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Cinematica relativistica

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!
"
!
"
v
z
#
x' = x
y'= y
z ' = γ ( z − β ct )
t ' = γ (t −
β=
γ=
β
c
z)
v
c
1
1− β 2
v
S’
S
t
$
!
"
%
&
#
x' = x
y ' = y cos α + z sin α
z ' = − y sin α + z cos α
!
"
r 2 = ( x1 − x2 ) + ( y1 − y2 ) + ( z1 − z2 ) = ( x '1 − x '2 ) + ( y '1 − y '2 ) + ( z '1 − z '2 )
2
2
2
2
2
2
invariante per rotazioni
'
'$
$
$
s 2 = ( ct1 − ct2 ) − ( x1 − x2 ) + ( y1 − y2 ) + ( z1 − z2 ) = ( ct '1 − ct '2 ) − ( x '1 − x '2 ) − ( y '1 − y '2 ) − ( z '1 − z '2 )
2
2
2
2
2
2
2
invariante per trasformazioni di Lorentz
#
$
$
$
(
(
)
'$
#
$
(
2
'$
$
$
'$
'$
(
&
$
$
'
'$
'
*
'$
'
+
'$
!
"
!
&$
"
#
(
v µ = v 0 , v1 , v 2 , v3
)
µ
/
,
&
,
.
-
.
( ) − (v ) − (v ) − (v )
v 2 = v0
2
1
2
!
#
2
2
3
2
" 0
12 3
'$
←
!
!
"
#
#
$
##
%
##
&
##
#
##
! '
(
)
#
# !
# !
! *
#
%
&
+
#
(v1 = vx , v2 = v y , v3 = vz , v4 = ivt )
v = v +v +v +v
2
. /0 /
'
2
1
2
2
2
3
2
4
,
-
!
1
!
#
v µ = v 0 , v1 , v 2 , v 3 = ( vt , vx , v y , vz )
(
)
vµ = ( v0 , v1 , v2 , v3 ) = ( vt , −vx , −v y , −vz )
µ
/
µ
v = v vµ = vµ v 4
2
'$
vu = v µ uµ = vµ u µ
'$
5
#
1
g µν = g µν =
0
0
0
0 −1
0
0
0
0
−1
0
0
0
0
−1
vν = g µν v µ , vν = g µν vµ
/
&$
#
!
"
2
-
2
-
.
)
%
.
$
3
)
%
4
,
6
45
,
%
$
'$
'$
/
'$
p µ = ( E , pc )
p
!
!
"
"
0
p = mγ v
E = m 2c 4 + p 2 c 2
m
'$
#
p 2 = E 2 − p2c2 = m2c4
c
1
'$
#
p⊥ ' = p⊥
p ' = γ ( p − β E)
E ' = γ (E − β p
)
- .
β
#
p⊥ = p⊥ '
p = γ ( p '+ β E ')
E = γ ( E '+ β p ' )
4
p
p⊥
p
'$
4
!
"
0
#
N particelle
i =1
piµ = costante
6
#
7
##
!
85
#
#
'
,.
+9
##
)
v = βc
#
)
#
#
'
β=
β8
p
p
E
, γ = , βγ =
E
m
m
,.
+9
:β;8
+9
,. )
7
+9
#
!
!
*4
#
p = ( E p + mb , p p )
%
<
β=
=
7 ,.
p* = ( E *p + Eb* , 0 ) &
<
p
E
=
mb + E p
=
%
=m pb = E *p + Eb*
<
β
+
pp
,.
β=
#
&
pp
mb + m 2p + p 2p
!
*
'$
7
7
)1
)1
" 8
p*
7
p*
!
:
p ⊥*
p*
"
p9
&$
#
#
*2
*2
2
p⊥*2 + p*2 = p*2
x + p y + pz = k
;
z#
)1
p⊥ = p⊥*
(
E = γ (E
)
+βp )
p = γ p* + β E *
*
*
#
p* =
1
γ
p − β E*
p
→p +
2
⊥
γ
2
−βE
= k2
*
(
p − βγ E *
p2
→ ⊥2 +
k
γ 2k 2
)
2
=1
)1
7 !
γ
β
#
"
z
β9
7
z
)1
βγE*. )
! ) 1"
&
! 7"
#
!"
β<β9 !
7
)1
"4
=
4
7
)1
p⊥
0
!
! "
β>
β?β9
!
"
"4
)1
7
7
)1
p⊥
! "
β>β9 !
"#
p⊥
" 8
)1
cos θ =
tan ϕ =
φ
p
p +p
2
2
⊥
p⊥
, sin θ =
p + p⊥2
2
py
px
θ
7
p*
cos θ * =
p*2 + p⊥*2
, sin θ =
p⊥*
p*2 + p⊥*2
p*y
tan ϕ =
p*x
:
γ ( p − β E)
cos θ * =
γ ( p − β E ) 2 + p⊥2
py
tan ϕ * =
px
,sin θ * =
p⊥
γ ( p − β E ) 2 + p⊥2
= tan ϕ
#
tan θ * =
p sin θ
sin θ
=
γ ( p cos θ − β E ) γ ( cos θ − β β part )
β part =
)1
β
7
) 1 @" /
@%
!
)1
"
γ=5
7
∗
β/β .
β>β∗
)1
Theta LAB vs Theta CM
180
150
γ=5
β/β∗=1.5,1.,0.5
Theta LAB
120
90
60
30
0
0
30
60
90
120
Theta CM
"/
)1
)1
7
150
p
!
E
180
&
p⊥ = p⊥*
(
E = γ (E
)
+βp )
p = γ p* + β E *
*
*
E * = −γβ p + γ E
→ E * = −γβ p cos θ + γ E
→ E * = −γβ p cos θ + γ m 2 + p 2
p
/
(
(
cos θ β β * ± 1 + γ 2 1 − β β *
p= p
*
γ (1 − β 2 cos 2 θ )
2
2
θ
)1
β/β∗
#
) ) tan
#
y ;
∗
β/β ? "
∗
!
7
!β/β ≤ "
) 14
#
"
:
-
M
'$
:
'$
7
.
m
m:
#
.
( M , 0, 0, 0 ) = ( E1 , p1 ) + ( E2 , p 2 )
M = E1 + E2 = m12 + p12 + m22 + p22
p1 + p 2 = 0 → p1 = −p 2 → p1 = p 2 = p
7
#
1
2M
p=
M 2 − ( m1 − m2 )
2
M 2 − ( m1 + m2 )
2
#
1
M 2 + m12 − m22
2M
1
E2 = m22 + p 2 =
M 2 + m22 − m12
2M
E1 = m12 + p 2 =
(
)
(
)
6
M ≥ m1 + m2 ,
p
"
:"
7
&" /
&
:
4
'" ;
4
A"
B
7
B
B
B
B
) 14
p*
"4
!
p1 =
(M
2
)
'$
+ m12 − m22 p cos θ1 ± 2 E M 2 p*2 − m12 p 2 sin 2 θ1
(
2 M 2 + p 2 sin 2 θ1
)
p1
C θ1
*
Mp
<1
m1 p
)1
p1
:
p2
sin θ1max =
*
Mp
>1
m1 p
Mp*
4
m1 p
:
p1
+
*
π,
3! >
-
%γ
0
π
,
,.!
M
E
0
p
0
p
1
M
E
p
1
massa
#
1
p
2
2
p
2
0
fotone
0
p
p
-
p
1
E
p
p
2
1
M
m2
p
E
p
2
2
p
0
2
# /
π5
,
)
# /
,.
!
?
π0
p
,
?'
γ:
7
π0;
2γ
0
E
p
E *1
1
p
1
p
p
1
M
E *1
p* 1
*
E
1
1
*
1
*
1
0
p*1
2
p* cos
p* 1
p
*
1
Casi estremi :
cos
*
1
1
p* 1
p*
cos
*
1
1
p* 1
p*
M
0
2
M
0
2
M
p
1
p
1
M
1
0
2
1
p
1
1
0
M
M
1
0
2
val. min
2
1
0
2
M
0
2
M
1
p
M
0
2
0
val.max.
2
1
##
1
M
1
1
M
0
2
0
2
)
?)
,.
+9
!&
)
#
!
#
π
#
,.
φ
dN
d *
)
A
,
5
#
#
dN
d cos * d
*
A
dN
d cos
*
B
θ
#
+9
θ@!
E
1
p
E *1
1
M
2
dE
M
0
cos
2
M
1
dN
dE 1
,
0
M
p* 1
2
dN
d cos
0
p* cos
2
M
*
0
2
d cos
2
M
*
0
1
*
*
cos
*
0
2B
M
0
#
!
π5
%! )
A
π5
<
#
γ
=
)
&
%
>
#
%
!
-
- π5
3
1
%!
p
1
p* 1
E *1
p
2
p* 2
E*2
p
p*
1
E *1
M
E*2
0
*
1
p cos
p* 2
p* cos
*
2
cos
*
1
sin
*
1
1
2
*
1
min
*
2
sin
tan
p
1
p
1
p* sin
p* cos 1*
p
2
p
2
*
2
sin 1*
cos 1*
p*
sin 2*
cos 2*
sin
cos
*
1
*
1
1
0
m
*
1
da' l'angolo minimo fra i due
2
2 arctan
p
p *1
*
2
cos
tan
p* 1
2
*
1
p
*
1
2 arctan
min
0
m
0
p
0
Bα
π5
π
! (
π4
C
C4
D!
?-
K
µ
,
(π )
Eµ = mµ + pµ ,π =
2
2
Eµ( ) = mµ2 + pµ2 , K =
K
,.
M π2 + mµ2
2M π
M K2 + mµ2
2M K
%
(π )
→ Eµ
( 0.139 ) + ( 0.106 )
=
2
2
0.110 GeV
2*0.139
→ Eµ( ) =
K
( 0.494 ) + ( 0.106 )
2
2*0.494
2
0.258 GeV
π
,
µ
!
C
&
E%
!"
#
π
%
C
##
! π
#
C
7 ,.
p
1
2M
2
-
M 2, K
m 2 M 2, K
56.5
m2
248.2
,K
π
%
#
MeV
C
5!
+9
dP
d cos
*
E*
E
dP
dE
dP
dE
1
2
p* cos
1
2
p*
dP
d cos
*
d cos
*
dE
*
p*d cos
dE
1+
E * max
1-
E * min
*
d cos
*
dE
p*
+9
#
&
p
p
p
p
p
p
m2
m2
2p
p
m2
m2
m2
2 E
) E
m2
p
2
p
p
2
p p
2 E E
E
p cos
m
E
m2
m2
m2
m2
4 E sin 2
2 E 1 cos
2
'
E
m2
E
m2
2
θ
"
&
4
)
:
7
!
" 0
F
?:
#
)
78
G
G
!&
!
787 8
#
←
G
!
'
G
G
N part
i =1
piµ = P µ G
#
-
#
#
*
%
*
8% →78H 8%
θeφ
#
8 →78I 8J
#
&
,.
&
!
,.
%
%
!
,.
7
#
4
:
p1
'$
#
p2
P µ = p1µ + p2µ
P
B
P 2 = ( E1 + E2 , p1 + p2 ) 2 = ( E1 + E2 ) − ( p1 + p2 ) = m12 + m22 + 2 E1 E2 − 2 p1 ⋅ p2
2
0
B
B
P
P
2
'$
4
P2
B
!
&
'$
#
P = p1 + p2 + p3
#
"
s = P2 = M 2
s1 ≡ s23 = ( P − p1 ) = ( p2 + p3 )
2
2
s2 ≡ s13 = ( P − p2 ) = ( p1 + p3 )
2
s3 ≡ s12 = ( P − p3 ) = ( p1 + p2 )
2
2
2
s B
4 s1, s2 e s3
&
&
B
!
B
" /&
4
B
i
si
s1, s2 e s3
#
s1 + s2 + s3 = M 2 + m12 + m22 + m32
si
0
s1
7
#
s1 = M 2 + m12 − 2 ME1
E1≥m1
s1 ≤ ( M − m1 )
2
D
s1 ≡ s23 ≥ ( m2 + m3 )
s2
0
2
s3
#
( m2 + m3 ) ≤ s1 ≤ ( M − m1 )
2
2
( m1 + m3 ) ≤ s2 ≤ ( M − m2 )
2
2
( m1 + m2 ) ≤ s3 ≤ ( M − m3 )
2
2
s2
4
0
s1
#
7
:
& !
E
3
#
"4
F
p ^2 = −p 3^
p1^ = P ^
(
= (E
)
)
s1 = E ^ − E1^
^
2
+ E3^
2
=
(
M 2 + p12 − m12 + p12
)
2
2
(
)
= λ (s ,m ,m )
= λ (s ,m ,m )
p1^2 = λ s1 , M 2 , m12
p2^2
p3^2
1
2
2
2
3
1
2
2
2
3
con
λ ( x,y,z ) = x 2 + y 2 + z 2 − 2 xy − 2 yz − 2 zx
s2#
6
(
s2 = ( p1 + p3 ) = m12 + m32 + 2 E1^ E3^ − p1^ ⋅ p 3^
2
& !p2^
s2
)
p3^
"
s2#
s2± = m12 + m32 + 2 ( E1^ E3^ ± p1^ p3^ )
E1^
E3^
s2± = m12 + m22 +
s1
1
2 s1
( s − s − m )( s − m
1
2
1
2
2
)
(
)
(
+ m12 ± λ1 2 s1 , s, m12 λ 1 2 s1 , m22 , m32
!s1
0
:"
#
s1 = M 2 + m12 − 2 ME1
E1max
s1min ≡ s1−
p1max
E1 , p1
#
p1max =
1
2M
M 2 − ( m1 + m2 + m3 )
2
M 2 − ( m2 + m3 − m1 )
2
)
" D
! "
!:"
)1
!&" !'"
:
P
/ '$
:
#
)1
P
+
,
,
C
G
p1
7
p2
:
p3
:
,
,
,
,
,
p4
&
'
C&
C'
G&
G'
&
'
9
C9
G9
9
P
9 + 9:
,
,
,
p1
9
p2
9:
p3
9&
p4
9'
,
,
9
,
,
$ 9
9C
9G
9
$ 9C
$ 9G
$ 9
7
!
:"
!& '"
#
"
:
'$
>
7
s = ( p1 + p2 ) = ( p3 + p3 ) = P 2
2
D
2
#
(
s = (E
)
+E )
s = E1* + E2*
*
3
2
* 2
4
= m12 + m22 + 2m2 E1
= m32 + m42 + 2 ( E3 E4 − p3 ⋅ p 4 )
E1 =
1
s − m12 − m22
2m2
p1 =
1 12
λ s, m12 , m22
2m2
(
)
(
m2
s
1
*
2
=
+ m2 E1
E1,2
m1,2
s
p* = p1
(
)
)
s,
7
#
m1,m2, ..,mn
7
s≥
#
n
i =1
s
"
pLAB
"
mi !
:
'$
#
(
t = ( p1 − p3 ) = m12 + m32 − 2 ( E1 E3 − p1 ⋅ p3 ) = m12 + m32 − 2 E1* E3* − p* ⋅ p ' *
2
(
)
u = ( p1 − p4 ) = m12 + m42 − 2 ( E1 E4 − p1 ⋅ p 4 ) = m12 + m42 − 2 E1* E4* + p* ⋅ p ' *
2
D
#
cos θ3 =
1
1
t − m12 − m32 + E1 E3
p1 p 3 2
cos θ3* =
1
1
t − m12 − m32 + E1* E3*
p * p '* 2
(
)
(
)
s + t + u = m12 + m22 + m32 + m42
/
/
#
w ∝ d 3p1d 3p 2 ...d 3p nδ 4 ( P − p1 − p2 − ... − pn )
1 1
1
...
Fin→out
2 E1 2 E2 2 En
#
•
d 3p i
2 Ei
•
& 4
H
)
•
δ.
!
#
"
4
7
R2 ( E ) = d 3p1d 3p 2δ 4 ( P − p1 − p2 )
1 1
2 E1 2 E2
D:
D:
!
"
"
:
#
B
B
:
:
B
!
# B
B
7
B B
4
7
"
7
5
δ
B
B
#
d3 p
= d 4 pϑ ( E ) δ ( p 2 − m 2 )
2E
θ! "
B
!
"
G
δ f ( x) =
δ ( x-x i )
zeri di f
∂f
∂x
xi
f ( x ) : p 2 − m2 = E 2 − p 2 − m 2
zeri : E± = ± p 2 + m 2
∂f
∂E
= 2E
E±
δ ( p 2 − m2 ) =
(
d pϑ ( E ) δ p − m
4
2
2
δ ( E − E+ ) + δ ( E − E− )
2E
)=d
3
p dEϑ ( E )
δ ( E − E+ ) + δ ( E − E− )
2E
d3 p
=
2E
)
R2 ( E ) =
:
#
d 3 p1 d 3 p2 4
δ ( P − p1 − p2 ) = δ 4 ( P − p1 − p2 ) δ p12 − m12 δ p22 − m22 d 4 p1d 4 p2
2 E1 2 E2
:
δ#
R2 ( E ) = δ 4 ( P − p1 − p2 ) δ p12 − m12 δ p22 − m22 d 4 p1d 4 p2
= δ p12 − m12 δ
( P − p1 )
− m22 d 4 p1
δ
B
B
!
#
'$
"
2
!
"
7
#
δ
( P − p1 )
2
(M − E )
(M + E
− m22 = δ
*
1
=δ
2
2
*2
1
− p1*2 − m22
)
− 2ME1* − p1*2 − m22
= δ M 2 − 2 ME1* + m12 − m22
)
#
R2 ( E ) = δ p12 − m12 δ
6
2
− m22 d 4 p1
=
d 3 p1*
δ M 2 − 2 ME1* + m12 − m22
2 E1*
=
d 3 p1*
δ M 2 − 2 ME1* + m12 − m22
*
2 E1
#
E2
d 3p
0
( P − p1 )
p2
m2
EdE
p 2 dpd
p2
pdp
E
dE
p
dp
E
dEd
p
pEdEd
#
R2 ( E ) =
1 *
p1 δ M 2 − 2 ME1* + m12 − m22 dE1*d Ω1*
2
δ
*
#
P1*
d Ω1*
4M
R2 ( E ) =
M 2 + m12 − m22 λ
=
2M
λ 1 2 M 2 , m12 , m22
→ R2 ( E ) =
8M 2
12
P1* = P2* =
(
)
(M
2
, m12 , m22
)
2M
d Ω1*
"* B
#
R2 ( E ) =
P* B
πλ 1 2 ( M 2 , m12 , m22 )
7
2M 2
=
π P*
2M
" %
B
7#
dR2 ( E )
d Ω1*
=
λ 1 2 ( M 2 , m12 , m22 )
8M 2
/
7#
dw
d 1*
"
1
4
&
#
3
R3 E
i 1
/
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Distribuzioni angolari
1.6
1.4
1.2
Distribuzione
1
0.8
β=0.9
0.6
0.4
0.2
β=0.1
0
-1
-0.5
0
0.5
1
cos θ
5 3!
#
%
5 I!
#
#
(
%
#
34%→ 4
s
p1
p2
t
p1
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p1
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s
t
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2
2
2
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m22
m32
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2
,
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%
%
K
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&
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t
p1
p3
u
p1
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m42
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p1 p 4
2
,.
cos
*
s t
!
u
m12
s, m12 , m22
m22 m32
s, m32 , m42
m42
12
"
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cos
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m12
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m32
s
s, m12 , m22
m12
m22 u
u , m22 , m32
m22
12
"
#
)
D
#
$
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&
M
8
//
A &, N ) ;
m32
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