Weak Interactions

Particle Physicslecture 4

-Weak Interactions (Continued)

Yazid Delenda

Departement des Sciences de la matiereFaculte des Sciences - UHLB

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

Batna, 12 October 2014

1/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

How do we build a theory of weak interactions with parityviolation? The most general form of the matrix element we canwrite is:

M∝ (uψf Ouψi)(uφ,f Ouφi) (1)

where O is a combination of γ matrices. It turns out that there areonly 5 bilinear covariant expressions that can be formed by thegamma matrices:

Name Symb Current N0 of comps Effect under parity

Scalar S ψψ 1 +

Vector V ψγµψ 4 (+,−,−,−)

Tensor T ψσµνψ 6

Axial-vector A ψγµγ5ψ 4 (+,+,+,+)

Pseudo-Scalar P ψγ5ψ 1 −

2/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

How do we build a theory of weak interactions with parityviolation? The most general form of the matrix element we canwrite is:

M∝ (uψf Ouψi)(uφ,f Ouφi) (1)

where O is a combination of γ matrices. It turns out that there areonly 5 bilinear covariant expressions that can be formed by thegamma matrices:

Name Symb Current N0 of comps Effect under parity

Scalar S ψψ 1 +

Vector V ψγµψ 4 (+,−,−,−)

Tensor T ψσµνψ 6

Axial-vector A ψγµγ5ψ 4 (+,+,+,+)

Pseudo-Scalar P ψγ5ψ 1 −

2/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

How do we build a theory of weak interactions with parityviolation? The most general form of the matrix element we canwrite is:

M∝ (uψf Ouψi)(uφ,f Ouφi) (1)

where O is a combination of γ matrices. It turns out that there areonly 5 bilinear covariant expressions that can be formed by thegamma matrices:

Name Symb Current N0 of comps Effect under parity

Scalar S ψψ 1 +

Vector V ψγµψ 4 (+,−,−,−)

Tensor T ψσµνψ 6

Axial-vector A ψγµγ5ψ 4 (+,+,+,+)

Pseudo-Scalar P ψγ5ψ 1 −

2/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

How do we build a theory of weak interactions with parityviolation? The most general form of the matrix element we canwrite is:

M∝ (uψf Ouψi)(uφ,f Ouφi) (1)

where O is a combination of γ matrices. It turns out that there areonly 5 bilinear covariant expressions that can be formed by thegamma matrices:

Name Symb Current N0 of comps Effect under parity

Scalar S ψψ 1 +

Vector V ψγµψ 4 (+,−,−,−)

Tensor T ψσµνψ 6

Axial-vector A ψγµγ5ψ 4 (+,+,+,+)

Pseudo-Scalar P ψγ5ψ 1 −

2/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

where σµν = i(γµγν − γνγµ)/2.You can show, for example, that the vector current ψγµψtransforms under the parity operation:

P =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

on the coordinates as shown in the table.

3/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

where σµν = i(γµγν − γνγµ)/2.You can show, for example, that the vector current ψγµψtransforms under the parity operation:

P =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

on the coordinates as shown in the table.

3/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

A series of experiments through the end of the 1950’s lead to anew form of the effective weak interaction:

Mβ =GF√

2[unγ

µ(1− γ5)up] [uνeγµ(1− γ5)ue]

The factor 1/√

2, introduced for historical reasons, maintains thevalue of the Fermi constant GF . The uγµu and uγµγ5u transform,under Lorentz transformations of the coordinates, respectively as avector (V) and an axial vector (A)1:

ψγµψ → Λµνψγνψ, vector transformation

ψγµγ5ψ → det(Λ)Λµνψγνγ5ψ, Axial vector transformation

from which the name V-A.1They are called so since a 4-vector under parity flips spacial components

(e.g. momentum) while axial vector (pseudovector) is unchanged, e.g. angularmomentum. Under parity these components change as described.

4/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

A series of experiments through the end of the 1950’s lead to anew form of the effective weak interaction:

Mβ =GF√

2[unγ

µ(1− γ5)up] [uνeγµ(1− γ5)ue]

The factor 1/√

2, introduced for historical reasons, maintains thevalue of the Fermi constant GF . The uγµu and uγµγ5u transform,under Lorentz transformations of the coordinates, respectively as avector (V) and an axial vector (A)1:

ψγµψ → Λµνψγνψ, vector transformation

ψγµγ5ψ → det(Λ)Λµνψγνγ5ψ, Axial vector transformation

from which the name V-A.1They are called so since a 4-vector under parity flips spacial components

(e.g. momentum) while axial vector (pseudovector) is unchanged, e.g. angularmomentum. Under parity these components change as described.

4/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

A series of experiments through the end of the 1950’s lead to anew form of the effective weak interaction:

Mβ =GF√

2[unγ

µ(1− γ5)up] [uνeγµ(1− γ5)ue]

The factor 1/√

2, introduced for historical reasons, maintains thevalue of the Fermi constant GF . The uγµu and uγµγ5u transform,under Lorentz transformations of the coordinates, respectively as avector (V) and an axial vector (A)1:

ψγµψ → Λµνψγνψ, vector transformation

ψγµγ5ψ → det(Λ)Λµνψγνγ5ψ, Axial vector transformation

from which the name V-A.1They are called so since a 4-vector under parity flips spacial components

(e.g. momentum) while axial vector (pseudovector) is unchanged, e.g. angularmomentum. Under parity these components change as described.

4/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

A series of experiments through the end of the 1950’s lead to anew form of the effective weak interaction:

Mβ =GF√

2[unγ

µ(1− γ5)up] [uνeγµ(1− γ5)ue]

The factor 1/√

2, introduced for historical reasons, maintains thevalue of the Fermi constant GF . The uγµu and uγµγ5u transform,under Lorentz transformations of the coordinates, respectively as avector (V) and an axial vector (A)1:

ψγµψ → Λµνψγνψ, vector transformation

ψγµγ5ψ → det(Λ)Λµνψγνγ5ψ, Axial vector transformation

from which the name V-A.1They are called so since a 4-vector under parity flips spacial components

(e.g. momentum) while axial vector (pseudovector) is unchanged, e.g. angularmomentum. Under parity these components change as described.

4/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

Parity violation comes from the fact that the behaviour of thevector and axial vector currents under a parity transformation aredifferent. As you can see from the table, the vector current flipssign under parity whereas the axial vector does not.Theinterference between these two terms creates the parityviolation.One can see this schematically by remembering that whatwe observe is usually the square of the amplitude.Suppose theamplitude is pure V-A. Then:

|M|2 ∼ (V −A)(V −A) = V V +AA− 2AV

If we apply a parity transformation then the sign of the V termflips, but the sign of the A term does not:

P{|M|2} ∼ (−V )(−V ) +AA− 2A(−V ) = V V +AA+ 2AV

so we see that the cross-term has flipped sign.5/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

Parity violation comes from the fact that the behaviour of thevector and axial vector currents under a parity transformation aredifferent. As you can see from the table, the vector current flipssign under parity whereas the axial vector does not.Theinterference between these two terms creates the parityviolation.One can see this schematically by remembering that whatwe observe is usually the square of the amplitude.Suppose theamplitude is pure V-A. Then:

|M|2 ∼ (V −A)(V −A) = V V +AA− 2AV

If we apply a parity transformation then the sign of the V termflips, but the sign of the A term does not:

P{|M|2} ∼ (−V )(−V ) +AA− 2A(−V ) = V V +AA+ 2AV

so we see that the cross-term has flipped sign.5/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

Parity violation comes from the fact that the behaviour of thevector and axial vector currents under a parity transformation aredifferent. As you can see from the table, the vector current flipssign under parity whereas the axial vector does not.Theinterference between these two terms creates the parityviolation.One can see this schematically by remembering that whatwe observe is usually the square of the amplitude.Suppose theamplitude is pure V-A. Then:

|M|2 ∼ (V −A)(V −A) = V V +AA− 2AV

If we apply a parity transformation then the sign of the V termflips, but the sign of the A term does not:

P{|M|2} ∼ (−V )(−V ) +AA− 2A(−V ) = V V +AA+ 2AV

so we see that the cross-term has flipped sign.5/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

Parity violation comes from the fact that the behaviour of thevector and axial vector currents under a parity transformation aredifferent. As you can see from the table, the vector current flipssign under parity whereas the axial vector does not.Theinterference between these two terms creates the parityviolation.One can see this schematically by remembering that whatwe observe is usually the square of the amplitude.Suppose theamplitude is pure V-A. Then:

|M|2 ∼ (V −A)(V −A) = V V +AA− 2AV

If we apply a parity transformation then the sign of the V termflips, but the sign of the A term does not:

P{|M|2} ∼ (−V )(−V ) +AA− 2A(−V ) = V V +AA+ 2AV

so we see that the cross-term has flipped sign.5/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

Parity violation comes from the fact that the behaviour of thevector and axial vector currents under a parity transformation aredifferent. As you can see from the table, the vector current flipssign under parity whereas the axial vector does not.Theinterference between these two terms creates the parityviolation.One can see this schematically by remembering that whatwe observe is usually the square of the amplitude.Suppose theamplitude is pure V-A. Then:

|M|2 ∼ (V −A)(V −A) = V V +AA− 2AV

If we apply a parity transformation then the sign of the V termflips, but the sign of the A term does not:

P{|M|2} ∼ (−V )(−V ) +AA− 2A(−V ) = V V +AA+ 2AV

so we see that the cross-term has flipped sign.5/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

Parity violation comes from the fact that the behaviour of thevector and axial vector currents under a parity transformation aredifferent. As you can see from the table, the vector current flipssign under parity whereas the axial vector does not.Theinterference between these two terms creates the parityviolation.One can see this schematically by remembering that whatwe observe is usually the square of the amplitude.Suppose theamplitude is pure V-A. Then:

|M|2 ∼ (V −A)(V −A) = V V +AA− 2AV

If we apply a parity transformation then the sign of the V termflips, but the sign of the A term does not:

P{|M|2} ∼ (−V )(−V ) +AA− 2A(−V ) = V V +AA+ 2AV

so we see that the cross-term has flipped sign.5/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

Parity violation comes from the fact that the behaviour of thevector and axial vector currents under a parity transformation aredifferent. As you can see from the table, the vector current flipssign under parity whereas the axial vector does not.Theinterference between these two terms creates the parityviolation.One can see this schematically by remembering that whatwe observe is usually the square of the amplitude.Suppose theamplitude is pure V-A. Then:

|M|2 ∼ (V −A)(V −A) = V V +AA− 2AV

If we apply a parity transformation then the sign of the V termflips, but the sign of the A term does not:

P{|M|2} ∼ (−V )(−V ) +AA− 2A(−V ) = V V +AA+ 2AV

so we see that the cross-term has flipped sign.5/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

Parity violation comes from the fact that the behaviour of thevector and axial vector currents under a parity transformation aredifferent. As you can see from the table, the vector current flipssign under parity whereas the axial vector does not.Theinterference between these two terms creates the parityviolation.One can see this schematically by remembering that whatwe observe is usually the square of the amplitude.Suppose theamplitude is pure V-A. Then:

|M|2 ∼ (V −A)(V −A) = V V +AA− 2AV

If we apply a parity transformation then the sign of the V termflips, but the sign of the A term does not:

P{|M|2} ∼ (−V )(−V ) +AA− 2A(−V ) = V V +AA+ 2AV

so we see that the cross-term has flipped sign.5/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The V-A interaction actually violates parity maximally as bothcurrents have the same strength.Parity is not just violated in asmall percentage of interactions, it is violated in all of them. Theform of the interaction suggests that we put it in the form of acurrent-current interaction in analogy with electromagnetism.Wewrite the effective lagrangian as:

L =GF√

2J+µ (x)Jµ+(x) + h.c.

whereJ+µ = (νee

−)µ + (pn)µ

where for instance:

(νee−)µ = ψ(νe)γµ(1− γ5)ψ(e−)

6/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The V-A interaction actually violates parity maximally as bothcurrents have the same strength.Parity is not just violated in asmall percentage of interactions, it is violated in all of them. Theform of the interaction suggests that we put it in the form of acurrent-current interaction in analogy with electromagnetism.Wewrite the effective lagrangian as:

L =GF√

2J+µ (x)Jµ+(x) + h.c.

whereJ+µ = (νee

−)µ + (pn)µ

where for instance:

(νee−)µ = ψ(νe)γµ(1− γ5)ψ(e−)

6/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The V-A interaction actually violates parity maximally as bothcurrents have the same strength.Parity is not just violated in asmall percentage of interactions, it is violated in all of them. Theform of the interaction suggests that we put it in the form of acurrent-current interaction in analogy with electromagnetism.Wewrite the effective lagrangian as:

L =GF√

2J+µ (x)Jµ+(x) + h.c.

whereJ+µ = (νee

−)µ + (pn)µ

where for instance:

(νee−)µ = ψ(νe)γµ(1− γ5)ψ(e−)

6/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The V-A interaction actually violates parity maximally as bothcurrents have the same strength.Parity is not just violated in asmall percentage of interactions, it is violated in all of them. Theform of the interaction suggests that we put it in the form of acurrent-current interaction in analogy with electromagnetism.Wewrite the effective lagrangian as:

L =GF√

2J+µ (x)Jµ+(x) + h.c.

whereJ+µ = (νee

−)µ + (pn)µ

where for instance:

(νee−)µ = ψ(νe)γµ(1− γ5)ψ(e−)

6/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The V-A interaction actually violates parity maximally as bothcurrents have the same strength.Parity is not just violated in asmall percentage of interactions, it is violated in all of them. Theform of the interaction suggests that we put it in the form of acurrent-current interaction in analogy with electromagnetism.Wewrite the effective lagrangian as:

L =GF√

2J+µ (x)Jµ+(x) + h.c.

whereJ+µ = (νee

−)µ + (pn)µ

where for instance:

(νee−)µ = ψ(νe)γµ(1− γ5)ψ(e−)

6/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The V-A interaction actually violates parity maximally as bothcurrents have the same strength.Parity is not just violated in asmall percentage of interactions, it is violated in all of them. Theform of the interaction suggests that we put it in the form of acurrent-current interaction in analogy with electromagnetism.Wewrite the effective lagrangian as:

L =GF√

2J+µ (x)Jµ+(x) + h.c.

whereJ+µ = (νee

−)µ + (pn)µ

where for instance:

(νee−)µ = ψ(νe)γµ(1− γ5)ψ(e−)

6/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The corresponding matrix element:

(νee−)µ = uνeγµ(1− γ5)ue

The superscript “+” reminds us that the current is acharge–raising current, corresponding to the transitions n→ p ande− → ν in beta decay. The two currents are taken at the samespace-time point x.

7/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The corresponding matrix element:

(νee−)µ = uνeγµ(1− γ5)ue

The superscript “+” reminds us that the current is acharge–raising current, corresponding to the transitions n→ p ande− → ν in beta decay. The two currents are taken at the samespace-time point x.

7/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The corresponding matrix element:

(νee−)µ = uνeγµ(1− γ5)ue

The superscript “+” reminds us that the current is acharge–raising current, corresponding to the transitions n→ p ande− → ν in beta decay. The two currents are taken at the samespace-time point x.

7/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The presence of the factor 1− γ5 in the current requires that allfermions participating in a weak process be left-handed and allanti-fermions be right-handed.For neutrino which are massless weexpect neutrino to always have negative helicity and anti-neutrinoto have positive helicity.This does not preclude the possibility of the existence of a neutrinowith right-handed helicity.It can be shown, however, that theprobability of generating a neutrino with right-handed helicity isproportional to (mν/Eν)2 and is therefore almost impossible.Weknow that the mass of the neutrino is of order of a few eV. For aneutrino with energy of, say, 10 MeV the probability of emitting awrong sign neutrino is around 4× 10−14.

8/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The presence of the factor 1− γ5 in the current requires that allfermions participating in a weak process be left-handed and allanti-fermions be right-handed.For neutrino which are massless weexpect neutrino to always have negative helicity and anti-neutrinoto have positive helicity.This does not preclude the possibility of the existence of a neutrinowith right-handed helicity.It can be shown, however, that theprobability of generating a neutrino with right-handed helicity isproportional to (mν/Eν)2 and is therefore almost impossible.Weknow that the mass of the neutrino is of order of a few eV. For aneutrino with energy of, say, 10 MeV the probability of emitting awrong sign neutrino is around 4× 10−14.

8/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The presence of the factor 1− γ5 in the current requires that allfermions participating in a weak process be left-handed and allanti-fermions be right-handed.For neutrino which are massless weexpect neutrino to always have negative helicity and anti-neutrinoto have positive helicity.This does not preclude the possibility of the existence of a neutrinowith right-handed helicity.It can be shown, however, that theprobability of generating a neutrino with right-handed helicity isproportional to (mν/Eν)2 and is therefore almost impossible.Weknow that the mass of the neutrino is of order of a few eV. For aneutrino with energy of, say, 10 MeV the probability of emitting awrong sign neutrino is around 4× 10−14.

8/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The presence of the factor 1− γ5 in the current requires that allfermions participating in a weak process be left-handed and allanti-fermions be right-handed.For neutrino which are massless weexpect neutrino to always have negative helicity and anti-neutrinoto have positive helicity.This does not preclude the possibility of the existence of a neutrinowith right-handed helicity.It can be shown, however, that theprobability of generating a neutrino with right-handed helicity isproportional to (mν/Eν)2 and is therefore almost impossible.Weknow that the mass of the neutrino is of order of a few eV. For aneutrino with energy of, say, 10 MeV the probability of emitting awrong sign neutrino is around 4× 10−14.

8/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The presence of the factor 1− γ5 in the current requires that allfermions participating in a weak process be left-handed and allanti-fermions be right-handed.For neutrino which are massless weexpect neutrino to always have negative helicity and anti-neutrinoto have positive helicity.This does not preclude the possibility of the existence of a neutrinowith right-handed helicity.It can be shown, however, that theprobability of generating a neutrino with right-handed helicity isproportional to (mν/Eν)2 and is therefore almost impossible.Weknow that the mass of the neutrino is of order of a few eV. For aneutrino with energy of, say, 10 MeV the probability of emitting awrong sign neutrino is around 4× 10−14.

8/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The V-A form of the weak interaction has been verifiedexperimentally both for neutrinos in β-decay, which we callelectron-neutrinos or νe as well as for neutrinos form π → µνdecays, called muon-neutrinos or νµ.These experimental resultshave greatly contributed to establishing the “V-A” interaction.

9/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

V-A interaction

The V-A form of the weak interaction has been verifiedexperimentally both for neutrinos in β-decay, which we callelectron-neutrinos or νe as well as for neutrinos form π → µνdecays, called muon-neutrinos or νµ.These experimental resultshave greatly contributed to establishing the “V-A” interaction.

9/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

The leptonic contribution

u(νe)γµ(1− γ5)u(e),

contains terms that resemble the electromagnetic current

jµ(x) = ψ(x)γµψ(x).

By analogy with the electromagnetic current, we thereforeintroduce the weak leptonic current:

jα(x) = u(νe)γα(1− γ5)u(e) + u(νµ)γα(1− γ5)u(µ)+

+ u(ντ )γα(1− γ5)u(τ) = jα(e) + jα(µ) + jα(τ). (2)

10/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

The leptonic contribution

u(νe)γµ(1− γ5)u(e),

contains terms that resemble the electromagnetic current

jµ(x) = ψ(x)γµψ(x).

By analogy with the electromagnetic current, we thereforeintroduce the weak leptonic current:

jα(x) = u(νe)γα(1− γ5)u(e) + u(νµ)γα(1− γ5)u(µ)+

+ u(ντ )γα(1− γ5)u(τ) = jα(e) + jα(µ) + jα(τ). (2)

10/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

To describe the mutual weak interaction of leptons we postulatethat each leptonic hierarchy interacts with itself as well as witheach of the other two. The following diagrams are some examplesfor such possible processes: Neutrino-electron scattering:

j(e)†α jα(e) = [ueγα(1− γ5)uνe ][uνeγα(1− γ5)ue],

11/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

To describe the mutual weak interaction of leptons we postulatethat each leptonic hierarchy interacts with itself as well as witheach of the other two. The following diagrams are some examplesfor such possible processes: Neutrino-electron scattering:

e−

νe

νe

e−

j(e)†α jα(e) = [ueγα(1− γ5)uνe ][uνeγα(1− γ5)ue],

11/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

Muon decay:

µ−

νµ

e−

νe

j(e)†α jα(µ) = [ueγα(1− γ5)uνe ][uνµγα(1− γ5)uµ],

12/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

and muon production in muon-neutrino-electron scattering:

µ−

νe

νµ

e−

j(µ)†α jα(e) = [uµγα(1− γ5)uνµ ][uνeγα(1− γ5)ue].

13/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

On the other hand, a process like:

νµνµ

e− e−

is not allowed.

14/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

This means that νµ and e− can interact only via the creation of amuon, which is an immediate consequence of the specific form ofthe currents jµ(i),allowing for a neutrino converting into a charged

lepton (or vice versa!),but prohibiting an interaction without aconversion of particles.This property of the interaction is usually expressed by calling thecurrents by charged currents (more accurate by charged transitioncurrents)since the charge of the particle of a particular leptonichierarchy changes by one unit.In the electromagnetic current the charge of the particle does notchange,it is therefore called a neutral current.We shall later seethat neutral currents also appear in the context of the gaugetheory of weak interaction.

15/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

This means that νµ and e− can interact only via the creation of amuon, which is an immediate consequence of the specific form ofthe currents jµ(i),allowing for a neutrino converting into a charged

lepton (or vice versa!),but prohibiting an interaction without aconversion of particles.This property of the interaction is usually expressed by calling thecurrents by charged currents (more accurate by charged transitioncurrents)since the charge of the particle of a particular leptonichierarchy changes by one unit.In the electromagnetic current the charge of the particle does notchange,it is therefore called a neutral current.We shall later seethat neutral currents also appear in the context of the gaugetheory of weak interaction.

15/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

This means that νµ and e− can interact only via the creation of amuon, which is an immediate consequence of the specific form ofthe currents jµ(i),allowing for a neutrino converting into a charged

lepton (or vice versa!),but prohibiting an interaction without aconversion of particles.This property of the interaction is usually expressed by calling thecurrents by charged currents (more accurate by charged transitioncurrents)since the charge of the particle of a particular leptonichierarchy changes by one unit.In the electromagnetic current the charge of the particle does notchange,it is therefore called a neutral current.We shall later seethat neutral currents also appear in the context of the gaugetheory of weak interaction.

15/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

This means that νµ and e− can interact only via the creation of amuon, which is an immediate consequence of the specific form ofthe currents jµ(i),allowing for a neutrino converting into a charged

lepton (or vice versa!),but prohibiting an interaction without aconversion of particles.This property of the interaction is usually expressed by calling thecurrents by charged currents (more accurate by charged transitioncurrents)since the charge of the particle of a particular leptonichierarchy changes by one unit.In the electromagnetic current the charge of the particle does notchange,it is therefore called a neutral current.We shall later seethat neutral currents also appear in the context of the gaugetheory of weak interaction.

15/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

This means that νµ and e− can interact only via the creation of amuon, which is an immediate consequence of the specific form ofthe currents jµ(i),allowing for a neutrino converting into a charged

lepton (or vice versa!),but prohibiting an interaction without aconversion of particles.This property of the interaction is usually expressed by calling thecurrents by charged currents (more accurate by charged transitioncurrents)since the charge of the particle of a particular leptonichierarchy changes by one unit.In the electromagnetic current the charge of the particle does notchange,it is therefore called a neutral current.We shall later seethat neutral currents also appear in the context of the gaugetheory of weak interaction.

15/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

This means that νµ and e− can interact only via the creation of amuon, which is an immediate consequence of the specific form ofthe currents jµ(i),allowing for a neutrino converting into a charged

lepton (or vice versa!),but prohibiting an interaction without aconversion of particles.This property of the interaction is usually expressed by calling thecurrents by charged currents (more accurate by charged transitioncurrents)since the charge of the particle of a particular leptonichierarchy changes by one unit.In the electromagnetic current the charge of the particle does notchange,it is therefore called a neutral current.We shall later seethat neutral currents also appear in the context of the gaugetheory of weak interaction.

15/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

This means that νµ and e− can interact only via the creation of amuon, which is an immediate consequence of the specific form ofthe currents jµ(i),allowing for a neutrino converting into a charged

lepton (or vice versa!),but prohibiting an interaction without aconversion of particles.This property of the interaction is usually expressed by calling thecurrents by charged currents (more accurate by charged transitioncurrents)since the charge of the particle of a particular leptonichierarchy changes by one unit.In the electromagnetic current the charge of the particle does notchange,it is therefore called a neutral current.We shall later seethat neutral currents also appear in the context of the gaugetheory of weak interaction.

15/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak-currents and allowed transitions

This means that νµ and e− can interact only via the creation of amuon, which is an immediate consequence of the specific form ofthe currents jµ(i),allowing for a neutrino converting into a charged

lepton (or vice versa!),but prohibiting an interaction without aconversion of particles.This property of the interaction is usually expressed by calling thecurrents by charged currents (more accurate by charged transitioncurrents)since the charge of the particle of a particular leptonichierarchy changes by one unit.In the electromagnetic current the charge of the particle does notchange,it is therefore called a neutral current.We shall later seethat neutral currents also appear in the context of the gaugetheory of weak interaction.

15/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We now know that the weak interaction is mediated by twomassive gauge bosons: the charged W± and the neutral Z0.Thepropagator term for the massive boson is:

1

M2W,Z − q2

where q2 is the square of the 4-momentum.If we assume that theFermi theory is the low energy limit of the weak interaction, thenwe can estimate the intrinsic coupling at high energy.In the Fermilimit, the coupling factor appears to be GF /

√2.At low energies,

with M2W,Z � q2, the propagator term reduces to just 1/M2

W andwe can make the identification:

16/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We now know that the weak interaction is mediated by twomassive gauge bosons: the charged W± and the neutral Z0.Thepropagator term for the massive boson is:

1

M2W,Z − q2

where q2 is the square of the 4-momentum.If we assume that theFermi theory is the low energy limit of the weak interaction, thenwe can estimate the intrinsic coupling at high energy.In the Fermilimit, the coupling factor appears to be GF /

√2.At low energies,

with M2W,Z � q2, the propagator term reduces to just 1/M2

W andwe can make the identification:

16/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We now know that the weak interaction is mediated by twomassive gauge bosons: the charged W± and the neutral Z0.Thepropagator term for the massive boson is:

1

M2W,Z − q2

where q2 is the square of the 4-momentum.If we assume that theFermi theory is the low energy limit of the weak interaction, thenwe can estimate the intrinsic coupling at high energy.In the Fermilimit, the coupling factor appears to be GF /

√2.At low energies,

with M2W,Z � q2, the propagator term reduces to just 1/M2

W andwe can make the identification:

16/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We now know that the weak interaction is mediated by twomassive gauge bosons: the charged W± and the neutral Z0.Thepropagator term for the massive boson is:

1

M2W,Z − q2

where q2 is the square of the 4-momentum.If we assume that theFermi theory is the low energy limit of the weak interaction, thenwe can estimate the intrinsic coupling at high energy.In the Fermilimit, the coupling factor appears to be GF /

√2.At low energies,

with M2W,Z � q2, the propagator term reduces to just 1/M2

W andwe can make the identification:

16/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We now know that the weak interaction is mediated by twomassive gauge bosons: the charged W± and the neutral Z0.Thepropagator term for the massive boson is:

1

M2W,Z − q2

where q2 is the square of the 4-momentum.If we assume that theFermi theory is the low energy limit of the weak interaction, thenwe can estimate the intrinsic coupling at high energy.In the Fermilimit, the coupling factor appears to be GF /

√2.At low energies,

with M2W,Z � q2, the propagator term reduces to just 1/M2

W andwe can make the identification:

16/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We now know that the weak interaction is mediated by twomassive gauge bosons: the charged W± and the neutral Z0.Thepropagator term for the massive boson is:

1

M2W,Z − q2

where q2 is the square of the 4-momentum.If we assume that theFermi theory is the low energy limit of the weak interaction, thenwe can estimate the intrinsic coupling at high energy.In the Fermilimit, the coupling factor appears to be GF /

√2.At low energies,

with M2W,Z � q2, the propagator term reduces to just 1/M2

W andwe can make the identification:

16/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We now know that the weak interaction is mediated by twomassive gauge bosons: the charged W± and the neutral Z0.Thepropagator term for the massive boson is:

1

M2W,Z − q2

where q2 is the square of the 4-momentum.If we assume that theFermi theory is the low energy limit of the weak interaction, thenwe can estimate the intrinsic coupling at high energy.In the Fermilimit, the coupling factor appears to be GF /

√2.At low energies,

with M2W,Z � q2, the propagator term reduces to just 1/M2

W andwe can make the identification:

16/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

GF√2

=g2w

8M2W

GF/√2

gW gW

1M2

W−q2∼ 1

M2W

Coupling ∼ GF√2

Coupling∼ g2W8M2

W

This allows us to compare the intrinsic couplings of the weakinteraction with the electromagnetic interaction.Experimentally themass of the W boson is 80.4 GeV and the Fermi constant is1.166× 10−5 GeV−2. 17/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

GF√2

=g2w

8M2W

GF/√2

gW gW

1M2

W−q2∼ 1

M2W

Coupling ∼ GF√2

Coupling∼ g2W8M2

W

This allows us to compare the intrinsic couplings of the weakinteraction with the electromagnetic interaction.Experimentally themass of the W boson is 80.4 GeV and the Fermi constant is1.166× 10−5 GeV−2. 17/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

GF√2

=g2w

8M2W

GF/√2

gW gW

1M2

W−q2∼ 1

M2W

Coupling ∼ GF√2

Coupling∼ g2W8M2

W

This allows us to compare the intrinsic couplings of the weakinteraction with the electromagnetic interaction.Experimentally themass of the W boson is 80.4 GeV and the Fermi constant is1.166× 10−5 GeV−2. 17/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

GF√2

=g2w

8M2W

GF/√2

gW gW

1M2

W−q2∼ 1

M2W

Coupling ∼ GF√2

Coupling∼ g2W8M2

W

This allows us to compare the intrinsic couplings of the weakinteraction with the electromagnetic interaction.Experimentally themass of the W boson is 80.4 GeV and the Fermi constant is1.166× 10−5 GeV−2. 17/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We get a weak coupling factor of gw = 0.65.Now, remember thatthe electromagnetic interaction coupling factor is the square rootof the fine structure constant, we have:

EM coupling: αEM =1

137, Weak coupling: αW =

g2W4π

=1

30.

The factor 1/8 comes from two 1/2 factors from the insertion of(1− γ5)/2 (projector operators) and two 1/

√2 factors coming

from the original definition of gw.

18/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We get a weak coupling factor of gw = 0.65.Now, remember thatthe electromagnetic interaction coupling factor is the square rootof the fine structure constant, we have:

EM coupling: αEM =1

137, Weak coupling: αW =

g2W4π

=1

30.

The factor 1/8 comes from two 1/2 factors from the insertion of(1− γ5)/2 (projector operators) and two 1/

√2 factors coming

from the original definition of gw.

18/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We get a weak coupling factor of gw = 0.65.Now, remember thatthe electromagnetic interaction coupling factor is the square rootof the fine structure constant, we have:

EM coupling: αEM =1

137, Weak coupling: αW =

g2W4π

=1

30.

The factor 1/8 comes from two 1/2 factors from the insertion of(1− γ5)/2 (projector operators) and two 1/

√2 factors coming

from the original definition of gw.

18/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

We get a weak coupling factor of gw = 0.65.Now, remember thatthe electromagnetic interaction coupling factor is the square rootof the fine structure constant, we have:

EM coupling: αEM =1

137, Weak coupling: αW =

g2W4π

=1

30.

The factor 1/8 comes from two 1/2 factors from the insertion of(1− γ5)/2 (projector operators) and two 1/

√2 factors coming

from the original definition of gw.

18/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

In fact the weak interaction is, intrinsically, about 4 times strongerthan the electromagnetic interaction.What makes the interactionso weak is the large mass of the relevant gauge bosons.In fact atvery high energies, where q2 ∼M2

W , the weak interaction iscomparable in strength to the electromagnetic interaction.At highenergies the mass of the W-boson suppresses the total crosssection and stops it going to infinity.

19/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

In fact the weak interaction is, intrinsically, about 4 times strongerthan the electromagnetic interaction.What makes the interactionso weak is the large mass of the relevant gauge bosons.In fact atvery high energies, where q2 ∼M2

W , the weak interaction iscomparable in strength to the electromagnetic interaction.At highenergies the mass of the W-boson suppresses the total crosssection and stops it going to infinity.

19/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

In fact the weak interaction is, intrinsically, about 4 times strongerthan the electromagnetic interaction.What makes the interactionso weak is the large mass of the relevant gauge bosons.In fact atvery high energies, where q2 ∼M2

W , the weak interaction iscomparable in strength to the electromagnetic interaction.At highenergies the mass of the W-boson suppresses the total crosssection and stops it going to infinity.

19/19 Particle Physics - lecture 4

Weak InteractionsV-A interactionWeak-currents and allowed transitionsWeak bosons and Fermi couplings

Weak bosons and Fermi couplings

In fact the weak interaction is, intrinsically, about 4 times strongerthan the electromagnetic interaction.What makes the interactionso weak is the large mass of the relevant gauge bosons.In fact atvery high energies, where q2 ∼M2

W , the weak interaction iscomparable in strength to the electromagnetic interaction.At highenergies the mass of the W-boson suppresses the total crosssection and stops it going to infinity.

19/19 Particle Physics - lecture 4