Tritium β decay in pionless EFT

Ohio University June 0, 207

Recipe for EFT(/π) For momenta p < m π pions can be integrated out as degrees of freedom and only nucleons and external currents are left. Write down all possible terms of nucleons and external currents that respect symmetries (rotational, isospin). Develop a power counting to organize terms by their relative importance. Organized by counting powers of momentum. Ensure order-by-order results are renormalization group invariant (converge to finite values for Λ ). Check that various sets of observables converge as expected. Calculate respective observables up to a given order in the power counting.

Two- and Three-Body Inputs of EFT(/π) Two-body inputs for EFT(/π): LO scattering lengths in a ( S ) and a 0 ( S 0 ) non-perturbative NLO range corrections r and r 0 perturbative N 2 LO SD-mixing term 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/990602 NNLO three-body energy dependent three-body force H 2 fit to triton binding energy perturbative Total of 6 NNLO parameters, ignoring SD-mixing.

The LO dressed deuteron propagator is given by a bubble sum c (0) 0t (LO) c () 0t S (S matrix) (NLO) Im[p] (N 2 LO) iγ t (γ t 45MeV) m π Re[p] (Z-parametrization) At LO coefficients are fit to reproduce the deuteron pole and at NLO to reproduce the residue about the deuteron pole (Phillips et al. (2000)) nucl-th/9908054.

LO Triton Vertex function Triton vertex function given by infinite sum of diagrams at LO Infinite sum is represented by integral equation that is solved numerically

NLO Triton Vertex function NLO triton vertex function given by sum of diagrams where Can also be solved by set of integral equations

LO Three-Nucleon Propagator Defining Σ 0. The dressed three-nucleon propagator is given by the sum of diagrams Σ 0 Σ 0 Σ 0 which yields i (E) = i Ω i Ω H LOΣ 0 (E) i Ω + = i Ω H LO Σ 0 (E),

Higher-Order Three-Nucleon Propagator Defining the functions Σ Σ 2 2 2 The NNLO three-nucleon propagator is Σ H NLO Σ 0 (NLO) Σ 2 H NLO Σ H NNLO Σ 0 Σ Σ 2H NLO Σ 0 Σ (H NLO ) 2 Σ 0 Σ 0 h 2 (NNLO)

Properly Renormalized Vertex Function Three-body forces are fit to ensure triton propagator has pole at triton binding energy. Three-nucleon wavefunction renormalization given by the residue of the three-nucleon propagator about the pole. LO three-nucleon wavefunction renormalization is Z LO ψ = π Σ 0 (B). NNLO three-body force h 2 fit to triton binding energy and NNLO correction to H 0 fit to doublet S-wave nd scattering length. Total of 2 three-body inputs at NNLO. Σ 0

Three-Nucleon generic Form Factor Three-nucleon LO generic form factor given by the three diagrams NLO generic form factor (a) (b) (c) (a) (b) (c) (d) (e)

Form Factor Couplings Generic form factor can be expanded as ( F (Q 2 ) = a r 2 ) Q 2 + 6 The couplings for the form factors of interest are given by Charge Magnetic Axial B LO e ˆN ˆNÂ 0 ˆN (κ 0 + τ κ )σ B ˆN 2 g A ˆN σ i τ + ˆNÂ + i 2B NLO ec 0t ˆt i ˆt i Â 0 e L 2 ˆt j ŝ B j e L 2 2 iɛ ijk ˆt i ˆt j B k l,a ŝ ˆt i Â i a Z µ GT Table: List of couplings for form factors of interest and their physical values at Q 2 = 0.

LO Form Factor The LO form factor for Q 2 = 0 is ) T π F 0 (0) = 2πM N ( Γ0 (q) 2 ( + q 2 l 2 (q 2 + l 2 M N B) 2 δ(q l) q 2 4 q2 M N B ( ) c + a c 2 c 2 c 22 + a 22 b 2a b 2 + (a + a 22 ) b 2 + (a + a 22 ) b 22 2a 22 The coefficients for various form factors are given by )} Γ 0 (l), Form factor a a 22 b b 2 b 2 b 22 c c 2 c 2 c 22 F H C (Q 2 2 ) 0-0 0 F He C (Q 2 ) 0 2 2-4 5 0 0 F GT W (Q2 ) - - 5 5 2 0 2 0 F F W (Q2 ) - - 5 0 0 0 4 Table: Values of coefficients for the LO H and He axial and charge form factors.

NLO Form Factor NLO correction to generic form factor at Q 2 = 0 ) T π F (0) = 2πM N ( Γ (q) 2 δ(q l) q 2 4 q2 M N B 0 ( c + a c 2 c 2 c 22 + a 22 ( )} b + 2a b 2 + (a + a 22 ) q 2 l 2 (q 2 + l 2 M N B 0 ) 2 b 2 + (a + a 22 ) b 22 2a Γ 0 (l) 22 ) T ( ) π δ(q l) c + a + 2πM N ( Γ0 (q) c 2 2 q 2 c 4 q2 M N B 2 c 22 + a 22 0 ( )} b + 2a b 2 + (a + a 22 ) q 2 l 2 (q 2 + l 2 M N B 0 ) 2 b 2 + (a + a 22 ) b 22 2a Γ (l) 22 ) { ( T π δ(q l) 4πM N ( Γ0 (q) 2 q 2 2 ρ )} ta + d d 2 d 2 2 ρ sa 22 + d Γ 0 (l), 22 )

NLO Form Factor (cont.) Form factor d d 2 d 2 d 22 F H C (Q 2 ) 2 ρ t 0 0 F He C (Q2 ) 2 ρ 5 t 0 0 F H M (Q 2 ) 2 L 2 L L 0 F He M (Q2 ) 2 L 2 L L 0 FW GT (Q2 ) 0 l,a l,a 0 Table: Values of coefficients for the NLO corrections to (d)-type diagrams for the H and He magnetic, charge, and axial form factors. 2 ρ s 2 ρ s

Radii Generic form factor given by ( F (Q 2 ) = a r 2 ) Q 2 + 6 Calculating only Q 2 contribution for diagram-(a) gives 2 2 Q 2 F n (a) (Q 2 ) = Z Q 2 ψ LO =0 i+j n i,j=0 { GT i (p) A n i j (p, k) G j (k) +2 G T i (p) A n i (p)δ j0 + A n δ i0 δ j0 },

Triton Charge Radius LO EFT(/π) r C = 2. ±.6 fm (Platter and Hammer (2005)) nucl-th/0509045 NLO EFT(/π) r C =.6 ±.2 fm (Kirscher et al. (200)) arxiv:090.558 NNLO EFT(/π) r C =.62 ±.0 fm (Vanasse (206)) arxiv:52.0805 Charge Radius δr C [fm] 2.8.6.4.2 LO NLO NNLO Exp-.5978 0.8 0 0 4 0 5 0 6 Cutoff [MeV]

The iso-scalar and iso-vector combination of magnetic moments are µ s = 2 (µ He + µ H), µ v = 2 (µ He µ H), µ s only depends on κ 0 and L 2, while µ v only depends on κ and L at NLO. µ s µ v L fit LO 0.440(52) -2.(78) N/A NLO 0.42(50) -2.20(26) σ np NLO 0.42(50) -2.56() µ H NLO 0.42(50) -2.50(0) σ np and µ H Exp 0.426-2.55 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 and are organized the same as the previous table.

NLO two-body magnetic currents given by ( L mag 2 = e L ) 2 ˆt j ŝ B j + H.c e L 2 2 iɛijk ˆt i ˆt j B k. L 2 is fit to deuteron magnetic moment and L is typically fit to cold np capture cross section (σ np ) Magnetic moments and polarizabilities also calculated to NLO by (Kirscher et al. (207)) arxiv:702.07268 µ H µ He r H M fm r He M fm σ np mb LO 2.75(95) -.87(7).40(24).49(26) 25.2 ± 225.6 NLO 2.62() -.78().8().92() 4.2 ± 79.7 NLO 2.98(6) -2.4(26).77().8() 70.47 ± 88.4 NLO 2.92(5) -2.08(25).78().85() 64.5 ± 87.0 Exp 2.979-2.27.84(8).97(5) 4.2(5) Table: Values of magnetic moments and magnet radii for three-nucleon systems and σ np to NLO compared to experiment. The first NLO row is for L fit to σ np, the second NLO row for L fit to the H magnetic moment (µ H), and the final NLO row is L fit to both σ np and µ H.

Bound State Observables for N Systems (Vanasse (206)+(207)). arxiv:52.0805 + arxiv:706.02665 Observable LO NLO NNLO Exp. H: r C [fm].4(9).59(8).62().5978(40) He: r C [fm].26(2).72(8).74().775(54) H: r m [fm].40(24).78().840(8) He: r m [fm].49(26).85().965(5) H: µ m [µ N ] 2.75(92) 2.92(5) 2.98 He: µ m [µ N ] -.87(7) -2.08(25) -2. Calculation of LO triton charge radius in unitary limit gives me B r 2 c = 0.224... Using analytical techniques in (Braaten and Hammer (2006)) cond-mat/04047 it can be shown that me B r 2 c = ( + s 2 0 )/9 = 0.224... in the unitary limit.

Tritium β-decay Half life t /2 of tritium given by ( + δ R )f V K/G 2 V The Gamow-Teller matrix element is t /2 = F 2 + f A /f V ga 2 GT 2 GT Exp = 0.955, GT 0 = 0.9807, GT 0+ = 0.995 Fitting L A to the GT-matrix element gives L A =.46 ±.9 fm Compares well to lattice prediction L A =.9(0.)(.0)(0.)(0.9) fm

Gamow-Teller and Wigner-symmetry The GT-matrix element is given by GT (P S + P D / P S /) In Wigner-SU(4) limit P S = 0 and P S = hence GT = and can also be seen by GT = He σ (i) τ (i) + H i = He σ (i) He i = He σ He =

Fermi matrix element Half life t /2 of tritium given by ( + δ R )f V K/G 2 V t /2 = F 2 + f A /f V ga 2 GT 2 In the isospin limit the Fermi matrix element reduces to Indeed, we find F =. F = He τ (i) + H i = He He =

Consequences of Wigner-symmetry and Unitarity Wigner-limit: a 0 = a and r 0 = r Unitary limit: a 0 = a = Wigner-breaking O(δ): δ = /a /a 0 /a +/a 0 Wigner-breaking all orders: Unitary Wigner O(δ) δ all orders LO EFT(/π).0.22.08/.9.4/.26 O(r).42.66.58/.70.59/.72 Experiment.5978(40)/.775(5) Table: H/ He charge radius in unitary and Wigner-limit (Vanasse and Phillips (206)) arxiv:607.08585 In Wigner-limit it can be shown both analytically and numerically µ ( H) = µ p = 2.79 e 2M N, µ ( He) = µ n =.9 e 2M N

Conclusions and Future directions Charge radii of H and He reproduced well at NNLO in EFT(/π). Magnetic moments and radii reproduced within errors at NLO in EFT(/π). L A prediction agrees with LQCD prediction. Better prediction for L A will further constrain EFT(/π) prediction for pp fusion. 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.