Pis'ma v ZhETF, vol.86, iss. 10, pp.722-724
© 2007 November 25
The running mass ms at low scale from the heavy-light meson decay
constants
A. M. Badalian+*1\ B. L. G. Bakker+V + State Research Center, Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia * Department of Physics and Astronomy, Vrije Universiteit, Amsterdam Submitted 22 October 2007
It is shown that a 25(20)% difference between the decay constants /d«(/b«) and /d(/b) occurs due to large differences in the pole masses of the s and d(u) quarks. The values t)D = fo./fo ~ 1.23(15), recently observed in the CLEO experiment, and t)b = fa,/fa « 1.20, obtained in unquenched lattice QCD, can be reached only if in the relativistic Hamiltonian the running mass ms at low scale is ms(~ 0.5 GeV)= 170 — 200 MeV. Our results follow from the analytical expression for the pseudoscalar decay constant fp based on the path-integral representation of the meson Green's function.
PACS: 12.38.Lg, 12.39.Ki, 14.40.^n
Relativistic potential models (RPM) have been successful in their description of light-light and heavy-light (HL) meson spectra [1, 2]. Still there exists a fundamental problem, which remains partly unsolved up to now. It concerns the choice of the quark masses in the kinetic term of the relativistic Hamiltonian, which in different RPMs vary in a wide range. For example, HL mesons were studied with the use of the Dirac equation, taking for the light quark mass mn(n = u,d) = 7MeV in [3] and 72 MeV in [4], and in the Salpeter equation for the strange quark mass the values ms = 419 MeV in [5] and ms = 180 MeV in [6] have been used. However, in contrast to constituent quark models, where the constituent mass can be considered as a fitting parameter, a fundamental relativistic Hamiltonian has to contain only conventional quark masses-the pole masses. These masses are now well established for heavy and light quarks [7]. They have been used in the QCD string approach giving a good description of meson spectra [6, 8, 9]. However, the strange quark mass ms is still not determined at low scale. At present, owing to the QCD sum rules calculations [10] and lattice QCD [11], ms is well established at a rather large scale: ms (p, = 2 GeV) = 90 ± 10 MeV, while in the Hamiltonian approach the mass ms enters at a lower scale, which is evidently smaller that the scale fic « 1.2 GeV for the c quark. Therefore it is very important to find physical quantities which are very sensitive to ms at low scale: /¿s < 1 GeV. In this letter we show that such information can be extracted from the analysis of the decay constants of HL mesons, namely, from the ratios fDJÎD and /B,//B.
Recently, direct measurements of the leptonic decay constants in the processes D(DS) —t pv^ [12, 13], and B tut [14, 15] have been reported. In Refs. [12, 13] the CLEO collaboration gives fDs = 274(20) MeV and fD = 222.6(20) MeV with Vd = fDJfD = 1.23(15), having reached an accuracy much better than in previous experiments [16]. This central value for rj£> appears to be larger than in many theoretical predictions which typically lie in the range 1.0 — 1.15 [17-20]. Therefore, one can expect that precise measurements of tjd and tjb in the future can become a very important criterium to distinguish different theoretical models and check their accuracy. In particular, relatively large values
f}D = 1.25(3), t]B = 1-20(3), (1)
have been obtained recently in lattice (unquenched) calculations [21, 22] and also in our paper [6]. In this letter we show that:
1. The running mass ms(p,i) at a low scale, /¿i « 0.5 GeV, can be extracted from the values tjd and tjb, if they are known with high accuracy, < 5%.
2. The values tjd and tjb, as given in Eq. (1), can be obtained only if the running mass ms(p,i) lies inside the range 170^200 MeV. In particular, in the chiral limit, fnd = mu = 0, as well as for to^ = 8 MeV, and for wg(/ii) = 180 MeV, the ratios tjd and tjb calculated here are
f}D = 1.25, î)B = 1-19.
3. The value tos(0.5 GeV) satisfies the relation tos(0.5 GeV)
e-mail: badalian8itep.ru; blg.bakker8few.vu.nl
TOS(2 GeV)
1.97.
(2)
(3)
The running mass ms at low scale ... 723
In our analysis we use the analytical expression for the leptonic decay constant in the pseudoscalar (P) channel, derived in Ref. [6] with the use of the path-integral representation for the correlator Gp of the currents jp(x): Gp(x) = {jp(x)jp(0))vac:
(Yp)n\fn(0)\2 -m„t
Jp =
J Gp(x)d3x = 2
UqnUQn
(4)
where Mn and <pn(r) are the eigenvalues and eigen-functions of the relativistic string Hamiltonian [23 - 25], while wqn(u)Qn) is the average kinetic energy of a quark q(Q) for a given nS state:
Uqn = {\Jfn2q + p2)nS, UQn = {^Jm2Q+ p2)„s. (5)
In Eq. (5), to,(toq) is the pole mass of the lighter (heavier) quark in a heavy-light meson. The matrix element (Fp)n refers to the P channel (with exception of the ir and K mesons where additional chiral terms occur) and was calculated in Ref. [6],
(Yp)n = TilqTilQ + U)qnU)Qn - (p2)nS- (6)
On the other hand, for the integral Jp (4) one can use the conventional spectral decomposition:
Jp = J Gp(x)d3x = ^T
2M,
-(/Pn)V
tn\2„-m„t
Then from Eqs. (4) and (7) one obtains that
2 2jVe(Fp)n|ij3n(0)|2 '•'p^ ~ —rr-;—m-•
UqnUQnMn
All necessary factors in Eq. (8) for the ground state (n = = 1) and the first radial excitation (n = 2) are calculated in Ref. [6] but here we consider only ground states and omit the index n everywhere. Our calculations are performed with the static potential Vq (r) = ar — |"B [9], and the hyperfine and self-energy contributions are considered as a perturbation. It is important that our input parameters contain only fundamental values: the string tension cr, the QCD constant A(n/ = 3) in «b(r), and the conventional pole quark masses. For the s-quark mass ms(p,i) one can expect that the scale /¿i is close to the characteristic momentum /¿i « ^J (p2) ~ ~ 0.5^0.6 GeV. This scale also corresponds to the r.m.s. radii Rm(1S) of the meson we consider. For HL mesons
Rd « Rd. = 0.55(1) fm, Rb « Rb. = 0.50(1) fm,
(9)
so that Hi ~ R^ « 0.4^0.5 GeV. We show here that this mass ms (p,\) is strongly correlated with the values of tjd and t]B■ For other quarks we take mc = 1.40 GeV and mb = 4.78 GeV [8].
It is of interest to notice that for HL mesons the ratios
Cb = fo. = ^^ = 0.347(3) (10) wqwc
are equal for the D and Ds mesons with an accuracy better than 1%, and also that these fractions for B and Bs mesons coincide with < 2% accuracy (<fii(0) = = R\{ 0)/4tt):
|iW0)|2
£b = =
wqwb
= 0.146(2). (11)
It is important that the equalities = and = = practically do not depend on the details of the interaction in HL mesons. Therefore, in the ratios ^d(^b) the factors £d(£b) cancel and one obtains
/ msmc(b) wswc{b) - {v2)Db{b,)\ Md{b)
2 _ / 7«V»te(6) wswc(b) ~ \P }D,
VD^~{(Yp)d{b)+ (Yp)d{b)
M,
d,(b,
(12)
In Eq. (12) the second term is close to 1.05, while the first term, proportional to ms, is not small, changing by 30-60% for different ms (below we take ms from the range 140±60MeV/c2). With an accuracy of < 2%
rfD = 2.708xtos(GeV) + 1.07(1), if md = mu = 0, rfD+ = 2.648xtos(GeV) + 1.05(1), if md = 8 MeV, (13) i.e., in the chiral limit
r]D = 1.14 (to, = 85 MeV), 1.25 (to, = 180 MeV), 1.27 (to, = 200 MeV),
(14)
and for TOd = 8 MeV, tjd = 1.13, 1.24, and 1.26, respectively, for the same values of to,, so decreasing only by ~ 1%.
For the B and Bs mesons
rj2B = 1-90 x tos + 1.07(1) (md = m = 0);
r]2Bo = 1.871 x tos + 1.07(1) (md = 8 MeV), (15)
which practically coincide, and in the chiral limit (md = = mu = 0)
tjb = 1.11 (to, = 85 MeV), 1.19 (to, = 180 MeV), 1.21 (to, = 200 MeV).
(16)
These values of tjb appear to be only by 3 — 5% smaller than t)d-
Thus for ms = 180 MeV and md = 8 MeV we have obtained
rjD+ = 1.25(2), tjb = 1.19(1), (17)
in good agreement with the CLEO data: i?n(exp) = = 1.23(15) [13].
724
A. M. Badalian, B. L. G. Bakker
To check our choice of ms = 180 MeV, we estimate the ratio tos(0.5 GeV)/ms(2 GeV) using the conventional perturbative (one-loop) formula for the running mass [26]
m(ß2) = too
1 ß 2 X^
1 ^d*
lnlnfr
In
(18)
Here too is an integration constant and the other constants are
dl = M
si 19 51 - y»/
ßo = 11 - ^n/,
dm~ ßo'
(19)
To calculate tos(2 GeV) we take n/ = 4, A(n/ = 4) = = 250 MeV (fa = 25/3, dm = 12/25, dx = 0.3548). We take n/ = 3, A(nf = 3) = 280 MeV to estimate tos(1 GeV) and TOs(0.5GeV) (for nf = 3, fa = 9, dm = 4/9, di = 0.3512). Then, from Eqs. (18,19) tos(2 GeV)= 0.618 to0, TOs(lGeV) = 0.7825 to0, and TOs(0.5GeV) = 1.217 too and therefore our estimates are
tos(1 GeV)
= 1.27,
tos(0.5 GeV)
= 1.97.
(20)
tos(2 GeV) ' ' tos(2 GeV) These estimates can be done because the QCD costant A (nf = 3) = 0.28 GeV is relatively small, so that the scale fii = lGeV, for which ln(/if/A2) = 2.55, still lies far from the Landau singularity.
The ratio (20) means that tos(0.5 GeV) = 180 MeV, which we have used in our calculations, corresponds to tos(2 GeV) = 91 MeV which is in agreement with the conventional value of tos(2 GeV) = 90±15 MeV [7]. Thus our estimate of tos(0.5 GeV) = 180 MeV supports our choice of this mass in the relativistic string Hamiltonian, which provides a good description of the HL meson spectra and decay constants, and gives rise to the relatively large values of tjd and tjb in Eq. (1).
The financial support of the NSh-843.2006.2 grant is aknowledged by A.M.B.
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7. Particle Data Gro
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