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CHIBA-EP-245, 2020.05.27
Complex poles and spectral functions of Landau gauge QCD and QCD-like theories
Yui Hayashi1, ∗ and Kei-Ichi Kondo1, 2, †
1 Department of Physics, Graduate School of Science and Engineering, Chiba University, Chiba 263-8522, Japan2 Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan
In view of the expectation that the existence of complex poles is a signal of confinement, weinvestigate the analytic structure of the gluon, quark, and ghost propagators in the Landau gaugeQCD and QCD-like theories by employing an effective model with a gluon mass term of the Yang-Mills theory, which we call the massive Yang-Mills model. In this model, we particularly investigatethe number of complex poles in the parameter space of the model consisting of gauge couplingconstant, gluon mass, and quark mass for the gauge group SU(3) and various numbers of quarkflavors NF within the asymptotic free region. Both the gluon and quark propagators at the best-fit parameters for NF = 2 QCD have one pair of complex conjugate poles, while the number ofcomplex poles in the gluon propagator varies between zero and four depending on the number ofquark flavors and quark mass. Moreover, as a general feature, we argue that the gluon spectralfunction of this model with nonzero quark mass is negative in the infrared limit. In sharp contrastto gluons, the quark and ghost propagators are insensitive to the number of quark flavors within thecurrent approximations adopted in this paper. These results suggest that details of the confinementmechanism may depend on the number of quark flavors and quark mass.
PACS numbers: 11.15.-q, 14.70.Dj, 12.38.Aw
I. INTRODUCTION
Color confinement, absence of color degrees of freedomfrom the physical spectrum, is one of the most funda-mental and significant features of strong interactions. Itis a long-standing and challenging problem in particleand nuclear physics to explain color confinement in theframework of quantum field theory (QFT). Understand-ing analytic structures of the correlation functions willbe of crucial importance to this end because a QFT de-scribing physical particles can be reformulated in termsof correlation functions [1], and there are some proposalsof confinement mechanisms whose criteria are expressedby them, e.g., [2]. In particular, the analytic structures ofpropagators encode kinematic information as the Kallen-Lehmann spectral representation [3], which will be usefultoward understanding confinement.
In the past decades, numerous studies of both thelattice and continuum approaches have focused on thegluon, quark, and ghost propagators in the Landau gaugeof the Yang-Mills theory and quantum chromodynamics(QCD). In the Yang-Mills theory, or the quenched limitof QCD, the so-called decoupling and scaling solutionsof the gluon and ghost propagators are observed basedon the continuum approaches [4]. The recent numericallattice results support the decoupling solution [5]. Thedecoupling solution has an impressing feature that therunning gauge coupling stays finite and nonzero for allnonvanishing momenta and eventually goes to zero in thelimit of vanishing momentum, which cannot be predictedfrom the standard perturbation theory that is plagued by
∗Electronic address: [email protected]†Electronic address: [email protected]
the Landau pole of the diverging running gauge coupling.
A low-energy effective model of the Yang-Mills the-ory is proposed following the decoupling behavior of thegluon propagator, which provides the gluon and ghostpropagators that show a striking agreement with the nu-merical lattice results by including quantum correctionsjust in the one-loop level [6, 7]. This effective model isgiven by the mass-deformed Faddeev-Popov Langrangianof the Yang-Mills theory in the Landau gauge, or theLandau gauge limit of the Curci-Ferarri model [8], whichcan be shown to be renormalizable due to the modifiedBecchi-Rouet-Stora-Tyutin (BRST) symmetry, and wecall it the massive Yang-Mills model for short. Thiseffective mass term could stem from the dimension-twogluon condensate [9–13] or could be taken as a (minimal)consequence of avoiding the Gribov ambiguity [14, 15].Moreover, it has been shown that the massive Yang-Millsmodel has “infrared safe” renormalization group (RG)flows on which the running gauge coupling remains finitefor all scales [7, 15, 16]. The three-point functions [17]and two-point correlation functions at finite temperature[18] in this model were compared to the numerical latticeresults, showing good agreements. Moreover, the two-loop corrections improve the accordance for the gluonand ghost propagators [19]. These works indicate thevalidity of the massive Yang-Mills model as an effectivemodel of the Yang-Mills theory.
The gluon and ghost propagators for unquenched lat-tice QCD with the number of quark flavors NF = 2, 2 +1, 2 + 1 + 1 have been studied, for instance [20], and ex-hibit the decoupling feature as well. The massive Yang-Mills model with dynamical quarks can reproduce thenumerical lattice gluon and ghost propagators for QCDas well [21]. However, it is argued that higher-loop cor-rections are important for the quark sector in this model[22, 23]. Despite this shortcoming, it appears that the ef-
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fective gluon mass term captures some nonperturbativeaspects of QCD. What is more, QCD phases have beenextensively studied in a similar model with the effectivegluon mass term of the Landau-DeWitt gauge [24].
Apart from the realistic QCD, it is also interesting tostudy gauge theories with many flavors of quarks. Formany quark flavors, the infrared conformality is predicted[25, 26] and well-studied in line with the walking tech-nicolor for physics beyond the standard model [27], forexample, [28–30]. Some argue that chiral symmetry re-stores while color degrees of freedom are “unconfined” inthe conformal window. For a better understanding of theconfinement mechanism, observing NF dependence willbe thus extremely valuable.
All works on the correlation functions described abovewere implemented in the Euclidean space. Considerableefforts have been devoted to reconstructing the spec-tral functions from the Euclidean data based on theKallen-Lehmann spectral representation, e.g., [31, 32].On the other hand, several models of the Yang-Mills the-ory [14, 33–39], including the (pure) massive Yang-Millsmodel [40, 41], and a way of the reconstruction from theEuclidean data [42] predict complex poles in the gluonpropagator that invalidate the Kallen-Lehmann spectralrepresentation. The existence of complex poles of thepropagators of the confined particles is a controversialissue, see, e.g., [43]. The complex singularities invali-date the standard reconstruction from a Euclidean fieldtheory to a relativistic QFT [1] and might correspondto unphysical degrees of freedom in an indefinite metricstate space [44]. Therefore, complex poles are expectedto be closely connected to the confinement mechanism.
In this paper, we investigate the analytic structure ofthe QCD propagators for various NF based on the mas-sive Yang-Mills model, mainly focusing on complex polesby utilizing the general relationship between the num-ber of complex poles and the propagator on timelike mo-menta from the argument principle [40]. This investi-gation extends the previous result [40] obtained for thepure Yang-Mills theory with no flavor of quarks NF = 0that the gluon propagator has a pair of complex conju-gate poles and the negative spectral function while theghost propagator has no complex pole. In this article, wewill see the following results. Both the gluon and quarkpropagators at the “realistic” parameters for NF = 2QCD have one pair of complex conjugate poles as wellas the gluon propagator in the zero flavor case. By in-creasing quark flavors, we find a new region in whichthe gluon propagator has two pairs of complex conjugatepoles for light quarks with the intermediate number offlavors 4 . NF < 10. However, the gluon propagator hasno complex poles if very light quarks have many flavorsNF ≥ 10 or both of the gauge coupling and quark massare small. In the other regions, the gluon propagator hasone pair of complex conjugate poles. On the other hand,the analytic structures of quark and ghost propagatorsare nearly independent of the number of quarks withinthis one-loop analysis.
This paper is organized as follows. In Sec. II, wepresent machinery to count the number of complex polesas an application of [40]. In Sec. III, we review the cal-culation of the massive Yang-Mills model with quarks of[21], consider the infrared safe trajectories of this model,and argue the infrared negativity of the gluon spectralfunction. Then, in Sec. IV, we analyze the analytic struc-tures of the gluon, quark, and ghost propagators at thebest-fit parameters of [21], and investigate the numberof complex poles for various parameters and the numberof quarks NF . Finally, Sec. V is devoted to conclusion,and Sec. VI contains further discussion. In Appendix A,we provide a generalization of the proposition of Sec. IIfor various infrared behaviors. Appendix B gives com-plementary one-loop analyses for Sec. IV A.
II. COMPLEX POLES IN PROPAGATORS
To elucidate the starting point, we review a general-ization of the spectral representation for a propagator soas to allow complex poles. We then develop a methodfor counting the number of complex poles from the dataon timelike momenta, e.g., the spectral function, as astraightforward application of the general relation [40].
A. A generalization of the spectral representation
We introduce some definitions and underlying assump-tions on propagators adopted in this article. Given apropagator defined in the Euclidean space, we analyti-cally continue the propagator to the whole complex k2
plane from the Euclidean momenta. In the complex k2
plane, we call points on the negative real axis Euclideanmomenta and points on the positive real axis timelikemomenta. We will study the gluon, quark, and ghostpropagators in the Landau gauge. We assume each prop-agator of the gluon, quark, and ghost has the followinggeneralized spectral representation allowing the presenceof complex poles:1
D(k2) =
∫ ∞0
dσ2 ρ(σ2)
σ2 − k2+
n∑`=1
Z`z` − k2
, (1)
where ρ(σ2) := 1π ImD(σ2 + iε) is its spectral function,
z` stands for the position of a complex pole, and Z` isthe residue associated with the complex pole.
1 This generalization can be related to the fact that complex spec-tra for confined particles need not be excluded in an indefinitemetric state space [44]. Such complex spectra of a Hamiltoniancan give rise to the complex poles. The kinematic aspects ofcomplex poles will be discussed elsewhere. Incidentally, anothergeneralization for the QCD propagators within the framework oftempered distribution is proposed in [45].
3
Im k2
Re k2
k2
C
γ1
γ2C1
C2
FIG. 1: Contour C on the complex k2 plane avoiding the sin-gularities on the positive real axis and the complex poles. Thecontour C consists of the large circle C1, the path wrappingaround timelike singularities C2, and the small contours γ`that clockwise surround the complex poles at z`. The prop-agator D(k2) is holomorphic in the region bounded by the
contour C = C∪{γ`}n`=1, where we denote the closed contourC1 ∪ C2 by C.
This representation can be derived under the follow-ing conditions using the Cauchy integral formula for theclosed contour C presented in Fig. 1 [40, 46]:
(I) D(z) is holomorphic except for singularities on thepositive real axis and a finite number of simplepoles.
(II) D(z)→ 0 as |z| → ∞.
(III) D(z) is real on the negative real axis.
If we replace the condition (I) with the more strict con-dition
(I’) D(z) is holomorphic except for singularities on thepositive real axis, namely has no complex poles,
the three conditions (I’), (II), and (III) lead to the Kallen-Lehmann form [3], which propagators for unconfined par-ticles are supposed to obey. In this sense, eq. (1) gives ageneralization of the Kallen-Lehmann spectral represen-tation.
B. Counting complex poles
We present a procedure to count complex poles fromthe propagator on the timelike momenta based on [40].
We apply the argument principle to a propagator onthe contour C presented in Fig. 1. Then the windingnumber NW (C) of the phase of the propagator D(k2)
along the contour C is equal to the difference betweenthe number of zeros NZ and the number of poles NP inthe region bounded by C,
NW (C) : =1
2πi
∮C
dk2D′(k2)
D(k2)
=1
2π
∮C
d(argD(k2))
= NZ −NP . (2)
The winding number NW (C) can be calculated from thepropagator on timelike momenta D(k2 + iε) and infrared(IR) and ultraviolet (UV) asymptotic forms.
In what follows, we assume the following asymptoticform for the propagator.
(i) 2 In the UV limit |k2| → ∞, D(k2) has the samephase as the free one, i.e., arg(−D(z)) → arg 1
z as|z| → ∞.
(ii) In the IR limit |k2| → 0, D(k2 = 0) > 0.
Let us comment on these assumptions. The first assump-tion is satisfied by the gluon, quark, and ghost propaga-tors in the Landau gauge, which follows from the RGanalysis for asymptotic free theories. The RG argumentof Oehme and Zimmermann [47] provides the followingUV asymptotic form for the propagators, as |k2| → ∞,
D(k2) ' − ZUVk2(ln |k2|)γ
, (3)
where ZUV is a positive constant and γ = γ0/β0 is theratio between the first coefficients γ0, β0 of the anomalousdimension and the beta function, respectively. For thegluon propagator of Yang-Mills theories with NF quarksin the Landau gauge, γ is computed as follows.
γ =γ0
β0,
γ0 = − 1
16π2
(13
6C2(G)− 4
3C(r)
),
β0 = − 1
16π2
(11
3C2(G)− 4
3C(r)
), (4)
where C2(G) and C(r) = NF /2 are the Casimir invari-ants of the adjoint and fundamental representations ofthe gauge group G, respectively.
We restrict ourselves to asymptotically free theories:β0 < 0, or NF < 11
2 C2(G), which is essential to derivethe above UV asymptotic expression for the propagator(3).
Note that the sign of γ is determined depending on thenumber of quarks as follows.
2 This assumption (i) is the same as assumption (i) of the asser-tions in Sec. III in [40].
4
• γ > 0 if C(r) < 138 C2(G). For the gauge group
SU(3) with C2(G) = 3, in particular, γ > 0 ifNF = 1, · · · , 9 < 13
4 C2(G) = 394 .
• γ < 0 if 138 C2(G) < C(r) < 11
4 C2(G). For thegauge group SU(3) with C2(G) = 3, in particular,γ < 0 if NF = 10, · · · , 16 < 11
2 C2(G) = 332 .
This determines the sign of the gluon spectral functionρ(σ2) in the ultraviolet region [47]. The spectral functionof the gluon for SU(3) in the UV limit takes negativevalues for NF = 1, · · · , 9 and positive values for NF =10, · · · , 16.
Additionally, the one-loop corrections give no contri-bution to the quark anomalous dimension γψ in the Lan-dau gauge: γψ = O(g4). Therefore, the quark propa-gator behaves in the UV limit as the free one becauseof the asymptotic freedom and the vanishing of the firstcoefficient γ0
ψ of the order g2 of the quark anomalous di-mension.
The second assumption (ii) indicates the massive-likebehavior for the propagator, which corresponds to thedecoupling solution for the Euclidean gluon propagator.For the general cases D(k2) → ZIR(−k2)α as |k2| → 0with a real exponent α, e.g., the scaling solution for thepure Yang-Mills gluon and a massless propagator, thereis an additional contribution to the expression of NW (C)described below (9). See Appendix A for the details ofthe additional term. From here on, we simply assume (ii)and will verify this assumption when we compute NW (C)for each propagator employed.
Let us add notes on the relationships between thenumber NW (C) and the spectral function [40]. Withthe assumptions (i) and (ii), the positive spectral func-tion implies NW (C) = 0 and the negative one impliesNW (C) = −2. Since the winding number is a topo-logical invariant, NW (C) is invariant under continuousdeformations. For example, if the spectral function is“quasinegative”, i.e., the spectral function is negativeρ(k2
0) < 0 at all real and positive zeros k20 of Re D(k2)
i.e., Re D(k20) = 0 (k2
0 > 0), then the propagator hasNW (C) = −2. Actually, this is the case of the massiveYang-Mills model with NF = 2 quarks at the realisticparameters analyzed below.
In order to calculate the winding number NW (C) ina numerical way, we divide the interval [δ2,Λ2] on thepositive real axis into (N + 1) segments x0, x1, · · · , xN+1
such that the following condition on {D(xn + iε)}Nn=1 atpoints {k2 = xn + iε}Nn=1 is satisfied.
(iii) {k2 = xn + iε}Nn=0 is sufficiently dense so thatD(k2 = x + iε) changes its phase at most half-winding (±π) between xn + iε and xn+1 + iε, i.e.,for n = 0, 1, · · · , N ,∣∣∣∣∫ xn+1
xn
dxd
dxargD(x+ iε)
∣∣∣∣ < π, (5)
where we denote sufficiently small x0 = δ2 > 0 andsufficiently large xN+1 = Λ2, on which we will takethe limits δ2 → +0 and Λ2 → +∞.
Let us now calculate NW (C) from the data {D(xn +iε)}Nn=1 under the above assumptions, by evaluatingNW (C1) and NW (C2) separately, where C1 stands forthe large circle and C2 for the path around the positivereal axis depicted in Fig. 1.
The first assumption (i) yields
NW (C1) = −1. (6)
For NW (C2), from the Schwarz reflection principle,[D(z)]∗ = D(z∗), we have,
NW (C2) = 2
∫ ∞0
dx
2π
d
dxargD(x+ iε). (7)
Notice that we have used the second assumption (ii) toeliminate the contribution from the small circle aroundthe origin. The third assumption (iii) transforms theintegral (7) into a discrete sum as
NW (C2) = 2
N∑n=0
1
2πArg
[D(xn+1 + iε)
D(xn + iε)
], (8)
where Arg is the principal value of the argument (−π <Arg z < π).
To sum up, we have the expression for NW (C),
NW (C) = −1 + 2
N∑n=0
1
2πArg
[D(xn+1 + iε)
D(xn + iε)
]. (9)
Though we have to know the number of complex zerosfor counting the exact number of complex poles NP fromthe winding number NW (C), note that it suffices to ver-ify NW (C) < 0 in order to show the existence of complexpoles because NP = NZ − NW (C) ≥ −NW (C). More-over, since NW (C) is invariant under continuous defor-mations, we can expect that NW (C) should be robustunder some approximations. We will consider complexpoles of the QCD propagators using an effective modeland the relation (9) based on this expectation.
III. MASSIVE YANG-MILLS MODEL AS ANEFFECTIVE MODEL
The massive Yang-Mills model [6, 7] to be definedshortly is an effective model of the Yang-Mills theory inthe Landau gauge, which captures some nonperturbativeaspects by introducing a phenomenological mass term forthe gluon. The mass term may be generated by the ef-fect of the dimension-two gluon condensate [9–13] or byavoiding the Gribov ambiguity [15]. For the latter effect,intuitively, some effect suppressing
∫dDx A A
µ A Aµ should
be taken into account due to the Gribov copies, and themost infrared relevant term of this effect will be the massterm. The massive Yang-Mills model reproduces the nu-merical lattice results of the gluon and ghost propaga-tors well and has a renormalization condition such that
5
there are RG trajectories whose running gauge couplingremains finite in all scales. Moreover, the massive Yang-Mills model gives a correct UV asymptotic behavior [47]by RG improvement, as we will see in Sec. III C. There-fore, although we expect the massive Yang-Mills modelis a low-energy effective model, this model can describeYang-Mills theory in all scales to some extent. One mightworry about the absence of the nilpotent BRST sym-metry. Although the massive Yang-Mills model suffersfrom the problem of physical unitarity [8, 48] as a con-sistent QFT, we insist that this model will suffice for ourpurpose of investigating analytic structures, as it givesthe well-approximating propagators with a sensible andstraightforward mass-deformation.
Moreover, by adding effective quark mass term, thismodel has good accordance with the numerical latticeresults for the unquenched gluon and ghost propagatorsand can also reproduce the quark mass function qualita-tively [21]. In this section, we review the results on theone-loop computations of the massive Yang-Mills modelwith quarks, prepare expressions that we will use in theinvestigation of the analytic structures of the propaga-tors in the next section, and study asymptotic behaviorsof the propagators of this model using RG.
In the Euclidean space, the Lagrangian of the model isgiven by [6, 7, 21]
LmYM = LYM + LGF + LFP + Lm + Lq, (10)
LYM =1
4FAµνF
Aµν ,
LGF = iN A∂µAAµ
LFP = CA∂µDµ[A ]ABCB
= CA∂µ(∂µCA + gbf
ABCA Bµ CC)
Lm =1
2M2bA A
µ A Aµ ,
Lq =
NF∑i=1
ψi(γµDµ[A ] + (mb)q,i)ψi
=
NF∑i=1
ψi(γµ(∂µ − igbA Aµ t
A) + (mb)q,i)ψi, (11)
where we have introduced the bare gluon, ghost, anti-ghost, Nakanishi-Lautrup, and quark fields denoted byA Aµ , CA, CA,N A, ψi respectively, the bare gauge cou-
pling constant gb, the bare quark mass (mb)q,i, and thebare effective gluon mass Mb, while fABC stands for thestructure constant associated with the generators tA ofthe fundamental representation of the group G.
We introduce the renormalization factors(ZA, ZC , ZC = ZC , Z
(i)ψ ), Zg, ZM2 , Zmq,i for the
gluon, ghost, anti-ghost, and quark fields (Aµ,C , C , ψi),the gauge coupling constant g, and the gluon and quark
mass parameters M2,mq,i respectively:
A µ =√ZAA µ
R , C =√ZCCR,
C =√ZCCR, ψi =
√Z
(i)ψ ψR,i,
gb = Zgg, M2b = ZM2M2, (mb)q,i = Zmq,imq,i (12)
Throughout this article, for simplicity, we employ thismodel with degenerate quark masses, mq := mq,i, and
therefore Zψ := Z(i)ψ and Zmq := Zmq,i .
A. Strict one-loop calculations
We review the strict one-loop results for the gluon,quark, and ghost propagators here and the RG functionsin the next subsection [21].
For the gluon, ghost, and quark, we introduce the two-
point vertex functions Γ(2)A , Γ
(2)gh , and Γ
(2)ψ , the trans-
verse gluon propagator DT , the ghost propagator ∆gh,the quark propagator S, dimensionless gluon and ghostvacuum polarizations Π and Πgh, and the scalar and vec-
tor part of the quark two-point vertex function Γ(2)s , Γ
(2)v
as
Γ(2)A (k2
E) := [DT (k2E)]−1
= M2[s+ 1 + Π(s) + sδZ + δM2 ]
=: M2[s+ 1 + Πren(s)], (13)
Γ(2)gh (k2
E) := −[∆gh(k2E)]−1
= M2[s+ Πgh(s) + sδC ]
=: M2[s+ Πrengh (s)], (14)
Γ(2)ψ (kE) := S(kE)−1
= i/kE(Γ(2)v (k2
E) + δψ) + (Γ(2)s (k2
E) +mqδmq )
= i/kEΓren.v (k2E) + Γren.s (k2
E), (15)
where kE is the Euclidean momentum,
s :=k2E
M2, (16)
and δZ := ZA − 1, δM2 := ZAZM2 − 1, δC := ZC − 1,δψ := Zψ − 1, δmq := ZψZmq − 1 are the counterterms.
The bare vacuum polarizations computed by the di-
6
mensional regularization read [7, 21], for gluons,
Π(s) = ΠYM (s) + Πq(s) (17)
ΠYM (s) =g2C2(G)
192π2s
{(9
s− 26
)[ε−1 + ln(
4π
M2eγ)
]
− 121
3+
63
s+ h(s)
}
Πq(s) = −g2C(r)
6π2s
{−1
2
[ε−1 + ln(
4π
m2qeγ
)
]
− 5
6+ hq
(ξ
s
)}, (18)
for ghosts,
Πgh(s) =g2C2(G)
64π2s
[−3
[ε−1 + ln(
4π
M2eγ)
]− 5 + f(s)
], (19)
where ε := 2−D/2, γ is the Euler-Mascheroni constant,C(r) = NF /2,
ξ :=m2q
M2, (20)
and,
h(s) := − 1
s2+
(1− s2
2
)ln s
+
(1 +
1
s
)3
(s2 − 10s+ 1) ln(s+ 1)
+1
2
(1 +
4
s
)3/2
(s2 − 20s+ 12) ln
(√4 + s−
√s√
4 + s+√s
),
hq(t) := 2t+ (1− 2t)√
4t+ 1 coth−1(√
4t+ 1),
f(s) := −1
s− s ln s+
(1 + s)3
s2ln(s+ 1), (21)
with
t :=ξ
s=m2q
k2E
. (22)
For the quark sector [21],
Γ(2)v (k2
E) = 1 + Σv(k2E),
Γ(2)s (k2
E) = mq + Σs(k2E) (23)
Σv(k2E) =
g2C2(r)
64π2M2k4E
[K2{
2M4 +M2(k2E −m2
q)
−(m2q + k2
E)2}Q− 2M2k2
E(−2M2 +m2q + k2
E)
− 2{2M6 + 3M4(k2E −m2
q) + (m2q + k2
E)3} ln(mq
M
)− 2(m2
q + k2E)3 ln
(m2q + k2
E
m2q
)], (24)
Σs(k2E) = −3g2C2(r)mq
8π2
[−2
ε+ ln
(Meγ/2√
4π
)− 2
3
− K2
4k2E
Q+1
2k2E
(M2 −m2q + k2
E) ln(mq
M
)],
(25)
where
K2 :=√M4 + 2M2(k2
E −m2q) + (m2
q + k2E)2,
Q := ln
((K2 − k2
E)2 − (m2q −M2)2
(K2 + k2E)2 − (m2
q −M2)2
), (26)
and C2(r) = N2−12N for G = SU(N) and fundamental
quarks.
Henceforth, we adopt the “infrared safe renormaliza-tion scheme” by Tissier and Wschebor [7], which respectsthe nonrenormalization theorem ZAZCZM2 = 1 [49], inthe one-loop level,
ZAZCZM2 = 1
Γ(2)A (kE = µ) = µ2 +M2
Γ(2)gh (kE = µ) = µ2
Γren.v (µ2) = 1
Γren.s (µ2) = mq
⇔
δC + δM2 = 0
Πren(s = ν) = 0
Πrengh (s = ν) = 0,
Γren.v (µ2) = 1
Γren.s (µ2) = mq
(27)
combined with the Taylor scheme [50] ZgZ1/2A ZC = 1 for
the coupling, where
ν :=µ2
M2. (28)
In this renormalization scheme, the running coupling ofsome RG flows turns out to be always finite, which im-plies that the perturbation theory will be valid to someextent.
By imposing the above renormalization condition, we
7
have the renormalized two-point vertex functions,
ΠTWren.(s) = ΠTW
YM,ren.(s) + ΠTWq,ren.(s), (29)
ΠTWYM,ren.(s) =
g2C2(G)
192π2s
[48
s+ h(s) +
3f(ν)
s− (s→ ν)
],
(30)
ΠTWq,ren.(s) = −g
2C(r)
6π2s
[hq
(ξ
s
)− hq
(ξ
ν
)], (31)
ΠTWgh,ren.(s) =
g2C2(G)
64π2s
[f(s)− f(ν)
], (32)
ΓTW,ren.v (k2E) = 1 + Σv(k
2E)− Σv(µ
2), (33)
ΓTW,ren.s (k2E) = mq + Σs(k
2E)− Σs(µ
2). (34)
Note that the gluon propagator exhibits the decouplingfeature and satisfies the condition (ii) of the previoussection,
DT (k2E = 0) =
1
M2[1 + ΠTWren.(0)]
> 0. (35)
Indeed,
ΠTWq,ren.(s = 0) = 0,
ΠTWYM,ren.(s = 0) =
g2C2(G)
192π2
[3f(ν)− 15
2
]> 0, (36)
where we have used hq(t → ∞) = O(1), h(s) = − 1112s +
O(ln s), f(0) = 5/2, and the fact that f(s) is a monoton-ically increasing function.
Finally, note that the one-loop expression of gluonpropagator will have disagreement at momenta far fromthe renormalization scale µ. Indeed, in the UV limit,the strict one-loop expression has the wrong asymptoticform:
DT (k2) ' −[g2γ0k
2 ln |k2|+O(k2)]−1
, (37)
while the RG analysis yields
DT (k2) ' − ZUVk2(ln |k2|)γ
, (38)
where we have analytically continued the gluon propaga-tor from Euclidean momentum k2 = −k2
E to complex k2,γ0 and γ = γ0/β0 are given in (4), and ZUV > 0.
However, for γ = γ0/β0 > 0, the phase of the gluonpropagator on UV momenta is qualitatively correct de-spite the wrong exponent of the logarithm. Furthermore,both of the propagators have negative spectral functionsρ(σ2) = 1
π Im DT (σ2+iε) < 0 in the UV limit. Therefore,we expect the one-loop expression will provide a goodapproximation of the phase of the gluon propagator forγ > 0 based on the robustness of the winding number.In Sec. IV, we indeed confirm that the RG improved andstrict one-loop gluon propagators yield the same NW (C)for qualitatively the same parameter region.
B. Renormalization group functions
Here we present the renormalization group functionsfor (g,M2,mq, ZA, ZC , Zψ) [7, 21]. For later convenience,we put
λ :=C2(G)g2
16π2. (39)
The beta functions βα and anomalous dimensions γΦ aredefined for the masses and coupling α = g,M2,mq, λ andfor the fields Φ = A ,C , ψi renormalized at a scale µ as
βα := µd
dµα, γΦ := µ
d
dµlnZΦ, (40)
where the bare masses and the bare coupling are fixed intaking the derivative.
From the nonrenormalization theorems, ZAZCZM2 =1 and Zg
√ZAZC = 1, βλ, or equivalently βg, and βM2
are expressed by γA and γC :
βλ = λ(γA + 2γC), βM2 = M2(γA + γC). (41)
The other RG functions, γC , γA, γψ and βmq are com-puted as follows within the one-loop approximation. Bydifferentiating the counterterms, γΦ = µ d
dµδΦ, we have
for ghosts, [7]
γC = − λ
2ν2[2ν2 + 2ν − ν3 ln ν
+ (ν − 2)(ν + 1)2 ln(ν + 1)], (42)
for gluons,[7, 21]
γA =γYMA + γquarkA (43)
γYMA =− λ
6ν3
[ (17ν2 − 74ν + 12
)ν − ν5 ln ν
+ (ν − 2)2(ν + 1)2(2ν − 3) ln(ν + 1)
+ ν3/2√ν + 4
(ν3 − 9ν2 + 20ν − 36
)× ln
(√ν + 4−
√ν√
ν + 4 +√ν
)],
γquarkA =16λC(r)
3C2(G)
6t2 ln(√
4t+1+1√4t+1−1
)√
4t+ 1− 3t+
1
2
, (44)
where
t :=m2q
µ2, (45)
8
and, for quarks, [21]
γψ =g2C2(r)
32π2µ4M2
{Q
K2
[µ8 + µ6
(m2q +M2
)+ 2
(M2 −m2
q
)3 (m2q + 2M2
)+ µ4
(−2M2m2
q − 3m4q + 3M4
)+ µ2
(M2 −m2
q
) (6M2m2
q + 5m4q + 7M4
)]+ 2µ2M2
(µ2 − 2m2
q + 4M2)
+ 2(µ6 − 3µ2m4
q − 2m6q
)ln
(µ2
m2q
+ 1
)+ 2[µ6 − 3µ2
(m4q +M4
)− 2
(−3M4m2
q +m6q + 2M6
)]ln(mq
M
)}. (46)
Since Zmq = 1+δmq−δψ in the one-loop level, we obtainthe beta function for the quark mass mq [21],
βmq = mqγψ +3g2C2(r)mq
16π2
[−2 +
2(M2 −m2
q
)µ2
ln(mq
M
)−
(µ2(m2q +M2
)+(M2 −m2
q
)2)Q
µ2K2
],
(47)
where Q and K2 are given in (26) with k2E = µ2.
The flow diagram of (λ = C2(G)g2
16π2 , u = M2/µ2, t =
m2q/µ
2) for NF = 3 is depicted in Fig. 2. In Fig. 2, theinfrared safe trajectories are shown as the blue curves,while the green curves terminate at an infrared Landaupole. The red one is not ultraviolet asymptotic free in thesense that the running coupling λ(µ) does not vanish inthe UV limit µ→∞ on the RG flow. As the pure Yang-Mills case, the diagram has the infrared safe trajectories,on which the running gauge coupling λ is always finitein the all scales. One can confirm that the qualitativefeature is the same for NF ≤ 16.
C. Asymptotic behaviors of infrared safetrajectories
From the flow diagram Fig. 2, the infrared safe tra-jectories possess the following features: (i) In the UVlimit µ → ∞, the parameters (λ, u, t) tend to (λ →0, u → 0, t → 0), (ii) In the IR limit µ → 0, they tendto (λ→ 0, u→∞, t→∞). In this subsection, we studytheir asymptotic behaviors within the one-loop level.
Beforehand, there is a caveat on this discussion. Inthe RG analysis of Oehme and Zimmermann [47], theparameter of the theory is only the gauge coupling g,which guarantees that the higher-loop effects are sup-pressed in the ultraviolet region precisely. Then, the
FIG. 2: RG flows in the parameter space (λ = C2(G)g2
16π2 , u =
M2/µ2, t = m2q/µ
2) for NF = 3. The arrows indicate infrareddirections µ→ 0. The blue trajectories are infrared safe. Thegreen ones end at an infrared Landau pole. The red one isnot ultraviolet asymptotic free.
one-loop RG gives a strong argument on asymptotic be-haviors and enables us to establish the UV negativityof the gluon spectral function for NF < 10. However,the massive Yang-Mills model has the mass parameter,which can potentially invalidate the perturbation theoryin λ even though λ→ 0 asymptotically. Here, we assumethat the perturbation theory “works well”. In particular,we assume that the spectral function is dominated by theone-loop contribution in both UV and IR limits.
In the pure Yang-Mills case, a similar discussion on theRG functions and Euclidean propagators can be found ine.g., [16]. In what follows, we study asymptotic behaviorsof the RG functions, Euclidean propagators, and spectralfunctions in the massive Yang-Mills model described inthe previous subsections.
1. UV limit
We consider the UV limit and confirm that the asymp-totic behaviors of the massive Yang-Mills model is consis-tent with those of the Faddeev-Popov Lagrangian. Tak-ing the limit u, t→ 0, we have
γA →(−13
3+
8C(r)
3C2(G)
)λ+O(u, t), (48)
γC → −3
2λ+O(u lnu), (49)
9
which reproduce the standard one-loop beta function:
βλ = λ(γA + 2γC)→ β0,λλ2, (50)
β0,λ := −22
3+
8C(r)
3C2(G)< 0, (51)
recovering the UV asymptotic behavior of the gauge cou-pling λ ∼ −β0,λ/ ln(µ/Λ) with some constant scale Λ. Onthe other hand, the beta functions for the masses read,for gluons,
βM2 = M2(γA + γC)→ β0,M2λM2,
β0,M2 := −35
6+
8C(r)
3C2(G), (52)
and for quarks,
βmq → −6C2(r)
C2(G)λmq +O(u lnu, t ln t). (53)
Notice that at G = SU(3), the gluon mass is suppressedlogarithmically for β0,M2 < 0, or NF < 35
8 C2(G) =105/8 ≈ 13.1 and enhanced logarithmically for 14 ≤NF ≤ 16, while the correction to the quark mass alwayssuppresses the quark mass in the logarithmic way.
Note that u = M2/µ2 → 0 and t = m2q/µ
2 → 0 ex-ponentially faster than λ→ 0. This justifies a posterioritaking the limit u, t→ 0 in the first step.
Next, let us consider the propagators on the Euclideanmomenta. From the nonrenormalization theorems, ZA =ZλZ
−2M2 and ZC = ZM2Z−1
λ , which yield together withthe renormalization conditions (27), [7],
DT (k2E , µ
20) =
λ0
M40
M4(k2E)
λ(k2E)
1
k2E +M2(k2
E),
∆(k2E , µ
20) = −M
20
λ0
λ(k2E)
M2(k2E)
1
k2E
, (54)
where µ0 is the renormalization scale, and M0 and λ0 arethe mass and coupling at µ0. In the UV limit k2
E →∞,
DT (k2E , µ
20) ∼ [λ(k2
E)]2β
0,M2
β0,λ−1 1
k2E
∼ 1
k2E(ln k2
E)γ0,Aβ0,λ
,
∆(k2E , µ
20) ∼ −[λ(k2
E)]−β0,M2
β0,λ+1 1
k2E
∼ − 1
k2E(ln k2
E)γ0,Cβ0,λ
,
(55)
in accordance with the asymptotic form [47], where we
have defined γ0,A := − 133 + 8C(r)
3C2(G) = 2β0,M2 − β0,λ and
γ0,C := − 32 = β0,λ − β0,M2 . These lead to the supercon-
vergence relations for the gluon spectral function whenγ0,A < 0, and also for the ghost spectral function. Forquarks, one can find γψ = O(λ2, u, t), which gives nologarithmic correction to the quark propagator.
One can see the UV negativity for the gluon spectralfunction and UV positivity for the ghost spectral function
from (55) combined with the analyticity or RG improve-ment with respect to |k2| in the complex k2 plane [47].
Thus, it turns out that the massive Yang-Mills modelcan describe the correct UV behavior by RG improve-ment from the above observations, although we have re-garded the massive Yang-Mills model as a low-energy ef-fective model.
2. IR limit
Next, we consider the IR limit of the infrared safetrajectories: λ → 0, u → ∞, t → ∞. Taking the limitu, t→∞, we have
γA →1
3λ+O(u−1, t−1),
γC → 0 +O(u−1 lnu), (56)
as those of the pure Yang-Mills case shown in [16]. Thisindicates that the infrared gluon and ghost sector is in-sensitive to the quark flavor effect except the case of“massless quark” mq = 0. As shown in [16], we findβλ/λ = βM2/M2 = λ/3, which leads as µ→ 0 to
M2 ∼ λ ∼ 1
lnµ−1. (57)
The quark mass mq runs according to
βmq → O(u−1 lnu, u−1 ln t, t−1), (58)
which indicates absence of logarithmic correction to thequark mass in the infrared. Since u = M2/µ2 →∞ andt = m2
q/µ2 → ∞ exponentially increase more rapidly
than λ ∼ 1lnµ → 0, the above evaluation is consistent.
From (57) and (54), the infrared safe trajectories de-scribe the decoupling solution, i.e., the massive gluon andthe massless free ghost in the IR limit as those in the puremassive Yang-Mills model of [16].
Finally, we examine the spectral functions. In thegluon vacuum polarization, the quark loop affects thegluon spectral function in the region of (2mq)
2 < k2 <∞as hq(t) has the branch cut only on −1/4 < t < 0. There-fore, the IR spectra of the gluon and ghost are the sameas those in the pure Yang-Mills case within the one-looplevel because the quark mass will be finite in the infraredas can be seen from (58).
As s→ 0, the vacuum polarizations are
ΠTWYM,ren.(s) =
λ
12
[3f(ν)− 15
2+ s ln s+O(s)
],
ΠTWgh,ren.(s) =
λ
4s
[(5
2− f(ν)
)− s ln s+O(s2)
]. (59)
From the assumption that the higher loop terms aresuppressed as µ → 0, i.e., the running u and t do not
10
invalidate the perturbation theory in λ, the following ap-proximation holds in the IR limit |k2| → 0,
DT (k2, µ20) ' ZA(|k2|, µ2
0)D1−loopT (k2, |k2|),
∆(k2, µ20) ' ZC(|k2|, µ2
0)∆1−loop(k2, |k2|) (60)
where ZA(|k2|, µ20) and ZC(|k2|, µ2
0) are the renormaliza-
tion factors and ZA(µ2, µ20) = λ0
M40
M4(µ2)λ(µ2) , ZC(µ2, µ2
0) =
M20
λ0
λ(µ2)M2(µ2) from the nonrenormalization theorems. As
µ2 → 0,
ZA(µ2, µ20) ∼M2(µ2), ZC(µ2, µ2
0) ∼ const. (61)
Therefore, the spectral functions originating from thebranch cut on timelike momenta have the IR asymptoticforms σ2 → +0: for gluons,
ρ(σ2) =1
πIm DT (σ2 + iε, µ2
0)
' ZA(σ2, µ20)
1
πIm D1−loop
T (σ2 + iε, σ2)
∼ Im1[
1 + s+ ΠTWren.(s)
]∣∣∣∣∣∣s=− σ2
M2−iε
∼ − Im ΠTWren.(−
σ2
M2− iε)
∼ − λ(σ2)
M2(σ2)σ2 ∼ −σ2 < 0, (62)
and for ghosts,
ρgh(σ2) =1
πIm ∆(σ2 + iε, µ2
0)
' ZC(σ2, µ20)
1
πIm ∆1−loop(σ2 + iε, σ2)
∼ − Im1
M2(σ)[s+ ΠTW
gh,ren.(s)]∣∣∣∣∣∣s=− σ2
M2−iε
∼ const. > 0. (63)
These results demonstrate the IR negativity of the gluonspectral function and the IR positivity of the ghost spectralfunction for the infrared safe trajectories with mq > 0.The ghost spectral function has a delta function at σ2 = 0with a negative coefficient associated to the negativenorm massless ghost state and shows the finite positivespectrum in the limit σ2 → +0. Although we assumethat the running u and t do not invalidate the asymp-totic perturbative expansion in λ, the IR negativity ofthe gluon spectral function will be a remarkable conse-quence from the infrared safety.
As an alternative evidence for the IR negativity of thegluon spectral function, we can utilize the IR behavior ofthe Euclidean propagator. For example, although this re-lation gives a trivial result in this model, it is worthwhileto note that they are related as limkE→+0
ddkE
DT (k2E) =
−π limσ→+0ddσρ(σ2) [32]. Let us consider the IR asymp-
totic behavior of the gluon propagator. The Euclideanpropagator (54) can be rewritten as [16],
DT (k2E) =
Z−1C (k2
E)
1 + k2E/M
2(k2E), (64)
from which we have
k2E
d
dk2E
DT (k2E) ' 1
2Z−1C (k2
E)[−γC − 2u−1]
' 1
2Z−1C (k2
E)u−1[λ
2u−1 lnu− 2]
∼ −k2E ln k2
E > 0, (65)
where we have used γC → −λ2u−1 lnu + O(λu) and
λ(µ) ' − 6lnµ2 as µ → 0. Therefore, as kE → 0, we
have
d
dk2E
DT (k2E) ∼ | ln k2
E | → +∞. (66)
This logarithmic divergence is precisely consistent withthe IR spectral negativity of ρ(σ2) ∼ −σ2 < 0. Indeed,if ρ(σ2) ∼ −σ2 < 0, then, from the (generalized) spectralrepresentation, as kE → +0,
d
dk2E
DT (k2E) ' −
∫ ∞0
dσ2 ρ(σ2)
(σ2 + k2E)2
∼ | ln k2E |, (67)
which shows the coherency between the approximation of(60) in the complex k2 momentum and the IR asymptoticbehavior of the Euclidean propagator of (54). Both ofthe two arguments imply the IR negativity of the gluonspectral function. As negativity of a spectral functionin a weak sense leads to complex poles [40], the IR andUV negativity of the gluon spectral function supports theexistence of complex poles in the gluon propagator.
For the quark propagator, the one-loop expressiongives no nontrivial IR spectrum from (25). This is ratherclear from the facts that the Feynman diagram indicatesIm Σ has nonvanishing value only for k2 > m2
q and thatmq(µ) is constant in the IR limit.
Incidentally, let us comment on the nontrivial infraredfixed point to give the Gribov-type scaling solution [16].In the massive Yang-Mills model of the pure Yang-Millstheory, a nontrivial fixed point in (λ, u) plane appears,where the gluon and ghost exhibit the Gribov-type be-havior. One can find a similar fixed point in the massiveYang-Mills model with “massless quarks” mq = 0. Forinstance, the fixed point lies at (λ, u, t) = (4.0, 1.6, 0)in NF = 3. However, this scaling-type fixed point hasan infrared unstable direction. Therefore, with quarksmq > 0, the unstable scaling-type fixed point disappearsdue to the running of the quark mass.
11
IV. NUMERICAL RESULTS ON ANALYTICSTRUCTURES
In this section, we present results on the number ofcomplex poles of the gluon and quark propagators inthe massive Yang-Mills model with quarks at the one-loop approximation and its RG improvements. With-out quarks, we found the gluon propagator has NP =−NW (C) = 2 for all parameters (g2,M2) due to thenegativity of the spectral function [40]. Surprisingly, itturns out that the model with many light quarks hasNP = 0, 2, 4 regions, depending on the number of quarksNF and the parameters (g2,M2,m2
q). In particular, forNF (4 . NF ≤ 9) light quarks, the NP = 4 region,where the gluon propagator has two pairs of complexpoles, covers a typical value of coupling g, e.g., g ∼ 4 inthe setting to be described shortly, and the models withNF (NF ≥ 10) very light quarks have the region with nocomplex poles around the typical value of coupling g.
From here on, we set G = SU(3) and the renormaliza-tion scale µ0 = 1 GeV. With the RG improvements, thebest fit parameters reported in [21] are g = 4.5,M = 0.42GeV, and the up and down quark masses mu = md =0.13 GeV for the case of NF = 2 while g = 5.3,M = 0.56GeV, and mu = md = 0.13 GeV for NF = 2+1+1 (withthe assumption on the strange and charm quark massesof ms = 2mu and mc = 20mu). Notice that the “quarkmass” mq of this model should not be confused with thecurrent quark mass. The “quark mass” parameter mq
will be chosen to reproduce the propagators. Rather, mq
will be of the same order of the constituent quark mass,since we renormalized this model at µ0 = 1 GeV. In par-ticular, the massless quarks will differ from the “masslessquarks” mq = 0 due to the spontaneous breakdown of thechiral symmetry.
First, we investigate NW (C) of the strict one-loopgluon propagator, mainly at fixed typical values of theparameters g = 4 and M2 = 0.2 GeV2 to see a quali-tative overview of the analytic structures of this model.Next, we consider the RG improvements of these resultsand evaluate the gluon spectral function at the best-fitparameters for NF = 2. Comparing the strict one-loopand RG-improved results could support the robustness ofNW (C). Furthermore, we discuss the complex poles ofthe quark propagator and comment on the ghost propa-gator.
A. Gluon propagator: Strict-one-loop analysis
Based on the strict one-loop gluon propagator (13)combined with (29), (30), and (31) of Sec. III, we com-pute the winding number NW (C) for NF ≤ 9 accordingto the procedure (9) of Sec. II. Notice that the strictone-loop gluon propagator satisfies the conditions (i)and (ii) in Sec. II. Since the two-point vertex function
Γ(2)A = [DT ]−1 is finite, the gluon propagator has no ze-
ros, NZ = 0. Thus we can count the number of complex
FIG. 3: Contour plot of NW (C) for the gluon propagatoron the (NF , ξ) plane at g = 4 and M2 = 0.2 GeV2, whichgives the number of complex poles through the relation NP =−NW (C). In the NP = 0, 2, 4 regions, the gluon propagatorhas zero complex pole, one pair, and two pairs of complexconjugate poles, respectively. The vertical dashed lines atNF = 3 and NF = 6 correspond to the top and bottomfigures of Fig. 5, respectively.
poles by computing NW (C) = −NP .
1. The number of complex poles at the typical (g,M)
As a first step, we investigate the (NF , ξ = m2q/M
2)dependence of NP at the fixed typical values of the pa-rameters to obtain an overview.
Figure 3 displays the contour plot of NW (C) on the
(NF , ξ =m2q
M2 ) plane at the fixed g = 4 and M2 =
0.2 GeV2. This figure is restricted to 0 ≤ NF < 10,since the one-loop approximation will be not reliable forNF ≥ 10. Indeed, the naive one-loop UV asymptoticform (37) for γ0 > 0, i.e., for NF ≥ 10, shows the ex-istence of a Euclidean pole together with the fact thatDT (0) > 0 and DT (k2) has no zeros.
This result illustrates that the gluon propagator hasfour complex poles in the region colored with dark gray(4 . NF ≤ 9, 0.2 . ξ . 0.6). It predicts that the tran-sition occurs from the NP = 2 region to NP = 4 regionby adding light quarks since NW (C) is a topological in-variant. In the NP = 4 region, the gluon propagator hastwo pairs of complex conjugate poles if we exclude thepossibility of Euclidean poles on the negative real axis.
To see details of the NP = 4 region and its boundary,let us look into positions of complex poles.
12
2. Location of complex poles
Here we take a further look into positions of complexpoles and the transition of NW (C). We will report theratio w/v for a position of a complex pole k2 = v + iwand trajectories of complex poles for varying ξ.
First, we focus on the ratio w/v at the complex polek2 = v + iw, on which we set w ≥ 0 without loss ofgenerality from the Schwarz reflection principle, in orderto see whether or not the gluon presents a particlelikeresonance. Since the gluon propagator has at most twopairs of complex conjugate poles, it is sufficient to findmaxw/v and minw/v.
The results are demonstrated in Fig. 4. In general, theratio w/v of a complex pole decreases as NF increases.This implies that the gluon exhibits more particlelike be-havior for larger NF . Note that the ratio minw/v rapidlydecreases near the boundary between NW (C) = −2 andNW (C) = −4.
Second, we examine trajectories of complex poles forNF = 3 and NF = 6. Figure 5 plots the pole locationwith varying ξ. In the case of NF = 3, no transitionchanging NP occurs; the pole moves gradually and low-ers its real part v as increasing ξ. On the other hand,the trajectory of complex poles is completely different atNF = 6 where the transition occurs. The pole movescontinuously from the position of ξ = 0, but is absorbedinto the branch cut at ξ ≈ 0.6. Beforehand, the new polearises at ξ ≈ 0.2 from the branch cut.
These observations demonstrate that (i) the gluonpropagator has a pole at timelike momentum, whichcould correspond to a physical particle, on the bound-ary between the NP = 2 and NP = 4 regions and that(ii) the pole bifurcates and becomes a new pair of com-plex conjugate poles in the NP = 4 region. This appear-ance of the new pair of complex conjugate poles from thebranch cut on the real positive axis is compatible withthe rapid decrease of minw/v on the boundary betweenthe NP = 2 and NP = 4 regions in Fig. 4.
3. (g,M) dependence
We have seen that the gluon propagator changes itsnumber of complex poles when many light quarks areincorporated at the fixed typical values of the parametersof g = 4 and M2 = 0.2 GeV2. Therefore, we have tocheck whether or not the existence of the transition isinsensitive to choices of the parameters (g,M).
One can verify that the NW (C) = −4 region in the pa-
rameter space (λ = C2(G)g2
16π2 , u = M2/µ20, ξ =
m2q
M2 ) largelyexpands by changing NF from NF = 3 to NF = 6. AtNF = 6, the NW (C) = −4 region dominates the parame-ter region around the typical value g ∼ 4, M2 ∼ 0.2 GeV2
with 0.2 .m2q
M2 . 0.6. See Appendix B for details.Thus our qualitative conclusion will be valid: this
model has the transition of NP , and the gluon propa-
FIG. 4: Contour plots of minw/v (top) and maxw/v (bot-tom) for a complex pole at k2 = v + iw, w ≥ 0 of the gluonpropagator on the (NF , ξ) plane. The regions of w/v > 1,1 > w/v > 0.1, 0.1 > w/v > 0.01, and 0.01 > w/v are rep-resented by different levels of gray. The vertical dashed linesat NF = 3 and NF = 6 correspond to the top and bottomfigures of Fig. 5, respectively. The region of minw/v < 0.01appears around the transition between NW (C) = −2 andNW (C) = −4. As NF increases, the ratio w/v tends to besmall.
gator has four complex poles in the presence of NF (4 .
NF ≤ 9) light quarks with mq satisfying 0.2 .m2q
M2 . 0.6.
Incidentally, to compare the results with those of theRG improved gluon propagator, we show in Fig. 6 con-tour plots of NW (C) at u = M2/µ2
0 = 0.2.
13
●●●●●
●●●●●●●●●●
●●●●●
●●●●●●●●●●
ξ = 0
ξ = 0.8
NF = 3
0.0 0.1 0.2 0.3 0.4 0.5 0.6v [GeV
2 ]0.0
0.1
0.2
0.3
0.4
0.5
w [GeV2 ]
●
●
●
●●
●●
●●
●
●●
●●●
●
●
●
●●●●● ● ●● ●
●●●
ξ = 0
ξ = 0.6ξ = 0.2
ξ = 0.8
NF = 6
0.0 0.2 0.4 0.6 0.8 1.0v [GeV
2 ]0.0
0.1
0.2
0.3
0.4
0.5
w [GeV2 ]
FIG. 5: Positions of poles of the gluon propagator at NF = 3
(top) and NF = 6 (bottom) for varying 0 < ξ =m2q
M2 < 0.8.As ξ increases, the poles move in the direction shown by thearrows. At NF = 3 (top), the pole moves continuously fromthe position of ξ = 0. In contrast, at NF = 6 (bottom), thepole from the position of ξ = 0 goes behind the branch cutat ξ ≈ 0.6. Moreover, the new pole (v ≈ 0.16 GeV2) appearsfrom the branch cut at ξ ≈ 0.2.
B. Gluon propagator: RG improved analysis
So far, we have studied the gluon propagator in thestrict one-loop level, relying on the robustness of thewinding number. To make this robustness more reliableand to study the structure of the gluon propagator forNF ≥ 10, it is important to survey the winding numberfor the RG improved gluon propagator. Moreover, thismodel reproduces the numerical lattice results with theRG improved propagators at the “realistic” parametersg = 4.5,M = 0.42 GeV, and mq = 0.13 GeV at µ0 = 1GeV and NF = 2 [21]. In this subsection, we investigatethe analytic structure of the one-loop RG improved gluonpropagator for the realistic parameters and its parameterdependence for each number of quark flavors NF .
The RG equation for the gluon propagator is
DT (k2E , α(µ2), µ2) = Z−1
A (µ2, µ20)DT (k2
E , α(µ20), µ2
0)(68)
where α denotes the set of gauge coupling and massesα = (λ, u = M2/µ2, t = m2
q/µ2) and ZA(µ2, µ2
0) is therenormalization factor computed by the anomalous di-mension. We then approximate the gluon propagator to
FIG. 6: Two-dimensional slice of equi-NW (C) volume of thestrict one-loop gluon propagator at NF = 3 (top) and NF = 6
(bottom) in the (λ = C2(G)g2
16π2 , u = M2/µ20 = 0.2, ξ =
m2q
M2 )space.
avoid the large logarithms as
DT (k2, α(µ20), µ2
0) ≈
ZA(|k2|, µ20)D1−loop
T (k2, α(|k2|), |k2|) (69)
Although the RG improvements only for the modulus|k2| on the complex k2 plane may break the analyticity,this will give a better approximation than the strict one-loop approximation for the propagator on the timelikemomenta.
First, let us see the gluon propagator for the realis-tic parameters. Figure 7 plots the real and imaginaryparts of the gluon propagator on the timelike momenta.The spectral function is quasinegative, i.e., the spectralfunction is negative ρ(k2
0) < 0 at all timelike zeros k20
of Re D(k2). Then, NW (C) = −2 from the invariance
14
1 2 3 4k
2 [GeV2 ]
-1
1
2
3
4
5
Re D(k2 + i ϵ), Im D(k2 + i ϵ)[GeV-2 ]
FIG. 7: Real part (blue) and imaginary part (orange) of thegluon propagator at the realistic parameters for NF = 2 [21]on the positive real axis. This shows its spectral function isquasinegative, from which NW (C) = −2. Notice that thespectral function exhibits the linear decrease with respect tok2 in agreement with (62) in the IR limit.
of NW (C) under continuous deformation [40]. Since thegluon propagator has no zeros NZ = 0 from the fact thatthe RG improvement only affects the renormalization fac-tor and the running couplings, we deduce the gluon prop-agator of this model has one pair of complex conjugatepoles as in the pure Yang-Mills case. Incidentally, noticethat the spectral function ρ(σ2) decreases linearly withrespect to σ2 in the IR limit σ2 → +0, which is a generalfeature for the infrared safe trajectories with mq > 0 inthis model as shown in (62).
Next, we investigate the number of complex poles NPin the whole parameter space for each NF . Since NZ = 0within this approximation as before, NP can be com-puted as NP = −NW (C). Note that different points ona same renormalization group trajectory in the three di-mensional parameter space, e.g., Fig. 2, provide the sameanalytic structure. Indeed, they are exactly connected bythe scale transformation from the dimensional analysisand the anomalous dimension: under a scaling κ,
DT (k2, α(κ2µ20), µ2
0)
= κ2Z−1A (κ2µ2
0, µ20)DT (κ2k2, α(µ2
0), µ20), (70)
which shows that DT (k2, α(κ2µ20), µ2
0) andDT (k2, α(µ2
0), µ20) have the same number of com-
plex poles. Therefore, it suffices to compute in thetwo-dimensional slice of the renormalization group flow,see Fig. 2. From here on, we employ the two dimensionalslice at u = M2/µ2 = 0.2.
Figures 8 and 9 reveal the results for NF = 3, 6, 9, 10.Note that, since the analytic structure of the gluon prop-agator approaches to that of the pure massive Yang-Millsmodel as mq →∞, the gluon propagator has one pair ofcomplex conjugate poles NP = 2 for large ξ. At NF = 3,the NP = 2 region dominates the region of the param-eters. At NF = 6, in the presence of light quarks, theNP = 4 region, on which the gluon propagator has twopairs of complex conjugate poles, occupies the larger re-gion, especially around a typical coupling λ ∼ 0.3. On
FIG. 8: Two-dimensional slice of equi-NW (C) volume of thegluon propagator at NF = 3 (top) and NF = 6 (bottom) in
the (λ = C2(G)g2
16π2 , u = M2/µ20 = 0.2, ξ =
m2q
M2 ) space. Besidesthe region with the Landau poles (white), these plots havealmost the same structure as Fig. 6.
the other hand, at NF = 9, the NP = 0 region, on whichthe gluon propagator has no complex poles, expands andthe NP = 4 region shrinks. At NF = 10, notably, theNP = 4 region disappears, and the NP = 0 region ap-pears in the larger region of λ for the very light quarkmass mq/M � 1.
Note that the strict one-loop gluon propagator andthe RG improved one provide almost the same result forNF = 3, 6 by a comparison between Fig. 6 and Fig.8. This supports the robustness of the winding numberNW (C).
It is remarkable that the parameter dependence of theanalytic structure drastically changes between NF = 9and NF = 10. The extension of NP = 0 region for verylight quark mass can be seen from the UV positivity of
15
FIG. 9: Two-dimensional slice of equi-NW (C) volume of thegluon propagator at NF = 9 (top) and NF = 10 (bottom)
in the (λ = C2(G)g2
16π2 , u = M2/µ20 = 0.2, ξ =
m2q
M2 ) space. Thewhite region shows RG trajectories with the Landau poles.
the spectral function for NF ≥ 10 [47], since the posi-tivity in a weak sense indicates NW (C) = 0 [40]. In asimilar sense, the disappearance of NP = 4 region couldbe related to the UV positivity of the spectral function,which might lower NW (C).
C. Quark propagator
As pointed out in [21], the one-loop RG computation isnot enough to reproduce the quark wave function renor-malization, although the one-loop quark mass functionexhibits a qualitative agreement with lattice results. Thehigher loop corrections or other ignored effects are thushighly significant to describe the quark sector well. Here,we try to examine the quark propagator within the frame-
1 2 3 4k
2 [GeV2 ]
-2
2
4
6
Re Ss(k2 + i ϵ), Im Ss(k
2 + i ϵ) [GeV-1 ]
1 2 3 4k
2 [GeV2 ]
-5
5
10
Re Sv (k2 + i ϵ), Im Sv (k
2 + i ϵ) [GeV-2 ]
FIG. 10: Real (blue) and imaginary (orange) parts of thescalar part (top) and the vector part (bottom) of the quarkpropagator on the positive real axis at the realistic parametersfor NF = 2 [21]. Their spectral function are negative, fromwhich NW (C) = −2 for both the scalar and vector parts.
1 2 3 4σ2 [GeV
2 ]
-0.10
-0.05
σ ρv (σ2) - ρs(σ
2) [GeV-1 ]
FIG. 11: The quark spectral function σρv(σ2) − ρs(σ2) is
plotted at the realistic parameters. The positivity conditionis violated below 0.8 GeV2.
work of the one-loop RG to catch the qualitative featureas a first attempt.
Let us consider the vector and scalar parts of the quarkpropagator:
Sv(k2E) =
Γren.v
k2E(Γren.v )2 + (Γren.s )2
,
Ss(k2E) =
Γren.s
k2E(Γren.v )2 + (Γren.s )2
. (71)
We define the spectral functions associated to these prop-agators,
ρv(σ2) =
1
πImSv(k2
E = −σ2 − iε),
ρs(σ2) =
1
πImSs(k2
E = −σ2 − iε). (72)
16
FIG. 12: Two-dimensional slice of equi-NW (C) volume of thequark propagator at NF = 3 (top) and NF = 12 (bottom) in
the (λ = C2(G)g2
16π2 ,M2/µ20 = 0.2, ξ =
m2q
M2 ) space. Both of thescalar and vector parts provide the same NW (C).
Note that the positivity of the state space would imply
ρv(σ2) > 0, σρv(σ
2)− ρs(σ2) > 0. (73)
We use the same improvement scheme as (69). Thescalar and vector parts of the quark propagator at therealistic values of the parameters are plotted in Fig. 10.Both of the propagators have negative spectral functionsand therefore have one pair of complex conjugate poleslike the gluon propagator. Notice that both the condi-tions for the positivity (73) are violated. The formercondition can be seen from the vector part of the quarkpropagator in Fig. 10. Figure 11 shows the violation ofthe latter one.
Next, we investigate the number of complex poles withvarious parameters (NF , g,M,mq). The results on thewinding number NW (C) at NF = 3 and NF = 12 are
shown in Fig. 12. We numerically checked that both ofthe scalar and vector parts of the quark propagator (71)yield the same NW (C) on the region shown in Fig. 12,from which NZ = 0 for the both parts.
The quark sector does not exhibit a new structure byadding NF quarks. The qualitative insensitivity of thequark sector to NF is evident in the sense that the strictone-loop expression is, of course, independent of NF andthat the quark propagator will only be influenced indi-rectly by NF via the running parameters. However, itis worthwhile to note that the region with NP = 0 ex-tends slightly as NF increases, in particular for very lightquarks ξ � 1, which is a common feature with the gluonone.
D. Ghost propagator
Finally, let us add some comments on the ghost prop-agator. The strict one-loop propagator is not affected bythe dynamical quarks. Therefore, from the proposition(Case III) of [40], the value NW (C) = 0 for ghosts willhold after the RG improvements unless the RG trajec-tory has a Landau pole. Therefore, we conclude that theanalytic structure of the ghost propagator within thisapproximation is insensitive to NF and that the ghostpropagator has no complex poles.
V. CONCLUSION
Let us summarize our findings. The argument principlefor a propagator relates the propagator on the timelikemomenta and the number of complex poles. The windingnumberNW (C) = NZ−NP can be computed numericallyfrom a propagator on timelike momenta according to (9),where NZ and NP stand for the number of complex zerosand poles respectively.
To study the analytic structures of the QCD propaga-tors, we have employed an effective model of QCD, themassive Yang-Mills model. We have found that the in-frared safe trajectories, on which the running coupling isfinite in all scales, reproduce the UV asymptotic behav-ior (55) originally obtained by Oehme and Zimmermann[47] and that the gluon spectral function is negative inthe IR limit ρ(σ2) ∼ −σ2 for mq > 0 for quarks with thequark mass parameter mq of this model (62). The UVnegativity of Oehme-Zimmermann and IR negativity ofthe gluon propagator argued in this paper supports theexistence of complex poles in the gluon propagator fromthe general relationship claiming that a negative spectralfunction in a weak sense leads to complex poles [40].
In the “realistic” parameters used in [21] to fit the nu-merical lattice results, both the gluon and quark propa-gators have quasinegative spectral functions and one pairof complex conjugate poles, while the ghost propagatorhas no complex poles.
17
Finally, we have investigated the number of complexpoles in this model with many quarks by computingNW (C) for various parameters. For the gluon, there areseveral interesting features. First, the NP = 2 regiondominates the parameter region if ξ = m2
q/M2 is not so
small or NF . 4. In this region, the gluon propagatorhas one pair of complex conjugate poles as in the pureYang-Mills case [40]. Second, the NP = 4 region, wherethe gluon propagator has two pairs of complex conjugatepoles, expands by adding light quarks. A typical set ofthe parameters (g ≈ 4,M2 ≈ 0.2 GeV2) is covered by theNP = 4 region when 4 . NF ≤ 9, 0.2 . ξ . 0.6. Thesefeatures hold in the computations both for the strict one-loop and the RG-improved one-loop results. Third, thestrict one-loop result on the locations of complex poles in-dicates that the gluon tends to be “particlelike” as NF in-creases. Fourth, for NF ≥ 10, the one-loop RG-improvedgluon propagator has no NP = 4 region. Moreover, thegluon propagator with NF (NF ≥ 10) very light quarksξ � 1 has no complex poles even for a typical value ofthe gauge coupling.
The existence of complex poles invalidates the Kallen-Lehmann spectral representation, which is a fundamentalconsequence of QFT describing physical particles. There-fore, the existence of complex poles of the gluon andquark propagators could be a signal of confinement ofthe elementary degrees of freedom in QCD.
In conclusion, the above results imply that the confine-ment mechanism may depend on the number of quarksand quark mass since complex poles represent a deviationfrom physical particles and will be related to confine-ment. Furthermore, the drastic change between NF = 9and NF = 10 quarks could be possibly related to the“deconfinement” in line with the conformal window.
VI. DISCUSSION AND FUTURE WORK
Several comments regarding these results are in order.First, let us mention a comparison with a similar ap-
proach [38, 39], where the analytic structures of gluonand quark propagators are investigated with a massive-type model in light of the variational principle and opti-mization. In there, the gluon propagator has two pairsof complex conjugate poles, while the quark propagatorhas a timelike pole and no complex poles in NF = 2QCD. These structures are different from ours, in whichthe gluon and quark propagators have one pair at the“realistic” parameter. Although the quark sector of bothmodels lacks accuracy, the difference will be relevant inlight of the confinement of quark degrees of freedom3; atimelike pole might correspond to a physical one-particle
3 The confinement of quark degrees of freedom, which stands forabsence of the quark one-particle state from the physical spec-trum, should not be confused with the well-studied “quark con-finement” that means an external quark source requires infinite
state even after some confinement mechanism works. Inthis sense, the absence of a timelike pole will be favored.
Second, we comment on the NF dependence of the con-densation. We have studied the mass-deformed model asan effective model for QCD or QCD-like theories basedon the facts that the gluon mass can minimally improvethe gauge fixing and that the effective potential for theoperator AµAµ calculated by the local composite opera-tor technique indicates the condensation of this operator[12, 13]. The former argument can give a mass-like ef-fect independently upon NF . However, the latter onewill be substantially affected by the presence of quarks.The effective potential of [13] appears to be “unbounded”for γ0 > 0, or NF ≥ 10 for G = SU(3). The “un-boundedness” follows from the fact that the coefficientof the Hubbard-Stratonovich transformation ζ runs intothe negative infinity ζ → −∞ in the UV limit µ → ∞due to additive counterterms for NF ≥ 10 even if it isset to be a positive value at some scale. This problemmight indicate the limitation of the perturbative treat-ment for the local composite operator. Therefore, wehave no cogent argument supporting the dimension-twogluon condensate for NF ≥ 10.
Third, related to the second remark, the validity of ourresults will be questionable for large NF . For instance,two-loop corrections will be important if the first coeffi-cient of the beta function is small as in the original argu-ment of the infrared conformality [25, 26]. Moreover, theresults from the truncated Schwinger-Dyson equation [29]show that the gluon and ghost propagators seem to obeya scaling-type power law in a wide range of momentumin the conformal window; the description by the massiveYang-Mills model will be inappropriate above the criticalvalue of NF . However, we can expect that the massiveYang-Mills model will be valid in the QCD-like phase, orbelow the critical value of NF . Therefore, the massiveYang-Mills model may capture some information on thetransition from the QCD-like phase.
Fourth, NF = 10, where the first coefficient of thegluon anomalous dimension changes its sign, is the valueat which the analytic structure of the gluon propagatorchanges drastically in our analysis. This value appears invarious perspectives. For example, there has been someproposal that the negativity of the gluon anomalous di-mension is crucial for confinement [51]. NF ≈ 10 can bethe critical value of the conformal phase transition [30].
Finally, the investigation of the analytic structures bymodel calculations is speculative and should be taken asan attempt toward capturing some aspects of the intri-cate dynamics of QCD. Many works of literature havedifferent claims. For example, reference [52] claims thatthe quark propagator has a pole at a timelike momentumwhile the gluon propagator has complex poles by usingsome parameterizations for the propagators and the nu-
energy or the linear rising quark-antiquark potential.
18
merical solution of truncated Dyson-Schwinger equation.The gluon propagator obtained by solving the truncatedDyson-Schwinger equation on the complex momentumplane in pure Yang-Mills theory is shown to have no com-plex poles [43]. A recent reconstruction technique indi-cates complex poles in the gluon propagator [42]. Pursu-ing the rich analytic structure of the QCD propagatorsdeserves further investigations because a QFT describingconfined particles is not yet well understood.
On a formal side, a local QFT cannot yield complexpoles in the standard perspective, see e.g., [53]. Onemight assert that complex poles correspond to short-livedexcitations and break the locality and unitarity in thelevel of propagators [34, 35]. However, if we analyticallycontinue4 a propagator with complex poles not in thecomplex momentum but in the complex time from theEuclidean space to the Minkowski one, the resulted prop-agator can be interpreted as a propagator of a QFT withan indefinite metric state space having complex spectrathat can satisfy local commutativity, e.g., [54]. Noticethat such theories with complex spectra and an indefi-nite metric are out of the scope of the axiomatic quan-tum field theory because their Wightman functions arenot tempered distribution due to complex energies, andthe theorems derived by assuming the temperedness arenot applicable to the “propagators with complex poles”.Then, complex poles would not lead to the nonlocalityand just represent unphysical degrees of freedom. Fur-ther discussion on this issue is reserved for future works.As discussed in [36, 55], it would also be interesting tostudy how the complex poles are “canceled” in the phys-ical propagator.
Acknowledgements
This work was supported by Grant-in-Aid for Sci-entific Research, JSPS KAKENHI Grant Number (C)No. 19K03840. Y. H. is supported by JSPS ResearchFellowship for Young Scientists.
Appendix A: Counting complex poles for variousinfrared behaviors
Here, we generalize the IR asymptotic condition (ii)D(k2 = 0) > 0 to derive the formula (9) in Sec. II. Thiswill be relevant for the scaling solution of the gluon prop-agator or a propagator having massless singularity.
Incidentally, in regard to the IR behavior of the gluonpropagator, there is an argument on the gluon propaga-
4 To our knowledge, a method to reconstruct a QFT from a givenEuclidean field theory has not been established in the presenceof complex poles. The standard reconstruction does not workdue to the violation of the reflection positivity. [1, 41]
tor in the Landau gauge [56]: If one restricts the con-figuration space inside the first Gribov region, the gluonpropagator satisfies limk→+0 k
d−2D(k) = 0, where d isthe spacetime dimension. This excludes the massless freebehavior for the gluon propagator irrespective of NF ford ≤ 4. Note that, however, this proposition is shown fortheories avoiding the Gribov ambiguity by the restric-tion; this proposition is not applicable to theories thatfix the gauge by schemes averaging over Gribov copieslike [15] or some justification of the standard Faddeev-Popov Lagrangian.
The generalization of (9) is as follows.Suppose that the propagator D(k2) and its data{D(xn + iε)}Nn=1 satisfies the following three conditions:
(i) In the UV limit |z| → ∞, D(k2) has the same phaseas the free propagator, i.e., arg(−D(z))→ arg 1
z as|z| → ∞.
(ii’) D(k2)→ ZIR(−k2)α as |k2| → 0, where α is a realnumber.
(iii) {k2 = xn + iε}Nn=0 is sufficiently dense so thatD(k2 = x + iε) changes its phase at most half-winding (±π) between xn+ iε and xn+1 + iε, wherewe denote sufficiently small x0 = δ2 > 0 and suffi-ciently large xN+1 = Λ2, on which we will take thelimits δ2 → +0 and Λ2 → +∞.
Then, the winding number NW (C) is expressed as
NW (C) = −α− 1 + 2
N∑n=0
1
2πArg
[D(xn+1 + iε)
D(xn + iε)
].
(A1)
Let us derive this expression. First, we decompose thepath around the positive real axis C2 into three piecesC2 = C2,+ ∪ Cδ ∪ C2,−, where C2,± stands for the pathalong the positive real axis of C2,± = {x ± iε; δ2 < x <Λ2} and Cδ for the small circle whose center is the ori-gin k2 = 0. Accordingly, the winding number can bedecomposed into the integrals
NW (C) = NW (Cδ) +NW (C1) +NW (C2,−) +NW (C2,+).(A2)
The integral NW (C1) + NW (C2,−) + NW (C2,+) can beevaluated as before,
NW (C1) +NW (C2,−) +NW (C2,+)
= −1 + 2
N∑n=0
1
2πArg
[D(xn+1 + iε)
D(xn + iε)
]. (A3)
For the contribution from the small circle, note thatD(k2±iε)|D(k2±iε)| = e∓iπα as k2 → +0. Therefore the phase
factor D/|D| varies from e+iπα to e−iπα, from which
NW (Cδ) = −α. (A4)
To sum up, we obtain (A1). Note that the IR suppres-sion contributes negatively to NW (C) = NZ −NP .
19
Appendix B: Detailed results on the one-loop gluonpropagator for various parameters
Here, we show detailed analyses on the results given inSec. IV A 3 on the strict one-loop gluon propagator. Tocheck the insensitivity of the transition, we shall computeNW (C) in the three dimensional parameter space (λ =C2(G)g2
16π2 , u = M2
µ20, ξ =
m2q
M2 ).
Figure 13 plots the boundary surfaces of equi-NW (C)volume at NF = 3 and NF = 6 in the three-dimensionalparameter space (λ, u, ξ). The gluon propagator has fourcomplex poles inside the green surface (0.2 . ξ . 0.6)and no complex poles inside the blue surface (with smallλ and ξ), and two complex poles except these regions.
Contour plots for selected ξ are displayed in Fig. 14for NF = 3 and in Fig. 15 for NF = 6. They give two-dimensional slices at fixed ξ = 0.1, 0.3, 0.5 of Fig. 13. Theregions colored with light gray (NP = 0) in Fig. 14 andFig. 15 are two-dimensional cross sections at fixed ξ ofthe volumes inside the blue surfaces in the top and bot-tom figures of Fig. 13, respectively. Similarly, the regionscolored with dark gray (NP = 4) in Fig. 14 and Fig. 15are those of the volumes surrounded by the green surfacesin the top and bottom figures of Fig. 13, respectively.
Figure 13 demonstrates the NW (C) = −4 region insidethe green surface expands from NF = 3 to NF = 6. Inthe figure of NF = 3 (top of Fig. 13), the set of the typicalvalues (g = 4, M2 = 0.2 GeV2, and ξ = 0.1) is shown asa black dot, of which Fig. 3 is computed at (g,M2). The
NW (C) = −4 region occupies the region 0.2 .m2q
M2 .0.6 around the typical values of the parameters (g ∼ 4,M2 ∼ 0.2 GeV2) at NF = 6. From these observations, wededuce that the existence of the transition is insensitiveto a detailed choice of g and M2. Therefore, the roughinvestigation in Sec. IV A will be qualitatively valid inthis model.
In addition, since the gluon propagator acquires a poleat timelike momentum on the boundary of the NP = 4region and its poles will move like Fig. 5, w/v for thepole of the gluon propagator decreases as NF increases.Hence, the gluon will become more particlelike in thepresence of many light quarks, irrespectively of a detailedchoice of the parameter (g,M).
FIG. 13: Boundary surfaces of equi-NW (C) volume atNF = 3
(top) and NF = 6 (bottom) in the (λ = C2(G)g2
16π2 , u = M2
µ20, ξ =
m2q
M2 ) space. Their cross sections at several values of ξ areshown in Fig. 14 and Fig. 15. The gluon propagator has fourcomplex poles inside the green surface and no complex polesinside the blue surface; otherwise the gluon propagator hastwo complex poles. At NF = 3, the black dot shows a typicalset of the parameters (g = 4, M2 = 0.2 GeV2, and ξ = 0.1)and the gluon propagator has two complex poles around there.On the other hand, at NF = 6, the gluon propagator has
NP = 4 for 0.2 .m2q
M2 . 0.6 around the typical values g ∼ 4,
M2 ∼ 0.2 GeV2.
20
FIG. 14: Contour plots of NW (C) = −NP of the gluonpropagator on the two-dimensional parameter space (λ =C2(G)g2
16π2 , u = M2
µ20
) at ξ = 0.1, 0.3, 0.5 for NF = 3.
FIG. 15: Contour plots of NW (C) = −NP of the gluonpropagator on the two-dimensional parameter space (λ =C2(G)g2
16π2 , u = M2
µ20
) at ξ = 0.1, 0.3, 0.5 for NF = 6.
21
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