P780.02 Spring 2003 L8 Richard KassQuarksOver the years inquiring minds have asked: “Can we describe the known physics with just a few building blocks ?”

Historically the answer has been yes.Elements of Mendeleev’s Periodic Table (chemistry)nucleus of atom made of protons, neutrons

proton and neutron really same “particle” (different isotopic spin) By 1950’s there was evidence for many new particles beyond , e, p, n It was realized that even these new particles fit certain patterns:

pions: +(140 MeV) -(140 MeV) o(135 MeV)kaons: k+(496 MeV) k-(496 MeV) ko(498 MeV)

Some sort of pattern was emerging, but ........... lots of questions If mass difference between proton neutrons, pions, and kaons is due

to electromagnetism then how come:Mn > Mp and Mko > Mk+ but M+ > Mo

Lots of models concocted to try to explain why these particles exist: Model of Fermi and Yang (late 1940’s-early 50’s):

pion is composed of nucleons and anti-nucleons (used SU(2) symmetry)

+ = pn, - = np, o = pp - n n note this model was proposed before discovery of anti-proton !

With the discovery of new unstable particles (, k) a new quantum number was invented:

strangeness

P780.02 Spring 2003 L8 Richard KassQuarksGell-Mann, Nakano, Nishijima realized that electric charge (Q) of all particles could be related to isospin (3rd component), Baryon number (B) and Strangeness (S):

Q = I3 +(S + B)/2= I3 +Y/2Coin the name hypercharge (Y) for (S+B)

1-1

1

-1

I3

Y k+

k-

ko

k o

+ - o

Interesting patterns started to emerge when I3 was plotted vs. Y:

Particle Model of Sakata (mid 50’s): used Q = I3 +(S + B)/2 assumed that all particles could be made from a combination of p,n, tried to use SU(3) symmetry

In this model: + = pn, - = np, o = pp - n n, k+= p, ko= nBaryons are made up of 3 at a time: +

= pnThis model obeys Fermi statistics and explains why:

Mn > Mp and Mko > Mk+ and M+ > Mo

Unfortunately, the model had major problems….

P780.02 Spring 2003 L8 Richard KassQuarksProblems with Sakata’s Model:Why should the p, n, and be the fundamental objects ?

why not pions and/or kaonsThis model did not have the proper group structure for SU(3)

What do we mean by “group structure” ?SU(n)= (nxn) Unitary matrices (MT*M=1) with determinant = 1 (=Special) and

n=simplest non-trivial matrix representationExample: With 2 fundamental objects obeying SU(2) (e.g. n and p)

We can combine these objects using 1 quantum number (e.g. isospin)Get 3 Isospin 1 states that are symmetric under interchange of n and p:

|11> =|1/2 1/2> |1/2 1/2>|1-1> =|1/2 -1/2> |1/2 -1/2>|10> = [1/2](|1/2 1/2> |1/2 -1/2> + |1/2 -1/2> |1/2 1/2>)

Get 1 Isospin state that is anti-symmetric under interchange of n and p|00> = [1/2](|1/2 1/2> |1/2 -1/2> - |1/2 -1/2> |1/2 1/2>)

In group theory we have 2 multiplets, a 3 and a 1:2 2 = 3 1

Back to Sakata's model: For SU(3) there are 2 quantum numbers and the group structure is more complicated:

3 3 3 = 1 8 8 10Expect 4 multiplets (groups of similar particles) with either 1, 8, or 10 members.Sakata’s model said that the p, n, and were a multiplet which does not fit into the above scheme of known particles! (e.g. could not account for o, +)

P780.02 Spring 2003 L8 Richard KassEarly 1960’s Quarks“Three Quarks for Muster Mark”, J. Joyce, Finnegan’s Wake

Model was developed by: Gell-Mann, Zweig, Okubo, and Ne’eman (Salam)Three fundamental building blocks 1960’s (p,n,) 1970’s (u,d,s)mesons are bound states of a of quark and anti-quark:Can make up "wavefunctions" by combing quarks:

+ = ud, - = du, o =12

(uu - d d), k+= ds, ko= ds

baryons are bound state of 3 quarks:proton = (uud), neutron = (udd), = (uds)

anti-baryons are bound states of 3 anti-quarks:

p u u d n u d d u d s

These quark objects are:point likespin 1/2 fermions parity = +1 (-1 for anti-quarks)two quarks are in isospin doublet (u and d), s is an iso-singlet (=0)Obey Q = I3 +1/2(S+B) = I3 +Y/2Group Structure is SU(3)For every quark there is an anti-quarkquarks feel all interactions (have mass, electric charge, etc)

P780.02 Spring 2003 L8 Richard KassEarly 1960’s QuarksThe additive quark quantum numbers are given below:Quantum # u d s c b telectric charge 2/3 -1/3 -1/3 2/3 -1/3 2/3I3 1/2 -1/2 0 0 0 0Strangeness 0 0 -1 0 0 0Charm 0 0 0 1 0 0bottom 0 0 0 0 -1 0top 0 0 0 0 0 1Baryon number 1/3 1/3 1/3 1/3 1/3 1/3Lepton number 0 0 0 0 0 0Successes of 1960’s Quark Model:

Classify all known (in the early 1960’s) particles in terms of 3 building blockspredict new particles (e.g. -)explain why certain particles don’t exist (e.g. baryons with S = +1)explain mass splitting between meson and baryonsexplain/predict magnetic moments of mesons and baryonsexplain/predict scattering cross sections (e.g. p/pp = 2/3)

Failures of the 1960's model:No evidence for free quarks (fixed up by QCD)Pauli principle violated (++= uuu wavefunction is totally symmetric) (fixed up by color)

What holds quarks together in a proton ? (gluons! ) How many different types of quarks exist ? (6?)

P780.02 Spring 2003 L8 Richard KassDynamic QuarksDynamic Quark Model (mid 70’s to now!)

Theory of quark-quark interaction QCDincludes gluons

Successes of QCD: “Real” Field Theory i.e. Gluons instead of photons Color instead of electric chargeexplains why no free quarks confinement of quarks

calculate lifetimes of baryons, mesonsFailures/problems of the model:

Hard to do calculations in QCD (non-perturbative) Polarization of hadrons (e.g. ’s) in high energy collisionsHow many quarks are there ?

calculate cross sections, e.g. e+e- qq

Historical note: Original quark model assumed approximate SU(3) for the quarks. Once charm quark was discovered SU(4) was considered. But SU(4) is a badly “broken” symmetry.Standard Model puts quarks in SU(2) doublet, COLOR exact SU(3) symmetry.

P780.02 Spring 2003 L8 Richard Kass

From Quarks to ParticlesHow do we "construct" baryons and mesons from quarks ?

Use SU(3) as the group (1960’s model)This group has 8 generators (n2-1, n=3)Each generator is a 3x3 linearly independent traceless hermitian matrixOnly 2 of the generators are diagonal 2 quantum numbers

Hypercharge = Strangeness + Baryon number = YIsospin (I3)

In this model (1960’s) there are 3 quarks, which are the eigenvectors (3 row column vector)

of the two diagonal generators (Y and I3)Baryons are made up of a bound state of 3 quarksMesons are a quark-antiquark bound stateThe quarks are added together to form mesons and baryons using the rules of SU(3).It is interesting to plot Y vs. I3 for quarks and anti-quarks:

M&S P133-140

1-1

1

-1

I3

Y

ud

s

1/3

2/3

1-1

1

-1

I3

Y

u d

s2/3

P780.02 Spring 2003 L8 Richard KassMaking Mesons with QuarksMaking mesons with (orbital angular momentum L=0)The properties of SU(3) tell us how many mesons to expect:

8133

Thus we expect an octet with 8 particles and a singlet with 1 particle.

If SU(3) were a perfectsymmetry then all particlesin a multiplet would have the same mass.

old Vs new pu nds

P780.02 Spring 2003 L8 Richard KassBaryon OctetMaking Baryons (orbital angular momentum L=0).Now must combine 3 quarks together:

10881333 Expect a singlet, 2 octets, and a decuplet (10 particles) 27 objects total.Octet with J=1/2:

P780.02 Spring 2003 L8 Richard KassBaryon DecupletBaryon Decuplet (J=3/2)Expect 10 states.Prediction of the (mass =1672 MeV/c2, S=-3)Use bubble chamber to find the event.

1969 Nobel Prize to Gell-Mann!

“Observation of a hyperon with strangeness minus 3”PRL V12, 1964.

P780.02 Spring 2003 L8 Richard KassQuarks and Vector MesonsLeptonic Decays of Vector MesonsWhat is the experimental evidence that quarks have non-integer charge ? Both the mass splitting of baryons and mesons and baryon magnetic moments depend on (e/m) not e. 

Some quark models with integer charge quarks (e.g. Han-Nambu) were also successful in explaining mass patterns of mesons and baryons.Need a quantity that can be measured that depends only on electric charge ! 

Consider the vector mesons (V=): quark-antiquark bound states with:mass 0electric charge = 0orbital angular momentum (L) =0spin = 1charge parity (C) = -1parity = -1strangeness = charm = bottom=top = 0

These particles have the same quantum numbers as the photon.The vector mesons can be produced by its coupling to a photon:

e+e- V e.g. : e+e- Y(1S) or The vector mesons can decay by its coupling to a photon:

V e+e- e.g. : e+e- (BR=6x10-5) or e+e- (BR=6.3x10-2)

P780.02 Spring 2003 L8 Richard KassQuarks and Vector MesonsThe decay rate (or partial width) for a vector meson to decay to leptons is:

222

2

)0(||16

i

iiV

QaM

em The Van Royen-Weisskopf Formula

In the above MV is the mass of the vector meson, the sum is over the amplitudes that make up the meson, Q is the charge of the quarks and (0) is the wavefunction for the two quarks to overlap each other.

12

(uu - d d) 12

(23 -(- 1

3) )

2 = 1

2 7 KeV

12

(uu + dd) 12

(23 +(-1

3) )

2 = 1

18 0.8 KeV

ss (- 13) 2

= 19

1.3 KeV

cc (23) 2

= 49

bb 1.2 KeV

13.2

12.8

11.8

4.7 KeV 10.5

Y (- 13) 2 = 1

910.6

meson quarks |aiQi|2 L(exp) L(exp) |aiQi|-2

If we assume that |(o)|2/M2 is the same for , (good assumption since masses are 770 MeV, 780 MeV, and 1020 MeV respectively) then:

expect: L() : L() : L() = 9 : 1 : 2measure: (8.8 ± 2.6) : 1 : (1.7 ± 0.4)

Good agreement!

P780.02 Spring 2003 L8 Richard KassFour QuarksOnce the charm quark was discovered SU(3) was extended to SU(4) !

P780.02 Spring 2003 L8 Richard KassMore Quarks

PDG listing of theknown mesons.

With the exception ofthe b, all ground statemesons (L=0) have beenobserved and are in goodagreement with thequark model.

A search for the b ispresently underway!

P780.02 Spring 2003 L8 Richard KassMagnet Moments of BaryonsMagnetic Moments of BaryonsThe magnetic moment of a spin 1/2 point like object in Dirac Theory is:

= (eh/2mc)s = (eh/2mc) /2, ( = Pauli matrix)The magnetic moment depends on the mass (m), spin (s), and electric charge (e) of a point like object.From QED we know the magnetic moment of the leptons is responsible for the energy difference between the 13S1 and 11So states of positronium (e-e+): 13S1 11So Energy splitting calculated = 203400±10 Mhz

measured = 203387±2 Mhz If baryons (s =1/2, 3/2...) are made up of point like spin = 1/2 fermions (i.e. quarks!) then we should be able to go from quark magnetic moments to baryon magnetic moments.Note: Long standing physics puzzle was the ratio of neutron and proton moments:

Experimentally: p/n -3/2In order to calculate we need to know the wavefunction of the particle.In the quark model the space, spin, and flavor (isotopic spin) part of the wavefunction is symmetric under the exchange of two quarks.The color part of the wavefunction must be anti-symmetric to satisfy the Pauli Principle (remember the ++). Thus we have:

= R(x,y,z) (Isotopic) (Spin) (Color) Since we are dealing with ground states (L=0), R(x,y,z) will be symmetric.

always anti-symmetric becausehadrons are colorless

P780.02 Spring 2003 L8 Richard KassMagnet Moments of Baryons Consider the spin of the proton. We must make a spin 1/2 object out of

3 spin 1/2 objects (proton = uud)From table of Clebsch-Gordon coefficients we find:

|1/2 1/2> = 23

|1 1> |1/2 -1/2> - 13

|1 0> |1/2 1/2>

Also we have: |1 1> = |1/2 1/2> |1/2 1/2>

|10> = 12

|1/2 1/2> |1/2 -1/2> + 12

|1/2 -1/2 > |1/2 1/2>

For convenience, switch notation to “spin up” and “spin down”: |1/2 1/2> = and |1/2-1/2> =

Thus the spin part of the wavefunction can be written as:

1 / 2 1 / 2 2 / 3 1 / 3 1 / 2 1/ 6 2 Note: the above is symmetric under the interchange of the first two spins.Consider the Isospin (flavor) part of the proton wavefunction. Since Isospin must have the two u quarks in a symmetric (I=1) state this means that spin must also have the u quarks in a symmetric state.

This implies that in the 2term in the spin function the two are the u quarks. But in the other terms the u’s have opposite sz’s.We need to make a symmetric spin and flavor (Isospin) proton wavefunction.

P780.02 Spring 2003 L8 Richard KassMagnet Moments of BaryonsWe can write the symmetric spin and flavor (Isospin) proton wavefunction as:

p 1 / 182u u d 2d u u 2u d u

u d u u u d u u d d u u u d u d u u

The above wavefunction is symmetric under the interchange of any two quarks.To calculate the magnetic moment of the proton we note that if is the magnetic moment operator:=1+2+3 Composite magnetic moment = sum of moments.

<u| |u> = u = magnet moment of u quark<d| |d> = d = magnet moment of d quark<usz| |usz> = u|sz = (2e/3)(1/mu)(sz)(h/2c), with sz = ±1/2<dsz| |dsz> = d|sz = (-e/3)(1/md)(sz)(h/2c), with sz = ±1/2

For the proton (uud) we have:<p||p> = (1/18) [24u,1/2 + 12 d,-1/2 + 3 d,1/2 + 3 d,1/2] <p||p> = (24/18)u,1/2 - (6/18)d,1/2 using d,1/2 = - d,-1/2 <p||p> = (4/3)u,1/2 - (1/3)d,1/2

For the neutron (udd) we find: <n||n> = (4/3)d,1/2 - (1/3)u,1/2

<usz=1/2| |usz=-1/2> =0, etc..

P780.02 Spring 2003 L8 Richard KassMagnet Moments of BaryonsLet’s assume that mu = md = m, then we find: <p| |p>=(4/3)(h/2c) (1/2)(2e/3)(1/m)-(1/3)(h/2c)(1/2)(-e/3)(1/m)<p| |p>=(he/4mc) [1]<n| |n>=( 4/3)(h/2c) (1/2)(-e/3)(1/m)-(1/3)(h/2c)(1/2)(2e/3)(1/m)<n| |n>=( he/4mc) [-2/3]Thus we find:

2

3

||

||

n

p

nn

pp

In general, the magnetic moments calculated from the quark model are ingood agreement with the experimental data!

-1.46

P780.02 Spring 2003 L8 Richard Kass

Are Quarks really inside the proton? Try to look inside a proton (or neutron) by shooting high energy electrons and muons at it and see how they scatter.

Review of scatterings and differential cross section.The cross section () gives the probability for a scattering to occur.

unit of cross section is area (barn=10-24 cm2)differential cross section= d/dnumber of scatters into a given amount of solid angle: d=ddcos

Total amount of solid angle ():

d0

2

1

1 d cos d 4

0

2

1

1

Cross section () and Impact parameter (b)and relationship between d and db:

d =|bdbd|Solid angle: d =|sindd|

P780.02 Spring 2003 L8 Richard Kass

Examples of scattering cross sectionsHard Sphere scattering: Two marbles of radius r and R with R>>r.

b=Rsin()=Rcos(/2)db= -1/2Rsin(/2)dd=|bdbd= [Rcos(/2)][1/2Rsin(/2)d]d d=|bdbd= [R2][1/4][sin()d] d

The differential cross section is:

4sin

sin)4/(

sin

22 R

dd

ddR

dd

bdbd

d

d

The total cross section is:

21

1

22

sin4

2sin

4Rd

Rdd

Rd

d

d

This result should not be too surprising since any “small” (r) marble within this areawill scatter and any marble at larger radius will not.

P780.02 Spring 2003 L8 Richard Kass

Examples of scattering cross sectionsRutherford Scattering:A spin-less, point particle with initial kinetic energy E and electric charge e scatters off a stationary point-like target with electric charge also=e:

dd

e2

4E sin2( / 2)

2

note: = which is not too surprising since the coloumb force is long range. This formula can be derived using either classical mechanics or non-relativistic QM.The quantum mechanics treatment usually uses the Born Approximation:

dd

f (q2)2

with f(q2) given by the Fourier transform of the scattering potential V:

f (q2) ei

q

r V(

r )d

r

Mott Scattering: A relativistic spin 1/2 point particle with mass m, initial momentum p and electric charge e scatters off a stationary point-like target with electric charge e:

dd

2p2 sin2 ( / 2)

2

(mc)2 p2 cos2 2

In the low energy limit, p<< mc2, this reduces to the Rutherford cross section.Kinetic Energy = E=p2/2m

stationary target hasM>>m

P780.02 Spring 2003 L8 Richard Kass

Examples of scattering cross sectionsMott Scattering: A relativistic spin 1/2 point particle with mass m, initial momentum p and electric charge e scatters off a stationary point-like target with electric charge e:

dd

2p2 sin2 ( / 2)

2

(mc)2 p2 cos2 2

In the high energy limit p>>mc2 and Ep we have:

dd

cos( / 2)

2Esin2( / 2)

2

“Dirac” proton: The scattering of a relativistic electron with initial energy E and final energy E' by a heavy point-like spin 1/2 particle with finite mass M and electric charge e is:

dd

cos( / 2)

2Esin2( / 2)

2E

E1

q2

2M2 tan 2( / 2)

scattering with recoil, neglect mass of electron,E >>me.

q2 is the electron four momentum transfer: (p-p)2 = -4EE'sin2(/2)The final electron energy E' depends on the scattering angle :

E E

1 2EM

sin2( / 2)

P780.02 Spring 2003 L8 Richard Kass

Examples of scattering cross sectionsWhat happens if we don’t have a point-like target, i.e. there is some structure insidethe target?The most common example is when the electric charge is spread out over space andis not just a “point” charge.

Example: Scattering off of a charge distribution. The Rutherford cross section is modified to be:

dd

e2

4E sin2( / 2)

2

F(q2)2

q2 4Esin2( / 2)with: E=E and

The new term |F(q2)| is often called the form factor.The form factor is related to Fourier transform of the charge distribution (r) by:

F(q2 ) (r )ei

q r d3r usually (r)d3r 1

In this simple model we could learn about an unknown charge distribution (structure) by measuring how many scatters occur in an angular region and comparing this measurement with what is expected for a "point charge" (|F(q2)|2=1 (what's the charge distribution here?) and our favorite theoretical mode of the charge distribution.

P780.02 Spring 2003 L8 Richard Kass

Elastic electron proton scattering (1950’s)Electron-proton scattering: We assume that the electron is a point particle. The "target" is a proton which is assumed to have some "size" (structure). Consider the case where the scattering does not break the proton apart (elastic scattering). Here everything is "known" about the electron and photon part of the scattering process since we are using QED.

As shown in Griffiths (8.3) and many other textbooks we can describe the proton in terms of two (theoretically) unknown (but measurable) functions or "form factors", K1, K2:

dd

4MpEsin2( / 2)

2E

E2K1(q2)sin2( / 2) K2(q2)cos2( / 2)

This is known as the Rosenbluth formula (1950). This formula assumes that scattering takes place due to interactions that involve both the electric charge and the magnetic moment of the proton. Thus by shooting electrons at protons at various energies and counting the number of electrons scattered into a given solid angle (d =|sindd|) one can measure K1 and K2.

note:q2 is the electron four momentum transfer: (p-p)2 = -4EE'sin2(/2), and:

E E

1 2EM

sin2( / 2)

P780.02 Spring 2003 L8 Richard Kass

Elastic electron proton scattering (1950’s)An extensive experimental program of electron nucleon (e.g. proton, neutron) scattering was carried out by Hofstadter (Nobel Prize 1961) and collaborators at Stanford. Here they measured the "size" of the proton by measuring the form factors. We can get information concerning the "size" of the charge distribution by noting that:

F(q2 ) (r )ei

q r d3r (r) 1 i

q

r

(q

r )2

2!

d3r

For a spherically symmetric charge distribution we have:

F(q2 )1 q2

6r2(r)dr 1

q2

6r2

Hofstadter et al. measured the root mean square radii of the proton charge to be:

r2

charge

1/2(0.74 0.24) 10 13 cm

McAllister and Hofstadter, PR, V102, May 1, 1956.Scattering of 188 MeV electronsfrom protons and helium.