Measurement of the cross section for hard exclusive $\pi^0$ leptoproduction

TL;DR

This paper reports the measurement of the cross section for exclusive $$ leptoproduction on the proton, providing data crucial for understanding Generalised Parton Distributions and offering evidence for the chiral-odd GPD $ar{E}_T$.

Contribution

First measurement of the cross section for exclusive $$ leptoproduction on the proton at COMPASS, with results supporting models of GPDs and revealing evidence for chiral-odd GPDs.

Findings

Measured cross sections as a function of $t$ and azimuthal angles.

Evidence for the chiral-odd GPD $ar{E}_T$ from interference contributions.

Provides input for GPD modeling and understanding nucleon structure.

Abstract

We report on a measurement of hard exclusive $\pi^0$ muoproduction on the proton by COMPASS using 160 GeV/$c$ polarised $\mu^+$ and $\mu^-$ beams of the CERN SPS impinging on a liquid hydrogen target. From the average of the measured $\mu^+$ and $\mu^-$ cross sections, the virtual-photon proton cross section is determined as a function of the squared four-momentum transfer between initial and final proton in the range $0.08\,(\text{GeV/}c)^2 < |t| < 0.64\,(\text{GeV/}c)^2$. The average kinematics of the measurement are $\langle Q^2 \rangle =2.0\; {(\text{GeV}/c)^2}$, $\langle \nu \rangle = 12.8\; {\text{GeV}}$, $\langle x_{Bj} \rangle = 0.093 $ and $\langle -t \rangle = 0.256\; {(\text{GeV}/c)^2} $. Fitting the azimuthal dependence reveals a combined contribution by transversely and longitudinally polarised photons of $(8.1 \ \pm \ 0.9_{\text{stat}}{}_{- \ 1.0}^{+ \ 1.1}\big\rvert_{\text{sys}})\,{\text{nb}}/{(\text{GeV}/c)^{2}}$, as well as transverse-transverse and longitudinal-transverse interference contributions of $(-6.0 \pm 1.3_{\text{stat}}{}_{- \ 0.7}^{+ \ 0.7}\big\rvert_{\text{sys}})\,{\text{nb}}/{(\text{GeV}/c)^{2}}$ and $(1.4 \pm 0.5_{\text{stat}}{}_{- \ 0.2}^{+ \ 0.3}\big\rvert_{\text{sys}})\,{\text{nb}}/{(\text{GeV}/c)^{2}}$, respectively. Our results provide important input for modelling Generalised Parton Distributions. In the context of the phenomenological Goloskokov-Kroll model, the statistically significant transverse-transverse interference contribution constitutes clear experimental evidence for the chiral-odd GPD $\overline{E}_T$.

Tables

Table 1. Table 1: Four-dimensional grid used for the calculation of the acceptance. The full width of the respective dimension is given in the bottom row of the table.

$\varphi$ /rad	$\left|t\right|$ /${\left(\text{GeV}/c\right)}^2$	$Q^2$ /${\left(\text{GeV}/c\right)}^2$	$\nu$ /GeV
$-\pi$  – $\frac{-3\pi }{4}$	0.08 – 0.15	1.00 – 1.5	08.50 – 11.45
$\frac{-3\pi }{4}$ – $\frac{-\pi }{2}$	0.15 – 0.22	1.50 – 2.24	11.45 – 15.43
.	0.22 – 0.36	2.24 – 3.34	15.43 – 0.78
.	0.36 – 0.5	3.34 – 5	20.78 – 28
.	0.50 – 0.64
$\frac{3\pi }{4}$   – $-\pi$
$\Delta \varphi$/rad	$\Delta \left|t\right|$/${\left(\text{GeV}/c\right)}^2$	$\Delta Q^2$/${\left(\text{GeV}/c\right)}^2$	$\Delta \nu$/GeV
$2\pi$	0.56	4	19.5

Table 2. Table 2: Summary of the estimated relative systematic uncertainties for the | t | 𝑡 |t| and ϕ italic-ϕ \phi -dependent cross sections and the integrated cross section. The values are given in percent. Note that the uni-directional uncertainty σ ↑ subscript 𝜎 ↑ \sigma_{\uparrow} ( σ ↓ subscript 𝜎 ↓ \sigma_{\downarrow} ) has to be used with positive (negative) sign.

source	${\sigma }_{\uparrow }^t$	${\sigma }_{\downarrow }^t$	${\sigma }_{\uparrow }^{\varphi }$	${\sigma }_{\downarrow }^{\varphi }$	${\sigma }_{\uparrow }$	${\sigma }_{\downarrow }$
${\mu }^{+}$ flux	2	2	2	2	2	2
${\mu }^{-}$ flux	2	2	2	2	2	2
ECal threshold	5	5	5	5	5	5
acceptance	4	7	4	7	4	7
kinem. fit	0	7	0	7	0	7
$\omega$ background	0	1	0	1	0	1
rad. corr.	2	5	2	5	2	5
${\mu }^{+}$ event loss	4–13	0	0–12	0–5	9	0
Lepto norm.	5–28	3–11	5–51	3–21	8	3
$\sum$	12–29	10–14	12–53	12–24	14	13

Table 3. Table 3: Numerical values of the average cross sections shown in Fig.  5 .

lower	$⟨\frac{d\sigma }{d\left|t\right|d\varphi }⟩/\frac{\text{nb}}{{\left(\text{GeV}/c\right)}^2}$	lower	$⟨\frac{d\sigma }{d\left|t\right|}⟩/\frac{\text{nb}}{{\left(\text{GeV}/c\right)}^2}$
$\varphi$ bin		$\left|t\right|$ bin
limit		limit
$-\pi$	${0.4{}_{-0.3}{}^{+0.4}|}_{\text{stat}}{}_{-0.1}{}^{+0.1}|{}_{\text{sys}}$	$0.08$	${16.3{}_{-3.1}{}^{+3.6}|}_{\text{stat}}{}_{-2.0}{}^{+2.0}|{}_{\text{sys}}$
$\frac{-3\pi }{4}$	${2.1{}_{-0.6}{}^{+0.7}|}_{\text{stat}}{}_{-0.2}{}^{+0.3}|{}_{\text{sys}}$	$0.15$	${16.2{}_{-3.2}{}^{+3.8}|}_{\text{stat}}{}_{-1.9}{}^{+2.1}|{}_{\text{sys}}$
$\frac{-\pi }{2}$	${2.1{}_{-0.4}{}^{+0.5}|}_{\text{stat}}{}_{-0.3}{}^{+0.3}|{}_{\text{sys}}$	$0.22$	${11.5{}_{-2.2}{}^{+2.6}|}_{\text{stat}}{}_{-1.4}{}^{+1.5}|{}_{\text{sys}}$
$\frac{-\pi }{4}$	${1.0{}_{-0.3}{}^{+0.4}|}_{\text{stat}}{}_{-0.1}{}^{+0.2}|{}_{\text{sys}}$	$0.36$	${03.4{}_{-1.2}{}^{+1.4}|}_{\text{stat}}{}_{-0.5}{}^{+0.8}|{}_{\text{sys}}$
$0$	${1.2{}_{-0.4}{}^{+0.5}|}_{\text{stat}}{}_{-0.2}{}^{+0.2}|{}_{\text{sys}}$	$0.5$	${01.5{}_{-0.8}{}^{+1.0}|}_{\text{stat}}{}_{-0.2}{}^{+0.4}|{}_{\text{sys}}$
$\frac{\pi }{4}$	${1.9{}_{-0.4}{}^{+0.5}|}_{\text{stat}}{}_{-0.2}{}^{+0.2}|{}_{\text{sys}}$
$\frac{\pi }{2}$	${1.5{}_{-0.4}{}^{+0.5}|}_{\text{stat}}{}_{-0.2}{}^{+0.2}|{}_{\text{sys}}$
$\frac{3\pi }{4}$	${0.1{}_{-0.1}{}^{+0.2}|}_{\text{stat}}{}_{-0.0}{}^{+0.1}|{}_{\text{sys}}$

Equations

$$\displaystyle\frac{\mathrm{d}^{2}\sigma_{\gamma^{*}p}^{\leftrightarrows}}{\mathrm{d}t\mathrm{d}\phi}=\frac{1}{2\pi}\Big{[}\frac{\mathrm{d}\sigma_{T}}{\mathrm{d}t}+\epsilon\frac{\mathrm{d}\sigma_{L}}{\mathrm{d}t}+\epsilon\cos\left(2\phi\right)\frac{\mathrm{d}\sigma_{TT}}{\mathrm{d}t}$$

$$\displaystyle+\sqrt{2\epsilon\left(1+\epsilon\right)}\cos\phi\frac{\mathrm{d}\sigma_{LT}}{\mathrm{d}t}\mp|P_{l}|\sqrt{2\epsilon(1-\epsilon)}\sin{\phi}\frac{\mathrm{d}\sigma_{LT}^{\prime}}{\mathrm{d}t}\Big{]},$$

$$ϵ=\frac{1-y-\frac{y^2{\gamma }^2}{4}}{1-y+\frac{y^2}{2}+\frac{y^2{\gamma }^2}{4}}$$

$$\frac{\mathrm{d}^{2}\sigma_{\gamma^{*}p}}{\mathrm{d}t\mathrm{d}\phi}=\frac{1}{2}\Bigl{(}\frac{\mathrm{d}^{2}\sigma_{\gamma^{*}p}^{\leftarrow}}{\mathrm{d}t\mathrm{d}\phi}+\frac{\mathrm{d}^{2}\sigma_{\gamma^{*}p}^{\rightarrow}}{\mathrm{d}t\mathrm{d}\phi}\Bigr{)}.$$

$$\frac{{\mathrm{d}}^2{\sigma }_{{\gamma }^{\ast }p}}{\mathrm{d}t\mathrm{d}\varphi }=$$

$$\displaystyle+\sqrt{2\epsilon\left(1+\epsilon\right)}\cos\left(\phi\right)\frac{\mathrm{d}\sigma_{LT}}{\mathrm{d}t}\Big{]}.$$

$$\frac{\mathrm{d}{\sigma }_T}{\mathrm{d}t}$$

$$\frac{\mathrm{d}{\sigma }_L}{\mathrm{d}t}$$

$$\displaystyle-2\xi^{2}Re\left[\langle\tilde{H}\rangle^{\ast}\langle\tilde{E}\rangle\right]-\frac{t^{\prime}}{4M^{2}}\xi^{2}\lvert\langle\tilde{E}\rangle\rvert^{2}\Big{]},$$

$$\frac{\mathrm{d}{\sigma }_{TT}}{\mathrm{d}t}$$

$$\frac{\mathrm{d}{\sigma }_{LT}}{\mathrm{d}t}$$

$$\xi \approx \frac{x_{\text{Bj}}}{2-x_{\text{Bj}}},$$

$$0.08(\text{GeV}/c{)}^2$$

$$1(\text{GeV}/c{)}^2$$

$$\text{and }8.5\text{GeV}$$

$$M_{\text{X}}^2=(k_{\mu }+p_p-k_{{\mu }^{\prime }}-p_{p^{\prime }}-p_{{\gamma }_1}-p_{{\gamma }_2}{)}^2$$

$$\frac{{\mathrm{d}}^2\sigma }{\mathrm{d}|t|\mathrm{d}\varphi }=\frac{1}{\Gamma (Q^2,\nu ,E_{\mu })}\frac{\mathrm{d}{\sigma }^{\mu p}}{\mathrm{d}{Q}^2\mathrm{d}\nu \mathrm{d}\varphi \mathrm{d}|t|},$$

$$\begin{split}\Gamma(Q^{2},\nu,E_{\mu})&=\frac{\alpha_{\text{em}}(1-x_{\text{Bj}})}{2\pi Q^{2}yE_{\mu}}\Bigg{[}y^{2}\bigg{(}1-\frac{2m_{\mu}^{2}}{Q^{2}}\bigg{)}\\ &+\frac{2}{1+Q^{2}/\nu^{2}}\bigg{(}1-y-\frac{Q^{2}}{4E_{\mu}^{2}}\bigg{)}\Bigg{]}.\end{split}$$

$$\begin{array}{rl}{\mathcal{Y}}_{klmn}^{\pm }& =\sum _{i=1}^{N_{\text{data}}^{\pm ,\Delta {\Omega }_{klmn}}}\frac{1}{\Gamma (Q_i^2,{\nu }_i,E_{\mu ,i})}\\ & -f^{\pm }\sum _{i=1}^{N_{\text{L}}^{\pm ,\Delta {\Omega }_{klmn}}}\frac{1}{\Gamma (Q_i^2,{\nu }_i,E_{\mu ,i})}.\end{array}$$

$$\Big{\langle}\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}|t|\mathrm{d}\phi}\Big{\rangle}_{\Delta\Omega_{kl}}^{\pm}=\frac{1}{\mathcal{L^{\pm}}\Delta\Omega_{kl}}\sum_{mn}\frac{\mathcal{Y}^{\pm}_{klmn}}{a(\Delta\Omega_{klmn})}.$$

$$\Big{\langle}\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}|t|\mathrm{d}\phi}\Big{\rangle}_{\Delta\Omega_{kl}}=\frac{1}{2}\Biggl{(}\Big{\langle}\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}|t|\mathrm{d}\phi}\Big{\rangle}_{\Delta\Omega_{kl}}^{+}+\Big{\langle}\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}|t|\mathrm{d}\phi}\Big{\rangle}_{\Delta\Omega_{kl}}^{-}\Biggr{)}.$$

$$\Big{\langle}\frac{\mathrm{d}\sigma}{\mathrm{d}|t|}\Big{\rangle}_{\Delta\Omega_{l}}=\sum_{k}\Delta\phi_{k}\Big{\langle}\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}|t|\mathrm{d}\phi}\Big{\rangle}_{\Delta\Omega_{kl}},$$

$$\Big{\langle}\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}|t|\mathrm{d}\phi}\Big{\rangle}_{\Delta\Omega_{k}}=\frac{1}{\Delta|t|}\sum_{l}\Delta|t|_{l}\Big{\langle}\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}|t|\mathrm{d}\phi}\Big{\rangle}_{\Delta\Omega_{kl}},$$

$$\Big{\langle}\frac{\mathrm{d}\sigma_{{T}}}{\mathrm{d}|t|}+\epsilon\frac{\mathrm{d}\sigma_{{L}}}{\mathrm{d}|t|}\Big{\rangle}=(8.1\pm 0.9_{\text{stat}}{}_{-\ 1.0}^{+\ 1.1}\big{\rvert}_{\text{sys}})\frac{\text{nb}}{(\text{GeV}/c)^{2}}.$$

$$\Big{\langle}\frac{\mathrm{d}\sigma_{{TT}}}{\mathrm{d}|t|}\Big{\rangle}=(-6.0\pm 1.3_{\text{stat}}{}_{-\ 0.7}^{+\ 0.7}\big{\rvert}_{\text{sys}})\frac{\text{nb}}{(\text{GeV}/c)^{2}}$$

$$\Big{\langle}\frac{\mathrm{d}\sigma_{{LT}}}{\mathrm{d}|t|}\Big{\rangle}=(1.4\pm 0.5_{\text{stat}}{}_{-\ 0.2}^{+\ 0.3}\big{\rvert}_{\text{sys}})\frac{\text{nb}}{(\text{GeV}/c)^{2}}.$$

Full text

††journal: Physics Letters B

1 Introduction

Measurements of pseudoscalar mesons produced in hard exclusive lepton-nucleon scattering provide important data for phenomenological parameterisations of Generalised Parton Distributions (GPDs) [1, 2, 3, 4, 5]. In the past two decades, GPDs have shown to be a very rich and useful construct for both experiment and theory as their determination allows for a detailed description of the parton structure of the nucleon. In particular, GPDs correlate transverse spatial positions and longitudinal momentum fractions of the partons in the nucleon. They embed parton distribution functions and nucleon form factors, and they give access to energy-momentum-tensor form factors. For each quark flavour, there exist four parton-helicity-conserving (chiral-even) GPDs, denoted $H$, $\tilde{H}$, $E$, and $\tilde{E}$, and four parton-helicity-flip (chiral-odd) GPDs, denoted $H_{T}$, $\tilde{H}_{T}$, $E_{T}$, and $\tilde{E}_{T}$. While hard production of vector mesons is sensitive primarily to the GPDs $H$ and $E$, the production of pseudoscalar mesons by longitudinally polarised virtual photons is sensitive to $\tilde{H}$ and $\tilde{E}$ in the leading-twist description.

Contributions from transversely polarised virtual photons to the production of spin-0 mesons are expected to be suppressed in the production amplitude by $1/Q$ [6], where $Q^{2}$ is the virtuality of the photon $\gamma^{*}$ that is exchanged between muon and proton. However, experimental data on exclusive $\pi^{+}$ production from HERMES [7] and on exclusive $\pi^{0}$ production from JLab CLAS [8, 9, 10, 11] and Hall A [12, 13, 14] suggest that such contributions are substantial. In the GPD formalism such contributions are possible if a quark helicity-flip GPD couples to a twist-3 wave function [15, 16]. In the framework of the phenomenological model of Ref. [15], pseudoscalar-meson production is described by the GPDs $\tilde{H}$, $\tilde{E}$, $H_{T}$ and $\overline{E}_{T}=2\tilde{H}_{T}+E_{T}$. Different sensitivities to these GPDs are expected when comparing $\pi^{+}$ vs. $\pi^{0}$ production. When taking into account the relative signs and sizes of these GPDs for $u$ and $d$ quarks, the different quark flavour contents of these mesons lead to different predictions for the $\lvert t\rvert$-dependence of the cross section, especially at small values of $|t|$. Here, $t$ is the square of the four-momentum transfer between initial and final nucleon. The production of $\pi^{+}$ mesons is dominated by the contributions from longitudinally polarised virtual photons, of which a major part originates from pion-pole exchange that is the main contributor to $\tilde{E}$. Also the contributions from $\tilde{H}$ and $H_{T}$ are significant, and there is a strong cancellation between the contributions from $\overline{E}_{T}$ for $u$ and $d$ quarks. On the contrary, in the case of $\pi^{0}$ production there is no pion-pole exchange, the contributions from $\tilde{H}$ and $H_{T}$ are small and a large contribution from transversely polarised photons is generated mainly by $\overline{E}_{T}$.

These differences between $\pi^{+}$ and $\pi^{0}$ production are experimentally supported. While for $\pi^{+}$ production a fast decrease of the cross section with increasing $|t|$ is predicted by theoretical models and confirmed by the experimental results from HERMES [7], for $\pi^{0}$ production a dip is expected as $|t|\rightarrow 0$ [15] and confirmed by results in the JLab kinematic domain [9, 10, 13]. Constraints for modelling the poorly known GPD $\overline{E}_{T}$ were obtained in a lattice-QCD study [17] of its moments. The COMPASS results on exclusive $\pi^{0}$ production in muon-proton scattering presented in this Letter provide new input for modelling this GPD and chiral-odd (‘transversity’) GPDs in general.

2 Formalism

The reduced cross section for hard exclusive meson production by scattering a polarised lepton beam off an unpolarised proton target reads:

[TABLE]

where the sign $\mp$ of the lepton beam polarisation $P_{l}$ corresponds to negative and positive helicity of the incoming lepton, respectively, denoted by $\leftrightarrows$. The conversion from the lepton-nucleon cross section to the virtual-photon nucleon cross section, using the one-photon-exchange approximation, is explained in Sect. 5. The contribution to the cross section from transversely (longitudinally) polarised virtual photons is denoted by $\sigma_{T}$ ($\sigma_{L}$). The symbols $\sigma_{TT}$ and $\sigma_{LT},\sigma^{\prime}_{LT}$ denote contributions from the interference between transversely and longitudinally polarised virtual photons, respectively, with transversely polarised ones The factor

[TABLE]

is the virtual-photon polarisation parameter and $\phi$ is the azimuthal angle between the lepton scattering plane and the hadron production plane, see Fig. 1. Here, $Q^{2}=-(k_{\mu}-k_{\mu^{\prime}})^{2}$ is the photon virtuality, $\nu=(k_{\mu}^{0}-k_{\mu^{\prime}}^{0})$ the energy of the virtual photon in the target rest frame, $y=\nu/k_{\mu}^{0}$ and $\gamma^{2}=Q^{2}/\nu^{2}$, where $k_{\mu}$ and $k_{\mu^{\prime}}$ denote the four-momenta of the incoming and the scattered muon in the laboratory system, respectively.

The spin-independent cross section can be obtained by averaging the two spin-dependent cross sections,

[TABLE]

When forming this average, the last term in Eq. (1) cancels if the magnitude $|P_{l}|$ of the beam polarisation is the same for measurements with $\mu^{+}$ and $\mu^{-}$ beam, so that

[TABLE]

The individual contributions appearing in Eq. (4) are related to convolutions of GPDs and meson distribution amplitudes with individual hard scattering amplitudes [15, 10]:

[TABLE]

Here, the aforementioned convolutions are denoted by triangular brackets, $t^{\prime}=t-t_{min}$ with $|t_{min}|$ being the kinematically smallest possible value of $|t|$, and $M$ is the mass of the proton. The quantity $\xi$ is equal to one half of the longitudinal momentum fraction transferred between the initial and final proton and can be approximated at COMPASS kinematics as

[TABLE]

where $x_{\text{Bj}}=Q^{2}/(2M\nu)$.

3 Experimental set-up and data selection

The main component of the COMPASS set-up is the two-stage magnetic spectrometer. Each spectrometer stage comprises a dipole magnet complemented by a variety of tracking detectors, a muon filter for muon identification and an electromagnetic (ECal) as well as a hadron calorimeter. A detailed description of the set-up can be found in Refs. [18, 19, 20].

The data used for this analysis were collected within four weeks in 2012, during which the COMPASS spectrometer was complemented by a 2.5 m long liquid-hydrogen target surrounded by a time-of-flight (TOF) system, and a third electromagnetic calorimeter that was placed directly downstream of the target. The TOF detector consisted of two cylinders, each made of 24 scintillating-counter slats, mounted concentrically around the target.

In order to determine the spin-independent cross section through Eq. (3), data with $\mu^{+}$ and $\mu^{-}$ beam were taken separately. The natural polarisation of the muon beam provided by the CERN SPS originates from the parity-violating decay in flight of the parent meson, which implies opposite polarisation for $\mu^{+}$ and $\mu^{-}$ beams. Within regular time intervals during the measurement, charge and polarisation of the muon beam were swapped simultaneously. In total, a luminosity of 18.9 pb*-1* was collected for the $\mu^{+}$ beam with negative polarisation and 23.5 pb*-1* for the $\mu^{-}$ beam with positive polarisation. For both beams, the absolute value of the average beam polarisation is about 0.8 with an uncertainty of about 0.04.

In the data analysis, $\pi^{0}$ mesons are selected by their dominant two-photon decay. At least two neutral clusters are required that had to be detected above the respective threshold in one of the electromagnetic calorimeters, in conjunction with an interaction vertex reconstructed within the target using the incoming and outgoing muon tracks. The outgoing muon is identified by requiring that it has the same charge as the beam particle and traverses more than 15 radiation lengths. As neutral cluster we denote a reconstructed calorimeter cluster that is not associated to a charged track, thereby including any cluster in case of the most upstream calorimeter that had no tracking system in front.

For each interaction vertex and each combination of two neutral clusters, the kinematics of the recoil proton are predicted from the four-momentum balance of the analysed process, $\mu p\rightarrow\mu^{\prime}p^{\prime}\pi^{0}$, $\pi^{0}\rightarrow\gamma\gamma$, by using the reconstructed spectrometer information, i.e. the vertex position, the momenta of the incoming and outgoing muons as well as the energy and position of the two clusters. The predicted properties of the recoil proton $p^{\prime}$ are compared to the properties of each track candidate as reconstructed by the TOF system. Note that the four-momentum of the recoil proton is determined by the target TOF system based on the assumption that the reconstructed track belongs to a proton. Figure 2 shows an example for the result of such a comparison, including also the corresponding constraints applied for the selection of events. The Monte Carlo yields shown in this figure, denoted as HEPGEN and LEPTO, will be explained in more detail in Sect. 4.

In the case that more than one combination of vertex, cluster pair and recoil-track candidate exist that satisfy the aforementioned selection criteria for a given event, this event is excluded from the analysis. Figure 3 shows the two-photon mass distribution in the region close to the nominal $\pi^{0}$ mass together with the constraints applied to select $\pi^{0}$ mesons.

In order to further enhance the purity of the selected data and to improve the precision of the particle kinematics at the interaction vertex, a kinematic fit for the exclusive reaction $\mu p\rightarrow\mu^{\prime}p^{\prime}\pi^{0}$ is performed, which requires a single $\pi^{0}$ to decay into the two photons selected as described above.

With the selection procedure described above, data are analysed in the kinematic range

[TABLE]

In addition, two reference samples are selected in a wider kinematic range, which are denoted as signal and background sample. Apart from the extended kinematic range given below, the signal sample corresponds to the aforementioned selections. In contrast to the signal sample, the background sample contains only events with more than one combination of vertex, cluster pair and recoil-track candidate. Apart from the small peak at zero, it contains non-exclusive events. The purpose of the reference samples is explained in the following section.

4 Estimation of the background contribution

The main background to exclusive $\pi^{0}$ muoproduction originates from non-exclusive deep-inelastic scattering processes. In such processes, low-energy hadrons are produced in addition to the $\pi^{0}$, which remain undetected in the apparatus. In order to estimate the background contribution, two Monte Carlo generators are employed.

First, the LEPTO 6.5.1 generator with the high-$p_{T}$ COMPASS tuning [21] is used to describe the non-exclusive fraction of events. Secondly, the HEPGEN++ $\pi^{0}$ generator is used to model the kinematics of single $\pi^{0}$ muoproduction [22, 23], in the following denoted by HEPGEN. The generated events from both generators are independently passed through a complete description of the COMPASS set-up [24], and the resulting simulated data are treated in the same way as it is done for real data.

As there exists essentially no information on the cross section of exclusive $\pi^{0}$ production in the kinematic domain of COMPASS, the two reference samples described in Sect. 3 are used to normalise the HEPGEN and LEPTO Monte Carlo yields. Using several variables, the kinematic information from beam and spectrometer measurements as well as that of the recoil-proton candidates are compared between experimental data and the two simulations in order to determine the best normalisation of each simulated data set relative to that of the experimental data. As an example of such a comparison, the undetected mass

[TABLE]

is shown in Fig. 4.

Here, the four-momenta are denoted by $p_{p}$ and $p_{p^{\prime}}$ for the target and recoil proton, respectively, and by $p_{\gamma_{1}}$ and $p_{\gamma_{2}}$ for the two produced photons. In addition to the measured data points, the HEPGEN simulation and the sum of the HEPGEN and LEPTO simulations are shown. In order to estimate the amount of non-exclusive background, the simulated data are scaled such that they describe the data for both reference samples. The scaling factor for the LEPTO Monte Carlo yield, which is denoted by $f^{\pm}$, will be used in Sect. 5 to normalise this simulation when correcting the data for background.

The resulting fraction of non-exclusive background in the data is estimated to be (29{}_{-\ 6}^{+\ 2}\big{\rvert}_{\text{sys}})\%. Here, the uncertainty is estimated by comparing the scaling factors extracted for various variables and by using several extraction methods for the scaling factors. Details are given in Ref. [25]. Contributions of other background sources are found to be negligible. For example, the production of single $\omega$ mesons, where the $\omega$ decays into a $\pi^{0}$ and a photon that remains undetected, was found in Monte Carlo studies to contribute at the level of 1% [25].

5 Determination of the cross section

The virtual-photon proton cross section is obtained from the measured muon-proton cross section using

[TABLE]

where the transverse virtual-photon flux is given by

[TABLE]

Here, $\alpha_{\text{em}}$ denotes the electromagnetic fine structure constant and $E_{\mu}$ the zero-th component of $k_{\mu}$.

For the cross section determination, the HEPGEN Monte Carlo simulation described in Sect. 4 is used. The acceptance $a(\Delta\Omega_{klmn})$ is calculated in a four-dimensional grid as the number of reconstructed events divided by the number of generated events using 8 bins in $\phi$, 5 in $|t|$, 4 in $Q^{2}$ and 4 in $\nu$. The phase-space element is given by $\Delta\Omega_{klmn}=\Delta\phi_{k}\Delta|t|_{l}\Delta Q^{2}_{m}\Delta\nu_{n}$. The spacing of the grid is given in Table 1.

In each four-dimensional bin, the experimental yield corrected for background according to the LEPTO simulations is obtained as

[TABLE]

Here, $N_{\text{data}}^{\pm,\Delta\Omega_{klmn}}$ is the number of measured events and $N_{\text{L}}^{\pm,\Delta\Omega_{klmn}}$ the number of LEPTO events within the phase-space element $\Delta\Omega_{klmn}$. The second sum represents the LEPTO simulations that are appropriately normalised by the factor $f^{\pm}$, which was introduced in Sect. 4. Each event is weighted with the transverse virtual-photon flux $\Gamma(Q^{2}_{i},\nu_{i},E_{\mu,i})$ to obtain the virtual-photon yield from the measured yields for muon-proton interactions.

The spin-dependent virtual-photon proton cross sections measured with positively or negatively charged muons are determined in each of the ($\phi_{k}$, $|t|_{l}$) bins as luminosity-normalised experimental yield averaged over the measured ranges $\Delta Q^{2}=4$ $(\text{GeV}/c)^{2}$and $\Delta\nu=19.5$ GeV as

[TABLE]

Here, $\Delta\Omega_{kl}=\Delta\phi_{k}\Delta|t|_{l}\Delta Q^{2}\Delta\nu$, $\mathcal{L}^{\pm}$ denotes the luminosity and $a(\Delta\Omega_{klmn})$ the acceptance in the phase-space element $\Delta\Omega_{klmn}$.

The spin-independent virtual-photon proton cross section is obtained according to Eq. (3) as average of the two spin-dependent cross sections given in Eq. (14):

[TABLE]

The cross section integrated over the full $2\pi$-range in $\phi$ is obtained as

[TABLE]

with $\Delta\Omega_{l}=\Delta|t|_{l}\Delta Q^{2}\Delta\nu$. Similarly, the $|t|$-averaged cross section in the measured range is given by

[TABLE]

with $\Delta\Omega_{k}=\Delta\phi_{k}\Delta|t|\Delta Q^{2}\Delta\nu$.

The systematic uncertainties on the extracted values of the cross section are shown in Table 2, arranged in three groups. The first group contains the systematic uncertainties on the determination of the beam flux, possible systematic effects related to the uncertainty on the energy thresholds for the detection of the low-energetic photon in the electromagnetic calorimeters, and the uncertainty on the determination of the acceptance. The second group contains the systematic uncertainties related to a variation of the energy and momentum balance of the kinematic fit, the influence of background originating from the production of $\omega$ mesons and the estimated influence of radiative corrections [29, 25]. The largest systematic effects appear in the third group, which contains the uncertainty related to the estimation of non-exclusive background as described in Sect. 4, and that related to an observed mismatch between the measured single-photon yield in the 2012 COMPASS data and a corresponding Monte Carlo simulation of the Bethe-Heitler process. The latter effect was observed in Refs. [26, 27] in a kinematic region where single-photon production is dominated by the Bethe-Heitler cross section, which is calculable at the percent level. The total systematic uncertainty $\Sigma$ is obtained by quadratic summation of its components for each bin separately.

6 Results

For the background corrected final data sample the average kinematics are $\langle Q^{2}\rangle=2.0\;{(\text{GeV}/c)^{2}}$, $\langle\nu\rangle=12.8\;{\text{GeV}}$, $\langle x_{Bj}\rangle=0.093$ and $\langle-t\rangle=0.256\;{(\text{GeV}/c)^{2}}$. The dependences of the measured cross section on $\lvert t\rvert$ and $\phi$ are shown in Fig. 5, with the numerical values given in Table 3. The cross section in bins of $|t|$ is shown in the top panel of Fig. 5. It appears to be consistent with an exponential decrease with increasing $|t|$ for values of $|t|$ larger than about 0.25 (GeV/$c$)2, while at smaller $|t|$ the $t$-dependence becomes weaker. Our result is compared to the predictions of two versions of the Goloskokov-Kroll (GK) model [15, 28]. The results of the GK model shown in this letter are obtained by integrating over the analysis range in the same way as it is done for the data. The dashed-dotted curve represents the cross section from the earlier version [15] as a function of $|t|$, while the upwards pointing triangles correspond to the cross section averaged over $|t|$ bins of the data. The mean cross sections for the full $t$-range are compared in the rightmost part of this panel. Analogously, the dotted curve and the downward pointing triangles correspond to the later version of the model [28], which was inspired by the results presented in this Letter. We observe that for the earlier version the magnitude of the predicted cross section overshoots our measurement by approximately a factor of two.

The cross section as a function of $\phi$ for the full measured $t$-range is shown in the bottom panel of Fig. 5 in eight $\phi$ bins of equal width. The full dots show the measured cross section for each bin and the solid curve represents the fit described below.

In order to extract the different contributions to the spin-independent cross section, a maximum-likelihood fit is applied to the data according to Eq. (4). In the fit, the measured average value of the virtual-photon polarisation parameter is used, $\epsilon=0.996$. The $\phi$-integrated cross section determined by the fit is obtained as

[TABLE]

The $TT$ and $LT$ interference terms are obtained as

[TABLE]

and

[TABLE]

We observe a large negative contribution by $\sigma_{TT}$ and a smaller positive one by $\sigma_{LT}$, which indicates a significant role of transversely polarised photons in exclusive $\pi^{0}$ production.

The $\phi$-dependence of the measured cross section is compared to the calculations of the GK model in the bottom panel of Fig. 5. Apart from the discrepancy in the magnitude of cross sections mentioned before, here we observe also different shapes for the measurement and the model predictions, which indicates that the relative contributions of the interference terms $\sigma_{TT}$ and $\sigma_{LT}$ are different when comparing measurement and model.

According to Refs. [15, 10], the different terms contributing two the cross section for exclusive pseudoscalar meson production, which appear in Eq. (4), depend on GPDs $\tilde{H}$, $\tilde{E}$, $H_{T}$ and $\overline{E}_{T}$ = 2$\tilde{H}_{T}$ + $E_{T}$. For $\pi^{0}$ production a large contribution from transversely polarised virtual photons is expected, which is mainly generated by the chiral-odd GPD $\overline{E}_{T}$. It manifests itself in a large contribution from $\sigma_{TT}$ and a dip in the differential cross section ${\rm d}\sigma/{\rm d}t$ as $|t|$ decreases to zero. These features are in qualitative agreement with our results, as also with earlier measurements at different kinematics [9, 10, 13]. The COMPASS results on exclusive $\pi^{0}$ production provide significant constraints on modelling the chiral-odd GPDs, in particular GPD $\overline{E}_{T}$.

7 Summary and conclusion

Using exclusive $\pi^{0}$ muoproduction we have measured the $t$-dependence of the virtual-photon proton cross section for hard exclusive $\pi^{0}$ production at $\langle Q^{2}\rangle=2.0\;{(\text{GeV}/c)^{2}}$ $\langle\nu\rangle=12.8\;{\text{GeV}}$, $\langle x_{Bj}\rangle=0.093$ and $\langle-t\rangle=0.256\;{(\text{GeV}/c)^{2}}$. Fitting the azimuthal dependence reveals a large negative contribution by $\sigma_{TT}$ and a smaller positive one by $\sigma_{LT}$, which indicates a significant role of transversely polarised photons in exclusive $\pi^{0}$ production. These results provide important input for modelling Generalised Parton Distributions. In the context of the phenomenological GK model, the statistically significant $TT$ contribution constitutes clear experimental evidence for the existence of the chiral-odd GPD $\overline{E}_{T}$.

Acknowledgements

We thank Sergey Goloskokov and Peter Kroll for their continuous support with model predictions, Pierre Guichon for the evaluation of the Bethe-Heitler contribution taking into account the muon mass. We gratefully acknowledge the support of the CERN management and staff and the skill and effort of the technicians of our collaborating institutes.