A. Goshaw Physics 846 - Duke Universitywebhome.phy.duke.edu/~goshaw/Lecture14_2017.pdf · 6 A. Goshaw Physics 846 Start with the free field fermion Lagrangian for a weak doublet with

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A. Goshaw Physics 846

Lecture 14 October 17 , 2017

The weak interaction (one generation)

Recap of the program

The weak interaction using SU(2) symmetry Ø Obtain the Lagrangian as was done for QCD Ø The weak interactions of fermions and gauge bosons

Inserting parity violation into the weak interaction Ø How to do it? Why use V-A ? Ø  Chiral operators, left handed weak doublets and SUL(2)

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The program for developing the SM’s Electroweak theory with one generation

1. Apply a procedure analogous to that used for QCD to develop a weak interaction theory using SU(2) as the gauge symmetry applied to weak isospin doublets.

2. Modify this weak interaction theory to incorporate parity violation. This results in the introduction of left-handed isospin doublets and a SUL(2) gauge symmetry. Discover this can be done only if all particle have mass zero. 3. Combine the weak and electromagnetic theories using an SUL(2) x UY(1) gauge symmetry. This will unify the EM and WI coupling strengths (with zero mass fermions and bosons).

4. Postulate a mechanism for introducing particle masses (referred to as electroweak symmetry breaking). This can be done at the expense of introducing a new scalar field (the Higgs boson). This leads to the SM’s electroweak theory.

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Step 1 The weak interaction from SU(2)

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●  Deriving the weak interaction Lagrangian from SU(2) is analogous to deriving the strong interaction from SUc(3). I will in fact use notation that makes the Lagrangians look the same. (see the QCD summary in L11-12, p7-11. )

●  QCD uses color triplets of quarks of the same flavor.

●  The weak interaction has vector doublets for each generation of lepton and quarks.

s =

2

4qrqgqb

3

5 =

2

4q1q2q3

3

5

w =

ud

�or

⌫ee�

�=

1

2

�

●  Each quark weak doublet has quarks of the same color but different flavor. As usual each particle symbol is 4-component Dirac fermion vector.

The weak interaction doublets

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The weak interaction from SU(2) The prescription

●  The WI Lagrangian Lweak is obtained by postulating invariance under the SU(2) transformation of the weak doublets Ψ :

●  As for QCD, the invariance requires that the weak interaction spin 1 boson fields Wa

µ simultaneously be transformed by: For the case of WI the fabc are the SU(2) group structure constants.

where gw is the weak coupling, the αa(x) are real functions of (ct,x,y,z) and with σa the 2x2 Pauli matrices (a =1,2,3).

!

0= exp [�igw↵a(x)Ta]

Ta = 12�a

W

µa ! W

0µa = W

µa + (~c)@µ

↵a(x) + gwfabc↵b(x)Wµc

[Ta,Tb] = i fabcTc

where fabc = ✏abc (✏123 = +1, ✏132 = �1, ✏113 = 0 etc.

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●  Start with the free field fermion Lagrangian for a weak doublet with upper member ψ1 and lower member ψ2 : The doublet index j = 1,2 and 4-vector index µ = 0,1,2,3) As usual there is an implied sum over repeated indices..

●  Here I have anticipated notation arising in the weak Lagrangian by introducing the δjk which just insures each of the terms in the brakets [ ] connects fermions of the same flavor.

The weak interaction from SU(2) the fermion part of the Lagrangian

Lfermion

free

= [i(~c) j

�µ

@µ k

- (mj

c2) j

k

] �jk

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● 

● 

● 


The fermion part of the weak Lagrangian is obtained by modifying the

free particle Lagrangians Lfermion

free

. This is done by replacing:

�jk@µ by Dµjk = �jk@µ + i gw~c [Ta]jkWµ

a

Again, gw = the weak interaction coupling =

p4⇡↵w

and Ta = �a/2 with �a = the 3 Pauli matrices of SU(2).

The [Ta]jk are simply the jk component of the 2x2 Ta matrices.

For example [T1]11 = [T1]22 = 0, [T1]12 = [T1]21 =

12 (check it out).

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●  Therefore the fermion part of the WI Lagrangian is:

 the free quark Lagrangian  flavor mixing between   two members of the   weak doublet


Lfermion

weak

= i(~c) j

�µ

Dµ

jk

k

- (mc2) j

k

�jk

= [i(~c) j�µ@µ k - (mc2) j k ]�jk - gw[Ta]jk j�µ k Wµa

●  Something entirely new. The weak interaction fields Waµ change the

flavor of fermions (quarks or leptons) within each doublet.

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●  By inserting the covariant derivative Djkµ (page 7) into the

the fermion free particle Lagrangian (page 6) produces a weak interaction Lagrangian (page 8) that is invariant under the SU(2) transformation of the weak doublet if the weak field W is simultaneously transformed as as given on page 5.

The weak interaction from SU(2) the fermion Lagrangian --comments

w = =

1

2

�

Lfermion

weak

[ 0,W 0µa

] = Lfermion

weak

[ ,Wµ

a

]

●  However, this can only be accomplished if the members of the weak doublet have the same mass. This is obviously a problem. Note that for QCD this was also the case but there the prediction of all three color states having the same mass was a success not a problem.

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The weak interaction from SU(2) the fermion Lagrangian --comments

●  Note that as for QED, where only one electromagnetic coupling e was needed for both leptons and quarks, here we assume the same weak coupling gw can be used for both lepton and quark doublets.

●  Ignoring the mass problem for now let’s look at the predictions of this weak interaction theory.

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●  The weak interaction fermion current interacting with the the three weak fields Wa

µ is (see page 8):

with T1 = 12

0 11 0

�T2 = 1

2

0 �ii 0

�T3 = 1

2

1 00 �1

�- gw[Ta]jk j�µ k Wµ

a

� gw2 [ 2�µ 1 (Wµ

1 + iWµ2 ) + 1�µ 2 (Wµ

1 � iWµ2 )

+ ( 1�µ 1 - 2�µ 2) Wµ3

●  Inserting the elements of the Ta matrices this becomes:

●  From this you can identify two charged and one neutral weak boson:

W±µ = 1p2(Wµ

1 ⌥ iWµ2 )

W 0µ = Wµ3

The weak interaction from SU(2) the fermion Lagrangian -- predictions

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●  In terms of the charged/neutral bosons the weak current is:

� gwp2[ 2�µ 1 W�µ + 1�µ 2 W+µ ] � gw

2 [ ( 1�µ 1 - 2�µ 2) W 0µ]

where

1

2

�=

ud

�or

⌫ee�

�

●  The four terms describe fermion-field interactions ov the type:

1 = u

W�µW+µ

W 0µ

2 = d 1 = u 1 = u 1 = u 2 = d

è flavor-changing transitions with vertex couplings ~ gw .

The weak interaction from SU(2) the fermion Lagrangian -- predictions

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The weak interaction from SU(2) the field part of the Lagrangian

●  Following the same procedure as for QCD, the weak interaction field term is:

Lfieldweak = - 1

4 Waµ⌫ Wµ⌫a

where Wµ⌫a = @µW ⌫

a - @⌫Wµa - gwfabcW

µb W

⌫c

●  This introduces the field-field self-couplings through the non-ablelian SU(2) structure (i.e., the generators do not commute è fabc ≠ 0).

is invariant as required (see page 5) under the field transformation: W

µa ! W

0µa = W

µa + (~c)@µ

↵a(x) + gwfabc↵b(x)Wµc

Lfieldweak

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The weak interaction from SU(2) the field part of the Lagrangian

●  As for QCD, this predicts triple and quartic gauge couplings among the weak gauge bosons.

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� These predict triple and quartic boson couplings of the form: '

The weak force in an SUL(2) model Summary

� The non-abelien (non commuting) character of the SUL(2) generators introduces a self-coupling of the 3 field bosons Wi µ in contrast to QED. The boson self-interaction is described by:

boson self-interaction ~ g [Wµν x Wν ]i Wi µ for i = 1,2,3

A unique prediction of the SM for the weak interaction

�observed experimentally �not yet observed

experimentally

●  After completing the development of electroweak unification, this will evolve into specific predictions for triple/quartic couplings among the photon and the W and Z bosons.

●  As for QED and QCD, the SU(2) invariance requires that the gauge bosons have zero mass.

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●  The weak interaction theory developed using this SU(2) gauge symmetry is perfectly self-consistent. But it does not include parity violation as experimentally observed in processes mediated by the weak interaction. ●  Therefore parity violation has to be patched into the theory by hand. This raises two questions:

1. How do you experimentally identify the nature of the parity violation? There are several options.

2. How can you modify the SU(2) weak interaction theory to include the correct parity violation?

Lweak = - 14 Waµ⌫ Wµ⌫

a + i(~c) j�µDµjk k - (mc2) j k�jk

Dµjk = �jk@µ + i gw~c [Ta]jkWµ

a

Wµ⌫a = @µW ⌫

a - @⌫Wµa - gwfabcW

µb W

⌫c

The weak interaction from SU(2) Summary

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The parity operator and parity violation in the weak interaction

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●  The parity operator =

The parity operator

●  Under vectors such as momentum change sign ( ) and axial vectors such as angular momentum do not ( )

P

●  It follows that 4-vectors such as momentum Pµ change under by ( changes as is done by gµν ). ●  The parity operator operating on a particle spinor Ψ reveres the sign of the momentum. This can be accomplished by multiplying by γ0 (check L9, p9 for spinors and the gamma matrix handout). If we make the choice that fermions have intrinsic parity +1, then anti-fermions have intrinsic parity -1.

~J ! ~J

Pµ ! Pµ

P =) ~x ! - ~x and t ! +t

PP

~p ! -~p

(E, ~p) ! (E,�~p) = �0 (E, ~p)

(E, ~p) ! (E,�~p) = (E, ~p) �0

~p ! -~p

�o = �o

=

2

664

1 0 0 00 1 0 00 0 �1 00 0 0 �1

3

775

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●  We also need some gamma matrix identities to manipulate the parity transformation of currents that appeared in the weak interaction Lagrangians. You can verify that:

�0�µ�0 = �µ

●  Now consider the transformation under the parity operator of the fermion-field currents that appeared in the weak (and QED and QCD) Lagrangians.

P

�µ Wµ ! �0�µ�0 Wµ

= �µ Wµ = + �µ Wµ

P

The parity operation on interactions

�0�µ�5�0 = ��µ�5

This vector current is even under parity.

�5 = i �0�1�2�3 =

2

664

0 0 1 00 0 0 11 0 0 00 1 0 0

3

775

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●  Replacing γµ by γ5γµ the current transformation under parity is: This axial vector current is odd under parity.

●  The vector or axial vector alone will not cause any parity violation in the interaction.

●  But if they are both included, the interaction is not invariant and parity violating effects will be created. For example either will generate (different) parity violating effects.

Parity violating interactions

�µ�5 Wµ ! �0�µ�5�0 Wµ

= - �µ�5 Wµ = - �µ�5 Wµ

P

¯ [�µ + �µ�5] Wµ (called ”V + A”)

or

¯ [�µ � �µ�5] Wµ (called ”V - A”)

P

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Parity violating weak interactions

●  Returning to considerations of the weak interaction, the correct choice of options for introducing parity violation is determined from experiment.

●  This is nicely done by studying parity violating terms that occur in muon decay. Produce muons polarized along the z axis. Then measure the energy and angular distribution of the decay electron relative to the muon spin direction ~pe ~J

muon

~Jmuon ~pe

✓Measure d

2�dcos✓dx

where x = 2Eemµc

2

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●  The muon decay properties can be predicted allowing for the possibility of contributions form vector, axial vector and tensor interactions. This is expressed in terms of so-called Michel parameters, named for the person first working out these distributions. (L. Michel Proc. Phys. Soc. A63, 514 (1950)) d

2�dcos✓dx

= Ax

2�3(1� x) + 2⇢( 43x� 1) - ⇠cos✓ [(1� x) + 2�( 43x� x)]

where ⇢, ⇠ and � are the Michel parameters that can be used to

distinguish between the various possible sources of parity violation.

Source ρ δ ξ V - A 3/4 3/4 + 1 V + A 3/4 3/4 - 1 experiment PDG 2017

0.74979 + 0.00026

0.75047 + 0.00034

+1.0009+ 0.0016

 the winner

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●  Therefore the interaction term in the weak interaction must be of the form ⇡ gw �µ 1

2 [1� �5] Wµ


●  The question remains how to accomplish this change in the weak interaction Lagrangian that was created requiring SU(2) invariance.

●  Before doing this we need to introduce chirality, the associated concepts of left and right handed Dirac spinors.

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The chirality operator, left and right handed spinors

and SUL(2)

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Chirality operators and left/right spinors

●  Define chirality operators: PL = 12 (1� �5)

PR = 12 (1 + �5)

●  Define left and right handed Dirac spinors

L = 12 (1� �5)

R = 12 (1 + �5)

●  Some properties of :

●  From this it follows that: L = 12 (1 + �5)

R = 12 (1� �5)

=) �5 = �5 and (�5)2 = 1

= L + R

= L R + R L

L L = R R = 0

�5 = i �0�1�2�3 =

2

664

0 0 1 00 0 0 11 0 0 00 1 0 0

3

775

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●  Parity violation will be introduced into the weak interaction by postulating that the SU(2) symmetry operator acts only on the left-handed projection of the fermion spinor states. Call this symmetry operator SUL(2).

●  The up-side (and of course the reason it is done) is that this introduces the correct V-A parity violation into the weak interaction.

●  The down-side can be immediately seen by looking at the the mass term in the Lagrangian:

Introduction of SUL(2)

(mc2) = (mc2) [ L R + R L]

If SUL(2) transforms only L leaving R unchanged,

this term in the Lagrangian can not be invariant under

the postulated gauge transformation.

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End Lecture 14 Next Lecture:

Development of the weak interaction with SUL(2)

Documents

A. Goshaw Physics 846 - Duke Universitywebhome.phy.duke.edu/~goshaw/Lecture14_2017.pdf · 6 A. Goshaw Physics 846 Start with the free field fermion Lagrangian for a weak doublet with