Covariant effective field theory (EFT!)

Transcription

1 Outline Covariant effective field theory (EFT!) (or, can the dinosaur learn anything from the cockroach?) Part I: The Covariant Spectator approach for two and three nucleon interactions at JLab momentum transfers (aside) Part II: Ideas for improvements -- toward a Covariant Effective field theory (EFT!) for GeV reactions

2 Acknowledgements CS collaborators (partial list): J. W. Van Orden and Karl Holinde (NN ) D. O. Riska (2N currents ) Alfred Stadler (3N ) John Tjon and Cetin Savkli (Feynman-Schwinger studies ) Alfred Stadler and Teresa Pena (3N currents ) CS teachers: Dick Arndt, Ruprecht Machleidt, Nijmegen group JLab EFT discussion group (E. Epelbaum, R. Higa, Jose Goity) and Vladimir Pascalutsa All the participants at this series of Workshops -- many thanks for very enjoyable and stimulating discussions. Its fun to argue, BUT let s not degenerate into a Fo News syndrome let s argue about real issues and not fabricated ones

3 Aside Typical JLab process: deuteron form factor at Q 2 1 GeV 2. In relativistic physics momenta and energies are NOT correlated! We go off the mass-shell, but remain on the energy-shell. In this case (in the Breit system) p p 1 D p 2 = m 2 p 1 2 = (D p) 2 = M d 2 + m 2 2 M d Q2 m 2 + p 2 + Q p (M d m) 2 D 2 = M d 2 If p is large, we are far off-shell and probe the short range structure (the important relative momenta are of order Q/4), but still the rest energy of the deuteron is FIXED at its mass; the only energy change is due to the Lorentz boost In nonrelativistic physics we go off the energy shell but remain on the mass shell. In this case E + E 1 = m 2 + p 2 + m Q p 2 and energy and momentum are correlated ( ) 2 2m

4 The true landscape #1 what does the dinosaur see? what does the cockroach see?

5 The true landscape #2 what do they see together?

6 Part I Covariant Spectator theory - philosophy Few body nuclear physics at JLab (GeV) energies (conventional EFT NOT an option - aside). What do we do? Preserve all symmetries Poincare invariance essential -- manifest covariance useful unitarity (conservation of flu) electromagnetic gauge invariance chiral invariance Microscopic dynamics OBE dynamics with point couplings, but form factors for the self energies of each hadron Organizational principle -- include echanges of all mesons and quantum numbers up to about 1 GeV. Cutoff at the nucleon mass scale. Mesons needed: π, 2π (σ 0, σ 1 ), η, ρ, ω plus short distance counter terms. Maintain consistency electromagnetic currents constrained by WT identities (but still not unique) three-body forces constrained by two-body forces

7 Covariant Spectator theory -- Definition The spectator theory starts from the n-body Bethe-Salpeter equation and restricts n-1 particles to their positive energy mass shells. The propagator for these particles is replaced by S αβ ( p) = ( m + p / ) αβ u β (p, s) m 2 p 2 iε 2πiδ + (m2 p 2 ) u α (p, s) s Integration over the n-1 internal energies (p 0 ) places these particles on their positive energy mass-shell. All 4-d integrations reduce to 3-d integrations. Remark: These on-shell particles do not propagate in intermediate states. The spinors are absorbed into matri elements, and the on-shell particles becomes part of the source for the single propagating off-shell particle. The two body scattering equation is, diagrammatically, M = + M

8 Both the BS and the CS theories have a close connection to field theory The Bethe-Salpeter amplitude is a well defined field theoretic matri element: Ψ( 1, 2 ) = 0 T ψ ( 1 )ψ ( 1 ) ( ) d The Covariant Spectator amplitude is also a well defined field theoretic amplitude: Ψ( 1 ) = N ψ ( 1 ) d Equations for the Bethe-Salpeter and the Spectator* amplitudes can be derived from field theory Both are manifestly covariant under all Poincaré transformations (advantage) Both incorporate negative energy (antiparticle) states (disadvantage?) *O. W. Greenberg s "n-quantum approimation"

9 Properties of the two-body Spectator amplitude from translational invariance: d 4 e ip from rotational invariance n ψ () d = (2π ) 4 δ 4 (p + n d) n ψ (0) d 1 n,λ ψ α (0) d,ξ = 2 M d ( 2π ) S(p)Γ µ (p) C 3 αβ u Τ β ( p,λ)ξ µ α λ ξ from transformations under boosts B(Λ) n,λ ψ α (0) d,ξ = B αα ' { } ξ µ + = ψ µ λ ' λ (p) u α (p,λ ') + ψ µ λ ' λ (p) v α ( p,λ ') positive energy spinor negative energy spinor Λn,λ' ψ α' (0) Λd,Λξ D (1/2) λ 'λ (ω) eact conservation of momentum and energy at the verte eact the most general form possible for the coupling of a spin 1 particle to two spin 1/2 particles, one off-shell eact obtained from Wigner rotations and Dirac boost matri boost matri for off-shell particle in Dirac space Wigner rotation of the spin of the on-shell particle

10 Dynamics: phenomenological OBE One boson echange diagram: Scalar: σ 0 NN (and σ 1 NN) coupling p 1 ' p 2 ' p 1 p 2 Λ(p 1 ', p 1 ) Λ(p 2 ', p 2 ) m m 2 (p 1 ' p 1 ) 2 Λ(p', p) = g + ν [ 2m 2m /p' /p ] zero on-shell Pseudoscalar: π NN (and η NN) coupling Λ(p', p) = g γ 5 1 ν 2m ( m p ' )γ 5 + γ 5 m p ( ) Vector: ρ NN (and ω NN) coupling Λ(p', p) = g γ µ + κ µν iσ p' p 2m ( ) ν + ν 2m ( m p ' )γ µ + γ µ m p ( )

11 Family of OBE models for NN scattering based on 1993 calculations Kernel of the integral equation was represented by OBE g π g η g σ g δ g ω g ρ = 13 Parameters π η σ δ ω = ρ ν π =1 =0 ν η =1 =0 ν σ ν δ κ ω κ ρ λω=1 λρ λ ρ = 1 ν ρ κ ρ σ 0 σ 1 spin I- mass g 2 /4π κ # of parity spin Para π η ± σ ± δ ± 0.4* -- 2 ω ± ρ ± ± cutoffs Λ π 2000 Λ m 1300 Λ N 1800 ρ miing λ ρ = 1.55 ± 0.4 We fied the ratio of the ν s { ν σ = 0.75 ν g 2 /4π ν δ = 2.60 ν g χ 2 /datum ~ /4π

12 Spectator equations for three-body systems* Define three-body verte functions for each possibility M M M M Γ this particle is the last spectator then three body Faddeev-like equations emerge automatically. For identical particles they are: Γ = 2 Γ M Γ this amplitude already known from the 2-body sector *Alfred Stadler, FG, and Michael Frank, Phys. Rev. C 56, 2396 (1997)

13 3N binding energy is very sensitive to ν (off shell coupling of the scalar mesons)* E T ( ) ν=1.6 ν σ = 1.2, ν δ = 4.16 is strongly favored, both by the 3N binding energy and the 2N data! eperimental value MeV eperimental binding energy at ν=1.6! χ 2 data best fit to the 2N data (minimum χ ) at ν=1.6! ν ν=1.6 *three body calculations done with Alfred Stadler, Phys. Rev. Letters 78, 26 (1997)

14 Recent results (in progress): OBE model for NN scattering Kernel of the integral equation is still represented by OBE Recent fit (still under development) with 21 Parameters Thanks to J. de Swart for helpful advice. spin parity I- spin χ 2 /datum = 1.26 (for the 2001 data set)! mass g 2 /4π κ ν # of Para γ π ± π η σ σ ω ρ Cutoffs (3) Λ π =1786 Λ m = 1192 Λ N = 1861 Thanks to Mart Rentmeester and Rob Timmermans for helpful discussions about data

15 Two body current operator in the spectator theory Inelastic Scattering Γ + M Γ + Γ + M Γ RIA FSI MEC Elastic Scattering Interaction current Γ Γ + Γ Γ = 1 ± 2 RIA MEC + +

16 Three body current operator in the spectator theory * The gauge invariant three-body breakup current in the spectator theory (with on-shell particles labeled by an ) requires many diagrams where the FSI term is = ζ +6ζ ζ +2ζ + RIA IAC FSI *Kvinikhidze & Blankleider, PRC 56, 2973 (1997) Adam & Van Orden PRC 71: (2005) FG, A. Stadler, & T. Pena PRC 69: (2004) ζ +6ζ

17 Conclusions to part I We have a covariant theory (CS theory) suitable for the calculation of 2 and 3 body electromagnetic observables when the ecitations are small but the momentum transfers are large. It has been (and is being) applied to NN (and 3N?) scattering, deuteron form factors, electrodisintegration of the deuteron, 3 He form factors, and 2 and 3 body electrodisintegration of 3 He. The goals are to eplain these interactions in terms of a consistent dynamics based on the CS theory using a covariant OBE model. determine the parameters of the OBE model and the OBE interaction currents that emerge. compare these effective interactions with QCD predictions!

18 Part II Can the ideas of EFT improve the CS theory? At present, regularization and short range physics are both contained in the form factors The most important of these is the nucleon form factor S( p) = f (p) m p / ; f ( p) = 2(Λ 2 m 2 ) 2 (Λ 2 p 2 ) 2 + (Λ 2 m 2 ) 2 The fits are very sensitive to Λ Use the ideas of EFT to separate these two roles: Regularize using the PDS of Kaplan, Savage, and Wise Parameterize short range physics using constants Assume that the physics is known up to echange masses of about 1 GeV. Short range physics is above 1 GeV

19 Overview -- Report on work in progress Assumptions: Ignore the known physics corresponding to echanges of bosons with masses less than 1 GeV (add this later). Parameterize the short range physics with contact interactions of the ψψ 2 type. Chose the mass scale M for the ψψ 2 interaction to be m (the nucleon mass) Regularize using power divergence subtraction (PDS) Eample: the 1 S 0 partial wave

20 Lagrangian for a 1 S 0 state Introduce the NN 1 S 0 verte function Γ 0+ () = (Cγ 5 ) ab ψ a ()ψ b () = ψ T ()Cγ 5 ψ () Then, the Lagrangian density for a 1 S 0 state is L() = ψ () i m ψ () λ Γ0+ () Γ 0+ () In d dimensions, λ has dimensions of λ = λ 0 where λ 0 is dimensionless. M 2 d 2-d, so the coupling is

21 Power counting (naive) Scales: there are only three scales in the problem: p, m, and M. First, assume p ~ m «M, so that δ p M m M 1 Naive dimensional counting gives a superficial divergence D for any NN scattering Feynman diagram equal to D = (d-2) (n-1) where n is the number of vertices (or the order) of the diagram. Hence we epect the size of any diagram V (n) to go like V (n) λ 0 M d 2 λ 0 δ d 2 ( ) n 1 Conclusion: if λ 0 ~ 1, and d >2, perturbation theory applies.

22 Power counting (nonrelativistic domain) Assume, p << m. Introduce a new scale α = p m δ 2 δ Then the n th order diagram has the form: V (n) λ 0 m λ M d 2 0 M Assume that the term of order α is of order 1. Then only the terms of order α 2 or smaller can be ignored. All other terms must be summed to all orders. Hence: d 2 large constant term 1+ c 1 α + c n α n 1 All diagrams contribute to the constant term. This depends on the scale, and will be fied phenomenologically. The task is to calculate the terms of order α, which are independent of the scale. ~1 n 1 small remainder term

23 Eample: the s-channel bubble 1 P + k 2 1 P k 2 The s-channel bubble B(s) is V (2) (s) = λ 2 B(s)(Cγ 5 ) a 'b ' (Cγ 5 ) ba where ε d =2-d/2, p 2 =m 2 -s/4, and λ λb(s) = i 0 d d k tr 5 Cγ ( m + 1 P + k 2 )Cγ 5 m M P k d 2 (2π ) d m 2 ( 1 P + k 2 ) 2 iε m2 1 P k 2 = λ Γ(ε )M 2ε 1 d d 4m dz (4π ) 2 ε d 2 6 4ε d ( p 2 + sz 2 ) ε d (1 ε 1 2 d )( p 2 + sz 2 ) ε d 1 lim λb(s) = λ 0 d 4 with the large constant and the small remainder µ p s 8π M + R 2 p2 ( ) µ = 1 2π + m 2 8π 2 M 2 R(p 2 ) = ( ) T ( ) 2 iε Γ '(1) log 4πm2 M log 2 p s 4π 2 M 2 arctan 2p s p 2 4π 2 M 2 Γ '(1) log 4πm2 M log 2

24 Summation of the leading bubble terms gives M 0+ (s) = λ 0 M 2 = = 1 + λb(s) + λb(s) M 2 λ 0 8π m p s λ 0 µm 2 p s 8π ( ) 2 + i i i i = 1 1 M 2 µ λ 0 + p s 8π Hence, λ 0 runs with M according to: λ 0 = and M has a pole (or resonance) at where m 0 is a parameter fied by the effective range epansion. s = 2m 2 1 ± 1 m 4 0 m 4 1 µ + m 2 0 8π M 2

25 The u- and t-channel bubbles New feature of relativistic theory is the presence of u- and t- channel bubbles We can show(?) that these are analytic in p 2 near p 2 ~ 0. Hence they contribute only to the constant and p 2 terms, and their effect can be absorbed into adjustable parameters Conclusion: the presence of u- and t- channel bubbles does not change the conclusions drawn from study of s-channel bubbles

26 Conclusion: Power counting for relativistic theory in a nonrelativistic domain Each diagram is of the form M = Constant + c 1 α + R(α 2 ) All diagrams contribute a constant term, which sometimes violates power counting. This does not matter since the constant term is simply a parameter describing the short range physics that must be fit to the data. Only diagrams with an elastic cut contribute non-analytic terms of order α, and these can be calculated and summed. The remainder terms R(α 2 ) are analytic and can be absorbed into derivative terms in the Lagrangian. These are then fit to the data. Not much predictive power, but divergences are handled without form factors. (Is this really an advantage?)

27 Connection with the Covariant Spectator theory If A ± iε = m 2 ( 1 P ± k 2 ) 2 iε = ( E k 1 P 2 0 k 0 iε )( E k + 1 P 2 0 ± k 0 iε ) A + A iε = 2P k iε the CS theory gives the following bubble contribution = i 4 B CS (s) = i d d k 4 (2π ) N 1 d (A A + iε)(a + iε) + 1 (A + A iε)(a iε) a b one pole LHP one pole UHP = i d d k 4 (2π ) N d 1 d 0 [(1 )A + + A i(1+ )ε ] (1 )A + A + i(1 + )ε d d 1 k =1/2+z 2 (2π ) N dz 1 d ( 1 z)a z ( 2 ) A i( 3 + z)ε 2 + dz 2 [ ] 1 + z 2 =1/2-z 1 ( ) A + ( 1 2 z)a + i( 3 2 z)ε

28 Connection with the Covariant Spectator theory (2) Hence B CS (s) = i 4 d d k N dz (2π ) d ( 1 z)a z 2 2 µ ' p s 8π M R ( ) p2 ( ) A i( z)ε B(s) Different constant and half the remainder term THEREFORE, in an EFT sense, equivalent to the full bubble term CS theory is equivalent (from an EFT! point of view) to the full field theory. (Is this useful?)

29 Etensions and reinterpretation: EFT! in a relativistic domain Suppose we are in the relativistic domain (p ~ m). Then either: power counting works, because λ 0 << (M/m) 2 and the physics is perturbative, or λ 0 ~ (M/m) 2, power counting does not work, and all diagrams are of equal size, and all must be summed Even if power counting does NOT work, we may separate the terms nonanalytic in p 2 from those which are analytic, and sum them using s-channel bubbles. These terms are predicted by the theory (up to the arbitrary constant!). Is this important?? adjust the size of the analytic terms by adding derivative terms to the Lagrangian. These cannot be predicted. There is also no longer an organizational principle for choosing derivatives -- all are of equal size. In either case, this relativistic effective theory has no content (i.e.) is purely phenomenological (in common with its nonrelativistic counterpart). But we have found a way to regularize and handle the short range physics.

30 Where do we go from here? One scenario: Step 1: Start with a kernel of the form V = µ p s 8π M 2 + c 2p 2 + c 4 p 4 + Fit data adjustable parameters Add the pion (and other meson echanges?) in an attempt to reduce the number of unknown short range parameters. For eample, get c 2 and c 4 from meson echange?? Calculate meson echange without form factors or cutoffs using a two potential formalism and a Pade series.

31 Where do we go from here? One scenario (2) the two potential series is M = M m + (1+ M m ) 0 λ 1+ λb(1+ M m ) 0 (1+ M m ) the Pade series is used to calculate M m M m = 2 p c n n = n=1 p n=1 p n '=1 d n 1 n d 2 n ' n '

32 Conclusions to Part II What has been learned from this eercise? A full power counting scheme, with accompaning organizational principle, seems to eist only in nonrelativistic situations. BUT it can be done with either a relativistic, or nonrelativistic formalism. From the EFT! point of view, CS theory is just as good as a full field theory (to be thought about some more) Does this help justify our original approach and calculations? Why do relativistic calculations? justified if fewer parameters are needed to fit data and a greater unity between dynamics and interaction currents can be achieved

33 END