# Exclusive \(\rho ^0\) meson photoproduction with a leading neutron at HERA

## Abstract

A first measurement is presented of exclusive photoproduction of \(\rho ^0\) mesons associated with leading neutrons at HERA. The data were taken with the H1 detector in the years 2006 and 2007 at a centre-of-mass energy of \(\sqrt{s}=319\) GeV and correspond to an integrated luminosity of 1.16 pb\(^{-1}\). The \(\rho ^0\) mesons with transverse momenta \(p_T<1\) GeV are reconstructed from their decays to charged pions, while leading neutrons carrying a large fraction of the incoming proton momentum, \(x_L>0.35\), are detected in the Forward Neutron Calorimeter. The phase space of the measurement is defined by the photon virtuality \(Q^2 < 2\) GeV\(^2\), the total energy of the photon–proton system \(20 < W_{\gamma p}< 100\) GeV and the polar angle of the leading neutron \(\theta _n < 0.75\) mrad. The cross section of the reaction \(\gamma p \rightarrow \rho ^0 n \pi ^+\) is measured as a function of several variables. The data are interpreted in terms of a double peripheral process, involving pion exchange at the proton vertex followed by elastic photoproduction of a \(\rho ^0\) meson on the virtual pion. In the framework of one-pion-exchange dominance the elastic cross section of photon-pion scattering, \(\sigma ^\mathrm{el}(\gamma \pi ^+ \rightarrow \rho ^0\pi ^+)\), is extracted. The value of this cross section indicates significant absorptive corrections for the exclusive reaction \(\gamma p \rightarrow \rho ^0 n \pi ^+\).

## 1 Introduction

Measurements of leading baryon production in high energy particle collisions, i.e. the production of protons and neutrons at very small polar angles with respect to the initial hadron beam direction (forward direction), are important inputs for the theoretical understanding of strong interactions in the soft, non-perturbative regime. In *ep* collisions at HERA, a hard scale may be present in such reactions if the photon virtuality, \(Q^2\), is large, or if objects with high transverse momenta, \(p_T\), are produced in addition to the leading baryon. In such cases the process usually can be factorised into short-distance and long-distance phenomena and perturbative QCD often is applicable for the description of the hard part of the process.

Previous HERA measurements [1, 2, 3, 4, 5, 6, 7] have demonstrated that in the semi-inclusive reaction \( e+p \rightarrow e+n+X\) the production of neutrons carrying a large fraction of the proton beam energy is dominated by the pion exchange process. In this picture a virtual photon, emitted from the beam electron, interacts with a pion from the proton cloud, thus giving access to the \(\gamma ^*\pi \) cross section and, in the deep-inelastic scattering regime, to the pion structure function.

The aim of the present analysis is to measure exclusive \(\rho ^0\) production on virtual pions in the photoproduction regime at HERA and to extract the quasi-elastic \(\gamma \pi \rightarrow \rho ^0\pi \) cross section for the first time. Since no hard scale is present, a phenomenological approach, such as Regge theory [8], is most appropriate to describe the reaction. In the Regge framework such events are explained by the diagram shown in Fig. 1a which involves an exchange of two Regge trajectories in the process \(2 \rightarrow 3\), known as a *Double Peripheral Process* (DPP), or Double-Regge-pole exchange reaction [9, 10, 11]. This process can also be seen as a proton dissociating into \((n,\pi ^+)\) system which scatters elastically on the \(\rho ^0\) via the exchange of the Regge trajectory with the vacuum quantum numbers, called the “Pomeron”.

In the past, similar reactions were studied at lower energies in nucleon–nucleon and meson–nucleon collisions [12, 13, 14, 15, 16, 17]. Most of the experimental properties of these reactions were successfully explained by the generalised Drell–Hiida–Deck model (DHD) [18, 19, 20, 21], in which in addition to the pion exchange (Fig. 1a) two further contributions (Fig. 1b, c) are included. The graphs depicted in Fig. 1b, c give contributions to the total scattering amplitude with similar magnitude but opposite sign [22, 23]. Therefore they largely cancel in most of the phase space, in particular at small momentum transfer squared at the proton vertex, \(t \rightarrow 0\), such that the pion exchange diagram dominates the cross section [21]. One of the specific features observed in these experiments is a characteristic \(t^{\prime }\) dependence at the ‘elastic’ vertex,^{1} with the slope dependent on the mass of the \((n\pi )\) system produced at the other, \(pn\pi ^+\), vertex, and changing in a wide range of approximately \(4<b(m)<22\) GeV\(^{-2}\). The Deck model in its original formulation cannot fully describe such a strong mass-slope correlation and interference between the amplitudes corresponding to the first three graphs in Fig. 1 has to be taken into account to explain the experimental data [24, 25].

*Y*containing a neutron (Fig. 1d). Using the H1 detector capabilities in the forward region such processes can be suppressed to a certain extent. The residual background contribution is estimated from a Monte Carlo model tuned to describe vector meson production in diffractive dissociation at HERA.

The analysis is based on a data sample corresponding to an integrated luminosity of 1.16 pb\(^{-1}\) collected with the H1 detector in the years 2006 and 2007. During this period HERA collided positrons and protons with energies of \(E_e=27.6~\mathrm {GeV}\) and \(E_p=920~\mathrm {GeV}\), respectively, corresponding to a centre-of-mass energy of \(\sqrt{s}=319~\mathrm {GeV}\). The photon virtuality is limited to \(Q^2<2~\mathrm {GeV}^2\) with an average value of \(0.04~\mathrm {GeV}^2\).

## 2 Cross sections definitions

the square of the

*ep*centre-of-mass energy \(s = (P+k)^2\),the modulus of the four-momentum transfer squared at the lepton vertex \(Q^2 = -q^2 = -(k-k')^2\),

the inelasticity \(y=(q \cdot P)/(k \cdot P)\),

the square of the \(\gamma p\) centre-of-mass energy \(W_{\gamma p}^2 = (q+P)^2 \simeq ys-Q^2\),

the fraction of the incoming proton beam energy carried by the leading neutron \(x_L = (q \cdot N)/(q \cdot P) \simeq E_n/E_p\),

the four-momentum transfer squared at the proton vertex \(t = (P-N)^2 \simeq -\frac{p_{T,n}^2}{x_L} - \frac{(1-x_L)(m_n^2-m_p^2x_L)}{x_L}\), and

the four-momentum transfer squared at the photon vertex \(t' = (q-V)^2\).

Experimentally, the kinematic variables at the photon vertex (the mass \(M_{\rho }\), the pseudorapidity \(\eta _{\rho }\) and the transverse momentum squared \(p_{T,\rho }^2\) of the \(\rho ^0\) meson) are determined from the \(\rho ^0\) decay pions, while those at the proton vertex (\(x_L\) and \(p_{T,n}^2\)) are deduced from the measured energy and scattering angle of the leading neutron.

In the limit of photoproduction, i.e. \(Q^2 \rightarrow 0\), the beam positron is scattered at small angles and escapes detection. In this regime the square of the \(\gamma p\) centre-of-mass energy can be reconstructed via the variable \(W_{\gamma p,rec}^2 = s \, y^{rec}\), where \(y^{rec}\) is the reconstructed inelasticity, measured as \(y^{rec} = (E_{\rho }-p_{z,\rho })/(2E_e)\). Here, \(E_{\rho }\) and \(p_{z,\rho }\) denote the reconstructed energy and the momentum along the proton beam direction (*z*-axis) of the \(\rho ^0\) meson and \(E_e\) is the positron beam energy. The variable \(t'\) can be estimated from the transverse momentum of the \(\rho ^0\) meson in the laboratory frame via the observable \(t'_{rec} = -p_{T,\rho }^2\) to a very good approximation.^{2}

## 3 Experimental procedure and data analysis

### 3.1 H1 detector

A detailed description of the H1 detector can be found elsewhere [41,42]. Only those components relevant for the present analysis are described here. The origin of the right-handed H1 coordinate system is the nominal *ep* interaction point. The direction of the proton beam defines the positive *z*-axis; the polar angle \(\theta \) is measured with respect to this axis. Transverse momenta are measured in the *x*–*y* plane. The pseudorapidity is defined by \(\eta = -\ln {[\tan (\theta /2)]}\) and is measured in the laboratory frame.

The central region of the detector is equipped with a tracking system. It included a set of two large coaxial cylindrical drift chambers (CJC), interleaved by a *z* chamber, and the central silicon tracker (CST) [42] operated in a solenoidal magnetic field of \(1.16~\mathrm{T}\). This provides a measurement of the transverse momentum of charged particles with resolution \(\sigma (p_T) / p_T \simeq 0.002 \ p_T \oplus 0.015\) (\(p_T\) measured in GeV), for particles emitted from the nominal interaction point with polar angle \( 20^{\circ } \le \theta \le 160^{\circ }\). The interaction vertex is reconstructed from the tracks. The five central inner proportional chambers (CIP) [43] are located between the inner CJC and the CST. The CIP has an angular acceptance in the range \(10^{\circ }<\theta <170^{\circ }\). The forward tracking detector is used to supplement track reconstruction in the region \(7^\circ <\theta <30^\circ \) and improves the hadronic final state reconstruction of forward going low momentum particles.

The tracking system is surrounded by a finely segmented liquid argon (LAr) calorimeter, which covers the polar angle range \(4^\circ <\theta <154^\circ \) with full azimuthal acceptance. The LAr calorimeter is used to measure the scattered electron and to reconstruct the energy of the hadronic final state. The backward region (\(153^\circ <\theta <177.8^\circ \)) is covered by a lead/scintillating-fibre calorimeter (SpaCal) [44]; its main purpose is the detection of scattered positrons.

A set of “forward detectors” is sensitive to the energy flow close to the outgoing proton beam direction. It consists of the forward muon detector (FMD), the Plug calorimeter and the forward tagging system (FTS). The lead–scintillator Plug calorimeter enables energy measurements to be made in the pseudorapidity range \(3.5 < \eta < 5.5\). It is positioned around the beam-pipe at \(z = 4.9\) m. The FMD is a system of six drift chambers which are grouped into two three-layer sections separated by a toroidal magnet. Although the nominal coverage of the FMD is \(1.9 < \eta < 3.7\), particles with pseudorapidity up to \(\eta \simeq 6.5\) can be detected indirectly through their interactions with the beam transport system and detector support structures. The very forward region, \(6.0 < \eta < 7.5\), is covered by an FTS station which is used in this analysis. It consists of scintillator detectors surrounding the beam pipe at \(z=28\) m. The forward detectors together with the LAr calorimeter are used here to suppress inelastic and proton dissociative background by requiring a large rapidity gap (LRG) void of activity between the leading neutron and the pions from the \(\rho ^0\) decay.

The absolute electromagnetic and hadronic energy scales of the FNC are known to 5 and \(2~\%\) precision, respectively [7]. The energy resolution of the FNC calorimeter for electromagnetic showers is \(\sigma (E)/E \approx 20~\%/\sqrt{E~[\mathrm GeV]} \oplus 2~\%\) and for hadronic showers \(\sigma (E)/E \approx 63~\%/\sqrt{E~[\mathrm GeV]} \oplus 3~\%\), as determined in test beam measurements. The spatial resolution is \(\sigma (x,y)\approx 10~\hbox {cm}/\sqrt{E~[\mathrm GeV]} \oplus 0.6~\hbox {cm}\) for hadronic showers starting in the Main Calorimeter. A better spatial resolution of about \(2~\hbox {mm}\) is achieved for electromagnetic showers and for those hadronic showers which start in the Preshower Calorimeter.

The instantaneous luminosity is monitored based on the rate of the Bethe–Heitler process \(ep \rightarrow ep\gamma \). The final state photon is detected in the photon detector located close to the beampipe at \(z=-103\) m. The precision of the integrated luminosity measurement is improved in a dedicated analysis of the elastic QED Compton process [46] in which both the scattered electron and the photon are detected in the SpaCal.

### 3.2 Event selection

The data sample of this analysis has been collected using a special low multiplicity trigger requiring two tracks with \(p_T>160\) MeV and originating from the nominal event vertex, and at most one extra track with \(p_T>100\) MeV. The tracks are found by the Fast Track Trigger (FTT) [47, 48], based on hit information provided by the CJCs. The trigger also contains a veto condition against non-*ep* background provided by the CIP. The average trigger efficiency is about \(75 ~\%\) for the analysis phase space. The trigger simulation has been verified and tuned to the data using an independently triggered data sample.

For the analysis, exclusive events are selected, containing two oppositely charged pion candidates in the central tracker, a leading neutron in the FNC and nothing else above noise level in the detector.^{3} The photoproduction regime is ensured by the absence of a high energy electromagnetic cluster consistent with a signal from a scattered beam positron in the calorimeters. This limits the photon virtuality to \(Q^2 \lesssim 2 \, \mathrm {GeV}^2 \), resulting in a mean value of \(\langle Q^2 \rangle = 0.04 \, \mathrm {GeV}^2 \).

The \(\rho ^0\) candidate selection requires the reconstruction of the trajectories of two, and only two, oppositely charged particles in the central tracking detector. They must originate from a common vertex lying within \(\pm 30\) cm in *z* of the nominal *ep* interaction point, and must have transverse momenta above 0.2 GeV and polar angles within the interval \( 20^{\circ } \le \theta \le 160^{\circ }\). The momentum of the \(\rho ^0\) meson is calculated as the vector sum of the two charged particle momenta. The two-pion invariant mass is required to be within the interval \(0.3 < M_{\pi \pi }<1.5\) GeV. Since no explicit hadron identification is used, events are discarded with \(M_{KK}<1.04\) GeV where \(M_{KK}\) is the invariant mass of two particles under the kaon mass hypothesis. This cut suppresses a possible background from exclusive production of \(\phi \) mesons.

Events containing a leading neutron are selected by requiring a hadronic cluster in the FNC with an energy above \(120~\mathrm {GeV}\) and a polar angle below \(0.75~\mathrm {mrad}\). The cut on polar angle, defined by the geometrical acceptance of the FNC, restricts the neutron transverse momenta to the range \(p_{T,n}< x_L \cdot 0.69~\mathrm {GeV}\).

Event selection criteria and the definition of the kinematic phase space (PS) of the measurements. The measured cross sections are determined at \(Q^2=0\) using the effective flux (3), based on the VDM

Event selection \((2006\!-\!2007,~e^+p)\) | Analysis PS | Measurement PS | |
---|---|---|---|

Trigger s14 (low multiplicity) | |||

No \(e'\) in the detector | \(Q^2 < 2\) GeV\(^2\) | \(Q^2 = 0\) GeV\(^2\) | |

2 tracks, net charge \(=0\), | |||

\(p_T\!>\!0.2\) GeV, \(20^o\!<\!\theta \!<\!160^o\), | \(20<W_{\gamma p}<100\) GeV | \(20<W_{\gamma p}<100\) GeV | |

from \(|z_\mathrm{vx}|<30\) cm | \(p_{T,\rho }<1.0\) GeV | \(-t^{\prime }<1.0\) GeV\(^2\) | |

\(0.3<M_{\pi \pi }<1.5\) GeV | \(0.6<M_{\pi \pi }<1.1\) GeV | \(2m_{\pi }<M_{\rho }< M_{\rho }\!+\!5\Gamma _{\rho }\) | |

LRG requirement | \({\sim } 637{,}000\) events | ||

\(E_n>120\) GeV | \(x_L > 0.2\) | \(0.35<x_L<0.95\) | |

\(\theta _n<0.75\) mrad | \(\theta _n<0.75\) mrad | \(p_{T,n}<x_L\cdot 0.69\) GeV | |

\({\sim } 7000\) events | \({\sim } 6100\) events | \({\sim } 5770\) events | |

(OPE dominated range) | OPE1 | \(p_{T,n}\!<\!0.2\) GeV (\({\sim } 3600\) events) | |

OPE2 | \(p_{T,n}\!<\!0.2\) GeV, \(0.65\!<\!x_L\!<\!0.95\) (\({\sim } 2200\) events) |

After these cuts the data sample contains about 7000 events. The event selection criteria together with the analysis and the measurement phase space definitions are summarised in Table 1. In order to better control migration effects and backgrounds most of the selection cuts are kept softer than the final measurement phase space limits. In the end, the \(\gamma p\) cross sections measured in the \(\theta _n<0.75\) mrad range are based on \({\sim }5770\) events. For the \(\gamma \pi \) cross section extraction additional cuts are applied in order to stay within a range where the validity of OPE can be safely expected. Two sub-samples are defined: OPE1 with \(p_{T,n}<200\) MeV, containing \({\sim }3600\) events and OPE2 with \(p_{T,n}<200\) MeV and \(x_L>0.65\), containing \({\sim }2200\) events.^{4}

### 3.3 Monte Carlo simulations and corrections to the data

Monte Carlo (MC) simulations are used to calculate acceptances and efficiencies for triggering, track reconstruction, event selection and background contributions and to account for migrations between measurement bins due to the finite detector resolution.

Since the exact shape of the \(p_{T,\rho }^2\) dependence is not *a priori* known, two extreme versions are generated. In the first version a simple exponential shape is assumed, as expected for elastic \(\rho ^0\) photoproduction on the pion, with the slope \(b=5\) GeV\(^{-2}\). For the second version a mass-dependent slope is taken, \(4\le b(M_{n\pi })\le 22\) GeV\(^{-2}\), typical for DPP processes as observed at lower energies [12, 13, 14, 21]. The difference in the correction factors obtained using these two versions of MC simulations is part of the model dependent systematic uncertainty.

The background events originating from diffractive \(\rho ^0\) production (Fig. 1d) are generated using the program DIFFVM [51], which is based on Regge theory and the Vector Dominance Model. All channels (elastic, single- and double-dissociation processes) are included, with the relative composition as measured in [52]. For the proton dissociative case the \(M_Y\) mass spectrum is parametrised as \(\mathrm {d} \sigma / \mathrm {d}M_Y^2 \propto 1/M_Y^{2.16}\), for \(M_Y^2 > 3.6~\mathrm {GeV}^2 \) with quark and diquark fragmentation using the JETSET program [53]. For the low mass dissociation the production of excited nucleon states at the proton vertex is taken into account explicitly. Signal events, corresponding to the diagram shown in Fig. 1c, are excluded from the generated background sample.

The DIFFVM program is also used to estimate possible contaminations from diffractive \(\omega (782),~ \phi (1020)\) and \(\rho ^{\prime }(1450{-}1700)\) production.

As discussed in Sect. 1, the pion exchange diagram dominates the cross section in the low *t* region where the contributions from the diagrams in Fig. 1b, c almost cancel. To check a possible influence of these terms on the MC correction factors, neutron exchange events (b) were generated using POMPYT and events of class (c) using DIFFVM. As expected, these events have kinematic distributions and selection efficiencies similar to those from the pion exchange process and do not alter the MC correction factors beyond the quoted systematic uncertainties.

Small, but non-zero values of \(Q^2\) cause \(|t'|\) to differ from \(p_{T,\rho }^2\) by less than \(Q^2\). To account for this effect a multiplicative correction factor determined with the Monte Carlo generators is applied to the bins of the \(p_{T,\rho }^2\) distribution; the correction is obtained by taking the ratio between the \(|t'|\) and \(p_{T,\rho }^2\) distributions at the generator level. This correction varies from 1.1 at \(p_{T,\rho }^2 = 0\) to 0.77 at \(p_{T,\rho }^2 = 1\) GeV\(^2\).

For all MC samples detector effects are simulated in detail with the GEANT program [57]. The MC description of the detector response, including trigger efficiencies, is adjusted using comparisons with independent data. Beam-induced backgrounds are taken into account by overlaying the simulated events with randomly triggered real events. The simulated MC events are passed through the same reconstruction and analysis chain as is used for the data.

The MC simulations are used to correct the distributions at the level of reconstructed particles back to the hadron level on a bin-by-bin basis. The size of the correction factors is 12 in average, corresponding to an efficiency of \({\sim }8~\%\), and varies between \({\sim }10\) and \({\sim }24\) for different parts of the covered phase space. The main contributions to the inefficiency are: the azimuthal acceptance of the FNC (\({\sim }30~\%\) on average), the \(\rho \) meson reconstruction efficiency which is zero if one of the tracks has low transverse momentum (\({\sim }60~\%\)), the LRG selection efficiency (\({\sim }60~\%\)) and the trigger efficiency (\({\sim }75~\%\)). The bin purity, defined as the fraction of events reconstructed in a particular bin that originate from the same bin on hadron level, varies between 70 and \(95~\%\) for one-dimensional distributions and between 45 and \(65~\%\) for two-dimensional ones. As an example, Fig. 2c illustrates the binning scheme used in the two-dimensional \((x_L,p_{T,n})\) distribution.

### 3.4 Extraction of the \(\rho ^0\) signal

A fit is performed in the range \(M_{\pi \pi }>0.4\) GeV using the Ross–Stodolsky parametrisation (7) for the \(\rho ^0\) meson mass shape and adding the contributions for the reflection from \(\omega \rightarrow \pi ^+\pi ^-\pi ^0\) and for the non-resonant background. Other sources of non-\(\rho ^0\) background, such as \(\omega (782) \rightarrow \pi ^+\pi ^-, ~~ \phi (1020)\rightarrow K_L^0 K_S^0, \, \pi ^+\pi ^-\pi ^0, ~~ \rho ^{\prime } \rightarrow \rho \pi \pi , 4\pi , \pi \pi \), which may be misidentified as \(\rho ^0\) candidates, are estimated using MC simulations with the relative yield normalisation fixed to previously measured and published values: \(\sigma _{\gamma p}(\omega )/\sigma _{\gamma p}(\rho ^0)=0.10(\pm 20~\%)\) [58], \(\sigma _{\gamma p}(\phi )/\sigma _{\gamma p}(\rho ^0)=0.07(\pm 20~\%)\) [59] and \(\sigma _{\gamma p}(\rho ^{\prime })/\sigma _{\gamma p}(\rho ^0)=0.20(\pm 50~\%)\) [60, 61, 62, 63]. The resulting overall background contamination in the analysis region \(0.6<M_{\pi \pi }<1.1\) GeV is found to be \((1.5 \pm 0.7)~\%\).

In summary, all properties of the selected \(\pi ^+\pi ^-\) sample investigated here are consistent with \(\rho ^0\) photoproduction.

### 3.5 Signal and background decomposition

The event selection described in Sect. 3.2 does not completely suppress non-DPP background. According to the MC simulations, the remaining part is mostly due to proton dissociation with some admixture of double dissociative events.

*t*dependencies and proton dissociation fraction in the overall diffractive cross section. Figure 4 illustrates this decomposition using the nominal DIFFVM parameters.

Control plots for the data description by the Monte Carlo models using this signal to background ratio are shown in Fig. 5. Since neither POMPYT nor DIFFVM are able to provide reliable absolute cross section predictions for such a final state, only a shape comparison is possible. The irregular shape of the azimuthal angle distribution, \(\varphi _n\), is due to the FNC aperture limitations, as shown in Fig. 2b.

In the fit described above the absolute normalisation for the DIFFVM prediction is left free. As a cross check, this normalisation has been fixed using an orthogonal, background dominated sample, obtained by requiring an ‘anti-LRG’ selection, i.e. \(\rho ^0 + n\) events with additional activity in the forward detectors. In this sample the background fraction is found to be \(0.58\pm 0.07\). Fixing the DIFFVM normalisation by a fit to the ‘anti-LRG’ sample results in a background contribution of \(F_{bg}=0.29\pm 0.05\) in the main sample. Since the signal-to-background decomposition fit in this cross check gives a worse \(\chi ^2\), the nominal value \(F_{bg}=0.34\) is used for the cross section determinations. The difference to the \(F_{bg}\) value determined in the nominal analysis, as described above, is well covered by systematic uncertainty of the LRG condition efficiency.

### 3.6 Cross section determination and systematic uncertainties

*ep*events, \(N_{data}\), the bin-integrated \(\gamma p\) cross section in bin

*i*is calculated as

*i*taking into account the overall normalisation fraction \(F_{bg}=0.34\), \(A \cdot \epsilon \) is the correction for detector acceptance and efficiency, \(\mathcal{L}\) is the integrated luminosity of the data, \(C_{\rho }\) is the extrapolation factor for the number of \(\rho ^0\) events from the \(M_{\pi \pi }\) measurement interval to the full \(\rho ^0\) mass range and \(\Phi _{\gamma }=0.1543\) is the value of the equivalent photon flux from Eq. (3) for the given \((W_{\gamma p},Q^2)\) range.

^{5}Since the statistics available does not allow for a reliable \(\rho ^0\) mass fit in every measurement bin, \(C_{\rho }^i\) is calculated using \(C_{\rho }=1.155\), obtained from the fit of the full sample and bin-dependent skewing correction factor derived from the fitted dependence of \(n_{RS}(p_{T,\rho }^2)\) in Eq. (9).

**Detector related sources.**The trigger efficiency is verified and tuned with the precision of \(3.4~\%\) using an independent monitoring sample. It is treated as correlated between different bins. The uncertainty due to the track finding and reconstruction efficiency in the central tracker is estimated to be \(1~\%\) per track [75] resulting in \(2~\%\) uncertainty in the cross section, taken to be correlated between bins. Several sources of uncertainties related to the measurement of the forward neutrons are considered. The uncertainty in the neutron detection efficiency which affects the measurement in a global way is \(2~\%\) [7]. The \(2~\%\) uncertainty on the absolute hadronic energy scale of the FNC [7] leads to a systematic error of \(1.1~\%\) for the \(x_L\)-integrated cross section and varying between 2 and \(19~\%\) in different \(x_L\) bins. The acceptance of the FNC calorimeter is defined by the interaction point and the geometry of the HERA magnets and is determined using MC simulations. The uncertainty of the impact position of the particle on the FNC, due to beam inclination and the uncertainty on the FNC position, is estimated to be 5 mm [7]. This results in an average uncertainty on the FNC acceptance determination of \(4.5~\%\) reaching up to \(10~\%\) for the \(p_{T,n}\) distribution. The systematics due to the exclusivity condition in the main part of the H1 detector is estimated to be \(2.1~\%\). It gets contributions from varying the LAr calorimeter noise cut between 400 MeV and 800 MeV (\(0.9~\%\)) and from the parameters of the algorithm connecting clusters with tracks (\(1.9~\%\)). This error influences only the overall normalisation. The uncertainty from the LRG condition is determined in the same manner as in the H1 inclusive diffraction analyses based on the large rapidity gap technique [76, 77]. It is further verified by comparing the cross sections obtained using different components of the forward detector apparatus for the LRG selection: FMD alone vs FMD\(+\)FTS vs FMD\(+\)Plug vs FMD\(+\)FTS\(+\)Plug. The resulting uncertainty is conservatively estimated to be \(9.0~\%\) affecting all bins in a correlated manner.**Backgrounds.**Three different types of background are considered. Non-*ep*background is estimated from the shape of the*z*-vertex distribution and from the analysis of non-colliding proton bunches to be \((1.2 \pm 0.7)~\%\). Background originating from random coincidences between \(\rho ^0\) photoproduction events and neutrons from*p*-gas interactions amounts to \((1.0 \pm 0.2)~\%\). This results in \(2.2~\%\) background which was statistically subtracted in all distributions with an uncertainty of \(0.8~\%\). Non-\(\rho ^0\) background, as discussed in Sect. 3.4, has an uncertainty of \(0.7~\%\) and affects the overall normalisation only. Diffractive background to the DPP signal events (Sect. 3.5) is estimated with a precision of \(7.6~\%\). This is one of the largest individual uncertainties in the analysis. It is correlated between the bins.**MC model uncertainties.**The uncertainty in the subtracted diffractive background due to the limited knowledge on \(\gamma p\) diffraction is evaluated by varying the \(M_Y\) and*t*dependencies in the DIFFVM simulation and the relative composition of diffractive channels within the limits allowed by previous HERA measurements. The resulting uncertainty is a part of the background subtraction systematics listed above. The systematic uncertainty of the MC correction factors for signal events is \(4.1~\%\), varying between \(1~\%\) and \(9~\%\) in different bins. It is evaluated from the difference between two versions of the POMPYT MC program with different \(p_T^2\) dependencies of the \(\rho ^0\) cross section, as described in Sect. 3.3. Here the uncertainty due to the POMPYT reweighting procedure is also accounted for.**Normalisation uncertainties.**The uncertainty related to the \(\rho ^0\) mass fit, extrapolating from the measurement domain \(0.6 \le M_{\pi \pi } \le 1.1~\,\text{ GeV }\) to the full mass range \(2m_{\pi }<M_{\pi \pi }<M_{\rho }+5\Gamma _{\rho }\), which implies a correction factor of \( C_{\rho }=1.155\) on average in Eq. (11) with an uncertainty of \(1.6~\%\) due to fit errors. The integrated luminosity of the data sample is known with \(2.7~\%\) precision [46]. Together with other normalisation errors listed above the resulting total normalisation uncertainty amounts to \(4.4~\%\).

The total systematic uncertainty for the integrated \(\gamma p\) cross section is \(14.6~\%\) including the global normalisation errors.

## 4 Results

Differential photoproduction cross sections \(\mathrm{d}\sigma _{\gamma p}/\mathrm{d}x_L\) for the exclusive process \(\gamma p \rightarrow \rho ^0 n \pi ^+\) in two regions of neutron transverse momentum and \(20\!<\!W_{\gamma p}\!<\!100\) GeV. The statistical, uncorrelated and correlated systematic uncertainties, \(\delta _{stat}\), \(\delta _{sys}^{unc}\) and \(\delta _{sys}^{cor}\) respectively, are given together with the total uncertainty \(\delta _{tot}\), which does not include the global normalisation error of \(4.4~\%\)

\(x_L\) | (\(p_{T,n}< x_L \cdot 0.69\) GeV) | (\(p_{T,n}< 0.2\) GeV) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

\(\mathrm{d}\sigma _{\gamma p}/\mathrm{d}x_L\)\([\upmu \mathrm{b}]\) | \(\delta _{stat}\)\([\%]\) | \(\delta _{sys}^{unc}\)\([\%]\) | \(\delta _{sys}^{cor}\)\([\%]\) | \(\delta _{tot}\)\([\%]\) | \(\mathrm{d}\sigma _{\gamma p}/\mathrm{d}x_L\)\([\mu \mathrm{b}]\) | \(\delta _{stat}\)\([\%]\) | \(\delta _{sys}^{unc}\)\([\%]\) | \(\delta _{sys}^{cor}\)\([\%]\) | \(\delta _{tot}\)\([\%]\) | |

\(0.35{-}0.45\) | 0.213 | 9.8 | 10.6 | 15.1 | 20.9 | 0.119 | 11.2 | 10.3 | 15.2 | 21.5 |

\(0.45{-}0.55\) | 0.398 | 7.0 | 9.8 | 15.4 | 19.5 | 0.164 | 8.6 | 7.5 | 15.3 | 19.1 |

\(0.55{-}0.65\) | 0.530 | 5.9 | 7.2 | 15.7 | 18.2 | 0.190 | 7.6 | 7.8 | 15.4 | 18.9 |

\(0.65{-}0.75\) | 0.761 | 4.1 | 6.9 | 12.8 | 15.1 | 0.274 | 5.1 | 9.5 | 12.0 | 16.2 |

\(0.75{-}0.85\) | 0.806 | 3.6 | 5.0 | 11.7 | 13.2 | 0.354 | 4.1 | 5.8 | 10.7 | 12.8 |

\(0.85{-}0.95\) | 0.402 | 5.4 | 19.4 | 12.8 | 23.9 | 0.204 | 6.3 | 15.0 | 11.2 | 19.7 |

Double differential photoproduction cross sections \(\mathrm{d^2}\sigma _{\gamma p}/\mathrm{d}x_L\mathrm{d}p_{T,n}^2\) in the range \(20\!<\!W_{\gamma p}\!<\!100\) GeV. The statistical, uncorrelated and correlated systematic uncertainties, \(\delta _{stat}\), \(\delta _{sys}^{unc}\) and \(\delta _{sys}^{cor}\) respectively, are given together with the total uncertainty \(\delta _{tot}\), which does not include the global normalisation error of \(4.4~\%\)

\(x_L\) range | \(\langle x_L \rangle \) | \(p_{T,n}^2\) range [GeV\(^2\)] | \(\langle p_{T,n}^2 \rangle \) [GeV\(^2\)] | \(\frac{\mathrm{d^2}\sigma _{\gamma p}}{\mathrm{d}x_L\mathrm{d}p_{T,n}^2}\)\([\upmu \mathrm{b}/\mathrm {GeV}^2]\) | \(\delta _{stat}\)\([\%]\) | \(\delta _{sys}^{unc}\)\([\%]\) | \(\delta _{sys}^{cor}\)\([\%]\) | \(\delta _{tot}\)\([\%]\) |
---|---|---|---|---|---|---|---|---|

\(0.35{-}0.50\) | 0.440 | \(0.00{-}0.01\) | 0.00499 | 3.178 | 13.9 | 6.3 | 14.8 | 21.3 |

\(0.01{-}0.03\) | 0.01998 | 3.545 | 12.1 | 5.4 | 12.7 | 18.4 | ||

\(0.03{-}0.06\) | 0.04495 | 2.974 | 13.7 | 6.1 | 12.7 | 19.7 | ||

\(0.50{-}0.65\) | 0.581 | \(0.00{-}0.01\) | 0.00492 | 5.242 | 10.5 | 4.3 | 14.0 | 18.0 |

\(0.01{-}0.03\) | 0.01969 | 4.925 | 8.6 | 4.1 | 12.9 | 16.0 | ||

\(0.03{-}0.06\) | 0.04429 | 3.344 | 11.7 | 4.7 | 13.9 | 18.8 | ||

\(0.06{-}0.12\) | 0.08719 | 2.775 | 11.2 | 7.3 | 13.7 | 19.1 | ||

\(0.65{-}0.80\) | 0.728 | \(0.00{-}0.01\) | 0.00489 | 9.623 | 6.3 | 4.5 | 11.4 | 13.8 |

\(0.01{-}0.03\) | 0.01957 | 7.229 | 5.5 | 5.5 | 12.0 | 14.3 | ||

\(0.03{-}0.06\) | 0.04403 | 5.333 | 7.3 | 5.7 | 12.2 | 15.3 | ||

\(0.06{-}0.12\) | 0.08617 | 2.927 | 8.4 | 4.8 | 13.7 | 16.8 | ||

\(0.12{-}0.20\) | 0.15324 | 1.494 | 14.7 | 6.3 | 17.9 | 24.0 | ||

\(0.80{-}0.95\) | 0.863 | \(0.00{-}0.01\) | 0.00484 | 7.990 | 7.6 | 8.5 | 11.2 | 16.0 |

\(0.01{-}0.03\) | 0.01935 | 6.457 | 5.7 | 7.1 | 10.9 | 14.2 | ||

\(0.03{-}0.06\) | 0.04354 | 3.850 | 7.9 | 7.4 | 12.3 | 16.4 | ||

\(0.06{-}0.12\) | 0.08425 | 1.580 | 11.3 | 7.8 | 15.7 | 20.8 | ||

\(0.12{-}0.30\) | 0.16558 | 0.520 | 14.1 | 9.3 | 18.7 | 25.2 |

The effective exponential slope, \(b_n\), obtained from the fit of double differential photoproduction cross sections \(\mathrm{d^2}\sigma _{\gamma p}/\mathrm{d}x_L\mathrm{d}p_{T,n}^2\) to a single exponential function in bins of \(x_L\). The first uncertainty represents the fit error from the statistical and uncorrelated systematic uncertainty and the second one is due to the correlated systematic uncertainty

\(x_L\) range | \(\langle x_L \rangle \) | \(b_n\) [GeV\(^{-2}\)] |
---|---|---|

\(0.35{-}0.50\) | 0.440 | \(~~2.23 \pm 4.57 \pm 2.10\) |

\(0.50{-}0.65\) | 0.581 | \(~~8.51 \pm 1.74 \pm 1.10\) |

\(0.65{-}0.80\) | 0.728 | \(13.17 \pm 0.90 \pm 0.65\) |

\(0.80{-}0.95\) | 0.863 | \(18.21 \pm 0.94 \pm 1.05\) |

Energy dependence of the exclusive photoproduction of a \(\rho ^0\) meson associated with a leading neutron, \(\gamma p \rightarrow \rho ^0 n \pi ^+\). The first uncertainty is statistical and the second is systematic. The global normalisation uncertainty of \(4.4~\%\) is not included. \(\Phi _{\gamma }\) is the integral of the photon flux (3) in a given \(W_{\gamma p}\) bin

\(W_{\gamma p}\) [GeV] | \(\Phi _{\gamma }\) | \(\sigma (\gamma p \rightarrow \rho ^0 n \pi ^+)\) [nb] |
---|---|---|

\(20{-}36\) | 0.06306 | \(343.7 \pm 10.1 \pm 45.4\) |

\(36{-}52\) | 0.03578 | \(308.7 \pm 12.3 \pm 43.5\) |

\(52{-}68\) | 0.02413 | \(294.2 \pm 15.8 \pm 45.2\) |

\(68{-}84\) | 0.01769 | \(260.0 \pm 23.1 \pm 44.9\) |

\(84{-}100\) | 0.01362 | \(214.5 \pm 50.2 \pm 45.0\) |

Differential photoproduction cross section \(\mathrm{d}\sigma _{\gamma p}/\mathrm{d}\eta \) for the exclusive process \(\gamma p \rightarrow \rho ^0 n \pi ^+\) as a function of the \(\rho ^0\) pseudorapidity in the kinematic range \(0.35\!<\!x_L\!<\!0.95\), \(\theta _n\!<\!0.75\) mrad and \(20\!<\!W_{\gamma p}\!<\!100\) GeV. The statistical, uncorrelated and correlated systematic uncertainties, \(\delta _{stat}\), \(\delta _{sys}^{unc}\) and \(\delta _{sys}^{cor}\) respectively, are given together with the total uncertainty \(\delta _{tot}\), which does not include the global normalisation error of \(4.4~\%\)

\(\eta _{\rho }\) | \(\mathrm{d}\sigma _{\gamma p}/\mathrm{d}\eta \) [nb] | \(\delta _{stat}\) [%] | \(\delta _{sys}^{unc}\) [%] | \(\delta _{sys}^{cor}\) [%] | \(\delta _{tot}\) [%] |
---|---|---|---|---|---|

\([-5.0;-4.5)\) | 0.9 | 68. | 28. | 12. | 75. |

\([-4.5;-4.0)\) | 5.1 | 27. | 18. | 11. | 34. |

\([-4.0;-3.5)\) | 8.8 | 22. | 11. | 12. | 27. |

\([-3.5;-3.0)\) | 23.7 | 14. | 6.1 | 12. | 20. |

\([-3.0;-2.5)\) | 44.0 | 9.7 | 4.0 | 13. | 17. |

\([-2.5;-2.0)\) | 45.2 | 9.3 | 3.1 | 16. | 18. |

\([-2.0;-1.5)\) | 47.5 | 8.6 | 3.5 | 17. | 19. |

\([-1.5;-1.0)\) | 48.2 | 7.6 | 2.9 | 15. | 17. |

\([-1.0;-0.5)\) | 45.9 | 7.1 | 5.9 | 13. | 16. |

\([-0.5;~~~0.0)\) | 38.9 | 8.0 | 3.2 | 14. | 16. |

\([~~~0.0;+0.5)\) | 46.2 | 6.9 | 5.7 | 13. | 16. |

\([+0.5;+1.0)\) | 52.1 | 6.7 | 7.1 | 13. | 16. |

\([+1.0;+1.5)\) | 63.8 | 6.0 | 5.4 | 13. | 15. |

\([+1.5;+2.0)\) | 86.2 | 5.8 | 4.4 | 13. | 14. |

\([+2.0;+2.5)\) | 39.8 | 7.7 | 3.1 | 12. | 15. |

\([+2.5;+3.0)\) | 17.7 | 11. | 4.0 | 12. | 17. |

\([+3.0;+3.5)\) | 7.8 | 17. | 6.8 | 12. | 22. |

\([+3.5;+4.0)\) | 3.4 | 26. | 11. | 12. | 30. |

\([+4.0;+4.5)\) | 1.0 | 55. | 21. | 11. | 60. |

\([+4.5;+5.0)\) | 0.7 | 64. | 33. | 11. | 73. |

Differential photoproduction cross section \(\mathrm{d}\sigma _{\gamma p}/\mathrm{d}t^{\prime }\) for the exclusive process \(\gamma p \rightarrow \rho ^0 n \pi ^+\) as a function of the \(\rho ^0\) four-momentum transfer squared, \(t^{\prime }\), in the kinematic range \(0.35\!<\!x_L\!<\!0.95\), \(\theta _n\!<\!0.75\) mrad and \(20\!<\!W_{\gamma p}\!<\!100\) GeV. The statistical, uncorrelated and correlated systematic uncertainties, \(\delta _{stat}\), \(\delta _{sys}^{unc}\) and \(\delta _{sys}^{cor}\) respectively, are given together with the total uncertainty \(\delta _{tot}\), which does not include the global normalisation error of \(4.4~\%\)

\(-t^{\prime }\) range [GeV\(^2\)] | \(\langle -t^{\prime } \rangle \) [GeV\(^2\)] | \(\mathrm{d}\sigma _{\gamma p}/\mathrm{d}t^{\prime }\)\([\mu \mathrm{b}/\mathrm {GeV}^2]\) | \(\delta _{stat}\)\([\%]\) | \(\delta _{sys}^{unc}\)\([\%]\) | \(\delta _{sys}^{cor}\)\([\%]\) | \(\delta _{tot}\)\([\%]\) |
---|---|---|---|---|---|---|

\(0.00{-}0.02\) | 0.0094 | 2.771 | 4.5 | 2.5 | 12.1 | 13.2 |

\(0.02{-}0.05\) | 0.0338 | 1.821 | 4.9 | 1.7 | 13.0 | 14.0 |

\(0.05{-}0.10\) | 0.0727 | 0.996 | 5.9 | 1.3 | 14.6 | 15.8 |

\(0.10{-}0.15\) | 0.1236 | 0.600 | 8.7 | 1.0 | 16.3 | 18.5 |

\(0.15{-}0.20\) | 0.1741 | 0.402 | 11.6 | 2.9 | 17.8 | 21.4 |

\(0.20{-}0.25\) | 0.2242 | 0.343 | 12.0 | 3.7 | 16.0 | 20.3 |

\(0.25{-}0.35\) | 0.2973 | 0.279 | 8.6 | 5.1 | 13.8 | 17.0 |

\(0.35{-}0.50\) | 0.4189 | 0.178 | 8.3 | 6.4 | 12.7 | 16.4 |

\(0.50{-}0.65\) | 0.5689 | 0.104 | 9.2 | 7.8 | 11.6 | 16.8 |

\(0.65{-}1.00\) | 0.7924 | 0.037 | 9.4 | 18.7 | 11.5 | 23.9 |

Exponential slopes, \(b_1\) and \(b_2\), and the ratio \(\sigma _1/\sigma _2\), obtained from the components of fit (14) to the differential cross section \(\mathrm{d}\sigma _{\gamma p}/\mathrm{d}t^{\prime }\) in bins of \(x_L\) and in bins of \(p_{T,n}^2\). The errors represent the statistical and systematic uncertainties added in quadrature

\(x_L\) range | \(\langle x_L \rangle \) | \(b_1\) [GeV\(^{-2}\)] | \(b_2\) [GeV\(^{-2}\)] | \(\sigma _1/\sigma _2\) |
---|---|---|---|---|

\(0.35{-}0.50\) | 0.440 | \(18.6 \pm 4.2\) | \(2.54 \pm 0.79\) | \(1.501 \pm 1.024\) |

\(0.50{-}0.65\) | 0.581 | \(26.0 \pm 5.5\) | \(2.79 \pm 0.43\) | \(0.782 \pm 0.316\) |

\(0.65{-}0.80\) | 0.728 | \(28.1 \pm 7.9\) | \(4.24 \pm 0.34\) | \(0.244 \pm 0.091\) |

\(0.80{-}0.95\) | 0.863 | \(27.9 \pm 6.5\) | \(4.42 \pm 0.50\) | \(0.394 \pm 0.142\) |

\(0.35{-}0.95\) | 0.686 | \(25.7 \pm 3.2\) | \(3.62 \pm 0.32\) | \(0.492 \pm 0.143\) |

\(p_{T,n}^2\) range [GeV\(^2\)] | \(\langle p_{T,n}^2 \rangle \) [GeV\(^2\)] | \(b_1\) [GeV\(^{-2}\)] | \(b_2\) [GeV\(^{-2}\)] | \(\sigma _1/\sigma _2\) |
---|---|---|---|---|

\(0.0{-}0.04\) | 0.015 | \(26.8 \pm 4.5\) | \(4.07 \pm 0.34\) | \(0.384 \pm 0.077\) |

\(0.04{-}0.30\) | 0.092 | \(26.6 \pm 4.4\) | \(3.08 \pm 0.46\) | \(0.635 \pm 0.423\) |

Energy dependence of elastic \(\rho ^0\) photoproduction cross section on the pion, \(\gamma \pi ^+ \rightarrow \rho ^0 \pi ^+\), extracted in the one-pion-exchange approximation using OPE1 sample. The first uncertainty represents the full experimental error and the second is the model error coming from the pion flux uncertainty (see text). \(\Gamma _{\pi }(x_L)\) represents the value of the pion flux (5-6) integrated over the \(p_{T,n}<0.2\) GeV range, at a given \(x_L\)

\(x_L\) range | \(\Gamma _{\pi }(x_L)\) | \(\langle W_{\gamma \pi }\rangle \) [GeV] | \(\sigma (\gamma \pi ^+ \rightarrow \rho ^0\pi ^+)\) [\(\mu \)b] |
---|---|---|---|

\(0.35{-}0.45\) | 0.04407 | 34.08 | \(2.71 \pm 0.58 ^{+0.82} _{-0.86} \) |

\(0.45{-}0.55\) | 0.07262 | 31.11 | \(2.25 \pm 0.43 ^{+0.62} _{-0.41} \) |

\(0.55{-}0.65\) | 0.10400 | 27.83 | \(1.83 \pm 0.35 ^{+0.41} _{-0.23} \) |

\(0.65{-}0.75\) | 0.13154 | 24.10 | \(2.09 \pm 0.34 ^{+0.38} _{-0.25} \) |

\(0.75{-}0.85\) | 0.13386 | 19.68 | \(2.65 \pm 0.34 ^{+0.41} _{-0.39} \) |

\(0.85{-}0.95\) | 0.07431 | 13.91 | \(2.74 \pm 0.54 ^{+0.46} _{-0.69} \) |

### 4.1 \(\gamma p\) cross sections

*t*(or \(p_{T,n}^2\)) of the leading neutron. The double differential cross section \(\mathrm{d^2}\sigma _{\gamma p}/\mathrm{d}x_L\mathrm{d}p_{T,n}^2\) is measured, and the results are presented in Table 3 and Fig. 7. The bins are chosen such, that the data are not affected by the polar angle cut (see Fig. 2c). Although neither the

*t*-, nor the \(p_{T,n}^2\)-dependence of the pion flux models are exactly exponential they can be approximated by a simple exponent in many cases. Such an approximation has been used already in other analyses [2, 6]. The \(p_{T,n}^2\)-distributions measured here for fixed \(x_L\) are compatible with an exponential shape within the statistical and uncorrelated systematic errors. Therefore, the same approach is used here. The cross sections are fitted by a single exponential function \(e^{-b_n(x_L)p_{T,n}^2}\) in each \(x_L\) bin. The quality of the fits is good, with \(P(\chi ^2)=0.35 \div 0.60\). The results are presented in Table 4 and Fig. 8. The measured

*b*-slopes are compared to those obtained from several pion flux parametrisations. Despite of the large experimental uncertainties none of the models is able to reproduce the data.

^{6}A possible reason for this discrepancy could be the effect of energy-momentum conservation affecting the proton vertex in this exclusive reaction more strongly than in inclusive production of a leading neutron in which an apparent factorisation of the proton vertex has been observed. Another explanation [21, 89] could be absorptive corrections which modify the

*t*dependence of the amplitude, leading to an increase of the effective

*b*-slope at large \(x_L\) as compared to the pure OPE model without absorption.

*a, b*and

*c*in Fig. 1, leading to a slope dependence on the invariant mass of the \((n\pi ^+)\) system produced at the proton vertex. Since the forward pion is not detected in this analysis the \((n\pi ^+)\) invariant mass cannot be determined with sufficient precision, which prevents explicit measurement of the

*b*(

*m*) dependence.

In order to investigate the presence of a possible factorisation between the proton and the photon vertices, the \(t^{\prime }\) distribution is studied in bins of \(x_L\). The result of the fit by Eq. (14) with \(x_L\) dependent parameters \(a_i(x_L), b_i(x_L)\) is presented in Table 8 and in Fig. 12 in comparison with the values given in Eq. (15) for the full \(x_L\) range. Also the evolution with \(x_L\) of the ratio of two components, \(\sigma _1/\sigma _2\), where \(\sigma _i = \frac{a_i}{b_i}(1-e^{-b_i})\), is shown. Given the large experimental uncertainties no strong conclusion about factorisation of the two vertices can be drawn.

### 4.2 \(\gamma \pi \) cross section

*ep*collisions [34, 35, 37] suggest that in addition to the pion exchange process other processes

^{7}may contribute at \(10{-}20~\%\) level. To suppress these contributions it is recommended to perform cross section measurements in the ‘OPE safe’ phase space region: low \(p_{T,n}\) and high \(x_L\). In order to investigate a possible influence of non-OPE contributions the extraction of the photon-pion cross section is repeated for two additional regions, in which the validity of pure OPE is assumed. The cross sections for the full FNC acceptance range \((\theta _n<0.75~\mathrm{mrad},~0.35<x_L<0.95)\) and for the OPE2 sub-sample \((p_{T,n}<200~\mathrm{MeV},~0.65<x_L<0.95)\) together with the value (16) obtained for the OPE1 sample are presented in Table 10. The values of \(\sigma (\gamma \pi ^+\rightarrow \rho ^0\pi ^+)\) extracted in these three different phase space regions agree well within the experimental errors. Thus no evidence for an extra contribution beyond the OPE is found in the full FNC acceptance range for the exclusive reaction studied here.

Cross section of elastic \(\rho ^0\) photoproduction on the pion, \(\gamma \pi ^+ \rightarrow \rho ^0 \pi ^+\), extracted in the one-pion-exchange approximation using three different samples: full sample, OPE1 and OPE2. The first uncertainty represents the full experimental error and the second is the model error coming from the pion flux uncertainty (see text). \(\Gamma _{\pi }\) represents the value of the pion flux (5) and (6) integrated over the corresponding \((x_L,p_{T,n})\) range

\(x_L\) range | \(p_{T,n}^\mathrm{max}\) [GeV] | \(\Gamma _{\pi }\) | \(\langle W_{\gamma \pi }\rangle \) [GeV] | \(\sigma (\gamma \pi ^+ \rightarrow \rho ^0\pi ^+)\) [\(\mu \)b] |
---|---|---|---|---|

\(0.35{-}0.95\) | \(x_L\cdot 0.69\) | 0.13815 | 23.65 | \(2.25 \pm 0.34 ^{+0.54} _{-0.50} \) |

\(0.35{-}0.95\) | 0.2 | 0.05604 | 23.65 | \(2.33 \pm 0.34 ^{+0.47} _{-0.40} \) |

\(0.65{-}0.95\) | 0.2 | 0.03397 | 19.73 | \(2.45 \pm 0.33 ^{+0.41} _{-0.40} \) |

Taking a value of \(\sigma (\gamma p \rightarrow \rho ^0 p) = (9.5 \pm 0.5)~\mu \mathrm{b}\) at the corresponding energy \(\langle W \rangle =24\) GeV, which is an interpolation between fixed target and HERA measurements (see e.g. figure 10 in [56]), one obtains for the ratio \(r_\mathrm{el} = \sigma _\mathrm{el}^{\gamma \pi }/\sigma _\mathrm{el}^{\gamma p} = 0.25 \pm 0.06\). A similar ratio, but for the total cross sections at \(\langle W \rangle =107\) GeV, has been estimated by the ZEUS collaboration as \(r_\mathrm{tot} = \sigma _\mathrm{tot}^{\gamma \pi }/\sigma _\mathrm{tot}^{\gamma p} = 0.32 \pm 0.03\) [2]. Both ratios are significantly smaller than their respective expectations, based on simple considerations. For \(r_\mathrm{tot}\), a value of 2 / 3 is predicted by the additive quark model [78, 79, 80], while \(r_\mathrm{el} = (\frac{b_{\gamma p}}{b_{\gamma \pi }}) \cdot (\sigma _\mathrm{tot}^{\gamma \pi }/\sigma _\mathrm{tot}^{\gamma p})^2 = 0.57 \pm 0.03\) can be deduced by combining the optical theorem, the eikonal approach [81] relating cross sections with elastic slope parameters [82] and the data on \(pp, \pi ^+p\) [83, 84, 85] and \(\gamma p\) [57] elastic scattering. Such a suppression of the cross section is usually attributed to rescattering, or absorptive corrections [86, 87, 88, 89], which are essential for leading neutron production. For the exclusive reaction \(\gamma p \rightarrow \rho ^0 n \pi ^+\) studied here this would imply an absorption factor of \(K_{abs} = 0.44 \pm 0.11\). It is interesting to note, that this value is similar to the somewhat different, but conceptually related damping factor in diffractive dijet photoproduction, the rapidity gap survival probability, \(\langle S^2 \rangle \simeq 0.5\), which has been determined by the H1 collaboration [90, 91, 92].

## 5 Summary

The photoproduction cross section for exclusive \(\rho ^0\) production associated with a leading neutron is measured for the first time at HERA. The integrated \(\gamma p\) cross section in the kinematic range \(20<W_{\gamma p}<100\) GeV, \(0.35<x_L<0.95\) and \(\theta _n<0.75\) mrad is determined with \(2~\%\) statistical and \(14.6~\%\) systematic precision. The elastic photon-pion cross section, \(\sigma (\gamma \pi ^+ \rightarrow \rho ^0\pi ^+)\), at \(\langle W_{\gamma \pi }\rangle = 24\) GeV is extracted in the one-pion-exchange approximation.

Single and double differential \(\gamma p\) cross sections are measured. The differential cross section d\(\sigma /\mathrm{d}t^{\prime }\) shows a behaviour typical for exclusive double peripheral exchange processes.

The differential cross sections for the leading neutron are sensitive to the pion flux models. While the shape of the \(x_L\) distribution is well reproduced by most of the pion flux parametrisations, the \(x_L\) dependence of the \(p_T\) slope of the leading neutron is not described by any of the existing models. This may indicate that the proton vertex factorisation hypothesis does not hold in exclusive photoproduction, e.g. due to large absorptive effects which are expected to play an essential rôle in soft peripheral processes. The estimated cross section ratio for the elastic photoproduction of \(\rho ^0\) mesons on the pion and on the proton, \(r_\mathrm{el} = \sigma _\mathrm{el}^{\gamma \pi }/\sigma _\mathrm{el}^{\gamma p} = 0.25 \pm 0.06\), suggests large absorption corrections, of the order of \(60~\%\), suppressing the rate of the studied reaction \(\gamma p \rightarrow \rho ^0 n \pi ^+\).

## Footnotes

- 1.
In the present analysis elastic vertex corresponds to the \(\rho ^0I\!\!P\) vertex, Fig. 1.

- 2.
A correction accounting for the small, but non-zero \(Q^2\) values is applied, based on the Monte Carlo generator information, as explained in Sect. 3.3.

- 3.
According to simulation, the forward going \(\pi ^+\) from the proton vertex is emitted in the range \(\eta > 5.7\) where it cannot be reliably measured or identified with the available apparatus.

- 4.
The OPE2 sample corresponds to the low \(|t|<0.2\) GeV\(^2\) region, see Fig. 2c.

- 5.
Note, that the effective VDM flux (3) converts the

*ep*cross section into a*real*\(\gamma p\) cross section at \(Q^2=0\), contrary to the EPA flux [71, 72, 73, 74] converting it to the transverse \(\gamma ^*p\) cross section, averaged over the measured \(Q^2\) range. The difference between the two approaches amounts to \({\approx } 6~\%\) integrated over the \((Q^2,y)\) range of the measurement. - 6.
Reweighting the signal MC using the measured \(b_n(x_L)\) slopes has only small effects on the cross section determination and is covered by the systematic uncertainties assigned to the pion flux models.

- 7.

## Notes

### Acknowledgments

We are grateful to the HERA machine group whose outstanding efforts have made this experiment possible. We thank the engineers and technicians for their work in constructing and maintaining the H1 detector, our funding agencies for financial support, the DESY technical staff for continual assistance and the DESY directorate for support and for the hospitality which they extend to the non DESY members of the collaboration. We would like to give credit to all partners contributing to the EGI computing infrastructure for their support for the H1 Collaboration.

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