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\begin{document}


\title{Kaon CP violation and radiative corrections}


\author{{\normalsize M. D. Scadron}\\
{\normalsize Physics Department, University of Arizona,}\\
{\normalsize Tucson, AZ 85721, USA}\normalsize }


\date{PACs \# 11.30.Er, 12.15.Lk, 13.20.Eb, 13.40.Ks}

\maketitle
\begin{abstract}
By recasting the recent KTeV measurement of \( \varepsilon '/\varepsilon \approx 28\times 10^{-4} \)
in its equivalent form \( \left| \eta _{+-}/\eta _{oo}\right| \approx 1.0084 \),
we show that the indirect CP-violation value \( \left| \eta _{+-}/\eta _{oo}\right| =1 \)
can be perturbatively enhanced to the above result via photonic radiative corrections.
This reformulation of new direct CP-violation \( \varepsilon '/\varepsilon  \)
data therefore lends further support to the 35-year-old conjecture suggesting
an (empirically accurate) \( \left| M_{K_{L}\pi \pi }/M_{K_{S}\pi \pi }\right|  \)
amplitude ratio \( \varepsilon =\alpha /\pi \approx 2.32\times 10^{-3} \) without
recourse to invoking a new direct CP-violating force.
\end{abstract}
\newpage

Ever since kaon CP violation (CPV) was discovered in 1964 {[}1{]}, physicists
have been trying to explain CPV as i) a first-order weak interaction effect
combined with electromagnetic radiative corrections, ii) a second-order weak
interaction indirect CPV \( K^{o}-\overline{K^{o}} \) mixing effect combined
with direct CPV, the latter due to perhaps a new exotic force such as generated
by QCD gluonic penguin graphs characterizing a nonzero \( \varepsilon '/\varepsilon  \)
value. In this note we blend the two approaches together by starting with a
standard model-violating small CPV term (due to a \( WW\gamma  \) interaction)
treated as a direct CPV perturbation of the lead \( K^{o}-\overline{K^{o}} \)
indirect mixing term. Then the first-order weak CPV \( K_{L}\rightarrow 2\pi  \)
decays are taken as a very small electromagnetic radiative correction to the
weak CP-conserving \( K_{S}\rightarrow 2\pi  \) decays. We downplay a nonzero
\( \varepsilon '/\varepsilon  \) value as a measure of a new CPV force.

The recently observed CP-violating (KTeV) measurement is {[}2{]}
\begin{equation}
\label{1}
Re\varepsilon '/\varepsilon =\frac{1}{6}\left[ \frac{\Gamma _{K_{L}\rightarrow \pi ^{+}\pi ^{-}}/\Gamma _{K_{S}\rightarrow \pi ^{+}\pi ^{-}}}{\Gamma _{K_{L}\rightarrow \pi ^{o}\pi ^{o}}/\Gamma _{K_{S}\rightarrow \pi ^{o}\pi ^{o}}}-1\right] =(28\pm 4)\times 10^{-4}.
\end{equation}


{\par\raggedright This result eq.(1) is mathematically equivalent to\par}


\begin{equation}
\label{2}
6\, Re\, \varepsilon '/\varepsilon =\: \left| \frac{\eta _{+-}}{\eta _{oo}}\right| ^{2}-1=\: 0.0168,\; or\; \; \left| \frac{\eta _{+-}}{\eta _{oo}}\right| \: =1.0084\pm 0.0006,
\end{equation}
 

{\par\raggedright and the latter eq.(2) suggests of a perturbative (radiative)
expansion. \par}

Many years ago {[}3{]}, the standard photon radiative correction

\begin{equation}
\label{3a}
\varepsilon =\alpha /\pi \approx 2.32\times 10^{-3}
\end{equation}


{\par\raggedright was suggested as the consequence of CPV, with \( \varepsilon \approx \eta _{+-}\approx \eta _{oo} \),
and to leading order\par}

\begin{equation}
\label{3b}
\left| \eta _{+-}/\eta _{oo}\right| =1.
\end{equation}
Here \( \eta _{+-,\, oo} \) is the amplitude ratio \( \mid M_{K_{L}\rightarrow \pi \pi }/M_{K_{S}\rightarrow \pi \pi }\mid  \),
with measured values {[}4{]} in fact close to the original theoretical conjecture
in eq.(3) above :


\begin{equation}
\label{4}
\eta _{+-}=\: (2.285\pm 0.019)\times 10^{-3},\; \; \eta _{oo}=\: (2.275\pm 0.019)\times 10^{-3}.
\end{equation}


In the SU(2)\( \times  \)U(1) electroweak standard model (SM), CP is manifestly
conserved; a chiral lepton or quark SM triangle loop cannot generate the needed
{[}3{]} CPV \( WW\gamma  \) chiral coupling necessary to produce eq.(3). Stated
a second way, any CPV \( \overline{\psi }\gamma _{5}\psi  \) SM bilinear covariant
can be rotated away {[}5{]} by the chiral transformation \( \psi \rightarrow e^{\pi /2\gamma _{5}}\psi  \).
Alternatively a V-A chiral current-current theory driving CP-conserving weak
decays pertains to the original derivation for \( \mu  \) decay \( \mu \rightarrow e\nu \overline{\nu } \)
{[}6{]} involving massless neutrinos. Then no CPV is initially allowed in any
SM scheme. Conversely, massive neutrinos can generate CPV, but in the SM?

However, due to recent observations {[}7{]} detecting neutrino mixing, one may
infer that at least one neutrino (say the muon neutrino) in fact has a nonvanishing
(but small) mass. This then generates different chiral transformations on the
muon and the muon neutrino, which in turn results in a small (but nonvanishing)
CP-violating term.

Specifically for a SM leptonic doublet \( \psi =\left( \begin{array}{c}
\nu \\
e
\end{array}\right)  \), an off-diagonal electromagnetic (em) field term \( \overline{\chi }\gamma \cdot A^{em}\psi  \)
will change under a different chiral rotation of \( \chi  \) and \( \psi  \)
(different because the associated neutrino mass is now assumed to be small but
nonvanishing). This generates a (small) CP-violating neutrino mass term \( \overline{\nu }m_{\nu }\gamma _{5}\nu  \).
Then the resulting CPV term can be allocated with the upper part of the lepton
SM doublet as a \( WW\gamma  \) \emph{perturbation} due to the now nonvanishing
lepton triangle graph of Fig 1. This will fix the scale of the \( WW\gamma  \)
vertex, with an effective CPV lagrangian density term 
\begin{equation}
\label{5}
L_{CPV}=ie\lambda _{w}W^{\dagger }_{\mu }W_{\nu }\varepsilon ^{\mu \upsilon \alpha \beta }\partial _{\alpha }A_{\beta }^{em}.
\end{equation}


{\par\raggedright Since the actual CPV neutrino mass is not yet known, we need
another approach to determine the \( \lambda _{w} \) scale in (6). Later we
shall use the CKM matrix to fix \( \lambda _{w} \) in (6) near \( \lambda _{w}\approx 1 \).\par}

To extend this CPV term (6) to \( \Delta I=1/2 \) \( \overline{K^{o}}\rightarrow 2\pi  \)
decays, we anticipate a small CPV \( \overline{s}\gamma _{5}s \) term (the
upper component of the \( \left( \begin{array}{c}
s\\
c
\end{array}\right)  \) doublet), as depicted in Fig. 2a. The darkened circles in Figs. 2 indicate
a \( \Delta I=1/2 \) \( s\rightarrow d \) transition, which also characterizes
the known {[}4{]} \( \Delta I=1/2 \) dominance of \( K^{o},\, \overline{K^{o}}\rightarrow 2\pi  \)
first-order weak decays. Then we combine the CPV vertex \( \overline{s}\gamma _{5}s \)
of Fig. 2a with the CP-conserving s\( \rightarrow  \)d transition by expliciting
denoting the photon propagator self-energy graphs in Figs. 2b. The resulting
\( sd\gamma  \) vertices in Fig. 2b are generated by the CPV vertex \( WW\gamma  \)
(ie. \( VA\gamma  \)) in eq. (6) above.

But first we comment on (leading order) indirect CPV. Since the observed {[}4{]}
CPV phase angle is (\( \phi _{+-,\: oo}\rightarrow 45^{o} \) in the CP-conserving
limit) 
\begin{equation}
\label{6a}
\phi _{+-}=(43.5+0.6)^{o},\; \; \phi _{oo}=(43.4\pm 1.0)^{o},
\end{equation}


{\par\raggedright compatible with the unitarity prediction {[}8{]}
\begin{equation}
\label{6b}
\phi =arctan\, (2\Delta m_{LS}/\Gamma _{S})\approx 43.45^{o}
\end{equation}
\par}

{\par\raggedright for {[}4{]} \( \Delta m_{LS}\approx 0.4736\Gamma _{S} \),
we employ the \( K^{o}-\overline{K^{o}} \) mixing graph of Fig. 3 to diagonalize
the neutral kaon mass matrix as
\begin{equation}
\label{7a}
\left( \begin{array}{cc}
m^{2}_{K^{o}} & \xi \\
\xi  & m_{\overline{K}^{o}}^{2}
\end{array}\right) \begin{array}{c}
\rightarrow \\
\phi 
\end{array}\left( \begin{array}{cc}
m_{S}^{2} & 0\\
0 & m_{L}^{2}
\end{array}\right) ,
\end{equation}
\par}

\begin{equation}
\label{7b}
sin2\phi =\frac{2\xi }{m_{L}^{2}-m_{S}^{2}}\approx \frac{\xi }{\Delta m_{LS}m_{K}}\approx 1.
\end{equation}


{\par\raggedright But Fig. 3 requires \( \xi \equiv \left\langle \overline{K^{o}}\left| H_{w}\right| K^{o}\right\rangle \approx 2b^{2}m^{2}_{K} \)
on the kaon mass shell, where \( b \) is the dimensionless s\( \rightarrow  \)d
weak scale (the darkened circles in Figs. 2 and Fig. 3). Substituting \( \xi  \)
above into eq. (10) then leads to {[}8{]}
\begin{equation}
\label{8a}
\Delta m_{LS}/m_{K}\approx 2b^{2}.
\end{equation}
 However present data {[}4{]} says \( \Delta m_{LS}/m_{K}=(0.701\pm 0.002)\times 10^{-14} \),
fixing the weak scale in (11) to
\begin{equation}
\label{8b}
\left| b\right| \approx 5.92\times 10^{-8}.
\end{equation}
 Note that the s\( \rightarrow  \)d and \( \overline{s}\rightarrow \overline{d} \)
second order weak graph of Fig. 3 is in the spirit of the SM and also indirect
CPV.\par}

Applying (12) to the long distance (LD) low energy chiral \( K_{S}\rightarrow 2\pi ^{o} \)
current algebra-PCAC amplitude {[}8{]} for \( f_{K}/f_{\pi }\approx 1.22 \),
\( f_{\pi }\approx 93MeV \), one predicts
\begin{equation}
\label{9a}
\left| M_{K_{S}\rightarrow \pi ^{o}\pi ^{o}}\right| =2b\left( m_{K}^{2}-m_{\pi }^{2}\right) f_{K}/f^{2}_{\pi }\approx 35.6\times 10^{-8}GeV,
\end{equation}
 near the observed amplitude {[}4{]}
\begin{equation}
\label{9b}
\left| M_{K_{S}\rightarrow \pi ^{o}\pi ^{o}}\right| =m_{K}\sqrt{16\pi \Gamma /p}=\left( 37.1\pm 0.2\right) \times 10^{-8}GeV.
\end{equation}
 The agreement is even better with (14) if we fold in the small (5\%) \( \Delta I=3/2 \)
\( K_{2\pi } \) amplitude in (13). Moreover the weak \( \Delta I=1/2 \) scale
in (12) was correctly predicted to within 5\% via a W-mediated s\( \rightarrow  \)d
quark transition in the GIM scheme {[}9{]}. To this leading order the \( \Delta I=1/2 \)
s\( \rightarrow  \)d radiative graphs of Figs. 2b sum to
\begin{equation}
\label{10a}
\varepsilon =\eta _{+-}=\eta _{oo}=\left| M_{K_{L}\rightarrow \pi \pi }/M_{K_{S}\rightarrow \pi \pi }\right| =\frac{b\alpha /\pi }{b}=\alpha /\pi ,
\end{equation}
 in agreement with eq. (3), and both \( K_{S}\rightarrow \pi \pi  \) and CPV
\( K_{L}\rightarrow \pi \pi  \) remain similarly \( \Delta I=1/2 \) dominated
{[}4, 8{]}:
\begin{equation}
\label{10b}
\frac{\Gamma _{+-}^{S}}{\Gamma _{oo}^{S}}=2.186\pm 0.009,\; \; \frac{\Gamma _{+-}^{L}}{\Gamma _{oo}^{L}}=2.208\pm 0.060.
\end{equation}
 

For this indirect CPV we can test for the \( \lambda _{w}=1 \) scale in the
\( WW\gamma  \) radiative vertex eq. (6) by studying the CKM matrix {[}10{]}
in the limit \( s_{1}>>s_{2}>s_{3} \):
\begin{equation}
\label{11}
V=\left( \begin{array}{ccc}
V_{ud} & V_{us} & V_{ub}\\
V_{cd} & V_{cs} & V_{cb}\\
V_{td} & V_{ts} & V_{tb}
\end{array}\right) \rightarrow \left( \begin{array}{ccc}
c_{1} & -s_{1} & 0\\
s_{1} & c_{1} & 0\\
0 & 0 & -(1+i\delta )
\end{array}\right) .
\end{equation}
 Here the complex CPV phase \( -e^{i\delta } \) is a small correction to \( V_{tb}=-1 \)
(from data {[}4{]} \( \left| V_{tb}\right| =0.9991 \) to \( 0.9994) \). The
CKM graphs of Figs. 4 for \( V_{tb} \) then give {[}8{]}
\begin{equation}
\label{12}
\delta \approx \lambda _{w}\frac{\alpha }{\pi }ln\left( 1+\Lambda ^{2}/m^{2}_{t}\right) \approx 3.39\times 10^{-3}\lambda _{w},
\end{equation}
 where the ultraviolet cutoff \( \Lambda  \) for Fig. (4b) can be taken as
the (heavy) SM Higgs scalar, now greater than {[}4{]} 77.5 GeV. Since the observed
t quark at 175 GeV is much heavier than all other fermion masses, the Veltman
condition {[}11, 12{]} becomes
\begin{equation}
\label{13}
M^{2}_{H}=4m_{t}^{2}-\left( 2M_{W}^{2}+M_{Z}^{2}\right) .
\end{equation}
 Then with \( M_{W}=80.4 \) GeV, \( M_{Z}=91.2 \) GeV, eq. (19) suggests {[}12{]}
\( M_{H}\approx 318 \) GeV. Taking the latter as the UV cutoff \( \Lambda  \)
then gives the numerical value in (18) assuming \( \lambda _{w} \) is still
undetermined.

The phase relation {[}13{]} is \( \delta =2\left| \varepsilon \right| cos\phi  \)
and the predicted or observed \( \varepsilon  \) and \( \phi  \) in eqs. (3,
5) and eqs. (7, 8) to leading order requires \( \delta  \) to be
\begin{equation}
\label{14a}
\delta =2\left| \varepsilon \right| cos\phi \approx 3.37\times 10^{-3}.
\end{equation}
 This CPV phase \( \delta  \) (measured from semileptonic \( K_{l3} \) decays)
is {[}4{]} 
\begin{equation}
\label{14b}
\delta =\left[ \frac{\Gamma \left( K_{L}\rightarrow \pi ^{-}l^{+}\nu \right) -\Gamma \left( K_{L}\rightarrow \pi ^{+}l^{-}\nu \right) }{\Gamma \left( K_{L}\rightarrow \pi ^{-}l^{+}\nu \right) +\Gamma \left( K_{L}\rightarrow \pi ^{+}l^{-}\nu \right) }\right] =(3.27\pm 0.12)\times 10^{-3}.
\end{equation}
 The nearness of (21) to eqs. (18) and (20) suggest \( \lambda _{w}\approx 1 \)
in eq. (18). Note that the CPV \( WW\gamma  \) vertex in Fig. 4b affects eqs.
(18)-(20) only in the CKM-SM spirit while recovering the lead CPV amplitude
ratios (4) and (15). 

This completes our leading order (indirect) CPV picture for \( K\rightarrow 2\pi  \)
decays. Now we examine higher-order (direct) CPV effects for charged final-state
pions in \( \overline{K^{o}}\rightarrow \pi ^{+}\pi ^{-} \), as depicted in
Figs. 5. Note that the small perturbative CPV \( \overline{s}\gamma _{5}s \)
term in Fig. 2a is already ``glued'' to the much larger \( \Delta I=1/2 \)
s\( \rightarrow  \)d darkened single quark line (radiative correction) \( \overline{K^{o}}\rightarrow \pi ^{+}\pi ^{-} \)
transitions in Figs. 2b and in Figs. 5. Note too that this direct CPV contact
term in Figs. 5 is not an s\( \rightarrow  \)d, \( \overline{u} \) spectator
version of Fig. 2b resulting in indirect CPV. Then the total (leading order)
indirect plus (first order) direct CPV graphs give
\begin{equation}
\label{15}
\frac{\eta _{+-}}{\eta _{oo}}=1+\left( \alpha /\pi \right) \cdot 2\cdot 2\cdot \left( 0.97/1.05\right) \approx 1.0086,
\end{equation}
 where the \( \alpha /\pi  \) factor is the standard photon radiative correction,
the first 2 on the rhs of (22) comes from \( \pi ^{+} \) as well as \( \pi ^{-} \)
radiative effects (or in quark language approximately due to the s\( \rightarrow  \)d
direct QED-type CPV vertex modification), and the second 2 on the rhs of (22)
comes from adding in \( K^{o}\rightarrow \pi ^{+}\pi ^{-} \) decay in \( K_{L}\rightarrow \pi ^{+}\pi ^{-} \)
\( \left( \sqrt{2}K_{L}=K^{o}+\overline{K^{o}}\right)  \). The 1.05 factor
in (22) is due to the small \( \Delta I=3/2 \) (5\%) amplitude addition in
\( K_{S}\rightarrow \pi ^{+}\pi ^{-}\, vs\, \pi ^{o}\pi ^{o} \). Lastly the
0.97 factor in (22) stems from binding the (free) s\( \overline{d} \) quarks
into bound neutral kaons as found using light-plane quark wave functions in
ref. {[}14{]}.

This predicted 0.8-0.9\% CPV net radiative correction for (chiral) nonleptonic
decays \( K_{2\pi }^{o} \) in (22) is quite typical of other chiral radiative
corrections. Specifically for semileptonic charged mu decay \( \mu ^{-}\rightarrow e^{-}\nu \overline{\nu } \),
the net radiative correction V-A enhancement long ago found by Kinoshita-Sirlin
{[}15{]} is


\begin{equation}
\label{16}
1+\left( \alpha /\pi \right) \left[ \pi ^{2}-\frac{25}{4}\right] \approx 1.0084.
\end{equation}


We note that the photonic CPV radiative amplitude ratio in (22) accurately reflects
the recent KTeV \( \varepsilon '/\varepsilon  \) measurement in eq. (2). The
alternative QCD gluonic penguin graph for direct CPV and \( \varepsilon '/\varepsilon  \)
is found to be much too small {[}16, 17{]} to fit either the leading order CP-conserving
\( K_{2\pi } \) decay amplitudes in eqs. (13, 14) or the measured {[}2{]} \( \varepsilon '/\varepsilon  \)
CPV ratio in eq. (1).

In summary, a) the first-order CP-conserving \( K_{S}\rightarrow \pi ^{o}\pi ^{o} \)
LD amplitude is given by (13), with the weak scale (12) also determining the
\( K^{o}-\overline{K^{o}} \) second-order weak mixing in (9, 10); b) the latter
\( K^{o}-\overline{K^{o}} \) mixing also generates the leading order CPV amplitude
ratio \( \eta _{+-}/\eta _{oo} \) of unity in (4) and is (photonic) perturbatively
enhanced to (22), the latter being compatible with the KTeV measurement of CPV
in (2); c) then the alternative penguin version of direct CPV appears not needed
and such graphs are evaluated as very small in any case {[}16, 17{]}.\\


Acknowledgments: The author appreciates recent discussions on CPV with R. Delbourgo
and D. Liu, and prior interactions with S.R. Choudhury.

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\end{thebibliography}

\title{Figure Captions}

\begin{description}
\item [Fig.\( \:  \)1]Lepton triangle \( \mu \mu \nu  \) with massive neutrino generating
a CPV vertex \( WW\gamma  \).
\item [Fig.\( \:  \)2]CPV \( \overline{s}\gamma _{5}s \) and CP-conserving s\( \rightarrow  \)d
transitions (a), combined CPV s\( \rightarrow  \)d transitions (b).
\item [Fig.\( \:  \)3]Second order weak \( \overline{K^{o}}-K^{o} \) transition.
\item [Fig.\( \:  \)4]Lead order CKM element \( V_{tb} \) (a), higher order CKM
element \( V_{tb} \) (b).
\item [Fig.\( \:  \)5]Direct CPV graphs for \( \overline{K^{o}}\rightarrow \pi ^{+}\pi ^{-} \).
\end{description}
\end{document}

