%Paper: 
%From: Ahmady Mohammad <ahmady@ictp.trieste.it>
%Date: Mon, 21 Dec 1992 12:59:05 +0100

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\begin{document}
\begin{center}{{\Large \tenbf ON-MASS-SHELL RENORMALIZATION OF THE $\langle
\bar{t}t
\rangle$\\
\vglue 3pt
CONTRIBUTION TO THE $\Delta I = 1/2$ $s \leftrightarrow d$ SELF-ENERGY
TRANSITION\\
\vglue 5pt
%{\ninerm (For 20\% Reduction to 8.5 $\times$ 6 in Trim Size)\\}
\vglue 1.0cm
{\tenrm M.R. AHMADY\\}
{\tenit International Centre for Theoretical Physics, Miramare P.O.
Box 586, 34100 Trieste,
Italy}}\\

\vglue 0.6cm
{\tenrm V. ELIAS and N.C.A. HILL\\  }
{\tenit Department of Applied Mathematics, University of Western
Ontario, London, Ontario
N6A 5B7, Canada\\}}
\vglue 3.0cm
{\tenrm ABSTRACT}
\end{center}
\vglue 0.3cm
{\rightskip=3pc
 \leftskip=3pc
 \tenrm\baselineskip=20pt
 \noindent
In $\langle \bar{t}t \rangle$-scenarios for dynamical electroweak
symmetry breaking, the
presence of a very
large t-quark condensate necessarily generates an $s \leftrightarrow
d$ self-energy transition
which
will contribute exclusively to $\Delta I = 1/2$ strangeness-changing
matrix elements.  This
contribution is calculated and compared to the purely perturbative
contribution, which is
well-known to be far too small to have an appreciable effect on
$\Delta I = 1/2$
strangeness-changing
matrix elements after mass-shell renormalization subtractions are
taken into account.  The ratio
between the t-quark condensate's contribution and the standard-model
perturbative contribution
to the mass-shell renormalized $s \leftrightarrow d$ self-energy
transition is shown to be of order
$\mid m_t \langle \bar{t} t \rangle \mid / M_W ^4$, suggesting that
the large t-quark condensate
proposed for
electroweak symmetry breaking may also be responsible for the $\Delta
I = 1/2$ enhancement
of
strangeness-changing decays.
\vglue 0.6cm}
\newpage
\baselineskip=18pt
\elevenrm
If electroweak symmetry breaking is driven dynamically by a t-quark
condensate $\langle \bar{t}t
\rangle$, such a condensate would necessarily have an enormous
magnitude.  A
symmetry-breaking
scenario based upon a Nambu-Jona-Lasinio mechanism would suggest a
condensate of magnitude
$-m_t\Lambda^2$, where $\Lambda$, the NJL cutoff, is certainly larger
than $m_t$ and may
even correspond to an order-Planck-mass unification mass
scale.$^{1,2}$  Such a large
condensate has
already been shown not to alter salient electroweak results, such as
the less-than-1\% deviation
of the electroweak $\rho$-parameter from unity.$^1$  Nevertheless, it
is of genuine importance
to
examine how such a very large condensate of {\elevenit standard-
model} fermions, as opposed
to
technifermions outside the purview of CKM mixing, might enter other
electroweak
results.  For example, the presence of a very large t-quark
condensate ($\langle \bar{t}t \rangle$)
necessarily generates an $s \leftrightarrow d$ self-energy transition
(via $\bar{t}s$ and $\bar{s}t$
CKM mixing angles$^3$) which will contribute exclusively to $\Delta I
= 1/2$ strangeness
-changing matrix elements.

Of course, there already exists a purely perturbative $s
\leftrightarrow d$ self-energy transition
within the standard model, but this contribution is known to be two
to three orders of magnitude
too small to have an appreciable effect on $\Delta I = 1/2$
strangeness-changing matrix elements
after mass-shell renormalization subtractions are taken into
account.$^4$ In unitary gauge, the
unrenormalized $s \leftrightarrow d$ self-energy amplitude is of the
form
\begin{equation}
\Sigma_{ds}(p)=H(p^2) \rlap/ p (1 - \gamma_5),
\end{equation}
where $H(p^2)$ contains both constant and momentum-dependent terms
(the former being
divergent).  However, the on-mass-shell renormalization conditions
$\bar{u}_d(p)[\Sigma_{ds}(p)]^{ren}$ $=0, [\Sigma_{ds}(p)]^{ren}
u_s(p)=0$, provide sufficient
information to determine algebraically the four allowed counterterms
contributing to
$[\Sigma_{ds}(p)]_{ren} = \Sigma_{ds}(p)+A \rlap/ p (1-\gamma_s)$ $+
B \rlap/ p(1+\gamma_s)
+
C
+ D \gamma_s$.$^5$  If we approximate $H(p^2)$ with $H(0) + p^2H '
(0)$, we find that$^4$
\begin{equation}
\Sigma_{ds}(p)^{ren} = H ' (0) \{[p^2 - m_s^2-m_d^2] \rlap/ p (1 -
\gamma_s) +
m_sm_d [- \rlap/ p (1 + \gamma_5) + (m_s+m_d) - \gamma_s(m_s-m_d)] \},
\end{equation}
where the t-quark's Fig. 1 (f1) contribution is given by$^4$
\begin{equation}
[H ' (0)] _{f1} = (G_F/ \sqrt{2}) (V_{td})^* V_{ts} (B_t + 5/12) / 4
\pi ^2,
\end{equation}
with $B_t$ between -1.0 and -1.3 for $m_t$ between 1.5 $M_W$ and 2.0
$M_W$. Using
the
present empirical range of CKM mixing angles $[0.003 \leq | V_{td} |
\leq 0.018, 0.030 \leq
| V_{ts}| \leq 0.054]$,$^6$ we see that $1 \cdot 10^{-11} GeV^{-2}
\leq [H ' (0)]_{f1} \leq 2
\cdot 10^{-10} GeV^{-2}$.  The renormalized transition amplitude (2)
can be incorporated into
the $\Delta I = 1/2$ component of the $K_s \rightarrow 2 \pi$
amplitude, denoted by $^{4,7}$
\begin{equation}
a_{1/2} = 3 \sqrt{2} g_{Kqq} (m_s-m_d) M^4 (1-m_\pi ^2 / m_K^2) H'
(0) / (64 \pi^2 f_\pi
^2)
\end{equation}
where M is a phenomenological cut-off of order 2 GeV and $g_{Kqq}
\approx 3.8$.  Using (3)
and (4) we find the t-quark's purely perturbative contribution to
$a_{1/2}$ to be between $2
\cdot 10^{-9}$ and $9 \cdot 10^{-11}$ GeV, a number at least two
orders of magnitude smaller
than the experimental value ($3.5 \cdot 10^{-7}$GeV) reflecting the
enhancement of the $\Delta
I = 1/2$ component of strangeness-changing nonleptonic decays. A more
careful treatment$^8$
of the hadronization of $\Sigma_{ds}(p)^{ren}$, based upon a direct
comparison of the weak
kaon
axial-vector current $-i \int d^4 x e^{iq \cdot x} \langle0|T[A_\mu^3
(x) H_W^{\Delta S =
1}]|\bar{K}^0 \rangle$ to the strong-interaction axial-vector current
$i \sqrt{2} f_K q_\mu = -i
\int d^4 x e^{iq \cdot x} \langle0|A_\mu ^{6+i7} (x)|\bar{K}^0
\rangle$, leads to virtually the
same conclusions:  $[H'(0)]_{f1}$ is between two and four orders of
magnitude too small to
account for the measured $K \rightarrow 2\pi$ decay rate.

If electroweak symmetry breaking is caused by condensation of a t-
quark-antiquark pair, residual
contributions to the Wick-Dyson expansion of Feynman amplitudes
involving the vacuum
expectation values of normal ordered t-quark and antiquark fields are
no longer seen to vanish.
The presence of such contributions is entirely analogous to how
$\langle u \bar{u} \rangle$ and
$\langle d \bar{d} \rangle$ condensates characterizing the chiral-
symmetry breakdown of QCD
enter Feynman amplitudes that are utilized in the construction of QCD
sum-rules.$^{8,9}$  In
particular, the unitary gauge contribution of Fig. 2, corresponding
to replacing the time ordered
vacuum expectation value of interaction-picture t-quark fields
$\langle 0 |T[t(x) \bar{t} (0)] | 0
\rangle$ with the corresponding normal-ordered contribution $\langle
0 |: t(x) \bar{t} (0) : | 0
\rangle$ in the Wick-Dyson expansion of $ \langle 0 | T[t_H (x)
\bar{t}_H (0)] | 0 \rangle$ [the
fermion two-point function of Heisenberg fields], is seen to be
\begin{eqnarray}
\Sigma_{ds} (p) & = & \frac{G_F M_W^2}{\sqrt{2}} V_{td}^* V_{ts} \int
\frac{d^4q}{(2
\pi)^4}
\int
d^4x e^{i(p-q) \cdot x} \nonumber\\
& \times & \left[ \frac{g^{\mu \nu} - q^\mu q^\nu/M_W^2}{q^2-M_W^2}
\right] \gamma_\mu
(1-\gamma_5) \langle
0 | : t(x)
\bar{t} (0): | 0 \rangle \gamma_\nu (1-\gamma_5).
\end{eqnarray}
The $ \langle \bar{t} t \rangle$ projection of $\langle 0 | : t(x)
\bar{0} (x) : | 0 \rangle$ is given
by$^{10}$
\begin{equation}
\langle 0 | : t(x) \bar{t} (0) : | 0 \rangle = - \frac{\langle
\bar{t}t \rangle}{6m_t^2} (i \gamma
\cdot \partial + m_t) [J_1 (m_t \sqrt{x^2})/ \sqrt{x^2} ].
\end{equation}
The theoretical consistency of eq. (6) has been examined in a number
of
places.  Insertion of this vacuum expectation value into flavour-
diagonal fermion two-point
functions has been shown to uphold the gauge invariance of the
fermion propagator pole in both
covariant $^{11}$ and noncovariant $^{12}$ gauges, despite strong
overall gauge dependence
of
the fermion self-energy.  Moreover, inclusion of the $\langle
\bar{f}f \rangle$ projection of
the $\langle 0 | : f(x) \bar{f}(y): | 0 \rangle$ contribution to the
Wick-Dyson expansion of the
vacuum polarization of a vector-coupled spin-1 field has been shown
to yield only transverse
contributions, consistent with gauge invariance and the vector-
current Ward identity.$^{13}$
Similar inclusion of the $ \langle \bar{f}f \rangle$ projection of
the
$ \langle 0 | : f(x) \bar{f} (y):|0 \rangle$-contributions to the
Wick-Dyson expansion of the
VVA-triangle amplitude has also been demonstrated to uphold the
anomalous axial-vector Ward
identity.$^{14}$

Equation (6) may be re-expressed as follows:$^{15}$
\begin{equation}
\langle 0 |: t(x) \bar{t}(0):| 0 \rangle \equiv \int d^4k e^{-ik
\cdot x} (\rlap/ k + m_t) F_t(k),
\end{equation}
\begin{equation}
\int d^4k e^{-ik \cdot x} F_t (k) = -\langle \bar{t}t \rangle J_1
(m_t \sqrt{x^2})/(6m_t^2
\sqrt{x^2}),
\end{equation}
\begin{equation}
\int d^4k e^{-ik \cdot x} g(k^2)F_t(k) = \int d^4k e^{-ik \cdot x}
g(m_t^2) F_t (k),
\end{equation}
where the last property [eq. (9)] is a consequence of $\langle 0|:
t(x) \bar{t}(0):| 0 \rangle$ being
a
solution of the free-field Dirac equation.  One finds from Eq.(5)
that the coefficient $H(p^2)$
in the unrenormalized Fig. 2 (f2) self-energy (1) is given by
\begin{eqnarray}
[H(p^2)]_{f2} = \frac{G_F}{\sqrt{2}}M_W^2 V_{td}^* V_{ts} \left\{
\left( -4 -
\frac{2(m_t^2+p^2)}{M_W^2} \right) \left[ \frac{\langle \bar{t}t
\rangle}{24m_tp^2} \right.
\right. \nonumber\\
+ \left. \left. \frac{p^2+m_t^2-M_W^2}{2p^2} I(p^2) \right] +
\frac{4m_t^2}{M_W^2} I(p^2)
\right\},
\end{eqnarray}
\begin{equation}
I(p^2) = \int d^4k F_t(k) / [(p-k)^2 - M_W^2 + i \epsilon].
\end{equation}
The integral in Eq. (11) may be evaluated first by exponentiating the
propagator and through
utilization of Eqs. (9) and (8):$^{15}$
\begin{eqnarray}
I(p^2) & = & -i \int_0^\infty d \eta e^{i \eta(p^2+m_t^2-M_W^2+i
\epsilon)} \int d^4k e^{-i(2
\eta
p) \cdot k} F_t(k)\nonumber \\
& = & \frac{i \langle \bar{t}t \rangle}{6m_t^2} \int_0^\infty d \eta
\: exp[i
\eta(p^2+m_t^2-M_W^2
+ i \epsilon)] \frac{J_1(2m_t \eta \sqrt{p^2})}{2 \eta
\sqrt{p^2}}\nonumber \\
& = & - \frac{ \langle \bar{t}t \rangle}{24m_t^3p^2} \left[(p^2+m_t^2-
M_W^2) - \sqrt{(p^2-
m_t^2)^2 - M_W^2(2p^2+2m_t^2-M_W^2)}\right],
\end{eqnarray}
where the final line of Eq.(12) is obtained by use of a tabulated
integral.$^{16}$  Substitution
of the final
line of (12) into (10) reveals that the fig. 2 contribution to
$H(p^2)$ is analytic at $p^2=0$:
\begin{equation}
[H(p^2)]_{f2} = [H(0)]_{f2} + p^2 [H'(0)]_{f2},
\end{equation}
\begin{equation}
[H(0)]_{f2} = -\frac{G_F V_{td}^* V_{ts} \langle \bar{t}t \rangle}{4
\sqrt{2} (m_t^2-
M_W^2)^2} [1-7M_W^2/3m_t^2-2M_W^4/3m_t^4],
\end{equation}
\begin{equation}
[H'(0)]_{f2} = \frac{G_F V_{td}^* V_{ts} \langle \bar{t}t \rangle
m_t^5}{12 \sqrt{2} (m_t^2-
M_W^2)^4} [1-4M_W^2/m_t^2+9M_W^4/m_t^4].
\end{equation}

The relative contributions of Fig. 2 and Fig. 1 to the on-shell-
renormalized self-energy transition,
as defined by (2), may be obtained via direct comparison of Eqs. (3)
and (15).  One finds for
$m_t$ between 120 GeV and 160 GeV that $|12m_t \langle \bar{t}t
\rangle/M_W^4| \geq |
[H'(0)]_{f2}/[H'(0)]_{f1}| \geq |0.4m_t \langle
\bar{t}t \rangle / M_W^4|$. As remarked earlier, the Fig. 1
contribution $[H'(0)]_{f1}$ to the
$\Delta I = 1/2$ component of
the $K \rightarrow 2 \pi$ decay amplitude is between two and four
orders of magnitude below
the experimental value.  Consequently, we
see that the
Fig. 2 contribution $[H'(0)]_{f2}$ yields a contribution {\elevenit
comparable to the
experimental
ampitude} provided $\langle \bar{t}t
\rangle \sim m_t^3$ for maximal mixing angles and $m_t \approx 120
GeV$. For
minimal mixing
angles and $m_t \approx 160 GeV$, the value $\langle \bar{t}t
\rangle \sim (10 m_t)^3$ is compatible with the experimental $K
\rightarrow 2\pi$ decay rate.

These results suggest, within the context of t-quark-condensate
driven electroweak symmetry
breaking, that the large magnitude of the t-quark condensate may be
responsible for the $\Delta
I = 1/2$ rule.  However, these results appear to rule out any
sensitivity of the t-quark condensate
to Planck-mass unification scales.
\newpage
{\elevenbf\noindent References \hfil}
\vglue 0.4cm
\begin{thebibliography}{9}%BIBLIOGRAPHY
\bibitem {vam/mt/ky}V.A. Miransky, M. Tanabishi and K. Yamawaki,
{\elevenit Phys. Lett.}
{\elevenbf B221} (1989) 177.
\bibitem {rrm}R.R. Mendel,Massachusetts Institute of Technology Ph.D.
Thesis, 1982
(unpublished); W.A. Bardeen, C.T. Hill and M. Lindner,{\elevenit
Phys. Rev.} {\elevenbf D41}
(1990) 1647.
\bibitem {mk/tm}M. Kobayashi and T. Maskawa, {\elevenit Prog. Theor.
Phys.}
{\elevenbf 49} (1973) 652.
\bibitem {spc}S.-P. Chia {\elevenit Phys. Lett.} {\elevenbf
B147}(1984) 361.
\bibitem {ngd/ge}N.G. Deshpande and G. Eilam {\elevenit Phys. Rev.}
{\elevenbf D26} (1982)
2463.
\bibitem {kh}K. Hikasa et. al. (Particle Data Group), {\elevenit
Phys. Rev.} {\elevenbf D45}
(1992) S1.
\bibitem {bhjm/mds} B.H.J. McKellar and M.D. Scadron, {\elevenit
Phys. Rev.} {\elevenbf D27}
(1983) 157.
\bibitem {ve/rrm/mds} V. Elias, R.R. Mendel and M.D. Scadron,
{\elevenit Phys. Rev.}
{\elevenbf D42} (1990) 1591.
\bibitem {mas/aiv/viz}M.A. Shifman, A.I. Vainshtein and V.I.
Zakharov, {\elevenit Nucl. Phys.}
{\elevenbf B147}
(1979) 385 and 448; P. Pascual and R. Tarrach, {\elevenit
Renormalization for the Practitioner}
(Springer-Verlag, Berlin, 1984), pp. 155-191.
\bibitem {ve/tgs/mds} V. Elias, T.G. Steele and M.D. Scadron,
{\elevenit Phys. Rev.} {\elevenbf
D38} (1989) 1584; E. Bagan, J.I. Latorre and P. Pascual, {\elevenit
Z. Phys.} {\elevenbf
C32}
(1986) 43; F.J. Yndurain, {\elevenit Z. Phys.} {\elevenbf C42}
(1989) 653.
\bibitem {ve/mds}V. Elias and M.D. Scadron {\elevenit Phys. Rev.}
{\elevenbf D30} (1984) 647;
L.J. Reinders and K. Stam, {\elevenit Phys. Lett.} {\elevenbf B180}
(1986) 125; M.R.
Ahmady,
V. Elias, R.R. Mendel, M.D. Scadron and T.G. Steele, {\elevenit Phys.
Rev.} {\elevenbf
D39}
(1989) 2764.
\bibitem {ve/tgs} V. Elias and T.G. Steele {\elevenit Phys. Rev.}
{\elevenbf D40} (1989) 2669;
M. Lavelle and M. Schaden, in {\elevenit Physical and Nonstandard
Gauges}, P. Gaigg, W.
Kummer and M. Schweda, eds. (Springer-Verlag, Berlin, 1990) p. 240.
\bibitem {tgs} T.G. Steele, {\elevenit Z. Phys.} {\elevenbf C42}
(1989) 499.
\bibitem {mra/asd/mt/ve} M.R. Ahmady, A.S. Deakin, Mong Tong and V.
Elias {\elevenit Z.
Phys.} {\elevenbf C56} (1992) 395.
\bibitem {eb/mra/ve/tgs} E. Bagan, M.R. Ahmady, V. Elias and T.G.
Steele, submitted to
{\elevenit Phys. Rev.}{\elevenbf D.}
\bibitem {isg/imr} I.S. Gradshteyn and I.M. Ryzhik, {\elevenit Table
of Integrals, Series and
Products} (Academic Press, Orlando, 1980) 712 [Eq. 6.623.3].
\end{thebibliography}
\pagebreak
{\bf \centerline{Figure Captions}}
\vglue 0.4cm
\noindent{\bf Figure 1:} Purely perturbative t-quark contribution to
the s-d transition amplitude.

\noindent{\bf Figure 2:} $\langle \bar{t}t \rangle$ contribution to
the s-d transition amplitude.
\end{document}
%{\elevenbf\noindent 7. Footnote}\footnote{\ninerm\baselineskip=11pt
%{\ninebf Footnotes} should be typeset in 9 point roman
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