Nucleon Nucleon Forces and Mesons

Transcription

1 Canada s ational Laboratory for Particle and uclear Physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules ucleon ucleon Forces and Mesons Sonia Bacca TRIUMF Theory Group Series of Lectures on Key Concepts Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the ational Research Council Canada Propriété d un consortium d universités canadiennes, géré en co-entreprise à partir d une contribution administrée par le Conseil national de recherches Canada

2 Content Properties of forces (from empirical observation) Properties of forces (from theoretical considerations) Construction of a phenomenological potential Chiral Effective Field Theory Sonia Bacca 2

3 Starting statements Protons and neutrons are bound states of quarks and gluons. When we put more of them together they interact due to the strong force. If we are interested in the low energy region, where nucleons do not get internally excited, we can treat the nucleons as inert (no internal structure). If the nucleons are non relativistic then we can introduce the concept of a potential V, Hamiltonian H=T+V with T kinetic energy If we knew V, we could then solve the Schroedinger equation to calculate properties of nuclei (deuteron, light nuclei, heavy nuclei) from first principle The thing is that we do not know V, we can only construct (many) models for nuclear forces Since protons and neutrons interact strongly, V should come from QCD, but this is very hard. For now we take a semi-phenomenological approach and then come back to QCD later Sonia Bacca 3

4 Properties of forces- Empirical What do we know about the nuclear forces from experimental observations? Short range Attractive at long range Repulsive at short range Spin dependent Charge independent Spin-orbit force Tensor force Conserve parity We can learn this looking at properties of nucleonic matter ( phase-shifts, spectra of nuclei etc...) The potential has a dominant strong and electromagnetic and a weak component. The em part is well known and the weak component is extremely small. When we talk about potentials, we refer to the strong interaction part. Sonia Bacca 4

5 BE(Z, )/A Properties of forces- Empirical Short range The range of the nuclear force must be smaller than the size of the atom. All atomic and molecular spectra can be explained by the electromagnetic (em) force alone. The binding energy/nucleon is almost constant If it was long range the energy should scale like the number of pairs A(A-1)~A 2 Thus BE/A ~ A BE/A is almost constant ~ MeV Sonia Bacca 5

6 The scattering from a short range potential k k z scattered spherical wave s / eikr r ~ k 0 d detector x z Incident plane wave elastic scattering ~ k 0 = k The wave function at large distances will be a linear combination of incident (plane wave) and scattered wave. After decomposing the plan wave into incoming and outcoming spherical waves, the final state wave function is f = 1X p 4 (2` + 1)i`Y`0 ( ) e i!t `=0 2ikr h `e 2i `e i (kr y ` 2 ) e i (kr ` 2 ) i } The outgoing wave can be changed in both amplitude and phase ` for each partial wave ` Incoming wave unchanged by causality phase shift 6

7 Phase Shifts What does the sign of a phase shift mean? A positive phase shift means the wave function is pulled towards the origin, i.e. an attractive potential. zero potential r sin kr attractive potential of radius R r sin(kr + ) wave function pulled towards the origin Sonia Bacca L11 PHYS

8 Phase Shifts A negative phase shift means the wave function is pushed away from the origin, i.e. a repulsive potential. zero potential r sin kr repulsive potential of radius R r sin(kr ) wave function pushed away from the origin Sonia Bacca L11 PHYS

9 ucleon-ucleon Phase Shifts from Introductory uclear Physics Wong Spectroscopic otation 3 (S=1) for triplet 1 (S=0) for singlet 2S+1 LJ ~1 2 + ~ 1 2 = 0 or 1 L=0 S-wave L=1 P-wave L=2 D-wave L=3 F-wave total angular momentum Sonia Bacca 9

10 Properties of forces- Empirical Attractive at long range uclear forces must be predominantly attractive, as nuclei are bound objects despite the Coulomb force The S-phase shifts at low energy, where they dominate, are both positive, i.e. the interaction is attractive Repulsive at short range Evidence of it from scattering: S-wave phase shift become negative at large momentum. This has to be like that to prevent nuclear matter from collapsing. uclear matter is highly incompressible as a liquid is. This incompressibility is, for example, what causes the in-falling material in a core-collapse supernovae to result in a strong outward-going shock wave that blows the star apart. Sonia Bacca 10

11 Properties of forces- Empirical Spin dependence Spin singlet and spin triplet phase shifts are a little different with the spin triplet being more attractive than the singlet (they both become negative at large momentum) Sonia Bacca 11

12 Properties of forces- Empirical Charge independence uclear forces do not distinguish between protons and neutrons (neglecting the Coulomb) as long as the two nucleons involved are in the same spin orientation Experimental evidence: consider the energy levels of 7 Li (3 protons, 4 neutrons) and 7 Be (4 protons, 3 neutrons). After subtracting the effect of the Coulomb, the energy levels are almost identical. It does not matter if you switch protons with neutrons! Sonia Bacca 12

13 Properties of forces- Empirical Charge independence If nuclear forces do not distinguish between proton and neutrons, so we could regard them as two different manifestations of the same particle. Example: electrons are spin 1/2 particles (S=1/2) with two possible spin states sz=1/2 spin up sz=-1/2 spin down By analogy: introduce isospin for nucleons nucleons are isospin 1/2 particles (T=1/2) with two possible isospin states tz=1/2 isospin up proton tz=-1/2 isospin down neutron B: This is the notation of nuclear physics and sometimes a different convention is used tz=1/2 isospin up neutron tz=-1/2 isospin down proton In particle physics isospin is denoted with I,Iz Isospin is a good quantum number for the strong Hamiltonian [H, ~ T ]=0 Strongly interacting particles can be given a pure isospin quantum number. Sonia Bacca 13

14 Properties of forces- Empirical Spin-orbit term This shows that potentials for 1 S0 (l=0) are different than for 1 D 2 (l=2). There must be an ` dependent term in the potentials, like a spin-orbit term analogous to the spin-orbit coupling in atomic physics. V ` s Sonia Bacca 14

15 Properties of forces- Empirical Spin-orbit term The spin-orbit term manifests itself as a left-right asymmetry scattered particles depending on their spin orientation. unpolarized beam (p,n) spin up spin down ~r ~` = ~r ~p ~p into page target p Spin up projectile will have spin antiparallel to the angular momentum So they will feel an attractive potential and scatter to the right Spin down projectile will have spin parallel to the angular momentum So they will feel an repulsive potential and scatter to the left V ` s V ` s negative positive Sonia Bacca 15

16 Properties of forces- Empirical Spin-orbit term So the scattering pattern will look like unpolarized beam (p,n) spin down spin up spin down target p spin up Particles with spin up scatter to the right and particles with spin down scatter to the left Sonia Bacca 16

17 Properties of forces- Empirical Spin-orbit term ow we change the position of the potential center with respect to the beam unpolarized beam (p,n) spin up spin down ~r target p ~p ~` = ~r ~p out of page Spin up projectile will have spin parallel to the angular momentum V ` s positive So they will feel an repulsive potential and scatter to the right Spin down projectile will have spin antiparallel to the angular momentum V ` s negative So they will feel an attractive potential and scatter to the left Sonia Bacca 17

18 Properties of forces- Empirical Spin-orbit term So the scattering pattern will look like (same as before) unpolarized beam (p,n) target p spin down spin up spin down spin up Particles with spin up scatter to the right and particles with spin down scatter to the left If you start from an unpolarized beam, after the scattering we will have polarized particles! This technique is used to create polarized beams. Sonia Bacca 18

19 Properties of forces- Empirical Spin-orbit term One can define the polarization of the scattered particles as detector P ( ) = (spin up) (spin down) (spin up) + (spin down) beam Count the number of spin up and down target From R. Machleidt Advances in uclear Physics Vol.19 The spin orbit term manifests as a non zero value of P Sonia Bacca 19

20 Properties of forces- Empirical Tensor force A tensor force is a non central force, because it does not just depend on the relative distance between two particles, but also on the relative orientation and on its orientation with respect to the spins of the nucleons V = V (~r ) Tensor force: depends on the angle between relative distance vector and the spins of the nucleons, in analogy to the force between two magnetic dipoles ~r repulsive ~r analogy attractive S S ~r ~r repulsive attractive S S S 12 [3(~ 1 ˆr)(~ 2 ˆr) ~ 1 ~ 2 ] U [3( ~µ 1 ˆr)( ~µ 2 ˆr) ~µ 1 ~µ 2 ] Sonia Bacca 20

21 Properties of forces- Empirical Tensor force Tensor force leads to mixing of orbital angular momentum states Evidence: the deuteron ground state is mostly ` =0(S-state) but there is a small admixture of ` =2 (D-state) which is necessary to describe the fact that the deuteron has a non zero quadrupole moment. hl ` =0 Y 2,0 l ` =0i =0 Sonia Bacca 21

22 Properties of forces- Theory Theoretical considerations Translational Invariance The Hamiltonian must be invariant with respect to translations in space of the two-body system as a whole. It can not depend only on the absolute position of one nucleon with respect to an arbitrary frame The potential can depend only on the relative distance ~r = ~r 2 ~r 1 Galilei Invariance The potential can depend also on the momenta ~p 1,~p 2 Since ~P = ~p 1 + ~p 2 ~p =(~p 2 ~p 1 )/2 is the momentum of the center of mass, then the potential can only depend on the relative momentum Rotational invariance The Hamiltonian is a scalar quantity with respect to rotations in spin and coordinate space So the potential cannot contain terms like ~r but can only have terms like r 2 = ~r ~r, f (r 2 ) ~ 1 ~ 2 Sonia Bacca 22

23 Properties of forces- Theory Theoretical considerations Parity Invariant Strong forces do not change parity V (~r, ~p, ~ 1,~ 2,...)=V ( ~r, ~p, ~ 1,~ 2,...) Hermiticity The Hamiltonian has to be an Hermitian operator, so that the energies are real H = H Symmetric with respect to particle exchange H(1, 2) = H(2, 1) Charge independence [H, ~ T ]=0 The Hamiltonian has to be a scalar in isospin space 1,~ 1 ~ 2 Sonia Bacca 23

24 Form of forces Adding these theoretical considerations (with the simplification of allowing only linear dependence in momentum) to the guidance obtained from experiment, one can construct the general form for nuclear forces to be: In spin-coordinate space In isospin space V = V 0 (r)+ + V s (r) ~ 1 ~ V`s (r) ~` ~S + V T (r) S 12 purely central spin-dependent spin-orbit tensor 8 terms [1, ~ 1 ~ 2 ] Actually it is found out that to better describe scattering phase shifts, one can add ~`2 +V`2 +V`2s ~`2 ~ 1 ~ 2 +V`2s 2 (~` ~S ~ ) 2 Sonia Bacca 6 terms by relaxing the linear momentum dependence [1, ~ 1 ~ 2 ] 8+6=14 terms AV terms that break charge symmetry small mass diff in p,n [H, T ~ ] 6= 0 PRC 29 (1984) 1207 AV18 PRC 51 (1995) 38 24

25 One-pion-exchange potential Yukawa postulated the existence of a massive particle m~130 MeV to be responsible of the interaction between nucleons. This idea came from the analogy with the one photon exchange in electromagnetism. Originally, Yukawa assumed that the particle exchanged was a scalar: J =0 + analogy r 1 m Hideki Yukawa obel prize in 1949 e e electromagnetic force: infinite range exchange of massless particle force: finite range exchange of massive particle Sonia Bacca 25

26 One-pion-exchange potential Can we estimate the mass of the exchanged particle, knowing that the range of the nuclear force is ~1.5 fm? We know that the exchanged particle is virtual and can violate the energy conservation within the uncertainty principle E t ~ t ~ E The distance travelled by the particle will be r c t c~ E Taking as energy its rest energy r c t c~ mc 2 E = mc 2 mc 2 ~c r = MeV fm 1.5 fm 130 MeVc 2 The pion was discovered as a free particle in 1947, after Yukawa had postulated his existence. Sonia Bacca 26

27 One-pion-exchange potential Originally, Yukawa assumed that the particle exchanged was a scalar: J =0 + Today, we know that the pion is a pseudo scalar particle, with and isovector nature T=1. J =0 ; T =1 The potential between two nucleons that are exchanging a pion as an pseudo scalar particle is named one-pion-exchange potential (OPEP): The OPEP is responsible for the long-range part of the interaction } OPEP Sonia Bacca 27

28 One-pion-exchange potential The one pion exchange potential in coordinate space looks like V OPEP = 3( 1 r)( 2 r) r V T (r)+ 1 2 ~ 1 ~ 2 V s (r) IMPORTAT: The OPEP gives rise to the tensor component in nuclear forces, but it also originates a part of the central force, with spin-isospin dependence In momentum space, it looks like below V OPEP = g2 A 1 q 2 q 4f m 2 + q 2 ~ 1 ~ 2 Pion propagator 1 m 2 + q 2 Pion ucleon vertex 1 q ~ 1 Sonia Bacca 28

29 One boson exchange potential The OPEP is at the basis of most potentials! For example the AV18 potential is PRC 51 (1995) 38 + phenomenological ansatz for the short range part Alternatively you can extend the OPEP to the one boson exchange potential (OBEP) +!,, Exchange of heavier mesons for the short range part. CD-Bonn These are named realistic potentials: fit scattering data with 2 1 Even though this is not pure phenomenology and there is a lot of theory in these potentials, one realizes that the connection to the underlying QCD is weak... Sonia Bacca 29

30 Chiral effective field theory ew vision H( )=T + V ( )+V 3 ( )+V 4 ( ) Separation of scales Limited resolution at low energy c (r 1 r 2 ) Construct an effective Lagrangian which respects the symmetry of the fundamental theory: L QCD (m q! 0) L = k c k Q b b k chiral symmetry in terms of p/n, pions (effective d.o.f.) and organize it in powers of Q Details of short distance physics not resolved, but captured in low energy coupling constants. Should come from QCD, but now are fit to scattering Systematic and can provide error estimates Sonia Bacca V >V 3 >V 4 Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Meissner, ogga, Machleidt, 30

31 Chiral effective field theory ew vision Sector - coupling constants fitted to scattering phase-shifts Lab. Energy [MeV] Important: fit scattering data with Sector - 3LO V >V 3 >V 4 fit coupling constants to A=3 data Theory Aim: Predict nuclear observables with these potentials Sonia Bacca 31

32 Chiral effective field theory ew vision uclear low energy spectra avratil et al. (2007) avratil 2007 Sonia Bacca 32