# (p.503) APPENDIX D THE DALITZ–FABRI PLOT

# (p.503) APPENDIX D THE DALITZ–FABRI PLOT

In this appendix a widely used construction for analysing three-body decays is discussed.

The Lorentz-invariant phase space element *d*(LIPS) for the three-body decay of a particle of mass *M* into three particles of masses *m _{i}*(

*i*= 1, 2, 3) is

*p**,*

_{i}*E*are the momenta and energies of the three particles,

_{i}*θ*,

_{i}*ϕ*their polar and azimuthal angles, and

_{i}**,**

*P**E*the momentum and energy of the decaying one. Due to energy-momentum conservation this expression actually depends on only two independent variables. The integration on one of the momenta is cancelled by the momentum delta function; the uniformly distributed variables Ω

_{1}and

*ϕ*

_{2}can be integrated over: the first gives the overall orientation of the decay in space and the second one the azimuthal orientation of particle 2 with respect to particle 1 (having defined the

*z*axis along the direction of particle 1) uniform for spinless particles. Therefore in this case

*θ*

_{2}is the angle between particles 1 and 2. Working in the centre of mass (but the same is true for any fixed

**) ${E}_{3}^{2}={m}_{3}^{2}+{\left|{p}_{1}\right|}^{2}+{\left|{p}_{2}\right|}^{2}-2\left|{p}_{1}\right|\left|{p}_{1}\right|\mathrm{cos}{\theta}_{2}$ so that for fixed |**

*P*

*p*_{1}|, |

*p*_{2}| one has

*E*

_{3}

*dE*

_{3}= –|

*p*_{1}||

*p*_{2}|

*d*cos

*θ*

_{2}and the integration over the angular variable is canceled by the energy delta function, resulting in

*E*

_{1},

*E*

_{2}).

The two independent variables describing the decay configuration can be chosen conveniently as the kinetic energies in the centre of mass frame ${T}_{i}^{*}={E}_{i}^{*}-{m}_{i}$
(p.504)
(where the star indicates variables in this frame), or some linear combination of them: a standard choice (explicitly Lorentz-invariant) is^{169}

*m*is one of the

*m*and

_{i}*m*=

_{i}*m*the

*s*range between (

_{i}*s*)

_{i}_{min}= 4

*m*

^{2}and (

*s*)

_{i}_{max}= (

*M*–

*m*)

_{i}^{2}.

A common application (for which this was invented) is to *K* → 3*π* decays: conventionally, in this case the subscript 3 is used for the ‘odd’ pion, i.e. the one with charge opposite to that of the decaying particle for *K* ^{±} → *π* ^{±} *π* ^{±} *π* ^{∓}, the charged one for *K* ^{±} → *π* ^{0} *π* ^{0} *π* ^{±}, the neutral one for *K* ^{0} → *π* ^{+} *π* ^{–} *π* ^{0}.

Plotting two independent variables on a plane (the ‘Dalitz–Fabri plane’), each decay configuration is represented by a point, constrained by energy-momentum conservation to lie within a given boundary. Given (D.3) a convenient construction (Dalitz, 1953; Fabri, 1954) is that in which each *s _{i}*–(

*s*)

_{i}_{min}represents the distance of the point from one side of an equilateral triangle (see Fig. D.1) of side $2\sqrt{2}\left({M}^{2}/3-5{m}^{2}\right)$; the centre of the Dalitz plane corresponds to a decay configuration symmetric in the three particles, with

*s*=

_{i}*s*

_{0}and

*u*= 0 =

*v*.

In the non-relativistic limit (*Q* ≡ *M* ∑_{i}*m _{i}* → 0) the physical boundary is the circumference inscribed in the triangle; using relativistic kinematics (which is a minor correction in the case of

*K*→ 3

*π*decays for which

*Q*≃ 80 MeV <

*m*) the actual physical boundary is distorted from a circumference, while still being tangential to the triangle at the centre of each side. The exact equation for the physical boundary is (in case of equal masses)

_{i}*u*,

*v*ranges are (assuming

*m*

_{1}=

*m*

_{2}).

All the points on the boundary correspond to collinear decay configurations, those on the sides of the triangle correspond to events in which one of the particles is at rest in the centre of mass frame, and those diametrically opposite to two particles with equal and parallel (minimum) momentum.

The Dalitz plane also has a uniform measure in the *u*, *v* variables: $du\text{}dv=8\left({M}^{2}/{m}_{\pi}^{4}\right){T}_{1}^{*}{T}_{2}^{*}$. Since the phase space measure element is proportional to the area in the plane, any departure from uniformity in the distribution reflects the nontrivial dynamics of the process; this is the usefulness of this representation, which factorizes the trivial phase space dependence. For example different spin-parity assignments for the decaying particle lead to different distributions in the (*u*, *v)* plane, and if e.g. particles 1 and 2 form a resonance an enhancement is expected for *s* _{3} = (*p* _{1} + *p* _{2})^{2} corresponding to its mass.

The decay distribution in the Dalitz plot has traditionally been parametrized in a generic way with the first few terms of a series expansion around its centre:

*i*= 1, 2) are identical (as for

*K*

^{±}→

*π*

^{±}

*π*

^{±}

*π*

^{0},

*π*

^{0}

*π*

^{0}

*π*

^{±}) odd terms in

*v*cannot appear, and in case of three identical particles (as for

*K*

^{0}→ 3

*π*

^{0})

*g*,

*j*,

*f*are not defined and a single quadratic term is present since

*u*

^{2}=

*v*

^{2}/3. With particles

*i*= 1, 2 being

*CP*-conjugate (as for

*K*

^{0}→

*π*

^{+}

*π*

^{–}

*π*

^{0}) a non-zero value for the parameters

*j*or

*f*indicates the presence of

*CP*violation.

The above parametrization is of course just an approximation, which works reasonably well for *K* → 3*π* in which the available phase space is small and no *ππ* resonances with mass below *m _{K}* exist, although even in this case accurate measurement shows that the actual matrix element is more complicated and cannot be written in such a form (Batley

*et al.*, 2006

*a*).

## Notes:

(169)
Sometimes in the literature *u* and *v* are denoted *y* and *x* respectively.