Holographic description of glueballs in a deformed AdS-dilaton background
aa r X i v : . [ h e p - ph ] S e p Holographic description of glueballs in adeformed AdS-dilaton background Frédéric Jugeau
INFN - Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy ([email protected])
Abstract.
We investigate the mass spectra of scalar and vector glueballs in the so-called bottom-upapproach of the AdS/QCD correspondence. The holographic model of QCD includes a static dilatonbackground field. We study the constraints on the masses coming from perturbing the dilaton fieldand the geometry of the bulk.
Keywords:
AdS/QCD correspondence, glueball spectroscopy, holographic constraints.
PACS:
INTRODUCTION
A breakthrough in the attempt to understand strongly coupled Yang-Mills theories camewith the AdS/CFT correspondence, proposed by Maldacena [1], that conjectures a con-nection between the large N limit of a maximally N = SU ( N ) gaugetheory defined in d dimensions and the supergravity limit of a superstring/M-theory liv-ing on a d + AdS ) space times a compact manifold [2, 3]. However, theapplication of this conjecture to a theory such as QCD is not straightforward, being QCDneither supersymmetric nor conformal so that its gravity dual theory remains unknown.Witten proposed a procedure to extend Maldacena’s duality to such gauge theories [4]:the conformal invariance is broken by compactification, while supersymmetry is brokenby appropriate boundary conditions on the compactified dimensions. The
AdS geometryof the dual theory is then deformed into an
AdS -Black-Hole geometry where the horizonplays the role of an IR brane.Adopting this (so-called up-down) approach, analyses of glueball spectroscopy in 3and 4 dimensions have been carried out, obtaining, for instance, that the scalar glueballwith J PC = ++ corresponds, in the supergravity side, to the massless dilaton fieldpropagating in the 10 dimension black-hole geometry [5]. Then, the glueball massis computable by solving the dilaton wave equation and gives results in reasonableagreement with the available lattice data [6].However, instead of trying to warp the original 11-dimensional AdS × S geometry inorder to obtain a 4-dimension gauge theory with similarities with QCD, one could adopta different strategy investigating what features the dual theory should have in order toreproduce known QCD properties. In this (so-called bottom-up) approach, or AdS/QCDcorrespondence, one attempts to construct a 5-dimensional holographic model able to Talk given at the International Workshop on Quantum Chromodynamics: QCD@Work 2007, MartinaFranca, Italy, 16-20 June 2007. eproduce the main features of QCD. As pioneered by Polchinski and Strassler [7],it turned out to be possible to modify the AdS/CFT duality, aimed at describing aconfining gauge theory, by considering a truncated
AdS holographic space-time onthe 4-dimensional boundary of which QCD is defined. In this so-called IR hard wallapproximation, the typical size of this AdS slice stands for an IR cutoff associated tothe QCD mass gap. The IR hard wall model has been widely used in order to investigate,for instance, light hadron spectrum and form factors [8]. Another holographic modelof QCD has been proposed, which consists in inserting a static dilaton field in the AdS space-time. This particular background allows one to recover the Regge behaviourbelieved to be satisfied by mesons [9], at odds with what happens with the IR hard wallmodel (which rather appears to be dual to a bag model of QCD) or starting from ageneral string theory and attempting to deform it [10].Even if somehow cumbersome shortcomings subsist when constructing holographicmodels of QCD, namely, for instance, the stringy corrections O ( / N ) , the role of theremaining compact manifold S or the accurate range of the holographic coordinate thatis effectively dual to the QCD energy scale, there is the hope that they do not spoil themain features of these dual models. THE d HOLOGRAPHIC MODEL DUAL TO QCD
Following [9], we consider a five dimensional conformally flat spacetime (the bulk)described by the metric: g MN = e A ( z ) h MN , ds = e A ( z ) ( h mn dx m dx n + dz ) , (1)where x M = ( x m , z ) and h MN = diag ( − , , , , ) . x m ( m = , . . .
3) represent the usualspace-time (the boundary) coordinates and z is the fifth holographic coordinate runningfrom zero to infinity. The metric function A ( z ) satisfies the condition : A ( z ) → z → ln (cid:18) Rz (cid:19) , (2)to reproduce the AdS metric close to the UV brane z →
0. In the following, we willtake the simplest choice compatible with the constraints displayed in [9]: A ( z ) = − ln z .Besides, we consider a background dilaton field f ( z ) = c z which only depends on theholographic coordinate z and vanishes at the UV brane. The large z dependence of thedilaton is chosen to reproduce the Regge behaviour of the low-lying mesons, and all themasses will be given with respect to the scale parameter c . Moreover, the introductionof this background dilaton allows one to avoid ambiguities in the choice of the bulk fieldboundary conditions at the IR wall.We construct a 5 d model that can be considered as a cut-off AdS space: a smoothcut-off in the IR replaces the hard-wall IR cutoff that would be obtained by allowing theholographic variable z to vary from zero to a maximum value z m ≃ L QCD . To investigate In the following, we put the AdS radius R to unity. he mass spectra of the QCD scalar and vector glueballs, we consider the two lowestdimension operators with the corresponding quantum numbers and defined in the fieldtheory living on the 4 d boundary [11]: ( O S = Tr ( F ) , O V = Tr ( F ( DF ) F ) , (3)(with D the covariant derivative) having conformal dimension D = D = p -form) operator on the boundary is related to the ( AdS mass ) of his dual field in the bulk as follows [2, 3]: ( AdS mass ) = ( D − p )( D + p − ) . (4)In the following, we assume that the mass m of the bulk fields is given by this expres-sion.A 5 d massless scalar field X ( x , z ) can be constructed as the correspondent of Tr ( F ) ,described by the action in the gravitational background: S = − Z d x √− g e − f ( z ) g MN ( ¶ M X )( ¶ N X ) , (5)with g = det ( g MN ) . Scalar glueballs are identified as the normalizable modes of X satisfying the equations of motion obtained from (5), corresponding to a finite action.For the spin 1 glueball, we introduce a 1-form A M described by the action: S = − Z d x √− g e − f ( z ) (cid:20) g MN g ST F MS F NT + m g ST A S A T (cid:21) , (6)with F MS = ¶ M A S − ¶ S A M and m =
24, and study its normalizable modes. Notice thatthe action (6), with a different value of m , describes a priori fields that are dual to otheroperators in QCD, namely those describing hybrid mesons with spin one, which is anexplicit example of different QCD operators having similar bulk fields as holographiccorrespondents. SCALAR AND VECTOR GLUEBALL SPECTROSCOPY
The field equations of motion obtained from the actions (5)-(6) can be reduced in theform of a one dimensional Schrödinger equation in the variable z : − y ′′ + V ( z ) y = − q y , (7)involving the function y ( z ) obtained applying a Bogoliubov transformation y ( z ) = e − B ( z ) / ˜ Q ( q , z ) to the Fourier transform ˜ Q of the field Q ( Q = X , A M ) with respect tothe boundary variables x m . The function B ( z ) is a combination of the dilaton and theetric function: B ( z ) = f ( z ) − a A ( z ) , with the parameter a given by: a = a = X and A M fields, respectively. The condition q = − m identifies the mass ofthe normalizable modes of the two fields.Eq. (7) is a one dimensional Schrödinger equation where V ( z ) plays the role of apotential. It reads as: V ( z ) = (cid:0) B ′ ( z ) (cid:1) − B ′′ ( z ) + m z = V ( z ) + m z , (8)with V ( z ) = c z + a + a z + c ( a − ) . (9)With this potential, eq. (7) can be analytically solved. Regular solutions at z → z → ¥ correspond to the spectrum: m n = c h n + + a + q ( a + ) + m i , (10)with n an integer (we identify it as a radial quantum number), while the correspondingeigenfunctions read as: y n ( z ) = A n e − c z / ( cz ) g ( a , m )+ / F (cid:0) − n , g ( a , m ) + , c z (cid:1) , (11)with F the Kummer confluent hypergeometric function, A n a normalization factor and g ( a , m ) = q ( a + ) + m . From these relations, we obtain the spectrum of scalar andvector glueballs, respectively: m n = c ( n + ) , (12) m n = c ( n + ) . (13)A few remarks are in order in respect to the results derived in [9]. First, both thespectra have the same dependence on the radial quantum number n as the mesons ofspin S : m n = c ( n + S ) . This is a consequence of the large z behaviour chosen forthe background dilaton. Second, both the lowest lying gueballs are heavier than the r mesons, the spectrum of which reads: m n = c ( n + ) . Finally, the vector glueball turnsout to be heavier than the scalar one.More precisely, comparing our result to the computed r mass, we obtain for thelightest scalar ( G ) and vector ( G ) glueballs m G m r = m G m r = , (14)which implies that these glueballs are expected to be lighter than as predicted by otherQCD approaches [6]. The result m G − m G = m r predicts indeed a lightest vectorglueball with mass below 2 GeV.It is interesting to investigate how it is possible to modify the z dependence of thebackground dilaton field and of the metric function A , and how the spectra change, anissue discussed in the following section. ERTURBED BACKGROUND
There are other choices for the background dilaton f and the metric function A whichallow us to reproduce the Regge behaviour of the low-lying mesons and to recover the AdS metric close to the UV brane when z →
0. As a matter of fact, it is possible to addto the background fields terms of the type z a with 0 ≤ a <
2. Considering the simplestcase: a =
1, this can be done in two different ways. Either we modify the dilaton fieldincluding a linear contribution which is subleading in the IR regime ( z → ¥ ) or wemodify the metric function which now acquires a linear term subleading in the UVregime ( z → ) : (cid:26) f ( z ) = c z + l czA ( z ) = − ln z , (cid:26) f ( z ) = c z A ( z ) = − ln z − l cz , (15)with l a real dimensionless parameter. The two choices produce different results. Mod-ifying the dilaton field, the potential (7) becomes: V ( z ) = V ( z ) + l V ( z ) + l c + m z f ( z , l ) with (cid:26) V ( z ) = c ( c z + a z ) f ( z , l ) = , (16)while modifying the metric in the IR, the potential term reads as: V ( z ) = V ( z ) + l ˜ V ( z ) + l c a + m z f ( z , l ) with (cid:26) ˜ V ( z ) = a c (cid:0) c z + a z (cid:1) f ( z , l ) = e − l cz , (17)with V ( z ) given in (9). Considering (15)-(17), one sees that the mass term is the mainresponsible of the difference between the scalar and vector cases when the geometryis perturbed, while its effect turns out to be the same when the background dilaton ismodified. Eq. (7) with the new potentials (16) and (17) can be solved perturbatively and,for small values of the parameter l , the spectra are modified: m n = m n , ( ) + l m n , ( ) . (18)The detailed analysis can be found in [11]. Different predictions at O ( l ) for the lowest-lying vector and scalar glueball mass difference are then obtained, modifying either thedilaton or the geometry: m G − m G = c (cid:0) − √ p l (cid:1) (modifying the dilaton) , (19) m G − m G = c (cid:0) − √ p l (cid:1) (modifying the metric function) . (20)Therefore, the mass splitting between vector and scalar glueballs increases if l isnegative, and the maximum effect is produced, for the same value of l , when the metricfunction is perturbed. This can be considered as an indication on the type of constraintsthe background fields in the bulk must satisfy. ONCLUSIONS
We have discussed how the QCD holographic model proposed in [9], with the hard IRwall replaced by a background dilaton field, allows one to predict the light glueballspectra [11]. Vector glueballs turn out to be heavier than the scalar ones, and thedependence of their masses on the radial quantum number is the same as obtained for r and higher spin mesons. Combining the calculations of the glueball and r masses inthe same holographic model, the glueballs turn out to be lighter than predicted in otherapproaches [6].We have also investigated how the masses change as a consequence of perturbingthe dilaton in the UV or the bulk geometry in the IR, finding that constraints in thebottom-up approach can be found if information on the spectra from other approachesis considered. Such constraints should be taken into account in the attempt to constructthe QCD gravitational dual. ACKNOWLEDGMENTS
I am very indebted to my collaborators at the INFN sezione di Bari who make my stayso pleasant and fruitful. I am also grateful to the organizers who gave me the opportunityto present my research activities during this workshop. This work was supported in partby the EU Contract No. MRTN-CT-2006-035482, "FLAVIAnet".
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