Strong couplings and form factors of charmed mesons in holographic QCD

# Strong couplings and form factors of charmed mesons in holographic QCD

## Abstract

We extend the two-flavor hard-wall holographic model of Erlich, Katz, Son and Stephanov [Phys. Rev. Lett. 95, 261602 (2005)] to four flavors to incorporate strange and charm quarks. The model incorporates chiral and flavor symmetry breaking and provides a reasonable description of masses and weak decay constants of a variety of scalar, pseudoscalar, vector and axial-vector strange and charmed mesons. In particular, we examine flavor symmetry breaking in the strong couplings of the meson to the charmed and mesons. We also compute electromagnetic form factors of the , , , , and mesons. We compare our results for the and mesons with lattice QCD data and other nonperturbative approaches.

## I Introduction

A major difficulty in the theoretical treatment of in-medium interactions of charmed hadrons is the lack of experimental information on the interactions in free space. For example, almost all knowledge on the interaction comes from calculations based on effective Lagrangians that are extensions of light-flavor chiral Lagrangians using flavor symmetry Mizutani:2006vq (); Lin:1999ve (); Hofmann:2005sw (); Haidenbauer:2007jq (); Haidenbauer:2008ff (); Haidenbauer:2010ch (); Fontoura:2012mz () and heavy quark symmetry Yasui:2009bz (); GarciaRecio:2008dp (). The Lagrangians involve coupling constants, like , , and , whose values are taken from flavor and heavy-quark symmetry relations. For instance, symmetry relates the couplings of the to the pseudoscalar mesons , and , namely . If in addition to flavor symmetry, heavy-quark spin symmetry is invoked, one has to leading order in the charm quark mass Casalbuoni:1996pg (); Manohar:2000dt (). The coupling is constrained by experimental data; the studies of the interaction in Refs Haidenbauer:2007jq (); Haidenbauer:2008ff (); Haidenbauer:2010ch () utilized such a relation, taking , which is the value used in a large body of work conducted within the Jülich model Haidenbauer:1991kt (); Hoffmann:1995ie () for light-flavor hadrons. This value of implies through symmetry , which is not very much different from predictions based on the vector meson dominance (VMD) model:  Mat98 (); Lin00a (). Moreover, to maintain unitarity in calculations of scattering phase shifts and cross sections, lowest-order Born diagrams need to be iterated with the use of a scattering equation, like the Lippmann-Schwinger equation, and phenomenological form factors are required to control ultraviolet divergences. Form factors involve cutoff parameters that also are subject to flavor dependence. Again, due to the lack of experimental information, they are also poorly constrained.

Flavor symmetry is strongly broken at the level of the QCD Lagrangian due to the widely different values of the quark masses; while in the light quark sector one has good symmetry, , thereby e.g. (up to a phase), in the heavy-flavor sector and symmetries are badly broken: . Given the importance of effective Lagrangians in the study of a great variety of phenomena involving mesons, in the present examine their properties in a holographic model of QCD. We extend the holographic QCD model of Refs. Erlich:2005qh (); Da Rold:2005zs () to the case of and investigate the implications of the widely different values of the quark masses on the effective three-meson couplings and and the electromagnetic form factors of the and mesons. The parameters of the model are the quark masses and condensates as well as the mass gap scale. Using experimental data for a selected set of meson masses to fix the model parameters, allows us to predict not only the strong couplings and electromagnetic form factors mentioned above but also many other observables not studied before with a holographic model.

The works in Refs. Erlich:2005qh (); Da Rold:2005zs () pioneered in the modeling of low energy QCD by incorporating features of dynamical chiral symmetry breaking in holographic QCD. They correctly identify the five dimensional gauge fields dual to the left and right currents associated with chiral symmetry as well as the five dimensional scalar field dual to the chiral condensate. The extension proposed in Ref. Abidin:2009aj () incorporated the strange quark and was able to identify the appearance of scalar modes associated with flavor symmetry breaking. In the present work, by extending the model of Refs. Erlich:2005qh (); Da Rold:2005zs () to the case , we are able to investigate the consequences of the dramatically different values of the quark masses on the phenomenology of charmed mesons. Moreover, by combining the formalism of Kaluza-Klein expansions and the AdS/CFT dictionary, we are able to directly extract the leptonic decay constants of mesons and find an expansion for the flavor currents that relate flavor symmetry breaking to the appearance of scalar modes. That relation bears a strong analogy with the generalized PCAC (partially conserved axial current relation) Holl:2004fr (); Dominguez:1976ut () that relates dynamical chiral symmetry breaking to the appearance of the pion and its resonances Ballon-Bayona:2014oma ().

In the model of Refs. Erlich:2005qh (); Da Rold:2005zs (), dynamical chiral symmetry breaking becomes manifest when considering fluctuations of the five dimensional gauge fields associated with the axial and vectorial sector. While the kinetic terms of the axial sector acquire a mass, signalizing chiral symmetry breaking, the vector sector remains massless. In our framework, it turns out that the vector sector also acquires a mass signalizing the breaking of flavor symmetry. The Kaluza-Klein decomposition of these fields allows us to obtain effective kinetic Lagrangians for the mesons from the five dimensional kinetic terms, with masses and decay constants obtained in terms of the wave functions representing the Kaluza-Klein modes. Moreover, expanding the five dimensional action to cubic order in the fluctuations and performing again a Kaluza-Klein decomposition allows us to obtain effective Lagrangians describing the three-meson interactions, with strong couplings given in terms of integrals involving the wave functions of the corresponding Kaluza-Klein modes.

It turns out that the symmetry breaking pattern in the strong couplings differs somewhat from previous studies in the literature. Calculations employing QCD sum rules found symmetry breaking in three-hadron couplings that vary within the range of 7% to 70%Bracco:2011pg (). In Ref. ElBennich:2011py (), using a model constrained by the Dyson-Schwinger equations of QCD, it was found that the relation is strongly violated at the level of 300% or more. In a recent follow up of that study within the same framework, Ref. El-Bennich:2016bno () finds that couplings between -, -mesons and -, -mesons can differ by almost an order-of-magnitude, and that the corresponding form factors also exhibit different momentum dependences. Our results calculations are more in line with calculations using the quark-pair creation model in the nonrelativistic quark model Krein:2012lra (); Fontoura:2017ujf ().

The organization of this paper is as follows. In Sec. II we describe how chiral and flavor symmetry breaking is realized in our model. Then in Sec. III we describe the five dimensional field equations and the AdS/CFT dictionary for the flavor and axial currents. In Sec. IV we describe the formalism of Kaluza-Klein expansions and obtain effective kinetic Lagrangians for the mesons. In Sec. V we use the prescription of our previous studies in Ref.  Ballon-Bayona:2014oma () for the leptonic decay constants and obtain relations describing flavor symmetry breaking and chiral symmetry breaking in terms of scalar and pseudoscalar modes respectively. In Sec. VI we obtain effective Lagrangians describing three-meson interactions with the holographic prescription for the strong couplings. Finally, in Sec. VII we fit the model parameters and present our numerical results for many observables, including the strong couplings and as well as the electromagnetic form factors of the and mesons. We compare the latter against lattice QCD data obtained in Ref. Can:2012tx (). Section VIII presents our conclusions.

## Ii Chiral symmetry and flavor symmetry in holographic QCD

Chiral symmetry for flavors holds in the massless limit of QCD and is described in terms of the left and right currents

 Jμ,aL/R=¯qL/RγμTaqL/R, (1)

where , are the generators of the group, and , with being the quark Dirac field. The generators are normalized by the trace condition , satisfying the Lie algebra . The generators are related to the Gell-Mann matrices by . Chiral symmetry is broken by the presence of the operator . This breaking can be explicit, when it appears in the QCD Lagragian associated with the nonzero quark masses, or dynamically, when it acquires a vacuum expectation value, giving rise to a condensate in limit of zero quark masses.

In the case , dynamical chiral symmetry breaking goes as , where is an exact vector symmetry and the broken symmetry occurs in the axial sector. This is described in terms of the vector and axial currents . The symmetry associated with the vector sector is known as isospin symmetry. When , both chiral and flavor symmetries are broken by the quark masses. We will describe how chiral and flavor symmetry breaking are implemented in a holographic model for .

In the pioneering work of Refs. Erlich:2005qh () and Da Rold:2005zs (), a simple holographic realization of chiral symmetry breaking (CSB) was proposed. They considered the simplest background in holographic QCD, known as the hard wall model Polchinski:2001tt (), consisting in a slice of anti-de-Sitter spacetime:

 ds2=1z2(ημνdxμdxν−dz2), (2)

with . The parameter determines an infrared (IR) scale at which conformal symmetry is broken. The action proposed in Ref. Erlich:2005qh () includes gauge fields and , corresponding to the left and right flavor currents , and a bifundamental field dual to the operator . The action can be written as

 S = ∫d5x√|g|Tr{(DmX)†(DmX)+3|X|2 (3) − (4)

where is the covariant derivative of the bifundamental field , and

 Lmn = ∂mLn−∂nLm−i[Lm,Ln], Rmn = ∂mRn−∂nRm−i[Rm,Rn], (5)

are non-Abelian field strengths. The 5-d squared mass of the field is fixed to , to match with the conformal dimension of the dual operator . The model of Ref. Erlich:2005qh () focused on and worked in the limit of exact flavor symmetry. In Ref. Abidin:2009aj (), Abidin and Carlson extended the model to , to incorporate the strange-quark sector. In the present paper we further extend that model to , with the aim of making predictions for charmed mesons. In our approach we use a Kaluza-Klein expansion for the 5-d fields in order to find a 4-d effective action for the mesons. This approach allows us to find directly the meson weak decay constants, couplings and expansions for the vector and axial currents. We find in particular a relation for the vector current describing flavor symmetry breaking (FSB).

 L0m= R0m=0,2X0=ζMz+Σζz3, (6)

where is the quark-mass matrix, , and is the matrix of the quark condensates, . The parameter is introduced to have consistency with the counting rules of large- QCD—for details, see Ref. Cherman:2008eh (). Note that we are assuming isospin symmetry in the light-quark sector, i.e. and , which is a very good approximate symmetry in QCD.

For the strange and charm quarks we will fit their masses and to the physical masses for the mesons. Note, however, that the model should not be valid for arbitrarily large quark masses. The reason is that, from the string theory perspective, the action in equation (4) is expected to arise from a small perturbation of coincident space-filling flavor branes. Specifically, the mass term appearing in Eq. (6) acts as a small source for the operator , responsible for the breaking of the chiral and flavor symmetries. A holographic description of quarks with very large masses requires the inclusion of long open strings and two sets of flavor branes distinguishing the heavy quarks from the light quarks (see e.g. Erdmenger:2006bg ()). In that framework, the string length is proportional to the quark mass and each set of flavor branes will carry a set of fields describing the dynamics of light and heavy mesons respectively. In this work we will show that the model described by the action in equation (4) is still a very good approximation for the dynamics of light and heavy-light charmed mesons, the reason being that the internal structure in both cases is governed by essentially the same nonperturbative physics, that occurs at the scale  Manohar:2000dt (). In heavy-heavy mesons, on the other hand, the internal dynamics is governed by short-distance physics. For recent holographic studies of mesons involving heavy quarks see Refs. Hashimoto:2014jua (); Braga:2015lck (); Liu:2016iqo (); Liu:2016urz ().

To investigate the consequences of chiral and flavor symmetry breaking it is convenient to rewrite the fluctuations of left and right gauge fields in terms of vector and axial fields , i.e. and . The bifundamental field X can be decomposed as

 X=eiπX0eiπ, (7)

where is the classical part and contains the fluctuations. The fields , and can be expanded as , and respectively.

It is important to remark that organizing the heavy-light D mesons together with the light pions and kaons in a 15-plet of fluctuation fields does not imply, automatically, that the heavy-light D mesons are being approximated by Nambu-Goldstone bosons. The reason is that the explicit breaking of chiral symmetry, driven by the heavy charm quark, is large and by no means its effects are neglected in the model. In the same way, the fact the mesons appear in the same multiplet of the SU(4) flavor group does not mean that flavor symmetry is exact; it is explicitly broken by the widely different values of the quark masses. The main advantage of using such a SU(4) representation with explicit symmetry-breaking terms is that it allows us to make contact with the four dimensional effective field theories describing the interactions of light and heavy-light mesons commonly used in phenomenological applications. This not only extends the work of Lin:1999ve (); Lin00a () but also leads to quantitative predictions for the strong couplings that can be tested against experiment or lattice QCD data, which is our main objective in the present paper. An alternative approach to describe the heavy-light mesons is to make contact with a particularly interesting class of four dimensional models that treat the light mesons as in the present paper, and treat heavy mesons by invoking heavy-quark symmetry. The Lagrangian in the heavy sector is written as an expansion in inverse powers of the heavy quark mass; Refs. Yasui:2009bz (); GarciaRecio:2008dp () are examples of such models. In holography the heavy quarks are realized in terms of long open strings, as described above in this section. For recent progress in the heavy quark approach to heavy-light mesons within holographic QCD see Liu:2016iqo (); Liu:2016urz ().

Expanding the action in Eq. (4) up to cubic order in the fields , and , we find

 S=S(2)+S(3)+…, (8)

where

 S(2) = ∫d5x√|g|{−14g25vmnavamn+12(MaV)2VmaVam−14g25amnaaamn+MabA2(∂mπa−Am,a)(∂mπb−Am,b)} (9)
 S(3) = ∫d5x√|g|{−12g25fabcvmna(VbmVcn+AbmAcn)−1g25fabcamnaVbmAcn−(MbV)22fabc(∂mπa−2Aam)Vm,bπc (10) + MaeAfebc(∂mπa−Aam)Vm,bπc}, (11)

and we have defined the Abelian field strengths and . In the kinetic term , the vector and axial symmetry breaking is dictated by the mass terms and , defined by the traces and . Note, however, that the axial sector in is invariant under the gauge transformation

 Aam→Aam−∂mλaA,πa→πa−λaA. (12)

Using Eq. (6) we find the following nonzero values for :

 (MaV)2=14(vs−vu)2fora=(4,5,6,7), (13) (MaV)2=14(vc−vu)2fora=(9,10,11,12), (14) (MaV)2=14(vc−vs)2fora=(13,14), (15)

and the nonzero values for :

 Ma,aA=v2ufora=(1,2,3), (16) Ma,aA=14(vs+vu)2fora=(4,5,6,7), (17) Ma,aA=14(vc+vu)2fora=(9,10,11,12), (18) Ma,aA=14(vc+vs)2fora=(13,14), (19) M8,8A=13(v2u+2v2s), (20) M15,15A=112(2v2u+v2s+9v2c), (21) M8,15A=M15,8A=13√2(v2u−v2s). (22)

In Eqs. (15) and (22) we have defined

 vq(z)=ζmqz+1ζσqz3,q=(u,s,c). (23)

In the interesting case where all the masses and condensates are equal we have that and the flavor symmetry is preserved. In this paper we consider quark masses and condensates that lead to a realistic spectrum for the mesons so that we could explore the consequences of flavor symmetry breaking. The kinetic term in Eq. (9) allows us to extract the meson spectrum and decay constants whereas the action in Eq. (11) leads to nontrivial predictions for three-meson couplings, including the heavy-light charmed mesons and .

Note that the mass term for the vectorial sector is zero not only for , corresponding to the light sector but also for , which implies that flavor symmetry has not been broken in the sector describing the dynamics of the and mesons. This is one clear example of heavy-heavy mesons (mesons composed by a heavy quark-antiquark pair), where we actually expect some corrections to appear in (4) describing flavor symmetry breaking. Those terms would arise from the dynamics of long open strings dual to heavy quarks, as explained above in this section.

## Iii Field equations and dual currents

Writing the kinetic action in Eq. (9) as , its variation takes the form where

 δS(2)Bulk = ∫d5x[(∂L(2)∂Vaℓ−∂mPmℓV,a)δVaℓ (24) + (∂L(2)∂Aaℓ−∂mPmℓA,a)δAaℓ (25) + (∂L(2)∂πa−∂mPmπ,a)δπa], (26)
 δS(2)Bdy=∫d5x∂m(PmℓV,aδVaℓ+PmℓA,aδAaℓ+Pmπ,aδπa), (27)

and

 PmℓV,a := ∂L(2)∂(∂mVaℓ)=−1g25√|g|vmℓa, (29) PmℓA,a := ∂L(2)∂(∂mAaℓ)=−1g25√|g|amℓa, (30) Pmπ,a := ∂L(2)∂(∂mπa)=MabA√|g|(∂mπb−Amb), (31)

are conjugate momenta to the 5-d fields , and respectively. The bulk term in Eq. (26) leads to the field equations

 ∂m[√|g|vmna]+g25(MaV)2√|g|Vna=0, (32) ∂m[√|g|amna]−g25MabA√|g|(∂nπb−Anb)=0, (33) ∂m[MabA√|g|(∂mπb−Amb)]=0. (34)

Imposing the boundary conditions and the boundary term (LABEL:deltaS2Bdy) reduces to

 δS(2)Bdy = −∫d4x[⟨J^μV,a⟩(δVa^μ)z=ϵ+⟨J^μA,a⟩(δAa^μ)z=ϵ (35) + ⟨Jπ,a⟩(δπa)z=ϵ], (36)

where we find the dual 4-d currents

 ⟨J^μV,a(x)⟩ = PzμV,a|z=ϵ=−1g25(√|g|vzμa)z=ϵ, (37) ⟨J^μA,a(x)⟩ = PzμA,a|z=ϵ=−1g25(√|g|azμa)z=ϵ, (38) ⟨Jπ,a(x)⟩ = Pzπ,a|z=ϵ=[√|g|MabA(∂zπb−Azb)]z=ϵ (39) = ∂^μ⟨J^μA,a(x)⟩. (40)

Note that we distinguish the vector Minkowski indices from the AdS indices . The results in Eqs. (37), (38) and (40) define the holographic prescription for expectation values of the 4-d vector, axial and pion current operators. Note from Eqs. (34) and (37) that when , i.e. the vector current is not conserved for those cases. Similarly, from Eqs. (34) and (38), one sees that for any (because ), i.e. the axial current is never conserved.

## Iv The 4-d effective action

After decomposing the vector and axial fields into their components and evaluating the metric in Eq. (2), the kinetic action in Eq. (9) takes the form

 S(2) = S(2)V+S(2)A, (41)

where

 S(2)V = ∫d4x∫dzz{−14g25[(va^μ^ν)2−2(vaz^μ)2] (42) + (MaV)22z2[(Va^μ)2−(Vaz)2]}, (43)

and

 S(2)A = ∫d4x∫dzz{−14g25[(aa^μ^ν)2−2(aaz^μ)2] (44) + MabA2z2[(∂^μπa−A^μ,a)(∂^μπb−Ab^μ) (45) − (∂zπa−Aaz)(∂zπb−Abz)]}. (46)

The vector and axial sectors admit a decomposition in irreducible representations of the Lorentz group. For the vector sector we find

 Va^μ = V⊥,a^μ+∂^μ(~ϕa−~πa), Vaz = −∂z~πa, (47)

where . The 5-d field describes an infinite tower of 4-d massive spin 1 fields, i.e. the vector mesons, whereas the 5-d fields and describe an infinite tower of massive scalar fields, i.e. scalar mesons associated with flavor symmetry breaking (FSB).

On the other hand, the gauge symmetry in Eq. (12) allows us to decompose the axial sector as

 Aa^μ → A⊥,a^μ,Aaz→−∂zϕa, (48) πa → πa−ϕa, (49)

where . This time the 5-d field will describe an infinite tower of 4-d massive axial spin 1 fields, i.e. the axial vector mesons. The 5-d fields and will describe an infinite tower of 4-d pseudoscalar fields, i.e the pions associated with chiral symmetry breaking (CSB).

Using Eqs. (47), (47) and (49) the actions in Eqs. (43) and (46) take the form

 S(2)V = ∫d4x∫dzz{−14g25[(v⊥,a^μν)2−2(∂zV⊥,a^μ)2 (50) − 2(∂z∂^μ~ϕa)2]+(MaV)22z2[(V⊥,a^μ)2+(∂^μ~πa−∂^μ~ϕa)2 (51) − (∂z~πa)2]+∂^μ(…)}, (52)
 S(2)A = ∫d4x∫dzz−14g25[(a⊥,a^μν)2−2(∂zA⊥,a^μ)2 (53) − 2(∂z∂^μϕa)2]+MabA2z2[A^μ,a⊥A⊥,b^μ (54) + (∂^μπa−∂^μϕa)(∂^μπb−∂^μϕb)−(∂zπa)(∂zπb)] (55) + ∂^μ(…)}, (56)

where the terms in are surface terms that vanish after choosing periodic boundary conditions for the fields.

The actions in Eqs. (52) and (56) are in a suitable form to perform a Kaluza-Klein expansion for the 5-d fields. Before performing this expansion note that the nondiagonal mass term induces a mixing in the axial sector for meson states with flavor indices . In this paper we are mainly interested in the axial sector states with and , corresponding to the heavy-light charmed mesons and the usual light mesons. Then from now on we will consider for the axial sector only those states where . The axial sector states corresponding to have an interesting physical interpretation, e.g mixing for the pseudoscalar sector, and deserve a further study that will be pursued in a future project.

The 5-d fields in the vector sector admit a Kaluza-Klein expansion of the form

 V⊥,a^μ(x,z) = g5va,n(z)^Va,n^μ(x), (57) ~πa(x,z) = g5~πa,n(z)^πa,nV(x), (58) ~ϕa(x,z) = g5~ϕa,n(z)^πa,nV(x),, (59)

where a sum from to is implicit. A similar decomposition holds for the 5-d fields in the axial sector:

 A⊥,a^μ(x,z) = g5aa,n(z)^Aa,n^μ(x), (60) πa(x,z) = g5πa,n(z)^πa,n(x), (61) ϕa(x,z) = g5ϕa,n(z)^πa,n(x). (62)

Using these expansions the actions in Eqs. (52) and (56) factorize into integrals and integrals and we find and , with the vector and axial 4-d Lagrangians given by

 LV = −14Δa,nmV^va,n^μ^ν^v^μ^νa,m+12Ma,nmV^Va,n^μ^V^μa,m (63) + 12Δa,nmπV(∂^μ^πa,nV)(∂^μπa,mV)−12Ma,nmπV^πa,nV^πa,mV, (64)
 LA = −14Δa,nmA^aa,n^μ^ν^a^μ^νa,m+12Ma,nmA^Aa,n^μ^A^μa,m (66) + 12Δa,nmπ(∂^μ^πa,n)(∂^μ^πa,m)−12Ma,nmπ^πa,n^πa,m, (67)

with coefficients defined by the integrals

 Δa,nmV = ∫dzzva,n(z)va,m(z), (69) Ma,nmV = ∫dzz{[∂zva,n(z)][∂zva,m(z)] (70) + βaV(z)va,n(z)va,m(z)}, (71) Δa,nmπV = ∫dzz{[∂z~ϕa,n(z)][∂z~ϕa,m(z)] (72) + βaV(z)[~πa,n(z)−~ϕa,n(z)][~πa,m(z)−~ϕa,m(z)]}, (73) Ma,nmπV = ∫dzzβaV(z)[∂z~πa,n][∂z~πa,m], (74)

for the vector sector and

 Δa,nmA = ∫dzzaa,n(z)aa,m(z), (75) Ma,nmA = ∫dzz{[∂zaa,n(z)][∂zaa,m(z)] (76) + βaA(z)aa,n(z)aa,m(z)}, (77) Δa,nmπ = ∫dzz{[∂zϕa,n(z)][∂zϕa,m(z)] (78) + βaA(z)[