Light by Light. Summer School on Reaction Theory. Michael Pennington Jefferson Lab
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1 Light by Light Summer School on Reaction Theory Michael Pennington Jefferson Lab
2 Amplitude Analysis g g R p p resonances have definite quantum numbers I, J, P (C) g p g p = S F Jl (s) Y Jl (J,j) J,l
3 Amplitude Analysis g g R p p resonances have definite quantum numbers I, J, P (C) resonances and backgrounds not separable within unitarity g p g p = S F Jl (s) Y Jl (J,j) J,l partial wave amplitudes
4 To perform a partial wave separation need to know the partial waves at low energy accurately
5 Morgan & P Amplitude Analysis Lingyun Dai & P 2014
6 M (pp) GeV M (pp) GeV
7 s (nb) gg p + p - : Mark II, Belle M (p + p - ) GeV
8 s (nb) gg p 0 p 0 : Crystal Ball, Belle M (p 0 p 0 ) GeV
9 Isospin decomposition gg pp One Pion Exchange B I=0 (s) = - B I=2 (s) = - / ` / ` 2/3 1/3 B (s) B (s) B +- (s) = B (s) B 00 (s) = 0
10 Isospin decomposition gg pp gg KK B I=0 (s) = - B I=2 (s) = - / ` / ` 2/3 1/3 B (s) B (s) } B +- (s) = B (s) B 00 (s) = 0
11 gg p p for each I, J, l calculable p unitarity Complex s-plane s (GeV) 2
12 Dispersive calculation of low energy partial waves M(pp) (GeV) M(pp) (GeV) Unusual feature: large D-waves near threshold, I=2 as large as I=0
13 along left hand cut For J = l = 0, consider ( F (s) B (s) ) W - 1 (s) with I = 0,2 with subtraction constants b I
14 Consider ( F (s) B (s) ) W - 1 (s) s n (s - 4m p2 ) J/2 with n = 2 l/2, and J > 0, l = 0,2, I = 0,2 /
15 s (nb) gg p + p - : Mark II, Belle M (p + p - ) GeV
16 s (nb) gg p 0 p 0 : Crystal Ball, Belle M (p 0 p 0 ) GeV
17 Mandelstam Plane p gp gp p t p -0.2 u s -0.4 gp gp gg pp p
18 Conservation of probability pp pp pp KK Sum of probabilities = S P i = 1 i
19 I = J = 0 s/f 0 (500) f 0 (980) f 0 (1370) 1 T M (pp) (GeV) pp pp pp KK pp
20 Unitarity PC definite J i = k i / E p p p p Im = p p p p p p p p = + + p p p p K K p p
21 Unitarity PC definite J i = k i / E p p p p Im = p p p p p p p p = + + p p p p K K p p
22 Elastic Unitarity Im T 11 (s) = 1 (s) T 11 *(s) T 11 (s) 1 T 11 = T 11 * T T 11 T 11 * Im = Im = - 1 T 11 2
23 Elastic Unitarity Im T 11 (s) = 1 (s) T 11 *(s) T 11 (s) 1 T 11 = T 11 * T T 11 T 11 * Im = Im = - 1 T 11 2 Re 1 T 11 = 1 where K is real K 11 for real s > 4m 2
24 Elastic Unitarity Im T 11 (s) = 1 (s) T 11 *(s) T 11 (s) 1 T 11 = T 11 * T T 11 T 11 * Im = Im = - 1 T 11 2 Re 1 T 11 = 1 where K is real K 11 for real s > 4m 2 1 T 11 = 1 K 11 - i 1
25 Elastic Unitarity Im T 11 (s) = 1 (s) T 11 *(s) T 11 (s) 1 T 11 = T 11 * T T 11 T 11 * Im = Im = - 1 T 11 2 Re 1 T 11 = 1 where K is real K 11 for real s > 4m 2 1 T 11 = 1 K 11 K 11 - i 1 T 11 = 1 i 1 K 11
26 Elastic Unitarity Im T 11 (s) = 1 (s) T 11 *(s) T 11 (s) 1 T 11 = T 11 * T T 11 T 11 * Im = Im = - 1 T 11 2 Re 1 T 11 = 1 where K is real K 11 for real s > 4m 2 1 T 11 = 1 K 11 K 11 - i 1 T 11 = 1 i 1 K 11 K 11 = 1 1 tan d
27 Elastic Unitarity K 11 T 11 = K 11 = tan d 1 i 1 K Physical particles are poles of the S-Matrix on nearby unphysical sheet (s). These are given by the zeros of 1 i 1 K 11
28 Elastic Unitarity K 11 T 11 = K 11 = tan d 1 i 1 K Physical particles are poles of the S-Matrix on nearby unphysical sheet (s). These are given by the zeros of 1 i 1 K 11 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
29 Elastic Unitarity K 11 T 11 = K 11 = tan d 1 i 1 K Example : K 11 = M G M 2 - s T 11 = M 2 - s M G - i M G Breit-Wigner form
30 Elastic Unitarity K 11 T 11 = K 11 = tan d 1 i 1 K Example : K 11 = M G M 2 - s T 11 = M 2 - s M G - i 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
31 Elastic Unitarity & Analyticity 1 T 11 T 11 * Im = Im = - 1 T 11 2 Re 1 T 11 = 1 where K is real K 11 for real s > 4m 2 1 T 11 = 1 K 11 - i 1
32 Elastic Unitarity & Analyticity 1 T 11 T 11 * Im = Im = - 1 T 11 2 Re 1 T 11 = 1 where K is real K 11 for real s > 4m 2 1 T 11 = 1 K 11 - i 1 1 T 11 = p K 11 ln ( ) 1-1 corrects the analyticity on right hand cut ONLY
33 Multi-channel Unitarity Amplitude with definite J PC Im i j = i j i j k 1 = pp 2 = KK Im T ij (s) = S k k (s) T ik *(s) T kj (s)
34 Multi-channel Unitarity Amplitude with definite J PC Im i j = i j i j k 1 = pp 2 = KK Im T ij (s) = S k k (s) T ik *(s) T kj (s) Im T 11 = 1 T * 11 T T * 12 T 21 Im T 12 = 1 T * 11 T T * 12 T 22 Im T 22 = 1 T * 21 T T * 22 T 22
35 Multi-channel Unitarity Amplitude with definite J PC 1 = pp 2 = KK Im T 11 = 1 T * 11 T T * 12 T 21 Im T 12 = 1 T * 11 T T * 12 T 22 Im T 22 = 1 T * 21 T T * 22 T 22 Im 1 = channel : n-channel : T 11 Im T -1 = - =
36 Multi-channel Unitarity Amplitude with definite J PC 1 = pp 2 = KK Im T 11 = 1 T * 11 T T * 12 T 21 Im T 12 = 1 T * 11 T T * 12 T 22 Im T 22 = 1 T * 21 T T * 22 T 22 Im 1 = channel : n-channel : T 11 Im T -1 = - Re T - 1 = K - 1 T - 1 = K i
37 Multi-channel Unitarity Amplitude with definite J PC 1 = pp 2 = KK Im T 11 = 1 T * 11 T T * 12 T 21 Im T 12 = 1 T * 11 T T * 12 T 22 Im T 22 = 1 T * 21 T T * 22 T 22 K 2-channel : T 11 i 2 det K 11 = D K T = D T 22 = D K 22 i 1 det K where D = 1 i 1 K 11 i 2 K det K
38 Unitarity Amplitude with definite J PC Im = S 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
39 Unitarity Amplitude with definite J PC Im = 1 = pp * Im F 1 (s) = 1 (s) F 1 *(s) T 11 (s) = 1 (s) F 1 (s) T 11 (s)
40 Unitarity Amplitude with definite J PC Im = 1 = pp * Im F 1 (s) = 1 (s) F 1 *(s) T 11 (s) = 1 (s) F 1 (s) T 11 (s) let F 1 = F 1 e ij recall T 11 = 1 1 sin d e id
41 Unitarity Amplitude with definite J PC Im = 1 = pp * Im F 1 (s) = 1 (s) F 1 *(s) T 11 (s) = 1 (s) F 1 (s) T 11 (s) let F 1 = F 1 e ij recall T 11 = 1 1 sin d e id sin j = sin d Watson s final state interaction theorem: j = d (+np)
42 Multi-channel Unitarity Amplitude with definite J PC k Im i = i 1 = pp 2 = KK Im F i (s) = S k k (s) F k *(s) T ki (s) Im F 1 = 1 F * 1 T F * 2 T 21 Im F 2 = 1 F * 1 T F * 2 T 22 e.g. F ( gg hadron channel i )
43 Single-channel Unitarity Amplitude with definite J PC * Im T 11 = 1 T 11 T 11 * Im F 1 = 1 F 1 T 11 K 11 T 11 = 1 i 1 K 11 P 1 F 1 = 1 i 1 K 11 Like K 11, P 1 must be real for real s > 4m 2 Resonances are the poles in T 11 and F 1, i.e. the common zeros of 1 i 1 K 11
44 Single-channel Unitarity * Im T 11 = 1 T 11 T 11 Amplitude with definite J PC * Im F 1 = 1 F 1 T 11 K 11 T 11 = 1 i 1 K 11 P 1 F 1 = 1 i 1 K 11 F 1 (s) = a 1 (s) T 11 (s) with a 1 (s) real for real s> 4m 2
45 Multi-channel Unitarity Amplitude with definite J PC Im F 1 = 1 F * 1 T F * 2 T 21 Im F 2 = 1 F * 1 T F * 2 T 22 1-channel : K 11 T 11 = F 1 = 1 i 1 K 11 1 i 1 K 11 P 1 2-channel : T 11 K 11 i 2 det K = D F 1 = with P i and Q i real for real s > 4m 2 P 1 i 2 Q 1 det K D etc.
46 Multi-channel Unitarity Amplitude with definite J PC Im F 1 = 1 F * 1 T F * 2 T 21 Im F 2 = 1 F * 1 T F * 2 T 22 1-channel : K 11 T 11 = F 1 = 1 i 1 K 11 1 i 1 K 11 P 1 2-channel : T 11 K 11 i 2 det K = D F 1 = P 1 i 2 Q 1 det K D etc. Solution : P 1 = K 11 Q 1 + K 12 Q 2, P 2 = K 12 Q 1 + K 22 Q 2
47 Multi-channel Unitarity Amplitude with definite J PC Im F 1 = 1 F * 1 T F * 2 T 21 Im F 2 = 1 F * 1 T F * 2 T 22 2-channel : F 1 = F 2 = Q 1 [ K 11 i 2 det K ] + Q 2 K 12 D Q 1 K 12 + Q 2 [ K 22 i 1 det K ] D where D = 1 i 1 K 11 i 2 K det K
48 Multi-channel Unitarity Amplitude with definite J PC 2-channel : F 1 = F 2 = Q 1 [ K 11 i 2 det K ] + Q 2 K 12 D Q 1 K 12 + Q 2 [ K 22 i 1 det K ] D K 2-channel : T 11 i 2 det K 11 = D K T = D T 22 = D K 22 i 1 det K
49 Multi-channel Unitarity Amplitude with definite J PC 2-channel : F 1 = Q 1 T 11 + Q 2 T 21 Q n a n F 2 = Q 1 T 12 + Q 2 T 22 K 2-channel : T 11 i 2 det K 11 = D K T = D T 22 = D K 22 i 1 det K
50 Multi-channel Unitarity PC definite J real coupling functions g p g p Im = g p g p on right hand cut g p p g K p 1 = pp 2 = KK = + g p p g K p +
51 analyticity & complex energy plane Im E E Re E resonance pole Universal: process independent
52 Amplitude Analysis g g R p p resonances have definite quantum numbers I, J, P (C) resonances and backgrounds not separable within unitarity g p g p = S F Jl (s) Y Jl (J,j) J,l partial wave amplitudes
53 Dispersive calculation of low energy partial waves M(pp) (GeV) M(pp) (GeV) Unusual feature: large D-waves near threshold, I=2 as large as I=0
54 cross-section (nb) Born amplitude modified by final state interactions p + p p 0 p E (GeV)
55 Datasets - 0.8
56 s (nb) gg p + p - : Mark II, Belle M (p + p - ) GeV
57 s (nb) gg p 0 p 0 : Crystal Ball, Belle M (p 0 p 0 ) GeV
58 gg p + p -, p 0 p 0 added
59 Multi-channel Unitarity Amplitude with definite J PC 2-channel : F 1 = a 1 T 11 + a 2 T 21 F 2 = a 1 T 12 + a 2 T 22 gg gg pp KK K 2-channel : T 11 i 2 det K 11 = D K T = D T 22 = D K 22 i 1 det K
60 Multi-channel Unitarity Amplitude with definite J PC 2-channel : F 1 = a 1 T 11 + a 2 T 21 F 2 = a 1 T 12 + a 2 T 22 a 1, a 2 have left hand cuts only K 2-channel : T 11 i 2 det K 11 = D K T = D T 22 = D K 22 i 1 det K
61 I=J=0 amplitudes pp pp pp KK pp KK KK KK
62 I=J=0 amplitudes pp pp pp KK pp KK KK KK Includes latest scattering information dispersive analyses (Roy-type) Garcia Martin et al., Descotes-Genon et al.,
63 I=0,2, J=0,2 phases & Omnes functions j S W S j D W D
64 Heavy flavour decays FOCUS D s (MM) p decay
65 pp S-wave in D s p(pp) decay M(p + p - ) GeV M(p + p - ) GeV
66 KK S-wave in D s p(kk) decay
67 gg p+p-
68 gg p + p - : angular distributions
69 gg p + p - : angular distributions
70 gg p + p - : angular distributions
71
72
73 gg p0p0
74 gg p 0 p 0 : angular distributions
75 gg p 0 p 0 : angular distributions
76 gg p + p -, p 0 p to 1050 MeV p + p - p 0 p 0
77 Multi-channel Unitarity Amplitude with definite J PC 2-channel : F 1 = a 1 T 11 + a 2 T 21 F 2 = a 1 T 12 + a 2 T 22 gg gg pp KK K 2-channel : T 11 i 2 det K 11 = D K T = D T 22 = D K 22 i 1 det K
78 gg K + K -, K s K s : cross-sections
79 gg K s K s : angular distributions
80 gg p + p -, p 0 p 0 Integrated cross-sections p + p - p 0 p 0
81 gg pp partial wave cross-sections I = 0 I = 2
82 gg pp partial wave cross-sections I = 0 I = 2 f 2 f 0 f 2 f 0 input into dispersion relation for LbL
83 Light by Light projector a m
84 Light by Light k n q 1 pp, p, KK, q 4 q 2 q 3 l m
85 gg pp S-waves I = 0 I = 2
86 gg pp S-waves I = 0 I = 2 f 0 (500)
87 gg pp S-waves I = 0 I = 2 f 0 (980)
88 gg pp S-waves I = 0 I = 2 f 0 (1370 )
89 analyticity & complex energy plane Im E E Re E F I Jl (s) = g pp g gg s R - s + model meaning of residue
90 gg couplings
91 gg couplings
92 G(0 ++ gg) model predictions
93 Shedding light on scalar mesons f (600) 0 f (980) 0 f (1370) 0 f (1500) 0
94 g qq q q (0) 2 Y g ( S < e q 2 > ) 2 q P R
95 g f 2 (1270) q q g ( S < e q 2 > ) 2 q P R
96 g q p + s/f 0 (600) q q q p - g
97 f 0 (980)? K K
98 f 0 (980) K? K
99 g q K + f 0 (980) q q q K - g ( S < e h 2 > ) 2 h h P R
100 G(0 ++ gg) model predictions
101 Scalar mesons gg f 0 ss f 0 sn nn K 0 a /f 0 0 ssnn snnn nnnn a /f 0 0 K 0 f 0 k s n = u,d
102 Scalar mesons s K K n gg f 0 ss sn f 0 K 0 n s Nathan Isgur nn a /f 0 0 ssnn snnn nnnn a /f 0 0 K 0 f 0 k s n = u,d
103 Scalar mesons gg f 0 ss f 0 sn nn K 0 a /f 0 0 ssnn snnn nnnn a /f 0 0 K 0 f 0 k s
104 gg p + p - m + m - contamination MeV
105 gg p + p -, p 0 p 0 Integrated cross-sections Dai & P
106 gg p + p -, p 0 p 0 Integrated cross-sections m + m - contamination
107 Integrated cross-section III II I
108 Light by Light g* g* g*
109 Light by Light k n q 1 pp, p, KK, q 4 q 2 q 3 l m
110 Light* by Light e PRIMEX: E g e Q 2 g* g* g
111 Light by Light* GlueX: E p + g g* p - Pb Hall D@JLab
112 Stanley Brodsky David Horn Peter Landshoff Michael Pennington
113
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