1. Hadron05, Cyprus, page 1 Excited and Exotic States on the Lattice Introduction –How to extract excited states ? –The  ′ ghost in quenched QCD Light baryons –Nucleon, Roper, and S 11 (1535),  (1405) Pentaquarks Magnetic moments and polarizabilities Frank Lee, The George Washington University Collaborators: K.F. Liu, N. Mathur, W. Wilcox, L. Zhou, R. Kelly Thanks also to: U.S. Department of Energy and computing resources from NERSC, PSC, and JLab

2. Hadron05, Cyprus, page 2 The Particle Zoo Mesons Mesons Baryons Excitation spectrum of QCD

3. Hadron05, Cyprus, page 3 The proton in QCD: t y z u d u u d u confinement chiral symmetry breaking asymptotic freedom m u, m d ~ 5 MeV, m s ~ 100 MeV

4. Hadron05, Cyprus, page 4 Baryons on the lattice Parity separation is exact. For a certain spin-parity channel, the entire mass spectrum is contained in the correlation function. M1M1 M2M2 M3M3 … 0 x M1M1 M2M2 M3M3 … ½+ ½-

5. Hadron05, Cyprus, page 5 Curve-Fitting in Lattice QCD Variational Method Bayesian Priors Maximum Entropy Method Basic Problem: Given a finite set of measurements {G(t)}, how to extract {A n, M n } for n=1,2,3,… ? Ground state is easy: look at large time (or the ‘plateau’ method) What about excited states?

6. Hadron05, Cyprus, page 6 Bayesian Statistical Inference Bayes’ Theorem: The conditional probability of X given Y is equal to the conditional probability of Y given X, multiplied by the probability distribution of X, divided by the probability distribution of Y. Or Translation into our problem H represents all prior knowledge about our model A. P(G|AH) is likelihood probability of the data P(A|H) is prior probability P(G|H) is a normalization factor independent of A. P(A|GH) is posterior probability Rev. T. Bayes (1702-1761) Basic idea: Find A by maximizing the posterior probability:

7. Hadron05, Cyprus, page 7 Likelihood For a large number of measurements, the data is expected to obey the Gaussian distribution according to the central limit theorem: Average data Covariance matrix

8. Hadron05, Cyprus, page 8 Two types of priors Likelihood (  2 ) For 2), maximize For 1), minimize 2) Entropy prior: 1) Bayesian  2 prior: Maximize: In practice

9. Hadron05, Cyprus, page 9 Constrained Curve Fitting with Bayesian Priors 2) Fit as many terms in G th. 3) Use prior knowledge, like 4) Un-constrain the term of interest to have conservative error bars. (See heplat/0208055) 1) All data points are used.

10. Hadron05, Cyprus, page 10 Example: fitting all time slices Un-constrained fitConstrained (Lepage, heplat/0110175)

11. Hadron05, Cyprus, page 11 Reconstructing Artificial Data (heplat/0405011, Y. Chen et el)

12. Hadron05, Cyprus, page 12 A ‘double blind test’ We could not reproduce the input initially. –Reason: error too big After reducing the error by a factor of 2, we could, but we found two solutions close to each other, one of them wrong. After reducing the error by a factor of 10, we could reproduce the input unambiguously. The details of this experience are in heplat/0405001 with relative error 1.1% at t=1 and 12% at t=16

13. Hadron05, Cyprus, page 13 Pion excited state (Nucleon channel later)

14. Hadron05, Cyprus, page 14 Maximum Entropy Method (MEM) unbiased entropy m(  ) is called the default model (real and positive) which incorporates our prior knowledge on the functional dependence of A(  ). For example, m(  )=m 0  n, where n=2 for mesons, n=5 for baryons. Maximize Key features of MEM: It makes no a priori assumptions on the input parameters. For given data, an unique solution is obtained if it exists. statistical uncertainty can be quantified. hep-lat/0011040 Y. Asakawa et al

15. Hadron05, Cyprus, page 15 Testing MEM: mock data

16. Hadron05, Cyprus, page 16 Reducing the errors is more effective than having more time steps. b=0.1 N t =10 N t =20 N t =30 b=0.01 b=0.001 Testing MEM: sharp peak +broad peak

17. Hadron05, Cyprus, page 17 Testing MEM: pole+continuum

18. Hadron05, Cyprus, page 18 Testing MEM: 3 Peaks width = 0.01, 0.02, 0.03 GeV spacing = 0.5 GeV spacing = 0.1 GeV spacing = 0.2 GeV spacing = 0.3 GeV spacing = 0.4 GeV

19. Hadron05, Cyprus, page 19 MEM fitting: nucleon channel, J P =1/2 +- (heplat/0504020, Sasaki et al)

20. Hadron05, Cyprus, page 20 We use overlap fermion action It’s a particular way of putting D+m q on the lattice that preserves exact chiral symmetry (H. Neuberger, PLB417 (1998) 141) –No O (a) error, O (a 2 ) is small –Numerically checked that there is no additive quark mass renormalization –Critical slowing down is gentle all the way to low pion mass –No exceptional configurations. Our results are based on –16 3 X 28, a = 0.200(3) fm. L = 3.2 fm (80 configurations analyzed, 300 more being added) –12 3 X 28, a = 0.200(3) fm. L = 2.4 fm (80 configurations, 200 more being added) –20 3 X 32, a ~ 0.171 fm. L ~ 3.4 fm (100 configurations) –pion mass down to about 180 MeV –quenched approximation ghosts

21. Hadron05, Cyprus, page 21 The  ′ ghost in quenched QCD Quenched QCD Full QCD (hairpin) ….. It becomes a light degree of freedom –with a mass degenerate with the pion mass. It is present in all hadron correlators G(t). It gives a negative-metric contribution to the G(t) –It is unphysical (thus the name ghost) –A pathology of the quenched approximation  ′ (958)

22. Hadron05, Cyprus, page 22 W > 0 W<0 - - - - ηη Evidence of  ′ N ghost state in S 11 The effect of the ghost state decreases as pion mass increases. It has a sensitive volume dependence. First time seen in a baryon channel. Phys. Lett. B605 (2005) 137-143

23. Hadron05, Cyprus, page 23 Decoupling of  ′ N ghost state The  ′N ghost state decouples from the nucleon correlator around m   300 MeV.

24. Hadron05, Cyprus, page 24 Baryon Two-point Function Nucleon interpolating field: Positive-parity channel: N +  ′ N ( p=2  /L ) + Roper + … Negative-parity channel:  ′ N (p=0) +  ′ N ( p=2  /L ) + S 11 +…

25. Hadron05, Cyprus, page 25 Cross-over occurs close to chiral limit. Nucleon, Roper and S 11 0.5 1.0 1.5 2.5 2.0 Mass (GeV) N(938) 1/2 + P 11 (1440) 1/2 + S 11 (1535) 1/2 - Phys.Lett. B605 (2005) 137 Lowest m  ~ 180 MeV Phys.Lett. B605 (2005) 137

26. Hadron05, Cyprus, page 26 Hyperfine Interaction of quarks in Baryons At higher quark mass (above 300 MeV pion mass) Color-Spin Interaction (one-gluon-exchange) dominates. Isgur and Karl, PRD18, 4187 (1978) At lower quark mass (below 300 MeV pion mass) Flavor-Spin interaction (meson-exchange) dominates. Chiral symmetry plays major role Glozman and Riska, Phys. Rep. 268,263 (1996)  (1600)1/2+

27. Hadron05, Cyprus, page 27 What about Hyperons? The  (1405)? … different story!! 0.5 1.0 1.5 2.5 2.0 Mass (GeV)  (1115) 1/2 +  (1405) 1/2 -  (1600) 1/2 +

28. Hadron05, Cyprus, page 28 S 11 (1535)1/2 - and  (1405)1/2 - Puzzle: the two states have the same spin-parity, but why  (1405)(uds), having a strange quark, is lighter than S 11 (1535)(uud)? Answer: it’s the flavor structure. The  (1405)1/2- was constructed as a flavor-singlet state.

29. Hadron05, Cyprus, page 29 Conclusions on light excited baryons The combination of overlap fermion action and Bayesian fitting algorithm has allowed access deep into the chiral regime and excited states in quenched QCD. –The  ′ ghost is clearly seen. –As along as the  ′ ghost is removed, useful physics can be explored in the chiral regime even in quenched QCD. Exploratory studies have shown that the basic ordering of low-lying baryons can be reproduced on the lattice with standard interpolating fields built from three QCD quarks. –cross-overs in the region around m  ~ 300 MeV where chiral dynamics starts to dominate. –This supports the notion that there is a transition from color-spin to flavor-spin in the hyperfine interactions between quarks. –a node is observed in Roper’s radial wavefunction.

30. Hadron05, Cyprus, page 30 Pentaquarks New Topic

31. Hadron05, Cyprus, page 31 (from J. Negele) See talk by A. Williams and F. Csikor at this workshop

32. Hadron05, Cyprus, page 32 Five-quark mass spectrum Pentaquark correlation function contains the entire 5-quark spectrum: KN scattering states + genuine pentaquark states, of both parities. M1M1 M2M2 M3M3 … 0 x On the lattice parity can be separated exactly: M1M1 M2M2 M3M3 … u u u d u u d u ½+ ½-

33. Hadron05, Cyprus, page 33 How to identify a pentaquark on the lattice? 1.43 1.54 GeV 0 Negative-parity channel Pentaquark, if it exists, is entangled with KN scattering states. Positive-parity is easier to identify than negative-parity. -KN threshold raised for positive-parity (p=n*2  /L) -at least two states have to be isolated for negative-parity. The nature of extracted states must be further tested. 1.43 1.54 GeV p=1 p=3 0 Positive-parity channel pentaquark p=2 p=4 p=1 p=3 p=2 p=0

34. Hadron05, Cyprus, page 34 P-wave (1/2 +, I= 0) Propagators turn negative: ground state is KNη' ghost state. In fitting function this ghost state, pentaquark and KN p-wave scattering state are the first three states. We find ghost and scattering states, but not pentaquark near 1.54 GeV. The volume dependence in E K (p L ) + E N (p L ) due to the P-wave nature is seen. Near chiral limit the scattering length is close to zero which is consistent with experiment.

35. Hadron05, Cyprus, page 35 S-wave (1/2¯, I = 0) No need to consider ghost state (propagators are positive). Near the chiral limit ground state mass is consistent with the threshold KN scattering state. Identification of this ground state with the scattering state implies vanishing scattering length, which is consistent with the experiment. The next state is an average of p=1, 2 and perhaps 3.

36. Hadron05, Cyprus, page 36 Test 1: volume dependence (squeeze it) For bound state, the spectral weight, as defined in G(t)=We -m t, will not show very weak volume dependence. For two particle scattering state, the spectral weight will show inverse volume dependence (1/V) We see (12)/W(16)=16 3 /12 3 =2.37, so the observed states are KN scattering states in both channels. Negative-parity channel Positive-parity channel

37. Hadron05, Cyprus, page 37 Test 2: hybrid boundary condition (twist it) Plateau raised, suggesting KN scattering state. Change b.c.: anti-periodic for u and d quarks, periodic for s quark Consequence: If KN state, mass will rise. If bound state, mass will not change (Ishii et al)

38. Hadron05, Cyprus, page 38 Test 3: KN open jaw (bite it) No sign of pentaquark of positive-parity near 1.54 GeV.

39. Hadron05, Cyprus, page 39 Conclusions on pentaquarks on the lattice We found no evidence for a pentaquark near 1.54 GeV for either parity. The states we found are all consistent with KN-states. The presence of KN scattering states is a major complication in the isolation of a true pentaquark signal, if it exists. –Positive-parity Almost consensus among different groups. The adjustable P-wave KN threshold is a great advantage. –Negative-parity The S-wave KN threshold is just below 1.54 GeV and is fixed. Very hard to tell apart a pentaquark from a KN-state. At least two lowest states in this channel need to be isolated reliably. Variational method based on multiple operators holds promise. After a state is isolated, it should be tested to reveal its true character. –Test 1: Squeeze it (inverse volume dependence of spectral weights) –Test 2: Twist it (hybrid boundary condition) –Test 3: Bite it (remove the KN scattering states explicitly from the correlation function, and examine the rest)

40. Hadron05, Cyprus, page 40 Pentaquarks Lesson 1: It’s very hard to establish its existence –pentaquark or KN scattering states? –Reliable isolation of two states in each parity channel –Must be put through tests (squeeze it, twist it, bite it) Lesson 2: It’s very hard to rule it out –One has to exhaust all possibilities. –operators –quenched approximation –…

41. Hadron05, Cyprus, page 41 Magnetic moments and polarizabilties in the background field method New Topic

42. Hadron05, Cyprus, page 42 Introduction of a static magnetic field on the lattice Minimal coupling in the QCD covariant derivative This suggests multiplying a U(1) phase factor to the SU(3) link variables : (Quark propagators are generated in the new background) To apply B in the z-direction, one can choose the vector potential, then the phase factor is

43. Hadron05, Cyprus, page 43 Lattice details Standard Wilson gauge action –24 4 lattice,  =6.0 (or a ≈ 0.1 fm) –150 configurations Standard Wilson fermion action –  =0.1515, 0.1525, 0.1535, 0.1540, 0.1545, 0.1555 –Pion mass about 1015, 908, 794, 732, 667, 522 MeV –Strange quark mass corresponds to  =0.1535 (or m  ~794 MeV) –Source location (x,y,z,t)=(12,1,1,2) –Boundary conditions: periodic in y and z, fixed in x and t The following 5 dimensionless numbers  ≡qBa 2 =+0.00036, -0.00072, +0.00144, -0.00288, +0.00576 correspond to 4 small B fields eBa 2 = -0.00108, 0.00216, -0.00432, 0.00864 for both u and d (or s) quarks. –Small in the sense that the mass shift is only a fraction of the proton mass:  B/m ~ 0.6 to 4.8% at the smallest pion mass. In absolute terms, B ~ 10 13 tesla. x z B

44. Hadron05, Cyprus, page 44 Polarizabilities on the Lattice l Polarizability is a measure of how tightly a hadron is bound in response to external probes. l In particular, they are related to the quadratic response in the interaction energy çwhere  is electric polarizability,  is magnetic polarizability l On the lattice, we determine the polarizabilities directly by calculating the mass shift of hadrons in the presence of progressively small E and B fields

45. Hadron05, Cyprus, page 45 A computational trick We generate two sets of quark propagators, one with the original set of fields, one with the fields reversed. The mass shift in the presence of small fields is At the cost of a factor of two, –by taking the average, [  m(B) +  m(-B)]/2, we get the leading quadratic response with the odd-powered terms eliminated. (magnetic polarizability) –by taking the difference, [  m(B) -  m(-B)]/2, we get the leading linear response with the even-powered terms eliminated. (magnetic moment) Our calculation is equivalent to 11 mass calculations. –5 original fields, 5 reversed, plus the zero-field to set the baseline

46. Hadron05, Cyprus, page 46 Magnetic Moment: two methods Form factor method: G M (Q 2 =0) –Since the minimum momentum on the lattice is non-zero (p=2  /L), extrapolation to zero momentum transfer is required. –Three-point function calculations Background field method –direct access –Two-point function calculations

47. Hadron05, Cyprus, page 47 Magnetic moment For a Dirac particle of spin s in small fields, where upper sign means spin-up and lower sign spin- down, and g factor is extracted from Look for the slope on the straight line of the form:

48. Hadron05, Cyprus, page 48 Proton mass shifts We use the 2 smallest fields to fit the line.

49. Hadron05, Cyprus, page 49 Neutron mass shifts

50. Hadron05, Cyprus, page 50 Proton magnetic moment

51. Hadron05, Cyprus, page 51 Neutron magnetic moment

52. Hadron05, Cyprus, page 52 Octet Sigma magnetic moments

53. Hadron05, Cyprus, page 53 Delta magnetic moments

54. Hadron05, Cyprus, page 54 Proton and  + magnetic moments

55. Hadron05, Cyprus, page 55 Baryon Magnetic Moments Phys. Lett. B, F.X. Lee et al

56. Hadron05, Cyprus, page 56 Polarizabilities New Topic

57. Hadron05, Cyprus, page 57 Experimental Information on Polarizabilities (a theorist’s summary) l Proton electric polarizability (  ) is around 12 in units of 10 -4 fm 3. l Proton magnetic polarizability (  ) is around 2 in units of 10 -4 fm 3. l Neutron  is about the same as proton  l Neutron  is about the same as proton  They are extracted from low-energy expansion of Compton scattering amplitude

58. Hadron05, Cyprus, page 58 Polarizabilities on the Lattice For polarizability (quadratic response), we average over mass shifts from B and –B to eliminate the odd-power terms. So we are expecting parabolas going through the origin. For magnetic moment (linear response), we take the difference in mass shifts from B and –B to eliminate the even-power terms. So we are expecting straight lines going through the origin. Mass shifts in response to background fields:

59. Hadron05, Cyprus, page 59 Effective-mass plot for neutron mass shifts

60. Hadron05, Cyprus, page 60 Neutron Mass Shifts in Magnetic Fields

61. Hadron05, Cyprus, page 61 Proton and Neutron

62. Hadron05, Cyprus, page 62 Octet Sigmas

63. Hadron05, Cyprus, page 63 Deltas --

64. Hadron05, Cyprus, page 64 Pions

65. Hadron05, Cyprus, page 65 Kaons

66. Hadron05, Cyprus, page 66 Rhos

67. Hadron05, Cyprus, page 67 Baryon Octet Magnetic Polarizabilities

68. Hadron05, Cyprus, page 68 Baryon Decuplet Magnetic Polarizabilities

69. Hadron05, Cyprus, page 69 Meson Magnetic Polarizabilities

70. Hadron05, Cyprus, page 70 Conclusion Using standard technology, we obtained the magnetic moments and polarizabilties, sweeping through 30 hadrons. –For magnetic moment, most of our results are consistent with experiments where available. Others are predictions. –For polarizabilities, most of our results are predictions. The background field method provides a good tool for to the form factor method for obtaining the magnetic moments using mass shifts. –The method is robust. No extrapolation to Q 2 =0 is needed. –For 4 non-zero field values, the cost is about a factor of 11 that of a standard mass spectrum calculation. –Polarizabilties can be obtained from the same data set with little overhead Further study: quantify systematic errors –Continuum extrapolation –Push deeper into the light mass region (tmQCD, or overlap fermions) –Chiral extrapolation (guidance from ChPT) –Quenching effects

71. Hadron05, Cyprus, page 71 Reserve Slides New Topic

72. Hadron05, Cyprus, page 72 Electric Polarizabilities of Neutral Particles

73. Hadron05, Cyprus, page 73 Scattering state and its volume dependence Scattering state and its volume dependence Box normalization on the lattice: Two point function : Lattice Continuum For one particle bound state there is no explicit volume dependence in the spectral weight W. For two particle state : Fitting fuction : Therefore, for two particle scattering state the spectral weight has inverse volume dependence.

74. Hadron05, Cyprus, page 74 The issue of interpolating fields KN type: Diquark-diquark-antiquark type: (minus sign for I=0, plus sign for I=1) However, the two types are related by a Fiertz re-arrangement.

75. Hadron05, Cyprus, page 75 Pentaquark interpolating fields The two types couple to the same physical spectrum, albeit with different strengths. The two types are explicitly related by a Fiertz re-arrangement: The complete set, including non-local operators, contains 19 operators. Need correlation matrix method to select the best operators.

76. Hadron05, Cyprus, page 76 Comparison of different operators At large time, they all project to the same state (KN scattering state). preliminary