1/m<sub>c</sub> Terms in lambda<sup>+</sup><sub>c</sub> Semileptonic Decays Page: 4 of 6
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Al - nlp' F 111, (16)
so that we may write, for I' - r7y,
- i K A(p)siFD y h4''IA7() > = u(p)(Fj + $F)rtu(v)
+ u(p)(Fs + $F6)ru(v). (17)
Fs and F6 umay be absorbed into the leading order form factors F1 and F2, so
that there are apparently two new form factors introduced by the local term to
this order. If we choose r P , then multiplying eqn. (15) by va and using the
equation of motion for h ', gives p * va3 = -0a.
--i < A(p)Isr D yah!lA'(v) >= (p)(F + F4)r(t - - p)u(v). (18)
At this point, if we could treat the s quark as heavy, we could arrive at a rela-
tionship between F1 and F1, and between F and F2, as has been done in the
second and third papers of [7[. Since there is no justification for doing this, we
exhibit F, and FT as independent form factors.
This local term introduces some right-handedness into the matrix element at
the baryon level. To see this right-handedness, we note that
-r < A(p)jsr,Al - ysC) yahA () >
-:t(p)( F3 + /F4)y,( I + ys)(t - v p)u(v). (19)
'Thus, at the haryon level, the full current will no longer be purely left-handed,
and we would expect departures from the leading order predictions of GA = -Cy,
The equation of motion of hi;' may be used to eliminate the (v - D)3 term.
The D2 term leads to a matrix element
- ! A p)IT rsr jrhF(m I ( &Dzhl (r)IAZ (v) >
-u(p)(lr, I fAdx~l - _____ ,
which simply renormalizes the leading order form factors Fr and F,.
Tis leaves time .T termm which is
- < i (p)lr t r" Cl r-$) sth' h ( (z)lAf v)
= rr(lp)(ar I J " rli ; ltr .,e( }M a",
where, MA = vap, is the most general tensor that ,an lie rum% niem t \ ".1111
examination shows us that this term vanishes. We may tlierebre wrt. thir tli
matrix element, up to terms of order 1/rn as
=ii(p)(F& +- $F2)-y,( - ys )u(v)
+ -t(p)(F + F4r,(1 + ys)(t - u p)u(v).
At this point it may appear that the use of HQET to this order in the l/m,
expansion does not lead to much of a gain since, in the general case, we needed six
form factors to describe the decays in which we are interested, and we find that
we now need four form factors. Nevertheless, this still represents some gain, and
we should keep in mind that the effects of two of these form factors are expected
to be small, as they arise at order !/m, in the HQET expansion. To illustrate
this more clearly, let us write the general form factors of eqn. (3) in terms of the
form factors of eqn. (22). We find
m 14 faF I '-r +1ma('ne - u -
r= Ft + F- +2m -mat Up) t+ F',
_ F + 1 F + ma + -n p 1
m t 2m ma+
__ mF +-! %iJ, -p -
= F F2 - + -- (ma - - P - Q+ F]
9"= 21m + m
l L[ ma+ -ma+v-p
gr-=--Fi -- 1F3 + FJ
ma+ 2mr ma J
l I mag+-mA +v-p
93=- F2------ F3 + F4 ,
ma+ 2m, ma+
where we have used q = Pa - pa+.
These forms suggest that there is a problem with the 1/m, expansion, since
some of the coefficients that are expected to be small, are not. Indeed, the
coefficients of F in the first and fourth of eqns. (23), are of order unity. To see
that in fact there is no problem, it is more instructive to look at the relations
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Roberts, Winston. 1/m<sub>c</sub> Terms in lambda<sup>+</sup><sub>c</sub> Semileptonic Decays, article, February 1, 1992; [Newport News, Virginia]. (https://digital.library.unt.edu/ark:/67531/metadc931989/m1/4/: accessed September 20, 2020), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.