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Файл:Совр эксперимент на ускорителях (4 сем, мага) / Lecture-4-5_HEP_29_02-7_03_2024
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Phase space in decays
Product over all
Energy must
Symmetry factor
outgoing particles
Lorentz Invariant |
M
| matrix element factor
(describes interaction)
Apart from matrix element, phase space is distributed evenly among all
Energy and
momentum must
be conserved
Outgoing
particles must
be on mass shell
be positive
Distributed
evenly in
phase space
particles subject to mass requirements, E/p conservation.
“Dynamics” like parity violation, etc. incorporated into matrix element.
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
11
The symmetry factor
Consider the integration of phase space for two particles in the final
state, where the particles are of the same species.
At some point, say, there is a configuration where
p
=K
1
and
1
p
= K
2
2
• Since the particles are identical, we should also have a reverse case:
•
p
=K
1
2 ,
p
= K
2
1
• the integral will contain both cases separately.
• However, in quantum mechanics, the identicalness of particles of the
same species means that these are the same state and we have
double counted. We need to add a factor of 1/2 to the phase space
• For n identical particles in the final state, we need a factor of 1/n!
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
12
, where:
Let’s integrate over overall outgoing particle phase space to get the
total decay rate. Start with the phase space factors:
Ignore the 2nd δ function since Θ(
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
0
p
) will be 0 whenever
0
p
is negative
13
Take into account energy and momentum conservation
Now integrate over
p
0
and
3
0
p
using the previous relations
2
Note
p
0
and
3
function
February 29 - March 7, 2024
0
are now set according to
p
2
Modern accelerator physics, Lecture 4-5
E/p
conservation by the δ
14
Consider decay at the rest frame
Decompose the product delta function (particle 1 at rest)
Perform the
d
3
integration:
p
3
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
15
assuming no
p
dependence of
|
M
|
The final integral over u sets
u = m
and makes
1
consistent with E conservation, and we get a final result
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
p
2
16
Invariant mass
Let us consider the decay of a particle in flight.
Let us suppose it decays in three particles:
pE,
),(
pEp
111
),(
pEp
222
The states 1,2,3 are observed in the spectrometer
Momenta get measured, a mass hypotesis is made:
Ingredients :
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
),(
pEp
333
,,,,, pmpmpm
332211
17
2
2
1
2
1
2
2
2
2
2
3
2
3
2
pppmpmpmpA
321
2
Which can also be written as :
But this is a Lorentz scalar. Then, we can compute it (for instance), in
321
the rest frame of the decaying particle :
2
Bump hunting in invariant mass distributions :
2
2
0 MMA
spectrum
2
pppEEEA
321
February 29 - March 7, 2024
Upsilon peaks
spectrum
B0 decay
Modern accelerator physics, Lecture 4-5
18
18
M
4
M
4
M
M
2
Example: masses are known
2
2
2
MpmpmEE
2
),(
pEp
111
ppP
21
pp
(in this section only) unstable particle decays
pp
21
121
0
2
2
1
),(
pEp
222
22
1
2
2
2
p
2
p
2 mpmMmM
1
4
4
2
1
4
2
pp
21
February 29 - March 7, 2024
22
Mpmpm
22
21
2
21
2
21
2
2
2
21
22
21
2
22
21
2
22
21
22
22
22)(
1
mMmMmmmmM
21
2
mmMmmM
)()(
21
2
2
2
22
1
mmMmmMmmmmM
• Possible only if
22
2
2
2121
2
)()()(
21
22
pmpmM
2)( pmMmmmmM
22
)(42)(
pmMmmmmMmmmmM
1
12121
22
mmM
21
• Momentum uniquely defined
Modern accelerator physics, Lecture 4-5
19
S-matrix
free particles
Interaction vanishes at t =
Conservation of probability unitarity:
interaction zone
±
-
T
transition matrix
free particles
so, we can write:
February 29 - March 7, 2024
Modern accelerator physics, Lecture 4-5
20
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