---
abstract: 'The fermions of the Standard Model are integrated out to obtain the effective Lagrangian in the sector violating $P$ and $CP$ at zero temperature. We confirm that no contributions arise for operators of dimension six or less and show that the leading operators are of dimension eight. To assert this we explicitly compute one such non-vanishing contribution, namely, that with three $Z^0$, two $W^+$ and two $W^-$. Terms involving just gluons and $W$’s are also considered, however, they turn out to vanish in the $P$-odd sector to eighth order. The analogous gluonic term in the $CP$-odd and $P$-even ($C$-odd) sector is non-vanishing and it is also computed. The expressions derived apply directly to Dirac massive neutrinos. All $CP$-violating results display the infrared enhancement already found at dimension six.'
author:
- 'L. L. Salcedo'
title: 'Leading order one-loop $CP$ and $P$ violating effective action in the Standard Model'
---

Introduction {#sec:1}
============

A full understanding of $CP$-violation remains a challenge and for this reason it is a fruitful field of research both in relation with the Standard Model of particle physics and in extensions thereof [@Xing:2003ez; @Buras:1997fb; @Neubert:1996qg; @Grossman:1997pa; @Winstein:1992sx; @Paschos:1989ur; @Wolfenstein:1987pe; @Donoghue:1987wu; @Charles:2004jd]. $CP$-violation enters in very different phenomena, like non vanishing of the electric dipole momentum of elementary particles, baryogenesis [@Sakharov:1967dj], or, assuming $CPT$ invariance, the puzzling $T$-violation. Yet, in the Standard Model, $CP$-violation is rather elusive. There is no trace of it in the QCD sector, while in the electroweak sector it enters through a small parameter in the CKM matrix for quarks [@Kobayashi:1973fv], and possibly also for leptons, for massive neutrinos [@Maki:1962mu]. Even in the electroweak sector manifestation of $CP$ breaking requires a subtle combination, the Jarlskog determinant $\Delta$, which requires order twelve in the quark (or leptons) masses and would vanish if two up-like or two down-like quarks were degenerated in mass [@Jarlskog:1985ht]. In any case only through fermions $CP$ can be broken in the Standard Model. The structure of the Standard Model action implies that integration of the fermions results in an effective Lagrangian of the form (we assume the unitary gauge throughout) $$\mathcal{L}^\text{eff}
{}
(x) = \sum_\alpha g_\alpha
\left(\frac{v}{\phi(x)}\right)^{d_\alpha-4}\mathcal{O}_\alpha(x) ,$$ where $\mathcal{O}_\alpha(x)$ represents any possible operator, of mass dimension $d_\alpha$, constructed as a Lorentz and gauge invariant product of the gauge fields, their derivatives and derivatives of the Higgs field. $g_\alpha$ is the operator coupling constant, with mass dimension $4-d_\alpha$. $\phi(x)$ denotes the Higgs field and $v$ its vacuum expectation value. The coupling constant (which may vanish for some operators) has two additive contributions, one from the quark loop and another from the lepton loop. In the $CP$-odd sector, $g_\alpha$ must contain the Jarlskog determinant. In terms of the Yukawa coupling this yields a tiny dimensionless number, $\Delta/v^{12}$, of the order of $10^{-24}$. This fact has occasionally been presented as an indication of an intrinsic limitation of the Standard Model to produce enough $CP$-breaking to account for observations, including the baryon asymmetry. While this might be true, qualitative arguments should eventually be supported by a detailed computation. Smit argued in [@Smit:2004kh] that the coupling $g_\alpha$ is just a homogeneous function of the quarks (or leptons) masses of the appropriate degree. This implies that $g_\alpha \sim \Delta \times I_\alpha$, where $I_\alpha$ has a large negative degree to compensate that of $\Delta$. Both $\Delta$ and $I_\alpha$ depend only on the fermion masses and do not involve $v$. On the other hand, the various quark masses are very different and widely different result can be obtained by combining them at random. Actual calculations have been carried out in [@Hernandez:2008db; @GarciaRecio:2009zp] for operators of dimension six, which is the first possible $CP$-violating contribution at one-loop. They show that $g_\alpha \sim J\kappa/m_c^2$ where is $m_c$ the charm quark mass, $J=2.9(2)\times 10^{-5}$ is the Jarlskog invariant [@Nakamura:2010zzi] and $\kappa$ is a dimensionless coefficient of the order of unity. Implications for cold electroweak baryogenesis have been considered in [@Tranberg:2009de; @Tranberg:2010af]. Unfortunately, these two references differ in that [@Hernandez:2008db] finds such a dimension six contribution in the $P$-odd sector whereas [@GarciaRecio:2009zp] finds a contribution in the $C$-odd sector but none in the $P$-odd one.

The purpose of this note is manyfold. First, to reduce to the simplest and more transparent terms the calculation of these couplings constants. Second, to confirm that, although dimension six $CP$-odd and $P$-odd operators do exist, their coupling vanish in the Standard Model. Third, to verify that the order six cancellation is accidental, and non vanishing contributions in the $CP$-odd and $P$-odd sector appear for the first time at dimension eight. The purely gluonic leading (eighth) order term is also computed since it is particularly simple. As it turns out, this term breaks $C$ but not $P$. Lastly, to verify that the enhancement (as compared to the naive estimate) found at order six is displayed also at higher orders.

The method {#sec:2}
==========

We will integrate out the fermions in the Standard Model to extract the $CP$ violating contribution of the resulting effective action. This is the one-loop approximation to the effective action with full one-particle irreducible bosonic lines and vertices. We work at zero temperature. Quarks will be explicitly considered. Leptons would not contribute to the $CP$-odd sector if neutrinos are assumed to be exactly massless. For massive Dirac neutrinos the contribution of the leptons will be completely analogous to the one obtained for quarks.

The quark-sector Lagrangian of the Standard Model, in its Euclidean version and in the unitary gauge, can be written as [@Huang:1992bk]: $${\cal L}(x) = 
\bar{q}(x){{\bf D}}q(x)
=
(\bar{q}_L,\bar{q}_R) \left(\begin{matrix}
 m & {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L  \\
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R & m
\end{matrix}
\right)
 \left(\begin{matrix}
 q_R  \\ q_L
\end{matrix}
\right).$$ Here $q_{L,R}$ carry Dirac, generation (family), $ud$ and color indices ($ud$ space distinguishes the up-like from down-like quarks in each generation). Expanding further the matrices in $ud$ space: $$\begin{aligned}
m &=&
\left(\begin{matrix}
\frac{\phi}{v} m_u &0
\\
0 & \frac{\phi}{v} m_d
\end{matrix}
\right)
,
\quad
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L = 
\left(\begin{matrix}
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_u + {\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}} + {\hbox{$ \mathrel{\mathop{G\!\!\!\!/}}$}} & {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+ C
\\
{\mathrel{\mathop{W\!\!\!\!\!/}}}{}^-C^{-1} & {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_d - {\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}} + {\hbox{$ \mathrel{\mathop{G\!\!\!\!/}}$}}
\end{matrix}
\right)
,
\quad
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R = 
\left(\begin{matrix}
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_u + {\hbox{$ \mathrel{\mathop{G\!\!\!\!/}}$}} & 0
\\
0 & {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_d + {\hbox{$ \mathrel{\mathop{G\!\!\!\!/}}$}}
\end{matrix}
\right)
.
\label{eq:2.2}\end{aligned}$$ Here $m_{u,d}$ are the diagonal matrices (in generation space) with the up-like and down-like quarks masses, respectively. $G_\mu$ the gluon field, $Z_\mu$ the $Z^0$ field, $W_\mu^\pm$ the $W$ boson fields, $C$ is the CKM matrix, finally $(D_\mu)_{u,d}= \partial_\mu+q_{u,d}B_\mu$ where $q_u=2/3$, $q_d=-1/3$, and $B_\mu$ is the weak hypercharge gauge connection. For convenience, in all cases the coupling constant has been included in the corresponding gauge connection. Further details can be found in [@GarciaRecio:2009zp].

After integration of the quark loop, the corresponding Euclidean effective action is just $$\Gamma = -{{\rm Tr}}\log{{\bf D}}.$$ $\Gamma$ is the sum of the all Feynman graphs with one quark loop and any number of bosonic legs, gauge fields and Higgs. This sum is written as a functional which will be expressed within a covariant derivative expansion of these bosonic fields.

Certainly, the effective action can be computed following the efficient method outlined in [@GarciaRecio:2009zp] and based on [@Salcedo:2008tc], applied there to sixth order in the derivative expansion. However, one of our goals here is to present a derivation as transparent as possible, and closer to the method introduced in [@Salcedo:2000hx] on which the calculation of [@Hernandez:2008db] is based. To this end, we will use the relation $$\delta\Gamma = 
-{{\rm Tr}}(\delta{{\bf D}}\, {{\bf D}}^{-1})
=
-\int d^4x \,{{\rm tr \,}}\!\!\left[\delta{{\bf D}}\langle x| {{\bf D}}^{-1}|x\rangle\right]
.
\label{eq:2.4}$$ In the second equality it has been used that the variation $\delta{{\bf D}}$, induced by the variation in the gauge and Higgs fields in ${{\bf D}}$, contains no derivatives. The method is then to choose a suitable variation of these fields, compute $\delta\Gamma$ in the desired sector, and subsequently seek a functional fulfilling such a variation. The virtues of this approach are i) the current $\langle x| {{\bf D}}^{-1}|x\rangle$ is easier to obtain than the ${{\rm Tr}}\log{{\bf D}}$ itself, ii) the condition on $\delta\Gamma$ of being a consistent variation provides a nontrivial check of the calculation, and iii) even if one where to compute $\Gamma$ directly, the simplest way to avoid integration by parts identities (i.e., redundant operators in the final expression) is to obtain its functional derivative, $\delta\Gamma/\delta{{\bf D}}=-\langle x| {{\bf D}}^{-1}|x\rangle$. This quantity is local and so free from $x$-integration by parts identities. This issue becomes increasingly important as the number of derivatives increases.

A convenient field to use as variation in [eq. (\[eq:2.4\])]{} is $Z$ which appears just in $D^L_\mu$. We adopt such a choice, namely, $$\delta{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L=\left(\begin{matrix}
 \delta {\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}} & 0  \\
0 & -\delta {\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}}
\end{matrix}
\right)
:=\delta {\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}},
\qquad
\delta{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R = \delta m = 0
.
\label{eq:2.5}$$

The Dirac operator can be written as ${{\bf D}}= P_L {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R P_R + P_R
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L P_L + P_R m P_R + P_L m P_L $ where $P_{R,L}=\frac{1}{2}(1\pm\gamma_5)$ project on the $R$ or $L$ spaces, respectively. ${{\bf D}}$ can be explicitly inverted in chiral blocks. In particular, for the $RL$ block, $P_R {{\bf D}}^{-1} P_L = P_R ({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L - m
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R\!\!^{-1} m)^{-1} P_L$. Hence, from [eq. (\[eq:2.4\])]{} and (\[eq:2.5\]), $$\delta\Gamma =
-{{\rm Tr}}(P_R \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}} \,
({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L - m {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R\!\!^{-1} m)^{-1} )
.$$ Therefore, if only the $P$-odd sector (i.e., that with a $\gamma_5$) is retained, we will have $$\begin{aligned}
\delta\Gamma^- &=&
-\frac{1}{2}{{\rm Tr}}\!\left[\gamma_5 \,\delta {\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}} \,
({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L- m {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R\!\!^{-1} m)^{-1}
\right]
\nonumber \\
&=&
-\frac{1}{2}{{\rm Tr}}\!\left[\gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}} 
\, (
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L\!\!^{-1}
+
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L\!\!^{-1}
m
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R\!\!^{-1}
m
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L\!\!^{-1}
+ \cdots
)\right]
.\end{aligned}$$

It can be noted that each term of the expansion between parenthesis starts and ends with a label $L$. Moreover, as the chiral label propagates through the term it flips (from $L$ to $R$ and vice versa) at $m$ but not at $D_R$ or $D_L$. Keeping these rules in mind we can simply write $$\begin{aligned}
\delta\Gamma^- &=&
-\frac{1}{2}{{\rm Tr}}\!\left[\gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}\,
(
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}\,{}^{-1}
+
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}\,{}^{-1}
m
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}\,{}^{-1}
m
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}\,{}^{-1}
+ \cdots
)\right]
\nonumber \\
&=&
-\frac{1}{2}{{\rm Tr}}\!\left[\gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}} \,
({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}+m)^{-1}
)\right]
.\end{aligned}$$ The second equality follows from the fact that terms with an odd number of $m$’s are automatically discarded since they cannot start and end with a label $L$.

A simple and convenient technique to compute $\langle x| ({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}+m)^{-1}
|x\rangle$ is the method of symbols [@Nepomechie:1984wt; @Salcedo:1996qy; @GarciaRecio:2009zp]. This method is suitable for computing one-loop Feynman graphs with external legs at zero momentum or more generally, for expansions around zero momentum. In the present case it takes the form $$\begin{aligned}
\delta\Gamma^- &=&
-\frac{1}{2}\int \frac{d^4x d^4p}{(2\pi)^4} \, {{\rm tr \,}}\!\!\left[
\gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}\,
(
{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}+i\!{\mathrel{\mathop{p\!\!\!/}}}+m)^{-1}
\right]
.
\label{eq:2.9}\end{aligned}$$ Two remarks apply here: i) the momentum variable $p_\mu$, like $D^{R,L}_\mu$, does not introduce a flip in the chiral label, and ii) after momentum integration all $D_\mu$ appear only in the form $[D_\mu,~]$ and so there are no longer differential (or pseudo differential) operators acting; only an ordinary function of $x$ survives. At the same time gauge invariance is ensured.

The covariant derivative expansion is just an expansion in powers of $D_\mu$. Only even orders contribute (the space-time dimension being even). Besides, in the $P$-odd sector the effective action starts at fourth order in four dimensions, since a Levi-Civita pseudo tensor must be present. Hence: $$\begin{aligned}
\delta\Gamma^- &=&
\frac{1}{2}\int \frac{d^4x d^4p}{(2\pi)^4} \, {{\rm tr \,}}\!\!\left[
\gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}\,
\sum_{n=0}^\infty
\left(
\frac{i\!{\mathrel{\mathop{p\!\!\!/}}}-m}{p^2+m^2}{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}
\right)^{2n+3}
\frac{i\!{\mathrel{\mathop{p\!\!\!/}}}-m}{p^2+m^2}
\right]
.
\label{eq:2.11}\end{aligned}$$

Sixth order $P$-odd terms {#sec:3}
=========================

Taking the appropriate values of $n$ in [eq. (\[eq:2.11\])]{} and the appropriate contributions in $D^{R,L}$ and $m$, one can select the desired terms of the effective action. At least two $W^+W^-$ pairs must be present in the $CP$-odd sector terms, since the quark loop must visit the three generations [@Kobayashi:1973fv].[^1] A term with just $(W^+W^-)^2$ would count as fourth order, however no $CP$-odd term can be constructed without introducing further fields or derivatives. To sixth order such a $P$-odd and $CP$-odd term can be written using $(W^+W^-)^2ZD$. ($D$ here refers to either $D_u$ or $D_d$.) The question is whether this operator appears with a non vanishing coefficient in the Standard Model or not. Ref. [@Hernandez:2008db] claims that it does whereas the calculation in [@GarciaRecio:2009zp] concludes that it does not. Therefore we will start by reconsidering such a term within our present approach.

Under a variation of $Z$, the contribution of the candidate term to be found in [eq. (\[eq:2.11\])]{} is of the form $\delta Z(W^+W^-)^2D$. We can set $\phi=v$, since we are not interested in contributions from Higgs, and likewise we can set $Z_\mu$ and $G_\mu$ to zero in $D^{R,L}_\mu$. Moreover, we can even set $D_\mu^u=D_\mu^d=\partial_\mu$ in $D^{R,L}_\mu$. The ordinary derivatives can be unambiguously replaced by covariant ones at the end without loss of information since no $F_{\mu\nu}$ tensor can be present in the term considered.

The computation is tedious but straightforward. Let us spell out the main steps in the calculation. We select terms with $n=1$ in [eq. (\[eq:2.11\])]{} and restore the $L,R$ labels. We keep only terms starting and ending with the label $L$ and only $m$ introduces a chiral label flip $L\leftrightarrow R$. No flip is introduced by $m^2$, $p$ or $D$. This yields terms of the type $$\delta\Gamma^- =
\int \frac{d^4x d^4p}{(2\pi)^4} \, {{\rm tr \,}}\!\!\left[
\frac{1}{2} \gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}\,
N \!{\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{\hat W\!\!\!\!\!/}}}
 N m {\mathrel{\mathop{\partial\!\!\!/}}} N m {\mathrel{\mathop{\hat W\!\!\!\!\!/}}}  N \!{\mathrel{\mathop{p\!\!\!/}}} \,
{\mathrel{\mathop{\hat W\!\!\!\!\!/}}} N \! {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{\hat W\!\!\!\!\!/}}} N \! {\mathrel{\mathop{p\!\!\!/}}}
+\cdots
\right]
+ \text{o.t.}
\label{eq:2.12}$$ Here we have set ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R={{\mathrel{\mathop{\partial\!\!\!/}}}}$ and ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L={\mathrel{\mathop{\partial\!\!\!/}}}+{\mathrel{\mathop{\hat W\!\!\!\!\!/}}}$ and ${\mathrel{\mathop{\hat W\!\!\!\!\!/}}}$ represents the off diagonal (charged) part of ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L$ as a matrix in $ud$ space. We have kept terms with four $\hat{W}$’s and one derivative. Also we have introduced the quantity $N= (p^2+m^2)^{-1}$. The dots in [eq. (\[eq:2.12\])]{} refer to further terms of the same type, while “o.t.” refers to other terms which cannot have a contribution to the pattern $\delta Z(W^+W^-)^2D$.

Next, we expand the $ud$ labels using [eq. (\[eq:2.2\])]{} for $m$ and ${\mathrel{\mathop{\hat W\!\!\!\!\!/}}}$, and [eq. (\[eq:2.5\])]{} for $\delta\hat{Z}$. This produces $$\delta\Gamma^- =
\int \frac{d^4x d^4p}{(2\pi)^4} \, {{\rm tr \,}}\!\!\left[
\frac{1}{2} \gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}}\,
N_u {\mathrel{\mathop{p\!\!\!/}}} \,
{\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+ C N_d m_d  
{\mathrel{\mathop{\partial\!\!\!/}}} N_d m_d  
{\mathrel{\mathop{W\!\!\!\!\!/}}}{}^- C^{-1} N_u {\mathrel{\mathop{p\!\!\!/}}} \,
{\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+ C N_d {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^- C^{-1}
N_u  \! {\mathrel{\mathop{p\!\!\!/}}} \,
+ 
\cdots
\right]
+  \text{o.t.}
,$$ where $ N_{u,d}= (p^2+m_{u,d}^2)^{-1} $.

At this point we can already factorize the trace between quantities which act only in generation space, namely, $N_{u,d}$, $m_{u,d}$ and $C$, and all the other quantities, which do not act on that space: $$\delta\Gamma^- =
\int \frac{d^4x d^4p}{(2\pi)^4} \, 
\!\Big(
\frac{1}{2}
{{\rm tr \,}}[N_u C N^2_d m_d^2 C^{-1} N_u C N_d C^{-1} N_u]
{{\rm tr \,}}\!\!\left[
 \gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}}\,
{\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+ \! {\mathrel{\mathop{\partial\!\!\!/}}} \,
{\mathrel{\mathop{W\!\!\!\!\!/}}}{}^-  \! {\mathrel{\mathop{p\!\!\!/}}} \,
{\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+  \! {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^- 
 \! {\mathrel{\mathop{p\!\!\!/}}} \,
\right]
+ 
\cdots
\Big)
+  \text{o.t.}
\label{eq:2.15}$$

It is a general rule that $m_u$ or $m_d$ can only appear raised to even powers and can be eliminated in favor of $N_u$ and $N_d$.[^2] Eventually, all required momentum integrals and traces on $3\times 3$ matrices in generation space can be cast in the form [@GarciaRecio:2009zp] $$I^k_{a,b,c,d}
=
 \int \frac{d^4p}{(2\pi)^4} (p^2)^k
\,{{\rm tr \,}}\!\!
\left[
N_u^a C N_d^b C^{-1} N_u^c C N_d^d
C^{-1}
\right]
,
\label{eq:2.16}$$ where the exponents $k,a,b,c,d$ are non negative integers. On the other hand, only the $CP$-odd contribution is of interest to us. This is the component antisymmetric under the exchange $C\to C^*$, $$\hat{I}^k_{a,b,c,d}
= i\,{{\rm Im \,}}I^k_{a,b,c,d}
.$$ Due to cyclic and hermiticity properties of the trace and matrices involved, these integral satisfy $$\hat{I}^k_{a,b,c,d}
 = 
-\hat{I}^k_{c,b,a,d}
 = 
-\hat{I}^k_{a,d,c,b}
\,.
\label{eq:2.18}$$ Such antisymmetry under exchange of the labels $a$ and $c$, or $b$ and $d$ implies that many terms in [eq. (\[eq:2.15\])]{} do not have a contribution to the $CP$-odd sector and this greatly alleviates the amount of subsequent computation.

Performing an angular average over the momentum and taking the color and Dirac traces in [eq. (\[eq:2.15\])]{} yields then, for $CP$-odd terms ($N_c=3$ is the number of colors) $$\delta\Gamma^- =
N_c\int d^4x \, 
\!\Big(
\frac{1}{3} \hat{I}^3_{1,1,2,2}
\, \epsilon_{\mu\nu\alpha\beta}
\,\delta Z_\mu
W^-_\nu
\partial_\alpha W^+_\beta W^-_\lambda W^+_\lambda
+ 
\cdots
\Big)
+  \text{o.t.}$$

It only remains to apply the derivative on all fields at its right. It can be checked that after this operation is carried out all terms so obtained cancel. So there is no term of the type $(W^+W^-)^2ZD$ in the $CP$-odd, $P$-odd sector of the Standard Model, in agreement with the alternative and more systematic calculation in [@GarciaRecio:2009zp].

Dimension eight operators {#sec:4}
=========================

In this section we show that at eighth order in the derivative expansion there are non vanishing contributions in the $CP$-odd and $P$-odd sector of the Standard Model. Concretely we consider terms of the form $(W^+W^-)^2 Z^3D$, with no gluons nor derivatives of the Higgs field, and apply the technique just described. In this case we select terms with $n=2$ in [eq. (\[eq:2.11\])]{} and seek terms of the type $\delta Z (W^+W^-)^2 Z^2D$. The calculation is analogous to the one shown previously, except that now $Z$ is not set to zero in ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L$, instead we use ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L={\mathrel{\mathop{\partial\!\!\!/}}}+{\mathrel{\mathop{\hat
  W\!\!\!\!\!/}}}+{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}$. Then we keep terms with precisely four $\hat{W}$’s, two $\hat{Z}$’s and one derivative. After restoring the $L,R$ labels and $u,d$ labels, and carrying out the momentum integration, the trace in color, Dirac and generation space, and applying the derivative to the right, one obtains: $$\delta\Gamma^- =
N_c\int d^4x \, 
\!\Big(
2 \hat{I}^4_{1,1,2,4}
\, \epsilon_{\mu\nu\alpha\beta}
\,\delta Z_\mu Z_\lambda Z_{\nu\alpha}
W^+_\beta W^+_\lambda W^-_\sigma W^-_\sigma
+ 
\cdots
\Big)
+  \text{o.t.}
\label{eq:2.20}$$ Here $Z_{\nu\alpha}$ stands for the $\nu$ derivative of $Z_\alpha$. We have eliminated the $\hat{I}^3_{a,b,c,d}$ in favor of $\hat{I}^4_{a,b,c,d}$ and have used identities involving $\delta_{\mu\nu}$ and $\epsilon_{\mu\nu\alpha\beta}$ to bring the expression to a canonical form. The expression contains 62 operators, each one weighted with various integrals $\hat{I}^4_{a,b,c,d}$.

It remains to find out the effective action from which the variation in [eq. (\[eq:2.20\])]{} derives. This serves also as a non trivial check of the computation. The method is just to propose all allowed independent terms of the form $(W^+W^-)^2Z^3D$ with arbitrary coefficients, and take a first order variation with respect to $Z$ to fix those coefficients. The Minkowski space result (see [@GarciaRecio:2009zp] for further details in the conventions) is $$\begin{aligned}
\mathcal{L}^{\rm eff}(x) &=&
\frac{N_c}{15}\frac{v^4}{\phi^4}
\epsilon_{\mu\nu\alpha\beta}\Big[
(12 \hat{I}_1 -16 \hat{I}_2) Z_\mu Z_\lambda^2   W^+_\nu W^+_\sigma  W^-_\alpha W^-_{\beta \sigma}
\nonumber\\&&
+ (4 \hat{I}_1 +23 \hat{I}_2) Z_\mu Z_\lambda^2  W^+_\sigma  W^+_{\nu \alpha} W^-_\beta W^-_\sigma 
+ (6 \hat{I}_1 -23 \hat{I}_2) Z_\mu Z_\lambda^2  W^+_\nu W^+_{\alpha \beta} W^-_\sigma{}^2
\nonumber\\&&
+(32 \hat{I}_1 +4 \hat{I}_2) Z_\mu Z_\lambda  Z_\sigma  W^+_\nu W^+_{\alpha \lambda } W^-_\beta W^-_\sigma 
+(16 \hat{I}_1 -38 \hat{I}_2) Z_\mu Z_\lambda  Z_\sigma  W^+_\lambda  W^+_{\nu \alpha} W^-_\beta W^-_\sigma 
\nonumber\\&&
+(16 \hat{I}_1 +22 \hat{I}_2) Z_\mu Z_\lambda  Z_\sigma  W^+_\lambda  W^+_\sigma  W^-_\nu W^-_{\alpha \beta}
+(-20 \hat{I}_1 -15 \hat{I}_2) Z_\lambda^2  Z_\sigma  W^+_\mu W^+_\sigma  W^-_\nu W^-_{\alpha \beta}
\nonumber\\&&
+10 \hat{I}_1 Z_\mu Z_\lambda  Z_{\nu \alpha} W^+_\sigma{}^2  W^-_\beta W^-_\lambda 
-20 \hat{I}_2 Z_\mu Z_\lambda  Z_{\nu \sigma} W^+_\alpha W^+_\lambda  W^-_\beta W^-_\sigma 
+\text{c.c}
\Big]
\label{eq:4.2}\end{aligned}$$ We have defined $\hat{I}_1= \hat{I}^4_{1,1,2,4}-\hat{I}^4_{1,1,4,2}$ and $\hat{I}_2= \hat{I}^4_{1,2,2,3}-\hat{I}^4_{2,1,3,2}$, and “c.c” refers to complex conjugate; $Z_\mu$ is real, $(W^{\pm}_\mu)^*=W^{\mp}_\mu$ and $\hat{I}_{1,2}$ are imaginary. The (underivated) Higgs field has been restored using that it scales as the mass dimension of $\hat{I}_{1,2}$. Also the derivative includes the field $B_\mu$ when it acts on the $W$’s [@GarciaRecio:2009zp]. Numerically,[^3] $$\hat{I}_{1,2} = \frac{iJ}{(4\pi)^2}\frac{\kappa_{1,2}}{m_s^2 m_c^2}, \quad
\kappa_1 = 0.226,\quad \kappa_2 = 0.456.
\label{eq:2.23}$$

We can see that the values of these coefficients are considerably larger than simple estimates based on the Jarlskog determinant divided by the appropriate power of $v$. At sixth order the enhancement is driven by the small mass of the light quarks and so this can be considered as a kind of chiral enhancement. The possibility of such an effect was first pointed out in [@Smit:2004kh] and confirmed in [@Hernandez:2008db; @GarciaRecio:2009zp].

The momentum integrals $\hat{I}^k_{a,b,c,d}$ are completely explicit but rather complicated homogeneous functions of the fermion masses [@GarciaRecio:2009zp]. At sixth order these integrals are not continuous at $\bar{m}_u,\bar{m}_d,m_s=0$, yet one can take the limit $\bar{m}_u,\bar{m}_d\to 0$ and subsequently $m_s\to 0$ and this approximation gives a value fairly close to the exact one [@GarciaRecio:2009zp]. To discuss the situation at eighth order we will consider the simpler case of $m_b,m_t\to\infty$ which is a quite good approximation for $\kappa_{1,2}$. At eighth order the momentum integrals are more ultraviolet convergent and also more infrared divergent than at sith order. Specifically, $\hat{I}_2$ diverges as $1/m_s^2$ when $\bar{m}_u=\bar{m}_d=0$, with $\kappa_2=1/2$. The other integral, $\hat{I}_1$, is more infrared divergent: in the same limit $\kappa_1$ depends on the ratio $\bar{m}_u/\bar{m}_d$, varying continuously between $-1/6$ for $\bar{m}_d\ll\bar{m}_u$ to $3/2$ for $\bar{m}_u\ll\bar{m}_d$ . In [eq. (\[eq:2.23\])]{} we have used $\bar{m}_u=2.55\,{\rm
  MeV}$ and $\bar{m}_d=5.04\,{\rm MeV}$.

We have also considered $CP$-violating terms containing only $W$’s and gluons. Such terms appear for the first time at eighth order since (at one loop) at least four $W$’s are needed to violate $CP$ and two $G_{\mu\nu}$ are required to make a color singlet. In the $P$-odd sector one can write three independent operators, however we find that they have zero coupling in the Standard Model. On the other hand, in the $P$-even ($C$-violating) sector, there are also three operators of which one has zero coupling while the other two terms result in the following effective Lagrangian (in Minkowski space)[^4] $$\mathcal{L}^{\rm eff}(x)
= -\frac{4}{3} \frac{v^4}{\phi^4}\hat{I}^2_{1,1,2,2} \left(
W^+_\lambda{}^2 W^-_\mu W^-_\nu G^a_{\mu\alpha} G^a_{\nu\alpha}
- \text{c.c.}
\right)
.
\label{eq:4.4}$$ Numerically, $\hat{I}^2_{1,1,2,2}=iJ \kappa_3/(4\pi)^2m_s^2 m_c^2$, with $\kappa_3=3.76$. This coefficient diverges logarithmically as $\bar{m}_u,\bar{m}_d\to 0$. Note that, at the order considered, the dimension four gluon condensate[^5] does not induce a $CP$-violating interaction between the four $W$’s. Such term vanishes identically, as it should be, since no $CP$-odd term can be written using just $W$’s without derivatives or other fields.

Conclusions {#sec:6}
===========

We have shown that, to one-loop and at zero temperature, the leading $P$-violating $CP$-odd operators in the effective action of the Standard Model are of dimension eight. We have computed explictly the couplings for the operators of the form $Z^3(W^+W^-)^2$ plus one covariant derivative, [eq. (\[eq:4.2\])]{}. These operators come from Feynman graphs with one quark-loop, four $W$ legs, three $Z$ legs and possibly one $B\sim Z+\gamma$ leg. In principle, dimension six operators could develop beyond one-loop or at finite temperature due to the breaking of Lorentz invariance. Purely gluonic operators of dimension eight have also been computed, [eq. (\[eq:4.4\])]{}, and they are $C$-odd and $P$-even. Remarkably the corresponding coupling constants we find are not vanishingly small, rather they have a natural scale related to intermediate mass quarks times the Jarlskog invariant. All formulas derived for quarks extend directly to massive Dirac leptons. This implies that even if the neutrino masses are small their contribution to the $CP$-violating couplings needs not be small, due to infrared sensitivity in the momentum integrals on which the couplings depend. As a consequence, such couplings will be strongly dependent on the mass ratios between neutrinos of the different generations.

I am grateful to C. Garcia-Recio for discussions. Research supported by DGI under contract FIS2008-01143, Junta de Andalucía grant FQM-225, the Spanish Consolider-Ingenio 2010 Programme CPAN contract CSD2007-00042, and the European Community-Research Infrastructure Integrating Activity [*Study of Strongly Interacting Matter*]{} (HadronPhysics2, Grant Agreement no. 227431) under the 7th Framework Programme of EU.

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[^1]: Of course, terms with a single $W^+W^-$ pair are allowed beyond one-loop.

[^2]: The label $u$ or $d$ does not change between two consecutive $W$’s. This implies two consecutive $D_L$ and so an even number of $m$’s. That the presence of $m_{u,d}$ can be obviated follows also from eq. (7.1) of [@GarciaRecio:2009zp].

[^3]: $\bar{m}_u,\bar{m}_d,m_c,m_s,m_t,m_b$ denote the quark masses.

[^4]: The gluon field strength tensor has been normalized according to $[D_\mu,D_\nu]=i(\lambda_a/2) G^a_{\mu\nu}$.

[^5]: In the presence of the gluon condensate $G^a_{\mu\nu} G^a_{\alpha\beta} = \frac{1}{12} (\delta_{\mu\alpha} \delta_{\nu\beta} -
      \delta_{\mu\beta} \delta_{\nu\alpha})\langle
      (G^a_{\lambda\sigma})^2\rangle + \text{fluctuations} $.
