William & Mary / Glasgow Michael Pennington William & Mary / Glasgow Thursday June 1st, 2017

q q q ( i D - m ) q - G G q = QCD q=u,d,s, c,b,t 1 4

q q g

q q q g

q q q q g

Can experiment distinguish between q q q q g Can experiment distinguish between these configurations ?

uc du c q

Light Meson Spectrum negative parity positive parity Mass (GeV) ` JPC 2.5 2.0 1.5 Mass (GeV) j 1.0 h ` w r h isoscalar isovector 0.5 p JPC 0++ 1++ 1+- 2++ 4++ 0-+ 1-- 2-+ 3--

0-+ 1-- Meson spectrum S = 0, 1 L=0 f w K*0 K*+ K*- r- r+ r0   K0 - + 0 0-+ 1-- s1 s2 L q S = 0, 1 1-- 0-+ L=0

S r f2 r3 p Events/20 MeV (2l +1) fl (s) Pl ( z ) (s,z) = M(pp) GeV F 0.2 0.6 1.0 1.4 1.8 2.2 35 30 25 20 15 10 5 40 Events/20 MeV M(pp) GeV r r3 f2 N B R p (s,z) (2l +1) fl (s) Pl ( z ) l =0 S = F CERN-Munich: Grayer et al.

. J r5 f4 r3 f2 r K5* K4* p K3* K2* K* M2 ideally mixed 5 4 3 2 1 1 2 3 4 5 6 GeV2 r r3 f2 f4 K* K3* K2* K4* 1 2 3 4 5 r5 K5* M2 ideally mixed N B R p exchange

. J 1 2 3 4 5 6 GeV2 r r3 f2 f4 K* K3* K2* K4* 1 2 3 4 5 r5 K5* M2 . q . . . . . . . . . . . . . g . . . . . . . . . . . .

. J 1 2 3 4 5 6 GeV2 r r3 f2 f4 K* K3* K2* K4* 1 2 3 4 5 r5 K5* M2 . q . . . . . . . . . . . . . . . . . . . . . . . . .

Hadron States E x Breit-Wigner M2 – s - iMG 1 s = E2

COMPASS @ CERN 2004 1- + M(3p) GeV

HUGS Hadron Spectrum: window on confinement Step One: spectrum of baryons, mesons quarks and QCD Step Two: tools for discovery, experiment and Amplitude Analyses Step Three: conserving probability and respecting causality Step Four: tools for discovery, S-matrix theory 2 Step Five: what’s new, computing QCD Step Six: what’s to come, what to watch for HUGS

S(p1,p2,...,pj; s1,s2…,sj; q1,…,qk; t1,…,tk) …. …... 2 2 2 3 3 3 …

Conservation of probability Unitarity Sum of probabilities = S Pi = 1 i

S S = I S = I + 2i T T T T – T = 2i r r Unitarity rn = ½ rn 2kn s  = T = transition matrix = diagonal matrix of phase-space factors T T T – T = 2i r matrix relation diagonalized by amplitudes of definite angular momentum

T T T – T = 2i Im T = T* r T r r is phase space for intermediate state Unitarity T T T – T = 2i r matrix relation diagonalized by amplitudes of definite angular momentum then Im T = T* r T where T has definite quantum numbers r is phase space for intermediate state

S Unitarity Im rn = Amplitude with definite JPC = 2kn s  = We call the partial waves for channel i j, Tij(s) the ij th element of the T-matrix with definite IJPC

S Unitarity Im rn Amplitude with definite JPC = A B C D 2kn s  We call the partial waves for channel i j, Tij(s) the ij th element of the T-matrix with definite IJPC

Unitarity Im p Im T11(s) = r1(s) T11*(s) T11(s) r1 = Amplitude with definite JPC Im = p Im T11(s) = r1(s) T11*(s) T11(s) 1 = pp r1 2k1 s  = s - 4m12 strictly for s < (4mp)2 in practice s < 1 GeV2

Unitarity Im p Im T11(s) = r1(s) T11*(s) T11(s) r1 = Amplitude with definite JPC Im = p Im T11(s) = r1(s) T11*(s) T11(s) 1 = pp T11 = | T11 | eid r1 1 let sin d = | T11 | 2k1 T11 = sin d eid r1 1 r1 =  s

Unitarity PC definite J ri = ki / E p = Im å p K = + 1 = pp 2 = KK

Unitarity Im T11(s) = r1(s) | T11(s) |2 + r2(s) | T12(s) |2 + … for each J, P, C, I, … Im T11(s) = r1(s) | T11(s) |2 + r2(s) | T12(s) |2 + … T11 = sin d eid r1 1

X Unitarity d phase shift h inelasticity h=1 elastic for each J, P, C, I, … Im T11(s) = r1(s) | T11(s) |2 + r2(s) | T12(s) |2 + … X T11 = sin d eid r1 1 d phase shift h inelasticity T11 = ( h e2id - 1 ) 2ir1 1 h=1 elastic

Unitarity d phase shift h inelasticity tl r1 T11 = ( h e2id - 1 ) for each J, P, C, I, … d phase shift h inelasticity r1 T11 = ( h e2id - 1 ) 2i 1 Im tl Re tl h=1 elastic tl tl

Unitarity d phase shift h inelasticity tl pK pK for each J, P, C, I, … d phase shift h inelasticity r1 T11 = ( h e2id - 1 ) 2i 1 Im tl tl h=1 elastic pK pK I = 1/2 S-wave example Re tl

Unitarity PC definite J ri = ki / E = p Im å K p K = + 1 = pp 2 = KK

Elastic Unitarity - i r1 Im T11(s) = r1(s) T11*(s) T11(s) Im - r1 Re K11 where K is real for real s > 4m2 T11 1 = K11 - i r1 T11 1 – i r1 K11 K11 = K11 = tan d r1 1

Elastic Unitarity K11 T11 = K11 = tan d r1 1 – i r1 K11 1 – i r1 K11 1 Physical particles are poles of the S-Matrix on nearby unphysical sheet (s). These are given by the zeros of 1 – i r1 K11 The K-matrix has no physical significance (except in models). It is just a convenient way to implement unitarity. In particular poles in K are not poles of the S-matrix K can involve sums of poles and polynomials. It’s just a matter of economy.

Elastic Unitarity K11 T11 = K11 = tan d r1 1 – i r1 K11 K11 = T11 = 1 Example : K11 = M2 - s M G T11 = M2 - s M G - i r M G Breit-Wigner form The K-matrix has no physical significance (except in models). It is just a convenient way to implement unitarity. In particular poles in K are not poles of the S-matrix

analyticity & complex energy plane Im E Re E resonance pole why sheets? see later

Elastic Unitarity & Analyticity 1 | T11 |2 * = Im - r1 T11 1 = Re K11 where K is real for real s > 4m2 T11 1 = K11 - i r1 = r1  s - 4m12 s = r1  s - 4m12 s = r1  s - 4m12 s T11 1 = K11 r1 r1+1 r1-1 ( ) ln p + corrects the analyticity on right hand cut ONLY

Energ = r  s - 4mK2 s Real KK Imaginary E (GeV)

Energ KK Real Real Imaginary E (GeV) Chew-Mandelstam function K-1

Multi-channel Unitarity Amplitude with definite JPC k Im i i i i j j j = i j j 1 = pp 2 = KK S k Im Tij(s) = rk(s) Tik*(s) Tkj(s) Im T11 = r1 T11 T11 + r2 T12 T21 * Im T12 = r1 T11 T12 + r2 T12 T22 Im T22 = r1 T21 T12 + r2 T22 T22

Multi-channel Unitarity Amplitude with definite JPC Im T11 = r1 T11 T11 + r2 T12 T21 * Im T12 = r1 T11 T12 + r2 T12 T22 Im T22 = r1 T21 T12 + r2 T22 T22 1 = pp 2 = KK T11 1 = Im - r1 Im T -1 = - r 1-channel : n-channel : r1 r2 r3 r4 r =

Multi-channel Unitarity Amplitude with definite JPC Im T11 = r1 T11 T11 + r2 T12 T21 * Im T12 = r1 T11 T12 + r2 T12 T22 Im T22 = r1 T21 T12 + r2 T22 T22 1 = pp 2 = KK T11 1 = Im - r1 Im T -1 = - r 1-channel : n-channel : Re T - 1 = K - 1 T - 1 = K - 1 - i r

Multi-channel Unitarity Amplitude with definite JPC Im T11 = r1 T11 T11 + r2 T12 T21 * Im T12 = r1 T11 T12 + r2 T12 T22 Im T22 = r1 T21 T12 + r2 T22 T22 1 = pp 2 = KK 2-channel : = T11 K11 – i r2 det K D T12 K12 T22 K22 – i r1 det K D = 1 – i r1 K11 – i r2 K22 – r1 r2 det K where

S Unitarity Im Amplitude with definite JPC = n n What happens when the final state particles are hadrons, but the incoming particles are not hadrons, eg e+e- , gg Sum n only need include hadrons as initial state particles are suppressed by powers of a

Unitarity Im p p p p p p Im F1 (s) = r1(s) F1*(s) T11(s) Amplitude with definite JPC example p p e+ e+ p Im = p 1 = pp e- p e- p Im F1 (s) = r1(s) F1*(s) T11(s) = r1(s) F1 (s) T11 (s) *

Unitarity Im p p p p p p Im F1 (s) = r1(s) F1*(s) T11(s) Amplitude with definite JPC example p p e+ e+ p Im = p 1 = pp e- p e- p Im F1 (s) = r1(s) F1*(s) T11(s) = r1(s) F1 (s) T11 (s) * T11 = sin d eid r1 1 recall F1 = | F1 | eij let

Unitarity Im j = d (+np) j = d (+np) p p p p p p Amplitude with definite JPC example p p e+ e+ p Im = p 1 = pp e- p e- p Im F1 (s) = r1(s) F1*(s) T11(s) = r1(s) F1 (s) T11 (s) * T11 = sin d eid r1 1 recall F1 = | F1 | eij let sin j = sin d sin j = sin d sin j = sin d j = d (+np) Watson’s final state interaction theorem: j = d (+np) Watson’s final state interaction theorem:

r in pp and e+e-

Multi-channel Unitarity Amplitude with definite JPC k Im i = i 1 = pp 2 = KK Im Fi (s) = rk(s) Fk*(s) Tki (s) S k Im F1 = r1 F1 T11 + r2 F2 T21 * Im F2 = r1 F1 T12 + r2 F2 T22 e.g. F ( gg hadron channel i )

Multi-channel Unitarity real coupling functions PC definite J real coupling functions g p g p Im å = g p g p on right hand cut g p g p p K = + + p K g 1 = pp 2 = KK p g p

analyticity & complex energy plane Im E Re E resonance pole Universal: process independent

Landscape to be explored

Unitarity Amplitude with definite JPC n Im = S n rn 2kn s  =

Unitarity Im Im T11(s) = r1(s) T11*(s) T11(s) Amplitude with definite JPC Im = Im T11(s) = r1(s) T11*(s) T11(s) 1 = pp for equal masses = r  s - 4m2 s m1 r1

f(s) = sth – s  f(s) = sth – s  Consider Consider sth = 4m2 if masses equal in complex s-plane Im s s Re s

f(s) = sth – s  f(s) = sth – s  Consider Consider in complex s-plane experiment X sth

Consider f(s) = sth – s  Consider f(s) = sth – s  B A s s2 X sth C D

Consider f(s) = sth – s  Consider f(s) = sth – s  B A s 2J X sth C D

 f(s) = sth – s  f(s) = sth – s  s = sth + s2 e f(s) = - s2 e Consider f(s) = sth – s  Consider f(s) = sth – s  A s s2 2J X X sth s = sth + s2 e 2iJ point A let -1 = e ip f(s) = - s2 e 2iJ  i(p/2+J) iJ = i s e = s e

 f(s) = sth – s  f(s) = sth – s  s = sth + s2 e f(s) = - s2 e Consider f(s) = sth – s  Consider f(s) = sth – s  B s s2 p - 2J X sth s = sth + s2 e i(p - 2J) point B f(s) = - s2 e i(p - 2J) i(p-J) -iJ = - s e = s e 

 f(s) = sth – s  f(s) = sth – s  s = sth + s2 e f(s) = - s2 e Consider f(s) = sth – s  Consider f(s) = sth – s  s = sth + s2 e i(p + 2J) point C s p + 2J X X sth s2 C f(s) = - s2 e i(p + 2J) i(p+J) +iJ = - s e = s e 

 f(s) = sth – s  f(s) = sth – s  s = sth + s2 e f(s) = - s2 e Consider f(s) = sth – s  Consider f(s) = sth – s  i(2p - 2J) s = sth + s2 e point D s 2p - 2J X s2 D f(s) = - s2 e i(2p - 2J) i(3p/2-J) -iJ = -i s e = s e 

Consider f(s) = sth – s  B A s 2J X sth C D

f(s) = sth – s  f(s) = sth – s  - s e i s e - s e - s e - i s e Consider f(s) = sth – s  Consider f(s) = sth – s  B A s -iJ - s e +iJ +iJ i s e X sth +iJ - s e +iJ - s e -iJ - i s e C D note f (s*) = f *(s) f (s) is real analytic

f(s) = sth – s  f(s) = sth – s  - s e i s e - s e - s e - i s e -iJ Consider f(s) = sth – s  Consider f(s) = sth – s  B A s -iJ - s e +iJ +iJ i s e X sth +iJ - s e +iJ - s e -iJ - i s e C D let’s see what happens when J 0

f(s) = sth – s  f(s) = sth – s  - s - i s + i s Consider Consider s A B s + i s - s X X sth - i s C D let’s see what happens when J 0

f(s) = sth – s  f(s) = sth – s  - s - i s + i s Consider Consider s A B s + i s - s X X cut sth - i s C D

Consider f(s) = sth – s  Consider f(s) = sth – s  Im f Im s Re s

Landscape to be explored

S i.e. Pi = 1 Scattering amplitudes exist in a world of cuts unitarity This is a consequence of the fundamental property of the conservation of probability i.e. Pi = 1 S i unitarity

f(s) = sth – s  f(s) = sth – s  - s e i s e - s e - s e - i s e -iJ chose -1 = e ip Sheet I Consider f(s) = sth – s  Consider f(s) = sth – s  B A s -iJ - s e +iJ +iJ i s e X sth +iJ - s e +iJ - s e -iJ - i s e C D

f(s) = sth – s  f(s) = sth – s  - s e - i s e - s e - s e -iJ choose -1 = e -ip Sheet II Consider f(s) = sth – s  Consider f(s) = sth – s  B A s -iJ + s e +iJ - s e - i s e 2J X X sth +iJ - s e +iJ + s e +iJ - s e -iJ + i s e C D

f(s) = sth – s  f(s) = sth – s  - s e i s e - s e - i s e - s e Consider f(s) = sth – s  Sheet I Consider f(s) = sth – s  Sheet II B A s -iJ - s e +iJ i s e -iJ + s e +iJ - s e - i s e X sth +iJ + s e +iJ - s e -iJ + i s e +iJ - s e +iJ - s e -iJ - i s e C D

analyticity & complex energy plane Im E Re E

analyticity & complex energy plane Im E +iJ i s e +iJ - s e - i s e +iJ - s e -iJ - i s e Re E +iJ - s e -iJ + i s e f(s) = sth – s  connections illustrated with where s2 = s – sth

analyticity & complex energy plane Im E Re E resonance pole resonance pole resonance pole

S Complex s-plane Im = Im s unitarity Re s for each I, J, l n Im s unitarity right hand cut Re s

Complex s-plane Im s Im s unitarity Re s Re s for each I, J, l t u left hand cut right hand cut Re s Re s

ds dW Scattering Amplitude, (s,t) for spinless particles J j describes dependence on energy and J ds dW k J j dW dW = d(cos J) dj p ds dW = K(s) | (s,z) |2 F K(s) = k 64p2 p s spinless flux factor depends on s & spin

Scattering Amplitude, (s,t) for spinless particles describes dependence on energy and J recall dW = d(cos J) dj let z = cos J ds d z = 2p K(s) | (s,z) |2 F K(s) = k 64p2 p s spinless S (s,z) (2l +1) fl (s) Pl ( z ) F = 16p l =0 fl (s) partial waves

S S Partial waves l l fl S P D G … l (2l +1) fl (s) Pl ( z ) (s,z) = notation:

stot = Im Fel (s, J=0) Unitarity optical theorem  s elastic amplitude 1 Im Fel (s, J=0) elastic amplitude Total cross-section for AB everything is related to the forward amplitude for AB AB