AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Low-Energy Theorems Expressed in terms of the quark currents S i ( x ) , P j ( y ) , Chiral Lagrangian ( L i – parameters of the NLO Lagrangian of order of O ( p 4 ) , B = G π / F π ), spectral density ρ ( λ, m ) and topological charge density Q ( x ) ∼ tr F µν ( x )˜ F µν ( x ) : G 2 B 2 π δ ij � d 4 x � � i δ ij S 0 ( x ) S 0 ( 0 ) − P i ( x ) P j ( 0 ) = − + δ ij 8 π 2 ( L 3 − 4 L 4 + 3 ) m 2 π m ∂ 2 m 2 ρ ( λ, m ) � � ∂ m ρ ( λ, m ) � = 2 δ ij d λ − , (1) ( λ 2 + m 2 ) ( λ 2 + m 2 ) 2 4 m 2 ρ ( λ, m ) d 4 x � Q ( x ) Q ( 0 ) � � � � d 4 x � � i S i ( x ) S j ( 0 ) − δ ij P 0 ( x ) P 0 ( 0 ) = δ ij d λ ( λ 2 + m 2 ) 2 − 2 δ ij , (2) m 2 V d 4 x � Q ( x ) Q ( 0 ) � � ρ ( λ, m ) � � d 4 x � P 3 ( x ) P 0 ( 0 ) � = 2 ( m u − m d ) m i d λ ( λ 2 + m 2 ) 2 − ( m u − m d ) . (3) m 3 V [J. Gasser and H. Leutwyler, 1984]
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Low-Energy Theorems Expressed in terms of the quark currents S i ( x ) , P j ( y ) , Chiral Lagrangian ( L i – parameters of the NLO Lagrangian of order of O ( p 4 ) , B = G π / F π ), spectral density ρ ( λ, m ) and topological charge density Q ( x ) ∼ tr F µν ( x )˜ F µν ( x ) : G 2 B 2 π δ ij � d 4 x � � i δ ij S 0 ( x ) S 0 ( 0 ) − P i ( x ) P j ( 0 ) = − + δ ij 8 π 2 ( L 3 − 4 L 4 + 3 ) m 2 π m ∂ 2 m 2 ρ ( λ, m ) � � ∂ m ρ ( λ, m ) � = 2 δ ij d λ − , (1) ( λ 2 + m 2 ) ( λ 2 + m 2 ) 2 4 m 2 ρ ( λ, m ) d 4 x � Q ( x ) Q ( 0 ) � � � � d 4 x � � i S i ( x ) S j ( 0 ) − δ ij P 0 ( x ) P 0 ( 0 ) = δ ij d λ ( λ 2 + m 2 ) 2 − 2 δ ij , (2) m 2 V d 4 x � Q ( x ) Q ( 0 ) � � ρ ( λ, m ) � � d 4 x � P 3 ( x ) P 0 ( 0 ) � = 2 ( m u − m d ) m i d λ ( λ 2 + m 2 ) 2 − ( m u − m d ) . (3) m 3 V [J. Gasser and H. Leutwyler, 1984]
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Five-Dimensional Effective Action � d 5 x √ ge − Φ tr � � | DX | 2 + 3 ℓ 2 | X | 2 + κ � g 2 ℓ 2 | X | 4 S 5 D = X with a metric ds 2 = ℓ 2 1 � z 2 ( − dz 2 + dx µ dx µ ) . ( F 2 L + F 2 − R ) 4 g 2 5 “Hard-wall”: Φ( z ) ≡ 0 , κ = 0 , 0 � z � z m , “Soft-wall”: Φ( z ) ∼ λ z 2 ( z → ∞ ) , κ � = 0 , 0 � z < ∞ . L a q L ( x ) γ µ t a q L ( x ) , µ ( x , z = 0 ) = source of ¯ R a source of ¯ q R ( x ) γ µ t a q R ( x ) , µ ( x , z = 0 ) = 2 z X αβ ( x , z ) q α L ( x ) q β source of ¯ lim = R ( x ) z → 0 m δ αβ in the absence of (pseudo)scalar currents . =
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Five-Dimensional Effective Action � d 5 x √ ge − Φ tr � � | DX | 2 + 3 ℓ 2 | X | 2 + κ � g 2 ℓ 2 | X | 4 S 5 D = X with a metric ds 2 = ℓ 2 1 � z 2 ( − dz 2 + dx µ dx µ ) . ( F 2 L + F 2 − R ) 4 g 2 5 “Hard-wall”: Φ( z ) ≡ 0 , κ = 0 , 0 � z � z m , “Soft-wall”: Φ( z ) ∼ λ z 2 ( z → ∞ ) , κ � = 0 , 0 � z < ∞ . L a q L ( x ) γ µ t a q L ( x ) , µ ( x , z = 0 ) = source of ¯ R a source of ¯ q R ( x ) γ µ t a q R ( x ) , µ ( x , z = 0 ) = 2 z X αβ ( x , z ) q α L ( x ) q β source of ¯ lim = R ( x ) z → 0 m δ αβ in the absence of (pseudo)scalar currents . =
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Five-Dimensional Effective Action � d 5 x √ ge − Φ tr � � | DX | 2 + 3 ℓ 2 | X | 2 + κ � g 2 ℓ 2 | X | 4 S 5 D = X with a metric ds 2 = ℓ 2 1 � z 2 ( − dz 2 + dx µ dx µ ) . ( F 2 L + F 2 − R ) 4 g 2 5 “Hard-wall”: Φ( z ) ≡ 0 , κ = 0 , 0 � z � z m , “Soft-wall”: Φ( z ) ∼ λ z 2 ( z → ∞ ) , κ � = 0 , 0 � z < ∞ . L a q L ( x ) γ µ t a q L ( x ) , µ ( x , z = 0 ) = source of ¯ R a source of ¯ q R ( x ) γ µ t a q R ( x ) , µ ( x , z = 0 ) = 2 z X αβ ( x , z ) q α L ( x ) q β source of ¯ lim = R ( x ) z → 0 m δ αβ in the absence of (pseudo)scalar currents . =
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Five-Dimensional Effective Action � d 5 x √ ge − Φ tr � � | DX | 2 + 3 ℓ 2 | X | 2 + κ � g 2 ℓ 2 | X | 4 S 5 D = X with a metric ds 2 = ℓ 2 1 � z 2 ( − dz 2 + dx µ dx µ ) . ( F 2 L + F 2 − R ) 4 g 2 5 “Hard-wall”: Φ( z ) ≡ 0 , κ = 0 , 0 � z � z m , “Soft-wall”: Φ( z ) ∼ λ z 2 ( z → ∞ ) , κ � = 0 , 0 � z < ∞ . L a q L ( x ) γ µ t a q L ( x ) , µ ( x , z = 0 ) = source of ¯ R a source of ¯ q R ( x ) γ µ t a q R ( x ) , µ ( x , z = 0 ) = 2 z X αβ ( x , z ) q α L ( x ) q β source of ¯ lim = R ( x ) z → 0 m δ αβ in the absence of (pseudo)scalar currents . =
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Zero momentum correlation functions of QCD currents Correlation functions are calculated via the AdS/CFT prescription: Z QCD [ J I ( x µ )] = exp ( iS 5 D classical ) | Φ I ( 0 , x µ )= J I ( x µ ) ⇒ � δ δ � �O I ( x ) O J ( y ) � conn = − δ Φ J ( 0 , y ) S 5 D classical � δ Φ I ( 0 , x ) � Φ I ( 0 , x µ )= 0 At zero momentum for N f quark flavors with equal masses G 2 C π i � P i ( 0 ) P j ( 0 ) � = δ ij m = δ ij , m 2 π = C δ αβ in the chiral limit, Σ = � ¯ � ¯ q α q β � q α q α � . One can where see that we obtain a singularity corresponding to the pion exchange. The pole residue is the same for the middle and left-hand sides of Eqn. (1).
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition N f = 2 case In the particular N f = 2 , m u � = m d case m − C ∆ m C � � i P i ( 0 ) P j ( 0 ) = δ ij 2 m 2 ( δ i 0 δ 3 j + δ j 0 δ 3 i − i δ i 1 δ 2 j − i δ j 1 δ 2 i ) , where m = m u + m d , ∆ m = m u − m d . 2 Scalar current correlators are calculated analogously and are regular in the chiral limit. Thus for a zero momentum i � δ ij S 0 ( 0 ) S 0 ( 0 ) − P i ( 0 ) P j ( 0 ) � C ∼ i � S i ( x ) S j ( 0 ) − δ ij P 0 ( x ) P 0 ( 0 ) � ∼ − δ ij m , i � P 3 ( 0 ) P 0 ( 0 ) � ∼ − C ∆ m 2 m 2 .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Chiral Lagrangian from Holography Another point of view on the holographic models: Five-dimensional fields on the AdS boundary are sources of the QCD currents. QCD currents are sources of the mesons. Kaluza–Klein modes ∝ meson wavefunctions with corresponding quantum numbers. If we integrate out the dynamics along the z axis, the 5D action will generate an effective chiral Lagrangian. Similar to the “top-down” models [T. Sakai, S. Sugimoto, 2005] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Chiral Lagrangian from Holography Another point of view on the holographic models: Five-dimensional fields on the AdS boundary are sources of the QCD currents. QCD currents are sources of the mesons. Kaluza–Klein modes ∝ meson wavefunctions with corresponding quantum numbers. If we integrate out the dynamics along the z axis, the 5D action will generate an effective chiral Lagrangian. Similar to the “top-down” models [T. Sakai, S. Sugimoto, 2005] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Chiral Lagrangian from Holography Another point of view on the holographic models: Five-dimensional fields on the AdS boundary are sources of the QCD currents. QCD currents are sources of the mesons. Kaluza–Klein modes ∝ meson wavefunctions with corresponding quantum numbers. If we integrate out the dynamics along the z axis, the 5D action will generate an effective chiral Lagrangian. Similar to the “top-down” models [T. Sakai, S. Sugimoto, 2005] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Chiral Lagrangian from Holography Another point of view on the holographic models: Five-dimensional fields on the AdS boundary are sources of the QCD currents. QCD currents are sources of the mesons. Kaluza–Klein modes ∝ meson wavefunctions with corresponding quantum numbers. If we integrate out the dynamics along the z axis, the 5D action will generate an effective chiral Lagrangian. Similar to the “top-down” models [T. Sakai, S. Sugimoto, 2005] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Chiral Lagrangian from Holography Another point of view on the holographic models: Five-dimensional fields on the AdS boundary are sources of the QCD currents. QCD currents are sources of the mesons. Kaluza–Klein modes ∝ meson wavefunctions with corresponding quantum numbers. If we integrate out the dynamics along the z axis, the 5D action will generate an effective chiral Lagrangian. Similar to the “top-down” models [T. Sakai, S. Sugimoto, 2005] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Kaluza–Klein expansion Gauge fields can be combined into V a µ = L a µ + R a µ , A a µ = L a µ − R a µ One can fix the gauge V z = A z = ∂ µ V µ = 0, so that A µ retains a longitudinal component: A µ = A ⊥ µ + ∂ µ φ . φ ∝ the source of the pseudoscalar current. KK expansion: f ( n ) φ ( z ) φ a ( n ) ( x ) , f ( n ) φ a ( z , x ) = � φ ( z ) n − E.o.M. solution in AdS , φ a ( 0 ) ( x ) ∝ π a ( x ) .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Kaluza–Klein expansion Gauge fields can be combined into V a µ = L a µ + R a µ , A a µ = L a µ − R a µ One can fix the gauge V z = A z = ∂ µ V µ = 0, so that A µ retains a longitudinal component: A µ = A ⊥ µ + ∂ µ φ . φ ∝ the source of the pseudoscalar current. KK expansion: f ( n ) φ ( z ) φ a ( n ) ( x ) , f ( n ) φ a ( z , x ) = � φ ( z ) n − E.o.M. solution in AdS , φ a ( 0 ) ( x ) ∝ π a ( x ) .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Kaluza–Klein expansion Gauge fields can be combined into V a µ = L a µ + R a µ , A a µ = L a µ − R a µ One can fix the gauge V z = A z = ∂ µ V µ = 0, so that A µ retains a longitudinal component: A µ = A ⊥ µ + ∂ µ φ . φ ∝ the source of the pseudoscalar current. KK expansion: f ( n ) φ ( z ) φ a ( n ) ( x ) , f ( n ) φ a ( z , x ) = � φ ( z ) n − E.o.M. solution in AdS , φ a ( 0 ) ( x ) ∝ π a ( x ) .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Kaluza–Klein expansion Gauge fields can be combined into V a µ = L a µ + R a µ , A a µ = L a µ − R a µ One can fix the gauge V z = A z = ∂ µ V µ = 0, so that A µ retains a longitudinal component: A µ = A ⊥ µ + ∂ µ φ . φ ∝ the source of the pseudoscalar current. KK expansion: f ( n ) φ ( z ) φ a ( n ) ( x ) , f ( n ) φ a ( z , x ) = � φ ( z ) n − E.o.M. solution in AdS , φ a ( 0 ) ( x ) ∝ π a ( x ) .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Parameters of the Chiral Lagrangian Having integrated out all z -dependence in the 5D action of the lowest KK-mode we obtain S 5 D → A 1 · 1 2 ∂ µ φ ( 0 ) ∂ µ φ ( 0 ) + A 2 · [ ∂ µ φ ( 0 ) , ∂ ν φ ( 0 ) ] 2 + A 3 · m ∂ µ φ ( 0 ) ∂ µ φ ( 0 ) . This allows to find the parameters explicitly L 1 , 2 , 3 ∝ A 2 , L 4 ∝ A 3 . The following (universal for holographic A 2 A 1 1 models) equation holds L 3 = − 3 L 2 = − 6 L 1 . Parameters L i are regular in the chiral limit.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition The Spectral Density of the Dirac operator The Dirac operator ˆ D ≡ γ µ ( ∂ µ + ig YM A µ ) has no dual AdS �� � description, and its spectral density ρ ( λ ) = 1 δ ( λ − λ n ) , V n A where { λ n } are the eigenvalues of i ˆ D , has no direct AdS/QCD interpretation similar to the Chiral Lagrangian and QCD partition function. However one can express ρ ( λ ) via a partition function of a QCD-like theory whith a dual description. [S. F. Edwards and P . W. Anderson, 1975; J. J. M. Verbaarschot and M. R. Zirnbauer, 1984; K. B. Efetov, 1983] �� � � � ρ ( λ ) = 1 1 µ � δ ( λ − λ n ) = lim µ 2 + ( λ − λ n ) 2 V π V µ → 0 n n A A 1 ∂ � � log Det [ i ˆ D − λ − i µ ] + log Det [ i ˆ D − λ + i µ ] = 2 π V lim ∂µ µ → 0 A
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition The Spectral Density ∂ ∂ n z n � We use a so-called “replica trick” : log z = � n = 0 1 ∂ ∂ � � Det n [ i ˆ ρ ( λ ) = π V lim ∂µ lim D − λ − i µ ] ∂ n Re µ → 0 n → 0 A 1 ∂ ∂ � DA e iS YM [ A ] × Z quarks ( m q = m , N f )[ A ] = π V lim ∂µ lim ∂ n Re µ → 0 n → 0 ×Z ghosts ( m q = λ + i µ, n · N f )[ A ] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Spectral Density in Holography This partition function can be calculated in "hard-wall" AdS/QCD, with a boundary condition on the scalar field: m 1 . ... . 0 . 2 m N f lim z X = λ + i µ N f + 1 z → 0 . ... . 0 . λ + i µ N f ( n + 1 ) Result is the following: ρ ( λ ) = − 1 � Σ( m ) + m d �� � dm Σ( m ) . � π � m = λ
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Spectral Density in Holography A QCD result [V. A. Novikov, M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, 1981] � � 1 − 3 m 2 π log m 2 π /µ 2 hadr Σ( m ) = Σ( 0 ) leads to a formula 32 π 2 F 2 π ρ ( λ ) = − 1 � 3 Σ( 0 ) 3 Σ( 0 ) � π Σ( 0 ) 1 − λ − λ log λ , λ > 0 . 8 π 2 N f F 4 4 π 2 N f F 4 π π So: the result agrees with the Casher–Banks identity ρ ( 0 ) = − Σ( 0 ) /π [T. Banks and A. Casher, 1980] , up to a ∝ N 2 f − 4 factor reproduces the result of Smilga and Stern [A. Smilga and J. Stern, 1993] for the term linear in λ , has terms ∝ λ log λ .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Spectral Density in Holography A QCD result [V. A. Novikov, M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, 1981] � � 1 − 3 m 2 π log m 2 π /µ 2 hadr Σ( m ) = Σ( 0 ) leads to a formula 32 π 2 F 2 π ρ ( λ ) = − 1 � 3 Σ( 0 ) 3 Σ( 0 ) � π Σ( 0 ) 1 − λ − λ log λ , λ > 0 . 8 π 2 N f F 4 4 π 2 N f F 4 π π So: the result agrees with the Casher–Banks identity ρ ( 0 ) = − Σ( 0 ) /π [T. Banks and A. Casher, 1980] , up to a ∝ N 2 f − 4 factor reproduces the result of Smilga and Stern [A. Smilga and J. Stern, 1993] for the term linear in λ , has terms ∝ λ log λ .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Spectral Density in Holography A QCD result [V. A. Novikov, M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, 1981] � � 1 − 3 m 2 π log m 2 π /µ 2 hadr Σ( m ) = Σ( 0 ) leads to a formula 32 π 2 F 2 π ρ ( λ ) = − 1 � 3 Σ( 0 ) 3 Σ( 0 ) � π Σ( 0 ) 1 − λ − λ log λ , λ > 0 . 8 π 2 N f F 4 4 π 2 N f F 4 π π So: the result agrees with the Casher–Banks identity ρ ( 0 ) = − Σ( 0 ) /π [T. Banks and A. Casher, 1980] , up to a ∝ N 2 f − 4 factor reproduces the result of Smilga and Stern [A. Smilga and J. Stern, 1993] for the term linear in λ , has terms ∝ λ log λ .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Spectral Density in Holography Drawbacks of this result are the following: No dependence on mass; No terms ∝ λ 2 and higher. It seems that we might obtain a more precise result for ρ ′ ( 0 ) if we either manage to formulate a consistent “soft-wall” model with different flavors or take into account the N f –dependence of the 5D metric ( ∼ to the flavor brane back-reaction).
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Spectral Density in Holography Drawbacks of this result are the following: No dependence on mass; No terms ∝ λ 2 and higher. It seems that we might obtain a more precise result for ρ ′ ( 0 ) if we either manage to formulate a consistent “soft-wall” model with different flavors or take into account the N f –dependence of the 5D metric ( ∼ to the flavor brane back-reaction).
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Spectral Density in Holography Drawbacks of this result are the following: No dependence on mass; No terms ∝ λ 2 and higher. It seems that we might obtain a more precise result for ρ ′ ( 0 ) if we either manage to formulate a consistent “soft-wall” model with different flavors or take into account the N f –dependence of the 5D metric ( ∼ to the flavor brane back-reaction).
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Spectral Density in Holography Drawbacks of this result are the following: No dependence on mass; No terms ∝ λ 2 and higher. It seems that we might obtain a more precise result for ρ ′ ( 0 ) if we either manage to formulate a consistent “soft-wall” model with different flavors or take into account the N f –dependence of the 5D metric ( ∼ to the flavor brane back-reaction).
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Compatibility of the AdS/QCD models with low-energy theorems Integrals with ρ ( λ ) in the right-hand sides of the theorems Eqn. (1 – 3) yield m 2 ρ ( λ ) � ( λ 2 + m 2 ) 2 ∼ C � d λ m ∆ m ρ ( λ ) ( λ 2 + m 2 ) 2 ∼ C ∆ m d λ m , m 2 , and in the m → 0 limit for each theorem Eqn. (1, 2, 3) we get equal pole residues on all sides. Thus, AdS/QCD models are compatible with low-energy theorems in the chiral limit.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary We have demonstrated the agreement of the AdS/QCD model in the chiral limit with the low-energy theorems. We have proposed a way to calculate the spectral density of the Dirac operator in AdS/QCD. Result in the “hard-wall” model agrees with the Casher–Banks identity and up to a numerical N f –depending factor reproduces the Smilga and Stern result. Testing the theorems beyond the chiral limit might probably bring us closer to understanding the structure of the AdS/QCD models.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary We have demonstrated the agreement of the AdS/QCD model in the chiral limit with the low-energy theorems. We have proposed a way to calculate the spectral density of the Dirac operator in AdS/QCD. Result in the “hard-wall” model agrees with the Casher–Banks identity and up to a numerical N f –depending factor reproduces the Smilga and Stern result. Testing the theorems beyond the chiral limit might probably bring us closer to understanding the structure of the AdS/QCD models.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary We have demonstrated the agreement of the AdS/QCD model in the chiral limit with the low-energy theorems. We have proposed a way to calculate the spectral density of the Dirac operator in AdS/QCD. Result in the “hard-wall” model agrees with the Casher–Banks identity and up to a numerical N f –depending factor reproduces the Smilga and Stern result. Testing the theorems beyond the chiral limit might probably bring us closer to understanding the structure of the AdS/QCD models.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary We have demonstrated the agreement of the AdS/QCD model in the chiral limit with the low-energy theorems. We have proposed a way to calculate the spectral density of the Dirac operator in AdS/QCD. Result in the “hard-wall” model agrees with the Casher–Banks identity and up to a numerical N f –depending factor reproduces the Smilga and Stern result. Testing the theorems beyond the chiral limit might probably bring us closer to understanding the structure of the AdS/QCD models.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Outline The Setup of Holographic QCD 1 Low-Energy Theorems of QCD 2 Chiral Magnetic Effect in Soft-Wall AdS/QCD 3 Anomalous QCD Contribution to the Debye Screening in an 4 External Field via Holography Magnetic Susceptibility of the Chiral Condensate in a Model 5 with a Tensor Field Effect of the Gluon Condensate on the Gross–Ooguri Phase 6 Transition
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Essence of CME Generation of an electric current parallel to a magnetic field in a topologically nontrivial background [D. E. Kharzeev, L. D. McLerran and H. J. Warringa, 2008; K. Fukushima, D. E. Kharzeev, and H. J. Warringa, 2008] . Chirally symmetric phase of QCD with massless quarks q L , q R of a unit electromagnetic charge q L → ( q + + 1 / 2 ) and q R → ( q + − 1 / 2 , q − + 1 / 2 , q − − 1 / 2 ) (charge and helicity), magnetic moment ∝ charge × spin External magnetic field aligns magnetic moments ⇒ spins and momenta are correlated with the magnetic field Components ( q + − 1 / 2 , q − + 1 / 2 ) move in the opposite direction to the field, ( q + + 1 / 2 , q − − 1 / 2 ) – along the field.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Essence of CME Generation of an electric current parallel to a magnetic field in a topologically nontrivial background [D. E. Kharzeev, L. D. McLerran and H. J. Warringa, 2008; K. Fukushima, D. E. Kharzeev, and H. J. Warringa, 2008] . Chirally symmetric phase of QCD with massless quarks q L , q R of a unit electromagnetic charge q L → ( q + + 1 / 2 ) and q R → ( q + − 1 / 2 , q − + 1 / 2 , q − − 1 / 2 ) (charge and helicity), magnetic moment ∝ charge × spin External magnetic field aligns magnetic moments ⇒ spins and momenta are correlated with the magnetic field Components ( q + − 1 / 2 , q − + 1 / 2 ) move in the opposite direction to the field, ( q + + 1 / 2 , q − − 1 / 2 ) – along the field.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Essence of CME Generation of an electric current parallel to a magnetic field in a topologically nontrivial background [D. E. Kharzeev, L. D. McLerran and H. J. Warringa, 2008; K. Fukushima, D. E. Kharzeev, and H. J. Warringa, 2008] . Chirally symmetric phase of QCD with massless quarks q L , q R of a unit electromagnetic charge q L → ( q + + 1 / 2 ) and q R → ( q + − 1 / 2 , q − + 1 / 2 , q − − 1 / 2 ) (charge and helicity), magnetic moment ∝ charge × spin External magnetic field aligns magnetic moments ⇒ spins and momenta are correlated with the magnetic field Components ( q + − 1 / 2 , q − + 1 / 2 ) move in the opposite direction to the field, ( q + + 1 / 2 , q − − 1 / 2 ) – along the field.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Essence of CME Generation of an electric current parallel to a magnetic field in a topologically nontrivial background [D. E. Kharzeev, L. D. McLerran and H. J. Warringa, 2008; K. Fukushima, D. E. Kharzeev, and H. J. Warringa, 2008] . Chirally symmetric phase of QCD with massless quarks q L , q R of a unit electromagnetic charge q L → ( q + + 1 / 2 ) and q R → ( q + − 1 / 2 , q − + 1 / 2 , q − − 1 / 2 ) (charge and helicity), magnetic moment ∝ charge × spin External magnetic field aligns magnetic moments ⇒ spins and momenta are correlated with the magnetic field Components ( q + − 1 / 2 , q − + 1 / 2 ) move in the opposite direction to the field, ( q + + 1 / 2 , q − − 1 / 2 ) – along the field.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition A U ( 1 ) A chemical potential µ 5 induced by a sphaleron produces a state with a positive helicity: + 1 / 2 ) + N ( q + + 1 / 2 ) > N ( q + N ( q − − 1 / 2 ) + N ( q − − 1 / 2 ) An electromagnetic current J ∝ l.h.s. - r.h.s. B u R d R - spin direction - magnitic moment - spatial momentum - electric current - nonzero density of the topological charge u L d L Figure: CME as a simple rearrangement of momenta.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition A U ( 1 ) A chemical potential µ 5 induced by a sphaleron produces a state with a positive helicity: + 1 / 2 ) + N ( q + + 1 / 2 ) > N ( q + N ( q − − 1 / 2 ) + N ( q − − 1 / 2 ) An electromagnetic current J ∝ l.h.s. - r.h.s. B u R d R - spin direction - magnitic moment - spatial momentum - electric current - nonzero density of the topological charge u L d L Figure: CME as a simple rearrangement of momenta.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition CME at weak coupling In the weak-coupling limit [K. Fukushima, D. E. Kharzeev, and H. J. Warringa, 2008] the resulting current is J V = µ 5 B 2 π 2 ≡ J FKW A temperature-dependent expression for the susceptibility χ ∝ T has also been obtained [K. Fukushima, D. E. Kharzeev, and H. J. Warringa, 2009] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition CME in the gauge/gravity framework The CME has been studied in the dual models to shed light into its properties in the strong-coupling limit: [H.-U. Yee, 2009] – in a model of Einstein gravity with a U ( 1 ) L × U ( 1 ) R Maxwell theory in the AdS 5 space and in the Sakai-Sugimoto model. Results agree with those in the weak-coupling limit. [A. Rebhan, A. Schmitt and S. A. Stricker, 2010] – in the Sakai-Sugimoto model. Result is 2 / 3 of the weak-coupling result in the absence of the Bardeen counterterm and is zero with the counterterm.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition CME in the gauge/gravity framework The CME has been studied in the dual models to shed light into its properties in the strong-coupling limit: [H.-U. Yee, 2009] – in a model of Einstein gravity with a U ( 1 ) L × U ( 1 ) R Maxwell theory in the AdS 5 space and in the Sakai-Sugimoto model. Results agree with those in the weak-coupling limit. [A. Rebhan, A. Schmitt and S. A. Stricker, 2010] – in the Sakai-Sugimoto model. Result is 2 / 3 of the weak-coupling result in the absence of the Bardeen counterterm and is zero with the counterterm.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Experimental status of CME and CME on lattices Experimental status is discussed in [ STAR collaboration and S. A. Voloshin, 2009, 2011; P HENIX] collaboration, 2010]. Lattice calculations by the ITEP Lattice group in the quenched approximation [P . V. Buividovich, M. N. Chernodub, E. V. Luschevskaya and M. I. Polikarpov, 2009; P . V. Buividovich, M. N. Chernodub, T. Kalaydzhyan, D. E. Kharzeev, E. V. Luschevskaya and M. I. Polikarpov, 2010] ... and with light domain wall fermions [ M. Abramczyk, T. Blum, G. Petropoulos, R. Zhou, 2009] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Experimental status of CME and CME on lattices Experimental status is discussed in [ STAR collaboration and S. A. Voloshin, 2009, 2011; P HENIX] collaboration, 2010]. Lattice calculations by the ITEP Lattice group in the quenched approximation [P . V. Buividovich, M. N. Chernodub, E. V. Luschevskaya and M. I. Polikarpov, 2009; P . V. Buividovich, M. N. Chernodub, T. Kalaydzhyan, D. E. Kharzeev, E. V. Luschevskaya and M. I. Polikarpov, 2010] ... and with light domain wall fermions [ M. Abramczyk, T. Blum, G. Petropoulos, R. Zhou, 2009] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Experimental status of CME and CME on lattices Experimental status is discussed in [ STAR collaboration and S. A. Voloshin, 2009, 2011; P HENIX] collaboration, 2010]. Lattice calculations by the ITEP Lattice group in the quenched approximation [P . V. Buividovich, M. N. Chernodub, E. V. Luschevskaya and M. I. Polikarpov, 2009; P . V. Buividovich, M. N. Chernodub, T. Kalaydzhyan, D. E. Kharzeev, E. V. Luschevskaya and M. I. Polikarpov, 2010] ... and with light domain wall fermions [ M. Abramczyk, T. Blum, G. Petropoulos, R. Zhou, 2009] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Gauge sector of the soft-wall AdS/QCD In the chirally symmetric phase we only need to consider the gauge sector, | X | = 0 (its phase – the 5D pion – is a more involved issue). S 5 D = S YM [ L ] + S YM [ R ] + S CS [ L ] − S CS [ R ] dz d 4 x e − φ √ gF MN F MN − 1 � e − φ F ∧ ∗ F = − 1 � S YM [ A ] = 4 g 2 4 g 2 5 5 − N c � A ∧ F ∧ F = − N c � dz d 4 x ǫ MNPQR A M F NP F QR S CS [ A ] = 24 π 2 24 π 2 with a metric tensor ds 2 = g MN dX M dX N = ℓ 2 z 2 η MN dX M dX N = ℓ 2 z 2 ( − dz 2 + dx µ dx µ ) .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Symmetry Currents from the On-shell Action Chemical potentials µ = 1 2 ( µ L + µ R ) , µ 5 = 1 2 ( µ L − µ R ) and a magnetic field are incorporated as boundary conditions for the gauge fields. Boundary condition L µ ( ∞ ) = R µ ( ∞ ) , ∂ z L µ ( ∞ ) = − ∂ z R µ ( ∞ ) at z = ∞ is an adaptation of an analytical continuation of an analogous condition in the chirally broken Sakai-Sugimoto model. Action, estimated on-shell for the solutions of the E.o.M.’s for the gauge fields in the chirally symmetric phase ( | X | = 0), yields: δ S [ L , R ] δ S [ L , R ] δ L 3 ( z = 0 ) = J L , δ R 3 ( z = 0 ) = J R , J = J L + J R = N c 3 π 2 B µ 5 .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Anomalies and the Bardeen counterterm In our setup there are two external gauge fields on the boundary – V µ ( z = 0 ) and A µ ( z = 0 ) . V µ ( z = 0 ) corresponds to e × an external electromagnetic field and provides µ , while a nonzero A µ ( z = 0 ) accounts for µ 5 . It has been pointed out in [A. Rebhan, A. Schmitt and S. A. Stricker, 2010] that the divergence of the vector current ∂ µ J µ = − N c µν ˜ 24 π 2 F V F A µν has to be compensated for by a local counterterm � d 4 x ǫ µνρσ L µ R ν ( F L ρσ + F R S Bardeen = c ρσ ) , (4) with an appropriate choice of the constant c . S Bardeen may be considered as a product of holographic renormalization. Whether it needs to be taken into account remains unclear. In our model c = − N c 12 π 2 and J subtracted = J + J Bardeen = N c � − N c � 3 π 2 B µ 5 + × 4 B µ 5 = 0 . (5) 12 π 2
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Discussion of the relevance of the Bardeen counterterm It was suggested by V. Rubakov [arXiv: 1005.1888[hep-ph]] that the counterterm has to be excluded from the calculation, since it fixes the anomaly of the vector current only in the presence of a real dynamical axial gauge field, while in our case we are dealing with a constant axial chemical potential, which is different from a constant temporal component of an axial gauge field. In the absence of this counterterm the CME current in the strong coupling regime agrees exactly with the weak coupling limit (as it will be demonstrated below). How to formally distinguish between the two aforementioned cases in holography is a problem still open to discussions.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Scalar sector The 5D scalar field X = | X | e i π/ f π interacts with the gauge fields via the covariant derivatives, thus inducing an interaction between π and the gauge field A M . � Usually the 4D pion is associated with a holonomy A z dz and the π → γγ decay is determined by a part of the CS dzd 4 x A z F V µν F V µν . In the A z = 0 gauge we have � action to reintroduce pion into the CS term. CS action is gauge invariant up to a surface term which is nonzero in our setup. In order to make it explicitly invariant we introduce 2 scalars: N c �� � � S CS = L ∧ dL ∧ dL − R ∧ dR ∧ dR 24 π 2 �� � N c � ( L + d φ L ) ∧ dL ∧ dL − ( R + d φ R ) ∧ dR ∧ dR → 24 π 2
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Pseudoscalar contribution to the effect which under gauge transformations L → L + d α L , R → R + d α R transform as φ L , R → φ L , R − α L , R . f π ( φ R − φ L ) may be associated with the five-dimensional pion in the gauge in which A z is set to zero. In the D3/D7 models an R-symmetry chemical potential causes the D7 branes to rotate with an angular speed µ R , so that the phase of the scalar field is i µ R t . φ L , R ( z = 0 ) = µ L , R t . This yields another contribution to the CME: J φ AA = N c 6 π 2 B µ 5 .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary table Here is a summary of all the contributions to the CME: Term in Yang–Mills Chern–Simons Scalars the action bulk boundary bulk boundary in CS − 1 N c 1 N c 1 N c 1 N c 1 N c Contribution 2 π 2 B µ 5 2 π 2 B µ 5 2 π 2 B µ 5 2 π 2 B µ 5 2 π 2 B µ 5 3 3 3 3 3 to the current Action taken Total Total without scalars into account Resulting current, in terms of N c 2 2 π 2 B µ 5 1 3
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary We have presented a calculation of the CME in soft-wall AdS/QCD. The results differ from those in the Sakai-Sugimoto model due to the presence of scalar fields. Those scalars act as ’catalysts’ for the effect, only triggering it, while its magnitude is defined by the Chern-Simons action. The nature of the effect remains topological and it does not depend either on the dilaton or on the metric or on the details of the scalar Lagrangian. CME in soft-wall AdS/QCD exactly agrees with the weak-coupling result if the scalars are accounted for.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary We have presented a calculation of the CME in soft-wall AdS/QCD. The results differ from those in the Sakai-Sugimoto model due to the presence of scalar fields. Those scalars act as ’catalysts’ for the effect, only triggering it, while its magnitude is defined by the Chern-Simons action. The nature of the effect remains topological and it does not depend either on the dilaton or on the metric or on the details of the scalar Lagrangian. CME in soft-wall AdS/QCD exactly agrees with the weak-coupling result if the scalars are accounted for.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary We have presented a calculation of the CME in soft-wall AdS/QCD. The results differ from those in the Sakai-Sugimoto model due to the presence of scalar fields. Those scalars act as ’catalysts’ for the effect, only triggering it, while its magnitude is defined by the Chern-Simons action. The nature of the effect remains topological and it does not depend either on the dilaton or on the metric or on the details of the scalar Lagrangian. CME in soft-wall AdS/QCD exactly agrees with the weak-coupling result if the scalars are accounted for.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary We have presented a calculation of the CME in soft-wall AdS/QCD. The results differ from those in the Sakai-Sugimoto model due to the presence of scalar fields. Those scalars act as ’catalysts’ for the effect, only triggering it, while its magnitude is defined by the Chern-Simons action. The nature of the effect remains topological and it does not depend either on the dilaton or on the metric or on the details of the scalar Lagrangian. CME in soft-wall AdS/QCD exactly agrees with the weak-coupling result if the scalars are accounted for.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Summary We have presented a calculation of the CME in soft-wall AdS/QCD. The results differ from those in the Sakai-Sugimoto model due to the presence of scalar fields. Those scalars act as ’catalysts’ for the effect, only triggering it, while its magnitude is defined by the Chern-Simons action. The nature of the effect remains topological and it does not depend either on the dilaton or on the metric or on the details of the scalar Lagrangian. CME in soft-wall AdS/QCD exactly agrees with the weak-coupling result if the scalars are accounted for.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Outline The Setup of Holographic QCD 1 Low-Energy Theorems of QCD 2 Chiral Magnetic Effect in Soft-Wall AdS/QCD 3 Anomalous QCD Contribution to the Debye Screening in an 4 External Field via Holography Magnetic Susceptibility of the Chiral Condensate in a Model 5 with a Tensor Field Effect of the Gluon Condensate on the Gross–Ooguri Phase 6 Transition
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition The Debye mass Let us consider a hot quark-gluon plasma in a deconfined regime but still at strong coupling: (1 GeV � T � 200 MeV ≈ T c ∼ Λ QCD ). The electric charges are screened: V ( r ) ∼ e − mDr . r D = e 2 T 2 One-loop QED result is m 2 [H.A. Weldon, 1982] . 3 We are interested in the contribution of QCD with accuracy up to α em but to all orders in the external magnetic field
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition The Debye mass Let us consider a hot quark-gluon plasma in a deconfined regime but still at strong coupling: (1 GeV � T � 200 MeV ≈ T c ∼ Λ QCD ). The electric charges are screened: V ( r ) ∼ e − mDr . r D = e 2 T 2 One-loop QED result is m 2 [H.A. Weldon, 1982] . 3 We are interested in the contribution of QCD with accuracy up to α em but to all orders in the external magnetic field
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition The Debye mass Let us consider a hot quark-gluon plasma in a deconfined regime but still at strong coupling: (1 GeV � T � 200 MeV ≈ T c ∼ Λ QCD ). The electric charges are screened: V ( r ) ∼ e − mDr . r D = e 2 T 2 One-loop QED result is m 2 [H.A. Weldon, 1982] . 3 We are interested in the contribution of QCD with accuracy up to α em but to all orders in the external magnetic field
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition The Debye mass Let us consider a hot quark-gluon plasma in a deconfined regime but still at strong coupling: (1 GeV � T � 200 MeV ≈ T c ∼ Λ QCD ). The electric charges are screened: V ( r ) ∼ e − mDr . r D = e 2 T 2 One-loop QED result is m 2 [H.A. Weldon, 1982] . 3 We are interested in the contribution of QCD with accuracy up to α em but to all orders in the external magnetic field
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Debye mass definition � d 4 x � J µ ( 0 ) J ν ( x ) � ret e i ω x 0 − i � Π µν ( ω,� k � x , k ) = i k 2 = − m 2 q Π 00 ( ω = 0 ,� m 2 e 2 = D ) , D q Π 33 ( ω = 0 ,� m 2 e 2 k = − m 2 = D Mag ) . D Mag Up to α em we are dealing with Π µν ( ω = 0 ,� k = 0 ) .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Nonzero temperatures in holography We will use the same action as before = S YM [ L ] + S YM [ R ] + S CS [ L ] − S CS [ R ] S 5 D − 1 � S CS [ A ] = − N c � e − φ F ∧ ∗ F ; A ∧ F ∧ F S YM [ A ] = 4 g 2 24 π 2 5 with a metric tensor ds 2 = r 2 − ℓ 2 dr 2 f BH ( r ) dt 2 − dx i dx i � � f BH ( r ) . ℓ 2 r 2 r 4 πℓ 2 , r = ℓ 2 r 0 Here f BH ( r ) = 1 − r 4 , T = 0 z .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Adjusting the model to nonzero temperatures We Wick-rotate the 4D space, make the time periodical with β = T − 1 , look at the geometry near the horizon r = r 0 , there will be a conical singularity unless β = πℓ 2 r 0 . Retarded Green function – only in-falling waves at the horizon: regular at r = r 0 in accompanying Eddington–Finkelstein coordinates. When ω = 0 for the solution – that is the same as imposing regularity conditions in coordinates at infinity. At the horizon L 0 ( r 0 ) = R 0 ( r 0 ) = 0 due to g 00 ( r 0 ) = 0.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Results at zero magnetic field If B = 0 the Chern–Simons part of the action is strictly zero and the only nonzero field is: � � 1 − r 2 V 0 ( r ) = 1 0 2 ( L 0 ( r ) + R 0 ( r )) = µ r 2 and D = N c m 2 3 e 2 q T 2 . The non-perturbative QCD calculation is similar to the leading term of the QED perturbative series expression. Similarly m D Mag = 0 – as it is in perturbative QED [J. P . Blaizot, E. Iancu, R. R. Parwani, 1995] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Results at zero magnetic field If B = 0 the Chern–Simons part of the action is strictly zero and the only nonzero field is: � � 1 − r 2 V 0 ( r ) = 1 0 2 ( L 0 ( r ) + R 0 ( r )) = µ r 2 and D = N c m 2 3 e 2 q T 2 . The non-perturbative QCD calculation is similar to the leading term of the QED perturbative series expression. Similarly m D Mag = 0 – as it is in perturbative QED [J. P . Blaizot, E. Iancu, R. R. Parwani, 1995] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Results at zero magnetic field If B = 0 the Chern–Simons part of the action is strictly zero and the only nonzero field is: � � 1 − r 2 V 0 ( r ) = 1 0 2 ( L 0 ( r ) + R 0 ( r )) = µ r 2 and D = N c m 2 3 e 2 q T 2 . The non-perturbative QCD calculation is similar to the leading term of the QED perturbative series expression. Similarly m D Mag = 0 – as it is in perturbative QED [J. P . Blaizot, E. Iancu, R. R. Parwani, 1995] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Results at zero magnetic field If B = 0 the Chern–Simons part of the action is strictly zero and the only nonzero field is: � � 1 − r 2 V 0 ( r ) = 1 0 2 ( L 0 ( r ) + R 0 ( r )) = µ r 2 and D = N c m 2 3 e 2 q T 2 . The non-perturbative QCD calculation is similar to the leading term of the QED perturbative series expression. Similarly m D Mag = 0 – as it is in perturbative QED [J. P . Blaizot, E. Iancu, R. R. Parwani, 1995] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Results at zero magnetic field If B = 0 the Chern–Simons part of the action is strictly zero and the only nonzero field is: � � 1 − r 2 V 0 ( r ) = 1 0 2 ( L 0 ( r ) + R 0 ( r )) = µ r 2 and D = N c m 2 3 e 2 q T 2 . The non-perturbative QCD calculation is similar to the leading term of the QED perturbative series expression. Similarly m D Mag = 0 – as it is in perturbative QED [J. P . Blaizot, E. Iancu, R. R. Parwani, 1995] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition The Debye mass in strong magnetic fields If B = F V 12 � = 0 the Chern–Simons part of the action plays its role. Due to its Lorentz structure: F V 12 ( r ) ≡ B is a solution to the e.o.m. There is a mixing between the pairs V 0 and A 3 , V 3 and A 0 . These mixings are ∝ B . The Chern–Simons action is effectively bilinear.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Solutions The solutions of the e.o.m. are expressed in terms of Legendre functions P ν ( x ) , Q ν ( x ) of the 1st and 2nd order respectively � 1 − 9 e 2 q B 2 ℓ 4 / r 4 0 − 1 with a parameter ν = that are regular and 2 single-valued at | x | < 1, although Q ν ( x ) has a branching point at x = 1. Accounting for the boundary conditions we get: � 0 / r 2 �� r 2 µ � 0 / r 2 � � 0 r 2 r 2 V 0 ( r ) = r 2 P ν − P ν + 1 , ν P − 1 ν ( 0 ) µ � � P ν ( r 2 0 / r 2 ) − P ν ( 0 ) A 3 ( r ) = . P − 1 � ν ( 0 ) − ν ( ν + 1 )
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition The Debye mass in a strong field Variation of the action with respect to the source µ gives � 1 − 9 e 2 q B 2 N c − 1 2 + 1 3 T 2 F m 2 D = e 2 , m D Mag = 0 q 2 π 4 T 4 F ( ν ) ≡ P ν ( 0 ) = 2 Γ ( 1 − ν/ 2 ) Γ ( 3 / 2 + ν/ 2 ) Γ ( 1 + ν/ 2 ) Γ ( 1 / 2 − ν/ 2 ) . P − 1 ν ( 0 ) In a strong magnetic field, e q B ≫ T 2 , N c m 2 D = e 2 2 π 2 | e q B | . q
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition The Graph � F 3.0 2.5 2.0 1.5 1.0 0.5 eB 5 10 15 20 T 2 � � � 9 e 2 q B 2 Figure: The function ˜ − 1 2 + 1 1 − F = F (solid) vs its strong 2 π 4 T 4 field asymptotics (dashed).
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Values at the LHC and RHIC Quark-gluon plasma that is created during heavy-ion collisions at RHIC and at the LHC. π ≈ 2 × 10 4 MeV 2 for T ≈ 2 T c = 330 ± 20 MeV and | e B | ≈ m 2 RHIC and T ≈ 4 − 5 · T c = 750 ± 120 MeV and π ≈ 3 × 10 5 MeV 2 for the LHC, we obtain | e B | ≈ 15 m 2 ( 82 ± 3 ) 2 MeV 2 at RHIC and m 2 = D ( 185 ± 35 ) 2 MeV 2 at the LHC. m 2 = D
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Discussion The Debye mass due to strong interactions has been calculated to all orders in the magnetic field. In the strong field limit holographic calculation of m D gives a result similar to the weak-coupling QED [J. Alexandre, 2001] . m D Mag = 0 even in the presence of the magnetic field to all orders in B . This is true, however, only up to 1 / N c corrections. The same holds for Π 11 and Π 22 . The similarity of the dynamics of strongly coupled QCD and weakly coupled QED in large external magnetic fields is a nontrivial phenomenon, which was observed also in [Son, Thompson, 2008] .
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Outline The Setup of Holographic QCD 1 Low-Energy Theorems of QCD 2 Chiral Magnetic Effect in Soft-Wall AdS/QCD 3 Anomalous QCD Contribution to the Debye Screening in an 4 External Field via Holography Magnetic Susceptibility of the Chiral Condensate in a Model 5 with a Tensor Field Effect of the Gluon Condensate on the Gross–Ooguri Phase 6 Transition
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Magnetic Susceptibility of the Chiral Condensate Introduced in the framework of QCD Sum Rules [B. L. Ioffe, A. V. Smilga] qq � F µν , where σ µν = i � ¯ q σ µν q � F = χ � ¯ 2 [ γ µ , γ ν ] . Measures induced tensor current in the QCD vacuum
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Different approaches to χ π = − 8 . 9 GeV − 2 - OPE of the � VVA � correlator N c χ = − 4 π 2 f 2 and pion dominance [A. Vainshtein] Sum rule fit χ = − 3 . 15 ± 0 . 30 GeV − 2 Vector dominance χ = − ( 3 . 38 ÷ 5 . 67 ) GeV − 2 [Balitsky, Yung, Kogan ...]
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Possible Alternative Derivations of χ There is a significant discrepancy in the numerical results. It is natural to start looking elsewhere, e.g. holography, to identify the missing ingredients Holographic calculation of � VVA � : χ ∼ − 11 . 5 GeV − 2 [A. Gorsky, A. Krikun]. Vainshtein relation isn’t exact, but fulfilled to good accuracy. Holographic Son–Yamamoto relations: χ agrees with Vainshtein. Assumed to be valid at any momentum transfer. No field-theoretical derivation. A direct holographic calculation - motivated by enhanced AdS/QCD models [Cappiello, Cata, D’Ambrosio; Domokos, Harvey, Royston; Alvares, Hoyos, Karch] that take into account the 1 + − mesons. Check them for self-consistency.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Possible Alternative Derivations of χ There is a significant discrepancy in the numerical results. It is natural to start looking elsewhere, e.g. holography, to identify the missing ingredients Holographic calculation of � VVA � : χ ∼ − 11 . 5 GeV − 2 [A. Gorsky, A. Krikun]. Vainshtein relation isn’t exact, but fulfilled to good accuracy. Holographic Son–Yamamoto relations: χ agrees with Vainshtein. Assumed to be valid at any momentum transfer. No field-theoretical derivation. A direct holographic calculation - motivated by enhanced AdS/QCD models [Cappiello, Cata, D’Ambrosio; Domokos, Harvey, Royston; Alvares, Hoyos, Karch] that take into account the 1 + − mesons. Check them for self-consistency.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Possible Alternative Derivations of χ There is a significant discrepancy in the numerical results. It is natural to start looking elsewhere, e.g. holography, to identify the missing ingredients Holographic calculation of � VVA � : χ ∼ − 11 . 5 GeV − 2 [A. Gorsky, A. Krikun]. Vainshtein relation isn’t exact, but fulfilled to good accuracy. Holographic Son–Yamamoto relations: χ agrees with Vainshtein. Assumed to be valid at any momentum transfer. No field-theoretical derivation. A direct holographic calculation - motivated by enhanced AdS/QCD models [Cappiello, Cata, D’Ambrosio; Domokos, Harvey, Royston; Alvares, Hoyos, Karch] that take into account the 1 + − mesons. Check them for self-consistency.
AdS/QCD Setup Low-Energy Theorems CME Debye Mass in Holography Magnetic Susceptibility Gross–Ooguri Transition Possible Alternative Derivations of χ There is a significant discrepancy in the numerical results. It is natural to start looking elsewhere, e.g. holography, to identify the missing ingredients Holographic calculation of � VVA � : χ ∼ − 11 . 5 GeV − 2 [A. Gorsky, A. Krikun]. Vainshtein relation isn’t exact, but fulfilled to good accuracy. Holographic Son–Yamamoto relations: χ agrees with Vainshtein. Assumed to be valid at any momentum transfer. No field-theoretical derivation. A direct holographic calculation - motivated by enhanced AdS/QCD models [Cappiello, Cata, D’Ambrosio; Domokos, Harvey, Royston; Alvares, Hoyos, Karch] that take into account the 1 + − mesons. Check them for self-consistency.
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