The Quark Model

Introduction to Elementary Particle Physics

Diego Bettoni Anno Accademico 2010-2011 Outline

• Symmetries and groups (overview) • The flavor symmetry

•q1q2 states: mesons – Pseudoscalar and vector mesons – Zweig rule

•q1q2q3 states: baryons • Color • Hadron masses and magnetic moments Symmetries and Groups

Illustrative example: the rotation group

• Two successive rotations R1 followed by R2 are equivalent to a single rotation R=R2R1. The group is closed under multiplication. • There is an identity element (no rotation). • Every rotation R has an inverse R-1 (rotate back again).

• The product is not necessarily commutative R1R2  R2R1, but the associative law always holds: R3(R2R1) = (R3R2)R1. • It is a continuous group: each rotation can be labeled by a set of

continuously varying parameters (1, 2, 3) which can be regarded as the components of a vector (1, 2, 3) directed along the axis of rotation with magnitude given by the angle of rotation. • The rotation group is a Lie group: every rotation can be expressed as the product of a succession of infinitesimal rotations (arbitrarily close to the identity). The group is then completely defined by the “neighborhood of the identity”. • Rotations are a subset of the Lorentz transformations and they form a symmetry group of a physical system: Physics is invariant under rotations. For example suppose  that under a rotation R the state of a system transforms as R    U  Probabilities must be unchanged by R: 2 2  2     U U 

U+U = 1, U must be a unitary operator. The operators U(R) form a group, with exactly the same structure as the

original group (R1,R2,…): they are said to form a unitary representation of the rotation group.  • The Hamiltonian is unchanged by a symmetry operation R of the system and the matrix elements are preserved    H   U  HU  H  U  HU  H [U, H]  0 • The transformation U has no explicit time dependance and the equation of motion  d i (t)  H  (t) dt is unchanged by the the symmetry operation. As a consequence the  expectation value of U is a constant of the motion  d i U  UH  HU   0 dt • All group properties follow from considering infinitesimal rotations in the neighborhood of the identity. Example rotation through  around the 3-axis: U 1 iJ  3 J is called the generator of rotations around the 3-axis. 3     1  U U  (1 i J 3 )(1 i J 3)  2  1 i (J 3  J 3 )  O( ) + J3 = J3 is therefore hermitian, and hence is an observable. • Consider the effect of a rotation R on the wave function. Invariance under rotations requires:  (r)  (R 1r) U(r) For an infinitesimal rotation  around the 3-axis: U (x, y, z)  (R 1r)  (x  y, y  x, z)      (x, y, z)  (y  x ) U  1 iJ3   x y    1 i(xpy  ypx )

J3 component along the 3-axis of the angular momentum. Invariance under rotations corresponds to the conservation of angular momentum. For a rotation through a finite angle :

 n

n    iJ3 U ( )  [U ( )]  1i J 3   e  n  n

The commutator algebra of the generators is:

Jm ,Jn  imnl Jl The J’s are said to form a Lie Algebra

ijk = structure constants of the group Nonlinear functions of the generators which commute with all the generators are called invariants or Casimir operators. For the rotation group the only Casimir operator is: 2 2 2 2 2 J  J1  J2  J3 [ J , J l]  0 l 1,2,3 It follows that we can construct simultaneous eigenstates of J2 and one of the generators, e.g. J3: J 2 j,m  j( j 1) j,m

J 3 j,m  m j,m The Group SU(2)

In the lowest-dimension  nontrivial representation of the rotation group (j=½) the generators may be written: 1 J  k 1,2,3  matrici di Pauli k 2 k k 0 1 0  i 1 0           1 1 0 2  i 0  3  0  1 1 0 The basis for this representation is given by the eigenvectors of  :    3 0 1 describing a spin ½ particle of spin projections up and down.

The transformation matrices: i ii 2 U(i )  e are unitary. The set of all unitary 22 matrices is U(2) (Unitary Group). However

U(2) is larger than the group U(i), since the i all have zero trace. For any hermitian traceless matrix  it can be shown that: det ei  eiTr ( ) 1 This property is preserved in matrix multiplication. The set of traceless unitary 22 matrices form a subgroup of U(2) called SU(2) (Special Unitary). The two-dimensional representation is the fundamental representation. For a composite system |jA,jB,mA,mB> the operator J=JA+JB satisfies the Lie 2 algebra and the eigenvalues J(J+1) and M of J and J3 are conserved quantum numbers. The product of the two irreducible representations (2jA+1) and (2jB+1) may be decomposed into the sum of irreducible representations of dimension (2J+1) with basis |jA,jB,J,M>, where

|jA-jB|  J  |jA+jB| M=mA+mB

jA jB JM  C(mA,mB , J , M ) jA jBmAmB mA ,mB

Clebsch-Gordan coefficients Example: from two two-dimensional representations (j=½) we obtain one 3-dimensional (J=1, triplet) and one 1-dimensional 1 (J=0, singolet) representation. 2 2  31 2 2 2  (3 2) (1 2)  4  2  2

quadruplet doublet spin 3/2 spin 1/2 SU(2) of Isospin

Im ,In  imnlIl Generators in the fundamental representation:  1 I k   k k  1,2,3  2 0 1 0  i 1 0           1 1 0 2  i 0  3  0  1 basis 1 0 p    n    0 1 1 1 1 1 I  2 , I3   2 I  2 , I3   2 Isospin for Antiparticles

 p  i 2 p   p  0 1 p 2 Cp  p Cn  n   e    i 2       n   n  n  1 0  n   p 0 1 p       n  1 0  n  In order for the antidoublet to transform in  n 0 1 p the same way as the doublet we must:       •Reorder the doublet p 1 0 n •Introduce a minus sign     

 I 1, I3 1   pn  1 For NN  I 1, I3  0  2 ( pp  nn)   I 1, I3  1  np 1 I  0, I3  0  2 ( pp  nn) The Group SU(3)

It is the group of unitary 33 matrices with detU=1. The generators may be taken to be any 32-1=8 linearly independent traceless hermitian 33 matrices. There are therefore 8 generators, of which 2 are diagonal. This is also the maximum number of mutually commuting generators : Rank of the group. It can be shown that the rank of the group is equal to the number of Casimir operators. The fundamental representation of SU(3) is a triplet (e.g. the three color charges of a quark ). The generators are 33 matrices: i, i=1,..,8 (Gell-Mann matrices).  λ8 1  1      3  1 8  1     1 (λ + i λ ) 0  2 2 1- 2     G(-1,1 ) R(1,1 ) 3 3 λ 1 0 0 3       R  0 G  1 B  0 1 λ + i λ 1 (λ + i λ5)       2 ( 6 - 7) 2 4- 0 0 1 2 B(0,- ) simultaneous eigenvectors of 3and 8. 3 Isospin and Strangeness: Flavor SU(3)

The introduction of a second additive quantum number S in addition to

I3 suggests to enlarge isospin symmetry to a larger group, a group of rank 2. In 1961 SU(3) was proposed. The assignment of particles to SU(3) multiplets is not straightforward due to the high mass differences between the various particles (strange and non strange). For example, the baryonic octet group particles with mass differences up to 400 MeV, over an average octet mass of 1100 MeV. SU(3) flavor symmetry is much more approximate than SU(2) of isospin. We will see that this is due to the fact that the strange quark s is much heavier than u and d. SU(3) symmetry forms the basis of the quark model and it turns out to be very useful to classify hadrons and to understand some of their properties. Color SU(3) on the other hand is an exact symmetry of fundamental origin. The Quark Model

Already in 1949 Fermi and Yang, noticed that the  meson has the same 1 quantum numbers of an NN pair in an S0 state: (B=0, JP=0-, S=0, Q=0,1). They suggested that the  could be considered as an NN bound state (with a very high binding energy). With the discovery of strange particles Sakata proposed to extend the model to include the . This triplet allows to “construct” some particles (all particles known in the 1960s), but there were several reasons to prefer a different set of more fundamental constituents: The regularity with which particle families occured in nature suggested an internal structure also for p,n, . These regularities could be well explained by an SU(3) symmetry. In 1964 Gell-Mann and, independently, Zweig proposed a set of 3 elementary fermions in terms of which all hadrons could be constructed. Gell-Mann called them quarks, Zweig called them aces. • They must be fermions in order to construct both fermions and bosons.

• In analogy to the idea of Fermi and Yang, mesons are q1q2 pairs; baryons (antibaryons) are q1q2q3 (q1q3q3) states. • In order to form nonstrange particles with charges 0,1 at least two quarks are needed. These must form an isospin doublet (in order to have both I=0 and I=1). • In order to form strange particles a third quark is needed, to which by convention is assigned S=-1. The minimal number of constituents is thus 3. • To properly account for baryon numbers quarks are assigned B=1/3. • The simplest spin-parity assignment is JP=½+. B  S • From the Gell-Mann and Nishijima formula Q  I  it follows 3 2 2 1 Q(u)   Q(d )  Q(s)   3 3 Quarks have a fractional electric charge. J B S Y I I3 Q 1 1 1 1 1 2 u 2 3 0 3 2 2 3 1 1 1 1 1 1 d 2 3 0 3 2  2  3 1 1 2 1 s 2 3 1  3 0  3 In SU(3) language: the fundamental multiplet, from which all others can be constructed, is a triplet. u, d, s form the fundamental multiplet. • B conservation implies that it is impossible to create or destroy a single quark. • Quark-antiquark pairs can be created or annihilated. • Flavor conservation in strong interactions implies that flavor-changing transitions can take place only at the weak level, for example: s  u  leptoni s  u  du etc q1q2 States: Mesons

u  d  Let us start with two quarks (u and d)     d   u  Y s  1,1  ud 1 du  3 1 0 1,0  uu  dd x  2 1 - 3  1,1  ud u d 2  - 3 s 0,0  1 uu  dd 2 =

Y 2 dus s Adding a third quark s there are 9 possible 3 combinations: 1 octet and 1 singlet (under 1 3 A transformations in SU(3) the 8 states 0 du ud C 1 - B transform among themselves, but they 3 2 never mix with the singlet). - 3 ssud

3 3 8 1 The singlet state, C, is symmetric in flavor:

1 C  3 uu  dd  ss

A is the neutral member of the isospin triplet:

1 du, A,ud  A  2 uu  dd 

Quarks have spin ½, therefore the total spin of the q1q2 pair can be S=0 or S=1. The spin J of the mesons results as the combination of S and of the relative angular momentum L. The parity P of the meson is thus: P (1)L  (1)L1

Product of the intrinsic parities of fermion and antifermion The value of C is obtained like in positronium :

C (1)S  1(1)L  (1)LS

Fermion interchange Quantum Numbers of Mesons with the Quarks u, d, s

nonet  L(qq) S(qq) J PC I  1 I  1 I  0 M (MeV )   2 L  0 S  0 0 K  ,  500  S  1 1 K *  , 800  L  1 S  0 1 B Q2 H,? 1250  * 2 A2 K f , f  1400   S  1 1 A1 Q1 D,? 1300 0 K , S* 1150 Pseudoscalar Mesons (JPC=0-+)    S I I3 S Mesone Dec M(MeV)     1 1 0   140 o +   K (ds) 1 K (us)      1 1 0   140 0 0 ππ+ 1 0 0  135 0 (ud) I  3 η     η 1 1 1 K K   494 - 2 2  π- (du) 1 1 0 0   -1 - 1 1 1  1 K K  498 2 2 2 2  1 1 1 K K  494 K- (su) 2 2  1 1  0 0   -1 1 K K  498 o 2 2  K (sd)  0 0 0    549  0 0 0   958  Vector Mesons (JPC=1--)

3 S The q1 andq2 are in a S1 state (L=0, J=1, S=1). The wave functions + (ds) 1 (us) for the singlet 0 and for the neutral *0 * K K member of the octet 8 are: +(ud) I   1 (dd  uu  ss) - 8 3 0 3 ρ- (du) φ ρ -1 - 1 0 1 1 φ ρ0 1 2 2 8  6 (dd  uu  2ss) -1 The physical states (i.e. the ones observed - (su) * in nature) are linear combinations of  and  : o 0 8 KK* (sd)  sin  cos    cos   sin   0 8 0 8 In the case of ideal mixing sin  1   350 3   ss

1   2 (uu  dd )  Meson Decay

 and the Zweig Rule    0  K  K  49.1%   88.8%  0 0 0  8.5%  K L KS 34.4%     0       15.3%   2.2% For the (1020) phase space would favor the 3 decay with respect to 2K:   Q3  M 2M  M 0  600 MeV  Q 0  M  2M 0  24 MeV Q   32 MeV 2K  K  2K

- + u s K π π + u - d - s s u u d d Φ d 0 Φ π - ω π - -s s d u u d d d -s + − π 0 K u π u- The diagram for the 3 decay is suppressed because it contains disconnected quark lines (Zweig Rule). The kinematical suppression of the 2K decays is

reflected in the small total width of the :  = (4.26  0.05) MeV to be compared, for example, with  = (150.3  1.6) MeV. Leptonic Decays of Vector Mesons

 0  1 (uu  dd ) ,, have similar masses 2 2   1 (uu  dd )  (0) 2 2  costante  MV

  ss   2 (V  l l )  Q 2 Consider the decays  0 : 1 ( 2 ( 1 ) 1  2 3   3  2   2 V  l l V  , , l  e, 1 2 1 1 1 2 1 : 2 ( 3  ( 3 )  18  : ( 3 )  9 The partial width is given by:   ( 0) : ( ) : () 9 :1: 2 2 2 16 Q 2 (V  l l )   (0)  2 Van Royen - Weisskopf MV 2 2 2 2 Q   aiQi ; MV  V mass; ml  MV

( 0  ee ) (  ee ) Experimentally 11.3  2.3 (  ee ) (  ee ) 3 Quark States: Baryons

Y

1 dudu 3 Let us now add the third quark 0 1 x - 3 2 - 3 Y s s ddd uuu 1 = 0 1 Y 1 2 dd2 (ud+du) uu (ud-du) 3 2 1 -1 3 0 1 1 1 (us+su) -2 - 3 (ds+sd) 2 2 1 1 - 2 (ds-sd) (us-su) 3 2 2 sss 4 - 3 ss 3 3 3  (6 3)  (3 3) 3 3 6  3 10881 The Baryon Decuplet

S −0+++ ddd ddu duu uuu ΔΔΔΔ 0 Δ (1232) −0+ Pauli principle ? The solution lies in the dds dus uus ΣΣΣ-1 I Σ (1384) −0 3 ΞΞ fact that quarks have a further internal Ξ dss uss -2 (1533) degree of freedom,color, which can − -3 Ω Ω(1672) sss take on three values R,G,B. Quarks  form the fundamental triplet of an P 3 • They have spin-parity J  SU(3) color symmetry. 2 • The wave function is symmetric Hadrons are color neutral, i.e. they with respect to the exchange of belong to a singlet representation of any quark pair. color SU(3). In this way the overall • They are in s wave, they have wave function is antisymmetric under parallel spins, therefore also the interchange of any quark pair. space and spin wave functions are symmetric. 1 (qqq) c. s. 6(RGB  RBG  BRG  BGR  GBR  GRB) The Baryon Octet

S

ddu duu N(939)  0 I 1 uds np0 Σ (1193) P dds uus− + Λ (1116) J  -1 2 Σ Σ 3 Σ Ξ dss − Λ 0 -2 ΞΞ (1318) uss

The octet states are completely symmetric under the simultaneous exchange of flavor and spin of any quark pair. S S

−0+++ ddd ddu duu uuu ddu duu N(939) ΔΔΔΔΔ (1232)1232 N(939) dds dus uus 0 I ΣΣΣ−0+ I uds Σ (1193) Σ (1384) dds 0 uus 1384 0 Λ -1 − (1116)+(1193) dss −0 3 np ΞΞ -1 Ξ (1533) Ξ dss 3 (1116) uss -2 1533 − Σ 0Σ Σ Ω− Ω -2 Λ -3 sss (1672)1672 ΞΞ(1318) (1318) uss

If mass differences were solely due to the fact that the s quark is heavier than the u and d quarks we should have:

P 3  J  2 (1384)  (1232)  (1533)  (1384)  (1672)  (1533) 152 MeV 149 MeV 139 MeV

P 1  J  2 (1193)  (1116) (1116)  N(939)  (1318)  (1116) 152 MeV 149 MeV The order of magnitude is correct, but discrepancies are still significant. A quantitative understanding of hadron masses must take into account the effects of the hyperfine splitting in quark interactions. Hadron Masses

If flavor SU(3) symmetry were exact all members of a given multiplet would have exactly the same mass. Yet it is not so.  m  m (uu )  0.78 GeV

P  m (ss)  1.02 GeV J  1

m * (su )  0.89 GeV K If we consider hadron masses as the sum of the masses of the constituent quarks we obtain: mu  md  0.39 GeV Effective masses of quarks bound in hadrons. Constituent masses. ms  0.51GeV There are further problems: 3 1 m (J  2 )  mN (J  2 )

m (J  1)  m (J  0)  and N contain the same quarks, as do  and . Since hadron masses cannot be explained solely in terms of the masses of the constituent quarks it is necessary to consider the effects of quark interaction. In the hydrogen atom the spin-spin interaction leads to the hyperfine structure of levels.   For two pointlike fermions of magnetic moments  i and  j the interaction    energy is  i j r3 ij   e  Dirac theory gives:   2m

The hyperfine separation is given by: 2   2 E      (0) hf  1 2 3    2  2  1 2  (0) 3 m1m2 It is a contact interaction: it contains the square of the wave function at zero separation and therefore it only applies to L=0 states. For quarks the magnetic interaction associated to charge and spin is of the order of the MeV. But quarks interact through their color charges with a potential of the form: 4  V(r)   s  kr 3 r

At small distances the term in 1/r dominates and s is small enought to make the strong hyperfine splitting important:   8s   2 4s   2 E(QQ)  1  2  (0) E(QQ)  1  2  (0) 9m1m2 9m1m2

In this scheme hadron masses are given by:

  a   m(q1q2 )  m1  m2  1  2 m1m2   a i  j m(q1q2q3 )  m1  m2  m3   2 i j mim j For two quarks (or for quark-antiquark):    i  2si      1 2 2 2 s  s1  s2  s1  s2  2 (S  s1  s2 )   Hence the eigenvalues of  1   2 are:     1 2  4s1 s2  2[S (S  1) s1 (s1  1) s2 (s2  1)]   1 S  1    3 S  0 Similarly for 3-quark systems:      i  j  4si  s j  2[S(S 1)  3s(s 1)] 3  3 S  2   1  3 S  2 3a 3 a m  mu  md  (S  0) (E)   2 mumd mu 2 a 3 a m *  m  m  (S  1) (E)   K u s N 2 mums mu 2 Using the experimentally measured mass values it is possible to fit the parameters

mu, md, ms, a and a’. The results are:

a mu  md  363 MeV 2  100 MeV mu a ms  538 MeV 2  160 MeV mu

In this way the agreement with experimental data is of the order of 1 % or better. Electromagnetic Mass Differences

A further contribution to hadron mass comes from the electromagnetic interaction. Let us take as an example the baryons in the octet and let us assume that the charge distributions are similar. We expect similar electromagnetic contributions m:  p(uud) m p  m   (uus)    (dds) m   m   (dss) 0  (uss) m 0  mn n(udd)

Let us add the bare hadron masses and sum these equations: m m m m m  m (m  m )  (m   m  )  (m   m 0 ) p   0   n p n     1.3 MeV  8 MeV  6.4 MeV   1.6 MeV Coleman-Glashow. Mass differences are associated with isospin symmetry breaking. Electromagnetic mass differences are due to three effects:

• Difference in mass of the u and d quarks; since mn>mp we expect md>mu. • Coulomb energy difference associated with the electrical energy between pairs of quarks, of the order of: e2  2 MeV R0 • Magnetic energy difference associated with the magnetic moment (hyperfine) interaction between quark pairs:

2  e  1   3 1 2 MeV  mc  R0 Fitting the exact forms of these terms to the data it is found that:

md mu  2 MeV

The approximate isospin invariance can be associated with the near equality of the u and d quark masses. Baryon Magnetic Moments

Baryon magnetic moments can be calculated as the vector sums of the moments of the constituent quarks. For a Dirac pointlike particle of mass m and charge e:  e     2m As and example let us calculate the magnetic moment of the proton (uud). The  two u quarks are in a triplet state. Combining with a further  1 ,  1  we get:   2 2 1 1 2 1 1 1 1 1 (2 ,2 )  3 (1,1) (2 , 2 )  3 (1,0) (2 ,2 )      u  u  d  u  u d  2 1 4 1 p  (2 u  d )  d    3 3 p 3 u 3 d e  p  2.79 2mp Comparison between predicted and measured magnetic moments for some baryons:   th exp p 4  1 2.79 2.793 3 u 3 d  4 1 n 3 d  3 u 1.86 1.913

  s  0.58  0.614  4  1  3 u  3 s 2.68 2.33  4 1  3 d  3 s 1.05 1.00 0 4  1 1.40 1.25 3 s 3 u  4 1  3 s  3 d  0.47 1.85