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\begin{document}
\preprint{LA-UR-98-3390}
\title{Comment on ``Regge Trajectories for All Flavors''}
\author{\\ L. Burakovsky\thanks{E-mail: BURAKOV@T5.LANL.GOV} \ and \ 
T. Goldman\thanks{E-mail: GOLDMAN@T5.LANL.GOV} \
\\  \\  Theoretical Division, MS B285 \\  Los Alamos National Laboratory \\ 
Los Alamos, NM 87545, USA }
\date{ }
\maketitle
%\begin{abstract}
%\end{abstract}
%\bigskip
%{\it Key words:} flavor symmetry, quark model, charmed mesons, 

%PACS: 11.30.Hv, 11.55.Jy, 12.39.-x, 12.40.Nn, 12.40.Yx, 14.40.Lb
\bigskip

%\section*{  }
In a recent Letter \cite{FPS} (and in ref. \cite{FS}), Filipponi, Pancheri and
Srivastava report on the construction of a formula for linear Regge 
trajectories for all quark flavors:
\beq
\alpha _{j\bar{i}}(t)=0.57-\frac{(m_i+m_j)}{{\rm GeV}}+\frac{0.9\;{\rm GeV}^2}{
1+0.2\;\!(\frac{m_i+m_j}{{\rm GeV}})^{3/2}}\;\!t,
\eeq
where $m_i,m_j$ are the corresponding constituent quark masses for the $j\bar{
i}$ trajectory. 
%They compare the results given by (1) for $j=c,i=n(=u,d)$ 
%trajectory with the experimental data on the inclusive $D$ meson production, 
%using the standard parametrization
%\beq
%\frac{d\sigma }{dx_F}\propto (1-x_F)^n,\;\;\;n=1-2\alpha (0),
%\eeq
%and conclude that their formula (1) is in good agreement with experiment, as
%far as the $D$ meson production is concerned. Here we wish to comment on both
%the formula (1) and comparison of this formula with the $D$ meson production 
%data. 

%As analysed in detail in our paper \cite{BG}, the formula (1) with given
%constant values of the constituent quark masses contradicts the heavy quark 
%limit $m_Q\gg m_n,$ in which one expects $M(Q\bar{Q})\simeq 2M(Q\bar{n}).$ We
%have shown in \cite{BG} that only additivity of inverse Regge slopes 
%satisfies the heavy quark limit (as well as the formal chiral limit $m_n
%\rightarrow 0).$ Such additivity results from string type models \cite{Kai}.
 
As the authors of \cite{FPS,FS} remark, no unique quark mass can be extracted
from (1), and each trajectory $\alpha _{j\bar{i}}(t)$ rather corresponds to 
its own set $(m_i,m_j).$ The values of $m_i$ can be extracted by using the
vector meson masses with hidden flavor into Eq. (1): $\alpha _{i\bar{i}}(M^2_{
i\bar{i}})=1.$ Such an extraction gives (in GeV, $n=u,d,$ and the superscript 
indicates the trajectory from which the corresponding value is extracted)
%\beq
$m_n^\rho =0.05,\;m_s^\phi =0.23,\;m_c^{J/\psi }=1.70,\;m_b^\Upsilon =5.12.$
%\eeq
%(We note that the corresponding values indicated in ref. \cite{FS} by two of
%the authors, $m_n=0.03$ GeV, $m_s=0.5$ GeV, are incorrect; with, e.g., $m_s=
%0.5$ GeV, $\alpha _{s\bar{s}}(M^2(\phi ))=0.35.)$ 
%Following a similar procedure in the $j\bar{i},\;i\neq j$ sectors, one finds 
%that $m_c+m_n=1.92$ GeV, $m_c+m_s=2.08$ GeV, and $m_s-m_n=0.16$ GeV, but $m_b+
%m_n=6.02$ GeV, $m_b+m_s=6.12$ GeV, and therefore $m_s-m_n=0.10$ GeV. These 
%results show that no unique quark mass may be extracted. 
Then, the values of $m_i$'s for the $j\bar{i},\;i\neq j$ trajectories should 
be related to the above hidden-flavor values by additivity of trajectory 
intercepts. This additivity is satisfied in 
%the dual-resonance models, 
%\cite{KKY},
%two-dimensional QCD,
% \cite{BESW}, 
%the quark bremsstrahlung model 
%\cite{DB} 
%and the $q\bar{q}$-string model for hadrons 
%\cite{Kai}
two-dimensional QCD and many QCD-motivated models
(\cite{BG} and references therein), %Also, the Gell-Mann--Okubo mass 
%formula in the light quark sector is only consistent with Regge relations in 
%the approximation of equal slopes in this sector provided that additivity of 
%intercepts holds. Thus, this additivity 
and therefore should be considered as a firmly 
established theoretical constraint on Regge trajectories. 
%
It is easily seen that in the case of the trajectories (1), this
constraint implies 
%the validity of the following relations:
%\bqry
$m_n^\rho +m_c^{J/\psi }=m_n^{D^\ast }+m_c^{D^\ast },$ %& 
$m_s^\phi +m_c^{J/\psi }=m_s^{D_s^\ast }+m_c^{D_s^\ast },$ %\NL
$m_n^\rho +m_b^\Upsilon =m_n^{B^\ast }+m_b^{B^\ast },$ %& 
$m_s^\phi +m_b^\Upsilon =m_s^{B_s^\ast }+m_b^{B_s^\ast }.$
%\eqry
Thus, e.g., the parameters $m_i$ of the $D^\ast $ and $D_s^\ast $ trajectories
must be related to those of the $\rho ,\phi $ and $J/\psi $ ones, even if no 
unique values of $m_i$ can be extracted. Using now these parameters as given 
by the above relations for calculating the vector meson masses, through 
$\alpha _{j\bar{i}}(M^2_{j\bar{i}})=1,$ one finds (in MeV) $M(D^\ast )=
1882.5,$ $M(D_s^\ast )=2007.1,$ $M(B^\ast )=4566.3,$ $M(B_s^\ast )=4724.1,$
in contrast to the measured values \cite{pdg} (in MeV) $M(D^\ast )=2008\pm 2,$
$M(D_s^\ast )=2112.4\pm 0.7,$ $M(B^\ast )=5324.8\pm 1.8,$ $M(B_s^\ast )=5416.3
\pm 3.3.$ In the last two cases, the discrepancy between the calculated and 
measured values is $\sim 700$ MeV which is an unsatisfactorily large 
inaccuracy. Thus, the trajectories (1) cannot combine both meson spectroscopy 
and additivity of intercepts; fixing the parameters $m_i$ to reproduce 
spectroscopy will necessarily result in violation of the intercept additivity
constraint. We note that simple constituent quark model relations, e.g., 
%$M(D^\ast )=(M(\rho )+M(J/\psi ))/2,$ $M(D_s^\ast )=(M(\phi )+M(J/\psi ))/2,$ 
$M(B^\ast )=(M(\rho )+M(\Upsilon ))/2,$ $M(B_s^\ast )=(M(\phi )+M(\Upsilon ))
/2,$ give better values than Eq. (1): (in MeV) 
%$M(D^\ast )=1933,$ $M(D_s^\ast )=2058,$ 
$M(B^\ast )=5114,$ $M(B_s^\ast )=5240.$  
%
Moreover, the numerical values of intercepts given by (1) in the light quark 
sector contradict data. Indeed, Eq. (1) gives $\alpha _\rho (0)=0.47,$ vs. 
$\alpha _\rho (0)=0.55,$ as extracted by Donnachie and Landshoff 
from the analysis of $pp$ and $p\bar{p}$ scattering data \cite{DL}, and 
$\alpha _{K^\ast }(0)=0.29,$ vs. $\alpha _{K^\ast }(0)\approx 0.40$ as follows
from the analysis of hypercharge exchange processes $\pi ^{+}p\rightarrow 
K^{+}\Sigma ^{+}$ and $K^{-}p\rightarrow \pi ^{-}\Sigma ^{+}$ \cite{VKKT}. 
%Note that $\alpha _{K^\ast }(0)=0.29$ violates, while $\alpha _{K^\ast }(0)
%\approx 0.40$ satisfies the unitarity constraint \cite{IY} $0.39\leq \alpha _{
%K^\ast }(0)\leq 0.46.$ 
Since the values of intercepts determine the $s$-dependence of the total 
cross-sections, $\sigma _{tot}\propto s^{\alpha (0)-1},$ and the differential 
cross-section profiles, $d\sigma/dx_F\propto (1-x_F)^{1-2\alpha (0)},$ it is 
among the requirements for the theory to predict the exact numerical values of
intercepts.   

%If the quark masses vary with the state, we, of course, can no longer claim a 
%contradiction with the heavy quark limit. 
%%(In fact, the numerical values of 
%the slopes predicted by (1) are marginally consistent with inverse additivity:
%e.g., $1/\alpha ^{'}_{n\bar{n}}+1/\alpha ^{'}_{c\bar{c}}\approx 3.6\simeq 3.4
%\approx 2/\alpha ^{'}_{c\bar{n}},$ $1/\alpha ^{'}_{s\bar{s}}+1/\alpha ^{'}_{c
%\bar{c}}\approx 3.7\simeq 3.6\approx 2/\alpha ^{'}_{c\bar{s}}.)$ 
%However, in this event, the other important property of (1), namely additivity
%of intercepts, firmly established theoretically (\cite{BG} and references 
%therein), is then lost.

%It also follows that (1) cannot be used for spectroscopic purposes:
%The masses of the $c$- and $b$-quarks extracted vary in the ranges $1.7-1.87$
%GeV and $5.12-5.97$ GeV, respectively. Then, the mass of, e.g., the $B_c^
%\ast $ meson, as yet undiscovered by experiment, derived from (1) is $M(B_c^
%\ast )=6.55\pm 0.5$ GeV, with an unsatisfactorily large inaccuracy. Thus, we 
%claim that (1) does not have sufficient predictive power to be useful. 

In ref. \cite{FS}, two of the authors notice that since the flavor dependent
Regge slope $\alpha ^{'}=\alpha ^{'}(0)/(1+A\tilde{m}),\;\tilde{m}=m_i+m_j$ 
has a large negative derivative for small $\tilde{m},$ it appears that the 
condition on all the slopes in the light quark sector $\alpha ^{'}\sim 0.8-0.9$
GeV$^{-2}$ can be satisfied only with almost exact mass degeneracy in this 
sector. This fact, as noticed in ref. \cite{FS}, prevented the authors from
constructing trajectories satisfying additivity of inverse slopes which is
another constraint provided by the heavy quark limit \cite{BG}, in addition to
intercept additivity, which the trajectories (1) do not meet. Although
their remark is correct, we disagree that $\alpha ^{'}=\alpha ^{'}(0)/(1+A
\tilde{m})$ is the only form that may be used in order to construct the 
trajectory. Indeed, as we discuss in \cite{BG}, the form
$\alpha ^{'}_{j\bar{i}}=\frac{4}{\pi }\;\!\frac{\alpha ^{'}}{1+\sqrt{
\alpha ^{'}}\;\!(m_i+m_j)/2},$
where $\alpha ^{'}=0.88$ GeV$^{-2}$ is the standard Regge slope in the light
quark sector, satisfies additivity of inverse slopes, and reproduces the 
values of the slopes in agreement with those extracted from data, for the 
following constituent quark masses (in GeV):
$m_n=0.29,\;\;m_s=0.46,\;\;m_c=1.65,\;\;m_b=4.80,$
which, in contrast to the above values given by (1), are not 
atypical of values used in phenomenological quark models. 

%A simple example of a trajectory which like (1) depends 
%only on the constituent quark masses, but unlike (1) possesses additivity of 
%both intercepts and inverse slopes is
%\beq
%\alpha _{j\bar{i}}(t)=1-\lambda \frac{(m_i+m_j)}{{\rm GeV}}+\frac{\lambda }{
%{\rm GeV}}\frac{t}{(m_i+m_j)}.
%\eeq
%This trajectory is taken to describe mesonic systems for which the lowest 
%physical state has $J=1.$ With the constituent quark masses defined as $M(j
%\bar{i})=m_i+m_j,$ where $M(j\bar{i})$ is a vector meson mass, one has $\alpha
%_{j\bar{i}}(M^2(j\bar{i}))=1.$ Close {\it et al.} \cite{Close} suggested 
%$\lambda =0.75,$ but we shall take $\lambda =1$ for ease comparison with ref.
%\cite{FPS}. With this value of $\lambda ,$ the properties of the trajectory 
%(6) are considerably improved over (1): The constituent quark masses 
%extracted, $m_c=1.55-1.62$ GeV and $m_b=4.73-4.94$ GeV, vary by 4.5\%, not by 
%10-15\% as for (1) (see above). The tensor meson masses as first Regge 
%recurrences are predicted with accuracy of better than 1\%: $\alpha _{c\bar{
%n}}(M^2(D_2^\ast ))=2$ leads to
%\beq
%M(D_2^\ast )=\sqrt{M(D^\ast )\left[ M(D^\ast )+1\;{\rm GeV}\right] }=2458\pm 2
%\;{\rm MeV}
%\eeq
%vs. $2459\pm 4$ MeV measured experimentally, and similarly (in MeV) $M(D_{s2}^
%\ast )=2564$ vs. 2573.5 and $M(\chi _{c2}(1P))=3562$ vs. 3556, in 
%contrast to (1) which predicts the tensor meson masses with accuracy of $\sim 
%2.5-3$\%: $M(D_2^\ast )\approx 2.39$ GeV, $M(D_{s2}^\ast )\approx 2.50$ GeV, 
%$M(\chi _{c2}(1P))\approx 3.48$ GeV. 
%
%Moreover, the trajectory (6) predicts $n=3.02$ for $D$ meson inclusive 
%production. This value agrees with experimental data by the Fermilab E515, 
%$n\approx 3$ \cite{E515}, WA75, $n=3.5\pm 0.5$ \cite{WA75}, and WA82, 
%$n=2.9\pm 0.3$ \cite{WA82} collaborations as well as the value $n=3.7$ given 
%by (1) agrees with data referred to in \cite{FPS}. This emphasizes that 
%present experimental data are not convergent enough to draw any firm 
%conclusion on the value of $n.$ Thus, experimental data on inclusive 
%production cannot be considered as the measure of consistency of the
%results, at least, at present.

%As to the inclusive $D$ meson production, we note that present experimental 
%data are not convergent enough to draw any firm conclusion on the value of 
%$n.$ Except for the data referred to in \cite{FPS} which support the value $n=
%3.7$ given by (1), another data exist: $n=2.5\pm 0.35$ \cite{NA32}, $n=2.9\pm 
%0.3$ \cite{WA82}, $n=4.25\pm 0.33$ \cite{E653}, $n=4.03\pm 0.18$ \cite{E769}. 
%These data would support any value of $n$ from a rather wide range of 
%$\sim 3.5\pm 1.0.$ Thus, experimental data on inclusive production cannot be 
%considered as the measure of consistency of the results, at least, at present.

We believe this analysis raises serious doubts as to the suitability of the 
formula (1) for the phenomenological description of quarkonia.

\bigskip
\bigskip
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\end{thebibliography}
\end{document}
\end







