Electroweak Probes of Three-Nucleon Systems

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1 Stetson University July 3, 08

2 Brief Outline Form factors of three-nucleon systems Hadronic parity-violation in three-nucleon systems

3 Pionless Effective Field Theory Ideally suited for momenta p < m π since pions can be integrated out as degrees of freedom and only nucleons and external currents are left. Power counting organizes terms by their relative importance and allows for error estimation of calculation. ( ) Organized by counting powers of Q n = p n. m π Ensure order-by-order results are renormalization group invariant (converge to finite values for Λ ). Power counting gives relative size of two- and three-body currents.

4 Dibaryon fields make three-body calculations easier ( L = ˆN i 0 + ) ( ˆN ˆt i M N ( ŝ a i 0 + ( S 0 ) 4M ( ) S 0 ) ( (0) N ] + y [ ŝ a ˆN T P a ˆN + H.c.. i 0 + (3 S ) 4M ( ) S ) (3 (0) N ) ŝ a + y [ˆt i ) ˆt i ] ˆN T P i ˆN + H.c. ˆN nucleon fields ˆt i (dibaryon field) two nucleons in 3 S channel ŝ a (dibaryon field) two nucleons in S 0 channel Can be matched to theory of only nucleons by integrating out dibaryon fields

5 Two- and Three-Body Inputs of EFT π Two-body inputs for EFT π : LO scattering lengths a ( 3 S ) and a 0 ( S 0 ) non-perturbative NLO range corrections r and r 0 perturbative Three-body inputs for EFT π : LO three-body force H 0 fit to doublet S-wave nd scattering length non-perturbative (Bedaque et al.) nucl-th/ NNLO three-body energy dependent three-body force H fit to triton binding energy perturbative Total of 6 NNLO parameters predicts observables to 3% two-body LECs fit using Z-parametrization: LO fits to 3 S and S 0 poles and NLO to their residues.

6 The LO dressed deuteron propagator is given by a sum of bubble diagrams (LO) (NLO) (NNLO) yielding the LO dibaryon propagator id LO t,s (p 0, p) = 4πi M N y γ t,s p 4 M Np 0 iɛ

7 LO and NLO vertex function LO three-nucleon vertex function NLO correction to three-nucleon vertex function Resulting integral equations are solved numerically

8 LO form factor (arbitrary probe) NLO form factor (a) (b) (c) (a) (b) (c) (d) (e)

9 Form Factor Couplings Generic form factor can be expanded as ( F (Q ) = a r ) Q + 6 The couplings for the form factors of interest are given by Charge Magnetic Axial B LO e ˆN ˆN 0 ˆN (κ 0 + τ 3 κ )σ B ˆN g A ˆN σ i τ + ˆN + i B NLO ec 0t ˆt i ˆt i  0 e L ˆt j ŝ 3 B j e L iɛ ijk ˆt i ˆt j B k l,a ŝ ˆt i  i a Z µ GT Table: List of couplings for form factors of interest and their physical values at Q = 0.

10 The iso-scalar and iso-vector combination of magnetic moments are µ s = (µ3 He + µ3 H), µ v = (µ3 He µ3 H), µ s only depends on κ 0 and L, while µ v only depends on κ and L at NLO. µ s µ v L fit LO 0.440(5) -.3(78) N/A NLO 0.4(50) -.0(6) σ np NLO 0.4(50) -.56(3) µ3 H NLO 0.4(50) -.50(30) σ np and µ3 H Exp N/A Table: Table of three-nucleon iso-scalar and iso-vector magnetic moments compared to experiment. The different NLO rows are different fits for L. Magnetic moments and polarizabilities also calculated to NLO by (Kirscher et al. (07)) arxiv:

11 Bound State Observables for 3N Systems (Vanasse (06)+(07)). arxiv: arxiv: Observable LO NLO NNLO Exp. 3 H: r C [fm].4(9).59(8).6(3).5978(40) 3 He: r C [fm].6().7(8).74(3).7753(54) 3 H: r m [fm].40(4).78().840(8) 3 He: r m [fm].49(6).85().965(53) 3 H: µ m [µ N ].75(9).9(35).98 3 He: µ m [µ N ] -.87(73) -.08(5) -.3 Calculation of LO triton charge radius in unitary limit gives me 3B r c = Using analytical techniques in (Braaten and Hammer (006)) cond-mat/04047 it can be shown that me 3B r c = ( + s 0 )/9 = in the unitary limit.

12 Tritium β-decay Half life t / of tritium given by ( + δ R )f V K/G V The Gamow-Teller matrix element is t / = F + f A /f V ga GT GT Exp 3 = 0.955, GT 0 3 = , GT 0+ 3 = Fitting L A to the GT-matrix element gives L A = 3.46 ±.9 fm 3 Compares well to lattice prediction L A = 3.9(0.)(.0)(0.3)(0.9) fm 3

13 Wigner-symmetry Wigner-limit: a 0 = a and r 0 = r Wigner-breaking O(δ): δ = /a /a 0 /a +/a 0 The GT-matrix element is given by GT 3(P S + P D /3 P S /3) In Wigner-SU(4) limit P S = 0 and P S = hence GT = 3 Wigner O(δ) LO EFT π..08/.9 O(r).66.58/.70 Experiment.5978(40)/.775(5) Table: 3 H/ 3 He charge radius in Wigner-limit (Vanasse and Phillips (06)) arxiv: In Wigner-limit it can be shown both analytically and numerically µ ( 3 H) = µ p µ ( 3 He) = µ n

14 Conclusions and Future directions Charge radii of 3 H and 3 He reproduced well at NNLO in EFT π. Magnetic moments and radii reproduced within errors at NLO in EFT π. Better prediction for L A will further constrain EFT π prediction for pp fusion. Slight inconsistencies with how LECs are fit in two and three-body, differences likely higher order. Wigner-symmetry gives good expansion for charge radii and is interesting limit for three-nucleon magnetic moments and GT-matrix element. Results should be used as benchmark. Reproduce analytical results in unitary limit for charge radii. Should be used as benchmark for all such calculations. An N 3 LO calculation will result in observables with roughly % uncertainty.

15 Quartet Channel (nd Scattering) 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.

16 Projecting spin and isospin in the quartet channel and projecting out in angular momentum gives t0,q(k, l p) = y t M N pk where + π Λ 0 Q l ( p + k M N E iɛ dqq t l 0,q(k, q) pk Q l ( p + q M N E iɛ pq Q l (a) = 3 q γ t ), dx P l(x) x + a. ) 4 M qp NE iɛ

17 Higher Orders NLO correction is NNLO corrections are Note the second diagram contains full off-shell scattering amplitude.

18 Partial Resummation Technique Denoting tnlo l = tl 0,q + tl,q, for the partial resummation technique one finds (Bedaque, Rupak, Grießhammer, and Hammer (003)) tnlo l (k, p) = Bl 0(k, p)+b(k, l p)+(k0(q, l p, E)+K(q, l p, E)) tnlo l (k, q), with the diagrammatic representation

19 New Full Perturbative technique Picking out only NLO pieces gives (Vanasse (03)) t l,q(k, p) = B l (k, p)+k l (q, p, E) t l 0,q(k, q)+k l 0(q, p, E) t l,q(k, q). Terms are reshuffled to inhomogeneous term. Kernel at each order is the same. Diagrammatically NLO correction is now given by NLO LO NLO Note all corrections are half off-shell.

20 Doublet Channel nd scattering At LO in the doublet channel, nd scattering is given by a coupled set of integral equations T S T S S T

21 Two-Body Parity Violation The LO Lagrangian in EFT π has five LECs (Girlanda (008)) [ ( L = g (3 S P ) t i N t σ τ i ) i N ( + g ( S 0 3 P 0 ) ( I =0) s a N T σ σ τ τ a i ) N ( + g ( S 0 3 P 0 ) ( I =) ɛ 3ab (s a ) N T σ σ τ τ b N ( + g ( S 0 3 P 0 ) ( I =) I ab (s a ) N T σ σ τ τ b i N ( )] + g (3 S 3 P ) ɛ ijk (t i ) N T σ σ k j τ τ 3 N + h.c., where I ab = diag(,, ) and a b = a( b) ( a)b. Contains all possible S P transition operators and isospin structures ) )

22 LO Nd Scattering LO Nd scattering given by sum of diagrams PC PC PC PC PC PC PC PC PC PC PC PC Diagrams with lower two-body vertex not shown

23 LO Nd Scattering (Alternative) Sum of diagrams can also be represented via integral equation PC PC PC PC Makes asymptotic analysis easier

24 The amplitude can be projected in partial waves of J = L + S t JM L S,LS(k, p) K(k, p) JM L S,LS t JM L S,LS(k, p) 0 ) ( dqq K(q, p) JM L S,LSD (E q, M ) l t JM PC LS,LS(k, q) N t JM L S,LS(k, p) 0 K(q, l) JM L S,LSD ( ) ( T dqq dll JM t PC L S,L S (p, l) D E 0 ) ( ) (E q, q t JM PC LS,LS(k, q) M N ) l, M l N

25 One term of projected K(k, p) JM L S,LS is given by [ ] K(k, p) J L S,LS where ) = y t (3g S 0 3 P 0 ( I =0) g S 0 3 P 0 ( I =) { L C 0,0,0 L,,L L S J S 4π 6( ) / L J δ S/ δ S / L } kp (kq L (a) + pq L(a)) and x = x + a = k + p M N E iɛ kp All Projections given in (Vanasse (0)). Agree with S-wave to P-wave projections in (Grießhammer, Schindler, and Springer (0)).

26 NLO 3-Body NLO amplitude is given by type of diagrams below. NLO box is the half off shell NLO amplitude. NLO LO NLO LO NLO LO LO LO LO NLO LO LO (Note not all diagrams given here) As shown by (Schindler and Grießhammer (00)) no NLO three-body force for Nd scattering should exist.

27 Using Fierz rearrangements only single derivative 3B forces in S / P / are (Grießhammer and Schindler (00)) im [ S ] Wigner ( P, p, q = A 3NI H 3NI tree-level diagrams are given by im [ S where ] ( Wigner P, p, q =A (a) NI NI ( I =0) + τ 3 H 0 0 S + T S T + A (b) NI ( I =) ) ) ( ( 0 S + T 0 S T S = 3g (3 S P ) + τ 3 g (3 S 3 P ), T = 3g ( S 0 3 P 0 ) ( I =0) + τ 3 g ( S 0 3 P 0 ) ( I =) PC scattering amplitudes are diagonal in Wigner basis, therefore NLO diagrams do not contain element in upper left of Wigner basis matrix. Hence, no NLO 3B force ) )

28 Asymptotic Behavior (Bedaque Numbers) Going to Wigner basis asymptotic form of nd scattering integral equation is t (l) 8λ ( λ (p) = ( ) l dq p 3π 0 q Q l q + q ) t (l) p λ (q) (λ = ): Wigner-symmetric combination (λ = /): Wigner antisymmetric combination Equation is scaleless and must have solution of form (Grießhammer 005) t (l) λ (p) = p sλ l partial wave l s l (λ = ) s l (λ = ) i

29 NLO Nd Scattering The NLO amplitude can be calculated with S S P S S P S P P S P P PC PC PC PC NLO 3BF PC PC PC PC Note, three-body force is PC

30 Asymptotic behavior of NLO S / P / scattering 6π { ( ( Λ s [(ρ t + ρ s s) CD P / Λ sin s 0 ln 0 + (s ) ( ( +B P / H P / Λ sin s 0 ln Λ ( +(ρ t ρ s)ce P / sin s 0 ln ) + arctan ( Λ Λ ( s0 ) + arctan Λ s + 4HNLO(Λ) 3π Λ CD P / + s 0 ( s ) Λ3 s sin where H NLO(Λ) is NLO PC 3B force H NLO(Λ) = Λ 3π( + s 0 ) (ρ t+ρ s) 8 ( s0 ) ( s0 + arctan ) + Arg(H P / ) s ( s 0 ln ))] ( ( sin s +4s 0 ln ( Λ 0 s )} )) ( ) ) Λ arctan(s 0) + b, Λ Λ ) + arctan ( )) ) s 0 sin ( s 0 ln ( Λ Λ ) arctan(s0) ) +

31 g g 3 S P 000 g,0 (k, k, E ) Λ s /t NLO Λ[MeV]

32 g g 3 S 3 P,0 (k, k, E ) Λ s /t NLO g Λ[MeV]

33 3 T g S 0 3 P 0 ( I =0) + 3 τ 3g S 0 3 P 0 ( I =) T,0 (k, k, E ) Λ s /t NLO Λ[MeV]

34 Asymptotic behavior of NLO S / 4 P / scattering 6π ( ( Λ s [(ρ t + ρ s s)cd 4 P / Λ sin s 0 ln 0 + (s ) ( ( +(ρ t ρ s)ce 4 P / Λ sin s 0 ln ( +4ρ tb 4 P / H 4 P / sin s 0 ln Λ ( Λ Λ ) + arctan ) + arctan ) ( s0 + arctan Λ ( s0 ( s0 + 4HNLO(Λ) 3π Λ CD 4 P / + s 0 ( s ) Λ3 s sin )) s )) s ) )] + Arg(H 4 P / ) s ( s 0 ln ( ) ) Λ arctan(s 0) + b Λ

35 g g 3 S P g 0 3,0 (k, k, E ) t NLO Λ[MeV]

36 g g 3 S 3 P 500 g 3,0 (k, k, E ) Λ s /t NLO Λ[MeV]

37 3 T g S 0 3 P 0 ( I =0) + 3 τ 3g S 0 3 P 0 ( I =) T 3,0 (k, k, E ) Λ s /t NLO Λ[MeV]

38 Large-N C Convenient to rescale LECs by dibaryon nucleon-nucleon coupling y g = g 3 S P y, g = g 3 S 3 P y, g 3 = g S 0 3 P 0 ( I =0) y, g 4 = g S 0 3 P 0 ( I =) y, g 5 = g S 0 3P 0 ( I =), y Large-N C basis of LECs (Schindler, Springer, and Vanasse (05)) (Gardner, Haxton, and Holstein (07)) g (N C ) = 4 g g 3, g (N C ) = g 5 LO g (N C ) 3 = 4 g 3 4 g 3, g (N C ) 4 = g, g (N C ) 5 = g 4 N LO,

39 pd Scattering Parity violating asymmetry in pd scattering in large-n C basis A pd L (E lab = 5 MeV) = [ 943g (N C ) 636g (N C ) g (N C ) g (N C ) 5 ]MeV Dominant contribution from only LO large-n C LEC. Experiment (Nagle et. al (979)) gives bound Gives bound A pd L (E lab = 5 MeV) = ( 3.5 ± 8.5) 0 8 g (N C ) = (3.7 ± 9.0) 0 MeV

40 pp scattering Asymmetry in pp scattering in large-n C basis (Phillips, Schindler, and Springer (009)) A pp L (E lab = 3.6 MeV) = [603g (N C ) +904g (N C ) 603g (N C ) g (N C ) 5 ]MeV Experiment (Eversheim et. al (99)) gives A pp L (E lab = 3.6 MeV) = ( 0.93 ± 0.) 0 7 Dropping subleading LECs in large-n C basis and using bound on g (N C ) gives g (N C ) = (.3 ± 0.83) 0 0 MeV Consistent with experimental bound (Knyaz kov (984)) and theoretical EFT π prediction (Schindler and Springer (00)) for P γ (np d γ)

41 np and nd spin rotation np spin rotation to NLO in the large-n C basis gives (Grießhammer, Schindler, and Springer (0)) dφ np dz = [ ] 6.0g (N C ) + 67g (N C ) + 38g (N C ) 3 + 6g (N C ) 4 rad cm Using prediction for g (N C ) gives dφ np rad cm dz LO nd spin rotation in large-n C basis is dφ nd dz = [ ] 50g (N C ) 7g (N C ) 3 38g (N C ) 4 + g (N C ) 5 rad cm

42 Conclusions and Future Directions Any observable of interest can be calculated for nd scattering at LO in EFT π. three-body force needed at NLO New perturbative technique can be used with external currents making calculations of in nd 3 H + γ and pd 3 He + γ feasible. Need to add Coulomb to investigate pd at lower energies