Two-Photon-Exchange effects in $ep\rightarrow en\pi^+$ and their corrections to separated cross sections $\sigma_{\textrm{L,T,LT,TT}}$ at small $-t$

In this work, the two-photon-exchange (TPE) effects in $ep\rightarrow en\pi^+$ at small $-t$ are discussed within a hadronic model. Under the pion dominance approximation the TPE contribution to the amplitude can be described by a scalar function in the limit $m_e\rightarrow 0$. The TPE contributions to the amplitude and the unpolarized differential cross section are both estimated when only the elastic intermediate state is considered. We find that the TPE corrections to the unpolarized differential cross section are about from $-4\%$ to $-20\%$ at $Q^2=1$--$ 1.6$ GeV$^2$. After considering the TPE corrections to the experimental data sets of unpolarized differential cross section, we analyze the TPE corrections to the separated cross sections $\sigma_{\textrm{L,T,LT,TT}}$. We find that the TPE corrections (at $Q^2=1$--$1.6$ GeV$^2$) to $\sigma_{\textrm{L}}$ are about from $-10\%$ to $-30\%$, to $\sigma_{\textrm{T}}$ are about $20\%$, and to $\sigma_{\textrm{LT,TT}}$ are much larger. By these analysis, we conclude that the TPE contributions in $ep\rightarrow en\pi^+$ at small $-t$ are important to extract the separated cross sections $\sigma_{\textrm{L,T,LT,TT}}$ and the electromagnetic magnetic form factor of $\pi^+$ in the experimental analysis.


I. INTRODUCTION
In the last two decades, the two-photon-exchange (TPE) effects in ep → ep have attracted much interest due to their importance in the extraction of the electromagnetic (EM) form factors of protons. Many model-dependent methods have been used to estimate the TPE contributions in ep → ep such as the hadronic model [1], GPD method [2], perturbative QCD (pQCD) calculation [3], dispersion-relation approach [4,5], SCEF * E-mail: zhouhq@seu.edu.cn method [6], and phenomenological parametrization [7]. Among all these methods, the dispersion-relation approach for ep scattering gives the most reliable results in the region with medium momentum transfer, and the cost is that it needs to continue the physical quantity analytically into the unphysical region and take some experimental data as inputs to fix the subtraction constant [5]. Furthermore, the difference between the dispersion-relation approach and the hadronic model can be expressed as a polynomial function on the squared center-of-mass energy. In some special cases, the two methods give the same results.
Due to the important contributions of the TPE corrections in ep → ep, similar TPE corrections in e + e − → pp [8], eπ → eπ [9], ep → eNπ [10], µp → µp [11], and ep → e∆ → epπ 0 [12] are studied, aiming at the precise extraction from the experimental data of the EM form factor of protons in the timelike region, the EM form factor of pions in the spacelike region, and the EM transition form factors of γ * N∆ in the spacelike region.
Experimentally, the extraction of the EM form factor of pion via eπ → eπ is limited at very small Q 2 with Q 2 ≡ −q 2 and with q being the four-momentum transfer because there is no free pion target. The electromagnetic production of pions in ep → enπ + is usually used to extract the EM form factor of pions [13][14][15][16][17]. It is a natural question that how large the TPE contributions in this process and how large their corrections to the extracted EM form factor of pion are. In this work, we estimate the TPE contributions in this process within the hadronic model and analyze the TPE corrections to the separated cross sections which are used to determine the EM form factor of pions.
We organize the paper as follows: In Sec. II we describe the basic formulas of our calculation under the pion-dominance approximation, in Sec. III we express the physical amplitude as a sum of two invariant amplitudes and discuss the infrared (IR) property of the TPE amplitude, in Sec. IV we express the unpolarized differential cross section by the coefficients of the invariant amplitudes, in Sec. V we present the numerical results for the TPE corrections to the amplitude, to the unpolarized differential cross section and to the separated cross sections σ L,T,LT,TT . A detailed discussion on these numerical results and the conclusion from these numerical results are also given.

II. BASIC FORMULA FOR ep → enπ +
Under the one-photon exchange (OPE) approximation, the ep → enπ + process can be separated into two subprocesses, e → eγ * and γ * p → nπ + , shown in Fig. 1 where we label the momenta of initial electron, initial proton, final electron, final pion, and final neutron as p 1 −p 5 , respectively, and for simplicity we define the following five independent Lorentz-invariant variables: The dynamics of the subprocess e → eγ * is clear while the dynamics of the subprocess γ * p → nπ + is very complex. In this work we limit our discussion on the momenta region with Q 2 small, −t ≈ 0 and W far away from the resonances. In this region, one can estimate the subprocess γ * p → nπ + in the hadronic level as an approximation and can expect that the π exchange diagram showed in Fig. 2(a) may give the most important contribution due to the large enhancement from the pion propagator. In Fig. 2, the schannel diagram is also presented to keep the gauge invariance. The sum of the t-channel and s-channel is gauge invariant. The contribution from the u-channel diagram with the neutron as the intermediate state is also gauge invariant independently and is small at low −t since the electric charge of neutron is zero and it couples to photon via F µν . We neglect this contribution in the following and only consider the sum of the t-channel and s-channel.
The unpolarized differential cross section at small −t is usually used to determine the µ ν e(p 1 ) EM form factor of pions. Different from eπ + → eπ + process where the EM form factor of pions can be extracted from the total cross section directly, the EM form factor can not be extracted directly from the total unpolarized cross section of ep → enπ + and should be extracted via the φ π -dependence of the unpolarized differential cross section. The TPE contributions may change the angle dependence of the unpolarized differential cross section and then effect the extraction of the EM form factor in an indirect and nontrivial way.
When we discuss the TPE effects, the contributions from the corresponding TPE diagrams showed in Fig. 3 should be considered where the TPE contributions from GeV 2 the contributions from the elastic state and the inelastic states such as 2π(ρ, σ), 3π between the two photons may both give the vital contributions. In this work, as a first step we limit our discussion on the contributions from the elastic state since naively the transition form factors γ * πρ, γ * πσ are much smaller than the EM form factor γ * ππ when When taking Feynman gauge and limiting the discussion on the small −t, the contri-  Under the above approximation, the ratio of the TPE contributions to the OPE contributions is not dependent on the interactions between pions, protons, and neutrons since the relative TPE corrections are only dependent on the subprocess eπ * → eπ. In the practical calculation we simply take the interaction between pions, protons, and neutrons as iso-scalar type. We use the interactions constructed in Ref. [18] to describe the interactions between the pion and the photon.
Taking the Feynman gauge, one has with and where e = −|e|, k ≡ p f − p i , and f (k 2 ) describes the EM form factor of pion F π (k 2 ) and has the relation In Ref. [5], the authors discussed and compared the TPE contributions calculated from the hadronic model and the dispersion-relation in detail in the ep case. Their discussions hint that the two methods give the same results when the interaction between the point-like particles is traditionally renormalized and give different results when the interaction between the point-like particles is traditionally nonrenormalized, respectively. In ep scattering, the pure electric interaction belongs to the former, and the pure magnetic interaction belong to the latter. This property is natural since a new contact interaction should be introduced when the interaction leads to a traditional nonrenormalized UV divergence. Such contact interaction includes an undetermined finite contribution in polynomial form and should be also considered in the hadronic model. This reason can also explain the behavior of the difference between the dispersion-relation approach and the hadronic model in the ep case. In the dispersion-relation approach, such finite contribution is fixed by the experimental data or by the asymptotic behavior in high energy predicted by other methods. In the eπ-interaction case, the situation is different from the ep case since now the interaction for a point-like charged pseudoscalar particle is renormalized. On the other hand the gauge invariance results in a contact term. Such a contact term gives a pure real contribution and does not appear in the ep case. These two properties prompts us to use the dynamical hadronic model to calculate the TPE contribution in ep → enπ + when only the elastic state is included.

III. THE IR DIVERGENCE OF THE AMPLITUDE
Generally, the amplitudes given in Eq. (1) can be expressed in the following simple form: with with Γ 5 ≡ g πN N γ 5 being the vertex of the πNN isoscalar interaction. The coefficients can be easily gotten and are expressed as with α e ≡ e 2 /4π. even the form factor f (k 2 ) is taken as a simple monopole form. A general property is that there is only IR divergence in c 1 . The detailed analysis shows that the IR divergence comes from the diagrams in Figs. 3(a) and 3(b), and the corresponding IR divergence [19] in c and where a ≡ 2p 14 + Q 2 + t − m 2 π , λ is the introduced infinitesimal mass of photon, µ c is an energy scale to keep the result dimensionless, and the properties that a > 0, p 14 > 0 in the physical region are used in the expansions on m e . The results show that the full IR divergence is free from m e . The above IR divergence should be included in any experimental data analysis when the real radiative corrections are included.
In ep → ep process, the contribution from the TPE diagrams under the soft momentum approximation which includes the IR divergence is usually estimated via the classical Mo-Tsai's soft approximation [20] in the experimental analysis. In this approximation the soft TPE contribution is calculated by taking the momentum of one photon as zero both in the numerator and one of the denominators of the propagators. In Ref. [21], Maximon and Tjon suggest another approximation to estimate the soft TPE contribution. In their estimation, the soft contribution is calculated by taking momentum of one photon as zero only in the numerator. The analytical expressions in the latter method can be get in ep → ep or eπ → eπ. In the ep → enπ + process, the intermediate pion is off-shell, which introduces an additional variable t, the analytical expressions under the above soft approximation are very complex and we do not go to show them. To show the TPE corrections from the finite momentum transform, we define the IR-free TPE contributions as follows: where the indexes Tsai and Tjon refer to the corresponding contributions by the Mo-Tsai's method and Maximon-Tjon's method, respectively. In the practical calculation with the experimental momenta as inputs, we find that the results c

IV. THE UNPOLARIZED CROSS SECTION
Using the general expression of the amplitudes (5) and (6), one can get the expressions of the unpolarized differential scattering cross sections as follows: where E e ′ is the energy of final electron in the laboratory frame, Ω e ′ is the angle of the final electron in the laboratory frame, Ω π is the angle of the pion in the center frame of the pion and final proton, and we have taken c can be neglected when making the approximation m e = 0.
The unpolarized cross sections above can be written as where X refers to 1γ or 2γ, J(t, φ π → Ω π ) = dt sin θπdθπ , and Γ ν = αe 1−ǫ is the virtual photon flux factor with E e being the energy of initial electron in the laboratory frame, m p the mass of the proton, and ǫ the longitudinal polarization of the virtual photon, whose definition can be found in the appendix. According to the dependence on φ π and ǫ, the OPE cross section d 2 σ 1γ un /dtdφ π can be separated into four terms as follows: 2π where the four separated cross sections dσ 1γ L,T,LT,TT /dt shortly written as σ 1γ L,T,LT,TT are only depend on Q 2 , W , and θ π .
When one takes m e = 0 in Eq. (12), one can see that the TPE cross section d 2 σ 2γ un /dtdφ π has the same form with OPE cross section. After using the variables Q 2 , W, ǫ, θ π , and φ π to express the cross section one can see that d 2 σ 2γ un /dtdφ π has the same φ π dependence with d 2 σ 1γ un /dtdφ π and can also be separated into the same form as Eq. (14) but now the four corresponding separated cross sections σ 2γ L,T,LT,TT are dependent on Q 2 , W , θ π , and ǫ.

V. THE NUMERICAL RESULTS AND DISCUSSION
In the practical calculation, we take the input form factor F π (q 2 ) as the monopole from which is used in [22] and [18].
with Λ = 0.77 GeV. We use the packages FEYNCALC [23] and LOOPTOOLS [24]  ] is presented in Fig. 4 where the left and right panels are corresponding to Q 2 = 1 GeV 2 and Q 2 = 1.6 GeV 2 , respec-tively. The (blue) dashed curves and the (olive) dash-dotted curves refer to the results at φ π = π/6 and φ π = π/3 with ǫ = 0.65 or 0.63, the (black) solid curves and the (red) dotted curves are associated with ǫ = 0.33 or 0.27. The results clearly show that the absolute magnitude of TPE corrections Re[c 1,Tsai /c 1 ] at φ π = π/6 increase when −t increases while the corrections at φ π = π/3 are not sensitive to −t. Another interesting property is that the TPE corrections at very small −t are not sensitive to φ π while the TPE corrections at large −t are sensitive to φ π .
At φ π = π/6, one can see that the TPE corrections at small ǫ range about from −4% to −6% at small −t and reach about from −7% to −10% at large −t at Q 2 = 1.0 and 1.6 GeV 2 , respectively. The magnitude at small −t and small ǫ is similar to the TPE corrections in eπ → eπ. These properties suggest that the −t dependence of the TPE corrections at small φ π is relatively important.
The −t dependence of the imaginary parts of the TPE corrections, Im[c is presented in Fig. 6 where θ π is taken as π/18, π/12 and −t is limited within the experimental data sets. The (black) solid curves and the (red) dotted curves refer to the results with θ π = π/18 at Q 2 = 1 ] reach about −9% and −12% at Q 2 = 1GeV 2 and 1.6 GeV 2 , respectively. The results in the right panel clearly show that the TPE corrections to the imaginary part are not sensitive to ǫ. vs ǫ at fixed Q 2 , W, θ π , and φ π . The left panel is the result for the real part and the right panel is the result for the imaginary part.  vs −t at fixed Q 2 , W, ǫ, and φ π where the index X refers to fin, Tsai, and Tjon, respectively.

C. Two-photon-exchange corrections to unpolarized differential cross section
To show the TPE corrections to the unpolarized differential scattering cross section, where we have used the property that c vs −t at fixed Q 2 , W, ǫ, and φ π where the index X refers to fin, Tsai, and Tjon, respectively. vs ǫ at fixed Q 2 , W, θ π , and φ π where the index X refers to fin, Tsai, and Tjon, respectively.
GeV 2 can reach about −10%, which is not small. Furthermore, the TPE corrections are sensitive to ǫ, φ π , and −t or θ π when Q 2 and W are fixed. Generally one can expect that these two properties may result in nontrivial effects when extracting some physical quantities from the angle dependence of the differential cross section.
When comparing with the TPE corrections in e + e − → pp [8], eπ → eπ [9], µp → µp [11], and ep → e∆ → epπ 0 at W = 1.232 GeV [12], we can see that the absolute magnitude of the TPE corrections in ep → enπ + are much larger. This property can be understood by the fact that the intermediate pion with four-momentum p t ≡ p 5 − p 2 is off-shell which is different from the other processes. Naively, if p 2 t = t goes to m 2 π , the TPE corrections to vs ǫ at fixed Q 2 , W, θ π , and φ π where the index X refers to fin, Tsai, and Tjon, respectively.
the coefficients should be same as the TPE corrections in the physical process eπ → eπ. D. Two-photon-exchange corrections to separated cross sections σ L , σ T , σ LT and σ TT Experimentally, the separated cross sections σ L , σ T , σ LT , and σ TT are usually extracted from the original experimental data dσ Ex un /dtdφ π via Eq. (14) and are then used to determine the EM form factor of π + . Since the TPE corrections to the unpolarized cross section are not small and sensitive on the angles, one should be careful in the separation.
In this section, we analyze the TPE corrections to the separated cross sections.
When considering the TPE contribution, one has dσ Ex un dtdφ π = dσ ph,1γ un dtdφ π (1 + δ ph,2γ un ), where dσ Ex un /dtdφ π refers to the experimentally observed cross section, dσ ph,1γ un /dtdφ π refers to the physical cross section via OPE, and δ ph,2γ un refers to the physical TPE correction to the cross section. Since actually we do not known all the dynamics of QCD, the physical The current experimental analysis is based the experimental cross section dσ Ex un /dtdφ π and Eq. (14). After considering the TPE contributions, in principle the analysis should be based on the corrected experimental cross section dσ Ex un /dtdφ π and Eq. (14). The comparison between the results from these two analyses can tell us how large are the TPE corrections to the separated cross sections σ L , σ T , σ LT , and σ TT .  In the practical analysis, we take two data sets named ExA and ExB as inputs to do the analysis. In the data sets ExA, we take the experimental extracted σ ExA L , σ ExA T , σ ExA LT , and σ ExA TT by JLab F π [16] as inputs to get dσ ExA un /dtdφ π at specific ǫ and φ π via Eq. (14). The corresponding values are listed in Table. I. We take ǫ as 0.33, 0.65 at low Q 2 , take ǫ as 0.27, 0.63 at high Q 2 , and take φ π from 5 • to 355 • with ∆φ π = 25 • . In the data sets ExB, we use the experimentally fitted formula [16] to produce dσ ExB un /dtdφ π . For comparison, the corresponding σ ExB L , σ ExB T , σ ExB LT , and σ ExB TT are listed in Table. II. In this data sets,  we take ǫ from 0.33 to 0.65 with ∆ǫ = 0.03 at low Q 2 , take ǫ from 0.27 to 0.63 with ∆ǫ = 0.035 at high Q 2 to produce more data points, and take φ π from 5 • to 355 • with ∆φ π = 25 • .
After getting the data sets dσ ExA,ExB un /dtdφ π , we use the estimated TPE corrections in the corresponding kinematics region to get dσ ExA,ExB un /dtdφ π . Then we use Eq. (14)    and are large and sensitive to the input data sets at large −t. The TPE corrections to σ ExA,ExB LT are always large and even become un-reliable and very sensitive to the input data sets at large −t. The experimentally extracted σ L is usually used to determine the pion form factor F π through the Chew-Low method (based on the Born term model [25]) or the Regge model [26]. Our results show that the relative TPE corrections to σ ExA,ExB L reach about from −10% to −30% at Q 2 = 1-1.6 GeV 2 . This means that the relative Q 2 (GeV 2 ) W (GeV) t(GeV 2 ) σ ExA   TPE corrections to the EM form factor of pions are about on the order of −5% to −15% and should be considered carefully. At high Q 2 , one can expect that the TPE corrections should be much more important and should be considered seriously to extract the EM form factor of pions reliablely.
In summary, in this work the TPE corrections to the amplitude and the unpolarized differential cross section of ep → enπ + are estimated in a hadronic model. The TPE corrections to the extracted four separated cross sections are also analyzed based on the experimental data sets. Our results show that at Q 2 = 1-1.6 GeV 2 , the TPE correction to σ L is about from −10% to −30% and about 20% to σ T .