van der Waals Interactions in Hadron Resonance Gas:
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- George Briggs
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1 van der Waals Interactions in Hadron Resonance Gas: From Nuclear Matter to Lattice QCD Volodymyr Vovchenko Collaborators: P. Alba, D. Anchishkin, M. Gorenstein, and H. Stoecker Based on Phys. Rev. Lett. 8, 823 (27) and ongoing work Palaver SS 27 ITP, Frankfurt May 22, 27 HGS-HIRe Helmholtz Graduate School for Hadron and Ion Research /2
2 Outline Motivation Nuclear matter as van der Waals system van der Waals in hadron resonance gas and lattice QCD Extensions Summary 2/2
3 Strongly interacting matter Theory of strong interactions: Quantum Chromodynamics (QCD) Basic degrees of freedom: quarks and gluons At smaller energies confined into hadrons: baryons (qqq) and mesons (q q) Big bang Quark Gluon Plasma Where is it relevant? Temperature T c hadron gas Heavy Ion Collisions Early universe Neutron stars Heavy-ion collisions nucleon gas nuclei neutron stars net-baryon density First principles of QCD are rather established, but direct calculations are problematic Phenomenological tools are very useful 3/2
4 QCD equation of state at µ = Lattice simulations provide equation of state at µ B =.2 B B χ 4/χ2 hadron resonance gas WB HotQCD stout cont. HISQ, N τ =6 8 data: BNL-Bielefeld-CCNU preliminary.2 free quark gas T [MeV] Common model for confined phase is ideal HRG: non-interacting gas of known hadrons and resonances Good description of thermodynamic functions up to 8 MeV Rapid breakdown in crossover region for description of susceptibilities 2 Often interpreted as clear signal of deconfinement... But what is the role of hadronic interactions beyond those in ideal HRG? Bazavov et al., PRD 9, 9453 (24); Borsanyi et al., PLB 73, 99 (24) 2 Ding, Karsch, Mukherjee, IJMPE 24, 537 (25) 4/2
5 QCD equation of state at µ = T c HRG Lattice simulations provide equation of state at µ B = 3p/T 4 ε/t 4 3s/4T 3 non-int. limit T [MeV] χ 4 B /χ2 B hadron resonance gas stout cont. HISQ, N τ =6 8 data: BNL-Bielefeld-CCNU preliminary.2 free quark gas T [MeV] Common model for confined phase is ideal HRG: non-interacting gas of known hadrons and resonances Good description of thermodynamic functions up to 8 MeV Rapid breakdown in crossover region for description of susceptibilities 2 Often interpreted as clear signal of deconfinement... But what is the role of hadronic interactions beyond those in ideal HRG? Bazavov et al., PRD 9, 9453 (24); Borsanyi et al., PLB 73, 99 (24) 2 Ding, Karsch, Mukherjee, IJMPE 24, 537 (25) 4/2
6 van der Waals (VDW) equation 5/2 P(T, V, N) = NT V bn a N2 V 2 Formulated in 873. Simplest model which contains attractive and repulsive interactions Contains st order phase transition and critical point Can elucidate role of fluctuations in phase transitions Two ingredients: ) Short-range repulsion: excluded volume (EV) procedure, V V bn, b = 4 4πr 3 3 2) Intermediate range attraction in mean-field approximation, P P a n 2 Nobel Prize in 9.
7 Nucleon-nucleon interaction Nucleon-nucleon potential: Repulsive core at small distances Attraction at intermediate distances Suggestive similarity to VDW interactions Could nuclear matter be described by VDW equation? Standard VDW equation is for canonical ensemble and Boltzmann statistics Nucleons are fermions, obey Pauli exclusion principle Unlike for classical fluids, quantum statistics is important 6/2
8 Quantum statistical van der Waals fluid Free energy of classical VDW fluid: F (T, V, N) = F id (T, V bn, N) a N2 V Ansatz: F id (T, V bn, N) is free energy of ideal quantum gas ( ) F Pressure: p = = p id (T, µ ) a n 2 V T,N ( ) p n id (T, µ ) Particle density: n = = µ + b n id (T, µ ) Shifted chemical potential: T µ = µ b p a b n a n Model properties: Reduces to classical VDW equation when quantum statistics are negligible Reduces to ideal quantum gas for a = and b = Entropy density non-negative and s with T V.V., Anchishkin, Gorenstein, JPA 5 and PRC 5; Redlich, Zalewski, APPB 6. a= excluded-volume model, D. Rischke et al., ZPC 9 7/2
9 VDW gas of nucleons: pressure isotherms P (MeV/fm 3 ) a and b fixed to reproduce saturation density and binding energy: n =.6 fm 3, E/A = 6 MeV a = 329 MeV fm 3 and b = 3.42 fm T=8 MeV T=5 MeV T=2 MeV T=9.7 MeV v (fm 3 ) (a) P (MeV/fm 3 ) T=2 MeV T=9.7 MeV T=8 MeV T=5 MeV n (fm -3 ) Behavior qualitatively same as for Boltzmann case Mixed phase results from Maxwell construction Critical point at T c = 9.7 MeV and nc =.7 fm 3 Experimental estimate : T c = 7.9 ±.4 MeV, n c =.6 ±. fm 3 (b) J.B. Elliot, P.T. Lake, L.G. Moretto, L. Phair, Phys. Rev. C 87, (23) 8/2
10 VDW gas of nucleons: (T, µ) plane (T, µ) plane: structure of critical fluctuations χi = i (p/t 4 )/ (µ/t )i 4 n (fm -3 4 ) ω[ N ] χ2 / χ g a s 5 liq u id g a s liq u id. (a ) χ4 / χ2-9 3 κσ S σ χ3 / χ2-9 2 µ(m e V ) µ(m e V ) g a s g a s liq u id µ(m e V ) liq u id µ(m e V ) V.V., D. Anchishkin, M. Gorenstein, R. Poberezhnyuk, PRC 9, 6434 (25) 9/2
11 van der Waals interactions in hadron resonance gas Let us now include nuclear matter physics into HRG... VDW-HRG model Identical VDW interactions between all baryons The baryon-antibaryon, meson-meson, and meson-baryon VDW interactions are neglected Baryon VDW parameters extracted from ground state of nuclear matter (a = 329 MeV fm 3, b = 3.42 fm 3 ) Three independent subsystems: mesons + baryons + antibaryons P M (T, µ) = j M p(t, µ) = P M (T, µ) + P B (T, µ) + P B(T, µ), p id j (T, µ j ) and P B (T, µ) = j B p id j (T, µ B j ) a nb 2 n B (T, µ) = ( b n B ) j B nj id (T, µ B j ). In this simplest setup model is essentially parameter-free V.V., M. Gorenstein, H. Stoecker, Phys. Rev. Lett. 8, 823 (27) /2
12 VDW-HRG at µ B = : thermodynamic functions Comparison of VDW-HRG with lattice QCD at µ B = Pressure p/t 4 L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G V D W -H R G p /T Energy density ε/t 4 L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G V D W -H R G ε/t VDW-HRG does not spoil existing agreement of Id-HRG with LQCD despite significant excluded-volume interactions between baryons Not surprising: matter meson-dominated at µ B = V.V., M. Gorenstein, H. Stoecker, Phys. Rev. Lett. 8, 823 (27) /2
13 VDW-HRG at µ B = : speed of sound c 2 S Speed of sound c 2 s = dp dε L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G V D W -H R G Monotonic decrease in Id-HRG, at odds with lattice Minimum for EV-HRG/VDW-HRG at 5-6 MeV No acausal behavior, often an issue in models with eigenvolumes V.V., M. Gorenstein, H. Stoecker, Phys. Rev. Lett. 8, 823 (27) 2/2
14 VDW-HRG at µ B = : baryon number fluctuations Susceptibilities: χ BSQ lmn = Net-baryon χ B 2 L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G V D W -H R G l+m+n p/t 4 (µ B /T ) l (µ S /T ) m (µ Q /T ) n Baryon-charge correlator χ BQ L a ttic e (W u p p e rta l-b u d a p e s t p re lim in a ry ) L a ttic e (H o tq C D ) Id -H R G E V -H R G V D W -H R G χ B 2. 5 χ B Q Very different qualitative behavior between Id-HRG and VDW-HRG For χ B 2 lattice data is between Id-HRG and VDW-HRG at high T For χ BQ lattice data is below all models, closer to EV-HRG V.V., M. Gorenstein, H. Stoecker, Phys. Rev. Lett. 8, 823 (27) 3/2
15 VDW-HRG at µ B = : baryon number fluctuations Higher-order of fluctuations are expected to be even more sensitive.4.2. χ B 4 /χb χ B 6 /χb 2 χ B 4 /χb χ B 6 /χb L a ttic e (W u p p e rta l-b u d a p e s t).2 Id -H R G E V -H R G V D W -H R G Id -H R G -.5 E V -H R G V D W -H R G χ B 4 deviates from χb 2 at high enough T, they stay equal in Id-HRG Cannot be related only to onset of deconfinement VDW-HRG predicts strong non-monotonic behavior for χ B 6 /χb 2 V.V., M. Gorenstein, H. Stoecker, Phys. Rev. Lett. 8, 823 (27) 4/2
16 VDW-HRG: influence on hadron ratios VDW interactions change relative hadron yields in HRG Thermal model fit to ALICE 2.76 TeV) yields: from π to Ω A L IC E -5 %, N d o f = 8 6 χ Id -H R G E V -H R G V D W -H R G Fit quality slightly better in EV-HRG/VDW-HRG vs Id-HRG but very different picture! All temperatures between 5 and 2 MeV yield similarly fair data description in VDW-HRG Results likely to be sensitive to further modifications, e.g for strangeness 5/2
17 VDW-HRG at finite µ B Net-baryon fluctuations in T -µ B plane: χ B 4 /χb 2 h e a v y -io n -c o llis io n s Id -H R G χ B 4 /χb µ B (M e V ) Almost no effect in Id-HRG, only Fermi statistics... V.V., M. Gorenstein, H. Stoecker, Phys. Rev. Lett. 8, 823 (27) 6/2
18 VDW-HRG at finite µ B Net-baryon fluctuations in T -µ B plane: χ B 4 /χb 2 la ttic e Q C D h e a v y -io n -c o llis io n s - V D W -H R G 2 n u c le a r m a tte r 2 χ B 4 /χb µ B (M e V ) Almost no effect in Id-HRG, only Fermi statistics... Rather rich structure for VDW-HRG, huge effect of VDW interactions! Fluctuations seen at RHIC are remnants of nuclear liquid-gas PT? V.V., M. Gorenstein, H. Stoecker, Phys. Rev. Lett. 8, 823 (27) 6/2
19 Beyond van der Waals From van der Waals equation to Clausius equation: p = nt bn a n2 p = nt bn a n2 + cn Nuclear incompressibility K : from 762 MeV in VDW to 3 MeV in Clausius Clausius-HRG: baryon-baryon interactions in HRG with Clausius equation net baryon χ B 2 L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G V D W -H R G, K = M e V C la u s iu s -H R G, K = 3 M e V µ B = µ B = net baryon χ B 4 /χb 2 L a ttic e (W u p p e rta l-b u d a p e s t) Id -H R G V D W -H R G, K = M e V C la u s iu s -H R G, K = 3 M e V χ B 2. 5 χ B 4 /χb Clausius-HRG yields improved K and improved description of LQCD Behavior of LQCD observables correlates with nuclear matter properties V.V., arxiv: /2
20 3 ) Hard-core repulsion: classical vs Beth-Uhlenbeck QM approach to NN hard-core repulsion: Beth-Uhlenbeck (BU) formula b (fm p(t, µ) T φ(t ) λ + T B 2 (T ) λ 2, λ = e µ/t, φ(t ) = n id (T, µ = ) B 2 (T ) = T 2π 3 n u c le o n -n u c le o n p ro to n -p ro to n p ro to n -n e u tro n dε ε 2 K 2 (ε/t ) (2J + ) (2T + ) δ J,T (ε) 2m N ε J,T { jl [2r c q(ε)] δ r c =.3 fm J,T (ε) = arctan y L [2r c q(ε)] c la s s ic a l Classical eigenvolume: p = p id (T, µ bp), } b = 6πr 3 c 3 Beth-Uhlenbeck eigenvolume: b = b(t ) = B 2 (T )/[φ(t )] 2 NN-scattering data : r c fm EV of nucleon-nucleon interaction is strongly T -dependent Classical approach underestimates EV by factor 2-3 at T 2 MeV R. B. Wiringa et al., Phys. Rev. C 5, 38 (995) 8/2
21 Hard-core repulsion: classical vs Beth-Uhlenbeck Now use this T -dependent eigenvolume b(t ) to model BB repulsion in HRG BU-HRG: P B (T, µ) = i B pi id (T, µ B ) T b(t ) i,j φ i (T ) φ j (T ) λ 2 B EV-HRG: P B (T, µ) = i B p id i [T, µ B b(t ) P B ] χ B net baryon χ B 2 L a ttic e (W u p p e rta l-b u d a p e s t) Id e a l H R G B U -H R G E V -H R G w ith b (T ) fro m B U.2 5 fm < r N N <.3 fm c χ B 4 /χb net baryon χ B 4 /χb 2.4 L a ttic e (W u p p e rta l-b u d a p e s t) Id e a l H R G.2 B U -H R G E V -H R G w ith b (T ) fro m B U Pure BU approach breaks down at T 6 7 MeV, higher orders matter! EV-HRG with BU-motivated b.5 fm 3 describes LQCD fairly well 9/2
22 Repulsive baryonic interactions and imaginary µ B Lattice QCD is problematic at real µ but tractable at imaginary µ E.g., net-baryon density is imaginary and has trigonometric series form µ B i µ B n B(T, i µ B ) T 3 = i b j (T ) sin (j µ B /T ) j= b j b b 2 b 3 b 4 n B /T 3 = b (T )* s in h (µ B /T ) + b 2 (T )* s in h (2 µ B /T ) + b 3 (T )* s in h (3 µ B /T ) + b 4 (T )* s in h (4 µ B /T ) +... B o rs a n y i e t a l. (Q M 2 7 ). -.2 E V -H R G, b =. 7 fm 3, a = Non-zero b j (T ) for j 2 signal deviations from ideal HRG Addition of EV interactions between baryons reproduces lattice trend V.V., H. Stoecker, and Wuppertal-Budapest collaboration, in preparation 2/2
23 Summary 2/2 Nuclear matter can be described as VDW equation with Fermi statistics VDW interactions between baryons have strong influence on fluctuations of conserved charges in the crossover region within HRG Behavior of lattice QCD observables at µ B = correlates strongly with nuclear matter properties such as incompressibility Nuclear liquid-gas transition manifests itself into non-trivial net-baryon fluctuations in regions of phase diagram probed by HIC Interpretation of results obtained within standard ideal HRG should be done with extreme care
24 Summary 2/2 Nuclear matter can be described as VDW equation with Fermi statistics VDW interactions between baryons have strong influence on fluctuations of conserved charges in the crossover region within HRG Behavior of lattice QCD observables at µ B = correlates strongly with nuclear matter properties such as incompressibility Nuclear liquid-gas transition manifests itself into non-trivial net-baryon fluctuations in regions of phase diagram probed by HIC Interpretation of results obtained within standard ideal HRG should be done with extreme care Thanks for your attention!
25 Backup slides 2/2
26 Van der Waals equation 2/2 VDW isotherms show irregular behavior below certain temperature T C Below T C isotherms are corrected by Maxwell s rule of equal areas Results in appearance of mixed phase 2. P T=. T= T=.9 T=.8 T gas mixed phase liquid v n Critical point p v =, 2 p v =, v = V /N 2 p C = a 27b, n 2 C = 3b, T C = 8a 27b Reduced variables p = p p C, ñ = n n C, T = T T C
27 Net-baryon fluctuations and nuclear matter 2/2.2. Are NN interactions relevant for heavy-ion collisions? Net-nucleon fluctuations within RMF (σ-ω model) of nuclear matter along line of chemical freeze-out Skewness Kurtosis BES/STAR HRG RMF.2. BES/STAR HRG RMF.8.8 S¾.6 κ¾ Collision Energy [GeV] Collision Energy [GeV] A notable effect in fluctuations even at µ B Reconciliation of HRG with nuclear matter can be interesting K. Fukushima, PRC 9, 449 (25)
28 VDW equation originally formulated in canonical ensemble How to transform CE pressure P(T, n) into GCE pressure P(T, µ)? Calculate µ(t, V, N) from standard TD relations Invert the relation to get N(T, V, µ) and put it back into P(T, V, N) Consistency due to thermodynamic equivalence of ensembles Result: transcendental equation for n(t, µ) N V n(t, µ) = n id (T, µ ) + b n id (T, µ ), n T µ = µ b b n + 2a n Implicit equation in GCE, in CE it was explicit May have multiple solutions below T C Choose one with largest pressure equivalent to Maxwell rule in CE Advantages of the GCE formulation ) Hadronic physics applications: number of hadrons usually not conserved. 2) CE cannot describe particle number fluctuations. N-fluctuations in a small (V V ) subsystem follow GCE results. 3) Good starting point to include effects of quantum statistics. 2/2
29 Scaled variance for classical VDW equation Transformation of VDW eq. to GCE leads to new physical applications Scaled variance of particle number fluctuations in VDW gas (pure phases) [ ω[n] = σ2 N = ( bn) 2 2an ] T T gas 2 liquid. mixed phase 2 3 n Repulsive interactions suppress N-fluctuations Attractive interactions enhance N-fluctuations.. V.V., R. Poberezhnyuk, D. Anchishkin, M.I. Gorenstein, J. Phys. A 35, 48 (25) 2/2
30 Scaled variance in VDW equation New application from GCE formulation: particle number fluctuations Scaled variance is an intensive measure of N-fluctuations σ 2 N = ω[n] N2 N 2 = T ( ) n = T ( 2 ) P N n µ n µ 2 In ideal Boltzmann gas fluctuations are Poissonian and ω id [N] =. ω[n] in VDW gas (pure phases) [ ω[n] = ( bn) 2 2an T T ] T Repulsive interactions suppress N-fluctuations Attractive interactions enhance N-fluctuations N-fluctuations are useful because they Carry information about finer details of EoS, e.g. phase transitions Measurable experimentally 2/2
31 Scaled variance 3 ω[n] = 9.5 [ (3 ñ) 2 ñ 4 T. ]. 2 T gas 2 liquid. mixed phase 2 3 Deviations from unity signal effects of interaction Fluctuations grow rapidly near critical point n V. Vovchenko et al., J. Phys. A 35, 48 (25).. 2/2
32 Skewness Higher-order (non-gaussian) fluctuations are even more sensitive Skewness: Sσ = ( N)3 σ 2 = ω[n] + T ( ) ω[n] asymmetry ω[n] µ 3 T S T gas metastable - unstable metastable liquid 2 3 Skewness is Positive (right-tailed) in gaseous phase Negative (left-tailed) in liquid phase n /2
33 Kurtosis Kurtosis: κσ 2 = ( N)4 3 ( N) σ 2 - peakedness 4.. T 2 gas metastable - - unstable 2 3 n metastable liquid Kurtosis is negative (flat) above critical point (crossover), positive (peaked) elsewhere and very sensitive to the proximity of the critical point V. Vovchenko et al., J. Phys. A 53, 49 (26) /2
34 VDW equation with quantum statistics in GCE Requirements for VDW equation with quantum statistics ) Reduce to ideal quantum gas at a = and b = 2) Reduce to classical VDW when quantum statistics are negligible 3) s and s as T Ansatz: Take pressure in the following form,2 p(t, µ) = p id (T, µ ) an 2, µ = µ b p a b n 2 + 2an where p id (T, µ ) is pressure of ideal quantum gas. ( ) p n id (T, µ ) n(t, µ) = = µ + b n id (T, µ ) T Algorithm for GCE ) Solve system of eqs. for p and n at given (T, µ) 2) Choose the solution with largest pressure V. Vovchenko, D. Anchishkin, M. Gorenstein, Phys. Rev. C 9, 6434 (25) 2 Alternative derivation: K. Redlich, K. Zalewski, arxiv: (26) 3 a= excluded-volume model, D. Rischke et al., Z.Phys. C5, 485 (99) 2/2
35 VDW gas of nucleons: zero temperature How to fix a and b? For classical fluid usually tied to CP location. Different approach: Reproduce saturation density and binding energy From E B = E/A = 6 MeV and n = n =.6 fm 3 at T = and p = a = 329 MeV fm 3 and b = 3.42 fm 3 E B (MeV) (b) n (fm -3 ) Mixed phase at T = is specific: A mix of vacuum (n = ) and liquid at n = n r (m ) C O 2 A rg o n W a te r N u c le a r m a tte r a /b (e V ) VDW eq. now at very different scale! 2/2
36 VDW gas of nucleons: scaled variance 2/2 Scaled variance in quantum VDW: [ ω[n] = ω id (T, µ ) ( bn) 2 2an T ω id(t, µ ) ] T (MeV) gas liquid (MeV).....
37 VDW gas of nucleons: skewness and kurtosis Skewness ω[n] T Sσ = ω[n] + ω[n] µ T Kurtosis [Sσ] 2 2 κσ = (Sσ) + T µ T S T (MeV) T (MeV) gas 5 liquid gas 5 liquid (MeV) (MeV) For skewness and kurtosis singularity is rather specific: sign depends on the path of approach V. Vovchenko et al., Phys. Rev. C 92, 549 (25) 2/2
38 VDW gas of nucleons: skewness and kurtosis VDW Skewness VDW Kurtosis S T (MeV) T (MeV) gas 5 liquid gas 5 liquid (MeV) NJL, J.W. Chen et al., PRD 93, 3437 (26) (MeV) PQM, V. Skokov, QM22 Fluctuation patterns in VDW very similar to effective QCD models 2/2
39 VDW-HRG at µ = : baryon number fluctuations 2/2 χ B 6 /χb 2 BNL-Bielefeld-CCNU VDW-HRG χ B 6 /χb Id -H R G -.5 E V -H R G V D W -H R G
40 VDW-HRG at µ = : net-light and net-strangeness χ L L a ttic e (W u p p e rta l-b u d a p e s t) Id -H R G E V -H R G V D W -H R G. 5 (a ) Net-strangeness χ S 2 Underestimated by HRG models, similar for χ BS Extra strange states? Net number of light quarks χ L 2 L = (u + d)/2 = (3B + S)/2 Improved description in VDW-HRG Weaker VDW interactions for. strange baryons? χ S L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G V D W -H R G Bazavov et al., PRL 3, 72 (24) 2 Alba, Vovchenko, Gorenstein, Stoecker, arxiv: /2
41 VDW-HRG at µ B = : net-light and net-strangeness Lattice shows peaked structures in crossover regions Not at all reproduced by Id-HRG, signal for deconfinement? Peaks at different T for net-l and net-s flavor hierarchy? 2 VDW-HRG also shows peaks and flavor hierarchy cannot be traced back directly to deconfinement S. Ejiri, F. Karsch, K. Redlich, PLB 633, 275 (26) 2 Bellwied et al., PRL, 2232 (23) 2/2 χ L 4 /χl Fluctuations of net-light L = (u + d)/2 = (3B + S)/2 and net-strangeness χ L 4 /χl 2 χ S 4 /χs L a ttic e (W u p p e rta l-b u d a p e s t) Id -H R G.2 E V -H R G V D W -H R G χ S 4 /χs L a ttic e (W u p p e rta l-b u d a p e s t).4 Id -H R G.2 E V -H R G V D W -H R G
42 VDW-HRG: extensions χ B 4 /χb 2 Effect of reducing VDW interactions involving strange hadrons 3 times smaller EV for strange baryons Small EV for mesons Illustrative calculation Most observables improved L a ttic e Q C D Id -H R G E V -H R G, b = fm V D W -H R G, b = fm V D W -H R G, b l = fm 3 3, a = M e V fm 3, a l = M e V fm , b S =. 7 fm 3, b M =. 3 fm 3 χ B 2 χ u s L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G, b = fm V D W -H R G, b = fm V D W -H R G, b l = fm 3 3, a = M e V fm 3, a l = M e V fm L a ttic e Q C D Id -H R G E V -H R G, b = fm V D W -H R G, b = fm V D W -H R G, b l = fm 3 3, a = M e V fm 3, a l = M e V fm 3 3, b S =. 7 fm 3, b M =. 3 fm , b S =. 7 fm 3, b M =. 3 fm 2/2 3 3
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