P780.02 Spring 2003 L10 Richard Kass

Weak InteractionsSome Weak Interaction basicsWeak force is responsible for decay e.g. n → pev (1930’s)interaction involves both quarks and leptonsnot all quantum numbers are conserved in weak interaction:

parity, charge conjugation, CPisospinflavor (strangeness, bottomness, charm)

Weak (+EM) are “completely” described by the Standard Model 

Weak interactions has a very rich history1930’s: Fermi’s theory described decay.  

1950’s: V-A (vector-axial vector) Theory: Yang & Lee describe parity violation Feynman and Gell-Mann describe muon decay and decay of strange mesons

 

1960’s: Cabibbo TheoryN. Cabibbo proposes “quark mixing” (1963)"explains" why rates for decays with S =0 > S =1

%9.99)(%5.63)( BRKBR

Quarks in strong interaction are not the same as the ones in the weak interaction: weak interaction basis different than strong interaction basis previous example: ),(),( Ls

oo KKKK

Chapter 8 M&S

Chapter 8.2.3 M&S

P780.02 Spring 2003 L10 Richard KassWeak InteractionsWeinberg-Salam-Glashow (Standard Model 1970’s-today)

unify Weak and EM forces

predict neutral current (Z) reactions

gives relationship between mass of W and Z

predict/explain lots of other stuff!..e.g. no flavor changing neutral currents

existence of Higgs (“generates”mass in Standard Model)

Renormalizable Gauge Theory

But the picture is still incomplete:

must input lots of parameters into the Standard Model (e.g. masses)

where’s the Higgs and how many are there ?

how many generations of quarks and leptons are there ?

mass pattern of quarks and leptons ?

neutrinos have mass!

CP violation observed with quarks!

is there CP violation with leptons?

P780.02 Spring 2003 L10 Richard KassWeak InteractionsClassification of Weak Interactions

Type Comment ExamplesLeptonic involves only leptons muon decay ( evv)

ee- ee- Semileptonic leptons and quarks neutron decay (s=0)

K+ + (s=1) (b=1)

Non-Leptonic involves only quarks -p & K+ +o

Some details of Weak Interactionsquarks and leptons are grouped into doublets (SU(2))

(sometimes called families or generations)For every quark doublet there is a lepton doublet

B Do

u

d

c

s

t

b

Q2 / 3|e|

Q 1/ 3|e|

e

e

Q |e|

Q 0

Charged Current Interactions (exchange of a W boson)W’s couple to leptons in the same doubletThe W coupling to leptons/quarks is a combination of vector and axial vector terms:

Ju uu(1-5 )u (parity violating charged current)

e

e-

W-

-

W-

-

W-

,

e-

W-

e,

-

W-

e,

-

W-

AllowedNOT Allowed

P780.02 Spring 2003 L10 Richard KassCabibbo ModelCabibbo’s conjecture was that the quarks that participate in the weak interaction are a mixture of the quarks that participate in the strong interaction. 

This mixing was originally postulated by Cabibbo (1963) to explain certain decay patterns in the weak interactions and originally had only to do with the d and s quarks.

d’ = d cos + s sinThus the form of the interaction (charged current) has an extra factor for d and s quarks

d quark: Ju u(1-5 )cosc s quark: Ju u(1-5 )sinc

u

d

u

d cosc ssin c

The Cabibbo angle is important for determining the rate of many reactions:

The Cabibbo angle can measured using data from the following reactions:

K oe e

BR(K+ +v)

BR(+ +v) = sin 2c

cos2c

mkm

1- (m/ mk)

2

1- (m/ m)2

2

From the above branching ratio’s we find:c= 0.27 radiansWe can check the above by measuring the rates for:

d

u

W-

s

u

W-

cosc sincPurely leptonic decays (e.g. muon decay) do notcontain the Cabibbo factor:

+

u

W+coscor sinc

d, s

eoe Find:c= 0.25 radians

P780.02 Spring 2003 L10 Richard Kass

The CKM model:In 1972 (2 years before discovery of charm!) Kobayashi and Maskawa extended Cabibbo’s idea to six quarks: 6 quarks (3 generations or families) 3x3 matrix that mixes the weak quarks and the strong quarks (instead of 2x2) The matrix is unitary3 angles (generalized Cabibbo angles), 1 phase (instead of 1 parameter) The phase allows for CP violation

Cabibbo’s ModelExtensions to the Cabibbo Model:Cabibbo’s model could easily be extended to 4 quarks:

Just like c had to be determined from experiment, the matrix elements of the CKM matrix must also be obtained from experiment.

Cabibbo’s namewas added tomake “CKM”

u

d

u

d cosc ssin c

cc ds

c

s

c

sincos

Adding a fourth quark actually solved a long standing puzzle in weak interactions, the“absence” (i.e. very small BR) of decays involving a “flavor” (e.g. strangeness) changingneutral current:

890

1064.0

107

)(

)(

KBR

KBR

However, Cabibbo’s model could NOT incorporate CP violation and by 1977 there was evidencefor 5 quarks!

s

d

s

d

cc

c

cossin

sincos

M&S section 8.3.1

P780.02 Spring 2003 L10 Richard KassThe GIM MechanismIn 1969-70 Glashow, Iliopoulos, and Maiani (GIM) proposed a solution to theto the K0 + - rate puzzle. 8

90

1064.0

107

)(

)(

KBR

KBR

The branching fraction for K0 + - was expected to be small as the first order diagram is forbidden (no allowed W coupling).

+

u

W+

s

+

d

??0

s

-

K+

allowed

K0

forbidden

The 2nd order diagram (“box”) was calculated & was found to give a rate higher than the experimental measurement!

amplitude sinccosc

GIM proposed that a 4th quark existed and its coupling to the s and d quark was:

s’ = scos - dsinThe new quark would produce a second “box” diagram with

amplitude sinccoscThese two diagrams almost cancel each other out. The amount of cancellation depends on the mass of the new quark A quark mass of 1.5GeV is necessary to get good agreement with the exp. data.

First “evidence” for Charm quark!

P780.02 Spring 2003 L10 Richard Kass

CKM Matrix

d'

s'

b'

=

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

d

s

b

The CKM matrix can be written in many forms:1) In terms of three angles and phase:

b

s

d

ccescsscesccss

csesssccessccs

escscc

b

s

d

ii

ii

i

132313231223121323122312

132313231223121323122312

1313121312

1313

1313

13

The four real parameters are , 12, 23, and 13.Here s=sin, c=cos, and the numbers refer to the quark generations, e.g. s12=sin12.

This matrix is not unique,many other 3X3 forms inthe literature. This one is from PDG2000.

2) In terms of coupling to charge 2/3 quarks (best for illustrating physics!)

3) In terms of the sine of the Cabibbo angle (12). This representation uses the fact that s12>>s23>>s13.

b

s

d

AiA

A

iA

b

s

d

1)1(

2/1

)(2/1

23

22

32

“Wolfenstein” representaton

Here =sin12, and A, , are all real and approximately one.This representation is very good for relating CP violation to specific decay rates.

P780.02 Spring 2003 L10 Richard Kass

CKM Matrix

9993.09990.010)3.45.3(10)4.14.0(

10)3.47.3(9749.09734.0225.0219.0

10)52(226.0219.09757.09742.0

22

2

3

tbtstd

cbcscd

ubusud

VVV

VVV

VVV

The magnitudes of the CKM elements, from experiment are (PDG2000):

There are several interesting patterns here:1) The CKM matrix is almost diagonal (off diagonal elements are small).2) The further away from a family, the smaller the matrix element (e.g. Vub<<Vud).3) Using 1) and 2), we see that certain decay chains are preferred:

c s over c d D0 K-+ over D0 -+ (exp. find 3.8% vs 0.15%)b c over b u B0 D-+ over B0 -+ (exp. find 3x10-3 vs 1x10-5)

4) Since the matrix is supposed to be unitary there are lots of constraints among the matrix elements:

0

1***

***

tdtbcdcbudub

tdtdcdcdudud

VVVVVV

VVVVVV

So far experimental results are consistent with expectations from a Unitary matrix.But as precision of experiments increases, we might see deviations from Unitarity.

P780.02 Spring 2003 L10 Richard KassMeasuring the CKM MatrixNo one knows how to calculate the values of the CKM matrix.Experimentally, the cleanest way to measure the CKM elements is by usinginteractions or decays involving leptons.

CKM factors are only present at one vertex in decays with leptons.Vud: neutron decay: npev duevVus: kaon decay: K0 +e-ve suevVbu: B-meson decay: B- ( or +)e-ve buevVbc: B-meson decay: B- D0e-ve bcevVcs: charm decay: D0 K-e+ve csevVcd: neutrino interactions: d -c dc

“Spectator” Model decay of D0 K-e+ve

c

u

s

u

e,

e,W

K-D0 Vcs

Amplitude Vcs

Decay rate Vcs

For massless neutrinos the lepton “CKM” matrixis diagonal

Called a “spectator” diagram because only one quark participatesin the decay, the other “stands around and watches”.