Particle Physics lecture 11 Gauge theories in particle ... Physics lecture 11 Gauge theories in...

Particle Physicslecture 11

Gauge theories in particle physics and thestandard model

Yazid Delenda

Departement des Sciences de la matiereFaculte des Sciences - UHLB

http://delenda.wordpress.com/teaching/particlephysics/

Batna, 04 January 2015

(http://delenda.wordpress.com) Particle Physics - lecture 11 1 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Consider the Lagrangian

L = ψ(iγµ∂µ −m)ψ, (1)

where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:

∂L∂ψ

= (iγµ∂µ −m)ψ,

∂L∂(∂µψ)

= 0. (2)

The Euler-Lagrange equation gives the Dirac equation:

(iγµ∂µ −m)ψ = 0. (3)

so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 2 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Consider the Lagrangian

L = ψ(iγµ∂µ −m)ψ, (1)

where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:

∂L∂ψ

= (iγµ∂µ −m)ψ,

∂L∂(∂µψ)

= 0. (2)

The Euler-Lagrange equation gives the Dirac equation:

(iγµ∂µ −m)ψ = 0. (3)

so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 2 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Consider the Lagrangian

L = ψ(iγµ∂µ −m)ψ, (1)

where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:

∂L∂ψ

= (iγµ∂µ −m)ψ,

∂L∂(∂µψ)

= 0. (2)

The Euler-Lagrange equation gives the Dirac equation:

(iγµ∂µ −m)ψ = 0. (3)

so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 2 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Consider the Lagrangian

L = ψ(iγµ∂µ −m)ψ, (1)

where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:

∂L∂ψ

= (iγµ∂µ −m)ψ,

∂L∂(∂µψ)

= 0. (2)

The Euler-Lagrange equation gives the Dirac equation:

(iγµ∂µ −m)ψ = 0. (3)

so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 2 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Consider the Lagrangian

L = ψ(iγµ∂µ −m)ψ, (1)

where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:

∂L∂ψ

= (iγµ∂µ −m)ψ,

∂L∂(∂µψ)

= 0. (2)

The Euler-Lagrange equation gives the Dirac equation:

(iγµ∂µ −m)ψ = 0. (3)

so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 2 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Consider the Lagrangian

L = ψ(iγµ∂µ −m)ψ, (1)

where ψ is a Dirac spinor and L(ψ, ∂ψ, ψ, ∂ψ) (L is Lorentz invariant).Wehave:

∂L∂ψ

= (iγµ∂µ −m)ψ,

∂L∂(∂µψ)

= 0. (2)

The Euler-Lagrange equation gives the Dirac equation:

(iγµ∂µ −m)ψ = 0. (3)

so the above Lagrangian describes free Dirac particles. We shall now seehow the interaction between Maxwell fields and Dirac fields is generatedby the powerful principle of gauge transformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 2 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Let us insist invariance of the Dirac Lagrangian under:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x), (4)

as a “local phase invariance” of L. This is clearly not a symmetry of theDirac Lagrangian.Indeed, under this transforation the term:

mψψ → mψ′ψ′ = mψψ, (5)

is invariant.However we have under this transformation:

ψγµ∂µψ → ψ

′γµ∂

µψ′ = ψγµ [(ie∂µΩ(x))ψ + ∂µψ] , (6)

where we have used ψ = ψ†γ0, so that ψ → ψ′= e−ieΩ(x)ψ.

(http://delenda.wordpress.com) Particle Physics - lecture 11 3 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Let us insist invariance of the Dirac Lagrangian under:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x), (4)

as a “local phase invariance” of L. This is clearly not a symmetry of theDirac Lagrangian.Indeed, under this transforation the term:

mψψ → mψ′ψ′ = mψψ, (5)

is invariant.However we have under this transformation:

ψγµ∂µψ → ψ

′γµ∂

µψ′ = ψγµ [(ie∂µΩ(x))ψ + ∂µψ] , (6)

where we have used ψ = ψ†γ0, so that ψ → ψ′= e−ieΩ(x)ψ.

(http://delenda.wordpress.com) Particle Physics - lecture 11 3 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Let us insist invariance of the Dirac Lagrangian under:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x), (4)

as a “local phase invariance” of L. This is clearly not a symmetry of theDirac Lagrangian.Indeed, under this transforation the term:

mψψ → mψ′ψ′ = mψψ, (5)

is invariant.However we have under this transformation:

ψγµ∂µψ → ψ

′γµ∂

µψ′ = ψγµ [(ie∂µΩ(x))ψ + ∂µψ] , (6)

where we have used ψ = ψ†γ0, so that ψ → ψ′= e−ieΩ(x)ψ.

(http://delenda.wordpress.com) Particle Physics - lecture 11 3 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Let us insist invariance of the Dirac Lagrangian under:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x), (4)

as a “local phase invariance” of L. This is clearly not a symmetry of theDirac Lagrangian.Indeed, under this transforation the term:

mψψ → mψ′ψ′ = mψψ, (5)

is invariant.However we have under this transformation:

ψγµ∂µψ → ψ

′γµ∂

µψ′ = ψγµ [(ie∂µΩ(x))ψ + ∂µψ] , (6)

where we have used ψ = ψ†γ0, so that ψ → ψ′= e−ieΩ(x)ψ.

(http://delenda.wordpress.com) Particle Physics - lecture 11 3 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Let us insist invariance of the Dirac Lagrangian under:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x), (4)

as a “local phase invariance” of L. This is clearly not a symmetry of theDirac Lagrangian.Indeed, under this transforation the term:

mψψ → mψ′ψ′ = mψψ, (5)

is invariant.However we have under this transformation:

ψγµ∂µψ → ψ

′γµ∂

µψ′ = ψγµ [(ie∂µΩ(x))ψ + ∂µψ] , (6)

where we have used ψ = ψ†γ0, so that ψ → ψ′= e−ieΩ(x)ψ.

(http://delenda.wordpress.com) Particle Physics - lecture 11 3 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Let us insist invariance of the Dirac Lagrangian under:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x), (4)

as a “local phase invariance” of L. This is clearly not a symmetry of theDirac Lagrangian.Indeed, under this transforation the term:

mψψ → mψ′ψ′ = mψψ, (5)

is invariant.However we have under this transformation:

ψγµ∂µψ → ψ

′γµ∂

µψ′ = ψγµ [(ie∂µΩ(x))ψ + ∂µψ] , (6)

where we have used ψ = ψ†γ0, so that ψ → ψ′= e−ieΩ(x)ψ.

(http://delenda.wordpress.com) Particle Physics - lecture 11 3 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

In order to make the Lagrangian invariant under such a transformation weshall need to introduce a new field Aµ which, under a certaintransformation, is capable of absorbing the extra term.We do this bymaking the replacement ∂µ → Dµ, such that Dµ transforms like:

Dµψ → Dµ′ψ′ = eieΩ(x)Dµψ, (7)

then:ψγµD

µψ → ψ′γµD

µ′ψ′ = ψγµDµψ. (8)

and this way the Dirac Lagrangian remains invariant under local phasetransformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 4 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

In order to make the Lagrangian invariant under such a transformation weshall need to introduce a new field Aµ which, under a certaintransformation, is capable of absorbing the extra term.We do this bymaking the replacement ∂µ → Dµ, such that Dµ transforms like:

Dµψ → Dµ′ψ′ = eieΩ(x)Dµψ, (7)

then:ψγµD

µψ → ψ′γµD

µ′ψ′ = ψγµDµψ. (8)

and this way the Dirac Lagrangian remains invariant under local phasetransformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 4 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

In order to make the Lagrangian invariant under such a transformation weshall need to introduce a new field Aµ which, under a certaintransformation, is capable of absorbing the extra term.We do this bymaking the replacement ∂µ → Dµ, such that Dµ transforms like:

Dµψ → Dµ′ψ′ = eieΩ(x)Dµψ, (7)

then:ψγµD

µψ → ψ′γµD

µ′ψ′ = ψγµDµψ. (8)

and this way the Dirac Lagrangian remains invariant under local phasetransformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 4 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

In order to make the Lagrangian invariant under such a transformation weshall need to introduce a new field Aµ which, under a certaintransformation, is capable of absorbing the extra term.We do this bymaking the replacement ∂µ → Dµ, such that Dµ transforms like:

Dµψ → Dµ′ψ′ = eieΩ(x)Dµψ, (7)

then:ψγµD

µψ → ψ′γµD

µ′ψ′ = ψγµDµψ. (8)

and this way the Dirac Lagrangian remains invariant under local phasetransformations.

(http://delenda.wordpress.com) Particle Physics - lecture 11 4 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′

(indicated above) and Aµ → Aµ′

(which is yet to be determined to be ableto absorb the extra term) we have:

Dµψ → D′µψ′ = (∂µ − ieA′µ)(eieΩψ) = eieΩ[ie(∂µΩ)ψ + ∂µψ − ieA′µψ].

(9)We require that this be equal to (see above):eieΩDµψ = eieΩ(∂µ − ieAµ)ψ.Thus we obtain:

ie(∂µΩ)ψ − ieA′µψ = −ieAµψ. (10)

(http://delenda.wordpress.com) Particle Physics - lecture 11 5 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′

(indicated above) and Aµ → Aµ′

(which is yet to be determined to be ableto absorb the extra term) we have:

Dµψ → D′µψ′ = (∂µ − ieA′µ)(eieΩψ) = eieΩ[ie(∂µΩ)ψ + ∂µψ − ieA′µψ].

(9)We require that this be equal to (see above):eieΩDµψ = eieΩ(∂µ − ieAµ)ψ.Thus we obtain:

ie(∂µΩ)ψ − ieA′µψ = −ieAµψ. (10)

(http://delenda.wordpress.com) Particle Physics - lecture 11 5 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′

(indicated above) and Aµ → Aµ′

(which is yet to be determined to be ableto absorb the extra term) we have:

Dµψ → D′µψ′ = (∂µ − ieA′µ)(eieΩψ) = eieΩ[ie(∂µΩ)ψ + ∂µψ − ieA′µψ].

(9)We require that this be equal to (see above):eieΩDµψ = eieΩ(∂µ − ieAµ)ψ.Thus we obtain:

ie(∂µΩ)ψ − ieA′µψ = −ieAµψ. (10)

(http://delenda.wordpress.com) Particle Physics - lecture 11 5 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′

(indicated above) and Aµ → Aµ′

(which is yet to be determined to be ableto absorb the extra term) we have:

Dµψ → D′µψ′ = (∂µ − ieA′µ)(eieΩψ) = eieΩ[ie(∂µΩ)ψ + ∂µψ − ieA′µψ].

(9)We require that this be equal to (see above):eieΩDµψ = eieΩ(∂µ − ieAµ)ψ.Thus we obtain:

ie(∂µΩ)ψ − ieA′µψ = −ieAµψ. (10)

(http://delenda.wordpress.com) Particle Physics - lecture 11 5 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′

(indicated above) and Aµ → Aµ′

(which is yet to be determined to be ableto absorb the extra term) we have:

Dµψ → D′µψ′ = (∂µ − ieA′µ)(eieΩψ) = eieΩ[ie(∂µΩ)ψ + ∂µψ − ieA′µψ].

(9)We require that this be equal to (see above):eieΩDµψ = eieΩ(∂µ − ieAµ)ψ.Thus we obtain:

ie(∂µΩ)ψ − ieA′µψ = −ieAµψ. (10)

(http://delenda.wordpress.com) Particle Physics - lecture 11 5 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

So we need to find an expression for Dµ in terms of the new field Aµ sothat Dµ has the desired transformation. This is a achieved if we setDµ ≡ ∂µ − ieAµ.Then under the simultaneous transformations of ψ → ψ′

(indicated above) and Aµ → Aµ′

(which is yet to be determined to be ableto absorb the extra term) we have:

Dµψ → D′µψ′ = (∂µ − ieA′µ)(eieΩψ) = eieΩ[ie(∂µΩ)ψ + ∂µψ − ieA′µψ].

(9)We require that this be equal to (see above):eieΩDµψ = eieΩ(∂µ − ieAµ)ψ.Thus we obtain:

ie(∂µΩ)ψ − ieA′µψ = −ieAµψ. (10)

(http://delenda.wordpress.com) Particle Physics - lecture 11 5 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Hence:−∂µΩ +A′µ = Aµ (11)

Hence we choose the transformation of Aµ as follows:

Aµ → A′µ = Aµ + ∂µΩ(x) (12)

We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):

L = ψ(iγµDµ −m)ψ − 1

4FµνF

µν , (13)

L = ψ(iγµ∂µ −m)ψ + eψγµA

µψ − 1

4FµνF

µν , (14)

(http://delenda.wordpress.com) Particle Physics - lecture 11 6 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Hence:−∂µΩ +A′µ = Aµ (11)

Hence we choose the transformation of Aµ as follows:

Aµ → A′µ = Aµ + ∂µΩ(x) (12)

We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):

L = ψ(iγµDµ −m)ψ − 1

4FµνF

µν , (13)

L = ψ(iγµ∂µ −m)ψ + eψγµA

µψ − 1

4FµνF

µν , (14)

(http://delenda.wordpress.com) Particle Physics - lecture 11 6 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Hence:−∂µΩ +A′µ = Aµ (11)

Hence we choose the transformation of Aµ as follows:

Aµ → A′µ = Aµ + ∂µΩ(x) (12)

We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):

L = ψ(iγµDµ −m)ψ − 1

4FµνF

µν , (13)

L = ψ(iγµ∂µ −m)ψ + eψγµA

µψ − 1

4FµνF

µν , (14)

(http://delenda.wordpress.com) Particle Physics - lecture 11 6 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Hence:−∂µΩ +A′µ = Aµ (11)

Hence we choose the transformation of Aµ as follows:

Aµ → A′µ = Aµ + ∂µΩ(x) (12)

We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):

L = ψ(iγµDµ −m)ψ − 1

4FµνF

µν , (13)

L = ψ(iγµ∂µ −m)ψ + eψγµA

µψ − 1

4FµνF

µν , (14)

(http://delenda.wordpress.com) Particle Physics - lecture 11 6 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Hence:−∂µΩ +A′µ = Aµ (11)

Hence we choose the transformation of Aµ as follows:

Aµ → A′µ = Aµ + ∂µΩ(x) (12)

We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):

L = ψ(iγµDµ −m)ψ − 1

4FµνF

µν , (13)

L = ψ(iγµ∂µ −m)ψ + eψγµA

µψ − 1

4FµνF

µν , (14)

(http://delenda.wordpress.com) Particle Physics - lecture 11 6 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Hence:−∂µΩ +A′µ = Aµ (11)

Hence we choose the transformation of Aµ as follows:

Aµ → A′µ = Aµ + ∂µΩ(x) (12)

We identify this as the gauge transformation for the Maxwell vector fieldAµ.Thus we have introduced a coupling between Dirac and Maxwell fields(i.e. interaction).Hence, the Lagrangian becomes (substituting):

L = ψ(iγµDµ −m)ψ − 1

4FµνF

µν , (13)

L = ψ(iγµ∂µ −m)ψ + eψγµA

µψ − 1

4FµνF

µν , (14)

(http://delenda.wordpress.com) Particle Physics - lecture 11 6 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

which is invariant under the simultaneous transformations:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x),

Aµ(x)→ A′µ(x) = Aµ(x) + ∂µΩ(x). (15)

This is a U(1) phase invariance of the Dirac field and gauge invariance ofthe photon field.Inverting this logic:

Local phase invariance of the free field Dirac Lagrangian

“demands” that we introduce a vector field Aµ which transforms

like Maxwell field and couples to the original Dirac Field.

Now we have:

∂L∂Aν

= eψγνψ,∂L

∂(∂µAν)= −Fµν , (16)

(http://delenda.wordpress.com) Particle Physics - lecture 11 7 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

which is invariant under the simultaneous transformations:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x),

Aµ(x)→ A′µ(x) = Aµ(x) + ∂µΩ(x). (15)

This is a U(1) phase invariance of the Dirac field and gauge invariance ofthe photon field.Inverting this logic:

Local phase invariance of the free field Dirac Lagrangian

“demands” that we introduce a vector field Aµ which transforms

like Maxwell field and couples to the original Dirac Field.

Now we have:

∂L∂Aν

= eψγνψ,∂L

∂(∂µAν)= −Fµν , (16)

(http://delenda.wordpress.com) Particle Physics - lecture 11 7 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

which is invariant under the simultaneous transformations:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x),

Aµ(x)→ A′µ(x) = Aµ(x) + ∂µΩ(x). (15)

This is a U(1) phase invariance of the Dirac field and gauge invariance ofthe photon field.Inverting this logic:

Local phase invariance of the free field Dirac Lagrangian

“demands” that we introduce a vector field Aµ which transforms

like Maxwell field and couples to the original Dirac Field.

Now we have:

∂L∂Aν

= eψγνψ,∂L

∂(∂µAν)= −Fµν , (16)

(http://delenda.wordpress.com) Particle Physics - lecture 11 7 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

which is invariant under the simultaneous transformations:

ψ(x)→ ψ′(x) = eieΩ(x)ψ(x),

Aµ(x)→ A′µ(x) = Aµ(x) + ∂µΩ(x). (15)

This is a U(1) phase invariance of the Dirac field and gauge invariance ofthe photon field.Inverting this logic:

Local phase invariance of the free field Dirac Lagrangian

“demands” that we introduce a vector field Aµ which transforms

like Maxwell field and couples to the original Dirac Field.

Now we have:

∂L∂Aν

= eψγνψ,∂L

∂(∂µAν)= −Fµν , (16)

(http://delenda.wordpress.com) Particle Physics - lecture 11 7 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

The Euler Lagrange equation now gives (for Aµ):

∂µFµν + eψγνψ = 0. (17)

We can interpret −eψγνψ ≡ jν as a current density.So:

∂µFµν = jν (18)

which is equivalent to:

~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E

∂t, (19)

where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.

(http://delenda.wordpress.com) Particle Physics - lecture 11 8 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

The Euler Lagrange equation now gives (for Aµ):

∂µFµν + eψγνψ = 0. (17)

We can interpret −eψγνψ ≡ jν as a current density.So:

∂µFµν = jν (18)

which is equivalent to:

~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E

∂t, (19)

where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.

(http://delenda.wordpress.com) Particle Physics - lecture 11 8 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

The Euler Lagrange equation now gives (for Aµ):

∂µFµν + eψγνψ = 0. (17)

We can interpret −eψγνψ ≡ jν as a current density.So:

∂µFµν = jν (18)

which is equivalent to:

~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E

∂t, (19)

where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.

(http://delenda.wordpress.com) Particle Physics - lecture 11 8 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

The Euler Lagrange equation now gives (for Aµ):

∂µFµν + eψγνψ = 0. (17)

We can interpret −eψγνψ ≡ jν as a current density.So:

∂µFµν = jν (18)

which is equivalent to:

~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E

∂t, (19)

where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.

(http://delenda.wordpress.com) Particle Physics - lecture 11 8 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

The Euler Lagrange equation now gives (for Aµ):

∂µFµν + eψγνψ = 0. (17)

We can interpret −eψγνψ ≡ jν as a current density.So:

∂µFµν = jν (18)

which is equivalent to:

~∇ · ~E = ρ, ~∇× ~B = ~j +∂ ~E

∂t, (19)

where jµ = (ρ,~j), and ~E and ~B are defined previously.This means thatwe have gone from a non-interacting (free) field to an interacting one byintroducing gauge invariance.

(http://delenda.wordpress.com) Particle Physics - lecture 11 8 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

The term +eψγµAµψ = −jµAµ generates the interaction and gives the

vertex:

∝ jµAµ = −eψγµAµψ

(http://delenda.wordpress.com) Particle Physics - lecture 11 9 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Note: We still need to add a gauge fixing term to the Lagrangian todefine a photon propagator:

Lgauge fixging =(∂µA

µ)2

2ξ

which leads to a propagator of the form:

Dµν = −i(gµν − (1− ξ)pµpν

p2

)1

p2

The ξ dependence drops out in physical calculations of cross-sectionshence preserving gauge invariance.

(http://delenda.wordpress.com) Particle Physics - lecture 11 10 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Note: We still need to add a gauge fixing term to the Lagrangian todefine a photon propagator:

Lgauge fixging =(∂µA

µ)2

2ξ

which leads to a propagator of the form:

Dµν = −i(gµν − (1− ξ)pµpν

p2

)1

p2

The ξ dependence drops out in physical calculations of cross-sectionshence preserving gauge invariance.

(http://delenda.wordpress.com) Particle Physics - lecture 11 10 / 35

Gauge field theory Gauge invariance and QED

Gauge invariance and QED

Note: We still need to add a gauge fixing term to the Lagrangian todefine a photon propagator:

Lgauge fixging =(∂µA

µ)2

2ξ

which leads to a propagator of the form:

Dµν = −i(gµν − (1− ξ)pµpν

p2

)1

p2

The ξ dependence drops out in physical calculations of cross-sectionshence preserving gauge invariance.

(http://delenda.wordpress.com) Particle Physics - lecture 11 10 / 35

Non-Abelian gauge invariance

Non-Abelian gauge invariance

The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).

(http://delenda.wordpress.com) Particle Physics - lecture 11 11 / 35

Non-Abelian gauge invariance

Non-Abelian gauge invariance

The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).

(http://delenda.wordpress.com) Particle Physics - lecture 11 11 / 35

Non-Abelian gauge invariance

Non-Abelian gauge invariance

The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).

(http://delenda.wordpress.com) Particle Physics - lecture 11 11 / 35

Non-Abelian gauge invariance

Non-Abelian gauge invariance

The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).

(http://delenda.wordpress.com) Particle Physics - lecture 11 11 / 35

Non-Abelian gauge invariance

Non-Abelian gauge invariance

The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).

(http://delenda.wordpress.com) Particle Physics - lecture 11 11 / 35

Non-Abelian gauge invariance

Non-Abelian gauge invariance

The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).

(http://delenda.wordpress.com) Particle Physics - lecture 11 11 / 35

Non-Abelian gauge invariance

Non-Abelian gauge invariance

The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).

(http://delenda.wordpress.com) Particle Physics - lecture 11 11 / 35

Non-Abelian gauge invariance

Non-Abelian gauge invariance

The transformation ψ(x)→ eieΩψ(x) is an Abelian gauge invariance.Letus generalise ψ to ψi, where i ∈ 1, 2, ..., N, i.e. we endow ψ with someinternal structure. An example of such a situation is the wavefunction ofthe quarks, which has 3 colour degrees of freedom (Nc = 3).Let us impose the invariance of the quark wave function under the phaserotation: ψi → Uijψj (compare this with with Abelian phaseinvariance),where U(x) is N ×N matrix.We want U †U = 1 to preservenorm of the wavefunction (ψiψi → ψiψi).We further require that the determinant of the matrix U be equal to 1(rotation matrix).The set of matrices U form a Lie Group, meaning thatthe set U is the set of all N ×N unitary matrices with unitdetermiunant.Uij furnish a matrix representation of the group SU(N).

(http://delenda.wordpress.com) Particle Physics - lecture 11 11 / 35

Non-Abelian gauge invariance Group theory basics

Non-Abelian gauge invarianceGroup theory basics

A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .

G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c

G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a

G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e

An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a

(http://delenda.wordpress.com) Particle Physics - lecture 11 12 / 35

Non-Abelian gauge invariance Group theory basics

Non-Abelian gauge invarianceGroup theory basics

A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .

G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c

G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a

G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e

An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a

(http://delenda.wordpress.com) Particle Physics - lecture 11 12 / 35

Non-Abelian gauge invariance Group theory basics

Non-Abelian gauge invarianceGroup theory basics

A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .

G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c

G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a

G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e

An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a

(http://delenda.wordpress.com) Particle Physics - lecture 11 12 / 35

Non-Abelian gauge invariance Group theory basics

Non-Abelian gauge invarianceGroup theory basics

A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .

G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c

G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a

G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e

An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a

(http://delenda.wordpress.com) Particle Physics - lecture 11 12 / 35

Non-Abelian gauge invariance Group theory basics

Non-Abelian gauge invarianceGroup theory basics

A group (G, .) is defined by the set of elements that are combinedtogether using a “multiplication law” such that certain axioms are obeyed:G = a, b, c, · · · .

G0: closure: ∀a, b ∈ G2 : a.b = c ∈ GG1: Associativity: ∀ (a, b, c) ∈ G3, a.(b.c) = (a.b).c

G2: Identity element: ∃! e ∈ G,∀ a ∈ G : a.e = e.a = a

G3: Inverse element:∀ a ∈ G, ∃ a−1 ∈ G : a.a−1 = a−1.a = e

An abelian group is one for which ∀ (a, b) ∈ G2 : a.b = b.a

(http://delenda.wordpress.com) Particle Physics - lecture 11 12 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:

D(g1.g2) = D(g1).D(g2)

Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.

(http://delenda.wordpress.com) Particle Physics - lecture 11 13 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:

D(g1.g2) = D(g1).D(g2)

Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.

(http://delenda.wordpress.com) Particle Physics - lecture 11 13 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:

D(g1.g2) = D(g1).D(g2)

Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.

(http://delenda.wordpress.com) Particle Physics - lecture 11 13 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:

D(g1.g2) = D(g1).D(g2)

Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.

(http://delenda.wordpress.com) Particle Physics - lecture 11 13 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:

D(g1.g2) = D(g1).D(g2)

Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.

(http://delenda.wordpress.com) Particle Physics - lecture 11 13 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Group Representations: An n dimentional representation of a group is amapping of elements of the group to non-singular n× nmatrices,g → D(g) such that group multiplication is preserved:

D(g1.g2) = D(g1).D(g2)

Continuous groups: Are groups with infinite number of elementscorresponding to infinite values of a continuous parameter (e.g. a rotationangle).Let us take an example:The full rotation group for rotations through any angle in 2 or 3dimensional space.We have two groups: SO(2) and SO(3),which are the“special orthogonal groups” representing rotations in 2D and 3D spaces.

(http://delenda.wordpress.com) Particle Physics - lecture 11 13 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).

(http://delenda.wordpress.com) Particle Physics - lecture 11 14 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).

(http://delenda.wordpress.com) Particle Physics - lecture 11 14 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).

(http://delenda.wordpress.com) Particle Physics - lecture 11 14 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).

(http://delenda.wordpress.com) Particle Physics - lecture 11 14 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).

(http://delenda.wordpress.com) Particle Physics - lecture 11 14 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

They are “special” because det(D) = 1 (where D are the grouprepresentation matrices), and orthogonal because DTD = 1.Generators: A key concept in the context of Lie groups.It is related to theinfinitesimal transformations.Let us take a simple example of SO(2) group,the group of rotations in 2D space.The elements of this group are rotations through an angle φ (about the zaxis) with multiplication law “successive application” (exercise: check thatthis is an Abelian group).

(http://delenda.wordpress.com) Particle Physics - lecture 11 14 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

A matrix representation for the element (rotation through an angle φ) isthe rotation matrix:

R(φ) =

(cosφ − sinφsinφ cosφ

)

Now for infinitesimal rotations we may write (φ 1):

R(φ) =

(1− φ2/2 −φ

φ 1− φ2

)=

(1 00 1

)+

(0 −11 0

)φ+O(φ2)

=1− iXφ+O(φ2)

The matrix X is the generator of this group and can be read off from theexpansion of the full rotation matrix.

(http://delenda.wordpress.com) Particle Physics - lecture 11 15 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

A matrix representation for the element (rotation through an angle φ) isthe rotation matrix:

R(φ) =

(cosφ − sinφsinφ cosφ

)

Now for infinitesimal rotations we may write (φ 1):

R(φ) =

(1− φ2/2 −φ

φ 1− φ2

)=

(1 00 1

)+

(0 −11 0

)φ+O(φ2)

=1− iXφ+O(φ2)

The matrix X is the generator of this group and can be read off from theexpansion of the full rotation matrix.

(http://delenda.wordpress.com) Particle Physics - lecture 11 15 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

A matrix representation for the element (rotation through an angle φ) isthe rotation matrix:

R(φ) =

(cosφ − sinφsinφ cosφ

)

Now for infinitesimal rotations we may write (φ 1):

R(φ) =

(1− φ2/2 −φ

φ 1− φ2

)=

(1 00 1

)+

(0 −11 0

)φ+O(φ2)

=1− iXφ+O(φ2)

The matrix X is the generator of this group and can be read off from theexpansion of the full rotation matrix.

(http://delenda.wordpress.com) Particle Physics - lecture 11 15 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

A matrix representation for the element (rotation through an angle φ) isthe rotation matrix:

R(φ) =

(cosφ − sinφsinφ cosφ

)

Now for infinitesimal rotations we may write (φ 1):

R(φ) =

(1− φ2/2 −φ

φ 1− φ2

)=

(1 00 1

)+

(0 −11 0

)φ+O(φ2)

=1− iXφ+O(φ2)

The matrix X is the generator of this group and can be read off from theexpansion of the full rotation matrix.

(http://delenda.wordpress.com) Particle Physics - lecture 11 15 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

A matrix representation for the element (rotation through an angle φ) isthe rotation matrix:

R(φ) =

(cosφ − sinφsinφ cosφ

)

Now for infinitesimal rotations we may write (φ 1):

R(φ) =

(1− φ2/2 −φ

φ 1− φ2

)=

(1 00 1

)+

(0 −11 0

)φ+O(φ2)

=1− iXφ+O(φ2)

The matrix X is the generator of this group and can be read off from theexpansion of the full rotation matrix.

(http://delenda.wordpress.com) Particle Physics - lecture 11 15 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Alternatively given the generator of the group X we can write:

R(φ) = e−iXφ

which means that all elements of the group can be built from thegenerator.Alternatively:

−iX =dR

dφ

∣∣∣∣φ=0

where we note that X is hermitian (because R is unitary) and thatdetR = 1⇒ Tr(X) = 0.A more interesting group is SO(3) which describes rotations in 3D spaceand ultimately angular momentum.

(http://delenda.wordpress.com) Particle Physics - lecture 11 16 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Alternatively given the generator of the group X we can write:

R(φ) = e−iXφ

which means that all elements of the group can be built from thegenerator.Alternatively:

−iX =dR

dφ

∣∣∣∣φ=0

where we note that X is hermitian (because R is unitary) and thatdetR = 1⇒ Tr(X) = 0.A more interesting group is SO(3) which describes rotations in 3D spaceand ultimately angular momentum.

(http://delenda.wordpress.com) Particle Physics - lecture 11 16 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Alternatively given the generator of the group X we can write:

R(φ) = e−iXφ

which means that all elements of the group can be built from thegenerator.Alternatively:

−iX =dR

dφ

∣∣∣∣φ=0

where we note that X is hermitian (because R is unitary) and thatdetR = 1⇒ Tr(X) = 0.A more interesting group is SO(3) which describes rotations in 3D spaceand ultimately angular momentum.

(http://delenda.wordpress.com) Particle Physics - lecture 11 16 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

Alternatively given the generator of the group X we can write:

R(φ) = e−iXφ

which means that all elements of the group can be built from thegenerator.Alternatively:

−iX =dR

dφ

∣∣∣∣φ=0

where we note that X is hermitian (because R is unitary) and thatdetR = 1⇒ Tr(X) = 0.A more interesting group is SO(3) which describes rotations in 3D spaceand ultimately angular momentum.

(http://delenda.wordpress.com) Particle Physics - lecture 11 16 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

In fact SO(3) generators satisfy commutation relations for angularmomentum in quantum mechanics:

[Xi, Xj ] = iεijkXj

The same algebra is also satisfied by 2× 2 hermitian matrices Xi = σi/2,with σi the Pauli matrices,which act on 2-component spinors,e.g theelectron spin-up and spin down-states :

|12,+

1

2〉 =

(10

), |1

2,−1

2〉 =

(01

)

which are eigenvalues of σ3.

(http://delenda.wordpress.com) Particle Physics - lecture 11 17 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

In fact SO(3) generators satisfy commutation relations for angularmomentum in quantum mechanics:

[Xi, Xj ] = iεijkXj

The same algebra is also satisfied by 2× 2 hermitian matrices Xi = σi/2,with σi the Pauli matrices,which act on 2-component spinors,e.g theelectron spin-up and spin down-states :

|12,+

1

2〉 =

(10

), |1

2,−1

2〉 =

(01

)

which are eigenvalues of σ3.

(http://delenda.wordpress.com) Particle Physics - lecture 11 17 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

In fact SO(3) generators satisfy commutation relations for angularmomentum in quantum mechanics:

[Xi, Xj ] = iεijkXj

The same algebra is also satisfied by 2× 2 hermitian matrices Xi = σi/2,with σi the Pauli matrices,which act on 2-component spinors,e.g theelectron spin-up and spin down-states :

|12,+

1

2〉 =

(10

), |1

2,−1

2〉 =

(01

)

which are eigenvalues of σ3.

(http://delenda.wordpress.com) Particle Physics - lecture 11 17 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

In fact SO(3) generators satisfy commutation relations for angularmomentum in quantum mechanics:

[Xi, Xj ] = iεijkXj

The same algebra is also satisfied by 2× 2 hermitian matrices Xi = σi/2,with σi the Pauli matrices,which act on 2-component spinors,e.g theelectron spin-up and spin down-states :

|12,+

1

2〉 =

(10

), |1

2,−1

2〉 =

(01

)

which are eigenvalues of σ3.

(http://delenda.wordpress.com) Particle Physics - lecture 11 17 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

The matrix of rotation through any angle θ about a direction n cantherefore be obtained by exponentiating:

U(θ) = e−i~X .n θ

where ~X = (σ1, σ2, σ3)/2.

(http://delenda.wordpress.com) Particle Physics - lecture 11 18 / 35

Non-Abelian gauge invariance Group representations and continuous or Lie groups

Non-Abelian gauge invarianceGroup representations and continuous or Lie groups

The matrix of rotation through any angle θ about a direction n cantherefore be obtained by exponentiating:

U(θ) = e−i~X .n θ

where ~X = (σ1, σ2, σ3)/2.

(http://delenda.wordpress.com) Particle Physics - lecture 11 18 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:

U = eiωata

where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).

(http://delenda.wordpress.com) Particle Physics - lecture 11 19 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:

U = eiωata

where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).

(http://delenda.wordpress.com) Particle Physics - lecture 11 19 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:

U = eiωata

where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).

(http://delenda.wordpress.com) Particle Physics - lecture 11 19 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:

U = eiωata

where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).

(http://delenda.wordpress.com) Particle Physics - lecture 11 19 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:

U = eiωata

where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).

(http://delenda.wordpress.com) Particle Physics - lecture 11 19 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:

U = eiωata

where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).

(http://delenda.wordpress.com) Particle Physics - lecture 11 19 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:

U = eiωata

where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).

(http://delenda.wordpress.com) Particle Physics - lecture 11 19 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:

U = eiωata

where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).

(http://delenda.wordpress.com) Particle Physics - lecture 11 19 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

SU(N) is the special unitary group,i.e. group of all N ×N unitary matriceswith the special property that their determinants are 1.There are 2N2 realparameters in an N ×N matrix (real + virtual parts).Imposing unitarityleaves only half of them independent.Further imposing det = 1 removesone free parameter.A general rotation matrix in SU(N) is written as:

U = eiωata

where ωa are rotation angles and ta the generators of the group.It is easy to show that the generators are hermitian and traceless (ta† = ta,trta = 0)(which follow from unitarity and unit determinant).

(http://delenda.wordpress.com) Particle Physics - lecture 11 19 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

For example to construct the generators of SU(2) we may expressinfinitesimal rotations as:

(a b− ic

b+ ic −a

)

which has 3 free parameters (N2 − 1). Hence:

a

(1 00 −1

)+ b

(0 11 0

)+ c

(0 −ii 0

)

So the Pauli matrices are the generators of SU(2).For SU(3) there are N2 − 1 = 8 generators which are called the Gell-Mannmatrices.

(http://delenda.wordpress.com) Particle Physics - lecture 11 20 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

For example to construct the generators of SU(2) we may expressinfinitesimal rotations as:

(a b− ic

b+ ic −a

)

which has 3 free parameters (N2 − 1). Hence:

a

(1 00 −1

)+ b

(0 11 0

)+ c

(0 −ii 0

)

So the Pauli matrices are the generators of SU(2).For SU(3) there are N2 − 1 = 8 generators which are called the Gell-Mannmatrices.

(http://delenda.wordpress.com) Particle Physics - lecture 11 20 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

For example to construct the generators of SU(2) we may expressinfinitesimal rotations as:

(a b− ic

b+ ic −a

)

which has 3 free parameters (N2 − 1). Hence:

a

(1 00 −1

)+ b

(0 11 0

)+ c

(0 −ii 0

)

So the Pauli matrices are the generators of SU(2).For SU(3) there are N2 − 1 = 8 generators which are called the Gell-Mannmatrices.

(http://delenda.wordpress.com) Particle Physics - lecture 11 20 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

For example to construct the generators of SU(2) we may expressinfinitesimal rotations as:

(a b− ic

b+ ic −a

)

which has 3 free parameters (N2 − 1). Hence:

a

(1 00 −1

)+ b

(0 11 0

)+ c

(0 −ii 0

)

So the Pauli matrices are the generators of SU(2).For SU(3) there are N2 − 1 = 8 generators which are called the Gell-Mannmatrices.

(http://delenda.wordpress.com) Particle Physics - lecture 11 20 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The generators of SU(Nc) form a lie Algebra:

[ta, tb] = ifabctc

where fabc are the structure constants of the group.The generators of thegroup are also normalised as:

tr[tatb] = TF δab

with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:

L(x) =∑

f

qif (x)(iγµ∂µ −mf )qjf (x)δij

where the sum runs over flavours and i and j are colour indices which runover colours.

(http://delenda.wordpress.com) Particle Physics - lecture 11 21 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The generators of SU(Nc) form a lie Algebra:

[ta, tb] = ifabctc

where fabc are the structure constants of the group.The generators of thegroup are also normalised as:

tr[tatb] = TF δab

with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:

L(x) =∑

f

qif (x)(iγµ∂µ −mf )qjf (x)δij

where the sum runs over flavours and i and j are colour indices which runover colours.

(http://delenda.wordpress.com) Particle Physics - lecture 11 21 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The generators of SU(Nc) form a lie Algebra:

[ta, tb] = ifabctc

where fabc are the structure constants of the group.The generators of thegroup are also normalised as:

tr[tatb] = TF δab

with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:

L(x) =∑

f

qif (x)(iγµ∂µ −mf )qjf (x)δij

where the sum runs over flavours and i and j are colour indices which runover colours.

(http://delenda.wordpress.com) Particle Physics - lecture 11 21 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The generators of SU(Nc) form a lie Algebra:

[ta, tb] = ifabctc

where fabc are the structure constants of the group.The generators of thegroup are also normalised as:

tr[tatb] = TF δab

with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:

L(x) =∑

f

qif (x)(iγµ∂µ −mf )qjf (x)δij

where the sum runs over flavours and i and j are colour indices which runover colours.

(http://delenda.wordpress.com) Particle Physics - lecture 11 21 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The generators of SU(Nc) form a lie Algebra:

[ta, tb] = ifabctc

where fabc are the structure constants of the group.The generators of thegroup are also normalised as:

tr[tatb] = TF δab

with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:

L(x) =∑

f

qif (x)(iγµ∂µ −mf )qjf (x)δij

where the sum runs over flavours and i and j are colour indices which runover colours.

(http://delenda.wordpress.com) Particle Physics - lecture 11 21 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The generators of SU(Nc) form a lie Algebra:

[ta, tb] = ifabctc

where fabc are the structure constants of the group.The generators of thegroup are also normalised as:

tr[tatb] = TF δab

with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:

L(x) =∑

f

qif (x)(iγµ∂µ −mf )qjf (x)δij

where the sum runs over flavours and i and j are colour indices which runover colours.

(http://delenda.wordpress.com) Particle Physics - lecture 11 21 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The generators of SU(Nc) form a lie Algebra:

[ta, tb] = ifabctc

where fabc are the structure constants of the group.The generators of thegroup are also normalised as:

tr[tatb] = TF δab

with TF = TR = 1/2.Let us now construct the QCD Lagrangian using theSU(3) group.We would like to construct a quantum field theory for quarks starting fromthe free Dirac Lagrangian for quarks:

L(x) =∑

f

qif (x)(iγµ∂µ −mf )qjf (x)δij

where the sum runs over flavours and i and j are colour indices which runover colours.

(http://delenda.wordpress.com) Particle Physics - lecture 11 21 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:

qi(x)→ Uijqj(x)

where U = eigωata and g and ωa are constants.Note that the sum over a is

in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:

qi(x)→ q′i(x) = Uij(x)qj(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 22 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:

qi(x)→ Uijqj(x)

where U = eigωata and g and ωa are constants.Note that the sum over a is

in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:

qi(x)→ q′i(x) = Uij(x)qj(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 22 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:

qi(x)→ Uijqj(x)

where U = eigωata and g and ωa are constants.Note that the sum over a is

in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:

qi(x)→ q′i(x) = Uij(x)qj(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 22 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:

qi(x)→ Uijqj(x)

where U = eigωata and g and ωa are constants.Note that the sum over a is

in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:

qi(x)→ q′i(x) = Uij(x)qj(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 22 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:

qi(x)→ Uijqj(x)

where U = eigωata and g and ωa are constants.Note that the sum over a is

in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:

qi(x)→ q′i(x) = Uij(x)qj(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 22 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Although not stated there are also spinor indices for the each colouredquark field (which are columns of 4 entries)and the Dirac matrices whichare 4 by 4 matrices.This Lagrangian is invariant under the global SU(Nc)rotation:

qi(x)→ Uijqj(x)

where U = eigωata and g and ωa are constants.Note that the sum over a is

in the exponent.We can gauge this symmetry by requiring the Lagrangianbe invariant under local gauge transformations of the quark fields:

qi(x)→ q′i(x) = Uij(x)qj(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 22 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

where now U = eigωa(x)ta .So under this transformation the Lagrangian as

it stands is not invariant and acquires 8 new terms:

δL =∑

f

qkf (x)U †ki(x)(iγµ∂µU(x)jl)qlf (x)δij

To restore gauge invariance we must introduce 8 new vector fields Aaµ (8gluons),and let us define Aµ = Aaµt

a, so Aµ are N ×N matrices.Thesefields transform as Aµ → A′µ (to be determined).We also replace ordinaryderivatives with covariant derivatives:

Dµ(x) = 1∂µ − igAµ(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 23 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

where now U = eigωa(x)ta .So under this transformation the Lagrangian as

it stands is not invariant and acquires 8 new terms:

δL =∑

f

qkf (x)U †ki(x)(iγµ∂µU(x)jl)qlf (x)δij

To restore gauge invariance we must introduce 8 new vector fields Aaµ (8gluons),and let us define Aµ = Aaµt

a, so Aµ are N ×N matrices.Thesefields transform as Aµ → A′µ (to be determined).We also replace ordinaryderivatives with covariant derivatives:

Dµ(x) = 1∂µ − igAµ(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 23 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

where now U = eigωa(x)ta .So under this transformation the Lagrangian as

it stands is not invariant and acquires 8 new terms:

δL =∑

f

qkf (x)U †ki(x)(iγµ∂µU(x)jl)qlf (x)δij

To restore gauge invariance we must introduce 8 new vector fields Aaµ (8gluons),and let us define Aµ = Aaµt

a, so Aµ are N ×N matrices.Thesefields transform as Aµ → A′µ (to be determined).We also replace ordinaryderivatives with covariant derivatives:

Dµ(x) = 1∂µ − igAµ(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 23 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

where now U = eigωa(x)ta .So under this transformation the Lagrangian as

it stands is not invariant and acquires 8 new terms:

δL =∑

f

qkf (x)U †ki(x)(iγµ∂µU(x)jl)qlf (x)δij

To restore gauge invariance we must introduce 8 new vector fields Aaµ (8gluons),and let us define Aµ = Aaµt

a, so Aµ are N ×N matrices.Thesefields transform as Aµ → A′µ (to be determined).We also replace ordinaryderivatives with covariant derivatives:

Dµ(x) = 1∂µ − igAµ(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 23 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

where now U = eigωa(x)ta .So under this transformation the Lagrangian as

it stands is not invariant and acquires 8 new terms:

δL =∑

f

qkf (x)U †ki(x)(iγµ∂µU(x)jl)qlf (x)δij

To restore gauge invariance we must introduce 8 new vector fields Aaµ (8gluons),and let us define Aµ = Aaµt

a, so Aµ are N ×N matrices.Thesefields transform as Aµ → A′µ (to be determined).We also replace ordinaryderivatives with covariant derivatives:

Dµ(x) = 1∂µ − igAµ(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 23 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

where now U = eigωa(x)ta .So under this transformation the Lagrangian as

it stands is not invariant and acquires 8 new terms:

δL =∑

f

qkf (x)U †ki(x)(iγµ∂µU(x)jl)qlf (x)δij

To restore gauge invariance we must introduce 8 new vector fields Aaµ (8gluons),and let us define Aµ = Aaµt

a, so Aµ are N ×N matrices.Thesefields transform as Aµ → A′µ (to be determined).We also replace ordinaryderivatives with covariant derivatives:

Dµ(x) = 1∂µ − igAµ(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 23 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Let us thus just concentrate on the transformation of the termq(x)Dµq(x) of the Lagrangian:

q(x)(1∂µ − igAµ)q(x)→ q′(x)(1∂µ − igA′µ)q′(x)

So

q′(x)(1∂µ − igA′µ)q′(x) = q(x)U †(1∂µ − igA′µ)Uq(x)

= q(x)1∂µq(x) + q(x)U †(∂µU)q(x) + q(x)U †(−igA′µ)Uq(x)

To keep gauge invariance we force this to equalq(x)(1∂µ − igAµ)q(x).Hence we find the expression for A′µ

−igAµ = U †(∂µU) + U †(−igA′µ(x))U

hence we have the transformation for the fields as:

Aµ → A′µ = UAµU† − i

g(∂µU(x))U †

(http://delenda.wordpress.com) Particle Physics - lecture 11 24 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Let us thus just concentrate on the transformation of the termq(x)Dµq(x) of the Lagrangian:

q(x)(1∂µ − igAµ)q(x)→ q′(x)(1∂µ − igA′µ)q′(x)

So

q′(x)(1∂µ − igA′µ)q′(x) = q(x)U †(1∂µ − igA′µ)Uq(x)

= q(x)1∂µq(x) + q(x)U †(∂µU)q(x) + q(x)U †(−igA′µ)Uq(x)

To keep gauge invariance we force this to equalq(x)(1∂µ − igAµ)q(x).Hence we find the expression for A′µ

−igAµ = U †(∂µU) + U †(−igA′µ(x))U

hence we have the transformation for the fields as:

Aµ → A′µ = UAµU† − i

g(∂µU(x))U †

(http://delenda.wordpress.com) Particle Physics - lecture 11 24 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Let us thus just concentrate on the transformation of the termq(x)Dµq(x) of the Lagrangian:

q(x)(1∂µ − igAµ)q(x)→ q′(x)(1∂µ − igA′µ)q′(x)

So

q′(x)(1∂µ − igA′µ)q′(x) = q(x)U †(1∂µ − igA′µ)Uq(x)

= q(x)1∂µq(x) + q(x)U †(∂µU)q(x) + q(x)U †(−igA′µ)Uq(x)

To keep gauge invariance we force this to equalq(x)(1∂µ − igAµ)q(x).Hence we find the expression for A′µ

−igAµ = U †(∂µU) + U †(−igA′µ(x))U

hence we have the transformation for the fields as:

Aµ → A′µ = UAµU† − i

g(∂µU(x))U †

(http://delenda.wordpress.com) Particle Physics - lecture 11 24 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Let us thus just concentrate on the transformation of the termq(x)Dµq(x) of the Lagrangian:

q(x)(1∂µ − igAµ)q(x)→ q′(x)(1∂µ − igA′µ)q′(x)

So

q′(x)(1∂µ − igA′µ)q′(x) = q(x)U †(1∂µ − igA′µ)Uq(x)

= q(x)1∂µq(x) + q(x)U †(∂µU)q(x) + q(x)U †(−igA′µ)Uq(x)

To keep gauge invariance we force this to equalq(x)(1∂µ − igAµ)q(x).Hence we find the expression for A′µ

−igAµ = U †(∂µU) + U †(−igA′µ(x))U

hence we have the transformation for the fields as:

Aµ → A′µ = UAµU† − i

g(∂µU(x))U †

(http://delenda.wordpress.com) Particle Physics - lecture 11 24 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Let us thus just concentrate on the transformation of the termq(x)Dµq(x) of the Lagrangian:

q(x)(1∂µ − igAµ)q(x)→ q′(x)(1∂µ − igA′µ)q′(x)

So

q′(x)(1∂µ − igA′µ)q′(x) = q(x)U †(1∂µ − igA′µ)Uq(x)

= q(x)1∂µq(x) + q(x)U †(∂µU)q(x) + q(x)U †(−igA′µ)Uq(x)

To keep gauge invariance we force this to equalq(x)(1∂µ − igAµ)q(x).Hence we find the expression for A′µ

−igAµ = U †(∂µU) + U †(−igA′µ(x))U

hence we have the transformation for the fields as:

Aµ → A′µ = UAµU† − i

g(∂µU(x))U †

(http://delenda.wordpress.com) Particle Physics - lecture 11 24 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Let us thus just concentrate on the transformation of the termq(x)Dµq(x) of the Lagrangian:

q(x)(1∂µ − igAµ)q(x)→ q′(x)(1∂µ − igA′µ)q′(x)

So

q′(x)(1∂µ − igA′µ)q′(x) = q(x)U †(1∂µ − igA′µ)Uq(x)

= q(x)1∂µq(x) + q(x)U †(∂µU)q(x) + q(x)U †(−igA′µ)Uq(x)

To keep gauge invariance we force this to equalq(x)(1∂µ − igAµ)q(x).Hence we find the expression for A′µ

−igAµ = U †(∂µU) + U †(−igA′µ(x))U

hence we have the transformation for the fields as:

Aµ → A′µ = UAµU† − i

g(∂µU(x))U †

(http://delenda.wordpress.com) Particle Physics - lecture 11 24 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Let us thus just concentrate on the transformation of the termq(x)Dµq(x) of the Lagrangian:

q(x)(1∂µ − igAµ)q(x)→ q′(x)(1∂µ − igA′µ)q′(x)

So

q′(x)(1∂µ − igA′µ)q′(x) = q(x)U †(1∂µ − igA′µ)Uq(x)

= q(x)1∂µq(x) + q(x)U †(∂µU)q(x) + q(x)U †(−igA′µ)Uq(x)

To keep gauge invariance we force this to equalq(x)(1∂µ − igAµ)q(x).Hence we find the expression for A′µ

−igAµ = U †(∂µU) + U †(−igA′µ(x))U

hence we have the transformation for the fields as:

Aµ → A′µ = UAµU† − i

g(∂µU(x))U †

(http://delenda.wordpress.com) Particle Physics - lecture 11 24 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

But how does Aaµ transform (recall Aµ = Aaµta).To find the generators of

the representation in which the gluons transform it suffices to considerinfinitesimal rotations:

U = 1 + igωc(x)tc

Then

Aaµta → Aa

′µ t

a =(1 + igωata)Abµtb(1− igωctc)−

− i

g[∂µ(1 + igωa(x)ta)][1− igωd(x)td]

Expanding to O(ωa) and simplifying and using the Lie algebra[ta, tb] = ifabctc we find:

Aaµta → Aa

′µ t

a = Abµtb + igωaAbµ(tatb − tbta) + ta∂µω

a(x)

= Abµtb − gωaAbµfabctc + ta∂µω

a(x) = Aaµta + gωcAbµf

abcta + ta∂µωa(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 25 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

But how does Aaµ transform (recall Aµ = Aaµta).To find the generators of

the representation in which the gluons transform it suffices to considerinfinitesimal rotations:

U = 1 + igωc(x)tc

Then

Aaµta → Aa

′µ t

a =(1 + igωata)Abµtb(1− igωctc)−

− i

g[∂µ(1 + igωa(x)ta)][1− igωd(x)td]

Expanding to O(ωa) and simplifying and using the Lie algebra[ta, tb] = ifabctc we find:

Aaµta → Aa

′µ t

a = Abµtb + igωaAbµ(tatb − tbta) + ta∂µω

a(x)

= Abµtb − gωaAbµfabctc + ta∂µω

a(x) = Aaµta + gωcAbµf

abcta + ta∂µωa(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 25 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

But how does Aaµ transform (recall Aµ = Aaµta).To find the generators of

the representation in which the gluons transform it suffices to considerinfinitesimal rotations:

U = 1 + igωc(x)tc

Then

Aaµta → Aa

′µ t

a =(1 + igωata)Abµtb(1− igωctc)−

− i

g[∂µ(1 + igωa(x)ta)][1− igωd(x)td]

Expanding to O(ωa) and simplifying and using the Lie algebra[ta, tb] = ifabctc we find:

Aaµta → Aa

′µ t

a = Abµtb + igωaAbµ(tatb − tbta) + ta∂µω

a(x)

= Abµtb − gωaAbµfabctc + ta∂µω

a(x) = Aaµta + gωcAbµf

abcta + ta∂µωa(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 25 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

But how does Aaµ transform (recall Aµ = Aaµta).To find the generators of

the representation in which the gluons transform it suffices to considerinfinitesimal rotations:

U = 1 + igωc(x)tc

Then

Aaµta → Aa

′µ t

a =(1 + igωata)Abµtb(1− igωctc)−

− i

g[∂µ(1 + igωa(x)ta)][1− igωd(x)td]

Expanding to O(ωa) and simplifying and using the Lie algebra[ta, tb] = ifabctc we find:

Aaµta → Aa

′µ t

a = Abµtb + igωaAbµ(tatb − tbta) + ta∂µω

a(x)

= Abµtb − gωaAbµfabctc + ta∂µω

a(x) = Aaµta + gωcAbµf

abcta + ta∂µωa(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 25 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

But how does Aaµ transform (recall Aµ = Aaµta).To find the generators of

the representation in which the gluons transform it suffices to considerinfinitesimal rotations:

U = 1 + igωc(x)tc

Then

Aaµta → Aa

′µ t

a =(1 + igωata)Abµtb(1− igωctc)−

− i

g[∂µ(1 + igωa(x)ta)][1− igωd(x)td]

Expanding to O(ωa) and simplifying and using the Lie algebra[ta, tb] = ifabctc we find:

Aaµta → Aa

′µ t

a = Abµtb + igωaAbµ(tatb − tbta) + ta∂µω

a(x)

= Abµtb − gωaAbµfabctc + ta∂µω

a(x) = Aaµta + gωcAbµf

abcta + ta∂µωa(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 25 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

But how does Aaµ transform (recall Aµ = Aaµta).To find the generators of

the representation in which the gluons transform it suffices to considerinfinitesimal rotations:

U = 1 + igωc(x)tc

Then

Aaµta → Aa

′µ t

a =(1 + igωata)Abµtb(1− igωctc)−

− i

g[∂µ(1 + igωa(x)ta)][1− igωd(x)td]

Expanding to O(ωa) and simplifying and using the Lie algebra[ta, tb] = ifabctc we find:

Aaµta → Aa

′µ t

a = Abµtb + igωaAbµ(tatb − tbta) + ta∂µω

a(x)

= Abµtb − gωaAbµfabctc + ta∂µω

a(x) = Aaµta + gωcAbµf

abcta + ta∂µωa(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 25 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

But how does Aaµ transform (recall Aµ = Aaµta).To find the generators of

the representation in which the gluons transform it suffices to considerinfinitesimal rotations:

U = 1 + igωc(x)tc

Then

Aaµta → Aa

′µ t

a =(1 + igωata)Abµtb(1− igωctc)−

− i

g[∂µ(1 + igωa(x)ta)][1− igωd(x)td]

Expanding to O(ωa) and simplifying and using the Lie algebra[ta, tb] = ifabctc we find:

Aaµta → Aa

′µ t

a = Abµtb + igωaAbµ(tatb − tbta) + ta∂µω

a(x)

= Abµtb − gωaAbµfabctc + ta∂µω

a(x) = Aaµta + gωcAbµf

abcta + ta∂µωa(x)

(http://delenda.wordpress.com) Particle Physics - lecture 11 25 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and we used the freedom to relabel indices and that the structureconstants are totally anti-symmetric.So we can write:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc(x) + ∂µω

a(x)

After a bit of Algebra we can cast this into:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) +Dab

µ ωb(x)

where the covariant derivative acting on the “adjoint” representation is:

Dabµ = δab∂µ − ig(T c)abAcµ

where (T c)ab = if cba.

(http://delenda.wordpress.com) Particle Physics - lecture 11 26 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and we used the freedom to relabel indices and that the structureconstants are totally anti-symmetric.So we can write:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc(x) + ∂µω

a(x)

After a bit of Algebra we can cast this into:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) +Dab

µ ωb(x)

where the covariant derivative acting on the “adjoint” representation is:

Dabµ = δab∂µ − ig(T c)abAcµ

where (T c)ab = if cba.

(http://delenda.wordpress.com) Particle Physics - lecture 11 26 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and we used the freedom to relabel indices and that the structureconstants are totally anti-symmetric.So we can write:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc(x) + ∂µω

a(x)

After a bit of Algebra we can cast this into:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) +Dab

µ ωb(x)

where the covariant derivative acting on the “adjoint” representation is:

Dabµ = δab∂µ − ig(T c)abAcµ

where (T c)ab = if cba.

(http://delenda.wordpress.com) Particle Physics - lecture 11 26 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and we used the freedom to relabel indices and that the structureconstants are totally anti-symmetric.So we can write:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc(x) + ∂µω

a(x)

After a bit of Algebra we can cast this into:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) +Dab

µ ωb(x)

where the covariant derivative acting on the “adjoint” representation is:

Dabµ = δab∂µ − ig(T c)abAcµ

where (T c)ab = if cba.

(http://delenda.wordpress.com) Particle Physics - lecture 11 26 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and we used the freedom to relabel indices and that the structureconstants are totally anti-symmetric.So we can write:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc(x) + ∂µω

a(x)

After a bit of Algebra we can cast this into:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) +Dab

µ ωb(x)

where the covariant derivative acting on the “adjoint” representation is:

Dabµ = δab∂µ − ig(T c)abAcµ

where (T c)ab = if cba.

(http://delenda.wordpress.com) Particle Physics - lecture 11 26 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Particularly if ωa are constants we get:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc

= Aaµ(x) + gf bacAcµ(x)ωb

which is the expansion of the full rotation:

Aaµ(x)→ Aa′µ (x) = (e−igT

bωb)acA

cµ(x)

(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).

(http://delenda.wordpress.com) Particle Physics - lecture 11 27 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Particularly if ωa are constants we get:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc

= Aaµ(x) + gf bacAcµ(x)ωb

which is the expansion of the full rotation:

Aaµ(x)→ Aa′µ (x) = (e−igT

bωb)acA

cµ(x)

(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).

(http://delenda.wordpress.com) Particle Physics - lecture 11 27 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Particularly if ωa are constants we get:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc

= Aaµ(x) + gf bacAcµ(x)ωb

which is the expansion of the full rotation:

Aaµ(x)→ Aa′µ (x) = (e−igT

bωb)acA

cµ(x)

(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).

(http://delenda.wordpress.com) Particle Physics - lecture 11 27 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Particularly if ωa are constants we get:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc

= Aaµ(x) + gf bacAcµ(x)ωb

which is the expansion of the full rotation:

Aaµ(x)→ Aa′µ (x) = (e−igT

bωb)acA

cµ(x)

(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).

(http://delenda.wordpress.com) Particle Physics - lecture 11 27 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Particularly if ωa are constants we get:

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ωc

= Aaµ(x) + gf bacAcµ(x)ωb

which is the expansion of the full rotation:

Aaµ(x)→ Aa′µ (x) = (e−igT

bωb)acA

cµ(x)

(check by expansion). So we have written an infinitesimal transformationfor the gluon fields that looks like that for the quark fields except that thegenerators are T c,related to the structure constants of the “fundamentalrepresentation” (the one defining the quark transformations).

(http://delenda.wordpress.com) Particle Physics - lecture 11 27 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.

(http://delenda.wordpress.com) Particle Physics - lecture 11 28 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.

(http://delenda.wordpress.com) Particle Physics - lecture 11 28 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.

(http://delenda.wordpress.com) Particle Physics - lecture 11 28 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.

(http://delenda.wordpress.com) Particle Physics - lecture 11 28 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

The structure constants thus form what is called the “adjointrepresentation” of SU(N) and the generators are eight-dimensionalmatrices that have the same Lie Algebra: [T a, T b] = ifabcT c.So quarks transform in the fundamental representation whose generatorsare ta,while gluons transform in the adjoint representation whosegenerators are the structure constants (T a)bc = fabc.So we have now imposed the invariance of the free quark Lagrangian andintroduced a new photon-like field into the Lagrangian.This new field alsomust satisfy a corresponding Euler-Lagrange equation.

(http://delenda.wordpress.com) Particle Physics - lecture 11 28 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:

F aµν = ∂µAaν − ∂νAaµ

However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:

Fµν = DµAν −DµAν = ∂µAν − ∂νAµ − ig[Aµ, Aν ]

and since Aµ = Aaµta we define F aµνt

a = Fµν .Thus we write:

F aµν = ∂µAaν − ∂νAaµ + gfabcAbµA

cν

(http://delenda.wordpress.com) Particle Physics - lecture 11 29 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:

F aµν = ∂µAaν − ∂νAaµ

However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:

Fµν = DµAν −DµAν = ∂µAν − ∂νAµ − ig[Aµ, Aν ]

and since Aµ = Aaµta we define F aµνt

a = Fµν .Thus we write:

F aµν = ∂µAaν − ∂νAaµ + gfabcAbµA

cν

(http://delenda.wordpress.com) Particle Physics - lecture 11 29 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:

F aµν = ∂µAaν − ∂νAaµ

However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:

Fµν = DµAν −DµAν = ∂µAν − ∂νAµ − ig[Aµ, Aν ]

and since Aµ = Aaµta we define F aµνt

a = Fµν .Thus we write:

F aµν = ∂µAaν − ∂νAaµ + gfabcAbµA

cν

(http://delenda.wordpress.com) Particle Physics - lecture 11 29 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:

F aµν = ∂µAaν − ∂νAaµ

However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:

Fµν = DµAν −DµAν = ∂µAν − ∂νAµ − ig[Aµ, Aν ]

and since Aµ = Aaµta we define F aµνt

a = Fµν .Thus we write:

F aµν = ∂µAaν − ∂νAaµ + gfabcAbµA

cν

(http://delenda.wordpress.com) Particle Physics - lecture 11 29 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:

F aµν = ∂µAaν − ∂νAaµ

However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:

Fµν = DµAν −DµAν = ∂µAν − ∂νAµ − ig[Aµ, Aν ]

and since Aµ = Aaµta we define F aµνt

a = Fµν .Thus we write:

F aµν = ∂µAaν − ∂νAaµ + gfabcAbµA

cν

(http://delenda.wordpress.com) Particle Physics - lecture 11 29 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

From our experience with the photons we must construct the “kinetic”term for the gluon field from the field strength tensor:

F aµν = ∂µAaν − ∂νAaµ

However we immediately hit a problem that this is not gauge invariant(check this)and thus does not correspond to a physical observable.Tomake it gauge invariant we aught to promote the derivatives to covariantderivatives and express the field strength tensor as:

Fµν = DµAν −DµAν = ∂µAν − ∂νAµ − ig[Aµ, Aν ]

and since Aµ = Aaµta we define F aµνt

a = Fµν .Thus we write:

F aµν = ∂µAaν − ∂νAaµ + gfabcAbµA

cν

(http://delenda.wordpress.com) Particle Physics - lecture 11 29 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).

We can thus express the full QCD Lagrangian as:

LQCD =∑

f

qif (δijiγµ∂µ+gsγ

µAaµtaij−mfδij)q

jf−

1

4F aµνF

µν,a+Lgauge+Lghost

and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.

(http://delenda.wordpress.com) Particle Physics - lecture 11 30 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).

We can thus express the full QCD Lagrangian as:

LQCD =∑

f

qif (δijiγµ∂µ+gsγ

µAaµtaij−mfδij)q

jf−

1

4F aµνF

µν,a+Lgauge+Lghost

and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.

(http://delenda.wordpress.com) Particle Physics - lecture 11 30 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).

We can thus express the full QCD Lagrangian as:

LQCD =∑

f

qif (δijiγµ∂µ+gsγ

µAaµtaij−mfδij)q

jf−

1

4F aµνF

µν,a+Lgauge+Lghost

and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.

(http://delenda.wordpress.com) Particle Physics - lecture 11 30 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).

We can thus express the full QCD Lagrangian as:

LQCD =∑

f

qif (δijiγµ∂µ+gsγ

µAaµtaij−mfδij)q

jf−

1

4F aµνF

µν,a+Lgauge+Lghost

and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.

(http://delenda.wordpress.com) Particle Physics - lecture 11 30 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).

We can thus express the full QCD Lagrangian as:

LQCD =∑

f

qif (δijiγµ∂µ+gsγ

µAaµtaij−mfδij)q

jf−

1

4F aµνF

µν,a+Lgauge+Lghost

and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.

(http://delenda.wordpress.com) Particle Physics - lecture 11 30 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).

We can thus express the full QCD Lagrangian as:

LQCD =∑

f

qif (δijiγµ∂µ+gsγ

µAaµtaij−mfδij)q

jf−

1

4F aµνF

µν,a+Lgauge+Lghost

and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.

(http://delenda.wordpress.com) Particle Physics - lecture 11 30 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

and the term F aµνFµνa is gauge invariant in the Lagrangian (check this).

We can thus express the full QCD Lagrangian as:

LQCD =∑

f

qif (δijiγµ∂µ+gsγ

µAaµtaij−mfδij)q

jf−

1

4F aµνF

µν,a+Lgauge+Lghost

and we have suppressed spinor indexes (contained in γ matrices andspinors).The gauge fixing part is identical to that for QED and is necessaryto invert the gluon propagator.The ghost term is added to removeunphysical polarisation states of the gluons,and this is beyond the scope ofthis course.This Lagrangian is invariant under the simultaneous gauge transformationsof the quark and gluon fields.

(http://delenda.wordpress.com) Particle Physics - lecture 11 30 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA

cν as well as AaµA

bνA

µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.

(http://delenda.wordpress.com) Particle Physics - lecture 11 31 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA

cν as well as AaµA

bνA

µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.

(http://delenda.wordpress.com) Particle Physics - lecture 11 31 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA

cν as well as AaµA

bνA

µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.

(http://delenda.wordpress.com) Particle Physics - lecture 11 31 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA

cν as well as AaµA

bνA

µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.

(http://delenda.wordpress.com) Particle Physics - lecture 11 31 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA

cν as well as AaµA

bνA

µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.

(http://delenda.wordpress.com) Particle Physics - lecture 11 31 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA

cν as well as AaµA

bνA

µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.

(http://delenda.wordpress.com) Particle Physics - lecture 11 31 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

If we expand out the kinetic term we notice the existence of the terms∂µAν,aAbµA

cν as well as AaµA

bνA

µcAνb in the gauge invariantLagrangian.This is a crucial result.In contrast to QED where the structureconstants vanish (since the generators commute, as there is only onegenerator, thus it is an Abelian group),there is no term of the above formsin the Lagrangian,so there are no photon self-interactions.However with QCD, being non-Abelian, the structure constants do notvanish in general because the generators do not commute (non-Abeliangroup),which means from the terms proportional to the structureconstants that gluon self-interactions are allowed.

(http://delenda.wordpress.com) Particle Physics - lecture 11 31 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.

(http://delenda.wordpress.com) Particle Physics - lecture 11 32 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.

(http://delenda.wordpress.com) Particle Physics - lecture 11 32 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.

(http://delenda.wordpress.com) Particle Physics - lecture 11 32 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.

(http://delenda.wordpress.com) Particle Physics - lecture 11 32 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.

(http://delenda.wordpress.com) Particle Physics - lecture 11 32 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.

(http://delenda.wordpress.com) Particle Physics - lecture 11 32 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.

(http://delenda.wordpress.com) Particle Physics - lecture 11 32 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Having such allowed vertices is a crucial property for QCD whichdistinguishes it from QED.The tripple and quadric gluon vertices lead tothe running of the QCD coupling αs.At high energy scales the coupling issmall and perturbative calculations are valid.This is asymptotic freedom.Atlow energy scales the coupling diverges and perturbation theory becomesnot valid. This is confinement.With QED the situation is opposite,thecoupling grows larger at higher energies.

(http://delenda.wordpress.com) Particle Physics - lecture 11 32 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

(http://delenda.wordpress.com) Particle Physics - lecture 11 33 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

We can deduce Feynman rules from the QCD Lagrangian.

u(p)

u(p)

v(p)

v(p)

ǫaµ(λ, p)

ǫ∗aµ (λ, p)

i/(γµpµ −m)

−i(gµν − (1− ξ)pµpν/p2

)/p2

−igsγµta +igsγµta

gsfabc(gµν(k1 − k2)

ρ + gνρ(k2 − k3)µ

+gρµ(k3 − k1)ν)

k1, a, µ k3, c, ρ

k2, b, ν

All momenta incoming

−ig2s(fabcf cde(gµρgνσ − gµσgνρ))−ig2s(facef bde(gµνgρσ − gµσgνρ))−ig2s(fadef bce(gµνgρσ − gµρgνσ))

a, µ b, ν

c, ρ d, σ

a, µ

a, µ

(http://delenda.wordpress.com) Particle Physics - lecture 11 34 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Finally we note that the generators satisfy the following algebra:

∑

a

taijtajk = δikCF

tr(T aT b) = CAδab

where CF = (N2c − 1)/(2Nc) = 4/3 and CA = Nc = 3 are the casimir

operators in the fundamental and adjoint representations respectively,theyrepresent the colour factors for the emission of a gluon from a quark, andof a gluon from a gluon.

(http://delenda.wordpress.com) Particle Physics - lecture 11 35 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Finally we note that the generators satisfy the following algebra:

∑

a

taijtajk = δikCF

tr(T aT b) = CAδab

where CF = (N2c − 1)/(2Nc) = 4/3 and CA = Nc = 3 are the casimir

operators in the fundamental and adjoint representations respectively,theyrepresent the colour factors for the emission of a gluon from a quark, andof a gluon from a gluon.

(http://delenda.wordpress.com) Particle Physics - lecture 11 35 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Finally we note that the generators satisfy the following algebra:

∑

a

taijtajk = δikCF

tr(T aT b) = CAδab

where CF = (N2c − 1)/(2Nc) = 4/3 and CA = Nc = 3 are the casimir

operators in the fundamental and adjoint representations respectively,theyrepresent the colour factors for the emission of a gluon from a quark, andof a gluon from a gluon.

(http://delenda.wordpress.com) Particle Physics - lecture 11 35 / 35

Non-Abelian gauge invariance SU(3) and QCD

Non-Abelian gauge invarianceSU(3) and QCD

Finally we note that the generators satisfy the following algebra:

∑

a

taijtajk = δikCF

tr(T aT b) = CAδab

where CF = (N2c − 1)/(2Nc) = 4/3 and CA = Nc = 3 are the casimir

operators in the fundamental and adjoint representations respectively,theyrepresent the colour factors for the emission of a gluon from a quark, andof a gluon from a gluon.

(http://delenda.wordpress.com) Particle Physics - lecture 11 35 / 35