Transcript Elementary Particle Physics

Lecture 9
Bound states
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Already done a lot to understand the basic particles of nature
Lepton
universality
Strong,
weak,em ?
Isospin
singlets
175000
Isospin
multiplets
small
small
C-parity
small
Parity
CP
G-parity
Neutrino
oscillations/
mass
Quark
composition
Understand most of the properties and decays of
the low lying particles with symmetry
invariance/violation arguments.
What about the excited states ?
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OZI rule
2
Ideal world.
Hadrons
We want to be able start with a qq or qqq and make calculations
with Feynman diagrams involving gluon exchange showing that
they form bound states with the masses and decays we observe.
s
Real world.
The Feynman diagram formalism (perturbation theory)
is powerful but only useful when dealing with particles
s
Not possible
interacting over distances  1016 m - (asymptotic freedom,
10 15m!! Non-perturbative techniques
(lattice gauge theory) try to numerically solve the equations
of the strong force but we still can't understand from first
principles why quarks are confined - this is one of the major
meson
to come!). Proton size
unanswered questions in particle physics.
Similarly, we can’t use
Feynman diagrams to show
that 3 quarks form a bound
baryon state.
We've observed that certain symmetries are respected/violated
and used this together with the fundamental principles of
Go beyond symmetry. Try to understand the masses of the
hadrons from strong force interactions and the quarks. Perform
rudimentary tests of the model that hadrons consist
of spin 1/2 quarks.
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baryons
quantum mechanics to explain much of the hadron behaviour.
3
Strategy for understanding hadron masses
●
Study the contributions to a hadron mass

Quark masses
●

Binding energy of the quark system from the strong
force
●
●
●

Definition of quark masses
Analogy with atomic physics
Spin-spin couplings
Energy levels
Extract a form for the strong potential.
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Start with quark masses
Quark mass is a nightmare to think about. Its not an observable.
Two ways to think about it.
(1) Bare/intrinsic mass.
Rough definition:
The mass a quark would if we could measure
it as a free particle - from calculations based on
hadrons masses.
Obs!!
2mu bare  md bare 2  4  7

 0.02 (9.01) tiny !!
mp
1000
(2) Effective/constituent mass.
Absorb interactions and motion of quarks into
171000
N/A
effective masses:
2mu eff  md eff
mp

2  360  360
 1 (9.02)
1000
Effective mass of a quark in a baryon is not the same as in a meson!
Health warnings come when quoting quark masses.
Different text-books will not always quote the same numbers.
The numbers are model-dependent.
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How does the proton get its mass and the quarks get
their effective masses,
Simple model: consider a system of N massless quarks confined in a region of
radius a by a mechanism which we don't understand.
Energy density of confining field in volume =B (9.03)

Total energy E =   quark energy
 quarks

 4 3 
a  (9.04)
+ B
 3


Uncertainty principle: xp  1 ; x  a,  quark momentum: p
p
1
(9.05)
a
a
1
N
 4 3 
 E   B
a  (9.06)
a
a
 3

Which radius gives a stable configuration ?
massless quark  quark energy 
1
4
E
4N
 N 
 4 4  4
0  a
:
Rewrite
(9.06):
Ea

N

B
a

N

E

(9.07)



a
4

B
3
3
3
a




E =mass of system; proton mass  1 GeV , N  3  a 1.6 fm. (9.08)
Good agreement with size obt ained from scattering experiments (to come).
 Hadrons get their masses (quarks get their effective masses) from the quark
kinetic energy and the confining potential energy (strong force).
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Contribution of the strong force to hadron mass
Light hadrons:
p (uud ), m p  1 GeV
Bare masses: 2mu  md  0.015 GeV  m p (9.15)
The strong force + quark motion account for
98% of the proton mass
Hadron with a heavy quark: c, b eg B  .
B  (bu ), mB  5 GeV
Bare masses: mb  mu  4.2  .004  4 GeV (9.16)
The strong force + quark motion account for only
20% of the B  mass.
Generally true that strong force contributes less as quark mass
increases (fairly obviously!)
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Story so far
●
●
●
●
Two types of quark mass

Bare mass

Effective mass
Quarks get their effective mass from their
motion and the confining force
Effective mass  bare mass for hadrons with c,b
quarks
What else contributes to mass ?
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Angular momentum
Eg    ud  ;    ud  , K   us  ; K *  us 
have the same quark composition.
J P :  1  ,    0  , K   0  ; K * 1 
but different angular momentum.
M   ud   800 MeV >> M   ud   140 MeV
M K *  su   892 MeV >>M K   su   500 MeV
Can this be understood ?
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ud system
• Ground states of most particle configurations are
stable (eg weak, electromagnetic decays).
• Ground state of ud – 
• Higher masses become indeterminate/uncertain
• Decay after  10-23s (resonances – lecture 4)
-1300
a1
a0
-1300
a2
b1
a2
 width
a0
b1


a1






JP
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JP
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Bound state behaviour – eg
electromagnetism
Electron in a potential well exists in energy states with increasing energy
indeterminacy for higher energy states.
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Angular momentum of a bound state
Hydrogen atom:
Consider an electron around a proton (at the origin).
Le
1 2
  (r )  V (r )  E  (9.17)
2me
e-
e 2
V (r ) 
; me  electron mass (9.18)
4 0 r
r
Solution wave function of electron:
 nlm  Rn ,l (r )Yl m  ,   (9.19)
p
Bound state  electron energy quantised: En 
me e 4
32  0  n
2
2
(9.20)
Electron orbital angular momentum quantised:
L2  l  l  1 Lz  ml (-l  ml  l ) (9.21)
Its also quite wrong for (at least) two reasons.
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The first reason – two body motion
The proton also has a (tiny) angular momentum.
We should talk about both particles motion about
Le
e-
their common centre of mass and orbital angular
momentum of the system.
 2
2
2
2
2
2 
 2  2  2  2  2  2
 x1 y1 z1 x2 y2 z2 
V (r )  x1 , y1 , z1 , x2 , y2 , z2   E   x1 , y1 , z1 , x2 , y2 , z2  (9.22)
r
1
2me
p
Lp
Classically in a two body system  reduce to a one body problem
 m1 , m2
rotating around a common centre-of-mass  to a one body problem .
Orbit of a reduced mass  
m1m2
(9.23) around origin at distance r.
m1  m2
r
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
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Using reduced mass
L
 2
2
2
2
2
2 
 2  2  2  2  2  2
 x1 y1 z1 x2 y2 z2 
V (r )  x1 , y1 , z1 , x2 , y2 , z2   E   x1 , y1 , z1 , x2 , y2 , z2  (9.22)
1
2me

r
Use reduced mass  
me m p
me  m p
(9.23)
O
Separate (9.22) into two equations:
one for the motion of the ce ntre-of-mass
(ignore here - work in the centre-of-mass system)
one for a particle of mass  moving around a fixed centre in exactly the same
way the electron is attracted to the proton:
1 2
  (r )  V (r )  E  (9.24)
2
 e4
Quantised energy levels: En  
(9.25)
2 2
32  0  n
Angular momentum quantised: L2  l  l  1 Lz  ml (-l  ml  l ) (9.21)
Of course very little effect since 
me (9.26)
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Three bodies with orbital angular momentum
Several different ways to combine angular
momentum for systems with more than two
bodies, eg, baryons.
Combine two particle system 1 and 2 in
"sub-system".
L12 =angular momentum of 1 and 2 in their
centre-of-mass frame.
L3 =angular momentum of 3 about the centre-of-mass
of 1 and 2 in the overall centre-of-mass frame.
Total orbital angular momentum Ltot  L12  L3 (9.27)
Conserved quantity in the absence of spin.
Systems with spin : total angular momentum is conserved: J  Ltot  S (9.28)
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The second reason – fine and hyperfine structure
Fine structure:
Fine
Hyperfine
p2
p2 p4
(1) relativistic correction T 
 T
(9.29)
2m
2 m 8m 3
p4
Perturbative piece to Hamiltonian H rel   3 (9.30)
8m
e2  1   L  S 
(2) spin-orbit coupling: H so 
 (9.31)


4 0  2me 2  r 3 
Frame where electron is stationary and nucleus orbits it.
Proton orbit generates a magnetic field which the electron's
magnetic momentum interacts with.
(3) Lamb shift (QED)
Hyperfine structure:
Magnetic moments of proton and electron interact (spin-spin)
E 
Hydrogen

me m p
( Se  S p ) (9.32)
Most relevant for this lecture! Breaks the
degeneracy of the ground state. Study simple l  0 states.
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Positronium
Look for a model of a qq system.
Try positronium e  e  - analogous to hydrogen e  p
In classical language the e and e  rotate around their centre-of-mass.
Use analogy and earlier mathematics to work out energy levels in positronium.
mm
m
 pos  e e  e (9.33)  H  me (9.26)
me  me
2
H : EnH 
H e
4
32  0  n 2
2
(9.25) Enpos
  pos e
4

H
e4
H
E
n
2



(9.34)
2 2
2 2
2
32  0  n
32  0  n
positronium
H
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Positronium energy levels
Energy level hydrogen
2  energy level positronium (reduced mass).
Hyperfine/spin-spin structure:
EH 

me m p
( S e  S p ) ; E P 
'
me me
( Se  S p )  EH  EP (9.35)
Not tiny/hyperfine for positronium since m p  me .
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More on positronium
State can have spin and orbital angular momentum.
Characterise state by principal quantum number n , orbital l and spin s, total
j angular momentum quantum numbers.
Total spin quantum number s  1  1,1 , 1, 0 , 1, 1
Parity: P  Pe Pe  1
C  parity : C   1
l
ls
(7.10)  P  1 1 .
l
 or
s0
 0, 0 
(9.36)
Pe Pe  1 (7.13)
(7.23)
Consider two states:
3
S1 (n  1) -orthopositronium ; 1S0 (n  1) - parapositronium
Spectroscopic formalism: ( 2 s 1L j ; j  total ang.momentum quantum number
s  total spin quantum number , 2s  1  total number of spin states.
L  S,P, D.. (standard notation). S : (l  0), P : (l  1), D : (l  2)
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Positronium states
n=1
n=2
1S
0
3S
1
1S
0
3S
1
1P
1
3P
2
3P
1
3P
0
J
0
1
0
1
1
2
1
0
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P
-1
-1
-1
-1
1
1
1
1
C
1
-1
1
-1
-1
1
1
1
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Question
●
●
Decide whether parapositronium or
othopositronium can decay into 3 photons.
Which of the two states would you expect to
have the longer lifetime ? With knowledge that
orthopositronium has a lifetime of 1.42x10-7 s
estimate the lifetime of parapositronium.
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Spin-spin coupling
Hydrogen atom:
Energy level shifted due to interaction of spin dipole moment of electron
with spin dipole moment of proton: E 

me m p
S
e
• Sp

(9.32)
Does it also occur in ground state hadrons (l  0) ?
M (meson)  m1  m2 
A( S1  S2 )
m1m2
(9.37)
( S1  S 2 ) 2  S12  S22  2S1  S2
3
1
 2S1  S 2 (s1  s2  , S 2  s(s  1 )) (9.38)
2
2
Vector meson: s  1  ( S1  S 2 ) 2  s( s  1)  2 (9.39)
 s1 ( s1  1)  s2 ( s2  1)  2S1  S 2 
Pseudoscalar mesons: s  0  ( S1  S 2 ) 2  s( s  1)  0 (9.40)
1
3
(vector mesons) S1  S2  
(pseudoscalar mesons) (9.41)
4
4
If there is a spin-spin coupling as in electromagnetism we could expect to use
(9.37) to predict the meson masses.
S1  S2 
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Spin-spin corrections
Meson
Spin
Calculated mass
(MeV)
Observed mass
(MeV)

K
h
w
K*
f
0
0
0
1
1
1
140
484
559
780
896
1032
138
496
549
783
892
1020
A  constant  (2mu )2159 MeV (9.42), use constituent masses
mu  md  308 MeV , ms  483 MeV (9.43)
Good agreement for most light mesons.
Spin-spin interactions in the strong force!
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Baryons
Baryon
Spin
Calculated
mass
(MeV)
Observed
mass
(MeV)
p,n
L
S
1/2
1/2
1/2
939
1114
1179
939
1116
1193
X
1/2
1327
1318

3/2
1239
1232
W
3/2
1682
1672
1 3
Similar strategy as for mesons. Consider l  0, s  , states.
2 2
 S1 • S2 S1 • S3 S 2 • S3 
M baryon  m1  m2  m3  A ' 


 (9.44)
m
m
m
m
m
m
1 3
2 3 
 1 2
A '   2mu  50 MeV , mu  md  363 MeV, ms  538 MeV (9.45)
2
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Another way of thinking about the mass
Two massless quarks connected by string of length 2 R.
Use practical units instead of natural units in this example.
 =energy density of string GeV/fm .
2R
Centre of string is stationary.
At a point r along th e string has speed v(r ) 
E  mc 2  2 
R
0
 dr
1
2
v
c2
Substitute: r  R cos 
 2
R
0

 dr
1
2
r
c (9.46)
R
(9.47)
r
R2
2
m 2
c


2
0
 Rd 
 R
c
2
. (9.48)
Angular momentum assumed to be orbital only: J  2 
 dr
R
0
c
2
v2
1 2
c
rv 
1
 R 2 (9.49)
2c
m2c3
J
(9.50) Ok - we've made quite assumptions here. Does it work ??
2
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It seems to work
j
J 2  j  j  1
2
(9.51)
m2c3
J
  0.18 GeV 2  0.9 GeV/fm (9.52)
2
This formed the basis for early ideas on string theory. Hadrons are
excitations of a string. Nowadays, quarks are considered to be
string excitations.
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Quarkonium
●
●
●
●
●
Time to extract rigorous results on mass states
and the strong force potential
qq bound state
Easiest to deal with charmonium, bottomium
(rest mass > quark kinetic energy)
Non-relativistic treatment possible –
Schrödinger’s equation
Compare with positronium
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Production of onium states
Simplest way is via e  e  bb , cc .
Enhanced reaction rate when centre-of-mass energy
Ecm mass of a charmonium/bottomium state.
Charmonium production measurements.
Rate(e e  hadrons)
R
Rate(e e     - )
resonances
(9.61)
4 2
Rate(e e    )  2 (9.62)
3ECM
 

-
Other charmonium states which don't
have J PC  1 areproduced via decays,
eg  (3686)   ci  
Narrow – OZI
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Heavy onium states
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Can we work backwards ?
●
We have the mass splittings.
●
What does it tell us about the strong force?
●
Is the form of the strong force the same for
bottomium and charmonium ?
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Positronium, charmonium and bottomium
mass/energy levels
Electromagnetic force
Strong force
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Expected differences when changing the potential
3s
3p
3s
2p
2s
3p
2s
2p
1s
1s
1
Coulomb V 
r
Oscillator V  r 2
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Quark-antiquark potential from
charmonium and bottomium
Data are consistent with
a
 br (9.63)
r
=Coulomb term + confining term
V(r)  
a  0.3, b  0.23GeV 2  10 tons!! (9.64)
or
V (r )  a ln(br ) (9.65)
a  0.75, b  0.80 GeV (9.66)
Both curves in good agreement between 0.2 and 1fm.
Need to study the strong force over distances >1fm to get good discrimination
between parameterisations. Nature doesn't grant us this!!
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Summary
●
●
●
●
Understanding hadron masses is messy
Two definitions of quark masses
Strong force responsible for observed mass of
hadrons containing light quarks
Hadron masses partially understood





Energy levels in analogy to atomic physics
Positronium
Hyperfine splitting
Quantitative study made with heavy onia states
Confining force consistent with a string like behaviour
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(Almost) finished studying the properties of the basic particles of
nature
Strong,
Lepton
universality
Quark
masses
weak,em ?
Isospin
singlets
175000
Isospin
multiplets
small
small
C-parity
small
Parity
CP
G-parity
Neutrino
oscillations/
mass
Mass
differences:
spin-spin
Quark
composition
Understand most of the properties and decays of
the low lying particles with symmetry
invariance/violation arguments and quantum
mechanics
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OZI rule
35