A

# Charge and Magnetic Properties of Three-Nucleon Systems in Pionless Effective Field Theory

## Abstract

A method to calculate the form factor for an external current with non-derivative coupling for the three-body system in an effective field theory (EFT) of short-range interactions is shown. Using this method the point charge radius of is calculated to next-to-next-to-leading order () in pionless EFT (), and the magnetic moment and magnetic radius of and are calculated to next-to-leading order (NLO). For the charge and magnetic form factors Coulomb interactions are ignored. The point charge radius is given by 1.74(4) fm at . This agrees well with the experimental point charge radius of 1.7753(54) fm Angeli and Marinova (2013). The () magnetic moment in units of nuclear magnetons is found to be 2.92(35) (-2.08(25)) at NLO in agreement with the experimental value of 2.979 (-2.127). For () the NLO magnetic radius is 1.78(11) fm (1.85(11) fm) which agrees with the experimental value of 1.840(182) fm (1.965(154) fm) Sick (2001). The fitting of the low-energy constant of the isovector two-body magnetic current and the consequences of Wigner-SU(4) symmetry for the three-nucleon magnetic moments are also discussed.

latex-community, revtex4, aps, papers

## I Introduction

When systems are probed at length scales much larger than the scale of their underlying interaction then those interactions can be expanded in a series of contact interactions known as short range effective field theory (srEFT). Systems with short range interactions (i.e. cold atom systems, halo nuclei, and low energy few-nucleon systems) exhibit such behavior at low energies. The applicability of srEFT to such a broad class of systems is known as universality Braaten and Hammer (2006). Importantly, srEFT possesses a power counting that allows for systematically improvable calculations with error estimates. The power counting is in powers of , where is the typical momentum scale of particles in the system, is the breakdown scale of srEFT, and using naive dimensional analysis van Kolck (1999) low energy constants (LECs) in the theory are assumed to scale dimensionally in powers of . However, for physical systems of interest it is observed that the scattering length scales unnaturally (). This leads to interactions in being treated non-perturbatively at leading order (LO) and the creation of relatively shallow two-body bound states Kaplan et al. (1998a, b). Higher order range corrections are then added perturbatively in a series of .

srEFT has been used successfully in the description of low-energy few-nucleon systems through the use of pionless EFT (), characterized by the breakdown scale and valid for energies . has been used in the two-body sector to calculate nucleon-nucleon () scattering Chen et al. (1999); Kong and Ravndal (1999, 2000); Ando et al. (2007), neutron-proton () capture Chen et al. (1999); Chen and Savage (1999); Ando and Hyun (2005) to (Rupak (2000), deuteron electromagnetic properties Chen and Savage (1999); Ando and Hyun (2005), proton-proton fusion Kong and Ravndal (2001); Ando et al. (2008); Chen et al. (2013), and neutrino-deuteron scattering Butler et al. (2001). In the three-body sector it has been used to calculate neutron-deuteron () scattering  Bedaque et al. (1998, 2000); Gabbiani et al. (2000); Bedaque et al. (2003); Grießhammer (2004); Vanasse (2013); Margaryan et al. (2016), proton-deuteron () scattering Rupak and Kong (2003); König and Hammer (2011, 2014); Vanasse et al. (2014); König et al. (2015); König (2016), and binding energies Bedaque et al. (2000); Ando and Birse (2010); König and Hammer (2011); König et al. (2016), three-nucleon electromagnetic Platter and Hammer (2006); Kirscher et al. (2017) and weak properties De-Leon et al. (2016), and capture Sadeghi et al. (2006); Arani et al. (2014).

Techniques to calculate scattering strictly perturbatively were introduced in Ref. Vanasse (2013). Ref. Vanasse (2017a) then extended this method to the calculation of perturbative corrections to three-body bound states. Using these methods, Ref. Vanasse (2017a) calculated the triton point charge radius to next-to-next-to leading-order () finding good agreement with experiment. This paper builds upon this work by considering the electric and magnetic properties of three-nucleon systems in the absence of Coulomb interactions. In fact the calculation of the general three-nucleon form factor, resulting moments (value at ), and radii for any external current with non-derivative coupling is considered in this work. This is possible since the form factors for such currents depend on the same integrals but with different constants in front of them.

In the charge form factor up to can be predicted using four two-body LECs and two three-body LECs encoding interactions between nuclei. The two-body LECs in this work are fit to the and poles for scattering and their associated residues, while the three-body LECs are fit to the triton binding energy and the doublet -wave scattering length. In this work Coulomb interactions and isospin breaking from strong interactions are ignored for , therefore next-to leading order (NLO) and Coulomb and isospin breaking corrections to the three-body force can be ignored Vanasse et al. (2014). The three-nucleon magnetic form factor to NLO requires the same LECs as the charge form factor with the exception of the energy dependent three-body force. In addition the NLO magnetic form factor will require an isoscalar and isovector two-body magnetic current.

The three-nucleon charge form factors are reproduced well using potential model calculations (PMCs) Schiavilla et al. (1990); Marcucci et al. (1998), whereas the magnetic form factor of is reasonably reproduced, but the magnetic form factor poorly describes the first observed diffraction minimum from experiment. Chiral EFT (EFT) Piarulli et al. (2013) reproduces the three-nucleon charge and magnetic form factors well for . The resulting charge radii, magnetic moments, and magnetic radii from PMCs and EFT agree reasonably well with experimental data.1 is only valid for momentum transfers of and thus cannot directly address the issues observed in PMCs and EFT for larger values. However, can garner insight into the importance of two- and three-body currents.

As shown in Ref. Vanasse and Phillips (2017), going to the Wigner-SU(4) symmetric limit in which the scattering lengths and effective ranges for the and channels are set equal reproduces properties (e.g. bound state energy and charge radii) of the three-nucleon systems well within expected errors. It was also shown that a dual perturbative expansion in and powers of a Wigner-SU(4) symmetry breaking parameter led to good convergence with experimental data for three-nucleon systems. Expanding on this, the values of the three-nucleon magnetic moments in the Wigner-SU(4) symmetric limit are calculated in this work. At LO in this limit the Schmidt-limit Schmidt (1937) is reproduced in which the magnetic moment of the three nucleon system is given by the magnetic moment of the unpaired nucleon. It is also demonstrated in the Wigner-SU(4) limit that the expressions for the NLO magnetic moments can be written entirely in terms of LO three-nucleon vertex functions.

This paper is organized as follows. Section II gives the Lagrangian and all necessary two-body physics, while Sec. III reviews relevant properties of the three-body system. In Sec. IV properties of the charge and magnetic form factor in are derived, and the consequences of Wigner-symmetry on the form factors discussed. Finally, in Sec. V results are given and conclusions are given in Sec. VI.

## Ii Lagrangian and Two-Body System

The two-body Lagrangian is

 L2= ^N†⎛⎝iD0+→D22MN⎞⎠^N+^t†i⎡⎣Δt−c0t⎛⎝iD0+→D24MN+γ2tMN⎞⎠⎤⎦^ti (1) +^s†a⎡⎣Δs−c0s⎛⎝iD0+→D24MN+γ2sMN⎞⎠⎤⎦^sa +yt[^t†i^NTPi^N+H.c.]+ys[^s†a^NT¯Pa^N+H.c.],

where () is the spin-triplet (spin-singlet) dibaryon field. Parameter () sets the interaction strength between the spin-triplet (spin-singlet) dibaryon and nucleons, while () projects out the spin-triplet iso-singlet (spin-singlet iso-triplet) combination of nucleons. The covariant derivative is defined by

 Dμ=∂μ+iQ^Aμ, (2)

where is the photon field, and is the charge operator given by , , and for the fields , , and respectively.2 is the bare spin-triplet dibaryon propagator which at LO is dressed by an infinite series of nucleon bubble diagrams as shown in Fig. 1. This series, a geometric series, yields the LO spin-triplet dibaryon propagator, which receives range corrections from at NLO and as shown in Fig. 1. The resulting parameters of the spin-triplet dibaryon propagator are then fit to give the deuteron pole at LO and its residue at higher orders. The same procedure can be carried out for the spin-singlet dibaryon propagator with parameters fit to the virtual bound state pole at LO and to its residue at NLO. This fitting procedure is known as the -parametrization Phillips et al. (2000); Grießhammer (2004) and has the advantage of giving the correct residue about the poles in the and channels at NLO instead of being approached perturbatively as in the effective range expansion (ERE) parametrization.

Using the -parametrization gives the coefficients Grießhammer (2004)

 y2t=4πMN,Δt=γt−μ,c(n)0t=(−1)n(Zt−1)n+1MN2γt, (3) y2s=4πMN,Δs=γs−μ,c(n)0s=(−1)n(Zs−1)n+1MN2γs,

where  MeV is the deuteron binding momentum, is the residue about the deuteron pole,  MeV is the virtual bound-state momentum, and is the residue about the pole de Swart et al. (1995). The scale comes from using dimensional regularization with the power-divergence subtraction scheme Kaplan et al. (1998a, b), and all physical observables do not depend on . Parameter () is split up into contributions () at each order to ensure the pole position is fixed and has the correct residue. The resulting spin-triplet (spin-singlet) dibaryon in the -parametrization up to is given by

 iDNNLO{t,s}(p0,→p)=iγ{t,s}−√→p24−MNp0−iϵ (4) ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣\vrulewidth0.0ptheight0.0ptdepth14.226378pt1LO+Z{t,s}−12γ{t,s}⎛⎝γ{t,s}+√→p24−MNp0−iϵ⎞⎠NLO +(Z{t,s}−12γ{t,s})2(→p24−MNp0−γ2{t,s})NNLO+⋯⎤⎥ ⎥ ⎥ ⎥ ⎥⎦.

LO interactions between nucleons and the magnetic field at the one-body level are given by the Lagrangian

 Lmag1,0=e2MN^N†(κ0+κ1τ3)→σ⋅B^N, (5)

where is the isoscalar magnetic moment of the nucleon and is the isovector magnetic moment of the nucleon in nuclear magnetons. At NLO there are two two-body magnetic currents,  Chen et al. (1999); Beane and Savage (2001) and  Kaplan et al. (1999); Chen et al. (1999) given by the Lagrangian

 Lmag2=(eL12^tj†^s3Bj+H.c.)−eL22iϵijk^t†i^tjBk. (6)

In the three-body system there will be a LO three-body force Bedaque et al. (2000) with non-derivative coupling, which receives corrections at higher orders to avoid refitting. At a new energy dependent three-body force is required in  Bedaque et al. (2003). These three-body forces are easily represented by the introduction of an interaction between  Bedaque et al. (2003); Vanasse (2017a), dibaryons, and nucleons via the Lagrangian

 L3= ^ψ†[Ω−h2(Λ)(iD0+→D26MN+γ2tMN)]^ψ+∞∑n=0[ω(n)t0^ψ†σi^N^ti−ω(n)s0^ψ†τa^N^sa] (7) +H.c.,

where is a three-nucleon iso-doublet field containing and . The energy dependent three-body force term is given by

 ˆH2=−3(ω(0)t0)2πΩ2MNh2(Λ)=−3(ω(0)s0)2πΩ2MNh2(Λ)=−3ω(0)t0ω(0)s0πΩ2MNh2(Λ). (8)

For further details of three-body forces and how they are fit consult Ref. Vanasse (2017a).

## Iii Three-Body System

Detailed methods for calculating the three-nucleon vertex function can be found in Ref. Vanasse (2017a) and a brief review of them, in order that this work is relatively self contained, is given below. The LO three-nucleon vertex function is the solution of an integral equation represented by the diagrams of Fig. 2.

Double dashed lines are spin-singlet dibaryons and the triple lines three-nucleon fields. In cluster-configuration (c.c.) space Grießhammer (2004) the LO three-nucleon vertex function is given by the integral equation

 G0(E,p)=˜B0+K0(q,p,E)⊗G0(E,q), (9)

where is a c.c. space vector given by

 G0(E,p)=(G0,ψ→Nt(E,p)G0,ψ→Ns(E,p)), (10)

and the inhomogeneous term is a c.c. space vector given by

 ˜B0=(1−1). (11)

() is the three-nucleon vertex function for a three-nucleon system going to a nucleon and deuteron (nucleon and spin-singlet dibaryon). The kernel of Eq. (9) is a c.c. space matrix given by

 K0(q,p,E)=R0(q,p,E)D(0)(E−q22MN,→q), (12)

where

 R0(q,p,E)=−2πqpQ0(q2+p2−MNE−iϵqp)(1−3−31), (13)

matrix multiplies

 D(0)(E,→q)=(D(0)t(E,→q)00D(0)s(E,→q)), (14)

which is a matrix of LO dibaryon propagators. is a Legendre function of the second kind defined as

 Q0(a)=12ln(1+a1−a), (15)

and the “” notation is defined by

 A(q)⊗B(q)=12π2∫Λ0dqq2A(q)B(q).

The NLO and three-nucleon vertex functions are given by integral equations represented in Figs. 3 and 4 respectively.

In c.c. space the NLO three-nucleon vertex function is

 Extra open brace or missing close brace (16)

where is a c.c. space matrix defined by

 R1(p0,→p)=⎛⎜ ⎜ ⎜⎝Zt−12γt(γt+√14→p2−MNp0−iϵ)00Zs−12γs(γs+√14→p2−MNp0−iϵ)⎞⎟ ⎟ ⎟⎠. (17)

In c.c. space the three-nucleon vertex function is given by

 G2(E,p)=R1(E−→p22MN,→p)[G1(E,p)−c1G0(E,p)]+K0(q,p,E)⊗G2(E,q), (18)

where

 c1=(Zt−100Zs−1), (19)

is a c.c. space matrix.

To properly normalize the three-nucleon vertex function the three-nucleon wavefunction renormalization is needed, which is obtained by calculating the residue about the three-nucleon propagator pole. This pole is fixed to the triton binding energy ,  MeV Wapstra and Audi (1985), by appropriate tuning of three-body forces. Further details of how this is done can be seen in Ref. Vanasse (2017a). The resulting three-nucleon wavefunction renormalization up to and including is given by

 Missing \left or extra \right (20) Missing or unrecognized delimiter for \left

where the functions are defined by

 Σn(E)=−πTr[D(0)(E−q22MN,q)⊗Gn(E,q)], (21)

and is the energy dependent three-body force Vanasse (2017a); Bedaque et al. (2003) from Eq. (8). Taking the square root of and expanding, the properly renormalized LO three-nucleon vertex function is given by

 Γ0(p)=√ZLOψG0(B,p), (22)

the properly renormalized NLO correction to the three-nucleon vertex function by

 Γ1(p)=√ZLOψ[G1(B,p)−12Σ′1(B)Σ′0(B)G0(B,p)], (23)

and the properly renormalized correction to the three-nucleon vertex function by

 Γ2(p) =√ZLOψ[G2(B,p)−12Σ′1(B)Σ′0(B)G1(B,p) (24) +⎧⎨⎩38(Σ′1(B)Σ′0(B))2−12Σ′2(B)Σ′0(B)−23MNˆH2Σ20(B)Σ′0(B)⎫⎬⎭G0(B,p)⎤⎦,

where

 ZLOψ=πΣ′0(B). (25)

## Iv Charge and Magnetic Form Factors

### iv.1 Charge and Magnetic Moments

In Ref. Vanasse (2017a) the charge form factor of the triton was calculated to in . Calculating the charge form factor, magnetic form factor, and the magnetic form factor in the absence of Coulomb interactions is essentially the same calculation as the charge form factor. The only difference between these calculations are the coefficients that appear in front of the same integrals. Both charge and magnetic form factors at LO are given by the sum of diagrams in Fig. 5, where all photons are either minimally coupled photons or magnetically coupled from Eq. (5).

Form factors are calculated in the Breit frame in which the photon only imparts momentum but no energy on the three-nucleon system, and all form factors are only functions of . Using the work of Ref. Vanasse (2017a) the LO “generic” form factor in the limit is given by

 F0(0)=2πMN(˜Γ0(q))T⊗⎧⎪ ⎪⎨⎪ ⎪⎩π2δ(q−ℓ)q2√34q2−MNB(c11+a11c12c21c22+a22) (26) +1q2ℓ2−(q2+ℓ2−MNB)2(b11−2a11b12+3(a11+a22)b21+3(a11+a22)b22−2a22)}⊗˜Γ0(ℓ),

where the c.c. space vector function is

 ˜Γn(q)=D(0)(B−q22MN,→q)Γn(q), (27)

and . The coefficients and come from the c.c. space matrix of diagram Fig. 5(a), the coefficients , , , and from the c.c. space matrix of diagram Fig. 5(b), and the coefficients , , , and from the c.c. space matrix of diagram Fig. 5(c). The only difference between the LO magnetic and charge form factors for and are the values of these coefficients shown in Table 1 for each.

Further details of how these coefficients are obtained are given in Appendix A.

Choosing the coefficients for the triton charge form factor gives

 F0(0)=2πMN(˜Γ0(q))T⊗⎧⎪ ⎪⎨⎪ ⎪⎩π2δ(q−ℓ)q2√34q2−MNB(1001) (28) −1q2ℓ2−(q2+ℓ2−MNB)2(1−3−31)}⊗˜Γ0(ℓ).

This expression is the same as the normalization condition in Ref. König and Hammer (2011), and therefore it follows automatically that for the triton charge form factor. Plugging in the charge form factor coefficients gives two times Eq. (28), and hence for the charge form factor.3

The NLO correction to the charge and magnetic form factors is given by the diagrams in Fig. 6.

Diagram-(d) for charge form factors comes from gauging the dibaryon kinetic term and for the magnetic form factor comes from the and term of Eq. (6). Not shown in Fig. 6 are diagrams related by time reversal symmetry. Diagram-(e) in the dashed box is subtracted from the other diagrams to avoid double counting from diagram-(a) and its time reversed version. The NLO correction to the “generic” form factor in the limit is

 F1(0)=2πMN(˜Γ1(q))T⊗⎧⎪ ⎪⎨⎪ ⎪⎩π2δ(q−ℓ)q2√34q2−MNB0(c11+a11c12c21c22+a22) (29) +1q2ℓ2−(q2+ℓ2−MNB0)2(b11−2a11b12+3(a11+a22)b21+3(a11+a22)b22−2a22)}⊗˜Γ0(ℓ) +2πMN(˜Γ0(q))T⊗⎧⎪ ⎪⎨⎪ ⎪⎩π2δ(q−ℓ)q2√34q2−MNB0(c11+a11c12c21c22+a22) +1q2ℓ2−(q2+ℓ2−MNB0)2(b11−2a11b12+3(a11+a22)b21+3(a11+a22)b22−2a22)}⊗˜Γ1(ℓ) −4πMN(˜Γ0(q))T⊗⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩π2δ(q−ℓ)q2⎛⎜ ⎜ ⎜⎝c(0)0tMNa11+d11d12d21c(0)0sMNa22+d22⎞⎟ ⎟ ⎟⎠⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭⊗˜Γ0(ℓ),

where the coefficients ,,, and are from the NLO c.c. space matrix for diagram Fig. 6(d) and are shown in Table 2. Again the derivation of these coefficients is given in Appendix A. For the first two terms simply come from replacing by in Eq. (26). The last term of has NLO corrections from diagrams (a),(d), and (e) of Fig. 6. For the three-nucleon charge form factor as a consequence of gauge symmetry.

### iv.2 Charge and Magnetic Radius

In general the form factor can be expanded in powers of yielding

 FAZX(Q2)=fAZX(1−16⟨δr2X⟩AZQ2+⋯), (30)

where () for the charge (magnetic) form factor, and or . () is the charge (magnetic moment) of the three-nucleon system, and () is the point charge (magnetic) radius of the three-nucleon system. Higher order terms in are not considered in this work, because for values of for which is valid form factors are dominated by the constant and pieces. Methods for calculating the form factor with all powers of can be seen in Refs. Vanasse (2017a); Hagen et al. (2013).

The coefficient of the contribution to the “generic” form factor to any order up to from type (a) diagrams is given by

 12∂2∂Q2F(a)n(Q2)∣∣Q2=0=ZLOψi+j≤n∑i,j=0{˜GTi(p)⊗An−i−j(p,k)⊗˜Gj(k) (31) +2˜GTi(p)⊗An−i(p)δj0+Anδi0δj0},

where the subscripts denote the order of the term in . is a c.c. space matrix, is a c.c. space vector, and is a c.c. space scalar. The detailed form of these functions is given in Appendix B and they all depend on the coefficients and . Note that the NLO diagram-(e) of Fig. 6 is absorbed into the NLO expression for diagram-(a) Vanasse (2017a). The c.c. space vector is defined by

 ˜Gn(p)=D(0)(B−p22MN,→p)Gn(B,p). (32)

Type-(b) diagrams to any order up to give a contribution of

 12∂2∂Q2F(b)n(Q2)∣∣Q2=0=ZLOψn∑i=0˜GTi(p)⊗B0(p,k)⊗˜Gn−i(k), (33)

where is a c.c. space matrix given in Appendix B. Functions for do not exist. The contribution from type-(c) diagrams to any order up to gives

 12∂2∂Q2F(c)n(Q2)∣∣Q2=0=ZLOψi+j≤n∑i,j=0{˜GTi(p)⊗Cn−i−j(p,k)⊗˜Gj(k)+Cn−i(k)⊗˜Gi(k)δj0}, (34)

where is c.c. space matrix and is a c.c. space vector both given in Appendix B. Finally, the contribution from type-(d) diagrams to any order up to gives

 12∂2∂Q2F(d)n(Q2)∣∣Q2=0=ZLOψi+j≤n−1∑i,j=0{˜GTi(p)⊗Dn−i−j(p,k)⊗˜Gj(k)+Dn−i(k)⊗˜Gi(k)δj0}, (35)

with a c.c. space matrix and a c.c. space vector both given in Appendix B.

Summing the contribution from all LO diagrams the part of the “generic” LO form factor is given by

 12∂2∂Q2F0(Q2)∣∣Q2=0=12∂2∂Q2(F(a)0(Q2)+F(b)0(Q2)+F(c)0(Q2))∣∣Q2=0. (36)

The NLO correction to the part of the “generic” form factor is

 12∂2∂Q2F1(Q2)∣∣Q2=0=12∂2∂Q2(F(a)1(Q2)+F(b)1(Q2)+F(c)1(Q2)+F(d)1(Q2))∣∣Q2=0 (37) −Σ′1(B)Σ′0(B)12∂2∂Q2F0(Q2)∣∣Q2=0,

where the NLO diagrams are summed together and the LO contribution is multiplied by the NLO three-nucleon wavefunction renormalization. Finally, including all contributions and multiplying the NLO term by the NLO three-nucleon wavefunction renormalization and the LO contribution by the three-nucleon wavefunction renormalization gives

 12∂2