aa r X i v : . [ nu c l - t h ] A ug Explicit derivation of the completeness condition in pseudoscalar mesonphotoproduction
K. Nakayama Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA
By exploiting the underlying symmetries of the relative phases of the pseudoscalar meson pho-toproduction amplitude, we provide a consistent and explicit mathematical derivation of the com-pleteness condition for the observables in this reaction. In particular, we determine all the possiblesets of four double-spin observables that resolve the phase ambiguity of the amplitude in transversitybasis up to an overall phase. The present work substantiates and corroborates the original findingsof Chiang and Tabakin [Phys. Rev. C , 2054 (1997)]. It is found, however, that the completenesscondition of four double-spin observables to resolve the phase ambiguity holds only when the relativephases do not meet the condition of equal magnitudes. In situations where this condition occurs, itis shown that one needs extra chosen observables, resulting in the minimum number of observablesrequired to resolve the phase ambiguity reaching up to eight, depending on the particular set of fourdouble-spin observables considered. Furthermore, a way of gauging when the condition of equalmagnitudes occurs is provided. PACS numbers: 13.60.Le, 25.20.Lj, 13.88.+e, 24.70.+s
I. INTRODUCTION
The issue of model-independent determination of thepseudoscalar meson photoroduction amplitude has at-tracted much attention since the early stage of investiga-tion of this reaction process. In particular, early paperson the minimum number of experimental observables re-quired to determine the pseudoscalar meson photopro-duction amplitude – the so-called complete experiments – have resulted in contradictory findings (for a brief ac-count on these, see Ref. [1]). Barker, Donnachie andStorrow [1] have cleared this situation, by deriving thenecessary and sufficient conditions for determining thefull photoproduction amplitude up to discrete ambigui-ties. They also provided the rules for choosing furthermeasurements to resolve these ambiguities. According tothese authors, for a given kinematics (total energy of thesystem and meson production angle), one requires nineobservables to determine the full reaction amplitude upto an arbitrary overall phase. Keaton and Workman [2],however, have realized that there are cases obeying therules given in Ref. [1] that still leave unsolved ambigui-ties. Finally, Chiang and Tabakin [3], have shown that,instead of nine observables as claimed in Ref. [1], one re-quires a minimum of eight carefully chosen observablesfor a complete experiment. Apart from solving for theamplitude magnitudes and phases directly, Chiang andTabakin [3] in their study, have also used a bilinear he-licity product formulation to map an algebra of measure-ments over to the well-known algebra of the 4x4 gammamatrices. This latter method leads to an alternate proofthat eight carefully chosen experiments suffice for deter-mining the transversity amplitudes completely. The is-sue of complete experiments has been also discussed byMoravcsik [4] in the context of a general reaction process.There, a very similar approach to that of Ref. [3] is usedfor resolving the discrete phase ambiguities of the reac- tion amplitude with a geometrical interpretation. San-dorf et al. [5] have concluded among other things that,while a mathematical solution to the problem of deter-mining an amplitude free of ambiguities may require eightobservables [3], experiments with realistically achievableuncertainties will require a significantly larger number ofobservables. Also, the Gent group has extended much ef-fort along this line [6–8]. Recently, with the advances inexperimental techniques, many spin-observables in pho-toproduction reactions became possible to be measuredand this has attracted much interest in constraints onpartial-wave analysis in the context of complete exper-iments [9–14]. Of particular interest in this connectionis the issue of whether the baryon resonances can be ex-tracted model independently or with minimal model in-puts. Efforts in this direction are currently in progress[12–14].In this work, we revisit the problem of complete ex-periments in pseudoscalar meson photoproduction froma mathematical point of view, i.e., under ideal experi-ments with zero uncertainties. Thus, it is most directlyrelated to the work of Ref. [3]. We tackle this problemby solving for the amplitude magnitudes and phases di-rectly, as has been done in Ref. [3]. In doing so, weshall reveal and exploit the underlying symmetries of therelative phases of the photoproduction amplitude, whichallows a consistent and explicit mathematical derivationof the completeness condition for the observables cover-ing all the relevant cases. The completeness conditionof a set of four double-spin observables to resolve thephase ambiguity of the transversity amplitude is shownto hold, except in situations where the equal relative-phase magnitudes relation - as specified in Eq.(48) laterin Sec. VI - occur. It will be shown that, when this sit-uation occurs, one needs up to seven chosen double-spinobservables, instead of four, to resolve the phase ambi-guity. Furthermore, in the particular situation where therelative phases vanish, eight chosen double-spin observ-ables are required to resolve the phase ambiguity.The paper is organized as follows. In Sec. II, we intro-duce the notations used throughout this work and expressthe observables as bilinear combinations of the four basictransversity amplitudes. In addition, we group the ob-servables and classify them in cases which are convenientfor determining the possible sets of four observables thatresolve the phase ambiguity. In Secs. III, IV and V, wedetermine these sets of four double-spin observables, ac-cording to the classification introduced in Sec. II. There,we also consider the cases where the restriction on therelative phases for the completeness condition of the fourobservables is not satisfied. In Sec. VI, we discuss howto identify when this restriction is violated. Finally, asummary is given in Sec. VII.
II. NOTATIONS
The basic four independent amplitudes, M j ( j =1 , · · · , M j = r j e iφ j , (cid:26) r j = magnitude ,φ j = phase . (1)Then, following Ref. [3], the 16 non-redundant observ-ables can be expressed in terms of these amplitudes M j in transversity basis and grouped according to S = dσ/d Ω = (cid:2) | r | + | r | + | r | + | r | (cid:3) , Σ = (cid:2) | r | + | r | − | r | − | r | (cid:3) ,T = (cid:2) | r | − | r | − | r | + | r | (cid:3) ,P = (cid:2) −| r | + | r | − | r | + | r | (cid:3) , (2) BT = O a ≡ − G = B sin φ + B sin φ ,O a − ≡ F = B sin φ − B sin φ ,O a ≡ E = B cos φ + B cos φ ,O a − ≡ H = B cos φ − B cos φ , (3) BR = O b ≡ O z = B sin φ + B sin φ ,O b − ≡ − C x = B sin φ − B sin φ ,O b ≡ − C z = B cos φ + B cos φ ,O b − ≡ − O x = B cos φ − B cos φ , (4) T R = O c ≡ − L x = B sin φ + B sin φ ,O c − ≡ − T z = B sin φ − B sin φ ,O c ≡ − L z = B cos φ + B cos φ ,O c − ≡ T x = B cos φ − B cos φ , (5) where B ij ≡ r i r j and φ ij ≡ φ i − φ j . (6)In the following we refer to φ ij as the relative phase.The observables in S include the unpolarized cross sec-tion, dσ/d Ω, and single-spin observables Σ (beam asym-metry), T (target asymmetry) and P (recoil asymmetry).It is clear from Eq.(2) that, together, they determineuniquely the magnitudes of the basic four amplitudes intransversity basis. Throughout this work, these four ob-servables are assumed to be measured, so that the mag-nitudes of the basic transversity amplitudes are known.The remaining observables given in Eqs.(3,4,5) are alldouble-spin observables and some combinations of themwill serve to determine the phases of the four transversityamplitudes up to an overall phase, i.e., the three relativephases φ ij involved. We refer to the observables in eachof BT (beam-target asymmetry), BR (beam-recoil asym-metry) and T R (target-recoil asymmetry) as a group.We use a = BT , b = BR and c = T R .In Ref. [3], the unnormalized spin asymmetries aredenoted by ˇΩ β , i.e., ˇΩ β ≡ ( dσ/d Ω)Ω β , where Ω β stands for a given spin asymmetry specified by theindex β . Throughout this work, we simply use thesame notation Ω β for the unnormalized spin asymmetries(( dσ/d Ω)Ω β → Ω β ) to avoid overloading the notations.For example, Σ in Eq.(2) actually stands for ( dσ/d Ω)Σ,and so on.From the above list of observables, one sees that allpossible sets of four double-spin observables can be ob-tained by considering the following cases:1) ( ) case: two pairs of observables, each pairfrom distinct groups.2) ( ) case: a pair of observables from onegroup and two other observables, one from each ofthe remaining two groups.3) ( ) case: three observables from one groupand one observable from another group.4) 4 case: all four observables from one group.In the following we shall consider each of the caseslisted above. III. PHASE FIXING FOR THE
CASE
We start by noticing that there are two basic typesof combination of a pair of observables ( O mnν , O mn ′ ν ′ ) in agiven group, one type with n = n ′ and the other with n = n ′ . Here, ( m = a, b, c ), ( n, n ′ = 1 ,
2) and ( ν, ν ′ = ± ). Apair of observables of the type ( O mn + , O mn − ) leads to a four-fold phase ambiguity, with two-fold ambiguity in each ofthe relative phases involved, φ ij and φ kl . There are twodistinct pairs of this type ( n = 1 ,
2) in each group. On theother hand, a pair of observables of the type ( O m ν , O m ν ′ ),leads only to a two-fold phase ambiguity. We have fourdistinct pairs of this type ( ν, ν ′ = ± ) in each group.To see the properties mentioned above, let us considerall the possible pairs one can form in a given group, say,group a = BT . For the pair ( O a , O a − ) = ( − G, F ), wehave from Eq.(3), O a = B sin φ + B sin φ ,O a − = B sin φ − B sin φ , (7)which leads tosin φ = O a + O a − B = ⇒ φ = (cid:26) α ,π − α , sin φ = O a − O a − B = ⇒ φ = (cid:26) α ,π − α , (8)where − π/ ≤ α , α ≤ + π/ α ij ’s are uniquely de-fined. In the following, we use the notation φ λij to desig-nate φ + ij = α ij , φ − ij = π − α ij . (9)Note that a (relative) phase is meaningful only modulo π .Analogously, for the pair ( O a , O a − ) = ( E, H ), wehave from Eq.(3), O a = B cos φ + B cos φ ,O a − = B cos φ − B cos φ , (10)which leads to the two-fold ambiguity φ + ij = α ij , φ − ij = − α ij , (11)where α ij is uniquely defined with 0 ≤ α ij ≤ π .Next we consider the pair ( O a , O a − ) = ( − G, H ).From Eq.(3), O a = B sin φ + B sin φ ,O a − = B cos φ − B cos φ . (12)We first combine the above two expressions into O a + O a − = B + B − B B cos( φ + φ ) . (13)Now, we define angle ζ ≡ ζ mnν,n ′ ν ′ through cos ζ ≡ O mnν N , sin ζ ≡ O mn ′ ν ′ N , (14) ζ mnν,n ′ ν ′ has a geometrical interpretation as the polar angle of avector in a 2-dimensional coordinate system, where O mnν definesthe x -coordinate and O mn ′ ν ′ , the y -coordinate. This provides anintuitive understanding of the fact that such an angle, ζ mnν,n ′ ν ′ ,can indeed always be found. with N ≡ N mnν,n ′ ν ′ ≡ p O mnν + O mn ′ ν ′ . In the followingwe simply use ζ and N to avoid the heavy notation, butit should be kept in mind that they depend on the givenpair of observables. For the pair under consideration, wehave cos ζ ≡ O a N , sin ζ ≡ O a − N , (15)with N ≡ p O a + O a − .Then, Eq.(12) can be expressed in terms of ζ as N cos ζ = B sin φ + B sin φ ,N sin ζ = B cos φ − B cos φ . (16)Multiplying the first equality in the above equation bysin φ and the second one by cos φ and subtractingthe second from the first, we arrive atcos( φ + φ ) = B + N sin( ζ − φ ) B . (17)Inserting the above result into Eq.(13) yieldssin( ζ − φ ) = B − B − N N B , (18)leading to the following two-fold ambiguity for φ : φ = (cid:26) ζ − α ,ζ − π + α . (19)Analogously, from Eqs.(13,16), we find thatsin( ζ + φ ) = B − B + N N B , (20)leading to the two-fold ambiguity φ = (cid:26) − ζ + α , − ζ + π − α . (21)Note that, in Eqs.(19,21), phases α and α areuniquely defined bysin( α ) = B − B − N N B , sin( α ) = B − B + N N B , (22)with − π/ ≤ α , α ≤ + π/ φ and φ have a two-fold ambiguity each. However, there is another constraintthat cos( φ + φ ) is uniquely defined by Eq.(13). Then,first we note that the sum of φ and φ should be of theform φ + φ = ± ˜ α . Combining this with Eqs.(19,21),it leads to the following possibilities for ˜ α :˜ α = ( λ (cid:0) φ λ + φ λ (cid:1) = ( α − α ) ,λ (cid:16) φ λ + φ λ ′ (cid:17) = ( α + α − π ) , (23)where the notation introduced in Eq.(9) has been used.Here, λ, λ ′ = ± and λ = λ ′ .Next, we calculate cos( φ + φ ) = cos( ± ˜ α ), with ˜ α given in Eq. (23). For ˜ α = α − α , we obtaincos( φ + φ ) = cos( ± ( α − α ))= cos α cos α + sin α sin α = q (1 − sin α )(1 − sin α )+ sin α sin α = B + B − N B B , (24)where Eq.(22) has been used. This result coincides withEq.(13). For ˜ α = α + α − π , on the other hand, it isimmediately seen that the result for cos( φ + φ ) doesnot agree with Eq.(13) since, in this case, apart from anoverall sign, all that changes from the ˜ α = α − α case is the change in the sign of the term sin α sin α - which is non-zero in general - in Eq. (24).Thus, we conclude that Eq.(13), together withEqs.(19,21), leads to φ + φ = ± ( α − α ) , (25)i.e., we end up with only two-fold ambiguity for φ and φ , viz., (cid:26) φ = − ζ + α ,φ = ζ − α , or (cid:26) φ = − ζ − α + π ,φ = ζ + α − π . (26)For the pair ( O a − , O a − ) = ( F, H ), O a − = B sin φ − B sin φ ,O a − = B cos φ − B cos φ , (27)the results can be readily obtained by simply changingthe sign of φ everywhere in the results of the previouscase of ( O a , O a − ). We obtain (cid:26) φ = − ζ + α ,φ = − ζ + α , or (cid:26) φ = − ζ − α + π ,φ = − ζ − α + π . (28)For the pair ( O a − , O a ) = ( F, E ), O a − = B sin φ − B sin φ ,O a = B cos φ + B cos φ , (29)the only change from the previous case of ( O a , O a − ), isin the sign of B . Thus, we can simply follow the stepsof the derivation for the case of ( O a , O a − ), making therethe replacement B → − B . This leads to the changein the constraint given by Eq.(25) to φ + φ = ± ( α − α + π ) . (30) Thus, we obtain the two-fold ambiguity (cid:26) φ = − ζ − α + π ,φ = ζ + α , or (cid:26) φ = − ζ + α ,φ = ζ − α + π . (31)For the pair ( O a , O a ) = ( − G, E ), O a = B sin φ + B sin φ ,O a = B cos φ + B cos φ , (32)we simply flip the sign of φ in Eq.(31). We have (cid:26) φ = − ζ − α + π ,φ = − ζ − α , or (cid:26) φ = − ζ + α ,φ = − ζ + α − π . (33)To avoid any confusion, we emphasize that, in all thecases discussed above, ( O a ± , O a ± ) (with the signs ± be-ing independent), the phases α and α are uniquelydefined and given by Eq.(22).From the preceding considerations in this section, weconclude thati) Any pair of observables of the form ( O m , O m − )leads to a four-fold phase ambiguity of the formgiven by Eq.(9), while any pair of the form( O m , O m − ) leads to a four-fold ambiguity of theform given by Eq.(11). These result in (in viewof the consistency relations given by Eq.(41) thatshall be used later on to help resolve the phase am-biguity)( O a , O a − ) : φ +13 − φ +24 = ( α − α ) ,φ +13 − φ − = [( α + α ) − π ] ,φ − − φ +24 = − [( α + α ) − π ] ,φ − − φ − = − ( α − α ) ,φ +13 + φ +24 = ( α + α ) ,φ +13 + φ − = ( α − α ) + π ,φ − + φ +24 = − ( α − α ) + π ,φ − + φ − = − ( α + α ) , (34)and( O a , O a − ) : φ +13 − φ +24 = ( α − α ) ,φ +13 − φ − = ( α + α ) ,φ − − φ +24 = − ( α + α ) ,φ − − φ − = − ( α − α ) ,φ +13 + φ +24 = ( α + α ) ,φ +13 + φ − = ( α − α ) ,φ − + φ +24 = − ( α − α ) ,φ − + φ − = − ( α + α ) , (35)ii) Any pair of observables of the form ( O m ± , O m ∓ ) =( O m , O m − ) or ( O m − , O m ), leads to a two-fold am-biguity of the form given by Eqs.(26,31), while anypair of the form ( O m ν , O m ν ), leads to a two-fold am-biguity of the form given by Eqs.(28,33). Theseresult in (recall that (relative) phases are modulo π )( O a − , O a − ) : (cid:26) φ λ − φ λ = λ ( α − α ) ,φ λ + φ λ = − ζ + λ ( α + α ) , (36)( O a , O a − ) : (cid:26) φ λ − φ λ = − ζ + λ ( α + α ) ,φ λ + φ λ = λ ( α − α ) , (37)with λ = ± , and( O a , O a ) : φ +13 − φ − = ( α − α ) + π ,φ − − φ +24 = − ( α − α ) + π ,φ +13 + φ − = − ζ + ( α + α ) − π ,φ − + φ +24 = − ζ − ( α + α ) + π , (38)( O a − , O a ) : φ +13 − φ − = − ζ + ( α + α ) − π ,φ − − φ +24 = − ζ − ( α + α ) + π ,φ +13 + φ − = − ( α − α ) + π ,φ − + φ +24 = ( α − α ) + π . (39)From the results obtained above for the pairs of ob-servables ( O anν , O an ′ ν ′ ) ( n, n ′ = 1 , ν, ν ′ = ± with( nν ) = ( n ′ ν ′ )) in group a = BT (cf. Eq.(3)), it isstraightforward to obtain the corresponding results forthe pairs of observables in other two groups b = BR and c = T R (cf. Eqs.(4,5)). All we have to do is to replace( O anν , O an ′ ν ′ ) by ( O mnν , O mn ′ ν ′ ) ( m = b, c ) and the relativephases φ and φ , respectively, by φ and φ in thecase m = b or by φ and φ in the case m = c .The discrete ambiguities exhibited by the relativephases so far in this section (cf. Eqs.(34,35,36,37,38,39))cannot be resolved without further constraint. This isprovided by the property obeyed by the relative phases( φ ij ≡ φ i − φ j ): φ + φ + φ = φ . (40)Here, it should be emphasized that this relation is satis-fied up to an addition of multiples of 2 π , because phasesare meaningful only modulo 2 π . We refer to the aboverelation as the consistency relation, because it is going tobe used to check on the ’consistency’ among the relative Equation (40) may be seen as a direct consequence of the factthat a complex number can be represented by a vector in thecomplex plane and that the sum of all angles between neighboringvectors in a given set of vectors is 2 π (or zero since phases are modulo π ). phases with discrete ambiguities as we have shown in ourconsiderations up to this point. As the reader shall see,the consistency relation allows us to resolve the discreteambiguities for certain sets of four chosen observables.Equation (40) can be rewritten as φ − φ = φ − φ ( a ←→ c ) , (41a) φ + φ = φ + φ ( a ←→ b ) , (41b) φ + φ = φ − φ ( c ←→ b ) . (41c)The first relation in the above equation is used to relatethe observables in group a = BT to those in group c = T R , while the second relation connects the observablesin group a to those in group b = BR . The third relationconnects the observables in group b to those in group c .Note that, apart from an irrelevant overall factor, Eq.(40)leads to a unique relation which connects the relativephases belonging to two specific groups of observables asexhibited in Eq.(41). Equation (41) has been also usedby the authors of Refs. [3, 4] in their analyses.The logic for determining whether a given set of fourobservables can or cannot resolve the phase ambiguity isas follows. From the chosen set of four observables, usingthe appropriate consistency relation in Eq.(41), form allpossible solutions due to the discrete ambiguities of therelative phases which, for the ( ) case , are given byEqs.(34,35,36,37,38,39)). Then, check if these solutionsare linearly independent (non-degenerated) or dependent(degenerated). If there is no degeneracy in the possiblesolutions (i.e., they are all linearly independent), then,only one of them will be satisfied, in general, once the setof unique values of the phases α ij ’s and ζ ′ s (= ζ mnν,n ′ ν ′ )is provided by the measurements of the four observablesin consideration. The precise relation of each α ij tothe corresponding φ ij is known once the correct solutionamong the possible solutions is identified, thus, resolvingthe ambiguity of φ ij . Hence, this set of four observablesresolves the phase ambiguity. If the degeneracy occursamong the possible solutions, then, this set of observablescannot resolve the ambiguity. The logic just describedapplies to all cases ( , , , ) specified at the end ofthe previous section. Only the discrete ambiguities ofthe relative phases are case-dependent, as shown later inSecs. IV,V.It should be clear from the above consideration that,whether a set of four observables resolves the phase am-biguity or not, rests on the linear independence of thepossible solutions provided by the consistency relation(cf. Eqs.(41)) for that set of four observables. Recall that the unpolarized cross section and single-spin observ-ables are assumed to be measured. They fix the magnitudes ofthe four basic transversity amplitudes which enter in the deter-mination of α ij ’s (cf. Eq.(22)). We are now prepared to identify the possible sets offour double-spin observables that resolve the phase am-biguity of the transversity amplitude in the ( ) case defined in item (1) of the preceding section. There arethree basic combinations of the pairs of observables to beconsidered:aa) two pairs from item (i) above with 4 × O mn + , O mn − ) and ( O m ′ n ′ + , O m ′ n ′ − )with m = m ′ .bb) two pairs from item (ii) above with 2 × O m ν , O m ν ′ ) and ( O m ′ µ , O m ′ µ ′ )with m = m ′ .ab) one pair from item (i) and one pair from item (ii)with 4 × O mn + , O mn − )and ( O m ′ µ , O m ′ µ ′ ) with m = m ′ . A. Case (aa)
First, consider case (aa). To be concrete, choose theset of pairs [( O a , O a − ) , ( O c , O c − )]. From Eqs.(3,4),the observables in group a contain relative phases φ and φ , while those in group c contain relative phases φ and φ . Then, using Eq.(41a), we have φ λ − φ λ ′ = φ λ ′′ − φ λ ′′′ , (42)where the indices on which these relative phases dependhave been written explicitly. Inserting the correspondingfour-fold phase ambiguity given by Eq.(35) into the aboverelation, we end up with 16 possible solutions ± α ± α = ± α ± α , (43)where all four signs ± are independent. The 16 possi-ble solutions given above are not all linearly indepen-dent. For example, consider the solution α + α = α + α corresponding to ( λ, λ ′ , λ ′′ , λ ′′′ ) = (+ , − , + , − )in Eq.(42). This solution is degenerated with the so-lution − ( α + α ) = − ( α + α ) corresponding to( λ, λ ′ , λ ′′ , λ ′′′ ) = ( − , + , − , +). Hence, the phase ambigu-ity cannot be resolved in this case. It is also straightfor-ward to see that none of the other combinations of thepairs of observables in case (aa) resolve the ambiguity.This includes the corresponding sets of pairs of observ-ables from group a and group b and from b and c , in whichcases we use the consistency relations given by Eqs.(41b)and (41c), respectively. B. Case (bb)
For case (bb), let’s start by considering the set of twopairs [( O a , O a − ) , ( O c − , O c − )]. From Eqs.(3,4), the rel- ative phases involved for this combination are ( φ , φ )and ( φ , φ ). Then, inserting Eqs.(37,36) into Eq.(41a),yields the following four possible solutions: − ζ + ( α + α ) = ( α − α ) , − ζ + ( α + α ) = − ( α − α ) , − ζ − ( α + α ) = ( α − α ) , − ζ − ( α + α ) = − ( α − α ) . (44)Since the above possible solutions are all linearly in-dependent, there will be only one solution satisfied, ingeneral, for the set of unique values of α , α , α , α and ζ (= ζ a , − ), once they are extracted from the mea-surements of the four observables in question. The cor-rect solution, then, will tell us the exact relation of each α ij ( ij = 13 , , ,
34) to the corresponding φ ij , resolv-ing the ambiguity of φ ij . Hence this set of four observ-ables will resolve the phase ambiguity.Consider now the set of pairs [( O a , O a − ) , ( O c , O c − )].Again, with the help of Eq.(37), Eq.(41a) leads to − ζ + ( α + α ) = − ζ ′ + ( α + α ) , − ζ + ( α + α ) = − ζ ′ − ( α + α ) , − ζ − ( α + α ) = − ζ ′ + ( α + α ) , − ζ − ( α + α ) = − ζ ′ − ( α + α ) . (45)Note that ζ is distinct from ζ ′ (cf. Eq.(14)). As in theprevious case just discussed above, since the four possi-ble solutions here are all linearly independent, the samereasoning to the previous case applies and we concludethat this set of four observables also resolves the phaseambiguity.Now, take the set [( O a − , O a − ) , ( O c − , O c − )]. In thiscase, we obtain the following results:( α − α ) = ( α − α ) , ( α − α ) = − ( α − α ) , − ( α − α ) = ( α − α ) , − ( α − α ) = − ( α − α ) , (46)and we see that this set of observables cannot resolve thephase ambiguity, since there are degenerated (or linearlydependent) solutions (first and fourth solutions and sec-ond and third solutions).Now, from Eqs.(36,37,38,39), we note that the two rel-ative phases, φ ij and φ kl , involved in a given pair of ob-servables from the same group, have the following prop-erties ( m = a, b, c ):( O m ± , O m ∓ ) = ( O m , O m − ) or ( O m − , O m ) −→ (cid:26) φ ij − φ kl −→ ζ − dependent ,φ ij + φ kl −→ ζ − independent , ( O m ± , O m ± ) = ( O m , O m ) or ( O m − , O m − ) −→ (cid:26) φ ij − φ kl −→ ζ − independent ,φ ij + φ kl −→ ζ − dependent . (47)Then, from the pattern exhibited by the above threesets of observables worked out explicitly and with thehelp of Eq.(47), we can easily determine those sets of twopairs of observables for case (bb) that cannot resolve thephase ambiguity. They are the sets which yield the phaserelations in Eq.(41) being ζ -independent. All the othersets do resolve the ambiguity. The results are displayedin Table.I.It should be noted, however, that there is a restric-tion to the fact that those sets of two pairs of observ-ables can resolve the phase ambiguity. For example,for the set [( O a , O a − ) , ( O c − , O c − )], from Eqs.(44,45),it is clear that when α = − α and/or α = α ,no ambiguity can be resolved since the possible solu-tions become degenerated. The same is true for theset [( O a , O a − ) , ( O c , O c − )] when α = − α and/or α = − α . It is easy to see that, had we consideredthe set [( O a , O a ) , ( O c , O c )] instead, we would havefound that when α = α and/or α = α no phaseambiguity can be resolved (cf. Eqs.(38,41a)). Thus, inthese situations, we need to measure one or two moreextra observables to be able to resolve the phase ambigu-ity. For example, for the set of two pairs of observables[( O a , O a − ) , ( O c − , O c − )], we require the extra observable O a − to resolve the ambiguity in the case α = − α and, the extra observable O c in the case α = α .If α = − α and α = α , simultaneously, then, werequire both extra observables O a − and O c . Note that O a differs by a sign of relative phase φ from O a − .This later feature is true for all the observables of theform O m ν . Thus, for the sets of two pairs of the form[( O a ± , O a ν ) , ( O c ± , O c ν ′ )], we need the extra observable O a ∓ and/or O c ∓ (here the ± signs are not independent)to completely resolve the phase ambiguity, depending onwhether α = ± α and/or α = ± α . This meansthat we need a minimum of five or six chosen observ-ables, instead of four, to resolve the phase ambiguityin these situations of equal magnitudes of the relativephases α ij ’s. It is straightforward to extended the aboveconsiderations to other sets of two pairs of observables in-volving groups a and b , and groups b and c . The resultsare given in Table. I. Explicitly, the equal relative-phasemagnitudes relations for the sets of two pairs of observ- ables, in general, are | α | = | α | and / or | α | = | α | ( a ←→ c ) , | α | = | α | and / or | α | = | α | ( a ←→ b ) , | α | = | α | and / or | α | = | α | ( c ←→ b ) . (48)Even with the additional observables as discussedabove, the ambiguity still will not be resolved if α = α = 0 and/or α = α = 0. The only way to resolvethe phase ambiguity in this case is to measure a set ofeight chosen double-spin observables to determine bothcos φ ij and sin φ ij for all four relative phases φ ij ’s asso-ciated with the four basic photoproduction amplitudes. C. Case (ab)
We now turn out attention to case (ab). In this case,it is straightforward to see that any pair of double-spinobservables belonging to item (ii) that leads to the cor-responding phase relations as given by Eq.(41) being ζ -dependent, resolves the phase ambiguity, irrespective ofthe pair of observables belonging to item (i). Otherwisethe phase ambiguity cannot be resolved. The results aredisplayed in Table.II.Analogous to the previous case ( bb ), here we have alsothe restriction of no equal relative-phase magnitudes, | α ij | ’s, for the sets of two pairs of double-spin observ-ables, as given in Table. II, to be able to resolve the phaseambiguity. This case involves the pairs of observables( O mn + , O mn − ) ( n = 1 , bb ).In the case of [( O a , O a − ) , ( O c , O c )], e.g., fromEqs.(34,41a), the extra observable required to resolve thephase ambiguity is either O a or O a − when | α | = | α | .Note that the relevant new pair of observables to helpresolve the phase ambiguity here is either ( O a , O a ) or( O a − , O a − ) (cf. Eqs.(38,36)). When | α | = | α | , theextra observable required is O c − as in case ( bb ).Now consider the set [( O a , O a − ) , ( O b , O b )]. In thiscase, from Eqs.(34,41b), it requires both O a and O a − ,in addition, to resolve the phase ambiguity when | α | = | α | . And, as above, extra observable O b − when | α | = | α | .For the set [( O c , O c − ) , ( O b , O b )], fromEqs.(34,41c), it requires both O c and O c − in addition,to resolve the phase ambiguity when | α | = | α | , and O b − in addition, when | α | = | α | .As for the two pairs of observables involving( O a , O a − ), from Eqs.(35,41), we see that it always re-quires both O a and O a − in addition, to resolve the phaseambiguity when | α | = | α | , irrespective of the otherpair of observables from item (ii). The latter, requires oneextra observable when the corresponding relative phaseshave equal magnitudes.We therefore see that in case ( ab ), the minimum num-ber of double-spin observables required to resolve thephase ambiguity - when the magnitudes of the relativephases α ij are equal - can be five, six or seven depend-ing of the set of two pairs of observables that, otherwise,resolves the phase ambiguity. Based on the above con-siderations, the additional observables required to resolvethe phase ambiguity are indicated in Table. II. IV. PHASE FIXING FOR THE
CASE
We start by considering two observables from a givengroup. For the sake of concreteness, consider the pair( O a , O a − ) = ( − G, F ). This pair of observables was ex-amined in the previous section with the phase ambiguitygiven in Eqs.(8,9). Note that these two observables de-termine sin φ and sin φ (cf. Eq.(8)):sin φ = O a + O a − B , sin φ = O a − O a − B . (49)Appropriate combination of φ λ and φ λ ′ result in (cf.Eq.(34))( O a , O a − ) : φ +24 − φ +13 = ( α − α ) ,φ +24 − φ − = ( α + α ) − π ,φ − − φ +13 = − ( α + α ) + π ,φ − − φ − = − ( α − α ) . (50)Now we consider two observables from the remainingtwo groups, b = BR and c = T R . For a given observablein one of these two groups, say c = T R , there will be fourpossible combinations of the pairs of observables one canform involving another observable from group b = BR (cfEqs(4,5)). For example, for the observable O c , we havethe combinations ( O b − , O c ), ( O b , O c ), ( O b − , O c ),and ( O b , O c ). A. ( O b ± , O c ± ) We start by considering the pair ( O b − , O c ) =( − C x , − L x ). From Eqs.(4,5), O b − = B sin φ − B sin φ ,O c = B sin φ + B sin φ . (51) Expressing φ and φ as φ = φ + φ ,φ = φ − φ , (52)we have O b − = A c sin φ + A s cos φ , (53)with A c ≡ B cos φ + B cos φ ,A s ≡ B sin φ − B sin φ . (54)Using cos φ ij = ± q − sin φ ij , we solve Eq.(53) forsin φ to obtainsin φ = A c O b − ± A s q D − (cid:0) O b − (cid:1) D , (55)with D ≡ A c + A s = B + B + 2 B B cos( φ + φ ) . (56)We now note that while A s is uniquely determined(cf. Eq.(49)), A c has a four-fold ambiguity because know-ing only sin φ ij implies that cos φ ij is known up to a sign.In particular, according to the notation of (9),knowing sin φ λij = ⇒ cos φ λij = λ cos α ij . (57)Since A c depends on cos φ λ and cos φ λ ′ (cf. Eq.(54)),we introduce the notations A λλ ′ c and D λλ ′ , such that, A λλ ′ c = B cos φ λ + B cos φ λ ′ ,D λλ ′ = B + B + 2 B B cos( φ λ + φ λ ′ ) . (58)and, from Eq.(55), we see that φ , in turn, depends on λ and λ ′ , i.e.,sin φ λλ ′ ( η ) = A λλ ′ c O b − + η A s q D λλ ′ − (cid:0) O b − (cid:1) D λλ ′ , (59)where η takes the values ± A ++ c = − A −− c and A + − c = − A − + c ,D ++ 2 = D −− and D + − = D − + 2 . (60)Then, we havesin φ ++12 ( η ) = A ++ c O b − + η A s q D ++ 2 − (cid:0) O b − (cid:1) D ++ 2 , sin φ + − ( η ) = A + − c O b − + η A s q D + − − (cid:0) O b − (cid:1) D + − , sin φ − +12 ( η ) = − A + − c O b − + η A s q D + − − (cid:0) O b − (cid:1) D + − , sin φ −− ( η ) = − A ++ c O b − + η A s q D ++ 2 − (cid:0) O b − (cid:1) D ++ 2 . (61)From the above results, we see that there are, in gen-eral, eight possible sin φ λλ ′ ( η )’s (recall that λ, λ ′ and η take two possible values each), and each of them leads toa two-fold ambiguity φ λλ ′ ( η ) = ( α λλ ′ ( η ) ,π − α λλ ′ ( η ) . (62)An inspection of Eq.(61) reveals thatsin φ ++12 ( ± ) = − sin φ −− ( ∓ ) , sin φ + − ( ± ) = − sin φ − +12 ( ∓ ) , (63)and, consequently, α ++12 ( ± ) = − α −− ( ∓ ) and α + − ( ± ) = − α − +12 ( ∓ ) . (64)Note that since all sin φ λλ ′ ( η )’s are distinct from eachother, so are α λλ ′ ( η )’s.Now, taking the equation for O c in (51) and solvingfor sin φ , yieldssin φ λλ ′ ( η ) = O c − B sin φ λλ ′ ( η ) B , (65)where we have displayed all the indices of the relativephases φ and φ explicitly. The above result leads tothe two-fold ambiguity φ λλ ′ ( η ) = ( α λλ ′ ( η ) ,π − α λλ ′ ( η ) , (66)with all eight α λλ ′ ( η ) being distinct from each otherto the extent that sin φ λλ ′ ( η )’s are. However, α λλ ′ ( η )lacks the symmetry exhibited by α λλ ′ ( η ) in Eq.(64), i.e., α λλ ′ ( η )’s are not related to each other in general.Appropriate combinations of the relative phases φ λλ ′ ( η ) and φ λλ ′ ( η ) involved in each pair contain, in gen-eral, a four-fold ambiguity of the form given by φ λλ ′ ( η ) − φ λλ ′ ( η ) = (cid:16) α λλ ′ ( η ) − α λλ ′ ( η ) (cid:17) , (cid:16) α λλ ′ ( η ) + α λλ ′ ( η ) (cid:17) − π , − (cid:16) α λλ ′ ( η ) + α λλ ′ ( η ) (cid:17) + π , − (cid:16) α λλ ′ ( η ) − α λλ ′ ( η ) (cid:17) , (67)for a given set of { λ, λ ′ , η } (note that λ, λ ′ and η taketwo possible values each).At this stage, in analogy to what we have done in the (2+2) case in the previous section, we invoke the con-sistency relation (40) reexpressed as (cf. Eq.(41a)) φ λ − φ λ ′ = φ λλ ′ ( η ) − φ λλ ′ ( η ) . (68) Inserting Eq.(67) into the above equation, we arrive atthe possible solutions φ λ − φ λ ′ = (cid:16) α λλ ′ ( η ) − α λλ ′ ( η ) (cid:17) , (cid:16) α λλ ′ ( η ) + α λλ ′ ( η ) (cid:17) − π , − (cid:16) α λλ ′ ( η ) + α λλ ′ ( η ) (cid:17) + π , − (cid:16) α λλ ′ ( η ) − α λλ ′ ( η ) (cid:17) , (69)for a given set of { λ, λ ′ , η } . The left-hand-side of theabove equation is given by Eq.(50). Since λ, λ ′ and η take two possible values each, we have 2 × × × φ λλ ′ ( η )’s - and, in turn, all α λλ ′ ( η )’s - are distinct from each other as pointed outpreviously (see below Eq.(66)). Thus, once the uniquevalues of α , α and the associated α λλ ′ ( η ) and α λλ ′ ( η )are provided by the measurements of the observables[( O a , O a ) , ( O b − , O c )], there will be only one solutionsatisfying the consistency relation (68). Therefore, weconclude that this set of observables will resolve the phaseambiguity.It is clear that the preceding results for the pair ofobservables ( O b − , O c ), actually holds for any of thepairs ( O b ± , O c ± ), with the signs ± being independent,since the only difference is the sign change of B and/or B according to the particular combination of the ob-servables in the pair considered. These sign changes donot affect any of the properties exhibited by the phases α λλ ′ ( η ) and α λλ ′ ( η ). Thus, any one of the pairs of ob-servables ( O b ± , O c ± ), together with the pair ( O a , O a − ),can resolve the phase ambiguity of the transversity am-plitude. B. ( O b ± , O c ± ) We now consider the pair ( O b − , O c ) = ( − O x , − L x ), O b − = B cos φ − B cos φ ,O c = B sin φ + B sin φ . (70)In this case, inserting Eq.(52) into the expression for O b − in the above equation, yields O b − = A c cos φ − A s sin φ , (71)with A c ≡ B cos φ − B cos φ ,A s ≡ B sin φ + B sin φ . (72)0Solving Eq.(71) for sin φ , we havesin φ = − O b − A s ± A c q D − (cid:0) O b − (cid:1) D , (73)where D ≡ A c + A s = B + B − B B cos( φ + φ ) . (74)Using the same notation introduced in Eq.(58), wewrite Eq.(73) assin φ λλ ′ ( η ) = − O b − A s + η A λλ ′ c q D λλ ′ − (cid:0) O b − (cid:1) D λλ ′ . (75)Noticing that both A λλ ′ c and D λλ ′ here have the samesymmetry as in Eq.(60), we can verify in this case thatsin φ ++12 ( ± ) = sin φ −− ( ∓ ) , sin φ + − ( ± ) = sin φ − +12 ( ∓ ) , (76)and, consequently, α ++12 ( ± ) = α −− ( ∓ ) and α + − ( ± ) = α − +12 ( ∓ ) . (77) Also, note that for a given set of { λ, λ ′ , η } , Eq.(75) leadsto a two-fold phase ambiguity as given by Eq.(62).Solving now the equation for O c in (70) for sin φ ,we have sin φ λλ ′ ( η ) = O c − B sin φ λλ ′ ( η ) B , (78)leading to a two-fold phase ambiguity as given byEq.(66). Here we note that, unlike in the case of thepair of observables ( O b − , O c ), where sin φ λλ ′ ( η ) has nosymmetry, this quantity given by Eq.(78) above exhibitsthe following symmetry:sin φ ++34 ( ± ) = sin φ −− ( ∓ ) , sin φ + − ( ± ) = sin φ − +34 ( ∓ ) , (79)where Eq.(76) has been used. Consequently, α ++34 ( ± ) = α −− ( ∓ ) and α + − ( ± ) = α − +34 ( ∓ ) . (80)The relative phases α λλ ′ ( η ) and α λλ ′ ( η ) derived here,with the symmetry properties given by Eqs.(77,80),should obey Eq.(69). It happens that the set of pairs[( O b − , O c ) , ( O a , O a − )] cannot resolve the phase ambi-guity. To see this, it suffices to consider the following twoparticular solutions from Eq.(69), φ +24 − φ +13 = α ++34 (+) − α ++12 (+) = ⇒ ( α − α ) = (cid:0) α ++34 (+) − α ++12 (+) (cid:1) ,φ − − φ − = α −− ( − ) − α −− ( − ) = ⇒ − ( α − α ) = − (cid:0) α ++34 (+) − α ++12 (+) (cid:1) , (81)where we have made use of Eqs.(50,77,80). This showsthat these solutions are linearly dependent (degenerated)and, consequently, the set of observables in considerationcannot resolve the phase ambiguity. Degeneracy of thesolutions involving α + − ij ( ± ) and α − + ij ( ∓ ) also occurs.The above consideration shows that any of thepairs of observables ( O b ± , O c ± ), together with the pair( O a , O a − ), cannot resolve the phase ambiguity of thetransversity amplitude. C. ( O b ± , O c ± ) For ( O b − , O c ) = ( − O x , − L z ), O b − = B cos φ − B cos φ ,O c = B cos φ + B cos φ , (82) proceeding analogously to the case of ( O b − , O c ), wehavecos φ λλ ′ ( η ) = A λλ ′ c O b − + η A s q D λλ ′ − (cid:0) O b − (cid:1) D λλ ′ , (83)where A s = B sin φ + B sin φ ,A λλ ′ c = B cos φ λ − B cos φ λ ′ ,D λλ ′ = B + B − B B cos( φ λ + φ λ ′ ) . (84)It is clear that cos λλ ′ ( η ) above exhibits the symmetrycos φ ++12 ( ± ) = − cos φ −− ( ∓ ) , cos φ + − ( ± ) = − cos φ − +12 ( ∓ ) , (85)and, consequently, α ++12 ( ± ) = π + α −− ( ∓ ) and α + − ( ± ) = π + α − +12 ( ∓ ) . (86)1Now, solving the equation for O c in (82) for cos φ ,yields cos φ λλ ′ ( η ) = O c − B cos φ λλ ′ ( η ) B , (87)which reveals that all eight possible values of it are dis- tinct. Consequently, all α λλ ′ ( η )’s are distinct, resultingin linear independence of all possible solutions from theconsistency relation (41a). Then, it follows that, any pairof observables of the form ( O b ± , O c ± ), together with thepair ( O a , O a − ) can resolve the phase ambiguity.Summarizing the results obtained in this section so far,we have( O a , O a − ) and ( O bn ± , O cn ± ) ( n = 1 , → do resolve the ambiguity , ( O a , O a − ) and ( O b ± , O c ± ) → do not resolve the ambiguity . (88)In the above relations, the ± signs are independent. D. ( O a , O a − ) We now turn our attention to the case of the pair ofobservables from group a being ( O a , O a − ) = ( E, H ), O a = B cos φ + B cos φ ,O a − = B cos φ − B cos φ . (89)The difference from the previous case of ( O a , O a − ) isthat ( O a , O a − ) determines cos φ and cos φ uniquely,instead of sin φ and sin φ . This implies that, for thepair ( O b − , O c ), the quantity A c defined in Eq.(54) be-comes uniquely determined, while A s will have a four-fold ambiguity and the quantity D in Eq.(56) depends on ( λλ ′ ), but remains unchanged otherwise, viz., A c = B cos φ + B cos φ ,A λλ ′ s = B sin φ λ − B sin φ λ ′ ,D λλ ′ = B + B + 2 B B cos( φ λ + φ λ ′ ) . (90)Then, Eq.(59) changes tosin φ λλ ′ ( η ) = A c O b − + η A λλ ′ s q D λλ ′ − (cid:0) O b − (cid:1) D λλ ′ . (91)Analogously, for the pair ( O b − , O c ), Eq. (75) changestosin φ λλ ′ ( η ) = − O b − A λλ ′ s + η A c q D λλ ′ − (cid:0) O b − (cid:1) D λλ ′ . (92)In the above equation A c , A λλ ′ c and D λλ ′ are given byEq.(90) except for the change in the sign of B .It, then, follows that the symmetry properties ofsin φ λλ ′ ( η ) given in the above two equations have inter-changed from the corresponding quantities in the case of( O a , O a − ). This, in turn, interchanges the property of α λλ ′ ( η ). We can now see that the role of ( O b ± , O c ± ) and( O b ± , O c ± ) interchanges in Eq.(88), i.e.,( O a , O a − ) and ( O bn ± , O cn ± ) ( n = 1 , → do not resolve the ambiguity , ( O a , O a − ) and ( O b ± , O c ± ) → do resolve the ambiguity . (93) E. ( O a ± , O a ± ) In the case of ( O a ± , O a ∓ ) (here the signs ± are notindependent), we note that φ λ − φ λ ′ is ζ -dependent (cf.Eqs.(37,39)). Therefore, in this case, the phase ambi- guity will be resolved because the possible solutions inEq.(69) will all be linearly independent. For the case of( O a ± , O a ± ) (not independent ± signs), however, φ λ − φ λ ′ is ζ -independent (cf. Eqs.(36,38)) and the above argu-ment valid for ( O a ± , O a ∓ ) does not apply. However,2it happens that the relative phases φ and φ in the( O a ± , O a ± ) case are given by (cf. Eqs(28,33)) ( φ = − ζ + α ,φ = − ζ + α − δ + π , (94)or ( φ = − ζ − α + π ,φ = − ζ − α + δ − π , (95)with two-fold ambiguity. δ + = 1 and δ − = 0 for( O a , O a ) and δ + = 0 and δ − = 1 for ( O a − , O a − ).It is then easy to see that all cos φ ij ( ij = 24 ,
34) aredistinct from each other. The same is true for sin φ ij .This implies that the quantities A c and A s entering intoEqs.(59,75) have all distinct values, in general, as canbe seen from their definitions in Eqs.(54,72) for the case( O b − , O c ) and ( O b − , O c ), respectively. Hence, all thephases α λλ ′ ( η ) and α λλ ′ ( η ) entering into Eq.(69) assumedistinct values in general, resulting in linearly indepen-dent possible solutions. Consequently, the phase ambi-guity can be resolved with the pairs ( O a ± , O a ± ) as well.We conclude that any pair of the form ( O a ± , O a ± ),together with any pair of the form ( O b ± , O c ± ) or( O b ± , O c ± ), will resolve the phase ambiguity. Here allthe signs ± are independent.This completes the analysis of all possible ( )cases . Collecting the results for all the possibilities, thefollowing sets of four observables will resolve the phaseambiguity in the ( ) case :i) ( O a , O a − ) and (cid:2) ( O b ± , O c ± ) or ( O b ± , O c ± ) (cid:3) .ii) ( O a ± , O a ± ) and (cid:2) ( O b ± , O c ± ) or ( O b ± , O c ± ) or( O b ± , O c ± ) (cid:3) .iii) ( O a , O a − ) and ( O b ± , O c ± ).with any permutation of a, b, c . Here, the ± signs are allindependent. The results are displayed in Table. III forthe case ( a ) + 1( b ) + 1( c ) ) . Other combinations canbe obtained by an appropriate permutation of a, b, c .As in the ( ) case discussed in preceding Sec. III,here we have also the restriction of no equal relative-phase magnitudes in order to enable the sets of two pairsof observables, as given in Table. III, to resolve the phaseambiguity. Analogous considerations for the ( ) case allows us to identify the additional observables requiredto resolve the phase ambiguity when this restriction isnot met. They are indicated also in Table. III for the case a ) + 1( b ) + 1( c ). V. PHASE FIXING FOR THE ( ) AND 4CASES
It is straightforward to show that no sets of observ-ables with the ( ) or ( ) cases can resolve the phaseambiguity.Consider the ( ) case of three observables from,say, group a = BT and one from group b = BR .Then, from Eqs.(3,4), we have the following possiblesets of four observables: [( O anν , O an ′ ν ′ ) , ( O an ′′ ν ′′ , O bn ′′′ ν ′′′ )],with [ n, n ′ , n ′′ , n ′′′ = 1 , ν, ν ′ , ν ′′ , ν ′′′ = ± and ( n, ν ) =( n ′ , ν ′ ) and ( n ′′ , ν ′′ ) = ( n, ν ) , ( n ′ , ν ′ )]. For concreteness,consider the set [( O a , O a − ) , ( O a , O b )]. The pair ofobservables ( O a , O a − ) determines sin φ and sin φ uniquely, yielding the two-fold ambiguity for each of therelative phases φ and φ as given by Eq.(8). This,then, leads to the following four possible expressions forthe observable O a : O a = B cos φ + B cos φ = B cos α + B cos α ,B cos α − B cos α , − ( B cos α + B cos α ) , − ( B cos α − B cos α ) , (96)where Eq.(57) has been used. Since these expressionsare all linearly independent, only one of them will besatisfied - except perhaps for a few special cases - once O a is measured. That is, O a should in principle beable to resolve the discrete ambiguities of φ and φ .The remaining observable O b , O b = B sin φ + B sin φ , (97)however, can determine neither φ nor φ , one of whichis needed, in addition to φ and φ , for resolving thephase ambiguity of the transversity amplitude up to anarbitrary phase. The analogous reasoning applies to allother sets of four observables in the ( ) case . Thereader may convince himself/herself that none of thesesets are capable of resolving the phase ambiguity.In the case of four observables from one given group ( ) case , say, [( O a , O a − ) , ( O a , O a − )], it is clear fromEq.(3) that they determine the relative phases φ and φ uniquely, but no information about a third relativephase is available for resolving the phase ambiguity. VI. IDENTIFYING WHEN THE EQUALRELATIVE-PHASE-MAGNITUDES CONDITIONOCCURS
As we have seen in Secs. III and IV, the completenesscondition for a set of four double-spin observables to re-solve the phase ambiguity of the transversity amplitudeholds, provided the equal relative-phase-magnitudes re-lation (cf. Eq.(48)) is not met. This restriction wouldn’t3cause a significant problem if this is a rarely occurringsituation. However, we find no reason a priori to expectthat this is indeed a rare case. This forces us to verify ifthe no equal relative-phase condition is met for each kine-matics (total energy of the system and meson productionangle) where the four double-spin observables are mea-sured, for the completeness argument that only four care-fully selected double-spin observables are needed. Canwe know when the equal-magnitudes relation are real-ized? The answer to this question is yes as we show inthe following.To be concrete, consider the pair of observables of theform ( O an ± , O an ∓ ) ( n = 1 , α = ± α , these ob-servables obey the relation B (cid:0) O an ± − O an ∓ (cid:1) = ± B (cid:0) O an ± + O an ∓ (cid:1) . (98)Hence, by measuring the cross section and single-spin ob-servables (which determine B and B ) and the double-spin observables in the above equation, we will be able togauge if the equal magnitudes relation, | α | = | α | , ismet. Note that in the particular case of α = α = 0,we have O a = O a − = 0 and O a − O a = B − B B + B . (99)For the pair of observables of the form ( O a ± , O a ± ) ( ± signs are independent), from Eq.(22), when α = ± α ,we have O a ± + O a ± = ( B ∓ B ) . (100)Note that the ± sign on the right-hand-side of the aboveequation goes with the ± sign of α . In the particularcase of α = α = 0, we have O a ± = O a ± = 0 and B = ± B . (101)For the pair ( O b − , O c ), when α λλ ′ ( η ) = ± α λλ ′ ( η ), wehave, from Eqs.(59,65), A λλ ′ c O b − + η A s q D λλ ′ − (cid:0) O b − (cid:1) D λλ ′ = O c B ± B , (102)where A λλ ′ c , A s and D λλ ′ are given by Eqs.(54,59). Inthe particular case of α λλ ′ ( η ) = α λλ ′ ( η ) = 0, we have O c = 0 and | O b − | = | A s | , (103)where Eq.(56) has been also used. Equations (102,103)hold for all the pairs of observables of the form( O bn ± , O cn ± ) ( n = 1 , ± signs are independent) withthe appropriate signs of B and B in A λλ ′ c , A s and D λλ ′ , and also of B and B . Analogously, for the pair ( O b − , O c ), fromEqs.(75,78), we obtain when α λλ ′ ( η ) = ± α λλ ′ ( η ), − A λλ ′ s O b − + η A c q D λλ ′ − (cid:0) O b − (cid:1) D λλ ′ = O c B ± B , (104)where A λλ ′ s , A c and D λλ ′ are given by Eqs.(72,74). Inthe particular case of α λλ ′ ( η ) = α λλ ′ ( η ) = 0, we have O c = 0 and | O b − | = | A c | . (105)Equations (104,105) hold for all the pairs of observablesof the form ( O b ± , O c ± ) ( ± signs are independent) withthe appropriate signs of B and B in A λλ ′ c , A s and D λλ ′ , and also of B and B .Equations (98,99,100,101,102,103,104,105) enable usto gauge when the equal relative phase magnitudes rela-tion is met for any of the sets of two pairs of observablesas listed in Tables. I, II and III, which - otherwise - canresolve the phase ambiguity. VII. SUMMARY
By revealing and exploiting the underlying symmetriesof the relative phases of the pseudoscalar photoproduc-tion amplitude, we have provided a consistent and ex-plicit mathematical derivation of the completeness con-dition for the observables in this reaction covering all therelevant cases. In particular, we have determine all thepossible sets of four observables that resolve the phaseambiguity of the transversity amplitude up to an overallphase. The present work substantiates and corroboratesthe original findings of Ref.[3]. However, the complete-ness condition of a set of four double-spin observablesto resolve the phase ambiguity holds only if the rela-tive phases do not have equal magnitudes as specified inEq.(48). In situations where the equal-magnitudes con-dition occur, we have shown that one or two or eventhree extra chosen observables are required, dependingon the particular set of two pairs of observables consid-ered as given in Tables. I, II and III, resulting in five orsix or seven as the minimum number of chosen double-spin observables required to resolve the phase ambiguity.In the particular case of vanishing relative phases, weneed, eight chosen observables to resolve the phase ambi-guity. This results in a minimum of up to twelve chosenobservables to determine the amplitude up to an overallphase: four, to determine the magnitudes of the basicfour transversity amplitudes that comprise the full pho-toproduction amplitude and, up to eight more to resolvethe phase ambiguity depending on the particular set offour double-spin observables.To apply the argument of the completeness condition ofa set of four double-spin observables to resolve the phaseambiguity of the photoproduction amplitude, we need to4make sure that the restriction of no equal relative-phasemagnitudes, as specified in Eq.(48), is satisfied. We haveshown that it is possible to gauge whether this restrictionis satisfied or not for each kinematics where the set offour double-spin observables is measured, because, theseobservables obey the well defined relationships that areunique to the case of equal relative-phase magnitudes, asseen in Sec.VI.We also remark that quantum mechanics does not al-low us to determine the overall phase of the reactionamplitude from experiment. For this, some physics in-put is required. This fact must have a strong impacton partial-wave analysis in the context of complete ex-periments for extracting the baryon resonances since, ifthe overall phase of the amplitude is unknown, the cor-responding partial-wave amplitude is an ill defined quan-tity. The issues related to the unknown overall phasehave been discussed earlier by several authors. In partic-ular, Omelaenko [15] mentioned the overall phase prob-lem for photoproduction in the summary section of his paper on discrete ambiguities in truncated partial-waveanalysis. In the classic review paper by Bowcock andBurkhardt [16], this problem is discussed as well. Deanand Lee [17] also investigated this problem mainly forthe formalism of πN -scattering. Two recent publications[18, 19] treat the same problem, but mostly in the simplercontext of spinless particle scattering.Finally, the present type of analysis may be applied toother reaction processes where the interest in determiningthe complete experiments exist. ACKNOWLEDGMENT
The author is indebted to Frank Tabakin and HarryLee for many invaluable discussions and for the encour-agement to publish the present results. The author isalso grateful to Frank Tabakin for a careful reading ofthe manuscript. [1] I. S. Barker, A. Donnachie, and J. K. Storrow, Nucl.Phys. B , 347 (1975).[2] G. Keaton and R. Workman, Phys. Rev. C , 1434(1996).[3] W.-T. Chiang and F. Tabakin, Phys. Rev. C , 2054(1997).[4] M.J. Moravcsik, J. Math. Phys. , 1 (1985).[5] A. Sandorfi, S. Hoblit, H. Kamano, and T.-S. H. Lee, J.Phys. G , 053001 (2011).[6] Tom Vrancx, Jan Ryckebusch, Tom Van Cuyck, andPieter Vancraeyveld, Phys. Rev. C , 055205 (2013).[7] Jannes Nys, Tom Vrancx, and Jan Ryckebusch, J. Phys.G: Nucl. Part. Phys. , 034016 (2015).[8] J. Nys, J. Ryckebusch, D. G. Ireland, and D. I. Glazier,Phys. Lett. B , 260 (2016).[9] R. L. Workman, Phys. Rev. C , 035201 (2011).[10] R. L. Workman, M. W. Paris, W. J. Briscoe, L. Tiator,S. Schumann, M. Ostrick and S. S. Kamalov, Eur. Phys. J. A , 143 (2011).[11] B. Dey, M. E. McCracken, D. G. Ireland, C. A. Meyer,Phys. Rev. C , 055208 (2011).[12] Y.Wunderlich, R. Beck, and L. Tiator, Phys. Rev. C ,055203 (2014).[13] R. L. Workman, L. Tiator, Y. Wunderlich, M. D¨oring,and H. Haberzettl, Phys. Rev. C , 015206 (2017).[14] F. Krinner, D. Greenwald, D. Ryabchikov, B. Grube, andS. Paul, Phys. Rev. D , 114008 (2018).[15] A. S. Omelaenko, Sov. J. Nucl. Phys. , 406 (1981).[16] J. E. Bowcock and H. Burkhardt, Rept. Prog. Phys. ,1099 (1975).[17] N. W. Dean and P. Lee, Phys. Rev. D , 2741 (1972).[18] Y. Wunderlich, A. ˇSvarc, R.L. Workman, L. Tiator, andR. Beck, Phys. Rev. C , 065202 (2017).[19] A. ˇSvarc, Y. Wunderlich, H. Osmanovi´c, M.Hadˇzimehmedovi´c, R. Omerovi´c, J. Stahov, V. Ka-shevarov, K. Nikonov, M. Ostrick, L. Tiator, and R.Workman, Phys. Rev. C , 054611 (2018). TABLE I. Sets of two pairs of double-spin observables for case (bb) mentioned in the text. √ = do resolve. X = do not resolve. Observables indicated outsidethe parentheses are the additional ones required in case the equal-relative-phase-magnitudes condition, as given by Eq.(48), is met for the pairs of observables (inparentheses) that do resolve the phase ambiguity otherwise.( O b , O b ) , O b − ( O b , O b − ) , O b − ( O b − , O b ) , O b ( O b − , O b − ) , O b ( O c , O c ) , O c − ( O c , O c − ) , O c − ( O c − , O c ) , O c
1+ ( O c − , O c − ) , O c O z , C z ) , C x ( O z , O x ) , C x ( C x , C z ) , O z ( C x , O x ) , O z ( L x , L z ) , T z ( L x , T x ) , T z ( T z , L z ) , L x ( T z , T x ) , L x ( O a , O a ) , O a − √ √ √ √ X √ √ X( G, E ) , F ( O a , O a − ) , O a − √ X X √ √ √ √ √ ( G, H ) , F ( O a − , O a ) , O a √ X X √ √ √ √ √ ( F, E ) , G ( O a − , O a − ) , O a √ √ √ √ X √ √ X( F, H ) , F ( O c , O c ) , O c − √ √ √ √ ( L x , L z ) , T z ( O c , O c − ) , O c − X √ √ X( L x , T x ) , T z ( O c − , O c ) , O c X √ √ X( T z , L z ) , L x ( O c − , O c − ) , O c √ √ √ √ ( T z , T x ) , L x TABLE II. Sets of two pairs of double-spin observables for case (ab) mentioned in the text. √ = do resolve. X = do not resolve. Observables indicated outsidethe parentheses are the additional ones required in case the equal-relative-phase-magnitudes condition, as given by Eq.(48), is met for the pairs of observables (inparenthese) that do resolve the phase ambiguity otherwise. The additional observable required is either one of the observables indicated for each pair, except for thoseindicated with ∗∗ , which require two additional observables.( O a , O a − ) , O a ± ( O a , O a − ) , O a ± ( O b , O b − ) , O b ± ( O b , O b − ) , O b ± ( O c , O c − ) , O c ± ( O c , O c − ) , O c ± ( G, F ) , E/H ( E, H ) , G/F ( O z , C x ) , C z /O x ( C z , O x ) , O z /C x ( L x , T z ) , L z /T x ( L z , T x ) , L x /T z ( O a , O a ) , O a − √ ** √ ** X X( G, E ) , F ( O a , O a − ) , O a − X X √ √ ( G, H ) , F ( O a − , O a ) , O a X X √ √ ( F, E ) , G ( O a − , O a − ) , O a √ ** √ ** X X( F, H ) , G ( O b , O b ) , O b − √ ** √ ** X X( O z , C z ) , C x ( O b , O b − ) , O b − X X √ ** √ **( O z , O x ) , C x ( O b − , O b ) , O b X X √ ** √ **( C x , C z ) , O z ( O b − , O b − ) , O b √ ** √ ** X X( C x , O x ) , O z ( O c , O c ) , O c − X X √ ** √ **( L x , L z ) , T z ( O c , O c − ) , O c − √ √ X X( L x , T x ) , T z ( O c − , O c ) , O c √ √ X X( T z , L z ) , L x ( O c − , O c − ) , O c X X √ ** √ **( T z , T x ) , L x TABLE III. Sets of two pairs of double-spin observables for case ( a ) + 1( b ) + 1( c ) ) . Other combinations can be obtainedby appropriate permutations of the indices a, b, c . √ = do resolve. X = do not resolve. Observables indicated outside theparentheses are the additional ones required in case the equal-relative-phase-magnitudes condition, as given by Eq.(48), is metfor the pairs of observables (in parentheses) that do resolve the phase ambiguity otherwise. The additional observable requiredis either one of the observables indicated for each pair, except for those marked with ∗∗ , which require any two additionalobservables from those indicated.( O a , O a − ) , O a ± ( O a , O a ) , O a − ( O a , O a − ) , O a − ( O a − , O a ) , O a ( O a − , O c − ) , O a ( O a , O a − ) , O a ± ( G, F ) , E/H ( G, E ) , F ( G, H ) , F ( F, E ) , G ( F, H ) , G ( E, H ) , G/FO b − /O c − , ( O b , O c ) √ √ √ √ √ X C x /T z , ( O z , L x ) O b − /O c , ( O b , O c − ) √ √ √ √ √ X C x /L x , ( O z , T z ) O b − , ( O b , O c ) X √ √ √ √ √ ** C x , ( O z , L z ) O b − , ( O b , O c − ) X √ √ √ √ √ ** C x , ( O z , T x ) O b /O c − , ( O b − , O c ) √ √ √ √ √ X O z /T z , ( C x , L x ) O b /O c , ( O b − , O c − ) √ √ √ √ √ X O z /L x , ( C x , T z ) O b , ( O b − , O c ) X √ √ √ √ √ ** O z , ( C x , L z ) O b , ( O b − , O c − ) X √ √ √ √ √ ** O z , ( C x , T x ) O c − , ( O b , O c ) X √ √ √ √ √ ** T z , ( C z , L x ) O c ( O b , O c − ) X √ √ √ √ √ ** L x , ( C z , T z ) O b ± /O c ± , ( O b , O c ) ** √ ** √ √ √ √ X( C z , L z ) O b ± /O c ± , ( O b , O c − ) ** √ ** √ √ √ √ X( C z , T x ) O c − , ( O b − , O c ) X √ √ √ √ √ ** T z , ( O x , L x ) O c , ( O b − , O c − ) X √ √ √ √ √ ** L x , ( O x , T z ) O b ± /O c ± , ( O b − , O c ) ** √ ** √ √ √ √ X( O x , L z ) O b ± /O c ± , ( O b − , O c − ) ** √ ** √ √ √ √ X( O x , T xx