Parity violation in low energy neutron deuteron scattering
aa r X i v : . [ nu c l - t h ] J a n Parity violation in low energy neutron deuteron scattering
Young-Ho Song, ∗ Rimantas Lazauskas, † and Vladimir Gudkov ‡ Department of Physics and Astronomy,University of South Carolina, Columbia, SC, 29208 IPHC, IN2P3-CNRS/Universit´e Louis Pasteur BP 28,F-67037 Strasbourg Cedex 2, France (Dated: August 28, 2018)
Abstract
Parity violating effects for low energy elastic neutron deuteron scattering are calculated forDDH and EFT-type of weak potentials in a Distorted Wave Born Approximation, using realistichadronic strong interaction wave functions, obtained by solving three-body Faddeev equations inconfiguration space. The results of relation between physical observables and low energy constantscan be used to fix low energy constants from experiments. Potential model dependencies of parityviolating effects are discussed.
PACS numbers: 24.80.+y, 25.10.+s, 11.30.Er, 13.75.Cs ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION The study of parity violating (PV) effects in low energy physics is a very sensitive toolto test methods of calculations both of weak and strong interactions in the Standard model.This also can be a way to search for a possible manifestations of new physics resulted indeviations from unambiguous and precise calculations of PV effects and experimental mea-surements. However, to use this approach, it is crucial to prove that implemented theoreticaltechniques are sufficient to describe experimental data with high accuracy which exceeds ex-perimental accuracy. There is a large amount of experimental data for different PV effects innuclear physics, each of which in general agrees with theoretical predictions. However, in thelast years it became clear (see, for example [1–4] and references therein) that the traditionalDDH [5] method for calculation of PV effects cannot reliably describe the whole available setof experimental data within the same set of parameters. If this is not the manifestation ofnew physics, which is very unlikely for the current accuracy of experimental measurementsand theoretical calculations, then this discrepancy could be blamed on systematic errorsin experimental data, theoretical uncertainties in calculations of strong interactions at lowenergy, or it might be that DDH approach is not adequate for the description of the set ofprecise experimental data because it is based on a number of models and assumptions. Toresolve this discrepancy and to eliminate nuclear model dependent factors in calculations, itis necessary to focus on the analysis of new and existing experimental data for different PVparameters in few-body systems, where calculations of nuclear related effects can be donewith a high precision. Recently new approach, based on the effective field theory (EFT),has been introduced for a model independent parametrization of PV effects (see, papers[1, 4] and references therein), and some calculations for two-body systems have been done[6]. The power of the EFT approach for parametrization of all PV effects in terms of a smallnumber of constants could be utilized if we can analyze a large enough number of PV effectsto be able to constrain all free parameters of the theory which are usually called low energyconstants (LEC). Thus, one can guarantee the adequate description (parametrization) ofthe strong interaction hadronic parts and weak interaction constants for symmetry violat-ing observables. Unfortunately, the number of experimentally measured (and independentin terms of unknown LECs) PV effects in two body systems is not enough to constrain allLECs. In spite of the fact that five independent observable parameters in a two body system2ould fix five unknown PV LECs [7–10], it is impossible to measure all of them using exist-ing experimental techniques. Therefore, one has to include into analysis few-body systemsand even heavier nuclei, the latter of which are actually preferable from the experimentalpoint of view, because as a rule, the measured effects in nuclei are much larger than innucleon-nucleon system due to nuclear enhancement factors [11–13].The natural and unambiguous way to verify the applicability of the EFT for the calcula-tion of symmetry violating effects in nuclear reactions requires a development of a regularand self consistent approach for calculation of PV amplitudes in three-body (few-body) sys-tems [14], with a hope to extend the formalism for the description of many body systems.This systematic approach for the solution of three-body PV scattering problem in EFTframework [14] requires additional numerical efforts and will be presented elsewhere. As afirst step for the clarification of the possible difference in contributions to PV effects fromDDH and EFT-type potentials, one can use a “hybrid” method (similar to the method usedin paper [15]) for the simplest process of neutron-deuteron scattering. We calculate three-body wave functions with realistic Hamiltonians of strong interaction using exact Faddeevequations in configuration space, and then, calculate PV effects in the first order of pertur-bation with DDH potential and potentials derived in EFT formalism. In the next section,we present our formalism for the calculation PV effects for elastic neutron-deuteron scatter-ing with different set of nucleon weak potentials, with DDH and weak potentials obtainedfrom pionless and pionful EFTs. Then, we present results of numerical calculations anddiscussions.
II. FORMALISM
We treat weak nucleon interactions as a perturbation and calculate three-body wave func-tions exactly using Faddeev equations with phenomenological potentials for strong interac-tions. Similar hybrid approach has been successfully applied to the weak and electromagneticprocesses involving three-body and four-body hadronic systems [16–21]. We consider threetypes of parity violating potentials. The first one is the standard DDH potential whichis based on meson exchange mechanism of nucleon-nucleon interactions. The second andthird potentials are derived from pionless and pionful versions of effective field theory withparity violating hadronic interactions. Instead of calculating parity violating amplitudes by3umming PV diagrams in EFT, we use these potentials to calculate PV effects. This is asimplification, which we call a “hybrid” approach.
A. Observables
Since PV effects in neutron-deuteron system are very small, we consider only coherentprocesses which are related to the propagation of neutrons through unpolarized deuterontarget and, therefore, do not have an additional suppression in low energy region. Then, twoPV observable parameters are the angle φ of rotation of neutron polarization around neutronmomentum and the relative difference of total cross sections P = ( σ + − σ − ) / ( σ + + σ − ) forneutrons with opposite helicities. The value of the angle of neutron spin rotation per unitlength of the target sample can be expressed in terms of elastic scattering amplitudes atzero angle for opposite helicities f + and f − as dφdz = − πNp Re ( f + − f − ) , (1)where N is a number of target nuclei per unit volume and p is a relative neutron momentum.Using optical theorem, one can write the relative difference of total cross sections P in termsof these amplitudes as P = Im ( f + − f − )Im ( f + + f − ) . (2)It is convenient to represent the amplitudes in terms of matrix ˆ R which is related toscattering matrix ˆ S as ˆ R = ˆ1 − ˆ S . With partial waves decomposition for the case ofneutron-deuteron scattering | p , m n , m d i = X l y l zy X SM,JJ z | p, ( l y S ) J J z ih J J z | l y l zy , S M ihS M | m n , m d i Y ∗ l y l zy (ˆ p ) , (3)where l y is an orbital angular momentum between neutron and deuteron, S is a sum ofneutron spin and deuteron total angular momentum, and J is the total angular momentumof the neutron-deuteron system, the above equations can be written at low energies as1 N dφdz = 2 π p Im h R , + R , − √ R , − √ R , +4 R , + 4 R , − √ R , − √ R , i (4)4nd P = 13 Re h R , + R , − √ R , − √ R , + 4 R , +4 R , − √ R , − √ R , i / Re h R , + 2 R , i , (5)where R Jl ′ S ′ ,l S = h l ′ S ′ | R J | l Si , unprimed and primed parameters correspond to initial andfinal states. Since we are interested in low energy neutron scattering, it would be sufficientto include only s - and p -waves contributions to parity violating amplitudes; for the totalcross section (the denominator in the last equation), we keep only dominant contributionsfrom s -wave neutrons. It should be noted that time-reversal invariance leads to the relation h S ′ | R J | Si = h S ′ | R J | S ′ i between matrix elements, therefore, only half of parity violatingamplitudes are independent.Nucleon-nucleon interaction can be written as a sum V = V pc + V pv of the parity con-serving ( V pc ) and weak parity violating ( V pv ) terms. Due to the weakness of parity violatinginteraction, one can use Distorted Wave Born Approximation (DWBA) to calculate PVamplitudes with a high level of accuracy as R Jl ′ y S ′ ,l y S ≃ i − l ′ y + l y +1 µp ( − ) pc h Ψ , ( l ′ y S ′ ) J J z | V pv | Ψ , ( l y S ) J J z i (+) pc , (6)where µ is a neutron-deuteron reduced mass and | Ψ , ( l ′ y S ′ ) J J z i ( ± ) pc are solutions of 3-bodyFaddeev equations in configuration space for parity conserving strong interaction Hamilto-nian, defined by V P C and normalized as described in section II C. The factor i − l ′ y + l y in thisexpression is introduced to match the R -matrix definition in the modified spherical harmon-ics convention [22] with the wave functions which are calculated in this paper using sphericalharmonics convention.In the rest of the paper, we use only wave functions calculated for parity conservingpotentials and, therefore, will omit subscript P C .As will be explained in section II C, we use jj-coupling scheme (with a basis states | l y j y i )when solving Faddeev equtions. One can transform jj -basis states into l y S -basis by meansof | [ l y ⊗ ( s k ⊗ j x ) S ] JJ z i = X j y | [ j x ⊗ ( l y ⊗ s k ) j y ] JJ z i× ( − j x + j y − J ( − l y + s k + j x + J [(2 j y + 1)(2 S + 1)] l y s k j y j x J S , (7)5ne interesting observation is that the neutron spin rotation, as well as parameter P, in | l y j y i basis involves potential matrix elements only between j y = states.It should be noted that at low energy the Im( R Jl ′ y S ′ ,l y S ) ∼ p l ′ y + l y +1 , and thus the expres-sion eq.(4) for the angle φ of neutron spin rotation is finite and well defined in the zeroenergy limit of the n-d scattering. Numerically, it is calculated by evaluating expressionIm( R Jl ′ y S ′ ,l y S ) /p l ′ y + l y +1 at zero energy. On the other hand, Re( R Jl ′ y S ′ ,l y S ) ∼ p · Im( R Jl ′ y S ′ ,l y S ) atlow energy, and thus the real part of this quantity vanishes in the zero energy limit. There-fore, the parameter P is calculated at 15 KeV neutron kinetic energy in the laboratorysystem, where both imaginary and real parts of the R-matrix elements become comparablein magnitude and thus can be discerned numerically. B. The parity violating potentials
To understand the possible difference in the description of parity violating effects by DDHand EFT-type for potentials, we compare calculations with the DDH potential[5] and twodifferent choices of EFT potentials: the potential derived from pionless EFT lagrangian [1]and the potential derived from pionful EFT Lagrangian [1]. It was shown [15] that all thesethree potentials can be expanded in terms of a set of O ( n ) ij operators as v αij = X n c αn O ( n ) ij , α = DDH or pionless EFT or pionful EFT (8)with parameters c αn and operators O ( n ) ij given in the Table I.One can see that operators O ( n ) ij are products of isospin, spin, and vector operators X ( n ) ij, ± defined as X ( n ) ij, + ≡ [ p ij , f n ( r ij )] + , X ( n ) ij, − ≡ i [ p ij , f n ( r ij )] − , (9)where p ij ≡ ( p i − p j )2 .For the DDH potential, radial functions f x ( r ), x = π, ρ , and ω are modified Yukawafunctions, f x ( r ) = 14 πr (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) . (10)6 ABLE I: Parameters and operators of parity violating potentials. πN N coupling g πNN can berepresented by g A by using Goldberger-Treiman relation, g π = g A m N /F π with F π = 92 . T ij ≡ (3 τ zi τ zj − τ i · τ j ). Scalar function ˜ L Λ ( r ) ≡ L Λ ( r ) − H Λ ( r ). n c DDHn f DDHn ( r ) c πn f πn ( r ) c πn f πn ( r ) O ( n ) ij g π √ m N h π f π ( r ) µ Λ χ C π f πµ ( r ) + g π √ m N h π f π ( r ) ( τ i × τ j ) z ( σ i + σ j ) · X (1) ij, − − g ρ m N h ρ f ρ ( r ) 0 0 0 0 ( τ i · τ j )( σ i − σ j ) · X (2) ij, + − g ρ (1+ κ ρ ) m N h ρ f ρ ( r ) 0 0 0 0 ( τ i · τ j )( σ i × σ j ) · X (3) ij, − − g ρ m N h ρ f ρ ( r ) µ Λ χ ( C π + C π ) f πµ ( r ) Λ Λ χ ( C π + C π ) f Λ ( r ) ( τ i + τ j ) z ( σ i − σ j ) · X (4) ij, + − g ρ (1+ κ ρ )2 m N h ρ f ρ ( r ) 0 0 √ πg A Λ Λ χ h π L Λ ( r ) ( τ i + τ j ) z ( σ i × σ j ) · X (5) ij, − − g ρ √ m N h ρ f ρ ( r ) − µ Λ χ C π f πµ ( r ) − Λ χ C π f Λ ( r ) T ij ( σ i − σ j ) · X (6) ij, + − g ρ (1+ κ ρ )2 √ m N h ρ f ρ ( r ) 0 0 0 0 T ij ( σ i × σ j ) · X (7) ij, − − g ω m N h ω f ω ( r ) µ Λ χ C π f πµ ( r ) Λ χ C π f Λ ( r ) ( σ i − σ j ) · X (8) ij, + − g ω (1+ κ ω ) m N h ω f ω ( r ) µ Λ χ ˜ C π f πµ ( r ) Λ χ ˜ C π f Λ ( r ) ( σ i × σ j ) · X (9) ij, − − g ω m N h ω f ω ( r ) 0 0 0 0 ( τ i + τ j ) z ( σ i − σ j ) · X (10) ij, + − g ω (1+ κ ω )2 m N h ω f ω ( r ) 0 0 0 0 ( τ i + τ j ) z ( σ i × σ j ) · X (11) ij, − − g ω h ω − g ρ h ρ m N f ρ ( r ) 0 0 0 0 ( τ i − τ j ) z ( σ i + σ j ) · X (12) ij, + − g ρ m N h ′ ρ f ρ ( r ) 0 0 − √ πg A Λ Λ χ h π L Λ ( r ) ( τ i × τ j ) z ( σ i + σ j ) · X (13) ij, −
14 0 0 0 0 Λ χ C π f Λ ( r ) ( τ i × τ j ) z ( σ i + σ j ) · X (14) ij, −
15 0 0 0 0 √ πg A Λ Λ χ h π ˜ L Λ ( r ) ( τ i × τ j ) z ( σ i + σ j ) · X (15) ij, − For pionless EFT ( π EFT) one, f n ( r ) are described by single function f µ ( r ), f µ ( r ) = 14 πr e − µr , (11)with µ ≃ m π .For the case of pionful EFT model ( π EFT), there are long range interactions from onepion exchange( V − ,LR ) and from their corrections ( V ,LR ), middle range interactions due totwo pion exchange ( V ,MR ), and short range interactions ( V ,SR ) due to nucleon contactterms. The radial part of the leading term of long range one pion exchange, V − ,LR , isdescribed by the function f π ( r ). Since one-pion exchange contribution is dominated by longrange part, we do not use a regulator for it, i.e. we assume that the long range interactionshave the same radial functions f π ( r ) as DDH potential with infinite cutoff. The short range7nteraction V ,SR in pionful theory has the same structure as for pionless EFT; however,in spite of the structural similarity, their meanings are rather different. One can ignorethe higher order corrections of long range interactions, V ,LR , because they can either beabsorbed by renormalization of low energy constants [6] or suppressed. The middle rangeinteractions V ,MR are described by functions L ( q ) and H ( q ) in momentum space L ( q ) ≡ p m π + q | q | ln p m π + q + | q | m π ! , H ( q ) ≡ m π m π + q L ( q ) , (12)where, q µ = ( q , q ) = p µ − p ′ µ = p ′ µ − p µ . To calculate two pion exchange functions (divergentat large q ) in spacial representation , we use regulators (Λ − m π ) (Λ + q ) . For the sake of simplicity,we use only one cutoff parameter with the same regulator, both for middle range and forshort range interactions. Then, one can write { L Λ ( r ) , H Λ ( r ) , f Λ ( r ) } = 1Λ Z d q (2 π ) e − i q · r (Λ − m π ) (Λ + q ) { L ( q ) , H ( q ) , } . (13)In the given representation, coefficients c αn have fm dimension and scalar functions f αn ( r ) havefm − dimension. One can see that only the new operator structure, which is not includedin DDH or pionless EFT, is due to V P V ,LR . Therefore, pionful EFT does not introduce newoperator structure, provided we neglect V P V ,LR term [6 ? ].To see a sensitivity to the choice of cutoffs for parity violating potentials, we used twoset of cutoff parameters for each models, which are listed in the Table II.Using the discussed above three potentials, one can represent parity violating amplitudesas a linear expansion in terms of given set of matrix elements for corresponding operators O ( n ) ij . Thus, the angle of neutron spin rotation can be written as1 N dφdz = X n =1 c αn I αn , (14)and the parameter P as P = X n =1 c αn ˜ I αn , (15)in terms of coefficients I αn and ˜ I αn with α = DDH-I,II , π EFT-I,II , π
EFT-I,II for differentpotentials and cutoff parameters. 8
ABLE II: Parameter of parity violating potentials in GeV units. We used masses of mesons m π , m ρ , and m ω , respectively, as 0 . . .
783 in DDH potential.Λ π Λ ρ Λ ω µ ΛDDH-I 1 .
72 1 .
31 1 . π EFT-I 0 . π EFT-I 0 . ∞ ∞ ∞ 6 π EFT-II 1 . π EFT-II 1 . C. Faddeev wave function
To obtain 3-body wave functions for neutron-deuteron scattering with parity conservinginteractions, we solve Faddeev equations (also often called Kowalski-Noyes equations) inconfiguration space [23, 24]. For isospin invariant interactions (with nucleon masses fixedto ~ /m = 41 .
471 MeV · fm), three Faddeev equations become formally identical, having theform ( E − H − V ij ) ψ k = V ij ( ψ i + ψ j ) , (16)where ( ijk ) are particle indices, H is kinetic energy operator, V ij is two body force betweenparticles i , and j , ψ k = ψ ij,k is Faddeev component.The wave function in Faddeev formalism is the sum of three Faddeev components,Ψ( x , y ) = ψ ( x , y ) + ψ ( x , y ) + ψ ( x , y ) . (17)Using relative Jacobi coordinates x k = ( r j − r i ) and y k = √ ( r k − r i + r j ), one can expandthese Faddeev components in bipolar harmonic basis: ψ k = X α F α ( x k , y k ) x k y k (cid:12)(cid:12)(cid:12)(cid:0) l x ( s i s j ) s x (cid:1) j x ( l y s k ) j y E JM ⊗ (cid:12)(cid:12) ( t i t j ) t x t k (cid:11) T T z , (18)where index α represents all allowed combinations of the quantum numbers presented inthe brackets: l x and l y are the partial angular momenta associated with respective Jacobicoordinates, s i and t i are the spins and isospins of the individual particles. Functions F α ( x k , y k ) are called partial Faddeev amplitudes. It should be noted that the total angularmomentum J as well as its projection M are conserved, but the total isospin T of the systemis not conserved due to the presence of charge dependent terms in nuclear interactions.Boundary conditions for Eq. (16) can be written in the Dirichlet form. Thus, Faddeevamplitudes satisfy the regularity conditions: F α (0 , y k ) = F α ( x k ,
0) = 0 . (19)9or neutron-deuteron scattering with energies below the break-up threshold, Faddeev com-ponents vanish for x k → ∞ . If y k → ∞ , then interactions between the particle k and thecluster ij are negligible, and Faddeev components ψ i and ψ j vanish. Then, for the compo-nent ψ k , which describes the plane wave of the particle k with respect to the bound particlepair ij ,lim y k →∞ ψ k ( x k , y k ) l n j n = 1 √ X j ′ n l ′ n (cid:12)(cid:12)(cid:12) { φ d ( x k ) } j d ⊗ (cid:8) Y l ′ n ( ˆy k ) ⊗ s k (cid:9) j ′ n E JM ⊗ (cid:12)(cid:12)(cid:12) ( t i t j ) t d t k E , − × i h δ l ′ n j ′ n ,l n j n h − l ′ n ( pr nd ) − S l ′ n j ′ n ,l n j n h + l ′ n ( pr nd ) i , (20)where deuteron, being formed from nucleons i and j , has quantum numbers s d = 1, j d = 1,and t d = 0, and its wave function φ d ( x k ) is normalized to unity. Here, r nd = ( √ / y k is relative distance between neutron and deuteron target, and h ± l n are the spherical Hankelfunctions. The expression (20) is normalized to satisfy a condition of unit flux for nd scattering wave function.For the cases where Urbana type three-nucleon interaction (TNI) is included, we modifythe Faddeev equation (16) into( E − H − V ij ) ψ k = V ij ( ψ i + ψ j ) + 12 ( V ijk + V jki )Ψ (21)by noting that the TNI among particles ijk can be written as sum of three terms: V ijk = V kij + V ijk + V jki . D. Evaluation of matrix elements
Due to anti-symmetry of the total wave function in isospin basis, one has h Ψ | V + V + V | Ψ i = 3 h Ψ | V ij | Ψ i for any pair i = j .Using decomposition of momentum p , p = − i ∇ x = − i (cid:18) ˆ x ∂∂x + 1 x ˆ ∇ Ω (cid:19) , (22)we can represent general matrix elements of local two-body parity violating potential oper-ators as ( − ) h Ψ f | O | Ψ i i (+) = ( √
32 ) X αβ "Z dxx dyy e F (+) f,α ( x, y ) xy ! ˆ X ( x ) e F (+) i,β ( x, y ) xy ! h α | ˆ O (ˆ x ) | β i , (23)10here ( ± ) means outgoing and incoming boundary conditions and ˆ X ( x ) is derivative ofscalar function or derivative of wave function with respect to x . (Note that we have usedthe fact that ( e F ( − ) ) ∗ = e F (+) .) The partial amplitudes e F i ( f ) ,α ( x, y ) represent the total systemswave function in one selected basis set among three possible angular momentum couplingsequences for three particle angular momenta:Ψ i ( f ) ( x, y ) = X α e F i ( f ) ,α ( x, y ) xy (cid:12)(cid:12)(cid:12)(cid:0) l x ( s i s j ) s x (cid:1) j x ( l y s k ) j y E JM ⊗ (cid:12)(cid:12) ( t i t j ) t x t k (cid:11) T T z . (24)The “angular” part of the matrix element is h α | ˆ O (ˆ x ) | β i ≡ Z d ˆ x Z d ˆ y Y † α (ˆ x, ˆ y ) ˆ O (ˆ x ) Y β (ˆ x, ˆ y ) , (25)where Y α (ˆ x, ˆ y ) is a tensor bipolar spherical harmonic with a quantum number α . One cansee that operators for “angular” matrix elements have the following structure:ˆ O (ˆ x ) = ( τ ⊙ τ )( σ ⊚ σ ) · ˆ x, or ( τ ⊙ τ )( σ ⊚ σ ) · ∇ Ω , (26)where ⊙ , ⊚ = ± , × . The explicit values of these matrix elements are summarized in theappendix. III. RESULTS AND DISCUSSIONS
As it was mentioned in the previous section, because of low energy property of R Jα ′ α , itis convenient to present results for elements R Jl ′ y ,l y in terms of a ratio, R Jα ′ α ( p )4 µi − l ′ y + l y +1 p l ′ y + l y +1 = 1 p l ′ y + l y ( − ) h Ψ , ( l ′ y S ′ ) J J z | V P Vn | Ψ , ( l y S ) J J z i (+) (27)For the case of parity violation, we fix l ′ y = 1 and l y = 0. To obtain the observableparameters when neutron energies are larger than thermal ones (which correspond to zeroenergy limit for neutron spin rotation), one can use a simple extrapolation based on theabove representation with a good accuracy up to hundreds KeV.The contributions to parity violating matrix elements π c n Im h R Jα ′ α ( p )4 µp i from different termsof parity violating potentials (see Table I) are presented in the Table III. These matrix ele-ments were calculated using strong AV18+UIX and weak DDH-II parity violating potentialsfor the case of low neutron energies (up to thermal ones). From this table, one can see that11 ABLE III: Contributions to π Im (cid:20) R J S′ , J ( p )4 µp (cid:21) at very low energy in fm units. We chose AV18+UIXas strong potential and DDH-II as parity violating potential. Matrix elements of n = 6 , S ′ = , J = S ′ = , J = S ′ = , J = S ′ = , J = . × +00 . × +00 − . × − − . × +00 − . × − − . × − . × − . × − . × − . × − − . × − − . × − . × − − . × − . × − . × − . × − − . × − . × − . × − . × − . × − − . × − − . × − − . × − . × − . × − . × −
10 0 . × − − . × − . × − . × −
11 0 . × − − . × − . × − . × −
12 0 . × − . × − − . × − − . × −
13 0 . × − . × − − . × − − . × − the main contribution to PV effects comes from J = 3 / n = 1) is dominant for DDH potential, slightly enhanced for pionfull potential, and aboutequal to other ones for pionless potential.The neutron spin asymmetry P was calculated for laboratory neutron energy E = 15KeV. The results are summarized in tables VII, VIII, and IX for DDH, pionless EFT, andpionful EFT weak interaction potentials with different sets of parameters, correspondingly.These results provide a pattern similar to that of the results for the angle of neutron spin12 ABLE IV: Coefficients I DDHn for AV18 and AV18+UIX strong potentials, and DDH-I and DDH-IIparameter sets for parity violating potentials. I DDH , = 0. n DDH-I/AV18 DDH-I/AV18+UIX DDH-II/AV18 DDH-II/AV18+UIX1 0 . × +02 . × +02 . × +02 . × +02 . × +00 . × +00 . × +01 . × +01 − . × +01 − . × +01 − . × +01 − . × +01 . × +00 . × +00 . × +01 . × +01 . × +00 . × +00 . × +01 . × +01 − . × +01 − . × +01 − . × +01 − . × +01 − . × +00 − . × +00 − . × +00 − . × +00
10 0 . × +00 . × +01 . × +01 . × +01
11 0 . × +01 . × +01 . × +01 . × +01
12 0 . × +00 . × +00 . × +01 . × +01
13 0 . × +01 . × +01 . × +01 . × +01 TABLE V: Coefficients I πn for AV18 and AV18+UIX strong potentials, and π EFT-I and π EFT-IIparameter sets for parity violating potentials. I π , , , , , , , , = 0. n π EFT-I/AV18 π EFT-I/AV18+UIX π EFT-II/AV18 π EFT-II/AV18+UIX1 0 . × +02 . × +02 . × +00 . × +00 . × +02 . × +02 . × +00 . × +00 − . × +02 − . × +02 − . × +00 − . × +00 − . × +01 − . × +01 − . × +00 − . × +00 rotation. The parameter J n in these tables is defined as J n ≡ c n π Re (cid:20) µp (cid:16) R , − √ R , + 4 R , − √ R , (cid:17)(cid:21) , (28)and is related to the parameter ˜ I n in the expression P = P c n ˜ I n by˜ I n = (2 πµp ) J n Re h R , + 2 R , i = 8 π µ J n σ tot , (29)13 ABLE VI: Coefficients I πn for AV18 and AV18+UIX strong potentials, and π EFT-I and π EFT-IIparameter sets for parity violating potentials. I π , , , , , , = 0. n π EFT-I/AV18 π EFT-I/AV18+UIX π EFT-II/AV18 π EFT-II/AV18+UIX1 0 . × +02 . × +02 . × +02 . × +02 . × +01 . × +01 . × +00 . × +00 . × +01 . × +01 . × +01 . × − − . × +01 − . × +01 − . × +00 − . × +00 − . × +00 − . × +00 − . × +00 − . × +00
13 0 . × +02 . × +02 . × +01 . × +01
14 0 . × +01 . × +01 . × +01 . × +01
15 0 . × +02 . × +02 . × +02 . × +02 where σ tot is the total n − d cross section. The total cross section σ tot can be calculated, orone can use its known experimental value.From the presented data, one can see that the results of our calculations are only slightlydifferent for the cases when we use AV18 and AV18+UIX strong Hamiltonians. This indi-cates stability of the results with respect to the three nucleon forces. Indeed, by analyzingthe DDH one-pion exchange matrix element (see Table III), one can see that for DDH-Iwith potentials AV18 and AV18+UIX, the contributions to the I n =1 are − . × +01 and − . × +01 for doublet channel ( J = 1 / J = 3 /
2) theyare 0 . × +02 and 0 . × +02 , correspondingly. The quartet channel is dominated bythe repulsive and long-range part of the strong interactions, but the doublet channel is de-fined by attractive part. Therefore the quartet channel is less sensitive to the off-energy shellstructure of the strong interactions compared to the doublet channel. Then, due to the dom-inant contribution from the quartet channel, the net result turns to be rather independenton the contribution from three nucleon forces. This fact demonstrates the independence ofour results on models of strong interactions. However, further investigations with differentstrong interaction potentials are desirable.It should be noted, that the dependence on cutoff parameters for the contributions frompotentials with short and middle range interactions, even though it appears large, does notlead to cutoff dependence for the observable parameters. Indeed, the renormalization of14 ABLE VII: Coefficients J DDHn for AV18 and AV18+UIX strong potentials, and DDH-I and DDH-II parameter sets for parity violating potentials at E = 15 KeV in the laboratory frame. J DDH , = 0. n DDH-I/AV18 DDH-I/AV18+UIX DDH-II/AV18 DDH-II/AV18+UIX1 0 . × +00 . × +00 . × +00 . × +00 . × − . × − . × − . × − − . × − − . × − − . × − − . × − . × − . × − . × − . × − . × − . × − . × − . × − − . × − − . × − − . × − − . × − − . × − . × − − . × − . × −
10 0 . × − . × − . × − . × −
11 0 . × − . × − . × − . × −
12 0 . × − . × − . × − . × −
13 0 . × − . × − . × − . × − TABLE VIII: Coefficients J πn for AV18 and AV18+UIX strong potentials, and π EFT-I and π EFT-II parameter sets for parity violating potentials. J π , , , , , , , , = 0. n π EFT-I/AV18 π EFT-I/AV18+UIX π EFT-II/AV18 π EFT-II/AV18+UIX1 0 . × +00 . × +00 . × − . × − . × − . × − . × − . × − − . × +00 − . × − − . × − − . × − − . × − . × − − . × − . × − low energy constants would cancel those cutoff dependencies by the cutoff dependencies ofLECs. Therefore, as a result, calculated PV observables are practically cutoff independent.All these tables present information about contributions of different PV operators to PVeffects, provided we know corresponding weak coupling constants. Then, to calculate parityviolating effects, we can use either DDH potential or one of the considered EFT potentials.However, for the case of EFT potentials, we need to know a set of LECs which cannot becalculated in the given theoretical framework but must be obtained from a number of inde-15 ABLE IX: Coefficients J πn for AV18 and AV18+UIX strong potentials, and π EFT-I and π EFT-IIparameter sets for parity violating potentials. J π , , , , , , = 0. n π EFT-I/AV18 π EFT-I/AV18+UIX π EFT-II/AV18 π EFT-II/AV18+UIX1 0 . × +00 . × +00 . × +00 . × +00 . × − . × − . × − . × − . × − . × − . × − . × − − . × − − . × − − . × − − . × − − . × − . × − − . × − . × −
13 0 . × − . × − . × − . × −
14 0 . × − . × − . × − . × −
15 0 . × +00 . × +00 . × +00 . × +00 pendent experiments. Unfortunately, currently available experimental data are not enoughto define the LECs with required precision. Even for pionless EFT, the estimated LECs[1] have large uncertainties preventing us from predicting the values of PV effects. For thepionful EFT, the situation with determination of LECs is even worse. Therefore, it is impos-sible to make reliable predictions for PV effects using EFT-type potentials at this time, andthe only reasonable way to estimate magnitudes of PV effects is to use the DDH potential.Taking into account the difficulty of the systematic description of PV effects using “stan-dard” DDH potentials (see discussions in the introduction), we estimate PV effects usingthe DDH potential for different sets of weak coupling constants: both for the “best value”coupling constants and for two possible sets of the values of the coupling constants recentlyobtained by Bowman [25] from the fit of reliable existing experimental data (see Table X).The results for these three sets of weak coupling constants are summarized in Tables XI andXII for the angle of spin rotation and for neutron spin asymmetry, correspondingly. One cansee that in contrast to the fact that the one-pion exchange dominates in the DDH-“best”coupling parameter set, the rho meson exchange dominates in the case of Bowman’s couplingparameter set. One can see that the angle of neutron spin rotation has almost the samemagnitude for all three sets of parameters, but it has opposite signs for the “best value”set and for the Bowman’s fits. The neutron spin asymmetry does not only have oppositesigns but also essentially different values for these two choices of parameters. This allows16 ABLE X: DDH PV coupling constants in units of 10 − . Strong couplings are g π π = 13 . g ρ π = 0 . g ω π = 20, κ ρ = 3 .
7, and κ ω = 0, h ′ ρ contribution is neglected. 4-paramter fir and 3-parameter fituses the same h ρ and h ω with DDH ‘best’.DDH Coupling DDH ‘best’ 4-parameter fit[25] 3-parameter fit[25] h π +4 . − . − . h ρ − . − . − h ρ − . . h ω − . . h ρ − . − . − . h ω − . − . − . − rad-cm − for the case of DDH-II potential withAV18+UIX strong potential for a liquid deuteron density N = 0 . × atoms per cm .DDH ’best’ 4-parameter fit[25] 3-parameter fit[25]1 0 . × +00 − . × − − . × − . × − . × − . × − − . × − − . × +00 − . × − . × − . × − . × − . × − . × − . × − − . × − . × − . × +00 − . × − . × − . × +00
10 0 . × − . × − . × −
11 0 . × − . × − . × −
12 0 . × − . × − . × − total 0 . × − − . × − − . × − one to choose between two possible sets of DDH parameters and, as a consequence, to testthe dominance of pion-meson contribution in PV effects in n − d scattering.Finally, we would like to mention that our results are quite different from the resultsobtained in paper [15]. For example, in paper [15], the values of I n for J = and J = have17 ABLE XII: Neutron spin asymmetry for the case of DDH-II potential with AV18+UIX strongpotential (the total cross section σ tot = 3 .
35 b at E = 15 KeV ).DDH ’best’ 4-parameter fit[25] 3-parameter fit[25]1 0 . × − − . × − − . × − . × − . × − . × − − . × − − . × − − . × − . × − . × − . × − . × − . × − . × − − . × − . × − . × +00 . × − − . × − − . × +00
10 0 . × − . × − . × −
11 0 . × − . × − . × −
12 0 . × − . × − . × − total 0 . × − − . × − − . × − the same signs for operator with n = 1, but our results show opposite signs for these matrixelements. Another discrepancy is related to the systematic difference between the values ofmatrix elements calculated [15] for AV18 and AV18+UIX potentials, which indicates a largewave function difference for AV18 and AV18+UIX potentials. Contrary to those, our resultsshow that these matrix elements are insensitive to the presence of the three nucleon force. IV. CONCLUSION
We have calculated parity violating angle of neutron spin rotation and asymmetry intransmission of neutrons with opposite helicities for low energy neutron deuteron scattering.Using Distorted Wave Born Approximation for weak interactions with realistic three-nucleonwave functions from Faddeev equations in configuration space, we have parameterized PVobservables in terms of matrix elements presented in the DDH weak potential and in weakpotentials derived from pionless and pionful EFTs. It is shown that our results practically We thank R. Schiavilla and M. Viviani for discussions which clarified that the reason for theses discrep-ancies is related to numerical errors in paper [15].
18o not depend on the choice for strong interaction potentials and on cutoff parameters.Based on the given analysis, one can see that for DDH potential, the dominant contri-bution to observable PV effects comes from the pion-exchange matrix element with n = 1.However, for pionless EFT potential, all types of matrix elements contribute almost equally,and for pionful EFT potential the pion-exchange matrix element is sightly enhanced ascompared to the other ones. Therefore, it would be interesting to compare the estimationof observable PV effects using appropriate LECs and coupling constants for DDH. Un-fortunately, due to insufficient data for LECs this is impossible at this time. However, acomparison of PV effects for two different sets of coupling constants shows that n − d scatter-ing experimental results can be used to distinguish between different sets of DDH couplingconstants and to help in clarification of the issue about the importance of the contributionof pion-exchange weak potential. Appendix: Explicit results of angular part of Matrix elements
Explicit values of matrix elements of iso-spin operators for two-body states are h T ′ T ′ z | τ · τ | T T z i = δ T ′ z ,T z δ T ′ ,T [1 δ T, − δ T, ] , h T ′ T ′ z | ( τ + τ ) z | T T z i = δ T ′ δ T δ T ′ z T z [2 T z ] , h T ′ T ′ z | ( τ − τ ) z | T T z i = δ T ′ ,T ± δ T z ,T ′ z δ T z , [2] , h T ′ T ′ z | i ( τ × τ ) z | T T z i = δ T ′ z ,T z δ T z , δ T ′ ,T ± [ ± , h T ′ T ′ z |T z | T T z i = δ T ′ , δ T, δ T ′ z ,T z [2 δ T z , − δ T z , + 2 δ T z , − ] , (A.1)and matrix elements of orbital and spin operators for two-body states | ( l x s x ) j x j zx i are h ( j x ± , j x j zx | ( σ + σ ) · ˆ x | ( j x , j x j zx i = h ( j x , j x j zx | ( σ + σ ) · ˆ x | ( j x ± , j x j zx i = − s j x + 1 / ∓ / j x + 1 (A.2) h ( j x ± , j x j zx | ( σ − σ ) · ˆ x | ( j x , j x j zx i = h ( j x , j x j zx | ( σ − σ ) · ˆ x | ( j x ± , j x j zx i = ∓ s j x + 1 / ± / j x + 1 (A.3)19 ( j x , j x j zx | i ( σ × σ ) · ˆ x | ( j x ± , j x j zx i = ( − ) h ( j x ± , j x j zx | i ( σ × σ ) · ˆ x | ( j x , j x j zx i = ± s j x + 1 / ± / j x + 1 (A.4) h ( j x ± j x j zx | ( σ + σ ) · ˆ ∇ Ω x | ( j x j x j zx i = ± j x + 1 / ∓ / p j x + 1 / ∓ / √ j x + 1 h ( j x j x j zx | ( σ + σ ) · ˆ ∇ Ω x | ( j x ± j x j zx i = ∓ j x + 1 / ± / p j x + 1 / ∓ / √ j x + 1 (A.5) h ( j x ± j x j zx | ( σ − σ ) · ˆ ∇ Ω x | ( j x j x j zx i = 2 ( j x + 1 / ∓ / p j x + 1 / ± / √ j x + 1 h ( j x j x j zx | ( σ − σ ) · ˆ ∇ Ω x | ( j x ± j x j zx i = − j x + 1 / ± / p j x + 1 / ± / √ j x + 1 (A.6) Acknowledgments
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