text: 9 (cid:90) (cid:88) In Eq. (54) use has been made of the results [19] (cid:90) 2 − D η = η , η = m dc c c (c), ijk(cid:96) s,ijk(cid:96) s,ijk(cid:96) s i j s,k(cid:96) L A M A −m m n s=1 dc m c ( + ) = 1 2 ν D , (50) 1 i 1 1,j 1 2,j ρ D ij (cid:90) (55) (cid:88) (cid:90) (cid:48)(cid:48) 2 (cid:48)(cid:48) (cid:48)(cid:48) − m A D = D , D = s dc c2c (c), ij s,ij s,ij 2T2 i s,j s=1 ρ ∂P(0) − ∂P(0) (51) dc m c A = 1 ij 1,ij , (56) 1 i 1,j ρ ∂x ∂x 1 1 (cid:90) (cid:88) 2 −m B where L = L , L = s dc c2c (c), (52) ij s,ij s,ij 2 i s,j s=1 (cid:88) (cid:90) ν = ρω12 µ (1 + α ). (57) D dρ 21 12 2 2 −m C λ = λ , λ = s dc c2c (c). (53) ij s,ij s,ij 2 i s,j In the hydrodynamic regime, the diﬀusion tensor can s=1 ∗ be written as D = D D where D = (ρT/m m ν ) ∗ ij 0 ij 0 1 2 0 Upon writing Eqs. (44)–(53) use has been made of the and D is a dimensionless function of the reduced shear A B C D ∗ij symmetry properties of r, r, r and r,k(cid:96). In general, rate a≡, the coeﬃcients of restitution αrs, the mass ra- the set of generalized transport coeﬃcients deﬁned above tio µ m1/m2, the ratio of diameters ∗σ1/σ2 and the are nonlinear functions of the shear rate, the coeﬃcients mole fraction x1. The dependence of Dij on the pres- of restitution and the parameters of the mixture (masses, su∗r∝e and−temperature is through the reduced shear rate sizes and concentration). a T1 β/p. Thus, IV. GENERALIZED TRANSPORT ∗ COEFFICIENTS (p∂ + T∂ ) D = (p∂ + T∂ ) D D p T ij p T 0 ij ∗ − − ∗ ∂D This Section is devoted to the evaluation of the gen- = (1 β)Dij βD0a ∂a∗ij . eralized transport coeﬃcients associated with the mass, (58) momentum and heat ﬂuxes. We consider each ﬂux sepa- rately. A. Mass ﬂux Consequently, in dimensionless form, Eq. (54) yields (cid:18) (cid:19) (cid:20) (cid:21) The constitutive form for the mass ﬂux to ﬁrst order in ∗ ∗ spatial gradients is given by Eq. (44). To illustrate with 2a ∗ ∗ − ∗ − (cid:18) ∗ ∂D − ∗ ∗(cid:19) some detail the evaluation of the transport coeﬃcients P + ζ (1 β)D βa ∗ij ν D d xy ij ∂a D ij of the mass ﬂux, let us consider the diﬀusion coeﬃcients ∗ ∗ ∗ ∗ ∗ − ∗ ∗ (cid:18) ρ ∂P − ∂P (cid:19)− 2a ∂P ∂ζ Dij, deﬁned by Eq. (47). These coeﬃcients can be ob- a D = 1 ij 1,ij xy + tained by multiplying both sides of Eq. (39) by m c and ik kj ρ ∂x ∂x d ∂x ∂x 1 j ∗ 1∗ 1∗ 1 1 inte(cid:18)grating over c.(cid:19)After some algebra, one arrives at × ∗ − 2a ∂P ∂ζ ∗ D xy + D . (59) p,ij d ∂x ∂x T,ij 1 1 2a (cid:32) − −(cid:33) P(0) + ζ(0) (p∂ + T∂ ) D ν D a D dp xy p T ij D ij ik kj ∗ ≡ ∗ ≡ ∗ ≡ ∗ ≡ Here, ζ ζ(0)/ν , P P(0))/p, ν ν /ν , a = ρ1 ∂Pi(j0) −(cid:32)∂P1(,0i)j − 2a ∂Px(0y) +(cid:33)p∂ζ(0) aij/ν0, Dp∗,ij ≡ Dp,i0j/Drp,0ij, andrD,ijT∗,ij ≡DDT,ij/DDp00 whijere ρ ∂x ∂x d ∂x ∂x D = (p/ρν ) and we recall that a = aδ δ . It must 1 1 1 1 p0 0 ∗ ∗ ij ix jy ∗ be noted that P and ζ depend also on x and a × ρ2Dp,ij − 2aT ∂Px(0y) + T ∂ζ(0) ρ2DT,ij . through their depr,einj dence on the temperature r1atio γ ≡ m m np dp ∂x ∂x m m nT T /T . 1 2 1 1 1 2 1 2 ∗ (54) ∗The equations deﬁning the (scaled) tensors Dp,ij and D can be obtained by following similar steps as those T,ij ∗ made before for D . After some algebra, the results are ij ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: Eq.

text: use

text: has

text: been

text: made

text: of

text: the

text: results

text: D

text: η

text: η

text: η

text: m

text: dc

text: c

text: c

text: (c),

text: s

text: i

text: j

text: L

text: A

text: M

text: A

text: −m

text: m

text: n

text: dc

text: m

text: c

text: ν

text: D

text: i

text: ρ

text: D

text: ij

text: m

text: A

text: D

text: D

text: D

text: s

text: dc

text: (c),

text: ij

text: s,ij

text: s,ij

text: i

text: s,j

text: ρ

text: dc

text: m

text: c

text: A

text: ij

text: i

text: ρ

text: ∂x

text: ∂x

text: −m

text: B

text: where

text: L

text: L

text: L

text: s

text: dc

text: (c),

text: ij

text: s,ij

text: s,ij

text: i

text: s,j

text: ν

text: ρω12

text: µ

text: α

text: D

text: dρ

text: −m

text: C

text: λ

text: λ

text: λ

text: s

text: dc

text: (c).

text: ij

text: s,ij

text: s,ij

text: i

text: s,j

text: In

text: the

text: hydrodynamic

text: regime,

text: the

text: diﬀusion

text: tensor

text: can

text: be

text: written

text: as

text: D

text: D

text: D

text: where

text: D

text: (ρT/m

text: m

text: ν

text: ij

text: ij

text: Upon

text: writing

text: Eqs.

text: use

text: has

text: been

text: made

text: of

text: the

text: and

text: D

text: is

text: a

text: dimensionless

text: function

text: of

text: the

text: reduced

text: shear

text: A

text: B

text: C

text: D

text: ∗ij

text: symmetry

text: properties

text: of

text: r,

text: r,

text: r

text: and

text: In

text: general,

text: rate

text: a≡,

text: the

text: coeﬃcients

text: of

text: restitution

text: αrs,

text: the

text: mass

text: ra-

text: the

text: set

text: of

text: generalized

text: transport

text: coeﬃcients

text: deﬁned

text: above

text: tio

text: µ

text: the

text: ratio

text: of

text: diameters

text: ∗σ1/σ2

text: and

text: the

text: are

text: nonlinear

text: functions

text: of

text: the

text: shear

text: rate,

text: the

text: coeﬃcients

text: mole

text: fraction

text: The

text: dependence

text: of

text: Dij

text: on

text: the

text: pres-

text: of

text: restitution

text: and

text: the

text: parameters

text: of

text: the

text: mixture

text: (masses,

text: su∗r∝e

text: and−temperature

text: is

text: through

text: the

text: reduced

text: shear

text: rate

text: sizes

text: and

text: concentration).

text: a

text: β/p.

text: Thus,

text: IV.

text: GENERALIZED

text: TRANSPORT

text: COEFFICIENTS

text: (p∂

text: T∂

text: D

text: (p∂

text: T∂

text: D

text: D

text: p

text: T

text: ij

text: p

text: T

text: ij

text: ∂D

text: This

text: Section

text: is

text: devoted

text: to

text: the

text: evaluation

text: of

text: the

text: gen-

text: β)Dij

text: ∂a∗ij

text: eralized

text: transport

text: coeﬃcients

text: associated

text: with

text: the

text: mass,

text: momentum

text: and

text: heat

text: ﬂuxes.

text: We

text: consider

text: each

text: ﬂux

text: sepa-

text: rately.

text: A.

text: Mass

text: ﬂux

text: Consequently,

text: in

text: dimensionless

text: form,

text: Eq.

text: yields

text: The

text: constitutive

text: form

text: for

text: the

text: mass

text: ﬂux

text: to

text: ﬁrst

text: order

text: in

text: spatial

text: gradients

text: is

text: given

text: by

text: Eq.

text: To

text: illustrate

text: with

text: ∂D

text: some

text: detail

text: the

text: evaluation

text: of

text: the

text: transport

text: coeﬃcients

text: P

text: ζ

text: β)D

text: βa

text: ∗ij

text: ν

text: D

text: d

text: xy

text: ij

text: ∂a

text: D

text: ij

text: of

text: the

text: mass

text: ﬂux,

text: let

text: us

text: consider

text: the

text: diﬀusion

text: coeﬃcients

text: ρ

text: ∂P

text: ∂P

text: ∂P

text: ∂ζ

text: Dij,

text: deﬁned

text: by

text: Eq.

text: These

text: coeﬃcients

text: can

text: be

text: ob-

text: a

text: D

text: ij

text: xy

text: tained

text: by

text: multiplying

text: both

text: sides

text: of

text: Eq.

text: by

text: m

text: c

text: and

text: ik

text: kj

text: ρ

text: ∂x

text: ∂x

text: d

text: ∂x

text: ∂x

text: j

text: 1∗

text: 1∗

text: over

text: some

text: algebra,

text: one

text: arrives

text: at

text: ∂P

text: ∂ζ

text: D

text: xy

text: D

text: p,ij

text: d

text: ∂x

text: ∂x

text: T,ij

text: (p∂

text: T∂

text: D

text: ν

text: D

text: a

text: D

text: dp

text: xy

text: p

text: T

text: ij

text: D

text: ij

text: ik

text: kj

text: Here,

text: ζ

text: P

text: ν

text: ν

text: /ν

text: a

text: ρ1

text: Dp∗,ij

text: andrD,ijT∗,ij

text: whijere

text: ρ

text: ∂x

text: ∂x

text: d

text: ∂x

text: ∂x

text: D

text: (p/ρν

text: and

text: we

text: recall

text: that

text: a

text: aδ

text: δ

text: It

text: must

text: ij

text: ix

text: jy

text: be

text: noted

text: that

text: P

text: and

text: ζ

text: depend

text: also

text: on

text: x

text: and

text: a

text: T

text: through

text: their

text: depr,einj

text: dence

text: on

text: the

text: temperature

text: γ

text: m

text: m

text: np

text: dp

text: ∂x

text: ∂x

text: m

text: m

text: nT

text: T

text: /T

text: ∗The

text: equations

text: deﬁning

text: the

text: (scaled)

text: tensors

text: Dp,ij

text: and

text: D

text: can

text: be

text: obtained

text: by

text: following

text: similar

text: steps

text: as

text: those

text: T,ij

text: made

text: before

text: for

text: D

text: After

text: some

text: algebra,

text: the

text: results

text: are

text: ij

text: ##LTLine##

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text: A gauge invariant Debye mass and the complex heavy-quark potential Yannis Burniera, Alexander Rothkopfb aInstitute of Theoretical Physics, EPFL, CH-1015 Lausanne, Switzerland bInstitute for Theoretical Physics, Universita¨t Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany Abstract Following the original idea of Debye, we deﬁne and extract a gauge-invariant screening mass from the complex static in-medium heavy-quark potential VQQ¯, recently obtained from lattice QCD. To this end we derive a ﬁeld theoretically motivated analytic formula that faithfully reproduces both the screened real- as well as the imaginary part of the lattice potential with a single temperature dependent ﬁt parameter mD(T). Using values of the real part of VQQ¯ in a gluonic medium, we obtain Debye masses compatible with predictions from HTL perturbation theory. Keywords: Debye mass, Heavy quarkonium, Complex potential 1. Introduction The concept of a screening mass helps us to intuitively understand the in- tricate interplay of interactions that take place, once a test particle is inserted into a medium of charge carriers. Originally Debye and Hu¨ckel [1] investigated the behavior of ions in electrolyte solutions and found that their interactions could be understood by an exponential suppression of the vacuum Coulomb potential due to the presence of classical Boltzmann distributed charges. This in-medium modiﬁcation, translated into the modern language of quantum ﬁeld theory, amounts to a medium induced dressing of the mediating gauge bosons, bestowing them with an otherwise absent mass. In the context of perturbative quantum electro-dynamics this mechanism of a thermal mass generation is well understood, as arising from the zero momentum limit of the gauge invariant longitudinal gluon self energies. In perturbative QCD, only the leading order and the logarithm at next to leading order (NLO) of the Debye mass can be computed, the constant at NLO already receives non-perturbative contributions [2]. On the lattice, its deﬁnition encounters another diﬃculty, unlike in QED where the Debye mass can be extracted from the electric ﬁeld correlator, in QCD the electric ﬁeld itself is not gauge invariant. Several approaches were proposed to circumvent this problem using e.g. eﬀective theories obtained by Preprint submitted to arXiv May 27, 2020 ##LTLine## ##LTLine##

text: A

label: title

text: gauge

text: invariant

text: Debye

text: mass

text: and

text: the

text: complex

text: heavy-quark

text: potential

text: Yannis

label: author

text: Burniera,

text: Alexander

text: Rothkopfb

text: aInstitute

text: of

text: Theoretical

text: Physics,

text: EPFL,

text: Lausanne,

text: Switzerland

text: bInstitute

text: for

text: Theoretical

text: Physics,

text: Universita¨t

text: Heidelberg,

text: Philosophenweg

text: Heidelberg,

text: Germany

text: Abstract

label: paragraph

text: Following

label: abstract

text: the

text: original

text: idea

text: of

text: Debye,

text: we

text: deﬁne

text: and

text: extract

text: a

text: gauge-invariant

text: screening

text: mass

text: from

text: the

text: complex

text: static

text: in-medium

text: heavy-quark

text: potential

text: VQQ¯,

text: recently

text: obtained

text: from

text: lattice

text: QCD.

text: To

text: this

text: end

text: we

text: derive

text: a

text: ﬁeld

text: theoretically

text: motivated

text: analytic

text: formula

text: that

text: faithfully

text: reproduces

text: both

text: the

text: screened

text: real-

text: as

text: well

text: as

text: the

text: imaginary

text: part

text: of

text: the

text: lattice

text: potential

text: with

text: a

text: single

text: temperature

text: dependent

text: ﬁt

text: parameter

text: mD(T).

text: Using

text: values

text: of

text: the

text: real

text: part

text: of

text: VQQ¯

text: in

text: a

text: gluonic

text: medium,

text: we

text: obtain

text: Debye

text: masses

text: compatible

text: with

text: predictions

text: from

text: HTL

text: perturbation

text: theory.

text: Keywords:

text: Debye

text: mass,

text: Heavy

text: quarkonium,

text: Complex

text: potential

text: Introduction

text: The

text: concept

text: of

text: a

text: screening

text: mass

text: helps

text: us

text: to

text: intuitively

text: understand

text: the

text: in-

text: tricate

text: interplay

text: of

text: interactions

text: that

text: take

text: place,

text: once

text: a

text: test

text: particle

text: is

text: inserted

text: into

text: a

text: medium

text: of

text: charge

text: carriers.

text: Originally

text: Debye

text: and

text: Hu¨ckel

text: investigated

text: the

text: behavior

text: of

text: ions

text: in

text: electrolyte

text: solutions

text: and

text: found

text: that

text: their

text: interactions

text: could

text: be

text: understood

text: by

text: an

text: exponential

text: suppression

text: of

text: the

text: vacuum

text: Coulomb

text: potential

text: due

text: to

text: the

text: presence

text: of

text: classical

text: Boltzmann

text: distributed

text: charges.

text: This

text: in-medium

text: modiﬁcation,

text: translated

text: into

text: the

text: modern

text: language

text: of

text: quantum

text: ﬁeld

text: theory,

text: amounts

text: to

text: a

text: medium

text: induced

text: dressing

text: of

text: the

text: mediating

text: gauge

text: bosons,

text: bestowing

text: them

text: with

text: an

text: otherwise

text: absent

text: mass.

text: In

text: the

text: context

text: of

text: perturbative

text: quantum

text: electro-dynamics

text: this

text: mechanism

text: of

text: a

text: thermal

text: mass

text: generation

text: is

text: well

text: understood,

text: as

text: arising

text: from

text: the

text: zero

text: momentum

text: limit

text: of

text: the

text: gauge

text: invariant

text: longitudinal

text: gluon

text: self

text: energies.

text: In

text: perturbative

text: QCD,

text: only

text: the

text: leading

text: order

text: and

text: the

text: logarithm

text: at

text: next

text: to

text: leading

text: order

text: (NLO)

text: of

text: the

text: Debye

text: mass

text: can

text: be

text: computed,

text: the

text: constant

text: at

text: NLO

text: already

text: receives

text: non-perturbative

text: contributions

text: On

text: the

text: lattice,

text: its

text: deﬁnition

text: encounters

text: another

text: diﬃculty,

text: unlike

text: in

text: QED

text: where

text: the

text: Debye

text: mass

text: can

text: be

text: extracted

text: from

text: the

text: electric

text: ﬁeld

text: correlator,

text: in

text: QCD

text: the

text: electric

text: ﬁeld

text: itself

text: is

text: not

text: gauge

text: invariant.

text: Several

text: approaches

text: were

text: proposed

text: to

text: circumvent

text: this

text: problem

text: using

text: e.g.

text: eﬀective

text: theories

text: obtained

text: by

text: Preprint

text: submitted

text: to

text: arXiv

text: May

text: ##LTLine##

text: ##LTLine##


text: expansion is expected to apply. We see however that including the lower tem- peratures close to T does not change the determination of the parameters c, d c in an signiﬁcant way. In previous studies in quenched QCD, Debye masses were e.g. obtained by ﬁtting not the proper heavy-quark potential but the color-singlet free-energies with a simple Coulombic Debye-screened form [6]. Comparing, we ﬁnd that the values obtained here lie consistently lower than these previous estimates. We would like to note that even though the lattices used to determine the heavy-quark potential deployed in this study are quite ﬁnely spaced, no contin- uum extrapolation has been carried out. Therefore it is doubtful whether the values of the Debye mass shown on the left of Fig.2 together with the values for α˜ and σ can be directly used in phenomenological models in the conti√nuum. s As a possible remedy we follow Ref. [7] in providing the ratio of m / σ on D the right of Fig.2, in which some of the systematic uncertainties arising from a ﬁnite lattice spacing might be expected to cancel. 6. Conclusion Based on a combination of the generalized Gauss law, introduced in [11] and the in-medium modiﬁcation of its point charge distribution by a weakly inter- acting medium of light quarks and gluons, described by the HTL permittivity, we derived an analytic expression for the real- and imaginary part of the static inter-quark potential at ﬁnite temperature. After ﬁxing the strong coupling and string tension at low temperature, we are able to qualitatively and quan- titatively reproduce the real-part of the potential measured in lattice QCD by ﬁtting a single temperature dependent parameter m (T), which is proposed as D gauge invariant screening mass in QCD. The temperature dependence of m we D obtained in a purely gluonic medium agrees well with the predictions of HTL perturbation theory. Using the ﬁtted values for m we furthermore ﬁnd that D a quite successful postdiction of ImV at high temperatures is possible. Agree- ment with ImV at smaller temperatures, currently hampered by uncertainties in the ﬁt of m to the real-part, should improve once higher statistics has been D collected. We hope that phenomenological modeling will beneﬁt from the derivation of a well motivated and lattice data validated parametrization of both ReV and ImV . In addition our study opens up the possibility to extract the imaginary part of the potential in full QCD simulations, in which up to date only the real part has been determined in a reliable fashion. Acknowledgments The authors thank O. Kaczmarek, P. Petreczky and R. Pisarski for stimu- lating discussions. YB is supported by SNF grant PZ00P2-142524. References References [1] P. Debye, H. Hu¨ckel, Physikalische Zeitschrift 24, 185-206 (1923). 11 ##LTLine##

text: expansion

text: is

text: expected

text: to

text: apply.

text: We

text: see

text: however

text: that

text: including

text: the

text: lower

text: tem-

text: peratures

text: close

text: to

text: T

text: does

text: not

text: change

text: the

text: determination

text: of

text: the

text: parameters

text: c,

text: d

text: c

text: in

text: an

text: signiﬁcant

text: way.

text: In

text: previous

text: studies

text: in

text: quenched

text: QCD,

text: Debye

text: masses

text: were

text: e.g.

text: obtained

text: by

text: ﬁtting

text: not

text: the

text: proper

text: heavy-quark

text: potential

text: but

text: the

text: color-singlet

text: free-energies

text: with

text: a

text: simple

text: Coulombic

text: Debye-screened

text: form

text: Comparing,

text: we

text: ﬁnd

text: that

text: the

text: values

text: obtained

text: here

text: lie

text: consistently

text: lower

text: than

text: these

text: previous

text: estimates.

text: We

text: would

text: like

text: to

text: note

text: that

text: even

text: though

text: the

text: lattices

text: used

text: to

text: determine

text: the

text: heavy-quark

text: potential

text: deployed

text: in

text: this

text: study

text: are

text: quite

text: ﬁnely

text: spaced,

text: no

text: contin-

text: uum

text: extrapolation

text: has

text: been

text: carried

text: out.

text: Therefore

text: it

text: is

text: doubtful

text: whether

text: the

text: values

text: of

text: the

text: Debye

text: mass

text: shown

text: on

text: the

text: left

text: of

text: together

text: with

text: the

text: values

text: for

text: α˜

text: and

text: σ

text: can

text: be

text: directly

text: used

text: in

text: phenomenological

text: models

text: in

text: the

text: conti√nuum.

text: s

text: As

text: a

text: possible

text: remedy

text: we

text: follow

text: Ref.

text: in

text: providing

text: the

text: ratio

text: of

text: m

text: σ

text: on

text: D

text: the

text: right

text: of

text: in

text: which

text: some

text: of

text: the

text: systematic

text: uncertainties

text: arising

text: from

text: a

text: ﬁnite

text: lattice

text: spacing

text: might

text: be

text: expected

text: to

text: cancel.

text: Conclusion

text: Based

text: on

text: a

text: combination

text: of

text: the

text: generalized

text: Gauss

text: law,

text: introduced

text: in

text: and

text: the

text: in-medium

text: modiﬁcation

text: of

text: its

text: point

text: charge

text: distribution

text: by

text: a

text: weakly

text: inter-

text: acting

text: medium

text: of

text: light

text: quarks

text: and

text: gluons,

text: described

text: by

text: the

text: HTL

text: permittivity,

text: we

text: derived

text: an

text: analytic

text: expression

text: for

text: the

text: real-

text: and

text: imaginary

text: part

text: of

text: the

text: static

text: inter-quark

text: potential

text: at

text: ﬁnite

text: temperature.

text: After

text: ﬁxing

text: the

text: strong

text: coupling

text: and

text: string

text: tension

text: at

text: low

text: temperature,

text: we

text: are

text: able

text: to

text: qualitatively

text: and

text: quan-

text: titatively

text: reproduce

text: the

text: real-part

text: of

text: the

text: potential

text: measured

text: in

text: lattice

text: QCD

text: by

text: ﬁtting

text: a

text: single

text: temperature

text: dependent

text: parameter

text: m

text: (T),

text: which

text: is

text: proposed

text: as

text: D

text: gauge

text: invariant

text: screening

text: mass

text: in

text: QCD.

text: The

text: temperature

text: dependence

text: of

text: m

text: we

text: D

text: obtained

text: in

text: a

text: purely

text: gluonic

text: medium

text: agrees

text: well

text: with

text: the

text: predictions

text: of

text: HTL

text: perturbation

text: theory.

text: Using

text: the

text: ﬁtted

text: values

text: for

text: m

text: we

text: furthermore

text: ﬁnd

text: that

text: D

text: a

text: quite

text: successful

text: postdiction

text: of

text: ImV

text: at

text: high

text: temperatures

text: is

text: possible.

text: Agree-

text: ment

text: with

text: ImV

text: at

text: smaller

text: temperatures,

text: currently

text: hampered

text: by

text: uncertainties

text: in

text: the

text: ﬁt

text: of

text: m

text: to

text: the

text: real-part,

text: should

text: improve

text: once

text: higher

text: statistics

text: has

text: been

text: D

text: collected.

text: We

text: hope

text: that

text: phenomenological

text: modeling

text: will

text: beneﬁt

text: from

text: the

text: derivation

text: of

text: a

text: well

text: motivated

text: and

text: lattice

text: data

text: validated

text: parametrization

text: of

text: both

text: ReV

text: and

text: ImV

text: In

text: addition

text: our

text: study

text: opens

text: up

text: the

text: possibility

text: to

text: extract

text: the

text: imaginary

text: part

text: of

text: the

text: potential

text: in

text: full

text: QCD

text: simulations,

text: in

text: which

text: up

text: to

text: date

text: only

text: the

text: real

text: part

text: has

text: been

text: determined

text: in

text: a

text: reliable

text: fashion.

text: Acknowledgments

text: The

text: authors

text: thank

text: O.

text: Kaczmarek,

text: P.

text: Petreczky

text: and

text: R.

text: Pisarski

text: for

text: stimu-

text: lating

text: discussions.

text: YB

text: is

text: supported

text: by

text: SNF

text: grant

text: References

text: References

text: P.

text: Debye,

text: H.

text: Hu¨ckel,

text: Physikalische

text: Zeitschrift

text: ##LTLine##


text: Similar to Ref.[12], we use the perturbative HTL expression − p2 − p m2 ε 1(p(cid:126), m ) = iπT D , (14) D p2 + m2 (p2 + m2 )2 D D henceforth assuming a medium of weakly coupled quarks and gluons, in which our test charge is immersed. Inserting this formula into equation (13) and multiplying by p2+m2D , we obtain: (cid:16) (cid:17) p2 − m2 p2V (p(cid:126)) + m2 V (p(cid:126)) = 4πα˜ 1 iπT D . (15) C D C s p(p2 + m2 ) D The inverse Fourier transform of the real part of Eq. (15) exactly reproduces the linear-response expression of (10), allowing us to identify A = m2 and in C D turn gives an expression for the charge density m2 T n = D . (16) 0 8πα˜ s The imaginary part arising on the RHS of Eq. (15) can also be Fourier trans- formed to coordinate space, which com(cid:16)pletes our generalized fo(cid:17)rmula for the in-medium modiﬁcation of a Coulombic test charge −∇ − 2VC(r) + m2DVC(r(cid:90)) = α˜s 4πδ((cid:126)r) iTm2Dg(mDr) , (17) with ∞ sin(px) p g(x) = 2 dp . (18) px p2 + 1 0 | The solution to Eq. (17) with the physical boundary condition ReV (r) ∞ = | | C r= 0, ImV (r) = 0 and ∂ ImV (r) ∞ = 0 coincides with the HTL potential C r=0 r C r= (2) obtained by Laine et a(cid:18)l. [10] (cid:19) − ReV (r) = −α˜s e mDr + m r , ImV (r) = −α˜ T φ(x), (19) C r r D C s where φ(x) has been deﬁned in Eq. (3). Now we turn to the string-like, i.e. linearly rising part of the test-charge potential. Note that in contrast to the Coulomb case, for a = 1 we cannot transform Eq.(5) into a simple relation for the Fourier space potential akin to Eq.(13). Instead, as a ﬁrst step, we return to Eq. (11) to obtain an explicit expression for the coeﬃcient A . Taking the charge density n to be the same, s 0 irrespective of the test charge being Coulombic or string-like one then obtains together with Eq. (16) σ A = µ4 = m2 . (20) s D α˜ s 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Similar

text: to

text: we

text: use

text: the

text: perturbative

text: HTL

text: expression

text: p

text: ε

text: m

text: iπT

text: D

text: D

text: D

text: D

text: henceforth

text: assuming

text: a

text: medium

text: of

text: weakly

text: coupled

text: quarks

text: and

text: gluons,

text: in

text: which

text: our

text: test

text: charge

text: is

text: immersed.

text: Inserting

text: this

text: formula

text: into

text: equation

text: and

text: multiplying

text: by

text: we

text: obtain:

text: V

text: 4πα˜

text: iπT

text: D

text: C

text: D

text: C

text: s

text: D

text: The

text: inverse

text: Fourier

text: transform

text: of

text: the

text: real

text: part

text: of

text: Eq.

text: exactly

text: reproduces

text: the

text: linear-response

text: expression

text: of

text: allowing

text: us

text: to

text: identify

text: A

text: and

text: in

text: C

text: D

text: turn

text: gives

text: an

text: expression

text: for

text: the

text: charge

text: density

text: T

text: n

text: D

text: 8πα˜

text: s

text: The

text: imaginary

text: part

text: arising

text: on

text: the

text: RHS

text: of

text: Eq.

text: can

text: also

text: be

text: Fourier

text: trans-

text: formed

text: to

text: coordinate

text: space,

text: which

text: our

text: generalized

text: for

text: the

text: in-medium

text: modiﬁcation

text: of

text: a

text: Coulombic

text: test

text: charge

text: α˜s

text: with

text: sin(px)

text: p

text: g(x)

text: dp

text: px

text: The

text: solution

text: to

text: Eq.

text: with

text: the

text: physical

text: boundary

text: condition

text: ReV

text: (r)

text: C

text: r=

text: ImV

text: (r)

text: and

text: ImV

text: (r)

text: coincides

text: with

text: the

text: HTL

text: potential

text: C

text: r

text: C

text: r=

text: obtained

text: by

text: Laine

text: et

text: ReV

text: (r)

text: −α˜s

text: e

text: mDr

text: m

text: r

text: ImV

text: (r)

text: −α˜

text: T

text: φ(x),

text: C

text: r

text: r

text: D

text: C

text: s

text: where

text: φ(x)

text: has

text: been

text: deﬁned

text: in

text: Eq.

text: Now

text: we

text: turn

text: to

text: the

text: string-like,

text: i.e.

text: linearly

text: rising

text: part

text: of

text: the

text: test-charge

text: potential.

text: Note

text: that

text: in

text: contrast

text: to

text: the

text: Coulomb

text: case,

text: for

text: a

text: we

text: cannot

text: transform

text: into

text: a

text: simple

text: relation

text: for

text: the

text: Fourier

text: space

text: potential

text: akin

text: to

text: Instead,

text: as

text: a

text: ﬁrst

text: step,

text: we

text: return

text: to

text: Eq.

text: to

text: obtain

text: an

text: explicit

text: expression

text: for

text: the

text: coeﬃcient

text: A

text: Taking

text: the

text: charge

text: density

text: n

text: to

text: be

text: the

text: same,

text: s

text: irrespective

text: of

text: the

text: test

text: charge

text: being

text: Coulombic

text: or

text: string-like

text: one

text: then

text: obtains

text: together

text: with

text: Eq.

text: σ

text: A

text: µ4

text: s

text: D

text: α˜

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##