Vector and baryon spectra via holography in an AdS deformed - PowerPoint PPT Presentation

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook Vector and baryon spectra via holography in an AdS deformed background Miguel ´ Angel Mart´ ın Contreras With A. Vega, E. Folco-Cappssoli, Danning Li, and H. Boschi-Filho Based on arXiv:1903.06269 Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile HADRON 2019, Guilin, China 2019 Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook Outline 1 Motivation: AdS and Confinement 2 AdS with quadratic deformations 3 Baryons spectra 4 Mesons Spectra 5 Conclusions and Outlook HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook AdS/CFT: in a nano-nutshell AdS/CFT establishes an equivalence between non-perturbative QFT and Gravity. Figure 1: AdS/CFT Figure 2: Field/Operator duality But this QFT, in the first approximation, is conformal. Real world is far away from conformal!!! HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook AdS and Confinement For example, the existence of confinement: for some energies, hadrons are bounded. For others, they break apart. One evidence of the presence of confinement is the Regge Trajectories. These trajectories can be defined as a systematic form to organize hadronic states according to their angular momentum and excitation level. (See Prof. S. Afonin’s talk). Figure 3: Regge Trajectory HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook AdS and Confinement in the bottom-up approach Bottom-up in this holographic context means fixing the gravity and background fields to mimic the QFT properties . If we said that this QFT is QCD, we call this approach, AdS/QCD (See Prof. A. Vega’s talk). Since AdS does not have an energy scale (so, there is no confinement in such geometry), we need to introduce one. This extra energy scale will induce confinement and as a consequence, the normalizable of the fields living in AdS will have a spectrum. If we do the the identification of these normalizable states with hadrons, we can construct Regge trajectories. This is the key point!. HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook AdS and Confinement In these bottom-up approaches, confinement is realized via the breaking of the conformal symmetry. This can be done in many forms. For example: explicitly, by introducing a cuto ff to the AdS/Space. This is the hardwall model (Braga and Boshi-Filho, 2005, Polchinski-Strassler 2006). softly by introducing a smooth quadratic and static dilaton field (Karch et. al. 2006). mixing both approaches: a UV cuto ff and a static and quadratic dilaton (Braga, M.A. Martin and Diles, 2014). This leads us to conclude that we can induce confinement by: Deforming the AdS background. Introducing a proper dilaton field. HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook How we construct hadrons: Hadronic Identity Hadrons are characterized by its scaling dimension, that is fixed to be the conformal one, ∆ , for the bulk fields. According to the original AdS/CFT, the bulk mass M 5 carries the information ∆ as follows: Mesons with ∆ = 3: 5 R 2 = ( ∆ − S )( ∆ + S − 4) M 2 (1) Baryons of spin 1 / 2 with ∆ = 3 / 2: m 5 = ∆ + 2 . (2) Thus, the bulk mass defines the hadronic identity of the state at hand. HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook Holographic algorithm Define a geometry background. Define an action for the bulk fields dual to hadronic states. Obtain equations of motion. Solve the associated Sturm-Liouville problem (Boundary Value Problem). Find the mass spectrum as the eigenvalues of the BVP. Evaluate the Regge Trajectory. HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook AdS deformed Background First consider a geometric background given by following the line-element: dS 2 = e 2( z ) 󰀆 dz 2 + η µ ν dx µ dx ν 󰀇 , (3) where η µ ν is a 4-dimensional Minkowski tensor. Since we want AdS-like geometries, we will impose that the warp factor behaves as 󰀖 R 󰀗 + 1 2 k z 2 A ( z ) = log (4) z such that, at the conformal boundary z → 0, we recover the usual AdS Poincare patch. HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook Hadronic States in the AdS deformed background Hadronic states will be given by fields living in this background. For each specie we can construct an action of the form 󰁞 d 5 x √− g L Hadron I = 1 (5) K with L Hadron given by: − 1 g m r g n p F mn F rp − 1 5 , V g m n A m A n , 2 M 2 L V = (6) 4 g 2 V − 1 g m n ∂ m S ∂ n S − M 2 5 , S S 2 , L S = (7) 2 g 2 S Ψ [ Γ m ∂ m − M 5 , B ] Ψ . ¯ L B = (8) HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook Equations of motion From the action for the bulk field we obtain: for mesons with β = − 3 + 2 S : 󰁬 󰁭 +( − q 2 ) e β A ( z ) ψ ( z , q ) − M 2 5 , β e ( β +2) A ( z ) ψ ( z , q ) = 0 . e β A ( z ) ∂ z ψ ( z , q ) ∂ z (9) for baryons 󰀆 − + 4 A ′ ψ ′ 4 A ′ 2 + 2 A ′′ ψ ′′ − + 5 e 2 A 󰀇 − m 5 A ′ e A − m 2 ψ − + ( − q 2 ) ψ − = 0 (10) where Ψ ( z , q ) = ψ + ( z , q ) + ψ − ( z , q ). From these equations, fixing the bulk mass ( M 5 , β for mesons and m 5 for baryons) we can construct the mass spectrum by solving the BVP. HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook Parameter fixing For the numerical calculation, the parameter choice was defined as: k (GeV 2 ) Hadronic state Bulk mass β − 0 . 613 2 Vector 0 − 1 − 0 . 332 2 Scalar − 3 − 3 0 . 205 2 1 / 2 baryon 5 / 2 X HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook Spin 1 / 2 baryons In the case of spin 1 / 2 baryons we obtain the following results: 6 5 4 m 2 (GeV 2 ) 3 2 1 Our Model PDG 0 0 1 2 3 4 5 6 7 n Figure 4: N (1 / 2 + ) radial trajectory obtained with the deformed AdS 5 space approach (dots) and PDG data (squares). HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook Spin 1 / 2 baryons In this case the radial Regge trajectories for both cases, experimental and theoretical data, are given by m 2 = (0 . 863 ± 0 . 029) n + (0 . 114 ± 0 . 111) , (11) Exp m 2 = (0 . 860 ± 0 . 042) n − (0 . 081 ± 0 . 164) . (12) th with a RMS error of 4 . 1%. HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile

Motivation: AdS and Confinement AdS with quadratic deformations Baryons spectra Mesons Spectra Conclusions and Outlook Higher fermionic spin baryons The high 1 / 2 spin equation for fermions has the same estructure as the Sturm–Liouville that we had obtained for the nucleon case. Thus, we can use the same equation but changing the bulk mass m 5 since the conformal dimension of the operator that creates these hadrons has a di ff erent dimension. We will discuss the 3 / 2 and 5 / 2 cases. The parameters in these cases are: k (GeV 2 ) Hadronic state Bulk mass 0 . 190 2 3 / 2 7 / 2 0 . 205 2 5 / 2 13 / 2 HADRON 2019, Guilin, China Instituto de F´ ısica y Astronom´ ıa, Universidad de Valpara´ ıso, Chile