ЯДЕРНАЯ ФИЗИКА, 2004, том 67, № 2, с. 443-445
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
ON THE DETERMINATION OF QUARK MASSES FROM THE DALITZ PLOT OF n — n+n-n0 DECAY
© 2004
B. V. Martemyanov , V. S. Sopov
Institute of Theoretical and Experimental Physics, Moscow, Russia Received January 17, 2003
Experimental Dalitz-plot distribution of n ^ n+
decay is fitted by the theoretical one obtained in
Chiral Perturbation Theory with unitarity corrections taken into account. The fit shows that the difference of light-quark masses is larger than it is expected from electromagnetic mass differences of neutral and charged kaons.
1. INTRODUCTION
Since n — 3n decay cannot conserve both C parity and isospin simultaneously, it can go due to either weak or electromagnetic or isospin-violating part of strong interactions. Weak interactions are too weak to explain the observed decay probability. Electromagnetic interactions are strongly suppressed in this decay by chiral symmetry, so the decay is due almost exclusively to the isospin-breaking part of QCD Hamiltonian [1].
In chiral perturbation theory (ChPT) the decay width r depends on the quark mass ratios and theoretically calculable factor r:
where
Q-2 =
Г
,2
Q
DT
Q
Г,
(1)
Щ ~ти ml — m2 :
md + mu ,оч m =---, (2)
mu, md, and ms are up, down, and strange-quark masses,
q-t =
(3)
m
m^Q - m
K+,
- [mzq - ml+))
2
m2 q
П
(m2K - m^o)
m
K
= (24.1)2,
with m2K = (m2K + + m2Ko - m2n+ + m2n0 )/2.
Note that Q = QDT and r = r if electromagnetic mass differences of kaons and pions are equal to each other, what is the case in the lowest order of ChPT [2] (Dashen theorem — DT).
E-mail: martemja@heron.itep.ru
Both experimentally known decay width r and theoretically calculable factor r have uncertainties. The experimental uncertainties of r are [3]:
r = 281 ± 28 eV. (4)
The uncertainties of r are not so definite. In the lowest order of ChPT [4]
r = 66 eV. (5)
The first corrections (one loop-corrections) move r to the value [2]
r = 167 ± 50 eV. (6)
Higher order corrections (taken into account by the dispersion method [5, 6]) give larger result with some uncertainties. In [5] they were estimated to be
r = 209 ± 20 eV. (7)
The difference between r and r means that Q is different from QDT and the ratio of quark masses mu/md can be measured by the deviation of r from r. Surely, the higher will be the accuracy of r and r, the clearer will be the difference between QDT and Q. The accuracy of r is purely of experimental origin. The accuracy of r is partly of theoretical origin (used order of ChPT, the inclusion of final-state interaction in the decay width), but partly of experimental origin — some unknown quantities in the theoretical scheme could be better adjusted if the Dalitz-plot distribution would be known better.
In [5, 6] the factor r was calculated by taking into account the results of one-loop order of ChPT and the unitarity corrections that allow to sum the effects of the rescatterings of two pions in all orders of ChPT. The subtraction polynomial was taken from the decomposition of one-loop-order amplitude. This polynomial had therefore uncertainties connected to higher orders of ChPT corrections to the amplitude.
0
П П
4
2
2
444
MARTEMYANOV, SOPOV
In what follows we will fix the uncertainties of the polynomial by fitting the experimental data on Dalitz-plot distribution. As a result, we obtain the improved value of r and, hence, the improved values of lightquark masses.
2. SIMULATION OF n — n+n-n0 DALITZ-PLOT DISTRIBUTION AND ITS FIT BY x2 METHOD
In order to simulate experimental Dalitz-plot distribution we have taken it in a form
1 + ay + by2 + cx2,
where
3T0
(8)
(9)
T = T+ + T_ + To,
P(s) = a + ßSa + YS2a + S(sb - Sc)2,
(11)
a = -1.28 ± 0.14, ß = 21.81 ± 1.52 GeV-2,
(12)
-4
7 = 4.09 ± 3.18 GeV"4, 5 = 4.19 ± 1.08 GeV
(the case of zero subtraction points [5]) the "Minuit" fit of experimental Dalitz-plot distribution has terminated on the values
70 = 7.02 GeV-4, So = 5.23 GeV-4, with x2/^d.o.f. = 134/(100 - 4).
Equally possible (with the same value of X2/Nd.o.f.) is the fit
a = aoC, 3 = ¡3oC [GeV-2], (14) 7 = 70C [GeV-4], S = SoC [GeV-4]
because the normalization factor of the amplitude is not defined by the Dalitz-plot distribution. Varying constant C in such a way as to stay within the above-mentioned limits on parameters a, 3, 7, and S we obtain C = 1+0 038. The undefiniteness of constant C can be translated to the undefiniteness of the width r:
(15)
r = 213+32 eV.
T+, T-, and T0 are the kinetic energies of pions in the rest frame of n — n+n-n0 decay. The parameters a, b, and c were taken from one of the best known experimental results [7]:
a = -1.17 ± 0.02, b = 0.21 ± 0.03, (10) c = 0.06 ± 0.04.
The Dalitz plot was divided into 100 = 10 x 10 bins (x x y) that have equal number of events for the distribution considered. Then the number of events n in each bin was simulated by Gaussian distribution with the variance equal to n. Fitting this distribution of events over the bins by the same form 1 + ay + + by2 + cx2 of Dalitz-plot distribution, we get the parameters a, b, and c equal to the experimental values within experimental errors for approximately N = 100n « 1000 000 events.
As a next step, we have taken an approximate solution of unitary equations for the amplitude of n — — n+n-n0 decay from Eq. (5.28) of [5]. It contains the subtraction polynomial
where sa, sb, and sc are invariant masses squared of n", n+n0, and n"n0 pairs, respectively. For the values of parameters a, /3, 7, and 5 within the regions
ao = -1.17, ßo = 21.74 GeV
2
(13)
The obtained result is very close to the result (7) of [5]. This coincidence might be the occasional one and future experimental data on Dalitz-plot distribution will give slightly different value of the width r.
3. CONCLUSION
We have fitted the amplitude of n — de-
cay calculated within ChPT in next-to-leading order and with unitarity corrections (final-state interaction of pions) taken into account [5] to the experimental data [7] on Dalitz-plot distribution. The obtained results forthewidth r (see Eq. (1)) mean that Q = QDT and the difference of u- and d-quark masses is slightly larger than it follows from the Dashen theorem for electromagnetic mass differences of kaons and of pions, in which case Q = QDT. The Dashen theorem says that mu/md = 0.57, while the result (7) of [5] on quantity f for n — n+n-n0 decay gives mu/md = = 0.52.
The work was partially supported by RFBR grant no.02-02-16957.
REFERENCES
1. D. G. Sutherland, Phys. Lett. 23,384(1966); J. S. Bell and D. G. Sutherland, Nucl. Phys. B 4, 315 (1968); R. Baur, J. Kambor, and D. Wyler, Nucl. Phys. B 460, 127(1996).
2. J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 539 (1985).
3. K. Hagiwara et at., Phys. Rev. D 66, 010001 (2002).
4. H. Osborn and D. J. Wallace, Nucl. Phys. B 20, 23 (1970); J. A. Cronin, Phys. Rev. 161, 1483(1967).
5. J. Kambor, C. Wiesendanger, and D. Wyler, Nucl. Phys. B 465,215(1996).
6. A. V. Anisovich and H. Leutwyler, Phys. Lett. B 375, 335(1996).
7. M. Gormley etat., Phys. Rev. D 2,501 (1970).
MEPHA^ OH3HKA TOM 67 № 2 2004
ON THE DETERMINATION OF QUARK MASSES
445
ОБ ОПРЕДЕЛЕНИИ МАСС КВАРКОВ ПО ДАЛИТЦ-ПЛОТУ п ^ п+ж-п°-РАСПАДА
Б. В. Мартемьянов, В. С. Сопов
Экспериментальный далитц-плот п ^ п+п-п0-распада фитируется теоретическими результатами, полученными в киральной теории возмущений с унитарными поправками. Фит показывает, что разность масс легких кварков больше, чем ожидаемая из электромагнитных разностей масс нейтральных и заряженных каонов.
ЯДЕРНАЯ ФИЗИКА том 67 № 2 2004
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