Effects of a quark chemical potential on the analytic structure of the gluon propagator
CCHIBA-EP-246, 2020.08.20
Effects of quark chemical potential on analytic structure of gluon propagator
Yui Hayashi ∗ and Kei-Ichi Kondo
1, 2, † Department of Physics, Graduate School of Science and Engineering, Chiba University, Chiba 263-8522, Japan Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan
We perform complex analyses of the gluon propagator at non-zero quark chemical potential inthe long-wavelength limit, using an effective model with a gluon mass term of the Landau-gaugeYang-Mills theory, which is a Landau-gauge limit of the Curci-Ferrari model with quantum correc-tions being included within the one-loop level. We mainly investigate complex poles of the gluonpropagator, which could be relevant to confinement. Around typical values of the model parameters,we show that the gluon propagator has one or two pairs of complex conjugate poles depending onthe value of the chemical potential. In addition to a pair similar to that in the case of zero chemicalpotential, a new pair appears near the real axis when the chemical potential is roughly between theeffective quark mass and the effective gluon mass of the model. We discuss possible interpretationsof these poles. Additionally, we prove the uniqueness of analytic continuation of the Matsubarapropagator to a class of functions that vanish at infinity and are holomorphic except for a finitenumber of complex poles and singularities on the real axis.
I. INTRODUCTION
For a long time, it has been expected that quarkdegrees of freedom would dominate in a highly densematter of quantum chromodynamics (QCD) rather thanhadrons, although details of the phase structure are stillunclear mainly due to the sign-problem [1]. Studyingthe analytic structure of the gluon propagator is of im-portance to this end since this structure provides infor-mation on the in-medium behavior, e.g., whether or nota quasi-particle description is appropriate. We thus ex-plore the analytic structure of the gluon propagator inthis article.The main difficulty in a continuum approach for thequark matter is the breakdown of the perturbation theoryin the infrared QCD. Indeed, perturbative calculations ofdense QCD matter suggest that the quark matter is al-ready strongly correlated at the quark chemical potential µ q (cid:46) the massive Yang-Millsmodel . This mass deformation could be a consequenceof generating the dimension-two gluon condensate [8–12]or avoiding the Gribov ambiguity [13, 14]. The massiveYang-Mills model has the modified Becchi-Rouet-Stora-Tyutin (BRST) symmetry and is multiplicatively renor-malizable to be proved through the modified Slavnov- ∗ Electronic address: [email protected] † Electronic address: [email protected]
Taylor identities (at all orders of the perturbation theory)[7]. Moreover, there exists the “infrared safe” renormal-ization scheme, which respects the non-renormalizationtheorems [15, 16], in that the running gauge couplingconstant g is finite at all scales on some renormalizationgroup (RG) flows [4, 14, 17].This model provides the gluon and ghost propagatorsthat agree strikingly with the numerical lattice resultsjust in the one-loop level [3, 4]. The three-point functions[18] and two-point correlation functions at finite temper-ature [19] were compared to the numerical lattice resultswith good accordance. Furthermore, the two-loop cor-rections improve the agreement for the gluon and ghostpropagators [20]. Therefore, the effective mass capturessome nonperturbative aspects of the Yang-Mills theory.For unquenched cases with the number of quark fla-vors N F = 2 , a r X i v : . [ h e p - t h ] A ug pared with numerical lattice data [27] for the gauge group SU (2). While the singlet diquark gap can improve theconsistency with the lattice results, the agreement withthe lattice results is not quite satisfactory for parameters( g, M ) that are independent of chemical potential. If oneenables the gluon mass parameter M to depend on thechemical potential, one can obtain a fair agreement be-tween the massive Yang-Mills model and the numericallattice results. Therefore, although this model may lacksome important aspects, it is still worthwhile studyingthe analytic structure of the gluon propagator at a finitechemical potential by utilizing the massive Yang-Millsmodel with various model parameters.In the vacuum case, i.e., vanishing chemical potential µ q = 0, we have investigated the analytic structures ofthe gluon, quark, and ghost propagators and revealedthat the gluon and quark propagators have one pair ofcomplex conjugate poles while the ghost propagator hasno complex poles [28–30]. Other several models and a re-construction method also predict such complex poles ofthe gluon propagator, e.g. [13, 31–39]. The DSE with theray technique had provided the gluon propagator holo-morphic except for timelike momenta [40], but the recentstudy [41] has updated this conclusion and strongly sug-gested a singularity on the complex momentum plane.Complex poles invalidate the K¨all´en-Lehmann spectralrepresentation [42] and might correspond to unphysicaldegrees of freedom in an indefinite metric state space [43].Therefore, the complex poles represent deviations fromobservable particles and are expected to be related to theconfinement mechanism. For example, the connectionbetween complex poles of the fermion propagator andconfining potential in three-dimensional quantum elec-trodynamics has been discussed in [44]. Incidentally, an-other generalization of the spectral representation takingunphysical degrees of freedom into account is proposedin [45].In this article, we employ the massive Yang-Millsmodel with quantum corrections being included withinthe one-loop level and investigate the analytic structureof its in-medium gluon propagator at finite quark chem-ical potential µ q . Since we are interested in the long-distance behavior and analytic structure of the gluonpropagator on the complex frequency plane, we performcomplex analyses on the gluon propagator in the long-wavelength limit (cid:126)k →
0. In addition, we consider theuniqueness of the analytic continuation in the presenceof complex poles, since we use the Matsubara propagatorin the zero-temperature limit and the analytic continua-tion is in principle not unique before taking the limit.This article is organized as follows. In the next section,the definition of complex poles of an in-medium propa-gator and the method for counting complex poles arepresented. A proof of the uniqueness in a class of func-tions having a finite number of complex poles is providedin Appendix A. The massive Yang-Mills model and itsone-loop expressions are presented in Sec. III. We detailthe vacuum part of the one-loop expressions in Appendix Im z Re zz (= k ) z = r w ℓ ( ~k ) z = iω n FIG. 1: Schematic picture of singularities on the complex z (= k ) plane. We analytically continue a Matsubara propagator D ( z = iω n , (cid:126)k ) defined at the Matsubara frequencies z = iω n (shown as the dots) to D ( z, (cid:126)k ) on the complex z plane. Thetwo sets of complex conjugate poles in the z plane representa pair of complex conjugate poles with respect to z in (1). B. In Sec. IV and Appendix C, we determine the num-ber of complex poles and their locations in the space ofthe model parameters and the spectral function at a spe-cific set of model parameters. It turns out that the gluonpropagator has one pair of almost real poles in additionto the other pair of complex conjugate poles similar tothe vacuum ones at intermediate quark chemical poten-tial. In Sec. V, we discuss possible interpretations ofthese almost real poles and estimates for slightly large µ q . In Sec. VI, a summary of these findings and futureprospects are given. II. COMPLEX POLES OF IN-MEDIUMPROPAGATORS
In this section, we define complex poles of propagatorsin medium. Then, we introduce a method to count thenumber of complex poles, which is utilized in the subse-quent sections.
A. Definition
In medium, we compute a Matsubara propagator D ( iω n , (cid:126)k ) within the imaginary-time formalism, where ω n is the Matsubara frequency and (cid:126)k is the spatial mo-mentum. We consider the analytic continuation D ( z, (cid:126)k )on the complex z plane for a fixed (cid:126)k from the Matsub-ara frequencies on the imaginary axis z = iω n . Thisprovides information on the spectrum and is useful forstudying linear response, in which the retarded propaga-tor, namely the propagator analytically continued to thereal axis from the upper-half plane, plays an importantrole [46].For a field describing a physical observable particle, theusual spectral representation holds. The spectral condi-tion forces analytically-continued Matsubara propagator D ( z, (cid:126)k ) to have singularities only on real axis z ∈ R .However, the spectral condition may be violated forconfined degrees of freedom, since not all states have tobe physical. Thus, we can consider the possibility of com-plex spectra, which need not be excluded in an indefinitemetric state space [43]. If a state with complex energyexists, this should correspond to a confined state. Fur-ther formal aspects will be discussed elsewhere.Here, we assume the following generalized spectral rep-resentation allowing complex poles for the gluon prop-agator D ( z, (cid:126)k ), which is a propagator obtained by theanalytic continuation from the Matsubara propagator D ( iω n , (cid:126)k ) defined at points on the pure imaginary axis ofthe complex z plane: D ( z, (cid:126)k ) = (cid:90) ∞ dσ ρ ( σ, (cid:126)k ) σ − z + n (cid:88) (cid:96) =1 Z (cid:96) ( (cid:126)k ) w (cid:96) ( (cid:126)k ) − z , (1) ρ ( σ, (cid:126)k ) = 1 π Im D ( σ + i(cid:15), (cid:126)k ) (2)where ρ ( σ, (cid:126)k ) is the spectral function, w (cid:96) ( (cid:126)k ) is a positionof a complex pole, and Z (cid:96) ( (cid:126)k ) is its residue for arbitrarybut fixed (cid:126)k .Notice that, in the vacuum case, there is a one-to-onecorrespondence between the propagator D ( z, (cid:126)k ) analyti-cally continued to the upper-half plane in z and the ana-lytic continuation in the complex k plane ˜ D ( k ), whichhas been considered in the previous articles [28, 30], inthe sense that D ( z, (cid:126)k ) = ˜ D ( z − (cid:126)k ).Since the set of Matsubara frequencies { ω n } has noaccumulation points, uniqueness of the analytic contin-uation is an important problem to be proved. Indeed,there is a well-known theorem saying that the unique-ness holds in a class of functions satisfying (i) D ( z ) → | z | → ∞ and (ii) D ( z ) is holomorphic except for thereal axis, i.e., these two conditions are sufficient to de-termine the unique continuation [47]. Although this the-orem cannot be applied to our case due to the existenceof complex poles, we can generalize this theorem in astraightforward way. In Appendix A, we present a proofof the uniqueness under the weaker conditions allowingcomplex poles:(i) D ( z ) → | z | → ∞ ,(ii) D ( z ) is holomorphic except for singularities on thereal axis and a finite number of complex poles. Therefore, the uniqueness of the analytic continuationfrom the Matsubara propagator is valid in a similar senseeven in the presence of complex poles.Note that complex poles defined here do not corre-spond to poles of quasi-particles. This is because thecomplex poles defined here yield poles in both of theupper-half and lower-half planes in z . While a quasi-particle pole is in the second Riemann sheet in z , thecomplex pole is in the first Riemann sheet. B. Counting complex poles
Let us introduce a method to count the number ofcomplex poles based on the argument principle [28, 30]to be used in the following sections. We can relate apropagator at real frequencies to complex poles and zeros.In the vacuum case, we have applied the method to apropagator on the complex k plane. For an in-mediumpropagator, we can take k as the squared complex fre-quency z . The statement is as follows.Suppose that a complex-valued propagator D ( z ) := D ( z, (cid:126)k ) with a fixed spatial momentum (cid:126)k and its data { D ( z = x n + i(cid:15) ) } for real frequencies z (namely, z > | z | → ∞ , D ( z ) has the same phaseas the free propagator, i.e., arg( − D ( z )) → arg z as | z | → ∞ .(ii) In the limit | z | → D ( z = 0) > { z = x n + i(cid:15) } Nn =0 is sufficiently denseso that D ( z = x + i(cid:15) ) changes its phase at mosthalf-winding ( ± π ) between x n + i(cid:15) and x n +1 + i(cid:15) ,i.e., for n = 0 , , · · · , N , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) x n +1 x n dx ddx arg D ( x + i(cid:15) ) (cid:12)(cid:12)(cid:12)(cid:12) < π, (3)where we denote sufficiently small x = δ > x N +1 = Λ , on which we will takethe limits δ → +0 and Λ → + ∞ .Then the winding number, which is the difference be-tween the number of complex zeros ( N Z ) and poles ( N P )with respect to z , reads N W ( C ) = N Z − N P = − N (cid:88) n =0 π Arg (cid:20) D ( x n +1 + i(cid:15) ) D ( x n + i(cid:15) ) (cid:21) . (4)Thus the number of complex poles N P is given by N P = N Z − N W ( C )= N Z + 1 − N (cid:88) n =0 π Arg (cid:20) D ( x n +1 + i(cid:15) ) D ( x n + i(cid:15) ) (cid:21) . (5)For details of the derivation, see [30]. When the threeconditions (i), (ii), and (iii) hold, we can numericallycompute the number of complex poles N P from thenumber of zeros ( N Z ) and data at the real frequencies { D ( x n + i(cid:15) ) } Throughout this article, N P denotes the number ofcomplex poles on the z plane, i.e., the number of poleson the (upper-)half plane on the z plane, and “complexconjugate poles” denote those on the z plane. The prop-agator has 2 N P complex poles on the whole z complexplane. III. MODEL
In this section, we introduce the massive Yang-Millsmodel, which is regarded as an effective model of theLandau gauge Yang-Mills theory, or the Landau gaugelimit of Curci-Ferrari model, and review the one-loop ex-pressions.
A. Massive Yang-Mills model
The Euclidean Lagrangian of the model at N colorswith N F flavors is given by [3, 4, 21] L mY M = L Y M + L GF + L F P + L m + L q , (6) L Y M = 14 F Aµν F Aµν , L GF = i N A ∂ µ A Aµ L F P = ¯ C A ∂ µ D µ [ A ] AB C B = ¯ C A ∂ µ ( ∂ µ C A + g b f ABC A Bµ C C ) L m = 12 M b A Aµ A Aµ , L q = N F (cid:88) i =1 ¯ ψ i ( γ µ D µ [ A ] + ( m b ) q,i ) ψ i = N F (cid:88) i =1 ¯ ψ i ( γ µ ( ∂ µ − ig b A Aµ t A ) + ( m b ) q,i ) ψ i , (7)where we have introduced the bare gluon, ghost, anti-ghost, Nakanishi-Lautrup, and quark fields denoted by A Aµ , C A , ¯ C A , N A , ( A = 1 , , · · · , N − ψ i ( i =1 , , · · · , N F ) respectively, the bare gauge coupling con-stant g b , the bare gluon mass M b , and the bare quarkmass ( m b ) q,i , while f ABC ( A, B, C = 1 , , · · · , N − t A of the fundamental representation of the group G = SU ( N ).The renormalization factors ( Z A , Z C , Z ¯ C = Z C , Z ψ i ) , Z g , Z M , Z m q,i for the gluon, ghost,anti-ghost, and quark fields ( A µ , C , ¯ C , ψ i ), the gaugecoupling constant g , and the gluon and quark massparameters M , m q,i are introduced respectively as follows: A µ = (cid:112) Z A A µR , C = (cid:112) Z C C R , ¯ C = (cid:112) Z C ¯ C R , ψ i = (cid:112) Z ψ i ψ R,i ,g b = Z g g, M b = Z M M , ( m b ) q,i = Z m q,i m q,i (8)In this article, we consider the two flavor case N F = 2 andemploy this model with degenerate quark masses, m q := m q,i , and therefore Z ψ := Z ψ i and Z m q := Z m q,i . Noticethat the quark mass parameter m q of this model is chosento fit the propagators obtained from other methods, e.g.,numerical lattice results. In particular, the quark massparameter m q is non-zero even for massless quarks dueto the spontaneous breakdown of the chiral symmetry.The general tensorial structure of the gluon propagator D µν ( k E ) reads, from the spatial rotational symmetry andthe transversality of the Landau gauge, D µν ( k E ) = D T ( k E ) P Tµν + D L ( k E ) P Lµν , (9)where k E = ( k , k , k , k ) = ( (cid:126)k, k ) is the Euclidean mo-mentum, P Tµν and P Lµν are the transverse and longitudinalprojectors respectively, i.e., P Tij = δ ij − k i k j (cid:126)k P T i = P Ti = P T = 0 ( i, j = 1 , ,
3) (10)and, P Lµν = P µν − P Tµν , P µν = δ µν − k E,µ k E,ν k E . (11)We define the vacuum part of the gluon and ghost two-point vertex functions Γ (2) A ,vac , Γ (2) gh,vac as the zero temper-ature T = 0 and the zero chemical potential µ = 0 limit, D µν ( k E ) | T = µ =0 = [Γ (2) A ,vac ( k E )] − P µν , ∆ gh ( k E ) | T = µ =0 = − [Γ (2) gh,vac ( k E )] − , (12)where ∆ gh is the ghost propagator.As a renormalization scheme, we adopt the “infraredsafe scheme” [4, 21] respecting the nonrenormalizationtheorem Z A Z C Z M = 1 [15]. For the gluon and ghostsector, we impose Z A Z C Z M = 1Γ (2) A ,vac ( k E = µ ) = µ + M Γ (2) gh,vac ( k E = µ ) = µ (13)combined with the Taylor scheme [16] Z g Z / A Z C = 1 forthe coupling. In this renormalization scheme, it turnsout that there exist RG flows on which the running cou-pling constant is always finite in a whole momentum re-gion, which implies that the perturbation theory is validto some extent. For the quark sector, we put Γ (2) s,vac ( k E = µ ) = m q and B. One-loop expressions
Here we review the results of one-loop calculations ofthe in-medium gluon propagator.Beforehand, we decompose the vacuum polarizationΠ µν ( k E ) into the vacuum part Π vacµν ( k E ) and the mat-ter part Π matµν ( k E ),Π µν ( k E ) = Π vacµν ( k E ) + Π matµν ( k E ) . (14)Π vacµν ( k E ) had been calculated in [3, 4, 21]. For complete-ness, the vacuum part is presented in Appendix B.The relation between Π µν ( k E ) and D µν ( k E ) is givenby the further decomposition of Π matµν ( k E ) as follows: ingeneral, the spatial rotational symmetry yieldsΠ matµν ( k E ) = Π matT ( k E ) P Tµν + Π matL ( k E ) P Lµν + δ Π µν , (15)where the last term δ Π µν is spanned by the tensorialstructures k E,µ k E,ν and ( P µρ t ρ ) k E,ν + ( P νρ t ρ ) k E,µ with t µ = ( (cid:126) ,
1) and does not contribute to the propagatordue to the transversality of the Landau gauge, while thevacuum part can be written as Π vacµν ( k E ) = Π vac ( k E ) P µν .The gluon propagator is thus of the form (9) with thecomponents of the vacuum polarization: D µν ( k E ) = D T ( k E ) P Tµν + D L ( k E ) P Lµν ,D T ( k E ) = 1 k E + Π vac ( k E ) + Π matT ( k E ) D L ( k E ) = 1 k E + Π vac ( k E ) + Π matL ( k E ) . (16)The matter part Π matµν ( k E ) at zero temperature T =0 and non-zero quark chemical potential µ q > µ q > m q , [46]Π matµν ( k E ) = 12 (cid:20) Π matρρ − k E (cid:126)k Π mat (cid:21) P Tµν + k E (cid:126)k Π mat P Lµν (17)Π matρρ = 2 g C ( r ) π Re (cid:90) p F dpp E p (cid:34) − m q − k E p | (cid:126)k | ln (cid:18) R + R − (cid:19)(cid:35) Π mat = g C ( r ) π Re (cid:90) p F dpp E p (cid:34) − k E + 4 E p + 4 iE p k p | (cid:126)k | ln (cid:18) R + R − (cid:19)(cid:35) , (18) Γ (2) v,vac ( k E = µ ) = 1, where the quark propagator S ( k E ) isparametrized as S − ( k E ) = i/k E Γ (2) v ( k E ) + Γ (2) s ( k E ). Note that,with the RG functions determined by these renormalization con-ditions, the parameter dependence of the analytic structure ofthe RG-improved gluon propagator is qualitatively the same asthat of the strict one-loop gluon propagator for N F ≤ N F = 3 ,
6, see Fig. 6 and Fig. 8 of [30]. where C ( r ) = N F / , p F = (cid:113) µ q − m q , E p = (cid:113) p + m q , R ± = − k E + 2 ik E p ± p | (cid:126)k | , (19)and Re denotes the real part when k is real, namely,Re f ( ik ) := ( f ( ik ) + f ( − ik )) for any function f ( ik ).Now, since we are interested in complex mass and long-distance behavior, let us take the long-wavelength limit (cid:126)k → (cid:126)k →
0, we have P ij = δ ij , P i = P i = P = 0 , ( i, j = 1 , , P Tµν → P µν , P Lµν → P µν , Π matµν ( k E ) = 13 Π matρρ P µν , (20)and,Π matµµ ( (cid:126)k → , k ) = g C ( r )4 π k θ ( µ q − m q ) (cid:104) k p F (cid:113) p F + m q + 2 k ln m q (cid:113) p F + m q + p F + (cid:0) m q − k (cid:1) (cid:113) k + 4 m q × ln (cid:113)(cid:0) k + 4 m q (cid:1) (cid:0) p F + m q (cid:1) − k p F (cid:113)(cid:0) k + 4 m q (cid:1) (cid:0) p F + m q (cid:1) + k p F (cid:105) , (21)where θ ( µ q − m q ) is the step function. Then, the gluonpropagator D µν ( (cid:126)k → , k ) can be written as D µν ( k ) = D T ( − k ) P µν , D T ( − k ) = 1 M ( s + 1 + ˆΠ vac ( s ) + ˆΠ mat ( s )) (22)where s = k M , (23)ˆΠ vac ( s ) is the vacuum part given in Appendix B (B8),and,ˆΠ mat ( s ) = g C ( r )12 π θ ( ζ − ξ ) (cid:104) (cid:112) ζ ( ζ − ξ )+ 2 s ln (cid:18) √ ξ √ ζ + √ ζ − ξ (cid:19) + 1 √ s (2 ξ − s ) (cid:112) s + 4 ξ × ln (cid:32) (cid:112) ζ ( s + 4 ξ ) − (cid:112) s ( ζ − ξ ) (cid:112) ζ ( s + 4 ξ ) + (cid:112) s ( ζ − ξ ) (cid:33)(cid:105) . (24)with ξ = m q M , ζ = µ q M . (25)Notice thatˆΠ mat ( s →
0) = g C ( r )3 π θ ( ζ − ξ ) (cid:104) ( ζ − ξ ) / √ ζ (cid:105) > , (26)and ˆΠ mat ( s → ∞ ) = O ( s ) . (27) IV. RESULTS
In this section, we study the analytic structure of thegluon propagator with the one-loop quantum correctionspresented in the previous section.From here on, we set G = SU (3) and the renormaliza-tion scale µ = 1 GeV. With the RG improvements, thebest-fit parameters reported in [21] are g = 4 . , M = 0 .
42 GeV , (28a)and the up and down quark mass parameters m q = 0 .
13 GeV , (28b)in the case of N F = 2.An important advantage of this model is the existenceof the infrared safe scheme, in which there exist RG tra-jectories whose running coupling constant is finite for allscales in the one-loop level. Thus, we can implement aone-loop RG improvement, which will give a better fit-ting result.However, in the vacuum case µ q = 0 [30], the pa-rameter dependence of the analytic structure of the RG-improved gluon propagator is qualitatively the same asthat of the strict one-loop gluon propagator. Therefore,while we adopt the infrared safe scheme, we employ onlythe strict one-loop gluon propagator to study its analyticstructure to simplify analyses. A. Number of complex poles
First, we compute the number of complex poles for theone-loop gluon propagator (22) at the parameters (28a),and N F = 2 by using the winding number N W ( C ) definedin (5) of Sec. IIB. We analytically continue the gluonpropagator D T ( − k ) from the Euclidean axis z = − k to the whole z plane. In terms of (1), D ( z, (cid:126)k →
0) = D T ( z ) . (29)Let us check the prerequisites for the claim of Sec. IIB.The gluon propagator takes the form (1), since it has nobranch cut except for the real axis as can be confirmedfrom (22). Thus, it can have only complex poles in thecomplex plane excluding the real axis. Also, this gluonpropagator satisfies the conditions (i) and (ii) in Sec. II Band has no zeros N Z = 0: N P = N P = ζξ FIG. 2: Contour plot of N W ( C ) for the gluon propagatoron the ( ζ = µ q M , ξ = m q M ) plane at the set of parame-ters (28a), which gives the number of complex poles throughthe relation N P = − N W ( C ). In the N P = 2 , N = 8 × , x n = ( n + 1) × − M ( n = 0 , · · · , N ), and x N +1 = 50 M for the discretization (5) and (cid:15) = 10 − M for the infinitesimalimaginary part. For larger ζ or ξ , the gluon propagator hasone pair of complex conjugate poles. • As | z | → ∞ , D T ( z ) (cid:39) [ g γ ( − z ) ln | z | + O ( z )] − , (30)from (B14) and (27) as desired. • As | z | → D T ( z ) > , (31)from (B11) and (26) as desired. • The gluon propagator has no zeros N Z = 0, sincethe inverse of the propagator (22) does not diverge.Therefore, the number of complex poles can be calculatedaccording to (5) and N P = − N W ( C ). For the condition(iii) in Sec. II B, we numerically check convergence of therefinement of the discretization.Figure 2 is a contour plot of N W ( C ) on the plane( µ q , m q ) normalized by the gluon mass M , i.e. ( ζ, ξ ) Although the naive one-loop asymptotic form has the wrong ex-ponent of the logarithm (ln | z | ), we can expect this does notchange N W ( C ) as it has similar phase to the correct one (for N F < FIG. 3: Modulus of the gluon propagator | D T ( k ) | with theset of parameters (32) on the complex k plane. The toppanel is written in the range of − < Re k /M < , − < Im k /M <
5. A pair of complex conjugate poles is clearlyillustrated. The other pair of complex conjugate poles existsat Re k /M ≈ . k , which is howeverdifficult to be identified in the top panel, and hence is enlargedto be visible in the range 1 . < Re k /M < . , − × − < Im k /M < × − in the bottom panel. plane with (25). At the vacuum case µ q = 0, the gluonpropagator has one pair of complex conjugate poles,namely two complex poles ( N P = 2), irrespective of thevalue m q . The novel N P = 4 region appears for lightquarks ( ξ (cid:46) .
5, or m q (cid:46) .
30 GeV). As the quark chem-ical potential µ q increases for such light quarks, the num-ber of complex poles becomes four ( N P = 4) at slightlyabove the quark mass m q and backs to two ( N P = 2) at ζ ≈ .
6, or µ q ≈ .
33 GeV. In the intermediate quarkchemical potential, the gluon propagator has four com-plex poles in complex z plane. For large m q or µ q , thegluon propagator has two complex poles as in the vacuumcase. B. Analytic structure at a specific set ofparameters
Next, we take a further look at the analytic structureof the gluon propagator at a specific set of parameters.As the N P = 4 region with intermediate µ q will be inter-esting, let us choose (28a), (28b), µ q = 0 .
25 GeV, i.e.,( g, M, m q , µ q ) = (4 . , .
42 GeV , .
13 GeV , .
25 GeV) , (32) - - k / M - M Re D T ( k + i ϵ ) , M Im D T ( k + i ϵ ) ω [ GeV ]- - ρ ( ω ) [ GeV - ] FIG. 4: (Top panel) Real (orange) and Imaginary (blue)parts of the gluon propagator (22) with real k at the set ofparameters (32) and N F = 2. The peak at k /M ≈ . ω ≈ . ω ≈ .
5, the positive peak lasts up to max ρ ∼ . − andthe negative one to min ρ ∼ −
29 GeV − . The purple dashedcurve plots the vacuum one µ q = 0. In the ω → ω → ∞ limit, both of them exhibit the similar behavior. and N F = 2.In what follows, we use k to denote the complex vari-able z : k := z. (33)To take a look at the analytic structure of the gluon prop-agator, let us see its modulus on the complex k plane.The modulus of the gluon propagator D T ( k ) is plottedin Fig. 3. We can observe that the gluon propagator atthe given parameters (32) has indeed two pairs of com-plex conjugate poles. One pair that is clearly visible inthe top panel of Fig. 3 is located at k /M ≈ . ± . i ,or k ≈ ± . ± . i GeV. The other pair of complexconjugate poles is at k /M ≈ . ± (1 . × − ) i , or k ≈ ± . ± (3 . × − ) i GeV.The latter pair has very small imaginary part, whilethe former one is similar to that in the vacuum case. Thissmallness of the imaginary part is a universal feature notonly around the transition, but also on the whole N P = 4region, as we will see in the next subsection.The gluon propagator (22) with real k and its spectralfunction ρ ( ω ) := ρ ( ω, (cid:126)k →
0) := 1 π Im D T ( ω + i(cid:15) ) (34)are displayed in Fig. 4. The propagator shows a rapid os-cillation at k /M = − k /M ≈ .
4, or k ≈ . ≈ (2 µ q ). The negative peak of the spectral function has alarger value than the positive one: max ρ ∼ . − and min ρ ∼ −
29 GeV − . The rapid change is consistentwith existence of almost real complex poles. Apart fromthe sharp peak, the gluon propagator is similar to thevacuum one. The quark chemical potential affects thegluon propagator significantly only around k ≈ µ q . C. Locations of complex poles
Let us investigate locations of complex poles of thegluon propagator for various parameters ( ζ = µ q M , ξ = m q M ) with fixed ( g, M ) of (28a). We present the ratio ω I /ω R of the real and imaginary parts of a complex pole k = ω R + iω I ∈ C (35)on the ( ζ, ξ ) plane and a trajectory of poles for varying µ q and at fixed m q .First, we compute the ratio ω I /ω R to obtain anoverview on positions of complex poles on the parameterspace ( ζ, ξ ). We can restrict ourselves to ω R > , ω I > k → − k . As the gluonpropagator has at most two pairs of complex conjugatepoles with respect to k , it is sufficient to find max ω I /ω R and min ω I /ω R .Contour plots of the ratios (max ω I /ω R andmin ω I /ω R ) are shown in Fig. 5. This result isconsistent with Fig. 2 as max ω I /ω R (cid:54) = min ω I /ω R onlyon N P = 4 region, where the gluon propagator D T ( k )has two pairs of complex conjugate poles with respectto k . These figures indicate that the gluon propagatorhas a pair of almost real complex poles in the N P = 4region shown, while the pair with max ω I /ω R is alwaysof the same order of magnitude.Moreover, in general, the ratio ω I /ω R tends to increaseas the quark chemical potential µ q increases, except forthe almost real poles. In other words, the gluon propa-gator becomes “less particlelike” for large µ q .In the previous subsection, we observed that both thesharp spectral peak and almost real poles appear atRe k ≈ µ q ( ≈ . µ q = 0 .
25 GeV. This featureis not limited to the specific parameter but universal. Letus examine locations of complex poles at the parameter(28a) and (28b) with varying µ q .Figure 6 plots a trajectory of complex poles on thecomplex k plane and µ q -dependence of the real parts ofthe complex poles. As µ q increases, a new pole appearsfrom the branch cut (at µ q ≈ .
16 GeV), then movesalong the real axis, and is finally absorbed into the branch ζξ ζξ FIG. 5: Contour plots of min ω I /ω R (top) and max ω I /ω R (bottom) for a complex pole k = ω R + iω I , ( ω R > , ω I > ζ = µ q M , ξ = m q M ) plane.The region of min ω I /ω R < − is represented by a blank,where the gluon propagator has two pairs of complex conju-gate poles. The horizontal dashed line is at ξ = 0 . m q = 0 .
13 GeV. cut (at µ q ≈ .
33 GeV). On the other hand, the otherpole increases its imaginary part gradually. This featureis consistent with the number of complex poles of Sec.IV A.The bottom panel of Fig. 6 clearly indicates that thereal part of the new almost real pole can be approximatedby 2 µ q : ω R ≈ µ q . We have also checked that the almostreal poles are at Re k ≈ µ q for different values of m q . ω R [ GeV ] ω I [ GeV ] μ q [ GeV ] ω R [ GeV ] FIG. 6: (Top panel) Trajectory of a complex pole k = ω R + iω I , ( ω R > , ω I >
0) of the gluon propagator in the plane( ω R , ω I ) at the parameter (28a) and (28b) with varying µ q from 0 to 1 GeV. As µ q increases, the poles move along thearrows. Note that the almost real pole ( ω I ≈
0) exists only forthe N P = 4 region while the other pole for any value of µ q .(Bottom panel) µ q dependence of the real part of locationof a complex pole. The data of the new complex poles areapproximated by the straight line ω R = 2 µ q (purple dashedline) well. This figure shows the almost real pole appears at µ q ≈ .
16 GeV and disappears at µ q ≈ .
33 GeV in agreementwith Fig. 2. D. ( g, M ) dependence Before concluding this section, let us consider ( g, M )dependence of the above results, especially, the numberof complex poles. For details of these analyses, see Ap-pendix C. We have found that the N P contour plot is notsensitive to a detailed choice of the parameters ( g, M ). E. Summary of results
In summary, we have observed the following points inthis section. • There is a N P = 4 region, where the gluon propa-gator has two pairs of complex conjugate poles withrespect to k . See Fig. 2. • In N P = 4 region, the gluon propagator has analmost real pair of complex conjugate poles atRe k ≈ µ q . See Fig. 6 • With almost real poles, the real part and imaginarypart (to be identified with the spectral function)have narrow peaks at k ≈ µ q . See Fig. 4 • The ratio ω I /ω R of a complex pole k = ω R + iω I , ( ω R > , ω I >
0) tends to increase as µ q in-creases, except for the almost real poles. See Fig. 5. V. DISCUSSION
In this section, we discuss implications of the resultsshown in the previous sections, especially the appearanceof the almost real pole in the N P = 4 region, and com-ment on estimates of the analytic structure of the gluonpropagator for relatively large µ q . A. Almost real complex poles and spectral function
For the gluon propagator, we found a new pair of com-plex conjugate poles at Re k ≈ µ q with quite smallimaginary parts (Im k ≈ k ≈ µ q .The importance of the scale 2 µ q can be understoodby the fact that 2 µ q is the lowest energy for the quarkpair creation to occur, which contributes to the spectrumof the gluon, due to the Fermi degeneracy. Moreover, inthe massive model, the gluon “decouples” at low energies.Thus, quark loop dominates the low-energy region of thegluon spectral function. On the other hand, in the highenergy region, the gluon and ghost loops win against thequark loop for the gluon spectral function to yield ρ < N F <
10 [48]. Therefore,2 µ q will be quite an important scale for relatively small µ q (but larger than m q ), while less important in the high-energy region. This might explain the appearance anddisappearance of the almost real complex poles as varying µ q .Since complex poles never appear in the physical spec-trum, they should correspond to confined degrees of free-dom. The transition between N P = 2 and N P = 4 re-gions indicates that timelike spectra transform to con-fined complex degrees of freedom, or vice versa. There-fore, the transition between N P = 2 and N P = 4 regionsmight have a physical significance on the dynamics of thestrong interaction.Note that, however, the appearance of the almost realpole may be an artifact of the approximation: D T ( − k ) ≈ k + M + Π − loop ( k ) , (36)where the vacuum polarization Π is replaced by the one-loop expression Π − loop . For example, in this approxi-mation, even the propagator of the Higgs field in U (1)Higgs model with the small gauge-fixing parameter hascomplex poles with tiny imaginary parts [49]. The new0 ω [ GeV ] ρ ( ω ) [ GeV - ] FIG. 7: An estimate of the spectral function if the almost realcomplex poles are artifacts of the approximation (36). Thisplots ρ ( ω ) = π Im D T ( k = ω + i(cid:15) (cid:48) ) with (cid:15) (cid:48) /M = 10 − ,which is larger than the imaginary part of the almost realpole. This shows that the spectral function has a long-livedquasi-particle peak, if the complex pole is an error. pole reported in the previous section may be similar tothis one. In this case, the almost real pole should be in-terpreted as a long-lived collective mode with frequency2 µ q .If the almost real pole is an artifact, an estimate of thespectral function will be given by ρ ( ω ) = π Im D T ( k = ω + i(cid:15) (cid:48) ), where (cid:15) (cid:48) is small but larger than the imaginarypart of the almost real pole ω I . This estimate is displayedin Fig. 7 at (32) and N F = 2. We take (cid:15) (cid:48) /M = 10 − because the complex poles are at k /M ≈ . ± (1 . × − ) i . This plot implies that the new “complex pole”may correspond to actually an long-lived quasi-particle.Finally, note that N P = 4 region is located in the re-gion less than µ q ≈ .
33 GeV, which is approximatelythe matter threshold, if exists. Therefore, in any case,the new complex pole or the quasi-particle pole will bein the confined dynamics. Although it might be acci-dental, it could be interesting that the right side of theboundary between N P = 2 and N P = 4 regions locatesapproximately at the liquid-gas threshold µ q ≈ .
33 GeVfor all m q (cid:46) .
33 GeV.In summary, we again emphasize the following points, • The chemical potential influences the gluon prop-agator significantly around k ≈ µ q . This can beexplained by the facts, (1) it is the least energyfor the quark pair production without momentumtransfer (cid:126)k = 0 and (2) the quark loop is importantin the energy scale less than the effective gluon massin this model. • If the new pair of complex conjugate poles indeedemerges as µ q increases, there may be a transitionon the boundary between N P = 2 and N P = 4phase. • On the other hand, the almost real pole may bean artifact of the approximation (36). Then, thegluon propagator would have a quasi-particle spec- - k [ GeV ]- D T ( k + i ϵ ) [ GeV - ] , Im D T ( k + i ϵ ) [ GeV - ] FIG. 8: The real and imaginary parts of the gluon propagatorat µ q = 0 . M = 0 .
42 GeV and M = 0 . M = 0 .
42 GeV, which was regardedas the effective gluon mass in the vacuum. Those of M =0 . g, m q ) = (4 . , .
13 GeV) and N F = 2. - k [ GeV ]- D T ( k + i ϵ ) [ GeV - ] , Im D T ( k + i ϵ ) [ GeV - ] FIG. 9: The real and imaginary parts of the gluon propagatorat M = µ q = 0 . , . , . g, m q ) = (4 . , .
13 GeV) and N F = 2. tral peak instead of the complex poles, which cor-respond to confined states. B. Slightly larger µ q To obtain a fair agreement with lattice results in two-color QCD, the effective gluon mass parameter M is cho-sen of order µ q for µ q ∼ . µ q , we investigateit at µ q = M .Beforehand, let us see how the in-medium modificationof the effective gluon mass affects the analytic structure.The real and imaginary parts of the gluon propagatorat µ q = 0 . M = 0 .
42 GeV and M = 0 . M does not largelymodify the location of the spectral peak, k ≈ (2 µ q ) ,while the direction of the peak is inverted.1 ω R [ GeV ] ω I [ GeV ] μ q [ GeV ] ω R [ GeV ] FIG. 10: (Top panel) Trajectory of a complex pole k = ω R + iω I , ( ω R > , ω I >
0) of the gluon propagator in the plane( ω R , ω I ) at ( g, m q ) = (4 . , .
13 GeV) with varying µ q = M from 0.6 to 1 GeV. As µ q increases, the pole moves along thearrow. (Bottom panel) µ q dependence of the real part ω R oflocation of a complex pole. The real and imaginary parts of the gluon propagatorat M = µ q = 0 . , . , . k ≈ (2 µ q ) . Themagnitude of this peak decreases as µ q increases. Thegluon propagator has one pair of complex conjugate polesas the vacuum one. The effect of the chemical potentialaround k ≈ µ q is less significant for large µ q in thismodel.For complex poles, we have numerically confirmed N P = 2 in this set up for µ q ∼ . k = ω R + iω I , ( ω R > , ω I >
0) are plotted in Fig. 10. The apparent linear-ity of ω R and ω I with respect to µ q (= M ) suggests that M and µ q are the dominating scales in the propagator.A comparison with Fig. 6 indicates that the in-mediummodification of the gluon mass makes ω R and ω I larger. VI. SUMMARY AND FUTURE PROSPECTS
Let us summarize our findings. We have performedcomplex analyses of the gluon propagator at non-zeroquark chemical potential µ q in the long-wavelength limit (cid:126)k →
0, by using the massive Yang-Mills model. We haveverified that the two conditions, (i) D ( z ) → | z | → ∞ and (ii) D ( z ) is holomorphic except for the real axis anda finite number of complex poles, are sufficient to sin- gle out the correct analytic continuation of a Matsubarapropagator. Therefore, the uniqueness of the analyticcontinuation is concluded even if we allow the existenceof complex poles. For the proof, see Appendix A.We have found that there is N P = 4 region, wherethe gluon propagator has two pairs of complex conjugatepoles with respect to the complex variable z = k . Inthis region, a new pair appears near the real axis in addi-tion to the other pair similar to that in the vacuum case.At the typical parameters (Fig. 2), the N P = 4 regionappears for light quarks ( m q (cid:46) .
30 GeV). As the quarkchemical potential µ q increases, the number of complexpoles becomes four ( N P = 4) at slightly above the quarkmass m q and backs to two ( N P = 2) at µ q ≈ . M ≈ . g, M ) as shown in Appendix C. More-over, in this N P = 4 region, the new pair of complexconjugate poles has quite small imaginary part, and itslocation is approximately Re k ≈ (2 µ q ) . On the otherhand, in the N P = 2 region, the gluon propagator be-haves less “particlelike” with larger ratio ω I /ω R of thecomplex pole at k = ω R + iω I , as µ q increases.The chemical potential influences the gluon propaga-tor significantly around k ≈ µ q , where the new poleappears and the spectral peak is observed. We can at-tribute this to the facts (i) it is the least energy for thequark pair production to occur at (cid:126)k = 0 and (ii) thequark loop dominates in the energy scale less than thegluon mass M .Finally, we can interpret the new almost real poles intwo ways. First, the results may imply that the gluonpropagator indeed has a new pair of complex poles. Thissuggests a transition in confined degrees of freedom in-volving the gluon. Second, the almost real pole may be anartifact of “the one-loop approximation” (36). Then, thegluon propagator would have a long-lived quasi-particlespectral peak instead of the confined complex pole, whichsuggests a quasi-particle picture of the in-medium gluon.To sum up, although the gluon propagator presentsonly mild changes on the Euclidean side [27], it mighthave a rich and interesting structure in the complex fre-quency plane.As future prospects, there is plenty of room for im-provement in the present work in many aspects. First,this work does not take into account the quark conden-sation, which is expected to be essential in the highlydense quark matter. The effect on the analytic struc-ture of the quark gap would be interesting. Second, asremarked in the introduction, the one-loop level is notenough in the quark sector of the massive Yang-Millsmodel. A possible improvement is the double expansionthat improves the quark mass function significantly [23].Third, while a fair agreement with lattice results can beobtained by making the gluon mass M depend on µ q [26], the medium modification of the effective gluon massshould be determined in a more systematic way. Fourth,since the massive Yang-Mills model has the infrared saferenormalization scheme, it would be important to com-2pare the RG improved Euclidean gluon propagator withthe lattice one. This could improve the current unsat-isfactory agreement. Lastly, when using lattice results,we have to keep in mind that the lattice gluon prop-agator has non-negligible systematic errors, e.g., finitelattice-spacing effect [50], at low momenta and how Gri-bov copies affect results because there is no reason of thecoincidence between the minimal Landau gauge and theEuclidean version of Landau gauge of the well-known co-variant operator formalism due to the Gribov ambiguity.For other directions, it would be interesting to intro-duce temperature and to consider the physical sectorand its transport properties in the massive Yang-Millsmodel and compare them with other approaches, e.g.,[51]. Although it is very difficult, it is important to dis-cuss implications of complex poles in the physical sector.The corresponding state should be confined and not it-self have any physical impact, but its composite statemight have physical significance [52]. Formal aspects ofcomplex poles will be discussed in a future work. Acknowledgements
Y. H. is supported by JSPS Research Fellowship forYoung Scientists Grant No. 20J20215, and K.-I. K. issupported by Grant-in-Aid for Scientific Research, JSPSKAKENHI Grant (C) No. 19K03840.
Appendix A: Uniqueness of analytic continuation ofthe Matsubara propagator with complex poles
In the absence of complex singularities, a theorem forthe uniqueness of analytic continuation of the Matsubarapropagator is well-known and proved in [47]. In this ap-pendix, we shall extend the theorem to propagators withcomplex poles.In practice, we have not faced the problem of theuniqueness of the analytic continuation as we haveemployed the gluon propagator at zero temperature T = 0. However, it is the low temperature limit T → Theorem
Let D ( z ) be a complex function whose values at Mat-subara frequencies z = iω n := i πnβ are given. Then, itsanalytic continuation D ( z ) to the whole complex z planeis unique provided that an analytic continuation satisfiesthe following conditions,(i) D ( z ) → | z | → ∞ (ii) D ( z ) is holomorphic except for the real axis and afinite number of complex poles. Proof Im z Re zzC ′ C R C ρ − R − ρ Rρ FIG. 11: The contour of the integral I ( R ) consisting of lines( − R, − ρ ) and ( ρ, R ) and semicircles C ρ and C R : C (cid:48) = ( ρ, R ) ∪ C R ∪ ( − R, − ρ ) ∪ C ρ . Let D ( z ) and D ( z ) be two analytic continuationssatisfying the above two conditions that coincide at allthe Matsubara frequencies: D ( iω n ) = D ( iω n ). Then, ϕ ( z ) := D ( z ) − D ( z ) satisfies • ϕ ( iω n ) = 0 for all Matsubara frequencies ω n , • ϕ ( z ) may have a finite number of poles, • ϕ ( z ) → | z | → ∞ .We shall show that ϕ ( z ) is identically zero, i.e., an as-sumption that ϕ ( z ) had only isolated zeros leads to acontradiction. The proof is a straightforward generaliza-tion of a proof of the Carleman theorem given in Titch-marsh’s book [53].Consider the integral I ( R ) := (cid:73) C (cid:48) dz πi (cid:18) R − z (cid:19) ln ϕ ( z + i(cid:15) ) , (A1)where the contour C (cid:48) = ( ρ, R ) ∪ C R ∪ ( − R, − ρ ) ∪ C ρ isdepicted in Fig. 11 and C R = { z ; Im z > , | z | = R } and C ρ = { z ; Im z > , | z | = ρ } are the semicircles withcounterclockwise and clockwise directions respectively.In this integral, we are going to keep ρ finite and take alimit R → ∞ . From here on, we omit + i(cid:15) for notationalsimplicity.We take a sufficiently small ρ (or appropriate choice ofbranch cuts of ln ϕ ( z )) so that C ρ does not intersect withany branch cut of the logarithm.We evaluate this integral Im I ( R ) in two ways to obtainthe contradiction.First, we decompose the integral I ( R ) into four piecesfollowing C (cid:48) = ( ρ, R ) ∪ C R ∪ ( − R, − ρ ) ∪ C ρ , I ( R ) = I ρ → R + I C R + I − R →− ρ + I C ρ . (A2)3Then, we have I ρ → R + I − R →− ρ = (cid:90) Rρ dx πi (cid:18) R − x (cid:19) ln[ ϕ ( x ) ϕ ( − x )] , (A3)and, I C R = (cid:90) C R dz πi (cid:18) R − z (cid:19) ln ϕ ( z )= iπR (cid:90) π dθ sin θ ln ϕ ( Re iθ ) . (A4)Thus, we obtainIm I ( R ) = Im I C ρ + (cid:90) Rρ dx π (cid:18) x − R (cid:19) ln | ϕ ( x ) ϕ ( − x ) | + 1 πR (cid:90) π dθ sin θ ln | ϕ ( Re iθ ) | . (A5)Note that Im I C ρ is O (1) as R → ∞ . The othertwo integrals could diverge as R → ∞ ; however, then,Im I ( R ) would be negative infinity, since ϕ ( z ) → | z | → ∞ and the other parts of the integrands are pos-itive, (cid:0) x − R (cid:1) > , sin θ >
0. Therefore, Im I ( R ) isbounded from above: Im I ( R ) ≤ M for some M ∈ R .On the other hand, the integral I ( R ) is closely relatedto zeros and poles inside C (cid:48) . I ( R ) = (cid:73) C (cid:48) dz πi ln ϕ ( z ) ddz (cid:18) zR + 1 z (cid:19) = (cid:73) C (cid:48) dz πi ddz (cid:20) ln ϕ ( z ) (cid:18) zR + 1 z (cid:19)(cid:21) − (cid:73) C (cid:48) dz πi ϕ (cid:48) ( z ) ϕ ( z ) (cid:18) zR + 1 z (cid:19) . (A6)The first integral sums up ‘discontinuities’ from thebranch cuts of the logarithm. Since we have assumedthat the branch cuts of the logarithm do not intersectwith C ρ , the first term contributes only from ( ρ, R ) ∪ C R ∪ ( − R, − ρ ), on which (cid:0) zR + z (cid:1) is real. Therefore,Im (cid:73) C (cid:48) dz πi ddz (cid:20) ln ϕ ( z ) (cid:18) zR + 1 z (cid:19)(cid:21) = 0 . (A7)Finally, the second term can be evaluated as a weightedsum of zeros and poles. The generalized argument prin-ciple yields − (cid:73) C (cid:48) dz πi ϕ (cid:48) ( z ) ϕ ( z ) (cid:18) zR + 1 z (cid:19) = − (cid:88) z j :zeros z j ∈D (cid:48) (cid:18) z j R + 1 z j (cid:19) + (cid:88) w k :poles w k ∈D (cid:48) (cid:18) w k R + 1 w k (cid:19) , (A8) where D (cid:48) is the region surrounded by C (cid:48) . To sum up,Im I ( R ) = (cid:88) z j :zeros z j ∈D (cid:48) (cid:18) r j − r j R (cid:19) sin θ j + O (1) , (A9)where we have defined r j e iθ j := z j , used the finiteness ofthe number of poles, and O (1) stands for a finite termfor all R . As ϕ ( iω n ) = 0 for all Matsubara frequenciesand (cid:16) r j − r j R (cid:17) > z j ∈ D (cid:48) , (cid:88) z j :zeros z j ∈D (cid:48) (cid:18) r j − r j R (cid:19) sin θ j ≥ (cid:88) niω n ∈D (cid:48) (cid:18) ω n − ω n R (cid:19) (A10)Moreover, as R → ∞ , (cid:88) niω n ∈D (cid:48) ω n R = (cid:88) <ω n Here, we present the one-loop expression forΠ vacµν ( k E ) = Π vac ( k E ) P µν = M ˆΠ vac ( k E M ) P µν .Beforehand, we rewrite the two-point vertex functionsΓ (2) A ,vac and Γ (2) gh,vac by dimensionless gluon and ghost vac-uum polarizations ˆΠ and ˆΠ gh asΓ (2) A ,vac ( k E ) = M [ s + 1 + ˆΠ( s ) + sδ Z + δ M ]=: M [ s + 1 + ˆΠ vac ( s )] , (B1)Γ (2) gh,vac ( k E ) := − [∆ gh ( k E )] − = M [ s + ˆΠ gh ( s ) + sδ C ]=: M [ s + ˆΠ rengh ( s )] , (B2)where k E is the Euclidean momentum, s = k E M , and δ Z := Z A − δ M := Z A Z M − 1, and δ C := Z C − s ) = ˆΠ Y M ( s ) + ˆΠ q ( s ) (B3)ˆΠ Y M ( s ) = g C ( G )192 π s (cid:40)(cid:18) s − (cid:19) (cid:20) ε − + ln( 4 πM e γ ) (cid:21) − s + h ( s ) (cid:41) ˆΠ q ( s ) = − g C ( r )6 π s (cid:40) − (cid:20) ε − + ln( 4 πm q e γ ) (cid:21) − 56 + h q (cid:18) ξs (cid:19)(cid:41) , (B4)for ghosts,ˆΠ gh ( s ) = g C ( G )64 π s (cid:20) − (cid:20) ε − + ln( 4 πM e γ ) (cid:21) − f ( s ) (cid:21) , (B5)where ε := 2 − D/ γ is the Euler-Mascheroni constant, C ( G ) and C ( r ) = N F / N F ) rep- resentations of the gauge group G , ξ = m q M and, h ( s ) := − s + (cid:18) − s (cid:19) ln s + (cid:18) s (cid:19) ( s − s + 1) ln( s + 1)+ 12 (cid:18) s (cid:19) / ( s − s + 12) ln (cid:18) √ s − √ s √ s + √ s (cid:19) ,h q (˜ t ) := 2˜ t + (1 − t ) (cid:112) t + 1 coth − ( (cid:112) t + 1) ,f ( s ) := − s − s ln s + (1 + s ) s ln( s + 1) , (B6)with ˜ t := ξs = m q k E .The renormalization conditions (13) for the gluon andghost sector can be cast into in the one-loop level, Z A Z C Z M = 1Γ (2) A ,vac ( k E = µ ) = µ + M Γ (2) gh,vac ( k E = µ ) = µ ⇔ δ C + δ M = 0ˆΠ vac ( s = ν ) = 0ˆΠ rengh ( s = ν ) = 0 , (B7)with ν := µ M .By imposing this renormalization condition, we havethe renormalized two-point vertex functions,ˆΠ vac ( s ) = ˆΠ ren.Y M ( s ) + ˆΠ ren.q ( s ) , (B8)ˆΠ ren.Y M ( s ) = g C ( G )192 π s (cid:20) s + h ( s ) + 3 f ( ν ) s − ( s → ν ) (cid:21) , (B9)ˆΠ ren.q ( s ) = − g C ( r )6 π s (cid:20) h q (cid:18) ξs (cid:19) − h q (cid:18) ξν (cid:19)(cid:21) . (B10)Note that the gluon propagator at T = µ = 0 exhibitsthe decoupling feature and satisfies the condition (ii) ofSec. II B:ˆΠ vac ( s = 0) > , (B11) ⇒ Γ (2) A ,vac ( k E = 0) = M [1 + ˆΠ vac (0)] > . (B12)Indeed, we haveˆΠ ren.Y M ( s = 0) = g C ( G )192 π (cid:34) f ( ν ) − (cid:35) > , ˆΠ ren.q ( s = 0) = 0 , (B13)where we have used h q (˜ t → ∞ ) = O (1), h ( s ) = − s + O (ln s ), f (0) = 5 / 2, and the fact that f ( s ) increasesmonotonically in s .Note also that the strict one-loop expression has thefollowing asymptotic form in the limit | k | → ∞ :Γ (2) A ,vac (cid:39) g γ ( − k ) ln | k | + O ( k ) , (B14)5while the asymptotic freedom and RG analysis yieldsΓ (2) A ,vac (cid:39) Z − UV ( − k )(ln | k | ) γ /β (B15)where we have analytically continued the gluon propaga-tor from the Euclidean momentum k = − k E to complex k , Z UV > γ and β arerespectively the first coefficients of the gluon anomalousdimension and the beta function: γ = − π (cid:18) C ( G ) − C ( r ) (cid:19) ,β = − π (cid:18) C ( G ) − C ( r ) (cid:19) . (B16)Both the strict one-loop gluon propagator and RG im-proved one satisfy the condition (i) of Sec. II B. In spiteof the wrong logarithmic exponent, the one-loop gluonpropagator has qualitatively the same phase as the RGimproved one (for N F < N W ( C ) = N Z − N P , and hence the strict one-loop expression maybe enough for our purpose. Appendix C: Number of complex poles with various ( g, M ) In the main text, we have investigated the analyticstructure of the gluon propagator with the fixed parame-ters g = 4 . M = 0 . 42 GeV, as they give best-fit pa-rameters to the lattice results [21]. In this appendix, wecheck that the qualitative features of the analytic struc-ture are not sensitive to the model parameters ( g, M ).We have confirmed that the contour plots of N P on the( ζ = µ q M , ξ = m q M ) plane are qualitatively same. Indeed,Fig. 12 gives contour plots of N P at M = 0 . 42 GeV and g = 3 (top) and g = 8 (bottom). Fig. 13 gives contourplots of N P at g = 4 . M = 0 . M = 0 . ζ side of the boundary) is near µ q ∼ m q and the right boundary (large- ζ side of theboundary) at µ q ≈ . 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