V. Hadron quantum numbers

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1 V. Hadron qantm nmbers Characteristics of a hadron: 1) Mass 2) Qantm nmbers arising from space-time symmetries : total spin J, parity P, charge conjgation C. Common notation: J P (e.g. for proton: ), or J PC if a particle is also an eigenstate of C-parity (e.g. for π 0 : 0 -+ ) 3) Internal qantm nmbers: charge Q and baryon nmber B (always conserved), S, C, B, T (conserved in e.m. and strong interactions) How do we know which are the qantm nmbers of a newly discovered hadron? How do we know that mesons consist of a qark-antiqark pair, and baryons - of three qarks? Pala Eerola Lnd University 123

2 Some a priori knowledge is needed: Particle Mass (GeV/c 2 ) Qark composition Q B S C B p d n dd K s D dc B b Considering the lightest 3 qarks (, d, s), possible 3-qark and 2-qark states will be (q i,j,k are - or d- qarks): sss ssq i sq i q j q i q j q k S Q -1 0; -1 1; 0; -1 2; 1; 0; -1 ss sq i sq i q i q i q i q j S Q 0 0; -1 1; 0 0-1; 1 Hence restrictions arise: for example, mesons with S= -1 and Q=1 are forbidden Pala Eerola Lnd University 124

3 Particles which fall ot of above restrictions are called exotic particles (like dds, ds etc.) From observations of strong interaction processes, qantm nmbers of many particles can be dedced: p + p p + n + π + Q= S= B= p + p p + p + π 0 Q= S= B= p + π - π 0 + n Q= S= B= Observations of pions confirm these predictions, proving that pions are non-exotic particles. Pala Eerola Lnd University 125

4 Assming that K - is a strange meson, one can predict qantm nmbers of Λ-baryon: K - + p π 0 +Λ Q= S= B= And frther, for K + -meson: π - + p K + + π - + Λ Q= S= B= Before 2003: all of the more than 200 observed hadrons satisfy this kind of predictions Jly 2003: Pentaqark dds observed in Japan! Simple qark model -- only qark-antiqark and 3-qark (or 3-antiqark) states exist -- has to be revised! Pala Eerola Lnd University 126

5 Figre 42: Schematic pictre of the pentaqark prodction. Pentaqark observation: qark composition dds, mass 1.54 GeV, B = +1, S = +1, spin = 1/2. Rssian theorists predicted in 1998 a pentaqark state with these qantm nmbers, having a mass 1.53 GeV! Observation in Japan: 19 events. The signal has been confirmed in other experiments. Pala Eerola Lnd University 127

6 Figre 43: Left: invariant mass distribtion of the photon-proton reactions, prodcing K + Λ(1520). Right: invariant mass distribtion of the photon-netron reactions, prodcing Z + (1540). Pala Eerola Lnd University 128

7 The reaction for prodcing the pentaqarks was: γ + n Z + (1540) + K - K + + K - + n Laser beam was shot to a target made of 12 C (n:p=1:1). A reference target of liqid hydrogen (only protons) was sed. In the p-target, the following reaction occrred: γ + p K + Λ(1520) K + + K - + p (visible in the Fig. 43, left) Figre 44: Qark diagram for pentaqark prodction. Pala Eerola Lnd University 129

8 Even more qantm nmbers... It is convenient to introdce some more qantm nmbers, which are conserved in strong and e.m. interactions: Sm of all internal qantm nmbers, except of Q, Instead of Q : hypercharge Y B + S + C + + T I 3 Q - Y/2 which is to be treated as a projection of a new vector: Isospin I (I 3 ) max so that I 3 takes 2I+1 vales from -I to I It appears that I 3 is a good qantm nmber to denote p- and down- qarks, and it is convenient to se notations for particles as I(J P ) or I(J PC ) B Pala Eerola Lnd University 130

9 B B S C T Y Q I 3 I 1/ /3 2/3 1/2 1/2 d 1/ /3-1/3-1/2 1/2 s 1/ /3-1/3 0 0 c 1/ /3 2/3 0 0 b 1/ /3-1/3 0 0 t 1/ /3 2/3 0 0 Hypercharge Y, isospin I and its projection I 3 are additive qantm nmbers, so that corresponding qantm nmbers for hadrons can be dedced from those of qarks: Y a I a + b Y a Y b ; Ia + b 3 = + = Ia 3 + Ib 3 + b I a I b I a I b 1 I a I b = +, +,, Proton and netron both have isospin of 1/ 2, and also very close masses: p(938) = d ; n(940) = dd : proton and netron are said to belong to an isospin doblet, p 1 2 n I3 = 1 2 IJ ( ) P 1 = I3 Pala Eerola Lnd University 131

10 Other examples of isospin mltiplets: K + (494) = s ; K 0 (498) = ds : IJ ( ) P 1 = -- ( 0) - 2 π + (140) = d ; π - (140) = d : π 0 (135) = (-dd)/ 2 : IJ ( ) P = 10 ( ) - IJ ( ) PC = 10 ( ) - + Principle of isospin symmetry: it is a good approximation to treat - and d-qarks as having same masses. Particles with I=0 are isosinglets : Λ(1116) = ds, IJ ( ) P = By introdcing isospin, we imply new criteria for non-exotic particles. S, Q and I for baryons are: sss ssq i sq i q j q i q j q k S Q -1 0; -1 1; 0; -1 2; 1; 0; -1 I 0 1/2 0; 1 3/2; 1/2 Pala Eerola Lnd University 132

11 S, Q and I for mesons are: ss sq i sq i q i q i q i q j S Q 0 0; -1 1; 0 0-1; 1 I 0 1/2 1/2 0; 1 0; 1 In pre-2003 observed interactions these criteria are satisfied as well, confirming the qark model. Pentaqark dds: S=+1, B= +1, Q= +1, Y B + S = +2, I 3 Q - Y/2 = 0 => I=0. On the other hand, pentaqarks are rare -- we can still predict new mltiplet members based on the simple qark model. Sppose we observe prodction of the Σ + baryon in a strong interaction: K - + p π - +Σ + Q= S= B= which then decays weakly : Σ + π + + n, OR Σ + π 0 + p It follows that Σ + baryon qantm nmbers are: B=1, Pala Eerola Lnd University 133

12 Q=1, S= -1 and hence Y=0 and I 3 =1. Since I 3 >0 I 0, so there are more mltiplet members! If a baryon has I 3 =1, the only possibility for isospin is I=1, and we have an isospin triplet: S +, S 0, S - Indeed, all these particles have been observed: K - + p π 0 + Σ 0 K - + p π + + Σ - Λ + γ π - + n Masses and qark composition of Σ-baryons are: Σ + (1189) = s ; Σ 0 (1193) = ds ; Σ - (1197) = dds It indicates that the d-qark is heavier than the -qark nder the following assmptions: a) Strong interactions between qarks do not depend on their flavor. Strong i.a. give a contribtion M o to the baryon mass. b) Electromagnetic i.a. contribte as δ, e i e j Pala Eerola Lnd University 134

13 where e i are qark charges and δ is a constant. The simplest attempt to calclate the mass difference of the p and down qarks: M(Σ - ) = M 0 + m s + 2m d + δ/3 M(Σ 0 ) = M 0 + m s + m d + m - δ/3 M(Σ + ) = M 0 + m s + 2m m d - m = [ M(Σ - ) + M(Σ 0 ) -2M(Σ + ) ] / 3 = 3.7 MeV NB : this is a very simple model, becase nder these assmptions M(Σ 0 ) = M(Λ), however, their mass difference M(Σ 0 ) - M(Λ) 77 MeV. Generally, combining other methods: 2 MeV m d - m 4 MeV which is negligible compared to hadron masses (bt not if compared to the estimated and d masses themselves: m = ( ) MeV, m d = ( ) MeV). Pala Eerola Lnd University 135

14 Resonances Resonances are highly nstable particles which decay by the strong interaction (lifetimes abot s). L=0 1 S 0 3 S 1 d d I(J P )=1(0 - ) I(J P )=1(1 - ) grond state resonance Figre 45: Example of a qq system in grond and first excited states If a grond state is a member of an isospin mltiplet, then resonant states will form a corresponding mltiplet too. Pala Eerola Lnd University 136

15 Since resonances have very short lifetimes, they can only be detected by registering their decay prodcts: π - + p n + X A + B Invariant mass W = m X of the resonance X is measred via energies and 3-momenta of its decay prodcts, particles A and B. By sing 4-vectors (see eqs. 10 and 11 for scalar prodct of two 4-vectors), W can be calclated as : p x 2 = ( + ) 2 2 = m x p A p B W 2 ( p A + p B ) 2 p A + p B = ( ) ( p A + p B ) = ( E A + E B ) 2 ( p A + p B ) 2 = E 2 2 X p X = m2 X = W 2 (82) Pala Eerola Lnd University 137

16 Figre 46: A typical resonance peak in K + K - invariant mass distribtion Pala Eerola Lnd University 138

17 Resonance peak shapes are approximated by the Breit-Wigner formla: NW ( ) = K ( W W 0 ) 2 + Γ 2 4 (83) N(W) N max N max /2 Γ W 0 W Figre 47: Breit-Wigner shape Mean vale of the Breit-Wigner shape is the mass of the resonance: M=W 0 Γ is the width of the resonance, and it is the inverse of the mean lifetime of the particle at rest: Γ 1/τ. Pala Eerola Lnd University 139

18 Internal qantm nmbers of resonances are derived from their decay prodcts. For example, if a resonance X 0 decays to 2 pions: X 0 π + + π - then B = 0 ; S = C = B = T = 0 ; Q = 0 Y=0 and I 3 =0 for X 0. To determine whether I=0 or I=1, searches for isospin mltiplet partners have to be done. Example: ρ 0 (769) and ρ 0 (1700) both decay to π + π - pair and have isospin partners ρ + and ρ - : π ± + p p + ρ ± π ± + π 0 By measring the anglar distribtion of the π + π - pair, the relative orbital anglar momentm of the pair, L, can be determined, and hence spin and parity of the resonance X 0 are (S=0): J = L ; P = P2 π ( 1) L = ( 1) L ; C = ( 1) L Pala Eerola Lnd University 140

19 Some excited states of pion: resonance I(J PC ) L ρ 0 (769) 1(1 - - ) 1 f 2 0 (1275) 0(2 ++ ) 2 ρ 0 (1700) 1(3 - - ) 3 Resonances with B=0 are meson resonances, and with B=1 baryon resonances. Many baryon resonances can be prodced in pion-ncleon scattering: π ± } X p R N Figre 48: Formation of a resonance R and its sbseqent inclsive decay into a ncleon N Peaks in the observed total cross-section of the π ± p-reaction correspond to resonance formation - see Fig. 49. Pala Eerola Lnd University 141

20 Figre 49: Scattering of π + and π - on proton Pala Eerola Lnd University 142

21 All resonances prodced in pion-ncleon scattering have the same internal qantm nmbers as the initial state: p+π +- R B = 1 ; S = C = B = T = 0 and ths Y=1 and Q=I 3 +1/2. Possible isospins are I=1/2 or I=3/2, since for pion I=1 and for ncleon I=1/2 I=1/2 N-resonances (N 0, N + ) I=3/2 -resonances (, 0, +, ++ ) At Figre 49, peaks at 1.2 GeV correspond to ++ and 0 resonances: π + + p ++ π + + p π - + p 0 π - + p π 0 + n Fits with the Breit-Wigner formla show that both ++ and 0 have approximately same mass of 1232 MeV and width 120 MeV. Pala Eerola Lnd University 143

22 Stdies of anglar distribtions of the decay prodcts show that IJ ( P 3 ) = Remaining members of the mltiplet are also observed: + and - There is no lighter state with these qantm nmbers is a grond state, althogh a resonance. Pala Eerola Lnd University 144

23 Qark diagrams Qark diagrams are convenient way of illstrating strong interaction processes. Consider an example: ++ p + π + The only 3-qark state consistent with ++ qantm nmbers is (), while p=(d) and π + =(d). ++ d d π + p Figre 50: Qark diagram of the reaction ++ p + π + Analogosly to Feinman diagrams: arrow pointing to the right denotes a particle, and to the left antiparticle time flows from left to right Pala Eerola Lnd University 145

24 Allowed resonance formation process: π + d ++ d π + p d d p Figre 51: Formation and decay of ++ resonance in π + p elastic scattering Hypothetical exotic resonance: Κ + s Ζ ++ s Κ + p d d p Figre 52: Formation and decay of an exotic resonance Z ++ in K + p elastic scattering Qantm nmbers of sch a particle Z ++ are exotic, moreover, there are no resonance peaks in the corresponding cross-section: Pala Eerola Lnd University 146

25 Center of mass energy (GeV) Figre 53: Cross-section for K + p scattering Pala Eerola Lnd University 147

26 SUMMARY Characteristics of a hadron: Mass; Space-time qantm nmbers total spin J, parity P, charge conjgation C; Internal qantm nmbers charge Q and baryon nmber B (always conserved), flavor qantm nmbers S, C, B, T (conserved in e.m. and strong interactions). The possible 3-qark and 2-qark states can be defined sing the conservation rles of the qantm nmbers. Other states are defined as exotic states - like pentaqarks. Qantm nmbers, which are conserved in strong and e.m. interactions: hypercharge Y B + S + C + B + T, and I 3 Q - Y/2, which is the third component of isospin I. I 3 is a good qantm nmber to denote - and d-qarks. Isospin symmetry: approximate symmetry which assmes that - and d-qarks have the same masses. Resonances are highly nstable particles which decay by the strong interaction (lifetimes Pala Eerola Lnd University 148

27 abot s). Resonances can only be detected by registering their decay prodcts: invariant mass W = mass of the resonance = invariant mass of the decay prodcts. Internal qantm nmbers of resonances are derived from their decay prodcts as well. Resonance peak shapes are approximated by the Breit-Wigner formla which gives N (nmber of events) as the fnction of W. The vale W=W 0 which gives N max is the mass of the resonance, and Γ, the width of the resonance, is the inverse of the mean lifetime of the particle at rest: Γ 1/τ. Qark diagrams = Feynman-like diagrams for strong interaction processes. Only qark lines are drawn, no glons, no vertices. Pala Eerola Lnd University 149