$\bm{\Lambda_{b}\to\Lambda_{c}^{*}(2595,2625)\ell^{-}\bar{\nu}}$ form factors from lattice QCD

Semileptonic $b\to c\ell^{-}\bar{\nu}$ decays are used to determine the CKM matrix element $V_{cb}$ and to search for deviations from lepton flavor universality Gambino:2020jvv ; Bifani:2018zmi ; Bernlochner:2021vlv . They also provide an important testing ground for heavy-quark effective theory Neubert:1993mb . In recent years, the operation of the Large Hadron Collider has provided new opportunities for measurements involving $b$ baryons. The simplest baryonic $b\to c\ell^{-}\bar{\nu}$ process is $\Lambda_{b}\to\Lambda_{c}\ell^{-}\bar{\nu}$, in which both the initial and final hadrons are the ground states with $J^{P}=\frac{1}{2}^{+}$. This mode has been used in combination with $\Lambda_{b}\to p\ell^{-}\bar{\nu}$ to determine $|V_{ub}/V_{cb}|$ Detmold:2015aaa ; Aaij:2015bfa and offers the prospect of measuring the $\tau$-versus-$\mu$ ratio $R(\Lambda_{c})$ and related observables Bifani:2018zmi ; Bernlochner:2021vlv . The baryonic decays can provide complementary information on physics beyond the Standard Model when compared with mesonic decays Dutta:2015ueb ; Li:2016pdv ; Albrecht:2017odf ; Datta:2017aue ; Alioli:2017ces ; Ray:2018hrx ; Boer:2019zmp ; Penalva:2019rgt ; Ferrillo:2019owd ; Mu:2019bin ; Hu:2020axt . The $\Lambda_{b}\to\Lambda_{c}$ form factors have been computed using lattice QCD Bowler:1997ej ; Gottlieb:2003yb ; Detmold:2015aaa ; Datta:2017aue , and the lattice results predict a shape for the $\Lambda_{b}\to\Lambda_{c}\mu^{-}\bar{\nu}$ differential decay rate in the Standard Model that is consistent with the LHCb measurement Aaij:2017svr . Heavy-quark symmetry provides strong constraints on $\Lambda_{b}\to\Lambda_{c}\mu^{-}\bar{\nu}$, in which the light hadronic degrees of freedom have total angular momentum zero. In the heavy-quark-effective-theory (HQET) description of this decay, no subleading order-$\Lambda_{\rm QCD}/m_{c,b}$ Isgur-Wise functions occur and only two sub-subleading Isgur-Wise functions enter at order $\Lambda_{\rm QCD}^{2}/m_{c}^{2}$; the available lattice and LHCb results are well described by a fit of this order Bernlochner:2018kxh ; Bernlochner:2018bfn .

In addition to $\Lambda_{b}\to\Lambda_{c}\mu^{-}\bar{\nu}$, the LHCb detector has also collected (and will continue to collect) large numbers of $\Lambda_{b}\to\Lambda_{c}^{*}(2595)\mu^{-}\bar{\nu}$ and $\Lambda_{b}\to\Lambda_{c}^{*}(2625)\mu^{-}\bar{\nu}$ samples Aaij:2017svr ; the relative branching fractions of these modes have been measured by the CDF Collaboration to be $\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{*}(2595)\mu^{-}\bar{\nu})/\mathcal{B}(\Lambda_{b}\to\Lambda_{c}\mu^{-}\bar{\nu})=0.126\pm 0.033^{+0.047}_{-0.038}$ and $\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{*}(2625)\mu^{-}\bar{\nu})/\mathcal{B}(\Lambda_{b}\to\Lambda_{c}\mu^{-}\bar{\nu})=0.210\pm 0.042^{+0.071}_{-0.050}$ Aaltonen:2008eu . The $\Lambda_{c}^{*}(2595)$ and $\Lambda_{c}^{*}(2625)$ are the lightest charm baryons with $J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{3}{2}^{-}$, respectively, and are very narrow resonances decaying to $\Lambda_{c}\pi\pi$ Zyla:2020zbs . It has been projected that $R(\Lambda_{c}^{*})=\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{*}\tau^{-}\bar{\nu})/\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{*}\mu^{-}\bar{\nu})$ can be measured using LHCb data with approximately 17 percent uncertainty at the end of LHC Run 3, and as low as 5 percent uncertainty at the end of Run 6 Bernlochner:2021vlv . To predict $R(\Lambda_{c}^{*})$ in the Standard Model and beyond, the $\Lambda_{b}\to\Lambda_{c}^{*}$ form factors are needed. A calculation of these form factors may also improve the control of the backgrounds in a measurement of $R(\Lambda_{c})$. Another potential impact will be on zero-recoil sum rules Mannel:2015osa ; Boer:2018vpx and on global analyses of $b\to c\ell^{-}\bar{\nu}$ form factors using dispersion relations Cohen:2019zev . The authors of Ref. Cohen:2019zev wrote “Given the large fractional saturation of the unitarity bounds by $\Lambda_{b}\to\Lambda_{c}$, the inclusion of $\Lambda_{b}\to\Lambda_{c}^{*}$ could be particularly fruitful once such data is available.” Finally, we note that there is significant interest in the structure and strong decays of the $\Lambda_{c}^{*}(2595)$ and $\Lambda_{c}^{*}(2625)$, in part due to the closeness of the $\Sigma_{c}^{(*)}\pi$ thresholds Blechman:2003mq ; Guo:2016wpy ; Arifi:2018yhr ; Nieves:2019kdh ; Nieves:2019nol .

In the limit of heavy charm quarks, the light degrees of freedom in $\Lambda_{c}^{*}(2595)$ and $\Lambda_{c}^{*}(2625)$ have total angular momentum 1 and these two baryons become degenerate. Note that there is no heavy-quark spin-symmetry relation between the $\Lambda_{c}^{*}$ and the $\Lambda_{c}$ due to the different quantum numbers of the light degrees of freedom. This difference also means that the normalization of the leading Isgur-Wise function for $\Lambda_{b}\to\Lambda_{c}^{*}$ remains unconstrained in the heavy-quark limit, and the matrix elements vanish at zero recoil Roberts:1992xm ; Leibovich:1997az . The HQET relations for the $\Lambda_{b}\to\Lambda_{c}^{*}(2595)$ and $\Lambda_{b}\to\Lambda_{c}^{*}(2625)$ vector and axial vector form factors including the subleading order-$\Lambda_{\rm QCD}/m_{c,b}$ contributions were derived in Refs. Roberts:1992xm , Leibovich:1997az , and Boer:2018vpx ; the authors of the latter reference specifically studied the possibility of using HQET fits to LHCb data for the muonic decay $\Lambda_{b}\to\Lambda_{c}^{*}\mu^{-}\bar{\nu}$ to make Standard-Model predictions for $R(\Lambda_{c}^{*})$. It is still an open question how well HQET at this order can describe these transitions.

Quark-model studies of the $\Lambda_{b}\to\Lambda_{c}^{*}(2595)$ and $\Lambda_{b}\to\Lambda_{c}^{*}(2625)$ form factors can be found in Refs. Pervin:2005ve ; Gutsche:2017wag ; Gutsche:2018nks ; Becirevic:2020nmb . In the following, we present the first lattice-QCD determination of these form factors. Our calculation follows that of the $\Lambda_{b}\to\Lambda^{*}(1520)$ form factors in Ref. Meinel:2020owd and uses the same ensembles of gauge-field configurations. We observe that the $\Lambda_{c}^{*}(2595)$ and $\Lambda_{c}^{*}(2625)$ energy levels for our simulation parameters are below all potential strong-decay thresholds, although they come quite close at the lowest pion mass. As in Ref. Meinel:2020owd , we work in the rest frame of the final-state baryon to avoid mixing between $J=\frac{3}{2}$ and $J=\frac{1}{2}$ and between negative and positive parity. This again limits the kinematic coverage to the region near $q^{2}_{\rm max}$.

Our definitions of the form factors are given in Sec. II. Following a brief summary of the lattice parameters in Sec. III, we discuss the baryon interpolating fields, two-point functions, and the results for the masses in Sec. IV. The extraction of the form factors from three-point functions is described in Sec. V, and their extrapolation to the physical pion mass and continuum limit is discussed in Sec. VI. We test the compatibility with zero-recoil sum rules in Sec. VII and present the Standard-Model predictions for $\Lambda_{b}\to\Lambda_{c}^{*}(2595)\ell^{-}\bar{\nu}$ and $\Lambda_{b}\to\Lambda_{c}^{*}(2625)\ell^{-}\bar{\nu}$ in Sec. VIII. Our conclusions are given in Sec. IX, and Appendix A contains relations to other form factor definitions used in the literature.

In the following, we denote the $\Lambda_{c}^{*}(2595)$ and $\Lambda_{c}^{*}(2625)$ as $\Lambda_{c,1/2}^{*}$ and $\Lambda_{c,3/2}^{*}$, respectively. The masses and total decay widths determined by experiments are $m_{\Lambda_{c,1/2}^{*}}=2592.25(28)\>{\rm MeV}$, $m_{\Lambda_{c,3/2}^{*}}=2628.11(19)\>{\rm MeV}$, $\Gamma_{\Lambda_{c,1/2}^{*}}=2.6(0.6)\>{\rm MeV}$, $\Gamma_{\Lambda_{c,3/2}^{*}}<0.97\>{\rm MeV}$ (${\rm CL}=90\%$) Zyla:2020zbs . We neglect the decay widths throughout this work. In our lattice calculations at heavier-than-physical pion masses, the strong decays are in fact kinematically forbidden, except perhaps at the lightest pion mass; the hadron masses we find on the lattice are given in Sec. IV.

We normalize the baryon states as

	$\displaystyle\langle\Lambda_{b}(\mathbf{k},r)|\Lambda_{b}(\mathbf{p},s)\rangle$	$\displaystyle=$	$\displaystyle\delta_{rs}2E_{\Lambda_{b}}(2\pi)^{3}\delta^{3}(\mathbf{k}-\mathbf{p}),$		(1)
	$\displaystyle\langle\Lambda_{c,1/2}^{*}(\mathbf{k}^{\prime},r^{\prime})|\Lambda_{c,1/2}^{*}(\mathbf{p}^{\prime},s^{\prime})\rangle$	$\displaystyle=$	$\displaystyle\delta_{r^{\prime}s^{\prime}}2E_{\Lambda_{c,1/2}^{*}}(2\pi)^{3}\delta^{3}(\mathbf{k}^{\prime}-\mathbf{p}^{\prime}),$		(2)
	$\displaystyle\langle\Lambda_{c,3/2}^{*}(\mathbf{k}^{\prime},r^{\prime})|\Lambda_{c,3/2}^{*}(\mathbf{p}^{\prime},s^{\prime})\rangle$	$\displaystyle=$	$\displaystyle\delta_{r^{\prime}s^{\prime}}2E_{\Lambda_{c,3/2}^{*}}(2\pi)^{3}\delta^{3}(\mathbf{k}^{\prime}-\mathbf{p}^{\prime}),$		(3)

and work with Dirac and Rarita-Schwinger spinors satisfying

	$\displaystyle\sum_{s}u(m_{\Lambda_{b}},\mathbf{p},s)\bar{u}(m_{\Lambda_{b}},\mathbf{p},s)$	$\displaystyle=$	$\displaystyle m_{\Lambda_{b}}+\not{p},$		(4)
	$\displaystyle\sum_{s^{\prime}}u(m_{\Lambda_{c,1/2}^{*}},\mathbf{p}^{\prime},s^{\prime})\bar{u}(m_{\Lambda_{c,1/2}^{*}},\mathbf{p}^{\prime},s^{\prime})$	$\displaystyle=$	$\displaystyle m_{\Lambda_{c,1/2}^{*}}+\not{p}^{\prime},$		(5)
	$\displaystyle\sum_{s^{\prime}}u_{\mu}(m_{\Lambda_{c,3/2}^{*}},\mathbf{p}^{\prime},s^{\prime})\bar{u}_{\nu}(m_{\Lambda_{c,3/2}^{*}},\mathbf{p}^{\prime},s^{\prime})$	$\displaystyle=$	$\displaystyle-(m_{\Lambda_{c,3/2}^{*}}+\not{p}^{\prime})\left(g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2}{3m_{\Lambda_{c,3/2}^{*}}^{2}}p^{\prime}_{\mu}p^{\prime}_{\nu}-\frac{1}{3m_{\Lambda_{c,3/2}^{*}}}(\gamma_{\mu}p^{\prime}_{\nu}-\gamma_{\nu}p^{\prime}_{\mu})\right).$

In the equations throughout this paper, Minkowski-space gamma matrices and the metric $(g_{\mu\nu})={\rm diag}(1,-1,-1,-1)$ are used, except where indicated otherwise. We introduce the notation

	$\displaystyle\langle\Lambda_{c,1/2}^{*}(\mathbf{p^{\prime}},s^{\prime})|\,\bar{c}\Gamma b\,|\Lambda_{b}(\mathbf{p},s)\rangle$	$\displaystyle=$	$\displaystyle\bar{u}(m_{\Lambda_{c,1/2}^{*}},\mathbf{p^{\prime}},s^{\prime})\>\gamma_{5}\>\mathscr{G}^{(\frac{1}{2}^{-})}[\Gamma]\>u(m_{\Lambda_{b}},\mathbf{p},s),$		(7)
	$\displaystyle\langle\Lambda_{c,3/2}^{*}(\mathbf{p^{\prime}},s^{\prime})|\,\bar{c}\Gamma b\,|\Lambda_{b}(\mathbf{p},s)\rangle$	$\displaystyle=$	$\displaystyle\bar{u}_{\lambda}(m_{\Lambda_{c,3/2}^{*}},\mathbf{p^{\prime}},s^{\prime})\>\mathscr{G}^{\lambda(\frac{3}{2}^{-})}[\Gamma]\>u(m_{\Lambda_{b}},\mathbf{p},s)$		(8)

and

$$s_{\pm}=(m_{\Lambda_{b}}\pm m_{\Lambda_{c}^{*}})^{2}-q^{2},$$

(9)

where $q=p-p^{\prime}$. We use a helicity basis for all form factors. For the $J^{P}=\frac{1}{2}^{-}$ final state, our definition follows the one introduced previously for $J^{P}=\frac{1}{2}^{+}$ final states Feldmann:2011xf except for the changes resulting from the opposite parity [note the $\gamma_{5}$ in Eq. (7)]:

	$\displaystyle\mathscr{G}^{(\frac{1}{2}^{-})}[\gamma^{\mu}]$	$\displaystyle=$	$\displaystyle f_{0}^{(\frac{1}{2}^{-})}\>(m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}})\frac{q^{\mu}}{q^{2}}$		(10)
			$\displaystyle+f_{+}^{(\frac{1}{2}^{-})}\frac{m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}}}{s_{-}}\left(p^{\mu}+p^{\prime\mu}-(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2})\frac{q^{\mu}}{q^{2}}\right)$
			$\displaystyle+f_{\perp}^{(\frac{1}{2}^{-})}\left(\gamma^{\mu}+\frac{2m_{\Lambda_{c,1/2}^{*}}}{s_{-}}p^{\mu}-\frac{2\,m_{\Lambda_{b}}}{s_{-}}p^{\prime\mu}\right),$

	$\displaystyle\mathscr{G}^{(\frac{1}{2}^{-})}[\gamma^{\mu}\gamma_{5}]$	$\displaystyle=$	$\displaystyle-g_{0}^{(\frac{1}{2}^{-})}\gamma_{5}\>(m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}})\frac{q^{\mu}}{q^{2}}$		(11)
			$\displaystyle-g_{+}^{(\frac{1}{2}^{-})}\gamma_{5}\frac{m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}}}{s_{+}}\left(p^{\mu}+p^{\prime\mu}-(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2})\frac{q^{\mu}}{q^{2}}\right)$
			$\displaystyle-g_{\perp}^{(\frac{1}{2}^{-})}\gamma_{5}\left(\gamma^{\mu}-\frac{2m_{\Lambda_{c,1/2}^{*}}}{s_{+}}p^{\mu}-\frac{2\,m_{\Lambda_{b}}}{s_{+}}p^{\prime\mu}\right),$

	$\displaystyle\mathscr{G}^{(\frac{1}{2}^{-})}[i\sigma^{\mu\nu}q_{\nu}]$	$\displaystyle=$	$\displaystyle-h_{+}^{(\frac{1}{2}^{-})}\,\frac{q^{2}}{s_{-}}\left(p^{\mu}+p^{\prime\mu}-(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2})\frac{q^{\mu}}{q^{2}}\right)$		(12)
			$\displaystyle-h_{\perp}^{(\frac{1}{2}^{-})}\,(m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}})\left(\gamma^{\mu}+\frac{2\,m_{\Lambda_{c,1/2}^{*}}}{s_{-}}\,p^{\mu}-\frac{2\,m_{\Lambda_{b}}}{s_{-}}\,p^{\prime\mu}\right),$

	$\displaystyle\mathscr{G}^{(\frac{1}{2}^{-})}[i\sigma^{\mu\nu}\gamma_{5}q_{\nu}]$	$\displaystyle=$	$\displaystyle-\widetilde{h}_{+}^{(\frac{1}{2}^{-})}\gamma_{5}\frac{q^{2}}{s_{+}}\left(p^{\mu}+p^{\prime\mu}-(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2})\frac{q^{\mu}}{q^{2}}\right)$		(13)
			$\displaystyle-\widetilde{h}_{\perp}^{(\frac{1}{2}^{-})}\gamma_{5}\,(m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}})\left(\gamma^{\mu}-\frac{2m_{\Lambda_{c,1/2}^{*}}}{s_{+}}\,p^{\mu}-\frac{2m_{\Lambda_{b}}}{s_{+}}\,p^{\prime\mu}\right).$

For the $J^{P}=\frac{3}{2}^{-}$ final state, we use the definition introduced by us in Ref. Meinel:2020owd , which reads

	$\displaystyle\mathscr{G}^{\lambda(\frac{3}{2}^{-})}[\gamma^{\mu}]$	$\displaystyle=$	$\displaystyle f_{0}^{(\frac{3}{2}^{-})}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{+}}\,\frac{(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})\,p^{\lambda}q^{\mu}}{q^{2}}$		(14)
			$\displaystyle+f_{+}^{(\frac{3}{2}^{-})}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{-}}\,\frac{(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})\,p^{\lambda}(q^{2}(p^{\mu}+p^{\prime\mu})-(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2})q^{\mu})}{q^{2}\,s_{+}}$
			$\displaystyle+f_{\perp}^{(\frac{3}{2}^{-})}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{-}}\left(p^{\lambda}\gamma^{\mu}-\frac{2\,p^{\lambda}(m_{\Lambda_{b}}p^{\prime\mu}+m_{\Lambda_{c,3/2}^{*}}p^{\mu})}{s_{+}}\right)$
			$\displaystyle+f_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{-}}\left(p^{\lambda}\gamma^{\mu}-\frac{2\,p^{\lambda}p^{\prime\mu}}{m_{\Lambda_{c,3/2}^{*}}}+\frac{2\,p^{\lambda}(m_{\Lambda_{b}}p^{\prime\mu}+m_{\Lambda_{c,3/2}^{*}}p^{\mu})}{s_{+}}+\frac{s_{-}\,g^{\lambda\mu}}{m_{\Lambda_{c,3/2}^{*}}}\right),$

	$\displaystyle\mathscr{G}^{\lambda(\frac{3}{2}^{-})}[\gamma^{\mu}\gamma_{5}]$	$\displaystyle=$	$\displaystyle-g_{0}^{(\frac{3}{2}^{-})}\gamma_{5}\,\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{-}}\frac{(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})\,p^{\lambda}q^{\mu}}{q^{2}}$		(15)
			$\displaystyle-g_{+}^{(\frac{3}{2}^{-})}\gamma_{5}\,\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{+}}\frac{(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})\,p^{\lambda}(q^{2}(p^{\mu}+p^{\prime\mu})-(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2})q^{\mu})}{q^{2}\,s_{-}}$
			$\displaystyle-g_{\perp}^{(\frac{3}{2}^{-})}\gamma_{5}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{+}}\left(p^{\lambda}\gamma^{\mu}-\frac{2\,p^{\lambda}(m_{\Lambda_{b}}p^{\prime\mu}-m_{\Lambda_{c,3/2}^{*}}p^{\mu})}{s_{-}}\right)$
			$\displaystyle-g_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\gamma_{5}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{+}}\left(p^{\lambda}\gamma^{\mu}+\frac{2\,p^{\lambda}p^{\prime\mu}}{m_{\Lambda_{c,3/2}^{*}}}+\frac{2\,p^{\lambda}(m_{\Lambda_{b}}p^{\prime\mu}-m_{\Lambda_{c,3/2}^{*}}p^{\mu})}{s_{-}}-\frac{s_{+}\,g^{\lambda\mu}}{m_{\Lambda_{c,3/2}^{*}}}\right),$

	$\displaystyle\mathscr{G}^{\lambda(\frac{3}{2}^{-})}[i\sigma^{\mu\nu}q_{\nu}]$	$\displaystyle=$	$\displaystyle-h_{+}^{(\frac{3}{2}^{-})}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{-}}\,\frac{p^{\lambda}(q^{2}(p^{\mu}+p^{\prime\mu})-(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2})q^{\mu})}{s_{+}}$
			$\displaystyle-h_{\perp}^{(\frac{3}{2}^{-})}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{-}}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})\left(p^{\lambda}\gamma^{\mu}-\frac{2\,p^{\lambda}(m_{\Lambda_{b}}p^{\prime\mu}+m_{\Lambda_{c,3/2}^{*}}p^{\mu})}{s_{+}}\right)$
			$\displaystyle-h_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{-}}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})\left(p^{\lambda}\gamma^{\mu}-\frac{2\,p^{\lambda}p^{\prime\mu}}{m_{\Lambda_{c,3/2}^{*}}}+\frac{2\,p^{\lambda}(m_{\Lambda_{b}}p^{\prime\mu}+m_{\Lambda_{c,3/2}^{*}}p^{\mu})}{s_{+}}+\frac{s_{-}\,g^{\lambda\mu}}{m_{\Lambda_{c,3/2}^{*}}}\right),$

	$\displaystyle\mathscr{G}^{\lambda(\frac{3}{2}^{-})}[i\sigma^{\mu\nu}q_{\nu}\gamma_{5}]$	$\displaystyle=$	$\displaystyle-\widetilde{h}_{+}^{(\frac{3}{2}^{-})}\gamma_{5}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{+}}\,\frac{p^{\lambda}(q^{2}(p^{\mu}+p^{\prime\mu})-(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2})q^{\mu})}{s_{-}}$
			$\displaystyle-\widetilde{h}_{\perp}^{(\frac{3}{2}^{-})}\gamma_{5}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{+}}(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})\left(p^{\lambda}\gamma^{\mu}-\frac{2\,p^{\lambda}(m_{\Lambda_{b}}p^{\prime\mu}-m_{\Lambda_{c,3/2}^{*}}p^{\mu})}{s_{-}}\right)$
			$\displaystyle-\widetilde{h}_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\gamma_{5}\frac{m_{\Lambda_{c,3/2}^{*}}}{s_{+}}(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})\left(p^{\lambda}\gamma^{\mu}+\frac{2\,p^{\lambda}p^{\prime\mu}}{m_{\Lambda_{c,3/2}^{*}}}+\frac{2\,p^{\lambda}(m_{\Lambda_{b}}p^{\prime\mu}-m_{\Lambda_{c,3/2}^{*}}p^{\mu})}{s_{-}}-\frac{s_{+}\,g^{\lambda\mu}}{m_{\Lambda_{c,3/2}^{*}}}\right).$

Only the vector and axial-vector form factors are needed to describe $\Lambda_{b}\to\Lambda_{c}^{*}\ell^{-}\bar{\nu}$ decays in the Standard Model, but we also compute the tensor form factors. Above, $\sigma^{\mu\nu}=\frac{i}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$. Note that the overall sign of the form factors for each decay mode depends on the phase conventions of the states. This means that also the relative overall sign between the two different final states is left undetermined. Relations between our form-factor definitions and alternative definitions used in the literature are given in Appendix A.

The lattice actions and parameters used in this work are the same as in our calculation of $\Lambda_{b}\to\Lambda^{*}(1520)$ form factors Meinel:2020owd , except that here the valence strange quark is replaced by a valence charm quark. For the latter, we employ the same form of action and analogous tuning conditions as for the bottom quark Aoki:2012xaa , i.e., an anisotropic clover action with bare parameters $am_{Q}^{(c)}$, $\nu^{(c)}$, $c_{E,B}^{(c)}$ tuned to obtain the correct $D_{s}$ meson kinetic mass, rest mass, and hyperfine splitting (our notation for the bare parameters follows Ref. Brown:2014ena , while Ref. Aoki:2012xaa uses $m_{0}=m_{Q}$, $\zeta=\nu$, $c_{P}=c_{E}=c_{B}$). The values of these parameters are given in Table 1. The gauge-field ensembles with $2+1$ flavors of domain-wall fermions were generated by the RBC and UKQCD Collaborations Aoki:2010dy ; Blum:2014tka . For the up and down valence quarks, we reuse the domain-wall propagators computed for Ref. Meinel:2020owd . Our computation utilizes all-mode averaging Blum:2012uh ; Shintani:2014vja , in which unbiased estimates with small statistical uncertainties are obtained at reduced cost by combining “exact” and “sloppy” samples.

We now move to the discussion of the baryon interpolating fields, two-point functions, and results for the masses. For the $\Lambda_{b}$, everything is identical to Ref. Meinel:2020owd . The $\Lambda_{c}^{*}(2625)$ has the same isospin and spin-parity quantum numbers as the $\Lambda^{*}(1520)$ ($I=0$, $J^{P}=\frac{3}{2}^{-}$), but with a charm quark instead of a strange quark. We therefore use the interpolating field

$$(O_{\Lambda_{c}^{*}})_{j\gamma}=\epsilon^{abc}\>(C\gamma_{5})_{\alpha\beta}\Big{(}\frac{1+\gamma_{0}}{2}\Big{)}_{\gamma\delta}\left[\widetilde{c}^{a}_{\alpha}\>\widetilde{d}^{b}_{\beta}\>(\widetilde{\nabla}_{j}\widetilde{u})^{c}_{\delta}-\widetilde{c}^{a}_{\alpha}\>\widetilde{u}^{b}_{\beta}\>(\widetilde{\nabla}_{j}\widetilde{d})^{c}_{\delta}+\widetilde{u}^{a}_{\alpha}\>(\widetilde{\nabla}_{j}\widetilde{d})^{b}_{\beta}\>\widetilde{c}^{c}_{\delta}-\widetilde{d}^{a}_{\alpha}\>(\widetilde{\nabla}_{j}\widetilde{u})^{b}_{\beta}\>\widetilde{c}^{c}_{\delta}\right],$$

(18)

which differs from Eq. (18) of Ref. Meinel:2020owd only by the replacement $s\to c$. As before, this form will work only at zero momentum. The tilde indicates gauge-covariant Gaussian smearing of the quark fields with the parameters given in Table 2. The field (18) actually has nonzero overlap with both the $\Lambda_{c}^{*}(2595)$ and the $\Lambda_{c}^{*}(2625)$,

	$\displaystyle\langle 0|(O_{\Lambda_{c}^{*}})_{j}|\Lambda_{c,1/2}^{*}(\mathbf{0},s^{\prime})\rangle$	$\displaystyle=$	$\displaystyle Z_{\Lambda_{c,1/2}^{*}}\>\frac{1+\gamma_{0}}{2}\,\gamma_{j}\gamma_{5}\,u(m_{\Lambda_{c,1/2}^{*}},\mathbf{0},s^{\prime}),$		(19)
	$\displaystyle\langle 0|(O_{\Lambda_{c}^{*}})_{j}|\Lambda_{c,3/2}^{*}(\mathbf{0},s^{\prime})\rangle$	$\displaystyle=$	$\displaystyle Z_{\Lambda_{c,3/2}^{*}}\>\frac{1+\gamma_{0}}{2}\,u_{j}(m_{\Lambda_{c,3/2}^{*}},\mathbf{0},s^{\prime}),$		(20)

and we can isolate the $J=\frac{1}{2}$ and $J=\frac{3}{2}$ components¹¹1At zero momentum, the continuum $J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{3}{2}^{-}$ irreducible representations subduce identically to the $G_{1}^{u}$ and $H^{u}$ irreducible representations of the double-cover of the cubic group Johnson:1982yq ; the next-higher values of $J^{P}$ that subduce to the same cubic irreps are $\frac{7}{2}^{-}$ and $J^{P}=\frac{5}{2}^{-}$, respectively, and such states will have higher energies. It is therefore safe to refer to only the continuum quantum numbers in this case. using the projectors

	$\displaystyle P^{kj}_{(1/2)}$	$\displaystyle=$	$\displaystyle\frac{1}{3}\gamma^{k}\gamma^{j},$		(21)
	$\displaystyle P^{kj}_{(3/2)}$	$\displaystyle=$	$\displaystyle g^{kj}-\frac{1}{3}\gamma^{k}\gamma^{j}.$		(22)

The zero-momentum $\Lambda_{c}^{*}$ two-point functions are defined like those for the $\Lambda^{*}$ in Ref. Meinel:2020owd , and after applying the above projectors their spectral decomposition reads

	$\displaystyle P^{jl}_{(1/2)}C^{(2,\Lambda_{c}^{*})}_{lk}(t)$	$\displaystyle=$	$\displaystyle-\frac{1}{2}Z_{\Lambda_{c,1/2}^{*}}^{2}(1+\gamma_{0})\,\gamma^{j}\gamma_{k}\,e^{-m_{\Lambda_{c,1/2}^{*}}t}$		(23)
			$\displaystyle+\>\>(\text{excited-state contributions}),$
	$\displaystyle P^{jl}_{(3/2)}C^{(2,\Lambda_{c}^{*})}_{lk}(t)$	$\displaystyle=$	$\displaystyle-\frac{1}{2}Z_{\Lambda_{c,3/2}^{*}}^{2}(1+\gamma_{0})\left(g^{j}_{\>\>k}-\frac{1}{3}\gamma^{j}\gamma_{k}\right)\,e^{-m_{\Lambda_{c,3/2}^{*}}t}$		(24)
			$\displaystyle+\>\>(\text{excited-state contributions}).$

At this point the reader may wonder why we did not analyze the $\Lambda^{*}(1405)$ with $J^{P}=\frac{1}{2}^{-}$ in Ref. Meinel:2020owd , despite being able to project to $J^{P}=\frac{1}{2}^{-}$ with the available data. The reason is that we do not trust the single-hadron/narrow-width approximation for the $\Lambda^{*}(1405)$, which has a larger decay width than the $\Lambda^{*}(1520)$ and likely a two-pole structure Oller:2000fj .

The masses extracted from single-exponential fits to our results for $P^{jl}_{(1/2)}C^{(2,\Lambda_{c}^{*})}$ and $P^{jl}_{(3/2)}C^{(2,\Lambda_{c}^{*})}$ in the plateau regions are given in Table 3, along with the masses of potential decay products. The latter are not used in our determination of the form factors but are included to assess whether the $\Lambda_{c}^{*}$ baryons are stable under the strong interactions for our quark masses. We find that both $m_{\Lambda_{c,1/2}^{*}}$ and $m_{\Lambda_{c,3/2}^{*}}$ are lower than all of the following: $m_{\Lambda_{c}}+m_{\pi}+m_{\pi}$, $m_{\Sigma_{c}}+m_{\pi}$, $m_{D}+m_{N}$, although the difference $m_{\Lambda_{c,3/2}^{*}}-m_{\Sigma_{c}}-m_{\pi}$ becomes consistent with zero for the F004 ensemble within the statistical uncertainties. The results are of course affected by the finite volume to some degree, but it appears likely that both the $\Lambda_{c,1/2}^{*}$ and the $\Lambda_{c,3/2}^{*}$ are stable hadrons at least on the C01 and C005 ensembles, where the energies are well below all thresholds.

We also performed simple chiral-continuum extrapolations of $m_{\Lambda_{c,1/2}^{*}}$ and $m_{\Lambda_{c,3/2}^{*}}$ of the form

$$m_{\Lambda_{c,J}^{*}}=m_{\Lambda_{c,J}^{*}}^{(\rm phys)}\left[1+c_{J}\>\frac{m_{\pi}^{2}-m_{\pi,\rm phys}^{2}}{(4\pi f_{\pi})^{2}}+d_{J}\>a^{2}\Lambda^{2}\right]$$

(25)

with fit parameters $m_{\Lambda_{c,J}^{*}}^{(\rm phys)}$, $c_{J}$, $d_{J}$, and constants $f_{\pi}=132\,\text{MeV}$, $\Lambda=300\,\text{MeV}$. These fits yield $m_{\Lambda_{c,1/2}^{*}}^{(\rm phys)}=2693(43)$ MeV, $m_{\Lambda_{c,3/2}^{*}}^{(\rm phys)}=2742(43)$ MeV. To estimate systematic uncertainties associated with the choice of fit model, we additionally performed higher-order fits of the form

$$m_{\Lambda_{c,J}^{*}}=m_{\Lambda_{c,J}^{*},{\rm HO}}^{(\rm phys)}\left[1+c_{J,{\rm HO}}\>\frac{m_{\pi}^{2}-m_{\pi,\rm phys}^{2}}{(4\pi f_{\pi})^{2}}+h_{J,{\rm HO}}\>\frac{m_{\pi}^{3}-m_{\pi,\rm phys}^{3}}{(4\pi f_{\pi})^{3}}+d_{J,{\rm HO}}\>a^{2}\Lambda^{2}+g_{J,{\rm HO}}\>a^{3}\Lambda^{3}\right],$$

(26)

with Gaussian priors $h_{J,{\rm HO}}=0\pm 10$ and $g_{J,{\rm HO}}=0\pm 10$, and computed the systematic uncertainties using

$$\sigma_{m,{\rm syst}}={\rm max}\left(|m_{\rm HO}-m|,\>\sqrt{|\sigma_{m,{\rm HO}}^{2}-\sigma_{m}^{2}|}\right),$$

(27)

where $m$, $\sigma_{m}$ denote the central value and uncertainty obtained using the parameter values and covariance matrix of the nominal fit and $m_{\rm HO}$, $\sigma_{m,{\rm HO}}^{2}$ denote the central value and uncertainty obtained using the parameter values and covariance matrix of the higher-order fit. In this way we finally obtain

	$\displaystyle m_{\Lambda_{c,1/2}^{*}}^{(\rm phys)}$	$\displaystyle=$	$\displaystyle(2693\pm 43_{\,\rm stat}\pm 95_{\,\rm syst})\>{\rm MeV},$		(28)
	$\displaystyle m_{\Lambda_{c,3/2}^{*}}^{(\rm phys)}$	$\displaystyle=$	$\displaystyle(2742\pm 43_{\,\rm stat}\pm 96_{\,\rm syst})\>{\rm MeV},$		(29)

which are consistent with the experimental values of $m_{\Lambda_{c,1/2}^{*}}=2592.25(28)\>{\rm MeV}$, $m_{\Lambda_{c,3/2}^{*}}=2628.11(19)\>{\rm MeV}$ Zyla:2020zbs . Plots of the extrapolations are shown in Fig. 1. Note that we do not use the chiral-continuum extrapolations of the baryon masses in our determination of the form factors; we use the lattice baryon masses when computing the form factors on each ensemble, and then extrapolate only the form factors themselves. The mass extrapolations merely provide a test of our methodology. Finally, in Table 3 we also list the hyperfine splittings $m_{\Lambda_{c,3/2}^{*}}-m_{\Lambda_{c,1/2}^{*}}$ computed on each ensemble. Their relative uncertainties are too large to obtain a useful chiral-continuum extrapolation, but the results are consistent within $<2\sigma$ with the experimental value of $35.86(34)$ MeV on each ensemble.