Measurement of the cross section for hard exclusive π0 muoproduction on the proton

(1)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Measurement

of

the

cross

section

for

hard

exclusive

π

0 muoproduction

on

the

proton

.

The

COMPASS

Collaboration

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received29March2019

Receivedinrevisedform22April2020 Accepted22April2020

Availableonline5May2020 Editor:M.Doser

Keywords:

Quantumchromodynamics Muoproduction

Hardexclusivemesonproduction GeneralisedPartonDistributions COMPASS

We reportonameasurementofhard exclusive

π

0 _{muoproduction} _on_the_proton _by_COMPASS _using

160 GeV/c polarised

μ

+ and

μ

− beamsoftheCERNSPS impingingonaliquidhydrogentarget.From the average of the measured

μ

+ and

μ

− 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(GeV/c)2_<_|_t_|_<₀_.₆₄₍_GeV/c₎2_. _The _average _kinematics _of _the _measurement _are Q2₌₂_.₀₍_GeV_/_c₎2_,

_ν

₌₁₂_.₈_GeV, _x

B j=0.093 and−t=0.256(GeV/c)2. Fitting the azimuthal

dependence reveals a combined contribution by transversely and longitudinally polarised photons of (8.2 ± 0.9stat+₋11..22sys)nb/(GeV/c)

2_, _as _well _as _{transverse-transverse} _and _{longitudinal-transverse}

interferencecontributionsof(−6.1±1.3stat+₋00..77sys)nb/(GeV/c)

2_and₍₁_.₅_±₀_.₅

stat+₋00..32sys)nb/(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-transverseinterferencecontributionconstitutesclearexperimentalevidenceforthechiral-oddGPDET.

1. Introduction

Measurements of pseudoscalar mesons produced in hard ex-clusivelepton-nucleonscatteringprovideimportantdata for phe-nomenological parameterisations of Generalised Parton Distribu-tions (GPDs) [1–5]. In the past two decades, GPDs have shown to be a very rich and useful construct for both experiment and theoryastheir determination allowsfora detaileddescriptionof theparton structureof the nucleon.In particular,GPDs correlate transversespatial positionsandlongitudinal momentumfractions of the partons in the nucleon. They embed parton distribution functionsandnucleonformfactors,andtheygiveaccessto energy-momentum-tensorformfactors.Foreachquarkﬂavour,thereexist fourparton-helicity-conserving(chiral-even)GPDs,denoted H ,H ,

˜

E, and E,

˜

andfourparton-helicity-ﬂip(chiral-odd)GPDs,denoted HT, H

˜

T, ET,and E

˜

T. While hard productionofvector mesons is

sensitiveprimarilytotheGPDs H and E, theproductionof pseu-doscalarmesonsbylongitudinallypolarisedvirtualphotonsis sen-sitiveto H and

˜

E in

˜

theleading-twistdescription.

Contributionsfromtransverselypolarisedvirtualphotonstothe productionofspin-0mesonsareexpectedtobesuppressedinthe productionamplitude by 1

/

Q [6], where Q2 is the virtuality of thephoton

γ

∗ thatisexchangedbetweenmuonandproton. How-ever, experimental data on exclusive

π

+ production from HER-MES [7] and on exclusive

π

0 _production_from _JLab _{CLAS [}₈_–₁₁_] andHall A [12–14] suggest that such contributions are substan-tial. In the GPD formalism such contributions are possible if a

quark helicity-ﬂip GPD couples to a twist-3 meson wave func-tion [15,16]. Intheframeworkofthe phenomenologicalmodelof Ref. [15],pseudoscalar-mesonproductionisdescribedbytheGPDs

˜

H ,E,

˜

HT andET

=

2H

˜

T

+

ET.DifferentsensitivitiestotheseGPDs

are expectedwhen comparing

π

+ vs.

π

0 _production. _When tak-ing intoaccount the relative signsandsizes of theseGPDs foru andd quarks,thedifferentquarkﬂavourcontentsofthesemesons lead to different predictions for the

|

t

|

-dependence of the cross section, especially at smallvalues of

|

t

|

. Here, t is thesquare of thefour-momentumtransferbetweeninitialandﬁnalnucleon.The productionof

π

+mesonsisdominatedbythecontributionsfrom longitudinally polarised virtual photons, of which a major part originates from pion-pole exchange that is the main contributor to E.

˜

Also the contributions from H and

˜

HT are signiﬁcant, and

there isastrong cancellationbetweenthecontributions from ET

for u and d quarks. On the contrary, in the caseof

π

0 produc-tionthereisnopion-poleexchange,thecontributionsfromH and

˜

HT aresmallandalargecontributionfromtransverselypolarised

photonsisgeneratedmainlyby ET.

These differences between

π

+ and

π

0 _production_are exper-imentally supported. While for

π

+ production a fastdecrease of the cross section with increasing

|

t

|

is predicted by theoretical models and conﬁrmed by the experimental results from HER-MES [7],for

π

0 _production_a_dip _is_expected_as

_|

_t

_|

_→

_{0 [}₁₅_{] and} conﬁrmedbyresultsintheJLabkinematicdomain [9,10,13]. Con-straintsformodellingthepoorlyknownGPDET wereobtainedin

alattice-QCDstudy [17] ofitsmoments.TheCOMPASS resultson https://doi.org/10.1016/j.physletb.2020.135454

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Fig. 1. Deﬁnition ofφ,theazimuthalanglebetweenthelepton-scatteringandπ0 -productionplanes.

exclusive

π

0 _production _in _muon-proton _scattering_presented _in thisLetter provide newinput formodelling thisGPDand chiral-odd(‘transversity’)GPDsingeneral.

2. Formalism

Thereducedcrosssectionforhardexclusivemesonproduction by scattering a polarised lepton beam off an unpolarised proton targetreads: d2

σ

_γ∗p dtd

φ

=

1 2

π

_d

_σ

_T dt

+

d

σ

L dt

+

cos

(

2

φ)

d

σ

T T dt (1)

+

2

(

1

+

)

cos

φ

d

σ

LT dt

∓ |

Pl

|

2

(

1

−

)

sin

φ

d

σ

LT dt

,

wherethesign

∓

oftheleptonbeampolarisation Pl corresponds

to negative andpositive helicity of the incoming lepton, respec-tively, denoted by

. The conversion from the lepton-nucleon crosssectiontothevirtual-photonnucleoncrosssection,usingthe one-photon-exchange approximation, is explained in Sect.5. The contributiontothecrosssectionfromtransversely(longitudinally) polarisedvirtualphotonsisdenotedby

σ

T (

σ

L).Thesymbols

σ

LT,

σ

_LT and

σ

T T denotecontributionsfromthe interferencebetween

longitudinallyandtransversely polarisedvirtual photons, and be-tween transversely polarised virtual photons of opposite helicity. Thefactor

=

1

−

y

−

y2_γ2 4 1

−

y

+

y₂2

+

y2₄γ2 (2) isthevirtual-photonpolarisationparameterand

φ

istheazimuthal angle between the lepton scattering plane and the hadron pro-ductionplane, see Fig.1.Here, Q2

= −(

kμ

−

kμ

)

2 is thephoton virtuality,

ν

= (

k0

μ

−

k0μ

)

the energyofthe virtual photon inthe target restframe, y

=

ν

/

k0

μ and

γ

2

=

Q2

/

ν

2,where kμ and kμ denotethefour-momentaoftheincomingandthescatteredmuon inthetargetrestframe,respectively.

Thespin-independentcrosssectioncanbeobtainedby averag-ingthetwospin-dependentcrosssections,

d2

_σ

γ∗p dtd

φ

=

1 2

d2

σ

_γ←∗_p dtd

φ

+

d2

σ

_γ→∗_p dtd

φ

.

(3)

Whenformingthisaverage,thelastterminEq. (1) cancelsifthe magnitude

|

Pl

|

ofthebeampolarisationisthesamefor

measure-mentswith

μ

+and

μ

− beam,sothat

d2

_σ

γ∗p dtd

φ

=

1 2

π

_d

_σ

_T dt

+

d

σ

L dt

+

cos

(

2

φ)

d

σ

T T dt (4)

+

2

(

1

+

)

cos

(φ)

d

σ

LT dt

.

Theindividual contributionsappearinginEq. (4) arerelatedto convolutions of GPDs andmeson wave functions withindividual hardscatteringamplitudes,seeRefs. [10,15]:

d

σ

T dt

∝

(

1

− ξ

2

)

|

HT

|

2

−

t 8M2

|

ET

|

2

_,

₍₅₎ d

σ

L dt

∝

(

1

− ξ

2

)

| ˜

H

|

2

−

2

ξ

2Re

˜

H

∗

˜

E

−

t 4M2

ξ

2

_{| ˜}

_E

_|

2

_,

₍₆₎ d

σ

T T dt

∝

t

_|

_E T

|

2

,

(7) d

σ

LT dt

∝ ξ

1

− ξ

2

√

₋

_t_Re

_H T

∗

˜

E

.

(8)

Here, the aforementioned convolutionsare denoted by triangular brackets,t

=

t

−

tmin with

|

tmin

|

beingthekinematically smallest

possiblevalue of

|

t

|

,andM isthemassoftheproton.The quan-tity

ξ

isequaltoonehalfofthelongitudinalmomentumfraction transferredbetweentheinitialandﬁnalprotonandcanbe approx-imatedatCOMPASSkinematicsas

ξ

≈

xBj

2

−

xBj

,

(9)

wherexBj

=

Q2

/(

2M

ν

)

.

3. Experimentalset-upanddataselection

The main component of the COMPASS set-up is the two-stagemagneticspectrometer.Eachspectrometerstagecomprisesa dipole magnetcomplementedbya varietyoftrackingdetectors,a muon ﬁlterformuonidentiﬁcationandanelectromagnetic(ECal) aswellasahadroncalorimeter.Adetaileddescriptionofthe set-upcanbefoundinRefs. [18–20].

Thedatausedforthisanalysiswerecollectedusinga160 GeV/c muon beam within four weeksin 2012, during which the COM-PASS spectrometer was complemented by a 2.5 m long liquid-hydrogentarget surroundedby atime-of-ﬂight(TOF) system,and athirdelectromagneticcalorimeterthatwasplaceddirectly down-stream of the target [20,21]. The TOF detector consisted of two cylindersmountedconcentrically aroundthetarget,eachmadeof 24 scintillating-counterslats withread-outatboth endsofevery slat. The read-out scheme allowed to measure the time-of-ﬂight betweentwolayers,determinethehitpositionsofaparticleupon traversalthroughalayerandmeasuretheenergylossineachlayer. Thisallowsustodeterminepolarangleandmomentumofthe par-ticle.

In order to determine the spin-independent cross section through Eq. (3), data with

μ

+ and

μ

− beam were taken sepa-rately.Thenaturalpolarisationofthemuonbeamprovidedbythe CERN SPS originates from the parity-violating decay in ﬂight of theparentmeson,whichimpliesoppositepolarisationfor

μ

+and

μ

−beams.Withinregulartimeintervalsduringthemeasurement, charge andpolarisation ofthemuon beamwere swapped simul-taneously. In order to equalize the spectrometer acceptance for thetwobeamcharges,alsothepolaritiesofthetwospectrometer magnet currentswere changed accordingly.In total, aluminosity of18.9 pb−1 was collected forthe

μ

+ beamwithnegative polar-isationand23.5 pb−1 _for_the

_μ

− _beam_with_positive_{polarisation.}

(3)

embedded in the standard COMPASS data taking based on a ra-dioactivesource[22] andisknownwithanuncertaintyof3%.For bothbeams,theabsolutevalueoftheaveragebeampolarisationis about0.8withanuncertaintyofabout0.04 [18,23].

Forthe analysis, only data taken withstable beamand spec-trometerconditions are used. The selection of

π

0 _mesons _is ac-complishedusingtheirdominanttwo-photondecay.Thethreshold forthedecayphotonwithlowerenergyis300 MeV,whilethatfor the photon with higher energy is 1 GeV for the most upstream calorimeter and 2 GeV for the calorimeter in the ﬁrst stage of the spectrometer. The mostdownstream calorimeteris not used inthis analysisasit contributes only very smallstatistics to the

π

0 _sample. _At _least _two _neutral _clusters _are _required _that _had to be detected in any of the electromagnetic calorimeters above therespectivethreshold,inconjunctionwithaninteractionvertex reconstructedwithin the target usingthe incomingandoutgoing muon tracks. The outgoing muon is identiﬁed by requiring that it hasthe same charge as the beamparticle and traversesmore than 15radiation lengths. As neutralcluster we denotea recon-structed calorimeter cluster that is not associated to a charged track,thereby includinganyclusterincaseofthemostupstream calorimeterthathadnotrackingsysteminfront.

Foreach interaction vertexandeach combinationoftwo neu-tral clusters, the kinematics of the recoil proton are predicted fromthefour-momentumbalanceoftheanalysedprocess,

μ

p

→

μ

p

π

0_,

_π

0

_→

_{γ γ}

_, _by _using _the _{reconstructed} _spectrometer in-formation,i.e.the vertexposition, themomenta of theincoming andoutgoingmuonsaswellastheenergyandpositionofthetwo clusters.Thepredictedpropertiesoftherecoilproton p are com-paredtothepropertiesofeachtrackcandidateasreconstructedby theTOFsystem.Notethatthefour-momentumoftherecoilproton isdeterminedby thetargetTOF systembasedontheassumption thatthereconstructedtrackbelongstoaproton.

Thefollowing exclusivityconstraints are usedto selectevents forthecross-sectiondetermination:

|

ϕ

| <

0

.

4 rad

,

|

pT

| <

0

.

3 GeV

/

c

,

|

z

| <

16 cm

,

|

M2_X

| <

0

.

3

(

GeV

/

c2

)

2

.

Here,

ϕ

is the difference between predictedand measured az-imuthalangleoftherecoilprotoncandidate;

pT isthedifference betweenpredictedandmeasuredtransversemomentumofthe re-coilprotoncandidate;

z isthedifferencebetweenpredictedand measured hit position in the inner ring of the TOF system. The quantity pT is deﬁned in the target rest frame. The undetected massisgivenby

M2_X

= (

kμ

+

pp

−

kμ

−

pp

−

pγ1

−

pγ2

)

2

_.

₍₁₀₎

Here,thefour-momentaaredenotedby pp and pp forthetarget

andrecoil proton, respectively, and by pγ1 and pγ2 for the two

producedphotons. Inaddition aconstrainton theinvariant mass Mγ γ isused:

0

.

1092

<

Mγ γ

/(

GeV

/

c2

) <

0

.

1576

.

Inthecasethatmorethanonecombinationofvertex,clusterpair andrecoil-trackcandidateexistthatsatisfytheaforementioned se-lectioncriteriafora givenevent, thiseventisexcluded fromthe analysis.

Fig. 2 shows an example for the result of a comparison be-tweenpredictedandmeasured kinematicsoftherecoilcandidate, andFigs. 3 and 4 show correspondingly the distributions of un-detected mass and two-photon mass. In these ﬁgures, the four

Fig. 2. Measured andsimulateddistributionof pT ofthe recoilproton forthe kinematicregiondescribedinthetext.Theverticallinesindicatetheconstraints appliedfortheselectionofevents.Errorbarsdenotestatisticaluncertainties.

Fig. 3. Distribution of the undetected mass M2

X. Otherwise as in Fig.2.

Fig. 4. Distribution oftheinvariantmassMγ γ ofthetwo-photonsystem.Otherwise

asinFig.2.

exclusivityconstraintsaswell asthe

|

Mγ γ

|

constraintandsingle vertex/recoil-proton-candidate selection are applied. Additionally applied are constraintson thepull distributions of incomingand outgoing muon,the positionof ECALclustersandthe

φ

-value of hits in the recoil-proton detector, as determined using the kine-matic fit described below. Here, a pull is defined asratio of the difference betweenthe measured andthe fittedvalue of a given quantity,andthestandarddeviationofthisdifference.

Notethat the quantity presentedina givenfigure isnot con-strainedandthat everyfigure displays numericvaluesbeforethe kinematicfit.

(4)

theresulting simulateddataare treatedinthesame wayasitis doneforrealdata.

Inordertoenhancethepurityoftheselecteddataandto im-prove the precision of the particle kinematics at the interaction vertex, akinematicﬁtforthe exclusivereaction

μ

p

→

μ

p

π

0 _is performed,whichrequiresasingle

π

0 _to_decay_into_the_two pho-tonsselectedasdescribedabove.

Togetherwiththeselectionproceduregivenabove,the require-ments

0

.

08

(

GeV

/

c

)

2

<

|

t

| <

0

.

64

(

GeV

/

c

)

2

,

1

(

GeV

/

c

)

2

<

Q2

<

5

(

GeV

/

c

)

2

,

8

.

5 GeV

<

ν

<

28 GeV

areusedtoobtaintheeventsforthedeterminationoftheexclusive

π

0 _cross _section _as_described _in _Sect.₅_. _The _resulting_minimum valueofW isabout3

.

5GeV

/

c2_.

4. Estimationofthebackgroundcontribution

In orderto obtain a larger event sample for the study of the background,tworeferencesamplesareselectedinthewider kine-matic range

|

t

|

>

0

.

08

(

GeV

/

c

)

2, Q2

>

1

(

GeV

/

c

)

2 and y

>

0

.

05. Thesetwosamples aredenotedassignal andbackgroundsample in thissection. In contrastto the signal sample, the background sample contains only eventswithmore thanone combinationof vertex,clusterpairandrecoil-trackcandidate.Apartfromthesmall peakatzero(seeFig.5bottom), itcontainsnon-exclusive events. Thepurposeofthereferencesamplesisexplainedinthefollowing section.

The main background to exclusive

π

0 _{muoproduction} origi-nates from non-exclusive deep-inelastic scattering processes. In such processes, low-energy hadrons are produced in addition to the

π

0_,_which_remain_undetected_in_the_apparatus._In_order_to es-timate the background contribution,two Monte Carlogenerators areemployed.

First,theLEPTO6.5.1 generatorwiththe COMPASStuning [25] isusedtodescribe thenon-exclusivefractionofevents.Secondly, the HEPGEN++

π

0 _generator, _which _is _denoted _HEPGEN _in _this paper,is usedto modelthe kinematicsof single

π

0 muoproduc-tion [26,27]. Note that events withthe topology of exclusive

π

0 productionwereremovedfromtheLEPTOsample.

Asthere existsessentially noinformationonthe crosssection ofexclusive

π

0 _production_in_the _kinematic_domain _of_COMPASS, thetworeferencesamplesdescribedabove areusedtonormalise theHEPGENandLEPTOMonteCarloyields.Usingseveralvariables, thekinematicinformationfrombeamandspectrometer measure-mentsaswellasthatoftherecoil-protoncandidatesarecompared between experimental data and the two simulations in order to determine thebest normalisationof each simulateddata set rel-ative to that of the experimental data. This comparison is done withoutapplyingthekinematicﬁt.Asanexampleofsucha com-parison,theundetectedmassisshowninFig.5.Inadditiontothe measureddatapoints,theHEPGENsimulationandthesumofthe HEPGEN and LEPTO simulations are shown. In order to estimate the amount ofnon-exclusive background, the simulateddata are scaled such that they describe the data for both reference sam-ples.Thescaling factorfortheLEPTOMonteCarloyield, whichis denotedby f± forthetwo beamcharges,willbe usedinSect.5

to normalise this simulation when correcting the data for back-ground.

Usingthescalingfactors f±,thefractionofnon-exclusive back-groundinthedata isestimatedtobe

(

29₋+2₆

_sys

)

%. Here, the un-certainty is estimatedby comparing the scaling factors extracted forvarious variablesandby using severalextractionmethods for

Fig. 5. Data usedforthedeterminationofthebackgroundcontribution: Distribu-tionsofthe undetectedmass M2X for the signal(top)and background(bottom)

referencesamples,whichareselectedintheextendedkinematicrange.Simulated dataarealsoshown(seetext).Errorbarsdenotestatisticaluncertainties.

the scalingfactors.Detailsare giveninRef. [28].Contributionsof other backgroundsourcesarefoundtobenegligible.Forexample, theproductionofsingle

ω

mesons,wherethe

ω

decaysintoa

π

0 andaphotonthat remainsundetected,wasfound inMonteCarlo studiestocontributeatthelevelof1% [28].

5. Determinationofthecrosssection

The virtual-photon proton cross section is obtained from the measuredmuon-protoncrosssectionusing

d2

σ

d

|

t

|

d

φ

=

1

(

Q2

_,

_ν

_,

_E μ

)

d

σ

μp dQ2_d

_ν

_d

_φ

_d

_|

_t

_|

,

(11)

wherethetransversevirtual-photonﬂuxisgivenby

(

Q2

,

ν

,

Eμ

)

=

α

em

(

1

−

xBj

)

2

π

Q2_{y E} μ

y2

1

−

2m 2 μ Q2

+

2 1

+

Q2

_/

_ν

2

1

−

y

−

Q 2 4E2 μ

.

(12)

Here,

α

em denotestheelectromagneticﬁnestructureconstantand Eμ

=

k0_μ.

For thecross section determination,the HEPGEN MonteCarlo simulationdescribed inSect.4isused.Theacceptancea

(

klmn

)

is calculated ina four-dimensionalgrid as thenumber of recon-structed eventsdivided bythe numberofgeneratedeventsusing 8 bins in

φ

, 5 in

|

t

|

, 4 in Q2 and 4 in

ν

, including bin-to-bin event migration. The phase-space element is given by

klmn

=

φ

k

|

t

|

l

Qm2

ν

n.ThespacingofthegridisgiveninTable1.

(5)

Table 1

Four-dimensionalgridusedforthecalculationoftheacceptance.The fullwidthoftherespectivedimensionisgiveninthebottomrowof thetable.

φ/rad |t|/(GeV/c)2 _Q2_/₍_GeV_/_c₎2 _ν_/GeV −π–−43π 0.08 – 0.15 1 – 1.5 8.5 – 11.45 −3π 4 –−π2 0.15 – 0.22 1.5 – 2.24 11.45 – 15.43 . 0.22 – 0.36 2.24 – 3.34 15.43 – 20.78 . 0.36 – 0.5 3.34 – 5 20.78 – 28 . 0.5 – 0.64 3π 4 – π φ/rad |t|/(GeV/c)2 Q2_/ (GeV/c)2 ν/GeV 2π 0.56 4 19.5

Y

_klmn±

=

N±, klmn data

i=1 1

(

Q_i2

,

ν

i

,

Eμ,i

)

−

f± N±, klmn L

i=1 1

(

Q_i2

,

ν

i

,

Eμ,i

)

.

(13) Here, N±, klmn

data is thenumberofmeasuredeventsandN

±, klmn

L the number of LEPTO events within the phase-space element

klmn.ThesecondsumrepresentstheLEPTOsimulationsthatare

appropriatelynormalisedby thefactor f±,whichwas introduced in Sect. 4. Each event is weighted with the transverse virtual-photon ﬂux

(

Q_i2

,

ν

i

,

Eμ,i

)

in order to obtain the virtual-photon

yieldfromthemeasuredyields formuon-protoninteractions, and withthe

π

0

_→

_{γ γ}

_branching_ratio._Radiative _corrections_are _not appliedbuttakenintoaccountassystematicuncertainty.

Thespin-dependent virtual-photonprotoncrosssections mea-suredwithpositivelyornegativelychargedmuonsaredetermined ineachofthe(

φ

k,

|

t

|

l)binsasluminosity-normalisedexperimental

yield averaged over the measured ranges

Q2

₌

₄

₍

_GeV

_/

_c

₎

2 _and

ν

=

19

.

5GeV as

_d2

_σ

d

|

t

|

d

φ

± kl

=

1

L

±

_kl

mn

Y

_klmn± a

(

klmn

)

.

(14)

Here,

kl

= φ

k

|

t

|

l

Q2

ν

,

L

± denotes the luminosity and

a

(

klmn

)

theacceptanceinthephase-spaceelement

klmn.

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

_d2

_σ

d

|

t

|

d

φ

kl

=

1 2

_d2

_σ

d

|

t

|

d

φ

+ kl

+

d2

σ

d

|

t

|

d

φ

− kl

.

(15)

Thecrosssectionintegratedoverthefull2

π

-rangein

φ

isobtained as

_d

_σ

d

|

t

|

l

=

k

φ

k

_d2

_σ

d

|

t

|

d

φ

kl

,

(16)

with

l

= |

t

|

l

Q2

ν

.Similarly, the

|

t

|

-averagedcrosssection

inthemeasuredrangeisgivenby

_d2

_σ

d

|

t

|

d

φ

k

=

1

|

t

|

l

|

t

|

l

_d2

_σ

d

|

t

|

d

φ

kl

,

(17) with

k

= φ

k

|

t

|

Q2

ν

.

The systematic uncertainties on the extracted values of the crosssection are shownin Table 2,arranged infour groups. The ﬁrst group contains the systematicuncertainties fromthe deter-minationofthe integratedbeamﬂux.The second group contains

Table 2

Summary ofthe estimatedrelative systematic uncertainties for the |t| and φ -dependentcrosssectionsandtheintegratedcrosssection.Thevaluesaregivenin percent.Notethattheuni-directionaluncertaintyσ↑isapositivenumber,andσ↓ isanegativenumber. Source σt ↑ −σ↓t σ↑φ −σ↓φ σ↑ −σ↓ μ+flux 2 2 2 2 2 2 μ−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 ωbackground 0 1 0 1 0 1 rad. corr. 2 5 2 5 2 5 Leptonorm. 5–28 3–11 5–51 3–21 8 3 yield mismatch 4–13 3–7 0–12 3–12 9 5 12–29 13–18 12–53 13–25 14 14

possiblesystematiceffectsstudiedbytheMonteCarlosimulation, which are related to the uncertainty on the energy thresholds forthe detection ofthe low-energetic photon inthe electromag-netic calorimeters, and the uncertainty on the determination of the acceptance. The third group contains the systematic uncer-tainties relatedto a variationof the energyandmomentum bal-ance ofthekinematicﬁt,the inﬂuenceofbackgroundoriginating fromtheproductionof

ω

mesonsandthe estimatedinﬂuence of radiative correctionsincluding thepossible impact ofa

φ

modu-lation [28,29]. The largestsystematiceffects appearinthe fourth group,whichcontainstwoelements:(i)theuncertaintyrelatedto theestimationofnon-exclusivebackgroundasdescribedinSect.4; (ii) the uncertainty due to an observed mismatch between the measured single-photonyield inthe2012COMPASS dataandthe correspondingMonteCarlosimulationoftheBethe-Heitlerprocess. It appears in akinematic region wheresingle-photon production is dominatedby theBethe-Heitler cross section, anditis related to different intensities of positive and negative muon beams for the data analysed in this paper. The mismatch is discussed in Refs. [21,30]. The total systematic uncertainty

is obtained by quadraticsummationofitscomponentsforeachbinseparately.

6. Results

For the background corrected ﬁnal data sample the average kinematicsare

Q2

₌

₂

_.

₀

₍

_GeV

_/

_c

₎

2_,

_ν

₌

₁₂

_.

₈_GeV,

_x

B j

=

0

.

093

and

−

t

=

0

.

256

(

GeV

/

c

)

2_. _The _dependences _of _the _measured cross section on

|

t

|

and

φ

are shownin Fig.6, withthe numer-ical values given in Table 3. The cross section in bins of

|

t

|

is showninthetoppanelofFig.6.Itappearstobe consistentwith anexponential decreasewithincreasing

|

t

|

forvaluesof

|

t

|

larger than about 0.25 (GeV/c)2_, _while _at _smaller

_|

_t

_|

_the _t-dependence becomesweaker.Ourresultiscomparedtothepredictionsoftwo versionsoftheGoloskokov-Kroll(GK)model [15,31].Theresultsof theGKmodelshowninthisletterareobtainedbyintegratingover theanalysisrangeinthesamewayasitisdone forthedata.The dashed-dotted curverepresentsthe crosssection fromtheearlier version [15] asafunctionof

|

t

|

,whiletheupwardspointing trian-glescorrespond tothecross sectionaveraged over

|

t

|

bins ofthe data.Themeancrosssectionsforthefullt-rangearecomparedin therightmostpartofthispanel.Analogously,thedottedcurveand the downward pointingtriangles correspond to the later version ofthemodel [31],whichwasinspiredby theresultspresentedin this Letter. We observe that forthe earlier version of the model themagnitudeofthepredictedcrosssectionovershootsour mea-surementbyapproximatelyafactoroftwo.

(6)

Fig. 6. Average valueofthedifferentialvirtual-photonprotoncrosssection dσ

d|t| asa functionof|t|(top)and dd|t2|σdφas afunctionofφ (bottom). Forthe top

panelthedatawasintegratedoverφ,whileforthebottompanelitwasaveraged overtherange0.08(GeV/c)2_<_|t|_<₀_.₆₄₍_GeV_/_c₎2_._The_result_of_an_integration_over φ and|t|isshowninthe right-mostpartofthetoppanel.Inner errorbars in-dicatethestatisticaluncertainty,outererrorbarsthequadraticsumofstatistical andsystematicuncertainties.ThedataiscomparedwithtwopredictionsoftheGK model [15,31].Radiativecorrectionsarenotappliedbutanestimateisincludedin thesystematicuncertainties.

Table 3

NumericalvaluesoftheaveragecrosssectionsshowninFig.6.Fordetailssee cap-tionofFig.6. Lowerφ binlimit _d2_σ d|t|dφ /₍_GeVnb_/_c₎2 Lower|t| binlimit _d σ d|t| /₍_GeVnb_/_c₎2 −π 0.4+₋00..43stat +0.1 −0.1sys 0.08 16.4 +3.6 −3.1stat +2.0 −2.3sys −3π 4 2.1+ 0.7 −0.6stat+ 0.3 −0.3sys 0.15 16.4+ 3.8 −3.2stat+ 2.1 −2.1sys −π 2 2.1+ 0.5 −0.4stat +0.3 −0.3sys 0.22 11.6 +2.6 −2.2stat +1.5 −1.5sys −π 4 1.1+ 0.4 −0.3stat+ 0.2 −0.1sys 0.36 3.4+ 1.4 −1.2stat+ 0.8 −0.5sys 0 1.2+0.5 −0.4stat+ 0.2 −0.2sys 0.5 1.5+ 1.0 −0.8stat+ 0.4 −0.3sys π 4 1.9+ 0.5 −0.4stat +0.3 −0.2sys π 2 1.6+ 0.5 −0.4stat+ 0.2 −0.2sys 3π 4 0.2 +0.2 −0.1stat +0.1 −0.0sys

binsofequalwidth.Thefulldotsshowthemeasuredcrosssection foreachbinandthesolidcurverepresentsthefitdescribedbelow. In order to extract the different contributions to the spin-independentcrosssection,abinnedmaximum-likelihoodfitis ap-plied to the data according to Eq. (4). In the fit, the measured averagevalueofthevirtual-photonpolarisationparameterisused,

=

0

.

996.The

φ

-integratedcrosssectiondeterminedby theﬁtis obtainedas

_d

_σ

T d

|

t

|

+

d

σ

L d

|

t

|

= (

8

.

2

±

0

.

9stat₋+₁1._.2₂

_sys

)

nb

(

GeV

/

c

)

2

.

(18) TheT T andLT interferencetermsareobtainedas

_d

_σ

_{T T} d

|

t

|

= (−

6

.

1

±

1

.

3stat₋+00..77

sys

)

nb

(

GeV

/

c

)

2 (19) and

_d

_σ

_LT d

|

t

|

= (

1

.

5

±

0

.

5stat₋+₀0._.3₂

_sys

)

nb

(

GeV

/

c

)

2

.

(20) We observe a large negative contribution by

σ

T T and a smaller

positive one by

σ

LT, which indicates a signiﬁcant role of

trans-verselypolarisedphotonsinexclusive

π

0_production.

The

φ

-dependenceofthemeasured cross sectionis compared tothecalculationsoftheGKmodelinthebottompanel ofFig.6. Apart from the discrepancy in the magnitude of cross sections mentioned before,here we observe also different shapesfor the measurementandthemodelpredictions,whichindicates thatthe relative contributions ofthe interference terms

σ

T T and

σ

LT are

differentwhencomparingmeasurementandmodel.

According to Eqs. (5) to (8), the different terms contributing to thecrosssection forexclusivepseudoscalarmesonproduction, which appear in Eq. (4), depend on GPDs H ,

˜

E,

˜

HT and ET

=

2H

˜

T

+

ET.For

π

0productionalargecontributionfromtransversely

polarised virtual photonsis expected, which is mainly generated by the chiral-oddGPD ET. It manifestsitself ina large

contribu-tionfrom

σ

T T andadipinthedifferentialcrosssection d

σ

/

dt as

|

t

|

decreasesto zero. Thesefeatures are in qualitative agreement with ourresults andalso withearlier measurements atdifferent kinematics [9,10,13]. The COMPASS results on exclusive

π

0 pro-ductionprovidesigniﬁcantconstraintsonmodellingthechiral-odd GPDs,inparticularGPD ET.

7. Summaryandconclusion

Using exclusive

π

0 _{muoproduction} _we _have _measured _the t-dependence of the virtual-photon proton cross section for hard exclusive

π

0 _production _at

_Q2

₌

₂

_.

₀

₍

_GeV

_/

_c

₎

2_,

_ν

₌

₁₂

_.

₈_GeV,

xB j

=

0

.

093 and

−

t

=

0

.

256

(

GeV

/

c

)

2. Fitting the azimuthal

dependence reveals a large negative contribution by

σ

T T and a

smaller positive one by

σ

LT, whichindicates a signiﬁcant role of

transversely polarised photons inexclusive

π

0 _production._These results provide importantinput formodelling Generalised Parton Distributions. In the context ofthe phenomenological GK model, thestatisticallysigniﬁcantT T contributionconstitutesclear exper-imentalevidencefortheexistenceofthechiral-oddGPDET.

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompeting ﬁnan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

Acknowledgements

WethankSergeyGoloskokovandPeter Krollfortheir continu-ous support withmodelpredictions,Pierre Guichonforthe eval-uation of the Bethe-Heitler contribution taking into account the muonmassandAndrei Afanasevforthediscussionontheradiative corrections. We gratefully acknowledge the support ofthe CERN managementandstaffandtheskillandeffortofthetechniciansof ourcollaboratinginstitutes.

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a_University_of_Aveiro,_I3N_{- Physics}_Department,_3810-193_Aveiro,_Portugal b_Universität_Bochum,_Institut_für_{Experimentalphysik,}₄₄₇₈₀_Bochum,_Germany18,19

c_Universität_Bonn,_{Helmholtz-Institut}_für_{Strahlen- und}_Kernphysik,₅₃₁₁₅_Bonn,_Germany18

d_Universität_Bonn,_{Physikalisches}_Institut,₅₃₁₁₅_Bonn,_Germany18

e_Institute_of_Scientiﬁc_Instruments_of_the_CAS,₆₁₂₆₄_Brno,_Czech_Republic20

f_Matrivani_Institute_of_Experimental_Research_&_Education,_Calcutta-700_030,_India21

g_Joint_Institute_for_Nuclear_Research,₁₄₁₉₈₀_Dubna,_Moscow_region,_Russia h_Universität_Freiburg,_{Physikalisches}_Institut,₇₉₁₀₄_Freiburg,_Germany18,19

i_CERN,₁₂₁₁_Geneva_23,_Switzerland

j_Technical_University_in_Liberec,₄₆₁₁₇_Liberec,_Czech_Republic20

(8)

l_Universität_Mainz,_Institut_für_Kernphysik,₅₅₀₉₉_Mainz,_Germany18

m

UniversityofMiyazaki,Miyazaki889-2192,Japan23 n_Lebedev_Physical_Institute,₁₁₉₉₉₁_Moscow,_Russia

o_Technische_Universität_München,_Physik_Dept.,₈₅₇₄₈_Garching,_Germany18,5

p_Nagoya_University,₄₆₄_Nagoya,_Japan23

q_Charles_University_in_Prague,_Faculty_of_Mathematics_and_Physics,₁₂₁₁₆_Prague,_Czech_Republic20

r_Czech_Technical_University_in_Prague,₁₆₆₃₆_Prague,_Czech_Republic20

s_State_Scientiﬁc_Center_Institute_for_High_Energy_Physics_of_National_Research_Center_‘Kurchatov_{Institute’,}₁₄₂₂₈₁_Protvino,_Russia t_IRFU,_CEA,_Université_{Paris-Saclay,}₉₁₁₉₁_{Gif-sur-Yvette,}_France19

u_Academia_Sinica,_Institute_of_Physics,_Taipei_11529,_Taiwan24

v_Tel_Aviv_University,_School_of_Physics_and_Astronomy,₆₉₉₇₈_Tel_Aviv,_Israel25

w_University_of_Trieste,_Dept._of_Physics,₃₄₁₂₇_Trieste,_Italy x_Trieste_Section_of_INFN,₃₄₁₂₇_Trieste,_Italy

y_University_of_Turin,_Dept._of_Physics,₁₀₁₂₅_Turin,_Italy z_Torino_Section_of_INFN,₁₀₁₂₅_Turin,_Italy

aa_Tomsk_Polytechnic_University,₆₃₄₀₅₀_Tomsk,_Russia26

ab_University_of_Illinois_at_{Urbana-Champaign,}_Dept._of_Physics,_Urbana,_IL_61801-3080,_USA27

ac_National_Centre_for_Nuclear_Research,_00-681_Warsaw,_Poland28

ad_University_of_Warsaw,_Faculty_of_Physics,_02-093_Warsaw,_Poland28

ae_Warsaw_University_of_Technology,_Institute_of_{Radioelectronics,}_00-665_Warsaw,_Poland28

af_Yamagata_University,_Yamagata_992-8510,_Japan23

*

Correspondingauthors.

E-mailaddresses:oleg.denisov@cern.ch(O.Yu. Denisov),jan.friedrich@cern.ch(J.M. Friedrich). 1 _Also_at_Instituto_Superior_Técnico,_Universidade_de_Lisboa,_Lisbon,_Portugal.

2 _Also_at_Dept._of_Physics,_Pusan_National_University,_Busan_609-735,_Republic_of_Korea. 3 _Also_{at Physics}_Dept.,_Brookhaven_National_Laboratory,_Upton,_NY_11973,_USA. 4 _Also_at_Abdus_Salam_ICTP,₃₄₁₅₁_Trieste,_Italy.

5 _Supported_by_the_DFG_cluster_of_excellence_‘Origin_and_Structure_of_the_Universe’_{(www.universe}_-cluster.de)_(Germany). 6 _Supported_by_the_DFG_Research_Training_Group_Programmes₁₁₀₂_and₂₀₄₄_(Germany).

7 _Also_at_Chubu_University,_Kasugai,_Aichi_487-8501,_Japan.

8 _Also_at_Dept._of_Physics,_National_Central_University,₃₀₀_Jhongda_Road,_Jhongli_32001,_Taiwan. 9 _Also_at_KEK,_1-1_Oho,_Tsukuba,_Ibaraki_305-0801,_Japan.

10 _Present_address:_Universität_Bonn,_{Physikalisches}_Institut,₅₃₁₁₅_Bonn,_Germany. 11 _Also_at_Moscow_Institute_of_Physics_and_Technology,_Moscow_Region,_141700,_Russia. 12 _Also_at_Yerevan_Physics_Institute,_Alikhanian_Br._Street,_Yerevan,_0036,_Armenia.

13 _Also_at_Dept._of_Physics,_National_Kaohsiung_Normal_University,_Kaohsiung_County_824,_Taiwan. 14 _Supported_by_the_P2IO_LabEx_Grant_No._{ANR-10-LABX-0038,}_Agence_Nationale_de_la_Recherche,_France. 15 _Also_at_Institut_für_Theoretische_Physik,_Universität_Tübingen,₇₂₀₇₆_Tübingen,_Germany.

16 _Also_at_University_of_Eastern_Piedmont,₁₅₁₀₀_Alessandria,_Italy.

17 _Present_address:_RWTH_Aachen_University,_III._{Physikalisches}_Institut,₅₂₀₅₆_Aachen,_Germany. 18 _Supported_by_BMBF_{- Bundesministerium}_für_Bildung_und_Forschung_(Germany).

19 _Supported_by_FP7,_{HadronPhysics3,}_Grant₂₈₃₂₈₆_(European_Union). 20 _Supported_by_MEYS,_Grant_LM20150581_(Czech_Republic). 21 _Supported_by_B.Sen_fund_(India).

22 _Supported_by_FCT,_COMPETE_and_QREN,_Grants_CERN/FP_116376/2010,_123600/2011_and_{CERN/FIS-NUC/0017/2015}_(Portugal). 23 _Supported_by_MEXT_and_JSPS,_Grants_18002006,_20540299,_18540281_and_26247032,_the_Daiko_and_Yamada_Foundations_(Japan). 24 _Supported_by_the_Ministry_of_Science_and_Technology,_Taiwan.

25 _Supported_by_the_Israel_Academy_of_Sciences_and_Humanities_(Israel).

26 _Supported_by_the_Russian_Federation_program_“Nauka”_(Contract_No._{0.1764.GZB.2017)}_(Russia). 27 _Supported_by_the_National_Science_Foundation,_Grant_no._PHY-1506416_(USA).

28 _Supported_by_NCN,_Grant_{2017/26/M/ST2/00498}_(Poland). 29 _Retired.