Applications of the chiral potential with the semi-local regularization in momentum space to the disintegration processes
V.Urbanevych, R.Skibiński, H.Witała, J.Golak, K.Topolnicki, A.Grassi, E.Epelbaum, H.Krebs
AApplications of the chiral potential with the semi-localregularization in momentum space to the disintegration processes
V. Urbanevych, R. Skibi´nski, H. Wita(cid:32)la, J. Golak, K. Topolnicki, and A. Grassi
M. Smoluchowski Institute of Physics,Jagiellonian University, PL-30348 Krak´ow, Poland
E. Epelbaum and H. Krebs
Ruhr-Universit¨at Bochum, Fakult¨at f¨ur Physik und Astronomie,Institut f¨ur Theoretische Physik II, D-44780 Bochum, Germany (Dated: July 30, 2020)
Abstract
We apply the chiral potential with the momentum space semi-local regularization to the H and He photodisintegration processes and to the (anti)neutrino induced deuteron breakup reactions.Specifically, the differential cross section, the photon analyzing power and the final proton polar-ization have been calculated for the deuteron photodisintegration at the photon energies 30 MeVand 100 MeV. For the He photodisintegration predictions for the semi-inclusive and exclusivedifferential cross sections are presented for the photon energies up to 120 MeV. The total crosssection is calculated for the (anti)neutrino disintegrations of the deuteron for the (anti)neutrinoenergies below 200 MeV. The predictions based on the Argonne V18 potential or on the older chi-ral force with regularization applied in coordinate space are used for comparison. Using the fifthorder chiral nucleon-nucleon potential supplemented with dominant contributions from the sixthorder allows us to obtain converged predictions for the regarded reactions and observables. Ourresults based on the newest semi-local chiral potentials show even smaller cutoff dependence forthe considered electroweak observables than the previously reported ones with a coordinate-spaceregulator. However, some of the studied polarization observables in the deuteron photodisinte-gration process reveal more sensitivity to the regulator value than the unpolarized cross section.The chiral potential regularized semi-locally in momentum space yields also fast convergence ofresults with the chiral order. These features make the used potential a high quality tool to studyelectroweak processes.
PACS numbers: 13.75.Cs, 21.45.-v, 25.10.+s, 25.20.x, 13.15.+g a r X i v : . [ nu c l - t h ] J u l . INTRODUCTION Chiral effective field theory ( χ EFT) is nowadays the most reliable approach to study low-energy nuclear forces. It’s continuous development for nearly 30 years resulted in advancedtwo-nucleon and many-nucleon interactions [1–7].In 2018 the Bochum group presented a new version of the chiral interaction [1], whichdiffers from the previous realization, among others, by the regularization scheme applieddirectly in momentum space. This semi-local momentum-space (SMS) regularized potentialhas been derived completely up to the fifth order (N LO) of the chiral expansion and somecontributions from the sixth order have been even included in its “N LO + ” version. Up tonow this potential has been used out of necessity in the two-nucleon (2N) system to fix itsfree parameters and later applied to nucleon-deuteron elastic scattering as well as to thenucleon induced deuteron breakup process [8, 9], delivering at the N LO a data descriptionof the similar quality as the non-chiral semi-phenomenological potentials. In this paper weextend applications of the SMS interaction beyond the purely strong processes to a fewelectroweak reactions.Investigations of the electroweak processes with the older chiral force with the non-localregularization resulted in predictions strongly dependent on the cut-off parameter (see e.g.[10–12]). Such a picture was observed with the single-nucleon current and with inclusionof some meson-exchange currents. In 2016 we showed [13] that semi-local regularization(applied in coordinate space) helps to avoid such big variations with respect to the regulator,but predictions were still clearly dependent on the cutoff parameter. In the present paperwe check if properties of the SMS potential match or possibly even surpass those of theformer version of the chiral potential. To this end we apply the SMS chiral potential to theselected electromagnetic and weak processes. Specifically we study the γ + H → p + n, ν e + H → ν e + p + n, ¯ ν e + H → ¯ ν e + p + n, ¯ ν e + H → e + + n + n and γ + He → p + p + nreactions. For the sake of comparison we also use the results of our previous research withthe semi-local coordinate-space (SCS) regularized potential [5, 14] and with the ArgonneV18 (AV18) potential [15].One of the most challenging problems with the application of a nucleon-nucleon (NN)potential to electroweak processes with nuclear systems is a construction of consistent two-nucleon (2N) and, more generally, many-nucleon electroweak current operators [16].Many-nucleon currents linked to various models of nuclear forces have been investigatedfor a long time (see e.g. [17–23]). No full 2N electromagnetic or weak current operatorconsistent with the SMS chiral force has been derived yet, see however [24] for the recentcalculation of the deuteron structure radius with the consistent 2N charge operator. Thuswe rely on the single-nucleon current (SNC) and use the Siegert theorem to take into accountmany-body contributions to the nuclear electromagnetic current. Many-nucleon currents,even included implicitly, are absolutely indispensable in a correct treatment of photonuclearreactions. It has been shown (see for example Ref. [25]) that their omission leads to incorrectpredictions, especially for polarization observables. In the case of weak reactions we use thenon-relativistic form of the single-nucleon weak current operator, whose components aredefined in Ref. [26]. The dominant role of the single-nucleon weak current in neutrinoscattering of the deuteron has been recently demonstrated by Baroni and Schiavilla [23]who have found only a few percent contribution from higher order current operators.The rest of this paper is structured as follows. In the next section we briefly describeour formalism and then in Sec. III we show our results for the deuteron photodisintegration2nd (anti)neutrino induced deuteron disintegration processes. Section IV comprises ourpredictions for He disintegration. We summarize and conclude in Sec. V.
II. THEORETICAL FORMALISM
Our approach, which is based on the Schr¨odinger and Lippmann-Schwinger equations(for 2N reactions) and on the Faddeev formalism (for 3N reactions) has been described indetail in [25–27]. In short, the path to the observables for the electromagnetic or weakdisintegrations leads through the appropriate nuclear matrix elements.For the deuteron photodisintegration process, in the nuclear matrix elements N µdeu ≡ (cid:104) Ψ Nscatt | j µ N | Ψ Nbound (cid:105) , (2.1)a full 2N electromagnetic current operator appears between the initial deuteron bound state | Ψ Nbound (cid:105) and the final 2N scattering state | Ψ Nscatt (cid:105) . In order to obtain the deuteron boundstate we solve the Schr¨odinger equation with a given 2N potential V . Further, the scatteringstate is constructed from a solution of the Lippmann-Schwinger equation for the t operator: t = V + tG V , where G is a free 2N propagator. Using this equation (2.1) takes the form: N µdeu = (cid:104) (cid:126)p | (1 + tG ) j µ N | Ψ Nbound (cid:105) , (2.2)where | (cid:126)p (cid:105) is the antisymmetrized eigenstate of the relative proton-neutron momentum.Our formalism for the (anti)neutrino induced deuteron disintegrations is essentially thesame [26, 28, 29]. Especially, for the neutral-current (NC) driven processes the isospin struc-ture of the current operator and the 2N final state are the same as for the photodisintegrationreaction. In the case of the charged-current (CC) driven reaction only some straightforwardmodifications are introduced in the corresponding weak single-nucleon current operator.For the He photodisintegration with three free nucleons in the final state, if the 3N forceis neglected, the nuclear matrix elements are given as [25]: N µ N = (cid:104) Φ N | (1 + P ) j µ N | Ψ Nbound (cid:105) + (cid:104) Φ N | (1 + P ) | U µ (cid:105) , (2.3)where | Ψ Nbound (cid:105) represents the initial 3N bound state and | Φ N (cid:105) is an antisymmetrizedstate which describes the free motion of three outgoing nucleons. Further, the permutationoperator P is built from transpositions P ij of particles i and j : P = P P + P P andthe auxiliary state | U µ (cid:105) allows us to include all the final state interactions among the threeoutgoing nucleons. It is a solution of the Faddeev-like equation [25] which reads | U µ (cid:105) = t ˜ G (1 + P ) j µ N | Ψ Nbound (cid:105) + t ˜ G P | U µ (cid:105) , (2.4)where ˜ G is a free 3N propagator and j µ N is the total 3N electromagnetic current operator.At the moment 2N currents, with the same regularization as used in the SMS interactioninvestigated here, are not available. While their operator form has been already derived [22,30–34], a consistent regularization of these currents is still under development. Thus forthe weak processes we use contributions from the single-nucleon currents only. In the caseof the electromagnetic reactions additional contributions are taken implicitly into accountusing the Siegert theorem, as described in detail in Ref. [12, 25, 27]. We partly substituteelectric multipoles by the Coulomb ones, calculated from the single-nucleon charge density3perator. To this end we carry out the multipole decomposition of the corresponding SNCmatrix elements.We perform our calculations in the momentum space, in the partial wave decompositionscheme. For the deuteron disintegration we take into account all partial waves in the 2Nsystem up to the total angular momentum j = 4. For He photodisintegrations we useall two-nucleon partial waves up to the total 2N angular momentum j = 3 and all three-nucleon partial waves up to the total 3N angular momentum J = . For further details onour computational scheme see Ref. [25]. III. RESULTS FOR H DISINTEGRATION
First we discuss our results for the deuteron photodisintegration process γ + H → p + nat two laboratory photon energies E γ = 30 MeV and E γ = 100 MeV. Figure 1 shows thedifferential cross section d σd Ω obtained using the chiral SMS potential for both energies. Forthe sake of comparison we also show predictions obtained using the AV18 NN potential [15].Both for the chiral SMS force and for the AV18 potential our predictions were obtainedwith the single-nucleon current supplemented implicitly by some 2N parts, using the Siegerttheorem [10].In the left column of Fig.1 we present results calculated at different chiral orders (from LOto N LO + ) with the regularization parameter Λ = 450 MeV. For the energy E γ = 30 MeV(top row) only the LO prediction is noticeably separated from all others and the differenceamong remaining predictions is very small ( ≈ .
06% at the maximum of the cross sectionbetween N LO and N LO results) and even less for all subsequent chiral orders. It showsthat for this photon energy the SMS potential based predictions converge rapidly and contri-butions from high orders are not crucial. At 100 MeV our predictions converge more slowly,but starting from N LO all the curves are very close to each other (the difference betweenthe lines remains below 3%). In the case of E γ = 100 MeV the data description is worse thanat E γ = 30 MeV, but based on the semi-phenomenological results by Arenh¨ovel et al. [35]it is expected that for higher energies 2N electromagnetic currents contribute substantiallyand we thus expect that our predictions will improve significantly when explicit 2N currentoperators, fully consistent with the 2N potential, are included.The middle column of Fig.1 presents the truncation errors arising, at a given chiral order,due to neglecting of higher-order contributions to the chiral potential. These theoreticalestimates were calculated employing the prescription advocated in [36] and later used alsofor electromagnetic reactions with the SCS potential in [13] . The observed picture demon-strates that only tiny contributions from the potential components above N LO should beexpected, as the band showing the truncation error for the highest presented chiral orderN LO + is quite narrow - its width in the maximum of the cross section is around 0.1%(2.5%) of the cross section magnitude at the photon energy E γ = 30 MeV (100 MeV). No-tice, however, that the estimates of truncation error may change upon performing a morecomplete treatment of the current operators.Finally, the right column of Fig.1 shows the dependence of predictions on the value of theregularization parameter Λ in the range Λ ∈ [400 , It is also possible to perform a more sophisticated estimation based on the Bayesian approach [8, 37],however, for the sake of comparison with [13] we use the prescription [36] in this work. E γ = 100 MeV. d σ / d Ω [ µ b s r - ] (b) (c)1357 0 30 60 90 120 150(d) Θ p [deg] 0 30 60 90 120 150(e) Θ p [deg] 0 30 60 90 120 150 180(f) Θ p [deg] FIG. 1. The differential cross section d σd Ω for the γ + H → p + n reaction at the laboratoryphoton energy E γ = 30 MeV (top) and E γ = 100 MeV (bottom) as a function of the protonemission angle θ p (the angle between the initial photon momentum and the proton momentum)in the center of mass frame. The left column shows the dependence of predictions on the order ofchiral expansion. The orange solid line with diamonds, the blue solid line with circles, the greendashed, the red dotted, the black solid, the cyan double-dot-dashed and the violet dot-dashed curvescorrespond to the LO, NLO, N LO, N LO, N LO, N LO+ and AV18 potential based predictions,respectively. Truncation errors for the different orders of chiral expansion are presented in themiddle column. The yellow band shows the truncation errors at the NLO, green - at the N LO,blue - at the N LO, red - at the N LO and black one at the N LO + orders. The right columnshows the chiral SMS predictions at N LO, calculated using different values of the cut-off parameterΛ. The cyan double-dot-dashed, the black solid, the green dashed, the red dotted and the violetdot-dashed curves correspond to Λ = 400 MeV, 450 MeV, 500 MeV, 550 MeV, and to the AV18based predictions respectively. For the predictions shown in the left and in the middle columns theregulator value Λ = 450 MeV is used. The data points at 30 MeV are the same as in [38] and at100 MeV are taken from [38] (open and filled circles, squares) and from [39](triangles).
Our previous investigations [13] were devoted to the application of the older version ofthe NN potential, namely the chiral SCS force, to some electroweak processes. In particular,this potential applied to the deuteron photodisintegration reaction yields predictions for the5ifferential cross section well converged with respect to the order of chiral expansion. Nowwe are in a position to compare the outcomes from the SMS and from the SCS potentials.Results of our calculations of the differential cross section obtained using higher chiral orders(starting from N LO) of these two forces are presented in Figs. 2 and 3 for the photon energy30 MeV and 100 MeV, respectively. It is interesting that despite the convergence of bothpotentials, curves approach different values of the cross section and quite a big gap betweenpredictions of the SMS and SCS potentials is visible on both figures including the inset inFig. 2. The difference between the N LO SMS and SCS cross sections at E γ = 30 MeVreaches 1.07 µ b sr − ( ≈ θ p = 79 ◦ and 0.467 µ b sr − ( ≈ θ p = 52 ◦ for E γ =100 MeV. The observed deviation can be caused by the fact that the potentials use differentvalues of low energy constants, which results also in different deuteron wave functions. Itis also possible that the lack of explicit 2N current contributions affects differently thepredictions based on the two potentials. Absence of such gaps for full calculations, that isones including an electromagnetic current, which is complete and consistent with the NNinteraction, will be a challenging test for the chiral approach.It is also worth mentioning that one has to be cautious about judging the agreementbetween predictions obtained with the two potentials as, beside the various regularizationschemes, they differ in other aspects [1]. Even regarding the regularization method itselfit is not possible to establish any one to one correspondence between particular values ofregulators in the two spaces. The prescription given in Ref. [40], R ↔ Λ, yields Λ ≈
438 MeVfor R = 0.9 fm. It means that Λ = 450 MeV and R = 0.9 fm deliver only approximately thesame regularization effect.Nevertheless, since for the SMS potential its version for Λ = 450 MeV is available, here andin other cases where results based on the two potentials are compared, we use the accessiblepair of regulators: R = 0.9 fm and Λ = 450 MeV. For the presented here calculations thecomparison with experimental data cannot be used to judge between both forces becausedepending on the photon energy or the proton scattering angle either one or the otherprediction is closer to the data. Moreover, the inclusion of consistent two-body currentsmay change predictions differently for both interactions.In Figs. 4 and 5 we compare predictions based on the two chiral potentials, SMS andSCS, more closely. We show the relative difference between the differential cross sectionfor all the presented chiral orders. To this end, we define δσ (c hiral order ) as a differencebetween the maximum and minimum values of d σd Ω among predictions at orders from LO toN LO(N LO + ) for the SCS(SMS) forces for each angle θ p and divide it by the mean valuefrom five(six) predictions for the SCS(SMS), correspondingly. The resulting quantity bothfor the SMS (solid green line) and the SCS (dashed violet line) potentials is presented as afunction of the proton detection angle in the left panels of Fig. 4 for E γ = 30 MeV, and Fig. 5for E γ = 100 MeV. In both cases the SCS result lies above the SMS one which means thatthe net spread with respect to chiral orders for the newer potential is smaller. Neverthelessthis observation can also be an effect of the leading order predictions, which for the SMS aswell as for the SCS case are far away from all the other results. Therefore it is interesting tocheck the absolute difference between differential cross sections at N LO and N LO for thetwo potentials, which is done in Figs. 4c and 5c. One can see that on both plots the SMSprediction is above the SCS one for nearly all scattering angles. Thus the contribution fromN LO is bigger for the new SMS potential.Figures 4b and 5b show δσ (Λ), which is an analogous quantity to this shown in Figs. 4aand 5a, but now defined with respect to the values of the regularization parameter. That6 d σ / d Ω [ µ b s r - ] θ p [deg] 34 35 36 37 38 70 80 90 FIG. 2. The differential cross section d σd Ω at E γ = 30 MeV as a function of the proton emissionangle θ p calculated using the SCS (with R = 0.9 fm) and the SMS (with Λ = 450 MeV) chiralpotentials at higher orders of chiral expansion (N LO, N LO and N LO+ (for the SMS force only)).The green dashed, violet dotted and blue double-dot-dashed lines represent results calculated usingthe SMS potential up to N LO, N LO and N LO + respectively. The orange solid and the yellowdash-dotted lines are obtained using the SCS potential at N LO and N LO, respectively. All datapoints (open and filled circles) are taken from [38]. means that δσ (Λ) is the difference between the largest and the smallest value of d σd Ω , calcu-lated using all values of the cut-off parameter ( R for the SCS and Λ for the SMS) divided byits average σ avrg (Λ) . From the figure it is clear that dependence on the cut-off parameter ismuch weaker for the new potential. Note that for the SCS interaction, which comprises alsosofter regulators, like R=1.2 fm, some artifacts may be introduced to the potential, leadingto a wider spread of the predictions.Figure 6 demonstrates convergence of the cross section with respect to the chiral order.Each of the panels represents a certain combination of the photon energy and the protonemission angle: E γ = 30 MeV, θ p = 60 ◦ at Fig. 6a, E γ = 100 MeV, θ p = 15 ◦ at Fig. 6b, andE γ = 100 MeV, θ p = 150 ◦ at Fig. 6c. The quantity presented in this figure is the absolutedifference between differential cross section d σd Ω at each two subsequent orders: the one givenby the corresponding value on the x -axis and the subsequent one. For example the valuewith x -coordinate NLO is nothing but (cid:12)(cid:12)(cid:12) d σd Ω | N LO − d σd Ω | NLO (cid:12)(cid:12)(cid:12) . We see that the SCS potentialtends to converge faster, at least at presented scattering angles, since the difference shownin Fig. 6 drops to zero earlier. This is in agreement with the results shown in Fig. 4c.Nevertheless the presence of an additional term N LO + in the SMS potential makes theSMS predictions converge as well, though with more terms included. The SMS potentialdoes not reveal a jump between N LO and N LO predictions as observed for the SCS force,what is caused by different off-shell behaviour of the potential. Again, as in the case ofthe comparison with data, it would be interesting to see the convergence pattern for the7 d σ / d Ω [ µ b s r - ] θ p [deg] FIG. 3. The same as in Fig. 2 but for E γ = 100 MeV. Data are taken from [39] (triangles) andfrom [38](open and filled circles and squares). predictions obtained with complete currents for both potentials.In Figs. 7 and 8 we give examples of the SMS chiral force predictions for the polarizationobservables in the deuteron photodisintegration process [35]. Figure 7 presents the photonanalyzing power A X as a function of the outgoing proton angle θ p at the photon laboratoryenergy E γ = 100 MeV. Figure 7a shows the dependence of predictions on the order of thechiral expansion with a fixed regulator Λ = 450 MeV. One can see that few lowest ordersof expansion are not sufficient to obtain convergence of the predictions, as only after N LOlines overlap. This can indicate that subsequent orders do not bring significant contributionsto the observable’s final value. In Fig. 7b each curve corresponds to the particular value ofΛ (taken as 400 MeV, 450 MeV, 500 MeV, and 550 MeV) used in N LO calculations. Itis clearly visible that for A X the dependence on the regulator value is much stronger thanfor the maximum of the differential cross section (Fig.1) and amounts up to 14%, so in thiscase a proper choice of the Λ parameter value can be important in order to obtain realisticpredictions. In Fig. 8 we present the outgoing proton polarization P y for the same reaction.The dependence of the predictions on the Λ value is slightly weaker for P y than for A X ,since at the minimal values of these observables the relative difference between predictionswith different regulator values is less than 5% for P y (at θ p = 131 ◦ ) and reaches nearly14% for A X (at θ p = 72 ◦ ). All our predictions for the deuteron photodisintegration are in agood agreement with the results obtained using the AV18 force and the observed differencesamount to approximately 12% (7%) for A X (P y ) at the minima.Now we turn to the neutrino and antineutrino induced deuteron disintegration processes.For obvious reasons we restrict ourselves to the total cross sections σ tot . These results areobtained from the nuclear response functions generated on dense rectilinear grids of the( E N , Q ) points, where E N is the internal 2N energy and Q is the magnitude of the three-momentum transfer [29]. Figure 9 presents predictions for the total cross section σ tot forthe ν e + H → ν e + p + n neutral-current driven reaction. Again, in the left panel we show8 θ p [deg] δσ (chiral order)[%] 0 1 2 3 4 5 0 60 120 180(b) θ p [deg] δσ ( Λ )[%] 0 0.02 0.04 0.06 0.08 0.1 0.12 0 60 120 180(c) θ p [deg] Δσ (N4LO, N3LO) [µb sr -1 ] FIG. 4. The relative spread of predictions for the differential cross section for the deuteronphotodisintegration reaction at the initial photon energy E γ = 30 MeV. Left panel (a) representsthe difference between maximum and minimum values of the differential cross section for all chiralorders used, divided by its average (from LO to N LO + for the SMS potential - green solid line,and from LO to N LO for the SCS - violet dashed line). The middle panel (b) shows analogousquantity, but measuring the spread with respect to the different cutoff values used (from 400 MeVto 550 MeV for the SMS force and from 0.8 to 1.2 fm for the SCS potential) at fixed chiral order(N LO). The right panel (c) shows absolute difference between d σd Ω calculated at N LO and N LOfor each of two potentials. results obtained using different orders of the chiral expansion (from LO up to N LO + , Λ=450MeV) and in Fig. 9b the variation of the results with respect to the cutoff parameter valueis presented. For the sake of comparison with the results based on a semi-phenomenologicalpotential we also give predictions based on the AV18 interaction. The same dependenciesbut for the antineutrino induced NC disintegration ¯ ν e + H → ¯ ν e + p + n are presented inFig. 10. The results for the charged-current induced process ¯ ν e + H → e + + n + n aredemonstrated in Fig. 11.As for our energy range σ tot takes a large spectrum of values, it is hard to see thedifferences between curves in Figs. 9, 10 and 11 with the naked eye. However, in the insetsone can see that in the left panels of Figs. 9, 10 and 11 the two curves, representing theLO and the AV18 predictions, are separated from all the others. It is interesting that therelative position of the different curves in the inset in Fig. 11a does not remain the sametrough all the energy range. It is so for E γ ∈ (0 , LO usingdifferent regulator values. To give some numerical examples: for the initial particle energyE ν (¯ ν ) =100 MeV the relative differences between values of total cross section σ tot calculatedwith the chiral SMS force up to the fifth order (N LO) and up to fifth order plus correctionsfrom the sixth order (N LO + ) are 0.093% for ν e + H → ν e + p + n, 0.092% for ¯ ν e + H → ¯ ν e + p + n and 0.029% for the ¯ ν e + H → e + + n + n reactions. The relative differencesfor the cross section obtained using different values of the cut-off parameter at the same9 θ p [deg] δσ (chiral order)[%] 0 5 10 15 20 0 60 120 180(b) θ p [deg] δσ ( Λ )[%] 0 0.02 0.04 0.06 0.08 0.1 0.12 0 60 120 180(c) θ p [deg] Δσ (N4LO, N3LO) [µb sr -1 ] FIG. 5. Same as in Fig. 4 but for E γ = 100 MeV. LO NLO N LO N LO N LO (a) Δ σ [ µ b s r - ] E = 30MeV, θ p = 60 o LO NLO N LO N LO N LO (b) E = 100MeV, θ p = 15 o LO NLO N LO N LO N LO (c) E = 100MeV, θ p = 150 o FIG. 6. The absolute difference between the values of the differential cross section d σd Ω taken ateach two subsequent chiral orders (the one marked on the x-axis and the next one) at fixed protonangle θ p and photon energy: θ p = 60 ◦ E γ = 30 MeV(a), θ p = 15 ◦ E γ = 100 MeV(b), θ p = 150 ◦ E γ = 100 MeV(c). The violet dashed (the green solid) line represents results obtained using theSCS (SMS) potential with the cutoff parameter R = 0 . energy, at N LO, are 0.96%, 0.98% and 0.90% for the same reactions, respectively. It isseen that the cutoff dependence is nearly one order bigger than difference between the lasttwo chiral orders. Nevertheless both uncertainties remain very small which reflects lowsensitivity of these inclusive observables to employed dynamics. The relative difference ofN LO and N LO + predictions for the last reaction is approximately three times smallerthan for the first two in both regarded cases. Our treatment of all above mentioned weakreactions is very similar. The common ingredients are the deuteron wave functions and thekinematics, which is only slightly modified due to the small but non-zero positron mass.The single-nucleon weak neutral and charged currents are potential independent so they10 A X θ p [deg] 0 30 60 90 120 150 180(b) θ p [deg] FIG. 7. The photon analyzing power A X as a function of the center-of-mass proton detectionangle θ p for the deuteron photodisintegration process at E γ = 100 MeV. The left panel (a) showsthe dependence of A X on the chiral order of the SMS potential at Λ=450 MeV. The right paneldemonstrates the dependence of A X on the value of the cutoff parameter Λ at N LO. Lines are asin Figs. 1a and 1c, respectively. -0.3-0.2-0.1 0 0.1 0 30 60 90 120 150(a) P y θ p [deg] 0 30 60 90 120 150 180(b) θ p [deg] FIG. 8. Same as in Fig. 7 but for the proton polarization P y . cannot explain the difference in N LO - N LO + spreads for the NC and CC driven reactions.However, the final states for the reactions are different. While in the first two reactionsdriven by the neutral current a neutron-proton pair emerges in the final state, in the thirdprocess, ¯ ν e + H → e + + n + n, a two-neutron final state is present. Thus the observedvariation in spreads stems from the difference between the neutron-neutron and neutron-proton potentials. From Figs. 9-11 it is visible that AV18 curve here is also detached11rom all the other predictions. However the difference between the predictions is small andacceptable as the potentials are constructed in quite different ways.
0 30 60 90 120 150 180(a) σ t o t [ f m ] E ν [MeV] 0 30 60 90 120 150 180(b) E ν [MeV]
90 92 94 96
90 92 94 96
FIG. 9. The total cross section σ tot for the ν e + H → ν e + p + n reaction as a function of theincoming neutrino energy in the laboratory system. The left panel (a) presents the dependenceof σ tot on the chiral order at Λ = 450 MeV. Dependence of σ tot on the cutoff parameter value atN LO is presented in panel (b). In the left panel the orange solid line with diamonds, the bluesolid line with circles, the green dashed, red dotted, black solid, cyan double-dot-dashed and violetdot-dashed curves correspond to the LO, NLO, N LO, N LO, N LO, N LO+ and AV18 potentialbased predictions, respectively. In the right panel the cyan double-dot-dashed, black solid, greendashed, red dotted and violet dot-dashed curves represent results with Λ = 400 MeV, 450 MeV,500 MeV, 550 MeV, and the AV18 based predictions respectively.
IV. HE PHOTODISINTEGRATION
In this section we discuss predictions obtained when applying the chiral force withthe semi-local regularization in momentum space to the He photodisintegration process γ + He → p + p + n. As in the H case we use the Siegert theorem to go beyond the SNCapproximation [25]. In the following we neglect the three-nucleon interaction. The semi-inclusive differential cross section d σd Ω p dE p for the photon laboratory energy E γ = 120 MeVis presented in Fig. 12. Each of the four columns corresponds to a particular angle of theoutgoing proton momentum with respect to the photon beam in the laboratory system (0 o ,60 o , 120 o , and 180 o , respectively). Top row shows the dependence of the predictions on theorder of chiral expansion. As in Figs. 1, 7 and 8 we see that it is not enough to take intoaccount only leading and next-to-leading orders to achieve convergence of the predictionsand one has to include higher orders of chiral expansion. It is interesting to note that theolder SCS potential seems to converge even faster as the NLO line in Fig. 12 is farther fromthe higher order curves than one in Fig. 9 of Ref. [13], where the predictions of the SCS chiralforce for the same observable are shown. This is similar to the already observed picture for12
0 30 60 90 120 150 180(a) σ t o t [ f m ] E ν - [MeV] 0 30 60 90 120 150 180(b) E ν - [MeV]
90 92 94 96
90 92 94 96
FIG. 10. Same as in Fig. 9 but for the ¯ ν e + H → ¯ ν e + p + n reaction.
0 30 60 90 120 150 180(a) σ t o t [ f m ] E ν - [MeV] 0 30 60 90 120 150 180(b) E ν - [MeV]
90 92 94 96
90 92 94 96
FIG. 11. Same as in Fig. 9 but for the ¯ ν e + H → e + + n + n reaction. the deuteron photodisintegration.The bottom row represents the cutoff dependence of the cross section which proves tobe weak. For most of the proton energies the maximum difference between all predictionsremains below 10%. There are only exceptions for the outgoing proton angle 0 ◦ and itsenergies greater than 80 MeV, where the difference amounts to 20% and for the angle 180 ◦ at the proton energies around 40 MeV, where it reaches 12%. The cut-off dependencerevealed by the SMS chiral force is weaker than it is observed for the SCS force (compareFig. 11 in Ref. [13]).The five-fold differential cross section d σd Ω d Ω dS for the same process is presented in Fig. 13for the photon laboratory energy E γ = 40 MeV for two protons detected at the following polar13 d σ / d Ω p d E p [ µ b s r - M e V - ] E p [MeV] θ = 0 deg p [MeV] θ = 60 deg p [MeV] θ = 120 deg p [MeV] θ = 180 deg p [MeV] θ = 0 deg p [MeV] θ = 60 deg p [MeV] θ = 120 deg p [MeV] θ = 180 deg FIG. 12. The semi-inclusive differential cross section d σd Ω p dE p for the γ + He → p + p + n reactionat E γ = 120 MeV as a function of the outgoing proton energy E p at different values of the polarangle of the outgoing proton momentum θ p . Top row shows the cross section dependence on theorder of chiral expansion (at Λ = 450 MeV), the bottom row shows the dependence on the valueof the cutoff parameter Λ. The curves are as in Fig. 1 but the AV18 prediction is not shown here. and azimuthal angles (assuming that the momentum of the initial photon (cid:126)p γ is parallel tothe z -axis) Θ = 15 ◦ , Φ = 0 ◦ and Θ = 15 ◦ , Φ = 180 ◦ . The arc-length S of the kinematicallocus in the E − E plane, where E and E are the kinetic energies of the two detectednucleons, is used to uniquely define the kinematics of the three-body breakup reaction [41].We observe that the NLO contribution is very important since it raises the LO cross sectionby a factor of two. An additional shift (around 9%) comes from N LO and only smallchanges are seen when N LO, N LO and N LO + force components are included. However,even for the highest orders (N LO and N LO + ) curves do not overlap, which suggests thatfull convergence is not achieved yet. With respect to the cut-off dependence the picture ismore stable, since it is very hard to distinguish individual predictions in the right panel.Figure 14 shows the cross section for the same choice of polar and azimuthal angles aspresented in Fig. 13 but for the higher incoming photon energy E γ = 120 MeV. The generaltrends do not change with increasing energy, except for the fact that the dependence onthe regularization parameter is slightly stronger which is visible at the maxima of the crosssections around S=15 MeV and S=100 MeV. But for all other values of the arc-lengthparameter S almost full agreement between all the lines exists.14 d σ / d Ω d Ω dS [ µ b s r - M e V - ] S [MeV] 0 5 10 15 20 25 30(b) S [MeV]
FIG. 13. The five-fold differential cross sections d σd Ω d Ω dS for the complete kinematical configurationwith the two protons detected at Θ = 15 o , Φ = 0 o , Θ = 15 o , Φ = 180 o angles for the Hephotodisintegration process at the photon energy E γ = 40 MeV in the laboratory frame. Thedependence on the chiral order (with Λ = 450 MeV) is presented in the left panel (a) and theresults for different cutoff values (at N LO) are displayed in the right one (b). Curves are as inFig.12. d σ / d Ω d Ω dS [ µ b s r - M e V - ] S [MeV] 0 20 40 60 80 100 120(b) S [MeV]
FIG. 14. Same as in Fig. 13 but for E γ = 120 MeV. V. SUMMARY AND CONCLUSIONS
We presented the results of the application of the chiral NN potential with the semi-localregularization in momentum space to a number of electroweak reactions: the H and Hephotodisintegrations and the (anti)neutrino induced deuteron breakup reactions: γ + H →
15 + n, γ + He → p + p + n, ν e + H → ν e + p + n, ¯ ν e + H → ¯ ν e + p + n and ¯ ν e + H → e + + n + n. In the case of the He photodisintegration we focus on the properties of theapplied NN potential and neglect the three-body force.All our results show a weaker, compared to the previous version of the Bochum-Bonnchiral potential (the SCS force), dependence on the cutoff parameter Λ. A small spreadof results obtained with different values of Λ make the predictions based on the currentinteraction model more unambiguous. We also observe good convergence of the predictionswith respect to the chiral order and the possibility to include some terms from the sixthorder (N LO + ) favorably distinguishes the SMS force from the SCS one.As we have no full 2N electromagnetic current consistent with the SMS interaction atour disposal, the Siegert theorem was used to take two-nucleon contributions in the electro-magnetic current operator at least partly into account for the photodisintegration processes.As a consequence, the incomplete electromagnetic current operator leads to some problemswith the data description, as seen in Fig. 1. We expect that future application of the electro-magnetic current operator fully consistent with the 2N potential will significantly improvethe agreement with the data.The presented polarization observables for the deuteron photodisintegration processes( A X and P y ) are the ones, where the slowest convergence with respect to the chiral orderand the strongest dependence on the regularization parameter is noticed. For instance forthe deuteron analyzing powers ( T , T , T and T ) the convergence is very fast (abovethe leading order) and the cutoff variation is negligible.The same picture is also valid for the investigated here total cross sections for the weak H disintegrations via neutral or charged currents: the convergence with respect to the chiralorder is quite rapid and the cutoff dependence is weak.The new SMS potential has a number of practical advantages in comparison to the olderchiral forces. Its predictions show weaker dependence on the cut-off parameter and goodconvergence with respect to the order of chiral NN potential. There is still room for im-provement in our calculations. The main drawback of the present formalism is the lack ofthe explicit electroweak current operator entirely consistent with the 2N SMS interaction aswell as the omission of the three-nucleon force for the He disintegration. In addition, thefuture complete studies should be supplemented with an analysis of truncation errors usingthe Bayesian approach, which would allow one to draw more reliable conclusions about theconvergence pattern of chiral EFT predictions for these reactions. Nevertheless, our resultswith the simplified Hamiltonian reveal the usefulness of the SMS chiral force for studies ofvarious electroweak processes in few-nucleon systems in the near future.
VI. ACKNOWLEDGEMENTS
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