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Chiral perturbation theory calculation for pn ->dpipi at threshold

We investigate the reaction pn ->dpipi in the framework of Chiral Perturbation Theory. For the first time a complete calculation of the leading order contributions is presented. We identify various diagrams that are of equal importance as compared to those recognized in earlier works. The diagrams at leading order behave as expected by the power counting. Also for the first time the nucleon-nucleon interaction in the initial, intermediate and final state is included consistently and found to be very important. This study provides a theoretical basis for a controlled evaluation of the non-resonant contributions in two-pion production reactions in nucleon-nucleon collisions.


I. INTRODUCTION
As the first strong inelastic threshold of the nucleon-nucleon system, pion production in NN collisions has attracted a large number of theoretical as well as experimental works.
Due to advances in the experimental methods measurements in the threshold region became possible in the beginning of the 1990ies, and it quickly became clear that the theoretical models existing at the time [1] were not able to describe these data -see Ref. [2] and references therein. They fell short by a factor of two for reactions with an isoscalar NN pair in the final state, while there was a discrepancy of even more than an order of magnitude for those with an isovector NN pair in the final state. The numerous efforts to improve the phenomenological approaches [3,4], although quite successful for various observables [5], did lack for systematics, gave contradictory answers, and involved a sensitivity to unobservable short range effects.
To overcome those deficiencies, in recent years considerable theoretical efforts went into the development of an effective field theory that can be applied to the reactions NN → NNπ.
In early studies it became clear, however, that the original power counting by Weinberg [6,7] needs modifications in order to arrive at a convergent expansion [8,9] (see also Ref. [10] where it was pointed out that the naive power counting using the heavy baryon formalism is not applicable above the pion production threshold -the necessary adaptions are outlined in Ref. [11]). Indeed, for neutral pion production in pp collisions, the corrections due to the next-to-leading order (NLO) increased the discrepancy with the data and, moreover, the next-to-next-to-leading order (NNLO) contributions turned out to be even larger than the NLO terms [12]. The origin of these difficulties was identified quite early by Cohen et al. [13], see also [14], who stressed that the additional new scale, inherent in reactions of the type of NN → NNπ, needs to be accounted for in the power counting. Since the two nucleons in the initial state need to have sufficiently high kinetic energy to produce the on-shell pion in the final state, the initial center-of-mass momentum needs to be larger than where m π and M N refer to the pion and nucleon mass, respectively, and Λ χ denotes the typical hadronic scale, here identified with M N to get a numerical estimate for the expansion parameter. The proper way to include this scale was presented in Ref. [15] and implemented in Ref. [16], see Ref. [2] for a review article. As a result p-wave pion production is governed by tree-level diagrams up to NNLO [15,17] in the modified power counting scheme. On the other hand, for s-wave pion production, pion loops, studied in detail in Ref. [18], start to contribute already at NLO. The individual loop contributions turned out to grow linearly with increasing NN relative momentum which resulted in a large sensitivity to the employed NN wave functions [19]. However, it was demonstrated in Refs. [18,20] that all irreducible loop contributions at NLO cancel altogether which was needed to keep the scheme selfconsistent. Furthermore, the proper separation of irreducible terms at NLO from reducible ones in the loops resulted in an increase of the most important LO operator for charged pion production, first investigated in Ref. [1], by a factor of 4/3. This increase was sufficient to overcome the apparent discrepancy with the data in the reaction pp → dπ + . The neutral pion channel is more challenging -it still calls for a calculation of subleading loop contributions.
First steps in this direction were taken in Refs. [11,21]. We further emphasize that the ∆(1232) isobar should be taken into account explicitly as a dynamical degree of freedom [13] because the Delta-nucleon mass difference, ∆M, is also of the order of p thr . This general argument was confirmed numerically in phenomenological calculations [5,22,23], see also Refs. [16,20] where the effect of the ∆ in NNπ was studied within ChPT.
The progress in the theory of isospin conserving pion production in nucleon-nucleon collisions allowed us to perform a complete leading-order calculation of charge symmetry breaking effects in pn → dπ 0 [24], see also Refs. [25,26] for related works.
Given the developments just discussed, one now should be in the position to investigate two-pion production in NN collisions using the same formalism. Such calculations do not exist yet, although in the pioneering works of Refs. [27,28] some constraints from chiral symmetry were already implemented. A more recent phenomenological analysis of this reaction can also be found in Ref. [29]. For two-pion production in NN collisions at threshold, the center-of-mass momentum of the initial nucleons is necessarily larger by a factor of √ 2 compared to the single-pion production. Thus we now have With this large momentum scale, the value of the expansion parameter is and one may question the applicability of ChPT for the two-pion production case. However, while the expansion may converge slowly, it is still meaningful at least close to the chiral limit and therefore should also provide some guideline on the hierarchy of diagrams. Thus it is still interesting and important to investigate the structure of diagrams that contribute at the lowest orders. Note that, since the NN momentum at the threshold for two-pion production is quite close to the one for single-pion production, for the purpose of power counting below we identify the two.
The amount of experimental data for NN → NNππ has increased greatly in recent years [30][31][32][33] and even a polarized measurement exists [34]. However, up to now there are no data available sufficiently close to threshold to allow for a straightforward comparison with our leading order calculation of the reaction pn → dππ. To extend our calculation to higher energies the inclusion of the ∆ as an explicit degree of freedom will be neces- ChPT studies [39,40] predict that the double-neutral-pion photoproduction is significantly enhanced near threshold as compared to the production of the charged pions due to the contribution of chiral (pion) loops at NLO (O(p 3 )). This prediction was confirmed experimentally in Ref. [44]. Also it was found in Ref. [39] that the ∆-isobar contribution, which is potentially important because the ∆−nucleon mass difference is of comparable magnitude to 2m π , turns out to be insignificant due to specific cancellations of the individual contributions. The Roper resonance contributes to this process at one order higher, i.e. at N 2 LO (O(p 4 )). The effect of the Roper resonance encoded in low-energy constants was found to be sizable although significantly smaller than the contribution of the leading chiral loops [40]. Similar conclusions about the role of the Roper resonance were drawn based on studies of πN → ππN. In this case, however, the loops were found to be negligible [42] and the leading contribution was provided by tree level terms. For two-neutral-pion production in NN collisions we expect the production mechanism to be in accord with the dimensional analysis, i.e it should be dominated by the tree level graphs at LO. However, an explicit study of loops at NLO will be necessary to support our uncertainty estimates.
The paper is structured in the following way: In Sect. II our theoretical framework is introduced. Results for the reaction pn → dπ 0 π 0 are presented and discussed in Sect. III.
A summary and some concluding remarks are provided in Sect. IV. Technical details of our calculations are summarized in two Appendices.

II. THEORETICAL FORMALISM
Our calculations are based on the effective chiral Lagrangian [45][46][47]: where the superscripts 0 and 1 denote the leading and next-to-leading order Lagrangian, respectively, f π denotes the pion decay constant in the chiral limit, g A is the axial-vector coupling of the nucleon, N and π are the nucleon and pion field, respectively, and σ (τ ) denotes the Pauli matrix in spin (isospin) space. The ellipses in the above equations represent terms which are not relevant for the present study.
As mentioned in the Introduction, the system is characterized by three scales, where we use The counting rules here are similar to those for one-pion production used in [15][16][17][18] (for a review see also [2]). For the Weinberg-Tomozawa (WT) πN → πN vertex, as discussed in detail in Ref. [18], the leading term should be proportional to 2ω π where ω π is the energy of the outgoing (on-shell) pion. The diagrams that contribute at the lowest order are shown in Subleading vertices are marked as ⊙. Fig. 1 and start to contribute at order χ ∼ m π /M N . Obviously, the number of diagrams is significantly larger for two-pion production than for one-pion production already at LOafter all the latter appears as building block of the former in diagrams a)-d). For Fig. 1d), we note that the naive dimensional analysis shows that this diagram contributes at nextto-leading order. However, due to the existence of the two-nucleon cut in the intermediate state, this diagram is promoted to leading order following the very same logic adopted by Weinberg [7]. In addition, Fig. 1 e)-i) show genuine two-pion production diagrams. Note that the only non-resonant diagrams included in the earlier studies of Refs. [27,28] are those depicted in the second line of Fig. 1, whereas in Ref. [29] such diagrams are not considered at all.
Another potentially important class of contributions arises from diagrams where one of the nucleons is excited to the ∆(1232)-resonance. However, in the reaction considered here the N∆ system can only couple to the NN system in the intermediate state. The NN systems in the initial and final states have to have isospin zero, cf. below, while the N∆ system has always isospin one. The ∆(1232) can only appear when the first pion is emitted and, thus, the power counting for the ∆(1232) in pn → dππ is similar to the one in single-pion production. In particular, the ∆(1232) starts to contribute at NLO due to the fact that the ∆ propagator is suppressed by 1/p as compared to 1/m π in the nucleon case. Moreover, the aforementioned isospin arguments prevent the potentially dangerous situation when there is an extremely enhanced ∆-propagator (∆M − 2m π − recoil) −1 in the N∆ intermediate state that would occur from a ∆ excitation prior to the first pion emission. It is interesting to note that a similar observation was made in Ref. [39] in the reaction γN → ππN where the corresponding term (∆M − 2m π ) −1 did not show up in the denominator because of an exact cancellation with the same term in the numerator.
Once the production operators are constructed, they should be convoluted with NN wave functions to account for the nonperturbative character of NN interaction [7]. In principle, the NN wave functions should be calculated on the same basis as the production operators.
However, up to now ChPT NN interactions are only available for energies below the onepion production threshold [48][49][50]. Therefore, in this work we adopt the so-called hybrid approach that consists in the convolution of the production operator calculated within ChPT with phenomenological NN wave functions from realistic NN models, in particular from the CCF model [51]. We utilize also wave functions generated from the CD-Bonn NN potential [52] so that we can examine in how far our results depend on the specific choice of the NN wave functions. The ∆(1232) resonance is included as explicit degree of freedom in the construction of the CCF potential [51] as well as in an extended version of the CD-Bonn potential [53]. This will allow us to extend in a future work the present calculation to higher orders in a straightforward manner. The ellipses in the Feynman diagrams of Fig Based on the Lagrangian given above, it is straightforward to calculate the amplitudes corresponding to the diagrams shown in Fig. 1. Furthermore, since we are only interested in the energy region very near to threshold, only a few partial waves contribute to the reaction amplitude. At threshold, the ππ system is in a relative S-wave and its isospin can be either zero or two as required by the symmetry of a system of identical bosons. However, since the isospin of the initial NN system can only be zero or one and the deuteron is an isosinglet the isospin of the produced ππ pair has to be zero and, consequently, also the isospin of the initial NN state is zero. Then the conservation of parity and angular momentum further requires that the total angular momentum of the system should be one and the NN pair in the initial state can only be in the 3 S 1 or 3 D 1 partial waves. The Pauli principle is also satisfied in this case. The explicit expressions for the amplitudes of these two partial waves can be found in Appendix A.

III. RESULTS AND DISCUSSION
As discussed in the introduction, some leading order diagrams depicted in Figs. 1a)-1e) are missing in Ref. [27]. First, we want to discuss the role of the diagrams 1a)-1d). As can be seen from the figure, these diagrams are closely related to the one-pion production case. There the leading order contributions come from the direct pion emission and the rescattering term. Thus, one may expect that the amplitudes 1a)-1d) reveal a pattern similar to the one-pion production case. For example, one may guess that the amplitude of  Table I. In addition, due to the fact that the NN amplitudes for the ISI appear in coupled channels ( 3 S 1 -3 D 1 ) the production amplitudes, as given in Table I two-pion production amplitudes listed in Tables II and III. We find that the values of these contributions are basically comparable with each other, as expected from the power counting.
The values of the amplitudes of Fig. 1b) and Fig. 1d) are about a factor of two to three larger than the other two amplitudes 1a) and 1c), which is still a reflection of the dynamics governing the one-pion production. We observe that the resulting values of the amplitudes 1a)-1d) are similar in size to those of 1f)-1i) considered in previous studies [27,28] and, thus, are important. We also find that the NN interaction in the initial and intermediate  Tables II and III . For example, the value of the S-wave amplitude of Fig. 1a) becomes a factor of two smaller than the Born amplitude after including the ISI. When the ImSI is also considered, the full amplitude is further reduced so that it is finally a factor of six smaller than the Born amplitude. This observation shows explicitly that the ISI and the ImSI are very important quantitatively.
Since they also strongly influence the phase of the amplitudes, their proper inclusion is compulsory, especially for studies of polarization observables.
The strength of the contact terms, that appear in Fig. 1e), is given by the low energy constants c i that can be determined from πN scattering. Because we are only interested in the amplitudes at threshold, we only need the values of c 1 , c 2 and c 3 . In this work, we take the values from Ref. [54] where the values of c i 's are obtained by fitting πN threshold parameters. To the order we are working, three solutions are offered in that paper. One corresponds to the results without considering the ∆(1232) explicitly, and the other two are results where the ∆(1232) is included but with different choices of the πN∆ coupling constant h A . It is well known that c 2 and c 3 can be largely accounted for by the contribution from the ∆(1232) [55], and therefore the values of c 2 and c 3 can change significantly between extractions with and without inclusion of the ∆(1232). One may expect that using different sets of parameters will affect the value of the amplitudes of Fig. 1e) strongly. However, it is interesting to note that the amplitude of Fig. 1e) is independent of those choices, because it only depends on the value of the linear combination with q 0 1 = q 0 2 = m π for the pion energies in the kinematics relevant here. From Table 1 of Ref. [54] one can see that the values of c 1 and c 2 + c 3 are not affected by the ∆(1232) up to order O(p 2 ). Thus, although the values of c 2 and c 3 can significantly change individually by considering the ∆(1232) explicitly, the sum of them is not affected up to order O(p 2 ), as shown explicitly in Ref. [55]. Hence at LO the contribution from contact terms are not influenced by the ∆(1232), which is also consistent with the power counting used. It is also interesting to note that the linear combination of low energy parameters displayed in Eq. (5) is large, for the individual terms interfere constructively -this is in contradistinction to πN scattering, where q 0 1 = −q 0 2 = m π holds at threshold and then physics is governed by the very small isoscalar scattering length [55], see Ref. [56] for a recent accurate extraction of a + from pionic atoms. Thus, once the reaction pn → dπ 0 π 0 can be studied with sufficient accuracy, it might be a good source of information for the c i individually. Note also that in Ref. [54] the 1/M N corrections are not included explicitly in the fits to πN data but absorbed effectively into the LECs c i . To be consistent with their treatment we also drop the 1/M N term in Eq. (5). Furthermore, the residual combination c 2 + c 3 which enters Eq. (5) is totally determined by the S-wave threshold parameters. Numerically the result for diagram 1e) turns out to be one of the largest individual contributions -cf. Table I. The diagrams 2 of Figs. 1f)-1i) are included in the previous study of Ref. [27], except for the initial state interaction which is not considered in that work. However, as can be seen from Tables II and III, the consistent inclusion of the ISI changes the value of the amplitudes 1f)-1i) significantly and is thus important.
Besides the interference between the individual amplitudes of each particular diagram, the interference between the contributions from different diagrams is also very significant as can be seen from Tables II and III. In order to examine the sensitivity of our results and especially of the interference pattern to the employed NN interaction we also performed calculations with a different NN model, namely with the CD-Bonn model [52]. We found that the values of the individual amplitudes vary in a reasonable range (up to 30%) due to is constant and therefore take the energy dependence as originating only from the threebody phase space. Our result is roughly a factor of two smaller than the central value of the data point at Q = 70 MeV. We want to stress, however, that the uncertainties of the experiment in question is quite large (even a factor of 25 larger than the data themselves).
For future reference, in Fig. 2 we show the central value for the cross section predicted under the assumption that the matrix element is a constant and that no higher partial waves contribute. We also indicate the uncertainty band based on a dimensional analysis.

IV. SUMMARY
We presented a complete leading order calculation for the reaction pn → dπ 0 π 0 at threshold within chiral perturbation theory. There is no free parameter in the calculation at this order. We included various additional diagrams as compared to previous investigations [27,28] and we also made several technical improvements. Our most important findings, summarized in Table I, are • all diagrams evaluated are of similar importance; • there are sizable interferences between the individual contributions; • the accurate inclusion of the NN interaction in both intermediate as well as initial states is very important.
We also stress that for two-pion production the expansion parameter where p (2) thr denotes the initial NN momentum at the two-pion production threshold, is rather large. It is therefore important to calculate higher order contributions to check the rate of convergence. We argue in this paper that despite of the proximity of the ∆(1232) to the ππN threshold the potentially most important N∆ intermediate state is not allowed for the reaction pn → dππ because of isospin conservation. Intermediate states containing the ∆(1232) must be of the kind πN∆, i.e. can only occur after one-pion emission. Therefore, the role of the ∆(1232) resonance in the reaction considered is expected to be analogous to that in one-pion production. In particular, the ∆(1232) starts to contribute at next-to-leading order. Furthermore, it is known from phenomenological studies of NN → NNππ [27,28] that the Roper resonance can play a significant role already near threshold and that it will become even more important when considering a larger range of energies. However, this resonance is not included explicitly in the present study. One may expect that its contribution is absorbed into some low energy constants. The contribution of the Roper resonance to the c i parameters, that scale the strength of the leading isoscalar πN scattering, has been discussed, e.g., in Ref. [55]. It seems that it plays only a minor role here. Similar conclusions were drawn from systematic studies within ChPT of the double-pion photoproduction process [39,40] and the reaction πN → ππN [41][42][43] near threshold. In particular, for the reaction γp → π 0 π 0 p the contribution of the Roper was found to be rather moderate as compared to the large contribution of chiral loops [40]. A recent model calculation of πN → ππN by S. Schneider et al. [35] suggests also that the Roper resonance plays a rather minor role. On the other hand, the Roper might contribute significantly to the (N † N) 2 ππ counterterms, which enter at NNLO in a ChPT calculation of NN → NNππ. Based on the discussions above, one can not expect that our current calculation can describe the experimental data well at higher energies. However, the presented calculation provides an estimate for the contribution of the non-resonant background near threshold. It therefore forms a basis for future studies and is thus a precondition to extract reliable information on the Roper resonance from near-threshold experiments. In this appendix, we present expressions for the amplitudes that we consider in this work.
To obtain the partial wave amplitudes, we follow the technics developed in Ref. [57].
For the reaction NN → dππ at threshold, the initial state can only be in the 3 S 1 or 3 D 1 partial waves and the total isospin is zero. Thus, the amplitudes for NN → dππ at threshold can be written in the general form where S denote the normalized spin-orbit structure of the initial nucleons for the 3 S 1 and 3 D 1 partial waves, respectively, f ull are the corresponding partial wave amplitudes, π is the isospin wave function of outgoing pions and ε is the deuteron polarization vector. Here χ 1 (χ 2 ) and ϕ 1 (ϕ 2 ) are spinors of the initial nucleons in spin and isospin space, respectively, andn denotes the unit vector of the relative momentum of the initial nucleons. The expressions for A 3 S 1 ( 3 D 1 ) f ull are obtained by projecting the amplitudes that contribute to a given order into corresponding partial waves using the technics developed in Ref. [57]. To leading order, the relevant diagrams in our work are shown in Fig. 1. Each diagram in Fig. 1 represents a set of diagrams, in which possible NN interaction in the initial and intermediate NN states are considered. For example, the partial wave amplitudes of Figs. 1a)-d) originate from several parts as follows: To compute diagrams with the NN interaction in the initial and intermediate states we take the NN scattering amplitudes from some potential models. In our calculation, instead of the commonly used T matrix, the M matrix is used. These quantities are related by shell M matrix, where P W → P W ′ represents the transition from the partial wave "P W " to "P W ′ " with "P W " and "P W ′ " being 3 S 1 or 3 D 1 in our case. For the intermediate state interaction, the only possible transition is 3 P 1 → 3 P 1 , so we just use M 3 P 1 → 3 P 1 N N to denote the fully off-shell NN M matrix. For the deuteron-NN vertex we adopt the notation and structures used in Ref. [58]. We use u(p) and w(p) to denote the NN components of the deuteron wave function corresponding to S-wave and D-wave respectively. Clearly, the bound (deuteron) wave functions u(p) and w(p) should be calculated using the same potential model as the NN scattering amplitudes in the continuum state. In this work we use the CCF model [51] to generate the NN amplitudes and the deuteron wave functions.
Since the ∆∆ channel is considered in the CCF model, that can also couple to the deuteron, the NN part of the deuteron wave functions, u(p) and w(p), are normalized as: In this work, we adopt the following values of the parameters: f π = 92.4 MeV, g A = 1.32, m π = 139.58 MeV and M N = 938.27 MeV. The loop integration is regularized using the cut-off method with p cut = Λ ChPT ≈ 1 GeV. We checked that the dependence of the results on this parameter is small -the results change by 10% when Λ is increased to 10 GeV. Fig. 1 a) To get the partial wave amplitudes of Fig. 1 a)

Amplitudes of
where π i is the isospin wave function of the π meson in the final state. Here we introduced Correspondingly, the expressions of the building blocks for Fig. 3(C) and (D) can be given in the following way: where B 3 S 1 and B 3 D 1 for the diagrams (C) and (D) read: Given the building blocks shown above and the corresponding NN scattering amplitudes, it is straightforward to construct the amplitudes of Fig. 1 a)-d). For example, the amplitudes corresponding to diagram 1a) are: A a,P W where p 2 = p  Fig. 1b) can be obtained by changing C to D in the above equations.
The amplitudes of Figs. 1c) and 1d) can be obtained in a similar manner. The amplitudes of Fig. 1c) are A c,P W ISI = N S · 3 S 1 , 3 D 1 A c,P W ImSI = N S · The corresponding amplitudes of Figs. 1d) can be obtained by changing A to B in the above equations.

Amplitudes of Fig. 1e)-h)
For the diagrams of Fig. 1e)-h) there is no intermediate state interaction so that the amplitude can be written as: Note that only the sum of the amplitudes of Figs. 1g)-i) is independent of the choice of the pion field. That is why below we give only the amplitude for the sum of them. The partial wave amplitudes of Figs. 1e)-i) without the ISI can be written as: The corresponding amplitudes with the ISI can be obtained through the following expression: where m can be 'e', 'f', or 'g+h+i'. The symmetry factors for these amplitudes are N e s = 8 where E CM and p CM are the energy and momentum of the initial nucleons in the c.m. frame.
Φ is the phase space factor defined as: The prefactor 1 4·2·2 is due to averaging over the initial spin states, the isospin wave function of the initial NN states and the identity factor for the final pions, respectively. The factor of 3 in Eq. (B1) is from summing up the spin states.