γ ∗ ρ 0 → π 0 Transition Form Factor in Extended AdS/QCD Models
aa r X i v : . [ h e p - ph ] J a n γ ∗ ρ → π Transition Form Factor in Extended AdS/QCD models
Fen Zuo a ∗ , Yu Jia b † , and Tao Huang b ‡ a Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China b Institute of High Energy Physics and Theoretical Physics Center for Science Facilities,Chinese Academy of Sciences, Beijing 100049, China
Abstract
The γ ∗ ρ → π transition form factor is extracted from recent result for the γ ∗ γ ∗ π form factorobtained in the extended hard-wall AdS/QCD model with a Chern-Simons term. In the largemomentum region, the form factor exhibits a 1 /Q behavior, in accordance with the perturbativeQCD analysis, and also with the Light-Cone Sum Rule (LCSR) result if the pion wave functionexhibits the same endpoint behavior as the asymptotic one. The appearance of this power behaviorfrom the AdS side and the LCSR approach seem to be rather similar: both of them come from the“soft” contributions. Comparing the expressions for the form factor in both sides, one can obtainthe duality relation z ∝ p u (1 − u ), which is compatible with one of the most important relationsof the Light-Front holography advocated by Brodsky and de Teramond. In the moderate Q region,the comparison of the numerical results from both approaches also supports a asymptotic-like pionwave function, in accordance with previous studies for the γ ∗ γ ∗ π form factor. The form factorat zero momentum transfer gives the γ ∗ ρ π coupling constant, from which one can determine thepartial width for the ρ ( ω ) → π γ decay. We also calculate the form factor in the time-like region,and study the corresponding Dalitz decays ρ ( ω ) → π e + e − , π µ + µ − . Although all these resultsare obtained in the chiral limit, numerical calculations with finite quark masses show that thecorrections are extremely small. Some of these calculations are repeated in the Hirn-Sanz modeland similar results are obtained. PACS numbers: 11.25.Tq, 11.10.Kk, 11.15.Tk 12.38.Lg ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] . INTRODUCTION In recent years, the phenomenological bottom-up approach to describing strong interac-tion based on the AdS/CFT correspondence [1–3], now known as AdS/QCD, has offeredmuch insight into various low-energy aspects of QCD. In the simplest setup, various hadronstates are considered to be dual to different string modes propagating in a slice of 5D AdSspace [4–6]. High-energy scattering of glueballs naturally exhibits QCD-like power behaviordue to the warped geometry of the dual theory [4]. Spectra of low-lying hadron states arewell reproduced [6, 7]. Chiral symmetry and its spontaneous breaking are also well imple-mented [8–10]. Up to now there have been extensive studies on various dynamical quantitiessuch as decay constants, coupling constants and form factors, e.g., [11–17]. Furthermore, anovel relation between the string modes and the Light-Cone wave functions of the mesonswas found in Ref. [7], from which the so-called Light-Front holography was established.To reproduce the Wess-Zumino-Witten term in the chiral Lagrangian, a Chern-Simons (CS) term must be added [18, 19]. The CS term naturally introduces baryondensity [20], since baryons are related to the instantons in the 5D model. The effect ofthis term to the baryon properties were later studied in Ref. [21]. Furthermore, with theCS term turned on, the anomalous form factor of the pion coupling to two virtual photoncan be well reproduced [22]. Interestingly, the predictions for the form factor in the limit oflarge photon virtualities coincide with those of perturbative QCD (pQCD) calculated usingthe asymptotic form of the pion distribution amplitude. In this paper we attempt to extendthis calculation to the form factor of γ ∗ ρ → π transition , and then compare the resultswith those of the traditional approaches.In pQCD, the asymptotic behavior of the γ ∗ ρπ form factor has been predicted to be1 /Q [23]. A simple expression for this form factor in large and moderate momentumregion can be obtained in the Light-Cone Sum Rules (LCSR) approach [24, 25], which givesthe same asymptotic behavior if the pion distribution amplitude is asymptotic-like at theendpoint. However, the dominant contribution is quite different from the pQCD analysis.At zero momentum transfer the form factor defines the γρπ coupling which determines thewidth of the radiative decay ρ → πγ . This coupling was extracted from the traditionalthree-point QCD sum rule [26], and also from QCD sum rules in the presence of externalfield [27]. In this paper we will try to give a unified description of the form factor in the2hole region. We will mainly focus on the results in the standard hard-wall model [8, 9],and repeat part of the calculations in the Hirn-Sanz model [10] as a check.The organization of the paper is as follows. In the next section we will briefly introducethe hard-wall AdS/QCD model, and review the calculation of the γ ∗ γ ∗ π form factor. Theextraction of γ ∗ ρ π form factor and comparison with other approaches will be presented inSec. III. In Sec. IV we give the γ ∗ ρ π form factor in the Hirn-Sanz model and compare it tothat in the hard-wall model. The last section is reserved for the summary. II. EXTENDED ADS/QCD MODEL WITH CHERN-SIMONS TERMA. hard-wall AdS/QCD model
In the hard-wall model [8], the background is given by a slice of AdS space with themetric: d s = g MN dx M dx N = 1 z (cid:0) η µν d x µ dx ν − d z (cid:1) , (1)where η µν = Diag (1 , − , − , −
1) and µ, ν = (0 , , , M, N = (0 , , , , z ). The SU ( N f ) × SU ( N f ) chiral symmetry is realized through the gauge symmetry of two sets of gauge fields A ( L ) and A ( R ) . To breaking the chiral symmetry to the vector part, an additional scalar field X is introduced. The whole action is then given by: S AdS = Tr Z d x (cid:20) z ( D M X ) † ( D M X ) + 3 z X † X − g z ( F MN ( L ) F ( L ) MN + F MN ( R ) F ( R ) MN ) (cid:21) , (2)where A = A a t a , F MN = ∂ M A N − ∂ N A M − i [ A M , A N ] , DX = ∂X − iA ( L ) X + iXA ( R ) ,and the generators are normalized as Tr t a t b = δ ab /
2. The vacuum solution h X ( x, z ) i = v ( z ) / m q z + σz ) / X ( x, z ) gives the pion field: X ( x, z ) = h X i e it a π a ( x,z ) . The vectorcombination V = ( A ( L ) + A ( R ) ) / V z = 0 and Fourier transforming to the 4D momentum space, the transverse components V T µ then satisfies the following equation: ∂ z (cid:18) z ∂ z V T µ ( q, z ) (cid:19) + q z V T µ ( q, z ) = 0 . (3)3ith Neumann boundary condition chosen at the cutoff, the normalized solution is simplygiven by ψ Vn ( z ) = √ z J ( γ ,n ) zJ ( M n z ) (4)with γ ,n being the n th zero of the Bessel function J ( x ) and M n = γ ,n /z . Matching to theexperimental ρ mass fixes z = (323 MeV) − . The coupling constants, which are defined by h | J aµ | ρ bn i = f n δ ab ε µ , can be obtained by analyzing the two-point correlation function derivedfrom the action. The results are expressed through ψ Vn ( z ) as f n = 1 g (cid:20) z ∂ z ψ Vn ( z ) (cid:21) z =0 = √ M n g z J ( γ ,n ) . (5)The non-normalized solution, or the bulk-to-boundary propagator, can also be derived an-alytically: J ( Q, z ) = Qz (cid:20) K ( Qz ) + I ( Qz ) K ( Qz ) I ( Qz ) (cid:21) , (6)where J ( Q, z ) is taken at a spacelike momentum q with q = − Q and satisfies the boundarycondition J ( Q,
0) = 1. I n and K n are the order- n modified Bessel functions of the first andsecond kind, respectively. It can be shown that J ( Q, z ) has the following decompositionformula [28, 29]: J ( Q, z ) = g ∞ X m =1 f m ψ Vm ( z ) Q + M m (7)From J ( Q, z ) the vector current correlator can be derived, whose asymptotic behavior de-termines the 5D coupling g = 2 π .The axial combination A = ( A ( L ) + A ( R ) ) / ∂ z (cid:18) z ∂ z A T µ (cid:19) + q z A T µ − g v z A T µ = 0; (8) ∂ z (cid:18) z ∂ z ϕ (cid:19) + g v z ( π − ϕ ) = 0; (9) − q ∂ z ϕ + g v z ∂ z π = 0 . (10)where ∂ µ ϕ = A µ − A T µ . The normalization of ϕ and π is fixed by the pion kinetic term: Z z d z (cid:18) ϕ ′ ( z ) g z + v ( z ) ( π − ϕ ) z (cid:19) = f π . (11)4his normalization naturally leads to the charge conservation constraint for the electromag-netic form factor of the pion, since the pion form factor is given by [14, 15] F π ( Q ) = 1 f π Z z d z J ( Q, z ) (cid:18) ϕ ′ ( z ) g z + v ( z ) ( π − ϕ ) z (cid:19) . (12)Notice that both the equations and the normalization condition is invariant if the ϕ and π fields are shifted by a constant simultaneously.In the chiral limit m q = 0, the pion decay constant can be derived from the residue ofthe axial current correlator at q = 0 [8]: f π = − g ∂ z A c (0 , z ) z (cid:12)(cid:12)(cid:12)(cid:12) z = ǫ , (13)where A c (0 , z ) is the nonnormalizable solution to Eq. (8) at q = 0, satisfying A ′ c (0 , z ) = 0and A c (0 ,
0) = 1. We use the subscript “c” to indicate that the solution is obtained in thechiral limit. The explicit form of A c (0 , z ) is given by A c (0 , z ) = z Γ (2 / (cid:16) α (cid:17) / (cid:20) I − / (cid:0) αz (cid:1) − I / (cid:0) αz (cid:1) I / ( αz ) I − / ( αz ) (cid:21) , (14)where α ≡ g σ/
3. Matching to the experimental value of f π , one obtains α =(424 MeV) [22], or σ = (332 MeV) . In this case π c ( z ) is just a constant and can beshifted to zero. Then ϕ c ( z ) satisfies the same equation as A c (0 , z ) with the same boundarycondition at z = z , so we have ϕ c ( z ) = ϕ c (0) A c (0 , z ). The normalization condition (11)finally fixes ϕ c (0) to be one.When m q = 0, A ( q , z ) will generally develop a z log z term, unless q = g m q . Thisshould not be identified with the pion pole. Away from this point, ∂ z A ( q , z ) /z is divergentas z →
0, which makes the generalization of Eq. (13) unfeasible. This can be overcome if wechoose ϕ ( z ), rather than A (0 , z ), to define the pion decay constant. These two are identicalin the chiral limit, but different now. Thus we have f π = − g ∂ z ϕ ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) z = ǫ . (15)One may worry that if this definition is consistent with the normalization (11). Notice thatin this case ( ϕ − π ) is forced to vanish at the ultraviolet boundary. Integration by parts andimposing the equation of motion with the boundary condition, Eq. (11) becomes m π g Z z zv ( z ) ϕ ′ ( z ) d z = f π , (16)5sing the same trick as in ref. [8] and ref. [17], one can show that the above conditiontogether with Eq. (15) leads to the Gell-Mann-Oakes-Renner relation.To solve the equations for ϕ ( z ) and π ( z ), one has to employ numerical methods. Wechoose the boundary value ϕ (0) = π (0) = 1, for the convenience of comparing with thechiral solution. Adjusting m π and f π to the experimental value m π = 135 . f π = 92 . m q = 2 .
14 MeV and σ = (329 MeV) . The explicit form of ϕ ( z )and π ( z ) are plotted in Fig. (1), together with the solution ϕ c ( z ). Surprisingly, the curvesof ϕ ( z ) and ϕ c ( z ) almost coincide, and one can hardly distinguish them. π ( z ) is also veryclose to π c ( z ) = 0, except for a peak near z = 0. Thus one may expect that the quark masscorrection to any physical observable would be small. To check this, we recalculate the pionform factor according to Eq. (12), which has already been done in ref. [14] for finite quarkmass, and in ref. [15] in the chiral limit. The result confirmed our expectation, see Fig. (2).We also calculated some other observable and obtained similar results in both cases. c (z) (z)(z) z/z FIG. 1:
Explicit form of the solution ϕ ( z ) and π ( z ) (solid curves), together with ϕ c ( z ) (dashedline), the solution in the chiral limit. Q F (Q )[GeV ] Q [GeV ] FIG. 2:
The pion electromagnetic form factor calculated in the hard-wall model, the solid linedenotes the result with finite quark mass and the dashed one in the chiral limit.
B. Extended hard-wall model with Chern-Simons term
Now let us turn to the derivation for the γ ∗ γ ∗ π form factor by adding a CS term tothe hard-wall model, pioneered by Grigoryan and Radyushkin [22]. First we should enlargethe previous considered SU (2) L ⊗ SU (2) R gauge group into U (2) L ⊗ U (2) R . To do this, wereplace the gauge fields t a A aµ in the action by A µ = t a A aµ + I ˆ A µ . The gauge field ˆ A µ willcouple to the isosinglet current in the boundary theory. The cubic part of the CS term canbe expressed in the axial gauge as S (3)CS [ A ] = k N c π ǫ µνρσ Tr Z d x d z ( ∂ z A µ ) (cid:20) F νρ A σ + A ν F ρσ (cid:21) , (17)with k an integer. For the U (2) L ⊗ U (2) R gauge group, the corresponding cubic action reads: S AdSCS [ A ( L ) , A ( R ) ] = S (3)CS [ A ( L ) ] − S (3)CS [ A ( R ) ] . (18)The relevant term for the anomalous π γ ∗ γ ∗ form factor can be found to be S anom = k N c π ǫ µνρσ Z d x Z z dz ( ∂ z ϕ a ) (cid:0) ∂ ρ V aµ (cid:1) (cid:16) ∂ σ ˆ V ν (cid:17) . (19)7ased on the holographic dictionary, one then obtains the bare form factor as F γ ∗ γ ∗ π ( Q , Q ) = − N c π f π · k Z z J ( Q , z ) J ( Q , z ) ∂ z ϕ ( z ) d z. (20)In QCD, the axial anomaly determines the value of the form factor with real photons tobe F γ ∗ γ ∗ π (0 ,
0) = N c π f π . To reproduce this result, a surface term must be added, and theinteger k must be taken to be 2. The final result for the normalized function K ( Q , Q ) isthen K ( Q , Q ) = F γ ∗ γ ∗ π ( Q , Q ) /F γ ∗ γ ∗ π (0 , ϕ ( z ) J ( Q , z ) J ( Q , z ) − Z z J ( Q , z ) J ( Q , z ) ∂ z ϕ ( z ) d z. (21)This result has very interesting properties [22]. When one photon is real, the form factorhas the following expansion at low momentum : K (0 , Q ) = 1 − a π Q m π , (22)with a π ≈ .
031 in perfect agreement with the experimental value: a | exp ≃ . ± . a = m π m ω ≃ .
03. At large virtuality for one or both photons, the asymptoticbehavior of the form factor can be found to be: K (0 , Q ) → ¯ s/Q ,K ( Q , Q ) → ¯ s Q Z x (1 − x ) dx ω (2 x − , (23)where ¯ s = 8 π f π . Both coincide with the leading-order pQCD results calculated for theasymptotic form of the pion distribution amplitude. However, the origins of the powerbehavior are quite different. The power behavior appears only after we have integrated outthe meson wave function in the holographic direction. This is very similar to the “soft”contributions described in the LCSR approach, which will be discussed in the followingsection. III. FORM FACTOR OF γ ∗ ρ → π TRANSITION
It will be illuminating to further study the ρ → π transition form factor based on theprevious result. One starts with the dispersion relation for the amplitude F γ ∗ γ ∗ π ( Q , Q ) in8he variable Q and at fixed Q . In the standard QCD sum rule approach, one assumes thatthe spectral density in the dispersion relation can be approximated by the ground states ρ , ω and the the higher states with an effective threshold s : F γ ∗ γ ∗ π ( Q , Q ) = √ f ρ F ρ π ( Q ) m ρ + Q + ∞ Z s d s ρ h ( Q , s ) s + Q . (24)Here, the γ ∗ ρ ( ω ) → π form factor is defined as13 h π | j µ ( q ) | ω ( q ) i = h π | j µ ( q ) | ρ ( q ) i = F ρ π ( Q ) m − ρ ǫ µναβ e ν q α q β , (25)and the decay constants have the relation:3 h ω | j µ | i = h ρ | j µ | i = f ρ √ m ρ e ∗ ν , (26) e µ being the polarization vector of the ρ ( ω ) meson. Since we are working in the U (2) V symmetric limit, the above relations are exact. On the other hand, the dispersion relationcan be carried out explicitly using the decomposition formula (7). Extracting the lowest ρ and ω pole contributions, one immediately obtains the expression for the γ ∗ ρ π form factor: F ρ π ( Q ) = N c π f π g m ρ (cid:20) J ( Q, z ) ψ V ( z ) ϕ ( z ) − Z z J ( Q, z ) ψ V ( z ) ∂ z ϕ ( z )d z (cid:21) (27)As discussed before, the quark mass correction is very small, so we mainly work in the chirallimit. The results for finite quark mass are listed only in Table I for comparison. A. Large Q region First let us focus on the large- Q asymptotic behavior of the form factor. The large- Q behavior of J ( Q, z ) is dominated by the term zQK ( zQ ) in Eq. (6), which behaves like e − Qz . Thus the first term in Eq. (27) will vanish exponentially ∼ e − Qz in the asymptoticregion, hence can be neglected. Due to the exponential factor of J ( Q, z ), only small valuesof z are important in the remaining integral, and the outcome is determined by the small- z behavior of the wave function ∂ z ϕ ( z ) and ψ V ( z ). From the previous discussion, we knowthat when z → ∂ z ϕ ( z ) ∼ − f π g z, (28)and ψ V ( z ) ∼ m ρ z √ z J ( γ ,n ) . (29)9tilizing all these facts one finds: F ρ π ( Q ) → N c m ρ πf π m ρ √ z J ( γ , ) ( − f π g ) Z z z ∗ zQK ( zQ )d z = πf π m ρ √ γ , J ( γ , ) R ∞ χ K ( χ ) dχQ = 8 √ πf π m ρ γ , J ( γ , ) 1 Q = 1 .
23 GeV Q (30)Using the holographic expression for m ρ and f ρ , one may further express the result as F ρ π ( Q ) → √ π f π f ρ m ρ Q . (31)Although this power behavior is the same as the pQCD prediction [23], the underlyingmechanism is rather different. The appearance of the power behavior in this way is similarto the LCSR analysis [24, 25], where the form factor is given by the following expression: F ρπ LC ( Q ) = f π f ρ Z Q s Q d uu ϕ π ( u ) + uQ d ϕ (4) ( u )d u ! exp (cid:18) − Q (1 − u ) uM + m ρ M (cid:19) (32)= f π f ρ ϕ ′ π (1) Q exp (cid:18) m ρ M (cid:19) Z s s e − s/M d s + O(1 /Q ) . (33)Here ϕ π ( u ) and ϕ (4) ( u ) are the leading twist and the twist-4 distribution amplitudes, and M the Borel parameter. In deriving the asymptotic behavior, we have assumed that φ π ( u ) u → ∼ ϕ ′ π (1)(1 − u ). That is to say, in order to obtain the same power behavior as in the holographicmodel, φ π ( u ) must has the same end-point behavior as the asymptotic one, namely φ as π ( u ) =6 u (1 − u ). This is also in accordance with the general analysis for the end-point behavior ofthe pion wave function [23].Moreover, in the holographic approach J ( Q, z ) → e − Qz tells us in the large Q limit weare actually probing the 0 < z < /Q interval of the AdS slice, while in the LCSR (32) theendpoint region, 0 < − u < /Q , or 0 < √ − u < /Q dominates. Taking into accountthe symmetry u ↔ − u of the light quark system, one may expect that z should be dual to p u (1 − u ) b with b a light-cone distance parameter, at least in the high energy region. Thisis just one of the key relations of the Light-Front holography [7, 13].Since the absolute normalization of the asymptotic behavior is not known in both thepQCD and LCSR approaches, one can only compare their predictions to the form fac-tor at moderate Q with ours. At Q ≃
10 GeV , direct calculation from Eq. (27) gives10 ρ π ( Q ) = 8 . × − , as shown in Fig. 3. This is much larger than the pQCD result F ρπ pQCD ( Q ) ≃ × − [23]. In the LCSR approach, the result strongly depends on theshape of the leading twist distribution amplitude of the pion meson, which can be seen fromFig. 4. For the asymptotic distribution amplitude, one obtains F ρπ as ( Q ) ≃ . × − . Forsome non-asymptotic distribution amplitudes, the results are much larger, e.g., input of theChernyak-Zhitnitsky (CZ) [23] and Braun-Filyanov (BF) [31] distribution amplitudes give: F ρπ CZ ( Q ) ≃ . F ρπ BF ( Q ) ≃ . γ ∗ γ ∗ π form factor [22]. Q [GeV ]Q F (Q )[GeV ] FIG. 3: γ ∗ ρ → π form factor calculated in the extended hard-wall AdS/QCD model, the resultfor finite quark mass (in solid curve) and that in the chiral limit (dashed line) almost coincide. In Ref. [24] an alternative light-cone sum rule was derived for this form factor, from which a much smallervalue F ρπ as ( Q ≃ ) ≃ × − (see Fig. 4) was obtained. However, with the aid of the techniquein Ref. [30] one can show that a boundary term was missing in their calculations. After including thisterm, a similar result F ρπ as ( Q ) ≃ . × − will be obtained. Q2[GeV2]Q4 F (cid:26)(cid:25)(Q2) [GeV4]
FIG. 4:
LCSR results for γ ∗ ρ → π form factor excerpted from [25]. The solid line corresponds tothe result calculated with the asymptotic pion wave function, while the long-dashed and the short-dashed lines with the CZ and BF wave functions respectively. In comparison, the predictions of thethree-point QCD sum rule (dotted) [32] and an alternative light-cone sum rule for the γ ∗ ρ → π form factor [24] (dash-dotted) were also plotted. B. Low Q region The γρ π form factor at zero momentum transfer defines the coupling constant g ρ π γ through the effective Lagrangian [33] L effρπγ = g ρπγ m − ρ ε µναβ ∂ µ ρ ν ∂ α A β π . (34)Since J ( Q,
0) = J (0 , z ) = 1, one immediately obtains the coupling constant: g ρ π γ = N c m ρ πf π (cid:20) ψ V ( z ) ϕ ( z ) − Z z ψ V ( z ) ∂ z ϕ ( z )d z (cid:21) = 0 . . (35)This result is very close to the value extracted from the analysis of ρ and ω photoproductionreactions through pseudoscalar exchange, which gives rise to g ρπγ = 0 .
54 [34] . Also it isconsistent with the QCD sum rule prediction g ρπγ = 0 . ± .
07 [26]. Based on the effective12 xperiment [35] m q = 0 m q = 0Γ( ρ → π γ ) 0 . ± .
013 0 . . ω → π γ ) 0 . ± .
03 0 . . ρ → π e + e − ) —— 6 . × − . × − Γ( ρ → π µ + µ − ) —— 6 . × − . × − Γ( ω → π e + e − ) (6 . ± . × − . × − . × − Γ( ω → π µ + µ − ) (8 . ± . × − . × − . × − TABLE I:
Predictions of the partial decay widths (in MeV) of ρ and ω in the present approach,both in the chiral limit and with finite quark mass. Lagrangian given in Eq. (34), the decay width for V → π γ can be readily deduced:Γ( V → π γ ) = α
24 ( m V − m π ) m V g V πγ . (36)Substituting the physical masses of the mesons and taking α = 1 / ρ → π γ ) and Γ( ω → π γ ) can be obtained. We can further extrapolate the form factorto the time-like region by analytically continuing J ( Q, z ) to the region q = − Q > J ( q, z ) = − π qz (cid:20) Y ( qz ) − J ( qz ) Y ( qz ) J ( qz ) (cid:21) (37)with Y n the second kind Bessel function. From the resulting γ ∗ ρ ( ω ) π form factor in thetime-like region, one obtains the decay widths for the ρ ( ω ) → π e + e − and ρ ( ω ) → π µ + µ − decays. In Table I we list these results together with those for the radiative decays. Theonly reason we keep four digits for our predictions is to show the corrections due to finitequark mass. IV. γ ∗ ρ π FORM FACTOR IN THE EXTENDED HIRN-SANZ MODEL
Spontaneous chiral symmetry breaking can also be implemented through the boundaryconditions at the IR cutoff, without employing the scalar field, as proposed by Hirn andSanz [10]. Specifically, the axial combination of the left-handed and right-handed vectorfields was chosen to satisfy Dirichlet boundary condition, rather than the Neumann one for13he vector part. That is to say, we require: F ( R ) zµ ( x, z = z ) + F ( L ) zµ ( x, z = z ) = 0 . (38) A ( R ) µ ( x, z = z ) − A ( L ) µ ( x, z = z ) = 0 . (39)Then the 5D gauge transformations for R µ and L µ at the point z = z must be equal. Thechiral field can be defined as U ( x ) ≡ ξ R ( x, z ) ξ † L ( x, z ) , (40)where the Wilson line is defined as ξ R ( L ) ( x, z ) ≡ P n e i R zz d zA ( R ) z ( A ( L ) z )( x,z ) o , (41)with P denoting path-ordered integral. The equality of the 5D gauge symmetry at z = z enforces the following transformation law for the chiral field U ( x ) g R ( x ) U ( x ) g † L ( x ) . (42)where ( g R , g L ) represent the 5D gauge symmetries located on the UV brane, which are theninterpreted as the 4D SU ( N f ) × SU ( N f ) chiral symmetry. A vacuum state with U = 1naturally leads to the spontaneous breaking of the symmetry group to the vector part.To separate the dynamical fields and external sources from A ( L ) and A ( R ) , one shouldfirst make a gauge transformation using the above Wilson lines:ˆ V M , ˆ A M ≡ i n ξ † L (cid:0) ∂ M − i A ( L ) M (cid:1) ξ L ± ( L → R ) o . (43)After making this transformation we are then working in the axial gauge, ˆ V z = ˆ A z = 0. Forthe vector part, one can simply make the following substraction V µ ( x, z ) ≡ ˆ V µ ( x, z ) − ˆ V µ ( x, z ) , (44)and the dynamics of V µ ( x, z ) is completely the same as in the original hard-wall model.However, to remove the effect of the UV source of the axial field on the IR, a function α ( z )has to be introduced with the boundary values α (0) = 1 , α ( z ) = 0 . (45)14ubsequently, the axial field can be decomposed as A µ ( x, z ) ≡ ˆ A µ ( x, z ) − α ( z ) ˆ A µ ( x, z ) . (46)Since ˆ A µ ( x, z ) contains the derivative of the chiral field U , α ( z ) will play the role of the 5Dwave function of the pion. Moreover, to eliminate the mixing of the dynamical axial fieldand the pion, α ( z ) must satisfy ∂ z (cid:18) z ∂ z α (cid:19) = 0 . (47)Together with the aforementioned boundary conditions, this fixes α ( z ) to be of the form α ( z ) = 1 − z /z . (48)Substituting these decompositions into the original action, one can naturally deduce thechiral lagrangian, with all the low energy constants given by simple integrals of α ( z ). Mostimportantly, one has f π = 1 g Z z d zz ( ∂ z α ) = 2 g z . (49)If g and z were fixed as before, we would have f π ≃ . γ ∗ γ ∗ π form factor in this model, which has already been derivedin Ref. [36] along the same line as in the hard-wall model: F γ ∗ γ ∗ π (cid:0) Q , Q (cid:1) = − N c π f π Z z J ( Q , z ) J ( Q , z ) ∂ z α ( z ) d z, (50)where the normalization constant k of the CS term has also been chosen to be 2. This isenough to ensure the anomaly relation since α ( z ) = 0. No surface term at the IR boundaryneeds to be introduced. From the above expression we see that α ( z ) indeed plays the roleof pion wave function, as Ψ( z ) does in the original hard-wall model. Moreover, the behaviorof these two functions near the UV boundary are also the same, since ∂ z α ( z ) = − z/z = − f π g z. (51)From this one can conclude that the asymptotic behavior of the γ ∗ γ ∗ π form factor must bethe same as in the hard-wall model, which was found in Ref. [36].15he γ ∗ ρ π form factor can derived as in previous sections, which is given by F ρ π ( Q ) = − N c π f π g m ρ (cid:20)Z z J ( Q, z ) ψ V ( z ) ∂ z α ( z )d z (cid:21) . (52)For the same reason as preceding discussion, its asymptotic behavior is the same as Eq. (31).The ρ π γ coupling can also be obtained g ρ π γ = − N c π f π g m ρ Z z ψ V ( z ) ∂ z α ( z )d z. (53)Substituting the experimental value of f π in the normalization factor, we get g ρ π γ = 0 .
65, inreasonable agreement with the hard-wall result and those derived from other approaches. Inref. [37] an exhaustive list of the three-point and four-point couplings was given for the Hirn-Sanz model. The corresponding value for the ρπγ vertex is g ρπγ = 0 . f − π GeV ≃ . V. SUMMARY
In this work, the γ ∗ ρ → π transition form factor has been extracted from the γ ∗ γ ∗ π form factor, which has been obtained in the extended hard-wall AdS/QCD model including aChern-Simons term. As expected from pQCD, the form factor exhibits the 1 /Q asymptoticbehavior, but with a rather different mechanism. It comes out only after we integrate themeson solution with the bulk-to-boundary propagator along the holographic direction. Thepower is then determined by the z → z = p u (1 − u ) b with b alight-cone distance parameter, which is just one of the important relations in the Light-Frontholography. Since the numerical results of the form factor in the LCSR approach stronglydepend on the profile of the pion distribution amplitude φ π ( u ), the present analysis canhelp to discriminate between various models for φ π ( u ). As in the discussion for the γ ∗ γ ∗ π form factor, our result favors an asymptotic-like pion distribution amplitude. From the formfactor at Q = 0 we obtains the partial width of the radiative decays ρ ( ω ) → π γ . Wealso extend our analysis by analytically continuing the bulk-to-boundary propagator to thetime-like region. The Dalitz decays ρ ( ω ) → π e + e − , π µ + µ − are then studied. All these16ecay rates are roughly consistent with the available measured values. The quark masscorrections are found to be very small, as expected.Some of the calculations have been performed in the Hirn-Sanz model, which successfullydescribes the spontaneous chiral symmetry breaking in a simple way. Just as in the caseof the γ ∗ γ ∗ π form factor, the asymptotic behavior of the γ ∗ ρ π form factor in this modelis exactly the same as in the standard hard-wall model. The γρ π coupling is also inreasonable agreement with the hard-wall result. Acknowledgments : This work was supported in part by Natural Science Foundationof China under Grant No. 10875130, No. 10935012, No. 10805082, and No. 10675132. Wewould like to thank Andrea Wulzer for good comments. [1] Juan Martin Maldacena. Adv. Theor. Math. Phys. (1998): 231-252 [arXiv: hep-th/9711200].[2] Edward Witten. Adv. Theor. Math. Phys. (1998): 253-291 [arXiv: hep-th/9802150].[3] S. S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov. Phys. Lett. B428 (1998): 105-114 [arXiv: hep-th/9802109].[4] Joseph Polchinski and Matthew J. Strassler. Phys. Rev. Lett. (2002): 031601[arXiv: hep-th/0109174].[5] Henrique Boschi-Filho and Nelson R. F. Braga. Eur. Phys. J. C32 (2004): 529-533[arXiv: hep-th/0209080].[6] Guy F. de Teramond and Stanley J. Brodsky. Phys. Rev. Lett. (2005): 201601[arXiv: hep-th/0501022].[7] Stanley J. Brodsky and Guy F. de Teramond. Phys. Rev. Lett. (2006): 201601[arXiv: hep-ph/0602252].[8] Joshua Erlich, Emanuel Katz, Dam T. Son, and Mikhail A. Stephanov. Phys. Rev. Lett. (2005): 261602 [arXiv: hep-ph/0501128].[9] Leandro Da Rold and Alex Pomarol. Nucl. Phys. B721 (2005): 79-97[arXiv: hep-ph/0501218].[10] Johannes Hirn and Veronica Sanz. JHEP (2005): 030 [arXiv: hep-ph/0507049].[11] Sungho Hong, Sukjin Yoon, and Matthew J. Strassler. JHEP (2006): 003 arXiv: hep-th/0409118].[12] Hovhannes R. Grigoryan and Anatoly V. Radyushkin. Phys. Lett. B650 (2007): 421-427[arXiv: hep-ph/0703069].[13] Stanley J. Brodsky and Guy F. de Teramond. Phys. Rev.
D77 (2008): 056007[arXiv: 0707.3859].[14] Herry J. Kwee and Richard F. Lebed. JHEP (2008): 027 [arXiv: 0708.4054].[15] H. R. Grigoryan and A. V. Radyushkin. Phys. Rev. D76 (2007): 115007 [arXiv: 0709.0500].[16] Hilmar Forkel. Phys. Rev.
D78 (2008): 025001 [arXiv: 0711.1179].[17] Zainul Abidin and Carl E. Carlson (2009): [arXiv: 0908.2452].[18] Tadakatsu Sakai and Shigeki Sugimoto. Prog. Theor. Phys. (2005): 843-882[arXiv: hep-th/0412141].[19] Giuliano Panico and Andrea Wulzer. JHEP (2007): 060 [arXiv: hep-th/0703287].[20] Sophia K. Domokos and Jeffrey A. Harvey. Phys. Rev. Lett. (2007): 141602[arXiv: 0704.1604].[21] Alex Pomarol and Andrea Wulzer. Nucl. Phys. B809 (2009): 347-361 [arXiv: 0807.0316].[22] H. R. Grigoryan and A. V. Radyushkin. Phys. Rev.
D77 (2008): 115024 [arXiv: 0803.1143].[23] V. L. Chernyak and A. R. Zhitnitsky. Phys. Rept. (1984): 173.[24] Vladimir M. Braun and Igor E. Halperin. Phys. Lett.
B328 (1994): 457-465[arXiv: hep-ph/9402270].[25] Alexander Khodjamirian. Eur. Phys. J. C6 (1999): 477-484 [arXiv: hep-ph/9712451].[26] A. Gokalp and O. Yilmaz. Eur. Phys. J. C24 (2002): 117-120 [arXiv: nucl-th/0103033].[27] Shi-lin Zhu, W. Y. P. Hwang, and Ze-sen Yang. Phys. Lett.
B420 (1998): 8-12[arXiv: nucl-th/9802043].[28] Sungho Hong, Sukjin Yoon, and Matthew J. Strassler. JHEP (2006): 003[arXiv: hep-th/0409118].[29] H. R. Grigoryan and A. V. Radyushkin. Phys. Rev. D76 (2007): 095007 [arXiv: 0706.1543].[30] Patricia Ball and Vladimir M. Braun. Phys. Rev.
D55 (1997): 5561-5576[arXiv: hep-ph/9701238].[31] Vladimir M. Braun and I. E. Filyanov. Z. Phys.
C44 (1989): 157.[32] V. L. Eletsky and Ya. I. Kogan. Z. Phys.
C20 (1983): 357.[33] A. I. Titov, T. S. H. Lee, H. Toki, and O. Streltsova. Phys. Rev.
C60 (1999): 035205.
34] Yong-seok Oh, Alexander I. Titov, and T. S. Harry Lee (2000): [arXiv: nucl-th/0004055].[35] C. Amsler et al. (Particle Data Group Collaboration). Phys. Lett.
B667 (2008): 1.[36] Hovhannes R. Grigoryan and Anatoly V. Radyushkin. Phys. Rev.
D78 (2008): 115008[arXiv: 0808.1243].[37] S. K. Domokos, H. R. Grigoryan, and J. A. Harvey (2009): [arXiv: 0905.1949].[38] Diego Becciolini, Michele Redi, and Andrea Wulzer (2009): [arXiv: 0906.4562].(2008): 115008[arXiv: 0808.1243].[37] S. K. Domokos, H. R. Grigoryan, and J. A. Harvey (2009): [arXiv: 0905.1949].[38] Diego Becciolini, Michele Redi, and Andrea Wulzer (2009): [arXiv: 0906.4562].