# Isovector Axial Vector Form Factors of Octet-Decuplet Hyperon Transition in
QCD

A. Kucukarslan  Physics Department, Canakkale Onsekiz Mart University, 17100
Canakkale, Turkey<EMAIL_ADDRESS>U. Ozdem  Physics Department,
Canakkale Onsekiz Mart University, 17100 Canakkale, Turkey
<EMAIL_ADDRESS>A. Ozpineci  Physics Department, Middle East Technical
University, 06531 Ankara, Turkey<EMAIL_ADDRESS>
###### Abstract

We calculate the isovector axial vector form factors of the octet-decuplet
hyperon transitions within the framework of the light-cone QCD sum rules
(LCSR) to leading in QCD and including higher-twist corrections. In
particular, we motivate the most recent version of the $\Sigma$ and $\Lambda$
baryons distribution amplitudes which are examined up to twist-6 based on
conformal symmetry of the massless QCD Lagrangian.

Keywords: Hyperon axial form factors, Octet-decuplet transition, light-cone
QCD sum rules

##  1 Introduction

Form factors are extremely important in the studies of the hadron physics.
They give an information about the internal structure of composite particles.
The isovector axial vector form factors are important to measure the axial
charge of the hadrons. The axial charge $g_{A}$ is also viewed as an indicator
of the phenomenon of spontaneous breaking of chiral symmetry of non-
perturbative QCD [1].

The axial form factors of N $\rightarrow\Delta$ have been studied using
Lattice QCD [2, 3], quark model [4], light cone QCD sum rules [5], chiral
perturbation theory ($\chi$PT) [6, 7] and weak pion production [8, 9]. The
experimental information on the weak axial form factors come from parity-
violating electron scattering experiments at JLAB [10]. For $Q^{2}$ = 0.34
$GeV^{2}$, they found that the value of axial form factors determined from the
hydrogen asymmetry was $G_{A}^{N\Delta}$ = -
$0.05\pm(0.35)_{stat}\pm(0.34)_{sys}\pm(0.06)_{theory}$ ($G_{A}^{N\Delta}$
linear combination of $C^{A}_{4,5}$ form factors). However recently there have
been attempts to extract the octet-octet and decuplet-decuplet hyperon axial
charges using lattice QCD, quark model, $\chi$PT and QCD sum rules [1, 11, 12,
13, 14, 15]. The hyperon sector is interesting because they provide for an
ideal system in which to study $SU(3)$ flavor symmetry breaking by replacement
of up or down quarks in nucleons by strange ones [16]. For hyperon the axial
charge is a major parameter for low-energy effective description of baryon
sector as they enter in the loop graphs of chiral perturbation theory [11].
Our information about the axial charge of the hyperons from experiment is also
limited since their experimental measurements are difficult due to their
unstable nature.

In the present work, we calculate the isovector axial vector transition form
factors of the $\Xi-\Xi^{*}$, $\Sigma-\Sigma^{*}$ and $\Lambda-\Sigma^{*}$.
The considered processes take place in low energies far from the perturbative
region, hence to calculate the form factors as the main ingredients, we need a
non-perturbative method. One of the most powerful non-perturbative methods is
traditional QCD sum rules (QCDSR), which is also a powerful tool to extract
hadron properties [17, 18, 19, 20]. An alternative to the traditional QCD sum
rules is the light cone QCD sum rules (LCSR) [21, 22, 23]. In this method, the
hadronic properties are expressed in terms of the properties of the vacuum and
the light cone distribution amplitudes of one of the hadrons in the process.
Since the form factors are expressed in terms of the properties of the QCD
vacuum and the distribution amplitudes, any uncertainty in these parameters
reflects in the uncertainty of the predictions of the form factors. This
method has been rather successful in determining hadron form factors at high
$Q^{2}$ (see e.g. [14, 24, 25, 26, 27, 28, 29, 30]).

This work is organized as follows: In the next section, we obtain the
formulation of the baryon axial form factors in LCSR. In Section III, we
present our numerical analysis and discussion.

##  2 Hyperon axial form factors

The baryon (hyperon) matrix element expressed in terms of four invariant form
factors can be written as [31, 32, 33];

$\displaystyle\langle
H^{*}(p^{\prime},s^{\prime})|A_{\nu}^{3}(x)|H(p,s)\rangle$
$\displaystyle=i\overline{\upsilon}^{\lambda}(p^{\prime},s^{\prime})\bigg{[}\left\\{\frac{C_{3}^{H^{*}}(q^{2})}{M_{H}}\gamma_{\mu}+\frac{C_{4}^{H^{*}}(q^{2})}{M_{H}^{2}}p^{\prime}_{\mu}\right\\}(g_{\lambda\nu}g_{\rho\mu}$
(1) $\displaystyle-
g_{\lambda\rho}g_{\mu\nu})q^{\rho}+C_{5}^{H^{*}}(q^{2})g_{\lambda\nu}+\frac{C_{6}^{H^{*}}(q^{2})}{M_{H}^{2}}q_{\lambda}q_{\nu}\bigg{]}H(p,s)$

where $H$ and $H^{*}$ denote the $\Xi$, $\Sigma$ or $\Lambda$ and $\Xi^{*}$ or
$\Sigma^{*}$, respectively, and $A^{3}_{\nu}$ is an axial vector-isovector
current defined as

$\displaystyle
A^{3}_{\nu}(x)=\frac{1}{2}\bigg{(}\bar{u}(x)\gamma_{\nu}\gamma_{5}u(x)-\bar{d}(x)\gamma_{\nu}\gamma_{5}d(x)\bigg{)}.$
(2)

Besides, $\upsilon^{\lambda}$ is a Rarita-Schwinger spinor describing spin 3/2
baryons. In order to obtain the correlation function, phenomenological
summation over spin of Rarita-Schwinger spinor have been performed, and the
following formula have been used;

$\displaystyle\upsilon_{\mu}(p^{\prime},s)\overline{\upsilon_{\nu}}(p^{\prime},s)$
$\displaystyle=-(p\\!\\!\\!/^{\prime}+m_{H^{*}})\\{g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2p^{\prime}_{\mu}p^{\prime}_{\nu}}{3m_{H^{*}}^{2}}+\frac{p^{\prime}_{\mu}\gamma_{\nu}-p^{\prime}_{\nu}\gamma_{\mu}}{3m_{H^{*}}}\\}$
(3)

In deriving Light cone QCD sum rules for the axial baryon transitions form
factors, we start our analysis with the following two-point correlation
function:

$\Pi_{\mu\nu}(p,q)=i\int d^{4}xe^{iqx}\langle
0|T[j_{\mu}^{H^{*}}(0)A_{\nu}^{3}(x)]|H(p)\rangle,$ (4)

where $J^{H^{*}}_{\mu}(0)$ denotes the interpolating current $\Xi^{*}$ or
$\Sigma^{*}$ baryon and T denotes the time ordering product. In order to
calculate this correlation function either phenomenologically in which we
insert a complete set of hadronic states into the correlator to represent in
terms of the hadronic parameters, or theoretically in which we calculate
correlation function via the Operator Product Expansion (OPE) in Euclidean
region $p^{2}\rightarrow-\infty$ and $(p+q)^{2}\rightarrow-\infty$ in terms of
QCD parameters. QCD sum rules for the considered form factors are obtained by
mathching these two correlation function expressions and applying Borel
transformation in order to suppress contribution of the higher states and
continuum. In this work, we choose the general form of the interpolating
currents for $\Sigma^{*}$ and $\Xi^{*}$ baryons as [34]:

$\displaystyle
J_{\mu}^{\Sigma^{*}}(0)=\frac{\varepsilon^{abc}}{\sqrt{3}}[2(u^{aT}(0)C\gamma_{\mu}s^{b}(0))u^{c}(0)+(u^{aT}(0)C\gamma_{\mu}u^{b}(0))s^{c}(0)]$
$\displaystyle
J_{\mu}^{\Xi^{*}}(0)=\frac{\varepsilon^{abc}}{\sqrt{3}}[2(s^{aT}(0)C\gamma_{\mu}u^{b}(0))s^{c}(0)+(s^{aT}(0)C\gamma_{\mu}s^{b}(0))u^{c}(0)]$
(5)

where $a$, $b$, $c$ are the color indices and $C$ is the charge conjugation
operator. Inserting a complete set of intermediate hadronic states with the
same quantum numbers as the corresponding interpolating currents, we determine
the phenomenological part of the correlation function as follows

$\displaystyle\Pi_{\mu\nu}(p,q)=\sum_{s^{\prime}}\frac{\langle
0|j_{\mu}^{H^{*}}|H^{*}(p^{\prime},s^{\prime})\rangle\langle
H^{*}(p^{\prime},s^{\prime})|A_{\nu}^{3}(x)|H(p,s)\rangle}{M_{H^{*}}^{2}-p^{\prime
2}}+...$ (6)

where $M_{H^{*}}$ is the $\Xi^{*}$ or $\Sigma^{*}$ mass and … represent the
contributions of the higher states and continuum. The matrix element of the
interpolating current between the vacuum and baryon state is determined as

$\displaystyle\langle
0|j^{H^{*}}_{\mu}|H^{*}(p^{\prime},s^{\prime})\rangle=\lambda_{H^{*}}\upsilon_{\mu}(p^{\prime},s^{\prime})$
(7)

where $\lambda_{H^{*}}$ is the baryon overlap amplitude and
$\upsilon_{\mu}(p^{\prime},s^{\prime})$ is the baryon spinor. Using the
Eqs.(1), (3) and (7) in Eq.(6) one determines the correlation function in
terms of the hadronic parameters as

$\displaystyle\Pi^{H^{*}}_{\mu\nu}(p,q)$
$\displaystyle=-i\frac{\lambda_{H^{*}}}{M_{H^{*}}^{2}-p^{\prime
2}}\bigg{[}C_{3}^{H^{*}}(q^{2})\left\\{1-\frac{M_{H^{*}}}{M_{H}}\right\\}(q_{\mu}\gamma_{\nu}-g_{\mu\nu}q\\!\\!\\!/)$
(8)
$\displaystyle+\left\\{C_{5}^{H^{*}}(q^{2})+C_{4}^{H^{*}}(q^{2})\frac{p^{\prime}.q}{M^{2}_{H}}\right\\}g_{\mu\nu}q\\!\\!\\!/-\frac{C_{4}^{H^{*}}(q^{2})}{M_{H}^{2}}q_{\mu}p^{\prime}_{\nu}q\\!\\!\\!/$
$\displaystyle+C_{3}^{H^{*}}(q^{2})(q_{\mu}\gamma_{\nu}-g_{\mu\nu}q\\!\\!\\!/)q\\!\\!\\!/+\left\\{C_{5}^{H^{*}}(q^{2})(M_{H^{*}}+M_{H})\right\\}g_{\mu\nu}$
$\displaystyle+\left\\{-\frac{2C_{3}^{H^{*}}(q^{2})}{M_{H}}-\frac{C_{4}^{H^{*}}(q^{2})}{M_{H}}(1+\frac{M_{H^{*}}}{M_{H}})\right\\}(q_{\mu}p^{\prime}_{\nu}-g_{\mu\nu}p^{\prime}.q)$
$\displaystyle+\left\\{C_{6}^{H^{*}}(q^{2})(\frac{M_{H^{*}}+M_{H}}{M_{H}^{2}})\right\\}q_{\mu}q_{\nu}+\frac{C_{6}^{H^{*}}(q^{2})}{M_{H}^{2}}q_{\mu}q_{\nu}q\\!\\!\\!/\bigg{]}$

In this expression, the interpolating current $J_{\mu}$ couples to the
$J^{P}=3/2^{+}$ states. However, the interpolating current $J_{\mu}$ couples
not only to $J^{P}=3/2^{+}$ states, but also to the $J^{P}=1/2^{-}$ states.
Actually, the current $J_{\mu}$ has nonzero overlap with spin -$1/2$ states.
Therefore, the matrix element of the current $J_{\mu}$ can be written as the
following relation

$\langle
0|j^{H^{*}}_{\mu}|1/2(p^{\prime})\rangle=\left(Ap^{\prime}_{\mu}+B\gamma_{\mu}\right)u(p^{\prime}).$
(9)

The condition $\gamma^{\mu}p_{\mu}=0$ has been used to determine the relation
in Eq.(9). Hence spin-$1/2$ states contribute only the structures which
include a $\gamma^{\mu}$ at the beginning or which are proportional to
$p^{\prime}_{\mu}$. By choosing the appropriate structures, the contributions
from these states are eliminated [35, 36]. In this work we will consider the
ordering $\gamma_{\mu}\gamma_{\nu}q\\!\\!\\!/p\\!\\!\\!/^{\prime}$.

The QCD side of the correlation function is obtained in terms of quark and
gluon degrees of freedom. For calculation of the theoretical part of the
correlation function from the QCD side the interpolating fields in Eq. (2) are
inserted into the correlation function in Eq. (4), we obtained

$\displaystyle\Pi_{\mu\nu}$ $\displaystyle=\frac{i}{16\sqrt{3}}\int
d^{4}xe^{iqx}~{}(C\gamma_{\mu})_{\alpha\beta}(\gamma_{\nu}\gamma_{5})_{\rho\sigma}\left.\
\bigg{[}2\delta_{\alpha}^{\eta}\delta_{\sigma}^{\theta}\delta_{\beta}^{\phi}S(-x)_{\lambda\rho}\right.$
$\displaystyle\left.+2\delta_{\lambda}^{\eta}\delta_{\sigma}^{\theta}\delta_{\beta}^{\phi}S(-x)_{\alpha\rho}\right.\left.+\delta_{\alpha}^{\eta}\delta_{\sigma\theta}\delta_{\lambda}^{\phi}S(-x)_{\beta\rho}\right.\left.+\delta_{\beta}^{\eta}\delta_{\sigma}^{\theta}\delta_{\phi}^{\lambda}S(-x)_{\alpha\rho}\bigg{]}\right.$
$\displaystyle\left.4\epsilon^{abc}\langle
0|{q_{1}}_{\sigma}^{a}(0){q_{2}}_{\theta}^{b}(x){q_{3}}_{\phi}^{c}(0)|H(p,s)\rangle\right.$

where $q_{1,2,3}$ denote the quark fields and $S(x)$ represents the light-
quark propagator as

$S(x)=\frac{ix\\!\\!\\!/}{2\pi^{2}x^{4}}-\frac{\langle
q\bar{q}\rangle}{12}\left(1+\frac{m_{0}^{2}x^{2}}{16}\right).$ (11)

In this expressions, the first term describes the hard-quark propagator. The
other term represents the contribution from non-perturbative structure of the
QCD vacuum. Using Borel transformations, these contributions are removed. In
the background field the hard-quark propagator has corrections, which are
expected to give negligible contributions as they are related to four- and
five-particle baryon distribution amplitudes [37]. In this work, we will not
take into account such contributions, hence only the first term in Eq. (11)
leaves for our discussion. The matrix elements of the local three-quark
operator is

$\displaystyle 4\epsilon^{abc}\langle
0|q_{1\alpha}^{a}(a_{1}x)q_{2\beta}^{b}(a_{2}x)q_{3\gamma}^{c}(a_{3}x)|H(p,s)\rangle.$

This expression can be written in terms of DAs using the Lorentz covariance,
the spin and the parity of the baryon. Based on a conformal expansion using
the approximate conformal invariance of the QCD Lagrangian up to one-loop
order, the DAs are then decomposed into local non-perturbative parameters,
which can be estimated using QCD sum rules or fitted so as to reproduce
experimental data. We refer the reader to Refs. [38, 39, 40, 41] for a
detailed analysis on DAs of hyperons, which we employ in our work to extract
the axial-vector form factors.

The QCD sum rules for axial form factors of the hyperon transitions are
determined by equating both representations of correlation function. To do
this, we choose the structures proportional to
$q_{\mu}\gamma_{\nu}q\\!\\!\\!/$, $q_{\mu}{p^{\prime}}_{\nu}q\\!\\!\\!/$,
$(g_{\mu\nu}q\\!\\!\\!/-q_{\mu}\gamma_{\nu}q\\!\\!\\!/)$ and
$q_{\mu}q_{\nu}q\\!\\!\\!/$ for the form factors $C_{3}^{HH^{*}}$,
$C_{4}^{HH^{*}}$, $C_{5}^{HH^{*}}$ and $C_{6}^{HH^{*}}$, respectively. After
that, applying Borel transformation, the QCD sum rules for the axial form
factors are obtained as

$\displaystyle
C_{3}^{\Xi\Xi^{*}}(q^{2})\frac{\lambda_{\Xi^{*}}}{M_{\Xi^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=\frac{1}{\sqrt{3}}\bigg{\\{}\frac{M_{\Xi}^{2}}{8}\int_{0}^{1}d\alpha\frac{1-\alpha}{(q-p\alpha)^{4}}F_{1}-\frac{M_{\Xi}^{2}}{4}\int_{0}^{1}d\beta\frac{1-\beta}{(q-p\beta)^{4}}F_{2}$
$\displaystyle+\frac{M_{\Xi}^{2}}{2}\int_{0}^{1}d\beta\frac{1}{(q-p\beta)^{4}}F_{3}+\frac{1}{4}\int_{0}^{1}dx_{2}\frac{1}{(q-px_{2})^{2}}F_{4}\bigg{\\}}$

$\displaystyle
C_{4}^{\Xi\Xi^{*}}(q^{2})\frac{\lambda_{\Xi^{*}}}{M_{\Xi^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=\frac{M_{\Xi}}{\sqrt{3}}\bigg{\\{}-\frac{M_{\Xi}^{2}}{2}\int_{0}^{1}d\beta\frac{1-\beta}{(q-p\beta)^{6}}F_{5}+\frac{M_{\Xi}}{2}\int_{0}^{1}d\alpha\frac{\alpha}{(q-p\alpha)^{4}}F_{6}$
$\displaystyle+\frac{M_{\Xi}}{2}\int_{0}^{1}d\alpha\frac{1-\alpha}{(q-p\alpha)^{4}}F_{7}\bigg{\\}}$

$\displaystyle
C_{5}^{\Xi\Xi^{*}}(q^{2})\frac{\lambda_{\Xi^{*}}}{M_{\Xi^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=\frac{M_{\Xi}}{\sqrt{3}}\bigg{\\{}\frac{M_{\Xi}^{2}}{4}\int_{0}^{1}d\beta\frac{1}{(q-p\beta)^{4}}F_{8}+\frac{1}{8}\int_{0}^{1}dx_{2}\frac{1}{(q-px_{2})^{2}}F_{9}$
$\displaystyle+\frac{M_{\Xi}^{2}}{8}\int_{0}^{1}d\beta\frac{1}{(q-p\beta)^{4}}F_{10}+\frac{1}{8}\int_{0}^{1}dx_{2}\frac{1-x_{2}}{(q-px_{2})^{2}}F_{11}$
$\displaystyle+\frac{M_{\Xi}^{2}}{4}\int_{0}^{1}d\beta\frac{\beta(1-\beta)}{(q-p\beta)^{4}}F_{12}+\frac{M_{\Xi}^{2}}{4}\int_{0}^{1}d\beta\frac{1-\beta}{(q-p\beta)^{4}}F_{13}$
$\displaystyle+\frac{M_{\Xi}^{2}}{8}\int_{0}^{1}d\alpha\frac{\alpha(1-\alpha)}{(q-p\alpha)^{4}}F_{14}-\frac{M_{\Xi}}{2}\int_{0}^{1}dx_{2}\frac{x_{2}}{(q-px_{2})^{2}}F_{15}$
$\displaystyle+\frac{M_{\Xi}^{2}}{2}\int_{0}^{1}d\beta\frac{\beta}{(q-p\beta)^{4}}F_{16}+\frac{M_{\Xi}^{2}}{8}\int_{0}^{1}d\beta\frac{1-\beta}{(q-p\beta)^{4}}F_{17}$
$\displaystyle-\frac{1}{4}\int_{0}^{1}d\alpha\frac{1}{(q-p\alpha)^{2}}F_{18}\bigg{\\}}$

$\displaystyle
C_{6}^{\Xi\Xi^{*}}(q^{2})\frac{\lambda_{\Xi^{*}}}{M_{\Xi^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=-\frac{M_{\Xi}^{2}}{2\sqrt{3}}\bigg{\\{}M_{\Xi}\int_{0}^{1}d\beta\frac{(1-\beta)^{2}}{(q-p\beta)^{6}}F_{19}+\int_{0}^{1}d\alpha\frac{1-\alpha}{(q-p\alpha)^{4}}F_{20}\bigg{\\}}$
(12)

for the $\Xi$-$\Xi^{*}$ transition,

$\displaystyle
C_{3}^{\Sigma\Sigma^{*}}(q^{2})\frac{\lambda_{\Sigma^{*}}}{M_{\Sigma^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=\bigg{\\{}\frac{M_{\Sigma}^{2}}{\sqrt{3}}\int_{0}^{1}d\alpha\frac{1-\alpha}{(q-p\alpha)^{4}}F_{21}+\frac{2}{\sqrt{3}}\int_{0}^{1}dx_{2}\frac{1}{(q-px_{2})^{2}}F_{22}$
$\displaystyle-\frac{2M_{\Sigma}^{2}}{\sqrt{3}}\int_{0}^{1}d\beta\frac{1}{(q-p\beta)^{4}}F_{23}\bigg{\\}}$
$\displaystyle
C_{4}^{\Sigma\Sigma^{*}}(q^{2})\frac{\lambda_{\Sigma^{*}}}{M_{\Sigma^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=\frac{M_{\Sigma}}{\sqrt{3}}\bigg{\\{}M_{\Sigma}^{2}\int_{0}^{1}d\beta\frac{\beta(1-\beta)}{(q-p\beta)^{6}}F_{24}-\int_{0}^{1}d\alpha\frac{\alpha}{(q-p\alpha)^{4}}F_{25}$
$\displaystyle-2\int_{0}^{1}d\alpha\frac{1-\alpha}{(q-p\alpha)^{4}}F_{26}\bigg{\\}}$
$\displaystyle
C_{5}^{\Sigma\Sigma^{*}}(q^{2})\frac{\lambda_{\Sigma^{*}}}{M_{\Sigma^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=\frac{M_{\Sigma}}{\sqrt{3}}\bigg{\\{}M_{\Sigma}^{2}\int_{0}^{1}d\beta\frac{1}{(q-p\beta)^{4}}F_{27}+\int_{0}^{1}dx_{2}\frac{1}{(q-px_{2})^{2}}F_{28}$
$\displaystyle-2M_{\Sigma}^{2}\int_{0}^{1}d\beta\frac{1}{(q-p\beta)^{4}}F_{29}+\int_{0}^{1}dx_{2}\frac{1-x_{2}}{(q-px_{2})^{2}}F_{30}(x_{2})$
$\displaystyle+M_{\Sigma}^{2}\int_{0}^{1}d\beta\frac{1-\beta}{(q-p\beta)^{4}}F_{31}+M_{\Sigma}^{2}\int_{0}^{1}d\alpha\frac{\alpha(1-\alpha)}{(q-p\alpha)^{4}}F_{32}$
$\displaystyle-2\int_{0}^{1}dx_{2}\frac{x_{2}}{(q-px_{2})^{2}}F_{33}+2M_{\Sigma}^{2}\int_{0}^{1}d\beta\frac{\beta}{(q-p\beta)^{4}}F_{34}$
$\displaystyle+2M_{\Sigma}^{2}\int_{0}^{1}d\beta\frac{1-\beta}{(q-p\beta)^{4}}F_{35}-\int_{0}^{1}d\alpha\frac{1}{(q-p\alpha)^{4}}F_{36}\bigg{\\}}$

$\displaystyle
C_{6}^{\Sigma\Sigma^{*}}(q^{2})\frac{\lambda_{\Sigma^{*}}}{M_{\Sigma^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=-\frac{4M_{\Sigma}}{\sqrt{3}}\bigg{\\{}M_{\Sigma}^{2}\int_{0}^{1}d\beta\frac{(1-\beta)^{2}}{(q-p\beta)^{6}}F_{37}-\int_{0}^{1}d\alpha\frac{1-\alpha}{(q-p\alpha)^{4}}F_{38}\bigg{\\}}$
(13)

for the $\Sigma$-$\Sigma^{*}$ transition,

$\displaystyle
C_{3}^{\Lambda\Sigma^{*}}(q^{2})\frac{\lambda_{\Sigma^{*}}}{M_{\Sigma^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=\bigg{\\{}\frac{M_{\Lambda}^{2}}{\sqrt{3}}\int_{0}^{1}d\alpha\frac{1-\alpha}{(q-p\alpha)^{4}}F_{21}+\frac{2}{\sqrt{3}}\int_{0}^{1}dx_{2}\frac{1}{(q-px_{2})^{2}}F_{22}$
$\displaystyle-\frac{2M_{\Lambda}^{2}}{\sqrt{3}}\int_{0}^{1}d\beta\frac{1}{(q-p\beta)^{4}}F_{23}\bigg{\\}}$
$\displaystyle
C_{4}^{\Lambda\Sigma^{*}}(q^{2})\frac{\lambda_{\Sigma^{*}}}{M_{\Sigma^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=\frac{M_{\Lambda}}{\sqrt{3}}\bigg{\\{}M_{\Lambda}^{2}\int_{0}^{1}d\beta\frac{\beta(1-\beta)}{(q-p\beta)^{6}}F_{24}-\int_{0}^{1}d\alpha\frac{\alpha}{(q-p\alpha)^{4}}F_{25}$
$\displaystyle-2\int_{0}^{1}d\alpha\frac{1-\alpha}{(q-p\alpha)^{4}}F_{26}\bigg{\\}}$
$\displaystyle
C_{5}^{\Lambda\Sigma^{*}}(q^{2})\frac{\lambda_{\Sigma^{*}}}{M_{\Sigma^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=\frac{M_{\Lambda}}{\sqrt{3}}\bigg{\\{}M_{\Lambda}^{2}\int_{0}^{1}d\beta\frac{1}{(q-p\beta)^{4}}F_{27}+\int_{0}^{1}dx_{2}\frac{1}{(q-px_{2})^{2}}F_{28}$
$\displaystyle-2M_{\Lambda}^{2}\int_{0}^{1}d\beta\frac{1}{(q-p\beta)^{4}}F_{29}+\int_{0}^{1}dx_{2}\frac{1-x_{2}}{(q-px_{2})^{2}}F_{30}(x_{2})$
$\displaystyle+M_{\Lambda}^{2}\int_{0}^{1}d\beta\frac{1-\beta}{(q-p\beta)^{4}}F_{31}+M_{\Lambda}^{2}\int_{0}^{1}d\alpha\frac{\alpha(1-\alpha)}{(q-p\alpha)^{4}}F_{32}$
$\displaystyle-2\int_{0}^{1}dx_{2}\frac{x_{2}}{(q-px_{2})^{2}}F_{33}+2M_{\Lambda}^{2}\int_{0}^{1}d\beta\frac{\beta}{(q-p\beta)^{4}}F_{34}$
$\displaystyle+2M_{\Lambda}^{2}\int_{0}^{1}d\beta\frac{1-\beta}{(q-p\beta)^{4}}F_{35}-\int_{0}^{1}d\alpha\frac{1}{(q-p\alpha)^{4}}F_{36}(\alpha)\bigg{\\}}$

$\displaystyle
C_{6}^{\Lambda\Sigma^{*}}(q^{2})\frac{\lambda_{\Sigma^{*}}}{M_{\Sigma^{*}}^{2}-p^{\prime
2}}$
$\displaystyle=-\frac{4M_{\Lambda}}{\sqrt{3}}\bigg{\\{}M_{\Lambda}^{2}\int_{0}^{1}d\beta\frac{(1-\beta)^{2}}{(q-p\beta)^{6}}F_{37}-\int_{0}^{1}d\alpha\frac{1-\alpha}{(q-p\alpha)^{4}}F_{38}(\alpha)\bigg{\\}}$
(14)

for the $\Lambda$-$\Sigma^{*}$ transition. The explicit form of the functions
that appear in Eqs. (12), (13) and (14) are given in Appendix A.

In order to eliminate the subtraction terms in the spectral representation of
the correlation function, the Borel transformation is performed. After the
transformation, contributions from excited and continuum states are also
exponentially suppressed. Then, using the quark-hadron duality and subtracted,
the contributions of the higher states and the continuum can be modelled. Both
of the Borel transformation and the subtraction of higher states are applied
by using following substitution rules (see e.g. [27]):

$\displaystyle\int
dx\frac{\rho(x)}{(q-xp)^{2}}\rightarrow-\int_{x_{0}}^{1}\frac{dx}{x}\rho(x)e^{-s(x)/M^{2}},$
$\displaystyle\int
dx\frac{\rho(x)}{(q-xp)^{4}}\rightarrow\frac{1}{M^{2}}\int_{x_{0}}^{1}\frac{dx}{x^{2}}\rho(x)e^{-s(x)/M^{2}}+\frac{\rho(x)}{Q^{2}+x_{0}^{2}m_{B}^{2}}e^{-s_{0}/M^{2}},$
$\displaystyle\int
dt\frac{\rho(t)}{(q-xp)^{6}}\rightarrow-\frac{1}{2M^{4}}\int_{x_{0}}^{1}\frac{dx}{x^{3}}\rho(x)e^{-s(x)/M^{2}}$
$\displaystyle-\frac{1}{2M^{2}}\frac{\rho(x)}{x_{0}(Q^{2}+x_{0}^{2}m_{B}^{2})}+\frac{1}{2}\frac{x_{0}^{2}}{Q^{2}+x_{0}^{2}m_{B}^{2}}\bigg{[}\frac{d}{dx_{0}}\frac{\rho(x_{0})}{x_{0}(Q^{2}+x_{0}^{2}m_{B}^{2})}\bigg{]},$

where

$s(x)=(1-x)m_{B}^{2}+\frac{1-x}{x}Q^{2},$

$M$ is the Borel mass, $m_{B}=M_{H}$ and $x_{0}$ is the solution of the
quadratic equation for $s=s_{0}$:

$x_{0}=\left[\sqrt{(Q^{2}+s_{0}-m_{B}^{2})^{2}+4m_{B}^{2}Q^{2}}-(Q^{2}+s_{0}-m_{B}^{2})\right]/(2m_{B}^{2}),$

where $s_{0}$ is the continuum threshold.

##  3 Numerical Analysis and Conclusion

In this section, we present LCSR results for the octet-decuplet hyperon
transition form factors. To obtain our numerical results the main input
parameters are the baryon DAs. The DAs of the hyperon depending on various
non-perturbative parameters are studied in [38, 39, 40, 41]. In Table.1 we
present the values of the input parameters using the DAs of each hyperon. In
this section, we will only consider the central values of these parameters.

Parameter | $\Sigma$ | $\Xi$ | $\Lambda$ |
---|---|---|---|---
$f_{B}$ (GeV2) | 0.0094 | 0.0099 | 0.0060 |
$\lambda_{1}$ (GeV2) | -0.025 | -0.028 | 0.0083 |
$\lambda_{2}$ (GeV2) | 0.044 | 0.052 | 0.0083 |
$\lambda_{3}$ (GeV2) | 0.02 | 0.017 | 0.010 |
| $\Sigma$ |  |  |
$V_{1}^{s}$ | $A_{1}^{u}$ | $f_{1}^{s}$ | $f_{2}^{s}$ |
0.39 | 0.29 | -0.15 | 9.9 |
$f_{3}^{s}$ | $f_{1}^{u}$ | $P_{2}^{0}$ | $S_{1}^{u}$ |
1.6 | -0.11 | 0.004 | -0.0014 |
| $\Lambda$ |  |  |
$A_{1}^{s}$ | $A_{1}^{q}$ | $f_{1}^{s}$ |  |
0.31 | 0.032 | 0.23 |  |
$f_{1}^{q}$ | $f_{3}^{q}$ | $f_{4}^{q}$ |  |
-0.23 | 0.43 | 1.07 |  |
Table 1: The values of the parameters are used in the DAs of $\Sigma$,
$\Lambda$ and $\Xi$. The first column includes the dimensionful parameters for
each baryon. In the other four columns we represent the list of the values of
parameters that determine the shape of the DAs, which have been extracted for
$\Sigma$ and $\Lambda$. For $\Xi$ these parameters are taken as zero.

We use the values of some parameters as follow; $M_{\Lambda}=1.11~{}GeV$,
$M_{\Sigma}=1.18~{}GeV$, $M_{\Sigma}*=1.38~{}GeV$, $M_{\Xi}=1.31GeV$,
$M_{\Xi}*=1.53~{}GeV$ [42]. To evaluate a numerical prediction for the form
factors, we need also specify the values of the residue of $\Sigma^{*}$ and
$\Xi^{*}$. The residues can be determined from the mass sum rules as
$\lambda_{\Sigma}*=0.043~{}GeV^{3}$ and $\lambda_{\Xi}*=0.053~{}GeV^{3}$ [43]
which are used in our calculations. The predictions for the form factors
depend on two auxiliary parameters: the Borel mass $M^{2}$, and the continuum
threshold $s_{0}$. The continuum threshold signals the scale at which, the
excited states and continuum start to contribute to the correlation function.
Hence, it is choosen as
$s_{0}\simeq(m_{\Sigma^{*}}+0.3~{}GeV)^{2}=2.82~{}GeV^{2}$ for $\Sigma^{*}$
and $s_{0}\simeq(m_{\Xi^{*}}+0.3~{}GeV)^{2}=3.34~{}GeV^{2}$ for $\Xi^{*}$. One
approach to determine the continuum threshold and the working region of the
Borel parameter $M^{2}$ is to plot the dependence of the predictions on
$M^{2}$ for a range of values of the continuum threshold and determine the
values of $s_{0}$ for which there is a stable region with respect to
variations of the Borel parameter $M^{2}$. For this reason, in Figs.(1), (2)
and (3) we plot the dependence of the form factors on $M^{2}$ for two fixed
values of $Q^{2}$ and for various values of $s_{0}$ in the range
$2.5~{}GeV^{2}<s_{0}<4,5~{}GeV^{2}$. As can be seen from these figures, for
$s_{0}=3.0\pm 0.5~{}GeV^{2}$ for $\Sigma-\Sigma^{*}$, $\Lambda-\Sigma^{*}$ and
$s_{0}=3.5\pm 0.5~{}GeV^{2}$ for $\Xi-\Xi^{*}$ the predictions are practically
independent of the value of $M^{2}$ for the related range. The uncertainty due
to variation of $s_{0}$ in this range is much larger than the uncertainty due
to variations with respect to $M^{2}$. Note that the determined range of
$s_{0}$ is in the range that one would expect from the physical interpretation
of $s_{0}$. In Figs.(4), (5) and (6) we also represent twist contributions of
DAs. In these figures, the form factors of $\Sigma-\Sigma^{*}$ and
$\Xi-\Xi^{*}$ are represent very good asymptotic behaviour. Twist-3 and
twist-4 give dominant contribution for these two transitions, respectively,
but twist-5 and 6 give very small contribution. In the form factors of
$\Lambda-\Sigma^{*}$ transition we expect dominant contribution coming from
twist-3 but it comes from higher twists. In Figs. (7), (8) and (9) we plot the
form factors of hyperons, $C_{3}^{HH^{*}}$, $C_{4}^{HH^{*}}$, $C_{5}^{HH^{*}}$
and $C_{6}^{HH^{*}}$, as a function of $Q^{2}$ in the region $1~{}GeV^{2}\leq
Q^{2}\leq 10$ $GeV^{2}$. We see that the behavior of the form factors agree
well with our expectations except $C_{5}^{\Lambda\Sigma^{*}}$. While the form
factor $C_{5}^{\Lambda\Sigma^{*}}(Q^{2})$ in the region of $1~{}GeV^{2}\leq
Q^{2}\leq 2$ $GeV^{2}$ has positive sign, in the region of $2~{}GeV^{2}\leq
Q^{2}\leq 10$ $GeV^{2}$ the form factor changes it’s sign. For this reason we
have not any prediction about the form factor
$C_{5}^{\Lambda\Sigma^{*}}(Q^{2})$. In Ref.[39] DAs of $\Sigma$ and $\Lambda$
baryons have been calculated by employed QCD sum rules but this DAs do not
contain higher order terms. In Refs.[40, 41] higher order corrections have
been calculated for these baryons. As seen second and third column of the
Table.I, calculated higher order corrections of $\Sigma$ and $\Lambda$ give
dominant contributions. For that reason the form factors of $\Sigma$ and
$\Lambda$ give higher values by comparison with the $\Xi$ form factors.

The form factor $C_{5}^{HH^{*}}$ is the dominant axial-vector form factor and
the only one that can be extracted directly from the matrix element at
$Q^{2}=0$, determining the axial charge of the transitions
$\Sigma-\Sigma^{*}$, $\Xi-\Xi^{*}$ and $\Lambda-\Sigma^{*}$. In order to
extrapolate, we have tried to fit LCSR results. To do this, we use an
exponential form for $C_{5}^{HH^{*}}$ as

$C_{5}^{HH^{*}}(Q^{2})=C_{5}^{HH^{*}}(0)\exp[-Q^{2}/m_{A}^{2}]$ (15)

with which we can make reasonable description of data with a two-parameter
fit. Using this from, we have studied three fit region $Q^{2}$ $\geq$
$1~{}GeV^{2}$, $Q^{2}$ $\geq$ $1.5~{}GeV^{2}$ and $Q^{2}$ $\geq$
$2~{}GeV^{2}$.

Our results for axial charges and axial masses are presented in Table 2.
Obtained values could not be compared any results that available recently. We
observe that the axial masses are very close to $\Sigma$ and $\Xi$. At this
point, it will be very interesting to compare our results to those from
different method in the near future. The mass of the lightest axial vector
meson is $m_{A}=1.23~{}GeV$ [42]; therefore the axial masses of the $\Sigma$
and $\Xi$ have close values with the predictions of the VMD model.

Baryon | Fit Region (GeV2) | $C_{5}^{HH^{*}}(0)$ |  $m_{A}$ (GeV) |
---|---|---|---|---
| [1.0-10] | -41.80 | 1.22 |
$\Sigma-\Sigma^{*}$ | [1.5-10] | -28.19 | 1.38 |
| [2.0-10] | -21.96 | 1.49 |
| [1.0-10] | 2.73 | 1.20 |
$\Xi-\Xi^{*}$ | [1.5-10] | 2.13 | 1.29 |
| [2.0-10] | 1.72 | 1.37 |
Table 2: The values of exponential fit parameters, $C_{5}^{HH^{*}}(0)$ and
$m_{A}$ for axial form factors. The results include the fits from three
region.

To summarize, we have extracted the isovector axial vector form factors of
octet-decuplet hyperons by applying the LCSR. Studied the form factors in this
work have been calculated for the first time in the literature. These form
factors bring information about the shape, size and axial charge of the
baryons. We also obtain the axial charges $C_{5}(0)$ for all transitions by
using the fit to an exponential form. Our axial charge results are
$C_{5}^{\Sigma\Sigma^{*}}(0)$ = - $30.65\pm 10$ and $C_{5}^{\Xi\Xi^{*}}(0)$ =
$2.19\pm 0.5$. We could not find any prediction for axial charge of
$\Lambda-\Sigma^{*}$ transition, because of unstable behaviour of
$C_{5}^{\Lambda\Sigma^{*}}(Q^{2})$. Unfortunately, there is no sufficient
experimental data yet to compare our results within this region. Maybe we
compare our axial charge result with $N-\Delta$ transition results. We
observed that the prediction of quark model results change from
$C_{5}^{N\Delta}(0)$ = $0.81$ to $1.53$ [4], in the case of chiral
perturbation theory is $C_{5}^{N\Delta}(0)$ = $1.16$ [6], in the case of
lattice QCD is $C_{5}^{N\Delta}(0)$ = $0.9\pm 0.02$ [3] and the results from
weak pion production $C_{5}^{N\Delta}(0)$ = $1.08\pm 0.1$ [9]. The
experimental result is $G_{A}^{N\Delta}$ = -
$0.05\pm(0.35)_{stat}\pm(0.34)_{sys}\pm(0.06)_{theory}$ [10]. We see that our
results different from other theoretical approaches and experimental results.

This work has been supported by The Scientific and Technological Research
Council of Turkey (TUBITAK) under project number 110T245. The work of A. O.
and A.K. is also partially supported by the European Union (HadronPhysics2
project Study of strongly interacting matter).

##  4 References

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1: The dependence of the form factors; on the Borel parameter squared
$M^{2}$ for the values of the continuum threshold $s_{0}=2.5~{}GeV^{2}$,
$s_{0}=3.0~{}GeV^{2}$, $s_{0}=3.5~{}GeV^{2}$, $s_{0}=4.0~{}GeV^{2}$,
$s_{0}=4.5~{}GeV^{2}$ and $Q^{2}=2~{}GeV^{2}$ and $4~{}GeV^{2}$ (a) and (b)
for $C_{3}^{\Sigma\Sigma^{*}}$ form factor, (c) and (d) for
$C_{4}^{\Sigma\Sigma^{*}}$ form factor, (e) and (f) for
$C_{5}^{\Sigma\Sigma^{*}}$ form factor and (g) and (h) for
$C_{6}^{\Sigma\Sigma^{*}}$ form factor.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 2: The dependence of the form factors; on the Borel parameter squared
$M^{2}$ for the values of the continuum threshold $s_{0}=2.5~{}GeV^{2}$,
$s_{0}=3.0~{}GeV^{2}$, $s_{0}=3.5~{}GeV^{2}$, $s_{0}=4.0~{}GeV^{2}$,
$s_{0}=4.5~{}GeV^{2}$ and $Q^{2}=2~{}GeV^{2}$, $4~{}GeV^{2}$ (a) and (b) for
$C_{3}^{\Lambda\Sigma^{*}}$ form factor, (c) and (d) for
$C_{4}^{\Lambda\Sigma^{*}}$ form factor, (e) and (f) for
$C_{5}^{\Lambda\Sigma^{*}}$ form factor and (g) and (h) for
$C_{6}^{\Lambda\Sigma^{*}}$ form factor.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3: The dependence of the form factors; on the Borel parameter squared
$M^{2}$ for the values of the continuum threshold $s_{0}=2.5~{}GeV^{2}$,
$s_{0}=3.0~{}GeV^{2}$, $s_{0}=3.5~{}GeV^{2}$, $s_{0}=4.0~{}GeV^{2}$,
$s_{0}=4.5~{}GeV^{2}$ and $Q^{2}=2~{}GeV^{2}$, $4~{}GeV^{2}$ (a) and (b) for
$C_{3}^{\Xi\Xi^{*}}$ form factor (c) and (d) for $C_{4}^{\Xi\Xi^{*}}$ form
factor, (e) and (f) for $C_{5}^{\Xi\Xi^{*}}$ form facto and (g) and (h) for
$C_{6}^{\Xi\Xi^{*}}$ form factor.

(a)

(b)

(c)

(d)

Figure 4:  The convergence of form factors at $Q^{2}=2~{}GeV^{2}$ (a) for
$C_{3}^{\Sigma\Sigma^{*}}$ form factors, (b) for $C_{4}^{\Sigma\Sigma^{*}}$
form factors, (c) for $C_{5}^{\Sigma\Sigma^{*}}$ form factors, (d) for
$C_{6}^{\Sigma\Sigma^{*}}$ form factors. The T = 3,4,5 and 6 are twist-3,
twist-4, twist-5 and twist-6 contributions, respectively.

(a)

(b)

(c)

(d)

Figure 5: The convergence of form factors at $Q^{2}=2~{}GeV^{2}$ (a) for
$C_{3}^{\Lambda\Sigma^{*}}$ form factor, (b) for $C_{4}^{\Lambda\Sigma^{*}}$
form factor, (c) for $C_{5}^{\Lambda\Sigma^{*}}$ form factor and (d) for
$C_{6}^{\Lambda\Sigma^{*}}$ form factor. The T = 3,4,5 and 6 are twist-3,
twist-4, twist-5 and twist-6 contributions, respectively.

(a)

(b)

(c)

(d)

Figure 6: The convergence of form factors at $Q^{2}=2~{}GeV^{2}$ (a) for
$C_{3}^{\Xi\Xi^{*}}$ form factor (b) for $C_{4}^{\Xi\Xi^{*}}$ form factor, (c)
for $C_{5}^{\Xi\Xi^{*}}$ form facto and (d) for $C_{6}^{\Xi\Xi^{*}}$ form
factor. The T = 3,4,5 and 6 are twist-3, twist-4, twist-5 and twist-6
contributions, respectively.

(a)

(b)

(c)

(d)

Figure 7: (a)The dependence of the form factors for the values of the
continuum threshold $s_{0}=2.5~{}GeV^{2}$, $s_{0}=3.0~{}GeV^{2}$,
$s_{0}=3.5~{}GeV^{2}$ and $M^{2}=3~{}GeV^{2}$ (a) for
$C_{3}^{\Sigma\Sigma^{*}}$ form factor, (b) for $C_{4}^{\Sigma\Sigma^{*}}$
form factor, (c) for $C_{5}^{\Sigma\Sigma^{*}}$ form factor and (d) for
$C_{6}^{\Sigma\Sigma^{*}}$ form factor.

(a)

(b)

(c)

(d)

Figure 8: The dependence of the form factors for the values of the continuum
threshold $s_{0}=2.5~{}GeV^{2}$, $s_{0}=3.0~{}GeV^{2}$, $s_{0}=3.5~{}GeV^{2}$
and $M^{2}=3~{}GeV^{2}$ (a) for $C_{3}^{\Lambda\Sigma^{*}}(Q^{2})$ form
factor, (b) for $C_{4}^{\Lambda\Sigma^{*}}(Q^{2})$ form factor, (c) for
$C_{5}^{\Lambda\Sigma^{*}}(Q^{2})$ form factor and (d) for
$C_{6}^{\Lambda\Sigma^{*}}(Q^{2})$ form factor.

(a)

(b)

(c)

(d)

Figure 9:  The dependence of the form factors for the values of the continuum
threshold $s_{0}=3.0~{}GeV^{2}$, $s_{0}=3.5~{}GeV^{2}$, $s_{0}=4.0~{}GeV^{2}$
and $M^{2}=3~{}GeV^{2}$ (a) for $C_{3}^{\Xi\Xi^{*}}$ form factor (b) for
$C_{4}^{\Xi\Xi^{*}}$ form factor, (c) for $C_{5}^{\Xi\Xi^{*}}$ form facto and
(d) for $C_{6}^{\Xi\Xi^{*}}$ form factor.

##  Appendix A Explicit forms of the functions $F_{i}$ for the hyperon
transitions

$\displaystyle F_{1}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(2S_{1}-2S_{2}+2P_{2}-2P_{1}-10V_{1}+10V_{2}$
$\displaystyle+4V_{3}+6V_{4}-2A_{1}+2A_{2}+4A_{3}-6A_{4}+4T_{1}-2T_{2}-2T_{5}$
$\displaystyle-4T_{7}-4T_{8})(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle
F_{2}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(T_{2}-T_{3}-T_{4}+T_{5}+T_{7}+T_{8})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{3}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(T_{2}-T_{3}-T_{4}+T_{5}+T_{7}+T_{8})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{4}$
$\displaystyle=\int_{0}^{1-x_{2}}dx_{1}(5V_{1}-T_{1}+A_{1})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{5}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(-V_{1}+V_{2}+V_{3}+V_{4}+V_{5}-V_{6}$
$\displaystyle+3A_{1}-3A_{2}+3A_{3}+3A_{4}-3A_{5}+3A_{6}+2T_{2}-2T_{3}-2T_{4}$
$\displaystyle+2T_{5}+2T_{7}+2T_{8})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{6}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(A_{1}-A_{2}+A_{3}-T_{1}-T_{2}+2T_{3}+3V_{1}$
$\displaystyle-3V_{2}-3V_{3})(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle
F_{7}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(A_{1}-A_{2}+A_{3}-V_{1}+V_{2}+V_{3})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{8}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(5A_{1}-5A_{2}+5A_{3}+5A_{4}-5_{A}5$
$\displaystyle+5A_{6}+V_{1}-V_{2}-V_{3}-V_{4}-V_{5}+V_{6}-T_{1}-T_{2}+2T_{3}+2T_{4}$
$\displaystyle-T_{5}-T_{6})(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{9}$
$\displaystyle=\int_{0}^{1-x_{2}}dx_{1}(2P_{1}-2S_{1}-14V_{1}+6V_{2}+A_{1}+A_{2}+5A_{3}-2T_{3}$
$\displaystyle-2T_{7})(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{10}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(-V_{1}+V_{2}+V_{3}+V_{4}+V_{5}-V_{6}$
$\displaystyle+3A_{1}-3A_{2}+3A_{3}+3A_{4}-3A_{5}+3A_{6}+2T_{2}-2T_{3}-2T_{4}$
$\displaystyle+2T_{5}+2T_{7}+2T_{8})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{11}$
$\displaystyle=\int_{0}^{1-x_{2}}dx_{1}(-2P_{1}+2S_{1}+4V_{1}-6V_{2}-2A_{2}-4A_{3}+2T_{3}$
$\displaystyle+2T_{7})(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{12}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(T_{2}-T_{3}-T_{4}+T_{5}+T_{7}+T_{8})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{13}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(-T_{1}+T_{2}T_{5}-T_{6}+2T_{7}+2T_{8})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{14}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(2S_{1}-2S_{2}+2P_{2}-2P_{1}-10V_{1}+10V_{2}$
$\displaystyle+4V_{3}+6V_{4}-2A_{1}+2A_{2}+4A_{3}-6A_{4}+4T_{1}-2T_{2}-2T_{5}$
$\displaystyle-4T_{7}-4T_{8})(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle
F_{15}$
$\displaystyle=\int_{0}^{1-x_{2}}dx_{1}(5V_{1}-T_{1}+A_{1})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{16}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(T_{2}-T_{3}-T_{4}+T_{5}+T_{7}+T_{8})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{17}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(-V_{1}+V_{2}+V_{3}+V_{4}+V_{5}-V_{6}$
$\displaystyle+3A_{1}-3A_{2}+3A_{3}+3A_{4}-3A_{5}+3A_{6}+2T_{2}-2T_{3}-2T_{4}$
$\displaystyle+2T_{5}+2T_{7}+2T_{8})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{18}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(-2V_{1}+2V_{2}+2V_{3}-T_{1}-3T_{2}+4T_{3}$
$\displaystyle-2T_{7})(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{19}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(-V_{1}+V_{2}+V_{3}+V_{4}+V_{5}-V_{6}$
$\displaystyle+3A_{1}-3A_{2}+3A_{3})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{20}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(3A_{1}-3A_{2}+3A_{3}-T_{1}-T_{2}+2T_{3}+V_{1}$
$\displaystyle-V_{2}-V_{3})(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{21}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(2T_{1}-T_{3}-T_{4}-T_{7}-T_{8}+A_{3}-A_{4}+S_{2}$
$\displaystyle-
S_{1}+P_{1}-P_{2}-2V_{1}+2V_{2}+V_{3}+V_{4})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{22}$
$\displaystyle=\int_{0}^{1-x_{2}}dx_{1}(V_{1}-T_{1})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{23}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(T_{2}-T_{3}-T_{4}+T_{5}+T_{7}+T_{8})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{24}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(T_{1}-T_{3}-T_{4}+T_{6}-T_{7}-T_{8}-A_{1}$
$\displaystyle+A_{2}-A_{3}-A_{4}+A_{5}-A_{6})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{25}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(2A_{1}-2A_{2}+2A_{3}-4T_{1}+4T_{3}+4T_{7}$
$\displaystyle+2V_{1}-2V_{2}-2V_{3})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{26}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(A_{1}-A_{2}+A_{3}-V_{1}+V_{2}+V_{3})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{27}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(-T_{1}-T_{2}+2T_{3}+T_{4}-T_{5}-T_{6}$
$\displaystyle+2A_{1}+2A_{3}-2A_{4}-2A_{5}-2A_{6})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{28}$
$\displaystyle=\int_{0}^{1-x_{2}}dx_{1}(P_{1}-S_{1}+V_{1}+V_{2}-A_{1}+A_{2}-T_{3}-T_{7})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{29}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(-T_{1}+T_{3}+T_{4}-T_{6}+T_{7}+T_{8}$
$\displaystyle+A_{1}-A_{2}+A_{3}+A_{4}-A_{5}+A_{6})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{30}$
$\displaystyle=\int_{0}^{1-x_{2}}dx_{1}(-P_{1}+S_{1}-2V_{2}-V_{3}-A_{3}+T_{3}+T_{7})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{31}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(-T_{1}+T_{2}+T_{5}-T_{6}+2T_{7}+2T_{8})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{32}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(2T_{1}-T_{3}-T_{4}-T_{7}-T_{8}+A_{3}-A_{4}+S_{2}$
$\displaystyle-
S_{1}+P_{1}-P_{2}-2V_{1}+2V_{2}+V_{3}+V_{4})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{33}$
$\displaystyle=\int_{0}^{1-x_{2}}dx_{1}(V_{1}-T_{1})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{34}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(T_{1}-T_{3}-T_{4}+T_{6}-T_{7}-T_{8}-A_{1}$
$\displaystyle+A_{2}-A_{3}-A_{4}+A_{5}-A_{6})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{35}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(T_{2}-T_{3}-T_{4}+T_{5}+T_{7}+T_{8})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{36}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(A_{1}-A_{2}+A_{3}-V_{1}+V_{2}+V_{3})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$ $\displaystyle F_{37}$
$\displaystyle=\int_{0}^{\beta}d\alpha\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(T_{1}-T_{3}-T_{4}+T_{6}-T_{7}-T_{8}$
$\displaystyle-
A_{1}+A_{2}-A_{3}-A_{4}+A_{5}-A_{6})(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle F_{38}$
$\displaystyle=\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}(A_{1}-A_{2}+A_{3}-T_{1}+T_{3}+T_{7})$
$\displaystyle(x_{1},x_{2},1-x_{1}-x_{2})$


Suppose $\omega=Z(P_{1})/Z(P_{0})$ lies in $(0,+\infty)$, which is the
interval of the first kind with $n=0$. A type A stability condition with
central charge $Z$ must be standard, which is determined uniquely by $\omega$,
as explained before.

Suppose $\omega=Z(P_{1})/Z(P_{0})$ lies in $(-1/2,0)$, which is the interval
of the first kind with $n=1$. Consider the central charge
$Z^{\prime}=Z\circ\sigma_{1}$. Note that $\sigma_{1}^{-1}$ takes stability
conditions of type A to those of type B. Also, note that
$\omega^{\prime}=Z^{\prime}(P_{0})/Z^{\prime}(P_{1})$ lies in the interval
$(0,+\infty)$. The previous argument applied to type B conditions shows that
there is a unique $\tau^{\prime}$ of type B with central charge $Z^{\prime}$.
Then $\tau=\sigma_{1}\tau^{\prime}$ is the unique stability condition of type
A with central charge $Z$.

Suppose $\omega=Z(P_{1})/Z(P_{0})$ lies in
$\left(-\frac{n}{n+1},-\frac{n-1}{n}\right)$ for $n\geq 2$, which are the rest
of the intervals of the first kind. We construct the corresponding stability
condition using the action of $\gamma=\sigma_{1}\sigma_{0}$. Note that
$\gamma$ preserves the set of stability conditions of type A. Let $m=\lfloor
n/2\rfloor$ and set $Z^{\prime}=Z\circ\gamma^{m}$. Then
$\omega^{\prime}=Z^{\prime}(P_{1})/Z^{\prime}(P_{0})$ lies in the interval
$(-1/2,0)$ or $(0,+\infty)$, depending on the parity of $n$. In either case,
the previous argument shows that there is a unique stability condition
$\tau^{\prime}$ of type A with central charge $Z^{\prime}$. Then
$\tau=\gamma^{m}\tau^{\prime}$ is the unique stability condition of type A
with central charge $Z$.

For $\omega=Z(P_{1})/Z(P_{0})$ in the intervals of the second kind, we use a
similar argument with $Z^{\prime}=Z\circ\gamma^{-m}$. We omit the details. ∎

Let $\overline{\Delta}$ be a closed disk, considered as the one-point
compactification of the closed upper half plane. Consider the map

$\delta\colon\Lambda\to\overline{\Delta}$

defined as follows. Let $\tau$ be a point of $\Lambda$ with central charge
$Z$. Set

$\delta_{\tau}=\frac{Z(P_{0})}{Z(P_{0})+Z(P_{1})}.$

By 6.13, the map $\delta$ is injective and a homeomorphism onto its image. It
is easily verified that the image of $\delta$ is

$\delta(\Lambda)=\overline{\Delta}\setminus(\mathbb{Z}\cup\\{\infty\\}).$

Figure 11 shows a sketch of $\Lambda$ as a subset of the closed disk
$\overline{\Delta}$.

$0$$-1$$-2$$-3$$-4$$\cdot$$\cdot$$\cdot$$1$$2$$3$$4$$\cdot$$\cdot$$\cdot$$\infty$
Figure 11. The set $\Lambda$ of stability conditions of type A for the CY2
category associated to $\widehat{A}_{1}$. The stability conditions are
labelled by the ratio $\frac{Z(P_{0})}{Z(P_{0})+Z(P_{1})}$. The interior of
the disk represents the upper half plane.

###  6.7. The closure

We use the embedding
$\Lambda\stackrel{{\scriptstyle\delta}}{{\hookrightarrow}}\overline{\Delta}$
to prove the following proposition.

######  Proposition 6.14 (Closure of $\Lambda$).

The closure of $\Lambda$ in $\mathbb{P}(\mathbb{R}^{\mathbf{S}})$ is precisely

$\overline{\Lambda}=\Lambda\cup\\{\operatorname{\overline{hom}}(P_{i})\mid
i\in\mathbb{Z}\\}\cup\\{\operatorname{\overline{hom}}(P_{1})-\operatorname{\overline{hom}}(P_{0})\\}.$

This closure is homeomorphic to $\overline{\Delta}$.

###### Proof.

Via the identification
$\Lambda\xrightarrow{\delta}\overline{\Delta}\setminus(\mathbb{Z}\cup\\{\infty\\})$,
the set $\overline{\Delta}\setminus(\mathbb{Z}\cup\\{\infty\\})$ maps to
$\mathbb{P}(\mathbb{R}^{\mathbf{S}})$ by the mass map. We now show that this
map extends continuously to $\overline{\Delta}$. By 6.12, we can express the
mass functional $m_{\tau}$ of $\tau\in\Lambda$ as an infinite linear
combination of the Gromov coordinates of $\tau$. We now compute the limit of
each Gromov coordinate of $\tau$ as $\delta_{\tau}$ approaches one of the
points in $\mathbb{Z}\cup\\{\infty\\}$, and thereby check that the map
$m_{\tau}$ extends continuously.

Let $Z_{\tau}$ be the central charge of $\tau$. We know that up to a complex
scalar, we have

$Z_{\tau}(P_{0})=1,\quad
Z_{\tau}(P_{1})=\frac{1-\delta_{\tau}}{\delta_{\tau}}.$

We see that

$Z_{\tau}(P_{i})=\frac{|i|(1-\delta_{\tau})+|i-1|\delta_{\tau}}{\delta_{\tau}}.$

First suppose that $\delta_{\tau}$ approaches some $n\in\mathbb{Z}$. In this
case,

$Z_{\tau}(P_{i})\to\frac{|i|(1-n)+|i-1|n}{n}.$

From 6.11, the limit of each Gromov coordinate is

$x_{j}(\tau)\to\begin{cases}0,&j\neq n,\\\ 1,&j=n\end{cases}.$

The sum in (23) obviously converges in the limit. Up to a simultaneous scalar,
the limit of the mass functional is

$m_{\tau}(-)\to\operatorname{\overline{hom}}(P_{n},-).$

Now suppose that $\delta_{\tau}$ approaches $\infty$. In this case,

$Z_{\tau}(P_{j})\to-|i|+|i-1|=\begin{cases}-1,&i>0,\\\ 1,&i\leq 0\end{cases}.$

We compute that

$x_{j}(\tau)\to\begin{cases}0,&i<0\text{ or }i>1,\\\ -1,&i=0,\\\
1,&i=1.\end{cases}$

Again, the sum in (23) obviously converges in the limit. Up to a simultaneous
scalar, the limit of the mass functional is

$m_{\tau}(-)\to\operatorname{\overline{hom}}(P_{1},-)-\operatorname{\overline{hom}}(P_{0},-).$

We have exhibited a factoring

${\Lambda}$${\mathbb{P}(\mathbb{R}^{\mathbf{S}})}$${\overline{\Delta}}$$\scriptstyle{\delta}$$\scriptstyle{m}$$\scriptstyle{d}$

and identified the additional points in the image of $\overline{\Delta}$.
Since $\overline{\Delta}$ is compact, no other points are in the closure. ∎

######  Remark 6.15.

The point $\infty$ is already a limit point of the subset
$\mathbb{Z}\cup\\{\infty\\}\subset\overline{\Delta}$. Therefore the additional
point in the closure, namely the functional
$\operatorname{\overline{hom}}(P_{1})-\operatorname{\overline{hom}}(P_{0})$,
is already a limit point of the set
$\\{\operatorname{\overline{hom}}(P_{i})\mid i\in\mathbb{Z}\\}$.

It is now easy to identify the closure of
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$, using the following contraction
property, analogous to 5.17. Fix a metric on
$\mathbb{P}^{1}(\mathbb{R})=S^{1}$.

######  Proposition 6.16.

Let $\epsilon>0$ and a compact subset $I$ of $\mathbb{P}^{1}(\mathbb{R})$ be
given. Then for all but finitely many elements $g\in G$, the image $g(I)$ has
diameter less than $\epsilon$.

###### Proof.

The action of $G$ on $\mathbb{P}^{1}(\mathbb{R})$ is via the inclusion
$G\subset\operatorname{PSL}_{2}(\mathbb{Z})$. The claim is true for
$\operatorname{PSL}_{2}(\mathbb{Z})$, as shown in the proof of 5.17, and hence
also for $G$. ∎

Let $M$ and $P$ be the images of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$
and $\mathbf{S}$ in $\mathbb{P}^{\mathbf{S}}$, respectively.

######  Proposition 6.17 (Closure of $M$).

The sets $M$ and $P$ are disjoint, and their union is the closure of $M$.

###### Proof.

The proof is entirely analogous to the proof of 5.18. ∎

We conjecture that the union $M\cup P$ is homeomorphic to a closed disk, as in
the $A_{2}$ case, but we do not yet have a complete proof.

##  Appendix A Geodesic filtrations

Our goal in this note is to understand filtrations that can be re-arranged to
the Harder–Narasimhan (HN) filtration. All the objects and morphisms are in a
triangulated category that is equipped with a stability condition. Denote by
$\lfloor-\rfloor$ (resp $\lceil-\rceil$) the bottom (resp. top) sub-quotient
in the HN filtration.

We begin by studying distinguished triangles. Consider a distinguished
triangle

$A\to C\to B\to A[1].$

When is the HN-filtration of $C$ composed of the HN-filtrations of $A$ and
$B$? The following property of the map $B\to A[1]$ gives an answer.

######  Definition A.1 (Rectifiable map).

We say that a map $B\to A[1]$ is _rectifiable_ if for every
$\alpha\in\operatorname{\mathbb{R}}$, the induced map

$B_{>\alpha}\to A_{\leq\alpha}[1]$

vanishes.

######  Example A.2.

  1. (1)

The zero map $B\to A[1]$) is rectifiable.

  2. (2)

If both $B\to A_{1}[1]$ and $B\to A_{2}[1]$ are rectifiable, then so is the
direct sum $B\to(A_{1}\oplus A_{2})[1]$.

  3. (3)

Suppose $\lfloor A\rfloor\geq\lceil B\rceil$. In this case, we say that $A$
and $B$ are _non-overlapping_. Then any map $B\to A[1]$ is rectifiable.
Indeed, either $B_{>\alpha}$ or $A_{\leq\alpha}$ is zero.

######  Proposition A.3.

Suppose $B\to A[1]$ is rectifiable. Then for every $C\to B$ and $A\to D$, the
induced map $C\to D[1]$ is rectifiable.

###### Proof.

The map $C_{>\alpha}\to D_{\leq\alpha}[1]$ factors as

$C_{>\alpha}\to B_{>\alpha}\to A_{\leq\alpha}[1]\to D_{\leq\alpha}[1].$

Since $B\to A[1]$ is rectifiable, the map $B_{>\alpha}\to A_{\leq\alpha}[1]$
vanishes. Hence, the map $C_{>\alpha}\to D_{\leq\alpha}[1]$ vanishes too. ∎

######  Proposition A.4.

Suppose $e\colon B\to A[1]$ is rectifiable. Set
$C=\operatorname{Cone}(e)[-1]$, so that we have the triangle

$A\to C\to B\xrightarrow{+1}.$

Then we have exact triangles

(25) $A_{\leq\alpha}\to C_{\leq\alpha}\to B_{\leq\alpha}\xrightarrow{+1},$

and

(26) $A_{>\alpha}\to C_{>\alpha}\to B_{>\alpha}\xrightarrow{+1},$

in which the connecting maps $B_{\leq\alpha}\to A_{\leq\alpha}[1]$ and
$B_{>\alpha}\to A_{>\alpha}[1]$ are induced from $e$. Furthermore, these
induced maps are also rectifiable.

###### Proof.

We refine the triangle

$A\to C\to B\xrightarrow{+1}$

to the filtration

${0}$${C_{1}}$${C_{2}}$${C_{3}}$${C_{4}=C.}$${A_{>\alpha}}$${A_{\leq\alpha}}$${B_{>\alpha}}$${B_{\leq\alpha}}$$\scriptstyle{+1}$$\scriptstyle{+1}$$\scriptstyle{+1}$$\scriptstyle{+1}$

Since $B\to A[1]$ is rectifiable, the middle map $B_{>\alpha}\to
A_{\leq\alpha}[1]$ vanishes. Hence, we can swap the middle two sub-quotients
to obtain a new filtration

${0}$${C_{1}}$${C^{\prime}_{2}}$${C^{\prime}_{3}}$${C_{4}=C.}$${A_{>\alpha}}$${B_{>\alpha}}$${A_{\leq\alpha}}$${B_{\leq\alpha}}$$\scriptstyle{+1}$$\scriptstyle{+1}$$\scriptstyle{+1}$$\scriptstyle{+1}$

We see that $C_{>\alpha}=C_{2}^{\prime}$ and
$C_{\leq\alpha}=\operatorname{Cone}(C_{2}^{\prime}\to C_{4})$, and these are
extensions of $A_{>\alpha},B_{>\alpha}$ and of
$A_{\leq\alpha},B_{\leq\alpha}$, as desired.

Let us describe the connecting maps of (25). Consider the composite of
$B_{>\alpha}\to B$ and $B\to A[1]$, say $e\colon B_{>\alpha}\to A[1]$. The
composite of $e$ with $A[1]\to A_{\leq\alpha}[1]$ vanishes, by assumption, and
hence $e$ induces the map

$B_{>\alpha}\to A_{>\alpha}[1].$

This is the connecting map of the triangle (25). The connecting map of the
triangle (26) is obtained similarly.

Let us check that the connecting map of (25) is rectifiable. We must check
that for every $\beta$, the map

$B_{(\beta,\alpha]}\to A_{\leq\min(\alpha,\beta)}[1]$

vanishes. We have the diagram

${B_{>\max(\alpha,\beta)}}$${B_{>\beta}}$${B_{(\beta,\alpha]}}$${A_{(\alpha,\beta]}[1]}$${A_{\leq\beta}[1]}$${A_{\leq\min(\alpha,\beta)}[1],}$

where the rows are distinguished triangles. Since $B\to A[1]$ is rectifiable,
the left and the middle vertical maps vanish. Hence, the right vertical map
also vanishes.

The proof that the connecting map in (26) is rectifiable is similar. ∎

######  Proposition A.5.

Let $A\to C\to B\xrightarrow{e}A[1]$ be a distinguished triangle where $e$ is
rectifiable. Then the HN filtration of $C$ is a rearrangement of the
concatenation of the HN filtrations of $A$ and $B$. In particular, we have

$\operatorname{mass}(C)=\operatorname{mass}(A)+\operatorname{mass}(B).$

###### Proof.

We induct on the HN length of $C$. The baes case—$C$ being semi-stable—is
easy. In general, choose an $\alpha$ such that $C_{>\alpha}$ and
$C_{\leq\alpha}$ have smaller HN lengths than $C$. The HN filtration of $C$ is
the concatenation of the HN filtrations of $C_{>\alpha}$ and $C_{\leq\alpha}$.
From the inductive assumption and the triangles (25) and (26) in A.4, we get
that the HN filtration of $C_{>\alpha}$ (resp. $C_{\leq\alpha}$) is a re-
arrangement of the concatenation of the HN filtrations of $A_{>\alpha}$ and
$B_{>\alpha}$ (resp. $A_{\leq\alpha}$ and $B_{\leq\alpha}$). The result
follows. ∎

We now take up filtrations.

######  Definition A.6 (Geodesic filtration).

Consider a filtration

$X_{0}\to X_{1}\to\dots\to X_{n}.$

We inductively define what it means for such a filtration to be _geodesic_.
All filtrations for $n=0$ and $1$ are geodesic. For $n\geq 2$, the filtration
is geodesic if

  1. (1)

the filtration $X_{1}\to\dots\to X_{n}$ is geodesic, and

  2. (2)

the map $\operatorname{Cone}(X_{1}\to X_{n})\to\operatorname{Cone}(X_{0}\to
X_{1})[1]$ is rectifiable.

######  Remark A.7.

Observe that Harder–Narasimhan filtrations are obviously geodesic.

######  Remark A.8.

The definition of a geodesic filtration is “translation invariant.” That is, a
filtration

$X_{0}\to X_{1}\to\dots\to X_{n}$

is geodesic if and only if the filtration

$0=X_{0}^{\prime}\to X_{1}^{\prime}\to\dots\to X_{n}^{\prime}$

is also geodesic, where $X_{i}^{\prime}=\operatorname{Cone}(X_{0}\to X_{i})$.

######  Remark A.9.

We comment on the name “geodesic”. A stability condition on a triangulated
category gives a translation invariant metric in the sense of [15], defined as
follows. We set the length of a morphism $f\colon A\to B$ to be the HN-mass of
$\operatorname{Cone}(f)$. We can view a filtration $X_{0}\dots\to X_{n}$ as a
path in the category from $X_{0}$ to $X_{n}$. By definition, its length is the
sum of the masses of the sub-quotients. Suppose we translate so that the path
begins at $0$, that is, we pass to $0=X^{\prime}_{0}\to\dots\to
X^{\prime}_{n}$, as above. If the result is the HN-filtration, or a re-
arrangement of it, then its length is the minimum possible. We show that the
A.6 implies that this is the case (A.5).

######  Proposition A.10.

Suppose

$X_{0}\to\dots\to X_{n}$

is geodesic. Then for every $i,j$ with $i<j$, the truncation

$X_{i}\to\dots\to X_{j}$

is geodesic

###### Proof.

This is an easy consequence of A.3. ∎

######  Proposition A.11.

Consider a filtration

$X_{0}\to X_{1}\to\dots\to X_{n}.$

The following are equivalent:

  1. (1)

$X_{0}\to X_{1}\to\dots\to X_{n}$ is geodesic.

  2. (2)

for all $i$, both $X_{0}\to\dots\to X_{i}$ and $X_{i}\to\dots\to X_{n}$ are
geodesic, and the connecting map

$\operatorname{Cone}(X_{i},X_{n})\to\operatorname{Cone}(X_{0},X_{i})[1]$

is rectifiable.

  3. (3)

for some $i$, both $X_{0}\to\dots\to X_{i}$ and $X_{i}\to\dots\to X_{n}$ are
geodesic, and the connecting map

$\operatorname{Cone}(X_{i},X_{n})\to\operatorname{Cone}(X_{0},X_{i})[1]$

is rectifiable.

###### Proof.

Let us show that (1) implies (2). We induct on $i$. The case $i=1$ follows
from the definition of a geodesic filtration. Assume $i\geq 2$. By A.10, we
get that both $X_{0}\to\dots\to X_{i}$ and $X_{i}\to\dots\to X_{n}$ are
geodesic. For brevity, set $X(i,j)=\operatorname{Cone}(X_{i}\to X_{j})$. We
have to prove that the map $X(i,n)\to X(0,i)[1]$ is rectifiable, that is the
map

$X(i,n)_{>\alpha}\to X(0,i)_{\leq\alpha}[1]$

vanishes. By A.4, we have the triangle

$X(0,1)_{\leq\alpha}\to X(0,i)_{\leq\alpha}\to X(1,i)_{\leq\alpha}.$

By the inductive hypothesis applied to the geodesic filtration
$X_{1}\to\dots\to X_{n}$ for $(i-1)$, we get that the map

(27) $X({i},n)\to X(1,i)[1]$

is rectifiable. Therefore, the composite of

$X(i,n)_{>\alpha}\to X(0,i)_{\leq\alpha}[1]$

with $X(0,i)_{\leq\alpha}[1]\to X(1,i)_{\leq\alpha}[1]$ already vanishes, and
hence we get an induced map

$X(i,n)_{>\alpha}\to X(0,1)_{\leq\alpha}[1].$

By A.4 applied to the rectifiable map in (27), we have the triangle

$X(1,i)_{>\alpha}\to X(1,n)_{>\alpha}\to X(i,n)_{>\alpha}\xrightarrow{+1}.$

The map $X(1,n)_{>\alpha}\to X(0,1)_{\leq\alpha}[1]$ vanishes since
$X_{0}\to\dots\to X_{n}$ is geodesic. The induced map $X(1,i)_{>\alpha}[1]\to
X(0,1)_{\leq\alpha}[1]$ vanishes for phase reasons. Therefore, the map
$X(i,n)_{>\alpha}\to X(0,1)_{\leq\alpha}[1]$ vanishes, as required.

That (2) implies (3) is a tautology.

Let us show that (3) implies (1). We again induct on $i$. The case $i=1$ is
the definition. Assume $i\geq 2$. By using A.10 and the inductive hypothesis,
we conclude that

$X_{1}\to\dots\to X_{n}$

is geodesic. It remains to show that the map $X(1,n)\to X(0,1)[1]$ is
rectifiable. We have the rectifiable map

$X(1,i)\to X(1,n)\to X(i,n)\xrightarrow{+1},$

which by A.4 gives the exact triangle

$X(1,i)_{>\alpha}\to X(1,n)_{>\alpha}\to X(i,n)_{>\alpha}\xrightarrow{+1}.$

Since $X_{0}\to\dots\to X_{i}$ is geodesic, the map $X(1,i)_{>\alpha}\to
X(0,1)_{\leq\alpha}[1]$ vanishes. We thus get an induced map

$X(i,n)_{>\alpha}\to X(0,1)_{\leq\alpha}[1],$

which we must show vanishes. Since $X_{0}\to\dots\to X_{i}$ is geodesic, we
have the triangle

$X(0,1)_{\leq\alpha}\to X(0,i)_{\leq\alpha}\to
X(1,i)_{\leq\alpha}\xrightarrow{+1}.$

The map $X(i,n)_{>\alpha}\to X(0,i)_{\leq\alpha}[1]$ vanishes since $X(i,n)\to
X(0,i)[1]$ is rectifiable. The induced map $X(i,n)_{>\alpha}\to
X(1,i)_{\leq\alpha}$ vanishes for phase reasons. Therefore, the map
$X(i,n)_{>\alpha}\to X(0,1)_{\leq\alpha}[1]$ vanishes. ∎

######  Proposition A.12.

Suppose

$0=X_{0}\to\dots\to X_{n}=X$

is geodesic. Set $A_{i}=\operatorname{Cone}(X_{i-1}\to X_{i})$. Then the HN
filtration of $X$ is a rearrangement of the concatenation of the HN
filtrations of the $A_{i}$. In particular, we have

$\operatorname{mass}(X)=\sum\operatorname{mass}(A_{i}).$

###### Proof.

Apply A.5 repeatedly. ∎

######  Proposition A.13.

Suppose

$X_{0}\to\dots\to X_{i-1}\to X_{i}\to X_{i+1}\to\dots\to X_{n}$

is geodesic, and the map
$\operatorname{Cone}(X_{i},X_{i+1})\to\operatorname{Cone}(X_{i-1},X_{i})[1]$
is zero; that is

$\operatorname{Cone}(X_{i-1},X_{i+1})=\operatorname{Cone}(X_{i-1},X_{i})\oplus\operatorname{Cone}(X_{i},X_{i+1}).$

Then the flipped filtration

$X_{0}\to\dots\to X_{i-1}\to X^{\prime}_{i}\to X_{i+1}\to\dots\to X_{n}=X,$

where
$X_{i}^{\prime}=\operatorname{Cone}(X_{i+1}\to\operatorname{Cone}(X_{i-1},X_{i}))[-1]$,
is also geodesic.

###### Proof.

We may assume $i-1=0$, and $X_{0}=0$. By A.11 applied with $i=2$, it suffices
to prove that the map

$\operatorname{Cone}(X_{2},X_{n})\to X_{2}[1]$

is rectifiable. Note that this map is unchanged after the swap. But this map
is rectifiable by the hypothesis. ∎

######  Proposition A.14.

Consider a filtration

$0=X_{0}\to\dots\to X_{n}=X.$

Set $A_{i}=\operatorname{Cone}(X_{i-1}\to X_{i})$. Suppose for every $i,j$
with $i<j$, we have

$\operatorname{Hom}(A_{j},A_{i}[1])=0\text{ or }\lfloor A_{i}\rfloor\geq\lceil
A_{j}\rceil.$

Then the filtration is geodesic.

###### Proof.

We induct on $n$. The base case $n=0$ is trivial. Assume $n\geq 1$. By
applying A.13 repeatedly, we may assume that

$\lceil A_{1}\rceil\geq\dots\geq\lceil A_{n}\rceil.$

As before, set $X(i,j)=\operatorname{Cone}(X_{i}\to X_{j})$. Let $i$ be the
largest such that for all $j>i$, we have

$\lfloor A_{1}\rfloor\geq\lceil A_{j}\rceil.$

By the assumption, we have $\operatorname{Hom}(A_{j},A_{1}[1])=0$ for all
$j\leq i$. Therefore, we get $X_{i}=A_{1}\oplus X(1,i)$. By the inductive
assumption and the translation invariance (A.8),

$X_{0}\to\dots\to X_{i}$

and

$X_{i}\to\dots\to X_{n}$

are both a geodesic filtrations with sub-quotients $A_{1},\dots A_{i}$ and
$A_{i+1},\dots,A_{n}$, respectively. By A.11, it suffices to show that the map

$X(i,n)\to X_{i}[1]$

is rectifiable. But $X_{i}=A_{1}\oplus X(1,i)$. By the inductive assumption,
the filtration

$0\to X(1,2)\to\dots\to X(1,n)$

is geodesic, and hence the map $X(i,n)\to X(1,i)[1]$ is rectifiable. The map
$X(i,n)\to A_{1}[1]$ is rectifiable by construction, since $\lfloor
A_{1}\rfloor\geq\lceil X(i,n)\rceil$. Hence, $X(i,n)\to X_{i}[1]=(A_{1}\oplus
X(1,i))[1]$ is rectifiable. The proof is now complete. ∎

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