Fermion spectral function in hot strongly interacting matter from the functional renormalization group
FFermion spectral function in hot strongly interacting matter from the functionalrenormalization group
Ziyue Wang ∗ and Lianyi He , Physics Department, Tsinghua University, Beijing 100084, China and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China (Dated: August 28, 2018)We present the first calculation of fermion spectral function at finite temperature in quark-mesonmodel in the framework of the functional renormalization group (FRG). We compare the results intwo truncations, after first evolving flow equation of effective potential, we investigate the spectralfunction either by taking the IR values as input to calculate one-loop self-energy or by taking thescale-dependent values as input to evolve the flow equation of the fermion two-point function. Thelatter one is a self-consistent procedure in the framework of FRG. In both truncations, we finda multi-peak structure in the spectral function, indicating quark collective excitations realized interms of the Landau damping. However, in contrast to fermion zero-mode in the one-loop truncation,we find a fermion soft-mode in the self-consistent truncation, which approaches the zero-mode astemperature increases.
I. INTRODUCTION
The Quantum Chromodynamics (QCD) phase transi-tions at finite temperature and density provide a deepinsight into the strong interacting matter created in highenergy nuclear collisions and compact stars. The prop-erties of the QGP phase near the critical temperature( T c ) acquire much interest, the heavy-ion collisions hassuggested that the QGP matter is an ideal fluid [1, 2],indicating that the created matter is a strongly coupledsystem. The spectral properties of quark and hadron inthis strongly interacting matter, are of fundamental im-portance for identifying the relevant degrees of freedomin the equation of state and respective transport proper-ties.Whether quark can be described by well-defined quasi-particles has long been investigated, quasi-particles cor-respond to peaks with a small width in the spectralfunction with relevant quantum numbers. Quenchedlattice QCD simulation indicates the existence of thequasi-particles of quarks with small decay width [3–5].Finite temperature gauge theory with Dyson-Schwingerequation also predicts quasi-particle properties of quarks[6, 7]. At high temperature, where the hard thermal loop(HTL) approximation applies, quarks still have some col-lective excitation known by the normal quasi-quark andthe plasmino branches in the spectral function [8–10]. Ithas also been investigated that, quarks in the QGP phasecan be described within a quasi-particle picture with amulti-peak spectral function [11–13], whenever the inter-action is mediated by scalar, pseudo-scalar, vector andaxial-vector meson, which may exist as bosonic excita-tions in the QGP phase [13].In the vicinity of T c of the chiral phase transition,non-perturbative effects are important, and one may ex-pect, that the quark spectral functions will possess novel ∗ [email protected] properties, when non-perturbative methods are adopted.In this paper, we employ the functional renormaliza-tion group (FRG) [14–19] approach to the quark-mesonmodel. As a non-perturbative method, FRG enablesus to incorporate fluctuation effects beyond mean fieldtheory, see Refs. [14]. The self-consistent treatment offluctuations is important towards the understanding ofphysics near a phase transition. Since the FRG allows adescription of scale transformation, it provides a deep in-sight into the system where scale dependence plays a cru-cial role. To calculate the spectral function in the usuallyused imaginary time formalism with FRG, an analyticalcontinuation is required to bring the imaginary time inthe Euclidean two point function at finite temperatureto the real time in the Minkowski space [20–26]. Thismethod has been applied to the study of real time ob-servables such as shear viscosity [26] and soft modes [27]near the QCD critical point. The consistent investigationof spectral function in the framework of the FRG hasbeen applied to various systems, including meson spec-tral function in a chiral phase transition [23, 24], quarkspectral function in vacuum [28], meson spectral functionin a pion superfluid system [29].When the FRG is put to use to investigate the quarkspectral function, the most crucial difference is that onetakes into account the scale dependence of the mesonmasses, and hence the thresholds of each decay, creationand scattering channel. The scale dependence is a highorder effect and brings about difference in spectral func-tion mainly in the following three aspects. First, it givesarise to novel structures in the imaginary part and realpart of the self-energy. Second, the Landau dampingwhich is the dominant effect at high temperature, is for-bidden at low energy when considering the scale depen-dence of the meson masses, and leads to more peaks atlow energy region at high temperature. Third, it is foundthat a fermion zero mode starts to appear when temper-ature is comparable to meson mass [13], which also orig-inates from the Landau damping effect. However, whenthe FRG is adopted, this zero mode becomes a soft mode, a r X i v : . [ h e p - ph ] A ug and approaches the origin when temperature increases.We organize the paper as follows. The FRG flow equa-tions for the effective potential and the two truncationsto calculate the quark self-energy are derived in SectionII. The procedure to solve the flow equations and thenumerical results are presented in Section III. We sum-marize in Section IV. II. FLOW EQUATIONS AND TRUNCATION
As an low energy effective model, the quark-mesonmodel comes from the partial bosonization of the four-fermion interaction model and exhibits many of theglobal symmetries of QCD. It is widely used as an effec-tive chiral model to demonstrate the spontaneous chiralsymmetry breaking in vacuum and its restoration at finitetemperature and density [30–32]. Here we take the two-flavor version of the model with pseudo-scalar mesons π and scalar meson σ as the dominant meson degrees offreedom at energy scale up to Λ ≈ T anddensity µ is given asΓ = (cid:90) x (cid:104) ¯ ψ ( ∂/ − γ µ ) ψ + g ¯ ψ ( σ + iγ (cid:126)τ · (cid:126)π ) ψ + 12 ( ∂ µ φ ) + U ( φ ) − cσ (cid:105) , (1)where the abbreviation (cid:82) x stands for (cid:82) β dx (cid:82) d x withthe inverse of temperature β = 1 /T , and (cid:126)τ are the Paulimatrices in flavor space. The Yukawa coupling is chosenas g = 3 . ψ and meson field φ are defined as ψ = ( u, d ) and φ = ( σ, π , π , π ). The explicit chiral symmetry break-ing term − cσ corresponds to a finite current quark mass m .Quantum and thermal fluctuations are of particularimportance in the vicinity of a phase transition and areconveniently included within the framework of FRG. Thecore quantity in this approach is the averaged effectiveaction Γ k at the RG scale k in Euclidean space, its scaledependence is described by the flow equation [14–19]˙Γ k = Tr (cid:90) p (cid:20) G φ,k ( p ) ˙ R φ,k ( p ) − G ψ,k ( p ) ˙ R ψ,k ( p ) (cid:21) , (2)where ˙Γ k = ∂ k Γ k and so as for ˙ R k . The symbol ’Tr’ rep-resents the summation over all inner degrees of freedomof mesons and quarks. G φ,k ( q ) = (cid:16) Γ (2) k [ φ ] + R φ,k ( q ) (cid:17) − ,G ψ,k ( q ) = (cid:16) Γ (2) k [ ψ ] + R ψ,k ( q ) (cid:17) − (3)are the FRG modified meson and quark propagatorswith the two-point functions Γ (2) k [ φ ] = δ Γ k /δφ andΓ (2) k [ ψ ] = δ Γ k /δψδ ¯ ψ and the two regulators R φ,k and R ψ,k . The evolution of the flow from the ultraviolet limit k = Λ to infrared limit k = 0 encodes in principle allthe quantum and thermal fluctuations in the action. Tosuppress the fluctuations with momentum smaller thanthe scale k during the evolution, an infrared regulator R is introduced in the flow equation. At finite tempera-ture and density where the Lorentz symmetry is broken,we employ the optimized regulator function which is thethree dimensional analogue of the 4-momentum regula-tor [33, 34]. The bosonic and fermionic regulators arechosen to be R φ,k ( p ) = (cid:126)p r B ( y ) ,R ψ,k ( p ) = (cid:126)γ · (cid:126)pr F ( y ) (4)in momentum space with y = (cid:126)p /k and r B ( y ) = (1 /y − − y ) and r F ( y ) = (1 / √ y − − y ). The regula-tors R φ,k and R ψ,k in the propagators G φ and G ψ amountto having regularized three-momenta (cid:126)p r = (cid:126)p (1 + r B ( y ))and (cid:126)p r = (cid:126)p (1 + r F ( y )) for bosons and fermions respec-tively. The three dimensional regulators break down theLorentz symmetry in vacuum. However, physical quan-tities are measured in the ground state at k = 0, wherethe regulators vanish and the Lorentz symmetry is guar-anteed.In order to derive the meson and quark propagators G φ,k and G ψ,k , we expand the effective potential aroundthe mean field (cid:104) σ (cid:105) k , which describes the chiral symmetrybreaking, and introduce the chiral invariant ρ k = (cid:104) σ (cid:105) k .The RG-modified propagators of mesons and quark are G − φ,k = p + (cid:126)p r + m φ,k ,G − ψ,k = − γ ( ip + µ ) + (cid:126)γ · (cid:126)p + m f,k , (5)with m σ,k = 2 U (cid:48) + 4 ρ k U (cid:48)(cid:48) and m π,k = 2 U (cid:48) , U (cid:48) and U (cid:48)(cid:48) are first and second order derivatives of effective potentialwith respect to ρ , and quark mass m f,k = g (cid:104) σ (cid:105) k .Assuming uniform field configurations, the integralover space and imaginary time becomes trivial, and theeffective action Γ k = βV U k is fully controlled by the po-tential U k with V and β being the space and time regionsof the system. The flow equation U k hence comes directlyfrom that of Γ k , namely ∂ k U k = ( T /V ) ∂ k Γ k . The flowequation of the effective potential is then calculated by ∂ k U k = 12 J φ ( E σ,k ) + 32 J φ ( E π,k ) − N c N f J ψ ( E ψ,k ) , (6)with J φ and J ψ are one-loop threshold functions, the ex-plicit expressions are presented in Appendix by Eq.(A2),and the energies are given by E φ,k = (cid:113) k + m φ,k and E ψ,k = (cid:113) k + m ψ,k .In the following, we are going to present two trunca-tions to calculate the self-energy and spectral function.In both truncations, we first evolve the flow of the ef-fective potential, and then take the masses as input tocalculate the self-energy. In truncation A, we take scale-dependent masses m σ,k and m π,k at IR-minimum ρ k =0 as input to integrate the flow of the two point function;while in truncation B, we take the IR masses m σ,k =0 and m π,k =0 to directly calculate the one-loop self-energyof quark. The diagrammatic description is presented inFIG.1 FIG. 1. The diagrammatic presentation of self-energy in twotruncations, ∂ k Γ (2) ψ,k is the flow equation for fermion two-pointfunction in truncation A, Σ is the one-loop self-energy in trun-cation B. A. Truncation A
In truncation A, we first evolve the flow of effectivepotential and prepare the scale-dependent meson massesas input. We then integrate the flow equation of the two-point function to obtain the self-energy in the infraredlimit. The flow equation of fermion two-point functionhas Dirac structure, namely is a 4 × ∂ k Γ (2)¯ uu ( p ) = − g (cid:101) ∂ k (cid:90) q (cid:104) G σ ( q − p ) G u ( q ) (7)+ 3 G π ( q − p )( iγ ) G u ( q )( iγ ) (cid:105) , where (cid:101) ∂ k is the derivative of RG-scale k that only acts onthe regulator R φ,k and R ψ,k in the propagators. In thiswork, we consider only the spectral function at zero exter-nal momentum (cid:126)p = 0. In the ultraviolet, the Euclideaninverse quark propagator at (cid:126)p = 0 at the IR expansionpoint is G − E, Λ ( ip ) = Γ (2)¯ uu, Λ ( ip ) = − γ ( ip + µ ) + g (cid:104) σ (cid:105) . (8)The inverse propagator at scale k is then an integral ofthe flow equation two-point function from the ultraviolet k = Λ down to kG − E,k ( ip ) = Γ (2)¯ uu,k ( ip ) (9)= − γ ( ip + µ ) + g (cid:104) σ (cid:105) + (cid:90) k Λ ∂ k Γ (2)¯ uu ( ip ) dk. Hence, the Euclidean inverse propagator can be writ-ten in terms of k-dependent self energy G − E,k ( ip ) =Γ (2)¯ uu,k ( ip ) = − γ ( ip + µ ) + g (cid:104) σ (cid:105) + Σ k ( ip ), with theinitial condition Σ Λ ( ip ) = 0. The scale-dependent self-energy in Euclidean space is thusΣ k ( ip ) − Σ Λ ( ip ) = (cid:90) k Λ ∂ k Γ (2)¯ uu ( ip ) dk. (10) At ultraviolet limit k = Λ, no fluctuation is included,Σ Λ ( ip ) = 0 agrees with the bare propagator. As thescale is lowered, quantum fluctuation are included, con-tributing to the self-energy of the quark. The spectralfunction is a real-time quantity which requires analyticalcontinuation to bring imaginary time to real time G − R ( ω ) = − G − E ( ip → ω + iη ) . (11)The inverse retarded propagator is G − R,k ( ω ) = γ ( ω + µ + iη ) − g (cid:104) σ (cid:105) − Σ R,k ( ω ) and the corresponding retardedself-energy isΣ R,k ( ω ) = (cid:90) k Λ ∂ k Γ (2)¯ uu ( ω + iη ) dk. (12)The quark propagator at zero momentum (cid:126)p = 0 can bedecomposed to the positive energy part and negative en-ergy part G R,k ( ω ) = G + ,k ( ω )Λ + γ + G − ,k ( ω )Λ − γ , (13)with projection operators Λ ± ≡ (1 ± γ ) / G ± ,k ( ω ) = 12 Tr (cid:2) G R,k ( ω ) γ Λ ± (cid:3) = (cid:2) ω + iη + µ ∓ m f − Σ ± ,k ( ω ) (cid:3) − , (14)with Σ ± ,k ( ω ) = 12 Tr (cid:2) Σ R,k ( ω ) γ Λ ± (cid:3) . (15)We here focus on the self-energy of the ’quark’ sector,the flow equation is given by ∂ k Σ ± ,k ( ω ) = 12 Tr (cid:104) ∂ k Γ (2)¯ uu ( ω ) γ Λ ± (cid:105) , (16)which, after integral over the three momentum, Matsub-ara sum and analytical continuation, has the followingstructure, ∂ k Σ + ,k ( ω ) = g (cid:16) J Sψσ ( ω ) + J Sσψ ( ω ) + 3 J P Sψπ ( ω ) + 3 J P Sπψ ( ω ) (cid:17) , (17)with J Sψσ , J
Sσψ , J
P Sψπ , J
P Sπψ are the threshold functions pre-sented in the appendix. At vanishing quark number, fromthe charge conjugation symmetry, we have a relation be-tween G ± , G + ( ω ) = − G ∗− ( − ω ) , (18)but finite chemical potential breaks the charge con-jugation symmetry. We limit our study to the casewith µ = 0 and focus on the spectral function ofquark sector only. The spectral function is definedthrough the imaginary part of the retarded propagator ρ k ( ω ) = − (1 /π )Im G R,k ( ω ). This can decomposed simi-larly, ρ k ( ω ) = ρ + ,k ( ω )Λ + γ + ρ − ,k ( ω )Λ − γ , with ρ ± ,k ( ω ) = − π Im G ± ,k (19)= − π ImΣ ± ,k ( ω ) (cid:0) ω ∓ m f − ReΣ ± ,k ( ω ) (cid:1) + ImΣ ± ,k ( ω ) . B. Truncation B
In another truncation, we first evolve the flowequation of the effective potential, and find theinfrared quark mass m f,k =0 ( T, µ ) and meson mass m σ,k =0 ( T, µ ) , m π,k =0 ( T, µ ) at certain temperature anddensity. We then take these quantities as input to cal-culate the self-energy of quark at one-loop order. Thismethod has been discussed in Ref. [11–13], here we takethe FRG result as an input. In Euclidean space, thefermion self-energyΣ E ( ip ) = − g (cid:90) q (cid:2) G ψ ( q ) G σ ( p + q )+ 3 G ψ ( q )( iγ ) G π ( p + q )( iγ ) (cid:3) . (20)The analytical continuation is taken by Σ R ( ω ) =Σ E ( ip ) (cid:12)(cid:12) ip → ω + iη . Taking the energy projection as be-fore, we have the self-energy of ’quark’ and ’anti-quark’sector Σ ± ( ω ) = Tr[Σ R ( ω ) γ Λ ± ]. The self-energy hascontribution from the scalar channel and the pseudo-scalar channel, where the interaction mediated by sigmameson and pion respectively,Σ ± ( ω ) = Σ S ± ( ω ) + 3Σ P S ± ( ω ) . (21)One may first deal with the imaginary part of the scalarchannelImΣ S + ( ω ) = − g π (cid:90) + ∞ p dpE σ (cid:110) (22)+ δ ( ω + E σ + E f ) (cid:16) m f E f (cid:17) (cid:104) n b ( E σ ) − n f ( E f ) (cid:105) + δ ( ω + E σ − E f ) (cid:16) − m f E f (cid:17) (cid:104) n b ( E σ ) + n f ( E f ) (cid:105) + δ ( ω − E σ + E f ) (cid:16) m f E f (cid:17) (cid:104) n b ( E σ ) + n f ( E f ) (cid:105) + δ ( ω − E σ − E f ) (cid:16) − m f E f (cid:17) (cid:104) n b ( E σ ) − n f ( E f ) (cid:105)(cid:111) . The momentum integral can be carried out analytically,giving [12]ImΣ S + ( ω ) = − g π ( ω + M + )( ω − M − ) ω × (cid:113) ( ω − M )( ω − M − ) × (cid:20) coth ω + M + M − ωT + tanh ω − M − M + ωT (cid:21) × (cid:0) Θ( ω − M ) − Θ( M − ω ) (cid:1) , (23) where M + = m σ + m f and M − = | m σ − m f | and Θ( x )is the step function. The self-energy of the pseudo-scalar channel can be obtain from that of the scalarchannel by taking the substitution m f → − m f and m σ → m π . The self-energy Σ R ( ω ) has an ultraviolet di-vergence which originates from the T-independent partΣ RT =0 ( ω ) ≡ lim T → + Σ R ( ω ). The divergence can be re-moved by imposing the on-shell renormalization condi-tion on the T-independent part of the quark propagator.The T-dependent part, Σ RT (cid:54) =0 ( ω ) = Σ R ( ω ) − Σ RT =0 ( ω ) isfree from divergence. The real part and imaginary partare related by the Krames-Kronig relation,ReΣ R,T (cid:54) =0 ( ω ) = P (cid:90) + ∞−∞ dzπ ImΣ
R,T (cid:54) =0 ( z ) z − ω . (24)The spectral function of quark and anti-quark are thengiven by ρ ± ( ω ) = − π Im G ± (25)= − π ImΣ ± ( ω ) (cid:0) ω ∓ m f − ReΣ ± ( ω ) (cid:1) + ImΣ ± ( ω ) . III. NUMERICAL METHOD AND RESULT
To numerically solve the flow equations for the effec-tive potential and the two-point functions, we adopt thegrid method, and assume the initial condition at the ul-traviolet limit, U Λ ( ρ ) = 12 m ρ + 14 λ Λ ρ , (26)for one-dimensional grid, and for the quark self-energyΣ + , Λ ( ω ) = 0 . (27)During the process of integrating the flow equations fromthe ultraviolet limit to the infrared limit, the conden-sates (cid:104) σ (cid:105) k is obtained by locating the minimum of the k -dependent effective potential U k . Choosing the quarkmass m q = 300 MeV, pion mass m π = 135 MeV andpion decay constant f π = 93 MeV in vacuum, the corre-sponding initial parameters are m / Λ = 0 . λ Λ = 1and c/ Λ = 0 . k = 0 cannotbe reached. Instead, the evolution of the flow equationis stopped at k IR < T c ∼ � �� ��� ��� ��� ��� ���������� � ( ��� ) � / � π � π � π � ψ � σ FIG. 2. Temperature dependence of chiral condensate andvarious masses. interpretation to each of the channels, (I) δ ( ω + E φ + E ψ )describes a annihilation of a quasi-quark with an on-shellfree anti-quark and boson, (IV) δ ( ω − E φ − E ψ ) is a de-cay of a quasi-quark to on-shell free quark and a boson,(II) δ ( ω + E φ − E ψ ) is the decay process of a quasi-quarkstate Q, into an on-shell quark via a collision with a ther-mally excited boson, and its inverse process Q + b ↔ q .(III) δ ( ω − E φ + E ψ ) corresponds to a pair annihilationprocess between the quasi-quark and a thermally excitedanti-quark with an emission of a boson into the thermalbath, and its inverse process Q + ¯ q ↔ b . The latter twoprocesses are called Landau damping, which vanishes at T = 0, as it involves thermally excited particles in theinitial states. The Landau damping plays an importantrole in the spectral function as temperature rises and isclosely related to the three peak structure at high tem-perature. Process (II) and (III) both cause a mixing be-tween the quark and anti-quark hole through coupling tothermally excited boson as discussed in Ref. [12].In truncation A, the real part and imaginary part of theself-energy are calculated separately, the real part is givenby the principle value integral. While for the imaginarypart of the two-point function, the integral over RG-scale k only has contribution from a few scales k i due to theappearance of aforementioned Dirac delta functions. Thefollowing structures are encountered in the integration ofthe imaginary part of the flow equation, where g ( k ) = ± E φ,k ± ( ∓ ) E ψ,k , and k i are zero points of the delta-function at a certain energy ω , with ω + g ( k i ) = 0, (cid:90) f ( k ) δ ( ω + g ( k )) dk = − (cid:88) i f ( k i ) | g (cid:48) ( k i ) | , (28) (cid:90) f ( k ) δ (cid:48) ( ω + g ( k )) dk = (cid:88) i (cid:18) f (cid:48) ( k i ) g (cid:48) ( k i ) − f ( k i ) g (cid:48)(cid:48) ( k i ) g (cid:48) ( k i ) (cid:19) | g (cid:48) ( k i ) | . It is required that g ( k ) is a continuously differentiablefunction with g (cid:48) nowhere zero. In the integral of the flowequation g ( k ) has certain points where the derivatives arezero, the domain must be broken up to exclude the g (cid:48) = 0 point. These g (cid:48) ( k i ) = 0 points are similar to the van-Hove singularities in the density of states in condensedmatter physics[35], g ( ω ) = (cid:80) n (cid:82) d k (2 π ) δ ( ω − ω n ( (cid:126)k )) = (cid:80) n (cid:82) dS ω (2 π ) |∇ ω n ( (cid:126)k ) | . The group velocity ∇ ω n ( (cid:126)k ) vanishesat certain momenta, resulting a divergent integrand. Thedivergence is integrable in 3-dimensions, in lower dimen-sions, the van-Hove singularity appears. The van Hovesingularities have also attracted interests in high energyphysics [36–39]. In our case, the integral over k is one-dimensional and gives divergence in the imaginary part atfinite temperature. This divergence exists only in trun-cation A, when scale dependence of various masses aretaken into consideration, and appears only when two con-ditions are satisfied at the same time, that ω + δE ( k ∗ ) = 0and δE (cid:48) ( k ∗ ) = 0. The scale dependence of meson massalso causes divergence in the real part of the self-energy.In FIG.3, we present the RG-scale dependence of thresh-old for channels (I) to (IV), ± E φ,k ± E ψ,k , ( φ = σ, π )for three temperature T = 50 , , ω ,the k-integral runs into points that δE (cid:48) ( k ∗ ) = 0, leadingto the divergence in imaginary part. At low temperature,when the Landau damping is suppressed, the diverge doesnot appear. In contrast, in truncation B, we take m φ,k =0 as input, the imaginary part and real part are alwaysfinite after remove the zero-temperature part. ��� � ���������������� � = ����� ( � ) ( �� )( �� ) ( ��� ) ��� � ���������������� � ( � � � ) � = ������ ( � ) ( �� )( �� ) ( ��� ) - ��� � ���������������� ω ( ��� ) + � ϕ + � ψ + � ϕ - � ψ - � ϕ + � ψ - � ϕ - � ψ � = ������ ( � ) ( �� )( �� ) ( ��� ) FIG. 3. The RG-scale dependence of threshold for channels(I) to (IV), ± E φ,k ± E ψ,k , ( φ = σ, π ) for three temperature T = 50 , , We first present the spectral function of quark sectorat relatively low temperature, In FIG.4, from top to bot-tom, are figures for the spectral function, the real partand imaginary part of the self-energy, with the black solidline represents truncation A, and red dashed line for trun-cation B. The zero-temperature result can be found inRef. [28], to which, we have also found the same result.At T=50 MeV, the system is still in the chiral symme-try breaking phase, with quark mass m f = 296MeV andmeson mass m σ = 477MeV and m π = 140MeV. In bothtruncations we have delta-peaks around fermion mass, ω = 315MeV for truncation A, and ω = 296MeV fortruncation B. The peak structure in ρ + ( ω ) emerges atthe ”quasi-pole” [12] which is defined as zero of the realpart of the inverse propagator ω − m f − ReΣ + ( ω ) = 0,providing that the imaginary part is small enough at thatpoint. In both truncations, the real part of the inversepropagator only has one ”quasi-pole”, which is the crosspoint of ω − m f (the blue dotted line) with ReΣ + ( ω ) inthe figure. The difference in the position of the peak inboth truncations comes the inclusion and subtraction ofdifferent fluctuation. The Landau damping is still wellsuppressed, giving small imaginary part and flat struc-tures in the spectral function at low energy. At largeenergy, when | ω | > m ψ + m φ the decay processes (I) and(IV) take place, giving finite imaginary part and the con-tinuous spectrum in the spectral function in truncationA. While in truncation B, this continuous spectrum issubtracted when performing the renormalization. ��� � ������������������������ ρ + ( � � � - � ) �� � = ����� - ������������� � � Σ + ( � � � ) - ��� � ��� - ��� - ��� - ���� ω ( ��� ) � � Σ + ( � � � ) FIG. 4. Quark spectral function at T = 50MeV, from top tobottom are spectral function, real and imaginary part of thequark self-energy. The black solid line stands for truncationA, while red dashed line for truncation B. The blue dottedline represent ω − m f , with m f = 296MeV. The chiral phase transition takes place at the tem-perature about T = 170MeV, pion and sigma mesonhave not been degenerate yet, with m f = 130MeV, m σ = 259MeV, m π = 212MeV. For truncation A, thethreshold of various channels is presented in FIG.3. Inthe scattering channel with the thermally excited boson(II) and the annihilation channel with the thermally ex-cited anti-quark (III), ± E φ,k ∓ E ψ,k has points where the derivative with RG-scale vanished. This brings about di-vergence in the imaginary part, and oscillation in the realpart, see black lines in Fig.5. The imaginary part is dis-continuous at the four van-Hove singularities, and is zerowhen ω ≤ | ± E φ,k ∓ E ψ,k | min , where process (II) (III)are forbidden. ω − m f − ReΣ + ( ω ) has several quasi-polesin truncation A, yet the spectral function has only onepeak, when the imaginary part is small ω = 21MeV indi-cating a quasi-particle mode here. For other quasi-poles,the imaginary parts are too large to form a peak, giv-ing several bumps instead. In truncation B, however, wehave a quite different case, the imaginary part is non-zerobut finite when process (II) and (III) are allowed. Thereal part has five quasi-poles, at three of them, the imag-inary part is small enough to give a peak in the spectralfunction. When process (II) and (III) are allowed, theimaginary part is large and gives only small bumps inthe spectral function, see the red dashed lines in FIG.5.The spectral function presents a three peak structure intruncation B, with one peak at the origin, and two quasi-pole where the imaginary part is small. This is the typicalstructure at T ∼ m b and has been discussed in-depth inRef.[12]. The Landau damping, which causes peaks inimaginary part, is essential in the three peak structurein the spectral function. In truncation A, the spectralfunction also presents a peak at small ω but not at atthe exact origin, namely, instead of a zero mode, we havea soft-mode in truncation A, which also arises from theLandau damping effect. ��� ��� � ��� �������� ρ + ( � � � - � ) �� � = ������ - ������� � � Σ + ( � � � ) - ��� - ��� � ��� ��� - ���� - ���� - ���� - ��� - ��� - ��� - ���� ω ( ��� ) � � Σ + ( � � � ) FIG. 5. Quark spectral function at T = 170MeV, from top tobottom are spectral function, real and imaginary part of thequark self-energy. The black solid line stands for truncationA, while red dashed line for truncation B. The blue dottedline represent ω − m f , with m f = 130MeV. Finally, we analyze the spectral function in both trun-cations at T = 300MeV, where the system has reachedthe chiral restored phase, with m f = 24MeV, and de-generate meson mass m σ ≈ m π = 490MeV. For trun-cation A, the threshold of each channel is presented inthe last figure in FIG.3. There is a large area wherechannel (II) (III) are forbidden, leading to ImΣ + = 0 atsmall energy. Channel (II) (III) both have points where ± E (cid:48) φ,k ∗ ∓ E (cid:48) ψ,k ∗ = 0, where the imaginary part diverges.The real part of the inverse propagator has 7 quasi-poles,at two of which, the imaginary parts are too large to givea peak at ω = 10 , ± ω , the Landaudamping effect gives two bumps in the spectral function.While in truncation B, the Landau damping effect againgives two peaks in the imaginary part of the self-energy,and an oscillation in real part. ω − m f − ReΣ + ( ω ) = 0has five quasi-poles, ImΣ + has relative large values atfour of the quasi-poles, leading to two peaks with finitewidth, and a delta-peak at the origin. ��� ��� ��� � ��� ��� ������� ρ + ( � � � - � ) �� � = ������ - ���� - ����������� � � Σ + ( � � � ) - ��� - ��� - ��� � ��� ��� ��� - ���� - ��� - ��� - ��� - ���� ω ( ��� ) � � Σ + ( � � � ) FIG. 6. Quark spectral function at T = 300MeV, from top tobottom are spectral function, real and imaginary part of thequark self-energy. The black solid line stands for truncationA, while red dashed line for truncation B. The blue dottedline represent ω − m f , with m f = 24MeV. Two scales are of crucial importance in the structureof the fermion spectral function, m f /m b and T /m b . Thethree peak structure is most obvious when m f /m b (cid:28) T /m b ∼
1. When these two factors are approached,the peaks become higher and sharper. For the chiralcrossover,
T /m b ∼ T /m b ≈ . m f /m b (cid:28) ω = 0. In the case of finite fermionmass, one always has ImΣ + ( ω = 0) = 0, ReΣ + ( ω ) canbe exactly calculated to give a quasi-pole very close tothe origin, thus a peak will appear at the origin. In trun-cation A, the multi-peak structure is also observed as T /m b approaches unit. However, the zero-mode becomesa soft-mode, where the quasi-pole is slightly away fromthe origin. The disappearance of this zero-mode resultsfrom the limited scale of momentum in the propagatorand the scale dependence of the meson masses. One mayexpect that with larger Λ, the soft-mode would be closerto the origin. IV. SUMMARY
We investigate the spectral function of quark in twotruncations in a quark-meson model with functionalrenormalization group. In both truncations, we first solvethe flow equation of the effective potential and find outthe temperature and scale dependence of fermion and me-son masses. In truncation A, we take the scale-dependentmasses in step one as the input, and evolve the flow equa-tion of the two-point function; in truncation B, we takethe masses in the infrared as input and calculate the one-loop self-energy. After the analytical continuation, wehave the spectral function.When one consistently integrates the flow equation ofthe two-point function, the RG-scale dependence of theenergy thresholds of decay and creation channels leads tovan Hove singularities at finite temperature, where Lan-dau damping plays an important role. This singularityleads to divergence in both the real and imaginary part.Another feature is that, at high temperature, the Lan-dau damping is forbidden at low energy, leading to zeroimaginary part and several peaks in the low energy area.In comparison, when directly calculating the one-loopself-energy, one gets a three-peak structure when tem-perature rises and becomes comparable to meson mass.The spectral function has a peak at the origin, namely afermion zero-mode, and the other peaks comes from theLandau damping effect.Our work presents a first calculation of finite tempera-ture quark spectral function, and supports quasi-particlepicture of quarks around the critical temperature.
Note Added:
During the writing of this manuscript,we became aware of the work by R. A. Tripolt et al. [28],where the fermion spectral function at zero temperaturewas investigated.
Acknowledgement:
The work is supported by theNational Natural Science Foundation of China (GrantNos. 11335005, 11575093, and 11775123), MOST (GrantNos. 2013CB922000 and 2014CB845400), and TsinghuaUniversity Initiative Scientific Research Program.
Appendix A: Loop functions
With the boson and fermion occupation numbers and their derivatives, n B ( x ) = 1 e x/T − , n F ( x ) = 1 e x/T + 1 ,n (cid:48) B ( x ) = dn B ( x ) dx , n (cid:48) F ( x ) = dn F ( x ) dx , (A1)the loop functions J φ and J ψ in the flow equation for effective potential are explicitly expressed as J φ = k π n B ( E φ )2 E φ ,J ψ = k π − n F ( E ψ − µ ) − n F ( E ψ + µ ) E ψ . (A2)The threshold function J Sψσ , J
Sσψ , J
P Sψπ , J
P Sπψ in the two-point function can be obtained taking derivatives with respectto the corresponding energy, J Sψσ ( ip ) = − E ψ ∂∂E ψ J S ( ip ) , J Sσψ ( ip ) = − E σ ∂∂E σ J S ( ip ) ,J P Sψπ ( ip ) = − E ψ ∂∂E ψ J P S ( ip ) , J P Sπψ ( ip ) = − E π ∂∂E π J P S ( ip ) . (A3)After the analytical continuation, J S in Minkovski space is J S ( ω ) = − k π E φ (cid:26) ω + µ + E φ + E ψ + iη (cid:18) − m f E ψ (cid:19) (1 + n B ( E φ ) − n F ( E ψ + µ ))+ 1 ω + µ + E φ − E ψ + iη (cid:18) m f E ψ (cid:19) ( n B ( E φ ) + n F ( E ψ − µ ))+ 1 ω + µ − E φ + E ψ + iη (cid:18) − m f E ψ (cid:19) ( n B ( E φ ) + n F ( E ψ + µ ))+ 1 ω + µ − E φ − E ψ + iη (cid:18) m f E ψ (cid:19) (1 + n B ( E φ ) − n F ( E ψ − µ )) (cid:27) , (A4)Making substitution by m f → − m f and E σ → E π then we can get threshold for pseudoscalar channel J P S ( ω ). Thereal part is given by the principle value, while the imaginary part of the threshold function is then,Im J S ( ω ) = k π E φ (cid:110) δ ( ω + µ + E φ + E ψ ) (cid:18) − m f E ψ (cid:19) (cid:16) n B ( E φ ) − n F ( E ψ + µ ) (cid:17) + δ ( ω + µ + E φ − E ψ ) (cid:18) m f E ψ (cid:19) (cid:16) n B ( E φ ) + n F ( E ψ − µ ) (cid:17) + δ ( ω + µ − E φ + E ψ ) (cid:18) − m f E ψ (cid:19) (cid:16) n B ( E φ ) + n F ( E ψ + µ ) (cid:17) + δ ( ω + µ − E φ − E ψ ) (cid:18) m f E ψ (cid:19) (cid:16) n B ( E φ ) − n F ( E ψ − µ ) (cid:17)(cid:111) . (A5)Following Eq.(A3), one have the imaginary part of the threshold functions Im J Sψφ , Im J Sφψ for the scalar channel.Im J Sψφ ( ω ) = − k π E φ E ψ (cid:26) δ (cid:48) ( ω + µ + E φ + E ψ ) (cid:18) − m f E ψ (cid:19) (cid:16) n B ( E φ ) − n F ( E ψ + µ ) (cid:17) − δ (cid:48) ( ω + µ + E φ − E ψ ) (cid:18) m f E ψ (cid:19) (cid:16) n B ( E φ ) + n F ( E ψ − µ ) (cid:17) + δ (cid:48) ( ω + µ − E φ + E ψ ) (cid:18) − m f E ψ (cid:19) (cid:16) n B ( E φ ) + n F ( E ψ + µ ) (cid:17) − δ (cid:48) ( ω + µ − E φ − E ψ ) (cid:18) m f E ψ (cid:19) (cid:16) n B ( E φ ) − n F ( E ψ − µ ) (cid:17)(cid:27) (A6) − k π m f E φ E ψ (cid:26) δ ( ω + µ + E φ + E ψ ) (cid:20) n B ( E φ ) − n F ( E ψ + µ ) − E ψ (cid:18) E ψ m f − (cid:19) n (cid:48) F ( E ψ + µ ) (cid:21) − δ ( ω + µ + E φ − E ψ ) (cid:20) n B ( E φ ) + n F ( E ψ − µ ) − E ψ (cid:18) E ψ m f + 1 (cid:19) n (cid:48) F ( E ψ − µ ) (cid:21) + δ ( ω + µ − E φ + E ψ ) (cid:20) n B ( E φ ) + n F ( E ψ + µ ) + E ψ (cid:18) E ψ m f − (cid:19) n (cid:48) F ( E ψ + µ ) (cid:21) − δ ( ω + µ − E φ − E ψ ) (cid:20) n B ( E φ ) − n F ( E ψ − µ ) + E ψ (cid:18) E ψ m f + 1 (cid:19) n (cid:48) F ( E ψ − µ ) (cid:21) (cid:27) Im J Sφψ ( ω ) = − k π E φ (cid:110) δ (cid:48) ( ω + µ + E φ + E ψ ) (cid:18) m f E ψ (cid:19) (cid:16) n B ( E φ ) − n F ( E ψ + µ ) (cid:17) + δ (cid:48) ( ω + µ + E φ − E ψ ) (cid:18) − m f E ψ (cid:19) (cid:16) n B ( E φ ) + n F ( E ψ − µ ) (cid:17) − δ (cid:48) ( ω + µ − E φ + E ψ ) (cid:18) m f E ψ (cid:19) (cid:16) n B ( E φ ) + n F ( E ψ + µ ) (cid:17) − δ (cid:48) ( ω + µ − E φ − E ψ ) (cid:18) − m f E ψ (cid:19) (cid:16) n B ( E φ ) − n F ( E ψ − µ ) (cid:17)(cid:111) (A7)+ k π E φ (cid:110) δ ( ω + µ + E φ + E ψ ) (cid:18) m f E ψ (cid:19) (cid:16) n B ( E φ ) − n F ( E ψ + µ ) − E φ n (cid:48) B ( E φ ) (cid:17) + δ ( ω + µ + E φ − E ψ ) (cid:18) − m f E ψ (cid:19) (cid:16) n B ( E φ ) + n F ( E ψ − µ ) − E φ n (cid:48) B ( E φ ) (cid:17) + δ ( ω + µ − E φ + E ψ ) (cid:18) m f E ψ (cid:19) (cid:16) n B ( E φ ) + n F ( E ψ + µ ) − E φ n (cid:48) B ( E φ ) (cid:17) + δ ( ω + µ − E φ − E ψ ) (cid:18) − m f E ψ (cid:19) (cid:16) n B ( E φ ) − n F ( E ψ − µ ) − E φ n (cid:48) B ( E φ ) (cid:17)(cid:111) [1] J. 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