Light baryons below and above the deconfinement transition: medium effects and parity doubling

Journal of High Energy Physics, Jun 2017

We study what happens to the N , Δ and Ω baryons in the hadronic gas and the quark-gluon plasma, with particular interest in parity doubling and its emergence as the plasma is heated. This is done using simulations of lattice QCD, employing the FASTSUM anisotropic N f = 2 + 1 ensembles, with four temperatures below and four above the deconfinement transition temperature. Below T c we find that the positive-parity groundstate masses are largely temperature independent, whereas the negative-parity ones are reduced considerably as the temperature increases. This may be of interest for heavy-ion phenomenology. Close to the transition, the masses are nearly degenerate, in line with the expectation from chiral symmetry restoration. Above T c we find a clear signal of parity doubling in all three channels, with the effect of the heavier s quark visible.

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Light baryons below and above the deconfinement transition: medium effects and parity doubling

Received: March Light baryons below and above the decon nement transition: medium e ects and parity doubling Gert Aarts 1 4 Chris Allton 1 4 Davide De Boni 1 4 Simon Hands 1 4 Benjamin Jager 1 2 4 Swansea SA 1 PP 1 U.K. 1 CH- 1 Zurich 1 Switzerland 1 Open Access 1 c The Authors. 1 Plasma, Thermal Field Theory 0 School of Mathematics, Trinity College Dublin 1 Maynooth , County Kildare , Ireland 2 Institute for Theoretical Physics, ETH Zurich 3 Department of Theoretical Physics, National University of Ireland Maynooth 4 Department of Physics, College of Science, Swansea University We study what happens to the N , and the quark-gluon plasma, with particular interest in parity doubling and its emergence as the plasma is heated. This is done using simulations of lattice QCD, employing the FASTSUM anisotropic Nf = 2 + 1 ensembles, with four temperatures below and four above the decon nement transition temperature. Below Tc we nd that the positive-parity groundstate masses are largely temperature independent, whereas the negative-parity ones are reduced considerably as the temperature increases. This may be of interest for heavyion phenomenology. Close to the transition, the masses are nearly degenerate, in line with the expectation from chiral symmetry restoration. Above Tc we nd a clear signal of parity doubling in all three channels, with the e ect of the heavier s quark visible. medium; e; Lattice Quantum Field Theory; Phase Diagram of QCD; Quark-Gluon 1 Introduction Baryonic correlators and spectral functions Baryonic operators Spectral relations Charge conjugation Chiral symmetry and parity doubling Nonzero chemical potential Lattice setup Thermal baryon correlators Hadronic gas Quark-gluon plasma Thermal baryon spectral functions Default model and operator dependence In the past decade the study of the quark-gluon plasma at the Large Hadron Collider at CERN and the Relativistic Heavy Ion Collider at BNL has matured into a quantitative area of research, in which more detailed questions can be asked and answered { see e.g. refs. [1, 2] and references therein. One topic of interest concerns the changes to the spectrum of QCD, which are expected as hadrons are immersed in a hadronic gas at temperatures below the decon nement transition, and in the quark-gluon plasma (QGP) at higher temperatures. This has been especially important for quarkonium, bound states of a heavy quark and anti-quark, as their melting/survival pattern can act as a thermometer for the temperatures reached in these collisions. Indeed, both the LHC [3, 4] and RHIC [5] have reported clear suppression patterns for bottomonium states at high temperature. Ref. [6] contains a recent comprehensive review and ref. [7] a discussion of open questions. For light hadrons on the other hand, the emphasis has been on the statistical properties of the hadrons emerging from the system and on the dilepton spectrum [1]. Dileptons are predominantly produced by the decay of vector mesons and hence properties of their spectrum provide a connection with chiral symmetry and its restoration at high temperature. This observation has led to substantial activity on the role of chiral symmetry at nite temperature in the mesonic sector [8].1 Due to the nature of the thermal transition in QCD, studies using lattice QCD can provide important nonperturbative insight. Probably the cleanest signal with respect to chiral symmetry comes from the analysis of mesonic screening masses, which are relatively easy to compute in lattice simulations, see e.g. ref. [9], even though their relation to phenomenologically relevant quantities is not immediately clear (see however ref. [10]). Direct computation of spectral quantities in a medium, such as thermal masses, is considerably harder, due to the need to consider analytical continuation on lattices with a poral extent. Recent interesting work on the pion in the hadronic gas can be found in refs. [11, 12]. The vector meson correlator has been analysed extensively, not only due its role in the dilepton rate but also in the context of the electrical conductivity and the charge di usion coe cient [13{18]. Concerning quarkonia, both charmonium [19{24] and, more recently, bottomonium [25{28] have been studied on the lattice. Surprisingly, even though light baryons are sensitive to chiral symmetry and play an important role in the analysis of heavy-ion data, corresponding studies in the baryonic sector are very limited. In the context of lattice QCD, baryon screening masses in a gluonic medium were studied a long time ago in refs. [29, 30] and, at small baryon chemical potential, in ref. [31]. More recently, screening and temporal correlators were analysed in ref. [32]. All these studies were carried out in the quenched approximation. In this work we aim to improve this situation substantially. We study the N (nucleon), atures below and four above the transition. This allows us to study the properties of these baryons and in particular in-medium modi cation in the hadronic gas. Chiral symmetry is closely linked to parity doubling and we analyse the emergence of parity doubling as the transition is approached. We nd a qualitative di erence in the response to the increasing temperature between positive- and negative-parity baryons, which may be of interest for heavy-ion phenomenology. We also contrast the behaviour in the channel with the N channel, to see the e ect of the heavier s quark. This paper is organised as follows. In section 2 we summarise the relations between baryon correlators and spectral functions, emphasising the di erences with the mesonic case. We discuss positivity of the spectral functions, the role of charge conjugation, and the connection between chiral symmetry and parity doubling, both at = 0 and Section 3 contains details of our lattice computation. The main results of our study are given in section 4: we analyse the euclidean correlators and draw conclusions for both the hadronic gas and the quark-gluon plasma. These results are supported by the spectral function analysis of section 5. The nal section summarises and contains an outlook. We note here that our previous work in the nucleon sector, with limited statistics, can be found 6= 0. in ref. [33] and preliminary results have appeared in refs. [34{36]. 1Note that throughout this paper chiral symmetry will refer to SU(2)A chiral symmetry, which is spontaneously broken in the vacuum (and explicitly by nonzero quark masses). We start with a brief discussion of baryonic operators and spectral relations for fermionic two-point functions. While for mesonic (bosonic) correlators the type of relations discussed below are very well known [37], for fermionic ones this is slightly less so. Moreover, it allows us to discuss how parity doubling manifests itself in correlators and spectral functions. Baryonic operators We consider two-point functions of fermionic operators, of the form 0 (x) = O (x) O (0) ; baryons are respectively [39, 40] ON (x) = abc ua (x) dbT (x)C 5uc(x) ; O ;i(x) = abc sa (x) sbT (x)C isc(x) ; where C corresponds to the charge conjugation matrix, satisfying CyC = 1; T = T = and hence 5T = C 5C 1 . We note here that as written eq. (2.3) describes the charged +(uud) channel. However, since QED interactions are not incorporated and the two light quarks are taken to be degenerate (isospin limit), the operator is also relevant for the 0(ddu) channel. The ++(uuu) and (ddd) states are in principle described by an operator of the form (2.4), with s ! u; d, but again in the degenerate limit one can show that Wick contractions coming from the latter are identical to the ones derived from Under parity, elementary quark elds transform as 1 = 4 (P x); P = diag ( 1; 1; 1; 1): It is straightforward to verify that this property is inherited by the baryonic operators, Hence one may introduce parity projectors and operators via 1 = 4O(P x): O (x) = P O(x); 1; : : : ; 4, and 5y = 5 = 1 2 3 4. 1 = O (P x): We refer to O as positive- and negative-parity operators. Similarly, under charge conjugation quark elds transform as (c) = C 1 T (c) = Again, this is inherited by the baryonic operators, and O(c) = C 1OT ; O(c) = Spectral relations We now derive some general spectral relations and properties of the two-point functions 0 (x). We work in spatial momentum space, 0 ( ; p) = denotes the euclidean time, 0 elds and operators satisfy anti-periodic boundary conditions in euclidean time. A Fourier (2n + 1) T , n 2 Z, which can be written as a spectral integral 0 (i!n; p) = 0 (!; p) is then given by twice the imaginary part of the retarded The spectral function Green function, or, in terms of the operators, by as always [37]. Transforming back to euclidean time yields the integral relation with the kernel, for 0 < < 1=T , K( ; !) = T X Fermi-Dirac distribution. We note that K( ; !) is neither even nor odd, but satis es 0 (!; p) = 2 Im G 0 (x) = 0 ( ; p) = ; !) = K( ; !): A decomposition of the kernel in terms of its even and odd parts yields K( ; !) = [Ke( ; !) + Ko( ; !)] ; Ke( ; !) = Ko( ; !) = = [1 nF (!)] e ! + nF (!)e! ; = [1 where ~ = in the zero-temperature limit (for positive !). These kernels should be contrasted with the kernel appearing in bosonic spectral relations, Kboson( ; !) = = [1 + nB(!)] e ! + nB(!)e! ; where nB(!) = 1=(e!=T 1) is the Bose-Einstein distribution. The di erent denominators, the problems associated with the singular behaviour of the bosonic kernel, Kboson( ; !) ! In order to resolve the Dirac indices, we use the decomposition (other tensor structures will not appear in our application) 0 (x) = 0 (x) = G (x) + 1 G (x) = Gm(x) = where the trace is over the Dirac indices, and similarly for Below we will specialise to zero spatial momentum, for which Gi and i vanish. It is convenient to combine the two remaining components with the help of positive- and negative-parity projectors (2.8) as (x) = tr P (x) = tr fO (x); O (0)g = 2 [ m(x) We will now prove a number of properties of (x) and 4;m(x). G ( ; p) = We start with positivity: we will show3 that not have a de nite sign, even when restricting to ! ? 0. rium, we can write Suppressing Dirac indices and using the KMS condition [37], valid in thermal equilib(p); 4(p) 0 for all !, while m(p) does (p) = G>(p) G<(p) K=MS 1 + e p0=T 1 + e p0=T d4x e ip x G>(x); where G7 are the usual Wightman functions [37], We rst consider 4(p) and take the trace with 4. This yields 4(p) = 1 + e p0=T O(x)Oy(0) : x0) = O(x)O(x0) ; x0) = O(x0)O(x) : plete sets of eigenstates jni of the translation operator K H with eigenvalues kn0). Recalling that the expectation value denotes the thermal average (here K0 is the Hamiltonian 4(p) = 1 + e p0=T e kn0=T + e km0=T X e kn0=T 1 tr jhnjO(0)jmij2 4 jhnjO (0)jmij2 (2 )4 (4)(p + kn where we have written the Dirac index explicitly again. It is easy to see that the terms added within the summation are nonnegative and hence we arrive at positivity: 4(p) for all p. Next we consider (p) and take the trace with P . We now encounter where we used 4 = P+ Proceeding as above then yields (p) = tr P O(x)O(0) = tr O (x)Oy (0); P ; P 2 = P ; P+P e kn0=T + e km0=T n O (0)jmi 2 (2 )4 (4)(p + kn i.e. we nd positivity of the spectral functions 0 for all p. Positivity of 4(p) also follows from 4(p) = [ +(p) 0; on the other hand, m(p) = [ +(p) + ! ? 0, already at leading order in perturbation theory [42, 43]. To contrast, we note that for bosonic operators the spectral decomposition takes the form as above, but with a minus sign between the two thermal factors [37]. In addition, if the operator satis es J y = is odd under ! ! since Oy 6= Charge conjugation J ,4 it follows that the corresponding spectral function !, and ! B(!; p) 0. This can be seen by swapping n $ m in the summation. For the fermionic operators we consider here, this argument does not apply, O. Hence in general fermionic spectral functions are neither even nor odd. Next we relate, in the case of vanishing baryon chemical potential (or baryon density), positive- and negative-parity correlators and spectral functions, i.e. we show that G ( ; p) = G (1=T ( !; p) = We follow ref. [46], where this is demonstrated at the level of the single-quark prop Here we consider baryonic (or fermionic in general) operators, transforming under charge conjugation as in eq. (2.11). We assume isotropy, i.e. invariance under p, throughout. The time-ordered correlation function is given by x0) = with the imaginary-time-ordered product A( )B( 0) 0)A( )B( 0) )B( 0)A( ): Here the minus (plus) sign applies to fermionic (bosonic) operators. conjugation. We hence nd, suppressing Dirac indices, At zero chemical potential, thermal expectation values are invariant under charge x0) = CO(x)O(x0)C iE = = C 1GT(x0 < 1=T . Using the cyclicity of thermal expectation values [37] then gives Applying this to eq. (2.37), we nd, in momentum space, x) = G( x) = G( ; p) = C 1GT(1=T , where is a Dirac matrix selecting the channel, since Jy = J [44, 45]. We can now take the trace with P , which yields G ( ; p) = tr P G( ; p) = tr P C 1GT(1=T ) G(1=T G (1=T ; p) = tr P G(1=T where we used that (CP C 1)T = P : We have now demonstrated the rst relation in eq. (2.34). The second relation immediately follows, when using the integral relations (2.27) as well as the property (2.18). Physically it re ects that positive-parity states propagate forward in euclidean time, when using G+, and backward in time when using G , and vice versa for negative-parity states. In terms of spectral functions, this relates the positive part of the spectrum of + with the negative part , and again vice versa. Explicitly, if the spectrum is dominated by single groundstates with masses m , this implies G ( ) = A e m + A e m (1=T ) Using the relation ( p) = (p), we can subsequently note that 4(p) = m(p) = (p)] = (p)] = [ +(p) + +( p)] ; and hence these are even, respectively odd under ! ! !. Their spectral relations hence involve the even and odd kernels Ke;o( ; !) respectively, see eq. ( 2.20). We remark that this only holds when there is no net density, i.e. when the density matrix is invariant under charge conjugation. Chiral symmetry and parity doubling The nal relations we derive are for the case of unbroken chiral symmetry. Here we work in the harmonious world of thermal eld theory in which chiral symmetry is simply expressed lattice QCD computations, to be discussed below. From the anti-commutation relation of the correlator with 5, it immediately fol Gm(x) = m(x) = 0; G+( ; p) = G ( ; p) = G+(1=T ; p) = 2G4( ; p); +(p) = (p) = +( p) = 2 4(p): These relations imply that the lattice correlators are symmetric around the centre of lattice information is contained in (p). We refer to this as parity doubling. We emphasise that any of these signatures are equivalent statements of parity doubling. An alternative proof for two massless avours goes as follows [38]. When chiral symmetry is unbroken, i.e. the quarks are massless and chiral symmetry is not broken spontaneously, the theory is unchanged when the following chiral rotation is performed on the quark elds, = (u; d)T and T3 = chiral transformations on the spinor elds: It is then easy to check that the N and operators, see eqs. (2.2), (2.3), transform as 3=2 acts in avour-space. Choosing = , we get distinct In both channels the correlator then transforms as ON = P ON ! i 5ON ; = P O G (x) = tr O (x)O (0) tr O (x)O (0) = which was to be shown. Nonzero chemical potential For completeness, we indicate here how the properties derived above are modi ed in presence of a nonzero baryon chemical potential , such that the Hamiltonian in the Boltzmann weight is changed from H ! H Q, with Q the baryon number. First we consider positivity. Following the same steps as in section 2.3, in which the KMS condition (2.31) is modi ed as p0 ! p , and using that the states jni are simultaneous eigenstates of H and Q (with eigenvalues qn), we arrive at 4(p) = qn)=T + e (km0 qm)=T tr jhnjO(0)jmij2 (2 )4 (4)(p + kn and similar for (p). Hence positivity holds, as before. At nonzero chemical potential, the density matrix is not invariant under charge conjugation, since baryon number changes sign. Therefore invariance is obtained by simultaneously changing , which yields the relations 1 X G ( ; p; ) = G (1=T ( !; p; ) = T =Tc G+( ; ) = A+( )e (m+ G ( ; ) = A ( )e (m )e (m + )(1=T ) )e (m++ )(1=T ) The lattice size is Ns3 N , with the temperature Nsrc. The sources were chosen = as=a = 3:5. G4;m are then no longer (anti)symmetric around = 1=2T , but satisfy instead G4(1=T Gm(1=T ; p; ) = G4( ; p; ; p; ) = Again explicitly, if the spectrum is dominated by single groundstates, eq. (2.42) is modim(x) = 0 still holds and G+( ; p; ) = G ( ; p; ) = G+(1=T ) = 2G4( ; p; ); +(p; ) = (p; ) = ) = 2 4(p; ): Lattice setup We have computed baryon correlators using the thermal ensembles of the FASTSUM collaboration [15, 16, 27]. These ensembles are generated with 2 + 1 avours of Wilson fermions on an anisotropic lattice, with a smaller temporal lattice spacing, a < as; the renormalised anisotropy is as=a anisotropic gauge action with tree-level mean- eld coe cients and a mean- eld-improved Wilson-clover fermion action with stout-smeared links and follows the Hadron Spectrum Collaboration [47]. Details of the action and parameter values can be found in refs. [16, 27]. The choice of masses for the degenerate u and d quarks yields a pion with a mass of tuned to its physical value. Con gurations and correlation functions have been generated using the CHROMA software package [40], via the SSE optimizations when possible [49]. We use a xed-scale approach, in which the temperature is varied by changing N , according to T = 1=(a N ). Table 1 gives an overview of the ensembles. Access to the \zero-temperature" con gurations (N the Hadron Spectrum Collaboration. An estimate for the pseudo-critical temperature, higher than in nature, due to the large pion mass. Note that there are four ensembles in the hadronic phase and four in the quark-gluon plasma. Concerning the baryonic correlators, Gaussian smearing [50] has been employed to increase the overlap with the groundstate. In order to have a positive spectral weight, we apply the smearing on both source and sink, i.e., 0 = where A is a normalisation factor and H is the spatial hopping part of the Dirac operator. The hopping term contains APE smeared links [51] using = 1:33 and one iteration. We plateau for the e ective mass of the groundstate at the lowest temperature. Smearing is applied only in the spatial directions, equally to all temperatures and ensembles. Thermal baryon correlators In this section we present the results for the baryon correlators at all temperatures. Based on the determination of the pseudo-critical temperature Tc via the renormalised Polyakov loop, the discussion is organised in terms of the hadronic gas (T < Tc) and the quark-gluon plasma (T > Tc). Since the transition is a crossover, it is not immediately obvious at which temperatures light and strange baryons cease to exist.5 However, below we will nd clear we consider. Hadronic gas We have computed the baryon two-point functions in the N , channels on the lattice from now on). The results are shown in gure 1, at all the eight temperatures available. The positive- and negative-parity channels are shown separately, i.e. the negative-parity channel is obtained using eq. (2.34), and the correlators are normalised to the rst Euclidean time point, =a = 1; N 1 respectively, such that G+( ) = G ( ) = G+(N a At low temperatures (open symbols), the correlators show exponential decay, indicating the presence of a well-de ned groundstate. As the temperature is increased, some temperature dependence on the positive-parity side is observed, but considerably more 10−2 10−2 10−2 positive parity positive parity positive parity in the N , sectors, on a logarithmic scale temperature dependence is visible on the negative-parity side. The correlators naturally bend upwards around the minima, which are, however, not in the centre of the lattice symbols), the correlators appear to drop slower than exponential, indicating the absence of a well-separated groundstate. exponentials, see eq. (2.42), To analyse this quantitatively, we have tted the correlators to a combination of simple G+( ) = A+e m+ + A e m (1=T ) the groundstate masses in both parity channels. While gure 1 shows the positiveand negative-parity channels separately, the t is carried out to the correlator G+( ) in one go. Around the minimum of the correlator, one might become susceptible to signalto-noise problems, but we found this to be relevant at the lowest temperature only. Here we excluded points around the minimum of the correlators from the analysis, based on the quality of the t and error analysis.6 In order to estimate the systematic uncertainties of the four t parameters, we have considered various Euclidean time intervals and, to suppress contributions from excited states, we have excluded very small times. We used the so-called Extended Frequentist Method [52, 53] to carry out the statistical analysis: this method considers all possible variations and weighs the nal results according to the obtained p-value, which measures how extreme an outcome is, see refs. [52, 53] for more details. In the con ned phase we found that it is possible to extract the mass parameters m , whereas above Tc the exponential ts are no longer adequate, as can be expected in the decon ned phase (see below). Table 2 lists the results for the masses m in all three channels, at the four temperatures below Tc. The results are shown in units of MeV, using the estimate for the temporal light quarks are heavier than in nature, the groundstate masses in the N and at the lowest temperature are larger as well. The splitting between the positive- and negative-parity groundstate masses, denoted with m, is of the right order, however. The strange quark mass is tuned to the physical value [48] and the result for the is consistent with the PDG value (within errors). Surprisingly, the particle has not been unambiguously identi ed in the PDG and there are three candidates. The value we analysis (continuum extrapolation and physical u and d quarks) is necessary to make a more stringent prediction. Our results for the spectrum at the lowest temperature are in agreement with those of the HadSpec collaboration for the positive-parity states [48]; for the negative-parity baryons the masses obtained in ref. [55] on a smaller spatial lattice (163 instead of 243) are somewhat lower, at the 2 level. As the temperature is increased, we nd that the groundstate mass in the positiveparity channels is largely una ected by temperature; the deviation from the results at the lowest temperature is always less than 5%. Very close to Tc, the values drops slightly T =Tc m+N [MeV] mN [MeV] m+ [MeV] m [MeV] m+ [MeV] m [MeV] mN [MeV] m [MeV] m [MeV] N− 2138(117) 1898(106) 1734(97) 1628(104) 1425(94) 0.155(35) 0.099(40) PDG (T = 0) 0.097(23) 0.050(23) -0.009(25) 0.147{0.175{0.192 in both parity sectors in the N , channels below Tc. Estimates for statistical and systematic uncertainties are included. The nal column shows the is de ned as m = m m+ and the dimensionless ratio N; ; as = (m m+)=(m of the N and (left, slightly shifted horizontally for clarity) and (right) baryon, below Tc. The masses are normalised by m+ at the lowest plotted normalised by m+ at the lowest temperature, in the channel under consideration. In the negative-parity channel we observe a stronger temperature dependence, which is remarkably similar in all three channels. Already at 0:75Tc, the masses have dropped noticeably (see again gure 2) and this trend continues towards Tc. Very close to Tc the parity channels are nearly degenerate. This is further quanti ed by the dimensionless ratio N− m+, but assuming the exponential decay of eq. (4.2), in the N (left) and (right) channels. also included in table 2. The smaller value of at all four temperatures is due to both m+ being larger and being smaller. Both of these e ects are presumably due to the s quark being heavier than the u and d quarks, which makes the contribution to the groundstate mass due to chiral symmetry breaking less important in the Quark-gluon plasma We now turn to the temperatures above the decon nement transition. To start, we have considered the same analysis as above, using exponential ts, assuming that the hypothesis of separated well-de ned groundstates still holds. The results are shown in gure 3, in the N (or reducing N from 32 to 28). The error on the would-be groundstate masses, obtained by combining systematic and statistical uncertainties, is substantially larger, which cannot be simply explained by the reduction in the number of time slices used in the ts. This, and other results presented below, lead us to conclude that bound states are absent at channel. Hence even though the transition is a crossover, we nd that the spectrum changes rather drastically between 0.95 and 1.09Tc. We hence focus on the signal for parity doubling, i.e. the emergent degeneracy in the positive- and negative parity channels. Following ref. [32], we study the ratio R( ) = G+(1=T G+( ) + G+(1=T which approaches 1 in the case that separated groundstates dominate, with m vanishes in the case of parity doubling. We have previously shown R( ) for all temperatures in the nucleon sector [35]. Here we present the outcome at two selected temperatures in gure 4 in the N , We note the clear qualitative and quantitative di erence: below Tc the ratio is signi cantly di erent from zero,7 while at the highest temperature it is much smaller. It should be emphasised that if chiral symmetry is exactly T/Tc = 0.76 T/Tc = 1.90 restored, complete degeneracy in the positive- and negative-parity channels is expected number of reasons. First of all we use Wilson fermions, which break chiral symmetry at short distances. We have found that smearing suppresses these contributions, yielding a better signal for parity doubling [35]. Moreover, the quarks are not massless, with the two light avours heavier than in nature. Hence this explicit symmetry breaking also a ects the signal. However, this is expected to become less important at higher temperature, of R( ) in gure 4 (right) between the N; channels and the channel at the highest temperature; this is most likely due to the larger s quark mass. In order to summarise the results for all temperatures, we show in gure 5 the summed ratio PN =2 1 R( n)= 2( n) ; −1 −1.25 −1.5 and N correlators, G ( )=GN ( ), for di erent ms=T ! 0. symmetric correlator and parity doubling. We observe clear crossover behaviour in all three channels. The location of this transition is consistent with Tc, which has been determined by an analysis of the renormalised Polyakov loop. Hence it is natural to associate the transition with the approximate restoration of chiral symmetry in the quark-gluon plasma and to interpret R as a quasi-order parameter. We also note that the e ect is less pronounced in channel, due to the larger s quark mass. It will therefore be interesting to study the e ect of strangeness on parity doubling. At the highest temperature available, R > 0 in channel; it is expected that the e ect of the quark mass will eventually disappear as The N and the baryon have the same quark content but di erent spin structure. In the con ned phase this results in the mass splittings listed in table 2. In the positive-parity channel the mass splitting is of the order of 300 MeV at all four temperatures, consistent with the PDG; in the negative-parity channel the mass di erence is larger than in the PDG, but so is the uncertainty. In the decon ned phase, however, the quarks are quasifree and the spin structure may become less important.8 To investigate this, we show in gure 6 the logarithm of the ratio of the and N correlators. All ratios are normalised exponentially below Tc, due to the (approximately constant) mass di erence between the N baryons (in both parity channels). Above Tc, however, we observe a attening of the ratio, approaching 1 at the highest temperature. We interpret this as an approximate degeneracy in the N and channels at very high temperature, which would be of interest to study further analytically. We also note the qualitative change in the ratio immediately 8We thank Thomas Cohen for raising this question. The information in the thermal correlators discussed above is also present in the corresponding spectral functions, via relation (2.27) G ( ) = K( ; !) = 1 + e !=T : As is well-known [56], a simple inversion of this type of relation, using numerically determined correlators, is not possible. Hence we use the Maximum Entropy Method MEM [56, 57], which extremises a combination of the standard likelihood ( 2) function, determined by the data, and an entropy function, S = encoding prior knowledge, via the default model m(!). The conditional probability to be extremised is of the form exp( 12 2 + a parameter balancing the relative importance of the data and the prior knowledge. Both m(!) and are further discussed below. In the past 15 years, this method, and related ones, have been used by a number of groups, mostly for mesonic correlators, i.e. charmonium, the dilepton rate and the electrical conductivity, see e.g. refs. [13{24]. Applications to bottomonium, in which some simpli cations occur, can be found in refs. [25{28]. Here we give the rst application Generic details of our implementation can be found in previous work [13, 16, 25, 27]. Here we brie y mention some di erences with the bosonic (mesonic) case. We are interested in the spectrum for both positive and negative !, since (!) = +( !). Hence the negative part of the spectrum of + informs us of , and vice versa. To bring the spectral relation (5.1) to a numerically tractable form, we employ a cuto !max < ! < !max, nite interval is discretised using analysis we used all the euclidean-time points, except for the time slices closest to the source and sink. At the lowest temperature, we have left out the points around the minimum of the correlators; this will be further discussed below. As default model, we use a featureless t to the correlation a similar normalisation for all temperatures. We come back to the choice of default model below as well. We now discuss the results. We have performed MEM on the normalised correlators G+( )=(a G+( We note that the normalisation only a ects the vertical scale but not the ! dependence. Figure 7 contains the spectral functions in the three channels below Tc (left) and above Tc (right). Spectral information for the positive-parity channel can be found at ! > 0, whereas ! < 0 refers to the negative-parity channel. Below Tc, the groundstate -16 -12 -8 -4 -16 -12 -8 -4 -16 -12 -8 -4 -16 -12 -8 -4 -16 -12 -8 -4 -16 -12 -8 -4 (bottom) channels. -16 -12 -8 -4 spectral functions at the lowest (left) and highest (right) temperatures. peaks on the positive-parity side are clearly visible and their positions agree with m+, discussed in the previous section. Excited states are suppressed, due to the choice of smearing parameters. Some broadening is observed as the temperature is increased, but given the data and resolution, it is not clear whether this is a physical e ect or due to the limitations of MEM. The negative-parity groundstates are visible as well, but are considerably less pronounced. The asymmetry between the positive- and negative-parity sides below Tc is, however, clearly visible. Above Tc, sharp groundstate peaks are no longer discernible. The broad peaks present above Tc are most likely a combination of physical spectral features for decon ned quarks, as seen at very high temperature in perturbation theory [42, 43], and lattice artefacts due nite Brillouin zone, similar to in the mesonic case. To make this statement more quantitative would require a repetition of the calculation on ner lattices, which is one of eq. (2.47). Hence the most important feature here is the emerging symmetry between the positive- and negative-parity sides as the temperature is increased. This is clearly visible for the N and channels, in which the position and height of the main features become comparable at positive and negative !. On the other hand, parity doubling is not yet complete in the channel, as the positive-parity side is still enhanced. Nevertheless, the di erence with spectral functions in the con ned phase is manifest. This is consistent with the analysis of the correlators above. We note that in these plots we have not shown error bands for clarity; these will be discussed below. The combined results in all three channels are shown in gure 8, at the lowest (left) and highest (right) temperature. The di erence between the spectral functions in the con ned and decon ned phase is clear. We also note that below Tc the negative-parity state is best visible in the nucleon sector. For clarity error bands are shown for one default model only. Default model and operator dependence We now discuss some systematic e ects in the construction of the spectral functions. We start with the default model dependence. The results above were obtained with a in each case m0 is determined by a t to the correlation function. The absolute value ensures positivity. In the continuum theory at leading order in weak coupling [42, 43], the spectral functions increase as j!j5 for large j!j T; mq, but this behaviour is modi ed nite lattice [42, 43]. Results are shown in gure 9. The error band indicates the variation with the parameter using Bryan's method [57] and is shown for one default model only, for clarity. We observe that even though the default models are widely di erent, the resulting spectral functions are consistent within the uncertainty. The second peaks in the con ned phase at both ! > 0 and ! < 0 are presumably a combination of excited states and lattice artefacts. Whether a structure is due to a nite lattice cuto or represents a physical feature can ultimately be tested by repeating the computation at smaller lattice spacings. One may also test the robustness with regard to the operators used, to which we The dependence on the operator and the amount of smearing requires some discussion. In previous studies in the mesonic sector, it has been common to use a xed local operator of the form , without smearing.9 Locality is well motivated when the problem under investigation is related to a symmetry, such as electromagnetism (electrical conductivity, charge di usion, dilepton production) and in refs. [15, 16] the conductivity and charge di usion coe cient were determined using the exactly conserved lattice vector current. For spectral questions at zero temperature, smearing and optimised operators aim to increase the overlap with the ground (or other) state, in such a way that the spectrum remains invariant, but spectral weight is redistributed. On the other hand, at nite tem9For charmonium, smearing has been employed in ref. [19]. T = 1.52Tc T = 0.76 Tc T = 0.76Tc -5 -4 -3 -2 -1 0 1 2 3 4 5 the correlators (above) and the corresponding spectral functions (below). For clarity, error bands are shown for operator 4 only. perature, where spectral functions are broadened and bound states eventually dissolve, spectral weight will potentially be nonzero at all energies. It is then less clear which features of the spectral function are invariant (and re ect the underlying physics) and which are e.g. operator dependent. Smearing was already discussed to some extent in ref. [33]. Here we study the role of di erent operators. We focus on the nucleon, with the interpolator chosen to be ON (x) = abcua (x) dbT (x)CYnuc(x) ; follows Chroma [40]). Note that in the main part of the paper we have used operator Y4. The operator dependence is shown in gure 10 for two temperatures. We observe that the correlators depend on the operator, as expected, since the overlap with ground- and excited states will di er. This manifests itself e.g. in the skewness of the correlator below Tc, while at high temperature approximate parity doubling is visible for all three operators. Below Tc, we can quantify the spectral properties more precisely by comparing the masses mN T =Tc m+N [MeV] mN [MeV] m+N [MeV] mN [MeV] 1628(104) from exponential ts, see table 3. We observe that the positive-parity mass m+N is stable and consistent within the error. The negative-parity mass mN is consistent for operator 4 and 5, while for operator 6 the error is twice as large. This can be explained by noting that in gure 10 (top, left) the correlator is most skewed for operator 6, which leads to the smallest temporal range available on the negative-parity side, which is then re ected in the larger uncertainty. we observe groundstate peaks on the positive-parity side for all three operators. position of the second peak at ! > 0 depends on the operator used; hence no physical relevance can be assigned to it. On the negative-parity side the overlap with the groundstate is less pronounced. In particular operator 6 seems to have especially poor overlap with lowenergy features on the negative-parity side. Just as above, this nding can be understood from the asymmetric shape of the correlator: the number of data points available for MEM is emerging, with the positive side still slightly enhanced, for all three operators. The fact that the overall area under the spectral curves appears di erent is related to the choice of normalisation. Yet the emerging symmetry, i.e. parity doubling, is present in all three cases, independent of the operator. At the lowest temperature, we left out the points around the minimum of the correlators, both in the mass ts and the spectral function analysis, to handle a (mild) signal-tonoise problem. The e ect of choosing various time ranges in the MEM analysis is shown gure 11 (left). For both ranges the groundstates are clearly distinguishable and in agreement, while di erences appear for the possible excited states, which is as expected. In the results presented above, smearing was used to single out the groundstate at low temperature and suppress contributions from highly excited states at all temperatures. As a nal result we show in gure 11 (right) the spectral function obtained at the lowest temperature in the nucleon channel, using local sources and sinks, i.e. without smearing. We observe a large contribution at higher energy, which is however not related to the lowenergy states discussed above. The groundstate in the positive-parity channel is in fact still visible, as indicated in the inset, albeit much suppressed. When taken at face value, the mass is larger than found above, which is presumably due to the di culty of extracting a signal from the local correlator. This gure therefore indicates the importance of smearing in this analysis, from a spectral function point of view. In conclusion, we nd that smearing and the choice of operator a ects the correlators and hence the associated spectral functions at all temperatures. This is expected. At zero ·10−2 -16 -12 -8 -4 -16 -12 -8 -4 = [6; 66] [ [90; 126] for range 2. Right: result obtained without smearing, using operator 4 as local sources and sinks. The inset shows a blow-up around the positive-parity groundstate. temperature, the masses of the groundstates are stable against these variations, as long as the groundstates are clearly identi able. At nonzero temperature, the information gleaned from spectral functions is at a more qualitative level. Nevertheless, the conclusions drawn from the correlators and spectral functions are in agreement. Conclusion We studied the fate of the N , baryons as the temperature is increased, using phase, we observed a strong temperature dependence of the groundstate masses for the negative-parity baryons, while the masses of the positive-parity baryons are stable up to the decon nement transition. The temperature dependence is such that the positive- and negative-parity groundstates become approximately degenerate close to this transition. Degeneracy, i.e. parity doubling, is expected to coincide with chiral symmetry restoration and hence the transition from the hadronic to the quark-gluon plasma, but the precise manner in which this occurs is not known a priori. It would therefore be interesting to compare and contrast our nonperturbative predictions with model approaches, such as those discussed in refs. [58{66], to reach further insight and understanding. In the decon ned phase, we found strong indications that the light baryons no longer exist. Here we study parity doubling directly from an analysis of the correlators, using the R parameter (4.5), relating the positive- and negative-parity channels. We nd a clear signal for the emergence of parity doubling, with the R parameter acting as a quasi-order parameter. In the case of the baryon, with the heavier s quark, we nd that parity doubling is not yet fully realised for the temperatures we considered. The e ect of the The conclusions from the correlator analysis are supported by the results obtained from the associated spectral functions. In the baryonic sector in vacuo, it is well understood that smearing and the use of optimised operators are essential to nd clear signals for the groundand other states. At nite temperature, with nonzero spectral weight at all energies, it is not immediately clear how to proceed with smearing and operator choice. In this paper we choose to optimise the smearing parameters and operators at zero temperature and keep xed as the temperature increases. With this prescription we found it is possible to obtain quantitative results from the correlator analysis and qualitative insight from the spectral functions, which are mutually consistent. It would be interesting to consider this question further and e.g. employ variational bases, widely used in vacuum, also at nite temperature, as suggested in ref. [67]. As an outlook, there are various directions in which this study can be taken further, in addition to those mentioned above. From the viewpoint of lattice QCD, an important role is played by chiral symmetry. Since the Wilson-clover quarks employed here break chiral symmetry at short distances (and the two light avours are still somewhat heavy), it would be interesting to repeat this calculation with manifestly chiral (domain wall/overlap) fermions. The signal for parity doubling should then be easily visible in the correlators, without the need to suppress short-distance contributions. A physical question is related to the role of strangeness, since a nite s quark mass breaks chiral symmetry explicitly. baryon, we indeed observed the e ect of the strange quark mass in the signal for parity doubling, but a more comprehensive study of strange baryons would enlighten this further. Finally, we observed strong in-medium e ects for the negative-parity baryons in the hadronic phase. It would hence be interesting to investigate whether and how this a ects heavy-ion phenomenology, e.g. in the context of the hadron resonance gas or the statistical hadronisation model [68]. Acknowledgments We thank Sinead Ryan, Tim Burns, Thomas Cohen, Claudia Ratti and Rene Bellwied for discussion. The work has been supported by STFC grant ST/L000369/1, SNF grant 200020-162515, ICHEC, the Royal Society, the Wolfson Foundation and the Leverhulme Trust, and has been performed in the framework of COST Action CA15213 THOR. We are grateful for the computing resources made available by HPC Wales. This work used the DiRAC Blue Gene Q Shared Peta op system at the University of Edinburgh, operated by the Edinburgh Parallel Computing Centre on behalf of the STFC DiRAC HPC Facility ( This equipment was funded by BIS National E-infrastructure capital grant ST/K000411/1, STFC capital grant ST/H008845/1, and STFC DiRAC Operations grants ST/K005804/1 and ST/K005790/1. DiRAC is part of the National E-Infrastructure. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [INSPIRE]. [arXiv:1404.2246] [INSPIRE]. 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Gert Aarts, Chris Allton, Davide De Boni, Simon Hands, Benjamin Jäger, Chrisanthi Praki, Jon-Ivar Skullerud. Light baryons below and above the deconfinement transition: medium effects and parity doubling, Journal of High Energy Physics, 2017, 1-30, DOI: 10.1007/JHEP06(2017)034