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Baryon wave function: Large-Nc QCD and lessons from models

The structure of the 1/Nc expansion for the baryon distribution amplitude in QCD is tested using quark models. Earlier conjectures about this structure based on the evolution equation and on the soft-pion theorem are confirmed by the model analysis. The problem of the calculation of the baryon wave function at large Nc is reduced to the analysis of equations of motion for an effective classical dynamical system.


I. INTRODUCTION
A. Large Nc: QCD and models The limit of the large number of colors N c and the 1/N c expansion [1]- [5] are important nonperturbative methods in QCD. However, in spite of the long studies of the large-N c limit, QCD is not solved even in the leading order of the 1/N c expansion. In the applications of the 1/N c expansion to QCD, one usually deals not with the order-by-order calculation of the 1/N c corrections but with the analysis of the structure of the 1/N c expansion.
The failure of the dynamical approach to the 1/N c expansion in QCD led to the redistribution of the effort towards various models imitating large-N c QCD. This approach proved to be rather fruitful. Instead of being trapped into the study of model artifacts (which often happens in the phenomenology of strong interactions), the analysis of various large-N c models has revealed a number of their common properties including the so-called large-N c spin-flavor symmetry [6]- [11] which is believed to be a symmetry of large-N c QCD as well.
A classical example of results, historically first derived in the Skyrme model but actually based on the spin-flavor symmetry of large-N c QCD, is the ratio of the pion-nucleon (g πN N ) and pion-nucleon-∆ (g πN ∆ ) constants [12] g πN ∆ g πN N = 3 2 1 + O(N −2 c ) (exp. 1.48) . (1.1) The same ratio 3/2 can be derived in the quark model [13] in the limit N c → ∞. This simple example shows that the study of large-N c models can be helpful for establishing the properties of large-N c QCD itself.

B. Baryon wave function in the large-Nc limit
This paper is devoted to the 1/N c expansion for the baryon wave function. This problem was raised in Refs. [14,15]. The construction of the 1/N c expansion for the baryon wave function is rather nontrivial, since one has to deal with the exponential large-N c behavior. In Refs. [16,17] it was shown that this exponential behavior can be consistently described using a specially designed generating functional for the baryon wave function. Strictly speaking, the results of Refs. [16,17] are based on conjectures, whose consistency was checked using various indirect methods like evolution equation [16] and soft-pion theorem [17]. In this paper we provide additional arguments supporting these conjectures, using models of large-N c QCD.
Before describing the main features of the 1/N c expansion for the baryon wave function we would like to agree about the definition of the baryon wave function. The concept of the wave function is natural for nonrelativistic quantum mechanics but its incorporation in QCD requires some extra specifications. Usually one keeps in mind the transition matrix element between baryon |B and vacuum |0 0| Nc c=1 ψ c (z c )|B , (1.2) where ψ c (z) is the quark field with the color index c. Strictly speaking one has to settle several questions: 1) ordering of quark operators ψ c (z c ), 2) insertion of Wilson lines providing the gauge invariance, 3) choice of the space-time or momentum kinematics (equal-time wave function, light-cone wave function etc.). From the point of view of QCD applications to hard exclusive processes the most interesting object is the baryon distribution amplitude [18,19,20] which corresponds to the choice z c = λ c n, n 2 = 0 (1.3) in Eq. (1.2). Here n is a fixed light-cone vector. The standard definition of the baryon distribution amplitude also involves the Fourier transformation from parameters λ c to variables x c corresponding to the momentum fractions of quarks in the infinite momentum frame: We have omitted various normalization factors which are not important for the discussion of the general properties of the 1/N c expansion.
Although the light-cone elements of the definition of the baryon distribution amplitude are important from the dynamical point of view, they are rather insignificant for the structure of the 1/N c expansion. What is really important for the construction of the 1/N c expansion is that the baryon wave function depends on N c quark variables. At this moment one meets a serious problem: how can one construct the 1/N c expansion for the function (1.4) which depends on N c variables x k ? This problem is much more general than the context of the baryon distribution amplitude. Below we will consider the general case without specifying the precise meaning of the wave function Ψ B (x 1 , x 2 , . . . , x Nc ) and of its arguments x 1 , x 2 , . . . , x Nc . For example, the variables x k can be composite objects carrying information about the spin and flavor of quarks in addition to their coordinates or momenta. Since the antisymmetric color tensor ε c1c2...cN c is factored out in Eq. (1.4), the function Ψ B (x 1 , x 2 , . . . , x Nc ) is symmetric in all its arguments x 1 , x 2 , . . . , x Nc .
In Refs. [16,17] the problem of the construction the 1/N c expansion for function depending on N c variables was solved by introducing a generating functional for the baryon wave function. Omitting irrelevant technical details (spin and flavor structure, light-cone kinematics, 1/N c rescaling of the quark momentum, etc.) and concentrating on the general structure which is important for the large-N c limit, we can define this generating functional as Φ B (g) = Ψ B (x 1 , x 2 , . . . , x Nc )g(x 1 )g(x 2 ) . . . g(x Nc )dx 1 dx 2 . . . dx Nc , (1.5) where g(x) is an arbitrary "source" function. Since function Ψ B (x 1 , x 2 , . . . , x Nc ) is symmetric in permutations of x 1 , x 2 , . . . , x Nc , the transition from the wave function Ψ B (x 1 , x 2 , . . . , x Nc ) to the functional Φ B (g) does not lead to any loss of information at finite N c . As was argued in Refs. [16,17], the functional Φ B (g) has an exponential behavior at large N c : (1. 6) where W (g) is some (N c independent) functional of g. An important property of the functional W (g) is its universality: all low-lying baryons [with O(N −1 c ) or O(N 0 c ) excitation energy] are described by the same functional W (g). The dependence on the type of the baryon B appears in the preexponential factor A B (g): Although the functional A B (g) depends on the baryon B, this dependence has simple factorization properties studied in Ref. [17]. Strictly speaking, these properties of functionals W (g) and A B (g) were not properly derived in Refs. [16,17]. Instead the consistency of these properties with a number of basic features of QCD was checked: -QCD evolution equation [16], -asymptotic behavior in terms of anomalous dimensions of leading twist operators [16], -spin-flavor symmetry of large-N c QCD [17], -soft-pion theorem for the baryon distribution amplitude [17]. Although all these consistency checks successfully passed, they cannot be considered as a final proof. In this paper we turn to the analysis of large-N c models in order to find additional arguments supporting the large-N c behavior (1.7) and illustrating the basic properties of functionals W (g) and A B (g).

C. Consequences of the spin-flavor symmetry
In the absence of the dynamical solution for large-N c QCD, one cannot expect to find much more than relations like the N -∆ ratio (1.1) in the pure large-N c approach. In Ref. [17] a relation was derived for the generating functionals Φ (N ) T3J3 (g) and Φ (∆) T3J3 (g) describing the baryon wave functions of nucleon and Delta resonance where T 3 and J 3 are projections of isospin and spin, respectively.
In order to write this relation let us first notice that Φ (N ) T3J3 (g) is a 2 × 2 matrix with indices T 3 , J 3 . Let us consider the 2 × 2 matrix (1.8) Obviously det R = 1 (1.9) so that matrix R belong to the group SL(2, C). The irreducible representations of SL(2, C) are described by Wigner functions D (j1,j2) (R) depending on two "spins" j 1 , j 2 . Let us consider the Wigner function corresponding to j 1 = 3/2, j 2 = 0. For brevity we will denote it simply D 3/2 T3J3 . Now we are ready to write the N -∆ relation derived in Ref. [17]: (1.10) Relation (1.10) was derived in Ref. [17] using the realization of the large-N c spin-flavor symmetry in the space of the preexponential functionals A B (g) appearing in Eq. (1.7). Due to the universality of the functional W (g), relation (1.10) is equivalent to the analogous relation for the preexponential functionals A It should be stressed that the construction of viable realistic models of the baryon distribution amplitude is not the aim of this paper. We are interested in another problem. We want to trace how the exponential large-N c behavior (1.7) appears within a certain class of models. The main problem of large-N c QCD is that its dynamics is not solved. Therefore the analysis of Refs. [16,17] was based on the assumption about the exponential large-N c behavior (1.7) with the universal functional W (g).
In QCD we cannot compute the functional W (g). On the contrary, in the models studied in this paper both the exponential behavior (1.7) and the calculation of W (g) are under good theoretical control. From the point of view of the aim of this paper (model check of the assumptions made in large-N c QCD), the result of this work is trivial: all assumptions involved in the theoretical construction [16,17] are confirmed in the models studied in this paper. This puts the results of Refs. [16,17] on more solid ground.
Several comments must be made about the status of the spin-flavor symmetry in our model analysis. Most of this work is based on the Hamiltonian (Schrödinger) analysis of models. As is well known, the mass difference M ∆ − M N between the ∆ resonance and nucleon is O(N −1 c ), whereas the masses M N and M ∆ grow as O(N c ). Therefore the mass splitting M ∆ − M N is an effect of the next-to-next-to-leading order of the 1/N c expansion. In this paper we do not go so deeply into the 1/N c expansion. Actually the main part is devoted to the calculation of the functional W (g) corresponding to the leading order. Therefore the subtleties of the realization of the spin-flavor symmetry are not important for the most of our analysis. Moreover, in our analysis we make no special assumptions about the symmetries of the Hamiltonian [only the SU (N c ) color symmetry is important for us].
On the other hand, the readers interested in model implementations of the spin-flavor symmetry can be addressed to Ref. [16] where functionals W (g) and A B (g) are computed in the naive quark model and the results explicitly agree with all constraints imposed by the spin-flavor symmetry. E. Mean field approximation and large-Nc limit In contrast to "unsolvable" large-N c QCD, many large-N c models allow for a straightforward construction of the 1/N c expansion. Although the precise form of the 1/N c expansion can be model dependent, many features are common since they are based on the mean field approximation which is justified in these models (but not in QCD) in the large-N c limit. The manifestations of the underlying mean field approximation can be different: saddle point approximation in the path integral approach, the Hartree equation in the Hamiltonian approach to the models with explicit quark degrees of freedom, classical equations of motion in models based on meson fields, etc.
In this paper we work with models explicitly containing quark degrees of freedom. In the leading order of the 1/N c expansion the solution of these models is described by the Hartree equation. This paper deals with the 1/N c expansion for the functional Φ B (g) (1.5) describing the baryon wave function. One can wonder how this approach is related to the mean field method. Note that the mean-field approach to fermion systems is based on equations for the density matrix whereas the functional Φ B (g) is constructed in terms of the wave function Ψ B (x 1 , x 2 , . . . , x Nc ). Strictly speaking the transition from the multiparticle wave function to the single-particle density matrix leads to a loss of information. Therefore the knowledge of the mean field solution in terms of the density matrix ρ(x ′ 1 , x 1 ) is not sufficient for the calculation of the functionals W (g) and A B (g) describing the large-N c behavior (1.7) of the baryon wave function. Nevertheless one can derive a closed equation for the functional W (g) [See Sec. IV and Eq. (4. 6)]. An essential part of this paper is devoted to the analysis of this equation and to the construction of its solutions.
In the leading order of the 1/N c expansion the density matrix ρ(x ′ , x) is described by the Hartree equation which leads to the representation for ρ(x ′ , x) in terms of single particle wave functions ψ n (x) of occupied states: In the simplest models with only one "valence" level ψ 0 occupied by N c quarks, the density matrix factorizes into the product (1.14) Here we assume that the trivial color term δ cc ′ is factored out. In the large-N c limit the factorization of the density matrix (1.14) becomes asymptotically exact. Naively one could think that the wave function Ψ(x 1 , x 2 , . . . , x Nc ) has a similar factorization at large N c : Unfortunately this statement is wrong. In order to see the violation of Eq. (1.15) let us multiply it by Nc k=1 g(x k ) and integrate over all x k . If Eq. (1.15) were correct, then we would obtain using Eq. (1.7)  (1.14) in the leading order of the 1/N c expansion.
To summarize, we cannot use Eq. (1.16) for the calculation of the functional W (g). Moreover, the knowledge of the solution of the Hartree equation is not sufficient for the calculation of W (g). In this paper we derive a differential equation for W (g) and construct its solution.

G. Factorization of the preexponential terms
The universality of the large-N c behavior (1.6) holds only with the exponential accuracy. The preexponential functional A B (g) in Eq. (1.7) depends on the baryon (baryon-meson) state B. In Refs. [16,17] several important factorization properties of these functionals were suggested and checked using the evolution equation, the spin-flavor symmetry and the soft-pion theorem. In this paper we verify these factorization properties by explicit calculations in large-N c models.
One should keep in mind that the factorization properties of functionals A B (g) are determined by two ingredients: 1) the breakdown of the spin and flavor symmetries by the mean field solution and the restoration of these symmetries via the standard method of "the quantization of collective coordinates", 2) the spectrum of the harmonic fluctuations above the mean field solution.
If the mean field solution for the baryon had the same symmetries as the vacuum mean field solution, then we would have a simple oscillator-like spectrum of baryons where the baryon excitations B are labeled by sets of integer numbers {n k } associated with elementary Ω k excitations. In this case the factorization of A B (g) is described by the formula In the case when the mean field solution breaks some symmetries (which are nevertheless restored by the collective quantization) the situation is more subtle and the precise expression for A B (g) depends on the involved symmetries. As is well known, the effects of broken symmetries in the mean field approach to large-N c models correspond to the manifestation of the spin-flavor symmetry in large-N c QCD. The structure of the functional A B (g) in large-N c QCD and the role of the spin-flavor symmetry were studied in Ref. [17].
In this paper we concentrate on models where the effects of broken symmetries and zero modes are absent so that one has the simple factorization formula (1.18). In Sec. VII we describe a model where the factorization relation (1.18) can be checked explicitly.

H. Schrödinger equation versus path integral
One can use two methods for the analysis of the models for the baryon wave function: the stationary Schrödinger equation or the path integral approach. In principle, these two methods should give equivalent results. It is well known that the large-N c limit has many common features with the semiclassical limit. In the path integral approach this leads to a representation for W (g) in terms of classical time-dependent trajectories. In the case of the stationary Schrödinger equation one works with the time-independent formalism.
In the simplest models (i.e. models with the trivial vacuum), the connection between the two representations can be easily seen. However, in the general case the equivalence of results obtained in the two approaches is less obvious. Therefore in this paper we use both methods. The first part of the paper (Secs. II-VIII) is based on the Schrödinger equation. In Sec. IX we describe the path integral approach and derive a representation for W (g) in terms of trajectories. After that we demonstrate the equivalence of the two representations for W (g) in Sec. X.

I. Large-Nc limit and classical dynamics
An essential part of this paper is devoted to the representation of W (g) in terms of classical trajectories described by an effective Hamiltonian. In fact, the functional W (g) can be interpreted as a classical action depending on the "coordinate" g. This action obeys the classical Hamilton-Jacobi equation. The method of trajectories can be used for the construction of the solution of this Hamilton-Jacobi equation.
The connection between the large-N systems and classical dynamics was extensively discussed in literature [21] (see also review [22] and references therein). In the path integral approach to the large-N systems, the 1/N expansion can be constructed using the saddle point method. In a straightforward approach the corresponding saddle point equations have a nonlocal form. Nevertheless in many cases it is possible to find new degrees of freedom which reduce the saddle point equations to a local form described by a local Hamiltonian dynamics.
In this paper we use similar methods and reduce the problem of the calculation of the functional W (g) to the analysis of trajectories described by an effective Hamiltonian. However, the problem studied in this paper is different from the "traditional" time-dependent problems in large-N systems [21]. Most of the works on the effective classical dynamics of large-N c systems concentrate on the time dependence of expectation values of observables so that the classical trajectories correspond to the time-dependent diagonal matrix elements of the corresponding quantum operators. On the contrary, the calculation of the baryon wave function is a problem of nondiagonal transition matrix elements. In spite of the difference between the two cases, we will see that the corresponding effective classical dynamical systems have many common features.
J. Large Nc models and traditional many-body physics In our analysis of the large-N c models we will meet many equations which are well known in the "traditional" manybody physics (see e.g. Ref. [23]). The machinery of the old many-body physics includes many interesting equations: stationary Hartree-Fock, time-dependent Hartree-Fock equation, random phase approximation (RPA) equations. A common feature of these equations is that their derivation is usually based on ad hoc approximations so that in a general case the validity of these approximations is a matter of luck. For example, the standard derivation of the Hartree-Fock equation is based on the variational principle with a special ansatz for the wave function. Nevertheless in some cases one can find a justification for these approximations. In particular, the 1/N expansion (N not necessarily being color) puts these equations on solid ground [24,25]. In the framework of the 1/N expansion, the Hartree equation corresponds to the leading order of the 1/N expansion whereas the contribution of the Fock term is 1/N suppressed. Similarly, in the next-to-leading order of the 1/N expansion one arrives at the RPA equation [24,26] with slight modifications caused by the large-N limit As was explained above, the functional W (g) cannot be expressed via the solution of the stationary Hartree equation. However, in Sec. VI we will see that W (g) can be expressed via solutions of the so-called time-dependent Hartree equation (TDHE). Indeed, as was mentioned above, the functional W (g) is just an action for some effective mechanical system. The equations of motion describing these trajectories of this mechanical system are similar to the TDHE. The properties and solutions of the TDHE were extensively studied in the context of the problems of the many-body physics (typically in its Hartree-Fock version [23]). Strictly speaking, the analogy between our equations and TDHE is not complete since our problem the large-N c wave functions differs from the traditional problems of the many-body physics. Nevertheless many interesting connections can be found. For example, our analysis of the effective large-N c dynamics makes use of the Hamiltonian structure of the classical equations of motion. In the context of the "traditional" approach to TDHE (with Fock term included), the classical Hamiltonian structure of the TDHE was studied in Ref. [27].

K. Antiquarks in large-Nc models
In the simplest models the baryon consists of N c quarks without any quark-antiquark pairs. This class of models is studied in the first part of the paper (Secs. III -VII). The second part of this paper (Secs. VIII-X) deals more complicated models where the quark-antiquark pairs are described using the well-known "old-fashioned" Dirac representation for antiparticles. In these models we have a nontrivial physical vacuum made of M N c quarks put into the bare vacuum (with some integer M ), whereas the baryon is considered as a state containing (M + 1)N c quarks above the bare vacuum. The baryon wave function [defined by Eq. (1.2) as a transition matrix element between the baryon and the physical vacuum] still can be used for the construction of the generating functional Φ B (g) (1.5). We show that in this class of models we also have the exponential behavior (1.6) of Φ B (g) and develop methods for the calculation of the functional W (g).

L. From quantum mechanics to quantum field theory
The general structure of the 1/N c expansion for the functional Φ B (g) is the same in quantum mechanics and in quantum field theory. For simplicity we use the discrete quantum mechanical notation in this paper. For example, instead of the continuous variables x k of Eq. (1.5) we write discrete indices i k : (1.19) This compact discrete notation allows us to simplify equations. By inertia we will often call Φ B (g) generating functional, although the word function could be more appropriate. One can ask whether our compact discrete notation is only a matter of language or some serious problems will come in the case of field theoretical models. The main problem is that our quantum-mechanical models are based on the four-fermion interaction. Their field theory analog of these models is represented by model of the Nambu-Jona-Lasinio (NJL) type. Formally our quantum-mechanical equations can be generalized for case of the field theory. However, there is a problem with the nonrenormalizability of the four-fermion interaction. From the physical point of view, models of the NJL type can be considered only as low-energy effective models which should be simply cut at high momenta. But problem is that our treatment is based on the Hamiltonian (or path integral) formalism which assumes the locality in time. Therefore applications of our formalism to NJL and similar models would require regularizations preserving the locality in time. Unfortunately most of regularization local in time conflict with the Lorentz invariance.
It should be stressed that the methods developed in this paper can be applied only to the large-N c models of QCD but not for QCD itself. Nevertheless some special problems of large-N c QCD these methods still can be used:: 1) the problem of heavy-quark baryons [4], 2) the problem of the diagonalization of the matrix of anomalous dimensions for the leading-twist baryon operators [16].
M. Structure of the paper As was already mentioned, the paper consists of two parts. The first part (Secs. III-VII) is devoted to models with the trivial vacuum (i.e. without antiquarks) whereas in Secs. VIII-X more powerful methods are developed for models imitating antiquarks in terms of the old Dirac picture. The general class of models is discussed in Sec. II. In Sec. III we describe the simplest models of the baryon wave function where the baryon appears as a bound state of N c quarks (without quark-antiquark pairs). In Sec. IV we derive a nonlinear differential equation for the functional W (g) and show that this equation agrees with the standard Hartree equation for large-N c systems. In Sec. V we turn to the analysis of the preexponential functional A(g) (1.7). In Sec. VI we show how the problem of the calculation of W (g) can be reduced to the analysis of trajectories in some effective classical dynamics. In Sec. VII we study a simple model where the functional W (g) can be computed analytically. In Sec. VIII we turn to systems with the nontrivial vacuum, using the Schrödinger equation. In Sec. IX we describe the path integral approach to the large-N c models of the baryon wave function. In Sec. X we check the agreement of the results based on the Schrödinger equation and on the path integral approach.

A. Choice of models
As was explained in the introduction, our aim is to check conjectures about the structure of the 1/N c expansion for the baryon wave function which were put forward in Refs. [16], [17]. We want to test these conjectures in simple large-N c models. Our choice of these models is determined by the balance between the intention to preserve the main features of the 1/N c expansion in QCD and the possibility to solve the model. As was explained in the introduction, the aspects of the phenomenological relevance will play a secondary role (if any) in our choice of models.
The minimal constraints on the models include two conditions: the model must have 1) quarks degrees of freedom, 2) SU (N c ) symmetry. The quark degrees of freedom are understood here in a rather weak sense. Neither full-fledged quantum field theory nor complete quantum mechanics is needed for our aims. For example, we can work with models ignoring the space motion of quarks where the dynamics of quarks is described only by color and some other discrete quantum numbers (e.g. spin and flavor). For the construction of the 1/N c expansion we need the global SU (N c ) symmetry but there is no need in the local gauge invariance. Now let us turn to the dynamics of the models. Most of our work will be done in the Hamiltonian approach based on the analysis of the stationary Schrödinger equation. In principle, we can include both quarks and antiquarks in our models using the old Dirac picture. In this case the nontrivial physical vacuum will contain the "Dirac" sea of quarks above the vacuum and we must solve the Schrödinger equation both for the vacuum |0 and for the baryon (2. 2) The dynamics of our models will be formulated in terms of the fermionic annihilation and creation operators a nc , is the color index and n = 1, . . . , K is some index which may have the meaning of spin, isospin, etc. We will be interested in the baryon wave function given by the transition matrix element At large N c we expect the exponential behavior (1.6) (2.7)

B. Hamiltonian
We will work with models described by the Hamiltonian Nc c=1 a + n1c a n2c (2.8) assuming that We we will be interested in color singlet states. The color singlet states can contain M N c "quarks" where M is an integer number. For any fixed M we can study the problem of the lowest color-singlet state containing M N c fermions. In our models of the baryon function we associate notation M with the vacuum whereas the baryon will correspond to M + 1 so that one has a nonzero matrix element (2.5) which will be interpreted as the baryon wave function.

C. Hartree equation
Below we will study the limit of large N c at fixed values of the parameters V n1n2n3n4 and L n1n2 in the Hamiltonian (2.8). It is well known (see e.g. [24,26]) that in the leading order of the 1/N c expansion the energy of the ground state can be found by solving the Hartree equation where the single particle Hamiltonian h n1n2 is given by (2.11) and the single particle eigenstates are normalized by the condition n φ r n φ s * n = δ rs . (2.12) In the leading order of the 1/N c expansion the energy of the corresponding state is where E 0 is given by (2.14) In the case of Hamiltonians with L n1n2 = 0 the expression for E 0 can be simplified using Eq. (2.10): Below we will see that the knowledge of the solutions of the Hartree equations for the vacuum |0 and for the baryon |B is not sufficient for the calculation of the generating functional W (g) appearing in Eq. (2.7). One has to derive a special equation for W (g).

D. Time-dependent Hartree equation
The time-dependent Hartree equation (TDHE) appears in various problems of the traditional many-body physics (where its Hartree-Fock modification is usually discussed, see e.g. [23]). In the case of the Hamiltonian (2.8) TDHE has the form Although the context of the traditional problems of the many-body physics where TDHE appears differs from our problem of the calculation of the functional W (g), we will see in Sec. VI B that the functional W (g) allows for an interesting representation in terms of the solutions of the TDHE (2.16), (2.17). The possibility of the Hamiltonian interpretation of the Hartree-Fock equations was discussed in Ref. [27]. This Hamiltonian interpretation plays an important role in our analysis.

A. Simplest models
As was explained in Sec. II A, we want to study the baryon wave function in models described by the Hamiltonian (2.8) using the states containing N c M and N c (M + 1) fermions for the vacuum and baryon, respectively.
We want to start from the simplest case M = 0 (the generalization to arbitrary M will be considered in Secs. VIII-X). In these M = 0 models the physical vacuum coincides with the bare one, and the baryon consists of N c quarks only. We will refer to the M = 0 models as models with the trivial vacuum. The main advantage of these models is that the solution Ψ bar of the Schrödinger equation (2.2) directly gives us the baryon wave function (2.5).
Our first aim is to define the generating functional Φ B (g) for the baryons in these models and to rewrite the stationary Schrödinger equation in terms of Φ B (g). This can be easily done using the well known holomorphic representation (also known as boson representation in the context of the many-body physics [23]).
In our simple models "baryons" are described by states containing N c quarks: Now we can construct the model analog of the generating functional (1.5) describing the baryon wave function: The correspondence between the states |ψ B and functions Φ B (g) is useful not only for the description of the eigenstates of the Hamiltonian (2.8) but for any color singlet states containing N c quarks. Let us consider operator on the state |ψ . In terms of the Φ B (g) representation for states |ψ B operator T mn takes the form C. Schrödinger equation for ΦB(g) Using the g representation (3.7) for the operators T mn , we can rewrite the Hamiltonian (2.8) in the form .
Then the Schrödinger equation

IV. LARGE-Nc LIMIT
A. Two approaches to the 1/Nc expansion In this section we want to study the large-N c limit in the simplest models of baryons introduced in Sec. III. As was mentioned in the introduction, one can use two approaches to the large-N c limit in fermion systems: 1) approach based on the single-particle density matrix obeying the Hartree equation, 2) approach based on the baryon wave function described in terms of the generating functional Φ B (g).
In principle these two methods are equivalent. For example, they lead to the same 1/N c expansion for the spectrum of states. The second method based on the functional Φ B (g) provides more information. However, one has to pay some price for the additional information: the large-N c equations derived for the functional Φ B (g) are more complicated than the Hartree equation for the density matrix.
Below we consider both approaches and demonstrate their equivalence.

B. Large Nc limit for functionals ΦB(g)
We want to study the limit of Φ B (g) at large N c . According to Eq. (3.4) we have Let us insert Eq. (1.7) into the Schrödinger equation (3.10). Taking into account the 1/N c expansion for the energy we obtain in the leading order In the next order of the 1/N c expansion we find (4.7)

C. Hartree equation
As was explained in Sec. II C, the energy of the lowest baryon state can be described in the leading order of the 1/N c expansion by the Hartree equations (2.10), (2.11). Our case corresponds to M = 1 in these equations: where φ n is the single-particle wave function of the level occupied with N c quarks. The leading order contribution N c E 0 to the energy E B (4.5) is given by Eq. (2.14) Now we can compare equation (4.6) for W (g) with the expression (4.10) for the Hartree energy E 0 . We see that the two expressions agree if we take In other words, if we know function W (g), then the Hartree solution φ n can be found by solving the equation (4.12) Note that the normalization condition (4.9) is automatically satisfied due to the property (4.3) of the function W (g).

E. Normalization integral
Let us consider two states (3.1) .
(4. 16) At large N c we can use the asymptotic expression (1.7) for Φ (k) (g). As a result, we find .

(4.17)
This integral can be computed using the saddle point method. The saddle point equation is nothing else Eq. (4.12) with the solution g n = φ * n given by the Hartree equation (4.8). Thus we find with the exponential accuracy We cannot use the saddle point method beyond this exponential accuracy because of the noncommutativity of the limits Taking a normalizable state we conclude from (4.18) that Re W (φ * ) = 0 .  [24,26].
On the other hand, in the problem of the baryon wave function the analysis of the next-leading-order of the 1/N c expansion is based on Eq. (4.7) for the functional A B (g). Therefore one can expect that there must be some connection between the RPA equations and Eq. (4.7). From our experience with the leading order equations [the Hartree equation (4.8) and Eq. (4.6) for W (g)] we know that this connection may be rather nontrivial. Indeed, we will see that the RPA equation really appears in the analysis of the 1/N c expansion for the basic functional Φ B (g) but in a modified version of the large-N c limit when the argument g of the functional Φ B (g) changes with N c in a special way. This new version of the large-N c limit is discussed in Sec. V B. In Sec. V D we turn to the derivation of the RPA equations corresponding to the modified large-N c limit.

B. Two versions of the large-Nc limit
Eq. (1.7) describes the asymptotic behavior of Φ(g) in the large-N c limit when g is fixed.
But we can also study another limit when g n approaches the Hartree solution φ n in the limit N c → ∞. In addition let us impose the condition on ∆ n n φ * n ∆ n = 0 .

C. Special choice of basis
It is convenient to choose the basis diagonalizing the Hartree Hamiltonian such that In models with one occupied level we will label this level with the index s = 0 and use the short notation Then the Hartree Hamiltonian (2.11) becomes and the Hartree equation (4.8) takes the form In particular, The Hartree energy is We introduce a special notation for the components g n with n > 0: Now we find from Eq. (5.6) On the LHS the differential operator g 0 ∂/∂g 0 acts on Φ B (g) whereas on the RHS we have the corresponding operator acting on C B (∆ n ). Equation (5.17) means that the following∆ representation is valid for the operator g 0 ∂/∂g 0 : This equation can be simplified using Eq. (5.11): corresponds in the∆ representation (5.24), (5.25) to the wave function Thus we have the Hamiltonian According to Eqs. (2.9) and (5.10) we have Now we find This is nothing else but the large-N c version of the RPA Hamiltonian.

E. Diagonalization of the RPA Hamiltonian
We can diagonalize the RPA Hamiltonian (5.33) using the Bogolyubov transformation Matrices α, β obey the conditions Eqs. (5.39) and (5.40) can be rewritten in the form Introducing the notation Now we can diagonalize the RPA Hamiltonian (5.33) using Eq. (5.45) where Ω is a diagonal matrix The spectrum of this Hamiltonian gives the O(N 0 c ) contribution ∆E B to the 1/N c expansion of the total energy (4.5): The corresponding eigenstates of H RPA are according to Eqs. (5.38) and (5.49) Here |0 is the b m vacuum (5.28) and the ground state of H RPA is Using Eqs. (5.37) and (5.28), we find Combining Eqs. (5.52) and (5.53), we obtain Functions C B (∆) were defined via the asymptotic expression (5.4). Now let us compute these functions. We start from the function C 0 (∆) corresponding to the ground state |0 RPA . The ground state |0 RPA obeys Eq. (5.54). In the∆ representation (5.24), (5.25), Eq. (5.54) takes the form The solution of this equation is and N is a normalization factor. According to Eq. (5.42) we have This means that matrix is symmetric The excited states (5.51) are described in the∆ representation (5.24), (5.25) by the wave functions Here P B (∆) is a polynomial of degree At large∆ n we have according to Eq. (5.62) Using Eqs. (5.60) and (5.41), we find Matching two types of the 1/Nc expansion In Sec. V B we have described two different asymptotic expressions for the large-N c limit. These two expressions are valid in different regions. However, we can match the two expressions in their common overlap area.
Let us insert Eq. (5.62) into Eq. (5.4) This expression must agree with with Eq. (1.7) In this region only the leading term of degree (5.64) survives in the polynomial P B (∆) so that Eq. (5.67) becomes Now we want to check the general factorization relation (1.18) If this factorization holds, then Eq. (5.68) takes the form Due to Eq. (4.11) and condition (5.3), the variation W φ + δφ has no linear term: Inserting this into Eq. (5.70), we see that Now we see thatÃ Combining this result with Eq. (4.1), we obtain Taking into account Eq. (5.7), we conclude that if In Sec. IV B we derived Eq. (4.6) for the function W (g). This is a partial differential equation which has many solutions. The asymptotic expansion (5.83) plays the role of the boundary condition for this differential equation. The matrix X mn appearing in this boundary condition is given by Eq. (5.60).
In the same way, using Eq. (4.2), we can rewrite relation (5.78) in the form This asymptotic behavior of A k (g) should be used as the boundary condition for the differential equation (4.7).

VI. CLASSICAL DYNAMICS OF LARGE-Nc SYSTEMS
A. Hamilton-Jacobi equation According to our previous results, function W (g) can be computed by solving the differential equation (4.6) with the boundary condition (5.83). This representation for W (g) is not quite convenient since one has to deal with the partial differential equation (4.6). In this section we show how the problem can be reduced to the analysis of ordinary differential equations. This reduction can be done using the well-known Hamilton-Jacobi method.
Eq. (4.6) has the structure of the classical Hamilton-Jacobi equation. In order to make this structure explicit, we change the notation g → q , (6.1) Then Eq. (4.6) takes the form where the Hamiltonian is One should keep in mind that g can be complex and W (g) is an analytical function of g (which may have singular and branch points). Therefore our classical mechanics is complex.
As is well known, the solutions of the Hamilton-Jacobi equation can be constructed in terms of the action for the trajectories obeying the Hamiltonian equations Solution S(q) of the Hamilton-Jacobi equation (6.3) describes the set of trajectories covering the coordinate space. All these trajectories have the same energy E 0 and obey the condition

B. Static and time-dependent Hartree equations
If we change the notation in the Hartree equation (4.8) and in the normalization condition (4.9) then we obtain 9) n p n q n = 1 , (6.10) Let us stress that we consider p n and q n as independent complex variables of our phase space. Conditions (6.8)-(6.11) define a point in this space. At this point the variables p n and q n are complex conjugate. But generally speaking p n = q * n . Strictly speaking, Eqs. (6.8)-(6.11) have more than one solution. For example, these equations are invariant under the phase transformation p n → e iα p n , q n → e −iα q n . (6.12) In principle, there can be additional symmetries which increase the number of solutions. If one chooses the basis (5.7), then the solution of the Hartree equations (6.8)-(6.11) is Parameter ε 0 is given by Eq. (5.11). The change of notation (6.7) in the Hamilton equations (6.5) transforms them into the form where These equations can be interpreted as a Euclidean version of TDHE (2.16), (2.17). It should be stressed that the variables φ n and φ * n are not complex conjugate in equations (6.14) in contrast to the standard TDHE.

C. Asymptotic conditions
In the vicinity of the "Hartree point" (6.8)-(6.11), the Hamilton equations for trajectories can be approximated by According to Eq. (6.4) this asymptotic behavior corresponds to the Hartree energy E 0 (5.12) The Hamiltonian (6.4) and equations of motion (6.5) are invariant under the transformations This identity can be rewritten in terms of the Poisson bracket n p n q n , H = 0 . E. Boundary condition Using notation (6.1), (6.2), we can rewrite the asymptotic expansion (5.83) in the form This expression is valid for Inserting this decomposition into Eq. (6.6), we find This agrees with the asymptotic behavior of trajectories (6.18), (6.19) at t → −∞. Now we understand that the action S(q) is associated with the configuration of trajectories p(t), q(t) which start at t → −∞ at points p, q obeying Eqs. (6.8)-(6.10) with the asymptotic behavior (6.18), (6.19). All these trajectories have the same energy E 0 and the same integral of motion (6.25).
Let p(t), q(t) be one of these trajectories with energy E 0 : Let us assume that at t = 0 this trajectory passes through the point p (0) , q (0) : and at t → −∞ it has the asymptotic behavior (6.18), (6.19) with some τ = τ (q (0) ) (6.32) depending on q 0 : Note that at t → −∞ we have along this trajectory This means that the integral is convergent at t → −∞. Now let us define the function and show that that this function obeys the differential equation (6.3). According to Eq. (6.29) we have Taking some large but fixed |T | on the RHS, we see that this definition of S(q (0) ) has the standard property n is the momentum p n (t) taken at t = 0 according to Eq. (6.30). Inserting Eq. (6.41) into Eq. (6.31) and taking into account that the point q (0) is arbitrary, we find that the function S(q) defined by Eq. (6.37) obeys the Hamilton-Jacobi equation (6.3).
Since the definition (6.37) of S(q) is based on the trajectories with the asymptotic behavior (6.33), (6.34), we automatically have the property (6.28). Therefore function S(q) defined by Eq. (6.37) also obeys the boundary condition (6.26). Since both differential equation (6.3) and the boundary condition (6.26) are satisfied, we conclude that Eq. (6.37) gives a correct representation for the functional which determines the large-N c behavior (1.7). The construction described in this section leads us to the following algorithm of the calculation of W (g). 1) Solve equations of motion (6.5) imposing boundary condition q(0) = g (6.43) with given g (6.60) and boundary conditions (6.33), (6.34) with unknown τ (g). In this way one finds the trajectory and τ (g).
2) Now W (g) is given by Eqs. (6.39), (6.42): This allows us to rewrite Eqs. (6.33) and (6.34) in the form Thus in the case (6.47) the action can be directly read from the asymptotic behavior of q n (t), p n (t) at t → −∞.
Remember that these trajectories are fixed by extra conditions (6.30) and (6.20), (6.48) Now we conclude from Eqs. (6.42), (6.53), (6.54) that in the case L mn = 0 the functional W (g) is given by the equation where I(g) is the parameter of the trajectories obeying the asymptotic conditions q n (0) = g n . (6.60) Let us summarize. In the case L mn = 0 the calculation of the functional W (g) reduces to the following steps: 1) Solve equations of motion (6.5) imposing boundary conditions (6.58) -(6.60) with given g and with unknown I(g). In this way one finds I(g).

A. Model
In the above sections we have considered the general theory of fermionic systems described by operators a nc , a + nc (2.3) with c = 1, 2, . . . , N c . Now we want to consider the simplest version of these models with only two values for the index n of a nc . It is convenient to interpret this two-valued index as the projection of spin 1/2. In other words, we want to consider the model described by the Hamiltonian where Obviously this Hamiltonian belongs to the class (2.8) with parameters 3) On the other hand, Hamiltonian (7.1) is nothing else but the rotator described by the angular momentum J a [J a , J b ] = iε abc J c . (7.5) We will consider the case of the asymmetric rotator (I ab = Iδ ab ) but the quantum number J 2 is still conserved: and has the standard eigenvalues We want to study the states made of N c quarks and antisymmetric in color. The Fermi statistics and the color antisymmetry lead to the complete symmetry of the spin wave function so that we deal with Thus the algebraic formulation of the problem of the baryon is obvious: take the sector with J = N c /2 and diagonalize the Hamiltonian (7.1) in this sector.

B. Direct semiclassical approach
In principle, we can find the spectrum of low-lying states of the Hamiltonian (7.1) at large N c solving the general Hartree equation (4.8) and diagonalizing the RPA Hamiltonian (5.33). However, the same results can be obtained by directly applying the semiclassical approximation to the large momentum J (7.8) in the Hamiltonian (7.1).
Instead of solving the Hartree equation (4.8) we can simply minimize the Hamiltonian (7.1) considering it as a function of the classical momentum J. This minimization should be performed assuming the classical version of the constraint (7.8). Introducing the Lagrange multiplier λ, we arrive at the extremum problem for the ground state energy (4.5) in the leading order of the 1/N c expansion: This leads to the equation Inserting this into Eq. (7.8), we find This equation determines λ. Knowing λ, we can find J from Eq. (7.11). The leading order of the 1/N c expansion (4.5) for the energy of the lowest states is determined by the parameter In principle, we may also have other solutions of Eq. (7.10) corresponding to det(K − λ) = 0 . (7.14)

C. Special case
Below we concentrate on the case when matrix K ab is diagonal K ab = K a δ ab (7.15) and We also assume that Y 3 > 0 , (7.17) In this case Eq. (7.12) yields and Eq. (7.13) results in According to Eq. (7.17) the minimal energy corresponds to the upper sign. Thus We find from Eq. (7.11) One can show that under conditions (7.17)- (7.19) this solution gives the true semiclassical ground state and solutions of Eq. (7.14) can be ignored.

D. Hartree equation
The semiclassical spin J a has the following interpretation in terms of the solution of the Hartree equation (4.8) Therefore the classical solution (7.23) corresponds to In the Hartree equation (4.8) we have the single-particle Hamiltonian Hence the single-particle energy ε 0 is It is easy to see that the general expression for the Hartree energy E 0 (4.10) reproduces the above result (7.22). The single nonoccupied Hartree state is and its energy is In the general RPA matrix R (5.34), the indices m, n correspond to the nonoccupied eigenstates of the Hartree equation. In our case we have only one nonoccupied state (7.28). We label the occupied Hartree state with m = 0 whereas the nonoccupied state is labeled by m = 1. As a result, matrix R (5.34) takes the form Using Eq. (7.3), we find for the diagonal tensor K ab (7.15) With these coefficients V n1n2n3n4 and with expression (7.29) for ε 1 − ε 0 , we obtain This leads to the Hamiltonian (5.33) we find where According to Eqs. (7.36) and (7.38) we have Comparing Eq. (5.49) with Eq. (7.35), we see that and the only RPA excitation energy is Comparing our result (7.41) for S with Eq. (5.45), we obtain Using Eq. (5.60), we find In the case of the Hamiltonian (7.1), Eq. (4.6) for the functional W (g) takes the form According to Eq. (4.1) function W (g 1 , g 2 ) has a nontrivial dependence only on the ration g 1 /g 2 so that we can represent W (g 1 , g 2 ) in the form with some function f . It is easy to see that Eq. (7.46) leads to an ordinary differential equation for the function f . In order to solve this equation it is convenient to change the variables: In terms of these variables representation (7.48) becomes A straightforward calculation of expressions (7.47) shows that In the case (7.15), (7.16) we find from Eq. (7.46) Inserting Eqs. (7.51)-(7.53), we obtain Let us find the next term of this small ζ expansion. Applying Eq. (5.83) to our case and taking X 11 from Eq. (7.45), we obtain (7.58) According to Eq. (4.22) we have for normalizable states (with the appropriately chosen phase) (7.60) Combining this with Eq. (7.50), we find Eq. (7.55) is a quadratic equation with respect to ∂R/∂z which can be easily solved. Using expression (7.22) for E 0 , we find from Eq. (7.55) Here we have introduced the compact notation C = 4Y 3 , (7.63) In order to fix the sign uncertainty on the RHS of (7.62), let us consider the limit ζ → 0 On the other hand, we find from Eq. (7.61) Comparing these two expressions, we conclude that we must have This identity holds if we choose the upper sign in the numerator of the RHS. Now Eq. (7.62) takes the form One can easily integrate this equation. The integration constant is fixed by Eq. (7.61). The calculation of the integral yields Now we insert Eqs. (7.70) and (7.49) into Eq. (7.50) Thus we have computed function W (g 1 , g 2 ) which determines the exponential part of the large-N c behavior (1.7) for the system with the Hamiltonian (7.1). One should keep in mind that function F (ζ) has branch points. At small ζ this function is regular. According to Eq. (7.61) Therefore for small |g 1 /g 2 | our result for W (g 1 , g 2 ) is unambiguous. However, at larger |g 1 /g 2 | one can meet cuts. In this case the determination of the relevant branch of W (g 1 , g 2 ) will require a special analysis.

A. Color singlet states with M Nc quarks
Our previous work was devoted to models of the baryon wave function based on systems with the trivial vacuum. In such systems the baryon wave function comes directly from the solution of the Schrödinger equation for the baryon (2.2). In other words, we worked with models which have quarks but no antiquarks. Within this class of models we could check the general properties of the large-N c limit like the exponential large-N c behavior (1.7) and the factorization of the preexponential factors A B (1.18). Now we want to show that these general large-N c properties also hold in more complicated models containing antiquark degrees of freedom in addition to quarks. It is convenient to formulate these models in terms of the old Dirac picture when both vacuum and baryons represented as a result of the occupation of the bare vacuum with quarks. In these models the physical vacuum contains M N c quarks whereas the baryon is made of (M + 1)N c quarks where M is some integer number. This allows us to define the quark wave function of baryons via the transition matrix element (2.5).
We want to start from the case M = 1 in order to concentrate on the idea of the method and to avoid unnecessary complications. The generalization for arbitrary M is discussed in Sec. VIII J in terms of the Schrödinger approach. In Secs. IX, X the case of arbitrary M will be analyzed using the path integral method. It should be stressed that the case M = 1 from which we want to start is rather nonphysical: the vacuum made of N c quarks is a fermion for odd N c whereas the baryon made of 2N c quarks is always a boson. Nevertheless the M = 1 case is a good starting point for understanding the dynamics of models including quark-antiquark pairs.

B. Color singlet states with 2Nc quarks
The color singlet states with 2N c quarks above the bare vacuum |Ω can be represented in the form Without any loss of generality we can assume that B The Fermi statistics of quarks

C. Large-Nc limit
At large N c we use the standard ansatz for the generating function corresponding to the 2N c quark state: Inserting this ansatz into Eq. (8.10), we find Combining the large-N c expression (8.14) with the Schrödinger equation (8.13) and using the 1/N c expansion for the energy we find in the leading order where In the next-to-leading order we obtain

D. Next-to-leading order
In the NLO we have equation (8.19). The structure of this equation allows for the solutions of the form and ξ k (γ) are solutions of the homogeneous equation

E. Overlap matrix elements
Now let us consider two baryon states |B 1 and |B 2 . Let us calculate the overlap matrix element for these two states This is nothing else but the expression for the Hartree energy (2.14) corresponding to the case M = 2, L mn = 0: Indeed, taking Inserting this expression into Eq. (8.27), we reproduce the Hartree expression for the energy (8.28).

G. Saddle point equation for the baryon wave function
We have explained above how to solve the Schrödinger equation for the baryon state |B (8.1) described by the generating functionΦ B (γ) (8.5). Now let us turn to the vacuum. In our model the physical vacuum |0 state contains N c quarks put into the bare vacuum |Ω . This means that the physical vacuum of this model is described by the same equations as the baryon in simpler models with the trivial vacuum which were studied earlier. Therefore we can apply the old baryon representation (3.3) to our new vacuum: Here we use notation k for the "source" argument of the function Φ vac instead of g as it was in Eq.  Note that the universal functional W bar (γ) describes the large N c behavior ofΦ B (γ) for all low-lying baryons B. In the same way the functional W vac (k) describes not only the exponential behavior of the vacuum Φ vac (k) but also the "meson" functionals Φ mes (γ) corresponding to the O(N 0 c ) excitations above the vacuum. Thus at large N c we have At large N c the integral over Q, Q * can be computed using the saddle point method. The deformation of the integration contour in the saddle point method can lead to the violation of the complex conjugation relation between Q ij and Q * ij . Therefore we introduce an independent notation for Q * The saddle point equations are Now we obtain from Eqs. (8.38), (8.39) These equations determine  where W (g) = [W vac (u * g * )] * + W bar (Q ij ) Q=Q(g),u=u(g) , (8.47) .

(8.48)
Using the factorization ofÃ B (Q ij ) (8.20), we arrive at the factorized form of A B (g) Here ε ij is the antisymmetric tensor We assume that the values i = 1, 2 correspond to the two occupied single-particle levels of the baryon state in the Hartree picture. The Hamilton equation describing the dynamics in the space of antisymmetric tensors Q bar ij , P bar ij will be discussed in Sec. X. By analogy with Eq. (8.57) we have The analog of Eq. (8.56) is the variables Q and k on the RHS become g (g * ) dependent. We use the label Sch in W Sch (g) in order to stress that we are dealing with the Schrödinger approach. Later we will show how the functional W (g) can be calculated in the path integral approach [using notation W pi (g)]. Certainly the result of the calculation should not depend on the method but in order to avoid confusion at the intermediate steps we prefer to use different notations for the two different methods. The dependence of k and Q on g is described by equations (8.40)-(8.42): The previous analysis was devoted to "baryon" states containing 2N c quarks above the bare vacuum. These states were described by functionsΦ B (γ) depending on antisymmetric tensors γ ij . The generalization to the case of states with M N c quarks is straightforward. In this case we must work with wave functions depending on antisymmetric tensors γ [i1i2...iM ] of rank k. The analog of Eq. (8.8) is where the derivative is normalized by the condition Our previous analysis of the large-N c models was based on the operator approach with the stationary Schrödinger equation (2.2) as a starting point. Now we want to study large-N c models using the path integral method. In principle, both Schrödinger and path integral approaches must lead to the same results. In Sec. X we will demonstrate the equivalence of the results derived using these two methods. However, in spite of this equivalence we will see that the path integral method allows us to obtain in a straightforward way rather interesting results which are not so obvious in the Schrödinger approach. The reason is that the path integral approach is based on time-dependent trajectories. On the other hand, as we know from Sec. VI, in the Schrödinger approach the trajectories appear rather indirectly. In the Schrödinger approach one first derives the Hamilton-Jacobi equation for the "action" W (g) and only after that W (g) can be interpreted in terms of trajectories. In the case of models with the nontrivial vacuum the situation is even more involved, since the trajectories appearing in the Schrödinger approach belong to the phase space of antisymmetric tensors. On the contrary, the classical dynamics coming from the path integral approach can be formulated in simpler terms.
Since the path integral approach allows for a straightforward derivation of the representation for the functional W (g) in terms of simple classical dynamics, we prefer to turn to the path integral formalism in this section. The equivalence of the results obtained in this section with the results based on the Schrödinger equation will be established in Sec. X.

B. Model
We want to apply the path integral approach to models described by the Hamiltonian (2.8). For simplicity we will consider the case when L n1n2 = 0 in this Hamiltonian. For our aims it is convenient to rewrite the Hamiltonian in the form The coefficients V αβ and Γ mn α are assumed to obey the conditions The expression (9.1) for the Hamiltonian corresponds to the choice in Eq. (2.8) .
In the path integral approach this system is described by the action We use the same notation a + , a for the Grassmann integration variables as for the operators in Eq. (9.1). Strictly speaking, the transition from the operator formalism to the path integral has to be accompanied by a careful treatment of the problems of the operator ordering. However, in the leading order of the 1/N c expansion these subtleties are not important.
In order to study Φ B (g) in the path integral approach we introduce the correlation function Z(g, g ′ , t 1 , t 2 ) = 0|T J(g, a, t 1 )J(g ′ , a + , t 2 ) |0 . The path integral representation for this correlation function is At large t 1 − t 2 → +∞ only the contribution of the lightest baryon |B survives in Eq. (9.15): Now we want to consider the limit of large N c . In this case we have with the exponential accuracy 0|J(g, a, 0)|B ∼ e NcW (g) . (9.18) The difference E B − E vac appearing in Eq. (9.17) has the order O(N c ): If we combine the large-time limit with the large-N c limit (9.21)

E. Calculation of the path integral
First we can compute the Gaussian integral over quarks in Eq. (9.16) At large N c the path integrals in the numerator and in the denominator can be computed using the saddle point method. The numerator and the denominator have different saddle points which will be denoted π cl and π vac , respectively. The result of the saddle-point integration is 1 N c ln Z(g, g ′ , t 1 , t 2 ) = S nonloc (π cl ) + ∆S bos π cl , (9.23) where S nonloc (π) = ln g · t 1 |K −1 (π)|t 2 · g ′ Det K(π) K(π vac ) , (9.24) The saddle point π cl is given by the equation The solution π cl is t dependent. It also depends on g, g ′ , t 1 , t 2 : π cl = π cl (t|g, g ′ , t 1 , t 2 ) . Combining Eqs. (9.21) and (9.23), we find

F. Fermion formulation of the effective large-Nc theory
In principle, Eq. (9.30) combined with the saddle point equation (9.26) contains everything what is needed for the calculation of the functional W (g). However, these equations involve the nonlocal action S nonloc (π cl ) defined in Eq. (9.24). In order to solve the saddle point equation (9.26) we would like to find a local formulation for our effective large-N c theory. To this aim we write the path integral representation for S nonloc (π) over auxiliary Grassmann variables b, b + : exp S nonloc (π) = g · t 1 |K −1 (π cl )|t 2 · g ′ Det K(π) K(π vac ) DbDb + expS(b, b + , π vac ) . (9.31) Note that the fermion fields b, b + appearing here have no color indices. The actionS is The path integral (9.31) can be also interpreted in terms of the operator approach to the theory of quantum fermions in the background classical field π For brevity we omit the in and out labels of the two "vacuum" states in this matrix element. One should also keep in mind that the Euclidean evolution violates the Hermitian conjugation of operator b(t) and b + (t). Moreover even the hermiticity of the Hamiltonian can be violated. Indeed, the background field π (1) comes from the saddle point equation (9.26) and the application of the saddle point method can be accompanied by the deformation of the integration contour. Combining Eqs. (9.31) and (9.33), we find Inserting Eq. (9.34) into Eq. (9.30), we obtain Using Eqs. (9.8), (9.9), and (9.34), we find from the saddle point equation (9.26) The variational derivative on the LHS has a simple interpretation in terms of matrix elements in the operator formulation of the same theory: Combining equations (9.36) and (9.37), we find This is a new form of the saddle point equation which determines π cl .
G. Separation of the t1 and t2 contributions Our aim is to compute the functional W (g). However, representation (9.35) is written for the sum W (g)+[W (g ′ * )] * . Now we want to extract information about W (g) from this equation. At finite t 1 , t 2 the RHS of Eq. (9.35) is a nontrivial functional of g and g ′ . Only in the limit t 1 − t 2 → +∞ the dependence on g and g ′ reduces to the sum of two independent terms W (g) + [W (g ′ * )] * . This additive dependence is a consequence of the properties of the classical solution π cl (t) at large t 1 and t 2 .
Eq. (9.35) was derived in the limit t 1 − t 2 → +∞. But if we look at the history of the derivation of this equation, then we find that the limit t 1 − t 2 → +∞ was used only on the RHS of this equation. In particular, the saddle point equation (9.38) does not know anything about the limit t 1 − t 2 → +∞.
According to Eq. (9.29) at large t → ±∞ (when t ≫ t 1 or t ≪ t 2 ) the field π cl (t) approaches the constant configuration π vac . At large t 1 − t 2 → +∞ we can also consider the intermediate region t 2 ≪ t ≪ t 1 . In this region the field π cl (t) is also asymptotically static but with another value π bar . Thus we have three regions where π cl (t) becomes asymptotically static: The constant configuration π bar corresponds to the "baryon" state |B of the effective fermion theory (remember that now we deal with the effective fermion theory where fermions have no color indices). Therefore the matrix elements appearing in Eq. (9.38) become Inserting these factorized expressions into the saddle point equation (9.38), we find in the region Let us take the limit t 2 → −∞ keeping t 1 and t fixed. In this limit Eq. (9.42) "forgets" about the behavior of π(t) at t ∼ t 2 and at t ≪ t 2 . Hence we can replace π cl by its t 1 component As a result, Eq. (9.42) becomes Working with the background field π (1) , we can consider the limits t → ±∞ at fixed t 1 Inserting these asymptotic expressions into Eq. (9.45), we obtain In Sec. IX H we will show that these are nothing else but the static Hartree equations for the vacuum and for the baryon in the large-N c theory. Note that the time-ordered product has a discontinuity at t = t 1 : Combining this with Eq. (9.45), we conclude that the field π (1) (t) has a discontinuity at t → t 1 : (1) .

H. Hartree equations from the path integral formalism
The saddle point field π (1) (t) is determined by equation (9.45). This equation contains the matrix elements of the effective theory with the b fermions that have no color. The matrix elements (9.57) and (9.58) are written for the theory in the background t-dependent field π (1) . The vacuum state 0| appears in these matrix elements as an out state corresponding to t → +∞. At t → +∞ we have π (1) (t) → π vac according to Eq. (9.56). Therefore the state 0| is the ground state of the theory with the Hamiltonian associated with the quadratic form (9.11) taken in the static background field π (1) : This means that the physical vacuum state 0| could be thought of as made of M quarks above the bare vacuum Ω| where the "single-particle" wave functions φ S vac are solutions of the equation Using expression (9.60), we can compute the matrix element on the RHS of Eq. (9.48): Inserting this result into Eq. (9.48), we find Now we see that Eqs. (9.61) and (9.63) are nothing else but the Hartree equations (2.10), (2.11) for the Hamiltonian (9.1). In the framework of these Hartree equations we have Instead of the second quantized representation for the physical vacuum |0 we can use the language the of M -particle wave function written in terms of the Slater determinant Similar equations can be written for the baryon state |B which is associated in the matrix elements (9.57), (9.58) with the limit t → +∞: We work with the pure 4-fermionic Hamiltonian (9.1) which has no quadratic piece. Therefore we have additional simplifications. Combining Eqs. (9.9), (9.61), (9.63), (9.64), (9.69), we find and similarly for the baryon Now we can simplify expression (9.55) dt L bos (π (1) ) − L bos (π vac ) . (9.73)

I. Time dependence
In the previous section we have shown how the static Hartree equations for the baryon and for the vacuum appear in the context of the calculation of the generating functional W (g) for the baryon wave function. These static equations correspond to the asymptotic limit (9.56) of the field π (1) (t) at t → ±∞.
Now we turn to the calculation of the time-dependent matrix elements (9.57), (9.58). These matrix elements are written in the Heisenberg representation with fixed states 0| and |B but time-dependent operators b(t) and b + (t). However, we can rewrite these matrix elements in terms of the Schrödinger representation. The Schrödinger representation is based on the time-dependent wave functions corresponding to the static Heisenberg wave functions 0| (9.60) and |B (9.66): We use notationφ s vac (t) instead of [φ s vac (t)] + because the Euclidean evolution is not unitary.
The time dependence of the wave functionsφ s vac (t) and φ s bar (t) is controlled by the quadratic form K(π (1) ) (9.11) which defines the b, b + theory (9.31), (9.32). Obviously the role of the Hamiltonian in the quadratic form K(π (1) ) is played by The corresponding evolution operator is On the RHS we have the time independent wave functions φ s vac and φ s bar which are determined by the Hartree equations (9.61), (9.63) for φ s vac and (9.67), (9.68) for φ s bar . For the matrix element (9.57), the reduction to the Schrödinger representation can be done as follows Let us define using the evolution operator (9.77). Next, let us introduce a special notation for the set of M + 1 functions including φ s vac (t) and g(t)φ Then we find from Eq. (9.81) where we have introduced the notation D rs (t) =φ r vac,g (t) · φ s bar (t) . We see that D rs (t) is t independent d dt D rs (t) = 0 , (9.87) since the t evolution (9.79), (9.82) ofφ r vac,g (t) compensates the t evolution (9.80) of φ s bar (t). By analogy with Eq. (9.81) we can write In the case t < t 1 this leads to According to Eq. (9.85) D rs is a local (in t) function of φ vac (t), φ bar (t) and g(t). Therefore Eq. (9.90) provides us a local expression for π α (t) via φ vac (t), φ bar (t) and g(t): Theφ vac,g is defined similarly to Eq. (9.83).
It is easy to see that the above argument can be repeated for the case t > t 1 : Inserting Eq. (9.84) into Eq. (9.73), we find According to Eq. (9.87) D rs is independent of t. However, D rs depends on t 1 . This t 1 dependence is implicit. Indeed, if we shift t 1 in the original path integral representation, then this will lead to the corresponding shift of the saddle point π (1) π (1) (t) → π (1) (t − ∆T ) (9.96) and to the shift in the evolution operator Therefore the matrix D rs (9.86) will change as follows: We see that the ∆T shifts of t 1 (9.95) and D rs (9.98) compensate each other in Eq. (9.94) so that the functional W (g) is independent of t 1 as it should be. Using this t 1 independence, we can simplify our equations by choosing Then Eq. (9.94) becomes In the rest of the calculation we will assume the choice (9.99). Using the expression (9.9) for L bos and the result (9.90) for π (1) at t < 0, we find We see that the RHS is t independent. Taking the limit t → −∞ and using Eq. (9.56), we obtain L bos (π bar ) = 1 2 We use the label p.i. in order to emphasize that this result is obtained in the path integral approach.

J. Closed system of differential equations
The U evolution (9.77) of functionsφ s vac (t), φ s bar (t), g(t) described by Eqs. (9.79), (9.80), and (9.82) can be represented in terms of the differential equations Using expressions (9.91) and (9.93) for π (1) α , we can rewrite these differential equations in the form d dtφ where Π (+) α should be used for t > 0 and Π (−) α for t < 0 (we work with t 1 = 0). This is a closed coupled system of differential equations for functions φ s vac (t), φ s bar (t), g(t). In addition we have the boundary conditions which follow from Eqs. Note that the g dependence appears via the boundary condition (9.116). Solving the system of differential equations (9.111)-(9.113) with the boundary conditions (9.114)-(9.116), inserting the solution into Eq. (9.117) and using Eq. (9.85), we can compute the functional W p.i. (g) (9.117): This construction solves the problem of the calculation of the functional W p.i. (g). However, one can wonder how this representation for W p.i. (g) derived in the framework of the path integral formalism is related to our results obtained in the Schrödinger approach. This question is studied in the next section.

A. Problem of equivalence
Our analysis of models with the nontrivial vacuum has resulted in two expressions for the generating functional W (g). One expression (8.68) was derived from the Schrödinger equation using the operator formalism. The other result (9.117) for W (g) was obtained in the path integral formalism. Certainly both approaches are equivalent and should lead to the same results. However, in order to see this equivalence explicitly some extra work is needed. In Sec. X B we demonstrate the equivalence for the simple case of models with the trivial vacuum and after that turn to the much more interesting models with the nontrivial vacuum. The baryon is described by the Hartree equations (9.67)-(9.69) corresponding to one occupied level: 3) Therefore the matrix element (9.88) vanishes at t > t 1 . According to Eq. (9.99) we work with t 1 = 0. Thus equation (9.88) results in On the other hand, Eq. (9.84) gives which is t independent. Now we insert Eqs. (10.8) and (10.9) into Eq. (9.45): . (10.10) According to Eqs. (9.109) and (9.110) we have the differential equations with the boundary conditions (9.115), (9.116) Let us show that this path integral result W p.i. (g) is equivalent to the Schrödinger approach discussed in Sec. VI. Note that our path integral treatment dealt with the case L mn = 0. In the Schrödinger approach this case was discussed in Sec. VI F.
In order to match the equations of Sec. VI F with the equations of the path integral approach one has to use the following "dictionary" connecting the two formalisms:  In the previous section we have demonstrated the equivalence of the Schrödinger and path integral approaches in the case of systems with the trivial vacuum. Now we want to consider the general case when the vacuum is nontrivial.
Our main lesson from the analysis of the systems with the trivial vacuum is that the equivalence between the two approaches can be established on the basis of the Hamiltonian description of trajectories used for the calculation of the functional W (g).
In the case of the nontrivial vacuum we already have the representation for W (g) in terms of trajectories obeying the differential equations (9.108)-(9.110). Now we want to rewrite these differential equations in the Hamiltonian form.
To this aim we must introduce the canonical variables. Note that the original differential equations (9.108)-(9.110) are written in terms of single-particle wave functionsφ s vac (t), φ s bar (t). As we will see below, the Hamiltonian formalism is based on M -and (M + 1)-particle wave functions including the Slater determinants associated with the states (9.74) and (9.75) We assume that the antisymmetrization operator Antisym is normalized by the condition Assuming the choice (9.99) t 1 = 0, we deal with the M -particle states at t > 0 and with the (M + 1)-particle states at t < 0. Therefore the M + 1 particle "wave function" P Now let us rewrite the differential equations (9.108 )-(9.110) in terms of variables Q (M) , P (M) for t > 0 and in terms of Q (M+1) , P (M+1) for t < 0: where The field π (1) α (t) is given by  where the Hamiltonian H is

I. Comparison of path integral and Schrödinger results
We have computed the functional W (g) using the operator (the Schrödinger) approach and the path integral method. The two results are given by Eqs. (8.68) and (10.56), respectively. Our path integral result (10.56) is more general since it deals with models where the physical ground state contains N c M quarks with arbitrary M whereas the Schrödinger result (8.68) corresponds to M = 1.
Comparing equations (8.68) and (10.56), we see that they give the same result for W (g) if I(g) = {I vac (k(g * ))} * I bar (Q(g)) . (10.57) The functional I(g) arises in the path integral approach whereas the functionals I vac (k(g * )) and I bar (Q(g))come from the Schrödinger approach. The aim of this section is to give an independent derivation of the identity (10.57) which directly establishes the equivalence of the path integral and Schrödinger methods. The functionals appearing in Eq. (10.57) can be represented in terms of trajectories. In the path integral approach the trajectory representation for I(g) arises via the saddle point method. The Schrödinger representation for W (g) was formulated in Sec. VIII I in terms of trajectories q vac i (t), p vac i (t) for the vacuum sector and trajectories Q bar ij (t), P bar ij (t) for the baryon sector. Let us show that the connection between the trajectories of the Schrödinger approach and the trajectories P (L) (t), Q (L) (t) (with L = 1, 2) of the path integral approach is given by equations Remember that the trajectories q vac i (t), p vac i (t) were introduced in Sec. VIII I for t < 0 whereas the trajectories P (1) (t), Q (1) (t) are defined at t > 0. Note that the equations for the trajectories are invariant under the combination of the time reversal transformation t → −t with the complex conjugation and with the exchange P (1) ↔ Q (1) . Therefore functions P (1) (t), Q (1)  Now we can turn to the proof of the equality of functionals W Sch (g) and W p.i. (g) associated with the Schrödinger and path integral methods. As was explained above, this proof reduces to the derivation of the identity (10.57). Using The equivalence of the Schrödinger and path integral representations for W (g) is checked.

J. Analyticity and time contour
The functional Φ B (g) is holomorphic in g by construction (2.6). This analyticity is inherited in the large-N c limit (2.7) by the functional W (g), although this limit can be accompanied by the appearance of singularities in W (g). The representation for W (g) in terms of classical trajectories is compatible with the analyticity of W (g). Indeed, the Hamiltonian (10.44) used in this representations is a holomorphic functions of coordinates and momenta. However, the configuration of trajectories used for the construction of W (g) can lead to the singularities in W (g). Examples of these singularities can be found in the analytical expressions for W (g) in the rotator model analyzed in Sec. VII [see Eq. (7.74)] and in the naive quark model studied in Ref. [16].
The analyticity of W (g) in g is closely related to the analyticity of the corresponding trajectories in t. In our path integral construction we used Euclidean time. However, the choice of the Euclidean time is not necessary. For example, in the Schrödinger approach the functional W (g) can be represented in a form which requires no time and no trajectories. The Schrödinger representation is formulated in terms of the action which is defined without explicit usage of the time variable. But if one wants to represent this action in terms of trajectories, then one has to use some version of the time variable. Naively the choice of the time contour (Euclidean, Minkowski or more complicated) is not important. However, even in simple systems we can meet singularities restricting the freedom of the choice of the time contour. Our general representation for the functional W (g) in terms of trajectories ignored this problem which requires an additional analysis in any particular model.

A. Main results
The main aim of this work was to check the consistency of the assumptions used for the construction of the 1/N c expansion for the baryon wave function in QCD. We have concentrated on those nonperturbative features which cannot be directly proved in QCD. On the other hand, these properties can be studied in detail in models preserving the basic structure of the 1/N c expansion in QCD. Our check is quite successful: in the analysis of models we could explicitly demonstrate all basic features: -exponential large-N c behavior of the generating functional for the baryon wave function, -universality of this behavior, -factorization of the preexponential terms.
In the absence of the direct proof of these properties in QCD, this model analysis combined with the independent arguments of Refs. [16,17] (based on the evolution equation, spin-flavor symmetry and soft-pion theorem) gives us a rather solid self-consistent picture of the baryon wave function in large-N c QCD.
The main criterion for our choice of models was the possibility to perform the direct calculation of the 1/N c expansion. In the simplest models we could obtain analytical results. In more general models the problem has been reduced to the solution of a coupled system of differential equations corresponding to the large-N c effective classical dynamics.

B. Phenomenological applications
Although the construction of realistic models was not the aim of this work, our analysis still has a direct relation to phenomenology.
1) The practical applications of the 1/N c expansion which can be of interest for phenomenology are based on the large-N c contracted spin-flavor SU (2N f ) symmetry. In the problem of the baryon wave function we have a rather nonstandard manifestation of this symmetry [17]. If we assume this realization of the spin-flavor symmetry, then we can derive interesting results like the connection between the distribution amplitudes of nucleon and ∆ resonance. The aim of this paper was to justify this assumption by checking the structure of the 1/N c expansion with a direct calculation.
2) The complete analysis of the effects related to the spin-flavor symmetry requires a thorough study of the nextto-next-to leading order (NNLO) of the 1/N c expansion. In the current analysis we did not go so far. Most of the work was concentrated on the leading order (LO). We could also check the factorization properties appearing in the next-to-leading order (NLO). However, the subtle NNLO contributions crucial for the understanding of the spin-flavor symmetry effects remained beyond the scope of this paper. Nevertheless the successful LO and NLO check of the picture suggested in Refs. [16,17] gives us additional evidence for the validity of this picture.
3) Although the results of this paper are presented in a simple quantum-mechanical form, they can be directly rewritten in field-theoretical terms relevant for the phenomenological models with four-fermionic interaction of the Nambu-Jona-Lasinio type.

C. Other aspects
The methods used in our analysis of large-N c models are also interesting from the point of view of the relation between the large-N systems and classical dynamics. Our study of models has also revealed a connection between the equations for the generating functional W (g) and such traditional equations of the many-body physics as the Hartree equation (including its time-dependent version) and RPA equation. Although the methods developed here cannot be applied directly to large-N c QCD, they can be used in some special QCD problems like the diagonalization of the matrix of anomalous dimensions of the leading twist baryon operators [16].