Few-nucleon contributions to π-nucleus scattering

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1 Mitglied der Helmholtz-Gemeinschaft Few-nucleon contributions to π-nucleus scattering Andreas Nogga, Forschungszentrum Jülich INT Program on Simulations and Symmetries: Cold Atoms, QCD, and Few-hadron Systems Motivation Q- versus W-counting: a closer look to the deuteron Numerical approach 3-nucleon corrections Results for π- 3 He 4-nucleon corrections Results for π He Conclusions & utlook S. Liebig, V. Baru, F. Ballout, C. Hanhart, AN, arxiv:

2 2 Motivation π-n scattering length pattern of chiral symmetry and its breaking L ChPT (PCAC, Weinberg, 1966 a (+ =0 a ( = 1 8π(1 + m π m N m π f π m π 1 Higher order results have been calculated Experimentally not so well known, requires very accurate determination of scattering lengths in two isospin channels, e.g. a ( = 1 2 ( aπ p a π n a (+ = 1 2 ( aπ p + a π n 0 Most accurate determination from pionic atoms using a π A = ( 1+mπ /m N 1+m π /Am N ( Aa (+ +2T 3 a ( + IV corrections + few-nucleon corrections (for IV corrections see Hoferichter et al., NPA 2010, Baru et al. arxiv: [nucl-th]

3 Motivation ChPT calculation of few-nucleon corrections (Weinberg, 1992 a πa Ψ A Ô Ψ A perturbative expansion of the transition operator Ô phenomenological few-nucleon wave functions or chiral wave functions (Beane at al., 1998 Ψ A Result (for A=2: some of the few-nucleon corrections are smaller than expected but they are in general somewhat larger than expected Is this a systematic deviation from the power counting? Beane at al., 2003 : consequence of a second scale entering Q-counting q d = m N E d 45 MeV m π consequences for error estimates, power counting in complex nuclei? April 14,

4 Contributions to π- 2 H scattering Weinberg-counting - typical momenta m π 1-nucleon contributions a π A = ( 1+mπ /m N 1+m π /Am N ( Aa (+ +2T 3 a ( m π f π 2 2-nucleon contributions (Beane et al., (Weinberg, 1992; Beane et al (isovector 4 4 (isoscalar no 3 contributions boost & dispersive corrections almost cancel each other (Lensky et al. 2007, Baru et al April 14,

5 5 Numerical results (Beane et al. 2-nucleon contributions have different sizes than expected by power counting m π < m π m π 1 4 Solution of Beane at al. assume that the typical momentum is the binding momenta of the deuteron q d = m N E d 45 MeV m π q ( mπ Λ Λ 2

6 6 Q-counting Numerical results seem to fit expectations! m π 1 2 (Q 3 + < m π 1 2 (Q m π 1 4 (Q 6? 4 (Q 7 First counter term is very much suppressed compared to few-nucleon contributions Boost corrections are of the order (Q 5 Is Weinberg counting restored for 3 He, 4 He? (Is the kinetic energy measurable?

7 7 Physical & unphysical deuterons L chiral interaction V L = ( ga 2F π 2 σ 1 q σ 2 q q 2 + m 2 π τ 1 τ 2 + C S + C T σ 1 σ 2 small deuteron binding energy is fine-tuning effect undo fine-tuning of CS and fit to range of binding energies (here CT=0 E d [MeV] C S [GeV 2 ] large range of cutoffs in L to study size of first ππ4n contribution CS is in general of natural size ( 100 GeV -2 except close to new spurious bound states Λ[fm 1 ] C S [GeV 2 ]

8 8 Cutoff dependence - π- 2 H a (1a π 2 H [10 3 m 1 π ] a ( 1bc π 2 H [10 3 m 1 π ] Λ [fm 1 ] Λ [fm 1 ] +... a (2 π 2 H [10 3 m 1 π ] Λ [fm 1 ] few-nucleon contributions are independent of the cutoff (see also AN, Hanhart, 2005, Pavon Valderrama, Ruiz Arriola, 2006 Platter, Phillips, 2006 cutoff variation most significant for double scattering estimate of short range contribution m π 1 naive estimate agrees (Weinberg

9 9 Energy dependence - π- 2 H m π 2 q 2 const Q-counting (W-counting +... q2 m π 2 m π q Q-counting: ratios should have distinct binding energy dependence +... E d 2 E d

10 10 Energy dependence - π- 2 H 0.10 E d r (d E d E d [MeV] Explicit calculation strongly disagrees with Q-counting expectations Q-counting is not realized in low energy pion scattering

11 11 Toy model - π- 2 H: Hulthén wf binding momentum is not the only scale entering the deuteron wf Hulthén wf allows one to incorporate second scale ψ( p =N(γ, β 1 1 p 2 + β 2 p 2 + γ 2 fit / to normalized s-wave part of the / L wave function second scale is binding energy independent β =1.7m π u(r[fm 1/2 ] Hulthén βcontact Λ = 4 fm 1 Λ = 10 fm 1 Λ = 20 fm 1 E d =2.23 MeV (a r [fm] u(r[fm 1/2 ] Hulthén β contact Λ = 20 fm 1 (b E d =0.01 MeV r [fm]

12 Toy model - π- 2 H: Hulthén wf ( a (1a π 2 H [10 3 m 1 π ] Hulthén -40 expansion of Hulthén -50 L (only S-wave E d [MeV] 0.07 (b a (1a π 2 H β ( (1 + x 4x = κ x (1 x 2 ln (1 + x 2 κ = β 1 8π 2 (1 + m π /2m N m 2 π f 4 π x = γ/β a (1a π 2 H,L = κ x ln(4x for double scattering diagram: Hulthén wf is a reasonable approximation r April 14, 2010 Hulthén L (only S-wave (a E d [MeV] energy dependence is logarithmic ratio is qualitatively reproduced (if restricted to S-wave contact wf is not a good approximation convergence of HPEFT breaks down much earlier than expected 12

13 Triple scattering enhancement Q-counting is not confirmed! How to explain deviations of the numerical results from W-counting expectations? +... probably accidental cancelations look at the corresponding loop function I 0 (ω,v Q, Q 2 = 1 i d d l 1 (2π d v l ω iɛ 1 m π2 l 2 iɛ 1 m π2 (l Q 2 iɛ (Dmitrasinovic et al., 1999 I 0 (m π, 0, q 2 = 1 8 q + δi 0 1 8π 2 q naive dimensional analysis π 2 enhancement compared to power counting from integrable singularities Similar topologies lead to similar enhancements in NN interaction, nucleon form factor, π 0 photoproduction (Friar et al., 2003; Becher et al., 1999; Bernard et al., 1991 April 14,

14 π scattering on A=3 and A=4 same strategy as before obtain wave functions from solutions of the Schrödinger equation (or Faddeev/Yakubovsky equations fold with π scattering operator (based on W-counting following calculations are based on a set of wf L NL/N 2 L models April 14, 2010 to check dependence on cutoffs Λ[fm 1 ] C S [GeV 2 ] C T [GeV 2 ] consistent with operator 3nf in N 2 L binding energies well described include 3nf s describe binding energies well Λ / Λ B( 3 He B( 4 He L 2.0 / L 3.0 / L 4.0 / L 5.0 / L 10.0 / L 20.0 / NL 400/ NL 550/ NL 550/ NL 400/ NL 550/ N 2 L 450/ N 2 L 600/ N 2 L 550/ N 2 L 450/ N 2 L 600/ CD-Bonn AV Expt

15 Numerical approach Most involved calculation: 4-nucleon operator in 4-nucleon system Avoid partial wave decompostion (tedious! Need to calculate expectation value Ô = αα = αα d 3 p 12 d 3 p 3 d 3 q 4 d 3 p 12d 3 p 3d 3 q 4 Ψ p 12 p 3 q 4 α... Ô... p 12 p 3 q 4 α Ψ d 3 p 12 d 3 p 3 d 3 q 4 d 3 p 12d 3 p 3d 3 q 4w(p 12,p 3,q 4 ; p 12,p 3,q 4 Ψ p 12 p 3 q 4 α... Ô... p 12 p 3 q 4 α Ψ w(p 12,p 3,q 4 ; p 12,p 3,q 4 2 p 12 p 3 q 4 4 Ψ ( p 12 p 3 q 4, α 4 He wave function spin-isospin channels 4NF matrix element p 12 p 3 q 4 α Ô p 12 p 3 q 4 α α m 1 m 2 m 3 m 4 m t 1m t 2m t 3m t generated using Maple/Mathematica Metropolis walk for evaluation based on weight function w(p 12, p 3, q 4 ; p 12, p 3, q 4 1 (p i + C i n i April 14, 2010 i=12,3,4, 12,3,4 15

16 Contributions to π- 3 He scattering Weinberg-counting - typical momenta m π 1-nucleon contributions ( 1+mπ /m N a π A = 1+m π /Am N 2-nucleon contributions ( Aa (+ +2T 3 a ( m π f π π 2 3 (isovector 4 (isoscalar 3-nucleon contributions more April 14, 2010 Does the explicit calculation support this power counting? 16

17 17 Cutoff dependence for π- 3 He a [10 3 m 1 π ] CD-Bonn, AV18 a (3 L dominant 3-nucleon contributions a (1a L NL N 2 L dominant 2-nucleon contributions Λ [ fm 1] (1a no unexpected cutoff dependences for L wave functions cutoff dependence seems to be relevant: much larger variation than for deuteron

18 18 Binding energy dependence Bulk of cutoff dependence is result of low order NN interaction binding energy dependence aπ 3 [10 3 m 1 ] He π a (3 a (1a a (2is 2N triple scattering 3N 2N double scattering B 3He [MeV] Estimate cutoff dependence from variation of N 2 L interactions

19 19 Results for π- 3 He results do not strongly dependent on cutoff when binding energies are K counter term estimate based on naive dimensional analysis larger cutoff dependence might result in low estimate since leading few-nucleon contribution is isoscalar Λ/ Λ a (??a a (??bc a (??is a (??iv a (?? a (?? CD Bonn 25.08( ( ( ( ( (6 AV ( ( ( ( ( (4 N 2 L 450/ ( ( ( ( ( (2 N 2 L 600/ ( ( ( ( ( (4 N 2 L 550/ ( ( ( ( ( (4 N 2 L 450/ ( ( ( ( ( (1 N 2 L 600/ ( ( ( ( ( (2 a (2N+3N =(( ( ± m 1 π =( 26.0 ± m 1 π. ã (+ = (1± m 1 π, a ( = (86.5± m 1 a (1N+2N+3N = (62 ± 4 ± m 1 π π (from Hoferichter et al., 2009 a [ 10 3 m 1 π R. Abela et al. 56 ± 6 G. R. Mason et al. 43 ± 5 I. Schwanner et al. 41 ± 4 ]

20 20 Contributions to π- 4 He scattering 1-nucleon contributions a π A = ( 1+mπ /m N 1+m π /Am N ( Aa (+ +2T 3 a ( 2-nucleon contributions π 2 3 (isovector 4 (isoscalar 3-nucleon contributions (isovector 4-nucleon contributions (incomplete 6

21 21 Results for π- 4 He Λ/ Λ AV (7-1.29(2 5.00(5 2.73(84 N 2 L 450/ (4 1.92(1 6.98(2 3.00(20 N 2 L 600/ (8-0.09(2 6.16(3 2.13(11 N 2 L 550/ (6 0.50(1 6.42(3 2.31(31 N 2 L 450/ (7 1.81(1 6.68(3 2.56(10 N 2 L 600/ (8 0.36(1 6.34(2 2.81(17 cutoff dependence in line with naive expectations (below 5 % of leading 2N contribution 2N contribution only twice as large as for the deuteron (np and pp/nn pairs enter with opposite sign 4N term finally comparable to counter term contribution (can be neglected few-nucleon contributions larger than naively expected

22 Relative scaling of few-nucleon diagrams relative suppression of two-nucleon contributions are completely in agreement with expansion parameter & naive dimensional analysis m π 1 5 (cutoff dependence, triple scattering few-nucleon contributions are suppressed by much less than expected suppression is comparable for 1N:2N:3N:4N Λ/ Λ a (2N /a(1n a (3N /a(2n a (4N /a (3N AV N 2 L 450/ N 2 L 600/ N 2 L 550/ N 2 L 450/ N 2 L 600/ systematically understandable? impact for other few-nucleon operators / three-nucleon interactions? April 14,

23 23 Conclusions & utlook systematic overview of few-nucleon contributions to πa scattering at low energies π- 2 H no significant cutoff dependence of the most prominent diagrams binding momentum is not driving the power counting (no Q-counting toy model based on Hulthén wf indicates early breakdown of Hπ-EFT enhancement of specific topolgy could be visible in other processes π- 3 He complete set of 3-nucleon diagrams leading 3-nucleon diagrams are isovector no significant cutoff dependence prediction of π- 3 He scattering length up to the accuracy possible π- 4 He studied the probably most important 4-nucleon contribution accurate results for all relevant few-nucleon contributions isospin violation needs to be take into account check systematics of π- 2 H, probably no further constraints on π-n naive counting good for estimate of subleading A-nucleon contributions relative suppression of A and A+1-nucleon contributions smaller than expected