Fate of the neutron-deuteron virtual state as an Efimov level

Transcription

1 Fate of the neutron-deuteron virtual state as an Efimov level Gautam Rupak Collaborators: R. Higa (USP), A. Vaghani (MSU), U. van Kolck (IPN-Orsay/UA) Jefferson Laboratory Theory Seminar, Mar 19,

2 Outline Background Efimov levels, experiments, 3-nucleon systems Effective Field Theory Pionless EFT as the fundamental theory Halo EFT of deuteron Results Conclusions 2

3 Efimov levels Two body interaction At gv (r) g = g 0, scattering length a Three-body bound states # 1 a ln r between r A+A+A g<g. -~1 ~ g>g, A+D 1 ma 2 and 1 mr 2 Efimov, PLB33, 563 (1970) geometrical scaling 3

4 Why does it happen? Efimov levels a 1/ V (R)/m 15 1 m 2 R 2 r mr 4

5 Cold Atom Experiments atom loss in Cs-133 He-4 trimers Kunitski et al. Science 348, 551 (2015) Kraemer et al. Nature 440, 315 (2006) Naidon and Endo review paper Rep. Prog. Phys. 80,56001 (2017) 5

6 Nuclear Systems? Triton binding energy ~ 8.5 MeV, deeper state ~ 4.4 GeV. Forget that! Coulomb force introduces new scale. Only very light systems However, neutron-deuteron scattering does have a virtual state Girard and Fuda (1979), Adhikari and Torreao (1983) 6

7 Neutron-Deuteron Virtual State pcotδ (MeV) van Oers & Seagrave (1967): p cot = 1/a + rp2 /2+ p 2 + p Data Kievsky et al. ERE van Oers & Seagrave p (MeV) Pole at p 2 = p MeV 2 Shallow virtual state B~ 0.5 MeV Data: Ref. [1] and [2] in Phys. Lett. 562 (1967) 7

8 Virtual State as Efimov Level? Accumulation of 3-body Efimov levels near unitarity:! a!1,r! 0 1. Achieve unitarity theoretically (not feasible experimentally) Want a model-independent method Universally applicable 2. Model-independent description of shallow virtual state Derive the modified ERE below deuteron breakup For first task: use pionless EFT that produces triton and virtual state as the fundamental theory to generate data For second task: formulate a low energy theory with fundamental deuteron fields (a halo EFT) 8

9 EFT: the long and short of it Identify degrees of freedom! L = c 0 O (0) +c 1 O (1) +c 2 O (2) + expansion in! Hide UV ignorance- short distance IR explicit- long distance Determine c n from data (elastic, inelastic) EFT : ERE + currents + relativistic corrections power counting Not just Ward-Takahashi identity 9

10 Pionless EFT EFT nucleon-nucleon scattering ia(p) = 2 µ 2 µ i p cot 0 ip = 2 µ i 1/a + ip Example: neutron-proton scattering i 1/a + r 2 p2 + ip apple 1+ rp2 /2 1/a + ip + 1 S 0 : a = 23.8 fm, r =2.73 fm, 3 S 1 : a =+5.42 fm, r =1.75 fm., for a ~ 1/p >> r 10

11 Construct EFT Non-relativistic nucleons Short ranged interaction point-like interaction ia(p) = C 0 Weinberg 90! Bedaque, van Kolck 97! Kaplan, Savage, Wise 98 i 1 C 0 + i µ 2 p ) C 0 2 a µ 1/a p Q 1/r m δ S 1 power-counting C 0 1/Q single fine-tuning (rho-pion physics) k (MeV) Chen, Rupak, Savage (1999) Phillips, Rupak, Savage (1999)

12 Neutron-Deuteron Scattering dimer-formulation (auxiliary field) C 0 $ g2 = Tnd = + Tnd Bedaque, Hammer, van Kolck Nucl. Phys. A 676, 357 (2000) + + Tnd h 0 ( ) 4 2 s sin s 0 ln( / ) tan 1 (s 0 ) sin s 0 ln( / ) + tan 1 (s 0 ), 12 3-nucleon coupling limit cycle, Wilson (1971) Phillips line (1968)

13 Limit Cycle, Phillips line λ 2 h0/ Numerical result Analytical form λ (MeV) Bedaque, Rupak, Grießhammer, Hammer Nucl. Phys. A 714, 589 (2003) 13

14 Neutron-Deuteron in pionless EFT 0 50 pionless EFT Input 100 pcotδ (MeV) Data Kievsky et al LO EFT NLO EFT NNLO EFT p (MeV) LO :, a s, a 3 NLO : LO + r t, r s NNLO : NLO + B 3 Bedaque, Rupak, Grießhammer, Hammer (2003) NLO: S. König, J. Vanesse Next proceed to derive a theory with fundamental deuteron fields below breakup 14

15 Phase Shift 0 g s = 1, g t = g s = 0, g t = a s!1, (p/γt)cotδ 4 6 a s = 24 fm, = 46 MeV (p/γt)cotδ 0 50 = 41 MeV Ramsauer-Townsend effect p/γ t p/γ t 8 g s = 0, g t = g s = 0, g t = 0.2 (p/γt)cotδ a s!1, = 27 MeV (p/γt)cotδ a s!1, =9MeV p/γ t p/γ t

16 Halo EFT and modified ERE L =n [i@ 0 + r2 2m N ]n + d a[i@ 0 + r2 2m d ]d a + + 2X i=1 r 2 µ [ (i) a p 3 nd a +h. c.], 2X i=1 (i) [ i + c i (i@ 0 + r2 2M Introduce two auxiliary fields! )] (i) neutron-deuteron amplitude: it t (p) = 2 µ h 1 1+c 1 p 2 /(2µ) i i 1 ip = 2 µ i p cot i Generate modified ERE: calculate as a 2-body amplitude p 2 0 =2µ c 1, 1 a = , 16 2 r =2µ c 1 2,

17 Halo EFT Power-Counting Breakdown scale Λ set by deuteron breakup momentum Zero of T-matrix at Q Virtual state momentum א Initially: Q MeV MeV MeV 2 We define: t g t 45.7g t MeV, s g s /a s = 8.3g s MeV Approach unitarity as: (g s =0,g t! 0) As we tune the pionless EFT, Q 2 gets smaller, changes sign and approaches Λ 2 and exceeds it. Power-counting has to account for the varying relative size of Q 2 (and other fine tunings) 17

18 Consider 3 intervals Power-Counting Continued 0.7. g t. 1 : small a Q 2 /(@ 2 ), large r 2 /(@Q 2 ) 1 2 and c 2 c 1 small shape parameter 0.3. g t. 0.7 : large a r 1/@ Q & 2 1 and still c 2 c 1 Second auxiliary field decouples: regular ERE 0.1. g t. 0.3 : large a 1/@ and r. 1/ Familiar unitary limit EFT with a single auxiliary field Continue on? 18

19 Phase Shift Again 0 g s = 1, g t = g s = 0, g t = (p/γt)cotδ 4 6 (p/γt)cotδ 0 50 (p/γt)cotδ p/γ t g s = 0, g t = 0.6 (p/γt)cotδ p/γ t g s = 0, g t = p/γ t p/γ t

20 Virtual, Bound and Resonance States Look at analytic structure of the S-matrix S t (p) =e 2i (p) =1+ i2p p cot = (p + i 1)(p + i 2 )(p + i 3 ) (p i 1 )(p i 2 )(p i 3 ) ip =1+iµp T t(p) =1+ i2p 1/a+rp 2 /2 p 2 +p 2 0 ip Interpretation of the three poles in halo EFT: = r 2 p2 0, = p 2 0, = p2 0 a 3 rd root not relevant as 3 1 st root is the shallow virtual state 2 nd root on positive imaginary axis triton? 20 No, a redundant pole.

21 Redundant Pole We look at the residue of the S-matrix near the poles S t (p) X i p R i i i + regular pieces, Normalization of bound and virtual states N 1 2 = ir 1 = 2 1( 2 1 p 2 0) ( 1 2 )( 1 3 ), N 2 2 = ir 2 = 2 2( 2 2 p 2 0) ( 2 1 )( 2 3 ) < 0. 2 > 0 is called a redundant/shadow pole Ma, Phys. Rev. 69, 668 (1946) 21

22 Virtual State to Efimov Level Ni 2 (MeV) π 2 π π i (MeV) 22

23 Efimov Levels Excited state, g s = 0 Triton, g s = 0 Triton, g s = 1 mnb3 (MeV) Real world (45.7 MeV) e /s g t γ t (MeV) 23

24 Conclusions Efimov level emerged from the n-d virtual state near unitarity Model-independent analysis using a halo EFT Claim the mechanism for emergence of Efimov levels is universal Atomic systems lattice QCD at unphysical quark masses radiative capture in n-d, p-d system for Big Bang Nucleosynthesis 24