Rotating strings confronting PDG mesons
Abstract
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 and planes, where and are the angular momentum and radial excitation number respectively. We start from states comprised of and quarks alone, move on to trajectories involving and quarks, and finally analyze the trajectories of the heaviest observed 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 trajectories involving , , , and quarks. We include a list of predictions for higher mesons in both and .
Keywords:
Tel Aviv University, Ramat Aviv 69978, Israel
1 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 centuryCollins:book (). 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 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 nonconformal and nonsupersymmetric. Prototype backgrounds of such a nature are that of a brane compactified on Witten:1998zw () (and its noncritical analogous modelKuperstein:2004yf ()). 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 incorporatedSakSug ().
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 probebranes which associate with scalar and vector mesons respectively. (See for instance SakSug (),Casero:2005se (),oai:arXiv.org:0806.0152 ()).
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 and the angular momentum nor the linearity between and the radial excitation number . 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 stringsPeeters:2006iu ().^{1}^{1}1The bottomup approach of the softwall model has been proposed in order to admit linearity oai:arXiv.org:hepph/0602229 ().This model suffers from certain other drawbacks and does not admit linearity. It seems fair to say that generically the spectra of the bulk and probe modes associated with confining backgrounds do not admit the Regge behavior.
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}^{2}2This approach was used also in Imoto:2010ef (). or folded closed strings for mesons. Regge trajectories of the latter in various confining backgrounds were analyzed in oai:arXiv.org:hepth/0311190 (). It is a very well known feature of rotating stringy configurations in flat spacetime.
The major difference between rotating open strings in holographic backgrounds and those in flat spacetime is that the former do not connect the two endpoints along the probe brane, but rather stretch along the “wall”^{3}^{3}3In top down models the “wall” refers to the minimal value of the holographic radial direction. and then connect vertically to flavor branes (See figure (1)). The figure depicts the special case of . In a similar manner we can have by attaching the “vertical” strings segments to different flavor branes.
In Kruczenski:2004me () it was shown that classically the holographic rotating string can be mapped into one in flat spacetime with massive endpoints.^{4}^{4}4In Kruczenski:2004me () the map was shown for a particular class of models. One can generalize this map to rotating open strings in any confining backgroundcobitalks (). 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 toKruczenski:2004me ()
(1) 
where is the string tension, is the holographic radial coordinate, is its minimal value (the “wall”), is the location of the flavor branes and and 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 decaysPeeters:2005fq () 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 spacetime with massive endpoints as a model for mesons and leave aside holography altogether.^{5}^{5}5Approximating the “vertical segment” with the massive endpoints is reminiscent of a similar approximation done with holographic Wilson lines oai:arXiv.org:hepth/9911123 (). A comparison between mesons and holographic rotating strings, rather than massive strings in flat spacetime, is deferred to future work.
The theoretical models we use are rather simple. We start from an action that includes a NambuGoto 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 and angular momentum of the system. Unlike the massless case, for massive endpoints there is no explicit relation between for instance and , but rather and can be written in terms of , and , where is the angular velocity and is the string length. For two limits of light massive endpoints where and heavy ones where one can eliminate (the two limits involve taking and respectively) and get approximated direct relations between and .
Going beyond the classical limit for rotating strings is a nontrivial 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
(2) 
where the slope , is the radial excitation number and is the intercept.
In a recent paperHellerman:2013kba () 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 dimensions, an open string with no radial excitation () indeed admits with . This is a nontrivial result since the calculation of the intercept (to order ) yields in D dimensions the result , where the first term is the usual “Casimir” term and the second is the PolchinskiStrominger 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 addressedChodos:1973gt ()Baker:2002km ()Zahn:2013yma ().
Falling short of the full quantum expression for the Regge trajectories one can use a WKB approximated determination of the trajectoriesSchreiber:2004ie (). 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

The “massive trajectory” which is based on the classical expressions for and where the latter includes assumed quantum correction, again in the form of . The trajectories then read
(3) (4) These expressions reduce to the linear trajectory equation in the limit .

The WKB approximation for the linear potential which takes the form
(5) where .
The parameters that we extract from the fits are the string tension (or the slope ), 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 nonrelativistic 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 as a function of the angular momentum and of the radial excitation . In both categories we discuss light quark mesons, strange mesons, charmed mesons and mesons containing 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 and the total angular momentum , 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.
2 Basic theoretical model
2.1 Classical rotating string with massive endpoints
We describe the string with massive endpoints (in flat spacetime) by adding to the NambuGoto action,
(6) 
a boundary term  the action of a massive chargeless point particle
(7) 
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 to be the boundaries, with an arbitrary constant with dimensions of length.
The variation of the action gives the bulk equations of motion
(8) 
and at the two boundaries the condition
(9) 
It can be shown that the rotating configuration
(10) 
solves the bulk equations (8) for any choice of . We will use the simplest choice, , from here on.^{6}^{6}6Another common choice is . Eq. (9) reduces then to the condition that at the boundary,
(11) 
with .^{7}^{7}7Notice that in addition to the usual term for the mass, the tension that balances the “centrifugal force” is . 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:
(12) 
(13) 
Solving the integrals, and defining  physically, the endpoint velocity  we write the expressions in the form
(14) 
(15) 
The terms proportional to are the contributions from the endpoint masses and the term proportional to is the string’s contribution. These expressions are supplemented by condition (11), which we rewrite as
(16) 
This last equation can be used to eliminate one of the parameters and from and . Eliminating the string length from the equations we arrive at the final form
(17) 
(18) 
These two equations are what define the Regge trajectories of the string with massive endpoints. They determine the functional dependence of on , where they are related through the parameter ( when ). 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 , we have an expansion in :
(19) 
from which we can easily see that the linear Regge behavior is restored in the limit , and that the first correction is proportional to . The Regge slope is related to the string tension by . The high mass limit, , holds when . Then the expansion is
(20) 
2.2 The WKB approximation
We follow here the approach of E. SchreiberSchreiber:2004ie (), 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:
(21) 
If the particle is at the end of a rotating rod of length , then . With the usual replacement of , we arrive at the one dimensional differential equation to be solved
(22) 
If we define
(23) 
then the spectrum of the system is obtained, in the WKB approximation, by the quantization condition
(24) 
The limits of the integral and are the classical “turning points”  those points where the integrand, , is zero. The condition that the integral be an integer multiple of implies the relation between and the energy eigenvalues . How we continue from here depends on our choice of the potential . Also, we have to decide how to relate the total angular momentum with the momentum carried by the point particles, , and the angular momentum carried by the string itself, which we’ll call .
If we treat the string as a classical rotating rod, then the (nonrelativistic) expression for its energy is
(25) 
Another option is the quantum mechanical expression for a string fixed at both endsArvis:1983fp ()
(26) 
More general potentials can also include a spinorbit 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 to zero. Namely, to take the linear potential . To solve the integral, we also have to assume , so the state has no angular momentum at all.
(27) 
with . 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 , and . The equation is now of the form
(28) 
with as always, and now redefined to be
(29) 
The high mass expansion () of the above expression is similar to the one obtained for the classical rotating string:
(30) 
Comparing this with the high mass limit for the classical rotating string, in eq. (20), 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 the classical rotating string:
(31) 
The leading order term now being proportional to , as opposed to the of the expansion in eq. (19).
3 Fitting models
3.1 Rotating string model
We define the linear fit by
(32) 
where the fitting parameters are the slope and the intercept, .
For the massive fit, we use the expressions for the mass and angular momentum of the rotating string, eqs. (17) and (18), generalized to the case of two different masses, and we add to them, by hand, an intercept and an extrapolated dependence, assuming the same replacement of .
(33) 
(34) 
We relate the velocities and can be related using the boundary condition (11), from which we have
(35) 
so the functional dependence between and is still through only one parameter . With the two additions of and , the two equations reduce to that of the linear fit in (32) in the limit where both masses are zero. Now the fitting parameters are and as before, as well as the the two endpoint masses and . For a lot of the cases we assume and retain only one free mass parameter, .
3.2 WKB model
The third fitting model is the WKB. It is defined by
(36) 
where 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 with . The angular momentum is then carried only by the quarks. We chose to identify with the orbital angular momentum . For those states with we solve the integral and use the resulting formula,
(37) 
where . If we can’t make that assumption we solve eq. (36) numerically. The fitting parameters are again , , and .
3.3 Fitting procedure
We measure the quality of a fit by the dimensionless quantity , which we define by
(38) 
and are, respectively, the measured and calculated value of the mass of the th particle, and the number of points in the trajectory. We will also use the subscripts , , or to denote which fitting model a given value of pertains to. So, for instance, is the ratio of the value of obtained by a linear fit to that of a massive fit of the same trajectory. A more common definition of would have the standard deviation in the denominator, but we have used . We do this mostly for reasons of practicality. The high accuracy to which some of the meson’s masses are known makes (when defined using as the denominator) vary greatly with very small changes in the fitting parameters. ^{8}^{8}8For example, the mass of the is MeV. Fixing the mass and slope at values near the minimum for as defined in (38), a change to the intercept from (the minimum using our definition) to takes (using the standard definition) from to , and going to takes us to . This type of behavior may also result in our fitting algorithms missing the optimum entirely. We feel the kind of precision required then in the fits is unnecessary for the purposes of our work. By using definition (38) for 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.
4 Fit results
This section discusses the results of our fits. The fits to the trajectories in the plane and the trajectories in the 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 or planes, can be found in appendix A.
A note on units and notation:
When units are not explicitly stated, they are GeV for and MeV for masses. The intercept is dimensionless. If the letters , , or , are used as subscripts, they will always refer to the linear, massive, and WKB fits respectively.
4.1 Trajectories in the plane
4.1.1 Light quark mesons
We begin by looking at mesons consisting only of light quarks  and . We assume for our analysis that the and quarks are equal in mass, as any difference between them would be too small to reveal itself in our fits. 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 , , , and .
Of the four trajectories, the two trajectories, of the and the , show a weak dependence of on . Endpoint masses anywhere between and MeV are nearly equal in terms of , and no clear optimum can be observed. For the two trajectories, of the and , the linear fit is optimal. If we allow an increase of up to in , we can add masses of only MeV or less. Figure (2) presents the plots of vs. and for the trajectories of the and and shows the difference in the allowed masses between them.
The slope for these trajectories is between for the two trajectories starting with a pseudoscalar ( and ), and for the trajectories beginning with a vector meson ( and ). The higher values for the slopes are obtained when we add masses, as increasing the mass generally requires an increase in to retain a good fit to a given trajectory. This can also be seen in figure (2), in the plot for the trajectory fit.
4.1.2 Strange and mesons
We analyze three trajectories in the involving the strange quark. One is for mesons composed of one quark and one light quark  the , the second is for mesons  the trajectory of the , and the last is for the charmed and strange , which is presented in the next subsection with the other charmed mesons.
The trajectory alone cannot be used to determine both the mass of the quark and the mass of the . The first correction to the linear Regge trajectory in the low mass range is proportional to . This is the result when eq. (19) is generalized to the case where there are two different (and small) masses. The plot on the left side of figure (3) shows as a function of the two masses.
The minimum for the trajectory resides along the curve . If we take a value of around MeV for the quark, that means the preferred value for the is around 220 MeV. The higher mass fits which are still better than the linear fit point to values for the quark mass as high as 350 MeV, again when is taken to be 60 MeV. The slope for the fits goes from in the linear fit to near the optimum to for the higher mass fits.
The trajectory of the 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 , . The value of 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 MeV still have or less, and the slope then has a value close to that of the other fits, around GeV.
4.1.3 Charmed and mesons
There are three trajectories we analyze involving a charm quark. The first is of the , comprised of a light quark and a quark, the second is the with a and an , and the third is  the . All trajectories have only three data points.
For the meson, the optimal fit has , and . In this case, unlike the result for the 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 has a good fit consistent with the previous and fits at , , and . The plots of vs. the two masses ( and /) can be seen in figure (4).
In the same figure, we have as a function of the single mass for the trajectory. The minimum there is obtained at MeV, where the slope is GeV.
It is worth noting that while the linear fit results in values for that are very far from the one obtained for the , , and quark trajectories  , , and for the , , and 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 .
4.1.4 mesons
The last of the trajectories is that of the meson, again a trajectory with only three data points. The fits point to an optimal value of , exactly half the mass of the lowest particle in the trajectory. The slope is significantly lower than that obtained for other mesons, at the optimum. The bottom plot in figure (4) shows for the trajectory.
4.1.5 Universal fit
Based on the combined results of the individual fits for the trajectories of the , , , and quark mesons, we assumed the values
(39) 
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 trajectory from this fit, but we can include the three trajectories involving a 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 and the intercepts that would give the best overall fit to the nine trajectories of the , , , , , , , , and mesons. The best fit of this sort, with the masses fixed to the above values, was
(40) 
and it is quite a good fit with . The trajectories and their fits are shown in figure (5). The values obtained for the masses vs. their experimental counterparts are in appendix B.
4.2 Trajectories in the plane
4.2.1 Light quark mesons
In the light quark sector we fit the trajectories of the and , the , the , and the and .
The has a very good linear fit with GeV, that can be improved upon slightly by adding a mass of 100 MeV, with the whole range MeV being nearly equal in .
The offers a similar picture, but with a higher and a wider range of available masses. Masses between and are all nearly equivalent, with the slope rising with the added mass from to GeV.
The and trajectories were fitted simultaneously, with a shared slope and mass between them and different intercepts. Again we have the range to MeV, GeV, with MeV being the optimum. The preference for the mass arises from nonlinearities in the trajectory, as the when fitted alone results in the linear fit with GeV being optimal.
The and trajectories were also fitted simultaneously. Here again the higher spin trajectory alone resulted in an optimal linear fit, with GeV. The two fitted simultaneously are best fitted with a high mass, , and high slope, GeV. Excluding the ground state from the fits eliminates the need for a mass and the linear fit with GeV is then optimal. The mass of the ground state from the resulting fit is MeV. This is odd, since we have no reason to expect the to have an abnormally low mass, especially since it fits in perfectly with its trajectory in the plane.
The fit for the with the ground state included is shown in figure (6), along with the fit for the , which has .
4.2.2 mesons
For the we have only one trajectory of three states, that of the . There are two ways to use these states. The first is to assign them the values . Then, the linear fit with the slope GeV 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 and for the highest state and obtained the values for the optimal fit. These are much closer to the values obtained in previous fits.
The missing state is predicted to have a mass of around MeV. Interestingly, there is a state with all the appropriate quantum numbers at exactly that mass  the , and that state lies somewhat below the line formed by the linear fit to the radial trajectory of the . Even if the is not the missing (or predominantly ) state itself, this could indicate the presence of a state near that mass.
4.2.3 mesons
Here we have the radial trajectory of the , consisting of four states.
The massive fits now point to the range MeV for the quark mass. The biggest difference between the fits obtained here and the fits obtained before, in the plane is not in the mass, but in the slope, which now is in the range GeV, around half the value obtained in the angular momentum trajectories involving a quark  .
It is also considerably lower than the slopes obtained in the 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 plane like the one we had in the plane.
4.2.4 mesons
There are two trajectories we use for the mesons.
The first is that of the meson, with six states in total, all with . For this trajectory we have an excellent fit with MeV and the slope GeV. It is notable for having a relatively large number of states and still pointing clearly to a single value for the mass.
The second trajectory is that of the  . Here we have only three states and the best fit has a slightly higher mass for the quark  MeV  and a higher value for the slope GeV.
4.3 WKB fits
The WKB fits are all done in the 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. (34), which was used for all the previous fits, we ultimately have a functional dependence of the form
(41) 
The contribution from the angular momentum, when fitting trajectories in the plane, amounts to nothing more than a shift of the axis, and can be fully absorbed into the intercept . Eq. (36), 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, in the notation of eq. (36), is the orbital angular momentum .
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. (20) and (30), 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 (). The WKB fits generally allow for higher masses for the light quarks, as can be seen in figure (9). For the trajectories we actually obtain a minimum around MeV, where before it was less than half that value. The trajectory now has an optimum at a mass of MeV, with masses lower than MeV now excluded. The trajectory again has an optimum at the high mass of MeV, and the trajectory now has an even wider range of nearly equivalent mass than before, MeV.
For the heavier quark trajectories we obtain the same masses as before. The fits for the trajectory of the result in a mass of MeV for the quark. The trajectory narrows down somewhat the mass of the quark to the range MeV. The bottomonium trajectories of the and indicate the value of the quark mass to be or MeV, respectively. The values of as a function of the mass for these four trajectories can be seen in figure (10).
4.4 Summary of results for the mesons
Table (1) summarizes the results of the fits for the mesons in the plane. Tables (2) and (3) likewise summarize the results of the two types of fits for the trajectories, that of the rotating string and of the WKB approximation. The higher values of and always correspond to higher values of the endpoint masses, and the ranges listed are those where is within 10% of its optimal value.
Traj.  

Traj.  

/  
Traj.  

/  
4.5 L vs. J and the values of the intercept
Table (4) offers a comparison between the values of the intercept when fitting to instead of to . In other words, they are the values obtained when identifying the on the left hand side of eq. (34) 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 . With the exception of the trajectory, all the trajectories have negative intercepts between and zero, with the intercept being closer to zero as the endpoint masses grow heavier.
4.6 The length of the mesonic strings
Lacking the basic string theory of QCD, one may revert to an effective low energy theory on long stringsAharony:2013ipa (). The effective theory is expanded in powers of . In such a framework, the semiclassical 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. (33) for the energy and the relation (35) between and we extract the two velocities given the total mass and the two endpoint masses. Then, again by using eq. (35) and , we have
(42) 
with the total string length between the two masses being .
In table (5) we present the values of for the fitted trajectories.
We can see that for the , , and mesons the lengths are not too small, with the ratio starting from for the low spin mesons and increasing as increases to values for which the string can be called a long string more confidently. For the mesons involving quarks the lowest spin states are short strings, but the higher states (the maximum we have for those is ) are getting to be long enough. For the meson, the lowest state’s string length tends to zero, and the highest spin state used (again with ) has of only about 2.
Traj.  

  3.6  5.2  6.4  7.4    
2.3  4.4  5.8  7.0  8.0  8.9  
1.8  4.3  5.8  7.0  8.0    
2.8  4.8  6.2  7.4  8.4  9.2  
2.4  4.3  5.7  6.9  7.9    
1.0  3.1  4.5        
0.6  3.0  4.3        
0.6  2.6  3.8        
0.4  2.1  3.2        
0.0  1.5  2.3       