II Russian-Spanish Congress “Particle and Nuclear Physics at all scales and Cosmology”, Saint Petersburg, Oct. 4, 2013 RECENT ADVANCES IN THE BOTTOM-UP HOLOGRAPHIC APPROACH TO QCD Sergey Afonin Saint Petersburg State University

A brief introduction AdS/CFT correspondence – the conjectured equivalence between a string theory defined on one space and a CFT without gravity defined on conformal boundary of this space. Maldacena example (1997): Type IIB string theory on in low-energy (i.e. supergravity) approximation YM theory on AdS boundary in the limit AdS/QCD correspondence – a program to implement such a duality for QCD following the principles of AdS/CFT correspondence Up down Bottom up String theory QCD We will discuss Basic property: Algebra of SO(4,2) group and that of isometries of AdS 5 coincide Equivalence of energy scales The 5-th coordinate – (inverse) energy scale

[Witten; Gubser, Polyakov, Klebanov (1998)] Essence of the holographic method generating functional effective action Operators in a 4D gauge theory Classical fields in 5D dual theory In the sence that the corresponding sources Boundary values One postulates: The correlation functions are given by Mass spectrum: Poles of the two-point correlator Alternative way for finding the mass spectrum is to solve e.o.m. The output of the holographic models: Correlators

An important example of dual fields for the QCD operators (R=1): Main assumption of AdS/QCD: There is an approximate 5D holographic dual for QCD Here The holographic correspondence dictates the relation

A typical model (Erlich et al., PRL (2005); Da Rold and Pomarol, NPB (2005)) For Hard wall model: At one imposes certain gauge invariant boundary conditions on the fields. Equation of motion for the scalar field Solution independent of usual 4 space-time coordinates bare quark mass quark condensate here As the holographic dictionary prescribes

Denoting the equation of motion for the vector fields are (in the axial gauge) where The spectrum of normalizable modes is given by zeros of Bessel function, thus the asymptotic behavior is that is not Regge like due to chiral symmetry breaking

Soft wall model (Karch et al., PRD (2005)) The IR boundary condition is that the action is finite at To have the Regge like spectrum: To have AdS space in UV asymptotics: The mesons of arbitrary spin J can be considered, the spectrum has the form But! No natural chiral symmetry breaking! Self-consistent extension to the arbitrary intercept: Afonin, PLB (2013)

Some applications  Meson, baryon and glueball spectra  Low-energy strong interactions (chiral dynamics)  Hadronic formfactors  Thermodynamic effects (QCD phase diagram)  Condensed matter (high temperature superconductivity etc.) ... Deep relations with other approaches  Light-front QCD  Soft wall models: QCD sum rules in the large-N c limit  Hard wall models: Chiral perturbation theory supplemented by infinite number of vector and axial-vector mesons

Holographic description of thermal and finite density effects Basic ansatz - corresponds to One uses the Reissner-Nordstrom AdS black hole solution whereis the charge of the gauge field. The hadron temperature is identified with the Hawking one: The chemical potential is defined by the condition

The critical temperature and density (deconfinement) can be found from the condition of complete dissociation of meson peaks in the correlators. The typical critical temperature at zero chemical potential for the light flavors lies about 200 MeV, for heavy ones does near 550 MeV. Some examples of phase diagrams He et al., JHEP (2013) Colangelo et al., EPJC (2013)

Hadronic formfactors Definition for mesons: Electromagnetic formfactor: In the holographic models for QCD: Brodsky, de Teramond, PRD (2008)

Linear spectrum and quark masses The dependence of A and B on the quark masses? Afonin, Pusenkov, PLB (2013)

Basic construction: The no-wall holographic model (Afonin, PLB (2009)) The result: From the ω-meson trajectory: From the holography: Charmonim: Bottomonium: In the heavy-quark limit: Interpretation: When a non-relativistic quark is created and moves with the velocity v in the c.m. frame, should compensate its kinetic energy The binding energy grows linearly with the quark mass! In the limit This coincides with a prediction of the Lovelace-Shapiro dual amplitude!

Thank you!