KKEK-CP-364, RBRC-1257
Topological Susceptibility in N f = QCD at Finite Temperature
Sinya
Aoki , Yasumichi
Aoki , ,(cid:63) , Guido
Cossu , Hidenori
Fukaya , Shoji
Hashimoto , , and Kei
Suzuki (JLQCD Collaboration) Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom Department of Physics, Osaka University, Osaka 560-0043, Japan SOKENDAI (The Graduate University for Advanced Studies), Tsukuba, Ibaraki 305-0801, Japan
Abstract.
We study the topological charge in N f = χ t of the topological charge definedeither by the index of overlap Dirac operator or a gluonic operator is investigated atseveral values of temperature T ( > T c ) varying the quark mass. A strong suppression ofthe susceptibility is observed below a certain value of the quark mass. The relation withthe restoration of U A (1) is discussed. Topological susceptibility in QCD at finite temperature has acquired much attention recently due toits phenomenological interest. Mass of the QCD axion, one of the candidates of dark matter, is givenby the topological susceptibility, and its dependence on temperature determines the abundance of theaxion in the universe. A quantitative estimate can in principle be provided by lattice QCD, and wasone of the topics of the panel discussion of this year’s lattice conference [1–4]. This study is not meantto provide some quantitative results at phenomenologically important temperatures 500 (cid:46) T (cid:46) U A (1) symmetry at and above the phase transition for vanishing u and d quarkmasses is one of the long standing and fundamental questions in QCD. While at any temperaturethe U A (1) chiral anomaly exists, manifestation of the U A (1) breaking is only possible if the gaugefield configurations with non-trivial topology actually have non-vanishing contribution. The non-trivial QCD configurations also produce the topological susceptibility. Thus there is naturally a linkin between these two physical quantities.One powerful theoretical approach for these problems is to use the properties of the spectrum ofthe Dirac operator [5–8]. Along this line Aoki, Fukaya and Taniguchi (AFT) revisited the problem as-suming the overlap fermions for the UV regulator for quarks [9]. They claim that the U A (1) symmetryin two flavor ( N f =
2) QCD is recovered in the chiral limit for temperatures at and above the criticalone. Furthermore, the derivatives of the topological susceptibility with respect to the quark mass m (cid:63) Speaker, e-mail: [email protected] a r X i v : . [ h e p - l a t ] N ov anish at any order. It means that the susceptibility, which is zero at the chiral limit, stays zero in thevicinity of m =
0. As the susceptibility is non-zero for infinitely heavy quarks, there must be a criticalmass which divides the regions with zero and non-zero topological susceptibility.The relation of the spectrum of the Dirac operator with U A (1) was also studied by Kanazawa andYamamoto (KY) more recently [10]. Assuming the U A (1) breaking they derived a relation betweenthe U A (1) susceptibility, which is a measure of the U A (1) breaking, and the topological susceptibilitythrough a low energy constant. According to their study, the topological susceptibility should beproportional to the squared quark mass, thus, should exhibit quite di ff erent mass dependence to that ofAFT. Kanazawa-Yamamoto claims the assumption that the spectral density is analytic near the originin AFT needs to be abandoned to have the U A (1) breaking. The analyticity, however, seems intactin the simulations with overlap fermions [11] and domain wall fermions with overlap-reweighting[12–14], which have exact chiral symmetry.Studying the topological susceptibility in depth would add another dimension for the understand-ing of the nature of the finite temperature transition in N f = U A (1) breaking. Also, understanding the fate of the U A (1) breaking should be important for the computation of the topological susceptibility to a requiredprecision necessary for phenomenology.Chiral symmetry plays a crucial role for the study of U A (1) [12, 13]. We use the Möbius domainwall fermion and reweighting method to the overlap fermion ensemble. In this report and the one forthe U A (1) breaking [14], the main lattice spacing used is finer than we have used in [12, 13]. Thishelps to reduce the residual chiral symmetry breaking of the domain wall fermions and to make thereweighting e ffi cient.The AFT scenario suggests a critical mass m c > U A (1) restoration. This could, then,change the widely believed phase diagram, called the Columbia plot at the upper-left corner. If similardynamics exists at the physical strange quark mass point, it would a ff ect the nature of the transition ofthe physical point depending on the value of m c .This report is organized as follows. In Sec. 2, the calculation set-up and methods are described.Starting with a discussion on the sampling of the topological charge, an elaborate estimate of the errorfor the topological susceptibility is explained in Sec. 3, followed by our main results. Sec. 4 is devotedto summary and outlook. We use a = Our simulation is carried out using Möbius domain wall fermions for two dynamical quark flavors[12]. A particular focus is placed on N t = β = . ff erent masses in thisreport. The corresponding temperature is T (cid:39)
220 MeV. At a fixed β value, two di ff erent temperatures N t = ff ects, acoarser lattice at β = . N t = T (cid:39)
220 MeV is examined. For all latticesreported here the spatial site number is L = ff as a function of β for these lattices is obtained with the Wilson flow scale t using the zero temperature results and an interpolation [17].Topological susceptibility is defined as χ t = V (cid:104) Q t (cid:105) , (1)where V is the four dimensional volume and Q t is the topological charge.e examine two definitions of the topological charge. One is the space-time sum of the gluonictopological charge density after the Symanzik flow at t =
5. The other is the index of the overlap-Diracoperator [17].As pointed out in [12, 17], it is essential to reweight to overlap ensemble from domain wall (cid:104)O(cid:105) OV = (cid:104)O R (cid:105) DW (cid:104) R (cid:105) DW , (2)where R is the reweighting factor defined on each gauge field configuration, to correctly take intoaccount the e ff ect of (near) zero modes of the overlap-Dirac operator. Partial quenching by the useof valence overlap operators on dynamical domain wall ensembles leads to an artificial enhancementof low modes. The topological charge defined through the zero mode counting su ff ers from suchartificial e ff ects, which can be eliminated by the reweighting.We investigate two definition of the topological charge on the original domain wall ensemble andon the overlap ensemble generated through the reweighting. Altogether, four values of topologicalsusceptibility are obtained at each parameter point as shown in the next section.We are aiming to acquire the data from 30,000 molecular dynamics time units with hybrid Monte-Carlo simulation for each ensemble. Some of the reported data here are still undergoing improvementof statistics. Q t GL-DWOV-DW β =4.3, 32 x12, m f =0.00375 -3 -2 -1 0 1 2 3Q t h i s t og r a m GL-DWGL-OV gluonic x12, β =4.3, m=0.00375 Figure 1.
Monte-Carlo time history of topological charge (left) and histogram for gluonic measurement at m = . (cid:39)
10 MeV) (right).
The left panel of Figure 1 shows the Monte-Carlo time history of the topological charge for β = . N t =
12 ( T (cid:39)
220 MeV) and bare mass m = . (cid:39)
10 MeV) sampled every 20th trajectory.One trajectory amounts to a unit time molecular dynamics evolution followed by an accept-rejectstep. The red line corresponds to the charge measured with the gluonic definition (“GL”), while cyanrepresents that with the overlap index (“OV”). The legends also show the ensemble on which the t h i s t og r a m OV-DWOV-OV
OV index x12, β =4.3, m=0.00375 (a) m = . (cid:39)
10 MeV). -3 -2 -1 0 1 2 3Q t h i s t og r a m OV-DWOV-OV
OV index x12, β =4.3, m=0.001 (b) m = .
001 ( (cid:39) . Figure 2.
Histogram of topological charge measured by the overlap index before (OV-DW) and after (OV-OV)the reweighting to overlap ensemble. calculations are based, which are domain wall (“DW”) for both. The right panel plots the histogramof the charge from “GL-DW” and that after the reweighting to the overlap ensemble “GL-OV”. Thebin size used can be read from the combined size of a pair of neighboring red and yellow bars. Itshows there is not much di ff erence between the data before and after the reweighting. Figure 2(a)shows the histogram of the topological charge measured through the overlap index before (OV-DW)and after (OV-OV) the reweighting. Here the width of the distribution shrinks significantly afterthe reweighting. This is due to the fact that the spurious zero modes on the domain wall ensemblegets suppressed. On the other hand, since such spurious zero modes are also suppressed by gaugefield smearing, there appeared less di ff erence between the gluonic measurements before and after thereweighting. From these data we calculate the topological susceptibility from Eq. (1).Special attention is required when there is no weight for the non-trivial topology, shown inFig. 2(b) as an example. The OV-OV histogram shows that all samples fall in the Q t = | Q t | = y axis shownbecause of the small reweighting factor. As a result, the topological susceptibility is consistent withzero, with a jackknife error χ t = . . × MeV . One should not take this as the sign of exactzero of χ t . This situation is similar to null measurements of rare processes in experiment. We estimatethe upper bound of (cid:104) Q t (cid:105) by imposing the condition that one measurement out of the full sample had | Q t | = N , then the upper bound of the topological susceptibility is ∆ (cid:48) χ t = N V . (3)With a reweighting, the e ff ective number of samples gets reduced. We use the following quantity forthe number of samples after reweighting: N e ff = (cid:104) R (cid:105) DW R max , (4)where R max is the maximum value of the reweighting factor in the ensemble [17]. As ∆ (cid:48) χ t can alsobe regarded as a resolution of the topological susceptibility given the number of samples – even ifountable | Q | > χ t forall the cases as ∆ χ t = max( ∆ JK χ t , ∆ (cid:48) χ t ) , (5)where ∆ JK χ t is the jackknife error of χ t . For the case of Fig. 2(b), N e ff =
32 out of a total of 1326samples measured every 20th trajectory. Now the error after this correction reads ∆ χ t = . × MeV . T (cid:39) MeV f [MeV]0.05.0 × × × × χ t [ M e V ] GL-DWGL-OVOV-DWOV-OV x12, β =4.3 a [fm ] × × × χ t [ M e V ] GL-DWOV-OV m=6.6MeV
Figure 3.
Topological susceptibility χ t at T (cid:39)
220 MeV as function of quark mass (left) and a dependence of χ t at m = . ma = . The left panel of Fig. 3 shows the quark mass dependence of topological susceptibility for N t = T (cid:39)
220 MeV. The color coding used here is the same as in the history and histogram shownin Figs. 1 and 2. As noted in the previous section, OV-DW can yield enhanced fictitious zero-modes.Indeed, the cyan points appear as outliers and the resulting χ t gets fictitious enhancements. Also, asmentioned for m (cid:39)
10 MeV, the histograms of GL-DW and GL-OV are similar. Because of this, χ t forGL-DW and GL-OV appear consistent. As the reweighting reduces the e ff ective number of statistics,we use GL-DW in comparison with GL-OV.The right panel of Fig. 3 shows χ t at m (cid:39) . T (cid:39)
220 MeV as a function of squaredlattice spacing a , where the finer lattice results are on the measured point and the coarser latticeresults are obtained by linear-interpolation from the nearest two points . The GL-DW result developsa large discretization error, and it gets close to OV-OV towards the continuum limit. The OV-OVresult is more stable against lattice spacing. All results suggest χ t is vanishing in the continuum limit.Focusing on the OV-OV result in the left panel the mass dependence of the topological suscepti-bility indicates two regions for mass: one is 0 < m (cid:46)
10 MeV where the observation of continuumscaling above strongly suggests χ t =
0. Actually, χ t with OV-OV is consistent with zero in this re-gion. The other is m (cid:38)
10 MeV where χ t is significantly non-zero. We note that the existence of theboundary at non-zero m is also suggested from GL-DW. While χ t > < m (cid:46)
10 MeV, it is The matching here is done with a constant bare mass in units of MeV. The logarithmic correction to an ideal matchingwith the renormalized mass should be negligible for this qualitative study, given that the mass dependence of topologicalsusceptibility is mild in the region in question. lmost constant. For χ t (cid:38)
10 MeV sudden development of χ t is observed. Due to its better precisionover OV-OV, GL-DW results may be useful to identify the location of the boundary.We note that a preliminary computation of the pion mass on the zero temperature configurationleads to an estimate of the physical ud quark mass as m = χ t = × × χ t [ M e V ] β =4.3 GL-DW β =4.3 OV-OVT=0 (N f =2+1)KY scenario from ∆ π−δ (m=3MeV)AFT scenario T=220 MeV
Figure 4.
Topological susceptibility χ t at T (cid:39)
220 MeVwith possible scenarios based onAoki-Fukaya-Taniguchi [9] (orange) andKanazawa-Yamamoto [10] (brown). Azero-temperature result [18] for N f = + Figure 4 shows a magnified view of the left panel of Fig. 3 without GL-OV and OV-DW. Thenewly added green line shows a zero temperature reference represented as a two-flavor ChPT fit with N f = + χ t with respect to quark mass vanish. With χ t = m = χ t = m < m c . The OV-OV result is consistent with thispicture with 10 (cid:46) m c (cid:46)
12 MeV. The AFT result is based on the analyticity of Dirac eigenvaluespectral density ρ ( λ ). On the other hand, Kanazawa-Yamamoto [10] (KY) claims that U A (1) shouldbe violated for T > T c due to its violation in the high enough temperature claimed in [19, 20]. Theyreported that the analyticity of ρ ( λ ) needs to be abandoned for the U A (1) violation. There is a KYscenario of χ t ( m ) given in [10]. To evaluate χ t ( m ), one needs to know the value of a low energyconstant, which may be extracted from the U A (1) order parameter measured with fixed topology. Atthe lightest mass where the topological charge is practically fixed at | Q t | = U A (1) breaking parameter ∆ π − δ [14] and obtain the browncurve ( ∝ m ) in the figure. This curve shows how χ t ( m ) behaves if the U A (1) symmetry were violatedin the thermodynamic limit. Comparing with our OV-OV result, it has a tension ( > σ ) at m (cid:39) T (cid:38) MeV
To check whether the jump of the topological susceptibility observed at T (cid:39)
220 MeV persists atother temperatures, ensembles with di ff erent N t with fixed β = . N t =
10 and 8 with fixed L =
32 as for N t =
12. The correspondingtemperatures are T (cid:39)
264 and 330 MeV respectively. Figure 5 shows the topological susceptibilityas a function of quark mass for three di ff erent temperatures, where only GL-DW data are shown. Asimilar jump of χ t at finite quark mass is observed also for T (cid:39)
264 and 330 MeV. The position of thejump shifts toward larger mass as T is increased.
20 40 60 80 100 m [MeV] χ t [ M e V ] T=220 MeVT=264 MeVT=330 MeV
GL-DW a =0.075 fm Figure 5.
Topological susceptibility χ t at T (cid:39) a − (cid:39) . Topological susceptibility χ t in N f = T (cid:39)
220 MeV ensembles with N t =
12. The preliminary results suggest that forthe range of bare mass 0 ≤ m (cid:46)
10 MeV (which includes physical ud mass m (cid:39) χ t = m (cid:38)
10 MeV a sudden development of χ t starts. It is consistent with the prediction of Aoki-Fukaya-Taniguchi [9] with U A (1) symmetry restoration in the chiral limit, thus consistent with thedirect measurement of the order parameter of U A (1) [14]. If that were due to finite volume e ff ectsand eventually we were to see the breaking in the thermodynamic limit, then Kanazawa-Yamamoto[9] explains how the U A (1) order parameter at finite volume is related to χ t . The result has a > σ tension.We have examined the stability of the observation of the χ t = ff erent. We are now examining the volumedependence on the finer lattice used in this report to check this.Higher temperatures T (cid:39)
264 and 330 MeV are also studied with fixed lattice spacing a − (cid:39) . χ t as a function of the quark mass is also observed for these temperatures.The point where the change occurs shifts towards larger mass for higher temperature. To get moreinsight for this observation, a systematic study for these high temperatures in conjunction with the U A (1) order parameter is planned. Acknowledgments
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