The Higgs-Kibble Mechanism and some Alternativesstruck/hp/wsem/higgs_talk.pdf · 2013. 12. 19. ·...

Historic evolution of gauge theoryReview of conventional gauge theory and the Higgs mechanism

Alternatives to the Higgs mechanism

The Higgs-Kibble Mechanismand some Alternatives

Jürgen Struckmeier

GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany

HICforFAIR-Workshop

“Current Topics in Accelerator-, Astro-, and Plasmaphysics”

Institute for Applied PhysicsGoethe University, Frankfurt am Main

Riezlern, 10–16 March 2013

Jürgen Struckmeier The Higgs-Kibble Mechanism and some Alternatives

Outline

1 Historic evolution of gauge theoryGauge invariance in classical physicsGauge invariance in quantum physics

2 Review of conventional gauge theory and the Higgsmechanism

Classical physicsQuantum physicsFormulation of the Higgs mechanism

3 Alternatives to the Higgs mechanismInhomogeneous gauge transformationGauge transformation with dynamic space-time

Gauge invariance in classical physicsGauge invariance in quantum physics

History of the gauge principle

The gauge principle emerged in the context of electromagnetics.1861/62 Maxwell equations1835/67 Formulation of EM theory with vector potentials (Gauß)

1904 Gauge invariance of EM theory (Lorentz)

A = a −∇Λ(x), V = v +∂Λ

1926 Form-invariance of the relativistic wave equation forquantum particles in an EM field (Fock) in conjunction witha simultaneous phase shift

Ψ = ψ eiΛ

1919/28 Perception of gauge-invariance as a basic principle (Weyl):“The electromagnetic field is a necessary accompanimentof the (charged) matter field”

A = a −∇Λ(x), V = v +∂Λ

Ψ = ψ eiΛ

A = a −∇Λ(x), V = v +∂Λ

Ψ = ψ eiΛ

A = a −∇Λ(x), V = v +∂Λ

Ψ = ψ eiΛ

A = a −∇Λ(x), V = v +∂Λ

Ψ = ψ eiΛ

1954 Generalization of the gauge principle to pairs of particles(proton/neutron symmetry?). “non-Abelian gauge theory”: numbers Λ(x)→ matrices

(Yang-Mills, Shaw, Salam)>1954 Protons/neutrons were identified to be composite particles

the Yang-Mills theory was dismissed>1960 The Yang-Mills theory was resumed with the theory of

electroweak interactions (Sheldon, Glashow, Salam,Weinberg) and also successfully applied to work out thetheory of strong interactions (QCD)

>1960 Yet, the conventional gauge principle fails to explain themasses of the weak interaction bosons W± and Z

1964 The Higgs mechanism was formulated independently byHiggs/Kibble and Englert/Brout to resolve this mass problem

Gauge invariance of the EM vector potential

We know from classical electrodynamics that the definition ofthe 4-vector potential (−V ,a) ≡ (a0,a) ≡ (aµ) from the electricand the magnetic fields, E and H, is not unique

E = −grad V − ∂a∂t

H = rot a.

The local shifting transformation of the 4-vector potential with qa coupling constant and Λ(x) an arbitrary differentiable function,

Aµ = aµ +1q∂Λ(x)

∂xµ

does not modify the equations of motion of the fields, hence theMaxwell equations.

Gauge invariance of the EM vector potential

We know from classical electrodynamics that the definition ofthe 4-vector potential (−V ,a) ≡ (a0,a) ≡ (aµ) from the electricand the magnetic fields, E and H, is not unique

E = −grad V − ∂a∂t

H = rot a.

The local shifting transformation of the 4-vector potential with qa coupling constant and Λ(x) an arbitrary differentiable function,

∂xµ

does not modify the equations of motion of the fields, hence theMaxwell equations.

Dirac equation

Let us now consider the Dirac equation that describes thedynamics of a spin-1

2 particle

iγα∂ψ

∂xα−mψ = 0, (~ = c = 1).

The phase of the spinor function ψ(x) is clearly arbitrary.

The physics and hence the form of the Dirac equationshould not depend on the phase definition.

But performing a local phase-shift transformation of the form

ψ(x) 7→ Ψ(x) = ψ(x) eiΛ(x)

actually changes the form of the Dirac equation

iγα∂Ψ

∂xα+

∂Λ(x)

∂xα−m

)Ψ = 0.

Dirac equation

Let us now consider the Dirac equation that describes thedynamics of a spin-1

2 particle

iγα∂ψ

∂xα−mψ = 0, (~ = c = 1).

The phase of the spinor function ψ(x) is clearly arbitrary.

The physics and hence the form of the Dirac equationshould not depend on the phase definition.

But performing a local phase-shift transformation of the form

ψ(x) 7→ Ψ(x) = ψ(x) eiΛ(x)

actually changes the form of the Dirac equation

iγα∂Ψ

∂xα+

∂Λ(x)

∂xα−m

)Ψ = 0.

Gauge invariance in quantum physics

It was discovered by V. Fock in 1926 that a shift of the vectorpotential in conjunction with a phase shift of the wave function

Aµ(x) = aµ(x) +1q∂Λ(x)

∂xµ, Ψ(x) = ψ(x) eiΛ(x)

applied to the amended Dirac equation

iγα∂Ψ

∂xα+ (qγαAα −m) Ψ = 0.

yields a form-invariant theory.Indeed, inserting the transformed fields yields Ψ,Aµ, we find

iγα∂ψ

∂xα−��

γα∂Λ

∂xαψ +

(qγαaα +

��γα∂Λ

∂xα−m

)ψ = 0.

The form of the amended Dirac equation is maintained.

Gauge invariance in quantum physics

It was discovered by V. Fock in 1926 that a shift of the vectorpotential in conjunction with a phase shift of the wave function

Aµ(x) = aµ(x) +1q∂Λ(x)

∂xµ, Ψ(x) = ψ(x) eiΛ(x)

applied to the amended Dirac equation

iγα∂Ψ

∂xα+ (qγαAα −m) Ψ = 0.

yields a form-invariant theory.Indeed, inserting the transformed fields yields Ψ,Aµ, we find

iγα∂ψ

∂xα−��

γα∂Λ

∂xαψ +

(qγαaα +

��γα∂Λ

∂xα−m

)ψ = 0.

The form of the amended Dirac equation is maintained.

U(1) gauge group

If the system Hamiltonian is the Dirac Hamiltonian describingthe relativistic dynamics of an electron, H = HD, then thegauge-invariant Hamiltonian H̃ is

H̃ = HD + iq(παψ − ψπα

)aα − 1

4pαβpαβ.

Then, aµ is nothing but the electromagnetic potential, and thelast two terms represent the Maxwell Hamiltonian provided thatwe identify its (conserved) 4-current density jµ with

jµ = iq(πµψ − ψπµ

)⇒ HM = −1

4pαβpαβ + jα aα.

The requirement of local phase invariance, applied to the freeDirac Hamiltonian HD, thus generates all of electrodynamicsand specifies the current produced by electrons.

U(1) gauge group

)aα − 1

4pαβpαβ.

)⇒ HM = −1

U(1) gauge group

)aα − 1

4pαβpαβ.

)⇒ HM = −1

SU(N) gauge group

The gauge principle has been generalized to systems withmore than one base field ψ1, ψ2, . . . by Young and Mills in 1954.We start with a known system Hamiltonian H describing anN-tuple of base fields ψI .Requiring now that this Hamiltonian be form-invariant underlocal gauge transformations of ψI 7→ ΨI of the form

ΨI = uIJψJ , ΨI = ψJu∗JI , uIK u∗KJ = δIJ

then forces us to introduce a Hermitian N × N matrix of gaugevector field, aKJµ.Furthermore, we learn how the ψI and the aKJ interact due tothe emerging of the coupling terms that involve both the basefields ψI and the gauge 4-vector fields aKJ . The gauge invariantHamiltonian H̃ has the form

H̃ = H+Ha +Hint.

SU(N) gauge group

H̃ = H+Ha +Hint.

SU(N) gauge group

H̃ = H+Ha +Hint.

Example: SU(3) gauge group

For a particular quark flavor, the respective three quark colorscarry the same mass. They are described by a generalizedDirac equation for the quark color triplet

ψ ≡

ψrψbψg

, ψ ≡(ψr ψb ψg

In the same sense as before, we now require the correspondingHamiltonian to be invariant under local transformations of theform (I, J = 1,2,3)

Ψ = U ψ, Ψ = ψU†

ΨI = uIJ ψJ , ΨI = ψJ u∗JI ,

with the matrix U being unitary in order to preserve the norm ψψ.

Example: SU(3) gauge group

For a particular quark flavor, the respective three quark colorscarry the same mass. They are described by a generalizedDirac equation for the quark color triplet

ψ ≡

ψrψbψg

, ψ ≡(ψr ψb ψg

In the same sense as before, we now require the correspondingHamiltonian to be invariant under local transformations of theform (I, J = 1,2,3)

Ψ = U ψ, Ψ = ψU†

ΨI = uIJ ψJ , ΨI = ψJ u∗JI ,

with the matrix U being unitary in order to preserve the norm ψψ.

Hamiltonian of the SU(3) gauge theoryHQCD = HD + iq

(παKψJ − ψKπ

aKJα − 14pαβ

IJ pJIαβ

− 12 iq pαβ

(aKIα aIJβ − aKIβ aIJα

For the SU(3) gauge group, aKJ is a Hermitian 3× 3 matrix ofcomplex fields in conjunction with the condition det U = +1. We have 32 − 1 = 8 independent real 4-vector gauge fields

that describe eight massless bosons, referred to as gluons.

With the field equations from HQCD, one can show that the vectors

jµJK = iq(πµKψJ − ψKπ

µJ + aJIαpαµ

IK − pαµJI aIKα

∂jαJK∂xα

then define conserved color currents, which act as sources ofthe eight color fields aKJ . In contrast to the Abelian case ofelectromagnetics, the fields aKJ themselves contribute to the jJK . The additional effect of self-coupling emerges. This renders

the dynamics of a quark-gluon plasma far more complicated.Jürgen Struckmeier The Higgs-Kibble Mechanism and some Alternatives

Hamiltonian of the SU(3) gauge theoryHQCD = HD + iq

(παKψJ − ψKπ

aKJα − 14pαβ

IJ pJIαβ

− 12 iq pαβ

(aKIα aIJβ − aKIβ aIJα

For the SU(3) gauge group, aKJ is a Hermitian 3× 3 matrix ofcomplex fields in conjunction with the condition det U = +1. We have 32 − 1 = 8 independent real 4-vector gauge fields

that describe eight massless bosons, referred to as gluons.

With the field equations from HQCD, one can show that the vectors

jµJK = iq(πµKψJ − ψKπ

µJ + aJIαpαµ

IK − pαµJI aIKα

∂jαJK∂xα

then define conserved color currents, which act as sources ofthe eight color fields aKJ . In contrast to the Abelian case ofelectromagnetics, the fields aKJ themselves contribute to the jJK . The additional effect of self-coupling emerges. This renders

the dynamics of a quark-gluon plasma far more complicated.Jürgen Struckmeier The Higgs-Kibble Mechanism and some Alternatives

The general principle of gauge invariance

The perception of gauge theory as a general principle datesback to H. Weyl (1928):

The invariance of a theory under combined transformationssuch as ψI 7→ ΨI , aJKµ 7→ AJKµ is referred to as a gaugeinvariance. This principle is the cornerstone in the creation ofthe Standard Model.

The principle yields the Lagrangian/Hamiltonian ofchromodynamics, which has been experimentally verified.This is an amazing achievement!

But there is one problem: in its actual form, the principle cannotexplain the occurrence of massive gauge bosons such as Z 0

and W±.Jürgen Struckmeier The Higgs-Kibble Mechanism and some Alternatives

Proca Hamiltonian

The dynamics of a real 4-vector field Aµ that is associated withthe mass m is described by the Proca Hamiltonian density HP

HP = −14pαβpαβ − 1

2m2aαaα, pµν = −pνµ.

The canonical field equations follow as

pνµ =∂aµ

∂xν− ∂aν

∂xµ,

∂pνα

∂xα= m2aν .

Under the transformation rule of the gauge fields

∂xµ,

this yields

Pνµ = pνµ, PαβPαβ = pαβpαβ, but AαAα 6= aαaα.

Thus, all gauge fields must be massless in order to preservethe property of local gauge invariance ( Higgs mechanism).

Proca Hamiltonian

pνµ =∂aµ

∂xν− ∂aν

∂xµ,

∂pνα

∂xα= m2aν .

∂xµ,

this yields

Proca Hamiltonian

pνµ =∂aµ

∂xν− ∂aν

∂xµ,

∂pνα

∂xα= m2aν .

∂xµ,

this yields

Higgs potential

Fig. 1: View of a hypothetical complex potential function of agauge boson

The idea behind the Higgs model

If the gauge boson is described by such a potential, aparticular ground state does not share the potential’ssymmetry.The Feynman formalism (which is a perturbation theory tocalculate interactions), requires to expand around theground state (“vacuum”).We are thus forced to start from non-symmetric state( “spontaneous symmetry breaking”).

Canonical shifting transformation

The starting point is the Hamiltonian of a massless boson, aµ

that interacts (coupling constant g) with some Dirac particle φ

Hg = −14pαβ pαβ + ig

(πα φ− φπα

)aα, pµν !

= −pνµ.

This Hamiltonian was shown to be form-invariant under a localgauge transformation φ 7→ φexp(iΛ(x)).

If we perform a shifting transformation into the ground state

Φ(x) = φ(x)− ϕ, ∂Φ

∂xµ=

∂xµ, ϕ = const,

then we must set up a corresponding generating function inorder to ensure the overall transformation to be physical

Fµ2 =

(Πµ − iq ϕaµ

)(φ− ϕ) +

(φ− ϕ

)(Πµ + iq aµϕ) + Pαµaα.

The action principle is then maintained.Jürgen Struckmeier The Higgs-Kibble Mechanism and some Alternatives

(πα φ− φπα

)aα, pµν !

= −pνµ.

Φ(x) = φ(x)− ϕ, ∂Φ

∂xµ=

∂xµ, ϕ = const,

Fµ2 =

(Πµ − iq ϕaµ

)(φ− ϕ) +

(φ− ϕ

(πα φ− φπα

)aα, pµν !

= −pνµ.

Φ(x) = φ(x)− ϕ, ∂Φ

∂xµ=

∂xµ, ϕ = const,

Fµ2 =

(Πµ − iq ϕaµ

)(φ− ϕ) +

(φ− ϕ

The complete transformation effects all fields described by Hg

φ = Φ + ϕ, φ = Φ + ϕ

πµ = Πµ − iq ϕaµ, πµ = Πµ + iq aµϕ

aµ = Aµ, pνµ = Pνµ + iq ηνµ(φϕ− ϕφ

The transformed Hamiltonian H′g is obtained by expressing Hgin terms of the transformed fields. This yields

H′g = −14PαβPαβ − 1

2 iq(Φϕ− ϕΦ

α + q2(Φϕ− ϕΦ)2

ΠαAαΦ− ΦAαΠα + Π

αAαϕ− ϕAαΠα)

+ 2q2AαAα

(ϕΦ + Φϕ+ 2ϕϕ

This seems meaningless at first sight, but now comes the trick:

φ = Φ + ϕ, φ = Φ + ϕ

2 iq(Φϕ− ϕΦ

+ 2q2AαAα

(ϕΦ + Φϕ+ 2ϕϕ

φ = Φ + ϕ, φ = Φ + ϕ

2 iq(Φϕ− ϕΦ

+ 2q2AαAα

(ϕΦ + Φϕ+ 2ϕϕ

Choosing a particular gauge

As the original Hamiltonian was form-invariant underφ 7→ φexp(iΛ(x)), we are allowed to choose Λ(x) appropriatelyfor φ(x) — and hence Φ, ϕ,Πµ — to become real. With thisparticular choice, the transformed Hamiltonian simplifies to

2 iq(((((((((

Φϕ− ϕΦ)

Pαα + q2

��

�(Φϕ− ϕΦ

+ iq(��

�ΠαAαΦ −��ΦAαΠα +��

�ΠαAαϕ −��ϕAαΠα

)+ 2q2AαAα

(ϕΦ + Φϕ+ 2ϕϕ

henceH′g = −1

4PαβPαβ + 4q2ϕAαAαΦ + 12ω

2AαAα.

Now, the real quantity 4q2ϕ2 = 12ω

2 defines a constant massterm pertaining to the quadratic gauge field term AαAα. Higgs mechanism: gauge principle + symmetry breaking

Choosing a particular gauge

As the original Hamiltonian was form-invariant underφ 7→ φexp(iΛ(x)), we are allowed to choose Λ(x) appropriatelyfor φ(x) — and hence Φ, ϕ,Πµ — to become real. With thisparticular choice, the transformed Hamiltonian simplifies to

2 iq(((((((((

Φϕ− ϕΦ)

Pαα + q2

��

�(Φϕ− ϕΦ

+ iq(��

�ΠαAαΦ −��ΦAαΠα +��

�ΠαAαϕ −��ϕAαΠα

)+ 2q2AαAα

(ϕΦ + Φϕ+ 2ϕϕ

henceH′g = −1

4PαβPαβ + 4q2ϕAαAαΦ + 12ω

2AαAα.

Now, the real quantity 4q2ϕ2 = 12ω

2 defines a constant massterm pertaining to the quadratic gauge field term AαAα. Higgs mechanism: gauge principle + symmetry breaking

Inhomogeneous gauge transformationGauge transformation with dynamic space-timeSummary

The principle of local gauge invariance revisited

Review: The principle of local gauge invariance was fullyestablished when it was recognized that

Dirac Lagrangian + requirement of local gauge invariance⇒ Maxwell Lagrangian + interaction of electrons/photons

As the photon is massless, the problem that the principle isnot consistent with a massive gauge boson did not exist atthe time.The principle of local gauge invariance was generalized byYang and Mills, but this particular generalization did notresolve the problem of massive gauge bosons. Instead of supplementing the conventional gauge

principle with the Higgs mechanism, one can think offurther generalizing the gauge principle.

Inhomogeneous local gauge transformation

A straightforward generalization of the conventional gaugeprinciple would be to require the given system to beform-invariant under the inhomogeneous gauge transformation

ΦI = uIJ φJ + ϕI , ΦI = φJ u∗JI + ϕI

Example: For the case N = 1, hence for a single base field φ,there exists a combined transformation of base field φ and twogauge fields, aµ,bµ that generalizes the conventional gaugetransformation and reduces to it for ϕ ≡ 0

φ 7→ Φ = φeiΛ + ϕ, aµ 7→ Aµ = aµ +1g∂Λ

∂xµ

bµ 7→ Bµ = bµ eiΛ − igm

(aµ +

1g∂Λ

∂xµ

1m∂ϕ

∂xµ.

Inhomogeneous local gauge transformation

A straightforward generalization of the conventional gaugeprinciple would be to require the given system to beform-invariant under the inhomogeneous gauge transformation

ΦI = uIJ φJ + ϕI , ΦI = φJ u∗JI + ϕI

Example: For the case N = 1, hence for a single base field φ,there exists a combined transformation of base field φ and twogauge fields, aµ,bµ that generalizes the conventional gaugetransformation and reduces to it for ϕ ≡ 0

φ 7→ Φ = φeiΛ + ϕ, aµ 7→ Aµ = aµ +1g∂Λ

∂xµ

bµ 7→ Bµ = bµ eiΛ − igm

(aµ +

1g∂Λ

∂xµ

1m∂ϕ

∂xµ.

Inhomogeneous local gauge transformationWe may easily verify that the following twofold amendedKlein-Gordon Lagrangian L̄KG is form-invariant under the abovecombined transformation of the fields φ,aµ,bµ.

L̄KG =

(∂φ

∂xα+ ig φaα −m b

∂xα− ig aαφ−m bα

)− 1

4pαβ pαβ − 12qαβqαβ

The field tensors for the case N = 1 (one base field φ) are

pµν =∂aν

∂xµ− ∂aµ

∂xν

qµν =∂bν

∂xµ− ∂bµ

∂xν+ ig (aν bµ − aµ bν) +

(∂aν

∂xµ− ∂aµ

∂xν

With m2 bαbα, this locally gauge-invariant Lagrangian contains

a mass term for the complex bosonic 4-vector field bµ(x).Jürgen Struckmeier The Higgs-Kibble Mechanism and some Alternatives

Inhomogeneous local gauge transformationWe may easily verify that the following twofold amendedKlein-Gordon Lagrangian L̄KG is form-invariant under the abovecombined transformation of the fields φ,aµ,bµ.

L̄KG =

(∂φ

∂xα+ ig φaα −m b

∂xα− ig aαφ−m bα

)− 1

4pαβ pαβ − 12qαβqαβ

The field tensors for the case N = 1 (one base field φ) are

pµν =∂aν

∂xµ− ∂aµ

∂xν

qµν =∂bν

∂xµ− ∂bµ

∂xν+ ig (aν bµ − aµ bν) +

(∂aν

∂xµ− ∂aµ

∂xν

With m2 bαbα, this locally gauge-invariant Lagrangian contains

a mass term for the complex bosonic 4-vector field bµ(x).Jürgen Struckmeier The Higgs-Kibble Mechanism and some Alternatives

Local gauge transformation in Riemannian spaceFrom the canonical equations of the gauge-invariant SU(N)Hamiltonian, the canonical momenta are obtained as

πIµ =∂φI

∂xµ+ ig aIJµφJ

pIJµν =∂aIJν

∂xµ−∂aIJµ

∂xν+ ig

(aIKν aKJµ − aIKµ aKJν

Compare this with the following quantities of General Relativity,namely the covariant derivative and the Riemann tensor

DaIJν

Dxµ=∂aIJν

∂xµ− Γα

νµaIJα

Rβαµν =

∂Γβαν

∂xµ− ∂Γβ

∂xν+ Γξ

ανΓβξµ − Γξ

αµΓβξν .

We observe that the connection coefficients Γανµ

(“Christoffel symbols”) on their part play the roles ofgauge fields for the gauge fields aIJ (dim Γ = dim a = length−1!)

πIµ =∂φI

∂xµ+ ig aIJµφJ

pIJµν =∂aIJν

∂xµ−∂aIJµ

∂xν+ ig

DaIJν

Dxµ=∂aIJν

∂xµ− Γα

νµaIJα

Rβαµν =

∂Γβαν

∂xµ− ∂Γβ

∂xν+ Γξ

αµΓβξν .

πIµ =∂φI

∂xµ+ ig aIJµφJ

pIJµν =∂aIJν

∂xµ−∂aIJµ

∂xν+ ig

DaIJν

Dxµ=∂aIJν

∂xµ− Γα

νµaIJα

Rβαµν =

∂Γβαν

∂xµ− ∂Γβ

∂xν+ Γξ

αµΓβξν .

Local gauge-invariance under dynamical space-time

A generalization of the gauge principle to also requireform-invariance under variation of the space-time metricalso exhibits field equations for massive gauge fields.The mechanism is: The gauge field causes anon-vanishing curvature tensor, and this curvature tensorappears in the field equations as a mass factor.This idea can only be formulated as an extended canonicaltransformation in the realm of covariant Hamiltonian fieldtheory.It was published recently in:J. Phys. G: Nucl. Part. Phys. 40 (2013) 015007

Summary

The requirement of local gauge invariance has turned outto be an extremely powerful principle.A problem is that the masses of gauge bosons cannot beexplained by the conventional gauge principle.Obviously the gauge principle that was deduced from theU(1) gauge theory cannot be transposed 1 : 1 to the caseof SU(N) if massive gauge bosons are involved.There are straightforward options to generalize the con-ventional gauge principle that resolve the mass problem.The gauge transformation theory is most efficiently workedout in the covariant Hamiltonian formalism of field theoryas a canonical transformation.

Summary

The Higgs-Kibble Mechanism and some Alternativesstruck/hp/wsem/higgs_talk.pdf · 2013. 12. 19. ·...

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