Pion photoproduction off nucleons in covariant chiral perturbation theory

Pion photoproduction off the nucleon close to threshold is studied in covariant baryon chiral perturbation theory at O($p^3$) in the extended-on-mass-shell scheme, with the explicit inclusion of the $\Delta(1232)$ resonance using the $\delta$ counting. The theory is compared to the available data of cross sections and polarization observables for all the charge channels. Most of the necessary low energy constants are well known from the analysis of other processes and the comparison with data strongly constrains some of the still unknown ones. The $\Delta(1232)$ contribution is significant in improving the agreement with data, even at the low energies considered.


I. INTRODUCTION
Single pion photoproduction off the nucleons has been a subject of strong and continuous theoretical and experimental efforts. Many have been dedicated to the investigation of the process at intermediate energies which allowed us to study the spectrum and properties of numerous baryon resonances [1,2]. Here, we address the near threshold region, where chiral perturbation theory (ChPT) [3][4][5][6], the effective field theory of QCD at low energies, should provide an adequate framework and only nucleons, pions, and the lowest lying resonances might play a role.
Early work attempted to describe this process through low-energy theorems (LET) obtained from gauge and Lorentz invariance [7] and later from current algebra and the partial conservation of the axial-current [8,9]. These theorems described well the production of charged pions, but failed for the case of the γ + p → p + π 0 process [10][11][12][13]. In one of the earliest successes of ChPT with baryons, Bernard et al. [14][15][16] could solve the discrepancies between theory and the data available at the time with corrections related to loop-diagram contributions. Still, the theoretical models showed their limitations with the more precise measurements of cross sections and polarization observables obtained at the Mainz Microtron (MAMI) in 2013 [17]. For instance, it was found in an O(p 4 ) heavy baryon (HB) calculation that the agreement with data was satisfactory only up to some 20 MeV above threshold [18]. This indicated the need of calculations at even higher orders. The situation was not better in other approaches to baryon ChPT, such as the extended-on-mass-shell (EOMS) scheme, which at O(p 4 ) also obtained a good agreement only for a very limited range of energies [19]. However, a higher order calculation, apart from the added technical complications, would lose its predictive power because of the new set of undetermined low energy contants (LECs) appearing in the Lagrangian. These difficulties could be intrinsic to this specific process, for instance because of the cancellation happening at the lowest order of the chiral expansion, but they may also signal the need for some revision of the theoretical approach.
In recent years, there has been a considerable advance in the qualitative and quantitative understanding of low energy hadron physics using ChPT. It provides a systematic framework to obtain a perturbative expansion in terms of small meson masses and external momenta and has a quite impressive record on its predictivity and the quality of its description of multiple observables involving mesons, nucleons and photons [20,21]. Nonetheless, ChPT, when applied to systems with baryons, contains some subtleties which may hinder its progress. As shown in Ref. [22], in the presence of baryon loops the naive power counting is broken because of the nonzero nucleon mass in the chiral limit, making difficult the development of a scheme that allows for a systematic evaluation of higher orders in the chiral expansion. This problem was first solved in the HB formalism (HBChPT) [23,24], at the expense of losing Lorentz covariance, and later by some covariant methods as the infrared regularization (IrChPT) [25] and the EOMS scheme [26] 1 . Here, we adhere to the latter, which, apart from providing a proper power counting, preserves the analytic structure of the calculated amplitudes. Furthermore, it usually leads to a faster chiral convergence than HBChPT or IrChPT [27][28][29]. This approach has been used with satisfactory results in the calculation of many baryon observables such as masses, magnetic moments, axial form factors, among others [28][29][30][31][32][33][34][35][36][37][38][39][40]. Moreover, it has been successfully applied to many processes, among which πN scattering [27,37,41,42] and the pion electromagnetic production on the nucleons [19,[43][44][45].
Both are directly related to the pion photoproduction on the nucleons investigated in this work.
In addition to the power counting problem in baryon ChPT, another issue arises due to the small mass difference between the nucleon and the ∆(1232) resonance. In fact, the mass of the latter is little above the pion production threshold. Due to this proximity of the resonance to the threshold and its large transition couplings to pions and photons, the ∆(1232) is crucial for the description of πN and γN processes even at very low energies [46]. These facts suggest the importance of the explicit inclusion of the ∆(1232) resonance in our effective theory. The hope is that the incorporation of the most relevant degrees of freedom, such as those associated to the ∆, could lead to a faster convergence of the chiral series. A price to pay is the emergence of a new small parameter, δ = m ∆ − m N ≈ 300 MeV that should be properly accounted for in the chiral expansion, where m ∆ and m N are the masses of the ∆ resonance and the nucleon, respectively.
In this work, we investigate the near threshold pion photoproduction off nucleons within the aforementioned effective theory approach, i.e. EOMS ChPT, at O(p 3 ). We also choose to include the ∆(1232) resonance explicitly. This, or a very similar approach has already been used for the analysis of Compton scattering [47,48], πN scattering [27,41,42] or the weak process νN → lN π [49,50] of high interest for neutrino detection. Moreover, the fact that we are using the same framework at the same chiral order as some of these works allows us to fix many of the LECs of the theoretical model.
Besides the general reasons given in the previous paragraphs, the inspection of the cross section data shows that the ∆(1232) resonance is conspicuously dominant for all pion photoproduction 1 See, e.g., Ref. [6] for a review of the three schemes and the discussion in the introduction of Ref. [27]. channels [46], and its tail could well be large even close to threshold. The possible importance of the ∆ resonance for the chiral analyses of pion photoproduction had already been suggested in Refs. [16][17][18]51]. Indeed, the convergence of the chiral series and the agreement with data was found to improve substantially with this inclusion in the investigation of the neutral pion photoproduction on the proton in Refs. [43,44]. Nevertheless, it is also clear from even a cursory perusal of data, that the energy dependence is very different for the γp → π 0 p and the γp → π + n processes. In the latter case there is a large nonresonant electric dipole contribution that produces s-wave pions and is strongly suppressed in the π 0 case. For this reason, the importance of including higher orders and the ∆ resonance is especially strong for the neutral pion channel. However, also the other ones will have noticeable corrections due to these inclusions. Furthermore, the different channels of pion production are sensitive to different ChPT LECs, leading to the need of studying carefully also the charged pion channels in the same framework.
Motivated by the sensitivity to different mechanisms of the various channels, here we extend the analysis of Refs. [43,44], restricted to the γp → π 0 p process, by incorporating the other channels, in which charged pions are produced. We perform a global study of all the data currently available in the low energy region. This amounts to measurements of angular distributions, total cross sections and spin observables such as beam and target asymmetries. Ultimately, these studies will benchmark the ability to improve upon the predictions for the weak pion production processes [49,50,52,53], for which the data are very scarce, and integrated over wide ranges of energies, thus making it impossible to constrain well the LECs or to make concrete statements about the behavior at specific energies. While the predictive power of ChPT calculations is limited to the threshold region, they should properly be taken into account in phenomenological models that aim to describe weak pion production in wider energy regions, see also Ref. [54] and references therein.
The inclusion of the charged channels requires the addition of a more extensive set of diagram topologies and also of some extra pieces of the chiral Lagrangian with their corresponding LECs.
Furthermore, we incorporate a more detailed analysis of the errors, estimating both the statistical uncertainty coming from the fits and the uncertainty related to the truncation of the chiral series.
The structure of the paper is as follows. In Sec. II, we present the basic formalism, the chiral Lagrangian and the theoretical model for the amplitude. Sec. III describes the experimental database and the fit method, including the procedure for the error estimation. Finally, results are presented in Sec. IV. We summarize in Sec. V. The pion photoproduction off the nucleon, depicted in Fig. 1, can occur in four possible charge channels: γp → π 0 p, γp → π + n, γn → π − p, γn → π 0 n. The differential cross section in the center of mass (c.m.) system can be written as where s ≡ (k + p) 2 is the Mandelstam variable, Λ(x, y, z) = (x − y − z) 2 − 4yz is the Källén function, m N and m π are the nucleon and pion masses, respectively. The modulus squared of the scattering amplitude T is averaged over the initial nucleon spin (s i ) and photon polarization (λ) and summed over the final nucleon spin (s f ). For practical purposes, it is convenient to use a representation of T in terms of the Chew-Goldberger-Low-Nambu (CGLN) amplitudes F i [55], which lead to simple expressions for multipoles, cross sections and the polarization observables. In the CGLN formalism, T can be written as where χ i and χ f are Pauli spinors of the initial and final nucleon states, respectively. For real photons and in the Coulomb gauge ( 0 = 0, · k = 0), the amplitude F may be decomposed as with σ the Pauli matrices, the photon polarization andq,k unit vectors in the direction of q and k, respectively. The explicit expressions for each F i amplitude are given in Appendix A 1, Eqs. (A8)-(A11). In this representation, the unpolarized angular cross section in the c.m. system in Eq. (1) is recast as where θ stands for the scattering angle between the incoming photon and the outgoing pion and with | q| and | k| evaluated in the c.m. system.
At the studied energies, apart from the unpolarized angular cross section, there are many data for the polarized photon asymmetry. This observable is defined by with dσ ⊥ and dσ the angular cross sections for photon polarizations perpendicular and parallel to the reaction plane, respectively. In the CGLN representation we have [56] In its turn, the target asymmetry, defined as the ratio where dσ + and dσ − correspond to the cross sections for target nucleons polarized up and down in the direction of k × q, can be written as Useful expressions for other polarization observables in terms of the F i amplitudes can be found, for instance, in Ref. [56].

B. Power counting and chiral Lagrangians
As was discussed in the Introduction, when the ∆(1232) resonance is explicitly included a new small parameter δ = m ∆ − m N ≈ 300 MeV appears, which must be taken into account in the chiral expansion. In this work, we use the δ counting, introduced in Ref. [57], in which δ ∼ O(p 1/2 ).
Therefore, the chiral order D of a diagram with L loops, V (k) vertices of O(p k ), N π internal pions, N N nucleon propagators and N ∆ ∆(1232) propagators is given by Here, we consider all contributions up through O(p 3 ). The following pieces of the chiral effective Lagrangian are required, where the superscripts represent the chiral order of each of the terms. The needed terms of the pionic interaction are given by [4,22] where the Goldstone pion fields are written in the isospin decomposition τ i (i = 1, 2, 3) are the Pauli matrices, F 0 indicates the pion decay constant in the chiral limit, and Here, e is the electric charge of the electron, Q = (τ 3 + 1 2×2 )/2 is the charge matrix, and A µ is the photon field. Tr[· · · ] denotes the trace in flavor space. We will be working in the isospin symmetric limit, and thus χ = m 2 π 1 2×2 , with m π the corresponding pion mass. The relevant terms that describe the interaction with nucleons at O(p 1 ) are given by [58] where N = (p, n) T is the nucleon doublet with mass m and axial charge g, both in the chiral limit. Furthermore, At the second order the only relevant terms are with where d j ( j = 8, 9,16,18,20,21,22) are new LECs appearing at O(p 3 ) in units of GeV −2 . The derivative operator D λν = D λ D ν + D ν D λ acts over the nucleon doublet 2 and The interaction between the nucleon and ∆ is described by a Lagrangian that decouples the spin-1/2 components from the spin-3/2 Rarita Schwinger field [59,60]. For a calculation up through O(p 3 ) in the δ counting the relevant terms are where T are the components of the spin-3/2 Rarita Schwinger field corresponding to the isospin multiplet for the ∆ resonance. The isospin transition matrices T a can be found in Ref. [60]. 2 The totally antisymmetric Levi-Civita tensor can be written as µναβ = − i 8 γ µ , γ ν , γ α , γ β γ 5 .  To restore the power counting, we apply the EOMS scheme. Therefore, after the cancellation of the UV divergences we proceed to perform the required finite shifts to the corresponding LECs, so that the transformed parameterX fulfills which in our case applies for X ∈ {m, g, c 1 , c 6 , c 7 }. For the parameters, m and g, from L (1) πN , we get is the MS-renormalized scalar 1-point Passarino-Veltman function with µ the renormalization scale introduced in the dimensional regularization. For the second order LECs we have [30] 5 Finally, the full amplitude, T , is related to the amputated one,T , via the Lehmann-Symanzik- where Z π and Z N are the wave function renormalization constants of the pion and nucleon, respectively. Their explicit expressions are given in Appendix B.

A. Experimental database
We compare our theoretical model to the data in the energy range from threshold √ s ∼ 1080 MeV up to √ s ∼ 1130 MeV. This choice guarantees that the momentum of the outgoing pion is small and that we stay well below the ∆ resonance peak. We should point out that we work in the isospin limit, both in the choice of the Lagrangian and in the further calculation of the loops.
Therefore, the framework is not well suited for the measurements corresponding to the first MeV's above threshold, where the mass splittings are quite relevant. We have checked, nonetheless, that our numerical results are not modified by the inclusion or exclusion of those data points.
The larger part of the database corresponds to the γp → π 0 p process. Furthermore, the experimental errors are relatively smaller when compared to the other channels. As a consequence, the neutral pion production has a preeminent weight in the fits. There have been extensive measurements in the near threshold region [64][65][66][67], although the largest contribution comes from the comprehensive set of data on angular cross sections and photon asymmetries obtained at MAMI [17]. 6 At the higher end of our energy range there are a few data points measured by the LEGS facility at the Brookhaven National Laboratory [68].
In comparison, at these energies data are scarce for the channels with charged pions and there are very few recent experiments on them. For the reaction γn → π − p, we use the angular distributions and total cross sections from Refs. [69][70][71][72]. There are no data on polarization observables.
The early experiments at Frascati [69] and DESY [70] actually measured the reaction on deuterium and then, the cross sections on the neutron were obtained using the spectator model. On the other hand, the experiments at TRIUMF [71,72] correspond to the inverse reaction: radiative pion capture on the proton. There are some later measurements from the early 1990s, also at TRIUMF, quoted by SAID [73], but they are unfortunately unpublished. Only recently, the π − photoproduction on the deuteron has been measured again at the MAX IV Laboratory [74], but the neutron cross section has not been derived yet.
There are some more data for the γp → π + n channel, which can be measured more directly.
They are mostly angular and total cross sections but they also include some photon asymmetries.
In total, the database contains 957 points. For most of them the total error estimation (statistic plus systematic) was given in the original references. A typical 5% systematic error has been added in quadrature for the few points where only the statistical error was provided [69][70][71].

B. Low energy constants
Most of the parameters required in the calculation are readily available as they have been ob-  In  8 The remaining four constants have been fitted to the experimental data minimizing the χ squared.

C. Error estimation
There are two sources of uncertainties in our prediction of any observable. First, there is the uncertainty propagated from the statistical errors of the LECs in the fit, which we take as where O refers to the observable, and the i and j indices are labels for a given LEC x i , withx i and δx i its corresponding mean and error values as obtained from the fit. Finally, Corr(i, j) is the (i, j)th matrix element of the correlation matrix.
Additionally, we consider the systematic errors due to the truncation of the chiral series. We have used the method from Refs. [42,82] where the uncertainty δO (n) th , at order n for any observable O is given by where Q = m π /Λ b , Λ b is the breakdown scale of the chiral expansion. We set Λ b = 4πF ∼ 1 GeV as in Ref. [80]. In our case, the lowest order considered is n LO = 1 and the upper order calculated is n = 3.

IV. RESULTS AND DISCUSSION
A. Fit with and without ∆ contribution We have fitted the free LECs of our model comparing our calculation with the experimental database and minimizing the χ squared. The results of the fit for several different options are 7 In some of the references, e.g. [78], the ∆ resonance was not explicitly included, but its contribution starts at a higher order in the δ counting. 8 The parameter d 16 has been investigated, within the current approach, studying the dependence on the pion mass of the axial coupling of the nucleon in lattice data. A value of d 16 = (−0.83 ± 0.03) GeV −2 was obtained in Ref. [80]. given in Table II. Fit I corresponds to our full model, as described in the previous sections. The LECs from Table I have been set to their central values except for d 18 , which has been left to vary within the quoted range, and g M that was left free. In the minimization procedure, we have chosen the combinations d 8 + d 9 and d 8 − d 9 because of the strong correlation existing between d 8 and d 9 which cannot be well determined independently. Furthermore, the channel γp → π 0 p, with the most accurate data, depends just on d 8 + d 9 , which leads to a quite precise value for this combination. With the current data, we are less sensitive to d 8 − d 9 that would benefit from better data on the other channels that depend only on d 9 for the γp → π + n and γn → π − p cases, and on Also, the O(p 3 ) constants d 20 and d 21 are less constrained, because they only affect the channels with charged pions. These latter channels are already relatively well described by lower order calculations and are not very sensitive to third order effects. Furthermore, the uncertainties in their data are comparatively larger than for the π 0 channel. We must also recall that, in pion photoproduction, d 21 is fully correlated with d 22 and only appears in the amplitudes in the 2d 21  A first remark is that g M takes a value consistent with that obtained from the electromagnetic ∆ decay width. This clearly shows the sensitivity of the pion photoproduction to the ∆ resonance even at the low energies investigated. In fact, removing the ∆ mechanisms we get Fit II, with a much worse agreement with data. The reshuffling of the free parameters is ineffective in describing the rapid growth of the cross section of the π 0 channel. The importance of the resonant mechanisms can be also appreciated in Fig. 5. The quality of the agreement decreases rapidly as a function of the maximum photon energy of the data included in the fit in the ∆-less case, whereas it is practically stable for the full model. This behavior (rapid growth of χ 2 as a function of energy) can also be seen even Comparing the absolute values of χ 2 /do f (χ squared per degree of freedom), we see that the O(p 3 ) calculation without ∆ (Fit II) gives χ 2 /do f = 29.5. This number is mostly driven by the contribution of the γp → π 0 p channel (Table II), whereas the contribution of the channels with charged pions to χ squared is barely modified. The χ 2 value is substantially reduced with the explicit inclusion of the ∆ (Fit I), still at O(p 3 ) and even when the corresponding LECs are previously fixed. A reduction can also be obtained without the ∆ by doing an O(p 4 ) calculation [18,19]. However, apart from requiring a number of extra parameters, in the ∆-less calculations the fit quality diminishes rapidly as a function of the photon energy.
In Figs. 6, 7 and 8, we compare the results from Fit I with data from the π 0 channel. The only free third order LEC is the d 8 + d 9 combination. The agreement is overall good for both cross section and beam asymmetries in the full range of energies considered. Only the total cross sections from Ref. [83] are systematically below the calculation from 165 to 205 MeV, see Fig. 8.
However, these data are incompatible with the differential cross sections measured at the same energies in Ref. [17]. Also, there is some overestimation (within the error bands but systematic) of the angular distributions at backward angles. The uncertainties due to the truncation of the chiral expansion are considerable. This fact reflects the large size of the ∆ contribution and the O(p 3 ) mechanisms to this observable.
The channel γp → π + n is sensitive to the LECs d 9 , d 20 and d 21 . As shown in Figs. 9, 10 and 11, the agreement is good for the cross sections and for the few data available on beam asymmetry.
The model also agrees well with the γn → π − p data as shown in Figs   π 0 channel and, in particular, the cross section at backward angles, Fig. 6. Notice, however, the slight worsening of the agreement with the charged pion channels. While this might be pointing to some issue with the neutral pion production data at backward angles [17], it could also be arising from the theory side, as detailed below.
Of the now unconstrained LECs, the c 6 and c 7 values remain in the range given in Table I  pion nucleon coupling constant g πN [27]. We have also examined how χ 2 changes when moving triangles: data from Ref. [83], not included in the fit. Description same as Fig. 6. need for a higher order calculation to reach a proper chiral convergence.
A first step would be the inclusion of O(p 7/2 ) mechanisms, which correspond to tree mechanisms with higher order ∆π or ∆γ coupling and a set of loop diagrams with ∆ propagators inside the loop. This approach was already explored in Ref. [44] for the π 0 channel and did not change much the results as compared with the third order calculation, remaining consistent with the preference of large positive d 18 values.
A full O(p 4 ) calculation would incorporate further extra terms. The fourth order Lagrangian, N , that contributes to the process entails fifteen additional couplings [45]. 11 We have estimated the importance of this order by considering the tree level amplitude generated by L (4) N . The explicit expression can be found in Appendix C from Ref. [45]. In particular, we have explored how d 18 is affected by the new terms and we have found that it is very sensitive to some of the parameters, as e 48 , e 50 or e 112 . 11 With the current dataset, the use of the full L (4) N Lagrangian with fifteen extra parameters and an already small χ 2 leads to many minima and obvious overfitting.
γp → π + n FIG. 9: Angular cross section for the γp → π + n channel at various energies. Going from low to high energy, the data from Ref. [84] are marked as red diamonds, [77] as black squares, [76] as blue triangles, [68] as violet squares. Finally, in the lowest right panel, data for different energies from Ref. [75] are marked as magenta circles and from [65] as dark-green squares. In this latter panel the theory has been calculated at precisely the energies and angles of the data points, and the lines and bands have been interpolated.

B. Convergence of the approach
In Table III,  which are directly connected to the magnetic moments of the neutron and proton. 12 Next, in the δ counting, comes the inclusion of the ∆ mechanisms which start contributing at O(p 5/2 ). Once more, there are no free constants. We already get a much better description in the three channels.
Still, the agreement is poor for the neutral pion channel. An even larger improvement is reached in a third order calculation, without ∆ but with some extra free parameters (Fit II of Table II). At this order, the loop diagrams start appearing. They are also important for improving the agreement of all channels. Finally, in the last column we show our Fit I results, incorporating both ∆ mechanisms and a full third order calculation. It leads to an overall good agreement with the data in all channels.
Altogether, the ∆ mechanisms and the third order contributions play a capital role in reaching a good description of the pion photoproduction process. This is especially the case for neutral pion photoproduction, but also the charged pion production channels feel the improvement. We remark γn → π − p FIG. 13: Angular cross section for the γn → π − p channel at various energies. Data from Ref. [88] are presented as red filled squares, [71] marked as black triangles, [69] as blue squares, [70] as violet filled triangles and data from [72] as magenta filled diamonds. Description same as Fig. 8.
here that these effects also play a significant role in weak pion production [49,50].

V. SUMMARY
In this work, we have investigated pion photoproduction on the nucleon close to threshold in covariant ChPT, following the EOMS renormalization scheme. Our approach includes explicitly the ∆(1232) resonance mechanisms. We have made a full calculation up through O(p 3 ) in the δ counting.
The model reproduces well the total cross section, angular distributions and polarization observables for all the channels. The agreement is better, and for a wider range of energies, than in the O(p 4 ) calculations in both, covariant [19] and HB [18] schemes, without explicit ∆. As Finally, the extension to the description of electro-and weak production data will advance these studies even further, while offering the possibility of making reliable and accurate predictions for weak processes where data are more scarce.
with M µ the factorized hadronic current and Note that in the c.m. system p · = 0. One can easily find the conversion between the two different representations: For practical purposes, as explained in Sec. II A, it is sometimes convenient to use the CGLN amplitudes [55]. In this way the scattering amplitude from Eq. (2) reads where W = √ s is the center-of-mass energy, and the amplitude F can be expressed as the decomposition of the F i (i = 1, . . . , 4) pieces as shown in Eq. (3). These pieces are given explicitly by where the coefficients C (1) i for i = {I, II, III, IV} are given in Table IV and u ≡ (p − q) 2 . Here m N is the physical nucleon mass coming from the external legs in the Feynman diagrams of Fig. 2, that in our case corresponds to the order O(p 3 ) nucleon mass, whose expression is derived in Eq. (B3). The inner nucleon propagator has the second order nucleon mass m 2 instead of m.
This automatically generates the O(p 2 ) and higher order contributions corresponding to c 1 mass insertions.
γn → nπ 0 0 0 0 0 In what follows, for the amplitudes of order O(p 2 ) and higher, the leading order bare constants, e.g. m, can be replaced by their corresponding physical ones since the difference is of higher order than our current accuracy [O(p 3 )]. This replacement is actually made in our calculation: The definitions for the constants C (2) II and C (2) III are presented in Table V.
where d 168 = 2d 16 − d 18 and d 212 = 2d 21 − d 22 . The coefficients C (3) j for j = {Ia, Ib, II, III, IV} are given in Table VI  For the following amplitudes the definitions of the constants D II and D III are given in Table VII.

Appendix B: Renormalization factors
The wave function renormalization of the external legs is written as where δ (2) Z N = − 3g 2 A 64π 2 F 2 m 2 π − 4m 2 4m 2 π A 0 m 2 + m 2 π − 3m 2 B 0 m 2 , m 2 π , m 2 − m 2 Furthermore, the mass corrections are given by with Finally, the corrections to the axial vector coupling and the pion decay constant read where Note here that l r 4 and d r 16 are MS-renormalized LECs.