An effective-field-theory analysis of low-energy parity-violation in nucleon-nucleon scattering
Daniel R. Phillips, Matthias R. Schindler, Roxanne P. Springer
AAn effective-field-theory analysis of low-energy parity-violation innucleon-nucleon scattering
Daniel R. Phillips , , ∗ Matthias R. Schindler , † and Roxanne P. Springer , ‡ Department of Physics and Astronomy,Ohio University, Athens, OH 45701, United States; Department of Physics, Box 90305,Duke University, Durham, NC, 27708, United States School of Physics and Astronomy, University of Manchester,Manchester, M13 9PL, United Kingdom (Dated: November 2, 2018)
Abstract
We analyze parity-violating nucleon-nucleon scattering at energies
E < m π /M using the effectivefield theory appropriate for this regime. The minimal Lagrangian for short-range parity-violating N N interactions is written in an operator basis that encodes the five partial-wave transitions thatdominate at these energies. We calculate the leading-order relationships between parity-violating
N N asymmetries and the coefficients in the Lagrangian and also discuss the size of sub-leadingcorrections. We conclude with a discussion of further observables needed to completely determinethe leading-order Lagrangian.
Keywords: Parity violation, nucleon-nucleon scattering, effective field theory ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ nu c l - t h ] M a r . INTRODUCTION The existence of parity violation in nuclear forces is a manifestation of the presence ofweak interactions between the quarks in the nucleon. In this paper we discuss the mostbasic observables that display this phenomenon:
N N scattering asymmetries that wouldbe zero were parity conserved in the
N N interaction. We do this using an effective fieldtheory (EFT) that is based on the existence of large ( (cid:29) /m π ) scattering lengths in the N N system [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The scattering experiments we will study allow us to usean EFT that contains only nucleon degrees of freedom, and treats all interactions betweenthose degrees of freedom as being short-ranged. We choose to consider only the scatteringlength anomalously large; higher order corrections are obtained as a systematic expansionin powers of r/a , where a is the two-body scattering length and r the effective range in thecorresponding partial wave.We find it convenient to examine such corrections to parity-violating N N observables us-ing a dynamical dibaryon field [11, 12, 13]. This treatment captures the dominant dynamicsin the situation where both the scattering length and effective range are unnaturally large.It corresponds to resumming an infinite subset of terms that are higher order in the powercounting we use, where r ∼ /m π (cid:28) a ∼ /p . Therefore, expanding the results obtainedfrom the dibaryon formalism in powers of r/a reproduces the case of interest to us here, andso we employ dibaryon fields as a calculational tool.There are a number of theoretical treatments based on hadronic degrees of freedom thathave been used to study parity violation (PV). Pioneering theoretical studies on PV innuclear forces were carried out by Danilov [14] and Desplanques and Missimer [15]. Formost of the last thirty years, the framework of single-meson exchange, most commonlyusing the taxonomy developed by Desplanques, Donoghue, and Holstein (DDH) [16], hasbeen the one used to interpret and motivate experiments [17, 18, 19]. But the lack of aconcordance region in the space of DDH PV parameters (see, e.g., the plot in Ref. [18]) maybe related to the model assumptions—such as the mediation of the PV N N interaction byvector mesons—in that approach. The consequent desire by the community to “...recast theDDH language....in terms of...effective short-range parity-violating N-N interactions” [20]has motivated EFT treatments of the PV
N N force.Recently the version of chiral perturbation theory appropriate for few-nucleon systems [1,21, 22, 23, 24, 25] ( χ ET hereafter) has been used to derive the long-range ( r ∼ /m π ) part ofthe PV N N force, and to classify the short-distance operators that appear in that force [26].(See also the original χ PT analysis of PV pion-nucleon operators in Ref. [27]). This hasthe advantage that—up to a given order in χ ET—one can guarantee that a complete setof parity-violating operators has been considered. The potential of Ref. [26] is formulatedin terms of the appropriate degrees of freedom for momenta of order m π : nucleons andpions. The use of heavier mesons to encode
N N interactions of different t-channel quantumnumbers is not necessary at energies below ∼
200 MeV. By fitting the constants that encodethe short-distance
N N interaction to data, the χ ET treatment of PV in
N N scatteringavoids any assumption about what dynamics is at work for r (cid:28) /m π . The consequences of χ ET for PV
N N scattering, as well as other PV few-nucleon-system observables, have beencomputed in Refs. [30, 31, 32]. However, the PV operators and strong-nuclear-force wavefunctions employed in these works were not consistent, since phenomenological models were For extensions to include the ∆ isobar see Refs. [28, 29]. χ ET for the former. Refs. [30, 32] also include what are referred toas “pionless-theory” results, and there the mismatch between operators and wave functions isa serious problem. The use of AV18 wave functions for one, and a pionless EFT for the other,involves a mismatch of roughly an order of magnitude in the resolution ( ∼ renormalization)scales of these two different calculational ingredients.The pionless theory, EFT( (cid:54) π ), is relevant to studies of parity violation because manyof the existing and planned experiments [33, 34, 35, 36, 37, 38, 39, 40] take place atenergies at or below 10’s of MeV. In this region the pion-exchange nature of the nuclearforce is not resolved. For energies E < m π /M the EFT in which interactions are encodedas N N contact operators contains all the relevant degrees of freedom. The convergenceof this EFT for two-nucleon-system observables is well demonstrated [9, 10, 41, 42] in thelow-energy regime. In contrast, there have been significant questions raised recently aboutthe appropriate power counting for short-distance operators in the
N N system in the χ ETwhere pions are explicit degrees of freedom [5, 43, 44, 45, 46, 47], as well as about whetherthe Delta(1232) needs to be included as an explicit degree of freedom in order to guaranteereasonable convergence [21, 48, 49].In EFT( (cid:54) π ), parity violation in N N scattering is described by contact operators thathave a lowest possible dimension of seven. As Girlanda has recently shown [50] there arefive such independent operators. These five leading-order EFT contact operators containthe same physics as the five Danilov amplitudes [14] that encode the mixing between Sand P-waves that becomes possible in the presence of parity violation. In Section II werewrite the Lagrangian of Ref. [50] in terms of five operators that each mediate a specific S-Ptransition. This makes the computation of longitudinal asymmetries in
N N scattering (givenin Sec. III) straightforward. We present analytic results for the longitudinal asymmetriesin nn , pp , and np scattering. We use a dynamical dibaryon field to obtain a portion of thehigher-order corrections in r/a and rp in the EFT( (cid:54) π ) expansion for the strong rescattering.Subsection III B computes the Coulomb effects that are present in pp scattering at lowenergy and so modify the expression in the pp case. We extract the scale dependence of theresult obtained when assuming that the r dependence is higher order, and then discuss the r dependent corrections. In Sec. IV we compare our results with the two existing pieces ofexperimental data [33, 34] that are within the range of validity of this EFT. We close inSec. V with a summary and a discussion of further N N system experiments that could pindown the LO PV EFT( (cid:54) π ) Lagrangian. Details of the conversion of one set of operators toanother are given in an Appendix. II. LAGRANGIANS
At low enough energies, the details of the gauge boson ( g , W ± , Z ) exchange betweeninteracting quarks in the two-nucleon system are not experimentally accessible. Instead,the system can be described by using nucleon interpolating fields and treating both strongand weak interactions as contact interactions. To a given order there are a finite number ofindependent operators that describe these interactions. While Ref. [26] lists ten operators, it is pointed out that only five different combinations are relevant atlow energies. L = N † ( i∂ + (cid:126) ∇ M ) N − C S ( N † N ) − C T ( N † (cid:126)σN ) + . . . , (1)where the ellipsis stands for terms that contain more derivatives, and the nucleon field N carries both isospin and spin indices. The σ i are the SU(2) Pauli matrices in spin spaceand the τ a will be the SU(2) Pauli matrices in isospin space. In Eq. (1) the N N effectiveranges are assumed to be “natural” compared to the expected scale of 1 /m π while the N N scattering lengths are large in the same units. This facilitates an expansion of the
N N amplitude in powers of the small parameter Q , where Q ∼ /a ∼ p (with p the N N relativemomentum) [2, 3, 6].The same physics can be expressed using an operator basis that makes the incoming andoutgoing partial waves explicit [51, 52], L = N † ( i∂ + (cid:126) ∇ M ) N − C ( S )0 ( N T τ τ a σ N ) † ( N T τ τ a σ N ) − C ( S )0 ( N T τ σ σ i N ) † ( N T τ σ σ i N ) + . . . , (2)where now C ( S )0 = C S − C T and C ( S )0 = C S + C T . The operator between nucleons is simplythe projector onto the relevant partial wave, with the normalized projectors being [51, 52] P a ( S ) = 1 √ τ τ a σ ; P i ( S ) = 1 √ τ σ σ i . A convenient form of this Lagrangian is provided by use of dibaryon fields [11, 12, 13].This form is equivalent to Eqs. (2) and (1) at leading order, but resums all the higher-ordercorrections in
N N scattering that are proportional to the effective range. It would thereforegive the exact
N N amplitude in both the S and S channel were the shape parameterand all higher-order terms in the effective-range expansion zero. Dibaryon fields, s a and t i ,respectively, for the S and S states, are included in the Lagrangian [13, 53]: L = N † (cid:32) i∂ + (cid:126) ∇ M (cid:33) N − t † i (cid:32) i∂ + (cid:126) ∇ M − ∆ ( S ) (cid:33) t i − g ( S ) (cid:104) t † i N T P i ( S ) N + h.c. (cid:105) − s † a (cid:32) i∂ + (cid:126) ∇ M − ∆ ( S ) (cid:33) s a − g ( S ) (cid:2) s † a N T P a ( S ) N + h.c. (cid:3) , (3)with a an isospin and i a spin index. If PDS (power divergence subtraction) [3, 4] is usedto compute loops then for both channels we have (in an obvious notation with channelsubscripts suppressed): g = 8 πM r ; ∆ = 2 M r (cid:18) a − µ (cid:19) . (4)The leading-order PV two-nucleon Lagrangian can also be expressed in a variety of bases.4aralleling the one used in Eq. (1), Ref. [50] writes: L GirP V = (cid:110) G ( N † (cid:126)σN · N † i ↔ ∇ N − N † N N † i ↔ ∇ · (cid:126)σN ) − ˜ G (cid:15) ijk N † σ i N ∇ j ( N † σ k N ) − G (cid:15) ijk (cid:2) N † τ σ i N ∇ j ( N † σ k N ) + N † σ i N ∇ j ( N † τ σ k N ) (cid:3) − ˜ G I ab (cid:15) ijk N † τ a σ i N ∇ j ( N † τ b σ k N )+ G (cid:15) ab (cid:126) ∇ ( N † τ a N ) · N † τ b (cid:126)σN (cid:111) , (5)where a ↔ ∇ b = a (cid:126) ∇ b − a ← ∇ b and I = − . Note that we have renamed the coefficients as compared to Ref. [50], in order to avoidconfusion with the parity-conserving Lagrangian. In doing so we absorbed into the G s theoverall normalization factor of 1 / Λ χ that multiplies the C i ’s and ˜ C i ’s in Ref. [50]. InEFT( (cid:54) π ) the coefficients are typically dependent on the renormalization point, µ , used in theevaluation of loop diagrams. Using the partial-wave basis, as in Eq. (2), we have: L P WP V = − (cid:104) C ( S − P ) (cid:0) N T σ (cid:126)στ N (cid:1) † · (cid:16) N T σ i ↔ ∇ τ N (cid:17) + C ( S − P )(∆ I =0) (cid:0) N T σ τ (cid:126)τ N (cid:1) † (cid:16) N T σ (cid:126)σ · i ↔ ∇ τ (cid:126)τ N (cid:17) + C ( S − P )(∆ I =1) (cid:15) ab (cid:0) N T σ τ τ a N (cid:1) † (cid:16) N T σ (cid:126)σ · ↔ ∇ τ τ b N (cid:17) + C ( S − P )(∆ I =2) I ab (cid:0) N T σ τ τ a N (cid:1) † (cid:16) N T σ (cid:126)σ · i ↔ ∇ τ τ b N (cid:17) + C ( S − P ) (cid:15) ijk (cid:0) N T σ σ i τ N (cid:1) † (cid:18) N T σ σ k τ τ ↔ ∇ j N (cid:19)(cid:21) + h.c.. (6)The two Lagrangians in Eq. (5) and (6) give the same results for physical observables ifthe low-energy constants obey the relationships (see Appendix): C ( S − P ) = 14 ( G − ˜ G ) , C ( S − P )(∆ I =0) = 14 ( G + ˜ G ) , C ( S − P )(∆ I =1) = 12 G , C ( S − P )(∆ I =2) = −
12 ˜ G , C ( S − P ) = 14 G . (7)Note that there are five independent coefficients at this order—as explained in Refs. [26, 50,54]. They dictate the only possible nucleon-nucleon scattering observables at low enough(i.e., non-dynamical pion) energies. From the partial-wave point of view, only the coefficients5 ( S − P ) , C ( S − P )(∆ I =0) , C ( S − P )(∆ I =2) , and C ( S − P ) are involved in parity-violating neutron-protonobservables; while C ( S − P )(∆ I =0) , C ( S − P )(∆ I =1) , and C ( S − P )(∆ I =2) are involved in parity-violating neutron-neutron (or proton-proton) observables.In the DDH approach, all but the S − P operator are considered only in terms ofvector-meson exchange. At low energies, the vector mesons are not dynamical, so the DDHdescription can be considered a way to “encode” the processes so that calculations and exper-iments can be compared, so long as the vector-meson interpretation is not taken literally. Inparticular, at very low energies only five independent parameters are relevant for the physicsof parity violating N N scattering—they are sufficient to encode all leading-order phenom-ena. The EFT parameterization presented here provides a model-independent language inwhich to compare experiments.
III. RESULTSA. Calculation of longitudinal analyzing power
Parity violation in the
N N interaction leads to mixing between odd and even partialwaves. To obtain the leading effects of this mixing in EFT( (cid:54) π ) it is sufficient to calculate theamplitude that mediates S -wave to P -wave transitions. The mixing of higher partial wavesis suppressed by additional powers of the small parameter Q .The leading diagrams contributing to this parity-violating N N scattering amplitude areshown in Fig. 1. Note that only the S -wave side receives an enhancement from the strong S -wave bubble sum. Diagrams with strong rescattering on the P -wave side are higher order.We evaluate the diagrams shown in Fig. 1 using the PDS [3, 4] renormalization scheme tocalculate the loops. Keeping in mind the issue of higher-order corrections (see Subsec. III C),we employ the dibaryon Lagrangian (3) to compute the strong rescattering. Our result forthe scattering amplitude is: T P Vnn/pp = ± p A nn/pp (cid:18) a S − r S p − µ (cid:19) (cid:18) a S − r S p + ip (cid:19) − , (8) P S + P S
FIG. 1: Diagrams contributing to parity-violating NN scattering. The shaded blob is the leadingorder parity-conserving amplitude, while the square denotes the vertex from the parity-violatingLagrangian. The nucleons are in a P-wave on the left of the parity-violating vertex, and in anS-wave on the right. T P Vnp = ± p (cid:34) A S np (cid:18) a S − r S p − µ (cid:19) (cid:18) a S − r S p + ip (cid:19) − + A S np (cid:18) a S − r S p − µ (cid:19) (cid:18) a S − r S p + ip (cid:19) − (cid:35) , (9)where − iT is the sum of diagrams in Fig. 1, p = | (cid:126)p | (see Fig. 2) and the upper (lower) signis for a beam of positive (negative) helicity. The (strong) parameters a S +1 L J and r S +1 L J are, respectively, the scattering length and effective range of a particular partial wave. Theweak N N interaction parameters are collected in amplitudes A , which are given by: A nn = G + ˜ G − G + ˜ G ) (10)= 4 (cid:16) C ( S − P )(∆ I =0) − C ( S − P )(∆ I =1) + C ( S − P )(∆ I =2) (cid:17) , (11) A pp = G + ˜ G + 2( G − ˜ G ) (12)= 4 (cid:16) C ( S − P )(∆ I =0) + C ( S − P )(∆ I =1) + C ( S − P )(∆ I =2) (cid:17) , (13) A S np = G + ˜ G + 4 G (14)= 4 (cid:16) C ( S − P )(∆ I =0) − C ( S − P )(∆ I =2) (cid:17) , (15) A S np = G − ˜ G − G (16)= 4 (cid:16) C ( S − P ) − C ( S − P ) (cid:17) , (17)in the notation of the Lagrangians (5) and (6), respectively. While the T P VNN expressionsappear to have an explicit subtraction point ( µ ) dependence, as physical observables eachmust be µ independent. This dictates the scaling of the A NN with respect to µ .The leading-order [ O ( Q )] amplitudes are obtained by setting the effective ranges r S and r S equal to zero in Eqs. (8) and (9). This yields, for example, T P Vnn ( r S = 0) = ± p A nn C ( S )0 πM a S + ip , (18)a result that already appeared in Section 4 of Ref. [26], but here we have also provided therelationship of the parity-violating amplitude A nn to the coefficients in the Lagrangian(s)(Eqs. (10) and (11)). The ratio of A nn to C ( S )0 must be independent of µ . Since, in PDS, C ( S )0 = 4 πM a S − µ , (19)we have 1 A nn ∂ A nn ∂µ = 1 a S − µ . (20)In N N scattering at these energies the weak interaction is about 10 − times the stronginteraction. Therefore feasible experiments involve observables that vanish under strong7nteractions. Relevant N N measurements have focused on longitudinal asymmetries in (cid:126)N + N scattering. Here, the interference terms between the strong and weak operators changesign when the longitudinal polarization of the incoming nucleon changes sign. The strong-interaction scattering is unaffected by a change in polarization, so an asymmetry is formedwhen the differential cross sections of the two different polarization states are subtracted.From the scattering amplitude calculated above we can determine the longitudinal asym-metry: A L = σ + − σ − σ + + σ − , (21)where σ ± is the total scattering cross section of a nucleon with helicity ± on an unpolarizednucleon target—unless integration over a restricted angular range (e.g. in pp scattering) isindicated. Neglecting, for the moment, the Coulomb interaction in the pp case we find A nnL = 2 Mπ p A nn (cid:18) a S − r S p − µ (cid:19) , (22) A ppL = 2 Mπ p A pp (cid:18) a S − r S p − µ (cid:19) , (23)and A npL = 2 Mπ p (cid:40) dσ S d Ω dσ S d Ω + 3 dσ S d Ω A S np (cid:18) a S − r S p − µ (cid:19) + dσ S d Ω dσ S d Ω + 3 dσ S d Ω A S np (cid:18) a S − r S p − µ (cid:19)(cid:41) . (24)The differential cross sections dσ S d Ω and dσ S d Ω only contain contributions from the parity-conserving Lagrangian (see Eqs. (1) and (2)): dσd Ω = (cid:34)(cid:18) a − rp (cid:19) + p (cid:35) − . (25)We have again suppressed the channels’ superscripts.Upon setting r S = 0 and using Eq. (19), Eqs. (22) and (23) recapture the form derivedin Refs. [26, 54]: A pp/nnL = 8 p A pp/nn C S . (26)This, and the more complex formula for A npL : A npL = 8 p (cid:32) dσ S d Ω dσ S d Ω + 3 dσ S d Ω A S np C S + dσ S d Ω dσ S d Ω + 3 dσ S d Ω A S np C S (cid:33) , (27)are the LO predictions of EFT( (cid:54) π ) for these asymmetries. The only unknown quantitiesin these predictions are the coefficients of the parity-violating Lagrangian. Eqs. (22)–(24)relate these coefficients (Eqs. (10)–(17)) to observable asymmetries. From these expressions8e see that a measurement of all three analyzing powers as a function of energy could pindown four different combinations of coefficients, since the two pre-factors in Eq. (24) havedistinct energy dependence—even if r S = r S = 0. However, the nn experiment is notfeasible in the foreseeable future, and so alternative strategies to access the combination A nn are probably necessary. B. Coulomb corrections for pp The result in Eq. (26) ignores the Coulomb interaction. Coulomb photons can be includedin EFT( (cid:54) π ), and the computation of Coulomb scattering was carried out to leading order forS-wave N N scattering in Ref. [55, 56].In the parity-violating case the computation proceeds as in Fig. 1, except now Coulombphotons must be added to the initial, final, and all intermediate states. Since the initial-state and final-state Coulomb scattering factorizes this yields the final result, quoted inRef. [26, 54]: T P VNC ( r S = 0) = ± p A pp C S C η exp( i ( σ ( η ) + σ ( η ))) 1 C S − J ( p ) , (28)where the purely Coulombic part of the scattering amplitude has been separated off thetotal amplitude [57], T = T NC + T Coul . (29) C η is the Sommerfeld factor: C η = 2 πηe πη − , (30)with the Coulomb parameter η ≡ Mα p , and σ l ( η ) = argΓ( l + 1 + iη ), where Γ is the Eulergamma function. J ( p ) is the Coulomb-modified bubble: J finite0 ( p ) = − αM π (cid:20) H ( η ) − ln (cid:18) µ √ παM (cid:19) − C E (cid:21) − µM π (31)once divergences in D = 4 and D = 3 have been subtracted, H ( η ) = ψ ( iη ) + 12 iη − log( iη ) , (32)with ψ the derivative of the Euler Gamma function, and C E = 0 . ... ) is Euler’s constant.(See also Refs. [58, 59].)Experimental asymmetries are typically measured over a finite angular range. This is aparticularly important detail in pp scattering, due to the infinite Coulomb cross section inthe forward direction. Implementing the integrals over a finite range θ ≤ θ ≤ θ we have: A ppL = (cid:82) θ θ dθ sin θ T P VNC )( T NC + T Coul ) † ] (cid:82) θ θ dθ sin θ | T NC + T Coul | ≈ (cid:20) T P VNC T NC (cid:18) − θ − cos θ (cid:90) θ θ dθ sin θ T Coul T NC (cid:19)(cid:21) , (33)9here we have used the fact that T NC,P V and T NC are angle independent at this order, andhave neglected | T Coul | . Ref. [55] finds that T NC = C η e iσ ( η )1 C S − J ( p ) , yielding A ppL ≈ p A pp C S Re (cid:20) e i [ σ ( η ) − σ ( η )] (cid:18) − θ − cos θ (cid:90) θ θ dθ sin θ T Coul T NC (cid:19)(cid:21) . (34)The factor in square brackets contains the Coulomb corrections to the result of the previoussection. It is a function of η , the scattering length a , and the angular range being examined.For the experiments of interest here we have η (cid:28)
1. For small η , T Coul = 2 παp − cos θ + O ( α ) , (35)and T NC = 4 πM a S ( µ ) + ip + O ( η ) , (36)with the strong pp scattering length, a S ( µ ), defined by:4 πM C S ( µ ) = 1 a S ( µ ) − µ. (37)A reliable extraction of a S ( µ ) from pp data appears to require a computation to severalorders in EFT( (cid:54) π ) [60]. Instead, for comparison to experiments in Sec. IV, we use isospinsymmetry, and take for a S ( µ ) the ‘recommended’ central value of the nn scattering length, a nn = − .
59 fm [61]. We obtain A ppL ≈ p A pp C S Re (cid:20) e i [ σ ( η ) − σ ( η )] (cid:26) η (cid:18) a S ( µ ) p + i (cid:19) θ − cos θ ln (cid:18) − cos θ − cos θ (cid:19)(cid:27)(cid:21) (38)= 8 p A pp C S (cid:20) η (cid:18) a S ( µ ) p (cid:19) θ − cos θ ln (cid:18) − cos θ − cos θ (cid:19) + O ( η ) (cid:21) , (39)so long as forward angles are avoided.Even for pp experiments at T lab = 0 . η ≈ .
22, so η should be a goodexpansion parameter. Since parity-violating asymmetries grow as √ T lab the extant measure-ments of A L were conducted at energies significantly higher than this, so in practice ignoringCoulomb (as was done in the pioneering study of Ref. [15]), or expanding in powers of η , isa good approximation.This suggests using a different expansion where effects proportional to M α/p are treatedperturbatively. However, numerically
M α ∼ /a in the S channel. If M α/p is treated asa small parameter, then 1 / ( ap ) should really also be treated as a perturbation. This resultsin a theory set up as an expansion around the unitary ( | a | → ∞ ) limit [62]. Attempts totreat Coulomb interactions in perturbation theory and retain the 1 / ( ap ) corrections in theunitary limit to all orders requires care since the divergences that are present in the Coulomb10ubble must still be absorbed [55, 63]. While the Coulomb contributions to the final result,Eq.(39), are small for all existing and proposed experiments, here we have retained themto all orders in the intermediate steps of the calculation of the pp scattering amplitude,and only performed the expansion in powers of η when that amplitude is inserted in theexpression for the asymmetry. C. Corrections proportional to the effective range
Here we discuss corrections to the leading-order result of Eq. (26). Since we only examinethe form of the NLO correction, and do not compare to experimental data, we consider nn scattering. The arguments are similar for pp scattering and np scattering. Only operatorswith the same space-spin structure as the leading-order ones of Eq. (6) are necessary for thisanalysis. Other space-spin structures, e.g, mixing between P - and D -waves, have the samenumber of derivatives as the operators we will consider in this section, but the resultingamplitudes are not enhanced by the strong S -wave rescattering. (E.g, effects of P - D mixingdo not enter until O ( Q ): three orders beyond leading.)The result in Eq. (22) is actually somewhat deceptive. The use of the dibaryon formalismin the strong Lagrangian seems to imply that the physics of the effective range has beenincluded to all orders in Eq. (22). This is not the case because the dibaryon formalismwas not used in the weak Lagrangian. Either scaling for the effective range (as Q − or Q )will lead to consistent results—but only if the choice is used uniformly in all aspects of thecalculation.In particular, demanding that ∂A nnL ∂µ = 0 (40)implies that the A nn of Eqs. (10) and (11) becomes energy dependent: A nn ∼ a − rp − µ , (41)where we have dropped the partial-wave specification on a and r . To obtain consistent resultsfor the case considered here, where the effective range is natural ( ∼ Q ), we must expandEq. (41) in powers of r . This allows us to estimate the impact of corrections proportionalto r in the weak-interaction piece of the Lagrangian. It yields: A nn ∼ a − µ (cid:18) rp a − µ + · · · (cid:19) . (42)Writing this as A nn = A LOnn + p A NLOnn + · · · , (43)makes it clear that there must be corrections to the leading-order weak-interaction La-grangian that have the same space-spin structure, but are proportional to the square of themomentum (equivalently, the energy) of the N N collision. Each term will be accompaniedby its own low-energy constant. These corrections to the A nn of Eqs. (10) and (11) are sup-pressed by one power of the small parameter Q . They are missing from the result (8), whichincludes only the leading-order part of A nn , and the effect of strong rescattering. Neglecting11LO contributions to the weak Lagrangian results in an inconsistent calculation as soon as r (cid:54) = 0.While A LOnn recaptures the scaling of C (Eq. (19)), as expected, A NLOnn runs like C , theNLO strong coefficient (see, for example, Eq. (2.26) of Ref. [4]): A NLOnn A LOnn ∼ r a − µ . (44)The necessity for the weak Lagrangian to have an O ( Q ) piece with coefficients scaling ac-cording to Eq. (44) can also be derived by considering the µ -invariance of the NLO amplitudefor PV N N scattering in a strictly perturbative calculation in powers of Q . Conversely, werewe to use a weak Lagrangian expressed using dibaryon fields for the S -channels the scaling(41) would emerge automatically. In either case, in order to maintain µ -independent resultsthe counting of r must remain consistent between L weak and L strong . Both Lagrangianscontain higher-order terms that are proportional to r , and the scaling of these contributionswith µ is correlated. If effects proportional to r are resummed using a dibaryon formalismconsistently in both weak and strong Lagrangians, naive dimensional analysis suggests thatadditional corrections in L P V , which are related to additional parameters, are suppressedby two powers of Q.
IV. COMPARISON WITH EXPERIMENT
In order to completely specify the leading-order PV
N N
Lagrangian in this EFT thecoefficients of the five dimension-7 operators must be determined. The only way to do thisin a model-independent fashion is to fit them to experiment. If the experiments are at lowenough energy the corrections from higher-dimensional operators that encode other partial-wave transitions, as well as energy-dependence of the S-P transitions, will presumably besmall. In practice a higher-order analysis, together with a variety of different measurements,will have to be employed to see if the EFT is complete and consistent.Here we pursue only a leading-order analysis of the two most recent low-energy measure-ments of the longitudinal asymmetry in pp scattering. These yielded [33] A (cid:126)ppL ( E = 13 . − . ± . × − (45)and [34] A (cid:126)ppL ( E = 45 MeV) = ( − . ± . × − (46)in the angular range 23 o < θ lab < o .Using (39) together with the lower-energy number (45) yields: A pp ( µ = m π ) = (1 . ± . × − MeV − . (47)(Here and below the errors are only experimental, and do not include the uncertainty dueto higher-order corrections.) For the µ -independent ratio this gives: A pp C S = ( − . ± . × − MeV − . (48)12t this value of p the Coulomb parameter η = 0 . η inEq. (39) also includes a factor of 1 / ( ap ) ≈ − .
13. The Coulomb correction is only 3 percent,smaller than the uncertainties in the measurement and higher-order effects in EFT( (cid:54) π ).Equation (48) may be used to predict the scattering asymmetry at the higher energy of45 MeV, yielding: A (cid:126)ppL ( E = 45 MeV) = ( − . ± . × − . (49)The two extant low-energy data are thus consistent with a leading-order EFT( (cid:54) π ) analysiswithin their combined uncertainties. This is really nothing more than the statement thatat these energies the asymmetry is scaling with the center-of-mass momentum—as alreadyobserved in Ref. [26].It should, however, be noted that the center-of-mass momentum for the second experimentis already larger than m π . Sub-leading corrections could therefore be large. A crude estimateof these effects can be obtained by using Eq. (23), which includes the effects proportional to r (but see also Sec. III C) due to strong rescattering. This yields (with r S = 2 .
73 fm): A pp ( µ = m π ) C S = ( − . ± . × − MeV − . (50)The shift of ∼
30% with respect to the leading-order value (48) is entirely consistent withthe expansion parameter of EFT( (cid:54) π ). The prediction for the higher-energy datum is now A (cid:126)ppL ( E = 45MeV) = ( − . ± . × − . (51)In this case the shift is more than 50% of the leading-order value (49), suggesting that thepoint at 45 MeV is indeed too high for profitable application of EFT( (cid:54) π ). The large (par-tial) O ( Q ) correction computed here suggests that we can anticipate significant additionalcorrections at next-to-next-to-leading order. Given the presence of these corrections, as wellas the experimental error, there is no real tension between (51) and (46). The large NLOcorrection would, though, seem to imply that the agreement between the datum of Ref. [34]and the LO prediction (49) is fortuitous.Measurements of the neutron’s spin rotation as it passes through parahydrogen have beenproposed, e.g., in Ref. [64]. The hope here is to extract the longitudinal analyzing powerof (cid:126)n + p scattering. The thermal energies at which these experiments take place are idealfor EFT( (cid:54) π ). The leading-order pionless EFT prediction is given in Eq. (24) and dependsupon the coefficients C ( S − P ) , C ( S − P )(∆ I =0) , C ( S − P )(∆ I =2) , and C ( S − P ) , so once low-energy datais available constraints on EFT( (cid:54) π ) coefficients will result. V. CONCLUSION AND OUTLOOK
We have presented the leading-order low-energy prediction for nn , pp , and np longitudinalasymmetries. They depend on five different parameters, but one asymmetry measurementeach in (cid:126)n + n and (cid:126)p + p , as well as two at different energies in (cid:126)n + p , would allow theextraction of four of the five parameters. We determined, in agreement with the findings ofearlier authors, that the Coulomb corrections to pp scattering are not significant at leadingorder for the energies at which these measurements are made. Finally, we showed that whenthe effective range is taken to scale as Q , the running of the leading order weak interaction13oefficients mimics that of C of the strong interaction, while the running of the next-to-leading-order coefficients is expected to mimic that of C of the strong interaction. This is asimple consequence of the mixing of the S -wave side of the parity violating operators withthe bubble-sum enhancement of strong S -wave scattering.Our EFT( (cid:54) π ) calculations presented here, which implement a systematic power countingscheme, show the consistency of the Danilov hypothesis that (at least for energies < m π /M )the dominant energy dependence in parity-violating N N observables arises from the large
N N scattering lengths in the parity-conserving sector. Effects of energy (or momentum)dependence in the parity-violating
N N interaction, as well as those due to PV mixing be-tween other partial waves, constitute higher-order effects in EFT( (cid:54) π ) which are accompaniedby additional unknown parameters.At leading order in EFT( (cid:54) π ) there are only five independent PV N N operators [50].This means that five independent measurements will serve to pin down the leading-orderLagrangian. Equation (6) is one way to write the five terms of the LO PV Lagrangian inEFT( (cid:54) π ). It is equivalent to the previously published form (5), with the matching computedin detail in the subsequent Appendix. The Lagrangian (6) has the advantage of beingwritten in an operator basis where each coefficient contributes to one and only one partial-wave transition.For instance, the longitudinal asymmetry in pp scattering probes the coefficients C ( S − P )(∆ I =0) , C ( S − P )(∆ I =1) , and C ( S − P )(∆ I =2) . Our formulae (13) and (23) encode the specific combination inwhich the coefficients associated with different isospin transitions appear. Existing experi-mental data can be used to extract this combination—admittedly with large error bars. Alower-energy pp experiment with high precision would be a useful development.Meanwhile the partial-wave transitions C ( S − P ) , C ( S − P )(∆ I =0) , C ( S − P )(∆ I =2) , and C ( S − P ) areprobed in the np longitudinal asymmetry. In principle a detailed study of the energy de-pendence of this asymmetry could allow the extraction of C ( S − P ) and the particular linearcombination of the other three coefficients relevant for np scattering.Finally, the LO EFT( (cid:54) π ) prediction for the longitudinal analyzing power of (cid:126)n + n scatter-ing depends upon the coefficients C ( S − P )(∆ I =0) , C ( S − P )(∆ I =1) , and C ( S − P )(∆ I =2) , but in a different linearcombination to that appearing in the prediction for A ppL . Given the difficulties inherent insuch an experiment it seems more productive to focus on the asymmetry in (cid:126)n + d scatteringat low energies (see also Ref. [32]) and perform the necessary three-body calculations forthe interpretation of that asymmetry (see Ref. [65, 66] for examples in the parity-conservingsector) within the consistent EFT( (cid:54) π ) framework for parity-violating N N scattering laid outhere.In any calculation of PV
N N observables it is important to treat the PV and PC
N N interactions consistently. In particular, care must be taken that operators and wave functionsused in the same calculation are evaluated using compatible schemes and subtraction points.Use of the AV18
N N potential to evaluate matrix elements of the short-range operators inEq. (5) [30, 32] represents a significant mismatch in this regard, and cannot be considereda systematic EFT( (cid:54) π ) calculation.In Sec. III C we emphasized the importance of maintaining a consistent power counting forboth the weak and strong parts of the Lagrangian. A consistent calculation in EFT( (cid:54) π )can becarried out assuming either that r scales as Q —in which case range corrections are treatedperturbatively— or that it scales as Q − , in which case a dibaryon formalism is necessary.The most appropriate choice should be revealed by seeing which (possibly higher-order)14redictions provide a better explanation of the data.The PV coefficients for N N asymmetries are presently experimentally underconstrained,so it is necessary to use electromagnetic reactions in the
N N system to probe additionallinear combinations of the five PV parameters. At lowest order in EFT( (cid:54) π ), both (cid:126)np → dγ and the anapole moment of the deuteron depend only upon a single coefficient ( C ( S − P ) ),and so serve to disentangle this coefficient from the linear combination involved in other np processes. These have been computed in Ref. [67]. Measurements of the asymmetryin (cid:126)np → dγ are presently consistent with zero [37, 68], but improvements by an order ofmagnitude are expected [38, 69].A further constraint on the five PV parameters is potentially available from circularlypolarized photon-deuteron breakup (or the inverse reaction). Experimentally, results arepresently consistent with zero [40]. The development of high intensity free electron lasersto produce circularly polarized photons has led to proposals (e.g., Ref. [39, 70]) to performthis measurement if the necessary luminosity can be achieved. The PV parameters involvedin the LO EFT( (cid:54) π )prediction are C ( S − P ) , C ( S − P )(∆ I =0) , and C ( S − P )(∆ I =2) . This system has beendiscussed in Ref. [71]. A partial LO calculation has recently been reported [72].We have presented a model-independent set of operators and coefficients with whichlow-energy PV observables can be described and compared, emphasizing the utility of thepartial-wave basis. Such a treatment conveys significant advantages in our efforts to under-stand manifestations of parity violation in few-nucleon systems. Further calculations andexperiments which use the EFT( (cid:54) π ) framework to map out the landscape of possible exper-iments in two-, three-, and four-body systems that are pertinent to parity violation wouldbe very useful. Acknowledgments
DRP gratefully acknowledges the hospitality of the Theoretical Physics group at theUniversity of Manchester and the Center for the Subatomic Structure of Matter at theUniversity of Adelaide during part of this work. MRS would like to thank the TheoreticalPhysics group at the University of Manchester and the Lattice and Effective Field Theorygroup at Duke University for their hospitality. RPS acknowledges the hospitality of OhioUniversity, where much of this work was performed. We are grateful for discussions withPil-Neyo Seo on the status of experiments. We would like to thank L. Tiator for helpobtaining Ref. [33]. We thank D. Eversheim for making the most recent analysis of the 13 . VI. APPENDIX
In this appendix we discuss the matching between the Weinberg basis Lagrange densityof Eq. (1) and the partial wave basis Lagrange density of Eq. (2). One method for matchingthe coefficients in these two different bases is to use Fierz rearrangement identities. Anothermethod is to use the Lorentz structures with their nucleon spin and isospin indices explicit,and employ orthogonality and completeness of the operators in order to isolate one set of15perators in terms of the other. The latter is easy to implement using a Mathematica [73]code with the HighEnergyPhysics ‘FeynCalc‘ package [74] to perform SU(2) manipulations.As a simple example we consider the lowest order terms of the strong interaction Lagrangedensity for the two nucleon system. A useful Fierz identity is( σ µ ) ij ( σ µ ) kl = 2 (cid:15) ik (cid:15) jl , where (cid:15) = iσ is the totally anti-symmetric Levi-Civita tensor, µ is summed over ( µ =0 , , , σ µ = (1 , (cid:126)σ ) and i, j, k, and l are summed over spin indices. An identical equationserves just as well for the τ matrices and the isospin indices (we will use a, b, c , and d forthese). Putting this together:( σ µ ) ij ( σ µ ) kl ( τ ν ) ab ( τ ν ) cd = (2 (cid:15) ik (cid:15) jl )(2 (cid:15) ac (cid:15) bd ) = 4( σ ) ik ( σ ) jl ( τ ) ac ( τ ) bd . To obtain an equality involving the C S operator, ( N † N )( N † N ), we want delta func-tions rather than σ ’s and τ ’s on the right hand side, so act on both sides with( σ ) i (cid:48) i ( σ ) ll (cid:48) ( τ ) a (cid:48) a ( τ ) dd (cid:48) and obtain( σ σ µ ) i (cid:48) j ( σ µ σ ) kl (cid:48) ( τ τ ν ) a (cid:48) b ( τ ν τ ) cd (cid:48) = 4 δ i (cid:48) k δ jl (cid:48) δ a (cid:48) c δ bd (cid:48) . (52)Contracting this with N † kc N i (cid:48) a (cid:48) N † l (cid:48) d (cid:48) N jb – the nucleon operators with their spin and isospinindices explicit – and noticing that diagonal terms with N T σ τ N and N T σ σ i τ τ j N aredisallowed by the Pauli principle, we obtain the decomposition of the operator associatedwith C S in terms of the operators in the partial wave basis:( N † N )( N † N ) = − N T P a ( S ) N ) † ( N T P a ( S ) N ) − N T P i ( S ) N ) † ( N T P i ( S ) N ) , where a and i are now summed over (1,2,3) as in the projection operators introduced insection II.To obtain the form ( N † σ i N )( N † σ i N ), associated with C T , out of the Fierz identity, weneed to appropriately insert not only σ and τ (to get to delta functions) but σ A ’s on theright-hand side as well. Contracting( σ A ) mi (cid:48) ( σ A ) l (cid:48) n N † ma (cid:48) N kc N † jb N nd (cid:48) on both sides of Eq. (52) yields,4( N † σ i N )( N † σ i N ) = ( N ∗ σ i σ τ N † )( N T σ σ i τ N ) − ( N ∗ σ i σ τ τ j N † )( N T σ σ i τ j τ N ) − ( N ∗ σ i σ σ j τ N † )( N T σ j σ σ i τ N ) + ( N ∗ σ i σ σ j τ τ k N † )( N T σ j σ σ i τ k τ N ) . (53)But this can be considerably simplified because( N ∗ σ i σ τ τ j N † )( N T σ σ i τ j τ N ) = 0 ;( N ∗ σ i σ σ j τ N † )( N T σ j σ σ i τ N ) = 2( N ∗ σ i σ τ N † )( N T σ σ i τ N ) ;( N ∗ σ i σ σ j τ τ k N † )( N T σ j σ σ i τ k τ N ) = 3( N † σ τ k τ N † )( N σ τ τ k N ) , (54)so that( N † σ i N )( N † σ i N ) = − N T P a ( S ) N ) † ( N T P a ( S ) N ) + 6( N T P i ( S ) N ) † ( N T P i ( S ) N ) , (55)16 , c, ~p i, a, ~p ′ l, d, − ~p j, b, − ~p ′ FIG. 2: Assignment of spin indices ( i, j, k, l ), isospin indices ( a, b, c, d ), and momentum labels forpurposes of matching the partial wave basis operators to the strong Weinberg and weak Girlandaoperators. Figure created using JaxoDraw [75]. which completes the decomposition of the C S and C T operators in terms of the C S and C S operators.An easier procedure for obtaining one set of basis coefficients in terms of the other is to useorthogonality and completeness of the operator sets. We will illustrate this using, again, theleading-order strong-interaction terms. Making the spin indices ( i, j... ) and isospin indices( a, b... ) explicit (referring to Fig. 2) and including all possible nucleon assignments, − C S δ ik δ ac δ jl δ bd + C S δ jk δ bc δ il δ ad − C T ( σ A ) ik δ ac ( σ A ) jl δ bd + C T ( σ A ) jk δ bc ( σ A ) il δ ad =12 C ( S )0 ( τ A τ ) ab ( σ ) ij ( τ τ A ) cd ( σ ) kl + 12 C ( S )0 ( τ ) ab ( σ B σ ) ij ( τ ) cd ( σ σ B ) kl , (56)where summation over A, B = 1 , , δ ik δ ac δ jl δ bd , yields: C ( S )0 + C ( S )0 = 2 C S − C T . A second equation is found by contracting both sides with the third structure,( σ A ) ik δ ac ( σ A ) jl δ bd , yielding: 3 C ( S )0 − C ( S )0 = 2 C S − C T . Solving for the partial wave coefficients yields the relationships given in Section II.Now consider the two weak-interaction bases from Eq. (5) and Eq. (6). Note that the op-erators in Eq. (5) are explicitly hermitian, but not symmetric under interchange of outgoing(or incoming) particles. On the other hand, the basis used in Eq. (6) is symmetric underinterchange of particles, but each is not its own hermitian conjugate. This is important toremember when comparing coefficients.Even without Fierzing, inspection of the operators suggests that not all of the partialwave operators are involved in the decomposition of, say, the operator associated with G in Eq. (5). But no orthogonality need be assumed because it is easy to verify. The startingpoint is (momenta from Fig. 2):4 C ( S − P ) ( σ A σ ) ij ( σ ) kl ( τ ) ab ( τ ) cd (2 p A ) + 4 C ( S − P ) ( σ σ A ) kl ( σ ) ij ( τ ) ab ( τ ) cd (2 p (cid:48) A )+ 4 C ( S − P )(∆ I =0) ( σ ) ij ( σ σ A ) kl ( τ B τ ) ab ( τ τ B ) cd (2 p A )+ 4 C ( S − P )(∆ I =0) ( σ ) kl ( σ A σ ) ij ( τ B τ ) ab ( τ τ B ) cd (2 p (cid:48) A )17 4 C ( S − P )(∆ I =1) ( σ ) ij ( σ σ A ) kl ( τ B τ ) ab ( τ τ C ) cd (cid:15) BC ( − ip A )+ 4 C ( S − P )(∆ I =1) ( σ ) kl ( σ A σ ) ij ( τ τ B ) cd ( τ C τ ) ab (cid:15) BC ( − ip (cid:48) A )+ 4 C ( S − P )(∆ I =2) ( σ ) ij ( σ σ A ) kl ( τ B τ ) ab ( τ τ C ) cd I BC (2 p A )+ 4 C ( S − P )(∆ I =2) ( σ ) kl ( σ A σ ) ij ( τ τ B ) cd ( τ C τ ) ab I BC (2 p (cid:48) A )+ 4 C ( S − P ) ( σ A σ ) ij ( σ σ C ) kl ( τ ) ab ( τ τ ) cd (cid:15) ABC ( − ip B )+ 4 C ( S − P ) ( σ σ A ) kl ( σ C σ ) ij ( τ ) cd ( τ τ ) ab (cid:15) ABC ( − ip (cid:48) B )= 2 G (( σ A ) ik δ jl − δ ik ( σ A ) jl )( p + p (cid:48) ) A δ ac δ bd − G (( σ A ) jk δ il − δ jk ( σ A ) il )( p − p (cid:48) ) A δ bc δ ad − G (cid:15) ABC ( σ A ) ik ( σ C ) jl ( ip (cid:48) − ip ) B δ ac δ bd + 2 ˜ G (cid:15) ABC ( σ A ) jk ( σ C ) il ( − ip (cid:48) − ip ) B δ bc δ ad − G (cid:15) ABC ( σ A ) ik ( σ C ) jl ( ip (cid:48) − ip ) B (( τ ) ac δ bd + δ ac ( τ ) bd )+ 2 G (cid:15) ABC ( σ A ) jk ( σ C ) il ( − ip (cid:48) − ip ) B (( τ ) bc δ ad + δ bc ( τ ) ad ) − G (cid:15) ABC ( σ A ) ik ( ip (cid:48) − ip ) B ( σ C ) jl I DE ( τ D ) ac ( τ E ) bd + 2 ˜ G (cid:15) ABC ( σ A ) jk ( − ip (cid:48) − ip ) B ( σ C ) il I DE ( τ D ) bc ( τ E ) ad + 2 G δ ik ( σ A ) jl ( − ip (cid:48) + ip ) A (cid:15) BC ( τ B ) ac ( τ C ) bd − G δ jk ( σ A ) il ( ip (cid:48) + ip ) A (cid:15) BC ( τ B ) bc ( τ C ) ad . 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