Rotating strings confronting PDG mesons

We revisit the model of mesons as rotating strings with massive endpoints and confront it with meson spectra. We look at Regge trajectories both in the (J,M2) and (n,M2) planes, where J and n are the angular momentum and radial excitation number respectively. We start from states comprised of u and d quarks alone, move on to trajectories involving s and c quarks, and finally analyze the trajectories of the heaviest observed b-bbar mesons. The endpoint masses provide the needed transition between the linear Regge trajectories of the light mesons to the deviations from linear behavior encountered for the heavier mesons, all in the confines of the same simple model. From our fits we extract the values of the quark endpoint masses, the Regge slope (string tension) and quantum intercept. The model also allows for a universal fit where with a single value of the Regge slope we fit all the (J,M2) trajectories involving u, d, s, and c quarks. We include a list of predictions for higher mesons in both J and n.


Introduction
The stringy description of mesons, which was one of the founding motivations of string theory, has been thoroughly investigated since the seventies of the last century [1]. In this note we reinvestigate this issue. What is the reason then to go back to "square one" and revisit this question? There are at least three reasons for reinvestigating the stringy nature of mesons: (i) Holography, or gauge/string duality, provides a bridge between the underlying theory of QCD (in certain limits) and a bosonic string model of mesons. (ii) There is a wide range of heavy mesonic resonances that have been discovered in recent years, and (iii) up to date we lack a full exact procedure of quantizing a rotating string with massive endpoints. In this note we will not add anything new about (iii) but rather combine points (i) and (ii). Namely, we describe a model of spinning bosonic strings with massive endpoints that follows from a model of spinning strings in holographic confining backgrounds. Leaving aside the regime where holography applies, we then confront this model with experimental data of meson spectra. We use χ 2 fits to check the validity of the model and to extract its defining parameters.
The passage from the original AdS/CFT duality to the holographic description of hadrons in the top down approach includes several steps. First one has to deform the background, namely the geometry and the bulk fields, so that the corresponding dual gauge field theory is non-conformal and non-supersymmetric. Prototype backgrounds of such a nature are that of a D 4 brane compactified on S 1 [2] (and its non-critical analogous model [3]). The fundamental quark degrees of freedom are then injected to the gravity models via "flavor probe branes". For instance for the compactified D4 brane model D8 anti D8 branes are incorporated [4]. The spectra of hadrons has been determined in these models by computing the spectra of the fluctuations of bulk fields corresponding to glueballs and scalar and vector fluctuations of the probe-branes which associate with scalar and vector mesons respectively. (See for instance [4] [5]).
Both for the glueballs and for the mesons the spectra deduced from the gravitational backgrounds and the probe branes do not admit Regge behavior, neither the linear relation between M 2 and the angular momentum J nor the linearity between M 2 and the radial excitation number n. In fact in terms of the bulk fields one can get also scalar and vector mesons. To get higher spin mesons one has to revert to a stringy configuration. There is an unavoidable big gap between the low spin mesons described by the gravity and probe modes and the high spin one described by holographic strings. [6] 1 An alternative approach to extract the spectra of mesons and glueballs, both low and high spin ones, is to study rotating open strings connected to the probe branes 2 , or folded closed strings for mesons. Regge trajectories of the latter in various confining backgrounds were analyzed in [9]. It is a very well known feature of rotating stringy configurations in flat space-time. The major difference between rotating open strings in holographic backgrounds and those in flat space-time is that the former do not connect the two endpoints along the probe brane, but rather stretch along the "wall" 3 and then connect vertically to flavor branes (See figure (1)). The figure depicts the special case of m 1 = m 2 . In a similar manner we can have m 1 = m 2 by attaching the "vertical" strings segments to different flavor branes.
In [10] it was shown that classically the holographic rotating string can be mapped into one in flat space-time with massive endpoints. 4 Basically it was shown that the equations of motion of the two systems are equivalent.
The string endpoint mass is given approximately by the string tension times the "length" of the string along one of the two "vertical" segments. This reduces to [10] where T is the string tension, u is the holographic radial coordinate, u Λ is its minimal value (the "wall"), u f is the location of the flavor branes and g 00 and g uu are the metric components along the time and holographic radial directions respectively. Obviously this mass is neither the QCD physical mass nor the constituent quark mass. We would like to argue that both for the spectra as well as for decays [12] of mesons this is the relevant physical mass parameter. In this note we assume this map, consider a bosonic string rotating in flat four dimensional space-time with massive endpoints as a model for mesons and leave aside holography altogether. 5 The theoretical models we use are rather simple. We start from an action that includes a Nambu-Goto term for the string and two terms that describe relativistic massive chargeless particles. We write down the corresponding classical equations of motion and the Noether charges associated with the energy E and angular momentum J of the system. Unlike the  massless case, for massive endpoints there is no explicit relation between for instance E and (J, m, T ), but rather E and J can be written in terms of T , m and ωl, where ω is the angular velocity and l is the string length. For two limits of light massive endpoints where 2m E 1 and heavy ones where E−2m 2m 1 one can eliminate ωl (the two limits involve taking ωl → 1 and ωl → 0 respectively) and get approximated direct relations between E and J.
Going beyond the classical limit for rotating strings is a non-trivial task. The common lore for strings with massless endpoints, namely the linear trajectories, is that the passage from classical to quantum trajectories is via the replacement where the slope α = (2πT ) −1 , n is the radial excitation number and a is the intercept. In a recent paper [14] a precise analysis of the quantum massless string has been performed. It was shown there that for a case of a single plane of angular momentum, in particular in D = 4 dimensions, an open string with no radial excitation (n = 0) indeed admits J − a = α E 2 with a = 1. This is a non-trivial result since the calculation of the intercept (to order J 0 ) yields in D dimensions the result a = D−2 24 + 26−D 24 = 1, where the first term is the usual "Casimir" term and the second is the Polchinski-Strominger term. For the rotating string with massive endpoints a similar determination of the intercept has not yet been written down even though certain aspects of the quantization of such a system have been addressed [15][16] [17].
Falling short of the full quantum expression for the Regge trajectories one can use a WKB approximated determination of the trajectories [18]. The latter depends on the choice of the corresponding potential.
The models used in this paper to fit that experimental data are the following: • The linear trajectory J + n = α E 2 + a • The "massive trajectory" which is based on the classical expressions for E and J where the latter includes assumed quantum correction, again in the form of J → J + n − a.
The trajectories then read These expressions reduce to the linear trajectory equation in the limit m → 0.
• The WKB approximation for the linear potential V = T l which takes the form The parameters that we extract from the fits are the string tension (or the slope α ), the string endpoint masses, and the intercept. The main idea of this paper is to investigate the possibility of constructing a unified description of mesons that covers mesons of light quarks as well as those built from heavy quarks. It is a common practice to view mesons of light quarks with the linear Regge trajectories (which correspond to rotating open strings with massless endpoints) and non-relativistic potential models for heavy quark mesons. Here we suggest and test a stringy model that interpolates between these two descriptions.
In a sequel paper we propose and confront with data in a similar manner a stringy rotating model for baryons.
The paper is organized as follows. In the next section we describe the basic theoretical model. We start with the action, equations of motion and Noether charges of the rotating bosonic string with massive endpoints. We then present a WKB approximation. Next we describe the fitting procedure. Section 4 is devoted to the results of the various fits. We separate the latter to fits of the M 2 as a function of the angular momentum (J, M 2 ) and of the radial excitation (n, M 2 ). In both categories we discuss light quark mesons, strange mesons, charmed mesons and mesons containing b quarks. We present also a universal fit. We then present our WKB fits. We discuss the issue of fits with respect to the orbital angular momentum L and the total angular momentum J, and calculate the string lengths to verify the validity of a long string approximation for the fitted mesons. Section 5 is devoted to a summary, conclusions and open questions.

Classical rotating string with massive endpoints
We describe the string with massive endpoints (in flat space-time) by adding to the Nambu-Goto action, a boundary term -the action of a massive chargeless point particle at both ends. There can be different masses at the ends, but here we assume, for simplicity's sake, that they are equal. We also define σ = ±l to be the boundaries, with l an arbitrary constant with dimensions of length. The variation of the action gives the bulk equations of motion and at the two boundaries the condition It can be shown that the rotating configuration solves the bulk equations (2.3) for any choice of R(σ). We will use the simplest choice, R(σ) = σ, from here on. 6 Eq. (2.4) reduces then to the condition that at the boundary, We then derive the Noether charges associated with the Poincaré invariance of the action, which include contributions both from the string and from the point particles at the boundaries. Calculating them for the rotating solution, we arrive at the expressions for the energy and angular momentum associated with this configuration: Solving the integrals, and defining q ≡ ωl -physically, the endpoint velocity -we write the expressions in the form The terms proportional to m are the contributions from the endpoint masses and the term proportional to T is the string's contribution. These expressions are supplemented by condition (2.6), which we rewrite as This last equation can be used to eliminate one of the parameters l, m, T, and q from J and E. Eliminating the string length from the equations we arrive at the final form These two equations are what define the Regge trajectories of the string with massive endpoints. They determine the functional dependence of J on E, where they are related through the parameter 0 ≤ q < 1 (q = 1 when m = 0). Since the expressions are hard to make sense of in their current form, we turn to two opposing limits -the low mass and the high mass approximations.
In the low mass limit where the endpoints move at a speed close to the speed of light, so q → 1, we have an expansion in (m/E): from which we can easily see that the linear Regge behavior is restored in the limit m → 0, and that the first correction is proportional to √ E. The Regge slope α is related to the string tension by α = (2πT ) −1 .

The WKB aprroximation
We follow here the approach of E. Schreiber [18], where the string is treated as a "fast" degree of freedom that can be replaced by an effective potential between the "slow" degrees of freedom -the string endpoints. Then, we treat the endpoints as spinless point particles in a potential well. As such, the relativistic energy carried by the quarks is: If the particle is at the end of a rotating rod of length x, then p 2 = p 2 With the usual replacement of p x → −i∂ x , we arrive at the one dimensional differential equation to be solved then the spectrum of the system is obtained, in the WKB approximation, by the quantization condition The limits of the integral x − and x + are the classical "turning points" -those points where the integrand, p(x), is zero. The condition that the integral be an integer multiple of π implies the relation between n and the energy eigenvalues E n . How we continue from here depends on our choice of the potential V (x). Also, we have to decide how to relate the total angular momentum J with the momentum carried by the point particles, J q , and the angular momentum carried by the string itself, which we'll call J s .
If we treat the string as a classical rotating rod, then the (non-relativistic) expression for its energy is Another option is the quantum mechanical expression for a string fixed at both ends [19] V (x) = (T x) 2 − T π(D − 2) 12 (2.21) More general potentials can also include a spin-orbin interaction term, or an added Coulomb potential.
The simplest option, and the one for which we can solve the integral analytically, is to set contributions from the string's angular momentum and the quantum corrections to the potential V (x) to zero. Namely, to take the linear potential V (x) = |T x|. To solve the integral, we also have to assume J q = 0, so the state has no angular momentum at all.
with b = m/E. Now this is a result for only one particle -half our system. We modify the result to apply it to the two particle system (assuming the two particles are identical in mass) by the simple replacement n → n/2, and E → E/2. The equation is now of the form with α = (2πT ) −1 as always, and b now redefined to be The high mass expansion (1−b 1) of the above expression is similar to the one obtained for the classical rotating string: Comparing this with the high mass limit for the classical rotating string, in eq. (2.15), we see that the only difference is in the expansion coefficients, a difference of about 10% in the coefficients for the leading term, and 16% in the next to leading order. The low mass expansion, on the other hand, results in a different kind of behavior from where the fitting parameters are the slope α and the intercept, a.
For the massive fit, we use the expressions for the mass and angular momentum of the rotating string, eqs. (2.12) and (2.13), generalized to the case of two different masses, and we add to them, by hand, an intercept and an extrapolated n dependence, assuming the same replacement of J → J + n − a.
We relate the velocities q 1 and q 2 can be related using the boundary condition (2.6), from which we have so the functional dependence between E and J is still through only one parameter 0 ≤ q i < 1.
With the two additions of n and a, the two equations reduce to that of the linear fit in (3.1) in the limit where both masses are zero. Now the fitting parameters are a and α as before, as well as the the two endpoint masses m 1 and m 2 . For a lot of the cases we assume m 1 = m 2 and retain only one free mass parameter, m.

WKB model
The third fitting model is the WKB. It is defined by where x ± are the points where the integrand is zero and again we have added an intercept as an independent parameter by hand. The potential we chose was simply the linear potential The angular momentum is then carried only by the quarks. We chose to identify J q with the orbital angular momentum L. For those states with J q = 0 we solve the integral and use the resulting formula, where b ≡ (2m/E). If we can't make that assumption we solve eq. (3.5) numerically. The fitting parameters are again m, α , and a.

Fitting procedure
We measure the quality of a fit by the dimensionless quantity χ 2 , which we define by M i and E i are, respectively, the measured and calculated value of the mass of the i-th particle, and N the number of points in the trajectory. A more common definition of χ 2 would have the standard deviation σ i = ∆M 2 i in the denominator, but we have used M 2 i . We do this mostly for reasons of practicality. The high accuracy to which some of the meson's masses are known makes χ 2 (when defined using σ i as the denominator) vary greatly with very small changes in the fitting parameters. 8 We feel the kind of precision required then in the fits is unnecessary for the purposes of our work. By using definition (3.7) for χ 2 we can still extract reasonably accurate values for the fitting parameters from the different trajectories, and identify those deviations from the linear Regge behavior which we will attribute to the presence of massive endpoints.

Fit results
This section discusses the results of our fits. The fits to the trajectories in the (J, M 2 ) plane and the trajectories in the (n, M 2 ) plane are presented separately. For the radial trajectories, where we have used both the massive model and the WKB model, the results are further separated between the two different types of fits. In each subsection, we describe the lightest quark trajectories first and move on gradually to the heaviest.
The details of the fits to each of the individual trajectories, including the specification of all the states used and the plots of each of the trajectories in the (J, M 2 ) or (n, M 2 ) planes, can be found in appendix A.
A note on units and notation: When units aren't explicitly stated, they are GeV −2 for α and MeV for masses. The intercept a is dimensionless. If the letters l, m, or w, are used as subscripts, they will always refer to the linear, massive, and WKB fits respectively.

Light quark mesons
We begin by looking at mesons consisting only of light quarks -u and d. We assume for our analysis that the u and d quarks are equal in mass, as any difference between them would be too small to reveal itself in our fits. The intercept a is optimized to get a best fit for each point in the (α , m) plane. χ 2 in these plots is normalized so that the value of the optimal linear fit (m = 0) is This sector is where we have the most data, but it is also where our fits are the least conclusive. The trajectories we have analyzed are those of the π/b, ρ/a, η/h, and ω/f . Of the four (J, M 2 ) trajectories, the two I = 1 trajectories, of the ρ and the π, show a weak dependence of χ 2 on m. Endpoint masses anywhere between 0 and 160 MeV are nearly equal in terms of χ 2 , and no clear optimum can be observed. For the two I = 0 trajectories, of the η and ω, the linear fit is optimal. If we allow an increase of up to 10% in χ 2 , we can add masses of only 60 MeV or less. Figure (2) presents the plots of χ 2 vs. α and m for the trajectories of the ω and ρ and shows the difference in the allowed masses between them.
The slope for these trajectories is between α = 0.81 − 0.86 for the two trajectories starting with a pseudo-scalar (η and π), and α = 0.88 − 0.93 for the trajectories beginning with a vector meson (ρ and ω). The higher values for the slopes are obtained when we add masses, as increasing the mass generally requires an increase in α to retain a good fit to a given trajectory. This can also be seen in figure (2), in the plot for the ρ trajectory fit.

Strange and ss mesons
We analyze three trajectories in the (J, M 2 ) involving the strange quark. One is for mesons composed of one s quark and one light quark -the K * , the second is for ss mesons -the trajectory of the φ, and the last is for the charmed and strange D * s , which is presented in the next subsection with the other charmed mesons.
The K * trajectory alone cannot be used to determine both the mass of the u/d quark and the mass of the s. The first correction to the linear Regge trajectory in the low mass range is proportional to α m E. This is the result when eq. (2.14) is generaziled to the case where there are two different (and small) masses. The plot on the left side of figure (3) shows χ 2 as a function of the two masses.
The minimum for the K * trajectory resides along the curve m If we take a value of around 60 MeV for the u/d quark, that means the preferred value for the m s is around 220 MeV. The higher mass fits which are still better than the linear fit point to values for the s quark mass as high as 350 MeV, again when m u/d is taken to be 60 MeV. The slope for the K * fits goes from α = 0.85 in the linear fit to 0.89 near the optimum to 0.93 for the higher mass fits. The trajectory of the ss mesons includes only three states, and as a result the optimum is much more pronounced than it was in previous trajectories. It is found at the point m s = 400, α = 1.07. The value of χ 2 near that point approaches zero. The range in which the massive fits offer an improvement over the linear fit is much larger than that, as can be seen in the right side plot of figure (3). Masses starting from around m s = 250 MeV still have χ 2 m /χ 2 l = 0.50 or less, and the slope then has a value close to that of the other fits, around 0.9 GeV −2 .

Charmed and cc mesons
There are three trajectories we analyze involving a charm quark. The first is of the D, comprised of a light quark and a c quark, the second is the D * s with a c and an s, and the third is cc -the Ψ. All trajectories have only three data points.
For the D meson, the optimal fit has m c = 1640, m u/d = 80 and α = 1.07. In this case, unlike the result for the K * trajectory, there is a preference for an imbalanced choice of the masses, although with four fitting parameters and three data points we can't claim this with certainty. The fit for the D * s has a good fit consistent with the previous s and c fits at m c = 1580, m s = 400, and α = 1.09. The plots of χ 2 vs. the two masses (m c and m u/d /m s ) can be seen in figure (4).
In the same figure, we have χ 2 as a function of the single mass m c for the cc Ψ trajectory. The minimum there is obtained at m c = 1500 MeV, where the slope is α = 0.98 GeV −2 . It is worth noting that while the linear fit results in values for α that are very far from the one obtained for the u, d, and s quark trajectories -0.42, 0.48, and 0.52 for the Ψ, D, and D * respectively -the massive fits point to a slope that is very similar to the one obtained for the previous trajectories. This is also true, to a lesser extent, of the values of the intercept a.

bb mesons
The last of the (J, M 2 ) trajectories is that of the bb Υ meson, again a trajectory with only three data points. The fits point to an optimal value of m b = 4730, exactly half the mass of the lowest particle in the trajectory. The slope is significantly lower than that obtained for other mesons, α = 0.64 at the optimum. The bottom plot in figure (4) shows χ 2 for the bb trajectory. for the endpoint masses and attempted to find a fit in which the slope is the same for all trajectories. This wish to use a universal slope forces us to exclude the bb trajectory from this fit, but we can include the three trajectories involving a c quark. For these, with added endpoint masses (and only with added masses), the slope is very similar to that of the light quark trajectories. The only thing that was allowed to change between different trajectories was the intercept. With the values of the masses fixed, we searched for the value of α and the intercepts that would give the best overall fit to the nine trajectories of the π/b, ρ/a, η/h, ω/f , K * , φ, D, D * s , and Ψ mesons. The best fit of this sort, with the masses fixed to the above values, was α = 0.884 (4.2)

Light quark mesons
In the light quark sector we fit the trajectories of the π and π 2 , the h 1 , the a 1 , and the ω and ω 3 .
The h 1 has a very good linear fit with α = 0.83 GeV −2 , that can be improved upon slightly by adding a mass of 100 MeV, with the whole range 0 − 130 MeV being nearly equal in χ 2 .
The a 1 offers a similar picture, but with a higher χ 2 and a wider range of available masses. Masses between 0 and 225 are all nearly equivalent, with the slope rising with the added mass from 0.78 to 0.80 GeV −2 .
The π and π 2 trajectories were fitted simultaneously, with a shared slope and mass between them and different intercepts. Again we have the range 0 to 130 MeV, α = 0.78−0.81 GeV −2 , with m u/d = 100 MeV being the optimum. The preference for the mass arises from non-linearities in the π trajectory, as the π 2 when fitted alone results in the linear fit with α = 0.84 GeV −2 being optimal.
The ω and ω 3 trajectories were also fitted simultaneously. Here again the higher spin trajectory alone resulted in an optimal linear fit, with α = 0.86 GeV −2 . The two fitted simultaneously are best fitted with a high mass, m u/d = 340, and high slope, α = 1.09 GeV −2 . Excluding the ground state ω(782) from the fits eliminates the need for a mass and the linear fit with α = 0.97 GeV −2 is then optimal. The mass of the ground state from the resulting fit is 950 MeV. This is odd, since we have no reason to expect the ω(782) to have an abnormally low mass, especially since it fits in perfectly with its trajectory in the (J, M 2 ) plane.
The fit for the J P C = 1 −− ω with the ground state included is shown in figure (6), along with the fit for the h 1 , which has J P C = 1 +− .

ss mesons
For the ss we have only one trajectory of three states, that of the φ. There are two ways to use these states. The first is to assign them the values n = 0, 1, 2. Then, the linear fit with the slope α = 0.54 GeV −2 is optimal.
Since this result in inconsistent both in terms of the low value of the slope, and the absence of a mass for the strange quark, we tried a different assignment. We assumed the values n = 0, 1, and 3 for the highest state and obtained the values α = 1.10, m s = 515 for the optimal fit. These are much closer to the values obtained in previous fits.
The missing n = 2 state is predicted to have a mass of around 1960 MeV. Interestingly, there is a state with all the appropriate quantum numbers at exactly that mass -the ω(1960), and that state lies somewhat below the line formed by the linear fit to the radial trajectory of the ω. Even if the ω(1960) isn't the missing ss (or predominantly ss) state itself, this could indicate the presence of a φ state near that mass.

cc mesons
Here we have the radial trajectory of the J/Ψ, consisting of four states.
The massive fits now point to the range 1350 − 1475 MeV for the c quark mass. The biggest difference between the fits obtained here and the fits obtained before, in the (J, M 2 ) Figure 8. Left: χ 2 as a function of α and m b for the Υ radial trajectory. The discontinuity in the plot arises from the condition that the intercept a ≤ 1, otherwise the mass of the ground state is undefined. The two areas in the plot are then where a is still allowed to change (left) and where a is blocked from increasing further and is fixed at a = 1 (oval shape on the right). Right: χ 2 as a function of m b for the χ b trajectory.
plane is not in the mass, but in the slope, which now is in the range 0.48 − 0.56 GeV −2 , around half the value obtained in the angular momentum trajectories involving a c quark -0.9 − 1.1.
It is also considerably lower than the slopes obtained in the (n, M 2 ) trajectories of the light quark mesons, which would make it difficult to repeat the achievement of having a fit with a universal slope in the (n, M 2 ) plane like the one we had in the (J, M 2 ) plane.

bb mesons
There are two trajectories we use for the bb mesons.
The first is that of the Υ meson, with six states in total, all with J P C = 1 −− . For this trajectory we have an excellent fit with m b = 4730 MeV and the slope α = 0.46 GeV −2 . It is notable for having a relatively large number of states and still pointing clearly to a single value for the mass.
The second bb trajectory is that of the χ b -J P C = 1 ++ . Here we have only three states and the best fit has a slightly higher mass for the b quark -m b = 4800 MeV -and a higher value for the slope α = 0.50 GeV −2 .

WKB fits
The WKB fits are all done in the (n, M 2 ) plane. The biggest difference between the WKB fits in and the fits done using the expressions obtained from the classically rotating string is the way in which the angular momentum is included. In eq. The contribution from the angular momentum, when fitting trajectories in the (n, M 2 ) plane, amounts to nothing more than a shift of the n axis, and can be fully absorbed into the intercept a. Eq. (3.5), on the other hand, carries out the contribution from the angular momentum in a different way.
The following fits are done assuming the angular momentum carried by the quarks, J q in the notation of eq. (3.5), is the orbital angular momentum L.
Another point of difference between the two fits is in the values of the slope, which tend to be lower in the WKB fits. For the heavy quark trajectories we can understand this by comparing the heavy mass expansions in eqs. (2.15) and (2.25), and the ratio between the massive fit slopes and the WKB slopes is usually close to the ratio between the leading term coefficients of each of the two expansions (2 √ 3/π ≈ 1.1).
The WKB fits generally allow for higher masses for the light quarks, as can be seen in figure (9). For the π/π 2 trajectories we actually obtain a minimum around m u/d = 230 MeV, where before it was less than half that value. The h 1 trajectory now has an optimum at a mass of 100 − 150 MeV, with masses lower than 100 MeV now excluded. The ω/ω 3 trajectory again has an optimum at the high mass of 350 MeV, and the a 1 trajectory now has an even wider range of nearly equivalent mass than before, m u/d = 0 − 250 MeV.    (2) and (3) likewise summarize the results of the two types of fits for the (n, M 2 ) trajectories, that of the rotating string and of the WKB approximation.

Summary of results for the mesons
The higher values of α and a always correspond to higher values of the endpoint masses, and the ranges listed are those where χ 2 is within 10% of its optimal value.  Table 3. The results of the meson WKB fits, all in the (n, M 2 ) plane. The ranges listed are those where χ 2 is within 10% of its optimal value. N is the number of data points in the trajectory.  Table 4. The ranges of the intercept from tables (1)-(3) adjusted to fits to n + L. The right-most column is for the WKB fits and the other two the regular massive fits.

L vs. J and the values of the intercept
Table (4) offers a comparison between the values of the intercept when fitting M 2 to (n + L) instead of to (n + J). In other words, they are the values obtained when identifying the J on the left hand side of eq. (3.3) with the orbital, as opposed to total, angular momentum. The advantage of this choice is that the results are made more uniform between the different trajectories when doing the fits to (n + L). With the exception of the χ b trajectory, all the trajectories have negative intercepts between (−0.55) and zero, with the intercept being closer to zero as the endpoint masses grow heavier.

The length of the mesonic strings
Lacking the basic string theory of QCD, one may revert to an effective low energy theory on long strings [20]. The effective theory is expanded in powers of √ α l . In such a framework, the semi-classical approximation describes the system more faithfully the longer the string is. To examine the issue of how long are the rotating strings with massive endpoints that describe the mesons we have computed the length of the strings associated with various mesons.
Using eqs. (3.2) for the energy and the relation (3.4) between q 1 and q 2 we exctract the two velocities given the total mass E and the two endpoint masses. Then, again by using eq. (3.4) and q i = ωl i , we have with the total string length between the two masses being l 1 + l 2 .
In table (5) we present the values of (l 1 + l 2 )/ √ α for the fitted (J, M 2 ) trajectories. We can see that for the u, d, and s mesons the lengths are not too small, with the ratio l/ √ α starting from 2 − 3 for the low spin mesons and increasing as J increases to values for which the string can be called a long string more confidently. For the mesons involving Traj. L = 0 L = 1 L = 2 L = 3 L = 4 L = 5 π/b 3.6 5.  Table 5. l s / √ α for all the states used in the (J, M 2 ) trajectories (arranged here by their orbital angular momentum L). For each trajectory, the length was calculated at the mass in the midpoint of the range given in summary table (1), except for the K * , where we used m u/d = 60, m s = 220.
c quarks the lowest spin states are short strings, but the higher L states (the maximum we have for those is L = 2) are getting to be long enough. For the bb meson, the lowest state's string length tends to zero, and the highest spin state used (again with L = 2) has l/ √ α of only about 2.

Summary
We have seen that the Regge trajectories of mesons involving the s, c, and b quarks are generally best fitted when introducing endpoint masses to the relativistic string. These masses help account for the deviations one can observe from the linear Regge trajectories.
The masses of the heaviest quarks, the c and the b, as obtained from the fits, are the values one would assign to them as constituent quarks: around 1500 MeV for the c quark, and 4730 for the b. This means that in our model the mass of the lowest cc and bb states is due only to the quark masses.
For the s quark we have a different picture, where the mass is somewhere between the QCD mass of 100 MeV and the constituent mass of around 500 MeV. The results for the s quark vary from 200-300 MeV in the best fits for the strange (i.e. sū or sd) K * meson to 400 MeV for the ss states of the φ trajectory. The radial trajectory for the ss gives the mass at an even higher value of 500 MeV. It is not clear if this discrepancy can be attributed to an actual physical feature of the mesons, that will result in different end-point masses for different physical configurations. We know, though, that we would have liked not to see a discrepancy between the mass obtained from the (J, M 2 ) fit and the one obtained from the (n, M 2 ) fit -the ground state in these two trajectories is the same, and naturally we don't expect the same physical state to have two different endpoint masses.
A similar discrepancy was found between the charmed D meson and the cc trajectories, but in this case the situation is reversed: the mass obtained when fitting the states with a single c quark was higher (1600-1700 MeV) than the 1400-1500 MeV mass of the c quark in the cc states. We also have the charmed/strange meson D * s , which points towards the higher masses for the s and c quarks, of around 400 MeV for the s and 1600 MeV for the c, where decreasing one of the masses would then require the other to increase even more.
One should note, however, that of all the relevant trajectories, involving s or c quarks, only two have more than three available data points. Those trajectories with only three points tend to pinpoint the mass at a very specific value, with a small margin of error, and it is hard to estimate the realistic value of the error in such a measurement. When we do not assume two equal endpoint masses, increasing the number of fitting parameters from three to four, we have to contend ourselves with the optimum lying along a curve in the m 1 , m 2 plane, rather than an accurate determination of both masses. This is also true of the 5-point The light quark (u and d) trajectories are the most problematic in terms of the mass, even though the light meson sector is the richest in data. In most cases, no real optimally fitting mass was found. A typically found range would have all the masses between 0 and 200 MeV as nearly equivalent. In those cases where an optimum is easily discernable from our fits, it is the massless linear fit.
For the radial trajectories, the masses obtained for the tend to be higher, as they seem to be more prone to deviations from the linear trajectories. These deviations sometimes result in optimal fits for relatively high masses, but this could be due to our simple model misinterpreting the more complex physical phenomena that are behind those non-linearities.
The slopes that were found, on the other hand, are quite uniform for the light quark trajectories. The (J, M 2 ) trajectories have a slope in the range 0.80 − 0.90 GeV −2 , and this slightly decreases to 0.78 − 0.84 GeV −2 in the (n, M 2 ) plane fits. The slope for the strange meson is also in this range, while for the ss states the optimum is found with a higher slope of around 1.1 GeV −2 , in both planes. The charmed and cc mesons are best fitted in the (J, M 2 ) plane with a similar value of approximately 1 GeV −2 for the slope -and only when adding the appropriate mass for the c quark. This is what allows the universal slope fit in section 4.1.5, which had an optimum for the slope α = 0.884 GeV −2 .
This uniformity of the slope is then broken. First, the bb trajectory was excluded from the universal (J, M 2 ) fit because its optimal slope is much lower, at 0.64 GeV −2 , and in the (n, M 2 ) plane both the cc and bb trajectories have a slope of 0.42 − 0.50 GeV −2 . For the bb the difference between the (J, M 2 ) slope and the (n, M 2 ) slope is not too large, or different from what we have seen for the u/d and s quark trajectories, but for the cc the slope is nearly halved when moving from the trajectory in the (J, M 2 ) plane to the trajectory beginning with the same ground state in the (n, M 2 ) plane.
We can then divide the trajectories into four main groups based on the approximate value of their best fitting slope. In the (J, M 2 ) plane we have α ≈ 0.9 GeV −2 for the u, d, s, and c quark trajectories, and α ≈ 0.6 GeV −2 for the single bb trajectory. In (n, M 2 ) we have lower values for the slope, around 0.8 GeV −2 for the u, d, and s, and α ≈ 0.5 GeV −2 for the last group which now includes both the c and the b. We had non a priori reason to anticipate a dependence of the slope (or equivalently, the string tension) on the mass, nor the behavior it seems to exhibit, with the slope being more or less constant for the lightest quark trajectories, and then dropping for the heavier mesons. The difference between the two slopes for the cc is especially puzzling.
The fact that the stringy description of mesons built from b quarks refuses to unify with the one that is associated with lighter quark mesons, is presumably related to the fact that for these mesons the string length, in units of the string basic length √ α , is not really very long as we have seen in table (5). This is true also of the cc and bb in the (n, M 2 ) plane, where the trajectories we have are comprised only of states with low angular momentum, and hence, short string lengths.
As for the intercept, the only assumption that was made regarding it in the fits was that it was constant (i.e. independent of J, Eand m). The results appear at first glance to be quite scattered, with both positive and negative values appearing in our results. This should not surprise us since the assumption is in fact not justified. It is clear that the intercept, that gets contributions from both the Polchinski-Strominger term and from the Casimir term, is some function of m 2 T and not a constant. In the picture of an effective string, the analog of expanding in √ α l is expanding in m √ T J [21]. In such an expansion the J 0 term of the intercept can for low values of J be contaminated by terms of negative powers of J.
One thing that can make it easier to compare the values for the intercept between different trajectories is moving from the fits to (n + J) to fits to (n + L) -from the total to the orbital angular momentum. Then we get all the light and strange (L, M 2 ) trajectories have a somewhere between (−0.5) and (−0.2), and this value increases to the upper limit of 0 for the heavier quarks. In the (n, M 2 ) plane the picture is similar, but there the ss trajectory already has a = 0.
The transition from the (J, M 2 ) plane to the (L, M 2 ) plane is easy to implement in practice, as it only requires the occasional shift of the intercept by one unit, but it requires us to do away completely with spin. Our model does not include spin in the first place, but it seems odd that ignoring it completely, by doing the fits to L, should be rewarded with the added consistency in a.
The WKB model was used in a way that distinguished between total and orbital momentum quite strongly, and the most significant change in the WKB fit results is in the values of the intercept. These are generally more negative, but there the values for the endpoint masses remain roughly the same in all trajectories, and there is always a small decrease in the values of the slope, relative to their values when fitting to the rotating string model. In terms of the goodness of the fits, the WKB model does not offer any significant improvement.
There are several questions and research directions that one can further investigate: • Our model does not incorporate spin degrees of freedom. It is well known that the spin and the spin-orbit interaction play an important role in the spectra of mesons. Thus the simple rotating string models that we are using have to be improved by introducing spin degrees of freedom to the endpoints. One way to achieve it is by replacing the spinless relativistic particle with one that carries spin or in the holographic framework associating spin to the vertical segments of the holographic string.
• Our model assumes chargeless massive endpoint particles. The endpoint of a string on a flavor brane carries a charge associated with the symmetry group of the flavor branes. Thus it is natural to add and interaction, for instance abelian interaction, between the two string endpoints. It is easy to check that this change will introduce a modification of the intercept.
• In our WKB analysis we have used only the simplest linear potential. One obvious generalization, which probably will work better for heavy mesons, is the Cornell potential where a 1 r potential term is added to the linear one.
• As was discussed in the introduction, the models we are using are not the outcome of a full quantization of the system. We have been either using a WKB approximation for the spectra in the (n, M 2 ) plane or using an ansatz or J → J +n−a for passing from the classical to the quantum model. In [14] the quantization of the rotating string without massive endpoints was determined. The quantum Regge trajectories associated with strings with massive endpoints require determining the contributions to the intercept to order J 0 from both the " Casimir" term and the Polchinski-Strominger term [21]. Once a determination of the intercept as a function of m 2 T is made, an improved fit and a re-examination of the deviations from a universal model should be made.
• We have looked in the present work into only one feature of meson physics -the Regge trajectories of the spectra. One additional property that can be explored is the width of the decay of a meson into two mesons. The stringy holographic width was computed in [12]. A detailed comparison with decay width of mesons can provide an additional way to extract string endpoint masses that can be compared to the one deduced from the spectra.
• There also remains the other sector of the hadronic spectrum -the baryons. As mentioned above the spectra of these hadrons could also be examined using a stringy model with or without massive endpoints. In addition, closed strings can be used to describe glueballs [9], and using a stringy model we can search for evidence of glueballs among the observed flavorless baryonless spectrum.
• Eventually we have in mind to perform "precise comparisons" using holographic rotating string models instead of the model of rotating string with massive endpoints in flat space-time.
Traj. I J P C State Traj. I J P C State

A Individual trajectory fits
In this appendix we present the results for the individual trajectory fits in detail. This includes a specification of the states used for each fit, the results for all fitting parameters (masses, slope, and intercept), and the values for χ 2 in each fit. The plots of all trajectories and their fits in the (J, M 2 ) or (n, M 2 ) planes are also presented here.
The experimental data is taken almost entirely from the 2012 Review of Particle Physics [22]. Other sources are indicated where relevant. The observation of linear Regge behavior in the hadron spectrum dates back to the 1970s [1] but has remained the subject of much more recent work as new states are continually discovered in experiment. The heavier sector of the meson spectrum in particular is getting richer and richer in data [23] [24]. [25] and [26] are examples of relatively recent analyses of the meson spectra using Regge trajectories, as is the work of Ebert, Faustov, and Galkin, which covers the spectrum from light [27], to light-heavy [28], to heavy-heavy mesons [29] using a relativistic quark model. The selection of trajectories was in part based on these later works, where we have included in our analysis only those trajectories with three or more data points.
Traj. I(J P C ) n State Traj. I(J P C ) n State π 1(0 −+ ) 1 π(1300) Table 7. The states used in the (n, M 2 ) trajectory fits. Note that we assign n = 0 to the ground state rather than n = 1.

A.1 Summary of states used in the fits
We begin by presenting, in tables (6) and (7), all the states we have used in our analysis. The two tables are for the trajectories in the (J, M 2 ) and (n, M 2 ) planes respectively.
A.2 Trajectories in the (J, M 2 ) plane

A.2.1 Light quark mesons
The states in this section are all comprised of u and d quarks only. We assume in our analysis that the two lightest quarks are equal in mass, and make no attempt to differentiate between them. with χ 2 l = 8.85 × 10 −4 . It is optimal. We can place the limit on the mass at m = 60, where χ 2 m /χ 2 l = 1.10.

A.2.2 Strange and ss mesons
Strange. The K * trajectory: This is the trajectory depicted on the left of figure (13). The states are K * (892)1 − , K * 2 (1430)2 + , K * 3 (1780)3 − , K * 4 (2045)4 + , and K * 5 (2380)5 − . These are comprised of one light u or d quark and one s quark. Since we expect a difference between the mass of the s quark and that of the light quarks we fit to a formula with two different masses, m s > m u/d .
The linear fit has α = 0.849, a = 0.33 and χ 2 l = 7.15 × 10 −4 . The optimal massive fits have m u/d + m s ≈ 300, but there is no way to determine the masses separately from these fits. It is not possible even to distinguish between the symmetric case where the two masses are equal and the other extreme where one of the endpoints is massless, or nearly massless, and the other is not. The optimal fits are obtained on a curve on the (m u/d , m s ) plane -m   with χ 2 l = 3.41 × 10 −4 , but the optimal fit is far from the linear, with endpoint masses in the range of the constituent c quark mass: m c = 1500, α = 0.979, a = −0.09 with χ 2 m = 5 × 10 −7 (χ 2 m /χ 2 l = 0.002). Aside from the improvement in χ 2 , by adding the mass we also get a value for the slope (and to a lesser extent, the intercept) that is much closer to that obtained in fits for the light meson trajectories.

A.3.2 ss mesons
ss. The φ trajectory: There are three φ states with J P C = 1 −− : φ(1020), φ(1680), and φ(2170). Assuming the assignment n = 0, 1, 2 for the three states we get the best fit is the linear fit with α = 0.543, a = 0.44 It has χ 2 l = 3.11 × 10 −4 . This is inconsistent with previous results, both in the resulting slope and s quark mass. We can get a fit with parameters closer to what we expect them to be if we make the assignment n = 0, 1, 3 for the three states. The fits with this assignment are what is depicted in figure (18).
The linear fit then is α = 0.724, a = 0.21 with χ 2 l = 129 × 10 −4 . The best massive fit has m s = 515, α = 1.098, a = 1.00 with χ 2 m /χ 2 l = 0.16. We can fit with a lower mass as well. The fit m s = 400, α = 0.909, a = 0.84 has χ 2 m /χ 2 l = 0.52. The best WKB fit, using this same assignment for n, is m w = 515, α = 1.027, a = 0.00 The mass of the missing n = 2 state is predicted to be in the range M = 1949−1963 MeV. Interestingly, there is a state with the appropriate quantum numbers I G (J P C ) = 0 − (1 −− ) at that exact mass -the ω(1960). If we add this state in this trajectory, the fits don't change much: α = 0.730, a = 0.22 is the optimal linear fit -χ 2 l = 86 × 10 −4 , and the optimal massive fit is still at the same mass  WKB fit plots: Lastly, we include, in figures (20) and (21), the plots of all the (n, M 2 ) trajectories and their respective WKB fits.