The equation of state in Standard Model Tomasz Westwa´nski Institute of Physics, University of Silesia Poland
The equation of state in Standard Model – p.1/24
Introduction. The tree level Lagrangian. The phenomenological basis for the formulation of the Standard Model (MS) is given by the following empirical facts:
The equation of state in Standard Model – p.2/24
Introduction. The tree level Lagrangian. The phenomenological basis for the formulation of the Standard Model (MS) is given by the following empirical facts: • The SU(2) ×U(1) family structure of the fermions:
The fermions appear as families with left-handed doublets and right-handed singlets: νµ ντ u c t νe , , , , , d s b e µ τ L
eR ,
L
µR ,
τR ,
L
uR ,
dR ,
L
cR ,
sR ,
L
tR ,
L
bR
The equation of state in Standard Model – p.2/24
Introduction. • They can be characterized by the quantum numbers
of weak isospin I, I3 and the weak hypercharge Y .
The equation of state in Standard Model – p.3/24
Introduction. • They can be characterized by the quantum numbers
of weak isospin I, I3 and the weak hypercharge Y . • Between the quantum numbers classifying the
fermions with respect to the group SU(2) ×U(1) and their electric charges Q the Gell-Mann-Nishijima relation is valid. Y Q = I3 + 2
The equation of state in Standard Model – p.3/24
Introduction. • They can be characterized by the quantum numbers
of weak isospin I, I3 and the weak hypercharge Y . • Between the quantum numbers classifying the
fermions with respect to the group SU(2) ×U(1) and their electric charges Q the Gell-Mann-Nishijima relation is valid. Y Q = I3 + 2 • The existence of vector bosons: γ, W + , W − , Z.
The equation of state in Standard Model – p.3/24
Introduction. • This empirical structure can be embedded in a gauge
invariant field theory of the unified electromagnetic and weak interactions by interpreting SU(2) ×U(1) as the group of gauge transformations under which the Lagrangian is invariant.
The equation of state in Standard Model – p.4/24
Introduction. • This empirical structure can be embedded in a gauge
invariant field theory of the unified electromagnetic and weak interactions by interpreting SU(2) ×U(1) as the group of gauge transformations under which the Lagrangian is invariant. • This full symmetry has to be broken by the Higgs
mechanism down to the electromagnetic gauge symmetry; otherwise the W ± , Z bosons would also be massless.
The equation of state in Standard Model – p.4/24
Introduction. • This empirical structure can be embedded in a gauge
invariant field theory of the unified electromagnetic and weak interactions by interpreting SU(2) ×U(1) as the group of gauge transformations under which the Lagrangian is invariant. • This full symmetry has to be broken by the Higgs
mechanism down to the electromagnetic gauge symmetry; otherwise the W ± , Z bosons would also be massless. • The Standard Model requires a single scalar field
(Higgs field) which is a doublet under SU(2).
The equation of state in Standard Model – p.4/24
The classical Lagrangian. The electroweak Lagrangian is given in the form: L = LG + LH + LF + LYukawa
The equation of state in Standard Model – p.5/24
The classical Lagrangian. The electroweak Lagrangian is given in the form: L = LG + LH + LF + LYukawa • Gauge fields.
• SU(2) ×U(1) is a non-Abelian group which is generated by the isospin operators I1 , I2 , I3 and the hypercharge Y .
The equation of state in Standard Model – p.5/24
The classical Lagrangian. The electroweak Lagrangian is given in the form: L = LG + LH + LF + LYukawa • Gauge fields.
• SU(2) ×U(1) is a non-Abelian group which is generated by the isospin operators I1 , I2 , I3 and the hypercharge Y . • Each of these charges is associated with a vector field: a isotriplet of vector fields Wµ1,2,3 with I 1,2,3 and a isosinglet field Bµ with Y .
The equation of state in Standard Model – p.5/24
The classical Lagrangian. • The field strength tensors: a Wµν = ∂µWνa − ∂νWµa + g2 εabcWµbWνc
Bµν = ∂µ Bν − ∂ν Bµ
The equation of state in Standard Model – p.6/24
The classical Lagrangian. • The field strength tensors: a Wµν = ∂µWνa − ∂νWµa + g2 εabcWµbWνc
Bµν = ∂µ Bν − ∂ν Bµ • Parameters g2 and g1 denote the non-Abelian SU(2)
gauge coupling constant and the Abelian U(1) coupling, respectively.
The equation of state in Standard Model – p.6/24
The classical Lagrangian. • The field strength tensors: a Wµν = ∂µWνa − ∂νWµa + g2 εabcWµbWνc
Bµν = ∂µ Bν − ∂ν Bµ • Parameters g2 and g1 denote the non-Abelian SU(2)
gauge coupling constant and the Abelian U(1) coupling, respectively. • The gauge field Lagrangian:
1 a µν,a 1 LG = − WµνW − Bµν Bµν 4 4
The equation of state in Standard Model – p.6/24
The classical Lagrangian. • Fermion fields and fermion-gauge interaction.
The equation of state in Standard Model – p.7/24
The classical Lagrangian. • Fermion fields and fermion-gauge interaction. • Lagrangian LF :
¯ L iγµ Dµ ψL + ∑ ψ ¯ R iγµ Dµ ψR LF = ∑ ψ
The equation of state in Standard Model – p.7/24
The classical Lagrangian. • Fermion fields and fermion-gauge interaction. • Lagrangian LF :
¯ L iγµ Dµ ψL + ∑ ψ ¯ R iγµ Dµ ψR LF = ∑ ψ • The covariant derivative
Y a Dµ = ∂µ − ig2 IaWµ + ig1
2
Bµ
The equation of state in Standard Model – p.7/24
The classical Lagrangian. • Fermion fields and fermion-gauge interaction. • Lagrangian LF :
¯ L iγµ Dµ ψL + ∑ ψ ¯ R iγµ Dµ ψR LF = ∑ ψ • The covariant derivative
Y a Dµ = ∂µ − ig2 IaWµ + ig1
2
Bµ
For singlets ⇒ Ia = 0.
The equation of state in Standard Model – p.7/24
The classical Lagrangian. • Higgs field, Higgs-gauge field and Yukawa coupling.
The equation of state in Standard Model – p.8/24
The classical Lagrangian. • Higgs field, Higgs-gauge field and Yukawa coupling. • A single complex scalar doublet field
Φ(x) =
φ+ (x) φ0 (x)
The equation of state in Standard Model – p.8/24
The classical Lagrangian. • Higgs field, Higgs-gauge field and Yukawa coupling. • A single complex scalar doublet field
• Lagrangian LH
Φ(x) =
φ+ (x) φ0 (x)
LH = (Dµ Φ)+ (Dµ Φ) −V (Φ)
The equation of state in Standard Model – p.8/24
The classical Lagrangian. • Higgs field, Higgs-gauge field and Yukawa coupling. • A single complex scalar doublet field
• Lagrangian LH
Φ(x) =
φ+ (x) φ0 (x)
LH = (Dµ Φ)+ (Dµ Φ) −V (Φ)
with the covariant derivative g1 a Dµ = ∂µ − ig2 IaWµ + i Bµ 2
The equation of state in Standard Model – p.8/24
The classical Lagrangian. • Potential V (Φ):
λ + 2 V (Φ) = −µ Φ Φ + (Φ Φ) 4 2
where v =
+
2µ √ . λ
The equation of state in Standard Model – p.9/24
The classical Lagrangian. • Potential V (Φ):
λ + 2 V (Φ) = −µ Φ Φ + (Φ Φ) 4 2
where v =
+
2µ √ . λ
• Field Φ(x) can be written as:
Φ(x) =
φ+ (x)
√ (v + h(x) + i χ(x))/ 2
The equation of state in Standard Model – p.9/24
The classical Lagrangian. • In the unitary gauge, the Higgs field has the simple
form:
0 1 Φ(x) = √ 2 v + h(x)
The equation of state in Standard Model – p.10/24
The classical Lagrangian. • In the unitary gauge, the Higgs field has the simple
form:
0 1 Φ(x) = √ 2 v + h(x)
• The real part of φ0 , h(x), describes physical neutral
scalar particles with mass
√
MH = µ 2.
The equation of state in Standard Model – p.10/24
The classical Lagrangian. • In the unitary gauge, the Higgs field has the simple
form:
0 1 Φ(x) = √ 2 v + h(x)
• The real part of φ0 , h(x), describes physical neutral
scalar particles with mass
√
MH = µ 2. • The Yukawa Lagrangian has a following form:
mf ¯ f ψf h ¯ f ψf −∑ ψ LYukawa = − ∑ m f ψ f f v
The equation of state in Standard Model – p.10/24
The equation of state. • The stress-energy density tensor Tµν :
∂L Tµν = 2 − gµν L ∂ gµν
The equation of state in Standard Model – p.11/24
The equation of state. • The stress-energy density tensor Tµν :
∂L Tµν = 2 − gµν L ∂ gµν • Hamiltonian H
H=
Z
d 3 x T00
The equation of state in Standard Model – p.11/24
The equation of state. • The stress-energy density tensor Tµν :
∂L Tµν = 2 − gµν L ∂ gµν • Hamiltonian H
H=
Z
d 3 x T00
Pi =
Z
d 3 x Tii
• Pressure Pi
The equation of state in Standard Model – p.11/24
The equation of state. • The stress-energy density tensor Tµν :
∂L Tµν = 2 − gµν L ∂ gµν • Hamiltonian H
H=
Z
d 3 x T00
Pi =
Z
d 3 x Tii
• Pressure Pi
• We have to calculate average energy density (ε) and
average pressure (P)
The equation of state in Standard Model – p.11/24
The equation of state. • The complete Lagrangian has the following form:
1 a µν,a 1 ¯ L iγµ Dµ ψL + ∑ ψ ¯ R iγµ Dµ ψR L = − WµνW − Bµν Bµν + ∑ ψ 4 4 mf + µ ¯ f ψf −∑ ¯ f ψf h +(Dµ Φ) (D Φ) −V (Φ) − ∑ m f ψ ψ f f v
The equation of state in Standard Model – p.12/24
The equation of state. • The complete Lagrangian has the following form:
1 a µν,a 1 ¯ L iγµ Dµ ψL + ∑ ψ ¯ R iγµ Dµ ψR L = − WµνW − Bµν Bµν + ∑ ψ 4 4 mf + µ ¯ f ψf −∑ ¯ f ψf h +(Dµ Φ) (D Φ) −V (Φ) − ∑ m f ψ ψ f f v • For electron and Higgs:
1 ¯ e (i γ ∂µ − me )ψe + (∂µ h(x))2 −V (Φ) Le,Higgs = ψ 2 µ
The equation of state in Standard Model – p.12/24
The equation of state. • The complete Lagrangian has the following form:
1 a µν,a 1 ¯ L iγµ Dµ ψL + ∑ ψ ¯ R iγµ Dµ ψR L = − WµνW − Bµν Bµν + ∑ ψ 4 4 mf + µ ¯ f ψf −∑ ¯ f ψf h +(Dµ Φ) (D Φ) −V (Φ) − ∑ m f ψ ψ f f v • For electron and Higgs:
1 ¯ e (i γ ∂µ − me )ψe + (∂µ h(x))2 −V (Φ) Le,Higgs = ψ 2 µ
with 2 4 2 2 2 φ φ φ m m V (Φ) ∼ m2H (− + 2 ) + H (3 2 − 1)h2 (x) + H2 h4 (x) 4 8v 4 v 8v
The equation of state in Standard Model – p.12/24
The equation of state. • Average value < A >
< A >= Tr(ρA),
1 −β(H−µN) ρ= e Z
1 β= KB T
The equation of state in Standard Model – p.13/24
The equation of state. • Average value < A >
< A >= Tr(ρA),
1 −β(H−µN) ρ= e Z
1 β= KB T
• Electron field:
ψe =
2
∑
α=1
Z
d3 k ¯ α) u(k, ¯ α) e−ikx c( k, (2π)3 2E
The equation of state in Standard Model – p.13/24
The equation of state. • Average value < A >
< A >= Tr(ρA),
1 −β(H−µN) ρ= e Z
1 β= KB T
• Electron field:
ψe =
2
∑
α=1
Z
d3 k ¯ α) u(k, ¯ α) e−ikx c( k, (2π)3 2E
• The Higgs field:
h(x) =
Z
d3 k ¯ e−ikx + a+ (k) ¯ eikx ] [a( k) (2π)3 2E
The equation of state in Standard Model – p.13/24
The equation of state. • The partition function for fermions (FD): +
< c c >∼
1 eβ(Ek −µ) + 1
The equation of state in Standard Model – p.14/24
The equation of state. • The partition function for fermions (FD): +
< c c >∼
1 eβ(Ek −µ) + 1
• The partition function for bosons (BE): +
< a a >∼
1 eβ(Ek −µ) − 1
The equation of state in Standard Model – p.14/24
The equation of state. • The partition function for fermions (FD): +
< c c >∼
1 eβ(Ek −µ) + 1
• The partition function for bosons (BE): +
< a a >∼
1 eβ(Ek −µ) − 1
• The mean field approximation:
< A2 > ∼ < A > 2
The equation of state in Standard Model – p.14/24
The equation of state. • Electron part of average energy density:
εe = 8π
Z ∞ 0
dk¯ k¯ 2 Ek
1 eβ(Ek −µ) + 1
The equation of state in Standard Model – p.15/24
The equation of state. • Electron part of average energy density:
εe = 8π
Z ∞ 0
dk¯ k¯ 2 Ek
1 eβ(Ek −µ) + 1
• The change of variable k¯ → x:
µ = me + µ¯ ,
x = β(Ek − me ),
q Ek = k¯ 2 + m2e ,
y = β¯µ
The equation of state in Standard Model – p.15/24
The equation of state. • Electron part of average energy density:
εe = 8π
Z ∞ 0
dk¯ k¯ 2 Ek
1 eβ(Ek −µ) + 1
• The change of variable k¯ → x:
µ = me + µ¯ ,
x = β(Ek − me ),
• Fermi-Dirac integrals:
Fj (y) =
Z ∞ 0
dt
q Ek = k¯ 2 + m2e , t
y = β¯µ
j 2
e(t−y) ± 1
The equation of state in Standard Model – p.15/24
The equation of state. • We can rewrite εe
εe = 8π(2me ) √
3/2
+8π 2(me )
(KB T )
5/2
5/2
(KB T )
Z ∞
dx
x
0
3/2
Z ∞ 0
dx
3 2
3 KB T x ( 2me + 1) 2 ex−ye + 1
x
1 2
1 KB T x ( 2me + 1) 2 ex−ye + 1
The equation of state in Standard Model – p.16/24
The equation of state. • We can rewrite εe
εe = 8π(2me ) √
3/2
+8π 2(me )
(KB T )
5/2
5/2
(KB T )
Z ∞
dx
x
0
3/2
Z ∞ 0
dx
3 2
3 KB T x ( 2me + 1) 2 ex−ye + 1
x
1 2
1 KB T x ( 2me + 1) 2 ex−ye + 1
• Let’s take very high temperature KB T >> me , then:
εe = 8π(KB T )
4
Z ∞ 0
x3 + 8π m2e (KB T )2 dx x−y e e +1
with (0) me = m e
Z ∞ 0
dx
x ex−ye + 1
φ v
The equation of state in Standard Model – p.16/24
The equation of state. • Similarly, we can calculate εHiggs
εHiggs = 2π(2mH ) √
3/2
+2π 2(mH ) +m2H (−
φ2 4
(KB T )
5/2
+ φ2
5/2
(KB T )
φ4 8v2
Z ∞
dx
x
3 2
0
3/2
Z ∞
dx
3 KB T x 2 ( 2mH + 1) ex−yH − 1
x
0
) Z ∞
1 2
1 KB T x ( 2mH + 1) 2 ex−yH − 1
1 2
1 KB T x 2 ( 2mH + 1) ex−yH − 1
x 3π 5/2 3/2 + √ mH ( 2 − 1)(KB T ) dx v 0 2 "Z 1 K Tx 1 #2 ∞ 2( B 2 x m7H 2mH + 1) 3 dx + 2 (KB T ) 4v ex−yH − 1 0
The equation of state in Standard Model – p.17/24
The equation of state. • Expanding expression in bracket into the series:
KB T x +1 2mH
n 2
n KB T ∼ 1+ x 2 2mH
The equation of state in Standard Model – p.18/24
The equation of state. • Expanding expression in bracket into the series:
KB T x +1 2mH
n 2
n KB T ∼ 1+ x 2 2mH
• We will get:
2 4 φ φ ( 12 ) ( 23 ) ( 25 ) 2 εHiggs = mH (− + 2 ) + (. . . )IH + (. . . )IH + (. . . )IH 4 8v
where
( 2j ) IH
denotes ( 2j )
IH =
Z ∞ 0
dx
x
j 2
ex−yH − 1
The equation of state in Standard Model – p.18/24
The equation of state. • The polylogarithm functions
Lin (z) =
∞
∑
zk /kn
k=1
The equation of state in Standard Model – p.19/24
The equation of state. • The polylogarithm functions
Lin (z) =
∞
∑
zk /kn
k=1
• Fermi-Dirac integrals have been calculated using
MATHEMATICA 5.1 1√ ( 21 ) IH = π Li3/2 (eyH ), 2 ( 25 ) IH
15 √ = π Li7/2 (eyH ), 8
( 23 ) IH
3√ = π Li5/2 (eyH ) 4
with yH = −mH /KB T
The equation of state in Standard Model – p.19/24
The equation of state. • The polylogarithm functions
Lin (z) =
∞
∑
zk /kn
k=1
• Fermi-Dirac integrals have been calculated using
MATHEMATICA 5.1 1√ ( 21 ) IH = π Li3/2 (eyH ), 2 ( 25 ) IH
15 √ = π Li7/2 (eyH ), 8
⇒ integrals
( 2j ) IH
( 23 ) IH
3√ = π Li5/2 (eyH ) 4
with yH = −mH /KB T
∼ 0
The equation of state in Standard Model – p.19/24
The equation of state. • Fermi-Dirac integrals for electron part: ( j)
Ie =
Z ∞ 0
dx
xj ex−ye + 1
The equation of state in Standard Model – p.20/24
The equation of state. • Fermi-Dirac integrals for electron part: ( j)
Ie = (1) Ie
Z ∞
ye
0
= −Li2 (−e ),
dx
xj ex−ye + 1 (3) Ie
= −Li4 (−eye )
The equation of state in Standard Model – p.20/24
The equation of state. • Fermi-Dirac integrals for electron part: ( j)
Ie = (1) Ie
Z ∞ 0
dx
xj ex−ye + 1
ye
= −Li2 (−e ), ye =
(0) µ − me
(3) Ie
= −Li4 (−eye )
φ /KB T v
The equation of state in Standard Model – p.20/24
The equation of state. • Average pressure Pi :
8 Pi = π(2me )3/2 (KB T )5/2 3
Z ∞
dx
0
Z ∞
x
3 2
3 KB T x ( 2me + 1) 2 ex−ye + 1
3 2
3 KB T x ( 2mH + 1) 2 ex−yH − 1
x 2 3/2 5/2 + π(2mH ) (KB T ) dx 3 0 2 4 φ 1 2 φ − mH (− + 2 ) 3 4 8v 1 K Tx 1 Z ∞ B 2 x 2 ( 2mH + 1) 2 π 5/2 φ 3/2 − √ mH ( 2 − 1)(KB T ) dx v ex−yH − 1 0 2 "Z 1 K Tx 1 #2 B ∞ 2( 2 x + 1) m7H 2m 3 H (K T ) dx − B 12v2 ex−yH − 1 0
The equation of state in Standard Model – p.21/24
Results. Energy density ε as a function of φ for mH = 115 GeV, v = 200 GeV, µe = 1 GeV and KB T = 0 GeV (left) and KB T = 20 GeV (right).
Ε
Ε
j j
The equation of state in Standard Model – p.22/24
Results. Energy density ε as a function of φ for mH = 115 GeV, v = 200 GeV, µe = 1 GeV and KB T = 0 GeV and KB T > 0 GeV .
Ε
j
The equation of state in Standard Model – p.23/24
Results. Equation of state P/ε.
The equation of state in Standard Model – p.24/24