Transcript Elementary Particle Physics

The quark model
Goal - to demonstrate that the basic properties of the observed
mesons and baryons can be understood with a quark model:
(a) Composition of multiplets
(b) Spins
(c) Magnetic moments
Show the theory and ideas which won Gell-Mann the Nobel
prize in 1969.
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1
The quark model
Start with ground state hadrons containing only u or d quarks
which have no orbital angular momentum.
Postulates:
(1) Mesons consist of qq and baryons of qqq.
(2) Quarks carry a quantum number: isospin. The u , d  u , d 
quarks represent different projections of the same quark (antiquark)
in an internal isospin space, as p, n.
When adding different isospins to form a hadron nature uses
SU(2) symmetry to determine the allowed flavour combinations.
(3) Quarks (antiquarks) carry a colour quantum number: r , g , b (r , g , b )
When adding colours nature has chosen an SU(3) symmetry to
determine the possible colour combinations. Nature always
chooses hadrons to be colour singlets.
With these postulates can we support the quark picture of hadrons ?
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Forming hadrons
When combining two particles to form a composite particle we need
to know what combinations could occur in nature given the various
conservation laws. Out of everything which can exist our theory should
predict only those do exist!!
How do the various properties add up ?
(1) Angular momentum
(2) Isospin
(3) Colour
Do we satisfy Pauli's exclusion principle ?
- is the wave function for a hadron consisting of identical particles
antisymmetric to the interchange of two fermions ?
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Combining particles (1) – angular momentum
Combine a quark and an antiquark
Angular momentum:
Add in usual quantum mechanics way (Clebsch-Gordon co-efficients)
Symmetric spin-1 triplet:
11 
11 11
= 
22 22
11 
1 1 1 1



2 2 2 2
1 11 1 1
1 1 11  1
   




 22 2 2

2 2 22 
2
2
Antisymmetric spin-0 singlet
10 
00 
1 11 1 1
1 1 11  1
   




 22 2 2

2 2 22 
2
2
An alternative way to do this is to use group theory:
Invariance to a rotation  SU(2) symmetry in nature
SU(2) group theory language: 2  2 1  3 gives the same result.
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Quarks and isospin
Same mathematics as angular momentum.
The up and down quarks form a doublet:
1 1
u
2 2
iso
1 1
; d
2 2
iso
Anti-up and down quarks form a doublet:
1 1
u
2 2
iso
1 1
; d 
2 2
iso
(- sign is a technical and (for us) unimportant detail)
The other quarks carry no isospin.
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Isospin of antiquarks (not for lecture or exam)
1 1
2 2
Light quarks form an isospin doublet: u 
iso
1 
 
 0 iso
d 
1 1
2 2
iso
 0
 
1 iso
From spin questions in lecture 5 and eqn. 523.
1 
 0
1 1
U ( )       Rˆ
2 2
 0 iso 1 iso
 0
1 
1 1
1 1
; U ( )        Rˆ

2 2 iso
2 2 iso
1 iso
 0 iso
iso
iso
Rˆ  rotation of  around "y"-axis (or 2-axis) in isospin space  Rˆ u  d ; Rˆ d   u (6.08)

1 1
2 2
Define charge conjugation phase factors:
Cˆ u  u
; Cˆ d  d
 Cˆ u  u
; Cˆ d  d
ˆˆ u  u
CC

Apply charge conjugation transformation to rotation operations:
Rˆ u  d ; Rˆ d   u  Rˆ u  d
; Rˆ d   u (6.10)
ˆ ˆ d  d  (6.09)
; CC

Use this info to define antiquark isospin doublet. Desire that antiquark doublet transforms in
the same way as the quark doublet (necessary when we combine quarks and antiquarks
together in mesons and want to transform the whole thing by a rotation).
i.e. Rˆ upper  lower
; Rˆ lower   upper
 1  
 0
Then Rˆ         
 1 iso
  0 iso 
 Rˆ d
Without the negative sign: d 
 u
 set: d  
1 1
2 2
iso
1 
  
 0 iso
 0  
1 
Rˆ       
 0 iso
 1 iso 
 0
1 1
u 
 
2 2
1 
, ok!
1 
1 1
 
2 2
0

u 
 Rˆ u
;

1 1
2 2
d
iso
 0
   (6.11)
1 iso
, ok!

(6.12)
 1    0 
 0  
1 
Then Rˆ      
Rˆ d  u ,not ok!
; Rˆ       
Rˆ u   d ,not ok! (6.13)
 0 iso
 0 iso  1 iso
 1 iso 
The minus signs are a way ensuring symmetry under charge conjugation and are defined after a convention.



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
6
Combining particles (2) – isospin
Combine a quark and an antiquark but only u , d quarks
Isospin: u 
11
22
,u 
iso
1 1

2 2
,d 
iso
1 1

2 2
,d  
iso
11
22
iso
Symmetric spin-1 triplet:
ud  11 iso
11

22
1
du  ud   10

2
iso
iso
11
22
du  1  1 iso
iso
1 11


2 22
1 1
 
2 2
iso
1 1

2 2
11

22
iso
iso
1 1

2 2
11

22
iso
iso
1 1

2 2
iso
iso
1 1

2 2


iso 
iso
1 1

2 2


iso 
Antisymmetric spin-0 singlet
1
du  ud   00

2
iso
1 11


2 22
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Combining particles (3) – colour
Combine a quark and antiquark with colour and anticolour 3  3  8  1.
Singlet:
 singlet 
Octet:
octet
1
 RR  BB  GG 
3
 RB ; RG ; BR ; GR ; BG ;
1
 RR  GG  ;
2
1
 RR  GG  2 BB  .
6
We've never observed a particle with naked colour so nature
clearly takes the singlet.
Obs if you're unconvinced - we also test SU(3) when combining
three quarks for baryons.
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Can we describe the observed
mesons
Need to make sure our model leads to
isomultiplets of light hadrons.
Spin 0:   ,0, (isospin 1) and  0 (isospin 0)
-1
0
+1
r
r0
r
Spin 1: r  ,0,  (isospin 1) and w 0 (isospin 0)
Can we form wave functions ?
w?
-1
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0
+1
9
Meson wave functions
The total wave function consists of different parts: spatial, spin, flavour and colour.
   space iso flav col
Consider ground state hadrons - no orbital angular momentum
Lq
2
q
Central potential:  space  R(r12 )Yl m  r12 
L  0  Yl m  r12   1, space  R (r12 )   space
 symmetric for exchange of quarks i.e. r12   r12
r12
1
(symmetry only important for baryons since q, q are not
identical particles).
  wave function:
 
 1

1
    ud 
 RR  BB  GG  
  space
2
 3

1 uR    d R     uB    d B     uG    d G    
=  space
6  uR    d R     u B    d B     uG    d    
G


q
Lq
1
of the time the up quark would be red and spin-up.
6
Its straightforward to form the meson wave functions and thus demonstrate that
these states are quantum mechanically possible.
 if you could pull apart a   ,
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Baryons
Same procedure as before. Which spin wave functions can be formed ?
3
S
Spin
 spin

 symmetric with change of a pair of quarks
2
33
3 3
   

   
22
2 2
31
1
3 1
1

     


     


22
2 2
3
3
1
PA
Spin
 spin
partially antisymmetric with change of a pair of quarks.


2
12
12
23
23
11
1
1 1
1

    


    


22
2 2
2
2
antisymmetric upon exchange of particles 1 and 2
11
1
1 1
1

     


     
22
2
2
2
2
antisymmetric upon exchange of particles 2 and 3.
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 12 
 23 
11
Can also have
13
13
11
1
1 1
1






   



22
2 2
2
2
antisymmetric upon exchange of particles 1 and 3.
 13 
Not independent of the other two
11
22
13
11

22
12
11

22
23
;
1 1

2 2
13
1 1
 
2 2
12
1 1
 
2 2
23
Could obtain the combined spin states either from
Clebsch-Gordon co-efficients or group theory/SU(2) symmetry.
222  422
 One quadruplet of symmetric spin combinations and
2 independent doublets of partially antisymmetric spin
combinations.
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Baryons
Now do the same for isospin.
Same as before - isospin and angular momentum have same algebra.
3
2
Isospin
33
22
S
iso
  uuu  ;
iso
3 3

2 2
31
22
  ddd  ;
iso

iso
1
3 1
 uud  udu  duu  ; 
2 2
3
12
12
1
1 1
1



 ud  du  u
 ud  du  d
2
2
2
2
iso
iso
antisymmetric upon exchange of particles 1 and 2
11
22

iso
1
 udd  dud  ddu 
3
1
PA
 iso

 partially antisymmetric with change of a pair of quarks.
2
Isospin
11
22
  symmetric with change of a pair of quarks
23
23
1
1 1
1

u  ud  du 


d  ud  du 
2
2
2
2
iso
iso
antisymmetric upon exchange of particles 2 and 3.
13
 
12
iso
 
23
iso
13
1
1 1
1
Also:


 uud  duu  ; 
 udd  ddu   iso13 
2 2 iso
2
2
iso
Antisymmetric upon exchange of particles 1 and 3. Not independent of the other two
11
22
11
22
13
iso
11

22
12
11

22
iso
23
;
iso
1 1

2 2
13
iso
1 1
 
2 2
12
1 1
 
2 2
iso
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23
iso
13
Combining colour
Follow the postulates - combine colours according to SU (3) symmetry.
We have 3 colours R, G, B  3 projections of one quark
3  3  3  10  8  8  1
According to the postulate, the colour singlet is chosen:
 col 
1
 rgb  rbg  gbr  grb  brg  bgr 
6
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Baryon wave function
   space spin iso col
Spatial wave function.
Depends on orbital angular momentum.
General: two contributions:
L12  angular momentum of 1 and 2 in "subsystem"
L3  angular momentum of 3 about c.m of 1 and 2
in the overall c.m. frame: Ltot  L12  L3
We only consider ground state baryons: angular momentum
comes only from spin: L12  L3  0
 space depends only the distance between quarks
no change if r
12
 r12 
  spin iso must be completely symmetric to satisfy Pauli's
exclusion principle.
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Can we describe the observed
baryons
Need to make sure our model leads to
isomultiplets of light hadrons.
-1/2
1
1
Spin : p, n (isospin )
2
2
3
3
 ,0,  , 
Spin : D
(isospin )
2
2
D
D0
0
+1/2
D
D
I3
Can we form wave functions ?
-3/2
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-1/2
+1/2
3/2
16
Test the quark model – Spin 3/2 baryon
3
baryons should be present in nature
2
with the same properties except those governed by quark content,eg charge.
If our theory is correct then all four spin-
S
S
 D   spin
 iso
(ignore colour and space p art)  symmetric !
Try and form a spin
 spin 
3
3 1
wave function for D   uud 

2
2 2
3 1
1


     

2 2
3
  D
 iso 
1
 uud  udu  duu 
3

 u    u    d     u   u    d    u   u   d  

1 
 u    d    u     u    d    u     u    d    u    
3

d   u   u    d   u   u    d   u   u   


1
?
2
No ! It is imposible to form a completely symmetric spin-flavour
Can we form a wave function for a D   uud  if it was spin
wave function and thus satisfy Pauli's exclusion principle.
1
 no multiplet of four baryons for spin- observed in data !
2
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17
Test the quark model – spin-1/2 baryons
Ignore the spatial and colour parts. Need a symmetric flavour - spin wave function.
12
12
13
13
23
23

  A  spin
 iso
 spin
 iso
 spin
 iso
 partially antisymmetric  partially antisymmetric  symmetric!
A  normalisation factor  23 is not independent of  13 , 12 
1
).
2
Consider spin-up projection and form the wave function.
Consider the proton (spin
1
1
1

  A        udu  duu         uud  udu        uud  duu  
2
2
2

Left as an exercise….
1  2u    d    u     u    d    u     u    d    u     d    u    u     2d   u   u   


18  d    u    u     u    u    d     u    u    d     2u    u    d   


Can form wave functions for two spin
1
states ( p, n).
2
3
;  iso is symmetric and  spin partially antisymmetric
2
3
 can't form doublet from spin
baryons. Agreement with data!
2
If baryons are spin
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18
So far so good
Can understand the existence and properties
of multiplets the very lightest hadrons with
SU(2)-isospin symmetry.
SU(2) symmetry is ”good” - all hadrons in an
isospin multiplet have similar masses since u,d
have similar masses.
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Extending the model
These multiplets of the lowest mass hadrons are observed.
Our model should able to predict these (and nothing more).
-1
-½
½
+1
-1
I3
Meson nonet
(spin 0)
Baryon decuplet
(spin 1/2)
-½
½
+1
I3
Meson nonet
(spin 1)
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Baryon decuplet
(spin 3/2)
20
Extending the quark model
Start with ground state hadrons containing only u , d , s quarks
which have no orbital angular momentum.
Postulates:
(1) Mesons consist of qq and baryons of qqq.
(2) Quarks carry a flavour quantum number. The u , d , s  u , d , s 
quarks represent different projections of the same quark (antiquark)
in an internal flavour space.
When adding different isospins to form a hadron nature uses
SU(3) symmetry to determine the allowed flavour combinations.
(3) Quarks (antiquarks) carry a colour quantum number: r , g , b (r , g , b )
When adding colours nature has chosen an SU(3) symmetry to
determine the possible colour combinations. Nature always
chooses hadrons to be colour singlets.
With these postulates can we support the quark picture of hadrons ?
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Combining quarks (3) - flavour
SU (3): 3  3  8  1
Octet:
1
1
dd  uu  2ss 
uu - dd  , us , ds , us, ds,
ud , du ,


6
2
1
dd  uu  ss 
Singlet:

3
Isospin multiplets
form part of pattern.
Meson nonet
(octet+singlet)
(spin 0)
-1
-½
+1
½
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I3
22
Identifying the states
Put together the flavour states into observed particles:
K 0  sd K -  su K 0  ds K   us (Mass  500 MeV)
1
 -  du  0 
(uu  dd )    du (Mass  140 MeV)
2
Two neutral combinations with same quantum numbers:
K 0  ds
1
1
8 
(uu  dd  2ss ) and 0 
(uu  dd  ss )
6
3
Nature uses a linear combination of them as observable
particles: (if it can happen, it will happen!)
  100
1
  8 cos   0 sin  
(uu  dd  2 ss )
6
1
 '  8 sin   0 cos  
(uu  dd  ss )
3
0
FK7003
(Mass  550 MeV)
(Mass  950 MeV)
23
SU(3) Flavour – mesons of spin 1 (vector)
K *0  sd K *-  su K *0  ds K *  us (Mass  890 MeV)
1
0
r  du r 
(uu  dd ) r   du (Mass  780 MeV)
2
  35o 8 , 0 in analogy with 0 ,8
K *0  ds
1
w  8 sin   0 cos  
(uu  dd ) (Mass  550 MeV)
2
  8 cos   0 sin   ss (Mass  950 MeV).
FK7003
24
Meson wave functions
The total wave function consists of different parts: spatial, spin, flavour and colour.
   space spin flav col
As earlier
K  wave function:
     space
 1

1
    us 
 RR  BB  GG  
2
 3

1 uR    sR     uB    sB     uG    sG    
=  space
6  uR    sR     uB    sB     uG    s    
G


1
 if you could pull apart a K  ,
of the time the antistrange quark would be antired
6
and spin-down.
FK7003
25
Baryons
SU (3) :
3  3  3  10  8  8 1
uuu
ddd
1
 ddu  dud  udd 
3
1
 dds  dsd  sdd 
3
1
 duu  udu  uud 
3
1
 uds  usd  dus  dsu  sud  sdu 
6
1
 uus  usu  suu 
3
1
 uss  sus  ssu 
3
1
 dss  sds  ssd 
3
sss
-3/2
-1
-1/2
0
1/2
1
Decuplet:  S - completely symmetric states.
FK7003
3/2
I3
26
1
 ud  du  u
2
1
 ud  du  d
2
1
 2  ud  du  s   us  su  d   ds  sd  u 
12
Octet:
 12 : antisymmetric in 1 and 2
1
 us  su  u
2
1
 ds  sd  d
2
1
 us  su  d   ds  sd  u 
2
1
 ds  sd  s
2
1
 us  su  s
2
1
d  ud  du 
2
1
u  ud  du 
2
1
 2s  ud  du   d  us  su   u  ds  sd  
12
Octet:
 23 : antisymmetric in 2 and 3
1
d  ds  sd 
2
1
u  us  su 
2
1
 d  us  su   u  ds  sd  
2
1
s  ds  sd 
2
1
s  us  su 
2
1
 uds  usd  dsu  dus  sud  sdu 
6
Singlet:
 A : completely antisymmetric
FK7003
27
1
 udd  ddu 
2
1
 uud  duu 
2
1
 2  usd  dsu    uds  sdu  dus  sud  
12
Octet:
13

: antisymmetric in 1 and 3
1
 uus  suu 
1
 dds  sdd 
2
2
1
 uds  sdu  dus  sud  
2
1
 dss  ssd 
2
1
 uss  ssu 
2
 13   12  23 - not independent of the other two.
FK7003
28
Test the quark model – decuplet
If our theory is correct then all ten decuplet baryons should be present in nature
with the same properties except those governed by quark content,eg charge.
S
 D   spin
 Sflavour
(ignore colour and space part)  symmetric !
Try and form a spin
 spin 
3
3 1
wave function for    dds 

2
2 2
3 1
1
1


      ;  flavour 
 dds  dsd  sdd 

2 2
3
3
  

 d    d   s    d   d   s    d   d   s   

1 
  d    s    d     d   s   d    d   s   d  
3

d    s    d    d   s   d    d   s   d   


1
?
2
No ! It is imposible to form a completely symmetric spin-flavour
wave function and thus satisfy Pauli's exclusion principle.
1
 no decuplet of baryons for spin- observed in data !
2
Can we form a wave function for a    dds  if it was spin
FK7003

29
The quark model works
Possible
mutliplets of
ground state
baryons with
similar properties
Spin 1/2 octet
Observed Predicted to
in nature ? exist by the
quark
model


Comments
PA
 spin
 PA
flav  (part antisym  part antisym)
 symmetric.
Spin 3/2 octet
X
X
S
 spin
 PA
flav  (sym  part. antisym)
 can't be symmetric.
Spin 1/2 singlet
X
X
PA
A
 spin
 flav
(part antisym  full antisym )
can't be symmetric.
Spin 3/2 singlet
X
X
S
A
 spin
 flav
(sym  full antisym)
 can't be symmetric.
Spin 1/2 decuplet
X
X
PA
 spin
 Sflav
part antisym  sym
 can't be symmetric.
Spin 3/2 decuplet


S
 spin
 Sflav  (sym  sym)
 symmetric.
FK7003
30
Success of SU(3)
With SU(3) symmetry and u,d,s quark model
can understand the existence and basic
properties of all light ground state hadrons
SU(2) symmetry is ”good” - all hadrons in an
isospin multiplet have similar masses since
u,d have similar masses.
SU(3) symmetry is ”approximate” – larger
mass differences owing to difference of s
and u,d, masses.
To include charm, use SU(4) symmetry –
useful to enumerate states but a poor
symmetry owing to the heavy charm mass.
FK7003
31
A historical note
Like all good theories, the theory of
quarks came with a prediction.
3
The   sss  (spin ) state had not
2
been measured when the theory was

developed. When measured this helped
to confirm the quark model.


If the    sss  was not present in nature
the whole theory would have been dead.
FK7003
32
Baryon magnetic moments
Need to check if quarks are not just a simple mathematical model.
Wave functions tell us combinations of quarks and spins.
 predict magnetic moments and test against octet baryons.
Magnetic dipole moment of a point-like particle:
qSˆz
1
1
ˆ
z 
; Spin- ; S z   .
m
2
2
Spin-up baryon B:  B  aq1    q 2    q 3    +bq1    q 2    q 3     ...
 ˆ z  B  a   1z   z2   z3  q1    q 2    q 3     b    1z   z2   z3  q1    q 2    q 3     .....
i.e. not an eigenstate
 need to find expectation value:  zB   B | ˆ z |  B
Proton:
1  2u    d    u     u    d    u     u    d    u     d    u    u     2d   u   u   


18  d    u    u     u    u    d     u    u    d     2u    u    d   


2
2 u
2 d
2 u
u
d
u
u
d
u
d
u
u
2 d
2 u
2 u

2


2


2




















2


2


2
z 
1


z
z
z
z
z
z
z
z
z
z
z
z
z
z
  zp  

  d
u
u
u
u
d
u
u
d
2 u
2 u
2 d
 18     z   z   z   z   z   z   z   z   z  2  z  2  z  2  z

1
1
 zp =  24 zu  6 zd    4 zu   zd 
18
3
FK7003
33
Baryon magnetic moments
qS z
q 
m
2S z
 u 
3mu
Sz
; d  
3md
Sz
; s  
3ms
Quark constituent masses in baryons: mu  md  363 MeV ; ms  538 MeV
 Predict all octet baryon magnetic moments.
Baryon
p
n
L

0


0

Moment
4 u 1 d
z  z
3
3
4 d 1 u
z  z
3
3
zs
4 u 1 s
z  z
3
3
2 u
1
 z   zd    zs

3
3
4 d 1 s
z  z
3
3
4 s 1 u
z  z
3
3
4 s 1 d
z  z
3
3
Prediction Experiment
2.79
2.793
-1.86
1.913
-0.58
-0.61
2.68
2.33 +/- 0.13
0.82
No measurement
-1.05
-1.41 +/- 0.25
-1.40
-1.253 +/- 0.014
-0.47
-0.69 +/- 0.04
Good agreement with data!
FK7003
34
Summary
●
●
●
●
Hadrons with similar properties occur in
multiplets in nature
The quark model is based on a few simple
postulates
The quark model successfully describes the
observed multiplet structure
Still to show evidence that ”quarks” are physical
objects (next lecture)
FK7003
35