The parity-violating asymmetry in the 3He(n,p)3H reaction
M. Viviani, R. Schiavilla, L. Girlanda, A. Kievsky, L.E. Marcucci
aa r X i v : . [ nu c l - t h ] J u l The parity-violating asymmetry in the He( ~n, p ) H reaction
M. Viviani a , R. Schiavilla b , c , L. Girlanda d , a , A. Kievsky a , and L.E. Marcucci d , a a INFN-Pisa, 56127 Pisa, Italy b Department of Physics, Old Dominion University, Norfolk, VA 23529, USA c Jefferson Lab, Newport News, VA 23606 d Department of Physics, University of Pisa, 56127 Pisa, Italy (ΩDated: November 5, 2018)The longitudinal asymmetry induced by parity-violating (PV) components in the nucleon-nucleonpotential is studied in the charge-exchange reaction He( ~n, p ) H at vanishing incident neutron en-ergies. An expression for the PV observable is derived in terms of T -matrix elements for transitionsfrom the S +1 L J = S and S states in the incoming n - He channel to states with J = 0 and1 in the outgoing p - H channel. The T -matrix elements involving PV transitions are obtained infirst-order perturbation theory in the hadronic weak-interaction potential, while those connectingstates of the same parity are derived from solutions of the strong-interaction Hamiltonian with thehyperspherical-harmonics method. The coupled-channel nature of the scattering problem is fullyaccounted for. Results are obtained corresponding to realistic or chiral two- and three-nucleonstrong-interaction potentials in combination with either the DDH or pionless EFT model for theweak-interaction potential. The asymmetries, predicted with PV pion and vector-meson couplingconstants corresponding (essentially) to the DDH “best values” set, range from –9.44 to –2.48 inunits of 10 − , depending on the input strong-interaction Hamiltonian. This large model depen-dence is a consequence of cancellations between long-range (pion) and short-range (vector-meson)contributions, and is of course sensitive to the assumed values for the PV coupling constants. PACS numbers: 21.30.-x,24.80.+y,25.10.+s,25.40.Kv
I. INTRODUCTION, RESULTS, ANDCONCLUSIONS
A number of experiments aimed at studying parity vio-lation in low-energy processes involving few nucleon sys-tems are being completed or are in an advanced stageof planning at cold neutron facilities, such as the LosAlamos Neutron Science Center, the NIST Center forNeutron Research, and the Spallation Neutron Source atOak Ridge. The primary objective of this program is todetermine the fundamental parameters of hadronic weakinteractions, in particular the strength of the long-rangepart of the parity-violating (PV) two-nucleon (
N N ) po-tential, mediated by one-pion exchange (OPE). Whilesuch a component is theoretically expected on the ba-sis of the weak interactions between quarks and thespontaneously-broken chiral symmetry of QCD, exper-imental evidence for its presence has proven to be elu-sive, and indeed current constraints are inconclusive, fora review see Ref. [1].In contrast, in the strong-interaction sector OPE domi-nates the
N N potential at internucleon separations largerthan 1.5 fm, and the spatial-spin-isospin correlations itinduces leave their imprint on many nuclear properties.These include, for example, i) the observed ordering oflevels in light nuclei and, in particular, the observed ab-sence of stable systems with mass number A = 8 [2], ii)the single-particle energy spacings and shell structure ofmedium- and heavy-weight nuclei [3] and, in particular,the observed change in the energy gap between the h / and g / orbits in tin isotopes [4], and iii) the relativemagnitude of the momentum distributions of pp versus np pairs in nuclei [5], which leads to the strong suppres-sion of ( e, e ′ pp ) relative to ( e, e ′ np ) knock-out cross sec-tions from C, recently measured at Jefferson Lab [6].The determination of the parameters that character-ize parity violation in nuclei requires evaluating matrixelements of hadronic weak-interaction operators betweeneigenstates of the strong-interaction Hamiltonian. Thus,experiments in this field are especially reliant on the-ory for their analysis and interpretation. For this rea-son, over the last several years, we have embarked ona program aimed at developing a systematic frameworkfor studying PV observables in few-nucleon systems, forwhich accurate—essentially exact—calculations are pos-sible. Two earlier papers [7, 8] dealt with the two-nucleonsystem, and provided a rather complete analysis of thelongitudinal asymmetry in ~p - p scattering [7] up to 300MeV lab energies, and of a variety of PV observables inthe np system [8], including, among others, the neutronspin rotation in ~n - p scattering and the photon angularasymmetry in the ~n - p radiative capture at thermal neu-tron energies. In the next phase, we have studied thespin rotation in ~n - d [9] and ~n - α [10] scattering at coldneutron energies.Measurements are available for the following PV ob-servables: the longitudinal analyzing power in ~p - p [11]–[14] and ~p - α [15] scattering, the photon asymmetryand photon circular polarization in, respectively, the H( ~n, γ ) H [16]–[17] and H( n, ~γ ) H [18] radiative cap-tures, and the neutron spin rotation in ~n - α scatter-ing [19, 20]. There is also a set of experiments whichare currently being planned, including measurementsof the neutron spin rotation in ~n - p [19] and ~n - d [21]scattering, and of the longitudinal asymmetry in thecharge-exchange reaction He( ~n, p ) H at cold neutron en-ergies [22], the subject of the present paper.At vanishing neutron energies, the only channels en-tering the incoming n - He scattering state have quan-tum numbers S +1 L J = S and S . In the out-going p - H scattering state, the relevant channels are: S +1 L J = S , S , D with positive parity, and P , P , P with negative parity. We show (in Sec. II) thatthe PV observable in this process, i.e. the longitudinal analyzing power A z , reads A z = a z cos θ , (1)where θ is the angle between the proton momentum andthe neutron beam direction, and the coefficient a z canbe expressed in terms of products of T -matrix elementsinvolving (three) parity-conserving (PC) and (three) PVtransitions as a z = −
4Σ Re (cid:16) √ T , , T , ∗ , − T , , T , ∗ , + √ T , , T , ∗ , + √ T , , T , ∗ , + √ T , , T , ∗ , (cid:17) , (2)and Σ = (cid:12)(cid:12)(cid:12) T , , (cid:12)(cid:12)(cid:12) + 3 (cid:12)(cid:12)(cid:12) T , , (cid:12)(cid:12)(cid:12) + 3 (cid:12)(cid:12)(cid:12) T , , (cid:12)(cid:12)(cid:12) . (3)In T ,JLS,L ′ S ′ the label J specifies the total angular mo-mentum, the superscripts 21 denote the charge-exchangetransition n - He to p - H (as opposed, for example, tothe elastic transition, which would be denoted by the su-perscripts 22), the subscripts LS ( L ′ S ′ ) are the relativeorbital angular momentum and channel spin of the n - He ( p - H) clusters, and lastly the overline is to note theinclusion of a convenient phase factor—see Eq. (16) be-low. The PC (PV) T -matrix elements have L + L ′ even(odd), and the sum Σ in Eq. (3) is proportional to the He( n, p ) H cross section. We observe that a z vanishes ifonly the channels S and P (with J = 0) are retained.The T -matrix elements are related to the (real) R -matrix elements (Sec. III and Appendix A), and thelatter for PC transitions are calculated via the Kohnvariational principle with the hyperspherical-harmonics(HH) method [23, 24] (Sec. V). We use strong-interactionHamiltonian models, consisting of the Argonne v (AV18) [25] or chiral (N3LO) [26] two-nucleon potentialin combination with the Urbana IX (UIX) [27] or chiral(N2LO) [28] three-nucleon potential. The HH calcula-tion is a challenging one, for two reasons. The first is thecoupled-channel nature of the scattering problem: evenat vanishing energies for the incident neutron, the elas-tic n - He and charge-exchange p - H channels are bothopen. The second is the presence of a J π = 0 + reso-nant state (of zero total isospin) between the p - H and n - He thresholds, which slows down the convergence ofthe expansion, and requires a large number of HH basisfunctions in order to achieve numerically stable results.Further discussion of this aspect of the calculations is inSec. V, where we also present current predictions for the n - He scattering lengths corresponding to the Hamilto- nian models mentioned earlier. They are in good agree-ment with the measured values.The R -matrix elements involving PV transitions arecomputed in first-order perturbation theory with Quan-tum Monte Carlo techniques (Sec. VI). We adopt as PVpotential the meson-exchange (DDH) model of Desplan-ques et al. [29] as well as the pionless effective-field-theory(EFT) model recently derived in Refs. [30, 31] (Sec. IV),and present results for the various components of theDDH and EFT potentials in combination with the AV18,AV18/UIX, N3LO, and N3LO/N2LO Hamiltonians inSec. VII. Additional results for the R - and T -matrixelements, and combinations thereof entering the PV ob-servable, are listed (for the AV18/UIX) in Appendix Bfor completeness. For the DDH model only, we alsopresent predictions for a z corresponding essentially—butsee Sec. IV for further details—to the “best values” ofthe π -, ρ -, and ω -meson weak-interaction coupling con-stants [29]. These predictions range from –9.44 to –2.48in units of 10 − depending on whether the N3LO/N2LOor AV18/UIX Hamiltonian is considered, and thus ex-hibit a significant model dependence due to cancellations(or lack thereof) between the pion and vector-meson con-tributions.It is useful to express the asymmetry as a z = h π C π + h ρ C ρ + h ρ C ρ + h ρ C ρ + h ω C ω + h ω C ω , (4)where the h iα ’s, α = π , ρ , ω and i = 0, 1, 2, denotethe PV coupling constants in the DDH model along withthe isospin content of the corresponding interaction. Thecoefficients C iα are listed in Table I, and depend on theinput Hamiltonian used to generate the continuum wavefunctions, as well as on the assumed values for the PCpion- and vector-meson coupling constants and associ-ated cutoffs (see Table IV). C π C ρ C ρ C ρ C ω C ω AV18 –0.1892(86) –0.0364(40) +0.0193(9) –0.0006(1) –0.0334(29) +0.0413(10)AV18/UIX –0.1853(150) –0.0380(70) +0.0230(18) –0.0011(1) –0.0231(56) +0.0500(20)N3LO –0.1989(87) –0.0120(49) +0.0242 (9) +0.0002(1) +0.0080(30) +0.0587(11)N3LO/N2LO –0.1110(75) +0.0379(56) +0.0194 (10) –0.0007(1) +0.0457(36) +0.0408(14)TABLE I: The coefficients C iα entering the PV observable a z , corresponding to the AV18, AV18/UIX, N3LO, and N3LO/N2LOstrong-interaction Hamiltonians. The statistical errors due to the Monte Carlo integrations are indicated in parentheses, andcorrespond to a sample consisting of ∼ The coefficients C iα follow from the linear combinationgiven in Eq. (2). Isotensor ρ -exchange ( C ρ ) is negligible.The isoscalar and isovector vector-meson exchanges givecontributions of the same magnitude, both of which aresmaller than OPE. However, the OPE contribution seemsto be significantly suppressed. For example, in the caseof the neutron spin rotation in ~n - d scattering this contri-bution is calculated to be at least a factor of ∼
30 largerthan that of any of the ρ and ω exchanges, which is notthe case for the process under consideration. This maybe due to the predominant isoscalar character of the S and P channels—see discussion in Appendix B. TheN3LO/N2LO results should be considered as preliminary,since the HH solution for the 0 + wave function has notyet fully converged (at least as far as the singlet scat-tering length is concerned, see Sec. V). This fact mayexplain why the inclusion of a three-nucleon potentiallike N2LO [28] should reduce C π by almost a factor oftwo relative to the other models. This point will be dis-cussed in Secs. V and VI. Finally we note that the “bestvalues” for the PV couplings constants of the pion and ρ -meson are (in units of 10 − ) respectively +4.56 and–16.4, and this leads to the large cancellation (and con-sequent model dependence) in the values predicted for a z and referred to earlier.We conclude by observing that the EFT analysis pre-sented in this work could be improved by employing chi-ral potentials in both the strong- and weak-interactionsectors. At order Q/ Λ χ , where Q is the low en-ergy/momentum scale that characterizes the particularprocess of interest, and Λ χ ≃ A = 2–5 systems, whichwould constrain, in fact over-constrain, these eight LECs.Some of these have been mentioned above, additionalones include, for example, measurements of the photonasymmetries in the radiative captures H( ~n, γ ) H and He( ~n, γ ) He. These processes are strongly suppressed:the experimental values for the corresponding (PC) crosssections [32, 33] are, respectively, almost 3 and 4 ordersof magnitude smaller than measured in H( n, γ ) H. Onewould naively expect relatively large PV asymmetries inthese cases, possibly orders of magnitude larger than inthe A =2 system. Clearly, accurate theoretical estimatesfor them could be useful in motivating our experimen-tal colleagues to carry out these extremely challengingmeasurements.From a theoretical perspective, most of the method-ological and technical developments needed to carry outthe calculations are already in place. We have recentlyreported results [34] for the A = 3 and 4 (PC) captures,using wave functions obtained from the N3LO/N2LOHamiltonian and electromagnetic currents derived in chi-ral EFT up to one loop [35], which are in excellent agree-ment with data. However, there is one aspect in the com-putation of the proposed PV threshold captures, whichstill needs to be addressed: the determination of thesmall admixtures induced by the PV potential into thebound and continuum wave functions. Even a first-orderperturbative treatment of those admixtures requires con-struction of the full Green’s function for the strong (PC)Hamiltonian, an impractical task. However, it may bepossible to generate them using correlated basis meth-ods, similar to those employed in Ref. [36]. II. THE PARITY-VIOLATING OBSERVABLE
The neutron energies in the reaction He( ~n, p ) H of in-terest here are in the meV range, and at these energiesonly two channels are open: the n - He elastic channel andthe p - H charge-exchange channel. In the following, theindex γ =1 (2) is used to identify the p - H ( n - He) clus-ters in the final (initial) state. In the absence of strongand weak interactions between the two clusters, the wavefunction in channel γ is written asΦ m m γ = 1 √ X p =1 Ψ m γ ( ijk ) χ m γ ( l ) φ q γ ( y p ) ≡ √ X p =1 Φ m m γ, p , (5)where Ψ m γ is the (antisymmetrized) trinucleon bound-state wave function in spin projection m , χ m γ is the nu-cleon spin-isospin state with spin and isospin projections m and p for γ =1 or n for γ =2, respectively, and φ is theinter-cluster wave function, i.e. a Coulomb wave func-tion for γ = 1 or simply a plane wave e i q · y p for γ = 2.The separation between the center-of-mass positions ofthe two clusters is denoted by y p with y p = r l − R ijk ,and their relative momentum is specified by q γ , so thatthe energy E is given by E = − B γ + q γ µ γ , µ γ = 1 m γ + 1 M γ . (6)Here B γ and M γ are the binding energy and mass of H( He) for γ =1 (2), and m γ is the proton (neutron) massfor γ =1 (2). Lastly, the wave functions in Eq. (5) areantisymmetrized by summing over the four permutations p with ( ijk, l ) ≡ (123 , , , , m m γ = 1 √ X p =1 X LSJ i L Z L SJJ z m m Ω JJ z γLS, p F FL ( q γ ; y p ) q γ y p , (7) where F FL ( q γ ; y p ) reduces to a regular Coulomb function F L ( q γ ; y p ) (multiplied by a phase factor we need not spec-ify here) for γ = 1 or a spherical Bessel function x j L ( x )for γ = 2, with x = q γ y p . The channel functions Ω JJ z γLS, p are defined asΩ JJ z γLS, p = h Y L (ˆ y p ) ⊗ (cid:2) Ψ γ ( ijk ) ⊗ χ γ ( l ) (cid:3) S i JJ z , (8)while the Clebsch-Gordan coefficients associated with there-coupling of the angular momenta (and other factors)are lumped into Z LMSJJ z m m = √ π √ L + 1 h / , m ; 1 / , m | S, S z i×h L, M ; S, S z | J, J z i . (9)The momentum q γ has been taken to define the spin-quantization axis, i.e. the z -axis.In the presence of inter-cluster interactions, the n - Hewave function in the asymptotic region readsΨ m m γ =2 ≃ √ X p =1 X LSJ i L Z L SJJ z m m " Ω JJ z LS, p j L ( q y p ) + X L ′ S ′ T ,JLS, L ′ S ′ Ω JJ z L ′ S ′ , p e i ( q y p − L ′ π/ y p + X L ′ S ′ T ,JLS, L ′ S ′ Ω JJ z L ′ S ′ , p e i [ q y p − L ′ π/ − η ln(2 q y p )+ σ L ′ ] y p , (10)and contains outgoing spherical waves in the n - He elas-tic channel ( γ = 2) as well as in the p - H charge-exchangechannel ( γ = 1) multiplied by corresponding T -matrix el-ements T γγ ′ ,JLS,L ′ S ′ . Here η = αµ /q , where α is the finestructure constant and µ is the p - H reduced mass de-fined above, and σ L is the Coulomb phase-shift. ThusCoulomb distortion in the p - H outgoing state is fullyaccounted for.The probability amplitude M m ′ m ′ , m m to observe a p - H final state with spin projections m ′ and m ′ , respec-tively, is obtained from h Φ m ′ m ′ γ =1 , p =1 | Ψ m m γ =2 i = 1 √ M m ′ m ′ , m m × e i [ q y − η ln(2 q y )] y , (11) where we have assumed that the p - H state is in partition(123,4) corresponding to permutation p = 1, namely thebound cluster consists of particles 123 and the proton isparticle 4. For brevity, we have also set y ≡ y p =1 . Usingthe orthonormality of the channel functions Ω JJ z γLS, p , wefind M m ′ m ′ , m m = 1 √ π X JLSL ′ S ′ i L ( − i ) L ′ e i σ L ′ √ L ′ + 1 Z L SJJ z m m × T ,JLS, L ′ S ′ Z L ′ M ′ S ′ JJ z m ′ m ′ Y L ′ M ′ (ˆ y ) , (12)where the Clebsch-Gordan coefficients require J z = S z = m + m , S ′ z = m ′ + m ′ , and M ′ = J z − S ′ z = m + m − ( m ′ + m ′ ).The spin-averaged cross section follows from σ ≡ d σ dΩ = 14 µ µ q q X m ,m X m ′ ,m ′ | M m ′ m ′ , m m | , (13)since (1 / q /µ ) | M m ′ m ′ , m ,m | dΩ is the flux ofoutgoing particles in the solid angle dΩ ≡ dˆ y , and(1 / q /µ ) is the incident flux, where the factors 1 / / √ / A z is defined as σ A z = 12 µ µ q q X m X m ′ ,m ′ (cid:20) | M m ′ m ′ , m m =+ | −| M m ′ m ′ , m m = − | (cid:21) . (14)At meV energies it suffices to keep only L = 0 in theentrance channel, so that M m ′ m ′ , m m = X J =0 , X L ′ S ′ h / , m ; 1 / , m | J, J z i× T ,J J,L ′ S ′ √ L ′ + 1 Z L ′ M ′ S ′ JJ z m ′ m ′ Y L ′ M ′ (ˆ y ) , (15)where we have defined T ,J J,L ′ S ′ = ( − i ) L ′ e i σ L ′ T ,J J,L ′ S ′ . (16)After inserting the expression for Z L ′ M ′ S ′ JJ z m ′ m ′ and carry-ing out the sums over m , m and m ′ , m ′ , we find theunpolarized cross section to be given by σ = 14 µ µ q q X J =0 , X L ′ S ′ (2 J + 1) (cid:12)(cid:12)(cid:12) T ,J J,L ′ S ′ (cid:12)(cid:12)(cid:12) = 14 µ µ q q " (cid:12)(cid:12)(cid:12) T , , (cid:12)(cid:12)(cid:12) + 3 (cid:12)(cid:12)(cid:12) T , , (cid:12)(cid:12)(cid:12) + 3 (cid:12)(cid:12)(cid:12) T , , (cid:12)(cid:12)(cid:12) , (17)where in the second line we have ignored T -matrix el-ements involving transitions to odd parity final states(and hence parity violating), since these are induced byhadronic weak interactions and consequently are muchsmaller than the parity-conserving T -matrices associatedwith strong interactions. We observe that the matrix ele-ments T ,J (and T ,J ) are finite in the limit q = 0, andtherefore σ is divergent as q goes to zero, as expectedfor a neutron capture reaction.The asymmetry A z can be written as σ A z = 12 µ µ q q X J ,J =0 , X L L S ǫ L L T ,J J ,L S h T ,J J ,L S i ∗ × X | M | C J J L L S ( | M | ) P | M | L ( θ ) P | M | L ( θ ) , (18) TABLE II: The coefficients C J J L L S ( | M | ) for the relevantchannels. J , J L L S | M | C J J L L S ( | M | )0 , −√
31 0 1 0 +11 2 1 0 −√
21 2 1 1 − p / , −√
30 1 1 0 +12 1 1 0 −√
22 1 1 1 − p / , −√
62 1 1 0 −√
32 1 1 1 − p /
41 0 1 0 −√
61 2 1 0 −√
31 2 1 1 − p / where the P | M | L ( θ )’s are associated Legendre functions, θ is the angle of the outgoing proton momentum relativeto the direction of the incident beam, the C J J L L S ( | M | )’sdenote combinations of Clebsch-Gordan coefficients, de-fined as C J J L L S ( | M | ) = 12 π X J z X µ = ±| M | s ( L −| M | )!( L −| M | )!( L + | M | )!( L + | M | )! × Z L µ SJ J z m m =+1 / Z L µ SJ J z m m =+1 / , (19)and lastly the phase factor ǫ L L , ǫ L L ≡ − ( − ) L + L , (20)ensures that either L or L must be odd, which in turnimplies that either T ,J J ,L S or T ,J J ,L S involves a parity-violating transition, i.e. a transition from an incomingpositive parity n - He state to an outgoing negative parity p - H state. The non-vanishing C ’s for the relevant chan-nels are listed in Table II, and evaluation of the sums inEq. (18) allows one to express the parity-violating asym-metry as in Eqs.(1)–(3). III. T -MATRIX ELEMENTS The calculation proceeds in two steps: we first deter-mine, via the Kohn variational principle, the R -matrixelements, and then relate these to the T -matrix elements.The wave function describing a scattering state with to-tal angular momentum JJ z in channel γLS is writtenasΨ JJ z γ,LS = Ψ C,JJ z γ,LS +Ψ F,JJ z γ,LS + X γ ′ L ′ S ′ R γγ ′ ,JLS,L ′ S ′ Ψ G,JJ z γ ′ ,L ′ S ′ , (21)where the asymptotic wave functions Ψ λ,JJ z γ,LS with λ = F, G are defined asΨ λ,JJ z γ,LS = D γ √ X p =1 Ω JJ z γLS,p F λL ( q γ ; y p ) q γ y p , (22)and the superscript λ = F is to denote the regular ra-dial functions introduced earlier in Eq. (7), and λ = G is to denote the irregular Coulomb or spherical Besselfunctions, namely γ = 1 : F GL ( x ) = e G L ( η , x ) ; γ = 2 : F GL ( x ) = − x e y L ( x ) . (23)The tilde over G L and y L indicates that they have beenmultiplied by short-range cutoffs in order to remove thesingularity at the origin. Thus F GL is well-behaved in allspace. The normalization factor D γ , D γ = r µ γ q γ κ (24)and κ = p /
2, is introduced for convenience— κ is anumerical factor relating the inter-cluster separation y p to the Jacobi variable x p , i.e. x p = κ y p (see Eq. (44)below).The wave functions Ψ C,JJ z γ,LS vanish in the asymptoticregion, and describe the dynamics of the interactingnucleons when they are close to each other, while the R γγ ′ ,JLS,L ′ S ′ ’s are the R -matrix elements. The latter, as wellas the coefficients entering the expansion of Ψ C,JJ z γ,LS interms of hyperspherical-harmonics functions, are deter-mined via the Kohn variational principle h R γγ ′ ,JLS,L ′ S ′ i = R γ ′ γ,JL ′ S ′ ,LS − h Ψ JJ z γ,LS | H − E | Ψ JJ z γ ′ ,L ′ S ′ i , (25)as discussed in Sec. V.The next step consists in relating the R - to the T -matrix elements. To this end, it is convenient to simplifythe notation by dropping the superscripts JJ z and byintroducing a single label α to denote the channel quan-tum numbers LS , so that the wave functions in Eq. (21)corresponding to γ = 1 and 2 are written asΨ ,α = Ψ C ,α +Ψ F ,α + X α ′ R α,α ′ Ψ G ,α ′ + X α ′ R α,α ′ Ψ G ,α ′ , (26)Ψ ,α = Ψ C ,α +Ψ F ,α + X α ′ R α,α ′ Ψ G ,α ′ + X α ′ R α,α ′ Ψ G ,α ′ . (27)From these we form the linear combinationΨ = X α ′ ( U α,α ′ Ψ ,α ′ + V α,α ′ Ψ ,α ′ ) , (28)where the matrices U and V are determined below. In-serting the expressions above for Ψ γ,α and rearrangingterms lead toΨ = Ψ C + X α ′ h U − i ( U R + V R ) i α,α ′ Ψ F ,α ′ + X α ′ (cid:0) U R + V R (cid:1) α,α ′ (cid:16) Ψ G ,α ′ + i Ψ F ,α ′ (cid:17) + X α ′ h V − i ( U R + V R ) i α,α ′ Ψ F ,α ′ + X α ′ (cid:0) U R + V R (cid:1) α,α ′ (cid:16) Ψ G ,α ′ + i Ψ F ,α ′ (cid:17) , (29)where Ψ C is a combination of internal parts of no interesthere. We now require Ψ to consist, in the asymptoticregion, of a plane wave in channel γ =2 (or n - He) and ofa purely outgoing wave in channel γ =1 (or p - H). Theserequirements are satisfied by demanding that U − i ( U R + V R ) = 0 , (30) V − i ( U R + V R ) = I , (31)where I is the identity matrix. Comparing the resulting Ψ with the wave function given in Eq. (10), specificallyits component in channel LSJ , allows one to express the T -matrix as T ,JLS,L ′ S ′ = D D q (cid:0) U J R ,J + V J R ,J (cid:1) LS,L ′ S ′ = − i D D q U JLS,L ′ S ′ , (32)where we have reinstated the LSJ labels. Finally thematrix U is obtained by solving the system in Eq. (31): T ,JLS,L ′ S ′ = D D q "h I − i R ,J + R ,J ( I − i R ,J ) − R ,J i − R ,J ( I − i R ,J ) − LS,L ′ S ′ . (33)In fact, we compute the R -matrix elements at zero en- ergy, i.e. in the limit q →
0, and define R ,JLS,L ′ S ′ = R ,JLS,L ′ S ′ q L ′ +1 / , R ,JLS,L ′ S ′ = R ,JLS,L ′ S ′ q L +1 / , n c DDH n f DDH n ( r ) c EFT n f EFT n ( r ) O ( n ) ij g π h π √ m f π ( r ) µ Λ χ C f µ ( r ) ( τ i × τ j ) z ( σ i + σ j ) · X (1) ij, − − g ρ h ρ m f ρ ( r ) 0 0 τ i · τ j ( σ i − σ j ) · X (2) ij, + − g ρ h ρ (1+ κ ρ ) m f ρ ( r ) 0 0 τ i · τ j ( σ i × σ j ) · X (3) ij, − − g ρ h ρ m f ρ ( r ) µ Λ χ ( C + C ) f µ ( r ) ( τ i + τ j ) z ( σ i − σ j ) · X (4) ij, + − g ρ h ρ (1+ κ ρ )2 m f ρ ( r ) 0 0 ( τ i + τ j ) z ( σ i × σ j ) · X (5) ij, − − g ρ h ρ √ m f ρ ( r ) − µ Λ χ C f µ ( r ) (3 τ i,z τ j,z − τ i · τ j ) ( σ i − σ j ) · X (6) ij, + − g ρ h ρ (1+ κ ρ )2 √ m f ρ ( r ) 0 0 (3 τ i,z τ j,z − τ i · τ j ) ( σ i × σ j ) · X (7) ij, − − g ω h ω m f ω ( r ) µ Λ χ C f µ ( r ) ( σ i − σ j ) · X (8) ij, + − g ω h ω (1+ κ ω ) m f ω ( r ) µ Λ χ ˜ C f µ ( r ) ( σ i × σ j ) · X (9) ij, − − g ω h ω m f ω ( r ) 0 0 ( τ i + τ j ) z ( σ i − σ j ) · X (10) ij, + − g ω h ω (1+ κ ω )2 m f ω ( r ) 0 0 ( τ i + τ j ) z ( σ i × σ j ) · X (11) ij, − − g ω h ω − g ρ h ρ m f ρ ( r ) 0 0 ( τ i − τ j ) z ( σ i + σ j ) · X (12) ij, + TABLE III: Components of the DDH and EFT models for the parity-violating potential. The vector operators X ( n ) ij, ∓ andfunctions f x ( r ), x = π, ρ, ω, µ , are defined in Eqs. (38)–(39) and Eqs. (40)–(41), respectively. Only 5 operators and low-energyconstants enter the pionless EFT interaction at the leading order, and in this paper they have been chosen to correspond tothe rows 1, 4, 6, 8 and 9. R ,JLS,L ′ S ′ = R ,JLS,L ′ S ′ q L + L ′ +12 , (34)and it can be shown that the R -matrix elements are finitein this limit. In particular, we note that the factor q L fol-lows from the small argument expansion of the sphericalBessel function j L in Ψ F,JJ z γ =2 ,LS , while the extra q / is dueto the normalization D . At zero energy, we have h I − i R ,J + R ,J ( I − i R ,J ) − R ,J i → I , (35)since R ,J and the product R ,J R ,J are proportionalto q or higher powers of q . Furthermore, the relevant T -matrix elements entering the expression for the asymme-try A z are those with quantum number L = 0 in channel γ = 2, and hence T ,J J,L ′ S ′ = 1 √ q X L ′′ S ′′ R ,J J,L ′′ S ′′ (cid:0) I − i R ,J (cid:1) − L ′′ S ′′ ,L ′ S ′ , (36)with J = 0 ,
1. Note that we have neglected the differencein the n - He and p - H reduced masses.
IV. THE PARITY-VIOLATING POTENTIAL
Two different models of the PV weak-interaction po-tentials are adopted in the calculations reported below. One is the model developed thirty years ago by Desplan-ques et al. [29] (and known as DDH): it is parametrizedin terms of π -, ρ -, and ω -meson exchanges, and involvesin practice six weak pion and vector-meson coupling con-stants to the nucleon [37]. These were estimated withina quark model approach incorporating symmetry argu-ments and current algebra requirements [29, 38]. Due tothe inherent limitations of such an analysis, however, thecoupling constants determined in this way have ratherwide ranges of allowed values.The other model for the PV potential considered inthe present work is that formulated by Zhu et al. [30] in2005, and reduced to its minimal form by Girlanda [31]in 2008, within an effective-field-theory (EFT) approachin which only nucleon degrees of freedom are retainedexplicitly. At lowest order Q/ Λ χ , where Q is the smallmomentum scale characterizing the low-energy PV pro-cess and Λ χ ≃ v αij = X n =1 c αn O ( n ) ij , α = DDH or EFT , (37)where the parameters c αn and operators O ( n ) ij , n =1 , . . . ,
12, are listed in Table III. In this table the vectoroperators X ( n ) ij, ± are defined as X ( n ) ij, + ≡ [ p ij , f n ( r ij )] + , (38) X ( n ) ij, − ≡ i [ p ij , f n ( r ij )] − , (39)where [ . . . , . . . ] ∓ denotes the commutator ( − ) or anti-commutator (+), and p ij is the relative momentum oper-ator, p ij ≡ ( p i − p j ) /
2. In the DDH model, the functions f x ( r ), x = π, ρ and ω , are Yukawa functions, suitablymodified by the inclusion of monopole form factors, f x ( r ) = 14 π r (cid:26) e − m x r − e − Λ x r (cid:20) x r (cid:18) − m x Λ x (cid:19)(cid:21)(cid:27) . (40)In the EFT model, however, the short-distance behavioris described by a single function f µ ( r ), which is itselftaken as a Yukawa function with mass parameter µ , f µ ( r ) = 14 π r e − µr , (41)with µ ≃ m π as appropriate in the present formulation,in which pion degrees of freedom are integrated out.In the potential v DDH ij , the strong-interaction cou-pling constants of the π -, ρ -, and ω -meson to the nu-cleon are denoted as g π , g ρ , κ ρ , g ω , κ ω , while the weak-interaction ones as h π , h ρ , h ρ , h ρ , h ω , h ω , where the su-perscripts 0, 1, and 2 specify the isoscalar, isovector, andisotensor content of the corresponding interaction compo-nents. In the EFT model, the five low-energy constants C , ˜ C , C + C , C and C completely characterize v EFT ij ,to lowest order Q/ Λ χ . g α / π κ α × h α × h α × h α Λ α (GeV/c) π ρ ω π -, ρ -, and ω -meson in the DDH potential. The values for the coupling constants and short-rangecutoffs in the DDH model are listed in Table IV, while themass µ in the EFT model is taken to be m π . These valuesfor coupling constants and cutoffs were also used in theDDH-based calculations of PV two-nucleon observablesin Refs. [7, 8] and neutron spin rotation in ~n d scatter-ing [9]. In particular, we note that the linear combination of ρ - and ω -meson weak coupling constants correspond-ing to pp states has been taken from an earlier analysisof ~p p elastic scattering experiments [7]. The remainingcouplings are the “best value” estimates, suggested inRef. [29].In the analysis of the a z observable to follow, we willreport results for the coefficients I DDH n and I EFT n in theexpansion a z = X n =1 c αn I αn . (42)Thus we will not need to consider specific values (or rangeof values) for the strength parameters c αn . However, the I αn depend on the masses (and short-range cutoffs Λ x for the DDH model) occurring in the Yukawa functions.Note that the coefficients C iα entering Eq. (4) are ob-tained from the I DDH n ’s and c DDH n ’s listed in Table IIIvia C π = + g π √ m I DDH1 ,C ρ = − g ρ m I DDH2 − g ρ (1 + κ ρ ) m I DDH3 ,C ρ = − g ρ m I DDH4 − g ρ (1 + κ ρ )2 m I DDH5 + g ρ m I DDH12 ,C ρ = − g ρ √ m I DDH6 − g ρ (1 + κ ρ )2 √ m I DDH7 , (43) C ω = − g ω m I DDH8 − g ω (1 + κ ω ) m I DDH9 ,C ω = − g ω m I DDH10 − g ω (1 + κ ω )2 m I DDH11 − g ω m I DDH12 . V. THE HH WAVE FUNCTIONS
The “internal”wave function Ψ
C,JJ z γ,LS , see Eq. (21), isexpanded in the HH basis. For four equal mass particles,a suitable choice for the Jacobi vectors is x p = r (cid:18) r l − r i + r j + r k (cid:19) , x p = r (cid:18) r k − r i + r j (cid:19) , (44) x p = r j − r i , where p specifies a given permutation corresponding tothe ordering ( ijkl ). By definition, the permutation p = 1is chosen to correspond to (1234).For the given Jacobi vectors, the hyperspherical coor-dinates include the so-called hyperradius ρ , defined by ρ = q x p + x p + x p (independent of p ) , (45)and a set of angular variables which in the Zernike andBrinkman [39, 40] representation are (i) the polar anglesˆ x ip ≡ ( θ ip , φ ip ) of each Jacobi vector, and (ii) the twoadditional “hyperspherical” angles φ p and φ p , definedascos φ p = x p q x p + x p , cos φ p = x p q x p + x p + x p , (46) where x jp is the magnitude of the Jacobi vector x jp . Theset of angular variables ˆ x p , ˆ x p , ˆ x p , φ p , and φ p is de-noted hereafter as Ω p . A generic HH function reads H K Λ Mℓ ℓ ℓ L n n (Ω p ) = N ℓ ℓ ℓ n n (cid:20)h Y ℓ (ˆ x p ) ⊗ Y ℓ (ˆ x p ) i L ⊗ Y ℓ (ˆ x p ) (cid:21) Λ M (sin φ p ) ℓ (cos φ p ) ℓ (sin φ p ) ℓ + ℓ +2 n × (cos φ p ) ℓ P ℓ +1 / , ℓ +1 / n (cos 2 φ p ) P ℓ + ℓ +2 n +2 , ℓ +1 / n (cos 2 φ p ) , (47)where P a,bn are Jacobi polynomials, and the coefficients N ℓ ℓ ℓ n n are normalization factors. The quantity K = ℓ + ℓ + ℓ + 2 ( n + n ) is the so-called grand angular quan-tum number. The HH functions are the eigenfunctionsof the hyperangular part of the kinetic energy operator.Another important property is that ρ K H K Λ Mℓ ℓ ℓ L n n (Ω p )are homogeneous polynomials of the particle coordinatesof degree K .A set of antisymmetrized hyperangular-spin-isospinstates of grand angular quantum number K , total orbital angular momentum Λ, total spin Σ, and total isospin T (for the given values of total angular momentum J andparity π ) can be constructed as follows:Ψ K ΛΣ Tµ = X p =1 Φ K ΛΣ Tµ ( ijkl ) , (48)where the sum is over the 12 even permutations p ≡ ijkl ,andΦ K ΛΣ Tµ ( ijkl ) = (cid:20) H K Λ Mℓ ℓ ℓ L n n (Ω p ) ⊗ (cid:20)h(cid:2) χ i ⊗ χ j (cid:3) S a ⊗ χ k i S b ⊗ χ l (cid:21) Σ (cid:21) JJ z (cid:20)h(cid:2) ξ i ⊗ ξ j (cid:3) T a ⊗ ξ k i T b ⊗ ξ l (cid:21) T T z . (49)Here, χ i ( ξ i ) denotes the spin (isospin) state of particle i .The total orbital angular momentum Λ of the HH func-tion is coupled to the total spin Σ to give the total angu-lar momentum JJ z , whereas the parity π is ( − ℓ + ℓ + ℓ .The quantum number T specifies the total isospin of thestate, and µ labels the possible choices of hyperangular,spin and isospin quantum numbers, namely µ ≡ { ℓ , ℓ , ℓ , L , n , n , S a , S b , T a , T b } , (50)compatible with the given values of K , Λ, Σ, T , J ,and π . Another important classification scheme for thestates is to group them in “channels”: states belong-ing to the same channel have the same values of angular( ℓ , ℓ , ℓ , L , Λ), spin ( S a , S b , Σ), and isospin ( T a , T b , T )quantum numbers, but different values of n and n .Each state Ψ K ΛΣ Tµ entering the expansion of the four-nucleon wave function must be antisymmetric under theexchange of any pair of particles. Consequently, it isnecessary to consider states such thatΦ K ΛΣ Tµ ( ijkl ) = − Φ K ΛΣ Tµ ( jikl ) , (51) which is fulfilled when the condition ℓ + S a + T a = odd , (52)is satisfied.The number M K ΛΣ T of antisymmetrized functionsΨ K ΛΣ Tµ having given values of K , Λ, Σ, and T , but differ-ent combinations of quantum numbers µ —see Eq.(50)—is in general very large. In addition to the degeneracyof the HH basis, the four spins (isospins) can be coupledin different ways to total Σ ( T ). However, many of thestates Ψ K ΛΣ Tµ , with µ ranging from 1 to M K ΛΣ T , arelinearly dependent. In the expansion of Ψ C,JJ z γ,LS , it is nec-essary to include only the subset of linearly independentstates, whose number is fortunately significantly smallerthan M K ΛΣ T .The internal part of the wave function can be finallywritten asΨ C,JJ z γ,LS = X K ΛΣ T X µ u γ,LSK ΛΣ T µ ( ρ )Ψ K ΛΣ Tµ , (53)where the sum is restricted only to the linearly indepen-dent states. We have found it convenient to expand the0“hyperradial” functions u γ,LSK ΛΣ T µ ( ρ ) in a complete set offunctions, namely u γ,LSK ΛΣ T µ ( ρ ) = M − X m =0 c γ,LSK ΛΣ T µ,m g m ( ρ ) , (54)and have chosen g m ( ρ ) = s m !( m + 8)! β / L (8) m ( βρ ) e − βρ/ , (55)where L (8) l ( βρ ) are Laguerre polynomials [41].The c coefficients of the expansion (54) and the R-matrix elements of Eq. (21) are determined variationallyvia the Kohn variational principle. This principle statesthat the functional h R γγ ′ ,JLS,L ′ S ′ i defined in Eq. (25) is sta-tionary with respect to variations in the R γγ ′ ,JLS,L ′ S ′ and c γ,LSK ΛΣ T µ,m . By applying this principle, a linear set ofequations for R γγ ′ ,JLS,L ′ S ′ and c γ,LSK ΛΣ T µ,m is obtained [23],then solved using the Lanczos algorithm. The other pa-rameter entering the expansion is the (non linear) pa-rameter β (see Eq. (55)), used to describe the hyperra-dial functions u γ,LSK ΛΣ T µ ( ρ ). We have checked that, oncea sufficient number M of functions g m ( ρ ) are employed( M ≈ β .In the present work we have used β = 4 fm − .The application of the method has two main difficul-ties. The first is the accurate computation of the matrixelements of the Hamiltonian. By exploiting the proper-ties of the HH functions, however, this task can be notice-ably simplified, as discussed in Refs. [23, 42]. The seconddifficulty is the slow convergence of the HH expansion.This problem has been overcome by dividing the set ofstates Ψ K ΛΣ Tµ defined in Eq. (48) (in the following re-ferred to simply as “HH states”) in classes , dependingon the value of L = ℓ + ℓ + ℓ , total isospin T , and n and n . In the present paper, we have considered fourdifferent classes. Since for n - He scattering the asymp-totic states do not have a definite total isospin (they area superposition of T = 0 and T = 1 components), it ismandatory to include HH states with both T = 0 and 1.The contribution of T = 2 states is expected to be tinyand consequently they have been ignored in the presentpaper.Following Refs. [42, 43], in the first class we have in-cluded the n = 0 HH states belonging to some specialchannels, for which the convergence has been found tobe critical. The radial part of these HH states dependsonly on cos φ p = r ij /ρ and thus they take into accounttwo-body correlations. The n > L ≤
2. The other classes arethen defined simply by grouping HH states belonging tochannels with an increasing value of L . In particular, for the construction of the positive (negative) parity “inter-nal”wave function Ψ C,JJ z γ,LS , classes 3 and 4 include all HHstates with L = 4 and 6 ( L = 3 and 5), respectively.The convergence of these last two classes is less critical,and consequently, only HH states with lower values ofgrand angular quantum number K need be considered.Moreover, the convergence with L is quite fast. In par-ticular, we have found that, at the energy considered, thecontribution of HH states with L > i = 1 , . . . , K ≤ K i , where K , . . . , K are a set of nonnegative integers. The con-vergence of a quantity of interest (for example, the phase-shifts, or the coefficient a z defining the PV asymmetry)is then studied by increasing the values of K i . A morecomplete study of the convergence will be presented else-where [24].To exhibit the convergence pattern, we report in Ta-ble V the calculated n - He scattering lengths. As is evi-dent from Eq. (10), they are defined as a J = − lim q → T ,J J, J , (56)with both incoming and outgoing n - He clusters inrelative S-wave. Note that in general this scatteringlength is complex, since the channel p - H is alwaysopen, and therefore the unitarity condition imposes thatIm a J <
0, since the total cross section is proportional to P J =0 , (2 J + 1) Im T ,J J, J . The results obtained for thesinglet ( J = 0) and triplet ( J = 1) scattering lengths arereported in Table V, for all four potential models usedin this work. The calculated n - He scattering lengthsare compared with experimental values and the resultsof other calculations available in the literature.Inspection of the table shows that the convergence forthe triplet scattering length is very good, and that thereis reasonable agreement with available experimental val-ues, and the results of other calculations, in particularthose of the AGS method. In the case of the singletscattering length, the situation is more delicate, since inthe channel J π = 0 + the n - He interaction is attrac-tive and the wave function must be orthogonal to the He bound state. Consequently, the convergence is moreproblematic, in particular for the N3LO/N2LO interac-tion model. In the row labeled “EXT”, we have reportedthe extrapolated values for this quantity obtained by an-alyzing the convergence pattern. For the AV18, N3LO,and AV18/UIX interaction models we observe reasonableagreement with the results of other calculations and theexperimental data. The N3LO/N2LO values are signifi-cantly different from those obtained with the other inter-action models, which is presumably related to the slowconvergence observed in this case. A complete study ofthe n - He scattering lengths is in progress [24].1
Triplet scattering length a (fm) K K K K AV18 N3LO AV18/UIX N3LO/N2LO28 28 20 20 3 . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . − i . . . a (fm) K K K K AV18 N3LO AV18/UIX N3LO/N2LO48 44 30 22 7 . − i .
27 7 . − i .
23 7 . − i .
65 4 . − i . . − i .
16 7 . − i .
20 7 . − i .
59 5 . − i . . − i .
70 7 . − i .
97 7 . − i .
44 6 . − i . . − i .
02 7 . − i . . − i .
97 7 . − i . . − i . . − i . . . n - He singlet and triplet scattering lengths corresponding to the inclusion, in the internal part ofthe wave function, of four different classes in which the HH basis has been subdivided. For the singlet scattering length, the linelabeled “EXT” reports the extrapolated values obtained by examining the convergence pattern with increasing number of HHfunctions in the expansion. The calculated scattering lengths are compared with results obtained using the Resonating GroupMethod (RGM), Faddeev-Yakubovsky (FY) equations, Alt-Grassberger-Sandhas (AGS) equations, as well as with results ofR-matrix analyses. The experimental values are reported in the rows labeled “EXP” (the imaginary parts are taken fromRef. [44]).
Recently, there has been a new measurement [50] forthe quantity ∆ a ′ = Re( a − a ) = − . a ′ with the AV18, N3LO, AV18/UIX,and N3LO/N2LO models are − . − . − .
50, and − .
65 fm, respectively. Again the N3LO/N2LO valuestands out: it is off that obtained with the other interac-tion models and the measured value.The convergence for the negative-parity states is sim-ilar to that discussed above. For the 0 − state, there isa close resonant state and the convergence is slow as inthe 0 + case. For the 1 − state, the resonance is far andwe observe good convergence, as for the 1 + state. Note,however, that in these cases the N3LO/N2LO conver-gence pattern is not different from that observed withthe other models. VI. CALCULATION
There is a total of two (four) states with J = 0( J = 1): one (two) with positive parity having LS = 00( LS = 01 ,
21) and one (two) with negative parity having LS = 11 ( LS = 10 , R -matrix elements R γγ ′ , LS,LS with LS = 00 or LS = 11 for J = 0, and R γγ ′ , LS,L ′ S ′ with LS, L ′ S ′ = 01 ,
21 or
LS, L ′ S ′ = 10 ,
11 for J = 1, involv-ing parity-conserving transitions induced by the stronginteractions are calculated with the HH method, as de-scribed in the previous section. However, the R -matrixelements involving parity-violating (PV) transitions areobtained in first-order perturbation theory as R γγ ′ ,JLS,L ′ S ′ = −h Ψ JJ z γ ′ ,L ′ S ′ | v P V | Ψ JJ z γ,LS i , (57)where L + L ′ must be odd. Specifically, the R -matrixelements relevant for the calculation of the asymmetryare: R , , and R , , for J = 0, and R , , , R , , , R , , , R , , , R , , , and R , , for J = 1. QuantumMonte Carlo (QMC) techniques are employed to evaluatethese matrix elements (see below).The asymmetry in Eq. (2) is expressed in terms of T -matrix elements, which are in turn derived from R -matrixelements via Eq. (36). This latter equation can be furthersimplified by retaining only linear terms in the PV R -matrix elements, and the resulting expressions for thePC T , , , T , , and T , , and PV T , , , T , , and T , , matrix elements are listed in Appendix A.The QMC techniques used to evaluate the matrix ele-ment in Eq. (57) are similar to those discussed in Ref. [9]for the neutron spin rotation in ~n d scattering. The wave2functions for an assigned spatial configuration specifiedby the set of Jacobi variables ( x , x , x ) are expanded ona basis of 16 × ψ ( x , x , x ) = X a =1 ψ a ( x , x , x ) | a i , (58)where the components ψ a ( x , x , x ) are generally com-plex functions, and the basis states | a i = | ( n ↓ ) ( p ↓ ) ( n ↓ ) ( p ↓ ) i , | ( n ↓ ) ( n ↓ ) ( p ↓ ) ( p ↓ ) i , and so on. Ma-trix elements of the PV potential components are writtenschematically as h f | O | i i = X a,b =1 Z d x d x d x ψ ∗ f,a ( x , x , x ) × [ O ( x , x , x )] ab ψ i,b ( x , x , x ) , (59)where [ O ( x , x , x )] ab denotes the matrix representingin configuration space any of the components in Table III.Note that the operators X ( n ) ij, ∓ occurring in v PV ij are con-veniently expressed as X ( n ) ij, + = − i [2 f n ( r ij ) ∇ ij + ˆ r ij f ′ n ( r ij )] , (60) X ( n ) ij, − = ˆ r ij f ′ n ( r ij ) , (61)where the gradient operator ∇ ij = ( ∇ i − ∇ j ) / f ′ ( x ) = d f ( x ) / d x .Gradients are discretized as ∇ i,α ψ ( x , x , x ) ≃ (cid:2) ψ ( . . . r i + δ ˆ e α . . . ) − ψ ( . . . r i − δ ˆ e α . . . ) (cid:3) / (2 δ ) , (62)where δ is a small increment and ˆ e α is a unit vector in the α -direction. Matrix multiplications in the spin-isospinspace are performed exactly with the techniques devel-oped in Ref. [51]. The problem is then reduced to theevaluation of the spatial integrals, which is efficiently car-ried out by a combination of MC and standard quadra-tures techniques. We write h f | O | i i = Z dˆ x d x d x F (ˆ x , x , x ) ≃ N c N c X c =1 F ( c ) W ( c ) , (63)where the c ’s denote collectively (uniformly sampled) di-rections ˆ x and Jacobi coordinates ( x , x ), and the prob-ability density W ( c ) = | Ψ( x , x ) | / (4 π )—Ψ( x , x ) isthe triton bound-state wave function normalized to one—is sampled via the Metropolis algorithm. For each suchconfiguration c (total number N c ), the function F is ob-tained by Gaussian integrations over the x variable, i.e. F ( c ) = X a,b =1 Z ∞ d x x ψ ∗ f,a ( x , x , x ) × [ O ( x , x , x )] ab ψ i,b ( x , x , x ) . (64)Convergence in the x integrations requires of the orderof 50 Gaussian points, distributed over a non-uniformgrid extending beyond 20 fm, while N c of the order ofa hundred thousand is sufficient to reduce the statisticalerrors in the MC integration on the PV T -matrix ele-ments at the few percent level. In this respect, we notethat these errors are computed directly, by accumulat-ing, in the course of the random walk, values—and theirsquares—for the appropriate linear combinations of R -matrix elements, as given in Eqs. (A5) and (A14)–(A15)of Appendix A. Because of correlations, the errors on the T -matrix elements obtained in this way are much smallerthan those that would be inferred from the R -matrix el-ements by naive error propagation.The present method turns out to be computationallyintensive, particularly because of the large number ofwave functions (and their derivatives) that have to begenerated at each configuration ( x , x , x ). The com-puter codes have been successfully tested by carrying outa calculation based on Gaussian wave functions for theinitial and final states, as described in the following sub-section. A. Test calculation
In order to test the computer programs based on QMCtechniques, we carried out a preliminary calculation usingwave functions for which it is possible to evaluate thematrix elements of the PV potential also analytically.These (antisymmetric) wave functions are written asΨ JJ z γ,LS = 14 π X p =1 e − βρ y L +2 n β p × (cid:20) Y L (ˆ y p ) ⊗ h φ γ ( ijk ) ⊗ χ γ ( l ) i S (cid:21) JJ z , (65)where φ γ ( χ ) represents a three-nucleon (single-nucleon)spin-isospin one-half state with isospin projection –1/2(+1/2) for γ = 1 ( p - H channel) and +1/2 (–1/2) for γ = 2 ( n - He channel). Thus, as in the realistic case,the wave functions above do not have a definite totalisospin T but, rather, are combinations of T = 0 and T = 1 states (having, of course, T z = 0). The wholeradial dependence is given by the factor y L +2 n β p e − βρ ,where ρ is the hyperradius. The non-negative integer n β and the real parameter β can be varied so as to obtain afamily of wave functions. For the purpose of computingmatrix elements of two-body operators, it is convenientto express the pieces in Eq. (65) corresponding to per-mutations p = 1 in terms of quantities relative to thepermutation p = 1 or (123 , JJ z γ,LS = e − βρ X µ C LSJγ n β ; µ x n x n x n "(cid:20)h Y ℓ (ˆ x ) ⊗ (cid:2) χ ⊗ χ (cid:3) S i j ⊗ h Y ℓ (ˆ x ) ⊗ χ i j (cid:21) J ⊗ h Y ℓ (ˆ x ) ⊗ χ i j JJ z × (cid:20)h(cid:2) ξ ⊗ ξ (cid:3) T ⊗ ξ i T ⊗ ξ (cid:21) T , µ ≡ { ℓ ℓ ℓ n n n j j j J S T T T } , (66)where χ i and ξ i are the spin and isospin states of nu-cleon i , x j are the Jacobi vectors corresponding to thepermutation p = 1 and n + n + n = L + 2 n β , andthe C ’s denote combinations of Wigner coefficients. It isnow relatively simple to evaluate the matrix of the PVpotential P i State J π LS n β β | i + 00 0 0.25 | i − 11 0 0.25TABLE VI: Values of the quantum numbers and parametersfor some of the test wave functions used in this work. Seetext for explanation. The values for the matrix elements −h | O ( n )12 | i cor-responding to the 12 components of the DDH potential(see Table III) are reported in Table VII. There is good n Analytical QMC1 − . − . . 000 0 . − . − . − . − . − . − . . 000 0 . . 000 0 . − . − . − . − . − . − . − . − . − . − . −h | O ( n )12 | i calculated analytically and by us-ing the QMC code. For the latter calculation, the statisticaluncertainties are reported in parentheses, and correspond toa (rather modest) set of 5k samples. The operators O ( n )12 arethose of the DDH potential, listed in Table III. agreement between the results of the two calculations.Note that the n = 2 contribution associated with anisoscalar operator as well as the n = 6 , T = 0 and T = 1 components, and therefore it is not immediatelyapparent why this should be so. The reason for this resultbecomes clear only after carrying out the decompositionof the wave functions as in Eq. (66). It comes aboutbecause of delicate cancellations among various terms.We find it reassuring that these same matrix elementsare seen to vanish (within machine precision) with theQMC code. We have verified explicitly that the closeagreement between the two calculations persists for thematrix elements involving other pairs of states, includingthose having J = 1. VII. FURTHER RESULTS The results for the coefficients I αn in Eq. (42), ob-tained with the (zero energy) n - He continuum wavefunctions corresponding to the AV18, AV18/UIX, N3LO,and N3LO/N2LO strong-interaction Hamiltonians, arereported for the DDH and pionless EFT PV potentialsin Tables VIII and IX, respectively. The subscript n in I αn specifies the operators as listed in Table III, and theset of cutoff parameters entering the modified Yukawafunctions are given in Table IV.A quick glance at Table VIII makes it clear that i) thecontribution of the long-range component of the DDH po-tential due to pion exchange is at least a factor 15 largerthan that of any of the short-range components inducedby vector-meson exchanges, and ii) among the vector-meson exchange contributions the isoscalar ( n = 2 , n = 8 , 9) and isovector ( n = 4 , n = 8–12) ones arecomparable in magnitude and much larger than thosedue to isotensor ρ -meson exchanges ( n = 6 , n - He even and odd par-ity states with J = 0 and 1. However, the N3LO/N2LOmodel stands out: the pion-range contribution is (in mag-nitude) substantially smaller than that calculated forthe other models. Moreover, the isoscalar ρ -meson ( ω -meson) contribution corresponding to n = 3 ( n = 8)has opposite sign than obtained for the other (AV18 and4 I DDH n n AV18 AV18/UIX N3LO N3LO/N2LO1 –0.186E+00 –0.189E+00 –0.203E+00 –0.113E+002 –0.826E–02 –0.577E–02 –0.608E–02 –0.622E–023 +0.811E–02 +0.864E–02 +0.333E–02 –0.693E–024 –0.620E–02 –0.794E–02 –0.970E–02 –0.753E–025 –0.800E–02 –0.976E–02 –0.102E–01 –0.781E–026 –0.359E–03 –0.170E–03 –0.942E–03 +0.322E–037 +0.631E–03 +0.115E–02 –0.641E–04 +0.703E–038 +0.605E–02 +0.404E–02 –0.699E–03 –0.794E–029 +0.314E–02 +0.289E–02 –0.171E–02 –0.577E–0210 –0.689E–02 –0.887E–02 –0.115E–01 –0.902E–0211 –0.930E–02 –0.113E–01 –0.123E–01 –0.940E–0212 –0.801E–02 –0.979E–02 –0.115E–01 –0.606E–02TABLE VIII: The coefficient I DDH n corresponding to the DDHpotential components O ( n ) in combination with the AV18,AV18/UIX, N3LO, N3LO/N2LO strong interaction Hamilto-nians. The statistical Monte Carlo errors are not shown, butare at the most 10% for the smallest contributions, and lessthan 2% for the largest. The I DDH n are in units of fm − . I EFT n n AV18/UIX N3LO/N2LO1 –0.195E+00 –0.119E+004 –0.606E+00 –0.391E+006 –0.639E–02 +0.179E–018 +0.608E+00 –0.515E–019 +0.301E+00 +0.426E–01TABLE IX: The coefficient I EFT n corresponding to the pion-less EFT potential components O ( n ) in combination with theAV18/UIX and N3LO/N2LO strong interaction Hamiltoni-ans. Note that there are no potential components with n =2,3, 5, 7, 10, 11, and 12. The statistical Monte Carlo errors arenot shown, but are typically less than 5%. The I EFT n are inunits of fm − . AV18/UIX) models.To investigate the stability of the AV18/UIX andN3LO/N2LO results with respect to convergence in theinternal part of the wave function, we present in Ta-ble X the coefficients C iα entering the PV observable a z in Eq. (4) for two different choices of wave functions.The results labeled “wf2” were listed earlier in Table I,except that those relative to the N3LO/N2LO modelare based here on a smaller number of configurations.These results are obtained by including in the expansionof the internal parts of the 0 ± and 1 ± wave functionsthe maximum number of HH functions we have consid-ered in the present work. The results corresponding tothe row “wf1” are obtained by reducing this number:in practice, for each of the classes K , . . . , K we set K i (wf1) = K i (wf2) − C π C ρ C ρ C ρ C ω C ω AV18/UIX-wf1 –0.2077(281) –0.0433(116) +0.0242(29) –0.0011(2) –0.0232(77) +0.0490(30)AV18/UIX-wf2 –0.1853(150) –0.0380(70) +0.0230(18) –0.0011(1) –0.0231(56) +0.0500(20)N3LO/N2LO-wf1 –0.1118(29) +0.0369(25) +0.0200(8) –0.0009(1) +0.0390(23) +0.0402(12)N3LO/N2LO-wf2 –0.1050(35) +0.0445(33) +0.0189(9) –0.0008(1) +0.0454(31) +0.0417(12)TABLE X: The coefficients C iα entering the PV observable a z , corresponding to the AV18/UIX and N3LO/N2LO strong-interaction Hamiltonians for two sets of wave functions (see text for details). The statistical errors due to the Monte Carlointegrations are indicated in parentheses. Therefore, the differences found between theN3LO/N2LO and the other models are presum-ably due to the fact that the HH expansion for theN3LO/N2LO wave functions (specifically the 0 + wavefunction) has not fully converged. Consequently, inthe following we restrict our discussion to the results obtained with the AV18, N3LO, and AV18/UIX models.In reference to the pion contribution, the calculated C π is rather insensitive to the choice of strong Hamil-tonian. However, there is still a considerable modeldependence in the results obtained for the individualcontributions due to vector-meson exchanges. This5model dependence, in turn, impacts very significantlypredictions for the PV asymmetry a z , as it can besurmised from Table XI. Of course, this is so underthe assumption that the values for the strong- andweak-interaction coupling constants characterizing theDDH potential are those listed in Table IV. For example,the combination of coupling constants corresponding topion-exchange ( n = 1) and isoscalar ρ -meson exchange( n = 2 and 3) are, respectively, c DDH1 = (4 . × − ) fm, c DDH2 = (11 . × − ) fm and c DDH3 = (79 . × − ) fm—note that c DDH3 = (1 + κ ρ ) c DDH2 and κ ρ = 6 . ρ -mesonto the nucleon [52]. Consequently, the contribution c DDH3 × I DDH3 is comparable in magnitude and oppositein sign to the pion-exchange contribution c DDH1 × I DDH1 .In this respect, we note that the asymmetry a z changesroughly from − × − to +13 × − as the six PVweak coupling constants entering the DDH model arevaried over their respective allowed ranges determinedin Ref. [29]. Thus, a z could potentially be large enoughto make its measurement (relatively) easy. × a DDH z n AV18 AV18/UIX N3LO N3LO/N2LO1 –8.33 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± a z and associated errors (rows 1–12), obtained for the DDH PV potential with valuesfor the coupling constants as listed in Table IV. The four columns correspond to the different combinations of strong-interactionHamiltonians adopted in the calculations. The last row shows the minimum and maximum (central) values that a z can attain,as the PV couplings are varied over the allowed ranges in the original DDH formulation [29]. The coefficients I EFT n for the operators entering the pi-onless EFT PV potential, that is n =1, 4, 6, 8, and 9, arereported in Table IX. The coefficients I EFT n for n =1, 4,8, and 9, corresponding to isoscalar and isovector struc-tures, are all of the same order of magnitude, while thatfor n =6 with isotensor character is much smaller. Notethat the radial functions are taken to be the same for all n , f EFT n ( r ) = f µ ( r ). Of course, the I EFT n ’s will dependsignificantly on the value of the mass µ —either µ = m π ,as appropriate in the present pionless EFT formulation,or µ = 1 GeV, the scale of chiral symmetry breaking,as appropriate in the formulation in which pion degreesof freedom are explicitly retained. Indeed, in this latterformulation the leading order component of v PV has thesame form as the pion-exchange term in DDH.Finally, rough estimates have been made for the rangeof values allowed for the low-energy constants C , C + C , C , ˜ C , and C in Ref. [30]. However, at the presenttime a systematic program for their determination is yetto be carried out. In view of this, we refrain here from making EFT-based predictions for the longitudinal asym-metry. Acknowledgments The authors would like to thank J.D. Bowman, C.B.Crawford, and M.T. Gericke for their continued interestin the present work and for correspondence in referenceto various aspects of the calculations.One of the authors (R.S.) would also like to thank thePhysics Department of the University of Pisa, the INFNPisa branch, and especially the Pisa group for the sup-port and warm hospitality extended to him on severaloccasions. The work of R.S. is supported by the U.S. De-partment of Energy, Office of Nuclear Physics, under con-tract DE-AC05-06OR23177. The calculations were madepossible by grants of computing time from the NationalEnergy Research Supercomputer Center.6 Appendix A: From R - to T -matrices Consider the case with J = 0 first. For the parity-conserving (PC) T -matrix we have: T , , = 1 √ q h R , , ( I − i R , ) − , + R , , ( I − i R , ) − , i , (A1)where I − i R is a 2 × i.e. I − i R , = (cid:18) a ǫǫ b (cid:19) , a = 1 − i R , , , ǫ = − i R , , , b = 1 − i R , , , (A2)with | a | , | b | ≫ | ǫ | . To first order in ǫ , we approximate( I − i R , ) − = (cid:18) /a − ǫ/ab − ǫ/ab /b (cid:19) , (A3)and hence T , , = 1 √ q R , , a . (A4)Similarly, for the parity-violating (PV) T -matrix elementwe find: T , , = 1 √ q h R , , ( I − i R , ) − , + R , , ( I − i R , ) − , i = 1 √ q " i R , , R , , a b + R , , b . (A5)The case J = 1 is somewhat more involved since thematrices are now 4 × 4. The matrix ( I − i R , ) − iswritten as I − i R , = (cid:18) A ǫǫ T B (cid:19) . (A6)where A , ǫ , and B are 2 × A = − i R , , − i R , , − i R , , − i R , , ! , (A7) B = − i R , , − i R , , − i R , , − i R , , ! , (A8) ǫ = − i R , , − i R , , − i R , , − i R , , ! . (A9)Note that A and B , as well as their inverse A − and B − ,are symmetric. To first order in ǫ , it follows that( I − i R , ) − = (cid:18) A − CC T B − (cid:19) , (A10)where the 2 × C and its transpose are definedas C = − A − ǫB − , C T = − B − ǫ T A − . (A11)This shows that ( I − i R , ) − is also symmetric in thisapproximation. The PC T , , and T , , and PV T , , and T , , matrix elements entering Eq. (36) are thengiven by T , , = 1 √ q h R , , ( A − ) , + R , , ( A − ) , i , (A12) T , , = 1 √ q h R , , ( A − ) , + R , , ( A − ) , i , (A13) T , , = 1 √ q h R , , C , + R , , C , + R , , × ( B − ) , + R , , ( B − ) , i , (A14) T , , = 1 √ q h R , , C , + R , , C , + R , , × ( B − ) , + R , , ( B − ) , i . (A15) Appendix B: Numerical values for R - and T -matrixelements The set of Tables XII–XV are all relative to theAV18/UIX+DDH model, and present results for the R -matrix elements involving PV transitions between stateswith J = 0 and J = 1, the corresponding T -matrix ele-ments which follow from them and the parity-conserving(PC) R -matrix elements via Eqs. (A5) and (A14)–(A15),and lastly the coefficients d ( n ) i , d ( n )1 = T , , ( n ) T , ∗ , , d ( n )2 = T , , ( n ) T , ∗ , ,d ( n )3 = T , , ( n ) T , ∗ , , d ( n )4 = T , , ( n ) T , ∗ , ,d ( n )5 = T , , ( n ) T , ∗ , , (B1)where the T -matrix elements are defined as in Eq. (16),and the label ( n ) on those involving PV transitions refers7 n R , , R , , R -matrix elements for J =0 corresponding to the DDH potential components O ( n ) incombination with the AV18/UIX strong interaction potentialsat vanishing n - He energy. The statistical Monte Carlo errorsare not shown, but are typically ∼ R -matrix elementwithout (with) overline is in units of fm − (fm − / ), see textfor explanation. to the operator component O ( n ) in Table III. The I n ’sdiscussed earlier follow from I n = − 4Σ Re h √ d ( n )1 − d ( n )2 + √ d ( n )3 + √ d ( n )4 + √ d ( n )5 i , (B2)where Σ has been defined in Eq. (3). A few wordson units: since the operators O ( n ) do not include the c n ’s, i.e. the combinations of nucleon mass and strong-and weak-interaction coupling constants, the resulting R -matrix ( T -matrix) elements involving PV transitions arein units of fm − (adimensional)—they would otherwisebe adimensional (in units of fm). Further, because ofthe definition in Eq. (34), the R -matrix elements havedimensions of fm − / . Note, however, that the T - and T -matrix elements only differ by a phase factor, and hencethe former are also adimensional. n R , , R , , R , , R , , R , , R , , J = 1. The R -matrix elements in J = 1 states (Table XIII) aretypically two orders of magnitude smaller than those in J = 0 states (Table XII). Among the former, those withorbital angular momentum L = 2 in channel p - H ( γ = 1)are much suppressed at the low energies of interest inthe present work. Inspection of Table XII also showsthat the (isovector) pion-exchange interaction ( n = 1) isdominant, which suggests that the J π = 0 + and 0 − statesin both n - He and p - H are not purely isoscalar, butrather have significant admixtures of isospin components T > 0. In order to compute the d i ’s in Table XV, one needs,in addition to the T -matrix elements listed in Ta-ble XIV, also the T -matrix elements associated withPC transitions. These have been calculated to be (atzero n - He energy): T , , = ( − . 356 + i . T , , = (0 . − i . T , , = (0 . − i . d ( n )1 and d ( n )2 combinations give the leading contributions to I n and that, in the case of pion exchange, d (1)1 is infact dominant. This fits in well with the expectation8 T , , T , , T , , n Re Im Re Im Re Im1 –0.104E+01 –0.302E+01 –0.133E+00 –0.134E–02 –0.316E–01 –0.168E–012 +0.219E+00 –0.996E–01 –0.830E–02 +0.143E–03 +0.210E–01 +0.123E–023 –0.289E+00 +0.108E+00 +0.129E–01 –0.269E–03 +0.136E–02 –0.293E–034 –0.767E–01 –0.971E–01 –0.442E–02 +0.335E–05 +0.401E–02 +0.330E–035 –0.771E–01 –0.111E+00 –0.573E–02 –0.682E–05 +0.463E–02 +0.385E–036 –0.226E–02 –0.285E–02 –0.625E–03 +0.453E–04 –0.104E–01 –0.224E–037 –0.110E–02 –0.210E–02 +0.150E–03 +0.194E–04 –0.126E–01 –0.277E–038 –0.393E+00 +0.159E+00 +0.124E–01 –0.253E–03 +0.189E–02 –0.239E–039 –0.161E+00 +0.559E–01 +0.637E–02 –0.144E–03 +0.110E–01 +0.364E–0310 –0.843E–01 –0.107E+00 –0.495E–02 +0.375E–05 +0.445E–02 +0.367E–0311 –0.887E–01 –0.126E+00 –0.667E–02 –0.757E–05 +0.533E–02 +0.443E–0312 –0.265E–01 –0.295E–01 –0.640E–02 –0.285E–04 –0.340E–03 –0.245E–03TABLE XIV: The parity-violating T -matrix elements corresponding to the DDH potential components O ( n ) in combinationwith the AV18/UIX strong interaction potentials at vanishing n - He energy. 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