Effects of quark chemical potential
on analytic structure of Landau-gauge gluon propagatorYui Hayashi in collaboration with Kei-Ichi Kondo
Department of Physics, Graduate School of Science and Engineering, Chiba University
arXiv:2008.09012 [hep-th]; see also PRD 99 074001 (2019), 101 074044 (2020).

Gluon in the cold quark matter: massive YM at finite µqIntroduction

The analytic structure of a propagator contains information on the spectrum
We investigate the analytic structure of the gluon propagator D(k0, ~k → 0) atand in-medium behavior.
T → 0, µq > 0 and NF = 2 quarks of mass mq at the best-fit parameter
Massive Yang-Mills model: an effective model of the Landau-gauge pureg = 4.5, M = 0.42 GeV with G = SU (3).
Yang-Mills theory [Tissier and Wschebor 2011] or QCD [Peláez et al. 2014].
An NP = 4 region appears between mq . µq . 0.8M ≈ 0.33 GeV. In this→ We study the analytic structure of the gluon propagator in dense QCD using theregion, the gluon propagator has two pairs of complex conjugate poles withmassive Yang-Mills model.
respect to k02.
Complex poles of in-medium propagators
In the NP = 4 region, the gluon propagator has almost real complex poles at
In-medium propagator with complex poles: an analytically continued
Re k0 ≈ 2µq .
propagator D(z, ~k) from Matsubara frequencies D(z = iωn, ~k) has theWith almost real poles, the real part and imaginary part (to be identified withfollowing form
the spectral function) of the gluon propagator D(k02 + i) on the real axis have
Z ∞nX~~narrow
peaks
atk≈
2µ.
ρ(σ,k)
Z(k)0q`2dσ 2D(z, ~k) =+,2The
ratio
ω
/ω
ofacomplexpole
k=ω+iω
,(ω>0,ω>0)
tends
to2~
σ−zI
R0RIRIw
(k)
−z0`
`=1
increase as µq increases, except for the almost real poles. (“less particlelike”)
1
ρ(σ, ~k) = Im D(σ + i, ~k).
0.8π
In the vacuum case, such complex poles in the Landau-gauge gluon propagator
have been widely discussed. Complex poles represent deviation from an0.6
observable particle and could be relevant to confinement, since they invalidate
N =2the Källén-Lehmann spectral representation.
m0.4
The analytic continuation is in principle not unique. However, as aM
straightforward generalization of the well-known theorem [Baym and Mermin 1961],
the two conditions
Pq

2

2

1
20.2

D(z) → 0 as |z| → ∞,D(z) is holomorphic except for the real axis and a finite number of poles.

suffice to determine the correct continuation.
Im z 2

Counting complex poles: Argument principle
I0D(z)
12dz= NZ − NP .
NW (C) : =2πi C
D(z)Some relations to spectral function: (under suitable0.00.0z20.20.40.60.8

Modulus of the gluon propagator at mq = 0.13
GeV and µq = 0.25 GeV on the complex k02
plane.

μ2M2C

The number of complex poles NP = −NW (C)computed numericallyRe z 2assumptions)

NP = 4

positive spectral function: NW (C) = 0 ⇒ NP = NZ .
negative spectral function:
NW (C) = −2 ⇒ NP = NZ + 2 > 0.

Massive Yang-Mills model
The Landau gauge Yang-Mills theory (α → 0) + gluon mass term
1 A A
1 2 A A1A 2AAB B
LmY M = Fµν Fµν + (∂µAµ ) + c̄ ∂µDµ[A] c + M Aµ Aµ42α2

ωI [GeV]ωR [GeV]

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0
0.0

0.2

0.4

0.6

0.8

1.0

ωR [GeV]

0.0
0.0

0.2

0.4

0.6

0.8

1.0μq [GeV]

Location of poles k0 = ωR + iωI at mq = 0.13 GeV. (Left) plots of (ωR, ωI ) (Right) ωR vs µq

The gluon and ghost propagator agree strikinglywith the lattice results even in the strict one-looplevel.
For some renormalization conditions andparameters, the running coupling has no Landaupole in all scales. [Tissier and Wschebor 2011]
This model with dynamical quarks reproduces theunquenched lattice gluon and ghost propagators
as well. [Peláez et al. 2014].
A similar model in the Landau-deWitt gauge predicts a sensible deconfinement
temperature. [Reinosa et al. 2014]
At finite µq , the gluon propagator has been compared to the lattice results in
QC2D. [Suenaga and Kojo 2019] (Suenaga-san’s poster)In the vacuum case (T = µq = 0), we find:the gluon propagator has a negative spectral function and therefore two
22complex poles (NP = 2) for any parameters (g , M ).
With NF = 2 dynamical quarks, the gluon propagator has two complex poles(NP = 2) at the best-fit parameter.

1.0

30

25

22

220.65

M Re DT (k + iϵ), M Im DT (k + iϵ)
0.8

0.64

20

C2 (G) g 2

0.63

0.5

1516 π20.60.62

ξ1052-4-2

210203040

321-42-24

k 2 @GeV2 D-1

The gluon propagator at g = 4.1, M = 0.45 GeV, G = SU (3) in a suitable renormalization condition (used in [Tissierand Wschebor 2011]) in the pure Yang-Mills case. It has one pair of complex poles at k 2 = 0.23 ± 0.42i GeV268100.60
0.59

- 0.5

M 2 /μ2

Re DHk 2 +iΕL, Im DHk 2 +iΕL@GeV-2 D
442k /M

0.2

000.61

0.4

0.58
0.0
0.0

0.2

0.4

0.6

0.8

Real and imaginary parts of the gluon propagatorζat mq = 0.13 GeV and µq = 0.25 GeVContour plot of max ωI /ωR of polek0 = ωR + iωI

1.0

The quark chemical potential significantly affects the gluon propagator aroundk0 ≈ 2µq , which is the least energy for the quark-pair production at ~k = 0.
The appearance of the new pair of almost real poles suggests a transition in
the confined dynamics or, if the almost real pole are artifacts, wouldcorrespond to a long-lived quasi-particle at ωR ≈ 2µq .

Summary
The uniqueness of analytic continuation of the Matsubara propagator holds ina class of functions that vanish at infinity and are holomorphic except for at
most a finite number of complex poles and singularities on the real axis.
At T = µq = 0, in the massive Yang-Mills model (with NF = 0 or with
NF = 2 quarks at the best-fit parameter), an effective theory of the Landau
gauge Yang-Mills theory, the gluon propagator has one pair of complex
conjugate poles in the one-loop level.
The quark chemical potential significantly affects the gluon propagator aroundk0 ≈ 2µq . In particular, for mq . µq . 0.8M ≈ 0.33 GeV, the gluon
propagator has a new pair of complex conjugate poles near the realaxis at Re k0 ≈ 2µq and Im k0 ≈ 0. This suggests a transition of confined
degrees of freedom or appearance of quasi-particle at ωR ≈ 2µq .