Measurement of beam asymmetry for π − Δ ++ photoproduction on the proton at E γ =8.5 GeV
GlueX Collaboration, S. Adhikari, C. S. Akondi, A. Ali, M. Amaryan, A. Asaturyan, A. Austregesilo, Z. Baldwin, F. Barbosa, J. Barlow, E. Barriga, R. Barsotti, T. D. Beattie, V. V. Berdnikov, T. Black, W. Boeglin, W. J. Briscoe, T. Britton, W. K. Brooks, B. E. Cannon, E. Chudakov, S. Cole, O. Cortes, V. Crede, M. M. Dalton, T. Daniels, A. Deur, S. Dobbs, A. Dolgolenko, R. Dotel, M. Dugger, R. Dzhygadlo, H. Egiyan, T. Erbora, A. Ernst, P. Eugenio, C. Fanelli, S. Fegan, J. Fitches, A. M. Foda, S. Furletov, L. Gan, H. Gao, A. Gasparian, C. Gleason, K. Goetzen, V. S. Goryachev, L. Guo, H. Hakobyan, A. Hamdi, G. M. Huber, A. Hurley, D. G. Ireland, M. M. Ito, I. Jaegle, N. S. Jarvis, R. T. Jones, V. Kakoyan, G. Kalicy, M. Kamel, V. Khachatryan, M. Khatchatryan, C. Kourkoumelis, S. Kuleshov, A. LaDuke, I. Larin, D. Lawrence, D. I. Lersch, H. Li, W. B. Li, B. Liu, K. Livingston, G. J. Lolos, K. Luckas, V. Lyubovitskij, D. Mack, H. Marukyan, V. Matveev, M. McCaughan, M. McCracken, W. McGinley, C. A. Meyer, R. Miskimen, R. E. Mitchell, K. Mizutani, V. Neelamana, F. Nerling, L. Ng, A. I. Ostrovidov, Z. Papandreou, C. Paudel, P. Pauli, R. Pedroni, L. Pentchev, K. J. Peters, W. Phelps, E. Pooser, J. Reinhold, B. G. Ritchie, J. Ritman, et al. (37 additional authors not shown)
MMeasurement of beam asymmetry for π − ∆ ++ photoproductionon the proton at E γ =8.5 GeV S. Adhikari, C. S. Akondi, A. Ali,
11, 12
M. Amaryan, A. Asaturyan, A. Austregesilo, Z. Baldwin, F. Barbosa, J. Barlow, E. Barriga, R. Barsotti, T. D. Beattie, V. V. Berdnikov, T. Black, W. Boeglin, W. J. Briscoe, T. Britton, W. K. Brooks, B. E. Cannon, E. Chudakov, S. Cole, O. Cortes, V. Crede, M. M. Dalton, T. Daniels, A. Deur, S. Dobbs, A. Dolgolenko, R. Dotel, M. Dugger, R. Dzhygadlo, H. Egiyan, T. Erbora, A. Ernst, P. Eugenio, C. Fanelli, S. Fegan, J. Fitches, A. M. Foda, S. Furletov, L. Gan, H. Gao, A. Gasparian, C. Gleason, K. Goetzen, V. S. Goryachev, L. Guo, H. Hakobyan, A. Hamdi,
11, 12
G. M. Huber, A. Hurley, D. G. Ireland, M. M. Ito, I. Jaegle, N. S. Jarvis, R. T. Jones, V. Kakoyan, G. Kalicy, M. Kamel, V. Khachatryan, M. Khatchatryan, C. Kourkoumelis, S. Kuleshov, A. LaDuke, I. Larin, D. Lawrence, D. I. Lersch, H. Li, W. B. Li, B. Liu, K. Livingston, G. J. Lolos, K. Luckas, V. Lyubovitskij,
28, 29
D. Mack, H. Marukyan, V. Matveev, M. McCaughan, M. McCracken, W. McGinley, C. A. Meyer, R. Miskimen, R. E. Mitchell, K. Mizutani, V. Neelamana, F. Nerling,
11, 12
L. Ng, A. I. Ostrovidov, Z. Papandreou, C. Paudel, P. Pauli, R. Pedroni, L. Pentchev, K. J. Peters,
11, 12
W. Phelps, E. Pooser, J. Reinhold, B. G. Ritchie, J. Ritman, G. Rodriguez, D. Romanov, C. Romero, C. Salgado, S. Schadmand, A. M. Schertz, A. Schick, R. A. Schumacher, J. Schwiening, X. Shen, M. R. Shepherd, ∗ A. Smith, E. S. Smith, D. I. Sober, A. Somov, S. Somov, O. Soto, J. R. Stevens, I. I. Strakovsky, B. Sumner, K. Suresh, V. V. Tarasov, S. Taylor, A. Teymurazyan, A. Thiel, G. Vasileiadis, T. Whitlatch, N. Wickramaarachchi, M. Williams, Y. Yang, J. Zarling, † Z. Zhang, Z. Zhao, J. Zhou, Q. Zhou, X. Zhou, and B. Zihlmann (The GlueX
Collaboration) Arizona State University, Tempe, Arizona 85287, USA National and Kapodistrian University of Athens, 15771 Athens, Greece Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA The Catholic University of America, Washington, D.C. 20064, USA University of Connecticut, Storrs, Connecticut 06269, USA Duke University, Durham, North Carolina 27708, USA Florida International University, Miami, Florida 33199, USA Florida State University, Tallahassee, Florida 32306, USA The George Washington University, Washington, D.C. 20052, USA University of Glasgow, Glasgow G12 8QQ, United Kingdom Goethe University Frankfurt, 60323 Frankfurt am Main, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, D-64291 Darmstadt, Germany Institute of High Energy Physics, Beijing 100049, People’s Republic of China Indiana University, Bloomington, Indiana 47405, USA Alikhanov Institute for Theoretical and Experimental Physics NRC Kurchatov Institute, Moscow 117218, Russia Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA Forschungszentrum Juelich Nuclear Physics Institute, 52425 Juelich, Germany University of Massachusetts, Amherst, Massachusetts 01003, USA Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA National Research Nuclear University Moscow Engineering Physics Institute, Moscow 115409, Russia Norfolk State University, Norfolk, Virginia 23504, USA North Carolina A&T State University, Greensboro, North Carolina 27411, USA University of North Carolina at Wilmington, Wilmington, North Carolina 28403, USA Northwestern University, Evanston, Illinois 60208, USA Old Dominion University, Norfolk, Virginia 23529, USA University of Regina, Regina, Saskatchewan, Canada S4S 0A2 Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V Valpara´ıso, Chile Tomsk State University, 634050 Tomsk, Russia Tomsk Polytechnic University, 634050 Tomsk, Russia A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute), 0036 Yerevan, Armenia College of William and Mary, Williamsburg, Virginia 23185, USA Wuhan University, Wuhan, Hubei 430072, People’s Republic of China (Dated: September 15, 2020)We report a measurement of the π − photoproduction beam asymmetry for the reaction (cid:126)γp → π − ∆ ++ using data from the GlueX experiment in the photon beam energy range 8.2–8.8 GeV. Theasymmetry Σ is measured as a function of four-momentum transfer t to the ∆ ++ and compared to a r X i v : . [ nu c l - e x ] S e p phenomenological models. We find that Σ varies as a function of t : negative at smaller values andpositive at higher values of | t | . The reaction can be described theoretically by t -channel particleexchange requiring pseudoscalar, vector, and tensor intermediaries. In particular, this reactionrequires charge exchange, allowing us to probe pion exchange and the significance of higher-ordercorrections to one-pion exchange at low momentum transfer. Constraining production mechanismsof conventional mesons may aid in the search for and study of unconventional mesons. This is thefirst measurement of the process at this energy. I. INTRODUCTION
Determining the types of mesons that emerge fromquantum chromodynamics (QCD) is a critical experimen-tal input to our understanding of how QCD generates theproperties of hadrons [1]. The
GlueX experiment at Jef-ferson Lab provides a unique opportunity to search fornon- q ¯ q mesons and, by using a linearly polarized pho-ton beam, study their production dynamics in additionto their decay properties. The GlueX photon beam en-ergy of 8-9 GeV is in a regime where photoproductionof hadrons can be described by t -channel exchange pro-cesses [2], and the properties of exchanged Reggeons canbe constrained by experimental data. In particular, thelinear polarization of the beam allows one to distinguishbetween exchange of particles with natural ( P ( − J = 1)and unnatural ( P ( − J = −
1) parity [3, 4]. Ultimately,this gives insight into the coupling of the produced me-son and the photon to particular sets of Reggeons. Thisknowledge of production mechanisms for known mesonscan be leveraged in the future search for exotic hybridmesons using
GlueX data.Measurements that constrain production mechanismsat photon beam energies relevant for the
GlueX exper-iment are sparse. Recent measurements on the photo-production of pseudoscalar mesons [5–7] have begun toprovide insight into into production mechanisms. In thispaper, we seek to extend this understanding by measur-ing the beam asymmetry Σ for the charge-exchange re-action (cid:126)γp → π − ∆ ++ , where Σ = 1 (Σ = −
1) is indica-tive of pure natural (unnatural) parity exchange. Wefind that the asymmetry varies significantly over Man-delstam t , demonstrating the need for unnatural pionexchange as well as natural exchanges such as ρ and a .This reaction has been of theoretical interest for sev-eral decades [8–10]; however, most prior measurementshave been made at lower energies [11–15]. At energiesof E γ = 1 − t -channel and s -channel pro-cesses contribute to single pseudoscalar photoproduction,and often the experimental focus is on s -channel baryonresonances. Our measurements at higher energy will con-strain the t -channel background for these investigations.We report the first measurement of beam asymmetryΣ for π − photoproduction at 8.5 GeV. The analysis uti-lizes 20 pb − of data collected by the GlueX experi- ∗ Corresponding author: [email protected] † Corresponding author: [email protected] ment in 2017 at the Hall D facility. We compare our re-sults to theoretical predictions at E γ =8.5 GeV providedby the JPAC Collaboration [16] and B.-G. Yu and K.-J. Kong [17]. These models are informed by cross sec-tion and asymmetry results for this reaction measured at E γ =16 GeV with data from SLAC [18], the only previousmeasurement in this energy regime. II. EXPERIMENTAL APPARATUS
The
GlueX experiment utilizes the 12 GeV Contin-uous Electron Beam Accelerator Facility (CEBAF) toproduce a beam of linearly polarized photons via co-herent bremsstrahlung radiation on a thin (50 µ m) di-amond wafer [19]. Measuring the momentum of the elec-tron after radiation using a hodoscope allows the energyof the radiated photon to be determined with a resolu-tion of 10 MeV in the beam energy range of interest.By orienting the radiator, one may tune the coherentbremsstrahlung peak energy and direction of linear po-larization. Four data sets of approximately equal statis-tics were collected with the coherent bremsstrahlung en-hancement in the 8.2–8.8 GeV region and polarizationoriented in four directions relative to the laboratory floorplane: -45 ◦ , 0 ◦ , 45 ◦ , and 90 ◦ . We group these indepen-dent data sets in pairs of orthogonal orientations andrefer to them as ‘0/90’ or ‘-45/45’, each of which is usedto make a measurement of the observable of interest.Within each set we label the 0 and -45 as (cid:107) and the90 and 45 as ⊥ .Beam photons travel 75 m from the radiator and passthrough a 5 mm diameter collimator to enhance the po-larization, as coherent bremsstrahlung photons are pref-erentially produced at small angles with respect to thebeam axis. A downstream 75 µ m beryllium converterallows for photon beam flux and polarization measure-ments. Flux is measured from e + e − pair productionmeasured in a pair spectrometer (PS) [20]. Polarizationis measured via detection of the recoil atomic electronof the triplet production process in the triplet polarime-ter (TPOL) [21]. The azimuthal angle of this electron issensitive to the photon polarization plane. The photonpolarization is measured independently for each polar-ization direction as a function of E γ , with polarizationvalues up to 40%, as shown in Fig. 1. The statistical un-certainty in polarization is determined by the number oftriplet production events detected. The systematic un-certainty of the instrument is 1.5%.The GlueX spectrometer is an azimuthally symmetric P o l a r i z a t i o n E γ [GeV] Figure 1. The degree of linear polarization for four differentorientations of diamond radiator as a function of beam photonenergy, as measured by the TPOL. Events between the dashedlines (8.2 GeV < E γ < detector located in Hall D of Jefferson Lab. The centralelements of the detector are housed in a 2 T supercon-ducting solenoid. Incident beam photons interact in a30 cm long target filled with liquid hydrogen. The targetis surrounded by the Start Counter (ST) [22], a scintil-lating detector which provides determination of the pri-mary interaction time and allows for matching to radiat-ing electrons in the upstream tagger.Charged particles exiting the target are measured bytwo drift chamber systems: the Central Drift Cham-ber (CDC) [23, 24] and the Forward Drift Chamber(FDC) [25, 26]. The CDC consists of 28 layers of strawtubes surrounding the target region arranged in stereoand axial layers, providing track reconstruction to be-yond 120 ◦ and allowing for proton-pion separation belowabout 1 GeV/ c based on energy loss ( dE/dx ). The FDC,located immediately downstream, consists of four planarpackages. Each FDC package contains anode wire andcathode strip readouts. These two tracking systems allowfor charged track reconstruction with uniform azimuthalcoverage, polar angle coverage from 1 ◦ to beyond 120 ◦ ,and a momentum resolution of about 1-7% depending onmomentum and direction. In the forward direction, aTime-of-Flight (TOF) scintillator wall [27, 28] providesadditional charged particle timing information.Photon detection with the GlueX spectrometer is per-formed with two distinct electromagnetic calorimeters:the Barrel Calorimeter (BCAL) [29] and the ForwardCalorimeter (FCAL) [30]. The BCAL surrounds the twodrift chambers and is composed of 48 azimuthal lead-scintillating fiber matrix segments. The BCAL providespolar angle coverage from 11 ◦ to 120 ◦ . The FCAL islocated approximately 6 m downstream from the tar-get and consists of 2,800 lead-glass blocks in a circulararrangement, providing azimuthally symmetric coveragefor polar angles 1 ◦ to 11 ◦ . Detector readout is triggeredbased upon energy deposition in the two calorimeters. III. EVENT SELECTION
We detect the ∆ ++ baryon via its dominant de-cay ∆ ++ → π + p , hence we reconstruct the final state (cid:126)γp → π + π − p . A beam energy satisfying 8 . < E γ < . ◦ and with momen-tum lower than 1 GeV/ c . In this case, energy loss dE/dx measured in the CDC is effective at further distinguishingproton and π + candidates.Each reconstructed event is also required to bematched to a suitable reconstructed radiating electronthat is a candidate for the electron that radiated thebeam photon. The momentum of this electron deter-mines the photon energy. The CEBAF accelerator de-livers the electron beam in bunches with a 4 ns period.Hit information from the ST determines the beam bunch,and a precise value of arrival time of the bunch at the tar-get center ( t bunch ) is provided by the accelerator radio-frequency clock. We require electron candidates have atime t e such that | t e − t bunch | < < | t e − t bunch | <
18 ns. This selects eightadditional beam bunches which, when scaled appropri-ately, can be used to remove the contribution of theseaccidentals to the analysis.We impose several constraints to ensure the purity ofthe exclusive reaction of interest. First, the measuredmissing mass squared is required to satisfy | p i − p f | < . to suppress the contribution from events withundetected massive particles, where p i and p f are the sumof all initial and final four-momenta respectively. Then,a kinematic fit is performed, enforcing conservation ofenergy and momentum and a common vertex, assumingthe exclusive topology (cid:126)γp → π + π − p . We require thatthe kinematic fit χ satisfies χ / NDF < . (cid:126)γp → π + π − p in addition to the desired π − ∆ ++ channel. In particular, the topology is dominated by pro-duction of the ρ meson. We require 1 .
10 GeV /c 45 GeV /c to reduce backgrounds, particu-larly from ρ and ∆ ∗ production. This selection removesmost of the ρ background, as shown in Fig. 2. Figure 2. Dalitz plot of products of the reaction (cid:126)γp → π + π − p . Candidates between dashed lines are selected. Datashown are not efficiency corrected. C a n d i d a t e s / M e V / c m( π + p ) [GeV/ c (a) C a n d i d a t e s / . G e V x 103 | t| [(GeV/ c )2] D e t e c t i o n E ffi c i e n c y (b) Figure 3. (a) The π + p invariant mass distribution of eventssatisfying all selection criteria. In addition to the ∆ ++ , ex-cited states around 1.9 GeV/ c are visible. (b) The distribu-tion of | t | for candidates between the dashed lines in panel (a)and the detection efficiency as a function of | t | . Data plottedare not efficiency corrected. IV. ANALYSIS The differential cross section for pseudoscalar produc-tion by a polarized photon beam is related to the totalcross section σ by dσdφ = σ π (cid:16) − P γ Σ cos [2( φ − φ lin )] (cid:17) , (1)where φ is the azimuthal angle of the production planein the lab, φ lin is the azimuthal angle of beam polariza-tion in the lab, P γ is the degree of linear polarization ofthe beam, and Σ is the observable to be measured [32].By using data collected with linear polarization in or-thogonal directions, the term Σ can be isolated withoutexplicitly determining the total cross section or any φ -dependent detector acceptance.As shown in Fig. 3(a), selecting a region of m ( π + p )invariant mass does not ensure a pure sample of ∆ ++ events. Previous analyses typically first select a puresample of events, and then produce a distribution in∆ φ . (Here ∆ φ ≡ φ − φ lin in Eq. 1.) The amplitudeof the cos(2∆ φ ) component is then extracted to obtainΣ. In what follows, we perform the steps in reverse or-der: we project the cos 2∆ φ component of all data andthen isolate the ∆ ++ contribution by using the knownlineshape of the ∆ ++ . The technique follows from thatused to determine coefficients of a Fourier expansion.One can weight individual events by cos( n ∆ φ ) and createweighted histograms in m ( π + p ), thereby integrating over∆ φ . The bin-by-bin contents of such histograms are thenproportional to the strength of the cos( n ∆ φ ) component.One can then fit these histograms, referred to later as H n ,to measure the ∆ ++ contribution to each, referred to as Y n , with the Y component being most sensitive to Σ.Practically, one must use orthogonal orientations of thebeam polarization to cancel detector acceptance in theformulation of Σ. The full prescription for implementingthis technique is documented in Refs. [31, 33].Following this prescription, we define a set of weightedinvariant mass m ( π + p ) histograms for each separate ori-entation of polarization H ⊥ / (cid:107) n , each with accidental beamphoton candidates subtracted as described above. Datain 0/90 orientations are given an event-by-event weight-ing of cos( nφ ), while − sin( nφ ) is used for data in the-45/45 orientations. The shape of the ∆ ++ in each t region can be described by a relativistic Breit-Wignerfunction multiplied by a phase space factor [34]: S ( m ) = | p | m (cid:12)(cid:12)(cid:12)(cid:12) Am − m − im Γ( m ) (cid:12)(cid:12)(cid:12)(cid:12) , (2)where A is a parameter determined by a maximum like-lihood fit, andΓ( m ) = Γ (cid:16) m m (cid:17) (cid:18) | p || p | (cid:19) (cid:18) | p | a | p | a (cid:19) . (3)Here, m and p refer to the invariant mass of the π + p system and the three-momentum of the proton (or pion)in the π + p rest frame. The values of m and Γ are ∆ ++ resonance parameters obtained from Ref. [35], and | p | is | p | computed at m = m . The interaction radius a is taken from Ref. [36]. Thus, the signal component ofthe fit contains a single free parameter A in the equa-tion above. We use a fourth order Bernstein polynomialset to describe the smoothly varying background in the m ( π + p ) spectra. By integrating the signal fit function,we extract the moment-weighted yield of ∆ ++ candidates Y n corresponding to a particular histogram H n .Following Ref. [33], Σ can then be expressed asΣ = Y ⊥ − F R Y (cid:107) P (cid:107) ( Y ⊥ + Y ⊥ ) + F R P ⊥ ( Y (cid:107) + Y (cid:107) ) , (4)where F R = N ⊥ /N (cid:107) is the ratio of measured photon fluxfor data sets with orthogonal linear polarizations. Whilethe GlueX detector was designed to be uniform in φ ,this need not be assumed: any non-uniform azimuthalacceptance effects are removed by taking the difference oftwo orthogonal polarization directions and by includingthe terms Y ⊥ and Y (cid:107) .In practice, rather than fit each individual histogram H ⊥ n and H (cid:107) n to extract the Y n , we note that the numera-tor and denominator in Eq. 4 are linear combinations ofterms Y n , and hence we can construct two histograms D and N , where the contents of the i th mass bin for eachhistogram (denoted D i and N i ) are given by the linearcombinations D i = P (cid:107) H ⊥ ,i + H ⊥ ,i ) + F R P ⊥ H (cid:107) ,i + H (cid:107) ,i ) , (5a) N i = H ⊥ ,i − F R H (cid:107) ,i + D i . (5b)Let the weighted yield of ∆ ++ events in histograms N and D be denoted as Y N and Y D respectively. In termsof these two quantities, the asymmetry is then given byΣ = Y N Y D − . (6)In this formulation, Y N and Y D must be positive inorder to be physical. This is advantageous, as likeli-hood fitting techniques can then be employed. We usethis method to fit the m ( π + p ) spectrum in the massranges 1 . 14 GeV/ c < m ( π + p ) < . 60 GeV/ c and2 . 60 GeV /c < m ( π + p ) < . 50 GeV/ c , where the lowermass region contains the majority of the ∆ ++ signal andthe higher mass region is used to further constrain back-grounds while avoiding ∆ ∗ contributions. Figure 4 showsa fit to N and D histograms obtained over a large t rangeto demonstrate the ability of the lineshape to describe thedata at high statistical precision. Data are segmentedinto 16 regions of t , and in each region the 0/90 and -45/45 data sets provide two independent measurementsof Σ.The triply-differential cross section that describes theproduction of the ∆ ++ in each bin of | t | can be writtenin terms of spin density matrix elements ρ αλλ (cid:48) (SDMEs).When the two angles related to the polarization of the∆ ++ are integrated over, one obtains the expression inEq. 1 with Σ = 2 (cid:2) ρ + ρ (cid:3) , where ρ αλλ (cid:48) are SDMEs asdefined in Ref. [37]. Experimentally, the non-uniform ef-ficiency of detecting the ∆ ++ decay results in a weightedintegration over the decay phase space. This leads to anon-equal weighting of ρ and ρ and the introductionof other SDMEs that may cause the measured value of Σto deviate from the above expression. To correct for thisbias, we use a Geant4 [38] Monte Carlo (MC) simula-tion to calculate the efficiency (cid:15) as a function of the twodecay angles in the ∆ ++ rest frame for each bin of | t | . Wethen introduce an additional event-by-event weight of 1/ (cid:15) down to a cutoff value of (cid:15) = 0 . | t | , the effect of this weighting π + p ) [GeV/ c S u m o f C a n d i d a t e W e i g h t s / M e V / c (a)(b) Y N = 1.56 x 106 Y D = 1.03 x 106 Figure 4. (color online) Fit to (a) numerator N and (b)denominator D defined in Eqs. 5b and 5a, in the extendedrange 0 . c ) < | t | < . c ) . The ∆ ++ compo-nent is shown in green (dashed), polynomial background inblue (dotted), and total fit in red. Data are fit in the shadedregions only, the integral of the green (dashed) curve in thelower shaded region is used to determine the yields Y N and Y D . modifies Σ by a magnitude of about 40% of its total un-certainty. After this procedure, we find any residual biasto be negligible. Separately, we use MC simulation toevaluate m ( π + p ) and t dependent modifications to the∆ ++ lineshape, a dimension in which acceptance is un-correlated with decay angles. We assess the systematicuncertainties in these corrections later.To validate the statistical properties of our technique,we analyze simulated data from many toy experimentsand find that our method for extracting Σ is unbiased.We estimate the statistical uncertainty in our measure-ment by examining the variance of large ensembles of toyexperiments modeled to match our data. With these un-certainties, the results from 0/90 and -45/45 data setsagree statistically with χ / NDF=0.35 (NDF=15). Wecombine measurements from the independent 0/90 and-45/45 data sets, which have comparable statistical pre-cision, by averaging the results. In constructing the un-certainty on this average, we assume that individual sys-tematic errors in the measurement technique (detailedbelow) are fully correlated.To study systematic uncertainty related to choice offitting scheme, we perform additional evaluations of Σwhile independently varying: background polynomialfrom fourth to eighth order, choice of fit range, whetherto allow individual ∆ ++ signal parameters to float, andremoval of efficiency correction to the ∆ ++ lineshape.To study the systematic uncertainty related to relianceon MC-determined corrections applied to the phase spaceof the ∆ ++ , we perform additional evaluations of Σ byvarying the efficiency cutoff and systematically deformingthe efficiency map. We also roughly describe ∆ ∗ contri-butions using a double Gaussian shape, fitting to the re-gion of 1 . 14 GeV /c < m ( π + p ) < . 50 GeV/ c as an ad-ditional study. Each fit variation produces changes thatare largely uncorrelated in t and provide similar fit qual-ity and results as the nominal scheme. It is important tonote that variations in fitting scheme often affect Y N and Y D in the same way, which reduces the dependence of theextracted value of Σ on the fit scheme. Nevertheless, wefind that systematic uncertainties are comparable to orlarger than statistical uncertainties in several regions of t .Other sources of uncertainty investigated include uncer-tainty in flux, uncertainty in polarization due to limitedtriplet statistics, variations in number of beam bunchesselected for accidental subtraction, varying φ lin withinexperimental uncertainties, and choice of binning. Thesepotential sources of systematic uncertainty are describedin greater detail in Ref. [31]. The systematic uncertaintyin P γ , the polarization as measured by the TPOL, pro-duces a relative uncertainty of 1.5% on the magnitude ofthe measured value of Σ that is fully correlated amongstall t regions.As an additional check, the analysis was repeated withvaried selections of m ( π + π − ) region to include greateramounts of ρ and ∆ ∗ backgrounds into the analysis. Thesame systematic variations as described above were thenalso repeated. We found consistent results, even whenall events with m ( π + π − ) < . c , i.e., all ρ back-grounds, were included.The asymmetry Σ of the background can similarly beevaluated by inserting background yields to Eqs. 5aand 5b. In the mass range 1 . 14 GeV /c < m ( π + p ) < . 60 GeV/ c , the background is found to have a negativeasymmetry without clearly discernible t dependence. V. DISCUSSION OF RESULTS The results of beam asymmetry Σ for π − ∆ ++ pho-toproduction are listed in Table I and displayed in Fig.5 with theoretical predictions at 8.5 GeV provided byNys et al. [16] and B.-G. Yu and K.-J. Kong [17]. Sev-eral trends are apparent from the data. The asym-metry is negative in the range of approximately | t | < . 45 (GeV/ c ) , demonstrating that negative naturalitypion exchange is favored at smaller | t | . In the range | t | < . 25 (GeV/ c ) , the asymmetry is negative anddownward sloped as magnitude | t | increases. This is con-sistent with mixed-naturality modifications to one-pionexchange, which are sharply peaked in the forward direc-tion. For | t | > . 45 (GeV/ c ) the asymmetry becomespositive, consistent with descriptions including positivenaturality vector ρ and tensor a exchanges.We find that the model of Nys et al. describes thegeneral shape of the asymmetry over | t | , though it pre-dicts an overall lower value of Σ. The model by Yu andKong appears to slightly better describe the asymmetryfor | t | larger than 0.5 (GeV/ c ) ; however, it predicts a Σ -1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0 Nys et al . (JPAC), PLB 779, 77 (2018)Yu and Kong, PLB 769, 262 (2017) | t| [(GeV/ c )2] Figure 5. Beam asymmetry Σ vs. | t | compared to theoret-ical predictions. The error bars indicate the statistical andsystematic uncertainties combined in quadrature.Table I. Table of results. The uncertainty on | t | is the RMSof values in the ∆ ++ signal region. The uncertainties on Σare statistical and systematic (uncorrelated across t bins) re-spectively. There is an additional fully correlated systematicuncertainty of 1.5% on the magnitude of Σ. | t | (GeV/ c ) Σ0.050 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± minimum value and upward rise at much lower | t | thanobserved.In summary, we have measured the beam asymmetryΣ as a function of t for the reaction (cid:126)γp → π − ∆ ++ at E γ = 8 . GlueX experiment.These measurements are the first in this energy rangeand are of higher precision than and complementary tothose made at higher photon beam energies [18]. In the t -channel particle exchange picture, our measurements in-dicate that the naturality of exchanged Reggeons changessignificantly as a function of | t | , consistent with pion ex-change at smaller | t | and natural exchange processes athigher | t | . These results constrain models for t -channelphotoproduction of pions, which will be useful for under-standing backgrounds in both hybrid meson searches andbaryon spectroscopy studies at lower energies. VI. ACKNOWLEDGMENTS This work was supported by the US Department ofEnergy Office of Nuclear Physics under award DE-FG02-05ER41374. We recognize additional support by theNatural Sciences and Engineering Research Council ofCanada grant SAPPJ-2018-00021. We acknowledge inparticular input from from V. Mathieu to this work,as well as productive discussions with A. Szczepaniak, J. Nys, V. Mathieu, M. Mikhasenko, and B.-G. Yu. 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