Meson effective mass in the isospin medium in hard-wall AdS/QCD model

We study a mass splitting of light vector, axial-vector and pseudoscalar mesons in isospin medium in the framework of hard-wall model. We write an effective mass definition for the interacting gauge fields and scalar field introduced in gauge field theory in the bulk of AdS space-time. Relying on holographic duality we obtain a formula for the effective mass of a boundary meson in terms of derivative operator over the extra bulk coordinate. The effective mass found in this way coincides with the one obtained from finding of poles of the two-point correlation function. In order to avoid introducing distinguished infrared boundaries in the quantisation formula for the different mesons from the same isotriplet we introduce extra action terms at this boundary, which reduces distinguished values of this boundary to the same value. Profile function solutions and effective mass expressions were found for the in-medium $\rho$, $a_1$ and $\pi$ mesons.


I. INTRODUCTION
Isospin medium is the simplification of the dense nuclear medium, where the net baryon charge of the medium is taken zero while it's isospin chemical potential remains to be non-zero. Such a simplified model in QCD was introduced in [1]. AdS/CFT correspondence conjecture developed in [2][3][4][5] further was applied to QCD problems and it were constructed AdS/QCD models for mesons [6][7][8][9][10]. Nucleons were included into AdS/QCD models in [11]. AdS/QCD idea was extended to finite temperature case in [12][13][14]19] and great number of works was devoted to holographic description of Quark-Gluon Plasma and dense nuclear matter [15]. In the framework of holographic QCD the studies in the dense nucleon and isospin mediums turned out effective as in top-down approach [24,[26][27][28] as in bottom-up one [22,25,[35][36][37][38].
In holographic QCD the two phases of nuclear matter -the confining and deconfining ones, in the dual gravity theory side are described by the different metrics. In the bottom-up approach, when the quark matter is absent the deconfining phase at the gravity hand is described by the Schwarzschild AdS black hole (SAdS BH) metric [12]. The confined phase of the nuclear matter at the low temperature limit in the dual gravity theory is described by the thermal AdS space (tAdS) [16]. In [18] it was shown on the background geometry of the dual gravity for the confinement phase containing the quark matter fields as well. This metric is named as thermal charged AdS space (tcAdS) and can be obtained from the Reissner-Nordstrom black hole (RNAdS BH) metric by taking zero of the mass of black hole and cutting of the fifth dimension at infrared (IR) boundary.
Deconfinement phase in dual gravity is described by the RNAdS BH metric. It occurs Hawking-Page transition between these geometries when takes place the confinement-deconfinement phase transition in the dual field theory [20,23].
One of simplifications in nuclear matter studies by use of holography is the considering a zero temperature limit. Another one is taking the quark number densities of the medium to be zero and leaving only dependence from the isospin chemical potential. In the result of these simplifications it is obtained the isospin medium for which the background geometry in the dual theory has no modification and metric remains to be ordinary AdS space [31,32] . Such a model is useful to separate the effects taking place due to isospin from the ones occurring under the influences of other quantities of dense medium [1, 22,36,37]. One of such effects is the splitting of the mass spectra of the meson states from the same isospin triplet in the medium due to their isospin interaction with the non-zero isospin of the medium. This effect was considered by number authors within the holographic QCD [22,24,25,[31][32][33][34][35]40]. In [31,32] in the hard-wall model framework it was considered the meson mass splitting effect for the triplets of light mesons ρ, π and a 1 in the dense nuclear matter due to the isospin interaction in connection with the study of the pion condensation in isospin medium, which had been carried out in [22]. In this paper we reconsider this effect because of the recent observation a relation between the definition of effective mass for a meson in the medium with non-zero isospin and a fixing infrared boundary of hard-wall model.
Here we define an effective mass following to standard definition in the field theory and show that this definition will request fixing the IR boundary differently for the mesons from the same meson isotriplet in the medium. In other words the splitting of meson mass leads to the splitting of the IR boundary of the model, since the mass spectrum in this model is directly defined by the value of IR boundary. It should be noticed, the IR boundary shift in hard-wall model already is known from the works [29,36], where the authors deal with the bulk interaction which contributes to boundary meson mass.
The paper was organised as follows: In the second section we give a description for the dense and isospin mediums in the hard-wall model. In third section we reproduce the equation of motion for the ρ mesons in isospin medium and show the necessity to distinguish the IR boundary for this mesons if the effective mass is defined as in field theory. In fourth section similar analysis was made for the π and a 1 mesons. We discuss the difference between two mass formulas in the last section.

II. ISOSPIN MEDIUM IN HARD-WALL MODEL
Let us present at first a bulk foundation for the boundary isospin medium following to earlier works [18,21,31,32]. The bulk field content for the isospin medium will be a simplified version of the one for the dense nuclear medium utilised in ref. [31]. Since this medium is described by the SU (2) group the bulk flavor gauge group should be chosen as SU (2) L × SU (2) R which then will broken to SU (2) V . We introduce in the bulk of AdS space two gauge fields A (L) and A (R) transforming under the 1 2 , 0 and 0, 1 2 representations of this flavor symmetry group respectively. Field strength tensor for these fields is with where the flavour index a runs a = 1, 2, 3 and T a are generators in spinor representation of the SU (2) L or SU (2) R group and are expressed in terms of Pauli matrices σ a As a gauge condition on A z we shall choose the axial gauge, A z = 0.
The action for the gravity and for the gauge fields will be written as where Λ = −6/R 2 is the cosmological constant. AdS/CFT correspondence establishes following relations between the constants κ 2 , g 2 , the number of colors N c and the radius R of space-time: are divided into the background parts L M , R M and the fluctuations l M , r M of these parts: In the holographic description the nuclear matter in the boundary QCD is given by the bulk gauge fields L M , R M , while the fluctuations l M , r M are needed for the description of the vector and axialvector mesons in this boundary theory. The homogenous and isotropic matter at the boundary is described by the dual bulk fields L M and R M which are not depend on space-time coordinates.
Moreover, the only time components L 0 and R 0 of these fields are taken non-zero, because the only flavor diagonal element of them (a = 3) corresponds to the physical quantity of the boundary matter. From these components we compose the vector and axial vector fields: The boundary value of the V 3 0 maps to the isospin chemical potential of the medium u and d quarks 1 [21]. Following [31,32] here we shall deal with the case L a 0 = R a 0 , which means Lagrangian invariance under changing the left and right flavor groups SU (2) L ↔ SU (2) R . Obviously, A 3 0 = 0 for this case. This L ↔ R invariance in the dual boundary theory means that the medium particles,i.e. the nucleons, are required to be in the parity-even states. As is known, the ground states of the nucleons are parity-even ones, and the first excited state of the nucleon can be parityeven or parity-odd state. In parity-even state the nucleons have less energy than in parity-odd ones [11].
The field strength tensor for V 3 0 is a flavor diagonal matrix as well and hence it gets a form as one for an Abelian field: In such a way the SU (2) symmetry of V a M part of gauge fields is broken down to two U (1) symmetries. Meanwhile, the fluctuations l M , r M remain to be non-abelian. Then the V 3 M part of the action (2) gets a form: 1 In confinement phase these quantities respect to the nucleons.
The holographic dual of the A (u),(d) 0 = ± 1 √ 2 V 3 0 field will be u-and d-quarks of the boundary medium. Imposing the hard-wall cut-off on the bulk radial coordinate z makes these quarks a confined ones in the boundary theory. It should be noted, the number densities of such defined quarks (isoquarks) are zero (see [31]).
The bulk geometry will be the thermal charged AdS space (tcAdS) with radius R, which is the non-black brane solution of both Einstein and Maxwell equations obtained from (8) and the metric in this case is the metric of ordinary AdS space: where the radial coordinate z ranges in the limited area 0 < z ≤ z IR due to the hard-wall cut-off.
Since the number densities of medium particles were taken to zero there is no modification of the gravitational background due to the backreaction of the isospin medium [31,32]. Solutions to the Maxwell equations satisfying the Einstein equations as well are following constants: where µ α is the quark chemical potential.
Here we are going to deal with the medium in the confining phase, where the fundamental excitations are nucleons but not quarks. So, the solution (10) should be expressed in terms of the chemical potentials of nucleons. Taking into account the quark content of nucleons as in [31] the isospin chemical potentials of nucleons may be defined as a sum of quark chemical potentials µ P = 2µ u + µ d and µ N = µ u + 2µ d . For isospin matter with two flavors the number densities of nucleons are zero and V 3 0 = √ 2π 2 (µ u − µ d ) in deconfinement phase and in confinement phase. Thus, V 3 0 in the dual boundary theory describes the homogenius and constant isospin background field of medium made from isonucleons.

III. ρ MESON MASS AND SPLITTING OF IR BOUNDARY IN ISOSPIN MEDIUM
A. Bulk vector field and in-medium ρ mesons As in known, in AdS/QCD models the mesons in dual boundary theory are described by the fluctuations of the bulk gauge and scalar fields. In this section we shall consider vector mesons, which will be introduced as a holographic dual of bulk vector field where µ -the boundary coordinates, µ = 0, 1, 2, 3 and i = 1, 2, 3 is the SU ( with the action: After taking the trace in (13) the action gets a form with the following non-zero terms 2 : Here the m, n = 1, 2, 3 indices denote the boundary spatial coordinates. The terms in the second square bracket in (14) describe the interaction of vector field fluctuations with the background field V 3 0 . In the dual boundary theory this interaction takes into account the interaction of vector meson with the isospin medium. The temporal component of vector fluctuations will correspond to the fluctuations in chemical potential (or number density in nuclear matter case) of medium particles.
Setting this component zero (v i 0 = 0) we shall consider a medium with a constant isospin chemical potential. We compose a complex ρ field from the vector field components v i m as following: The boundary values of the ρ a m bulk fields introduced in such a way correspond to the neutral and charged vector mesons respectively. In order to get a mass spectrum of boundary ρ mesons in isospin medium we choose it's rest frame. Then, the bulk ρ field does not depend on spatial coordinates x m and the Fourier transformation for this will have a form: Since the ω i are the kinetic energies of the ρ fields they do not include the interaction of these fields with the background V 3 0 and therefore do not depend on chemical potentials µ P and µ N . As is known in the vacuum there is a mass degeneracy of these mesons, i.e. the vacuum masses of the charged and neutral ρ mesons are equal. So, we can take the equality ω + = ω − = ω 0 for the vacuum masses of the ρ fields as well. The constant background gauge field V 3 0 does not change the background geometry (9) and so, the kinetic energies ω ± will not be changed by this gauge field. Therefore, the equality of ω i masses remains to be right for the isospin in-medium case as well.
The equations of motion for the ρ (ω, z) fields are obtained from the action (14) taking into account Eq.s (15) and (16) [31,32]: In the background geometry the equations of motion (17) with (11) have got an explicit form: which are same with the ones obtained in [22] for the pions. The equations (18) can be obtained in an alternative way of the introducing v i fields interacting with the field of isospin V 3 0 [32]. Lagrangian for this system will be constructed by means of covariant derivative D M of the fluctuations v i M and has a form: The T i kj = −iε ikj are the generators of isospin group in adjoint representation. Taking into account the transverseness of v iM D M v iM = 0 in (19) the action for this Lagrangian will be reduced to the following form: which in terms of the ρ fields will be written as: M ρ a n * D (a)M ρ a n .
Here the derivative D (a) M has a shift for the ρ a field D (a) In the action (20) the terms of cubic and forth order of the fluctuations were neglected as was made in the (14). The equation of motion obtained from the action (21) has a form: This is a five-dimensional D'Alembert equation in a curved space-time for a vector field having zero fifth component and zero five-dimensional mass M 5 . Boundary terms which arise on obtaining (22) lead to boundary conditions ∂ z ρ a n | z IR = 0 and ρ a n (z) | ε = 0, which are same with free field ones.
It is useful to rewrite the equation (22) as a sum of it's four-dimensional part and the z coordinate where we divided it by √ −G. Now, let us write the D'Alembert equation in a four-dimensional curved space-time for the massive vector field in the constant external background gauge field: where the m a is the mass of the field ρ a n . In our case the four dimensional spacetime is the UV boundary of the AdS space (9) and the massive vector field is the ρ meson. In AdS/CFT correspondence here the UV values of the ρ a fields correspond to the ρ mesons defined on this boundary and the equations of motion for these fields at UV boundary correspond to the equations of motion for these mesons. Hence, we have to correspond the (23) equations at UV limit to the ones in (24). From this correspondence it is seen that the eigenvalues of the operator The equation (25) is the determining formula of effective mass for the vector field in isospin medium by means of derivative operator over the extra dimension. For the background geometry (9) the equation (23) will be simplified as: The equation (26) is, in fact, the equation (18). The eigenvalues of operator D (a) µ D (a)µ are given explicitly which afterwards were denoted by (m * a ) 2 . Now we apply Kaluza-Klein decomposition on the solution of equation of motion (18): Since the equation (18) has the same form as in free case ( [29,39]) the solutions also are same to the ones in vacuum case. However, here in solutions we should write the square root of the coefficients in last terms in the equations (18) instead of mass m in vacuum case: and an analysis carried out in [29] for the free case is available for the in-medium case as well. The (29) are Bessel functions of the first kind. Here by the m s a we denote a square root of the coefficients in front of last terms in the equations (18) for the s mode: As is seen from the last equation all excited and ground states of the charged ρ mesons have the same shift −V 3 0 or +V 3 0 . The Dirichlet boundary condition on solution (29) at UV boundary z = ε relates the normalisation constant c 2 with c 1 goes to zero [29]:

B. Mass splitting and splitting of IR boundary
Now discuss the holographic definition of effective mass of ρ meson in isospin medium. To this end let us return to the zero background field case. In this case mass of a vector meson in the boundary QCD is defined as a one, which is obtained by identifying it with the mass of dual vector field in the bulk. In this case the equation of motion for the bulk vector field is obtained from the equation (26) by setting V 3 0 = 0: Applying the known solution zJ 1 (m s a z) in (32) and using known properties of Bessel functions we get following equality for the mass of bulk ρ field in free case: This definition agrees with the definition of masses ω i in the four dimensional flat Minkowski The comparison of (34) with the (33) proves that the coefficient m s in the argument of Bessel function is the mass of the s state of the vector filed. Now we can extend this definition of the mass to the case of presence of external field V 3 0 in the bulk. It is obvious such defined mass in the dual boundary theory will be a mass of the vector meson interacting with the isospin medium, i.e. an effective mass m * of the ρ meson in this medium. Since the correspondence between the equations of motions in the bulk and boundary theories still holds, it is right to establish a correspondence between the eigenvalues of these equations as well. In other words, we suggest that the eigenvalue of the operator −∂ z 1 z ∂ z corresponds to the mass in an external field case also. This is equivalent to that we extend the definition of a mass in the background field case from the  (30) and we shall consider them as the effective masses of the in-medium ρ mesons. Such defined holographic effective mass also agrees with the effective mass m * defined in the field theory framework in flat space-time: since the eigenvalues (m * a ) 2 of the D Thus, the effective masses of ρ mesons in isospin medium are equal: Although the vacuum masses ω i of ρ mesons are equal, the in-medium masses m * are different.
Thus, the splitting in the mass spectrum of ρ mesons takes place due to interaction with the isospin medium. After taking into account in (26) the equality of vacuum mass (ω ± = ω 0 ) the splitting formula will look like: This splitting of meson mass in the holographic approach was studied in [22,25,31,32], but the sign of splitting obtained here is different from the one obtained in [22,31,32] and agrees with the one obtained in [25] if to take the value of V 3 0 in (38) equal to the µ I introduced in [25]. This distinction between Eq.s (38) and (39)  Applying the Neumann boundary condition at IR boundary ∂ z ρ a (z) | z IR = 0 on solution (29) (with c a 2 = 0) yields the following formula for the mass spectra m s a m s which has same form as one in the vacuum [29] and corresponds to mass spectra of excited states of the ρ mesons in the medium. Here we have distinguished the value of IR boundary for the differently charged meson denoting them by the z 0 IR and z ± IR respectively. Otherwise, the equality of the right hand sides of first and second equalities in (35) s − 1 4 π/z ± IR = s − 1 4 π/z 0 IR would mean the equality of their left hand sides also ω s 0 ∓ √ 2π 2 (µ P − µ N ) = ω s 0 which is nonsense in non-zero medium isospin case. To avoid this nonsense we should adopt that the z IR gets different values for the differently charged ρ fields in the isospin background V 3 0 . Thus, we have came to conclusion that in the medium with non-zero isospin the IR boundary for the vector mesons is subjected to splitting also due to the mass splitting of these mesons. Using (40) in (30) one may easily obtain the splitting formula for the z −1 IR : From (41) where L (a) is the Lagrangian term producing the equation of motion (17) for the isocomponent a: Notice, in this approach we should write the Lagrangian (4), then make the definitions (5), (6), (12) and (15) reducing this Lagrangian to the sum of three independent Lagrangians (43) and finally write the action (42) for these Lagrangians considering isocomponents as independent vector fields defined in different spacetimes.
When we derive the equations of motion (17) from the (43) the boundary terms arising on integration by parts have the same form (ρ a ) * ∂ z ρ a | z IR z U V for the different ρ a components. However, the value of z IR in these terms now will be different. These boundary terms lead to the boundary conditions imposed on the solutions of (18) and therefore, the IR boundary conditions imposed on these solutions also have same form for all ρ a fields, but the values of the z IR at which are imposed this condition will be different for the different component a: The values of z a IR coincide when turning off medium and equals to the vacuum one, i.e. to the z 0 IR . It should be noted, the mass splitting formula (38) (30) and (39)). The splitting formula (39) was obtained in the hard-wall model which has one IR boundary for the all isocomponents of the vector field. In this case all three equations in (18) should be solved under the same boundary condition at IR and with the same value of z IR .
This dictates that all coefficients in front of last terms in the equations (18) should be the same.
This leads to the splitting formula (39). In fact, if we put back (39) into (18) we shall get the same expression for these coefficients, which equals to ω 0 . Obviously, in this case the all m s a coefficients in Bessel function solutions are the same m s a = ω 0 and it does not arise the necessity to distinguish z IR in the mass spectrum formula (40).
Alternatively, here we don't suppose at the beginning that z IR is unique for the each field isocomponent ρ a and define the effective mass as above. In this case the coefficients m s a are different and this leads to distinguishing the z IR for the different Lagrangian terms L (a) in (43).
Since the values of the z IR at which the IR boundary condition is imposed on the solution are different, it cannot be claimed the equality of coefficients at last terms in (18). Consequently, in this case the previous splitting formula (39) does not take place. Thus, we have came to conclusion that in the medium with non-zero isospin in hard-wall model the definition of the effective mass and the fixation of the IR boundary reciprocally determine each-other.
The first state (s = 1) of charged vector field tower (40) is the ground state of ρ ± meson, which has a mass and for the neutral ρ meson we have the value of mass in the vacuum [29]: Finally, the solution (29) for the ρ a meson accepts the form same with one in vacuum case: As we noted above, in the vacuum all ρ mesons have same mass m (vac)s = ω s 0 . A mass difference m s − m (vac)s = ∆m s will determine the additional mass acquired by the ρ meson due to the interaction with the isospin medium: In the isospin medium with V 3 0 > 0 (or with V 3 0 < 0) the mass of all excited states of ρ ± meson get the same contribution ∓ √ 2π 2 (µ P − µ N ) (or ± √ 2π 2 |µ P − µ N |) respectively.

IV. A MASS AND IR BOUNDARY SPLITTING FOR THE a 1 AND π MESONS
In this section we shall define the effective mass for the axial-vector and pion fields as was done for the ρ mesons in previous section and will study the z IR splitting for these fields. Let us briefly present here the action and the equations of motion for the axial-vector and pion fields, all details of which can be found in [31].
The non-zero fluctuations of axial-vector field, which is defined as a i µ = 1 √ 2 l i µ − r i µ , were introduced into the model in order to describe the axial-vector mesons in the boundary QCD. In addition, it is introduced a complex scalar field Φ, which performs the chiral symmetry breaking SU (2) L × SU (2) R → SU (2) V of the model. The action for the Φ field has a form: Here The Φ field is written in the form: Φ = φ exp i √ 2π a T a . The fluctuations π i describe the pions in the dual theory. In the nuclear matter for the complex scalar field φ which is dual to chiral condensate it was found a solution: where m q is the mass of light quarks and the σ is the value of the condensate. In isospin medium case the number densities Q and D were turned off Q = D = 0 and the background geometry is given (9). The solution for the φ field in this metric has a simple form: The action (49) contains an interaction of Φ field with the axial-vector field a i µ and the total action of these fields will be the sum of S φ and the action S a for the fluctuations a i µ . The action S a , which is obtained from the action (4) has a form : and includes the interaction of a i µ with the background field V 3 0 . In the dual boundary theory the axial-vector mesons a 1 are described by the transverse field and hence, we have to divide the bulk axial-vector field a i µ into the transverseā i µ and the longitudinal χ parts: a i µ =ā i µ + ∂ µ χ i (∂ µā µ = 0). Longitudinal parts of the l i µ and r i µ fluctuations are the + 1 2 ∂ µ χ i and − 1 2 ∂ µ χ i respectively and the vector field fluctuations v i µ are transverse. The action for the transverse axial-vector fluctuationsā i µ obtained from (52) has a form [31]: It is recommended to join the action for the longitudinal χ field derived from the (52) with the action for the pseudoscalar field π derived from (49) in the following summary action [31]: which includes also the interaction term ofā i field with the φ field.

A. the a 1 mesons
Similarly to the ρ mesons it is useful to introduce neutral a 0 1m and charged a ± 1m components for the axial-vector fieldā m , which will correspond to the respective components of the axial-vector meson a 1 in the holography: The Fourier expansion for this field is whereω 0 andω ± are masses of free a 1 fields, the equations of motion for these fields have got a form similar to one for the ρ meson's case. The equations of motion for the a a 1 field are obtained from the summary action of (53) and (54) and given by [31]: Explicit form of the equations (57) after the replacement a a 1 (z) = za ′a 1 (z) will be written as The equations of motion (57) written in terms of D (a) M will be obtained by use of (52) and will be written as following: This is a five-dimensional D'Alembert equation in for a axial-vector field interacting with the φ(z) field. Obviously, the fifth component and five-dimensional mass M 5 of this field is taken zero. The boundary conditions, which will be imposed on a 1n are obtained from the boundary terms arising on deriving the equation (59) and are usual ones: ∂ z a a 1n | z IR = 0 and a a 1n (z) | ε = 0. It will be useful to rewrite the equation (59) as the sum of it's four-dimensional part and the fifth dimension part: At QCD side there is a massive axial-vector meson, which is described by the a 1n field and has a shift in the mass due to the quark condensate background. We may take into account this mass shift by adding to the QCD Lagrangian additional mass term proportional to the value of condensate, which will be denoted by the 4g 2 φ 2 0 . The constant isospin background of the medium can be taken into account by introducing a constant gauge field V 3 0 . The interaction with this background will be taken into account by minimal interaction. Let us write the D'Alembert equation for such a 1n field in a four-dimensional curved space-time, which is, in fact, the UV boundary of AdS space (9): Herem a is the mass of the a a 1n field including the condensate shift and the φ 0 is the value of φ(z) field at some fixed value of z. Since in AdS/QCD a mass spectrum is obtained from the IR boundary condition we take as a fixed value of z the z IR , i.e. φ 0 = φ (z IR ), although the AdS/CFT correspondence of fields takes place at UV boundary. In order to correspond the equations (60) to (61) we should accept the equality of mass term in (61) and z-dependent part of the (60) The equation (62) is the main formula determining the effective mass of axial-vector meson in isospin medium by means of derivative operator over the extra dimension defined in dual bulk theory. The equations (58) can be solved at UV (z → 0) and at IR (z → z IR ) limits. At UV limit the equations (58) are usual Bessel equations and their solutions are expressed in terms of Bessel functions J 1 and Y 1 as a a 1s (z) = c 1 zJ 1 (m s a z) + c 2 zY 1 (m s a z) .
According to (62) the eigenvaluesm s a will be effective mass of the a 1 field. After imposing the Dirichlet boundary condition on the Kaluza-Klein state a a 1s (z) at UV boundary only J 1 part of it remains: The mass spectram s a of the s states arē andω s ± =ω s 0 are the vacuum masses of the excited axial-vector meson. From (65) the mass splitting formula for the a ± 1 mesons will be written in the following form: which predicts a contribution of condensate due to isospin medium. However, the mass spectrum of mesons in AdS/QCD is usually found from the IR boundary condition. It is reasonable to apply the IR boundary condition to the solution found at IR limit. For the IR asymptotic solution we shall take z →z IR limit from (58) and set the z =z IR in the condensate term. Before doing this approximation let us compare numerically the last two terms in equations (58) when z →z IR .
Similar approximation for the axial-vector field was done in [29] for the V 3 0 = 0 case. The mass spectrumm s a in (68) is expressed in terms ofz a IR : The mass splitting formula has form similar to one for for the UV solution and has splitting sign opposite to one in earlier works. The mass spectrum formula (40) Consequently, the conclusion about the distinguishing of value ofz IR for different isocomponents a of the a a 1 field takes place here as well and the necessity fixing of the IR boundary in the action terms for the different isocomponents by the differentz IR remains to be right for the axial vector mesons as well.

B. π mesons in isospin medium
As is known from the free case, when there are no background or other fields interacting with π fields the equation of motion for this field can be solved analytically either in the chiral limit (m q = 0) or in the zero gluon condensate limit (σ = 0). In the isospin medium this equation will be solved in the zero condensate limit σ = 0, which suits to approximation in UV limit.
Following [31], we introduce the charged π a and χ a fields by means of π and χ fields respectively as below: In AdS/CFT correspondence the UV boundary values of the π a fields (a = o, ±) will be mapped to the pion fields of the dual QCD. Fourier components for the π a and χ a fields are written as where ω a are masses of π a (and χ a ) fields in the absence isospin background V 3 0 , i.e. are the vacuum masses and do not include the contribution of condensate. Obviously, the claim done in previous section for the ρ a field's vacuum masses ω a is valid for the π a masses ω a as well, i.e. the ω a do not depend on µ P and µ N and ω ± = ω 0 is right in the isospin medium case. The equations of motion for the π and χ fields were obtained from the action (54) in [31]: Equations of motion for the scalar fields are summarized as and Comparing the first and second equations one can establish following relations between the ∂ z derivatives of the π and χ fields Taking derivative form the equations (74) and (75) and taking into account the relations (76) the equation of motion for the π a field will be separated from the one for the χ a field: The physical situation with the π fields is more complicated in comparison with other mesons, which were considered in previous sections. The π fields interact with three other fields: background condensate field φ, longitudinal axial-vector field χ and isospin background of the medium V 3 0 . As it can be seen from the action (54) the interaction with the φ field gives a contribution to the mass term of the π a fields and this contribution is the same for all these fields. The interaction with the isospin field V 3 0 gives different contributions to the masses of π fields in equations of motion and thus, splits the mass of these fields, which can be seen from (77). Interaction with the χ field lifts up the order in equations of motion, that can be seen taking χ field to be zero in the action (54) or in equations (74) and (75). In this case we have second order differential equations, instead of third one in (77) Another observation is that the φ(z) field modifies the AdS metric (9). To see this let us consider at first χ = 0 case of the equations for the π fields in (74) and (75). Making following notation in the metric tensor: we get modified background geometry: Then and the equations for the π fields in (74) and (75) get a form: The equation (81) is the five dimensional Klein-Gordon equation for the free π a field in the background geometry (79). Consequently, the equation of motion for the π a field interacting with the φ field equivalent to the one for the free π a field in the modified geometry (79). It is obvious this equivalence does not depend on isospin medium and is right for zero medium case. It should be noticed, that the background geometry modification by scalar field φ was observed in [30] also.
Using (81) we can derive a formula determining the effective mass of the π fields. In order to make correspondence similarly to previous sections let us write the Klein-Gordon equation in a fourdimensional curved space-time with the new metric G for the massive scalar field π a interacting with the constant external gauge field V 3 0 and with the field of condensate denoted by φ 0 : where the m a is the mass of the π a field in the background of both condensate and gauge fields. If to consider the equations (82) as the UV limit of the (81) ones, then the eigenvalues of the operator (81) will correspond to the squared mass (m a ) 2 of the 4d vector field π a in (82). So, we may admit the equality of the eigenvalues of this operator to the effective mass of the π field defined at this boundary: The equation (83) is the one determining the effective mass of the π field in isospin medium by means of eigenvalue of the derivative operator over the extra dimension.
So, in the case of presence of χ field we can establish the correspondence between π fields in the bulk of AdS space and π mesons at the UV boundary of this space making some approximation.
Let us solve the equations (77) at the UV and IR limits of z for the background geometry (9). The equations (77) at z → 0 limit have got a simple form: Denoting ∂ z π a = zP a (z) we shall get the Bessel equation for the P a (z) and for the Kaluza-Klein mode π a s the solutions of (85) will be expressed in terms of Bessel functions J 0 and Y 0 : Here them s a denote the square root of coefficients of last terms in the equations (85): From the (87) we find following relation between them s a : Using the differentiation formula for the Bessel functions we find the solution π a s in terms of Bessel functions J 1 and Y 1 Boundary conditions which were imposed on π a s on deriving (77) are usual ones: ∂ z π a s | z IR = 0 and π a s | 0 = 0. Similarly to the previous section, the application of UV boundary condition on (90) gives c 2 = 0.
The eigenvaluesm s a can be considered as an effective mass of the s-th mode of pions in isospin medium. In order to compare with earlier results it is recommended to ignore the m q (m q ≪ ω a ) in them s a . In this case the equations (85) coincide with the equations of motion for the pions in isospin medium obtained in [22,31,32]. The splitting formula (38) for the effective mass in this work was obtained from the request of equality of the coefficients in last terms in (85), since these equations were imposed by the same boundary condition at the same IR boundary. In our approach here we don't suppose the equality of z IR for all π fields and define the effective mass for the pions in the approximation m q ≪ ω a as it was done for the ρ mesons in previous section, i.e. by the formula: with the same D The splitting formula for the effective mass in this case again is linear on V 3 0 and for definition (91) has a form: Since the solution (90) is the same as one for the ρ mesons and the mass splitting formula (93) is the same to the one of them there is no need to repeat the analysis done for the ρ meson case in previous section concerning the IR boundary splitting. We can regard the main results obtained for the ρ mesons in previous section and list them for the pions as well: 1) For the s state of the π meson we have mass spectrum similar to ρ meson's one: The interaction with the φ field changes the mass of pions and in the result we have the mass ω s a which ignore this interaction and the massm s a which includes the contribution of it. However, both these masses include the contribution of isospin medium. Consequently, we have to introduce two distinguished IR boundaries thez a IR corresponding tom s a and thez ′a IR corresponding to the ω s a .
Note, this distinguish of values of z IR is valid for the neutral pion also, since it's mass also gets a contribution due to the interaction with the φ field. Difference between the inverse values of these z IR can be found from the equality (86) by taking into account there the mass spectrum formulas (94) and (95): 2) Beside above shift due to condensate field there is a shift of z IR due to interaction with isospin medium and IR boundary splitting formula (41) due to this interaction takes place for the pions as well. This formula is valid for the both boundariesz a IR andz ′a IR . Obviously, the z ′0 The shift ofz −1 IR this time depends on m 2 q as well, which can be inferred from using (94) in (92). 3) Similarly to the (42), the integrals over the z in the action terms for the different π a fields will be supplied with the corresponding upper limitsz a IR : where L (0) = 1 4g 2 G zz G 00 ω 2 0 ∂ z χ 0 2 + φ 2 G zz ∂ z π 0 2 + φ 2 ω 2 0 G 00 χ 0 − π 0 2 , L (±) = 1 4g 2 G zz G 00 ω 2 ± ∂ z χ ± 2 + φ 2 G 00 ω 2 ± χ ± 2 + φ 2 G zz ∂ z π ± 2 +φ 2 G 00 π ± ω ± ∓ V 3 0 π ± ω ± ∓ V 3 0 − 2χ ± ω ± .
The IR boundary conditions ∂ z π a (z) |z a IR = 0 are applied at the correspondingz a IR values.
This equation has a form of the the equations in (82) and the second term may be considered as an effective mass including the condensate contribution in the background geometry with induced metric. This proves that the derivative operator over the extra dimension z determines the effective mass through eigenvaluesM s a .

V. DISCUSSION
The alternative way of definition of the effective mass offered here predicts different physical situation than in earlier works. This difference is an increase of mass of the negatively charged meson and decrease of one for the positively charged meson in the medium having positive isospin value. When it is applied the effective mass formula (38) to the equations (18) they become the same, i.e. these equations stop to carry an information about mass splitting. Meanwhile this does not occur when is applied the splitting formula (37). Another result of the formula (37) is distinguishing the IR boundary value for the different mesons from isotriplet. This can be easily understood if one remembers the similar problem arises when in the hard wall model in vacuum case two or more kind of particles are included into the model. It is well known that even in this vacuum case it is problematic to fit the mass spectrum of both meson and nucleon to their experimental values simultaneously, having only one z IR for the model. As we mentioned already the interaction with the bulk scalar field φ modifies the boundary condition and gives a contribution to the meson mass [29,36]. Indeed, the z IR shift in the isospin medium observed here is the result of the interaction which also contributes to the mass. These examples shows it may be correct to fix the IR boundary by the different z IR in the bulk actions for the fields, which corresponds to the particle states with the different masses on the boundary.