Quark Model

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Quark Model

• Kihyeon Cho

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???? (Standard Model)

• What does world made of?
• 6 quarks
• u, d, c, s, t, b
• Meson (q qbar)
• Baryon (qqq)
• 6 leptons
• e, muon, tau
• ?e, ??, ??

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Standard Model

• b, c are heavier than other quarks
• - heavy flavor quarks
• W, Z, top are stand out from the rest.

2/3e
-1/3e
0
-1
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Matter

• Hadron (Quark) - size
• Baryon (qqq) proton, neutron
• Meson (q qbar) pion, kaon
• Lepton no size
• Point particle

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Quarks
Over the years inquiring minds have asked Can
we describe the known physics with just a few
building blocks ? Þ Historically the answer has
been yes. Elements of Mendeleevs Periodic Table
(chemistry) nucleus of atom made of protons,
neutrons proton and neutron really same
particle (different isotopic spin)
By 1950s there was evidence for many new
particles beyond g, e, p, n It was realized that
even these new particles fit certain
patterns pions p(140 MeV) p-(140
MeV) po(135 MeV) kaons k(496 MeV) k-(496
MeV) ko(498 MeV)
Some sort of pattern was emerging, but
........... lots of questions Þ If mass
difference between proton neutrons, pions, and
kaons is due to electromagnetism then how
come Mn gt Mp and Mko gt Mk but Mp gt Mpo
Lots of models concocted to try to explain why
these particles exist Þ Model of Fermi and Yang
(late 1940s-early 50s) pion is composed of
nucleons and anti-nucleons (used SU(2) symmetry)
note this model was proposed before discovery of
anti-proton !
With the discovery of new unstable particles (L,
k) a new quantum number was invented Þ
strangeness
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Quarks
Gell-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/2 Coin the name hypercharge (Y) for (SB)
Interesting patterns started to emerge when I3
was plotted vs. Y
Particle Model of Sakata (mid 50s) used Q
I3 (S B)/2 assumed that all particles could
be made from a combination of p,n, L tried to
use SU(3) symmetry In this model
This model obeys Fermi statistics and explains
why Mn gt Mp and Mko gt Mk and Mp gt Mpo
Unfortunately, the model had major problems.
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Quarks
Problems with Sakatas Model Why should the p,
n, and L be the fundamental objects ? why not
pions and/or kaons This model did not have the
proper group structure for SU(3) What do we
mean by group structure ? SU(n) (nxn) Unitary
matrices (MTM1) with determinant 1
(Special) and nsimplest non-trivial matrix
representation
Example 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
11gt 1/2 1/2gt 1/2 1/2gt 1-1gt 1/2 -1/2gt
1/2 -1/2gt 10gt 1/Ö2(1/2 1/2gt 1/2 -1/2gt
1/2 -1/2gt 1/2 1/2gt) Get 1 Isospin state that is
anti-symmetric under interchange of n and
p 00gt 1/Ö2(1/2 1/2gt 1/2 -1/2gt - 1/2
-1/2gt 1/2 1/2gt) In group theory we have 2
multiplets, a 3 and a 1 2 Ä 2 3 Å1 Back to
Sakata's model For SU(3) there are 2 quantum
numbers and the group structure is more
complicated 3 Ä 3 Ä 3 1 Å 8 Å 8 Å
10 Expect 4 multiplets (groups of similar
particles) with either 1, 8, or 10
members. Sakatas model said that the p, n, and L
were a multiplet which does not fit into the
above scheme of known particles! (e.g. could not
account for So, S)
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Early 1960s Quarks
Three Quarks for Muster Mark, J. Joyce,
Finnegans Wake Model was developed by
Gell-Mann, Zweig, Okubo, and Neeman
(Salam) Three fundamental building blocks 1960s
(p,n,l) Þ 1970s (u,d,s) mesons are bound states
of a of quark and anti-quark Can make up
"wavefunctions" by combing quarks
baryons are bound state of 3 quarks proton
(uud), neutron (udd), L (uds) anti-baryons
are bound states of 3 anti-quarks
These quark objects are point like spin 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(SB) I3 Y/2 Group Structure is
SU(3) For every quark there is an
anti-quark quarks feel all interactions (have
mass, electric charge, etc)
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Early 1960s Quarks
The additive quark quantum numbers are given
below Quantum u d s c b t electric
charge 2/3 -1/3 -1/3 2/3 -1/3 2/3 I3 1/2 -1/2 0
0 0 0 Strangeness 0 0 -1 0 0 0 Charm 0 0 0 1 0 0
bottom 0 0 0 0 -1 0 top 0 0 0 0 0 1 Baryon
number 1/3 1/3 1/3 1/3 1/3 1/3 Lepton
number 0 0 0 0 0 0
Successes of 1960s Quark Model Classify all
known (in the early 1960s) particles in terms of
3 building blocks predict new particles (e.g.
W-) explain why certain particles dont exist
(e.g. baryons with S 1) explain mass
splitting between meson and baryons explain/predi
ct magnetic moments of mesons and
baryons explain/predict scattering cross
sections (e.g. spp/spp 2/3) Failures of the
1960's model No evidence for free quarks (fixed
up by QCD) Pauli principle violated (D 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?)
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Dynamic Quarks
Dynamic Quark Model (mid 70s to now!) Theory of
quark-quark interaction Þ QCD includes
gluons Successes of QCD Real Field Theory
i.e. Gluons instead of photons Color
instead of electric charge explains why no free
quarks Þ confinement of quarks calculate
lifetimes of baryons, mesons Failures/problems of
the model Hard to do calculations in QCD
(non-perturbative) Polarization of hadrons
(e.g. Ls) in high energy collisions How many
quarks are there ?
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.
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From Quarks to Particles
How do we "construct" baryons and mesons from
quarks ? Use SU(3) as the group (1960s
model) This group has 8 generators (n2-1,
n3) Each generator is a 3x3 linearly independent
traceless hermitian matrix Only 2 of the
generators are diagonal Þ 2 quantum
numbers Hypercharge Strangeness Baryon
number Y Isospin (I3) In this model (1960s)
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 quarks Mesons are a
quark-antiquark bound state The quarks are added
together to form mesons and baryons using the
rules of SU(3).
MS P133-140
It is interesting to plot Y vs. I3 for quarks and
anti-quarks
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Making Mesons with Quarks
Making mesons with (orbital angular momentum
L0) The properties of SU(3) tell us how many
mesons to expect
old Vs new pu nd ls
Thus we expect an octet with 8 particles and a
singlet with 1 particle.
If SU(3) were a perfect symmetry then all
particles in a multiplet would have the same
mass.
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Baryon Octet
Making Baryons (orbital angular momentum
L0). Now must combine 3 quarks together
Expect a singlet, 2 octets, and a decuplet (10
particles) Þ 27 objects total. Octet with J1/2
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Baryon Decuplet
Baryon Decuplet (J3/2) Expect 10
states. Prediction of the W- (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.
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Four Quarks
Once the charm quark was discovered SU(3) was
extended to SU(4) !
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More Quarks
PDG listing of the known mesons.
With the exception of the hb, all ground
state mesons (L0) have been observed and are in
good agreement with the quark model. A search
for the hb is presently underway!
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Reference

• Richard Cass HEP Class (2003)

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Quarks and Vector Mesons
Leptonic Decays of Vector Mesons What is the
experimental evidence that quarks have
non-integer charge ? Þ 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 (Vr, w, f, y, U)
quark-antiquark bound states with mass ¹
0 electric charge 0 orbital angular
momentum (L) 0 spin 1 charge parity (C)
-1 parity -1 strangeness charm
bottomtop 0 These particles have the same
quantum numbers as the photon.
The vector mesons can be produced by its
coupling to a photon ee- g V e.g. ee-
g Y(1S) or y The vector mesons can decay by
its coupling to a photon V g ee- e.g. r
g ee- (BR6x10-5) or yg ee- (BR6.3x10-2)
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Quarks and Vector Mesons
The decay rate (or partial width) for a vector
meson to decay to leptons is
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 y(0) is
the wavefunction for the two quarks to overlap
each other.
SaiQi2
GL(exp) SaiQi-2
meson
quarks
GL(exp)
If we assume that y(o)2/M2 is the same for r,
w, f, (good assumption since masses are 770 MeV,
780 MeV, and 1020 MeV respectively)
then expect GL(r) GL(w) GL(f) 9 1
2 measure (8.8 2.6) 1 (1.7
0.4) Good agreement!
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Magnet Moments of Baryons
Magnetic Moments of Baryons The magnetic moment
of a spin 1/2 point like object in Dirac Theory
is m (eh/2pmc)s (eh/2pmc) s/2, (s 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 20340010 Mhz measured
2033872 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 mp/mn -3/2 In order
to calculate m 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 D). Thus we have y
R(x,y,z) (Isotopic) (Spin) (Color) Þ Since we
are dealing with ground states (L0), R(x,y,z)
will be symmetric.
always anti-symmetric because hadrons are
colorless
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Magnet Moments of Baryons
Þ 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
Also we have 1 1gt 1/2 1/2gt 1/2 1/2gt
For convenience, switch notation to spin up and
spin down 1/2 1/2gt
and 1/2-1/2gt
Thus the spin part of the wavefunction can be
written as
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 (I1) state this means that
spin must also have the u quarks in a symmetric
state.
This implies that in the 2 term in the spin
function the two are the u quarks. But in the
other terms the us have opposite szs. We need
to make a symmetric spin and flavor (Isospin)
proton wavefunction.
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Magnet Moments of Baryons
We can write the symmetric spin and flavor
(Isospin) proton wavefunction as
The above wavefunction is symmetric under the
interchange of any two quarks. To calculate the
magnetic moment of the proton we note that if m
is the magnetic moment operator
mm1m2m3 Composite magnetic moment sum of
moments. ltu m ugt mu magnet moment of u
quark ltd m dgt md magnet moment of d
quark ltusz m uszgt musz (2e/3)(1/mu)(sz)(h/
2pc), with sz 1/2 ltdsz m dszgt mdsz
(-e/3)(1/md)(sz)(h/2pc), with sz 1/2
ltusz1/2 m usz-1/2gt 0, etc..
For the proton (uud) we have
ltypmypgt (1/18) 24mu,1/2 12 m d,-1/2 3
md,1/2 3 m d,1/2 ltypmypgt (24/18)mu,1/2 -
(6/18)md,1/2 using md,1/2 - md,-1/2
ltypmypgt (4/3)mu,1/2 - (1/3)md,1/2
For the neutron (udd) we find ltynmyngt
(4/3)md,1/2 - (1/3)mu,1/2
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Magnet Moments of Baryons
Lets assume that mu md m, then we find
ltyp m ypgt(4/3)(h/2pc) (1/2)(2e/3)(1/m)-(1/3)(h/
2pc)(1/2)(-e/3)(1/m) ltyp m ypgt(he/4pmc)
1 ltyn m yngt( 4/3)(h/2pc) (1/2)(-e/3)(1/m)-(1/
3)(h/2pc)(1/2)(2e/3)(1/m) ltyn m yngt( he/4pmc)
-2/3 Thus we find
In general, the magnetic moments calculated from
the quark model are in good agreement with the
experimental data!
-1.46
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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
(s) gives the probability for a scattering to
occur. unit of cross section is area (barn10-24
cm2) differential cross section ds/dW number
of scatters into a given amount of solid angle
dWdfdcosq Total amount of solid angle (W)
Cross section (s) and Impact parameter (b) and
relationship between ds and db ds
bdbdf Solid angle dW sinqdqdf
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Examples of scattering cross sections
Hard Sphere scattering Two marbles of radius r
and R with Rgtgtr. bRsin(a)Rcos(q/2) db
-1/2Rsin(q/2)dq dsbdbdf Rcos(q/2)1/2Rsin(q
/2)dqdf dsbdbdf R21/4sin(q)dq df The
differential cross section is
The total cross section is
This result should not be too surprising since
any small (r) marble within this area will
scatter and any marble at larger radius will not.
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Examples of scattering cross sections
Rutherford 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 alsoe
note s 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
with f(q2) given by the Fourier transform of the
scattering potential V
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
stationary target has Mgtgtm
In the low energy limit, pltlt mc2, this reduces to
the Rutherford cross section. Kinetic Energy
Ep2/2m
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Examples of scattering cross sections
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
In the high energy limit pgtgtmc2 and Ep we have
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
scattering with recoil, neglect mass of
electron, E gtgtme.
q2 is the electron four momentum transfer
(p-p)2 -4EE'sin2(q/2) The final electron
energy E' depends on the scattering angle q
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Examples of scattering cross sections
What happens if we dont have a point-like
target, i.e. there is some structure inside the
target? The most common example is when the
electric charge is spread out over space and is
not just a point charge.
Example Scattering off of a charge distribution.
The Rutherford cross section is modified to be
with EE 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(r) by
usually
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)21
(what's the charge distribution here?) and our
favorite theoretical mode of the charge
distribution.
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Elastic electron proton scattering (1950s)
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
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 (dW
sinqdqdf) one can measure K1 and K2. noteq2
is the electron four momentum transfer (p-p)2
-4EE'sin2(q/2), and
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Elastic electron proton scattering (1950s)
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
For a spherically symmetric charge distribution
we have
Hofstadter et al. measured the root mean square
radii of the proton charge to be
McAllister and Hofstadter, PR, V102, May 1,
1956. Scattering of 188 MeV electrons from
protons and helium.