Interpolating 't Hooft model between instant and front forms

The 't Hooft model, i.e. the two-dimensional quantum chromodynamics in the limit of infinite number of colors, is interpolated by an angle parameter $\delta$ between $\delta=0$ for the instant form dynamics (IFD) and $\delta=\pi/4$ for the light-front dynamics (LFD). With this parameter $\delta$, we formulate the interpolating mass gap equation which takes into account the non-trivial vacuum effect on the bare fermion mass to find the dressed fermion mass. Our interpolating mass gap solutions link the IFD and LFD results with the $\delta$ parameter. We find the interpolation angle independent characteristic energy function which satisfies the energy-momentum dispersion relation of the dressed fermion, identifying the renormalized fermion mass function and the wave function renormalization factor. Using the dressed fermion propagator interpolating between IFD and LFD, we derive the corresponding quark-antiquark bound-state equation in the interpolating formulation. The mass spectra of mesons bearing the feature of the Regge trajectories are found independent of the $\delta$-parameter for the equal mass quark and antiquark bound-states. The Gell-Mann - Oakes - Renner relation for the pionic ground-state in the zero fermion mass limit is confirmed indicating that the spontaneous breaking of the chiral symmetry occurs regardless of the quantization for $0 \le \delta \le \pi/4$. We obtain the corresponding bound-state wave functions and discuss their reference frame dependence. Applying them for the computation of the so-called quasi parton distribution functions, we note a possibility of utilizing not only the reference frame dependence but also the interpolation angle dependence to get an alternative effective approach to the LFD-like results.


I. INTRODUCTION
The two-dimensional quantum chromodynamics (QCD 2 ) with the number of colors N c → ∞ has served as a theoretical laboratory for the study of strong interactions. In 't Hooft's seminal paper in 1974 [1], the power of 1/N c expansion [2] was demonstrated in solving QCD 2 in the limit of N c → ∞, which was then widely studied also in relation to the string model and dual theories with the idea of 1/N c expansion as a topological expansion in the motion of physical strings (e.g. by Witten [3]). Under the large N c approximation, nonplanar diagrams are negligible and thus, for example, only the rainbow diagrams need to be summed over for the computation of the quark's self-mass. The two other parameters in QCD 2 besides N c , are the dimensionful coupling constant g and the quark mass m. Sharing the same mass dimension, g and m play an important role in determining the phase of QCD 2 [4]. Depending on the value of the dimensionless coupling g 2 N c /m 2 , it is known that there are at least two phases in QCD 2 [5]. While the regime of the strong coupling phase which doesn't require the finiteness condition on the dimensionless coupling g 2 N c /m 2 [6] can be studied by the bosonization method [7], the regime of the weak coupling phase which keeps the so-called "'t Hooft coupling" λ ∼ g 2 N c finite in the limit of not only N c → ∞ but also g → 0 is investigated typically in QCD 2 . Although the strong coupling regime of QCD 2 is interesting and deserves further study, the scope of the present work is limited to the weak coupling regime of QCD 2 . Yet, we notice that solving QCD 2 in the weak coupling regime, i.e. 't Hooft model, is still highly nontrivial as the theory captures the property of quark confinement and involves the infrared-cutoff procedures discussed in two-dimensional gauge field theories [8,9].
In particular, we notice that the 't Hooft model was originally formulated and solved in the Light Front Dynamics (LFD) [1] well before it was re-derived and discussed in the Instant Form Dynamcis (IFD) [10]. The numerical solution of the 't Hooft model in the IFD was presented in the rest frame of the meson [11] and more recently also in the moving frames [12]. While a particular family of the axial gauges interpolated between the IFD and the LFD was explored, the principal-value prescription for regulating the infrared divergences was shown to be inconsistent with the interpolated general axial gauges [9]. Since then, however, this issue involving the interpolation between the IFD and the LFD has not yet been examined any further although the nontrivial vacua in two-dimensional models including QED 2 have been extensively discussed [13]. It thus motivates us to explore the interpolation of the 't Hooft model between the IFD and the LFD fully beyond the gauge sector and discuss the outcome of the full interpolation which naturally remedies the previous issue on novel inconsistency with the interpolating gauge [9].
Before we get into the details and specific discussions on the 1+1 dimensional nature of the 't Hooft model, we first briefly summarize the general remarks on distin-guished features of the IFD and the LFD proposed originally by Dirac in 1949 [14] and the efforts of interpolating them together [13,[15][16][17][18][19][20][21]. The LFD has the advantage of having the maximum number (seven) of the kinematic operators among the ten Poincarè operators. More kinematic operators provide more symmetries that effectively save the efforts of solving dynamic equations. The conversion of the dynamic operator in one form of the dynamics into the kinematic operator in another form of the dynamics can be traced by introducing an interpolation angle parameter spanning between the two different forms of the dynamics. This in fact motivates the study of the interpolation between the IFD and the LFD.
In our previous works, we have applied the interpolation method to the scattering amplitude of two scalar particles [17], the electromagnetic gauge fields [18], as well as the helicity spinors [19] and established the interpolating QED theory between the IFD and LFD [15]. In particular, we presented [15] the formal derivation of the interpolating QED in the canonical field theory approach and discussed the constraint fermion degrees of freedom, which appear uniquely in the LFD. The constraint component of the fermion degrees of freedom in LFD results in the instantaneous contribution to the fermion propagator, which is genuinely distinguished from the ordinary equal-time forward and backward propagation of the relativistic fermion degrees of freedom. The helicity of the on-mass-shell fermion spinors in LFD is also distinguished from the ordinary Jacob-Wick helicity in the IFD with respect to whether the helicity depends on the reference frame or not. Our analyses clarified any conceivable confusion in the prevailing notion of the equivalence between the infinite momentum frame (IMF) approach and the LFD.
To link the 1 + 1 dimensional IFD space-time coordinates with the LFD ones, we introduce the "hat notation" for the interpolating variables, as we have done in Refs. [15,[17][18][19]]: x + x − = cos δ sin δ sin δ − cos δ where the interpolation angle δ is allowed to be in the region of 0 ≤ δ ≤ π 4 . When δ = 0, we recover the IFD coordinates (x 0 , −x 1 ), and when δ = π/4, we arrive at the LFD coordinates denoted typically by x ± = (x 0 ± x 1 )/ √ 2 without the "hat". In this coordinate system, the metric becomes where we use the short-hand notation C = cos 2δ and S = sin 2δ. Apparently, the interpolating gμν goes to the IFD metric 1 0 0 −1 when δ = 0, and the LFD metric and the inner product of two vectors a and b can be written as The same transformation as shown in Eq.(1) applies to momentum variables as well, i.e. p+ = p 0 cos δ + p 1 sin δ; p− = p 0 sin δ − p 1 cos δ.
A useful relationship for the energy-momentum of the on-mass-shell particle with mass m and two-momentum vector pμ can be found as follows With the "hat notation", the theory of QCD 1+1 in the interpolating quantization is then given by the Lagrangian density 1 where Dμ = ∂μ − igA â µ t a (10) and In this work, we start from the interpolating Lagrangian density, Eq.(9), derive the corresponding Hamiltonian and solve the mass gap equation which interpolates between the IFD and the LFD. We then apply the solutions of mass gap equation to the calculations of chiral condensates and the quark-antiquark bound-states to , i.e., when δ → π 4 , (quantities with a superscript "t" denote the notation used in Ref. [1], and the ones without are ours) find the meson mass spectra and the corresponding wavefunctions. As expected for any physical observables, the meson mass spectra are found to be independent of the interpolation angle parameter. Since we obtain the meson wavefunctions in terms of the interpolation angle parameter δ, we use these δ-dependent wavefunctions to compute the corresponding parton distribution functions (PDFs), comparing them with the PDFs in the LFD and the so-called quasi-PDFs based on the IMF approach in IFD [22]. The paper is organized as follows. In Section II, we derive the fermion mass gap equation in QCD 1+1 (N c → ∞) in the quantization interpolating between the IFD and the LFD, using a couple of different methods, namely, the Hamiltonian method and the Feynman-diagram method. In Section III, we present the solutions of the mass gap equation numerically spanning the interpolation angle between δ = 0 (IFD) and δ = π/4 (LFD). In Section IV, we apply the mass gap solutions to the calculations of the chiral condensates and the constituent quark mass defined in the full fermion propagator. In Section V, we derive the quark-antiquark bound-state equations in the interpolating dynamics and present their solutions in Section VI, including the meson mass spectra, wavefunctions, and (quasi-)PDFs in subsections VI A, VI B and VI C, respectively. The summary and conclusions follow in Section VII. In Appendix A, we describe in detail the derivation of the interacting quark/anti-quark spinor representation using the Bogoliubov transformation. In Appendix B, we show the method of minimizing the vacuum energy with respect to the Bogoliubov angle in getting the mass gap equation. In Appendix C, we discuss the interpolating mass gap equation and solution in terms of the rescaled variables with respect to the mass dimension √ 2λ and its treatment associated with the λ = 0 (Free) case vs. the λ = 0 (Interacting) case. In Appendix D, we present additional numerical solutions of the mesonic wavefunction for a few different quark masses beyond the ones presented in subsection VI B. The corresponding quasi-PDFs are discussed in Appendix E. In Appendix F, we present the quark-antiquark bound-state equations and solutions in the rest frame of the meson.

II. THE MASS GAP EQUATION
In this section, we will derive the quark self-energy equation in QCD 1+1 (N c → ∞) in the interpolating dynamics between the IFD and the LFD. While we use two different methods, i.e. the Hamiltonian method in Sec. II A and the Feynman-diagram method in Sec. II B, we show that both methods provide exactly the same set of equations. When δ → 0, C → 1 and p− → p 1 , these equations become the IFD mass gap equations presented in Ref. [10] (i.e. Eqs. (3.18) and (3.19) of Ref [10]). The agreement to Ref. [1] of the δ → π 4 limit is discussed in Sec. II C.

A. The Hamiltonian method
Before we start, we need to choose a gauge as in the case of any gauge field theory. We adopt here the interpolating axial gauge, i.e. A â − = 0, as explored previously in Ref. [9]. In this gauge, the gluon self-couplings are absent. With the gauge condition, Eq. (9) reduces to L = 1 2 ∂−A â + 2 +ψ(iγ+D+ + iγ−∂− − m)ψ. (12) As no interpolation-time derivative of A â + , i.e. ∂+A â + , appears in Eq. (12), A â + is not a dynamical variable but a constrained degree of freedom. We substitute this constrained degree of freedom using the equation of motion for the gluon field A â + given by The general solution of Eq.(13) is given by where F a and B a are constants. While B a is irrelevant as it can always be eliminated by a gauge transformation, F a is a background electric field which can provide some interesting physical effect such as the axial anomaly in Abelian gauge field theory [23]. For the color-singlet sector in the non-Abelian gauge field theory, however, the background field F a has no effect, e.g. on the spectrum of hadrons in the qq channel. For this reason, we drop the background field F a and take the first term of Eq. (14) as the solution of A â + (x+, x−) in this work. More details of the discussion on the effect from dropping the background field in the 't Hooft model can be found in Ref. [10].
The energy-momentum tensor in the interpolation form is Thus, the interpolating Hamiltonian is As we have shown in [15], all components of the ψ field are dynamical degrees of freedom for 0 ≤ δ < π/4, while half of the components become constrained for δ = π/4. The field operator conjugate to ψ(x) is The anti-commutation relation at x+ = x + is Consequently, The Dirac field ψ can be expanded in terms of the quark creation and annihilation operators = dp− 2π 2p+ When the form goes to the limit of IFD, i.e., δ → 0, p− → −p 1 = p 1 , and x− → −x 1 , and Eq.(20) becomes the ordinary field operator expansion in IFD 2 The non-trivial vacuum, |Ω >, which is defined by is different from the trivial vacuum |0 > defined by The trivial and non-trivial sets of creation and annihilation operators are related by a Bogoliubov transformation The non-trivial set of operators, just like the trivial ones, satisfy the canonical anti-commutation relations at and all others are zero.
The spinors can be defined through a combination of boost and Bogoliubov transformation, which can be represented by 2 It differs from the expression in Ref. [10] by a normalization factor 1 2p+ , which was inserted in order to be consistent with standard textbook [24].
where θ f (p−) is the boost part given by and ζ(p−) is the Bogoliubov angle defined in Eq. (23). While the details of the derivation are given in Appendix A, the results of the spinors are given by and In the case of free particles, Eqs. (28) and (29) simplify with θ(p−) = θ f (p−) given by Eq. (27) as and Now, plugging the gluon field solution Eq. (14) without the background field into Eq. (16), we obtain the interpolating Hamiltonian as where the kinetic energy and the potential energy Now, if we define the 't Hooft coupling as then λ has the dimension of mass squared in 1+1 dimension. By normal-ordering the Hamiltonian, we can write it in three pieces Here, the vacuum energy density E v for the onedimensional volume L is given by the two-body interaction term including the kinetic energy is given by with and : and the four-body interaction term is given by Although the mass gap equation can be obtained either by minimizing E v with respect to the Bogoliubov angle or by requiring : H 2 : to be diagonal in the quark anti-quark creation and annihilation operator basis, both methods provide the same resulting equations. While we present the derivation of minimizing E v in Appendix B, we derive here the mass gap equations by requiring : H 2 : to be diagonal. The requirement of : H 2 : to be diagonal means that it must take the form dp− The divergent piece that comes out during the normal-ordering process is regulated removing the infinite energy [10], and using the principal value prescription as was done in Ref. [11], Thus, the eigenvalue conditions on the spinors are given by where Γ ± is defined by By using Eqs. (28) and (29), one may see that the matrices on the right hand side of Eqs. (45) and (46) can be obtained by direct computation. Now, let us define Then, by subtracting Eqs.(44a) and (44b) as well as plugging in Eqs. (45) and (46), we arrive at On the other hand, by adding them, we get Also, we know that the off-diagonal elements of : H 2 : have to vanish . (50b) From either Eq.(50a) or Eq.(50b), we get Eqs.(48) and (51) are the mass gap equations in the interpolating dynamics. The same set of mass gap equations can be derived using the Feynman diagram method as we present in the following subsection II B.

B. The Feynman diagram method
The self-energy equation in the large N c approximation is drawn pictorially in Fig. 1. Following the Feynman rules for the gluon propagator, the free quark propagator and the vertex as 1 (52) Writing the self-energy as we express the dressed quark propagator as This dressed quark propagator can be obtained from the bare quark propagator with the replacement given by Then, Eq. (52) becomes where we have used the algebra for the interpolating γ matrices γ+ 2 = C · I 2×2 , γ− 2 = −C · I 2×2 , and γ+, γ− = 2S · I 2×2 . Now, the two poles of k+ in the denominator of Eq. (56) are given by After doing the k+ pole integration using Cauchy's theorem, we get where θ(k−) is defined by By comparing Eq.(58) with Eq.(53), we can identify and From Eq.(59), we may geometrically represent the effective mass and longitudinal momentum of the particle moving in non-trivial vacuum by drawing the triangle picture shown in Fig. 2. From Fig. 2, we identify the energy E(p−) as and find the mass gap equations In Sec. III, we numerically solve Eq. (51) to obtain θ(p−) and plug it into Eq. (48) to find E(p−). We note that the solution of E(p−) is not always positive and thus the geometric interpretation of Fig. 2 should be regarded as an pictorial device to represent Eqs. (62)-(64) without assuming that the lengths of the triangle sides are positive. We also recognize Eq. (57) as E u and E v mentioned in the previous subsection, Sec. II A, i.e. We note that the interpolating mass gap equations are greatly simplified in the limit to the LFD, i.e. C → 0. In this limit, Eq. (51) becomes where one should note p− = p − = p + as C → 0. The solution of Eq. (66) is analytically given by Likewise, Eq. (48) is simplified as Moreover, Eqs. (60) and (61) are now given by where the light-front zero-mode p + = 0 contribution should be taken into account separately with the form of A(p + ) = A(0)δ(p + ) solution in mind. Besides the p + = 0 contribution, it is interesting to note a remarkable simplification of the self-energy in the LFD given by Σ(p + ) = γ + B(p + ) due to the absence of the scalar part, i.e. A(p + ) = 0. For the computation of B(p + ) in Eq. (70), 't Hooft [1] didn't use the principal value but discussed how to make the infrared region finite by introducing the infrared cutoff parameter ε as summaried below for p + > 0 and p + < 0, respectively, i.e. in the limit ε → 0, for p + > 0, while for p + < 0, Thus, the 't Hooft's solution for B(p + ) is given by With these solutions, the replacement given by Eq. (55) in the LFD limit (δ → π 4 ) becomes and it leads to the dressed fermion propagator given by where p − on = m 2 2p + and p + on = p + . Here, we note the splitting of the instantaneous contribution (∼ γ + ) and the socalled on-mass-shell part ∼ ( / p on +m) of the fermion propagator in LFD. While the interpolating fermion propagator can split into the forward moving part with the energy denominator 1/(p+ − E u (p−)) and the backward moving part with the energy denominator 1/(p+ − E v (p−)) as we discuss the details of the fermion propagator in Sec. IV B, one can notice the behaviors of E u (p−) and 2p + ) for p + < 0 using Eqs. (62) and (65). Taking p + > 0 for the sake of discussion, one can rather easily identify that the on-mass-shell part ∼ ( / p on + m) of the dressed fermion propagator corresponds to the forward moving part from the 't Hooft solution for B(p + ) given by Eq.(73) as well as the correspondence / p on + m with the spinor biproduct u(p + )ū(p + ) of the dressed fermion. Likewise, one can identify the instantaneous contribution (∼ γ + ) to the backward moving part by noting the cancellation of 1/C factor in E v (p−) ∼ − 2p + C with the 1/C factor in the backward moving spinor biproduct v(−p + )v(−p + ) ∼ 2p + γ + C . However, the instantaneous contribution effectively vanishes in the actual calculation of the rainbow and ladder diagrams as every quark line is multiplied by the vertex factor gγ + from both sides and (γ + ) 2 = 0. Moreover, as {γ + , γ − } = 2, the dressed quark propagator can be effectively given by with the effective vertex factor 2g. We can now see that, due to the 1/ε infrared divergence, the on-mass-shell pole of this dressed quark propagator moves towards the infinity from the on-mass-shell pole p − on = m 2 2p + . This disappearance of the on-mass-shell pole due to the infrared cutoff term was interpreted as the confinement of the fermions in the 't Hooft model [1].
As shown in Ref. [1], however, the infrared cutoff terms cancel themselves in the bound-state spectroscopy calculation. Thus, one can use the principal value prescription as defined in Eq. (43) to regulate the infinite piece. With the same principal value prescription, B(p + ) in the limit of ε → 0 is given by The reduced quark propagator without the 1/ε infrared divergence factor in the denominator is then given by which leads to the same effective planar Feynman rule of the fermion propagator in the light-front gauge presented in Ref.
[1] besides the notation difference explained in the footnote in Sec. I.

III. THE MASS GAP SOLUTION
The mass gap equation given by Eq. (51) is solved numerically using a generalized Newton method mentioned in Ref. [11] as well as in Ref. [12], and we use the same numerical method elaborated in Refs. [11] and [12]. Using the same 200 grid points and choosing the same quark mass values (in unit of √ 2λ) as in Ref. [11] and in Ref. [12], we obtain the numerical solutions of Eq. (51). To cover the entire interpolating longitudinal  in Ref. [11]. The profiles of θ(ξ) for δ = 0 coincide with the IFD results provided in Ref. [11]'s Figure 4. Also, the profiles of θ(ξ) for δ → π/4 approach to the analytic solution given by Eq. (67) in LFD. It is interesting to note that the IFD results exhibit a convex profile for the lighter quark mass (m = 0.18), and as the quark mass increases (m = 1.00 and 2.11) the profile gets more and more concave. For the quark mass values presented  in Ref. [12], the numerical solutions of θ(p−) for several interpolation angles are plotted in Fig. 4. The authors of Ref. [12] chose the 't Hooft coupling λ as πλ = 0.18 GeV 2 in conformity to the value of the string tension in the realistic QCD 4 , and then determined the light quark mass m u/d = 0.045 to obtain the physical pion mass M π = 0.41 in the unit of √ 2λ by solving the boundstate equation, which we present in Sec. V. Likewise, the heavy quark mass m c = 4.23 was determined to get the physical J/ψ mass M J/ψ = 9.03 in the same unit. The strange quark mass m s = 0.749 was taken to provide a threshold, below (above) which is called light (heavy) flavor, by minimizing the relative distance between the θ(ξ) solution and the straight line θ(ξ) = ξ in IFD as the profile of θ(ξ) in IFD (i.e. δ = 0) passes from the convex to concave with the increasing quark mass as discussed in Fig. 3. The chiral limit with m u = 0 was also considered and the Gell-Mann -Oakes -Renner relation (GOR) was discussed in Ref. [12]. For those quark masses, m = 0, 0.045, 0.749 and 4.23, we obtain the interpolating mass gap solutions for several δ values between δ = 0 (IFD) and δ = π/4 (LFD) as plotted in Fig. 4. Our numerical solutions for δ = 0 coincide with the IFD results provided in Ref. [12] Examining the results of E(p−) themselves in the limit δ → π/4, we also find that all of them approach to the analytic result of E(p + ) given by Eq. (68) independent of quark mass m. Moreover, the LFD result of E(p + ) given by Eq. (68) is always positive regardless of p + while the IFD (δ = 0) results of E(p 1 ) for small quark mass values, e.g. m = 0.18 in Fig. 5a and m = 0 and 0.045 in Figs. 6a and 6b, respectively, are negative for small momentum regions. Indeed, we notice that the region of small momentum for the negative value of E(p 1 ) gets shrunken as m gets larger but persists up to m ≈ 0.56. It was argued in Ref. [10] that the existence of negative quark self-energy does not cause any concern though, due to the lack of observability for the energy of a confined single quark. While similar aspect of the negative quark energy exists for other interpolation angle values unless δ = π/4, the corresponding range of the small momentum gets reduced as δ approaches to π/4 as depicted in Figs. 5a, 6a and 6b. It is interesting to note that our observation of diminishing the negative quark self-energy region in the limit δ → π/4 (LFD) appears consistent with the result that the single particle energies do not change sign due to the additional constant term appearing in converting the light-front longitudinal momentum sum to a principal value integral discussed in Ref. [25] with the formulation of a finite light-front x − coordinate interval.

IV. CHIRAL CONDENSATE AND CONSTITUENT QUARK MASS
As we have obtained the mass gap solutions, we now utilize them to discuss the chiral condensate and the constituent quark mass in this section.

A. The chiral condensate
While the Coleman's theorem [26] prohibits the SBCS in two-dimensional theories with a finite number of degrees of freedom, the large N c limit of the 't Hooft model does not contradict with the Coleman's theorem. The exact result for the chiral condensate was found in the chiral limit (m → 0) for the weak coupling regime of in the unit of the mass dimension √ 2λ with the definition of λ given by Eq. (35). This indicates that the spontaneous breaking of the chiral symmetry (SBCS) occurs in the 't Hooft model, in contrast to the strong coupling regime of QCD 2 in which the SBCS does not occur according to the Coleman's theorem [26]. Ramifications of the nontrivial chiral condensates with respect to the vacuum in the LFD as well as its non-analytic behavior were discussed for nonzero quark masses and its chiral limit [27]. For nonzero quark masses, the renormalized quark condensation was defined by subtracting the free field expectation value to render the quark condensation finite [27], In the chiral limit, m → 0, <ψψ > | g=0 = 0 and thus <ψψ > | ren =<ψψ >. The numerical computation verifying Eq.(79) was presented in IFD [28] substituting the mass gap solution for θ(p 1 ) in the chiral limit m → 0 as (81) With the interpolating quark field between IFD and where we have used the interpolating spinors given by Eqs. (29) and (31). We numerically compute Eq. (82) in the chiral limit, m → 0, for different interpolation angles, and the results are shown in Table I. We observe that the closer one gets to the LFD (δ = π 4 ), the higher accuracy one needs for the numerical computation and thus we list the results obtained by increasing the number of grid points used. One can see that when δ is away from π/4, the coarser grid is already good enough to obtain an accurate result. For δ closer to π/4, the number of grid points should be increased to improve the numerical accuary. Table I indicates an eventual agreement to the analytical value −1/ √ 12 ≈ −0.29 regardless of the interpolation angle δ with the enhancement of the numerical accuracy.
Even if the chiral limit is not taken, we notice that the renormalized chiral condensate must be independent of the interpolating angle δ as it must be the characteristic quantity determining the vacuum property for a given phase of the theory. In fact, the interpolating longitudinal momentum variable p− can be scaled out by the interpolating parameter √ C and Eq. (82) can be given by the rescaled variable The same variable change can be applied to the interpolating mass gap equation given by Eq.(51) as we illustrate in Appendix C to obtain the rescaled mass gap equation without any apparent interpolation angle dependence as given by Eq. (C6). With θ(p − ) being the solution of Eq. (C6) as well as θ f (p − ) = tan −1 (p − /m) from Eq. (27), we can confirm that the chiral condensate is indeed independent of the interpolation angle δ. Thus, the result of the chiral condensate must be identical whichever dynamics is chosen for the computation between δ = 0 (IFD) and δ = π/4 (LFD). Computing <ψψ > | ren in Eq.(82) for any given interpolation angle between δ = 0 and δ = π/4, one can numerically verify the uniqueness of the result for each and every quark mass values that one takes. Varying the quark masses between m = 0 and m = 4 in the unit of √ 2λ, we computed <ψψ > | ren from Eq.(83) and obtained the numerical result shown in Fig. 7. Our numerical result for m = 0 in Fig. 7 confirms the results obtained in Table I reproducing the analytic value −1/ √ 12 ≈ −0.29. Likewise, our numerical results for 0 < m ≤ 4 coincide with the analytic result given by Eq.(2.19) of Ref. [27] which was also numerically confirmed in Ref. [12].
In the formulation of a finite light-front x − coordinate interval [25], a phase transition was reported in the weak coupling limit as a dramatic change of the quark condensate value to zero was observed. In the continuum limit, however, there is no such phase transition and the non-zero condensate value is intact regardless of the form of dynamics between δ = 0 (IFD) and δ = π/4 (LFD) as discussed in this subsection. In this respect, it is important to realize that the phase characterized by the SBCS is uniquely viable in the continuum 't Hooft model. While there is a simple analytic step function solution of Eq.(51) in IFD (δ = 0) given by θ(p 1 ) = π 2 sgn(p 1 ) in the chiral limit [10], it should be clearly distinguished from the step function solution in LFD given by Eq.(67). The step function solution in IFD leads to the zero chiral condensate <ψψ >= 0 in contrast to the non-trivial chiral condensate given by Eq.(79). It was indeed demonstrated in Ref. [29] that this chirally symmetric step function solution in IFD reveals the possession of infinite vacuum energy compared to the vacuum energy of the SBCS solution. Thus, the step function solution in IFD should be clearly distinguished from the physically viable solution characterized by the SBCS [30] discussed in this subsection.

B. The Fermion Propagator and Constituent Mass
The interpolating free fermion propagator has been discussed at length in our previous work [15] and can be readily applied for the interpolating bare quark propagator in 1+1 dimension with the notation of the on-massshell two-momenta P aμ = (P a+ , p−), P bμ = (−P b+ , −p−) taking P a+ and −P b+ as the positive and negative onshell interpolating energies of the bare quark, i.e. and The interpolating bare quark propagator is then given by can be easily verified using the interpolating free spinors given by Eqs. (30) and (31). In Eq.(86), we call m as the bare quark mass. As discussed in Sec. II B, the interpolating dressed quark propagator can be obtained from the interpolating bare quark propagator with the replacement given by Eq.(55). Effectively, P a+ and P b+ are replaced by E u and E v in Eq.(65) as well as the free spinors, u (0) (p−) and v (0) (p−), are replaced by the spinors given by Eqs. (28) and (29), u(p−) and v(p−), respectively. The dressed quark propagator is then obtained as where the first and second terms correspond to the forward and backward moving propagators, respectively.
The equivalence of Eq.(87) to Eq.(54) can be verified using the relations where Γ + and Γ − are given by Eqs. (45) and (46), respectively. As discussed in Sec. II C, one can verify that the forward and backward moving parts in the limit C → 0 correspond to the on-mass-shell part ∼ ( / p on +m) and the instantaneous contribution (∼ γ + ) in LFD, respectively.
To discuss the dressed fermion propagator in more physical terms [31], one can express the dressed quark propagator given by Eq.(54) in terms of the mass function M (p−) and the wave function renormalization factor F (p−), i.e.
and identify M (p−) and F (p−) respectively as and We then numerically compute M (p−) and F (p−) using the mass gap solutions θ(p−) and E(p−) obtained in Sec. III. In Figs. 8 and 9, the results of the mass function M (p−) are shown as a function of the variable ξ = tan −1 p− for the bare quark mass values in Ref. [11] (m = 0.18, 1.00, 2.11) and Ref. [12] (m = 0, 0.045, 0.749, 4.23), respectively. As we can see in Figs. 8 and 9, the mass function M (p−) approaches the respective bare quark mass value m as ξ → π/2 or p− → ∞ while it gets to the respective characteristic mass value M (0) at ξ = 0 or p− = 0 regardless of the interpolation angle δ, although the profile of M (p−) in 0 < ξ < π/2 does depend on the value of δ. The characteristic mass value M (0) may be regarded as the dressed quark mass acquired from the dynamical self-energy interaction depicted in Fig. 1. We note that the profile of M (p−) as δ → π/4 approaches to the shape of the step-function which drops from M (0) to m away from p + = 0.
In Figs. 10 and 11, the results of the wavefunction renormalization factor F (p−) are plotted with the same variable ξ for the bare quark mass values in Ref. [11] (m =   and (90), as and so that we may rewrite Eq.(62) associated with the tri-  angle diagram shown in Fig. 2 as or In contrast to Eq.(62), we can now express E(p−) itself without squaring it as E(p−) 2 with the support from the wavefunction renormalization factor F (p−) as well as the mass function M (p−). This is rather remarkable because the issue of E(p−) not being always positive for m 0.56, which was discussed in Sec. III, is now resolved by expressing the dressed quark propagator S(p) in terms of F (p−) and M (p−) as given by Eq.(88). While E(p−) can be negative, F (p−)E(p−) is always positive due to the sign correlation between E(p−) and F (p−) as one can see from Eq.(90) or equivalently from Eq.(92) due to the sign correlation between θ(p−) and p−. This allows us more physically transparent interpretation of the energy-momentum dispersion relation for the interpolating dressed quark with its self-energy. Moreover, using the rescaled variable p − = p−/ √ C introduced in Eq.(83) for the renormalized chiral condensate, we can assert the interpolation angle independence of the rescaled energymomentum dispersion given by where we define the interpolation angle independent energy functionẼ(p − ) which extends the interpolating energy-momentum dispersion relation of the on-massshell particle given by Eq. (7) to the case of the dressed quark with the rescaled variable, i.e.
As the solution ofẼ(p − ) is always positive in contrast to E(p−), we can now promote the mere pictorial device of geometric interpretation depicted in Fig. 2 to the more physically meaningful geometric interpretation withẼ(p − ), M (p − ) and p − as shown in Fig. 12 with all the positive lengths of the triangle sides. From Eqs. (7) and (96), we also note the correspondence m ↔ M (p − ) and p+/ √ C ↔Ẽ(p − ) between the bare quark and the dressed quark. As an illustration of this correspondence, we plot the profiles ofẼ as a function of p − for the two cases of m = 0 and m = 0.18 in Fig. 13. It is evident that Now, between the two limits (p − → 0 and p − → ∞), bothẼ(p − ) and M (p − ) are running with the variable p − and their profiles of the p − -dependence are completely independent of the interpolation angle δ. The invariance ofẼ(p − ) and M (p − ) under the interpolation between IFD and LFD indicates their universal nature as physically meaningful quantities. In this respect, one may callẼ(p − ) and M (p − ) as the effective energy and the constituent mass of the dressed quark moving with the scaled longitudinal momentum p − . In principle, these physical quantities can be also computed in the well-known Euclidean approaches [32] such as the lattice QCD [33] and the manifestly covariant Dyson-Schwinger formulation. In the Euclidean formulation, which can in principle be applied to the interpolation dynamics as far as 0 ≤ δ < π/4 except δ = π/4 due to the light-like nature of LFD, the effective energy and the constituent quark mass would be given in terms of the Lorentz-invariant Euclidean variableP 2 < 0, i.e. E(P 2 ) and M (P 2 ). In IFD, one may correspond the Wick rotated energyP 0 with the imaginary effective energy, i.e.P 0 = iẼ(P 2 ) (purely imaginary value), and  the longitudinal momentumP 1 with p 1 of the dressed quark, so In the same token, as generally in the interpolating dynamics, the Wick rotated interpolating energyP+ with the imaginary effective energy, i.e. i.e.
This correspondence is supported not only from the relativistic form invariance ofẼ(p − ) and M (p − ) but also from the matching condition between the Minkowsky space and the Euclidean space. Namely, when the real energy value in the Minkowsky space is converted into the purely imaginary value in the Euclidean space, the matching between the real value and the purely imaginary value occurs precisely where the value itself is zero. For instance, in IFD, the Wick rotation of the energy p 0 →P 0 = ip 0 has the common energy value p 0 =P 0 = 0 so that the Lorentz-invariant momentumsquared value in the Minkowsky space can be matched with the corresponding Euclidean momentum-squared value,P 2 , by taking simultaneously both the real value p 0 = 0 in the Minkowsky space and the purely imaginary valueP 0 = ip 0 = 0. Likewise, in the interpolation of the relativistic dynamics for 0 ≤ δ < π/4, one can match the Minkowsky space and the Euclidean space by taking p+/ √ C = ip+/ √ C =P+/ √ C = 0, confirming the correspondence given by Eq.(97).

V. THE BOUND-STATE EQUATION
Having solved the mass gap equation and obtained the dressed quark propagator interpolating between IFD and LFD, we now derive the quark-antiquark bound-state equation in the interpolating form. While we consider the quark-antiquark bound-state with the equal bare mass m in this work, one may generalize it rather straightforwardly to the unequal quark and antiquark mass cases. Bound-states for the unequal quark and antiquark mass cases were analyzed in Refs. [1,34], and the quark-hadron duality, analytical heavy quark expansion and chiral symmetry breaking effects in the heavy-light mesons have been discussed respectively in Refs. [35][36][37]. Denoting the covariant Bethe-Salpeter amplitude Γ(r, p) of the two-body bound-state with the two-momentum rμ of the bound-state and the two-momentum pμ of one of the two constituents depicted in Fig. 14, we first write down the Bethe-Salpeter equation following the Feynman rules of the gluon propagator and the dressed quark propagator in the 't Hooft model To project the covariant Bethe-Salpeter amplitude Γ(r, p) into the interpolating equal-time wavefunction, we first integrate out the interpolating energy p+ and define the wave function φ(r−, p−) as φ(r−, p−) = dp+Γ(r, p) with the on-mass-shell condition to fix the external interpolating energy r+. Then, Eq.(98) is converted to the equation for φ(r−, p−) Now, we also write the fermion propagator S(p) given by Eq. (87) as where we define Substituting then Eq.(100) to Eq.(99) and performing the p+ integration, we get Here, we noteT (−p−)T (p−) = I and further defineφ(r−, Examining the general structure ofφ in Eq.(107), we realize that it can be split into the forward moving partφ + and the backward moving partφ − with the two 2 × 2 matricesM + andM − , i.e.φ =φ +M + +φ −M − , wherê We note here thatM + andM − coincide with the two 2 × 2 matrices M + and M − in the IFD given by Eq.(4.9) of Ref. [10] as For the interpolation angle 0 ≤ δ ≤ π/4 in general, from the direct calculation of the matrix multiplications, we find where and With Eqs.(110)-(113), we finally split Eq.(107) into the two coupled bound-state equations ofφ + andφ− We again note here that Eqs.(116a)-(116b) coincide with Eq.(4.12) of Ref. [10] in the IFD as δ → 0 because not only the energies of particle and anti-particle become −→ −E(p 1 ) but also the rest of the variables correspond to their IFD counterparts, e.g. r+ δ→0 −→ r 0 , etc..
In the LFD limit C → 0 (or δ → π/4) on the other hand, as discussed earlier in Sec. II C, E u (p−) −→ − 2p + C for the bound-state kinematics 0 < p + < r + . In this limit then, noting C(p + , k + , r + ) → 1 and S(p + , k + , r + ) → 0 from Eq.(67), we note that Eq.(116a) gets reduced to where φ(r + , p + ) corresponds toφ + (r−, p−) in the LFD limit. Also, the solution forφ − → 0 can be attained rather immediately from Eq.(116b) by dividing it by the energy denominator factor r+ + E u (p− − r−) − E v (p−) and noting the correspondence 1/(r+ + E u (p− − r−) − E v (p−)) → C/(2r + + C(r − + B(r + − p + ) + B(p + ) + m 2 2(r + −p + ) + m 2 2(p + ) ) → 0 as C → 0. Substituting the on-mass-shell condition r − = M 2 /(2r + ) for the bound-state meson of mass M, defining the manifestly boost-invariant light-front momentum fraction variables x = p + r + (0 ≤ x ≤ 1) and y = k + r + (0 ≤ y ≤ 1) and multiplying both sides of the equation by (−2r + ), we obtain where the r + independence of φ(r + , p + ) and φ(r + , k + ) is imposed in φ(x) and φ(y) due to the boost-invariance of the light-front bound-state equation correctly reproducing 't Hooft's bound-state equation, Eq.(25) in Ref. [1]. In general, relating E u 's and E v 's in Eq.(116) to the solutions of E in Eq.(65), we summarize the interpolating coupled bound-state equations forφ + (r−, p−) andφ − (r−, p−) between IFD and LFD as follows: In the next section, Sec. VI, we solve Eqs.(119a)-(119b) using the solutions of Eqs. (48) and (51) to determine the bound-state mass spectrum as well as the corresponding bound-state wavefunctionsφ + (r−, p−) andφ − (r−, p−) interpolating between IFD and LFD. Noting the conversion between the upper and lower indices of the momentum variable, we rewrite Eqs.(119a) and (119b) respectively as follows, and One may note here that the ordinary longitudinal momentum r 1 is related to the interpolating longitudinal momentum r− as  ± (r−, x) for the nth bound-state as we discuss in subsection VI A and subsection VI B, respectively. Then, we also apply thê φ (n) ± (r−, x) solutions to obtain the interpolating PDFs and discuss the comparison with the quasi-PDFs in subsection VI C.

A. Spectroscopy
As discussed in Ref. [30], the bound-state wavefunctionsφ (n) ± (r−, x) correspond to the coefficients of the compound operators creating/annihilating color singlet quark and antiquark bound-states in the generalized Bogoliubov transformation. In contrast to the Bogoliubov transformation for the fermion operators given by Eq.(23) with the coefficients satisfying the normalization cos 2 ζ + sin 2 ζ = 1, the coefficients of the generalized Bogoliubov transformation for the meson states satisfy the normalization condition [12] dx |φ (n) In conjunction with the compound operators [30], it is interesting to note that the scalar or fermionic matter fields transforming in the adjoint representation of SU (N ) have been discussed in the literature [38].
In solving numerically the coupled integral equations, Eqs. (120a) and (120b), we use the spectrum method illustrated in Refs. [11] and [12]. While the the rest frame basis wavefunction [11] was generalized to the moving frame basis wavefunction [12] in IFD, we further generalize it to the interpolating basis wavefunction applicable to any interpolation angle δ between IFD and LFD, i.e.
where H m is the m-th Hermite polynomial and α is the variational parameter which can be tuned to minimize the mass of the ground state. The reason for the Hermite polynomial basis can be traced back to the similarity between the λ = 0 free Hamiltonian expressed in terms of the quark-antiquark compound operators [30] and the simple harmonic oscillator Hamiltonian. Due to the charge conjugation symmetry of the bound-states, the wavefunctions of the mesonsφ + andφ − are then the superpositions of these basis functionŝ where the momentum fraction x = p−/r− goes to the familiar light-front momentum fraction p + /r + as δ → π/4. While x = p−/r− is unbounded, x ∈ (−∞, ∞), for 0 ≤ δ < π/4 in solving the interpolating coupled boundstate equations given by Eqs. (120a) -(120b), x gets bounded to be in [0, 1] consistently satisfying the boundstate kinematics 0 ≤ p + ≤ r + in LFD at δ = π/4 (or C = 0) and the two coupled equations, Eqs. (120a) and (120b), get reduced to the single light-front bound-state equation given by Eq.(118). We check this LFD reduction by confirming the correspondence of the interpolating solutionsφ + (r−, p−) → φ(x) andφ − (r−, p−) → 0 in the limit δ → π/4 (or C → 0).
In solving the coupled equations, Eqs. (120a) -(120b), for the ground state n = 0 as well as the n even excited states, we use the orthonormal basis of Ψ 2m (α, r−, x) with m = 0, 1, 2, ...N − 1 due to the x ↔ 1 − x symmetry. In contrast, for the n odd excited states, we use the orthonormal basis of Ψ 2m+1 (α, r−, x) with m = 0, 1, 2, ...N − 1 due to the x ↔ 1 − x antisymmetry dictated by the charge conjugation symmetry. By projecting Eqs. (120a) and (120b) onto the given set of orthonormal basis functions and integrating over the momentum fraction x on both sides, Eqs. (120a)-(120b) are then transformed into a matrix eigenvalue equation in the orthonormal basis given by Eq. (126) with m = 0, 1, 2, ...N −1, i.e. Ψ 2m or Ψ 2m+1 depending on whether n is even or odd. Due to the coupling betweenφ + andφ − , the size of the matrix to be diagonalized is then 2N × 2N for either n even or n odd states. While we get 2N of r+ In the numerical computation, we note that the numerical accuracy of the results depends not only on the number of basis functions but also the number of grid points for the numerical integrations. In practice, we take N = 20 in Eq.(126) to get the mass spectrum of gets smaller, the eigenvalues r+ (n) that we obtain from Eqs.(120a) and (120b) gets closer to the value r− that we take for our numerical computation so that it may lead to the cancellation of the two large values in getting the small value. The demand of numerical accuracy gets even more enhanced when δ gets close to π/4 (or C gets close to zero) and r− gets large as both the values in the numerator and the denomination get closer to zero.
In particular, as the bare quark mass m → 0, we need to check our numerical accuracy paying attention to the GOR relation which was explicitly derived in the 't Hooft model [30] and identifying the pionic boundstate M 2 (0) ∼ m where we used Eq.(79) for the vacuum condensation <ψψ > and the pion decay constant f π = N c /π derived [30] from the matrix element of the axial vector current between the pionic ground state and the non-trivial vacuum. Here, the pionic bound-state in the chiral limit corresponds to the zero mass bound-state that occurs when both quarks have mass zero as pointed out by 't Hooft [1]. The corresponding bound-state wavefunction in LFD is given by φ(x) = 1 for x ∈ [0, 1] as we will discuss in the next subsection, Sec. VI B. These results are consistent with the discussions [4,12,30,37] Table III. As δ gets close to π/4, our numerical computation demands much higher numerical accuracy. Although we haven't increased the number of grid points beyond 600, we anticipate that our numerical results would get closer and closer to zero as we increase the number of grid points even if δ gets close to π/4. In Table IV, we list the results of the meson mass spectra M (n) up to n = 7 for the bare quark mass m = 0 including M (0) obtained with the 600 grid points for δ = 0, 0.6 and 0.78. For the mass spectra of excited states, M (n) (n = 1, 2, ..., 7), the results were obtained with the number of grid points just 200 for δ = 0 and 0.6 although the number of grid points for the δ = 0.78 case was still kept as 600. Similarly, we present the results of the meson mass spectra for the bare quark mass m = 0.18 in Table V. Here, all the results M (n) (n = 0, 1, 2, ..., 7)  In Table VI, we summarize these stable numerical results up to the second digit after the decimal point. Here, we take M (0) = 0 for the case m = 0 from the theoretical SBCS ground. The values in Table VI are also shown in Fig. 16 depicting the feature of "Regge trajectories" for the quark-antiquark bound-states each with the corresponding equal bare mass m [1,8,11,12,34]. It is interesting to note that the Regge trajectory gets slightly modified from just the linear trajectory behavior, developing a bit of curvature in the trajectory for the ground-state and the low-lying excited states. For the small mass, in particular m = 0, the trajectory looks a little concave shape while for the larger mass the curvature turns somewhat into a convex shape. This seems to reflect the fact that the GOR works in the chiral limit as shown in Fig. 15 but the chiral symmetry gets broken as the quark mass gets larger. The convex feature of the Regge trajectory for the heavy quarkonia model was shown in Ref. [40].

B. Wavefunctions
We present here our numerical solutions of the boundstate wavefunctionsφ (n) ± (r−, x) interpolating between IFD and LFD for the ground state n = 0 and the first excited state n = 1 with r− = 0 in terms of the interpolating longitudinal momentum fraction variable x = p−/r−. The results for r− = 0 are presented separately in Appendix F in terms of the variable ξ = tan −1 (p−) without scaling the interpolating momentum variable p− with respect to r−. In the chiral limit, where the GOR relation M 2 (0) ∼ m √ λ → 0 is satisfied, the analytic solution of the pionic ground-state wavefunction φ ± in IFD, i.e. φ (0) ± for δ = 0, is known [30] in terms of the IFD longitudinal momentum variables p 1 and r 1 of the quark and the pionic meson, respectively. Corresponding the IFD longitudinal momentum p 1 and r 1 to the interpolat-ing longitudinal momentum p− and r− and confirming the consistency with the LFD analytic mass gap solution discussed in Sec. II C, we note that the corresponding analytic solution in the interpolating formulation is given byφ where the normalization is taken to satisfy Eq.(124) [12]. In Fig. 17, our numerical results of the plus and minus components of the bound-state wavefunction for the ground state, i.e.φ  − (r−, x) results tend to diminish although for very small momentum, e.g., r 1 = 0.2M 0.18 , it is still of comparable order of magnitude toφ (0) + (r−, x) as noted also in Ref. [12]. Ref. [22] also noted that for light mesons the large-momentum IFD numerical results approach the exact light-front solution very slowly. While the LFD solution for m = 0 exhibits an infinite slope at the endpoints x = 0, 1, such feature is not achieved in the IFD large momentum method [12,22]. As δ gets closer to π/4, however, the resemblance to the LFD solutions is attained even in the smaller longitudinal momentum (e.g. r− = 2M 0.18 ) and thus the large r− (e.g. r− = 5M 0.18 ) does not need to be taken for the confirmation of the LFD solutions. The similar behavior of resemblance to the LFD results depending on the values of δ and r− is also found in the first excited statesφ     plots of Figs. 17a,17b,18a,18b appear to be consistent with the results in Ref. [12] although only a qualitative comparison can be made as different momentum values are taken for the moving frames in Ref. [12] compared to what we present here.
In contrast to the case of m = 0, there are no known analytic solutions of the bound-state wavefunctions for m = 0. While we present the numerical results of φ        Fig. 19b that the large-momentum IFD (δ = 0) results approach to the LFD (δ = π/4) result again very slow forφ       They also share the similar behavior of resemblance to the LFD results depending on the values of δ and r−, which we have discussed for the ground state previously.

C. Quasi-PDFs
Since we obtained the bound-state wavefunctions, we now apply them to compute the so-called quasi-PDFs which have been discussed extensively even in the 't Hooft model application [22] due to the possibility of computing directly the longitudinal momentum fraction x-dependence of the parton distributions in Euclidean lattice approach using the large momentum effective field theory (LaMET) program [41]. In our interpolating 't Hooft model computation, the "quasi-PDFs" may be de- fined as the following matrix element for the n-th state of the meson with the interpolating longitudinal momentum r−: where r+ (n) = r 2 − + CM 2 (n) as obtained from Eq. (7), and the range of the longitudinal momentum fraction x = p−/r− is unconstrained, −∞ < x < +∞, for 0 ≤ δ < π/4, while bounded, 0 ≤ x ≤ 1, for δ = π/4. One should note that this definition of the "quasi-PDFs" is not unique, e.g. taking γ+ instead of γ− in front of the interpolating gauge link W[x−, 0] in Eq.(129), but still uniquely approach the PDF defined in the LFD as δ → π/4, whichever definition is taken. While one may consider the so-called universality class [42] of the interpolating "quasi-PDFs", we note that the definition given by Eq.(129) coincides with the canonical definition of the quasi-PDFs in IFD (δ = 0) [22]. While it has been discussed which definition approaches the PDFs in LFD faster for the perspectives of the LaMET program [22,41], we will take the definition given by Eq.(129) in this work and discuss our numerical results      corresponding to this definition. The interpolating gauge link inserted in Eq. (129) assures the gauge invariance of the interpolating "quasi-PDFs". The subscript C in Eq. (129) indicates the removal of the disconnected contribution discussed [43] for the forward matrix element computation: As we adopted the axial gauge in the interpolation form, i.e. A â − = 0, the gauge link becomes an identity and the quantization procedure illustrated in Sec. II A yields For δ = 0, i.e. IFD, Eq.(132) coincides with Eq.(74) of Ref. [22]. Based on this formula, we compute the interpolating "quasi-PDFs" for the cases of δ = 0, 0.6 and 0.78. In Figs. 21 and 22, the interpolating "quasi-PDFs" of the ground state (n = 0) and the first excited state (n = 1) are shown for the case of δ = 0, respectively. The two panels in each of these figures exhibit our numerical results for the bare quark mass m = 0 and m = 0.18, respectively. Numerical results for other mass cases (m =0.045, 1.0 and 2.11) are summarized in Appendix E. In each panel, the results for the meson longitudinal momentum r− = r 1 = 0.2M 0.18 , 2M 0.18 , 5M 0.18 are depicted by blue, yellow and green solid lines, respectively. The general behaviors of our numerical results with respect to the variation of r 1 values from small (0.2M 0.18 ) to large (5M 0.18 ) agree with the results presented in Ref. [22], although different longitudinal momentum values were taken between ours and Ref. [22]. As r 1 gets larger, the numerical results of the quasi-PDFs resemble the PDFs in LFD more closely fitting in x ∈ [0, 1] and [−1, 0]. It is our interest to take a look at the rate of achieving the resemblance to the PDFs in LFD as δ gets away from the IFD (δ = 0) and r− gets larger. The numerical results of the ground state (n = 0) and the first excited state (n = 1) are shown for the case of δ = 0.   Fig. 25. Besides such numerical sensitivity for the very large value of r−, it is apparent that the δ = 0.78 results are rather immediately close to the LFD result.
Interestingly, our numerical results of the interpolating "quasi-PDFs" in the moving frames indicate a possibility to utilize both variation of δ and r− to attain the LFD result more effectively. Namely, one may not need to boost the longitudinal momentum r− too large but search for a "sweet spot" by varying both δ and r− together to obtain the "LFD-like" result. In IFD, δ = 0 is fixed and thus the boost to the large longitudinal momentum is necessary for a successful approach to the LFD result. However, in the interpolating formulation between the IFD and the LFD, the LFD result can be approached even at rather small r−. Moreover, the application to the lattice formulation may be also possible with the existing technique of Wick rotation replacing the ordinary instant form time x 0 by the interpolating time x+ in the process of taking the "imaginary time" in the lattice as far as δ remains in the region 0 ≤ δ < π/4 avoiding the lightlike surface δ = π/4. As discussed in the later part of Sec. IV B, one can match the Minkowsky space and the Euclidean space confirming the correspondence given by Eq. (97).
For an illustration of the δ variation for a given finite r−, we take r− = 2M 0.18 for the case of m = 0 and show the "quasi-PDFs" of the ground-state and the first excited-state for the variation of δ parameter as δ = 0, δ = 0.6 and δ = 0.78 in Fig. 27. It indicates that a pretty slow approach to the LFD result in the large-momentum IFD can be fairly well expedited by taking δ away from the IFD (δ = 0) and getting closer to the LFD (δ = π/4) while the same value of the longitudinal momentum r− = 2M 0.18 is taken. The numerical sensitivity arising near δ = π/4 for the large r−, e.g. the "wiggle and bulge" mentioned for r− = 5M 0.18 in Figs. 25 and 26 for δ = 0.78, is also dodged by taking the smaller value of r−, e.g. r− = 2M 0.18 for this illustration. We think it would be worthwhile to explore this idea of utilizing the interpolating formulation between IFD and LFD for the application to the lattice computation.
For further application of the interpolating boundstate wavefunctions, one can also consider the interpolating "quasi-distribution amplitude (quasi-DA)" which may be written as where W is the gauge link introduced in Eq. (130), and f (n) is the decay constant mentioned in Sec. VI A. As mentioned in the definition of the interpolating "quasi-PDFs" given by Eq.(129), the definition of the interpolat- ing "quasi-DAs" is also not unique, e.g. γ+ can be taken instead of γ− in front of the interpolating gauge link . Whichever definition is taken, they all uniquely approach the DA defined in the LFD as δ → π/4, belonging to the same universality class [42] as mentioned for the case of interpolating "quasi-PDFs". Using the definition given by Eq.(133) which coincides with the canonical definition of the quasi-DAs in IFD (δ = 0) [22], we note that the interpolating "quasi-DAs" of the even-n mesonic states can be formulated as whereφ (n) ± (r−, x) denote the interpolating mesonic wave functions associated with the n-th excited mesonic state. The normalization condition of the interpolating "quasi-DAs" given by is consistent with the explicit form of the decay constant f (n) given by (136) While the corresponding results for the IFD (δ = 0) have been worked out in Refs. [12,22], it is interesting to note that the analytic result for the pion decay constant f π = N c /π [30] can be immediately obtained by taking the LFD solution in the chiral limit, i.e. φ As the DAs in LFD are directly involved with the QCD factorization theorem for the hard exclusive reactions involving hadrons, it would be useful to explore the utility of the interpolating "quasi-DAs" further in the future works.

VII. CONCLUSION AND OUTLOOK
In this work, we interpolated the 't Hooft model (i.e. QCD 2 in the large N c limit) with the interpolation angle δ between IFD (δ = 0) and LFD (δ = π/4) and analyzed its nontrivial vacuum effects on the quark mass and wavefunction renormalization as well as the corresponding meson mass and wavefunction properties taking the meson as the quark-antiquark bound-state. We derived the interpolating mass gap equation between IFD and LFD using not only the algebraic method based on the Bogoliubov transformation between the trivial and nontrivial vacuum as well as the bare and dressed quark but also the diagrammatic method based on the self-consistent embodiment of the quark self-energy. Our mass gap solutions agree not only with the LFD result in Ref. [1] for δ = π/4 but also with the IFD results in Refs. [11,12] for δ = 0. The renormalized chiral condensate was computed and the agreement of the result in the chiral limit was verified with the exact result in Ref. [4]. Its invariance regardless of the δ values between IFD and LFD was also confirmed. Taking into account the wavefunction renormalization factor F (p−) as well as the mass function M (p−) and expressing the dressed quark propagator S(p) in terms of F (p−) and M (p−) as given by Eq. (88), we resolved the issue of E(p−) not being always positive for m 0.56 discussed in Ref. [10]. Extending the interpolating energy-momentum dispersion relation of the on-mass-shell particle given by Eq. (7) to the case of the dressed quark with the rescaled variable given by Eq. (96), we obtained the interpolation angle indepen-dent energy functionẼ(p − ). Typical profiles ofẼ(p − ) were exemplified in Fig. 13.
Utilizing the dressed fermion propagators, we then derived the quark-antiquark bound-state equation interpolating between IFD and LFD for the equal bare quark and antiquark mass m and solved numerically the corresponding bound-state equations. From the numerical solutions of the spectroscopy, we find that the meson mass spectrum is independent of interpolation angle between the IFD and LFD as expected for physical observables. In particular, for the bare quark mass m → 0, we confirmed the GOR behavior of the pionic ground-state mass square M 2 (0) ∼ m √ λ → 0 as shown in Fig. 15. Our result is consistent with the discussions [4,12,30,37] on the SBCS in the 't Hooft model (N c → ∞). Plotting the meson mass spectra M (n) (n = 0, 1, 2, ..., 7) for various m values as summarized in in Table VI, we also observe the Regge trajectory feature as shown in Fig. 16. The corresponding bound-state wavefunctionsφ (n) ± (r−, x) were obtained, in particular, for the low-lying states, i.e., n = 0 and 1 states, and were applied to the interpolating formulation of "quasi-PDFs". The results of the wavefunctions clearly dictate the charge conjugation symmetry, exhibiting the symmetric and antisymmetric behaviors ofφ It is also interesting to note that the massless Goldstone boson cannot exist in the rest frame due to the null normalization from the equivalence betweenφ  Applying the bound-state wavefunctionsφ (n) ± (r−, x) for the computation of the interpolating "quasi-PDFs" given by Eq. (132), we note the consistency with the observation made in Ref. [22] for the quasi-PDFs at δ = 0 (IFD) that there exists considerable difference between the shapes of the LFD result and the IFD quasi-PDF result for the light mesons. Our results indicate that the slow approach to the LFD-like results may be remedied by varying the interpolation parameter δ as well as the interpolating longitudinal meson momentum r−. For the future work, one may explore such idea to search for the "sweet spot" of δ and r− to attain most effective computation with the least sensitive numerical errors in getting the LFD result. Extending the Wick rotation technique to the interpolating time x+, the idea of searching for the "sweet spot" may be applicable to the usual lattice formulation in the Eunclidean space. This would be in good contrast to the recent application of the present interpolating formulation to the two-dimensional φ 4 theory using the discretization technique in Minkowsky space consistent with the discrete light-cone quantization (DLCQ) approach [20,44]. It will be interesting to explore both "Euclidean" and "Minkowsky" numerical approaches implementing the interpolating formulation between IFD and LFD. In this Appendix, we summarize the interpolating spinors and γ matrices in the 1+1 dimensional chiral representation and the Bogoliubov transformation between the free and interacting spinors as well as the corresponding creation/annihilation operators.
For the representation of the spinors [19], we adopt the chiral representation (CR), under which the usual γ matrices in IFD for the 1 + 1 dimensions are given by the Pauli matrices: These can be transformed into standard representation (SR) easily by a transformation matrix In the SR representation, the free spinors in the rest frame in IFD are typically given by where we take the normalization factorū (0) SR u (0) SR = 2m in conformity with the standard textbooks [24]. The corresponding free spinors in the chiral representation are used in this work without denoting the "CR" specification: The IFD spinors in the moving frame with the energy E p = (p 1 ) 2 + m 2 are then obtained as and where the usual boost operator B(η) with the rapidity η = tanh −1 p 1 Ep and the longitudinal boost generator K 1 is given by (A8) In terms of the interpolating momentum variables, the rapidity η can be written [19] as where one can note the following equality as well The boost operator B(η) can then be written in terms of the interpolating momentum variables as so that the boosted interpolating spinors are given by and Although Eqs.(A12)-(A13) are expressed in terms of the interpolating momentum variables while Eqs.(A6)-(A7) are written in terms of the IFD momentum variables, one should note that they are intrinsically the same spinors with respect to each other as we have shown in the above derivation. As δ → π 4 , i.e., in the LFD limit, Eqs.(A12)-(A13) coincide with the LFD spinors [19] and where one may note the correspondence with p+ → p + , and p− → p + . Thus, the interpolating spinors given by Eqs.(A12)-(A13) are nothing but the same spinors as given by Eqs.(A6)-(A7) in IFD and Eqs.(A14)-(A15) in LFD, respectively. There are no differences in the spinors except the expression difference in terms of the momentum variables taken in each different form of the dynamics. Dropping the "CR" specification again for the γ matrices, we follow the link given by Eq.(1) to get the following interpolating γ matrices in the chiral representation as Lowering the indices with the interpolating metric given by Eq.
(2), we also get where √ 1 ± S can be identically given by cos δ ± sin δ with cos δ ≥ sin δ always due to 0 ≤ δ ≤ π 4 . One can explicitly check that the interpolating γ matrices satisfy γμ, γν = 2gμν · I 2×2 , in particular, (A19) As δ → π/4, one gets the usual LFD γ matrices given by where {γ µ , γ ν } = 2g µν · I 2×2 and (γ + ) 2 = (γ − ) 2 = 0. Now, using the Bogoliubov transformation given by Eq.(23) for the creation/annihilation operators of the quark/anti-quark fields as well as the boost operation for the free quark/anti-quark fields, we get the following relationship between the interacting spinors and the free spinors at rest: and Note here that the free spinors in the moving frame are related to each other as , and that the free spinors in the moving frame and the rest frame are related to each other where θ f (p−) is given by Eq. (27) and is related in the IFD case to the rapidity η as sin θ f = tanh η, while in general they are related by tanh η = sin θ f −tan δ 1−sin θ f tan δ , or in other words, sin θ f = tanh η+tan δ 1+tan δ tanh η . From Eq.(A21), one can get the interacting quark/anti-quark spinors given by Eqs. (28) and (29) in Sec.II A as well as Eqs. (45) where Γ ± is defined by Eqs.(45) and (46). Let us now compute the small variation of E v as In the second term of the above equation, we are able to swap variables p− and k−, i.e.
where we used the fact that the sum of Γ + (p−) and Γ − (p−) is independent of p− from the second line to the third line. Thus, we obtain The functional differentiation of E v relative to θ(p−) for a given p− is then given by i.e.
The computation of the trace leads to the gap equation given by Eq.(51).
Appendix C: Treatment of the λ = 0 (Free) case vs. the λ = 0 (Interacting) case with respect to the mass dimension √ 2λ As the 't Hooft coupling λ given by Eq.(35) has the mass-square dimension, we scaled out the mass dimension √ 2λ and used the dimensionless mass m, longitudinal momentum p−, etc., in presenting all the figures and tables of our work. Of course, we could have explicitly defined the dimensionless variables, e.g., denoted bȳ and rewrite, for example, Eqs. (48) and (51) as respectively. Similarly, we can also scale out the interpolation angle dependence by defining the following rescaled variablesp to reduce Eqs. (C2) and (C3) as respectively. Indeed, we used such rescaled variables in Eqs.(83), (95) and (96) to present the corresponding results without any dependence of δ ∈ [0, π/4], confirming that the physical results are indeed invariant regardless of the interpolation angle δ as they must be. However, one should note the contrast between the scaling by the  dimensionful parameter √ 2λ and the scaling by the dimensionless parameter √ C. While the rescaling over the dimensionless variable C includes the limit of C = 0, i.e. LFD, the rescaling over the dimensionful variable √ 2λ cannot include the limit to λ = 0. Namely, the free theory without any interaction must be distinguished from the interacting theory and should be discussed separately. For λ = 0, in fact, the mass gap solution θ(p−) = θ f (p−) can be immediately found even analytically by taking the right-hand-side of Eq.(51) to be zero, i.e. In Sec. VI B, we discussed the wavefunctions of the quark-antiquark bound-states for the cases of m = 0 and m = 0.18 in the unit of √ 2λ, i.e.m = 0 and 0.18. In this Appendix, we summarize the numerical results ofφ     Table VI) is the ground state meson mass for the quark mass valuem = 0.045, are depicted by the solid lines in blue, yellow, and green, respectively. We note that the IFD (δ = 0) results shown in Figs. 29a, 29b, 30a and 30b coincide with the corresponding plots in Fig. 9 of Ref. [12]. The results for the case ofm = 0.045 ex-   hibit the similar features that we discussed for the case ofm = 0 in Sec. VI B. Namely, the large-momentum IFD (δ = 0) numerical results approach to the LFD results quite slowly [22]       For the case ofm = 1.0 which we noted above its straight line profile for θ f (p − ) in Fig. 28, the numeri-    Table VI) is the ground state meson mass for the quark mass valuem = 1.0. The essential features that we discussed for the lower masses (m = 0, 0.045, 0.18) in Sec. VI B and above remain without much change. As mentioned earlier, we haven't increased the number of grid points beyond 600 while the numerical accuracy is much more demanded as C gets close to zero and r− gets large. However, we are not alarmed by the appearance of "rabbit ear" for δ = 0.78 and r− = 5M 1.0 = 13.5 in Fig. 31e as we are convinced from our numerical analyses that such numerical noise would disappear as we keep pushing the number of grid points even higher.     θ(p − ) solution gets close to the free mass gap solution θ f (p − ) as shown in Fig. 28, which may indicate that the binding effect gets lesser while the quark mass effect gets larger. In fact, the extreme heavy quark mass limit would yield the non-relativistic δ-function type of ground-state meson wavefunction peaked highly at x = 1/2 to share the longitudinal momentum equally between the two equal mass quark and antiquark. In Fig. 33, we see a kind of precursor for such tendency toward the heavy quarkantiquark bound-state system. In the case ofm = 2.11, we take our frames as r− = 0.2M 2.11 , M 2.11 and 2M 2.11 , where M 2.11 = 4.91 (see Table VI). As 2M 2.11 is already large enough for our numerical computation, we do not go beyond r− = 2M 2.11 . Besides the tendency toward the heavy quark-antiquark bound-state system, the essential features that we discussed previously including the charge conjugation symmetry under the exchange of x ↔ 1 − x for the ground state and the first excited state appear similar in Figs. 33 and 34. We notice some wiggles in theφ − component of the wavefunction solution in e.g. Fig. 33f, but the overall magnitude of theφ − wavefunction is always negligible compared toφ + whenever this occurs, thus it does not cause concern to us.
Appendix E: "Quasi-PDFs" corresponding to mesonic wavefunctions for m = 0.045, 1.0 and 2.11 in the unit of √ 2λ Starting from the definition of the "quasi-PDFs" interpolating between IFD and LFD given by Eq. (129), we obtained the "quasi-PDFs" using the mesonic wavefunc-tions of the quark-antiquark bound-states in the interpolating axial gauge, A â − = 0, as given by Eq. (132) and discussed the "quasi-PDFs" for the cases of m = 0 and m = 0.18 in Sec. VI C. As the mesonic wavefunctions for m = 0.045, 1.0 and 2.11 were presented in the previous appendix, Appendix D, we now discuss the corresponding "quasi-PDFs" in this Appendix E.
First, the δ = 0 (IFD) results of the ground state and first excited state mesonic quasi-PDFs are shown in Figs. 35 and 36, respectively, for a few different quark       Table VI) is the ground state meson mass for the quark mass value m = 0.045, are depicted by the solid lines in blue, yellow, and green, respectively. As noted in Ref. [22], the large-momentum IFD (δ = 0) numerical results approach to the LFD results quite slowly as the momentum r− = r 1 gets large. i.e. taking δ away from the IFD (δ = 0), the resemblance to the PDFs in LFD may appear more swiftly achieved with the boost of the meson longitudinal momentum r− to the larger value. Nevertheless, one should note here a numerical caveat demanding much higher numerical accuracy as the meson momentum r− gets larger while the δ value gets close to π/4 (e.g. δ = 0.78). In such situation, the numerical sensitivity kicks in so strongly that the results cannot be trusted unless they get tested for the improvement with much higher numerical accuracy. However, as discussed in Sec. VI C, one doesn't need to boost the longitudinal momentum r− too large if the δ value gets close to π/4. As the δ value gets close to π/4, relatively smaller r− value can do the job. Thus, it would be worthwhile to search for a "sweet spot" by varying both δ and r− together to obtain the "LFD-like" result.
where one should note the caveat of n = 0 solution for m = 0 that becomes null in the rest frame due to the relativity.
In the frame r− = 0, the meson mass square M 2 (n) = (r+ (n) ) 2 C and the corresponding wavefunctionsφ (n) ± (p−) are obtained by solving the coupled bound-state equations, Eqs.(F1a) and (F1b), using essentially the same technique that we described in Sec.VI A.
Due to the GOR relation M 2 (0) ∼ m √ λ → 0, the ground-state meson mass M (0) → 0 as m → 0. As mentioned earlier, the massless Goldstone boson then moves with the speed of light and can't exist in the rest frame according to the relativity. How can one understand this distinction of the massless Goldstone boson in the rest frame? To realize it, one may take a look more closely the analytic solution of the ground-state (n = 0) wavefunction (φ          Fig. 41. Here, we use the variable of the horizontal axis ξ = tan −1 (p−). In Fig. 41, the interpolation angle δ = 0, 0.6, and 0.78 results are depicted in solid, dashed, and dotted lines, respectively.
In contrast to the ground state, the excited states (n = 0) for m = 0 acquire the non-zero bound-state mass as shown in Table VI      For the non-zero mass cases, we plot n = 0, 1, and 2 together for the rest frame wavefunctionsφ For m = 0.045, although we have compared our results for the moving frames shown in Figs. 29 and 30 with the corresponding results in Ref. [12] as discussed in Appendix D, we couldn't compare our results shown in Fig. 43 with Ref. [12] as the rest frame was not considered in Ref. [12]. However, the results shown in Fig. 43 are not much different from the corresponding results for m = 0 discussed above as the difference in the mass m is rather marginal.
For m = 0.18, 1.0 and 2.11, all of our δ = 0 results shown in Figs. 44, 45, and 46 can be respectively compared with the corresponding rest frame results shown in Figs. 7, 8, 9 and 10 of Ref. [11]. Although the normalization of the bound-state wavefunction was mistaken in Ref. [11] by taking the + sign between |φ