1. William & Mary / Glasgow
Michael Pennington
William & Mary / Glasgow
Wednesday May 31st, 2017

2. 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

3. QCD asymptotic freedom confinement r (m) strong coupling 1 10
strong QCD 1 10
-15
r (m)
strong coupling
pQCD
strong QCD

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

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

6. Theme: what are the relevant degrees of freedom?

7. Theme: what are the relevant degrees of freedom?

8. Confinement Physics

9. accurate modelling
precision research
precision tools
QCD

10. S Basic tools of -matrix theory relativity causality
conservation of probability
causality

11. Hadron Physics A
hadrons } B

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

13. pN scattering
E (GeV)
s (mb)
p- p
p+ p
p+p p N p N
energy E

14. DELTA RESONANCE DISCOVERED!
MARCH 12, 1952
DELTA RESONANCE DISCOVERED!
Cross-section for positive pions on hydrogen peaks above 1200 MeV
positive
pion
MeV
negative .

15. Baryon resonances (N*s and D*s)
p- p X
p+ p
P33(1232)
s (mb)
E (GeV) 15

16. Hadron States E M
G ~ 1/ t
lifetime
G/2

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

18. pN scattering
E (GeV)
s (mb)
p- p
p+ p
p+p p N p N
energy E p N
L2I 2J

19. Baryon resonances (N*s and D*s)
s (mb)
W (GeV)
p- p X
p+ p
P33(1232)
S11(1650)
P31(1620)
S11(1535)
D13(1520)
P11(1440)
D15(1675)
F15(1680)
D33(1700)
P13(1720)
P31(1910)
F35(1905)
F37(1950)
G17(2190)
H19(2220)
G19(2250)
H31(2420) 19

20. Spectrum of hadrons

21. pN pN scattering ds/dW p p p p p p p n 1234 MeV 1449 MeV 1678 MeVq
p p p p + -
p p p n
1234 MeV
1449 MeV
1678 MeV
1900 MeV

22. pN pN scattering P p p p p p p p n 1234 MeV 1449 MeV 1678 MeV 1900 MeVq
p p p p + -
p p p n P
1234 MeV
1449 MeV
1678 MeV
1900 MeV

23. Relativistic kinematicsp4 p3 p1 p2 A C B D
2-to-2 scattering

24. C A B D Center of momentum frame: J, j p3 p1 p2 p4z x y D p4
p1 = ( E1, 0, 0, p )
p2 = ( E2, 0, 0, -p )

25. C A B D Center of momentum frame: J, j p3 p1 p2 p4z x y D p4
p3 = ( E3, q sinJ cosj, q sinJ sinj, q cos J )
p4 = ( E4, -q sinJ cosj, -q sinJ sinj, -q cos J )

26. E1 = + E2 = - Center of momentum frame: h = c = 1 (m12 – m22)
 s
E2 =
p12 = m12, p22 =m22
s = (p1 + p2)2
= (E1 + E2)2
4s p2 = s2 – 2 (m12 + m22) s + (m12 – m22)2
p1 = ( E1, 0, 0, p )
p2 = ( E2, 0, 0, -p )

27. C A B D Center of momentum frame: J p3 p1 p2 p4z x D p4
p3 = ( E3, q sinJ, 0 , q cos J )
p4 = ( E4, -q sinJ, 0 , -q cos J )

28. Center of momentum frame:
s = (p1 + p2)2
= (p3 + p4)2
t = (p1 – p3)2
= (p4 - p2)2
u = (p1 – p4)2
= (p3 - p2)2
s + t + u = m12 + m22 + m32 + m42
t = m12 + m32 -2(E1 E3 –p q cos J)
= m22 + m42 -2(E2 E4 –p q cos J)
p1 = ( E1, 0, 0, p )
p2 = ( E2, 0, 0, -p )
p3 = ( E3, q sinJ, 0 , q cos J )
p4 = ( E4, -q sinJ, 0 , -q cos J )

29. Center of momentum frame:
simplest case all masses equal, m
p = q
p1 = (
 s
/2, 0,0,p)
p2 = (
/2, 0,0, -p)
p3 = (
 s
/2, p sin J,0, p cos J)
p4 = (
/2, -p sin J,0, -p cos J)
s = 4 (p2 + m2)
t = -2 p2 (1 – cos J)
u = -2 p2 (1 + cos J)

30. scattering region AB CD s-channel Js
Physical region: p2 > 0, < Js < p
s = 4 (p2 + m2)
t = -2p2 (1 – cos Js )
Recall: if all masses equal (m)
s > 4m2

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

32. 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
(s,z)
(2l +1) fl (s) Pl ( z )
l =0 S = F
fl (s) partial waves

33. S S Partial waves Pl (z) are the corresponding eigenfunctions s l
(s,z)
(2l +1) fl (s) Pl ( z )
l =0 S = F
l are the eigenvalues of angular momentum
Pl (z) are the corresponding eigenfunctions

34. S S Partial waves fl (s) s l (2l +1) fl (s) Pl ( z ) (s,z) = (s,J) F
with z = cos J
fl (s) J F
(s,J)
Pl (cos J )

35. Legendre polynomials d l j dz Pl (z) Pj (z) = 0 if l = j 2
P0(z) = 1
P1(z) = z
P2(z) = (3z2 - 1)/2
P3(z) = (5z3 - 3z)/2
P4(z) = (35z4 –30 z2 + 3)/8
Pl (1) = 1
Pl (-z) = (-1)l Pl (z)
dz Pl (z) Pj (z) = 0
if l = j / -1 +1
dz Pl (z) Pj (z) = -1 +1 2
2l + 1
d l j

36. Pl (z) z Legendre polynomials P0(z) =1 P1(z) = z P2(z) = (3z2-1)/2 1 2 3 4
Pl (z) z
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8 -1
P0(z) =1
P1(z) = z
P2(z) = (3z2-1)/2
P3(z) = (5z3-3z)/2
Pl (1) = 1
Pl (-z) = (-1)l Pl (z)

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

38. Spin analysis M 1 q M 2
cos J
L = 0
L = 1
L = 2
ds/dW

39. pN pN scattering ds/dW p p p p p p p n 1234 MeV 1449 MeV 1678 MeVq
p p p p + -
p p p n
1234 MeV
1449 MeV
1678 MeV
1900 MeV

40. ds/dW
p 0p
q (deg.)

41. S
p 0p
q (deg.)

42. S
p 0p A N M1 M2 B
L2I 2J
q (deg.)

43. pN scattering
W (GeV)
s (mb)
p- p
p+ p
p+p p N p N
L2I 2J

44. p p p pN scattering L2I 2J N N N I = 1/2, 3/2 S = 1/2
J = L + S
= L - ½, L + ½ p N p p
L2I 2J L N N

45. pN amplitudes
Isospin 1/2
Imaginary T
SAID: Workman et al

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

47. S31
F15
D15
P31
S11
P33
D13
P11
F35
D35

48. N*(1520) D13
D13

49. Hadron States Breit-Wigner 1 s = E2 M2 – s - iMGx
Breit-Wigner
M2 – s - iMG 1
s = E2
definite quantum numbers J, P, C, I, ….

50. s = E2 Breit-Wigner 1 M2 – s - iMG 1 M2 (s) – s x
merely an approximation
valid in the region of the pole
M2 (s) – s 1

51. S Scattering Amplitude, (s,t) for spinless particlesds
d z
= 2p K(s) | (s,z) |2 F
with z = cos J
(s,z)
(2l +1) fl (s) Pl ( z )
l =0 S = F
Is F (s,z) and its partial wave decomposition unique?

52. S Scattering Amplitude, (s,t) for spinless particlesds
d z
= 2p g(s) | (s,z) |2 F
with z = cos J
(s,z)
(2l +1) fl (s) Pl ( z )
l =0 S = F
Is F (s,z) and its partial wave decomposition unique?
Clearly can multiply F (s,z) by a phase at every energy
and it describes the same ds/dz

53. S Scattering Amplitude, (s,t) for spinless particlesds
d z
= 2p K(s) | (s,z) |2 F
with z = cos J
(s,z)
(2l +1) fl (s) Pl ( z )
l =0 S = F
Is F (s,z) and its partial wave decomposition unique?
Clearly can multiply F (s,z) by a phase at every energy
and it describes the same ds/dz
For an elastic process,
imaginary part is directly proportional to the total cross-section,
so phase in the forward direction (z=1) can be fixed from experiment

54. stot = Im Fel (s, J=0) Unitarity optical theorem  s elastic amplitude1
Im Fel (s, J=0)
elastic amplitude
Total cross-section for AB everything
is related to the forward amplitude for AB AB
Measuring total cross-section
determines phase of elastic scattering amplitude

55. how to find the amplitudes
q (deg.)
ds/dcos q
absorb flux factor
into amplitude

56. how to find the amplitudes
q (deg.)
ds/dcos q
absorb flux factor
into amplitude

57. how to find the amplitudes
q (deg.)
ds/dcos q
absorb flux factor
into amplitude 2J
Truncate j < 2J

60. let z = cos 

61. let z = cos 

62. F(s,z) how to find the amplitudes exp (i ) 2J amplitudes
Barrelet ambiguity

63. F(s,z) how to find the amplitudes exp (i ) 2J amplitudes
continuum ambiguity

64. Barrelet ambiguity
cos  D P S

65. Barrelet ambiguity
cos  D P P´ S´ S D´

66. Barrelet zeros
Im zi
continuity

67. Barrelet zeros
dynamics and zeros, dynamics and poles

69. Hadroproduction M1 p R M2
exchange
M(K) GeV N B

70. Hadroproduction p R M1
exchange M2 N B