N-d 3-body scattering and NN interactions

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1 N- 3-boy scattering an NN interactions Collaboration o: H. Witała, J. Golak, R. Skibiński, K. Topolnicki, Jagiellonian University, Kraków W. Gloeckle, E. Epelbaum, H. Krebs, Bochum Ul-G. Meibner, A.Nogga, Forschungszentrum Juelich H. Kamaa, Kyushu Institute o Technology W.N. Polyzou, the University o Iowa LENPIC collaboration

2 Outline: 1. Some history 2. Aim o investigations 3. Formalism an calculational scheme 4. Results or nucleon-euteron reactions: - stanar orces - relativistic eects - chiral orces 5. Summary an conclusions

3 State o investigations in eighties: Moels o nucleon-nucleon potentials (Paris, Bonn, their separable approximations) Goo escritpion o 2-nucleon boun state (euteron) Goo escription o NN scattering ata Boun state o three nucleons ( 3 H, 3 He) calculate with NN interactions only is unerboun IMPORTANT IMFORMATION: three-nucleon Hamiltonian H must contain also a three-nucleon potential a term which cannot be reuce to interactions o pairs o nucleons : H H V V V V V V V V (1) (2) (3) splitting o 3N potential to three parts with proper symmetry

4 First moel o three-nucleon orce (3NF) came rom Japan: J.Fujita an H.Miyazawa, Prog. Theor. Phys. 17, 360 (1957)). Other important 3N potential moels: - Moel Tucson-Melbourne base on pion-nucleon scattering amplitue (S.A.Coon, W.Glöckle, PRC23, 1790 (1981)) - Urbana IX 3NF moel is base on 2-exchange with -resonance exctitation in intermeiate state; there is also spin- an isospin-inepenent shortrange term (B.S.Puliner et al., PRC56, 1720 (1997))

5 What equations must be solve? For 3N boun state one solves Faeev equation or Faeev component: G tg V (1) P G t P Faeev component; ull wave unction is given by: Free 3N propagator P P P P P 2N transition operator t generate rom L-S equation V 23 V 23 G 2N 0 t Part o V 4 symmetrical with respect to exchange o nucleons 2 an 3

6 Theoretical escription o 3N continuum (elastic nucleoneuteron scattering an the euteron breakup reaction) requires solution o the ollowing Faeev-type equation or T > state: T tp 1 tg V (1) 1 P 0 4 tpg T 0 q 0 1 tg V (1) 1 P G T, where is compose o the euteron internal wave unction an the state o the relative nucleon-euteron motion. The transition amplitue or elastic scaterring is given by: ' U ' PG 1 0 ' V (1) 4 1 P ' PT ' V (1) 4 1 PG T o

7 Transition amplitue or the euteron breakup is given by: 0 U PT Faeev equations or the boun state (negative 3N energy): G tg V (1) P G t P an or scattering states (positive 3N energy): T tp 1 tg V (1) 1 P 0 4 tpg T 0 1 tg V (1) 1 PG T contain the same operators. However, ue to singularities o operator G 0 an pole o operator t in calculations or positive energies, treatment o the 3N continuum is much more iicult than o the 3N boun state!

8 Basis states in momentum representation: We nee two relative momenta (Jacobi momenta) to escribe the coniguration o three-nucleons: p 1 2 p Simple choice: p q Pm q m m p 2 p3, q p1 p2 p3, P p1 p2 p3 The corresponing momentum space partial wave basis: Spin projection Izospin projection, with pq pq l s j I ( ji) JM t T M T

9 Faeev equations in case without 3NF T tp tpg 0 T Ater projecting on partial wave states: pq T pq tp ' " ' '' pq t p' q' ' p' q' ' P p" q" " p" q" " G T 0 It is a set o couple integral equations or matrix elements pq T Finite numer o equations ollows rom short-range o nuclear orces.. Metho o solution: Generation o von Neumann series an sumation by Paé metho! H. Witala et al., Few-Boy Syst. 3, 123 (1988). W. Glöckle et al., Phys. Rept. 274, 107 (1996)

10 Construction o coes or 3N continuum calculations opene the way to stuy properties o ierent moels or nuclear orces. Perorme stuies prove, that it is possible to solve numerically precisely 3N Faeev equations without any aitonal approximations or ANY 2N an 3N potentials. The main aim o investigations was an is to stuy properties o 2N an 3N potentials. It means one is looking or answers to the ollowing questions : What role plays 3N potential in escription o 3N systems? Does moern moels o nuclear orces properly escribe experimental ata? Where to look or large eects o 3N potential?

11 Nine ties o XX an begin o XXI: new 2N potentials (AV18 - R.B.Wiringa et al. PRC51,38(1995), CD Bonn - R.Machleit. PRC63,024001(2001), Nijm1, Nijm2 - V.G.J.Stoks et al., PRC49,2950(1994) ) escribe experimental ata in 2N system with 2 /N ~ 1 new 3N potentials (correcte TM - S.A.Coon,H.K.Han, FBS 30,131(2001), Urbana IX - B.S.Puliner et al.,prc56,1720(1997) ) erivation o new generation o nuclear orces (2N i 3N) in the ramework o chiral eective iel theory up to N4LO orer o chiral expansion: E.Epelbaum, Prog. Part. Nucl. Phys. 57,654(2006), E.Epelbaum, H.Krebs an U.-G.Meibner, PRL 115,122301(2015), R.Machleit, D.R.Entem, Phys. Rep. 503,1(2011), D.R.Entem, R.Machleit, Y.Nosyk, nucl-th arxiv:

14 A y puzzle Sensitivity to 3 P j NN orce components:

15 A y puzzle ierent eects o TM an UIX 3NF s inication on inconsistency between NN an 3N orce?

16 3N boun state: missing bining B ~ MeV bining energy B ~ 8 MeV 3NF eects ~ 10 % but <V> ~ -50 MeV -> V(3NF)/V ~ 1 % B = <T> + <V> 3N continuum: S ~ V expecte eects ~ 1% - Changing incoming energy increases V(3NF)/V 3NF s expecte to play increasing role at higher energies

17 For elastic N+ scattering 3N orce eects grow with energy! H.Witała et al., PRL 81, 1183 (1998) only 3N potential 2N+3N potential only 2N potential

18 Rather small 3NF eects at low energies Large 3NF eects start to appear at energies above ~ 60 MeV Region o large energies seems thus to be the proper one or stuy o 3NF eects

19 Elastic scattering (p,p) Higher energy iscrepancies NN only (AV18, CD Bonn, Nijm1, Nijm2) NN+3NF TM99 Total n cross section: (W.P.Abalterer et al. PRL 81(1998)57) up to ~ 50 MeV goo agreement with preictions base on 2N orces aing 3NF provies explanation o the isagreement up to ~ 150 MeV at even larger energies a clear isagreement which increases with energy ata 70: K.Sekiguchi et al., PR C65, (2002) ata 250: x n Y.Maea et al., PR C76, (2007) o p K.Hatanaka et al., PR C 66, (2002)

20 E=135 MeV (p,p) - NN ban (AV18,CDB,Nijm1,2) - NN+TM99 ban - AV18+UIX K.Sekiguchi et al., PRC C70,014001(2004)

21 p(,pp)n E =270 MeV K.Sekiguchi et al., PR C79,054008(2009) A y A yy A xx K yy y 2 =31 o, 12 =180 o

22 what is responsible or large ierences between theory an ata in 250 MeV cross section an in the total n cross section even ater inclusion o 2-exchange 3NF? it is evient, that in the applie ynamics something is wrong or missing PT (an stanar meson-exchange picture) provies multitue o aitional, short-range components to 3NF (in the mesonexchange picture connecte to exchanges o more or heavier mesons), which shoul be more important with increasing energy however, increasing energy means also a transition to a region, where relativity coul be important relativistic Faeev calculations

23 Relativistic Faeev calculations The ormal structure o the equations remains the same but the ingreients change ( PR C71, (2005), PR C77, (2008), PR C83, (2011) ) Form o the ree Hamiltonian H 0 (an G 0 ) changes: H0 2 m k q m q Interacting 2N subsystem (2-3) has nonzero total momentum -q in the 3N c.m. system, what leas to the booste potential V: V ( q) 2 m k v q 2 m k q V(q=0) reuces to the relativistic potential v eine in the 2N c.m. system. From V(q) the booste t-matrix is obtaine The Lorentz transormation rom 2N to 3N c.m. is perorme along the total momentum o the 2N subsystem, which in general is not parallel to momenta o these nucleons. This leas to Wigner rotation o spin states. When eining 3N partial wave states care must be taken about spin states

24 Faeev approach with both 3NF an relativity inclue elastic scattering: (p,p) ata K.Sekiguchi et al., Phys. Rev. Lett. 95, (2005) only NN orces (CDBonn): nrel soli re rel ashe blue --- NN + 3NF (TM99): nrel otte blue rel ashe-otte brown -.- ata 250: x n Y.Maea et al., PR C76, (2007) eects o relativity seen only in N elastic scattering backwar cross section! relativistic eects are not responsible or large iscrepancies in elastic N scattering cross section interplay o relativity an 3NF s leas to a slight increase o the cross section at angles larger than cm ~ 100 o o p K.Hatanaka et al., PR C 66, (2002)

25 elastic scattering (p,p) p ata: K.Sekiguchi et al., Phys. Rev. C65, (2002) only NN orces (CDBonn): nrel soli re rel ashe blue NN + 3NF (TM99): nrel otte blue rel ashe-otte brown -. - p ata: IUCF, Phys. Rev. C74, (2006) small rel. eect on spin observables

26 Large relativistic eects or breakup cross sections in some regions o phase-space! H.Witała et al.,phys.lett.b634,374(2006) R.Skibiński et al.,eur.phys.j.,a30,369(2006) 26

27 Supporte by ata! 27

28 relativistic eects are not responsible or large iscrepancies in elastic N scattering very probably those iscrepancies come rom neglection o short-range 3NF components, which become active at higher energies such short-range 3NF s are in the meson-exchange picture given by - an - exchanges an in PT comes a lot o short-range contributions in N 3 LO orer o chiral expansion (without ree parameters) Challenge: apply NN an 3NF s erive consistently in the ramework o the chiral perturbation theory

29 LENPIC (Low Energy Nuclear Physics International Collaboration): to unerstan nuclear structure an reactions with chiral orces perturbation in (Q/)

30 Few remarks on chiral orces: In orer to reprouce properly 2N ata up to about 250 MeV N3LO orer o chiral expansion is require About 2 years ago: NN interaction up to N3LO, 3NF interaction up to N4LO (N3LO use in preliminary calculations at low energies) nonlocal momentum space regularization has been applie: 6 6 p / V (p,) V(p,p) (p,) with ( p, ) e V 4 (p,q,) V ( p 0.75 q ) / (p,q,p,q) (p,q,) with ( p, q, ) e what leas to inite cut-o arteacts (problems when applie to higher energy N scattering) New, improve chiral orce, presente by Bochum-Bonn group in 2014: E. Epelbaum, H. Krebs, U.-G. Meißner, Eur. Phys. J. A51 (2015) 3,26 up to N3LO E. Epelbaum, H. Krebs, U.-G. Meißner arxiv: [nucl-th] up to N4LO 2 2 / Local regularization in the coorinate space V lr (r) V lr (r)(r) with ( r) (1 e r R ) n R= m what correspons to = MeV Such regularization preserves more long-range OPE an TPE physics All LECs in the long-range part are taken rom pion-nucleon scattering without ine tuning Very goo escription o the euteron properties, phase shits etc.

31 Nonlocal regularization Local regularization

32 Estimation o theoretical uncertainties (E.Epelbaum et al., arxiv: [nucl-th], arxiv: [nucl.th]): (i) (i) X(p) - observable an X, i 0, 2,3, preiction at orer Q in the chiral expansion (i) Orer-Q correction : The chiral expansion or X takes the orm: X X + X + + X (i) (0) (2) (i) X X X (2) (2) (0) ( i) ( i) ( i 1) X X X i (i) (i) (0) Size o corrections is expecte to be X =O(Q X ) (i) ( i) Quantitative estimates o the theoretical uncertainty X o the preiction X is mae using the expecte an actual sizes o higher-orer contributions: X Q X (0) 2 (0) (2) 3 (0) 1 (2) X max( Q X, Q X ) 3 i X Q X Q X Q X ( i) i1 (0) i1 (2) i2 (3) 3: max(,, ) X Q X, X Q X (2) (0) ( i3) ( i1) Q=max(p/,m / ) with 600,500 an 400 MeV b b b or regulator R= m, R=1.1 m an R=1.2 m, respectively.

33 NN evelope up to N4LO: E.Epelbaum et al. arxiv: [nucl-th] Novel way o quantiying the theoetical uncertainty ue to the truncation o the chiral expansion: E.Epelbaum et al. arxiv: [nucl-th] Theoretical uncertainty grows with energy an ecreases with increasing orer: one thus expects precise preictions starting rom N3LO For many observables the results at N2LO an higher orers ier rom ata well outsie the range o quantiie observables, thus proviing a clear evience or missing three-nucleon orces

34 Exclusive breakup reaction: symmetric space-star coniguration Big challenge: application o ull N3LO chiral orce: NN + 3NF

35 Various topologies contributing to the 3NF up to an incluing N 4 LO N 2 LO: (a) + () + () (E.Epelbaum et al., PR C66, (2002)) N 3 LO: (a) + (b) + (c) + () + (e) + () + rel V.Bernar et al., PR C77, (2008) - long range contributions (a), (b), (c) V.Bernar et al., PR C84, (2011) - short range terms (e) an leaing relativistic corrections N 3 LO contributions o not involve any unknown low energy constants! The ull N 3 LO 3NF epens on two parameters c D an c E coming with () an () terms, respectively. They are ajuste to two chosen 3N observables. N 4 LO (longest range contributions): (a) + (b) + (c) + () + (e) + () (H.Krebs et al., arxiv: )

36 Chiral 3N potential in N 2 LO orer: E.Epelbaum, Prog.Part.Nucl.Phys. 57, 654 (2006) V V V V V (3) (3) (3) ,cont cont 1 g ( q)( q ) ( ) F 2 2 ( )( ) (3) A 2 i i j j ijk F qi M q j M q p ' p i i i b b ijk i j 2 b b 4c1M 2c 3 c4 b Fijk q 2 2 i q j 2 k k qi q j F F F (3) g j q A j V 1, cont D i j i q j 8F q M ijk j (3) 1 cont j k 2 jk V E Two ree parameters : D i E

37 Fixing D an E values: The epenence o E on D or (D,E) pairs proviing proper triton bining energy

38 perorming 2 -itting only statistical errors were taken into account higher energy means importance o higher orers that leas to larger D values when itting at higher energies (3NF eects grow with energy)

39 - similar picture o 3NF eects or stanar an N2LO chiral 3NF s

40 small eects o N2LO 3N orce eects are practically inepenent on D (E) values use will 3N orce explain A y puzzle (N3LO 3NF)? alternative: wrong low energy NN phase-shits (in P-waves)

41 - practically no 3NF eects (at N2LO) or that coniguration

42 - practically no pp Coulomb orce eects or low energy space star coniguration!

43 - only 1 S 0 an 3 S 1 contribute at this energy - is something wrong with 1 S 0 (nn or pp) at low energies? - making 1 S 0 nn stronger (positive scattering length - boun 1 S 0 nn state) coul explain p ata (but not n ata!)

44 Summary: N elastic scattering an euteron breakup reaction reveal large sensitivity to unerlying nuclear orces --> goo tools to test nuclear Hamiltonian it is clear, that base on the present ay NN orces aitional term in nuclear Hamiltonian is neee --> 3NF 3NF moels erive inepenently rom NN potentials, can account in some cases or iscrepancies between theory an ata call or consistency between 2N an 3N orces: support an guiance --> chiral perturbation theory approach small relativistic eects in N elastic scattering (not in speciic exclusive breakup conigurations) Big challenge: application o chiral N 3 LO orces (2- an 3-boy) to 3N continuum (especially at higher energies)

Elastic nucleon-deuteron scattering and breakup with chiral forces

Elastic nucleon-deuteron scattering and breakup with chiral forces Elastic nucleon-euteron scattering an breakup with chiral orces H.Witała, J.Golak, R.Skibiński, K.Topolnicki JAGIELLONIAN UNIVERSITY LENPIC Collaboration Jagiellonian University, Kraków Ruhr-Universität,