The NN system: why and how we iterate

Transcription

1 The NN system: why and how we iterate Daniel Phillips Ohio University Research supported by the US department of energy

2 Plan Why we iterate I: contact interactions Why we iterate II: pion exchange How we iterate: The Lippmann-Schwinger equation Solution of the LSE for the LO potential Some words on higher orders

3 TNN in a very-low-energy EFT Lowest-order LNN L (0) NN = C T 2 (B σb)(b σb) C S 2 (B B)(B B) Short-range S-wave force, c.f. L>2 yesterday Weinberg NN force described by contact interaction im (1) = ic ic Description valid provided that λ 1/k>>R Tree level, C=4 π a/m C 1/(M m π )? Validity of loop expansion if a is large?

4 Order of iterate Let s calculate! E Loop integral for E=-B<0, i.e. B >0. d im (2) 4 p i = (2π) 4 ( ic)2 B p 0 p 2 2M + iη d im (2) = (ic) 2 3 p i (2 ) 3 B p 2 M d M (2) = MC 2 3 p 1 (2 ) 3 MB + p 2 ) MB = MC2 2 2 ( Λ 2 Exercise with a cutoff Piece proportional to Λ violates power counting. Either remove it ( subtraction scheme ) or use regularization + subtraction that does not produce it i p 0 p 2 2M + iη

5 Alternative reg. schemes For Dim. Reg. with Minimal Subtraction d M (2) = MC 2 d p 1 (2π) d MB + p 2 ( ) 2 d = MC2 (4π) d 2 Or use power-law divergence subtraction Regardless, key result is that loop C 2 M k Γ 2 = C 2 M MB 4π M (2) = MC2 4π (MB) d 2 2 (µ MB) Exercise Kaplan, Savage, Wise, PLB (1998)

6 Compare for E>0 For E>0 need to analytically continue to B=-E-iη M (2) = imc2 k 4π Loop C 2 M k, Tree Graph C ; k = ME Size of C? If C 1/(M m π ) loop down by k/m π. For natural scattering length, loops are suppressed. If C a/m then loop ka. Need to include to describe k 1/a. Formally, assign C to be of order P -1, so that loop is of order P -1 Kaplan, Savage, Wise, PLB (1998)

7 Why we iterate I: summary M (1) =-C; M (2) = imc2 k 4π ; k = ME Loop C 2 M k, Tree Graph C. van Kolck; Kaplan, Savage, Wise; Gegelia; Richardson, Birse, McGovern ( ) For k 1/a<<1/R, C a 1/k P -1 loop corrections matter Strongly interacting Quantum System T=V + VG0V + VG0VG0V +..., all terms O(P -1 ) if C P -1 T = C = 4π M ( 1 C ( 1 a µ d d p 1 (2π) d E + p 2 /M ) 1 T = 4πa M ) iak Exercise

8 Iteration: Argument I We know that the NN system has a low-energy bound state: the deuteron with B= MeV: γ=45 MeV In 1 S0: virtual bound state, a=-23 fm momentum=8mev These states will not be produced by perturbation theory Need to choose C a/m NOT C 1/(M m π ) Non-trivial infra-red fixed point associated with fine tuning How does fine tuning affect coupling constants of higher dimension? van Kolck; Kaplan, Savage, Wise; Gegelia; Richardson, Birse, McGovern

9 Applications of EFT(π) This LO result can be extended to arbitrarily high precision, provided that kr << 1 t = 4πa M 1 1+iak + O ( R Extension to electroweak reactions straightforward. See, e.g. Rupak np dγ up to 10 MeV to 1%; Butler & Chen, pp de + ν cross sections for SNO Extended to NNN system, calculations at 3% accuracy Leading-order prediction for alpha particle Works well, but restricted to p<mπ a, kr ) See, e.g. DP, Rupak, Savage (1999) Bedaque, Griesshammer, Hammer, Rupak; Platter, DP Platter, Hammer, Meissner

10 Are pion loops suppressed? Let s calculate! p 0 = p2 2M ; p 0 = p 2 2M ; E = p2 M = p 2 M q = p p im T P E = ( ) 4 iga d 4 p σ 1 (p p )σ 2 (p p ) 2f π (2π) 4 (p p ) 2 m 2 π i i σ 1 (p p)σ 2 (p p) E p 0 p 2 2M + iη p 0 p 2 2M + iη (p p) 2 m 2 π Dominant contributions: p 0 p 0,p 0 p2 2M 1 1 (p p ) 2 m 2 π (p p ) 2 + m 2 π

11 Instantaneous potential etc. Now can do p0 integral analytically dp 0 2π i E p 0 p 2 2M + iη i p 0 p 2 2M + iη = i E p 2 M + iη Recover usual non-relativistic propagator and potential im TPE = d 3 p (2π) 3 ( i)v (p, p ) Question 1: what is order of this, relative to tree-level? Question 2: What is order of V OPE relative to C? i E p 2 M + iη ( i)v (p, p) T=T (-1) + V OPE. Iterated Cs P -1, and V OPE P 0 Kaplan, Savage, Wise, PLB (1998)

12 Compare M (TPE) to M (OPE) im TPE = d 3 p (2π) 3 ( i)v (p, p ) i E p 2 M + iη ( i)v (p, p) V (p, p )=C τ a 1 τ a 2 g 2 A 4f 2 π σ 1 qσ 2 q q 2 + m 2 π ; q = p p For OPE have M (TPE) ( ga 2f π ) 4 q 2 q 2 + m 2 π Mk 4π q 2 q 2 + m 2 π Tree ( ga ) 2 q 2 2f π q 2 + m 2 π Comparable for k Λ NN 16πf 2 π Mg 2 A 300 MeV Breakdown of perturbation theory for pions can occur at k=m π Fleming, Mehen, Stewart; Taurus, Soto

13 Iteration: Argument II Iterates of one-pion exchange become comparable with treelevel OPE for momenta of order ΛNN That is the momentum at which OPE becomes strong This is somewhat associated with the tensor part of OPE (which is known to be large) If we wish to describe processes for p ΛNN we need to iterate one-pion exchange (although maybe not in 1 S0) Sum up V + VG0V + VG0VG0V +... Beane et al. (2002); Pavon Valderrama, Ruiz Arriola (2003)

14 χet: a theory for NN χpt, low scales: m π, p; high scales: m ρ, M, Δ Integrate out pion production to get theory of potentials Also account for appearance of new light scales: γ, Λ NN χet, low scales: γ, m π, p, Λ NN ; high scales: m ρ, M, Δ, (Mm π ) 1/2 Neither Perturbative Nor a Field Theory EFT(π), low scales: γ, p; high scales: (Mm π ) 1/2, Δ, m ρ, M, m π, Λ NN

15 Why & how we iterate In the NN system χpt becomes χet: the use of potentials becomes valid Due to Presence of Two Heavy Particles Large scattering lengths + OPE is strong for p ΛNN iterate Compute scattering amplitude by identifying LO part of twoparticle-irreducible kernel as V and iterating that T=V + VG0V + VG0VG0V +...=V + VG0T V=OPEP + C at leading order in χet V=C at leading order in EFT(π)

16 The Lippmann-Schwinger eqn k T k = 4π M T=V + VG0T T k = V ψ k (2L + 1)P L (ˆk ˆk ) L=0 ( E ˆp2 M ) ψ k = V ψ k 1 ψ k = k + E ˆp 2 /M V ψ k 1 T k = V k + V E ˆp 2 /M T k 1 k cot δ L ik (-it)=(-iv) + (-iv)ig0 (-iv) +... ; M=-T

17 Partial-wave expanded LSE Insert partial-wave representation of V and T V (p, p) = (2L + 1)P L (ˆp ˆp)V L (p,p) Use Obtain M L=0 Y LM (ˆp)Y LM (ˆp )= T L (p,p; k) =V L (p,p)+ M 2π 2 2L +1 4π P L(ˆp ˆp ) dp V L (p,p ) Exercise p 2 k 2 p 2 + iη T L Compute VL for L=0 ( 1 S0). Result available upon application... Exercise

18 LSE with one-pion exchange For Coupled Channels: t l l(p,p; E) =v l l(p,p)+ 1 2π 2 For one-pion exchange: Frederico, Timoteo, Tomio, NPA (1999); Yang, Elster, DP, PRC (2007) l M Note: v00 and v20 do not vanish as p and pʼ, want C present to absorb divergences 0 dp p 2 v ll (p,p ) t l l(p, p, E) k 2 + iη p 2, ( 1 v00(p t,p) = g2 A m 2 ) π 8fπ 2 2 dx 1 p 2 + p 2 2p px + m 2 π ( v20(p t,p) = g2 A 2 1 4fπ 2 dx p 2 2p px + p 2 ( 3 2 x2 1 2 ) ) 1 p 2 + p 2 2p px + m 2 π ( 1 v22(p t,p) = g2 A 8fπ 2 dx 2pp x (p 2 + p 2 )( 3 2 x2 1 2 ) ) 1 p 2 + p 2 2p px + m 2. π

19 Results for OPEP alone Yang, Elster, Phillips, PRC 77, (2007) Unrenormalized!( 1 S 0 ) [deg] Unrenormalized!( 3 S 1 ) [deg] T lab [MeV] T lab [MeV] Renormalized!( 1 S 0 ) [deg] Renormalized!( 3 S 1 ) [deg] "=500 MeV "=1 GeV "=5 GeV "=10 GeV "=50 GeV T lab [MeV] T lab [MeV]

20 Results: S waves at leading order Yang, Elster, Phillips, PRC 77, (2007)!( 1 S 0 ) [deg] Subtraction method: LO CD-Bonn!( 3 S 1 ) [deg] Need two input parameters Converges to definite result as Λ " 1 [deg] !( 3 D 1 ) [deg] Agreement with earlier results Beane, Bedaque, Savage, van Kolck; Pavon Valderrama, Ruiz Arriola T lab [MeV] T lab (MeV) 2π exchange needed, especially in 1 S0

21 To O(P 4 ) (?) in low partial waves L NN with two derivatives appears in tree diagrams at O(P 2 ) Renormalizes irreducible one-loop graphs at O(P 2 ): 7 new parameters O(P 3 ): loop with vertices from second-order L πn,c i s appear, no (logarithmic) divergences, no new NN parameters Three-pion exchange and additional NN parameters at O(P 4 ) Do you want to treat these higher-order parts in perturbation theory or put them in V and use the LSE to iterate? c.f. Weinberg ( )

22 Higher orders in V Two-pion exchange, additional contact terms at NLO (Ordonez, Ray, van Kolck; Kaiser, Brockmann, Weise; Epelbaum, Meissner, Gloeckle; Entem, Machleidt) 3NF at NNLO, free parameters fit in A=3: BE, and Consistent 3NFs, 4NFs Courtesy E. Epelbaum

23 Improving L=0 at O(P 3 ) Epelbaum, Meissner, Gloeckle, NPA (2000)

24 State of the Art: O(P 4 ) Epelbaum, Meissner, Gloeckle, Nucl.Phys.A747, (2005) But Limited Range of Cutoffs...

25 Issues for other L Eiras, Soto (2002); Nogga, Timmermans, van Kolck (2005) Attractive singular potentials need renormalization Pavon Valderrama, Ruiz Arriola If Λ>600 MeV need additional contact terms already at LO, to renormalize L>0 Or should we consider Λ>1 GeV? Are the appropriate contact terms present at, e.g. NLO? Epelbaum & Meissner RG analysis indicates which contact terms are promoted Birse Ideas not practically implemented

26 Triplet channel at NNLO Yang, Elster, Phillips, PRC 80, (2009) O(P 3 ) [NNLO] χet NN potential: calculated with DR Limited to 0.6 GeV< Λ< 1 GeV Increasingly singular potentials as chiral order increases Momentum-dependent short-distance part has limited effect as Λ

27 Successes in A=2-4 ev MeV MeV NLO NNLO Exp. 3 H He Courtesy E. Epelbaum N 3 LO potential, χ 2 /dof good c.f. AV18. Entem, Machleidt (2003) χpt TPE in pp PSA gives ci s, m π =128(9) MeV Rentmeester et al. (1999) Reproduce A=3 and 4 binding energies Epelbaum, Nogga, et al.(2002) nd scattering works well

28 Frontiers Is there a perturbation theory for corrections beyond LO. If so, what is/are the small parameter(s)? If we use the LSE with a cutoff then how do we estimate uncertainties associated with higher-order effects? Convergence pattern: is it better with an explicit Δ(1232)? Pandharipande, DP, van Kolck (2005); Krebs, Epelbaum, Meissner (2008-9) What is the sub-leading three-body force? Where does it show up in experiments? How reliably can πn LECs be extracted? Where? What is the m π dependence of nuclear forces? Epelbaum, Bernard, Meissner (2007-???) Rentmeester et al. (1999); Entem, Machleidt (2001) Beane, Savage (2002); Epelbaum, Meissner, Gloeckle (2003)

29 Conclusion χet is a non-perturbative theory because we are interested in describing few-nucleon physics at momenta of order ΛNN, which is larger than γ EFT(π): only interested in p γ, but still non-perturbative Use of potentials justified at LO: therefore use Lippmann- Schwinger equation to solve for T (LO) Applying naive EFT power counting to potential in other partial waves does not work for Λ>1 GeV Either use perturbation theory to treat corrections beyond LO or be restricted to low cutoffs, where iteration is (somewhat) justified