HAL Id: tel-00391727
https://tel.archives-ouvertes.fr/tel-00391727
Submitted on 4 Jun 2009
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Potentiel effectif non-perturbatif, limites sur la masse du
boson de Higgs et applications dynamiques
Hugo Faivre
To cite this version:
Hugo Faivre. Potentiel effectif non-perturbatif, limites sur la masse du boson de Higgs et applications
dynamiques. Physique mathématique [math-ph]. Université Louis Pasteur - Strasbourg I, 2006.
Français. �tel-00391727�
◦
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¯h = c = 1 .
[
] = [
] = [
]
−1
= [
]
−1
.
(+, −, −, −)
$$ *(+, +, +, +)
x
p
x
µ
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0
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p
µ
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0
, ~p)
θ(x)
%δ(x)
,θ(x) =
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x < 0
1
x > 0
,
δ(x) =
d
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θ(x) .
d
δ
(d)
(x) =
Z
d
d
k
(2π)
d
e
i k·x
.
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Z
d
d
k
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d
e
i k·x
f (k)
e
,
f (k) =
e
Z
d
d
x e
−i k·x
f (x)
f (x) =
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1
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d
X
k
e
f
k
e
i k·x
,
f (k) =
e
q
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d
X
x
f
x
e
−i k·x
,
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d
=
R
d
d
x
d
e
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Z
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0
dt e
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t
z−1
.
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Y
n
i=1
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i
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1
2
n
X
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i
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x
j
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n
2
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0
E
1
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λ
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k
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k
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2
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λ
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2
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,φ
t
v
,λ = 5 · 10
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m
2
= −10
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m
2
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−3
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cl
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1l
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µ = 1.1 · 10
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λ = 2 · 10
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m
2
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V
1l
V
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% - " ,V
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k=0
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V
1l
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φ
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0
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0
, t
1
]
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m
2
= −1
λ = 1.1
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J
(q)
ψ
g
(q)
.q(t)
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J
(q)
ψ
g
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q − p
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λ = 0.15
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J
(q)
ψ
g
(q)
.q(t)
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J
(q)
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g
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J
(q)
ψ
g
(q)
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ψ
J
(q)
ψ
g
(q)
# # %q − p
,λ = 0.07
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ef f
V
1l
V
RG
q(0) = 0.5
λ =
0.15
V
RG
V
1l
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ef f
V
1l
V
RG
q(0) = 0.7
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V
RG
V
1l
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ef f
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1
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L =
1
2
∂
µ
φ∂
µ
φ −
1
2
m
2
φ
2
−
24
1
λ φ
4
.
(),' ' ,$ '(&(& +)&( )() +)$ ' '(&(&
L =
R
d
3
x L
'( +) +'(&(&
J(x)
!Z[J] = h0, +∞|0, −∞i
J
= N
Z
Dφ e
i
R
d
4
x
(
L(φ,∂
µ
φ)+J(x)φ(x)
) .
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, ,Dφ
, 2 3 %λ = 0
Z[J]
Z
0
[J] = exp
i
2
Z
d
4
x d
4
y J(x)G
0
(x − y)J(y)
,
Z
0
[0] = 1
G
0
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2
)G
0
(x − y) = δ(x − y) .
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G
(n)
(x
1
, ... , x
n
)
,n
G
(n)
(x
1
, ... , x
n
) = h0| T [φ(x
1
) ... φ(x
n
)] |0i .
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(n)
5 " !n
Z[J]
G
(n)
δ
n
Z[J]
δJ(x
1
) ... δJ(x
n
)
J=0
= (i)
n
G
(n)
(x
1
, ..., x
n
) .
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G
(n)
0
G
0
(x − y)
i
!G
0
(x − y)
G
0
(x − y) = lim
→0+
Z
d
3
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0
(2π)
4
e
i k
0
(x
0
−y
0
)−i ~k·(~x−~y)
k
0
2
− ~k
2
− m
2
+ i
,
/
±(~k
2
+ m
2
)
1
2
/ " , 2 3 ! / " ,Z[J]
( $x
0
→ −i x
4
k
0
→ −i k
4
2 3 $$ ,−k
E
2
= −(k
4
2
+ ~k
2
)
−x
E
2
= −(x
4
2
+ ~x
2
)
−k
E
· x
E
= −(k
4
x
4
+ ~k · ~x)
, !G
0E
(x
E
− y
E
) =
Z
d
4
k
E
(2π)
4
e
−k
E
·(x
E
−y
E
)
k
2
E
+ m
2
,
G
e
0E
(p
E
) =
1
p
2
E
+ m
2
,
# ) / $$G
0
(x)|
x
0
→−i x
4
→ i G
0E
(x
E
) ;
G
0E
(x
E
)|
x
4
→i x
0
→ −i G
0
(x) .
( $ ,
Z
E
[J] = N
Z
Dφ e
−
R
d
4
x
E
(
L
E
(φ,∂
µ
φ)+Jφ
) ,
L
E
=
1
2
(∂
µ
φ)
2
+
1
2
m
2
φ
2
+
1
24
λ φ
4
.
E
vr rruw rswxs~wq rw {qrsrurz
L
int
= λφ
4
/24
Z[J]
Z[J] = e
−
R
d
4
x L
int
(
δJ(x)
δ
) Z
0
[J]
=
∞
X
n=0
(−1)
n
n!
"Z
d
4
x L
int
δ
δJ(x)
!#
n
Z
0
[J] .
% ! 'φ
L
int
J
Z
J
λ
!n
) , - ! "Z[J]
J
G
(2)
(x − y)
G
0
(x−y)
λ
φ
4
G
(2)
(x − y) = G
0
(x − y) −
λ
2
G
0
(0)
Z
dz G
0
(z − x) G
0
(z − y)
e
G
(2)
(p) =
G
e
0
(p) −
λ
2
G
0
(0)
G
e
0
(p) .
2 # 3G
(2)
(x − y) =
1
(2π)
4
Z
d
4
p
e
−p(x−y)
p
2
+ m
2
+
λ
2
G
0
(0)
.
# / ,m
2
R
$ % ') $) )() ,()) +) +&) )$ &(', )) ,() $%) $%) $$) (% $) ,())) )('() +() ( $+% &('
m
2
B
Σ
0
m
2
R
= m
2
B
+ Σ
0
Σ
0
=
λ
2
G
0
(0) .
,G
0
(0) = G
0
(x − x)
" #G
0
(0) =
Z
+∞
−∞
d
4
k
1
k
2
+ m
2
∝
Z
Λ
0
dp
p
3
p
2
+ m
2
∝ Λ
2
,
# "Λ
m
2
B
"Λ
, 4* )λ
Z[J]
!Γ
(n)
# 2 # 3e
Γ
(4)
(p
1
, p
2
, p
3
, p
4
) = λ +
3
X
i=1
λ
2
2
Z
d
d
p
(2π)
4
1
p
2
+ m
2
·
1
(p − q
i
)
2
+ m
2
,
q
2
i
's = p
1
2
+ p
2
2
t = p
1
2
+ p
3
2
u = p
1
2
+ p
4
2
,ln (Λ/m
2
)
rus ~ qz ~w qu rswxs ~ wqr , ,!
δm
2
δλ
δZ
" "d
d = 4 − 2
!Γ(n)
d → 4
→ 0
, /1
%MS
/ ,MS
,−γ
ln(4π)
µ
,p
2
= µ
2
2 3e
Γ
(2)
(p)
p
2
=µ
2
= m
2
d
2
dp
2
Γ
e
(2)
(p)
p
2
=µ
2
= 1
e
Γ
(4)
(p
1
, p
2
, p
3
, p
4
)
s=t=u=µ
2
= λ .
, , )δm
2
m
2
B
+ Σ
0
= m
2
R
+ δm
2
+ Σ
0
= m
2
R
+ Σ
R
,
m
2
R
Σ
R
,Σ
0
d
2 3Σ
0
=
Z
d
d
p
(2π)
d
1
p
2
+ m
2
= m
2
λ
(4π)
2
m
2
4πµ
2
!
d
2
−2
1
2
Γ
1 −
d
2
!
.
#
δm
2
= −m
2
λ
(4π)
2
1
MS
→ 0
Σ
R
=
1
2
m
2
λ
(4π)
2
ln
m
2
4πµ
2
!
+ γ − 1
!
,
µ
% )δλ
δZ
) ,φ
4
φ
6
, }sx r {r srus~qz~wqu ,µ
Λ
" 4* " #< Λ
−1
4* " , # 1 ! + +!µ
g(µ)
µ
)β
,µ
∂g
∂µ
= β
g
({g
i
}, µ) ,
{g
i
}
µ
.g(µ)
β
µ
g(µ)
µ
"µ
0
= µ + δµ
g(µ
0
)
δµ = 0
β
, ,γ
m
µ
∂m
∂µ
= m γ
m
({g
i
}) .
,µ
∂
∂µ
ln
√
Z = γ({g
i
}) ,
#γ
φ
4
. " , ! )µ
) ,µ → µ
0
Γ
(n)
)µ
d
dµ
Γ
(n)
(x
1
, ... , x
n
; µ, g
i
) = 0 .
φ
4
"
µ
∂
∂µ
+ β
λ
(λ)
∂
∂λ
− nγ(λ) + m
2
γ
m
2
(λ)
∂
∂m
2
#
Γ
(n)
= 0 .
$ 2 3 !t = ln(µ/µ
0
)
2 3Γ
(n)
({x
i
}, t, λ, m
2
) = exp
−n
Z
t
0
dt
0
γ
λ(t
0
)
Γ
(n)
({x
i
}, t, λ(t), m
2
(t)) ,
g
i
(t)
#g
i
(µ)
,µ
g
i
(t)
t
g
i
(t)
µ
0
= µ
t = 0
∂g
∂t
= β
g
({g
i
})
g
i
(t = 0) = g(µ) .
% 5λ
β
β
1l
λ
(λ) =
16π
3
2
λ
2
µ
0
λ(µ
0
) = λ(µ)
1 +
3λ(µ)
16π
2
ln
µ
0
µ
!
.
$λ(µ
0
) =
λ(µ)
1 −
3λ(µ)
16π
2
ln
µ
0
µ
.
% "V
ef f
$V
ef f
+!V
RGI
"3λ(µ)
16π
2
ln
µ
0
µ
)µ
0
5 +!µ
0
$,(')($ $% ,$+! " (%' #(