Fully Perturbative Calculation of nd Scattering to Next-to-next. Next-to-next-to-leading-order

Transcription

1 Fully Perturbative Calculation of nd Scattering to Next-to-next-to-leading-order Duke University April 15, 14

2 Ingredients of Pionless Effective Field Theory For momenta p < m π pions can be integrated out as degrees of freedom and only nucleons and external currents are left. For any effective (field) theory one writes down all terms with degrees of freedom that respect symmetries. Develop a power counting to organize terms by their relative importance. Calculate respective observables up to a given order in the power counting.

3 Lagrangian Building Blocks Lagrangian Dibaryon Propagators The Lagrangian in EFT π is ( L = ˆN i + ) ( ˆN ˆt i M N ( ŝ a i + (1 S ) 4M ( 1) S ) (1 () N ] + y s [ŝ a ˆN T P a ˆN + H.c.. i + (3 S 1 ) 4M ( 1) S 1 ) (3 () N ) ŝ a + y t [ˆt i ) ˆt i ] ˆN T P i ˆN + H.c. The projector P i = 1 8 σ σ i τ ( P a = 1 8 τ a τ σ ) projects out the spin-triplet iso-singlet (spin-singlet iso-triplet) combination of nucleons.

4 Lagrangian Dibaryon Propagators The LO dressed deuteron propagator is given by a bubble sum (LO) (NLO) (NNLO) (Z-parametrization) At LO coefficients are fit to reproduce the deuteron pole and at NLO to reproduce the residue about the deuteron pole. 1 y t = M N 8πγ t Z t (Z t 1), (3 S 1 ) ( 1) = y t M N γ t µ Z t 1, (3 S 1 ) () = γ t M N

5 Lagrangian Dibaryon Propagators The spin-triplet ( deuteron ) and spin-singlet dibaryon propagator to NNLO are given by idt,s NNLO (p, p) = 4πi 1 M N yt,s p γ t,s 4 M Np iɛ ( 1 + Z ) t,s 1 p γ t,s + γ t,s 4 M Np iɛ }{{}}{{} LO NLO ( ) Zt,s 1 ( ) p + γ t,s 4 M Np γt +. }{{} NNLO

6 (nd Scattering) Higher Orders Partial Resummation Technique New Technique At LO in the Quartet channel, nd scattering is given by an infinite sum of diagrams. This infinite sum of diagrams can be represented by an integral equation.

7 Higher Orders Partial Resummation Technique New Technique The integral equation gives ( it ji ) βb αa ( k, p, h) = y t (σi σ j ) β αδ b a + y t (σi σ k ) β γδc b ( k id () t k 4M N k 4M N d 4 q (π) 4 (itjk ) γc 4M N γ t M N + h + q, q γ t M N i + h + q ( q+ p) M N i + γ t M N + h ( k+ p) M N + iɛ αa( k, q, h + q ) ) k M N. + iɛ i h q q M N + iɛ

8 Higher Orders Partial Resummation Technique New Technique Projecting spin and isospin in the Quartet channel and projecting out in angular momentum gives tq(k, l p) = y t M N pk where + π Λ Q l ( p + k M N E iɛ dqq t l q(k, q) pk 3 q γ t Q l ( p + q M N E iɛ pq Q l (a) = ) M qp NE iɛ ), dx P l(x) x + a.

9 Higher Orders NLO correction is Building Blocks Higher Orders Partial Resummation Technique New Technique NNLO corrections are Note the second diagram contains full off-shell scattering amplitude.

10 The NNLO scattering amplitude is Higher Orders Partial Resummation Technique New Technique t l,q(k, p) + t l 1,q(k, p) + t l,q(k, p) = B l (k, p) + B l 1(k, p) + B l (k, p)+ + (K l (q, p, E) + K l 1(q, p, E) + K l (q, p, E)) (t l,q(q, k) + t l 1,q(q, k) + t l,q(q, k)), where Λ A(q) B(q) = dqq A(q)B(q). π The inhomogeneous and homogeneous terms are B(k, l p) = y t M N pk Q l K l n(q, p, E) = M Ny t 4π ( p + k M N E iɛ D(n) t pk ) (E q 1, q M N qp Q l ), B l 1(k, p) = B l (k, p) = ( q + p ) M N E iɛ pq

11 Higher Orders Partial Resummation Technique New Technique Partial Resummation Technique Denoting tnlo l = tl,q + tl 1,q, for the partial resummation technique one finds tnlo l (k, p) = Bl (k, p)+b1(k, l p)+(k(q, l p, E)+K1(q, l p, E)) tnlo l (k, q), with the diagrammatic representation

12 Higher Orders Partial Resummation Technique New Technique New Full Perturbative technique Picking out only LO pieces gives t,q(k, l p) = B(k, l p) + K(q, l p, E) t,q(k, l q), only NLO pieces t l 1,q(k, p) = B l 1(k, p)+k l 1(q, p, E) t l,q(k, q)+k l (q, p, E) t l 1,q(k, q), and only NNLO pieces t l,q(k, p) = B l (k, p)+k l t l,q(k, q) + K l 1(q, p, E) t l 1,q(k, q) + K l (q, p, E) t l,q(k, q). Terms are reshuffled to inhomogeneous term. Kernel at each order is the same

13 Higher Orders Partial Resummation Technique New Technique Diagrammatically NLO correction is now given by NLO LO NLO, and NNLO correction by NNLO LO NLO NNLO Note all corrections are half off-shell.

14 nd scattering At LO in the Doublet channel, nd scattering is given by a coupled set of integral equations

15 Using the perturbative technique, the Doublet nd scattering amplitude integral equation at LO is at NLO t l,d(k, p) = B l (k, p) + K l (q, p, E) t l,d(k, q), t l 1,d(k, p) = B l 1(k, p) + K l 1(q, p, E) t l,d(k, q) + K l (q, p, E) t l 1,d(k, q), and at NNLO t l,d(k, p) = B l (k, p) + K l (q, p, E) t l,d(k, q)+ + K l 1(q, p, E) t l 1,d(k, q) + K l (q, p, E) t l,d(k, q). The Equations are the same as in Quartet case but are now matrix equations in cluster configuration space.

16 The vector t l n,d (k, q) is The inhomogeneous term is B l (k, p) = ( t t l l n,d (k, q) = n,nt Nt (k, q) tn,nt Ns l (k, q) yt M N pk Q l 3ytysM N pk ( B l H1 (E, Λ)δ 1(k, p) = l H 1 (E, Λ)δ l ( ) p +k M N E iɛ ( pk Q p +k M N E iɛ l pk ), B l (k, p) = ). + H (E, Λ)δ l ) H (E, Λ)δ l ( H (E, Λ)δ l H (E, Λ)δ l ).

17 The homogeneous term is Building Blocks K l n(q, p, E) =D (n) (E, q) 1 ( q qp Q + p ) ( M N E iɛ 1 3 l qp 3 1 n ( ) + δ l D (j) 1 1 (E, q)h n j (E, Λ), 1 1 j= where ( ) D n D (n) t (E, q) (E, q) = D s (n), (E, q) and the three-body force terms defined by ) H(E, Λ) = HLO (Λ) Λ (Λ) + HNLO Λ (Λ) + HNLO Λ (Λ) Λ 4 (M N E+γd ). + HNLO

18 The SD-mixing Lagrangian is [ L SD Nd = y SD ˆd i ˆN (( T ) i ( ) j 13 ) ] δij ( ) P j ˆN +H.c. The SD-mixing amplitude is given by the sum of diagrams SD

19 The sum of all diagrams gives the amplitude (tsd xw )βb αa ( k, p) = 4M N vp T (K xw ) βb αa ( k, p)v p 8 4M N 8 4M N 8 + 4M N 8 where d 3 q (π) 3 vt p (K xy ) βb γc ( q, p)d d 3 q ( (π) 3 (t xy ) βb d 3 q (π) 3 (K zy ) δd γc( q, l)d (E q M N, q ) ( ) (t yw ) γc αa ( k, q) ) ( ) T γc ( q, p) D E q, q (K yw ) γc M αa( k, q)v p N d 3 l ( ) ) T (π) 3 (t xz ) βb δd ( l, p, ) D (E q, q M N (E l ) (, M ) l (t yw ) γc αa ( k, q), N v p = ( 1 ).

20 All angular dependence is contained in (K xw ) βb αa( q, 1 l) = q + q l + l M N E iɛ ( ( K 11 xw) βb SD αa ( q, ( l) K 1 xw) βb SD αa ( q, ) l) ( K 1 xw) βb SD αa ( q, ( l) K xw) βb SD αa ( q,, l) where ( K 11 PV xw ) βb ( k, p) = y αa t y SD (σ y σ x ) β αδa [( p b + k) w ( p + k) y 13 ] δ yw ( p + k) ( K 1 xa) βb PV αa ( k, p) = y s y SD (σ y ) β α(τ A ) b a [( k + p) x ( k + p) y 13 ] δ yx( k + p) ( K 1 PV Bw ) βb αa ( k, p) = y t y SD (σ y ) β α(τ) b a ( K BA) βb PV αa ( k, p) =. [( p + k) w ( p + k) y 13 δ yw ( p + k) ]

21 The amplitude can be projected in partial waves of J = L + S JM t SD L S,LS(k, p) = 1 ( dω k dω p Y M 4π J,L S (ˆp)) tsd ( k, p)y J,LS(ˆk). M The projected amplitude is t SD JM L S,LS(k, p) = M N 8π v T p K(k, p) J L S,LSv p+ M N 8π 3 M N 8π 3 + M N 4 8π 5 dqq v T p K(q, p) J L S,LSD dqq ( t JM L S,L S (q, p) ) T D ( E dqq K(q, l) JM L S,LSD ( ) ( (E q, q t JM M N ) q M N, q dll ( t JM L S,L S (p, l) ) T D ( E E ) l (, M ) l t JM LS,LS(k, q). N LS,LS(k, q) ) K(k, q) J L S,LSv p ) q, q M N

22 [ ] K(k, p) J L S,LS = 4π S S L 1 ( δs 11 3,1/ + δ S,3/) C,, L,,L ( 1) S +S+L J { } { } 1 1 S S 1/ S S L J L ] 1 kp (k Q L (a) + 4p Q L (a)) ( ) + 8π S S L δs,1/ + δ S,3/ C,, L,1,L C,, L,1,L L ( 1) 3/ S L L 1/ 1 S L 1 L L S L Q L (a) J ( ) + 8π S S L δs,1/ + δ S,3/ C,, L,1,L C,, L,1,L ( 1) 1/+S +L+L L { } { } 1/ L 1 S L 1 L L J L Q 1/ J S L (a) 16π 3 1 L ( ) δs,1/ + δ S,3/ ( 1) 1/ S δ L,L δ S,S + (S S )(L L )(k p) C,, L,1,L C,, 1,L,L L Q L (a) L

23 [ ] K(k, p) J L S,LS = 8π 5 S Lδ S 1/C,, L,,L ( 1) 1+S+L J 1 { } { } 1 1 S S 1 1/ S S L J L kp (k Q L (a) + 4p Q L (a)) + 8π L 1/ 1 S 6 L SδS 1/ C,, L,1,L C,, L,1,L 1 L L L S L Q L (a) J + 8π 6 L Sδ S 1/ L C,, L,1,L C,, L,1,L L ( 1) 1+L+L { L 1 L } { L 1 L 1/ J S 1/ J S + 16π ( 1) L L 1 3 L L } Q L (a) L C,, L,1,L C,, L,1,L δ S 1/δ S Sδ L LQ L (a)

24 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles The S matrix can be decomposed into irreducible representations of J and parity. For J = 1 / it is a unitary matrix S 1 + = S 1 S 1 3, 3 3, 1, S 1 = S 1 S 1 1 1, ,1 3, S 1 1, 3 which is decomposed by u 1 + = S 1 1, 1 S Jπ = ( u Jπ) T e iδ Jπ u Jπ S 1 1 3,1 1 S 1 1 3,1 3 δ 1 + = δ 1 3, δ 1 δ 1 = δ 1 1 1, δ ( cos η 1 + sin η 1 + sin η 1 + cos η 1 + ) ( 1, u 1 cos ɛ = sin ɛ 1 sin ɛ 1 cos ɛ 1 )

25 For J 3 / S is a unitary 3 3 matrix S Jπ = S J J 3 3 S J J± 1 1,J 3 3,J 3 3 S J J± 1 3,J 3 3 S J J 3 3 S J J± 1 1 which is decomposed by δ Jπ = v Jπ = δ J J 3 3 δ J J± 1 1 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles,j± 1 1,J± 1 1 S J J± 1 3,J± 1 δ J J± cos ɛ Jπ sin ɛ Jπ sin ɛ Jπ cos ɛ Jπ x Jπ =, w Jπ = 1 S J J 3 3 S J J± 1 1,J± 1 3,J± 1 3 S J J± 1 3,J± 1 3, u Jπ = v Jπ w Jπ x Jπ sin η Jπ cos η Jπ cos ηjπ sin η Jπ 1 1 cos ζjπ sin ζ Jπ sin ζ Jπ cos ζ Jπ,

26 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles quartet S-wave phase shift at NNLO. Crosses are AV-18+UIX with hyperspherical harmonics method, and x s Bonn-B with Faddeev equations. - quartet-s (a) Re(δ)[deg]

27 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles imaginary part of quartet S-wave phase shift at NNLO. Note here imaginary part is positive unlike in partial resummation technique quartet-s (b) Im(δ)[deg]

28 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles The bands are due to cutoff variation from Λ = 16 MeV - doublet-s (c) LO NLO NNLO Re(δ)[deg]

29 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles for l = 1: j =l 1, l + 1, l + 3 :: for l > 1: j =l 3, l 1, l + 1, l + 3 Re(δ)[deg] 3 1 quartet-p (a) Re(δ)[deg] quartet-d (b) quartet-f (c) -.5 quartet-g (d) Re(δ)[deg] Re(δ)[deg]

30 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles 3 quartet-p (a) Re(δ)[deg]

31 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles Re(δ)[deg] quartet-d (b)

32 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles 1. 1 quartet-f (c) Re(δ)[deg]

33 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles Re(δ)[deg] quartet-g (d)

34 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles Re(δ)[deg] Re(δ)[deg] 7 6 doublet-p (a) doublet-f (e) Re(δ)[deg] Re(δ)[deg] 9 8 doublet-d (c) doublet-g (g)

35 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles Re(η 1/+ )[deg] 3.5 η 3 1/+ (a) Re(η 3/+ )[deg] η 3/+ (c) Re(η 5/+ )[deg] -1 η 5/+ (e) Re(η 7/+ )[deg] η 7/+ (g)

36 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles 1 5 η 3/- (a) Re(η 3/- )[deg] Re(η 5/- )[deg] η 5/- (c)

37 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles Re(ζ 3/+ )[deg] ζ -4 3/+ (a) Re(ζ 5/+ )[deg] 6 ζ 5/+ (c) Re(ζ 7/+ )[deg] ζ 7/+ (e)

38 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles 5 4 ζ 3/- (a) Re(ζ 3/- )[deg] 3 1 Re(ζ 5/- )[deg] ζ 5/- (c)

39 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles Re(ε 3/+ )[deg] Re(ε 7/+ )[deg] ε 3/+ (a) ε 7/+ (e) Re(ε 5/+ )[deg] Re(ε 9/+ )[deg] ε 5/ ε 5/+ (c) ε 9/+ (g)

40 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles -5 ε 1/- (a) 5 ε 3/- (c) Re(ε 1/- )[deg] Re(ε 3/- )[deg] Re(ε 5/- )[deg] ε 5/- (e) Re(ε 7/- )[deg] ε 7/- (g)

41 Quartet S-wave Doublet S-wave Higher Quartet Partial Waves Higher Doublet Partial Waves mixing angles Conclusions and Future Directions Most phase shifts and mixing angles are described well by NNLO EFT π at low energies. N 3 LO contributions will be important as they contain two-body P wave contact interactions, which may help fix certain mixing-angles and phase shifts. New technique makes higher order perturbative calculations much easier, which is important in order to calculate polarizaiton observables such as A y. The new technique can be used to calculate any diagram with full off-shell scattering amplitudes. This makes calculations involving external currents much simpler to calculate (e.g. 3 H and 3 He Compton scattering and photodisintegration both parity conserving and violating).

42 Numerical Techniques t i = B i + j K ij w j t j (δ ij K ij w j ) t j = B i j Integral equation is discretized and solved along contour in complex plane (Hetherington-Schick Method)