Polarization of the neutron induced from hadronic weak interactions in the photo-disintegration of the deuteron

Polarization of the neutron induced from hadronic weak interactions in the photo-disintegration of the deuteron

Abstract

New observables with which we can study the two-nucleon weak interactions at low energies are considered. In the breakup of the deuteron by photons, polarization of outgoing neutrons can depend on the parity-violating component of two-nucleon interactions. We express the parity-violating polarization in general forms, and perform numerical calculations with a pionless effective field theory. The theory has unknown parity-violating low energy constants, and the results are expressed in linear combination of them. We discuss the results and their implication to the understanding of the hadronic weak interactions.

pacs:
13.75.Cs, 23.20.-g, 24.70.+s, 24.80.+y

I Introduction

Present knowledge about the most fundamental interactions indicate that parity is not conserved only in the weak interactions. Such a nature of the weak interaction has been successfully probed in leptonic and semi-leptonic processes in high-energy experiments as well as in decay. In principle parity-violating (PV) aspects of the weak interaction can emerge in the pure hadronic processes too at both high and low energies. However our understanding of PV aspects of the weak interaction in the low energy region is still very poor even though more than 50 years have passed since the first observation of the parity violation in nuclear phenomena.

Nevertheless efforts in both experiments and theories have been continued. Especially there has been significant progress in the low energy few-nucleon systems in the last three decades. In Ref. DDH (), the authors wrote down the two-nucleon PV interactions in terms of -, - and -meson exchanges (DDH potential), which contain seven weak meson-nucleon coupling constants. Several PV observables in nuclear and hadronic processes have been calculated in terms of the DDH potential, and experiments were attempted to determine the values of the seven weak meson-nucleon coupling constants. In the last decade, calculations have been improved by the use of modern nucleon-nucleon () phenomenological potentials such as Argonne v18, CD Bonn and Nijmegen93. The relevant PV observables are the asymmetry in plb01 (); schi04 (), the anapole moment of the deuteron plb03 (); prc03 (), the polarization in epja05np (), asymmetry in prc04 (); fuji04 (); epja05dg (), longitudinal asymmetry in scattering prc06 (); prc08 (). The PV observables were expressed in terms of the seven meson-nucleon coupling constants in the DDH potential. Longitudinal asymmetries were measured with good accuracies at low energies ever91 (). Polarization in was measured in the late 70’s, but the experiments could provide only the upper limit knya84 (). Asymmetry in is under measurement at SNS in Oak Ridge geri11 (). There was an experimental trial for the deuteron anapole moment and .

In the mean time, there was a reformulation in the theory for the PV interactions in the frame work of effective field theory (EFT) zhu05 (). Authors in zhu05 () derived the PV interactions from the theory with pions (pionful theory) and also without pions (pionless theory). In the pionless theory where all the interactions are described in terms of contact terms only, it was shown that only five PV low energy constants (LECs) are independent after removing redundancy in the DDH potential girl08 (). PV observables in the two-nucleon systems were recalculated with the EFT PV potentials with and without pions savage01 (); plb07 (); liu07 (); phil09 (); prc10 (); schin10 (). Nowadays, effort to determine the PV LECs in the pionless theory is getting more attention in the field. Asymmetry in is now becoming a potential candidate for the measurements in the two-nucleon processes. Measurements have been proposed at JLab, SPring-8, Shanghai Synchrotron, and most recently at TUNL. The aimed accuracy in the experiment at TUNL is of the order of , with which one can obtain stringent constraint to pin down the values of either meson-nucleon coupling constants in the DDH potential or the PV LECs in the pionless theory. For precise determination of the coupling constants or LECs, however, it is necessary to have additional observables.

In this work, we calculate polarization of the neutron in at low energies. There is a long history of discrepancy between theory and experiment for the polarization rus60 (); sch05 (); kukulin08 (), which is a parity-conserving (PC) quantity. The problem with a pionless EFT with dibaryon fields as auxiliary fields for the two-nucleon states was revisited ando11 (). With dibaryon fields, the calculation becomes simple and the convergence is especially efficient at low energies. In fact, we applied the theory successfully to various quantities such as the electromagnetic moments of the deuteron ando05 (), capture at the big-bang nucleosynthesis energies ando06b (), and fusion in the Sun ando08 (). Also, we observed good agreement with other theoretical results for at low energies, but the discrepancy with the measurements still remains unresolved. is the polarization along axis (convention for the coordinate system will be shown later). One can also think of the polarization along and directions, but they vanish if only PC interactions are considered. As will be shown in the following section, however, PV interactions cause non-zero contributions to and . Motivated by this simple observation, we calculate and with a pionless EFT with dibaryon fields. Assuming the first order approximation, the observables are obtained in the linear combination of PV LECs. Since the values of PV LECs are completely unknown, we cannot determine the numerical values of the polarizations. Instead, the coefficients of the LECs can be calculated easily. We compare the resultant coefficients with those appearing in other PV observables such as the asymmetry in and the polarization in . By this comparison, we can roughly estimate the order of the physical quantity, and discuss the feasibility of the measurement.

The paper is organized as follows. In Sect. II, we present the basic Lagrangians. In Sect. III, we obtain the diagrams at leading order (LO), and calculate the amplitudes. In Sect. IV, the numerical results are discussed. We summarize the work in Sect. V, and give detailed forms of complicated equations are given in Appendix A.

Ii Effective Lagrangian

In the pionless theory, pions are treated as heavy degrees of freedom, and thus the typical scale of expansion parameter is , where is a physical or exchange momentum. In the system where scattering length is unusually long or binding energy is very shallow, one can also treat these small scales as expansion parameters. It is natural to assign order to the quantities such as , , , and , where , are the scattering length in the and states, respectively and is the effective range in the state. where is the binding energy of the deuteron and is the effective range corresponding to the deuteron. In a diagram, propagators of a single nucleon and a dibaryon field are counted as and integration of a nucleon loop generates .

ii.1 Parity-conserving part

PC part of the Lagrangian consists of strong and electromagnetic (EM) interactions. PC Lagrangian with dibaryon fields can be written as ando05 ()

 LPC=LN+Ls+Lt+Lst, (1)

where , , and include interactions for nucleons, dibaryon in state, dibaryon in state, and EM transition between and states, respectively. Retaining terms that are relevant to the present work, we have

 LN = N†⎧⎨⎩iD0+→D22mN−e2mN12(μS+μVτ3)→σ⋅→B⎫⎬⎭N, (2) Ls = −s†a⎧⎨⎩iD0+→D24mN+Δs⎫⎬⎭sa−ys{s†a[NTP(1S0)aN]+h.c.}, (3) Lt = −t†i⎧⎨⎩iD0+→D24mN+Δt⎫⎬⎭ti−yt{t†i[NTP(3S1)iN]+h.c.}−2L2mNρd(i)ϵijkt†itjBk, (4) Lst = L1mN√r0ρd[t†is3Bi+h.c.], (5)

where the projection operators for the and states are respectively defined as

 P(1S0)a=1√8σ2τ2τa,   P(3S1)i=1√8σ2σiτ2. (6)

The covariant derivative is defined as where represents the external vector field. For the nucleon, we use , , where is the charge operator. For the dibaryon fields, we have , . Dibaryon fields in and states are denoted by and , respectively, and is the external magnetic field given by . are defined by the mass difference between the dibaryon and two nucleon states, i.e. .

LECs and represent the strength of the coupling between a two-nucleon state and a dibaryon field. They are determined from the empirical values of effective range parameters, and . LECs and can be determined from the capture cross section at threshold and the deuteron magnetic moment, respectively ando05 ().

ii.2 Parity-violating part

It was shown that the insertion of a nucleon loop in the propagator of a dibaryon field leaves the order of the diagram the same as that of a single dibaryon propagator bean01 (). As a result, LO diagrams have only dibaryon- () vertices for the strong interaction, and other types of strong vertices, e.g. four-nucleon contact terms belong to sub-leading contributions. If we are to consider the weak effects, we have to include PV interactions in a diagram. This can be easily achieved by simply replacing one PC vertex in a diagram for PC transition with a PV interaction. Even with this replacement, remaining part of the diagram is unchanged, so the ordering of the diagram is not affected by the insertion of a PV vertex. Therefore, it may suffice to represent the weak interactions in terms of only PV vertices at LO.

At low energies, two-nucleon systems are dominantly occupied by -wave states, i.e., and . PV interactions change the spatial parity of the -wave states to the next low lying opposite parity states such as and . is isosinglet, and thus it is allowed to system only. On the other hand, are isotriplet, and thus and as well as can occupy the states. If we consider the change of the states from -wave to -wave by the PV interaction, we have the following selections: to (, , ), to (), and to (). As a result, we have five terms for the PV interactions as

 L0\tiny PV = 3∑a=1h0sad2√2ρdr0m5/2Ns†aNTσ2σiτ2τai2(\lx@stackrel←∇−\lx@stackrel→∇)iN+h.c. (8) L1\tiny PV = ih1d2√2ρdm5/2Nϵijkt†iNTσ2σjτ2τ3i2(\lx@stackrel←∇−\lx@stackrel→∇)kN+h.c.. (9)

Superscript in denotes the change of the isospin accompanied in the interaction. In Eq. (8), and give isospin operator proportional to and identity matrix, respectively. With the isodoublet of the proton and the neutron, these matrices give mixture of and states. These terms are irrelevant in this work, and the term corresponding to generates isotriplet state of the system. For the sake of simplicity, we disregard the constants and , and replace with . Consequently we have three unknown LECs , and for the coupling constants of PV interactions.

Iii Amplitude

With the counting rules, we can arrange the pertinent Feynman diagrams order by order. We have verified in former works that applications to the PC processes were successful ando05 (); ando06b (); ando08 (); ando11 () already at the next-to-leading order (NLO). PC amplitude up to NLO of reaction is written in the form as ando11 ()

 APC = χ†1→σσ2τ2χT†2⋅{[→ϵ(d)×(^k×→ϵ(γ))]XMS+→ϵ(d)→ϵ(γ)⋅^pYE} (10) +χ†1σ2τ3τ2χT†2i→ϵ(d)⋅(^k×→ϵ(γ))XMV +χ†1→σσ2τ3τ2χT†2⋅{→ϵ(d)→ϵ(γ)⋅^pXE+[→ϵ(d)×(^k×→ϵ(γ))]YMV} +χ†1σ2τ2χT†2i→ϵ(d)⋅(^k×→ϵ(γ))YMS,

where and are the spin polarization vectors for the incoming deuteron and photon, respectively, while and are the spinors of the outgoing nucleons. is the momentum of an incoming photon, is the relative three-momentum of the two nucleons in the final state, and unit vectors and . Details for ’s and ’s can be found in Appendix A.1.

PV vertices have a spatial derivative as shown in Eqs. (8), (8), (9), and thus they are linear in momentum. It is natural to count the order of a PV vertex as . When a photon is coupled to a PV vertex minimal, it is equivalent to replacing the derivative to a photon field, and thus the order of PV minimal coupled vertices becomes . With the additional counting rules for the PV vertices, the LO diagrams for are obtained and depicted in Fig. 1.

If we neglect the orders of the propagators for incoming dibaryon and outgoing nucleons, the diagrams are of . PV amplitudes obtained from the diagrams can be written as

 APV=h∑i=aAPV(i). (11)

Detailed expressions for are summarized in Appendix A.2. The sum of both PC and PV contributions is

 A=APC+APV. (12)

The polarization is defined as

 Pi≡σi+−σi−σi++σi−, (13)

where and are the differential cross sections with the neutron spin up and down along a specific direction , respectively. Polarization of neutrons can be expressed by introducing the projection operator

 P±=12(1−τ3)12(1±→σ⋅^n), (14)

where denotes the direction of the neutron spin. Squaring the amplitude given by Eq. (12) with the polarized neutrons, we obtain

 S−1P∑spin|A|2=4(|XMS|2+|YMV|2−2YMVReXMS) (15) +2(|XMV|2+|YMS|2−2YMSReXMV)+3[1−(^k⋅^p)2](|XE|2+|YE|2−2XEYE) ∓2^n⋅(^k×^p)(XE−YE)ImXMV∓2(^k⋅^n)Im~f1∓2(^p⋅^k)(^k⋅^n)Im~f2 ∓2(^p⋅^n)Im~f3∓2(^p⋅^k)(^p⋅^n)Im~f4,

where is a symmetry factor for spin average, , and ’s are the PV-PC interference terms, whose details can be found in Appendix A.3.

Conventions for the coordinate systems are quoted from rus60 (). We have the incoming photons along , relative momentum of the nucleons along , and orthogonal basis vectors are definded as , and . If we align the neutron spin along , scalar products of the unit vectors for the PV-PC interference terms vanish. In this case, we obtain the PC polarization ando11 (). If we polarize the neutrons along , while , and thus and terms are non-vanishing. With , all the terms contribute to the polarization . With and , one can easily check that PC interference term proportional to becomes null. Consequently, we can obtain hadronic weak effects by calculating the polarizations and which are not interfered by the PC components of the interactions.

Iv Result and discussion

iv.1 Polarization along ^z′

In this section, we present and discuss the results for . With Eqs. (13), (15), we obtain as

 Pz′=(−2)Im[(~f1+~f4)cosθ+~f2cos2θ+~f3]/ΣPC, (16)

where

 ΣPC ≡ 4(|XMS|2+|YMV|2−2YMVReXMS)+2(|XMV|2+|YMS|2−2YMSReXMV) (17) +3(1−cos2θ)(|XE|2+|YE|2−2XEYE).

Since ’s contain linear combinations of ’s () whose values are not known, we may rewrite the polarization in the form

 Pz′≡c0tzh0td+c0szh0sd+c1zh1d. (18)

Coefficients are functions of the colatitude angle and the relative momentum (or equivalently photon energy in the lab frame ), and they take into account the characteristics of PV as well as PC interactions of the theory. Explicit forms of can be found in Appendix A.4.

In Fig. 2, we plot the numerical results for ’s as functions of photon energies in the lab frame. Angle dependences are examined by picking up three angles, and . A common feature in is that there is a minimum in the range  MeV regardless of angle and the isospin structure of the PV vertex, i.e. the superscript in . At higher energies, tends to converge to a value, while and show a linear increase. Another noticeable behavior at high energies is that and show distinct dependence on the angle, but is almost independent of the angle, and the magnitude of the coefficients and is greater than that of by an order.

In Fig. 3, we investigate in more detail the dependence on the angle in the center of mass frame at  MeV. It was seen in Fig. 2 that ’s have the largest magnitude in  MeV, and thus the value 2.75 MeV was chosen arbitrarily. is almost zero regardless of the angle. Indeed, the order of magnitude of and does not exceed in the energy range considered, and thus the dominance of is very clear at forward or backward angles, where the magnitude is greater than . Since the values of are not known, we cannot determine the value of . However, we can roughly estimate the order of magnitude of in comparison with other PV observables.

In Ref. savage01 (), in was calculated with the dibaryon fields and the result was obtained as

 Aγ = −m3/2N2√2πh(1)331−γa1/3κ1(1−γa0)−γ2a0L1/2, (19)

where is the convention for the PV LEC in the work. and are related through

 h(1)33=h1dρ1/2dm2N, (20)

which allows us to write in terms of as

 Aγ=−3.2×10−3h1d. (21)

The measured value of is alberi88 (), and NPDGamma collaboration aims at determining the value unambiguously at the order of . Since can be larger than by an order of magnitude, it can be an observable experimentally advantageous in measurement to determine PV LEC with a better accuracy than one can achieve with .

iv.2 Polarization along ^x′

Polarization along the axis is obtained as

 Px′=2sinθIm[~f1+~f2cosθ]/ΣPC. (22)

We can cast it in the form

 Px′≡c0txh0td+c0sxh0sd+c1xh1d, (23)

and investigate ’s. Complete expressions for are shown in Appendix A.4.

Figure 4 shows the energy dependence of ’s at the angles , and . and show the behavior and order of magnitudes similar to those of and , respectively. The magnitude of is smaller than by a factor of , and shows a structure with a maximum and a minimum slightly below and above 3 MeV, respectively. Similar to ’s, ’s have a maximum magnitude for  MeV in the low energy region.

In Fig. 5, we plot with respect to at  MeV. While monotonically dominates over other contributions, ’s contain more information. forms a maximum at with the value . The values of and at this angle are and , respectively, and thus the contribution from can be safely ruled out in the consideration. In Ref. prc10 (), we calculated PV polarization in at threshold with the theory as employed in this work, and obtained the result

 Pγ=−(2.59h0td−1.01h0sd)×10−2. (24)

The coefficient of in is larger than in at by a factor of 3, but they are roughly similar in order. Therefore, measurement of at the , in addition to PV in , can provide a complementary constraints to determine and . At , and are of the order of or less, while . At this angle, we can neglect the contributions from and , and thus have chances to determine with a minor uncertainty. This value is comparable to in . With the measurements of in , at forward or backward angles, and at around , we can check the consistency of the theory and determine the value of . At the angles around , all the ’s become maximum with , and the ratios of and to are 0.76 and 0.27, respectively. By determining from at , and from , at or , and at , we may pin down the value of through the measurement of at the angles in the backward direction.

V Summary

In this work, we have considered the polarization of the neutron in with a pionless EFT incorporating dibaryon fields. Polarization along the azimuthal direction imposes the information about the interactions that conserve parity. Along the radial () and colatitude () directions, on the other hand, non-vanishing contributions reflect the effect of PV interactions. We focused on the PV components of the polarization, and calculated and as functions of the incident photon energies up to 15 MeV. Since the coefficients ’s and ’s can be evaluated at different angles and energies, one can determine the unknown PV LECs by comparing the calculated and with the experimental values.

In the dependence on the energy, both and show a peak structure slightly above the threshold regardless of the angle . At these energies, pionless EFT with dibaryon fields was verified to a good accuracy for many observables in the two-nucleon processes, but care should be taken as we increase energy near 15 MeV prc12 (). Therefore, if measurement is performed, it may be most desirable to concentrate on the low energies.

By fixing the photon energy to 2.75 MeV, we explored the dependence of and on the angle . Concerning , the coefficients of the iso-scalar components of the PV interaction, and , are more or less constant in the angle, but the coefficient of the iso-vector component, , changes drastically in the forward and backward angles. In these directions, is exclusively dominated by the PV iso-vector interaction, and thus the measurement of along with in will provide a chance for a unique determination of . is expected to give more information about the PV LECs. At the angles close to the forward direction, the values of are more significant than those of and , and thus is expected to be dominated by . On the other hand, in the backward directions, contributions from , and terms are expected to be of the same order. By combining the measurements of and at various angles, we can determine the PV LECs in the system, , and .

Enhancement of compared to in by an order of magnitude is a striking result. Since the prediction can imply significant impact on the experiments, the results in this work should be counter-checked by other calculations. One possibility is to adopt the DDH potential, and obtain the results for and in terms of the PV meson-nucleon coupling constants. Many PV observables in the two-nucleon systems were already calculated in terms of the DDH potential. The work is in progress for evaluating and liu13 ().

Acknowledgments

Work of CHH and SIA is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology Grant No. 2010-0023661, the work of JWS by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology Grant No. 2011-0025116 and the work of SWH by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology Grant No. 2012R1A1A2007826.

Appendix A Summary of lengthy expressions

a.1 PC terms

For the PC amplitude given in Eq. (10), we have the PC terms

 XMV = −√πγ1−γρd11a0+ip−12r0p212mN (25) ×⎧⎪ ⎪⎨⎪ ⎪⎩μV⎡⎢ ⎢⎣arccos⎛⎜ ⎜⎝mN√(mN+12ωγ)2−p2⎞⎟ ⎟⎠+iln⎛⎜ ⎜⎝mN+12ωγ+p√(mN+12ωγ)2−p2⎞⎟ ⎟⎠⎤⎥ ⎥⎦ −μVmN(1a0+ip−12r0p2)F++ωγL1}, XMS = −√πγ1−γρd1γ+ip−12ρd(γ2+p2)12mN (26) ×⎧⎪ ⎪⎨⎪ ⎪⎩μS⎡⎢ ⎢⎣arccos⎛⎜ ⎜⎝mN√(mN+12ωγ)2−p2⎞⎟ ⎟⎠+iln⎛⎜ ⎜⎝mN+12ωγ+p√(mN+12ωγ)2−p2⎞⎟ ⎟⎠⎤⎥ ⎥⎦ −μSmN[γ+ip−12ρd(γ2+p2)]F++2ωγL2}, XE = √πγ1−γρd1m2NpωγF+,   YE=√πγ1−γρd1m2NpωγF−, (27) YMV = √πγ1−γρdμV2m2NF−,   YMS=√πγ1−γρdμS2m2NF−, (28)

where is the incident photon energy in the c.m. frame, and

 F± = 12⎡⎢ ⎢⎣11+ωγ2mN−→p⋅^kmN±11+ωγ2mN+→p⋅^kmN⎤⎥ ⎥⎦. (29)

a.2 PV amplitudes

We calculated the amplitude for the diagrams (a) to (h) in Fig. 1. and are the spinors of the nucleons in the final state.

 iAPV(a) = Ch0td[(i)χ†1σ2τ2χT†2ϵ(d)iϵ(γ)j^ki^pjpmNF+ (30) −(i)χ†1σ2τ2χT†2ϵ(d)iϵ(γ)j^pi^pj2p2mNωγF− +(i)χ†1σ2τ3τ2χT†2ϵ(d)iϵ(γ)j^ki^pjpmNF− −(i)χ†1σ2τ3τ2χT†2ϵ(d)iϵ(γ)j^pi^pj2p2mNωγF+ +χ†1σiσ2τ2χT†2ϵijkϵ(d)aϵ(γ)k^kj^paμSmNpF− −χ†1σiσ2τ2χT†2ϵijkϵ(d)aϵ(γ)k^kj^kaμS2mNωγF+ +χ†1σiσ2τ3τ2χT†2ϵijkϵ(d)aϵ(γ)k^kj^paμVmNpF+ −χ†1σiσ2τ3τ2χT†2ϵijkϵ(d)aϵ(γ)k^kj^k