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Progress report : Holographic chiral symmetry

Progress report : Holographic chiral symmetry. O.Bergman, S. Seki B. Burrington V. Mazo also O. Aharony and S.Yankielowicz and K. Peeters and M. Zamaklar. Introduction. QCD admits at low energies: Confinement and chiral symmetry breaking.

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Progress report : Holographic chiral symmetry

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  1. Progress report : Holographic chiral symmetry O.Bergman, S. Seki B. Burrington V. Mazo also O. Aharony and S.Yankielowicz and K. Peeters and M. Zamaklar

  2. Introduction • QCD admits at low energies: Confinement and chiral symmetry breaking. • Apriori there is no relation between the two phenomena. • Except of lattice simulations the arsenal of non perturbative field theory tools is quite limited. • Gauge/gravity duality is a powerful method to deal with strongly coupled gauge theories. • There are several stringy ( gravitational) models with a dual field theory in the same universality class as QCD • Confinement is easily realized, flavor chiral symmetry is not.

  3. Fundamental quarks can be incorporated via probe branes. First were introduced in duals of coulomb phase. (Karch Katz). • Quarks in confining scenarios were introduced into the KS confining background using D7 braens ( SakaiSonnenscehin) • Models based on Witten’s near extremal D4 branes with D6 ( Kruczenski et al). Also Erdmengeret al • A model that admits full flavor chiral symmetry breaking by incorporating D8 and anti D8 branes to Witten’s model. Sakai and Sugimoto • In this work we examine the issues of • (i) chiral symmetry breaking, current algebra quark mass and the GOR relation via a “tachyonic DBI” action • (ii)Back-reaction on the background and the stability of the SS model. • (iii) Thermal phase structure using non critical string background. In particular we discuss the transitions • confinement/deconfinement • chiral symmetrybreaking/ restoring • We determine the thermal spectra of meson and their dissociation • (no) drag on mesons and breakup due to velocity

  4. Witten’s model of near extremal D4 branes • The periodicity of x4 • The high temperature background • The periodicity of t

  5. The parameters of the gauge theory are given in the sugra • 5d coupling 4d coupling glueball mass • String tension • The gravity picture is valid only provided that l5 >> R • At energies E<< 1/R the theory is effectively 4d. • However it is not really QCD since Mgb ~ MKK • In the opposite limit of l5 << R we approach QCD

  6. The Sakai Sugimoto model • The basic underlying brane configuration is -|------|- • In the limit of Nf < <Nc the Sugra background is that of the near horizon limit of the near extremal D4 branes with Nf probe D8 branes and Nf probe anti D8 branes. • The strings between the D4 branes and the D8 and anti D8 branes D4- D8 stringsyL – left chiral fermions in ( Nf, 1 ,Nc) of U(Nf )x U(Nf )x U(Nc) D4- anti-D8 stringsyR – right chiral fermions in (1, Nf ,Nc) of U(Nf )x U(Nf)xU(Nc) • Note that it is a chiral symmetry and not an U(Nf)xU(Nf) of Dirac fermions. This is due to the fact that the there is no transverse direction to the D8 branes. • The same applies to D4 branes in 6d non critical model ( Casero Paredes J.S) D4 D8 anti D8

  7. Outline • Quark mass, condensate from tachyonic DBI • Back reaction of the flavor branes • Phases of thermal QCD from Non critical strings

  8. 1. Quark mass, chrial symmetry breaking and tachyonic DBI In the Sakai Sugimoto model the quarks are massless and there is no apparent way to add a current algebra mass. Even in the generalized model the pion mass is zero and hence so is the C.A quark mass u0 -u L corresponds to constituent quark mass It is not clear what is the source of the chiral symmetry breaking and in particular it is not associated with an expectation value of a bifundamental One would like to have a holographic dual of the GOR relation That states that

  9. To understand the dynamics of the chiral symmetry • Breaking we incorporate a complex bi-fundamental • “Tachyon”. Discussed also by Casero Kiritsis Paredes • We start with an action proposed by Garousi • for a separated parrallel Dp and anti- D p branes • This action is obtained by generalizing Sen’s action • for non-BPS branes. • The action reads

  10. Setting the gauge fields to zero in the non compact case leaves us with the following tachyonic DBI action and The brane locations where For small u the bi-fundamental T will become tachyonic The filed T has a localized tachyonic mode at small u

  11. The corresponding EOM for L and T are For T=0 the EOM are those of the Sakai Sugimoto model From the point of view of the potential of T this is a non stable solution. We are after a solution with T(u) and L(u) such that when the tachyon condenses the brane anti brane separation vanishes Profile of T(u) T diverges

  12. IR asymptotic solution We expand the equations around u=u0 and find As expected The tachyon diverges at u=u0 where the brane anti brane merge UV asymptotic solution

  13. We put a cutoff and expand the solution around it. We than compute “physical quantities” and check that they are independent of this cutoff. Strictly we cannot take it to infinity since at this region the string coupling becomes large. Requiring that the perturbation are small implies We identify the non-normalizable solution with the mass

  14. The Hamiltonian density has the form then the condensate is determined from by differentiating with respect to mq And since the condensate is given by

  15. To compute the spectrum of the vector mesons and of the pions we analyze the spectrum of fluctuations of the flavor gauge fields that live on the probe branes We go over to vector and axial gauge fields and use the unitary gauge where the tachyon Is real

  16. The A+ sector We dimensionally reduce the 9d action to a 5 one, and plug the background We parameterize the guage fields After solving the eigenvalues problem and reducing to 4d we get Thus we find a spectrum of massive vector mesons

  17. The A- sector In a similar manner the 5d action now is We make the following decompositions In our gauge the w0 are the pions

  18. The 4d action of the A- now reads Thus we see that the pions are massive The mass of the pion is determined by eigenvalue equation

  19. The pion decay constant and the GOR relation We evaluate the pion decay constant fp by computing the correlator of axial Vector currentsErlich, Katz,Son,Stefanov The effective action is By taking we find

  20. Finally we get which in leading order in mq/<qq> translates into the Gell-Mann Oakes Rener relation

  21. 2. From probe to fully back reaction • In the Sakai Sugimoto model flavor is introduced by incorporating Nf D8 anti-D8 branes. This is done in the probe approximation based on having Nf<<Nc . • The profile of the probe branes is determined by solving the DBI EOM. In the probe approximation the configuration is stable ( no tachyonic modes). • The motivation to go beyond the probe approximation is tow follded: (a) To check that the back-reaction on the background does not destroy the stability. (b) To determine the flavor dependence on properties that are extracted from the background like string tension, beta function, the viscosity /entropy density etc.

  22. The action of the back-reacted system is Action of A9form Massive IIA action DBI action [d (x4)+d(x4-p)] Induced metric

  23. The EOM of the metric, dilaton, and RR forms are

  24. The solution for the parameter M and F10 The jumps are at the locations of the branes Our basic assumption is that the back-reaction is a small perturbation controled by the small parameter The bulk term in the massive IIA can be neglected since it is of order • The delta functions are codimension one which leads to • an harmonic function ( absolute value x4) which is finite at the • location of the delta function

  25. We take the following ansatz for the background We plug it to the EOM and assume an expansion perturbation Original background

  26. To simplify the computation we replace the cigar • Of the (u,x4) directions with a cylinder that asymptote to it Which means that we omitted the factor f(u)= 1-(uL/u)^3 from the background Note that for large u this factor is irrelevant

  27. We separate the equations using the following variables F1 and F2 are invariant under x4 and u coordinate transformations We Fourier decompose in x4

  28. Finally defining the equations read The general solution after enforcing the convergence of the Fourier sum takes the form

  29. The behavior at large u The general behavior is of spikes around the locations of the branes.

  30. The spike structure ocours here only for much larger u

  31. The perturbed dilaton at small u Note that the solution developes a duble notch behavior between the cusp solution of very small u (red) and the spikes of large u

  32. The perturbed A at small u

  33. We were able to solve the equations also without the • Fourier decomposition. It can be shown that the general • structure of the of the solution is of the following form where In particular for F2 we get • The implications of the solution on the stability • Is still not clear to us.

  34. 3. Thermal phases of QCD from non critcal string model In 2006 with O. Aharony and S. Yankielowicz we have analyzed the hologrphic thermal phase structure of QCD based on thermalizing the Sakai Sugimoto model. With K. Peeters and M. Zamaklar we analyzed the spectrum of thermal mesons and the ( no ) drag of mesons. With V. Mazo we have done a similar analysis based on A model of non critical D4 color branes with Nf D4 and anti D4 flavor branes.

  35. The color and flavor barnes are The non critical near extremal D4 brane The flavor probe action is

  36. Review of the bulkthermodynamics • We introduce temperature by compactifying the Euclidean time direction with periodicity b=1/T and imposing anti-periodic boundary conditions on the fermions. • We use amodel of near extremal D4 branes (either ciritical or non critical). • There is already a compact direction x4 so in our thermal model ( t, x4) are compact. • There are only two such smooth SUGRA backgrounds

  37. At any given T the background that dominates is the one that has a lower free energy, namely, lower classical SUGRA action ( times T). • The classical actions diverge. We regulate them by computing the difference between the two actions. • It is obvious that the two actions are equal for b =2p R , thus at Td =1/2p R there is a first order phase transition . • The transition is first order since the two solutions continue to exist both below and above the transition. • It is easy to see that for T<1/2 pR the background with a thermal factor on X4 dominates, and above it the one with the thermal factor on t.

  38. The interpretation of the phase transition is clear. The order parameters are: • (i)low temperature the string tensionTst ~gtt gxx(um=uL ) >0 confinement • high temperature the string tensionTst ~gtt gxx(um=uT ) =0 deconfinement • (ii) Discrete spectrum versus continuum and dissociation of glueballs. • (iii) Free energy ~ Nc2 at high temperature Nc0 at low temperature • (iv) Vanishing/non vanishing Polyakov loop ( string wrapping the time direction) • The dominant phase for small l /R, due to the symmetry under • T > 1/2p R, is a symmetric phase Aharony, Minwalla Weismann T< 1/2 pR

  39. The phase diagram of the pure glue theory Symmetric phase ----------------

  40. Low temperature phase • At the UV the D8 and anti D8 are separated  U(Nf)L x U(Nf)R • In the IR they merge together  spontaneous breakingU(Nf)D • To verify this we analyze the DBI probe brane action • The solution of the corresponding equation of motion is

  41. The low temperature phase with flavor

  42. The high temperature deconfining phase • Recall that the action has now the thermal factor on the t direction • The equation of motion admits a solution similar to the one • of the low temperature domain, namely with chiral symmetry breaking • However there is an additional stable solution of two disconnected • stacks of branes. • This obviously corresponds to chiral restoration. • This is possible since at u=uT the t cycle shrinks to zero and the • D8 branes can smoothly end there.

  43. Chiral symmetry breaking/restoring

  44. The configuration with the lower free energy ~DBI action dominates • The action diverges but can be regulated by computing the difference • between the chiral symmetry breaking and restoring solutions where y=u/u0 • We solve it numerically and find • For yT > yTc~ 0.735DS > 0 | | • For yT < yTc~ 0.735DS < 0 U

  45. The action difference DS as a function of yT (~LT) DS c symmetry restored yT c symmetry broken • The c symmetry breaking/restoring phase transition, just like the conf/decon Is a first order phase transition

  46. Phase diagram- • We express the critical point in terms of the physical quantities T,L

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