On the strength of the U^sub A^(1) anomaly at the

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Published for SISSA by Springer

Received: September 1, 2016 Revised: December 9, 2016

Accepted: December 20, 2016 Published: December 30, 2016

Bastian B. Brandt,a,b Anthony Francis,c Harvey B. Meyer,d Owe Philipsen,a Daniel Robainad and Hartmut Wittigd

aInstitut fur Theoretische Physik, Goethe-Universitat,

D-60438 Frankfurt am Main, Germany

bInstitut fur theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany

cDepartment of Physics & Astronomy, York University,

4700 Keele St, Toronto, ON M3J 1P3, Canada

dPRISMA Cluster of Excellence, Institut fur Kernphysik and Helmholtz Institut Mainz, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study the thermal transition of QCD with two degenerate light avours by lattice simulations using O(a)-improved Wilson quarks. Temperature scans are performed at a xed value of Nt = (aT )1 = 16, where a is the lattice spacing and T the temperature, at three xed zero-temperature pion masses between 200 MeV and 540 MeV. In this range we nd that the transition is consistent with a broad crossover. As a probe of the restoration of chiral symmetry, we study the static screening spectrum. We observe a degeneracy between the transverse isovector vector and axial-vector channels starting from the transition temperature. Particularly striking is the strong reduction of the splitting between isovector scalar and pseudoscalar screening masses around the chiral phase transition by at least a factor of three compared to its value at zero temperature. In fact, the splitting is consistent with zero within our uncertainties. This disfavours a chiral phase transition in the O(4) universality class.

Keywords: Global Symmetries, Lattice QCD, Phase Diagram of QCD, Spontaneous Symmetry Breaking

ArXiv ePrint: 1608.06882

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2016)158

Web End =10.1007/JHEP12(2016)158

On the strength of the UA(1) anomaly at the chiral phase transition in Nf = 2 QCD

JHEP12(2016)158

Contents

1 Introduction 1

2 Lattice simulations 62.1 Simulation and scan setup 62.2 Scale setting and lines of constant physics 72.3 Observables and renormalisation 72.3.1 Decon nement and the Polyakov loop 72.3.2 The chiral condensate 82.3.3 Mesonic correlation functions and screening masses 102.4 Investigating the order of the transition in the chiral limit 112.4.1 Critical scaling 112.4.2 UA(1) symmetry restoration 12

3 Results 123.1 Ensembles and measurement setup 123.2 The pseudocritical temperature 143.2.1 Polyakov loops 143.2.2 Chiral condensate and its susceptibility 143.2.3 Scaling in the approach to the chiral limit 163.3 Screening masses and chiral symmetry restoration pattern 193.4 On the relative size of the UA(1) breaking e ects around TC 24

4 Conclusions 27

A Simulation and analysis details 28A.1 Simulation algorithms and associated constraints 28A.2 Error analysis 29A.3 Interpolation of zero temperature quantities 30A.4 Estimating lines of constant physics 31A.5 Interpolation of the zero temperature chiral condensate 33

B Simulation parameters and results 35

1 Introduction

Nuclear matter under extreme conditions of high temperatures T and/or baryon chemical potential B is the subject of intense experimental and theoretical studies in nuclear, particle and astro-physics. One of the salient features of strongly interacting matter is the

{ 1 {

JHEP12(2016)158

high-temperature transition from the hadronic phase to the decon ned quark-gluon plasma (QGP). The transition takes place in a temperature regime between 100 and 300 MeV, where the QCD running coupling is strong. Thus, a non-perturbative investigation of the transition and the properties of the QGP is necessary and a lot of e ort has been invested by the lattice community in the past decades (for recent reviews see [1{5]).

Because lattice studies of the QCD transition at nite baryon chemical potentials are severely hampered by the sign problem, the QCD phase diagram remains largely unknown. Even at zero baryon density, the nature of the thermal transition with light quark masses approaching the chiral limit is not yet determined in the continuum. Knowledge of this important limit would also help to constrain the phase diagram at non-zero B. Figure 1 summarises the current knowledge about the order of the thermal transition for vanishing baryon density in the (mud; ms)-plane, where mud is the mass of the degenerate up and down quarks and ms the strange quark mass. In the opposite limits of pure gauge theory and QCD with three massless quarks, there are true rst-order phase transitions associated with the breaking of centre symmetry [6], and the restoration of the SU(3) chiral symmetry [7], respectively. These get weakened by the explicit breaking of those symmetries by nite fermion masses, until they disappear along second order critical lines. For intermediate quark masses, the nite temperature transition is then merely an analytic crossover.

There is plenty of evidence that the physical quark mass con guration realised in nature is in the crossover region. Early results based on the staggered fermion discretisation [8, 9] have been con rmed by domain wall fermions [10, 11] and simulations with Wilson fermions are approaching the physical point as well [12{15]. The critical line separating the rst order chiral transitions from the crossover region, the chiral critical line, has been mapped out on coarse lattices and is in the Z(2) universality class of the 3d Ising model [16, 17]. The critical line in the heavy quark region, the decon nement critical line, is in the same universality class and was mapped out on coarse lattices simulating a hopping expanded determinant [18] and a 3d e ective lattice theory [19]. However, the location of the critical lines in the quark mass plane is subject to severe cut-o e ects. With standard staggered fermions the Nf = 3 critical pion mass is at around two to three times the physical quark mass [16, 17], yet one nds that it shrinks to nearly half that value when going from Nt = 4 to Nt = 6 [20]. Improved staggered fermions can only give an upper bound for the critical mass which is around 0.1 times the physical quark mass [21, 22]. First results with Wilson fermions on the other hand see the Nf = 3 critical endpoint at around ve times the physical quark mass [23]. While the latter result will still change when going towards ner lattices, it highlights the importance of taking the continuum limit before discussing critical behaviour.

While all current results indicate that the critical line passes the physical point to the left, its detailed continuation is still largely unknown [2]. There are two possible scenarios [7, 24, 25]: in scenario (1), depicted in the left panel of gure 1, the chiral critical line reaches the mud = 0 axis at some tri-critical point ms = mtrics, implying a second order transition for Nf = 2. In the alternative scenario (2) the chiral critical line never reaches the mud = 0 axis, so that the chiral transition at mud = 0 is rst order for all values of the strange quark mass.

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Nf = 2

pure gauge

Nf = 2

pure gauge

mtrics

N f=1

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mud

scenario (1):

scenario (2):

1st crossover 1st2nd

Z(2)

Figure 1. The two possible scenarios for the quark mass dependence of the phase structure of QCD at zero chemical potential. The lower lines highlight the dependence of the phase structure in the Nf = 2 case of the u and d quark masses.

Many past studies have investigated the nature of the Nf = 2 transition using staggered [26{32], O(a) improved [33{35] or twisted mass [36] Wilson fermions and domain wall and overlap fermions [37{40], without being able to provide a conclusive answer in the chiral limit. The main problem in investigating scaling properties is the similarity of the critical exponents of the universality classes in question (cf. section 2.4.1). A new method to exploit the known tricritical scaling when coming from the plane of imaginary chemical potential has been proposed in [41]. First results on coarse lattices with staggered [42] and Wilson fermions [43] agree on scenario (2), but show enormous quantitative discrepancies. A similar approach proposed recently is to look at the extension of the chiral critical line in the plane with NF additional degenerate heavy quarks [44, 45].

Which of these scenarios is realised depends crucially on the strength of the anomalous breaking of the UA(1) symmetry at the critical temperature in the chiral limit when the mass of the strange quark is sent to in nity, i.e. in the Nf = 2 case. If the breaking of UA(1)

remains strong, the transition will be of second order in the O(4) universality class [7], so that scenario (1) will be realised. If, on the other hand, the symmetry is su ciently restored (see the discussion below about di erent restoration criteria), either scenario is possible [24, 25]. For scenario (1) the breaking would then likely be in the U(2) U(2) !

U(2) universality class [24, 25] (another symmetry breaking pattern/universality class of the form SU(2) SU(2) Z4 ! SU(2) has also been proposed [46, 47]) and the O(4)

universality class would be disfavoured. To be able to distinguish between the di erent

{ 3 {

Nf = 2

2nd

[braceleftBig]

O(4)? U(2)?

crossover 1st2nd

Z(2)

scenarios by looking at the UA(1) symmetry, it is thus crucial to have a measure for the strength of the breaking.

First, we recall that the UA(1) classical symmetry does not imply the existence of a conserved current: the divergence of the singlet axial current A0[notdef](x) is proportional to the gluonic operator G = [notdef] G[notdef] G in the chiral limit, an equality valid in any on-shell correlation function. For instance, the static correlator (@2=@x23)

dx0dx1dx2hA03(x)A03(0)i

is proportional to the corresponding static two-point function of G, which is certainly non-vanishing at any nite temperature. By contrast, the corresponding two-point function of the non-singlet axial current Aa3(x) vanishes in the chiral limit. In this particular sense, the UA(1) symmetry is never restored.

In this paper we use static correlators and the associated screening spectrum to probe the UA(1) e ects. A thermal state with an exactly restored UA(1) symmetry implies that the correlators h

(x)a (x) (0)a (0)i and h

(x) 5a (x) (0) 5a (0)i are equal in the

massless theory. However, we expect the restoration of UA(1) symmetry in this sense to only be partial, improving as the temperature is increased (see e.g. [48]).

In the literature, the restoration of UA(1) symmetry has also been discussed in a di erent, more restrictive sense. The observables considered are e.g. the correlation functions mentioned in the previous paragraph projected onto zero-momentum (

d4x). Sim

ilar to the Banks-Casher relation for the chiral condensate, the di erence of the zero-momentum correlators can be expressed in terms of the spectral density of the Dirac operator alone [49, 50]. For a density ( ) j j with > 1, the di erence vanishes

exactly in the chiral limit. In such a scenario, it is said that UA(1) symmetry is restored. More generally, if the SU(2) symmetry is restored at TC (which we will nd to be the case later), the e ective restoration of the UA(1) symmetry is indicated by the degeneracy of correlation functions belonging to a U(2) U(2) multiplet (e.g. [49]). Using the

restoration of the SU(2) symmetry, it was shown by Cohen [49, 50] that the degeneracy of zero-momentum correlation functions in the multiplets is directly linked to the eigenvalue density ( ) in the vicinity of zero. In particular, if the spectrum of the Dirac operator develops a gap around = 0 at TC, the UA(1) symmetry becomes e ectively restored.

Using QCD inequalities, it has been argued that the UA(1) symmetry is expected to be e ectively restored as soon as the SU(2) symmetry is intact [49]. Later it was noted that the result using the inequalities was incorrect since the contributions from sectors with non-zero topology had not been taken into account properly [51, 52]. However, the nonzero topology sectors only contribute away from the thermodynamic limit. Only a bit later it was shown that the eigenvalue density in the chiral limit behaves like ( ) j j

with > 1 [50]. Using mild assumptions and Ward-Takahashi identities for higher order susceptibilities in the framework of overlap fermions on the lattice, it has recently been shown that in fact > 2 [53], meaning that not only the eigenvalue density, but also its rst and second derivative vanish at the origin, speaking strongly in favour of a restoration of the UA(1) symmetry.

The relation between degeneracy of correlators and the behaviour of the low modes of the Dirac operator triggered a number of numerical studies of the eigenvalue spectrum in the vicinity of TC [10, 37{40, 54]. Some groups [37, 38] see a restoration of UA(1) at

TC in the chiral limit (in particular, in [37] the behaviour of ( ) j j3 was observed),

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while others claim that UA(1) will still be broken in the chiral limit [10, 54]. The di erence between these studies is (apart from lattice spacings and volumes) the fermion action in use, in particular, how well the actions preserve chiral symmetry. In fact [53], to be able to show that the breaking of UA(1) still a ects zero-momentum correlation functions via the low eigenvalues of the Dirac operator it is mandatory to ful ll the requirements of: (a) restored chiral symmetry on the lattice; (b) extrapolation to the in nite volume limit; (c) extrapolation to the chiral limit. In particular, condition (a) appears to play a crucial role. The reason is that any explicit breaking of the symmetry in the chiral limit by the lattice interferes with the e ective restoration. Furthermore, it is mandatory to use the same fermion action also for the sea quarks, as shown in [39, 40]. Here the authors looked at the small eigenvalues with domain-wall fermions and overlap fermions on the domain-wall ensembles and observed a broken UA(1) symmetry, actually made worse by the use of \quenched" overlap quarks. Only after reweighting of the con gurations to the overlap ensemble an actual restoration of UA(1) in the chiral limit was observed. A possible explanation for this e ect is that the \quenching" of the overlap operator leads to the appearance of non-physical near zero modes in the overlap spectrum, much like the appearance of exceptional con gurations in quenched QCD. Indeed, the cases where a residual breaking was observed were those where either or both, valence and sea quarks, might not have a fully restored chiral symmetry on the lattice. Similar conclusions have been found using chiral susceptibilities, which can also be related to the eigenvalue spectrum, e.g. [10, 40, 55]. However, when computed on the lattice, the susceptibilities su er from contact terms, which need to be carefully subtracted to obtain conclusive results.

In this article we present a study of the phase transition in two- avour QCD using nonperturbatively O(a) improved Wilson fermions [56] and the Wilson plaquette action [57]. We work with a large temporal lattice extent of Nt = 16 throughout, which at the chiral transition corresponds to a lattice spacing a 0:07 to 0.08 fm. Our pion masses range from

about 200 to 500 MeV. In particular, we study the pseudo-critical temperatures de ned by the change in the Polyakov loop and the chiral condensate, pertaining to decon nement and chiral symmetry restoration, respectively, and check for the associated scaling in the approach to the chiral limit. As already discussed in [58], such a scaling analysis is not su cient to distinguish between the universality classes in question. We thus direct our attention to the strength of the UA(1) breaking by investigating the degeneracy pattern of screening masses. This is complementary to other studies of the UA(1) symmetry in the literature described above, e.g. [10, 11, 38{40], which are based on the eigenvalue structure of the Dirac operator. The screening masses probe the long-distance properties the correlators and are free of contaminations from contact terms, unlike chiral susceptibilities. We propose a measure for the strength of the UA(1)-breaking in the vicinity of TC and extrapolate it to the chiral limit. There we nd it to be consistent with zero and 3 standard deviations away from its non-zero value at zero temperature. This suggests an e ective restoration of the UA(1) symmetry around the critical temperature and thus a strengthening of the chiral transition for the lattice spacing considered.

As discussed above, at nite lattice spacing an exact chiral symmetry is mandatory in order to study the eigenvalue spectrum of the Dirac operator reliably. In this study we use

{ 5 {

JHEP12(2016)158

O(a) improved Wilson fermions. While the action breaks chiral symmetry at nite lattice spacing, the static screening masses that we study approach their continuum limit with O(a2) corrections. Therefore, as long as we work in a regime where these corrections are small compared to the physical mass splittings induced by the UA(1) anomaly, we should obtain qualitatively correct conclusions. If, at a given lattice spacing, a UA(1)-breaking mass splitting turns out to be small, a continuum extrapolation is required to determine how small exactly the splitting is.

Parts of our results have already been presented at conferences [58{61] and were used to investigate the properties of the pion quasiparticle in the vicinity of the transition [62{65].

The article is organised as follows: in the next section we introduce our observables, the details of our simulations and discuss the renormalisation and scale-setting procedures. In section 3, we present the numerical results. We rst discuss the extraction of the pseudo-critical temperatures in section 3.2 and try to compare the results to the scaling predictions in the approach to the chiral limit. We also compare our results with those from di erent fermion discretisations in the literature. In section 3.3 we discuss the screening masses in the di erent channels, before we come to the investigation of the strength of the breaking in the chiral limit in section 3.4. Finally we present our conclusions in section 4. Detailed tables collecting simulations parameters and results can be found in the appendices.

2 Lattice simulations

2.1 Simulation and scan setup

Our simulations are performed using two avours of non-perturbatively O(a) improved Wilson fermions [56] and the unimproved Wilson plaquette action [57]. We use the clover coe cient determined non-perturbatively in ref. [66]. The simulations are done employing de ation accelerated versions of the Schwarz [67, 68] (DD) and mass [69] (MP) preconditioned algorithms, the latter in the implementation of ref. [70]. Both algorithms make use of the Schwarz preconditioned and de ation accelerated solver introduced in [71, 72]. As discussed in detail appendix A.1, the algorithms o er a signi cant speedup for large volumes and low quark masses, but also pose constraints on the available lattice sizes.

In general there are two known procedures to vary the temperature T = 1=(Nta). The

rst option is to vary the temporal extent whilst keeping the lattice spacing xed. The advantage of this procedure is that all physical parameters and renormalisation constants remain xed, making it the optimal tool for spectroscopy (see [73], for instance). The disadvantage of this procedure is that the resolution around TC is limited, made even worse by the use of improved algorithms (cf. appendix A.1). The second option, used in this study, is to vary the lattice spacing a by varying the coupling = 6=g20, known as -scans. This o ers the possibility to obtain a ne resolution around TC, but requires a good tuning of the bare quark mass to scan along lines of constant physics (LCPs), as well as the interpolation of quantities needed for scale setting and renormalisation. This is particularly demanding for Wilson fermions due to the additive quark mass renormalisation.

We will use throughout a comparatively large temporal extent of Nt = 16 for two reasons. First, Wilson fermions break chiral symmetry explicitly at nite lattice spacing.

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JHEP12(2016)158

Being as close as possible to the continuum helps to reduce the resulting e ects as much as possible. Second, for our choice of lattice action the non-perturbative determination of the improvement coe cient cSW in the two- avour theory extends only down to = 5:2 [74].

Since a 0:08 fm at = 5:2 this means that Nt = 16 is necessary to allow for scans in the

desired temperature range.

2.2 Scale setting and lines of constant physics

To convert our results to physical units we use the Sommer scale [75] r0 with the interpolation of the CLS results from [74] discussed in appendix A.3. To convert to physical units we use the continuum result r0 = 0:503(10) fm [74]. The temperature scans are done along LCPs, for which we estimate the values for the bare parameter corresponding to a particular quark mass by inverting the analytic relation mud( ; ) discussed in appendix A.4. We test the validity of this relation a posteriori by computing mud along the -scan. Conventionally, quark masses will be quoted in the MS scheme at a renormalisation scale of 2 GeV.

Whenever we quote pion masses for our temperature scans, we imply that these are zero temperature pion masses which correspond to the quark masses of the respective ensemble. We estimate the pion masses from our results for mud using chiral perturbation theory to next-to-next-to leading order as given in [76]. For this we use the low-energy constants from [77] obtained by the t denoted as NNLO F; m2 with a mass cut of 560 MeV. The associated low-energy constants are given in table 6 of [77]. This procedure serves the purpose of enabling comparisons with the literature and should not be taken as a precision computation of the zero temperature pion mass.

2.3 Observables and renormalisation

2.3.1 Decon nement and the Polyakov loop

To investigate the decon nement properties of the transition we look at the associated order parameter, the Polyakov loop

L = 1

VX[vector]x

JHEP12(2016)158

(

Yn0=1U0(n0 a; ~x)

); (2.1)

a nonzero value of which signals the spontaneous breaking of centre symmetry. Dynamical fermions explicitly break the centre symmetry of the gauge action and favour the centre sector of the Polyakov loop on the real axis, so that it is su cient to look at hRe(L)i. In [78]

it was found that the use of smeared links can enhance the signal in investigations of phase transitions. We have thus also computed the real part of the APE-smeared [79] Polyakov loop, hRe(L)Si, using 5 steps of APE smearing with a parameter of 0.5 multiplying the

staples. The Polyakov loop susceptibility is given by

~L = V

Re(L)2

hRe(L)i2

(2.2)

and similarly for the smeared Polyakov loop. In order to have a quantity with a well de ned continuum limit the Polyakov loop requires multiplicative renormalisation [80]. Here we

{ 7 {

will ignore this issue and work with the unrenormalised Polyakov loop. Since the renormalisation factor is expected to behave monotonically at the values corresponding to the critical region, we do not expect the typical S-shape of the Polyakov loop vs. temperature graph to be a ected.

2.3.2 The chiral condensate

A second aspect of the transition to the quark gluon plasma is the restoration of chiral symmetry. In the chiral limit, the associated order parameter is the chiral condensate

JHEP12(2016)158

i =

T V

@ ln(Z)

@mud : (2.3)

It governs the response of the system with respect to the external eld which breaks the symmetry explicitly, i.e., the quark mass mud. The bare chiral condensate is given by

U ; (2.4)

where D is the Dirac operator and the expectation value on the right-hand side is taken with respect to the gauge eld.

The associated susceptibility

~[angbracketleft]

ibare =

NfT

[angbracketright] =

T V

@2 ln(Z) @m2ud

(2.5)

consists of a disconnected (the terms in the curly brackets) and a connected part,

~bare[angbracketleft] [angbracketright] =

T Nf V

Tr D1

D1 2 U

D1D1

U : (2.6)

In the region around Tc the disconnected part has been found to dominate the transition signal in the susceptibility [9, 30, 81]. Close to the chiral limit, however, this statement does not necessarily hold. The connected part only receives contributions from isovector states, while the disconnected part receives contributions both from isovector and isoscalar states. Since an unbroken UA(1) symmetry would imply light isovector scalar states, the relevant magnitude of the two contributions is an important indicator of the nature of the transition. Here we will focus on the disconnected part of the susceptibility and leave the comparison between the connected and the disconnected susceptibility for future publications.

Due to mixings with operators of lower dimension, the condensate contains cubic, quadratic and linear divergences, and therefore requires additive renormalisation [82, 83]. In addition, it renormalises multiplicative with the renormalisation factor associated with the scalar density, ZS, which is equivalent to the inverse of the mass renormalisation factor for Wilson fermions, ZS = Z1m [84], where Zm is de ned in appendix A.4. Neither the additive nor the multiplicative renormalisation depends on the temperature, so that we can

{ 8 {

= ~bare[angbracketleft] [angbracketright]jdisc

T Nf 2V

cancel the divergent terms by subtracting the chiral condensate at T = 0. The associated

di erence renormalises multiplicatively,

iren(T ) = Z1m hh

ibare(T ) h

ibare(0)i

: (2.7)

For the determination of Zm we can use eqs. (A.5), (A.6) and (A.7) to obtain

Zm = ZAZP ZPCAC( )(1 + bm a

m) ; (2.8)

which is correct up to O(a2). ZPCAC( ) can be taken from the t in appendix A.4.

Using axial Ward identities (AWIs) one can also de ne an observable which is free of cubic divergences and reproduces the chiral condensate in the chiral limit [83, 84]. The axial Ward identity in the form integrated over spacetime (and up to a contact term at y = x) is given by

1 Nf

hh ibare(x) b0i= 2mPCACa4

Xy@[notdef]y hP (x)A[notdef](y)i ; (2.9)

where mPCAC is the bare PCAC quark mass and b0 represents the cubic divergence in the

bare condensate. At nite quark mass the second term on the r.h.s. vanishes due to the absence of a true Goldstone boson [84] and we can use the rst term as the de nition of a bare subtracted condensate

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hP (x)P (y)i ZAa4

ibaresub = 2NfmPCAC

P P

(2.10)

with

P P = a4

NtN3s

Xx,y

P (x)P (y) = TV Tr D1 5D1 5

: (2.11)

ibaresub still su ers from additive quadratic and linear divergences and renormalises mul

tiplicatively with ZP [84]. Again we can subtract the residual additive divergences using the T = 0 counterpart, so that we obtain an alternative renormalised chiral condensate

irensub(T ) = ZP hh

ibaresub(T ) h

ibaresub(0)i

: (2.12)

We de ne the susceptibility of the quantity (2.10) as

~bare[angbracketleft] [angbracketright]sub = 4N2fV m2PCAC hD

P P 2

E P

2 : (2.13)

~bare[angbracketleft] [angbracketright]sub is not equivalent to the disconnected chiral susceptibility in

eq. (2.6), it is expected to show a peak at the position of the chiral transition. The subtraction of the condensates at T = 0 requires the measurement of h

Note that, while

ibare and

P P

on zero temperature ensembles. Here we use the set of Nf = 2 ensembles generated within the CLS e ort and the interpolation discussed in appendix A.5.

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channel S P V A 1 5 i i 5

Table 1. Bilinear operators used for the screening correlators. Here i = 1; 2 for screening masses in x3-direction.

2.3.3 Mesonic correlation functions and screening masses

Mesonic correlation functions are a valuable probe of the properties of the QGP [85, 86]. Let

CXY (x[notdef]) =

d3x? hX(x[notdef]; x?)Y (0)i ; (2.14)

be the correlation function of two operators X and Y . The equality of two correlation functions in channels of di erent quantum numbers signals the restoration of the associated chiral symmetry. Here x[notdef] is the coordinate of the direction in which the correlation function is evaluated and x? is the coordinate vector in the orthogonal subspace. The isovector

correlation functions of interest for the chiral transition are the vector (V ) vs. axial-vector (A), and the pseudoscalar (P ) vs. scalar (S) channels, related by

CV V

SU(2)

! CAA and CPP

! CSS : (2.15)

We choose the isovector channels as observables because they are free of disconnected diagrams, and the correlation functions can therefore be obtained with greater accuracy. The bilinear operators for the di erent channels are listed in table 1.

While temporal correlation functions CXY (x0) can be related to the real-time spectral densities (see [87]), here we are interested in spatial correlation functions CXY (x3), which

are related to the screening states of the plasma. In particular, the leading exponential decay of the correlator CXX(x3) de nes the lowest-lying screening mass MX associated with the quantum numbers of the operator X. Screening masses can be interpreted as the inverse length scale over which a perturbation is screened by the plasma. If a symmetry imposes the equality of two correlation functions, it must also imply the degeneracy of the corresponding screening masses. The latter are thus quantities sensitive to the restoration of the symmetry. Consequently, the screening masses in the V and A channels provide an alternative way of de ning the chiral symmetry restoration temperature. In contrast to susceptibilities, de ned by the integrated correlation function, screening masses probe the long-distance properties of the correlation functions and thus do not su er from contact terms.

Apart from their relation to chiral symmetry, mesonic screening masses are valuable quantities to probe the medium e ects of the plasma and, at high temperatures, to test the applicability of perturbation theory. They have been studied in lattice QCD for a long time (for a review of early results see [88] and for more recent studies [89, 90]). In the high-temperature limit, all screening masses approach the asymptotic value M1 = 2T [91, 92].

The leading order correction from the interactions has been computed in perturbation theory and is known to be positive [93, 94]. Static and non-static screening masses can also be computed within an e ective theory approach and provide an indirect probe for real-time physics in the Euclidean lattice setup [95, 96].

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JHEP12(2016)158

UA(1)

UC Ref. Z(2) 0.6301( 4) 1.2372( 5) 0.3265( 3) 4.789( 2) [97] O(4) 0.7479(90) 1.477 (18) 0.3836(46) 4.851(22) [99] U(2) 0.76(10)(5) 1.4(2)(1) 0.42(6)(2) 4.4(3)(1) [25]

Table 2. Critical exponents for the universality classes (UC) relevant for the chiral transition. The critical exponents of the U(2) U(2)!U(2) universality class, we have taken the re

sults from [25] (from the MS scheme). The rst error is statistical while the second quoted error denotes a systematic uncertainty arising from the scheme dependence. The critical exponents of the U(2) U(2)!U(2) universality class have also been obtained recently using the bootstrap

method [100].

2.4 Investigating the order of the transition in the chiral limit

2.4.1 Critical scaling

The main question driving the present study is the nature of the phase transition in the chiral limit. Simulations with vanishing quark masses are currently impossible; in order to extract information on the order of the transition, it is customary to investigate the scaling of various observables in the approach to the critical point (0; 0) in the parameter space of reduced temperature = (T Tc)=Tc and external eld h. The scaling laws can

be derived from the scaling of the free energy F (see [97]). The variable playing the role of the external eld depends on the particular scenario (see section 1). If the second order scenario (scenario (1)) is realised, no matter whether the universality class is O(4) or the one from the U(2) U(2)!U(2) scenario,1 the critical point is located in the chiral limit

and the chiral condensate constitutes a true order parameter. In this case the external eld h is proportional to the (renormalised) quark mass mud. In the rst-order scenario, depicted in the right panel of gure 1, the critical point is located at a nite quark mass mcrud, so that chiral symmetry is broken explicitly at the critical point. One must in general expect that the external eld is given by a linear combination of m = mud mcrud and .

Furthermore, h

i no longer constitutes a true order parameter. The situation is analogous

to the approach of the chiral critical line along the Nf = 3 axis [16].

Here we will perform an analysis based on the scaling of the order parameter or the transition temperature TC with the external eld. In the vicinity of the critical point a true order parameter satis es the scaling relation (see e.g. [9, 33, 97, 98])

h1/ f

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h1/( )

+ s:v: : (2.16)

Here f is a function depending on the universality class of the transition and s.v. stands for scaling violations which constitute terms that are regular in [9, 97, 98]. A number of studies have looked at the scaling of the chiral condensate in the approach to the chiral limit [9, 30, 33, 36, 98] and found consistency with O(4) scaling.

1We will denote the U(2)[notdef]U(2)!U(2) scenario in short as the U(2) scenario from now on.

From the scaling in eq. (2.16) one can also derive a scaling law for the critical temperature as a function of the external eld [26]. The resulting scaling relation is

TC(h) = TC(0)

h1 + Ch1/( )i+ s:v: ; (2.17)

where C is an unknown constant. It must be stressed that these scaling laws are only valid after the thermodynamic and continuum limits have been taken.

Another problem for any study of the scaling laws is the similarity of the critical exponents in the three potentially relevant universality classes. They are summarised in table 2. For the di erent scenarios the combination is given by 1:56 for Z(2), 1:86

for O(4) and 1:85 for U(2). Even between the Z(2) and the O(4) scenarios the di erence

is so small that very accurate results are needed to be able to distinguish between the two. Thus one cannot draw conclusions from the agreement of lattice data with the scaling of one universality class alone; instead one needs to demonstrate the ability to distinguish between the scenarios.

2.4.2 UA(1) symmetry restorationThe strength of the anomalous breaking of the UA(1) symmetry at the transition temperature in the chiral limit is thought to be crucial for the order of the chiral transition [7, 101]. However, this raises the question of how to quantify the strength of the UA(1)-breaking.

As a possible reference value we suggest the screening mass gap between the isovector pseudoscalar and scalar channels at T = 0,

MPS = MP MS = ma0 ; (2.18)

since the pion mass vanishes in the chiral limit. Ultimately, one would like to obtain the chirally extrapolated value of ma0 from lattice QCD, since this would give a result valid for the Nf = 2 case in the range of relevant lattice spacings. Unfortunately, the scalar correlation function in the iso-vector channel is rather noisy, so that a reliable extraction is currently not possible. We will discuss a phenomenological estimate for the chiral limit in section 3.4.

We note that in the two- avour theory, the a0 meson is expected to be stable or almost stable, since the

KK and decay channels known from experiment are missing. Indeed, in Nf = 2 QCD only a avour-singlet pseudoscalar exists, sometimes called 2, whose nature is closer to the physical [prime] meson, and whose mass has been estimated at about 800MeV at the physical pion mass [102]. The lightest isovector scalar state was found to lie between the physical a0(980) and a0(1450) states [103]. The splitting between the pion and the lightest isovector scalar state thus provides a convenient measure for the breaking of UA(1).

3 Results

3.1 Ensembles and measurement setup

In this paper we present results for the chiral transition obtained from the scans on 16 323

(and rst results from 16 483) lattices summarised in table 3. More details can be

{ 12 {

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scan Lattice Algorithm /mud [MeV] m [MeV] T [MeV] UP [MDU] MDUs

B1 16 323 DD-HMC 0.136500 190 275 20 20000

C1 16 323 DD-HMC 17:5 300 150 250 28 12000

D1 16 323 MP-HMC 8:7 220 150 250 16 12000

Table 3. -scans at Nt = 16. Listed is the temperature range in MeV, the integrated autocorrelation time of the plaquette UP and the number of molecular dynamics units (MDUs) used for the analysis. For scan B1 con gurations have been saved each 200 MDUs, for scan C1 each 40 MDUs and for scan D1 each 20 MDUs. The measurements of the Polyakov loop and the chiral condensate have been done each 4 MDUs. The autocorrelation times and numbers of measurements quoted here correspond to the ones at the location of the transition.

Figure 2. Simulation points in the fmud; T g parameter space. The grey areas mark the estimates

for the crossover regions.

found in table 10 in appendix B. We consider three di erent values of the quark mass, corresponding to pion masses between 200 to 540 MeV. The scan corresponding to the largest pion mass at the critical point, denoted as scan B1 (in our naming convention the letter labels quark/pion masses while the number labels volumes), has been done at xed hopping parameter, indicated by the subscript . Due to renormalisation and the change in the scale, the quark mass changes with the temperature in this scan. The scans at lighter quark masses, scans C1 and D1 with m 300 and 220 MeV, are done along

LCPs. To check the tuning of the quark masses we have measured the renormalised PCAC quark mass using the PCAC relation evaluated in the x3 z-direction (see [104]). The

simulation points in the (T; mud) parameter space are shown in gure 2. The plot shows that the tuning of the quark mass works well in the region below TC, while we see that we get smaller quark masses than expected above TC. It is unclear to us whether this is a cuto e ect or if our interpolation just becomes worse in this region (cf. appendix A.4).

The quantities relevant for the transition temperature, i.e. the Polyakov loop and the chiral condensate, have been measured during the generation of the con gurations with a separation of 4 MDUs. For the measurement of the condensate we have used 4 hits with a Z2 Z2 volume source. The exception is the B1 scan, where the chiral condensate has

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m = 480 MeV

m = 300 MeV

m = 220 MeV

m ud[MeV]

150 170 190 210 230 250 270

T [MeV]

only been measured on the stored con gurations, using 100 hits. The screening masses have been measured on the stored con gurations. For scan B1 , con gurations have been saved every 200 MDUs, for scan C1 every 40 MDUs and for scan D1 every 20 MDUs. The results for the expectation values of Polyakov loops, the chiral condensates and screening masses are tabulated in tables 11 and 12 in appendix B.

3.2 The pseudocritical temperature

The rst step of our investigation of the thermodynamics of QCD is the extraction of the pseudocritical temperatures. Since we are dealing with a crossover rather than a true phase transition there is no unique de nition of the critical temperature and estimates for TC will depend on the de ning observable. To determine TC we will primarily look at the Polyakov loop and the chiral condensate. In particular, the in ection point of the Polyakov loop will de ne the decon nement transition temperature T dcC, while the peak in the susceptibility of the chiral condensate de nes the temperature of chiral symmetry restoration, which will be our main estimate for TC.

3.2.1 Polyakov loops

We start with the extraction of T dcC using the (unrenormalised) smeared Polyakov loop. We use hRe(L)Si since it typically shows stronger signals for the transition. We have, however,

checked the agreement with the results for the unsmeared Polyakov loop explicitly (see also [59]). The results are shown in gure 3. The observable hRe(L)Si in scans C1 and D1

develops the S-shape characteristic for a phase transition, with some uctuations in the vicinity of the in ection point. For scan B1 the S-shape is not as prominent, possibly due to the limited temperature range explored. To extract the in ection point we have tted

hRe(L)Si to the form of an arctangent. To check the model dependence of the results

we have performed alternative ts using the Gaussian error function. Both ts tend to describe the data reasonably well and give similar ~2=dof values. The resulting curves from the arctangent ts are shown as black lines in gure 3. The results for the associated transition temperatures T dcC are given in table 4. Evidently the estimate for the uncertainty of the in ection point from the t cannot be reliable due to the strong uctuations in its vicinity. To account for this additional uncertainty we have assigned another systematic error of 10 MeV to the result from the t, re ecting the size of the interval where we observe deviations from the smooth behaviour of the Polyakov loop. The shaded areas in gure 3 represent the estimates for the transition region. In the vicinity of T dcC the Polyakov loop susceptibility increases and shows uctuations that can be interpreted as the onset of peak-like behaviour. Owing to this typical behaviour, we suspect that T dcC could be somewhat overestimated for scan D1.

3.2.2 Chiral condensate and its susceptibility

To estimate the chiral symmetry restoration temperatures, TC, we use the renormalised disconnected susceptibility, of the chiral condensate without T = 0 subtractions. In particular, we have Z2m~bare[angbracketleft] [angbracketright]jdisc and Z2P

~bare[angbracketleft] [angbracketright]sub for condensate and subtracted condensate,

respectively. Note that the additive renormalisation discussed in section 2.3.2 has not been

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0.08

B1 , V = 323

0.06

[angbracketrightbig]

Re(L) S

0.04

~ L SM

[angbracketleftbig]

0.02

0 140 160 180 200 220 240 260 280

T [MeV]

JHEP12(2016)158

0.08

C1, V 3

C1, V = 323

0.06

[angbracketrightbig]

Re(L) S

0.04

~ L SM

[angbracketleftbig]

0.02

0 140 160 180 200 220 240 260 280

T [MeV]

0.08

D1, V = 323

0.06

[angbracketrightbig]

Re(L) S

0.04

~ L SM

[angbracketleftbig]

0.02

0 140 160 180 200 220 240 260 280

T [MeV]

Figure 3. Results for the (unrenormalised) APE smeared Polyakov loop (left) and its susceptibility (right) for scans B1 , C1 and D1 (from top to bottom). The shaded areas indicate the estimates for the transition regions and the black lines are the results from the t of the Polyakov loop expectation value to the arctangent form.

taken into account. However, the position of the peak in the susceptibility should not be a ected, since the additive renormalisation gives regular contributions around TC.

Figure 4 displays the results for the disconnected susceptibilities for the unsubtracted and subtracted bare condensates. We de ne the pseudocritical temperature for chiral symmetry restoration through the position of the peak in the susceptibility of the subtracted condensate. To determine TC we t the susceptibility to a Gaussian. Since the error estimate for TC from the t will likely underestimate the true uncertainty we take the full

{ 15 {

scan T dcC [MeV] TC [MeV] mCud [MeV] mC [MeV] B1 241(5)(6)(10) 232(18)(6) 41 (11)(2) 485(55)(20) C1 214(9)(3)(10) 211( 5)(3) 16.8(30)(7) 300(27)( 9)

D1 210(6)(3)(10) 190(10)(5) 8.1(12)(4) 214(14)( 8)

Table 4. Results for the pseudocritical decon nement, T dcC, and chiral symmetry restoration, TC, temperatures and the associated critical value of the quark mass with its zero-temperature pion mass pendant. The rst error re ects the uncertainty of the extraction of the pseudocritical temperatures due to the t, the second error accounts for scale setting (and renormalisation). For T dcC the third error is the associated systematic error as explained in the text.

spread of points included in the t as a conservative error estimate. The resulting values for TC are given in table 4 and are shown as shaded areas in gure 4. The black curves correspond to the t. Scan B1 is a problematic case, since, due to the change of the quark mass, the scan remains longer in the vicinity of the critical region, TC(mud). Consequently, the peak is broad and we obtain large uncertainties for both, TC and mCud. Comparing the results for T dcC with TC, we see that they mostly agree within errors. The exception is D1, where we nd that T dcC lies somewhat above TC. Like other studies in the literature we see that TC decreases with the quark mass [26{36], which is a general feature, persisting even when dynamical heavy quarks are included (e.g. [14, 15]).

In gure 5 we show the results for the fully renormalised condensate and the fully renormalised subtracted version for scans C1 and D1. As expected, the condensates start close to zero and show a rapid decrease in the approach to TC. Around TC both condensates show uctuations, especially the standard condensate uctuates quite strongly. We note that TC, as de ned by the in ection point of the renormalised condensate, does not necessarily have to agree with the peak of the susceptibility for a broad crossover.

We also have preliminary data for the condensates from simulations on 16 483 lattices.

These were mainly used to check nite size e ects on the screening masses below and are insu cient for a precise nite size scaling analysis. However, they are fully consistent with saturating susceptibilities and a crossover, as expected.

3.2.3 Scaling in the approach to the chiral limit

Following section 2.4.1 we will now try to extract information about the order of the transition in the chiral limit by looking at the scaling of the temperatures with the quark mass. With three transition temperatures at our disposal and their relatively large uncertainties we are not in the position to extract the critical exponents from a t of the data for TC.

Instead, we x the critical exponents to the ones from the di erent universality classes and check whether any particular scenario is favoured by our data. We start by tting the results for TC from table 4 to eq. (2.17) with the critical exponents from the O(4) and

U(2) scenarios. The results are shown in gure 6. The plot highlights the similarity of the two curves, indicating that the scaling of the transition temperatures alone will not be su cient to distinguish between the two scenarios. We note that this is likely to remain

{ 16 {

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0.48

0.8

B1 , V = 323

0.4

0.7

B1 , V = 323

0.6

0 0.32

0.5

[angbracketright]

sub

0.24

[angbracketright]

0.4

[angbracketleft]

0.16

0.3

[angbracketleft]

0.2

0.08

0.1

0 140 160 180 200 220 240 260 280

T [MeV]

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0.48

0.9

C1, V = 323

0.8

0.4

0.7

0 0.32

0 0.6

[angbracketright]

sub

0.24

0.5

[angbracketright]

0.4

[angbracketleft]

0.16

[angbracketleft]

0.3

0.08

0.2

0.1

0 140 160 180 200 220 240 260 280

T [MeV]

0.48

D1, V = 323

1.5

D1, V = 323

0.4

1.2

0 0.32

[angbracketright]

sub

0.9

0.24

[angbracketright]

[angbracketleft]

0.16

~bare

[angbracketleft]

0.6

0.08

0.3

0 140 160 180 200 220 240 260 280

T [MeV]

Figure 4. Results for the disconnected susceptibility of the condensate (left) and the subtracted condensate (right) for scans B1 , C1 and D1 (from top to bottom). The shaded areas indicate the estimates for the transition regions and the black lines are the results from the t of the susceptibility to a Gaussian.

true even when the error bars are reduced by an order of magnitude. Potentially, a scaling analysis of the order parameter and its susceptibility (see [9]) might help in this respect. However, this would demand the knowledge of the scaling function from eq. (2.16) for the U(2) case. A distinction will only be possible if the scaling functions di er signi cantly.

In the rst order scenario we also have to x the value of the critical quark mass mcrud. Since mcrud is poorly constrained by the t, we can try ts for di erent xed values of mcrud and look for minima in ~2=dof. We nd that, not unexpectedly, ~2=dof is very at and does

{ 17 {

-0.01

C1, V = 323

D1, V = 323

C1, V = 323

D1, V = 323

-0.02

-0.04

0 -0.03

3 0 -0.06

ren r 3

-0.04

ren

subr

-0.08

[angbracketrightbig]

-0.05

-0.1

[angbracketleftbig]

-0.06

[angbracketleftbig]

-0.12

-0.07

-0.14

-0.08

-0.16

-0.09

-0.18

140 160 180 200 220 240 260 280

JHEP12(2016)158

T [MeV]

Figure 5. Results for the renormalised standard (left) and subtracted (right) condensate in scans C1 and D1. The shaded areas display the estimates for TC.

O(4) scaling

U(2) scaling

160

260

240

T C[MeV]

220

200

180

160

0 10 20 30 40 50 60

mud [MeV]

Figure 6. Results from the scaling ts to TC using the critical exponents from the O(4) (left) and U(2) (right) scenarios.

Z(2) mcrud = 1.7 MeV

scaling,

160

260

240

T C[MeV]

220

200

180

0 10 20 30 40 50 60

mud [MeV]

Figure 7. Results for the scaling t for the rst order scenario, i.e. the Z(2) universality class, with mcrud = 1:7 MeV.

not exhibit a minimum. Furthermore, ~2=dof is always of the same order as for the second order ts. As a typical case we show the curve obtained for mcrud = 1:7 MeV in gure 7. As for the scaling with the O(4) and U(2) critical exponents, the Z(2) curve agrees very well with the data and is hardly distinct from the curves of gure 6.

{ 18 {

Obviously, none of the scenarios is ruled out by the above analysis. These ndings are in agreement with the results from the tmfT collaboration [36]. When we assume that the data shows consistency with one of the second order scenarios and extract the associated critical temperatures in the chiral limit we obtain

TC(0)jO(4) = 163(27) MeV and TC(0)jU(2) = 167(25) MeV : (3.1)

Both results are consistent with the ndings from the tmfT collaboration for O(4) scaling, TC(0) = 152(26) MeV [36], and are on the lower side of the results for Wilson fermions at Nt = 4, TC(0) = 171(4) MeV [33], and of the study of the QCDSF-DIK collaboration with di erent Nt values TC(0) = 172(7) MeV [35]. Calculations using staggered fermions only quote values for the critical coupling in the chiral limit, without providing results for the lattice spacing. The two results in eq. (3.1) indicate that the result for TC(0) is not sensitive to the universality class used for the extrapolation. This property is just another manifestation of the di culty to distinguish between the two scenarios and shows that even a reduction of the error bars by an order of magnitude, in combination with results at much smaller quark masses, might not be su cient using the scaling of the transition temperatures alone.

3.3 Screening masses and chiral symmetry restoration pattern

We now turn to the investigation of screening masses and the chiral symmetry restoration pattern. Since the behaviour of screening masses below and close to TC in general depends on the quark mass we will focus on the scans along LCPs in this section, i.e. on scans C1 and D1.

The screening correlators have been measured in the x3 direction on the stored con gu-rations using unsmeared point sources. To make e cient use of the generated con gurations we have computed the correlation functions for 48 randomly chosen source positions (see also [64]). Compared to ref. [58], we have thus enlarged the statistics by another factor of three. Screening masses are extracted from the e ective mass,2 aMe X, de ned by the formula for the inverse hyperbolic cosine,

aMe X(z) = ln

; (3.2)

where C = CXX. Since in scans C1 and D1 the spatial extent is rather small we could not nd reasonable plateaus in most of the cases due to contaminations from excited states (see also [62]). To take the contaminations into account we have tted the results for the e ective mass to the form

aMe X(z) = aMX + aA exp z

; (3.3)

2Note, that the procedure di ers from the one used in [62], where we have used a direct t to the correlator. In general, the two sets of results are consistent within errors. The current set of measurements supersedes the ones from [62].

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C(z + a) + C(z a)

2C(z) +

C(z + a) + C(z a)

2C(z)

2 1

0.5

0.8

0.84 TC

1.07 TC

0.4

0.6

aMe

0.3

aMe

0.4

0.2

0.1

0.2

3 6 9 12 15

z/a

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0.5

0.8

0.4

0.6

aMe

0.3

0.4

0.2

0.1

0.2

3 6 9 12 15

z/a

Figure 8. E ective masses for T = 0:84 and 1:07 TC in P (top left), S (bottom left), V (top right) and A (bottom right) channels in lattice units from scan C1. The solid lines are the results from a constant plus exponential t to the points that are within the area for which the curves are shown and the shaded area indicates the result for the e ective mass.

where A and are additional t parameters. In gure 8 we show examples for the e ective masses in the di erent channels above and below TC, together with the results from the associated ts.

The formula for the e ective mass follows to leading order when the contamination from the rst excited state is included in the correlation function. In this case represents the energy gap between groundstate and rst excited state in this channel. However, when contaminations from higher excited states become important (and A) will also contain contributions from those. In both cases a t to the form (3.3) is known to improve the extraction of the groundstate energy signi cantly and to remove most of the contaminations of the excited states. Note that neither A nor are of direct importance for the following analysis. The ts to the form (3.3) typically work well for a variety of tranges with starting points zmin in the range from 4a to 9a for P and S channels and between 6a and 11a for V and A channels.3 In these regions the results for MX do not depend signi cantly on the particular choice of trange and also the change of A and is only signi cant in a few cases. We show the results for MX, A and from two representative cases in P and

S channels close to the critical temperature for D1 (where contaminations from excited

3For ts with yet larger zmin the information in the data is not su cient to constrain all three t parameters su ciently. In this case one could try a t to a constant to extract MX, but this procedure is less reliable than the use of eq. (3.3).

{ 20 {

0.3

0.25

T = 187 MeV

0.25

T = 193 MeV

aM X

0.2

0.15

aM X

0.15

0.1

aA X

JHEP12(2016)158

0.8

a X

0.6

a X

0.6

0.4

0.2

0 3 4 5 6 7 8 9 10

zmin/a

Figure 9. Result for the t parameters of the ts for the extraction of screening masses versus the starting point zmin of the t. Shown are the results for the temperatures of 187 MeV (left) and 193 MeV (right) from scan D1. The vertical line corresponds to the choice for zmin used to obtain

the nal results.

states are typically expected to be most pronounced due to the small quark mass in the scan) in gure 9. Note, that the S channel is the more problematic one, both due to large statistical uncertainties and large contaminations from excited states (see also gures 8 and 12). In fact, the results shown in the right panel of gure 9 for the S channels are an example for the case where the ts do not work starting from zmin=a = 8. Given these

results, we conclude that for these values of zmin the main contribution to comes from the rst excited state. Within these regions where the tparameters are insensitive to the particular choice for zmin we can choose to work with any value of zmin. We decided to use zmin = 6a for P and S channels and zmin = 8a for V and A channels. For these values all of the ts give a reasonable ~2=dof around 1.4

The results for the screening masses are shown in gure 10. At T=TC 0:7 the screening masses show the expected splitting from the zero temperature meson masses [88]. While the masses in the P and V channels initially remain constant, indicated by a slight decrease of M=T , the screening masses in S and A channels decrease drastically in the approach to TC. Around T=TC 0:9 all screening masses start to increase. In particular, the screening

masses in P and S channels are drastically enhanced. Around TC the screening masses in the V and A channels are mostly degenerate and around 85 to 90% of the asymptotic 2T limit, independent of the quark mass in the scan. This is consistent with the ndings in simulations with staggered fermions [89, 90].

4Note that there were a few cases for the S and A channels for T < TC where the tranges needed to be adapted due to large uncertainties for larger z values. However, those cases have no in uence on the further analysis.

{ 21 {

1.4

1.2

M/(2T)

0.8

M/(2T)

0.8

0.6

0.4

0.2

0.7 0.8 0.9 1 1.1 1.2 1.3

JHEP12(2016)158

T/TC

Figure 10. Results for the screening masses in the di erent channels for scans C1 (left) and D1 (right). The screening masses are normalised to the asymptotic limit M1.

0.2

MPS

MV A

MP S

MV A

MPS

MV A

MP S

MV A

M/(2T)

-0.2

M/(2T)

-0.4

-0.6

-0.8

-1 0.7 0.8 0.9 1 1.1 1.2 1.3

T/TC

Figure 11. Results for the screening mass di erences in the P and S channels, MPS, and in V and A channels, MV A, for scans C1 (left) and D1 (right). The di erences are normalised to 2T .

The screening masses in P and S channels move closer together and uctuate strongly around TC. For the P channel the screening masses are only around 40 to 50% of the 2T limit for scan C1 with a pion mass of around 300 MeV and 35 to 40% for scan D1 with a pion mass around 200 MeV, indicating a quark mass dependence of the properties of pseudoscalar (and scalar) states around TC. The screening mass in the S channel is typically around 10% larger than MP . In the temperature interval covered by our calculations above TC, all screening masses are below the asymptotic high temperature limit. Note that weak-coupling calculations [93, 94] predict the asymptotic approach to occur from above, implying that the screening masses must cross the value 2T at a certain temperature.

Our ndings are in good agreement with the results for screening masses obtained with staggered fermions [89, 90]. In simulations within the quenched approximation, nite size e ects have been found to be signi cant up to aspect ratios of Ns=Nt = 4 [105, 106]. To get an idea about their magnitude, we compare the e ective masses in the di erent channels at

{ 22 {

T = 150 MeV from scan C1 (with a volume of 323) with those obtained from a simulation with the same parameters but an increased spatial volume of 483. The comparison is shown in gure 12, no signi cant nite size e ects are visible in the data. Note that the comparison is done for the lattice with the smallest value of MP L (which is the relevant quantity governing the size of nite volume e ects in the con ned phase) in the scan C1, being smaller than MP L for all simulation points in the transition region. We thus conclude that our nal results are not strongly a ected by nite size e ects.

The chiral symmetry restoration pattern can be investigated by the degeneracies of the screening masses. To extract the di erences,

MY X = MY MX ; (3.4)

we have used the plateau in the e ective masses of the ratios of the two correlation functions. In these ratios some of the uctuations between di erent ensembles, evident in gure 10, and statistical uctuations cancel. For these di erences, some of the contaminations of the excited states cancel as well, so that we could use a t to a constant, where we reduced the trange to the last few points. As for the ts to extract the screening masses, we have explicitly checked that our results do not depend signi cantly on the particular choice for the trange. This is in particular true when we consider that the main uncertainties in the following analysis are coming from the uctuation between di erent ensembles in the region close to TC. On top of these checks for the ts, we also checked the extraction of the mass splittings using a t to the plateau in the ratio of e ective masses MX=MY to obtain another estimate for the di erence via

MY X MY

MY 1 : (3.5)

In addition, we have also compared the results from the direct di erences of screening masses. Note that in a very few cases the t for MPS failed, and in one case this also happened within the transition region ( = 5:37 for scan D1). We have excluded this datapoint from the following analysis, but we have checked with the mass di erence of the independently determined screening masses that MPS for this value indeed lies within the uncertainty of the nal estimate.

In gure 11 we show the results for the mass di erences. The plot indicates the degeneracy of MV and MA and the associated restoration of the SUA(2) symmetry around

TC for both scans. As already noted above, this is in agreement with the results from staggered fermion simulations [89, 90]. This also adds to the con dence concerning the extracted transition temperatures. In particular, both estimates for the mass di erence agree very well over the whole temperature range for these channels (the ratio t does not work for the lowest temperatures so that we have no reliable results in that region). For MP and MS there is still a signi cant mass splitting in the region around TC, which, however, appears to become weaker with decreasing quark mass. For the following analysis we will conventionally use the direct results for the mass di erences, i.e. the results labeled by MXY . The persistent mass splitting in P and S channels implies a residual breaking of UA(1) around TC in agreement with what has been found with staggered [89], domain

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0.25

0.6

V = 323 V = 483

0.5

0.2

aMe

V 0.4

0.15

0.3

0.1

0.2

3 6 9 12 15 18 21 24

z/a

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0.8

0.7

0.6

aMe

0.5

aMe

0.4

0.3

0.4

0.2

0.1

0.2

3 6 9 12 15 18 21 24

z/a

Figure 12. E ective masses for T = 150 MeV for scan C1 with a 323 volume and for the future scan C2 with a 483 volume. The solid lines are the results from a constant plus exponential t to the points for scan C1 that are within the area for which the curves are shown and the shaded area indicates the result for the e ective mass. For the scalar screening masses the excited states ts did not work, so that we could not include any t result in the plot.

wall [10, 11, 107] and overlap [38, 39] fermion formulations. To make any statement about the fate and strength of the breaking in the chiral limit, however, we still need to perform a chiral extrapolation, which is the topic of the next section.

3.4 On the relative size of the UA(1) breaking e ects around TC

An important question is which amount of symmetry breaking is \strong" or \weak". When looking at the domain wall fermion results [10, 11] for chiral susceptibilities, one might be led to the conclusion that the breaking is signi cant in the chiral limit. The same is true if one looks at the eigenvalues of the associated Dirac operator [107]. However, the chiral susceptibilities include contributions from contact terms which might give an additional contribution that overwhelms the e ect of chiral symmetry breaking. The eigenvalues are a more sensitive probe of the UA(1) breaking. In this case, studies with the overlap operator [38{40] have indicated that the residual breaking in the chiral limit might be weak in contrast to the breaking implied by the eigenvalues of the domain wall operator at nite extent of the fth dimension.

In contrast to the studies discussed above, we use non-perturbatively O(a)-improved Wilson fermions, which break chiral symmetry explicitly. The symmetry only becomes restored in the continuum limit. Since we use the mesonic screening spectrum as a probe,

{ 24 {

B1 C1 D1 mud = 0 linear mud = 0 sqrt

MPS [MeV] -272(33)(479) -172(24)(158) -121(23)(127) -82(282) 5(590)

Table 5. Results for MPS at TC for the di erent scans. The rst error is statistical, the second re ects the systematic error owing to the uctuations in the transition region. The value labeled with \mud = 0 linear" is the result from the linear chiral extrapolation to all three points, as explained in the text, and the value labeled with \mud = 0 sqrt" assumes a quark mass dependence proportional to pmud. The uncertainty on the result of the chiral extrapolations contains the statistical as well as the systematical uncertainty.

the results are expected to approach the continuum with O(a2) corrections. On our relatively ne lattices (Nt = 16), we expect these e ects to be numerically small. We saw in the previous section that we do observe | to a good accuracy | the expected degeneracy between the vector and axial-vector screening above TC, signalling the restoration of the non-anomalous chiral symmetry. Similarly, if the UA(1) symmetry becomes e ectively restored, we expect to obtain UA(1)-breaking mass splittings of O(a2). In particular, since both the anomalous and the non-anomalous chiral symmetry are broken explicitly by the Wilson term, we expect the lattice artefacts to be of the same order of magnitude. Owing to the results for the di erence between vector and axial-vector screening masses we expect this e ect to contribute corrections of order 10 MeV. Furthermore, any accidental cancellation between lattice artefacts and a mass splitting in the continuum can only happen on this scale. Determining the strength of the breaking with a relative precision better than a few MeV requires taking the continuum limit.

We will now look at the chiral extrapolation of the mass di erence MPS in physical units. First of all, using the value ma0 = 980(20) MeV from the particle data group [108] we obtain as an estimate for the zero temperature reference value at the physical point

MT=0PS = 845(20) MeV : (3.6) This value might serve as an estimate for the e ect in the mass di erence when the breaking of UA(1) is substantial. However, this estimate is valid for physical, non-zero light quark mass, while we are interested in its value in the chiral limit. The pion mass vanishes in the chiral limit, so that MT=0,mud=0PS = mmud=0a0. Next we need an estimate for mmud=0a0, for

which we use the following ansatz: we assume that the di erence between chiral limit and physical point is solely due to the change of the constituent quark masses mconstq in the meson and compare with another \iso-vector" scalar particle in the review of the Particle

Data Group (PDG) [109], namely the K 0, where one of the u/d quarks is replaced by a strange quark. This results in the ansatz

ma0 mK

0 = C(mconstu,d mconsts) ; (3.7)

where C is a proportionality constant, which can be determined from eq. (3.7). Using C we can estimate the mass of the scalar in the chiral limit following

ma0 mmud=0a0 = 2Cmu,d ; (3.8)

{ 25 {

JHEP12(2016)158

200

-200

-400

-600

-800

-1000

-1200

Figure 13. Chiral extrapolation of MPS in physical units in comparison to the zero temperature pendant estimated as explained in the text. The red line and the associated red point at mud = 0

indicate the result from a linear chiral extrapolation.

where the factor 2 comes from the fact that we need to send the masses of two quarks to zero. Using the numbers from the PDG [109] we obtain the nal estimate mmud=0a0 =

945(41) MeV. The error estimate follows from the uncertainties associated with the masses of the a0 and the K 0 mesons. This is a rather crude estimate, but it is unlikely that it underestimates the e ect by an order of magnitude (even then mmud=0a0 600 MeV, which does not change the picture dramatically). Our nal estimate for the chiral limit is

MT=0,mud=0PS = 945(41) MeV : (3.9) The width of the transition region must be taken into account when we extract an estimate for MPS from our simulations. We thus compute the di erence from a t to a constant to the data points in the grey bands in gure 11. The spread of the results in the region is taken as a systematic uncertainty on top of the statistical uncertainty of the average. The results from this procedure are listed in table 5. Here we have also included a result for scan B1 to be able to perform a sensible chiral extrapolation. Unfortunately,

B1 is not at xed quark mass and thus remains longer in the vicinity of TC, since the latter increases with the quark mass. This accounts for the rather large error bars for the associated MPS.

To perform the chiral extrapolation for MPS we need to deduce its quark mass dependence. Since the pion is a Goldstone boson, its mass is expected to be proportional to pmud, at least at small temperatures, T < TC. On the other hand, the mass of the scalar should depend linearly on the quark mass which might also be the case for the pion at TC, where chiral perturbation theory breaks down. Given that we have only three data points at our disposal with relatively large uncertainties, our data clearly does not allow for a detailed investigation of the quark mass dependence of MPS. We thus perform two types of ts; (i) linear in mud, (ii) proportional to pmud. The results for the two di erent types of ts including all three data points are listed in table 5. We see that both results are consistent with zero within the relatively large error bars. As our nal estimate we will thus use the linear t. The associated result is shown in gure 13. We have also checked

{ 26 {

T = TC linear chiral extrapolation

T = 0 physical point

T = 0 chiral limit

M P S[MeV]

0 10 20 30 40 50

mud [MeV]

JHEP12(2016)158

the robustness of the result with a linear t using only the data from scans C1 and D1. For this chiral extrapolation the central value remains within the error bar of the result quoted in table 5, but the uncertainty of the result increases.

Figure 13 indicates that the UA(1)-breaking screening-mass di erence is a fairly small e ect and has a mild quark-mass dependence only. In fact, at mud = 0 the breaking e ects are strongly suppressed compared to the e ect in the same quantity at zero temperature and consistent with zero. Our result is thus in qualitative agreement with the results from the spectrum of overlap fermions [39, 40].

4 Conclusions

In this paper we studied the nite temperature transition of two- avour QCD on 16 323

(and 483) lattices. In particular, we have presented our results for the decon nement and chiral symmetry restoration temperatures, extracted from the in ection point of the Polyakov loop and the peak in the susceptibility of the subtracted chiral condensate, respectively. In agreement with previous studies in the literature, we nd both temperatures to decrease with the quark mass [26{36]. Our results for the chiral symmetry restoration temperatures, reported in table 4, are within uncertainties consistent with the values quoted by the tmfT collaboration [36, 110].

The ultimate goal of our ongoing e orts is to determine the order of the transition in the chiral limit. In an attempt to extract information about the chiral limit, we tested for scaling of the transition temperatures in the approach to the chiral limit. As already reported in [58], the scaling behaviour alone is not su ciently constrained to distinguish between a second order O(4) chiral transition or a rst order transition with a Z(2) endpoint. This is consistent with earlier ndings of the tmfT collaboration [36], even though we were able to reduce the pion masses in our study down to 200 MeV. Extrapolations of the critical temperatures to the chiral limit are not very sensitive to the universality class. Our results, when interpreted in terms of the O(4) and U(2) scenarios are consistent with the ndings from the tmfT collaboration for O(4) scaling [36], and somewhat smaller than those for Wilson fermions at Nt = 4 [33] and the QCDSF-DIK collaboration with di erent Nt values [35]. The fact that the chiral transition temperature does not appear to be very sensitive to the universality class used for the chiral extrapolation is one particular manifestation of the di culty to distinguish between di erent scenarios. Even a reduction of the error bars by an order of magnitude in combination with results from smaller quark masses might not be su cient to allow to distinguish between the universality classes when using the scaling of the transition temperatures alone.

As an alternative, we have investigated the strength of the anomalous breaking of the UA(1) symmetry in the chiral limit by computing the symmetry restoration pattern of screening masses in various isovector channels. At T=TC 0:7 the screening masses assume

values close to the zero temperature meson masses. Initially the masses in the scalar and axial-vector channels decrease, before at T=TC 0:9 all masses start to increase. Around

TC the screening masses in the vector and axial-vector channels are degenerate and about 85 to 90% of the asymptotic 2T limit, while the screening masses in the pseudoscalar channel

{ 27 {

JHEP12(2016)158

are about 35 to 50% of this limit and exhibit a signi cant dependence on the quark mass. The screening mass in the scalar channel is typically around 10% larger. When going to higher temperatures all screening masses approach the 2T limit from below, while leading-order weak-coupling calculations [93, 94] predict an asymptotic approach from above. All these ndings are in qualitative agreement with the results found in Nf = 2+1 simulations with staggered fermions [89, 90].

To quantify the strength of the UA(1)-anomaly in the chiral limit, we use the di erence between scalar and pseudoscalar screening masses, MPS. Unlike susceptibilities, screening masses probe exclusively the long distance properties of the correlation functions and thus do not su er from contact terms. The comparison of the chirally extrapolated value, Mmud=0PS = 81(282) MeV to its zero temperature analogue MT=0,mud=0PS =

945(41) MeV (cf. eq. (3.9)) suggests that the UA(1)-breaking is strongly reduced at the transition temperature (see also gure 13). If this e ect persists in the continuum limit, it disfavours a chiral transition in the O(4) universality class.

Acknowledgments

We thank our colleagues from CLS for the access to the zero-temperature ensembles. B.B. would like to thank Bastian Knippschild for sharing his routine for APE smearing and Gergely Endr}odi for many helpful discussions. We acknowledge the use of computer time for the generation of the gauge con gurations on the JUROPA, JUGENE and JUQUEEN computers of the Gauss Centre for Supercomputing at Forschungszentrum Julich, allocated through the John von Neumann Institute for Computing (NIC) within project HMZ21. The correlation functions and part of the con gurations were computed on the dedicated QCD platforms \Wilson" at the Institute for Nuclear Physics, University of Mainz, and \Clover" at the Helmholtz-Institut Mainz. We also acknowledge computer time on the FUCHS cluster at the Centre for Scienti c Computing of the University of Frankfurt. This work was supported by the Center for Computational Sciences in Mainz as part of the Rhineland-Palatine Research Initiative and by DFG grant ME 3622/2-1 Static and dynamic properties of QCD at nite temperature. B.B. has also received funding by the DFG via SFB/TRR 55 and the Emmy Noether Programme EN 1064/2-1.

A Simulation and analysis details

A.1 Simulation algorithms and associated constraints

The simulations have been done using the de ation accelerated versions of the Schwarz [67, 68] (DD) and mass [69, 70] (MP) preconditioned HMC algorithms. Both algorithms employ the Schwarz preconditioned and de ation accelerated generalised conjugate residual (DFLSAP-GCR) solver introduced by Luscher in [71, 72]. Due to the e cient solver, both algorithms exhibit an improved scaling behaviour when lowering the quark mass and the lattice spacing.

The block structures used in the solver and the preconditioning of the HMC algorithm impose constraints on the size of the local lattices allocated to the single processors. The

{ 28 {

JHEP12(2016)158

main limitation follows from the constraint that Schwarz preconditioning blocks in the solver (SAP-blocks) have a minimal size of 44 [71]. At least two of these blocks need to t into a sublattice, so that the minimal sublattice size when using a SAP solver is 8 43.

This restriction translates directly to the version of the MP-HMC algorithm from [70]. The restrictions for the sizes of the DD-blocks are essentially the same as for the SAP-blocks, but it also a ects the e ciency of the HMC preconditioning, since the separation of modes works most e ciently if the DD-blocks have a minimal size of about half a fermi in each direction [67]. The inverse critical temperature corresponds to a temporal extent of about 1 fm, meaning that the size of the DD-blocks should be half the size of the temporal extent. For our scans the optimal sublattice size for the DD-HMC is thus given by 16 83. Further

constraints follow from the use of even/odd preconditioning in the di erent levels of the solver (see [71, 72] for the details). Summarizing, the algorithm restricts one to use lattice sizes of Nt/s = 8; 12; 16; 20; : : :.

In our initial runs (see [59{61]) we used the DD-HMC algorithm. A drawback of the algorithm is that a sizeable fraction R of the links remain xed during the molecular dynamics evolution [67]. While the ergodicity can be restored by shifting the gauge eld between two HMC trajectories, the autocorrelation between two trajectories increases signi cantly. For typical blocksizes of 44 and 84 one obtains R = 0:09 and 0.37. For Nt = 16 the optimal block size is 84, for which the autocorrelations are expected to be enhanced by about a factor of 3 compared to the MP-HMC algorithm. In particular on large scale machines, one is often forced to use a large number of processors and decrease the local system size, meaning that autocorrelations increase drastically and the algorithm becomes ine cient. An example is the B3 lattice discussed in [58].

Due to the suppression of low modes above TC accompanying the e ective restoration of the chiral symmetry, the main motivation for using de ation gradually disappears. The reduced bene t of de ation can be observed in practice. However, for the lattices and quark masses in this study, there is still a signi cant acceleration even above TC. In the

region around and below TC we had rare appearances of problems with de ation, most likely stemming from the fact that the \little" Dirac operator (see [72] for the details) has large condition numbers. Mostly these issues could be solved by changing the parameters of the de ation subspace.

A.2 Error analysis

For the error analysis we employ the bootstrap procedure [111] with 1000 bins. The data for di erent time slices of correlation functions is correlated, which means that in ts the full covariance matrix has to be taken into account. In practice, however, the uncertainties for the entries of the correlation matrix are often not precise enough for a stable least-square minimisation (see [112] for instance). This is in particular true for screening masses that typically have a bad signal-to-noise ratio. All our ts to correlation functions have thus neglected the non-diagonal terms of the correlation matrices.

For scale setting and renormalisation we have used interpolations of quantities that have their own uncertainties and for which we have no access to the raw data. To take these uncertainties into account, we have generated uncorrelated pseudo-bootstrap distri-

{ 29 {

JHEP12(2016)158

5.20 5.30 5.50 r0=a 6.15( 6) 7.26( 7) 10.00(11)

c 0.136055(4) 0.136457(4) 0.1367749(8)

Table 6. Input from zero temperature for scale setting and the estimation of LCPs taken from [74].

(A.3)(A.4)

(A.1)(A.2)

0.1368

r 0/a

0.1365

JHEP12(2016)158

0.1362

0.1359

5.1 5.2 5.3 5.4 5.5 5.6

Figure 14. Interpolations of r0=a (left) and c (right).

butions with 1000 bins, whose width reproduces the quoted uncertainties. The interpolations have then been done for each bin, giving bins for the desired quantities at each value of the coupling.

A.3 Interpolation of zero temperature quantities

In this appendix we discuss the interpolations of several quantities, such as lattice spacing and quark masses. For scale setting we perform an interpolation of r0=a obtained from the CLS lattices [74, 113], summarised in table 6. The nal result is obtained using the ansatz [114]

ln[a=r0( )] = c0 + c1 + c2 2 ; (A.1)

motivated by the solution of the renormalisation group equation for the bare coupling to two-loops. To check the model dependence of the result we use a simple polynomial of second order,

r0=a( ) =

c0 +

c1 +

c2 2 : (A.2)

The results for the coe cients are tabulated in table 7 and the interpolation is visualised in gure 14 (left). The plot displays the good agreement between the two interpolations, indicating that in the region 5:20 5:50, systematic errors due to the ansatz for

r0=a( ) are negligible.

Another important quantity is the critical hopping parameter c. We interpolate the result from [74], listed in table 6, using the two di erent ansatze,

c( ) = 18

1 + d0= + d1= 2

1 + d2= ; (A.3)

c( ) = 18 +

d0= +

d1= 2 +

d2= 3 : (A.4)

{ 30 {

c0 c1 c2

-13(17) 3.9(62) -0.21(58) 184(120) -79(45) 8.6(42) d0 d1 d2

-4.089(19) -2.967( 8) -4.703(16) -0.82(4) 10.2(4) -29.0(9) z0 z1 z2 ~2=dof

55(37) -20(14) 1.9(13) 0.2

Table 7. Results of the ts for the interpolations of zero temperature results and the analytic relation amPCAC( ; ). Note, that the for the interpolations there are as many parameters as data points, so that it is a parameterisation rather than a t.

The results are also listed in table 7 and the interpolations are shown in gure 14 (right). There is a slight model dependence in the region 5:35 [lessorsimilar] [lessorsimilar] 5:45. However, the interpolation mostly concerns the estimation of LCPs and does not enter the nal analysis.

For the other renormalisation factors and improvement coe cients we have used known interpolation formulas from the literature: for ZV we have used the interpolation formula from [115]; for cA the one from [116]; for ZA and ZP we have used the results from [74]; for bm and [bA bP ] we have used the non-perturbative result from [117].

A.4 Estimating lines of constant physics

LCPs can be realised once we have an analytic relation between the bare parameter and the renormalised quark mass mud. This relation can be obtained by using the two di erent de nitions for mud in the case of Wilson fermions. The rst de nition uses the bare quark mass

m [118],

mbareud = Zm (1 + bm a

m) mPCAC : (A.6)

Here ZA and ZP are the renormalisation factors for the axial current and the pseudoscalar density (ZP is scheme dependent) and bA and bP are due to improvement. mPCAC is the bare quark mass de ned by the PCAC relation (see [74, 77]).

Both quantities in eqs. (A.5) and (A.6) are estimates for the renormalized quark mass mud and di er only in lattice artifacts. From eqs. (A.5) and (A.6) we can thus infer (see also [117])

amPCAC = ZPCAC( ) (1 + [bm + bP bA] a

m) a

m : (A.7)

eq. (A.7) is valid up to O(a2) corrections and ZPCAC( ) only depends on the regularisation scheme (i.e. the lattice discretisation), but not on the renormalisation scheme. Inserting

{ 31 {

JHEP12(2016)158

m with a

m = 12

1 c

; (A.5)

where Zm is the scheme dependent mass renormalisation factor and bm is an O(a) improvement coe cient. Another possibility is to use the PCAC quark mass [119],

mPCACud = ZA

ZP (1 + [bA bP ] a

Figure 15. Comparison between mud( ; ) and the simulation results.

the relation between a

m and we obtain

amPCAC( ; ) = ZPCAC( )

m is small

as well. Since the factor Z2 contains the combination of b-factors from eq. (A.7), which is dominated by bm, and is a number smaller than one, we can expect that the second term in eq. (A.8) is negligible for small quark masses. We thus have to obtain only the -dependence of the factor ZPCAC. To this end we use the data for mPCAC at T = 0

from [74, 77], listed in table 8. Since data is available only for three couplings, we will use a simple second order polynomial,

ZPCAC( ) = z0 + z1 + z2 2 : (A.9)

The t with this ansatz works reasonably well, giving a small ~2=dof of 0.2. In fact, ~2=dof does not make any statements about the goodness of the ansatz for ZPCAC( ), but it shows that for the given range of quark masses amPCAC and a

For c( ) we take the known relation from appendix A.3. When mud is small a

m are indeed linearly related. The results for the coe cients zi are listed in table 7. For renormalisation we can then use eq. (A.6) with the constants from appendix A.3 which are valid for the Schrodinger functional scheme. To convert to the MS scheme we use the conversion factor mMSud=mSFud = 0:968(20) [62, 77].

Note, that the nal error bars on mud obtained from the relation mud( ; ) are much smaller than the uncertainties of the coe cients suggest, due to compensations of changes in one parameter by the others. Figure 15 shows a comparison between the predictions from mud( ; ) with the results obtained from the actual simulations. The plot displays the good agreement for temperatures below and up to TC. Note, that the tuning for scan C1 has been done using an early version of the matching with less information from the T = 0 side. This explains the fact that the black points do not lie on a constant mud line in that case. The good agreement between the measured quark masses and the predictions from the matching reported here indicates that this updated form of the matching works well.

{ 32 {

mud( , )

m = 300 MeV

m = 220 MeV

m ud[MeV]

160 180 200 220 240 260

T [MeV]

JHEP12(2016)158

1 1 c( ) + Z2( )

1 1 c( )

2+ : : : : (A.8)

size # con g amPCAC a3h

ibare a3

P P

5.20 0.13565 64 323 289 0.0158( 2) 0.2727633(18) 0.2064( 1)0.13580 265 0.0098 ( 3) 0.2727369(19) 0.2203( 2)

0.13590 400 0.0057 ( 4) 0.2727055(15) 0.2428( 5)

0.13594 211 0.0044 ( 2) 0.2726914(23) 0.2609(10)

0.13597 96 483 400 0.2726827( 7) 0.2878( 9)5.30 0.13610 64 323 162 0.2711093(16) 0.1883( 1)0.13625 508 0.0072 ( 3) 0.2711386(11) 0.2004( 2)

0.13635 0.00375(11)

0.13638 96 483 250 0.00267( 9) 0.2711641( 9) 0.2438( 7)0.13642 128 643 205 0.2711734( 5) 0.3162(22)5.50 0.13650 96 483 83 0.0091 ( 2) 0.2685038( 8) 0.1645( 1)0.13660 188 0.0059 ( 2) 0.2685683( 6) 0.1702( 1)

0.13667 221 0.00343(12) 0.2686063( 5) 0.1782( 2)

0.13671 128 643 137 0.00213( 6) 0.2686293( 5) 0.1912( 3) Table 8. Set of CLS T = 0 lattices used for the determination of the relation mud( ; ) and the

computation of the condensate at zero temperature and the associated results. For the measurements of amPCAC and more details on the lattices we refer to [74, 77].

Above TC, mud is typically lower than expected. It is unclear to us whether this result is an e ect due to markedly di erent cuto e ects above TC, or if the interpolation becomes worse at the higher values. Note that our zero temperature ensembles are located at the -values corresponding to 150, 180 and 245 MeV, so that the systematic error associated with the interpolation is expected to be most severe around 210 MeV.

A.5 Interpolation of the zero temperature chiral condensate

For the computation of the renormalised condensate the computation of the T = 0 subtraction terms is mandatory. Here we have computed h

P P

since the condensate, which is nite at a m = 0, is proportional to m

described by a polynomial. We have thus used

JHEP12(2016)158

P P on the CLS lattices, summarised with the measurement details in table 8. For the computation we have used12 Z2 Z2 volume (for h

ibare and

ibare) and wall (for

) sources on each con guration.

We have tested several di erent functional forms for the interpolation and found simple polynomials in a

m to work best. Note, that

P P is expected to diverge linearly in a m,

P P

, which can be

a3h

ibare( ; ) = p1( ) + p2( )(2a

P P ( ; ) = p1( )=(2a m) + p2( ) + p3( )(2a m) ;

{ 33 {

m) + p3( )(2a

m)2

(A.10)

0.273

0.33

0.272

0.3

bare

= 5.20 = 5.30 = 5.50

[angbracketrightbig]

0.27

[angbracketrightbig]

0.271

0.24

[angbracketleftbig]

0.27

0.21

0.269

0.18

0.268

0 0.005 0.01 0.015 0.02

2a m

Figure 16. Interpolation of the results for the bare (left) and subtracted (right) condensates obtained on the CLS lattices, as described in the text.

q11 q12 q13 q21 q22 q23 q31 q32 q33

0.539(4) -0.087(2) 0.0069(2) 9.6(8) -3.5(3) 0.32(3) -207(34) 77(13) -7.2(12)

q11

q12

q13

q21

q22

q23

q31

q32

q33

0.084(49) -0.030(19) 0.0027(17) 6.1(55) -2.1(21) 0.19(19) -80(170) 30(64) -2.8(60)

Table 9. Results for the t parameters for the interpolation of the chiral condensate at T = 0.

with

pi( ) = qi1 + qi2 + qi3 2 (A.11)

(similarly for

pi with q !

JHEP12(2016)158

q) and as usual 2a

m = 1= 1= c( ) with c( ) obtained from

the standard interpolation (A.3). Altogether, each t has 9 t parameters. The results are given in table 9 and the interpolations for the di erent -values are shown in gure 16. The ~2=dof values are 3.3 and 5.5 for h

ibare and

P P

, respectively. The tiny error

bars on h ibare and

explain the bad values for ~2=dof, despite the t providing a satisfactory description of the data. Once more, the uncertainties of the t parameters do not re ect the uncertainty on the results for the condensate. Note that we have neglected nite volume e ects for the condensates, which appears to be justi ed for mL [greaterorsimilar] 4.

{ 34 {

P P

B Simulation parameters and results

Scan MDU T [MeV] mud [MeV] Up [MDU]

B1 5.375 0.136500 10000 201(5) 17.0(1.4) 26(5)5.400 22000 209(5) 25.5(2.4) 56(8)5.425 20200 218(5) 32.3(3.4) 32(4)5.450 20600 227(5) 39.6(2.0) 20(2)5.475 21600 236(5) 39.3(2.0) 22(3)5.48125 20200 238(5) 46.2(2.2) 18(2)5.4875 19600 240(6) 46.9(2.1) 21(3)5.49375 21400 243(6) 45.3(2.1) 19(3)5.496875 16200 244(6) 47.9(2.2) 16(2)5.500 21600 245(6) 45.5(2.0) 19(2)5.503125 16600 246(6) 43.6(2.1) 16(2)5.50625 21000 248(6) 49.7(2.5) 17(2)5.5125 21600 250(6) 47.7(2.4) 17(2)5.51875 19800 252(6) 50.6(2.2) 14(2)5.525 19400 255(6) 53.5(2.5) 17(2)5.550 19200 265(7) 51.3(2.4) 14(2)5.575 12200 275(8) 55.6(2.7) 13(2)

C1 5.20 0.135940 12320 151(4) 15.5(6) 96(28)5.30 0.136356 12260 178(4) 15.1(6) 24(3)5.355 0.136500 13040 195(5) 15.6(7) 34(7)5.37 0.136523 12560 199(5) 15.5(7) 19(3)5.38 0.136545 12240 203(5) 15.8(7) 25(4)5.39 0.136565 12240 206(5) 15.9(7) 20(3)5.40 0.136575 12080 209(5) 16.9(7) 24(5)5.41 0.136603 12320 213(5) 16.8(7) 31(5)5.42 0.136619 12080 216(5) 13.9(7) 23(4)5.43 0.136635 12480 220(5) 12.1(7) 22(4)5.44 0.136649 12166 223(5) 14.5(8) 15(2)5.45 0.136662 12000 227(5) 8.8(6) 15(2)5.50 0.136700 12480 245(6) 10.4(8) 14(2)

D1 5.20 0.135998 4800 151(4) 7.4(5) 28(9)5.30 0.136404 8600 178(4) 9.0(6) 12(2)5.32 0.136460 8040 185(4) 8.9(5) 10(2)5.33 0.136486 12000 187(4) 7.2(6) 14(3)5.34 0.136510 14280 190(4) 8.1(4) 14(2)5.35 0.136532 14320 193(5) 8.9(5) 16(3)5.36 0.136553 14200 196(5) 9.4(4) 9(1)5.37 0.136573 13440 199(5) 8.7(5) 9(2)5.38 0.136592 10520 203(5) 9.0(5) 9(2)5.39 0.136609 8200 206(5) 8.2(6) 9(2)5.40 0.136625 8440 209(5) 8.5(5) 8(1)5.45 0.136691 8560 227(5) 5.9(7) 8(2)5.50 0.136735 7840 245(6) 5.6(6) 5(1)

Table 10. Run parameters of the simulations. We list the bare lattice coupling , the hopping parameter , the number of molecular dynamics units (MDU), temperature, renormalised quark mass in the MS-scheme at a renormalisation scale of 2 GeV and the autocorrelation time of the plaquette (Up).

{ 35 {

JHEP12(2016)158

T [MeV] hLi hLSi h

irensub

104 102 r20 r30 102 r20 r30 102

ibare

~bare[angbracketleft] [angbracketright] h

iren h

ibaresub

~bare[angbracketleft] [angbracketright]sub h

201 4.6(6) 3.1(3)209 5.1(5) 2.8(2)218 6.8(4) 3.7(3) 0.269397(9) 0.28(9) -2.5(7) 0.1691(13) 0.47(10) -7.2(1.3) 227 6.9(4) 4.0(2) 0.269085(6) 0.21(5) -1.8(5) 0.1649( 6) 0.52(11) -8.0(9) 236 10.0(4) 5.6(2) 0.268751(6) 0.24(6) -4.5(6) 0.1592(5) 0.27(11) -12.9(8) 238 9.3(6) 5.0(3)249 8.7(3) 4.7(2) 0.268614(7) 0.21(5) -3.6(7) 0.1598(6) 0.44(10) -10.5(1.1) 243 9.9(6) 5.3(3) 0.268534(5) 0.22(5) -4.3(5) 0.1584(5) 0.46(16) -12.3(9) 244 10.2(6) 5.4(3)245 9.9(7) 5.3(3) 0.268459(7) 0.23(7) -4.8(8) 0.1586(7) 0.42(6) -11.3(1.3) 246 11.3(5) 6.1(2)248 11.3(4) 5.9(2) 0.268378(6) 0.13(4) -5.9(7) 0.1568(4) 0.19(5) -14.1(9) 250 11.5(3) 6.0(2) 0.268304(8) 0.18(4) -6.5(9) 0.1570(5) 0.33(10) -13.0(1.1) 252 11.5(4) 6.1(2)255 12.9(3) 6.5(2) 0.268156(4) 0.12(4) -8.0(6) 0.1551(3) 0.15(3) -16.1(1.0) 265 13.5(5) 6.9(2) 0.267863(3) 0.16(6) -12.5(10) 0.1527(1) 0.04(2) -20.7(1.4) 275 15.9(4) 7.7(2) 0.267589(4) 0.11(7) -17.2(18) 0.1515(1) 0.08(3) -23.9(2.1) C1151 0.7(4) 0.59(3) 0.272657(4) 0.38(3) -0.8(9) 0.2849(8) 2.23(20) -2.7(3) 178 2.0(4) 1.21(4) 0.271101(3) 0.34(3) -2.3(2) 0.2050(7) 1.05(9) -6.3(4) 195 4.9(3) 2.76(6) 0.270331(3) 0.31(2) -3.3(3) 0.1831(5) 0.59(7) -9.5(7) 199 4.8(4) 2.76(6) 0.270120(3) 0.24(2) -4.1(3) 0.1769(4) 0.33(4) -10.5(8) 203 5.7(4) 3.52(6) 0.269991(3) 0.29(3) -4.3(3) 0.1723(4) 0.26(4) -11.8(9) 206 5.7(4) 3.45(6) 0.269869(2) 0.21(2) -4.1(3) 0.1716(3) 0.20(3) -11.6(9) 209 5.4(4) 3.40(6) 0.269754(3) 0.24(2) -3.2(2) 0.1724(4) 0.33(4) -10.6(9) 213 5.5(4) 3.31(8) 0.269627(3) 0.29(2) -3.9(3) 0.1709(5) 0.35(4) -11.1(10) 216 6.9(4) 3.85(6) 0.269503(2) 0.21(2) -4.1(3) 0.1679(3) 0.15(2) -11.9(11) 220 7.8(4) 4.49(7) 0.269375(2) 0.16(2) -4.9(3) 0.1646(3) 0.09(2) -12.8(11) 223 8.1(4) 4.80(7) 0.269267(2) 0.22(2) -4.1(3) 0.1657(4) 0.20(3) -12.1(11) 227 9.7(4) 5.38(6) 0.269133(2) 0.13(2) -5.7(3) 0.1608(1) 0.01(1) -13.7(10) 245 11.6(4) 6.21(7) 0.268564(2) 0.15(2) -6.3(3) 0.1574(3) 0.09(4) -15.0(6) D1151 0.8(7) 0.89(6) 0.2725611(11) 0.64(8) -2.1(2) 0.2859(50) 3.89(76) -6.1(4) 178 2.5(4) 1.89(6) 0.2710744( 4) 0.43(4) -3.6(3) 0.2068(17) 1.54(32) -9.7(6) 185 3.3(3) 2.15(6) 0.2708011( 4) 0.37(3) -3.9(3) 0.1963(11) 0.67(12) -10.6(7) 187 3.1(3) 2.47(6) 0.2706568( 3) 0.35(3) -4.4(3) 0.1909(10) 0.71(15) -11.1(9) 190 3.4(3) 2.36(5) 0.2705272( 3) 0.37(3) -4.3(3) 0.1901(10) 0.93(29) -10.8(9) 193 4.0(3) 2.78(5) 0.2703920( 2) 0.33(2) -4.5(3) 0.1850( 9) 0.69(16) -11.3(10) 196 4.1(3) 2.86(5) 0.2702744( 3) 0.32(3) -3.9(3) 0.1858(11) 0.83(27) -10.8(11) 199 4.7(3) 2.98(5) 0.2701396( 2) 0.30(2) -4.3(3) 0.1807( 7) 0.42(7) -11.3(12) 203 4.9(3) 3.14(6) 0.2700152( 3) 0.29(2) -4.1(3) 0.1778( 7) 0.28(5) -11.5(12) 206 5.4(4) 4.54(7) 0.2698810( 3) 0.24(2) -4.7(3) 0.1727( 7) 0.27(13) -12.2(13) 209 6.5(3) 3.96(7) 0.2697557( 3) 0.23(2) -4.8(3) 0.1705( 6) 0.16(4) -12.3(13) 227 8.5(4) 4.93(8) 0.2691539( 2) 0.15(2) -5.2(3) 0.1619( 3) 0.02(7) -13.2(12) 245 10.8(4) 5.85(9) 0.2685848( 2) 0.15(2) -6.0(3) 0.1580( 2) 0.02(5) -14.2(6)

Table 11. Simulation results for the (smeared) real part of the Polyakov loop, the bare chiral condensate and its disconnected susceptibility (h

JHEP12(2016)158

ibare and

~bare[angbracketleft] [angbracketright]), the associated renormalised

condensate (h

iren) in units of r0, and the subtracted versions (h

ibaresub,

~bare[angbracketleft] [angbracketright]sub and h

irensub).

{ 36 {

T [MeV] aMP aMS aMV aMA a MPS a MV A 102

C1151 0.136(2) | 0.348(15) 0.482(35) | -12.42(12) 178 0.127(2) 0.201(38) 0.311(13) 0.370(33) -0.102(25) -4.94(56) 195 0.153(4) 0.164(31) 0.332( 9) 0.344(19) -0.048(16) -1.32(24) 199 0.146(6) 0.172(13) 0.304(10) 0.317( 9) -0.041(11) -0.70(21) 203 0.186(5) 0.225(17) 0.351( 4) 0.344( 8) -0.049(13) -0.47(13) 206 0.189(4) 0.210(15) 0.324( 6) 0.315(11) -0.036( 9) -0.57(12) 209 0.164(5) 0.220(30) 0.332( 5) 0.336(11) -0.075(22) -0.84(17) 213 0.152(6) 0.243(23) 0.332( 6) 0.331( 9) -0.097(20) -0.76(14) 216 0.196(5) 0.252(15) 0.338( 5) 0.341( 7) -0.065(15) -0.38(11) 220 0.205(7) 0.240(18) 0.345( 4) 0.343( 5) -0.051(12) -0.96( 9) 223 0.180(7) 0.234(41) 0.336( 6) 0.329(11) -0.087(23) -0.45(11) 227 0.248(6) 0.261( 7) 0.359( 3) 0.358( 4) -0.014( 5) 0.01( 3) 245 0.264(6) 0.282(10) 0.356( 3) 0.358( 3) -0.028( 9) -0.03( 3) D1151 0.104(3) 0.365(107) 0.349(21) 0.412(25) | -5.0(1.7) 178 0.136(3) 0.289(45) 0.338(10) 0.324(30) -0.190(38) -1.8(5) 185 0.134(5) 0.207(30) 0.322( 9) 0.323(25) -0.086(28) -1.8(4) 187 0.150(5) 0.142(29) 0.339(14) 0.375(12) -0.037(13) -0.5(3) 190 0.136(4) 0.150(19) 0.310( 9) 0.333(15) -0.041(12) -1.0(3) 193 0.142(4) 0.145(28) 0.319( 8) 0.342(12) -0.031(15) -0.4(2) 196 0.136(4) 0.186(37) 0.311( 7) 0.315(12) -0.080(28) -0.9(2) 199 0.155(5) 0.271(44) 0.321( 7) 0.323(14) | -0.9(3) 203 0.161(5) 0.222(27) 0.332( 6) 0.324( 9) -0.068(24) -0.5(2) 206 0.173(9) 0.272(30) 0.324( 4) 0.326( 8) -0.104(22) 0.0(2) 209 0.170(6) 0.218(21) 0.328( 6) 0.349( 7) -0.061(19) -0.3(2) 227 0.239(6) 0.251(15) 0.343( 4) 0.349( 5) -0.039(21) -0.1(2) 245 0.220(7) 0.223(10) 0.347( 4) 0.348( 4) -0.012( 6) 0.1(1)

Table 12. Simulation results for screening masses M in P , S, V and A channels and the direct measurements for screening mass di erences M. The results for scan B1 are not listed.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

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Abstract

We study the thermal transition of QCD with two degenerate light flavours by lattice simulations using O(a)-improved Wilson quarks. Temperature scans are performed at a fixed value of N _t = (aT)^-1 = 16, where a is the lattice spacing and T the temperature, at three fixed zero-temperature pion masses between 200 MeV and 540 MeV. In this range we find that the transition is consistent with a broad crossover. As a probe of the restoration of chiral symmetry, we study the static screening spectrum. We observe a degeneracy between the transverse isovector vector and axial-vector channels starting from the transition temperature. Particularly striking is the strong reduction of the splitting between isovector scalar and pseudoscalar screening masses around the chiral phase transition by at least a factor of three compared to its value at zero temperature. In fact, the splitting is consistent with zero within our uncertainties. This disfavours a chiral phase transition in the O(4) universality class.

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Title

On the strength of the U^sub A^(1) anomaly at the chiral phase transition in N ^sub f^ = 2 QCD

Author

Brandt, Bastian B; Francis, Anthony; Meyer, Harvey B; Philipsen, Owe; Robaina, Daniel; Wittig, Hartmut

Pages

1-45

Publication year

2016

Publication date

Dec 2016

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP12(2016)158

ProQuest document ID

1854389269

On the strength of the U^sub A^(1) anomaly at the chiral phase transition in N ^sub f^ = 2 QCD

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