DataAnalysisTech11-ch04, Study notes of Advanced Data Analysis

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 (^) Mass of the proton ~1.7× 10 -27kg

 (^) Max. momentum of electrons @ LEP ~5.5× 10 -

17 kg∙m∕sec

 (^) Speed of muon in pion decay (at rest) ~8.1× 107 m∕sec

More practical/intuitive: ħ = c =1; this means

energy, momentum, mass have same units

 (^) E^2 = p^2 c^2 + m^2 c^4  E^2 = p^2 + m^2

 (^) E.g. m p

=0.938GeV, max. p LEP

=104.5GeV

 (^) Also:

 Time and length have units of inverse energy!

 (^) 1GeV-1^ =1.973× 10 -16m 1GeV-1^ =6.582× 10 -25sec

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hc

E h E

Monte CarloMonte Carlo 第第第第

 Random number generator

 Particle generator/decayer:

 Pythia, FakeEvt, BGenerator,

QQ, EvtGen

 Detector simulator:

GEANT

Monte CarloMonte Carlo 第第第第

 (^) Monte Carlo methods are statistical simulation methods

that utilize a sequences of random numbers to perform

simulations. As seen from the figure below the ratio of

the area of the circle to the area of the square can be

determined by this method (The ratio = A 1

/ A 2

= π/4).

2

1

2

2

4

r

A

A r







What do we study?What do we study?

 Particle Decays (AB+C+…)

 (^) Lifetimes, branching ratios etc…

 Reactions (A+BC+D+…)

 (^) Cross sections, scattering angles

etc…

 Bound States

 (^) Mass spectra etc…

第第第第 - - 第第第第第第第第

m = 10

nm

第第第第第第第第 第第第第第第第第 bunchbunch 第第第第第第

BESIII 第第第第 : 380 *

5.7 * 15000  m (x y

z)

第第第第第第第第第第 ns

380

 m

5.

 m

15000



m

第第第第第第 (( Interaction PointInteraction Point ))

 第第第 IP

 第第第第第第第第第第第第第第第第 0,0,0 第

Event by event IP

Run by run IP

AcceleratorAccelerator LuminosityLuminosity (( 第第第第 ))

2

b rep

D

n N f

L H

A



where:

n

b

= bunches / train

N = particles per bunch

f

rep

= repetition frequency

A = beam cross-section at IP

H

D

= beam-beam enhancement

factor

2

4

b rep

D

x y

n N f

L H

  

For Gaussian beam distribution:^ 

Collider luminosity (cm

• 2

s

• 1

) is

approximately given by

第第第第第第第第第第第第 sidebandsideband

Study of Decays (Study of Decays (AA  B+C+…B+C+…))

 Decay rate : “The probability per unit

time that a particle decays”

 (^) Lifetime : “The average time it takes to

decay” (at particle’s rest frame!)

 Usually several decay modes

 Branching ratio BR

 (^) We measure  tot

(or ) and BRs; we

calculate  i

tot

i

tot i

   and   1 



i tot

BR (decaymode i) 

t

N N t N(t) N( 0 )



-

d d e

N

N L

L

  



    



N: number of signal events;

L: luminosity ;

: signal cross-section;

: reconstruction efficiency

(typically determined from

Monte Carlo);

Cross-section Cross-section 第第第第

i i

i

i i

i

N

R

N

N N B  BR



    



N: Number of particles;

N

i

: Number of observed particles in the decay

channel i;

BR

i

: Branching ratio for the decay channel i;

i

: reconstruction efficiency for decay

channel i (typically determined from Monte

Carlo);

Branching Ratio Branching Ratio 第第第第