Dispersive Analysis of Low Energy $\gamma N\to\pi N$ Process and Studies on the $N^*(890)$ Resonance

We present a dispersive representation of the $\gamma N\rightarrow \pi N$ partial-wave amplitude based on unitarity and analyticity. In this representation, the right-hand-cut contribution responsible for $\pi N$ final-state-interaction effect are taken into account via an Omn\'es formalism with elastic $\pi N$ phase shifts as inputs, while the left-hand-cut contribution is estimated by invoking chiral perturbation theory. Numerical fits are performed in order to pin down the involved subtraction constants. It is found that good fit quality can be achieved with only one free parameter and the experimental data of the multipole amplitude $E_{0}^+$ in the energy region below the $\Delta(1232)$ are well described. Furthermore, we extend the $\gamma N\rightarrow \pi N$ partial-wave amplitude to the second Riemann sheet so as to extract the couplings of the $N^\ast(890)$. The modulus of the residue of the multipole amplitude $E_{0}^+$ ($S_{11pE}$) is $2.41\rm{mfm\cdot GeV^2}$ and the partial width of $N^*(890)\to\gamma N$ at the pole is about $0.369\ {\rm MeV}$, which is almost the same as the one of $N^*(1535)$, indicating that $N^\ast(890)$ strongly couples to $\pi N$ system.


Introduction
Single pion photoproduction off the nucleon has been extensively studied for its importance in determining the spectrum and properties of the nucleon resonances [1][2][3][4]. There have been many measurements on this process, accumulating a wealth of experimental data on, e.g., cross section, photon asymmetry, target asymmetry, etc; see e.g. Refs [5][6][7][8][9]. Based on the dataset, partial wave analyses were performed to anatomize the underlying structure of the reaction amplitude and justify the existence of the nucleon resonances theoretically. At low energies, it has been successful to explore the photoproduction processes in chiral perturbation theory (ChPT) [10][11][12][13][14][15][16][17]. In combination of unitarization approaches, the valid region of the chiral amplitudes is extended and physical states behave themselves as pole singularities of the unitarized amplitudes. Nevertheless, most of the unitarization methods only take the unitary cut into account, while the rest lefthand cuts (l.h.c.s) are left out, leading to the drawback that the proper analytic property of the amplitude is not faithfully guaranteed. In consequence, spurious poles arise to pretend the contribution of the left-hand cuts, or even worse, prevent us from discovering certain truly existent poles, e.g. virtual poles or subthreshold resonances.
In Refs. [18][19][20], a novel subthreshold resonance named N * (890) was found in the S 11 wave through a prudent analysis of the covariant chiral amplitude of πN scattering [21][22][23][24] by applying the method of Peking University (PKU) representation [25][26][27][28][29][30]. The PKU representation respects causality honestly and has already been used to establish the existences of the σ and κ states [25,27]. The discovery of the N * (890) resonance is nothing but an improved implement of analyticity compared to other unitarization methods. For instance, it is pointed out in Ref. [31] that the N * (890) resonance still exists even in a K-matrix parametrization if a better treatment of analyticity is executed. In this paper, we intend to explore the N * (890) resonance in the γN → πN to gain more information on its properties.
Our γN → πN amplitudes are obtained through a dispersive representation, which is set up with the help of unitarity and analyticity [32][33][34][35]. The inputs of the dispersive representation are πN final-state-interaction amplitude and chiral tree-level γN → πN amplitude estimating the left hand singularities of pion photoproduction. In a single channel approximation, the former can be achieved by Omnès solution with πN scattering phase as input. The left hand amplitudes are calculated based on chiral Lagrangian with pion and nucleon fields truncated at order q 2 . We review the analytic structures of pion photoproduction amplitudes in Ref. [36] and analyze the singularities to be involved in our calculation. In addition, we find that kinematical singularities in this inelastic process are rather complicated. Cuts coming from kinematical structure depend on how to organize the analytic functions in the amplitudes. These cuts could be in the complex plane and may affect the residues of N * (890). To avoid such complexity, we deform these cuts in a particular way to make sure that they are lying on the real axis, below the pseudo threshold of πN scattering.
We fit the multipole amplitudes E 0+ (S 11 pE and S 11 nE) from Ref. [37] below the ∆(1232) peak in order to determine the subtraction polynomial in the dispersive representation. The residue couplings of N * (890) can be computed by analytic continuation of the amplitude to second sheet, in which the PKU representation of πN S matrix is employed. We compare the residues of N * (890) extracted from multipole amplitudes with the ones of N * (1535) obtained in Ref. [38] to see if we could learn from N * (1535) about the properties of N * (890) and get some information of structures by analogy with the analysis of N * (1535).
The structure of this paper is organized as follows. In Section 2 we set up the dispersive formalism for γN → πN process. Then the left-hand-cuts are estimated based on chiral perturbation theory in Section 3, and we also make an analysis about the singularities which will appear in this pion photoproduction process. In last two sections, numerical results are carried out and summary is presented.

Dispersive representation
The unitarity relation for partial wave γN → πN amplitude reads where T is pion-nucleon scattering amplitude in S 11 wave. The function ρ(s) is defined by where S(s) = 1 + 2iρ(s)T (s) which is the πN scattering S matrix in single channel case. Apparently, the scattering amplitude M may be separated into two parts, i.e., M = M R + M L . The former part M R only contains the right hand cut (RHC) starting at s R , while the latter part M L is free of RHC singularity. Substituting M = M R + M L into Eq. (3), one gets For convenience, the abbreviations M ± (s) = lim →0 M(s ± i ) have been used. To proceed, we introduce a helper function D(s) which is analytic throughout the complex s plane but encounters the same unitarity singularity as M(s). Namely, it satisfies the same unitarity condition as M(s) along the unitary cut: By expressing the S matrix in Eq. (4) by D(s), it is ready to obtain the following spectral function Straightforwardly, a dispersive representation for M R can be written down, where n means proper subtraction order and P(s) is subtraction polynomial. Eventually, Thus, the pion photoproduction amplitude M(s) is determined up to a polynomial, once D(s) and M L (s) are known. Based on the unitarity condition in Eq. (5), one can write a spectral representation for the auxiliary function D(s), The above representation yields an integral equation for D(s), which has the so-called Omnés solution [39] withP standing for zero points in complex plane and δ(s) being the elastic πN phase shift, in accordance with the Watson final state interaction (FSI) theorem [40].
3 Estimate on the left-hand-cut contribution in ChPT 3.1 Basics of single one-pion photoproduction off the nucleon Single one-pion photoproduction off the nucleon (γN -1π) is the process as described by where a is the isospin index of the pion and momenta of the particles are indicated in the parentheses. The isospin structure of the scattering amplitude can be written as where τ a (a = 1, 2, 3) are Pauli matrices in isospin space. Amplitudes with definite isospin I = 1 2 , 3 2 can be obtained from M ± and M 0 via 1 2 The isospin amplitudes M I with either I = 1 2 , 3 2 or I = ±, 0 can be further decomposed in terms of four independent Lorentz operators as with Note that the operators L i µ obey Ward identity [1]. Here µ is the polarization vector of the photon, u(p) andū(p ) are the spinors of the nucleons.

Calculation of chiral amplitudes at tree level
The effective Lagrangian for our calculation of the chiral amplitude up to O(p 2 ) reads (18) with the superscripts referring to chiral orders. The terms in the above equation are given by [42] L (1) 1 The convention in Ref. [41] is adopted for the CG coefficients and for the physical pion states we use π + = − 1 √ 2 (π 1 − iπ 2 ) and π − = 1 √ 2 (π 1 + iπ 2 ). 2 M and P are actually vectors with two components in isospin space of I = 1 2 channel due to target asymmetry caused by electromagnetic interaction.
where m, g and F are nucleon mass, nucleon axial coupling and pion decay constant in the chiral limit, in order. Given our working accuracy, they are set equal to their physical counterparts, m N , g A and F π , the physical nucleon mass, physical axial charge and pion decay constants. Namely, m = m N , g = g A and F = F π . Here c 6 and c 7 are O(p 2 ) low energy constants (LECs) which are known parameters to be determined by experimental data; See Ref. [42] for definitions of the chiral blocks.
The relevant pieces extracted from the expanded form of the Lagrangians in Eq. (19) are Tree-level Feynman diagrams up to O(q 2 ) are displayed in Figures 1 and 2.  The full amplitude of reads and where superscript stands for chiral order. Now those invariant scalar functions can be extracted from the above amplitudes:

Partial wave projection
It is convenient to perform partial wave projection using the helicity formalism proposed in Ref. [43]. To that end, one can substitute the photon polarization vector µ (q), the nucleon spinors u(p) andū(p ) in Eq. (16) by their helicity eigenstates µ (q, λ 2 ), u(p, λ 1 ) andū(p , λ 3 ) in the center of mass frame 3 , where λ i (i = 1, 2, 3) stand for helicity quantum numbers of initial nucleon, photon and final nucleon, in order. For each set of helicity quantum numbers, denoted by H s ≡ {λ 1 λ 2 λ 3 }, there is a helicity amplitude M I Hs , which can be expanded as 4 By imposing the orthonormal properties of the d J functions, the partial wave helicity amplitudes M IJ Hs (s) in the above equation may be projected, i.e.
The partial wave amplitude with I = 1 2 , J = 1 2 and L = 0 (denoted by S 11 in L 2I2J convention) is obtained via which carry certain parity 5 and the helicity indices λ i = ± 1 2 or ±1 are abbreviated by ±.

The analytic structure of partial wave amplitudes
To illustrate the analytic structure of the partial wave amplitudes, we rewrite the partial wave projection formula in Eq. (29) in the following form where the invariant amplitude M I Hs has replaced by its Lorentz-decomposed expression given in Eq. (16). Furthermore, the scalar functions (G J Hs ) i (i = 1, · · · 4) are defined by where L i µ can be found in Eq. (17). In what follows, we proceed to discuss the analytic structure with the help of Eq. (31). Note here that the Mandelstam variable t is related to the cosine of the scattering angle θ via 3 The spinor satisfies Σ · p | p| u ± (p) = ±u ± (p), and polarization vector satisfies ± (q) = 1 √ 2 ( 1 (q) ± i 2 (q)). 4 It is worth stressing that there are in total 8 helicity amplitudes, nevertheless, only 4 of them are independent thanks to symmetries under parity and time reversion transformation. 5 The positive direction particle is the direction of nucleon and the negative direction state is defined through |−pz, λ = e −iπJz e −iπJy |pz, λ , which differs in a phase with the case in Ref. [43].
On the one hand, It should be emphasized that the functions (G J Hs ) i=...ts,4 rely merely on the kinematical structures of the scattering amplitudes, regardless of the dynamics of the system under consideration. Therefore, they are model-independent and can be calculated straightforwardly for any partial wave quantum numbers of J. In Appendix A, for J = 1/2 and H s ={++-,+++}, all the explicit expressions of (G J Hs ) i are listed for the sake of easy reference. It can be observed that (G J= 1 2 Hs ) i in S 11 channel are just polynomials of t.
On the other hand, information on the dynamics are completely encoded in the scalar amplitudes A I i (s, t). In our tree-level ChPT calculation, they are represented by the results shown in Subsection 3.2, which are comprised of contact, t-channel pion-pole, s-and u-channel nucleonexchange contributions. The contact and s-channel nucleon exchange terms are polynomials of t, while t-and u-channel pole terms 6 can be unified to a single type, 1/(t − c), with c a function of s. Restricted to our tree-level calculation and with the above discussions, one can conclude there exist only one master integral: All other integrals are either trivial in the sense that they are integrations over polynomials of t, being able to be reduced to the above integral by making use of the identity t n t−c = t n−1 + ct n−1 t−c with n a positive integer. In our current case, the constant c has three options, i.e., c ∈ {m 2 π , s − m 2 N −m 2 π , 2s−2m 2 N −m 2 π }, which result in three typical logarithms D i (s) after applying Eq. (34). We refer the readers to Appendix A for their explicit expressions. For the D i (s) except D 3 (s), which comes from kinematical decomposition, it is worth to reminding ourselves that those logarithms are stemmed from the dynamics term 1/(t − c), while their composite arguments could be square root functions originated from the kinematical limits of the integrations. The logarithms and square root functions give rise to the partial-wave singularities to be discussed in the following subsections.

Dynamical singularities
The generic dynamical singularities of partial-wave photoproduction amplitude have been discussed in detail in Ref. [36]. All possible singularities are displayed in Fig. 3 and are briefly illustrated in the following.
• unitarity cut: s ∈ [s R , ∞) on account of the s-channel continuous spectrum.
• Discrete term: located at s = m 2 N ≡ s N and induced by the t-channel single pion exchange as well as the u-channel single nucleon exchange. 8 .
Let us come back to our special case under consideration. Since the continuous spectrums are absent for a tree-level calculation, we meet only with the dynamical singularities of the trivial cut and the discrete term.

Kinematical singularities
Aside from the above-mentioned dynamical singularities, there exist additional kinematical singularities for an inelastic scattering process with spinors. The kinematical singularities are caused by the square-root and/or logarithmic functions appearing in the partial wave amplitudes. Kinematical cuts are introduced when the arguments of those two kinds of functions are negative. All the involved arguments together with their corresponding negative domains are listed in Table 1.
It should be pointed out that how these functions are organized in the way that does not affect the value in the physical region but may affect the values in complex plane. Here we give an example to illustrate it: (s − s R ) (s − s L ) Case: There are two cuts. One goes from s L to s R and the other is an infinitely-long line, which is perpendicular to the real axis and passes the midpoint of s L and s R .
There is just one cut stretching from s L to s R with the cuts below s L cancelling each other.
Meanwhile the values in the physical region in the above two cases are the same. In practice, we choose to expand the root functions in terms of power series and then continue them to full complex plane. In this way, all the kinematical singularities represent themselves as cuts lying on real axis. And the logarithm functions in form of ln a b , whose arguments contain root functions, are recast to lna − lnb in order to avoid circular cut in complex plane.
For the S 11 channel, the cut between s L and s R disappears since M As the result of kinematical singularities, we should include s channel and contact diagrams besides t and u channel resonance exchange in the estimation of M L at tree level.

Numerical results and discussions
We are now in the position to compare the dispersive representation of photoproduction amplitude given in Eq. (8) with experimental multipole amplitude data from Ref. [37] in S 11 channel. Based on our fitting results, the couplings of N * (890) to γN and πN can be extracted. . Secondly, we setP(s) = 1 and compute D(s) by using the S 11 -wave phase shift extracted from the πN S matrix given in Ref. [18]. Two solutions of the πN S matrix are adopted: one corresponding to s c = −1 GeV 2 and the other to s c = −9 GeV 2 with s c being a cut off parameter therein. Note that it should be a good approximation for a single-channel case that the integrations in Eqs. (10) and (8) are performed up to 2.095GeV 2 rather than to infinity. Lastly, the constants in the P are left as fitting parameters. 9 Here we only consider two fit cases: Fit I with P(s) = a and Fit II P(s) = a + b s while subtraction points are set to be zero.

The fitting procedure
We perform fit to the data points on the multipole amplitudes 10 , which are traditionally denoted by S 11 with suffixes of target type (n or p) and electromagnetic transition (E:electric, M :magnetic), from πN threshold to 1.440 GeV 2 just below the appearance of ∆(1232). The fit results of p target and n target are displayed in Figures 5 and 6, respectively. For comparison, in Fig. 5 and Fig. 6, we also show the O(q 2 ) chiral results of the real parts of multipole amplitudes. As expected, the chiral results only describe the data very well at low energies close to threshold. The values of the fit parameters are collected in Table 2. For Fit I, our results are in good agreement with the experimental data in that sense that the averaged chisqs are close to one, χ 2 /d.o.f = 1.58 for the p target and χ 2 /d.o.f = 1.22 for the n target. Compared to Fit I, the qualities of Fit II are improved, which is under our expectation since one more free parameter is involved in the fit procedure. However, the fitting parameters 9P can always be chosen to be 1 in Eq. (6). 10 The relation between multipole amplitudes and our amplitudes can be established through traditional CGLN convention, which can be found in Appendix B.

Analytic continuation and extraction of the N * (890) couplings
In the above subsection, all the involved parameters in the dispersive S-wave photoproduction amplitude M(s) have been determined. Since N * (890), as a crazy resonance, is located in the second Riemann sheet (RS), one needs to perform analytic continuation in order to extract its couplings to γN and πN systems.
The amplitude on the second RS can be deduced via where M(s) is the partial-wave photoproduction amplitude given in Eq. (8) and S(s) corresponds to the S matrix of πN scattering with same quantum numbers as M(s). If there exists a second RS pole located at z R , the S matrix can be approximated by  in the vicinity of z II . Thus, On the other hand, the couplings of this second RS pole to the γN and πN systems are defined as the residue via with g γ and g π denoting the γN and πN couplings, respectively. Compared to Eq. (37), one obtains The πN coupling can also be extracted from elastic πN scattering, i.e., where T is the corresponding partial-wave πN scattering amplitude. Now we proceed with the numerical calculation of the couplings of N * (890). According to Eqs. (39) and (67), the pion photoproduction N * (890) residue couplings, i.e. g γ g π , can be extracted from multipole amplitudes. In the meantime, g 2 π can be computed by using Eq. (40), which was already done in Ref. [18]. Results of the couplings are listed in Table 3. The results based on Fit II are also shown to check the stability of the obtained values. We employed two solutions of the pole position of the N * (890), √ s = 0.882 − 0.190i corresponding to the cutoff s c = −1 GeV and √ s = 0.960 − 0.192i to s c = −9 GeV; See Ref. [18] for detailed explanation. Table 3: Results of g γ g π and g 2 π . Pole position, moduli and phase are in GeV, 10 −2 × GeV 2 and degree, in order. g 2 π are the same for p target and n target due to the isospin symmetry. In extraction of g γ g π and g 2 π of N * (890), z R is treated as N * (890) pole position in s plane, M(z R ) can be obtained from dispersion relation in Eq. (8) once P is determined, T (z R ) can be obtained through S(z R ) = 1 + 2iρ πN T = 0 and 1 S (z R ) is just the residue of S II from Ref. [18]. But in order to compare the results of N * (1535), which are extracted directly from multipole amplitudes parameterized in √ s plane in Ref. [38], the conventions should be consistent. In S 11 channel following equation can be used to translate these residues from different conventions into residues directly extracted from multipole amplitudes in s plane.
where R stands for N * (890) or N * (1535). In S 11 pE the moduli of residue is 2.41mfm · GeV 2 with phase 120°, meanwhile the magnitude of N * (1535) residue coupling from Ref. [38] is about 0.736mfm · GeV 2 and phase is −27°. One can see the magnitude of the N * (890) residue is larger than that of the N * (1535) residue. The |g 2 π | of N * (890) is 0.2GeV 2 , and the one of N * (1535), which is obtained by the value in Ref. [44], is 0.08GeV 2 . The g 2 π of these two resonances may account for part of the reason why N * (890) photoproduction residue is large, and using above results g γ of these two resonances can be obtained. The |g γ | of N * (890) is 0.032GeV meanwhile the one of N * (1535) is 0.024GeV and one can see the magnitudes are almost the same. One should notice that the results of n target are quiet unstable. The fact that data points are few and they have large error bars may count for the main reason.
We can also calculate the decay amplitudes A 1 2 at the N * (890) pole position, which is related to the coupling g γ , using the formula given in Ref. [38]: where q r is modulus of the photon momentum calculated at the resonance pole position. Furthermore, we can obtain the partial widths of N * (890) → γN channel at the pole by following formula, which is from Ref. [45] and converted to our convention.
where z R is treated as N * (890) pole position. The values of the decay amplitudes A 1 2 and the partial decay width at the pole Γ γN are collected in Table 4. The |A 1 2 | of N * (890) is larger than the one of N * (1535), which is 0.074GeV − 1 2 with phase being −17°in S 11 pE from Ref. [38] but the decay widths at the pole are almost the same regardless of the instability of n target results.

Summary
In this paper, we have performed a careful dispersive analysis of the process of single pion photon production off the nucleon, in the S 11 wave of the final pion-nucleon system. In such a dispersive representation, the right-hand cut contribution can be related to an Omnés solution, which takes the elastic πN phase shifts as inputs, and hence is known up to a polynomial. On the other hand, we estimate the left-hand cut contribution by making use of the O(q 2 ) tree amplitudes taken from chiral perturbation theory. A detailed discussion on how to establish a proper analytic structure of the partial-wave pion photon production amplitude is also presented for easy reference in future. To pin down the free parameters in the dispersive amplitude, we perform fits to experimental data of multipole amplitudes in the channels indicated by S 11 pE and S 11 nE for the energies ranging from πN threshold to 1.440 GeV 2 .
It is found that the experimental data can be well described by the dispersive amplitude with only one free subtraction parameter. We then continue the dispersive amplitude to the second Riemann sheet for the purpose of being able to extract the couplings of N * to the γN and πN systems, which are denoted by g γ and g π , respectively. Based on the obtained value of g γ g π , the modulus of the corresponding residue of the multipole amplitude (S 11pE ) at the N * (890) pole position turns out to be 2.41mfm · GeV 2 , which is much larger than the modulus of the residue of N * (1535), i.e. 0.736mfm · GeV 2 [38]. That means the strength of the interaction of N * (890) with πN system is stronger, compared to the one regarding N * (1535). It is physically reasonable and within expectation, since N * (890) is supposed to be composed of πN system while N * (1535) has tiny coupling with πN as it is well-known. The results provides us further evidence of the existence of N * (890). As byproducts, the decay amplitude and the decay width at the N * (890) pole position A h and the Γ γN are obtained for future reference.

B CGLN Amplitudes
Traditional pion photoproduction partial wave analysis is in CGLN amplitudes (F) with where χ i(f ) are Pauli spinor and F = i σ · F 1 + σ · q σ · ( q × ) F 2 + i ( σ · q) q · F 3 + i q · σ q · F 4 , where there are four independent amplitudes. The connection of our scattering amplitudes to F can be obtained: where subscript f, i means initial and final states are substituted into Eq. 57 and we will omit it in the following discussion. Further, the partial wave amplitude F J is defined in Ref. [3]: where ± mean the final nucleon helicity and λ r = 1 2 or 3 2 , which is the moduli of initial helicity. Also, definite parity amplitudes can be obtained: is conventional multipole amplitude with 0 and + refers to S wave and minus parity respectively.