Pos(ICHEP 2010)467 ∗ [email protected] Speaker

PoS(ICHEP 2010)467 http://pos.sissa.it/ ∗ [email protected] Speaker. D-branes with a U-shaped geometry,QCD, like the are D8 encountered flavor frequently braneswhich in in the is holographic Sakai-Sugimoto commonly backgrounds. model used of Theenergies, as is Dirac-Born-Infeld an not action, effective appropriate field in this theoryscalar case, mode. description because it of We misses review open a thedescription string dynamics recent that of theory attempt extends a to at the (nonlocal) incorporate low DBI complex this action.of mode chiral Our into symmetry results an breaking are effective and relevant field bare forin quark theory curved the mass backgrounds. holographic in QCD description and open string tachyon condensation ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

35th International Conference of High EnergyJuly Physics 22-28, - 2010 ICHEP2010, Paris France Vasilis Niarchos Author affiliation E-mail: Towards a novel description of flavorholographic dynamics QCD in PoS(ICHEP 2010)467 (1.1) and form gauge theories Vasilis Niarchos N reconnect 1 D4-branes dissolve into must be present in the bi-  D8 pairs remain as D-branes c T N D8- f N ]). The corresponding setup comprises ]. At strong ’t Hooft coupling the IR 1 2 , the D8 system in type IIA string theory (also c N  f 2 N 4 : 08 1 2 3 : 4 01238 : 56789 0123 56789 anti-D8-branes oriented in the following way flavors. In this regime, the D D D f f f f c N N N N N . D8 open strings. We conclude that this setup realizes at low energies a theory D8-branes and etc f QCD itself is not expected to admit a weakly curved gravitational dual; there are, N N D4-branes, As a concrete example, let us consider the D4-D8- One can show that the IR dynamics of this model is captured at weak ’t Hooft coupling by The formulation of a proper effective field theory description of the low-energy open string Usually, in BPS D-brane configurations the degrees of freedom that dominate the low-energy The theoretical and phenomenological interest in analytic descriptions of the strong coupling c U-shaped D8-branes. The open string theory on these branes captures holographically the flavor N f a closed string background capturedbrane. by In the the supergravity quenched solution approximation, of where the Wick-rotated black D4- The 4-direction is compactified withfor supersymmetry-breaking the anti-periodic fermions. boundary conditions degrees Hence, of at freedom low of energiesfrom four-dimensional D4-D8 the and Yang-Mills open D4- theory. string theory Chiral on and the anti-chiral D4-branes quarks arise realizes the dynamics is captured by QCD with a non-local version of the Nambu-Jona-Lasinio model [ dynamics on these branes is athe largely ability open to problem. find this Typical problems effective fieldof whose theory solution flavor description depends chiral are: on symmetry the breaking, computationthe of the mesonic the incorporation spectra order of parameter bare quark masses, the determination of physics are the low-lying masslessfields. modes of The effective the field open theory string: ofThis these the modes is transverse is scalars captured not and by the the thereasons. Dirac-Born-Infeld gauge case, (DBI) First, action. however, large for accelerationsand, the develop by in above-mentioned definition, general U-shaped the near D8-branes approximations theregion. that for turning validate the point Second, the of following when use aimplies of applied U-shaped that the to brane DBI there the action has above break todescription holographic down be of in QCD more the that setup than U-shaped the just D8-branes. transverse holographic scalars dictionary and A gauge complex fields scalar in field the low-energy whose strong coupling limit can bemethods. studied Large- in the supergravity regimehowever, with theories the with use similar of IR holographic dynamicsdescription. as It is QCD interesting that to have examine a the dual lessons weakly that curved we gravitational learn from such descriptions. of with features very similar to those of QCD. embedded in this new background, butN the non-trivial geometry forces them to sector dynamics of the dual strongly coupled QCD-like gauge theory. known as the Sakai-Sugimoto holographic model of QCD [ dynamics of gauge theories,devoted like in QCD, the is context obvious. of the In AdS/CFT recent correspondence years to considerable the effort analysis has of been large- Towards a novel description of flavor dynamics in HQCD 1. The problem PoS(ICHEP 2010)467 and (2.2) T , µ X . BPS branes Vasilis Niarchos T . L , ) T b . Even for large values ) ∂ L T T ( a ∂ V ) . There are several technical X + ( T D branes in flat space Φ ab − F e + type II string theory (2.1) I = , -brane including the effects of the open ) X p T b , ∂ X I ( global symmetry group. The normalizable Φ X ) a − ], which focused on D-branes in flat space, is 1 ∂ e ( 5 + , U for bosonic 3 our experience with effective actions of such scalar 2 ab × arises in the open string theory of the U-shaped D8- ) i η 2 ) ( ( dT 1 1 T √ ( + , appears as an extra transverse scalar and the tachyon , det U 1 ν T sector of an open string stretching between the asymptotic − = dX 1) q − ] for a related discussion). µ α ) 3 = ). [ , T 0 dX . the scalar field, whose effects we want to capture, is a non-local ( g α 2.2 T . ) V µν e ii p α g ( σ scalar field τ 1 = + cos 2 p D8 pairs that capture the chiral symmetry breaking. d real ds -dependent contribution to the dilaton. Taking this analogy a step further . This action has been derived in open string theory from first principles in ) = Z ]. The starting point of Ref. [ T T T this field can play an important role in the low-energy dynamics of the brane. 5 − ( L V = as a S ) ]. 4 T ] verified explicitly the validity of this picture in flat space. Non-BPS branes are auto- ( 5 V We propose a new way to make progress in this problem. This is a brief review of the results Ref. [ In this action, the Our approach relies heavily on an effective field theory description of brane-antibrane dynam- A complex bi-fundamental scalar field We conclude that the proper description of the reconnected D8-brane requires an extension of ]. 7 , 6 metric and dilaton where BPS, non-BPS and brane-antibrane pairsformulated arise on as the different background solution ( curves of the DBI action obtained in [ fields is in general limitedlast in string decade, theory see (although below) significant and field progress that has arises been from achieved a in long the open string stretching along a macroscopic distance potential one can imagine an extended eleven-dimensional ‘fictitious’ spacetime with coordinates matically recovered as planar solutions oriented transverse to the extra coordinate complications in formulating such an action: which describes the low-energy dynamics of astring non-BPS tachyon D field [ ics put forward in [ of the separation With a non-trivial wavefunction in theimportant holographic radial in direction the the vicinity effectsreconnection of of of this the the field D8- become turning point where they are closely related with the geometric 2. A unified effective description of BPS, non-BPS and D the tachyon-DBI action (we set the DBI action that incorporates the bi-fundamental scalar mode brane as a low-lying mode inD8 the and NS anti-D8-branches of the brane which lie at an arbitrary distance Towards a novel description of flavor dynamics in HQCD fundamental representation of the flavor branch of this field captures the vacuum expectationacts value as (vev) of the the order dual meson-like parameter operatorthe for that dual chiral bare symmetry quark mass breaking. (see The non-normalizable branch captures PoS(ICHEP 2010)467 , to * Ρ crit 2 L (3.1) 100 < , which -oriented L T T 80 Vasilis Niarchos 2 is the critical √ space (see the left π ) 60 as a function the radial . Finally, one can re- = T ) D1 and anti-D1 semi- T , T f µ X crit ( ( N L . To recover the DBI de- V suggests that the dynamics 40 T (namely, the right degrees of 2 L -direction) 20 6 x D system ( + + ] applied the approach of section 4 0 , is incorporated naturally. When NS5-branes and L T T k Μ H 80 60 40 log 160 140 120 100 4 gauge fields arise from the left and right ) 1 ( 5 : 01 1 2 3 : 4 5 01 : 0 6 6 U D D f f k NS N N in the presence of the tachyon potential T the left and right . system g . p is massless). e denotes a brane oriented along the positive D - T + p ). In this description, different degrees of freedom arise from geometrically different 1 Left Figure: the tachyon-paperclip solution. Right Figure: a plot of log U-shaped branes, like the ones discussed above in the context of the Sakai-Sugimoto model, This is a particularly interesting case that has many similar features to the Sakai-Sugimoto infinite branes (6 The 4-direction is again compactifiedthe with near-horizon anti-periodic background boundary of conditions the for Wick the rotated fermions. black NS5 In branes, the D1-branes reconnect with pieces of the paperclip. The non-localarises nature as of an the emergent bi-fundamental degree of complexthe freedom scalar from field solution the is real time-dependent and describes a condensing D of U-shaped branes should bethe described tachyon-DBI by action. a ‘radially In condensing’ order tachyon-paperclip solution to of test this idea, Ref. [ parts of the solution, appear frequently in holographic backgrounds. The picture of section model. It is based on the following configuration of Towards a novel description of flavor dynamics in HQCD sciption of BPS branesindependent it of is the crucial direction tocover notice the key that properties the of the normalizable brane-antibranefreedom worldvolume system and at excitations their separation are interactions) asplot a in closed Fig. ‘tachyon-paperclip’ curve in separation where 3. The NS5-D describe U-shaped (hairpin) D-branes in the background of NS5-branes. Figure 1: turning point of thethe U-shaped analytically brane expected in result the andmotion the background following data from of points the NS5-branes. are action based (3.2). The on a solid numerical curve evaluation is of a thearise equations fit of as based planar on solutions with one worldvolume direction parallel to PoS(ICHEP 2010)467 ] 0 = 4 ρ z (3.2) (3.4)  ρ depicts a at 10 and 1 T = Vasilis Niarchos y . ]. In terms of the 8 . 2 [ ) ) 0 T 1 ρ ..., θ and coincides with the − ∂ 1 k + ( k ( ρ σ 2 2 100 is an expected feature of sin = 0 ∼ z coth 0 (3.3) ρ 0 k 1 , -dependent tachyon potential with ρ − = k k . The right plot in Fig. + 2 ρ arises in the open string spectrum of 2 T A . = ) 2 p ρ y T T B + ρ , ∂ 0 = ) ( ρ 0 0 is the same for all ) k 1 θ -dependent elliptical solution ρ 10 and a fit based on the analytic result with θ y = ρ at large , + ( tanh = ρ 1 T 5 2 ( ... k ..., B r 2 ) + + A T ρ σ ( logsinh − this relation translates to − z e V 0 T T cos ρ e − ρ . 0 2 M x with the use of the RNS formalism. In particular, we can see T 05. The analytically expected parameters are ρ of 0 k . 1 + √ 0 α sinh T − , but this requires a modified ) = tanh − µ 0 ... ρ e 2 = θ θ ρ z d ( + p ∼ kB ρ T is left as a free undetermined function. T A 2 ) µ µ ) T ( = dtd T logcos -branes intersecting NS5-branes. ∼ 5 and ( log ) V p . Z T k T V 6 ( e − c = + = y a 0556. The apparent deviation in the vicinity of S . behavior 0 ∼ T ∼ T ) for the randomly chosen value of is the D1-brane turning point) with a tachyon-paperclip solution of the ‘rolling tachyon’ 18 0 is a constant that depends on the normalization of / 3.2 ρ The slope of the tachyon potential around value of the flat space result. We can reproduce the asymptotic behavior of the bi-fundamental scalar field leading branch coefficient parameters In the vicinity of the turning point we find a ( form large- parametrized by two independent parameters. Similarly, there are two independent parameters controlling theof leading the and subleading solution branches set of data points determined( numerically by solving the equations of motion of the1 action the approximations employed to obtain the numerical results. x Bootstrap methods in worldsheet CFTmological imply constant a of the specific standard relation hairpin-brane between and the its boundary turning point cos- Comparing with the exactly known solution in string theory one finds that The starting point is the tachyon-DBI action for a non-BPS D2-brane that wraps the back- Since there are only NSNS-fluxes turned on in this case, it is possible to solve (both closed D1-branes to form U-shaped D1-branes. A trivial T-duality along any of the directions (1235) • • • • • The tachyon potential ground cigar-shaped geometry parametrized by was to reproduce these results using the tachyon-DBI effective field theory. and open) string theory exactly in relates this system to D Towards a novel description of flavor dynamics in HQCD the explicitly how a half-winding bi-fundamentalthe scalar reconnected field D1-branes and how it controls the open string theory on them. Our goal in Ref. [ PoS(ICHEP 2010)467 , 113 , 56 , 666 69 ). . Vasilis Niarchos etc , 061 (2006) 0612 6 of the U-shaped D8-branes is a modulus-dependent quantity L , 98 (2007) [arXiv:hep-th/0702155]. . 787 1 to reproduce this and other related features? ) T ( the asymptotic separation . V g , 268 (2010) [arXiv:1005.1650 [hep-th]]. . e (2003) [arXiv:hep-th/0304045]. 106009 (2004) [arXiv:hep-th/0401066]. [arXiv:hep-th/0408172]. arXiv:0901.4368 [hep-th]. 841 condensation,” Nucl. Phys. B arXiv:hep-th/0604017. 843 (2005) [arXiv:hep-th/0412141]. A qualitatively similar picture is anticipated in the Sakai-Sugimoto model. This model has The good qualitative and quantitative agreement of the tachyon-DBI-based results with the Beyond the very interesting applications to gauge theory we believe that this exercise will also It would be very interesting to determine appropriate tachyon-paperclip solutions in the Sakai- [7] V. Niarchos, “Notes on tachyon effective actions and Veneziano amplitudes,” Phys. Rev. D [8] K. Hosomichi, “N=2 Liouville Theory with Boundary,” JHEP [6] D. Kutasov and V. Niarchos, “Tachyon effective actions in open string theory,” Nucl. Phys. B [5] D. Erkal, D. Kutasov and O. Lunin, “Brane-Antibrane Dynamics From the Tachyon DBI Action,” [4] V. Niarchos, “Hairpin-Branes and Tachyon-Paperclips in Holographic Backgrounds,” Nucl. Phys. B [3] R. Casero, E. Kiritsis and A. Paredes, “Chiral symmetry breaking as open string tachyon [2] E. Antonyan, J. A. Harvey, S. Jensen and D. Kutasov, “NJL and QCD from string theory,” [1] T. Sakai and S. Sugimoto, “Low energy hadron physics in holographic QCD,” Prog. Theor. Phys. many similar features with the NS5-braneences, model discussed above, but also some interesting differ- results expected from string theory suggestthe that problem this outlined in is section a promising route towards the resolution of be useful in uncovering new informationcurved about backgrounds open where string an dynamics exact and string tachyon theory condensation treatment in is currently out ofReferences reach. Towards a novel description of flavor dynamics in HQCD 4. Towards QCD in this case. A naturalpotential question is: does one need a more drastic modification of the tachyon-DBI Sugimoto model, to put constraintsimplications on of the this corresponding tachyon formalismdictionary, potential, for compute and the holographic to bare QCD determine quark mass the (namely dependence, fix determine mesonic the spectra details of the holographic