Partial wave analysis for the in-hadron condensate

In-hadron condensates, defined as the scalar form factors at zero-momentum transfer, are investigated for flavor-symmetric mesons in pseudoscalar and vector channels under the rainbow-ladder truncation within the Dyson-Schwinger equations framework. We confirm the efficiency of the in-hadron condensates in describing the effects of dynamical chiral symmetry breaking from both global and structural perspectives by comparing the meson masses, the dimensionless in-hadron condensates, and the partial wave decompositions of in-hadron condensates as functions of current-quark mass. From partial wave analysis, we infer $\pi(1300)$ is a radial excitation dominated by $s$ waves and $\rho(1450)$ is not a $p$ wave-dominated excitation. This work provides a new insight into the studies of hadron properties with partial wave analysis for the in-hadron condensates.


I. INTRODUCTION
The chiral symmetry and its breaking are essential in quantum chromo-dynamics (QCD) and play an important role in hadron physics [1][2][3][4]. Due to the non-vanishing current-quark mass, the chiral symmetry is broken explicitly. The current mass of u/d quark is 3-5 MeV.
However, the constitute mass of u/d quark inside the nucleon is 300-500 MeV. It is believed that this large mass gap stems from the so-called dynamical chiral symmetry breaking [5,6].
With the DSEs approach, the effects of dynamical chiral symmetry breaking can be reflected in observable quantities such as quark mass function [29,30], sigma-term [31,32], and hadron decay constants [33]. The vacuum quark condensate is the order parameter of dynamical chiral symmetry breaking, which can measure the effects more directly [34,35].
However, the vacuum quark condensate is not well defined since it badly diverges at finite current-quark mass [35,36]. Besides, quarks are confined inside hadrons. The confinement effect is not considered in vacuum quark condensate, which might be related to the large cosmological constant [37,38].
In Refs. [33,34], the in-hadron condensate is first proposed from the Gell-Mann-Oakes-Renner (GMOR) relation [39]. This in-hadron order parameter of dynamical chiral symmetry breaking is finite at non-vanishing current-quark mass and might reduce the cosmological constant to the observed value [40]. However, this original definition is only valid in the pseudoscalar channel and is hard to be extended to other channels. In Ref. [41], the scalar form factor at zero-momentum transfer is proved as an equivalent quantity to measure the effects of dynamical chiral symmetry breaking, which can be easily extended to all hadrons [32].
Besides, it can also describe the responses of hadron masses to the current-quark mass, which can help to place constraints on the fundamental constants of nature by using observational data [42].
The scalar form factor at zero-momentum transfer, named in-hadron condensate hereafter, can not only measure the global effects of dynamical chiral symmetry breaking, but also provide a structural perspective for the effects of dynamical chiral symmetry breaking.
It is known that hadrons have rich and complicated Dirac tensor structures constrained by Poincaré covariance [43][44][45]. By organizing the Dirac tensors with respect to spin and orbital angular momentum of quarks, partial wave analysis can be used to distinguish the contributions of corresponding partial waves to the in-hadron condensates [46,47].
In this work, we solve the gap equation for quark propagator, the homogeneous Bethe-Salpeter (BS) equation for BS amplitude, and the inhomogeneous BS equation for scalar vertex within the DSEs framework under the rainbow-ladder (RL) truncation scheme. To measure the global effects of dynamical chiral symmetry breaking, the hadron masses and the in-hadron condensates are calculated for the ground and first-excited states of flavorsymmetric mesons in pseudoscalar and vector channels. Afterwards, the partial wave analyses are performed to the in-hadron condensates to provide a structural perspective for the effects of dynamical chiral symmetry breaking. This paper is organized as follows. In Sec. II and III , the definition of in-hadron condensate and the theoretical framework of DSEs approach are introduced respectively. In Sec. IV, the calculated results and discussions are presented. Finally, a summary is given in Sec. V.

II. QUARK CONDENSATES: FROM VACUUM TO IN-HADRON
In the chiral limit, the vacuum quark condensate is defined as where Sm =0 (q) is the quark propagator in the chiral limit with q the momentum and Z 4 is the corresponding renormalization constant.
Λ q represents a Poincaré invariant regularization of the four-dimensional integral with Λ the regularization mass-scale. The trace is over color and Dirac space.
Many successes have been achieved in describing the effects of dynamical chiral symmetry breaking by using the vacuum quark condensate. Nevertheless, it suffers from some problems. For example, the vacuum quark condensate is badly divergent for finite current-quark mass. Although this divergence can be partly eliminated by using subtraction schemes, ambiguities still exist especially at large current-quark mass [1,28,36,[48][49][50][51]. Besides, the cosmological constant derived from the vacuum quark condensate is about 10 46 times larger than the observed value [40]. It is argued that the quark condensate can be modified by the strong interaction inside the hadrons, where the vacuum quark condensate fails to provide a reasonable and realistic measure [38].
To settle these problems, it is natural to generalize the definition of quark condensate from vacuum to in-hadron. As pion is the lightest hadron and the Goldstone boson associated with dynamical chiral symmetry breaking, the in-pion condensate is first investigated. In Ref. [33], the Gell-Mann-Oakes-Renner (GMOR) relation [39] is reconsidered, which connects the pion decay constant f π and mass M π with the vacuum quark condensate where m u/d is the current-quark mass. f π is related to 0|qγ 5 γ µ q|Π with | Π the pion state.
Alternatively, the quantity in the left hand side of Eq. (2) can be obtained with the decay constant ρ π by equating pole terms of the corresponding axial and pseudoscalar vertices in axial-vector Ward Takahashi identity (AV-WTI) [33], where ρ π is defined as iρ π = − 0|qiγ 5 q|Π .
This definition is valid for non-vanishing current-quark mass and can return to the vacuum quark condensate in the chiral limit as long as the chiral limit residue of the bound state pole in the pseudoscalar vertex is defined as [33] ρ 0 where f 0 π is the value of f π in the chiral limit. The in-pion condensate has been used to measure the effects of dynamical chiral symmetry breaking inside the pion [33,34,37]. This definition in Eq. (4) is valid only for pseudoscalar mesons and is hard to be extended to other hadrons [34].
In Ref. [41], it is proved that the in-pion condensate can also be represented through the scalar form factor at zero-momentum transfer Q 2 = 0 Besides, the scalar form factor at zero-momentum transfer can be extended to other channels to measure the effects of dynamical chiral symmetry breaking inside the corresponding hadrons. Generally, the scalar form factor at zero-momentum transfer for the flavorsymmetric meson can be represented with the expectation value of the operatorqq in the where P is the meson momentum and is related to the meson mass M P via the on-shell condition P 2 = −M 2 P . Constrained by the AV-WTI, the scalar form factor is related to the hadron mass M P and the current-quark mass m q by [32,52] This relation can be viewed as a consequence of the Feynman-Hellmann theorem [53]. It is found that the dimensionless quantity S P (0)/M P is equivalent to the response of hadron mass to the current-quark mass, therefore it is a measure of the global effects of dynamical chiral symmetry breaking.
Within the DSEs framework, the quantity S P (0) can also be calculated from the loop integration where S(q ± ) are the renormalized dressed quark propagators with momentum q ± = q ± 1 2 P . Γ P (q; P ) represents the BS amplitude of the meson and Γ s (q + ; 0) is the scalar vertex with zero momentum. The corresponding Feynman diagram is shown in Fig. 1. It is obvious that Eq. (9) can be used to analyze the partial wave contributions of the BS amplitudes to the in-hadron condensates, which makes it feasible to probe the effects of dynamical chiral symmetry breaking from a structural perspective.

III. DYSON-SCHWINGER EQUATION APPROACH
The equations of motion for Green functions of QCD fields are described by a set of infinite coupled equations, i.e., DSEs. The accurate solutions of the DSEs are impossible to obtain. To solve the DSEs in practice, the truncation schemes as well as the interaction models are necessary. In this work, the RL truncation in Ref. [54,55] and a simplified version of the interaction in Ref. [56] are employed to solve the quark propagator S(q ± ), the BS amplitude Γ P (q; P ), and the scalar vertex Γ s (q + ; 0) in Eq. (9).

A. The quark propagator
The Dyson-Schwinger equation for quark propagator is known as the gap equation, which reads [19] where Σ(q) is the self-energy The renormalization constants Z 1 , Z 2 , and Z m correspond to the quark-gluon vertex, quark propagator, and current-quark mass, respectively. λ a 2 is the fundamental representation of SU(3) color symmetry. D µν (l) with l = q − k is the renormalized dressed gluon propagator and Γ ν (k, q) the renormalized dressed quark-gluon vertex. g is the coupling constant. The corresponding Feynman diagram of Eqs. (10) and (11) is shown in Fig. 2. For the dressed gluon propagator and the dressed quark-gluon vertex, the following Ansätz in Ref. [56] is used where the dressed quark-gluon vertex Γ ν (k, q) in the quark self-energy in Eq. (11) is truncated to the tree level γ ν . D free µν (l) = (δ µν − l µ l ν /l 2 ) /l 2 is the free gluon propagator in Landau gauge. The non-perturbative dressing effect is absorbed in the effective interaction function G(l 2 ), which is supposed to compensate for all the missing pieces in the quark-gluon vertex [56,57]. As the non-perturbative features are insensitive to the ultraviolet behavior of the interaction, in this work we adopt a simplified version of the interaction in Ref. [56], where only the infrared part is kept In this case the renormalization procedure can be skipped. In Eq. (13) K αγ,δβ (q, k; P ) is the quark-antiquark scattering kernel with α, β, γ, δ the Dirac and color indices. For this eigen equation, solutions exist only for particular, separated values of P 2 .
The corresponding Feynman diagram is shown in Fig. 3. To solve the homogeneous BS equation, the truncation of the quark-antiquark scattering kernel K is performed with respect to the constraints of AV-WTI [58][59][60]. Considering that the quark-gluon vertex is truncated to the tree level in Eq. (12), here we choose the one-gluon exchange for the scattering kernel The united truncation scheme combining Eq. (12) for the quark-gluon vertex and Eq. (15) for the scattering kernel is the so-called RL truncation [61]. It is the first term in a nonperturbative, systematic, and symmetry-preserving approximation. RL truncation preserves the one-loop renormalization group properties of QCD, and has provided a uniformly accurate description and prediction for a wide range of hadron properties [12,[62][63][64].
For mesons with spin-parity J P , the BS amplitude Γ P (q; P ) can be expanded in the corresponding tensor basis τ i P (q; P ) where F i (q 2 , q · P ) is the scalar coefficient function. N is the dimension of the tensor basis  Tab. I and Tab. II for pseudoscalar and vector mesons respectively [43]. The basis elements are organized with respect to their quark spin S and orbital angular momentum L in the meson's rest frame, where L = 0, 1, 2 correspond to s, p, d waves respectively [47]. In the Dirac bases,P = P/|P | is the unit vector of the meson momentum, q µ ⊥ = q µ − q ·PP µ is the relative momentum transverse to the corresponding meson momentum P . The Dirac matrix γ µ ⊥ = γ µ − γ ·PP µ is also transverse to P .

C. The scalar vertex
The scalar vertex can be obtained by solving the in-homogeneous BS equation. With RL truncation, it reads where D µν is the dressed gluon propagator with Ansätz given in Eq. (12). S(k ± ) are the quark propagator as the solutions of Eqs. (10) and (11). The Feynman diagram is shown in Since the in-hadron condensate is defined as the scalar form factor at zero-momentum transfer Q 2 = 0 in Eqs. (7) and (9), the Dirac basis is reduced to {1 , γ · q}.

IV. RESULTS AND DISCUSSION
p-channel p'-channel r-channel r'-channel To measure the total effects of chiral symmetry breaking including both the dynamical and explicit parts, the masses of the pseudoscalar and vector mesons at ground and firstexcited states are investigated as shown in Fig. 5. It is obvious that the meson masses M P increase monotonically with the increase of the current-quark mass m q . In the heavy quark region shown in the right panel, similar linear tendencies are found for all the channels. At m q 4 GeV, the relative difference between M P and 2m q are much smaller than that at m q 1 GeV for all these channels, which indicates the effects of dynamical chiral symmetry breaking are weaker compared to that of explicit chiral symmetry breaking for heavy quarks.
In the chiral limit shown in the left panel, the mass of the Goldstone boson π is vanished.
With the increase of m q , M π increases approximately proportional to √ m q according to Eq. (2). This nonlinearity indicates the interactions between the quarks inside π are significant, which clearly demonstrates the non-perturbative nature of the light quarks. Besides, in the chiral limit, the vector meson ρ and the excited states π ′ , ρ ′ possess non-vanishing masses, which are generated totally from the dynamical chiral symmetry breaking.
It is expected that the vector meson is heavier than the pseudoscalar meson for both ground and excited states. This is the case for heavy quarks as shown in the right panel in To probe the effects of dynamical chiral symmetry breaking from a structural perspective, the relative partial wave contributions to the in-hadron condensates S It should be noticed that the meson masses, the in-hadron condensates and the partial wave contributions are all obtained with the DSEs approach under the RL truncation, which truncates the quark-gluon vertex and the quark-antiquark scattering kernel to the tree level and preserves the AV-WTI. As the pseudoscalar mesons are strongly constrained by the symmetry embodied in AV-WTI, the calculated properties of the first-excited state in pseudoscalar channel is reliable qualitatively. While for the first-excited state of vector meson, the missing contribution from the higher orders might lead to non-negligible effects on spin-orbital interaction and thus on the partial waves, especially when the current-quark mass is close to the chiral limit. Thus the relative partial wave contributions to in-hadron condensates might be changed beyond the RL truncation.
Attempts have been made beyond the RL truncation [59,[65][66][67][68]. In Ref. [59], the dressedquark anomalous chromomagnetic moment is included and the mass ordering of the firstexcited states in pseudoscalar and vector channels has been corrected. In future work, it is interesting to study the meson mass spectra and the relative partial wave contributions to in-hadron condensates with a more realistic truncation scheme. In future, we plan to extend this analysis with a more realistic scheme beyond the RL truncation.