Physics 222 UCSD/225b UCSB

Lecture 2

• Weak Interactions• Intro and Overview• V-A nature of weak current• Nuclear beta decay

Weak Interactions• Some of the most surprising & mysterious

phenomena in particle physics:– Violates fundamental symmetries

• C, T, P, CP

– Changes flavor of quarks and leptons• Heavy flavor decay• Neutrino oscillations

– Matter - Antimatter Oscillations• K0, B0, Bs

0, D0 oscillations all observed• Dazzlingly complex and beautiful phenomena

– Matter - Antimatter symmetry violation• Decay width asymmetries• Symmetry violations as a function of proper time of decay• Symmetry violations as a function of angular correlations

Charged weak current• Leptonic:

– Conserves flavor.– Coupling independent

of flavor.

• Hadronic:– Flavor changing– Coupling = leptonic

coupling x Vqq’

Cabbibo-Kobayashi-Maskawa (CKM)

• Couplings within family dominate.

• The more off-axial the weaker the coupling.

CKM Matrix

+ 1 phasec = cos; s = sin; x,y,z are angles

CKM Matrix Phase Convention

• Is admittedly arbitrary.– See http://arxiv.org/abs/hep-ph/9708366 if you

really want to know the details.

• KISS principle for choice of phase:– Dominant processes are chosen to have zero

phase.

Crudely Categorize Charged Current by theoretical complexity

• Purely leptonic

• Semi-leptonic

• Hadronic

There’s also Weak Neutral Currents

• No flavor changing neutral currents at LO in EWK (FCNC):– E.g. BR(K0 -> e+ e-) < 1.4 10-7

BR(Bd -> mu+mu-) < 1.8 10-8

• Limits on FCNC impose some of the most stringent limits on beyond the standard model physics model building.

e- e-

νμ νμ

First observed in 1973.Z

Aside: FCNC Observation

• The first observation of “Penguin” decays (a 2nd order EWK process that is FCNC) is one of the highest cited papers in experimental particle physics in the last 15 years.– Phys.Rev.Lett.71:674-678,1993. 577 citations– Phys.Rev.Lett.74:2885-2889,1995. 793 citations

• Superceded by Phys.Rev.Lett.87:251807,2001. 517 citations.

And there’s boson self-coupling

and EWK symmetry breaking

We’ll walk through this in roughly the order outlined here.

Example: WZ production involves WWZ triple gauge coupling.

W,Z

Example: W+W-W+ production involves WWWW quartic gauge coupling.

Triple Gauge couplings are well studied, whileexperimental knowledge of quartic couplings is limited.

Historical Interlude

• Fermi proposed to explain nuclear beta-decay in analogy to electron-proton scattering.

€

n → pe−ν e

pe− → nν e

€

M = G u nγμ up( ) u ν e

γ μ ue( )

He thus envisioned a vector current with a weak coupling constant, G, that we now call “Fermi constant”.There was no propagator, nor parity violation in his theory.

We now know:

€

M ∝G u nγμ up( ) u ν e

γ μ ue( )

M ∝g

2Jμ 1( )

gμν + qμqν

MW2

MW2 − q2

Jν 2( )

At low q2, we have G/sqrt(2) = g2/(8Mw2)

For G=1.2 10-5 GeV-2 we thus get g=0.36 .

Weak interactions is weak because MW is large compared to, say the mass of the proton.

Form of Charged Current

• Charge Raising Current:

• Charge Lowering Current:

€

J μ = u ν γ μ 1

21− γ 5( )ue

€

JμT * = u eγ μ

1

21− γ 5( )uν

Any Matrix element needs to be a product of raising and lowering current in order to conserve charge !!!

weak interaction violates parity

• Basic structure of the weak interaction Matrix Element:

€

vertex( )μpropagator( )

μνvertex( )ν

€

γμ 1− γ 5( )γ μ 1− γ 5

( ) = γ μγ μ + γ μγ μγ 5γ 5 − 2γ μγ μγ 5

scalar pseudoscalar

Matrix element has mixed parity.Parity is thus not conserved.

Nuclear beta decay• 14O -> 14N* + e+ + electron-neutrino• I.e., u -> d + e+ + electron-neutrino• First Q:

– Can we successfully describe a nuclear transition using our formalism derived for partons?

• Answer: “Conserved Vector Current” (CVC)– Isospin symmetry guarantees that QCD does not

modify the weak vector currents because they are in isospin triplet with EM current, whose charge does not get modified by QCD, after all.

€

ψ nγμψ p

€

ψ pγμψ n

€

ψ pγμψ p

Axial Vector part of current

• Initial and final nuclear states have JP = 0+

– Both nuclei have J=0– Both nuclei have same parityWe can safely assume that the wave function of the nucleus is

unchanged, and thus ignore the axial vector part of the weak current in this transition.

• This turns out to be important because axial vectors receive ~20% modification of effective current from nuclear physics QCD, while vector currents don’t.

• CVC = conserved vector current• PCAC = partially conserved axial vector current

• Better use vector current transitions when trying to measure G !!!

Calculating Tfi

€

Tfi =−i4GVud

2ψ n x( )γ μ

1

21− γ 5

( )ψ p x( ) ⎡ ⎣ ⎢

⎤ ⎦ ⎥∫ ψ ν x( )γ μ 1

21− γ 5

( )ψ e x( ) ⎡ ⎣ ⎢

⎤ ⎦ ⎥d

4 x

Nuclean spinors are non-relativistic:

Now simplify:

€

up = 2mχ

0

⎛

⎝ ⎜

⎞

⎠ ⎟

χ =1

0

⎛

⎝ ⎜

⎞

⎠ ⎟or

0

1

⎛

⎝ ⎜

⎞

⎠ ⎟

Leptonic current has free particle wave function:

€

ψ ν x( )γ μ 1

21− γ 5( )ψ e x( ) = u ν pν( )γ μ 1

21− γ 5( )ve pe( )e−i pν + pe( )x

Positron spinor

€

γμ → γ0

Consider energy release

• Energy of e is O(1MeV).– > de Broglie wavelength ~ 10-11cm >> Rnucleus

– > we can consider x-dependence of leptonic current to be trivially integrable.

• We then end up with:

€

Tfi ≈−iG

2u ν pν( )γ 0 1

21− γ 5( )ve pe( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ψ n

T * x( )ψ p x( )∫ e−i pν + pe( )xd4 x

e−i pν + pe( )x ≈1⇒ ψ nT * x( )ψ p x( )∫ e−i pν + pe( )xd4 x ≈ 2me−i E p −En( ) 2

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟

Isospin factor (see homework)

Following the usual procedures

• We then follow the usual procedure to go from Tfi to M to d and get:

€

dΓ ≈ G2 Vud

2u ν pν( )γ 0 1

21− γ 5( )ve pe( )

2

spins

∑

×d3 pe

2π( )32Ee

d3 pν

2π( )32Eν

2πδ E0 − Ee − Eν( )

u ν pν( )γ 0 1

21− γ 5( )ve pe( )

2

spins

∑ = 8Ee Eν 1+ ve cosθ( )

dΓ

dpe

=G2 Vud

2

π 3pe

2 E0 − Ee( )2 For more detail,

See H&M p.260

Kurie plot and neutrino mass

€

1

pe

dΓ

dpe

=GVud

π 3E0 − Ee( )

The endpoint of this plot does not reach E0 if the neutrino is massive. This has been used in tritium beta decay to set

limits on neutrino masses. (See link to 17keV Neutrino story on last quarter’s website)

E0

Measuring Vud

• Comparisson of beta decay and muon decay allows for precision measurement of |Vud|

|Vud| = 0.9736 +- 0.0010

Precision of 1/1000 => challenge in nuclear physics.