Improved analysis of neutral pion electroproduction off deuterium in chiral perturbation theory ^{†}^{†}thanks: Work supported in part by Deutsche Forschungsgemeinschaft grants Me 864/162 and GL 87/341.
Abstract
Near threshold neutral pion electroproduction on the deuteron is studied in the framework of heavy baryon chiral perturbation theory. We include the next–to–leading order corrections to the threebody contributions. We find an improved description of the total and differential cross section data measured at MAMI. We also obtain more precise values for the threshold Swave multipoles. We discuss in detail the theoretical uncertainties of the calculation.
pacs:
25.30.RwElectroproduction reactions and 12.39FeChiral Lagrangians and 14.20.DhProtons and neutrons1 Introduction
Pion electroproduction off deuterium, , allows to extract the elusive elementary neutron amplitude. First data at small photon virtuality GeV and small values of the pion excess energy (MeV) have been taken at the Mainz Mikrotron MAMIII Ewald . In the threshold region, this reaction can be analyzed in the framework of chiral perturbation theory KBM1 ; KBM2 . In Ref. KBM2 , a (partial) nexttoleading order calculation was presented, which led to a satisfactory description of the differential and total cross section data. That analysis can be improved in two respects. First, the socalled threebody corrections (meson exchange currents) were only considered at third order in that paper. Second, a similar remark holds for the single scattering proton and neutron Pwaves. In this paper, we are concerned with the first issue, namely the nexttoleading (fourth) order corrections to the threebody corrections. As will be shown, this introduces (in principle) no new parameters and with that extension, the calculation is done with the same theoretical precision as the fairly successful analysis of coherent neutral pion photoproduction off the deuteron BBLMvK . It also leads to an improved description of the data as shown in this paper. The second extension can not be done so straightforwardly. This is because so far only the fourth order corrections to the photoproduction Pwave multipoles are available in the literature BKMp . For the electroproduction case, similar calculations are underway BKuM , triggered in particular by the rather unexpected results for neutral pion electroproduction off the proton at the photon virtuality GeV Merkel . These data can not be described within chiral perturbation theory at one–loop accuracy. This is in stunning contrast to the fairly good description of the data at higher photon virtuality from NIKHEF vandenBrink and MAMI Distler that was obtained in BKMel . Note that remeasurements at the low photon virtuality seem to be more in line with expectations from chiral perturbation theory or sophisticated models MerkelCD . As we will argue, the effect of such single scattering Pwave contributions can be effectively included in a refit of some fournucleonphoton operators, which leads to a much improved description of the data. Given the precision of the data, a more elaborate treatment appears inappropriate.
The manuscript is organized as follows. In Section 2, we briefly discuss the pertinent formalism. We heavily borrow from Ref. KBM2 and only discuss the new fourth order threebody contributions in some detail. Our results are presented and discussed in Section 3 and we end with a short summary and outlook in Section 4. Some technicalities are relegated to the appendices.
2 Effective field theory description
2.1 General remarks
In Ref. KBM2 , we developed the multipole formalism for near threshold pion electroproduction off deuterium (see also Aren1 ; Aren2 ) and calculated the pertinent transition matrix elements in chiral perturbation theory. This is based on an effective chiral Lagrangian of pions and nucleons chirally coupled to external sources like the photon,
(1) 
where the first term subsumes the interactions between the Goldstone bosons, the second the pionnucleon interactions and the third term the short range part of the twonucleon interaction. All of these Lagragians are a series of terms with increasing chiral dimension,
(2)  
(3)  
(4) 
where the ellipsis stand for terms of yet higher order. The explicit expressions for the various terms are well documented in the literature, see e.g. BKMrev . The transition operators derived from this effective Lagrangian are then sandwiched between wave functions that were consistently generated from chiral nuclear effective field theory. The latter we take from the recent work of Ref. EGMZ . The various contributions can be organized in terms of a consistent power counting in terms of a small parameter , like e.g. a meson mass, energy or nucleon threemomentum (with respect to the typical hadronic scale, say the mass of the rho meson). For a generic matrix element one has,
(5) 
where is a renormalization scale and the function is of order one. Furthermore, is a counting index, i.e. the chiral dimension of any Feynman graph, for the case of pion production off nuclei, see e.g. BLvK . In terms of this counting index, we include all terms with and (see also KBM2 for details).
The transition matrix elements are generated by two very different types of contributions,
(6) 
where “ss” and “tb” denote the singlescattering and the threebody contribution, respectively. Here, single scattering means that the pion is emitted from the same nucleon to which the photon couples with the other nucleon acting as a mere spectator. Processes involving both nucleons are often called exchange currents, but we follow here the notation due to Weinberg Weind . At third order, all tb diagrams involve graphs with one pion in flight, whereas at the order considered here, fournucleon contact terms with the photon absorption and pion emission from different nucleon legs come in (as discussed in more detail below). These terms can be understood from integrating out heavier mesons, see e.g. Friar ; EGME . In such a picture, such diagrams would thus correspond to heavy meson exchange currents.
The ss terms are of course sensitive to the elementary proton and neutron electroproduction multipoles, properly boosted to the piondeuteron centerofmass frame (for details, see KBM2 ). The proton amplitude is fixed from the chiral perturbation theory study of Ref. BKMel . As in our earlier work, we perform two types of fits, where we have one respectively two undetermined parameters related to the elementary amplitude. In the fits of type 1, we have one free fifth order parameter, called , and we include the constraint from a leading order lowenergy theorem to the fourth order counterterms (as explained in BKMel ). The numerical value for the LEC from resonance saturation is 4.11 GeV (comprised of the contribution from the in static approximation and vector mesons). For the fits of type 2, we relax this constraint and thus have two fit parameters, the lowenergy constants and . These two fitting procedures give us a measure of the theoretical uncertainty at the order we are considering. We also get another measure of the theoretical uncertainty by considering chiral EFT at NNLO and varying the cut–off in the LippmannSchwinger equation. This will lead to bands for the various observables rather than to lines as e.g. in KBM2 . We will come back to this topic when discussing the results. We note already here that the uncertainty related to the two fit procedures is sizeably bigger than the one induced by the cutoff variation.
In what follows, we will use as kinematical quantities the virtuality of the photon, , which is negative in electron scattering (in the literature one often uses the positive quantity ), the photon polarization and the pion excess energy, , that is the energy of the produced pion above threshold in the centerofmass system. For a more detailed discussion of the kinematics, see e.g. Ewald .
2.2 Fourthorder threebody contributions
At third order, there are 8 tree graphs contributing to the threebody corrections, see Fig. 4 of Ref. KBM2 (using Coulomb gauge). At fourth order, there are altogether 59 nonvanishing diagrams, as shown in Fig. 1. These are tree graphs with exactly one insertion from the dimension two chiral pion–nucleon Lagrangian, . More precisely, in that figure we have only shown the topologically inequivalent diagrams, the numbers under certain graphs denote how many diagrams can be generated if one attaches the pion emission vertex on the left or the right nucleon line above or below the pion exchange line. There is also a whole new class of shortdistance diagrams including the leading (momentumindependent) fournucleon interactions (the last four diagrams in Fig. 1 and the corresponding Okubocorrections, i.e. the diagrams with two closeby energy denominators, are not shown. For details on this point, see KBMokubo . A more general discussion one these reorthonormalization diagrams can be found e.g. in EG ; EGM1 .). The corresponding fournucleon LECs have already been determined in the fits to nucleonnucleon scattering data. We use here the values collected in table 1 for the corresponding NNLO wave functions.
LEC  MeV  MeV 

[GeV]  0.151  0.149 
[GeV]  0.168  0.130 
However, these values stem from a fit to the NN phase shifts at low energies. A fit including pion production data (which is from the kinematical point of view closer to the process considered here) can lead to an increased theoretical uncertainty in the determination of these LECs. To make that point more transparent, we briefly discuss the determination of the leading LEC related to the D–term in EGM34 . From lowenergy observables one obtains for this coupling in dimensionless units, , whereas its determination from P–waves in the reaction gives somewhat lower (but still consistent) values, HvKM ^{1}^{1}1Note that we have a different convention for the sign of the axialvector coupling . The relation between the coupling constant used in that paper and is given by .. Also, at fourth order we will have additional P–wave LECs from the single nucleon amplitudes. With these two observations in mind, we will also perform fits were we leave and as free parameters. This will be discussed in more detail when we present the results. Furthermore, contains some terms with fixed couplings (due to the constraints of Lorentz invariance) and other terms with finite and scaleindependent lowenergy constants, denoted . These LECs can e.g. be determined from the analysis of lowenergy elastic pion–nucleon scattering data utilizing chiral perturbation theory. From the hadronic couplings only the term contributes, we use the value GeV BuM .
We note that all the diagrams shown in Fig. 1 are irreducible, where we use the following definition for irreducible diagrams: A diagram is called irreducible, if it contains no contributions from the twonucleon potential (see e.g. the related discussion in BMPK ). This definitions becomes obvious if one considers the LippmannSchwinger equation,
(7) 
and the deuteron wave function is obtained from the solution of the Schrödinger equation,
(8) 
Here, , , are the pertinent pionnucleon, photonnucleon, transition potentials subject to the chiral expansion as explained above. This concept of reducibility is depicted in Fig. 2, where two typical reducible diagrams are shown, which include either the lowest order onepionexchange or the shortdistance parts of the twonucleon interaction. All diagrams depicted in Fig. 1 fall essentially in four classes, labeled a), b),c) and d). The first two classes were already present at third order KBM2 , these are the seagulltype and the pioninflight type graphs, respectively. Class c) collects the socalled timeordered graphs, were the photon couples to a nucleon line and the pion is emitted from a nucleon while the exchanged pion is in flight. All diagrams build from the fournucleon contact terms are collected in class d). Representative diagrams for these four classes are shown in Fig. 3. In fact, a) falls into two subclasses, depending on the value of the energy of the exchanged pion. One either has or , with the energy of the produced pion. The diagrams are evaluated using Fouriertransformation techniques, as briefly discussed in App. A. We also remark that all our coordinate space integrals are finite due to the exponential falloff of the chiral EFT wave functions. In contrast to what was done in BBLMvK , we thus do not need to introduce an additional cutoff in the fourth order tb terms.
3 Results and discussion
In this section, we display the results for the multipoles, differential and total cross sections and the S–wave cross section for the two fit strategies. We have performed calculations with the chiral EFT wave functions at NNLO EGMZ for cutoffs in the range from 450 to 650 MeV (with the spectral function cut–off fixed at 650 MeV, for details see EGMZ ). Consequently, for the observables we will obtain bands rather than single line. This method of estimating the theoretical uncertainty is one of the advantages of the chiral effective field theory approach employed here. We note that at threshold we have performed the calculations in momentum and in coordinate space, which serves as a good check on the nontrivial numerical evaluation of the various integrals.
3.1 Results with fixed fournucleon LECs
We first discuss the result obtained keeping the four–nucleon LECs and fixed at the values given in section 2.2. In table 2 we collect the fitted LECs related to the elementary neutron amplitude. As discussed before, we have performed two types of fits. We see from the table that the wave function dependence of the LECs is very weak, which points towards the conclusion that the pertinent matrix elements are dominated by the contributions from the long–ranged pion exchange. The resulting values are also not very different from the ones obtained in KBM2 , which are also given in the table.
LECs  MeV  MeV  

[GeV]  4.148  4.140  4.767 
[GeV]  7.085  6.730  5.644 
[GeV]  38.71  37.05  27.51 
Before showing results for the pertinent observables, we discuss a few general features of the new contributions. First, we observe that the 4N contact interactions contribute mostly to the Pwaves, their Swave contribution is very small. Further, these Pwave contributions mostly feed into the magnetic multipole and cancel to some extent the corresponding contribution from the third order tb terms. Second, the other new diagrams also lead to more changes in the Pwave multipoles, simply since at this order there are no free parameters. We also note that the value of the constant in the fits 2 agrees nicely in magnitude with the resonance saturation estimate, but comes out with opposite sign.
In Figs. 4,5 we show the differential cross sections for fits 1 and 2 employing the NNLO wave functions in comparison to the MAMI data Ewald . These two bands (which are generated by utilizing the NNLO wave functions with the cutoff MeV and 650 MeV) corresponding to the two fit procedures can be considered as a measure of the theoretical uncertainty at this order. We note that the bands due to the wave function dependence are very thin, which shows that the effective field theory calculation of these wave functions is of sufficient accuracy. The uncertainty generated from the two fit procedures is comparable to the experimental errors. We remark that the differences between the two fit procedures is somewhat smaller as it was the case when only the third order tb corrections were included KBM2 , see the dashed lines in Figs. 4,5. Consequently, as it should be in a converging effective field theory, the theoretical uncertainty has become smaller as compared to Ref. KBM2 , although this improvement is moderate.
The corresponding total cross sections as a function of the excess energy and of the photon polarization are shown in Figs. 6 for fit 1 and in Figs. 7 for fit 2 and the NNLO wave function with ranging from 450 to 650 MeV, respectively. We notice that for fit 1 with increasing excess energy and, in particular, with increasing photon polarization the data are systematically below the chiral prediction. Due to the fitting procedure, the slopes of the various curves for the Rosenbluth separation shown in the right panel of Fig. 6 are of course correct. These results are very similar to the ones obtained in KBM2 for fit 1. We note that the predictions for fit procedure 2 are visibly improved compared to the third order calculation. This is further seen by looking at the transverse and the longitudinal threshold Swave multipole shown in Fig. 8. The prediction for for fit 2 is now within the error bar of the MAMI result (which was not the case in KBM2 ). We note that the real part of the predicted longitudinal amplitude is negative, consistent with our findings in KBM2 . The resulting Swave cross section is shown in Fig. 9. The theoretical uncertainty is a bit smaller than in case of the third order tb calculation KBM2 .
3.2 Results with fitted fournucleon LECs
As argued before, we will also perform fits leaving the values of the two fournucleon LECs as free parameters. Since as we noted before these operators contribute very little to the Swaves, and not at all at threshold, the fits of type 1 are performed in two stages. First, is adjusted to give the proper value of at GeV. Then, the two fournucleon LECs are fitted to the total cross sections. The fits of type 2 are performed as before, all parameters are fixed on the total cross sections. Having said this, we collect in table 3 the values of the various LECs for the two types of fits. We note that the four–nucleon LECs come out much larger than from the investigation of nucleonnucleon scattering, cf. table 1. However, as we discussed earlier, leaving these LECs as free parameters effectively subsumes some effects from the single nucleon Pwave contributions, so that these fits should be considered indicative only.
LECs  MeV  MeV 

[GeV]  4.500  4.751 
[GeV]  8.549  8.573 
[GeV]  0.589  0.664 
[GeV]  1.168  1.105 
[GeV]  38.71  37.05 
[GeV]  0.325  0.387 
[GeV]  0.762  0.890 
The resulting total cross sections are shown in Fig. 10. We observe a clear improvement for fit 1 and a very good description for fit 2. We remark, however, that the corresponding differential cross sections for MeV are more symmetric around (bellshaped) than the ones shown in the preceding section, whereas the data indicate a peaking into the backward direction. This points towards an insufficient accuracy in the description of some of the Pwave multipoles. The resulting longitudinal deuteron S–wave multipole is shown in Fig. 11. While is almost unaffected, for fit 2 agrees with the empirical value. Consequently, the band for the S–wave cross section is much narrower than before, cf. the right panel of Fig. 11. This is because the cross sections are much sensitive to the longitudinal Swave amplitude due to the kinematical enhancement proportional to the longitudinal polarization , see Ewald .
4 Summary and outlook
In this manuscript, we have considered neutral pion electroproduction off deuterium in the framework of heavy baryon chiral perturbation theory, extending and improving upon the work presented in KBM2 . The salient results of this study are:

We have calculated the fourth order threebody corrections. These consist of 51 onepion exchange diagrams with exactly one insertion from the dimension two pion–nucleon Lagrangian and 8 diagrams with one insertion from the lowest order four–nucleon interaction Lagrangian. In principle, all parameters are fixed from earlier studies of pion–nucleon and nucleon–nucleon scattering. The deuteron wave functions are taken consistently from the recent chiral EFT study of EGMZ .

As in the earlier work KBM2 , in which the threebody corrections were only considered at third order, we have performed two types of fits to the MAMI data Ewald . The results are similar to the ones found there, although we observe a moderate improvement in the theoretical uncertainty. This is most pronounced for the longitudinal Swave multipole in fit 2. We have also considered the dependence on the deuteron wave functions related to the cut–off in the LippmannSchwinger equation. It turned out that this dependence is very weak. Therefore, the process is dominated by long–range pion physics and thus sensitive to the elementary amplitude.

We have also performed fits where we have left the four–nucleon LECs as free parameters, based on the argument that in that way one can effectively subsume new Pwave LECs from the single scattering contribution. This leads to a visibly improved description of the total cross sections and the longitudinal S–wave multipole. However, the differential cross sections come out bellshaped, in contrast to the experimental findings Ewald .
In conclusion, we have further sharpened the theoretical framework to analyse neutral pion production off deuterium in a model–independent framework. It remains to be seen whether a further improved description of the data will be possible when the fourth order pion–nucleon P–wave multipoles at fourth order become available.
Acknowledgements
We thank Evgeny Epelbaum for useful comments and supplying us with the EFT wave functions.
Appendix A Fourier transformations
The diagrams are evaluated in momentum space. To calculate the pertinent matrix elements, we perform Fourier integrations, see e.g. BBLMvK . The typical structures to be evaluated take the form
(A.1)  
where are initial (final) state relative nucleon momenta, is the coordinate space deuteron wave function and similarly denotes the momentum space wave function. As stated, we have to consider 5 classes of diagrams,

Diagrams of type a) with ,

Diagrams of type a) with ,

Diagrams of type b),

Timeordered diagrams of type c),

The NNdiagrams of type d).
In the following, symbolizes an arbitrary spinoperator.
Consider first the diagrams 1). The momentum of the exchanged pion is
(A.2) 
From the momentum space expression for this class of diagrams, one deduces the following pertinent Fouriertransform
(A.3) 
with
(A.4) 
Here, is the physical value of the and the threshold, respectively. So then we have to evaluate the following integrals
(A.5) 
Next, we evaluate the diagrams 2). The momentum of the exchanged pion now is
(A.6) 
and the pertinent Fourier integral is
(A.7) 
leading to the structures
(A.8) 
We now consider the diagrams 3). Here, the photon couples to the pion in flight, thus we have two propagators. The momentum of the exchanged pion before the photon absorption is
(A.9) 
and after the photon absorption this momentum is changed to
(A.10) 
We have two types of Fouriertransforms, these are:
(A.11) 
with
(A.12) 
and
(A.13) 
Note that this function is still square integrable. Higher order derivatives, however, can not be treated in this fashion. The corresponding integrals are thus
(A.14) 
We now discuss the diagrams 4), which have the momentum transfer given either by Eq. (A.2) or by Eq. (A.6). Due to the cancellation with the corresponding Okubotype diagrams, no new structures as compared to the other type of diagrams appears (for a more detailed discussion on this point see KBMokubo )
Finally, we turn to the NN diagrams 5). Since these only involve contact interactions, we have to consider simple structures of the form
(A.15)  
We end this appendix with a note on how we treat the contributions arising from the exchange of nucleon 1 with nucleon 2 when calculating the pertinent amplitude, . The corresponding momenta of the nucleons in terms of the centerofmass and the photon and pion momenta are
(A.16) 
Under a parity transformation and , the nucleon momenta behave as , , , , which gives
Thus, matrix elements are invariant under operations of the form
Consequently, the operation need only be applied to the spin and isospin operators and leave all other quantities unchanged. Furthermore, because of isospin symmetry we only have the operators and , which are symmetric under the interchange . Therefore, we finally only need to change the spin indices when calculating the contributions from the exchange diagrams.
Appendix B Angular integrations
Here, we briefly discuss how to efficiently perform the angular integrations of the Fourier integrals discussed in App. A. We define the following matrices constructed from the momentum space deuteron wave functions,
where and are the momentumspace S and Dwave components of the deuteron wave function, and the are spherical Bessel functions. The usual coordinate space expressions can be obtained using matrix elements and , i.e.
(B.2) 
This notations is particularly useful since we have to deal with derivatives of the wave functions. Using the properties of the spherical Bessel functions, one obtains the following recursion relation
(B.3) 
This allows one to express the derivatives of in the following way
(B.4) 
Similarly, the derivatives of can be expressed as
(B.5)  
Note that one only has to calculate diagonal matrix elements.
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