Two-Photon-Exchange effect in $ep\rightarrow en\pi^+$ at small $-t$ with the hadronic model and dispersion relation approach

In this work, the two-photon-exchange (TPE) effect in $ep\rightarrow en\pi^+$ at small $-t$ is discussed. In the previous work, the TPE contribution with one $\pi$ intermediate state is estimated numerically within a hadronic model under the pion-dominance approximation. Here we extend the discussion to include one $\rho$ intermediate state. The TPE contribution can be described by one scalar function in the limit $m_e\rightarrow 0$, the dispersion relation (DR) satisfied by this scalar function is analysed. The analytic expressions for the imaginary parts of the TPE contributions from one $\pi$ or one $\rho$ intermediate state are given within the hadronic model. Combining these analytic expressions and the DR, the corresponding real parts of the TPE contributions can be estimated easily at any available region. This can help the further experimental analysis to include the TPE contributions in a convenient way. The numeric results show that the TPE correction with one $\rho$ intermediate state is much smaller than that with one $\pi$ intermediate state in the current energy region. These results suggest that the TPE contribution with an elastic state is the main TPE contribution in $ep\rightarrow en\pi^+$ at small $-t$.

Comparing with the proton case, the discussions on how to extract the EM FF of π precisely are relatively fewer. Experimentally, the EM form factor of π is usually extracted via the process ep → enπ + [13][14][15][16][17]. Theoretically, such extraction of pion's FF is much more complex than that of the proton's FFs via the elastic ep scattering. The corresponding theoretical analysis on the experimental data sets should be done more carefully. Up to now, the discussions on the TPE effect in ep → enπ + are limited [18,19]. In the previous work [19], the TPE contributions with an elastic intermediate state are discussed, in this work we extent the discussion to include one ρ meson intermediate state. Furthermore, the DR for the TPE contributions and the analytic expressions for the imaginary parts are both given.
We organize the paper as follows. In Sec. II we describe the basic frame of our discussion under the pion-dominance approximation, in Sec. III we show some analytic properties of the TPE contributions and the DR relation they satisfied. in Sec. IV we present some numerical results for the TPE corrections and give our conclusion.
When go to discuss the TPE effect, the contribution from the corresponding TPE diagram showed in Fig. 2 should be considered.
Physically, the dynamics of the sub-processes γ * p → nπ + and γ * γ * p → nπ + are very complex. At the small energy scale, one can expect that the chiral perturbative theory (ChpT) works well for these two sub-processes. For example, in the leading order of ChpT the Feynman diagrams for γ * p → nπ + can be described by Fig.3 where the notations 0 ○ and 1 ○ refer to the vertexes with corresponding orders.
When the energy scales −t, Q 2 and W increase, one can expect that ChpT is not valid anymore and the contributions beyond ChpT such as the diagrams with ρ meson exchange and with N * intermediate states showed as Fig. 4 should be considered.
When −t is kept as small, W is a little far away from the masses of the narrow resonances  and only Q 2 increases, the contribution from one pion-exchange is still dominant among all these contributions although ChpT is not valid. The reasons are due to two facts: (1) the mass of pion is close to zero which results in a strong enhancement from the pion propagator, (2) the couplings of γ * pp,γ * pN * decrease much fast than the coupling γ * ππ when Q 2 increases. These properties means the pion-dominance is a good approximation when t → 0 and W is a little far away from the narrow resonances. This also greatly simplifies the dynamics of the process γ * γ * p → nπ + in this region. In this work, we limit our discussion on the TPE contributions under this approximation. In the practical calculation, one can combine the contributions beyond the pion-dominance under the OPE approximation and the TPE contributions together since their contributions are independent.
Under the pion-dominance approximation, the corresponding TPE contributions can be described as Fig. 5(a, b, c) where the contributions from Fig. 6(d, e, f ) are neglected since they are much smaller.   In the previous work [19], the TPE contributions from an elastic state π showed in  e(p 1 ) Taking Feynamn gauge, one has and where e = −|e|, k ≡ p f − p i ,q γ,V are the incoming momenta of photon and ρ meson, f (k 2 ) describes the EM form factor of pion F π (k 2 ) and has the relation[20] Here 2γ,π corresponding to Fig. 7(a, b, c) can be found in Ref. [19]. In the practical calculation, one can find that the relative TPE corrections are not dependent on the form of Γ 5 (iso-scalar form or iso-vector form), so we do not present its form here.
Generally, the amplitudes given in Eq. (1) can be written as the following simple form. with The coefficients c (1γ) 1,2 can be easily gotten which are expressed as with α e ≡ e 2 /4π.

III. SOME ANALYTIC PROPERTIES OF THE TPE CONTRIBUTIONS IN
ep → enπ + A. General properties due to the symmetry When taking the limit m e → 0, one has the following exact property due to the symmetry.
Our manifest calculation also shows such property.
In the literature, the approximation m e = 0 is often used before the loop integration since m e is much smaller than the other scales in the experimental region. In the elastic ep scatting and elastic eπ scattering cases, one can find that such approach works well since the full TPE contributions are not dependent on m e at the leading order of m e . In ep → enπ + , we find that such approach is good for c when taking m e → 0 after the loop calculation. Such term means that the usual Taylor series is not valid in the calculation. This is natural since the loop integration and the Taylor series is not commutated in some cases. Our numerical results also show such property and such log enhancement should be dealt carefully.
To keep this term, in the following calculation we at first take m e as non-zero and then expand the results on m e . The packages FEYNCALC [22], PackageX [23] and LOOPTOOL [24] are used in the practical calculations.
Under the pion-dominance approximation, although the cross sections are dependent on five variables but the TPE contributions c (a,b) 1π,1ρ are only dependent on three variables t, Q 2 and ν. Due to the crossing symmetry, one has the following general relation when Q 2 and t are fixed in the physical region: where ν + = ν + i0 + .

B. TPE contributions in the point-like particle case
To show the analytic properties of the TPE contribution in a clear form, at first we take the point-like interaction as example. In this case, one has where we have used the index I to refer to the point-like interaction. The same index is used for other quantities in the following expressions.
After the loop integration, we find the following analytic properties: (1) There are no kinematic poles in c 1π,1ρ on ν are showed as Fig. 9.
(3) The asymptotic behaviors of c I,(a) 1π,1ρ are expressed as follows: and Re[c where µ IR is the IR scale, and The asymptotical behaviors of c 1π,1ρ in the limit ν → −∞ can be got easily via Eq. (9).
Based on the above properties and Eqs. (9,11,12), one can easily check that c I,(a) 1π,1ρ satisfy the once-subtracted DR while c I,(a+b) 1π,1ρ satisfy the following non-subtracted DR: where ν (π) th = m 2 π + 4m e m π − Q 2 − t, ν (ρ) th = 2m 2 ρ − m 2 π + 4m e m ρ − Q 2 − t, and the manifest expressions for Im[c I,(a) 1π,1ρ (ν + , Q 2 , t)] are written as and with By the expressions of these imaginary parts and the DR, one can easily reproduce the real parts of c I,(a+b) 1π,1ρ (ν, Q 2 , t). We want to emphasize a general property that Im[c (a) 1π (ν, Q 2 , t)] has IR divergence and is dependent on the IR scale µ IR . This is natural since the DR Eq.(13) means that 1π (ν, Q 2 , t)] and the former has IR divergence. This property hints that Im[c (a) 1π (ν, Q 2 , t)] can not be determined by experimental data directly. This is very different with the case in the forward angle limit. On the contrary, the contribution Im[c (a) 1ρ (ν, Q 2 , t)] has no IR divergence.

C. TPE contributions with EM FFs
Physically, the EM FFs F γππ and F γπρ are not constants and the momentum dependence of the EM FFs should be considered when Q 2 increases. In the practical calculation, for simplicity the following monopole form FF is used [20,25].
After the loop integration with this FF as inputs, we find the properties on the kinematic poles and the branch cuts of c II, (a,b) 1π,1ρ are the same with those of c I, (a,b) 1π,1ρ . The asymptotic behaviors of c II,(a) 1π,1ρ are expressed as follows: and where a II 1π,1ρ are functions only dependent on m π , m ρ , t, Q 2 and Λ. Comparing the asymptotic behaviors Eqs. (18,19) and Eqs. (11,12), one can find an interesting property: the asymptotic behaviors of c the imaginary parts of the TPE contributions from one π, one ρ intermediate state are expressed as follows: where g ρ,5 = 1 8y 4 Q 2 x 4 (y 3 − m 2 π x 5 ) + Λ 2 (h 2 + m 2 π y 5 ) + 2Λ 4 y 6 − 4Λ 6 (Q 2 + ν) , and with We also want to point out that the contributions c Usually, the experimental quantities Q 2 , W, ǫ, θ π and φ π are chosen as variables to express the differential cross section where ǫ is the virtual photon polarization, θ π and φ π are the angles between the three-momentum of π and the ep scattering plane. Their detailed definitions can be found in the Appendix of Ref. [19]. In the poin-exchange dominance approximation, as discussed above the coefficients of the invariant amplitudes are only dependent on ν, t and Q 2 when taking Q 2 , W, ν, t and s as five independent variables. This property means it is much simpler to show the TPE contributions by choosing the latter quantities as independent variables. In the following, at first we present the numeric results with the latter choice, and then present the numeric results with the experimental variables as inputs.
Another very interesting property is that the TPE corrections are not sensitive on the variable t when ν and Q 2 are fixed.
The results clearly show that the absolute magnitude of TPE corrections Re[c II,(a+b) 1ρ /c (1γ) 1 ] are smaller than 10 −4 at Q 2 = 1 GeV 2 and smaller than 10 −3 at Q 2 = 1.6 GeV 2 . These corrections are much smaller than the results with one π meson intermediate state [19].
In the practical estimation, one can use Eq. (13) and Eq. can get the real parts of the TPE contributions at any available kinematic region easily.
We think these expression can help the further experimental analysis to include the TPE contributions conveniently. On the numerical part, we find the contributions from one ρ intermediate state are much smaller than those from one π intermediate state. This suggests that the estimation only with one π intermediate state can be applied to higher Q 2 and higher ν safely. ] vs. −t at fixed Q 2 , W, ǫ and φ π . This is the result with Q 2 = 1.6 GeV 2 .

V. ACKNOWLEDGMENTS
The author Hai-Qing Zhou would like to thank Hiren. Pate for his kind help in PackageX.