String Models, Stability and Regge Trajectories for Hadron States
aa r X i v : . [ h e p - ph ] M a y String Models, Stability and Regge Trajectories for Hadron States
German Sharov ∗ Tver state university, 170002, Sadovyj per. 35, Tver, Russia
Various string models of mesons and baryons include a string carrying 2 or 3 massive points (quarksor antiquarks). Rotational states (planar uniform rotations) of these systems generate quasilinearRegge trajectories and may be used for describing excited hadron states on these trajectories. Fordifferent string models of baryon we are to solve the problem of choice between them and the stabilityproblem for their rotational states. An unexpected result is that for the Y string baryon model theserotations are unstable with respect to small disturbances on the classical level. This instability hasspecific feature, disturbances grow linearly, whereas for the linear string baryon model they growexponentially and may increase predictions for baryon’s width Γ.The classical instability of rotational states and nonstandard Regge slope are the argumentsin favor of the stable simplest model of string with massive ends both for baryons and mesons.Rotational states of this model with two types of spin-orbit correction are used to describe Reggetrajectories for light, strange, charmed, bottom mesons and for N , ∆, Σ, Λ and Λ c baryons.Keywords: String hadron models, rotational states, instability, Regge trajectories. PACS numbers:
I. INTRODUCTION
In string models of mesons and baryons [1–11] shown inFig. 1 the Nambu-Goto string (relativistic string) simu-lates strong interaction between quarks at large distancesand QCD confinement mechanism. This string has lin-early growing energy with constant energy density equalto the string tension γ .Such a string with massive ends [2] may be regarded asthe meson string model in Fig. 1 a or the quark-diquarkmodel q - qq [5] in Fig. 1 b (on the classic level these modelscoincide). Other string models of baryons are [3]: ( c )the linear configuration q - q - q [10], ( d ) the “three-string”model or Y configuration [3, 9, 11], and (e) the “triangle”model or ∆ configuration [6, 9].For all cited string hadron models one can use rota-tional states of these systems (classical planar uniformrotations) to describe quasilinear Regge trajectories formesons and baryons [4–8]. In the limit of large energies E for a rotational state the angular momentum J of thisstate behaves as J ≃ α ′ E for any model in Fig. 1. Forthe meson and baryon models in Fig. 1 a , b and c the slope α ′ and the string tension γ are connected by Nambu re-lation [1] α ′ = (2 πγ ) − . So, if we use these models withthe same type of strings (the fundamental string), we cannaturally describe baryonic and mesonic Regge trajecto-ries with the same experimental slope α ′ ≃ . − .Rotational states of the string baryon model Y(Fig. 1 d ) demonstrate the Regge asymptotics with theslope [4] α ′ = 1 / (3 πγ ). To obtain the experimentalvalue α ′ ≃ . − we are to assume that the ef-fective string tension γ Y in this model differs from thefundamental string tension γ in models in Figs. 1 a – c and ∗ Electronic address: [email protected] q qq −− q q q q q a b c q q q d q q q e FIG. 1: String models of mesons and baryons equals γ Y = γ [4, 7].The string baryon model “triangle” or ∆ encountersthe similar problem. For describing Regge trajectorieswith the so called triangle rotational states [6] we are totake another effective string tension γ ∆ = γ [4, 7].To choose the most adequate string model of a baryonone should analyze the stability problem for rotationalstates of these models. Stability of classical rotationalstates with respect to small disturbances for the modelsin Fig. 1 was studied in numerical experiments [9] andanalytically [10, 11]. Rotational states for some models,in particular, for the linear model and the Y configurationappeared to be unstable. This fact is very important forapplications of these models in hadron spectroscopy.Note that instability of classical rotations for somestring configuration does not mean that the consideredstring model must be totally prohibited. All excitedhadron states (objects of modelling) are resonances, theyare unstable with respect to strong decays. So they haverather large width Γ. If classical rotations of a stringconfiguration are unstable and this instability has a char-acteristic time scale t inst , it gives the additional contri-bution Γ inst ≃ /t inst to width Γ. This effect can re-strict applicability of some string models, if the valueΓ inst predicted by this model essentially exceeds experi-mental data for Γ [10].In this paper we describe dynamics of the mentionedstring hadron models and the stability problem for theirrotations in Sect. II. In Sect. III string models with sta-ble rotational states are applied to Regge trajectories formesons and baryons. II. DYNAMICS AND STABILITY OF STRINGHADRON MODELS
Dynamics of an open or closed string carrying n = 2or 3 point-like masses m , m , . . . m n (the models inFigs. 1 a – c or e ) is determined by the action [4, 6, 10] A = − γ Z D √− g dτ dσ − n X j =1 m j Z q ˙ x j ( τ ) dτ. (2.1)Here γ is the string tension, g is the determinant ofthe induced metric g ab = η µν ∂ a X µ ∂ b X ν on the stringworld surface X µ ( τ, σ ) embedded in Minkowski space R , , η µν = diag(1 , − . − , − c = 1,the domain D = (cid:8) ( τ, σ ) : τ ∈ R, σ ( τ ) < σ < σ n ( τ ) (cid:9) for the models in Figs. 1 a – c (or with σ < σ < σ n forthe closed string), world lines of massive points are x µj ( τ ) = X µ ( τ, σ j ( τ )) , j = 1 , . . . , n. Equations of motion for these string models result fromthe action (2.1). Without loss of generality [4, 6] wechoose coordinates τ , σ satisfying the orthonormalityconditions on the world surface( ∂ τ X ± ∂ σ X ) = 0 . (2.2)Under these conditions the equations of motion are re-duced to the string motion equation ∂ X µ ∂τ − ∂ X µ ∂σ = 0 , (2.3)and equations for two types of massive points: for end-points m j ddτ ˙ x µj ( τ ) q ˙ x j ( τ ) + ǫ j γ (cid:2) X ′ µ + ˙ σ j ( τ ) ˙ X µ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) σ = σ j = 0 (2.4)(here ǫ = − ǫ n = 1) and for the middle point in themodel q - q - q or for all points on a closed string m j ddτ ˙ x µj ( τ ) q ˙ x j ( τ ) + γ h X ′ µ + ˙ σ j ( τ ) ˙ X µ i(cid:12)(cid:12)(cid:12) σ = σ j − − γ h X ′ µ + ˙ σ j ( τ ) ˙ X µ i(cid:12)(cid:12)(cid:12) σ = σ j +0 = 0 , (2.5) Here ˙ X µ ≡ ∂ τ X µ , X ′ µ ≡ ∂ σ X µ , the scalar product( ξ, ζ ) = η µν ξ µ ζ ν .For the open string with n = 2 or 3 masses (the modelsin Figs. 1 a – c ) rotational states are uniform rotations ofa rectilinear string segment. The correspondent solutionof Eqs. (2.2) – (2.5) may be presented in the form [9, 10]: X µ ( τ, σ ) = Ω − (cid:2) θτ e µ + cos( θσ + φ ) · e µ ( τ ) (cid:3) . (2.6)Here we fixed conditions at the ends in Eqs. (2.4) [4]: σ ( τ ) = 0 , σ n ( τ ) = π, σ ∈ [0 , π ] , (2.7)Ω is the angular velocity, e , e , e , e is the orthonormaltetrade in Minkowski space R , , e µ ( τ ) = e µ cos θτ + e µ sin θτ (2.8)is the unit rotating vector directed along the string. Val-ues θ (dimensionless frequency) and φ are connectedwith the constant speeds v j of the ends v = cos φ , v n = − cos( πθ + φ ) , m j Ω γ = 1 − v j v j . In the q - q - q system the central massive point is at rest (inthe comoving frame) at the rotational center. Its innercoordinate is σ = ( π − φ ) / (2 θ ) = const . In Refs. [9, 12] we analyzed stability of the rotationalstates (2.6) for the string with massive ends (Fig. 1 a, b ).These states appeared to be stable with respect to smalldisturbances.For the linear string baryon model q - q - q the stabil-ity problem for the states (2.6) was solved in numer-ical experiments [9] and analytically [10]. Analysis inRef. [10] demonstrated that the rotational states (2.6)for this system are unstable, because an arbitrary dis-turbed rotation has complex (imaginary) frequencies inits spectrum. They correspond to exponentially growingmodes of small disturbances: δX µ ∼ exp( t/t inst ) . (2.9)Calculations of the characteristic time t inst and its recip-rocal Γ inst ≃ /t inst in Ref. [10] showed that the valueΓ inst for the model q - q - q strongly depends on energy E ofthe rotational state. We are to compare this value withthe experimental width Γ corresponding to strong decaysof a baryon. In string models these decays are describedas string breaking with probability or width Γ = Γ br ,proportional to the string length ℓ [13, 14].In Ref. [10] we estimated the total width, predicted bythe baryon model q - q - q as Γ = Γ br + Γ inst in comparisonwith experimental data for N , ∆ and strange baryonsin the mass (or energy) range 1 – 3 GeV. For E ≃ inst in total width Γ appeared tobe essentially exceeding experimental data. So we con-cluded, that the linear string model q - q - q is unacceptablefor describing these baryon states and we should refusethis model in favor of the quark-diquark or Y models.For the string baryon model Y (Fig. 1 d ) threeworld sheets (swept up by three string segments) areparametrized with three different functions X µj ( τ j , σ )[9, 11]. It is convenient to use different notations τ , τ , τ for “time-like” parameters and the same symbol σ for “space-like” parameters. These three world sheets arejoined along the world line of the junction that may beset as σ = 0 for all sheets without loss of generality. Atthis junction parameters τ j are connected as follows [9] τ = τ ( τ ) , τ = τ ( τ ) , τ ≡ τ. So at the junction we have the condition X µ (cid:0) τ, (cid:1) = X µ (cid:0) τ ( τ ) , (cid:1) = X µ (cid:0) τ ( τ ) , (cid:1) . (2.10)The action of the Y configuration [9, 11] looks likeEq. (2.1), but the first term includes three integrals alongthe mentioned world sheets. So dynamical equations forthis model include the same equations (2.3) ∂ X µj ∂τ j − ∂ X µj ∂σ = 0 (2.11)under the conditions (2.2) ( ∂ τ j X j ± ∂ σ X j ) = 0 on threeworld sheets and also conditions (2.7) 0 ≤ σ ≤ π andequations (2.4) for massive endpoints with ǫ j = 1, σ j = π for all j = 1 , ,
3. One should substitute τ → τ j , X µ → X µj in Eq. (2.4) and add the relation at the junction X j =1 X ′ µj (cid:0) τ j ( τ ) , (cid:1) dτ j ( τ ) dτ = 0 . (2.12)Rotational states of the Y configuration correspond toplanar uniform rotation of three rectangular string seg-ments connected at the junction at angles of 120 ◦ [4, 5, 9].These states may be described as Eq. (2.6) [11] X µj ( τ j , σ ) = Ω − (cid:2) θτ j e µ + sin( θσ ) · e µ ( τ j + ∆ j ) (cid:3) . (2.13)Here τ = τ = τ , ∆ j = 2 π ( j − / (3 θ ), e µ ( τ ) is theunit rotating vector (2.8) directed along the first stringsegment. Below we consider the symmetric case [11] m = m = m , v = v = v (2.14)Expression (2.13) satisfies Eq. (2.11) and conditions(2.2), (2.4), (2.10), (2.12), if angular velocity Ω, the value θ , constant velocities v j of the massive points are con-nected by the relations [4] v j = sin( πθ ) = (cid:20)(cid:16) Ω m j γ (cid:17) + 1 (cid:21) / − Ω m j γ . (2.15)In Ref. [9] we demonstrated in numerical experiments,that rotational states (2.13) of the Y configuration areunstable with respect to small disturbances. This insta-bility was investigated by G. t Hooft [15]. Here we testthis stability problem analytically. Let us consider a slightly disturbed motion of themodel Y with a world surface X µj ( τ j , σ ) close to the sur-face X µj ( τ j , σ ) of the rotational state (2.13) (below weunderline values, describing rotational states). For thisdisturbed motion we use the general solution of Eq. (2.11) X µj ( τ j , σ ) = 12 (cid:2) Ψ µj + ( τ j + σ ) + Ψ µj − ( τ j − σ ) (cid:3) , (2.16)for every world sheet. Functions Ψ µj ± ( τ ) have isotropicderivatives with respect to their argumetns˙Ψ j + = ˙Ψ j − = 0 . (2.17)as a consequence of the orthonormality conditions (2.2).If we substitute Eq. (2.16) into conditions (2.10), (2.12)and (2.4), they may be reduced to the form [11]Ψ µ ( τ ) + Ψ µ − ( τ ) = Ψ µj + ( τ j ) + Ψ µj − ( τ j ) , (2.18) P j =1 (cid:2) ˙Ψ µj + ( τ j ) − ˙Ψ µj − ( τ j ) (cid:3) ˙ τ j ( τ ) = 0 , (2.19)˙Ψ µj ± ( τ j ± π ) = m j γ hq − ˙ U j ( τ j ) U µj ( τ j ) ∓ ˙ U µj ( τ j ) i . (2.20)Here U µj ( τ j ) = ˙Ψ µj + ( τ j + π ) + ˙Ψ µj − ( τ j − π ) (cid:2) (cid:0) ˙Ψ j + ( τ j + π ) , ˙Ψ j − ( τ j − π ) (cid:1)(cid:3) / are velocities of massive ends. Equations (2.20) and(2.16) determine functions X µj ( τ j , σ ) for world sheets ifwe know velocities U µj ( τ j ). So we search vectors U µj fordisturbed motion as small corrections to velocities U µj forrotational states (2.13): U µj ( τ j ) = U µj ( τ j ) + u µj ( τ j ) . (2.21)Here the vectors U µj are U µj ( τ j ) = (1 − v j ) − / (cid:2) e µ + v j ´ e µ ( τ j + ∆ j ) (cid:3) , (2.22)rotating vector ´ e µ ( τ ) = − e µ sin( θτ ) + e µ cos( θτ ) is or-thogonal to e µ ( τ ) (2.8).We suppose that for a disturbed motions the “time”parameters τ j ( τ ) = τ + δ j ( τ ) , j = 2 , , | δ j ( τ ) | ≪ , (2.23)have small deviations δ ( τ ) and δ ( τ ) from τ ≡ τ , andalso suppose disturbances u µj ( τ j ) in Eq. (2.21) to be small( | u µj | ≪ u j and δ j when wesubstitute the expressions (2.21) and (2.23) for disturbedmotion into dynamical equations (2.18) – (2.20) and intoequalities U j = U j = 1, resulting in relations (cid:0) U j ( τ j ) , u j ( τ j ) (cid:1) = 0 . (2.24)After this substitution we have the linearized system(with respect to u µj , δ j ) including Eqs. (2.24) and thefollowing vector equations: P ± h Qu µ ( ± ) ± ˙ u µ ( ± ) + U µ ( ± ) (cid:0) e ( ± ) , ˙ u ( ± ) (cid:1)i == P ± n Qu µj ( ± ) ± ˙ u µj ( ± ) + U µj ( ± ) (cid:0) e ( ± j ) , ˙ u j ( ± ) (cid:1) ++ γm h ˙ δ j ( τ ) ˙Ψ µj ± ( τ ) + δ j ( τ ) ¨Ψ µj ± ( τ ) io , X j =1 X ± n ∓ γm h ˙ δ j ˙Ψ µj ± ( τ ) + δ j ¨Ψ µj ± ( τ ) i ++ ˙ u µj ( ± ) ± Qu µj ( ± ) ± U j ( ± ) (cid:0) e ( ± j ) , ˙ u j ( ± ) (cid:1)o = 0 . (2.25)Here ( ± ) ≡ ( τ ± π ), ( ± j ) ≡ ( τ ± π + ∆ j ), the functions˙Ψ µj ± ( τ ) = m Qγc (cid:2) e µ + v ´ e µ ( ∓ j ) ± c e µ ( ∓ j ) (cid:3) (2.26)correspond to rotational state (2.13), Q = θv /c , c = cos( πθ ) = q − v . (2.27)We search oscillatory solutions of this system andsubstitute the following disturbances with u µj satisfyingEqs. (2.24) u µj ( τ ) = (cid:2) A j e µ + A zj e µ + c A j e µ ( τ + ∆ j ) ++ v − A j ´ e µ ( τ + ∆ j ) (cid:3) exp( − iξτ ) , (2.28) δ j ( τ ) = δ j exp( − iξτ ) , j = 2 , e µ , e µ ( τ ), ´ e µ ( τ ), e µ form the systemof algebraic equations with respect to the small complexamplitudes A j , A j , A zj , δ j .These projections onto the vector e µ are( ξ ˜ c + Q ˜ s )( A z + A z + A z ) = 0 , ( ξ ˜ s − Q ˜ c )( A z − A zj ) = 0 , j = 2 , . Here ˜ c = cos πξ , ˜ s = sin πξ. Solutions of these equationsdescribe 2 types of small oscillations of rotating Y con-figuration (in e -direction). Corresponding frequencies ξ of these oscillations are roots of the equations ξ/Q = cot πξ, ξ/Q = − tan πξ. (2.30)All roots of Eqs. (2.30) are simple roots and real numbers,therefore amplitudes of such fluctuations do not grow.Small disturbances in the rotational plane ( e , e ) aredescribed by projections of Eqs. (2.25) onto 3 vectors e , e ( τ ), ´ e ( τ ). These projections form the system of 9linear equations (with 8 independent ones among them)with respect to 8 unknown values A j , A j , δ j . Nontrivialsolutions of this system exist if and only if its determinantequals zero. This equality after simplification, is reducedto the following equation Ref. [11]:( ξ − θ ) (cid:18) ξ − q Qξ + tan πξ (cid:19)(cid:18) ξ − q Qξ − cot πξ (cid:19) . (2.31) Here q = Q (1 + v − ) = θ (1 + v ) / (1 − v ).We analyzed roots of this equation for complex values ξ = ξ + iξ in Ref. [11] and concluded that all rootsof Eq. (2.31) are real numbers and form a countable set.This behavior differs from that for the linear string model q - q - q . In the latter case the corresponding spectral equa-tion has complex roots (frequencies of disturbances) [10],so these disturbances grow exponentially in accordancewith Eq. (2.9).For the model Y the observed in Ref. [9] instability ofrotational states (2.13) has another nature. This insta-bility results from existence of double roots ξ = ± θ inEq. (2.31). If we put ξ = ± θ in Eq. (2.31), not only thefirst factor ( ξ − θ ), but also the second factor vanishes: θ − q Qθ + tan πθ = 0 . This equality results from Eqs. (2.15), (2.27).Double roots of Eq. (2.31) correspond to oscillatorymodes with linearly growing amplitude. If we fix fre-quency ξ = θ and substitute small disturbances in theform (2.28), (2.29) but with A j + ˜ A j τ and δ j + ˜ δA j τ in-stead of A j and δ j into Eqs. (2.25), we can find nontrivialsolutions with linearly growing amplitude: | u µj | ≃ | ˜ A | c τ | δ j | ≃ √ c Q − | ˜ A | τ. (2.32) τ τ j − τ FIG. 2: Shape of the Y configuration and dependence of τ ( τ ) − τ (solid line) and τ ( τ ) − τ (dashed line) on τ fordisturbed rotation In Ref. [9] we investigated numerically disturbed rota-tional states of the string configuration Y and observedinstability of the states (2.13). Omitting details of nu-merical modelling (described in Ref. [9]), we demonstratein Fig. 2 some “photographs” of the rotating Y config-uration with constant time intervals and dependence ofdeviations τ ( τ ) − τ (solid line) and τ ( τ ) − τ (dashedline) on the time parameter τ for disturbed rotationalstates (2.13). Here we test the state with masses (2.14)for θ = 0 . ± ( τ ). During further evolution small dis-turbances grow, the junction moves, lengthes of threearms vary and at last one of massive points merge withthe junction. Numerical experiments demonstrate thatevolution of small disturbances for velocities U µj or val-ues τ j ( τ ) corresponds to expression (2.32), amplitudes ofdisturbances linearly grow and frequency of oscillations(with respect to τ ) is equal θ .This behavior lets us to conclude, that rotational states(2.13) of the string model Y are unstable, because anarbitrary small disturbance contains linearly growingmodes of the type (2.32) in its spectrum.If we compare these two types of instability: expo-nential growth (2.9) of small disturbances for the linearstring model q - q - q and linear growth (2.32) for the Ystring model, we are to make the following conclusion.Instability of the model q - q - q has the characteristic time t inst and correspondent width Γ inst ≃ /t inst . But forthe Y model with linear growth (2.32) we have no anycharacteristic time, this corresponds to zero contributionΓ inst = 0 in the increment of instability. So instability ofrotational states (2.13) does not give an additional con-tribution in width Γ of baryons, described with the Ystring model.This rotational instability is not a weighty argumentagainst application of the Y configuration. But thismodel has another drawback mentioned above, it pre-dicts the slope α ′ = (3 πγ ) − for Regge trajectories, thatdiffers from the value α ′ = (2 πγ ) − for the string withmassive ends [4, 5]. The experimental value of Reggeslope α ′ ≃ . − is close for mesons and baryons.So the effective value of string tension γ is to be differentfor the baryon models Y and the meson model q - q .These arguments work in favor of the quark-diquarkmodel (Fig. 1 b ) for describing baryons on Regge trajecto-ries. In the next section we consider the common schemefor describing these trajectories for mesons and baryons. III. STRINGS AND REGGE TRAJECTORIESFOR MESONS AND BARYONS
Rotational states (2.6) of the string with massive endswere applied for describing excited mesons and baryonson Regge trajectories in Refs. [4–8]. All mentioned au-thors used quasilinear dependence between angular mo-mentum J and square of energy E of a state (2.6).Expressions for energy E (or mass M = E ) and angu-lar momentum J of a rotational state (2.6) for the stringwith massive ends have the following form [4, 7]: M = E = πγθ Ω + X j =1 m j q − v j + ∆ E SL , (3.1) J = L + S = 12Ω (cid:18) πγθ Ω + X j =1 m j v j q − v j (cid:19) + n X j =1 s j . (3.2)Here s j are spin projections of massive points, ∆ E SL isthe spin-orbit contribution to the energy in the following form [4, 7]:∆ E SL = X j =1 β ( v j )(Ω · s j ) , β ( v j ) = 1 − (1 − v j ) / . (3.3)This form of the spin-orbit contribution results from theassumption about pure chromoelectric field in the rota-tional center rest frame [4, 5, 16]. The authors of Ref. [5]used the alternative expression β ( v j ) = 1 − (1 − v i ) − / , (3.4)corresponding to the Thomas precession of the spins s j .If the string tension γ , values m j and s j are fixed, weobtain an one-parameter set of rotational states (2.6).Values J and E for these states form the quasilinearRegge trajectory with asymptotic behavior J ≃ α ′ E forlarge E and J [4] with the slope α ′ = 1 / (2 πγ ).We use the model of string with massive ends (consid-ered as the model q - q of a meson and the quark-diquarkmodel q - qq of a baryon) to describe experimental data forexcited states of mesons and baryons on Regge trajecto-ries. For this purpose we are to choose free parameters ofthe model: effective value of string tension γ and effectivemasses of quarks and diquarks m j for all flavors.This approach was developed in Refs. [4, 7], but in thepresent paper we study both types of spin-orbit correc-tion (3.3) and (3.4), use the optimization procedure forchoosing the mentioned effective values γ , m j and alsoinclude charmed and bottom hadrons. The main prin-ciple of this choice is to describe the whole totality ofexperimental data on excited mesons and baryons [17].At the first stage we describe main Regge trajectoriesfor light unflavored mesons and choose effective values oftension γ and mass m ud of the lightest quarks u and d (wesuppose below that they are equal: m u = m d = m ud ).The results for such isovector and isoscalar mesons withspin-orbit corrections (3.3) and (3.4) are shown in Fig. 3. πρ (770) b (1235)a (1320) π (1670) ρ (1690)a (2040) ρ (2350)a (2450)M , GeV J S = 1S = 0 0 2 4 60123456 ηω (782) h (1170)f (1270) η (1645) ω (1670)f (2050) ω (2250)f (2510)M , GeV J FIG. 3: Regge trajectories with corrections (3.3) (heavy lines)and (3.4) (thin lines) (a) for isovector mesons ρ , a , π ; (b) forisoscalar mesons ω , f , η . Here parameters for models with Eqs. (3.3), (3.4) are
TABLE I: Effective values of parameters γ , m ud , m s for spin-orbit corrections (3.3) and (3.4).Correction γ (GeV ) m ud MeV m s MeV(3.3) 0.154 231.5 369.0(3.4) 0.1767 320.0 436.0
These values are obtained in the following optimizationprocedure.We fix a set of n mesons with masses M k and angularmomenta J k , k = 1 , . . . , n , with the definite quark com-position. These mesons may lie on one or on a few Reggetrajectories differing in quark spin S or isospin I . Sucha set is shown in Fig. 3, it includes 4 Regge trajectories.For the best fitting between the model dependence J = J ( M ) (3.1), (3.2) and the experimental values M k , J k from the table [17] for this set of mesons we use theleast-squares method and minimize the sum of squareddeviations with positive weights ρ k : F ( m , m , γ ) = n X k =1 ρ k h J k − J ( M k ) i . (3.5)The weights ρ k correspond to data or model errors,they are fixed below in the following manner: ρ k = 1for reliable meson states from summary tables [17] withorbital momenta L ≥ ρ k = 0 . J omitted from summary tables [17], in par-ticular, for ρ (2350) in Fig. 3, but ρ k = 0 . ω , a , f ); ρ k = 0 . L = 0. States with L = 0, in particular, π , ρ (770), η , ω (782) in Fig. 3 should not be described bystring models, because the string shape may correspondonly to extended hadron states with high J [1–5]. . . . . . . . . . . . . m ud γ F(m ud , γ ), ∆ E SL (3.3)0.18 0.2 0.22 0.24 0.260.1450.150.1550.160.1650.170.175 . . . . . . . . m ud F(m ud , γ ), ∆ E SL (3.4)0.26 0.28 0.3 0.32 0.340.150.160.170.180.190.20.210.22 FIG. 4: Level lines of the sum (3.5) F ( m ud , γ ) for spin-orbitcorrection (3.3) (left) and (3.4) (right). To determine theoretical values of angular momenta J ( M k ), corresponding to masses M k from the table[17], we are to invert numerically the function M (Ω)(3.1), (2.15) and substitute the function Ω = Ω( M ) intoEq. (3.2): J ( M k ) = J (cid:0) Ω( M k ) (cid:1) . We calculate the sum(3.5) for the mesons in Fig. 3 with both types of spin- orbit correction (3.3) and (3.4) for different effective val-ues γ and m ud (here m = m = m ud ). The results ofthis calculation are presented in Fig. 4 as level lines ofthe function (3.5) in the ( m ud , γ ) plane.One can see that the sum (3.5) for the model (3.3)reaches its minimum F min ≃ .
18, if the effective pa-rameters m ud , γ are close to the values in Table I. Thesimilar minimum F min ≃ .
94 for the model with correc-tion (3.4) is 5 times larger. So this model is less successfulin describing these meson states (thin lines in Fig. 3).The results in Table I are obtained with taking into ac-count 8 leading Regge trajectories of mesons: to 4 men-tioned above trajectories we add 4 sets of mesons with thestrange quark s . They include 2 sets of K mesons with m = m ud and m = m s , in particular, K , K (1270), K (1770) with summary quark spin S = 0 and K ∗ (892), K ∗ (1430), K ∗ (1780), K ∗ (2045), K ∗ (2382) with S = 1.The sets η , h (1380), η (1870) with S = 0 and φ (1020), f ′ (1525), φ (1850) with S = 1 are supposed to be s - s mesons: m = m = m s . The weights ρ k in Eq. (3.5) aredetermined as mentioned above, for example, ρ k = 0 . K , K ∗ (892) , . . . h (1380), ρ k = 0 . K ∗ and η .The sum (3.5) for these 8 Regge trajectories with 32mesons depends on 3 parameters: F = F ( m ud , m s , γ ).The minimal values F min ≃ .
301 for the model (3.3)and F min ≃ .
267 (4 times larger) for the case (3.4) arereached, if these 3 parameters take the values in Table I.for the model (3.3) these values differ from the effectiveparameters used in Ref. [4] γ = 0 .
175 GeV , m ud = 130MeV, m s = 270 MeV. The values from Table I essentiallydiminish the sum F ( m ud , m s , γ ).Description of 4 Regge trajectories of mesons with s quark is presented in Fig. 5. Here we use the same nota-tions: heavy solid lines for the case (3.3) S = 1, dashedlines for S = 1, thin dash-dotted and dotted lines for thecase (3.4). One can see, that the model with spin-orbitcontribution (3.3) demonstrate better agreement.If parameters γ , m ud , m s of the model are fixed inTable I, we can determine the best value m c of c quark fordescribing D mesons and charmonium states. They form6 Regge trajectories: D , D (2420), D ∗ , D ∗ (2460) arecharmed analogs of K and K ∗ mesons; D s , D s (2536), D ∗ s , D ∗ s (2573) are their partners cs or sc ; η c , h c (1 P ), J/ψ , χ c are cc states. As stated above we take ρ k = 0 . L = 0 and ρ k = 1 for L = 1. Underthese circumstances we minimize the function (3.5) F = F ( m c ) of one argument and obtain the optimal values m c in Table II for both considered models with spin-orbitcorrections (3.3) and (3.4). TABLE II: Effective values of parameters m c , m b .Correction m c GeV m b GeV(3.3) 1.5372 4.818(3.4) 1.5322 4.8198
Six Regge trajectories for charmed mesons are pre-sented in Fig. 5. Note that for them we used only onefitting parameter m c , but all trajectories are describedrather well in both models with Eqs. (3.3) and (3.4).The similar approach to bottom mesons B , B ∗ , B s , B ∗ s , Υ, χ b (1 P ) with substitution c for b quark results inthe optimal values m b for bottom quark in Table II andcorresponding Regge trajectories for bottom mesons inFig. 5. Here we use the notations of Fig. 3, in particular,thick lines correspond to spin-orbit correction (3.3). Thismodel has advantage in comparison with the case (3.4)(thin lines) for mesons with strange quark s . For charmedand bottom mesons this advantage becomes inessential.Both models are successful in describing charmoniumstates, but they work worse for bottomonium. J S = 1S = 0K K (1270) K (1770) K * K (1430)K (1780)K (2045)K (2382) 0 1 2 3 401234 J η h (1380) η (1870) φ (1020)f ’(1525) φ (1850)3 4 5 6 7012 J D D (2420) D * D (2460) 3 4 5 6 7012 D s D s* D s2* (2573) D s1 η c h c (1P)J/ ψχ c2 (3556)26 28 30 32 34012 M J B B (5721) B * B (5747) 28 30 32 34 36012 M B s B s1 (5830) B s* B s2* (5840) 90 95 100012 M h b (1P) ϒ (1S) χ b2 (1P) FIG. 5: Regge trajectories for strange, charmed, bottommesons with model parameters γ , m q from Tables I, II. The similar approach may be applied to baryons. Wementioned above that for describing baryonic Regge tra-jectories we are to choose the quark-diquark string model.Only this model predicts rotational stability and naturalRegge slope α ′ = (2 πγ ) − corresponding to equal exper-imental Regge slopes α ′ both for mesons and baryons.When can apply to baryons the model of a string withmassive ends, if we determine the effective diquark mass m d = m (assume that m is the quark mass). In the sim-plest approach [4] we suppose that diquarks are weaklybound systems, so a diquark mass is close to sum of twoconstituent quark masses, in particular, for N and ∆baryons m d ≃ m ud .Another approach is widely used in string and poten-tial quark-diquark models of baryons [5, 18, 19] and sup-poses different diquark masses for scalar diquarks withtotal spin S d = 0 and for vector diquarks with S d = 1.Essential difference of these masses corresponds to strong coupling between two quarks in a diquark. Howeverthis mechanism remains vague, the diquark masses inthe mentioned models are used as fitting parameters.In particular, masses m d of the scalar [ u, d ] diquarkin N baryons and m d for the vector { u, d } diquark in∆ baryons in the potential model [19] are correspond-ingly 710 and 909 MeV. The similar masses in the stringmodel [5] are m d = 220 and m d = 550 MeV. This differ-ence requires special explanation.The last approach may be applied to our string modelwith two forms (3.3) and (3.4) of spin-orbit correction.For this purpose we use two sets of baryonic Regge tra-jectories with scalar and vector diquarks correspond-ingly. For scalar diquarks we choose the main Reggetrajectory for N baryons N , N (1520), N (1680) . . . andthe corresponding trajectory for Λ baryons: Λ, Λ(1520),Λ(1820) . . . If we use data for mesons from Table I, fixtension γ , the mass m = m ud or m s of a single quarkand vary the free parameter m d = m d , we determine theoptimal value m d for describing the mentioned Regge tra-jectories with the minimal sum (3.5). The similar opti-mal values m d for the vector { u, d } diquark with S d = 1is calculated on the base of trajectories with ∆ baryons∆(1232), ∆(1930), ∆(1930) . . . and Σ baryons Σ(1385),Σ(1775), Σ(2030). Optimal values of these scalar andvector diquark masses MeV are presented in Table III. TABLE III: Diquark masses m d for models (3.3) and (3.4).Correction m d for S d = 0 m d for S d = 1(3.3) 412.6 588(3.4) 352.9 702 One can see that in the model with spin-orbit correc-tion (3.4) optimal masses of scalar and vector diquarksare essentially different: m d ≃ m d . This difference cor-responds to so the similar relation between m d and m d in Ref. [5], where the same correction (3.4) in the quark-diquark model was used.This difference is not so large in the model with spin-orbit correction (3.3), the value m d in Table III is close to2 m ud = 463 MeV. So in this model we can use not onlyoptimal parameters from Table III, but also m d = 2 m ud .Regge trajectories for the mentioned baryons areshown in Fig. 6. The model parameters for both models(3.3) and (3.4) are from Tables I – III. Notations are sim-ilar to Fig. 5, but for the model (3.3) we draw predictionswith m d = 412 . m d = 588 MeV from Table III asthick solid lines (for Σ baryons the solid line describesthe case with quark’s spin S = 3 / S = 1 / s = − / s = 1).Predictions of the model (3.3) with m d = 2 m ud = 463MeV are described with dotted lines, predictions of themodel (3.4) with m d and m d from Table III are describedwith dash-dotted lines.We see that for main (parent) Regge trajectories for N ,Λ baryons with S = 1 / S = 3 / J p N(1520)N(1680) N(2190)N(2220) N(2600)N(2700)S = 1/20 2 4 6012345 J Λ Λ (1520) Λ (1820) Λ (2100) Λ (2350) Λ (2585)S = 1/2 0 2 4 6 80246 ∆ (1232) ∆ (1930) ∆ (1950) ∆ (2400) ∆ (2420) ∆ (2750) ∆ (2950)S = 3/20 2 4 601234 M S = 1/2 ∆ (1700) ∆ (1905) ∆ (2200) ∆ (2300)0 2 4 601234 Σ (1385) Σ (1775) Σ (2030) Σ Σ (1670) Σ (1915) Σ (2250)S = 3/2S = 1/24 5 6 7 8 90123 M Λ c Λ c (2625) Λ c (2880)S = 1/2 FIG. 6: Regge trajectories for baryons in models (3.3) (solidand dotted lines) and (3.4) (thin dash-dotted lines). m jd from Table III are very close. For ∆ and Σ baryonswith S = 3 / c baryons (only these charmed baryons form an apprecia-ble Regge trajectory). The trajectory for Λ c baryons isused here as a test for these models, all their parameters γ , m c , m d were determined previously.But for Σ baryons Σ, Σ(1670), Σ(1915) and ∆ baryons∆(1700), ∆(1905) . . . the models (3.3) and (3.4) predictdifferent Regge trajectories, if we interpret these hadronsas states with s = − / s = S d = 1. In this approachonly the model with correction (3.3) works successfully,in the case (3.4) the mass correction ∆ E SL appears to bepositive and too large. So we can describe these Reggetrajectories in the model (3.4) only if we suppose that diquarks in these hadrons are scalar ones. In this casetrajectories for ∆ and Σ baryons with S = 1 / N and Λ baryons.Dotted lines for all baryons in Fig. 6 show that themodel with correction (3.3) admits the diquark mass m d = 2 m ud . This assumption works rather good for N , Λ and Λ c baryons and it works worse for ∆ and Σbaryons. Note that the model with correction (3.4) isincompatible with the assumption m d = 2 m ud . IV. CONCLUSION
Different string hadron models are considered from thepoint of view of their application to describing Reggetrajectories for mesons and baryons. For this purpose westudy the stability problem for classical rotational statesof these models. It is shown that these states are unstablefor the Y string baryon model (Fig. 1 d ). The type of thisinstability differs from that for the linear string baryonmodel q - q - q . For the Y configuration small disturbancesgrow linearly, whereas for the linear model they growexponentially. This results in too large additional widthΓ of excited baryons in the linear model [10].For the Y string baryon model we have no additionalwidth, but this model predicts the slope α ′ = (3 πγ ) − for Regge trajectories, that differs from α ′ = (2 πγ ) − for the string with massive ends [4, 5]. So for describingboth mesons and baryons with almost equal experimentalvalue of α ′ we have to use the string with massive endsas the meson model q - q and as the quark-diquark baryonmodel q - qq .These models with spin-orbit correction in two forms(3.3) and (3.4) can describe main Regge trajectories forlight unflavored mesons, for K , D , D s , B , B s mesons,charmonium and bottomonium states, and also for N ,∆, Σ, Λ and Λ c baryons. In this approach we use theoptimization procedure with choosing the effective stringtension γ and effective masses of quarks and diquarks m j for all flavors (see Tables I – III). [1] Y. Nambu, Phys. Rev. D (1974) 4262-4268.[2] A. Chodos, C.B. Thorn, Nucl. Phys. B (1974) 509-522.[3] X. Artru, Nucl. Phys. B (1975) 442-460.[4] G.S. Sharov, Phys. Atom. Nucl. (1999) 1705-1715.[5] I.Yu. Kobzarev, B.V. Martemyanov, M.G. Shchepkin,Sov. Phys. Usp. (1992) 257.[6] G.S. Sharov, Phys. Rev. D (1998) 114009.[7] A. Inopin, G.S. Sharov, Phys. Rev. D63 (2001) 054023.[8] L.D. Soloviev, Phys. Rev. D (1999) 015009.[9] G.S. Sharov, Phys. Rev. D (2000) 094015,hep-ph/0004003.[10] G.S. Sharov, Phys. Rev. D. (2009) 114025.[11] G.S. Sharov, Phys. Atom. Nucl. (2010) 2027-2034.[12] G.S. Sharov, Theor. Math. Phys. (2004) 242. [13] I.Yu. Kobzarev, B.V. Martemyanov, M.G. Shchepkin,Sov. J. Nucl. Phys. (1988) 344.[14] K.S. Gupta, C. Rosenzweig. Phys. Rev. D (1994)3368.[15] G. t Hooft, Minimal strings for baryons, arXiv:hep-th/0408148.[16] T.J. Allen, M.C. Olsson, S. Veseli, K. Williams, Phys.Rev. D (1997) 5408.[17] J. Beringer et al. (Particle Data Group), Phys. Rev. D (2012) 010001, http://pdg.lbl.gov.[18] E. Klempt, J.M. Richard, Rev. Mod. Phys. (2010)1095-1153, arXiv:0901.2055 [hep-ph].[19] D. Ebert, R. N. Faustov, V. O. Galkin, Phys. Rev. D (2011) 014025, arXiv:1105.0583 [hep-ph]. πρ (770) b (1235)a (1320) π (1670) ρ (1690)a (2040) ρ (2350)a (2450)M , GeV J S = 1S = 0 0 2 4 60123456 ηω (782) h (1170)f (1270) η (1645) ω (1670)f (2050) ω (2250)f (2510)M , GeV2