Simulated Meson Decays

Drew Silcock

Saturday 1st February, 2014

1 Introduction

The aim of this project is to examine the productparticles of various meson decays, in particular B, Dand ψ(3770) decays. To this end, the invariant massdistributions of three decay chains were examined:

1. Two body ψ(3770)→ D0D0,

2. Three body B+ → D0D0K+, and

3. Four body D0 → K+K−K−π+

In addition, the decay angles between the result-ing particles in the lab frame were examined for thefollowing decays:

1. ψ(3770)→ D0D0 and

2. B →{ψ(3770)→ D0D0

}K

2 Background

2.1 Purpose

The ψ(3770) decays are of particular interest becausethe resultant D0 and D0 are produced in a quan-tum correlated state. This allows for better access toD0 phase information introduced by the strong force.This in turn is critical information in the accuratemeasurement of CP violation in beauty decays (e.g.B → DK), which is a large part of the Universityof Bristol’s LHCb programme. These ψ(3770) decayscan also be used to measure symmetry violations inthe charm system.

CP violation contributes to the matter anti-matter asymmetry that we see today, i.e. why theuniverse is full of matter and not antimatter. Atpresent the amount of observed CP violation cannot

account for the extent of matter-antimatter asymme-try present. Analysis of the CP violation of both thebeauty and charm decays involved with the ψ(3770)decays could both lead to a better explanation of thismatter-antimatter asymmetry, or to the discovery ofNew Physics.

2.2 ROOT Framework

The ROOT C++ framework and libraries were usedfor toy Monte Carlo (MC) decays and to anal-yse the results find the invariant mass dis-tributions and decay angles in both the cen-tre of mass frame and the laboratory frame.In particular the GetDecay()::TGenPhaseSpace,M()::TLorentzVector and Angle()::TLorentzVectormethods were used to generate toy MC decays, getthe invariant mass of a 4-vector and get the anglebetween two 4-vectors respectively.

3 Invariant Mass Distributions

Here the distribution of the invariant masses of theproduct particle of each decay chain were plotted andexamined.

3.1 Two body ψ(3770)→ D0D0 decay

The plot of the invariant mass distribution of theD0D0, showing in Figure 1, matches exactly withthe mass of the parent ψ(3770) particle1 , which isexactly as would be expected for a two body decay:

1Mass of ψ(3770) = 3773.15± 0.33 MeV[1]

1

]2

) [GeV/c0

D0

m(D

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 40

200

400

600

800

1000

310×

(3770) Meson Decayψ

(3770) Decayψ

Entries 1000000

Mean 3.773

RMS 1.828e­05

(3770) Meson Decayψ

Figure 1: The invariant mass distribution of the D0D0

corresponds exactly to the mass of the parent ψ(3770).

3.2 Three body B+ → D0D0K+ decay

For the three body decay, the invariant mass distribu-tions produces a 3-dimensional plot of the invariantmass combinations. These weighted distributions areshown below in Figures 2 and 3.

]2)2

) [(GeV/c

+K0

(D2 m5

67

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]2)

2

) [(GeV/c

+K0D(

2m

56

78

910

1112

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DK­DKDK­DK

Figure 2: 3D plot showing the invariant mass distribu-tion of D0K+ against D0K+. Note that the top of thedistribution is roughly flat.

]2)2

) [(GeV/c

0D0

(D2m

1214

1618

2022

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]2)

2

) [(GeV/c

+K0(D

2m

56

78

910

1112

0

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DK­DDDK­DD

Figure 3: 3D plot showing the invariant mass distribu-tion of D0K+ against D0D0. Note that the top of thisdistribution is also roughly flat.

Three body decay has a two-dimensional phasespace, so these data are best visualised as 3-dimensional density plots of the invariant mass distri-butions of two of the three combinations of the parti-cles. Note that all of the essential information aboutthe decay is contained in a single plot and the otherscan be derived from the one plot due to momentumconservation laws.

This simulation does not take into account inter-mediate resonances, which have the effect of alter-ing the topology of the top of the plot so that it isno longer roughly flat-topped. One such resonance,which will be discussed in Section 4.2, is of particularinterest.

3.3 Four body D0 → K+K−K−π+ de-cay

The four body decay produces a five dimensionalphase space, which is projected onto various axes sothat they can be easily analysed and understood.

The following projected plots were produced:

2

]2 m(K K) [GeV/c1 1.05 1.1 1.15 1.2 1.25

0

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K KK K

]2) [GeV/cπ m(K 0.65 0.7 0.75 0.8 0.85 0.9

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πK πK

]2 m(K K K) [GeV/c1.5 1.55 1.6 1.65 1.7

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K K KK K K

]2) [GeV/cπ m(K K 1.15 1.2 1.25 1.3 1.35

0

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πK K πK K

Figure 4: The projected plots of various invariantmass combinations for the four body decay D0 →K+K−K−π+

Simulating intermediate resonances in this decaywould produce rich strong phase information arisingfrom the interference between with the intermedi-ate decays. The possible intermediate resonances aretabulated below:

3

Light Unflavored Mesons (S = C = B = 0)Particle Name Mass [MeV] Full width, Γ [MeV] JP Decay Channelf0(980) 980 ± 10 40 → 100 0+ f0(980)→ KK

a0(980) 980 ± 20 50 → 100 0+ a0(980)→ KKφ(1020) 1019.45 ± 0.02 4.26 ± 0.04 1− φ(1020)→ K+K−

b1(1235) 1229.5 ± 3.2 142 ± 9 1+ b1(1235)→ [φ→ K+K−]π andb1(1235)→ [K∗(892)± → Kπ]K±

a1(1260) 1230 ± 40 250 → 600 1+ a1(1260)→ [f0(1370)→ KK]π anda1(1260)→ [f2(1270)→ KK]π anda1(1260)→ K[K∗(892)→ Kπ] + c.c.

f2(1270) 1275.1 ± 1.2 185.1+2.9−2.4 2+ f2(1270)→ KK

f1(1285) 1281.8 ± 0.6 24.3 ± 1.1 1+ f1(1285)→ KKπη(1295) 1294 ± 4 55 ± 5 0− η(1295)→ a0(980)π

a2(1320) 1318.3 ± 0.6 107 ± 5 2+ a2(1320)→ KK

f0(1370) 1200 → 1500 200 → 500 0+ f0(1370)→ KK

η(1405) 1409.8 ± 2.5 51.1 ± 3.4 0− η(1405)→ [a0(980)→ KK]π andη(1405)→ KKπ andη(1405)→ [K∗(892)→ Kπ]K

f1(1420) 1426.4 ± 0.9 54.9 ± 2.6 1+ f1(1420)→ KKπ andf1(1420)→ K[K∗(892)→ Kπ] + c.c.

a0(1450) 1474 ± 19 265 ± 13 0+ a0(1450)→ KK

ρ(1450) 1465 ± 13 265 ± 13 0+ ρ(1450)→ K[K∗(892)→ Kπ] + c.c.

η(1475) 1476 ± 4 85 ± 9 0− η(1475)→ KKπ andη(1475)→ [a0(980)→ KK]π andη(1475)→ K[K∗(892)→ Kπ] + c.c.

f0(1500) 1505 ± 6 109 ± 7 0+ f0(1500)→ KK

f ′2(1525) 1525 ± 5 73+6−5 2+ f ′2(1525)→ KK

η2(1645) 1617 ± 5 181 ± 11 2− η2(1645)→ [a2(1320)→ KK]π andη2(1645)→ KKπ andη2(1645)→ [K∗(892)→ Kπ]K andη2(1645)→ [a0(980)→ KK]π

π2(1670) 1672.2 ± 3.0 260 ± 9 2− π2(1670)→ [f2(1270)→ KK]π andπ2(1670)→ K[K∗(892)→ Kπ] + c.c.

φ(1680) 1680 ± 20 150 ± 50 1− φ(1680)→ KK andφ(1680)→ K[K∗(892)→ Kπ] + c.c.

ρ3(1690) 1688.8 ± 2.1 161 ± 10 3− ρ3(1690)→ KKπ andρ3(1690)→ KK andρ3(1690)→ [a2(1320)→ KK]π

ρ(1700) 1720 ± 20(ηρ0 & π+π−)

250 ± 100(ηρ0 & π+π−)

1− ρ(1700)→ KK andρ(1700)→ K[K∗(892)→ Kπ] + c.c.

f0(1710) 1720 ± 6 135 ± 8 0+ f0(1710)→ KKπ(1800) 1812 ± 12 208 ± 12 0− π(1800)→ [K∗(892)→ Kπ]K−

φ3(1850) 1854 ± 7 87+28−23 3− φ3(1850)→ KK and

φ3(1850)→ K[K∗(892)→ Kπ] + c.c.f2(1950) 1944 ± 12 472 ± 18 2+ f2(1950)→ KK

f2(2010) 2011+60−80 202 ± 60 2+ f2(2010)→ KK

a4(2040) 1996+10−9 255+28

−24 4+ a4(2040)→ KK anda4(2040)→ [f2(1270)→ KK]π

f4(2050) 2018 ± 11 237 ± 18 4+ f4(2050)→ KK andf4(2050)→ [a2(1320)→ KK]π

f2(2300) 2297 ± 28 149 ± 40 2+ f2(2300)→ KK

Table 1: Table of possible resonances for the light unflavored mesons.4

Strange Mesons (S = ±1, C = B = 0)Particle Name Mass [MeV] Full width, Γ [MeV] JP Decay ChannelK∗(892) 895.5 ± 0.8 46.2 ± 1.3 1−

K∗(892)→ KπK∗(892)± 891.66 ± 0.26 50.8 ± 0.9 1−

K∗(892)0 895.94 ± 0.22 48.7 ± 0.8 1−

K1(1270) 1272 ± 7 90 ± 20 1+ K1(1270)→ K[f0(1370)→ KK]

K1(1400) 1403 ± 7 174 ± 13 1+ K1(1400)→ K[f0(1370)→ KK]K∗(1410) 1414 ± 15 232 ± 21 1− K∗(1410)→ KπK∗0 (1430) 1425 ± 50 270 ± 80 0+ K∗0 (1430)→ KπK∗2 (1430)± 1425.6 ± 1.5 98.5 ± 2.7 2+

K∗2 (1430)→ KπK∗2 (1430)0 1432.4 ± 1.3 109 ± 5 2+

K∗(1680) 1717 ± 27 322 ± 110 1− K∗(1680)→ Kπ

K2(1770) 1773 ± 8 186 ± 14 2− K2(1770)→ K[f2(1270→ KK] andK2(1770)→ K[φ→ K+K−]

K∗3 (1780) 1776 ± 7 159 ± 21 3− K∗3 (1780)→ Kπ

K2(1820) 1816 ± 13 276 ± 35 2− K2(1820)→ K[f2(1270)→ KK]K∗4 (2045) 1045 ± 9 198 ± 30 4+ K∗4 (2045)→ Kπ

Table 2: Table of possible resonances for the strange mesons.

4 Decay Angles

In the next section we are interested in examiningthe distribution of the decay angle between productpairs in various simulated decays. By examining thedistribution of decay angles in the laboratory framethis angle can be used to distinguish the background(which should not have a strongly peaked angle de-pendence) from the decays in which we are interested.This is due to the ultra-relativistic parent particlebeaming the decay products in the direction of move-ment, while the background should not follow such astrongly beamed angle distribution.

The decay angles were obtained by relativisticallyboosting the MC simulated Lorentz vectors using theBoost()::TLorentzVector method. The TVector3vectors used to boost the parent particles were ob-tained from fits to the momentum component distri-butions of previously calculated LHCb MC simula-tion data. The GetRandom()::TF1 method was thenused to provide a random distribution following thatof the LHCb MC data. The boost was repeated alarge number (106) of times, each time boosted bya different 3-vector provided by the random momen-tum distribution following the LHCb MC data, andeach time the angle between the resulting particleswas calculated and plotted to provide final angle dis-tribution functions in the lab frame. This angle dis-tribution can then be used to distinguish these vitaldecays from the background, provided it is sharplypeaked enough.

4.1 ψ(3770)→ D0D0 decay

The first decay for which this was done was theψ(3770)→ D0D0 decay.

Before the LHCb MC momentum component dis-tribution data could individually be used to provide3-vectors to boost the ψ(3770), it had to be estab-lished that there was no correspondence or correlationbetween the individual momentum components. Ta-ble 3 shows the calculated correlation factors betweenthe individual direction components, whilst Figure 5visualises the dependence of the components on oneanother. Both clearly show no correlation betweenindividual momentum components. Thus, they canbe considered independent distributions and the ran-dom component generation can be done separately.

px py pzpx 1 0.00640212 0.0150375py 0.00640212 1 0.0167604pz 0.0150375 0.0167604 1

Table 3: The correlation factors between each momen-tum component. Note that these are all extremely low,thus the components can be considered independent ofone another.

5

[MeV/c]x

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(3770) (Contour)ψ for z

Versus py

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Versus py

p

Figure 5: Different visualisations of the correlation be-tween different momentum components in the LHCb MCdata. These clearly show no correlation. Note that thex−z and y−z plots are skewed to the right as z is alwayspositive.

The x and y components of the momentum werefitted to Gaussian distributions, whilst the z compo-nent of the momentum was fitted to a Landau upuntil 62 GeV/(̧, after which it was fitted to a Gaus-sian, as shown in Figure 6. Note that the fits arenot perfect, but follow the general distribution satis-factorily for the purposes of our random momentumgeneration.

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p in Lab Framez

p

Figure 6: Fits to the momentum component distributions.ROOT’s GetRandom()::TF1 method was then used to pro-vide a random distribution based on these fits.

The final decay angle distribution between the D0

and D0 is shown in Figure 7. Note that all decay an-gles are < 5◦, with a high peak at around ∼ 0.5◦,meaning that the decay angle can be used to sepa-rate the ψ(3770) decays from the background.

]° Angle [ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

20

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60

80

100

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310×

0D and 0Boosted Angle Between D

0D and 0Boosted Angle Between D

Figure 7: The final lab frame decay angle distribution be-tween D0 and D0. Note that the distribution is sharplypeaked about ∼ 5◦, with no angles > 5◦.

4.2 B → (ψ(3770)→ D0D0)K decay

As with the ψ(3770) decay, LHCb MC momentumdata was used to provide a distribution to use for theLorentz boost. However, only B → D Bach (wherethe bachelor particle is either a kaon or a pion) mo-mentum data was available. Hence conservation ofmomentum was used to determine the parent B mo-mentum from the D and bachelor particle momenta.

The corresponding correlation factor table isshown below in Table 4, while the correlation visual-isations are shown in Figure 8:

px py pzpx 1 0.0165178 -0.00486466py 0.0165178 1 0.017408pz -0.00486466 0.017408 1

Table 4: The correlation factors of the momentum compo-nents of the B meson, obtained from LHCb B → D BachMC data.

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Figure 8: Visualisations of the correlation between mo-mentum components of the B meson, obtained fromB → D Bach LHCb MC data. Note the hole presentin the x− y graphs.

The momentum components were once again fit-ted to mathematical functions, again a Gaussian forx and y components and for the z component a Lan-dau up until 130 GeV/(̧, after which a Gaussian. Notethat the x and to a lesser extent the y component dis-tributions cut off at low values. This is most likelydue to specific detector design and is thus ignoredfor the purposes of the mathematical fit, which in-tends only to find the general distribution of B me-son momentum components. The resulting fits areillustrated in Figure 9.

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zB Meson p

zB Meson p

Figure 9: The momentum component distributions forthe B meson, along with their mathematical fits. Notethe gap in the x and y component distributions near the0 point.

The angle distributions between the ψ(3770) andK and between the D0 and D0 product particles wasthen plotted both in the rest frame of the parent Bmeson and in the lab frame, i.e. the Lorentz boostedframe. The plots are shown below in Figures ??:

]° Angle [ 179 179.2179.4179.6179.8 180 180.2180.4180.6180.8 1810

2

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610×

(3770) and KψNon­Boosted Angle Between (3770) and KψNon­Boosted Angle Between

]° Angle [ 0 10 20 30 40 50

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0.5

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610×

0D and

0Non­Boosted Angle Between D

0D and

0Non­Boosted Angle Between D

Figure 10: The decay angle distributions in the rest frameof the parent B meson. Note the sharp cut off at ∼ 50◦

for the D0-D0 decay angle distribution.

]° Angle [ 0 1 2 3 4 5 6

0

0.05

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(3770) and KψBoosted Angle Between (3770) and KψBoosted Angle Between

]° Angle [ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

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0D and

0Boosted Angle Between D

0D and

0Boosted Angle Between D

Figure 11: The decay angle distributions in the lab frame.Note that both distributions are contained within a smallrange of angles, . 6◦ for the ψ(3770)-K angle and . 1◦

for the D0-D0 angle.

Both of these distributions are contained in asmall range of angles and are sharply peaked, withthe ψ(3770)-K angle at around ' 1.5◦, and the D0-D0 angle at around ' 0.4◦. This confirms thatB → (ψ(3770) → D0D0)K decays can be easily dis-tinguished from the background due to the sharp an-gular peaking in the ψ(3770)→ D0D0 product decay,allowing for its use in isolating ψ(3770) decays for thestudy of symmetry violation.

7

References[1] J. Beringer et al. “Review of Particle Physics”. In: Phys. Rev. D 86 (1 July 2012), p. 010001. doi:

10.1103/PhysRevD.86.010001. url: http://link.aps.org/doi/10.1103/PhysRevD.86.010001.

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