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Wednesday May 31 st , 2017

Michael Pennington William & Mary / Glasgow. Wednesday May 31 st , 2017. Hadron Spectrum: window on confinement. Step One: spectrum of baryons, mesons quarks and QCD Step Two: tools for discovery, experiment and Amplitude Analyses

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Wednesday May 31 st , 2017

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

  2. 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 confinement strong coupling 1 strong QCD 0 -15 0 10 r (m) asymptotic freedom strong QCD pQCD

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

  5. q q q q q Resonances inQCD 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. precision research QCD accurate modelling precision tools

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

  11. Hadron Physics } hadrons A B

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

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

  14. positive pion positive pion . . . negative pion . . 0 100 200 MeV MARCH 12, 1952 DELTA RESONANCE DISCOVERED! Cross-section for positive pions on hydrogen peaks above 1200 MeV

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

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

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

  18. pN scattering p+ p p+p p- p p- p s (mb) p p E (GeV) N N p p p p L2I2J energy E N N N N

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

  20. Spectrum of hadrons

  21. ds/dW 1234 MeV p p p p + + - 0 p p p n pN pNscattering 1449 MeV 1678 MeV - - p p p p 1900 MeV q q q

  22. P 1234 MeV p p p p + + - 0 p p p n pN pNscattering 1449 MeV 1678 MeV - - p p p p 1900 MeV q q q

  23. Relativistic kinematics C A t p1 p3 s D B p4 p2 2-to-2 scattering

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

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

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

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

  28. Center of momentum frame: 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 ) u = (p1 – p4)2 = (p3- p2)2 t = (p1 – p3)2 = (p4- p2)2 s = (p1 + p2)2 = (p3 + p4)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)

  29. Center of momentum frame: p1 = ( p2 = ( /2, 0,0,p) /2, 0,0, -p) p3 = ( /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)  s  s  s  s simplest case all masses equal, m p = q

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

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

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

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

  34. Partial waves S s l l S (2l +1)fl (s)Pl ( z ) (s,z) = F l =0 with z = cos J fl(s) (s,J) Pl(cosJ) J F

  35. Legendre polynomials +1 +1 dz Pl (z) Pj (z) = 0 -1 -1 2 dlj ifl = j / dz Pl (z) Pj (z) = 2l + 1 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)

  36. Legendre polynomials 1 0 0.8 0.6 1 0.4 2 2 3 0.2 4 4 Pl(z) 0 -0.2 3 -0.4 -0.6 1 -0.8 -1 -1 -0.5 0 0.5 1 z 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. Partial waves S (2l +1)fl (s)Pl ( z ) (s,z) = F l =0 l 0 1 2 3 … fl S P D G … S l l l notation:

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

  39. ds/dW 1234 MeV p p p p + + - 0 p p p n 1449 MeV 1678 MeV - - p p p p 1900 MeV q q q pN pNscattering

  40. p 0p ds/dW q (deg.)

  41. p 0p S q (deg.)

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

  43. pN scattering p+ p p+p p- p p- p s (mb) p p W (GeV) N N p p L2I2J N N

  44. pN scattering p p I = 1/2, 3/2 S = 1/2 L = 0, 1, 2, 3, … J = L + S = L - ½, L + ½ N N L2I2J p p L N N

  45. pNamplitudes Isospin 1/2 Imaginary T SAID: Workmanet al

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

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

  48. N*(1520) D13 D13

  49. Hadron States E E Breit-Wigner x 1 M2 – s - iMG s = E2 definite quantum numbersJ, P, C, I, ….

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

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