Algunos principios

Being a three-body decay, the ω decay is a more complicated process to simulate. The three pions must lie in a plane, due to momentum conserva- tion, but the allowed kinematic configurations are spread into a continuous distribution which may be calculated from the decay matrix element.

It can be shown [SAMacR62] that for vector mesons (JP _{= 1}−_{) decaying}

to three pseudoscalar mesons, the decay amplitude assumes the form: A=|p₀×p₁+p₁×p₂+p₂×p₀|2, (6.3) wherep_i,i= 0,1,2, are the momenta of the three pions in theω rest frame. A nice method to reproduce the correct probability distribution for the decay kinematic makes use of the Dalitz plot. Already in the original paper [MacARS61], in which the vector nature of theωmeson was proven, informa- tion about spin and parity were gained inspecting the intensity point(event)- distribution in a Dalitz plot. A particular decay configuration in a three- body decay can be represented by a point in a plane region. Several versions of a Dalitz plot can be used: we have chosen the so calledsymmetric repre- sentation, in which thex and y coordinates in the plane are defined as:

x= T1−T2 Q√3 , (6.4) y= T0 Q − 1 3, (6.5)

whereT_i,i= 0,1,2, are the kinetic energies of the three pions in theω rest frame, calculated from the formula

p2_i =Ti(2mi+Ti), (6.6)

and the quantityQ, defined as

Q=mω−3mπ =T0+T1+T2, (6.7)

is theQ-value of the decay reaction. In the last equation, as in the following of this section, we have neglected the mass difference between the charged and the neutral pions1_{. The same Dalitz plot and the same quantities x}

and y can be used to study the kinematic of other three-body decays, like K→3π. For different assignments of JP quantum numbers, different point distributions in the Dalitz plot are expected (and also observed). This is the feature that makes Dalitz plots so useful for our purpose: if one is able to generate events which reproduce the correct point distribution in the Dalitz plot, this is equivalent to have a set of events with the correct decay distribution.

1_{Without this assumption, the previous formula would read: Q=m}

80 The Monte Carlo simulation

Phase space alone implies a uniform distribution of points, and it must also be pointed out that not the whole xy-plane represents kinematically allowed configurations: the momentum conservation law must be taken into account. From the relation (6.6) the following equation can be derived (for the pair of pions (1,2); analogous relations are valid for the other two possible pairs):

cosθ12= p 2

0−p21−p22

2p1p2 . (6.8)

Imposing the usual constraint for the existence of the cosinus,

−1≤cosθ≤1, (6.9)

the kinematically allowed region is delimited by a closed curve, which tends to a circle centered in x = y = 0 in the non-relativistic case (Q → 0, i.e. mω → 3mπ) and degenerates to an equilateral triangle in the ultra-

relativistic case (Q→ ∞). Already forω, withQ≈369 MeV/c2, a deviation from the circular shape is easily observed (Figure 6.3).

We need at this point a recipe to reproduce the correct point distribution in the allowed region. It turns out, that for a decay amplitude as in (6.3), the density of points is proportional to the quantity

λ= |pi×pj|

2

|p_i×p_j|2

max

, (6.10)

(compare with equation (5.2)) where the denominator represents the max- imum value assumed by the numerator, thus leading to the normalization ofλ, now constrained to vary in the interval [0,1]. The value 0 is obtained when two pions are collinear or at least one of them at rest, and it corre- sponds to the border of the Dalitz plot; the maximum value 1 is achieved only in the center of the plot (x = y = 0), and corresponds to the unique completely symmetric decay configuration, where the three pions have the same kinetic energy and move away from each other forming three equal anglesθij = 6 (pi,pj) = 120◦. Intermediate values of λ correspond to con-

centric closed curved lines spanning the plot smoothly, with decreasingλ, from the center to the border. The generated event sample can be seen in Figure 6.3. A three dimensional representation in the xyλ-space is often calledDalitz-Stevenson plot (Figure 6.4).

To reproduce the desired linear increase, the value of λwas calculated for every generated point uniformly distributed in the allowed region of the Dalitz plot, and compared with a second quantityλ0_{, randomly chosen from}

a constant distribution in the interval [0,1]. The requirement

λ0< λ (6.11)

reproduces the linear increase ofλand the correct point distribution in the Dalitz plot. The reason to adopt (6.11) can be understood this way: if a

6.1 The event generator 81 x -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 y -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0 200 400 600 800 1000 1200

Figure 6.3: Dalitz plot for the generated ω → π+_π−_π0 _{decay. The correct}

point distribution for a vector meson (JP _{= 1}−_{) decay in three pseudoscalar}

is reproduced: the density reaches the maximum in the center of the plot, and goes to zero approaching the border of the kinematically allowed region (in color). x -0.4-0.3-0.2 -0.1 -0 0.10.2 0.3 0.4 y -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 λ 0 200 400 600 800 1000 1200

Figure 6.4: Dalitz-Stevenson plot for the generated ω → π+_π−_π0 _{decay (the}

82 The Monte Carlo simulation λ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 counts 0 10000 20000 30000 40000 50000 60000 70000 80000

Figure 6.5: Distribution of λfrom equation 6.10, for the generated event sam- ple. A linear fit has been performed to show the correctness of the generating algorithm.

quantityλ0 _{is uniformly distributed on the interval [0,1], the probability to}

obtain, after a random choice, a value lower than a fixed one, sayλ0, is then

proportional toλ0 itself, whereas the probability to have a higher value will

be proportional to 1−λ0. If we then compare our λ from (6.10) and the

chosenλ0, we see that high values of λ satisfies (6.11) more often than the case with low values, and this happens followingexactly a linear behaviour; this is thus obtained forλin the final sample (Figure 6.5).

Once a particular event has been accepted because satisfying the con- dition (6.11), the absolute values of the pion momenta can be calculated inverting relations (6.4) and (6.5), and using (6.6) and (6.8) to define the direction of the momenta in the decay plane of theω. The unit vector

n= pi×pj

|p_i×p_j| (6.12)

defines the orientation of the plane in space: a series of arbitrary rotations ofn(together with the corresponding momentap_i), performed on an event- by-event basis in such a way, to have the variousnuniformly distributed on a unit sphere, leads to the desired isotropic decay distribution for theω in its rest frame.

The 4-momenta of the three pions can be now calculated as seen by an observer moving with the ωπ0 _{system simply performing a boost L(−β}

ω),

where the boost vector β_ω corresponds to the velocity of the ω in the ωπ0

6.1 The event generator 83 (GeV/c) beam p 120 130 140 150 160 170 180 190 200 counts 0 5000 10000 15000 20000 25000 30000 35000 40000 45000

Figure 6.6: Generated beam momentumpbeam : 160±7 GeV/c normal distri-

bution. (GeV) ν 0 20 40 60 80 100 120 140 160 180 counts 0 5000 10000 15000 20000 25000 30000 ) 2 (GeV 2 Q 10 log -5 -4 -3 -2 -1 0 1 2 3 counts 0 5000 10000 15000 20000 25000 30000

Figure 6.7: Generated gaussian distributions for the vitual photon energy ν = E−E0 _{(left) and for its squared 4-momentumQ}2 _(right).