aa r X i v : . [ h e p - t h ] D ec Higher Derivative Corrections in Holographic QCD
Anirban Basu Institute for Advanced Study, Princeton, NJ 08540, USA
Abstract
We consider the effect of the R term in type IIA string theory on the supergrav-ity background dual to N c D4 branes compactified on a circle with supersymmetrybreaking boundary conditions. We study the dynamics of D8 branes in this perturbedgeometry in the probe approximation. This leads to an analysis of higher derivativecorrections in holographic QCD beyond the supergravity approximation. We make arough estimate of the corrections to the masses of some of the lightest (axial) vectormesons. The corrections are suppressed by a factor of ( g Y M N c ) − compared to theirsupergravity values. We find that the masses of these mesons increase from theirsupergravity values. email: [email protected] Introduction
It is a challenging problem to understand strong coupling phenomena such as confinementand chiral symmetry breaking in QCD at low energies. Using the gauge/string duality, amodel of the holographic dual of pure QCD without matter was proposed in [1]. Wittenconsidered N c D4 branes with one of the world volume directions compactified on a circle,with anti–periodic boundary conditions for the fermions. This configuration breaks super-symmetry, and the fermions and the scalars on the world volume theory of the D4 branesbecome massive at tree level and at one–loop level respectively. Thus at low energies, thisreduces to a theory of pure Yang–Mills in four dimensions which is confining. Flavor wasadded to this model in the probe approximation [2], by which one means that N f flavorbranes are placed in the background geometry dual to N c color branes, such that N f ≪ N c .More recently, Sakai and Sugimoto considered the dynamics of flavor D8 branes in thebackground geometry of color D4 branes in the probe approximation [3, 4]. This gives amodel of holographic QCD with matter, which differs from QCD at energies comparableto the Kaluza–Klein mass scale of the theory, which is determined by the radius of thecircle on which the D4 branes are compactified. Also the theory has an SO (5) symme-try transverse to the color branes, unlike QCD. Nevertheless, this theory is an interestingmodel in trying to understand QCD at energy scales below the Kaluza–Klein mass scale.Sakai and Sugimoto demonstrated chiral symmetry breaking in this theory by analyzingthe dynamics of the flavor branes in the background geometry of the color branes in thesupergravity approximation. For related work, see [5–9]. Letting the radius of the circle onwhich the D4 branes are compactified to go to infinity, one obtains a theory with brokenchiral symmetry, but which is unconfined. Such systems have been studied in [10–15] (alsosee [16] for a related discussion).Now understanding various aspects of holographic QCD has been so far done at the levelof supergravity. In this paper, we attempt to go beyond the supergravity approximation,and include the effects of stringy corrections. In particular, we shall include the effect ofthe R term in the effective action of type IIA string theory. When considering only thesupergravity contributions, we are working in the large N c limit, with g Y M N c → ∞ , where g Y M is the four dimensional gauge coupling. When we consider the contribution due to the R term, which is α ′ suppressed compared to the supergravity contributions, we considerthe leading correction to supergravity where g Y M N c is kept large but finite.We first consider the effect of the R term on the supergravity background dual to theD4 branes. We analyze the perturbed geometry, and consider the dynamics of the D8 branes1n this geometry, still in the probe approximation. In particular, we focus on the dynamicsof the gauge fields on the world volume theory of the D8 branes. The fluctuations whichare along the (3 + 1) directions are massive, and are interpreted as (axial) vector mesons [3]in holographic QCD. Using the perturbed geometry, we make a very rough estimate of thecorrections to the masses of the lightest mesons, due to the higher derivative corrections. Atthe level of approximation we use, we find that the masses increase from their supergravityvalues, with δm ∼ ( g Y M N c ) − . The analysis of the perturbed geometry is a generalizationof the method in [17, 18] to non–conformal cases, where the effect of the R term on thenear–horizon geometry dual to D3 branes in type IIB string theory was considered (alsosee [19]). We shall see that the analysis gets considerably more complicated, essentiallybecause of the loss of conformality. Though we obtain the exact perturbed geometry, we cansolve for the coefficients in the various expressions only recursively as we shall demonstratebelow. Coupled with the fact that the mesons masses can only be calculated roughly atthe supergravity level, the complexity of the equations allows us to obtain a rather roughestimate of the corrections to the meson masses.It should be noted that our analysis does not give the complete answer due to the α ′ corrections to supergravity. We consider only the R term in the entire supermultiplet at O ( α ′ ), and there are many other terms in this supermultiplet that will also contribute (forexample, R F , R ( ∂φ ) , and so on ). The entire R supermultiplet is not well understood(see [20, 21] for a relevant discussion in type IIB string theory), and so we restrict ourselvesto the R term only. Adding the other contributions, it is possible that the values of variousobservables like meson masses will change. In order to study holographic QCD, we consider N c D4 branes extending along the directions0 , ,
2, and 3, and compactified along the direction x , with antiperiodic boundary conditionsfor the fermions to break supersymmetry. The number of colors N c is taken to be very largein the entire discussion.The dual supergravity background in the string frame is given by ds string = (cid:16) UR (cid:17) / (cid:16) η µν dx µ dx ν + f ( U ) dx (cid:17) + (cid:16) RU (cid:17) / (cid:16) dU f ( U ) + U d Ω (cid:17) , (1) F and φ are the R–R four form and the dilaton respectively. This is obtained from the near extremal black 4—brane solution [22] by interchanging the role of thetime and x coordinates. e φ = g s (cid:16) UR (cid:17) / , R = πg s N c α ′ / , f ( U ) = 1 − U KK U , (2)and U is the radial coordinate transverse to the 4–brane world volume. The four formfield strength is given by F = Qω , where ω is the volume form on the unit four sphere.Using the Dirac quantization condition [23], Z S F = κ N c √ πα ′ , (3)we get that Q = 3 N c πα ′ / . (4)Including the R correction to the supergravity action, the relevant action in the stringframe is given by [24–27] S string = 12 κ Z d x √− g h e − φ (cid:16) R + 4( ∇ φ ) + γW (cid:17) − . F i , (5)where γ = ζ (3)8 α ′ , (6)and W = C HMNK C P MNQ C RSPH C QRSK + 12 C HKMN C P QMN C RSPH C QRSK , (7)where C HMNK is the Weyl tensor. We now want to compute the perturbed backgrounddue to the introduction of the R interaction. To do so, we find it convenient to go to theEinstein frame which gives the action S Einstein = 12 κ Z d x p − ˆ g h ˆ R −
12 ( ˆ ∇ φ ) − g / s e φ/ .
4! ˆ F + g / s γe − φ/ ˆ W i , (8)where ˆ g is the Einstein frame metric, and κ = κ g s . The metric ˆ g MN is given by ds Einstein = (cid:16) UR (cid:17) / (cid:16) η µν dx µ dx ν + f ( U ) dx (cid:17) + (cid:16) RU (cid:17) / (cid:16) dU f ( U ) + U d Ω (cid:17) . (9)We make the ansatz for the perturbed metric See [28–30] for the analysis using conformal sigma model techniques. s Einstein = H ( U ) h K ( U ) dx + P ( U ) dU + η µν dx µ dx ν i + L ( U ) d Ω . (10)Thus translational invariance along the (3 + 1) directions and the x direction, as wellas the transverse SO (5) rotational invariance is preserved. We now want to construct theperturbed metric and the dilaton in the Einstein frame to leading order in γ . One mightthink that F gets perturbed to F = Qλ ( U ) ω , where λ ( U ) = 1 + O ( γ ). However, theDirac quantization condition (3) prevents this and F remains unperturbed.Using the symmetries of the ansatz (10), we see that the action (8) is given by S Einstein = V Z ∞ U KK dU q ˜ˆ g h ˆ R −
12 ( ˆ ∇ φ ) − g s · (cid:16) UR (cid:17) / e ( φ + φ + ... ) / ˆ F + γ (cid:16) UR (cid:17) − / e − φ + φ + ... ) / ˆ W + i , (11)where q ˜ˆ g = H KP L , (12)and V = Vol(S ) V , Vol(S )2 κ . (13)In (13), Vol(S ) = 8 π / ) is thecircumference of the circle along x . Also we have expressed φ = φ + 34 ln (cid:16) UR (cid:17) + φ + φ + . . . , (14)where g s = e φ , φ ∼ O ( γ ), φ ∼ O ( γ ), and the remaining terms in . . . are of O ( γ ).Thus in (11), we have that( ˆ ∇ φ ) = 1 H P (cid:16) U + 32 U ( ∂ U φ ) + ( ∂ U φ ) + 32 U ( ∂ U φ ) + O ( γ ) (cid:17) . (15)We describe the various details for obtaining the perturbed metric and the dilaton inthe appendices. Note that even though we do not have closed form expressions for themetric or the dilaton perturbations in (78) or (90), we see that these perturbations take asimple form: they are given by sums of positive integer powers of harmonic functions (uptoa factor of U − / ). This is similar to what happens in the conformal case of the 3–branegeometry [17, 18]. 4 The flavor branes in the background geometry
Having obtained the perturbed background dual to the color D4 branes, we now analyzethe dynamics of N f probe D8 branes in this background geometry. We work in the probeapproximation so that N f ≪ N c . Before taking the dynamics into account, the D braneconfiguration is given by0 1 2 3 4 5 6 7 8 9D4 : x x x x x D8 , D8 : x x x x x x x x x
Thus the flavor D8 and D8 branes intersect the color D4 branes along (3 + 1) dimensionsand are separated along the x direction. This configuration becomes very different whenthe dynamics are considered [3]. The D8 and D8 branes get connected which is interpretedas chiral symmetry breaking. Our aim is to consider some aspects of the dynamics of theflavor branes in the perturbed background geometry dual to the color branes. First let ussee how the flavor branes deform in this background geometry. The background metric isgiven by ds string = (cid:16) UR (cid:17) / h e γ ˆ φ / η µν dx µ dx ν + f ( U ) e γ ( ˆ φ / a +18 b ) dx i + (cid:16) RU (cid:17) / h e γ ( ˆ φ / b ) dU f ( U ) + U e γ ( ˆ φ / c ) d Ω i . (16)Considering probe D8 branes in this geometry where U = U ( x ), we see that the inducedmetric on the D8 brane world volume is given by ds D = h(cid:16) UR (cid:17) / f ( U ) e γ ( ˆ φ / a +18 b ) + (cid:16) RU (cid:17) / e γ ( ˆ φ / b ) U ′ f ( U ) i dx + (cid:16) UR (cid:17) / e γ ˆ φ / η µν dx µ dx ν + (cid:16) RU (cid:17) / U e γ ( ˆ φ / c ) d Ω , (17)where U ′ = ( ∂U/∂x ). In the expressions (16) and (17) and in the ones that follow, theexponentials are only to be expanded to O ( γ ). Inserting the induced metric (17) into theDBI action of the D8 branes, we get that 5 D ∼ Z d xdx e γ ( ˆ φ / c )( U ) U s e γ ( ˆ φ / a +18 b )( U ) f ( U ) + (cid:16) RU (cid:17) e γ ( ˆ φ / b )( U ) U ′ f ( U ) . (18)Following Sakai and Sugimoto, we look for a solution to the classical equation of motionfor the D8 brane resulting from (18) which asymptotes as U → ∞ to a fixed value of x . Thisis the position of the flavor brane along the x direction in the naive picture. To considera configuration of D8 and D8 branes which connect together leading to chiral symmetrybreaking, we also want this configuration to satisfy U ( x = 0) = U , U ′ ( x = 0) = 0,such that it is symmetric about x = 0. Thus naively, without considering the dynamics,this corresponds to D8 and D8 branes placed symmetrically about x = 0. Including thedynamics, they get connected. This solution which has the interpretation of a wormholesolution connecting the D8 and D8 branes asymptotically exhibits chiral symmetry breakingin holographic QCD. The throat of this wormhole has its minimum radius U at x = 0.This solution is given by x ( U ) = U s f ( U )Θ ( U ) Z UU dU e γ ( ˆ φ / b )( U ) / f ( U ) (cid:16) UR (cid:17) / q U f ( U ) Θ ( U )Θ ( U )) − U f ( U ) Θ ( U )Θ ( U ) , (19)where Θ ( U ) = e γ (3 ˆ φ / a +18 b +4 c )( U ) , Θ ( U ) = e γ ( ˆ φ / a +18 b )( U ) . (20)Thus chiral symmetry continues to be broken in the presence of the higher derivativecorrections. In (19), U is an arbitrary parameter satisfying U ≥ U KK . For the sake ofsimplicity, we shall analyze the spectrum for the case U = U KK as done by Sakai andSugimoto. Taking only the supergravity action into account, they showed that the D8 andD8 branes are placed at antipodal points on the circle parametrized by x when U = U KK .Now even when the higher derivative corrections are turned on, the ansatz we have madefor the various perturbations preserves the same symmetries, in particular, translationalinvariance along the x circle is preserved. Thus when U = U KK , we expect the branes toremain at antipodal points on the x circle. We now show this is the case.Given the perturbed metric (16), in order to avoid a conical singularity at U = U KK , x must be a periodic variable satisfying 6 ∼ x + 4 πR / e − γ ( a +8 b )( U KK ) √ U KK . (21)Thus the circumference of the x circle is given by δx = 4 πR / e − γ ( a +8 b )( U KK ) √ U KK . (22)From (19), we can calculate the position in x where the D8 brane is placed which isgiven by x ( ∞ ), and the D8 and D8 branes are separated by twice this distance. However,for U = U KK , f ( U ) = 0, and the integrand in (19) diverges at U = U . Thus the expressionfor x ( ∞ ) needs to be regularized. We regularize it by setting σ = ( U /U KK ) = 1 + ǫ , andtaking the limit ǫ →
0. We need to pick out the O (1 / √ ǫ ) terms from the integral whichcancel the O ( √ ǫ ) term from p f ( U ). We have that x ( ∞ ) | U = U KK = R / √ U KK s g ( σ )Θ ( σ ) Z ∞ σ =1+ ǫ du e γ ( ˆ φ / b )( u ) / g ( u ) u / q u g ( u ) Θ ( u )Θ ( σ ) − σ g ( σ ) Θ ( u )Θ ( σ ) , (23)where g ( u ) = 1 − u . (24)It is straightforward to show that the relevant contribution is contained in the expressiongiven by x ( ∞ ) | U = U KK = R / √ U KK s ǫ Θ (1) Z ǫ dx e γ ( ˆ φ / b )(1) / x / (cid:16) − ǫx (cid:17) − / = R / √ U KK s e γ ( ˆ φ / b )(1) Θ (1) ∞ X k =0 Γ( k + 1 / / k !( k + 1 / πR / e − γ ( a +8 b )( U KK ) √ U KK = δx , (25)where by equality in the first two lines we mean that we only keep the relevant termswhich are non–vanishing in the limit ǫ →
0. Thus on adding the higher derivative cor-rections, the radius of x changes, but the D8 and D x circle. 7 Estimating the corrections to (axial) vector meson masses
Having discussed the flavor brane configuration, we now consider the effect of the higherderivative corrections on the masses of the (axial) vector mesons in holographic QCD. Thesevector mesons are obtained from the fluctuations of the gauge fields on the D8 brane worldvolume in the background geometry. In fact, the fluctuations along the directions x µ yieldthe vector mesons, while the fluctuations along U yield the pions. Expanding the DBIaction to quadratic order in the gauge fields, we get that S D = − T Z d xdU e γ ( ˆ φ / b +4 c )( U ) Tr h R / p U f ( U ) ˆ F µν +2 e − γb ( U ) U / R / p f ( U ) ˆ F µ U F µU i , (26)where T = Vol( S )(2 π ) α ′ / g s , (27)and ˆ F means that the four dimensional indices are raised with the Minkowski metric.We focus only on the vector mesons and drop the terms involving the pions. Thus defining A µ ( x, U ) = ∞ X n =1 B ( n ) µ ( x ) ψ n ( U ) , (28)we get that F µν ( x, U ) = ∞ X n =1 F ( n ) µν ( x ) ψ n ( U ) ,F µU ( x, U ) = − ∞ X n =1 B ( n ) µ ( x ) ∂ U ψ n ( U ) , (29)where F ( n ) µν = ∂ µ B ( n ) ν − ∂ ν B ( n ) µ . Thus we get that S D = − T Z d xdU e γ ( ˆ φ / b +4 c )( U ) ∞ X m,n =1 Tr h R / p U f ( U ) ˆ F µν ( m ) ( x ) F ( n ) µν ( x ) ψ m ( U ) ψ n ( U )+2 e − γb ( U ) U / R / p f ( U ) ˆ B µ ( m ) ( x ) B ( n ) µ ( x )( ∂ U ψ m )( U )( ∂ U ψ n )( U ) i . (30)Note that (30) is independent of a . Defining8 = U U KK − , K ( Z ) = 1 + Z , (31)we get that S D = − ˆ T Z d xdZe γ ( ˆ φ / b +4 c )( Z ) ∞ X m,n =1 Tr h K − / ( Z ) ˆ F µν ( m ) ( x ) F ( n ) µν ( x ) ψ m ( Z ) ψ n ( Z )+ 9 U KK R e − γb ( Z ) K ( Z ) ˆ B µ ( m ) ( x ) B ( n ) µ ( x )( ∂ Z ψ m )( Z )( ∂ Z ψ n )( Z ) i , (32)where ˆ T = 2 R / √ U KK Vol( S )3(2 π ) α ′ / g s . (33)Canonical normalization of the kinetic term in (32) leads toˆ T Z dZe γ ( ˆ φ / b +4 c )( Z ) K − / ( Z ) ψ m ( Z ) ψ n ( Z ) = δ m,n , (34)while canonical normalization of the mass term givesˆ T Z dZe γ ( ˆ φ / − b +4 c )( Z ) K ( Z )( ∂ Z ψ m )( Z )( ∂ Z ψ n )( Z ) = λ n δ m,n , (35)where the mass is given by m n = (9 U KK λ n ) / (4 R ). Thus (32) reduces to S D = − Z d x ∞ X m =1 Tr h
14 ˆ F µν ( n ) F ( n ) µν + m n B µ ( n ) B ( n ) µ i . (36)From (34) and (35) we obtain the eigenvalue equation for ψ n e − γ ( ˆ φ / b +4 c ) K / ∂ Z h e γ ( ˆ φ / − b +4 c ) K ( ∂ Z ψ n ) i = − λ n ψ n . (37)Now because the perturbations ˆ φ , b and c are functions of Z , note that the action(32) is invariant under ( x µ , Z ) → ( − x µ , − Z ), which has the interpretation of space timeparity [3]. Thus from (37) we see that ψ n has definite parity under Z → − Z . The aim isto construct ψ n which has even (odd) parity for n odd (even). Thus B ( n ) µ is a vector (axialvector) if n is odd (even). Under charge conjugation, B ( n ) µ is even (odd) if n is even (odd) [3].We shall focus on the lowest lying modes with n = 1 , C, P ) = ( − , − ) and(+ , +) respectively. In holographic QCD, these modes are to be identified with the ρ mesonand the a (1260) meson respectively. 9n order to find the correction to the vector meson masses due to the higher derivativecorrections, we write ψ n = ψ (0) n + γ p R U KK ψ (1) n ,λ n = λ (0) n + γ p R U KK λ (1) n , (38)where ψ (0) n and λ (0) n are the supergravity expressions, while ψ (1) n and λ (1) n are the higherderivative corrections.Using first order perturbation theory, (34), and (37), we get that λ (1) n = − ˆ T Z dZψ (0) n ( Z ) ˆ H n ( Z ) ψ (0) n ( Z ) , (39)where ˆ H n ( Z ) = K (cid:16) ˆ φ ′ − b ′ + 4 c ′ (cid:17) ∂∂Z − b (cid:16) Z ∂∂Z + K ∂ ∂Z (cid:17) = K (cid:16) ˆ φ ′ − b ′ + 4 c ′ (cid:17) ∂∂Z + 2 λ (0) n b K − / , (40)where we have used the O (1) relation from (37) K / ( K∂ Z ψ (0) n + 2 Z∂ Z ψ (0) n ) = − λ (0) n ψ (0) n . (41)Integrating (39) by parts, one can also express λ (1) n as λ (1) n = ˆ T Z dZK (cid:16) ˆ φ − b + 4 c (cid:17) ( ∂ Z ψ (0) n ) − λ (0) n ˆ T Z dZK − / (cid:16) ˆ φ b + 4 c (cid:17) ( ψ (0) n ) . (42)In (39) and the equations that follow, we have removed an overall factor of 1 / p R U KK from ˆ φ , b and c for notational simplicity. Now it is difficult to calculate λ (1) n from (39)(or (42)) exactly because the unperturbed wavefunction ψ (0) n ( Z ) is not known exactly. Alsoalthough we have recursion relations for the coefficients describing the perturbed geometry,we do not have closed form expressions for them. So we will make a very rough estimateof the correction to the meson masses, which we turn to now.Sakai and Sugimoto considered normalizable wavefunctions satisfying the Schrodingerequation (41), and the normalization condition (35), and obtained λ (0) n numerically usingthe shooting technique. They obtained 10 (0)1 ≈ . ( − , − ) , λ (0)2 ≈ . (+ , +) , (43)for the two lightest modes that satisfy (41). Now for low values of n , these normalizablewave functions must be concentrated around Z = 0, while they spread out more and moreto larger values of Z as n increases. Since we will focus on the two lowest lying normalizableeigenstates of (41), the correction to the meson masses given by (39) (or (42)) should receivethe maximum contribution from the neighbourhood of Z = 0 in the integral. Thus in orderto make a rough estimate of λ (1) n , we shall focus on this region only. Of course, as largerand larger values of Z are considered, the approximation gets better and better.So in the various expressions, we focus on the region of integration around Z = 0. Infact, we shall make the crudest approximation, and restrict ourselves to terms only upto O ( Z ) in the various expressions. Now it is easy to construct an approximate normalizedwavefunction which solves (41) at small Z . Keeping terms only upto O ( Z ), (41) reduces to ∂ Z χ n ( Z ) ≈ (1 − λ (0) n ) χ n ( Z ) , (44)where ψ (0) n ( Z ) = e − Z / χ n ( Z ). Thus the approximate normalized wave functions for thetwo lowest modes are given by ψ (0)1 ( Z ) ≈ . p ˆ T e − Z / cosh (cid:16) Z √ (cid:17) , (45)and ψ (0)2 ( Z ) ≈ . p ˆ T e − Z / sin (cid:16)r Z (cid:17) . (46)In order to estimate λ (1) n at this order, we also need approximate expressions for ˆ φ ( Z ) , b ( Z )and c ( Z ). Because ˆ φ ( Z ) , b ( Z ) and c ( Z ) are functions of Z only, and we are restrictingto terms of O ( Z ), we can replace ˆ φ ( Z ) , b ( Z ) and c ( Z ) by the constant modes ˆ φ ⋆ , b ⋆ and c ⋆ respectively, where ˆ φ ( Z ) = ˆ φ ⋆ + O ( Z ) , b ( Z ) = b ⋆ + O ( Z ) , c ( Z ) = c ⋆ + O ( Z ) . (47) In fact, we can look at the system of equations (69) at small Z , keeping terms only to O ( Z ). Notingfrom the exact expressions for the perturbations in terms of η that their behavior is similar at small Z , onecan solve them directly, and obtain the solutions for ˆ φ ( Z ) , b ( Z ) and c ( Z ) as described above. Note thatthe Z dependence of the perturbations obtained by solving (69) at small Z , is very different from that inthe wave functions (45) and (46) obtained by solving (41) at small Z . λ (1) n ≈ − λ (0) n b ⋆ . (48)Thus at this order δm δm ≈ λ (0)2 λ (0)1 ≈ . . (49)To obtain the masses, we estimate b ⋆ using (69) directly. Setting a ( Z ) = a ⋆ , b ( Z ) = b ⋆ and c ( Z ) = c ⋆ and equating terms of O (1), we get that b ⋆ ≈ − . , c ⋆ ≈ − . . (50)Note that the zero mode a ⋆ is undetermined by the equations.In order to get better estimates, one has to keep terms at higher orders in Z , and solvefor the wavefunction, as well as the metric and dilaton perturbations using (69) and (41).The Z dependence of these quantities is going to be different, and the values of the variouscoefficients (for example, the constant terms in b and c ) are going to change too. However,on including the various contributions, λ (1) n should not change by a large amount as the lowlying states are localized around Z = 0. It would be interesting to include higher powersof Z in this analysis, and try to get a better estimate. Presumably keeping a reasonablysmall number of terms in the expansion in Z will make the estimates converge to a sharpvalue of λ (1) n . Using our rough estimates, we get that δm M KK ≈ g Y M N c ) − ,δm M KK ≈ g Y M N c ) − , (51)where M KK ≡ π/δx | ( γ =0) = (3 √ U KK ) / (2 R / ), and we have used the relations [7] R = g Y M N c α ′ M KK , U KK = 2 g Y M N c M KK α ′ . (52)Thus at the level of the approximations we have made, we see that the masses of thetwo lightest (axial) vector mesons increase from their supergravity values. In fact, using Of the three resulting equations from (69), only two are linearly independent: the first two equationsare the same. g Y M N c ≈ δm M KK ≈ , δm M KK ≈ . (53)Trying to analyze the effect of the higher derivative corrections in other applicationsof holographic QCD is an important problem in general. It would be nice to have exactexpressions for the metric and dilaton perturbations, as that will make calculations moreconcrete and predictive. In order to make precise quantitative predictions in holographicQCD due to corrections to supergravity at O ( α ′ ), it is also important to understand thedetailed structure of the R supermultiplet in type IIA string theory. Acknowledgements
I would like to thank I. Klebanov, J. Maldacena, A. Maloney, D. Mateos, and S. Sugimotofor useful comments. I am particularly thankful to A. Maharana for many useful discussions,and for technical assistance in computing ˆ W . The work of A. B. is supported by NSF GrantNo. PHY-0503584 and the William D. Loughlin membership. In this appendix, we describe the construction of the perturbed metric and the dilaton dueto the higher derivative corrections.
A Obtaining the perturbed metric in the Einstein frame
The aim is to first solve (11) to leading order in γ and find the perturbed metric in theEinstein frame. Considering (11) at O ( γ ), we get that S O ( γ ) Einstein = S γ + S γ , (54)where S γ contains φ , and S γ is independent of it. Thus S γ = V Z ∞ U KK dU q ˜ˆ g h ˆ R − U H P − g s Q L (cid:16) UR (cid:17) / + γ (cid:16) UR (cid:17) − / ˆ W i , (55)and S γ = V Z ∞ U KK dU q ˜ˆ g h − ∂ U φ U H P − g s Q L (cid:16) UR (cid:17) / φ i . (56)13ince φ ∼ O ( γ ), we can replace the other fields in (56) by their supergravity values.Thus we get S γ = V Z ∞ U KK dU h − f ( U ) U ∂ U φ − g s Q U φ R i , (57)which vanishes on integrating by parts the first term, and using (2) and (4). So in orderto find the metric perturbation to O ( γ ), we only need to consider (55). Note that the dilatonperturbation to leading order is undetermined at O ( γ ) in the perturbative expansion.To evaluate (55), we use the parametrizations H ( U ) = (cid:16) UR (cid:17) / , K ( U ) = e a ( U )+ λb ( U ) , P ( U ) = e b ( U ) , L ( U ) = e c ( U ) U (cid:16) RU (cid:17) / , (58)where λ is a constant, which we now fix to simplify calculations. So in (58), we have c ( U ) ∼ O ( γ ). Using (58), we calculate q ˜ˆ g ˆ R and get q ˜ˆ g ˆ R = − e a +( λ − b +4 c √ U R h R − U R ( c ′ ) − e b − c ) U (cid:16) UR (cid:17) +5 U Ra ′ + 5( λ − U Rb ′ − U Rc ′ − U Rc ′ ( a ′ + λb ′ ) i − R / ddU h U / (cid:16) a ′ + λb ′ + 4 c ′ + 14 U (cid:17) e a +( λ − b +4 c i . (59)Note that for λ = 9, the coefficient of the U Rb ′ term vanishes, and this is the value wechoose. Thus (55) yields S γ = V Z ∞ U KK dU h l ( a, a ′ , b, b ′ , c, c ′ ) + γw ( a, a ′ , a ′′ , b, b ′ , b ′′ , c, c ′ , c ′′ ) i , (60)where l ( a, a ′ , b, b ′ , c, c ′ ) = − e a +8 b +4 c √ U R h R − U R ( c ′ ) − e b − c ) U (cid:16) UR (cid:17) +5 U Ra ′ − U Rc ′ − U Rc ′ ( a ′ + 9 b ′ ) i − U e a +8 b +4 c h(cid:16) RU (cid:17) / + 16 (cid:16) UR (cid:17) / e b − c i , (61)where we have dropped the total derivative, and w ( a, a ′ , a ′′ , b, b ′ , b ′′ , c, c ′ , c ′′ ) = U ( U R ) / e a +10 b +4 c ˆ W ( a, a ′ , a ′′ , b, b ′ , b ′′ , c, c ′ , c ′′ ) . (62)14o we need to solve the Euler–Lagrange equations of motion arising from (60) whichare given by ∂l∂ξ i − ddU (cid:16) ∂l∂ξ ′ i (cid:17) = − γ h ∂w∂ξ i − ddU (cid:16) ∂w∂ξ ′ i (cid:17) + d dU (cid:16) ∂w∂ξ ′′ i (cid:17)i , (63)where ξ i = a, b, c . The equations of motion for a, b and c are given by (cid:16) UR (cid:17) / h e b U (8 e c − − R e c (cid:16) U c ′′ − b ′ (5 + 8 U c ′ ) + 4 c ′ (4 + 5 U c ′ ) (cid:17)i = − γe a +8 b − c h ∂w∂a − ddU (cid:16) ∂w∂a ′ (cid:17) + d dU (cid:16) ∂w∂a ′′ (cid:17)i , r UR h R e c (cid:16) R + 2 U Ra ′ (5 + 8 U c ′ ) + 48 U Rc ′ (7 + 8 U c ′ ) + 144 U Rc ′′ (cid:17) +10 e b (cid:16) − e c (cid:17) U i = 2 Rγe a +8 b − c h ∂w∂b − ddU (cid:16) ∂w∂b ′ (cid:17) + d dU (cid:16) ∂w∂b ′′ (cid:17)i , r UR h U e b (3 + 4 e c ) − R e c n U (cid:16) U a ′ + 15 c ′ + 2 a ′ (3 + 17 U b ′ + 3 U c ′ )+2[72 U b ′ + 24 b ′ (1 + U c ′ ) + U (6 c ′ + a ′′ + 9 b ′′ + 3 c ′′ )] (cid:17)oi = − Rγe a +8 b − c h ∂w∂c − ddU (cid:16) ∂w∂c ′ (cid:17) + d dU (cid:16) ∂w∂c ′′ (cid:17)i , (64)respectively. We now expand (64) to O ( γ ). Since the right hand side is already of O ( γ ),we simply substitute the values in the supergravity solution. Defining U KK U ≡ η, (65)the relevant expressions are ∂w∂a − ddU (cid:16) ∂w∂a ′ (cid:17) + d dU (cid:16) ∂w∂a ′′ (cid:17) = 3 √ U R R h − η + 222212 η − η + 10728692 η i ,∂w∂b − ddU (cid:16) ∂w∂b ′ (cid:17) + d dU (cid:16) ∂w∂b ′′ (cid:17) = 3 √ U R R h − η + 105831 η − η + 4899321 η i ,∂w∂c − ddU (cid:16) ∂w∂c ′ (cid:17) + d dU (cid:16) ∂w∂c ′′ (cid:17) = − √ U R R h
945 + 216 η − η + 37581 η − η i . (66)15n the left hand side of (64), the contributions due to terms of O (1) vanish, and the O ( γ ) terms are the leading effects. Defining the order γ perturbations to the metric by a ( U ) = −
272 ln (cid:16) RU (cid:17) + 5ln f ( U ) + γa ( U ) ,b ( U ) = 32 ln (cid:16) RU (cid:17) −
12 ln f ( U ) + γb ( U ) ,c ( U ) = γc ( U ) , (67)where a , b , and c are O (1), substituting them into (64), and equating terms of O ( γ )we get the equations satisfied by the metric perturbations. Defining the dimensionlessvariables A = ( R U KK ) / a , B = ( R U KK ) / b , C = ( R U KK ) / c , (68)and changing coordinates to η , from (64) we get that − − η ) C ′′ + 12 C ′ η − − η ) B ′ η + 15 B + 12 C η = − √ η h − η + 222212 η − η + 10728692 η i ≡ f ( η ) √ η , − η ) C ′′ + 2( η − C ′ η − − η ) A ′ η − B + 4 C ) η = 11024 √ η h − η + 105831 η − η + 4899321 η i ≡ f ( η ) √ η , − − η )( A ′′ + 9 B ′′ + 3 C ′′ ) + 3(7 B − C ) η + (4 η + 5) A ′ η + 39( η + 1) B ′ η + 6( η + 2) C ′ η = 31024 √ η h
945 + 216 η − η + 37581 η − η i ≡ f ( η ) √ η . (69)From the equations (69), it follows that a has a zero mode given by a = const, whichis not fixed by the equations of motion. We shall fix its value shortly.Ignoring the issue of the zero mode of A for the time being, we now solve the equations(69) in order to find the metric perturbations in the Einstein frame. We find it convenientto further redefine variables 16 = η / A , B = η / B , C = η / C , (70)so that the metric perturbations are given by a ( U ) = A ( U ) √ R U , b ( U ) = B ( U ) √ R U , c ( U ) = C ( U ) √ R U , (71)Using this, we see that equations (69) reduce to − − η ) η C ′′ + 12(2 η − η C ′ − − η ) η B ′ + 15(1 + η )2 B + 6( η + 4) C = f ( η ) , − η ) η C ′′ + 2(7 − η ) η C ′ − − η )6 η A ′ − − η )4 A − B − η ) C = f ( η ) , − − η ) η ( A ′′ + 9 B ′′ + 3 C ′′ )+(10 η − η A ′ + 3(31 η − η B ′ + 6(4 η − η C ′ +3( 5 η A + 6(12 η + 11) B + 3(9 η − C = f ( η ) . (72)Though the system of equations (72) looks complicated, it is easy to see that they aresolved by sums of powers of harmonic functions η . So we make the ansatz A ( η ) = ∞ X k =0 ˆ a k η k , B ( η ) = ∞ X k =0 ˆ b k η k , C ( η ) = ∞ X k =0 ˆ c k η k , (73)where ˆ a k , ˆ b k and ˆ c k are numbers. Demanding regularity of the solution as U KK → k in (73). Thus including the zero mode, a ( U ) is givenby a ( U ) = const + 1 √ R U ∞ X k =0 ˆ a k (cid:16) U KK U (cid:17) k . (74)In the supersymmetric limit U KK → a ( U ) is given by a ( U ) = const + ˆ a √ R U . (75)We set the zero mode to zero so that the x coordinate is canonically normalized [17].Solving (72) boils down to solving a system of coupled difference equations obtained byequating terms involving the same powers of η . The equations are17ˆ b c = − , a b + 44ˆ c = − , a + 22ˆ b − c , (76)and − (cid:16) k − (cid:17)(cid:16) k + 2)ˆ c k − (6 k − c k − (cid:17) + 5(1 − k )2 (ˆ b k − ˆ b k − )= 2106 δ k, − δ k, + 474174 δ k, − δ k, , − k + 1)4 ˆ a k + 5(2 k − a k − + 6 (cid:16) k + k − (cid:17) ˆ c k − (cid:16) k − k + 8 (cid:17) ˆ c k − − b k = − δ k, + 1058312 δ k, − δ k, + 48993212 δ k, , (cid:16) − k + k + 2 (cid:1) ˆ a k + (cid:16) k − k + 12 (cid:17) ˆ a k − + (cid:16) − k + 3 k + 22 (cid:17) ˆ b k +3 (cid:16) k − k + 1 (cid:17) ˆ b k − − (cid:16) k + 232 (cid:17) ˆ c k + 9 (cid:16) k − k + 12 (cid:17) ˆ c k − = 108 δ k, − δ k, + 375812 δ k, − δ k, , (77)for k ≥
1. These equations can be solved recursively and yield the solution A ( η ) = 1512 h η − η + 536095621 η − η + . . . i , B ( η ) = 1512 h − − η + 45401435 η − η + 39126718351975 η + . . . i , C ( η ) = 1512 h − − η − η − η − η + . . . i , (78)where . . . are the terms higher order in η , starting from η . We do not have a closedform expression for the metric perturbations. Thus, for example, we cannot determine theprecise nature of the perturbations near U = U KK . Note that the metric perturbationsvanish as U → ∞ , i.e., far away from the color branes. B Obtaining the perturbed dilaton
We next calculate the dilaton perturbation at O ( γ ), for which we need to consider theaction (11) at O ( γ ). Including only the terms that depend on the dilaton, this gives18 γ φ = S γ φ + S γ φ , (79)where S γ φ = V Z ∞ U KK dU q ˜ˆ g h − ∂ U φ U H P − g s Q L (cid:16) UR (cid:17) / φ i , (80)which depends on φ , and S γ φ = V Z ∞ U KK dU q ˜ˆ g h − ∂ U φ U H P − ( ∂ U φ ) H P − g s · (cid:16) UR (cid:17) / (cid:16) φ + φ (cid:17) ˆ F − γ (cid:16) UR (cid:17) − / ˆ W φ i , (81)which depends on φ . In (80), φ ∼ O ( γ ), and so we substitute the supergravity valuesof the metric and the four form flux. Thus S γ φ vanishes for the same reason as in (56). In(81), we substitute the supergravity values of the various fields in the terms involving φ and in the term involving ˆ W φ , while in the remaining terms we substitute the values ofthe fields at O ( γ ). Now ˆ W for the supergravity metric (9) is given byˆ W = 9 (cid:16) U + 819 U U KK − U U KK + 4739 U KK (cid:17) U / R / . (82)Note that (82) does not vanish for U KK = 0, as the theory is not conformal. Thusdefining φ = γ ˆ φ , we get that S γ φ = γ V Z ∞ U KK dU h − f ( U ) U ( ∂ U ˆ φ ) − U ˆ φ − f ( U ) U (cid:16) a + 8 b + 4 c (cid:17) ( ∂ U ˆ φ ) − U (cid:16) a + 10 b − c (cid:17) ˆ φ − (cid:16) U + 819 U U KK − U U KK + 4739 U KK (cid:17) ˆ φ U / R / i . (83)This leads to the equation of motion(1 − η ) ϕ ′′ − ( η + 2) η ϕ ′ − ϕ η = 94 √ η (cid:16) A + 10 B − C (cid:17) − η + 1)8 √ η (cid:0) A + 8 B + 4 C (cid:17)
19 3(1 − η ) √ η (cid:0) A ′ + 8 B ′ + 4 C ′ (cid:17) + 272048 √ η (cid:16)
135 + 819 η − η + 4739 η (cid:17) , (84)where ϕ is the dimensionless dilaton perturbation defined by ϕ = ( R U KK ) / ˆ φ , (85)and all derivatives in (84) are with respect to η . Just like the metric perturbations,defining ϕ = η / ˆ ϕ , (84) leads to(1 − η ) η ˆ ϕ ′′ + (1 − η ) η ˆ ϕ ′ − η + )4 ˆ ϕ = 94 (cid:16) A + 10 B − C (cid:17) − η + 1)8 (cid:0) A + 8 B + 4 C (cid:17) + 3(1 − η ) η (cid:0) A ′ + 8 B ′ + 4 C ′ (cid:17) + 272048 (cid:16)
135 + 819 η − η + 4739 η (cid:17) . (86)which we solve by making the ansatzˆ ϕ ( η ) = ∞ X k =0 ˜ ϕ k η k . (87)The difference equations are˜ ϕ = − ˆ a − b + 4ˆ c − , (88)and (cid:16) k − (cid:17) ˜ ϕ k − (cid:16) k − k + 14 (cid:17) ˜ ϕ k − = k + a k − k − a k − + (2 k + 32 )ˆ b k − (2 k − b k − + (cid:16) k − (cid:17) ˆ c k − (cid:16) k − (cid:17) ˆ c k − + 34 (cid:16) δ k, − δ k, + 4739 δ k, (cid:17) , (89)for k ≥
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