C.A. Bertulani, Texas A&M University-Commerce

Transcription

1 C.A. Bertulani, Texas A&M University-Commerce in collaboration with: Renato Higa (University of Sao Paulo) Bira van Kolck (Orsay/U. of Arizona) INT/Seattle, Universality in Few-Body Systems, March 10, 014 1

2 probe: f(x) monopole: dipole: quadrupole: controlled precision

3 Scattering amplitude f( θ) = ( l +1)P l ( cosθ) l Bethe, Peierls, 1935 Bethe, 1949 T-matrix T l ( k) T l ( k) = 1 k eiδ l ( k ) sinδ l ( k) = 1 kcot δ l k ( ) ik k l+1 cot δ l = 1 a l + 1 r l k l 4 k 4 +! Effective Range Expansion (ERE) monopole : (scattering length) dipole : (effective range) quadrupole : (shape parameter) u( r) ~ k( r a) a ~ R 1+ 1 L R ( ) 1 r l ~ u asymp ( r) u r [ ( )] dr! sharp surface (L = logarith. der.) 3

4 4

5 1 S 0 nn channel 1 S 0 np channel 3 S 1 np channel u 0 ( r) ~ e ikr S 0 ( k)e ikr S 0 ~ i { { { a 0 nn = 18.8fm r 0 nn =1.7 fm ( ) = 3.7fm ( ) =.73 fm a 0 np I =1 r 0 np I =1 ( ) = 5.74fm ( ) =1.73 fm a 0 nn I = 0 r 0 nn I = 0 1/a + r 0 k / ik 1/a r 0 κ / + κ = 0 κ = fm -1 ; E B =! κ pole on imaginary axis (k = iκ) m N =.3 MeV a > 0 particularly large scattering lengths a >> r NN ~ 1fm (unnatural) existence of a bound state (deuteron) (deuteron) 5

6 small difference from a 0 nn = fm due to m n m p 6 Bethe, 1949 Jackson, Blatt, 1950 e ikr, e ikr F( kr), G( kr) F ~ C [ η 1 r /a B +! ] ( ) 1/kr + η( h η + γ 1+ lnr /a ) B +! G ~ 1/C η [ ] match logarithimic derivative: kcot δ C η + a B ~ 1 R 1+ 1 L R ( ) h η ln a B R = 1 a S + γ 1 definition of pp-scatttering length, a C : k cot δ C η + h a η = 1 +! B a C γ = , a B =1/mα, η =1/ka B C η = πη/ e πη 1 ( ) ( ), h η = ReH iη H( x) = ψ(x) +1/x ln x 1 a S = 1 a C a B ln a B R +1 γ a C = a 0 pp = 7.8fm r 0 pp =.83 fm a S = 17 fm ~ a 0 nn

7 Invariance: (a) parity, (b) Galilean, (c) time reversal, (d) particle number L EFT ~ N + i t + m N N + µ π-less EFT N T = ( p n) = isopin doublet; 4 D C 0 ( N + N) + C! = " " # + # [ ( ) + h.c.+" ] 8 N+ N N! + N δ( r) + higher derivatives of δ( r) µ = arbitrary mass to make C n n same dimension for any D Feynman rules: C 0 C = i ( µ /) 4 D C 0 = i ( µ /) 4 D C k { is N = i q 0 q /m + iε Weinberg, 1991 it tree = i ( µ /) 4 D C 0 i ( µ /) 4 D C k +! ( ) 4 D C n k n = i µ / n =0 Short-range physics (quarks, gluons) encoded in C 0, C, 7

8 (E/; k) (E/+q 0 ; k+q) C i q C j = i ( µ /) 4 D C i C j I i+ j ( ) 4 D dd 1 q I n = µ / (E/; -k) ( π) D 1 (E/-q 0 ; -k-q) q n residues + minimal subtraction scheme* E q /m N + iε ; m NE = k I MS n = i m N k 4π n !!! T = T tree + T loops = C l k l 1+ l m i m N 4π C m k m + i m C n k n N C 4π m k m C n k n +! m,n = n 1+ i m N 4π k C k n n n * MS = subtract any 1/(D - 4) pole before taking the D " 4 limit. 8

9 exp( iθ) = cosθ + isinθ mathematical jewel for physicists (Feynman Lectures of Physics) 1+ x + x +! = 1 1 x mathematical jewel for quantum field theorists M hi ~ m π C n ~ 4π 1 m N n +1 M hi higher derivative contact terms suppressed naturalness: physical parameters with dimension (mass) d scale as (M hi ) d. C 0 I 0 ~ C 0 m N 4π k ~ k M hi loops also suppressed EFT series perturbative: can be organized in powers of k/m hi 9

10 T EFT = C 0 1 i m N 4π C 0k T ERE = 4π a 1 iak a ar 0 k +! m N m N 4π C 0 C C 0 k +! C 0 ~ 4π m N a valid for C ~ C 0 ar 0 a, r n ~ 1 M hi (natural case) in general C n ~ 4π 1 m N M hi n M hi 10

11 deuteron, halo nuclei T expansion in terms of ka fails for k ~ 1/a " use ERE keeping all orders in ka : T eff.range = 4π 1 m N 1/a + ik 1+ r 0 / 1/a + ik k + van Kolck, 1997 Gegelia, 1998 Kaplan, Savage, Wise, 1998 ( r 0 /) k 4 +! ( 1/a + ik) To match with above EFT expansion scale as (k -1, k 0, k 1, ) 1 Expansion should be: T = Tn ; T n ~ O(k n ) Use PDS regularization* scheme (Kaplan, Savage, Wise): - subtract not only 1/(D-4) poles corresponding to log divergences but also poles of lower dimension D (e.g., I n has a pole at D = 3) by adding a counterterm: δi n = m N m N E ( ) n µ 4π(D 3) n = 1 I PDS n = I n + δi n = k m N 4π µ + ik ( ) * PDS = Power Divergence Subtraction. 11

12 Using subtraction scheme " T 1 = C 0 1+ C m 0 N 4π ( µ + ik ) 1 T 0 = C k 1+ C m 0 N 4π ( µ + ik ) 1

13 C 0 ( µ ) = 4π 1 ; C m N µ +1/a µ ( ) = 4π 1 r0 m N µ +1/a ;! T 1 = C k ( ) /4π ( ) m N µ + ik 1+ C m 0 N 4π ( µ + ik ) 3 C 4 k 4 1+ C m 0 N 4π µ + ik ( ) ;!! 4 power counting: C n ~ 4π 1 m N M n hi µ n +1 T-matrix for physics at k ~ 1/a scale: has a pole in k = iκ corresponding to real or virtual bound states κ ~ i/a + higher order corrections T EFT should not depend on µ e.g. µ d dµ 1 T = 0 renormalization group equations µ d dµ C n = m N 4π n m =0 C m C (n m ) with the boundary condition that C 0 (0) = 4πa/m N 13

14 C 0 ( µ ) = 4π 1 ; C m N µ +1/a µ ( ) = 4π 1 r0 m N µ +1/a ;! T EFT should not depend on µ e.g. µ d dµ 1 T = 0 renormalization group equations µ d dµ C n = m N 4π n m =0 C m C (n m ) with the boundary condition that C 0 (0) = 4πa/m N T 1 = C k ( ) /4π ( ) m N µ + ik 1+ C m 0 N 4π ( µ + ik ) 3 C 4 k 4 1+ C m 0 N 4π µ + ik ( ) ;!! 4 power counting: C n ~ 4π 1 m N M n hi µ n +1 T-matrix for physics at k ~ 1/a scale: has a pole in k = iκ corresponding to real or virtual bound states κ ~ i/a + higher order corrections 14

15 11 Be 15

16 J π kev ~ 5 fm 1.9 fm 6 He M hi 4 He J π MeV M lo ~ 1 threshold n + 4 He 16

17 Goal: use EFT to reproduce ERE for p-wave T ( 1 k, cosθ) = 1πa 1 CB, Hammer, van Kolck, 00 m k cosθ 1+ a 1r 1 k ia 1 k 3 + a 1 ( 4 a r 1 1 l )k 4 +! Natural case, assuming spinless particles most general p-wave interaction: L EFT ~ N + i 0 + N + C p m 8 N! N! = " # Galilean derivative is( p 0, p) = C C 4 ( ) + ( N! N) C p 4 i p 0 p /m + iε 64 N!! i N ( ) + ( N! i N) + h.c. propagator +" it 1(a ) = ic p k k' it 1(b) = ic 4 p k k k' 17

18 C C Natural case: it 1(c) = ic p = C p ( ) d 4 q ( π) 4 ( ) d 3 q imk' i k j iq k E / q 0 q /m + iε ( π) 3 q i q j q k + iε iq k' E / q 0 q /m + iε it 1(c) = ( p C ) im 6π k k' L + k 3 L 1 + π ik 3 L 1 and L 3 are (ultraviolet) infinities that can be absorbed in C and C 4 a Matching to ERE " T T 1(b) C p 4 = C p r 1 1(a) C p =1π a 1 m 1 + T 1(c) reproduces 3 rd term of ERE expansion 18

19 Unnatural case (shallow state) p waves: with C p and Cp 4 to all orders Kaplan, 1997 introduce auxiliary field d (dimeron) which reproduces same physics as original EFT Lagrangian L EFT ~ N + i 0 + N + η m 1 d + i i 0 + 4m Δ 1 d i + g 1 4 d + i N! N [ ( ) + h.c. ] +" 1- parameters η 1 = ±1, g 1 and Δ 1 fixed from matching - advantage: get quicker to the answer, appropriate for large a s dimeron propagator: id 0 ( 1 p 0,p) ij = Feynman rules: nucleon-dimeron vertex: V N d = g 1 ( 4 p 1 p ) iη 1 δ ij p 0 p /4m Δ 1 + iε 19

20 self-energy Σ 1 full dimeron propagator: mg iσ 1 = iδ 1 ij 1π π L 3 + π L 1( mp 0 k /4) + i( mp 0 k /4) 3 / infinite constants absorbed into g 1 and Δ 1 id 0 1 = id id 0 1 ( Σ 1 )id 0 1 +! = id Σ 1 D 1 0 attach external legs to full dimeron propagator " T (p wave) EFT = 1π R m k 1πΔ η 1 1 R m g 1 ( ) η 1 1πΔ 1 R ( ) k ik m g 1 R With renormalized parameters g R, Δ R Now match to T (p wave) EFE = 1π m k 1 + r 1 a 1 k ik 3 1 to get η 1R, g 1 R and Δ 1 R 0

21 for a 1, r 1 < 0 (e.g., n + 4 He) ( ) 1 r 1κ a 1 iκ3 = 0; κ 1 = iγ 1 ; κ ± = i γ ± i γ bound-state resonance δ 1 = 1 EB Γ( E) arctan arctan E B E E 0 E = k m ; ( ) ; E 0 = γ + γ m ; Γ( E) = 4γ E m ; B = γ 1 m 1

22 CB, Hammer, van Kolck, 00 Partial wave l ± a i± [fm 1+l ] r i± [fm 1-l ] P l± [fm 3-l ] (37) 1.385(41) (68) (16) (3) (11) -3.00(6) n + 4 He: p 3/ resonance - shallow: ~ 1 MeV - has to be treated nonperturbatively - p 1/ weak " perturbatively - s 1/ also perturbatively neutron spin ( ) ; T = π F + i σ! n ˆ G m 0 Arndt, Roper, 1973 dσ dω = F( θ) + G( θ) F(k,θ) = l 0 [(l +1)f l+ (k) + lf l (k)] P l ( cosθ) G(k,θ) = [ f l+ (k) f l (k)] l 1 P 1 l ( cosθ) f l± = 1 k cot δ l± ik

23 parity- and time-reversal-invariant Lagrangians: L LO = φ + i 0 + φ + N + i m 0 + N + η α m 1+ t + i 0 + N m α + m N [ ] + h.c. r t + S + Nφ + g 1+ t + S + N φ ( N)φ [ ( ) + h.c. ] { } [ ] + g' 1+ t + i 0 + L NLO = η 0+ s + [ Δ 0+ ]s + +g 0+ s + Nφ + φ + N + s notation: s, d, t = s 1/, p 1/, p 3/ ϕ = 4 He scalar field S i = x 4 spin-transition matrices ( ) Δ 1+ ( ) m α + m N t t S i S j + = 3 δ ij i 3 ε ijk σ k, S i + S j = 3 4 δ ij 1 6 J i { 3 / 3 /,J } j + i 3 ε J 3 / ijk k J i 3 /,J j 3 / [ ] = iε ijk J k 3 / generators of the J = 3/ representation of the rotation group 3

24 α and nucleon propagators (,a,b = spin, α,β = isospin) is φ ( p 0,p) = Dimeron propagators 1 p 0 p /m α + iε, is p,p ab iδ N( 0 ) αβ δ ab αβ = p 0 p /m N + iε 0 id 1+ 0 id 1 ab iη ( p 0,p) 1+ δ αβ δ ab αβ = p 0 p /(m N + m α ) Δ 1+ + iε, id 0 ab 0+ ( 0,0) αβ ab ( p 0,p) αβ = 0 id0+ ( ab ) αβ = iη 0+δ αβ δ ab Δ 0+ Leading contribution for k ~ M lo is ~ 1π/mM lo solely from the 1+ partial wave with the scattering-length and effective-range terms included to all orders. NLO order correction suppressed by M lo /M hi and fully perturbative. 1 partial wave still vanishes at NLO. 4

25 1+ dimeron propagator, using 0 id 1+ ab ( p 0,p) αβ = iη1+ δ αβ δ p ab p 0 m α + m N id = id 1 Σ 1 D + attaching the external particle lines (m 0 = reduced mass) ( ) Δ 1+ + η µg π ( m 0 ) 3 / p p 0 + m α + m N ( ) iε 3 / + iε 1 T LO = π k cosθ + i! 6πΔ ( σ n ˆ )sinθ η 1+ 3π m 1+ η 0 m 0 g m 0 g k ik Matching to ERE: m a 1+ = η 0 g 1+ 6π 1+ ; r 6πΔ 1+ = η m 0 g 1+ g 1+, Δ 1+, sign η 1+ in terms of a 1+ and r 1+ F LO = k cosθ 1/a 1+ + r 1+ k / ik 3, G LO = k cosθ 1/a 1+ + r 1+ k / ik 3 5

26 NLO contributions come from the shape parameter P 1+ and the s-wave scattering length a 0+ T NLO = η g π ' g 1+ πδ 0+ m 4 0 g 1+ m a 0+ = η 0 g ; P πδ 1+ = 6πg ' m 0 g 1+ k 6 ( cosθ + i σ! n ˆ )sinθ ( 1/a 1+ r 1+ k / ik 3 ) F NLO = a 0+ + P 1+ 4 k 6 cosθ ( 1/a 1+ + r 1+ k / ik 3 ) G NLO = P 1+ 4 k 6 cosθ ( 1/a 1+ + r 1+ k / ik 3 ) 6

27 PSA, Arndt et al. 73 scatt length only scatt length only 7

28 NNDC, BNL Haesner et al. 83 8

29 δt = C 0 e.g., pp-scattering L EFT ~ N + i t + N C m 0 N + N N d 3 q ( π) 3 ( ) e 1 k + λ E k q = C 0 η π + iln k + O λ λ Kong, Ravndal, 000 ( ) /m N + iε ~ C 0 αm N k = C 0 η ( ) non perturbative for k < αm N external legs strongly influenced by Coulomb repulsion δi 0 ~ η m N 8π 1 ε + ln µ π k +# non perturbative for k < αm N pole at D = 4 " need renorm of C 0 strong interaction also much modified by Coulomb interaction 9

30 + T = C 0 C η e iσ 0 + C 0C η e iσ 0 J ( k) +! = C C 0 e iσ 0 0 η 1 C 0 J 0 ( k) d ( ) 3 q = m N ( π) 3 J 0 k = αm N 4π ( ) πη q e πη q 1 ε + H η ( ) 1 ( ) + ln µ π 1 k q + iε 3 αm N ( ) µm N 4π pole at D = 4 " need renorm of C 0 use PDS and get rid of pole at D = 3, too. 30

31 1 a pp µ ( ) = 1 now add: a pp a B C ln µ π +1 3 αm N γ 1 µr 0 [ ] + L = C N+ NN! + N + h.c.+" = Jackson, Blatt, 1950 with renormalization of C 0 : 1 a pp µ ( ) = 4π m N C 0 ( µ ) + µ /a 1/a pp pp (MeV) C 0 + C C µ (MeV) µ (MeV) a ( pp µ = m ) π = 9.9 fm a exp ( pn µ = m ) π = 3.7 fm Kong, Ravndal,

32 L Nα Higa, CB, van Kolck, 014 LO = φ + id 0 + D φ + N + id m 0 + D N + ς α m 0+ s + [ Δ 0+ ]s α D + ς 1+ t + id 0 + ( m α + m N ) Δ 1+ t 1 4 F µνf µν { [ ] + h.c. r[ t + S + D( Nφ) + h.c. ]} + g 1+ t + S + ndφ ( DN)φ [ ] + g 0+ s + Nφ + φ + N + s - t and s = dimeron fields coupling Nα in P 3/ and S 1/ - with leading-order coupling constants g 1+ and g 0+ - S i = x 4 spin-transition matrices between J = 1/ and J = 3/ - sign variables ζ 0+, ζ 1+ = ±1 adjusted to reproduce the signs of the respective effective ranges. r = m α m N m α + m N D µ = µ + iez 1+ τ 3 A µ 3

33 NLO = ς 0+ s + id 0 + L Nα D ( ) Δ 1+ m α + m N s + g ' 1+ t + id 0 + D ( ) m α + m N t 33

34 k r (MeV) k i (MeV) E R (MeV) Γ R / (MeV) p 3/ pα resonance parameters Csoto, Hale, 1997 k p = k r ik i k r ~ M lo ~ 50 MeV, k i ~ M lo /M hi ~ 10 MeV a 0+ (fm) r 0+ (fm) 4.97 ± ± 0.08 S 1/ pα ERE parameters Arndt, Long, Roper, 1997 a 0+ (fm 3 ) r 1+ (fm -1 ) P 1+ (fm) ± ± ± 0.15 P 1/ pα ERE parameters P 1+ /4 ~ r 0+ / ~ 1/M hi " natural a 0+ ~ 1/M lo, a 1+ ~ 1/M lo 3 and r 1+ / ~ M lo " fine tuned 34

35 T 0+ = π µ C η e iσ 0 1/a 0+ k C H η ( ) 1 r 0+k / ( ) 1/a 0+ k C H η now pα reduced mass k C = Z p Z α α em µ T 1+ = π µ C (1) η e iσ 1 k P 1+ ( θ) 1/a 1+ + r 1+ k / k C H η ( ) 1+ P k 4 1+ /4 ( ) 1/a 1+ + r 1+ k / k C H (1) η C (1) η = ( 1+ η )C η H (1) ( η) = k ( 1+ η )H( η) k C = Z α Z p µα P 1+ ( θ) = cosθ + σ! n ˆ sinθ k θ k T (LO) = T 0+ + T 01 n = k k' k k' 35

36 T NLO pα = π µ r 0+k (0) ( C η ) e iσ 0 + P 1+k [ 1/a 0+ k C H( η) ] 4 4 (1) ( C η ) e iσ 1 k P 1+ θ ( ) 1/a 1+ + r 1+ k / k C k ( + k C )H η [ ( )] Matching to ERE: (R) µg a 1+ = ζ πΔ, r = ζ (R) π µ g, and P = 6πg ' 1+ (R) µ 3 g 1+ (R) p 3/ channel (R) µg a 0+ = ζ 0+ π 0+ (R), and r πδ 0+ = ζ µ g 0+ s 1/ channel (R) 36

37 S 1/ pα ERE fits with EFT S 1/ a 0+ (fm) r 0+ (fm) LO NLO P 1/ pα ERE fits with EFT P 3/ a 1+ (fm 3 ) r 1+ (fm -1 ) P 1+ (fm) LO NLO P 3/ pα ERE resonance fits with EFT P 3/ k r (MeV) k i (MeV) E R (MeV) Γ R / ( MeV) LO NLO

38 EFT results for S 1/ and P 3/ scattering phase shifts at LO (dotted) and NLO (solid), compared against the partial wave analysis (diamonds). 38

39 EFT at LO (dotted) and NLO (thick solid) for pα elastic crosssection at θ lab = 140 0, compared against the partial wave analysis (thin solid) and measured data points. 39

40 Recent results for pα system: 1 We include P-waves with resonance and Coulomb interactions. - We perform an expansion of the P 3/ amplitude around the resonance pole to extract the resonance properties directly from a fit to the phase shift. 3- Our results at LO and NLO exhibit good convergence and the resonance energy and width are consistent with the ones using the extended R-matrix analysis. 4- Comparison with the differential cross-section at 140 o reassures the consistency of the power counting, with P 1/ contribution showing up only for proton energies beyond 3.5 MeV. 5 Final adjustments necessary: a shallow bound state appears in the s-wave "unitary correction necessary in the absence of Coulomb might need test. 40

41 41

42 T 1+ = π µ For P 3/ resonance amplitude " resonance pole expansion 1+ r 1+ k k p r 1+ ( ) C (1) η e iσ 1 k P 1+ θ H (1) k k ( ) k C H (1) C k k p ( ) /4 P 1+ k k p k ( ) k C H (1) C k H (1) k C k p k C k p r 1+ = k r { L i ik i P 1+ } { k C H (1) k C /k p LO { NLO L i, L r defined at the pole from ( ) = k r 3 L r + ik r k i L i = k r L i k rp 1+ r 1+ 1 k i k k r L i k rp 1+ r { { LO NLO 1 = k 3 a r L r + L i 1 k i 1+ k r k rp k i k 3 k r L r + L i k rp 1+ r 4 { LO { NLO 4