Modern Hadron Spectroscopy : Challenges and Opportunities Adam - PowerPoint PPT Presentation

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

Probing QCD resonances (using physical states) 2 • When (color neutral) mesons and baryons a smashed, their quarks overlap, “stick together” to form resonances (quasi QCD eigenstates). They are short lived and decay to lowest energy, asymptotic states (pions, K’s, proton,…) • Resonances are fundamental to our understanding of QCD dynamics since they appear beyond perturbation theory. • (QCD) Resonances challenge QFT practitioners to develop all orders calculations (still ways to go). • (QCD) Resonance lead to extremely rich phenomenology (e.g. XYZ states). • In practice, one requires tools that relate asymptotic states before collision to asymptotic states after collision that include flexible parametrization of microscopic dynamics. This is often referred to as amplitude analysis. The rest of these lectures will focus on this topic. INDIANA UNIVERSITY

Bound states/Resonances/Asymptotic states 3  p 2 1 � α = α QED = − α 137 ψ ( r ) = E ψ ( r ) 2 m e r ψ ( r ) = e − ikr − S ( E, θ ) e ikr ψ ( r ) ∝ e − α m e r r r S ( E, θ ) = 1 + O ( α ) Bound states: compact wave function contains interaction to all orders. Born approximation : lowest order perturbation on free motion Resonances: particles interact to all orders (like bound states) but eventually decay (connect with asymptotically free states). Their effect appears in the S-matrix INDIANA UNIVERSITY

Amplitude analyticity: it is much about complex functions 4 Resonances Asymptotic states Scattering Amplitude s = E 2 c.m A ( s + i ✏ ) = A physical( s = real and above threshold) Bound states Scattering amplitude describes evolution between asymptotic states. The information related to formation about resonances is “hidden” in unphysical domains (sheets) of the kinematical variables. This “bump” is an indication of a “hidden” phenomenon. To uncover it one needs to analytically continue outset the physical sheet INDIANA UNIVERSITY

Introduction to Scattering 5 V → V ( t ) = V e − ✏ | t | H = H kin + V → H 0 + V ( t ) Interaction is switched on adiabatically at t=0 • Time evolution pictures: Schrodinger, Heisenberg, Interaction H 0 ,I ( t ) = H 0 O I ( t ) = e iH 0 O (0) e − iH 0 t V I ( t ) = e iH 0 V e − iH 0 t e − ✏ | t | i d | t i I = e iH 0 t | t i S dt | t i I = V I ( t ) | t i I • As t → ± ∞ interaction picture states evolve to eigenstates of H kin, i.e. to free particles • At t=0 interactions picture states are solution of the full Hamiltonian INDIANA UNIVERSITY

S-matrix and T-matrix 6 i d dt | t i I = V I ( t ) | t i I | t i I = U ( t, �1 ) | initial i Evolution operator • S-matrix S fi = h f ( t = + 1 ) | i ( t = �1i = h f, ( out ) | i, ( in ) i = h f | U (+ 1 , �1 ) | i i Z + ∞ ✓ ◆ = I − 2 π i δ ( E f − E i ) T U (+ ∞ , −∞ ) = P exp dtV I ( t ) − i −∞ • T-matrix 1 T = V + V G 0 V + · · · G 0 = E − H 0 E = E i = E f INDIANA UNIVERSITY

T matrix : Example 7 T = V + V G 0 V + · · · Example dim λ = − 1 Nonrelativistic particle scarring in external potential λ V = 2 µa 2 δ ( r − a ) H = p 2 2 µ + V ε V(r) 1/ ε Method 1: In coordinate space Method 2: Lippmann-Schwinger (see above) a r • It has ∞ number of zeros (this is related to ∞ number of poles when calculated to all INDIANA UNIVERSITY orders)

Solution 8 From method 1 h i − λ sin 2 ( ka ) ( ka ) 2 f ( k ) = h i h i − λ sin 2 ( ka ) sin( ka ) cos( ka ) 1 + λ − ik ( ka ) 2 a ka E = k 2 /2µ K ( E ) 1 f ( k ) = 1 − iK ( E ) k = K − 1 ( E ) − ik ∞ of zeros = P ( k ) Q ( k ) ∞ of zeros → Poles INDIANA UNIVERSITY

Solution 9 From method 2 − λ sin 2 ( ka ) ( ∞ ) zeros of K ! ( ka ) 2 f ( k ) = dE 0 k 0 � λ sin2( k 0 a ) R 1 ( k 0 a )2 1 − 1 E 0 � E ( k ) 0 π k 0 = k ( E 0 ) = √ 2 µE 0 � λ sin2( ka ) ( ka )2 K ( E ) = 1 � 1 R π < ··· INDIANA UNIVERSITY

Analyticity 10 K ( k = k R + ik I ) → const. + O ( e − 2 k I a ) = ik + O ( e − 2 k I a ) 1 K ( E ) 1 − iK ( E ) k = O ( e +2 ik I a ) f ( k ) = Essential singularity at infinity in the physical sheet ! “Conspiracy” between zeros and poles !!! E.g. ∞ number of zeros of K(s) are “fixed” by geometry of the sphere (“dynamics”) and this specific “physics” fixes all the poles. In more general case (no fixed scattering radius) correlation between zeroes and poles persist”, an infinite number poles requires infinite number of zeroes (and vice versa) INDIANA UNIVERSITY

S-matrix properties (in relativistic theory) 22 • Related to transition probability P fi = | h f | S | i i | 2 = h i | S † | f ih f | S | i i • Conservation of Probability = Unitarity X P fi = 1 f X 2 ImT ft = 2 πδ ( E i − E n ) T ∗ fn T ni S † S = I n • Lorentz symmetry: T is a product of Lorentz scalars and covariant factors representing wave functions of external states, e.g for π ( k 1 ) + N ( p 1 , λ 1 ) → π ( k 2 ) + N ( p 2 , λ 2 ) u ( p 1 , λ 1 )[ A ( s, t ) + ( k 1 + k 2 ) µ γ µ B ( s, t )] u ( p 2 , λ 2 ) ¯ • Crossing symmetry: the same scalar functions describe all process related by permutation of legs between initial and final states (only the wave function change) π ( k 1 ) + π ( − k 2 ) → ¯ N ( − p 1 , µ 1 ) + N ( p 2 , µ 2 ) v ( p 1 , µ 1 )[ A ( s, t ) + ( k 1 + k 2 ) µ γ µ B ( s, t )] u ( p 2 , µ 2 ) ¯ • Analyticity: The scalar functions are analytical functions of invariants INDIANA UNIVERSITY

Lorentz symmetry 23 N-to-M scattering depends on 4(N+M)-4-10 = 3(N+M)-10 invariants e.g for 2-to-2: 2 invariants related to the c.m. energy and scattering angle s = ( p 1 + p 2 ) 2 > ( m a + m b ) 2 = ( E 1 ,cm + E 2 ,cm ) 2 p 1 p 3 c a t = ( p 1 − p 3 ) 2 < 0 t = m 2 1 + m 2 2 − 2 E 1 ,cm E 2 ,cm + 2 | p 1 ,cm || p 2 ,cm | z s u = ( p 1 − p 4 ) 2 < 0 X m 2 s + t + u = i i b d u = m 2 1 + m 2 4 − 2 E 1 ,cm E 4 ,cm − 2 | p 1 ,cm || p 4 ,cm | z s p 2 p 4 2 πδ ( E f � E i ) iT = h c, d | ( S � 1) | a, b i Dimensions h p 0 , β | p, α i = 2 E ( p ) δ ( p f � p i ) δ α , β T = (2 π ) 3 δ ( p f − p i ) A ( s, t, u ) r.h.s has dim = -4 A(s,t,u) is a scalar function of mass dimension =0 INDIANA UNIVERSITY

Question 24 How many independent variables describe • Decay proces A → a + b +c • Three particle production A +B → a + b + c INDIANA UNIVERSITY

Helicity amplitudes 25 ~ S · ~ p p | | p, � i = � | p, � i We work in the c.m. frame | ~ h p 3 , λ 3 ; p 4 , λ 4 | A | p 1 , λ 1 ; p 2 , λ 2 i = A λ 1 , λ 2 , λ 3 , λ 4 ( s, t, u ) S z | p, m i z = m | p, m i z Helicity states vs canonical spin states: | p, m i z = Λ ( ~ p 0) | 0 , m i z | p, � i = R (ˆ p ) Λ ( | ~ p | ˆ z 0) | 0 , m i z S X | p, m i z D S | p, λ i z = m, λ (ˆ p ) Exercise show this: m = − S • Even though this looks non relativistic it is relativistic. Notion of LS amplitudes, LS vs. helicity relations are relativistic A λ 1 , λ 2 , λ 3 , λ 4 ( s, t, u ) = η A − λ 1 , − λ 2 , − λ 3 , − λ 4 ( s, t, u ) Parity INDIANA UNIVERSITY

Question 26 How many independent scalar functions describe J/ ψ → π + π - π 0 Ɣ p-> π 0 p INDIANA UNIVERSITY

Crossing symmetry p i = − p i = ( − ~ ¯ p i , − E i ) 27 _ _ _ _ a(p 1 ) + c(p 3 ) → b(p 2 ) + d(p 4 ) _ 1 3 1 1 3 3 s _ t _ u _ 2 4 2 4 2 4 _ _ _ _ a(p 1 ) + b(p 2 ) → c(p 3 ) + d(p 4 ) a(p 1 ) + d(p 4 ) → c(p 3 ) + b(p 2 ) 4 ) 2 s = ( p 1 + p 2 ) 2 3 ) 2 u = ( p 1 + p ¯ t = ( p 1 + p ¯ E c.m t = ( p 1 − p 3 ) 2 2 ) 2 t = ( p 1 − p 3 ) 2 s = ( p 1 − p ¯ Cos( θ ) u = ( p 1 − p 4 ) 2 u = ( p 1 − p 4 ) 2 2 ) 2 s = ( p 1 − p ¯ Cos( θ ) A ( s ) 1 · · · ] A ( t ) X [ D S 1 λ 1 , ··· ( s + i ✏ , t, u ) → 1 , ··· ( s, t + i ✏ , u ) → · · · λ 1 , λ 0 λ 0 λ 0 1 , ··· • The i ε is important. Function values at, e.g. s + i ε vs s - i ε are different ! INDIANA UNIVERSITY

Crossing Symmetry : Decays M 1 > m 2 + m 3 + m 4 28 1 3 1 3 _ 2 4 2 4 _ _ a(p 1 ) + b(p 2 ) → c(p 3 ) + d(p 4 ) a(p 1 ) → b(p 2 ) + c(p 3 ) + d(p 4 ) A ( s, t, u ) → A ( M 2 1 + i ✏ , s + i ✏ , t + i ✏ , u + i ✏ ) • In decay kinematics, the decaying mass becomes a dynamical variable, (i ε important) • Crossing from one kinematical region (e.g. s-channel) to another (e.g. t-channel) requires taking the corresponding variables off the real axis and to the complex plane : analytical continuation. INDIANA UNIVERSITY