PHYSICAL REVIEW D 101, 074044 (2020)
Complex poles and spectral functions of Landau gauge QCD and QCD-like theories
† Yui Hayashi 1,* and Kei-Ichi Kondo1,2, 1Department of Physics, Graduate School of Science and Engineering, Chiba University, Chiba 263-8522, Japan 2Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan
(Received 21 January 2020; accepted 16 March 2020; published 29 April 2020)
In view of the expectation that the existence of complex poles is a signal of confinement, we investigate the analytic structure of the gluon, quark, and ghost propagators in the Landau gauge QCD 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 investigate the number of complex poles in the parameter space of the model consisting of gauge coupling constant, gluon mass, and quark mass for the gauge group SUð3Þ and various numbers of quark flavors NF within the asymptotic free region. Both the gluon and quark propagators at the best-fit parameters for NF ¼ 2 QCD have one pair of complex conjugate poles, while the number of complex poles in the gluon propagator varies between zero and four depending on the number of quark flavors and quark mass. Moreover, as a general feature, we argue that the gluon spectral function of this model with nonzero quark mass is negative in the infrared limit. In sharp contrast to gluons, the quark and ghost propagators are insensitive to the number of quark flavors within the current approximations adopted in this paper. These results suggest that details of the confinement mechanism may depend on the number of quark flavors and quark mass.
DOI: 10.1103/PhysRevD.101.074044
I. INTRODUCTION quark, and ghost propagators in the Landau gauge of the Yang-Mills theory and quantum chromodynamics (QCD). Color confinement, absence of color degrees of freedom In the Yang-Mills theory, or the quenched limit of QCD, the from the physical spectrum, is one of the most fundamental so-called decoupling and scaling solutions of the gluon and and significant features of strong interactions. It is a long- ghost propagators are observed based on the continuum standing and challenging problem in particle and nuclear approaches [4]. The recent numerical lattice results support physics to explain color confinement in the framework of the decoupling solution [5]. The decoupling solution has an quantum field theory (QFT). Understanding analytic struc- impressing feature that the running gauge coupling stays tures of the correlation functions will be of crucial finite and nonzero for all nonvanishing momenta and importance to this end because a QFT describing physical eventually goes to zero in the limit of vanishing momen- particles can be reformulated in terms of correlation tum, which cannot be predicted from the standard pertur- functions [1], and there are some proposals of confinement bation theory that is plagued by the Landau pole of the mechanisms whose criteria are expressed by them, e.g., [2]. diverging running gauge coupling. In particular, the analytic structures of propagators encode A low-energy effective model of the Yang-Mills theory is kinematic information as the Käll´en-Lehmann spectral proposed following the decoupling behavior of the gluon representation [3], which will be useful toward under- propagator, which provides the gluon and ghost propaga- standing confinement. tors that show a striking agreement with the numerical In the past decades, numerous studies of both the lattice lattice results by including quantum corrections just in the and continuum approaches have focused on the gluon, one-loop level [6,7]. This effective model is given by the mass-deformed Faddeev-Popov Langrangian of the Yang- *[email protected] Mills theory in the Landau gauge, or the Landau gauge † [email protected] limit of the Curci-Ferarri model [8], which can be shown to be renormalizable due to the modified Becchi-Rouet-Stora- Published by the American Physical Society under the terms of Tyutin (BRST) symmetry, and we call it the massive Yang- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Mills model for short. This effective mass term could stem the author(s) and the published article’s title, journal citation, from the dimension-two gluon condensate [9–13] or could and DOI. Funded by SCOAP3. be taken as a (minimal) consequence of avoiding the
2470-0010=2020=101(7)=074044(22) 074044-1 Published by the American Physical Society YUI HAYASHI and KEI-ICHI KONDO PHYS. REV. D 101, 074044 (2020)
Gribov ambiguity [14,15]. Moreover, it has been shown complex poles and the propagator on timelike momenta that the massive Yang-Mills model has “infrared safe” from the argument principle [40]. This investigation renormalization group (RG) flows on which the running extends the previous result [40] obtained for the pure gauge coupling remains finite for all scales [7,15,16]. The Yang-Mills theory with no flavor of quarks NF ¼ 0 that the three-point functions [17] and two-point correlation func- gluon propagator has a pair of complex conjugate poles tions at finite temperature [18] in this model were compared and the negative spectral function while the ghost propa- to the numerical lattice results, showing good agreements. gator has no complex pole. In this article, we will see the Moreover, the two-loop corrections improve the accor- following results. Both the gluon and quark propagators at dance for the gluon and ghost propagators [19]. These the “realistic” parameters for NF ¼ 2 QCD have one pair of works indicate the validity of the massive Yang-Mills complex conjugate poles as well as the gluon propagator in model as an effective model of the Yang-Mills theory. the zero flavor case. By increasing quark flavors, we find a The gluon and ghost propagators for unquenched lattice new region in which the gluon propagator has two pairs of QCD with the number of quark flavors N ¼ 2; 2 þ 1; 2 þ complex conjugate poles for light quarks with the inter- F 4 ≲ 10 1 þ 1 have been studied, for instance [20], and exhibit the mediate number of flavors NF < .However,the decoupling feature as well. The massive Yang-Mills model gluon propagator has no complex poles if very light quarks ≥ 10 with dynamical quarks can reproduce the numerical lattice have many flavors NF or both of the gauge coupling gluon and ghost propagators for QCD as well [21]. and quark mass are small. In the other regions, the gluon However, it is argued that higher-loop corrections are propagator has one pair of complex conjugate poles. On the important for the quark sector in this model [22,23]. other hand, the analytic structures of quark and ghost Despite this shortcoming, it appears that the effective gluon propagators are nearly independent of the number of quarks mass term captures some nonperturbative aspects of QCD. within this one-loop analysis. What is more, QCD phases have been extensively studied This paper is organized as follows. In Sec. II,we in a similar model with the effective gluon mass term of the present machinery to count the number of complex poles Landau-DeWitt gauge [24]. as an application of [40].InSec.III, we review the calculation Apart from the realistic QCD, it is also interesting to study of the massive Yang-Mills model with quarks of [21], gauge theories with many flavors of quarks. For many quark consider the infrared safe trajectories of this model, and flavors, the infrared conformality is predicted [25,26] and argue the infrared negativity of the gluon spectral function. well-studied in line with the walking technicolor for physics Then, in Sec. IV, we analyze the analytic structures of the beyond the standard model [27], for example, [28–30]. gluon, quark, and ghost propagators at the best-fit parameters Some argue that chiral symmetry restores while color of [21], and investigate the number of complex poles for degrees of freedom are “unconfined” in the conformal various parameters and the number of quarks NF.Finally, window. For a better understanding of the confinement Sec. V is devoted to conclusion, and Sec. VI contains further discussion. In Appendix A, we provide a generalization of the mechanism, observing NF dependence will be thus extremely valuable. proposition of Sec. II for various infrared behaviors. All works on the correlation functions described above Appendix B gives complementary one-loop analyses for were implemented in the Euclidean space. Considerable Sec. IVA. efforts have been devoted to reconstructing the spectral functions from the Euclidean data based on the Käll´en- II. COMPLEX POLES IN PROPAGATORS Lehmann spectral representation, e.g., [31,32]. On the other – To elucidate the starting point, we review a generaliza- hand, several models of the Yang-Mills theory [14,33 39], tion of the spectral representation for a propagator so as to including the (pure) massive Yang-Mills model [40,41], and a allow complex poles. We then develop a method for way of the reconstruction from the Euclidean data [42] counting the number of complex poles from the data on predict complex poles in the gluon propagator that invalidate timelike momenta, e.g., the spectral function, as a straight- the Käll´en-Lehmann spectral representation. The existence of forward application of the general relation [40]. complex poles of the propagators of the confined particles is a controversial issue, see, e.g., [43]. The complex singularities invalidate the standard reconstruction from a Euclidean field A. A generalization of the spectral representation theory to a relativistic QFT [1] and might correspond to We introduce some definitions and underlying assump- unphysical degrees of freedom in an indefinite metric state tions on propagators adopted in this article. Given a space [44]. Therefore, complex poles are expected to be propagator defined in the Euclidean space, we analytically closely connected to the confinement mechanism. continue the propagator to the whole complex k2 plane In this paper, we investigate the analytic structure of the from the Euclidean momenta. In the complex k2 plane, we QCD propagators for various NF based on the massive call points on the negative real axis Euclidean momenta Yang-Mills model, mainly focusing on complex poles by and points on the positive real axis timelike momenta.We utilizing the general relationship between the number of will study the gluon, quark, and ghost propagators in the
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Landau gauge. We assume each propagator of the gluon, quark, and ghost has the following generalized spectral representation allowing the presence of complex poles1: Z ∞ ρ σ2 Xn 2 σ2 ð Þ Zl Dðk Þ¼ d 2 2 þ 2 ; ð1Þ 0 σ − k zl − k l¼1
2 1 2 where ρðσ Þ ≔ π ImDðσ þ iϵÞ is its spectral function, zl stands for the position of a complex pole, and Zl is the residue associated with the complex pole. This representation can be derived under the following conditions using the Cauchy integral formula for the closed contour C˜ presented in Fig. 1 [40,46]: (I) DðzÞ is holomorphic except for singularities on the positive real axis and a finite number of simple poles. FIG. 1. Contour C˜ on the complex k2 plane avoiding the → 0 → ∞ (II) DðzÞ as jzj . singularities on the positive real axis and the complex poles. ˜ (III) DðzÞ is real on the negative real axis. The contour C consists of the large circle C1, the path wrapping If we replace the condition (I) with the more strict condition around timelike singularities C2,andthesmallcontoursγl that ’ 2 (I ) DðzÞ is holomorphic except for singularities on the clockwise surround the complex poles at zl. The propagator Dðk Þ positive real axis, namely has no complex poles, is holomorphic in the region bounded by the contour C˜ ¼ ’ ∪ γ n ∪ the three conditions (I ), (II), and (III) lead to the Käll´en- C f lgl¼1, where we denote the closed contour C1 C2 by C. Lehmann form [3], which propagators for unconfined particles are supposed to obey. In this sense, Eq. (1) gives a generali- In what follows, we assume the following asymptotic zation of the Käll´en-Lehmann spectral representation. form for the propagator. (i) 2 In the UV limit jk2j→∞, Dðk2Þ has the same phase B. Counting complex poles − → 1 → ∞ as the free one, i.e., argð DðzÞÞ argz as jzj . We present a procedure to count complex poles from the (ii) In the IR limit jk2j → 0, Dðk2 ¼ 0Þ > 0. propagator on the timelike momenta based on [40]. Let us comment on these assumptions. The first assumption We apply the argument principle to a propagator on the is satisfied by the gluon, quark, and ghost propagators in contour C presented in Fig. 1. Then the winding number the Landau gauge, which follows from the RG analysis for 2 NWðCÞ of the phase of the propagator Dðk Þ along the asymptotic free theories. The RG argument of Oehme and contour C is equal to the difference between the number of Zimmermann [47] provides the following UV asymptotic 2 → ∞ zeros NZ and the number of poles NP in the region bounded form for the propagators, as jk j , by C, 2 ≃− ZUV I Dðk Þ 2 2 γ ; ð3Þ 2 k ln k 1 D0ðk Þ ð j jÞ N ðCÞ ≔ dk2 W 2πi Dðk2Þ γ γ β I C where ZUV is a positive constant and ¼ 0= 0 is the ratio γ β 1 2 between the first coefficients 0, 0 of the anomalous ¼ dðarg Dðk ÞÞ dimension and the beta function, respectively. For the gluon 2π C − propagator of Yang-Mills theories with NF quarks in the ¼ NZ NP: ð2Þ Landau gauge, γ is computed as follows.
N C γ0 The winding number Wð Þ can be calculated from the γ ; 2 ϵ ¼ β propagator on timelike momenta Dðk þ i Þ and infrared 0 (IR) and ultraviolet (UV) asymptotic forms. 1 13 4 γ0 − C2 G − C r ; ¼ 16π2 6 ð Þ 3 ð Þ 1 This generalization can be related to the fact that complex 1 11 4 spectra for confined particles need not be excluded in an β − − 0 ¼ 2 C2ðGÞ CðrÞ ; ð4Þ indefinite metric state space [44]. Such complex spectra of a 16π 3 3 Hamiltonian can give rise to the complex poles. The kinematic aspects of complex poles will be discussed elsewhere. Inciden- tally, another generalization for the QCD propagators within the 2This assumption (i) is the same as assumption (i) of the framework of tempered distribution is proposed in [45]. assertions in Sec. III in [40].
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2 π ϵ ϵ where C2ðGÞ and CðrÞ¼NF= are the Casimir invariants half-winding ( ) between xn þ i and xnþ1 þ i , of the adjoint and fundamental representations of the gauge i.e., for n ¼ 0; 1; …;N, Z group G, respectively. xnþ1 d We restrict ourselves to asymptotically free theories: dx arg Dðx þ iϵÞ < π; ð5Þ 11 dx β0 < 0,orNF < 2 C2ðGÞ, which is essential to derive the xn above UV asymptotic expression for the propagator (3). δ2 0 γ where we denote sufficiently small x0 ¼ > and Note that the sign of is determined depending on the Λ2 sufficiently large xNþ1 ¼ , on which we will take number of quarks as follows. 2 2 13 the limits δ → þ0 and Λ → þ∞. (a) γ > 0 if CðrÞ < C2ðGÞ. For the gauge group 8 Let us now calculate N ðCÞ from the data fDðx þ SUð3Þ with C2ðGÞ¼3, in particular, γ > 0 if W n ϵ N 13 39 i Þgn¼1 under the above assumptions, by evaluating NF ¼ 1; …; 9 < 4 C2ðGÞ¼ 4 . 13 11 NWðC1Þ and NWðC2Þ separately, where C1 stands for the (b) γ < 0 if 8 C2ðGÞ
074044-4 COMPLEX POLES AND SPECTRAL FUNCTIONS … PHYS. REV. D 101, 074044 (2020) the gluon. The mass term may be generated by the effect of constant gb, the bare quark mass ðmbÞq;i, and the bare – ABC the dimension-two gluon condensate [9 13] or by avoiding effective gluon mass Mb, while f stands for the the Gribov ambiguity [15]R . For the latter effect, intuitively, structure constant associated with the generators tA of D A A some effect suppressing d xAμ Aμ should be taken into the fundamental representation of the group G. account due to the Gribov copies, and the most infrared We introduce the renormalization factors ðZA;ZC;ZC¯ ¼ relevant term of this effect will be the mass term. The ðiÞ ψ 2 massive Yang-Mills model reproduces the numerical lattice ZC;Z Þ;Zg;ZM ;Zmq;i for the gluon, ghost, anti-ghost, and ¯ results of the gluon and ghost propagators well and has a quark fields ðAμ; C; C; ψ iÞ, the gauge coupling constant g, 2 renormalization condition such that there are RG trajecto- and the gluon and quark mass parameters M ;mq;i respec- ries whose running gauge coupling remains finite in all tively: scales. Moreover, the massive Yang-Mills model gives a pffiffiffiffiffiffi pffiffiffiffiffiffi correct UV asymptotic behavior [47] by RG improvement, Aμ Aμ C C ¼ ZA R; ¼ ZC R; as we will see in Sec. III C. Therefore, although we expect pffiffiffiffiffiffi qffiffiffiffiffiffiffi the massive Yang-Mills model is a low-energy effective ¯ ¯ ðiÞ C ¼ ZCCR; ψ i ¼ Zψ ψ R;i; model, this model can describe Yang-Mills theory in all 2 2 2 scales to some extent. One might worry about the absence gb ¼ Zgg; Mb ¼ ZM M ; ðmbÞq;i ¼ Zmq;i mq;i of the nilpotent BRST symmetry. Although the massive ð12Þ Yang-Mills model suffers from the problem of physical unitarity [8,48] as a consistent QFT, we insist that this Throughout this article, for simplicity, we employ this model model will suffice for our purpose of investigating analytic with degenerate quark masses, mq ≔ mq;i, and therefore structures, as it gives the well-approximating propagators ðiÞ Zψ ≔ Zψ and Z ≔ Z . with a sensible and straightforward mass-deformation. mq mq;i Moreover, by adding effective quark mass term, this model has good accordance with the numerical lattice A. Strict one-loop calculations results for the unquenched gluon and ghost propagators and We review the strict one-loop results for the gluon, can also reproduce the quark mass function qualitatively quark, and ghost propagators here and the RG functions in [21]. In this section, we review the results on the one-loop the next subsection [21]. computations of the massive Yang-Mills model with For the gluon, ghost, and quark, we introduce the two- quarks, prepare expressions that we will use in the inves- Γð2Þ Γð2Þ Γð2Þ tigation of the analytic structures of the propagators in the point vertex functions A , gh , and ψ , the transverse D Δ next section, and study asymptotic behaviors of the gluon propagator T, the ghost propagator gh, the quark propagators of this model using RG. propagator S, dimensionless gluon and ghost vacuum ˆ ˆ In the Euclidean space, the Lagrangian of the model is polarizations Π and Πgh, and the scalar and vector part given by [6,7,21] ð2Þ ð2Þ of the quark two-point vertex function Γs ; Γv as L L L L L L mYM ¼ YM þ GF þ FP þ m þ q; ð10Þ Γð2Þ 2 ≔ D 2 −1 A ðkEÞ ½ TðkEÞ 2 1 1 Πˆ δ δ 2 L FA FA ¼ M ½s þ þ ðsÞþs Z þ M YM ¼ 4 μν μν; ≕ M2½s þ 1 þ Πˆ renðsÞ ; ð13Þ A A LGF ¼ iN ∂μAμ L C¯ A∂ D A ABCB Γð2Þ 2 ≔−Δ 2 −1 FP ¼ μ μ½ gh ðkEÞ ½ ghðkEÞ C¯ A∂ ∂ CA ABCABCC 2 ˆ ¼ μð μ þ gbf μ Þ ¼ M ½s þ ΠghðsÞþsδC 1 2 ren 2 A A ≕ Πˆ L M Aμ Aμ ; M ½s þ gh ðsÞ ; ð14Þ m ¼ 2 b 2 XNF Γð Þ ≔ S −1 L ψ¯ γ D A ψ ψ ðkEÞ ðkEÞ q ¼ ið μ μ½ þðmbÞq;iÞ i ð2Þ 2 ð2Þ 2 i¼1 Γ δ Γ δ ¼ i=kEð v ðkEÞþ ψ Þþð s ðkEÞþmq mq Þ XNF Γren 2 Γren 2 ψ¯ γ ∂ − AA A ψ ¼ i=kE v ðkEÞþ s ðkEÞ; ð15Þ ¼ ið μð μ igb μ t ÞþðmbÞq;iÞ i; ð11Þ i¼1 where kE is the Euclidean momentum, where we have introduced the bare gluon, ghost, anti- 2 ghost, Nakanishi-Lautrup, and quark fields denoted by ≔ kE A A ¯ A A s 2 ; ð16Þ Aμ ; C ; C ; N ; ψ i respectively, the bare gauge coupling M
074044-5 YUI HAYASHI and KEI-ICHI KONDO PHYS. REV. D 101, 074044 (2020) 2 and δ ≔ Z − 1, δ 2 ≔ Z Z 2 − 1, δ ≔ Z − 1, g C2ðrÞ Z A M A M C C Σ ðk2 Þ¼ K2f2M4 þM2ðk2 −m2Þ δψ ≔ Zψ − 1, δ ≔ Zψ Z − 1 are the counterterms. v E 64π2 2 4 E q mq mq M kE The bare vacuum polarizations computed by the dimen- 2 2 2 2 2 2 2 2 −ðm þk Þ gQ−2M k ð−2M þm þk Þ sional regularization read [7,21], for gluons, q E E q E m −2 2 6 3 4 2 − 2 2 2 3 q Πˆ Πˆ Πˆ f M þ M ðkE mqÞþðmq þkEÞ gln ðsÞ¼ YMðsÞþ qðsÞð17Þ M 2 2 2 9 4π 2 2 3 mq þkE ˆ g C2ðGÞ −1 −2 Π ðsÞ¼ s − 26 ε þ ln ðmq þkEÞ ln 2 ; ð24Þ YM 192π2 s M2eγ mq 121 63 3 2 γ=2 − þ þ hðsÞ 2 g C2ðrÞmq 2 Me 2 3 s Σ ðk Þ¼− − þ ln pffiffiffiffiffiffi − s E 8π2 ϵ 4π 3 2 1 4π ˆ g CðrÞ −1 2 Π ðsÞ¼− s − ε þ ln K 1 2 2 2 mq q 6π2 2 m2eγ − Q þ ðM − m þ k Þ ln ; ð25Þ q 4k2 2k2 q E M 5 ξ E E − þ h ; ð18Þ 6 q s where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for ghosts, 2 4 2 2 2 2 2 2 K ≔ M þ 2M ðk − m Þþðm þ k Þ ; E q q E 2 g C2ðGÞ 4π 2 − 2 2 − 2 − 2 2 Πˆ −3 ε−1 ðK kEÞ ðmq M Þ ghðsÞ¼ 2 s þ ln 2 γ Q ≔ ; 64π M e ln 2 2 2 − 2 − 2 2 ð26Þ ðK þ kEÞ ðmq M Þ
− 5 2 þ fðsÞ ; ð19Þ N −1 and C2ðrÞ¼ 2N for G ¼ SUðNÞ and fundamental quarks. Henceforth, we adopt the “infrared safe renormalization where ε ≔ 2 − D=2, γ is the Euler-Mascheroni constant, scheme” by Tissier and Wschebor [7], which respects the 2 CðrÞ¼N = , 2 1 F nonrenormalization theorem ZAZCZM ¼ [49], in the one-loop level, m2 ξ ≔ q 8 8 2 ; ð20Þ M > Z Z Z 2 1 > δ δ 2 0 > A C M ¼ > C þ M ¼ > > and, > Γð2Þ μ μ2 2 > Πˆ ren ν 0 <> A ðkE ¼ Þ¼ þ M <> ðs ¼ Þ¼ Γð2Þ μ μ2 ⇔ Πˆ renðs ¼ νÞ¼0; 27 1 2 > gh ðkE ¼ Þ¼ > gh ð Þ ≔− 1 − s > > hðsÞ 2 þ ln s > 2 > Γren μ2 1 s 2 > Γrenðμ Þ¼1 > v ð Þ¼ > v > 3 : 2 : Γren μ2 1 Γrenðμ Þ¼m s ð Þ¼mq þ 1 þ ðs2 − 10s þ 1Þ lnðs þ 1Þ s q s pffiffiffiffiffiffiffiffiffiffi pffiffiffi 1=2 3=2 combined with the Taylor scheme [50] ZgZ ZC ¼ 1 for 1 4 2 4 þ s − s A þ 1 þ ðs − 20s þ 12Þ ln pffiffiffiffiffiffiffiffiffiffi pffiffiffi ; the coupling, where 2 s 4 þ s þ s pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ˜ ≔ 2˜ 1 − 2˜ 4˜ 1 −1 4˜ 1 μ2 hqðtÞ t þð tÞ t þ coth ð t þ Þ; ν ≔ 2 : ð28Þ 1 1 3 M ≔− − ð þ sÞ 1 fðsÞ s ln s þ 2 lnðs þ Þ; ð21Þ s s In this renormalization scheme, the running coupling of some RG flows turns out to be always finite, which implies with that the perturbation theory will be valid to some extent. By imposing the above renormalization condition, we ξ m2 ˜ ≔ q t ¼ 2 : ð22Þ have the renormalized two-point vertex functions, s kE Πˆ TWðsÞ¼Πˆ TW ðsÞþΠˆ TW ðsÞ; ð29Þ For the quark sector [21], ren YM;ren q;ren 2 48 3 ν ð2Þ 2 2 ˆ TW g C2ðGÞ fð Þ Γv ðk Þ¼1 þ Σ ðk Þ; Π ðsÞ¼ s þ hðsÞþ − ðs → νÞ ; E v E YM;ren 192π2 s s 2 Γð Þ k2 m Σ k2 23 s ð EÞ¼ q þ sð EÞðÞ ð30Þ
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2 ξ ξ B. Renormalization group functions Πˆ TW − g CðrÞ − q;renðsÞ¼ 6π2 s hq hq ν ; ð31Þ Here we present the renormalization group functions s 2 for ðg; M ;mq;ZA;ZC;Zψ Þ [7,21]. For later convenience, 2 we put Πˆ TW g C2ðGÞ − ν gh;renðsÞ¼ 2 s½fðsÞ fð Þ ; ð32Þ 64π 2 C2ðGÞg λ ≔ : ð39Þ ΓTW;ren 2 1 Σ 2 − Σ μ2 16π2 v ðkEÞ¼ þ vðkEÞ vð Þ; ð33Þ
The beta functions βα and anomalous dimensions γΦ are TW;ren 2 2 2 Γs ðk Þ¼m þ Σ ðk Þ − Σ ðμ Þ: ð34Þ 2 E q s E s defined for the masses and coupling α ¼ g; M ;mq; λ and for the fields Φ ¼ A; C; ψ renormalized at a scale μ as Note that the gluon propagator exhibits the decoupling i feature and satisfies the condition (ii) of the previous d d βα ≔ μ α; γΦ ≔ μ ln ZΦ; ð40Þ section, dμ dμ 1 D ðk2 ¼ 0Þ¼ > 0: ð35Þ where the bare masses and the bare coupling are fixed in T E 2 1 Πˆ TW 0 M ½ þ ren ð Þ taking the derivative. From the nonrenormalization theorems, Z Z Z 2 1 pffiffiffiffiffiffi A C M ¼ 1 β β β 2 Indeed, and Zg ZAZC ¼ , λ, or equivalently g, and M are expressed by γA and γC: Πˆ TW 0 0 q;renðs ¼ Þ¼ ; 2 β λ γ 2γ β 2 γ γ 2 15 λ ¼ ð A þ CÞ; M ¼ M ð A þ CÞ: ð41Þ Πˆ TW 0 g C2ðGÞ 3 ν − 0 YM;renðs ¼ Þ¼ 2 fð Þ > ; ð36Þ 192π 2 γ γ γ β The other RG functions, C, A, ψ and mq are computed where we have used h ðt˜ → ∞Þ¼Oð1Þ, hðsÞ¼ as follows within the one-loop approximation. By differ- q γ μ d δ − 111 0 5 2 entiating the counterterms, Φ ¼ dμ Φ, we have for 2s þ Oðln sÞ, fð Þ¼ = , and the fact that fðsÞ is a monotonically increasing function. ghosts, [7] Finally, note that the one-loop expression of gluon λ propagator will have disagreement at momenta far from γ − 2ν2 2ν − ν3 ν C ¼ 2ν2 ½ þ ln the renormalization scale μ. Indeed, in the UV limit, the 2 strict one-loop expression has the wrong asymptotic form: þðν − 2Þðν þ 1Þ lnðν þ 1Þ ; ð42Þ
2 2 2 2 2 −1 for gluons, [7,21] DTðk Þ ≃−½g γ0k ln jk jþOðk Þ ; ð37Þ γ γYM γquark while the RG analysis yields A ¼ A þ A ð43Þ λ 2 ZUV γYM − 17ν2 − 74ν 12 ν − ν5 ν D ðk Þ ≃− ; ð38Þ A ¼ 3 ð þ Þ ln T k2ðln jk2jÞγ 6ν þðν − 2Þ2ðν þ 1Þ2ð2ν − 3Þ lnðν þ 1Þ where we have analytically continued the gluon propagator pffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ν3=2 ν 4 ν3 − 9ν2 20ν − 36 from Euclidean momentum k ¼ −k to complex k , γ0 þ þ ð þ Þ E pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi and γ ¼ γ0=β0 are given in (4), and Z > 0. ν 4 − ν UV ffiffiffiffiffiffiffiffiffiffiffiþ However, for γ ¼ γ0=β0 > 0, the phase of the gluon ×ln p pffiffiffi ; ν þ 4 þ ν propagator on UV momenta is qualitatively correct despite pffiffiffiffiffiffiffiffi 2 4 1 1 6 pffiffiffiffiffiffiffiffitþ þ the wrong exponent of the logarithm. Furthermore, both of 16λCðrÞ t lnð 4 1−1Þ 1 2 γquark ffiffiffiffiffiffiffiffiffiffiffiffiffitþ − 3 the propagators have negative spectral functions ρðσ Þ¼ A ¼ p t þ ; ð44Þ 3C2 G 4 1 2 1 2 ð Þ t þ π ImDTðσ þ iϵÞ < 0 in the UV limit. Therefore, we expect the one-loop expression will provide a good approximation where of the phase of the gluon propagator for γ > 0 based on the robustness of the winding number. In Sec. IV, we indeed m2 t ≔ q ; confirm that the RG improved and strict one-loop gluon μ2 ð45Þ propagators yield the same NWðCÞ for qualitatively the same parameter region. and, for quarks, [21]
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2 g C2ðrÞ Q 8 6 2 2 2 2 3 2 2 4 2 2 4 4 γψ ¼ μ þ μ ðm þ M Þþ2ðM − m Þ ðm þ 2M Þþμ ð−2M m − 3m þ 3M Þ 32π2μ4M2 K2 q q q q q μ2 þ μ2ðM2 − m2Þð6M2m2 þ 5m4 þ 7M4Þ þ 2μ2M2ðμ2 − 2m2 þ 4M2Þþ2ðμ6 − 3μ2m4 − 2m6Þ ln þ 1 q q q q q q m2 q m þ 2 μ6 − 3μ2ðm4 þ M4Þ − 2ð−3M4m2 þ m6 þ 2M6Þ ln q : ð46Þ q q q M
1 δ − δ λ μ Since Zmq ¼ þ mq ψ in the one-loop level, we obtain running coupling ð Þ does not vanish in the UV limit the beta function for the quark mass mq [21], μ → ∞ on the RG flow. As the pure Yang-Mills case, the diagram has the infrared safe trajectories, on which the 2 2 2 λ 3g C2ðrÞmq 2ðM − mqÞ mq running gauge coupling is always finite in the all scales. β ¼ m γψ þ −2 þ ln mq q 16π2 μ2 M One can confirm that the qualitative feature is the same for N ≤ 16. ðμ2ðm2 þ M2ÞþðM2 − m2Þ2ÞQ F − q q ; ð47Þ μ2K2 C. Asymptotic behaviors of infrared safe trajectories 2 2 μ2 From the flow diagram Fig. 2, the infrared safe trajecto- where Q and K are given in (26) with kE ¼ . 2 C2ðGÞg 2 2 2 2 ries possess the following features: (i) In the UV limit The flow diagram of ðλ ¼ 2 ;u¼ M =μ ;t¼ m =μ Þ 16π q μ → ∞, the parameters ðλ;u;tÞ tend to ðλ → 0;u→ 0; for N ¼ 3 is depicted in Fig. 2. In Fig. 2, the infrared F t → 0Þ, (ii) In the IR limit μ → 0, they tend to safe trajectories are shown as the blue curves, while the ðλ → 0;u→ ∞;t→ ∞Þ. In this subsection, we study their green curves terminate at an infrared Landau pole. The red asymptotic behaviors within the one-loop level. one is not ultraviolet asymptotic free in the sense that the Beforehand, there is a caveat on this discussion. In the RG analysis of Oehme and Zimmermann [47], the param- eter of the theory is only the gauge coupling g, which guarantees that the higher-loop effects are suppressed in the ultraviolet region precisely. Then, the one-loop RG gives a strong argument on asymptotic behaviors and enables us to establish the UV negativity of the gluon spectral function for NF < 10. However, the massive Yang-Mills model has the mass parameter, which can potentially invalidate the perturbation theory in λ even though λ → 0 asymptotically. Here, we assume that the perturbation theory “works well.” In particular, we assume that the spectral function is dominated by the one-loop contribution in both UV and IR limits. In the pure Yang-Mills case, a similar discussion on the RG functions and Euclidean propagators can be found in e.g., [16]. In what follows, we study asymptotic behaviors of the RG functions, Euclidean propagators, and spectral functions in the massive Yang-Mills model described in the previous subsections.
1. UV limit We consider the UV limit and confirm that the asymp- totic behaviors of the massive Yang-Mills model is con- 2 sistent with those of the Faddeev-Popov Lagrangian. λ C2ðGÞg FIG. 2. RG flows in the parameter space ð ¼ 16π2 ; 2 2 2 2 Taking the limit u; t → 0, we have u ¼ M =μ ;t¼ mq=μ Þ for NF ¼ 3. The arrows indicate infra- μ → 0 red directions . The blue trajectories are infrared safe. The 13 8CðrÞ green ones end at an infrared Landau pole. The red one is not γA → − þ λ þ Oðu; tÞ; ð48Þ ultraviolet asymptotic free. 3 3C2ðGÞ
074044-8 COMPLEX POLES AND SPECTRAL FUNCTIONS … PHYS. REV. D 101, 074044 (2020)
3 3 γ0 ≔− ¼ β0 λ − β0 2 . These lead to the superconver- γ → − λ ;C 2 ; ;M C 2 þ Oðu ln uÞ; ð49Þ gence relations for the gluon spectral function when γ0;A < 0, and also for the ghost spectral function. For 2 which reproduce the standard one-loop beta function: quarks, one can find γψ ¼ Oðλ ;u;tÞ, which gives no logarithmic correction to the quark propagator. 2 βλ ¼ λðγA þ 2γCÞ → β0;λλ ; ð50Þ One can see the UV negativity for the gluon spectral function and UV positivity for the ghost spectral function 22 8 CðrÞ from (55) combined with the analyticity or RG improve- β0;λ ≔− þ < 0; ð51Þ 3 3C2ðGÞ ment with respect to jk2j in the complex k2 plane [47]. Thus, it turns out that the massive Yang-Mills model can recovering the UV asymptotic behavior of the gauge describe the correct UV behavior by RG improvement from coupling λ ∼−β0 λ= lnðμ=ΛÞ with some constant scale Λ. ; the above observations, although we have regarded the On the other hand, the beta functions for the masses read, massive Yang-Mills model as a low-energy effective model. for gluons,
2 2 β 2 γ γ → β 2 λ M ¼ M ð A þ CÞ 0;M M ; 2. IR limit 35 8CðrÞ Next, we consider the IR limit of the infrared safe β 2 ≔− 0;M þ ; ð52Þ 6 3C2ðGÞ trajectories: λ → 0;u→ ∞;t→ ∞. Taking the limit u; t → ∞,wehave and for quarks, 1 6 γ → λ −1 −1 C2ðrÞ A 3 þ Oðu ;t Þ; βm → − λmq þ Oðu ln u; t ln tÞ: ð53Þ q C2ðGÞ −1 γC → 0 þ Oðu ln uÞ; ð56Þ Notice that at G ¼ SUð3Þ, the gluon mass is suppressed 35 as those of the pure Yang-Mills case shown in [16]. This logarithmically for β0 2 < 0,orN < C2ðGÞ¼ ;M F 8 indicates that the infrared gluon and ghost sector is 105=8 ≈ 13.1 and enhanced logarithmically for 14 ≤ N ≤ F insensitive to the quark flavor effect except the case of 16, while the correction to the quark mass always “massless quark” mq ¼ 0. As shown in [16], we find suppresses the quark mass in the logarithmic way. 2 β λ β 2 λ 3 μ → 0 2 2 2 2 λ= ¼ M =M ¼ = , which leads as to Note that u ¼ M =μ → 0 and t ¼ mq=μ → 0 expo- λ → 0 nentially faster than . This justifies a posteriori taking 1 the limit u; t → 0 in the first step. M2 ∼ λ ∼ : μ−1 ð57Þ Next, let us consider the propagators on the Euclidean ln momenta. From the nonrenormalization theorems, ZA ¼ −2 −1 The quark mass m runs according to 2 q ZλZM2 and ZC ¼ ZM Zλ , which yield together with the renormalization conditions (27), [7], β → −1 −1 −1 mq Oðu ln u; u ln t; t Þ; ð58Þ 4 2 λ0 M ðk Þ 1 D k2 ; μ2 E ; Tð E 0Þ¼ 4 λ 2 2 2 2 which indicates absence of logarithmic correction to the M0 ðkEÞ kE þ M ðkEÞ quark mass in the infrared. Since u ¼ M2=μ2 → ∞ and 2 λ 2 1 Δ 2 μ2 − M0 ðkEÞ t m2=μ2 → ∞ exponentially increase more rapidly than ðk ; 0Þ¼ 2 2 2 ; ð54Þ ¼ q E λ 1 0 M ðkEÞ kE λ ∼ → 0 ln μ , the above evaluation is consistent. From (57) and (54), the infrared safe trajectories describe where μ0 is the renormalization scale, and M0 and λ0 are the 2 the decoupling solution, i.e., the massive gluon and the mass and coupling at μ0. In the UV limit k → ∞, E massless free ghost in the IR limit as those in the pure
2β 2 massive Yang-Mills model of [16]. 0;M −1 1 1 2 2 2 β0 λ D μ ∼ λ ; ∼ γ Finally, we examine the spectral functions. In the gluon TðkE; 0Þ ½ ðkEÞ 2 0;A ; 2 2 β kE 0;λ kEðln kEÞ vacuum polarization, the quark loop affects the gluon 2 2 β 2 spectral function in the region of ð2m Þ 074044-9 YUI HAYASHI and KEI-ICHI KONDO PHYS. REV. D 101, 074044 (2020) As s → 0, the vacuum polarizations are negative coefficient associated to the negative norm mass- less ghost state and shows the finite positive spectrum in the λ 15 limit σ2 → þ0. Although we assume that the running u and Πˆ TW ðsÞ¼ 3fðνÞ − þ s ln s þ OðsÞ ; YM;ren 12 2 t do not invalidate the asymptotic perturbative expansion in λ 5 λ, the IR negativity of the gluon spectral function will be a Πˆ TW − ν − 2 gh;renðsÞ¼4 s 2 fð Þ s ln s þ Oðs Þ : ð59Þ remarkable consequence from the infrared safety. As an alternative evidence for the IR negativity of the gluon spectral function, we can utilize the IR From the assumption that the higher loop terms are behavior of the Euclidean propagator. For example, suppressed as μ → 0, i.e., the running u and t do not although this relation gives a trivial result in this invalidate the perturbation theory in λ, the following 2 → 0 model, it is worthwhile to note that they are related as approximation holds in the IR limit jk j , d 2 d 2 lim → 0 D ðk Þ¼−π limσ→ 0 ρðσ Þ [32]. Let us kE þ dkE T E þ dσ 2 2 2 2 1−loop 2 2 consider the IR asymptotic behavior of the gluon propa- D ðk ; μ0Þ ≃ Z ðjk j; μ0ÞD ðk ; jk jÞ; T A T gator. The Euclidean propagator (54) can be rewritten Δ 2 μ2 ≃ 2 μ2 Δ1−loop 2 2 ðk ; 0Þ ZCðjk j; 0Þ ðk ; jk jÞ ð60Þ as [16], 2 μ2 2 μ2 −1 2 where ZAðjk j; 0Þ and ZCðjk j; 0Þ are the renormalization 2 ZC ðkEÞ 4 μ2 2 λ μ2 D ðk Þ¼ ; ð64Þ 2 2 λ0 M ð Þ 2 2 M0 ð Þ T E 1 2 2 2 factors and ZAðμ ; μ0Þ¼ 4 2 , ZCðμ ; μ0Þ¼ λ 2 2 þ kE=M ðkEÞ M0 λðμ Þ 0 M ðμ Þ μ2 → 0 from the nonrenormalization theorems. As , from which we have μ2 μ2 ∼ 2 μ2 μ2 μ2 ∼ ZAð ; 0Þ M ð Þ;ZCð ; 0Þ const: ð61Þ d 1 k2 D ðk2 Þ ≃ Z−1ðk2 Þ½−γ − 2u−1 E dk2 T E 2 C E C Therefore, the spectral functions originating from the E 1 λ branch cut on timelike momenta have the IR asymptotic ≃ Z−1ðk2 Þu−1 u−1 ln u − 2 forms σ2 → þ0: for gluons, 2 C E 2 ∼− 2 2 0 1 kE ln kE > ; ð65Þ ρ σ2 D σ2 ϵ μ2 ð Þ¼π Im Tð þ i ; 0Þ where we have used γ → − λ u−1 ln u þ OðλuÞ and λðμÞ ≃ 1 C 2 2 2 1−loop 2 2 6 ≃ Z ðσ ; μ Þ ImD ðσ þ iϵ; σ Þ − 2 as μ → 0. Therefore, as k → 0, we have A 0 π T ln μ E 1 ∼ Im d 2 2 ˆ TW 2 D k ∼ k → ∞: ½1 þ s þ Π ðsÞ − σ − ϵ 2 Tð EÞ j ln Ej þ ð66Þ ren s¼ 2 i dk M E σ2 ∼− Πˆ TW − − ϵ Im ren 2 i This logarithmic divergence is precisely consistent with the M IR spectral negativity of ρðσ2Þ ∼−σ2 < 0. Indeed, if λ σ2 ρ σ2 ∼−σ2 0 ∼− ð Þ σ2 ∼−σ2 0 ð Þ < , then, from the (generalized) spectral 2 2 < ; ð62Þ → 0 M ðσ Þ representation, as kE þ , Z ∞ ρ σ2 and for ghosts, d D 2 ≃− σ2 ð Þ 2 TðkEÞ d 2 2 2 dk 0 ðσ þ k Þ 1 E E ρ ðσ2Þ¼ ImΔðσ2 þ iϵ; μ2Þ ∼ 2 gh π 0 j ln kEj; ð67Þ 1 ≃ σ2 μ2 Δ1−loop σ2 ϵ σ2 which shows the coherency between the approximation of ZCð ; 0Þ π Im ð þ i ; Þ (60) in the complex k2 momentum and the IR asymptotic 1 ∼− behavior of the Euclidean propagator of (54). Both of the Im 2 ˆ TW M ðσÞ½s þ Π ðsÞ σ2 two arguments imply the IR negativity of the gluon spectral gh;ren s¼− 2−iϵ M function. As negativity of a spectral function in a weak ∼ const > 0: ð63Þ sense leads to complex poles [40], the IR and UV negativity of the gluon spectral function supports the existence of These results demonstrate the IR negativity of the gluon complex poles in the gluon propagator. spectral function and the IR positivity of the ghost spectral For the quark propagator, the one-loop expression gives 0 function for the infrared safe trajectories with mq > . The no nontrivial IR spectrum from (25). This is rather clear ghost spectral function has a delta function at σ2 ¼ 0 with a from the facts that the Feynman diagram indicates ImΣ has 074044-10 COMPLEX POLES AND SPECTRAL FUNCTIONS … PHYS. REV. D 101, 074044 (2020) 2 2 nonvanishing value only for k >mq and that mqðμÞ is Comparing the strict one-loop and RG-improved results constant in the IR limit. could support the robustness of NWðCÞ. Further- Incidentally, let us comment on the nontrivial infrared more, we discuss the complex poles of the quark propa- fixed point to give the Gribov-type scaling solution [16].In gator and comment on the ghost propagator. the massive Yang-Mills model of the pure Yang-Mills theory, λ a non-trivial fixed point in ð ;uÞ plane appears, where the A. Gluon propagator: Strict-one-loop analysis gluon and ghost exhibit the Gribov-type behavior. One can find a similar fixed point in the massive Yang-Mills model Based on the strict one-loop gluon propagator (13) combined with (29), (30), and (31) of Sec. III, we compute with “massless quarks” mq ¼ 0. For instance, the fixed point the winding number NWðCÞ for NF ≤ 9 according to the lies at ðλ;u;tÞ¼ð4.0; 1.6; 0Þ in NF ¼ 3. However, this scaling-type fixed point has an infrared unstable direction. procedure (9) of Sec. II. Notice that the strict one-loop Therefore, with quarks m > 0, the unstable scaling-type gluon propagator satisfies the conditions (i) and (ii) in q Γð2Þ D −1 fixed point disappears due to the running of the quark mass. Sec. II. Since the two-point vertex function A ¼½ T is finite, the gluon propagator has no zeros, NZ ¼ 0. Thus IV. NUMERICAL RESULTS ON ANALYTIC we can count the number of complex poles by computing − STRUCTURES NWðCÞ¼ NP. In this section, we present results on the number of complex poles of the gluon and quark propagators in the 1. The number of complex poles at the typical ðg;MÞ 2 2 massive Yang-Mills model with quarks at the one-loop As a first step, we investigate the ðNF; ξ ¼ mq=M Þ approximation and its RG improvements. Without quarks, dependence of NP at the fixed typical values of the − 2 we found the gluon propagator has NP ¼ NWðCÞ¼ for parameters to obtain an overview. 2 2 all parameters ðg ;M Þ due to the negativity of the spectral Figure 3 displays the contour plot of NWðCÞ on the 2 function [40]. Surprisingly, it turns out that the model with mq 2 2 ðN ; ξ ¼ 2Þ plane at the fixed g ¼ 4 and M ¼ 0.2 GeV . many light quarks has N ¼ 0, 2, 4 regions, depending on F M P This figure is restricted to 0 ≤ N < 10, since the one-loop the number of quarks N and the parameters ðg2;M2;m2Þ. F F q approximation will be not reliable for N ≥ 10. Indeed, the In particular, for N ð4 ≲ N ≤ 9Þ light quarks, the N ¼ 4 F F F P naive one-loop UVasymptotic form (37) for γ0 > 0, i.e., for region, where the gluon propagator has two pairs of complex poles, covers a typical value of coupling g, e.g., g ∼ 4 in the setting to be described shortly, and the model with NFðNF ≥ 10Þ very light quarks has the region with no complex poles around the typical value of coupling g. From here on, we set G ¼ SUð3Þ and the renormaliza- tion scale μ0 ¼ 1 GeV. With the RG improvements, the best fit parameters reported in [21] are g ¼ 4.5, M ¼ 0.42 GeV, and the up and down quark masses mu ¼ md ¼ 0.13 GeV for the case of NF ¼ 2 while g ¼ 5.3, M ¼ 0.56 GeV, and mu ¼ md ¼ 0.13 GeV for NF ¼ 2þ 1þ1 (with the assumption on the strange and charm quark masses of ms ¼ 2mu and mc ¼ 20mu). Notice that the “quark mass” mq of this model should not be confused with the current 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 ¼ 1 GeV. In particular, the massless quarks will differ from the “massless quarks” mq ¼ 0 due to the spontaneous breakdown of the chiral symmetry. FIG. 3. Contour plot of NWðCÞ for the gluon propagator on the First, we investigate N ðCÞ of the strict one-loop gluon 2 2 W ðN ; ξÞ plane at g ¼ 4 and M ¼ 0.2 GeV , which gives the propagator, mainly at fixed typical values of the parameters F number of complex poles through the relation NP ¼ −NWðCÞ.In g ¼ 4 and M2 ¼ 0.2 GeV2 to see a qualitative overview of the NP ¼ 0, 2, 4 regions, the gluon propagator has zero complex the analytic structures of this model. Next, we consider the pole, one pair, and two pairs of complex conjugate poles, RG improvements of these results and evaluate the gluon respectively. The vertical dashed lines at NF ¼ 3 and NF ¼ 6 spectral function at the best-fit parameters for NF ¼ 2. correspond to the top and bottom figures of Fig. 5, respectively. 074044-11 YUI HAYASHI and KEI-ICHI KONDO PHYS. REV. D 101, 074044 (2020) NF ≥ 10, shows the existence of a Euclidean pole together 2 with the fact that DTð0Þ > 0 and DTðk Þ has no zeros. This result illustrates that the gluon propagator has four complex poles in the region colored with dark gray (4 ≲ NF ≤ 9, 0.2 ≲ ξ ≲ 0.6). It predicts that the transition occurs from the NP ¼ 2 region to NP ¼ 4 region by adding light quarks since NWðCÞ is a topological invariant. In the NP ¼ 4 region, the gluon propagator has two pairs of complex conjugate poles if we exclude the possibility of Euclidean poles on the negative real axis. To see details of the NP ¼ 4 region and its boundary, let us look into positions of complex poles. 2. Location of complex poles Here we take a further look into positions of complex poles and the transition of NWðCÞ. We will report the ratio w=v for a position of a complex pole k2 ¼ v þ iw and trajectories of complex poles for varying ξ. First, we focus on the ratio w=v at the complex pole k2 ¼ v þ iw, on which we set w ≥ 0 without loss of generality from the Schwarz reflection principle, in order to see whether or not the gluon presents a particlelike resonance. Since the gluon propagator has at most two pairs of complex conjugate poles, it is sufficient to find max w=v and min w=v. The results are demonstrated in Fig. 4. In general, the ratio w=v of a complex pole decreases as NF increases. This implies that the gluon exhibits more particlelike behavior for larger NF. Note that the ratio min w=v rapidly decreases near the boundary between NWðCÞ¼−2 and NWðCÞ¼−4. Second, we examine trajectories of complex poles for NF ¼ 3 and NF ¼ 6. Figure 5 plots the pole location with varying ξ. In the case of N ¼ 3, no transition changing N F P FIG. 4. Contour plots of min w=v (top) and max w=v (bottom) occurs; the pole moves gradually and lowers its real part v as 2 ≥ 0 ξ for a complex pole at k ¼ v þ iw; w of the gluon propagator increasing . On the other hand, the trajectory of complex on the ðN ; ξÞ plane. The regions of w=v > 1, 1 > w=v > 0.1, 6 F poles is completely different at NF ¼ where the transition 0.1 > w=v > 0.01, and 0.01 > w=v are represented by different occurs. The pole moves continuously from the position of levels of gray. The vertical dashed lines at NF ¼ 3 and NF ¼ 6 ξ ¼ 0, but is absorbed into the branch cut at ξ ≈ 0.6. correspond to the top and bottom figures of Fig. 5, respectively. Beforehand, the new pole arises at ξ ≈ 0.2 from the branch cut. The region of min w=v < 0.01 appears around the transition −2 −4 These observations demonstrate that (i) the gluon propa- between NWðCÞ¼ and NWðCÞ¼ .AsNF increases, the gator has a pole at timelike momentum, which could ratio w=v tends to be small. correspond to a physical particle, on the boundary between the NP ¼ 2 and NP ¼ 4 regions and that (ii) the pole bifurcates and becomes a new pair of complex conjugate incorporated at the fixed typical values of the parameters of g 4 and M2 0.2 GeV2. Therefore, we have to check poles in the NP ¼ 4 region. This appearance of the new ¼ ¼ pair of complex conjugate poles from the branch cut on the whether or not the existence of the transition is insensitive real positive axis is compatible with the rapid decrease of to choices of the parameters ðg; MÞ. −4 min w=v on the boundary between the N ¼ 2 and N ¼ 4 One can verify that the NWðCÞ¼ region in the P P 2 2 regions in Fig. 4. λ C2ðGÞg 2 μ2 ξ mq parameter space ð ¼ 16π2 ;u¼ M = 0; ¼ M2Þ largely expands by changing NF from NF ¼ 3 to NF ¼ 6.At 3. (g,M) dependence NF ¼ 6, the NWðCÞ¼−4 region dominates the parameter region around the typical value g ∼ 4, M2 ∼ 0.2 GeV2 with We have seen that the gluon propagator changes its 2 0 2 ≲ mq ≲ 0 6 number of complex poles when many light quarks are . M2 . . See Appendix B for details. 074044-12 COMPLEX POLES AND SPECTRAL FUNCTIONS … PHYS. REV. D 101, 074044 (2020) FIG. 5. Positions of poles of the gluon propagator at NF ¼ 3 2 6 0 ξ mq 0 8 (top) and NF ¼ (bottom) for varying < ¼ M2 < . . As ξ increases, the poles move in the direction shown by the arrows. At NF ¼ 3 (top), the pole moves continuously from the position of ξ ¼ 0. In contrast, at NF ¼ 6 (bottom), the pole from the position of ξ ¼ 0 goes behind the branch cut at ξ ≈ 0.6. Moreover, the new pole (v ≈ 0.16 GeV2) appears from the branch cut at ξ ≈ 0.2. Thus our qualitative conclusion will be valid: this model has the transition of NP, and the gluon propagator has four FIG. 6. Two-dimensional slice of equi-NWðCÞ volume of the complex poles in the presence of NFð4 ≲ NF ≤ 9Þ light 2 strict one-loop gluon propagator at N ¼ 3 (top) and N ¼ 6 mq F F 0 2 ≲ ≲ 0 6 2 2 quarks with mq satisfying . 2 . . C2ðGÞg 2 2 mq M (bottom) in the ðλ ¼ 2 ;u¼ M =μ0 ¼ 0.2; ξ ¼ 2Þ space. Incidentally, to compare the results with those of the RG 16π M improved gluon propagator, we show in Fig. 6 contour 2 2 plots of NWðCÞ at u ¼ M =μ0 ¼ 0.2. The RG equation for the gluon propagator is D 2 α μ2 μ2 −1 μ2 μ2 D 2 α μ2 μ2 B. Gluon propagator: RG improved analysis TðkE; ð Þ; Þ¼ZA ð ; 0Þ TðkE; ð 0Þ; 0Þð68Þ So far, we have studied the gluon propagator in the strict one-loop level, relying on the robustness of the winding where α denotes the set of gauge coupling and masses α ¼ 2 2 2 2 2 2 number. To make this robustness more reliable and to study ðλ;u¼ M =μ ;t¼ mq=μ Þ and ZAðμ ; μ0Þ is the renorm- the structure of the gluon propagator for NF ≥ 10,itis alization factor computed by the anomalous dimension. We important to survey the winding number for the RG improved then approximate the gluon propagator to avoid the large gluon propagator. Moreover, this model reproduces the logarithms as numerical lattice results with the RG improved propagators at the “realistic” parameters g 4.5, M 0.42 GeV, and 2 2 2 ¼ ¼ DTðk ; αðμ0Þ; μ0Þ m ¼ 0.13 GeV at μ0 ¼ 1 GeV and N ¼ 2 [21]. In this q F ≈ 2 μ2 D1−loop 2 α 2 2 subsection, we investigate the analytic structure of the one- ZAðjk j; 0Þ T ðk ; ðjk jÞ; jk jÞ ð69Þ loop RG improved gluon propagator for the realistic param- eters and its parameter dependence for each number of quark Although the RG improvements only for the modulus jk2j 2 flavors NF. on the complex k plane may break the analyticity, this will 074044-13 YUI HAYASHI and KEI-ICHI KONDO PHYS. REV. D 101, 074044 (2020) propagator approaches to that of the pure massive Yang- Mills model as mq → ∞, the gluon propagator has one pair of complex conjugate poles NP ¼ 2 for large ξ.AtNF ¼ 3, the NP ¼ 2 region dominates the region of the parameters. At NF ¼ 6, in the presence of light quarks, the NP ¼ 4 region, on which the gluon propagator has two pairs of complex conjugate poles, occupies the larger region, especially around a typical coupling λ ∼ 0.3. On the other hand, at NF ¼ 9, the NP ¼ 0 region, on which the gluon propagator has no complex poles, expands and the NP ¼ 4 FIG. 7. Real part (blue) and imaginary part (orange) of the region shrinks. At N ¼ 10, notably, the N ¼ 4 region 2 F P gluon propagator at the realistic parameters for NF ¼ [21] on disappears, and the N ¼ 0 region appears in the larger the positive real axis. This shows its spectral function is quasi- P region of λ for the very light quark mass mq=M ≪ 1. negative, from which NWðCÞ¼−2. Notice that the spectral function exhibits the linear decrease with respect to k2 in Note that the strict one-loop gluon propagator and the agreement with (62) in the IR limit. RG improved one provide almost the same result for give a better approximation than the strict one-loop approximation for the propagator on the timelike momenta. First, let us see the gluon propagator for the realistic parameters. Figure 7 plots the real and imaginary parts of the gluon propagator on the timelike momenta. The spectral function is quasinegative, i.e., the spectral function is 2 2 2 negative ρðk0Þ < 0 at all timelike zeros k0 of ReDðk Þ. Then, NWðCÞ¼−2 from the invariance of NWðCÞ under continuous deformation [40]. Since the gluon propagator has no zeros NZ ¼ 0 from the fact that the RG improvement only affects the renormalization factor and the running couplings, we deduce the gluon propagator of this model has one pair of complex conjugate poles as in the pure Yang- Mills case. Incidentally, notice that the spectral function ρðσ2Þ decreases linearly with respect to σ2 in the IR limit σ2 → þ0, which is a general feature for the infrared safe trajectories with mq > 0 in this model as shown in (62). Next, we investigate the number of complex poles NP in the whole parameter space for each NF. Since NZ ¼ 0 within this approximation as before, NP can be computed as NP ¼ −NWðCÞ. Note that different points on a same renormalization group trajectory in the three dimensional parameter space, e.g., Fig. 2, provide the same analytic structure. Indeed, they are exactly connected by the scale transformation from the dimensional analysis and the anomalous dimension: under a scaling κ, 2 2 2 2 DTðk ; αðκ μ0Þ; μ0Þ κ2 −1 κ2μ2 μ2 D κ2 2 α μ2 μ2 ¼ ZA ð 0; 0Þ Tð k ; ð 0Þ; 0Þ; ð70Þ 2 2 2 2 2 2 2 which shows that DTðk ;αðκ μ0Þ;μ0Þ and DTðk ;αðμ0Þ;μ0Þ have the same number of complex poles. Therefore, it suffices to compute in the two-dimensional slice of the FIG. 8. Two-dimensional slice of equi-NWðCÞ volume of the 3 6 renormalization group flow, see Fig. 2. From here on, we gluon propagator at NF ¼ (top) and NF ¼ (bottom) in the 2 2 2 2 C2ðGÞg 2 2 mq employ the two dimensional slice at u M =μ 0.2. λ 2 μ 0 2 ξ 2 ¼ ¼ ð ¼ 16π ;u¼ M = 0 ¼ . ; ¼ M Þ space. Besides the re- Figures 8 and 9 reveal the results for NF ¼ 3,6,9,10. gion with the Landau poles (white), these plots have almost the Note that, since the analytic structure of the gluon same structure as Fig. 6. 074044-14 COMPLEX POLES AND SPECTRAL FUNCTIONS … PHYS. REV. D 101, 074044 (2020) higher loop corrections or other ignored effects are thus highly significant to describe the quark sector well. Here, we try to examine the quark propagator within the framework of the one-loop RG to catch the qualitative feature as a first attempt. Let us consider the vector and scalar parts of the quark propagator: Γren S ðk2 Þ¼ v ; v E 2 Γren 2 Γren 2 kEð v Þ þð s Þ Γren S ðk2 Þ¼ s : ð71Þ s E 2 Γren 2 Γren 2 kEð v Þ þð s Þ We define the spectral functions associated to these propagators, 1 ρ σ2 S 2 −σ2 − ϵ vð Þ¼π Im vðkE ¼ i Þ; 1 ρ σ2 S 2 −σ2 − ϵ sð Þ¼π Im sðkE ¼ i Þ: ð72Þ Note that the positivity of the state space would imply 2 2 2 ρvðσ Þ > 0; σρvðσ Þ − ρsðσ Þ > 0: ð73Þ We use the same improvement scheme as (69). The scalar and vector parts of the quark propagator at the realistic values of the parameters are plotted in Fig. 10. Both of the propagators have negative spectral functions and therefore have one pair of complex conjugate poles like the gluon propagator. Notice that both the conditions for the positivity (73) are violated. The former condition can be seen from the FIG. 9. Two-dimensional slice of equi-NWðCÞ volume of the gluon propagator at NF ¼ 9 (top) and NF ¼ 10 (bottom) in the 2 2 λ C2ðGÞg 2 μ2 0 2 ξ mq ð ¼ 16π2 ;u¼ M = 0 ¼ . ; ¼ M2Þ space. The white region shows RG trajectories with the Landau poles. NF ¼ 3, 6 by a comparison between Figs. 6 and 8. This supports the robustness of the winding number NWðCÞ. It is remarkable that the parameter dependence of the analytic structure drastically changes between NF ¼ 9 and NF ¼ 10. The extension of NP ¼ 0 regionforverylight quark mass can be seen from the UV positivity of the spectral function for NF ≥ 10 [47], since the positivity in a weak sense indicates NWðCÞ¼0 [40]. In a similar sense, the disappear- ance of NP ¼ 4 region could be related to the UV positivity of the spectral function, which might lower NWðCÞ. C. Quark propagator FIG. 10. Real (blue) and imaginary (orange) parts of the scalar As pointed out in [21], the one-loop RG computation is part (top) and the vector part (bottom) of the quark propagator on not enough to reproduce the quark wave function renorm- the positive real axis at the realistic parameters for NF ¼ 2 [21]. alization, although the one-loop quark mass function Their spectral function are negative, from which NWðCÞ¼−2 exhibits a qualitative agreement with lattice results. The for both the scalar and vector parts. 074044-15 YUI HAYASHI and KEI-ICHI KONDO PHYS. REV. D 101, 074044 (2020) vector part of the quark propagator in Fig. 10. Figure 11 shows the violation of the latter one. Next, we investigate the number of complex poles with various parameters ðNF;g;M;mqÞ. The results on the winding number NWðCÞ at NF ¼ 3 and NF ¼ 12 are shown in Fig. 12. We numerically checked that both of the scalar and vector parts of the quark propagator (71) yield the same NWðCÞ on the region shown in Fig. 12, from which NZ ¼ 0 for the both parts. The quark sector does not exhibit a new structure by 2 2 adding NF quarks. The qualitative insensitivity of the quark FIG. 11. The quark spectral function σρvðσ Þ − ρsðσ Þ is plotted at the realistic parameters. The positivity condition is sector to NF is evident in the sense that the strict one-loop violated below 0.8 GeV2. expression is, of course, independent of NF and that the quark propagator will only be influenced indirectly by NF via the running parameters. However, it is worthwhile to note that the region with NP ¼ 0 extends slightly as NF increases, in particular for very light quarks ξ ≪ 1, which is a common feature with the gluon one. D. Ghost propagator Finally, let us add some comments on the ghost propa- gator. The strict one-loop propagator is not affected by the dynamical quarks. Therefore, from the proposition (Case III) of [40], the value NWðCÞ¼0 for ghosts will hold after the RG improvements unless the RG trajectory has a Landau pole. Therefore, we conclude that the analytic structure of the ghost propagator within this approximation is insensitive to NF and that the ghost propagator has no complex poles. V. CONCLUSION Let us summarize our findings. The argument principle for a propagator relates the propagator on the timelike momenta and the number of complex poles. The winding number NWðCÞ¼NZ − NP can be computed numerically from a propagator on timelike momenta according to (9), where NZ and NP stand for the number of complex zeros and poles respectively. To study the analytic structures of the QCD propagators, we have employed an effective model of QCD, the massive Yang-Mills model. We have found that the infrared safe trajectories, on which the running coupling is finite in all scales, reproduce the UV asymptotic behavior (55) origi- nally obtained by Oehme and Zimmermann [47] and that the gluon spectral function is negative in the IR limit ρðσ2Þ ∼−σ2 for quarks with the quark mass parameter of this model for mq > 0 (62). The UV negativity of Oehme- Zimmermann and IR negativity of the gluon propagator argued in this paper supports the existence of complex poles in the gluon propagator from the general relationship claiming that a negative spectral function in a weak sense FIG. 12. Two-dimensional slice of equi-NWðCÞ volume of the leads to complex poles [40]. 3 12 quark propagator at NF ¼ (top) and NF ¼ (bottom) in the In the “realistic” parameters used in [21] to fit the 2 2 λ C2ðGÞg 2 μ2 0 2 ξ mq ð ¼ 16π2 ;M = 0 ¼ . ; ¼ M2Þ space. Both of the scalar and numerical lattice results, both the gluon and quark propa- vector parts provide the same NWðCÞ. gators have quasinegative spectral functions and one pair of 074044-16 COMPLEX POLES AND SPECTRAL FUNCTIONS … PHYS. REV. D 101, 074044 (2020) complex conjugate poles, while the ghost propagator has no might correspond to a physical one-particle state even after complex poles. some confinement mechanism works. In this sense, the Finally, we have investigated the number of complex poles absence of a timelike pole will be favored. in this model with many quarks by computing NWðCÞ for Second, we comment on the NF dependence of the various parameters. For the gluon, there are several interesting condensation. We have studied the mass-deformed model features. First, the NP ¼ 2 region dominates the parameter as an effective model for QCD or QCD-like theories based 2 2 region if ξ ¼ mq=M is not so small or NF ≲ 4.Inthisregion, on the facts that the gluon mass can minimally improve the the gluon propagator has one pair of complex conjugate poles gauge fixing procedure and that the effective potential for the A A as in the pure Yang-Mills case [40].Second,theNP ¼ 4 operator μ μ calculated by the local composite operator region, where the gluon propagator has two pairs of complex technique indicates the condensation of this operator conjugate poles, expands by adding light quarks. A typical set [12,13]. The former argument can give a masslike effect 2 2 of the parameters (g ≈ 4, M ≈ 0.2 GeV )iscoveredbythe independently upon NF. However, the latter one will be NP ¼ 4 region when 4 ≲ NF ≤ 9; 0.2 ≲ ξ ≲ 0.6.These substantially affected by the presence of quarks. The features hold in the computations both for the strict one-loop effective potential of [13] appears to be “unbounded” for and the RG-improved one-loop results. Third, the strict one- γ0 > 0,orNF ≥ 10 for G ¼ SUð3Þ. The “unboundedness” loop result on the locations of complex poles indicates that the follows from the fact that the coefficient of the Hubbard- gluon tends to be “particlelike” as NF increases. Fourth, for Stratonovich transformation ζ runs into the negative infinity NF ≥10, the one-loop RG-improved gluon pro- ζ → −∞ in the UV limit μ → ∞ due to additive counter- 4 pagator has no NP ¼ region. Moreover, the gluon propa- terms for NF ≥ 10 even if it is set to be a positive value at gator with NFðNF ≥ 10Þ very light quarks ξ ≪ 1 has no some scale. This problem might indicate the limitation of the complex poles even for a typical value of the gauge coupling. perturbative treatment for the local composite operator. The existence of complex poles invalidates the Käll´en- Therefore, we have no cogent argument supporting the Lehmann spectral representation, which is a fundamental dimension-two gluon condensate for NF ≥ 10. consequence of QFT describing physical particles. Third, related to the second remark, the validity of our Therefore, the existence of complex poles of the gluon results will be questionable for large NF. For instance, two- and quark propagators could be a signal of confinement of loop corrections will be important if the first coefficient of the elementary degrees of freedom in QCD. the beta function is small as in the original argument of the In conclusion, the above results imply that the confine- infrared conformality [25,26]. Moreover, the results from ment mechanism may depend on the number of quarks and the truncated Schwinger-Dyson equation [29] show that the quark mass since complex poles represent a deviation from gluon and ghost propagators seem to obey a scaling-type physical particles and will be related to confinement. power law in a wide range of momentum in the conformal Furthermore, the drastic change between NF ¼ 9 and window; the description by the massive Yang-Mills model 10 “ NF ¼ quarks could be possibly related to the decon- will be inappropriate above the critical value of NF. finement” in line with the conformal window. However, we can expect that the massive Yang-Mills model will be valid in the QCD-like phase, or below the VI. DISCUSSION AND FUTURE WORK critical value of NF. Therefore, the massive Yang-Mills model may capture some information on the transition from Several comments regarding these results are in order. the QCD-like phase. First, let us mention a comparison with a similar Fourth, N ¼ 10, where the first coefficient of the gluon approach [38,39], where the analytic structures of gluon F anomalous dimension changes its sign, is the value at which and quark propagators are investigated with a massive-type the analytic structure of the gluon propagator changes model in light of the variational principle and optimization. drastically in our analysis. This value appears in various In there, the gluon propagator has two pairs of complex perspectives. For example, there has been some proposal conjugate poles, while the quark propagator has a timelike that the negativity of the gluon anomalous dimension is pole and no complex poles in N ¼ 2 QCD. These F crucial for confinement [51]. N ≈ 10 can be the critical structures are different from ours, in which the gluon F value of the conformal phase transition [30]. and quark propagators have one pair at the “realistic” Finally, the investigation of the analytic structures by parameter. Although the quark sector of both models lacks model calculations is speculative and should be taken as accuracy, the difference will be relevant in light of the 3 an attempt toward capturing some aspects of the intricate confinement of quark degrees of freedom ; a timelike pole dynamics of QCD. Many works of literature have different claims. For example, Ref. [52] claims that the quark propa- 3 The confinement of quark degrees of freedom, which stands gator has a pole at a timelike momentum while the gluon for absence of the quark one-particle state from the physical spectrum, should not be confused with the well-studied “quark propagator has complex poles by using some parametriza- confinement” that means an external quark source requires tions for the propagators and the numerical solution of infinite energy or the linear rising quark-antiquark potential. truncated Dyson-Schwinger equation. The gluon propagator 074044-17 YUI HAYASHI and KEI-ICHI KONDO PHYS. REV. D 101, 074044 (2020) obtained by solving the truncated Dyson-Schwinger equation however, this proposition is shown for theories avoiding the on the complex momentum plane in pure Yang-Mills theory is Gribov ambiguity by the restriction; this proposition is not shown to have no complex poles [43]. A recent reconstruction applicable to theories that fix the gauge by schemes technique indicates complex poles in the gluon propagator averaging over Gribov copies like [15] or some justification [42]. Pursuing the rich analytic structure of the QCD of the standard Faddeev-Popov Lagrangian. propagators deserves further investigations because a QFT The generalization of (9) is as follows. 2 describing confined particles is not yet well understood. Suppose that the propagator Dðk Þ and its data fDðxn þ ϵ N On a formal side, a local QFT cannot yield complex i Þgn¼1 satisfies the following three conditions: poles in the standard perspective, see, e.g., [53]. One might (i) In the UV limit jzj → ∞, Dðk2Þ has the same phase assert that complex poles correspond to short-lived exci- − → 1 as the free propagator, i.e., argð DðzÞÞ arg z as tations and break the locality and unitarity in the level of jzj → ∞. propagators [34,35]. However, if we analytically continue4 ’ 2 → − 2 α 2 → 0 α (ii ) Dðk Þ ZIRð k Þ as jk j , where is a real a propagator with complex poles not in the complex number. momentum but in the complex time from the Euclidean 2 ϵ N (iii) fk ¼ xn þ i gn¼0 is sufficiently dense so that space to the Minkowski one, the resulted propagator can be Dðk2 ¼ x þ iϵÞ changes its phase at most half- interpreted as a propagator of a QFT with an indefinite π ϵ ϵ winding ( ) between xn þ i and xnþ1 þ i , where 2 metric state space having complex spectra that can satisfy we denote sufficiently small x0 ¼ δ > 0 and suffi- local commutativity, e.g., [54]. Notice that such theories Λ2 ciently large xNþ1 ¼ , on which we will take the with complex spectra and an indefinite metric are out of the limits δ2 → þ0 and Λ2 → þ∞. scope of the axiomatic quantum field theory because their Then, the winding number NWðCÞ is expressed as Wightman functions are not tempered distribution due to XN 1 ϵ complex energies, and the theorems derived by assuming −α − 1 2 Dðxnþ1 þ i Þ the temperedness are not applicable to the “propagators NWðCÞ¼ þ 2π Arg ϵ : ðA1Þ 0 Dðxn þ i Þ with complex poles.” Then, complex poles would not lead n¼ to the nonlocality and just represent unphysical degrees of Let us derive this expression. First, we decompose the freedom. Further discussion on this issue is reserved for path around the positive real axis C2 into three pieces ∪ ∪ future works. As discussed in [36,55], it would also be C2 ¼ C2;þ Cδ C2;−, where C2; stands for the path ϵ; δ2 interesting to study how the complex poles are “canceled” along the positive real axis of C2; ¼fx i This work was supported by Grant-in-Aid for Scientific NWðCÞ¼NWðCδÞþNWðC1ÞþNWðC2;−ÞþNWðC2;þÞ: Research, JSPS KAKENHI Grant Number (C) ðA2Þ No. 19K03840. Y. H. is supported by JSPS Research Fellowship for Young Scientists. The integral NWðC1ÞþNWðC2;−ÞþNWðC2;þÞ can be evaluated as before, APPENDIX A: COUNTING COMPLEX POLES NWðC1ÞþNWðC2;−ÞþNWðC2;þÞ FOR VARIOUS INFRARED BEHAVIORS XN 1 ϵ −1 2 Dðxnþ1 þ i Þ Here, we generalize the IR asymptotic condition ¼ þ 2π Arg ϵ : ðA3Þ 0 Dðxn þ i Þ (ii) Dðk2 ¼ 0Þ > 0 to derive the formula (9) in Sec. II. n¼ This will be relevant for the scaling solution of the gluon For the contribution from the small circle, note that 2 Dðk iϵÞ ∓iπα 2 propagator or a propagator having massless singularity. 2 e k → 0 jDðk iϵÞj ¼ as þ . Therefore the phase factor Incidentally, in regard to the IR behavior of the gluon D=jDj varies from eþiπα to e−iπα, from which propagator, there is an argument on the gluon propagator in the Landau gauge [56]: If one restricts the configuration NWðCδÞ¼−α: ðA4Þ space inside the first Gribov region, the gluon propagator d−2 To sum up, we obtain (A1). Note that the IR suppression satisfies lim → 0 k DðkÞ¼0, where d is the spacetime k þ contributes negatively to N ðCÞ¼N − N . dimension. This excludes the massless free behavior for the W Z P ≤ 4 gluon propagator irrespective of NF for d . Note that, APPENDIX B: DETAILED RESULTS ON THE ONE-LOOP GLUON PROPAGATOR FOR 4To our knowledge, a method to reconstruct a QFT from a VARIOUS PARAMETERS given Euclidean field theory has not been established in the presence of complex poles. The standard reconstruction does not Here, we show detailed analyses on the results given work due to the violation of the reflection positivity. [1,41] in Sec. IVA 3 on the strict one-loop gluon propagator. To 074044-18 COMPLEX POLES AND SPECTRAL FUNCTIONS … PHYS. REV. D 101, 074044 (2020) check the insensitivity of the transition, we shall compute NWðCÞ in the three dimensional parameter space 2 2 2 C2ðGÞg M mq ðλ¼ 2 ;u¼ 2 ;ξ¼ 2Þ. 16π μ0 M FIG. 13. Boundary surfaces of equi-NWðCÞ volume at NF ¼ 3 2 2 2 C2ðGÞg M mq (top) and NF ¼ 6 (bottom) in the ðλ ¼ 2 ;u¼ 2 ; ξ ¼ 2Þ 16π μ0 M space. Their cross sections at several values of ξ are shown in Figs. 14 and 15. The gluon propagator has four complex poles inside the green surface and no complex poles inside the blue surface; otherwise the gluon propagator has two complex poles. At NF ¼ 3, the black dot shows a typical set of the parameters (g ¼ 4, M2 ¼ 0.2 GeV2, and ξ ¼ 0.1) and the gluon propagator − has two complex poles around there. On the other hand, at FIG. 14. Contour plots of NWðCÞ¼ NP of the gluon propa- 2 2 C2ðGÞg mq λ 6 4 0 2 ≲ ≲ 0 6 gator on the two-dimensional parameter space ð ¼ 16π2 ;u¼ NF ¼ , the gluon propagator has NP ¼ for . M2 . ∼ 4 2 ∼ 0 2 2 M2 ξ 0 1 3 around the typical values g , M . GeV . μ2 Þ at ¼ . , 0.3, 0.5 for NF ¼ . 0 074044-19 YUI HAYASHI and KEI-ICHI KONDO PHYS. REV. D 101, 074044 (2020) Figure 13 plots the boundary surfaces of equi-NWðCÞ volume at NF ¼ 3 and NF ¼ 6 in the three-dimensional parameter space ðλ;u;ξÞ. The gluon propagator has four complex poles inside the green surface (0.2 ≲ ξ ≲ 0.6) and no complex poles inside the blue surface (with small λ and ξ), and two complex poles except these regions. Contour plots for selected ξ are displayed in Fig. 14 for NF ¼ 3 and in Fig. 15 for NF ¼ 6.Theygivetwo- dimensional slices at fixed ξ ¼ 0.1,0.3,0.5ofFig.13. The regions colored with light gray (NP ¼ 0)inFigs.14 and 15 are two-dimensional cross sections at fixed ξ of the volumes inside the blue surfaces in the top and bottom figures of Fig. 13, respectively. Similarly, the regions colored with dark gray (NP ¼ 4)inFigs.14 and 15 are those of the volumes surrounded by the green surfaces in the top and bottom figures of Fig. 13, respectively. Figure 13 demonstrates the NWðCÞ¼−4 region inside the green surface expands from NF ¼ 3 to NF ¼ 6. In the figure of NF ¼ 3 (top of Fig. 13), the set of the typical values (g ¼ 4, M2 ¼ 0.2 GeV2, and ξ ¼ 0.1) is shown as a black dot, of which Fig. 3 is 2 computed at ðg; M Þ. The NWðCÞ¼−4 region occupies 2 0 2 ≲ mq ≲ 0 6 the region . M2 . around the typical values of the 2 2 parameters (g ∼ 4, M ∼ 0.2 GeV )atNF ¼ 6. From these observations, we deduce that the existence of the transition is insensitive to a detailed choice of g and M2. Therefore, the rough investigation in Sec. 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