Lattice calculation of quarkonium decay matrix elements Page: 4 of 12
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decay rate can be written as a sum of terms. Each term is the product of a long-
distance matrix element of a four-fermion operator in the quarkonium state with a
short-distance coefficient. For example, decay rates for S-wave quarkonia through
next-to-leading order in v2 are given by
r(2s+1T - X)=g(2s+1lS) 2Imfi(28+1S,)/M
+,1(2'+1s)2 Im g1(2s+1Sj)/M4. (1)
Decay rates for P-wave quarkonia, to the lowest non-trivial order in v2, are given
by
r(2 ++1Pj -> X)=1x1(2s+1 PT) 2 Im fl(2s+1p1MQ
+7a(2s+1PT)2Im f8(2a+1ST)/M2. (2)
The f's and g's are the short-distance coefficients and are proportional to the rates
for the annihilation of a QQ pair from the 2s+1lL state. 91, 1, 71, and 7ls are
the long-distance matrix elements, which we calculate in this paper. The subscripts
I and 8 indicate the color state of the QQ pair. (An octet state, in lowest order in
v, consists of a QQg state.)
1.2. Matria elements
The long-distance matrix elements are defined by
1=(1s0i4txxt( 2Z a)2 1160,
7x1=(1P111Gt(i/2) D X"Xt (i/2) D 011i},
7ta={1P11 P1Ta t 7 j1P1) (3)
Here, 0 and xt are two-component spinors that annihilate a heavy quark and a
heavy antiquark, respectively.
The 91 and 7i1 terms in the decay rates (1) and (2) appear in the conven-
tional, color-singlet model.4 For the color-singlet matrix elements we can, to good
approximation, take only the vacuum in the intermediate state.3 Then we have
01= ('S x0[~O)f2 (1 + O(v4)),
1= (1S0 [&xiO)(Oxt( 22 )2Y'10) (1 + O(v4)),
xf1= j(1P1j - D x10)12 (1 + O(v4)). (4)
It then follows that, in the vacuum-saturation approximation,
3
91-- 2NRs(0)12,
9
7x 2 R'P(O)12, (5)
2
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Bodwin, G. T.; Sinclair, D. K. & Kim, S. Lattice calculation of quarkonium decay matrix elements, report, September 1996; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc685013/m1/4/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.