Yukawa Kopplung&CKM Matrix

Yukawa coupling and the CKM-Matrix

Nicolas Kaiser

November 23, 2010

Nicolas Kaiser () Yukawa coupling and the CKM-Matrix November 23, 2010 1 / 25

Table of contents

1 IntroductionDirac equationDirac mass terms

2 Yukawa couplingMotivationLagrangian for one generationGauge invarianceGeneralization to three generations

3 CKM-MatrixDerivationProperties

Two generationsThree generations

Experimental values


An overview of the Dirac equation

The Dirac equation is

(i 6 m) = 0, ( 6 = )With the Weyl spinors + and of helicity 12

~ ~p2 |~p|(p) =

1

2(p)

and the spinors for particles and anti-particles

u(p, ) =

(E + |~p|(p)E |~p|(p)

)v(p, ) =

(E |~p|(p)

E + |~p|(p))

we can write down the most general solution

(x) =

d3~p

(2pi)32p0(a(~p, )u(~p, )eipx + b(~p, )v(~p, )e ipx)


The right- and left-handed spinors mix

The Lagrangian for Dirac fields is

L = i 6 m, = 0

With the projectors

PL =1

2(1 5) PR = 1

2(1+ 5)

the mass term can be written in terms of the chiral(!) right- andleft-handed spinors L and R

m = m(PL + PR) = m(P2L + P

2R)

= m(PLL + PRR) = m(RL + LR)

because with{5,

}= 0 we get PL =

PR0 = (PR)0 = R


Table of contents





Experimental values


Why we need Yukawa coupling

We can describe the weak interaction with a SU(2) U(1) symmetrygroup. The W bosons couple only to chiral(!) left-handed states ofquarks and leptons.

We can assign the left-handed fermion fields to doublets of SU(2) withweak isospin T 3 = 12 .

EL =

(ee

)L

QL =

(ud

)L

The right handed fermion fields are singletts under this group.

(e)R (u)R (d)R

Their weak isospin is T 3 = 0 and Q = Yw since

Q = T 3 + Yw


Why we need Yukawa coupling

We can describe the weak interaction with a SU(2) U(1) symmetrygroup. The W bosons couple only to chiral(!) left-handed states ofquarks and leptons.We can assign the left-handed fermion fields to doublets of SU(2) withweak isospin T 3 = 12 .

EL =

(ee

)L

QL =

(ud

)L

The right handed fermion fields are singletts under this group.

(e)R (u)R (d)R

Their weak isospin is T 3 = 0 and Q = Yw since

Q = T 3 + Yw


Weak isospin T 3 and hypercharge Yw

It becomes obvious that the left- and right-handed fermions belong todifferent SU(2) representations and have different U(1) charges.


Soulution: The Yukawa coupling

A simple mass term like the one in the Dirac equation

L = me(eLeR + eReL)

violates gauge invariance!

In order to solve that problem we introduce the Yukawa coupling for thefirst particle generation:

LY = ye ELeR yd QLdR yuQLcuR + h.c .

The Higgs field in the unitary gauge is

=12

(0

v + h(x)

)c = i2

=12

(v + h(x)

0

)

YW () =1

2YW (

c) = 12


Soulution: The Yukawa coupling

A simple mass term like the one in the Dirac equation

L = me(eLeR + eReL)

violates gauge invariance!In order to solve that problem we introduce the Yukawa coupling for thefirst particle generation:


The Higgs field in the unitary gauge is

=12

(0

v + h(x)

)c = i2

=12

(v + h(x)

0

)

YW () =1

2YW (

c) = 12


Gauge invariance under SU(2)

The Yukawa coupling is invariant under SU(2) gauge transformations. TheHiggs doublet transforms under SU(2) like

(x) U(x)(x) = [1 + i aa(x)](x)

All fermion doublets transform like the first lepton generation

EL(x) EL(x)[1 i aa(x)]

All fermion singlets transform like the first lepton generation

eR eR

It is easy to see that the Yukawa coupling


is invariant under these transformaions.


Gauge invariance under SU(2)

The Yukawa coupling is invariant under SU(2) gauge transformations. TheHiggs doublet transforms under SU(2) like

(x) U(x)(x) = [1 + i aa(x)](x)

All fermion doublets transform like the first lepton generation

EL(x) EL(x)[1 i aa(x)]

All fermion singlets transform like the first lepton generation

eR eRIt is easy to see that the Yukawa coupling


is invariant under these transformaions.Nicolas Kaiser () Yukawa coupling and the CKM-Matrix November 23, 2010 9 / 25

Gauge invariance under U(1)Y 1/2

The Yukawa coupling is invariant under U(1) gauge transformations:

(x) U(x)(x) = [1 iYY (x)](x) Y = 12

EL(x) EL(x)[1 + iYELY (x)] YEL = 1

2

eR [1 iYeR Y (x)]eR YeR = 1

Lets look at the electron mass term

LY ,e = ye ELeR ye EL[1 i 12][1 i 1

2][1 + i]eR

= ye ELeR [1 + (i i2 i

2)] + O(2) = LY ,e



The Yukawa coupling is invariant under U(1) gauge transformations:

(x) U(x)(x) = [1 iYY (x)](x) Y = 12

EL(x) EL(x)[1 + iYELY (x)] YEL = 1

2

eR [1 iYeR Y (x)]eR YeR = 1Lets look at the electron mass term

LY ,e = ye ELeR ye EL[1 i 12][1 i 1

2][1 + i]eR

= ye ELeR [1 + (i i2 i

2)] + O(2) = LY ,e



The sum of the hypercharges is 0 for each term in the Lagrangian:

LY = ye EL1/2

1/2

eR1yd QL

1/61/2

dR1/3yu QL

1/6c1/2

uR2/3

+h.c.

Therefore the Yukawa coupling is invariant under U(1)Y


Spontaneous symmetrie breaking yields fermion masses

When we insert the expressions

=12

(0

v + h(x)

)c = i2

=12

(v + h(x)

0

)into the mass term we obtain

LY = ye v2eLeR(1 +

h

v)yd v

2dLdR(1 +

h

v)yu v

2uLuR(1 +

h

v) +h.c .

These mass terms look like the ordinary Dirac mass terms. With thevacuum expectation value h(x) = 0 we get the relations

me = ye12v md = yd

12v mu = yu

12v

It is possible to construct mass terms for dirac neutrinos exactly that way.


More general: In three quark generations

When we introduce additional quark generations there can be additionalcoupling terms that mix generations. With

Q iL =

(ui

d i

)L

=

((ud

)L

,

(cs

)L

,

(tb

)L

)and

uiR =(uR , cR , tR

)d iR =

(dR , sR , bR

)

the Yukawa coupling reads

LY ,q = y ijd Q iLd jR y iju Q iLcujR + h.c .

y ijd and yiju are general complex-valued matrices. We can diagonalize the

Higgs couplings by choosing a new basis for the quark fields.


More general: In three quark generations

When we introduce additional quark generations there can be additionalcoupling terms that mix generations. With

Q iL =

(ui

d i

)L

=

((ud

)L

,

(cs

)L

,

(tb

)L

)and

uiR =(uR , cR , tR

)d iR =

(dR , sR , bR

)the Yukawa coupling reads

LY ,q = y ijd Q iLd jR y iju Q iLcujR + h.c .

y ijd and yiju are general complex-valued matrices. We can diagonalize the

Higgs couplings by choosing a new basis for the quark fields.


How to diaganolize the coupling

It is always possible to diagonalize these matrices with a bi-unitarytransformation like

yd ULd ydURd

In order to find these transformation matrices U we diagonalize thehermitian matrices obtained by squaring yd . Since ydy

d and y

dyd are

hermitian we can diagonalize them with a unitary transformation

ULdydy

dU

Ld = D

2d U

Rdy dydU

Rd = D

2d

ydy d = ULdD2dULd y dyd = URd D2dURd Here ULd and U

Rd are unitary matrices and D

2d is a diagonal matrix with

real, positive eigenvalues. Then

yd = ULdDdU

Rd, ULd

ydURd = Dd with (Dd)ii = +

(D2d)

ii

and analogously

yu = ULuDuU

Ru, ULu

yuURu = Du




yd ULd ydURdIn order to find these transformation matrices U we diagonalize thehermitian matrices obtained by squaring yd . Since ydy

d and y

dyd are


ULdydy

dU

Ld = D

2d U

Rdy dydU

Rd = D

2d




real, positive eigenvalues.

Then

yd = ULdDdU

Rd, ULd


(D2d)

ii

and analogously

yu = ULuDuU

Ru, ULu

yuURu = Du




yd ULd ydURdIn order to find these transformation matrices U we diagonalize thehermitian matrices obtained by squaring yd . Since ydy

d and y

dyd are


ULdydy

dU

Ld = D

2d U

Rdy dydU

Rd = D

2d




real, positive eigenvalues. Then

yd = ULdDdU

Rd, ULd


(D2d)

ii

and analogously

yu = ULuDuU

Ru, ULu

yuURu = DuNicolas Kaiser () Yukawa coupling and the CKM-Matrix November 23, 2010 14 / 25

Change of variables

We insert the unitary transformations

d iR (URd )ijd jR uiR (URu )ijujRd iL (ULd )ijd jL uiL (ULu )ijujL

and the explicit form of in our Lagrangian LY ,q.

LY ,q = d iLy ijd d jRv2

(1 +h

v) uiLy iju ujR

v2

(1 +h

v) + h.c .

[d iL (ULd )ijy jkd (URd )kl (ULd

ydURd )il

d lR + uiL (U

Lu)ijy jku (U

Ru )

kl (ULu

yuURu )il

ulR ]v2

(1 +h

v)

In our new basis the Higgs couplings are diagonal:

ULdydURd = Dd and U

LuyuURu = Du


The mass matrices are now diagonal, too

We can relate the elements of Dd and Du to quark masses

mid =12D iid v m

iu =

12D iiu v

Now we are also in the basis of mass eigenstates and the mass terms are

LY ,q = mid d iLd iR(1 +h

v)miuuLiuiR(1 +

h

v)

The corresponding Feynman rule is:

In higher theories it is in general not possible to diagonalize the massmatrix and the higgs coupling at the same time!


Table of contents





Experimental values


The CKM-Matrix

The Lagragian for the electroweak interaction is

L = Lkin + g(W+ J+W + W JW + Z 0JZ ) + eAJEM

The unitary transformation matrices URu ,URd ,U

Lu and U

Ld cancel out of the

kinetic terms, the electromagnetic current and the Z 0 boson current. Forexample

uiLuiL uiL(ULu )ij(ULu )jkukL = uiLuiL

However, in the W boson current, we find

J+ =12uiL

d iL 12uiL

(ULuULd )

ijd jL

The unitary matrixV = ULu

ULd 6= 1is called the Cabibbo-Kobayashi-Maskawa mixing matrix.


The CKM-Matrix

The Lagragian for the electroweak interaction is

L = Lkin + g(W+ J+W + W JW + Z 0JZ ) + eAJEMThe unitary transformation matrices URu ,U

Rd ,U

Lu and U

Ld cancel out of the

kinetic terms, the electromagnetic current and the Z 0 boson current. Forexample

uiLuiL uiL(ULu )ij(ULu )jkukL = uiLuiL

However, in the W boson current, we find

J+ =12uiL

d iL 12uiL

(ULuULd )

ijd jL

The unitary matrixV = ULu

ULd 6= 1is called the Cabibbo-Kobayashi-Maskawa mixing matrix.


Parameters of the CKM-Matrix

A complex N N matrix has 2N2 real parameters.We want V to be unitary. The condition VikV

jk = jk reduces the

number of free parameters to N2:N(N-1)/2 rotation angles and N(N+1)/2 phases

Each quark field can absorb a phase and a global phase is irrelevant.This reduces the number of parameters by 2N-1 to (N 1)2:N(N-1)/2 rotation angles and (N-1)(N-2)/2 phases

In two dimensions there is only one angle left which is the Cabibbo anglec .

V =

(Vud VusVcd Vcs

)=

(cosc sincsinc cosc

)


A graphic makes things more vivid

V =

(cosc sincsinc cosc

)

Abbildung: Source: Prof. Drexlin, Physics VI SS10, Lecture 23

The off-diagonal terms allow weak-interaction transitions between quarkgenerations! They are rather small compared to the diagonal elements(cosc 0, 98 vs. sinc 0, 22).Furthermore we can see already here that the W bosons do not couple tomass eigenstates but rotated quark flavour states.


Parameters in three generations

In 3 dimensions the CKM matrix is

V =

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

We have 3 (euler) rotation angles ij and a complex phase e

i.Essentially the CKM-Matrix is a rotation in 3 dimensions. The phase isresponsible for CP-violation.

V =

1 0 00 c23 s230 s23 c23

c13 0 s13ei0 1 0s13ei 0 c13

c12 s12 0s12 c12 00 0 1

=

c12c13 s12c13 s13eis12c23 c12s23s13ei c12c23 s12s23s13ei s23c13s12c23 c12s23s13ei c12s23 s12c23s13ei c23c13


Consequences

The CKM-matrix describes transitions between all three quark generations.


Experimental values

One can determine experimentally the absolute values of the matrixelements

V =

0, 97459 0, 2257 0, 003590, 2256 0, 97334 0, 04150, 00874 0, 0407 0, 99913

Once again the off-diagonal elements are suppressed. Especially in thethird generation the probability of transitions into quarks of othergenerations is extremely small.


Where the experimental values come from

This example shows how one can determine the absolute value of onematrix element. We look at the following reaction:

(B D0ee) |Vbc |2

We can measure the decay rate and then easily calculate the absolutevalue of Vbc .


The end

Thank you for your attention!

References:

Michael E. Peskin Daniel V. Schroeder - An Introduction toQuantum Field Theory

Manfred Boehm, Ansgar Denner, Hans Joos - Gauge Theories of theStrong and Electroweak Interaction

Prof. Dr. D. Zeppenfeld

Prof. Drexlin, Physics VI SS10, Lecture 23


IntroductionDirac equationDirac mass terms

Yukawa couplingMotivationLagrangian for one generationGauge invarianceGeneralization to three generations

CKM-Matrix DerivationPropertiesExperimental values

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