Kaon form factor in holographic QCD
AAPCTP Pre2019-010Kaon form factor in holographic QCD
Zainul Abidin ∗ and Parada T. P. Hutauruk † Sekolah Tinggi Keguruan dan Ilmu Pendidikan Surya, Tangerang, Jawa Barat 15115, Indonesia Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 37673, Korea (Dated: September 5, 2019)The kaon form factor in the spacelike region is calculated using a holographic QCD model with the “bottom-up” approach. We found that our result for the kaon form factor in low Q has a remarkable agreement with theexisting data, where Q is the four-momentum transfer squared. The charge radius of the kaon as well as the kaondecay constant are found to be in a good agreement with the experiment data. We then predict the kaon formfactor in the asymptotic region (larger Q ) showing 1 / Q behavior, which is consistent with the perturbativeQCD prediction. I. INTRODUCTION
A theory of quantum chromodynamics (QCD), which is anon-Abelian gauge theory, is believed so far as a correct theoryof hadrons, where hadrons are the composite particles made ofquarks and gluons [1]. QCD has the essential features, namely,confinement and chiral symmetry breaking [1, 2]. However,the form factor, which is one of the nonperturbative quanti-ties, is very difficult to compute directly from QCD. Severaltheoretical and phenomenological models [3–7] as well as alattice QCD calculation [8] have been used to calculate thisnonperturbative quantity of QCD.Apart from those models, during the past few years, holo-graphic QCD models, which are the complementary modelof QCD, have also been applied to describe the structure ofhadrons, namely, meson [9–14] and nucleon [14] form factorsas well as charmed meson [15], in order to gain a deep under-standing of the structure of hadrons, from a different substan-tially point of view. Surprisingly, these holographic modelswork well in predicting other hadron observables, namely thedecay constant and mass spectrum. Also, one can argue thatQCD approximately behaves as a conformal over a particularkinematic region [13, 16]. Those holographic QCD modelsare able to preserve confinement [11, 17, 18] and chiral sym-metry breaking [12, 19–21], which is in many ways similar tothe main properties of QCD in low energy, after a few yearssince the holographic model was proposed [22, 23].The original AdS/CFT correspondence [22] has been firstused to connect a strongly coupled 4D conformal theory forlarge N c , where N c is the color number, and a weakly coupledgravity theory on AdS space. It then has been reconstructedstarting from QCD and its 5D gravity dual theory to reproducethe properties of QCD [16, 24–26].However, among those holographic models with variousapproaches [9–15, 17–23], only a few models have been usedto calculate the kaon form factor in a holographic QCD modelwith different approaches [15]. K + ( u ¯ s ) is a very interestingobject, because it consists of a strange quark, beside an up ∗ [email protected] † [email protected] quark, where the mass of the strange quark is heavier thanthe u quark. Experimentally, the existing data on the kaonform factor are very poor in higher Q , only old data for low Q are available [27], where Q is the four-momentum transfersquared. In the future, experiments will measure the kaon formfactor in higher Q [28, 29]. It would be interesting to see howour complementary model, which is inspired by this AdS/CFTcorrespondence, predicts the kaon form factor in higher Q .This work may pave the way to understand the strange quarkproperties as well as the strange quark form factor.In the present paper, we calculate the kaon form factorin holographic QCD, which is a complementary approach ofQCD. In this work, we adopt a “bottom-up” approach of theAdS/CFT correspondence, instead of a “top-down” approach,where we employ the properties of QCD to construct its 5Dgravity dual theory as performed in Refs. [24, 25, 30]. Webegin to describe the AdS/CFT correspondence formalism,describing a correspondence between 4D operators O (x) andfields in the 5D bulk φ (x,z). We then calculate the kaon formfactor in holographic QCD. We find the result on the kaonform factor is in good agreement compared to the existing datain low Q [27]. We then predict the kaon form factor in higher Q . Experimentally, the experimental data are really poor inhigher Q . We find that the kaon form factor in higher Q isconsistent with the perturbation QCD prediction [31]. Next,we calculate the charge radius of the kaon in holography. Wefind that our result on the charge radius is an excellent agree-ment with the data [27] as well as the Particle Data Group(PDG) [32].This paper is organized as follows. In Sec. II, we brieflyreview the AdS/CFT correspondence, two- and three-pointfunctions, and how to extract the form factor of the kaon fromholographic QCD in Sec. III. In Sec. IV, we present thecalculation of the charge radius of the kaon. In Sec. V, numer-ical results are presented and their implications are discussed.Section VI is devoted to a summary. a r X i v : . [ h e p - ph ] S e p II. FORMALISMA. The AdS/QCD correspondence
In this section, we briefly review the calculation of the vac-uum expectation values of the operators based on a generatingfunction Z D in the 4D space, which is defined by Z D [ φ ] = (cid:28) exp (cid:18) iS D + i ∫ d x O( x ) φ ( x ) (cid:19)(cid:29) , (1)where S D is the action for the 4 D theory and φ is a sourcefunction together with a specific operator O (x), which corre-sponds to the expectation value. It then can be written by (cid:104) |T O( x ) . . . O( x n )| (cid:105) = (− i ) n δ Z D δφ ( x ) . . . φ ( x n ) . (2)The following AdS/CFT correspondence provides the equiv-alence between the generating functional of the connected cor-relation for the 4D theory and the effective partition functionfor the 5D theory: Z D [ φ ] = exp ( iS D ( φ cl )) , (3)where φ cl is a solution of the 5D equation of motion with aboundary, as defined in Eq. (5).We consider only the tree-level diagram on the 5D theory,and we choose the following metric for the 5D space-time: ds = g M N dx M dx N = z (cid:16) η µν dx µ dx ν − dz (cid:17) , ε < z < z , (4)where x is the 4D space-time coordinate, η µν = diag ( , − , − , − ) is the flat space metric, and z is the fifthcoordinate, which corresponds to the energy scale ( Q ∼ / z ).We set z = ε → z = z = / Λ QCD is the infrared boundary, which is usedfor the conformal symmetry breaking of QCD.The UV boundary value of the 5D field is the source ofthe corresponding 4D operator O . One can write the classicalsolution of the 5D field as φ cl ( x , z ) = φ ( x , z ) φ ( x ) . (5)The value of φ ( x , ε ) → ε ∆ ).Hence, φ ( x ) is identified as the UV-boundary value of the φ cl ( x , z ) field. B. The 5D AdS model
The action in 5D theory is written as S = ∫ d x √ g Tr (cid:40) | DX | + | X | − g (cid:16) F L + F R (cid:17)(cid:41) , (6)where g = | det g M N | is the determinant of metric tensor, g is agauge coupling parameter, which is fixed by the QCD operatorproduct expansion, and the bifundamental scalar field X inEq. (6) is expressed by X ( x , z ) = exp ( i π a ( x , z ) t a ) X ( z ) exp (− i π a ( x , z ) t a ) , (7)where t a = σ a / [ t a t b ] = δ ab /
2, where σ a are the Pauli matrices. The covariant deriva-tive is defined as D M X = ∂ M X − iL M X + iX R M , (8)where the 5D space-time is denoted by the lowercase index of M = ( µ, z ) and F LM N is written as F LM N = ∂ M L N − ∂ N L M − i [ L M , L N ] , (9)Analogously for F RM N .The L and the R fields can be written as vector field V andthe axial-vector field A : L M = V M + A M (10) R M = V M − A M . (11)In this work, we consider the following 4D operators thatare defined by the current operators J aL µ = ¯ ψ qL γ µ t a ψ qL and J aR µ = ¯ ψ qR γ µ t a ψ qR that correspond to the gauge fields L a µ and R a µ in the 5D theory, respectively. The operator of ¯ ψ q R ψ q L corresponds to a bifundamental scalar field X a in Eq. (7),where the index a = , , . . . , µ = , , , C. Two-point functions
Here we consider only the scalar parts of the action X ( z ) ,up to second order, it gives S scalar = ∫ d x √ g Tr (cid:16) g M N ∂ M X ∂ N X + | X | (cid:17) , (12)where g M N is defined in Eq. (4), which is the nontrivial 5Dmetric.The UV boundary of the scalar field X is proportional to thequark mass matrix M , which can be considered as the sourcefor the operator of ¯ ψ R ψ L . Solving the equation of motion forthe scalar field, it then gives X ( z ) = a z + a z , (13)where a is defined as in Ref. [30] by a = M = (cid:169)(cid:173)(cid:171) m q m q m s (cid:170)(cid:174)(cid:172) , (14)where we consider the SU(2) isospin symmetry, where themass for the up and down quarks are identical.Using the AdS/CFT correspondence, we then calculatethe quark condensate (cid:10) ¯ ψψ (cid:11) = Σ by performing a functionalderivative of the action in Eq. (12), evaluated on the classicalsolution, over δ M and identify a = Σ = (cid:169)(cid:173)(cid:171) σ q σ q σ s (cid:170)(cid:174)(cid:172) . (15)We also assume that σ q = σ s = σ and define v q ( z ) = m q z + σ z and v s ( z ) = m s z + σ z . D. Transverse vector
We now consider only vector parts of the action up to secondorder. It gives S vector = ∫ d x (cid:213) a = g z (cid:16) − (cid:0) ∂ M V aN − ∂ N V aM (cid:1) + α a ( z )( V aM ) (cid:17) . (16)A contraction over 5D metric η M L is implied. We then define α a ( z ) = a = , , g (cid:0) v s − v q (cid:1) /( z ) a = , , , a = , (17)We have gauge choice to set V az = a = , , , V a ⊥ ,µ ( q , z ) iswritten as (cid:18) ∂ z z ∂ z + q − α a z (cid:19) V a ⊥ ,µ ( q , z ) = , (18)where α a = g ( m s − m q ) / σ s = σ q for a = , , , V a ⊥ ,µ ( q , z ) = V , a ⊥ ,µ ( q ) V a ( q , z ) with the so-called bulk-to-boundary propagator V a ( q , z ) , which is normalized to V a ( q , ε ) = z =
0, and V , a ⊥ ,µ ( q ) is the Fourier transform of the source of the vector current J aV ,µ = ¯ ψ q v γ µ t a ψ q v at the UV boundary z = ε . We also im-pose a Neumann boundary condition ∂ z V ( q , z ) =
0. The solution for the bulk-to-boundary propagator is written as V a ( q , z ) = π qz (cid:18) Y ( ˜ qz ) J ( ˜ qz ) J ( ˜ qz ) − Y ( ˜ qz ) (cid:19) , (19)where ˜ q = q − α a , and Y ( x ) and J ( x ) are the Bessel func-tions, respectively.For spacelike four-momentum transferred q = − Q < V a ( Q , z ) = ˜ Qz (cid:18) K ( ˜ Qz ) I ( ˜ Qz ) I ( ˜ Qz ) + K ( ˜ Qz ) (cid:19) , (20)where ˜ Q = (cid:112) Q + α a , and K ( x ) and I ( x ) are the modifiedBessel functions, respectively.The action on the solution in Eq. (20) is evaluated withapplying transverse projector η µν → (cid:16) η µν − q µ q ν q (cid:17) = P µν T ,since ∂ µ V a ,µ ⊥ =
0, one has the form S vector = − g ∫ d q ( π ) V , a µ ( q ) V , a ν ( q ) P µν T ∂ z V a ( q , ε ) z . (21)After solving the differential part, by the AdS/CFT corre-spondence, we obtain the current-current two-point functions (cid:68) (cid:12)(cid:12)(cid:12) T J a ,µ ⊥ ( x ) J b ,ν ⊥ ( y ) (cid:12)(cid:12)(cid:12) (cid:69) = i δ S D i δ V , a ⊥ ,µ ( x ) δ V , b ⊥ ,ν ( y ) , (22)where V , a µ ( q ) = ∫ d xe iqx V , a µ ( x ) , (23)and this leads to i ∫ d x e iqx (cid:68) (cid:12)(cid:12)(cid:12) T J a ,µ ⊥ ( x ) J b ,ν ⊥ ( ) (cid:12)(cid:12)(cid:12) (cid:69) = − g P µν T δ ab ∂ z V a ( q , ε ) z , (24)where T is the time-ordering operator.The bulk-to-boundary propagator can be written as V a ( q , z ) = ∞ (cid:213) n = c an ( q ) ψ n ( z ) , (25)where the wave function of ψ n satisfies the eigenvalue equation (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ∂ z z ∂ z + (cid:18)(cid:16) M aV , n (cid:17) − α a (cid:19) z (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) ψ an ( z ) = ∫ dz z ψ an ψ am = δ mn , (27)with boundary condition ψ an ( ε ) = = ∂ z ψ an ( z ) and the solu-tion is ψ an ( z ) = √ z J (cid:16) z (cid:113) ( M aV , n ) − α a (cid:17) z J (cid:16) z (cid:113) ( M aV , n ) − α a (cid:17) (28)where the eigenvalues of M aV , n (Kaluza-Klein tower of themass of the vector mesons: ρ meson for a = , , K ∗ mesonfor a = , , ,
7, and ω meson for a =
8) are obtained from J (cid:16) z (cid:113) ( M aV , n ) − α a (cid:17) = c an ( q ) = − ε ∂ z ψ n ( ε ) q − ( M aV , n ) . (29)Since i ∫ d x e iqx (cid:68) (cid:12)(cid:12)(cid:12) T J a ,µ ⊥ ( x ) J b ,ν ⊥ ( ) (cid:12)(cid:12)(cid:12) (cid:69) = (cid:213) ( f aV , n ) δ ab q − ( M aV , n ) (cid:18) η µν − q µ q ν q (cid:19) + ( nonpole terms ) , (30)where the definition of f aV , n is given by the matrix elementof current, (cid:10) | J a µ | V cn ( q , λ ) (cid:11) = f aV , n δ ac ε µ ( q , λ ) . We identify f aV , n = ∂ z ψ an ( ε )/ ε . Also, the parameter g = π / N c = π is fixed from the quark bubble diagram in the leading order,with N c = E. Axial-vector and pseudoscalar
The action for the axial-vector and pseudoscalar sector partsup to second order is written as S axial = ∫ d x (cid:213) a = × g z (cid:16) − (cid:0) ∂ M A aN − ∂ N A M (cid:1) + β a ( z ) (cid:0) ∂ M π a − A aM (cid:1) (cid:17) (31)A contraction over 5D metric η M L is implied. We have gaugechoice A aM → A aM − ∂ M λ a , and π a → π a − λ a and A az = β a ( z ) = g v q / z a = , , g ( v q + v s ) /( z ) a = , , , g ( v q + v s )/( z ) a = , (32)For the field φ that comes from the longitudinal part, we define A (cid:107) ,µ = ∂ µ φ a . We then write the Fourier transform ofthe fields in terms of the bulk-to-boundary propagators thatgives φ a ( p , z ) = φ a ( p , z ) φ a ( p ) = φ a ( p , z ) ip α p A a (cid:107) α ( p ) ,π a ( p , z ) = π a ( p , z ) ip α p A a (cid:107) α ( p ) , A b ⊥ µ ( q , z ) = A b ( q , z ) A b ⊥ µ ( q ) (33)where A a (cid:107) α ( p ) is the Fourier transform of the source functionof the 4D axial current operator J a ,α A , (cid:107) and A b ⊥ µ ( q ) is the Fouriertransform of the source function of the 4D axial current oper-ator J a ,α A , ⊥ .We obtain the coupled differential equations for the lon-gitudinal part of the axial-vector and pseudoscalar fields asfollows − q ∂ z φ a ( q , z ) + β a ( z ) ∂ z π a ( q , z ) = , (34) ∂ z (cid:18) z ∂ z φ a ( q , z ) (cid:19) − β a ( z ) z (cid:16) φ a ( q , z ) − π a ( q , z ) (cid:17) = , (35)with the boundary conditions φ a ( q , ε ) = , π a ( q , ε ) = − ∂ z φ a ( q , z ) = = ∂ z π a ( q , z ) . The expression for thetransverse part of the axial-vector field is analogous to thevector field. It then gives (cid:18) ∂ z z ∂ z + q − β a ( z ) z (cid:19) A a ⊥ ( q , z ) = . (36)We then substitute the coupled differential in Eqs. (34)and (35) into a second-order equation, and we obtain ∂ z (cid:18) z β a ( z ) ∂ z y a ( q , z ) (cid:19) + z (cid:18) q β a ( z ) − (cid:19) y a ( q , z ) = . (37)where y a ( q , z ) = ∂ z φ a ( q , z )/ z . In this form the boundarycondition is y ( q , z ) = ε∂ z y a ( q , ε )/ β a ( ε ) =
1. Wethen have the solution as y a ( q , z ) = (cid:213) ( M a π, n ) y an ( ε ) y an ( z ) q − ( M a π, n ) (38)where y an ( z ) is a normalized solution of the eigenvalue inEq. (37) with q = M a π, n , boundary conditions y n ( z ) = ε∂ z y an ( ε )/ β ( ε ) =
0. The normalization is ∫ z β a ( z ) y an ( z ) y am ( z ) = δ mn ( M a π, n ) (39)The eigenvalues of ( M a π, m ) are the Kaluza-Klein (KK)masses for the pseudoscalar mesons: the pions for a = , , a = , , ,
7, and η ’s for a =
8. The eigenvaluesare obtained from the transverse part of the axial vector inEq. (36), giving us the KK mass of the a and K mesons.As noted above, for the vector sector, we do not have thefreedom to set V az =
0, for a = , , ,
7. However, if wedefine V az = − ∂ z ˜ π a , V a (cid:107) ,µ = ∂ µ ( ˜ φ a − ˜ π a ) , we obtain analogousequations as in Eqs. (34) and (35) with α a ( z ) in place of β a ( z ) .We may proceed as above to obtain the eigenvalues of the KKmass of the scalar mesons K ∗ .A current-current correlator for the axial sector is written as i ∫ x e iqx (cid:10) (cid:12)(cid:12) T J a µ A ⊥ ( x ) J b ν A ⊥ ( ) (cid:12)(cid:12) (cid:11) = − P µν T δ ab ∂ z A a ⊥ ( q , ε ) g ε , i ∫ x e iqx (cid:10) (cid:12)(cid:12) T J a µ A (cid:107) ( x ) J b ν A (cid:107) ( ) (cid:12)(cid:12) (cid:11) = − P µν L δ ab ∂ z φ a ( q , ε ) g ε , (40)where P µν L = q µ q ν / q . Using the completeness relation (cid:205) n ∫ d q ( π ) q | n ( q )(cid:105)(cid:104) n ( q )| = q − m n , and taking a limit q → m n , oneidentifies the decay constant of the pseudoscalar mesons fromAdS/QCD correspondence: f aA , n = − y an ( ε ) g = − ∂ z φ an ( ε ) g ε , (41)where the decay constants are defined by (cid:68) (cid:12)(cid:12)(cid:12) J aA µ (cid:107) ( ) (cid:12)(cid:12)(cid:12) π bn ( q ) (cid:69) = i f aA , n q µ δ ab , (42)where the states of | π bn ( q ) (cid:11) are also considered for the pions( b = , ,
3) as well as the kaons ( b = , , , III. KAON ELECTROMAGNETIC FORM FACTOR
The electromagnetic form factors of the pion and kaon arepresented in this section. The relevant parts of the action are S A (cid:107) V ⊥ A (cid:107) = ∫ d x (cid:18) g z ∂ µ φ a ∂ µ V b ν ∂ ν φ c f abc + z ( ∂ µ π a − ∂ µ φ a ) V b µ π c g abc + z (cid:18) − ∂ µ ( π a π c ) + ∂ µ φ a π c (cid:19) V b µ h abc (cid:19) , (43)where the first term in Eq. (43) that contains f abc arises fromthe gauge part of the original action and other terms come fromthe chiral part. We then define g abc = − i Tr { t a , X } (cid:2) t b , { t c , X } (cid:3) , h abc = − i Tr (cid:2) t b , X (cid:3) { t a , { t c , X }} . (44)If g abc and h abc in Eq. (44) do not have a , b , or c , which do not equal “8”, it then gives g abc = f abc v a v c , h abc = f abc ( v c − v a ) v c , (45)where for X = c + c t , the v a is defined as v a = c + c d aa = (cid:26) v q , a = , , (cid:0) v q + v s (cid:1) , a = , , , . (46)where f abc and d abc are the structure constants of the SU(3)algebra.For three-point functions, we calculate three current oper-ators by taking the functional derivative of Eq.(43). One hasthe form (cid:104) |T J a ,α A (cid:107) ( x ) J µ ⊥ ( y ) J c ,β A (cid:107) ( w )| (cid:105) = i δ S A (cid:107) V ⊥ A (cid:107) i δ A a (cid:107) α ( x ) δ V b ⊥ µ ( y ) δ A c (cid:107) β ( w ) (47)From Eq. (47), we then extract the form factor using thefollowing matrix element − f a ∗ n f bm p β k α (cid:68) π an ( p ) (cid:12)(cid:12)(cid:12) J b ,µ ⊥ (cid:12)(cid:12)(cid:12) π cm ( k ) (cid:69) ( π ) δ ( p − q − k ) = lim p →( M a π n ) k →( M c π m ) (cid:16) p − ( M a π n ) (cid:17) (cid:16) k − ( M c π m ) (cid:17) × ∫ d xd y d w e i ( px − qy − kw ) ×(cid:104) | T J a α A (cid:107) ( x ) J b µ ⊥ ( y ) J c β A (cid:107) ( w )| (cid:105) . (48)We then obtain (cid:68) π an ( p ) (cid:12)(cid:12)(cid:12) J b ,µ ⊥ (cid:12)(cid:12)(cid:12) π cm ( k ) (cid:69) = i ( p + k ) µ ∫ dzV b ( q , z ) z (cid:16) ( ∂ z φ an )( ∂ z φ cm ) + g v a v c z (cid:0) π an − φ an (cid:1) (cid:0) π cm − φ cm (cid:1) (cid:17) f abc (49)For three quark flavors, the electromagnetic current operatoris defined as J E M ,µ = J µ + √ J µ (50)The current matrix element for the kaons | K + n (cid:105) = | π n + i π n (cid:105) is written as (cid:104) K + n ( p B ) (cid:12)(cid:12) J E M ,µ (cid:12)(cid:12) K + n ( p A )(cid:105) = ( p A + p B ) µ F Knn ( Q ) (51)where Q = − q = −( p A − p B ) and a final expression for thekaon form factor is obtained by F Knn ( Q ) = ∫ dzV ( Q , z ) z (cid:16) ( ∂ z φ n )( ∂ z φ m ) + g v z (cid:16) π n − φ n (cid:17) (cid:16) π m − φ m (cid:17) (cid:17) . (52) IV. KAON CHARGE RADIUS
In this section, we present the charge radius of the kaon inlow Q as well as in higher Q . For doing so, we recall thekaon form factor in Eq. (52) that is F Knn ( Q ) = ∫ z zV a ( Q , z ) ρ bnn ( z ) , (53)where a = , , b = , , ,
7, and ρ bnn ( z ) is defined by ρ bnn ( z ) = ( ∂ z φ bn ) z + g v b z (cid:16) π bn − φ bn (cid:17) . (54)In the limit of Q →
0, the bulk-to-boundary propagator inEq.(20) is written as, V a ( Q , z ) = − Q z (cid:18) − (cid:18) zz (cid:19)(cid:19) . (55)Using the expansion of Eq.(20), we obtain the radius of thekaon as follows (cid:104) r Kn (cid:105) = − dF Knn ( Q ) d ( Q ) = ∫ z z (cid:18) − (cid:18) zz (cid:19)(cid:19) ρ bnn ( z ) . (56) V. NUMERICAL RESULTS
Our numerical results for the kaon masses, decay constants,and kaon form factors are presented in this section. FollowingRef. [13], we fix the parameter values of the hard-wall cutoff at z = ( . ) − , which is chosen to fit the lightest ρ mesonmass M aV , = . a = , ,
3. Parameters m q and σ q is chosen to reproduce the pion mass and decay constant,respectively. Given the values of the pion mass M a π, = . f a = . a = , , m q = . σ q = ( . ) . We then fix σ s = σ q . The strange current quark mass m s = .
96 MeV is chosento fit the kaon mass M aK , = . a = , , , K + , K − , K , and ¯ K , respectively). We simplyconsider m q , m s , and σ as model parameters, not the (realistic)physical values of the quark mass and quark condensate. Forgetting a better connection between the light current quarkmass and condensate, we redefine the parameters by taking m q → √ N c / π , and σ → π / √ N c without modifying theabove results for the two-point and three-point functions. With this redefinition, we obtain m q = .
31 MeV, m s = . σ = ( . ) .Using the obtained parameters above, we determine the de-cay constant of the lightest KK of the kaons f K + =
104 MeV,and the mass and decay constant of the K ∗ are m K ∗ = f K ∗ =
28 MeV, respectively. The decay constant ofthe ρ meson f / ρ =
329 MeV. The mass and decay constantof the lightest KK of the vector mesons K ∗ are m K ∗ = f / K ∗ =
329 MeV, respectively. For the axial vectormesons, the mass and decay constant of the a are m a = f / a =
489 MeV, respectively. For the K , we obtain m K = f / K =
511 MeV. The values of the de-cay constant and the mass of the kaon obtained are consistentwith PDG [32].Results for the kaon form factor are shown in Figs. 1- 3. Fig-ure 1 shows our prediction for the kaon form factor comparedto the existing data [27] in low Q . We find that our predictionis in excellent agreement with the data [27]. We then calculatethe kaon form factor up to Q = Q data which will collect soon [28, 29], as in Fig. 2, however,experimentally, the kaon form factor is poorly known.Figure 3 shows the same results as in Fig. 2, but for Q F K ( Q ) . For larger Q (asymptotic region), the bulk-to-boundary propagator is written as V a ( Q , z ) Q →∞ = ( Qz ) K ( Qz ) ≈ (cid:114) π Qz e − Qz , (57)which goes to zero unless z is infinitesimal, z ∼ / Q . Note thatthe first term in Eq. (54) goes to g ( f an ) when z →
0, whilethe second term goes like ε →
0. The quantity zV a ( Q , z ) behaves like a delta function picking up ρ ann ( z ) at z →
0. Theupper limit of the form factor integral can be set to infinity asthe integrated vanish at large z . Then, the kaon form factor inhigher Q is defined by F a ( Q ) Q →∞ = g ( f an ) Q ∫ ∞ d w w K ( w ) = g ( f an ) Q = π ( f an ) Q . (58)We find that the kaon form factor for larger Q agrees wellwith the perturbative QCD prediction [31].Using Eq. (56), we obtain the charge radius for the lightestkaon r K + = ψ R ψ L inthe AdS boundary, which is consistent with the AdS/CFT rule, .
02 0 .
04 0 .
06 0 .
08 0 . Q (GeV )0 . . . F K ( Q ) FIG. 1. The kaon form factor (solid line) compared to the existingdata taken from Ref. [27]. F K + Q (GeV )00 . . . . F K ( Q ) FIG. 2. . The kaon form factor (solid line) compared with data at low Q . The experiment data are taken from Ref. [27]. and it appears as a coefficient in the background field, X .Consequently, the quark mass parameter appears differentlyin the effective potential, compared to the work of Ref. [33].In addition, they identify the light-front wave function, wherehadron properties are encoded, by comparing the electromag-netic form factor in AdS and the LF QCD form factor.In comparing our obtained results with their results on thekaon form factors, their results for the charge radius of the K + are slightly larger than our result in the low Q regime, wherethe charge radius is 0 .
615 fm for the dynamical spin parameter B = B >
0. However, the behaviorprediction of the kaon form factor in the large Q regime,which goes like 1 / Q , is similar to our obtained result.We note that, in this paper, we started with an AdS La-grangian that has SU ( ) L × SU ( ) R symmetry and it repro-duces a chiral symmetry breaking of QCD. An approximate relation due to a chiral-symmetry-breaking-like, Gell-Mann-Oakes-Renner relation is preserved in our approach. Q F K + Q (GeV )00 . . . . Q F K ( Q ) FIG. 3. . The same as in Fig. 1 but for Q F + K ( Q ) . The experimentdata are taken from Ref. [27]. VI. SUMMARY
In summary, we have computed the kaon form factor in holo-graphic QCD, which is a complementary approach of QCD.We adopt a “bottom-up” approach of the AdS/CFT correspon-dence, instead of a “top-down” approach, where we employthe properties of QCD to construct its 5D gravity dual theory.We begin to describe the AdS/CFT correspondence formalism,describing a correspondence between 4D operators O (x) andfields in the 5D bulk φ (x,z). We calculate the kaon form factorin holographic QCD.The result for the kaon form factor is in good agreement withthe existing data in low Q . We then predict the kaon formfactor in higher Q . We found that the kaon form factor inhigher Q is consistent with the perturbation QCD prediction.We finally calculate the charge radius of the kaon in holog-raphy QCD. We obtained r + K = ACKNOWLEDGMENTS
The work of P.T.P.H. was supported by the Ministry of Sci-ence, Information Communication and Technology and Fu-ture Planning, Gyeongsangbuk-do and Pohang City throughthe Young Scientist Training Asia-Pacific Economic Cooper-ation program of Asia Pacific Center for Theoretical Physics(APCTP). [1] C. G. Callan, Jr., R. F. Dashen and D. J. Gross, Toward a theoryof the strong interactions, Phys. Rev. D , 2717 (1978).[2] W. J. Marciano and H. Pagels, Quantum chromodynamics: A review, Phys. Rep. , 137 (1978).[3] P. T. P. Hutauruk, I. C. Cloet and A. W. Thomas, Flavor depen-dence of the pion and kaon form factors and parton distribution functions, Phys. Rev. C , 035201 (2016).[4] W. W. Buck, R. A. Williams and H. Ito, Elastic charge form-factors for K mesons, Phys. Lett. B , 24 (1995).[5] P. C. Tandy, Hadron physics from the global color model ofQCD, Prog. Part. Nucl. Phys. , 117 (1997).[6] E. O. da Silva, J. P. B. C. de Melo, B. El-Bennich and V. S. Filho,Pion and kaon elastic form factors in a refined light-front model,Phys. Rev. C , 038202 (2012).[7] A. F. Krutov, S. V. Troitsky and V. E. Troitsky, The K -mesonform factor and charge radius: Linking low-energy data to futureJefferson Laboratory measurements, Eur. Phys. J. C , 464(2017).[8] J. Koponen, A. Zimermmane-Santos, C. Davies, G. P. Lepageand A. Lytle, Light meson form factors at high Q from latticeQCD, EPJ Web Conf. , 06015 (2018).[9] H. J. Kwee and R. F. Lebed, Pion form factor in improved holo-graphic QCD backgrounds, Phys. Rev. D , 115007 (2008).[10] H. R. Grigoryan and A. V. Radyushkin, Pion form-factor inchiral limit of hard-wall AdS/QCD model, Phys. Rev. D ,115007 (2007).[11] H. R. Grigoryan and A. V. Radyushkin, Structure of vectormesons in holographic model with linear confinement, Phys.Rev. D , 095007 (2007).[12] J. Babington, J. Erdmenger, N. J. Evans, Z. Guralnik andI. Kirsch, Chiral symmetry breaking and pions in nonsuper-symmetric gauge / gravity duals, Phys. Rev. D , 066007(2004).[13] Z. Abidin and C. E. Carlson, Gravitational form factors of vectormesons in an AdS/QCD model, Phys. Rev. D , 095007 (2008).[14] T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Dilatonin a soft-wall holographic approach to mesons and baryons,Phys. Rev. D , 076003 (2012).[15] A. Ballon-Bayona, G. Krein and C. Miller, Strong couplings andform factors of charmed mesons in holographic QCD, Phys. Rev.D , 014017 (2017).[16] G. F. de Teramond and S. J. Brodsky, Hadronic Spectrum of aHolographic Dual of QCD, Phys. Rev. Lett. , 201601 (2005).[17] T. Sakai and J. Sonnenschein, Probing flavored mesons of con-fining gauge theories by supergravity, J. High Energy Phys. (2003) 047. [18] A. M. Polyakov, String theory and quark confinement, Nucl.Phys. Proc. Suppl. , 1 (1998).[19] T. Gherghetta, J. I. Kapusta and T. M. Kelley, Chiral symmetrybreaking in the soft-wall AdS/QCD model, Phys. Rev. D ,076003 (2009).[20] T. Sakai and S. Sugimoto, Low energy hadron physics in holo-graphic QCD, Prog. Theor. Phys. , 843 (2005).[21] K. Ghoroku and M. Yahiro, Chiral symmetry breaking drivenby dilaton, Phys. Lett. B , 235 (2004).[22] J. M. Maldacena, The large N limit of superconformal fieldtheories and supergravity, Int. J. Theor. Phys. , 1113 (1999).[Adv. Theor. Math. Phys. , 231 (1998)].[23] E. Witten, Anti-de Sitter space and holography, Adv. Theor.Math. Phys. , 253 (1998).[24] J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, QCD anda Holographic Model of Hadrons, Phys. Rev. Lett. , 261602(2005).[25] L. Da Rold and A. Pomarol, Chiral symmetry breaking fromfive dimensional spaces, Nucl. Phys. B721 , 79 (2005).[26] S. J. Brodsky and G. F. de Teramond, Hadronic Spectra andLight-Front Wavefunctions in Holographic QCD, Phys. Rev.Lett. , 201601 (2006).[27] S. R. Amendolia et al. , A measurement of the kaon chargeradius, Phys. Lett. B , 435 (1986).[28] M. Carmignotto et al. , Separated kaon electroproduction crosscection and the kaon form factor from 6 GeV JLab data, Phys.Rev. C , 025204 (2018).[29] T. Horn, Meson Form Factors and Deep Exclusive Meson Pro-duction Experiments, EPJ Web Conf. , 05005 (2017).[30] Z. Abidin and C. E. Carlson, Strange hadrons and kaon-to-pion transition form factors from holography, Phys. Rev. D ,115010 (2009).[31] G. P. Lepage and S. J. Brodsky, Exclusive processes in per-turbative quantum chromodynamics, Phys. Rev. D , 2157(1980).[32] K. A. Olive et al. (Particle Data Group), Review of ParticlePhysics, Chin. Phys. C , 090001 (2014).[33] M. Ahmady, C. Mondal and R. Sandapen, Dynamical spin ef-fects in the holographic light-front wavefunctions of light pseu-doscalar mesons, Phys. Rev. D98