Magnetic susceptibility of the quark condensate via holography
aa r X i v : . [ h e p - ph ] J a n Magnetic susceptibility of the quark condensate via holography
A. Gorsky ∗ and A. Krikun † Institute for Theoretical and Experimental PhysicsB. Cheremushkinskaya ul. 25, 117259 Moscow, Russia (Dated: October 22, 2018)We discuss the holographic derivation of the magnetic susceptibility of the quark condensate. It isfound that the susceptibility emerges upon the account of the Chern-Simons term in the holographicaction. We demonstrate that Vainshtein’s relation is not exact in the hard wall dual model but isfulfilled with high accuracy. Some comments concerning the spectral density of the Dirac operatorare presented.
PACS numbers: 11.25.Tq, 11.40.Ha, 11.15.Ex, 12.38.Aw
I. INTRODUCTION
The analysis of the QCD properties by holographicmethods is one of the most promising approaches to thedescription of the strong coupling region. The uniqueholographic model for QCD has not been found yet hencethere is no hope to get the generic quantitative predic-tions at present. However there are some QCD resultswhich seem to be independent on the details of the dualgeometry hence one could consider these universal ob-jects or relations to test the holographic picture. On theother hand it is instructive to analyze if some relationis universal indeed testing it in the different holographicgeometries.The simplest relation to be tested is the Gell-Mann-Rennes-Oaks one which was shown to be true in all holo-graphic models of QCD like hard wall models [1, 20],soft wall model [2] or Sakai-Sugimoto model [3]. Themain focus in our paper is the magnetic susceptibility ofthe quark condensate describing the response of the QCDvacuum on the external magnetic field. It was introducedin [4] in the context of the sum rules and investigationof its numerical value was performed in [5, 6, 16, 17].More recently using QPE arguments Vainshtein [7] ob-tained the expression for the susceptibility in terms ofthe known QCD quantities . However the status of thisrelation is questionable since both sides of the correspon-dence have different anomalous dimensions and it is notclear if the higher states could influence the answer. (see[18, 19])In this paper we shall analyze the magnetic suscepti-bility in the holographic setting and shall focus mostly atthe simplest hard wall model [1, 8](introduced in [21–23]).We shall consider the calculation of the three-point func-tion similar the two-point calculations in [8, 9] and form-factor calculations in [10, 11]. We shall consider the spe-cial kinematics of the three-point function related to thesusceptibility. It turns out that the only nontrivial con-tribution to the correlator comes from the Chern-Simons ∗ [email protected] † [email protected] term in the dual action and substituting the solutions tothe classical equations of motion we get the result for thesusceptibility which is close to the Vainshtein’s relation.The paper is organized as follows. First in SectionII we calculate the three-point function and the mag-netic susceptibility of quark condensate in the AdS/QCDhard wall model. In Section III we survey some otherapproaches to the calculation of this value, namely viaChiral Perturbation theory and via the relation with theDirac operator spectrum density. The conclusion is givenin Section IV. To make the paper self consistent, we statein the Appendix some results of [1, 9], which we will use. II. HARD WALL ADS/QCD MODEL.A. Chern-Simons action and 3-point function
Our aim is to calculate the correlation function of twovector and one axial currents. To make the holographiccalculation we take the simple ”hard wall” AdS/QCDmodel [1, 8]. We will work in notation of [9] and use someresults, calculated in [1, 9] (see also Appendix). The holo-graphic action involves kinetic and Chern-Simons term : S = S Y M ( A L .A R ) + S CS ( A L ) − S CS ( A R )where S CS ( A ) = N C π Z T r (cid:18) AF − A F + 110 A (cid:19) It is clear that only terms containing 3 gauge fields(
A, V, V )are relevant for the calculation of correlator:
T r ( A L F L F L − A R F R F R ) →−−−→ AV V T r ( V F V F A + V F A F V + AF V F V )The classical solutions for the fields have the form: V aµ ( z, Q ) = ˆ V aν ( Q ) · V µν ( Q , z ) (1) A aµ ( z, Q ) = ˆ A aν ( Q ) · A µν ( Q , z ) A µν ( Q , z ) = P ( Q ) ⊥ µν a ⊥ ( Q , z ) + P ( Q ) k µν a k ( Q , z ) V µν ( Q , z ) = P ( Q ) ⊥ µν v ( Q , z )where ˆ V aν ( Q ) and ˆ A aν ( Q ) provide the sources for the oper-ators. We assume the currents to have arbitrary chargeswith respect to SU (2) group, so the gauge group struc-ture of the result is: h AV ˜ V i = δ δ ˆ Aδ ˆ V δ ˆ˜ V S CS == N C π Z h T A T V T ˜ V i · ( V ˜ F V F A + ˜ V F A F V + AF V ˜ F V )+ h T A T V T ˜ V i · ( ˜ V F V F A + V F A ˜ F V + A ˜ F V F V )We will work in the gauge A z = V z = 0. To figure outthe integral over z , we rewrite it as: Z V ˜ F V F A + ˜ V F A F V + AF V ˜ F V == 4 ǫ µνρσ Z ∂ z A µ [ ∂ σ V ν ˜ V ρ − V ν ∂ σ ˜ V ρ ]+ A µ [ ∂ z V ν ∂ σ ˜ V ρ − ∂ σ V ν ∂ z ˜ V ρ ]+ ∂ σ A µ [ V ν ∂ z ˜ V ρ − ∂ z V ν ˜ V ρ ] Substituting Fourier components of fields (1) and in-troducing the tensors ǫ ⊥ µνρσ = ǫ αβγσ P ⊥ αµ ( k ) P ⊥ βν ( k ) P ⊥ γρ ( k ) ǫ k µνρσ = ǫ αβγσ P k αµ ( k ) P ⊥ βν ( k ) P ⊥ γρ ( k )we get (denote ˜ v = v ( k ), ˙ v = ∂ z v ) : h A ⊥ µ ( k ) V ν ( k ) ˜ V ρ ( k ) i == N C π h T A T V T ˜ V i δ ( k + k + k ) ǫ ⊥ µνρσ × Z dz ( ik σ )[ ˙ a ⊥ v ˜ v + a ⊥ ˙ v ˜ v − a ⊥ v ˙˜ v ] − ( ik σ )[ ˙ a ⊥ v ˜ v + a ⊥ v ˙˜ v − a ⊥ ˙ v ˜ v ]We can add a surface term ( − ik + ik ) ∂ z ( a ⊥ v ˜ v ) in theaction, in order to make the 3-point function vanish, ifone of vector momenta tends to zero. This will lead usto the expression: h A ⊥ µ ( k ) V ν ( k ) ˜ V ρ ( k ) i = − N C π h T A T V T ˜ V i δ ( k + k + k ) ǫ ⊥ µνρσ Z dz ( ik σ ) av ˙˜ v − ( ik σ ) a ˙ v ˜ v (2)Similarly h A k µ ( k ) V ν ( k ) ˜ V ρ ( k ) i = − N C π h T A T V T ˜ V i δ ( k + k + k ) ǫ k µνρσ Z dz ( ik σ ) av ˙˜ v − ( ik σ ) a ˙ v ˜ v (3) B. Solution for A ⊥ Let us consider the equation of motion for A ⊥ at small Q similarly to calculation in [10]. The equation is [9]: (cid:20) ∂ z (cid:18) z ∂ z A aµ (cid:19) + q z A aµ − R g Λ v z A aµ (cid:21) ⊥ = 0We will work with the bulk-to-boundary propagator a ⊥ ( z ), which is defined in (1) and denote R g Λ = k =3 (A.4). Using variable y = kσ z = α z , we get theequation ∂ y a + 13 y ∂ y a − a = Q α y − / a + 2 k mσ α y − / a + m α y − / a which is an inhomogeneous modified Bessel equation. Weintroduce here the dimension parameter α which equals (cid:0) kσ (cid:1) / = 395 M ev (see (A.3),(A.4)). One can argue thatthe last term is negligible and solution to homogeneous part is a (0) ( y ) = F y / [ AI / ( y ) + BK / ( y )] , (4)where constants are fixed by the conditions on the IRboundary y m = α z m = 1 .
82 (see (A.2)): ∂ z a ( z ) | z = z m = 3 αy / ∂ y a ( y ) | y m = 0 A = K / ( y m ); B = I − / ( y m )and UV boundary: a ( z ) | z = ǫ = F y / By − / Γ(1 / / = 1 F = 2 / B Γ(1 / . Given this solution (which corresponds to Q=0), wecan compute f π , using the recipe, described in [1] (see(A.1)). f π = − Rg ∂ z a ( z ) z | z =0 ,Q =0 = Rg . α ∼ (85 M ev ) (5) C. Solution for A k To obtain the longitudinal part of the 3-point func-tion we need to find bulk-to-boundary propagator in thepseudoscalar sector. It is the solution to equations [9]: ∂ z (cid:18) z ∂ z ϕ a (cid:19) + R g v z ( π a − ϕ a ) = 0 (6) Q ∂ z ϕ a + R g v z ∂ z π a = 0 , (7)where ϕ is related to the longitudinal part of A µ as A k µ = ∂ µ ϕ . We introduce the function ψ ( z ) = ϕ ( z ) − π ( z ), andeliminate π ( z ) from the system to get an equation on ψ with the dimensionless variable t = αzt∂ t (cid:18) t ∂ t ψ (cid:19) − k v t ψ − t∂ t t q k v t + q ∂ t ψ ! = 0 , where q = Q /α . Now we can substitute v ( t ) = σα t + mα t and write down terms up to the first order in m/α and q , assuming Q to be small enough. ∂ y ψ + 13 ∂ y ψy − ψ = 29 k mσα y − / ψ + q y − / (cid:20) − ∂ y y (cid:21) ψ + O (cid:16) q , q mσα (cid:17) , (8) where y = t .The homogeneous solution, subject to the boundaryconditions π ( ǫ ) = 0 , ϕ ( ǫ ) = 1 , ∂ z π ( z m ) = ∂ z ϕ ( z m ) = 0 isthe same as for a ⊥ (4) as expected at Q = 0. The Greenfunction of this equation is: G ( u, v ) = u / v / AD − BC [ AI / ( u ) + BK / ( u )] × [ CI / ( v ) + DK / ( v )]with C and D defined by the condition: G ( y, y ′ ) | y = ǫ = 0 C = − K / ( ǫ ); D = I / ( ǫ ) . It satisfies the equation (cid:20) ∂ y + 13 y ∂ y − (cid:21) G ( y, y ′ ) = δ ( y − y ′ ) 1 y / . We can compute the correction due to the quark massin (8). It is obtained by the integral: ψ ( m ) ( y ) = 2 k mσ α Z y m ǫ y ′ / G ( y, y ′ ) y ′− / ψ (0) ( y ′ ) == 2 k mσ α F y / [ AI / ( y ) + BK / ( y )] AD − BC Z yǫ y ′ / [ CI / ( y ′ ) + DK / ( y ′ )][ AI / ( y ′ ) + BK / ( y ′ )]+ 2 k mσ α F y / [ CI / ( y ) + DK / ( y )] AD − BC Z y m y y ′ / [ AI / ( y ′ ) + BK / ( y ′ )][ AI / ( y ′ ) + BK / ( y ′ )]Using the value of y m = 1 .
82, which corresponds tothe IR boundary in our model, we get for the correction: ψ ( m ) ( y ) = − . · k mσ α z Consequently, the solution for ϕ with correction dueto the quark mass is: ϕ ( z ) = ψ ( z ) + O ( q ) == F αz [ AI / ( y ) + BK / ( y )] − k mσα z (9)we neglect here the correction due to the q term assumingthat Q ≪ α = 1180 M ev . D. Magnetic susceptibility of the quark condensate
In this Subsection we calculate the magnetic suscepti-bility χ of the quark condensate defined as h ¯ qσ µν q i F = χ h ¯ qq i F νµ (10)In order to find magnetic susceptibility , we study the3-point function h A k µ ( − Q ) V ν ( Q − k ) ˜ V ρ ( k ) i in the limit k →
0, according to [7] where the followingexpression for the susceptibility has been obtained χ = − N c π f π . (11)Consider the classical solutions for the vector fields,calculated in [1, 9]. v ( Q, z ) = Qz (cid:18) K ( Qz ) + K ( Qz m ) I ( Qz m ) I ( Qz ) (cid:19) −−−→ Q → v ( k, z ) = k z (cid:18) K ( kz ) + K ( kz m ) I ( kz m ) I ( kz ) (cid:19) ∼ k ln ( k )and substitute them into the correlator (3) : h A k µ ( − Q ) V ν ( Q − k ) ˜ V ρ ( k ) i == N C π h T A T V T ˜ V i ǫ k µνρσ ( ik σ ) Z dz a k ( z ) ˙ v ( Q, z ) == N C π h T A T V T ˜ V i ǫ k µνρσ ( ik σ ) × Z dz h ϕ (0) ( z ) + ϕ ( m ) ( z ) i [ Q zK ( Qz )]In this integral due to the fast fall of the vector prop-agator K ( Qz ) we can take the boundary value of φ (0) in the first term. The second term can be calculatedexplicitly Z dzϕ (0) ( z )[ Q zK ( Qz )] = ϕ (0) Z dzQ zK ( Qz ) = 12 Z dzϕ ( m ) ( z )[ Q zK ( Qz )] == 29 k mσα Z dzz [ Q zK ( Qz )] = 1 . m h ¯ qq i Q f π where we’ve used the result (5) and the relation (A.3).Finally, we get the expression for 3-point function withcorrections: h A k µ ( − Q ) V ν ( Q − k ) ˜ V ρ ( k ) i = h T A T V T ˜ V i ǫ k µνρσ ( ik σ ) (cid:20) N C π − . N c π m h ¯ qq i Q f π + O (1 /Q ) (cid:21) which can be matched the OPE of [7]: h A k µ ( − Q ) V ν ( Q ) ˜ V ρ (0) i = h T A T V T ˜ V i ǫ k µνρσ ( ik σ ) (cid:20) N C π + 4 m f h ¯ qq i χQ + O (1 /Q ) (cid:21) This comparison allows us to determine the magnetic sus-ceptibility of quark condensate χχ = − . N c π f π (12)in close agreement with the result of Vainshtein (11).This agreement is parametrical, but not numerical, be-cause due to the small value of f π (5) in our model, weget χ mod = 11 . − which is too large. Anyway, tun-ing the parameters of the model (mainly h ¯ qq i ) can helpfix f π to its real value and get the reasonable numericalagreement with Vainshtein’s χ vain = 8 . − . We’vechecked, that such variation of parameters do not affectcoefficient in (12) significantly. Namely, its value changesless than 5%, then the parameter y m which is propor-tional to h ¯ qq i /m ρ is varied in the wide range from 1 to 8. III. OTHER APPROACHESA. 3-point function in ChPT
We can compute the same 3-point function in the Chi-ral Perturbation theory [13] and compare the result withAdS/QCD. Note that the chiral Lagrangian is derived inthe Sakai-Sugimoto [3] model hence the comment belowcan be considered as the justification of the Vainshtein[7] relation. To obtain the h AV V i correlator, we con-sider the Wess-Zumino-Witten term in ChPT action andturning on axial and vector external currents, Z χP T = Z d xL + Z W ZW L = F h D µ U D µ U † i , with D µ = ∂ µ U − ir µ U + iU l µ Z [ U, l, r ] W ZW == − iN c π Z M d xǫ ijklm h Σ Li Σ Lj Σ Lk Σ Ll Σ Lm i− iN c π Z d xǫ µναβ (cid:0) W ( U, l, r ) µναβ − W (1 , l, r ) µναβ (cid:1) W ( U, l, r ) µναβ == h U l µ l ν l α U † r β + 14 U l µ U † r ν U l α U † r β + iU ∂ µ l ν l α U † r β + i∂ µ r ν U l α U † r β − i Σ Lµ l ν U † r α U l β + Σ Lµ U † ∂ ν r α U l β − Σ Lµ Σ Lν U † r α U l β + Σ Lµ l ν ∂ α l β + Σ Lµ ∂ ν l α l β − i Σ Lµ l ν l α l β + 12 Σ Lµ l ν Σ Lα l β − i Σ Lµ Σ Lν Σ Lα l β i− ( L ↔ R ) . Here U = exp i √ F π a t a ! , Σ Lµ = U † ∂ µ U, Σ Rµ = U ∂ µ U † and ( L ↔ R ) stands for U ↔ U † , l µ ↔ r µ , Σ Lµ ↔ Σ Rµ The leading contribution is given by the tree diagram,including ( aπ ) vertex from L and ( πvv ) vertex from L W ZW ∆ L [ aπ ] = √ F h ∂ µ πa µ i ∆ L W ZW [ πvv ] = N c π √ F ǫ µναβ × h ∂ µ π∂ ν v α v β − ∂ µ πv α ∂ ν v β i We can check, that the longitudinal part is exactly thesame as in [7], if we, formally, expand it in M /Q : h A k µ ( − Q ) V ν ( Q ) ˜ V ρ ( k ) i == h T A T V T ˜ V i ǫ k µνρσ ( ik σ ) N C π (cid:20) − M Q (cid:21) ( Q is Euclidean momentum) B. Relation with the Dirac operator spectrum
Let us also briefly comment on the different calculationof the magnetic susceptibility via the spectral density ofthe Dirac operator and introduce eigenfunctions of theDirac operator in the external gluon field Aˆ D ( A ) u λ ( x ) = λu λ ( x ) Then the standard definition of the spectral density readsas ρ ( λ ) = h V − X n δ ( λ − λ n ) i A where V is Euclidean volume and the averaging over thegluon ensemble is assumed. The value of the spectraldensity at the origin is fixed by the Casher-Banks rela-tion [14] while the linear term was determined comparingthe different calculations of the correlator of the scalarcurrents [15]. In the perturbation theory the spectraldensity behaves as ρ ( λ ) ∝ λ that is starting from thethird order the universality is lost because of the mixingwith the perturbative modes. We would like to note thatthe magnetic susceptibility is sensitive to the last ”non-perturbative” quadratic λ term in the spectral density.To explain this point let us consider the ” two-pointloop diagram” with tensor and vector vertexes in termsof the eigenfunctions and eigenvalues of the Dirac opera-tor. The simple inspection shows that the susceptibilityis expressed in terms of two different contributions. Thefirst ”diagonal” contribution reads as m Z dλ ρ ( λ )( λ + m ) while the second ”nondiagonal” contributions involvesthe following integrals Z d x ¯ u λ ( x ) x ν u λ ′ ( x )and double integrals over the eigenvalues R dλ R dλ ′ . The”diagonal” contribution is IR divergent and this diver-gence is expected to be canceled by the ”nondiagonal”terms amounting to a kind of sum rules. On the otherhand it is clear that quadratic term in the spectral densityyields the finite contribution. It is not clear if the ”non-diagonal” terms yield the IR finite contribution as well.This point does not allow us to write down the coefficientin front of the λ in the spectral density immediately.One can not also exclude that more careful treatment ofthe IR divergences should involve the derivation a kind ofthe effective action with the tensor insertion . We hopeto discuss these issues elsewhere. IV. CONCLUSION
In this paper we have derived the expression for themagnetic susceptibility of the quark condensate in theholographic QCD model. We have demonstrated thatthis object captures nontrivial anomalous properties ofthe dual model encoded in the Chern-Simons term. Itvanishes if the CS term is not taken into account. Thesecond important lesson concerns the validity of Vain-shtein’s relation which is not exact but is fulfilled withthe high accuracy.The numerical value of the susceptibility do not co-incide with recent estimations from the instanton liquidmodel [16, 17],sum rules fit [18] and phenomenology ofD-meson decays [19]. But it is calculated at significantlyless energy scale: for our calculation Q ≪ Q ∼ . ACKNOWLEDGMENTS
We are grateful to I. Denisenko and P. Kopnin for theuseful discussions. The work of A.G. was supported inpart by grants, INTAS-1000008-7865, PICS- 07-0292165and of A.K. by Russian President’s Grant for Supportof Scientific Schools NSh-3036.2008.2, by RFBR grant09-02-00308 and Dynasty Foundation.
Appendix
In Appendix, we state some results of [1, 9] concerningthe ”hard wall” AdS/QCD model. The 5D coupling con-stant g is fixed by the 2-point function of vector currentsin [1] g R = 12 π N c (A.1)The position of the IR boundary z m is related to the ρ -meson mass [1]. z m = 1323 M ev (A.2)The parameter σ is coupled with the value of quark con-densate (we take the value h ¯ qq i = (230Mev) ) and equals[9]: σ = N f h ¯ qq i R Λ = (460 M ev ) (A.3)We shall also fix the constant Λ correcting calculationmade in [9]. First, compute the leading order solutions tothe equation of motion for the pseudoscalar fields. Theseare solutions to the equations of motion (6), (7) withfixed boundary value of φ at z = ǫ . Differentiating (7)and substituting ∂ z φ from (6) we get : ∂ z v z ∂ z π − (cid:18) ∂ z v z (cid:19) z v ∂ z v z ∂ z π − Q v z ∂ z π − g R Λ v z ∂ z π = 0 . We need to solve it near the boundary, so substituteasymptotic value v ( z ) = mz | z → and denoting x = Qz ittakes the form: ∂ x x ∂ x π + 1 x ∂ x x ∂ x π − x ∂ x π − g R Λ m Q x ∂ x π = 0 . At large Q we neglect the last term and obtain the mod-ified Bessel equation with λ = 0 . Hence the solution for π ( z ) reads as : π ( z ) = A ′ QzI ( Qz ) + B ′ QzK ( Qz ) − C ′ . and using (6) we immediately obtain the solution for φ : φ ( z ) = − g R Λ m Q Qz [ A ′ I ( Qz ) + B ′ K ( Qz )] + C ′ . The boundary condition on φ at z = ǫ fixes the constant B ′ : φ (0 , q ) = φ ( q ) = − g R Λ m Q B ′ + B ′ B ′ = 11 − g R Λ m Q φ ( q )therefore we finally get : φ ( z ) | z = ǫ = φ ( q ) ∂ z φ ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) z = ǫ = − g R Λ m Q B ′ φ ( q ) Q ln ( Q ǫ ) π ( z ) | z = ǫ = 0We can compute the 2-point function of pseudoscalarcurrents, using the relation: ∂ µ (¯ qγ γ µ q ) = 2 m q (¯ qγ q ) . which yields us the source for pseudoscalar current2 m q φ ↔ (¯ qγ q ) . In order to obtain the 2-point function, we vary theaction twice with respect to 2 m q φ (0) and find: δS π = Z d x Rg (cid:20) δ∂ µ φ ∂ z ∂ µ φz (cid:21) z = ǫ − Λ R (cid:20) δπ v z ∂ z π (cid:21) ǫ = Z ( x,q ,q ) e ı ( q + q ) x (cid:18) RQ g (cid:20) δφ ( q , z ) ∂ z φ ( q , z ) z (cid:21) z = ǫ − Λ R m (cid:20) δπ ( q , z ) ∂ z π ( q , z ) z (cid:21) z = ǫ (cid:19) , hence, the pseudoscalar correlator is: h J aπ ( q ) , J π ( q ) b i == 2 δ ab m RQ g (cid:20) − g R Λ m Q B ′ Q ln ( Q ǫ ) (cid:21) →−−−→ m =0 δ ab R Λ Q ln ( Q ǫ ) Comparing with the QCD value [12]: h J aπ ( q ) , J π ( q ) b i = δ ab N c π Q ln ( Q ǫ )we find Λ R = N c π = R g k = R Λ g = 3 (A.4) [1] J. Erlich, E. Katz, D. T. Son and M. A. Stephanov,“QCD and a Holographic Model of Hadrons,” Phys. Rev.Lett. , 261602 (2005) [arXiv:hep-ph/0501128].[2] A. Karch, E. Katz, D. T. Son and M. A. Stephanov,“Linear Confinement and AdS/QCD,” Phys. Rev. D ,015005 (2006) [arXiv:hep-ph/0602229].[3] T. Sakai and S. 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