Using an amplitude analysis to measure the photon polarisation in $B \to K\pi\pi\gamma$ decays

A method is proposed to measure the photon polarisation parameter $\lambda_\gamma$ in $b\to s\gamma$ transitions using an amplitude analysis of $B\to K\pi\pi\gamma$ decays. Simplified models of the $K\pi\pi$ i system are used to simulate $B^+ \to K^+\pi^-\pi^+\gamma$ and $B^0\to K^+\pi^-\pi^0$ decays, validate the amplitude analysis method, and demonstrate the feasibility of a measurement of the $\lambda_\gamma$ parameter irrespective of the model parameters. Similar sensitivities to $\lambda_\gamma$ are obtained with both the charged and neutral hadronic systems. In the absence of any background and distortion due to experimental effects, the statistical uncertainty expected from an analysis of $B^+ \to K^+\pi^-\pi^+\gamma$ decays in an LHCb data set corresponding to an integrated luminosity of $9\,$fb$^{-1}$ is estimated to be $0.009$. A similar measurement using $B^0\to K^+\pi^-\pi^0$ decays in a Belle II data sample corresponding to an integrated luminosity of $5\,$ab$^{-1}$ would lead to a statistical uncertainty of $0.018$


Introduction
Rare b → sγ flavour-changing neutral-current transitions are expected to be sensitive to New Physics (NP) effects. These transitions are allowed only at loop level, and NP could arise from the exchange of a heavy particle in the electroweak penguin loop. In the Standard Model (SM), the recoil s quark that couples to a W boson is left-handed, causing the photon emitted in b → sγ transitions to be almost completely left-handed. Several theories beyond the SM predict a significant right-handed component for the photon polarisation: in the minimal supersymmetric model (MSSM), left-right squark mixing causes a chirality flip along the gluino line in the electroweak penguin loop [1], while in some grand unification models right-handed neutrinos (and the associated right-handed quark coupling) are expected to enhance the right-handed photon component [2].
Various complementary approaches have been proposed for the determination of the polarisation of the photon in b → sγ transitions. An indirect method consists in studying the time-dependent decay rate of B 0 (s) → f CP γ decays, where f CP is a particle or system of particles in a CP eigenstate [3]. An alternative approach involves the study of angular distributions of the four-body final state in B 0 → K * 0 + − decays [4]. Yet another proposed method involves exploiting the angular distributions of the photon and the proton in the final state of Λ b → Λ X (→ ph)γ decays, where Λ X is either the ground state or an excited state of the Λ hyperon and h is a kaon or a pion [5].
Information on the photon polarisation can also be obtained from B decays to three hadrons and a photon. This approach is enabled by the fact that the three final-state hadrons allow the construction of a parity-odd triple product that inverts its sign with a change in the photon chirality, and by the existence of interference between the amplitudes of the hadronic system.
In B → K res γ decays, where K res is a kaonic resonance decaying to a Kππ final state, the required interference in the Kππ system can arise from several sources. In the case of a single K res state, the helicity amplitudes must contain at least two terms with a non-vanishing relative phase. This can occur between intermediate resonance amplitudes in the decay K res → Kππ, between S and D wave amplitudes in the decay, or between two intermediate K * π states with different charges, related by isospin symmetry. * Interference can also appear in the presence of different overlapping K res states; in fact, the presence of a multitude of interfering resonances makes it very difficult to distinguish them, thus complicating the interpretation of the observed distributions.
A simplified approach to the study of the photon polarisation consists in exploiting the distribution of the polar angle of the photon with respect to the hadronic decay plane integrating over the resonance content of the Kππ system [6]. Using 3 fb −1 of pp collisions at the LHC, the LHCb collaboration determined the shape of this distribution and the up-down asymmetry between the number of events with photons emitted on either side of the plane [7]. The updown asymmetry was found to differ from zero by 5.2 standard deviations. As this asymmetry is expected to be proportional to the photon polarisation parameter λ γ , this result represents the first observation of a parity-violating nonzero photon polarisation in b → sγ transitions. The proportionality coefficient between the up-down asymmetry and λ γ depends on the resonance content of the Kππ system, and in particular on the interference pattern between the various decay modes. Without precise knowledge of these amplitudes, a measurement of the up-down asymmetry cannot be translated into a photon polarisation value.
In this paper, a method to determine the value of the photon polarisation parameter by means of an amplitude analysis of the Kππγ system is proposed. It is organised as follows: a description of the up-down asymmetry and its limitations in extracting a value for the photon polarisation parameter are detailed in Sec. 2. In Sec. 3, a general expression for the B → Kππγ decay rate in terms of a photon polarisation parameter is derived, the amplitude formalism is described, and the fit method used for the amplitude analysis is explained. In Sec. 4, results for simulated data sets with assumed models of B + → K + π − π + γ and B 0 → K + π − π 0 γ decays are presented. Statistical sensitivities on the photon polarisation parameter are quoted for these models, assuming no background and no experimental effect. Conclusions are drawn in Sec. 5.

Motivation
B + → K + π − π + γ and B 0 → K + π − π 0 γ decays can be described in terms of five independent variables: two angles (cos θ and χ) that describe the direction of the photon in the rest frame of the kaonic resonance K res , and three squared invariant masses (s 123 , s 12 , s 23 ), where the indices 1, 2 and 3 refer respectively to the final-state π + , π − and K + for the charged decay mode, and to π − , π 0 and K + for the neutral decay mode.
As illustrated in Fig. 1 for B + → K + π − π + γ decays, in the rest frame of the kaonic resonance K res , the normal to the hadronic decay plane is denoted byn = ( p 1 × p 2 )/| p 1 × p 2 |. The polar angle θ is the angle betweenn and the opposite of the photon momentum, so that cos θ = −n · p γ /| p γ | * . The angle χ is defined from The B → Kππγ differential branching fraction has the following dependence on cos θ [8]: a j (s 123 , s 12 , s 23 , χ) cos j θ .
Integrating Eq. 3 over the squared invariant masses and χ, the up-down asymmetry (A ud ) is defined as [6,8] where the terms in even powers of cos θ disappear, and the resulting asymmetry is directly proportional to λ γ with a proportionality coefficient that depends on the resonance content of the Kππ system. Figure 1 Definitions of the angular variables used to describe the Kππγ system. The indices 1, 2 and 3 refer respectively to the final-state π + , π − and K + in B + → K + π − π + γ decays, and to π − , π 0 and K + in B 0 → K + π − π 0 γ decays.
The effects of the resonant structure of the Kππ system on A ud can be illustrated using a simplified B + → K + res γ model containing only two amplitudes corresponding to the decays K 1 (1270) + → K + ρ(770) 0 → K + π − π + and K 1 (1270) + → K * (892) 0 π + → K + π − π + . Simulated samples of decays containing only right-handed photons are generated with different relative fractions (as defined in Eq. 18) and phase differences between these amplitudes, and the updown asymmetry is computed for each of them. The results in Fig. 2 show that the up-down asymmetry varies widely depending on the phase difference between the amplitudes, while it is less dependent on the relative fraction. This implies that, even in this simple model, the proportionality coefficient that relates the up-down asymmetry to the photon polarisation parameter depends strongly on the phase difference between the amplitudes, making the knowledge of this phase essential to measure the value of λ γ ; additionally, for some values of the relative phase, the proportionality coefficient is null, indicating that the measurement of the up-down asymmetry is not sensitive to λ γ in such configurations.
To overcome these difficulties and measure the photon polarisation, we propose an analysis that combines information from the angular variables and the squared invariant-mass distributions in order to characterise the interferences between decay processes and their effect on λ γ .

Photon polarisation parameter
The differential decay rate for B → Kππγ decays that proceed through a single resonance K i res can be written as [8] where s 123 is the invariant mass of the Kππ system, c i R and c i L are the right-and left-handed weak radiative decay amplitudes, T i (s 123 ) is the propagator associated to resonance K i res , and A i R and A i L are the strong decay amplitudes for K i res, R/L → Kππ. The right-and left-handed amplitudes do not interfere since the photon polarisation is an observable quantity. For a given Figure 2 Up-down asymmetry A ud for simulated samples of B + → K 1 (1270) + γ decays governed by two amplitudes only, K 1 (1270) + → K + ρ(770) 0 and K 1 (1270) + → K * (892) 0 π + , shown as a function of the generated ratio of fractions (radial coordinate, from 0.1 to 9.0) and phase difference between the two amplitudes (polar coordinate).
resonance K i res , a photon polarisation parameter λ i γ is defined in terms of the weak radiative decay amplitudes, Using an argument of parity invariance in strong interactions, detailed in Ref. [2], the weak radiative decay amplitudes associated with a resonance K i res in decays of a B + or B 0 meson can be written as [8,9] where G F is the Fermi constant, V tb and V * ts are CKM matrix elements, P i and J i are the parity and spin of the K i res resonance, g i (0) is the process-dependent hadronic form factor, C eff 7 and C 7 are the radiative Wilson coefficients, and the quantities h i R/L encode remaining contributions from the Q 1−6 and Q 8 hadronic operators (see Ref. [9] for more details). The coefficient C eff 7 includes "effective" linear contributions from the other coefficients C 1−6 in order to make it regularisation-and renormalisation-scheme independent, as discussed in Ref. [10]. Assuming that the h i R/L terms are small enough to be neglected in the expressions of c i R and c i L , the photon polarisation parameter reduces to i.e., the photon polarisation in the weak decay B + → K i+ res γ is the same for all kaonic resonances K i res and it can be expressed only as a function of Wilson coefficients. * In the SM, the value of λ γ is expected to be +1 (up to corrections of the order of m 2 s /m 2 b ) for decays of a B + or B 0 meson while it is expected to be −1 for decays of a B − orB 0 meson.

Amplitude formalism
To develop our formalism, decays of B mesons to Kππγ are assumed to proceed through a cascade of quasi-independent two-body decays, an approximation known as the isobar model [11,12]. In this study, decay topologies of the form B → R i γ, R i → R j P 1 , and R j → P 2 P 3 are considered, where R i is a Kππ intermediate state, R j is either a Kπ or ππ resonant state and P α is a final-state kaon or pion. The function used to describe B → Kππγ decays with the above topologies is therefore written as where amplitudes for various decay modes associated with right-handed (or left-handed) photons are summed coherently, The decay amplitude A k,R/L (x) corresponds to a B → Kππγ process k involving resonances R i and R j and a right-or left-handed photon, and x is the set of four-vectors associated with the final-state particles in the rest frame of the B meson. The complex coefficient f k = a k e iφ k accounts for the magnitude a k and phase φ k of decay amplitude k and is assumed to be the same for decays with right-or left-handed photons. The amplitude for a given decay mode k is a product of resonance propagators T for each intermediate two-body decay with relative angular momentum L, a normalised Blatt-Weisskopf coefficient B L B for the two-body decay of the B characterised by relative angular momentum L B and breakup momentum q B , and an overall spin factor S ij that encodes the dependence of the amplitudes on angular momenta, and Resonances are described by the product of a normalised Blatt-Weisskopf coefficient and a relativistic Breit-Wigner [13] lineshape, † * It is sufficient to assume that the ratio h i R/L /g i (0) is process independent to enable the definition of a photon polarisation parameter that does not depend on the kaonic resonance K i res . Actually, differences in g i (0) and h i R/L between the considered kaonic resonances should be small, as spectator scattering and weak annihilation corrections are expected to be similar amongst the considered resonances, leaving mainly soft gluon corrections to quark loop spectator scattering as the main source of differences. These latter corrections would need to be taken into account when translating the measurement of the photon polarisation to constraints on the Wilson coefficients.
† Alternative lineshapes, such as the Gounaris-Sakurai one [14], may be more adequate to describe certain resonances, but for simplicity only Breit-Wigner lineshapes are used in the study presented here. Table 1 Normalised Blatt-Weisskopf centrifugal barrier factors for angular momentum L. The meson radial parameter R is set to 1.5 (GeV/c) −1 following a measurement by Belle [15].
where m 0 is the nominal mass of the resonance, q denotes the breakup momentum of the outgoing particle pair in the rest frame of the resonance and Γ(s, q, L) is its energy-dependent width. The normalisation constant reduces correlations between the coupling to the decay channel and the mass and width of the resonance. The width of the resonance for a decay into two particles is parametrised as where q 0 is the value of the breakup momentum at the resonance pole s = m 0 2 , and B L (q, q 0 ) is the normalised Blatt-Weisskopf barrier factor, listed in Table 1.
The spin factors S ij,R/L are constructed using the Rarita-Schwinger (covariant tensor) formalism, following the method described in Ref. [16]. The spin factors used in this study, as well as a brief description of their computation, are given in Appendix A.

Amplitude fit
The proposed method to determine the photon polarisation parameter λ γ utilises all the degrees of freedom of the system to perform a maximum likelihood fit to the data using a probability density function (PDF) that depends explicitly on λ γ . This amplitude fit allows the direct measurement of λ γ , as well as of the relative magnitudes and phases of the different decaychain amplitudes included in the model. The PDF is computed using the function P s given in Eq. 9 as where Ω = λ γ , {a k }, {φ k } is the set of fit parameters, Φ 4 (x) is the four-body phase-space density, and ξ(x) is the efficiency, which accounts for effects related to detector acceptance, reconstruction, and event selection. The magnitude and phase of each amplitude k (a k and φ k ) are measured with respect to those of amplitude 1, for which a 1 and φ 1 are fixed to 1 and 0, respectively.
The normalisation integral of Eq. 16 is computed numerically using a large sample of simulated events, generated according to an approximate model P gen . The signal acceptance ξ(x) is inherently taken into account by applying the event selection used in data to these simulated events; the normalisation integral can then be estimated as where N sel is the total number of generated events that pass the selection criteria. Note that I gen does not depend on the parameters of the fit, and therefore does not need to be evaluated to perform the maximisation. For the studies presented here, the effect of the application of a selection is not considered, i.e., ξ(x) = 1.
The fraction of a decay mode k is defined as the ratio of the phase-space integral of the sum of right-and left-handed contributions over the phase-space integral of the function P s , Due to interferences between the decay modes, the sum of these fractions may not be equal to unity. The interference term between the decay modes k and l, where k > l, can be expressed as such that the sum of all the fractions and interference terms is equal to unity:

Sensitivity
The amplitude formalism described in Sec. 3 is implemented in a generator and fitter software framework developed for the amplitude analysis of D 0 → K + K − π + π − decays at CLEO [16,17]. The performance of the amplitude fitter is studied initially by generating and subsequently fitting simulated data sets of B → Kππγ decays using models containing two or three amplitudes. Once the methodology is validated, more realistic models of the Kππ system are used in order to obtain prospects for measurements of the photon polarisation parameter in B-physics experiments.

Proof-of-concept using simplified models
As illustrated in Fig. 2, the sensitivity to the photon polarisation parameter obtained from the up-down asymmetry depends primarily on the relative phase. The same set of simplified models of the B + → K 1 (1270) + γ channel, which include only two decay modes of the kaonic resonance (K 1 (1270) + → K + ρ(770) 0 and K 1 (1270) + → K * (892) 0 π + ) is used to test the performance of the full amplitude fit, as well as its stability and the accuracy of the obtained uncertainties. The free parameters of the fit are the photon polarisation parameter λ γ , and the modulus and phase associated with the K 1 (1270) + → K + ρ(770) 0 channel, hereafter referred to as the relative magnitude and phase, where the K 1 (1270) + → K * (892) 0 π + channel is chosen as a reference.
For several pairs of relative magnitude and phase, 10 simulated data sets of 8 000 events are generated with λ γ = +1 (close to the SM value) and fitted independently. The average uncertainty on λ γ as a function of relative fraction (as defined in Eq. 18) and phase is shown in Fig. 3, where areas of higher colour saturation indicate regions with higher sensitivity to λ γ : unlike A ud , the amplitude analysis is sensitive to λ γ for all values of relative fractions and phases, with statistical uncertainties ranging from 0.01 to 0.05.
A higher average uncertainty on λ γ is seen for models in which the fraction of one amplitude is much larger than the other, and the maximum sensitivity is obtained for a phase difference of around 3π/2 and a relative fraction of 1.5.
To evaluate the performance of the fit as a function of the photon polarisation parameter, the study is repeated for various generated values of λ γ , and the results are shown in Fig. 4. The highest sensitivities to λ γ are obtained for λ γ = ±1, with increasing uncertainties observed as the generated absolute value of λ γ decreases.
To study the fit accuracy and error estimation, 100 simulated data sets are generated and Figure 4 Uncertainty on λ γ obtained from the average of 10 amplitude fits of simulated samples of B → K 1 (1270) + γ decays governed by two amplitudes only, shown as a function of the generated λ γ value and the relative fraction (left) or phase difference (right) of the two amplitudes.
fitted for selected values of the model parameters (relative magnitude, relative phase and λ γ ).
As asymmetric errors are used in these fits, the quality of the parameter estimation is evaluated by checking that the distribution of the pull variable g is compatible with a standard normal distribution, where g is defined as: The mean values and standard deviations of the fitted parameters and the associated pull parameters can be found in Tables B.1, B.2, and B.3 of Appendix B. For all models, each fit parameter has a Gaussian distribution centered on the generated value with a pull distribution of width consistent with unity, resulting in an unbiased measurement and correct error estimation.
As a final test, we study decays of B mesons to Kππγ with a π 0 in the final state, which can have an additional source of interference from intermediate states that include a K * (892) resonance. It has been claimed that the presence of these additional interference terms results in a higher maximum possible up-down asymmetry [8], and thus that the analysis of B 0 → K + π − π 0 γ decays could be potentially more sensitive to the photon polarisation than that of B + → K + π − π + γ decays.
The effect of an additional decay amplitude (and therefore additional interference terms) is studied using a B 0 → K 1 (1270) 0 γ model with three different K 1 (1270) 0 decay channels, K 1 (1270) 0 → K + ρ(770) − , K 1 (1270) 0 → K * (892) + π − , and K 1 (1270) 0 → K * (892) 0 π 0 . Ten simulated data sets with 8 000 events each are generated for different values of the phase differences of the K 1 (1270) 0 → K + ρ(770) − and K 1 (1270) 0 → K * (892) 0 π 0 modes relative to the K 1 (1270) 0 → K * (892) + π − mode; all samples are generated with λ γ = +1, with the decay rate for all amplitudes being equal. The uncertainty on the photon polarisation parameter for all models studied, shown in Fig. 5, is within the same range as seen in the two-amplitude B + → K + π − π + γ model, showing that the amplitude analysis is not very sensitive to the number of interference terms in the Kππ system. We conclude that this amplitude analysis is sensitive to the photon polarisation parameter for all simplified models studied, for both charged and neutral decay modes.

Prospects for future measurements
In light of the results of the proof-of-concept model, the most promising measurement of the photon polarisation parameter is expected to come from B + → K + π − π + γ decays, which are the most abundantly reconstructed at LHCb and Belle II.
An estimate of the statistical sensitivity of a measurement of the photon polarisation from an amplitude analysis of B + → K + π − π + γ decays is obtained by studying the model described in Table 2, which provides a good approximation to the Kπ, ππ and Kππ invariant mass spectra observed in a data sample of 3 fb −1 collected by LHCb during Run 1 of the LHC [7,18]. A total of 100 data sets of 14 000 events each, corresponding to the LHCb signal yield of Run 1 [7], are generated with λ γ = +1. The fits of these samples yield a mean uncertainty on λ γ of 0.014. Figure 6 shows the distributions for the five variables for one of these simulated data sets along with the corresponding projections of the fit PDF. While the pull means (µ pull ) and widths (σ pull ) of the complex coefficients a k and φ k , listed in Table 3, show that the fit is unbiased and the errors are well estimated, the pull distribution associated with λ γ has a mean of 0.22 ± 0.12 and a width of 1.22 ± 0.08, indicating that the obtained uncertainty on λ γ is underestimated by about 20%.
Taking into account a corrected uncertainty of 0.017, the comparison of this result with the simplified models discussed in the previous section suggests that the model complexity does Table 2 Model used to describe the K res → K + π − π + hadronic system in the B + → K + π − π + γ decays.
The table is divided in sections according to the spin-parity J P of the K res resonance. The amplitude with the S-wave decay K 1 (1270) + → K * (892) 0 π + is chosen as a reference for the magnitudes and phases.
not have a large effect on the sensitivity to λ γ . * This fact can be used to evaluate the gain in sensitivity that could be obtained by exploiting the additional 6 fb −1 of data that have been recorded by LHCb at a pp energy of 13 TeV in Run 2, where the B production cross-section is almost twice that at the Run 1 energy of 7 − 8 TeV: assuming that a total of 70 000 signal decays are selected using the LHCb Run 1 and Run 2 data sets, the resulting corrected statistical uncertainty on the measurement of the photon polarisation parameter could reach 0.009.

B
As discussed in Sec. 4.1, B 0 → K + π − π 0 γ decays can also be used for a measurement of the photon polarisation parameter. The main difference with the B + → K + π − π + γ decays used above is that the hadronic part of the decays is a priori more complex due to an additional source of interference involving K * (892) 0 π 0 and K * (892) + π − intermediate states in the decays of the heavy kaonic resonances K res → K + π − π 0 . In order to evaluate the sensitivity of a measurement of the photon polarisation parameter using B 0 → K + π − π 0 γ decays, samples of 10 000 simulated signal events (corresponding to the number of expected B 0 → K + π − π 0 γ decays to be reconstructed by Belle II with 5 ab −1 of integrated luminosity) are used. As little is known about the hadronic system in such decays, a model of the Kππ system is obtained from the model used for the charged modes, assuming the relative magnitudes and phases of all allowed decay modes without a K * (892)π  Figure 6 Squared invariant-mass (m 2 K + π − π + , m 2 K + π − , m 2 π − π + ) and angular (cos θ and χ) distributions for a single data set of 14 000 B + → K + π − π + γ decays generated with the 14 amplitudes listed in Table 2. The red histograms represent the projections of the PDF obtained from the fit. Table 3 Pull parameters of the fit to B + → K + π − π + γ samples for all magnitudes and phases relative to the amplitude with the S-wave decay K 1 (1270) + → K * (892) 0 π + is chosen as a reference for the magnitudes and phases.

Conclusions
A new method to measure the photon polarisation parameter in B → Kππγ decays from an amplitude analysis is presented. Using simplified models of the hadronic part of the decay, it is Table 4 Model used to describe the K res → K + π − π 0 hadronic system in B 0 → K + π − π 0 γ decays.
The table is divided in sections according to the spin-parity J P of the K res resonance. The amplitude with the S-wave decay K 1 (1270) 0 → K * (892) 0 π 0 is chosen as a reference for the magnitudes and phases.
1.01 0.00 shown that the sensitivity of the photon polarisation parameter measurement does not depend strongly on the configuration or complexity of the Kππ system. The performed studies demonstrate that, in the ideal case of a background-free sample without distortions due to experimental effects, and ignoring the differences between non-factorisable hadronic parameters between the resonances in the Kππ system, this method allows the measurement of the photon polarisation with a statistical uncertainty of around 0.009 on a sample of 70 000 B + → K + π − π + γ decays corresponding to the signal statistics assumed for LHCb in Runs 1 and 2. Belle II is assumed to reconstruct about 10 000 B 0 → K + π − π 0 γ decays with a data set corresponding to an integrated luminosity of 5 ab −1 . The analysis of these data could also determine independently the photon polarisation with a statistical uncertainty of the order of 0.018, again ignoring background and experimental effects, as well as non factorisable hadronic uncertainties.
The uncertainty on the measurement of the photon polarisation parameter λ γ can be trans- Table 5 Pull parameters of the fit to B 0 → K + π − π 0 γ samples for all magnitudes and phases relative to the amplitude with the S-wave decay K 1 (1270) 0 → K * (892) 0 π 0 .
These constraints could then be compared to those set by other relevant observables such as the B 0 → K * 0 e + e − angular observables, the time-dependent decay rate of B 0 s → φγ decays, the CP asymmetry in B 0 → K * 0 γ decays or the inclusive B → X s γ branching fraction, which are discussed extensively in Ref. [9]. While the particular dependence of λ γ on the Wilson coefficients makes this observable a priori less interesting to size non-SM effects, the statistical power of the studies shown here will compensate this limitation. Additionally, since the dependence of λ γ on the Wilson coefficients is different from that of the other observables, its measurement provides complementary information; in particular, a measurement of λ γ in B → Kππγ decays could help break an ambiguity that arises in the determination of Re(C 7 ) when constraints from all radiative observables are combined assuming both Wilson coefficients to be real [9].
However, as already mentioned, theory calculations of the hadronic contributions are crucial to be able to perform this intepretation of λ γ in terms of the Wilson coefficients. Additionally, the effect of the process dependent corrections, which are disregarded at the moment, should be estimated or taken into account as nuisance parameters.
In summary, the measurement of the photon polarisation parameter through an amplitude analysis of B → Kππγ decays is a very promising method that could exploit the large data samples available at LHCb and Belle II in the near future. If the current shortcomings in the interpretation are overcome, the proposed approach will allow to set very competitive and complementary new constraints on the Wilson coefficients C eff 7 and C 7 , and will pave the way to a new array of measurements involving decays of b hadrons to three hadrons and a photon.  Figure 7 Squared invariant-mass (m 2 K + π 0 π − , m 2 K + π 0 , m 2 π 0 π − ) and angular (cos θ and χ) distributions for a single data set of 10 000 B 0 → K + π − π 0 γ decays generated with the 23 amplitudes listed in Table 4. The red histograms represent the projections of the PDF obtained from the fit.

A. Spin factors
The description of the spin structure of B → Kππγ decays is encoded in spin factors that are determined using the covariant-tensor formalism. The spin factors are constructed such that they satisfy Lorentz invariance, angular momentum conservation, and, when applicable, parity conservation. The three objects from which spin factors are built, namely polarisation vectors, spin projectors and angular momentum tensors, are presented briefly here. More details can be found in Refs. [19,20].
Massive particles of mass M , four-momentum p, spin 1 and spin projection m are represented in momentum space by a polarisation vector µ (p, m) that is orthogonal to the four-momentum p, leaving three degrees of freedom (hence three polarisation states m = −1, 0, 1). In the case of a massless particle, a particular choice of gauge is made by requiring 0 = 0, leaving only two polarisation states (m = −1, 1). Spin-2 polarisation tensors are then obtained by coupling spin-1 polarisation vectors, where 1m 1 , 1m 2 |2m are Clebsch-Gordon coefficients. By construction, the polarisation tensors satisfy the Rarita-Schwinger conditions: they are traceless, symmetric and orthogonal to p.
To project any tensor on the subspace spanned by a set of these polarisation tensors, operators called spin projectors are used. The spin-1 projection operator associated with a massive particle is defined as where g µν = diag(+1, −1, −1, −1) is the Minkowski metric. The spin-2 projection operator can then be obtained from the spin-1 projection operator as Finally, the angular momentum tensor that describes a two-particle state of pure angular momentum L is obtained from the total four-momentum p R = p 1 + p 2 and the relative fourmomentum q R = p 1 − p 2 , where p 1 and p 2 are the final-state four-momenta. The angular momentum tensor is built by projecting the rank-L tensor of relative momenta q ν 1 R q ν 2 R ...q ν L R on the spin-L subspace where the spin projection tensor reduces the number of degrees of freedom from 4 L to 2L + 1.
The spin factors considered in the present study are those that describe decays of the type B → γR i , R i → P 1 R j , R j → P 2 P 3 , where P 1 , P 2 and P 3 are the pseudoscalar particles corresponding to the final-state kaon and pions. The spin projection of the photon is denoted m γ . A right-handed photon corresponds to m γ = +1 and a left-handed photon to m γ = −1. In gen-eral, the spin factor for such a decay can be written as a sum over the allowed spin projections of the resonances R i and R j where M is the matrix element of the relevant decay. Each of the terms associated with a two-body process R → AB with a spin-orbit configuration (L AB , S AB ) is expressed as where The term ε (S R ) (R) is a polarisation tensor assigned to the decaying particle and ε * (S A ) (A) and ε * (S B ) (B) are conjugated polarisation tensors assigned to the children particles. The spin projector P (S AB ) (R) and the angular momentum tensor L (L AB ) (R) describe the spin and angular momentum coupling, respectively. All tensors are contracted to give a scalar, requiring in some cases the inclusion of the tensor ε αβγδ u δ R through where u R is the momentum of resonance R divided by its invariant mass, To obtain the spin factor associated with a given decay chain, the various two-body processes are combined and all the allowed spin projections that are not distinguishable are summed. This implies that the sum is performed on all the spin projections of the hadrons present in the decay chains, but not on the spin projections of the photon. In the end, the expression of the spin factor only depends on the spin projection of the photon and on the spin-parity of the resonances R i and R j . The spin factors obtained for the decay chains used in this paper are shown in Table A.1. Decay chain Spin factor L (1)α (B) * β (γ)L αβ (2) (T − ) B → T + γ, T + → V P 1 , V → P 2 P 3 κλµν u κ T + L (1)α (B) * β (γ)P αβλξ (2) (T + )L µ (2)ξ (T + )P νρ (1) (T + )L (1)ρ (V )

B. Additional sensitivity studies
The tables below present results of fits performed on simulated B + → K + π − π + γ samples generated with two amplitudes: K 1 (1270) + → K + ρ(770) 0 and K 1 (1270) + → K * (892) 0 π + . Table B.1 lists results of fits on simulated data sets generated with different relative magnitudes and phases. Results of fits for models generated with various values of λ γ are shown for two sets of two-amplitude samples corresponding respectively to a region of high up-down asymmetry (relative phase of −0.91) in Table B.2 and a region of low up-down asymmetry (relative phase of 0.82) in Table B.3. Table B.1 Results of unbinned maximum likelihood fits for 100 pseudo-experiments, for simplified twoamplitude models generated with λ γ = 1, for various relative magnitudes and phases. The parameters a and φ stand respectively for the relative magnitude and phase between the decay with K 1 (1270) + → K + ρ(770) 0 and the decay with K 1 (1270) + → K * (892) 0 π + . The value φ = −0.91 corresponds to a region of high up-down asymmetry while the values 0.82 and −2.32 correspond to a region of low up-down asymmetry. The magnitudes a = 1.01, 2.02 and 3.03 correspond to ratios of fractions between the two amplitudes of 0.62, 2.47 and 5.57, respectively.  Table B.2 Results of unbinned maximum likelihood fits for 100 generated data sets simulated according to the simplified two-amplitude model with relative magnitude and phase (2.02, −0.91) corresponding to a ratio of fractions of 2.47, and various values of λ γ . The parameters a and φ stand respectively for the relative magnitude and phase between the decay with K 1 (1270) + → K + ρ(770) 0 and the decay with K 1 (1270) + → K * (892) 0 π + .