Low-Energy Theorems from Holography
Johanna Erdmenger, A. Gorsky, P.N. Kopnin, A. Krikun, A.V. Zayakin
aa r X i v : . [ h e p - t h ] J a n ITEP-TH-32/10MPP-2010-167
Low-Energy Theorems from Holography
Johanna Erdmenger † , Alexander Gorsky ∗ , Petr N. Kopnin ∗‡ , Alexander Krikun ∗‡ ,and Andrew V. Zayakin ♭ ∗ ♮ ∗ Institute of Theoretical and Experimental Physics,B. Cheremushkinskaya ul. 25, 117259 Moscow, Russia † Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut),F¨ohringer Ring 6, D-80805 M¨unchen, Germany ‡ Moscow Institute of Physics and Technology,Institutsky per. 9, 141 700 Dolgoprudny, Russia ♭ Fakult¨at f¨ur Physik der Ludwig-Maximilians-Universit¨at M¨unchen undMaier-Leibniz-Laboratory, Am Coulombwall 1, 85748 Garching, Germany ♮ Dipartimento di Fisica, Universit`a di Perugia, I.N.F.N. Sezione di Perugia,Via Pascoli, I-06123 Perugia, Italy
Abstract
In the context of gauge/gravity duality, we verify two types of gauge theory low-energytheorems, the dilation Ward identities and the decoupling of heavy flavor. First, weprovide an analytic proof of non-trivial dilation Ward identities for a theory holo-graphically dual to a background with gluon condensate (the self-dual Liu–Tseytlinbackground). In this way an important class of low-energy theorems for correlatorsof different operators with the trace of the energy-momentum tensor is established,which so far has been studied in field theory only. Another low-energy relationship,the so-called decoupling theorem, is numerically shown to hold universally in threeholographic models involving both the quark and the gluon condensate. We showthis by comparing the ratio of the quark and gluon condensates in three differentexamples of gravity backgrounds with non-trivial dilaton flow. As a by-product ofour study, we also obtain gauge field condensate contributions to meson transportcoefficients. email addresses: [email protected], [email protected], [email protected], [email protected],[email protected] ontents A.1 Self-Dual Background at Zero Temperature . . . . . . . . . . . . . . . 23A.2 Self-Dual Background at Finite Temperature . . . . . . . . . . . . . . 24
B Equations of Motion 27
On the long road towards a holographic description of QCD, there are some milestonescorresponding to exact relations which have to be satisfied also in any holographicmodel. These are so called low-energy theorems [1] (see e.g. [2] for review). In fieldtheory these are statements which impose restrictions on the various correlators. Thepurpose of this work is to compare holography to field theory by considering the low-energy theorems concerning one- and two-point functions of a strongly coupled gaugetheory on both sides of the correspondence. We report nice non-trivial agreement intwo important cases: the dilation Ward identities and the decoupling theorem for theheavy flavor. Recently the validity of a related class of theorems (QCD sum rules)was shown holographically in [3] at finite temperature. Apart from demonstrating1he validity of low-energy theorems, a particular result of our analysis is a statementon the IR universality of theories dual to three scale-dependent backgrounds withnon-trivial dilaton flow.First, we aim at realizing the QCD low-energy theorems explicitly, for instance Z d x h T ( x ) O (0) i = − dim( O ) h O i , (1)where T = T µµ is energy-momentum trace on the boundary. This is trivially satisfiedin the conformal case: The right-hand side is expected to be zero in a conformal fieldtheory where all condensates vanish. For an explicit expression for the correlatorsof energy-momentum components see e.g. [4]. Thus for a nontrivial test we need abackground which is different from AdS in the IR, dual to a non-conformal field theory,for instance with a gluon condensate. There are a number of models which generalizethe original AdS/CFT correspondence to the backgrounds corresponding to non-vacuum states of N = 4 SYM or to non-conformal and non-supersymmetric theories.We use the self-dual background by Liu and Tseytlin [5] with non-zero expectationvalue of the gluon operator h tr G i in this part of our work. To perform the testof dilation Ward identities, we calculate the two-point correlators h tr G ( x )tr G (0) i , h tr G ( x )tr G ˜ G (0) i , h T ( x )tr G (0) i , h T ( x )tr G ˜ G (0) i , h T µν ( x ) T αβ (0) i in this background.The analysis of correlators is easily performed for non-zero frequency. In this waywe reproduce the results for transport coefficients, extending the analysis to the caseof the non-conformal backgrounds considered. First of all, we calculate the η/s ratioof shear viscosity over entropy via h T xy T xy i , which was performed for the conformalcase in [6, 7, 8, 9]. Here we find using suitable holographic renormalization thatcondensate corrections to ηs | T → = π are absent in the Liu-Tseytlin background, i.e.the nonzero VEV of the gluon field strength h tr G i does not affect the value of η/s .Secondly, we check the relationship between two-point and one point functionsin gauge theory with fundamental fermions, known as decoupling relation h α s π tr G i = − m h qq i . (2)Fundamental fermions are introduced in our system via probe D D SU ( N f )flavor symmetry in the Maldacena limit. The length of the strings corresponds to thequark mass, and the subleading term in the asymptotics of the embedding coordinates2o the condensate. A non-trivial test of the theorem considered is possible only foran IR-non-trivial metric. For that purpose we use three different dilaton flow back-grounds with gluon condensate: the self-dual Liu-Tseytlin background mentionedabove [5], the Gubser–Kehagias–Sfetsos background [11], [12] and the Constable–Myers background [13]. All of these are examples for non-trivial dilaton flows. Aremarkable universality among the three models and agreement with standard fieldtheory is observed.Let us now present the two low-energy theorems discussed in this paper. Dilation Ward Identity.
It was argued in [1] that the following dilation Wardidentity holds within field theory,lim q → i Z e iqx d x (cid:28) T (cid:26) O ( x ) , β ( α s )4 α s tr G (0) (cid:27)(cid:29) = ( − d ) hOi [1 + mass-dependent terms] , (3)where d is the canonical dimension of the operator O , T {· , · } stands for the timeordered product and the one-loop beta-function is normalized as β ( α s ) = − bα s π , b = N c − N f . Identities for higher correlators are also available: i Z d xd y (cid:28) T (cid:26) O ( x ) , β ( α s )4 α s tr G ( y ) , β ( α s )4 α s tr G (0) (cid:27)(cid:29) = ( − d ) hOi [1 + mass-dep.] . (4)For the gluon field strength operators we obtain: i Z (cid:28) T (cid:26) α s π tr G ( x ) , α s π tr G (0) (cid:27)(cid:29) = 18 b D α s π tr G E . (5) Decoupling Theorem.
Novikov, Shifman, Vainshtein and Zakharov derived in[1] the following equation for light quarks by considering the regularity of the betafunction ddm q D α s π tr G E = − b h qq i . (6)This low-energy theorem for heavy quarks is recovered also in an independent mannerin [14]. Besides, for heavy quarks the following relation due to Shifman, Vainshteinand Zakharov holds m h qq i = − D α s π tr G E . (7)The derivation of this equation is found in [15]. It expresses the continuity of theenergy-momentum trace at the flavor number thresholds of the beta-function. The3actors 12 and 24 in the equations above are universal, they do not contain N c or N f . In this paper, we shown that relation (7) holds holographically in the threedilaton-flow backgrounds to great accuracy.A related calculation, the holographic derivation of the Veneziano-Witten for-mula relating the mass of the η ′ meson and the topological susceptibility of pureYang-Mills theory, was performed in [10]. The holographic conformal anomaly waspreviously considered under finite temperature in the 5-dimensional model with adilaton potential adjusted in such way that both confinement and the correct UVbehaviour of the coupling are reproduced [16].This paper is organized as follows. In Section 2 we describe technicalities relatedto finding correlators. In Section 3 we describe the holographic description of thedilation Ward identities. Section 4 contains our main result – the derivation of thenonperurbative decoupling of the heavy flavor in the different dilaton flow models. Inthe last section we discuss the importance of having established the decoupling andscaling theorems holographically. Several necessary facts concerning the models arecollected in Appendix A, while Appendix B concerns the derivation of the transportproperties of the models under consideration. For later use, let us briefly review the AdS/CFT prescription for calculating two-point functions, emphasizing in particular the derivation of the gauge-fixing and theGibbons-Hawking term. In the analysis of the boundary term we follow here veryclosely the analysis of [4]. A reader familiar with these technicalities can proceeddirectly to the next section.We consider the general rules for two-point functions and calculate the matrixof correlators M ij = h O i O j i| ( p ) = δ S full δ ¯Φ i ( p ) δ ¯Φ j ( − p ) . (8)The standard wisdom on finding Green function of the fields present is to set theaction of the type S bulk = Z d xdzφ ′ g zz √ g (9)4ut onto the boundary as S boundary = Z d xφφ ′ g zz √ g | z → . (10)The correlator in terms of bulk-to-boundary Green functions G ( x, z ) of the field φ isgiven by h O ( x ) O (0) i = G ( x, z ) ∂ z G (0 , z ) | z =0 . (11)In our case two additional difficulties arise. First, the correct boundary term shouldbe supplemented by the Gibbons–Hawking term [4], which makes a theory definedon manifold with boundary globally diffeomorphism-invariant. Second, the bilinearaction of fields’ fluctuations is non-diagonal, this means that we shall be dealing witha matrix of Green functions rather than with separately-treatable ones.Let us define Green function matrix. Namely, if field Φ i has a bulk solution Φ i ( z ),satisfying z δ i Φ i ( z ) | z → = ¯Φ i , then by definition K ij ( z ) = δ Φ j ( z ) δ ¯Φ i . (12)Let us establish the correct boundary term. The full action of our bulk theory isactually [4] S full = S d + S div + S d (13)where the Gibbons–Hawking term S d = − ∂ z Z d x √− g − c Z d x √− g , (14)is here given by g = det( g ij ) , i = 0 , , , . (15)The constant c can be fixed arbitrarily to our convenience, e.g. as in eq. (4.15) in [4].The other piece which one has to take into account is the full divergence term S div ,which does not affect equations of motion, but does change the appearance of theaction and makes it diagonal in terms of physical degrees of freedom of the graviton.It is the well-known fluctuation term S div = 32 ∂ µ W µ , (16)the vector W µ is (see [17], Vol.II, § W µ = √− g (cid:16) g αβ δ Γ µαβ − g αµ δ Γ βαβ (cid:17) , (17)5here δ Γ µαβ = Γ µαβ ( g + h ) − Γ µαβ ( g ). This constitutes the gauge-fixing prescription forour problem.Consider now the second variation of these actions in fluctuation fields; denote thesesecond-order expressions as S (2)10 d , S (2) div , S (2)4 d respectively; they contain both fields andtheir derivatives. The two-point correlator is then h O i O j i = K ik ∂ L ∂ Φ ′ k ∂ Φ ′ m ∂ z K jm + K ik ∂ S (2)4 d ∂ Φ k ∂ Φ ′ m ∂ z K jm + K ik ∂ S (2)4 d ∂ Φ k ∂ Φ m K jm , (18)here L is Lagrangian density of the bulk action: S bulk = S (2)10 d + S (2) div = Z dz L . (19)The above structure is obvious from the following reasons. Consider the bulk action δ S bulk = δ Φ m ( z ) δ ¯Φ j δ S bulk δ Φ m δ Φ k δ Φ k ( z ) δ ¯Φ i , (20)where δ S bulk δ Φ m δ Φ k = Z dz (cid:20) ∂ L∂ Φ ′ m ∂ Φ ′ k ∂ z δ Φ m ∂ z δ Φ k + ∂ L∂ Φ m ∂ Φ ′ k δ Φ m ∂ z δ Φ k + ∂ L∂ Φ m ∂ Φ k δ Φ m δ Φ k (cid:21) . (21)Taking into account that Green functions of field fluctuations by definition satisfyequations: (cid:20) − ∂ z ∂ L∂ Φ ′ m ∂ Φ ′ k ∂ z + ∂ L∂ Φ m ∂ Φ ′ k ∂ z + ∂ L∂ Φ m ∂ Φ k (cid:21) δ Φ k ( z ) = 0 , (22)one sees that the only contribution of S bulk into the correlator will be, after taking offthe derivative and integration, the term: δ S bulk = δ Φ m ( z ) ∂ L∂ Φ ′ m ∂ Φ ′ k ∂ z δ Φ k ( z ) . (23)Now remembering the definition of Green function matrix K mj = δ Φ m ( z ) δ ¯Φ j , (24)6e arrive exactly at (18). Then there is the purely boundary term (Hawking-Gibbonsterm). It does not require the above procedure, since it already sits on 4d. Then itcontributes the following: δ S d = ∂ S d ∂ Φ ′ m ∂ Φ k ∂ z δ Φ m δ Φ k + ∂ S d ∂ Φ m ∂ Φ k δ Φ m δ Φ k . (25)The action S d contains no more than one derivative term, which is due to normaldifferentiating of extrinsic curvature, thus ∂ L∂ Φ ′ = 0. This contributes the other twoterms into the correlator (18). In this Section we calculate the matrix of the two-point correlators for the gluonicoperators and components of the energy-momentum tensor. Then we compare theseto one-point correlators and find that the correct scaling relations from field theoryare satisfied on the gravity side. We begin by introducing the Liu–Tseytlin model inwhich we will perform our calculations in this section.
Liu–Tseytlin model.
In the Einstein frame the bulk action of
IIB superstringtheory is [5] S = 1 g s (2 π ) α ′ Z d x √ g (cid:18) R −
12 ( ∂ µ φ ) − e φ ( ∂ µ C ) − | F | (cid:19) , (26)where R is the curvature, φ is dilaton, F is 5-form and C is axion.The Liu–Tseytlin model is a generalized background for holography those whichpossesses self-duality. It describes a field-theory flow from a strongly-coupled con-formal theory in the UV to a theory with condensate tr G in the IR. By virtue ofself-duality it is still supersymmetric. However, it possesses a scale parameter, whichmakes it closer to real-world physics. The self-duality is provided by the presenceof a non-trivial axion field. Despite the presence of the scale, it is conformal in theUV; in the IR the dilaton singularity is determined by the gluon condensate tr G .Within supergravity this background is understood as “smeared” D ( −
1) brane witha usual stack of D D ( −
1) brane is an instanton in 10D, the resulting4d theory can be considered as having an instanton-gas type of vacuum, which is ad-vantageous for QCD purposes. Moreover, this background is confining (in the sense7f Wilson loop linear behavior at large temporal separation), and the string tensionis proportional to the condensate. Of course, we do not claim to produce any realQCD results in this framework, but we believe it to be a very useful toy model.For the Liu–Tseytlin background [5] metric in Einstein frame looks like thestandard conformal solution ds = g µν dx µ dx ν = R (cid:18) dx µ √ h + p h dz + z d Ω z (cid:19) , (27)but the dilaton is modified by the smeared instanton (nonzero density of D ( − e φ = h − , (28)and an axion is present C = 1 h − −
1; (29)the D D ( −
1) form-factors are: h = z , (30)and h − = 1 + qz . (31)The parameter q is the crucial quantity for us, since it measures the degree of IR-non-conformality of the theory (remember that in the UV, the theory is conformaland its β -function is zero).The Tseytlin-Liu background has been successfully used for a number of appli-cations, e.g. calculating meson spectra [18, 19, 20, 21]. In all these applications, itsrelevance to QCD has been demonstrated. In [22] a finite-temperature extension ofthe [5] solution has been found, which has been a further motivation to apply it torealistic high-energy quark-gluon plasmas. We shall employ Liu-Tseytlin backgroundto test dilation Ward identities in Section 1 and decoupling relation in Section 4. Holographic normalization of the operators
Here we consider normalization of the gluon field strength operator; the normal-ization of the quark operators will be considered in the next Section. According tothe AdS/CFT dictionary we state that the fluctuation δφ ( z, Q ) of dilaton field φ ( z, Q ) = φ ( z ) + δφ ( z, Q ) (32)8s dual to the operator O φ , proportional to the QCD scalar gluonic operatortr( G ) ≡ c φ O φ . (33)We can fix the normalization constant c φ by comparing the two-point functions h O φ O φ i = c φ h tr( G )tr( G ) i . (34)At large momenta the leading behavior of gluonic correlator in QCD is [23]: h tr( G )( Q )tr( G )( Q ) i = N c − π Q ln( Q ǫ ) . (35)To obtain a two-point function from holography we take the second variation of theaction computed on a classical solution. In the vicinity of the boundary of AdS theaction (26) for the fluctuation is: S = π R g s (2 π ) α ′ Z d xdz z (cid:2) − ( ∂ z δφ ) − ∂ µ δφ∂ µ δφ + 2 e φ δφ ( ∂ z C ) (cid:3) . (36)Here we have taken the near boundary limit r ≫ L (so that r ≃ ρ ) and changedcoordinates z = R r . π is the volume of the S sphere, R came from the determinantof the metric ( √ g = R z ). The last term containing the profile of axion field isnegligible at the boundary (small z) because ∂ z C ( z ) ∼ z . We can find the bulk-to-boundary propagator of φ ( z, Q ) at small z and large Q . It is ϕ ( z, Q ) = Q z K ( Qz ) , ϕ (0 , Q ) = 1 , (37)where K i is McDonald function of the second kind. Now we can compute the secondvariation of the action. It is h O φ O φ i = δ S cl δφ δφ = π R g s (2 π ) α ′ ϕ ( z, Q ) ∂ z ϕ ( z, Q ) z (cid:12)(cid:12)(cid:12)(cid:12) z = ǫ = N c π ) Q ln( Q ǫ ) , (38)where we used the definition R = 4 πg s α ′ N c and the asymptotic of McDonald func-tion. Comparing this result with the expression of QCD we find O φ = 14 √ G ) . (39)9o establish a relation between gluon condensate and the expansion coefficient ofthe dilaton field we compute the vacuum expectation value of O φ at zero momentumtaking the first variation of the action with respect to the boundary value of the field φ . At zero momentum near the boundary the dilaton field behaves as φ ( z ) = φ + φ z . (40)For the dual operator given by (39) we find h O φ i = δS cl δφ = π R g s (2 π ) α ′ ϕ ( z, Q ) ∂ z φ ( z, Q ) z (cid:12)(cid:12)(cid:12)(cid:12) z = ǫ = N c π ) φ . (41)From (39) and (40) we get the expression for the gluon condensate h tr( G ) i ≡ √ O φ = N c √ π ) φ . (42)In the Liu-Tseytlin model the infinitesimal fluctuations of the fields on the bulkcouple to the operators tr G , tr G ˜ G , T µν in the boundary N = 4 SYM theory. More-over, in the Liu-Tseytlin model the dilaton field behaves as e φ = 1 + qz , so theparameter of solution φ in (42) equals q and the scalar and pseudoscalar gluon con-densates are nontrivial and equal to the value given in (42), i.e. h tr G i = h tr G ˜ G i = N c √ π ) q. (43) Correlators at Zero Frequency
Fluctuation terms are defined as φ = φ c + ϕ,C = C + ξ,g = g µν + h µν . (44)We consider the following interaction term to provide a correspondence with theboundary theory: S int = Z d x (cid:20) T µν ¯ h µν − e − φ c (cid:16) ¯ ϕ tr G √ + ¯ ξ tr G ˜ G √ (cid:17)(cid:21) , (45)which, after introduction of useful self-dual and anti-self-dual components G ± = G ± ˜ G η ± = ϕ ± ξ, (47)becomes S int = Z d x (cid:20) T µν ¯ h µν − e − φ c √ (cid:0) ¯ η + tr G +2 + ¯ η − tr G − (cid:1)(cid:21) . (48)Here bars denote four-dimensional sources, which are proportional to boundary valuesof five-dimensional fields:¯ h µν = z h µν | z =0 , ¯ η ± = η ± | z =0 , ¯ ϕ = ϕ | z =0 . (49)Fluctuations of F are fully determined by h µµ , thus there is no independent sourcefor them.Let us choose the gauge h µ = 0, k µ h µν = 0, u µ h µν = 0, where wave-vector k = ( ω, , , k ), constant vector u is u = (1 , , , i = ( η + , ¯ h + ¯ h , ¯ h − ¯ h , ¯ h , η − ) , (50) i = 1 , . . .
5, each coupled to the corresponding O i operator O i = (cid:18) tr G +2 √ , T µµ , T − T − T − T , T xy , tr G − √ (cid:19) , (51)with G + and G − the self-dual and anti-self-dual parts of G , respectively, via S int = Z d xdz X i =1 O i Φ i . (52)The relevant part of the fluctuation action in the bulk is S (2) , double deriv. d + div = Z d xdz (cid:18) z Φ ′ Φ ′ + z ′ + z ′ + z ′ (cid:19) . (53)One should not be mislead by its diagonal structure; besides the diagonal terms withdouble derivatives, the full bilinear action contains terms which make it non-diagonal.The boundary Gibbons-Hawking action term is S (2) , derivatives4 d = Z d x (cid:0) c h xy ( z ) + 16 zh ′ xy ( z ) h xy ( z ) + Φ ( z ) ( c Φ ( z ) + 4 z Φ ′ ( z )) (cid:1) . (54) Some of these operators, e.g. the O are not of immediate interest; however, it costs no additionaleffort to incorporate them into the calculation, so we work the correlators out for them as well. F form we always have δF = − /r Φ , which solves automatically the equationsof motion for this field and at the same time retains the constancy of the Ramond-Ramond flow R S F = N c .It is instructive to start with zero-frequency correlators (setting ω = 0 in (119)in Appendix B). Subsequently, we introduce finite frequencies ω . In this case we findoscillatory solutions (Bessel functions) (121) instead of the rational ones (120). Thelimit ω → ω = 0 and thus provides an additional check of the validity for our procedure.The solutions (121) contain ten modes labelled by coefficients C i , i = 1 . . . C , C , C , C , C , C ), and the remainingfour are infinite. An extra constraint is therefore necessary to make the Green func-tion matrix (12) a well-defined 5 × C = C /
2, which removesexactly one redundant degree of freedom.The Green function matrix is then (recall that c φ = √ ): K ij = c φ tr G +2 18 T µµ O O c φ tr G − c φ tr G +2 qz − qǫ + 1 0 0 0 0 T µµ z O z O z c φ tr G − − q ( ǫ − z ) 0 0 0 qz − qǫ + 1 (55)with the O i as given by (51). q is the non-conformality parameter defined in (31).As a result, combining our knowledge of Green function matrix (55), the bound-ary action (54) and the derivative piece of the bulk action (53) we obtain the matrix: M = c φ tr G +2 18 T µµ O O c φ tr G − c φ tr G +2 − q − q − q T µµ − q − ǫ O − ǫ O − ǫ c φ tr G − − q , (56)12hich contains information on the correlators of O i , O j via the following relation hO i O j i = N c π M ij . (57)Some comments are due here. The singular terms ǫ are expected due to thedivergencies on the field theory side; they are subtracted by a holographic renormal-ization procedure, analogously to field-theoretical subtraction. The asymmetry in O ↔ O is also expected: what we consider is a self-dual configuration, therefore,the self-dual and the anti-self-dual operators have different properties.Using the matrix elements obtained above, we can now establish the low-energytheorems. After normalization according to (39) we have Z d x (cid:10) tr G +2 ( x ) T (0) (cid:11) = 4 (cid:10) tr G +2 (0) (cid:11) , Z d x (cid:10) tr G − ( x ) T (0) (cid:11) = 0 , Z d x (cid:10) tr G ( x )tr G (0) (cid:11) = 12 (cid:10) tr G (cid:11) , Z d x D tr G ˜ G ( x )tr G ˜ G (0) E = 0 . (58)where T = T νν . Here we see that the first and the second lines of the equationsabove (58) constitute exactly the statement of the low-energy theorems h ˆ OT i = dim( O ) h ˆ O i . (59)Note that h tr ( G − ) i = 0.The third line of (58) must be compared to the field-theoretical result Z h tr G tr G i ∼ /β h tr G i , (60)where in standard perturbation theory, β is the one-loop coefficient of the beta-function. This equation reflects a breaking of the conformal symmetry. For theLiu–Tseytlin model the standard beta function vanishes. Nevertheless, the massiveparameter q generates additional terms in the effective action. This gives rise to thecontribution T µµ ∼ tr G − to the trace of the energy-momentum tensor at the operator13evel. This is consistent with the low-energy theorem given by the third line of (58).On the other hand, h tr G − i = 0, thus the expectation value of the energy-momentumtensor and the vacuum energy vanish, ensuring consistency with supersymmetry.The fourth relation in (58) implies that the topological susceptibility of thevacuum, which is proportional to this correlator [24], vanishes in the Liu–Tseytlinmodel, which is in the agreement with the fact that the model is supersymmetric . Correlators at Finite Frequency
Now let us analyze the finite-frequency solu-tions. The solutions are given in Appendix, eq. (121); only relevant modes shown.Unlike the ω = 0 solutions, which were exact solutions, here Φ ( z ) and Φ ( z ) arepowerlog expansions in ω and r . Since we are interested in the near-UV behaviourof Green functions, and eventually expand correlator matrix in powers of ω , thisapproximation is reasonable. The matrix of correlators becomes: M = c φ tr G +2 18 T µµ O O c φ tr G − c φ tr G +2 − q − q log( ωe ) ω − q T µµ − q − log( ωe ) ω O − log( ωe ) ω O − log( ωe ) ω c φ tr G − ωe ) ω − q h T xy T xy i element. It is proportional to ηs | T =0 , and here we observe its independence of q . This fact is not trivial from dimensional considerations, since we possess anotherdimensionful parameter, the frequency ω . Thus we have established ηs ( q, ω ) (cid:12)(cid:12)(cid:12) T =0 = 14 π . (62)As a bonus of this calculation, in the Appendix A we easily elaborate the matrix ofquarkonium transport coefficient based on the above correlator matrix. Note that in the D4/D6 model [10] the topological susceptibility does not vanish. However thereis no contradiction between these facts, since the model of [10] breaks supersymmetry (similarly toSakai-Sugimoto model), whereas Liu-Tseytlin model retains supersymmetry. Holographic Decoupling of the Heavy Flavor
In this Section we holographically derive the central result of this paper, which isknown as “decoupling relation”. In can be found in [15]: α s π (cid:10) G aµν G aµν (cid:11) = − m q h qq i . (63)The derivation of this relation is somewhat intuitive, but let us still restate the argu-ments by Shifman, Vainshtein and Zakharov. For vacuum expectation values of thedifferent operators pertinent to light quarks the parameter of expansion is quark mass.For heavy quarks we expand in the inverse quark mass and set external momentumto Q ∼
0. Let us suppose there exists a quark for which both expansions, small andlarge m are true. As it is in particular a “heavy” quark, the quark condensate canbe done perturbatively from the triangle diagram with gluons as “vacuum sources”,shown in Fig. (1). G mn a G mn a m q Figure 1: Vacuum diagram with heavy quarks depicting h qq i as gluon-driven quantity.One can understand the argument from which the relation (7) emerges as follows.Consider the trace of energy-momentum tensor of a gauge theory. For low quarkmass there is beta-function contribution from the quark, for heavy quark there is onlythe gluonic contribution to the beta-function, yet there is quark chiral condensate ispresent: θ µµ = (cid:18) N c − (cid:19) α s π tr G , above threshold , (cid:18) N c (cid:19) α s π tr G + mqq, below threshold . (64)15hen the two are equated at some intermediate scale, the necessary relation (7) ap-pears. Equating small and large m domains happens on the ground that we select thescale at which the heavy quarks “decouple” from the one-loop polarization operator.Hence this theorem is also known as decoupling relation. A picture of condensate asfunction of quark mass is given in [1]. We now establish relation (63) holographically by considering different backgrounds,those of Constable and Myers [13], of Gubser [11] and of Liu and Tseytlin. The Liuand Tseytlin background (27) was already discussed above in the Section 1. TheConstable—Myers background in the Einstein frame has the metric ds = (cid:16) b + r r − b (cid:17) b √ h dx µ + ( r − b ) (cid:16) b + r r − b (cid:17) ( − b ) q(cid:0) b + r r − b (cid:1) b − r (cid:0) dr + r d Ω (cid:1) , (65)where h = (cid:18) b + r r − b (cid:19) b − , (66)and the dilaton is e φ = (cid:18) b + r r − b (cid:19) q − b , (67)axion is zero, and F = ǫ h , where ǫ is the unitary antisymmetric tensor in the S directions.The chiral condensate and meson spectrum involving a Goldstone boson wereobtained in [25] by embedding a D7 brane probe into a Constable–Myers background.Masses of heavy-light mesons in this background in D7 model were obtained in [26].The quark condensate, pion decay constant and the higher order Gasser- Leutwylercoefficients were calculated for D7 model in this background in [27]. D7 embeddingswere argued to be stable in this background [28, 29].One of the first non-conformal backgrounds introduced into AdS/CFT was con-sidered by Gubser [11]: ds = r − b r r dx µ + 1 r (cid:0) dr + r d Ω (cid:1) , (68)16ilaton in this background is e φ = r b + 1 r b − ! √ , (69)and the axion is zero. Originally it was intended to model confinement, yet itbecame also useful for introducing the gluon condensate. Shortly before Gubser,this background was also obtained by Kehagias and Sfetsos [12] in a less convenientparametrization. Introduction of fundamental fields.
We are modelling the fundamental fermionicdegrees of freedom by embedding the D7 brane into one of the three backgrounds de-scribed above. The Dirac–Born–Infeld action for the D S D = 1 g s (2 π ) α ′ Z d ξ e φ q det αβ ( ∂ α X µ ∂ β X ν g µν ) . (70)The embedding of D AdS × S D − − . (71)One can get an image of the corresponding physics in Fig. (4.2), where string modesgenerating specific sectors of the spectrum are shown. D7 D3 N Figure 2: Scheme of the D3-D7 geometry and the corresponding string/field modes.We look for embeddings of the form X = w ( ρ ) , X = 0 , (72)17here embedding function w , worldsheet coordinates ξ i and target space coordinates r, ρ are related as follows w ( ρ ) = r − ρ ,ρ = p ξ + ξ + ξ + ξ . (73) N f quark flavours can be considered introducing N f corresponding D w i , i = 1 . . . N f . If the quark masses are equal, D N f . In the following werestrict ourselves to the case of just one flavour for simplicity, considering only oneembedding coordinate w ( ρ ). Using these definitions we easily construct the equationsof motion for w ( ρ ),2 ρg ( r ) w ′ ( ρ ) ( w ′ ( ρ ) + 1) g ′ ( r ) − w ( ρ ) ( w ′ ( ρ ) + 1) ( g ( r ) g ′ ( r ) + g ( r ) g ′ ( r )) ++ g ( r ) (2 ρg ′ ( r ) w ′ ( ρ ) + 2 ρg ′ ( r ) w ′ ( ρ ) + rg ( r ) w ′′ ( ρ )) = 0 , (74)where the corresponding g ii should be taken for each respective metric. We solve themnumerically at different values of the vacuum parameters and fields, corresponding tothe boundary conditions at ρ → ∞ ; a typical embedding is shown in Fig. (3). (a) Constable-Myers (b)
Gubser-Kehagias-Sfetsos (c)
Liu-Tseytlin
Figure 3: Typical embeddings of D Following the same steps as in Section 3 we explore the scalar field w dual to theoperator ¯ qq , where q is the quark field. It is described by the action of the D7 brane(70), for which w i is embedding coordinate. Here and after we are dealing only withflavour i and will omit this index where it is possible. The action for the fluctuations18f w is S = − π R g s (2 π ) α ′ Z d xdze φ (cid:20) z ( ∂ z w ) + 12 z ∂ µ w∂ µ w (cid:21) . (75)Here we change coordinates the same way as in (36), 2 π is a volume of 3-sphere R comes again from the determinant of the metric p g (8) = R z . In the limit of largemomenta near the boundary the bulk-to-boundary propagator is˜ w ( z, Q ) = Qz K ( Qz ) , ˜ w (0 , Q ) = 1 . (76)The scalar field is dual to the operator O w , which is proportional to ¯ qq = c w O w . Wecompute two-point function of O w to fix the normalization h O w O w i = δ S cl δw δw = 2 π R g s (2 π ) α ′ e φ
12 ˜ w ( z, Q ) ∂ z ˜ w ( z, Q ) z | z = ǫ = N c π ) α ′ Q ln( Q ǫ ) | z = ǫ . (77)Here the fact is used that e φ | boundary = 1 [5] and again R = 4 πg s α ′ N c . We comparethis result with the QCD calculation (see eq. 4.27 in [15]), h ¯ qq ¯ qq i = N c π Q ln ( Q ǫ ) , (78)and find O w = 12 πα ′ ¯ qq. (79)At this stage we can identify the boundary value of the field w = w | z =0 . It is thesource of O w = c w (¯ qq ), so it is proportional to the quark mass w = c w M . Thus wehave M = 12 πα ′ w . (80)To identify quark condensate h ¯ qq i we compute the expectation value of O w at Q = 0.In this limit near the boundary the supergravity field takes the asymptotic form w ( z ) = w + w z . (81)The result is h O w i = δS cl δw = 2 π R g s (2 π ) α ′ e φ
12 ˜ w ( z, Q ) ∂ z w ( z, Q ) z (cid:12)(cid:12)(cid:12)(cid:12) z = ǫ = N c π ) α ′ w . (82)19he quark condensate is normalized as follows h ¯ qq i = 1 c w h O w i = N c (2 π ) α ′ w . (83)To check the decoupling theorem, we have to study the relation M h ¯ qq i g Y M π h tr( G ) i for onespecific quark flavour, i.e. N f = 1. Expressed via the parameters of our model(42),(80),(83), it turns out to be M h ¯ qq i g Y M π h tr( G ) i = 1 N c √ α ′ g Y M w w φ , (84)with coefficients w , w , φ defined in (40), (81). It is convenient to express all coef-ficients via the expansion parameters in the coordinate r = R z . We denote them by φ = φ + b r , w = a + cr . Obviously, these are related to the former defined in (40), (81)by φ = b R , ω = cR . Recalling that R = 4 πg s α ′ N c and g Y M = 4 πg s , we obtain M h ¯ qq i g Y M π h tr( G ) i = 14 √ acb . (85)For the theorem M h ¯ qq i g Y M π h tr( G ) i = −
112 (86)to hold, the parameters a, b , c must satisfy acb = − √ . (87)This relation, equivalent to the decoupling theorem, will be tested numerically below. We obtain numerically the dimensionless ratio of the solution coefficients acb , linearlyrelated to the condensate ratio by (85). The ratio of the condensates obtained numer-ically is shown in Fig. (4) for the Gubser background. Similar pictures are obtainedfor the other two backgrounds. Each point in the parameter space represents anindividual “measurement”, that is, a solution for a D ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ tr G ( ) / m < y>y tr G m Figure 4: Dependence of the ratio m h qq ih tr G i on quark massWe obtain numerically the following results for the dimensionless ratio of thecondensates we are looking for: − g Y M π h tr( G ) i M h ¯ qq i = Constable–Myers 12 . ± . . ± . . ± . − g Y M π h tr( G ) i M h ¯ qq i = 12, we see agree-ment with good accuracy. The obvious universality of the three different metrics mightsignal that the decoupling theorem is insensitive to the details of the IR physics. Let us restate the main results of this study: • We have established a universal constant value for the ratio m h qq ih tr G i in the holo-graphic duality with a good precision (0 . We have obtained a version of the low-energy theorem R h T Oi = dim( O ) hOi satisfied in holography with condensates for the pure glue sector.In addition we also find the following results • A non-trivial relation between two-point and one-point functions R h G G i = const h G i has been established. • Shear and bulk viscosities have been shown to be independent of condensates. • The quarkonium diffusion coefficient has been obtained both at non-zero tem-perature and for a non-vanishing condensate in the Appendix A.The significance of establishing the decoupling ratio is its relevance to justifyingphenomenological approaches to QCD via holography. We hope to find an analyticexplanation of the amazing agreement which appears not to be a coincidence. Herewe provide demonstration for a very important example of a statement which relatesthe quark and gluon sectors. This should encourage further development of realisticAdS/QCD constructions based on geometries with broken scale-invariance.
Acknowledgements
A.Z. is extremely grateful to Prof. Dr. Dietrich Habs for his generosity, encourage-ment and a fruitful atmosphere of Theory vs. Experiment interaction he has createdin Garching providing a powerful inspiration for this work. Thanks to the Organizersof the International School on Strings and Fundamental Physics in Munich/Garching.A.Z. thanks Derek Teaney for correspondence on transport coefficients. The work wasalso supported by the DFG Cluster of Excellence MAP (Munich Centre of AdvancedPhotonics) (A.Z.), and by the Cluster of Excellence “Origin and Structure of the Uni-verse” (J.E.). The work of A.G. is supported in part by the grants PICS- 07-0292165,RFBR-09-02-00308 and CRDF - RUP2-2961-MO-09. The work of P.K. is supportedin part by the grant RFBR-09-02-00308 and by the Dynasty foundation. The workof A.K. is supported in part by RFBR grant no. 10-02-01483 and by the Dynastyfoundation. The work of A.Z. is supported in part by the RFBR grant 10-01-00836supported by Ministry of Education and Science of the Russian Federation under thecontract 14.740.11.0081. 22
Quarkonium Transport in Self-Dual Background
A.1 Self-Dual Background at Zero Temperature
Here we review the method of [30] for calculating quarkonium transport properties.The basic result of this discussion is a decoupled structure, in which the contributionsof the fermionic part of the action will be separated from those of the gluonic partaccording to the patternmeson kin. coeff. = (cid:20) meson mass shift(D7 contribution) (cid:21) × (cid:20) two-point correlator(D3 contribution) (cid:21) . (89)Consider a complex field ϕ of a slowly moving meson of velocity v , coupled to someoperators of gluonic sector, L = ϕ + v∂ t ϕ + X n c n ϕ + O n ϕ, (90)where coefficients c n are defined e.g. from D D C n are then introduced as “susceptibility” of mass with regard toswitching on the operator O n : δM = − c n hO n i . (91)Considering one-particle dynamics we can obtain from (90) dp i dt = F i , (92)where F i = Z d xϕ + ∇ i c n O n ϕ, (93)while correlator of two forces is directly related to transport coefficient κ = 13 Z dt hF ( t ) F (0) i . (94)One can integrate field ϕ out of these relations and obtain finally κ = 13 Z k d kc n Tω Im hO n O n i| k , (95)23here hO n O n i| k = Z d xθ ( t ) e i ( ωt − ~k~x ) hO n ( x ) O n (0) i . (96)Here the contributions of flavor dynamics and pure gluodynamics are decoupled;below we proceed in calculating the gluodynamical part (the two-point correlator);the coefficients c n being responsible for mass shifts are known in literature. A.2 Self-Dual Background at Finite Temperature
It is possible to obtain quarkonium diffusion and relaxation coefficients at finite tem-perature and condensate, extending the work [30] to the background of [22]. Thisbackground has the metric ds = R (cid:18) − r π T r dt + dx r + dr r (1 − r π T ) (cid:19) + R d Ω , (97)dilaton is e φ = 1 + qπ T log (cid:18) − r π T (cid:19) , (98)axion is related to dilaton in the same way as in the zero-temperature Liu-Tseytlinbackground C = e − φ − . (99)Quarkonium transport coefficients are quantities which feel both the fermionic pieceof the action (some embedded brane) and the gluodynamics. From the former comesmass susceptibility to condensate, from the latter – correlators of interest. In prin-ciple, it would make a good sense to work in a back-reacted metric, however thiswe postpone till the method is fully technically developed for the well-controllableGhoroku–Liu–Tseytlin metric.For convenience we further use the variable u = r π T , (100)which lives in the interval (0 , η + , η − , h + h . The equations of motion are given in AppendixA (122). We thank Derek Teaney for providing us with his unpublished Notes.
24e see now that the problem of fields coupling to each other is additionallyburdened by presence of finite temperature. Yet diagonalization of these equations ispossible by means of the following functional transformation ¯ η + ( u ) = η + ( u )¯ h ( u ) = h ( u ) + q ( C − π log (1 − u )) η + ( u )¯ η − ( u ) = qh ( u ) (cid:18) F − log ( − u ) π (cid:19) + η − ( u )++ q (cid:18) q log (1 − u ) − qC log ( − u ) π + C (cid:19) η + ( u ) (101)Now for each of the variables we can write down an equation similar to that for thesimple dilaton modes: ϕ ′′ ( u ) + u ( u + 6 u + 4 ω + 4 k ( u − − u ( u − ϕ ( u ) = 0 , (102)for which transport coefficient is known; we calculated it independently, and found itto be in agreement with the previous results [30]2 ωT G φ,φ = π k e − C γ k/T , (103)where C γ = 4 q π Γ (cid:0) (cid:1) ≈ .
62. Knowledge of diagonalization matrix allows us totransform these results (at q = 0) into non-zero-condensate background: h Φ i Φ j i = (ˆ1 + qA ) h Φ ′ i Φ ′ j i q =0 (ˆ1 + qA ) + , (104)where zero-condensate solutions are rotated to non-zero-condensate by the followingrotation matrix in mode space: A = π / /π , (105)and the non-perturbed matrix of finite-temperature correlators is diagonal h Φ i Φ j i q =0 = h tr G +2 tr G +2 i h T T i
00 0 h tr G − tr G − i , (106)whence one easily gets the mesonic transport coefficient by use of the following for-mula: κ = X O c O π Z k d k (2 π ) ωT h Φ O Φ O i , (107)25here the respective mass susceptibility coefficients are obtained from considering thefermionic fluctuations coming from the embedded D δM = − c O hOi , (108)where M refers to the mass of quarkonium.The correlators themselves are obtained in the following way, which we illustrateon the example of dilaton. We consider three domains: UV, IR and the intermedi-ate domain (we denote the latter QC for semiclassics, since semiclassical approximatesolutions will be valid therein). The physical limitations are infalling boundary condi-tion on the horizon and reflected wave in the UV, which reduces number of unknowncoefficients from 6 to 4. Then, we have matching conditions separate for each of themodes in the matching regions between UV and QC, an between QC and IR. Thisprovides additional 4 constraints, thus the system is fully defined. In the UV thegeneral solution to EOM is φ = 2 uI (cid:0) √ u √ k − ω (cid:1) C k − ω + 2 u (cid:0) k − ω (cid:1) K (cid:16) √ u √ k − ω (cid:17) C . (109)Taking the UV asymptotic ( u →
0) of φ , we see that physical boundary conditionsare C = B, C = 1, where B is related to correlator straightforwardly:2 ωT G φφ = ImBω . (110)On the contrary, expanding it for large k , we get the form appropriate for matchingwith QC: φ = e − k √ u √ πk − / u − / − Be k √ u k − / u − / √ π . (111)The semiclassical equation has the approximate potential V QC = k u (1 − u ) , (112)which allows to obtain the wave-functions in the standard way ψ , = e ± R pdx √ p , (113)where p = p V QC − E. (114)26he semiclassical solution near u = 0 and u = 1 is φ QC,u → = − ie − k √ u (cid:0) e k √ u A + A (cid:1) √ k √ uφ QC,u → = − ie −√ k ( √ − u +1 ) (cid:16) e √ k A + e k √ − u ) A (cid:17) √ k √ − u . (115)The IR solution with infalling boundary condition has only one degree of freedom: φ IR = (cid:16) e √ k √ − u csc( πω + e −√ k √ − u (cid:17) √ πC / √ k √ − u . (116)Equating the QC solution branches with those of IR and UV solutions, we getIm B = π k e − C γ k/T , (117)as already stated above. Taking the integral over phase space (107) and performinglinear transformation of correlator matrix (104), we get for transport coefficient κ = 13 T (cid:0) (cid:1) π Γ (cid:0) (cid:1) (cid:2) c tr G +2 (1 + 2 qπ ) + c T (1 + qπ ) + c tr G − (cid:3) , (118)where c i are found in [30], c tr G = π (cid:16) πM (cid:17) , c T = π (cid:16) πM (cid:17) , M being the mesonmass. B Equations of Motion
Here we shown the equations of motion for Liu–Tseytlin background in the graviton,axion and dilaton sector, corresponding to the pure glue sector on the boundary. Thedefinitions of the fields are contained in eqs. (51)- (53).27 z (cid:16)(cid:0) qωz + ω (cid:1) − q z (cid:17) η + ( z )++ ( qz + 1) (cid:0) (11 qz + 3) η ′ + ( z ) − z ( qz + 1) η ′′ + ( z ) (cid:1) = 0 , q η + ( z ) z + (cid:0) qz + 1 (cid:1) (cid:0)(cid:0) qz + 1 (cid:1) (cid:0) z ω + 4 (cid:1) Φ ( z ) −− z (cid:0) qη ′ + ( z ) z + ( qz + 1) (Φ ′ ( z ) + z Φ ′′ ( z )) (cid:1)(cid:1) = 0 , (cid:0) z ω + 4 (cid:1) Φ ( z ) − z (Φ ′ ( z ) + z Φ ′′ ( z )) = 0 , (cid:0) z ω + 4 (cid:1) h xy ( z ) − z (cid:0) h ′ xy ( z ) + zh ′′ xy ( z ) (cid:1) = 0 , − q η + ( z ) z + (cid:0) qωz + ω (cid:1) η − ( z ) z −− ( qz + 1) (cid:0) q Φ ( z ) z + (cid:0) q Φ ′ ( z ) z + ( qz + 1) η ′′− ( z ) (cid:1) z + (5 qz − η ′− ( z ) (cid:1) = 0 . (119)Solutions for the EOM in the Liu–Tseytlin case at zero frequency ω = 0 are: Φ Φ Φ Φ Φ = C ( qz + 1) + C ( qz + 1) − q C z + C z + C z C z + C z C z + C z qC − C q + ( q ( qz +2 ) z +2 ) (4 q ( C + C )+2 C )4 q ( qz +1) (120)Solution modes for a non-zero frequency:28 = qω K ( zω ) C z + ω K ( zω ) C z , Φ = C h γqω z − qω z ++ qω log( z ) z + qω log( ω ) z − qω log(16) z −− qω log(8) z − qω log(4) z + γqω z − qω z ++ qω log( z ) z + qω log( ω ) z − qω log(16) z −− qω log(4) z + γqω z − qω z + qω log( z ) z ++ qω log( ω ) z − qω log(4) z + qω z (cid:3) + ω K ( zω ) C , Φ = 12 ω K ( zω ) C , Φ = 12 ω K ( zω ) C , Φ = − qω C z + 16 qω C z − qC z − qI ( zω ) C z ( qz + 1) ω − ω K ( zω ) C z qz + 1 ++ 4 q I ( zω ) C z ( qz + 1) ω + qω K ( zω ) C z qz + 1) . (121)The thermal version of the Liu–Tseytlin backgrounds leads to the following equationsof motion: ( u ( u +6 u +4 ω +4 k ( u − )) − ) η + u ( u − + η + ′′ = 0 , − q ( u + 1) h ( u ) u + 4 ( u − (cid:0) quh ′ + π ( u − η − ′′ (cid:1) u ++ π ( u ( u + 6 u + 4 ω + 4 k ( u − − η − = 0 , (cid:16) h ′′ ( u − + 2 π q (cid:0) u ( u − η + ′ − ( u + 1) η + ( u ) (cid:1)(cid:17) u ++ ( u ( u + 6 u + 4 ω + 4 k ( u − − h = 0 . (122) References [1] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Are AllHadrons Alike?,”
Nucl. Phys.
B191 (1981) 301.292] M. A. Shifman, “Anomalies and Low-Energy Theorems of QuantumChromodynamics,”
Phys. Rept. (1991) 341–378.[3] D. R. Gulotta, C. P. Herzog, and M. Kaminski, “Sum Rules from an ExtraDimension,” .[4] H. Liu and A. A. Tseytlin, “D = 4 super Yang-Mills, D = 5 gaugedsupergravity, and D = 4 conformal supergravity,”
Nucl. Phys.
B533 (1998)88–108, hep-th/9804083 .[5] H. Liu and A. A. Tseytlin, “D3-brane D-instanton configuration and N = 4super YM theory in constant self-dual background,”
Nucl. Phys.
B553 (1999)231–249, hep-th/9903091 .[6] G. Policastro, D. T. Son, and A. O. Starinets, “From AdS/CFT correspondenceto hydrodynamics,”
JHEP (2002) 043, hep-th/0205052 .[7] G. Policastro, D. T. Son, and A. O. Starinets, “The shear viscosity of stronglycoupled N = 4 supersymmetric Yang-Mills plasma,” Phys. Rev. Lett. (2001)081601, hep-th/0104066 .[8] D. T. Son and A. O. Starinets, “Minkowski-space correlators in AdS/CFTcorrespondence: Recipe and applications,” JHEP (2002) 042, hep-th/0205051 .[9] G. Policastro, D. T. Son, and A. O. Starinets, “From AdS/CFT correspondenceto hydrodynamics. II: Sound waves,” JHEP (2002) 054, hep-th/0210220 .[10] J. L. F. Barbon, C. Hoyos-Badajoz, D. Mateos, and R. C. Myers, “Theholographic life of the eta’,” JHEP (2004) 029, hep-th/0404260 .[11] S. S. Gubser, “Dilaton-driven confinement,” hep-th/9902155 .[12] A. Kehagias and K. Sfetsos, “On running couplings in gauge theories fromtype-IIB supergravity,” Phys. Lett.
B454 (1999) 270–276, hep-th/9902125 .[13] N. R. Constable and R. C. Myers, “Exotic scalar states in the AdS/CFTcorrespondence,”
JHEP (1999) 020, hep-th/9905081 .3014] E. V. Gorbar and A. A. Natale, “Relating the quark and gluon condensatesthrough the QCD vacuum energy,” Phys. Rev.
D61 (2000) 054012, hep-ph/9906299 .[15] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “QCD and ResonancePhysics. Sum Rules,”
Nucl. Phys.
B147 (1979) 385–447.[16] U. Gursoy, E. Kiritsis, L. Mazzanti, and F. Nitti, “Holography andThermodynamics of 5D Dilaton-gravity,”
JHEP (2009) 033, .[17] L. D. Landau and E. M. Lifshitz, “Textbook On Theoretical Physics. Vol. 5:Statistical Physics,”. Moscow, 1986, 484 pp.[18] I. H. Brevik, K. Ghoroku, and A. Nakamura, “Meson mass and confinementforce driven by dilaton,” Int. J. Mod. Phys.
D15 (2006) 57–68, hep-th/0505057 .[19] J. Erdmenger, K. Ghoroku, and I. Kirsch, “Holographic heavy-light mesonsfrom non-Abelian DBI,”
JHEP (2007) 111, .[20] J. Erdmenger, N. Evans, I. Kirsch, and E. Threlfall, “Mesons in Gauge/GravityDuals - A Review,” Eur. Phys. J.
A35 (2008) 81–133, .[21] F. Rust, “In-medium effects in the holographic quark-gluon plasma,” .[22] K. Ghoroku, T. Sakaguchi, N. Uekusa, and M. Yahiro, “Flavor quark at hightemperature from a holographic model,”
Phys. Rev.
D71 (2005) 106002, hep-th/0502088 .[23] A. L. Kataev, N. V. Krasnikov, and A. A. Pivovarov, “Two Loop Calculationsfor the Propagators of Gluonic Currents,”
Nucl. Phys.
B198 (1982) 508–518, hep-ph/9612326 .[24] E. Witten, “Current Algebra Theorems for the U(1) Goldstone Boson,”
Nucl.Phys.
B156 (1979) 269.[25] J. Babington, J. Erdmenger, N. J. Evans, Z. Guralnik, and I. Kirsch, “Chiralsymmetry breaking and pions in non-supersymmetric gauge / gravity duals,”
Phys. Rev.
D69 (2004) 066007, hep-th/0306018 .3126] J. Erdmenger, N. Evans, and J. Grosse, “Heavy-light mesons from theAdS/CFT correspondence,”
JHEP (2007) 098, hep-th/0605241 .[27] N. J. Evans and J. P. Shock, “Chiral dynamics from AdS space,” Phys. Rev.
D70 (2004) 046002, hep-th/0403279 .[28] R. Apreda, J. Erdmenger, and N. Evans, “Scalar effective potential for D7brane probes which break chiral symmetry,”
JHEP (2006) 011, hep-th/0509219 .[29] R. Apreda, J. Erdmenger, N. Evans, J. Grosse, and Z. Guralnik, “Instantons onD7 brane probes and AdS/CFT with flavour,” Fortsch. Phys. (2006)266–274, hep-th/0601130 .[30] K. Dusling et al. , “Quarkonium transport in thermal AdS/CFT,” JHEP (2008) 098,0808.0957