Heavy particle non-decoupling in flavor-changing gravitational interactions

The discoveries of the gravitational waves at frequencies $f>10\>{\rm Hz}$ by LIGO and Virgo collaboration via a binary black hole merger and a binary neutron star inspiral have been hailed as a major milestone of gravitational wave astronomy abbott0 ; abbott1 ; abbott2 ; abbott3 . The gravitational wave is now expected to be an exquisite tool not only to study astronomical objects such as black holes and neutron stars, but also to probe viable extension of general relativity as well as what lies Beyond the Standard Model (BSM) of elementary particles. It would be extremely interesting if we could look into the early Universe before the time of last scattering by searching for gravitational waves.

The recent analyses of the 12.5-year pulsar timing array data at frequencies $f\sim 1/{\rm yr}$ by NANOGrav Collaboration nanograv in search for a stochastic gravitational wave background lommen ; tiburzi ; burkespolaor are also of particular importance and are encouraging enough for us to speculate much about BSM: cosmic strings or super-massive black holes as possible sources of the gravitational wave, first order phase transitions in the dark sector, new scenarios of leptogenesis induced by gravitational backgrounds and so on so forth. In search for new avenues of BSM with the help of stochastic gravitational waves, it would sometimes happen that one has to deal with gravitational interactions of heavy unknown particles, in particular, on the quantum level. In such a case we are necessarily forced to pay attention to heavy particle mass effects on physical observables.

Bearing these new directions in our mind, we would like to present in this paper an example in which heavy particles running along internal loops in the gravitational backgrounds induce potentially large and important new type of interactions. Recall that an important issue in particle physics incorporating possible heavy particles has been whether heavy particles have power-suppressed and therefore negligiblly small effects in low-energy processes (decoupling), or their effects may be observable in the form of new induced interactions in the limit of very large mass (non-decoupling). To keep our investigation within a reasonable size, we study specifically the loop-induced flavor-changing process

$\displaystyle d\to s+{\rm graviton}$

(1)

in the standard electroweak interactions, instead of launching into the BSM studies. In our case the top quark is supposed to be the heavy particle as opposed to all the other light quarks. The reason for computing (1) is that the process (1) is analogous to $d\to s+\gamma$ and $d\to s+{\rm gluon}$ (Penguin) processes and that the latter two processes are known to exhibit top quark non-decoupling effects in low-energy decay phenomena. It is quite natural to expect that similar non-decoupling phenomena would take place in (1) and we will argue in the present paper that this expectation is in fact the case. So far as we know, this is the first example of the non-decoupling of the internal heavy top quark in gravitational interactions of light quarks.

One of the sources of the non-decoupling may be searched for in the unphysical scalar field coupling to quarks with the strength proportional to the quark masses. This can be seen most apparently in the Feynman rules in the ’t Hooft-Feynman gauge, which we will use throughout. We are particularly interested in whether or not the process (1) would be enhanced by the large factor $m_{t}^{2}/M_{W}^{2}$, where $m_{t}$ and $M_{W}$ are the top quark and $W$-boson masses, respectively. A quick glance over the Feynman rules, in fact, tells us that apparently this large factor comes into the Feynman amplitude as the coupling of exchanged unphysical scalar field to internal top quark. However we will show in the present paper by explicit calculation that this enhancement factor disappears due to cancellation among the terms of ${\cal O}(m_{t}^{2}/M_{W}^{2})$ in the renormalized transition amplitude ¹¹1 When we say “terms of ${\cal O}(m_{t}^{2}/M_{W}^{2})$”, it is implicitly assumed that logarithmically corrected terms such as $(m_{t}^{2}/M_{W}^{2}){\rm log}(m_{t}^{2}/M_{W}^{2})$ are also included. Likewise, ${\cal O}(1)$ terms are assumed to include ${\rm log}(m_{t}^{2}/M_{W}^{2})$ terms as well. . The breaking of the top quark decoupling thus takes place mildly on the ${\cal O}(1)$ level. We will confirm that the cancellation of the ${\cal O}(m_{t}^{2}/M_{W}^{2})$ terms is consistent with the Ward-Takahashi identity associated with the invariance under the general coordinate transformation.

Appelquist and Carazzone AC once pointed out in the mid 1970’s that virtual effects of heavy unknown particles can be safely neglected in low-energy phenomena, provided that coupling constants are all independent of heavy particle masses. This fact is often referred to as the decoupling theorem, which provides us with an effective strategy to handle low-energy experimental data without worrying much about unknown new physics. In the course of the development of particle physics towards the end of the last century, however, the table has been turned around: we now believe that non-decoupling phenomena are much more interesting than decoupled cases and that we would perhaps be able to have a glimpse of high energy contents of the future theory of elementary particles by investigating non-decoupling phenomena.

In the standard electroweak theory, the Higgs boson and unphysical scalar fields are coupled to quarks with the strength proportional to the quark masses. For large quark masses as for the case of the top quark, the breaking of the decoupling theorem is naturally expected and in fact non-decoupling phenomena are ubiquitous in the Standard Model. They include Higgs boson production in the $pp$-collision via gluon fusion process through a top quark loop georgi ; inamikubotaokada ; spira , the various decay processes of the Higgs boson involving heavy quarks wilczek ; shifman1 ; shifman2 ; inamikubota ; sakai ; braatenleveille , heavy quark effects on the $K_{0}-\overline{K}_{0}$ and $B_{0}-\overline{B}_{0}$ mixings gaillardlee ; inamilim ; buras ; deshpande1 ; deshpande2 ; botella , etc. It is very interesting to see whether a similar explanation for non-decoupling of heavy top quark effects would work as well in gravitational interactions of light quarks. This is actually a strong motive force for us to examine (1).

After submitting the present paper for publication, we learned that the process (1) had once been computed in the ’t Hooft-Feynman gauge and in the unitary gauge by Degrassi et al. degrasse and was investigated by Corianò et al coriano1 ; coriano2 for a different purpose from ours. Their elaborate calculations, however, are not quite suitable for our use since they put external quarks on the mass shell, while we would like to make the non-decoupling phenomena manifest by studying off-shell effective interactions in the large top quark mass limit. Our one-loop calculation is made to this end.

The present paper is organized as follows. First of all we explain in Section 2 the method of putting “weight” on the Fermion fields in the curved background to render the Feynman rules to be discussed in Section 3 a little simpler. The self-energy type $d\to s$ transition in Minkowski space is evaluated in Section 4, the result of which is closely connected with the counter terms eliminating the divergencies associated with (1) as argued in Section 5. In Section 6 we compute all the one-loop Feynman diagrams associated with (1). The renormalization constants prepared in Section 5 are shown in Section 7 to be instrumental to eliminate all the ultraviolet divergences in (1). It is argued in Section 8 that unrenormalized and renormalized quantities associated with (1) satisfy the same Ward-Takahashi identity. The terms in the renormalized transition amplitude behaving asymptotically as ${\cal O}(m_{t}^{2}/M_{W}^{2})$ in the large top quark mass limit are investigated in Section 9 and are shown to vanish via mutual cancellation. The ${\cal O}(1)$ terms for the large top quark mass are also discussed in Section 10, highlighting those that can be expressed by the operator of quark bilinear form coupled to the scalar curvature. Section 11 is devoted to summarizing the present paper. Various definitions of Feynman parameters’ integrations are collected in Appendix A and some combinations thereof are defined in Appendix B.

Techniques of loop calculations involving Dirac Fermions in the curved spacetime, which is our central concern in studying (1), were discussed long time ago by Delbourgo and Salam delbourgosalam ; delbourgosalam2 in connection with anomalies kimura ; eguchifreund . They took a due account of “the weight factors” of Fermions ishamsalamstrathdee , which we now recapitulate while setting up our notations. Hereafter in this Section we will use Greek indices $\mu$, $\nu$ etc. for labeling general coordinates and indices $a$, $b$ etc. for labeling the coordinates in a locally inertial coordinate system. The latter indices are raised and lowered by the Minkowski metric $\eta^{ab}$ and $\eta_{ab}$, respectively.

The Lagrangian of Fermions in the curved spacetime is as usual given by

$\displaystyle{\cal L}_{\rm Dirac}$

$\displaystyle=$

$\displaystyle\sqrt{-g}\left\{\frac{i}{2}\left(\overline{\psi}\;\gamma^{\mu}\>\nabla_{\mu}\psi-\nabla_{\mu}\overline{\psi}\>\gamma^{\mu}\>\psi\right)-\overline{\psi}\>m\>\psi\right\}\>,$

(2)

where our notations are

	$\displaystyle\gamma^{\mu}={e^{\mu}}_{a}\gamma^{a}\>,$		(3)
	$\displaystyle\nabla_{\mu}\psi=\partial_{\mu}\psi-\frac{i}{4}{\omega_{\mu ab}}\sigma^{ab}\>\psi\>,\hskip 28.45274pt\nabla_{\mu}\overline{\psi}=\partial_{\mu}\overline{\psi}+\frac{i}{4}\overline{\psi}\>{\omega_{\mu ab}}\sigma^{ab}\>,$		(4)
	$\displaystyle\sigma^{ab}=\frac{i}{2}(\gamma^{a}\gamma^{b}-\gamma^{b}\gamma^{a})\>,$		(5)

and $g={\rm det}\>(g_{\mu\nu})$. The relation between the spacetime metric $g_{\mu\nu}$ and the vierbein ${e_{\mu}}^{a}$ is given as usual by $g_{\mu\nu}={e_{\mu}}^{a}\>{e_{\nu}}^{b}\>\eta_{ab}$. The spin connection $\omega_{\mu ab}$ is expressed in terms of the vierbein as

$\displaystyle\omega_{\mu ab}$

$\displaystyle=$

$\displaystyle\frac{1}{2}\>{e^{\nu}}_{a}\left(\partial_{\mu}e_{\nu b}-\partial_{\nu}e_{\mu b}\right)-\frac{1}{2}\>{e^{\nu}}_{b}\left(\partial_{\mu}e_{\nu a}-\partial_{\nu}e_{\mu a}\right)-\frac{1}{2}\>{e^{\rho}}_{a}{e^{\sigma}}_{b}\left(\partial_{\rho}e_{\sigma c}-\partial_{\sigma}e_{\rho c}\right){e_{\mu}}^{c}\>.$

Noting the identity of gamma matrices

$\displaystyle\gamma^{\mu}\sigma^{ab}+\sigma^{ab}\gamma^{\mu}={e^{\mu}}_{c}\left(\gamma^{c}\sigma^{ab}+\sigma^{ab}\gamma^{c}\right)=-2{e^{\mu}}_{c}\>\varepsilon^{abcd}\>\gamma_{d}\gamma^{5}\;,$

(7)

we are able to cast the Dirac Lagrangian (2) into

$\displaystyle{\cal L}_{\rm Dirac}$

$\displaystyle=$

$\displaystyle\sqrt{-g}\left\{\frac{i}{2}\left(\overline{\psi}\;\gamma^{\mu}\partial_{\mu}\psi-\partial_{\mu}\overline{\psi}\gamma^{\mu}\psi\right)-\overline{\psi}m\psi\right\}-\frac{1}{4}\sqrt{-g}\left(\overline{\psi}\>{e^{\mu}}_{a}\omega_{\mu bc}\>\varepsilon^{abcd}\gamma_{d}\>\gamma^{5}\psi\right)\>.$

In order to facilitate perturbative calculations in Section 6 we would like to absorb $\sqrt{-g}$ on the right hand side of (2) into dynamical fields as much as possible, putting a weight factor $(-e)^{1/4}$ on the Dirac fields

$\displaystyle\Psi\equiv(-e)^{1/4}\>\psi\>,\hskip 28.45274pt\overline{\Psi}\equiv(-e)^{1/4}\>\overline{\psi}\>,$

(9)

where

$\displaystyle(-e)={\rm det}\>({e_{\mu}}^{a})=\sqrt{-g}\;.$

(10)

In terms of the weighted Dirac fields (9), the Dirac Lagrangian (2) turns out to be

$\displaystyle{\cal L}_{\rm Dirac}$

$\displaystyle=$

$\displaystyle\frac{i}{2}\>{\tilde{e}^{\mu}}_{\>\>\>a}\left(\overline{\Psi}\;\gamma^{a}\partial_{\mu}\Psi-\partial_{\mu}\overline{\Psi}\gamma^{a}\Psi\right)-\sqrt{-e}\>\overline{\Psi}m\Psi-\frac{1}{4}\overline{\Psi}\>{\tilde{e}^{\mu}}_{\>\>\>a}\>\tilde{\omega}_{\mu bc}\>\varepsilon^{abcd}\gamma_{d}\>\gamma^{5}\Psi\>.$

Here we have introduced weighted vierbein

$\displaystyle\tilde{e}^{\mu}_{\>\>\>a}=\sqrt{-e}\>{e^{\mu}}_{a}$

(12)

and $\tilde{\omega}_{\mu ab}$ is defined analogously to (LABEL:eq:spinconnection) by

$\displaystyle\tilde{\omega}_{\mu ab}$

$\displaystyle=$

$\displaystyle\frac{1}{2}\>{\tilde{e}^{\nu}}_{\>\>\>a}\left(\partial_{\mu}\tilde{e}_{\nu b}-\partial_{\nu}\tilde{e}_{\mu b}\right)-\frac{1}{2}\>{\tilde{e}^{\nu}}_{\>\>\>b}\left(\partial_{\mu}\tilde{e}_{\nu a}-\partial_{\nu}\tilde{e}_{\mu a}\right)-\frac{1}{2}\>{\tilde{e}^{\rho}}_{\>\>\>a}{\tilde{e}^{\sigma}}_{\>\>\>b}\left(\partial_{\rho}\tilde{e}_{\sigma c}-\partial_{\sigma}\tilde{e}_{\rho c}\right){\tilde{e}_{\mu}}^{\>\>\>c}\>.$

To arrive at (LABEL:eq:dirac), use has been made of an identity

$\displaystyle{e^{\mu}}_{a}\>\omega_{\mu bc}\>\varepsilon^{abcd}=\frac{1}{\sqrt{-e}}\>\tilde{e}^{\mu}_{\>\>\>a}\>\tilde{\omega}_{\mu bc}\>\varepsilon^{abcd}\>.$

(14)

As we see in (LABEL:eq:dirac), the factor $\sqrt{-e}$ appears only in the mass term. Also note the relation

$\displaystyle\sqrt{-e}=\left\{{\rm det}({\tilde{e}^{\mu}}_{\>\>a})\right\}^{{\color[rgb]{0,0,0}{1/(D-2)}}}\>,$

(15)

where $D$ is the number of spacetime dimensions. We will use the dimensional method for regularization and we do not set $D=4$ .

Putting the weight on the fields as in (9) and (12) changes the choice of dynamical variables and will lead us to a different set of Feynman rules. It has been known, however, that the point transformation of dynamical variables does not alter the structure of S-matrix kamefuchi1 ; chisholm ; borchers and therefore we need not worry much about the choice of variables. In the meanwhile although the weighted field method renders loop calculation a little simpler, it hinders us from comparing our calculation directly with the preceding ones by Degrassi et al. degrasse and by Corianò et al. coriano1 ; coriano2 who did not put weight on the vierbein or Fermion fields, either.

We are going to work with the standard $SU(2)_{L}\times U(1)_{Y}$ electroweak theory embedded in the curved background field with the metric $g_{\mu\nu}$. Deviation from the Minkowski spacetime is described, in terms of the vierbein, as

$\displaystyle{{\tilde{e}}^{\mu}}_{\>\>\>a}={\eta^{\mu}}_{a}+\kappa\>{h^{\mu}}_{a}\>,$

(16)

where $\kappa=\sqrt{8\pi G}$, $G$ being the Newton’s constant. In terms of the metric, fluctuations are expressed as

$\displaystyle\tilde{g}^{\mu\nu}={{\tilde{e}}^{\mu}}_{\>\>\>a}{{\tilde{e}}^{\nu}}_{\>\>\>b}\>\eta^{ab}=\eta^{\mu\nu}+\kappa\>\left(h^{\mu\nu}+h^{\nu\mu}\right)+\kappa^{2}h^{\mu\lambda}{h^{\nu}}_{\lambda}\>,$

(17)

where Greek and Latin indices are no more distinguished and indices of $h^{\mu\nu}$ are raised and lowered by the Minkowski metric. Also from here we assume that $h^{\mu\nu}$ is symmetric i.e., $h^{\mu\nu}=h^{\nu\mu}$. Also note that (15) gives rise to the formula

$\displaystyle\sqrt{-e}=1+\frac{\kappa}{{\color[rgb]{0,0,0}{D-2}}}{\eta_{\mu}}^{a}{h^{\mu}}_{a}+\cdots\cdots\>\>.$

(18)

In the $R_{\xi}$-gauge in the curved background, we add the following gauge-fixing terms to the action

	$\displaystyle{\cal L}_{\rm g.f.}$	$\displaystyle=$	$\displaystyle-\frac{1}{\xi}\sqrt{-g}\>\Big{|}g^{\lambda\rho}\nabla_{\lambda}W_{\rho}{\color[rgb]{0,0,0}{-}}i\>\xi\;M_{W}\>\chi\Big{|}^{2}-\frac{1}{2\xi^{\prime}}\sqrt{-g}\left(g^{\mu\nu}\>\nabla_{\mu}Z_{\nu}+\xi^{\prime}\;M_{Z}\>\chi_{0}\right)^{2}$		(19)
			$\displaystyle-\frac{1}{2\alpha}\>\sqrt{-g}\;\left(g^{\mu\nu}\>\nabla_{\mu}A_{\nu}\right)^{2}\>,$

where $\xi$, $\xi^{\prime}$ and $\alpha$ are gauge parameters. The masses of $W$- and $Z$-bosons are denoted by $M_{W}$ and $M_{Z}$, respectively. The electromagnetic field is denoted by $A_{\mu}$ and $\chi$ and $\chi_{0}$ are charged and neutral unphysical scalar fields, respectively. In our actual calculations we will use the $\xi=1$ ’t Hooft-Feynman gauge, in which the $W$-boson propagator is very much simplified and is most convenient to deal with. The second and third terms in (19) are not relevant to our later calculations but are included here just for completeness. The gravitational field is an external field and therefore the general covariance is not gauge-fixed.

The electroweak Lagrangian in the curved space is given in the power-series expansion in $\kappa$, namely,

	$\displaystyle{\cal L}$	$\displaystyle=$	$\displaystyle{\cal L}_{\rm SM}+\kappa\>h^{\mu\nu}T_{\mu\nu}+{\cal O}(\kappa^{2})\>,$		(20)
	$\displaystyle T_{\mu\nu}$	$\displaystyle=$	$\displaystyle T_{\mu\nu}^{(W)}+T_{\mu\nu}^{(\chi)}+\sum_{q}T_{\mu\nu}^{(\rm q)}+T_{\mu\nu}^{\rm(qW)}+T_{\mu\nu}^{{\rm(q}\chi)}\>,$		(21)

where ${\cal L}_{\rm SM}$ is the standard electroweak Lagrangian in the flat Minkowski space and the second term in the expansion in $\kappa$ in (20) corresponds to the one-graviton emission. Since we will not consider graviton loops, the Einstein-Hilbert gravitational action is not included in (20) . The summation in the third term of (21) is taken over all quark flavors $(q=u,d,s,\cdots)$ and each term in (21) is respectively given by

	$\displaystyle T^{(W)}_{\mu\nu}$	$\displaystyle=$	$\displaystyle{{V}_{\mu\nu}}^{\sigma\tau\lambda\rho}\>\>(\partial_{\sigma}W_{\tau}^{{\dagger}})\>(\partial_{\lambda}W_{\rho})+2\>M_{W}^{2}\>{\eta^{\tau}}_{(\mu}{\eta_{\>\nu)}}^{\rho}\>W_{\tau}^{{\dagger}}W_{\rho}$		(22)
			$\displaystyle+\frac{2}{\xi}\>{\eta^{\sigma}}_{(\mu}\>{\eta_{\nu)}}^{\tau}\>\eta^{\lambda\rho}\>\left(W^{{\dagger}}_{\tau}\>\partial_{\sigma}\partial_{\lambda}W_{\rho}\right)+\frac{2}{\xi}\>{\eta^{\sigma}}_{(\mu}\>{\eta_{\nu)}}^{\rho}\>\eta^{\lambda\tau}\>\left(W_{\rho}\>\partial_{\lambda}\partial_{\sigma}W^{{\dagger}}_{\tau}\right)\>,$
	$\displaystyle T^{(\chi)}_{\mu\nu}$	$\displaystyle=$	$\displaystyle\partial_{\mu}\chi^{{\dagger}}\>\partial_{\nu}\chi+\partial_{\nu}\chi^{{\dagger}}\>\partial_{\mu}\chi-\eta_{\mu\nu}\>{\color[rgb]{0,0,0}{\frac{2}{D-2}}}\>\xi\>M_{W}^{2}\chi^{{\dagger}}\chi\>,$		(23)
	$\displaystyle T_{\mu\nu}^{\rm(q)}$	$\displaystyle=$	$\displaystyle\frac{i}{2}\>\overline{\Psi}_{q}\left(\gamma_{\mu}\overleftrightarrow{\partial_{\nu}}+\gamma_{\nu}\overleftrightarrow{\partial_{\mu}}\right)\>\Psi_{q}-{\color[rgb]{0,0,0}{\frac{1}{D-2}}}\>\eta_{\mu\nu}\overline{\Psi}_{q}\>m_{q}\Psi_{q}\>,$		(24)
	$\displaystyle T^{\rm(qW)}_{\mu\nu}$	$\displaystyle=$	$\displaystyle-\>\frac{g}{2\sqrt{2}\>}\left(\overline{U_{L}}\gamma_{\mu}V_{\rm CKM}D_{L}W^{{\dagger}}_{\nu}+\overline{U_{L}}\gamma_{\nu}V_{\rm CKM}D_{L}W^{{\dagger}}_{\mu}\right)+({\rm h.c.})\>,$		(25)
	$\displaystyle T^{{\rm(q}\chi)}_{\mu\nu}$	$\displaystyle=$	$\displaystyle\eta_{\mu\nu}\>{\color[rgb]{0,0,0}{\frac{1}{D-2}}}\frac{\sqrt{2}}{v}\>\chi^{{\dagger}}\>\left(\overline{U_{R}}\>{\cal M}_{u}\>V_{\rm CKM}\>D_{L}-\overline{U_{L}}\>V_{\rm CKM}{\cal M}_{d}\>D_{R}\right)+({\rm h.c.})\>.$		(26)

The quantity ${{V}_{\mu\nu}}^{\sigma\tau\lambda\rho}$ in (22) is defined by

	$\displaystyle{{V}_{\mu\nu}}^{\sigma\tau\lambda\rho}$	$\displaystyle=$	$\displaystyle-2\>{\eta^{\sigma}}_{(\mu}\>{\eta_{\nu)}}^{\lambda}\>\eta^{\tau\rho}-2\>{\eta^{\tau}}_{(\mu}\>{\eta_{\nu)}}^{\rho}\>\eta^{\sigma\lambda}+2\>{\eta^{\tau}}_{(\mu}\>{\eta_{\nu)}}^{\lambda}\>\eta^{\sigma\rho}+2\>{\eta^{\sigma}}_{(\mu}\>{\eta_{\nu)}}^{\rho}\>\eta^{\tau\lambda}$		(27)
			$\displaystyle+{\color[rgb]{0,0,0}{\frac{2}{D-2}}}\>\eta_{\mu\nu}\>\left(\eta^{\sigma\lambda}\>\eta^{\tau\rho}-\>\eta^{\tau\lambda}\>\eta^{\sigma\rho}+\frac{1}{\xi}\>\eta^{\sigma\tau}\>\eta^{\lambda\rho}\right)\;,$

and the symmetrization with respect to indices in a pair of parentheses in (27) is done in the following manner

$\displaystyle A_{(\sigma}B_{\tau)}\equiv\frac{1}{2}\left(A_{\sigma}B_{\tau}+A_{\tau}B_{\sigma}\right)\>.$

(28)

The symbol of left-right derivative in (24) is defined by

$\displaystyle\overleftrightarrow{\partial_{\mu}}=\frac{1}{2}\left(\overrightarrow{\partial_{\mu}}-\overleftarrow{\partial_{\mu}}\right)\>.$

(29)

The Cabibbo-Kobayashi-Maskawa (CKM) matrix is denoted by $V_{\rm CKM}$ in (25) and (26) and the diagonal mass matrices of up- and down-type quarks are given, respectively, by

$\displaystyle{\cal M}_{u}=\left(\begin{tabular}[]{ccc}$m_{u}$&0&0\\ 0&$m_{c}$&0\\ 0&0&$m_{t}$\end{tabular}\right)\>,\hskip 28.45274pt{\cal M}_{d}=\left(\begin{tabular}[]{ccc}$m_{d}$&0&0\\ 0&$m_{s}$&0\\ 0&0&$m_{b}$\end{tabular}\right)\>.$

(36)

The left ($L$)- and right ($R$)-handed quarks are projected as usual by

$\displaystyle L=\frac{1-\gamma^{5}}{2},\hskip 28.45274ptR=\frac{1+\gamma^{5}}{2},$

(37)

and the projected up- and down-type quarks are expressed as

	$\displaystyle U_{L}=L\left(\begin{tabular}[]{c}$\Psi_{u}$\\ $\Psi_{c}$\\ $\Psi_{t}$\end{tabular}\right)\>,\hskip 34.14322ptD_{L}=L\left(\begin{tabular}[]{c}$\Psi_{d}$\\ $\Psi_{s}$\\ $\Psi_{b}$\end{tabular}\right)\>,$		(44)
	$\displaystyle U_{R}=R\left(\begin{tabular}[]{c}$\Psi_{u}$\\ $\Psi_{c}$\\ $\Psi_{t}$\end{tabular}\right)\>,\hskip 34.14322ptD_{R}=R\left(\begin{tabular}[]{c}$\Psi_{d}$\\ $\Psi_{s}$\\ $\Psi_{b}$\end{tabular}\right)\>,$		(51)

in (25) and (26) . The $SU(2)_{L}$ gauge coupling is denoted by $g$ in (25) and $v$ in (26) is the vacuum expectation value.

Before closing this section, let us add a comment on the relation between the energy-momentum tensor and our $T_{\mu\nu}$. The conventional energy-momentum tensor is defined as the functional derivative of the action under the variation $\delta e^{\mu a}$, while $T_{\mu\nu}$ of (21) is the functional derivative of the action under $\delta{\tilde{e}}^{\mu a}$. The connection between the two types of functional derivatives is given by

$\displaystyle\frac{\delta}{\delta e^{\>\mu a}}=\sqrt{-e}\frac{\delta}{\delta{\tilde{e}}^{\>\mu a}}-\frac{1}{2}\sqrt{-e}e_{\>\mu a}\>e^{\>\lambda b}\frac{\delta}{\delta{\tilde{e}}^{\>\lambda b}}\;,$

(52)

and therefore the conserved energy-momentum tensor in the flat-space limit is a linear combination of $T_{\mu\nu}$ given by

$\displaystyle T_{\mu\nu}-\frac{1}{2}\>\eta_{\mu\nu}\>\eta^{\lambda\rho}\>T_{\lambda\rho}\>.$

(53)

The Ward-Takahashi identity which will be discussed later in Section 8 is associated with (53).

The purpose of the present work is to uncover the non-decoupling nature of the internal heavy top quark in the low-energy process (1), which is induced at the loop levels. There are eight one-loop diagrams of two different types, which will be shown later in Section 6 (i.e., Figure 3 and Figure 4). There the gravitons ($h_{\mu\nu}$) are expressed by double-wavy lines and are attached to vertices in Figure 3 and to internal propagators in Figure 4. The internal quark propagator consists of $j=$ top ($t$), charm ($c$) and up ($u$) quarks and we are interested in the large top quark mass behavior of the amplitudes of Figure 3 and Figure 4.

These one-loop contributions to (1) will be computed in Section 6 and they turn out to be ultraviolet divergent. The divergences should be subtracted by using the corresponding counter term Lagrangian $\widehat{\cal L}_{\rm c.t.}$ in the curved spacetime, which should be diffeomorphism invariant and will be given in Section 5. Diagrammatically the counter term in $\widehat{\cal L}_{\rm c.t.}$ with one external graviton will be denoted by a cross in Figure 2 (b). As it turns out, the flat spacetime limit ${\cal L}_{\rm c.t.}$ of $\widehat{\cal L}_{\rm c.t.}$ should serve as the counter term Lagrangian that is supposed to eliminate the divergences associated with the self-energy type $d\to s$ transition $\Sigma(p)$ in the Minkowski space (without graviton emission). The vertex associated with ${\cal L}_{\rm c.t.}$ will be denoted by a cross in Figure 2 (a).

Now by turning the other way reversed, we may proceed along the following way: namely, after computing $\Sigma(p)$ in this Section 4, we first work out in Section 5.1 the renormalization constants contained in ${\cal L}_{\rm c.t.}$, and then we deduce in Section 5.2 an explicit form of $\widehat{\cal L}_{\rm c.t.}$ by employing the diffeomorphism invariance argument. We will confirm in Section 7 that our $\widehat{\cal L}_{\rm c.t.}$ thus obtained is necessary and sufficient to eliminate all the divergences that appear in the one-loop induced $d$-$s$-graviton vertex to be computed in Section 6. Keeping these procedures in our mind, we would like to start with the calculation of $\Sigma(p)$ in the flat Minkowski space. The $d\to s$ transition takes place at the one-loop level via $W$-and charged unphysical scalar boson ($\chi$) exchanges as depicted in Figure 1. The Feynman rules in the ’t Hooft-Feynman gauge ($\xi=1$) lead us to

$\displaystyle\Sigma(p)$

$\displaystyle=$

$\displaystyle\sum_{j=t,c,u}(V_{\rm CKM})_{js}^{*}(V_{\rm CKM})_{jd}\left\{{\cal S}^{(a)}(p)+{\cal S}^{(b)}(p)\right\}\>,$

(54)

where we have defined the integrations of the following forms

	$\displaystyle{\cal S}^{(a)}(p)$	$\displaystyle\equiv$	$\displaystyle-i\left(\frac{-ig}{\sqrt{2}}\right)^{2}\mu^{4-D}\int\frac{d^{D}q}{(2\pi)^{D}}\>\gamma_{\alpha}L\frac{i}{\gamma\cdot(p-q)-m_{j}}\>\gamma_{\beta}L\frac{-i\>\eta^{\alpha\beta}}{q^{2}-M_{W}^{2}}\>,$		(55)
	$\displaystyle{\cal S}^{(b)}(p)$	$\displaystyle\equiv$	$\displaystyle-i\left(\frac{-ig}{\sqrt{2}}\right)^{2}\frac{\mu^{4-D}}{M_{W}^{2}}\int\frac{d^{D}q}{(2\pi)^{D}}\left(m_{j}R-m_{s}L\right)\frac{i}{\gamma\cdot(p-q)-m_{j}}\left(m_{j}L-m_{d}R\right)$		(56)
			$\displaystyle\hskip 85.35826pt\times\frac{i}{q^{2}-M_{W}^{2}}\>,$

in correspondence to Figure 1 (a) and Figure 1 (b), respectively. Here $\mu$ is the mass scale of the $D$-dimensional regularization method.

The integrations in (55) and (56) are rather standard and we find

	$\displaystyle{\cal S}^{(a)}(p)$	$\displaystyle=$	$\displaystyle\frac{-g^{2}}{(4\pi)^{2}}\left[\frac{1}{D-4}+\frac{1}{2}+\>f_{1}(p^{2})\right]\gamma\cdot p\>L\>,$		(57)
	$\displaystyle{\cal S}^{(b)}(p)$	$\displaystyle=$	$\displaystyle\frac{-g^{2}}{(4\pi)^{2}}\bigg{[}\frac{1}{D-4}\left\{\frac{1}{2M_{W}^{2}}\>\gamma\cdot p\>(m_{j}^{2}\>L+m_{s}m_{d}R)-\frac{m_{j}^{2}}{M_{W}^{2}}(m_{s}L+m_{d}R)\right\}$
			$\displaystyle+\frac{1}{2M_{W}^{2}}\left\{f_{1}(p^{2})\>\gamma\cdot p\>(m_{j}^{2}\>L+m_{s}m_{d}\>R)-f_{2}(p^{2})\>m_{j}^{2}(m_{s}L+m_{d}R)\right\}\bigg{]}\>,$
					(58)

where $f_{1}(p^{2})$ and $f_{2}(p^{2})$ are defined respectively by (168) and (169) in Appendix A . Note that terms independent of $m_{j}$ that are present in ${\cal S}^{(a)}$ and ${\cal S}^{(b)}$ will disappear after the $j$-summation in (54) because of the unitarity of the CKM matrix, $V_{\rm CKM}$, i.e.,

$\displaystyle\sum_{j=t,c,u}(V_{\rm CKM})_{js}^{*}(V_{\rm CKM})_{jd}=0\>.$

(59)

Also remember that both of $f_{1}(p^{2})$ and $f_{2}(p^{2})$ contain $m_{j}^{2}$ in their definitions, (168) and (169). Therefore putting (57) and (58) together, we end up with the formula of the self-energy type $d\to s$ transition

	$\displaystyle\Sigma(p)$	$\displaystyle=$	$\displaystyle\frac{-g^{2}}{(4\pi)^{2}}\sum_{j=t,c,u}(V_{\rm CKM})^{*}_{js}(V_{\rm CKM})_{jd}\>\>\bigg{[}f_{1}(p^{2})\;\gamma\cdot p\>L$
			$\displaystyle+\frac{1}{D-4}\left\{\frac{m_{j}^{2}}{2M_{W}^{2}}\>\gamma\cdot p\>L-\frac{m_{j}^{2}}{M_{W}^{2}}(m_{s}L+m_{d}R)\right\}$
			$\displaystyle+\frac{1}{2M_{W}^{2}}\left\{f_{1}(p^{2})\>\gamma\cdot p\left(m_{j}^{2}\>L+m_{s}m_{d}\>R\right)-f_{2}\left(p^{2})\>m_{j}^{2}(m_{s}L+m_{d}R\right)\right\}\bigg{]}\>.$