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Spectral function of the Bloch-Nordsieck model at finite temperature A. Jakovác∗ Institute of Physics, Eo¨tvös University, H-1117 Pa´zmány Péter sétány 1/A, Budapest, Hungary P. Mati† Institute of Physics, Budapest University of Technology and Economics, H-1111 Budafoki ut 8, Budapest, Hungary (Dated: January 10, 2013) InthispaperwedeterminetheexactfermionicspectralfunctionoftheBloch-Nordsieckmodelat finite temperature. Analytic results are presented for some special parameters, for other values we havenumericalresults. Thespectralfunctionisfiniteandnormalizableforanynonzerotemperature values. TherealtimedependenceoftheretardedGreen’sfunctionispower-likeforsmalltimesand exhibits exponential damping for large times. Treating the temperature as an infrared regulator, we can also givea safe interpretation of thezero temperature result. 3 1 0 2 n I. INTRODUCTION a J Thebehavioroftheultra-softregimeofmasslessfieldtheoriespresentsaseriouschallengewhich,ontheotherhand, 9 iscrucialforunderstandingofthemost,physicallyrelevanttheories. Thesoftnatureoftheexcitationsperturbatively leads to infrared (IR) divergences in various physical quantities such as the self-energy near the mass shell. To have ] h reliable results, one has to resum the most sensitive part of the IR physics. The identification of the sources of these -t divergences,theelaborationoftheappropriatemathematicaltoolsandfinallytherealisationoftheresummationitself p is a formidable task. Moreover,the details can depend on the environment,that is the resummationin deep inelastic e scatteringand atfinite temperature equilibriummay requiredifferent approaches. It is nota surprise,therefore,that h so many resummation methods exist, working at different circumstances. [ Inthissensitivefield,wherethephysicalreliabilityofaresummationmaycruciallydependonthecorrectidentifica- 1 tionoftherelevantsourcesoftheIRdivergences,itisquitevaluabletofindamodelwhichisphysicallymotivatedand v exactly solvable. This is the reason why so many two-dimensional conformal and integrable theories have relevance. 3 In four dimensions, however, exactly solvable models are much rarer. 0 Aphysicallywell-motivatedmodelisthe Bloch-Nordsieck(BN)model[1]. Initslonghistoryitbecameatext-book 8 1 material [2, 3]. Physically it corresponds to the deep IR limit of QED, where the photons have no energy even for a . fermion spin-flip. In particular it can be used to prove QED theorems in this energy regime [4]. The BN model can 1 be solved exactly, the photon contributions can be fully summed up. The fermion propagator has been calculated at 0 3 zero temperature both based on functional methods [2, 3], and with help of Dyson-Schwinger equations [5, 6], where 1 also a detailed renormalization analysis is possible. The spectral function of the model at zero temperature reads in : Feynman gauge as v Xi ̺(w,T =0) w−1−απ, where w =uµpµ m, α= e2. (1) r ∼ − 4π a Here u is a 4-vector parameter of the model, loosely identifiable with the four-velocity of the fermion. This function, however, has a singular behavior: it is not normalizable, therefore the sum rule ρ = 2π can be satisfied only with zero wave function renormalization factor. Moreover, the naive inverse Fourier transform of this function is tα/π describing growthof correlationin time. The correct physicalinterpretation oRf these results requires some IR ∼ regulator, which can be, for example, the temperature. At finite temperature the model is much less studied. In the seminal papers of Blaizotand Iancu [7, 8] the authors studied the large time behavior of the fermion propagator with the Hard Thermal Loop (HTL) improved photon propagator. Using this result, Weldon worked out a spectral function which is valid in the vicinity of the mass shell [9]. With a different approach, Fried et. al. studied the time dependence of the momentum loss of a hard incoming fermionic particle [10]. We have severalgoals in this paper. The main goal is to work out the complete spectral function of the BN model for all momenta, and see how the short time dynamics, resembling the T 0 limit, goes over to the long time → ∗ [email protected] † [email protected] 2 damping. Because of the relative simplicity of the model we can even give analytic solutions for certain parameters, while for other, analytically not reachable parameter values we used a well controlled numerical procedure. Another goal is to extend our Dyson-Schwinger formalism combined with Ward identities [6], which works excellently at zero temperature,tofinitetemperatures. Withthehelpofit,thecompleterenormalizationprocessremainsfullycontrolled. Our paper will be organized as follows. First, we define the Bloch-Nordsieck model in Section II. We review the Dyson-Schwinger equations and Ward identities in finite temperature real time formalism, and apply them to the Bloch-Nordsieck model. In Section III we solve these equations. At zero velocity (Subsection IIIC) we provide an analytic formula for the fermion propagator, supported by a numerical verification. At nonzero velocity (Subsection IIID) we solve them numerically. In Subsection IIIE we compare our results with previous works in the literature. In Section IV we give the conclusions of the paper. II. THE BLOCH NORDSIECK MODEL AT FINITE TEMPERATURE TheBloch-NordsieckmodelisthelowenergylimitofQED,wherewetakeintoaccountonlyasinglespinorientation. Its Lagrangianis related to the QED Lagrangianby changing the Dirac matrices γµ for a four-vector uµ: 1 = F Fµν +Ψ†(iu Dµ m)Ψ, iD =i∂ eA , F =∂ A ∂ A . (2) µν µ µ µ µ µν µ ν ν µ L −4 − − − Wecanchooseutobeafour-velocity,oritcanbeu=(1,v): the twoarerelatedbyasimplefieldandmassrescaling, since by replacing Ψ Ψ/√u0 and m mu0, we can reach the u0 = 1 scenario. The quantity v = u/u0 can be → → interpreted as the velocity of the fermion. We are interested in the finite temperature fermion propagator. To determine it, we use the real time formalism (for details, see [11]). Here the time variable runs over a contour containing forward and backward running sections (C andC ). ThepropagatorsaresubjecttoboundaryconditionswhichcanbeexpressedastheKMS(Kubo-Martin- 1 2 Schwinger) relations. The physical time can be expressed through the contour time t = (τ). This makes possible T to work with fields living on a definite branch of the contour, Ψ (t,x) = Ψ(τ ,x) where (τ ) = t, and τ C for a a a a a T ∈ a=1,2; and similarly for the gauge fields. The propagators are matrices in this notation: i (x)= T Ψ (x)Ψ†(0) and iG (x)= T A (x)A (0) , (3) Gab C a b µν,ab h C µa νb i D E where T denotes ordering with respect to the contour variable (contour time ordering). G corresponds to the C 11 Feynman propagator, and, since the C contour times are always larger than the C contour times, G = G> and 2 1 21 G = G< are the Wightman functions. The KMS relation for a bosonic/fermionic propagator reads G (t,x) = 12 12 G (t iβ,x) which has the following solution in Fourier space 21 ± − iG (k)= n (k )̺(k), iG (k)=(1 n )(k )̺(k), (4) 12 ± 0 21 ± 0 ± ± where 1 n (k )= and ̺(k)=iG (k) iG (k) (5) ± 0 eβk0 1 21 − 12 ∓ arethedistributionfunctions(Bose-Einstein(+)andFermi-Dirac(-)statistics),andthespectralfunction,respectively. Itis sometimesadvantageoustochangeto the R/AformalismwithfieldassignmentΨ =Ψ Ψ /2. Thenone has 1,2 r a ± G = 0 for both the fermion and the photon propagators. The relation between the 12 and the R/A propagators aa reads G +G 21 12 G = , G =G +G , ̺=iG iG . (6) rr 11 ra 12 ra ar 2 − TheG propagatoristheretarded,theG istheadvancedpropagator,G isusuallycalledtheKeldyshpropagator. ra ar rr At zero temperature the fermionic Feynman-propagatorreads: 1 (p)= . (7) G0 u pµ m+iε µ − It has a single pole which means that there is no antiparticles in the model. Consequently, all closed fermion loops are zero, thus there is no self-energy correctionto the photon propagatorat zero temperature. Physically this means that the energy is not enough to excite the antiparticles. In fact, if we interpret the u parameter as the four-velocity 3 ofthe fermion,the Bloch-Nordsieckmodeldescribesthatregimewherethe softphotonfields donothaveenergyeven for changing the velocity of the fermion (no fermion recoil). This leads to the interpretation that the fermion is a hard probe of the soft photon fields, and as such it is not part of the thermal medium [8]. So we will set = 0, 12 G therefore the closed fermion loops as well as the photon self energy remain zero even at finite temperature. Another, mathematicalreason,why we mustnotconsiderdynamicalfermions– whichcouldshowupin fermionloops –is that the spin-statistics theorem [13] forbids a one-component dynamical fermion field. This means that now the exact photon propagator reads in Feynman gauge 1 G (k)= g G (k), G = , ̺(k)=2πsgn(k )δ(k2), (8) ab,µν − µν ab ra k2 0 (cid:12)k0→k0+iε (cid:12) (cid:12) all other propagators can be expressed using identities (4) a(cid:12)nd (6). A. Dyson-Schwinger equations The operator equations of motion give relations of the different Green’s functions, formulated as the Dyson- Schwinger equations. These equations are local, and so they are valid in generic non-equilibrium situations, and, of course, in a thermal medium, too. The generating form of the Dyson-Schwinger equations for generic fields Φ reads [12] i n δS Φ (x )...Φ (x ) =i δ δ(y x ) Φ (x )...Φ (x )Φ (x )...Φ (x ) . (9) δΦ (y) a1 1 an n iak − k a1 1 ak−1 k−1 ak+1 k+1 an n (cid:28) i (cid:29) k=1 X (cid:10) (cid:11) Inrealtime formalismthe time variableisthe contourtime (usually itis the variableofthe pathintegral). We define the fermionic self energy in the usual way (x,y)= (0)(x,y)+ d4x′d4y′ (0)(x,x′)Σ(x′,y′) (y′,y), (10) G G G G ZC where the symbol means time integration over the contour. Then we find in the Bloch-Nordsieck model C R Σ(x,y)=iα(x )e2u d4wd4z (x,w)Gµν(x,z)Γ (z;w,y), (11) 0 µ ν G ZC where the tree level vertex is eu , the proper vertex is denoted by eΓ , and α(x ) is 1 if x C and 1 if x C . µ µ 0 0 1 0 2 ∈ − ∈ This factor appears because we expressed the functional derivative δS through the derivatives of the Lagrangian, δΦi(y) which, however,changes sign on C . 2 Wecanalsoexpressthisequationwiththetwo-componentnotationasitcanbeseenonFig.1. Intermsofanalytic FIG. 1. The Dyson-Schwingerequations in real time formalism formulas it reads: 2 Σ (x,y)=iα e2u d4wd4z (x,w)Gµν(x,z)Γ (z;w,y), (12) ab a µ Gac ad ν;dcb c,d=1Z X where α =( 1)a+1. In Fourier space it reads: a − 2 d4k Σ (p)=iα e2u (p k)Gµν(k)Γ (k;p k,p). (13) ab a µ (2π)4 Gac − ad ν;dcb − c,d=1Z X 4 B. The vertex function in the Bloch-Nordsieck model The second use of the Dyson-Schwinger equation is to have a form for the vertex function. From (9) we find for any gauge theories δS O(Ψ¯,Ψ) =0, (14) δAµ(x) (cid:28) (cid:29) where O is any local operator containing Ψ¯ and Ψ. This implies, in particular A (x)Ψ(y)Ψ¯(z) = d4x′G (x,x′) jν(x′)Ψ(y)Ψ¯(z) , (15) µ µν ZC (cid:10) (cid:11) (cid:10) (cid:11) where j is the conserved current. The vertex function shows up in the AΨΨ† correlator as µ A (x)Ψ(y)Ψ¯(z) = d4x′d4y′d4z′iG (x,x′)i (y,y′)( ie)Γν(x′,y′,z′)i (z′,z). (16) µ µν G − G ZC (cid:10) (cid:11) From here we find d4y′d4z′i (y,u)eΓµ(x;u,v)i (v,z)= jµ(x)Ψ(y)Ψ¯(z) . (17) G G ZC (cid:10) (cid:11) In the BN model the fermion propagator is a scalar, moreover j = eu Ψ†Ψ is proportional to u . Therefore the µ µ µ vertex function is proportional to uµ, too. This is written in the Fourier space as Γµ(k;p,q)=uµΓ(k;p,q)(2π)4δ(k+p q), (18) − where we also used the energy-momentum conservation. C. Ward identities The local equations expressing current conservation can be used in a similar manner. The generating form reads n ∂ jµ(x)Φ (x )...Φ (x ) = i δ δ(x x ) Φ (x )...Φ (x )∆Φ (y)Φ (x )...Φ (x ) , ∂xµ h a1 1 an n i − iak − k a1 1 ak−1 k−1 i ak+1 k+1 an n k=1 X (cid:10) (cid:11)(19) where ∆Φ is the transformation of the ith field generated by the conserved charge Q = d3xj0(t,x). This means, i in particular R ∂ jµ(x)Ψ(y)Ψ¯(z) =eδ(x z) (y,z) eδ(x y) (y,z). (20) ∂xµ − G − − G (cid:10) (cid:11) We can write the corresponding equation for the vertex function, using (17): ∂ d4ud4vi (y,u)Γµ(x;u,v)i (v,z)=δ(x z) (y z) δ(x y) (y z). (21) ∂xµ G G − G − − − G − ZC This form is easy to rewrite in the two-component formalism, taking into account that to satisfy the delta function the time arguments must be on the same contour. One finds in Fourier space k Γµ (k;p,q)= δ −1(q) δ −1(p) (2π)4δ(k+p q). (22) µ abc abGbc − acGbc − (cid:2) (cid:3) In the Bloch-Nordsieck model, because of the special property of the vertex function expressed in (18), the vertex function is completely determined by the fermion propagator: 1 Γ (k;p,q)= δ −1(q) δ −1(p) . (23) abc uk abGbc − acGbc (cid:12)p=q−k (cid:2) (cid:3)(cid:12) (cid:12) (cid:12) 5 III. SOLUTION OF THE DYSON-SCHWINGER EQUATIONS Since the vertex function in the Bloch-Nordsieck model can be expressed with the fermion propagator,the Dyson- Schwinger equations for the fermion propagator become closed. At zero temperature it can be shown [5, 6] that the solution of this equation yields the same result as the functional techniques, moreover, renormalization can be fully controlled here. In this Section we discuss the solution at finite temperature. We will use Feynman gauge, and denote the photon propagator as G = g G. Then this closed equation can µν µν − be written as 2 d4k 1 Σac(p)=−ie2U2αa (2π)4 ukGaa′(k)Gab′(p−k) δa′b′(G−1)b′c(p)−δa′c(G−1)b′c(p−k) = a′,b′=1Z (cid:20) (cid:21) X 2 d4k 1 d4k 1 =−ie2U2αa"a′=1(G−1)a′c(p)Z (2π)4 ukGaa′(k)Gaa′(p−k)−δacZ (2π)4 ukGaa(k)#, (24) X where U2 =u2 u2. In particular 0− 2 d4k 1 d4k G (k) Σ11(p)=−ie2U2"a′=1(G−1)a′1(p)Z (2π)4 ukG1a′(k)G1a′(p−k)−Z (2π)4 1u1k # X 2 d4k 1 Σ12(p)=−ie2U2 (G−1)a′2(p) (2π)4 ukG1a′(k)G1a′(p−k). (25) a′=1 Z X Instead of G and G it is more aesthetic to work with the retarded and advanced propagators (the relations 11 22 are given in (6)). Since in the R/A formalism G = 0, the retarded propagator satisfies a homogeneous self-energy aa relation G (p)=G(0)(p)+G(0)(p)Σ (p)G (p), (26) ra ra ra ar ra while the propagatorsin the 1,2 components mix. From the definitions we easily find Σ =Σ +Σ , = . (27) ar 11 12 11 12 ar G −G G Therefore we have, using (25) and (6) Σ (p)= (p) −1(p) ∆M, (28) ar J Gra − where d4k 1 (p)= ie2U2 (G (k) (p k) G (k) (p k)), J − (2π)4 uk 21 Gra − − ra G12 − Z d4k G (k) ∆M = ie2U2 11 . (29) − (2π)4 uk Z It is easy to see that ∆M = 0. The 11 photon propagator G is even for k k, which is true in general, but 11 → − now we can prove by inspecting the free propagator which is exact in our case i iG (k)= +(n(k )+Θ( k ))2πsgn(k )δ(k2). (30) 11 k2+iε 0 − 0 0 For the first term the k k symmetry is evident, in the second we should use the identity n(k )+n( k )+1=0. 0 0 →− − Thereforewiththechangek k oftheintegrationvariable,theG propagatorremainsthesamewhileuk changes 11 →− sign, so ∆M changes sign, too. As a consequence ∆M =0. The Bloch-Nordsieck model, as all 4D interacting quantum field theories, contains divergences. To obtain finite result,weneedwavefunction,massandcouplingconstantrenormalization. Sincetheaboveexpressionshavecontained the originalparametersofthe Lagrangian,weshouldrewritethemintermsofthe renormalizedquantities. Fromnow ontheparametersmandewilldenotetherenormalizedones,whilem ande arethebarequantities. Renormalization 0 0 goes like in the zero temperature case [6]: assuming that the renormalized mass m = Zm where Z is the fermion 0 wavefunctionrenormalizationconstant(thisis ensuredbytheWardidentities)wecanwrite −1 =Z(up m) Σ , Gra − − ar and from (28) we find ζ(p) 1+ (p) (p)= , where ζ(p)= J . (31) ra G up m Z − 6 A. Calculation of J In the expression of in eq. (29) there appears (k). As we discussed earlier, for the sake of physical and 12 J G mathematical consistency of the model, we must assume that the fermion describes a hard probe, itself is not a dynamicalfield,whichmeansthatwemustset (k)=0. Thenfrom(29)wecaneasilyrecoverthezerotemperature 12 G result [6]. At finite temperature we have d4k 1 (p)= ie2U2 G (k) (p k). (32) J − (2π)4 uk 21 Gra − Z Next we prove by recursion that the solution for depends solely on w = up m. It is true at tree level where ra −1 =up m. So let us assume that (p)= ¯ (uGp m). Then − Gra − Gra Gra − d4k 1 (p)= ie2U2 G (k)¯ (up m uk), (33) J − (2π)4 uk 21 Gra − − Z implying (p) = ¯(up m). Equation (31) tells us that if depends only on up m, then also depends only ra J J − J − G on up m. With this statement the recursion is closed. − Since in the BN model the free photon propagator is exact, we shall write it into eq. (32). Using (4) for the G 21 propagator,and applying the Landau prescription (w w+iε) we find → d4k 1 2π ¯(w)=e2U2 (1+n(k )) (δ(k k) δ(k +k)) ¯ (w uk). (34) J (2π)4 uk 0 2k 0− − 0 Gra − Z This result, as we shall show in Section IIIE, is consistent with the results of [7, 8]. The k integration can be performed, apart from the single component q = ku. We find after a straightforward calculation: ∞ α ¯(w)= − dqf(q,u) ¯ (w q), (35) ra J π G − Z −∞ where α=e2/(4π) and u0(1+v) u0(1 v2) ds q u0(1 v2) eβq/(u0(1−v)) 1 f(q,u)= − (1+n( ))= − ln − , (36) 2v us2 s 2vqβ eβq/(u0(1+v)) 1 Z − u0(1−v) where u=u (1,v) and v = v (i.e. v is the velocity v=u/u ). 0 0 | | At zero temperature f(q)=Θ(q). At v =0 we find f(q,u=0)=1+n(q). (37) B. Renormalization In (35) we find ultraviolet (UV) divergences. From the expression of f(q,u) (eq. (36)) we see that for large momenta the thermal distribution functions always decrease exponentially, thus yielding UV finite result. So all the UV singularity is in the T =0 part, discussed already in [6]. To apply the renormalized treatment at finite temperature, we recall some results from [6]. At T = 0 (35) can be written in spectral representation and with dimensional regularizationas ∞ ∞ ∞ ∞ α α dw′ 1 α dw′ w′ w iε ¯(w)= − dq ¯ (w q)= ̺¯(w′) dq = ̺¯(w′) ln − − , (38) J0 π Gra − π 2π q+w′ w iε π 2π Dε− µ Z0 −Z∞ Z0 − − −Z∞ (cid:20) (cid:21) where 1 1 1 = + ln(4π)+ P (39) Dε 2ε 2 2 1/2 7 (P = 1.96351is the value of the polygamma function with 0,1/2 arguments). 1/2 − Aswediscussedin[6],the divergenttermisnecessaryforthecouplingconstantandwavefunctionrenormalization. We can write, assuming normalizability of ̺¯ 4π2 + + ¯ (w) 1+ ¯(w) e2 Dε Jfin ζ¯(w)= J = 0 , (40) Z 4π2Z e2 0 where ¯ (w) is finite. We introduce fin J 4π2 4π2 4π2Z 4π2z r + = , = , (41) e2 Dε e2 e2 e2 0 0 where z and e now are finite (renormalized) values. Using renormalization group invariance we can write for the r complete finite temperature contribution ∞ ∞ e¯2 dw′ Λ ζ¯(w)= ̺¯(w′)ln dq(f(q,u) Θ(q)) ¯ (w q) , (42) 4π2  2π w′ w iε − − Gra −  −Z∞ − − −Z∞   where e¯is a RG invariantcoupling, Λ=µexp(4π2) is the momentum scale of the Landau-pole. The derivative of the e2 first term reads ∞ ∞ dw′ Λ dw′ ̺¯(w′) I(w)= ̺¯(w′)ln , I′(w)= = ¯ (w). (43) 2π w′ w iε − 2π w w′+iε −Gra −Z∞ − − −Z∞ − The imaginary part of I(w) term is zero for w <0, moreover for w =0 it is negative (at least for large Λ), while for w it is positive. So there exists a value w= M for which it is zero. Then we can write: →−∞ − w I(w)= dq ¯ (q). (44) ra − G Z −M The scale M replaces the scale Λ. Assuming that M T we can change the integration limits to M M in the ≫ − → second part, too. Then we find e¯2 ζ¯(w)= dqf(q,u) ¯(w q), (45) −4π2 G − Z M wheretheintegralsymbolmeans = . Ifitdoesnotcauseproblem,wewillsendM . Thezerotemperature −M →∞ part is the same as in our earlier publication [6]. R R Summarizing, the renormalized equation reads now: α w¯(w)= dqf(q,u) ¯(w q), (46) G −π G − Z where f(q,u) is given by (36). Since this equation is linear, the same will be true for the spectral function (with different normalization conditions) α w̺¯(w)= dqf(q,u)̺¯(w q), (47) −π − Z C. Zero velocity case For v =0, u =1 we find for (47) 0 α w̺¯(w)= dq(1+n(q))̺¯(w q). (48) −π − Z 8 By sending the limits of the integration to infinity, we realize that the right hand side is a convolution. Therefore we change to Fourier space where it becomes a product, and the left hand side will be i∂ ̺¯(t). Using the Fourier t transform of 1+n(q) dw eβw iT e−iwt = − (49) 2π eβw 1 2tanh(πtT) Z − we obtain the differential equation iTα i∂ ̺¯(t)= ̺¯(t). (50) t tanh(πtT) This has the following solution: ̺¯(t)=̺¯ (sinhπtT)α/π. (51) 0 Before we proceed, we shall discuss this result. First we can easily recover the T = 0 result, since for t 1 the ≪ T sinh function can be approximated linearly, and we get ̺¯(t) tα/π. On the other hand this result is rather weird, it ∼ describes forever increasing correlation instead the physically sensible loss of correlation. Since this happens also at zero temperature, this is not an artifact of the finite temperature calculation. In accordance with Blaizot and Iancu [7,8], we shouldnotconsiderthis expressionas the physicalresponsefunction. Mathematically we canarguethat we are notin the physically sensible analytic domain,the time dependent spectralfunction is not square-integrablefor a realαvalue,asitshould. Wemustthereforegoovertothephysicalanalyticdomain,wheretheFourier-transformation is well defined. For the analytic continuation we think equation (51) valid as long as it yields sensible formulae, which is the case ofimaginary αvalues. Withthis assumptionthespectralfunctionintheFourierspacewillbeananalyticfunctionin α. For real α values the spectral function will be interpreted as an analytic continuation. We will see that it indeed provides sensible results. To perform the inverse Fourier transformation we apply Laplace transformation. With s = iw we find ± ± ∞ ∞ ∞ dteiwt̺¯(t)= dte−s−t̺¯(t)+ dte−s+t̺¯( t)=̺¯+(s−)+( 1)α/π̺¯+(s+), (52) − − Z Z Z −∞ 0 0 where α βs α ∞ Γ 1+ Γ ̺¯ (s)= dte−st(sinhπtT)α/π = π (cid:18)2π − 2π(cid:19) . (53) + 21(cid:16)+α/ππT(cid:17) βs α Z Γ 1+ + 0 2π 2π (cid:18) (cid:19) Since the Γ-function satisfies Γ(1 z)Γ(z)=π/sinπz, we can write with s= iw: − ± βs α Γ 2π − 2π π (cid:18) (cid:19) (cid:12) = − . (54) βs α (cid:12) α βw 2 α βw Γ 1+ + (cid:12) Γ 1+ +i sin i (cid:18) 2π 2π(cid:19)(cid:12)(cid:12)(cid:12)s=±iw (cid:12) (cid:18) 2π 2π(cid:19)(cid:12) (cid:18)2 ∓ 2 (cid:19) (cid:12) (cid:12) (cid:12) Then we get for the spectral function (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) N βsinαeβw/2 1 α ̺¯(w)= , (55) cosh(βw) cosα α βw 2 − Γ 1+ +i 2π 2π (cid:12) (cid:18) (cid:19)(cid:12) (cid:12) (cid:12) where N is a numerically determined normalization consta(cid:12)nt required by the su(cid:12)m rule α (cid:12) (cid:12) dw ̺¯(w)=1 (56) 2π Z to be satisfied. On Fig. 2 we can see the shape of the spectral function for different α values and for different temperatures. To discuss this result we make the following observations: 9 0.5 0.5 α=0.2 T=0.2 0.45 α=0.5 T=1.0 0.4 α=1.0 0.4 T=3.0 0.35 α=3.0 0.3 0.3 ρT 0.25 ρ 0.2 0.2 0.15 0.1 0.1 0.05 0 0 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 5 w/T w a.) b.) FIG. 2. a.) The exact, normalized spectral function at v =0. Common features are the dominantly exponential decrease for w→−∞,power-law decrease ∼w−1−α/π for w→∞ and at thepeak a finitecurvature∼α. b.) Temperaturedependenceof thespectral function at v=0. In thelimit T →0 it is singular at w=0 point. ̺¯(w) is a function of βw only, which is understandable, since there is no other scale in the system which could • form a dimensionless combination. For α 0, we find • → eβw/2sinα 2πδ(w), (57) cosh(βw) cosα → − so we recover the free case. It is interesting, that this behavior periodically returns for α=2πn. For large values of w which is equivalent to the small temperature case we can use the asymptotic form of the • Γ function for complex arguments with large absolute value: Γ(x)=e−xxx x−1/2+ (x−3/2) . (58) O (cid:16) (cid:17) Then we find, up to normalization factors eβw 1 ̺¯(βw ≫1)∼ cosh(βw) w1+απ T−→→0 Θ(w)w−1−απ. (59) This is the well-known exact solution by Bloch and Nordsieck at zero temperature. Note that the Θ function came out correctly from the formula. At finite but small temperatures, for negative arguments we observe exponential decrease. This form also shows how at zero temperature we obtain zero wave function renormalization factor. The normalizationfactor(c.f. (55))isproportionaltoβ,whilethe asymptoticformis(βw)−1−απ. Thenapproaching zero temperature we obtain Tαπw−1−απ, which means a renormalizationfactor vanishing as Tαπ for T 0. ∼ → Now let us consider the w 0 limit, i.e. the vicinity of the mass shell. We can expand ̺¯into power series • → 4̺¯(0)CT2 ̺¯(w)= , (60) 4 (w CT)2+ 1 C2T2+ (w3) − C − O (cid:18) (cid:19) where 1 1 2 1 α = + + Ψ2(1+ ), (61) C 2 1 cosα π2 2π − andΨ(a)is the digammafunction. The maximumofthis functionis atCT,the widthis CT√4C−1 1. Since, − however,the function is not symmetric, these parameters cannotbe interpretedas a thermalmass and thermal width. For that we need to examine the real time dependence. 10 For the realtime dependence we use the fact that, accordingto (55), ̺(p)=βf (β(p m)), which means that 0 0 • − ̺(t) = e−imtf˜(Tt). Omitting the oscillating phase (i.e. if we consider the envelope of ̺(t)), we recover the 0 Fourier transform of f . 0 The real time dependence obtained from the inverse Fourier transformation of (55) differs from (51). This is becauseweperformedananalyticcontinuationtothephysicallysensibleanalyticdomain. Thenumericalinverse Fourier transform of the normalized spectral function (and, because iG (t) = Θ(t)̺(t), for t > 0 this is also ra the real time dependence of the retarded Green’s function) can be seen on Fig. 3. 0.8 data 0.7 1-A(Tt)α/π 0.6 Be-αTt 0.5 ρ(t) 0.4 0.3 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Tt FIG.3. Timedependenceofthe(envelopeofthe)retardedGreen’sfunction(or,equivalently,thespectralfunction)forv=0at α=0.5onalogarithmicy-scale. Forsmalltimeswefind1−A(Tt)α/π,correspondingtothezerotemperaturetimedependence. For large times it turnsinto an exponential exp(−αTt) damping. Atsmalltimesweexpecttorecoverthezerotemperatureresult. Indeed,weobserve̺¯(t)=(1 A(Tt)α/π)e−imt − asymptoticform(forα=0.5thisisvaliduptoTt<0.4),thepowerlawtimedependenceischaracteristictothe zero temperature result. At t=0 the value of the spectral function is 1, this is because of normalization. Note however,that naively at zero temperature we would obtain tα/πe−imt time dependence, describing growthof ∼ correlationand violating the normalization condition. Interpreting the zero temperature result as T 0 limit, → we could cure this apparent inconsistency of the model. At strictly T = 0 we get back the physically sensible oscillating solution ̺(t)=e−imt. For large times (for tT >1) the time dependence is e−αTt, which agreeswith [7, 8]. Comparing it to (51) we ∼ see that instead of an exponential rise we found an exponential decay, but with the same coefficient. This can be understoodby noting that if we havea pole at w=w0 in the momentum space, meaning e−iw0t exponential timedependence,this poleispresentinthespectralfunctioninpositionw∗,too. The physicalretardedGreen’s 0 function can have poles in the lower half plane, therefore we find in our case only the w = iαT pole, giving 0 − exponential damping. For the justification of the analytic continuation we also used a different method. We expanded the t-dependent result (51) into power series using (sinhx)α/π = 1 απ ∞ Θ(x)( 1)k απ ex(απ−k)e−xk+Θ( x)( 1)απ−k απ e−x(απ−k)exk (62) 2 − k − − k (cid:18) (cid:19) k=0(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21) X Now the inverse Fourier transformation acts on a pure exponential function. We use the formula ∞ 1 dte±iwt−st = (63) s iw Z ∓ 0 which is true, of course, if s > 0, but this is the formula for the analytic continuation, too. Then the result of the Fourier transformation, with appropriate normalization to ensure reality of ̺: ∞ α ( 1)−α/2π ( 1)α/2π ̺(w) ( 1)k π − + − , (64) ∼ − k s +iw s iw k=0 (cid:18) (cid:19)(cid:20) k k− (cid:21) X