Modern Hadron Spectroscopy : Challenges and Opportunities Adam Szczepaniak, Indiana University/Jefferson Lab Lecture 1: Hadrons as laboratory for QCD: • Introduction to QCD • Bare vs effective effective quarks and gluons • Phenomenology of Hadrons Lecture 2: Phenomenology of hadron reactions • Kinematics and observables • Space time picture of Parton interactions and Regge phenomena • Properties of reaction amplitudes Lecture 3: Complex analysis Lecture 4: How to extract resonance information from the data • Partial waves and resonance properties • Amplitude analysis methods (spin complications) INDIANA UNIVERSITY
Constructing partial waves (from singularities) 2 K-matrix K-matrix with Chew-Mandelstam “phase space” N/D Coupled channels Flatte formula Isobar model Kinematical singularities (example) Veneziano Amplitude INDIANA UNIVERSITY
Accessing QCD resonances 3 Effective potential between meson Z V eff ( R ) ∼ ψ 1 ψ 2 V Q ¯ Q ψ 3 ψ 4 | n ih n | High spin/mass resonances generated by X V eff + · · · T = V eff + V eff confining V QQ interact with multi-hadron E � E n intermediate states and screen the quark- n antiquark potential Both potentials produce ∞ number of poles ! (poles cannot disappear) Confined : ∞ n number of bound states in the continuum. No parton production thresholds Screened : Large decay withers to mupltipartilce finals states (cover rapidity gap in inclusive production) INDIANA UNIVERSITY
Accessing QCD resonances 4 e.g. for harmonic oscillator confinement E = 2n + l INDIANA UNIVERSITY
Accessing QCD resonances 5 Coupling to multi-hadron asymptotic states increases with mass/spin of the resonance INDIANA UNIVERSITY
Scattering through resonances 6 As energy increases Γ (E) receives ∞ number of poles contributions from intermediate states with an increasing number of hadrons. INDIANA UNIVERSITY
7 ⇣ g 2 ⌘ Π r r − E − E r f ( E ) = ⇣ ⌘ − g 2 1 − i Π r k r r E − E r One resonance, (e.g. ρ ) decaying to 2 π (µ=0.134) g 2 p f ( s ) = (0 . 77 2 − i 0 . 12 4 × 0 . 134 2 − s ) − s gives a pole at √ s=0.767 − 0.056 (on II nd sheet) K(s) has ( ∞ number of) poles 1 K − 1 = Π r ( E − E r ) f ( s ) = K − 1 ( s ) − i Γ ( s ) Γ (s) Contains effects of multi particles thresholds INDIANA UNIVERSITY
8 With ∞ number of resonances this formula doesn’t make sense f(s) → 0 K -1 (s) needs to have ∞ number of poles (K(s) needs zeros) Example Quadratically spaced radial trajectories r 2 − s ∼ cos( π √ s ) ∞ g 2 1 X X K ( s ) = r sin( π √ s ) r − s → m 2 r =1 r Linearly spaced radial trajectories (Veneziano) K ( s ) ∼ Γ ( a − s ) Γ ( b − s ) INDIANA UNIVERSITY
Unitarization 9 K(s) has poles and zeros K ( s ) f ( s ) = Phase space, besides unitarity branch point 1 − iK ( s ) ρ ( s ) has spurious singularities K ( s ) = P ( s ) Zeros Q ( s ) Poles P ( s ) cos( π s ) f ( s ) = f ( s ) = s tr ds 0 ρ ( s 0 ) P ( s 0 ) s tr ds 0 ρ ( s 0 ) cos( π s 0 ) Q ( s ) − 1 sin( π s ) − 1 R R s 0 � s s 0 � s π π If P(s) has ∞ of zeros it is necessary to 1 Γ ( b � s ) f ( s ) = “divide” out asymptotic behavior Γ ( a � s ) − 1 1 1 R s tr ds 0 ρ ( s 0 ) Γ ( b � s 0 )( s 0 � s ) π 1 Γ ( b � s ) f ( s ) = ( s 0 ) b � s 0 Γ ( a � s ) − s � b + s 1 R s tr ds 0 ρ ( s 0 ) Γ ( b � s 0 )( s 0 � s ) π “Almost correct”, need to remove phases) INDIANA UNIVERSITY
Other effects 10 Non-quark model resonances (tetraquarks) Yukawa exchange (possibly relevant of when pion exchange) INDIANA UNIVERSITY
Relativistic case 12 q q s = ( p 1 + p 2 ) 2 = ( p 2 + m 2 p 2 + m 2 2 ) 2 s 1 + s tr = ( m 1 + m 2 ) 2 E From u/t channel s INDIANA UNIVERSITY
Other effects of partial wave analyticity 13 Scalar particle scattering 1+2 -> 3 + 4 Z A l ( s ) = dz s A ( s, t ( s, z s ) , u ( s, z s )) P l (cos θ ) Partial waves have “right hand” singularity (from s) and “left hand” (from t and u) For example assume equal masses t = − ( s − 4 m 2 ) (1 − z s ) ∝ ( m 2 e − t ( s, z s )) − 1 2 Z 1 1 A 0 ( s ) ∼ dz s e + ( s − 4 m 2 ) m 2 (1 − z s ) − 1 2 For s>4m 2 integral is finite For s<4m 2 - m e2 the detonator crosses 0 within integration limi, implying A 0 (s) has a cut for negative s Scalar amplitudes have simple singularity structure, but partial waves a much more complicated. They also have kinematical singularities when spin and/or unequal masses are involved INDIANA UNIVERSITY
Bound states and Virtual States • f0(980), 14 • a0(980), • a1(1420), Deuteron the np molecule bound by meson exchange • Lambda(1405), V(r) forces • XYZ, • … r 3 S 1 S • Thresholds are “windows” to singularities (particles, visual 3 S 1 S states, forces” ) located on the nearby unphysical sheet. • They appear as cusps (if below Threshold threshold) or bumps (is above) bound state : pole on the thresholds “cut” physical energy plane the physical energy plane virtual state : pole on “unphysical sheet” closest the physical region II(-) INDIANA UNIVERSITY
Example : B-> J/psi K pi 15 J/ Ψ (1) B(2) → K(3) π (4) : s-channel _ _ 1 3 J/ Ψ (1) K(3) → B(2) π (4) : t-channel _ _ J/ Ψ (1) π (4) → K(3) B(2) : u-channel 2 4 _ B(2) → J/ Ψ (1) K(3) π (4) : decay INDIANA UNIVERSITY
Veneziano model and application to Dalitz plot analysis Adam Szczepaniak, Indiana U./JLab Historical role Properties Application to J/ ψ → 3 π decays Generalizations (dual models) INDIANA UNIVERSITY
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J/ ψ dual model “standard” (isobar) J/ ψ BESIII, Phys.Lett. B710 (2012) 594-599 ψ ’ dual model ψ ’ manifestation of PRELIMINARY force - particle duality ? A ( s, t ) = Γ ( − J ( s )) Γ ( − J ( t )) Γ ( − J ( s ) − J ( t )) ω → 3 π INDIANA UNIVERSITY
Properties: • Duality: resonances in direct channel dual to reggeons in cross channels and backgrounds are dual to the pomeron • All resonances are “connected”: resonances belong to Regge trajectories (reggeons) • Asymptotics: determined by Regge poles • Unitarity: imaginary parts determined by decay thresholds Veneziano amplitude satisfies all of the above except unitarity, which implemented in the Szczepaniak- Pennington model INDIANA UNIVERSITY
Veneziano amplitude: “compact” expression for the full amplitude A ( s, t ) = Γ ( − α ( s )) Γ ( − α ( t )) α ( s ) = a + bs Γ ( − α ( s ) − α ( t )) resonance/reggeon in s=m 122 β (s) _______ [k - α (t)] β (t) _______ ~ B.W. propagator [k - α (s)] resonance/reggeon in t=m 232 A(s,t) can be written as sum over resonances in ether channel. β k ( t ) β k ( s ) X X A ( s, t ) = k − α ( s ) = k − α ( t ) k k V-model resonance couplings, β , are fixed! Note: in β k ( t ) ∝ (1 + α ( t ))(2 + α ( t )) · · · ( k + α ( t )) INDIANA UNIVERSITY
or meson with momentum p and , V ( p, � ) → ⇡ i ( p 1 ) ⇡ j ( p 2 ) ⇡ k ( p 3 ) 2 p β A ( s, t, u ) = ✏ ijk ✏ µ ναβ ✏ µ ( p, � ) p ν 1 p α 3 × [ A n,m ( s, t ) + A n,m ( s, u ) + A n,m ( t, u )] l and confinement pr leading 1 st 2 nd 3 rd ↵ ( s ) = ↵ 0 + ↵ 0 s . 5 singularities of A 4 α ( s ) = 1 3 2 + s 2 1 no-double poles n ≥ m ≥ 1 s 1 s 2 s 3 s 4 s 5 s 6 s Regge limit A n,m ( s, t ) ≡ Γ ( n − ↵ s ) Γ ( n − ↵ t ) Γ ( n + m − ↵ s − ↵ t ) . INDIANA UNIVERSITY
Resonances couplings, β , should depend on final state particles: a linear superposition of Veneziano amplitudes can be used to suppress or enhance individual resonances or trajectories 1 p ν 2 p α 3 ✏ β A ( s, t, u ) M = ✏ µ ναβ p µ Γ ( n − α ( s )) Γ ( n − α ( t )) � X A = Γ ( n + m − α ( s ) − α ( t )) + ( s, u ) + ( t, u ) Re α (s) c n,m l Re α (s) = a + b s leading n,m 1 st 2 nd 3 rd ρ 5 5 4 • even-spin ρ ’s do not couple ρ 3 ρ 3 (1690 to π π and should decouple 3 ρ 3 (1990) in J/ ψ→ 3 π 2 • coupling of odd-spin ρ ’s ρ (770 ρ (1570 ρ depend of can depend vary 1 depending on trajectory ρ ρ (1450) s s 5 s s 1 s 2 s 3 s 4 s 6 INDIANA UNIVERSITY
∞ ∞ β ( t ) β ( s ) X X A n,m ( s, t ) = k − α ( s ) = k − α ( t ) k k how to remove (infinite) number of poles? t n ≥ m ≥ 1 A 1 , 1 = Γ (1 − α s ) Γ (1 − α t ) has poles at α s =1,2,3,... Γ (2 − α s − α t ) A 2 , 1 = Γ (2 − α s ) Γ (2 − α t ) Γ (3 − α s − α t ) have poles at α s =2,3,4,... s Use a linear combination of A 2,1 A 2 , 2 = Γ (2 − α s ) Γ (2 − α t ) and A 2,2 to remove pole at α s =2 Γ (4 − α s − α t ) Use a linear combination of A 3,1 , A 3,2 have poles at α s =3,4,5,... ,A 3,3 , to remove pole at α s =3, A 3 , 1 , A 3 , 2 , A 3 , 3 etc. A 4 , 1 , A 4 , 2 , A 4 , 3 , A 4 , 4 have poles at α s =4,5,6,... INDIANA UNIVERSITY
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