Bled Workshops in Physics Vol. 12, No. 1 P. 7

0++ states in a large-N c Regge approach*
Enrique Ruiz Arriolaa and Wojciech Broniowski b
a Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain b The H. Niewodniczanski Institute of Nuclear Physics PAN, PL-31342 Krakow, Poland, and Institute of Physics, Jan Kochanowski University, PL-25406 Kielce, Poland
Abstract. Scalar-isoscalar states (JPC = 0++) are discussed within the large-N c Regge approach. We find that the lightest f° (600) scalar-isoscalar state fits very well into the pattern of the radial Regge trajectory where the resonance nature of the states is advantageously used. We confirm the obtained mass values from an analysis of the pion and nucleon spin-
0	gravitational form factors, recently measured on the lattice. We provide arguments suggesting an alternating meson-glueball pattern of the 0++ states, which is supported by the pseudoscalar-isovector 0+- excited spectrum and asymptotic chiral symmetry. Finally, matching to the OPE requires a fine-tuned mass condition of the vanishing dimension-2 condensate in the Regge approach with infinitely many scalar-isoscalar states.
1	Introduction
The goal of this talk is to discuss various intriguing aspects of the spectrum of scalar-isoscalar states. Approaches developed in recent years may shed new light on this long-elaborated problem in hadronic physics.
Subsequent hadron resonances listed in the Particle Data Group (PDG) tables increase their mass up to the upper experimental limit of 2.5 GeV, while their width remains bound within 500 MeV. In Fig. 1 we show separately the widths of all baryons and mesons listed in the PDG tables [1] as functions of the mass of the state. One naturally expects that broad resonances, i.e., with r ~ m, escape phenomenological analysis; even if they existed, they might be missing from the PDG as difficult to assess experimentally. Note, however, that with the exception of the notorious f0(600) resonance and a few baryon and meson states, the ratio is bound by the line r/m ~ 1 /3 (the dashed line in Fig. 2).
A natural and model-independent framework to understand this feature is provided by the limit of large number of colors in QCD. Indeed, in this large-Nc limit, with g2Nc fixed, baryons are heavy with mass m = O(Nc) and width r = O(N°) [2,3], while mesons and glueballs are stable, with mass independent of Nc, namely m = O(N°), and width r suppressed as 1 /Nc and 1 /N;:, respectively. This means that r/m is suppressed (see, e.g., [4] for a review). In particular, one has r/m ~ N-1 for mesons and baryons, while r/m ~ N-2 for glueballs.1
* Talk delivered by Enrique Ruiz Arriola
1 Fig. 1 suggests that it is reasonable to assume that excited states in the spectrum follow a more accurate large-N c pattern than the ground state.
>
<U
O O-4
• y
.• • V. ••
>
<U
o n-4
• •• •
m |GcV|
m |GcV|
Fig. 1. Width of all baryons (left panel) and mesons (right panel) listed in the PDG tables [1] as a function of the hadron mass (in GeV). The surface of each point is proportional to the (2J + 1) spin degeneracy, while the intensity is proportional to the isospin degeneracy
(2I + 1).
1.0-		i.o-		• -
0.8 -	all baryons	0.8 -		all mesons -
c 0-6		c 0-6		
U " 0.4 -	•	U " 0.4 -		
0.2 o.o - , , , ,	• » * • . . , 0 . ^p 8	0.2 o.o- • ,	. . 1	
in |GcV|
in |GcV|
Fig. 2. Same as Fig. 1 for the width/mass ratio. The large-N c limit predicts F/m ~ O(N c 1 ). The dashed horizontal line corresponds to 1/3. The gray bands reflect the uncertainties of the fit F/m = 0.12(8).
On the other hand, at high excitations mesons are expected to resemble strings of length I ~ m/ff, with ct denoting the string tension (see, e.g., Ref. [5] and references therein). Actually, the decay rate of a string per unit time, r, is estimated to be proportional to the length [6,7], which immediately yields constant F/m.2 In Fig. 2 we show the ratio of widths to masses of all baryons and mesons listed in the PDG tables [1], plotted as functions of the mass of the state. If we compute the average weighted with the (2J + 1) spin degeneracy and its spread, we find (a are the remaining quantum numbers, including isospin)
— = 5J2T + 1)-^ =0.12(8)	(1)
m Ja J
for both baryons and mesons! This rather small ratio, complying to the large-Nc and string-model arguments, suggests that this is a generic feature of the hadronic spectrum rather than a lack of experimental ability to resolve too broad states. One may thus assert in this regard that the PDG spectrum is fairly complete.
The argument directly carries over to baryons, treated as a quark-diquark string.
2 Masses of resonances
A resonance may be interpreted as a superposition of states with a given mass distribution, approximately spanning the m±F/2 interval. Of course, the shape of the distribution depends on the particular process where the resonance is produced, and thus on the background. The rigorous quantum-mechanical definition of the resonance corresponds to a pole in the second Riemann sheet in the partial-wave amplitude of the corresponding decay channel, which becomes independent of the background. However, although quoting the pole is highly desirable, with a few exceptions this is not what one typically finds in the PDG.
As a matter of fact, several definitions are used: pole in the second Riemann sheet, pole in the K-matrix, Breit-Wigner resonance, maximum in the speed plot, time delay, etc. (see, e.g., [8,9]). Clearly, while all these definitions converge for narrow resonances, even for broad states we expect the masses to be compatible within their corresponding m ± r/2 intervals. As mentioned, the values listed in the PDG for a given resonance correspond to different choices and/or processes, but mostly the results are compatible within the estimated width differences.
The lowest resonance in QCD is the 0++ state fo(600) or the a— meson. It appears as a complex pole in the second Riemann sheet of the nn scattering amplitude at y/s^ = mff - irff/2 with mff = 441+J6 and Tff = 544+7® MeV t10] (see also Ref. [11]). While these are remarkably accurate, it is unclear whether these numbers can be directly used in hadronic physics. An analysis of the role played by the a as a correlated 2n exchange in the central component to the NN force shows that the complex-pole exchange does not accurately describe this effect in the range 1 fm < r < 5 fm, but prefers a value in between the pole and the Breit-Wigner approximation, mCT = 600(50) MeV [12]. The Breit-Wigner parameters are (mCT,TCT) = (841 (5),820(20)) MeV [13]. For the time delay method we obtain (mCT, rCT) ~ (475,630) MeV. As we see, the different determinations agree within the wide mCT ± TCT/2 interval.
3 Scalar Regge spectrum
Higher 0++ states listed in the PDG [1] follow roughly the general pattern of increasing mass but not their width. Radial and rotational Regge trajectories were analyzed in Ref. [14]. For scalar states [15] two parallel radial trajectories could then be identified, including three states per trajectory. In a recent work [16,17] (see also [18]) we have analyzed all the 0++ states which appear in the PDG tables (see Fig. 3) and found that all fit into a single radial Regge trajectory of the form
a.
Ms(u)2 = -u + m2.	(2)
The mass of the a state can be deduced from this trajectory as the mass of the lowest state. The resonance nature of these states suggests to use the corresponding half-width as the mass uncertainty in the x2 fit:
2 v- ( Mf n — Ms(n)
X=M TW2 J'	<3>
M2 (GeV2)
Fig. 3. Radial Regge trajectory corresponding to the squared mass of all JPC = 0++ scalar-isoscalar and JPC = 0- + pseudoscalar-isovector states listed in the PDG tables [1]. The four heaviest 0++ and two 0- + states are not yet well established and are omitted from the PDG summary tables. The error bars correspond to the errors in the determination of the square of mass as Am2 = mF with F from [1]. The straight line is the result of our joined fit. Labels of 0++ states are above their mark whereas labels of 0+- states are below their mark.
Minimization yields x2/DOF = 0.12, with
a = 1.31(12) GeV2, m2 = 556(127) MeV.
(4)
Formula (2) is actually equivalent to two parallel radial Regge trajectories with the standard slope,
Ms,_(n)2 = an + m;
Ms,+ (n)2 = an + m2 +
2
(5)
(6)
where a = 2na, and ct is the string tension associated to the potential V(r) = or between heavy colored sources. The value a/o1 = 456(21) MeV obtained from our fit agrees well with lattice determinations of = 420 MeV [19]. Of course, one expects some of these states to correspond eventually to glueballs. However, there seems to be no obvious difference between mesons and glueballs, as far as the radial Regge spectrum is concerned. Note that Casimir scaling suggests that the string tension is o"giuebaii = f Cmeson/ but this holds in the case of fixed and heavy sources. The fact that we have light quarks might explain why we cannot allocate easily the Casimir scaling pattern in the light-quark scalar-isoscalar spectrum.
a
4 Interpolating fields
For scalar states a measure of the spectrum is given in terms of the (gauge and renorm invariant) trace of the energy momentum tensor [20]
©[; = © = J^Ig^G^ + Y_ ™q [1 + Ym(a)] qq.	(7)
q
Here (3(a) = |x2da/d^.2 denotes the beta function, a = g2/(4n) is the running coupling constant, Ym(a) = dlogm/dlog |j.2 is the anomalous dimension of the current quark mass mq, and is the field strength tensor of the gluon field.
It is interesting to consider the situation of massless quarks, where things become somewhat simpler. Then, we have in principle two scalar operators with smallest canonical dimensions, the gluon G2 and the quark qq. While these two operators are both scalars, they are chirally even and odd, respectively, i.e., under the q —> Y5 q transformation. Because the chiral symmetry is spontaneously broken, there is some mixing between G2 and qq. These operators connect scalar states to the vacuum through the matrix element
(0|©|n) = mf , (0|qq|n) = m^c^ .	(8)
The two-point correlators read
n00(q) = i n@s(q) = i nss(q) = i
f2 m4
d4xeiqx(O|T{0(x)0(O)}|O) = Y 7n n 7+c.t.,	(9)
V mn - q2 f2 m2 c m
d4xeiq x(O|T{0(x)qq(O)}|0) =	+c t > (10)
n	n
22
d4xeiq "(0|T{qq(x)qq(0)}|0) = £ "T^ +c-t- - (H)
mn q
where in the r.h.s. we saturate with scalar states and c.t. stands for subtraction constants which can chosen as to replace m2 —> q2 in the numerator. In that scheme, in the large — q2 > Aqcd limit, a comparison with the Operator Product Expansion (OPE) [21-23] leads to the matching conditions
n00(q2) = q4 Co log(—q2)+ ..., nss(q2) = q2 CO log(—q2) + ...,
n0s(q2) = CO' (q q) log(—q2) + ...,	(12)
where C0 = — (2p(a)/an)2, CO = —3/(8n2) and CO' = —2p(a)/an. As we see, qq and G2 do not mix at high q2 values, a consequence of asymptotic chiral symmetry [20]. In these limits the sums over n can be replaced by integrals, whence the following asymptotic conditions are found:
f2/(dm2/dn) Co , c2/(dm2/dn) CO , Cnfnm2/(dm2/dn) C'(q($3)
We see that the first two conditions are incompatible with the third one if m2 increases for large n, as is the case of the data. However, if we group the states in two families, as suggested by Eqs. (6) and Fig. 3, we get a compatible solution
c2)-,f2)+—> const c2)+,f2>_—> const/m^.	(14)
This is equivalent to assuming an asymptotically alternating pattern of mesons and glueballs, coupling to chirally odd and even operators, q q and G2, respec-
1	3
tively. Since asymptotically m2 - an/2, we find cn _/cni+ and fn,+/"fn,- ^ n.?. Of course, this is not the only solution.
The situation described above suggests the existence of a hidden symmetry in the 0++ sector. In our case we could think of the Y5-parity (which becomes a good quantum number for excited states) as the relevant symmetry which makes the difference between the chirally even and odd states. This, however, only explains the fact that asymptotically the slopes of the + and — branches are the same, but not why the intercepts accurately differ by half the slope.
5 The holographic connection
To further elaborate on this intriguing point of the accidental degeneracy, let us consider the one-dimensional harmonic oscillator with frequency as an example; all states ^n(z) with the energy En = h^(n + 1 /2) can be separated into parity even and parity odd states, satisfying the conditions ^n,± (z) = ^2n(z) and ^±,n( — z) = ±^±,n(z), respectively, and having the energies E+,n =	/4)
and E_,n =	+ 3/4). These formulas display twice the slope of En. Thus,
given the states with energies E+,n and E_,n, we might infer that parity was a hidden symmetry of a Hamiltonian explaining the correlation between the slope and intercepts.
In the relativistic case the argument can also be made in a suggestive manner. Let us consider the Klein-Gordon action for infinitely many bosons in four dimensions, described with fields ^n(x) of masses mn:

d4x ^	- mM •	(15)
We assume the spectrum of the form m^ = an + m0. Next, we can introduce the five-dimensional field ^(x,z) = Y.n ^n(x)^n(z), with ^n(z) fulfilling the auxiliary Sturm-Liouville problem in the variable 0 < z < oo,
—9z [p(z)9z^n(z)l + q(z)^n(z) = mipM^nM,	(16)
where the functions are orthogonal with respect to the weight function p(z), provided suitable boundary conditions
P(z) (^(z)^m(z) — ^niz^m(z)) |z=0 = 0	(17)
and ^n(oo) —> 0 are fulfilled. The action can then be written as
1
S 2
d4 x
dz [p(z)9^9^ - p(z)(9^)2 - q(z)^2]	(18)
after some integration by parts in the variable z. This action can be written as a five-dimensional action with a non-trivial metric [24], featuring the AdS/CFT (soft-wall) approach (see [25] and references therein), with the extra dimension z playing the role of a holographic variable and the orthogonal set of functions ^n(z) denoting the corresponding Kaluza-Klein modes. Clearly, z has the dimension of length, suggesting that z 0 corresponds to the ultraviolet and z to the infrared regime.
0
Turning to Eq. (16), we may take the standard Harmonic oscillator Schrodinger-like equation (p(z) = p(z) = 1, q(z) = U(z) = a2z2/16)
+ la2z2^n(z) = m2^n(z)	(19)
and obtain for the regular solutions at infinity the result
K(O)
WO) v r (!-■=£)
(20)
where r (x) is the Euler Gamma function, which is meromorphic and have simple
poles at x = 0, — 1, -2,____The solutions fulfilling the Dirichlet, ^n(0) = 0, and
Neumann, ^n (0) = 0, boundary conditions, respectively, have the masses
2	a	2	3a
m-,n = arL+4>	m+,n = arL + -j- .	(21)
which can be merged into one single formula
a
m^ = -(2n + 1).	(22)
This yields mCT = m,f0/\/3 = 566 MeV and, for the string tension, ct = m,f0 \flfZri = 450 MeV with mf0 = 980 MeV, quite reasonable values.
In this approach the symmetry in the scalar spectrum corresponds to a parity symmetry in the holographic z variable z) = ±^n(z). Note that usually the holographic variable z is taken to be positive3, but if we extend it to —«> < z < oo, we may define a holographic superfield containing two different and orthogonal modes. Otherwise, in the interval 0 < z < oo the Dirichlet and Neumann modes are not orthogonal to each other.
6 Pseudoscalar mesons and chiral symmetry
Discerning the nature of the ct state has been a recurrent pastime for many years. As is well known, glueballs are more weakly coupled to mesons, O( 1/NC), than other mesons, 0( 1 /a/Nc)- The minimum number of states, allowed by certain sum rules and low energy theorems, is just two. In Ref. [16] we undertake such an analysis which suggests that fo(600) (denoted as ct) is a qq meson, while fo(980) (denoted as f0) is a glueball. This is supported by the rather small width ratio,
3 This is supported by the light-front interpretation of Brodsky and de Teramond [26], where the holographic variable is the polar coordinate of a two dimensional vector, z = |C| and £ = b^/xTT — x), with b denoting the impact parameter and x the momentum fraction of the quark. This interpretation yields a two dimensional potential U(Z) = k2 Z2 + 2k2 (L + S — 1) with J = L + S which, when passing to the polar variable z, generates the usual centrifugal term (L2 —1 /4)/z2 not present in our discussion, yielding Min.l.s = 4k2 (n + L + S/2) which for J = 0 and L = 1 resembles Eq. (22).
which yields Ti/Ta - {g2nnm3f)/{gl„„m3J - 1/NC, thus for mff - 0.8 MeV - mf the ratio gamt/9inn - %/Nc is obtained.
A further piece of evidence for the alternating meson-glueball pattern is provided by looking at the excited pion spectrum, which we show in Fig. 3. The alternating pattern was unveiled by Glozman [27], suggesting that states degenerate with the pion might not be identified with glueballs. Remarkably, the states generating doublets with pion states are f0(600) n0(140), f0(1370) n0(1300), f0(1710) ^ n0(1800), f0(2100) ^ n0(2070), and f0(2330) ^ n0(2360), whereas the other scalar states f0(980), f0(1500), f0(2020) and f0(2200) are not degenerate with other mesons with light u and d quarks. Our analysis is reinforced by this observation.
As a matter of fact, fitting the pion n(140) as the n = 0 state of the Regge spectrum requires strong departure from a simple linear trajectory, m^ = an + m0. One may improve on this by using the holographic connection and a mixed boundary condition at z = 0 determined by fixing the mass of the ground state m0 using Eq. (17) together with Eq. (20) for the harmonic oscillator case, Eq. (19). This procedure ensures the orthogonality between all states and implements linearity for large n. This can be done for the ground states m0 = mn, mCT, mfo. The fit to all states yields a = 1.37GeV2 and the mass spectra (in GeV)
n (Regge) (0.140,1.260,1.730, 2.092, 2.400,...) n (PDG) (0.140,1.300,1.812, 2.070, 2.360) a- (Regge) (0.527,1.297,1.750, 2.106, 2.411,...) a (PDG) (0.600,1.350,1.724, 2.103, 2.321) f0 (Regge) (0.977,1.513,1.906, 2.232, 2.517,...) f(PDG) (0.980,1.505,1.992, 2.189)
yielding l/,/aij>o(0)/'4,o(0) = —3.1, —14.9, and 0.2, respectively. Note the large and small values for the a and f0 cases, which suggests that these boundary conditions are very close to the Dirichlet and Neumann cases. Chiral symmetry breaking corresponds to the different n and a values.
7 Scalar dominance and heavy pions
Hadronic matrix elements of the energy-momentum tensor, the so-called gravitational form factors (GFF) of the pion and nucleon, correspond to a dominance of scalar states in the large-Nc picture, as (u(p) is a Dirac spinor)
where the sum rules Y.n gnnnfn = 1 [28] and MN = Y.n gnNNfn [29] hold. Unfortunately, the lattice QCD data for the pion [30] and nucleon (LHPC [31] and QCDSF [32] collaborations), picking the valence quark contribution, are too noisy as to pin down the coupling of the excited scalar-isoscalar states to the energy-momentum tensor. Nevertheless, useful information confirming the (Regge) mass estimates for the a-meson can be extracted [16] through the use of the multiplicative QCD evolution of the GFF in the valence quark momentum fraction, (x)u+d,
(23)
m^MeV]	ttw [MeV]	ttw [MeV]	ttw [MeV]	ttw [MeV]
	GFF	GFF	( qq q ) 2 -dynam.	(q q)2 -quench.
230	580(190)	620(100)	-	400(30)
342	630(190)	660(90)	-	720(20)
478	710(200)	730(90)	-	1000(20)
318	620(190)	650(90)	468(50)	-
469	700(190)	720(90)	936(13)	-
526	739(200)	750(90)	1066(13)	-
Table 1. Scalar monopole mass obtained from the nucleon gravitational form factors, extracted from the (q q) components obtained by LHPC [31] and QCDSF [32] and compared to the lattice calculation using the tetraquark (q q)2 probing fields, both for the dynamical and quenched fermions [33] .
as seen in deep inelastic scattering or on the lattice at the scale = 2 GeV. For the pion and nucleon GFFs we obtain the fits
(x)U+d = 0.52(2), mCT = 445(32) MeV,	(25)
<x)N+d = 0.447(14), mCT = 550-280 MeV.	(26)
Assuming a simple dependence of mCT on mn,
ma (mn) = ma + c	- ^phys) >	(27)
yields mff = 550-180MeV and c = 0.95-0.70, or mff = 600-800MeV and c = 0.8(2), depending on the choice of the lattice data [31] or [32], respectively. Note that c is close to unity. Higher quark masses might possibly clarify whether or not the state evolves into a glueball or a meson. For a (qq)n system one expects mCT —> 2nmq + const at large current quark mass mq. The data from [31] or [32] are too noisy to see the difference, although for the largest pion masses we see that mCT ~ mn, as it simply corresponds to the q q-component of 0. We observe, however, that for mn ~ 500MeV our results are not far away from the recent lattice calculation using the tetraquark probes, (qq)2 [33], which provide mCT ~ 2mn for the largest pion masses as they should (see Table 1). From this viewpoint, and unless operator mixing is implemented, the nature of the state is predetermined by the probing operator.
8 Dimension-2 condensates
One of the problems of the large-Nc Regge models [35] and their holographic relatives [36,37] is that they may contradict expectations from the OPE, as they involve dimension-2 condensates. For instance, the OPE for the n@@(q2) correlator in Eq.(12) gives corrections O(q0), while the O(q2) terms are missing [21].
This yields a one to one comparison:
Co = - lim	= -™LlL (ß^V	(28)
oo dm^/ dn	2n2 \ a J '
C2 = X ^	=	(29)
■
C4=If^	=	(30)
Equation (28) requires infinitely many states, while Eq. (29) suggests a positive and non-vanishing gauge-invariant dimension-2 object, C2 = i J" d4xx2(0(x)0), which is generally non-local, as it should not appear in the OPE. Note that C2 > 0, hence is non-vanishing for a finite number of states. The infinite Regge spectrum of Eq. (2) with Eq. (28) may be modeled with a constant ffo = fn,+ = O(Nc) whereas fn,_ = 0(\/ÑQ goes as Eq. (14) and yields a convergent and hence positive contribution. Naively, we get C2 = oo. However, C2 may vanish, as required by standard OPE, when infinitely many states are considered after regularization. The use of the Z-function regularization [16,34] gives
C2 = limH f^MsM25 = X fi- + ffo (1/2 - mf /a) . (31)
Then C2 = 0 for mfo > ^fafl = 810(40) MeV, a reasonable value to 0(1 /Nc) In any case, the important remark is that while at the OPE level C2 = 0 vanishes for trivial reasons, at the Regge spectrum level some fine tuning must be at work.
Acknowledgments This work is partially supported by the Polish Ministry of Science and Higher Education (grants N N202 263438 and N N202 249235), Spanish DGI and FEDER funds (grant FIS2008-01143/FIS), Junta de Andalucía (grant FQM225-05).
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