A Holographic Model of Heavy-light Mesons
We construct a holographic model of heavy-light mesons by extending the AdS/QCD to incorporate the behavior of the heavy quark limit. In that limit, the QCD dynamics is governed by the light quark and the heavy quark simply plays the role of a static color source. The heavy quark spin symmetry can be treated as a global symmetry in the AdS bulk. As a consequence, the heavy-light mesons are mapped to “fermions” in the AdS theory. The light flavor chiral symmetry is naturally built in by this construction, and its breaking produces the splitting of the parity-doubled heavy-light meson states. The scaling dependences of physical quantities on the heavy quark mass in the heavy quark effective theory are reproduced. The mass spectra and decay constants of the and mesons can be well fit by suitable choices of model parameters. The couplings between the heavy-light mesons and the pions are also calculated. The holographic model may capture the essence of the long distance effects of QCD and can serve as a useful tool for studying the non-perturbative hadronic matrix elements involving heavy-light mesons.
The bottom-up AdS/QCD [1, 2, 3] attempts to approximate the low-energy Quantum Chromodynamics (QCD) by a five-dimensional (5D) theory living in a slice of anti-de Sitter (AdS) space using the AdS/CFT correspondence [4, 5, 6]. Even though it is not derived from the first principle and the real QCD is neither conformal nor possessing large number of colors (), it has worked reasonably well in describing the low-energy mesons made of light quarks. Many features of the low-energy QCD, such as Vector Meson Dominance  and Hidden Local Symmetry , are built in the AdS/QCD. Its success may be viewed as that it captures some essence of the strong dynamics of QCD.
The simplest version of AdS/QCD describes the vector mesons, axial vector mesons, and the Nambu-Goldstone bosons (pions) associated with the chiral symmetry for light quarks. The action is given by
with the AdS metric
between the UV-boundary () and the IR-boundary (), where is the AdS curvature radius. The and are the field strength tensors of the and gauge symmetry in the bulk which are associated with the corresponding and current operators of QCD. The scalar field transforms as which corresponds to the operator in QCD. is the gauge covariant derivative, . is related to the 5D gauge coupling, , and is taken as the 5D fundamental scale. The solution of in the bulk takes the form:
where is the scaling dimension of the operator and is related to the field bulk mass by . The first term is associated with the light quark mass which corresponds to an explicit chiral (and conformal) symmetry breaking effect, and the coefficient of the second term is related to the vacuum expectation value (VEV) of the operator, which spontaneously breaks the chiral (and conformal) symmetry. The model has few parameters and can be used to fit a wide range of light meson data. The number of parameters can be further reduced if one matches them to the perturbative QCD results as was done in the original AdS/QCD papers [1, 2, 3]. There, the scaling dimension was taken to be the naïve dimension of the operator, . As a consequence, the predictions depend on the combination but not on or separately. If one further matches the two-point function in the UV to the perturbative QCD result, one finds
After taking the position of the UV-boundary to 0, the predictions of this simplest model only depend on three parameters: the light quark mass , the position of the IR-boundary which corresponds to the confinement scale, and which represents the ratio of chiral symmetry breaking and the confinement scale, both of which are related to a common QCD scale . They can be chosen to fit the light meson spectrum. Specifically, to fit the and masses it was found that MeV and , and can be obtained by fitting the mass . The theory can then be used to calculate a variety of low-energy quantities, including the mass spectrum of the excited meson states, decay constants, couplings among meson states, and coefficients of the chiral Lagrangian. A reasonable agreement with the experimental measurements has been found for the ground states. The spectrum of the higher excited meson states does not follow the Regge trajectory in this simple hard-wall model where there is a sharp IR cutoff at , but it can be improved by introducing a soft-wall potential in the bulk . Given the simplicity of the model and its crude approximation to the real QCD, the extent of the agreement with the real QCD data is quite impressive.
The success faces challenges when one tries to include the -like mesons. They are created by the dimension-3 tensor operator, which are associated with a two-form field in the AdS bulk [10, 11, 12]. In particular, if one also requires the new parameters related to the two-form field sector to be matched to the perturbative QCD values, the predictions of the AdS/QCD do not match well with the actual data and even the success of meson sector is ruined due to mixing of the vector and tensor operators . However, it was argued in Ref.  that there is no reason to insist that the parameters in AdS/QCD should be matched to the perturbative QCD values. The two theories have different UV limits and the renormalization group (RG) running in the real QCD can change the parameters in the IR. Therefore, it was advocated in Ref.  that the parameters other than those protected by symmetries should be treated as free parameters to be fit from the experimental data. It turns out that the best-fit values for the parameters in the original AdS/QCD are close to the old values matched perturbative QCD, while the new parameters involving the two-form field need to take different values . In that case, at least the success of the original hard-wall AdS/QCD is preserved though the predictions of the sector are not as good.
Because AdS/QCD and the real QCD have different UV limits, one should not expect AdS/QCD to be a good model for QCD at high energies far above . As shown in Ref. , the event shape of the AdS/QCD in high energy collisions is more spherical with high multiplicities, unlike the jetty structure in the real QCD. Indeed, at high energies the QCD coupling is perturbative and there is no need to choose a dual theory where the coupling is strong and perform calculations there. For the same reason, AdS/QCD may not be a good approximation when applied to heavy quarkonium states [15, 16, 17, 18]. An interesting question is whether AdS/QCD can provide a good approximation to QCD bound states made of both heavy and light quarks, in particular, the heavy-light mesons such as and mesons. In the heavy quark limit, the heavy quark in a heavy-light meson just plays the role of a static color source and the dynamics is governed by the light quark. From this point of view, one might expect that the success of the AdS/QCD for the light mesons could be carried over to the heavy-light meson system. There have been studies of AdS/QCD for the heavy-light mesons in the top-down approach with string and brane constructions as well as the light-front holography [19, 15, 16, 20, 21]. In this paper we follow the bottom-up approach of Ref. [1, 2] and extend it to the heavy-light meson system. We try to fit the real experimental or lattice and meson data and hope that such a model can reproduce the qualitative feature of the non-perturbative aspects of the heavy-light mesons.
In the heavy quark limit, the heavy-light mesons exhibit the heavy quark spin symmetry . The scalar and vector mesons related by the spin symmetry become degenerate in that limit. It is convenient and commonly done in the heavy quark effective theory (HQET) to express them as a bi-spinor field where the spin symmetry can be made manifest. (For a review of the HQET, please see Ref. .) Since the heavy quark is static, its fermionic nature plays no role other than providing the multiplicity of the spin states. One might as well treat the heavy quark as a boson and the heavy quark spin symmetry as a global symmetry. The light quark component, on the other hand, participates in the strong dynamics which may be modeled by AdS/QCD. This suggests that in AdS/QCD, the heavy-light mesons should be mapped to “fermions” in the AdS bulk, with the heavy quark spin symmetry treated as a bulk flavor symmetry. We show that in such a setup, which we dub AdS-HQET, many heavy-light meson data can be described in the AdS/QCD model with suitable parameters. It may provide qualitative insights of nonperturbative effects of processes involving heavy-light mesons. Since AdS/QCD is at best a crude approximation for the real QCD, we only focus on the leading effects in the heavy quark limit. Effects suppressed by the heavy quark mass such as the mass splitting between the spin-0 and spin-1 mesons from the hyperfine interaction will not be considered in this paper.
This paper is organized as follows. In Sec. 2 we review the HQET formalism for heavy-light mesons and set up our notations and convention. We then derive the fermionic Lagrangian in the static heavy quark limit, which serves as the starting point to construct the holographic AdS-HQET model. In Sec. 3, we incorporate the heavy-light mesons into AdS/QCD in the chiral limit as an illustration of the construction and calculation techniques. In Sec. 4 the chiral symmetry breaking and the splitting between the parity doublets of the heavy-light mesons are introduced. We perform fits of the spectrum and decay constants to the experimental and lattice data to determine the model parameters. We also calculate the coupling of the heavy-light mesons to the pions. The future applications of the AdS-HQET model, such as computations of weak-interaction processes involving heavy-light mesons, are discussed in Sec. 5.
2 Effective Lagrangian for Heavy-light Mesons
In this section we review the effective Lagrangian for the heavy-light mesons and show that they can be put in a form of the fermion Lagrangian which will be our starting point to incorporate them into AdS/QCD. We follow the notation of HQET in Ref.  by Bardeen, Eichten and Hill (BEH), in which the spin-zero and spin-one mesons are combined to be written as a velocity-dependent bi-spinor field
Here, () and () represent spin-zero and spin-one mesons, respectively, and the velocity-dependent field is related to the original field by
where represents the heavy quark mass.111This definition differs from that of BEH by for later convenience. We have chosen the first index in to be the light quark spinor index and the second index to be the heavy quark spinor index. The field satisfies and using the relation for physical spin-one particles. It was shown in the Appendix of Ref.  that to order the free Lagrangian of can be written as
where represents the difference between the meson mass and the heavy quark mass. The division between and is somewhat arbitrary and for convenience we can “gauge away” . Similarly, we have the bi-spinor for parity-even states constructed from () and (). Combining and we can form linear representations of the light flavor chiral symmetry ,
The Lagrangian in Eq. (7) can also be written equivalently as
if we define . It looks like a fermion Lagrangian except that the adjoint of the bi-spinor is defined with multiplying on both spinor indices, . In the Pauli-Dirac representation, is given by
If we treat the heavy quark spinor index as a flavor index, this Lagrangian simply describe four species of fermions with the last two fermions having the opposite sign in the Lagrangian. Since the bi-spinor fields always appear in pairs, we can redefine the field to absorb the minus sign in the path integral, i.e., treating and as independent fields and absorb the multiplied on the heavy quark spinor index into , then it takes the standard form of the fermion Lagrangian. The reason that we can describe the heavy-light mesons by fermion fields simply reflects the fact that the heavy quark just plays the role of a static color source and whether it is a fermion or a boson does not affect the dynamics, as long as we do not include heavy quark loops in the calculation.
In the fermionic theory, we introduce a global flavor symmetry to match the heavy spin symmetry . Specifically, we consider two copies of four Weyl fermions, , , , , where “” is the flavor index which represents the degrees of freedom coming from the heavy quark. Each Weyl fermion of course has a Lorentz spinor index “” which corresponds to the spin degrees of freedom of the light quark. These Weyl fermions can be put into a matrix form:
where the minus signs are just a convention. Just like the bi-spinor in the HQET, the first index of is spinor index of the light quark (except that it is in the Weyl representation), while the second index corresponds to the global flavor symmetry which is matched to the heavy spin symmetry in the HQET. If we identify with and expand the Lagrangian of Eq. (9) (with only on the light quark spinor side in the adjoint) in terms of the Weyl fermion components, the kinetic term and mass term are given by
where the flavor indices are implicitly summed over. We see that it is indeed a standard Lagrangian describing four massive Dirac fermions.
To match to the meson fields, it is more convenient to transform the fermions from the Weyl representation to the Pauli-Dirac representation using the transformation relation in Appendix A,222Here the transformation between the Weyl and Pauli-Dirac (PD) representations acts on the light spinor index only. For the heavy quark spinor index, the Pauli-Dirac representation is always used.
On the other hand, in the rest frame of the heavy quark, , the projection operator in the Pauli-Dirac representation takes the form
and in terms of the spin-0 and spin-1 meson fields, can be written as
The number of degrees of freedom in the Weyl fermion combination, , are , which matches to that of one spin-zero meson plus one physical spin-one meson .
3 The AdS/QCD Model for Heavy-light Mesons in the Chiral Limit
We are now ready to write down the 5D AdS/QCD model for the heavy-light mesons. For simplicity we first consider the chiral limit and focus on the sector. The effects of chiral symmetry breaking will be studied in the next section. The formalism developed in the previous section suggests that the heavy-light mesons should be represented by fermions in AdS/QCD. To include the heavy quark spin symmetry, we introduce two pairs of Dirac fermions in the AdS bulk,
where corresponds the heavy quark spin degree of freedom. For notational simplicity, the index will be suppressed in the rest of the paper. The quadratic action for these fermions in the 5D AdS space between the UV cutoff and IR cutoff is given by
The “mass” terms for and determine the scaling dimensions of the CFT operators. They are chosen to be of opposite signs because and should have the same scaling dimension as we see from the previous section that will correspond to the physical mesons. Their scaling dimension is [27, 28, 29, 30, 31].
To incorporate the heavy quark mass we introduce the following term
where corresponds to the heavy quark scalar bilinear operator . Its VEV takes the form
which corresponds to the heavy quark mass term (neglecting the heavy quark condensate). Plugging the VEV into Eq. (21), we obtain a constant mass term in the AdS bulk between and ,
For one might worry about the validity of the effective theory. However, this term only lifts the whole spectrum by . The relevant momentum scale is still controlled by which is of order . It is just like in the heavy-light meson system: the heavy quark simply provides a static color source and the dynamics is governed by the light quark with the relevant energy scale .
The bulk equations of motions (EOM’s) are calculated to be
If we want to calculate the spectrum and the -dependent wave functions of the meson states, we need to choose boundary conditions such that the boundary terms vanish at the UV () and IR () boundaries. On the other hand, if we want to calculate the bulk-to-boundary propagators of the fields with which we can study the correlation functions of the HQET operators, then we need to fix , at the UV boundary and introduce the following term on the UV boundary,
so that the total action is invariant under the variations of and fields. and play the role of the sources for the operators which create the mesons. They will be discussed in more details later in subsection 3.2 when we compute decay constants for the heavy-light mesons.
To solve the EOM’s, it is convenient to first perform a Fourier transformation
where the additional power of is introduced for convenience of imposing boundary conditions. In the rest frame , the equations can be recombined and separated into two sets of first order differential equation in . Define
They are coupled through their EOM’s:
The first order equations can be combined to give the second order differential equation for :
The second order differential equation for can be obtained by changing to . The other two combinations of fields, and , have the same EOM’s by changing , and , but are not relevant for our discussion. From Sec. 2 we know that the physical meson fields map to , so we will focus on the system of and only.
The solutions of and are Bessel functions:
where and . If is an integer, the two independent solutions should be taken as and instead. The power of the dependence of the in the limit of determines the scaling dimension of the corresponding heavy-light current operator. For small , we have
The scaling dimension of the operator corresponding to (sourced by ) is for . For there is another CFT which can be obtained by a Legendre transformation exchanging the source and the operator [32, 30, 31]. The operator would correspond to in that case and has the scaling dimension , but it is not of our concern. The naïve dimension of the heavy-light current is 3, which would correspond to and . However, this current is not conserved as the corresponding symmetry is badly broken by the heavy quark mass. Therefore, there is no reason to expect to remain 3. (If the heavy quark were a scalar as we have pretended it to be, the naïve dimension of the heavy-light current operator would be 5/2, which corresponds to and .) The unitarity bound requires the scaling dimension to be above the free particle limit, . So, in general one may expect that which translates to , or .
To obtain the spectrum of the heavy-light mesons or the corresponding 5D fermion Kaluza-Klein (KK) modes, we need to impose appropriate boundary conditions for the 5D wave functions. For the wave function to be normalizable when the UV cutoff is taken to zero, the wave function near should have a scaling power bigger than [or equivalently, has a scaling power bigger than ]. However, for , this is always satisfied and the normalizability condition does not impose any extra constraint on the solutions. This is related to the fact that in this range of there are two possible CFT’s discussed earlier. To pick out the CFT of our interest, we impose a stronger condition that has a positive power of dependence near , which is equivalent to the Dirichlet condition on the UV boundary for . The boundary conditions for and are333If we switch the IR boundary conditions for and , there would be a “zero mode” where , , and . However, this is a special solution for the hard-wall model. If we imagine that the hard wall is an approximation to a soft wall, there is no solution with a soft wall which resembles that zero-mode solution.
where the mixed boundary condition is the generalization of the Neumann condition for the case of a warped extra dimension, which is consistent with the EOM’s and the Dirichlet condition on the other component of the fermion field. The Dirichlet condition of on the UV boundary sets for in the solutions. The KK spectrum is then determined from the IR boundary condition:
The spectrum of the KK-modes can be expressed as
where means the ’th positive zero of the Bessel function . In the heavy quark limit of , the meson mass is linear in :
Right now there is not much experimental information for higher excited modes of heavy-light mesons. As in the case of light mesons, one may not expect that the spectrum has the correct behavior for very high KK modes in this simple hard-wall model.
3.2 Decay constants
The decay constants of meson fields can be obtained from the two-point function of the current operators which create the mesons. The two-point functions have poles corresponding to the meson masses and the decay constant of a meson is related to the residue of the corresponding pole. In AdS/QCD the two-point function can be obtained from the boundary effective action by including a source field on the UV brane and integrating out the AdS bulk using the EOM’s.
Starting from the UV boundary term in Eq. (28) and rewriting it in terms of and (ignoring , ), we have
As from Eq. (36), in order to have a finite limit when is taken to zero, the source for the heavy-light current is related to the UV boundary value by
where the additional factor accounts for that the source has engineering dimension one and is expected to be an number.
To solve for for the given boundary condition , it is convenient to define
then the bulk EOM’s can be written as
It is easy to show that the operator is Hermitian with the weight function if the boundary terms vanish. The eigenfunctions of are just the KK wave function solutions discussed in the previous subsection with eigenvalues . If we normalize the wave functions by
then they form an orthonormal basis which can be used to expand any function in the interval between and . (The eigenfunctions and has dimension one in this normalization.) The solution of in the presence of the source term can be written as
and the coefficient can be computed by
using the Hermiticity of .
Substituting into the boundary term in Eq. (41), we obtain the boundary effective action:
Matching the source of Eq. (42), the current-current correlator is given by
from which one can easily obtain the decay constant for the th excited meson state:
The normalization condition Eq. (45) for implies that444For (), the normalization integral is dominated by the UV region and the result will be sensitive to the UV cutoff if it is kept finite. In reality we can not expect so the applicability of the holographic model in this range of may be questionable.
Expanding for small , one obtains
Substituting it into Eq. (51), the decay constant is
For the heavy-light meson, we have , . The decay constant scales as
The scaling agrees with the general expectation in the heavy quark limit.
4 Parity Doubling and Chiral Symmetry Breaking
In the chiral symmetric limit of the light flavors, the “left-handed” and “right-handed” heavy-light mesons are degenerate and form a parity doublet. After the chiral symmetry breaking effect is included, the mass eigenstates are the parity eigenstates which are linear combinations of the left-handed and right-handed fields, and the mass splitting between the parity-odd and parity-even states is of the order of . To discuss the parity-doublet states, we introduce two sets of bi-spinors in the Weyl representation
which are and under the light flavor chiral symmetry . Using the relation in Eq. (8) between and and rotating to the Pauli-Dirac representation, we have
Similar to the simplest case in Eq. (18), the dictionary for relating and to the physical spin-0 and spin-1 states is
The parity symmetry on the and fields is defined as
One can check that this is consistent with the parity of the meson fields.
The chiral symmetry breaking in AdS/QCD is parametrized by the VEV of a bi-fundamental scalar field, , under ,
where we followed the notation in Ref.  and assumed the naïve scaling dimension for the corresponding operator. can be matched to the bare quark mass in the chiral Lagrangian but will be neglected in the rest of this paper. Under the parity transformation, we have . The chiral symmetry breaking in the holographic model of heavy-light mesons can be induced by the coupling of and fields to :