Real-Time Static Octet Potential in hot QCD
Nina Gausmann
Institut für Theoretische Physik
WWU Münster
2nd February 2009
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Outline
Motivation
Heavy Quarkonium
Non-Relativistic Schrödinger Equation
qq-Potential
Decay Mechanisms at vanishing temperature
J/ψ Suppression by QGP
Static Potential
Real-Time Static Singlet Potential
Dilepton Production
Real-Time Static Octet Potential
Scales of Thermal QCD
Effective theories
NRQCD
pNRQCD
Leading Thermal Effects to Potential
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Heavy Quarkonia
Definition
Quarkonium, a flavour-neutral meson, consists of a heavy quark
and its antiquark.
just for heavy quarks
high masses (much larger than ΛQCD )
non-relativistic speed
assumption: movement of quarks in a static potential
analogue: hydrogen atom
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Non-Relativistic Schrödinger Equation
discovery of heavy quarkonium
description of bound states by Schrödinger Equation:
Non-Relativistic Schrödinger Equation
∆
− + V (r ) ψ = E ψ
M
M: heavy quark mass
V (r ): static potential
E : binding energy
goal: derivation of a finite-temperature potential by
generalising SE
spectra of potential not purely Coulombic
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
qq-Potential
Popular potential model:
Cornell Potential
V (r ) = a/r + br
r : effective radius
a, b: parameters
first part:
1/r form equivalent to Coulomb potential
exchange of gluons
r → 0: pure Coulombic
second part:
potential increases linearly, development of flux tube in
between qq-pair
non-perturbative effects of QCD
r → ∞ ⇒ V → ∞ confinement!
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
qq-Potential
Experimentally obtained potential:
Potential
V (r ) = −
4 αs (r )
+ kr
3 r
αs : strong coupling constant (r dependence!)
k: string tension
string tension of cc is of magnitude k ≈ 1 GeV/fm
asymptotic behaviour
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Decay Mechanisms at vanishing temperature
Decay of Quarkonia
1
2
Decay through weak interaction
String Breaking: production of a light qq-pair resulting in a
break up of original bound state
strong interaction
3
Annihilation of heavy q and q
electromagnetic and strong interaction
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Decay Mechanisms at vanishing temperature
Decay of Quarkonia
1
Decay through weak interaction
strong and electromagnetic much faster than weak decay
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Decay Mechanisms at vanishing temperature
Decay of Quarkonia
2
String Breaking: production of a light qq-pair resulting
in a break up of original bound state
strong interaction
for static heavy qq-pair just strong interaction
minimum excitation energy necessary
creation of light qq-pair
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Decay Mechanisms at vanishing temperature
Decay of Quarkonia
3
Annihilation of heavy q and q
electromagnetic and strong interaction
strong: conservation of colour and parity → 3 gluons
electromagnetic: real or virtual photons
production of dilepton pairs
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
J/ψ Suppression
Quark-Gluon Plasma(QGP)
hot temperatures
quarks, gluons move freely over distances more than 1 fm
deconfinement
J/ψ Suppression by QGP[1]
J/ψ suppression: unambigous (model-independent) indicator
of QGP-formation
quarkonium dissociation due to colour screening
also J/ψ suppression in nuclear matter phase
assumption: medium effects can be explained by
temperature-dependent potential
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Real-Time Static Singlet Potential
potential in Minkowski time at finite temperature
first non-trivial order in Hard Thermal Loop (HTL) resummed
perturbation theory
defined by the time evolution of mesonic correlator
Mesonic Correlator
21
iC (t, r) ≡
Z
E
D
r
r
d 3 x ψ̄(t, x + )γ µ W ψ(t, x − )ψ̄(0, 0)γµ ψ(0, 0)
2
2
Generalising Schrödinger equation:
Modified Schrödinger Equation
4r
21
i∂t C (t, r) = 2M −
+ V (t, r) C 21 (t, r)
M
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Real-Time Static Singlet Potential
infinitely heavy quark propagators represented by Wilson lines
apart from phase factor e −iMt
Wilson Line (series expansion)
Z
z1
dx µ Aµ (x)
Z z1
Z x
+ (ig )2
dx µ
dy µ Aµ (x)Aν (y ) + · · ·
W [z1 , z0 ] = 1 + ig
z0
z0
z0
gauge fields: Aµ = Aaµ T a
Wilson loop in temporal gauge:
Figure: Symbolic representation of the Wilson loop to O g 2
Nina Gausmann
[5]
Real-Time Static Octet Potential in hot QCD
Real-Time Static Singlet Potential
potential is complex quantity
real part: screened Coulomb potential
imaginary part: medium-induced damping effects (Landau
damping)
classical non-perturbative approximation
definition of potential between two static sources
derivation of energy and thermal decay width of static quarks
Real-Time Static Singlet Potential
V (r) = lim
t→∞
Nina Gausmann
i∂t C21 (t, r)
C21 (t, r)
Real-Time Static Octet Potential in hot QCD
Spectral Function
modification of spectral function by imaginary part of
potential
Spectral Function
Z∞
ρV (Q) ≡
Z
dt
3
d xe
iQx
h
i
1 ˆµ
ˆ
J (x), Jµ (0)
2
−∞
colour current: Jˆµ ≡ ψ̄ˆγ µ ψ̂
heavy quark field operator: ψ̂
h
i
thermal expectation value: h...i ≡ Z −1 Tr (...) e −β Ĥ
inverse temperature: β ≡ 1/T
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Melting of Heavy Quarkonium
at threshold (Q 2 ∼
= 2M 2 ): resonance peak
height and width functions of temperature (in collision)
expected widening of peak with increasing temperature
Figure: Phenomenological results for the spectral function ρV [6]
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Dilepton Production
leptons: small cross section with QGP
consider µ+ µ−-pairs
Production Rate
dNµ− µ+
−2e 4 Z 2
=
d4 xd4 Q
3 (2π)5 Q 2
2mµ2
1+ 2
Q
!
4mµ2
1− 2
Q
!1
2
nB q 0 ρV (Q)
Z : heavy quark electric charge
nB : Bose-Einstein distribution function
Q = Pµ+ + Pµ− : total four-momentum
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Dilepton Production Rate
M: pole mass of bottom quark
Figure: Phenomenological results for dilepton production rate
Nina Gausmann
dNµ− µ+ [7]
d4 xd4 Q
Real-Time Static Octet Potential in hot QCD
Melting of Heavy Quarkonium
interval for bottom quark mass
uncertainties of several hundred MeV
practical limitations
non-equilibrium features
background effects
energy resolution of detector
but complete vanishing of peak still recognizable
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Real-Time Static Octet Potential
above deconfinement region
imaginary part of potential from thermal fluctuations
break up of colour singlet bound state into octet qq state +
gluons
forbidden at zero temperature
Static Octet Potential
o (t, r)
i∂t C21
t→∞ C o (t, r)
21
V o (r) = lim
Octet Meson
r r r
r
ψ − W 0, − T a W 0,
ψ
2
2
2
2
singlet meson interrupted by generator T a
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Real-Time Static Octet Potential
Correlation Function
o
C21
(t, r)
=
h r r i
1
Tr T̂ W t,
, 0,
2
Nc
2
h
r i
h
i
r
×W (t, 0) , t,
T a W t, − , (t, 0)
2
2
h
r r i
×W 0, − , t, −
2
2
h
i
r i b h r ×W (0, 0) , 0, −
T W 0,
, (0, 0)
2
2
T̂ : time-ordering
T a : Hermitean generators of SU(Nc )
normalised by Tr[T a T b ] = 21 δab
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Ansatz
Wilson Line
Z
z1
dx µ Aµ (x)
Z z1
Z x
+ (ig )2
dx µ
dy µ Aµ (x)Aν (y ) + · · ·
W [z1 , z0 ] = 1 + ig
z0
z0
z0
11 (k):
Fourier transform of time-ordered gluon propagator G̃µν
11
hT̂ {Aaµ (x)Abν (y )}i = δab iGµν
(x − y )
Z
d4 k −ik(x−y ) 11
= δab
e
i G̃µν (k)
(2π)4
related to retarded propagator G̃ R (k) and Wightman
propagator G̃ 21 (k) by:
11
G̃33
(k) = G̃ R (k) + G̃ 12 (k) = G̃ R (k) + e −βω G̃ 21 (k)
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Resummed Propagator
Schwinger-Dyson Equation
G̃
G̃
−1
⇒ G̃ =
= G + G · Π · G̃
= G −1 − Π
1
1
∼
= 2
G −1 − Π
k −Π
G∼
= k12 : bare gluon propagator
G̃ : resummed gluon propagator
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Resummed Propagator
Kubo-Martin-Schwinger (KMS) Condition
i G̃ 21 (k) = −2 (nB (ω) + 1) ImG̃R (k)
static case: nB (ω) =
ω→0 1
1
≈ βω
e βω −1
Gluon Propagator
G̃ijR (k) = −
1
1
k2
A
(k)
−
Bij (k)
ij
k 2 − ΠT (k)
k2 − ΠL (k) ω 2
Π: gluon self-energy
A, B: transversal and longitudinal projectors
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Debye Screening
Transversal and Longitudinal Projectors
Aij (k) = δij −
Bij (k) =
ki kj
k2
ki kj
k2
Transversal and Longitudinal Self Energies
2
mD
ω2
ω2 − k 2
ω+k
ΠT (k) =
1−
log
2 k2
2ωk
ω−k
ω
ω+k
2
log
−1
ΠL (k) = mD
2k
ω−k
Static Limit
2
lim Π(ω, k) = mD
∼ g 2T 2
ω→0
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Scales of Thermal QCD
Bound states at finite temperature are characterized by
Scales of Thermal QCD
inverse distance 1/r
temperature scale T
Debye mass
inverse length scale of chromoelectric screening
lower energy scales (gT ,g 2 T )
characteristic QCD scale ΛQCD (as for T = 0)
heavy quarkonia: M large in comparison to ΛQCD
near threshold: momenta ν small compared to masses M
ν/M 1
perturbation expansion breaks down when αs ∼ ν
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Scales of Thermal QCD
Characterisation of heavy quarkonia by three widely seperated
scales:
Scales
M - hard scale
mass M of heavy quarks
description in perturbation theory possible → asymptotic
freedom
Mν - soft scale
relative momentum of heavy qq
Mν 2
- ultrasoft scale
typical kinetic energy of heavy qq
Effective Field Theories
describing observables of particular energy range
integrating out degrees of freedom
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Non-relativistic QCD
heavy quarkonium → possibility to use NR picture
interested in physics at low energy scale E
integrating out energy scales larger than E
Degrees of Freedom
energy scales smaller than masses of static : no further pairs
ψ † (x) creates a quark
χ(x) creates an antiquark
gluon fields in covariant derivative Dµ
Leading-order Lagrangian density
1 2
1 2
†
†
L = ψ iD0 +
D ψ + χ iD0 −
D χ
2m
2m
principle: consistently integrating out mass from QCD and
expanding order by order in 1/M
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
potential Non-Relativistic QCD
Integrating out all energy scales but Mν 2 → pNRQCD
Differences from NRQCD
for NRQCD m must be larger than remaining scales
(|p|, E , ΛQCD , ...)
at scale of binding energy E : unphysical degrees of freedom
light degrees of freedom and heavy quarks with fluctuation of
energy ∼ |p| E
integrating out → new effective theory: pNRQCD
depends on relative size of ΛQCD compared to |p| and E
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
pNRQCD - Weak-Coupling Regime
Weak-Coupling Regime
|p| ΛQCD
perturbation theory applicable
Q − Q state can be decomposed into singlet and octet state
colour gauge transformation
in QED: no analogue to octet state
if E & ΛQCD : EFT in weak coupling regime
if E ΛQCD : necessary to integratep
out energy scale ΛQCD
and corresponding momentum scale ΛQCD m
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
pNRQCD - Strong-Coupling Regime
Strong-Coupling Regime
ΛQCD E
|p| & ΛQCD
cannot integrate out energy degrees of freedom at scale |p|
perturbatively in αs
ultrasoft scales → integrating out scales of larger energies
ground state energy: static QCD potential
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Leading Thermal Effects to Potential[4]
T ≤E
no thermal contributions to potential
Coulomb potential
thermal effects: loop corrections induced by low-energy gluons
thermal width due to thermal fluctuation
at short range: colour singlet qq-state breaks up into octet
state and gluons
dominant contribution for T = E
T : temperature
E : bound state energy
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Leading Thermal Effects to Potential[4]
1/r > T > E and mD ≤ E
potential develops real and imaginary part
contribution to imaginary part
singlet and octet thermal breakup
imaginary part of gluon self-energy
induced by Landau damping
mD < E
dominant contribution: singlet to octet thermal breakup
1/r : inverse distance
mD : Debye mass
1/r > T > E and mD > E
additional contribution to potential
HTL resummed gluon propagators
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
References
1
T. Matsui and H. Satz, Phys. Lett. B 178 (1986) 416
2
M. Laine, O. Philipsen, P. Romatschke and M. Tassler JHEP
03 (2007) 054
3
M. Laine, arXiv:0810.1112 [hep-ph]
4
N. Brambilla, J. Ghiglieri, A. Vairo and P. Petreczky, Phys.
Rev. D 78 (2008) 014017
5
M. Tassler, arXiv:0812.3225 [hep-lat]
6
M. Laine, JHEP 05 (2007) 028
7
Y. Burnier, M. Laine and M. Vepsäläinen, JHEP 01 (2008)
043
8
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Rev. Mod.
Phys. 77 (2005) 1423 [arXiv:hep-ph/0410047]
Nina Gausmann
Real-Time Static Octet Potential in hot QCD
Thank you for your attention!
Nina Gausmann
Real-Time Static Octet Potential in hot QCD