Covariant spectator theory of np scattering

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1 Departamento de Física da Universidade de Évora and Centro de Física Nuclear da Universidade de Lisboa Portugal October 20, 2009

2 Collaborators Franz Gross (Jlab) Sérgio Aleandre Pinto (PhD student, University of Évora and CFNUL Lisbon)

3 Outline 1 Principles of the Covariant Spectator Theory (CST) 2 CST two-body equations and NN scattering 3 CST of the 3N bound state 4 Fits of NN models in CST 5 Electromagnetic 3N form factors

4 Principles of the CST Main objective: relativistic theory of few-hadron systems (replace Schrödinger theory) Practical solutions should be feasible for realistic problems Start from relativistic quantum field theory: manifest covariance built in from the start Maintain manifest covariance in all modifications Bound states, unitarity: perturbation theory not enough, infinite sum of Feynman diagrams needed Formulate integral equations to sum infinite series of diagrams

5 Bethe-Salpeter series Assumption: hadronic interaction through boson echange ladder and crossed ladder diagrams dominate others treated effectively through form factors and physical masses Infinite series of ladder and crossed ladder diagrams generated by iterating an irreducible kernel V with propagator G 0 Bethe-Salpeter equation M = V + V G 0 M Kernel itself is infinite series Truncation after first diagram (OBE) ladder approimation

6 Equivalence of two-body equations The same total amplitude can be obtained when choosing different propagators and corresponding irreducible kernels Solution Suppose M is obtained from kernel V 1 and propagator G 1 M = V 1 + V 1 G 1 M = V 1 + MG 1 V 1 Choose different propagator G 2 and determine V 2 such that M = V 2 + V 2 G 2 M = V 2 + MG 2 V 2 This requires M = (1 V 1 G 1 ) 1 V 1 = V 2 (1 G 2 V 2 ) 1 V 2 = V 1 + V 1 (G 1 G 2 )V 2 Useful if equation with G 2 is easier to solve than with G 1 and if V 2 V 1 is reasonable (but eact equivalence is lost!) Basis of quasi-potential equations (dimensional reduction of BSE)

7 Two-Body Covariant Spectator Equation (CSE) I G BS G G Rewrite bo diagram G = G BS G Gross propagator G: one particle on mass shell( ) keep only positive-energy pole in energy loop integration Equivalence relation determines corresponding kernel V V = V BS + V BS G V = V BS + V BS G V BS + One obtains a reorganization of the full BS series tend to cancel Cancellations in some cases (scalar theory) Eact solution in one-body limit (M heavy ) not small not small

8 Two-Body Covariant Spectator Equation (CSE) II Cancellations in all orders: full BS series is reproduced (not BS ladder series!) M = V + VGM where V is OBE kernel, G Gross propagator Even when cancellations are not eact, remainder may be described in terms of effective OBE kernels

9 Importance of cancellations For CST with OBE kernels to work, are cancellations necessary? Depends on the problem to be solved! 1 Solution of a given field theory (original motivation) Fields, interaction vertices, coupling constants, masses given Find (approimately) sum of complete BS series If CST with OBE kernels is used, cancellations are important 2 Construction of an interaction model to fit data Coupling constants, particle masses, cut-off masses (form factors) determined through fitting data Echanged bosons are effective particles More complicated processes may be effectively described through OBE How can we know if this a good approimation? Efficiency of the fit number of needed adjustable parameters!

10 CST equation for NN scattering One intermediate nucleon on mass shell For closed set of equations, all eternal nucleons ecept one also placed on mass shell: M = + M = I We keep only OBE kernel ( CST ladder approimation ) Identical particles kernel has to be antisymmetrized guarantees antisymmetric amplitudes Advantages of OBE form Simplicity Consistency (construction of em currents, 3N forces,...)

11 Spurious singularities Antisymmetrization leads to singularities in the echange term of the kernel, when initial and final state off mass shell W E p E k W E p E k ω ω E p W E k E p W E k Real meson production Spurious Production singularity (when W > 2m + µ) is physical, but not useful here (further processes have to be included for realistic description of meson production) Spurious singularity not physical: would be canceled by term in net order kernel Both singularities should be removed

12 Removing singularities from the OBE kernel Different methods to eliminate spurious singularities Type A Replace denominator of propagator in echange term by denominator of direct term (eliminates all singularities) Disadvantage: difficult to construct consistent e.m. interaction current Type B Subtract most singular part (meson and nucleons on shell) eliminates imaginary part, leaves integrable singularity (principal value) Disadvantage: complicated in practice Type C New: redefine denominator of boson propagator µ 2 q 2 iɛ µ 2 + q 2 iɛ Does not change direct term (always q 2 < 0) We know how to construct consistent e.m. interaction currents

13 Structure of the new OBE kernels p 1 p 2 Λ b 1(p 1, k 1 ) q Λ b 2(p 2, k 2 ) V αα ;ββ (p, k; P) = b k 1 k 2 V b 12(p, k; P) Total four-momentum P P = k 1 + k 2 = p 1 + p 2 k 1 = 1 2 P + k, k 2 = 1 2 P k Relative momenta p and k k = (k 1 k 2 )/2, p = (p 1 p 2 )/2 sum of OBE kernels General form of OBE kernels V12(p, b k; P) = ɛ b δ Λb 1 (p 1, k 1 ) Λ b 2 (p 2, k 2 ) mb 2 + f (Λ b, q) q2 Boson-nucleon verte functions Λ b

14 Structure of boson-nucleon vertices 1/2 p gλ q = p k k General form of boson-nucleon vertices Terms with /p m or /k m contribute only if respective nucleon is off mass shell Scalar (σ, δ) or (σ 0, σ 1 ) scalar off-shell coupling if ν s 0 g s Λ s = g s + ν s ( p / m + /k m ) + κ s ( / 2m 4m 2 p m ) (/k m) Pseudoscalar (π, η) { g p Λ p = g p γ 5 + ν p [( p / m ) γ 5 + γ 5 (/k m) ] } + { 2m = g p (1 ν p )γ 5 + ν } p 2m γ5 /q + Pseudoscalar off-shell coupling corresponds to PS-PV miing

15 Structure of boson-nucleon vertices 2/2 Vector (ρ, ω) [ g v Λ µ v = g v γ µ + κ v 2m iσµν q ν + ν v [( p / m ) γ µ + γ µ (/k m) ] ] + 2m Aial vector (a 1, h 1 ) g a Λ µ a = g a γ 5 γ µ treated as contact interaction (in propagator, m 2 a + q 2 m 2 a) Verte form factors in product form: h(p 2 )h(k 2 )f (q 2 ) Meson form factor ( f (q 2 Λ 2 b ) = Λ 2 b + q2 Λ b boson cut-off mass ) 2 Form factor for off-shell nucleon ( λ h(p 2 2 ) = N m 2) 2 ( λ 2 N m 2) + (m 2 p 2 ) 2 m nucleon mass λ N nucleon cut-off mass

16 CST of 3N scattering Prescription: spectator is always on mass shell (follows from considerations analogous to two-body case) A typical 3N scattering diagram Red blobs are 2N scattering amplitudes solutions of 2N CST equations Consistency between two- and three-body sectors (cluster separability!) In all intermediate states two particles on mass shell Loop integrations are 3-dimensional

17 CST 3N bound state equation Bound state: pole in 3N scattering amplitude below threshold M = Γ Γ Mt 2 P 2 + R Homogenious Faddeev-type integral equation for verte function Γ q q' = 2 M in verte function indicates spectator during last interaction Γ 1 = 2M 1 G 0 P 12 Γ 1 Full Dirac structure of nucleons taken into account Permutation P 12 : change of reference frame boost effects Solution by decomposition into 3-body helicity partial waves

18 Older NN potentials χ 2 /N data for np scattering at MeV (N data = 2510) From: Stoks and de Swart, PRC 52, 1698 (1995) Potential χ 2 /N data HJ Reid TRS Paris Urb Arg Bonn Nijm OBE form First CST OBE potentials Gross, Van Orden and Holinde, PRC 45, 2094 (1992) IA, IB: 4 mesons, πnn PV-PS miing (10 parameters) IIA, IIB: 6 mesons, πnn pure PV (13 parameters) χ 2 /N data 3 4 Problem: 3N binding energy E t 6 MeV (models type A)

19 Improved CST models Inclusion of scalar off-shell coupling into model type IIA Scalar off-shell coupling parameter ν ν σ = 0.75ν, ν δ = 2.6ν Family of models with different ν (W00, W05, W10,...) 13 adjustable parameters fit to Nijmegen phase shift analysis (1993) χ 2 /N data to np observables calculated with SAID (code by Arndt) Best NN fit (W16) best 3N binding energy (without 3NF!)

20 Realistic NN potentials Nijmegen PSA (1993) achieved χ 2 /N data 1 (with 39 parameters) after eliminating some data sets Nijmegen potentials: with pure OBE form could not get below χ 2 /N data 1.9 Is some important physics missing? Nijmegen abandoned strict OBE form: parameters can vary independently in each partial wave increase in number of parameters Similarly CD-Bonn: scalar meson parameters vary per partial wave Argonne AV18: π echange + phenomenology Potential # parameters χ 2 /N data 1 Nijm I (nonlocal) 41 Nijm II (local) 47 Reid93 (update of Reid68) CD-Bonn Argonne AV

21 New models Several important improvements Type C (spurious singularities removed) Very general OBE kernel structure: 1 isoscalar + 1 isovector meson of each type (S, PS, V, AV) Ecept for pions, meson masses are adjustabe (effective mesons) Masses different for π 0 and π ± Independent off-shell couplings for σ and δ E.m. interaction included (photon echange between n and p) Fit directly to np observables instead of phase shifts 3788 np data updated 2007 world data base! Details: Gross and Stadler, PLB 660, 161 (2008); PRC 78, (2008)

22 Parameters of new potentials WJC-1 and WJC-2

23 Comparison to data base of Nijmegen PSA93 and of CD-Bonn WJC models 1993 and 2000 without refitting

24 Data selection Database includes world np data up to 2006 Nijmegen selection criterion applied (based on χ 2 distribution): 3788 data kept, 1180 ecluded z = χ 2 /n PO82 DE73 FR00 RA98 z ME n (number of data points in set)

25 Is there an A y problem in np scattering? New measurements of A y at TUNL E lab = 12 MeV

26 Scaling of data sets If normalization error δ sys given, scaling factor Z is determined by the fit n χ 2 (Z o i t i ) 2 (Z 1)2 t = (Z δo i ) 2 + (δ sys ) 2 i=1 Published TUNL data include normalization error added in quadrature After etracting the normalization error and scaling, χ 2 drops from 6.62 to 3.85

27 A y with WJC-1 and WJC-2 WJC-1 and WJC-2 yield better agreement to TUNL data Still, data set is (marginally) ecluded by the Nijmegen criterion (demands χ 2 /n < 2.26 for n = 16) 20 other A y sets below 20 MeV: only 1 ecluded (because its χ 2 /n too small!) No compelling evidence for serious A y problem χ 2 /n Nijm93PSA 3.85 WJC WJC

28 Detail of A y Detailed view around the maimum

29 Phase shifts 1/3

30 Phase shifts 2/3

31 Phase shifts 3/3

32 Deuteron

33 3N bound state 1.30 Scalar off-shell coupling ν σ is fied at different values, other parameters refit Obtain similar behavior as in case of older models W Again, best NN fit yields also best 3N binding energy, without additional 3N forces WJC WJC /N data E t (MeV) 2 /N data E t (MeV)

34 Three-nucleon forces What constitutes an irreducible 3NF depends on the framework CST generates reducible 3NF s, that would appear as irreducible 3NF s in nonrelativistic QM Off-shell nucleon propagator decomposed into positive and negative energy component `m + /p αβ m 2 p 2 iɛ = m " uα(p, λ)ū β (p, λ) E p E λ p p 0 iɛ vα( p, λ) v # β ( p, λ) E p + p 0 iɛ 3NF due to intermediate negative-energy states and difference between relativistic positive-energy and nonrelativistic propagator ρ =

35 Off-shell coupling and three-body forces Off-shell coupling can lead to effective contact interactions g σ ν σ (/p m) 1 /p m g σ g 2 σν σ This mechanism automatically creates a number of new diagrams Eamples Two-body sector Three-body sector

36 Electromagnetic three-nucleon current in CST = 3 (A) + 3 (B) + 3 (C) + 6ζ + 6ζ + 6ζ (D) (E) (F) ζ +6ζ (G) (H) (I) (J) Conserved current, consistent with CST 2N and 3N equations (see Gross, Stadler, Peña, PRC 69, (2004)) Complete Impulse Approimation (CIA): diagrams (A) to (F) Interaction Currents: diagrams (G) to (J)

37 Calculations with models W First CST calculations of the 3N e.m. form factors (PhD work of Sérgio A. Pinto) PRC 79, (2009) Family of W models in CIA W00 W10 W16 W19 W26 ν gσ 2 /4π ν σ m σ gδ 2 /4π ν δ m δ χ 2 /N data E t Description of data in CIA not epected (interaction currents!) Study model dependence Compare to nonrelativistic Impulse Approimation with Relativistic Corrections (IARC) Greens Function Monte Carlo calculations (AV18 + UI 3NF) by Marcucci et al. Same wave functions for 3 H and 3 He (no Coulomb)

38 Nucleon current p q p F 1N,2N (Q 2 ) on-shell form factors F 3N (Q 2 ) off-shell form factor (unknown) Λ (p) = m /p 2m Nucleon current used with conserved electron current S N (p) = [h(p2 )] 2 m /p j µ N (p, p) =f 0 (p 2, p 2 ) F 1N (Q 2 ) γ µ +f 0(p 2, p 2 ) F 2N (Q 2 ) i σµν q ν 2m +g 0 (p 2, p 2 )F 3N (Q 2 )Λ (p )γ µ Λ (p) dressed N propagator Ward-Takahashi identity q µ j µ N (p, p) = S 1 1 N (p) SN (p ) determines functions f 0 and g 0 ; f 0 (m2, m2) = 1 Effect of off-shell form factors found to be small The following results are with on-shell form factors only

39 Charge form factors of 3 He and 3 H F C /F scale He F C /F scale H

40 Isoscalar and isovector charge 3N form factors F S C /F scale F V C /F scale

41 Magnetic form factors of 3 He and 3 H F M /F scale He F M /F scale H

42 Isoscalar and isovector magnetic 3N form factors 4 F S M /F scale F V M /F scale

43 Results for W in CIA Results for CST models very reasonable W16 is remarkably close to IARC (for Q 4 6 fm 1 ), despite very different frameworks and dynamical input Looks like E t determines form factors to relatively large Q Good agreement with data in IS magnetic form factor (leading pion current suppressed) Z-graphs are included in CST to all orders Z-graph in πnn PS contact current in PV coupling When added to IARC, contact gives big effect. But contact current not included in IARC! So why do W16 and IARC agree so well?

44 Negative energy states W models use pure PV coupling Z-graphs are suppressed! Full result for W16 vs result without 3N negative-energy states F C /F scale 0 3 He F C /F scale H F M /F scale 0 3 He F M /F scale 0 3 H -1-1

45 Magnetic moments Magnetic moments are known to be sensitive to interaction currents impulse approimation not close to data µ( 3 He) µ( 3 H) µ S µ V W16/MMD W16/Galster IARC/Galster Eperiment

46 Rms radii r 2 ch [fm 2 ] r.m.s. radius (fm) charge magnetic Model E t (MeV) 3 He 3 H 3 He 3 H W W W W W Known relation r 2 ch 1/Et holds for CST models: 5 3 He 3 H /E t [MeV 61 ]

47 First results with WJC-1 and WJC-2 First calculations with new models WJC-1 and WJC-2 Not yet ready to calculate verte functions with 2 nucleons off mass shell, as required by 3N current, e.g. Simple approimation: Γ(p 0 ) Γ(E p ) and only positive energy We can test this approimation with W16 WJC-2 πnn pure PV, but WJC-1 has mied PS-PV coupling

48 3N charge form factors in CIA-0 F C H F C /F scale H -2-3 F C He F C /F scale He -1.5

49 3N IS and IV charge form factors in CIA-0 F S C F S C /F scale F V C F V C /F scale

50 3N magnetic form factors in CIA-0 F M H 10-6 F M /F scale H F M He 10-6 F M /F scale He -1

51 3N IS and IV magnetic form factors in CIA-0 F S M F S M /F scale F V M F V M /F scale

52 Summary New high-precision np potentials within CST Ecellent χ 2 to most complete data base new phase shift analysis (phases can be used outside CST!) OBE mechanism works very well 3N binding energy reproduced without additional 3-body forces First calculation of 3N e.m. form factors in CST 3N form factors in CIA in remarkable agreement with Argonne-Pisa IARC Results are very encouraging

53 NICKLAUSSE What was that? SPALANZANI Nothing! Physics! Ah, sir, physics! COCHENILLE S-s-supper s ready! (Jacques Offenbach, The Tales of Hoffmann)