The equation of state in Standard Model

The equation of state in Standard Model Tomasz Westwa´nski Institute of Physics, University of Silesia Poland The equation of state in Standard Model...
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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



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



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

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