Scattering vector mesons in D4/D8 model

arXiv:0912.0191v1 [hep-th] 1 Dec 2009 Scattering vector mesons in D4/D8 model C. A. Ballon Bayonaa ∗ , Henrique Boschi-Filhob† , Nelson R.F. Bragab ‡ and Marcus A. C. Torresb § a Centro Brasileiro de Pesquisas Fı́sicas, Rua Dr. Xavier Sigaud 150, Urca, 22290-180 Rio de Janeiro, RJ, Brazil b Instituto de Fı́sica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro, RJ, Brazil We review in this proceedings some recent results for vector meson form factors obtained using the holographic D4-D8 brane model. The D4-D8 brane model, proposed by Sakai and Sugimoto, is a holographic dual of a semi-realistic strongly coupled large Nc QCD since it breaks supersymmetry and incorporates chiral symmetry breaking. We analyze the vector meson wave functions and Regge trajectories as well. 1. Introduction vector meson wave functions ψn (z), masses and couplings and then we discuss our results for the vector meson form factor. In particular, we analyze the elastic case in which we extract the magnetic and quadrupole moments. Form factors have also been calculated using other holographic models like the hard and soft wall model [8,9,10] and the D3/D7 brane model [11]. Sakai and Sugimoto proposed an elegant string model dual to large Nc QCD at strong coupling [1]. This model consists on the intersection of Nc D4-branes and Nf D8-D8 pair of branes in type IIA string theory in the limit Nf ≪ Nc where Nc and Nf are interpreted as the color and flavor number of strongly coupled QCD. The principal characteristic of the Sakai-Sugimoto model is the holographic description of chiral symmetry breaking U (Nf )L × U (Nf )R → U (Nf ) from the merging of the D8-D8 branes. This model has been used in the recent years to describe various aspects of hadron physics [1,2,3,4,5]. In this proceedings we discuss the scattering of a photon with a vector meson using the D4D8 brane model, summarizing the results of ref [6]. One remarkable result in the D4-D8 model is the realization of an important property of hadron physics known as vector meson dominance (VMD) [7] where a hadron-photon interaction is mediated by vector mesons. As a consequence of VMD, the vector meson form factor takes the form of a sum involving vector meson masses and couplings. We first present some results for the 2. Vector mesons in the D4/D8 Model The induced metric in the probe D8-brane embedded in a D4 background can be written as [1] ds2ßD8 = h(Uz ) ηµν dxµ dxν + + R3/2 Uz1/2 dΩ24 , 4 UßKK dz 2 9 Uz h(Uz ) (1) 2 where Uz = UßKK (1 + z 2 /UßKK )1/3 , h(Uz ) = 3/2 (Uz /R)√ , the constant R is related to the string length α′ and the string coupling gs by R3 = πgs Nc α′3/2 and UßKK is related to the Kaluza1/2 Klein mass scale by MßKK = 3UßKK /2R3/2 . From the DBI action for U (Nf ) gauge fields in the D8-brane, we obtain a four dimensional effective lagrangian that can be written as [2] ∗ email: [email protected] [email protected] ‡ email: [email protected] § email: [email protected] L † email: 1 1 1 Tr(∂µ ṽνn − ∂ν ṽµn )2 + Tr(∂µ ãnν − ∂ν ãnµ )2 2 2 gVvn + Mv2n Tr(ṽµn − 2 Vµ )2 Mv n = 2 gAan Aµ )2 Ma2n X + Tr(i∂µ Π + fπ Aµ )2 + Lj , + Ma2n Tr(ãnµ − Κ Ψ1 0.6 (2) 0.4 j≥3 0.2 where ṽµn and ãnµ represent the vector and axial vector mesons, Vµ and Aµ are external vector and axial vector gauge fields from gauged chiral symmetry U (Nf )L × U (Nf )R , Π is a massless pion field and Lj represent interaction terms of order j. Later on we turn Aµ off and turn on a single abelian subgroup of U (Nf ) in Vµ that will be the source of electromagnetic interaction. The masses and couplings are defined by Mv2n 2 = λ2n−1 MßKK Z 2 n gVv = κMvn gAan = κMa2n Z 2 , = λ2n MßKK dz̃ K(z̃)−1/3 ψ2n−1 (z̃) dz̃K(z̃) −(K(z̃)) 20 40 20 40 20 40 20 40 20 40 Ž z -0.4 -0.6 Κ Ψ2 0.6 0.4 0.2 -20 Ž z -0.2 -0.4 -0.6 ψ2n (z̃)ψ0 (z̃) (3) Κ Ψ3 0.6 where the wave functions ψn are subjected to the conditions Z κ dz̃(K(z̃))−1/3 ψn (z̃)ψm (z̃) = δnm , (4) 1/3 -20 -0.2 -40 Ma2n −1/3 -40 0.4 0.2 -40 -20 -0.2 -0.4 ∂z̃ [K(z̃)∂z̃ ψn (z̃)] = λn ψn (z̃) , (5) where z̃ = z/UßKK , K(z̃) = 1 + z̃ 2 and (6π)3 κ = gY2 M Nc2 . Note that the constant gVvn in (2) is the coupling of the interaction between a vector meson ṽµn and an external U (1) field Vµ . -0.6 Κ Ψ4 0.6 0.4 0.2 -40 -20 Ž z -0.2 Wave functions. A regularity condition for the wave functions ψ2n (z̃) and ψ2n−1 (z̃) at the origin z̃ = 0, together with their parity ψn (−z̃) = (−1)n ψn (z̃) Ž z -0.4 -0.6 Κ Ψ 30 (6) 0.6 leads to the conditions 0.4 0.2 ∂z̃ ψ2n (0) = 0 , ψ2n−1 (0) = 0 . (7) We solve numerically the equations of motion for the vector and axial-vector modes using the shooting-method. From the normalization condition (4) and equation of motion (5) one sees that ψn decrease as z̃ −1 when z̃ → ±∞. Defining ψ̃n ≡ z̃ψn , the equation of motion takes the form h i z̃∂z̃ z̃∂z̃ ψ̃n + A(z̃) z̃∂z̃ ψ̃n + B(z̃) ψ̃n = 0 , (8) -40 -20 Ž z -0.2 -0.4 -0.6 Figure 1. Wave functions ψn (z̃) multiplied by for the cases n = 1, 2, 3, 4 and n = 30. √ κ 3 n √ gvn2 κMKK λ2n−1 √ Λn 700 κgvn v1 v1 λ2n 600 500 1 2 3 4 5 6 7 8 9 0.66931 2.87432 6.59118 11.79669 18.48972 26.67017 36.33796 47.49318 60.1312 2.10936 9.10785 20.7957 37.1502 58.1701 83.834 114.152 148.103 188.695 0.44658 −0.14654 1.8434ß×10ß−2 −3.6885ß×10ß−4 2.6953ß×10ß−4 3.0775ß×10ß−5 1.8572ß×10ß−5 6.9961ß×10ß−6 3.5081ß×10ß−6 1.56877 4.5461 9.00797 14.9573 22.394 31.3182 41.7297 53.6285 67.0149 Table 1 Dimensionless squared masses and coupling constants, where κ = (gY M Nc )2 /216π 3 . 400 300 200 100 ** ********* **** *** 10 *** * * *** ** 20 ** ** 30 * ** ** ** * * ** 40 ** ** * * ** * * * 50 * * * * * * 60 n Log Λn 6 4 2 * * * * ** ** ****** ******* ****** ***** * * * * * **** **** *** *** * * ** ** * * * where A(z̃) and B(z̃) are well behaved for large z̃: ∞ X 1 + 3z̃ −2 ≡ Aℓ z̃ −2ℓ/3 , A(z̃) = − 1 + z̃ −2 ℓ=0 z̃ −2 B(z̃) = 2 + λn z̃ −2/3 (1 + z̃ −2 )−4/3 1 + z̃ −2 ∞ X Bℓ z̃ −2ℓ/3 . (9) ≡ ℓ=0 P∞ −2ℓ/3 Using the ansatz ψ̃n (z̃) = one ℓ=0 αℓ z̃ finds  −1 4 2 2 αℓ = ℓ + ℓ # " 9 ℓ−1 3 ℓ−1 X 2X kαk Aℓ−k − αk Bℓ−k (10) × 3 k=1 k=0 with α0 = 1 and α1 = −(9/10)B1. Imposing the parity condition and the asymptotic behavior, we find numerically the wave functions ψn (z) and their eigenvalues λn . If we √ go to the√limit of small z, ψn simplifies to sin λn z or cos λn z, according to their parity. For large values of λn , there will be plenty of oscillations before the wave functions ψn (z) reach the z −1 behavior. We plot some wave functions in Fig. 1. Masses and couplings. Using numerical methods, we calculated eigenvalues λn for n = 1 1 2 3 4 Log n Figure 2. Meson Regge trajectory for D4/D8 model. The first graph shows the dependence of λn with the radial number n. In the logarithmic graph we fit our results, the dashed line is −0.46 + 1.42 log n corresponding to n = {1, 5} while the solid line is −1.28 + 1.91 log n corresponding to n = {6, 60}. to 60. The data for several eigenvalues are shown in Table 1 together with the calculated coupling constants for the vector mesons. With such data we obtain the corresponding Regge trajectory for the vector and axial vector mesons in the D4-D8 model. The result is shown in Figure 2. This Regge trajectory is different from that found for hadrons in the hard and soft wall holographic models [12,13,14,15]. 3. Form Factors The Sakai-Sugimoto D4/D8 model realizes vector meson dominance (VMD) in electromagnetic scattering. In [2] they were able to show that a 4 vm ❅ ❅ ❅ ❅ ❅ ❅ ❅① ❅ × γ vn ∞ X n=1  qµ qσ Mv2n Mv2n η µσ + gv n gv m v n v ℓ  q2 +   (12) where f abc is the structure constant of U (Nf ) and Mvn is the mass of the vector meson v n . Using the sum rule [2] ① ∞ X gv n gv n v m v ℓ = δmℓ Mv2n n=1 vℓ (13) we find Figure 3. Feynman diagram for vector meson form factor. photon-meson-meson couplings are all cancelled, and only a dipole interaction photon-vector meson coupling survives. Exploring further the work done in [2], we calculate the first excitation vector meson, ρ(770), elastic and non-elastic form factors derived from photon-meson scattering. We also calculated the axial vector mesons form factors [6]. 3.1. Generalized form factors ρ hv m a (p), ǫ|J µc (0)|v ℓ b (p′ ), ǫ′ i = ǫν ǫ′ f abc  ×  ησν (q − p)ρ + ηνρ (2p + q)σ − ηρσ (p + 2q) ν  qµqσ qµ qσ Fvm vℓ (q 2 ) + δmℓ 2 (14) × η µσ − 2 q q where the generalized vector meson form factor is defined by Fvm vℓ (q 2 ) = ∞ X gv n gv n v m v ℓ . q 2 + Mv2n n=1 (15) Taking into account the transversality of the vector meson polarizations: ǫ · p = 0 = ǫ′ · p′ , we find hv m a (p), ǫ|J µc (0)|v ℓ b (p′ ), ǫ′ i The form factors are calculated from the matrix elements of the eletromagnetic current. The interaction of a vector meson v m with momentum p and polarization ǫ with an off-shell photon with momentum q = p′ − p is described by the matrix element hv m a (p), ǫ|J˜µc (q)|v ℓ b (p′ ), ǫ′ i = δ 4 (p′ − p − q) ×(2π)4 hv m a (p), ǫ|J µc (0)|v ℓ b (p′ ), ǫ′ i (11) where J˜µ is the Fourier transform of the electromagnetic current J µ (x). This matrix element is calculated from the corresponding Feynman diagram shown in Figure 3. We find ρ hv m a (p), ǫ|J µc (0)|v ℓ b (p′ ), ǫ′ i = ǫν ǫ′ f abc × [ησν (q − p)ρ + ηνρ (2p + q)σ − ηρσ (p + 2q)ν ]   ρ q)σ + 2(ησν qρ − ηρσ qν ) = ǫν ǫ′f abc ηνρ (2p +  qµ qσ × η µσ − 2 Fvm vℓ (q 2 ) . (16) q Note that the term involving the factor δmℓ in eq. (14) did not contribute since in the elastic case (m = ℓ) we have: 2p · q + q 2 = 0. In a similiar way, for axial vector mesons we can calculate the form factors from the matrix element ham a (p), ǫ|J µc (0)|aℓ b (p′ ), ǫ′ i. This corresponds to evaluating Feynman diagrams similar to Fig. 3, but with the external vector meson lines replaced by the axial vector mesons am and aℓ . Note that the internal vector meson line v n , representing vector meson dominance, is unchanged. Thus, the generalized axial vector meson form factor is Fam aℓ (q 2 ) = ∞ X g v n g v n am aℓ . q 2 + Mv2n n=1 (17) 5 3.2. Elastic case The elastic form factor for vector mesons can be obtained considering the previous calculation with the same vector meson v m in the initial and final states. Then, from eq. (16) we find Fv1 1.0 0.8 0.6 0.4 µc mb ′ ′ hv m a (p),  ǫ|J′ (0)|v µ(p ), ǫ i abc = f  (ǫ · ǫ )(2p + q)  µ +2 ǫµ (ǫ′ · q) − ǫ′ (ǫ · q) Fvm (q 2 ) , 0.2 (18) where Fvm (q 2 ) is the elastic form factor: ∞ X gv n gv n v m v m Fvm (q 2 ) = . q 2 + Mv2n n=1 Fvm (q 2 ) = gv n gv n v m v q2 n=1  m 1− Mv2n q2 (19) gv n gv n v m v m = 0 . 1  + O( 4 ) . q 2 4 6 8 10 q2 HGeV2 L 0.2 0 gvn gvn v1 v1 ≈ −0.0007889(MKK)2 . (21) These results indicate that the conditions (20) are satisfied in the D4-D8 model. 8 10 q2 HGeV2 L Figure 4. Elastic form factor for the ρ meson. Electric, magnetic and quadrupole form factors. The matrix element of the electromagnetic current for a spin one particle in the elastic case can be decomposed as[9] µ hp, ǫ|JEM (0)|p′ , ǫ′ i = (ǫ · ǫ′ )(2p + q)µ F1 (q 2 )    µ + ǫµ (ǫ′ · q) − ǫ′ (ǫ · q) F1 (q 2 ) + F2 (q 2 ) 1 (22) + 2 (q · ǫ′ )(q · ǫ)(2p + q)µ F3 (q 2 ) . p From F1 , F2 and F3 we can define the electric, magnetic, and quadrupole form factors: (20) These conditions are known as superconvergence relations. For the case of v 1 we found numerically n=1 6 0.6 i q2 h q2 )F F − (1 − 3 2 6p2 4p2 FE = F1 + FM = FQ = F1 + F2  q2  −F2 + 1 − 2 F3 4p n=1 9 X 4 0.4 From this expansion we see the that the q −4 term dominates for large q if the following conditions hold ∞ X 2 0.8 Note that eqs. (18) and (19) are also valid for the axial vector mesons, replacing v m by am . Using the data from Table 1, we calculate the elastic form factors for the vector meson ρ(770) (v 1 ) and axial vector meson a1 (1260) (a1 ). We plot the results in Figure 4. Note that when q 2 → 0, the vector and axial vector form factors go to one, thanks to the sum rule (13). For large values of q 2 we found that the form factors decrease approximately as q −4 . This can be seen in the second panel in Figure 4 where we plot the form factors multiplied by q 4 . In order to explain this behavior, we expand the elastic form factors in powers of q −2 : ∞ X 0 q4 Fv1 1.0 (23) Then, using eqs. (18) and (22) we find that for a vector meson v m (v m ) F1 (v m ) = F2 = Fvm , (v m ) F3 = 0, (24) 6 where Fvm is given by eq. (19). Hence the electric, magnetic and quadrupole form factors predicted by the D4-D8 brane model are (v m ) FE (v m ) FQ q2 (v m ) )Fvm , FM = 2Fvm , 6p2 = −Fvm . = (1 + (25) We can now estimate three important physical quantities associated with the vector mesons: the electric radius, the magnetic and quadrupole moments. The electric radius for the vector mesons are given by hrv2m i = −6 d (vm ) 2 (q )|q2 =0 . F dq 2 E (26) Using our numerical results for the form factors of the lowest excited state ρ, we find its electric radius: hrρ2 i = 0.5739 fm2 . (27) The magnetic and quadrupole moments are defined by µ ≡ FM (q 2 )|q2 =0 , D ≡ − 1 FQ (q 2 )|q2 =0 . (28) p2 Using the fact that Fvm goes to one when q 2 → 0, we obtain 1 µv m = 2 , D v m = − 2 . (29) Mv m Our results for electric radius, magnetic and quadrupole moments for the vector meson ρ are in agreement with the hard wall model results found in [9]. 4. 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