69. Review of Multibody Charm Analyses

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69. Review of Multibody Charm Analyses
Revised August 2023 by J. Rademacker (Bristol U.); written with D.M. Asner (BNL)

69.1 Overview
The study of multibody charm decays is a vibrant field, with vast and fast-increasing datasets,
new developments in theory and experimental technique, and implications reaching beyond charm.
This review is structured as follows: Sec. 69.2 summarizes key aspects of amplitude models that
describe multibody charm decays, their application to data, and theory progress; Sec. 69.3 reviews
the special role of charm threshold data in model-independent approaches; and Sec. 69.4 describes
applications of multi-body charm analyses, focusing on Charge-Parity (CP) violation, charm mixing
and the role of threshold data in this context. In Sec. 69.5, we conclude.

69.2 Kinematics & Models
The differential decay rate to a point s = (s1 , . . . , sn ) in n dimensional phase space can be
expressed as∂nφdΓ = |M(s)|2dn s(69.1)∂(s1 . . . sn )where |∂ n φ/∂(s1 . . . sn )| represents the density of states at s, and M the matrix element for the
decay at that point in phase space, which is 2, 5, 8, . . . dimensional for D decays to 3, 4, 5, . . .
spinless particles. Additional parameters are required to fully describe decays involving particles
with non-zero spin in the initial or final state.
For the important case of D decays to three pseudoscalars, the decay kinematics can be represented in a two-dimensional Dalitz plot [1]. This is usually parameterized in terms of s12 ≡ (p1 +p2 )2and s23 ≡ (p2 + p3 )2 , where p1 , p2 , and p3 are the four-momenta of the final-state particles. In
terms of these variables, phase-space density is constant across the kinematically allowed region,
so that any structure seen in the Dalitz plot is a direct consequence of the dynamics encoded in|M|2 . Note that decays to four or more particles cannot be unambiguously described in termsof analogously-defined variables sij and sijk , which are all parity-even. Parity-odd kinematics areabsent in three-body decays because the final-state momenta are confined to a plane; the use ofparity-odd observables in four body decays is discussed in Sec. 69.4.4.
In the widely-used isobar approach, the matrix element M is modeled as a sum of interfering
decay amplitudes, each proceeding through resonant two-body decays [2–6]. See [2, 7, 8] for areview of resonance phenomenology. In most analyses, each resonance is described by a relativistic
Breit-Wigner [9] or Flatté [10] lineshape, and the model includes a non-resonant term with aconstant phase and magnitude. This approach has well-known theoretical limitations, such as theviolation of unitarity and analyticity, which can break the relationship between magnitude andphase across phase space. This motivates the use of more sophisticated descriptions, especially forbroad, overlapping resonances (frequently found in S-wave components) where these limitations
are particularly problematic. In charm analyses, these approaches have included the K-matrix [11]which respects two-body unitarity; the use of LASS scattering data [12]; dispersive methods [13–16];
multi-meson models using chiral Lagrangians [17–20]; and methods based on chiral unitarity [21–24]; QCD factorization [25–28]; and quasi model-independent parameterizations that use genericlineshapes, with minimal theory input and many free parameters, for a subset of resonances [29–37].
An important example for a multibody charm decay is D0 → KS0 π + π − , which is a key channel
in CP violation and charm-mixing analyses. The latest analysis, published jointly by BaBar andBelle [38], describes the Dalitz plot with 1.1M D0 → KS0 π + π − events with 18 resonant components,
including four doubly Cabibbo suppressed ones. The model uses a K-matrix description for the
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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69. Review of Multibody Charm Analyses

ππ S-wave based on [39] and input from LASS scattering data [12, 40] for the Kπ S-wave, withno need to add a non-resonant component to describe the data. These data were re-analyzed
by [25] in a QCD factorization framework, using line-shape parameterizations for the S [41, 42]
and P wave [14] contributions that preserve two-body unitarity and analyticity. The analyses givecompatible results for the components they share.
The field of charm amplitude analyses remains very active. Publications since the last update of this review two years ago include Dalitz plot analyses of D0 → KL0 π + π − , D0 → KS0 π + π − ,Ds+ → KS0 K + π 0 , Ds+ → KS0 KS0 π + , Ds+ → K + π − π + , Ds+ → π + π 0 η 0 , and Ds+ → π + π 0 π 0 by BESIII[43–48]; Ds+ → π + π − π + by LHCb [36] and BESIII [35]; and D+ → π + π − π + by LHCb [37]. LHCb
studied the amplitude structure of Λc → pK − π + [49] and BESIII that of Λc → Λπ + π 0 [50]. BESIII
continued its series of four-body amplitude analyses with D+ → KS0 π + π 0 π 0 and Ds+ → K + π − π + π 0[51,52], and performed a five-body amplitude analysis of Ds+ → K + K − π + π + π − [53]. These analyses provide valuable insight into hadronic dynamics and have wider impact beyond that. The amplitude structure of Λc → pK − π + for example is found to have a high sensitivity to the Λc polarization,
which can prove valuable in polarization measurements of Λb baryons decaying to Λc [49, 54]. The
importance of charm amplitude analyses for CP violation and charm mixing is further discussed
below. Comparing the D0 → KL0 π + π − amplitude structure with that of D0 → KS0 π + π − reveals
U-spin breaking effects [43]. The Ds+ and D+ → π + π − π + analyses mentioned above, as well an
earlier one by BaBar [31], analyse the π + π − S-wave using a quasi-model-independent approach.
They observe substantial differences between the π + π + S-wave components observed in the two decays, and between the associated phase shift seen in either compared to that observed in scattering
data [55–57]. These observations suggest that broad scalar meson resonances in these decays might
be generated from meson-meson rescattering [37]; more broadly, they emphasize the importance of
final state interactions in multibody decays.
Noteworthy is the increasing sophistication of recent amplitude analyses, most of which go
substantially beyond the isobar model with Breit Wigner and Flatté lineshapes. However, with
the notable exceptions of [15, 16, 23, 24, 58, 59], they remain within the isobar framework which
describes the decay as a series of two-body processes, ignores long-range hadronic effects such as
re-scattering, and does not respect three (or four)-body unitarity and analyticity.
Several groups work on improved models. Dispersive techniques, which respect three-bodyunitarity and analyticity, have been applied to regions of the D+ → K − π + π + and D+ → KS0 π 0 π +Dalitz plots below the η 0 K threshold [15, 16], where they provide a good description of the datawith fewer fit parameters than the isobar approach. Reference [58] uses a unitary coupled channel
approach to describe D+ → K − π + π + , which has no restrictions on the kinematic range, but
requires additional parameters to describe the Dalitz plot above the η 0 K threshold. Using aneffective chiral Lagrangian, the authors of reference [19] provide a description of the annihilation
contribution to the decay amplitude which respects three-body unitarity. The approach provides
a good description of LHCb’s D+ → K + K − K + data, with fewer parameters than an equivalentisobar model [59]. The same channel has more recently been re-analyzed by the authors of [23],who argue that the internal emission diagram should dominate and use a chiral unitarity-based
approach to achieve a reasonable description of the data with two free parameters.
Limitations in the theoretical description of interfering resonances are the leading source ofsystematic uncertainty in many analyses. In some cases, the model uncertainty can be removed
through model-independent methods, often relying on input from the charm threshold, as discussed
in Sec. 69.3. The authors of [60] expand the scope and applicability of the quasi model-independent
approach in amplitude fits. At the same time, increasingly sophisticated models are being developed, and applied to data. The authors of [20] and [61] provide valuable practical frameworks forsophisticated amplitude analyses.
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69.3 Model Independent Methods and the Charm Threshold
Precision measurements of the CP-violation phase γ/φ3 using B − → DK − , D → KS0 π + π −and related decay modes, as well as precision measurements of charm-mixing parameters in decay
modes such as D0 → KS0 π + π − , require input on the relative decay amplitudes of D0 and D0 . In
the above examples, KS0 π + π − can be replaced with other final state accessible to both D0 andD0 [62–70]. While the magnitudes of the decay amplitudes can be measured precisely using thevast charm samples at the B factories and LHCb, obtaining the relative phases requires either
amplitude models with reliable phase motion, or model-independent approaches. More details on
the measurements of γ/φ3 and charm mixing can be found in sections 69.4.1 and 69.4.2.
Model-independent measurements of relative phases between D0 and D0 decay amplitudes rely
on interference effects in the decays of well-defined coherent superpositions of D0 and D0 . These
are accessible at the charm threshold [65, 68, 70–75], where CLEO–c and BESIII operate. There,
quantum-correlated D-meson pairs are produced, which have opposite CP and flavour content.
For example if one (the tag) is identified as a CP eigenstate through its decay (e.g. to K + K −or KS π 0 ) the other D mesonstate with the opposite CP eigenvalue. The decay
 is an eigenstate

1
0
0
√
rate of a CP eigenstate 2 D ± D is proportional to |AD |2 + |AD |2 ± 2|AD ||AD | cos(δD ), where
AD and AD are the D0 and D0 decay amplitudes, and δD is their sought-after phase difference.
Other quantum correlations give access to sin(δD ). For multibody decays, AD , AD and δD varyacross phase space. Model-independent phase information is measured either integrated over theentire phase space of the decay, or in sub-regions/bins. The results can be expressed in terms of
one complex parameter Z = Re−iδ = c + is per pair of CP-conjugate bins; c and s are the averageof cos(δD ) and sin(δD ) over that region of phase space. Amplitude models can be used to optimize
the binning for sensitivity to γ/φ3 , without introducing a model-dependent bias in the result [76];novel unbinned approaches have the potential to further increase the precision on γ/φ3 in future
analyses [77–79].
0 π+π−,
CLEO–c data have been analyzed to provide binned Z for the self-conjugate decays D0 → KS,L
0
+
−
+
−
+
−
0
−
+
0
0
0
0
KS,L K K , π π π π , and KS π π π and phase space-integrated values for D , D → KS K + π − ,K + π − π 0 and K + π − π + π + [80–85]. BESIII significantly improved the precision for the KS0 π + π − ,KS0 K + K − , K + π − π 0 and K + π − π + π − final states [86–89]. CLEO–c and BESIII K + π − π + π − data
were also analysed in four bins of five-dimensional phase space [89,90] with a binning based on [33].
For self-conjugate decays such as D0 → π + π − π 0 , one can define the CP-even fraction F+ whichis +1 for a CP-even eigenstate and 0 for a CP-odd one [73]. F+ is related to Re(Z) = c, definedfor a single CP-conjugate bin pair. For the purpose of γ/φ3 measurements, this information allows
the decay to be treated equivalently to a two-body decay, where γ/φ3 is determined from relative
decay rates in multiple decay modes, without the need to resolve the phase-space distribution. In
the mathematical expressions for the decay rates, a factor (2F+ − 1) multiplies the γ/φ3 -sensitive
term, leading to best sensitivity for F+ = 1 or F+ = 0. D0 → π + π − π 0 is found to be, with F+ =(97.3 ± 1.7)%, compatible with being completely CP-even (CLEO–c data); a recent BESIII analysisof D0 → KL,S π + π − finds F+ = (35.3 ± 0.6 ± 1.4)% and (55.6 ± 0.6 ± 1.2)%, respectively; D0 →π + π − π + π − (BESIII, CLEO–c data), D0 → K + K − π + π − (BESIII), D0 → K + K − π 0 (CLEO–cdata), are all predominantly CP even with F+ ranging from 73% to 77%; D0 → KS0 π + π − π 0(BESIII, CLEO–c data), is predominantly CP odd with F+ = (23.5 ± 10)% [43, 74, 82, 83, 91–93].
In cases where results from both BESIII and CLEO–c data exist, BESIII’s more precise results arequoted.
Comparing measured parameters like F+ and Z with those predicted from amplitude models
provides a test of the models’ phase motion. Model-predicted values for F+ in D0 → KL,S π + π −agree, with (35.3 ± 0.6 ± 1.4)% and (55.6 ± 0.6 ± 1.2)% [38, 43] well with the measured values given
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above; as does the measured F+ for D0 → π + π − π + π − , F+ = 0.769 ± 0.023 [92], with 0.729 ± 0.020
calculated from [32]; and the coherence factor |Z| = R measured in D0 → K ± π ∓ π ± π ∓ , 0.44+0.10−0.09[85, 94], with 0.459 ± 0.025 predicted by [33]. The latest bin-wise comparisons between modeland data for D0 → KL,S π + π − , K ± π ∓ π ± π ∓ , K ± π ∓ π 0 , KS0 K + K − , and π + π − π + π − can be foundin [43,82,86–89,89] and generally show good agreement. These results are a welcome surprise given
the preceding discussion on the theoretical shortcomings of amplitude models.

69.4 Applications of multibody charm analyses for CP violation and mixing
Amplitude analyses provide sensitivity to both relative magnitudes and phases of the interfering
decay amplitudes. It is especially the sensitivity to phases that makes amplitude analyses such auniquely powerful tool for studying a wide range of phenomena. Here we concentrate on their criticalrole in CP violation and mixing measurements. Properties of light-meson resonances determinedin D amplitude analyses are reported in the light-unflavored-meson section of this Review.
The closely related topics of multibody charm decays in measurements of CP violation in beauty
decays to charm and charm mixing will be discussed in turn, followed by a review of searches fortime-integrated CP violation in multibody charm decays.
69.4.1 CP violation in decays of Beauty to Charm
Neutral D mesons originating from B − → DK − (here denoted as DB − ) are a superposition of0
D and D0 with a relative phase that depends on the CKM unitarity triangle parameter γ/φ3 ,DB − ∝ D0 + rB ei(δB −γ) D0 ,where δB is a CP conserving strong phase, and rB ∼ 0.1 is the magnitude of the ratio of the
B − → D0 K − and B − → D0 K − decay amplitudes. In the corresponding CP-conjugate expression,
γ/φ3 changes sign. The amplitude analysis of the subsequent DB ± decay to a state accessible toboth D0 and D0 allows the measurement of γ/φ3 [62–69]. The method generalizes to similar B
hadron decays, such as B 0 → DK ∗0 , and, albeit with reduced sensitivity, to decays where the kaon
is replaced by a pion.
Measurements based on this technique have been reported by BaBar [95, 96], Belle [97–99],
Belle II [100], and LHCb [101–114]. Since it relies on D0 − D0 interference, the phase differences
between the relevant D0 and D0 decay amplitudes are required in order to extract γ/φ3 . Because thetheoretical shortcomings of amplitude models discussed above make their phases unreliable, mostrecent γ/φ3 measurements use amplitude model-independent approaches to obtain this information[98–105, 107–114]. These depend critically on information from the charm threshold described inSec. 69.3. Charm mixing also results in a (time-dependent) D0 − D0 superposition, that can beused to measure the relevant phase information as input to γ/φ3 measurements. This method isparticularly powerful in doubly Cabibbo-suppressed decays such as D0 → K + π − π + π − , and whenused in combination with threshold data [115, 116]. Under some circumstances, with large data
sets, the relevant strong phases and γ/φ3 can be extracted simultaneously without external input,
for example in simultaneous analysis of the B 0 → DK + π − Dalitz plot and that of the subsequent
D → KS0 π + π − decay [117]. However, the global effort to measure γ/φ3 to sub-degree precision willcontinue to rely critically on input from the charm threshold and BESIII’s increasing datasets [118].
The most precise individual γ/φ3 measurement with an uncertainty of ∼ 5◦ results from LHCb’s
amplitude model-independent study of DB − → KS0 π + π − and DB − → KS0 K + K − [119], using input
from the charm threshold [80, 87]. LHCb’s analysis of DB − → K + π − π + π − decays leads to the◦◦ +6.7◦second most precise individual γ/φ3 measurement with γ/φ3 = 55.8◦ +6.0
−5.8◦ ± 0.6 −4.3◦ [113]. Here,the first uncertainty is due to B ± → DK ± statistics, the second systematic, and the third dueto the uncertainty on charm-threshold input. This illustrates the importance of charm-threshold
data, and the need for the analysis of larger threshold datasets to keep pace with the sharp increase
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in B ± → DK ± data expected at the upgraded LHCb detector and Belle II. Further improvementsin this mode are expected from including information on relative phases between D0 and D0
amplitudes obtained from charm mixing [85, 94, 116].
The current world average on γ/φ3 is dominated by LHCb who finds, combining its results◦with input from CLEO–c and BESIII in an amplitude-model independent approach, γ = 63.8◦ +3.5
−3.7◦
[111, 120]. Belle II, which is expected to reach excellent precision in γ/φ3 , recently released its firstγ/φ3 result, based on Belle, Belle II and charm threshold data [100].
The interference between mixing and decay in B 0 → D0 h0 with h0 = π 0 , η, ω provides sensitivity
to β, which can be extracted from the Dalitz plot of the subsequent D0 → KS0 π + π − decay [38,121–124]. The combined BaBar/Belle analysis based on this technique resolved the ambiguity in βpresent in other measurements [38].
Further details on CP violation in beauty and charm can be found in [125, 126].
69.4.2 Charm Mixing and CP violation
Time-dependent amplitude analyses in decays to final states that are accessible to both D0 andD0 have unique sensitivity to mixing parameters. A Dalitz plot analysis of a self-conjugate final
state, such as KS0 π + π − and KS0 K + K − , allows a direct measurement of x and y, the normalized
mass and width difference of the D0 − D0 system’s mass eigenstates [127]. This is in contrast to
decays like D0 → K + π − which only provide access to decay-specific parameters x02 , y 0 .
The phase differences between D0 and D0 amplitudes across the Dalitz plot affect these measurements in the same way as those of γ/φ3 in B + → DK + , and can be taken into account in an
amplitude model-independent way using the same charm threshold results [65, 70–75, 125]. Thisapproach recently resulted in the first observation of a non-zero mass difference between the neutral charm mass eigenstates by LHCb [128], using a model-independent analysis of D0 → KS0 π + π − .
The “bin flip” technique applied in this measurements exploits that CP-conjugate regions of theD0 → KS0 π + π − Dalitz plot have near-identical experimental efficiencies [129]. This measurement
is also sensitive to CP violation in mixing and in the interference between mixing and decay, which
is discussed further in [125, 126].
69.4.3 Everything, Everywhere, All at Once
The discussion above shows how closely related the measurements of γ/φ3 , charm mixing, andthe analysis of charm threshold data are, due to the shared charm parameters. While above, theemphasis was on the importance of charm as input to γ/φ3 , and threshold data as input to charm
mixing and γ/φ3 , these relationships are in fact omnidirectional. This is demonstrated by LHCb’s
combined γ/φ3 fit to γ/φ3 -sensitive beauty decays to charm such as B − → DK − , charm mixing,
and charm threshold data. This combined analysis has a notable effect on charm parameters dueto the input from beauty decays to charm [111].
69.4.4 Searches for time-integrated CPV in charm
The recent observation of direct CP violation in LHCb’s analysis of D0 → π + π − and D0 →K + K − decays [130, 131], and the open question as to its Standard Model or beyond the Standard
Model nature, add renewed interest in the search for CP violation in other decays.
Multibody decays, with their rich structure and varying strong phase across phase space, could
potentially have particularly high sensitivity to CP violation. Comparing the results of amplitude fits for CP-conjugate decay modes provides a measure of CP violation, as for example
done for D0 → KS0 K ± π ∓ , D0 → K + K − π + π − , and D0 → π + π − π + π − using LHCb and CLEO–
c data [32, 132–134]. A widely-used amplitude model-independent technique to search for local
CP violation in multibody phase space is based on performing a χ2 comparison of CP-conjugate
phase-space distributions. This method was pioneered by BaBar [135] and developed further
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in [136–138] with results reported by BaBar [135,139] and LHCb in D± → K + K − π ± [140,141], CDF+
in D0 → KS0 π + π − [142], and LHCb in D(s)→ K + K − K + , D+ → π− π+ π+ , D0 → K + K − π+ π− ,D0 → π + π − π + π − , and Ξc+ → pK − π + [138, 143–145]. Unbinned methods can increase sensitivity [146,147] and have been applied by LHCb to D+ → π − π + π + , D0 → π + π − π 0 , D0 → π + π − π + π,
and Ξc+ → pK − π + [144, 145, 148–150].
An alternative model-independent approach is based on observables in four body decays that are
odd under motion reversal (“naïve T ”) [151–159], which is equivalent to P for scalar particles [159].
One such observable is CT = p~2 · (~p3 × p~4 ) where p~i are the decay products’ three momenta in the)−Γ (CT <0)decay’s center of mass frame. Comparing the P violating asymmetry AT ≡ ΓΓ ((CCT >0
with
T >0)+Γ (CT <0)
its C-conjugate provides sensitivity to CP violation. Additional variables are proposed in [159],
which also provides an excellent review of this topic. Searches for CP violation using P -odd variables have been carried out by BaBar, Belle, FOCUS, and LHCb in D0 → K + K − π + π − [160–163],
D+ → K + KS0 π + π − , Ds+ → K + KS0 π + π − [164], D0 → KS0 π + π − π 0 [165], and D0 → π + π − π + π −[149].
Recently, LHCb performed the first full angular analysis of the rare decays D0 → π + π − µ+ µ−
and D0 → K + K − µ+ µ− . From this, several observables with sensitivity to physics beyond theStandard Model are constructed, including CP-violating ones. The results are in agreement withStandard-Model predictions [166, 167].
All results of analyses mentioned in this section are compatible with CP conservation. The
observation of CP violation in two body charm decays [130, 131], and the vast data samples about
to be collected, provide grounds for optimism that this may change in the foreseeable future.

69.5 Summary
Multibody charm decays offer a rich phenomenology, including unique sensitivity to CP violation and charm mixing. This is a highly dynamic field with many new results (some of which
we presented here) and rapidly increasing, high quality datasets. These datasets constitute a huge
opportunity, but also a challenge to improve the theoretical descriptions of soft hadronic effects in
multibody decays. For some measurements, model-independent methods, many relying on input
from the charm threshold, provide a way of removing model-induced uncertainties. At the sametime, substantial progress in the theoretical description of multibody decays is being made.
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