High Energy Photons from Quark-Gluon Plamsa
The 1991 paper from Kapusta,Lichard and Seibert, Phys. Rev. D44, 2774
Moritz Greif - HIC Seminar
Goethe University Frankfurt
10.06.2013
Moritz Greif (Proseminar)
10.06.2013
1
Motivation
Outline
1
Motivation
2
Rates of photon production
In general kinetic theory
Derivation of Self-energy- Dependence of the rate
3
Some details about given statements
4
For interest: self energy in imaginary time formalism (!?)
Moritz Greif (Proseminar)
10.06.2013
2
Motivation
Sources of direct photons
1
γ 0 s from initial nucleon-nucleon interactions (prompt photons; pQCD):
qa + gb → qc + γ or qa + q̄b → gc + γ
2
3
Interaction of q/q̄ -Jets with the medium, e.g.: q jet + q̄ medium → g + γ .
Jet momentum transferred to the photon! (Phys.Rev.Lett.90:132301,2003)
Thermal photons from medium in hadronic or QGP phase
(Data: around 2009)
Moritz Greif (Proseminar)
10.06.2013
3
Rates of photon production
Outline
1
Motivation
2
Rates of photon production
In general kinetic theory
Derivation of Self-energy- Dependence of the rate
3
Some details about given statements
4
For interest: self energy in imaginary time formalism (!?)
Moritz Greif (Proseminar)
10.06.2013
4
Rates of photon production
In general kinetic theory
Rate in relativistic kinetic theory, Eq.(2)/(3)
number of reactions per unit time per unit
7
volume that produce a photon E d4dxdn3 k
Maybe many reactions possible: R = R1 + R2 + ...
Contribution from Ri , e.g. Particle(p1 )+Particle(p2 ) ⇒ Particle(p3 ) +
Photon(p), as denition:
Photon production rate:
Ri = N
R
P
d3 p1
d3 p2
d3 p3
d3 p
(2π)4 δ ( pin
2E1 (2π)3 2E2 (2π)3 2E3 (2π)3 2E(2π)3
−
P
pout )
×f1 f2 (1 ± f3 ) |Mi |
Moritz Greif (Proseminar)
10.06.2013
5
Rates of photon production
Derivation of Self-energy- Dependence of the rate
Rate in eld theory, Eq.(1)
Initial hadronic state | ii ⇒ nal hadronic state | f i
Transition rate: Rf i =
|Sf i |
(observation time) V
Write Sf i = hf | d4 xJˆµ (x)Aµ (x) | ii
R
All QCD inside Jˆµ (x), Aµ EM-current operator
A few steps later


Rf i = −
g µν (2π)4
ωV


δ(k + pi − pf ) + δ(k + pf − pi )
|
{z
}
Emission!
D
ED
E
× f | Jˆµ (0) | i i | Jˆν (0) | f
see e.g. Appendix A from Nucl. Phys. B357 (1991), 65-89, "Vector dominance model at nite temperature",
Kapusta, Gale; or Charles Gale's review
Moritz Greif (Proseminar)
arXiv:0904.2184
10.06.2013
6
Rates of photon production
Derivation of Self-energy- Dependence of the rate
Rate depends on current-current correlator
Keep only emission term!
Average over initial state with Boltzmann weight
Sum over nal states
≡Z
dR
g µν V 1
E 3 =−
d k
V (2π)3 Z
z X }|
e
{ X
−βHi
initial st. i
(2π)4 δ(k + pf − pi )
nal st.f
D
ED
E
× f | Jˆµ (0) | i i | Jˆν (0) | f
|
{z
}
F
Our aim: Find F in terms of the self energy!
Moritz Greif (Proseminar)
10.06.2013
7
Rates of photon production
Derivation of Self-energy- Dependence of the rate
Some denitions of nite temperature QFT
Finite-Temperature expectation value:
h...i ≡
1
Tr {exp (−βH) ...}
Z
Often:
Lesser/Greater quantities: Index <, >,time ordered on the countour
e.g. propagators:
i∆> (x, y) = hφ(x)φ(y)i
i∆< (x, y) = hφ(y)φ(x)i
Important: (KMS-relation):
∆> (t − iβ, ~x) = ∆< (t, ~x)
Moritz Greif (Proseminar)
10.06.2013
8
Rates of photon production
Derivation of Self-energy- Dependence of the rate
Self Energy: deniton as current-current-corellator!
Greater/Lesser self energies:
D
E
†
Π>
(x,
y)
=
j
(x)j
(y)
µ
µν
ν
D
Eirreducibel
<
†
Πµν (x, y) = jµ (y)jµ (x)
irreducibel
complete self energy:
Π(x, y) = Π(x, y)singular + Θ(tx , ty )Π> (x, y) + Θ(ty , tx )Π< (x, y)
(Θ(t1 , t2 )
=1
if t1 later on contour than t2 ...)
Retarded self energy
Πret (x) ≡ Θ(t) Π> − Π<
(1)
Read e.g. Hendriks manuscript on transport theory: http: // fias. uni-frankfurt. de/ ~hees/ , or P.Danielewicz,
"Quantum Theory of Nonequilibrium Processes"
Moritz Greif (Proseminar)
10.06.2013
9
Rates of photon production
Derivation of Self-energy- Dependence of the rate
Some useful and well known equations
Fourier-transformed: Π̃ret (k), and special Θ-representation (residuum):
Kramers-Kronig relation
Π̃ret (k)
Z∞
=
dz A0 (z, ~k)
2π k 0 − z + iη
(2)
−∞
Spectral function
h
i
A0 (z, ~k) = i Π̃< − Π̃>
(3)
KMS-relation in Fourier space
0
Π̃> (k) = ek β Π̃< → Π̃< =
Moritz Greif (Proseminar)
−iA0
1 − ek0 β
(4)
10.06.2013
10
Rates of photon production
Derivation of Self-energy- Dependence of the rate
How we get the imaginary part of Π̃ret into play
1
1
δ(a − b) = − Im
π
a − b + iη
(
)
Z∞
~k)
A
(z,
dz
0
Im
ImΠ̃ret (k) =
2π
k 0 − z + iη
−∞
Z∞
=
−∞
dz
Re A0 (z, ~k) (−π)δ(k0 − z)
2π |
{z
}
=A0
⇒ A0 (k , ~k) = −2ImΠ̃ret (k)
0
Take home: Spectral function is (prop. to) imaginary part of the
self energy!!!
Moritz Greif (Proseminar)
10.06.2013
11
Rates of photon production
Derivation of Self-energy- Dependence of the rate
Identify Π̃< (k) with the rate !
Using the KMS-relation:
Π̃< (k) =
−iΠ<
µν (x) =
=
−2
ek0 β−1
ImΠ̃ret (k)
(5)
o
1 n −βH †
Tr e
jµ (0)jν (x)
Z
E
D
1 X
exp (−βHi ) i | jµ† (0) | f hf | jν (0) | ii
Z
|
{z
}
f,i
F
⇒ iΠ̃<
µν (k) = −
Z
d4 x
X
e−ikx ei(pi −pf )x e−βHi F
f,i
1 X −βHi
= −
e
(2π)4 δ(pi − pf − k)F
Z
f,i
Take home: Several expressions like that for the lesser self energy exist.
Moritz Greif (Proseminar)
10.06.2013
12
Rates of photon production
Derivation of Self-energy- Dependence of the rate
Identify Π̃< (k) with the rate !
Remember:
E
dR
g µν V 1
=
−
d3 k
V (2π)3 Z
X
initial st. i
e−βHi
X
(2π)4 δ(k + pf − pi )F
nal st.f
And
Π̃<
µν (k) = i
−2
1 X −βHi
e
(2π)4 δ(pi − pf − k)F = k0 β−1 ImΠ̃ret (k)
Z
e
f,i
(6)
yields:
Thermal emission rate of photons with (E, p~):
E
dR
2
1
=−
g µν Π̃ret
µν
3
3
Eβ
d p
(2π) e − 1
(7)
Take home: Photon emission rate prop. to retarded self energy!
Moritz Greif (Proseminar)
10.06.2013
13
Some details about given statements
Outline
1
Motivation
2
Rates of photon production
In general kinetic theory
Derivation of Self-energy- Dependence of the rate
3
Some details about given statements
4
For interest: self energy in imaginary time formalism (!?)
Moritz Greif (Proseminar)
10.06.2013
14
Some details about given statements
The cutting of diagrams...
Imaginary part of self Energy ∼ Sum of cut diagrams
ImC =
C − C?
2i
Good papers:
H. Arthur Weldon, "Simple rules for discontinuities in nite
temperature eld theory", 1983, Phys. rev. D28
Details about cutting procedure for higher loops: M.E. Carrington,
Hou Defu, R. Kobes, "Scattering amplitudes at nite temperatures"
Moritz Greif (Proseminar)
10.06.2013
15
Some details about given statements
Integration, eq.(11),eq.(12)
s = (p1 + p2 )2
t = (p1 − p)2
−s + kc2
≤ t ≤ −kc2
2kc2 ≤ s < ∞
s = ∞ =⇒ −∞ ≤ t ≤ −kc2
s = 2kc2 =⇒ −kc2
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≤ t ≤ −kc2 ⇒ 0
10.06.2013
16
Some details about given statements
Legendre-functions of the second kind
Q0 (x) =
Q1 (x) =
Moritz Greif (Proseminar)
1
1+x
ln
2
1−x
x 1+x
−1
2 1−x
(8)
10.06.2013
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Some details about given statements
Invariant momentum distribution of the photon (52)
simply isotropic.
dN
d3 p
dN
⇒
dΩ
dN
⇒
dE
E
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=
=
=
EdN
dE E 2 dΩ
1
4π
E
δ(E − E0 )
E0
10.06.2013
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Some details about given statements
Idea behind equation (62)
Recall: Fugacity λ ≡ eµ/T is also λ ≡ n/neq Rate equation in general:
∂
n = nproduction − ndestruction
∂t
(9)
And ndestruction = λnproduction
chemical potential µ = 0 ⇒ chemical equilibrium, λ = 1, no change
of photon number distribution
Large fugacity λ, lots of 3 → 2 processes (destruction), relaxation
towards λ = 1
Fugacity below 1: a lot of production-processes
R
Behind that stands: ∂µ N µ = dΓC[n] = source
Read e.g. chapter 7 of the PHD thesis of Andrej El.
Moritz Greif (Proseminar)
10.06.2013
19
For interest: self energy in imaginary time formalism (!?)
Outline
1
Motivation
2
Rates of photon production
In general kinetic theory
Derivation of Self-energy- Dependence of the rate
3
Some details about given statements
4
For interest: self energy in imaginary time formalism (!?)
Moritz Greif (Proseminar)
10.06.2013
20
For interest: self energy in imaginary time formalism (!?)
Finite temperature QFT, imaginary time
QM-path integral:
E
D
x2 | e−H(τ2 −τ1 ) | x1 =
q(τZ
2 )=x2
Dq e−SE (q)
q(τ1 )=x1
QM-Thermo-part sum:
Z = T r e−βH =
Z
D
E
dq q | e−βH | q =
0
q 0 (β)=q
Z (0)
Dq e−SE (q)
q 0 (0)=q 0 (0)
Generating functional Z[j],..., all with q(0) = q(β)
Matsubara frequencies: q(τ ) =
Moritz Greif (Proseminar)
q
β
m
∞
P
n=−∞
qn e−ωn τ
⇒
ωn =
2πn
β
10.06.2013
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For interest: self energy in imaginary time formalism (!?)
Self Energy
connected diagrams, 2-point-function: Gc2 (x1 , x2 ) =
δ 2 ln(Z[J]) δJ(x1 )δJ(x2 ) J=0
Proper function = 1PI-n-point-function Γn (x1 , ..., xn )
Is inverse to connected n-point function. In Fourier-space, n = 2:
1 = G̃c2 (p)Γ̃2 (p)
Dene Self-Energy: Σ̃(p) = p2 + m2 − Γ̃2 (p)
We have: Gc2 (x1 , x2 ) = p2 +m21−Σ̃(p)
Moritz Greif (Proseminar)
10.06.2013
22
For interest: self energy in imaginary time formalism (!?)
Self Energy-calculation via functional integrals
calculate proper-functions via
Γ2 (x1 , x2 ) =
δ 2 (Legendre-Trafo of ln(Z[J])) δJ(x1 )δJ(x2 )
J=0
Fourier-transform it
calculate self-energy via Σ̃2 (p) = p2 + m2 − Γ2 (x1 , x2 )
⇒ Interactions: inside the generating functional
Moritz Greif (Proseminar)
10.06.2013
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