Mathematical aspects of phase rotation ambiguities in partial wave analyses
MMathematical aspects of phase rotationambiguities in partial wave analyses
Yannick WunderlichHelmholtz-Institut f¨ur Strahlen- und Kernphysik,Universit¨at BonnNussallee 14-16, 53115 Bonn, GermanySeptember 7, 2017
Abstract
The observables in a single-channel 2-body scattering problem remaininvariant once the amplitude is multiplied by an overall energy- and angle-dependent phase. This invariance is known as the continuum ambiguity.Also, mostly in truncated partial wave analyses (TPWAs), discrete ambigu-ities originating from complex conjugation of roots are known to occur. Inthis note, it is shown that the general continuum ambiguity mixes partialwaves and that for scalar particles, discrete ambiguities are just a subsetof continuum ambiguities with a specific phase. A numerical method isoutlined briefly, which can determine the relevant connecting phases.
We assume the well-known partial wave decomposition of the amplitude A ( W, θ )for a 2 → A ( W, θ ) = ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) A (cid:96) ( W ) P (cid:96) (cos θ ) . (1)1 a r X i v : . [ nu c l - t h ] S e p he data out of which partial waves shall be extracted are given by the differ-ential cross section, which is (ignoring phase-space factors) σ ( W, θ ) = | A ( W, θ ) | . (2)Making a complete experiment analysis [1] for this simple example, we see thatthe cross section constrains the amplitude to a circle for each energy and angle: | A ( W, θ ) | = + (cid:112) σ ( W, θ ). Thus, one energy- and angle-dependent phase is inprinciple unknown when based on data alone. The other side of the medal in thiscase is given by the fact that the amplitude itself can be rotated by an arbitraryenergy- and angle-dependent phase and the cross section does not change. Thisinvariance is known as the continuum ambiguity [2]: A ( W, θ ) → ˜ A ( W, θ ) := e i Φ( W,θ ) A ( W, θ ) . (3)Another concept known in the literature on partial wave analyses is that of so-called discrete ambiguities [2, 3, 4]. Suppose the full amplitude A ( W, θ ) can besplit into a product of a linear-factor of the angular variable, for instance cos θ ,and a remainder-amplitude ˆ A ( W, θ ) [3]: A ( W, θ ) = ˆ A ( W, θ ) (cos θ − α ) . (4)This is generally the case whenever the amplitude is a polynomial (i.e. the series(1) is truncated), but it may also be possible for infinite partial wave models.Then, it is seen quickly that the cross section (2) is invariant under complexconjugation of the root α , which causes the discrete ambiguity α −→ α ∗ . (5)Figure 1 shows a schematic illustration of the meaning of the terms continuum- vs. discrete ambiguities . In this proceeding, the purely mathematical mechanisms(3) and (5) are investigated. Of course, constraints from physics may reducethe amount of ambiguity encountered. For instance, unitarity is a very powerfulconstraint which, for elastic scatterings, leaves only one remaining non-trivial so-called Crichton -ambiguity [5]. This is believed to be true independent of anytruncation-order L of the partial wave expansion [2]. However, in energy-regimeswhere the scattering becomes inelastic, so-called islands of ambiguity are knownto exist [6].Although here we focus just on the scalar example, ambiguities have become atopic of interest in the quest for so-called complete experiments in reactions withspin, for instance photoproduction of pseudoscalar mesons [1, 7].2igure 1: Three schematic pictures are shown in order to distinguish the terms discrete- and continuum ambiguities. The grey colored box depicts in each case thehigher-dimensional parameter-space composed by the partial wave amplitudes, beit for infinite partial wave models, or for truncated ones. Left:
One-dimensional (for instance circular) arcs can be traced out by continuumambiguity transformations, both for infinite and truncated models.
Center:
Connected continua in amplitude space, containing an infinite number ofpoints with identical cross section, can be generated by use of angle-dependentrotations (3) (however, only in case the partial wave series goes to infinity). Theconnected patches are also called islands of ambiguity [2, 6].
Right:
Discrete ambiguities refer to cases where the cross section is the samefor discretely located points in amplitude space. These ambiguities are mostprominent in TPWAs [2, 4]. However, two-fold discrete ambiguities can alsoappear for infinite partial wave models, once elastic unitarity is valid [2].These figures have been published in reference [8].This proceeding is a briefer version of the more detailed publication [8]. ThearXiv-reference [9] also treats very similar issues, as does the contribution of AlfredˇSvarc to these Bled-proceedings.
We let the general transformation (3) act on A ( W, θ ) and assume a partial wavedecomposition for the original as well as the rotated amplitude A ( W, θ ) −→ ˜ A ( W, θ ) = e i Φ( W,θ ) A ( W, θ ) = e i Φ( W,θ ) ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) A (cid:96) ( W ) P (cid:96) (cos θ ) ≡ ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) ˜ A (cid:96) ( W ) P (cid:96) (cos θ ) . (6)3ut of the infinitely many possibilities to parametrize the angular dependenceof the phase-rotation, the convenient choice of a Legendre-series is employed e i Φ( W,θ ) = ∞ (cid:88) k =0 L k ( W ) P k (cos θ ) . (7)In case this form of the rotation is inserted into the partial wave projection in-tegrals of the general rotated waves ˜ A (cid:96) (cf. equation (6)), the following mixingformula emerges [10]˜ A (cid:96) ( W ) = ∞ (cid:88) k =0 L k ( W ) k + (cid:96) (cid:88) m = | k − (cid:96) | (cid:104) k, (cid:96), | m, (cid:105) A m ( W ) . (8)Here, (cid:104) j , m ; j , m | J, M (cid:105) is just a usual Glebsch-Gordan coefficient.Some more remarks should be made on the formula (8): first of all, although it’sderivation is not difficult, this author has (at least up to this point) not foundthis expression in the literature, at least in this particular form. However, mixing-phenomena have been pointed out for πN -scattering [11] and for photoproduction[12].Secondly, in can be seen quickly from the mixing formula that for angle- in dependentphases, i.e. when only the coefficient L survives in the parametrization (7) ofthe rotation-functions, partial waves do not mix. Rather, in this case each partialwave is multiplied by L ( W ) = e i Φ( W ) . However, once the phase Φ( W, θ ) car-ries even a weak angle-dependence, the expansion (7) directly becomes infiniteand thus introduces contributions to an infinite partial wave set via the mixing-formula. There may be (a lot of) cases where the series (7) converges quickly andin these instances, it is safe to truncate the infinite equation-system (8) at somepoint.It has to be stated that the mixing under very general continuum ambiguitytransformations may lead to the mis-identification of resonance quantum num-bers (reference [9] illustrates this fact on a toy-model example).
In case of a polynomial-amplitude, i.e. a truncation of the infinite series (1) atsome finite cutoff L , the amplitude decomposes into a product of linear factors [4]4 ( W, θ ) = L (cid:88) (cid:96) =0 (2 (cid:96) + 1) A (cid:96) ( W ) P (cid:96) (cos θ ) ≡ λ L (cid:89) i =1 (cos θ − α i ) , (9)with a complex normalization proportional to the highest wave λ ∝ A L ( W ). Incase of a TPWA, one energy-dependent overall phase has to be fixed. This couldbe done, for instance, by choosing λ real and positive: λ = | λ | . Sometimes it isalso customary to fix the phase of the S -wave.Gersten [4] showed that discrete ambiguities in the TPWA can occur in case sub-sets of the roots { α i } are complex conjugated. All combinatorial possibilities canbe parametrized by a set of mappings π p , the number of which rises exponentiallywith L : π p ( α i ) := (cid:40) α i , µ i ( p ) = 0 α ∗ i , µ i ( p ) = 1 , p = L (cid:88) i =1 µ i ( p ) 2 ( i − , p = 0 , . . . , (2 L − . (10)In case these maps are applied, they yield a set of 2 L polynomial-amplitudes,which all have identical cross section: A ( p ) ( W, θ ) = λ L (cid:89) i =1 (cos θ − π p [ α i ]) ≡ L (cid:88) (cid:96) =0 (2 (cid:96) + 1) A ( p ) (cid:96) ( W ) P (cid:96) (cos θ ) . (11)Since σ is invariant under the discrete Gersten-ambiguities, these transforma-tions can effectively only be rotations (because of | A | = √ σ ). More precisely,because one overall phase is fixed for all partial waves, discrete ambiguities canonly be angle-dependent rotations. The corresponding rotation-functions are justfractions of two polynomial amplitudes e iϕ p ( W,θ ) = A ( p ) ( W, θ ) A ( W, θ ) = (cos θ − π p [ α ]) . . . (cos θ − π p [ α L ])(cos θ − α ) . . . (cos θ − α L ) . (12)Therefore, discrete ambiguities mix partial waves, just as the general continuumambiguities do. Furthermore, the expression on the right-hand-side of (12) is ex-plicitly an infinite series in cos θ . Thus, one may expect an infinite tower of rotatedpartial waves ˜ A (cid:96) to be non-vanishing upon consideration of the mixing-formula(8). However, in this case of course the rotation fine-tunes exact cancellations inthe results of the mixing for all higher partial waves ˜ A (cid:96)>L .Furthermore, Gersten [4] claims (without proof) that the root-conjugations ex-haust all possibilities for discrete ambiguities of the TPWA. We have to state thatwe believe him. 5he remainder of this proceeding is used to outline a numerical method that is orthogonal to the Gersten-formalism, but which can also substantiate this claim. We use the notation x = cos θ , introduce the complex rotation function F ( W, x ) := e i Φ( W,x ) and from now on drop the explicit energy W . The proposed numericalmethod assumes a truncated full amplitude A ( x ) as a known input. Then, allpossible functions F ( x ) are scanned numerically for only those that satisfy thefollowing two conditions:(I) The complex solution-function F ( x ) has to have modulus 1 for each valueof x . | F ( x ) | = 1 , ∀ x ∈ [ − , . (13)(II) The rotated amplitude ˜ A ( x ), coming out of an amplitude A ( x ) truncated at L , has to be truncated as well, i.e.˜ A L + k = 0 , ∀ k = 1 , . . . , ∞ . (14)Formally, this scanning -procedure can be implemented by minimizing a suitablydefined functional of F ( x ): W [ F ( x )] := (cid:88) x (cid:0) Re [ F ( x )] + Im [ F ( x )] − (cid:1) + Im (cid:20) (cid:90) +1 − dxF ( x ) A ( x ) (cid:21) + (cid:88) k ≥ (cid:40) Re (cid:20) (cid:90) +1 − dxF ( x ) A ( x ) P L + k ( x ) (cid:21) + Im (cid:20) (cid:90) +1 − dxF ( x ) A ( x ) P L + k ( x ) (cid:21) (cid:41) −→ min . (15)Here, the first term ensures the unimodularity of F ( x ) (i.e. condition (I)), thesecond fixes a phase-convention on the S -wave ˜ A and the big sum over k sets allhigher partial wave of the rotated amplitude to zero.It has to be clear that for practical numerical applications, the sums over k and x have to be finite, i.e. the former is cut off and the latter is defined on a grid of x -values. Also, a general function F ( x ) is defined by an infinite amount of realdegrees of freedom, which has to be made finite as well.6his can be achieved for instance by using a finite Legendre-expansion, i.e. atruncated version of equation (7) (with possibly large cutoff L cut ), or by discretiz-ing F ( x ) on a finite grid of points { x n } ∈ [ − , A ( x ) = (cid:88) (cid:96) =0 (2 (cid:96) + 1) A (cid:96) P (cid:96) ( x ) = A + 3 A P ( x ) + 5 A P ( x )= 5 + 3(0 . . i ) x + 52 (0 .
02 + 0 . i )(3 x − . (16)This model is truncated at L = 2. Thus it has two roots ( α , α ) and 2 = 4Gersten-ambiguities. The latter are generated by four phase-rotation functions: e iϕ ( x ) = 1, e iϕ ( x ) , e iϕ ( x ) and e iϕ ( x ) . Figures 2 and 3 demonstrate the convergence-process of the functional minimization towards a particular Gersten-rotation, forvery general initial functions. The fact that always one of the four Gersten-rotations is found is in dependent of the choice of the initial function. We have seen that general continuum ambiguity transformations, as well as dis-crete Gersten-ambiguities, are in the end manifestations of the same thing: angle-dependent phase-rotations. Therefore, they both mix partial waves.The rotations belonging to the Gersten-symmetries have the following definingproperty: they are the only rotations which, if applied to an original truncatedmodel, leave the truncation order L untouched. In order to demonstrate this fact,a (possibly) new numerical method has been outlined capable of determining allcontinuum ambiguity transformations satisfying pre-defined constraints.A possible further avenue of reserach may consist off the generalization of thesefindings to reactions with spin, for instance pseudoscalar meson photoproduction.Here, the massive amount of new polarization data gathered over the last yearshave renewed interest in questions of the uniqueness of partial wave decomposi-tions. However, once one transitions to the case with spin, some open issues stillexist, as have already been discussed during the workshop.7 iϕ ( x ) - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - cos ( θ ) R e / I m [ F ( x ) ] N max = e iϕ ( x ) - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = Figure 2: The convergence of the functional minimization procedure is illustratedin these plots. For the discrete ambiguities e iϕ ( x ) and e iϕ ( x ) of the toy-model (16),two randomly drawn initial functions have been chosen from an applied ensembleof initial conditions in the search. These initial conditions then converged to thesetwo respective Gersten-rotations.Results are shown for different values of the maximal number of iterations N max of the minimizer, as indicated. Numbers range from N max = 5 up to N max = 500.In all plots, the real- and imaginary parts of the precise Gersten-ambiguity aredrawn as blue and red solid lines. The results of the functional minimizations upto N max are drawn as thick dashed lines, having the same color-coding for real-and imaginary parts. (color online)These figures have already been published in reference [8].8 iϕ ( x ) - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = e iϕ ( x ) - - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = - - - cos ( θ ) R e / I m [ F ( x ) ] N max = Figure 3: These plots are the continuation of Figure 2. The convergence of thenumerical minimization of the functional (15) is shown for the phases e iϕ ( x ) and e iϕ ( x ) , which generate discrete ambiguities of the toy-model (16).9 CKNOWLEDGMENTS
The author (again, as in 2015) wishes to thank the organizers for the hospi-tality, as well as for providing a very relaxed and friendly atmosphere during theworkshop.This particular Bled-workshop takes a special place in this author’s biography,since after 4 months of battle with a very bad knee-injury, the participation inthe workshop marked one of the first careful steps back into the world. Fur-thermore, the wonderful nature and environment of Bled itself turned out to beinstrumental on the way of healing. By making the room on the ground floor ofthe Villa Plemelj available, the organizers have provided a key to make partici-pation possible at all, and the author wishes to express deep gratitude for that.The author’s wife also wishes to thank the organizers for the possibility to stayin Bled, as well as the nice hikes she made with the other participant’s spouses.In fact, one early morning she was very brave and made a balloon ride over thelake of Bled. This author decided to include one of her aerial photographies intothe proceeding.This work was supported by the
Deutsche Forschungsgemeinschaft within theSFB/TR16. 10 eferences [1] R.L. Workman, L. Tiator, Y. Wunderlich, M. D¨oring, H. Haberzettl, Phys.Rev. C no.1, 015206 (2017).[2] J. E. Bowcock and H. Burkhardt, Rep. Prog. Phys. , 1099 (1975).[3] L.P. Kok., Ambiguities in Phase Shift Analysis ,In ∗ Delhi 1976, Conference On Few Body Dynamics ∗ , Amsterdam 1976, 43-46.[4] A. Gersten, Nucl. Phys. B , p. 537 (1969).[5] J. H. Crichton, Nuovo Cimento , A , 256 (1966).[6] D. Atkinson, L. P. Kok, M. de Roo and P. W. Johnson, Nucl. Phys. B , 109(1974).[7] Y. Wunderlich, R. Beck and L. Tiator, Phys. Rev. C , no. 5, 055203 (2014).[8] Y. Wunderlich, A. ˇSvarc, R. L. Workman, L. Tiator and R. Beck,arXiv:1708.06840 [nucl-th].[9] A. ˇSvarc, Y. Wunderlich, H. Osmanovi´c, M. Hadˇzimehmedovi´c, R. Omerovi´c,J. Stahov, V. Kashevarov, K. Nikonov, M. Ostrick, L. Tiator, and R. Work-man, arXiv:1706.03211 [nucl-th].[10] One just has to use the re-expansion P k ( x ) P (cid:96) ( x ) = k + (cid:96) (cid:88) m = | k − (cid:96) | (cid:18) k l m (cid:19) (2 m + 1) P m ( x )= k + (cid:96) (cid:88) m = | k − (cid:96) | (cid:104) k, (cid:96), | m, (cid:105) P m ( x ) . (17)For the first equality, see the reference:W. J. Thompson, Angular Momentum , John Wiley & Sons (2008).The second equality uses a well-known relation between 3 j -symbols andClebsch-Gordan coefficients.[11] N. W. Dean and P. Lee, Phys. Rev. D , 2741 (1972).[12] A. S. Omelaenko, Sov. J. Nucl. Phys.34