Gauge Corrections to Strong Coupling Lattice QCD on Anisotropic Lattices
GGauge Corrections to Strong Coupling Lattice QCDon Anisotropic Lattices
Jangho Kim
Institut für Theoretische Physik, Goethe-Universität Frankfurt am Main, Max-von-Laue-Str. 1,60438 Frankfurt am Main, GermanyE-mail: [email protected]
Marc Klegrewe
Fakultät für Physik, Universität Bielefeld, Universitätstasse 25, D33619 Bielefeld, GermanyE-mail: [email protected]
Wolfgang Unger ∗ Fakultät für Physik, Universität Bielefeld, Universitätstasse 25, D33619 Bielefeld, GermanyE-mail: [email protected]
Lattice QCD with staggered fermions can be formulated in dual variables to address the finitebaryon density sign problem. In the past we have performed simulations in the strong couplingregime, including leading order gauge corrections. In order to vary the temperature for fixed β it was necessary to introduce a bare anisotropy. In this talk we will extend our work to includeresults from a non-perturbative determination of the physical anisotropy a σ / a τ = ξ ( γ , β ) , whichis necessary to unambiguously locate the critical end point and the first order line of the chiraltransition. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J a n auge Correctinos on Anisotropic Lattices Wolfgang Unger
1. Introduction
Despite many attempts and partial successes to address the finite density sign problem in latticeQCD, a solution applicable to the full parameter space (temperature T , baryon chemical potential µ B , quark mass m q and lattice gauge coupling β ) has not yet been established. Here we reporton the incremental progress to unravel the phase diagram in the strong coupling regime of latticeQCD with staggered fermions, based on a leading order strong coupling expansion valid to O ( β ) [1, 2, 3]. The recent progress to address higher order corrections [4] are not yet considered in fullMonte Carlo simulations.The phase diagram of lattice QCD in the strong coupling limit has been investigated sincemore than 30 years [5, 6, 7, 8] and is by now well known, with the Worm algorithm as a mainMonte Carlo tool to investigate its features [9, 10, 11]. Beyond the strong coupling limit, theleading order gauge corrections have been included as well, but ambiguities on the phase boundaryarising when using different N τ have not yet been addressed. These ambiguities have so far onlybeen successfully resolved in the strong coupling limit (both in the chiral limit [12] and at finitequark mass [13]).The long-term goal is to extend the validity of the strong coupling expansion to answer animportant question on the existence of the critical end point (CEP): At strong coupling, the CEPhas been located at ( a µ cB , aT c ) = ( . ( ) , . ( )) in the chiral limit (where the CEP turns into atri-critical point TCP), and its quark mass dependence has been investigated, with tri-critical scaling ∝ m / q for small quark masses [14]. The dependence of the location of the CEP as a function of β has not yet been determined. Whether the CEP also exists in the continuum limit remains an openquestion. First hints can be obtained by monitoring the β -dependence of the CEP for small β : ifit moves to smaller µ B (and if this behaviour is monotonous), it may exist; if it moves to larger µ B , it may even vanish in the continuum limit and the chiral transition is for all values of µ B just acrossover.The main difficulty when mapping out the phase diagram is that we need to introduce a bareanisotropy γ in the strong coupling regime in order to vary the temperature continuously at fixedvalues of β . The temperature and chemical potential are however determined by the physicalanisotropy ξ ≡ a σ a τ , which depends non-perturbatively on γ and the lattice gauge coupling β . Herewe will report on how the β -dependence of ξ is determined, and present preliminary results whenapplied to the phase diagram in the strong coupling regime.
2. Dual formulation of lattice QCD
The strong coupling regime of lattice QCD can be formulated in a dual representation and itwas generalized recently to include in principle any order in β [4]. In this proceedings however,we only incorporate the leading order gauge correction O ( β ) as outlined in [2] and re-derivedin the appendix of [4]. It is based on a series expansion in terms of the (anti-) quark hopping¯ d µ ( x ) from the staggered Dirac operator, and plaquette occupation numbers n p , ¯ n p on plaquettecoordinates p = ( x , µ , ν ) from the Wilson gauge action. In contrast to previous formulations of the1 auge Correctinos on Anisotropic Lattices Wolfgang Unger dual partition sum, we now adopt the notation: k µ ( x ) = min (cid:8) d µ ( x ) , ¯ d µ ( x ) (cid:9) , f µ ( x ) = d µ ( x ) − ¯ d µ ( x ) , (2.1)where k µ ( x ) ∈ { , . . . N c } is the dimer number and f µ ( x ) ∈ {− N c , . . . N c } is the net quark flux . The k µ ( x ) are always quark-antiquark combinations, and color singlets formed by a quark and gluon areno longer regarded as dimers (in contrast to our previous formulation - the new convention is advan-tageous when higher order corrections are considered). The dual degrees of freedom { k , f , m , ¯ n , n } fulfill the gauge constraint at each link: f µ ( x ) + ∑ ν > µ (cid:20) δ n µ , ν ( x ) − δ n µ , ν ( x − ν ) (cid:21) − ∑ ν < µ (cid:20) µ ↔ ν (cid:21) = N c q µ ( x ) , (2.2)where for the O ( β ) partition function, q µ ( x ) ∈ {− , , } and δ n µ , ν ( x ) ≡ δ n p = n p − ¯ n p ∈ {− , , } .The Grassmann constraint at each lattice site is: m x + ∑ ± µ (cid:18) k µ ( x ) + | f µ ( x ) | (cid:19) = N c , ∑ ± µ f µ ( x ) = . (2.3)In terms of the above dual variables, and including a bare anisotropy γ , the partition functioncan be rewritten as: Z ( β , γ , µ q , ˆ m q ) = ∑ C = { n p , ¯ n p , k (cid:96) , f (cid:96) , m x } σ ( C ) ∏ p ˜ β n p + ¯ n p n p ! ¯ n p ! ∏ (cid:96) =( x , µ ) e µ q δ µ , f µ ( x ) γ δ µ , ( | f µ ( x ) | + k µ ( x ) ) k (cid:96) ! ( k (cid:96) + | f (cid:96) | ) ! ∏ x ( m q ) m x m x ! T i ( C x ) (2.4)with ˜ β = β N c , the quark chemical potential µ q = N c µ B . The three non-trivial vertex weights T = N c ! √ N c , T = ( N c − ) ! , T = N c ! √ N c (2.5)depend on the local degrees of freedom C x = { m x , k µ ( x ) , f µ ( x ) , n µν ( x ) , ¯ n µν ( x ) } and are employedwhenever some n µν ( x ) > n µν ( x ) >
0) and some f µ ( x ) >
1. For N c =
3, the sign σ ( C ) = ∏ (cid:96) σ ( (cid:96) ) ∏ (cid:96) σ ( (cid:96) ) , σ ( (cid:96) ) = ( − ) + w ( (cid:96) )+ N − ( (cid:96) ) ∏ ˜ (cid:96) η µ ( x ) (2.6)factorizes into single fermion ( | f µ ( x ) | =
1) and triple fermion loops ( | f µ ( x ) | = O ( β ) , see [4]. The dual degrees of freedom are color singlets which are nolonger just baryons and mesons as in the strong coupling limit: the gauge corrections will resolvethe quark structure of the point-like baryons and mesons, making them effectively spread out overone or more lattice spacings. The reason why the sign problem is mild in the strong coupling limitis that baryons are heavy, where ∆ f (cid:39) − . This is still approximately true for β (cid:46)
1, where thesign problem remains manageable. For details see [3].In the following we will consider the chiral limit of the partition function Eq. (2.4), whichimplies m x = U ( ) V × U ( ) : χ ( x ) (cid:55)→ e i ε ( x ) θ A + i θ V χ ( x ) , ε ( x ) = ( − ) x + x + x + x , (2.7)with U ( ) V the baryon number conservation and U ( ) the remnant chiral symmetry which isbroken spontaneously at low temperatures and densities. In Sec. 4 we will address the chiral criticalline that terminates in a tri-critical point before turning first order.2 auge Correctinos on Anisotropic Lattices Wolfgang Unger
3. Anisotropy Calibration at finite β It is crucial to understand the relationship between the bare anisotropy γ and the non-perturbativeanisotropy ξ ≡ a σ a τ (with a ≡ a σ the spatial and a τ the temporal lattice spacing) in order set the tem-perature and chemical potential consistently for various N τ . Anisotropic lattices are necessary inthe strong coupling regime since at fixed β this is the only way to vary the temperature continu-ously [15, 16]. The precise correspondence between ξ and γ has been established in the strongcoupling limit and in the chiral limit [12], resulting in ξ ( γ ) ≈ κγ + γ + λ γ , κ = . ( ) , (3.1)and at finite quark mass in [13], where it was shown that κ ( m q ) = lim ξ → ∞ ξγ has a simple massdependence in the strong coupling limit. The basic idea of the anisotropy calibration is to identifya conserved current and scan in γ such that the lattice is physically isotropic for a fixed aspect ratio: N σ a σ ! = N τ a τ ⇒ ξ = N τ N σ . (3.2)The conserved current is related to the pion [17] j µ ( x ) = ε ( x ) (cid:18) k µ ( x ) − | f µ ( x ) | (cid:19) , (3.3)with ε ( x ) = ± x . Eq. (3.3) is the generalization of the strong coupling limit(where f µ ( x ) ∈ {− N c , , N c } is the baryon flux through that link) to incorporate gauge corrections.This allows us to extend the anisotropy calibration to finite β to obtain ξ ( γ , β ) . Away from thestrong coupling limit it is in principle necessary to include a second bare anisotropy γ G in thegauge part β n p + ¯ n p → β n p σ + ¯ n p σ σ β n p τ + ¯ n p τ τ , γ G = (cid:115) β τ β σ (3.4)and then scan in both the fermionic and gauge anisotropy to obtain ξ ( γ , γ G , β ) . On finer latticesthis is indeed necessary [7], but in the strong coupling regime, where we cannot set a scale, it is anunnecessary complication: as β is increased, the lattices needed to study the chiral phase transitionwill eventually become isotropic, and beyond this point, the temperature is varied via a ( β ) . In thisproceedings, we will always set γ G = left ) we show the anisotropy calibration for various fixed β : On lattices N σ × N τ with aspect ratios ξ = , , , , , γ ( ξ ) where the ratio of the temporal andspatial fluctuations of the conserved charge Q t , Q s are equal. This is repeated for various β . Sincethe partition function Eq. (2.4) depends on γ and N τ , the bare (mean field) temperature [ aT ] mf = γ N τ needs to be corrected by the non-perturbative factor [ ξ / γ ] β , shown in Fig. 1 ( right ), to yield thecorrect temperature aT = ξ ( γ ) N τ . Our result allows to define the Euclidean continuous time limit a τ → β : the temperature and chemical potential are then defined as aT = κ ( β )[ aT ] mf , a µ B = κ ( β )[ a µ B ] mf with κ ( β ) = lim ξ → ∞ [ ξ / γ ] β . (3.5)3 auge Correctinos on Anisotropic Lattices Wolfgang Unger γ Ratio of spatial and temporal variances /
ξ =2 β =0.0, γ =1.557(2) β =0.1, γ =1.558(1) β =0.2, γ =1.558(2) β =0.5, γ =1.559(2) β =1.0, γ =1.561(1) ξ / γ ξ Extrapolation N t → ∞ ( ξ → ∞ ) β =0.0, κ =0.783 β =0.1, κ =0.781 β =0.2, κ =0.779 β =0.5, κ =0.774 β =1.0, κ =0.766 β =2.0, κ =0.750 β =3.0, κ =0.735 Figure 1:
Left:
Determination of γ for various β by requiring the ratio of charge fluctuations to be equal,shown for ξ = Right:
Extrapolation of the correction factor ξ / γ towards continuous time to yield κ ( β ) .
4. Gauge Corrections to the Phase Diagram and Density of States
We will now focus on a particularly important application of the previous result: the modifi-cation of the chiral transition within the grand-canonical phase diagram, when taking into accountthe non-perturbative definition of temperature and chemical potential Eq. (3.5). In Fig. 2 we showthe effect of applying the β -dependent correction factor [ ξ / γ ] β to the phase boundary, for thevarious β in a regime where the sign problem is manageable. All data have been measured viathe Worm algorithm in combination with plaquette updates, on lattices N σ × N τ > f µ ( x ) form world lines, and the total number ofquark fluxes wrapping around in temporal direction is a multiple of N c due to the gauge constraintEq. (2.2), it is possible to define baryon number sectors N B ∈ {− N σ , . . . , N σ } and allow updatesthat modify the baryon number by one unit. We will explain the details of the canonical simulationsand the resulting canonical phase diagram is in the n B − T plane in a forthcoming publication. Theanalysis of the density of states in N B as shown in Fig. 3 can yield additional insights concerningthe first order phase boundary below the TCP: the density of states is weighted with e N B µ B / T forvarious β to the critical chemical potential µ st B , where the peak heights are equal. We observe thatthe first order transition weakens with β , and that the the critical chemical potential µ st B increasesonly slightly with β . This is in agreement with the findings of the β -dependence of the nucleartransition at low temperatures on isotropic lattices [3].4 auge Correctinos on Anisotropic Lattices Wolfgang Unger st and 2 nd order transitionaT MF a µ B,MF st order β =0 β =0 β =0.3 β =0.6 β =0.9TCP at β =0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6chiral 1 st and 2 nd order transitionaT NP a µ B,NP st order β =0 β =0 β =0.3 β =0.6 β =0.9TCP at β =0 Figure 2:
Comparison of the phase boundary with the mean field definition of the temperature (left) andits non-perturbative counterpart (right) , resulting in a collapse of the first order line for all values of β considered. µ c β =0.1 / µ c β =0 =1.0038 µ c β =0.2 / µ c β =0 =1.0104 P ( n B ) n B T/T
TCP =1.0 β=0.0β=0.1β=0.2 µ c β =0.1 / µ c β =0 =1.0036 µ c β =0.2 / µ c β =0 =1.0111 P ( n B ) n B T/T
TCP =0.9 β=0.0β=0.1β=0.2 µ c β =0.1 / µ c β =0 =1.0031 µ c β =0.2 / µ c β =0 =1.0091 P ( n B ) n B T/T
TCP =0.8 β=0.0β=0.1β=0.2
Figure 3:
The density of states weighted to the critical chemical potential µ cB , showing a double peakstructure for aT < aT TCP . The value of µ st B only very mildly grows with β .
5. Conclusions
We determined the non-perturbative relation between the bare anisotropy γ and the latticeanisotropy ξ = aa t at finite β in the range of validity β ≤
1, based on the leading order partitionfunction. The results have been used to define the temperature and baryon chemical potentialunambiguously. The extrapolation a t → β in the future.The main (still preliminary) finding on the phase boundary of lattice QCD in in the chiral limitis that the first order line is not β -dependent after the non-perturbative correction of the temperatureand chemical potential. This is consistent with mean-field theory [18] and results on isotropiclattices. Whether the first order line is β -dependent for β > auge Correctinos on Anisotropic Lattices Wolfgang Unger moves to larger or smaller chemical potential when β is increased requires further investigation.Most likely higher order corrections need to be included, as outlined in [4].We have also presented first results on the β -dependence of the density of states in the baryonnumber, from which the canonical phase diagram can be determined. Even though this dependenceis very weak, this method has the potential to discriminate between the chiral and nuclear transitionand address the question whether they split, as is expected: in the continuum, chiral symmetryshould still be broken in the nuclear phase, resulting in two separate first order transitions at lowtemperatures. Acknowledgments
We thank Aaron von Kamen for his contributions to the Wang-Landau method, and GiuseppeGagliardi for discussions on the partition function. Numerical simulations were performed onthe OCuLUS cluster at PC2 (Universität Paderborn). This work is supported by the DeutscheForschungsgemeinschaft (DFG) through the Emmy Noether Program under grant No. UN 370/1and through the CRC-TR 211 ’Strong-interaction matter under extreme conditions’– project num-ber 315477589 – TRR 211.
References [1] P. de Forcrand, J. Langelage, O. Philipsen, and W. Unger
Phys. Rev. Lett. (2014), no. 15 152002,[ ].[2] G. Gagliardi, J. Kim, and W. Unger
EPJ Web Conf. (2018) 07047, [ ].[3] J. Kim, Philipsen, Owe, and W. Unger
PoS
LATTICE2019 (2019).[4] G. Gagliardi and W. Unger .[5] N. Kawamoto and J. Smit
Nucl. Phys.
B192 (1981) 100. [,556(1981)].[6] P. Rossi and U. Wolff
Nucl. Phys.
B248 (1984) 105–122.[7] F. Karsch and K. H. Mutter
Nucl. Phys.
B313 (1989) 541–559.[8] N. Kawamoto, K. Miura, A. Ohnishi, and T. Ohnuma
Phys. Rev.
D75 (2007) 014502,[ hep-lat/0512023 ].[9] D. H. Adams and S. Chandrasekharan
Nucl. Phys.
B662 (2003) 220–246, [ hep-lat/0303003 ].[10] P. de Forcrand and M. Fromm
Phys. Rev. Lett. (2010) 112005, [ ].[11] W. Unger and P. de Forcrand
PoS
LATTICE2011 (2011) 218, [ ].[12] P. de Forcrand, W. Unger, and H. Vairinhos
Phys. Rev.
D97 (2018), no. 3 034512, [ ].[13] W. Unger, D. Bollweg, and M. Klegrewe
PoS
LATTICE2018 (2018) 181, [ ].[14] J. Kim and W. Unger
PoS
LATTICE2016 (2016) 035, [ ].[15] L. Levkova, T. Manke, and R. Mawhinney
Phys. Rev.
D73 (2006) 074504, [ hep-lat/0603031 ].[16] M. Klegrewe and W. Unger
PoS
LATTICE2018 (2018) 182, [ ].[17] S. Chandrasekharan and F.-J. Jiang
Phys. Rev.
D68 (2003) 091501, [ hep-lat/0309025 ].[18] K. Miura, N. Kawamoto, T. Z. Nakano, and A. Ohnishi
Phys. Rev.
D95 (2017), no. 11 114505,[ ].].