Coulomb effects in pionless effective field theory

Transcription

1 Coulomb effects in pionless effective field theory Sebastian König in collaboration with Hans-Werner Hammer Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn, Germany 6th Vienna Central European Seminar Vienna, 28/11/2009

2 Outline Introduction n-d system: Strong interaction only p-d system: Including Coulomb effects Application and Results Summary

3 Introduction Pionless effective field theory Effective Lagrangian

4 Pionless effective field theory Construct an effective field theory for low-energy few-nucleon systems Concepts and experimental input: At very low energies even pions can be integrated out only nucleons left as effective degrees of freedom Nonrelativistic framework demand Galilei-invariance of effective Lagrangian Large scattering lengths in N N scattering additional low-energy scale This program has been carried out quite successfully, already. e.g. Bedaque, Hammer, van Kolck 2000 Interesting: Study Coulomb effects in the 3-body system Rupak, Kong 2003 Use nucleon-deuteron system as an example!

5 Effective Lagrangian ( L = N D id 0 + ) 2 N + L photon 2M N ( d [σ i D d + id 0 + [ ( )]d 2 i t A σ t + id 0 + )] D 2 t A 4M N 4M N [ y ( d d i N T PdN ) i + h.c. ] [ y ( t t A N T Pt A N ) + h.c. ] Contents: Nucleon field N, doublet in spin and isospin space Two auxiliary dibaryon fields d i (spin triplet, isospin singlet) and t A (spin singlet, isospin triplet) corresponding to the respective channels in NN scattering Coupling constants y d and y t, dibaryon propagators are just constants (σ d, σ t ) to first order Couple to photons via covariant derivative: D µ = µ + ie ˆQ em A µ

6 n-d system: Strong interaction only With the lights out, it s less dangerous... Power counting Integral equation

7 Power counting Identify scales: Low-energy scale: p γ d O(Q) Cut-off Λ O(m π ), m π /M N O(Q/Λ) Assume yd 2 y2 t 1/Λ and σ d σ t Q Consequences: Integration measure d 3 q O(Q 3 ) Nucleon propagator O(M N /Q 2 ) (non-relativistic!) Leading order dibaryon propagator = i/σ O(1/Q) O(1) Re-sum propagator!

8 Power counting Identify scales: Low-energy scale: p γ d O(Q) Cut-off Λ O(m π ), m π /M N O(Q/Λ) Assume yd 2 y2 t 1/Λ and σ d σ t Q Consequences: Integration measure d 3 q O(Q 3 ) Nucleon propagator O(M N /Q 2 ) (non-relativistic!) Leading order dibaryon propagator = i/σ O(1/Q) O(1) Re-sum propagator! Bubble chain: d = = Fix parameters from N N scattering Dibaryon kinetic energy range corrections

9 Integral equation... all of same order Integral equation!

10 Integral equation... all of same order Integral equation! Formal structure: T (E; k, p) = K(E; k, p) + 1 π Λ = + + = dq K(E; q, p) D(E; q) T (E; k, q) for S-waves, incoming momentum k, outgoing momentum p Quartet channel quite simple

11 Integral equation... all of same order Integral equation! Formal structure: T (E; k, p) = K(E; k, p) + 1 π Λ = + + = dq K(E; q, p) D(E; q) T (E; k, q) for S-waves, incoming momentum k, outgoing momentum p Quartet channel quite simple Doublet channel coupled channels, 3N-force H(Λ) at leading order

12 p-d system: Including Coulomb effects...here we are now, entertain us! Coulomb photons Modified power counting New integral equations Numerical methods

13 Coulomb photons Consider L photon = 1 4 F µνf µν 1 2ξ quantization in Coulomb gauge ( ) µ A µ η µ η ν ν A µ 2 e jµ A µ }{{} = A for η µ =(1,0,0,0) One finds: Field component A 0 does not propagate eliminate with equation of motion Poisson equation: A 0 = e j 0 (ik) 2 A 0 = e j 0 Re-insert into Lagrangian: il int (k) (ie) j 0 (k) i k 2 (ie) j 0 (k) exchange of Coulomb photons ie 2 1 (ik) 2 = (ie) i k 2 (ie).

14 Coulomb photons Consider L photon = 1 4 F µνf µν 1 2ξ quantization in Coulomb gauge ( ) µ A µ η µ η ν ν A µ 2 e jµ A µ }{{} = A for η µ =(1,0,0,0) One finds: Field component A 0 does not propagate eliminate with equation of motion Poisson equation: A 0 = e j 0 (ik) 2 A 0 = e j 0 Re-insert into Lagrangian: il int (k) (ie) j 0 (k) i k 2 (ie) j 0 (k) exchange of Coulomb photons ie 2 1 (ik) 2 = (ie) i k 2 (ie). Compared to this, transverse photons are suppressed by powers of momenta and/or α/m N.

15 Power counting revisited Coulomb effects αm N /p are dominant at very low momenta! we can no longer assume p γ d, γ t Q Need simultaneous expansion in Q/Λ and p/(αm N )!

16 Power counting revisited Coulomb effects αm N /p are dominant at very low momenta! we can no longer assume p γ d, γ t Q Need simultaneous expansion in Q/Λ and p/(αm N )! Two scales in the loop integrations: Either d 3 q O(Q 3 ) or d 3 q O(p 3 ) Full dibaryon propagator either O(1/Q) or O(Q/p 2 ) Same for the photon propagator: O(1/Q 2 ) or O(1/p 2 ) depending on which contribution is picked up after the dq 0 -integration.

17 Power counting revisited Coulomb effects αm N /p are dominant at very low momenta! we can no longer assume p γ d, γ t Q Need simultaneous expansion in Q/Λ and p/(αm N )! Two scales in the loop integrations: Either d 3 q O(Q 3 ) or d 3 q O(p 3 ) Full dibaryon propagator either O(1/Q) or O(Q/p 2 ) Same for the photon propagator: O(1/Q 2 ) or O(1/p 2 ) depending on which contribution is picked up after the dq 0 -integration. Bottom line: Iterate and! Rupak, Kong 2003

18 Integral equation with Coulomb photons Include photons into the integral equation: = + + ( + + ) + full scattering amplitude T full Leave out nucleon exchange diarams Coulomb amplitude T c 1. Solve both equations 2. Get phase shifts from forward amplitude: δ = 1 2i log ( 1 + 2ikM N 3π Z 0T ) 3. Compare to experiment: δ diff (k) δ full (k) δ c (k) c.f. Jackson, Blatt 1950 Harrington 1965 This means that we remove the initial and final state Coulomb interaction!

19 Numerical methods (I) There are two types of singularities in the integral equations: Poles in the propagators (deuteron bound state!) D d,t (E; q) = 1 γ d,t + 3q 2 /4 M N E iǫ Logarithmic singularities in the kernels (from S-wave projection) K(E; q, p) Q 0 ( q 2 + p 2 M N E iǫ qp ) with Q 0 (a) = 1 ( ) a log a 1 Could deform integration contour, but not very convenient...

20 Numerical methods (I) There are two types of singularities in the integral equations: Poles in the propagators (deuteron bound state!) D d,t (E; q) = 1 γ d,t + 3q 2 /4 M N E iǫ Logarithmic singularities in the kernels (from S-wave projection) K(E; q, p) Q 0 ( q 2 + p 2 M N E iǫ qp ) with Q 0 (a) = 1 ( ) a log a 1 Could deform integration contour, but not very convenient... Instead: Principal value integration: 1 x±iǫ = PV 1 x iπδ(x) Log-singularities are integrable! However, numerically delicate... Discretisation yields a simple matrix equation!

21 Numerical methods (II) Photon propagator is singular at zero momentum transfer! Regularize this with a small photon mass: i q 2 i q 2 +λ 2 In the integral equation: K(E; k, p) Q 0 ( k2 +p 2 +λ 2 Major numerical difficulty turns out to be the peak in the inhom. terms! 2kp Re-shuffle integration mesh points! Then: Extrapolate to λ = 0 screening limit )

22 Numerical methods (II) Photon propagator is singular at zero momentum transfer! Regularize this with a small photon mass: i q 2 i q 2 +λ 2 In the integral equation: K(E; k, p) Q 0 ( k2 +p 2 +λ 2 Major numerical difficulty turns out to be the peak in the inhom. terms! 2kp Re-shuffle integration mesh points! Then: Extrapolate to λ = 0 screening limit ) MeV MeV δ [deg] MeV δ [deg] MeV λ [MeV] λ [MeV]

23 Application and Results Quartet channel Doublet channel He-3 binding energy

24 Quartet channel Couple deuteron and nucleon spins to spin 3 /2 All three nucleon spins aligned Pauli principle Consequences: Not very sensitive to short-range physics No bound state Only one channel, simple integral equation We can include the deuteron kinetic energy operator to all orders! D d (E; q) = 4π M N yd 2 1 γ d + 3q 2 /4 M N E iǫ ρ d 2 (3q 2 /4 M N E γ 2 d ) Cut-off stays below unphysical second pole in the propagator.

25 Scattering phase shifts - Quartet channel 0-25 αm N p 1 3 at p = 20 MeV n-d (full N2LO) p-d (full N2LO) Arvieux Christian et al. δ [deg] k [MeV] data from Arvieux 1973; Christian, Gammel 1953

26 Doublet channel Now couple deuteron and nucleon spins to spin 1 /2 Consequences: Pauli principle does not apply here need higher cut-off Singlet dibaryon in the intermediate state coupled channels Cannot easily use re-summed N2LO propagators... Furthermore: Fix leading-order 3-Nucleon force from bound-state equation! We can: Fix H(Λ) from triton binding energy in the strong system then calculate bound state from equation with Coulomb effects Predict He-3 binding energy!

27 He-3 binding energy 7.5 Exp. value: E He-3 B = 7.71 MeV LO NLO N2LO EB [MeV] Λ [MeV]

28 Scattering phase shifts - Doublet channel 0 n-d (N2LO) p-d (N2LO) Arvieux -20 δ [deg] k [MeV] data from Arvieux 1973

29 Summary It was demonstrated that... Coulomb effects can be included in pionless effective field theory the static Coulomb potential is dominant at low momenta power counting needs to be modified in the presence of Coulomb effects one needs to implement numerical methods carefully taking the screening limit is possible we can predict p-d observables from n-d experimental input *** Thanks for your attention!

30 Spares

31 Spin- and S-wave projection Use P i d = 1 8 σ 2 σ i τ 2 and P A t = 1 8 σ 2 τ 2 τ A to project onto the 3 S 1, I = 0 and 1 S 0, I = 1 states Couple the spins in the nucleon-deuteron system according to = = 2 channels for amplitude (T ij ) βb αa(e;k,p) Quartet: Set i, j = (1 i2)/ 2, α = β = 1, a = b = 1 T q Doublet: T d = 1 3 (σi ) α α (T ij ) β b α a (σj ) β β with α = β = 1, a = b = 1 Project onto S-waves with T l=0 (E; k, p) = dcosθ T (E;k,p)

32 Dibaryon propagators We have to re-sum the dibaryon propagators to all orders: d = = t = = Fix parameters from N N scattering: ia d,t (k) = y 2 d,t d,t (p 0 = k2 2M N,p = 0) = 4π M N i k cotδ d,t ik k cotδ d = γ d + ρ d 2 (k 2 + γ 2 d ) +... = y d, σ d k cotδ t = γ t + r 0t 2 k with γ t 1 a t = y t, σ t Residue of pole in d deuteron wave function renormalization Z 0 Include dibaryon kinetic energy operator effective range corrections

33 Effective range corrections Include dibaryon kinetic energy operators: i LO (p) i (p 0 p2 4M N ) i LO (p) Treat this as a perturbation NLO, N2LO Possible to re-sum geometric series, e.g. i ij d (p) = 4πi M N y 2 d 4πσ d M N y 2 d µ + δ ij p 2 4 M Np 0 iǫ + 4π M N y 2 d ( p 0 ) p2 4M N but still only N2LO (other contributions neglected!) Fix parameters by reproducing effective range expansions up to O(k 2 ) Unphysical second pole in re-summed propagator!

34 + + Three-nucleon force One finds: Strong cut-off dependence in the doublet channel! Renormalize with leading order three-nucleon force (SU(4)-symmetric) L 3 = M N H(Λ) Λ 2 ( y 2 dn ( d σ) ( d σ)n +... ) Substitute: H(Λ) e+05 Λ [MeV] Fix H(Λ) with three-body input e.g. Triton binding energy

35 Bound state equation Bound state pole in T-matrix T (E; k, p) = B(k)B(p) E + E B for E E B homogeneous integral equation for B(p) B(E, p) = 1 π Λ 0 dq K(E; q, p) D(E; q) B(E, q) Fix E = E Diagrammatic: = + B and cut-off Λ, find suitable H(Λ)

36 Scaling of the deuteron propagator Consider a diagram of the form Integrate over q 0... (Ed + q0,q) d ( E d + E N (E N q 0, q) ) q2,q 2M N γ d + e.g (q2 p 2 ) + γ 2 d 4 (q2 p 2 ) Q q 2 Deuteron pole enhanced for q p but typically suppressed by d 3 q p 3 Except when we also have a Coulomb photon propagator 1/p 2! Rupak, Kong 2003