Bled Workshops in Physics Vol. 8, No. 1 p. 10
Large-Nc Regge models and the (A2) condensate*
Wojciech Broniowskia,b and Enrique Ruiz Arriolac
a Institute of Nuclear Physics PAN, PL-31342 Cracow, Poland b Institute of Physics, Swietokrzyska Academy, PL-25406 Kielce, Kielce, Poland c Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada E-18071 Granada, Spain
Abstract. We explore the role of the (A2} gluon condensate in matching Regge models to the operator product expansion of meson correlators.
This talk is based on Ref. [1], where the details may be found. The idea of implementing the principle of parton-hadron duality in Regge models has been discussed in Refs. [2-8]. Here we carry out this analysis with the dimension-2 gluon condensate present. The dimension-two gluon condensate, (A2), was originally proposed by Celenza and Shakin [9] more than twenty years ago. Chetyrkin, Nar-ison and Zakharov [10] pointed out its sound phenomenological as well as theoretical [11-15] consequences. Its value can be estimated by matching to results of lattice calculations in the Landau gauge [16,17], and their significance for non-perturbative signatures above the deconfinement phase transition was analyzed in [18]. Chiral quark-model calculations were made in [19] where (A2) seems related to constituent quark masses. In spite of all this flagrant need for these unconventional condensates the dynamical origin of (A2) remains still somewhat unclear; for recent reviews see, e.g., [20,21].
For large Q2 and fixed Nc the modified OPE (with the 1/Q2 term present) for the chiral combinations of the transverse parts of the vector and axial currents is
T ,	1 f Nc , Q2 aS A2 n (aSG2) 1
= + (1)
On the other hand, at large-Nc and any Q2 these correlators may be saturated by infinitely many mesonic states,
oo f2	f2 00 f2
* Talk delivered by Wojciech Broniowski
The basic idea of parton-hadron duality is to match Eq. (1) and (2) for both large Q2 and Nc (assuming that both limits commute). We use the radial Regge spectra, which are well supported experimentally [22]
Mv,n = mv + av n,	= MA + aau, n = 0,1,...	(3)
The vector part, nV, satisfies the once-subtracted dispersion relation
We need to reproduce the log Q2 in OPE, for which only the asymptotic part of the meson spectrum matters. This leads to the condition that at large n the residues become independent of n, Fv,n ~ Fv and FA,n ~ FA. Thus all the highly-excited radial states are coupled to the current with equal strength! Or: asymptotic dependence of Fv,n or FA,n on n would damage OPE. Next, we carry out the sum explicitly (the dilog function is ^(z) = r'(z)/r(z))
y ( H___H \ F2
VM2 + am+Q2 M2 + atnj at

Z?
at
Qi^ i tI, \ I ai ~ 2M2 6Mf — 6ajM2 + a2 a J+W{aiJ+ 2Q2 +	12Q4
(5)
where i = V, A. nv-A satisfies the unsubtracted dispersion relation (no log Q2 term), hence
Fv/av = FA/aA.	(6)
This complies to the chiral symmetry restoration in the high-lying spectra [23,24]. Further, we assume av = aA = a, or Fv = FA = F, which is well-founded experimentally, as v/ox = 464MeV, ,/ov = 470MeV [22],
The simplest model we consider has strictly linear trajectories all the way down,
F
2
Mv + Q2\ , , ZM2A + Q
fiv_a(Q2)=- (,vLy:v ) ' ^
a

f_
Q
2
^(Mi - My) - f2) + (Mi — My)(a — Mi — M*)) -L + .
Matching to OPE yields the two Weinberg sum rules: F2
f2 = — (Mi-M2,),	(WSR I)
0 = (MA - Mv)(a - MA - Mv). (WSR II)
2
The V + A channel needs regularization. We proceed as follows: carry d/dQ2, compute the convergent sum, and integrate back over Q2. The result is
n;+A(Q2) = Q

V a
F2	1
+— (a2 - 3a(Mi + My) + 3(M4 + My)) ^ + ...
Matching of the coefficient of log Q2 to OPE gives the relation
24n2F2
a = 2tcu = — ,	(7)
where ct denotes the (long-distance) string tension. From the p —> 2n decay one extracts F = 154 MeV [25] which gives a/ct = 546 MeV, compatible to the value obtained in lattice simulations: a/ct = 420 MeV [26]. Moreover, from the Weinberg sum rules
t T 24n2 T	T	T 24n2
M^ = My + ——f , a = M^ + My = 2My + ——f .	(8)
Matching higher twists fixes the dimension-2 and 4 gluon condensates:
_ asA2 = 2 as(G2) = M4-4M^Mj + M4
4tt3 ' 12«	48«2	'	1 1
Numerically, it gives —= 0.3 GeV2 as compared to 0.12GeV2 from Ref. [10,20]. The short-distance string tension is ct0 = —2asA2/Nc = 782 MeV, which is twice as much as ct. The major problem of the strictly linear model is that the dimension-4 gluon condensate is negative for My > 0.46 GeV. Actually, it never reaches the QCD sum-rules value. Thus, the strictly linear radial Regge model is too restrictive!
We therefore consider a modified Regge model where for low-lying states both their residues and positions may depart from the linear trajectories. The OPE condensates are expressed in terms of the parameters of the spectra. A very simple modification moves only the position of the lowest vector state, the p meson.
My,o = mp, My,n = MV + an, n > 1
MA,n = MA + an, n > 0.	(10)
For the Weinberg sum rules (we use Nc = 3 from now on)
8n2f2 (4«2f2 + My2)
Mi = My + 87t2f2, a = 87t2F2 =-^-f.	(11)
A V	'	47T2f2 — mp2 + My	V ;
f2
+ q2 + const.
Fig. 1. Dimension-2 (solid line, in GeV2) and -4 (dashed line, in GeV4) gluon condensates plotted as functions of the square root of the string tension. The straight lines indicate phenomenological estimates. The fiducial region in s/a for which both condensates are positive is in the acceptable range compared to the values of Ref. [22] and other studies.
We fix mp = 0.77 GeV, and ct is the only free parameter of the model. Then — 16n3f4 + 4n2of2 - mp2CT aSA2 16n3f4 - no-2 + mp2CT
M
V	4f27t — (X	' 4tt3	16f27T3 -4tt2c
£__? ™ 2
as(G2) 2 4 2 M^v^r^ a^
]2n~ =	+-87t2-+ tt	(12)
The window for which both condensates are positive yields very acceptable values of ct. The consistency check of near equality of the long- and short-distance string tensions, ct ~ Co, holds for Jo ~ 500MeV. The magnitude of the condensates is in the ball park of the "physical" values. The value of MV in the "fiducial" range is around 820 MeV. The experimental spectrum in the p channel is has states at 770,1450,1700,1900*, and 2150* MeV, while the model gives 770,1355,1795, 2147 MeV (for ct = (0.47 GeV2). In the ai channel the experimental states are at 1260 and 1640 MeV, whereas the model yields 1015 and 1555 MeV.
We note that the V — A channel well reproduced with radial Regge models. The Das-Mathur-Okubo sum rule gives the Gasser-Leutwyler constant L10, while the Das-Guralnik-Mathur-Low-Yuong sum rule yields the pion electromagnetic mass splitting. In the strictly linear model with M^ = 2My and My = Y/247t2/Ncf = 764 MeV we have Jo = j3/2nM.v = 532 MeV, F = J~3f = 150 MeV, Lio = -Nc/[96j37t) = -5.74 x 10-3 (-5.5 ± 0.7 x 10-3)exp, m2± -mno = (31.4 MeV)2 (35.5 MeV)2xp. In our second model with ct = (0.48 GeV)2 we find Lio = —5.2 x 10-3 and m^± — m^0 = (34.4 MeV)2.
les.
To conclude, let us summarize our results and list some further related stud-
Matching OPE to the radial Regge models produces in a natural way the 1 /Q2 correction to the V and A correlators. Appropriate conditions are satisfied by the asymptotic spectra, while the parameters of the low-lying states are tuned to reproduce the values of the condensates.
In principle, these parameters of the spectra are measurable, hence the information encoded in the low-lying states is the same as the information in the condensates.
Yet, sensitivity of the values of the condensates to the parameters of the spectra, as seen by comparing the two explicit models considered in this paper, makes such a study difficult or impossible at a more precise level. Regge models work very well in the V — A channel. In [28] it is shown how the spectral (in fact chiral) asymmetry between vector and axial channel is generated via the use of Z-function regularization for each channel separately. We comment that effective low-energy chiral models produce 1 /Q2 corrections (i.e. provide a scale of dimension 2), e.g., the instanton-based chiral quark model gives [19]
A2 = _2Nc
n
-v7 M (u) M' (u) ~ 0.2 GeV2. (13)
u + M(u)2	v ;
In the presented Regge approach the pion distribution amplitude is constant, ^ (x) = 1, at the low-energy hadronic scale, similarly as in chiral quark models [27].
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