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
probe: f(x) monopole: dipole: quadrupole: controlled precision
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
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 = 0.137 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
small difference from a 0 nn = -18.8 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 = 0.57715, 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
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
(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 +1 + +!!! 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
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
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
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
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
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
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
11 Be 15
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
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
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
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
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 0 1 + id 0 1 ( Σ 1 )id 0 1 +! = id 1 0 1 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
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
CB, Hammer, van Kolck, 00 Partial wave l ± a i± [fm 1+l ] r i± [fm 1-l ] P l± [fm 3-l ] 0 +.4641(37) 1.385(41) -- 1 - -13.81(68) -0.419(16) -- 1 + -6.951(3) -0.8819(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
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
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+ ( 0 + 1 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
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 1+ 1+ 6π ( 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 1+ 1+ m 0 g k ik 3 1+ 1 Matching to ERE: m a 1+ = η 0 g 1+ 6π 1+ ; r 6πΔ 1+ = η 1+ 1+ 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
NLO contributions come from the shape parameter P 1+ and the s-wave scattering length a 0+ T NLO = η g 0+ 0+ + 6π ' g 1+ πδ 0+ m 4 0 g 1+ m a 0+ = η 0 g 0+ 0+ ; P πδ 1+ = 6πg ' 1+ 0+ 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
PSA, Arndt et al. 73 scatt length only scatt length only 7
NNDC, BNL Haesner et al. 83 8
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
+ 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 (0.577..) µm N 4π pole at D = 4 " need renorm of C 0 use PDS and get rid of pole at D = 3, too. 30
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 15.0-1/a 1/a pp pp (MeV) 10.0 5.0 0.0 C 0 + C C 0 5.0 0.0 80.0 140.0 00.0 µ (MeV) µ (MeV) a ( pp µ = m ) π = 9.9 fm a exp ( pn µ = m ) π = 3.7 fm Kong, Ravndal, 000 31
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
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
k r (MeV) k i (MeV) E R (MeV) Γ R / (MeV) 51.1 9.0 1.69 0.61 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 ± 0.1 1.95 ± 0.08 S 1/ pα ERE parameters Arndt, Long, Roper, 1997 a 0+ (fm 3 ) r 1+ (fm -1 ) P 1+ (fm) -44.83 ± 0.51-0.365 ± 0.013 -.39 ± 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
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
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 1+ 1+ 6πΔ, r = ζ (R) 1+ 1+ 1+ 6π µ g, and P = 6πg ' 1+ (R) 1+ 1+ µ 3 g 1+ (R) p 3/ channel (R) µg a 0+ = ζ 0+ π 0+ (R), and r πδ 0+ = ζ 0+ 01+ µ g 0+ s 1/ channel (R) 36
S 1/ pα ERE fits with EFT S 1/ a 0+ (fm) r 0+ (fm) LO 7.4 + 8.0. - NLO 4.81 + 0.05 0.1 1.7 + 1.3 0.8 P 1/ pα ERE fits with EFT P 3/ a 1+ (fm 3 ) r 1+ (fm -1 ) P 1+ (fm) LO -58.0 + 11.0 9.0-0.15 + 0.14 0.09 - NLO -44.5 + 1.6 0.1-0.40 + 0.04 0.10 -.8 + 1.0 1.8 P 3/ pα ERE resonance fits with EFT P 3/ k r (MeV) k i (MeV) E R (MeV) Γ R / ( MeV) LO -50.6 + 1..5-10.3 + 1.4 0.8 1.64 + 0.09 0.18 0.70 + 0.1 0.09 NLO -50.7 + 0.5 0.6 9.40 + 0.01 0.10 1.66 + 0.04 0.04 0.63 + 0.01 0.0 37
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
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
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
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 1+ 4 1 k i k 3 k r L r + L i k rp 1+ r 4 { LO { NLO 4