The asymptotic D-state to S-state ratio of triton

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— journal of November 2013

physics pp. 799–806

The asymptotic D-state to S-state ratio of triton

HOSSEIN SADEGHI^∗, REZA POURIMANI and HASSAN KHALILI

Department of Physics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

∗Corresponding author. E-mail: H-Sadeghi@Araku.ac.ir

MS received 4 April 2013; revised 14 August 2013; accepted 29 August 2013 DOI: 10.1007/s12043-013-0619-z; ePublication: 30 October 2013

Abstract. At low energies, an effective field theory (EFT) with only contact interactions as well as three-body forces allow a detailed analysis of renormalization in a non-perturbative context and uncovers novel asymptotic behaviour. Triton as a three-body system, based on the EFT have been previously shown to provide representative binding energies, charge radii, S-wave scattering amplitude and asymptotic normalization constants for the³H bound state system. Herein, EFT predictions of the asymptotic D-state to S-state ratio of triton are calculated to more fully evaluate the adequacy of the EFT model. Manifestly model-independent calculations can be carried out to high orders, leading to high precision.

Keywords. Few-body systems; radiative capture; effective field theory; three-body forces.

PACS Nos 21.30.−Fe; 27.10.+h; 21.45.+V; 25.40.Lw; 11.80.Jy

1. Introduction

Asymptotically, the ground state of³H can be considered to be composed of a spin-1 deuteron and a spin-1/2 neutron bound with−6.257 MeV energy. The total spin and relative angular momentum of these two clusters can be either S =1/2,L =0 or S = 3/2,L =2 and still form J^π = ¹₂⁺. These two states are referred to as asymptotic S- and D-states, respectively. Given that the energy and angular momenta of these states are known, the mathematical forms of the asymptotic radial functions are known. The only unknowns are the normalization constants of the asymptotic S- and D-states, CSand CD. The ratio CD/CSis called the triton asymptotic ratio,ηt.

There are several ways by whichηt can be experimentally determined. One of them is illustrated in refs [1–4]. In these works, neutron pick-up reactions using polarized deuterons at sub-Coulomb energies were performed on medium weight nuclei. Differen- tial cross-sections and tensor analysing powers (TAP) were measured and analysed using finite range distorted wave Born approximation (DWBA). The calculated analysing powers are quite sensitive to changes inηt, which makes it possible to obtain reasonably

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accurate values ofηt. The weighted average of the results forηt, from George and Knutson [2] who obtainedηt = −0.0431±0.0025 and from Kozlowska et al [3] who obtained ηt = −0.0411±0.0018 is,ηt(ave.) = −0.0418±0.0015; the weights used were the inverse of the squares of the errors. Earlier experimental values forηtincluding values obtained by different techniques are given in refs [5,6]. In these works, the authors discuss the physical origin of the opposite signs ofηdandηt.

An early theoretical calculation of this ratio for several models using the Faddeev method is reported in refs [7,8]. They also discussed the possible presence of an additional phase factor which givesηta positive sign in some formalisms.

The pionless EFT would be an ideal tool to calculate low-energy cross-sections in a model-independent way and to possibly reduce the theoretical errors in a precision calculation. Recently developed pionless EFT is particularly suited for high-order precision calculation. There are many processes at low energies which are both interesting in their own right and important for nuclear physics.

We have suggested a method for the computation of neutron–deuteron radiative capture in very extremely low energy with pionless EFT [9–14], where with this formalism, we can estimate errors in perturbative expansion up to N²LO with few percent in comparison with experimental data and other theoretical methods. A variety of observables have been calculated to test the present model of nuclear current operator. In particular, for the A = 3 nuclear systems, observables such as cross-sections as well as electromagnetic form factors have been calculated and compared with the corresponding experimental results in low energies. Recently, we have calculated the cross-section of radiative capture process nd → ³Hγ by using pionless EFT and isospin-violating effect of this reaction.

The cross-section is determined to beσtot= [0.503±0.003]mb [9,10] and isospin effect correctionσtot = [0.505±0.003]mb [11]. Photon polarization parameter of neutron–

deuteron radiative capture was also calculated at thermal neutron energies, using pionless EFT up to N²LO. The result is determined to be R_c= −0.412±0.003 [12].

The present study investigates a low-energy observable in three-body systems, namely, the asymptotic D-state to S-state ratio of tritonηtusing pionless EFT up to N²LO. The emphasis is on constructing three-body currents with model-independent theory corresponding to three-nucleon interactions and comparing our model’s result with those of other model-dependent theory and experimental data.

This article is organized as follows. In the next section, a brief description of the relevant Lagrangian, neutron–deuteron scattering and triton form factors are reported.

Then, we discuss the theoretical errors, tabulation of the calculated result in comparison with the other theoretical approaches and the available experimental data in §3. Finally, summary and conclusions are given in §4.

2. Brief formulation of neutron–deuteron interaction and triton form factors The effective Lagrangian of pionless EFT, its power-counting rules, the parameter fixing necessary for this work and the derivation of the integral equation describing neutron–

deuteron scattering have been discussed extensively by Bedaque et al [15,16]. We repeat them here only briefly. We present here only the result, including the new term

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generated by the two-derivative three-body force. The three-nucleon Lagrangian is given by [14]

L = N^†

iD0+ D² 2M + e

2M(κ0+κ1τ3)σ· B

N−d_a^†

iD0+ D² 4M −s

sa

−ys

d_a^†N^TP_a⁽¹^S⁰⁾N+h.c.

−d_i^†

iD0+ D² 4M −t

ti

−yt

d_i^†N^TP_i⁽³^S¹⁾N+h.c.

− Csd

√Mρd

δi xδj y−1 3δi jδx y

×

d_i^†(N^TOx y,jN)+h.c.

− CQ

Mρd

d_i^† iD0,Oi j

tj + eL1

M√ r0ρd

d_i^†s3Bi+h.c.

+t^†

i∂0+ ∇² 6M +γ²

M +

t

−ωs

t^†(τ^AN)ds^A+h.c. −ωt

t^†(σⁱN)dtⁱ+h.c. + · · ·, (1) where the one-body Lagrangian (the first line) of the nucleon isodoublet N =_p

n with

isospin-averaged mass M =938.9 MeV consists of the kinetic term part with minimal substitution, D_μ = ∂_μ+ieQ A_μ(Q = ¹₂(1+τ3)), and the interaction via the isoscalar (isovector) magnetic moments,κ0 = 0.44 (κ1 = 2.35) in nuclear magnetons. The spin (isospin) Pauli matrices with vector (isovector) index i=1,2,3 (a=1,2,3) are denoted byσi(τa).

The two-body Lagrangian for the auxiliary dibaryon field da(di)in the¹S0(³S1)channel and four-nucleon operators coupling to the magnetic field are described by the Lagrange density involving dibaryon fields which are probing only the np system, D_μ=∂μ+ie A_μ for both dibaryons. The projectors are P_a⁽¹^S⁰⁾=(1/√

8)σ2τ2τa, P_i⁽³^S¹⁾ =(1/√ 8)σ2σiτ2

and

ys =

√8π M√

r0

, s= 2 Mr0

1 a0

−μ

, yt =

√8π

M√ρd , t= 2 Mρd

γ−ρd

2 γ²−μ

, (2)

where μ is calculated in the power divergence subtraction scheme (PDS) version of dimensional regularization and the physical amplitudes are independent of it. The scat- tering length and effective range are a0 = −23.71 fm, r0 = 2.73 fm. γ = √

M B = 45.70 MeV, where B = 2.225 MeV is the deuteron binding energy and effective range ρd=1.764 fm.

The operators in eq. (1) which parametrize SD-wave mixing in the spin-triplet, are Ox y,j = −1

4(DxDyP⁽_j³^S¹⁾+P_j⁽³^S¹⁾DxDy−DxP_j⁽³^S¹⁾Dy

−DyP_j⁽³^S¹⁾Dx), Oi j = −(DiDj−1

3δi jD²). (3)

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The asymptotic ratio of D- and S-wave components of the deuteron wave function, ηsd=0.0254, and the deuteron quadrupole momentμQ=0.2859 fm²[17]

Csd= 6√ πηsd

√Mγ² , μQ=2Z

yt

C_sd

√Mρd

M² 32π

2 3γ + C_Q

Mρd

. (4)

C_sd contributes at LO and C_Qprovides a next-to-leading order (NLO) correction. The parameter L₁= −4.0 fm in Lagrangian enters at NLO correction [17].

We use the power-counting rules of pionless EFT in the version in which the effective- range parameters are treated as unnaturally large and thus are kept at LO together with the scattering lengths [17]. SD mixing is suppressed by Q²with respect to pure S-wave amplitudes because of the two derivatives in eq. (2). Finally, in the last line of eq. (1), the auxiliary field t carries the quantum numbers of the³H spin and isospin doublet.

A convenient expression to calculate the charge form factors of a J = 1/2 nucleus, such as the triton, is obtained by the matrix element of the electric current, and is given by [14]

p,j|J_em⁰ |p,i = e

FC(q²)δi j + 1

2M_d²F_Q(q²)

qiqj− 1 n−1q²δi j

×

E+E 2Md

, p,j|J^k_em|p,i = e

2Md

FC(q²)δi j(p+p)^k+FM(q²)

δ^kjqi−δ^kiqj

+ 1

2M_d²F_Q(q²)

qiqj− 1 n−1q²δi j

(p+p)^k

, (5)

Figure 1. Diagrams of the contribution to the triton charge form factor at LO. The line indicates the nucleon and the thick solid line is a propagator of the two intermediate auxiliary fields d_sand d_t. The three-body interaction is denoted by the dot. Half-blobs show the three-body bound state and the wavy line shows the photon.

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where q=p −p, q² =q₀²− |q|²is the square of the four-momentum transfer and n is the number of space-time dimensions.

We now turn our attention to reactions where triton is present only in the final or ini- tial state. The diagrams of the contribution to the triton charge form factor in pionless EFT are shown in figure1. This figure represents contributions of electromagnetic interactions with a nucleon, deuteron and four-nucleon-magnetic-photon operator described by a coupling between the³S1-dibaryon and ¹S0-dibaryon and a magnetic photon, up to NLO.

In addition to diagrams where the photon couples to the nucleon, there are also couplings to the dibaryon field obtained by gauging the Lagrange density in eq. (1). At this order, the structure of the nucleons does not contribute to the calculation and the charge form factors of the proton and neutron are simply 1 and 0, respectively. At NLO, we have contributions coming only from insertions of the kinetic energy in the deuteron propagator. The NLO corrections to neutron–deuteron scattering are simply the range corrections in the first-order perturbation theory (for more details, see refs [9,14]).

At higher orders, there are contributions from higher-dimension operators involving more derivatives on the nucleon field, such as the nucleon charge radius operator, and also from higher-dimension couplings of the dibaryon field, that is, the dibaryon charge radius.

The first correction from the nucleon structure comes from the nucleon radius which would contribute at N²LO in our counting scheme. At N²LO order there are corrections to neutron–deuteron scattering, which are simply two insertions of the kinetic energy in the deuteron propagator as well as correction from the nucleon structure coming from the nucleon radius [14]. The N²LO contributions come from SD-mixing operators. The diagrams of the contribution to the triton charge form factor is shown in figure2. In particular, we wish to see if the z-parametrization improves the convergence of these reactions, in which only one of the asymptotic states involves a triton [18].

Figure 2. Diagrams of the contribution to the triton charge form factor at N²LO. The SD-mixing vertex proportional to C_SDis denoted by a square. The remaining notation is as in figure1.

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Table 1. Theoretical and experimental determinations ofηt. Last digit in parenthesis shows the uncertainty.

Experiment ηt

Pole extrapolation,²H(d,p)³H [19] −0.048(7)

Pole extrapolation,²H(d,p)³H [20] −0.051(5) Pole extrapolation,⁴He(d,⁴He)³H [21] −0.050(6)

DWBA analysis [22] −0.044(4)

Sub-Coulomb DWBA [23] −0.043(4)

Sub-Coulomb TAP [1] −0.043(1)

Theory ηt

Correlation ofηtand B_t[3] −0.043(2)

Correlation ofηtand B_t[24] −0.046(1)

Correlation ofηt/ηdand B_t[19] −0.043(0)

This calculation −0.043(1)

The F_Q(q²)form factor involves a transition between the S- and D-wave components of triton. At LO its value is determined by a D- to S-wave transition operator whose coefficient is extracted from the asymptotic D/S ratio of triton,ηD/S. At NLO there is a new two-body term

L= −eCQ(N PiN)^†(N PjN)

∇ⁱ∇^j−1 3∇δ^{i j}

A⁰, (6)

whose coefficient CQcan be fitted to the experimental deuteron quadrupole moment [17].

At N²LO the only contribution comes from the finite size of the nucleon charge distribu- tionr²N. The value of FQ(0)is then a fit, but the momentum dependence is an EFT prediction.

3. Results and discussion

We solve integral equation by inserting Q in integral equation and iteration of the kernel.

First H0is determined from the²S1/2scattering length a3 =(0.65±0.04)fm [25]. At LO and NLO, this is the only three-body force entering, but at N²LO, where we saw that H2 is required, it is determined by the triton binding energy B3 = 8.48 MeV. We introduced H2 such that it does not contribute at zero-momentum scattering. So, we adjust the three-body force to fit the triton binding energy. We numerically solved the Faddeev integral equation up to N²LO. We usedc=197.327 MeV fm, 938.272 MeV (939.566 MeV) for proton (neutron) mass, a deuteron binding energy (momentum), B= 2.225 MeV (γd=45.7066 MeV), for the N N triplet channel, a residue of Z_d=1.690(3), r_0t = 2.73 fm effective range and ¹S₀ scattering length a_t = −23.714 fm for the N N singlet channel. For the spin-triplet S-wave channel, one replaces the two-boson binding momentumγ and effective range ρ by the deuteron binding momentum γt =45.7025

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MeV and effective rangeγt=1.764 fm. Because there is no real bound state in the spin- singlet channel of the two-nucleon system, its free parameters are better determined by the scattering length as= 1/γs = −23.714 fm and the effective range rs=2.73 fm at zero momentum (for more details, see refs [9,10,16]). The LB and LE are obtained by fixing, at leading non-vanishing order, by the thermal neutron–proton radiative capture and deuteron photodisintegration cross-sections, respectively [25].

Further information on the structure of the three-nucleon system can be obtained by studying the asymptotic constants. It is well known that the J^π = ¹₂⁺ elastic scattering amplitude and the properties of the three-nucleon bound states are closely related. It is therefore of interest to compare the predictions of the AV18+Urbana IX potential model with recent experimental results [2–4]. By choosing slightly different constraints to deter- mine the coefficients in the Lagrange density, we have been able to write observables in terms of the normalization constants andηt, rigorously defined in terms of the residue of the scattering amplitude at the deuteron pole. If the coefficients are fit in this way then the complete result in effective range theory is obtained for elastic processes at NLO inηt. For the absolute values of the S-state asymptotic constant CS, the D-state asymptotic constant CDand the ratioηt=CD/CS, we obtainηt=0.0431 for triton. From the above results, it can be seen that the ratioηis lower than previous calculations [17] and agree well with the experimental results:ηt=0.0411±0.0013±0.0012 [2] and 0.0431±0.0025 [4] (see table1). This result is commensurate with the good agreement between modern potential model calculations and experimental results.

4. Summary and conclusion

EFT is the most general theory in the three-nucleon sector which is consistent with the symmetries of the strong and electroweak interactions in the very low-energy regime.

We calculate the asymptotic D-state to S-state ratio of triton, ηt, which is determined to be 0.0431. The calculated D-state to S-state ratio for ³H is in excellent agreement with recent measurements and other theoretical calculations. For nearly 50 years, both theorists and experimentalists were struggling to understand the three-nucleon forces and their manifestations in nuclear physics. Despite a great deal of effort on both fronts, there still are many problems. For this reason it has recently been advocated to employ effective field theory approach to the low-energy observables of light nuclei. Above we have compared Faddeev calculations as well as experimental results with EFT results in triton.

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