Universality and Regge-Like Spectroscopy for Orbitally-Excited

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II. THE LIGHT QUARK DYNAMICS AND AUXILIARY FIELDS

We begin with the dynamics of the relativized quark model -1 [12, 32–35] with the spin-dependent interactions ignored. It is given by the spinless Salpeter Hamiltonian

-2

2 2 -3 H0 = pi + mi + V, (2) = Xi 1 q

-4 where pi (p = p = p) is the particle momentum of the 0.8/ (r+ / ) 1 − 2

V quark i, and V the interquarkinteraction given by the usual lin- Coulomb ear conﬁning potential ar plus the short-range color-Coulomb -5

potential ks(r)/r, 0.0 0.5 1.0 1.5 2.0

− r k (r) V = ar s + V . (3) − r 0 FIG. 1: Comparison of the color-Coulomb interaction with the regu- Here ks(r) = 4αs(r)/3, with αs(r) the strong coupling, defined larized Coulomb potential. as the Fourier transformation of the running QCD coupling 2 αs(Q ), which depends on the relative coordinate r = x x | 1 − 2| of the quark 1 and antiquark 2 with the bare masses m1 and m , respectively. The mass m are that of bare quarks, 0.8 corresponds to αs( ) = 0.6 used in Ref. [12]. 2 i ∞ mi=u,d = 3MeV and mi=s = 96MeV, which differs our ap- The deviation produced by the approximation (5), Err = proach from most of the quark models with m the valence ks(r)/r k /(r + λ/Λ) Φ, was listed explicitly in Table I(a) i h − ∞ i quark masses, see Section 5 for discussions. As emphasized for the weighted function of the harmonic oscillator Φ = = in Ref. [36], V0 is a parameter as fundamental and indispens- Rnl(r) and λ 0.1. able as the quark masses and slope of the linear potential a. For the lattice evidences for the interaction (3) in the heavy- TABLE I(a): The averaged deviation Err = ks(r)/r k /(r + λ/Λ) h − ∞ i flavor sector, see [37, 38]. caused by the approximation (5) of the color-Coulomb interaction for k = 0.8,λ = 0.1 and Λ= 0.301GeV. The weighted function is that At short distance (high energy), the coupling αs in (3) de- ∞ pends on energy scale Q along the renormalization group of the harmonic oscillator Φ = Rnl(r) with the harmonic oscillator = equations in a known way [39, 40]. At long distance(or in length aH 0.18. the infrared region), the actual value of αs at a given Q re- l 0123456 lies mainly on experiment [1], and remains to be explored 2 [41–43]. The authors of Ref. [12] use a functional (sum Err(10− ) 2.59 1.93 1.49 1.18 0.96 0.81 0.69 Q2/4s 2 − − − − − − − of e− k ) to mimic the running of αs(Q ) and its possible critical saturation [41, 42, 44, 45] at some critical value αs = 2 2 αs(Q 0)(the infrared fixed point) when Q becomes low To explore the orbitally-excited spectrum of the light and the→ confinement emerges. Written in the position space, mesons, we extendthe analysisin Ref. [46]tothecaseofmas- this functional has the erf form sive strange quark: ms = 96MeV. Following [46, 47], we employ the auxiliary field (AF) method [48–50] to formally en- 3 large problem of the Hamiltonian (2) to a family of Hamiltoni- αs(r) = αk erf(skr), (4) ans parameterized by three auxiliary fields µ1, µ2, ν and solve k=1 X them in the enlarged Hilbert space. The{ eigenvalues} of the where erf(x) is the error function, αk = 0.25, 0.15, 0.2 , Hamiltonian (2) follows from the parameterized energy lev- critical { 3 } sk = 1/2, √10/2, √1000/2 , and αs = k=1 αk = 0.6. els by shrinking the parameterized space back to the original { } B λ A nontrivial IR-fixed point around αs( ) = 0.7 is suggested space. The point is to employ the relation √B = minλ + ∞ P { 2λ 2 } recently with respect to the confinement scale Λ = 345MeV (the minimization achieved when λ = √B > 0) to reformulate [42]. the Hamiltonian (2) as H = minµ ,ν H(µ1,2, ν) , where Lacking adequate knowledge of the strong coupling, we ap- 1,2 2 2 2 2 2 proximate, for simplicity, the color-Coulomb interaction in (3) p + m j µ j a r ν by H(µ1,2, ν) = + + + 2µ 2 2ν 2 j=1  j  4 α (r) k X   s ks(r)  ∞ , (5)  + V ,  (6) 3 r ≃ r + λ/Λ − r 0 which well fits the color-Coulomb interaction in the long- with the auxiliary fields µ , µ , ν being operators quantum- { 1 2 } distance region, as shown in FIG. 1. Here, k = 4αs( )/3 = mechanically. These fields has to be eliminated as the La- ∞ ∞ 3 grange multipliers eventually. One can show that H(µ1,2, ν) is where r∗ is some intermediate distance governed by the aver- equivalent to (2) up to the elimination of (µ1, µ2, ν) through age size of the bound system of the quarks. Choosing r∗ to be the constraints the expectation value x x , one has h| 1 − 2|i ks(r) k = = = 2 + 2 ∞ . (11) δµ j H(µ1,2, ν) 0 µ j µ j,0 p j m j , ⇒ → r ≃ x1 x2 + λ/Λ * + h| − |i δνH(µ , , ν) = 0 = ν ν = a xq x = ar. (7) 1 2 ⇒ → 0 | 1 − 2| In TABLE I(b), the estimations (10) and (11) are checked by averaging both sides of the equations for n = 0(N = L). Assuming that the quantum average of the AFs µi,0 mi One sees that (11) is valid, more accurately, when L is larger. (i = 1, 2), which is the case for the light quarks inh thei≫ ex- 2 cited mesons, for which the averaged momentum p j is large TABLE I(b): The color Coulomb term averaged with the harmonic enough compared to the bare masses m , one canh view,i us- i oscillator Φ= Rnl(r) is compared to the estimations (10) and (11) for ing the Born-Oppenheimer approximation, the average µi,0 Λ= = h i L from 0 to 7, with 0.301 and λ 0.1. The harmonic oscillator = p2 + m2 as slow variables, being the effective dynam- length aH = 0.18, and the unit of the averages is GeV. h i i i icalq mass of the quark i, and thereby treat them as real c- L 01234567 numbers [31]. As such, the relativized Hamiltonian (2) has k (r)/r 0.342 0.247 0.202 0.174 0.155 0.141 0.131 0.122 been reduced to that of nonrelativistic (6) formally. As (6) in- h s i dicated, one can view the quantum average ν0 as the static k ∞ 0.374 0.252 0.202 0.174 0.155 0.141 0.130 0.121 energy of the flux-tube (QCD string) linkingh thei quark 1 and r+λ/Λ 2[31, 47]. For more details of the AF method applied to the D ks( ) E r +∞λ/Λ 0.297 0.223 0.187 0.163 0.147 0.135 0.125 0.118 mesons, see [31, 46, 47, 51]. h| |i In the static systems of quark and antiquark, where the total momentum vanishes (p1+p2 = 0), the Hamiltonian (6) be- With the help of the relation x1 x2 = ν/a in (7),Eq. (9) comes becomes h| − |i

2 2 a 3 µm + ν p µ a 2 µm + ν EN (µ1,2, ν) = N + + H(µ1,2, ν) = + r + √µν 2 2 2µ 2 √µν 2 ! ! 2 2 2 2 k a m1 m2 ks(r) m1 m2 ∞ + + + V0. (12) + + + V0 (8) − ν + aL 2µ1 2µ2 − r 2µ1 2µ2 where aL λa/Λ. This is the quantized energy of (6) in the in which 2p = p p defines the relative momentum p be- enlarged Hilbert≡ space parameterized by the auxiliary fields 1− 2 tween quarks, µ = µ1µ2/µm is the reduced effective mass and and it will give, according to the AF method, the mass spec- µm = µ1 + µ2. trum of the quark-antiquark system considered, provided that Given that the AFs µ and ν are slow variable and thereby EN (µ1,2, ν) is minimized in the space of the auxiliary fields. keep constant effectively, one can diagonalize the first line In Eq. (9), we write the band quantum number of the har- of (8), which is exactly the Hamiltonian of harmonic oscilla- monic oscillator in the form N = n + L, instead of N = 2n + L. tor. For the whole Hamiltonian (8), one can choose the color This is so because when the color Coulomb term ignored a Coulomb term in the second line of (8) as a perturbation. This superfluous (SU(3)) dynamical symmetry enters in the re- approximation applies for high excited states for which the formulated Hamiltonian (8) which is originally absent in the confining force dominates. In the basis of harmonic oscillator Hamiltonian (2) before the AF method applied: r r2. → nLm , the quantized energy EN (µ1,2, ν) = H(µ1,2, ν) nLm of Such a SU(3) symmetry, known to exist in three-dimensional |(8) becomesi then h| |i isotropic harmonic oscillator (see [52, 53]), brings some unphysical ”accidental” degeneracy in the radially motion and should be removed. a 3 µm + ν ks(r) One simple way to remove the above unphysical symmetry EN(µ1,2, ν) = N + + √µν 2 2 − r ! * +N is to go back to the Hamiltonian (2) to consider a one dimen- 2 2 sional problem of a massless quark 1 moving in the force field m1 m2 + + + V0, (9) a x along the radial direction, with x = r/2 the radial co- 2µ 2µ 1 2 ordinate| | of the quark 1 in the CM system. In this case, the where N = n+ L, with n and L the radial quantum number and dynamics simplifies the orbital angular momentum of the bound system, respec- 1 L 2 tively. 2 q Hc H0 = p + + a x . (13) ≡ 2 s x x | | For expectation of the color-Coulomb interaction in (9), we ! estimate it by using (5), giving When Lq = 0 the WKB quantization condition for Eq. (13) gives

x+ x+ ks(r) k k ∞ = ∞ , (10) (n + b)π = pxdx = dx[Mc a x ]. (14) r ≃ r + λ/Λ r∗ + λ/Λ x x − | | * + * + Z − Z − 4

Here, x = Mc/a are two classical turning points given Given that the bare masses mi µi, onecan solveEqs. (16) ± ± ≪ 2 2 2 by the condition Mc = a x , the constant b depends on the and (17). Up to the leading order of ∆m /µm, where ∆m | ±| 2 2 ≡ boundary conditions, and 2Mc = Mn is the mass of the quark- m1 m2 and µm is the sum of two effective masses, the results antiquark system. Up integration of (14), one has (n + b)π = are− 2 Mc /a. Same analysis applies for quark 2 (x = r/2) so that 2 − µm a one can find a quantization condition for the whole quark- µ = = N , 3 2 (21) 4 ν [1 + 2ak /Nν] antiquark system, only that the range of quark motion need ∞ to be halved since x x reflection makes no difference to the meson spectrum.→ One − finds then, by mapping n n/2, µ ∆m2 = m + → µ1 1 2 , (22) 2 2 µm Mn = 2πa(n + 2b). (15) ! ∆ 2 2 µm m This confirms the linear relation M n + L claimed in (1) µ2 = 1 . ∼ 2 − µ2 by simply comparing (15) with the well-known linear relation m ! M2 2πaL that is derived from the rotating string picture 2 J ∝ where Nν (ν + aL) . Here, we always assume quark 1 is [21, 22]. heavier than≡ antiquark 2 if the quark 1 is strange while the The relation M2 n + L has also been suggested by Afonin ∝ quark 2 is nonstrange. et. al. [23, 24] and Bicudo [25]. Experimental evidences in Putting the relations (21)and (22) into (12), one has favor of this relation were given in [7, 24]. We will show, in the following section, that the formal discrepancy of the 2 harmonic-oscillator-like energy (12) with the linear Regge re- 3 3ak 2aN ak E (ν) = + ∞ + ν ∞ N 4 2 1/2 lation (1) can be removed by showing √µν √N in the large 2 Nν ν χ − N ∼  N  ν N limit. m¯ 2  +  ν3χ2 + V .  2 N 0 (23) 2aN III. MASS FORMULA AND QUASI-LINEAR REGGE in which TRAJECTORIES ak χN = 1 + 2 ∞ , As stated earlier, to solve the model (2) with the AF method, Nν 1 one has to minimize the energy (12) in the space of the auxil- m¯ 2 (m2 + m2). (24) iary fields. This amounts to solving simultaneously the three ≡ 2 1 2 constraints ∂AEN (µ1,2, ν) = 0 (A = µ1, µ2, ν), which are ex- Minimizing of the energy (23) by the constraint equation, plicitly δνEN (µ0, ν) = 0, yields 2 2 aN ν µ2 m1 2 2 = 1 , (16) 3 3ak 6aN 4ak ν 16ak aN 2 + ∞ ∞ + ∞ 3 µm − µ 4 2 3/2 3/2 (µν) 1 2 Nν − ν χ − 3 3 ! N Nν ν χN Nν 2 m2 paN ν µ1 2 m¯ 2 ν3χ = 1 , (17) = ak N ν2χ2 , 3 µ 2 2 8 / 3 N (25) (µν) m − µ2 2a ∞ 3 2 − ! N Nν ! aN µ 2k a p = 1 + ∞ , (18) Eq. (25) is nonlinear and quite involved for analytical treat- 3 (ν + a )2 (µν) L ment. What is more involved here is that the knowledgeof the interquark interaction (3) is not complete. Bearing in mind of with k 4αs( )/3, andp ∞ ≡ ∞ this limitation in the interquark interaction (3), we firstly solve aN a(N + 3/2). (19) (25) in the large N limit using the nonperturbative method of ≡ For unflavored meson nn¯ (n stands for u or d quarks) the homotopic analysis (HA) [54], and then extend the Regge- bare quark mass m should be small, much smaller than the like solution obtained thereby to the low-N case by quasi- i linearizing the ensuing mass formula, up to the leading order effective mass µi. Notice that the average interquark distance = of 1/N. l r is about ν/a, one can estimate, for the high excited 2 statesh (iν is large), In the large N limit, we assume ν aN 1, which can be shown by solving (25) numerically (see∼ FIG.≫ 2 and Table II). 2 2 a a rG Taking a/ν 0, Eq. (25) simplifies = 1, (20) → ν2 ∼ (la)2 l ≪ 3 6a2 m¯ 2 N = 8ak 3ν2 , (26) where rG √1/a is the characteristic size of the ground-state 2 − ν4 2 ∞ − 1 ∼ 2aN meson .

mined by the balancing two terms of the potential energy ar and 1/r. This 1 The characteristic size of a meson in the ground state can be roughly deter- gives r2 1/a. G ∼ 5

TABLE II(a): The analytical (solid line) solutions to (25) and numerical (dots) solutions to the auxiliary ﬁeld equations (16) through (18) 4 for ν2 . The deviations between two solutions are also listed in the { }

Numerical Solution fourth row. The accelerating factor h = 0.88. −

Analytical Solution

3 L 0123456

Numerical 0.107 0.483 0.844 1.203 1.562 1.921 2.280

2 2 ν2[GeV2]

Analytical 0.169 0.516 0.869 1.225 1.582 1.940 2.298 1 ν2[GeV2]

Ana. Num. 0.063 0.033 0.025 0.022 0.020 0.019 0.018

0 −

0 2 4 6 8 10

TABLE II(b): The eﬀective masses µ1,µ2 solved numerically from (16) through (17), compared to their{ analytical} values given by (22). FIG. 2: The analytical solution (solid line) (28) to the equation (25) 2 compared to the numerical solution for the L-dependence of ν . L 0123456

µ1 0.483 0.527 0.601 0.670 0.733 0.791 0.846 Num. 2 where Nν ν and χN 1 have been applied. It follows µ2 0.492 0.536 0.609 0.676 0.739 0.797 0.851 from (26),→ by treating the→ mass term as a perturbation, that µ1 0.402 0.506 0.590 0.663 0.728 0.787 0.842 Ana. 2 2 3 2 2 ν = 2aN 2¯m = 2a N + m m , (27) µ2 0.391 0.497 0.582 0.656 0.721 0.781 0.837 − 2 − 1 − 2 ! which agrees qualitatively, in the massless limit (m1 m2 = 0), with the Regge phenomenology: ν2 N = n + L. ∼ ∝ Since νN is solved from (16) through (18) in the rela- Given the solution (27), onecan use the methodof HA [54] tively large-N region, and the approximation (11) for the to solve Eq. (25). The result is (Appendix A) color Coulomb interaction applies better in the large distance regime, the prediction (30), obtained by the quark model (2) 2 4aL 2 4eN νN = 2aN + ak h 2 + 2¯m 1 , (28) combined with AF method, should be more reliable for the ∞ − √2a − − 3 N ! " # high excitedqq ¯ mesons. When N is very large, Nν as well as ν2 2a (N + 3/2), χ 1 and w 2, which leads, by in which N N N Eq.→ (30), to → → ak 2k eN ∞ = ∞ . (29) 2 ≡ a 2N + 3 Mqq¯ = 2νN + 2¯m ak /νN + V0 ,when N 1, (33) N − ∞ ≫ Here, h is the accelerating factor [54] remained to be fixed or empirically. We fix h simply by comparing ν2 in (28) with the 2 numerical solution to (25). The results are shown in FIG. 2 (Mqq¯ V0) 8aN, when N 1, − ∝ ≫ and Table II(a). One sees, quite remarkably, that the solution 1 ν2 to Eq. (25) rises almost linearly with L, both for that of This corresponds to the slope (8a)− for the linear Regge tra- 2 jectory on the (N, (Mqq V ) ) plot that is predicted by the analytical (solid line) and of numerical (dots in FIG. 2). ¯ − 0 Hence, Eqs. (16) through (18) are solved by (22) and (28). relativized quark model [28, 30]. It is to be compared with Putting them into (23) yields the slope 1/(2πa) predicted by the relativistic string model [21, 22]. AN We remark that when N is large Regge linearity stems from Mqq¯ = wN νN + + V0 (30) νN the first and second term in (12), which appears to be of the harmonic-oscillator form: a(N + 3/2) + const, as the most in which νN is given by (28) and quark models with harmonic-oscillator-like confinement pre-

2 dicted. This changes, however, in our model due to the con- 3 3ak 2aN w = + ∞ + (31) straints (16)-(17) of the AF ﬁelds. The solutions (21)and(22) N 4 2 2 Nν ν χ indicate µ = µm/4 √N and ν √N (namely, √µν √N) N ∼ ∼ ∼ 2 in the large N limit. Thus, when N is large the ﬁrst and second ν2 ak 2 2 N √ √ AN = 2¯m χN ∞ (32) terms in (12) scale as 2 2aN N, hence the linear Regge 2aN − 1 + aL/νN 2 ∼   behavior: EN N.   ∝   6

The mass relation (30) is inadequatein the low-N region for The similar relation for the inverse slope (37) is two reasons rigorously. The ﬁrst is obviously that Eqs. (28), 2 2 (21) and (22) are not valid in the low-N region, as seen in 1 1 3k h 2aL m¯ = + ∞ + Table II (a,b), and that the approximation (11) does not apply 8a 1 1 . (42) α′N 2 2N + 3 − 3 − √2aN 2aN in the low-lying states, as roughly shown in Table II (a,b). The " ! # second, more serious, is that the short distance behavior of the When N is very large Eq. (42) tends to a inverse slope: 1 interquark interaction is far from established [37, 38], e.g., limN (α′−N ) = 8a, in consistent with claims in Refs. the running of strong coupling remains unclear [41, 42, 44, [28, 30→∞]. 45]. Thus, to ﬁnd the mass relation for the low-N region, we One sees from(41)and(42) that the slope dependsupon the resort to the approximate linearity of the Regge trajectories dimensional parameters a and upon the dimensionless param- that is established experimentally in meson spectrums [7] to eters k , √a/Λ andm ¯ 2/a weakly (suppressed by N or √N) constraint the prediction (30). By squaring (30), it follows while the∞ intercept depends upon k strongly and also upon that √a/Λ andm ¯ 2/a weakly. Given (41)and(∞ 42), one rewrite (36)

2 in the front of analytical mass formula for light mesons, 2 2 3 m¯ 4eN Mqq¯ V0 = 2awN N + 1 − 2 − a − 3 Mqq¯ = (1 + KN ) 8a [N αN (0)] + V0, (43) " ! − k 4aL + ∞ h 2 + + DN , (34) where p 2 − √2aN ! # 2 2 3ak h 2λ √a m1 + m2 where KN = ∞ 1 + , (44) 4aN − 3 − Λ √ + 2a(2N + 3) 2N 3! A a A 2 D = N + N . N 2 (35) awN 2ν awN 2 N ! 3 k 5λ √a k m¯ 5h 3 αN (0) = + ∞ h 3 + + ∞ − . − 2 2 − Λ √ + 2a 2N + 3 2 2N 3! The N-dependence of Mqq V in (34) is nonlinear for- (45) ¯ − 0 mally when compared to the Chew-Frautschi plot [6]. The The formula (43) is the main result in this work. We see that constraining of (30) and extrapolating it to the relatively low- the flavor dependence enters explicitly through the mass term 2 2 N region can be done by making (34) quasi-linear in N. We (m1 + m2)/a. The following remarks are in order: note firstly that (34) is comparable to the linear Regge trajec- (i) The Hamiltonian (2), H = µ1 + µ2 + V, becomes almost tories (1), provided that the V is small when compared to the independentof the quark masses in the light-light limit m , 0 1 2 → meson scale, V0/Mqq¯ 1. If we rewrite (34) in the form 2 2 0 for which µi = p + m p . The same it true when ≪ | | i → | | 2 L is large since p 2qhas the expectation N in the harmonic α′ Mqq¯ V0 = N αN (0), (36) | | ∼ N − − basis nL . See (21) and (22). The m1,2-dependence of the system| massi is thereby suppressed by N. This accounts for in which the trajectory parameters α and α (0) are, ′N N the asymptotic flavor independence happened in Table II(b) that µ1,2 tend to be same with L increases. 2 2 2 2aN/ν (ii) In spite of assumption aN N 1 in obtaining (43) 1 2 9a N 2ak = aw = + + ∞ , ∝ ≫ 2 N 1 2 (37) and (30) from the Hamiltonian (2), the quasi-linearizing of the α′ 2  3χ Nν  N  N  model prediction (34) makes it applicable in the low-excited     states, thanks to the Regge phenomenology for light mesons.   (iii) The mass formula (43) goes beyond the native predic- 2 3 m¯ 4eN k 4aL tion of the relativized quark model in that it employs merely αN (0) = 1 + ∞ h 2 + + DN , − 2 − a − 3 2 − √2a the large-N asymptotic behaviors of the model spectrum that ! N ! (38) is implied in relativistic quark model. respectively, then one can quasi-linearize (34) by expanding (iv) The ultra-relativistic contributions from QCD string ro- (37) and (38) on1/N. To order of 1/N, the last term DN in tating to the orbital angular momentum and to the energy of (38) becomes meson has not taken into account in (43), which can otherwise enhance the confining parameter a by a factor of 8a/(2πa) = 2 2 m¯ 9¯m k aL 4/π in the high excited states of mesons, which will be dis- DN 1 + ∞ 1 (39) ≃ a − 4aN 2 √2a − cussed in the section 4 and section 5. ! N ! k 2 2 + ∞ ak (7 2h) 4aL + 4¯m (5h 3) , (40) 16a ∞ − − − N IV. NUMERICAL RESULTS AND DISCUSSIONS which leads to, when putting to (38), In this section, we confront the mass formula (43) with 2 3 k 5aL k m¯ the experiments and other approaches. As explained in the αN (0) = + ∞ h 3 + + ∞ (5h 3). (41) − 2 2 − √2a 4aN − text, we are of mainly interest in the orbitally excited states, N ! 7

TABLE III: The members in six families and their linear ﬁt to the observed masses squared in [1]. The squared bracket is used to indicate that ′ ++ η/h family has an abnormal slope in nonstrange sector. The mark EF in the last row for the meson f4 (4 ) indicates that the data comes from the quark model prediction in Ref. [13].

Linear ﬁt Slope 1/(2πα′) Traj. mesons (JPC) Intercept 2 2 2 2 M (GeV ) α′[GeV− ] GeV

+ + b1(1 −), π2(2− ), π/b 0.3770 + 1.199L 0.834 0.314 0.191  + + −  b3(3 −), π4(4− )   ++  ρ(1−−), a2(2 ),  ρ/a  ρ (3 ), a (4++), 0.6328 + 1.120L 0.893 0.565 0.178  3 −− 4 −   ++  ρ5(5−−), a6(6 )    η(0 +), h (1+ ),  − 1 −  [η/h]  η (2 +), h (3+ ), 0.1668 + 1.297L 0.771 0.128 0.206  2 − 3 − −   +  η4(4− )    ω(1 ), f (2++),  −− 2  ω/ f  ω (3 ), f (4++), 0.5877 + 1.115L 0.897 0.527 0.177  3 −− 4 −   ++  ω5(5−−), f6(6 )    K (1 ), K (2+),  ∗ − 2∗  + K∗  K (3 ), K (4 ), 0.7868 + 1.191L 0.840 0.661 0.189  3∗ − 4∗ −    K∗(5−)  5   φ(1 ), f (2++),  −− 2′ φ/ f ′ 0.9838 + 1.325L 0.755 0.742 0.211  ′ ++ EF −  φ3(3−−), f4 (4 )    where our approximation is expected to work best. For this, roughly. we choose six families of light mesons, marked by π/b, ρ/a, We use the mass formula (43), with KN and αN (0) given − η/h, ω/ f , K∗ and φ/ f ′. The members we pick for the tra- by (44) and (45), to map the observed masses in each fam- jectories are always the lightest known states with appropriate ily in Table III, guided globally by corresponding linear fit quantum numbers. In each family, the quantum numbers of P in the Table III. For the family of η/h, with the ideal mixing and C alternate their values across the trajectory provided that Φ(η/h) = √2nn 2¯ss / √6 assumed for the flavor content, L+1 − they are mainly made ofqq ¯ system with the parity P = ( ) we use the mass formula and C = ( )L+S . Furthermore, the J = 0 state in the−π/b- − 1 2 trajectory, which is actually the pion, has been excluded due Mη/h = Mnn¯ + Mss¯, (46) to its abnormally low mass. 3 3 with the masses M and M given by (43) and 2¯m2(= m2 + In Table III, the selected family members are shown explic- nn¯ ss¯ 1 2 = 2 2 itly, together with the linear fit for the observed mass squared m2) 2mud and 2ms, respectively. The results for the optimal M2 v.s. L, with the data taken entirely from the Particle Data parameters (a, Λ,V0) determined are shown in Table IV. Due Group’s (PDG) 2016 Review of Particle Physics [1]. We also to its abnormal nature, we always take the family of η/h to be abnormal in this work. list the corresponding slope α′, intercept and the MS error for 2 Th Exp 2 linear fit defined by χMS = L(ML ML ) /Lmax where the As can seen in theTable IV,the valuesof V0 are all negative, index L runs from 0 (1 for the π/b trajectory)− to the maximal as argued in [55] for the non-relativistic limit of the Bethe- value Lmax of the orbital angularP momentum. From the linear Salpeter equation. Furthermore, apart from the family of η/h fit shown in Table III one sees that the linear relation (1) ap- the model parameters (a, Λ) are taken values around plies for the trajectories of ρ/a and ω/ f for which intercept is 2 about 0.5, but is violated for the π/b, η/h and φ/ f ′ trajecto- a = 0.147 0.005GeV , ries for− which the intercepts are about 0.3, 0.1 and 0.74, ± (47) respectively. The K trajectory can fit− the relation− (1−) very Λ= 0.184 0.072GeV. ∗ ± 8

TABLE IV: The parameters in the mass formula (43) mapping the ex- to that predicted by Veseli and Olsson [30] and the linearly perimental spectrum of the meson families considered, including the confining parameter for the heavy-quarkonia in lattice QCD confining parameter (a), the low-energy cutoff (Λ) averaged and the [38], whereas it is smaller, about 20%, than that in the other vacuum constant V0. The squared bracket indicates that η/h family has an abnormal slope in nonstrange sector. quark models cited. In the end of Section 4 and in the Section 5, we address this issue in details. 2 Traj. h a(GeV ) Λ(GeV) V0(GeV) We list the mass predictions by the formula (43) and the experimentally observed masses in Table VI. As seen there, π/b 0.88 0.153 0.303 0.237 − − an good agreement is achieved between the mass predictions ρ/a 0.48 0.145 0.152 0.200 and the observed data for all families, if considering the spin- − − dependent interaction is ignored in present work. In spite of [η/h] 0.48 0.178 0.239 0.431 deviations about 8% for some states the mass formula (43) is − − confirmed qualitatively at the level of the average deviation ω/ f 0.28 0.146 0.133 0.245 − − less than 5%. The best agreement occurs for the ω/ f trajec-

K∗ 0.08 0.140 0.203 0.103 tory for which the average mass deviation is about 30MeV. − − In Table VII, we list the slopes given by the formula (42), φ/ f ′ 0.18 0.152 0.131 0.073 other calculations and the analysis of the experimental data − − cited. To answer why the determined value of the confining pa- The small fluctuation implies, especially for a, that a and Λ rameter a in (47) is relatively smaller than that in the other keep the same approximately. In contrast, the linear fit in Ta- quark models cited in Table V, we would like emphasize that ble III (the last column) implies an averaged string tension our method to solve the model differs from that in the most (assuming the QCD string picture) quark models in that it requires the light quark mass to be quite small (close to the bare mass). As stated in the introduc- a(Linear fit) = 0.189 0.0135GeV2, (48) tion, ourmass formulais obtainednot only by applyingthe AF ± method to solve the model (2) in the relatively large-N case, with much larger fluctuation than that in (47). We note that but also using the empirical Regge linearity that is confirmed in the above analysis η/h trajectory is excluded as an excep- experimentally in the large-N states. The later purpose is ful- tional case. In this sense, the confining parameter a is almost filled by quasi-linearizing the quark model formula for the universal. A detailed comparison of the result (47) with other mass squared around the high excited states. When mapping predictions in the literatures is shown in Table V. the observed masses, the parameters in our model, guided also by the linear fit in Table III, are mainly fixed so that the behav- TABLE V: Comparison of predictions for the confining parameter 2 ior of the high excited states is highlighted, where Mqq¯ 8aL. a. Here, QM stands for quark model, Rel. for relativistic, SSE for In the most of relativistic quark models cited, however,≃ Spinless Salpeter equation. LGT stands for lattice QCD. the parameter setup crucially relies on the low-lying spec- Reference a (GeV2) Method trum in which the quark masses are heavy, roughly around 200 300MeV. This setup of the quark mass will violate Present work 0.147 Rel. QM + AFM + quasi-linearizing the linearity∼ of Regge trajectories of the low-lying states, provided that no further relativistic treatment similar to that in [30] 0.142 Rel. QM + Bjorken Sum rule [12] is made to the potential V in (3) correspondingly. This [28] 0.30 Rel. QM + Scaling can be shown using the semiclassical approximation in the Section 5. [34] 0.180 Semi-relativistic QM

[12] 0.18 Relativized QM One sees that our Regge-like mass relation (43) for the orbitally-excited light mesons agrees well with the observed [35] 0.192 Semi-relativistic QM mass spectrum. Moreover, the relation has a feature that supports a universality underlying in formation dynamics of the [36] 0.211 SSE + Smooth transition potential light meson in the following sense: [56] 0.191 Rel. Kinematics + Linear + Coulomb (i) The quark mass dependence of the light meson spectroscopy is suppressed doubly bym ¯ 2 and 1/N in the high ex- [57] 0.183 Semi-relativistic QM cited states. (ii) The confining parameter a and the cutoff Λ is nearly [13] 0.24 Rel. pseudopotential QM same for all families of the mesons considered, except for the [16] 0.18 Massive Rel. string η/h trajectory. We remark that Table IV does not indicate flavor- [38] 0.155(19) Unquenched LGT independence of V0 though their dependence on the flavor contents is weak. This is so because V0 in practice accounts for all residual contributions including the averaged spin- One can observe from Table V that the values of a is close dependent interaction which depends spin nature of mesons 9

TABLE VI: The masses (MeV) computed by the formula (43), compared to the observed data from the experiment (PDG) [1]. The mean squared (MS) errors comparing with the observed masses are shown 7

Experiment for each trajectory. 6

Linear Fit

2 2 PC Formula Traj. χ (GeV ) Mesons J Exp. This work 5

MS (2250)

4 + b1 1 − 1229 1228 4

b (2030) 2

3 + M π2 2− 1672 1672 3

π/b MS (1670) + 2 b3 3 − 2030 2003 2 (I = 1) 0.00175 + b (1235)

1 π 4− 2250 2280 1 4 (a) / b +

b5 5 − 2524 0

−− 0 1 2 3 4 5 6 ρ 1−− 775 716 L

a2 2 1318 1324 8

Experiment ρ3 3−− 1689 1706 7 ρ/a MS Linear Fit

++ 6 Formula

a4 4 1995 2009 a (2450) (I = 1) 0.00345 6

(2350)

ρ5 5−− 2330 2270 5

4 2

++ M a6 6 2450 2503 a (2040) 4

(1690)

ρ7 7−− 2715 3

−− 2 + η 0− 548 459 a (1320) 2

1 (b) / a

h 1+ 1170 1233 (770) 1 − 0

0 1 2 3 4 5 6 7 + η/h MS η2 2− 1617 1673 L + (I = 0) 0.01807 h3 3 − 2025 2014

+ Experiment η4 4− 2328 2305

Linear Fit + 6 h5 5 − 2563 Formula

− (2330) 5

4 ω 1−− 783 772

2 ++ h (2025) M f2 2 1275 1315 3

ω 3 1667 1684 (1645) 3 −− 2 ω/ f MS 2 ++ f4 4 2018 1981 (c) / h 1 h (1170) (I = 0) 0.00806 1 ω5 5−− 2250 2239 0 ++ 0 1 2 3 4 5 6 f6 6 2469 2471 L

ω7 7−− 2682 − 8

Experiment

K∗ 1− 892 863 7

Linear Fit + K 2 1426 1430 6 2∗ Formula f (2510)

K∗ MS K∗ 3− 1776 1795 5 3 (2250)

+ 2

(S = 1) 0.01672 K∗ 4 2045 2087 M 4 f (2050) − 4 K 5 2382 2339 3 5∗ − (1670) 3

2 + K∗ 6 2565 6 f (1270)

(d) / f −− 1 2

φ 1−− 1019 986 (782)

++ 0 1 2 3 4 5 6 7 f2′ 2 1525 1538 L φ/ f ′ MS φ3 3−− 1854 1911 (S = 0) 0.02890 ++ f4′ 4 2255 2213 FIG. 3: The mass squared M2 vs. orbital angular momentum L for the nonstrange mesons. Four trajectories are the π/b family (a), the φ5 5−− 2475 −− ρ/a family (b), the η/h family (c) and the ω/ f family (d). 10

TABLE VII: The slopes computed by Eq. (42) where L varies from 0 to 5 and by the linear ﬁt in Table III. Some other typical predictions cited are also listed for comparison.

7 2 Experiment Reference α′(GeV− ) Methods

Linear Fit 6

This work 0.47-0.75 Rel. QM + AFM + Quasi-linearizing Formula K* (2380) 5

5 [13] 0.887/0.839 Rel. QM + Pseudo-Potential 4

2 K* (2045)

4 M . + [16] 0 884 Massive quark Rel. string 3

K* (1780)

[18] 0.88 Massive quark + Rel. string 2

K* (1430)

1 This work 0.832 0.059 Linear ﬁt (data in PDG 2016) (a) K* ± K*(892)

[7] 0.80 0.10 Linear ﬁt (data in PDG 1998) 0 ± 0 1 2 3 4 5 6

eventually in net value. 7 Accepting the above, one infers that the members of the Experiment η/h trajectory should contain exotic components which makes 6 Linear Fit

them exceptional, as seen in Table IV. The usual mixing 5 Formula

( √2nn 2¯ss)/ √6 for η/h trajectory is not adequate for ac- 4 counting− for their abnormal feature. 4 2

(1850)

3 3

2 V. SEMICLASSICAL APPROXIMATION f' (1525)

1 (b) / f' Before arguing why the quark mass mi should be small in (1020)

our model, let us check numerically what value the confin- 0 ing parameter will be equivalent to when the relativistic cor- 0 1 2 3 4 5 rection due to the rotating of QCD string tied to quarks is L taken account into. In the quark model, this correction has been ignored intrinsically by the potential assumption of the interquark interaction. Nevertheless, one can check how the FIG. 4: The mass squared M2 vs. orbital angular momentum L for data mapping of model makes up the deficit by comparing the the strange mesons. Two trajectories are the K∗ family (a) and the slopes between them. Our model predicts the slope 1/(8a) in φ/ f ′ family (b). the high excited limit of (42), while the rotating string model predicts the Regge slope 1/(2πσ), where σ is the string tension. The result a = 0.147 in (47) is equivalent, under the that following correspondence, H 2 p + ar,2L = ar2, (50) 1 1 → | | , (49) 8a 2πσ which yields (when using p = L/r and eliminating r) ←→ | | to the string tension σ = 0.187GeV2, which agrees well with L 1 α′LL(L 1) = = , (51) that in the most of relativistic quark models. ≫ H2 8a While it has been already known in the past [30, 58–61] This is in consistent with (42). we would like to reemphasize the connection between the lin- It is of heuristic to ”derive” the Regge relation (51) with the ear confinement, linear trajectories and relativistic dynamics help of Bohr-like argument for Hydrogen atom. From (50) [62], which is helpful to understand the results in Table II one has, for the large-L state 0L , (a,b), Table V and VI. Firstly, we note that the leading Regge | i 2 slope follows from the correspondence (classical) limit. Let H L = 2a r L, 2L = a r L, (52) us consider the radially-lowest state of (2) but with a given h i h i h i with n = 0 assumed. Usage of (27) yields r L = νN /a = large L which corresponds to the circular orbital motion of h i quark at large r and large p. The minimal energy condition √2L/a. It follows that 1 1 2 (∂H/∂r) L 0 for H in (2) implies that L p [µ− + µ− ] = ar , 1 2 2 L | ≡ | | H L = √8aL, r L , (53) 2 2 with µi = p + m . If one takes m , 0 (the light-light h i h i ∼ a | | i 1 2 → 2 2 2 limit), thenqµ , p and p L /r (pr 1 ). It follows as required to have (51). 1 2 →| | | | → ≪ 11

a2r2 ν + min + , (56) ν 2ν 2 1.0 ( )

F (2m/M) d where the color-Coulomb term and V0 are ignored for simplic- 0.8 ity. Transforming to the center-of-mass system, one has

2 2 2

0.6 m hl p µ a 2 µ2 + ν q H mQ = + r + + , (57) − 2µ 2 µν 2 2µ2 (2m/M)

d ! F 0.4 with µ = µ2/(1 + µ2/mQ) the reduced mass of the quarks. The similar analysis as in section 2 yields

0.2 2 aN µ2 + ν mq EN (µ2, ν) = + + . (58) 0.0 √µν 2 2µ2 0.0 0.2 0.4 0.6 0.8 1.0

2m/M The minimization of (58) with respect to µ2 and ν gives 1/3 1 1/3 2 1/3 4 1/3 µ2 = bN [1 3 bN /mQ] and aN = bN 1 3 bN /mQ with 2 − − FIG. 5: Fd as a function of 2m/M. bN a /ν. Assuming mQ 1, one can show ≡ N ≫ h i 2 2/3 ν = aN = (aNν) = µν, 1/3 On the other hand, if we assume m1,2/M is not small, with µ2 = bN = ν, M the mass of the quark-antiquark system, a similar analysis that leads to (14) yields (in the case that m1,2 = m, L = 0) a which enables us to rewrite (58) as WKB quantization condition 2 3 mq (M m)/a c EN mQ = 2 a N + + , (59) − 2 2 (n + b′)π = dx (Mc a x ) m . − s 2 2 √aN (M m)/a − | | − ! Z− c− p or equivalently, as the linear Regge relation (E m )2 = with M = M/2. It follows that N Q c 4a (N + 3/2) in the heavy-light limit. The later relation− has an inverse slope 4a, being a half of that of the light-meson 2 2m 2aπ(n + 2b′) = M Fd , (54) trajectories at light-light limit: α′ = 2α′ . This feature has M HL LL ! been pointed out in Refs. [14, 18, 30, 63]. Comparing with the case of the light mesons, the above argument using the AF + √ 2 method differs in that only one auxiliary field (µ2) is intro- 2 2 1 1 τ Fd(τ) = √1 τ τ ln − (55) duced for the kinematic of light quark, with the heavy quark − − τ   treated nonrelativistically. This is in consistent with the ob-     servation [14] that linear Regge trajectories result from light where the procedure n n/2 was used to remove the reflection (x x) degeneration→ (double counting of the radial quark kinematics and linear confinement. space of→ quark − motion). One sees from FIG. 5 that the deviation from the linear Regge trajectory, given by 1 F (2m/M), d VI. SUMMARY AND CONCLUDING REMARKS canbeupto25%for m = 220MeV and 40% for m− = 300MeV ++ if choosing M to be mass of a2(1318) 2 . This explains why the quark mass mi should be small in our model, compared to Up to date, mesons remain to be the ideal subjects for the the most of the quark models cited in Table V. study of strong interactions in the strongly coupled regime. Itisof interestto notethatthe methodin Section2 and3can Even though we have a theory of the strong interactions be extended to the case of heavy-light mesons which consist (QCD), we still know very few about the physical states of of a heavy quark and a light antiquark, though this is not the the theory which are crucial to understand QCD eventually. main issue in this work. Starting again from the Hamiltonian To a large extent our knowledge of hadron physics relies on (2), with m1 = mQ being the heavy quark mass and m2 = mq phenomenologicalmodels, for instance, the quark models and the mass of light antiquark, and assuming mQ to be heavy, one others. Though successful, the quark models manifest them- has, for the Hamiltonian of heavy-light mesons, selves in various forms, and their predictions can differ appre- ciably [4], in particular, for the excited states. So it entails 2 constraining of the model predictions by experiment in the hl p 2 2 H = mQ + + p + mq + ar 2mQ case of the excited states. q For the excited mesons, which can be generated abundantly, 2 p2 + m2 p q µ2 the issues such as whether non-conventional states (exotics) = mQ + + min + 2mQ µ2  2µ2 2  emergebecomesof great interest in the light sector of hadrons.     12

However, to have hope of distinguishing between conven- APPENDIX A tional and exotic mesons, it is crucial for us to understand conventional meson spectroscopy well [4, 5]. Great efforts In order to solve (25) using the method of homotopic anal- have been made to describe light meson spectrum [9–13] in ysis (HA) [54], we rewrite two nonlinear equations (26) with 2 an universal way, in which model parameters are assumed to m¯ = 0 and (25) in the form of Lν(q) = 0 and NLν(q) = 0, be more or less universal. These descriptions succeeded re- where markably in describing the most observed states. Meanwhile, 2 question as to whether the universality persists remains to be 3 6aN Lν(q) , (A-1) of important when new discovered states are considered. ≡ 2 − ν4 In this work, we addressed the orbitally-excited Regge 2 2 2 2 2 3 3ak q 6aN 4ak νq 16ak aN q NL q = + ∞ ∞ + ∞ spectrum of the light mesons and their universality using rela- ν( ) 2 3/2 3/2 2 Nν − ν4χ − 3 3 tivized quark model combined with approximated linearity of N Nν ν χN Nν Regge trajectory. By solving the model with the auxiliary field q2m¯ 2 ν3χ + ν2χ2 q 2 ak N method and quasi-linearizing the solution near the asymptotic 2 3 N − 8 3/2 (A-2) 2a − ∞ N limit of the large orbital angular momentum, an new Regge- N ν ! like mass relation is proposed for the orbitally excited mesons are two functionals for defining the two equations (26) and which supports the universality that the quark mass depen- (25). The sole difference is that a new and real artificial pa- dence of the light meson spectroscopy is suppressed signifi- rameter q (0 q 1) is introduced in the above two equations cantly and the confining parameters a is almost same for all to indicate the≤ order≤ of smallness (when N becomes large) by 2 2 families except for η/h-trajectory. The resulted predictions following scaling rules (based on ν N q− ) are found to be in good agreement with the observed data of ∝ ∝ 2 light mesons. ν ν/q √N, aN aN/q → ∝ → An explicit expressions for the Regge slope and intercept 2 2 Nν Nq/q , Nq (ν + qaL) , are obtained and one mass of the high exciton is predicted for → ≡ each family in Table VI. We suggest that the members of the χN χq 1 + 2ak /Nq. (A-3) → ≡ ∞ η/h trajectory may contain components of exotic. We have also discussed our results in comparison with the The idea of the homotopic analysis (HA) [54] is to solve results from the string (flux-tube) picture of mesons [21, 22] the functional equation G(Lν(q), NLν(q), q) = 0 with and from the other quark models. By the way, we presented a G(Lν(q), NLν(q), q) (1 q)Lν(q) hqNLν(q), (A-4a) semiclassical argument that the inverse slopes on the radial ≡ − − and angular-momentum Regge trajectories are equal in the or equivalently, massless limit of quarks, being in consistent with the sugges- tions in the literatures. (1 q)Lν(q) = hqNLν(q), (A-4b) −

before solving the nonlinear equation NLν(q = 1) = 0 which is (25). Here, h is the accelerating factor [54] remained to be ﬁxed either by the platform in the plot window for the characteristic quantity in the model or comparing with the numerical solution. When q = 0, G(Lν(q), NLν(q), q) = 0 becomes (A- 1) while it becomes (25) when q = 1. If one solves (A-4b) (helpful if using the computer) up to the third order of q, one ACKNOWLEDGMENTS ﬁnds

2 2 4aL 3 D. J is grateful to Atsushi Hosaka and Xiang Liu for many ν = 2aN 2ak q + ak h + q − ∞ ∞ √2a discussions. D. J is supported by the National Natural Science N ! Foundation of China under the no. 11565023 and the Feitian 2 ak 2 2¯m 1 + ∞ (3 + 4h)q , (A-5) Distinguished Professor Program of Gansu (2014-2016). W. − 2aN D is supported by Undergraduate Innovative Ability Program ! 2018 (Grants No. CX2018B338). which gives (28) when putting q = 1.

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