A magnetic instability of the Sakai-Sugimoto model Nele Callebaut - PowerPoint PPT Presentation

Introduction Holographic set-up The ρ meson mass T χ Chiral symmetry • Massless QCD-Lagrangian ψ i γ µ D µ ψ − 1 4 F 2 µν invariant under chiral symmetry transformations ( g L , g R ) ∈ U ( N f ) L × U ( N f ) R ψ L → g L ψ L , ψ R → g R ψ R with ψ L = 1 ψ R = 1 2 ( 1 − γ 5 ) ψ , 2 ( 1 + γ 5 ) ψ . • But chiral symmetry U ( N f ) L × U ( N f ) R not reflected in mass spectrum of the mesons...

Introduction Holographic set-up The ρ meson mass T χ Chiral symmetry • Massless QCD-Lagrangian ψ i γ µ D µ ψ − 1 4 F 2 µν invariant under chiral symmetry transformations ( g L , g R ) ∈ U ( N f ) L × U ( N f ) R ψ L → g L ψ L , ψ R → g R ψ R with ψ L = 1 ψ R = 1 2 ( 1 − γ 5 ) ψ , 2 ( 1 + γ 5 ) ψ . • But chiral symmetry U ( N f ) L × U ( N f ) R not reflected in mass spectrum of the mesons... Explanation: spontaneous chiral symmetry breaking U ( N f ) L × U ( N f ) R → U ( N f )

Introduction Holographic set-up The ρ meson mass T χ Chiral symmetry in the dual picture “On the stack of N f coinciding pairs of D8-D8 flavour branes lives a U ( N f ) L × U ( N f ) R theory, to be interpreted as the chiral symmetry in QCD.”

Introduction Holographic set-up The ρ meson mass T χ Chiral symmetry in the dual picture “On the stack of N f coinciding pairs of D8-D8 flavour branes lives a U ( N f ) L × U ( N f ) R theory, to be interpreted as the chiral symmetry in QCD.” The U-shaped embedding of the flavour branes models spontaneous chiral symmetry breaking U ( N f ) L × U ( N f ) R → U ( N f ) .

Introduction Holographic set-up The ρ meson mass T χ The flavour gauge field The U ( N f ) gauge field A µ ( x µ , u ) that lives on the flavour branes describes a tower of vector mesons v µ , n ( x µ ) in the dual QCD-like theory: U ( N f ) gauge field A µ ( x µ , u ) = ∑ v µ , n ( x µ ) ψ n ( u ) n ≥ 1 with v µ , n ( x µ ) a tower of vector mesons with masses m n , and { ψ n ( u ) } n ≥ 1 a complete set of functions of u , satisfying the eigenvalue equation � � u 1 / 2 γ − 1 / 2 u 5 / 2 γ − 1 / 2 = − R 3 m 2 ( u ) ∂ u ( u ) ∂ u ψ n ( u ) n ψ n ( u ) , B B

Introduction Holographic set-up The ρ meson mass T χ Flavour gauge field and mesons • the way it works: dynamics of the flavour D8/D8-branes: 5D YM theory S DBI [ A µ ] = · · · , A µ ( x µ , u ) = ∑ n ≥ 1 v µ , n ( x µ ) ψ n ( u ) ↓ integrate out the extra radial dimension u effective 4D meson theory for v n µ ( x µ )

Introduction Holographic set-up The ρ meson mass T χ Flavour gauge field and mesons • the way it works: dynamics of the flavour D8/D8-branes: 5D YM theory S DBI [ A µ ] = · · · , A µ ( x µ , u ) = ∑ n ≥ 1 v µ , n ( x µ ) ψ n ( u ) ↓ integrate out the extra radial dimension u effective 4D meson theory for v n µ ( x µ ) • ideal holographic QCD model to study low-energy QCD confinement and chiral symmetry breaking effective low-energy QCD models drop out: Skyrme ( π , also: baryons as skyrmions), HLS ( π , ρ coupling), VMD

Introduction Holographic set-up The ρ meson mass T χ Approximations of the model Duality is valid in the limit N c → ∞ and large ’t Hooft coupling λ = g 2 YM N c ≫ 1, and at low energies (where redundant massive d.o.f. decouple). Approximations (inherent to the model): quenched approximation ( N f ≪ N c ) chiral limit ( m π = 0, bare quark masses zero) Choices of parameters: N c = 3 N f = 2 to model charged mesons

Introduction Holographic set-up The ρ meson mass T χ How to turn on the magnetic field A non-zero value of the flavour gauge field A m ( x µ , z ) on the boundary, A m ( x µ , u → ∞ ) = A µ , corresponds to an external gauge field in the boundary field theory that couples to the quarks ψ i γ µ D µ ψ with D µ = ∂ µ + A µ .

Introduction Holographic set-up The ρ meson mass T χ How to turn on the magnetic field A non-zero value of the flavour gauge field A m ( x µ , z ) on the boundary, A m ( x µ , u → ∞ ) = A µ , corresponds to an external gauge field in the boundary field theory that couples to the quarks ψ i γ µ D µ ψ with D µ = ∂ µ + A µ . To apply an external electromagnetic field A em µ , put A µ ( u → + ∞ ) = − iQ em A em = A µ µ [Sakai and Sugimoto hep-th/0507073]

Introduction Holographic set-up The ρ meson mass T χ How to turn on the magnetic field A non-zero value of the flavour gauge field A m ( x µ , z ) on the boundary, A m ( x µ , u → ∞ ) = A µ , corresponds to an external gauge field in the boundary field theory that couples to the quarks ψ i γ µ D µ ψ with D µ = ∂ µ + A µ . To apply an external electromagnetic field A em µ , put A µ ( u → + ∞ ) = − iQ em A em = A µ µ A em [Sakai and Sugimoto hep-th/0507073] = x 1 B 2 � 2 / 3 � 0 = 1 6 1 2 + 1 Q em = 2 σ 3 − 1 / 3 0

Introduction Holographic set-up The ρ meson mass T χ Overview Introduction 1 Holographic set-up 2 The Sakai-Sugimoto model Introducing the magnetic field The ρ meson mass 3 Taking into account constituents Full DBI-action Effect of Chern-Simons action and mixing with pions Chiral temperature 4

Introduction Holographic set-up The ρ meson mass T χ Plan • Action: � ∞ � � � ǫ 4 e − φ STr d 4 x 2 − det [ g D 8 mn + ( 2 πα ′ ) iF mn ] , S DBI = − T 8 du u 0 with STr ( F 1 · · · F n ) = 1 n ! Tr ( F 1 · · · F n + all permutations ) the symmetrized trace, g D 8 mn = g mn + g ττ ( D m τ ) 2 the induced metric on the D8-branes (with covariant derivative D m τ = ∂ m τ + [ A m , τ ] ), and F mn = ∂ m A n − ∂ n A m + [ A m , A n ] = F a mn t a the field strength

Introduction Holographic set-up The ρ meson mass T χ Plan • Action: � ∞ � � � ǫ 4 e − φ STr d 4 x 2 − det [ g D 8 mn + ( 2 πα ′ ) iF mn ] , S DBI = − T 8 du u 0 • Gauge field ansatz: � A m = A m + ˜ A m τ = τ + ˜ τ Determine embedding τ ( u ) as a function of A µ (put 1 ˜ A m = ˜ τ = 0) Determine EOM for ρ µ : 2

Introduction Holographic set-up The ρ meson mass T χ Plan • Action: � ∞ � � � ǫ 4 e − φ STr d 4 x 2 − det [ g D 8 mn + ( 2 πα ′ ) iF mn ] , S DBI = − T 8 du u 0 • Gauge field ansatz: � A m = A m + ˜ A m τ = τ + ˜ τ Determine embedding τ ( u ) as a function of A µ (put 1 ˜ A m = ˜ τ = 0) Determine EOM for ρ µ : 2 Plug total gauge field ansatz into S DBI , expand to 2nd order in the fluctuations and integrate out u -dependence

Introduction Holographic set-up The ρ meson mass T χ Plan • Action: � ∞ � � � ǫ 4 e − φ STr d 4 x 2 − det [ g D 8 mn + ( 2 πα ′ ) iF mn ] , S DBI = − T 8 du u 0 • Gauge field ansatz: � A m = A m + ˜ A m τ = τ + ˜ τ Determine embedding τ ( u ) as a function of A µ (put 1 ˜ A m = ˜ τ = 0) Determine EOM for ρ µ : 2 Plug total gauge field ansatz into S DBI , expand to 2nd order in the fluctuations and integrate out u -dependence Expand to order ( 2 πα ′ ) 2 ∼ 1 ( λ ≫ 1 ) vs use full DBI-action λ 2

Introduction Holographic set-up The ρ meson mass T χ General embedding u 0 > u K u 0 > u K to model non-zero constituent quark mass which is related to the distance between u 0 and u K . [Aharony et.al. hep-th/0604161]

Introduction Holographic set-up The ρ meson mass T χ Numerical fixing of holographic parameters There are three unknown free parameters ( u K , u 0 and κ ( ∼ λ N c )). In order to get results in physical units, we fix the free parameters by matching to the constituent quark mass m q = 0.310 GeV, the pion decay constant f π = 0.093 GeV and the rho meson mass in absence of magnetic field m ρ = 0.776 GeV. Results: u 0 = 1.92 GeV − 1 and u K = 1.39 GeV − 1 , κ = 0.00678

Introduction Holographic set-up The ρ meson mass T χ B -dependent embedding for u 0 > u K Keep L fixed: u 0 ( B ) rises with B . This models magnetic catalysis of chiral symmetry breaking [Bergman 0802.3720; Johnson and Kundu 0803.0038] .

Introduction Holographic set-up The ρ meson mass T χ B -dependent embedding for u 0 > u K Keep L fixed: u 0 ( B ) rises with B . This models magnetic catalysis of chiral symmetry breaking [Bergman 0802.3720; Johnson and Kundu 0803.0038] . Non-Abelian: u 0, u ( B ) > u 0, d ( B ) ! U ( 2 ) → U ( 1 ) u × U ( 1 ) d

Introduction Holographic set-up The ρ meson mass T χ B -dependent embedding for u 0 > u K Change in embedding models: chiral magnetic catalysis ⇒ m u ( B ) and m d ( B ) ր � B explicitly breaks global U ( 2 ) → U ( 1 ) u × U ( 1 ) d

Introduction Holographic set-up The ρ meson mass T χ B -dependent embedding for u 0 > u K Change in embedding models: chiral magnetic catalysis ⇒ m u ( B ) and m d ( B ) ր � B explicitly breaks global U ( 2 ) → U ( 1 ) u × U ( 1 ) d Effect on ρ mass?

Introduction Holographic set-up The ρ meson mass T χ B -dependent embedding for u 0 > u K Change in embedding models: chiral magnetic catalysis ⇒ m u ( B ) and m d ( B ) ր � B explicitly breaks global U ( 2 ) → U ( 1 ) u × U ( 1 ) d Effect on ρ mass? expect m ρ ( B ) ր as constituents get heavier

Introduction Holographic set-up The ρ meson mass T χ B -dependent embedding for u 0 > u K Change in embedding models: chiral magnetic catalysis ⇒ m u ( B ) and m d ( B ) ր � B explicitly breaks global U ( 2 ) → U ( 1 ) u × U ( 1 ) d Effect on ρ mass? expect m ρ ( B ) ր as constituents get heavier split between branes generates other mass mechanism: 5D gauge field gains mass through holographic Higgs mechanism

Introduction Holographic set-up The ρ meson mass T χ B -induced Higgs mechanism The string associated with a charged ρ meson ( ud , du ) stretches between the now separated up- and down brane ⇒ because a string has tension it gets a mass.

Introduction Holographic set-up The ρ meson mass T χ EOM for ρ for u 0 > u K ? Non-trivial embedding � τ u ( u ) θ ( u − u 0, u ) � 0 τ ( u ) = �∼ 1 , τ d ( u ) θ ( u − u 0, d ) 0 describing the splitting of the branes, severely complicates the analysis.

Introduction Holographic set-up The ρ meson mass T χ EOM for ρ for u 0 > u K ? Non-trivial embedding � τ u ( u ) θ ( u − u 0, u ) � 0 τ ( u ) = �∼ 1 , τ d ( u ) θ ( u − u 0, d ) 0 describing the splitting of the branes, severely complicates the analysis. � � 2 + .. ( F µν ) 2 + .. ( F µ u ) 2 + .. F µν [ ˜ � [ ˜ A µ , ˜ L 5 D = STr .. A m , τ ] + D m ˜ A ν ] τ � [ ˜ � � + .. ( ∂ u τ ) F A , τ ] + D ˜ τ F with all the .. different functions H ( ∂ u τ , F ; u ) of the background fields ∂ u τ , F .

Introduction Holographic set-up The ρ meson mass T χ Fixing the gauge to disentangle ˜ A and ˜ τ Faddeev-Popov gauge fixing: The functional integral D A D τ e i � L [ A , τ ] � Z = D A D τ e i � ( L [ A , τ ] − 1 � δ G [ A α , τ α ] � � 2 G 2 ) det = C ′ δα is restricted to physically inequivalent field configurations, by imposing the gauge-fixing condition G [ fields ] = 0.

Introduction Holographic set-up The ρ meson mass T χ Fixing the gauge to disentangle ˜ A and ˜ τ We choose the gauge condition on the fields 1 � G a [ ˜ √ ξ H m ( ∂ u τ , F ; u ) D m ˜ A a τ b τ c A , ˜ τ ] = m + ξǫ abc ˜ ( a = 1, 2 ) such that the gauge fixed Lagrangian τ ] − 1 L [ ˜ 2 G 2 A , ˜ no longer contains ˜ A ˜ τ mixing terms. τ 1,2 decouple. Then we choose ξ → ∞ (”unitary gauge”): ˜ Remaining gauge freedom in Abelian direction fixed by A a u = 0 ( a = 0, 3 ) .

Introduction Holographic set-up The ρ meson mass T χ Fixing the gauge to disentangle ˜ A and ˜ τ In the chosen gauge the Higgs-mechanism is more visible: τ 1,2 are ’eaten’ = Goldstone bosons ˜ τ 1,2 = massive gauge bosons (mass ∼ τ 2 ) A 1,2 ˜ eating the ˜ µ τ 0,3 in the direction of the vev τ = Higgs bosons ˜

Introduction Holographic set-up The ρ meson mass T χ Fixing the gauge to disentangle ˜ A and ˜ τ In the chosen gauge the Higgs-mechanism is more visible: τ 1,2 are ’eaten’ = Goldstone bosons ˜ τ 1,2 = massive gauge bosons (mass ∼ τ 2 ) A 1,2 ˜ eating the ˜ µ τ 0,3 in the direction of the vev τ = Higgs bosons ˜ We are left with τ ] + L [ ˜ L 5 D = L [ ˜ A ]

Introduction Holographic set-up The ρ meson mass T χ L [ ˜ τ ] : Stability of the embedding L [ ˜ τ ] � stability of the embedding: energy density δ L H = τ − L τ ∂ 0 ˜ δ∂ 0 ˜ τ 0,3 must fulfill associated with fluctuations ˜ � ∞ E = H > 0 u 0, d We checked that this is the case.

Introduction Holographic set-up The ρ meson mass T χ L [ ˜ A ] : back to the ρ meson EOM � A m , τ ] 2 + .. ( F µν ) 2 + .. ( F µ u ) 2 + .. F µν [ ˜ � .. [ ˜ A µ , ˜ L 5 D = STr A ν ] with all the .. different functions H ( ∂ u τ , F ; u ) of the background fields ∂ u τ , F .

Introduction Holographic set-up The ρ meson mass T χ L [ ˜ A ] : back to the ρ meson EOM � A m , τ ] 2 + .. ( F µν ) 2 + .. ( F µ u ) 2 + .. F µν [ ˜ � .. [ ˜ A µ , ˜ L 5 D = STr A ν ] with all the .. different functions H ( ∂ u τ , F ; u ) of the background fields ∂ u τ , F . STr-prescription [Myers, Hashimoto and Taylor, Denef et.al.] 2 � H ( F ) F 2 � = − 1 F 2 I ( H ) + ∑ ∑ · · · STr a 2 a = 1 a = 0,3 with � 1 0 d α H ( F 0 + α F 3 ) + � 1 0 d α H ( F 0 − α F 3 ) I ( H ) = 2

Introduction Holographic set-up The ρ meson mass T χ L [ ˜ A ] : back to the ρ meson EOM � A m , τ ] 2 + .. ( F µν ) 2 + .. ( F µ u ) 2 + .. F µν [ ˜ � .. [ ˜ A µ , ˜ L 5 D = STr A ν ] with all the .. different functions H ( ∂ u τ , F ; u ) of the background fields ∂ u τ , F . STr-prescription [Myers, Hashimoto and Taylor, Denef et.al.] 2 � H ( F ) F 2 � = − 1 F 2 I ( H ) + ∑ ∑ · · · STr a 2 a = 1 a = 0,3 with � 1 0 d α H ( F 0 + α F 3 ) + � 1 0 d α H ( F 0 − α F 3 ) I ( H ) = 2 L 5 D = − 1 µν ) 2 − 1 µ u ) 2 − 1 2 f 3 ( B ) F 3 4 f 1 ( B )( F a 2 f 2 ( B )( F a ij ǫ 3 ab ˜ A a i ˜ A b j − 1 µ ) 2 ( τ 3 ) 2 − 1 2 f 4 ( B )( ˜ A a 2 f 5 ( B )( ˜ A a i ) 2 ( τ 3 ) 2

Introduction Holographic set-up The ρ meson mass T χ EOM for ρ for u 0 > u K     � �  − 1 − 1 − 1 2 f 3 ( B ) F 3 d 4 x µν ) 2 µ u ) 2 4 f 1 ( B ) ( F a 2 f 2 ( B ) ( F a ij ǫ 3 ab ˜ A a i ˜ A b S 5 D = du j  � �� �  � �� � � �� �   ( F a µν ) 2 ψ 2 ( ρ a µ ) 2 ( ∂ u ψ ) 2 ρ a i ρ b j ψ 2     − 1 ( τ 3 ) 2 − 1  2 f 4 ( B ) ( ˜ A a µ ) 2 2 f 5 ( B ) ( ˜ A a i ) 2 ( τ 3 ) 2 with ˜ A µ = ρ µ ( x ) ψ ( u )  � �� � � �� �   ( ρ a i ) 2 ψ 2  µ ) 2 ψ 2 ( ρ a

Introduction Holographic set-up The ρ meson mass T χ EOM for ρ for u 0 > u K     � �  − 1 − 1 − 1 2 f 3 ( B ) F 3 d 4 x µν ) 2 µ u ) 2 4 f 1 ( B ) ( F a 2 f 2 ( B ) ( F a ij ǫ 3 ab ˜ A a i ˜ A b S 5 D = du j  � �� �  � �� � � �� �   ( F a µν ) 2 ψ 2 ( ρ a µ ) 2 ( ∂ u ψ ) 2 ρ a i ρ b j ψ 2     − 1 ( τ 3 ) 2 − 1  2 f 4 ( B ) ( ˜ A a µ ) 2 2 f 5 ( B ) ( ˜ A a i ) 2 ( τ 3 ) 2 with ˜ A µ = ρ µ ( x ) ψ ( u )  � �� � � �� �   ( ρ a i ) 2 ψ 2  µ ) 2 ψ 2 ( ρ a demand � du f 1 ( B ) ψ 2 = 1 and � du f 2 ( B )( ∂ u ψ ) 2 + f 4 ( B )( τ 3 ) 2 ψ 2 = m 2 ρ ( B ) , then � du f 3 ( B ) ψ 2 = k ( B ) � = 1 and � du f 5 ( B )( τ 3 ) 2 ψ 2 = m 2 + ( B ) � � � − 1 µν ) 2 − 1 µ ) 2 − 1 j − 1 2 k ( B ) F 3 d 4 x 2 m 2 2 m 2 i ) 2 4 ( F a ρ ( B )( ρ a ij ǫ 3 ab ρ a i ρ b + ( B )( ρ a S 4 D = (with F a µν = D µ ρ a ν − D ν ρ a µ )

Introduction Holographic set-up The ρ meson mass T χ EOM for ρ for u 0 > u K     � �  − 1 − 1 − 1 2 f 3 ( B ) F 3 d 4 x µν ) 2 µ u ) 2 4 f 1 ( B ) ( F a 2 f 2 ( B ) ( F a ij ǫ 3 ab ˜ A a i ˜ A b S 5 D = du j  � �� �  � �� � � �� �   ( F a µν ) 2 ψ 2 ( ρ a µ ) 2 ( ∂ u ψ ) 2 ρ a i ρ b j ψ 2     − 1 ( τ 3 ) 2 − 1  2 f 4 ( B ) ( ˜ A a µ ) 2 2 f 5 ( B ) ( ˜ A a i ) 2 ( τ 3 ) 2 with ˜ A µ = ρ µ ( x ) ψ ( u )  � �� � � �� �   ( ρ a i ) 2 ψ 2  µ ) 2 ψ 2 ( ρ a demand � du f 1 ( B ) ψ 2 = 1 and � du f 2 ( B )( ∂ u ψ ) 2 + f 4 ( B )( τ 3 ) 2 ψ 2 = m 2 ρ ( B ) , then � du f 3 ( B ) ψ 2 = k ( B ) � = 1 and � du f 5 ( B )( τ 3 ) 2 ψ 2 = m 2 + ( B ) � � � − 1 µν ) 2 − 1 µ ) 2 − 1 j − 1 2 k ( B ) F 3 d 4 x 2 m 2 2 m 2 i ) 2 4 ( F a ρ ( B )( ρ a ij ǫ 3 ab ρ a i ρ b + ( B )( ρ a S 4 D = (with F a µν = D µ ρ a ν − D ν ρ a µ ) modified 4D Lagrangian for a vector field in an external EM field

Introduction Holographic set-up The ρ meson mass T χ Solve the eigenvalue problem The normalization condition and mass condition on the ψ combine to the eigenvalue equation f − 1 ∂ u ( f 2 ∂ u ψ ) − f − 1 f 4 ( τ 3 ) 2 ψ = − m 2 ρ ψ 1 1 with b.c. ψ ( x = ± π / 2 ) = 0, ψ ′ ( x = 0 ) = 0 which we solve with a numerical shooting method to obtain m 2 ρ ( B ) . 2 � GeV 2 � k m Ρ 0.80 1.015 0.75 1.010 0.70 1.005 0.65 B � GeV 2 � B � GeV 2 � 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

Introduction Holographic set-up The ρ meson mass T χ Solve the eigenvalue problem 2 � GeV 2 � m � B � GeV 2 � 0.2 0.4 0.6 0.8 � 0.0002 � 0.0004 � 0.0006 � 0.0008 � 0.0010 � 0.0012 2 � GeV 2 � k m Ρ 0.80 1.015 0.75 1.010 0.70 1.005 0.65 B � GeV 2 � B � GeV 2 � 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

Introduction Holographic set-up The ρ meson mass T χ Landau vs Sakai-Sugimoto u 0 > u K Modified 4D Lagrangian for a vector field in an external EM field with k ( B ) � = 1 � modified Landau levels and m 2 ρ , eff ( B ) = m 2 ρ ( B )+ m 2 + ( B ) − k ( B ) B

Introduction Holographic set-up The ρ meson mass T χ Landau vs Sakai-Sugimoto u 0 > u K Modified 4D Lagrangian for a vector field in an external EM field with k ( B ) � = 1 � modified Landau levels and m 2 ρ , eff ( B ) = m 2 ρ ( B )+ m 2 + ( B ) − k ( B ) B 2 � GeV 2 � m Ρ ,eff Landau 0.6 Sakai � Sugimoto 0.4 0.2 B � GeV 2 � 0.2 0.4 0.6 0.8 � 0.2

Introduction Holographic set-up The ρ meson mass T χ ρ meson condensation in Sakai-Sugimoto Antipodal embedding ( u 0 = u K ) ⇒ Landau levels 2 � GeV 2 � m Ρ ,eff 0.6 0.4 0.2 B � GeV 2 � 0.2 0.4 0.6 0.8 � 0.2 Non-antipodal embedding ( u 0 > u K ) ⇒ modified Landau levels 2 � GeV 2 � m Ρ ,eff Landau 0.6 Sakai � Sugimoto 0.4 0.2 B � GeV 2 � 0.2 0.4 0.6 0.8 � 0.2

Introduction Holographic set-up The ρ meson mass T χ Full DBI-action Reasons for considering full DBI-action:

Introduction Holographic set-up The ρ meson mass T χ Full DBI-action Reasons for considering full DBI-action: Expansion parameter in action det ( g + iF ) = det g × det ( 1 + g − 1 iF ) is g − 1 iF ⇒ most strict condition � u 0, d ( B = 0 ) � 3 / 2 ( 2 πα ′ ) − 1 ≡ 0.45 GeV 2 eB ≪ 3 2 R

Introduction Holographic set-up The ρ meson mass T χ Full DBI-action Reasons for considering full DBI-action: Expansion parameter in action det ( g + iF ) = det g × det ( 1 + g − 1 iF ) is g − 1 iF ⇒ most strict condition � u 0, d ( B = 0 ) � 3 / 2 ( 2 πα ′ ) − 1 ≡ 0.45 GeV 2 eB ≪ 3 2 R α ′ -corrections can cause magnetically induced tachyonic instabilities of W -boson strings, stretching between separated D3-branes, to disappear; the Landau level spectrum for the W -boson receives large α ′ -corrections in general [Bolognesi 1210.4170; Ferrara hep-th/9306048] .

Introduction Holographic set-up The ρ meson mass T χ Full DBI-action Reasons for considering full DBI-action: Expansion parameter in action det ( g + iF ) = det g × det ( 1 + g − 1 iF ) is g − 1 iF ⇒ most strict condition � u 0, d ( B = 0 ) � 3 / 2 ( 2 πα ′ ) − 1 ≡ 0.45 GeV 2 eB ≪ 3 2 R α ′ -corrections can cause magnetically induced tachyonic instabilities of W -boson strings, stretching between separated D3-branes, to disappear; the Landau level spectrum for the W -boson receives large α ′ -corrections in general [Bolognesi 1210.4170; Ferrara hep-th/9306048] . � � − 1 µν ) 2 − 1 µ ) 2 − 1 d 4 x 4 ( F a 2 m 2 ρ ( B )( ρ a 2 b ( B )( F a 12 ) 2 S 4 D = � − 1 j − 1 i ) 2 − 1 2 k ( B ) F 3 i 3 ) 2 + ( F a ij ǫ 3 ab ρ a i ρ b 2 m 2 + ( B )( ρ a 2 a ( B )(( F a i 0 ) 2 )

Introduction Holographic set-up The ρ meson mass T χ Full DBI-action Reasons for considering full DBI-action: Expansion parameter in action det ( g + iF ) = det g × det ( 1 + g − 1 iF ) is g − 1 iF ⇒ most strict condition � u 0, d ( B = 0 ) � 3 / 2 ( 2 πα ′ ) − 1 ≡ 0.45 GeV 2 eB ≪ 3 2 R α ′ -corrections can cause magnetically induced tachyonic instabilities of W -boson strings, stretching between separated D3-branes, to disappear; the Landau level spectrum for the W -boson receives large α ′ -corrections in general [Bolognesi 1210.4170; Ferrara hep-th/9306048] . � � − 1 µν ) 2 − 1 µ ) 2 − 1 d 4 x 4 ( F a 2 m 2 ρ ( B )( ρ a 2 b ( B )( F a 12 ) 2 S 4 D = � − 1 j − 1 i ) 2 − 1 2 k ( B ) F 3 i 3 ) 2 + ( F a ij ǫ 3 ab ρ a i ρ b 2 m 2 + ( B )( ρ a 2 a ( B )(( F a i 0 ) 2 ) Further modified 4D Lagrangian for a vector field in an external EM field

Introduction Holographic set-up The ρ meson mass T χ 4-dimensional EOM √ Standard Proca EOM for charged rho meson ρ µ = ( ρ 1 µ + i ρ 2 µ ) / 2 3 D 2 µν ρ µ − D ν D µ ρ µ − m 2 µ ρ ν − 2 iF ρ ρ ν = 0, D ν ρ ν = 0 with D µ = ∂ µ + iA 3 µ and F µν = D µ ρ ν − D ν ρ µ

Introduction Holographic set-up The ρ meson mass T χ 4-dimensional EOM √ Standard Proca EOM for charged rho meson ρ µ = ( ρ 1 µ + i ρ 2 µ ) / 2 3 D 2 µν ρ µ − D ν D µ ρ µ − m 2 µ ρ ν − 2 iF ρ ρ ν = 0, D ν ρ ν = 0 with D µ = ∂ µ + iA 3 µ and F µν = D µ ρ ν − D ν ρ µ replaced by 3 ( 1 + a ) D 2 µ ρ ν − i ( 1 + b + k ) F µν ρ µ − ( 1 + a ) D ν D µ ρ µ − ( m 2 ρ + m 2 + ) ρ ν + ( b − a )( D 2 j ρ ν − D ν D j ρ j ) = 0, µν D ν ρ µ − m 2 i 3 + D ν ρ ν = ( 1 + b − k ) F D i ρ i m 2 m 2 ρ ρ

Introduction Holographic set-up The ρ meson mass T χ Generalized Landau levels Landau levels ǫ 2 n , s z ( p z ) = p 2 z + m 2 ρ + ( 2 n − 2 s z + 1 ) B

Introduction Holographic set-up The ρ meson mass T χ Generalized Landau levels Landau levels ǫ 2 n , s z ( p z ) = p 2 z + m 2 ρ + ( 2 n − 2 s z + 1 ) B replaced by m 2 ρ + m 2 B 2 + ( 2 n + 1 ) B ( B − M 2 ) + ( 1 + b − k ) + ǫ 2 n ( p z ) = B p 2 z + m 2 1 + a 2 ρ � � ( 2 n + 1 ) 2 � + ( K − 2 B ) 2 ± B M + K − 2 B 4 � 1 / 2 2 ) + ( 1 + b − k ) 2 − ( 1 + b − k )( 2 n + 1 ) ξ ( K − 2 B + M ξ 2 4 with 1 + a − m 2 B = 1 + b K = 1 + b + k M = b − a ξ = B + 1 + a , , and m 2 m 2 1 + a ρ ρ

Introduction Holographic set-up The ρ meson mass T χ Effective ρ meson mass from full DBI-action Condensing solution n = 0, p z = 0 for transverse charged ρ y ) and ρ † = ( ρ + mesons ρ = ( ρ − x − i ρ − x + i ρ + y ) m 2 ρ , eff ( B ) = m 2 ρ − B

Introduction Holographic set-up The ρ meson mass T χ Effective ρ meson mass from full DBI-action Condensing solution n = 0, p z = 0 for transverse charged ρ y ) and ρ † = ( ρ + mesons ρ = ( ρ − x − i ρ − x + i ρ + y ) m 2 ρ , eff ( B ) = m 2 ρ − B becomes m 2 ρ ( B ) + m 2 + ( B ) k ( B ) m 2 ρ , eff ( B ) = − 1 + a ( B ) B 1 + a ( B )

Introduction Holographic set-up The ρ meson mass T χ ρ meson condensation in Sakai-Sugimoto Antipodal embedding ( u 0 = u K ) ⇒ Landau levels 2 � GeV 2 � m Ρ ,eff Landau 0.6 full DBI 0.4 0.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 B � GeV 2 � � 0.2 � 0.4 � 0.6 � 0.8 Non-antipodal embedding ( u 0 > u K ) ⇒ modified Landau levels 2 � GeV 2 � m Ρ ,eff Landau 0.6 F 2 approximation full DBI 0.4 0.2 B � GeV 2 � 0.2 0.4 0.6 0.8 � 0.2

Introduction Holographic set-up The ρ meson mass T χ Effect of Chern-Simons action and mixing with pions S = S DBI + S CS with � � � A 3 ) ǫ mnpqr A m F np F qr + O ( ˜ S CS ∼ Tr ρπ B mixing terms in the Chern-Simons action: � � �� 3 ] + 1 � ∂ [ 0 π 0 ρ 0 ∂ [ 0 π + ρ − 3 ] + ∂ [ 0 π − ρ + S CS ∼ B + · · · , 3 ] 2 but only between pions and longitudinal ρ meson components so no influence of pions on condensation of transversal ρ meson A 2 analysis) components (in order ˜

Introduction Holographic set-up The ρ meson mass T χ Conclusion: back to objectives Studied effect: ρ meson condensation phenomenological models: B c = m 2 ρ = 0.6 GeV 2 lattice simulation: slightly higher value of B c ≈ 0.9 GeV 2 � holographic approach: can the ρ meson condensation be modeled? yes can this approach deliver new insights? e.g. taking into account constituents, effect on B c Up and down quark constituents of the ρ meson can be modeled as separate branes, each responding to the magnetic field by changing their embedding. This is a modeling of the chiral magnetic catalysis effect. We take this into account and find also a string effect on the mass, leading to a B c ≈ 0.8 GeV 2 . Effect of full DBI is further increase of B c .

Introduction Holographic set-up The ρ meson mass T χ Overview Introduction 1 Holographic set-up 2 The Sakai-Sugimoto model Introducing the magnetic field The ρ meson mass 3 Taking into account constituents Full DBI-action Effect of Chern-Simons action and mixing with pions Chiral temperature 4

Introduction Holographic set-up The ρ meson mass T χ Chiral symmetry • Massless QCD-Lagrangian ψ i γ µ D µ ψ − 1 4 F 2 µν invariant under chiral symmetry transformations ( g L , g R ) ∈ U ( N f ) L × U ( N f ) R ψ L → g L ψ L , ψ R → g R ψ R with ψ L = 1 ψ R = 1 2 ( 1 − γ 5 ) ψ , 2 ( 1 + γ 5 ) ψ .

Introduction Holographic set-up The ρ meson mass T χ Chiral symmetry • Massless QCD-Lagrangian ψ i γ µ D µ ψ − 1 4 F 2 µν invariant under chiral symmetry transformations ( g L , g R ) ∈ U ( N f ) L × U ( N f ) R ψ L → g L ψ L , ψ R → g R ψ R with ψ L = 1 ψ R = 1 2 ( 1 − γ 5 ) ψ , 2 ( 1 + γ 5 ) ψ . • But chiral symmetry U ( N f ) L × U ( N f ) R not reflected in mass spectrum of the mesons...

Introduction Holographic set-up The ρ meson mass T χ Chiral symmetry • Massless QCD-Lagrangian ψ i γ µ D µ ψ − 1 4 F 2 µν invariant under chiral symmetry transformations ( g L , g R ) ∈ U ( N f ) L × U ( N f ) R ψ L → g L ψ L , ψ R → g R ψ R with ψ L = 1 ψ R = 1 2 ( 1 − γ 5 ) ψ , 2 ( 1 + γ 5 ) ψ . • But chiral symmetry U ( N f ) L × U ( N f ) R not reflected in mass spectrum of the mesons... Explanation: spontaneous chiral symmetry breaking U ( N f ) L × U ( N f ) R → U ( N f )

Introduction Holographic set-up The ρ meson mass T χ Chiral temperature T χ = temperature at which chiral symmetry is restored T χ U ( N f ) → U ( N f ) L × U ( N f ) R

Introduction Holographic set-up The ρ meson mass T χ Chiral temperature T χ = temperature at which chiral symmetry is restored T χ U ( N f ) → U ( N f ) L × U ( N f ) R Studied effect: possible split between T c and T χ ( B )

Introduction Holographic set-up The ρ meson mass T χ Split between T c and T χ Expected behaviour ( Fig from ’08 ): T χ ( B ) ր : “chiral magnetic catalysis” seen in chirally driven models (e.g. NJL) [hep-ph/0205348] T c ( B ) ց : paramagnetic gas of quarks thermodynamically favoured [0803.3156] (e.g. bag model [1201.5881] )

Introduction Holographic set-up The ρ meson mass T χ Some results in different models PLSM q model [Mizher et.al., 1004.2712] Lattice [D’Elia et.al., 1005.5365] Different PNJL models [Gatto and Ruggieri, 1012.1291]

Introduction Holographic set-up The ρ meson mass T χ Sakai-Sugimoto at finite temperature “Black D4-brane background” � R � 3 / 2 � du 2 � u � � 3 / 2 f ( u ) dt 2 + δ ij dx i dx j + d τ 2 ) + ds 2 ( ˆ f ( u ) + u 2 d Ω 2 = 4 ˆ R u f ( u ) = 1 − u 3 ˆ T u T ∼ T 2 u 3 ,

Introduction Holographic set-up The ρ meson mass T χ Numerical fixing of holographic parameters Input parameters f π = 0.093 GeV and m ρ = 0.776 GeV fix all holographic parameters except L . Choice of L left free, determines the choice of holographic theory: L very small ∼ NJL-type boundary field theory L = δτ / 2 maximal ∼ maximal probing of the gluon background (original antipodal Sakai-Sugimoto)

Introduction Holographic set-up The ρ meson mass T χ Sakai-Sugimoto at finite T and B no backreaction ⇒ T c independent of B B -dependent embedding of flavour branes ⇒ T χ ( B ) : S merged − S separated = 0 ⇒ T χ

Introduction Holographic set-up The ρ meson mass T χ Conclusion on T χ ( B ) The appearance of a split between T χ (GeV) (blue) and T c (GeV) (purple) depends on the choice of L : T Χ T Χ T Χ 0.15 0.25 0.125 0.14 0.20 0.120 0.13 0.15 0.115 0.10 0.12 0.110 0.05 0.11 0.105 eB eB eB 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Plots for fixed L (from small to large) respectively corresponding to m q ( eB = 0 ) = 0.357, 0.310 and 0.272 GeV and T c = 0.103, 0.115 and 0.123 GeV [N.C. and Dudal, 1303.5674] Left: split for L small enough ∼ NJL results [1012.1291] Middle and right: split only at large B or no split at all for parameter values that match best to QCD ∼ lattice data of [Ilgenfritz et.al.,1203.3360] (no split, also quenched)

Introduction Holographic set-up The ρ meson mass T χ Chiral transition in Sakai-Sugimoto Antipodal embedding ( u 0 = u K ) ⇒ no split T Χ 0.25 0.20 0.15 0.10 0.05 eB 0.5 1.0 1.5 2.0 2.5 3.0 Non-antipodal embedding ( u 0 > u K ) ⇒ appearance split depending on L T Χ T Χ T Χ 0.15 0.25 0.125 0.14 0.20 0.120 0.13 0.15 0.115 0.10 0.12 0.110 0.05 0.11 0.105 eB eB eB 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0