Light Meson Mass Spectra with Massive Quarks

We investigate light meson mass spectra with massive u, d, and s quarks and with a spin effect under a bound system in 3 + 1 dimensional QCD by using the first order perturbation correction. In the process of determining charged kaon and neutral kaonmasses, we obtain masses of u, d, and s quarks that are slightly smaller than the currently accepted values. Using these masses, we obtain light meson mass spectra that includes mass splitting of charged and neutral kaons and ρ mesons. The most interesting of our results is that the pion mass remains unchanged even though u, d, and s quarks become massive.


Introduction
It is well understood that properly explaining meson properties such as mass spectra, decay constants, and the pion form factor (the pion wave function in momentum space) is fundamental to understanding hadron physics because a meson is the simplest composite particle system. It is especially important to explain meson properties within a quantum chromodynamics (QCD) framework because it is widely accepted that the interactions between quarks (or antiquarks) are described by QCD. Currently, there are two main approaches to explaining meson properties. One is based on the consideration that covariance should be the first priority when describing mesons. The other approach of describing mesons without setting the covariance as the first priority was developed mainly to investigate mass spectra. The mass spectra and pion wave functions in momentum space (pion form factor) resulting from these two descriptions are different. Although it is well-known that the former description cannot explain the et al. [5]. By contrast, the author [6] showed the existence of zeromass simultaneously with a nonzero mass spectrum, which is consistent with the results of t'Hooft. For a pion wave function in momentum space, it is also well known that the former description gives Reggae-like profile functions. For example, inMello et al. [7], a pion is described by three simple pole solutions in r space (Reggae-like functions in momentum space),whereas in the latter case, the description for satisfying light meson mass spectra (including a pion mass spectrum) is Gaussian in momentum space, as mentioned before. In addition, for considering the pion electromagnetic form factor, the following three methods are proposed.
The first is the covariant spectator theory (CST) by Biernat et al., but this wave function is Reggae-like [8]. The second is Dyson-Schwinger equation method, but this wave function is not Gaussian either, as shown in Chang et al. [9]. In addition, according to Arrington et al. [10], the pion valence quark distribu- . Here x is a light-front fraction of the system's total momentum at resolving scale ζ . The apparent β exponent can range between ~1 and ~2. 5 . Thus , this DFs derive Reggae-like wave functions. The third is the Drell-Yan frame shown by Li et al. [11]. The Drell-Yan frame restores dynamical covariance but does not include zero-mode contribution which needs to fit to the pion electromagnetic form factor. To include zero-mode contribution lose the dynamical covariance.
The most recent experimental results by the Jefferson Lab Hall A collaboration [12] show that the t-dependence of the cross section, usually parametrized by Reggae-like profile functions, is no longer valid at typical values of descriptions give satisfactory results, as shown in Aoki et al. [13] for the former (although the wave function is not Gaussian) and in Ref [2] for the latter (a Gaussian description). Based on these comparisons, the Gaussian description seems to be better for mesons, or at least for light mesons. However, the Gaussian description has not been given a global physics rule like covariance for the former description. This problem should be considered seriously. Fitting of the light meson mass spectra in Ref. [3] is not good enough. Instead, these results are taken for granted because these are obtained under the chiral limit condition, which sets the masses of u, d, and s quarks at zero without considering the spin effect for vector mesons. In this paper, we investigate light meson mass spectra under the conditions that u, d, and s quarks are massive and that vector mesons have a spin effect.

Formulation
We previously showed the chiral limit of light meson mass spectra [3]. Here we extend our method to the non-chiral limit case in which the masses of u, d, and s quarks are non-zero and there is a spin effect for vector mesons. To do this, we recall the Dirac equation in QCD with a mass term. The Dirac equation is expressed as k Ak The Dirac equation of the complex conjugate † q becomes as the following. † † † k Ak The 2 a λ components are generators of the adjoint representation of the color gauge group.
We employ the metric system and γ matrices as follows, according to Weinberg [14]. 1, where 0 σ is a unit matrix of a 2 × 2 matrix and k σ is the 2 × 2 Pauli-matrix specified by (k = 1, 2, 3) First, we briefly describe our formalism and the equation of motion we obtained previously [3]. Suura [15] (3) is not dependent of gauge fields explicitly, as shown in Ref. [17]. In this sense, the defined operator is gauge invariant but path dependent. Because the physical properties of an observable color singlet should be the path independent, Suura chose a straight line for the zeroth order [15].
We also adopt this choice to investigate chiral limit light meson mass splitting [3]. For the chiral limit case, the starting equation of motion is the following as given in Ref. [3].
Thus, for the non-chiral limit case, from Equation (1) and Equation (2) Here we adopt the center of mass of the system and relative coordinates as where ( ) ( ) 2 , 1 r r   denote the point 2 and 1, respectively.
In the relative coordinates and in the rest frame, we obtain the kinetic term as follows.
T. Kurai Then Equation (6) is expressed in relative coordinate as below. i q t r i r q t r m q t r q t r m t g z q t r z r z q t z Except for the mass terms, Equation (11) was previously obtained in Ref. [3].
We decompose ( ) q r to a Lorentz invariant description as follows.
The following kinetic terms are derived after sandwiching  For an evaluation leading to the electric terms, we follow the argument in Ref. [3]. After sandwiching it with the vacuum state 0 and the physical state P , the electric term becomes as follows.
In Ref. [3] we showed that the Hermitian conjugate of ( ) 1, 2 q in relative coordinates, i.e., ( ) ; q t r , is equal to taking ˆr r → − . The decomposition of the Hermitian conjugate of ( ) ; q t r q t r i r q t r q t r i r q t r T  ; ; 0 r g P z q t z i r q t z q t z i r q t z r z q t r z i r q t r z q t r z i r q t r z Here, we consider the contribution of each component. As an example, we show the detailed calculation of the β component. ; ; ; 0 d ; 0 0 ; 0 2 r r r g P z r z q t z q t r z q t z q t r z q t z q t r z q t z q t r z g P z r z q t r z q t z q t r z q t z q t r z q t z q t r z q t z g z P q t r z q t z In the second line of Equation (16), we commute fields because these are scalar quantities. In the third line, we insert a 0 0 term. Rigorously, this should be 1 n n

= ∑
where n denotes all states including the vacuum state. The expression ( ) 0 q r P represents a real meson as a bound system but ( ) ( ) 0 n q r P n ≠ represents an unbound state such as q q − jet state (refer Ref. [3]). Thus we neglect all states except the vacuum state given by the 0 0 term.
In the fourth line, we use the condition from Ref.
The reason of this choice of conditions is given in Ref. [17].
obtain the electric term as follows.
Unit matrix component: Here, we use the definition of the amplitude given in Equation (3).
For the magnetic term, using the same argument as the electric term, we obtain: Unit matrix component: Β component: Remembering that we work in the center of mass of the system and in the rest frame in relative coordinates, Then, a time integral is carried out as shown in Ref. [3] and we obtain in the following.
In the case of the mass term, from Equation (11), after sandwiching ( ) ; q t r ν by the vacuum state 0 and physical state P , each component becomes the following.
If we consider that these terms describe the mass terms of a meson particle, those of an anti-meson particle would be described as follows, because Remembering that the masses of a particle and an anti-particle are the same, the mean value must be as below.
( ) 1 mass of particle mass of antiparticle 2 + Then, the actual mass terms become the following.
, the equations of motion for the wave functions become as follows.
Here we introduce new notation for 0 P .
The corresponding eigenvalue is the following, as given in Ref. [3].
where 2 χ κ is a positive half integer.
The corresponding eigenvalue is the following.

Evaluation
In this section, we show how to perform the first order perturbation. Before proceeding with this argument, we insist on the fact that pion solutions are un- Equation (53) Using the following formula from Ref. [18] for the derivative of the confluent    r P' r g L r r P m m P P' g L P' g L ' For the first term, changing the variable to For the second term, changing the variable to Here we use the fact that ( ) ( ) Remembering the fact that Notice that the factor is cancelled out so that the dimension of ( ) ( ) Note that z' is dimensionless. From Equation (52), the difference of integral of Here, we use the notation z instead of z'. We use this integral notation from here on. The essential integral in the denominator of Equation (45) is expressed as below.
Then, the dimension of ( ) ( )( ) From this description, we can notice the obvious fact that the main contribution of ( ) 3 1 J χ comes from the first term because the integral appears to have a singularity. Thus, we must evaluate this integral carefully.
To perform the integration of the first term, we modify it as follows.
After changing the variable with z z′ = − , the first term becomes as follows.
The second term becomes the following.
( ) Thus, integral 1 T is expressed as below.
Notice that Equation (68) does not have singularity. When is sufficiently large, 1 T becomes as follows.
The derivation of this form is shown in Appendix B.

T. Kurai
In this paper, we use this expression for the calculation of correction masses except for the case of a kaon, an 0 f meson and anη meson. For the kaon and T χ , we use the following integral formula [19].
Then, the second term of Equation (64) is expressed as follows.
We neglect the ( ) 3 2 T χ terms because they are small compared to the 1 T term multiplied by

Results
To determine masses of u, d, and s quarks, we use the evaluation of the masses of for 2 χ . The reasoning for this is that 0 f meson appears always appears with a kaon and that kaon mass is close to 0 f meson mass. Table 1 shows the obtained masses of u, d, and s quarks in the process of kaon mass evaluation. To evaluate each mass term, we use quark and antiquark constitu-  To do this we define the spin contribution as below.
where Q and Q denote the charges of a quark and an antiquark, respectively.
In addition, s α is a spin parameter, q m and q m denote masses of a quark and an antiquark, respectively. This description of spin contribution is based on that of Choi et al. [2].
The total perturbative energy ((mass) 2 correction))can then be described as below. Notice that we use the fact that the quark and antiquark masses are equivalent for all cases. This yields the results in Table 2 for the pseudo-scaler case and in Table 3 for the vector meson case.
Here, the estimated mass est M is obtained by following equation. Recall that these eigenvalues are the same in Ref. [3]. The calculation is performed in the GeV region. Our corrected light meson mass spectra is better Table 2. Pseudo-scalar mesons.

Conclusion
We obtain plausible light meson mass spectra by invoking the masses of u, d, and s quarks and the contribution of a spin effect. There is a discrepancy between our values and those of the Particle data group values. However, the Particle data group uses a Lattice QCD approach that shows a Reggae-like function (for examples, see Ref. [7]). As mentioned in the Introduction, the Jefferson Lab Hall A collaboration showed that a Gaussian function is better for fitting to experimental data. Because our estimation is based on a Gaussian-like wave function, our values are still meaningful despite this discrepancy. We consider nonzero quark masses and a spin effect as perturbative corrections of the chiral limit mass spectra given in Ref. [3]. By invoking the masses of u, d, and s quarks, we can obtain the mass difference between charged kaons κ κ as well as the mass difference between charged ρ mesons ± ρ and neutral ρ mesons 0 ρ . The significant point is that the pion mass is unchanged even though quarks become massive. The corresponding pion wave functions are unchanged for 3 2 , χ χ and 1 χ , but 0 χ is no longer zero because it is expressed as . We then notice the following interesting correspondence to the results from lattice QCD approach. Broniowski et al. [22] showed in NJL model that the pseudo-scaler wave function of pions corresponding part of a pion wave function is Gaussian as shown in Ref. [11] (Gaussian in r space is Gaussian in momentum space), Our pion wave function is very plausible. We next argue for the value of our proposition stated in the Introduction that the first priority governing meson evaluation should be the use of a gauge invariance system instead of a covariance system.

Discussion
We estimate the masses of η and ′ η and those of ρ and ω independently. Usually, the assumption of a mixed state is used to obtain those masses, as in Ref. [2]. Instead, we estimate the masses from different chiral limit masses and calculate each of them within a closed process. However, from the view point of quark contents, η and ′ η and also 0 ρ and 0 ω are not linearly independent. Precisely, as quark contents, η and ′ η are usually described as  Table 2 and Table 3 show that η and ′ η and also 0 ρ and 0 ω are not linearly independent. At this time, we cannot interpret the meaning of these results. Also, as previously mentioned in Results, the mass of a pion is unchanged so we cannot obtain the mass difference between ± π and 0 π . We consider that the investigation of mass difference between ± π and 0 π might address the question of why a zero mass meson is still unobserved. In other words, we consider that this might clarify whether a pion is a Goldstone boson or not. We can at least say, from our previous results in Ref. [3] and this paper, that a pion is unique among mesons because it has a singularity in its wave function and its mass is unchanged even though its constituent quarks become massive.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this pa-  3  2  2  2  1  2  1  2  3  3  1  2  3  1  2  3  1  2  3   3  2  x x x x x x x r x r x r r r r Together with the second term of Equation (A4), the lower term of Equation (A5) becomes as below. e 1, ; For the last line, we use the fact that ( ) Note that Equation (67) is obtained by multiplying this form by a factor 2.
To obtain an approximation form of Equation (67), we use the following integral representation form of a confluent hypergeometric series [19].

Appendix C. Evaluation of Mass of κ ± and 0 κ
The nonperturbative wave function of a kaon is expressed as follows: ( ) ( )