# Hyperon semileptonic decays in QCD sum rules

Sheng-Qi Zhang1, Xuan-Heng Zhang1 and Cong-Feng<EMAIL_ADDRESS>

1School of Physical Sciences, University of Chinese Academy of Sciences
YuQuan Road 19A, Beijing 100049, China

###### Abstract

We investigate the hyperon semileptonic decays within the framework of QCD sum
rules. The flavor $SU(3)$ symmetry breaking effects are analyzed via the
relevant form factors and corresponding branching fractions. Employing the
$z$-series parameterization to capture the $q^{2}$ dependence of form factors,
we calculate the hyperon semileptonic decay rates and confront them with the
recent experimental measurements. Moreover, we calculate as well the non-
standard tensor form factors, which involve certain new physics beyond the
standard model.

##  1 Introduction

Hyperon semileptonic decays (HSDs) play an important role in the exploration
of strong and weak interactions in light baryon sector, where the former
involves the hadronization of partons, the latter associate with the flavor-
changing processes. The precise measurement of the Cabibbo-Kobayashi-Maskawa
(CKM) matrix element $|V_{us}|$ can offer substantial test for the standard
model (SM) and potential evidence for new physics beyond the SM. Furthermore,
it is anticipated that the decays of light baryons with strange quarks
(hyperons) may stand as potential probes for $CP$ violation as well [1, 2].

HSDs were ever described by the Cabibbo model featuring the exact $SU(3)$
symmetry [3]. However, the experimental data for HSDs suggest the potential
breaking of flavor $SU(3)$ symmetry. For example, processes $n\rightarrow
pe\bar{\nu}_{e}$ and $\Xi^{0}\rightarrow\Sigma^{+}e\bar{\nu}_{e}$ find that
the ratio of the axial-vector over vector form factors $g_{1}(0)/f_{1}(0)$ are
$1.2754\pm 0.0013$ and $1.22\pm 0.05$ [4, 5, 6], respectively. The difference
of these results implies certain deviation from the $SU(3)$ symmetry since
they should be equal in exact $SU(3)$ symmetry. More comprehensive
understanding of flavor $SU(3)$ symmetry breaking necessitates more precise
experimental measurements. However, the measurement of HSD process remains to
be an experimental challenge due to its tiny branching fractions [4] and
complexity in distinguishing semileptonic decays from dominant two-body decay
backgrounds [7]. This underscores the considerable scope for the enhancement
on both the experimental and theoretical fronts. Hopefully, the proposed super
$\tau$-charm facility (STCF) could offer a distinctive opportunity for hyperon
studies, leveraging high-statistics samples of hyperon-antihyperon pair events
and time-like form factor measurements [8].

Moreover, HSDs offer an exceptional opportunity for extracting the CKM matrix
element $|V_{us}|$, while the current determination of $|V_{us}|$ heavily
depends on kaon decays [4]. In recently, BESIII collaboration reported a new
result of $\Lambda\rightarrow p\mu^{-}\bar{\nu}_{\mu}$ [9]. The absolute
branching fraction is determined to be $\mathcal{B}(\Lambda\to
p\mu^{-}\bar{\nu}_{\mu})=(1.48\pm 0.21\pm 0.08)\times 10^{-4}$. This result
along with earlier ones provide an opportunity to extract a precise
$|V_{us}|$, playing an ancillary role to test the quark-flavor mixing in the
hyperon sector. Theoretically, a model-independent assessment of $SU(3)$
breaking effects is pivotal for the extraction of $|V_{us}|$ [10, 7], although
it appears that the HSDs data still align well with the unbroken $SU(3)$
predictions due to the smallness of hyperons mass splitting ($\sim$$15\%$).

Up to now, various approaches have been employed to the analysis of $SU(3)$
breaking effects of HSDs, including different quark models [11, 12, 13, 14],
soliton model [15, 16, 17, 18], chiral perturbation theory ($\chi$PT) [19, 20,
21, 22], $1/N_{c}$ and large $N_{c}$ expansion [23, 24], $SU(3)$ Skyrme model
[25, 26], and lattice QCD [27, 28, 29, 31, 30]. Additionally, the technique of
three-point QCD sum rules (QCDSR), initially introduced by Ioffe et al. to
investigate the pion electromagnetic form factors [32, 33], has been
extensively employed to the study of baryon semileptonic decays [34, 35, 36,
37, 38, 39, 40, 41, 42]. In contrast to a phenomenological model, QCDSR is a
theoretical framework based on the QCD first principle. QCDSR starts by
formulating three-point correlation functions through appropriate
interpolating current operators, where nonperturbative effects are
parameterized as vacuum expectation values at each dimension. Once the
equivalence between the QCD and phenomenological representation of the three-
point correlation functions is established via quark-hadron duality, the form
factors arising in weak transitions are formally ascertained. The foundational
physical assumptions of QCDSR are more closely associated with field theory
[43], a connection that may align more effectively with the argument of
”model-independence” in Ref. [7]. In this study, we employ QCDSR to evaluate
the transition form factors of HSDs, afterwards the $SU(3)$ breaking effects
and the branching fractions of HSDs can be analyzed. Moreover, we also present
the results of the non-standard tensor form factors, which may stand as probes
for the new physics beyond the SM [44].

The rest of the paper is organized as follows: in Sec. 2 we interpret the
fundamental concept of QCDSR for the three-point correlation functions. The
numerical results and analysis are detailed in Sec. 3. The conclusions and
discussions are presented in the final section.

##  2 Formalism

There are a total six transition modes of HSDs, which are $\Lambda\rightarrow
p,\,\Sigma^{0}\rightarrow
p,\,\Xi^{-}\rightarrow\Lambda,\,\Xi^{-}\rightarrow\Sigma^{0},\,\Xi^{0}\rightarrow\Sigma^{+}$,
and $\Sigma^{-}\rightarrow n$. One is only bothered to consider the first four
of them, and other two may be determined through isospin relations [17, 18].
The transition matrix element for the generic HSDs process can be formulated
as

$\mathcal{M}_{B_{1}\rightarrow
B_{2}\ell\nu_{\ell}}=\frac{G_{F}}{\sqrt{2}}V_{us}[\bar{\ell}\gamma^{\mu}\left(1-\gamma_{5}\right)\nu_{\ell}][\bar{u}\gamma_{\mu}\left(1-\gamma_{5}\right)s]\;,$
(1)

where $B_{1}$ and $B_{2}$ are the initial and final state baryons. $G_{F}$
represents the Fermi constant, and $V_{us}$ is the CKM matrix element. On the
other hand, the hadronic matrix element can be described in terms of
transition form factors:

$\displaystyle\bra{B_{2}(q_{2})}\bar{u}\gamma_{\mu}\left(1-\gamma_{5}\right)s\ket{B_{1}(q_{1})}$
$\displaystyle=\bar{u}_{2}(q_{2})\bigg{[}f_{1}(q^{2})\gamma_{\mu}+if_{2}(q^{2})\sigma_{\mu\nu}\frac{q^{\nu}}{M_{1}}+f_{3}(q^{2})\frac{q_{\mu}}{M_{1}}\bigg{]}u_{1}(q_{1})$
$\displaystyle-\bar{u}_{2}(q_{2})\bigg{[}g_{1}(q^{2})\gamma_{\mu}+ig_{2}(q^{2})\sigma_{\mu\nu}\frac{q^{\nu}}{M_{1}}+g_{3}(q^{2})\frac{q_{\mu}}{M_{1}}\bigg{]}\gamma_{5}u_{1}(q_{1})\;.$
(2)

Here, $q_{1}$ and $q_{2}$ are the four-vector momentum of the initial and
final state baryons, respectively. The momentum transfer $q=q_{1}-q_{2}$ and
$M_{1}$ is the mass of the initial state baryon. These real-valued form
factors are functions of Lorentz invariant $q^{2}$, due to the requirement of
time reversal invariance [10, 12]. They are known as the vector and axial-
vector form factors $f_{1}$ and $g_{1}$, the weak magnetism and electric form
factors $f_{2}$ and $g_{2}$, and the induced scalar and pseudoscalar form
factors $f_{3}$ and $g_{3}$. The weak electric form factor $g_{2}$, classified
as the second-class form factor by Weinberg [45], is often assumed to be very
small [46, 12, 17]. The induced scalar and pseudoscalar form factors $f_{3}$
and $g_{3}$ can be safely neglected in the calculation since their
contributions to hyperon semileptonic decays are suppressed significantly by a
factor of $(m_{\ell}/M_{1})^{2}$ [12, 29], where $m_{\ell}(\ell=e,\mu)$ denote
the lepton masses.

Through the above analysis, and considering the relatively more abundant
experimental data on $f_{1}$, $f_{2}$, and $g_{1}$, we will focus merely on
these three form factors. Note that in the Cabibbo model, $f_{1}(0)$ is well
determined, $f_{2}(0)$ can be determined in terms of anomalous magnetic
moments of proton and neutron, and $g_{1}(0)$ appears to be a function of $F$
and $D$ [10]:

$\displaystyle g_{1}(0)$ $\displaystyle=a\cdot F+b\cdot D\;,$ (3)

where $a$ and $b$ are generalised Clebsch-Gordan coefficients of $SU(3)$
group. Here, $F$ and $D$ denote the coupling constants, resulting from
Cabibbo’s fundamental assumption that the weak hadronic currents belong to a
single self-conjugate representation of $SU(3)$. The Cabibbo model predictions
for these three form factors are given in Table 1.

Table 1: Form factors of HSDs predicted by the Cabibbo model. $\mu_{p}$ and $\mu_{n}$ stand for the anomalous magnetic moments of the proton and neutron, respectively. | $f_{1}(0)$ | $g_{1}(0)$ | $g_{1}(0)/f_{1}(0)$ | $f_{2}(0)/f_{1}(0)$
---|---|---|---|---
$\Lambda\rightarrow p$ | $-\sqrt{\frac{3}{2}}$ | $-\sqrt{\frac{3}{2}}(F+\frac{D}{3})$ | $F+\frac{D}{3}$ | $\frac{M_{\Lambda}}{M_{p}}\frac{\mu_{p}}{2}$
$\Sigma^{0}\rightarrow p$ | $-\sqrt{\frac{1}{2}}$ | $-\sqrt{\frac{1}{2}}(F-D)$ | $F-D$ | $\frac{M_{\Sigma^{0}}}{M_{p}}\frac{(\mu_{p}+2\mu_{n})}{2}$
$\Xi^{-}\rightarrow\Lambda$ | $\sqrt{\frac{3}{2}}$ | $\sqrt{\frac{3}{2}}(F-\frac{D}{3})$ | $F-\frac{D}{3}$ | $-\frac{M_{\Xi^{-}}}{M_{p}}\frac{(\mu_{p}+\mu_{n})}{2}$
$\Xi^{-}\rightarrow\Sigma^{0}$ | $\sqrt{\frac{1}{2}}$ | $\sqrt{\frac{1}{2}}(F+D)$ | $F+D$ | $\frac{M_{\Xi^{-}}}{M_{p}}\frac{(\mu_{p}-\mu_{n})}{2}$

To calculate the above three form factors in the framework of QCDSR, the
three-point correlation functions could be constructed, as

$\Pi_{\mu}(q_{1}^{2},q_{2}^{2},q^{2})=i^{2}\int
d^{4}x\;d^{4}y\;e^{i(-q_{1}x+q_{2}y)}<0|T\\{j_{2}(y)j_{\mu}(0)j^{\dagger}_{1}(x)\\}|0>\;.$
(4)

The weak transition current
$j_{\mu}=\bar{u}\gamma_{\mu}\left(1-\gamma_{5}\right)s$. $j_{1}$ and $j_{2}$
represent the interpolating currents for the initial and final state baryons,
formally defined as: [47, 36, 37]:

$\displaystyle j$
$\displaystyle=\epsilon^{abc}\left(q^{aT}_{i}C\sigma^{\mu\nu}q^{b}_{j}\right)\sigma_{\mu\nu}\gamma_{5}q^{c}_{k}\;,$
(5)

where the superscripts $a$, $b$, and $c$ represent the color indices and $C$
is the charge conjugation matrix. The subscripts $i$, $j$, and $k$ denote
different flavor of light quarks. In our calculation,
$(i,j,k)=(u,u,d),(u,s,d),(u,d,s),$ and $(s,s,d)$ for
$p,\,\Lambda,\,\Sigma^{0},$ and $\Xi^{-}$, respectively.

On the QCD side, the three-point correlation functions presented in Eq. (4)
can be formulated using operator-product expansion (OPE) and double dispersion
relations:

$\displaystyle\Pi_{\mu}^{\text{QCD}}(q_{1}^{2},q_{2}^{2},q^{2})=\int_{s_{1}^{\text{min}}}^{\infty}ds_{1}\int_{s_{2}^{\text{min}}}^{\infty}ds_{2}\frac{\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})}{(s_{1}-q_{1}^{2})(s_{2}-q_{2}^{2})}\;,$
(6)

where $s_{1(2)}^{\text{min}}$ represents the kinematic limit. The spectral
density $\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})$ can be determined using
Cutkosky cutting rules [40, 38, 41, 39, 42, 48, 49, 50]. In this study,
contributions up to dimension 6 are considered in
$\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})$, expressed as:

$\displaystyle\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})$
$\displaystyle=\rho^{\text{pert}}_{\mu}(s_{1},s_{2},q^{2})+\rho^{\langle\bar{q}q\rangle}_{\mu}(s_{1},s_{2},q^{2})+\rho^{\langle
g_{s}^{2}G^{2}\rangle}_{\mu}(s_{1},s_{2},q^{2})$ $\displaystyle+\rho^{\langle
g_{s}\bar{q}\sigma\cdot
Gq\rangle}_{\mu}(s_{1},s_{2},q^{2})+\rho^{\langle\bar{q}q\rangle^{2}}_{\mu}(s_{1},s_{2},q^{2})\;.$
(7)

Here the first term corresponds to the perturbative contribution, whereas
$\langle\bar{q}q\rangle$, $\langle g_{s}^{2}G^{2}\rangle$, $\langle
g_{s}\bar{q}\sigma\cdot Gq\rangle$, and $\langle\bar{q}q\rangle^{2}$
characterize nonperturbative effects as different orders of vacuum expected
value that are known as the condensates.

On the phenomenological side, by incorporating a complete set of intermediate
hadronic states and leveraging double dispersion relations, the three-point
correlation functions of Eq. (4) can be expressed as follows:

$\displaystyle\Pi_{\mu}^{\text{phe}}(q_{1}^{2},q_{2}^{2},q^{2})=$
$\displaystyle\sum_{\text{spins}}\frac{\bra{0}j_{2}\ket{B_{2}(q_{2})}\bra{B_{2}(q_{2})}j_{\mu}\ket{B_{1}(q_{1})}\bra{B_{1}(q_{1})}j_{1}\ket{0}}{(q_{1}^{2}-M_{1}^{2})(q_{2}^{2}-M_{2}^{2})}$
$\displaystyle+$ $\displaystyle\text{higher resonances and continuum
states}\;,$ (8)

where $M_{2}$ denotes the mass of the final state baryon, and the transition
amplitudes from vacuum to baryon states can be parameterized by introducing
the decay constants,

$\displaystyle\bra{0}j_{i}\ket{B_{i}(p)}$
$\displaystyle=\lambda_{i}u_{i}(p)\;.$ (9)

Here, the subscript $i=1,2$ represents the notation of initial and final state
baryons, respectively. $p$ is the four-vertor momentum carried by the baryons,
and $u_{i}$ denotes the Dirac spinor. $\lambda_{i}$ represents the decay
constant of baryon octet states, which can be calculated through two-point sum
rules [51, 52, 53, 54, 41]. Utilizing the hadronic matrix element presented in
Eq. (2) and the spin sum completeness relations $\sum
u_{i}(p)\bar{u}_{i}(p)=\not{p}+M_{i}$, one can derive the phenomenological
representation of the three-point correlation functions of Eq. (4):

$\displaystyle\Pi_{\mu}^{\text{phe}}(q_{1}^{2},q_{2}^{2},q^{2})$
$\displaystyle=\frac{\lambda_{2}\left(\not{q}_{2}+M_{2}\right)\bigg{[}f_{1}(q^{2})\gamma_{\mu}+if_{2}(q^{2})\sigma_{\mu\nu}\frac{q^{\nu}}{M_{1}}+f_{3}(q^{2})\frac{q_{\mu}}{M_{1}}\bigg{]}\lambda_{1}\left(\not{q}_{1}+M_{1}\right)}{(q_{1}^{2}-M_{1}^{2})(q_{2}^{2}-M_{2}^{2})}$
$\displaystyle-\frac{\lambda_{2}\left(\not{q}_{2}+M_{2}\right)\bigg{[}g_{1}(q^{2})\gamma_{\mu}+ig_{2}(q^{2})\sigma_{\mu\nu}\frac{q^{\nu}}{M_{1}}+g_{3}(q^{2})\frac{q_{\mu}}{M_{1}}\bigg{]}\gamma_{5}\lambda_{1}\left(\not{q}_{1}+M_{1}\right)}{(q_{1}^{2}-M_{1}^{2})(q_{2}^{2}-M_{2}^{2})}$
$\displaystyle+\text{higher resonances and continuum states}\;.$ (10)

It should be noted that despite the presence of twenty-four Lorentz structures
in Eq. (2), only six form factors need to be determined. The redundant Lorentz
structures of the form factors are addressed in the contribution of negative-
parity baryons [55, 56, 40], in the ”higher resonances and continuum states”
term in Eq. (2).

Adopting quark-hadron duality and double Borel transform, we can establish an
equivalence between the two representations in Eqs. (6) and (2). Consequently,
the general form of the form factors can be obtained:

$\displaystyle f^{B_{1}\rightarrow
B_{2}}(q^{2})=\frac{e^{M_{1}^{2}/\tau_{1}^{2}}e^{M_{2}^{2}/\tau_{2}^{2}}}{\lambda_{1}\lambda_{2}}\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}\;\rho(s_{1},s_{2},q^{2})e^{-s_{1}/\tau_{1}^{2}}e^{-s_{2}/\tau_{2}^{2}}+\Pi_{0}\;,$
(11)

where $s_{1}^{0}$ and $s_{2}^{0}$ represent the threshold parameters of the
initial and final state baryons, respectively. $\Pi_{0}$ comprises
contributions in the correlation function devoid of an imaginary part yet
manifesting nontrivial features after the Borel transformation. Additionally,
$\tau_{1}^{2}$ and $\tau_{2}^{2}$ denote the Borel parameters that emerge
after the double Borel transform. For convenience, we showcase the $f_{1}(0)$
expression of the transition $\Lambda\rightarrow p$:

$\displaystyle f_{1}^{\Lambda\rightarrow p}(0)$
$\displaystyle=\frac{e^{M_{1}^{2}/\tau_{1}^{2}}e^{M_{2}^{2}/\tau_{2}^{2}}}{\lambda_{1}\lambda_{2}M_{1}^{\prime}M_{2}^{\prime}}\bigg{[}\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}\;\frac{3\,s_{2}^{2}\,m_{s}^{2}\left(m_{s}^{2}-\delta_{s}\right)e^{-s_{1}/\tau_{1}^{2}}e^{-s_{2}/\tau_{2}^{2}}}{64\,\pi^{4}\delta_{s}^{5}}$
$\displaystyle\times\big{(}4\,m_{s}^{2}\,\delta_{s}\left(2M_{1}^{-}M_{2}^{-}-s_{2}\right)+3\,m_{s}^{4}\left(s_{2}-3M_{1}^{-}M_{2}^{-}\right)+\delta_{s}^{2}\left(M_{1}^{-}M_{2}^{-}+3\,s_{2}\right)\big{)}$
$\displaystyle-\frac{2\,\langle\bar{q}q\rangle^{2}\,m_{s}(M_{1}^{-}+5M_{2}^{-})}{3}e^{-m_{s}^{2}/\tau_{1}^{2}}\bigg{]}\;.$
(12)

Here, we define $\delta_{s}=s_{1}-s_{2}$ and $M^{\prime}=M^{+}+M^{-}$. $M^{+}$
and $M^{-}$ stand for the masses of the corresponding positive and negative
parity baryons, where the values are given in the next section.

##  3 Numerical results and discussions

Table 2: Input parameters of baryon octet. | $p$ | $\Lambda$ | $\Sigma^{0}$ | $\Xi^{-}$
---|---|---|---|---
$M(\frac{1}{2}^{+})(\text{GeV})$ | 0.938 | 1.116 | 1.193 | 1.322
$M(\frac{1}{2}^{-})(\text{GeV})$ | 1.535 | 1.405 | 1.750 | 1.630
$\lambda(\text{GeV}^{3})$ | 0.0255 | 0.0324 | 0.0347 | 0.0364

In our numerical calculations, related parameters of baryon octet are shown in
Table 2 and other inputs are adopted as follows [4, 49, 55, 52, 54, 41, 57,
58, 53, 56, 59, 60, 50, 51]:

$\displaystyle\langle\bar{q}q\rangle$ $\displaystyle=-(0.24\pm
0.01)^{3}\text{GeV}^{3},\,\langle\bar{s}s\rangle=(0.8\pm
0.1)\langle\bar{q}q\rangle,$ $\displaystyle m_{s}$
$\displaystyle=93.4^{+8.6}_{-3.4}\,\text{MeV},\,s_{1,2}^{0}=(2.0\sim
3.0)\,\text{GeV}^{2}\;.$ (13)

The masses of $u,d$ quark are ignored. Moreover, two additional free
parameters are introduced according to Eq. (11), i.e. two Borel parameters
$\tau_{1}^{2}$ and $\tau_{2}^{2}$. For simplicity, we first emloy the
following relation of Borel parameters [39, 38, 61],

$\displaystyle\frac{\tau_{1}^{2}}{\tau_{2}^{2}}=\frac{M_{1}^{2}-m_{s}^{2}}{M_{2}^{2}-m_{u}^{2}}\;,$
(14)

where $m_{s}$ and $m_{u}$ denote the the mass of strange and up quark,
respectively.

Typically, two criteria are utilized to ascertain the values of Borel
parameters. The first criterion involves the pole contribution. To examine the
contribution of ground-state hadrons, it is essential for the pole
contribution to dominate the spectrum. Therefore, a selection of the pole
contribution larger than $50\%$ for the transition form factors is often made,
formulated as follows:

$\displaystyle
R_{1}^{\text{PC}}=\frac{\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}}{\int_{s_{1}^{\text{min}}}^{\infty}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}},\quad
R^{\text{PC}}_{2}=\frac{\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}}{\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{\infty}ds_{2}}\;.$
(15)

These two ratios can be interpreted as the pole contributions from the initial
and final baryon channel, respectively.

The second criterion is the convergence of OPE, ensuring the validity of the
truncated OPE. In Eq. (2), nonperturbative effects up to dimension 6 are taken
into account, implying that the relative contribution from the highest
dimension of the condensate $\langle\bar{q}q\rangle^{2}$ needs to be below
20%. Furthermore, as physical observables, the form factors should in
principle be independent of any artificial parameters. Hence, a reliable
result should be obtained within a optimal region where the form factors
exhibit minimal dependence on $\tau_{1}^{2}$ and $\tau_{2}^{2}$. In practice,
we adjust the threshold parameters $s_{1,2}^{0}$ by $0.1\,\text{GeV}^{2}$ to
find acceptable range of the Borel parameters, where the form factors exhibit
variations of no more than approximately 10%.

With the above preparation, the form factors of HSDs can be numerically
derived. The form factor results directly evaluated by QCDSR are shown in
Table 3, where the uncertainties stem from the variation of the Borel
parameters and the input parameters listed in Eq. (3). We have identified that
the most significant errors arise from the threshold parameters $s_{1,2}^{0}$,
while a recently proposed inverse problem method holds promise in mitigating
the errors introduced by these threshold parameters [62, 63]. The
corresponding ratios are also listed in Table 4 and compared with other
methods, where $N$, $\Lambda$, $\Sigma$, and $\Xi$ denote different isospin
multiplet in the baryon octet.

Table 3: Form factor predictions at $q^{2}=0$ in QCD sum rules.

| $f_{1}(0)$ | $\tau_{2}^{2}$ | $f_{2}(0)$ | $\tau_{2}^{2}$ | $g_{1}(0)$ | $\tau_{2}^{2}$
---|---|---|---|---|---|---
$\Lambda\rightarrow p$ | $-1.179\pm 0.075$ | $1.4\sim 1.8$ | $-0.888\pm 0.074$ | $2\sim 3$ | $-0.843\pm 0.010$ | $10\sim 13$
$\Sigma^{0}\rightarrow p$ | $-0.702\pm 0.042$ | $4\sim 5$ | $0.725\pm 0.041$ | $7\sim 10$ | $0.229\pm 0.029$ | $1.5\sim 2$
$\Xi^{-}\rightarrow\Lambda$ | $1.216\pm 0.075$ | $10\sim 20$ | $0.118\pm 0.029$ | $10\sim 20$ | $0.328\pm 0.036$ | $10\sim 20$
$\Xi^{-}\rightarrow\Sigma^{0}$ | $0.676\pm 0.035$ | $15\sim 20$ | $1.394\pm 0.150$ | $15\sim 20$ | $0.874\pm 0.048$ | $1.5\sim 2$

Table 4: Corresponding ratios of form factors. | $\Lambda\rightarrow N$ | $\Sigma\rightarrow N$ | $\Xi\rightarrow\Lambda$ | $\Xi\rightarrow\Sigma$
---|---|---|---|---
$f_{1}(0)/f_{1}^{SU(3)}$ |  |  |  |
This work | $0.963\pm 0.061$ | $0.993\pm 0.059$ | $0.993\pm 0.061$ | $0.956\pm 0.049$
Quark model [11] | $0.987$ | $0.987$ | $0.987$ | $0.987$
Quark model [12] | $0.976$ | $0.975$ | $0.976$ | $0.976$
$\chi$PT [20] | $1.027$ | $1.041$ | $1.043$ | $1.009$
$\chi$PT [22] | $1.001^{+0.013}_{-0.010}$ | $1.087^{+0.042}_{-0.031}$ | $1.040^{+0.028}_{-0.021}$ | $1.017^{+0.022}_{-0.016}$
$1/N_{c}$ expansion [23] | $1.02\pm 0.02$ | $1.04\pm 0.02$ | $1.10\pm 0.04$ | $1.12\pm 0.05$
lattice QCD [31] |  | $0.975\pm 0.01$ |  | $0.973\pm 0.007$
$g_{1}(0)/f_{1}(0)$ |  |  |  |
This work | $0.708\pm 0.047$ | $-0.327\pm 0.046$ | $0.271\pm 0.045$ | $1.293\pm 0.100$
Cabibbo model [7] | $0.731$ | $-0.341$ | $0.195$ | $1.267$
Quark model [13] | $0.724$ | $-0.260$ | $0.265$ | $1.20$
Soliton model [17] | $0.718\pm 0.003$ | $-0.340\pm 0.003$ | $0.250\pm 0.002$ | $1.210\pm 0.005$
Soliton model [18] | $0.68$ | $-0.27$ | $0.21$ | $1.16$
$1/N_{c}$ expansion [23] | $0.73$ | $-0.34$ | $0.22$ | $1.03$
lattice QCD [30, 29] |  | $-0.287\pm 0.052$ |  | $1.248\pm 0.029$
Exp [4] | $0.718\pm 0.015$ | $-0.340\pm 0.017$ | $0.25\pm 0.05$ | $1.22\pm 0.05$
$f_{2}(0)/f_{1}(0)$ |  |  |  |
This work | $0.752\pm 0.074$ | $-1.042\pm 0.090$ | $0.118\pm 0.032$ | $1.957\pm 0.255$
Cabibbo model [7] | $1.066$ | $-1.292$ | $0.085$ | $2.609$
Quark model [13] | $1$ | $-0.962$ | $0.129$ | $2.402$
Soliton model [17] | $0.637\pm 0.041$ | $-0.709\pm 0.036$ | $-0.069\pm 0.027$ | $1.143\pm 0.061$
Soliton model [18] | $0.71$ | $-0.96$ | $-0.02$ | $2.02$
$1/N_{c}$ expansion [23] | $0.90$ | $-1.02$ | $-0.06$ | $1.85$
lattice QCD [29] |  | $-1.52\pm 0.81$ |  |
Exp [4] |  $1.32\pm 0.81$ [64] | $-0.97\pm 0.14$ |  $-0.24\pm 0.25$ [64] | $2.0\pm 0.9$

The ratio $f_{1}(0)/f_{1}^{SU(3)}$ describes the flavor $SU(3)$ symmetry
breaking effects of $f_{1}$, where the values of $f_{1}^{SU(3)}$ are taken
from Cabibbo theory [7]. Early results of $f_{1}(0)/f_{1}^{SU(3)}$ from two
quark models [11, 12] exhibit remarkable similarity across the four decay
channels, displaying a consistent negative sign for the $SU(3)$ corrections of
$f_{1}$. In contrast, analyses employing chiral perturbation theory [20, 22]
and $1/N_{c}$ expansion [23] generally favor positive signs. Recent lattice
QCD results [31] support a negative correction for $f_{1}$, aligning with our
central values. Additionally, we note that the $f_{1}(0)/f_{1}^{SU(3)}$ value
for $\Xi\rightarrow\Sigma$ is the smallest among the four decay channels,
suggesting relatively larger breaking effects of $f_{1}$ in the
$\Xi\rightarrow\Sigma$ mode. This tendency is also in agreement with lattice
results.

The ratio $g_{1}(0)/f_{1}(0)$ is characterized by $F$ and $D$ in Eq. (3),
where a fit of these two constants using old experimental data yields
$F\approx 0.463$ and $D\approx 0.804$ [7]. In Table 4, it can be observed that
the results of $g_{1}(0)/f_{1}(0)$ from various theoretical methods, along
with experimental data, generally maintain overall consistency within the
margin of error. It is noteworthy that the $\Xi\rightarrow\Sigma$ mode holds a
particularly unique status as it is identical to the well measured neutron
beta decay under the exchange of the down quark with the strange quark [29,
5]. In this case, only the CKM matrix elements become important to distinguish
these decay amplitudes [18], and the flavor $SU(3)$ breaking effects can be
easily exposed through a comparison with results of neutron beta decay [29].
The experimental result of $g_{1}(0)/f_{1}(0)$ for $\Xi\rightarrow\Sigma$ mode
given in Table 4 represents the world average taken from the
$\Xi^{0}\rightarrow\Sigma^{+}$ process in Particle Data Group [4], while our
central value is more closely aligned with the results announced by KTeV
collaboration with $g_{1}(0)/f_{1}(0)=1.32_{-0.17}^{+0.21}\pm 0.05$ [5]. The
precise experimental value of $g_{1}(0)/f_{1}(0)$ for neutron beta decay is
$g_{1}(0)/f_{1}(0)=1.2754\pm 0.013$ [4], and it slightly deviates from the
current data as well as our calculated results of the
$\Xi^{0}\rightarrow\Sigma^{+}$ process. Furthermore, the isospin partner
process $\Xi^{-}\rightarrow\Sigma^{0}$ was measured in Ref. [64], and they
reported a ratio of $g_{1}(0)/f_{1}(0)=1.25^{+0.14}_{-0.16}$. If the form
factors for the $\Xi^{-}\rightarrow\Sigma^{0}$ decay can be measured with
greater precision, it may provide insights into isospin symmetry breaking.

As for the ratio $f_{2}(0)/f_{1}(0)$, considering the relatively large
uncertainties of the experimental data presented in Table 4, it is challenging
to make a direct comparison between the theoretical results and the existing
data. Notice, results from different theoretical approaches still exhibit
significant disparities, further investigations are needed to determine the
precise value of the weak magnetism form factor $f_{2}$.

To obtain the $q^{2}$ dependence of the form factors, we calculate them at a
small space-like interval $q^{2}\in[-0.05,0]$ $\text{GeV}^{2}$. Then we fit
the data by employing the $z$-series parameterization [65], which is widely
used in the fitting of form factors [42, 66, 67, 40, 68, 61]. The expression
of $z$-series parameterization is as follows:

$\displaystyle f_{i}(q^{2})$
$\displaystyle=\frac{f_{i}(0)}{1-q^{2}/(m_{pole})^{2}}\Bigl{\\{}1+a_{1}(z(q^{2},t_{0})-z(0,t_{0}))\Bigr{\\}},$
$\displaystyle z(q^{2},t_{0})$
$\displaystyle=\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}}}\;.$
(16)

Here, $a_{1}$ is a fitting parameter, and $f_{i}(0)$ represents the value of
form factors at $q^{2}=0$, which is also treated as a fitted parameter. The
variables $t_{\pm}=(M_{1}\pm M_{2})^{2}$, and
$t_{0}=t_{+}-\sqrt{t_{+}-t_{-}}\sqrt{t_{+}-t_{min}}$. In the numerical
analysis, we choose $t_{min}=-0.05\,\text{GeV}^{2}$ and
$m_{pole}=m_{K}=0.493\,\text{GeV}$ [4] for the HSDs process. The nonlinear
least squares ($\chi^{2}$) fitting method are employed in our analysis. The
fitting results are listed in Table 5, where the values of $f_{1}(0)$ and
$g_{1}(0)$ obtained from the fitting procedure are in agreement with our
directly calculated results provided in Table 3. The $q^{2}$ dependence of
form factors for all HSDs process are shown in Fig. 1. It should be mentioned
that the $q^{2}$ dependence of $f_{2}$ can be neglected since its contribution
to the decay amplitude is already $O(q)$ [7]. Besides, the results of
$\Sigma^{-}\rightarrow n$ and $\Xi^{0}\rightarrow\Sigma^{+}$ can be derived
using isospin relation [17, 18]:

$\displaystyle(\Sigma^{-}\rightarrow n)=\sqrt{2}\,(\Sigma^{0}\rightarrow
p),\,\,\,(\Xi^{0}\to\Sigma^{+})=\sqrt{2}\,(\Xi^{-}\to\Sigma^{0})\;.$ (17)

Table 5: Summary of the fitted parameters for the form factors. | $f_{1}(0)$ | $a_{1}$ | $g_{1}(0)$ | $a_{1}$
---|---|---|---|---
$\Lambda\rightarrow p$ | $-1.175\pm 0.075$ | $7.87\pm 1.42$ | $-0.825\pm 0.007$ | $-30.48\pm 1.71$
$\Sigma^{0}\rightarrow p$ | $-0.701\pm 0.042$ | $15.56\pm 3.18$ | $0.210\pm 0.028$ | $-89.46\pm 8.46$
$\Xi^{-}\rightarrow\Lambda$ | $1.207\pm 0.073$ | $-22.18\pm 13.11$ | $0.312\pm 0.034$ | $-117.31\pm 2.53$
$\Xi^{-}\rightarrow\Sigma^{0}$ | $0.678\pm 0.027$ | $28.31\pm 35.90$ | $0.840\pm 0.034$ | $-91.22\pm 29.90$

Figure 1: The $q^{2}$ dependence of form factors with the central value of
fitting parameters listed in Table 5. Moreover, the symbol point, as well as
the triangle, represent the fitted points for each of the form factors.

Equipped with the $q^{2}$ dependence of the form factors, the branching
fractions of HSDs can be systematically analyzed. For convenience, we
introduce the following helicity amplitudes [39, 66, 69]:

$\displaystyle
H_{\frac{1}{2},0}^{V}=\frac{\sqrt{Q_{-}}}{\sqrt{q^{2}}}\big{(}M_{+}f_{1}(q^{2})-\frac{q^{2}}{M_{1}}f_{2}(q^{2})\big{)},\quad
H_{\frac{1}{2},0}^{A}=\frac{\sqrt{Q_{+}}}{\sqrt{q^{2}}}\big{(}M_{-}g_{1}(q^{2})+\frac{q^{2}}{M_{1}}g_{2}(q^{2})\big{)}\
,$ $\displaystyle
H_{\frac{1}{2},1}^{V}=\sqrt{2Q_{-}}\big{(}-f_{1}(q^{2})+\frac{M_{+}}{M_{1}}f_{2}(q^{2})\big{)},\quad
H_{\frac{1}{2},1}^{A}=\sqrt{2Q_{+}}\big{(}-g_{1}(q^{2})-\frac{M_{-}}{M_{1}}g_{2}(q^{2})\big{)}\
,$ $\displaystyle
H_{\frac{1}{2},t}^{V}=\frac{\sqrt{Q_{+}}}{\sqrt{q^{2}}}\big{(}M_{-}f_{1}(q^{2})+\frac{q^{2}}{M_{1}}f_{3}(q^{2})\big{)},\quad
H_{\frac{1}{2},t}^{A}=\frac{\sqrt{Q_{-}}}{\sqrt{q^{2}}}\big{(}M_{+}g_{1}(q^{2})-\frac{q^{2}}{M_{1}}g_{3}(q^{2})\big{)}\;.$
(18)

Here, $H_{\lambda^{\prime},\lambda_{W}}^{V(A)}$ is the helicity amplitudes for
weak transitions induced by vector and axial-vector currents, where
$\lambda^{\prime}$ and $\lambda_{W}$ are the helicity of the final state
baryon and the virtual $W$ boson, respectively. $Q_{\pm}$ is defined as
$Q_{\pm}=M_{\pm}^{2}-q^{2}$ and $M_{\pm}=M_{1}\pm M_{2}$. With the above
helicity amplitudes, the differential distribution of HSDs process
$B_{1}\rightarrow B_{2}\ell\nu_{\ell}$ can be written as:

$\displaystyle\frac{d\Gamma\left(B_{1}\rightarrow
B_{2}\ell\nu_{\ell}\right)}{dq^{2}}=\frac{G_{F}^{2}\left|V_{us}\right|^{2}q^{2}\sqrt{Q_{+}Q_{-}}}{384\,\pi^{3}\,M_{1}^{3}}(1-\frac{m_{\ell}^{2}}{q^{2}})^{2}H_{\text{tot}}\;,$
(19)

where $m_{\ell}$ represents the lepton mass ($\ell=e,\mu$) and
$H_{\text{tot}}$ is defined as

$\displaystyle H_{\text{tot}}$
$\displaystyle=\big{(}1+\frac{m_{\ell}^{2}}{2q^{2}}\big{)}\big{(}H_{\frac{1}{2},1}^{2}+H_{-\frac{1}{2},-1}^{2}+H_{\frac{1}{2},0}^{2}+H_{-\frac{1}{2},0}^{2}\big{)}+\frac{3\,m_{\ell}^{2}}{2q^{2}}\big{(}H_{\frac{1}{2},t}^{2}+H_{-\frac{1}{2},t}^{2}\big{)}\;.$
(20)

Here, the total helicity amplitudes, $H_{\lambda^{\prime},\lambda_{W}}$, can
be derived using the following relations:

$\displaystyle H_{-\lambda^{\prime},-\lambda_{W}}^{V}$
$\displaystyle=H_{\lambda^{\prime},\lambda_{W}}^{V},\quad\quad
H_{-\lambda^{\prime},-\lambda_{W}}^{A}=-H_{\lambda^{\prime},\lambda_{W}}^{A},$
$\displaystyle H_{\lambda^{\prime},\lambda_{W}}$
$\displaystyle=H_{\lambda^{\prime},\lambda_{W}}^{V}-H_{\lambda^{\prime},\lambda_{W}}^{A}\;.$
(21)

From the definition of $H_{\text{tot}}$ in Eq. (20), one may notice that the
contribution to the differential decay width from $f_{3}(q^{2})$ and
$g_{3}(q^{2})$ is presented in the term $H_{\frac{1}{2},t}^{2}$ and
$H_{-\frac{1}{2},t}^{2}$, which is evidently suppressed by a factor of
$m_{\ell}^{2}$. Therefore, we opt to neglect the effect of $f_{3}(q^{2})$ and
$g_{3}(q^{2})$ in Eq.(2). To compute the numerical results for the
differential decay width, we utilize the following input parameters relevant
to the decay analysis [4]:

$\displaystyle m_{e}=0.511\,\text{MeV},\quad m_{\mu}=0.106\,\text{GeV},\quad
G_{F}=1.166\times 10^{-5}\,\text{GeV}^{-2},\quad|V_{us}|=0.2243,$
$\displaystyle\tau_{\Lambda}=2.632\times
10^{-10}s,\quad\tau_{\Sigma^{+}}=8.018\times
10^{-10}s,\quad\tau_{\Sigma^{0}}=74\times 10^{-21}s,$
$\displaystyle\tau_{\Sigma^{-}}=1.479\times
10^{-10}s,\quad\tau_{\Xi^{0}}=2.90\times
10^{-10}s,\quad\tau_{\Xi^{-}}=1.639\times 10^{-10}s\;.$ (22)

Figure 2: The $q^{2}$ dependence of the differential decay width for HSDs
process $B_{1}\rightarrow B_{2}\ell\nu_{\ell}$. The red solid line denotes
$\ell=e^{-}$, while the blue dashed line denotes $\ell=\mu^{-}$.

Here, the mean lifetime of hyperons, noted as $\tau$, is introduced to derive
the branching fractions. To determine the total decay width, we must first
calculate the differential decay width. Eq. (19) enables us to compute the
differential decay width for electronic and muonic mode, and the corresponding
$q^{2}$ dependence of the differential decay width are plotted in Fig. 2.
These are merely theoretical predictions of the $q^{2}$ dependence for the
differential decay width in HSDs, and as of now, there is no experimental
measurement available due to the extremely limited kinematic regions. The
global behavior of the curve closely resembles that in Ref. [14] using a
relativistic quark model, profoundly impacted by the factor
$q^{2}\sqrt{Q_{+}Q_{-}}$. Moreover, the differential decay width of electronic
mode does not vanish at $q^{2}=m_{e}^{2}$ since the limit behavior of the
helicity amplitudes, while it becomes zero at $q^{2}=m_{\mu}^{2}$ for muonic
mode.

Integrating over the $q^{2}$ of those curves in Fig. 2 we obtain the total
decay width, ready for the determination of the decay branching fractions. The
normal branching fraction results for HSDs are presented in the second column
of Table 6. By disregarding the $q^{2}$ dependence of the form factors and
employing the directly calculated values at $q^{2}=0$ from Table 3, we have
included the corresponding results for the branching fractions in the third
column of Table 6. In Table 6, it is evident that the inclusion of $q^{2}$
dependence of the form factors has a notable impact on the branching
fractions, and enhances the consistency with the experimental data.
Specifically, the $q^{2}$ dependence of the form factors results in a
contribution of approximately 10% to 30% to the branching fractions.
Additionally, the influence of $q^{2}$ dependence of the form factors is
typically more pronounced in the muonic mode compared to the electronic mode.
These characters agrees with the conclusions in Ref. [13] exploiting a Lorentz
covariant quark model approach.

Note that the results of $\Sigma^{0}\rightarrow p$ process present in the
fourth column of Table 6 are provided in Ref. [69] using $SU(3)$ flavor
symmetry due to the lack of experimental data of this transition. It should be
emphasised that the experimental data extracted from the Particle Data Group
mostly has a long history with large uncertainties, and many of them are not
direct measurements but rely on corresponding two-body hadronic decays [4].
Besides, the number of events of the muonic mode is scarce, especially for the
transitions $\Xi^{-}\rightarrow\Lambda\mu^{-}\bar{\nu}_{\mu}$ and
$\Xi^{-}\rightarrow\Sigma^{0}\mu^{-}\bar{\nu}_{\mu}$. The former exhibits
significant uncertainties, and the latter only yileds a upper limit.
Therefore, more precise measurements for HSDs are highly necessary.

Table 6: Branching fractions for HSDs process $B_{1}\rightarrow B_{2}\ell\nu_{\ell}$ in QCD sum rules. The superscript 0 in the third column represents the results without $q^{2}$ dependence of the form factors. |  |  | QCDSR |  | QCDSR0 |  | Exp [4]
---|---|---|---|---|---|---|---
$\mathcal{B}(\Lambda\rightarrow pe^{-}\bar{\nu}_{e})\times 10^{-4}$ |  |  | $8.09\pm 0.73$ |  | $7.12\pm 0.65$ |  | $8.34\pm 0.14$
$\mathcal{B}(\Lambda\rightarrow p\mu^{-}\bar{\nu}_{\mu})\times 10^{-4}$ |  |  | $1.44\pm 0.13$ |  | $1.15\pm 0.11$ |  | $1.51\pm 0.19$
$\mathcal{B}(\Sigma^{-}\rightarrow ne^{-}\bar{\nu}_{e})\times 10^{-3}$ |  |  | $1.06\pm 0.20$ |  | $0.87\pm 0.14$ |  | $1.017\pm 0.034$
$\mathcal{B}(\Sigma^{-}\rightarrow n\mu^{-}\bar{\nu}_{\mu})\times 10^{-4}$ |  |  | $5.17\pm 1.03$ |  | $3.74\pm 0.59$ |  | $4.5\pm 0.4$
$\mathcal{B}(\Sigma^{0}\rightarrow pe^{-}\bar{\nu}_{e})\times 10^{-13}$ |  |  | $2.67\pm 0.49$ |  | $2.18\pm 0.34$ |  |  $2.46\pm 0.32$ [69]
$\mathcal{B}(\Sigma^{0}\rightarrow p\mu^{-}\bar{\nu}_{\mu})\times 10^{-13}$ |  |  | $1.30\pm 0.25$ |  | $0.94\pm 0.15$ |  |  $1.13\pm 0.15$ [69]
$\mathcal{B}(\Xi^{-}\rightarrow\Lambda e^{-}\bar{\nu}_{e})\times 10^{-4}$ |  |  | $5.41\pm 0.79$ |  | $4.67\pm 0.63$ |  | $5.63\pm 0.31$
$\mathcal{B}(\Xi^{-}\rightarrow\Lambda\mu^{-}\bar{\nu}_{\mu})\times 10^{-4}$ |  |  | $1.66\pm 0.26$ |  | $1.24\pm 0.17$ |  | $3.5^{+3.5}_{-2.2}$
$\mathcal{B}(\Xi^{0}\rightarrow\Sigma^{+}e^{-}\bar{\nu}_{e})\times 10^{-4}$ |  |  | $2.48\pm 0.24$ |  | $2.39\pm 0.25$ |  | $2.52\pm 0.08$
$\mathcal{B}(\Xi^{0}\rightarrow\Sigma^{+}\mu^{-}\bar{\nu}_{\mu})\times 10^{-6}$ |  |  | $2.18\pm 0.24$ |  | $1.91\pm 0.20$ |  | $2.33\pm 0.35$
$\mathcal{B}(\Xi^{-}\rightarrow\Sigma^{0}e^{-}\bar{\nu}_{e})\times 10^{-5}$ |  |  | $8.07\pm 0.77$ |  | $7.72\pm 0.81$ |  | $8.7\pm 1.7$
$\mathcal{B}(\Xi^{-}\rightarrow\Sigma^{0}\mu^{-}\bar{\nu}_{\mu})\times 10^{-6}$ |  |  | $1.12\pm 0.12$ |  | $0.97\pm 0.10$ |  | $<800$

In conclusion, we focus on the non-standard form factor introduced by the
tensor interaction [45, 70]:

$\displaystyle\bra{B_{2}(q_{2})}\bar{u}\,\sigma_{\mu\nu}\,s\ket{B_{1}(q_{1})}\simeq
f_{T}(q^{2})\,\bar{u}_{2}(q_{2})\,\sigma_{\mu\nu}\,u_{1}(q_{1})\;,$ (23)

where $f_{T}$ represents the tensor form factor. The matrix element in Eq.
(23) is essential for calculating the production of the massless dark photon
induced by the magnetic dipole moment (MDM) [70]. Furthermore, other terms to
the matrix element of the tensor current are neglected since they are
kinematically suppressed [44]. By applying the same procedure, the values of
the tensor form factors $f_{T}(0)$ as well as the ratio $f_{T}(0)/f_{1}(0)$
can be determined, which are presented in Table 7. It can be observed that the
values of $f_{T}(0)/f_{1}(0)$ in each channel align consistently with Ref.
[44], whihc are very close to the experimental data of $g_{1}(0)/f_{1}(0)$.

Table 7: Determination of $f_{T}(0)$ and $f_{T}(0)/f_{1}(0)$ in QCDSR. | $\Lambda\rightarrow p$ | $\Sigma^{0}\rightarrow p$ | $\Xi^{-}\rightarrow\Lambda$ | $\Xi^{-}\rightarrow\Sigma^{0}$
---|---|---|---|---
$f_{T}(0)$ | $-0.777\pm 0.060$ | $0.195\pm 0.018$ | $0.349\pm 0.037$ | $0.861\pm 0.053$
$f_{T}(0)/f_{1}(0)$ | $0.663\pm 0.070$ | $-0.284\pm 0.046$ | $0.288\pm 0.051$ | $1.278\pm 0.112$
$f_{T}(0)/f_{1}(0)$ [44] | $0.72$ | $-0.28$ | $0.22$ | $1.22$
$g_{1}(0)/f_{1}(0)$ [4] | $0.718\pm 0.015$ | $-0.340\pm 0.017$ | $0.25\pm 0.05$ | $1.22\pm 0.05$

##  4 Conclusions

In this work, we utilize QCD sum rules to investigate the hyperon semileptonic
decays. Particularly, we compute the vector form factor $f_{1}$, the weak
magnetism form factor $f_{2}$, and the axial-vector form factor $g_{1}$ for
all the pertinent hyperon semileptonic decays. The Cutkosky cutting rules is
employed to derive the analytic results of the relevant form factors. We
neglect $f_{3}$ and $g_{3}$ due to their substantially suppressed
contributions to the decays and also neglect the second-class form factor
$g_{2}$ which is often assumed to be small.

We first derive the numerical value for the form factors at $q^{2}=0$, and
then discuss the corresponding ratios of these form factors. Our findings
regarding $f_{1}(0)/f_{1}^{SU(3)}$ exhibit a consistent trend with the quark
model [11, 12] and lattice QCD methods [31], and there are disparities
compared to the results derived from chiral perturbation theory and $1/N_{c}$
expansion [20, 22, 23]. For the value of $g_{1}(0)/f_{1}(0)$, different
theoretical methods as well as the experimental data yield approximately the
same results. The variation of $g_{1}(0)/f_{1}(0)$ between
$\Xi^{0}\rightarrow\Sigma^{+}\ell\nu_{\ell}$ and neutron beta decay indicates
potential flavor $SU(3)$ symmetry breaking in HSDs. The last ratio,
$f_{2}(0)/f_{1}(0)$, exhibits substantial variations among different
theoretical predictions and experimental data. Therefore, additional efforts
are required to thoroughly investigate the weak magnetism form factor $f_{2}$.

Based on the $q^{2}$ dependence of the form factors, we present the analysis
of differential decay width and branching fractions. Our numerical results for
HSDs could provide useful information to reliably extract the value of the CKM
matrix element $V_{us}$. Finally, we conduct explorations on the tensor form
factor $f_{T}$ related to the new physics beyond the standard model, and find
that there is still room for further refinement in the QCD sum rule method. As
previously discussed, the sensitivity of the results to threshold parameters
may be alleviated through the application of inverse problem methods. Besides,
error reduction could be achieved through the implementation of radiative
corrections.

Acknowledgments

The authors acknowledge the meaningful discussion with Q.M. Feng. This work
was supported in part by the National Key Research and Development Program of
China under Contracts No. 2020YFA0406400, by the National Natural Science
Foundation of China(NSFC) under the Grants 11975236 and 12235008.

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# HE-LHC prospects for diboson resonance searches and electroweak $WW/WZ$
production via vector boson scattering in the semi-leptonic final states

Viviana Cavaliere  Brookhaven National Laboratory  Robert Les  University of
Toronto  Tatsumi Nitta  University of Tokyo  Koji Terashi  University of Tokyo

###### Abstract

This note presents the prospects of searches for new heavy resonances decaying
to diboson ($WW$) and measurements of electroweak $WW/WZ$ production via
vector boson scattering (VBS) in association with a high-mass dijet system in
the $\ell\nu qq$ final states ($W\to\ell\nu$, $W/Z\to qq$). The prospects are
presented for an integrated luminosity of 15 ab-1 of proton-proton ($pp$)
collisions at $\sqrt{s}=27$ TeV with an ATLAS-like detector simulated in the
Delphes framework. The cross-section measurement of the electroweak $WW/WZ$
production in VBS processes is expected to reach the precision of $\sim$2-3%,
improving the expected accuracy at the HL-LHC by a factor of 2. Prospects are
presented also for the separation of the longitudinal component of the
electroweak $WW/WZ$ production, showing the expected significance of $\sim
3\sigma$ is reached with 3 ab-1 for the single $\ell\nu qq$ channel and $\sim
5\sigma$ for all the semi-leptonic channels combined. The diboson resonance
searches are interpreted for sensitivity to a simplified phenomenological
model with a heavy gauge boson. With 15 ab-1 of $pp$ data, the discovery reach
for the new resonance is extended to 8 TeV.

###### Contents

  1. 1 Introduction
  2. 2 Simulation samples
    1. 2.1 Signal simulation
    2. 2.2 Background simulation
  3. 3 Event selection
    1. 3.1 Resonance analysis
    2. 3.2 VBS analysis
  4. 4 Pileup effects on boson identification
  5. 5 Systematic uncertainties
  6. 6 Extraction of analysis sensitivities
    1. 6.1 Resonance analysis
    2. 6.2 VBS Analysis
    3. 6.3 Longitudinally polarized signal
  7. 7 Conclusion

##  1 Introduction

Since the discovery of a Higgs boson at the Large Hadron Collider (LHC) at
CERN, the properties of the discovered particle have been measured, indirectly
probing the existence of physics beyond the Standard Model (SM) through Higgs
production and decay processes. The measured properties of the Higgs boson
have been so far in good agreement with SM predictions; the coupling strength
to SM fermions and bosons are confirmed within the level of 10-20%. The spin-0
nature of the Higgs boson casts however a big challenge on particle physics,
requiring a natural explanation to the stability of the measured Higgs mass
under the large hierarchy between the electroweak and fundamental gravity
scales (so-called hierarchy problem). Many scenarios beyond the Standard Model
(BSM) to explain the hierarchy problem, such as the Randall–Sundrum (RS) model
with a warped extra dimension [1], models with extended Higgs sectors in the
two-Higgs-doublet model (2HDM) [2], models with composite Higgs bosons [3] or
extended gauge sectors in Grand Unified Theories [4, 5, 6], postulate the
existence of new heavy particles. Search for new massive particles has been an
important part of the physics program at the LHC, performed extensively over a
broad range of final states at the centre-of-mass energies $\sqrt{s}$ of 7, 8
and 13 TeV. However, no clear hint of such new particles has been observed
yet.

While the presence of resonances is the most dramatic signal for new
phenomena, they may be too heavy or broad to be clearly observed. The study of
high-energy scattering between the longitudinal components of the vector
bosons (vector boson scattering or VBS) is a perfect probe for the presence of
new particles or interactions behind the electroweak (EW) symmetry breaking.
In fact, the scattering amplitude of VBS processes could exhibit anomalous
behaviour at high energy, in presence of any small deviation of Higgs boson
couplings from SM predictions. Such behaviour could give an additional insight
to the presence of new resonances at TeV scale, that may play a role in
restoring the SM scattering amplitudes. Experimental test for the high-energy
behaviour of VBS processes is one of the most important measurements at the
HL-LHC.

The previous VBS measurements and searches for anomalous quartic gauge
couplings (aQGCs) have focused on channels involving leptonic boson decays
($W\to\ell\nu$ and $Z\to\ell\ell$) 111Unless otherwise noted, $\ell$ stands
for electron or muon in this note. and photons. The semi-leptonic channels,
i.e, $V(\to qq)Z(\to\nu\nu)$, $V(\to qq)W(\to\ell\nu)$ and $V(\to
qq)Z(\to\ell\ell)$ where $V=W$ or $Z$, however, can offer an interesting
advantage over the leptonic channels to being able to probe VBS processes with
high-$p_{\text{T}}$ vector bosons due to larger $V(\to qq)$ branching
fractions. The hadronically-decaying $V$ boson can be fully reconstructed and
identified using the jet substructure techniques. Exploring the VBS topology
at a TeV or higher scale will provide an unique opportunity to directly test
the SM description of the EW symmetry breaking and the role of Higgs particle.

The quest for new heavy particles and anomalous VBS processes will continue in
future runs of the LHC and high-luminosity LHC (HL-LHC). However, the
improvement on the mass reach for new particles is going to be marginal due to
available parton-parton energy limited by the maximum $\sqrt{s}$ of 14 TeV at
the LHC and HL-LHC. A proposal to extend the LHC beam energy by almost factor
2, called high-energy LHC (HE-LHC), will significantly improve sensitivity to
new massive particles and VBS processes at multi-TeV scale.

This note presents expected sensitivities, at the integrated luminosity of 15
ab-1 of $pp$ collisions at $\sqrt{s}=27$ TeV, of the search for resonances
decaying to diboson ($WW$) and VBS measurements in the semi-leptonic channel,
where one $W$ boson decays leptonically and the other $W$ or $Z$ boson decays
to quarks ($\ell\nu qq$ channel). Both analyses are based on event selection
and classification similar to those used in the Run 2 ATLAS searches for $VV$
resonances and VBS at $\sqrt{s}=13$ TeV [7, 8]. The main signal and background
processes are reconstructed using a generic detector in the Delphes simulation
framework [9], modeled from the ATLAS detector in the HL-LHC. It is assumed
that resonance searches in other semi-leptonic and fully hadronic decay
channels have similar sensitivities to the $\ell\nu qq$ channel at high
masses, as observed in the ATLAS $VV$ resonance search [8]. ATLAS has
presented results of VBS searches in the leptonic final states of the
$W^{\pm}W^{\pm}$ [10] and $WZ$ [11] channels with the observation of
$6.9\sigma$ and $5.6\sigma$ significance, respectively. ATLAS has also
recently presented prospects for resonance and VBS searches in the $\ell\nu
qq$ channel, assuming an integrated luminosity of 300 or 3000 fb-1 of $pp$
collisions at $\sqrt{s}=14$ TeV [12].

##  2 Simulation samples

###  2.1 Signal simulation

The prospect for resonance searches presented in this article is interpreted
in the context of the heavy vector triplet (HVT) model [13, 14]. The HVT model
provides a broad phenomenological framework to test a range of different
scenarios involving new heavy gauge bosons and their couplings to SM fermions
and bosons. In this model, a triplet $\mathcal{W}$ of colorless vector bosons
is introduced with zero hypercharge. This leads to a set of nearly-degenerate
charged $W^{\prime\pm}$ and neutral $Z^{\prime}$ states, collectively denoted
by $V^{\prime}$, whose masses are assumed to be the same. A HVT scenario with
Drell-Yan production of $V^{\prime}$ boson, referred to as model A, represents
the phenomenology of weakly coupled models based on an extended gauge symmetry
[15], and is used as a benchmark for interpreting results.

The HVT signal events are generated with MadGraph5_aMC@NLO v2.6.3 [16] at
leading order (LO) using the NNPDF23LO parton distribution function (PDF) set
[17]. The generated events are interfaced to Pythia 8.23 [18] for parton
showering, hadronization, and the underlying event.

The electroweak $VVjj$ production is modeled with MadGraph5_aMC@NLO v2.6.3
[19], interfaced to Pythia 8.23 [20] for parton showering and hadronization.
The NNPDF30LO PDF set [17] is used. The electroweak $VVjj$ samples are
generated with two on-shell $V$ bosons, with one $W$ boson decaying
leptonically ($W\to\ell\nu$ with $\ell=e,\mu,\tau$), and the other $V$ boson
decaying hadronically. For each sample, all of the purely electroweak tree-
level diagrams at $\mathcal{O}(\alpha_{EW}^{6})$ that contribute to the final
state are included, i.e, VBS diagrams, non-VBS electroweak diagrams with and
without $b$-quarks in the initial or final states. The non-VBS diagrams, e.g,
diagrams including a $Wtb$ vertex, are suppressed by requiring the tagging
jets not to be $b$-tagged ($b$-veto) in the analysis, as described in Section.
3. For electroweak $WWjj$ production, the electroweak $t\bar{t}$ processes
have a significant contribution ($\sim$ 70%). Such contribution can be
effectively removed by applying a $b$-jet veto. Diagrams that contain a
mixture of electroweak and QCD vertices
($\mathcal{O}(\alpha_{EW}^{4}\alpha_{S}^{2})$ diagrams) are not included in
these samples, and are not part of the signal definition. Such processes are
accounted for by the background samples of $t\bar{t}$, single-top, and diboson
production.

For the extraction of the longitudinal components in VBS processes, the
electroweak $WWjj$ samples are generated with the DECAY program to identify
the polarization state of the produced $V$ bosons. The generated events are
then classified according to the polarization state: both $V$ bosons are
longitudinally (LL) or transversely (TT) polarized, or in the mixed state
(LT). Each event is showered using Pythia 8.23 [20] and then processed through
the Delphes simulation.

###  2.2 Background simulation

The main background in this analysis is $W$ boson produced in association with
jets ($W$+jets), with a significant contribution from top-quark pair
production. The $W$+jets events are simulated using MadGraph5_aMC@NLO v2.6.2
at LO using the NNPDF30NLO PDF set, interfaced to Pythia 8.23 for parton
showering and hadronization. The $W\to\tau\nu$ events are included in the
$W$+jets sample. For the generation of top-quark pairs, the aMC@NLO event
generator is used with the NNPDF30NLO PDF set, interfaced to Pythia 8.23for
parton showering and hadronization. The top quark mass is set to 172.5 GeV.
The $Z$+jets, single-top and non-resonant diboson ($ZZ$, $WZ$ and $WW$)
processes are not simulated and are expected to contribute at most 10% to the
total background.

##  3 Event selection

Search for diboson resonances and the measurement of VBS signatures are
performed separately using dedicated event selections. The resonance search
focuses on high-invariant mass region of the diboson system, where the
hadronically-decaying $V$ boson has large transverse momentum and is
reconstructed as a single large-radius ($R$) jet. The VBS search considers a
wide range of the diboson-system mass, exploiting the techniques to
reconstruct the hadronic decay of the $V$ boson as either a large-radius jet
or two small-radius jets.

###  3.1 Resonance analysis

Events are required to have exactly one electron or muon with
$p_{\text{T}}\>>30$ GeV and $|\eta|<4.0$. Events are further required to
contain a hadronically-decaying $W/Z$ candidate reconstructed as a large-
radius jet (denoted by $J$) with a distance parameter $R=1.0$, and a
leptonically-decaying $W/Z$ candidate. The large-$R$ jet is required to have
$p_{\text{T}}\>^{J}>200$ GeV and $|\eta_{J}|<2.0$. If two or more large
large-$R$ jets are found, the one with the highest $p_{\text{T}}$ is chosen as
a hadronically-decaying $W/Z$ boson candidate. The missing transverse energy
$E_{\text{T}}^{\text{miss}}$ has to be greater than 80 GeV, which suppresses
multijet background to a negligible level. By constraining the
$E_{\text{T}}^{\text{miss}}$ \+ lepton system to be consistent with the
$W$-boson mass, the $z$ component of the neutrino momentum can be
reconstructed by solving a quadratic equation. The smallest solution is chosen
and in the case where the solution is imaginary, only the real part is taken.

The selected events are required to satisfy the mass-window cut of
$|m_{J}-m_{W/Z}|<60$ GeV and have $D_{2}<1.5$, where $D_{2}$ is a jet
substructure variable [21, 22] sensitive to a two-prong structure from
hadronically-decaying $V$ boson. If the selected large-$R$ jet contains a
$b$-quark, then event is rejected to reduce contribution from $t\bar{t}$
background.

The distributions of the relevant kinematic properties of the large-$R$ jets
are shown in Fig. 1. The invariant mass of the diboson system ($m_{\ell\nu
J}$) is most sensitive to resonant diboson production and hence used as a
discriminant variable.

The background shape and normalization are constrained using dedicated control
regions for systematic uncertainties associated with the background modeling.
The following control regions are used in the final fit:

  * •

If an event satisfies all the selection criteria except the $W/Z$-boson mass-
window cut and has no $b$-jets ($b$-veto), then the event is categorized as a
$W$ control region event.

  * •

If an event satisfies all the selection criteria and has additional $b$-jets
outside the large-$R$ jet, then the event is categorized as a top control
region event.

These regions are used to constrain the $W$+jets and top background
normalization and shape uncertainties.

Figure 1: Kinematic distributions of HVT signal and background ($W$+jets and
$t\bar{t}$) events at 15 ab-1 in the resonance analysis. The HVT signal with a
2 TeV resonance is shown after multiplying the expected yield by $10^{4}$. The
invariant mass of the diboson system $m_{lvJ}$ is used as a discriminant
variable. The bottom-right plot shows jet mass distributions for the HVT
signals with resonance masses of 3 and 9 TeV, illustrating the need of future
improvement to mitigate the effect of energy merging within the limited
detector granularity.

###  3.2 VBS analysis

Experimentally, VBS is characterized by the presence of a pair of vector
bosons ($W$, $Z$, or $\gamma$) and two forward jets with large separation in
pseudo-rapidity and a large invariant mass. Therefore the VBS search requires
events to have two additional forward jets characteristic of the VBS topology
(called tagging jets) in addition to jets associated with the boson decay.

The tagging jets are required to be non-$b$-tagged in order to suppress
contribution from diagrams with a $Wtb$ vertex (especially the electroweak
$t\bar{t}$ process) in the electroweak $VVjj$ production. The tagging jets
must be in the opposite hemispheres,
$\eta(j_{1}^{\mathrm{tag}})\cdot\eta(j_{2}^{\mathrm{tag}})<0$, and have the
highest dijet invariant mass among all pairs of jets remaining in the event
after selecting jets from $V\to qq$ decay (called signal jets). After the
tagging jet pair is selected, it is required that both tagging jets have
$p_{\text{T}}\>>30$ GeV to suppress contribution from pile-up interactions,
and that the invariant mass of the two tagging jets is greater than 200 GeV.
In the VBS search the tagging jets are selected after identifying the signal
jets (as described below) and are required to be $\Delta R>1$ from the
large-$R$ signal jets.

In contrast to the resonance analysis, the VBS analysis first selects events
based on the large-$R$ jets using the same criteria as those in the resonance
analysis (called merged selection), then apply the selection based on the
small-$R$ jets (called resolved selection) if the event fails the merged
selection. In the resolved selection events are required to have at least two
small-$R$ jets with $R=0.4$, and the hadronically-decaying $W/Z$ candidate is
reconstructed from the pair of small-$R$ jets with the mass $m_{jj}$ closest
to the $W/Z$ mass among all combinations of jets with $p_{\text{T}}\>>20$ GeV.
The leading (sub-leading) signal jet is further required to have
$p_{\text{T}}\>>40$ (30) GeV after the jet pairs are selected to improve
separation between the signal and background. The invariant mass of the signal
jets is then required to fall in the mass window of $60<m_{jj}<110$ GeV. To
suppress backgrounds containing top quarks, events are vetoed if they contain
any $b$-tagged jets away from the hadronic $W/Z$ boson candidate by $\Delta
R>1.0$. In addition, if the $m_{jj}$ is within the $W$-mass window and both
jets are $b$-tagged, the event is removed. This cut is not applied to the
$Z$-boson candidate in order to retain signal with $Z\rightarrow b\bar{b}$
decay.

To optimize the signal sensitivity, Boosted Decision Trees (BDT) are trained
on the background and signal simulation samples separately for the resolved
and merged selections. Five variables are included in the merged and resolved
BDT: the invariant mass of the diboson system ($m_{\ell\nu J}$ for the merged
and $m_{\ell\nu jj}$ for the resolved selections), the lepton $\eta$, the
leading and sub-leading tagging jet $p_{\text{T}}$ and the boson centrality
$\zeta_{V}$. The boson centrality is defined as
$\zeta_{V}=\text{min}(\Delta\eta_{+},\Delta\eta_{-})$ where
$\Delta\eta_{+}=\text{max}(\eta(j_{1}^{\mathrm{tag}}),\eta(j_{2}^{\mathrm{tag}}))-\text{max}(\eta(\ell\nu),\eta(J))$
and
$\Delta\eta_{-}=\text{min}(\eta(\ell\nu),\eta(J))-\text{min}(\eta(j_{1}^{\mathrm{tag}}),\eta(j_{2}^{\mathrm{tag}}))$,
as shown in Fig. 2. These variables are chosen as they are the minimal subset
of variables with the greatest separation between the signal and background.
The BDT are trained using a gradient descent BDT algorithm, maximizing the
Gini index, in the TMVA package [23]. The BDT output is chosen as a final
discriminant and the distribution is used in the final fit for the VBS search.
Similarly to the resonance search, if an event fails either the $W/Z$ mass-
window cut or the $b$-veto requirement but passes all the other selection
criteria, then the event is categorized into a $W$\- or top-control region.
The BDT response used for the VBS search is shown in Fig. 2.

Figure 2: Distributions of lepton $\eta$, invariant mass of the tagging jets,
leading and sub-leading tagging jet $p_{\text{T}}$ and boson centrality for
VBS signal and background ($W$+jets and $t\bar{t}$) events at 15 ab-1 in the
VBS analysis. The VBS signal with both $WW$ and $WZ$ processes is shown after
multiplying the expected yields by 500. Shown for the merged selection. Bottom
right distribution of the Boosted Decision Tree discriminant to separate VBS
(electroweak $WWjj$) signal from $W$+jets and $t\bar{t}$ backgrounds in the
VBS analysis.

##  4 Pileup effects on boson identification

The unprecedented energy of $pp$ collisions at the HE-LHC will significantly
improve sensitivity to new multi-TeV particles over LHC and HL-LHC. However,
the experimental environment is expected to be challenging at the HE-LHC,
primarily due to a significant increase of the number of $pp$ collisions in a
same and nearby bunch crossings (pile-up). The HE-LHC is planned to be
operated at a centre-of-mass energy of 27 TeV with a beam intensity of about
$2\times 10^{11}$ protons per bunch, resulting in a peak luminosity of about
$3\times 10^{35}$ cm-2s-1 and 800 pile-up collisions at the peak luminosity.
Such extreme pile-up condition is expected to be particularly challenging for
identifying hadronically-decaying $W$/$Z$ boson as the extra contribution of
particles produced from pile-up collisions into jets could degrade the
performance of $W$/$Z$-boson tagger significantly. It is therefore important
to assess the performance of pile-up mitigation technique at the HE-LHC in
order to have a reliable estimate of the search sensitivity.

The study presented here focuses on the performance of pile-up mitigation
techniques and $W$/$Z$-boson tagging. The VBS-signal events are produced with
the overlay of minimum-bias $pp$ interactions generated using Pythia 8.219.
The minimum-bias interactions are overlaid onto hard scattering event using
Poisson probability distribution with the mean number of interactions
($\mu_{\text{pileup}}$) varied from 0 to 100, 200, 400 and 800. Furthermore,
the minimum-bias interactions are distributed randomly in $z$ and timing using
Gaussian profiles of $\sigma_{z}=5.3$ cm and $\sigma_{\text{t}}=160$ ps,
respectively ($z$=0 at the detector center and t=0 for hard scattering event).
The overlaid VBS-signal events are processed through Delphes with two pile-up
mitigation techniques: the Pile-up Per Particle Identification (PUPPI)
algorithm [24] used in CMS and the trimming procedure used in ATLAS. The
trimming parameters of the $p_{\text{T}}$ fraction cut and the sub-jet
reclustering radius are chosen to be the same as those used in ATLAS. For the
PUPPI algorithm the standard Delphes implementation is used. The Delphes
detector configuration is identical regardless of whether the pile-up overlay
is included or not.

Figure 3 shows the $p_{\text{T}}$ and $\eta$ of the leading large-$R$ jet with
$p_{\text{T}}\>>200$ GeV in VBS events with the five pile-up overlay
conditions. The jets are processed with the PUPPI algorithm but not the
trimming in the figure. The jet $p_{\text{T}}$ spectra are similar while the
$\eta$ distributions show increasing jet rate at large $|\eta|$ above 2 when
the $\mu_{\text{pileup}}$ is 200 or higher. This behaviour is not
significantly altered even when the trimming procedure is applied in addition
to PUPPI. Figure 4 shows the leading large-$R$ jet mass ($m_{J}$) and $D_{2}$
distributions for the PUPPI-only jets and the PUPPI+trimmed jets, both
required to have $p_{\text{T}}\>>200$ GeV. The mass-window cut of
$|m_{J}-m_{W/Z}|<15$ GeV is applied for events used in the $D_{2}$
distributions. Both $m_{J}$ and $D_{2}$ distributions get shifted towards
lower values with the trimming applied, enhancing the peak around $m_{W}$ and
the characteristics of two-prong structure for the $W$-boson decay. The
residual pile-up effect is still visible as a shift towards larger values with
increasing $\mu_{\text{pileup}}$, but the overall signal yield after the mass-
window and $D_{2}$ requirements (e.g, $D_{2}<1.5$) is largely stable. This
indicates that an impact to the $W$/$Z$-boson tagging performance from
expected pile-up collisions at the HE-LHC can be mitigated to the level where
the tagging performance at the HE-LHC is similar to that at Run 2 or HL-LHC.
Therefore, the study presented in the rest of this paper is based on the
$W$/$Z$-boson tagging performance at Run 2.

Figure 3: Leading large-$R$ jet $p_{\text{T}}$ (left) and $\eta$ (right)
distributions after applying the PUPPI algorithm at an integrated luminosity
of 15 ab-1 at $\sqrt{s}=27$ TeV with five different pile-up overlay conditions
of $\mu_{\text{pileup}}=0$, 100, 200, 400 and 800.

Figure 4: Leading large-$R$ jet mass (top-left) and $D_{2}$ (top-right)
distributions with $p_{\text{T}}\>>200$ GeV after applying the PUPPI algorithm
at an integrated luminosity of 15 ab-1 at $\sqrt{s}=27$ TeV with five
different pile-up overlay conditions of $\mu_{\text{pileup}}=0$, 100, 200, 400
and 800. The bottom two plots show the same distributions but after
additionally requiring that the jets are trimmed with the condition described
in the text.

##  5 Systematic uncertainties

Two categories of systematic uncertainties are considered in the analysis:
experimental uncertainties associated with the detector response and
calibration of reconstructed objects, and uncertainties on the background
modeling. Among those uncertainties, most dominant systematic sources are
considered in each of the two categories.

For experimental sources, the jet energy scale and resolution uncertainties
are considered for the small (large)-$R$ jets used in the resolved VBS
selection (merged resonance and VBS selections). In addition, the jet mass
scale and resolution uncertainties are considered for the merged selections.
The scale and resolution uncertainties are considered to be 10%.

For theoretical uncertainties, the normalization uncertainties are considered
for the major backgrounds, i.e, $\pm 10$% for $W$+jets and $t\bar{t}$
backgrounds. The uncertainties on the background distribution shapes are taken
into account for $W$+jets and $t\bar{t}$ processes by taking the variation of
the diboson mass and BDT score distributions from different MC generators in
Run 2 analysis. The uncertainties due to limited statistical accuracy of
simulation predictions are not considered in the analysis.

##  6 Extraction of analysis sensitivities

The analysis sensitivities to the resonance and VBS signals are extracted by
performing a simultaneous binned maximum-likelihood fit to the $m_{\ell\nu J}$
or $m_{\ell\nu jj}$ (BDT) distributions for the resonance (VBS) analysis in
the signal regions and the $W$+jets and $t\bar{t}$ control regions. A test
statistic based on the profile likelihood ratio [25] is used to test
hypothesized values of the global signal-strength factor ($\mu$), separately
for each signal model considered. The likelihood is defined as the product of
the Poisson likelihoods for all signal and control regions for a given signal
category. The fit includes two main background contributions from $W$+jets and
$t\bar{t}$. The $W$+jets and $t\bar{t}$ backgrounds are constrained by the
corresponding control regions and are treated as uncorrelated among the
resolved and merged signal regions.

Systematic uncertainties are taken into account as constrained nuisance
parameters with Gaussian or log-normal distributions. For each source of
systematic uncertainty, the correlations across bins in the $m_{\ell\nu J}$
and $m_{\ell\nu jj}$ distributions and between different kinematic regions as
well as those between signal and background are taken into account.

###  6.1 Resonance analysis

A statistical analysis is performed using the CLs method to determine the
expected upper limits on the signal cross section in the absence of signal.
The $m_{\ell\nu J}$ distribution is constructed in the resonance analysis for
the background-only hypothesis and for a signal-plus-background hypothesis
with varying signal strength $\mu$, where $\mu=0$ corresponds to the
background-only model and $\mu=1$ is the prediction of the signal model. A
likelihood function based on the binned $m_{\ell\nu J}$ distribution is used
to exclude values of $\mu$ with 95% confidence. The expected upper limit set
on the signal cross section is the greatest value of $\mu$ that is not
excluded with 95% confidence. This procedure is repeated for each hypothetical
signal mass. The expected upper limits set on the HVT $Z^{\prime}$ signal
cross section times branching fraction to $WW$ as a function of the
$Z^{\prime}$ mass are shown in Figure 5 for the integrated luminosities of 3
and 15 ab-1 at $\sqrt{s}=27$ TeV. Based on the $Z^{\prime}$ production cross
section from the HVT signal model the exclusion mass reach is extracted to be
9 and 11 TeV for 3 and 15 ab-1. Also shown in Fig. 5 is the sensitivity
expected at the HL-LHC with 3 ab-1 at $\sqrt{s}=14$ TeV [12]. A nearly factor
2 improvement in the mass reach is foreseen at the HE-LHC.

The discovery sensitivity is also extracted as the luminosity required to
achieve a $5\sigma$ observation of the signal. Here the signal significance is
defined as the square sum of $s/\sqrt{s+b}$ calculated over the binned
$m_{\ell\nu J}$ distributions at a given luminosity, where $s$ ($b$) is the
signal (background) yield in each $m_{\ell\nu J}$ bin. Figure 6 shows the
expected discovery significance as a function of integrated luminosity for the
resonance search. With 15 fb-1 of data the mass reach is extended to about 8
TeV, more than a factor 2 increase from the sensitivity expected at the HL-LHC
with 3 ab-1 at $\sqrt{s}=14$ TeV [12].

Figure 5: 95% CL upper limits on the $Z^{\prime}$ signal production cross
section times branching fraction to $WW$ for the HVT model A with integrated
luminosities of 15 (left) and 3 ab-1 (right) at $\sqrt{s}=27$ TeV. The
expected limits at the HL-LHC with 3 ab-1 at $\sqrt{s}=14$ TeV are also shown
in the right figure. Figure 6: Expected luminosity required to observe HVT
$Z^{\prime}$ signal with a $5\sigma$ significance as a function of the
$Z^{\prime}$ mass at $\sqrt{s}=27$ TeV.

###  6.2 VBS Analysis

For the VBS search, the statistical analysis is performed using the BDT
distributions and the upper limits on the signal strength for the SM VBS
($WW/WZ\to\ell\nu qq$) processes are extracted.

The expected significance for the SM VBS processes is $\sim$ 7$\sigma$ at 300
fb-1. The expected cross-section uncertainties are 18% at 300 fb-1 and 6.5% at
3 ab-1. The effects from the detector reconstruction efficiency and migration
due to detector resolution are not considered for the estimates of cross-
section uncertainties. Figure 7 shows the expected signal sensitivity and
cross-section uncertainty as a function of integrated luminosity. In addition
to the $\ell\nu qq$ channel, curves representing the estimated combined
sensitivity including the other semi-leptonic channels, $\ell\ell qq$ and
$\nu\nu qq$, are shown assuming that they have equal sensitivity as the
$\ell\nu qq$ channel.

Figure 7: Observed significance as a function of integrated luminosity (left)
and expected cross-section uncertainty (right) for the VBS signal in the
$\ell\nu qq$ channel at $\sqrt{s}=27$ TeV. The dashed line on the left shows
expected significance from the combination of all the three semi-leptonic
channels assumed to have sensitivity similar to the $\ell\nu qq$ channel. Also
shown on the right is the expected cross-section uncertainty from the HL-LHC
at $\sqrt{s}=14$ TeV.

###  6.3 Longitudinally polarized signal

The VBS events are expected to be produced with three different polarization
states of the final-state vector bosons: both of them being longitudinally
(LL) or transversely polarized (TT), or in the mixed polarization state (LT).
To determine the expected sensitivity to the longitudinally-polarized
electroweak $WW\rightarrow lvqq$ scattering (denoted by $W_{L}W_{L}$), a large
set of kinematic variables has been investigated to provide the best
separation from the TT or LT component. The normalized distributions of the
selected variables used for the merged analysis are shown in Fig. 8,
illustrating the differences between the LL or TT component from the inclusive
VBS events. The single most powerful discriminating variable is the total
invariant mass of the system $m_{WWjj}$ composed of the diboson and the two
tagging jets. A BDT is built using the variables shown in Fig. 8 to take full
advantage of the kinematic differences. The BDT is trained such that the LL
component is discriminated as a signal from the LT and TT components as well
as the rest of the background sources.The discriminating power for the LL
component is potentially improved if the different background components are
treated separately with dedicated multivariate techniques.

The expected significance of the extracted electroweak $W_{L}W_{L}\rightarrow
lvqq$ signal is extracted by performing a simultaneous binned maximum-
likelihood fit to the BDT distributions in the signal regions and the $W$+jets
and $t\bar{t}$ control regions. A test statistic based on the profile
likelihood ratio is used to test hypothesized values of the longitudinal
scattering cross-sections. The likelihood is defined as the product of the
Poisson likelihoods for all signal and control regions for a given analysis
channel. The fit includes four background contributions from $W$+jets,
$t\bar{t}$, QCD diboson and the electroweak LT and TT $WW$ processes. The
treatment of systematic uncertainties is similar to that used in the VBS
significance estimate above. The result of the expected significance is shown
in Fig. 9. Also shown is the significance obtained by fitting only to the
$m_{WWjj}$ variable. The figure shows that the BDT will bring a significant
improvement in the analysis sensitivity, resulting in the expected
significance of about 3 (5)$\sigma$ with an integrated luminosity of 3 (12)
ab-1. In addition to the $\ell\nu qq$ channel, the expected combined
sensitivity with the other semi-leptonic channels, $\ell\ell qq$ and $\nu\nu
qq$, is shown as well, assuming an equal sensitivity for the individual
channel to the $\ell\nu qq$. Figure 9 also shows the expected cross-section
uncertainty of the electroweak $W_{L}W_{L}$ scattering as a function of
integrated luminosity.

Figure 8: Normalized distributions of $D_{2}$, trimmed large-$R$ jet mass, the
mass of the VBS system $m_{WWjj}$ and the mass of the tagging jet system for
the longitudinally-polarized (LL) and transversely-polarized (TT) VBS $WW$
events as well as the inclusive VBS $WW$ events. Only the distributions for
the merged analysis are shown.

Figure 9: Observed significance as a function of integrated luminosity (left)
and expected cross-section uncertainty (right) for the VBS $W_{L}W_{L}$
signal, assuming a 10% $W_{L}W_{L}$ fraction predicted by the MadGraph
generator, in the $\ell\nu qq$ channel at $\sqrt{s}=27$ TeV. The solid and
dashed lines on the left shows the expected significance obtained by fitting
to the total invariant mass of the VBS system and the BDT output,
respectively. The dot-dashed line shows the expected significance from the
combination of all the three semi-leptonic channels assumed to have
sensitivity similar to the $\ell\nu qq$ channel.

##  7 Conclusion

The prospects of searches for new heavy resonances decaying to diboson ($WW$)
and measurements of electroweak $WW/WZ$ production via vector boson scattering
(VBS) in association with a high-mass dijet system in the $\ell\nu qq$ final
states are presented. The results are based on an integrated luminosity of 15
ab-1 of proton-proton collisions at $\sqrt{s}=27$ TeV with an ATLAS-like
detector simulated in the Delphes framework. The cross-section measurement of
the electroweak $WW/WZ$ production in VBS processes is expected to reach the
precision of $\sim$2-3%, improving the expected accuracy at the HL-LHC by a
factor of 2. Probing the longitudinal component of the electroweak $WW/WZ$
production is of paramount importance at future colliders.With 3 ab-1 of data
at $\sqrt{s}=27$ TeV, the HE-LHC could separate the longitudinal component
with the significance of $\sim 3\sigma$ for the single $\ell\nu qq$ channel
and $\sim 5\sigma$ for all the semi-leptonic channels combined. The diboson
resonance searches are interpreted for sensitivity to a simplified
phenomenological model with a heavy gauge boson. The discovery reach at the
HE-LHC is extended to 8 TeV with 15 ab-1 of data at $\sqrt{s}=27$ TeV, more
than two-fold increase from the expectation at the HL-LHC with 3 ab-1 at
$\sqrt{s}=14$ TeV.

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