A holographic description of scalar mesons is presented, in which two- and
three-point functions are holographically reconstructed. Mass spectrum, decay
constants, eigenfunctions and the coupling of the scalar states with two pseu-
doscalars are found. A comparison of the results with current phenomenology is
discussed.Comment: 4 pages, talk given at IFAE 2009 (Incontri di Fisica delle Alte
  Energie 2009), Bari, Italy, 15-17 Apr 200

Nicotri, Stefano

English

arXiv

A holographic description of scalar mesons is presented, in which two- and three-point functions are holographically reconstructed. Mass spectrum, decay constants, eigenfunctions and the coupling of the scalar states with two pseudoscalars are found. A comparison of the results with current phenomenology is discussed

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DOI 10.1393/ncc/i2009-10415-7Colloquia: IFAE 2009IL NUOVO CIMENTO Vol. 32 C, N. 3-4 Maggio-Agosto 2009Holographic scalar mesonsS. Nicotri(∗)Institute for Particle Physics Phenomenology, Department of Physics, Durham UniversityDurham DH1 3LE, United Kingdom(ricevuto il 19 Settembre 2009; pubblicato online il 6 Ottobre 2009)Summary. — A holographic description of scalar mesons is presented, in whichtwo- and three-point functions are holographically reconstructed. Mass spectrum,decay constants, eigenfunctions and the coupling of the scalar states with two pseu-doscalars are found. A comparison of the results with current phenomenology isdiscussed.PACS 11.25.Tq – Gauge/string duality.PACS 12.38.Lg – Other nonperturbative calculations.PACS 12.40.Yx – Hadron mass models and calculations.PACS 14.40.Cs – Other mesons with S = C = 0, mass < 2.5 GeV.1. – IntroductionMany approaches and techniques have been developed to understand QCD in its non-perturbative regime, but, up to now, no one has been completely satisfactory, leavingstrongly coupled theories still a mystery. Recently, the possibility to apply AdS/CFTcorrespondence [1, 2] methods to (the large N limit of) strongly coupled gauge theorieshas been pointed out. This direction, known as AdS/QCD [3], has been followed alongtwo main approaches. The first is a top-down approach, consisting in the attempt toobtain QCD-like theories as gravity duals of certain limits of well-defined superstringframeworks [4]. The second is a phenomenological approach, consisting in building ahigher-dimensional model able to describe certain relevant degrees of freedom of stronginteraction, assuming its validity as QCD dual. This is the approach followed in thepresent discussion. Many aspects of chromodynamics have been studied in this frame-work, like chiral symmetry breaking [5, 6], deep inelastic scattering [7], deconfinementtransition [8, 9] and Q¯Q potential [10, 11], form factors [12] and spectra [13-18]. In thispaper, the model for chiral symmetry breaking introduced in [6] and, with a differentaim, in [19] is used to investigate the scalar meson sector [15], which is still debatednowadays, due to its features at large Nc [20].(∗) E-mail: stefano.nicotri@durham.ac.ukc© Societa` Italiana di Fisica 7374 S. NICOTRI2. – ModelThe model is defined by the five-dimensional action(1) S = −1k∫d5x√−g e−Φ(z) Tr{|DX|2 + m25X2 +12g25(F 2V + F2A)}in the five-dimensional Anti-de Sitter spacetime (the bulk), defined by the metric gMN =(R2/z2)ηMN , where ηMN is the Minkowski metric tensor with signature −++++, R isthe AdS radius and z is the fifth holographic coordinate 0  z < ∞. Every field is dualto a QCD operator defined on the boundary z = 0. X = (X0 + S)e2iπ is a scalar field,whose (negative) mass is fixed by the formula m25R2 = (Δ−p)(Δ+p−4), where Δ is thedimension of the corresponding operator and p is the order of the p-form (i.e. p = 0). πis the chiral field and X0 = v(z)/2 is dual to 〈q¯q〉 and is responsible for chiral symmetrybreaking. S = SATA = S1T 0 + Sa8Ta with T 0 = (1/√6)1 and T a the generators ofSU(3)F , (A = {0, a}, with a = 1, . . . 8). SA is dual to the QCD operator OAS = q¯TAqrepresenting the scalar mesons. FMNV = ∂MV N − ∂NV M − i[V M , V N ]− i[AM , AN ] andFMNA = ∂MAN − ∂NAM − i[V M , AN ]− i[AM , V N ] are the strength tensors of the fieldsV aM and AaM , obtained rotating AaL,R, which are inserted to gauge, in the bulk, the globalchiral symmetry SU(3)L ⊗ SU(3)R, broken to SU(3)V by 〈q¯q〉. AaL,R are dual to theq¯L,RγμT aqL,R currents. The field Φ(z) = c2z2, which in fact defines the model, is anon-dynamical field, inserted to break the conformal symmetry in the UV (c being amass parameter).Assuming for QCD the validity of the AdS/CFT relation(2)〈exp[i∫d4x (L+ ϕ0(x)O(x))]〉QCD= eiS[ϕ(x,z)] ,where the l.h.s. is the QCD generating functional and ϕ0(x) is the boundary (z → 0)value of the 5d field ϕ(x, z), the effective action (1) is the only ingredient needed toevaluate correlation functions.3. – SpectrumTo evaluate the spectrum of the scalar mesons, consider the quadratic action for thefield SA:(3) Seff = − 12k∫d5x√−g e−Φ (gMN∂MSA∂NSA + m25SASA) .Looking for a solution of the equation of motion which is a plane wave in the 4d coordi-nates, SA(x, z) = eiq·xS˜(z), the masses are found solving a second-order linear differentialequation, whose normalizable solutions represent the wave functions of the scalar mesons.The spectrum is discrete and the eigensystem is [21](4) m2n = c2(4n + 6) S˜n(z) =√2n + 1c3z3L1n(c2z2)with L1n the generalized Laguerre polynomials. The scalar mesons are then organized ina Regge trajectory and, fixing the parameter c with the ρ-meson mass, c = mρ/2 [6], theyHOLOGRAPHIC SCALAR MESONS 75turn out to be heavier than vector mesons, with m0 = 943MeV, and in good agreementwith the experimental masses, considering a0(980) and f0(980) as the lightest scalarstates.4. – Two-point correlation functionThe next step is to evaluate the two-point correlation function, defined in QCD as(5) ΠABQCD(q2) = i∫d4x eiq·x〈0|T [OAS (x)OBS (0)] |0〉.Writing S˜(q2, z2) = S(q2, z2)S˜0(q2), with S˜0(q2) the Fourier transform of the sourceof the operator OS in the QCD generating functional and S(q2, z2) a function calledbulk-to-boundary propagator [2], and using (2), one can evaluate (5) deriving twice theeffective action (1) with respect to S˜0, obtainingΠABAdS(q2) = δABR3kS(q2, z2)e−c2z2z3∂zS(q2, z2)∣∣∣∣z=1/ν→0(6)= δAB4c2Rk[(q24c2+12)ln(c2z2)+(γ − 12)+q24c2(2γ − 12)+(q24c2+12)ψ(q2/4c2 + 3/2)]∣∣∣∣z=1/ν.This function has poles at −q2n = m2n = c2(4n+ 6), in agreement with (4), with residuesF 2n = 16Rc4(n + 1)/k corresponding to the decay constants of the scalar mesons. Thefactor R/k can be fixed with a comparison of (6) with the known QCD result, obtainingR/k = Nc/(16π2). The AdS prediction F0 = 0.08GeV2 can be compared to QCDdeterminations Fa0 = (0.21± 0.05)GeV2 and Ff0 = 0.18 ± 0.015GeV2 [22, 23], showinga difference of about a factor of two.5. – Three-point correlation function and interaction with two pseudoscalarsThe three-point correlation function describing the interaction between a scalar mesonand two pseudoscalars is defined in QCD by(7) ΠabcQCDαβ(p1, p2) = dabc p1α p2βp21 p22f2π∞∑n=0Fn gSnPPq2 + m2nwith q = −(p1 + p2), fπ the pion decay constant and gSnPP the coupling.To evaluate it on the AdS side, consider the corresponding interaction term in (1):(8) SSPPeff = −R3k∫d5xe−Φ(z)z3v(z)[2√6S1(∂ψ)2 + dabcSa8ηMN(∂Mψb)(∂Nψc)]76 S. NICOTRIwhere AM = A⊥M +∂Mφ and ψa = φa−πa is the pseudoscalar dual field. Differentiatingwith respect to the sources, the coupling for the n = 0 state is given by(9) gS0PP =m2S0Rc√Nc4πf2π∫ ∞0du e−u2v(u)with u = cz. The numerical result is of O(10)MeV, at odds with experimental valuesga0ηπ = 12 ± 6GeV and phenomenological determinations gf0K+K− 	 7GeV [24]. Thisis an issue of the model and it is due to the fact that the integral in (9) is dominated bythe small quark mass parameter. This is related to the difficulty of the model in correctlyreproducing both spontaneous and explicit chiral symmetry breaking, difficulty alreadyanalyzed in [6].∗ ∗ ∗The author gratefully acknowledges an Early Stage Researcher position supported bythe EU-RTN Programme, Contract No. MRTN–CT-2006-035482, “Flavianet”. It is apleasure to thank P. Colangelo, F. De Fazio, F. Giannuzzi and F. Jugeau forcollaboration.REFERENCES[1] Maldacena J. M., Adv. Theor. Math. Phys., 2 (1998) 231.[2] Witten E., Adv. Theor. Math. Phys., 2 (1998) 253.[3] Polchinski J. and Strassler M. J., Phys. Rev. Lett., 88 (2002) 031601.[4] See e.g. Gursoy U. and Kiritsis E., JHEP, 0802 (2008) 032 and references therein.[5] Erlich J. et al., Phys. Rev. Lett., 95 (2005) 261602.[6] Karch A. et al., Phys. Rev. D, 74 (2006) 015005.[7] Ballon Bayona C. A., Boschi-Filho H. and Braga N. R. F., JHEP, 0803 (2008)064.[8] Ballon Bayona C. A. et al., Phys. Rev. D, 77 (2008) 046002.[9] Herzog C. P., Phys. Rev. Lett., 98 (2007) 091601.[10] Andreev O. and Zakharov V. I., JHEP, 0704 (2007) 100; Phys. Lett. B, 645 (2007)437; Phys. Rev. D, 74 (2006) 025023.[11] Boschi-Filho H., Braga N. R. F. and Ferreira C. N., Phys. Rev. D, 74 (2006) 086001.[12] Brodsky S. J. and de Teramond G. F., Phys. Rev. D, 77 (2008) 056007.[13] Carlucci M. V. et al., Eur. Phys. J. C, 57 (2008) 569; Giannuzzi F., Phys. Rev. D, 78(2008) 117501; 79 (2009) 094002.[14] de Teramond G. F. and Brodsky S. J., Phys. Rev. Lett., 102 (2009) 081601; 94 (2005)201601.[15] Colangelo P., De Fazio F., Giannuzzi F., Jugeau F. and Nicotri S., Phys. Rev. D,78 (2008) 055009.[16] Colangelo P., De Fazio F., Jugeau F. and Nicotri S., Phys. Lett. B, 652 (2007) 73;Int. J. Mod. Phys. 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Holographic scalar mesons

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Holographic scalar mesons

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