Measuring the Neutral Pion Polarizability Proposal Submitted to PAC 48 Draft to Gluex Collaboration

Measuring the Neutral Pion Polarizability Proposal Submitted to PAC 48 Draft to GlueX Collaboration

M.M. Ito∗1, D. Lawrence1, E.S. Smith†1, B. Zihlmann∗1, R. Miskimen∗3, I. Larin∗3, A. Austregesilo5, D. Hornidge6, and P. Martel7

1Thomas Jeﬀerson National Accelerator Facility, Newport News, VA 3University of Massachusetts, Amherst, MA 5Carnegie Mellon University, Pittsburgh, PA 6Mount Allison University, Sackville, New Brunswick, Canada 7Institute for Nuclear Physics, Johannes Gutenberg University, Mainz, Germany

March 6, 2020

Theoretical Support

J. Goity, Thomas Jeﬀerson National Accelerator Facility, Newport News, Virginia and Hampton University, Hampton, Virginia A. Aleksejevs, Memorial University of Newfoundland, Corner Brook, Canada S. Barkanova, Memorial University of Newfoundland, Corner Brook, Canada S. Gevorkyan, Joint Institute for Nuclear Research, Dubna, Russia L. Dai, Hunan University, Changsha, Hunan, China

∗Spokesperson †Contact and Spokesperson, email: [email protected] Abstract This proposal presents our plan to make a precision measurement of the cross section for γγ π0π0 via the Primakoff effect using the GlueX detector in Hall D. → The aim is to significantly improve the data in the low π0π0 invariant mass domain, which is essential for understanding the low-energy regime of Compton scattering on the π0. In particular, the aim is to obtain a first ever experimental determination of the neutral pion polarizability απ βπ, which is one of the important predictions of chiral − perturbation theory and a key test of chiral dynamics on the π0. Our goal is to measure σ(γγ π0π0) to a precision of about 6%, which would determine the combination of → απ0 βπ0 to a precision of 50%. We expect this experiment to run concurrently with the previously− approved experiment to measure the charged pion polarizability (CPP) [1] in Hall D.

2 Contents

1 Introduction 4

2 Theoretical predictions for the neutral pion polarizability 5

3 Past Measurements 7

4 Experimental conditions 9 4.1 Expected signal ...... 9 4.2 Detector resolution ...... 12 4.3 Trigger and acceptance ...... 15

5 Backgrounds 16 5.1 Coherent backgrounds ...... 18 5.2 Fits to the scattering angle ...... 19 5.3 Incoherent backgrounds ...... 21 5.4 Analysis of existing data ...... 21 5.4.1 Hydrogen target ...... 22 5.4.2 Helium and beryllium targets ...... 22

6 Photon beam flux 26 6.1 Photon beam flux accounting with the GlueX pair spectrometer ...... 26 6.2 Cross section verification with the exclusive single π0 photoproduction ...... 27 6.3 Muon pair production ...... 29

7 Errors and Sensitivity 30

8 Summary and beam request 31

A More on theoretical predictions 36

B Parameterization of the nuclear coherent production 39 B.1 Parameterization of the s-wave amplitude ...... 40

C Angular distribution in the helicity basis 41 C.1 Photon density matrix in the helicity basis ...... 41 C.2 Parity constraints ...... 41 C.3 S-wave production ...... 41

3 1 Introduction

Electromagnetic polarizabilities are fundamental properties of composite systems such as molecules, atoms, nuclei, and hadrons [2]. Whereas form factors provide information about the ground state properties of a system, polarizabilities provide information about the excited states of the system, and are therefore determined by the system’s dynamics. Measurements of hadron polarizabilities provide an important test point for Chiral Perturbation Theory, dispersion relation approaches, and lattice calculations. Among the hadron polarizabilities, the neutral pion polarizability is important because it tests fundamental symmetries, in particular chiral symmetry and its realization in QCD. Indeed, the non-trivial (non-perturbative) vacuum properties of QCD result in the phenomenon of spontaneous chiral symmetry breaking, giving rise to the Goldstone boson nature of the pions. In particular, the Goldstone boson nature of the π0 manifests itself most notably in its decay into γγ and also in its electromagnetic polarizability, which according to ChPT can be predicted to leading order in the expansion in quark masses. Hadron polarizabilities are best measured in Compton scattering experiments where, in the case of nucleon polarizabilities, one looks for a deviation of the cross section from the prediction of Compton scattering from a structureless Dirac particle. In the case of pions, the long lifetime of the charged pion permits experiments of low energy Compton scattering using a beam of high energy pions scattering on atomic electrons. On the other hand, the short lifetime of the neutral pion requires an indirect study of low energy Compton scattering via measurements of the process γγ π0π0, a method that can also be applied to the charged pion (CPP) and for which a proposal → in Hall D is already approved [1]. Measurements of hadron polarizabilities are among the most difficult experiments performed in photo-nuclear physics. For charged hadrons, because of the Born term, the polarizability effect in the cross section can range from 10 to 20% depending on the kinematics. For neutral hadrons, where the Born term is absent, the polarizability effect will be much less than this. To set rea- sonable expectations for what can be accomplished in a measurement of this type, it is important to recognize that after 30 years of dedicated experiments using tagged photons at facilities across North America and Europe, the error on the proton electric polarizability is 4%, without doubt the paramount experimental achievement in this field. However, the error on the proton magnetic polarizability is 16% [3]. Absolute uncertainties provide a better gauge of a measurements sensitivity; for proton electric and magnetic polarizabilities the uncertainty in both is 0.4 10−4 fm3. ± × Another level of precision to consider for setting expectations is the result COMPASS obtained for charged pion α β. COMPASS is also a Primakoff measurement. COMPASS achieved a relative − error of 46% in α β and an absolute error of 0.9 10−4 fm3. COMPASS cannot measure the − ± × neutral pion polarizability.

This proposal presents a plan to make a precision measurement of the γγ∗ π0π0 cross → section using the GlueX detector in Hall D. The measurement is based on the Primakoﬀ eﬀect

4 ∗ which allows one to access the low Wπ0π0 invariant mass regime with the virtual photon γ provided by the Coulomb ﬁeld of the target. The central aim of the measurement is to drastically improve the determination of the cross section in this domain, which is key for constraining the low energy Compton amplitude of the π0 and thus for extracting its polarizability. At present, the only accurate measurements exist for invariant masses of the two π0s above 0.7 GeV, far above the threshold 0.27 GeV. The existing data at low energy were obtained in e+e− π0π0 scattering in the early 1990’s → with the Crystal Ball detector at the DORIS-II storage ring at DESY [4]. Meanwhile, theory has made signiﬁcant progress over time, with studies of higher chiral corrections [5, 6, 7] and with the implementation of dispersion theory analyses which serve to make use of the higher energy data [8, 9, 10, 11]. It is expected that the experimental data from this proposal, together with these theoretical frameworks, will allow for the most accurate extraction of the π0 polarizabilities to date.

2 Theoretical predictions for the neutral pion polarizability

The low energy properties of pions are largely determined by their nature as the Goldstone Bosons of spontaneously broken chiral symmetry in QCD, and are described in a model independent way by the framework of Chiral Perturbation Theory (ChPT) (Gasser and Leutwyler [12]), which implements a systematic expansion in low energy/momentum and in quark masses. In particular the pions’ low energy electromagnetic properties can serve as tests of their Goldstone Boson (GB) nature. One such a case is the π0 γγ decay, which at the same time tests its GB nature and → the chiral anomaly. Another case is low energy Compton scattering on pions: at low energy the Compton diﬀerential cross section can be expanded in powers of the photon energy and expressed in terms of the corresponding pion charge form factor (for charged pion) and the electric and magnetic polarizabilities, where the latter give the order ω2 terms in the Compton cross section, being ω the photon energy. The polarizabilities appear as deviations of the pions from point like particles, and thus result from carrying out the chiral expansion to the next-to-leading order. For both charged an neutral pions the polarizabilities are fully predicted at leading order in quark masses, and thus represent a sensitive test of chiral dynamics. For the charged pions, at O(p4) ChPT predicts that the electric and magnetic polarizabilities (απ+ and βπ+ ) are related to the charged pion weak form + + factors FV and FA in the decay π e νγ → FA 1 απ+ = βπ+ = (l6 l5), (1) − ∝ FV 6 − where l5 and l6 are low energy constants in the Gasser and Leutwyler eﬀective Lagrangian [12]. 4 Using recent results from the PIBETA collaboration for FA and FV [13], the O(p ) ChPT prediction

5 for the charged pion electric and magnetic polarizabilities is given by:

−4 3 α + = β + = (2.78 0.1) 10 fm . (2) π − π ± ×

Figure 1: Diagrams contributing to Compton scattering oﬀ the π0.

In the case of the neutral pion, the polarizabilities are determined by the one loop chiral contributions (see Fig. 1) which are calculable, free of unknown parameters, and given only in terms of the ﬁne structure constant, the pion mass and the pion decay constant:

απ0 + βπ0 = 0 α −4 3 απ0 βπ0 = 2 2 1.1 10 fm (3) − −48π MπFπ ' − × However, there is a range of predictions beyond NLO and the experimental test of these important predictions is very challenging. In the first place, the polarizabilities drive the very low energy regime of Compton scattering on the π0 as there is no Thomson term, so one would expect that it would be easier to determine them than in the charged pion case. However, in the first place Compton scattering on the π0 is experimentally inaccessible due to its short lifetime, and therefore it is necessary to resort to the process γγ π0π0 of this proposal. In addition, ChPT indicates → that the polarizabilities are smaller in the case of the neutral pion, about a third of their value for the charged pion, i.e. somewhere between 1.7 10−4 fm3 and 1.9 10−4 fm3, depending on − × − × the model used to estimate higher order effects in the chiral expansion. The challenge is therefore 0 0 to measure the cross section for γγ π π with sufficient accuracy at low invariant mass Wππ → so that one can infer the low-energy Compton amplitude and extract the polarizabilities. The demand for accuracy is required in order to allow for the extrapolation of the Compton amplitude from the kinematics of γγ π0π0 to low energy Compton scattering, something that is at present → impossible with the poor accuracy of the only available data from the Crystal Ball experiment [].

6 For this purpose, the theoretical foundations have been laid in works studying γγ π0π0 both → using ChPT (Bellucci et al [14, 5], Gasser et al [6], Aleksejevs and Barkanova [7]) and dispersion theory (Oller and Roca [8], Dai and Pennington [9, 10], Moussallam [11]). In particular, in ChPT at the next-to-next to leading order, which provides the higher order quark mass corrections to the polarizabilities, some of the low energy constants need to be fixed and for that a significantly more accurate measurement of the γγ π0π0 cross section is needed than available presently. → Accurate measurements of the cross section near threshold combined with data for Wππ > 0.6 GeV will provide the necessary input for performing a full theoretical analysis, combining dispersion theory with and without inputs from ChPT at low energy. This is a well established method which has been used to analyze ππ scattering and also to the very problem of the γγ π0π0 process, → where numerous works have been steadily improved the theoretical dispersive analysis, to mention a few [15, 16, 8, 11, 17]. Through such an analysis it will be possible to determine, via combination with ChPT, the low energy Compton amplitude and extract the combination απ βπ. The latter − extraction represents a challenge as shown in Fig. 2, where the polarizabilities have a small direct effect on γγ ππ. Calculations by Dai and Pennington (Table II) [17] indicate that a 1.3% → 0 0 determination of σ(γγ π π ) will determine the combination of α 0 β 0 to a precision of 10%. → π − π In general, the determination of the accuracy one can get for απ βπ based on a more accurate − measurement as the one proposed here is still an issue being currently studied theoretically, with J. L. Goity and A. Aleksejevs and S. Barkanova forming a group to take a lead on the project. At present a theoretical study based on the S-wave dominance below Wππ 0.8GeV and dispersion ∼ theory has allowed to represent the two Compton amplitudes A and B in the physical domain of the experiment. The study of the extrapolation to low energy Compton kinematics is under study, in particular the issues related to the stability of the dispersive analysis. This study is expected to provide a more accurate estimate on the sensitivity with which the experiment will allow for the determination of the polarizability α β. −

3 Past Measurements

Past measurements of the γγ π0π0 cross section are shown Fig. 3 and with theoretical curves in → Fig. 2. The data can be summarized as follows:

1. In the early 1990’s measurements were made in e+e− collisions at DESY with the Crystal Ball detector at the DORIS-II storage ring [4], which are the only available data for Wππ < 0.6 GeV.

2. In 2008-2009, measurements were carried out by BELLE for 0.6 GeV < Wππ < 4.0 GeV [18, 19, 20]. Two data sets were produced with diﬀerent selection cuts on cos θ∗ . | | As mentioned above, several works have made use of dispersion theory methods (Oller and Roca [8],

7 Page 18 of 28 Eur. Phys. J. C (2011) 71:1743 where the increase in uncertainty compared to (113) is due to π0 the ChPT uncertainty in (α2 β2) .Intheremainderofthe − paper we will make use of the improved value (114)when Page 14 of 25 Eur. Phys. J. C (2013) 73:2539 referring to the ChPT predictions for pion polarizabilities. Note that, as expected given the results of Table 6,oursum- π very low-energy region. This analysis favoured the follow- into the expression of the amplitude (76). They were defined rule value of (α2 β2) ± is consistent with the first ChPT − 9 number quoted in Table 4, but it is not consistent with the ing value for the neutral pion polarisability difference: from the relevant matrix elements of the electromagnetic larger number found when the LECs of [68]aretakenas current operator by Eqs. (27), (38). We will employ usual input. 4 3 0 0 (α 0 β 0 ) (1.25 0.16) 10− fm (87) Problem with γγ π π π − π = − ± · phenomenological descriptions based on superposition of Breit–Wigner-type functions associated with the light vector 8.2 Total cross section while the charged polarisability difference was constrained → resonances. We give some details on these in Appendix D. need206 forBefore higher performing the analytic orderJ.A. continuation Oller et al. toin / Physics the σChPTpole, Letters B 659 (2008) 201–208 to lie in the range predicted by the two-loop calculation plus Page 18 of 28 Eur. Phys. J. C (2011) 71:1743 The pion form factor, of course, is known rather precisely we wish to make sure that the amplitude on the real axis resonance modelling of the LEC’s performed in Ref. [64]. is reasonably well described—at least up to √t 1GeV, from experiment. Some experimental data exist also for the = Both numbers are veryThe similar data despite favoured that values the imaginary in the lowerparts part of that range where the increasewhich in uncertainty we assess compared to be the region to (113 which) is willdue influenceto the ωπ form factor in two kinematical regions surrounding the 0 [Oller, Roca & Schat] 1/2 the ChPT uncertaintyanalytic in (α continuationβ )π .Intheremainderofthe to the σ pole. The results for the cross of the two sσ differ by 20%. The result of [17],withwhich 2 2 (α∼ β ) 4.7 10 4 fm3. (88) peak of the ρ meson. The data in these two ranges are com- section are− depicted in Figs. 5 and 6.Belowthematchingwe shall compare our resultsπ+ later,π+ corresponds− to the ratio in paper we will make use of the improved value (114)when − " · 3 patible with the simple model used except, possibly, in a point, the results for the once- and twice-subtracted formu- Eq. (3.10) being 20% bigger at (2.53 0.09) 10− with sσ of referring to the ChPT predictions for pion polarizabilities. Using the determinations± × (87), (88)forthecouplingsC small energy region (see Appendix D for more details). The lation are provided for both ChPT and GMM polarizabil- Ref. [46]. ˜V Note that, as expectedities. given The uncertainty the results due of to Tableππ input,6,oursum- represented by the F π These ratios of residuagives at the the followingσ pole position values are for the the well subtractions functions at ρπ form factor, finally, is more difficult to isolate experi- rule value of (α2 greyβ2) band,± is is consistent estimated by with the variation the first between ChPT CCL and 2 − defined predictions thatq follow0: from our improved dispersive mentally than Fωπ , because of the width of the ρ.Weused number quoted inGKPRY Table 4 phases, but itand is proves not consistent to be very small. with The the low-energy treatment of γγ (ππ) =.However,theradiativewidthtoγγ region is totally dominated by the Born terms in the charged I the same type of modelling together with symmetry argu- larger number found when the LECs of [68]aretakenas for a wide resonance→ like0 the σ ,thoughmoreintuitive,hasex-2 process, but it is very sensitive to the σ in the neutral reac- b (0) (0.66 0.20) GeV− , ments to fix the parameters. input. tion. The prediction of the twice-subtracted dispersion rela- perimental determinations that= are− parameterization± dependent. (89) 2 2 tion is in especially good agreement with γγ π 0π 0 data This is due to the non-trivialb (0) interplay(0.54 between0.14 background) GeV− and → 2 8.2 Total cross section(see Fig. 6), with the level of agreement comparable to that the broad resonant signal. An= unambiguous− ± definition is then 5.4 Case q 0: subtraction functions Fig. 5 (Color online) Total cross section for γγ π 0π 0 [6, 11]and %= obtained in the coupled-channel fit of [32]. required [17,19].Weemploy,asinRef.→[17],thestandardnar- 4 γγ π +π − [7–9]for cos(weθ have0.8and ascribedcos θ 0.6, an respectively error 1.4 10− to the charged polar- Above the matching point, we exploit the fact that the → | | ≤ | | ≤ ± · 0 2 2 2 2 Before performing the analytic continuation to the σ pole, row resonance width formulaisabilities in terms difference). of gσγγ determined from The values of b (q ), b (q ) when q 0areapriori cross section is dominated by the f2(1270),andthuscanbeEq. (3.9) by calculating the residue at s , %= we wish to make sure that the amplitude on the real axis The result forσ the γγ π 0π 0 cross section derived from not known and must thus be determined from experiment. [XBall MarkIIwellapproximated ’90] by employing a Breit–Wigner descrip- 0 0 is reasonably well described—at least upI 0 to √t 1GeV, 2 → I Given detailed experimental data on e e γπ π and Fig. 5. Final resultstion for theof thisγγ resonanceπ0π0 cross in h2 section.,= (t) and Experimental putting all otherdata are partial gσγγ our amplitudes using the values (89)forb (0) is shown on + − → − = Γ (σ γγ) . (3.11) → whichfrom we theassess Crystal Ball towaves be Collaboration. the to zero. region In this[1] which,scaledby1 way, (108 will)aloneyieldsagooddescrip-/0 influence.8, as cos θ the< 0.8is | | e e γ γπ π ,onecoulddeterminethesefunc- → = 16πMFig.σ 8 and compared to the experimental measurements from + − ∗ + − | | [Hoferichter et al] → → analyticleadingmeasured continuation and order S-wavetion dominates.to of the thepoorσ neutralpole. The solid cross Theeven line section results corresponds abovenear for the theto CGL cross matchingthreshold and the point. tions for each q2 by performing a fit of the differential dashed one to PY. TheIn contrast,dot-dot-dashed in the line charged results aftercase removing an additional the axial background vec- Nevertheless, the determinationsRefs. [54, 66 of]. the Note radiative that the widths cross from section displays a cusp at section are depicted in Figs. 5 and 6.Belowthematching 2 0 dσ/ds cross sections. In practice, one expects that a sim- tor exchange contributions.is necessary. The band As along observed each in line [26 represents], this can the be theoretical most easily this expression and those√s from4m common,duetothe experimentalπ analysesπ + mass difference, which point, the results for the once- and twice-subtracted formu- = π+ − 2 uncertainty. The dottedachieved line is by the adding one loop theχPT Born result terms[4,5] andand the the off-shell dot-dashed contri- can differ substantially.was The discussed following example in Ref. makes[67]usingChPT. this point ‘---,Iple parametrisation of q dependence should be adequate. lationneedone are the two providedpolarizabilities loop calculationbutions for both dropped[6]. ChPT in the and transition GMM @ from polarizabil- 2 (107 )to(loops108)backclear. We adopted the following form, which involves two arbi- ities. The uncertaintyinto due the charged-channel to ππ input, amplitude represented for h by2, (t) the. Moreover, − From Ref. [45] one obtains gσππ2 2.97–3v .01 GeV, corre- trary parameters: grey band,I 0 is estimatedafterI the0 by transition the variation to the isospin between basis, CCL we andadd a constant 5.3 Case| q |=0: Fπ , Fωπ , Fρπ form factors TI[Bellucci,= (s i% )GasserTII= &(s Sainio;i%).Then,Eq. Gasser, Ivanov(3.6) can & beSainio] rewrit- sponding to the square root of the residua%= of the I 0S-wave − =background+ phase to ensure matching with the ππ phase = GKPRYten as phases and proves to be very small. The low-energyγ ππ amplitude, as in Eq. (3.8).IfsimilarlytoEq.(3.11),one n 2 n 2 2 below the matching point. However, if Cπ C is chosen to 2 b q b (0)F q βρ GSρ q 1 f2 f2 In order to address the case with q 0wemustspecify0.3 0.4 0.5 0.6 0.7 region is totally dominated by theI 0 Born terms in the charged uses the formula, = + − F0(s) F0(s) 1be negative,2iρ(s)T the= mismatch(s) . of the phases is very(3.7) small: we 2 %= ˜ II σ the q dependence of the three form factors whichW enter (GeV) 6546A5 2 process, but= it is veryfind+ sensitive a correction to of theδcorr in the0.09 neutral and δcorr reac-0.04 in or- 2 ! " βω BW! ω" q !1 , ! " " (90) = − = − gσππ β(Mσ ) + − tion.Around The prediction the σ !pole,der of tos theσ obtain, twice-subtracted agreement" with the dispersion CCL and GKPRY rela- phases, Γσ | | , 0 0 (3.12) Fig. 6 (Color online) Total cross section for γγ π π for Fig. 7 c 2 c 2 2 respectively. 0 0 = 16πMσ → b q b (0) !β GS! q" "1 β BW q 1 tion is in especially2 good agreement with γγ π π data cos θ 0.8inthelow-energyregion ρ ρ ω ω I 0 gσππ Finally, we commentgσγγ ong theσππ analyticity→ properties of the | | ≤ = + − + − (seeT Fig.= 6), with the, levelF0(s) of agreement√2 comparable, to that(3.8) the resulting width lies in the range 309–319 MeV, that is II = s s partial waves˜ = at the matchings s point. As shown in theFig. ap- 5 (Color online) Total cross section for γγ π 0π 0 [6, 11]and ! " ! ! " " ! ! " " obtained in theσ coupled-channel fit of [32σ ]. around a 30% smaller than Γσ 430 MeV from the pole po- with − pendix of [52], the solutions− in terms of Omnès functionsγγ π theπ background[7–9]for cos in theθ charged0.8and reactioncos& θ is→ dropped,0.6, respectively the neu- with g the σ coupling to two pions such that Γ sition+ − of Ref. [45].Thisisduetothelargewidthoftheσ Above theσππ matching point, we exploit the fact that the → tral cross section| above| ≤ tm is still| correctly| ≤ reproduced, but 2 2 2 2 automatically fulfill continuity at the matching point,= but m (Jπ (q ) Gπ (q )) gσππ β/16πM,foranarrowenoughscalarresonanceof mesonthe input which for makes the I the0componentchanges,whichaffectsgσππ extracted from the residue of 2 2 π ¯ ¯ cross section is dominatedthe derivative by the at tmfis2( not1270 determined.),andthuscanbe Therefore, in general, | | F q 192π − . (91) | | T I 0,Eq.(3.8),besmallerbyarounda15%thanthevalue= I 0 2 mass M.Noticeaswellthestrong cusps can√ occur2factorin at the matchingF0(s) point.to match For example, with if II=the neutral cross section below tm:theresultfor h = (t) = q well approximated by employing a Breit–Wigner˜ descrip- | 2, | the g normalization used (the so-called unitary normaliza- needed in Eq. (3.12) to obtain Γσ 430 MeV. Similar− effects tion of thisσππ resonance in hI 0(t) and putting all other partial & ! " tion [42–44]). Then from2,= Eqs. (3.7) and (3.8) it follows that are then also expected in order to extract Γ (σ γγ) from The relation between b0, b2 and bn, bc is given in Eq. (78). waves to zero. In this way, (108− )aloneyieldsagooddescrip- → the Eq. (3.11).Equationssimilartothisareusuallyemployed This form (90)ismotivatedbythediscussionconcerning tion of2 the neutral cross section2 above the matching point. gσγγ 1 β(sσ ) 2 in phenomenological fits to data, e.g. see Ref. [47],butwith F0(sσ ) . (3.9) the chiral limit. Assuming that the parameters βρ , βω are In contrast,g2 = − in2 the8 chargedπ case an additional background g determined along the real axis. As a result of this dis- σππ # $ σγγ 2 n 2 c 2 is necessary. As observed in [26], this can be most easily cussion,| | one should allow a (20–30)%variationbetweenthe O(mπ ) ensures that b (q ), b (q ) have the correct chiral Let us stress that this equation gives the ratio between the 2 2 achieved by adding the Born terms and the off-shell contri- results obtained from Eq. (3.11) and those from standard exper- limit behaviour at q 0aswellasq 0(seeSect.4.3). residua of the S-wave I 0 γγ ππ and ππ ππ am- %= = butions dropped in the transition= from→ (107)to(108→)back imental analyses that still could deliver a γγ ππ amplitude We consider the experimental data in the region √s plitudes at the σ pole position. → ≤ into the charged-channel amplitude for h2, (t). Moreover, in agreement with our more theoretical treatment for physical 0.95 GeV where it is an acceptable approximation to ignore In order to derive specific numbers for− the previous ratio in ı after the transition to the isospin basis, we add a constant values of s. the effect of inelasticity in ππ scattering. We also ignore the terms of our dispersive approach one needs to introduce sσ .We Fig. 8 Comparison of the γγ π 0π 0 cross sections using the am- background phase to ensure matching with the ππ2phase We shall employ the following values for gσππ . First we take two different values for sσ (Mσ iΓσ /2) .Fromthe n | →| n n,V effect of ππ rescattering in D or higher partial waves, since = π −γ take gσππ 2.97–3.01plitude GeVH[42–45]as derived.Withthisvaluethere- from Eq. (76)andH H with experi- belowstudies the matching of unitary point.χPT [42–45] However,one if hasCfMCσf andis chosenΓσ around to the | |= ++ +− = +− the corresponding ππ phase-shifts are small in this region. 2 2 Figure 2:sulting Left two panel: photon width experimentalment. from The Eqs. influence(3.10) status; of and varying (3.11) righttheis polarisability panel: difference resultsαπ from0 βπ0 the 1990 XBall experi- be negative,interval 425–440 the mismatch MeV. The of the other phases values isvery that we small: will we use are is shown − Note, however, that J 2partialwavesintheγ γππ ccl 16 ccl 18 0 ≥ ∗ → findfrom a correction Ref. [46] of, Mδσcorr 4410+.809MeV and δ andcorr Γσ 0.04544 in+25 or-ment.MeV, TheΓ (σ lowerγγ) panel(1.8 shows0.4) KeV the. effect of π polarizabilities(3.13) on the cross section (√s = Wππ) == − − = −= − amplitudes are not necessarily small, except very close to derwhere to obtain the superscriptagreement withccl indicates, the CCL in and the GKPRY following, phases, values that → = ± Born [11]Fig. . 6 (Color online) Total9 cross section for γγ π 0π 0 cclfor 0 the ππ threshold. They are included via H (for charged employ the σ pole position of Ref. [46].Thecorrespondingra- We also consider a largerIn the value fit, forthe dipolegσππ andsince quadrupoleΓσ [46] polarisabilitiesis of the π were λλ respectively. cos θ 0.8inthelow-energyregion | |→ V ) tios of the residua given in Eq. (3.9) are: | |larger≤ by a factor 1.3thanallowedΓ toσ from vary subject Ref. [45] to.Onevalueis the constraint that the combination 6(απ0 amplitudes) and Hλλ .Theresultsofperformingfitstothe Finally, we comment on the analyticity properties of the ∼ 2 − ) βπ0 )dipole mπ (απ0 βπ0 )quadrupole is given by a chiral sum rule. data of Refs. [4]and[5] are shown in Table 1 and illustrated partial waves at the matching point. As shown in the ap- Γ ccl(σ +ππ) 1/2 − gσγγ 3 Dai and Penningtonccl [17], and in particular Moussallam [11] who performed the dispersive analysis (2.10 0.25) 10− ,sσ from Ref. [45], the backgroundgσππ ing theσππ charged reaction→ is dropped, the neu- pendixg of [=52], the± solutions× in terms of Omnès functions | |(1) & | | Γ (σ ππ) % σππ % wheretral one cross of section the photons above#t is has still→ non correctly vanishing$ reproduced, virtuality, but which is particularly important for our case.) automatically% % fulfill continuity at the matching point, but (1.127 m0.022) g % gσγγ % 3 σππ the% derivative% ( at2.t06m is not0.14 determined.) 10− ,s Therefore,σ from Ref. in[46] general,. with(3.10)the those input available for the= I data.0componentchanges,whichaffects± In| particular| these methods give results for the cross section at small gσππ = ± × (3.35= 0.08) GeV. I 0 (3.14) strong% cusps% can occur at the matching point. For example, if the neutral cross section below tm:theresultfor h = (t) % % Wππ, but the poor= accuracy± of the data in| 2 that, | region does not serve as a useful constraint that % % − % % could improve those analyses. On the other hand, the ChPT calculations carried out at NNLO (Bellucci et al [14, 5] ) can only be fitted to the low Wππ data, and thus the uncertainty in the determination of low energy constants is rather large. It is therefore expected that accurate data at low Wππ < 0.6 GeV will have a very big impact on both theoretical approaches, which together may allow for an accurate description of the low energy Compton amplitude, and for a first time experimental determination of the polarizability.

8 Threshold Region 250 40

25 150

Cross section (nb) Cross section (nb) 20

100 15

10 50 5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wππ (GeV) Wππ (GeV)

Figure 3: Data from Crystal Ball and BELLE. There are two data samples from BELLE ∗ experimental with diﬀerent selection on cos θ . Left) Full range of Wππ. Right) Threshold region. | |

4 Experimental conditions

The measurement of the neutral pion polarizability is expected to run concurrently with the experiment to measure the charged pion polarizability (CPP) [1] in Hall D. Essentially all the optimiza- tions for that experiment are expected to improve the sensitivity of this experiment also. We briefly summarize the configuration for CPP, which is compared in Table 1 to nominal GlueX running. The diamond radiator will be adjusted to set the coherent peak of the photon beam between 5.5 and 6 GeV. This enhances the polarization significantly and also the tagging ratio. The experimental target will be placed upstream of the nominal GlueX target by 64 cm (z=1 cm in the Hall D coordinate system). These changes benefit the present experiment. In addition, the CPP experiment will add multi-wire proportional chambers downstream for muon identification, but these do not impact this measurement one way or another.

4.1 Expected signal

In order to estimate rates, resolution and acceptance due to the Primakoﬀ reaction on lead, γ208Pb π0π0 Pb, we take the reaction process to be the same as for charged pion produc- → tion and given in Eq. 8 of the Proposal for the Charged Pion Polarizability experiment [1], which is reproduced here for convenience:

2 2 4 2 2 d σ 2αZ Eγβ sin θππ 2 2 0 0 = 2 4 F (Q ) σ(γγ π π )(1 + Pγ cos 2φππ). (4) dΩππdWππ π Wππ Q | | →

9 Table 1: Configuration of the CPP experiment compared to nominal GlueX. We propose that this experiment run concurrently with CPP. Detectors not identified in the table are assumed to be operated under the same conditions as in the nominal configuration.

Conﬁguration GlueX I CPP/NPP Electron beam energy 11.6 GeV 11.6 GeV Emittance 10−8m rad 10−8m rad Electron current 150 nA 20 nA Radiator thickness 50µm 50 µm diamond Coherent peak 8.4 – 9.0 GeV 5.5 – 6.0 GeV Collimator aperture 5 mm 5 mm Peak polarization 35% 72% Tagging ratio 0.6 0.72 Flux 5.5-6.0 GeV 11 MHz Flux 8.4-9.0 GeV 20 MHz Flux 0.3-11.3 GeV 367 MHz 74 MHz Target position 65 cm 1 cm Target, length H, 30 cm 208Pb, 0.028 cm Start counter nominal removed Muon identiﬁcation None Behind FCAL

The γγ cross section for charged pions has been substituted with the cross section for neutral 0 0 pions, namely σ(γγ π π ). In this expression, Ωππ is the solid angle in the laboratory frame → for the emission of the ππ system, Wππ is the ππ invariant mass, Z is the atomic number of the 2 target, β is the velocity of the ππ system, Eγ is the energy of the incident photon, F (Q ) is the electromagnetic form factor for the target with final-state-interactions (FSI) corrections applied, θππ is the lab angle for the ππ system, φππ is the azimuthal angle of the ππ system relative to the 1 incident photon polarization, and Pγ is the incident photon polarization. The cross section for σ(γγ π0π0) has been measured by the Crystal Ball Collaboration [4], → albeit with limited statistical precision. We have parameterized the cross section for Wππ < 0.8 GeV, which is of specific interest to this program as shown in Fig.4. Using this parameterization and Eq.4, we can calculate the photoproduction cross section on lead, which is shown in Fig.5. The integrated cross section is 0.30 0.05µb/nucleus. The uncertainty comes from the model ± dependence and was obtained by comparing two different calculations using completely different parameterizations for the nuclear form factor on lead, F (Q2). For reference, we note that the cross

1The expression for the cross section in terms of invariant quantities can be found in Ref. [21].

10 γ γ → π0 π0 20

18 (nb) Crystal Ball 1990 / 0.8 σ

16 Parameterization

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Mππ (GeV)

Figure 4: Parameterization of the σ(γγ π0π0) cross section as a function of the 2π invariant mass compared to the data from→ Crystal Ball [4].

γ Pb → Pb π0 π0 106

105 (nb/GeV) σ 104

103

102

1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Mππ (GeV)

Figure 5: Primakoﬀ cross section for γP b P b π0π0 using the parameterization of σ(γγ π0π0) in the previous ﬁgure. The integrated→ cross section is 0.3 µb/nucleus. →

11 section for charged pions (π+π−) production is 10.9 µb, a factor of 30 larger. The number of neutral-pion-Primakoﬀ-signal events produced during 20 PAC days is shown in Fig.6. The impacts of detector trigger, acceptance and resolution are discussed in the next section.

γ Pb → Pb π0 π0 104 Accepted Events=2572 Flux=1e+07/s 2 3 t= 0.318 g/cm 10 σ tot= 296 nb/nucleus

Counts/50 MeV Time=20 PAC days

102

1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Mππ (GeV)

Figure 6: Estimated production rate for γPb π0π0 Pb as a function 2π mass. For this calculation, it is assumed the detector has perfect→ resolution and has a linearly increasing eﬃciency from zero at threshold up to 0.6 at 0.34 GeV (see top right of Fig. 12 ).

4.2 Detector resolution

The response of the GlueX detector to neutral pion Primakoff events was simulated using the standard GlueX generation and reconstruction tools, but with the specific geometry for the CPP experiment. The schematic of the detector configuration is shown in Fig. 7. The primary differences between the standard GlueX geometry and CPP are summarized in Table 1. For this experiment, the main differences include a) coherent peak position at 5.5-6 GeV and re-positioning of the microscope to cover the coherent peak, b) solid 208Pb at z=1cm, and c) Start counter removed. For the CPP experiment, the addition of muon identification chambers behind the FCAL is critical. However, for neutral pions this addition plays no role because the photons are detected in the FCAL. The GEANT3 simulation,2 which is used for these studies, includes most changes except for the 2The event simulation has been updated to run with GEANT4 and yields similar results.

12 addition of the muon chambers, which are not needed. In addition, the microscope geometry has not been modiﬁed and we use the tagger hodoscope for that region in the simulation. The slightly reduced energy of the hodoscope relative to the microscope has little impact and the gaps between counters is ignored by simulating the tagged ﬂux.

200 solenoid

Y (cm) 150

100 TOF FCAL BCAL MWPC FDC FDC FDC FDC MWPC 50 CDC Iron π+ γ 0 target beam CDC π- -50 BCAL -100

-150

-200 -100 0 100 200 300 400 500 600 700 800 900 Z (cm)

Figure 7: Diagram of the GlueX detector including the additional muon chambers for the CPP experiment.

The Primakoff signal was generated according to the cross section described in the previous chapter, using the gen 2pi0 primakoff, which is a modified version of the CPP event generator. By default, the production amplitudes are symmetrized between the two identical π0’s by AmpTools. One hundred thousand events were generated with Mππ <0.5 GeV and with no background, fed to GEANT3 to track particles, and subsequently processed using mcsmear to simulate the detector response. The simulated events were then analyzed using the GlueX event filter to analyze the reaction γP b π0π0 with a missing Pb nucleus and constraining the detected photon pairs to → the π0 mass. Energy and momentum conservation is imposed on the reaction as well as the requirement that all photons originate from a common vertex (i.e. “vertex-P4”). The output of the reconstruction, both kinematically fit and “measured” quantities, were available for inspection. In the following we show various reconstructed quantities as well as estimated resolutions. The distribution of generated photon energy and the unconstrained reconstructed momenta of the two pions are shown in Fig. 8. The missing mass, 2π mass and t distributions are shown in Fig. 9. The − reconstructed momentum relative to its generated value is shown in Fig.10. The central peak of the kinematicall fit momentum is about 2%, similar to that for charged pions. However, there are long uniform tails that will effect the final reconstruction. The resolution of the azimuthal angle, φππ, between the production and the photon polarization planes is quite poor owing to the fact that the pion pairs are produced at very shallow angles. Nevertheless it is sufficient to measure the

13 BeamEnergy pi01MomentumMeasured pi02MomentumMeasured 2200 160 Entries 15351 160 Entries 21199 Entries 21199 2000 Mean 5.777 Mean 2.879 140 Mean 2.89 140 1800 Std Dev 0.1513 Std Dev 1.049 Std Dev 1.048 120 1600 Underflow 0 120 Underflow 0 Underflow 0

1400 Overflow 0 Overflow 0 100 Overflow 0 100 1200 80 1000 80

800 60 60 600 40 40 400 20 20 200

0 0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 π0 π0 Energy (GeV) 1 Momentum Measured (GeV) 2 Momentum Measured (GeV)

Figure 8: Left: Generated photon energies. Center: Reconstructed momentum distribution of one π0. Right: Reconstructed momentum distribution of the second π0.

DSelector_Z2pi0_trees_p1_signal_100000 DSelector_Z2pi0_trees_p1_signal_100000 DSelector_Z2pi0_trees_p1_signal_100000 MissingMassSquared M2pikin tkin

Entries 15728 Entries 15728 3 Entries 15728 10 120 1000 Mean −0.003071 Mean 0.3711 Mean 0.001444

Std Dev 0.006519 Std Dev 0.05636 Std Dev 0.001535 100 800 Underflow 0 Underflow 0 Underflow 0

Overflow 0 Overflow 0 2 Overflow 304 80 10 600 60

400 40 10

200 20

0 0 1 −0.2−0.15 −0.1−0.05 0 0.05 0.1 0.15 0.2 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0 0.002 0.004 0.006 0.008 0.01 2 2 Missing Mass (GeV ) Mππ Kin (GeV) -t Kin (GeV )

Figure 9: Left: Missing mass distribution minus the mass of the recoil nucleus. Center: Kinematically ﬁt 2π mass distribution. Right: Kinematically ﬁt -t distribution.

14 asymmetry due to the photon beam polarization. The resolution of the 2π invariant mass is shown

pi01Deltap_Measured pi01Deltap Phidiff Entries 42398 Entries 42398 Entries 15728 600 Mean 0.0101 1400 Mean 0.006513 350 Mean −0.1377 Std Dev 0.07763 Std Dev 0.07147 Std Dev 18.65 χ2 / ndf 1057 / 96 500 1200 p0 969.3 ± 15.0 300 p1 0.00188 ± 0.00021 p2 0.01987 ± 0.00029 1000 p3 ± 250 400 82.57 1.09

800 200 300 600 150

200 400 100

100 200 50

0 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −50 −40 −30 −20 −10 0 10 20 30 40 50 π0 π0 1: Thrown p - Measured p/Thrown p 1: Thrown p - KinFit p/Thrown p Phi Kin - Gen (degrees)

Figure 10: Left: Difference between measured and generated momentum. Center: Difference between kinematically fit and generated momentum. The central peak has a width of about 2%. Right: Difference between the kinematically fit azimuthal angle φππ and its generated value. in Fig. 11, along with the resolution of Mandelstam t, and the reconstructed time resolution. The − mass resolution is about 8 MeV.

DSelector_Z2pi0_trees_p1_signal_100000 DSelector_Z2pi0_trees_p1_signal_100000 DSelector_Z2pi0_trees_p1_signal_100000 M2pidiff tdiff thetapipidiff 240 Entries 15728 3000 Entries 15728 Entries 15728 220 Mean −0.002733 Mean 0.0004726 3500 Mean 0.06902 Std Dev 0.01296 Std Dev 0.001396 Std Dev 0.1852 200 Underflow 1115 2500 Underflow 7 3000 Underflow 1 180 Overflow 23 Overflow 223 Overflow 0

χ 2 / ndf 2334 / 372 χ 2 / ndf 2589 / 73 χ2 / ndf 1916 / 44 160 2000 2500 Prob 0 Prob 0 Prob 0 140 Constant 158.7 ± 2.4 Constant 2369 ± 33.2 Constant 3277 ± 40.2

− ± − ± − 2000 ± 120 Mean 0.0004631 0.0000721 1500 Mean 9.112e 05 4.064e 06 Mean 0.02258 0.00078 Sigma 0.007703 ± 0.000093 Sigma 0.0004348 ± 0.0000047 Sigma 0.08408 ± 0.00074 100 1500 80 1000 60 1000

40 500 500 20 0 0 0 −0.04 −0.02 0 0.02 0.04 −0.01 −0.005 0 0.005 0.01 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 2 2 Θ Mππ Kin - Gen (GeV/c ) |t| Kin - Gen (GeV/c) Lab ππ Kin-Gen (deg)

Figure 11: Left: Difference between kinematically fit and generated 2π mass. The central 2π-mass σ is about 8 MeV. Center: Difference between kinematically fit and generated -t. Right: Difference between kinematically fit and generated 2π polar angle. The resolution σ of the reconstructed angle is less than 0.1 degrees.

4.3 Trigger and acceptance

The Primakoﬀ reaction will transfer all the energy of the beam into four photons, which are going forward. All this energy will be deposited in the FCAL, except for leakage down the beampipe.

15 We expect a simple trigger with an energy threshold in the FCAL should have very high efficiency for any events that can be reconstructed. The acceptance of the signal events can be determined by comparing the kinematically fit to the generated distributions. The generated and kinematically fit 2π mass, φππ and t distributions − are shown in Fig. 12. The reconstruction was described in the previous section. The acceptance is quite high at about 60%. However, there is also significant slewing due to resolution in most variables of interest. The main effect of resolution in the 2π mass happens at threshold and the distortions above that are not very great. The relatively poor resolution in φππ results in dilution of the measured azimuthal dependence, which will need to be adjusted based on simulation. Finally the t resolution softens the measured t-slope due to the smearing of high rate regions down to − low rate regions.

5 Backgrounds

We ﬁrst classify the various backgrounds and then describe them in more detail one at a time. We note that two important backgrounds for the charged-pion polarizability experiment do not contribute in this experiment: First, coherent ρ0 photo-production is absent in this experiment because the ρ0 decay into the π0π0 channel is prohibited by I-spin conservation. Second, µ+µ− production is also not a factor for the neutral pion case. We have the following categories of backgrounds:

Coherent production: In this case, the target remains intact. Generically, one may classify the • two-pion production according to the sketches in Fig. 13. The left-hand diagram represents the exchange of a virtual photon with the nucleus, i.e. the Primakoff mechanism. This mechanism is very long range, approximately 100 fm, and is affected minimally by the effects of shadowing or absorption. This is the signal for the experiment and our goal is to determine its cross section. The right-hand diagram represents the exchange of a strongly interacting particle (or trajectory) and effectively results in the production of pions at the surface of the nucleus. We note that for the neutral pion production, pion exchange is not allowed due to charge conjugation conservation, while for the charged pion case, single pion exchange is related to the axial anomaly (γπ0 π+π−). When the interaction leaves the nuclear → target intact, the reaction is referred to as “nuclear coherent” and this is our most serious background.

Incoherent production: When the interaction produces two pions in the quasi-elastic scatter- • ing oﬀ a single nucleon, the scattered target usually fragments into particles that range out in the target and are unobservable experimentally. This reaction occurs at larger t and is − generally kinematically distinct from the signal. The π0π0 momentum relative to the photon

16 twopi_primakoff_DSelect_p1_W_100000 Acceptance 1000 M2pigen 1.2 M2piAcceptance

Entries 200 Entries 6529 900 Mean 0.3687 1 Mean 0.3992 800 Std Dev 0.07223 Std Dev 0.06035

700 Underflow 0 Underflow 0 0.8 600 Overflow 0 Overflow 0

500 0.6

400 0.4 300

200 0.2 100

0 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 M(π0π0) (GeV) M(π0π0) (GeV)

twopi_primakoff_DSelect_p1_W_100000 twopi_primakoff_DSelect_p1_W_100000

2000 Phigen 105 tdat Entries 50 Entries 100 1800 Mean 0.02269 Mean 0.001464 Std Dev 1.902 Std Dev 0.00166 1600 Underflow 0 104 Underflow 0 Overflow 0 Overflow 0 1400 χ2 / ndf 46.4 / 48 χ2 / ndf 79.91 / 14 Prob 0.5385 Prob 2.946e−11 1200 3 p0 805.7 ± 8.7 10 Constant 8.264 ± 0.046 1000 p1 0.7057 ± 0.0076 Slope −565.9 ± 11.7

800 102 600

400 10 200

0 −3 −2 −1 0 1 2 3 0 0.002 0.004 0.006 0.008 0.01 0.012 Φ (rad) -t (GeV2)

Figure 12: Top left: Generated and kinematically fit 2π mass distribution. Top right: Acceptance as a function of 2π mass. Events above Mππ >0.5 GeV were eliminated from the analysis. The acceptance is about 60%. Bottom left: Generated and kinematically fit azimuthal angle φππ. Bottom right: Generated and kinematically fit t distribution. There is significant slewing of the measured distribution to large t. − −

17 Figure 13: Sketch of coherent two-pion production. Left) Signal: Primakoﬀ mechanism, Right) Backgrounds: Other production mechanisms.

polarization plane does diﬀerentiate between the Primakoﬀ and incoherent production.

Any reaction that may be confused with the signal within the experimental resolution or • limited acceptance: An example of this type of reaction is Primakoﬀ production of η mesons, where the η π0π0π0 is mis-reconstructed as a two-pion ﬁnal state. →

5.1 Coherent backgrounds

PC ++ The largest coherent background is from the f0(500)(J = 0 ), also called the σ meson, and the f0(980) photo-production. The width of the f0(980) is fairly narrow and does not contribute directly to the strength near threshold. The f0(500)width is much broader, from 400 to 700 MeV, with significant overlap in the invariant mass region of interest. Since the f0(500) is a scalar particle with the same spin-parity as the γγ π0π0 final state near threshold, the azimuthal distribution of → the π0 momentum or the π0π0 c.m. momentum relative to the photon polarization plane does not differentiate between coherent f0(500) production and the Primakoff reaction. This is similar to the Primex-π0 experiment, where the dominant background was nuclear coherent π0 photo-production. The approach used in the Primex analysis was to measure the π0 angular distribution, effectively the t-distribution, then use theoretical calculations of the angular distributions to separate out contributions from Primakoff and nuclear coherent. The analysis of the π0π0 (NPP) reaction will approximately parallel what was done for the Primex-π0 analysis. We parameterize the σ meson as detailed in Appendix B and assume that the production amplitude can be factorized as

= t(t) W (mππ) τ (Φ, φ, θ), (5) A A A A

18 where the last factor represents the angular distribution that results in a dependence on the dipion azimuthal angle, φππ, of the form τ (1 + cos 2φππ). The mass dependence is given by A ∝ P the S-wave phase shifts that dominate the mass region below 0.8 GeV. We use the approximate description given in Appendix B.1.3

Figure 14: Left) Approximation to strong form factor for lead, Right) Figure 6 from Ref. [22] showing the calculated strong form factor for single π0 production oﬀ a lead target.

We assume the t dependence of the σ has a similar form as for single π0 production, namely − t(t) sin θππ Fst(t). The sin θππ comes from the spin-flip required at forward angles to produce A + ∝ × a 0 system from a spin-zero target. The factor Fst(t) is the strong form factor for the target, which is approximated to match calculations for the single π0 production (Fig. 6 from Ref.[22]). Our Gaussian approximation to the form factor is shown in Fig. 14 along side the calculation for single π0 production. Efforts are underway to calculate the strong form factor for this reaction.4. The Primex data showed that the nuclear coherent process is highly suppressed for heavy nuclei, as shown in Fig.23. The reason for the suppression is π0 absorption in the nuclear interior, making the coherent production primarily a surface effect, i.e. proportional to A and not A2. For NPP it is expected that suppression of the nuclear coherent will be approximately twice stronger than that seen in Primex because two pions are produced in NPP as compared to a single π0 in Primex.

5.2 Fits to the scattering angle

2 2 ◦ The Primakoﬀ reaction is peaked at very small scattering angle θππ m /2E 0.1 0.2 for ∼ ππ γ ∼ − masses between 0.35 and 0.5 GeV, as shown in Fig. 15a. Nevertheless it is still an order-of-magnitude larger than the scattering angle for single π0 production. The angular distribution for the nuclear

3More detailed studies may require including contributions from the D-wave and S-wave, I=2, amplitudes. 4S. Gevorkyan, private communication.

19 a) Primakoff only b) Nuclear Coherent only thetapipigen thetapipigen 500 Entries 4245 90 Entries 1444 Mean 0.3533 Mean 0.7688 Std Dev 0.2037 80 Std Dev 0.3232 Events Events 400 Underflow 0 Underflow 0 Overflow 1 70 Overflow 0 χ2 / ndf 36.97 / 25 χ2 / ndf 42.15 / 33 Prob 0.05818 60 300 Prob 0.132 offset 0.01783 ± 0.00157 norm 1444 ± 182.1 50 rise 0.06893 ± 0.00658 mean 0.09594 ± 0.07322 norm ± ± 4212 215.4 40 sigma 0.4557 0.0209 200 mean 0.1133 ± 0.0122 30

100 20

0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θππ (degrees) θππ (degrees)

c) Primakoff+Nuclear Coherent d) Primakoff+4xNuclear Coherent thetapipigen thetapipigen 400 Entries 4243 Entries 4127 Mean 0.4383 Mean 0.5198 350 300 Std Dev 0.2693 Std Dev 0.3171 Events Events Underflow 0 Underflow 0 300 Overflow 0 250 Overflow 1 χ2 / ndf 50.65 / 34 χ2 / ndf 51.24 / 38 250 Prob 0.03299 200 Prob 0.07409 norm1 2631 ± 122.4 norm1 1950 ± 119.4 200 norm2 369.3 ± 60.3 norm2 1338 ± 87.7 norm3 0.872 ± 0.189 150 norm3 0.3621 ± 0.0967 150 Sum Sum 100 Primakoff Primakoff 100 Nuclear Coherent Nuclear Coherent 50 50 Interference Interference

0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 θππ (degrees) θππ (degrees)

Figure 15: Angular distributions for a) Primakoff reaction only, b) Nuclear coherent reaction only, c) Sum of Primakoff and nuclear coherent, d) Same as c) with the nuclear coherent amplitude multiplied by 2. The curves represent fits to the coherent sum of the two components.

20 ◦ coherent process on lead is shown in Fig. 15b, which peaks at an angle of about θππ 0.7 . The ∼ coherent sum of both processes, e.g. Fig. 15c and d, is a distribution that is considerably wider than the Primakoff distribution alone, but a fit is required to determine the Primakoff signal. The amplitude of the NC component in Fig. 15d was generated with an amplitude that is twice that in Fig. 15c. In order to extract the Primakoff component, we proceed as follows: We first parameterize the shapes of the Primakoff and NC components separately, as shown with the curves in Fig. 15a and b and the parameters for those shapes are fixed. Two distributions generated using both the Primakoff and NC amplitudes are shown in the bottom plots of Fig. 15c and d and then fit to the sum of the two distributions plus an interference term. We let the normalization of each component vary as well as the size of the interference term. The resulting fitted parameters are shown on the figures. From this exercise we conclude that the determination of 2000 signal Primakoff events can be determined with a statistical uncertainty of about 5%.

5.3 Incoherent backgrounds

The inelastic and incoherent reactions that might contribute to the data include

(i) nuclear coherent production of η followed by η π0π0π0 γγγγ(γγ), where two of the six → → decay photons go unobserved

(ii) γN Nπ0π0 →

The first reaction is an inelastic, coherent process, and as such could produce a significant rate for a heavy nuclear target. Rejecting events that do not satisfy our elastic selection and the constraint that the four photons must reconstruct to two π0’s mass should eliminate most of these events. Additional gammas in the final state can also be used to suppress this background. The second reaction is an incoherent process, and is small relative to coherent processes. The Primex analysis showed that incoherent reactions generally peak at angles larger than about 1◦ and had a small effect on the extraction of the Primakoff π0 cross sections. Likewise, we expect that incoherent processes to be a small issue for this experiment. The π0π0 momentum relative to the photon polarization plane does differentiate between Primakoff and incoherent production.

5.4 Analysis of existing data

We investigated the challenges of reconstructing 2π0 ﬁnal states with a missing recoil proton using the 2017 GlueX data taken with a Hydrogen target and the 2019 PrimEx-Eta data with Beryllium and Helium targets.

21 5.4.1 Hydrogen target

We selected and reconstructed events that matched the topology of the reaction γp γγγγ (p) → with a missing proton. A kinematic fit was performed that conserved energy and momentum and imposed a vertex constraint at z=65 cm (CL > 10−6). We note that even though the vertex was fixed at 65 cm to perform the fit, the actual target extends from 50 to 80 cm. Several other nominal selections were imposed to clean up the event sample, including no charged tracks and no missing energy. No constraints were imposed on the π0 mass in order to study backgrounds. Accidental background subtractions were performed to obtain the resulting mass distributions.

The invariant mass distributions of two photon pairs each show a strong π0 peak, as shown in the top of Fig. 16. There are background events that fall under the two π0 peaks, which requires further study, nevertheless, using the selection of photon pairs that reconstruct to the π0, we can plot the 2π0 mass spectrum (bottom of Fig. 16). The mass spectrum has recognizable features, in 0 0 particular the prominent f2(1270) that decays to π π 85% of the time. The structure at Mππ 0.8 ∼ GeV appears too low for the f0(980) and is present in a location where the Crystal Ball data [4] shows a low yield. The yield for Mππ <0.5 GeV is consistent within a factor of two of the relative yield compared to the f2(1270 peak in the Crystal Ball data. This analysis demonstrates that these neutral events can be analyzed in our detector under significant more challenging circumstances than we anticipate for the Primakoff experiment. In particular, for the Primakoff experiment, we will have a point nuclear target that will allow valid geometrical constraints and limit the amount of missing momentum in the reaction. This will make the kinematic fitting more effective.

It is evident from top plot in fig. 16 that a cut on the invariant mass of one reconstructed π0 will reduce the background on the other π0 significantly. This is shown in fig. 17 where a cut on the invariant of one π0 significantly reduces the background in the other while keeping the main signal mostly undisturbed. These photons are detected by the lead-glass calorimeter and are the main contribution to the resolution of the reconstructed pi0 mass. A lead-tungstate calorimeter with a substantially better energy resolution would yield a significant improvement in the signal to noise ration as the width of the reconstructed π0 would be smaller by about a factor of 2.

5.4.2 Helium and beryllium targets

The PrimEx-Eta experiment collected valuable data on light nuclear targets (4He and 9Be) in 2019. Analysis of the two neutral pion system photoproduction on these nuclei gives a good estimation of the main background sources, signal to background levels, and the Hall-D detector resolution for the main kinematic variables of two neutral pion photoproduction process. The total PrimEx-Eta luminosity corresponds to approximately one day on 5 % rad. len. beryllium target and 18 days on a 4 % rad. len. helium target at 200 nA electron beam current and a 10−4 rad. len. thick amorphous

22 π0 vs π0 pi01_vs_pi021 0.3 Entries 267183 Mean x 0.1294 Mean y 0.1295 0.25 Std Dev x 0.02322 Std Dev y 0.0232 102 0.2

0.15

10 0.1

0.05

1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 2 0 Mpi [GeV/c ] 2-π0 System Invariant Mass at θ < 1deg. twopi0_Mass2 600 Entries 23710 Mean 1.146 Std Dev 0.4194 500

400

300

200

100

0 0 0.5 1 1.5 2 2.5 3 3.5 4 2 M2π0 [GeV/c ]

Figure 16: Experimental distributions from the 2017 GlueX data set analyzed as γp γγγγ (p) with a missing proton. Top: Two photon invariant mass of one pair vs the two→ photon invariant mass of the second pair. Bottom: 2π mass distribution selecting events with the reconstructed photon pair masses close to the π0 mass as shown above. The plot also requires that the angle of the two pion system be less than 1 degree.

23 Invariant Mass of First Pi0 pi0_1Mass0 Entries 1050412 20000 Mean 0.1319 Std Dev 0.02596

15000

10000

5000

0 0 0.05 0.1 0.15 0.2 0.25 0.3 Inv. Mass {GeV}

Figure 17: Invariant mass of the two photon system with(red) and without(blue) a cut on the invariant mass of the second pair of photons. tagger radiator. The Beryllium target has a thickness of only 1.5 cm (compare to 30 cm liquid Helium and Hydrogen targets), which allows constraining interaction point (important for the neutral pions reconstruction without any additional vertex information from the tracking system). First we identified the two neutral pion exclusive photoproduction process using the energy ratio of two pions to the initial beam energy with the expected recoil energy subtracted. Fig. 18 shows this distribution for pions detected in FCAL and time accidentals and out-of-target beam interaction subtracted. We first required exactly four showers to be detected in FCAL and no extra showers in BCAL and COMCAL, a minimum shower energy of 0.5 GeV, and no neutral signals in TOF. The number of the signal events here is about 900, the width of the observed signal with pion kinematic fit to the mass is about 3 %, and the signal to background ratio value is promising. Fig. 19 shows two dimensional distribution of those events: elasticity vs invariant mass. One can 0 0 see the horizontal line of the exclusive production events and vertical line of Kshort π π decays, 0 0 → which are separated from each other. Presence of Kshort π π decays in the data is really → beneficial for the Primakoff analysis since it allows tuning the detector resolution in Monte-Carlo and make an assessment of the level of this value agreement with the data, which is essential for the successful cross-section fitting procedure and systematic uncertainty control. Including BCAL showers in the neutral pion reconstruction increases the acceptance (especially for large invariant mass region) and number of observed events by an order of magnitude. For the beryllium target this increases the number of exclusive events to 10 K and for the helium target ∼ to 200 K events. Fig. 20 shows two π0 invariant mass distribution with the energy within 10% ∼

24 200

Events / 0.01 150

100

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Two π0 elasticity

Figure 18: Two neutral pion elasticity (energy ratio to the expected value for the exclusive production) for the Beryllium target.

1.3 1.2

elasticity 1.1 0 π 1

Two 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.4 0.6 0.8 1 1.2 1.4 Two π0 mass, (GeV)

Figure 19: Two neutral pion elasticity (energy ratio to the expected value for the exclusive production) vs invariant mass of two pions for the Beryllium target.

25 of the expected for the exclusive production with BCAL included for low production angle events (below one degree). The f2 meson peak is clearly seen. Fig. 21 shows elasticity distribution for both helium and beryllium targets with BCAL reconstructions included (time accidentals and “empty” target background subtracted).

m4c 80

Events / 50MeV 50

0 0 0.5 1 1.5 2 2.5 3 3.5 4 Two π0 mass, (GeV)

Figure 20: Two neutral pion invariant mass for the exclusive events (within 10% energy), the Beryllium target and production angle below one degree.

To conclude this section, we wish to highlight the good detector resolution for two π0 production kinematics variables, the presence of the calibration process (Kshort) in the data and controllable level of backgrounds observed for light nuclear targets exposition.

6 Photon beam ﬂux

6.1 Photon beam ﬂux accounting with the GlueX pair spectrometer

The photon beam ﬂux can be directly extracted by analyzing the pair spectrometer (PS) data with the thin beryllium converter installed in the beam in from of it. The absolute normalization of the PS performed with the total absorption counter (TAC) during the dedicated run. The systematics from the photon beam ﬂux accounting by pair spectrometer is originated from

26 20000

Events / 0.01 15000

10000

5000

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Two π0 elasticity

Figure 21: Two pion elasticity distribution with BCAL included in the analysis. ”Empty” target and time accidentals are subtracted. Open histogram - Helium target, 200 K events in the elastic peak; solid histogram - Beryllium target, 10 K events in the peak.∼ ∼ few main contributions: overall spectrometer calibration with TAC quality; accuracy of the Monte- Carlo simulation of this process; long term stability of the spectrometer performance; and change of conditions between low intensity beam (TAC calibration) and production intensity. There are few other less signiﬁcant contributions. GlueX PS acceptance [23] shown on Fig. 22. For the proposed experiment PS magnetic ﬁeld should be reduced to cover the beam energy range 5 6 GeV . The − methodology and accuracy of the PS analysis is the same as in PrimEx-D experiment, currently running in Hall-D, and has value 1 1.5 % [24]. ∼ −

6.2 Cross section veriﬁcation with the exclusive single π0 photoproduction

The extracted cross section can also be normalized on or independently from PS analysis verified with the π0 radiative decay width extraction. Fig. 23 shows exclusive single π0 photoproduction yield at forward angle obtained by the PrimEx experiment and used for π0 radiative decay width extraction. The photon beam flux in PrimEx was 0.725 1012 for 4.9-5.5 GeV bremsstrahlung × spectrum part on 5% rad. len. lead target. The distance between calorimiter and target was 7.3 m and the central square part of the calorimeter, used in analysis was 70 70 cm. These ∼ ∼ × conditions have to be compared with the proposed experiment conditions: 20 days of 107 collimated beam photon/sec (i.e. 20 times more than PrimEx lead target beam flux), the distance between

27 Figure 22: GlueX PS acceptance extracted from TAC data analysis (blue points); red points – Monte-Carlo simulation

350 123.2 / 121 Γ 7.719 0.1866

CS 0.6914 0.4137E-01 φ 1.254 0.7449E-01 300 CI 0.6843 0.1208

250 15 , Events / 0.02 deg. θ 200 10 dN/d 150 5 4 100 total 0 1 1.5 2 2.5 50 1 3 2 0 0 0.5 1 1.5 2 2.5 θ(π0), deg.

Figure 23: Exclusive π0 production yield at forward angle on lead target observed in the PrimEx experiment [25]. Curves show production mechanisms input: 1 – Primakoﬀ, 2 – strong coherent, 3 – interference of ﬁrst two mechanisms, 4 – strong incoherent

28 target and FCAL 6.2 m and active calorimeter part diameter 2 m. The central hole with ∼ ∼ one calorimeter modules layer around which should be excluded from the analysis for PrimEx case was 8 8 cm and for FCAL 20 20 cm, which is decreasing FCAL acceptance at forward ∼ × ∼ × angle. Comparison of these experimental conditions allows us expecting an order of magnitude higher exclusive single π0 photoproduction statistics. Thus PrimEx statistical uncertainty for lead will be decreased from 2.5 % down to 1 %. For the systematical uncertainty, in PrimEx ∼ ∼ it was 2.1 % and has two major contributions: yield extraction ( 1.6 %) and photon beam ∼ ∼ flux accounting ( 1.0 %). The first contribution is partly statistically driven and reduces with ∼ increasing of statistics; and the second one cancels out since it is the same photon beam flux for the single and double exclusive π0 photoproduction. The main factors increasing systematics for the proposed experiment are: the angular resolution of FCAL is about a factor of two worse than for PWO crystals used in the PrimEx analysis; and the magnetic field is not swiping out charged background like it was in PrimEx. As a result we can expect slightly worse systematical uncertainty than in PrimEx and statistical precision of 1 %, i.e. total error 2.5 3.5 % for π0 radiative width ∼ − extraction (excluding absolute photon beam flux accounting, target number of atoms and partly FCAL trigger efficiency contributions to the systematics which are canceling out). The expected total beam flux uncertainty for such a normalization should also include the PrimEx total error of the π0 radiative width, which was recently reported as 1.5% [26]. All this gives 3 4 % error for ∼ − photon beam flux from normalization to the re-extracted π0 radiative decay width.

6.3 Muon pair production

In addition to these normalization channels, production of muon pairs, which has a known cross section, can be used as a measurement of photon flux. Since the experiment will be running concurrently with the Charged Pion Polarizability (CPP) experiment, the photon flux on target will be the same by definition. CPP plans to use muon pair creation by beam photons as its main normalization channel, and so those measurements will be available for normalization of the neutral pion channel as well. In the case of CPP, the GlueX track finding and fitting efficiency will have to be determined for muon pairs, but any systematic error in that determination will largely cancel when applied to charged pion pairs. That will not be the case for the neutral pion channel and will have to be taken into account when evaluating systematic errors due to this method of normalization. In any case, muon pair production should provide a useful check on the other methods mentioned above.

29 0 Table 2: Uncertainties in the extraction of π polarizabilities α 0 β 0 . π − π Source Uncertainty 1 Signal extraction 5 % 2 Flux normalization 1.5 % 3 Background subtraction <1 % 4 Detector acceptance and eﬃciency 3.5 % 5 Total systematic error 3.9% 6 Total error on cross section 6.3% 7 Projected error in α β 49% −

7 Errors and Sensitivity

We summarize the anticipated errors in the determination of the π0 polarizability. We assume 20 days of running on a 5% radiation length 208Pb target, 107 photons/s, and nominal acceptance for π0π0. Table 2 summarizes the estimated statistical and systematic errors. In the following we describe each of these contributions in detail:

1. Primakoﬀ signal extraction from the angular distribution (Fig. 15), which also contains nuclear coherent contributions to the cross section. This uncertainty is approximately 5%.

2. Flux normalization. We have several methods for determining the ﬂux (Section 6). The Primex-D experiment expects an uncertainty of 1.5% and we use that as our estimate here.

3. Subtraction of backgrounds other than nuclear coherent. These backgrounds are not expected to be signiﬁcant (< 1%) and studies are on-going.

4. Detector acceptance and eﬃciency. We can measure the detector acceptance times eﬃciency for the process γPb π0Pb with an accuracy of 3.5% (Section 6.2), which should allow us → to reduce the systematic uncertainty in the acceptance calculation for the process of interest to this level.

5. Total systematic error (items 2-4): combining the systematic errors in quadrature gives 3.9%.

6. Error on cross section (quadrature sum of items 1 and 5): 6.3%.

7. The current estimate by Dai and Pennington (Table II in Ref. [17]) indicates that a 13% 0 0 determination of σ(γγ π π ) will determine the combination α 0 β 0 to a precision → π − π of 100%, i.e., ∆(α 0 β 0 ) 7.7∆(σ) . From here we estimate that our uncertainty on π − π ∼

30 ∆(α 0 β 0 ) 49%. We note that the basis for extracting the polarizabilities may be π − π ∼ improved in the near future and theoretical eﬀort is being directed speciﬁcally toward this goal.

Table 3: Approved beam request and running conditions for CPP. NPP would run concurrently.

Running condition Days for production running 20 Days for calibrations 5 Target 208Pb Photon intensity in coherent peak 107 photons/s Edge of coherent peak 6 GeV

8 Summary and beam request

We have investigated the possibility of determining the neutral pion polarizabilities α 0 β 0 , a π − π quantity for which there are no existing measurements. Our proposal is to extract the polarizability from a measurement of the cross section of the Primakoff reaction γPb π0π0Pb. We → propose to make this measurement using data taken simultaneously with the CPP[1] experiment in Hall D. Table 3 summarizes the approved beam request for the CPP experiment. The existing GlueX detector has sufficient resolution and high acceptance for this process. We expect to collect approximately 2500 signal events during the approved 20 PAC days. The anticipated statistical uncertainties on the signal represent a significant improvement over existing data as shown in Fig. 24. Using the estimate by Dai and Pennington [17] we expect to be able to make the first extraction of the α 0 β 0 polarizability with an uncertainty of 49%. π − π

31 γ γ → π0 π0 20 Crystal Ball 1990 / 0.8 18 (nb)

σ Expected Uncertainties 16

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Mππ (GeV)

Figure 24: Estimated statistical uncertainties on determining σ(γγ π0π0) during 20 PAC days running simultaneously with the approved CPP experiment. The→ data points from the single previous Cristal Ball measurement [4] are shown for comparison.

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35 A More on theoretical predictions

The scattering amplitude for γγ∗ π0π0 is given in terms of the Compton tensor, whose low → energy expansion in the Compton scattering channel γπ0 γπ0 is given in terms of the electric → and magnetic polarizabilities of the π0. For the case of interest with one real photon, the Compton tensor is given in by two amplitudes, namely:

1 1 Tµν = (A(s, t, u) + B(s, t, u))( s gµν kνqµ) (6) − 4 2 − 1 2 + B(s, t, u)((s q )p−µp−ν 2(k p− qµp−ν + q p−kνp−µ gµνk p− q p−) (7) 4s − − · · − · ·

2 0 Here s = Wππ is the invariant mass squared of the two π s, k the momentum of the beam photon, q the momentum of the virtual photon, and p− the p− = p1 p2 the momentum diﬀerence − between the two pions. The limit of interest for the polarizabilities is:

α 2 2 = ( ( ) ( )) 2 απ A s, t, u Mπ B s, t, u s=0,t=u=Mπ −2Mπ − s | α = 2 (8) βπ A s=0,t=u=Mπ 2Mπ | where απ βπ are the electric and magnetic polarizabilities respectively. The low energy limit is analyzed in ChPT. At the lowest signiﬁcant order, i.e., one loop, the π0 polarizabilities are entirely given in terms of known quantities, namely:

α −4 3 απ0 = βπ0 = 2 2 0.55 10 fm (9) − −96π MπFπ ' − ×

The positive magnetic susceptibility indicates that the π0 is diamagnetic, and naturally the negative electric polarizability tells that it behaves as a dielectric. There are higher order corrections in the chiral expansion to the above prediction corresponding to a two-loop calculation, which is undeﬁned up to two low energy constants h± in the notation of Ref. [5], expected to be signiﬁcant for the corrections.

The amplitudes A and B are constrained by unitarity and analiticity to satisfy dispersion relations. In particular below s 0.8 GeV2 the dominant contributions are for the pair of pions in ∼ an S-wave. The rather well established S-wave phase shifts thus allow for implementing dispersion relations [15, 8, 11, 9, 10, 17]. In this proposal the model by Donoghue and Holstein [15] for implementing the dispersive representation using S-wave ﬁnal state interaction was adopted. The model implements twice subtracted dispersion relations for the isospin 0 and 2 components of the

36 amplitude A with the addition of t- and u-channel resonance exchanges for both A and B. The four subtraction constants require the experimental input of the cross section to be measured by the proposed experiment. A summary of useful theory results is the following: 1) representation of the Compton amplitudes:

2 2 2 2 s X t + Mπ u + Mπ s A(s, t, u) = (f0(s) f2(s)) + (p0(s) p2(s)) RV ( + ) −3 − 3 − − 2 t M 2 u M 2 V =ρ,ω − V − V 1 X 1 1 B(s, t, u) = RV ( + ) −8 t M 2 u M 2 V =ρ,ω − V − V 2 6MV Γ(V πγ) RV = → (10) α (M 2 M 2)3 V − π where V = ρ, ω,

Born A ρ ω pI (s) = fI (s) + pI (s) + pI (s) + pI (s) r r 2 2 A A L9 + L10 MA Mπ 1 + β(s) + sA/s p0 (s) = p2 (s) = 2 s + − log F β(s) 1 β(s) + sA/s π − 2 ! ρ 3 Mρ 1 + β(s) + sρ/s p0(s) = Rρ log 2 β(s) 1 β(s) + sρ/s − ρ p2(s) = 0 2 ω 1 ω 1 Mω 1 + β(s) + sω/s p0 (s) = p0 (s) = Rω log s , (11) −2 −2 β(s) 1 β(s) + sω/s − −

q 2 s−4Mπ where β(s) = s , MA the mass of the A1 resonance. The fI s are given by the dispersive representation:

2 Z ∞ 0 ! s 0 −1 0 ds fI (s) = pI (s) + ΩI (s) cI + dI s pI (s )Im(ΩI (s )) 0 02 , (12) − π 2 (s s)s 4Mπ − with the Omn`esfunction: ! Z ∞ ( 0) ( ) 0 ( ) 4 2 2 iφI (s) s φI s φI s ds φI s Mπ ΩI (s > 4Mπ ) = e exp 0 − 0 + log 2 . (13) π 2 s s s π s 4M 4Mπ − − π the phases φI are related to the corresponding ππ S-wave phase shifts according to:

0 0 φ0(s) = θ(M √s)δ (s) + θ(√s M)(π δ (s)) − 0 − − 0 2 φ2(s) = δ0(s), (14)

37 where M is the mass of the f0 resonance. The values used for the parameters entering the representations above are:

Lr + Lr = 1.43 0.27 10−3 9 10 ± × 2 2 si = 2(M M ) i − π 2 2 Rω = 1.35/GeV ; Rρ = 0.12/GeV (15)

and the ππ phase shifts are well approximated up to √s 1.5 GeV by the parametrization: ∼   N I ΓI X n δ (s) = arcsin + an (√s) (16) 0  q 2  2 ΓI 2 (√s MI ) + n=0 − 4 where we include one single resonance for each I = 0, 2. For the available data we need only up to N = 3 for I = 0, with the result:

M0 = 0.994 GeV; Γ0 = 0.0624 GeV 2 3 a0 = 1.439; a1 = 6.461/GeV; a2 = 5.529/GeV ; a3 = 2.022/GeV (17) − − For the case I = 2 one ﬁnds that the resonance term is not needed at all and a good ﬁt is provided with N = 3 with the result:

2 3 a0 = 0.878; a1 = 0.611/GeV; a2 = 0.083/GeV ; a3 = 0.115/GeV (18) − − −

The γγ π0π0 in the S-wave approximation valid up to about √s 0.9 GeV is given by: → ∼ 2 παEM Z p 2 σ 0 0 ( cos θ < Z)(s) = s(s 4M ) (19) γγ→π π | | s2 2 − π ( A(s)s M 2B(s) 2 × | − π | 1 1 Z2 + M 4 ( s(4M 2 s) + 4(s 2M 2)2) B(s) 2) s2 π − 16 3 π − − π | |

Fitting to the Cristal Ball data[4] the parameters c0, d0, c2, d2 can be estimated, giving in the corresponding units:

c0 = 0.529 − d0 = 2.033 − c2 = 0.953

d2 = 1.271. (20) −

38 B Parameterization of the nuclear coherent production

We consider the reaction γA mππA, where mππ ππ is a dipion system. The 2π system is → → treated as a particle with mass mππ, which is produced with four-momentum transfer t. The cross section for a three-body ﬁnal state can be written as [27]:

1 2 dσ = dφ3 (21) 4 |A| cmF = pγ √s (22) F cm 4 pσ cm σ σ dφ3 = dΩ p dmππdΩ (23) (4π)5 √s σ π π cm cm dt dt 2 pγ pσ cm = cm cm = cm (24) dΩσ d cos θσ dφσ dφσ The center-of-mass energy (cm) energy and the momentum transfer are represented by the commonly used variables s and t. Other variables are subscripted by particle name and their su- cm cm perscripts indicate the reference frame. Thus pγ is the incident photon momentum, pσ is the cm scattered momentum, and Ωσ corresponds to the solid angle of the σ, all in the cm frame. The σ σ momentum of the pions in the σ rest frame is denoted by pπ and Ωπ denotes the solid angle of one of them. Thus the cross section can be written as dσ 1 pσ X = π i 2, (25) dtdm dφcmdΩσ 2(4π)5 (pcm)2s| A | ππ σ π γ i where the index i runs over the number of resonances or mechanisms included in the calculation. We will assume that we can parameterize each production amplitude as a factorized product

i i i i = t(t) W (mππ) τ (Φ, φ, θ) . (26) A A A A For simplicity, we will drop the superscript i since for the moment we are considering single production mechanism. The function τ (φππ, φπ, θπ) contains the angular dependence of the produced A pions, where (θπ, φπ) are the decay angles in the rest frame of the 2π system, which is flat for S-wave production. Azimuthal symmetry is broken by the photon polarization, where φππ is the angle between the plane of photon polarization and the production plane. The amplitudes are given by Eq. 41 and lead to a cross section dependence of the form τ (1 + cos 2φππ). A ∝ P PC ++ The primary background in this mass region is given by the f0(500)(J = 0 ) also called the σ. The σ has the same angular structure as the Primakoff reaction and can only be identified through its dependence on t and mππ. Our parameterization of the mass dependence for the σ meson is described in Section B.1.

39 We assume the t dependence of the σ has a similar form as for single π0 production, namely − t(t) sin θππ Fst(t). The sin θππ comes from the spin-ﬂip required at forward angles to produce A + ∝ × a 0 system from a spin-zero target. The factor Fst(t) is the strong form factor for the target, which is approximated to match calculations for the single π0 production (Fig. 6 from Ref.[22]).

B.1 Parameterization of the s-wave amplitude

There is considerable strength in the 2π channel coming from s-wave production, which is due to the now established f0(500) meson. It is also commonly referred to as the σ meson. We assume the amplitude for σ production is governed by the ππ J=0, I=0 phase shifts. We parameterize the mππ dependence as

mππ iδ0 2 iδ0 2 W (mππ) sin δ0e α1 + α2m + cos δ0e α3 + α4m , (27) A ∼ 2k ππ ππ where δ0 is the s-wave phase shift for I = 0 and αi (i=1, 2, 3, 4) are empirical constants to be obtained from data. The ﬁrst term is due to “compact source” production of the pion pair (see Eq. 5 from Ref. [28]) and the second term is due to production due to an “extended source,” for example pion rescattering (see Eq. 5 from Ref. [29] and Eq. 9 from Ref. [30]). We use the parameterization for the s-wave phase shifts from Appendix D of Ref. [31]:5

2 0 2k 0 0 2 0 4 0 6 4mπ s0 tan δ0 = A0 + B0 k + C0 k + D0k 2 − 0 , (28) mππ M s ππ − 0 0 0 −2 0 where we use the same notation as the reference with A0 =0.225, B0 =12.651 GeV , C0 = - −4 0 −6 0 2 43.8454 GeV , D0=-87.1632 GeV , and s0=0.715311 GeV . We have converted the constants to units of GeV and evaluated the parameters for a0 = 0.225 m−1, and a2 = 0.0371 m−1. These ﬁts 0 π 0 − π are only valid below mππ < 0.9 GeV because they do not properly include the f0(980).

2 The empirical constants in Eq. 27 were determined by fitting W to the S-wave contribution 6 |A | to the photoproduction cross section measured by CLAS for Eγ = 3 3.8 GeV [32] for t = 0.4 0.5 2 − − − GeV . The fits are for mππ = 0.3 0.95 GeV, which is our region of interest. All four parameters are − needed to obtain a good representation to the central values of the data, although the uncertainty band in the data allow for a wide range of parameters. Assuming that the constants are real and relatively independent of energy and t, we take the average of the fitted constants for our − parameterization (α1 = 8.4 1.4, α2 = 4.1 2.2, α3 = 2 1.1, α4 = 8 1.1). ± − ± ± ± 5See also Eq. 44 of Ref. [28]. 6The data are available through the Durham HEP Databases, http://durpdg.dur.ac.uk/.

40 C Angular distribution in the helicity basis

C.1 Photon density matrix in the helicity basis

The linear polarization of the photon can be expressed as (Ref.[33] Eq. 18-19):

1 1 ρ(γ) = I + P~γ ~σ, where (29) 2 2 · P~γ = ( cos 2φππ, sin 2φππ, 0) (30) P − − and ~σ are the Pauli matrices. The angle φππ is the angle between the polarization vector of the photon and the production plane and represents the degree of linear polarization. Multiplying P out these factors gives the expression for the photon density matrix in the helicity frame as (Ref.[34] Eq. 219):

−2iφππ 1 1 e ρ,0 (γ) = −P (31) 2 e2iφππ 1 −P

C.2 Parity constraints

We consider the reaction a + b c + d, where the spin of each particle is denoted by sj, their → helicity by λj and their intrinsic parity by ηj. If parity is conserved, there are relations between amplitudes with opposite helicities, which are given in Jacob and Wick [35] Eq. 43 and Ref.[33] Eq. 20 7 (see also Ref. [36] Eq. 4.2.3): ηcηd λa λdλb = ( 1)sc+sd−sa−sb ( 1)(λc−λd)−(λa−λb) −λa −λd−λb (32) Vλc V−λc ηaηb − −

C.3 S-wave production

For the case of S-wave production of two pions via the f0(500) or σ meson oﬀ an spinless target we have the following constraint: λγ λZ λZ 00 ηc+ 1 −1 − 00 − 00 V = V = ( 1) ( 1) V = ηc V (33) λσ 0 + − − 0 − 0 − For convenience, we have separated out the parity of the scattered state ηc. The 2π intensity distribution (see Ref.[34] Eq. 220-223 and also Eqs. 264) is given by the following expression after

7We thank Adam Szczepaniak for clarifying the connection between these papers.

41 dropping the superscripts related to the target helicities and collapsing the sums over external and internal spins because both the target and resonance are 0+ objects:

X 0 0 ∗ 0∗ = V0 Y ρ0 V Y (34) I 0 0 0 0 −2iφππ 1 ∗ 1 0 2 1 −1 1 e V0 = Y V0 V0 −P (35) 2 | 0 | e2iφππ 1 −1V ∗ −P 0 1 0 2 1 2 1 −1 ∗ −2iφππ 1 ∗ −1 2iφππ −1 2 = Y V0 V0 V e V V0 e + V0 (36) 2 | 0 | | | − P 0 − P 0 | | 1 −1 Noting that V0 = ηc V0, we obtain the following expression: − 1 1 2 = V0 (1 + ηc cos 2φππ) , (37) I 4π | | P where φππ is the angle of the polarization vector relative to the production plane. For the case of 0 σ production, ηc = +1, but for the case of π production we have the opposite sign, ηc = 1. For + − 0 − the Primakoﬀ production of π π in S-wave, ηc = ( 1)( 1)( 1) = +1. See Ref. [1] Eq. 8. − − − The intensity distribution in Eq. 34 may be written in a more convenient form for use with AmpTools, namely

1−P 2 1+P 2 = ( ) A+ + ( ) A− (38) I 4 | | 4 | | 0 1 −1 2iφππ A± = Y ( V0 V0 e ) (39) 0 ± 0 1 2iφππ A± = Y V0 (1 ηc e ), (40) 0 ∓ which can be written more symmetrically taking advantage of an arbitrary phase as

0 1 −iφππ iφππ A± = Y V0 (e ηc e ). (41) 0 ∓