Theoretical and experimental results for W → P + P ◮ For the coupling constant ˜ g ≡ g 2 , experimental results ˜ g i and errors on them g i we define χ 2 ≡ � N g i ) 2 (˜ g − ˜ ∆ g , ∆˜ i =1 g 2 ∆˜ i ◮ Minimizing χ 2 with respect to coupling d χ 2 g = 0 leads to d ˜ � N ˜ g i � 1 i =1 g 2 ∆˜ � , → g 2 WPP = (1 . 5 ± 0 . 1) · 10 − 10 (MeV) − 4 ˜ g = i , ∆˜ g = � � N � N 1 � 1 j =1 g 2 j =1 ∆ g 2 ∆˜ j j Decay process (in model) Theory (MeV) Experiment (MeV) ρ 3 (1690) − → π π 32 . 7 ± 2 . 3 38 . 0 ± 3 . 2 ↔ (23 . 6 ± 1 . 3)% → ¯ ρ 3 (1690) − 4 . 0 ± 0 . 3 2 . 54 ± 0 . 45 ↔ (1 . 58 ± 0 . 26)% K K → π ¯ K ∗ 3 (1780) − K 18 . 5 ± 1 . 3 29 . 9 ± 4 . 3 ↔ (18 . 8 ± 1 . 0)% → ¯ K ∗ 3 (1780) − 7 . 4 ± 0 . 6 47 . 7 ± 21 . 6 ↔ (30 ± 13)% K η → ¯ K ∗ K η ′ (958) 3 (1780) − 0 . 02 ± 0 . 01 → ¯ ω 3 (1670) − K K 3 . 0 ± 0 . 2 → ¯ φ 3 (1850) − K K 18 . 8 ± 1 . 4 seen Shahriyar Jafarzade (Jan Kochanowski University of Kielce, Poland) 12 / 19
Theoretical and experimental results for W → V + P ◮ Repeating the same calculations in the previous section g 2 WVP = (9 . 2 ± 1 . 9) · 10 − 16 (MeV) − 6 Decay process (in model) Theory (MeV) Experiment (MeV) ρ 3 (1690) − → ρ (770) η 3 . 8 ± 0 . 8 seen → ρ (770) η ′ (958) ρ 3 (1690) − 0 → ¯ K ∗ (892) K + c . c . ρ 3 (1690) − 3 . 4 ± 0 . 7 ρ 3 (1690) − → ω (782) π 35 . 8 ± 7 . 4 25 . 8 ± 9 . 8 ↔ (16 ± 6)% ρ 3 (1690) − → φ (1020) π 0 . 17 ± 0 . 04 K ∗ 3 (1780) − → ρ (770) K 16 . 8 ± 3 . 4 49 . 3 ± 15 . 7 ↔ (31 ± 9)% → ¯ K ∗ K ∗ (892) π 3 (1780) − 27 . 2 ± 5 . 6 31 . 8 ± 9 . 0 ↔ (20 ± 5)% → ¯ K ∗ K ∗ (892) η 3 (1780) − 0 . 09 ± 0 . 02 → ¯ K ∗ K ∗ (892) η ′ (958) 3 (1780) − 0 → ω (782) ¯ K ∗ 3 (1780) − K 4 . 3 ± 0 . 9 → φ (1020) ¯ K ∗ 3 (1780) − K 1 . 2 ± 0 . 3 Shahriyar Jafarzade (Jan Kochanowski University of Kielce, Poland) 13 / 19
Decay process (in model) Theory (MeV) Experiment (MeV) ω 3 (1670) − → ρ (770) π 96 . 9 ± 19 . 9 seen → ¯ K ∗ (892) K + c . c . ω 3 (1670) − 2 . 9 ± 0 . 6 ω 3 (1670) − → ω (782) η 2 . 8 ± 0 . 6 → ω (782) η ′ (958) ω 3 (1670) − 0 ω 3 (1670) − → φ (1020) η ≈ 0 → φ (1020) η ′ (958) ω 3 (1670) − 0 φ 3 (1850) − → ρ (770) π 1 . 1 ± 0 . 3 → ¯ K ∗ (892) K + c . c . φ 3 (1850) − 35 . 5 ± 7 . 3 seen φ 3 (1850) − → ω (782) η 0 . 01 ± 0 . 01 → ω (782) η ′ (958) φ 3 (1850) − ≈ 0 φ 3 (1850) − → φ (1020) η 3 . 8 ± 0 . 8 → φ (1020) η ′ (958) φ 3 (1850) − 0 Table: The total decay widths are Γ tot ρ 3 (1690) = (161 ± 10) MeV, Γ tot 3 (1780) = (159 ± 21) MeV, Γ tot ω 3 (1670) = (168 ± 10) MeV and Γ tot φ 3 (1850) = (87 +28 − 23 ) MeV K ∗ Shahriyar Jafarzade (Jan Kochanowski University of Kielce, Poland) 14 / 19
Results for W → X + P ◮ We used the following PDG data for defining the coupling constant Γ( ρ 3 → a 2 π ) Γ( ρ 3 → ρη ) = 5 . 5 ± 2 . 0 MeV ◮ g 2 WXP = (2 . 83 ± 1 . 18) · 10 − 9 (MeV) − 4 Decay process (in model) Theory (MeV) Experiment (MeV) ρ 3 (1690) − → a 2 (1320) π 20 . 9 ± 8 . 8 seen → ¯ K ∗ ρ 3 (1690) − 2 (1430) K + c . c . 0 K ∗ 3 (1780) − → a 2 (1320) K 0 → ¯ K ∗ K ∗ 3 (1780) − 2 (1430) π 5 . 9 ± 2 . 5 < 25 . 4 ↔ < 16% → ¯ K ∗ K ∗ 3 (1780) − 2 (1430) η 0 → ¯ K ∗ K ∗ 2 (1430) η ′ (958) 3 (1780) − 0 → f 2 (1270) ¯ K ∗ 3 (1780) − K ≈ 0 2 (1525) ¯ K ∗ → f ′ 3 (1780) − K 0 → ¯ K ∗ ω 3 (1670) − 2 (1430) K + c . c . 0 → ¯ K ∗ φ 3 (1850) − 2 (1430) K + c . c . 0 Shahriyar Jafarzade (Jan Kochanowski University of Kielce, Poland) 15 / 19
Theoretical limits for W → V + V ◮ For this channel there is no enough information in PDG for defining the coupling constant ◮ We could only find the boundary for it g 2 WVV ≤ (1036 ± 41) Decay process (in model) Theory (MeV) Experiment (MeV) ρ 3 (1690) − → ρ (770) ρ (770) ≤ 107 . 9 ± 36 . 0 seen → ¯ K ∗ (892) K ∗ (892) ρ 3 (1690) − 0 → ρ (770) ¯ K ∗ K ∗ (892) 3 (1780) − ≤ 44 . 5 ± 1 . 8 → ¯ K ∗ K ∗ (892) ω (782) 3 (1780) − ≤ 13 . 3 ± 0 . 5 → ¯ K ∗ K ∗ (892) φ (1020) 3 (1780) − 0 → ¯ K ∗ (892) K ∗ (892) ω 3 (1670) − 0 → ¯ K ∗ (892) K ∗ (892) φ 3 (1850) − ≤ 31 . 1 ± 1 . 2 Table: The total decay widths are Γ tot ρ 3 (1690) = (161 ± 10) MeV, Γ tot 3 (1780) = (159 ± 21) MeV, Γ tot ω 3 (1670) = (168 ± 10) MeV and Γ tot φ 3 (1850) = (87 +28 − 23 ) MeV K ∗ Shahriyar Jafarzade (Jan Kochanowski University of Kielce, Poland) 16 / 19
Theoretical predictions for Glueballs ◮ Glueballs are not experimentally observed yet ◮ Lattice results for the mass of J PC = 3 −− glueballs are about m G 3 = 4 . 2 GeV ◮ Decay of the tensor-glueball to the vector and pseudo-scalar mesons ◮ Interaction lagrangian L I = c GVP G µαβ ε µνρσ � ( ∂ ν V ρ ) , ( ∂ α ∂ β ∂ σ P ) � ◮ We can only calculate the decay ratios BR for G 3 (4200) Theory Γ( G 3 → ρ 1 π ) 5 . 74 Γ( G 3 → ω 1 η ) Γ( G 3 → K 1 K ) 6 . 36 Γ( G 3 → ω 1 η ) Γ( G 3 → K 1 K ) 10 . 44 Γ( G 3 → φ 1 η ′ ) Γ( G 3 → K 1 K ) 11 . 28 Γ( G 3 → φ 1 η ) Γ( G 3 → K 1 K ) 12 . 43 Γ( G 3 → ω 1 η ′ ) Table: Decay ratios for tensor Glueballs J PC = 3 −− Shahriyar Jafarzade (Jan Kochanowski University of Kielce, Poland) 17 / 19
Conclusion ◮ Phenomenology of the spin-3 mesons is studied ◮ PDG data is explained using the effective lagrangian description for tree level ◮ SU ( N f = 3) V approximate symmetry is considered main symmetry for effective model ◮ Other symmetries of QCD such as SU ( N c = 3) and U (1) V are still exist within this model since mesons are colorless and B = 0 objects ◮ U (1) A symmetry does not exist because of the quantum effects ◮ SU ( N f = 3) A is also broken since the model is not in chiral regime ( m i = 0) ◮ We predict some values for decay rates which can be tested in future ◮ We do not have enough experimental information for theoretical calculations of 3 −− → [0 − + + 1 ++ ] & [0 − + + 1 + − ] decay products ◮ Some theoretical predictions for decay ratios for 3 −− glueballs are presented ◮ Only allowed decay channels for 3 −− glueball are S + P and V + P Shahriyar Jafarzade (Jan Kochanowski University of Kielce, Poland) 18 / 19
Thank you for your attention! Shahriyar Jafarzade (Jan Kochanowski University of Kielce, Poland) 19 / 19
On the phase structure and equation of state of strongly-interacting matter Mario Motta in collaboration with W.Alberico, A.Beraudo and R. Stiele University of Turin Frontiers in Nuclear and Hadronic Physics 2020 March 4 2020,Florence Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 1 / 23
Contents Exploration of the QCD phase diagram Phase Diagram Observables PNJL Thermodynamic and Fluctuations Results Conclusion a outlook Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 2 / 23
Phase Diagram Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 3 / 23
Why we explore the QCD Phase Diagram? The description of nuclear matter and the interaction between nucleons in the nuclei should ultimately be provided by QCD. This theory contains two important features: confinement spontaneous chiral symmetry breaking The knowledge of these two QCD features is not complete. the explorations of the phase diagram (in particular the region where chiral symmetry is restored and confinement does not occur) may provide a full understanding of these two phenomena Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 4 / 23
Observables Which Observables are important for the explorations of the Phase Diagram? isentropic trajectories Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 5 / 23
Observables Which Observables are important for the explorations of the Phase Diagram? isentropic trajectories speed of sound Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 5 / 23
Observables Which Observables are important for the explorations of the Phase Diagram? isentropic trajectories speed of sound order parameters Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 5 / 23
Observables Which Observables are important for the explorations of the Phase Diagram? isentropic trajectories speed of sound order parameters fluctuations of conserved charge of QCD Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 5 / 23
Isentropic trajectories Along isentropic trajectories, the system evolves in time keeping constant entropy. The QGP expands, during this expansion baryon number density and entropy density change (pure dilution), but s / n = cost . Then the isentropic trajectories is replaced by ”Iso- s / n ” trajectories Figure: isentropic trajectories from N.Guenther et al Nucl.Phys. A967 (2017) 720-723 Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 6 / 23
Speed of sound The speed of sound is one of the most important characteristics in hydrodynamics: it is responsible for the collective acceleration of the fireball and it governs the evolution of the fireball produced in the heavy-ion collision as well as one of the most important observables for describing of QGP formation: the elliptic flow. Figure: the square of speed of sound (Szabolcs Borsanyi et all JHEP 1011 (2010) 077) and a picture of HIC Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 7 / 23
Speed of sound The speed of sound is one of the most important characteristics in hydrodynamics: it is responsible for the collective acceleration of the fireball and it governs the evolution of the fireball produced in the heavy-ion collision as well as one of the most important observables for describing of QGP formation: the elliptic flow. Figure: the square of speed of sound and the Elliptic Flow Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 8 / 23
Speed of sound The speed of sound is one of the most important characteristics in hydrodynamics: it is responsible for the collective acceleration of the fireball and it governs the evolution of the fireball produced in the heavy-ion collision as well as one of the most important observables for describing of QGP formation: the elliptic flow. Figure: the square of speed of sound and the Elliptic Flow ( ǫ + P ) dv i d ǫ dt = − c 2 (1) s dx i Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 8 / 23
Order Parameters The order parameters signal the transition lines and their behaviour near the transition fixes the order of transition (cross-over, 1 st -order, 2 st -order,ect.) Figure: the order parameters for deconfinament and chiral symmetry restoration transition W.Weise,174,JPS,10.1143/PTPS.174.1 Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 9 / 23
Fluctuations of QCD conserved charges In many different fields, the study of fluctuations can provide physical insights into the underlying microscopic physics. The fluctuations can become invaluable physical observable in spite of their difficult character. Fluctuations are powerful tools to diagnose microscopic physics, to trace back the history of the system and the nature of its elementary degrees of freedom (see M. Asakawa and M. Kitazawa, Prog. Part. Nucl.Phys. 90, 299 (2016) doi:10.1016/j.ppnp.2016.04.002). Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 10 / 23
Fluctuations of QCD conserved charges In many different fields, the study of fluctuations can provide physical insights into the underlying microscopic physics. The fluctuations can become invaluable physical observable in spite of their difficult character. Fluctuations are powerful tools to diagnose microscopic physics, to trace back the history of the system and the nature of its elementary degrees of freedom (see M. Asakawa and M. Kitazawa, Prog. Part. Nucl.Phys. 90, 299 (2016) doi:10.1016/j.ppnp.2016.04.002). My work is focused on fluctuations of conserved charges in the QCD Phase Diagram (B, Q, S) explored through their cumulants. Figure: Combinations of Cumulants in HIC N.R.Sahoo and the Star Collaboration 2014 J.Phys.:Conf. Ser. 535 012007 Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 10 / 23
How can we explore the Phase Diagram? Effective Field Theories The basic idea of an Effective Field Theories (EFT) is that, if one is interested in describing phenomena occurring at a certain (low) energy scale, one does not need to solve the exact microscopic theory in order to provide useful predictions. Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 11 / 23
How can we explore the Phase Diagram? Effective Field Theories The basic idea of an Effective Field Theories (EFT) is that, if one is interested in describing phenomena occurring at a certain (low) energy scale, one does not need to solve the exact microscopic theory in order to provide useful predictions. In general, Effective Field Theories are low-energy approximations of more fundamental theories. Instead of solving the underlying theory,low-energy physics is described with a set of variables that are suited to the particular energy region one is interested in. Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 11 / 23
PNJL m ) q + 1 τ q ) 2 + (¯ qi γ 5 � τ q ) 2 ]+ q ( i γ µ D µ − ˆ L PNJL = ¯ 2 G [(¯ q � (2) q (1 − γ 5 ) q ] } − U (Φ[ A ] , ¯ q (1 + γ 5 ) q ] + det [¯ + K { det [¯ Φ[ A ] , T ) Here D µ = ∂ µ − iA µ , A µ = δ µ 0 A 0 , the fields Φ and ¯ Φ are Polyakov fields defined as: Φ ≡ 1 Φ ≡ 1 ¯ Tr �� L † �� Tr �� L �� (3) N c N c Where L is the Polyakov loop defined in terms of the gauge field A 4 ,after Wick rotation: � β � � L ( � x ) ≡ P d τ A 4 ( τ, � x ) (4) i 0 Due to the second term in (2) the PNJL isn’t renormalizable and I introduce a cut-off (Λ) for regularizing the integrals. Below I indicate the quark chiral condensate as: � ¯ q i q i � ≡ ϕ i i = u , d , s (5) Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 12 / 23
Polyakov Potential The Polyakov Potential replaces the gluonic interaction of QCD in this EFT. I’m using these two parametrizations: Polynomial: T 4 = − b 2 ( T ) U ΦΦ − b 3 Φ 3 + Φ 3 ) + b 4 ¯ 6 (¯ 4 (¯ ΦΦ) 2 (6) 2 � 2 � 3 � T 0 � T 0 T 0 b 2 ( T ) = a 0 + a 1 T + a 2 + a 3 T T Logarithmic: U Φ 3 + Φ 3 ) − 3(¯ T 4 = a ( T )¯ ΦΦ + b ( T ) ln [1 − 6¯ ΦΦ + 4(¯ ΦΦ) 2 ] (7) � 2 � 3 � T 0 � T 0 T 0 a ( T ) = a 0 + a 1 T + a 2 , b ( T ) = b 3 T T The Parameters are fixed for reproducing the lattice data for pure YM-Theory (C.Ratti,et al in Phys.Rev. D 73,014019 (2006) and Nuc.Phys A Vol 814, 1-4 (2008)) Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 13 / 23
Thermodynamics From PNJL Lagrangian one obtains the Thermodynamic Grand Potential per unit volume( ω = Ω / V ) in Mean Fields Approximation: ω (Φ , ¯ Φ , T , M j , µ j ) = U (Φ , ¯ � ϕ 2 Φ , T ) + G i + 4 K ϕ u ϕ d ϕ s + i = u , d , s � Λ d 3 p � Λ d 3 p � � � z i + Φ ( E i , µ i ) + z i − � − 2 N c (2 π ) 3 E i + T Φ ( E i , µ i ) } (2 π ) 3 i = u , d , s Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 14 / 23
Thermodynamics From PNJL Lagrangian one obtains the Thermodynamic Grand Potential per unit volume( ω = Ω / V ) in Mean Fields Approximation: ω (Φ , ¯ Φ , T , M j , µ j ) = U (Φ , ¯ � ϕ 2 Φ , T ) + G i + 4 K ϕ u ϕ d ϕ s + i = u , d , s � Λ d 3 p � Λ d 3 p � � � z i + Φ ( E i , µ i ) + z i − � − 2 N c (2 π ) 3 E i + T Φ ( E i , µ i ) } (2 π ) 3 i = u , d , s From ω it is possible to obtain all thermodynamics quantities of interest: n = − ∂ n ω s = − ∂ω n i = − ∂ω χ i P = − ω, ∂ T , , (8) ∂µ n ∂µ i i � ǫ = − P + sT + µ i n i (9) i Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 14 / 23
Cumulants To describe the fluctuations of conserved charge we can use the cumulants. The relations between the first four moments and first cumulants read: Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 15 / 23
Cumulants To describe the fluctuations of conserved charge we can use the cumulants. The relations between the first four moments and first cumulants read: � x � c = � x � ≡ M � x 2 � c = � x 2 � − � x � 2 ≡ σ 2 (10) � x 3 � c = � ( x − � x � ) 3 � ≡ γσ 3 � x 4 � c = � ( x − � x � ) 4 � − 3 σ 4 ≡ κσ 4 Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 15 / 23
Cumulants To describe the fluctuations of conserved charge we can use the cumulants. The relations between the first four moments and first cumulants read: � x � c = � x � ≡ M � x 2 � c = � x 2 � − � x � 2 ≡ σ 2 (10) � x 3 � c = � ( x − � x � ) 3 � ≡ γσ 3 � x 4 � c = � ( x − � x � ) 4 � − 3 σ 4 ≡ κσ 4 κ is called kurtosis and γ is called skewness. This two quantities represent respectively the ”sharpness” and asymmetry of the distribution. Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 15 / 23
Cumulants To describe the fluctuations of conserved charge we can use the cumulants. The relations between the first four moments and first cumulants read: � x � c = � x � ≡ M � x 2 � c = � x 2 � − � x � 2 ≡ σ 2 (10) � x 3 � c = � ( x − � x � ) 3 � ≡ γσ 3 � x 4 � c = � ( x − � x � ) 4 � − 3 σ 4 ≡ κσ 4 κ is called kurtosis and γ is called skewness. This two quantities represent respectively the ”sharpness” and asymmetry of the distribution. The cumulants are more convenient than moments, e.g, when one consider the products of distributions, the cumulant of the product distribution is the product of cumulants. Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 15 / 23
Cumulants To describe the fluctuations of conserved charge we can use the cumulants. The relations between the first four moments and first cumulants read: � x � c = � x � ≡ M � x 2 � c = � x 2 � − � x � 2 ≡ σ 2 (10) � x 3 � c = � ( x − � x � ) 3 � ≡ γσ 3 � x 4 � c = � ( x − � x � ) 4 � − 3 σ 4 ≡ κσ 4 κ is called kurtosis and γ is called skewness. This two quantities represent respectively the ”sharpness” and asymmetry of the distribution. The cumulants are more convenient than moments, e.g, when one consider the products of distributions, the cumulant of the product distribution is the product of cumulants. In my work I consider the following combinations of cumulants: κσ 2 = � x 4 � c γσ 3 M = � x 3 � c (11) � x 2 � c � x � c Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 15 / 23
Results In the last part of my presentation I show the numerical results obtained with PNJL for the following observables: isentropic trajectories speed of sound fluctuations of net Baryon number Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 16 / 23
Results In the last part of my presentation I show the numerical results obtained with PNJL for the following observables: isentropic trajectories speed of sound fluctuations of net Baryon number From a general point of view the phase diagram of QCD is a 4-dimension space: 3 dimensions for the quark chemical potentials and 1 for temperature.I perform my calculations in the following scenarios: Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 16 / 23
Results In the last part of my presentation I show the numerical results obtained with PNJL for the following observables: isentropic trajectories speed of sound fluctuations of net Baryon number From a general point of view the phase diagram of QCD is a 4-dimension space: 3 dimensions for the quark chemical potentials and 1 for temperature.I perform my calculations in the following scenarios: Symmetric chemical potential: µ u = µ d = µ s = 1 3 µ B (Quasi-)Neutral Strangeness: µ u = µ d = 1 3 µ B , µ s = 0 HIC: n Q n B = 0 . 4, n s = 0 Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 16 / 23
Results:isentropic trajectories s/n B = 33.6 PNJL s/n B = 277.6 s/n B = 123.9 s/n B = 84.3 300 s/n B = 56.8 Temperature T [ MeV ] s/n B = 45.8 s/n B = 26.7 s/n B = 17.5 s/n B = 5 s/n B = 2 200 100 0 0 100 200 300 400 quark chemical potential μ q [ MeV ] Figure: Isentropic Trajectories in the (Quasi)-neutral strangeness (M.Motta et al in prep.(March 2020) Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 17 / 23
Results:EoS ǫ SB = 3 P SB (12) 1e+011 PNJL 1e+010 4 ] ε [MeV 1e+009 SB s/n B =331.6 s/n B =56.8 1e+008 s/n B =7 s/n B =5 1e+007 1e+008 1e+010 4 ] P [MeV Figure: Equation of State in the QNS scenaio on the isentropic trajectories Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 18 / 23
Results:speed of sound s = ∂ P c 2 (13) ∂ǫ PNJL 0.3 2 squared speed of sound c s s/n B = 331.6 s/n B = 277.6 s/n B = 123.9 0.2 s/n B = 84.3 s/n B = 56.8 s/n B = 45.8 s/n B = 26.7 0.1 s/n B = 17.5 s/n B = 5 s/n B = 2 0 0 100 200 300 400 Temperature T [ MeV ] Figure: Speed of sound in the (Quasi)-neutral strangeness on the isentropic trajectories (M.Motta eta al in prep.(2020)) Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 19 / 23
Results:fluctuations in Symmetric scenario ΚΣ 2 Γ B Σ 3 � M Μ B � 540 MeV 6 1.4 Μ B � 810 MeV 4 1.2 Μ B � 862 MeV 2 Μ B � 900 MeV 1.0 0 T � MeV � 0.8 100 150 200 250 � 2 0.6 � 4 0.4 0.2 � 6 0.0 T � MeV � � 8 100 150 200 250 300 Κ B Σ 2 0 T � MeV � 100 150 200 250 � 5 Μ B � 540 MeV � 10 Μ B � 810 MeV � 15 Μ B � 862 MeV Μ B � 900 MeV � 20 � 25 Figure: M.Motta et al [1909.05037] Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 20 / 23
Results:fluctuations in (Quasi-)Neutral strangness scenario Γ B Σ 3 � M 1st Order 200 000 CEP Crosoverx10 3 100 000 0 1200 Μ B � MeV � 800 900 1000 1100 � 100 000 � 200 000 Κ B Σ 2 1st Order 15 000 CEP 10 000 Crosover x20 5000 0 1200 Μ B � MeV � 800 900 1000 1100 � 5000 � 10 000 � 15 000 Figure: ( T 1 = 122 . 9 , T 2 = 132 . 9 , T 3 = 142 . 9) MeV,M.Motta et al [1909.05037] Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 21 / 23
Conclusions and outlook Conclusions : PNJL model provides a good qualitative and semi-quantitative guidance to describe the chiral and deconfinement QCD transition Nature of the active degrees of freedom is displayed by high order cumulants (e.g. kurtosis) outlook : Calculation in the HIC scenarios are in progress and other thermodynamic quantities are currently under investigation I’m going to perform the calculation of mixed flavours susceptibility for the comparison with Lattice QCD and experimental results. Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 22 / 23
Thank you for your attention! Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 23 / 23
Polyakov Fields The value of Polyakov Fields is connected to the energy for produce a free quark from the vacuum: Φ ∼ e − β F q (14) In the confined region, where it is not possible to create a sigle quarks from vacuum, F q → + ∞ then Φ → 0. In the deconfined region F q is finite and Φ � = 0. At extremely high temperature Φ → 1. In any case Φ is smaller than unity Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 1 / 3
Generators of Fermi Functions In PNJL lagrangian appears the functions z i ± Φ Φ e − β ( E i − µ i ) ) e − β ( E i − µ i ) + e − 3 β ( E i − µ i ) ] z i + Φ ( E i , µ i ) ≡ ln[1 + N c (Φ + ¯ (15) Φ + Φ e − β ( E i + µ i ) ) e − β ( E i + µ i ) + e − 3 β ( E i + µ i ) ] z i − Φ ( E i , µ i ) ≡ ln[1 + N c (¯ (16) The derivative of this function on chemical potential µ i are the Fermi modified function: ∂ z i ± Φ ( E i , µ i ) ≡ ± T f i ± ∂µ i N c (17) Φ e − β ( E i ± µ i ) ) e − β ( E i ± µ i ) + e − 3 β ( E i ± µ i ) (Φ + 2¯ = 1 + N c (Φ + ¯ Φ e − β ( E i ± µ i ) ) e − β ( E i ± µ i ) + e − 3 β ( E i ± µ i ) Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 2 / 3
Mass Gap Equation The PNJL Lagrangian is chiral symmetric if m i = 0. For non vanishing current mass chiral symmetry is explicitly broken. Moreover, at low temperature and chemical potential, chiral symmetry is also dynamically broken by self-interaction of quarks: the chiral condensate is negative and large. Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 3 / 3
Mass Gap Equation The PNJL Lagrangian is chiral symmetric if m i = 0. For non vanishing current mass chiral symmetry is explicitly broken. Moreover, at low temperature and chemical potential, chiral symmetry is also dynamically broken by self-interaction of quarks: the chiral condensate is negative and large. The mass gap equation for the quark of species i reads: M i = m i − 2 G ϕ i − 2 K ϕ j ϕ k , i � = j � = k (18) Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 3 / 3
Mass Gap Equation The PNJL Lagrangian is chiral symmetric if m i = 0. For non vanishing current mass chiral symmetry is explicitly broken. Moreover, at low temperature and chemical potential, chiral symmetry is also dynamically broken by self-interaction of quarks: the chiral condensate is negative and large. The mass gap equation for the quark of species i reads: M i = m i − 2 G ϕ i − 2 K ϕ j ϕ k , i � = j � = k (18) The second term of the RHS of the equation is due to the 4-fermion interaction vertex and the third term is due to the 6-fermion interaction vertex. This vertex mixes the chiral condensates one with an others. Mario Motta (UniTo) On the phase structure and equation of state of strongly-interacting matter March 4 2020, Florence 3 / 3
Introduction Holographic Black Hole Model Results Conclusions QCD Phase Diagram from Holographic Black Holes Joaquin Grefa with: Claudia Ratti & Israel Portillo(UH), Romulo Rougemont (UFRN), Jacquelyn Noronha-Hostler & Jorge Norhona (UIUC) Frontiers in Nuclear and Hadronic Physics 2020, February 24 - March 6, 2020 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 1 / 20
Introduction Holographic Black Hole Model The QCD Phase Diagram Results Limitations Conclusions Table of Contents Introduction 1 The QCD Phase Diagram Limitations Holographic Black Hole Model 2 Holography (Gauge/String duality) Fixing the model at µ B = 0 Thermodynamics at µ B = 0 Results 3 Mapping the QCD phase diagram and CEP Thermodynamics at finite µ B Equation of State Conclusions 4 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 2 / 20
Introduction Holographic Black Hole Model The QCD Phase Diagram Results Limitations Conclusions QCD Phase Diagram QCD is a nonabelian gauge theory: strongly interacting at low energies, and weakly interacting at large energies. Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 3 / 20
Introduction Holographic Black Hole Model The QCD Phase Diagram Results Limitations Conclusions QCD Phase Diagram QCD is a nonabelian gauge theory: strongly interacting at low energies, and weakly interacting at large energies. We can explore the QCD phase diagram by changing √ s in relativistic heavy ion collisions Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 3 / 20
Introduction Holographic Black Hole Model The QCD Phase Diagram Results Limitations Conclusions QCD Phase Diagram QCD is a nonabelian gauge theory: strongly interacting at low energies, and weakly interacting at large energies. We can explore the QCD phase diagram by changing √ s in relativistic heavy ion collisions We can solve the theory by using Lattice QCD. Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 3 / 20
Introduction Holographic Black Hole Model The QCD Phase Diagram Results Limitations Conclusions Limitations of Lattice QCD Fermi sign problem: It only provides the Equation of State (EoS) at µ B = 0. Taylor Expansion for small µ B P ( T , µ B ) − P ( T , µ B = 0) = T 4 � µ B � 2 n 1 Σ ∞ (2 n )! χ 2 n ( T ) n =1 T χ n ( T , µ B ) = ∂ n ( P / T 4 ) where ∂ ( µ B / T ) n As a consequence, a large part of the QCD phase diagram remains unknown. Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 4 / 20
Introduction Holographic Black Hole Model The QCD Phase Diagram Results Limitations Conclusions Model Requirements The model should exhibit: -Deconfinement How can we fulfill these conditions? -Nearly perfect fluidity -Agreement with Lattice EoS at µ B = 0 -Agreement with baryon susceptibilities at µ B = 0 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 5 / 20
Introduction Holographic Black Hole Model The QCD Phase Diagram Results Limitations Conclusions Model Requirements How can we fulfill these conditions? ...BLACK HOLES...!!! The model should exhibit: -Deconfinement -Nearly perfect fluidity -Agreement with Lattice EoS at µ B = 0 -Agreement with baryon susceptibilities at µ B = 0 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 5 / 20
Introduction Holography (Gauge/String duality) Holographic Black Hole Model Fixing the model at µ B = 0 Results Thermodynamics at µ B = 0 Conclusions Table of Contents Introduction 1 The QCD Phase Diagram Limitations Holographic Black Hole Model 2 Holography (Gauge/String duality) Fixing the model at µ B = 0 Thermodynamics at µ B = 0 Results 3 Mapping the QCD phase diagram and CEP Thermodynamics at finite µ B Equation of State Conclusions 4 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 6 / 20
Introduction Holography (Gauge/String duality) Holographic Black Hole Model Fixing the model at µ B = 0 Results Thermodynamics at µ B = 0 Conclusions Holography (Gauge/String duality) Holographic gauge/gravity correspondance String Theory/Classical Gravity Quantum Field Theory ⇐ ⇒ in 5-dimensions in 4-dimensions Maldacena 1997; Witten 1998; Gubser, Polyakov, Klebanov 1998 Near Perfect fluidity vanishing coupling in GR → strong coupling in QFT BH solutions → T and µ B in QFT Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 7 / 20
Introduction Holography (Gauge/String duality) Holographic Black Hole Model Fixing the model at µ B = 0 Results Thermodynamics at µ B = 0 Conclusions Gravitational Action � R − ( ∂ µ φ ) 2 − f ( φ ) F 2 1 d 5 x √− g µν S = V ( φ ) − 2 κ 2 2 4 � �� � 5 M 5 � �� � nonconformal µ B � =0 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 8 / 20
Introduction Holography (Gauge/String duality) Holographic Black Hole Model Fixing the model at µ B = 0 Results Thermodynamics at µ B = 0 Conclusions V ( φ ) and f ( φ ) The free parameters of the EMD holografic model are fixed to match the holografic results for the entropy density ( s / T 3 ) and second order baryon susceptibility ( χ 2 ) to state-of-the-art lattice QCD results for these quantities. Free Parameters for the Holographic Model κ 2 5 = 8 π G 5 = 8 π (0 . 46) , Λ = 1053 . 83 MeV , V ( φ ) = − 12 cosh(0 . 63 φ ) + 0 . 65 φ 2 − 0 . 05 φ 4 + 0 . 003 φ 6 , f ( φ ) = sech( c 1 φ + c 2 φ 2 ) c 3 + 1 + c 3 sech( c 4 φ ) , 1 + c 3 where c 1 = − 0 . 27 , c 2 = 0 . 4 , c 3 = 1 . 7 , c 4 = 100 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 9 / 20
Introduction Holography (Gauge/String duality) Holographic Black Hole Model Fixing the model at µ B = 0 Results Thermodynamics at µ B = 0 Conclusions Equations of Motion � − f ( φ ) F 2 R − ( ∂ µ φ ) 2 1 d 5 x √− g µν S = V ( φ ) − 2 κ 2 2 4 � �� � 5 M 5 � �� � nonconformal µ B � =0 e 2 A ( r ) [ − h ( r ) dt 2 + d − x 2 ] + e 2 B ( r ) dr 2 → ds 2 = h ( r ) φ = φ ( r ) A µ dx µ = Φ( r ) dt Equations of Motion � h ′ ( r ) � � ∂ V ( φ ) � − e − 2 A ( r ) Φ ′ ( r ) 2 1 ∂ f ( φ ) φ ′′ ( r ) + h ( r ) + 4 A ′ ( r ) φ ′ ( r ) − = 0 h ( r ) ∂φ 2 ∂φ � � 2 A ′ ( r ) + d [ln f ( φ )] Φ ′′ ( r ) + φ ′ ( r ) Φ ′ ( r ) = 0 d φ A ′′ ( r ) + φ ′ ( r ) 2 = 0 6 h ′′ ( r ) + 4 A ′ ( r ) h ′ ( r ) − e − 2 A ( r ) f ( φ )Φ ′ ( r ) 2 = 0 h ( r )[24 A ′ ( r ) 2 − φ ′ ( r ) 2 ] + 6 A ′ ( r ) h ′ ( r ) + 2 V ( φ ) + e − 2 A ( r ) f ( φ )Φ ′ ( r ) 2 = 0 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 10 / 20
Introduction Holography (Gauge/String duality) Holographic Black Hole Model Fixing the model at µ B = 0 Results Thermodynamics at µ B = 0 Conclusions Solutions Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 11 / 20
Introduction Holography (Gauge/String duality) Holographic Black Hole Model Fixing the model at µ B = 0 Results Thermodynamics at µ B = 0 Conclusions Thermodynamics at µ B = 0 4 3 10 2 5 1 0 0 100 200 300 400 500 100 200 300 400 500 0.4 4 0.3 3 2 0.2 1 0.1 0 100 200 300 400 500 100 200 300 400 500 Lattice Results: [WB] S Borsanyi et al.Phys. Lett. B730.99. Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 12 / 20
Introduction Mapping the QCD phase diagram and CEP Holographic Black Hole Model Thermodynamics at finite µ B Results Equation of State Conclusions Table of Contents Introduction 1 The QCD Phase Diagram Limitations Holographic Black Hole Model 2 Holography (Gauge/String duality) Fixing the model at µ B = 0 Thermodynamics at µ B = 0 Results 3 Mapping the QCD phase diagram and CEP Thermodynamics at finite µ B Equation of State Conclusions 4 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 13 / 20
Introduction Mapping the QCD phase diagram and CEP Holographic Black Hole Model Thermodynamics at finite µ B Results Equation of State Conclusions Mapping the QCD phase diagram from Black Hole solutions The BH solutions are parametrized by ( φ 0 , Φ 1 ), where φ 0 → value of the scalar field at the horizon, and Φ 1 → electric field in the radial direction at the horizon 5 200 4.5 180 4 160 140 3.5 120 3 100 2.5 80 2 60 1.5 40 1 20 0 0.5 0 200 400 600 800 1000 1200 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 14 / 20
Introduction Mapping the QCD phase diagram and CEP Holographic Black Hole Model Thermodynamics at finite µ B Results Equation of State Conclusions Locating the Critical End Point (CEP) T CEP = 89 MeV and µ CEP = 724 MeV B Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 15 / 20
Introduction Mapping the QCD phase diagram and CEP Holographic Black Hole Model Thermodynamics at finite µ B Results Equation of State Conclusions Thermodynamics at finite µ B Lattice Results: [WB] S Borsanyi et al. arXiv:1805.04445v1. BH curves: R. Critelli et al., Phys.Rev.D96(2017). Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 16 / 20
Introduction Mapping the QCD phase diagram and CEP Holographic Black Hole Model Thermodynamics at finite µ B Results Equation of State Conclusions Equation of State Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 17 / 20
Introduction Holographic Black Hole Model Results Conclusions Table of Contents Introduction 1 The QCD Phase Diagram Limitations Holographic Black Hole Model 2 Holography (Gauge/String duality) Fixing the model at µ B = 0 Thermodynamics at µ B = 0 Results 3 Mapping the QCD phase diagram and CEP Thermodynamics at finite µ B Equation of State Conclusions 4 Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 18 / 20
Introduction Holographic Black Hole Model Results Conclusions Conclusions The Lattice EoS was reproduced for small µ B . Holographic Black Holes predict a CEP: T CEP = 89 MeV, µ CEP = 724 MeV B The first order transition line was located in the QCD phase diagram. We are able to compute higher order susceptibilities. We obtained state variables for a larger region in the QCD phase diagram. Future work: -Calculate critical exponents. -Obtain other state variables. -Use QCD EoS to better understand structure of compact stellar objects. -Extend the model to include other conserved quantities. Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 19 / 20
Introduction Holographic Black Hole Model Results Conclusions Thanks THANKS..! Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 20 / 20
Collision Energy Estimates Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 1 / 6
χ measurements Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 2 / 6
Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 3 / 6
Critical Exponents Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 4 / 6
constant value lines Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 5 / 6
Points on First Order transition Line Joaquin Grefa QCD Phase Diagram from Holographic Black Holes 6 / 6
Femtoscopy of the D meson and nucleon interaction Isabela Maietto*, Gastao Krein, Sandra Padula Institute of Theoretical Physics - IFT - UNESP
Outline Motivation Femtoscopy and Correlations ¯ DN observables Results Summary 04/MAR/2020 Isabela M. Silverio 2
Motivation Femtoscopy: correlation function of two particles as a function of relative momentum q • Obtain the source size • Sensitive to the effects of the final-state interaction ◦ Coulomb interaction ◦ Strong Interaction ◦ Isospin 04/MAR/2020 Isabela M. Silverio 3
Motivation Femtoscopy: correlation function of two particles as a function of relative momentum q • Obtain the source size • Sensitive to the effects of the final-state interaction ◦ Coulomb interaction ◦ Strong Interaction ◦ Isospin Here, discuss DN interaction , no experimental data available yet • Important for the quest of possible existence of D-mesic nuclei (an exotic nuclear state) • Through D-mesic nuclei, one can possibly access chiral symmetry restoration effects ◦ Properties of light quarks in D mesons are sensitive to temperature and density 04/MAR/2020 Isabela M. Silverio 3
Correlation Function Two-particle correlation function: P ( p 1 , p 2 ) C ( p 1 , p 2 ) = (1) P ( p 1 ) P ( p 2 ) Experimentally can be obtained as: C ( q ) ∝ N same ( q ) (2) N mixed ( q ) 04/MAR/2020 Isabela M. Silverio 4
Correlation Function Two-particle correlation function: P ( p 1 , p 2 ) C ( p 1 , p 2 ) = (1) P ( p 1 ) P ( p 2 ) Experimentally can be obtained as: C ( q ) ∝ N same ( q ) (2) N mixed ( q ) If C ( p 1 , p 2 ) → 1 no particle correlation If C ( p 1 , p 2 ) � = 1 particles are correlated 04/MAR/2020 Isabela M. Silverio 4
Correlation Function The Correlation Function can be written with an equal-time approximation, e.g., the particles states are emitted simultaneously in the pair rest frame t 1 = t 2 and p 1 + p 2 = 0 : P ( p 1 , p 2 ) � d r S 12 ( r ) | Ψ( r , q ) | 2 C ( p 1 , p 2 ) = P ( p 1 ) P ( p 2 ) ≈ (3) 04/MAR/2020 Isabela M. Silverio 5
Correlation Function The Correlation Function can be written with an equal-time approximation, e.g., the particles states are emitted simultaneously in the pair rest frame t 1 = t 2 and p 1 + p 2 = 0 : P ( p 1 , p 2 ) � d r S 12 ( r ) | Ψ( r , q ) | 2 C ( p 1 , p 2 ) = P ( p 1 ) P ( p 2 ) ≈ (3) where, • S ( r ) is the source function. Tipically, this can be represented with a spherical Gaussian source: − r 2 1 � � S 12 ( r ) = 2 exp (4) 3 4 R 2 (4 πR 2 ) ◦ R is the width of the source. • Ψ( r , q ) is the wave function, where q is the relative momentum: q = | p 1 − p 2 | 04/MAR/2020 Isabela M. Silverio 5
Correlation Function Partial Wave Decomposition: ∞ � (2 l + 1) i l ψ l ( r ) P l ( cosθ ) Ψ( r, q ) = (5) l =0 04/MAR/2020 Isabela M. Silverio 6
Correlation Function Partial Wave Decomposition: ∞ � (2 l + 1) i l ψ l ( r ) P l ( cosθ ) Ψ( r, q ) = (5) l =0 Supose now, that only the s-wave is affected by the interaction: ∞ � (2 l + 1) i l ψ free Ψ( r, q ) = ψ 0 ( r, q ) + ( r, q ) P l (cos θ ) l l =1 04/MAR/2020 Isabela M. Silverio 6
Correlation Function Partial Wave Decomposition: ∞ � (2 l + 1) i l ψ l ( r ) P l ( cosθ ) Ψ( r, q ) = (5) l =0 Supose now, that only the s-wave is affected by the interaction: ∞ � (2 l + 1) i l ψ free Ψ( r, q ) = ψ 0 ( r, q ) + ( r, q ) P l (cos θ ) l l =1 ∞ (2 l + 1) i l ψ free ( r, q ) P l (cos θ ) − ψ free � = ψ 0 ( r, q ) + (6) l 0 l =0 • ψ 0 is the wave function for l = 0 ; = j 0 ( qr ) = sin ( qr ) • ψ free � ψ free ( r, q ) P l (cos θ ) = e iqr ; 0 l qr l 04/MAR/2020 Isabela M. Silverio 6
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