SScreening masses towards chiral limit ∗ Simon Dentinger † , Olaf Kaczmarek, Anirban Lahiri Fakult¨at f¨ur Physik, Universit¨at Bielefeld, D-33615 Bielefeld, GermanyA possible eﬀective restoration of the anomalous U A (1) symmetry wouldhave a non-trivial eﬀect on the global phase diagram of QCD. In this workwe investigate the eﬀective restoration of the U A (1) through the calculationof scalar and pseudo-scalar screening masses and corresponding susceptibil-ities, for physical and lower than physical pion masses. Calculations havebeen performed in (2+1)-ﬂavor HISQ discretization scheme with a physicalvalue of the strange quark mass. Preliminary calculations of the contin-uum extrapolated scalar and pseudo-scalar masses are presented, based onlattices with three diﬀerent temporal extent. Non-trivial structure of thediﬀerence between scalar and pseudo-scalar susceptibilites are discussed for N τ = 8 lattices. U A (1) symmetry and screening masses One important aspect for understanding the global QCD phase diagramis the fate of the anomalous U A (1) symmetry at the chiral phase transition,in the chiral limit m l →
0, for vanishing chemical potential µ B = 0. If U A (1)remains broken in the chiral limit then the chiral transition is expected to beof second-order belonging to the O (4) universality class . On the otherhand if the U A (1) gets eﬀectively restored at the chiral phase transitionthen the later can be of ﬁrst-order  or second-order belonging to otheruniversality classes . Therefore the study of eﬀective restoration of theanomalous U A (1) towards the chiral limit is very important to decide thenature of the chiral phase transition. Existing calculations using staggeredfermions [3–5], overlap and M¨obius domain wall fermions [6–8] as well asWilson fermions  are still inconclusive on the eﬀective restoration of theanomalous U A (1) symmetry. In this contribution we report on an attemptto study the breaking or restoration of the anomalous U A (1) through thedegeneracy between pseudo-scalar (iso-vector) and scalar (iso-vector) meson ∗ Presented at Criticality in QCD and the Hadron Resonance Gas, 29-31 July 2020,Wroc(cid:32)law, Poland. † Speaker (1) a r X i v : . [ h e p - l a t ] F e b S. Dentinger et al. channels which are related by U A (1) symmetry, towards the chiral limit.The explicit breaking of U A (1) through the quark mass term provides abackground which is supposed to vanish in the chiral limit, i.e. m l →
2. Screening masses for lower than physical quark masses
We have calculated screening correlators in (2+1)-ﬂavor QCD using thehighly improved staggered quark (HISQ) action and tree level improvedSymanzik gauge action. Screening mass calculations have been done follow-ing ref. , where details about the staggered correlation functions and ﬁtansatz for diﬀerent channels can be found. To approach the chiral limit,we have decreased the light quark masses keeping the strange quark massﬁxed at its physical value. For details of the lattice setup see . Since adirect calculation at the chiral limit m l = 0 is not possible on the lattice,to obtain screening masses and the susceptibilities in the chiral limit, anextrapolation in mass apart from thermodynamic and continuum limit ex-trapolations would be required. It is known that the staggered formulationat any ﬁnite lattice spacing preserves only a subgroup of the continuumchiral symmetry, thus it is necessary to take the continuum limit prior tothe chiral extrapolation. To control the potential ﬁnite volume eﬀects weneed to take the thermodynamic limit ﬁrst. In the following subsections wediscuss these limits in detail. The thermodynamic limit of screening masses can be taken using thefollowing form  m N s /N τ = m N s →∞ /N τ (cid:18) b N τ (cid:18) N τ N s (cid:19) c (cid:19) , (1)where m N s /N τ is the screening mass calculated on a N s × N τ lattice. m N s →∞ /N τ ,being the mass in the thermodynamic limit, is a ﬁt parameter along with c and b N τ . It has been argued that  c = 3 for T = 0 and c = 1 for T → ∞ .Therefore it is assumed here that c ∈ [1 ,
3] for any T . Assuming that c onlydepends on temperature T and the number of temporal lattice points N τ ,while m N s →∞ /N τ and b N τ depend on N τ , channels ( e.g. PS, S) and temper-ature T , a combined ﬁt with a shared parameter c between pseudo scalarand scalar particles at ﬁxed N τ and T is possible.In Fig. 1 we show the screening masses in the scalar (S) and pseudo-scalar(PS) channels for three volumes at diﬀerent N τ together with the combinedthermodynamic limit extrapolations at two temperatures. For the lowertemperature volume eﬀects are visible, but the screening masses for the creening masses towards chiral limit .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . H o t Q C D p r e l i m i n a r y T = 148 MeV m s /m l = 80 N τ /N s m [MeV] S N τ = 6 S N τ = 8 S N τ = 12 PS N τ = 12 PS N τ = 8 PS N τ = 6 0 .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . H o t Q C D p r e li m i n a r y T = 166 MeV m s /m l = 80 N τ /N s m [MeV] S N τ = 8 PS N τ = 8 Fig. 1: Inﬁnite volume extrapolations of scalar (S, red) and pseudo scalar(PS, blue) channels at T ≈
148 MeV (left) and T ≈
166 MeV (right) formass ratio m s /m l = 80.largest volumes in the plot agree with the inﬁnite volume extrapolated valueswithin errors in most cases. For this temperature we have shown the volumeextrapolation for 3 diﬀerent N τ . For each N τ we have performed the volumeextrapolation jointly to S and PS masses. For the higher temperature casethe two largest volumes agree very well with the extrapolation results whichin turn shows that volume eﬀects are smaller for higher T as also seen forother chiral observables . From the ﬁts the parameter c comes outto be 1 . . . T ≈
148 MeV at N τ = 6 , , . T ≈
166 MeV. Fig. 1 suggests that the use ofthe screening masses from the largest available volumes is well justiﬁed fortemperatures close to the pseudo critical temperature and above. This isfurther conﬁrmed in the following section where we perform the continuumextrapolation at lower temperature of Fig. 1 based on the screening massresults on the largest volume in comparison to the screening masses obtainedin the thermodynamic limit.
In Fig. 2 we show the continuum limit extrapolation for T ≈
148 MeVwith a linear ansatz in 1 /N τ . In the left panel we show the continuumlimit extrapolation of the inﬁnite volume extrapolated screening masses,obtained from Fig. 1 and in the right panel we show the continuum ex-trapolation of the screening masses from the largest available volumes. Ourpreliminary calculations give m S = 191(19) MeV and m P S = 120(5) MeV,in the continuum limit when we use the inﬁnite volume extrapolated masses(left panel). Using the masses from the highest volumes (right panel) the
S. Dentinger et al. .
000 0 .
005 0 .
010 0 .
015 0 .
020 0 . T = 148 MeV m s /m l = 80 H o t Q C D p r e li m i n a r y /N τ m [MeV] S N s = ∞ PS N s = ∞ .
000 0 .
005 0 .
010 0 .
015 0 .
020 0 . T = 148 MeV m s /m l = 80 H o t Q C D p r e li m i n a r y N s = 48 N s = 56 N s = 72 1 /N τ m [MeV] SPS
Fig. 2: Continuum extrapolation of scalar (S, red, upper) and pseudo scalar(PS, blue, lower) channels for T ≈
148 MeV for mass ratio m s /m l = 80.Left: Thermodynamic limit of the screening masses have been taken be-forehand for each N τ , see left plot of Fig. 1. Right: Screening masses fromlargest volume only.continuum results are m S = 173(5) MeV and m P S = 129(1) MeV. Sincethe two continuum extrapolations give results which agree within 95% con-ﬁdence interval, for further analyses we use the screening masses from thelargest available volumes.Since the temperature values for diﬀerent N τ do not exactly match always,temperature interpolations are needed to have the continuum extrapolationfor the entire temperature range available. Here we have used cubic splineswith node positions determined by the density of the data points . Thesetemperature interpolations for N τ = 6 , /N τ . Detailsabout the continuum extrapolations can be found in ref. .In Fig. 3 we show preliminary results of this continuum limit extrapolationin the full available temperature range for mass ratio m s /m l = 80. At lowtemperatures, T = 25 MeV and T = 50 MeV, the ﬁt is constrained on theﬁrst derivative to be equal to zero. For high temperature, well above theavailable temperature region of our study, a constraint at T = 1 . π has been used. The gray band showsa continuum estimate of the pseudo-critical temperature for m s /m l = 80 of146 . ± . creening masses towards chiral limit
140 145 150 155 160 165150175200225250275300325350
Scalar continuum T [MeV] m [MeV] H o t Q C D p r e li m i n a r y N τ = 6 N τ = 8 N τ = 12cont.
140 145 150 155 160 165100125150175200225250275
Pseudo scalar continuum T [MeV] m [MeV] H o t Q C D p r e li m i n a r y N τ = 6 N τ = 8 N τ = 12cont. Fig. 3: Continuum extrapolation for screening masses over the entire avail-able temperature range for m s /m l = 80. Left: Scalar (S). Right: Pseudo-scalar (PS). .
95 1 .
00 1 .
05 1 .
10 1 .
15 1 .
20 1 .
25 1 .
30 1 . ud H o t Q C D p r e li m i n a r y m s /m l T/T pc , cont . ( m s /m l ) m scr . [MeV] S PS S PS Fig. 4: Comparison of the continuum extrapolated screening masses for m s /m l = 27 and m s /m l = 80 as function of temperature scaled by therespective pseudo-critical temperatures in the continuum limit.It is well known that for low temperatures ( T < T pc ) the scalar channelsuﬀers from unphysical decay modes in the staggered discretization. Fora detailed discussion we refer the reader to ref.  and references therein.Above T pc this problem is expected to be mild or absent and one may usethe degeneracy between the S and PS channel masses to explore the eﬀec-tive restoration of the anomalous U A (1) symmetry. We show the continuumextrapolated results for the scalar (S) and pseudo-scalar (PS) channels fortwo quark mass ratios, m s /m l = 27 (taken from ) and m s /m l = 80, inFig. 4. One can see that the scalar and pseudo-scalar meson masses become S. Dentinger et al. degenerate which indicates that U A (1) is eﬀectively restored around the tem-perature region of T ≈ −
175 MeV corresponding to
T /T pc ∼ . − . m s /m l = 80. Although a better controlled continuum extrapolation and re-sults at higher temperatures would be helpful in this respect, at this point itis interesting to compare the situation to physical pion mass, m s /m l = 27, inFig. 4. One can see that the scalar and pseudo-scalar masses start becomingdegenerate at about 200 MeV corresponding to T /T pc ∼ . m s /m l = 27.Primarily it might be tempting to conclude that with decreasing pion massthe degeneracy between S and PS channel shifts to lower T /T pc and resultsfor lighter quark masses and a chiral extrapolation will be important to fullystudy this question in the future.
Keeping in mind the issue in the scalar channel mass related to thestaggered discretization, it might be a better way to investigate U A (1) byusing the integrated correlation functions (or susceptibilities) for scalar andpseudo-scalar mesons directly. Susceptibilities for pseudo-scalar ( π ) andscalar ( a ) channel for staggered discretization are deﬁned as χ π = N s − (cid:88) n =0 G M ( n ) and χ a = − N s − (cid:88) n =0 ( − ) n G M ( n ) (2)respectively. Here n denotes the distance between source and sink in latticeunits and for the deﬁnition of the staggered correlators G M and G M , werefer the reader to ref. . An advantage of susceptibilities is that theyare independent of multiple state ﬁts which are used for extracting screen-ing masses. It has been shown earlier  that with the HISQ action forphysical pion mass the degeneracy between the pseudo-scalar ( χ π ) and thescalar ( χ a ) susceptibilities happens around the degeneracy temperaturefor masses, which may not be too surprising because the restoration of asymmetry should be imprinted in the degeneracy of the corresponding cor-relators and therefore also in their large distance decay that determines thescreening masses. In Fig. 5 we show the diﬀerence between the pseudo-scalar ( χ π ) and scalar ( χ a ) as a function of temperature for various valuesof m l /m s for lattices with N τ = 8. The susceptibilities are renormalized by m s and normalized by the kaon decay constant f K to make it dimension-less. At low temperatures, in the chirally broken phase, the rapid increaseof the diﬀerence between χ π and χ a can be expected owing to the wardidentity χ π = (cid:104) ¯ ψψ (cid:105) l /m l and to the fact that in the chiral limit below thetransition temperature the chiral order parameter (cid:104) ¯ ψψ (cid:105) l is a constant dueto the spontaneous breaking of the chiral symmetry. Thus the leading mass creening masses towards chiral limit
135 140 145 150 155 160 165 170 175020040060080010001200140016001800200022002400 H o t Q C D p r e li m i n a r y N s m s /m l T [MeV] m s ( χ π − χ a ) /f K N s × Fig. 5: Temperature variation of χ π and χ a for various values of m l /m s for N τ = 8 lattices.dependence is expected to be proportional to 1 /m l in χ π . On the contraryfor χ a , the leading contribution is expected to come due to the well knownGoldstone eﬀect which vanishes in the continuum limit for 2 light ﬂavors ofthe staggered quarks . Although at ﬁnite lattice spacing there will be aﬁnite contribution proportional to 1 / √ m l in χ a .For temperatures around and above the chiral phase transition temper-ature, the mass dependence of the diﬀerence seems to be quite non-trivial.To show this more clearly, in Fig. 6 we show the variation of the diﬀerencebetween χ π and χ a for various temperatures. For the two highest temper-atures the diﬀerences tend to approach zero towards the chiral limit, quitesmoothly. As can be seen from Fig. 6, for T = 156 MeV the diﬀerenceslightly increases and then seems to decrease towards chiral limit. This in-crease becomes more evident when the temperature further decreases andapproaches T c and then this turning around happens for even lower masses.So the most non-trivial task is to ﬁnd that after turning whether the diﬀer-ence vanishes or stays ﬁnite in the chiral limit. In this respect it is worthto mention that in a chirally symmetric background the diﬀerence between χ π and χ a is exactly equal to the disconnected part of the chiral suscep-tibility  and the behaviour of the later is found to be almost the samearound the temperature regime shown in Fig. 6. Of course, it will be aninteresting task to repeat the analyses of Fig. 5 and Fig. 6 for other N τ andthen to take a continuum limit. Work is ongoing in that direction and willbe reported elsewhere. S. Dentinger et al. .
00 0 .
01 0 .
02 0 .
03 0 . H o t Q C D p r e li m i n a r y
55 80 110 140 m π [MeV] / N s = 321 / N s = 401 / N s = 561 / N s = 56 N s × m l /m s T [MeV] m s ( χ π − χ a ) /f K Fig. 6: Variation of the diﬀerence between χ π and χ a with respect to quarkmass for various values of temperatures for N τ = 8 lattices. Dotted linesare cubic splines just for guiding the eye.
4. Summary and Outlook
In this contribution we have presented the calculations of screeningmasses in the pseudo-scalar (iso-vector) and scalar (iso-vector) channelswithin (2+1)-ﬂavor HISQ discretization scheme with strange quark massbeing ﬁxed at its physical value and for the physical and lower than physi-cal values of the (degenerate) light quark masses. Since for the lighter thanphysical pions, the ﬁnite volume eﬀect is one of the main concern, we ﬁrstdiscussed the ﬁnite volume eﬀects for screening masses and also discussedthe extrapolation to the thermodynamic limit. It is shown that for most ofthe temperatures under consideration the screening masses for the largestavailable volumes agree with the same in the thermodynamic limit withinuncertainties. Finite volume eﬀects for higher (compared to T pc ) tempera-tures are found to be even smaller. Furthermore we show that the PS andS masses in the continuum limit with inﬁnite volume extrapolated massesor using masses from the highest available volumes for diﬀerent N τ , matchwithin 95% conﬁdence interval. Due to this small systematic uncertaintyoriginating from the thermodynamic extrapolations we can safely proceedwith continuum extrapolation of masses available on largest volumes. Thenwe compare the continuum extrapolated screening masses in PS and S chan-nels for m s /m l = 27 and preliminary results for m s /m l = 80 to understandwhether there exists any trend in degeneracy temperature with decreasing creening masses towards chiral limit pion mass. Results for even lighter quark mass will help to understand thesituation better in the future. As the staggered scalar channel suﬀers fromunphysical decay modes, we also presented calculations of susceptibilites(for N τ = 8) for PS and S channels and their diﬀerence as a measure of the U A (1) breaking. Preliminary calculations show that the degeneracy of thePS and S susceptibilites occur around the degeneracy temperature of thecorresponding screening masses. Although a non-trivial behaviour of thediﬀerence between scalar and pseudo-scalar susceptibilities makes it harderto make deﬁnite comments about chiral limit. More exploration in thisdirection is needed and planned for the future. Acknowledgments
This work was supported by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) - Project number 315477589-TRR 211 andthe grant number 05P18PBCA1 of the German Bundesministerium f¨ur Bil-dung und Forschung. REFERENCES  R. D. Pisarski and F. Wilczek.
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