QCD sum rules analysis of weak decays of doubly heavy baryons: the $b\to c$ processes

The pole residue of the doubly heavy baryon $\mathcal{B}_{Q_{1}Q_{2}q_{3}}$ can be obtained by calculating the following two-point correlation function

$$\Pi(q)=i\int d^{4}xe^{iq\cdot x}\langle 0|T\left[J_{\mathcal{B}_{Q_{1}Q_{2}q_{3}}}(x)\bar{J}_{\mathcal{B}_{Q_{1}Q_{2}q_{3}}}(0)\right]|0\rangle.$$

(2)

The interpolating currents of doubly heavy baryons are

$$J_{\mathcal{B}_{QQq}}(y)=\epsilon_{abc}(Q_{a}^{T}C\gamma^{\mu}Q_{b})\gamma_{\mu}\gamma_{5}q_{c},\quad J_{\mathcal{B}_{Q_{1}Q_{2}q}}(y)=\epsilon_{abc}\frac{1}{\sqrt{2}}(Q_{1a}^{T}C\gamma^{\mu}Q_{2b}+Q_{2a}^{T}C\gamma^{\mu}Q_{1b})\gamma_{\mu}\gamma_{5}q_{c}.$$

(3)

At the hadron level, by inserting the complete set of baryons in Eq. (2), one can obtain

$$\Pi^{{\rm had}}(q)=\lambda_{+}^{2}\frac{\not\!q+M_{+}}{M_{+}^{2}-q^{2}}+\lambda_{-}^{2}\frac{\not\!q-M_{-}}{M_{-}^{2}-q^{2}}+\cdots,$$

(4)

where we have also considered the contribution from the negative-parity baryon, and $M_{\pm}$ ($\lambda_{\pm}$) are respectively the masses (pole residues) of positive- and negative-parity baryons. The pole residues are introduced as

	$\displaystyle\langle 0|J_{+}(0)|\mathcal{B}_{+}(p,s)\rangle$	$\displaystyle=u(p,s)\lambda_{+},$
	$\displaystyle\langle 0|J_{+}(0)|\mathcal{B}_{-}(p,s)\rangle$	$\displaystyle=(i\gamma_{5})u(p,s)\lambda_{-}.$		(5)

At the QCD level, the correlation functions are calculated using the operator product expansion (OPE) technique. Contributions from up to dimension-5 operators are considered in this work. The result can be formally written as

$$\Pi(q)=\not\!q\Pi_{1}(q^{2})+\Pi_{2}(q^{2}).$$

(6)

$\Pi_{i}$ can be written in terms of dispersion relation for practical purpose

$$\Pi_{i}(q^{2})=\int_{0}^{\infty}ds\frac{\rho_{i}(s)}{s-q^{2}}.$$

(7)

Assuming quark-hadron duality and performing the Borel transformation, one can obtain the following sum rule for $1/2^{+}$ baryon

$$(M_{+}+M_{-})\lambda_{+}^{2}e^{-M_{+}^{2}/T_{+}^{2}}=\int^{s_{+}}ds(M_{-}\rho_{1}(s)+\rho_{2}(s))e^{-s/T_{+}^{2}},$$

(8)

where $T_{+}^{2}$ and $s_{+}$ are respectively the Borel parameter and continuum threshold parameter. From Eq. (8), one can obtain the squared mass for $1/2^{+}$ baryon

$$M_{+}^{2}=\frac{\int^{s_{+}}ds\ (M_{-}\rho_{1}+\rho_{2})\ s\ e^{-s/T_{+}^{2}}}{\int^{s_{+}}ds\ (M_{-}\rho_{1}+\rho_{2})\ e^{-s/T_{+}^{2}}}.$$

(9)

The leading logarithmic (LL) corrections are considered in this work. The Wilson coefficients of OPE should be multiplied by

$$\Bigg{(}\frac{{\rm log}(\mu_{0}/\Lambda_{{\rm QCD}}^{(n_{f})})}{{\rm log}(\mu/\Lambda_{{\rm QCD}}^{(n_{f})})}\Bigg{)}^{2\gamma_{J}-\gamma_{O}},$$

(10)

where $\gamma_{J}$ and $\gamma_{O}$ are anomalous dimensions of the interpolating current and the local operator respectively. $\Lambda_{{\rm QCD}}^{(n_{f})}$ is given by $\Lambda_{{\rm QCD}}^{(3)}=223\ {\rm MeV}$ and $\Lambda_{{\rm QCD}}^{(4)}=170\ {\rm MeV}$ Buras:1998raa ; Zhao:2020mod . The renormalization scale $\mu_{0}\sim 1\ {\rm GeV}$, and $\mu$ is chosen as $m_{c}$ for doubly charmed baryons and $m_{b}$ for doubly bottom and bottom-charmed baryons. The masses and pole residues of doubly heavy baryons are also be considered in Wang:2018lhz .

The following three-point correlation functions are adopted to extract the transition form factors of ${\cal B}_{bQq}\to{\cal B}_{cQq}$

$$\Pi_{\mu}^{V,A}(P_{1},P_{2})=i^{2}\int d^{4}xd^{4}ye^{-iP_{1}\cdot x+iP_{2}\cdot y}\langle 0|T\{J_{\mathcal{B}_{cQq}}(y)(V_{\mu},A_{\mu})(0)\bar{J}_{\mathcal{B}_{bQq}}(x)\}|0\rangle.$$

(11)

At the hadron level, the complete sets of baryon states are inserted to the correlation function to obtain for the vector current correlation function

	$\displaystyle\Pi_{\mu}^{V,{\rm had}}(P_{1},P_{2})$	$\displaystyle=$	$\displaystyle\frac{\lambda_{f}^{+}\lambda_{i}^{+}}{(P_{2}^{2}-M_{2}^{+2})(P_{1}^{2}-M_{1}^{+2})}(\not\!P_{2}+M_{2}^{+}){\cal V}_{\mu}^{++}(\not\!P_{1}+M_{1}^{+})$		(12)
		$\displaystyle+$	$\displaystyle\frac{\lambda_{f}^{+}\lambda_{i}^{-}}{(P_{2}^{2}-M_{2}^{+2})(P_{1}^{2}-M_{1}^{-2})}(\not\!P_{2}+M_{2}^{+}){\cal V}_{\mu}^{+-}(\not\!P_{1}-M_{1}^{-})$
		$\displaystyle+$	$\displaystyle\frac{\lambda_{f}^{-}\lambda_{i}^{+}}{(P_{2}^{2}-M_{2}^{-2})(P_{1}^{2}-M_{1}^{+2})}(\not\!P_{2}-M_{2}^{-}){\cal V}_{\mu}^{-+}(\not\!P_{1}+M_{1}^{+})$
		$\displaystyle+$	$\displaystyle\frac{\lambda_{f}^{-}\lambda_{i}^{-}}{(P_{2}^{2}-M_{2}^{-2})(P_{1}^{2}-M_{1}^{-2})}(\not\!P_{2}-M_{2}^{-}){\cal V}_{\mu}^{--}(\not\!P_{1}-M_{1}^{-})$
		$\displaystyle+$	$\displaystyle...$

where

	$\displaystyle{\cal V}_{\mu}^{ij}$	$\displaystyle\equiv$	$\displaystyle\frac{q_{\mu}}{q^{2}}(M_{1}^{j}-M_{2}^{i})f_{0}^{ij}(q^{2})+\frac{M_{1}^{j}+M_{2}^{i}}{Q_{+}}((P_{1}+P_{2})_{\mu}-(M_{1}^{j2}-M_{2}^{i2})\frac{q_{\mu}}{q^{2}})f_{+}^{ij}(q^{2})$		(13)
			$\displaystyle+(\gamma_{\mu}-\frac{2M_{2}^{j}}{Q_{+}}P_{1\mu}-\frac{2M_{1}^{i}}{Q_{+}}P_{2\mu})f_{\perp}^{ij}(q^{2})$

with $i,j=+,-$. In this step, both of the contributions from positive- and negative-parity baryons are considered. $M_{1(2)}^{+(-)}$ and $\lambda_{i(f)}^{+(-)}$ respectively denote the mass and pole residue of the baryon in the initial (final) state with positive (negative) parity, and $f_{0,+,\perp}^{ij}(q^{2})$ are 12 form factors defined by:

	$\displaystyle\langle{\cal B}_{f}^{+}(p_{2},s_{2})|V_{\mu}|{\cal B}_{i}^{+}(p_{1},s_{1})\rangle$	$\displaystyle=\bar{u}_{{\cal B}_{f}^{+}}(p_{2},s_{2}){\cal V}_{\mu}^{++}u_{{\cal B}_{i}^{+}}(p_{1},s_{1}),$
	$\displaystyle\langle{\cal B}_{f}^{+}(p_{2},s_{2})|V_{\mu}|{\cal B}_{i}^{-}(p_{1},s_{1})\rangle$	$\displaystyle=\bar{u}_{{\cal B}_{f}^{+}}(p_{2},s_{2}){\cal V}_{\mu}^{+-}(i\gamma_{5})u_{{\cal B}_{i}^{-}}(p_{1},s_{1}),$
	$\displaystyle\langle{\cal B}_{f}^{-}(p_{2},s_{2})|V_{\mu}|{\cal B}_{i}^{+}(p_{1},s_{1})\rangle$	$\displaystyle=\bar{u}_{{\cal B}_{f}^{-}}(p_{2},s_{2})(i\gamma_{5}){\cal V}_{\mu}^{-+}u_{{\cal B}_{i}^{+}}(p_{1},s_{1}),$
	$\displaystyle\langle{\cal B}_{f}^{-}(p_{2},s_{2})|V_{\mu}|{\cal B}_{i}^{-}(p_{1},s_{1})\rangle$	$\displaystyle=\bar{u}_{{\cal B}_{f}^{-}}(p_{2},s_{2})(i\gamma_{5}){\cal V}_{\mu}^{--}(i\gamma_{5})u_{{\cal B}_{i}^{-}}(p_{1},s_{1}).$		(14)

At the quark level, the correlation functions in Eq. (11) are calculated using OPE technique. In this work, contributions from the perturbative term (dim-0), quark condensate term (dim-3), mixed quark-gluon condensate term (dim-5), and four-quark condensate term (dim-6) are considered, as can be seen in Fig. 1. The vector current correlation function is further written into the double dispersion relation

$$\Pi_{\mu}^{V}(P_{1},P_{2})=\int^{\infty}ds_{1}\int^{\infty}ds_{2}\frac{\rho_{\mu}^{V}(s_{1},s_{2},q^{2})}{(s_{1}-P_{1}^{2})(s_{2}-P_{2}^{2})},\quad q=P_{1}-P_{2},$$

(15)

where the spectral density functions $\rho_{\mu}^{V}(s_{1},s_{2},q^{2})$ are obtained by taking discontinuities for $s_{1}$ and $s_{2}$. Our method is further illustrated by the calculation of perturbative diagram below.

The spectral density of the perturbative diagram in Fig. 1(a) can be obtained as

	$\displaystyle\rho_{\mu}^{V,{\rm pert}}(P_{1}^{2},P_{2}^{2},q^{2})$	$\displaystyle=$	$\displaystyle\frac{1}{(2\pi i)^{2}}\int\frac{d^{4}p_{b}d^{4}p_{c}d^{4}p_{Q}d^{4}p_{q}}{(2\pi)^{16}}N_{\mu}$		(16)
		$\displaystyle\times$	$\displaystyle(2\pi)^{8}\delta^{4}(p_{b}+p_{Q}+p_{q}-P_{1})\delta^{4}(p_{c}+p_{Q}+p_{q}-P_{2})$
		$\displaystyle\times$	$\displaystyle(-2\pi i)^{4}\delta(p_{b}^{2}-m_{b}^{2})\delta(p_{c}^{2}-m_{c}^{2})\delta(p_{Q}^{2}-m_{Q}^{2})\delta(p_{q}^{2}-m_{q}^{2}).$

with

$$N_{\mu}={\rm Tr}\left[(\not\!p_{Q}+m_{Q})\gamma^{\rho}(\not\!p_{c}-m_{c})(V_{\mu},A_{\mu})(\not\!p_{b}+m_{b})\gamma^{\nu}\right]\frac{6}{\sqrt{2}}(\gamma_{\rho}\gamma_{5}(\not\!p_{q}+m_{q})\gamma_{5}\gamma_{\nu}).$$

(17)

The integral in Eq. (16) can be written as a two-body phase space integral followed by a “triangle” phase space integral Shi:2019hbf .

Equating Eq. (15) with Eq. (12), assuming quark-hadron duality, and performing the Borel transformation, one can obtain

$${\cal B}\Pi_{\mu}^{V,{\rm pole}}(T_{1}^{2},T_{2}^{2})=\int^{s_{1}^{0}}ds_{1}\int^{s_{2}^{0}}ds_{2}\ \rho_{\mu}^{V}(s_{1},s_{2},q^{2})e^{-s_{1}/T_{1}^{2}}e^{-s_{2}/T_{2}^{2}},$$

(18)

where the left-hand side denotes the Borel transformed four pole terms in Eq. (12) and $s_{1,2}^{0}$ are the continuum threshold parameters. Equating the coefficients of the same Dirac structures on both sides of Eq. (18), one can arrive at 12 equations, from which, one can further extract the form factors. More details can be found in Shi:2019hbf ; Zhao:2020mod .

In addition, similar as the situation of the two-point correlation function, the LL corrections are also considered.

Our predictioins of pole residues and masses are respectively collected in Table 2 and Table 3, and the pole residues as functions of the Borel parameters are plotted in Fig 2. Both of the results without and with the LL corrections are shown. In Table 3, our predictions for the masses are also compared with those from Lattice QCD Brown:2014ena .

We take the process of $\Xi_{bb}\to\Xi_{bc}$ as an example to illustrate the selection of Borel windows. In Fig. 3, the transition form factors of $\Xi_{bb}\to\Xi_{bc}$ are plotted as functions of the Borel parameters $T_{1,2}^{2}$. Relatively flat regions are selected as the working Borel windows.

We also consider the uncertainties of the form factors caused by the Borel parameters $T_{1,2}^{2}$ and the continuum threshold parameter $s_{1,2}^{0}$, as can be seen in Table 4. To access the $q^{2}$ dependence, we calculate the form factors at small $q^{2}$, and then fit the data with the following formula

$$f(q^{2})=\frac{1}{1-q^{2}/(m_{{\rm pole}})^{2}}(a+bz(q^{2}))$$

(19)

with

$\displaystyle z(q^{2})=\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}}}.$

(20)

Here $t_{0}=q_{{\rm max}}^{2}=(m_{\mathcal{B}_{i}}-m_{\mathcal{B}_{f}})^{2}$ and $t_{+}=(m_{{\rm pole}})^{2}$ are chosen to be equal or below the location of any remaining singularity after factoring out the leading pole contribution Detmold:2015aaa . The nonlinear least-$\chi^{2}$ (lsq) method is used in our analysis Peter:2020 . The fitted results are shown in Table 4 and Fig. 4. The contributions of each local operator in the OPE are evaluated as

	$\displaystyle f_{+,0}^{\Xi_{bb}\to\Xi_{bc}}(0)$	$\displaystyle=$	$\displaystyle 0.332(73.309\%)_{dim0}+0.101(22.300\%)_{dim3}+0.014(3.063\%)_{dim5}-0.006(1.329\%)_{dim6},$
	$\displaystyle f_{\perp}^{\Xi_{bb}\to\Xi_{bc}}(0)$	$\displaystyle=$	$\displaystyle 0.630(75.700\%)_{dim0}+0.194(23.326\%)_{dim3}-0.003(0.310\%)_{dim5}-0.006(0.663\%)_{dim6}.$		(21)

It can be seen that the OPE has excellent convergence and the perturbative term dominates.

In addition, we have investigated the heavy quark limit of our full QCD results for the form factors, and close results are obtained.