of the electroweak theory of Glashow,

Salam, and Weinberg. Experimental veriﬁcation of the gauge bosons pre-

dicted by their theory as carrier particles of the weak force earned them the

Nobel Prize for physics in 1979.

We take the base M to be a 4-dimensional pseudo-Riemannian manifold.

Since we work in a local coordinate chart we may in fact consider the theory

to be based on R

in the Euclidean case and on the Minkowski space M

the case of Lorentz signature. The principal bundle P over M with the gauge

group SU(2)×U(1) and all the associated bundles are trivial in this case. We

choose a ﬁxed trivialization of these bundles and omit its explicit mention in

the rest of this section. Let us suppose that the dynamics is governed by the

free ﬁeld Lagrangian

L :=

(∂

†

)(∂

ψ) − V (ψ), (8.51)

where ψ

†

is the Hermitian conjugate of ψ and the potential V is given by

V (ψ)=

(|ψ|

− a

)

−

. (8.52)

Let us write the ﬁeld ψ as a pair of complex ﬁelds

ψ =



+ iπ

+ iν



. (8.53)

264 8 Yang–Mills–Higgs Fields

Then we have

|ψ|

= π

+ π

+ ν

We want to minimally couple the ﬁeld ψ to an (SU(2) × U(1))-gauge ﬁeld.

The gauge potential ω takesvaluesintheLiealgebrasu(2) ⊕ u(1) of the

gauge group and the pull-back of ω to the base splits naturally into a pair

(A, B) of gauge potentials; A with values in su(2) and B with values in u(1).

Let F (resp., G) denote the gauge ﬁeld corresponding to the gauge potential

A (resp., B). Then the Lagrangian of the coupled ﬁelds is given by

(ω,ψ)=c(M)[|F |

+ |G|

+ c

(|∇

ψ|

− V (ψ))] , (8.54)

where c(M) is a normalizing constant that depends on the dimension of M

(c(M)=1/(8π

) for a 4-dimensional manifold) and the constant c

measures

the relative strengths of the gauge ﬁeld and its interaction with the Higgs

ﬁeld. In what follows we put c

= 1. The covariant derivative operator ∇

deﬁned by

∇

ψ := (∂

+ ig

ι)ψ, (8.55)

where the subscript j corresponds to the coordinate x

, constants g

are

the gauge coupling constants, while the σ

,a=1, 2, 3, are the Pauli spin

matrices, which form a basis for the Lie algebra su(2) and ι is the 2 × 2

identity matrix. It is easy to see that E

min

, the absolute minimum of the

energy of the coupled system, is given by

min

= −

The manifold of states of lowest energy is the 3-sphere

=0,B=0,π

+ π

+ ν

= a

A vacuum state

corresponding to the lowest energy state ψ

has expecta-

tion value given by

|

|

= a

Thus we have a degenerate vacuum. Choosing a vacuum state breaks the

SU(2) × U(1) symmetry. Let us choose





. (8.56)

It is easy to verify that Qψ

=0,where

Q :=

(ι + σ

). (8.57)

Thus we see that the symmetry is not completely broken. The stability group

of ψ

, generated by the charge operator Q deﬁned in (8.57), is isomorphic to

8.7 Electroweak Theory 265

U(1). Thus, in the free ﬁeld situation, there would be three Goldstone bosons.

However, in coupling to the gauge ﬁeld, these Goldstone bosons disappear as

we now show. As before, we consider small oscillations about the vacuum to

determine the particle spectrum. The shifted ﬁelds are A, B,andφ = ψ −ψ

We look for quadratic terms in these shifted ﬁelds in the Lagrangian. The

potential contributes a quadratic term in ν

, which corresponds to a massive

scalar boson of mass

√

2m. The only other quadratic terms arise from the

term involving the covariant derivative. Since the charge operator annihilates

the ground state ψ

, we rewrite the covariant derivative as follows:

∇

:= ∂

+ ig

(−g

+ g

)(ι −σ

)

+ i(g

+ ig

)Q. (8.58)

Using this expression, it is easy to see that quadratic terms in the gauge ﬁelds

are

(|A

+ |A

)+a

|(−g

+ g

. (8.59)

To remove the mixed terms in expression (8.59) we introduce two new ﬁelds

,N by rotating in the A

,B-plane by an angle θ as follows:

= A

cos θ − B

sin θ, (8.60)

= A

sin θ + B

cos θ, (8.61)

where the angle θ is deﬁned in terms of the gauge coupling constants by

= g cos θ, g

= g sin θ, (8.62)

or, equivalently,

tan θ = g

= g

+ g

. (8.63)

The angle θ (also denoted by θ

or θ

) is called the mixing angle of the

electroweak theory. Using the ﬁelds Z

,N the quadratic terms in the gauge

ﬁelds can be rewritten as

(|A

+ |A

)+g

. (8.64)

It is customary to deﬁne two complex ﬁelds W

√

+ iA

),W

−

√

− iA

) (8.65)

and to write the quadratic terms in the Lagrangian as follows:

quad

=2m

+ g

(|W

+ |W

−

)+g

. (8.66)

The terms in equation (8.66) allow us to identify the complete particle spec-

trum of the theory. The absence of a term in N indicates that it corresponds

266 8 Yang–Mills–Higgs Fields

to a massless ﬁeld, which can be interpreted as the electromagnetic ﬁeld. In

the covariant derivative N is coupled to the charge operator Q by the term

g sin 2θQN

. In the usual electromagnetic theory, this term is eQN

,wheree

is the electric charge. Comparing these, we get the following relation for the

mixing angle θ:

g sin 2θ = e. (8.67)

Theelectricchargee is related to the ﬁne structure constant α ≈ 1/137 by

the relation e =

√

4πα. There are three massive vector bosons, two W

−

of equal mass m

= g

a and the third Z

of mass m

= ga.Inthe

electroweak theory the carrier particles W

, W

−

are associated with the

charged intermediate bosons, which mediate the β-decay processes, while the

is associated with the neutral intermediate boson, which is supposed to

mediate the neutral current processes. The masses of the W and Z bosons

are related to the mixing angle by the relation

= m

cos θ. (8.68)

The term containing ν

in equation (10.39) represents the massive Higgs par-

ticle. The mass of this particle is called the Higgs mass and is denoted by

. It can be expressed, in terms of the parameters in the Higgs poten-

tial (10.24), as

√

2m.

We note that, while there is strong experimental evidence for the existence

of the W and the Z particles, there is no evidence for the existence of the

Higgs particle.

Up to this point our calculations have been based on the Lagrangian

of equation (8.54). This Lagrangian is part of the standard model La-

grangian, which forms the basis of the standard model of electroweak

interactions. We conclude this section with a brief discussion of this model.

The phenomenological basis for the formulation of the standard model of

the electroweak theory evolved over a period of time. The four most important

features of the theory, based on experimental observations, are the following:

1. The family structure of the fermions under the electroweak gauge

group SU

(2) ×U

(1) consists of left-handed doublets and right-handed

singlets characterized by the quantum numbers I,I

of the weak

isospin group SU

(2) and the weak hypercharge Y . For the leptons

we have













,μ

,τ

and for the quarks we have













, ... .

8.7 Electroweak Theory 267

The subscript L (resp., R) in these terms is used to denote the left-handed

fermions (resp., right-handed fermions).

2. The physical ﬁelds are also subject to ﬁnite symmetry groups. Three fun-

damental symmetries of order 2 are the charge conjugation C, parity

P ,andtime reversal T . These symmetries may be preserved or vio-

lated individually or in combination. The electroweak theory is left-right

asymmetric. This result is called the parity violation. Thus, contrary to

what many physicists believed, nature does distinguish between left and

right. Lee and Yang had put forth an argument predicting that parity

could be violated in weak interactions. They also suggested a number of

experiments to test this violation. This parity violation was veriﬁed exper-

imentally by Chien-Shiung Wu in 1957. Lee and Yang were awarded the

Nobel Prize for physics in the year 1957 for their fundamental work on

the topic. However, it is Yang’s work on the matrix-valued generalization

of the Maxwell equations to the Yang–Mills equations and the relation of

these equations with the theory of connections in a principal bundle over

the space-time manifold that has had the most profound eﬀect on recent

developments in geometric topology.

If Φ

(resp., Φ

) denotes the set of all left-handed (resp., right-handed)

fermions, then the assignment of the hypercharge quantum number must

satisfy the following conditions:



Y =0,



where Φ

denotes the set of left-handed fermion doublets.

3. The quantum numbers of the fermions with respect to the electroweak

gauge group and their electric charge Q satisfy the Gell-Mann–

Nishijima relation

Q = I

Y. (8.69)

4. There are four vector bosons;

γ, W

−

These act as carriers of the electroweak force. The ﬁrst of these is identiﬁed

as the photon, which is massless, while the other three, collectively called

the weak intermediate bosons, are massive.

Taking into account the above features and applying the general princi-

ples of constructing gauge-invariant ﬁeld theory with spontaneous symmetry

breaking, one arrives at the following standard model Lagrangian for elec-

troweak theory:

= L

+ L

is the Yukawa Lagrangian, and L

, the electroweak gauge ﬁeld La-

grangian, is given by