A model for QCD at High Density and large Quark Mass

Transcription

1 A model for QCD at High Density and large Quark Mass R. De Pietri, A. Feo, E. Seiler, I.-O. Stamatescu based on arxiv: [hep-lat]

2 Introduction The exploration of the phase diagram at non-zero baryon density is a challenging and interesting problem. Quark matter at extremely high density may behave as a color superconductor see the review, Alford, hep-lat/646 The phase diagram in the temperature-density plane shows multiple phases separated by various critical lines and, except for high T and small µ region, not much is known about their position and nature. Lattice gauge theory calculations in various implementations that try to evade the sign problem generated by non-zero chemical potential have been mostly performed at high temperature and small baryon density, where they agree quite well with each other.

3 Introduction-2 At large baryon density there are only few numerical results which need to be corroborated by using different methods The aim of this work is to understand the phase structure of high density, strongly interacting matter. Most works on QCD at non-zero density proceeds from attempts to go in the In the spirit of the µ = quenched approx. one can consider a nonzero density quenched approx. for µ =, T T c µ > domain. region and µ > κ, µ, ζ κe µ see the review, Karsch, hep-lat/683 based on the double limit fixed it represents a static, charged background. which influences the gluonic dynamics (Bender et al., Nucl. Phys. Proc. Suppl. 992; Engels et al. hep-lat/9933; Blum et al. hep-lat/9592)

4 Features In the present model the gluonic dynamic is enriched by the effects of dynamical quarks of large (but not infinite) mass. Aarts, Kaczmarek, Karsch, Laermann, hep-lat/9933; Hofmann, Stamatescu, hep-lat/3979 It is an effective model obtained by an expansion in the hopping parameter κ of the fermionic determinant up to the next-to leading order κ 2 in. Determinant contributions The main ingredient of the model are Polyakov-type loops capturing the effects of heavy quarks with low mobility. n # ~ # " # e $ N %# ) " ' C # e $$ N ) " ~ N e $ " # 2 %# ) '# % C } Contributions to order# % T = / N a " " " %

5 Features-2 We approach the large mass density problems using our model as: - An approximation to QCD near the large mass/large µ limit - As a model by itself at any µ, β, κ and N τ as independent parameters - Allows simulations in the large density, small temperature region Here the model shows the sign problem but due to the factorization of the fermionic determinant it permits to develop local algorithms with reweighting and achieve large statistics

6 QCD at non zero µ Z(β, κ, γ G, γ F, µ) = [DU] e S G(β,γ G,{U}) Z F (κ, γ F, µ, {U}), ( S G (β, γ G, {U}) = β N c Re Tr γ G 3 j>i= P ij + γ G i ( The exponential prescription for mu Z F (κ, γ F, µ, {U}) = Det W (κ, γ F, µ, {U}), ( ensures cancelling of divergences in the 3 small a limit [Hasenfratz and Karsch, Phys. Lett. B W ff = δ ff [ κ f (Γ +i U i T i + Γ i Ti Ui ) (983)] i= P i4 κ f γ F ( e µ f Γ +4 U 4 T 4 + e µ f Γ 4 T 4 U 4 ) ], Γ ±µ = ± γ µ, γ µ = γ µ, γ 2 µ =, κ = 2(M γ F cosh µ) = 2(M γ F ), f : flavor index; U: links, T: lattice translations We use the grand canonical formulation of QCD, i.e., we introduce the chemical potential µ as a (bare) parameter and use Wilson fermions M: bare mass ; : bare mass at µ = M

7 Sign problem The fermionic coupling matrix W fulfills: U n,ν = U (n+ν), ν At γ 5 W (µ)γ 5 = W ( µ) ; DetW (µ) = DetW ( µ) µ the determinant is complex. Numerical simulations are based on the important sampling of the configurations. To work they need that the Boltzmann factor being real If it is not real and positive definite it does not define a probability measure for the Yang-Mills integration. There have been different methods devised to cope with this problem which all involve simulating a different ensemble and correcting the results either by continuing in µ or by redefining the observables. B = e S G(β,{U}) Z F (κ, µ, {U}) O B The reweighting method proceeds by choosing a positive definite measure B using the modulus of the fermion determinant inside the Boltzman factor. B = e S G(β,{U}) Z(κ, µ, {U}) w = e i arg(z(κ,µ,{u})) O B = w O B w B

8 Hopping parameter expansion Det W = exp(tr ln W ) = exp = l= {C l } f l= {C l } s= (κ l f gf C l ) s Det Dirac,Color ( (κ f ) l g f C l L Cl ) s Tr D,C L s C l (4) C l are distinguishable, non-exactly-self-repeating closed paths of length l, s is the number of times a loop L Cl covers C l, g f C l = ( ɛ e ±N τ µ f ) r if Cl = P olyakov r path, (5) = otherwise. A P olyakov r path closes over the lattice in the ±4 direction with winding number r and periodic(antiperiodic) b.c. [ɛ = +( )]. 7 C. Alexandrou et al., hep-lat/9828 PRD; De Forcrand, Laliena, hep-lat/9974 PRD; Alford et al. hep-lat/2 NPB, Schaefer et al., arxiv: ; Hasenfratz et al., Phys Lett. B 983.

9 The quenched limit at µ > Z [] Z F [] F κ, µ, (C, {U}) = exp (C, {U}) = exp = { x} = { x} { x} κ e µ ζ : fixed Det C ( ɛ CP x ) 2, 2 { x} (ɛc) s s Tr C (P x ) s s= s= s C = (2 ζ) N τ [Bender et al, Nucl. Phys. B (Proc.Suppl.) 26 (992) 323; Blum et al., Nucl. Phys. B (Proc.Suppl.) 47 (996) 543; Engels et al, Nucl. Phys. B 558 (999) 37] (ɛc) s Tr (P x ) s Det (I ɛ CP x ) 2, C = (2 ζ) N τ, This double limit produces a static, dense, charged background on the lattice and have been studied as a non-zero quenched approximation P x N τ t= U ( x,t),µ Effects expected to be due to the mobility of charges, in particular new phases in dependence of the chemical potential cannot be studied here

10 Next order corrections Z [2] F (κ, µ, {U}) = exp Tr (P x ) s + κ 2 = Z [] F (C, {U}) 2 { x} s= (ɛ C) s r,q,i,t,t (ɛ C) s(r ) (P r,q x,r,q,i,t,t Det U n,4 =, except for U ( x,n4 =N τ ),4 V x : free, P r,q x,i,t,t = (V x ) r q U ( x,t),i (V x+î ) q U ( x,t ),i r > q, i = ±, ±2, ±3, t t N τ (t < t for q = ) (3) s x,i,t,t ) s ( I (ɛ C) r κ 2 P r,q x,i,t,t ) 2. For easy bookkeeping we use the temporal gauge Determinant contributions n # ~ # " # Aarts, Kacmarek, Karsch, Stamatescu, hep-lat/45 e $ N %# ) " ' C N e $ " # 2 %# ) '# % C } T = / N a " " # e $$ N ) " ~ " % Contributions to order# %

11 Strong Coupling/Hopping parameter expansion "# "#%$ *+#, *+#,- *+#, *+#,- "# β = 3 β = 5 "#%$ *+#, *+#,- *+#, *+#,- "#% "#% "#"$ "#"$ " " "#'$ "#( "#($ "#) "#)$ % %#"$ "#'$ "#( "#($ "#) "#)$ % %#"$ FIG. 2: Comparison with strong coupling at β = 3 (upper plot) and β = 5 (lower plot), 4 4 lattice. Full symbols denote ReP, empty symbols ReP, the lines show the corresponding strong coupling results.

12 Tentative phase diagram Small mass QCD Large mass QCD T Quenched QCD T ~ 25 MeV c T ~ c µ =, full QCD (infinity) 5 MeV µ " µ c ~ log(m) T=, full QCD ~ / Mass µ c ~ m

13 Observables Polyakov loop ( ) n B = f n b,f T 3, Baryonic density χ nb = n 2 B n B 2, P = 3 Nσ 3 χ P = y n b T 3 = N 3 τ 3N 3 σ x Tr P x = N 3 σ ( P x P y P x P y ), ˆn, ˆn = ˆn + ˆn, ˆn = µ Z[] F 2C x ˆn = ( [2] Z F µ Z [] F Tr P x ) 2Cκ 2 x x P x, Tr P x,i,t,t,

14 Data analyzed µ FIG. : Data taken in the plane T vs. µ for fixed κ =.2.

15 Baryonic density ' ("# * % ) ("# µ 2 "# "$ ## #"% FIG. 3: Landscape of the baryonic density. The color scale (right) is based on log (n B ). ' % "' "# "$ )

16 B. Observables where the contribution of each flavour is: l observables under the variation of µ perties of the different phases for small ollowing we specialize to N c = 3. The olyakov loop, x Landscape of baryon density n b susceptibility Tr 5.9P x = N 3 σ x P x, (4.) µ ˆn.5 = with the corresponding.5 susceptibility 5.7 FIG. 3: Landscape of the baryonic density. The color scale (right) FIG. 5: 3d view of Fig. 4. is based on log (n B ) ( P x P y P x P y ), (4.2) 6.5 β. Here the parameters are such that we should observe only the transition χ 5.95 nb between = the n 2 hadronic and plasma phases. The indication for this again that theb n B 2, real parts touch the origin for 5.9 β 5.65, whereas for β > 5.65 they increase to positive.5 values, but staying below Both Fig. 23 and Fig. 24 show that the distribution is to good accuracy even in y, as required for the reality of P. In 5.75 Figs 25 and 26 we show the imaginary parts of the distributions T. The qualitative signal of the transitions/crossovers 3 Tr P σ ryon number 5.5 density n B, the spatial and temporal plaquettes is similar to that of ReT. It should be noted that now the 2 µ the topological 5.65 distributions susceptibility are, to very good precision, χ top odd in= y, again Q 2 in top agreement with the reality of P Polyakov loops and charge density (and their susceptibilities), have been the primary quantities used to uncover the FIG. 4: Landscape of the baryon density susceptibility. The color phase structure. We also have measured plaquette averages 5.5 B = topological charge was measured using an im n b,f (for both temporal and spatial plaquettes), the topological scale (right) T 3, (4.3) oretical formula based on five Wilson loop is based on log (χ nb ). µ f to check the character of the conjectured th charge density (using the field definition in conjunction with smoothing) and quark and di-quark correlators (in maximal T 3 = N 3 τ.5.5 ˆn = N 3 σ µ Z[] µ ˆn, ˆn = ˆn + ˆn, F 2C x ( Z [2] F Z [] F 2 Tr P x ) 2Cκ 2 x T

17 : Polyakov loop susceptibility vs. β at fixed µ. # ) ) % % " " Polyakov loop susceptibility "# "$ # #"% "# "$ # #"% "# "$ # #"% "# "$ #"% FIG. 3: Polyakov loop susceptibility vs. β at fixed FIG. 3: Polyakov loop susceptibility vs. at fixed *+' *+' (,"," %" %" % % '" '" ' ' " " "# "# "$ "$ # # #"% #"% *+' *+' ),"-,"$ '% % ' '" $ ' # " ) % "# "# "$ "$ # # #"% #"% *+' *+' ) ) *+' )," *+' ),"#," ' ' ' ' ' ' ' ' ") ") "# "# "$ "# "$"# # "$ "$ #"% #"% ' %" *+'( *+' ),"-( *+' ),"#.,"#. % '% '% ' ' '" $ ' # " ) % ") "# ") "$ "# "# # "$ #"% ',"#%,"#% '% *+') '% *+') ' ' $ # ) % ") ") "# "# "$ "$ ' *+' ) *+' ),".,". ') ') '% '% ' ' $ # ) ") "# "$ '

18 color superconducting or color-flavor locked phas quark-gluon plasma as well as the confined hadroni has been a long standing challenge for lattice QCD Landscape of the Polyakov this loop region. susceptibility µ FIG. 8: Landscape of the Polyakov loop susceptibility. The color scale (left) is based on log (χ P ) B. Observables We measure several observables under the vari and T, to check the properties of the different phase 2.6 T and large µ. In the following we specialize to N c observables are: the Polyakov loop, P = 3 Nσ 3 x Tr P x = N 3 σ P x, x and its susceptibility FIG. 9: 3d view of Fig. 8. at slightly larger µ, which already at β = 5.6 depart considerably from strong coupling estimates; this is an indication of a possible phase transition. ( PNext x P y we obtained P x a P phase y ) diagram, χ P = y µ FIG. 8: Landscape of the Polyakov loop susceptibility. The color scale (left) is based on log (χ P ) 4 the (dimensionless) baryon number density n B, [] M. G. Alford, hep-lat/646. [2] I. Bender, T. Hashimoto, F. Karsch, V. Linke, A. Nakamura, M. Plewnia, I.-O. Stamatescu, W. Wetzel, Nucl. Phys. Proc. Suppl. 26 (992) 323. [3] J. Engels, O. Kaczmarek, F. Karsch, ne. b,f Laermann, Nucl. Phys. B558 (999) 37 [hep-lat/9933]. [4] T. C. Blum, J. E. Hetrick and D. Toussaint, Phys. Rev. Lett. 76 n B = f T 3,

19 T/T c ". ()-+ N = 6, n =3 f FIG. 7: Phase diagram in the β (or T/T c ) - µ phys /T c QCD plane. The constant β. The blobs, shadowing and other features are explained in the te.9.8 (),( (),+ % phys #" +)/ /T = ) ( 2.4 ) ( 3. ) +)( +)$ ( 3.6 ) Finally in Figs. 28 and +)* 29 we present the dependence on µ and ( 4.2 ) on β of the diquark susceptibility obtained by integrating the diquark-correlators Eq.(4.6) for ξ =.5; here we only show +)*( ( 4.5 ) the contribution to this susceptibility from the κ 2 terms. This +), ( 4.8 ) co ity to ze.7.6 ()*( ()* ()*+ ()$* B +),( ( 5. ) +)- ( 5.4 ) +)-( ( 5.7 ).)+ ( 6. ) ()$( ()$.5 ()$+ ()(( A.4 ()(+ /3 ' phys /T c "#$ T/T c

20 Polyakov loop susceptibility x 4 =5.55 x 4 =5.6 x 4 = T/T c ". ()-+ N = 6, n f =3 #" +)/ +)( +)$ % phys /T = ) ( 2.4 ) ( 3. ) ( 3.6 ).9 (),( +)* ( 4.2 ) x 4 = x 4 = x = (),+ ()*( ()* ()*+ ()$* B +)*( ( 4.5 ) +), ( 4.8 ) +),( ( 5. ) +)- ( 5.4 ) +)-( ( 5.7 ).)+ ( 6. ) x 3 = x = x 3 = /3 ()$( ()$ ()$+ ()(( ()(+ A ' χ P vs. µ at fixed β phys /T c "#$ T/T c

21 Polyakov loop susceptibility x 3 µ= x 3 µ= x 3 µ= T/T c ". ()-+.9 (),( % phys N = 6, n f =3 #" +)/ +)( +)$ /T = ) ( 2.4 ) ( 3. ) ( 3.6 ) +)* ( 4.2 ) x 4 µ= x 4 µ= x 4 µ= (),+ ()*( ()* ()*+ ()$* B +)*( ( 4.5 ) +), ( 4.8 ) +),( ( 5. ) +)- ( 5.4 ) +)-( ( 5.7 ).)+ ( 6. ) x 4 µ=.85 2 x 3 µ= x µ= ()$( ()$ ()$ /3 ()(( ()(+ A ' χ P vs. β at fixed µ phys /T c "#$ T/T c

22 H (x, y) = ( ) Re(w P x ) Θ,x w P x B = w P x B = w B T/T c " ()-+ (),( (),+ ()*( ()* ()*+ ()$* ()$( ()$ ()$+ ()(( ()(+ % phys ( ) Im(w P x ) Θ,y w dxdy (x + iy)h (x, y) w B N = 6, n f =3 #" +)/ +)( +)$ /T = ) ( 2.4 ) ( 3. ) ( 3.6 ) B A +)* ( 4.2 ) +)*( ( 4.5 ) +), ( 4.8 ) +),( ( 5. ) +)- ( 5.4 ) +)-( ( 5.7 ).)+ ( 6. ) =5.65 µ = =5.65 µ = (4.9) µ =.4 µ = =5.65 µ = =5.65 µ = =5.65 µ =.9 =5.65 µ =.93 µ =.9 5 µ = FIG. 2: Polyakov loop histogram H (x, y) of eq. (4.7) vs. µ at β = =5.55 µ =.7 =5.62 µ =.7 =5.65 µ =.7 β = β = β = µ = =5.67 µ =.7 =5.7 µ =.7 =5.75 µ =.7 β = β = β = µ = β = /3 ' phys /T c "#$ T/T c.4 FIG. 7: Phase diagram in the β (or T/T c) - µ.4 phys/t c QCD plane. The dotted straight lines correspond to constant µ, the dashed ones to constant β. The blobs, shadowing and other features are explained in the text. Finally in Figs. 28 and 29 we present the dependence on µ and.3 corresponds to quarks showing a (limited) amount of mobil FIG. 22: Polyakov loop histogram H (x, y) of eq. (4.7) vs. β at µ =.7. µ =.7

23 confined to a deconfined phase. A distribution independent of the choice of B fined by considering +#% Reweighting at work +#' T (x, y) = Θ,x(RePx ) Θ,y (ImPx )" 5 +#) "#$%#"$' µ =.4 $( $) "#$%#"$% $) $% $% $' $' $* $* $* $* $' $' $% $% $) $) $( $( $# $# $( $(# µ =.8 $( $* $* $* $* $* $' $' $% $% $) $) $( $( β = 5.65 $(# $* $*# $* $% µ =.4 $% $( "#$##"$+ $( $% $(# $# $( $(# $# $( $(# $# $* "#$%+"$+ $% $( $(# µ =.9 $# $( $(# $) $* $% $% $' $* $' $* $* $% $% $# $) $( $* $*# $% $( $) $(# $* $* $) $# µ =.93 $( $(# $* T (x, y) = Θ,x(RePx ) Θ,y (ImPx )", $* "#$+#"$+ $ $' $* $% $' $* $#+ $) $% $* $) $' $( $(# $( where the sum runs over a lattice with lattice co the xy-plane. Sinceadding the expectation value of P is which means the weights of all configurations value in a given binin ReP a Pxand has to being even ImT odd y. x x /2, R oddthefor expectation y that goes to -y." r (T Because (x, y)) y /2. now value We give some representative figures showing t $% $' $( $# +#) "#$%#"$,- $( $( $' $) $' $* $' $% $* $( $' $% $' $* $) fined by considering x,y +#' $( $' $% $* $% $) $) $) $* $' ( $% $* $' $) "#$%#"$+ $% $* "#$+"$+ eptibility. The color $( $' +#% $% $# $% $) $* $- $' $( $* $) $% $* $' $' $* $% $( $( $'$( $*# "#$%#"$+ "#$%#"$, $) $) $* $( $' $% µ =.8 $(# $) $* $( $' "#$%#"$) $( (x, y) of eq. (4.) vs. µ at β = 5.65 fixed. % $- "#$%*"$+ $* $( $( $) $*# $) $* $# $* 6 ( ( $) $) $) $( $% $% $% $( $' $( $' $' $) $' $* $% $* $* $' $) $* $( "#* $(# $* $) $% $( $* $( $% $* $# "#$%#"$,- "#$%#"$% $) $' $' $* $(# $% $( $) $' $( $( $' $* $* $# $) $% $' $* $) "#$%#"$' $) $' $' FIG. 2: Real part of the Polyakov loop distribution T $( $% $( $' even for y that goes to -y which means adding the weights of all configurati ing a Px value in a given bin ReP x x /2 $#+ Real and imaginary part of the y µ /2. Because now the expectation value µ =.9 =.93 (x,2.7), y) Polyakov loopfactor distribution $#% the complex Boltzmann B (seet Eq. vs. at β = 5.65 µ plex and does not represent a probability distribut $#' small we have positive real parts; we interpret this as the transition $#) µ =.6 µ =.7 phase. confined to a deconfined P " independent (x + iy)t y)of, B ca (x, A distribution of the choice $) $% (T (x, y)) $) $% "#$%#"$, $' $( $) ( $' $# $% $# µ =.7 $( $ "#$%#"$) $) "#$%#"$+ µ =.6 $( ( $) $( $(# $* $*# $- $% $) $( $(# $* $*# $- $( $(eq. (4.) $#% FIG. 22: Imaginary part of the Polyakov loop distribution T (x, y) of vs. µ at β = 5.65 fixed. (

24 "#$, "# "#$+ "#$* "#%$ "#$) "#$$ "#% "#$( "#$' "#"$ "#$ "#$% "#) "#)$ "#* "#*$ "#+ "#+$ "#, "#,$ % " "#) "#)$ "#* "#*$ "#+ "#+$ "#, "#,$ % FIG. 24: Plaquette averages vs. µ at fixed β = FIG. 26: Topological susceptibility average vs. µ at fixed β = "#$, "# "#$+ "#$* "#%$ "#$) "#$$ "#% "#$( "#$' "#"$ "#$ "#$% $#$$ $#) $#)$ $#* $#*$ $#+ $#+$ $#, " $#$$ $#) $#)$ $#* $#*$ $#+ $#+$ $#, FIG. 25: Plaquette averages vs. β at fixed µ =.7. FIG. 27: Topological susceptibility average vs. β at fixed µ =.7.

25 "# "#%$ "#% "#"$ " "#' "#'$ "#( "#($ "#) "#)$ "#* "#*$ % FIG. 28: Diquark susceptibility average vs. µ at fixed β = "# "#%$ "#% "#"$ " $#$$ $#' $#'$ $#( $#($ $#) $#)$ $#* FIG. 29: Diquark susceptibility average vs. β at fixed µ =.7.

26 Conclusions We obtain a phase structure by the numerical simulations at nf=3. The signal for the deconfining transition (or narrow crossover) on the line connecting A and B is rather good. A second transition at large baryonic density could only be identified tentatively. In this region the di-quark susceptibility grows strongly. This region need further study to reach conclusions The algorithm works reasonably well over a wide range of parameters and lattice up to 6^4, 8^4. We obtain large densities for temperature ~/2 Tc and rations mu_phys/t ~5.

27 Diquark-diquark correlators C (qq) (τ) = (δ a i δb j + ξδa j δb i )(δc k δd l + ξδc l δd k ) x,y,t [ψ a i Cψb j (x, t)][ψc l Cψd k (y, t + τ)] = (δi a δb j + ξδa j δb i )(δc k δd l + ξδc l δd k ) { W ik;ac (x, t; y, t + τ)ct W,T jl;bd (x, t; y, t + τ)c x,y,t } W il;ad (x, t; y, t + τ)ct W,T jk;bc (x, t; y, t + τ)c, (4.6)