1 Introduction
\counterwithin

*equationsection \NewDocumentCommand\arxiv r [: u [ u]] [arXiv:#2 [#3]] \NewDocumentCommand\arxivold r[][#1] \NewDocumentCommand\arXiv r [: u [ u]] [arXiv:#2 [#3]] \NewDocumentCommand\arXivold r[][#1]

3 April 2018

Elastic Compton Scattering from

and the Role of the Delta

Arman Margaryan111Email: am343@duke.edu Bruno Strandberg222Email: b.strandberg@nikhef.nl Harald W. Grießhammer333Email: hgrie@gwu.edu,

Judith A. McGovern444Email: judith.mcgovern@manchester.ac.uk Daniel R. Phillips555Email: phillid1@ohio.edu and Deepshikha Shukla666Email: dshukla@rockford.edu

L/EFT Group, Department of Physics,

Duke University, Box 90305, Durham, NC 27708, USA

[1ex] School of Physics and Astronomy, University of Glasgow,

Glasgow G12 8QQ, Scotland, UK

[1ex] Nikhef, Science Park 105, 1098 XG Amsterdam, Netherlands

[1ex] Institute for Nuclear Studies, Department of Physics,

The George Washington University, Washington DC 20052, USA

[1ex] School of Physics and Astronomy, The University of Manchester,

Manchester M13 9PL, UK

[1ex] Department of Physics and Astronomy and Institute of Nuclear and Particle Physics, Ohio University, Athens, OH 45701, USA

[1ex] Department of Mathematics, Computer Science and Physics, Rockford University, Rockford, IL 61108, USA

We report observables for elastic Compton scattering from in Chiral Effective Field Theory with an explicit degree of freedom (EFT) for energies between and . The amplitude is complete at NLO, , and in general converges well order by order. It includes the dominant pion-loop and two-body currents, as well as the Delta excitation in the single-nucleon amplitude. Since the cross section is two to three times that for deuterium and the spin of polarised is predominantly carried by its constituent neutron, elastic Compton scattering promises information on both the scalar and spin polarisabilities of the neutron. We study in detail the sensitivities of observables to the neutron polarisabilities: the cross section, the beam asymmetry and two double asymmetries resulting from circularly polarised photons and a longitudinally or transversely polarised target. Including the Delta enhances those asymmetries from which neutron spin polarisabilities could be extracted. We also correct previous, erroneous results at NLO, i.e. without an explicit Delta, and compare to the same observables on proton, neutron and deuterium targets. An interactive Mathematica notebook of our results is available from hgrie@gwu.edu.

Suggested Keywords: Compton scattering, Helium-3, Effective Field Theory, neutron polarisabilities, spin polarisabilities, resonance

1 Introduction

Elastic Compton scattering from a Helium-3 target has been identified as a promising means to access neutron electromagnetic polarisabilities. In refs. [1, 2, 3, 4], Shukla et al. showed that the differential cross section in the energy range of to is sensitive to the electric and magnetic dipole polarisabilities of the neutron, and , and that scattering on polarised provides good sensitivity to the neutron spin polarisabilities. These calculations were carried out at in the Chiral Effective Field Theory expansion and led to proposals at MAMI and HIS to exploit this opportunity to extract neutron polarisabilities from elastic scattering [5, 6, 7, 8, 9].

Here, we extend the calculation of refs. [1, 2, 3] by one order in the chiral counting by incorporating the leading effects of the . As discussed in an erratum published simultaneously with this paper [4], these first results were obtained from a computer code which contained mistakes, and we take the opportunity to correct some of the results here as well.

EFT is the low-energy Effective Field Theory of QCD; see, e.g., refs. [10, 11, 12] for reviews of the mesonic and one-nucleon sector, and refs. [13, 14, 15] for summaries of the few-nucleon sector. It respects the spontaneously and dynamically broken symmetry of QCD and has nucleons and pions as explicit degrees of freedom. In this work, we consider a variant which also includes the Delta [16, 17, 18, 19]. Since EFT provides a perturbative expansion of observables in a small, dimension-less parameter, one can calculate observables to a given order, which in turns provides a way to estimate the residual theoretical uncertainties from the truncation [20, 21].

More details on the EFT expansion are given in sect. 2.1; for now, we summarise our calculation as follows. We employ both the and amplitudes of refs. [22, 10, 23] and supplement those with the -pole and loop graphs [24, 25, 26, 27]. The different pieces of the photonuclear operator are organised in a perturbative expansion which is complete to NLO []. Hence, it includes not only the Thomson term for the protons, as well as magnetic moment couplings and dynamical single-nucleon effects such as virtual pion loops and the Delta excitation, but also significant contributions from the coupling of external photons to the charged pions that are exchanged between neutrons and protons (referred to hereafter as “two-nucleon/two-body currents”). The photonuclear operator is evaluated between wave functions, which are calculated from and potentials derived using the same perturbative EFT expansion [28, 29, 30].

While experiments are only in the planning stages [5, 6, 7, 8, 9], the past two decades have seen significant progress in measurements of Compton scattering on the deuteron. However, deuteron data only provides access to the isoscalar polarisabilities; a target provides complementary information on neutron polarisabilities. A naïve model of the nucleus as two protons paired with total spin zero plus a neutron suggests that observables should depend on the combinations , , and that dependence on proton spin polarisabilities should be minimal. Such a picture is somewhat over-simplified; see sect. 5. Still, relative to the deuteron, data offers the promise of stronger signals and of cross-validation of the theory used to subtract binding effects and extract nucleon polarisabilities.

Most recently, Myers et al. [31, 32] measured the differential cross section on the deuteron at energies ranging from to , doubling the world data for elastic scattering. In combination with the proton results quoted below, a fit using the EFT amplitude at the same order, , as the current work, yields

(1.1)

for the electric and magnetic dipole polarisabilities of the neutron. The canonical units of are employed. Here, the Baldin sum rule  [33, 34] was used as a constraint, and the third error listed is an estimate of the theory uncertainty. Equation (1.1) is consistent with the extraction of neutron polarisabilities from quasi-free Compton scattering on the neutron in deuterium [35, 36, 37]. Further refinement of extractions from deuterium data is expected thanks to ongoing experiments at HI[5, 7, 8, 38] and the ongoing extension of the EFT calculation to  [39]. A comprehensive review of experimental and EFT work on Compton scattering from deuterium can be found in ref. [40], which also summarises work on the proton in EFT.

Concurrently, refs. [41, 21] extracted the electric dipole polarisabilities of the proton using the data set of elastic differential cross section data compiled in ref. [40] and the EFT amplitude:

(1.2)

Here, the Baldin sum rule value was used. It is fully consistent with a more recent determination of  [42]. Compared to the neutron values, the uncertainties are much smaller, for two reasons. The proton extraction used the EFT amplitude at , i.e. at one order higher than the deuteron extraction, leading to smaller theoretical uncertainties. The main difference is however that the deuteron data set is of lesser quality than that of the proton, contains fewer points, and is restricted to a much smaller energy range. This results in statistical uncertainties which are nearly four times those of the proton. Therefore, these extractions provide only a hint that the proton and neutron polarisabilities may be different. Reducing the experimental error bars is imperative to conclude whether short-range effects lead to small proton-neutron differences; such differences would have potentially important implications for the proton-neutron mass splitting, see, e.g., ref. [21] and references therein. We argue that Compton scattering on can serve to improve the neutron values.

In addition to the scalar polarisabilities, four spin polarisabilities , , and parametrise the response of the nucleon’s spin degrees of freedom to electromagnetic fields of particular multipolarities. Intuitively interpreted, the electromagnetic field of the spin degrees causes bi-refringence in the nucleon, just as in the classical Faraday-effect [43]. The type of experiment that will be most sensitive to the spin polarisabilities involves polarised photons and targets. A comprehensive exploration of such sensitivities for the proton was recently published by some of the current authors [44]. Some data exist, and in Ref. [45] it was used to fit a subset of the spin polarisabilities. The values obtained agree well within the respective uncertainties with EFT predictions [21]. However, it was argued in Ref. [44] that much of the data is at energies that are too high for any extraction to be independent of the theoretical framework employed.

No experiments have yet probed the individual neutron spin polarisabilities, which are also predicted by EFT at the same order [21]. They can be measured with a polarised deuteron target [46, 47, 48, 49], but that has not yet been attempted. Refs. [1, 2, 3] identified polarised as a promising candidate because the dominant (%) wave function component in consists of two protons in a spin-singlet pair. The spin of the nucleus is then carried by the neutron and observables are about times more sensitive to neutron spin polarisabilities than to their proton counterparts. Indeed, we will confirm again in sects. 4.5 and 4.6 that EFT predicts only small corrections to this expectation, which are slightly different for each observable and both energy- and angle-dependent. Even if Compton scattering from a free neutron were feasible, cross sections for coherent Compton scattering from are markedly larger than those for on a (quasi-)free neutron in this energy range. This is because in , the neutron contributions interfere with the proton ones and with those from two-body currents; see sect. 5 for comparisons to proton, neutron and deuteron targets.

In this presentation, we therefore examine the influence of the Delta on the cross section and asymmetries, using the same photonuclear kernel as in the extraction of the scalar polarisabilities of the neutron from d data, eq. (1.1[40, 31, 32]. The Delta degree of freedom plays a significant role in some of the spin polarisabilities, especially in . In scattering, its inclusion markedly enhances the pertinent asymmetries [47, 48, 49]. It is therefore important to examine its impact on the asymmetries.

All observables presented are available via a Mathematica notebook from hgrie@gwu.edu. It contains results for cross sections, rates and asymmetries from to about  MeV in the lab frame, and allows the scalar and spin polarisabilities to be varied, including variations constrained by sum rules.

This article is organised as follows. In sect. 2, we summarise the amplitude that constitutes the Compton-scattering operator in our calculation and sketch details of computing matrix elements of that operator between wave functions. Section 3 then defines the different observables, before sect. 4 presents the results of our study. Section 5 offers a summary and comparisons of EFT predictions for Compton scattering on the proton, neutron, deuteron, and .

2 Compton Scattering in EFT with the Delta

2.1 Chiral Effective Field Theory

Compton scattering on nucleons and light nuclei in EFT has been reviewed in refs. [40, 41], to which we refer the reader for notation, relevant parts of the chiral Lagrangian, and full references to the literature. Here, we first briefly summarise the power counting; then we sketch the content of the amplitude for Compton scattering on , which at this order only consists of one- and two-nucleon contributions to a photonuclear kernel evaluated between nuclear wave functions.

In EFT with a dynamical Delta, Compton scattering exhibits three low-momentum scales: the pion mass , the Delta-nucleon mass splitting , and the photon energy . Each provides a small, dimensionless expansion parameter when it is measured in units of the “high” scale at which the theory breaks down because new degrees of freedom enter.

While the two ratios and have very different chiral behaviour, we follow Pascalutsa and Phillips [19] and take a common breakdown scale , which is consistent with the masses of the and as the next-lightest exchange mesons, and then exploit a numerical coincidence for the physical pion masses to define a single expansion parameter:

(2.3)

We also count . Since is not very small, order-by-order convergence must be verified carefully; see our discussion for each observable in sect. 4.

The treatment of the scale depends on the experiment [40, 41]. In this paper, we concentrate on energies for which counts like a chiral scale and pion-cloud physics therefore dominates. As reviewed below, pion-cloud effects enter at [NLO], while the Delta appears at [NLO]. Including it is simply equivalent to adding one order. We note that the version of EFT without a dynamical Delta (often called Heavy Baryon Chiral Perturbation Theory) is limited to momenta well below the resonance. We denote its expansion parameter by , and its NLO is identical to ours, ; see ref. [40, sect. 4.2.7] for more details. The counting changes at both higher photon energies, where , and also at lower energies, where is comparable to the typical nuclear binding momentum. The latter limit is more relevant to the current paper, since, as we will see in sect. 2.4.2, it sets a lower limit on the energy for which our calculations may be considered reliable. In particular, calculations consistent with the low-energy counting are required to have the correct nuclear Thomson limit; the current calculation would overshoot considerably as . We will address both extensions to higher and lower energies in future publications [50].

Our focus is therefore on energies in the overlap of the single-nucleon regime (i) of refs. [40, 41] and the few-nucleon regime (II) of ref. [40] : .

2.2 Compton Kernel: Single-Nucleon Contributions

Contributions to the single-nucleon amplitude up to or NLO in the regime are sketched in fig. 1:

Figure 1: (Colour on-line) The single-nucleon contributions in EFT up to for . The vertices are from: (no symbol), (green square), (green triangle), (red disc) [10]; the green disc of graph (f) stands for variations of the polarisabilities, cf. eq. (4.36). Permuted and crossed diagrams are not displayed.
  • LO []: The single-nucleon (proton) Thomson term.

  • NLO [] non-structure/Born terms: photon couplings to the nucleon charge beyond LO, to its magnetic moment, or to the -channel exchange of a meson. scattering is sensitive to the latter, in contradistinction to scattering, where its expectation value is zero, as the deuteron is isoscalar.

  • NLO [] structure/non-Born terms: photon couplings to the pion cloud around the nucleon, the source of the LO contributions to the polarisabilities as first reported in refs. [22, 10].

  • NLO [] structure/non-Born terms: photon couplings to the pion cloud around the Delta (d) or directly exciting the Delta (e), as calculated in refs. [51, 52]; these give NLO contributions to the polarisabilities. As detailed in ref. [40], we use a heavy-baryon propagator and a zero width, with , . The non-relativistic version of the coupling is and tuned to give the same effective strength as the relativistic value of found by fitting to the proton data set up to the Delta resonance [40]. In practice, the loops produce almost-energy-independent contributions to and for .

  • Short-distance/low-energy coefficients (LECs), which encode those contributions to the nucleon polarisabilities which stem from neither pion-cloud nor Delta effects; see eq. (4.36) for the precise form. These offsets to the polarisabilities are formally of higher order. For the proton, we fix these LECs to reproduce the values of the scalar polarisabilities. For the neutron, we exploit the freedom to vary the LECs in order to assess the sensitivity of cross sections and asymmetries to the neutron scalar polarisabilities. We refer to sect. 4.1 for the detailed procedure and values.

There is no contribution at NLO [], and only Delta contributions at NLO []. The difference between the previous  [1, 2, 3] and our new calculation is hence the inclusion of Delta effects. Covariant kinematics for the fermion propagators, a nonzero Delta width, vertex corrections, etc. are just some examples of corrections which are of higher order than the last one we retain, NLO []; they are parametrically small.

2.3 Compton Kernel: Two-Nucleon Contributions

The leading two-body currents in EFT occur at and do not involve Delta excitations. They are the two-body analogue of the loop graphs depicted in fig. 1 and thus denoted as in EFT without dynamical Delta. They were first computed in ref. [23], where full expressions can be found. We depict them in fig. 2 below.

We note that the diagrams only contribute for pairs, i.e. they all contain an isospin factor of . However, one distinction between and the deuteron is that in the tri-nucleon case both isospin zero and one pairs are present.

No corrections enter at the next order, . Boost corrections and corrections with a nucleon propagating between the pion-exchange and a photon-nucleon interaction only enter at  [23, 53, 40], and those with an intermediate Delta at  [40]. Note that the pieces of the pion-exchange currents that are suppressed and must be derived consistently with the potential [54, 55, 56] only enter at orders higher than we consider here; see discussion in sect. 2.4.2.

Figure 2: (Colour on-line) contributions to the (irreducible) amplitude (no additional contributions at ). Notation as in fig. 1. Permuted and crossed diagrams not displayed.

2.4 From the Compton Kernel to Compton Amplitudes

Here, we provide a brief description of the computation of the integrals for matrix elements of the Compton operators in the centre-of-mass frame; full details can be found in refs. [3, 1].

2.4.1 Formulae for Matrix Elements

We seek a -Compton amplitude which depends on the spin projections of the incoming and outgoing nucleus and on the helicities of the incident and outgoing photon. Using permutations and symmetries, this amplitude can be written as

(2.4)

where nucleons are numbered and is an anti-symmetrised state of . Since we are concerned with the nucleus, we also take this state to have isospin quantum numbers . The operator with represents the one-body amplitude of sect. 2.2, where external photons of incoming (outgoing) helicity () interact with nucleon “”. represents the corresponding two-body current of sect. 2.3, where the interaction is with the nucleon pair “”. Thus in eq. (2.4), nucleon “” serves as a spectator to the scattering process. Three-nucleon currents (i.e. contributions of instantaneous interactions between three nucleons and at least one photon) do not enter before chiral order .

Figure 3: (Colour on-line) Kinematics of the calculation in the centre-of-mass frame. The vectors () and () are the three-momenta of the incoming and outgoing photon (). The nucleon loop momenta , and primed versions are defined and numbered as in the text.

We use the approach of Kotlyar et al. [57] to calculate matrix elements between three-nucleon basis states. The loop momenta and are the Jacobi momenta of the pair “” and the spectator nucleon “” respectively; see fig. 3 for kinematics. By introducing a complete set of states, we can write the matrix element as

(2.5)

where accounts for both one- and two-nucleon currents. Note that the helicity dependence is entirely carried by this operator. The total angular momentum of the nucleus is a result of coupling between and , the total angular momentum of the “” subsystem, and the spectator nucleon “”, respectively. The orbital angular momentum and the spin of “” combine to give . Similarly, and combine to give . Hence, in eq. (2.5), we defined

(2.6)

and

(2.7)

where the component of the wave function is parametrised by the quantum numbers . Similarly, the isospin quantum numbers are , where the isospin of the two-nucleon subsystem is (projection ) and combines with the isospin of the spectator nucleon to give the total isospin and its projection of the nucleus. The Pauli principle guarantees that is odd.

The function can be used to perform the integral over , and so eq. (2.5) can be recast as

(2.8)

In this expression, the integral

(2.9)

is computed numerically, as is

(2.10)

It turns out that a rather small number of partial waves is sufficient to achieve convergence. We test this by comparing to results with one more unit of . The slowest convergence is at the extremes of energies and momentum transfer (, ). When one includes all partial waves with , the one-nucleon matrix elements are converged to within there, and better at lower energies and less-backward angles. The large and medium-sized two-nucleon matrix elements are converged to better than for , and better than that for lower and . Higher numerical accuracy is only limited by computational cost: two-nucleon runs with take about times longer. Radial and angular integrations are converged at the level of a few per-mille. With these parameters, at , the cross section is numerically converged at about or  nb/sr. At lower energies and smaller angles (and hence smaller momentum transfers), convergence is substantially better. Results for the other observables are similar. Increased numerical accuracy is not really useful here, since the dominant uncertainty comes instead from the truncation of the EFT expansion at , translating roughly into a truncation error of of the LO result (see sect. 2.4.3).

2.4.2 Choice of Wave Functions

Following Weinberg’s “hybrid approach” [28], we finally convolute the EFT photonuclear kernels with wave functions which are obtained from three choices for the nuclear interaction: the chiral Idaho NLO interaction for the system at cutoff  [58] with the EFT interaction of variants “b” (our “standard”) and “a” as described in refs. [59, 60], and the AV18 model interaction [61], supplemented by the Urbana-IX interaction (3NI) [62]. (Unlike refs. [1, 2, 3], we do not consider wave functions found without interactions.) All choices capture the correct long-distance physics of one-pion exchange, give the right He binding energy and reproduce the scattering data to a degree that is superior to the accuracy aimed for in this article. They also all reproduce the experimental value of the triton and binding energies. The two EFT variants are parametrised differently and lead to different spectra in other light nuclei. All wave functions are fully anti-symmetrised and obtained from Faddeev calculations in momentum space [63, 64].

The chiral wave functions claim a higher accuracy than that of our Compton kernels. For the purposes of this article, it is not necessary to enter the ongoing debate about correct implementations of the chiral power counting or the range of cutoff variations, etc.; see ref. [65] for a concise summary. Similarly, even though the Compton Ward identities are violated because the one-pion-exchange potentials are regulated, any inconsistencies between currents, wave functions and nuclear potentials will be compensated by operators which enter at higher orders in EFT than the last order we fully retain, or NLO. We therefore take the difference between results with the three wave functions as indicative of the present residual theoretical uncertainties. Since the “chiral” wave functions match the chiral kernel better than the phenomenological wave function, we take the former as the preferred ones and consider differences between AV18+UIX and Idaho NLO results as indicative of “worst-case” discrepancies. For sensitivity studies, we do not see these discrepancies affecting our conclusions, but it is undoubtedly true that better extractions of polarisabilities from data will need a reduced wave function spread.

In particular, we expect that including terms in the amplitude which restore the Thomson limit should significantly reduces the wave function and interaction dependence even at nonzero energies. For the deuteron, this was seen at energies as high as  [26, 27, 40, 48, 49]. It is also likely to decrease the cross section at the low-energy end of our region of interest. As detailed in refs. [23, 53] and [40, sect. 5.2], the coherent-propagation process necessary to restore the Thomson limit becomes important for photon energies lower than the inverse target size. For , that scale is  [66]. Refs. [26, 27, 40] discuss in detail how the power counting for low energies, , leads to the restoration of the Thomson limit by inclusion of coherent propagation of the system in the intermediate state between absorption and emission of photons.

However, the present formulation of Compton scattering is not applicable in the Thomson-limit region, since it organises contributions under the assumption . If used for , where it does not apply, it would not yield the Thomson limit for but would be several times too large. Indeed, at the energies we study here, , the power counting predicts that incoherent propagation of the intermediate three-nucleon system dominates. This is supported by the deuteron case, where including the effects that restore the nuclear Thomson limit leads to a reduction of at MeV [26, 27]. For , it is plausible that the corrections by coherent-nuclear effects may suppress the cross section at the low end of our energy range somewhat more: the mismatch between the Thomson-limit and the amplitude in our calculation is larger than for the deuteron, and has a larger binding energy, so coherent propagation of the three-nucleon system may be important up to higher energies than in the two-nucleon case.s While work in this direction is in progress [50], the present approach suffices for reasonable rate estimates at .

2.4.3 A Note on Estimates of Theory Uncertainties

Since the Compton amplitudes are complete at NLO [], they carry an a-priori uncertainty estimate of roughly of the LO result, or twice that for observables since they involve amplitude-squares. This translates to for generic cross sections and beam asymmetries because they are nonzero at LO. (At lower energies, the restoration of the Thomson limit may lead to an additional reduction of the cross section, as discussed above.) The first nonzero contributions to the double asymmetries enter at NLO, so their a-priori accuracy is estimated as .

Here, we do not explore convergence with a statistically rigorous interpretation. We nonetheless briefly mention that two post-facto criteria (order-by-order convergence and residual wave function dependence) shown in sect. 4 are roughly commensurate with these estimates. An exception may be at the highest energies , where the convergence pattern discussed in sect. 4 indicates that NLO corrections might amount to changes by roughly of the magnitude of the cross section.

Reassuringly, we see that the sensitivities of observables to variations of polarisabilities are typically much less affected by convergence issues than are their overall magnitudes. We therefore judge that our sensitivity investigations are sufficiently reliable to be useful for current planning of experiments We reiterate that our goal here is an exploratory study of magnitudes and sensitivities of observables to the nucleon polarisabilities. Once data are available, a polarisability extraction will of course need to address residual theoretical uncertainties with more diligence, as was already done for the proton and deuteron in refs. [40, 41, 31, 32, 21].

3 A Catalogue of Compton-Scattering Observables

3.1 Observables for Polarised Cross Sections

This presentation follows reviews on polarised scattering by Arenhövel and Sanzone [67, 68], by Paetz [69], and the presentation of Babusci et al. [70] which addresses Compton scattering from a spin- target.

We start by summarising the kinematics and coordinate system; see fig. 4. Unless specified otherwise, we work in the laboratory frame. The incoming photon momentum, , defines the -axis. The scattered photon momentum, , lies in the -plane. The energies of the two photons are and , and the scattering angle is the angle between the two momenta. The -axis is chosen to form a right-handed triplet with and , so that finally

(3.11)

The linear-polarisation angle of the photon is the angle from the scattering plane to the linear photon polarisation plane111This definition varies from that of [67], whose angle is counted from the polarisation plane to the normal of the scattering plane, i.e. ., i.e. . The polarisation vector is , where is the degree of (vector-)polarisation and its direction is with angle from the -axis to and angle from the -axis to the projection of onto the -plane.

Figure 4: (Colour on-line) Kinematics of Compton scattering.

The cross section for Compton scattering of a polarised photon beam with density matrix from a polarised target of spin with density matrix , without detection of the final state polarisations, is found from the Compton tensor of eq. (2.4) via

(3.12)

where the trace is taken over the polarisation states and is the frame-dependent flux factor. In the lab frame [40, sect. 2.3]:

(3.13)

Two popular bases exist for each density matrix. In the helicity basis, the photon polarisation is described by222We correct a notational inconsistency for the bra-ket notation in ref. [48] which did not have any influence on the final result.

(3.14)

Here, positive/negative helicities are defined by333This corrects an inconsequential misprint in ref. [48]. . is the degree of right-circular polarisation, i.e. the difference between right- and left-circular polarisation, with describing a fully right-/left-circularly polarised photon. The degree of linear polarisation is .

In the Cartesian basis, the Stokes parameters parametrise the and components of the density matrix of the incident photon. With :

(3.15)

The degree of right-circular polarisation is then , and that of linear polarisation is , with angle and , so that . The combination , describes a beam which is linearly polarised in/perpendicular to the scattering plane, and with a beam which is linearly polarised at angle relative to the scattering plane.

The multipole-decomposition of the density matrix of a polarised spin- target is [68]:

(3.16)

where and . The conventions for -symbols and reduced Wigner- matrices are those of Rose [71], also listed in the Review of Particle Physics [72].

Another variant for spin- particles in the same basis uses Pauli spin matrices :

(3.17)

These variants also lead to different parametrisations of the differential cross section with definite beam and target polarisations (and no detection of the final-state polarisations), after parity invariance has been taken into account. In our analysis, both variants will be used side-by-side:

The first one, eq. (3.1), is based on the general multipole decomposition and helicity basis, and was adapted to deuteron Compton scattering in refs. [48, 49]. Its independent observables carry the target multipolarity as subscript and the beam polarisation as superscript, and naturally extend to arbitrary-spin targets.

The second one, eq. (3.1), by Babusci et al. [70], uses a Cartesian basis and the components444Babusci et al. denote them by  [70]. of the polarisation vector . It uses the indices of the Stokes parameter and the target polarisation direction as the labels of the asymmetries . This form is more convenient to translate rate-difference experiments since typically only one of the parameters is nonzero.

In either case, the cross section for Compton processes on spin- targets without detecting final-state polarisations is fully parametrised by linearly independent functions listed below. Here, is shorthand for ; superscripts refer to photon polarisations (“” for polarisation in the scattering plane, “” for perpendicular to it); subscripts to target polarisations; and the absence of either means unpolarised. The observables are:

  • differential cross section of unpolarised photons on an unpolarised target.

  • beam asymmetry of a linearly polarised beam on an unpolarised target:

    (3.20)
  • vector target asymmetry for a target polarised out of the scattering plane along the direction and an unpolarised beam:

    (3.21)
  • double asymmetries of right-/left-circularly-polarised photons on a target polarised along the or directions:

    (3.22)
  • double asymmetries of linearly-polarised photons on a vector-polarised target:

    (3.23)

The decompositions of eqs. (3.1) and (3.1) hold in any frame, but the functions are frame-dependent.

The differences of the rates, , for the different orientations associated with each asymmetry are important to facilitate runtime estimates. These are the numerators in eqs. (3.20) to (3.23) and can most conveniently be expressed in the Babusci basis:

(3.24)

with for and for all other asymmetries [44].

Finally, we provide the translations to the notations used by Hildebrandt et al. [26, 73] and Shukla et al. [1, 2, 3]. The asymmetries are

(3.25)

and the corresponding rates are

(3.26)

3.2 Translating Helicity Amplitudes into Observables

The Compton matrix elements of sect. 2.4.1 are provided in the helicity basis (dependencies on and other parameters are dropped for brevity in this section). For ease of presentation, we abbreviate the sum over all polarisations of the squared amplitude:

(3.27)

Note that we also suppress the indices and summations for straightforward final-state-polarisation sums, as indicated in the second half of Eq. (3.27).

By inserting the density matrices of eqs. (3.14) and (3.16) into eq. (3.12), one obtains the cross section in terms of the amplitudes, as a function of the photon polarisations and with polarisation angle and polarisation with orientation . The functional dependence of the result on these parameters is easily matched to the parametrisation in eq. (3.1). For the unpolarised part, the result is:

(3.28)

The factor is familiar from averaging over initial spins and helicities. The asymmetries555We correct a factor of in and for in the corresponding spin- results of ref. [48]; see erratum [49].

(3.29)