Determining pseudoscalar meson photo-production amplitudes from complete experiments
aa r X i v : . [ nu c l - t h ] M a r Determining pseudoscalar meson photo-productionamplitudes from complete experiments
A. M. Sandorfi , S. Hoblit , , H. Kamano and T.-S. H. Lee , Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA Department of Physics, University of Virginia, Charlottesville, VA 22901, USA National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY 11973,USA Physics Division, Argonne National Laboratory, IL 60439, USA
Abstract.
A new generation of complete experiments is focused on a high precisionextraction of pseudoscalar meson photo-production amplitudes. Here, we review thedevelopment of the most general analytic form of the cross section, dependent uponthe three polarization vectors of the beam, target and recoil baryon, including allsingle, double and triple-polarization terms involving 16 spin-dependent observables.We examine the different conventions that have been used by different authors, andwe present expressions that allow the direct numerical calculation of any pseudoscalarmeson photo-production observables with arbitrary spin projections from the Chew-Goldberger-Low-Nambu (CGLN) amplitudes. We use this numerical tool to clarifyapparent sign differences that exist in the literature, in particular with the definitionsof six double-polarization observables. We also present analytic expressions thatdetermine the recoil baryon polarization, together with examples of their potentialuse with quasi-4 π detectors to deduce observables. As an illustration of the use ofthe consistent machinery presented in this review, we carry out a multipole analysisof the γp → K + Λ reaction and examine the impact of recently published polarizationmeasurements. When combining data from different experiments, we utilize the Fierzidentities to fit a consistent set of scales. In fitting multipoles, we use a combined MonteCarlo sampling of the amplitude space, with gradient minimization, and find a shallow χ valley pitted with a very large number of local minima. This results in broad bandsof multipole solutions that are experimentally indistinguishable. While these bandshave been noticeably narrowed by the inclusion of new polarization measurements,many of the multipoles remain very poorly determined, even in sign, despite theinclusion of data on 8 different observables. We have compared multipoles fromrecent PWA codes with our model-independent solution bands, and found that suchcomparisons provide useful consistency tests which clarify model interpretations. Thepotential accuracy of amplitudes that could be extracted from measurements of all16 polarization observables has been studied with mock data using the statisticalvariations that are expected from ongoing experiments. We conclude that, while amathematical solution to the problem of determining an amplitude free of ambiguitiesmay require 8 observables, as has been pointed out in the literature, experiments withrealistically achievable uncertainties will require a significantly larger number.PACS numbers: 13.60.Le, 13.75.Gx, 13.75.Jz etermining pseudoscalar meson photo-production amplitudes ...
1. Introduction
As a consequence of dynamic chiral symmetry breaking, the Goldstone bosons ( π, η, K )dress the nucleon and alter its spectrum. Not surprisingly, pseudoscalar mesonproduction has been a powerful tool in studying the spectrum of excited nucleon states.However, such states are short lived and broad so that above the energy of the firstresonance, the P ∆(1232), the excitation spectrum is a complicated overlap of manyresonances. Isolating any one and separating it from backgrounds has been a long-standing problem in the literature.The spin degrees of freedom in meson photoproduction provide signatures ofinterfering partial wave strength that are often dramatic and have been useful fordifferentiating between models of meson production amplitudes. Models that mustaccount for interfering resonance amplitudes and non-resonant contributions are oftenseverely challenged by new polarization data. Ideally, one would like to partition theproblem by first determining the amplitudes from experiment, at least to within a phase,and then relying upon a model to separate resonances from non-resonant processes.Single-pseudoscalar photoproduction is described by 4 complex amplitudes (two forthe spin states of the photon, two for the nucleon target and two for the baryonrecoil, which parity considerations reduce to a total of 4). They are most commonlyexpressed in terms of the Chew-Goldberger-Low-Nambu (CGLN) [1] amplitudes. Toavoid ambiguities, it has been shown [2] that angular distribution measurements of atleast 8 carefully chosen observables at each energy for both proton and neutron targetsmust be performed. While such experimental information has not yet been available,even after 50 years of photoproduction experiments, a sequence of complete experimentsare now underway at Jefferson Lab [3, 4], as well as complementary experiments fromthe GRAAL backscattering source in Grenoble [5, 6] and the electron facilities in Bonnand Mainz, with the goal of obtaining a direct determination of the amplitude to withina phase, for at least a few production channels, notably K Λ and possibly πN .The four CGLN amplitudes can be expressed in Cartesian ( F i ), Spherical or Helicity( H i ), or Transversity ( b i ) representations. While the latter two choices afford sometheoretical simplifications when predicting asymmetries from models [7], when workingin the reverse direction, fitting asymmetries to extract amplitudes, such simplificationsare largely moot. The four amplitudes in each of these representations are angledependent. Extracting them directly from experiment would require separate fits at eachangle, which greatly limits the data that can be used and requires some model-dependentscheme to constrain an arbitrary phase that could be angle-dependent. The solutionto this intractable situation is a Wigner-Eckhart style factorization into reduced matrixelements, multipoles, and simple angle-dependent coefficients from angular momentumalgebra. One can then fit the multipoles directly, which both facilitates the searchfor resonance behavior and allows the use of full angular distribution data at a fixedenergy to constrain angle-independent quantities. The price is a significant increasein the number of fitting parameters, but since the excited states of the nucleon are etermining pseudoscalar meson photo-production amplitudes ... F i representation, whichhas the simplest decomposition into multipoles [1], equations (15)-(18) below.In single-pseudoscalar meson photoproduction there are 16 possible observables, theunpolarized differential cross section ( dσ ), three asymmetries which to leading orderenter the general cross section scaled by a single polarization of either beam, targetor recoil (Σ, T , P ), and three sets of four asymmetries whose leading polarizationdependence in the general cross section involves two polarizations of either beam-target (BT), beam-recoil (BR), or target-recoil (TR), as in [7]. Expressions for at leastsome of these observables in terms of the CGLN F i appear already in earlier papers[8, 9, 11, 12, 13]. In all cases we have found in the literature, the magnitudes of theexpressions relating the CGLN F i to experimental observables are identical, but thesigns of some appear to differ. This is only now becoming a significant issue since thesign differences occur in double-polarization observables for which little data have beenavailable until very recently. There is also a set of Fierz identities interrelating the 16polarization observables, the most complete list being given in [2]. We have found manyof the signs in the expressions of this list appear to be incompatible with several of thesepapers. As we will see below, much of this confusion has its origin in the same symbol,or observable name, being used by different authors to represent different experimentalquantities.Our purpose here is two fold. First we assemble a complete set of relations, definingobservables in terms of specific pairs of measurable quantities and providing the mostgeneral form of the cross sections in terms of all observables. We then give a consistentset of relations between these experimental observables and the CGLN amplitudes andelectromagnetic multipoles [1]. Next, as an illustration of the use of these relations alongthe path to determining an amplitude, we use recently published results on 8 differentobservables to carry out a multipole analysis of the γp → K + Λ reaction, free of modelassumptions, and examine the uniqueness of the resulting solutions. Finally, we usemock data to study the potential uniqueness of amplitudes that could be extractedfrom complete sets of all 16 observables.There are several coordinate systems in use in the literature and in section 2 wedefine ours, which is the same as used in the seminal paper by Barker, Donnachie andStorrow (BDS) [7]. In section 3 we present explicit and complete formulae that allowthe direct calculation of matrix elements with arbitrary spin projections from CGLNamplitudes or multipoles. In section 4 we present the most general analytic form ofthe cross section, dependent on the three polarization vectors of the beam, the targetand the recoil baryon. The derivation of this cross section expression is summarizedin Appendix A, and the experimental definitions of the observables in terms of crosssections with explicit polarization orientations is tabulated in Appendix B. Using thesedefinitions, we give in section 5 a summary of the variations in similar formulae thatappear in literature. While the beam and target polarizations can be controlled inan experiment, the recoil polarization is on a very different footing, in that it arises etermining pseudoscalar meson photo-production amplitudes ... Figure 1.
Kinematic variables in meson photoproduction in Lab and c.m. frames. as a consequence of the angular momentum of the entrance channel and the reactionphysics, neither of which is under experimental control. Expressions that determinethe recoil baryon polarization are developed in section 6. To evaluate the analyticrelations between observables and amplitudes we next use numerical calculations of theexpressions in section 3 to fix signs and present the complete set of equations in section 7that determine the 16 observables from the CGLN amplitudes. The 37 Fierz identitiesthat interrelate the observables are discussed in section 8 and presented with consistentsigns in Appendix C. In section 9 we utilize the machinery we have assembled to carryout a multipole analysis of the γp → K + Λ reaction. (Born terms for this process aresummarized in Appendix D.) In so doing we test the nature of the χ valley, discuss therole of the arbitrary phase and examine the impact of recently published polarizationdata and the uniqueness of the multipole solutions from resent data. The accuracyof the data needed for a precise model independent extraction of amplitudes is theninvestigated in section 10 from a study with mock data on all possible 16 observableswith varying levels of statistical precision. Section 11 concludes with a brief summary.
2. Kinematics and coordinate definitions
The kinematic variables of meson photoproduction used in our derivations are specifiedin figure 1. Some useful relations are : • The total center of mass (c.m.) energy: W = √ s = q m tgt ( m tgt + 2 E Lab γ ) . (1) • The laboratory (Lab) energy needed to excite the hadronic system with total c.m.energy W : E Lab γ = W − m m tgt . (2) • The energy of the photon in the c.m. frame: E c . m .γ = W − m W = q. (3) etermining pseudoscalar meson photo-production amplitudes ... Figure 2. (Color online) The c.m. coordinate system and angles used to specifypolarizations in the reaction, γ ( ~q, ~P γ ) + N ( − ~q, ~P T ) → K ( ~k ) + Λ( − ~k, ~P R Λ ). The left(right) side is for the initial γN (final K Λ) system; ˆ z is along the photon beam direction;ˆ y is perpendicular to the h ˆ x − ˆ z i reaction plane and ˆ x = ˆ y × ˆ z ; ˆ z ′ is along the mesonmomentum and ˆ x ′ is in the h ˆ x − ˆ z i plane, rotated down from ˆ z by θ K + π/ • The magnitude of the 3-momentum of the meson in the c.m. frame: (cid:12)(cid:12) p c . m .π,η,K (cid:12)(cid:12) = W (" − (cid:18) m π,η,K + m R W (cid:19) − (cid:18) m π,η,K − m R W (cid:19) / = k. (4) • The density of state factor: ρ = (cid:12)(cid:12) p c . m .π,η,K (cid:12)(cid:12) /E c . m .γ = k/q. (5)The definitions of polarization angles used in our derivation are shown in figure 2,using the case of K Λ production as an example. The h ˆ x − ˆ z i plane is the reactionplane in the center of mass. The figure illustrates the case of linear γ polarization,with the alignment direction P γL (parallel to the oscillating electric field of the photon)in the h ˆ x − ˆ y i plane at an angle φ γ , rotating from ˆ x towards ˆ y . The target nucleonpolarization ~P T is specified by polar angle θ p measured from ˆ z , and azimuthal angle φ p in the h ˆ x − ˆ y i plane, rotating from ˆ x towards ˆ y . The recoil Λ baryon is in the h ˆ x − ˆ z i plane; its polarization ~P R Λ is at polar θ p ′ , measured from ˆ z , and azimuthal φ p ′ in the h ˆ x − ˆ y i plane, rotating from ˆ x to ˆ y . Following BDS [7], observables involving recoilpolarization are specified in the rotated coordinate system with ˆ z ′ = +ˆ k , along themeson c.m. momentum and opposite to the recoil momentum, ˆ y ′ = ˆ y , and ˆ x ′ = ˆ y ′ × ˆ z ′ in the scattering plane at a polar angle of θ K + ( π/
2) relative to ˆ z .The case of circular photon polarization can potentially lead to some confusion.Most particle physics literature designates circular states as r , for right circular (or l ,for left circular), referring to the fact that with r polarization the electric vector of thephoton appears to rotate clockwise when the photon is traveling away from the observer . etermining pseudoscalar meson photo-production amplitudes ... l circularly polarized. Nonetheless, bothconventions agree on the value of the photon helicity [14] h = ~S · ~P / | ~P | = ± ~P γc = +1( −
1) when 100% of the photon spinsare parallel (anti-parallel) to the photon momentum vector.
3. Calculation of polarization observables
As discussed in section 1, all publications give similar formulae for polarizationobservables, but conflicting signs occur in some terms with very lengthy expressions.It is very difficult, if not impossible, to resolve this problem by repeating the samealgebraic procedures used in previous works. To resolve these sign problems, it isnecessary to develop completely different and yet simple formulae which can be used tocalculate numerically all spin observables of pseudoscalar meson photoproduction. Thisnumerical tool will then allow us to check unambiguously the analytic expressions forspin observables in all previous publications. In this section, we present the derivationof such formulae using the case of K Λ photoproduction as an example.Let us first consider the case when all beam, target, and recoil polarizationsare 100% polarized in certain directions. With variables specified as in figure 2, thedifferential cross section for γ ( ~q, ˆ P γ ) + N ( − ~q, m s N ) → K ( ~k ) + Λ( − ~k, m s Λ ) in the centerof mass frame can be written as dσd Ω ( ˆ P γ , m s N , m s Λ ) = 1(4 π ) kq m N m Λ W | ¯ u Λ ( − ~k, m s Λ ) I µ ǫ µ u N ( − ~q, m s N ) | , (6)where W = q + E N ( q ) = E K ( k ) + E Λ ( k ); ǫ µ = (0 , ˆ P γ ) with | ˆ P γ | = 1 is thephoton polarization vector; m s Λ and m s N are the spin substate quantum numbers ofthe Λ and the nucleon along the z -direction, respectively; ¯ u Λ I µ ǫ µ u N is normalizedto the usual invariant amplitude calculated from a Lagrangian in the convention ofBjorken and Drell [15]. For example, for a simplified Lagrangian density L ( x ) = − ( f K Λ N /m K ) ¯ ψ Λ ( x ) γ γ µ ψ N ( x ) ∂ µ φ K ( x ) + e N ¯ ψ N ( x ) γ µ ψ N ( x ) A µ ( x ), the s -channel γ ( q ) + N ( p ) → N ( p ′ + k ) → K + ( k ) + Λ( p ′ ) contribution to I µ is i e N ( f K Λ N /m K ) kγ [( k + p ′ ) − m N ] − γ µ . By averaging over all initial state polarizations and summing over final statepolarizations in (6), we can obtain the unpolarized cross section: dσ ≡ X m sN = ± / X m s Λ = ± / X γ − spins dσd Ω ( ˆ P γ , m s N , m s Λ ) , (7)where the symbol P γ − spins implies taking summation over two photon polarizationstates, with polarization vectors perpendicular to each other for linearly polarizedphotons and with helicity ± u Λ ( − ~k, m s Λ ) I µ ǫ µ u N ( − ~q, m s N ) = − πW √ m N m Λ h m s Λ | F CGLN | m s N i , (8) etermining pseudoscalar meson photo-production amplitudes ... | m s i is the usual eigenstate of the Pauli operator σ z , and F CGLN = X i =1 , O i F i ( θ K , E ) , (9)with O = − i ~σ · ˆ P γ , (10) O = − [ ~σ · ˆ k ][ ~σ · (ˆ q × ˆ P γ )] , (11) O = − i[ ~σ · ˆ q ][ˆ k · ˆ P γ ] , (12) O = − i[ ~σ · ˆ k ][ˆ k · ˆ P γ ] . (13)Here we have defined ˆ k = ~k/ | ~k | and ˆ q = ~q/ | ~q | . We then obtain dσd Ω ( ˆ P γ , m s N , m s Λ ) = kq |h m s Λ | F CGLN | m s N i| . (14)The formulae for calculating CGLN amplitudes from multipoles are well known [1]and are given below: F = X l =0 [ P ′ l +1 ( x ) E l + + P ′ l − ( x ) E l − + lP ′ l +1 ( x ) M l + + ( l + 1) P ′ l − ( x ) M l − ] , (15) F = X l =0 [( l + 1) P ′ l ( x ) M l + + lP ′ l ( x ) M l − ] , (16) F = X l =0 [ P ′′ l +1 ( x ) E l + + P ′′ l − ( x ) E l − − P ′′ l +1 ( x ) M l + + P ′′ l − ( x ) M l − ] , (17) F = X l =0 [ − P ′′ l ( x ) E l + − P ′′ l ( x ) E l − + P ′′ l ( x ) M l + − P ′′ l ( x ) M l − ] . (18)where x = ˆ k · ˆ q = cos θ K , l is the orbital angular momentum of the K Λ system, and P ′ l ( x ) = dP l ( x ) /dx and P ′′ l ( x ) = d P l ( x ) /dx are the derivatives of the Legendre function P l ( x ), with the understanding that P ′− = P ′′− = 0. In practice, the sum runs to alimiting value of l max which depends on the energy.In order to calculate the 16 polarization observables in an arbitrary experimentalgeometry, we develop a form for the cross section with arbitrary spin projections forinitial and final baryon states, γ ( ~q, ˆ P γ ) + N ( − ~q, ˆ P T ) → K ( ~k ) + Λ( − ~k, ˆ P R ), as specifiedin figure 2, where ˆ P T ( ˆ P R ) is the unit vector specifying the direction of the target(recoil) spin polarization. Here linear photon polarization must be in the h ˆ x − ˆ y i planeand circular photon polarization must be aligned with ˆ z , while ˆ P T and ˆ P R can bein any directions. The corresponding cross section is obtained by simply replacing |h m s Λ | F CGLN | m s N i| in (14) with |h ˆ P R | F CGLN | ˆ P T i| : dσ B , T , R ( ˆ P γ , ˆ P T , ˆ P R ) ≡ dσd Ω ( ˆ P γ , ˆ P T , ˆ P R ) = kq |h ˆ P R | F CGLN | ˆ P T i| , (19)where | ˆ P T i ( h ˆ P R | ) is a state of the initial (final) spin-1 / P T ( ˆ P R ) direction. We note that if ˆ P T ( ˆ P R ) is in the direction of the momentumof the initial (final) baryon, then | ˆ P T i ( h ˆ P R | ) is the usual helicity state as defined, forexample, by Jacob and Wick [16]. We need to consider more general spin orientations etermining pseudoscalar meson photo-production amplitudes ... | ˆ s i quantized in the directionof an arbitrary vector ˆ s = (1 , θ, φ ) is defined by ~S · ˆ s | ˆ s i = + 12 | ˆ s i , (20)where ~S is the spin operator. For the considered spin-1/2 baryons, ~S is expressed withthe Pauli matrix: ~S = ~σ/ h ˆ P R | F CGLN | ˆ P T i in terms of the CGLN amplitudes F i in (15)-(18). We note that the spin state | ˆ s i isrelated to the usual eigenstate of z -axis quantization by rotations: | ˆ s i = X m = ± / D (1 / m, +1 / ( φ, θ, − φ ) | m i , (21)where | m i is defined as S z |± / i = ( ± / |± / i , and D (1 / m,λ ( φ, θ, − φ ) = exp[ − i( m − λ ) φ ] d / m,λ ( θ ) . (22)We use the phase convention of Brink and Satchler [17] where, d / / , +1 / ( θ ) = d / − / , − / ( θ ) = cos θ d / − / , +1 / ( θ ) = − d / / , − / ( θ ) = sin θ . (23)Equation (21) can be easily verified by explicit calculations using the definition (20)and the properties (22) and (23) for the special cases where ˆ s = ˆ x , ˆ y , ˆ z , together withthe usual definition of the Pauli matrices, ( σ i ) mm ′ [ i = x, y, z and m (row), m ′ (column)= ± / , ± / σ x = ! , σ y = − ii 0 ! , σ z = − ! . (24)From figure 2, the momenta and linear photon polarization are expressed as ~q = q (0 , , , (25) ~k = k (sin θ K , , cos θ K ) , (26)ˆ P γL = (cos φ γ , sin φ γ , . (27)Circular photon polarizations of helicity λ γ are expressed as( ˆ P γc ) λ γ = ± = ∓ √ x ± iˆ y ) . (28)For the initial and final baryon polarizations, we use the spherical variables, as in figure 2:ˆ P T = (1 , θ p , φ p ) , (29)ˆ P R = (1 , θ p ′ , φ p ′ ) . (30)By using (25)-(27), we can rewrite O i in (10)-(13) as O i = X n =0 , C i,n ( θ K , φ γ ) σ n , (31) etermining pseudoscalar meson photo-production amplitudes ... Table 1. C i,n ( θ K , φ γ ) of (31) and (33). n = 0 n = 1 n = 2 n = 3 i = 1 0 − i cos φ γ − i sin φ γ i = 2 sin θ K sin φ γ i cos θ K cos φ γ i cos θ K sin φ γ − i sin θ K cos φ γ i = 3 0 0 0 − i sin θ K cos φ γ i = 4 0 − i sin θ K cos φ γ − i sin θ K cos θ K cos φ γ where σ = , σ = σ x , σ = σ y , σ = σ z . The explicit form of C i,n is given in table 1.By using (21) and (9) and (31), the photoproduction matrix element can then becalculated as h ˆ P R | F CGLN | ˆ P T i = X n =0 , G n ( θ K , φ γ ) h ˆ P R | σ n | ˆ P T i , (32)with G n ( θ K , φ γ ) = X i =1 , F i ( θ K , E ) C i,n ( θ K , φ γ ) , (33)and h ˆ P R | σ n | ˆ P T i = X m s Λ ,m sN = ± / D (1 / ∗ m s Λ , +1 / ( φ p ′ , θ p ′ , − φ p ′ ) D (1 / m sN , +1 / ( φ p , θ p , − φ p ) h m s Λ | σ n | m s N i , (34)where h m s Λ | σ n | m s N i = ( σ n ) m s Λ ,m sN are the elements of the Pauli matrices of (24).We may now start with any set of multipoles and use (15)-(18) to calculate theCGLN amplitudes, which are then used to calculate the matrix element h ˆ P R | F CGLN | ˆ P T i with the help of (32)-(34). Equation (19) then allows us to calculate all possiblepolarization observables, for the case of unit polarization vectors with arbitraryorientation.With non-unit polarization vectors, the general cross section can be expressed interms of (19) as, (see also Appendix A), dσ B , T , R ( ~P γ , ~P T , ~P R ) = X ˆ P = ˆ P γ , ˆ P γ X ˆ Q = ± ˆ P T X ˆ R = ± ˆ P R p γ ˆ P p T ˆ Q p R ˆ R dσ B , T , R ( ˆ P , ˆ Q, ˆ R ) . (35)Here the vector ~P X specifies the degree and direction of the polarization of particle X = γ, T, R . For the target (T) and recoil (R) baryons, this is just ~P X = ( p X + ˆ P X − p X − ˆ P X ) ˆ P X ,where p X ± ˆ P X ( X = T, R ) is the probability of observing X with its polarization vectorpointing in the ± ˆ P X direction. For the photons ( γ ), however, the non-unit polarizationvector can be expressed as ~P γ = ( p γ ˆ P γ − p γ ˆ P γ ) ˆ P γ . Here, ˆ P γ ( ≡ ˆ P γ ) and ˆ P γ are orthogonalpolarization directions, 90 ◦ apart for linear polarization, and opposite helicity statesfor circular polarization. Then p γ ˆ P γ ( p γ ˆ P γ ) is a probability observing photons with itspolarization vector pointing in the ˆ P γ ( ˆ P γ ) direction. To clarify (35), consider the casethat all beam, target, and recoil particles are unpolarized as an example. In this case etermining pseudoscalar meson photo-production amplitudes ... p T,R ± ˆ P T,R = p γ ˆ P γ , ˆ P γ = 1 /
2, which leads to ~P γ,T,R = ~
0. Then we have dσ B , T , R ( ~ , ~ , ~
0) = 18 h dσ B , T , R ( ˆ P γ , + ˆ P T , + ˆ P R ) + dσ B , T , R ( ˆ P γ , + ˆ P T , − ˆ P R )+ dσ B , T , R ( ˆ P γ , − ˆ P T , + ˆ P R ) + dσ B , T , R ( ˆ P γ , − ˆ P T , − ˆ P R )+ dσ B , T , R ( ˆ P γ , + ˆ P T , + ˆ P R ) + dσ B , T , R ( ˆ P γ , + ˆ P T , − ˆ P R )+ dσ B , T , R ( ˆ P γ , − ˆ P T , + ˆ P R ) + dσ B , T , R ( ˆ P γ , − ˆ P T , − ˆ P R ) i = 12 dσ , (36)where dσ is the unpolarized cross section defined in (7). The factor (1 /
2) in the lastequation appears because the polarization of the final recoil particles is also averaged in(36).
4. General cross section
While the formulae presented in the previous section can be used numerically to calculateany observable of pseudoscalar meson photoproduction, it is more convenient to analyzethe data using an analytic expression for the general cross section of equation (35). Interms of the polarization vectors of figure 2, and with signs verified numerically using(35) of section 3, the most general form of the cross section can be written as, dσ B , T , R ( ~P γ , ~P T , ~P R ) = 12 (cid:8) dσ (cid:2) − P γL P Ty P Ry ′ cos(2 φ γ ) (cid:3) + ˆΣ (cid:2) − P γL cos(2 φ γ ) + P Ty P Ry ′ (cid:3) + ˆ T (cid:2) P Ty − P γL P Ry ′ cos(2 φ γ ) (cid:3) + ˆ P (cid:2) P Ry ′ − P γL P Ty cos(2 φ γ ) (cid:3) + ˆ E (cid:2) − P γc P Tz + P γL P Tx P Ry ′ sin(2 φ γ ) (cid:3) + ˆ G (cid:2) P γL P Tz sin(2 φ γ ) + P γc P Tx P Ry ′ (cid:3) + ˆ F (cid:2) P γc P Tx + P γL P Tz P Ry ′ sin(2 φ γ ) (cid:3) + ˆ H (cid:2) P γL P Tx sin(2 φ γ ) − P γc P Tz P Ry ′ (cid:3) + ˆ C x ′ (cid:2) P γc P Rx ′ − P γL P Ty P Rz ′ sin(2 φ γ ) (cid:3) + ˆ C z ′ (cid:2) P γc P Rz ′ + P γL P Ty P Rx ′ sin(2 φ γ ) (cid:3) + ˆ O x ′ (cid:2) P γL P Rx ′ sin(2 φ γ ) + P γc P Ty P Rz ′ (cid:3) + ˆ O z ′ (cid:2) P γL P Rz ′ sin(2 φ γ ) − P γc P Ty P Rx ′ (cid:3) + ˆ L x ′ (cid:2) P Tz P Rx ′ + P γL P Tx P Rz ′ cos(2 φ γ ) (cid:3) + ˆ L z ′ (cid:2) P Tz P Rz ′ − P γL P Tx P Rx ′ cos(2 φ γ ) (cid:3) + ˆ T x ′ (cid:2) P Tx P Rx ′ − P γL P Tz P Rz ′ cos(2 φ γ ) (cid:3) + ˆ T z ′ (cid:2) P Tx P Rz ′ + P γL P Tz P Rx ′ cos(2 φ γ ) (cid:3)o . (37) etermining pseudoscalar meson photo-production amplitudes ... dσ with a caret, so that ˆ A = Adσ .These products are referred to as profile functions in [2, 11]. One can of course pulla common factor of dσ out in front of the above expression, in which case all theprofile functions are replaced by their corresponding asymmetries. However, we keepthe above form since it is the profile functions that are most simply determined bythe CGLN amplitudes. (The definition of each of these profile functions in terms ofmeasurable quantities is given by Appendix B.) The second, third and fourth terms ( ˆΣ,ˆ T , ˆ P ) are commonly referred to as single-polarization observables, since their leadingcoefficients contain only a single polarization vector. The subsequent 12 terms aregrouped into 3 sets, each containing four terms, referred to as { BT, BR, TR } accordingto the combination of polarization vectors appearing in their leading coefficients. Twoof the leading terms have negative coefficients. The first arises because we have takenfor the numerator of the beam asymmetry (Σ) the somewhat more common definitionof ( σ ⊥ − σ k ), rather than its negative. [Here ⊥ ( k ) corresponds to ~P γL = ˆ y ( ~P γL = ˆ x )in the left panel of figure 2.] In the second leading term with a negative coefficient, wehave taken the numerator of the E asymmetry as the difference of cross sections withanti-parallel and parallel photon and target spin alignments ( σ A − σ P ). This followsa convention first introduced by Worden [18] and propagated through many (thoughnot all) subsequent papers, and has been used in recent experimental evaluations ofthe GDH sum rules [19]. The specific measurements needed to construct each of theseobservables are tabulated in Appendix B.Recoil observables are generally specified in the rotated coordinate system withˆ z ′ = +ˆ k . Occasionally, a particular recoil observable will have a more transparentinterpretation in the unprimed coordinate system of figure 2 [20]. Since a baryonpolarization transforms as a standard three vector, the unprimed and primed observablesare simply related: A x = + A x ′ cos θ K + A z ′ sin θ K A z = − A x ′ sin θ K + A z ′ cos θ K , (38)and A x ′ = + A x cos θ K − A z sin θ K A z ′ = + A x sin θ K + A z cos θ K , (39)where A represents any one of the BR or TR observables.It is convenient to arrange the observables entering the general cross section intabular form, as in table 2. The four rows correspond to different states of beampolarization, either ignoring the incident polarization entirely (labeled unpolarized intable 2), or in one of three standard Stokes vector components that characterize anensemble of photons with polarization P γ , linear at ± ◦ to the reaction plane (whichenters the cross section with a sin(2 φ ) dependence), linear either in or perpendicular etermining pseudoscalar meson photo-production amplitudes ... φ ) dependence), orcircular. The columns of the table give the polarization of the target, recoil, or target+ recoil combination. One can readily construct from this table the terms that enterthe general cross section for any given combination of polarization conditions. Theseconsist of the terms involving all applicable polarization vectors, as well as those thatsurvive when initial states are averaged and/or final states are summed. We considertwo examples as an illustration. First, for a circularly polarized beam on an unpolarizedtarget with an analysis of the three components of recoil polarization, the generalcross section contains terms from the average of initial states (first row of table 2)and from the polarized initial state (forth row). Contributing terms come from onlythose columns that do not require knowing the target polarization state. Thus thecross section for this condition becomes (1 / dσ + P Ry ′ ˆ P ) + P γc ( P Rx ′ ˆ C x ′ + P Rz ′ ˆ C z ′ )].Alternatively, with linear beam polarization in or perpendicular to the reaction plane,a longitudinally polarized target (along ˆ z ) and an analysis of recoil polarization alongthe meson (kaon) momentum (ˆ z ′ ), the general cross section is given by the terms inthe first ( unpolarized ) and third rows that are either independent of target and recoilpolarization ( dσ , − Σ) or in columns associated with polarization along ˆ z and/or ˆ z ′ ,namely (1 / dσ + P Tz P Rz ′ ˆ L z ′ ) + P γL cos(2 φ γ )( − ˆΣ − P Tz P Rz ′ ˆ T x ′ )]. e t e r m i n i n g p s e u do s c a l a r m e s o n pho t o - p r od u c t i o n a m p l i t u d e s ... Table 2.
Polarization observables in pseudoscalar meson photoproduction. Each observable appears twice in the table. The 16 entriesin italics indicate the leading polarization dependence of each observable in the general cross section. The three underlined entries ( ˆ P ,ˆ T , ˆΣ) are nominal single-polarization quantities that can be measured with double-polarization. Those in bold are the unpolarized crosssection and 12 nominal double-polarization quantities that can be measured with triple-polarization. (See text.)Beam ( P γ ) Target ( P T ) Recoil ( P R ) Target ( P T ) + Recoil ( P R ) x ′ y ′ z ′ x ′ x ′ x ′ y ′ y ′ y ′ z ′ z ′ z ′ x y z x y z x y z x y z unpolarized dσ ˆ T ˆ P ˆ T x ′ ˆ L x ′ ˆΣ ˆ T z ′ ˆ L z ′ P γL sin(2 φ γ ) ˆ H ˆ G ˆ O x ′ ˆ O z ′ ˆC z ′ ˆE ˆF − ˆC x ′ P γL cos(2 φ γ ) − ˆΣ − ˆ P − ˆ T − ˆL z ′ ˆT z ′ − d σ ˆL x ′ − ˆT x ′ circular P γc ˆ F − ˆ E ˆ C x ′ ˆ C z ′ − ˆO z ′ ˆG − ˆH ˆO x ′ etermining pseudoscalar meson photo-production amplitudes ...
5. Variations within the existing literature
The form of the general cross section expression in equation (37) has been derivedanalytically in Appendix A and checked numerically with the tools of section 3. At thispoint, it is instructive to summarize the variations in similar formulae in the literaturewhich have already caused some confusions in analyzing recent data and must be resolvedfor future development. The most frequently quoted works that discuss the relationbetween observables and CGLN amplitudes are the following four: Barker-Donnachie-Storrow (BDS) [7], Adelseck-Saghai (AS) [10], Fasano-Tabakin-Saghai (FTS) [11], andKn¨ochlein-Drechsel-Tiator (KDT) [13]. A few of the differences between them aresummarized in the following subsections.
The coordinate system of the BDS paper is the same as ours in figure 2 above. Thephoton beam momentum is along +ˆ z ; h ˆ z − ˆ x i is the reaction plane containing the mesonmomentum ~p m emerging at a center of mass angle measured from ˆ z rotating towardsˆ x ; ( ~p γ × ~p m ) / | ~p γ × ~p m | = +ˆ y and [( ~p γ × ~p m ) × ~p γ ] / | ( ~p γ × ~p m ) × ~p γ | = +ˆ x . The recoilbaryon polarization is specified in a rotated primed -coordinate system, with +ˆ z ′ alongthe meson momentum, ~p m ; ˆ y ′ = ˆ y and ˆ x ′ lies in the h ˆ z − ˆ x i plane, rotated down from ˆ x by θ c . m . . It has since become common to indicate the use of this rotated system by includinga prime in the symbol of observables that involve recoil, e.g., C z ′ , O x ′ , etc., althoughthe prime is not used in the BDS paper. The BDS paper is certainly a seminal work onthis subject but, in its published form, it contains an unfortunate piece of typesettingthat has lead to some confusion. Page 348 of that journal article ends with the sentence,“ The precise relation between observables and the experiments we consider is as follows. ”The next page 349 contains table I with several columns, the “Usual symbol” for theobservables, their decomposition into “Helicity” and “Transversity” amplitudes, and inthe fourth column the “Experiment required” to measure each observable. This forthcolumn utilizes a notation that is somewhat condensed, but at least appears clear forlinear polarization at 45 ◦ to the reaction plane. For example the experiment required to determine the H asymmetry is listed as { L ( ± / x ; −} , which would imply thefollowing ratio of cross sections with polarized beam, target and recoil, dσ B , T , R ( φ Lγ = + π/ , ~P T = +ˆ x, sum f . s . ) − dσ B , T , R ( φ Lγ = − π/ , ~P T = +ˆ x, sum f . s . ) dσ B , T , R ( φ Lγ = + π/ , ~P T = +ˆ x, sum f . s . ) + dσ B , T , R ( φ Lγ = − π/ , ~P T = +ˆ x, sum f . s . ) , where unobserved final recoil polarization states are summed. However, equation (2) inBDS [7] at the top of the following page 350 gives the H -dependence of the cross sectionas, dσ B , T , R = dσ (cid:8) P Tx (cid:2) − P γL H sin(2 φ Lγ ) (cid:3) + · · · (cid:9) , (40)and using this to evaluate the above ratio results in − H . The sense of rotation forthe angle φ Lγ is not defined, but we assume it is measured from the x axis rotating etermining pseudoscalar meson photo-production amplitudes ... y axis. [The opposite sense would introduce another negative sign in termsproportional to sin(2 φ Lγ ).] We regard the equation for the cross section as the mostdefinitive. Thus, one should take the “Helicity” and “Transversity” expansions of theobservables in the second and third columns of table I in BDS [7] literally, but therequired experiment in column four as schematic only, leaving the sign of the specificcombination of measurements to be determined from their equations (2)-(4). Whilerather convoluted, we believe this represents the correct reading of the BDS paper.Finally, we note that equations (3) and (4) in BDS [7], which give their cross sectionsfor polarized beam and recoil, and for polarized target and recoil, respectively, are bothmissing a factor of 1 /
2. This is easily seen by averaging over initial states and summingover final states, which for the equations as written results in twice the unpolarized crosssection, 2 σ . While the BDS coordinates were focused on the meson, the coordinate system of theAS paper is focused on the final state baryon. The photon beam momentum is along − ˆ z . In the h ˆ z − ˆ x i reaction plane the recoil baryon emerges at a center of mass anglemeasured from ˆ z rotating towards ˆ x ; ˆ y is still defined with the meson momentum as( ~p γ × ~p m ) / | ~p γ × ~p m | = +ˆ y , but now [( ~p γ × ~p m ) × ( − ~p γ )] / | ( ~p γ × ~p m ) × ( − ~p γ ) | = +ˆ x .The sense of rotation for the linear photon polarization angle φ Lγ is defined from theˆ x axis rotating toward ˆ y . Relative to ( ~p γ × ~p m ) × ~p γ , a linear polarization orientationof + π/ − π/ primed -coordinate system is taken with +ˆ z ′ along the baryon momentum; ˆ y ′ = ˆ y andˆ x ′ lies in the h ˆ z − ˆ x i plane, rotated up from ˆ x by θ c . m . . The observables involvingcomponents of the recoil polarization refer to the primed coordinates, although primes are not included in their notation. The AS paper includes a general expression for thecross section in terms of beam, target and recoil polarizations. However, as discussedin section 6 below, their expression has at least one misprint in its last line, with twoterms involving P Rz and O z but none with P Rz and O x . As evident in our equation (37),each of these observables appears in the general cross section with two coefficients, onedependent upon P Rx and the other upon P Rz . The coordinate system of FTS is the same as that of BDS and of the present work.Circular polarization states are designated as r and l . Although the Stokes vector for thephoton beam is taken from optics (which associates r circular polarization with helicity − r beam polarization with helicity +1. The sense of rotationfor the linear photon polarization angle φ Lγ is defined from the ˆ x axis rotating towardˆ y . The observables involving components of the recoil polarization are designated withprimed symbols. FTS does not give an explicit expression for the cross section in terms ofobservables and polarizations, but the paper does list explicit definitions of observables etermining pseudoscalar meson photo-production amplitudes ... The coordinate system of KDT is the same as that of BDS and of the present work.Circular photon polarization ( P ⊙ ) is referred to as right handed; although helicity isnot discussed, we have assumed (in Table 3 below) that their right-handed state cor-responds to h = +1. The direction of rotation for the angle φ Lγ is not defined; in theevaluation below we have assumed their azimuthal polarization angle rotates from ˆ x toward ˆ y . Cross section equations are given for the cases of beam + target polarization, beam + recoil polarization and target + recoil polarization. As in BDS, the latter two aremissing a factor of 1 /
2, as is easily verified by averaging over initial states and sum-ming over final states. KDT provides explicit equations to relate each observable to theCGLN amplitudes.To completely define an observable in terms of measurable quantities one needs aspecification of the coordinate system and either the equation for the cross section interms of polarizations and observables, or an explicit definition of the observables interms of measurable cross sections. As an example of some of the variations that haveresulted from different conventions, consider the beam+target asymmetries. We candefine these as coordinate-independent ratios with directions specified by only photon( ~p γ ) and meson ( ~p m ) momenta. R E = h dσ B , T , R1 ( P γh = +1 , ~P T = − ˆ p γ , sum f . s . ) − dσ B , T , R2 ( P γh = +1 , ~P T = +ˆ p γ , sum f . s . ) i / [ dσ + dσ ] , (41) R F = h dσ B , T , R1 ( P γh = +1 , ~P T = ˆ p , sum f . s . ) − dσ B , T , R2 ( P γh = − , ~P T = ˆ p , sum f . s . ) i / [ dσ + dσ ] , (42) R G = h dσ B , T , R1 ( φ Lγ = + π/ p toward ˆ p , ~P T = +ˆ p γ , sum f . s . ) − dσ B , T , R2 ( φ Lγ = + π/ p toward ˆ p , ~P T = − ˆ p γ , sum f . s . ) i / [ dσ + dσ ] , (43) R H = h dσ B , T , R1 ( φ Lγ = + π/ p toward ˆ p , ~P T = +ˆ p , sum f . s . ) − dσ B , T , R2 ( φ Lγ = + π/ p toward ˆ p , ~P T = − ˆ p , sum f . s . ) i / [ dσ + dσ ] , (44) etermining pseudoscalar meson photo-production amplitudes ... Table 3.
Ratios of cross sections involving beam and target polarizations and thenames given these quantities by different authors.BDS [7] AS [10] FTS [11] KDT [13] Present work ~P γ +ˆ z − ˆ z +ˆ z +ˆ z +ˆ zR E E E − E E ER F F − F F F FR G G G G G GR H − H H H − H H with ˆ p ≡ ( ~p γ × ~p m ) × ~p γ | ( ~p γ × ~p m ) × ~p γ | , ˆ p ≡ ~p γ × ~p m | ~p γ × ~p m | . The variable names of these ratios as used by different authors are listed in table I.As evident there, the same symbol has been used in different papers to refer to differentquantities, with common magnitudes but varying signs. This creates the potential forspiraling confusion when a third party combines equations from different papers.The present work has avoided the confusions associated with variations in formulaefrom different papers by developing a consistent and self-contained set of expressionsthat (a) define each observable in terms of measurable cross sections (Appendix B),(b) provide the most general expression for the cross section in terms of the 16observables and the beam, target and recoil polarization states, both derived analytically(Appendix A) and checked numerically (sections 3 and 4), and (c) provide the definingrelations between the 16 spin observables and the CGLN amplitudes (section 7 below).
6. Recoil polarization
As a first application of the consistent expression of the general cross section presentedin section 4, we analyze the potential of experiments measuring recoil polarization. Thegeneral expression in (37) displays a level of symmetry in the three polarization vectors, ~P γ , ~P T and ~P R . However, while the first two are parameters that are under experimentalcontrol, the recoil polarization is not. Rather, ~P R is a consequence of the angularmomentum brought into the entrance channel through ~P γ and ~P T , and the reactionphysics. The relations determining ~P R are readily derived. We start by regroupingterms in the general cross section expression to display the explicit dependence on ~P R and recast (37) as, dσ B , T , R ( ~P γ , ~P T , ~P R ) = 12 h A + ( P Rx ′ ) A x ′ + ( P Ry ′ ) A y ′ + ( P Rz ′ ) A z ′ i , (45)where A = dσ − P γL cos(2 φ γ ) ˆΣ + P Ty ˆ T − P γL P Ty cos(2 φ γ ) ˆ P − P γc P Tz ˆ E + P γL P Tz sin(2 φ γ ) ˆ G + P γc P Tx ˆ F + P γL P Tx sin(2 φ γ ) ˆ H, etermining pseudoscalar meson photo-production amplitudes ... A x ′ = P γc ˆ C x ′ + P γL sin(2 φ γ ) ˆ O x ′ + P Tz ˆ L x ′ + P Tx ˆ T x ′ + P γL P Ty sin(2 φ γ ) ˆ C z ′ − P γc P Ty ˆ O z ′ − P γL P Tx cos(2 φ γ ) ˆ L z ′ + P γL P Tz cos(2 φ γ ) ˆ T z ′ ,A y ′ = ˆ P + P Ty ˆΣ − P γL cos(2 φ γ ) ˆ T − P γL P Ty cos(2 φ γ ) dσ + P γL P Tx sin(2 φ γ ) ˆ E + P γc P Tx ˆ G + P γL P Tz sin(2 φ γ ) ˆ F − P γc P Tz ˆ H,A z ′ = P γc ˆ C z ′ + P γL sin(2 φ γ ) ˆ O z ′ + P Tz ˆ L z ′ + P Tx ˆ T z ′ − P γL P Ty sin(2 φ γ ) ˆ C x ′ + P γc P Ty ˆ O x ′ + P γL P Tx cos(2 φ γ ) ˆ L x ′ − P γL P Tz cos(2 φ γ ) ˆ T x ′ . The recoil polarization ~P R can be resolved as the vector sum of three component vectors, P Rx ′ ˆ x ′ , P Ry ′ ˆ y ′ , P Rz ′ ˆ z ′ . Considering first P Rx ′ ˆ x ′ , this is the degree of polarization along ˆ x ′ andis given by P Rx ′ = p Rx ′ , + − p Rx ′ , − , (46)where p Rx ′ , ± is the probability for observing the recoil with spin along ± ˆ x ′ ≡ ( ± , , ′ .Using (45), we evaluate this as the ratio of cross sections, P Rx ′ = dσ B , T , R ( ~P γ , ~P T , +1ˆ x ′ ) − dσ B , T , R ( ~P γ , ~P T , − x ′ ) dσ B , T , R ( ~P γ , ~P T , +1ˆ x ′ ) + dσ B , T , R ( ~P γ , ~P T , − x ′ ) = A x ′ A . (47)The ˆ y ′ and ˆ z ′ recoil components are evaluated in a similar manner. Thus, thecomponents of the recoil polarization are determined from (45), in terms of combinationsof the profile functions and initial polarizations, as P Rx ′ = A x ′ A , P Ry ′ = A y ′ A , P Rz ′ = A z ′ A . (48)These recoil components determine the orientation of the recoil vector, ~P R , and itsmagnitude, | ~P R | = 1 A p ( A x ′ ) + ( A y ′ ) + ( A z ′ ) . (49)It is worth clarifying the relationship between (37) or (45) and (48). Equations (37)and (45) display the general dependence of the cross section upon the three polarizationvectors, each of which is in a superposition of two spin states. If any one polarization isnot observed, either by not experimentally preparing it ( ~P γ or ~P T ) or by not detectingit ( ~P R ), then the terms proportional to that polarization average or sum to zero anddrop out of the cross section. The action of preparing or detecting a polarization forcesthe corresponding magnetic substate population into a particular distribution, which inthe case of the recoil polarization is given by (48). A particular consequence of this isthat one may not substitute (48) back into (45) to obtain a cross section that appearsto be independent of recoil polarization.An expression similar in spirit to (45) but different in form is given by Adelseckand Saghai in [10]. However, the coordinate system is very different and there is at least etermining pseudoscalar meson photo-production amplitudes ... P Rz and O z but none with P Rz and O x ,as discussed in section 5.2.In practice, the recoil polarization is measured either following a secondaryscattering or, in the case of hyperon channels, through the angular distribution of theirweak decays. K Λ → Kπ − p production provides a particularly efficient channel forrecoil measurements. In the rest frame of the decaying Λ, the angular distributionof the decay proton follows (1 / α | ~P Λ | cos(Θ p )], where Θ p is the angle betweenthe proton momentum and the lambda polarization direction [21]. Since the analyzingpower in this decay is quite high, α = 0 . ± .
013 [22], recoil measurements in modernquasi-4 π detectors can be carried out without significant penalty in statistics. As aresult, such measurements provide information on combinations of observables through(48). It is instructive to consider a few examples.(i) Unpolarized beam and target, P γL,c = P T = 0: In this case, A = dσ , A x ′ = 0, A y ′ = ˆ P and A z ′ = 0, so that ~P R = (0 , P = ˆ P /dσ , . (50)Thus, even when the initial state is completely unpolarized, a measured recoilpolarization will be perpendicular to the reaction plane.(ii) Unpolarized beam and longitudinally polarized target, P γL,c = 0 and ~P T =(0 , , P Tz ): In this case, A = dσ , A x ′ = P Tz ˆ L x ′ , A y ′ = ˆ P , and A z ′ = P Tz ˆ L z ′ ,so that ~P R = ( P Tz L x ′ , P, P Tz L z ′ ) . (51)Thus a measurement of the components of the recoil polarization determine the L x ′ , P and L z ′ asymmetries.(iii) Circularly polarized beam ( P γc ) and unpolarized target ( P T = 0): In this case, A = dσ , A x ′ = P γc ˆ C x ′ , A y ′ = ˆ P , and A z ′ = P γc ˆ C z ′ , so that ~P R = ( P γc C x ′ , P, P γc C z ′ ) . (52)This is the form assumed in the analysis of the CLAS-g1c data in [20].(iv) Linearly polarized beam ( P γL ) and unpolarized target ( P T = 0): In this case, A = dσ − P γL cos(2 φ γ ) ˆΣ, A x ′ = P γL sin(2 φ γ ) ˆ O x ′ , A y ′ = ˆ P − P γL cos(2 φ γ ) ˆ T , and A z ′ = P γL sin(2 φ γ ) ˆ O z ′ , so that ~P R = (cid:18) P γL sin(2 φ γ ) O x ′ − P γL cos(2 φ γ )Σ , P − P γL cos(2 φ γ ) T − P γL cos(2 φ γ )Σ , P γL sin(2 φ γ ) O z ′ − P γL cos(2 φ γ )Σ (cid:19) , (53)which is the form assumed in the analysis of the GRAAL data in [6], although thecoordinate system is different.(v) Circularly polarized beam ( P γc ) and longitudinally polarized target [ ~P T =(0 , , P Tz )]: In this case, A = dσ − P γc P Tz ˆ E , A x ′ = P γc ˆ C x ′ + P Tz ˆ L x ′ , A y ′ =ˆ P − P γc P Tz ˆ H , and A z ′ = P γc ˆ C z ′ + P Tz ˆ L z ′ , so that ~P R = (cid:18) P γc C x ′ + P Tz L x ′ − P γc P Tz E , P − P γc P Tz H − P γc P Tz E , P γc C z ′ + P Tz L z ′ − P γc P Tz E (cid:19) . (54) etermining pseudoscalar meson photo-production amplitudes ... A , which determines the E asymmetry and hence the denominator in (54). In an analysis averaging overinitial target polarizations ± P Tz , measurements of the recoil polarization vectorthen determine the C x ′ , P and C z ′ asymmetries. Another pass through the data,averaging instead over initial beam polarization states, ± P γc , and with an analysisof the P Rx ′ and P Rz ′ recoil components, gives the L x ′ and L z ′ asymmetries. Finally,by keeping track of both beam and target polarization states, a measurement ofthe P Ry ′ recoil component gives the H asymmetry. Although the uncertainty in thisdetermination of H will include the propagation of errors from P and E , this isexpected to be held to a reasonable level in the modern set of experiments that arenow under way. The significance of this determination is that it does not requirethe use of a transversely polarized target, as would otherwise be required by theleading polarization dependence of H in (37). In general, the latter would requirea completely separate experiment with different systematics.(vi) Linearly polarized beam ( P γL ) and longitudinally polarized target [ ~P T = (0 , , P Tz )]:In this case, A = dσ − P γL cos(2 φ γ ) ˆΣ + P γL P Tz sin(2 φ γ ) ˆ G , A x ′ = P γL sin(2 φ γ ) ˆ O x ′ + P Tz ˆ L x ′ + P γL P Tz cos(2 φ γ ) ˆ T z ′ , A y ′ = ˆ P − P γL cos(2 φ γ ) ˆ T + P γL P Tz sin(2 φ γ ) ˆ F , and A z ′ = P γL sin(2 φ γ ) ˆ O z ′ + P Tz ˆ L z ′ − P γL P Tz cos(2 φ γ ) ˆ T x ′ , so that ~P R = (cid:18) P γL sin(2 φ γ ) O x ′ + P Tz L x ′ + P γL P Tz cos(2 φ γ ) T z ′ − P γL cos(2 φ γ )Σ + P γL P Tz sin(2 φ γ ) G ,P − P γL cos(2 φ γ ) T + P γL P Tz sin(2 φ γ ) F − P γL cos(2 φ γ )Σ + P γL P Tz sin(2 φ γ ) G ,P γL sin(2 φ γ ) O z ′ + P Tz L z ′ − P γL P Tz cos(2 φ γ ) T x ′ − P γL cos(2 φ γ )Σ + P γL P Tz sin(2 φ γ ) G (cid:19) . (55)With such data a beam-target analysis summing over final states (i.e., ignoring therecoil) determines the cross section A , and hence the Σ and G asymmetries froma Fourier analysis of the φ γ dependence. This fixes the denominators in (55). Withanother analysis pass, averaging over initial target polarizations, measurements ofthe recoil polarization vector provide a determination of the O x ′ , P and T , and O z ′ asymmetries. Another pass through the same data, integrating over φ γ , gives the L x ′ , P and L z ′ asymmetries from measurements of the recoil polarization vector.Finally, a Fourier analysis of beam polarization states, using the difference betweenopposing target orientations, P Tz − P T − z , together with a measurement of recoilpolarization allows the separation of L x ′ and T z ′ , F (which would otherwise requirea transversely polarized target), and L z ′ and T x ′ .Thus, by judicious use of recoil polarization and a polarized beam, all 16 observablescan be determined with a longitudinally polarized target (often in several ways) and indoing so, with largely common systematics. etermining pseudoscalar meson photo-production amplitudes ...
21A corresponding set of expressions can be developed for a transversely polarizedtarget, although they are inherently more complicated since, for fixed target polarizationperpendicular to +ˆ z , any reaction plane will generally involve both transverse targetcomponents P Tx and P Ty .(vii) Unpolarized beam ( P γL,c = 0) with a transversely polarized target and [ ~P T =( P Tx , P Ty , A = dσ + P Ty ˆ T , A x ′ = P Tx ˆ T x ′ , A y ′ = ˆ P + P Ty ˆΣ, and A z ′ = P Tx ˆ T z ′ , so that ~P R = P Tx T x ′ P Ty T , P + P Ty Σ1 + P Ty T , P Tx T z ′ P Ty T ! . (56)Here an analysis summing over final states (i.e., ignoring the recoil) results in thecross section A , and a fit varying P Ty as the reaction plane tilts relative to thedirection of the target polarization determines the T asymmetry. A subsequentanalysis of the recoil polarization components then determines T x ′ , P , Σ, and T z ′ .(viii) Circularly polarized beam ( P γc ) and transverse target polarization [ ~P T =( P Tx , P Ty , A = dσ + P Ty ˆ T + P γc P Tx ˆ F , A x ′ = P γc ˆ C x ′ + P Tx ˆ T x ′ − P γc P Ty ˆ O z ′ , A y ′ = ˆ P + P Ty ˆΣ + P γc P Tx ˆ G , and A z ′ = P γc ˆ C z ′ + P Tx ˆ T z ′ + P γc P Ty ˆ O x ′ , so that ~P R = P γc C x ′ + P Tx T x ′ − P γc P Ty O z ′ P Ty T + P γc P Tx F , P + P Ty Σ + P γc P Tx G P Ty T + P γc P Tx F ,P γc C z ′ + P Tx T z ′ + P γc P Ty O x ′ P Ty T + P γc P Tx F ! . (57)In this case, a beam-target analysis summing over final states (i.e., ignoringthe recoil) results in the cross section A containing the terms in the T and F asymmetries, and these can be separated by first averaging over initial photonstates, which removes F . A subsequent analysis, reconstructing the recoilpolarization while averaging over initial circular photon states allows one to deduce T x ′ and T z ′ from P Rx ′ and P Rz ′ . Alternatively, with fixed beam polarization and recoilanalysis, a fit varying P Tx and P Ty as the reaction plane tilts in azimuth relative tothe direction of the transversely polarized target determines all of the asymmetriesin the numerators of (57).We leave it to the reader to write out the final combination of linearly polarizedbeam and transverse target polarization. There the recoil polarization componentsinvolve ratios of 4 to 5 terms each. It remains to be seen if sequential analyses of suchdata are of practical use, given limitations on statistics.
7. Relating observables to CGLN amplitudes
To extract nucleon resonances, one needs to extract amplitudes from observables.Because of the apparent variations in the available literature, as summarized in section5, there exists sign differences in formula relating observables to CGLN amplitudes. etermining pseudoscalar meson photo-production amplitudes ... dσ = + ℜ e (cid:8) F ∗ F + F ∗ F + sin θ ( F ∗ F + F ∗ F ) /
2+ sin θ ( F ∗ F + F ∗ F + cos θF ∗ F ) − θF ∗ F (cid:9) ρ , (58 a )ˆΣ = − sin θ ℜ e { ( F ∗ F + F ∗ F ) / F ∗ F + F ∗ F + cos θF ∗ F } ρ , (58 b )ˆ T = + sin θ ℑ m (cid:8) F ∗ F − F ∗ F + cos θ ( F ∗ F − F ∗ F ) − sin θF ∗ F (cid:9) ρ , (58 c )ˆ P = − sin θ ℑ m (cid:8) F ∗ F + F ∗ F − F ∗ F − cos θ ( F ∗ F − F ∗ F ) − sin θF ∗ F (cid:9) ρ , (58 d )ˆ E = + ℜ e (cid:8) F ∗ F + F ∗ F − θF ∗ F + sin θ ( F ∗ F + F ∗ F ) (cid:9) ρ , (58 e )ˆ G = + sin θ ℑ m { F ∗ F + F ∗ F } ρ , (58 f )ˆ F = + sin θ ℜ e { F ∗ F − F ∗ F − cos θ ( F ∗ F − F ∗ F ) } ρ , (58 g )ˆ H = − sin θ ℑ m { F ∗ F + F ∗ F − F ∗ F + cos θ ( F ∗ F − F ∗ F ) } ρ , (58 h )ˆ C x ′ = − sin θ ℜ e { F ∗ F − F ∗ F − F ∗ F + F ∗ F − cos θ ( F ∗ F − F ∗ F ) } ρ , (58 i )ˆ C z ′ = −ℜ e (cid:8) F ∗ F − cos θ ( F ∗ F + F ∗ F ) + sin θ ( F ∗ F + F ∗ F ) (cid:9) ρ , (58 j )ˆ O x ′ = − sin θ ℑ m { F ∗ F − F ∗ F + cos θ ( F ∗ F − F ∗ F ) } ρ , (58 k )ˆ O z ′ = + sin θ ℑ m { F ∗ F + F ∗ F } ρ , (58 l )ˆ L x ′ = + sin θ ℜ e (cid:8) F ∗ F − F ∗ F − F ∗ F + F ∗ F + sin θ ( F ∗ F − F ∗ F ) /
2+ cos θ ( F ∗ F − F ∗ F ) } ρ , (58 m )ˆ L z ′ = + ℜ e (cid:8) F ∗ F − cos θ ( F ∗ F + F ∗ F ) + sin θ ( F ∗ F + F ∗ F + F ∗ F )+ cos θ sin θ ( F ∗ F + F ∗ F ) / (cid:9) ρ , (58 n )ˆ T x ′ = − sin θ ℜ e { F ∗ F + F ∗ F + F ∗ F + cos θ ( F ∗ F + F ∗ F ) / } ρ , (58 o )ˆ T z ′ = + sin θ ℜ e (cid:8) F ∗ F − F ∗ F + cos θ ( F ∗ F − F ∗ F ) + sin θ ( F ∗ F − F ∗ F ) / (cid:9) ρ . (58 p )A comparable set of expressions are given by Fasano, Tabakin and Saghai (FTS)in [11]. With the conventions discussed in section 5.3, and allowing for their differentdefinition of the E beam-target asymmetry (as in table 3), the above expressions areconsistent with those of [11].Comparing the above relations to those given by Kn¨ochlein, Drechsel and Tiator(KDT) (Appendix B and C of [13]), six of these equations have different signs, theBT observable H , the TR observable L x ′ and all four of the BR observables C x ′ , C z ′ , O x ′ and O z ′ . The KDT paper [13] is listed in the MAID on-line meson productionanalysis [23, 24, 25] as the defining reference for their connection between CGLN etermining pseudoscalar meson photo-production amplitudes ... F i amplitudes, and then used our equations (58 a )-(58 p ) above to construct observables. Comparing the results to direct predictions ofobservables from the MAID code, we find the same six sign differences. However, KDTgive a form of the general cross section with leading polarization terms in [13] and there,the equations for these six observables appear with a negative coefficient, as opposedto our form of the cross section in (37). This is equivalent to interchanging the σ and σ measurements of Appendix B that are needed to construct these six quantities.(Such differences were already discussed in section 5 above, with the H asymmetry asan example.) Thus, KDT use the same six observable names as the present work torefer measurable quantities of the same magnitude but opposite sign.We have conducted a similar test with the GWU/VPI SAID on-line analysiscode [26, 27], downloading SAID multipoles, using the relations in (15)-(18) to constructfrom these the four CGLN F i amplitudes, and then using our equations (58 a )-(58 p )above to construct observables. When the results are compared to direct predictions ofobservables from the SAID code, again the same 6 observables { H, C x ′ , C z ′ , O x ′ , O z ′ , L x ′ } differ in sign. For the definition of observables, SAID refers to the Barker, Donnachieand Storrow paper [7]. As discussed in section 5, the BDS definitions of asymmetriesshould be deduced from their equations (2)-(4) and these have signs consistent withKDT. Thus SAID also uses the same six observable names as the present work to referto quantities of the same magnitude but opposite sign.We have repeated this same test with the Bonn-Gatchina (BoGa) on-line PWA [28],downloading BoGa multipoles, using the relations of (15)-(18) to construct the fourCGLN amplitudes, and then using our (58 a )-(58 p ) to construct observables. Comparingthese to direct predictions of observables from the BoGa code, the results are identical,except for the E asymmetry which is of opposite sign. However, for the definition ofobservables the BoGa on-line site refers to FTS of [11], whose definitions are the sameas in our Appendix B except for a sign change in the E asymmetry, as in table 3. Thus,we conclude that the relations between observables and amplitudes used in the BoGaanalysis is completely consistent with the present work.New data are emerging from the current generation of polarization experimentswhich make these sign differences an important issue. In [20], recent results for the C x ′ and C z ′ asymmetries have been compared with the direct predictions of the Kaon-MAID code, with predictions from an earlier version of the BoGa multipoles and withpredictions from Juli´a-D´ıaz, Saghai, Lee and Tabakin (JLST) [29]. As an illustration,in figure 3 we have replotted figures 8 and 9 from [20] for two energies, transformed tothe primed kaon axes using equation (39), and added the predictions from SAID. TheMAID (black dashed) and SAID (black, dotted) curves for C z ′ approach − θ K = 0 ◦ ,while the BoGa (blue, dot-dashed) and JSLT (blue, solid) curves approach +1, alongwith the data (green circles) from [20].The behavior of C z ′ at θ K = 0 ◦ is a simple reflection of angular momentum etermining pseudoscalar meson photo-production amplitudes ... Figure 3. (Color online) C x ′ (left) and C z ′ (right) for the γp → K + Λ reactionat W = 1680 MeV (top) and W = 1940 MeV (bottom). Kaon-MAID predictionsare dashed (black) [23, 24, 25], SAID predictions are dotted (black) [26, 27], BoGapredictions are dot-dashed (blue) [28] and predictions from JSLT [29] are solid (blue).The green circles are from [20]. conservation. Using the definition from Appendix B, C z ′ = { σ (+1 , , + z ′ ) − σ (+1 , , − z ′ ) } / { σ + σ } . When the incident photon spin is oriented along +ˆ z , onlythose target nucleons with anti-parallel spin can contribute to the production of spinzero mesons at θ K = 0, and the projection of the total angular momentum along ˆ z is+1 /
2. Thus, the recoil baryon must have its spin oriented along +ˆ z = +ˆ z ′ at θ K = 0,so that σ must vanish. The recent measurements on K + Λ production [20] clearly showthis asymmetry approaching +1 at θ K = 0 ◦ .The MAID and SAID predictions appear to have the wrong limits for C z ′ at 0 and180 degrees. Also shown are predictions using the multipoles of Juli´a-D´ıaz, Saghai, Leeand Tabakin (JSLT) from [29], used with our expressions to construct observables (solidblue curves). The MAID and SAID sign differences are also evident in C x ′ , particularlyat low energies where only a few partial waves are contributing - top panels of figure 3.There it is clear that the predictions of the different partial wave solutions are essentiallyvery similar, differing only in sign. The comparisons of [20], repeated here in figure 3,illustrate the confusion that arises from the use of the same symbol to mean differentexperimental quantities by different authors (section 5). etermining pseudoscalar meson photo-production amplitudes ...
8. Relations between observables
Since photo-production is characterized by 4 complex amplitudes, equation (9), the16 observables of equations (58 a )-(58 p ) are not independent. There are in fact manyrelations between them. The profile functions of (58 a )-(58 p ) are bilinear products of theCGLN amplitudes, and one of the more extensive sets of equalities interrelating themhas been derived by Chiang and Tabakin from the Fierz identities that relate bilinearproducts of currents [2]. Such relations are particularly useful, since they allow thecomparison of data on one observable with an evaluation in terms of products of otherobservables. Any determination of the amplitude will invariably require combining dataon different polarization observables which in general come from different experiments,each having different systematic scale uncertainties. The Fierz identities provide ameans of enforcing consistency provided, of course, that they are consistent with theexpressions of general cross sections given in section 4.The Fierz identities as derived by Chiang and Tabakin (CT) are given in terms of16 quantities, ˇΩ i in [2], and the first column of table I in that paper gives the relationbetween these quantities and the conventional single, BT, BR and TR observable names.CT quote FTS for the definition of these observables. We have numerically checked the37 Fierz identities of Appendix D in [2]. Assuming the definitions of observables as givenin our Appendix B, or in FTS, a large number (more than half of them) require revisionsin signs. If the signs of { H, C x ′ , C z ′ , O x ′ , O z ′ , L x ′ } are reversed, as in BDS and KDT,still many of the equations of [2] require revision. A set of identities that are consistentwith our definitions of observables in terms of measurable quantities, Appendix B, andwith the form of our general cross section in equation (37), is given below in the firstthree sections of Appendix C. As a practical example, in the next section we use twoof the identities in a multipole analysis to fix the scales of different data sets in a fitweighted by their systematic errors.Another set of relations has been given by Artru, Richard and Soffer (ARS) [30, 31].These are different in form but can be derived from our Fierz identities, although withsome differences in signs. A consistent set is listed in Appendix C.4.In addition to identities, there are a number of inequalities, such as ( P ) + ( C x ′ ) +( C z ′ ) ≤
1, which are often referred to as positivity constraints [30]. These involvethe sums of the squares of asymmetries, and as such are immune to sign issues. Theycan be particularly useful when extracting sets of asymmetries from fits to experimentaldata [32], as in the examples discussed in section 6. But since our focus here is amplitudereduction from cross sections and asymmetries, we refer the reader to a recent reviewof such inequalities [31].
9. Multipole analyses
The ultimate goal of the new generation of experiments now under way is a completeexperimental determination of the multipole decomposition of the full amplitude in etermining pseudoscalar meson photo-production amplitudes ... γp → K + Λchannel, which so far has provided the largest number of different observables, assummarized in table 4.To avoid bias, the first stage in any multipole decomposition is a single-energyanalysis, one beam/ W energy at a time without any assumptions on energy-dependentbehavior. The range of recent published K + Λ measurements is summarized in table 4.[Cross section data from the SAPHIR detector at Bonn [33] have an appreciable (20%)angle- and energy-dependent difference from the CLAS experiments. This level ofincompatibility makes it impossible to include them in the present analyses.] Whilesome of the data sets span the full nucleon resonance region in extremely fine steps,single-energy analyses are limited by the observables with the coarsest granularity, whichin this case are the C x ′ , C z ′ measurements (data group 3 [20]). The only published O x ′ , O z ′ and T data are from GRAAL (data groups 5 and 8 [6]). The combination of thesedata sets allows us to combine groups 1-8 at 5 different beam energies, with roughly 100MeV steps in beam energy, for which 8 different observables are now available. There are several different choices for coordinate systems in use and before data fromthe different experiments can be combined in a common analysis we transformed themto the system defined in figure 2. The beam-recoil data of group 3 [20] were reportedin unprimed c.m. coordinates relative to the beam direction. These are related to theprimed system of figure 2 by the relations in equation (39). The GRAAL papers usethe coordinates of Adelseck and Saghai [10]. Relative to ˆ y ′ = ˆ y , their ˆ x and ˆ z axesare reversed from those of figure 2, so that, although Σ, T and P are unchanged intransferring to our coordinates, O x ′ ,z ′ become the negative of what GRAAL refer to as O x,z . Thus, O x ′ ,z ′ = − O GRAAL x,z . (59) Each experiment has reported systematic errors that reflect an uncertainty in the scale ofthe entire data set. We use a procedure of imposing self-consistence within a collectionof data sets by including their measurement scales as parameters in a fit minimizing χ [37]. To fix first the scales of the polarization observables, data groups (2,3,5,6,7,8)of table 4, we use the Fierz identities (L.BR) and (S.br) of Appendix C to construct thequantities, F L . BR = Σ P − C x ′ O z ′ + C z ′ O x ′ − T, etermining pseudoscalar meson photo-production amplitudes ... Table 4.
Summary of recent published results on K + Λ photoproduction. (Systematicuncertainties on the CLAS data are taken from the indicated references. Thesystematic errors on the GRAAL measurements reflect their reported uncertainty inbeam polarization, in the assumed weak-Λ-decay parameter and in the resulting errorpropagation through the extraction of O x ′ , O z ′ and T .) Data Experiment Observables E γ range (MeV) ∆ E γ /∆ W Systematicgroup W range (MeV) binning scale error1 CLAS-g11a [34] dσ ± E γ dep.)2 CLAS-g11a [34] P ± . C x ′ , C z ′ ± . dσ ± E γ dep.)5 GRAAL [6] O x ′ , O z ′ ± P ± ± T ± ± F S . br = O x ′ + O z ′ + C x ′ + C z ′ + Σ − T + P − , (60)both of which have the expectation value of 0 at each angle and energy. Our fittingprocedure then minimizes the χ function, χ = X E γ X θ K ((cid:20) F L . BR ( f i x exp iθ ) δF L . BR ( f i σ x iθ ) (cid:21) i =2 , , , , , + (cid:20) F S . br ( f i x exp iθ ) δF S . br ( f i σ x iθ ) (cid:21) i =2 , , , , , ) + X i (cid:20) f i − σ f i (cid:21) , (61)where the index i ≡ (2 , , , , ,
8) runs through each of the data groups of asymmetries( x exp .iθ ) needed to construct the Fierz relations of (60). All data from a set i havinga systematic scale error ( σ f i ) are multiplied by a common factor ( f i ) while adding( f i − /σ f i to the χ . This last term weights the penalty for choosing a normalizationscale different from unity by the reported systematic uncertainty of the experiment.In this procedure polynomial fits are used, where needed, to interpolate the data etermining pseudoscalar meson photo-production amplitudes ... Table 5.
Fitted scales for the data sets of table 4 that are used to construct therelations in (61).Data group Experiment Observables Fitted scale ( f i ) Scale error ( σ f i )2 CLAS-g11a P C x ′ , C z ′ O x ′ , O z ′ P T of table 4 to a common angle and energy. There are two measurements of the recoilpolarization asymmetry ( P ), from groups 2 and 6 in table 4, and a weighted mean ofthese data, including their scale factors, is used in evaluating (61). The scale factorsresulting from this fit are listed in table 5. All are close to unity. The resultingevaluations of the Fierz relation, using the scaled data, are shown in figure 4.While the results in figure 4 scatter around zero as expected, the fluctuations aresometimes appreciable. These cannot readily be removed with an energy- and angle-independent scale factor. It is likely this results from combining data from differentdetectors. While global uncertainties such as flux normalization and target thickness canbe readily estimated and easily fitted away in this type of procedure, angle-dependentvariations in detector efficiencies tend to be the most problematic to control and quantify. The observables of table 4 are determined by the CGLN amplitudes through (58 a )-(58 p ), and these are in turn determined by the multipoles through (15)-(18). Since themultipoles are reduced matrix elements and independent of angle, fitting them directlyallows the use of complete angular distributions for each observable. We fix the scales( f i ) of the polarization observables { Σ , T, P, C x ′ , C z ′ , O x ′ , O z ′ } to their fitted values intable 5, and now vary the multipoles, as well as the scales f and f for the unpolarizedcross section ( dσ ) measurements (groups 1 and 4 in table 4) to minimize the χ function, χ = N s X i =1 N i X j =1 " f i x exp ij − x fit ij ( ~ζ ) f i σ x ij + X i =1 , (cid:20) f i − σ f i (cid:21) , (62)where N s is the number of independent data sets, each having N i points. x exp ij and σ x ij are the j -th experimental datum from the i -th data set and its associated measurementerror, respectively, x fit ij ( ~ζ ) is the value predicted from the ~ζ multipole set being fit, and f i is the global scale parameter associated with the i -th data set. As before, the lastterm weights the penalty for choosing a cross section scale different from unity by thereported systematic uncertainties for data groups 1 and 4 [37].Thus our fitting procedure is a two-step process, first minimizing (61) by varyingthe scale factors of the polarization data, and then minimizing (62) in a second step by etermining pseudoscalar meson photo-production amplitudes ... Figure 4. (Color online) Evaluations of the two Fierz relations (L.BR) (solid redcircles) and (S.br) (open blue squares) of (60), using the data of table 4 and the fittedscales of table 5. varying the multipoles and the cross section scales. These two cannot be combined intoa single step in which Fierz relations such as (60) are minimized by varying multipoles,since all properly constructed multipoles will automatically satisfy the Fierz identities.While the cross section experiments report the global systematic uncertainties listedin table 4, comparisons given in [34] show a clear energy dependence to the scaledifference between them, which is most pronounced at low energies. Accordingly, wehave fitted separate cross section scales at each energy and the results are plotted infigure 5.Cross sections for any reaction generally fall with increasing angular momentum,which guarantees the ultimate convergence of a multipole expansion. However, inpractice such expansions must be truncated to limit the maximum angular momentumto a value that is essentially determined by the statistical precision and breadth of etermining pseudoscalar meson photo-production amplitudes ... Figure 5. (Color online) Fitted scales for the cross section ( dσ ) measurements of[34], f as red circles, and [35], f as green diamonds. kinematic coverage of the data sets. The ultimate goal of such work will be theidentification of the excited states of the nucleon, and this will require, as a minimum,accurate multipole information up to at least L = 2 to be useful. As has been shownby Bowcock and Burkhardt [38], the highest multipole fitted in any analysis alwaystends to accumulate the systematic errors stemming from truncation and is essentiallyguaranteed to be the most uncertain. Thus, when focusing on multipoles up to L = 2we must vary up to L = 3 and fix the multipoles for 4 ≤ L ≤ ≤ L ≤ χ comparison to the data of table 4, scaled by the fitted constants in table 5 and figure 5,are calculated. Whenever the resulting χ is within 10 times the current best value, agradient minimization is carried out. We have repeated this procedure for a wide rangeof Monte Carlo samples, up to 10 per energy, and have found a band of solutions withtightly clustered χ that cannot be distinguished by the existing data. In figures 6 and 7we plot the real and imaginary parts of 300 multipole solutions for which the gradientsearch has converged to a minimum. The χ /point of each solution within these bandsis always within 0.2 of the best, and is even more tightly clustered at low energies.The best and largest values of the χ /point for these bands are listed in table 6.(The corresponding multipole solutions are shown as the solid black and blue dashedcurves in figures 6 and 7, respectively.) The fact that most of the χ /point values aresubstantially less than one is a sign that fitting multipoles up to L = 3 provides morefreedom than the present collection of data warrant, even though the desired physicsdemands it.The bands in figures 6 and 7 reflect a relatively shallow valley in the χ space. Tounderstand if this valley is smooth, indicating a simple broad minimum, or is pittedwith many local minima, we have tracked solutions across χ . This can be done by etermining pseudoscalar meson photo-production amplitudes ... Table 6.
Best and largest values of the χ /point for the solutions in the bands plottedin figures 6 and 7. E γ / W (MeV) Best χ /point Largest χ /point1027 / 1676 0.49 0.541122 / 1728 0.59 0.621222 / 1781 0.52 0.621321 / 1833 0.74 0.921421 / 1883 0.97 1.15 forming a hybrid amplitude A h ( x ) from two solutions A and A : A h ( x ) = A × (cid:16) − x (cid:17) + A × (cid:16) x (cid:17) , x ∈ [0 , . (63)Here x is an effective distance in amplitude-space. For x = 0, A h is just A while for x = 100, A h becomes A . At each value of x between 0 and 100 the hybrid set ofmultipoles is used to predict observables and the χ relative to the data is calculated.If the valley between A and A is smooth and featureless the resulting χ map willbe similarly featureless. We have carried out this exercise for many pairs of solutionsand always found pronounced peaks in χ for any choice of A and A . As an example,the χ /point that results from forming a hybrid amplitude out of the best and largest(worst) solutions of figures 6 and 7 is shown in figure 8 for two of the energy bins oftable 6. (Similar results are obtained at other energies.) At E γ = 1122 MeV ( W = 1728MeV), in the bottom panel of figure 8, the peak in χ between the two is huge. At E γ = 1421 MeV ( W = 1883 MeV) the intermediate peak is still present, though not sotall, probably due to the presence of another local minimum that is nearby but off thedirect trajectory between the two solutions.Evidently the bands in figures 6 and 7 are created by clusters of local minima in χ which, for the present collection of data, are completely degenerate and experimentallyindistinguishable. The 8 observables in table 4 do not yet satisfy the Chiang and Tabakin(CT) criteria as a minimal set that would determine the photoproduction amplitude freeof ambiguities [2]. Nonetheless, from studies with mock data, as will be described insection 10, we have found that the presence of multiple local minima is essentiallyuniversal, even when the CT criteria are satisfied. But, as more observables are addedwith increasing statistical accuracy the degeneracy is broken and a global minimumemerges. The difficulty then becomes finding it among the pitted landscape in χ . In determining an amplitude there is one overall phase that can never be constrained,and so in fitting the solutions of figures 6 and 7 we have chosen to fix the phase of the E multipole to zero (which sets its imaginary part to zero). The consequence of notfixing a phase is illustrated in figure 9, where we plot as an example the S and P wavemultipoles from fits with an unconstrained phase angle. Again, the solutions within etermining pseudoscalar meson photo-production amplitudes ... Figure 6. (Color online) Real parts of multipoles for L = 0 to 3, fitted to the data oftable 4 with the phase of the E fixed to 0. The bands show variations in the χ /pointof less than 0.2, as in table 6. Solutions with the best and largest χ , correspondingto the columns of table 6, are shown as solid (black) and long-dashed (blue) curves,respectively. etermining pseudoscalar meson photo-production amplitudes ... Figure 7. (Color online) Imaginary parts of multipoles for L = 0 to 3, fitted to thedata of table 4 with the phase of the E fixed to 0. The bands show variations in the χ /point of less than 0.2, as in table 6. Curves are as in figure 6. etermining pseudoscalar meson photo-production amplitudes ... Figure 8.
The χ /point calculated by comparing the data of table 4 to predictions asa hybrid amplitude (63) is tracked between the solutions with the best and largest χ in table 6 (solid black and dashed blue curves in figures 6 and 7, respectively). Resultsare shown for E γ ( W ) energies of 1122 (1728) MeV in the bottom panel and 1421(1883) MeV in the top. these bands have values for the χ /point that differ by less than 0.2 from that of thebest solution. While these bands appear to be substantially broader, they are in factjust the bands of figures 6 and 7, expanded by rotating with a random phase angle. Thebehavior of the L = 2 ( D ) and L = 3 ( F ) waves show a similar broadening.In practice, the utility of determining a set of multipoles is not diminished by fixingone phase. Ultimately, such experimentally determined multipoles will be comparedto model predictions. For this, one only has to rotate the model phase to the samereference point, e.g., a real E in the analysis of figures 6 and 7. (The result of suchan exercise is shown in figures 12 and 13.)The choice of which multipole phase to fix at zero is somewhat arbitrary. Fromstudies with mock data, we have found that it is sufficient to fix the phase of any one ofthe larger multipoles ( L = 0 ,
1) when the data to be fit have modest statistical accuracy.Ultimately, if the data precision is very high, just fixing the higher L multipoles at theirreal Born values is enough to recover the amplitude. etermining pseudoscalar meson photo-production amplitudes ... Figure 9. (Color online) Real parts (top four panels in red) and imaginary parts(bottom four panels in green) of the S and P wave multipoles, fitted to the data oftable 4 without any phase constraints. The bands show variations in the χ /point ofless than 0.2. Predictions of the fitted multipole solutions are compared to the data of table 4 infigures 10 and 11 for two beam energies, 1122 and 1421 MeV. The best and worstsolutions from the bands of figures 6 and 7, in terms of the χ /point values of table 6, areshown as the solid (black) and dashed (blue) curves, respectively. The behavior at otherenergies is very similar. Based on such comparisons with existing published data, themultipole solutions within the bands of figures 6 and 7 are completely indistinguishable.Clearly, despite the presence of 8 polarization observables, the multipoles are still verypoorly constrained. For many of the higher multipoles not even the sign is known.In figures 12 and 13 we compare the S , P and D wave multipoles from existing PWAresults (BoGa [28], MAID [23], SAID [26] and JSLT [29]) with the bands of figures 6and 7, respectively. Here we have rotated all multipoles to our common reference point etermining pseudoscalar meson photo-production amplitudes ... Figure 10. (Color online) Predictions at E γ = 1122 MeV ( W = 1728 MeV) comparedto the data of table 4 for the multipole solutions of figures 6 and 7 having the minimum(solid black curves) and largest (long-dashed blue curves) χ /point (table 6). Datapoints are from CLAS-g11a [34] shown in red, CLAS-g1c [35, 20] shown in green, andGRAAL [5, 6] shown in blue. of a real E . (Each set of multipoles has been multiplied by exp( − iδ ), where δ is thephase of the E multipole of the PWA set.) For the most part, these PWA lie within ourexperimental solution bands. However, there are a few exceptions at the higher energies,in particular the M − multipole from Kaon-MAID (black dashed curve in figure 13) andthe E − and M − multipoles from JSLT (blue solid curves in figure 12). The upperend of our analysis range is near a potentially new N ∗ ( ∼ K Λ cross section near1.9 GeV with the D partial wave, which should resonate in either the E − or M − multipoles. However, our model-independent analysis excludes such conclusions, sincetheir solutions lie outside the experimental bands in these partial waves. On the otherhand, the BoGa analysis [39] has recently modeled the N ∗ ( ∼ P resonance,which should manifest itself in either the E or M multipoles. The BoGa solution etermining pseudoscalar meson photo-production amplitudes ... Figure 11. (Color online) Predictions at E γ = 1421 MeV ( W = 1883 MeV) comparedto the data of table 4 for the multipole solutions of figures 6 and 7 having the minimum(solid black curves) and largest (long-dashed blue curves) χ /point (table 6). Datapoints are plotted as in figure 10. is within the experimental solution bands of figures 12 and 13. (It is also the onlyPWA analysis that included the CLAS-g1c and GRAAL data sets in fitting their modelparameters.) We can conclude that their assignment is consistent with the experimentalsolution bands, but cannot yet confirm it due to the significant width of these bands.We have investigated a number of possible ways in which additional data may leadto narrower multipole bands and improved amplitude determination. For the most part,existing data does not extend to extreme angles (near 0 ◦ and 180 ◦ ), which in generaltend to be more sensitive to interfering multipoles of opposite parities. In fact, the bestand worst solutions at E γ = 1122 MeV ( W = 1728 MeV) exhibit a dramatic differencein the predicted unpolarized cross section at 180 ◦ – compare the solid (black) and dashed(blue) curves in figure 10. (The extreme angles of the asymmetries are constrained bysymmetry to either 0 or ±
1, and so contain little additional information.) As a test,we have created mock cross section data at 0 ◦ and 180 ◦ , centered on the best solutions etermining pseudoscalar meson photo-production amplitudes ... Figure 12. (Color online) The solution bands of figure 6, compared to the real partsof PWA multipoles of BoGa [28] (blue dashed-dot), Kaon-MAID [23] (black dashed),SAID [26] (black dotted) and JSLT [29] (blue solid). For this comparison, each PWAhas been rotated so that their E is real – see text. of table 6 with a statistical error of ± . µ b / sr. When the fits are repeated with thesemock points added to the CLAS and GRAAL data sets, variations such as seen infigure 10 disappear, but few of the resulting bands of multipole solutions are improved.While the M , M − and E − are slightly narrowed at low energies, generally, there islittle improvement over the trends seen in figures 6 and 7.The data in table 4 span a significant range in statistical precision. Frompreliminary analyses of data from an ongoing generation of new CLAS experimentswe can anticipate result on the Σ, T , O x ′ and O z ′ asymmetries that will have at least anorder of magnitude improvement over the GRAAL data set. To simulate the effect ofsuch an improvement, we have arbitrarily reduced the statistical errors on the GRAALΣ, T , O x ′ and O z ′ asymmetries by a factor of 3 and repeated the fits. Apart from anincrease in χ , due to undulations in the angular distributions that are now artificially etermining pseudoscalar meson photo-production amplitudes ... Figure 13. (Color online) The solution bands of figure 7, compared to the imaginaryparts of PWA multipoles of BoGa [28] (blue dashed-dot), Kaon-MAID [23] (blackdashed), SAID [26] (black dotted) and JSLT [29] (blue solid). For this comparison,each PWA has been rotated so that their E is real – see text. beyond the level of statistical fluctuations, there are no significant changes in any of themultipole bands of figures 6 and 7.Ongoing analyses of new experiments are expected to yield data on all 16observables. The potential impact of such an extensive set is simulated in the nextsection; here we can already study the expected trends by examining the impact thatthe GRAAL measurements of { Σ , T, O x ′ , O z ′ } have made so far. In figure 14 we showthe S and P wave multipoles obtained if the GRAAL data are removed from the fittingprocedure. Comparing these results to figures 6 and 7, it is clear that the M band hasdramatically narrowed with the inclusion of the GRAAL polarization results. Lesser butstill significant gains occur in the determination of most of the multipoles. The range ofvalues for the χ /point within these bands are similar to those of table 6. In figure 15we show the predictions of the band at 1421 MeV beam energy ( W = 1883 MeV),as represented by the solutions with the minimum χ / point = 1 .
07 and the maximum etermining pseudoscalar meson photo-production amplitudes ... Figure 14. (Color online) Real parts (top 4 panels in red) and imaginary parts(bottom 4 panels in green) of the S and P wave multipoles, fitted to the CLAS dataof table 4 (excluding the GRAAL measurements). Solutions with the best (1.07) andlargest (1.18) χ are shown as solid (black) and long-dashed (blue) curves, respectively. χ / point = 1 .
18. Not surprisingly, predictions for the observables where data have beenremoved from the fit are now wildly varied.There are several conclusions that can be drawn from this analysis, along withreasons for genuine hope. When the χ /point is near or even better than 1, solutionsdiffering in the χ /point by something like 0.2 are not experimentally distinguishable.The existence of bands of multipole solutions, each with small χ /point, indicates ashallow χ surface, pitted with many local minima. Certainly the width of the bandsevident in figures 6 and 7 precludes using the existing data to hunt for resonances .However, a comparison of figure 14 with figures 6 and 7 indicates the gains resultingfrom the GRAAL polarization observables are significant, even though the GRAALerrors are substantially larger than most of the CLAS data. CLAS data on all 16photoproduction observables are now under analysis. The fact that such data have all etermining pseudoscalar meson photo-production amplitudes ... Figure 15. (Color online) Predictions at E γ = 1421 MeV ( W = 1883 MeV) froma multipole fit to the CLAS data from CLAS-g11a [34] shown in red and CLAS-g1c [35, 20] shown in green, excluding the GRAAL results. The solid black and long-dashed blue curves show the solutions (figure 14) having the minimum (1.07) andlargest (1.18) χ /point. been accumulated within a single detector is likely to minimize the problems evident infigure 4. Furthermore, with a large number of different observables will come a largenumber of the Fierz identities, which can be used to constrain and essentially eliminatethe effects of systematic scale uncertainties.
10. The potential of complete experiments – studies with mock data
To further investigate the potential impact of measuring a complete set of all 16observables on the determination of multipole amplitudes, we have created mock data using predictions of the BoGa multipoles, Gaussian-smeared to reflect different levels ofuncertainty. Fitting such mock data with the same procedure described in the previoussection, i.e., Monte Carlo sampling combined with gradient minimization and a real E etermining pseudoscalar meson photo-production amplitudes ... Figure 16. (Color online) Real (top four panels in red) and imaginary (bottom fourpanels in green) parts of the S and P wave multipoles resulting from fits to mock datawith 5% errors on all 16 observables, with mock data points every 10 ◦ c.m. Solutionswith the best (0.7) and largest (1.3) χ / point are shown as solid (black) and long-dashed (blue) curves, respectively. multipole, leads to the following conclusions. • With data points at every 10 degrees and with 0.1% errors on each point for everyobservable, a level of accuracy that will never be achieved at any facility, two minimaare always found, one with a χ /point near 1 and the other substantially larger –e.g., greater than 50. Thus, a unique solution is easily identifiable. • When the uncertainties on the mock data are increased to 1% on each point every10 degrees, a few minima appear. Nonetheless, with the exception of the lowestenergies, these are still widely spaced in χ so that in general the true solution canstill be identified. • When the uncertainties on the mock data are increased to 5%, bands ofindistinguishable solutions from multiple χ minima begin to appear, although the etermining pseudoscalar meson photo-production amplitudes ... Figure 17. (Color online) Mock data on the single- and BT-polarization observablesat W = 1900 MeV, with uncertainties expected from the CLAS set of K + Λ experiment.The curves are predictions of multipole solutions fitted to these data with the best (0.6)and largest (1.4) χ /point, as shown by the solid (red) and long-dashed (blue) curves,respectively. bands are considerably narrowed from those of figures 6 and 7. As an example,the resulting real and imaginary parts of the E to M − multipoles are shown infigure 16 for the c.m. energy range from 1650 to 2200 MeV. With small errors andbands as narrow as in figure 16, there are typically a few local minima for eachenergy. However, the positions of such minima depend on the particular statisticaldistribution of the mock data, due to the complicated structure of the χ space. Toremove this dependence we have repeated the exercise of creating Gaussian smearedmock 5% data and searching for local minima 300 times, with a different randomseed to distribute the mock data each time. This is the result plotted in figure 16.It should be noted that a real experiment will not have the luxury of being repeatedso many times, if at all, and so fits to the particular statistical distribution of datathat is accumulated will have a narrower band width, which will not represent the etermining pseudoscalar meson photo-production amplitudes ... Figure 18. (Color online) Mock data on the TR- and BR-polarization observables at W = 1900 MeV, with uncertainties expected from the CLAS set of K + Λ experiment.The curves are predictions of multipole solutions fitted to these data with the best(0.6) and largest (1.4) χ /point, as shown by the solid (red) and long-dashed (blue)curves, respectively. true uncertainty. Nonetheless, the full uncertainty can readily be determined bysimulation. • While actual data sets may attain 5% uncertainties on some observables, others willbe considerably larger, notably those involving polarized targets which are alwayssignificantly shorter than liquid targets. To consider a more realistic collectionof data, we have created Gaussian-smeared mock data with a kinematic coveragetypical of the CLAS detector at Jefferson Lab, using uncertainties on liquid targetmeasurements taken from the CLAS g1c [20], g8 [32] and g11a [34] data sets,and with polarized target data errors estimated for the g9-FROST running period.As an example, the resulting mock data with expected CLAS uncertainties at W = 1900 MeV are shown in figures 17 and 18. The multipole bands resultingfrom fits to these mock data are plotted in figures 19 and 20. As with the 5% etermining pseudoscalar meson photo-production amplitudes ... Figure 19. (Color online) Real parts of the S , P and D wave multipoles resultingfrom fits to mock K + Λ data with the precision and kinematic coverage expected fromthe complete set of CLAS experiments on all 16 observables. Solutions with the best(typically 0.6) and largest (typically 1.2) χ /point are shown as solid (black) and long-dashed (blue) curves, respectively. error study, the Gaussian smearing followed by Monte Carlo and minimization tosearch for local minima has been repeated 300 times to avoid the dependence on thestarting distribution of the data. This had a smaller effect in the resulting multipolebands of figures 19 and 20, since the errors are somewhat larger than the 5% caseof figure 16. Compared to figures 6 and 7, the multipole bands of figures 19 and 20are dramatically narrower. Almost all multipoles are well determined. Some, likethe imaginary part of the M − , remain broad at low energies. But all are welldefined above above 1.9 GeV where unobserved N ∗ states are predicted in variouscalculations. From extensive studies we attribute this mainly to the larger numberof observables rather than to increased statistics on any specific asymmetry. Thesestudies give us confidence in the expectation of a well determined amplitude fromcomplete experiments, such as those from CLAS. This will be a truly significant etermining pseudoscalar meson photo-production amplitudes ... Figure 20. (Color online) Imaginary parts of the S , P and D wave multipoles resultingfrom fits to mock K + Λ data with the precision and kinematic coverage expected fromthe complete set of CLAS experiments on all 16 observables. Solutions with the best(typically 0.6) and largest (typically 1.2) χ /point are shown as solid (black) and long-dashed (blue) curves, respectively. milestone after over fifty years of photo-production experiments.
11. Summary
It is anticipated that data will soon be available on all 16 pseudoscalar mesonphotoproduction observables from a new generation of ongoing experiments, certainlyfor K Λ final states and possibly for πN channels as well. This will significantly reducethe model dependence in the study of excited baryon structure by providing a totalamplitude that is experimentally determined to within a phase. Such an experimentalamplitude can be utilized at two levels, first as a test to validate total amplitudesassociated with different models and second as a starting point that can be analyticallycontinued into the complex plane to search for poles. Here we have laid the ground work etermining pseudoscalar meson photo-production amplitudes ... a )-(58 p ), and from these to electromagneticmultipoles (15)-(18). From a review of some of the more frequently quoted works inthis field, we have found that the same symbol for a polarization asymmetry has beenused by different authors to refer to different experimental quantities; the magnitudesremain the same across published works, but their signs vary (section 5). For example,the definitions of the six observables H , C x ′ , C z ′ , O x ′ , O z ′ and L x ′ in the MAID andSAID on-line PWA codes is the negative of that used by BoGa and the present work.This has already lead to confusion in the analysis of recent double-polarization data(figure 3).We have used the assembled machinery to carry out a multipole analysis of the γp → K + Λ reaction, free of model assumptions, and examined the impact of recentlypublished measurements on 8 different observables. We have used a combined MonteCarlo sampling of the amplitude space, with gradient minimization, and have founda shallow χ valley pitted with a very large number of local minima. This results inbroad bands of multipole solutions, which are experimentally indistinguishable (figures 6and 7). Comparing to models that have recently reported a new N ∗ ( ∼ D since their amplitudes lie outside the model-independent solution bands in the associated multipoles. (These PWA were carried outbefore most of the data used in our analysis were available.) Recent BoGa analyseshave modeled the N ∗ ( ∼ P resonance. While their solution lies within ourexperimental multipole bands, we cannot yet validate it due to the significant width ofthe bands.From our studies with published measurements, as well as simulations with mockdata, we have seen that clusters of local minima in χ are often present. With thecurrent collection of results on 8 observables, these minima are completely degenerateand experimentally indistinguishable. In studies with mock data we have seen that thisdegeneracy can be removed with high precision data on a large number of observables(section 10). As determined in the present analysis, a greater number of differentobservables tend to be more effective in creating a global minimum than higher precision.We conclude that, while a general solution to the problem of determining an amplitudefree of ambiguities may require 8 observables, as has been discussed by CT [2], suchrequirements assume data of arbitrarily high precision. Experiments with realistically etermining pseudoscalar meson photo-production amplitudes ... Acknowledgments
This work was supported by the U.S. Department of Energy, Office of Nuclear PhysicsDivision, under Contract No. DE-AC05-060R23177 under which Jefferson ScienceAssociates operates Jefferson Laboratory, and also by the U.S. Department of Energy,Office of Nuclear Physics Division, under Contract No. DE-AC02-06CH11357 andContract No. DE-FG02-97ER41025.
Appendix A. General expression for the differential cross section with fixedpolarizations
We summarize here the derivation of an analytic expression for the differential crosssection in pseudoscalar meson photoproduction with general values of the beam, targetand recoil polarization. Following the formalism of the spin density matrices describedby FTS [11], one can write the general cross section (35) as, dσ B , T , R ( ~P γ , ~P T , ~P R ) = ρ ( ρ R ) kn ( F µ ) nm ( ρ T ) ml ( F † λ ) lk ( ρ γ ) µλ . (A.1)(Throughout this appendix the same indices in equations imply taking summation.)Here ρ = k/q ; ( F λ ) m s Λ m sN = h m s Λ | F CGLN | m s N i , in which the spin states of the initialand final baryons are quantized in the z -direction and the (unit) photon polarizationvector is taken to be circularly polarized with the helicity λ .The 2 × ρ X for X = γ, T, R is given by ρ γ = 12 [ + ~ P γ · ~σ ] , (A.2) ρ T = 12 [ + ~P T · ~σ ] , (A.3) ρ R = 12 [ + ~P R · ~σ ] , (A.4)where ~σ is the Pauli spin vector, as in (24), and ~ P γ is the so-called Stokes vectorfor the photon polarizations [11]. Note that in the x - y - z coordinate (see figure 2), ~ P γ = ( − P γL cos 2 φ γ , − P γL sin 2 φ γ , P γc ).Substituting (A.2)-(A.4) into (A.1), we have dσ B , T , R ( ~P γ , ~P T , ~P R ) = ρ
12 ( + ~P R · ~σ ) kn ( F µ ) nm
12 ( + ~P T · ~σ ) ml ( F † λ ) lk ×
12 ( + ~ P γ · ~σ ) µλ = ρ + ~P R · ~σ ) kn h ( F λ ) nm ( F † λ ) mk etermining pseudoscalar meson photo-production amplitudes ...
49+ ( F µ ) nm ( F † λ ) mk ~ P γ · ~σ µλ + ( F λ ) nm ~P T · ~σ ml ( F † λ ) lk + ( F µ ) nm ~P T · ~σ ml ( F † λ ) lk ~ P γ · ~σ µλ i . (A.5)Noting that dσ = ( ρ / N where N = ( F λ ) nm ( F † λ ) mn , we can further expand the aboveequation as dσ B , T , R ( ~P γ , ~P T , ~P R ) = dσ ( ~ P γ ) a ( F µ ) kn ( F † λ ) nk σ aµλ N + ( ~P T ) a ( F λ ) kn σ anm ( F † λ ) mk N + ( ~P R ) a ′ σ a ′ kn ( F λ ) nm ( F † λ ) mk N + ( ~P T ) a ( ~ P γ ) b ( F µ ) km σ aml ( F † λ ) lk σ bµλ N + ( ~P R ) a ′ ( ~ P γ ) b σ a ′ kn ( F µ ) nl ( F † λ ) lk σ bµλ N + ( ~P R ) a ′ ( ~P T ) a σ a ′ kn ( F λ ) nm σ aml ( F † λ ) lk N + ( ~P R ) a ′ ( ~P T ) a ( ~ P γ ) b σ a ′ kn ( F µ ) nm σ aml ( F † λ ) lk σ bµλ N ) = dσ n ~ P γ ) a Σ a + ( ~P T ) a T a + ( ~P R ) a ′ P a ′ + ( ~P T ) a ( ~ P γ ) b C BT ab + ( ~P R ) a ′ ( ~ P γ ) a C BR ab +( ~P R ) a ′ ( ~P T ) a C TR a ′ b + ( ~P R ) a ′ ( ~P T ) a ( ~ P γ ) b C BTR a ′ ab o . (A.6)In the last step we have introduced Σ a = ( F µ ) mn ( F † λ ) nm σ aµλ N , (A.7) T a = ( F λ ) kn σ anm ( F † λ ) mk N , (A.8) P a ′ = σ a ′ kn ( F λ ) nm ( F † λ ) mk N , (A.9) C BT ab = ( F µ ) kn σ anm ( F † λ ) mk σ bµλ N , (A.10) C BR a ′ b = σ a ′ kn ( F µ ) nm ( F † λ ) mk σ aµλ N , (A.11) C TR a ′ b = σ a ′ kn ( F λ ) nm σ aml ( F † λ ) lk N , (A.12) C BTR a ′ ab = σ a ′ kn ( F µ ) nm σ aml ( F † λ ) lk σ bµλ N . (A.13)In (A.6)-(A.13) the Pauli matrices that are combined in products with beam and targetpolarizations are defined in reference to the unprimed x, y, z coordinates of figure 2 with etermining pseudoscalar meson photo-production amplitudes ... z , so that ~ P γ · ~σ = ( ~ P γ ) a σ a ≡ ( ~ P γ ) x σ x + ( ~ P γ ) y σ y + ( ~ P γ ) z σ z ,~P T · ~σ = ( ~P T ) a σ a ≡ ( P Tx ) σ x + ( P Ty ) σ y + ( P Tz ) σ z . However, the Pauli matrices appearing in products with the recoil polarization vectorare defined in reference to the primed x ′ , y ′ , z ′ coordinates of figure 2 with the mesonmomentum along +ˆ z ′ , so that ~P R · ~σ = ( ~P R ) a ′ σ a ′ ≡ ( P Rx ′ ) σ x ′ + ( P Ry ′ ) σ y ′ + ( P Rz ′ ) σ z ′ . [If the unprimed x, y, z coordinates were also used to expand ~P R · ~σ , then one wouldobtain a corresponding set of unprimed observables that are related via equation (38).]We note that Σ a , T a , P a , C BT ab , C BR a ′ b , and C TR a ′ b are exactly the same as those definedin [11]. The C BTR a ′ ab term was not included in [11], which did not consider the triplepolarization case. Each component in (A.7)-(A.13) can be related with 16 observablesdefined in tables B1-B4 of Appendix B: Σ x B = Σ , T y T = T, P y ′ R = P, (A.14) C BT z T z B = − E, C BT z T y B = − G, C BT x T z B = F,C BT x T y B = − H, C BT y T x B = P, (A.15) C BR z ′ R z B = C z ′ , C BR z ′ R y B = − O z ′ , C BR x ′ R z B = C x ′ ,C BR x ′ R y B = − O x ′ , C BR y ′ R x B = T, (A.16) C TR z ′ R z T = L z ′ , C TR z ′ R x T = T z ′ , C TR x ′ R z T = L x ′ ,C TR x ′ R x T = T x ′ , C TR y ′ R y T = T, (A.17) C BTR y ′ R x T y B = − E, C
BTR y ′ R x T z B = G, C
BTR y ′ R z T y B = − F,C
BTR y ′ R z T z B = − H, C
BTR x ′ R y T y B = − C z ′ , C BTR x ′ R y T z B = − O z ′ ,C BTR z ′ R y T y B = C x ′ , C BTR z ′ R y T z B = O x ′ , C BTR x ′ R x T x B = L z ′ ,C BTR x ′ R z T x B = − T z ′ , C BTR z ′ R x T x B = − L x ′ , C BTR z ′ R z T x B = T x ′ ,C BTR y ′ R y T x B = 1 . (A.18)Here, all other components not explicitly shown are identically zero, due to symmetryconstraints.Finally, we also note that the spin density matrices (A.2)-(A.4) can be expressedas ρ γ = X ˆ P = ˆ P γ , ˆ P γ p γ ˆ P
12 [ + ˆ P ˆ P · ~σ ] , (A.19) ρ T = X ˆ Q = ± ˆ P T p T ˆ Q
12 [ + ˆ Q · ~σ ] , (A.20) ρ R = X ˆ R = ± ˆ P R p R ˆ R
12 [ + ˆ R · ~σ ] . (A.21) etermining pseudoscalar meson photo-production amplitudes ... p X ˆ P is the probability of observing particle X polarized in the ˆ P direction; ˆ P ˆ P is the Stokes vector specified by the unit photon polarization vector ˆ P ; ˆ P γ is a unitphoton polarization vector perpendicular to ˆ P γ ≡ ˆ P γ for linearly polarized photons,while ˆ P γ and ˆ P γ express two different helicity states for circularly polarized photons.The non-unit polarization vectors can be expressed with the unit polarization vectors as ~P γ = ( p γ ˆ P γ − p γ ˆ P γ ) ˆ P γ , ~P T = ( p T + ˆ P T − p T − ˆ P T ) ˆ P T , and ~P R = ( p R + ˆ P R − p R − ˆ P R ) ˆ P R . Substituting(A.19)-(A.21) into (A.1), one obtain the relation between the general cross sections withunit and non-unit polarization vectors (35). Appendix B. Constructing observables from measurements
We tabulate here the pairs of measurements needed to construct each of the 16 transversephotoproduction observables in terms of the polarization orientation angles of figure 2.The photon beam is characterized either by its helicity, h γ for circular polarization, or by φ Lγ for linear polarization. Assuming 100% polarizations, each observable ˆ A = Adσ isdetermined by a pair of measurements, each denoted as σ ( B, T, R ); “ unp ” indicates theneed to average over the initial spin states of the target and/or beam, and to sum overthe final spin states of the recoil baryon. For observables involving only beam and/ortarget polarizations, dσ = (1 / σ + σ ) and ˆ A = (1 / σ − σ ). For observablesinvolving the final state recoil polarization, dσ = ( σ + σ ) and ˆ A = ( σ − σ ). Table B1.
The cross section and the observables involving only one polarization intheir leading terms in equation (37); dσ = β ( σ + σ ) and ˆ A = β ( σ − σ ), where β = 1 ( β = 1 /
2) if recoil polarization is (is not) observed. dσ , Σ, T , P Beam Target RecoilObservable ( σ − σ ) h γ φ Lγ θ p φ p θ p ′ φ p ′ dσ unp unp unp unp unp unp σ = σ ( ⊥ , ,
0) - π/ unp unp unp unp σ = σ ( k , ,
0) - 0 unp unp unp unp
T σ = σ (0 , + y, unp unp π/ π/ unp unp σ = σ (0 , − y, unp unp π/ π/ unp unp ˆ P σ = σ (0 , , + y ′ ) unp unp unp unp π/ π/ σ = σ (0 , , − y ′ ) unp unp unp unp π/ π/ etermining pseudoscalar meson photo-production amplitudes ... Table B2.
Observables involving both beam and target polarizations in their leadingterms in equation (37); dσ = (1 / σ + σ ) and ˆ A = (1 / σ − σ ). B - T Beam Target RecoilObservable ( σ − σ ) h γ φ Lγ θ p φ p θ p ′ φ p ′ E σ = σ (+1 , − z,
0) +1 - π unp unp σ = σ (+1 , + z,
0) +1 - 0 0 unp unp
E σ = σ (+1 , − z,
0) +1 - π unp unp σ = σ ( − , − z, − π unp unp G σ = σ (+ π/ , + z,
0) - π/ unp unp σ = σ (+ π/ , − z,
0) - π/ π unp unp G σ = σ (+ π/ , + z,
0) - π/ unp unp σ = σ ( − π/ , + z,
0) - 3 π/ unp unp F σ = σ (+1 , + x,
0) +1 - π/ unp unp σ = σ ( − , + x, − π/ unp unp F σ = σ (+1 , + x,
0) +1 - π/ unp unp σ = σ (+1 , − x,
0) +1 - π/ π unp unp H σ = σ (+ π/ , + x,
0) - π/ π/ unp unp σ = σ ( − π/ , + x,
0) - 3 π/ π/ unp unp H σ = σ (+ π/ , + x,
0) - π/ π/ unp unp σ = σ (+ π/ , − x,
0) - π/ π/ π unp unp Table B3.
Observables involving both beam and recoil polarizations in their leadingterms in equation (37); dσ = ( σ + σ ) and ˆ A = ( σ − σ ). B - R Beam Target RecoilObservable ( σ − σ ) h γ φ Lγ θ p φ p θ p ′ φ p ′ ˆ C x ′ σ = σ (+1 , , + x ′ ) +1 - unp unp π/ θ K σ = σ ( − , , + x ′ ) − unp unp π/ θ K C x ′ σ = σ (+1 , , + x ′ ) +1 - unp unp π/ θ K σ = σ (+1 , , − x ′ ) +1 - unp unp π/ θ K C z ′ σ = σ (+1 , , + z ′ ) +1 - unp unp θ K σ = σ ( − , , + z ′ ) − unp unp θ K C z ′ σ = σ (+1 , , + z ′ ) +1 - unp unp θ K σ = σ (+1 , , − z ′ ) +1 - unp unp π + θ K O x ′ σ = σ (+ π/ , , + x ′ ) - π/ unp unp π/ θ K σ = σ ( − π/ , , + x ′ ) - 3 π/ unp unp π/ θ K O x ′ σ = σ (+ π/ , , + x ′ ) - π/ unp unp π/ θ K σ = σ (+ π/ , , − x ′ ) - π/ unp unp π/ θ K O z ′ σ = σ (+ π/ , , + z ′ ) - π/ unp unp θ K σ = σ ( − π/ , , + z ′ ) - 3 π/ unp unp θ K O z ′ σ = σ (+ π/ , , + z ′ ) - π/ unp unp θ K σ = σ (+ π/ , , − z ′ ) - π/ unp unp π + θ K etermining pseudoscalar meson photo-production amplitudes ... Table B4.
Observables involving both target and recoil polarizations in their leadingterms in equation (37); dσ = ( σ + σ ) and ˆ A = ( σ − σ ). T - R Beam Target RecoilObservable ( σ − σ ) h γ φ Lγ θ p φ p θ p ′ φ p ′ ˆ L x ′ σ = σ (0 , + z, + x ′ ) unp unp π/ θ K σ = σ (0 , − z, + x ′ ) unp unp π π/ θ K L x ′ σ = σ (0 , + z, + x ′ ) unp unp π/ θ K σ = σ (0 , + z, − x ′ ) unp unp π/ θ K L z ′ σ = σ (0 , + z, + z ′ ) unp unp θ K σ = σ (0 , − z, + z ′ ) unp unp π θ K L z ′ σ = σ (0 , + z, + z ′ ) unp unp θ K σ = σ (0 , + z, − z ′ ) unp unp π + θ K T x ′ σ = σ (0 , + x, + x ′ ) unp unp π/ π/ θ K σ = σ (0 , − x, + x ′ ) unp unp π/ π π/ θ K T x ′ σ = σ (0 , + x, + x ′ ) unp unp π/ π/ θ K σ = σ (0 , + x, − x ′ ) unp unp π/ π/ θ K T z ′ σ = σ (0 , + x, + z ′ ) unp unp π/ θ K σ = σ (0 , − x, + z ′ ) unp unp π/ π θ K T z ′ σ = σ (0 , + x, + z ′ ) unp unp π/ θ K σ = σ (0 , + x, − z ′ ) unp unp π/ π + θ K etermining pseudoscalar meson photo-production amplitudes ... Appendix C. The Fierz identities
We list here the Fierz identities relating asymmetries , with signs consistent with thedefinition of observables in Appendix B and with the form of the general cross sectionsin equation (37). The equation numbering sequence in Appendix C.1-Appendix C.3 isthat of Chiang and Tabakin [2]. Compared to the results given in [2], our equation (L.0)differs by a factor 4 / Squared relationsare the same. Sign changes in eight of the equations can be attributed to the differentdefinition for the E asymmetry used by Fasano, Tabakin and Saghai [11], to whichChiang and Tabakin refer. Appendix C.1. Linear-quadratic relations { Σ + T + P + E + G + F + H + O x ′ + O z ′ + C x ′ + C z ′ + L x ′ + L z ′ + T x ′ + T z ′ } / . (L.0)Σ = + T P + T x ′ L z ′ − T z ′ L x ′ . (L.TR) T = +Σ P − C x ′ O z ′ + C z ′ O x ′ . (L.BR) P = +Σ T + GF + EH. (L.BT) G = + P F + O x ′ L x ′ + O z ′ L z ′ . (L.1) H = + P E + O x ′ T x ′ + O z ′ T z ′ . (L.2) E = + P H − C x ′ L x ′ − C z ′ L z ′ . (L.3) F = + P G + C x ′ T x ′ + C z ′ T z ′ . (L.4) O x ′ = + T C z ′ + GL x ′ + HT x ′ . (L.5) O z ′ = − T C x ′ + GL z ′ + HT z ′ . (L.6) C x ′ = − T O z ′ − EL x ′ + F T x ′ . (L.7) C z ′ = + T O x ′ − EL z ′ + F T z ′ . (L.8) T x ′ = +Σ L z ′ + HO x ′ + F C x ′ . (L.9) T z ′ = − Σ L x ′ + HO z ′ + F C z ′ . (L.10) L x ′ = − Σ T z ′ + GO x ′ − EC x ′ . (L.11) L z ′ = +Σ T x ′ + GO z ′ − EC z ′ . (L.12) etermining pseudoscalar meson photo-production amplitudes ... Appendix C.2. Quadratic relations C x ′ O x ′ + C z ′ O z ′ + EG − F H = 0 . (Q.b) GH − EF − L x ′ T x ′ − L z ′ T z ′ = 0 . (Q.t) C x ′ C z ′ + O x ′ O z ′ − L x ′ L z ′ − T x ′ T z ′ = 0 . (Q.r)Σ G − T F − O z ′ T x ′ + O x ′ T z ′ = 0 . (Q.bt.1)Σ H − T E + O z ′ L x ′ − O x ′ L z ′ = 0 . (Q.bt.2)Σ E − T H + C z ′ T x ′ − C x ′ T z ′ = 0 . (Q.bt.3)Σ F − T G + C z ′ L x ′ − C x ′ L z ′ = 0 . (Q.bt.4)Σ O x ′ − P C z ′ + GT z ′ − HL z ′ = 0 . (Q.br.1)Σ O z ′ + P C x ′ − GT x ′ + HL x ′ = 0 . (Q.br.2)Σ C x ′ + P O z ′ − ET z ′ − F L z ′ = 0 . (Q.br.3)Σ C z ′ − P O x ′ + ET x ′ + F L x ′ = 0 . (Q.br.4) T T x ′ − P L z ′ − HC z ′ + F O z ′ = 0 . (Q.tr.1) T T z ′ + P L x ′ + HC x ′ − F O x ′ = 0 . (Q.tr.2) T L x ′ + P T z ′ − GC z ′ − EO z ′ = 0 . (Q.tr.3) T L z ′ − P T x ′ + GC x ′ + EO x ′ = 0 . (Q.tr.4) Appendix C.3. Squared relations G + H + E + F + Σ + T − P = 1 . (S.bt) O x ′ + O z ′ + C x ′ + C z ′ + Σ − T + P = 1 . (S.br) T x ′ + T z ′ + L x ′ + L z ′ − Σ + T + P = 1 . (S.tr) G + H − E − F − O x ′ − O z ′ + C x ′ + C z ′ = 0 . (S.b) G − H + E − F + T x ′ + T z ′ − L x ′ − L z ′ = 0 . (S.t) O x ′ − O z ′ + C x ′ − C z ′ − T x ′ + T z ′ − L x ′ + L z ′ = 0 . (S.r) etermining pseudoscalar meson photo-production amplitudes ... Appendix C.4. ARS-squared relations
Here we include a set of squared relations discussed in Artru, Richard and Soffer(ARS) [31]. These can be derived from combinations of relations in the precedingsections. For example, the first, (ARS.S.bt), can be obtained by combining (S.bt)and (L.BT). Our relations differ in sign from ARS in those terms involving F , C x ′ and C z ′ , and as a result there are sign differences in (ARS.S.bt), (ARS.S.br) and (ARS.btr1).(1 ± P ) = ( T ± Σ) + ( E ± H ) + ( G ± F ) . (ARS.S.bt)(1 ± T ) = ( P ± Σ) + ( C x ′ ∓ O z ′ ) + ( C z ′ ± O x ′ ) . (ARS.S.br)(1 ± Σ) = ( P ± T ) + ( L x ′ ∓ T z ′ ) + ( L z ′ ± T x ′ ) . (ARS.S.tr)(1 ± L z ′ ) = (Σ ± T x ′ ) + ( E ∓ C z ′ ) + ( G ± O z ′ ) . (ARS.btr1)(1 ± T x ′ ) = (Σ ± L z ′ ) + ( F ± C x ′ ) + ( H ± O x ′ ) . (ARS.btr2) Appendix D. Born amplitudes for γN → K ΛIn this Appendix, we summarize the Born amplitudes for γ ( q ) + p ( p ) → K + ( k ′ ) + Λ( p ′ )in the center of mass energy ( ~p = − ~q , ~p ′ = − ~k ′ ), which are used to fix high partial waves(4 ≤ L ≤
8) in the multipole analyses presented in section 9. We consider the followingBorn terms for I µ ǫ µ [see the paragraph including (6) for the description of I µ ǫ µ ]: I µ ǫ µ = I a + I b + I c + I d + I e + I f , (D.1)where I a = i f KN Λ m K k ′ γ p ′ + k ′ − m N Γ N ( q ) F ( | ~k ′ | , Λ KN Λ ) , (D.2) I b = i f KN Λ m K Γ Λ ( q ) 1 p − 6 k ′ − m Λ k ′ γ F ( | ~k ′ | , Λ KN Λ ) , (D.3) I c = i f KN Σ m K Γ ΛΣ ( q ) 1 p − 6 k ′ − m Σ k ′ γ F ( | ~k ′ | , Λ KN Σ ) , (D.4) I d = − ie f KN Λ m K ǫ γ γ F ( | ~k ′ | , Λ KN Λ ) , (D.5) I e = ie f KN Λ m K ˜ kγ ˜ k − m K (˜ k + k ′ ) · ǫ γ F ( | ~ ˜ k | , Λ KN Λ ) , (D.6) I f = − e g K ∗ N Λ g K ∗ K + γ m K [ γ δ + κ K ∗ N Λ m N + m Λ ) ( γ δ ˜ k − 6 ˜ kγ δ )] × ǫ αβηδ ˜ k η q α ǫ βγ k − m K ∗ F ( | ~ ˜ k | , Λ K ∗ N Λ ) , (D.7)with ˜ k = p − p ′ andΓ N = e {6 ǫ γ − κ N m N [ ǫ γ q − 6 q ǫ γ ] } , (D.8)Γ Λ = − e κ Λ m N [ ǫ γ q − 6 q ǫ γ ] , (D.9)Γ ΛΣ = − e κ ΛΣ m N [ ǫ γ q − 6 q ǫ γ ] . (D.10) etermining pseudoscalar meson photo-production amplitudes ... F ( | ~k | , Λ) for the hadronic vertex definedas F ( | ~k | , Λ) = Λ | ~k | + Λ ! . (D.11)We make use of the SU(3) relation for the coupling constants, f KN Λ m K = f πNN m π − d √ , (D.12) f KN Σ m K = f πNN m π − d √ , (D.13) g K ∗ N Λ = g ρNN − d √ , (D.14) κ K ∗ N Λ m N + m Λ = κ ρ m N , (D.15)and take parameters as f πNN = √ . × π [41], κ p = µ p − .
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635 [42], g ρNN = 8 .
72 [41], κ ρ = 2 .
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61 [42], and κ ΛΣ = − .
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