lecture 11 weak interactions, cabbibo - angle

11

Lecture 11Lecture 11

Weak interactions, Weak interactions, CabbiboCabbibo--angleangle

SS2011SS2011: : ‚‚Introduction to Nuclear and Particle Physics, Part 2Introduction to Nuclear and Particle Physics, Part 2 ‘‘

2

NeutrinoNeutrino--lepton reactionslepton reactions

Consider the reaction of neutrino-electron scattering:

Feynman diagram with W-boson exchange (OBE)

Matrix element (cf. Lecture N 10):

(2)

(3)

(1)

q For small q:

5

b) due to the odd number of g‘s

a)

indeed, for any component

Trace with Trace with gg55--matrixmatrix

Consider Tr (Spur) with g5-matrix:

0)(Spurfor

0Spur)ba(Spur1for

5

55

=-¹

==//Þ±=-=nm

nm

gggnm

ggggnm

c)

total antisymmetric tensor

E.g.

{ } 0,,1)( 525 == mggg

6

due to averaging over spin


Mmn - nm- m-tensor :

E mn - e-ne -tensor:

(12)

(13)

Consider the different terms in

this term is equal zero since is a symmetric tensor,whereas - is antisymmetric in m,n

Consider, e.g., a component with which can be 1 or 3

7


Finally,

(14)

(15)

(16)

Matrix element squared:

The matrix element squared doesn‘t depend on the scattering angle q !

8


Differential cross section in the cms:

(17)

(18)

for s>>mm2

Total (angle integrated) cross section:

(19)for s>>mm2

In the Lab. system

(20)

10

For we have

According to (14) the matrix element for is

Thus, the martix element for using the crossing symmetry for (23):

AntineutrinoAntineutrino--lepton reactionslepton reactions

q Use the ‚Crossing‘ symmetry for matrix elements:

(23)

(24)

q Kinematics in the CMS:

(25)

(26)

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The differential cross section in the CMS reads:

(27)

Angular momentum J=0 J=1JZ=1

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(28)

(29)

The total cross section then reads

q Neutrino, antineutrino reactions with antileptons (e+) can be calculated in full analogy to the reactions with leptons (e-)

Factor 1/3 since for the reaction the total angular momentum is J=1and only the JZ=1 state is realized from 3 possible (JZ =-1,0,1) combinations, whereas for

the total angular momentum is J=0 .

13

Quark constituent model: the reactions on a nucleon è reactions on (3 valence) quarks

One would expect that for valence quarks

From experiment è

(Anti(Anti--)neutrino)neutrino--nucleon reactionsnucleon reactions

è scattering on other constituents, not only on valence quarks è interaction with the q-qbar ‚sea‘

14

Weak interaction with hadronsWeak interaction with hadrons

q Consider the weak decay of strange baryons:

(30)total isospin

strangeness

èRule: Quantum numbers are changed by the weak hadronic decay : DI=1/2, DS=1

ès-quark transforms to a light quark (u or d)

)dduu(2

1:

du:

udd:n

uud:p

uds:

0 -

-

p

p

L

Weak interaction changes the quark flavor !

- from experiment

The ratio:

- from Clebsch-Gordon-Coefficient

15


q Consider the weak decay of strange mesons:

)dduu(2

1:

su:K

su:K

0 -

+

-

p

q Rule: in semileptonic weak decays the strangeness and charge are changed by the same value: DQ=1, DS=1

èWeak interaction changes the quark flavor !

strangeness charge

Experimental proof for the rule (32):

1) Branching ratio for the decay is 1.08.10-3, here DS=DQ

2) Branching ratio for the decay is <5.0.10-6, here DS=-DQsince one has to change two quarks uusàudd

(31)

(32)

udd:n

uus:

dds:+

-

S

S

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q for semileptonic weak decay :DQ=1, DS=1q for weak hadronic decay : DI=1/2, DS=1

Thus, for the weak decay of strange hadrons:

From experiment we know that the weak decay processes with changes of strangeness DS=1 are supressed by a factor of 20 compared to the non-strange hadronic decay, i.e. is supressed compared to

q Changes of the quark flavor can be interpreted as an emission of a W-boson

DQ=1, DS=1 DQ=1, S=0

E.g.:

evpen -®0ppL ®

è Cabbibo model

17

Weak interaction with hadrons, Cabbibo modelWeak interaction with hadrons, Cabbibo model

weekstrong HHH +=

èCabbibo model (1963):

Hamiltonian for strong+weak interaction:

÷÷ø

öççè

æ=

s

sstrong H0

0HH ÷÷

ø

öççè

æ=

0H

H0H

W

Wweek ÷÷

ø

öççè

æµ

sW

Ws

HH

HHH

èEigenstates (in flavor space) of the ‚strong‘ hamiltonian HS have to be rotated by an angle qC in order to be eigenstates of the total strong+weak interaction hamiltonian H.The rotation is done by the unitary matrix U:

i.e. the ‚physical‘ states (d‘,s‘) are a superposition of d and s quarks:

(33)

(34)

(35)

U

qC - Cabbibo angle

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Angle qC - Cabbibo angle - is a measure of the amplitude that one flavor of quark (either down or strange) will change into another flavor (up) under the action of the weak force .

Thus, the weak current for the d‘àu+W is

(36)

u(d‘)

19


Cabbibo angle qC from the weak meson decay:

From experiment:

(37)

lecture 11 weak interactions, cabbibo - angle

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