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Evidence for a J/ψpp¯ Pauli Strong Coupling ? T.Barnes,a,b∗ X.Lib† and W.Robertsc‡§ aPhysics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6373, USA bDepartment of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200, USA cDepartment of Physics and Astronomy, Florida State University, Tallahassee, FL 32306-4350, USA (Dated: February 5, 2008) 8 The couplings of charmonia and charmonium hybrids (generically Ψ) to pp¯are of great interest 0 in view of future plans to study these states using an antiproton storage ring at GSI. These low 0 to moderate energy Ψpp¯ couplings are not well understood theoretically, and currently must be 2 determined from experiment. In this letter we note that the two independent Dirac (γµ) and n Pe+aeu−li→(σµ(νJ)/ψp,p¯ψc′o)u→plipnp¯gsonofretshoneaJn/cψe. Aancdomψp′acraisnonbeofcoounrsttrhaeionreedticbaylrtehseultasntgoulraercednitsturnibpuotliaornizeodf a dataallowsestimatesofthepp¯couplings;inthebetterdeterminedJ/ψcasethedataisinconsistent J with a pure Dirac (γµ) coupling, and can be explained by the presence of a σµν term. This Pauli 1 couplingmaysignificantlyaffectthecrosssectionofthePANDAprocesspp¯→π0J/ψnearthreshold. There is a phase ambiguity that makes it impossible to uniquely determine the magnitudes and ] h relativephaseoftheDiracandPaulicouplingsfromtheunpolarizedangulardistributionsalone;we p show in detail how this can be resolved through a study of the polarized reactions. - p PACSnumbers: 11.80.-m,13.66.Bc,13.88.+e,14.40.Gx e h [ I. INTRODUCTION JPC =1−−.) 2 Unfortunately these low to moderate energy produc- v tion reactions involve obscure and presumably rather Charmonium is usually studied experimentally 1 through e+e− annihilation or hadronic production, complicated QCD processes, so for the present they are 9 best inferred from experiment. In Ref.[4] we carried out 4 notably in pp¯annihilation. The pp¯annihilation process this exercise by using the measured pp¯partial widths to 4 was employed by the fixed target experiments E760 and estimate the coupling constants of the J/ψ,ψ′,η ,η′,χ . E835 at Fermilab, which despite small production cross c c 0 9 andχ topp¯,assumingthatthesimplestDiraccouplings 0 sections succeeded in giving very accurate results for 1 were dominant. These Ψpp¯couplings were then used in 7 the masses and total widths of the narrow charmonium 0 states J/ψ, ψ′, χ and χ . A future experimental aPCAC-likemodeltogivenumericalpredictionsforsev- 1 2 : program of charmonium and charmonium hybrid pro- eral associated charmonium production cross sections of v the type pp¯ π0Ψ. i duction using pp¯ annihilation that is planned by the → X PANDA collaboration [1] at GSI is one of the principal In this paper we generalize these results for the J/ψ r motivations for this study. and ψ′ by relaxing the assumption of γµ dominance of a theψpp¯vertex. Weassumeaψpp¯vertexwithbothDirac Obviouslythestrengthsanddetailedformsofthecou- (γ ) and Pauli (σ ) couplings, and derive the differen- plings of charmonium states to pp¯ are crucial questions µ µν tial and total cross sections for e+e− ψ pp¯ given for any experimental program that uses pp¯ annihilation → → thismoregeneralvertex. Bothunpolarizedandpolarized to study charmonium; see for example the predictions processes are treated. for the associated production processes pp¯ π0Ψ in → A comparison of our theoretical unpolarized angular Refs.[2, 3, 4]. (We use Ψ to denote a generic charmo- distributions to recent experimental J/ψ results allows niumorcharmoniumhybridstate,andψ ifthe state has estimates of both the Dirac and Pauli J/ψpp¯couplings. There is a phase ambiguity that precludes a precise de- terminationofthe(complex)ratioofthePauliandDirac ∗Email: [email protected] J/ψpp¯couplings from the unpolarized data; we shall see †Email: [email protected] that an importance interference effect between the Pauli ‡Email: [email protected] and Dirac terms leads to a strong dependence of the un- §Notice: AuthoredbyJeffersonScienceAssociates,LLCunderU.S. polarized pp¯ π0J/ψ cross section near threshold on DOE Contract No. DE-AC05-06OR23177. The U.S. Government → the currently unknown phase between these terms. retainsanon-exclusive,paid-up,irrevocable,world-widelicenseto Determining these couplings is evidently quite impor- publish or reproduce this manuscript for U.S. Government pur- poses. tantforPANDA,andcanbeaccomplishedthroughstud- 2 ies of the polarized process e+e− J/ψ pp¯. The where we have assumed that we are on a narrow reso- angular distribution of the unpola→rized, se→lf-analyzing nance, so we can replace s by M2. process e+e− J/ψ ΛΛ¯ may also provide comple- We will first considerthe unpolarizedprocess e+e− mentaryinform→ationre→gardingthecloselyrelatedJ/ψΛΛ¯ ψ pp¯, and establish what the differential and tot→al → vertex. Both of these processes should be accessible at cross sections imply regarding the ψpp¯ vertex. The un- the upgraded BES-III facility. polarized total cross section predicted by Fig.1 is 4πα2M4 (1 4m2/s)1/2 σ = − (2m2 2+s 2). II. UNPOLARIZED CROSS SECTION h i 3f2 s2 [(s M2)2+Γ2M2] |GE| |GM| ψ − (4) The Feynman diagram used to model this process is (Weuseanglebracketstodenoteapolarizationaveraged shown in Fig.1. We assume a vertex for the coupling of quantity.) Exactlyonresonance(ats=M2)this canbe a generic 1−− vector charmonium state ψ to pp¯ of the expressed in terms of the ψ partial widths form iκ 4πα2M Γµ(ψpp¯) =g(cid:16)γµ+ 2mσµνqν(cid:17). (1) Γψ→e+e− = 3fψ2 (5) In this paper m and M are the proton and charmonium and mass, Γ is the charmonium total width, and we assume massless initial leptons. Following DIS conventions, qν (1 4m2/M2)1/2 is the four momentum transfer from the nucleon to the Γψ→pp¯= − (2m2 E 2+M2 M 2), (6) 12πM |G | |G | electron; thus in our reaction e+e− pp¯ in the c.m. frame, we have q =( √s,~0). The cou→plings g and κ are which gives the familiar result − actually momentum dependent form factors, but since we only access them very close to the kinematic point 12π q2 = M2 in the reactions e+e− (J/ψ,ψ′) pp¯, we hσi s=M2 = M2 Be+e−Bpp¯. (7) → → (cid:12) will treat them as constants. (cid:12) Here Be+e− and Bpp¯ are the ψ e+e− and ψ pp¯ e− p branching fractions. → → Since the (unpolarized) pp¯ width and total cross sec- tiononresonanceinvolveonly the single linearcombina- γ ψ tion (2m2 E 2+M2 M 2), separatingthese twostrong −ieγµ −iΓν form facto|rGs r|equires|aGdd|itional information, such as the +ieMfψ2gαβ angular distribution. The unpolarized e+e− → ψ → pp¯ differential cross section in the c.m. frame is given by e+ p¯ dσ α2 M4 (1 4m2/s)1/2 = − hdΩi 4f2 s2 [(s M2)2+Γ2M2] FIG. 1: The Feynman diagram assumed in this model of the ψ − generic reaction e+e− →ψ→pp¯. Theunpolarizeddifferentialandtotalcrosssectionsfor 4m2 E 2(1 µ2)+s M 2(1+µ2) , (8) · (cid:20) |G | − |G | (cid:21) e+e− ψ pp¯may be expressedsuccinctly intermsof the str→ong→ψpp¯Sachs form factors = g(1+κs/4m2) E where µ = cos(θ ). This angular distribution is often G c.m. and M =g(1+κ). Both E and M arecomplexabove expressed as 1+αµ2, where G G G pp¯ threshold, in part because phases are induced by pp¯ rescattering. If we assume that the lowest-order Feyn- 1 (4m2/s) / 2 E M man diagram of Fig.1 is dominant, the phase of g itself α= − G G . (9) (cid:12) (cid:12)2 is irrelevant, so here we take g to be realand positive. κ 1+(4m2/s)(cid:12) E/ M(cid:12) G G howeverhasanontrivialphase. Weexpressthisbyintro- (cid:12) (cid:12) (cid:12) (cid:12) ducing a Sachs form factor ratio, with magnitude ρ 0 InspectionofEqs.(8,9)showsthatonecandeterminethe and phase χ; ≥ magnitude ρ = E/ M of the Sachs form factor ratio |G G | from the unpolarized differential cross section, but that GE/GM ≡ρeiχ. (2) the phase χ of GE/GM is unconstrained. Theundeterminedphaseχimpliesanunavoidableam- The corresponding relation between the Pauli coupling biguity in determining the magnitude and phase of the constant κ and this Sachs form factor ratio is Dirac and Pauli ψpp¯ couplings g and κ from the unpo- ρeiχ 1 larizede+e− (J/ψ,ψ′) pp¯angulardistribution. We κ κ eiφκ = − (3) → → ≡| | (M2/4m2 ρeiχ) will discuss this ambiguity in the next section. − 3 III. COMPARISON WITH EXPERIMENT coupling at present accuracy, but the better determined J/ψangulardistributionisinconsistentwithapureDirac A. Summary of the data J/ψpp¯coupling at the 4σ level. The discrepancy evident in Fig.2 may imply the pres- enceofaPauliterm(κ=0)intheJ/ψpp¯vertex. Inspec- Experimentalvaluesofαhavebeenreportedbyseveral 6 tion of our result for α in the general case (Eq.9) shows collaborations. The results for the J/ψ are that one can certainly accommodate this discrepancy by 1.45 0.56, MarkI [5] introducing a Pauli term. ± 1.7 1.7, DASP [6] ± α=00..6516±00..2134,, MMaarrkkIIIII[7[8]] (10) C. Determining ρ=|GE/GM| from α ± 0.62 0.11, DM2 [9] ± 0.676±0.036±0.042, BES [10]. maTshse(fdroempenEdqe.n9c)eisofshtohwenpriendiFcitge.d3.αTohneρexaptertihmeeJn/taψl and for the ψ′ value α = 0.676 0.055 (shown) is consistent with the ± Sachs form factor magnitude ratio of 0.67 0.15 0.04, E835 [11] α= ± ± (11) (cid:26)0.85 0.24 0.04, BES [12]. ± ± ρ= E/ M =0.726 0.074. (12) |G G | ± For our comparison with experiment we use the sta- tistically most accurate measurement for each charmonium state, and combine the errors in quadrature. This gives experimental estimates for α of 0.676 0.055 and 1.0 0.67 0.155 for the J/ψ and ψ′ respectively.± ± 0.5 α B. Testing the pure Dirac hypothesis 0.0 Wefirstexaminetheseexperimentalnumbersusingthe “null hypothesis” of no Pauli term, κ=0, in which case α = (1 r)/(1+r), where r = 4m2/M2. This κ = 0 -0.5 − formula was previously givenby Claudson, Glashow and Wise [13] and by Carimalo [14]; the value of α under -1.0 various theoretical assumptions has been discussed by 0.0 0.5 1ρ.0 1.5 2.0 these references and by Brodsky and LePage [15], who predicted α = 1. Fig.2 shows these two experimental FIG. 3: The experimental value of the unpolarized e+e− → values together with the pure Dirac (γ ) formula for α. µ J/ψ → pp¯ angular coefficient, α = 0.676±0.055 (shaded), Theψ′caseisevidentlyconsistentwithaDirac(γ )ψ′pp¯ µ and the resulting Sachs J/ψpp¯strong form factor magnitude ratio ρ=|G /G | (Eq.12). E M In terms of ρ and χ this completes our discussion: 1.0 Given the unpolarized angular distribution, one obtains ψ’ a result for ρ = / from Eq.9, but the phase χ of 0.8 J/ψ |GE GM| α E/ M isundetermined. Howeveronemayaskthemore G G fundamental question of what values of the Dirac and 0.6 Pauli coupling constants g and κ in Eq.1 are consistent with a given experimental unpolarized angular distribu- 0.4 tion. 0.2 D. Determining κ 0.0 0.0 0.2 0.4 0.6 0.8 1.0 r = 4m2/ M2 First we consider the experimentally allowed values of κ. Theunpolarizedangulardistributionprovidesuswith FIG.2: Thecoefficientαobservedintheunpolarizede+e− → arangeofvaluesofρ(Eq.12),butχisunconstrained;we (J/ψ,ψ′) → pp¯angular distributions, together with the the- maycombinethisinformationthroughEq.2todetermine oretical result α=(1−r)/(1+r) predicted by a pure Dirac thelocusofallowed(complex)valuesofκ. Thisisshown (γ ) ψpp¯coupling. µ in Fig.4. 4 0.004 0.2 0.003 0.1 g ) κ m( 0 0.002 I -0.1 0.001 -0.2 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0-180 -90 0 90 180 Re(κ) χ [deg.] FIG.4: Thelocusofcomplexκ(theJ/ψpp¯Paulicoupling)al- FIG. 5: The value of the overall J/ψpp¯ vertex strength g lowedbytheexperimentalconstraintρ=0.726±0.074,taken implied by the experimental Γ and ρ as a function of from the unpolarized differential cross section for e+e− → theunknown J/ψpp¯SachsphaJs/eψχ→.pp¯ J/ψ→pp¯. previousestimate ofg =(1.62 0.03) 10−3[4]assuming Forχ=0,Eq.2impliesthatκisrealandnegative,and ± · only a Dirac J/ψpp¯ coupling, as a result of destructive takesonthe smallestallowedmagnitude. As we increase interference between the Pauli and Dirac terms. χfrom0,the allowedκvaluesproceedclockwise,since κ initiallyacquiresanegativeimaginarypart. Theextreme values of κ on the real axis in Fig.4 are for χ=0,π, and IV. EFFECT ON σ(pp¯→π0J/ψ) are 0.137 0.032, χ=0 κ= − ± (13) The effect of a J/ψpp¯Pauli term on the pp¯ π0J/ψ 0.500 0.011, χ=π. → n− ± cross section may be of considerable interest for the PANDA project, since one might use this as a “cali- bration”reactionforassociatedcharmoniumproduction, E. Determining g and the Pauli term may be numerically important. Al- thoughwehavecarriedoutthis calculationwiththe ver- Next we consider the determination of the overall tex of Eq.1 for general masses, the full result is rather J/ψ pp¯ vertex strength g. Since the differential and total→cross sections for e+e− J/ψ pp¯ only involve complicated; here for illustration we discuss the much → → simpler massless pion limit. g through the ratio g/f , we must introduce additional ψ For a massless pion the ratio of the unpolarized cross experimental data to constrain g. The partial width for section σ(pp¯ π0J/ψ) with a Pauli term to the pure J/ψ pp¯is especially convenientin this regard,since it h → i only→involvesthe strongJ/ψpp¯vertex,andthus depends Dirac result (γµ only, denoted by D) is only on g and κ (and kinematic factors). This partial width was given in terms of the strong Sachs form factors in Eq.6; as a function of g and κ it is 4.0 φ = 0ο 1 g2 r 1 κ Γ = M√1 r 1+ +3 (κ)+ 1+ κ2 . ψ→pp¯ 34π − (cid:20) 2 ℜ (cid:16) 2r(cid:17)| | (cid:21) D3.0 (14) σ> φ = 90ο ReTf.h[4i]s tgoenaeranloiznezsertohePaκul=i c0ourpelsiunlgt. giUvesningintEheq.2P7DoGf > / < 2.0 κ σ values [16] of Γ = 93.4 2.1 keV and B = < J/ψ J/ψ→pp¯ ± (2.17 0.07) 10−3, Eq.14imples a rangeofvaluesofthe 1.0 φ = 180ο ± · κ overallvertexstrengthg foreachvalueofthe(unknown) phase χ. This is shown in Fig.5. There is a range of un- 0.0 certaintying ateachχ (not shownin the figure),due to 0.0 0.2 0.4 0.6 0.8 1.0 |κ| the experimental errors in Γ , B and ρ, which J/ψ J/ψ→pp¯ is at most 5%. Note tha∼t g±is bounded by the limits at χ = 0 and π, FIG. 6: The dependence of the unpolarized, near-threshold for which g 2.0 10−3 and 3.4 10−3 respectively. cross section hσ(pp¯→ π0J/ψ)i on the (complex) Pauli cou- ≈ · ≈ · pling κ=|κ|eiφκ (from Eq.15). The allowed values of g are somewhat larger than our 5 ^y _ hσ(pp¯→π0J/ψ)i = 1+2 (κ)+ 1 + M2 κ2 p ^zp_ hσ(pp¯→π0J/ψ)iD(cid:12)(cid:12)mπ=0 (cid:20) ℜ (cid:16)2 8m2(cid:17)| | _p (cid:12)(cid:12) y^ ^y e− θ e+ + (s−M2) β κ2 , (15) e− zê− ^ze+ e+ 4m2 ln (1+β)/(1 β) | | (cid:21) p − (cid:2) (cid:3) where β = 1 4m2/s is the velocity of the annihilat- − y^ ing p and p¯pin the c.m. frame. The limit of this cross sectionratioatthresholdis showninFig.6forarangeof ^z p p complex κ. Evidently there is destructiveinterference fora κ with adominantnegativerealpart,asissuggestedbythe un- FIG. 7: Axesused to definethepolarization observables. polarizeddata. Forthevalueκ= 0.50(the largersolu- − tioninEq.13)thereisroughlyafactoroftwosuppression in the cross section over the prediction for a pure Dirac observables for this process. Of course there is consider- coupling. The suppression however depends strongly on able redundancy,since they are alldetermined by the 16 thephaseofκ,andforimaginaryκhasbecomeamoder- helicityamplitudes. Theconstraintsofparityandcharge ate enhancement. Thus, the near-thresholdcross section conjugationreducethissetto6independenthelicityam- for pp¯ π0J/ψ is quite sensitive to the strength and plitudes, and for massless leptons (as we assume here) → phase of the Pauli coupling; it will therefore be impor- this is further reduced to 3 independent nonzero helicity tant for PANDA to have an accurate estimate of this amplitudes. quantity. In the next section we will show how both the We introduce the normalized polarization observables magnitudeandphaseofκcanbedeterminedinpolarized Qǫe+ǫe−ǫp¯ǫp ≡ Q(ǫe+,ǫe−,ǫp¯,ǫp) / Q(0,0,0,0), where e+e− J/ψ pp¯scattering, and may be accessible at Q(0,0,0,0) is the unpolarized differential cross section. → → BES. The (nonzero) polarization observables for this process satisfy the relations (a) = = = =1, V. POLARIZATION OBSERVABLES Q0000 Qxxyy Qyyyy −Qzz00 (b) = = = = 00y0 xx0y yy0y zz0y Q Q Q Q The relative phase χ of the J/ψpp¯Sachs strong form = = = , factors and may be determined experimentally 000y xxy0 yyy0 zzy0 E M −Q −Q −Q −Q throughGastudyGofthepolarizedprocesse+e− J/ψ → → (c) xx00 = yy00 = 00yy = zzyy, pp¯. As each of the external particles in this reaction has Q Q Q −Q two possible helicity states, there are 16 helicity ampli- (d) = = = = z0x0 z00x yxyz yxzy Q Q Q Q tudes in total. All the helicity amplitudes to the final pp¯ helicitystates|p(±)p¯(±)iareproportionaltoGE,andall −Q0zx0 =−Q0z0x =−Qxyyz =−Qxyzy, to p( )p¯( ) are proportional to . In the unpolarized| c±ase t∓hesie are squared and sumGmMed, which leads to (e) Qz0z0 = Q0z0z = Qxyxy = Qyxyx = aancdross se2c.tiAons pwreopstorretsiosendaletaorliaerw,etihgihstiemdpsluiems tohfa|tGEth|e2 −Q0zz0 =−Qz00z =−Qyxxy =−Qxyyx, M phas|eGχ o|f GE/GM is not determined by the unpolarized (f) Qxxxx = Qyyxx = Qzzzz =−Q00zz, data. (g) = = = , To show how χ can be measured in polarized scat- Q00xx −Qzzxx −Qxxzz −Qyyzz tering, it is useful to introduce the polarization observ- (h) = = = = 00xz xxzx yyzx zzzx ables discussed by Paschke and Quinn [17]. These are Q Q Q Q angular asymmetries that arise when the polarizations xxxz = yyxz = zzxz = 00zx, −Q −Q −Q −Q of particles are aligned or anti-aligned along particular directions. For example, for our reaction e+e− pp¯, (i) Qxyx0 = Qxy0x = Qz0yz = Qz0zy = → Q(0,0,z,0)isthe differenceoftwoangulardistributions, = = = . (16) yxx0 yx0x 0zyz 0zzy (dσ/dΩ) (dσ/dΩ) . Here wewilluse xandy forthe −Q −Q −Q −Q p↑ p↓ − two transverse axes and z for the longitudinal axis (see Explicit expressions for these observables are given in Fig.7). xândzˆvarywiththe particle,andyîschosento Table I. be common to all. An entry of 0 signifies anunpolarized The results in Table I suggest how we may determine particle. Sincetherearefourpossibleargumentsforeach χexperimentally. Inspectionofthetableshowsthatonly particle, 0,x,y and z, there are 44 = 256 polarization four of the independent polarization observables depend 6 wouldsuggestplausibleJ/ψpp¯couplings. This approach TABLE I: Nonzero inequivalent polarization observables in hassomeexperimentaladvantages;astheΛandΛ¯ decays e+e− →J/ψ→pp¯. The function F is 4−2(1−ρ2)sin2θ. are self-analyzing, no rescattering of the final baryons is required to determine their polarization. In addition no Pol. Observable Result beampolarizationisrequired,sinceitsufficestomeasure Q0000 1 Q00y0 4ρ sinχsinθcosθ/F the (odd-ρ) polarization observables Q00y0 and Q00xz. Qxx00 2(1−ρ2)sin2θ/F One may alsomeasurethe even-ρ observablesQ00xx and Qz0x0 4ρ cosχsinθ/F 00zz as a cross-check of the result for ρ. Q Qz0z0 4cosθ/F Finally,wenoteinpassingthatitmayalsobepossible Q00xz −4ρ cosχsinθcosθ/F to measure the appropriate polarization observables in Qxyx0 −4ρ sinχsinθ/F the time-reversed reaction pp¯ J/ψ e+e−. Q [4−2(1+ρ2)sin2θ]/F → → xxxx Q −2(1+ρ2)sin2θ/F zzxx VI. SUMMARY AND CONCLUSIONS onχ; twoareproportionalto sinχ andtwo to cosχ. As- The unpolarized angular distribution for the process suming that one knows ρ with sufficient accuracy from e+e− J/ψ pp¯, measured recently by the BES the unpolarized data, one may then determine χ unam- → → Collaboration, is inconsistent with theoretical expecta- biguously by extracting sinχ and cosχ from the mea- tions for a pure Dirac J/ψpp¯ coupling. In this paper surement of two of these polarization observables. we have derived the effect of an additional Pauli-type Thedeterminationofsinχisthemoststraightforward, J/ψpp¯coupling,andfindthatthiscanaccommodatethe since it only requires the detection of a single final po- observedangulardistribution. TheJ/ψpp¯Paulicoupling larizedparticle(forexample the proton,through ). 00y0 may significantly affect the cross section for the charmo- Q If κ is close to real, which corresponds to χ 0 or π, nium production reaction pp¯ π0J/ψ, which will be ≈ ≈ thisobservablemayberelativelysmall. Theotherpolar- → studied at PANDA. There is an ambiguity in determin- ization observablesthat are proportionalto sinχ involve ing the relative Dirac and Pauli J/ψpp¯ couplings from asymmetries with either one or three particles polarized; the unpolarized e+e− J/ψ pp¯data; we noted that these are given in relations (b) and (i) of Eq.16. → → this ambiguity can be fully resolved through measure- Determining cosχ involves measuring double or ments ofthe polarizedreaction. The mostattractive po- quadruple polarization observables, which are given in larized process to study initially appears to be the case relations (d) and (h) in Eq.16. In the double polariza- of unpolarized initial e+e− beams, with only the final p tioncase,either oneinitial andone finalpolarizationare (transversely) polarized. Alternatively, measurement of measured(suchase− andp)orthepolarizationsofboth the required polarization observables may also be possi- final particles (p and p¯) are measured. In the first case ble using the time-reversed reaction pp¯ J/ψ e+e−. the relevant observables (such as ) require the ini- → → z0x0 It may also be possible to use self-analyzing processes Q tial lepton to have longitudinal ( zˆ) polarization,which such as e+e− J/ψ ΛΛ¯ to estimate the Dirac and ± is difficult to achieve experimentally. In the second case Pauli couplings→in the →closely related J/ψΛΛ¯ vertex. the initial e+e− beams are unpolarized, and the longitu- dinalpolarizationofonefinalparticleandthe transverse polarization of the other must be measured. Determin- VII. ACKNOWLEDGEMENTS ing the p¯polarization may prove to be an experimental challenge. Of these two general possibilities, the most attractive We are happy to acknowledge useful commu- “next experiment” beyond unpolarized e+e− J/ψ nications with W.M.Bugg, V.Ciancolo, F.E.Close, → → pp¯ scattering may be a measurement of the differential S.Olsen, J.-M.Richard, K.Seth, S.Spanier, E.S.Swanson, cross section with unpolarized leptons and only the final U.Wiedner,C.Y.Wong and Q.Zhao regarding this re- p polarizationdetected. This will determine sinχ, which search. Wealsogratefullyacknowledgethesupportofthe specifies χ up to the usual trigonometric ambiguities. 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