Overview of low energy NN interaction and few nucleon systems

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1 1 Overview of low energy NN interaction and few nucleon systems Renato Higa Theory Group, Jefferson Lab Cebaf Center, A3 (ext6363) Lecture II Basics on chiral EFT π EFT

2 Chiral effective field theory Chiral symmetry plays a major role mid 60s - 70s: current algebra Low-lying meson spectra suggest spontaneous chiral symmetry breaking, with pions acting as Goldstone bosons Massless QCD: Theoretical Paradise - exhibits explicit, global, χ-symmetry L (0) QCD = 1 4 Ga µνg µν,a + q i D q (1) L (v,a,s,p) QCD = 1 4 Ga µνg µν,a + q i D q+ qγ µ (v µ + a µ γ 5 )q q(s iγ 5 p)q () v, a, p 0, s M, L (v,a,s,p) QCD L true QCD (3)

3 3 promote a global symmetry, with external sources, to a local one non-trivial relations among Green functions e iz[v,a,s,p] = = Dq D q DG e i L (v,a,s,p) QCD (4) DU e i L eff (v,a,s,p,u) (5) H Leutwyler, Ann Phys 35, 165 (1994) L eff : includes all terms allowed by the relevant symmetries organization: derivative (momentum) expansion Q/Λ χ where Λ χ 4πf π πλ QCD power counting

4 4 Callan etal, Coleman, Wess, and Zumino (69): Effective Lagrangians χ - symmetry is spontaneously broken: SU(N f ) L SU(N f ) R }{{} G U G g R Ug L SU(N f ) V l µ = v µ a µ G gl l µ g L + ig L µ g L (6) r µ = v µ + a µ G gr l µ g R + ig R µ g R χ = B(s + ip) G g R χg L D µ (U, χ) = µ (U, χ) ir µ (U, χ) + i(u, χ)l µ (Same transformations of v, a, s, p which keeps L (v,a,s,p) QCD invariant) F L,R µν = µ (l ν,rν ) ν (l µ,rµ ) i[(l µ,rµ ), (l ν,rν )] (7)

5 5 Invariants: A where A g L Ag L L () π = F 4 (D µu) D µ U + U χ + χ U (8) { F } with U = 1 F π + iτ π and v, a, p 0, s M one gets L () π = 1 µπ µ π m π π + 1 F ( µπ π)( µ π π) m π 8F (π ) + O(π 3 ) (9) (non-linear σ model) Observables are independent of the parametrization of U!

6 6 Electroweak interactions [SU(3)] v µ eqa µ, a µ Q = 1 3 e sin θw W T +, T + = diag(, 1, 1), 0 V ud V us (10) (D µ U) D µ U (iu r µ il µ U ) µ U + µ U ( ir µ U + iul µ ) (11)

7 7 L (4) π = L 1 { Dµ U(D µ U) } + L D µ U(D ν U) D µ U(D ν U) +L 3 D µ U(D µ U) D ν U(D ν U) + L 4 D µ U(D µ U) χu + Uχ +L 5 D µ U(D µ U) (χu + Uχ ) + L 6 [ χu + Uχ ] +L 7 [ χu Uχ ] + L8 Uχ Uχ + χu χu il 9 F R µνd µ U(D ν U) + F L µν(d µ U) D ν U + L 10 UF L µνu F µν R +H 1 F R µνf µν R + F L µνf µν L + H χχ (1)

8 8 Lagrangian expansion starts at O(q ) L (n) π vertices of O(q n ) pion propagator: O(q ) each loop: O(q 4 ) ν chiral order of a particular Feynman diagram N L number of loops, N i number of pion propagators, d number of derivatives on a particular vertex, N d number of vertices with d derivatives (13)

9 9 Lagrangian expansion starts at O(q ) L (n) π vertices of O(q n ) pion propagator: O(q ) each loop: O(q 4 ) ν chiral order of a particular Feynman diagram (1) (1) (1) (3) (1) () d dn d = = 1 (3)

10 10 Lagrangian expansion starts at O(q ) L (n) π vertices of O(q n ) pion propagator: O(q ) each loop: O(q 4 ) ν chiral order of a particular Feynman diagram N L number of loops, N i number of pion propagators, d number of derivatives on a particular vertex, N d number of vertices with d derivatives ν = d dn d + 4N L N i = ] d dn d + 4N L [ d N d + (N L 1) = + N L + d N d(d ) (13)

11 11 EM form factors F φ (t) = r φ t + (14) r π ± = 1 Lr 9(µ) F 1 3π F [ ln = 0439 ± 0008 fm (exp) ( ) m π µ + ln ( m K µ ) ] + 3 (15) L r 9 dependence on µ cancels explicit dependence in the logs fixing L r 9 at µ = m ρ determines the LO parameters of L res in χpt saturates almost all the values of the other LECs (resonance saturation) r K 0 = 1 ( ) 16π F ln mk m π = 004 ± 003 fm exp = 0054 ± 006 fm (16)

12 1 Baryon χpt N G hn, h = h(g L, g R, U) u G g R uh = hug L, u = U (17) Invariants: N A N where A G h A h Building blocks: u µ = iu D µ Uu, χ ± = u χu ± uχ u, f ± µν = uf L µνu ± u F R µνu (18)

13 13 L (1) πn = ψ(i D m)ψ + g A ψ uγ 5 ψ (19) [ ] τ π U = u = exp, u µ = i{u, µ u}, D µ = µ + Γ µ, f π Γ µ = 1 [u, µ ] i u (v µ + a µ )u i u(v µ a µ )u exercise: expanding U in the pion fields obtain the result L (1) πn = ψ(i m)ψ 1 4f π ψγ µ τ (π µ π)ψ + g A f π ψγ5 τ µ πψ + O(π 3 ) (0) LO = + +

14 14 Loops in Baryon χpt nucleon mass doesn t vanish in the χ-limit introduces another (large) scale on the theory the power counting is not transparent considering the baryon infinitely heavy (HB-χPT) restores the power counting alternative: infrared regularization - keeps the covariant aspect of the theory power counting: ν = 1 + N L + d N π d (d )+ d N N d (d 1)

15 15 nucleons or more nucleons: χpt power counting is different from expected! p1 p1 + q = d 4 k (π) 4 1 k m π 1 (k q) m π 1 (p 1 +k) m N 1 (p k) m N k p p k q q = d 3 k dk 0 (π) 3 π 1 k 0 ω 1 1 (p (1) 0 +k 0) σ 1 (k 0 q 0 ) ω 1 (p () 0 k 0) σ (1) ω 1 = k + m π, ω = ( k q) + m π, σ 1 = ( p 1 + k) + m N, σ = ( p k) + m N poles: ±ω 1, ±ω, p (1) 0 + σ 1, p (1) 0 σ 1, p () 0 + σ, p () 0 σ

16 16 nucleons k 0 poles: ±ω 1, ±ω, p (1) 0 + σ 1, p (1) 0 σ 1, p () 0 + σ, p () 0 σ

17 17 Weinberg (90, 91): (and more) nucleons non-perturbative problem solution: χpt power counting rules applied to the potential, and solve the scattering with a LS equation T = V + V G T T = V + V T Ordoñez, Ray, van Kolck (9-96) - potential up to O(q ) 0-present: NN potential up to O(q 4 ) DR Entem, R Machleidt, PRC 66, RH, MR Robilotta, PRC 68, 04004, + CA Rocha, PRC 69, E Epelbaum, W Glöckle, U-G Meißner, NPA 747, 36

18 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) p p p p (k) (l) (m) (n) p p (o)

19 F F 10 Phase Shift (deg) 4 Phase Shift (deg) 3 1 Phase Shift (deg) ε E LAB (MeV) E LAB (MeV) E LAB (MeV) 0 3 F F 4 4 Phase Shift (deg) Phase Shift (deg) E LAB (MeV) E LAB (MeV)

20 0 5 1 G G 3 0 ε 4 Phase Shift (deg) Phase Shift (deg) 4 Phase Shift (deg) E LAB (MeV) E LAB (MeV) E LAB (MeV) 3 G 4 3 G Phase Shift (deg) 6 4 Phase Shift (deg) E LAB (MeV) E LAB (MeV)

21 1 0 1 H H 4 3 H 5 Phase Shift (deg) 05 1 Phase Shift (deg) 04 0 Phase Shift (deg) E LAB (MeV) E LAB (MeV) E LAB (MeV) 3 H H 6 Phase Shift (deg) 0 05 Phase Shift (deg) E LAB (MeV) E LAB (MeV)

22 V cont = V (0) cont + V () cont + V (4) cont, V (0) cont = C S + C T σ (1) σ (), V () cont = C 1 q +C z +(C 3 q +C 4 z )(σ (1) σ () )+ic 5 1 (σ(1) + σ () ) (q z) V (4) +C 6 (q σ (1) )(q σ () )+C 7 (z σ (1) )(z σ () ), cont = D 1 q 4 +D z 4 +D 3 q z +D 4 (q z) ( + D 5 q 4 +D 6 z 4 +D 7 q z +D 8 (q z) ) (σ (1) σ () ) +i (D 9 q +D 10 z ) σ (1) +σ () ( (q z)+ D 11 q +D 1 z ) (σ (1) q)(σ () q) + (D 13 q +D 14 z ) ( ) (σ (1) z)(σ () z)+d 15 σ (q z) σ () (q z) with q = p p and z = (p + p )/ ()

23 3 π EFT very low energies : m π Λ T = 4π 1 m N k cot δ ik k cot δ = 1 a + Λ n=0 r n ( k Λ (3) ) n+1 (ERE) (4) ( ) L = N i 0 + N m N { +(µ/) 4 D ( C 0 N N ) C + [(NN) N ( ] } ) N +hc + (5) 8 ( ) ( ) + ( ), (6)

24 4 tree graphs: C 0 = i(µ/) 4 D ( C 0 ) = i(µ/) 4 D C 0, (7) C = i(µ/) 4 D C { [ ( ik) ( ik) (ik) + (ik) ] 8 + [( ik ) ( ik ) (ik ) + (ik ) ]} = i(µ/) 4 D C ( ) k + k = i(µ/) 4 D C k (8) it tree = i(µ/) 4 D C 0 i(µ/) 4 D C k + = i(µ/) 4 D C n k n (9) n=0

25 5 loops: (E/ ; k ) ( E/+q 0 ; k+ q) C i C j q (E/ ; k) (E/ q 0; k q) = = [ i(µ/) 4 D C i ][ i(µ/) 4 D C j ] d D 1 q dq 0 (π) D 1 π (k + q)i+j is F (E/ + q 0 ; k + q) is F (E/ q 0 ; k q) [ ][ ] i(µ/) 4 D C i i(µ/) 4 D d D 1 q dq 0 C j (π) D 1 π qi+j is F (E/ + q 0 ; q) is F (E/ q 0 ; q) = i(µ/) 4 D C i C j I i+j (30)

26 6 d I n = i(µ/) 4 D D 1 q dq 0 (π) D 1 π [ q n i E/ + q 0 q /m N + iɛ q n ] [ ] i E/ q 0 q /m N + iɛ d = (µ/) 4 D D 1 q (π) D 1 E q /m N + iɛ ( ) 3 D (µ/) = m N (m N E) n ( m N E iɛ) (D 3)/ 4 D Γ ( mn ) = i k n+1 4π (4π) (D 1)/ (31)

27 7 + + = it loops = i(µ/) 4 D l,m C l C m I l+m i(µ/) 4 D l,m,n C l C m C n I l+m I m+n + = i(µ/) 4 D { l,m + l,m,n C l C m ( i m N 4π k ) k l+m } ( C l C m C n i m ) N 4π k k l+m+n + (3) T = T tree + T loops = n C nk n 1 + i m N 4π k n C (33) n nk

28 8 the natural case Scales : M lo, M hi, variables: k, a, r n a, r n 1/Λ C n 4π m N 1 M n+1 hi (34) C 0 I 0 C 0 m N 4π Q Q M hi (35) loops: suppressed at least by Q/M hi higher order contact terms: suppressed by (Q/M hi )

29 9 T = C 0 } {{ } T (0) + C 0 C 0 } {{ } T (1) + + C 0 C 0 C 0 + } {{ } T () C 0 C 0 C 0 C 0 C } {{ } T (3) + C 0 C + } {{ } T (3) C C 0 (36)

30 30 T ERE = 4π a + i 4π a k + 4π m N m N + T EF T = C 0 +i m N + [ a m N [ 4π m N a 5 + 6π m N r 0 a 4 [ ( mn [ (mn 4π C 0 k+ 4π ) 4 C π ( mn 4π a ar 0 π m N r 0a 3 ) C 3 0 C ] k +i m N ] k + i 4π a [ a r 0 a 3] k 3 m N π ] Λ r 1 a k 4 +, (37) m N [ ( mn ) ] C C0 k 3 4π 4π C 0 ) ] C C0 C 4 k 4 +, (38) C 0 = 4π m N a, C = C 0 ar 0, C 4 = C C 0 + C 0 Λ ar 1 (39) a, r n 1/Λ 1/M hi consistent with C n 1/M n+1 hi

31 31 the unnatural case new scale : 1/a ℵ M lo, higher order contact interactions still suppressed, C n 4π m N 1 ℵ(ℵM hi ) n (40) C (n+1) k (n+1) C n k n k ℵΛ Q M hi (41) loops: C n I n kn+1 ℵ(ℵΛ) n ( Q M hi ) n (4) loops with C 0 only are not suppressed have to be resummed to all orders!

32 3 LO: bubble sum with C 0 vertices NLO: tree graph with C, T ( 1) = plus bubble diagrams with all C 0 vertices but one with C exercise: obtain the above result C ik m N 4π C 0 (43) T (0) C k = [ ] (44) 1 + ik m N 4π C 0

33 33 ( 1) T = C 0 + C 0 C 0 + C 0 C 0 C 0 + (45) (0) T = C C ( 1) + T ( 1) + T C T ( 1) C ( 1) (46) T

34 34 [ T ERE = 4π ik m N }{{ a} r 0k }{{} O(ℵ) O(ℵ /M hi ) = 4π m N [ 1 + (1/a + ik) }{{} T ( 1) k r 1 }{{ Λ + } O(ℵ 4 /M hi 3 ) r 0 k (1/a + ik) }{{ + } T (0) ] 1 r0k 4 4(1/a + ik) }{{ 3 + } T (1) ] r 1 k 4 Λ (1/a + ik) }{{ + } T ()

35 35 Summary χpt for mesons: most successful application of EFT ideas χpt for baryons: slight complication due to the baryon mass (doesn t vanish in the χ-limit) but can still be treated perturbatively χeft for nuclear systems: χpt rules applied to the potential (non-perturbative problem: LS equation) π EFT: much simpler theory, 1st step towards χeft