known by Grojnowski-Lusztig [GL] that $\mathcal{B}$ coincides with
$\mathcal{B}^{\natural}$.

The following result was proved by Varagnolo-Vasserot [VV], and Rouquier [R2],
by using the Ext algebra obtained from Lusztig’s category $\mathcal{Q}_{V}$
related to the geometry of quivers (see Introduction).

######  Theorem 5.2 ([VV], [R2]).

Under the setup in 5.1, the isomorphisms
$\widetilde{\varTheta}_{0}:{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,K_{\operatorname{{\bf gp}}}$ sends
$\mathcal{B}^{\natural}$ to $\mathbf{B}$.

5.3. Since $\mathcal{B}^{\natural}$ is invariant under $*$,
$\mathcal{B}^{\natural}$ corresponds to $\mathbf{B}$ under the isomorphism
$\widetilde{\varTheta}:{}_{\mathbf{A}}\mathbf{U}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,K_{\operatorname{{\bf gp}}}$ in
Proposition 2.9. Since the basis $\mathcal{B}^{\natural}$ is almost
orthonormal, $\mathbf{B}$ is also almost orthonormal in the sense of 1.4,
i.e., for $b,b^{\prime}\in\mathbf{B}$,

(5.3.1) $(b,b^{\prime})\in\begin{cases}1+q\mathbf{Z}[[q]]&\quad\text{ if
$b=b^{\prime}$, }\\\ q\mathbf{Z}[[q]]&\quad\text{ if $b\neq
b^{\prime}$.}\end{cases}$

We want to show that $\underline{\mathbf{B}}$ has a similar property.

######  Proposition 5.4.

Assume that $b\in\underline{\mathbf{B}}$. Then we have $\mathbf{D}b=b$, and

(5.4.1) $(b,b^{\prime})\in\begin{cases}1+q\mathbf{Z}[[q]]&\quad\text{ if
$b=b^{\prime}$, }\\\ q\mathbf{Z}[[q]]&\quad\text{ if $b\neq b^{\prime}$.
}\end{cases}$

###### Proof.

Take $b,b^{\prime}\in\underline{\mathbf{B}}$, and write it as
$b=[P],b^{\prime}=[Q]$ for
$P=(P_{0},\phi),Q=(Q_{0},\phi^{\prime})\in\mathscr{P}_{\beta}$. The relation
$\mathbf{D}b=b$ is clear from the definition. We show (5.4.1). By Lemma 4.6,
$P_{0}$ is an indecomposable projective $R(\beta)$-module. Since
$(P_{0},\phi)$ is self-dual, $P_{0}$ is also self-dual. Thus $[P_{0}]$
corresponds to some element in $\mathbf{B}$. Hence by (5.3.1), $P_{0}$
satisfies the relation $([P_{0}],[P_{0}])\in 1+q\mathbf{Z}[[q]]$. We compare
the inner product (2.5.3) for $K_{\operatorname{{\bf gp}}}(\beta)$ and the
inner product (3.5.4) for $K(\mathscr{P}_{\beta})$. We show that

(5.4.2)
$([P],[Q])\in\begin{cases}1+q\mathbf{Z}[\zeta_{\mathbf{n}}][[q]]&\quad\text{
if $[P]=[Q]$, }\\\ q\mathbf{Z}[\zeta_{\mathbf{n}}][[q]]&\quad\text{ if
$[P]\neq[Q]$. }\end{cases}$

Since $\mathbf{D}P_{0}\simeq P_{0}$, by using the relation (3.5.2) (it also
holds if $M$ is projective), in the definition of the inner product on
$K_{\operatorname{{\bf gp}}}(\beta)$ and on $K(\mathscr{P}_{\beta})$, one can
replace the tensor product $P_{0}^{\psi}\otimes_{R(\beta)}P_{0}$ by the Hom
space $\operatorname{Hom}_{R(\beta)}(P_{0},P_{0})$. Then (5.3.1) shows that
$\dim_{\mathbf{k}}\operatorname{Hom}_{R(\beta)}(P_{0},P_{0})_{0}=1$, and
$\dim_{\mathbf{k}}\operatorname{Hom}_{R(\beta)}(P_{0},q^{k}P_{0})_{0}=0$ for
$k>0$. In particular, the grading preserving homomorphisms $P_{0}\to P_{0}$
are only the scalar multiplications. The action of $\phi\otimes\phi$ on
$(P_{0}^{\psi}\otimes_{R(\beta)}P_{0})_{0}$ corresponds to the action
$f\mapsto\phi^{-1}\circ f\circ\phi$ for
$\operatorname{Hom}_{\mathbf{R}}(P_{0},P_{0})_{0}$, hence it induces the
identity map on $\operatorname{Hom}_{R(\beta)}(P_{0},P_{0})_{0}$. This shows
the first formula in (5.4.2). The second formula is obtained similarly from
the relation $\dim\operatorname{Hom}_{R(\beta)}(P_{0},q^{k}Q_{0})=0$ for
$k>0$.

Now by Corollary 4.15 (i), we have an isometry
$\gamma:{}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}K(\mathscr{P})$. Hence
$([P],[Q])\in\mathbf{Q}((q))$. Combined with (5.4.2), we obtain (5.4.1). The
proposition is proved. ∎

######  Corollary 5.5.

Let $\underline{\mathcal{B}}$ be the lower global basis of
${}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$, and
$\underline{\mathcal{B}}^{*}$ the upper global basis of
${}_{\mathbf{A}}(\underline{\mathbf{U}}_{q}^{-})^{*}$. Then
$\gamma^{-1}(\underline{\mathbf{B}})$ coincides with $\underline{\mathcal{B}}$
or with $\underline{\mathcal{B}}^{\natural}$, up to sign, and
$\gamma^{*}(\underline{\mathbf{B}}^{*})$ coincides with
$\underline{\mathcal{B}}^{*}$, up to sign.

###### Proof.

By Proposition 5.4, $\gamma^{-1}(\underline{\mathbf{B}})$ gives an almost
orthonormal basis in ${}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$. Since
$\gamma^{-1}(\underline{\mathbf{B}})\subset\widetilde{\mathcal{B}}$, we have
$\widetilde{\mathcal{B}}=\gamma^{-1}(\underline{\mathbf{B}})\sqcup-\gamma^{-1}(\underline{\mathbf{B}})$.
Hence $\gamma^{-1}(\underline{\mathbf{B}})$ coincides with
$\underline{\mathcal{B}}$ or with $\underline{\mathcal{B}}^{\natural}$, up to
sign. The second statement follows easily from this. ∎

5.6. Take $j\in J$. Consider the operator
$E_{j}:\mathscr{L}_{\beta}\to\mathscr{L}_{\beta-\alpha_{j}}$, for $\beta\in
Q_{+}^{\sigma}$ such that $\beta-\alpha_{j}\in Q_{+}^{\sigma}$, defined as in
(3.9.3) ($E_{j}=E_{j}^{(n)}$ with $n=1$). $E_{j}$ induces an operator
$e_{j}^{\prime}:K(\mathscr{L}_{\beta})\to K(\mathscr{L}_{\beta-\alpha_{j}})$.
For $\beta\in Q_{+}^{\sigma}$, we define a functor
$F_{j}^{\prime}:\mathscr{L}_{\beta}\to\mathscr{L}_{\beta+\alpha_{j}}$ by

$F_{j}^{\prime}:M\mapsto M\circ L_{j}.$

Since the convolution product is an exact functor, $F_{j}^{\prime}$ is an
exact functor. Thus it induces an operator $f_{j}:K(\mathscr{L}_{\beta})\to
K(\mathscr{L}_{\beta+\alpha_{j}})$.

For a simple object $M\in\mathscr{L}_{\beta}$, we set

(5.6.1) $\varepsilon_{j}(M)=\max\\{k\geq 0\mid E_{j}^{k}M\neq 0\\}.$

We define crystal operators
$\widetilde{\mathbf{E}}_{j}M\in\mathscr{L}_{\beta-\alpha_{j}}$,
$\widetilde{\mathbf{F}}_{j}M\in\mathscr{L}_{\beta+\alpha_{j}}$ by

(5.6.2) $\displaystyle\widetilde{\mathbf{E}}_{j}M$
$\displaystyle=q_{j}^{1-\varepsilon_{j}(M)}\operatorname{soc}E_{j}M,$
$\displaystyle\widetilde{\mathbf{F}}_{j}M$
$\displaystyle=q_{j}^{\varepsilon_{j}(M)}\operatorname{hd}F^{\prime}_{j}M.$

If we use the restriction functor defined in 3.6, $E_{j}M$ is written as

$E_{j}M\simeq\operatorname{Hom}_{R(\alpha_{j})}(L(j),\operatorname{Res}_{\beta-\alpha_{j},\alpha_{j}}M).$

Thus $\widetilde{\mathbf{E}}_{j}M$ (resp. $\widetilde{\mathbf{F}}_{j}M$)
coincides with $\widetilde{e}_{j}^{*}M$ (resp. $\widetilde{f}_{j}^{*}M$) given
in [M, 9], where

$\displaystyle\widetilde{e}^{*}_{j}M$
$\displaystyle=q_{j}^{1-\varepsilon_{j}(M)}\operatorname{soc}\operatorname{Hom}_{R(\alpha_{j})}(L(j),\operatorname{Res}_{\beta-\alpha_{j},\alpha_{j}}M),$
$\displaystyle\widetilde{f}^{*}_{j}M$
$\displaystyle=q_{j}^{\varepsilon_{j}(M)}\operatorname{hd}(M\circ L(j)).$

(Note that we use $\varepsilon_{j}(M)$ as in (5.6.1). This corresponds to
$\varepsilon_{j}^{*}(M)$ in the notation of [M].)

The following result is immediate from [M, Lemma 8.3], though this formula is
not used in later discussions.

######  Lemma 5.7.

Let $M\in\mathscr{L}_{\beta}$ be a simple object. Then
$\widetilde{\mathbf{F}}_{j}M$ is a simple object such that
$\varepsilon_{j}(\widetilde{\mathbf{F}}_{j}M)=\varepsilon_{j}(M)+1$, and
$[F_{j}^{\prime}M]$ is written in the Grothendieck group
$K(\mathscr{L}_{\beta+\alpha_{j}})$,

$[F_{j}^{\prime}M]=q_{j}^{-\varepsilon_{j}(M)}[\widetilde{\mathbf{F}}_{j}M]+\sum_{k}[L_{k}],$

where $L_{k}\in\mathscr{L}_{\beta+\alpha_{j}}$ are simple objects such that
$\varepsilon_{j}(L_{k})<\varepsilon_{j}(M)+1$.

In order to consider the action of $E_{j}$, we need a lemma, which is obtained
from Lemma 8.1 and Lemma 8.2 in [M].

######  Lemma 5.8 ([M]).

  1. (i)

Let $N$ be a simple object in $\mathscr{C}_{\beta}$ such that
$\varepsilon_{j}(N)=0$. Set $M=N\circ L(j)^{(m)}$ for any $m\geq 1$. Then
$\operatorname{hd}M$ is irreducible, and $\varepsilon_{j}(M)=m$.

  2. (ii)

Let $N$ be a simple object in $\mathscr{C}_{\beta}$ such that
$\varepsilon_{j}(N)=m$. Then there exists a simple object $X$ in
$\mathscr{C}_{\beta-m\alpha_{j}}$ with $\varepsilon_{j}(X)=0$ such that

$\operatorname{Res}_{\beta-m\alpha_{j},m\alpha_{j}}(N)\simeq X\otimes
L(j)^{(m)}.$

We show the following.

######  Proposition 5.9.

Let $M\in\mathscr{L}_{\beta}$ be a simple object such that
$\varepsilon_{j}(M)=m>0$. Then $\widetilde{\mathbf{E}}_{j}M$ is a self-dual
simple object in $\mathscr{L}_{\beta-\alpha_{j}}$, and in the Grothendieck
group $K(\mathscr{L})$, $[E_{j}M]$ is written as

(5.9.1) $[E_{j}M]=G(q)[\widetilde{\mathbf{E}}_{j}M]+\sum_{1\leq k\leq
p}[L_{k}],$

where $G(q)\in\mathbf{N}[q,q^{-1}]$,
$\varepsilon_{j}(\widetilde{\mathbf{E}}_{j}M)=m-1$, and
$L_{k}\in\mathscr{L}_{\beta-\alpha_{j}}$ are simple objects such that
$\varepsilon_{j}(L_{k})<m-1$.

###### Proof.

The proof is done by a similar argument as in the proof of [M, Lemma 9.4]. We
give an outline of the proof below. It is known by [M, Lemma 9.4],
$\widetilde{\mathbf{E}}_{j}M$ is a self-dual simple object in
$\mathscr{L}_{\beta-\alpha_{j}}$. We fix a composition series of $E_{j}M$, and
let $N$ be a simple object such that $\varepsilon_{j}(N)=m-1$, appearing as a
composition factor. We show

(5.9.2) $N$ is isomorphic to $\widetilde{\mathbf{E}}_{j}M$, up to degree
shift.

Note that
$E_{j}M\simeq\operatorname{Hom}_{R(\alpha_{j})}(L(j),\operatorname{Res}_{\beta-\alpha_{j},\alpha_{j}}M)$.
Let $\widetilde{N}$ be a subobject of $E_{j}M$ such that $N$ is a simple
quotient of $\widetilde{N}$. By the adjunction isomorphism, we have

$\operatorname{Hom}_{\mathscr{C}_{\beta}}(\widetilde{N}\circ
L(j),M)\simeq\operatorname{Hom}_{\mathscr{C}_{\beta-\alpha_{j}}}(\widetilde{N},\operatorname{Hom}_{R(\alpha_{j})}(L(j),\operatorname{Res}_{\beta-\alpha_{j},\alpha_{j}}M)).$

Since the right hand side of this equality is non-zero, $M$ is a simple
quotient of $\widetilde{N}\circ L(j)$. By Lemma 5.8 (ii), there exists a
simple $X$ such that

(5.9.3) $\operatorname{Res}_{\beta-m\alpha_{j},m\alpha_{j}}(M)\simeq X\otimes
L(j)^{(m)}.$

Since the restriction functor is exact, we have a surjective map

$\operatorname{Res}_{\beta-m\alpha_{j},m\alpha_{j}}(\widetilde{N}\circ
L(j))\to X\otimes L(j)^{(m)}.$

We consider the Mackey filtration (see [M, Thm. 4.5]) for

$\operatorname{Res}_{\beta-m\alpha_{j},m\alpha_{j}}(\widetilde{N}\circ
L(j))=\operatorname{Res}_{\beta-m\alpha_{j},m\alpha_{j}}\circ\operatorname{Ind}_{\beta-\alpha_{j},\alpha_{j}}(\widetilde{N}\otimes
L(j))$

Since $\varepsilon_{j}(N)=m-1$, this Mackey filtration contains a non-zero
factor which involves
$\operatorname{Res}_{\beta-m\alpha_{j},(m-1)\alpha_{j}}N$, up to degree shift.
By Lemma 5.8 (ii), there exists a simple object $Y$ such that

(5.9.4) $\operatorname{Res}_{\beta-m\alpha_{j},(m-1)\alpha_{j}}N\simeq
Y\otimes L(j)^{(m-1)}.$

We have a surjective homomorphism $Y\otimes L(j)^{(m)}\to X\otimes L(j)^{(m)}$
with degree shift. Since $X,Y$ are simple, we have $X\simeq Y$, up to degree
shift.

By considering the adjunction isomorphism in (5.9.4), we have a non-zero
homomorphism $Y\circ L(j)^{(m-1)}\to N$. Since $\varepsilon_{j}(N)=m-1$, we
have $\varepsilon_{j}(Y)=0$. By Lemma 5.8 (i), $\operatorname{hd}(Y\circ
L(j)^{(m-1)})$ is irreducible. It follows that
$N\simeq\operatorname{hd}(Y\circ L(j)^{(m-1)})$. By applying this argument to
$\widetilde{\mathbf{E}}_{j}M$ (the original setup in [M]), we see that
$\widetilde{\mathbf{E}}_{j}M\simeq\operatorname{hd}(X\circ L(j)^{(m-1)})$.
Hence $N\simeq\widetilde{\mathbf{E}}_{j}M$, up to degree shift, and (5.9.2)
holds.

Since $\varepsilon_{j}(M)=m$, all the composition factors $L$ of $E_{j}M$
satisfy the relation $\varepsilon_{j}(L)\leq m-1$. The proposition is proved.
∎

5.10. Let $\underline{\mathbf{B}}=\bigsqcup_{\beta\in
Q_{+}^{\sigma}}\underline{\mathbf{B}}_{\beta}$ be the basis of
$K(\mathscr{P})$, and $\underline{\mathbf{B}}^{*}=\bigcup_{\beta\in
Q_{+}^{\sigma}}\underline{\mathbf{B}}_{\beta}^{*}$ be the basis of
$K(\mathscr{L})$ defined as before. The functors $\widetilde{\mathbf{E}}_{j}$
and $\widetilde{\mathbf{F}}_{j}$ induce operators on $K(\mathscr{L})$, which
we denote by $\widetilde{\mathbf{e}}_{j}$ and $\widetilde{\mathbf{f}}_{j}$.
The following result was proved by Lemma 9.1, Lemma 9.3 and Lemma 9.4,
together with the definition of $\underline{\mathbf{B}}^{*}$ in [M].

######  Proposition 5.11.

  1. (i)

$\widetilde{\mathbf{f}}_{j}$ sends $\underline{\mathbf{B}}^{*}_{\beta}$ to
$\underline{\mathbf{B}}^{*}_{\beta+\alpha_{j}}$.

  2. (ii)

$\widetilde{\mathbf{e}}_{j}$ sends $\underline{\mathbf{B}}^{*}_{\beta}$ to
$\underline{\mathbf{B}}^{*}_{\beta-\alpha_{j}}\cup\\{0\\}$.

  3. (iii)

For $b,b^{\prime}\in\underline{\mathbf{B}}^{*}$,
$b=\widetilde{\mathbf{f}}_{j}b^{\prime}$ if and only if
$b^{\prime}=\widetilde{\mathbf{e}}_{j}b$.

The following result is a generalization of Theorem 5.2.

######  Theorem 5.12.

Let $\gamma:{}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}K(\mathscr{P})$,
$\gamma^{*}:{}_{\mathbf{A}}K(\mathscr{L})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}(\underline{\mathbf{U}}_{q}^{-})^{*}$
be the isomorphisms as in Corollary 4.15.

  1. (i)

$\gamma^{*}$ sends $\underline{\mathbf{B}}^{*}$ to
$\underline{\mathcal{B}}^{*}$.

  2. (ii)

$\gamma$ sends $\underline{\mathcal{B}}$ to $\underline{\mathbf{B}}$.

###### Proof.

We show (i). We prove, by induction on $|\beta|$, that

(5.12.1)
$\gamma^{*}(\underline{\mathbf{B}}^{*}_{\beta})=\underline{\mathcal{B}}^{*}_{-\beta}\qquad(\beta\in
Q_{+}^{\sigma}).$

Since $\gamma^{*}$ maps
$[1]=[\mathbf{k},\operatorname{id}]\in{}_{\mathbf{A}}K(\mathscr{L}_{0})$ to
$1^{*}\in{}_{\mathbf{A}}(\underline{\mathbf{U}}_{q}^{-})^{*}_{0}$, we have
$\gamma^{*}(\underline{\mathbf{B}}^{*}_{0})=\underline{\mathcal{B}}^{*}_{0}$.
Hence (5.12.1) holds for $\beta=0$. Take $0\neq\beta\in Q_{+}^{\sigma}$, and
assume that (5.12.1) holds for $\beta^{\prime}\in Q_{+}^{\sigma}$ such that
$|\beta^{\prime}|<|\beta|$. Let $b\in\underline{\mathbf{B}}^{*}_{\beta}$.
There exists $j\in J$ such that $e_{j}^{\prime}b\neq 0$. Then by Proposition
5.9 and by Proposition 5.11, $e_{j}^{\prime}b$ can be written as

(5.12.2)
$e_{j}^{\prime}b=G(q)\widetilde{\mathbf{e}}_{j}b+\sum_{b^{\prime}}a_{bb^{\prime}}b^{\prime},\qquad(G(q)\in\mathbf{N}[q,q^{-1}],a_{bb^{\prime}}\in\mathbf{A}),$

where $\varepsilon_{j}(\widetilde{\mathbf{e}}_{j}b)=\varepsilon_{j}(b)-1$, and
$b^{\prime}$ runs over the elements in
$\underline{\mathbf{B}}^{*}_{\beta-\alpha_{j}}$ such that
$\varepsilon_{j}(b^{\prime})<\varepsilon_{j}(b)-1$. Let $b_{0}=\gamma^{*}(b)$.
By Corollary 5.5, $b_{0}\in\underline{\mathcal{B}}^{*}$, up to sign. Since the
map $\gamma^{*}$ commutes with the action of $e_{j}^{\prime}$, we have

(5.12.3)
$e_{j}^{\prime}(b_{0})=G(q)\gamma^{*}(\widetilde{\mathbf{e}}_{j}b)+\sum_{b^{\prime}}a_{bb^{\prime}}\gamma^{*}(b^{\prime}).$

Here by induction,
$\gamma^{*}(\widetilde{\mathbf{e}}_{j}b),\gamma^{*}(b^{\prime})\in\underline{\mathcal{B}}^{*}_{-\beta+\alpha_{j}}$.
Moreover, since $\gamma^{*}$ commutes with $e_{j}^{\prime}$,
$\varepsilon_{j}(\gamma^{*}(\widetilde{\mathbf{e}}_{j}b))=\varepsilon_{j}(b_{0})-1$,
and $\varepsilon_{j}(\gamma^{*}(b^{\prime}))<\varepsilon_{j}(b_{0})-1$.
Comparing (5.12.3) with Theorem 1.10 (iii) (applied for
${}_{\mathbf{A}}(\underline{\mathbf{U}}_{q}^{-})^{*}$), we see that
$b_{0}=\gamma^{*}(b)\in\underline{\mathcal{B}}^{*}_{-\beta}$. This proves
(5.12.1). Hence (i) holds. (ii) follows from (i). ∎

Remarks 5.13. (i) The quantum group of arbitrary type is obtained as
$\underline{\mathbf{U}}_{q}^{-}$ from some $\mathbf{U}_{q}^{-}$ of symmetric
type, through an admissible automorphism $\sigma$. Hence our result shows that
the global crystal basis of the quantum group of arbitrary type is obtained
from the KLR algebras through foldings.

(ii) In general, the natural basis $\mathbf{B}$ of $K_{\operatorname{{\bf
gp}}}$ has the positivity property, i.e., for $b,b^{\prime}\in\mathbf{B}$,
$bb^{\prime}$ is a linear combination of the basis with coefficients in
$\mathbf{N}[q,q^{-1}]$. This property does not hold for the global crystal
basis $\mathcal{B}$ of non-symmetric type. Note that the basis
$\underline{\mathbf{B}}$ of $K(\mathscr{P})$ not necessarily has the
positivity property.

5.14. By Theorem 5.2, the map $\widetilde{\varTheta}_{0}$ sends the canonical
basis $\mathcal{B}^{\natural}$ to $\mathbf{B}$. By applying Theorem 5.12 for
the case where $\sigma=\operatorname{id}$, we rediscover the result of [GL]
that $\mathcal{B}=\mathcal{B}^{\natural}$. In the following, we show that this
also holds in the non-symmetric case.

For $b\in\underline{\mathbf{B}},b^{\prime}\in\underline{\mathbf{B}}^{*}$ and
$j\in J$, we set

$\displaystyle\varepsilon_{j}(b)$ $\displaystyle=\max\\{k\geq 0\mid b\in
f_{j}^{k}K(\mathscr{P})\\},$ $\displaystyle\varepsilon_{j}(b^{\prime})$
$\displaystyle=\max\\{k\geq 0\mid e_{j}^{\prime k}b^{\prime}\neq 0\\}.$

Note that the definition $\varepsilon_{j}(b^{\prime})$ is compatible with the
definition $\varepsilon_{j}(M)$ in (5.6.1).

Take $b\in\underline{\mathbf{B}}$, and let
$b_{*}\in\underline{\mathbf{B}}^{*}$ be the basis dual to $b$. Then we have

(5.14.1) $\varepsilon_{j}(b)=\varepsilon_{j}(b_{*}).$

In fact, assume that $b\in f_{j}^{n}K(\mathscr{P})$. Then
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}f_{j}^{n}K(\mathscr{P}),b_{*}\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}\neq
0$, and there exists $b^{\prime}\in\underline{\mathbf{B}}$ such that
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}f_{j}^{n}b^{\prime},b_{*}\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}=\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}b^{\prime},e_{j}^{\prime
n}b_{*}\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}\neq
0$, hence $e_{j}^{\prime n}b_{*}\neq 0$. On the contrary, assume that
$e_{j}^{\prime n}b_{*}\neq 0$. Then there exists
$b^{\prime}\in\underline{\mathbf{B}}$ such that
$\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}b^{\prime},e_{j}^{\prime
n}b_{*}\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}=\operatorname{\operatorname{\langle}\\!\\!\operatorname{\langle}\,}f_{j}^{n}b^{\prime},b_{*}\operatorname{\operatorname{\rangle}\\!\\!\operatorname{\rangle}}\neq
0$. it is known ([L2, Thm. 14.3.2]) that a subset of $\mathcal{B}^{\natural}$
gives a basis of $f_{j}^{n}\underline{\mathbf{U}}_{q}^{-}$ for each $n\geq 0$.
Then by Corollary 5.5, a subset of $\underline{\mathbf{B}}$ gives a basis of
$f_{j}^{n}K(\mathscr{P})$ for each $n\geq 0$. It follows that $b\in
f_{j}^{n}K(\mathscr{P})$. Thus (5.14.1) holds.

######  Theorem 5.15.

$\gamma^{-1}$ sends $\underline{\mathbf{B}}$ to
$\underline{\mathcal{B}}^{\natural}$. Hence the canonical basis
$\underline{\mathcal{B}}^{\natural}$ and the global crystal basis
$\underline{\mathcal{B}}$ of $\underline{\mathbf{U}}_{q}^{-}$ coincide with.

###### Proof.

The proof is done just by applying some properties of $K(\mathscr{P})$ and
$K(\mathscr{L})$, together with Theorem 5.12. Below, we give an outline of the
proof.

By Proposition 5.11, Theorem 5.12, together with Theorem 1.10 (iii), we have,
for $b\in\underline{\mathbf{B}}^{*}$ such that $\varepsilon_{j}(b)=m>0$,

(5.15.1)
$e_{j}^{\prime}b=[\varepsilon_{j}(b)]_{j}\widetilde{\mathbf{e}}_{j}b+\sum_{b^{\prime}\in\underline{\mathbf{B}}^{*}}a_{bb^{\prime}}b^{\prime},\qquad(a_{bb^{\prime}}\in\mathbf{A}),$

where $\varepsilon_{j}(\widetilde{\mathbf{e}}_{j}b)=\varepsilon_{j}(b)-1$, and
$a_{bb^{\prime}}=0$ unless $\varepsilon_{j}(b^{\prime})<\varepsilon_{j}(b)-1$.

We define
$\widetilde{\mathbf{f}}_{j}:\underline{\mathbf{B}}_{\beta-\alpha_{j}}\to\underline{\mathbf{B}}_{\beta}$
as the transpose of the map
$\widetilde{\mathbf{e}}_{j}:\underline{\mathbf{B}}^{*}_{\beta}\to\underline{\mathbf{B}}^{*}_{\beta-\alpha_{j}}$,
i.e., $\widetilde{\mathbf{f}}_{j}(b)=b^{\prime}$ if and only if
$\widetilde{\mathbf{e}}_{j}((b^{\prime})_{*})=b_{*}$, where
$b_{*},(b^{\prime})_{*}$ are dual of $b,b^{\prime}$, respectively.
$\widetilde{\mathbf{f}}_{j}$ gives a bijection
$\underline{\mathbf{B}}_{\beta-\alpha_{j}}\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,\underline{\mathbf{B}}_{\beta}$. Since
$\underline{\mathbf{B}}$ is a dual basis of $\underline{\mathbf{B}}^{*}$,
(5.15.1) and (5.14.1) implies, for $b\in\underline{\mathbf{B}}$, that

(5.15.2)
$f_{j}b=[\varepsilon_{j}(b)+1]_{j}\widetilde{\mathbf{f}}_{j}b+\sum_{b^{\prime}\in\underline{\mathbf{B}}}c_{bb^{\prime}}b^{\prime},\qquad(c_{bb^{\prime}}\in\mathbf{A}),$

where $\varepsilon_{j}(\widetilde{\mathbf{f}}_{j}b)=\varepsilon_{j}(b)+1$, and
$c_{bb^{\prime}}=0$ unless $\varepsilon_{j}(b^{\prime})>\varepsilon_{j}(b)+1$.
By (5.15.2), we have the following.

(5.15.3) Assume that $b\in\underline{\mathbf{B}}$ is such that
$\varepsilon_{j}(b)=0$. Then for any $n\geq 1$, there exists
$b_{1}\in\underline{\mathbf{B}}$ such that

$f_{j}^{(n)}b=b_{1}+\sum_{b^{\prime}\in\underline{\mathbf{B}}}c_{bb^{\prime}}b^{\prime},\qquad(c_{bb^{\prime}}\in\mathbf{A}),$

where $\varepsilon_{j}(b_{1})=n$, and $c_{bb^{\prime}}=0$ unless
$\varepsilon_{j}(b^{\prime})>n$. Note that $c_{bb^{\prime}}\in\mathbf{A}$
follows from Corollary 4.15.

For each $j\in J,n\in\mathbf{N}$, set
$\underline{\mathbf{B}}_{n;j}=\\{b\in\underline{\mathbf{B}}\mid\varepsilon_{j}(b)=n\\}$.
We define a map
$\pi_{n;j}:\underline{\mathbf{B}}_{0;j}\to\underline{\mathbf{B}}_{n;j}$ by
$b\mapsto b_{1}$ in (5.15.3). $\pi_{n;j}$ is a bijection since
$\widetilde{\mathbf{f}}_{j}$ is a bijection.

By using Proposition 5.11, we have

$\bigcap_{j\in
J}\\{b\in\underline{\mathbf{B}}^{*}\mid\varepsilon_{j}(b)=0\\}=[1]\in{}_{\mathbf{A}}K(\mathscr{L}_{0})$

This implies, for each $\beta\in Q_{+}^{\sigma}$, that

(5.15.4) $\underline{\mathbf{B}}_{\beta}=\bigcup_{\begin{subarray}{c}j\in
J,n>0\\\ \beta-n\alpha_{j}\in
Q_{+}^{\sigma}\end{subarray}}\pi_{n;j}(\underline{\mathbf{B}}_{0;j}\cap\underline{\mathbf{B}}_{\beta-n\alpha_{j}}).$

We now consider the isomorphism
$\gamma^{-1}:{}_{\mathbf{A}}K(\mathscr{P})\,\raise
2.0pt\hbox{$\underrightarrow{\sim}$}\,{}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$,
and act $\gamma^{-1}$ on both sides of (5.15.4). Since $\gamma$ commutes with
the action of $f_{j}^{(n)}$, the right hand side of (5.15.4) corresponds
exactly an inductive construction of canonical basis (see [L2, 14.4.2, Thm.
14.4.3]). Thus $\gamma^{-1}(\underline{\mathbf{B}})$ coincides with the
canonical basis $\underline{\mathcal{B}}^{\natural}$ of
${}_{\mathbf{A}}\underline{\mathbf{U}}_{q}^{-}$. The theorem now follows from
Theorem 5.12. ∎

Remark 5.17. We have deduced the coincidence of $\underline{\mathcal{B}}$ and
$\underline{\mathcal{B}}^{\natural}$ by making use of the crystal structure of
the basis $\underline{\mathbf{B}}^{*}$ of $K(\mathscr{L})$. However, if we
assume the properties of canonical basis as in (5.15.4), then the coincidence
$\underline{\mathcal{B}}=\underline{\mathcal{B}}^{\natural}$ is shown,
directly without appealing KLR algebras, by using a similar formula as
(5.15.3) for $\underline{\mathcal{B}}$.

## References

  * [GL] I. Grojnowski and G. Lusztig; A comparison of bases of quantized enveloping algebras, Linear algebraic groups and their representations, Contemp. Math. 153 (1993), pp. 11-19.
  * [K1] M. Kashiwara; On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465-516.
  * [K2] M. Kashiwara; Global crystal bases of quantum groups, Duke Math. J. 69 (1993), 455-485.
  * [KL] M. Khovanov and A. D. Lauda; A diagrammatic approach to categorification of quantum groups, I, Represent. Theory, 13 (2009), 309-347.
  * [L1] G. Lusztig; Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365-421.
  * [L2] G. Lusztig; Introduction to quantum groups, Progress in Math. Vol. 110 Birkhauser, Boston/Basel/Berlin, 1993.
  * [M] P. J. McNamara; Folding KLR algebras, J. London Math. Soc (2), 100 (2019), 447-469.
  * [MSZ] Y. Ma, T. Shoji and Z. Zhou; Diagram automorphisms and canonical bases for quantized enveloping algebras, to appear in J. Algebra.
  * [R1] R. Rouquier; 2-Kac-Moody algebras, Preprint, 2008, arXiv:0812.5023.
  * [R2] R. Rouquier; Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq. 19 (2012), 359-410.
  * [VV] M. Varagnolo and E. Vasserot; Canonical bases and KLR-algebras, J. reine angew. Math. 659 (2011), 67-100.

Y. Ma
School of Mathematical Sciences, Tongji University
1239 Siping Road, Shanghai 200092, P.R. China
E-mail<EMAIL_ADDRESS>
T. Shoji
School of Mathematica Sciences, Tongji University
1239 Siping Road, Shanghai 200092, P.R. China
E-mail<EMAIL_ADDRESS>
Z. Zhou
School of Mathematical Sciences, Tongji University
1239 Siping Road, Shanghai 200092, P.R. China
E-mail<EMAIL_ADDRESS>


[a]Mary Hall Reno

#  EAS optical Cherenkov signatures of tau neutrinos for space and suborbital
detectors

Tonia M. Venters  John F Krizmanic

###### Abstract

Multi-messenger observations of transient astrophysical sources have the
potential to characterize the highest energy accelerators and the most extreme
environments in the Universe. Detection of neutrinos, in particular tau
neutrinos generated by neutrino oscillations in transit from their sources to
Earth, is possible for neutrino energies above 10 PeV using optical Cherenkov
detectors imaging upward-moving extensive air showers (EAS). These EAS are
produced from Earth-interacting tau neutrinos leading to tau leptons that
subsequently decay in the atmosphere. We compare neutrino detection
sensitivities for generic short- and long-burst transient neutrino sources and
sensitivities to the diffuse neutrino flux for the second generation Extreme
Universe Space Observatory on a Super-Pressure Balloon (EUSO-SPB2) balloon-
borne mission and the proposed space-based Probe of Extreme Multi-Messenger
Astrophysics (POEMMA) mission.

##  1 Introduction

Transient astrophysical sources such as tidal disruption events, binary
neutron star mergers and the merger of black hole pairs are expected to
generate signals in gravitational waves, cosmic rays, photons and in
neutrinos. Over cosmic distance scales, the flavor ratio of two muon neutrinos
to one electron neutrino will oscillate to approximately equal numbers of all
three flavors of neutrinos (and anti-neutrinos). At high energies, signals of
tau neutrino fluxes may be observed through tau neutrino induced up-going air
showers that produce an optical Cherenkov signal, the focus of this work.
Using the Earth as a tau neutrino converter, taus are produced in $\nu_{\tau}$
charged current interactions [1]. Taus may exit directly or proceed through a
series of decays and neutrino reinteractions called regeneration. Exiting taus
that decay into a shower in the atmosphere can produce optical and radio
Cherenkov signals, and at the highest energies, air fluorescence.

The optical Cherenkov signal of up-going tau air showers is the target of the
two missions describe here: the Probe of Extreme MultiMessenger Astrophysics
(POEMMA) [2] with a pair of satellites proposed for a 2028 launch, and the
second generation Extreme Universe Space Observatory on a Super-Pressure
Balloon (EUSO-SPB2) [3] pathfinder telescope scheduled to launch in May 2023.
Both missions also have a fluorescence telescope as well, discussed elsewhere
in these proceedings. Parameters of these satellite and sub-orbital telescopes
are listed in table 1. With reference to fig. 1, table 1 shows the altitude
$h$, the angular coverage below the limb $\Delta\alpha$ and the corresponding
maximum Earth emergence angle (relative to the surface plane) $\beta_{\rm
tr}$, the azimuthal coverage $\Delta\phi$ and the minimum photon number
density at the respective detectors $\rho_{\gamma}^{\rm min}$ consistent with
a reduction of the air glow background to less than 1% [4]. POEMMA has two
observing modes: stereo with both telescopes pointing to the same spot on the
ground, and dual, where the telescopes point to two different locations.
Stereo mode has the advantage of a lower photon number density threshold, most
important at lower energies, while the dual mode increases the observing area
by a factor of 2. Telescopes can slew over time frames of order 500 s [5].

Mission | POEMMA Stereo (Dual) | EUSO-SPB2
---|---|---
$h$ | 525 km | 33 km
$\Delta\alpha$ | $7^{\circ}$ | $6.4^{\circ}$
$\Delta\phi$ | $30^{\circ}$ | $12.8^{\circ}$
$\rho_{\gamma}^{\rm min}$ | 20 (40) $\gamma$/m2 | 200 $\gamma$/m2
$\beta_{\rm tr}^{\rm max}$ | 19.6∘ | 10.8∘
Table 1: Altitude $h$, angular coverage below the limb $\Delta\alpha$,
azimuthal coverage $\Delta\phi$, minimum photon number density on the
telescope surface $\rho_{\gamma}^{\rm min}$ and maximum Earth emergence angle
$\beta_{\rm tr}^{\rm max}$,  Figure 1: Definition of angles and distances for
diffuse flux and point source fluence sensitivities.

POEMMA’s altitude $h=525$ km is a clear advantage over the EUSO-SPB2 sub-
orbital altitude in coverage of the Earth’s surface for detection of the
diffuse astrophysical neutrino flux, a factor of $\sim 70$ enhanced geometric
effective aperture of the Earth. For point source detection, POEMMA’s
advantage is not in effective area, but instead that it has full sky coverage
over a year, during which time the orbital plane precesses through almost
seven periods. In the next section, we compare the diffuse neutrino flux
sensitivities of POEMMA and EUSO-SPB2. The target of opportunity (ToO)
sensitivities are compared in the following section. We conclude with a short
discussion of some of the modeling uncertainties. For the range of neutrino
energies relevant to these detectors, above $10^{7}$ GeV, neutrinos and
antineutrinos have equal cross sections, so in our discussion, “neutrino”
refers to both particle and antiparticle.

##  2 Diffuse neutrino flux

The sensitivity to the diffuse neutrino flux for a detector with effective
aperture $\langle A\Omega\rangle$ and observing time including duty cycle
$f_{\rm obs}$ for three neutrino flavors is

$F_{\rm sens}=\frac{2.44\times 3}{\ln(10)\times E_{\nu_{\tau}}\times\langle
A\Omega\rangle\times t_{\rm obs}}\,,$ (1)

assuming no background, at a 90% confidence level over a decade of energy.
With the variables defined in fig. 1, a detector’s effective aperture is given
by [6]

$\left<A\Omega\right>\left(E_{\nu_{\tau}}\right)=\int_{S}\int_{\Delta\Omega_{\rm
tr}}P_{\rm obs}\ \hat{r}\cdot\hat{n}\,dS\,d\Omega_{\rm tr}\ ,$ (2)

where the observation probability $P_{\rm obs}$ depends on the probability
$p_{\rm exit}$ for $\nu_{\tau}\to\tau$ with an exiting $\tau$, the probability
for the tau to decay in the atmosphere $p_{\rm decay}$, and the detection
probability $p_{\rm det}$:

$\displaystyle P_{\rm obs}$ $\displaystyle=\int p_{\rm
exit}\left(E_{\tau}|E_{\nu_{\tau}},\beta_{\rm tr}\right)$
$\displaystyle\times\left[\int ds^{\prime}\,{p}_{\rm decay}(s^{\prime})p_{\rm
det}\left(E_{\tau},\theta_{E},\beta_{\rm
tr},s^{\prime}\right)\right]\,dE_{\tau}\,,$ (3)

A full discussion of the diffuse neutrino flux sensitivity can be found in
ref. [4].

Figure 2:  Comparison of differential effective aperture for POEMMA and EUSO-
SPB2 for $E_{\nu}=10^{9}$ GeV, as a function of $\theta_{E}$. The dotted
vertical lines are labeled with the corresponding value of Earth emergence
angle $\beta_{\rm tr}$.

Figure 2 shows part of the integrand in eq. (2): the product of the geometric
portion of the effective area $\hat{r}\cdot\hat{n}dS=\cos\theta_{v}dS$ with
the tau exit probability $p_{\rm exit}$ for $E_{\nu}=10^{9}$ GeV, accounting
for the angular ranges $\Delta\alpha$ and $\Delta\phi$ for POEMMA and EUSO-
SPB2 shown in table 1. The shaded regions are outside of the respective
$\Delta\alpha$ ranges. For reference, the Earth emergence angles $\beta_{\rm
tr}=1^{\circ},\ 10^{\circ}$ are shown by vertical lines. Both telescopes cover
the most important range of $\theta_{E}$ ($\beta_{\rm tr}$).

Figure 3: The sensitivity of POEMMA and EUSO-SPB2 to the all-flavor diffuse
neutrino flux.

The sensitivity of both instruments to Cherenkov signals of up-going air
showers degrades at low energies because $p_{\rm exit}$ is smaller for low
energies than high energies. Tau decays dominate tau propagation, and the
decay length of the tau in vacuum is $500$ m$\times(E_{\tau}/10^{7}$ GeV).
Figure 3 shows the diffuse all-flavor neutrino flux sensitivity of POEMMA,
EUSO-SPB2 and the sensitivity of a $\Delta\phi=360^{\circ}$ version of POEMMA
[4]. Without $\Delta\phi=360^{\circ}$, POEMMA will not be competitive with
other instruments. The 5-year planned running time, a factor of $\sim 18$
times longer than EUSO-SPB2, and the advantage in geometric effective aperture
account are only partly offset by the enhanced detectability of EUSO-SPB2 that
comes from being closer to the showers as they develop in the atmosphere.
Overall, POEMMA with $\Delta\phi=30^{\circ}$ is a factor of $\sim 200$ more
sensitive than EUSO-SPB2 to the diffuse astrophysical neutrino flux over most
of the neutrino energy range.

##  3 Target of opportunity neutrino fluence

Figure 4: Long (left) and short (right) burst all-flavor neutrino fluence
sensitivities for POEMMA (blue) and EUSO-SPB2 (violet) with parameters of
Table 1 assuming instruments can slew over the $360^{\circ}$ in azimuth.

The sensitivity to neutrino point sources depends on the instantaneous
effective area [5]

$A_{\rm Ch}(s)\simeq\pi(v-s)^{2}\times\left(\theta_{\rm Ch}^{\rm
eff}\right)^{2}\ ,$ (4)

the area subtended by the Cherenkov cone over the distance between the tau
decay point and the telescope, $v-s$. For long neutrino bursts, the relevant
quantity is the instantaneous effective area weighted by the observation
probability, time averaged over the period $T_{0}$ of the satellite orbit (90
minutes for POEMMA) or the Earth’s rotation of approximately one day (for
EUSO-SPB2):

$\displaystyle A(\beta_{\rm tr}(t),E_{\nu},\theta,\phi)$ $\displaystyle\simeq$
$\displaystyle\int dP_{\rm obs}(E_{\nu},\beta_{\rm tr},s)A_{\rm Ch}(s)$ (5)
$\displaystyle\langle A(E_{\nu},\theta,\phi)\rangle_{T_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{T_{0}}\int_{t_{0}}^{t_{0}+T_{0}}\,dtA(\beta_{\rm
tr}(t),E_{\nu},\theta,\phi)\ .$ (6)

The sensitivity is determined from

$\phi_{\rm sens}=\frac{2.44\times 3}{\ln(10)\times E_{\nu_{\tau}}\times\langle
A(E_{\nu},\theta,\phi)\rangle_{T_{0}}\times f_{t}}\,,$ (7)

for the all-flavor fluence $\phi_{\nu}$.

The left panel of fig. 4 shows the long neutrino burst sensitivities for
POEMMA and EUSO-SPB2. For POEMMA, the sensitivity is evaluated assuming stereo
viewing, and an overall duty cycle $f_{t}$ of 30% is used to account for the
time that the Sun and Moon prohibit observing. Dual viewing mode for POEMMA
yields the best long neutrino burst sensitivity a factor $\sim 5$ higher
(weaker) than in stereo mode for $10^{7}$ GeV. Stereo and dual viewing modes
have very similar sensitivities above $E_{\nu}\sim 3\times 10^{8}$ GeV. For
EUSO-SPB2, the sensitivity to a long neutrino burst of 30 days $=10^{6.4}$ s
comes from the 30 day average of the effective area, assuming a 20% duty cycle
because of the Sun and Moon. Since there are some regions of the sky
inaccessible to EUSO-SPB2, we do not plot an upper bound to the sensitivity
band. POEMMA has the advantage of full sky coverage over time. We also show
the prediction of Fang and Metzger’s binary neutron star all-flavor neutrino
fluence for a merger at 3 Mpc distance over a time interval of
$10^{4.5}-10^{6.5}$ s [7].

For short neutrino bursts, we assume that a $10^{3}$ s neutrino burst begins
when the source is in an optimal position for observation by either
instrument. The best short neutrino burst sensitivities are shown in the right
panel of fig. 4, with the duty cycle $f_{t}=1$, part of the definition of
“best.” For a short neutrino burst, POEMMA does not have time to slew into
stereo mode, so the sensitivity is determined in dual mode. Also shown in the
right panel is the on-axis prediction for a short gamma ray burst at 40 Mpc
with moderate extended emission predicted by Kimura, Murase, Meszaros and
Kiuchi (KMMK) [8]. The short neutrino burst sensitivity of EUSO-SPB2 is much
closer to the POEMMA short neutrino burst sensitivity than for their
respective long neutrino burst sensitivities, and even better than POEMMA at
$E_{\nu}=10^{7}$ GeV. The low energy results for EUSO-SPB2 come from the
advantage of being closer to the developing shower. There is boost in the
effective area through $A_{\rm ch}$ with larger distances $(v-s)$ from the tau
decay, at the cost of a lower photon density at the telescope. At low
energies, the photon density threshold is the determining factor.

Both telescopes will require external (GCN) alerts for short and long neutrino
bursts. With five years of viewing, POEMMA has greater potential for transient
observations from tidal disruption events, binary black hole mergers and
binary neutron star mergers [5]. EUSO-SPB2 will be a pathfinder instrument.
The sensitivities shown here indicate that sources in nearby galaxies are
accessible for detection if those sources are located in optimal regions of
the sky [9].

##  4 Modeling uncertainties

The evaluations of the sensitivities to the diffuse neutrino flux and the
neutrino point source fluence rely on modeling the tau neutrino and tau
propagation in the Earth, the up-going tau air shower and its detection. In
this section, we illustrate the impact of modeling uncertainties in tau
propagation in the Earth (in $p_{\rm exit}$) and the shower modeling, which
impacts $p_{\rm det}$.

Neutrino propagation in the Earth has a primary uncertainty in the ultrahigh
energy extrapolation of the neutrino cross section. Once a tau is produced, it
propagates with electromagnetic energy loss dominated by photonuclear
processes. Again, the high energy behavior of tau photonuclear scattering
requires extrapolations. Here, we focus on electromagnetic energy loss. Two
examples of parameterizations of the photonuclear energy loss are those of
Abramowitz et al. (ALLM) [10] and Block et al. (BDHM) [11]. The blue region
between the solid black (ALLM) and blue (BDHM) curves shows an approximate
theoretical uncertainty range for photonuclear energy loss. Both solid curves
come from an Earth model based on the Preliminary Reference Earth Model [12]
with an outer layer of water of depth 3 km. The black dashed line shows the
result when the final layer of Earth is rock. At low energies, most of the
neutrino production of taus is in the final layer, so the rock density
enhances the probability for the neutrino to interact. At high energies, tau
can propagate farther. Propagation in water entails less energy loss than in
rock, so the sensitivity is best over water at high energies [13].

Figure 5:  Comparison of POEMMA ($\Delta\phi=360^{\circ}$) sensitivity
evaluations for the diffuse all-flavor neutrino flux with ALLM and BDHM tau
photonuclear energy loss extrapolations with 3 km water in the outer layer of
the Earth (solid curves) and with a final layer of all rock (dashed). The
figure is updated from ref. [4].

In refs. [4] and [5], the photon density as a function of angle and altitude
of the decay was determined using the Greisen parameterization of the EAS and
a composite atmosphere model (see ref. [4] for details) for $E_{\rm
shr}=10^{8}$ GeV, then rescaled by a factor of $E_{\rm shr}/10^{8}$ GeV. In
figs. 3 and 4, the shower energy is taken to be $0.5E_{\tau}$. Recently,
improvements to the air shower modeling have been reported in ref. [14].
There, the air shower modeling is based on CORSIKA, modified for upward
showers. The longitudinal profile as a function of shower age shows longer
shower tails than the Greisen parameterization, resulting in an increased
effective aperture at high energies. Figure 6 show a comparison of our results
(black) for the sensitivity of the diffuse flux of POEMMA [4] (left) and for
EUSO-SPB2 (right) with the results from the improved treatment of the showers
(red) evident at high energies, illustrating the projected improved
sensitivity at high energies for both instruments. At low energies, the
diffuse neutrino flux sensitivity of EUSO-SPB2 shows little change with the
improved shower treatment. For POEMMA, the results are in qualitative
agreement at low energies. The initial shower energy is expected to have a
larger impact close to the sharper threshold of the higher altitude telescope.
The quantitative discrepancy between two results at low energy for POEMMA is
under investigation.

Figure 6:  Comparison of sensitivity evaluations for the all flavor diffuse
neutrino flux $E_{\nu}^{2}\phi_{\nu+\bar{\nu}}$ following ref. [4] in black
and with the more detailed shower modeling of ref. [14] (black, labeled CAK
(2021)).

Further studies of the impact of different models for the inputs to $P_{\rm
obs}$ are in progress. The end-to-end simulation of neutrino to optical
Cherenkov and geomagnetic radio signals from EAS in the nuSpaceSim package is
under development [15], with a component nuPyProp that evaluates the tau exit
probabilities and energy distributions [16].

## Acknowledgments

This work is supported in part by NASA grants 80NSSC19K0626 at the University
of Maryland, Baltimore County, 80NSSC19K0460 at the Colorado School of Mines,
80NSSC19K0484 at the University of Iowa, and 80NSSC19K0485 at the University
of Utah, 80NSSC18K0464 at Lehman College, and under proposal 17-APRA17-0066 at
NASA/GSFC and JPL, and by the US Department of Energy grant DE-SC-0010113. The
conceptual design of POEMMA was supported by NASA Probe Mission Concept Study
grant NNX17AJ82G for the 2020 Decadal Survey Planning. Contributors to this
work were also supported in part by NASA awards 16-APROBES16-0023, NNX17AJ82G,
NNX13AH54G, 80NSSC18K0246, and 80NSSC18K0473.

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## +POEMMA Collaboration

A. V. Olinto,1 J. Krizmanic,2,3 J. H. Adams,4 R. Aloisio,5 L. A. Anchordoqui,6
A. Anzalone,7,8 M. Bagheri,9 D. Barghini,10 M. Battisti,10 D. R. Bergman,11 M.
E. Bertaina,10 P. F. Bertone,12 F. Bisconti,13 M. Bustamante,14 F. Cafagna,15
R. Caruso,16,8 M. Casolino,17,18 K. Černý,19 M. J. Christl,12 A. L. Cummings,5
I. De Mitri,5 R. Diesing,1 R. Engel,20 J. Eser,1 K. Fang,21 F. Fenu,10 G.
Filippatos,22 E. Gazda,9 C. Guepin,23 A. Haungs,20 E. A. Hays,2 E. G. Judd,24
P. Klimov,25 V. Kungel,22 E. Kuznetsov,4 Š. Mackovjak,26 D. Mandát,27 L.
Marcelli,18 J. McEnery,2 G. Medina-Tanco,28 K.-D. Merenda,22 S. S. Meyer,1 J.
W. Mitchell,2 H. Miyamoto,10 J. M. Nachtman,29 A. Neronov,30 F. Oikonomou,31
Y. Onel,29 G. Osteria,32 A. N. Otte,9 E. Parizot,33 T. Paul,6 M. Pech,27 J. S.
Perkins,2 P. Picozza,18,34 L.W. Piotrowski,35 Z. Plebaniak,10 G. Prévôt,33 P.
Reardon,4 M. H. Reno,29 M. Ricci,36 O. Romero Matamala,9 F. Sarazin,22 P.
Schovánek,27 V. Scotti,32,37 K. Shinozaki,38 J. F. Soriano,6 F. Stecker,2 Y.
Takizawa,17 R. Ulrich,20 M. Unger,20 T. M. Venters,2 L. Wiencke,22 D. Winn,29
R. M. Young,12 M. Zotov2

1The University of Chicago, Chicago, IL, USA 2NASA Goddard Space Flight
Center, Greenbelt, MD, USA 3Center for Space Science & Technology, University
of Maryland, Baltimore County, Baltimore, MD, USA 4University of Alabama in
Huntsville, Huntsville, AL, USA 5Gran Sasso Science Institute, L’Aquila, Italy
6City University of New York, Lehman College, NY, USA 7Istituto Nazionale di
Astrofisica INAF-IASF, Palermo, Italy 8Istituto Nazionale di Fisica Nucleare,
Catania, Italy 9Georgia Institute of Technology, Atlanta, GA, USA
10Universita’ di Torino, Torino, Italy 11University of Utah, Salt Lake City,
Utah, USA 12NASA Marshall Space Flight Center, Huntsville, AL, USA 13Istituto
Nazionale di Fisica Nucleare, Turin, Italy 14Niels Bohr Institute, University
of Copenhagen, DK-2100 Copenhagen, Denmark 15Istituto Nazionale di Fisica
Nucleare, Bari, Italy 16Universita’ di Catania, Catania Italy 17RIKEN, Wako,
Japan 18Istituto Nazionale di Fisica Nucleare, Section of Roma Tor Vergata,
Italy 19Joint Laboratory of Optics, Faculty of Science, Palacký University,
Olomouc, Czech Republic 20Karlsruhe Institute of Technology, Karlsruhe,
Germany 21Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Stanford, CA94305, USA 22Colorado School of Mines, Golden, CO, USA
23Department of Astronomy, University of Maryland, College Park, MD, USA
24Space Sciences Laboratory, University of California, Berkeley, CA, USA
25Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
Moscow,Russia 26Institute of Experimental Physics, Slovak Academy of Sciences,
Kosice, Slovakia 27Institute of Physics of the Czech Academy of Sciences,
Prague, Czech Republic 28Instituto de Ciencias Nucleares, UNAM, CDMX, Mexico
29University of Iowa, Iowa City, IA, USA 30University of Geneva, Geneva,
Switzerland 31Institutt for fysikk, NTNU, Trondheim, Norway 32Istituto
Nazionale di Fisica Nucleare, Napoli, Italy 33Université de Paris, CNRS,
Astroparticule et Cosmologie, F-75013 Paris, France 34Universita di Roma Tor
Vergata, Italy 35Faculty of Physics, University of Warsaw, Warsaw, Poland
36Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati,
Frascati, Italy 37Universita’ di Napoli Federico II, Napoli, Italy 38National
Centre for Nuclear Research, Lodz, Poland.

## +JEM-EUSO Collaboration

G. Abdellaouiah, S. Abefq, J.H. Adams Jr.pd, D. Allardcb, G. Alonsomd, L.
Anchordoquipe, A. Anzaloneeh,ed, E. Arnoneek,el, K. Asanofe, R. Attallahac, H.
Attouiaa, M. Ave Pernasmc, M. Bagheriph, J. Balázla, M. Bakiriaa, D.
Barghiniel,ek, S. Bartocciei,ej, M. Battistiek,el, J. Bayerdd, B.
Beldjilaliah, T. Belenguermb, N. Belkhalfaaa, R. Bellottiea,eb, A.A. Belovkb,
K. Benmessaiaa, M. Bertainaek,el, P.F. Bertonepf, P.L. Biermanndb, F.
Biscontiel,ek, C. Blaksleyft, N. Blancoa, S. Blin-Bondilca,cb, P. Bobikla, M.
Bogomilovba, E. Bozzoob, S. Brizpb, A. Brunoeh,ed, K.S. Caballerohd, F.
Cafagnaea, G. Cambiéei,ej, D. Campanaef, J-N. Capdeviellecb, F. Capelde, A.
Carameteja, L. Carameteja, P. Carlsonna, R. Carusoec,ed, M. Casolinoft,ei, C.
Cassardoek,el, A. Castellinaek,em, O. Catalanoeh,ed, A. Cellinoek,em, K.
Černýbb, M. Chikawafc, G. Chiritoija, M.J. Christlpf, R. Colalilloef,eg, L.
Contien,ei, G. Cottoek,el, H.J. Crawfordpa, R. Cremoniniel, A. Creusotcb, A.
de Castro Gónzalezpb, C. de la Tailleca, L. del Peralmc, A. Diaz Damiancc, R.
Diesingpb, P. Dinaucourtca, A. Djakonowia, T. Djemilac, A. Ebersoldtdb, T.
Ebisuzakift, L. Eliassonna, J. Eserpb, F. Fenuek,el, S. Fernández-Gonzálezma,
S. Ferrareseek,el, G. Filippatospc, W.I. Finchpc, C. Fornaroen,ei, M. Foukaab,
A. Franceschiee, S. Franchinimd, C. Fuglesangna, T. Fujiifg, M. Fukushimafe,
P. Galeottiek,el, E. García-Ortegama, D. Gardiolek,em, G.K. Garipovkb, E.
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A. Zuccaro Marchift

aa Centre for Development of Advanced Technologies (CDTA), Algiers, Algeria ab
Dep. Astronomy, Centre Res. Astronomy, Astrophysics and Geophysics (CRAAG),
Algiers, Algeria ac LPR at Dept. of Physics, Faculty of Sciences, University
Badji Mokhtar, Annaba, Algeria ad Lab. of Math. and Sub-Atomic Phys. (LPMPS),
Univ. Constantine I, Constantine, Algeria af Department of Physics, Faculty of
Sciences, University of M’sila, M’sila, Algeria ag Research Unit on Optics and
Photonics, UROP-CDTA, Sétif, Algeria ah Telecom Lab., Faculty of Technology,
University Abou Bekr Belkaid, Tlemcen, Algeria ba St. Kliment Ohridski
University of Sofia, Bulgaria bb Joint Laboratory of Optics, Faculty of
Science, Palacký University, Olomouc, Czech Republic bc Institute of Physics
of the Czech Academy of Sciences, Prague, Czech Republic ca Omega, Ecole
Polytechnique, CNRS/IN2P3, Palaiseau, France cb APC, Université de Paris,
CNRS, AstroParticule et Cosmologie, F-75013 Paris, France cc IRAP, Université
de Toulouse, CNRS, Toulouse, France da ECAP, University of Erlangen-Nuremberg,
Germany db Karlsruhe Institute of Technology (KIT), Germany dc Experimental
Physics Institute, Kepler Center, University of Tübingen, Germany dd Institute
for Astronomy and Astrophysics, Kepler Center, University of Tübingen, Germany
de Technical University of Munich, Munich, Germany ea Istituto Nazionale di
Fisica Nucleare - Sezione di Bari, Italy eb Università degli Studi di Bari
Aldo Moro and INFN - Sezione di Bari, Italy ec Dipartimento di Fisica e
Astronomia “Ettore Majorana", Università di Catania, Italy ed Istituto
Nazionale di Fisica Nucleare - Sezione di Catania, Italy ee Istituto Nazionale
di Fisica Nucleare - Laboratori Nazionali di Frascati, Italy ef Istituto
Nazionale di Fisica Nucleare - Sezione di Napoli, Italy eg Università di
Napoli Federico II - Dipartimento di Fisica “Ettore Pancini”, Italy eh INAF -
Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo, Italy ei
Istituto Nazionale di Fisica Nucleare - Sezione di Roma Tor Vergata, Italy ej
Università di Roma Tor Vergata - Dipartimento di Fisica, Roma, Italy ek
Istituto Nazionale di Fisica Nucleare - Sezione di Torino, Italy el
Dipartimento di Fisica, Università di Torino, Italy em Osservatorio
Astrofisico di Torino, Istituto Nazionale di Astrofisica, Italy en Uninettuno
University, Rome, Italy fa Chiba University, Chiba, Japan fb National
Institutes for Quantum and Radiological Science and Technology (QST), Chiba,
Japan fc Kindai University, Higashi-Osaka, Japan fd Kanazawa University,
Kanazawa, Japan fe Institute for Cosmic Ray Research, University of Tokyo,
Kashiwa, Japan ff Konan University, Kobe, Japan fg Kyoto University, Kyoto,
Japan fh Shinshu University, Nagano, Japan fi National Astronomical
Observatory, Mitaka, Japan fj Nagoya University, Nagoya, Japan fk Institute
for Space-Earth Environmental Research, Nagoya University, Nagoya, Japan fl
Graduate School of Science, Osaka City University, Japan fm Institute of Space
and Astronautical Science/JAXA, Sagamihara, Japan fn Saitama University,
Saitama, Japan fo Hokkaido University, Sapporo, Japan fp Osaka Electro-
Communication University, Neyagawa, Japan fq Nihon University Chiyoda, Tokyo,
Japan fr University of Tokyo, Tokyo, Japan fs High Energy Accelerator Research
Organization (KEK), Tsukuba, Japan ft RIKEN, Wako, Japan ga Korea Astronomy
and Space Science Institute (KASI), Daejeon, Republic of Korea gb Sungkyunkwan
University, Seoul, Republic of Korea ha Universidad Nacional Autónoma de
México (UNAM), Mexico hb Universidad Michoacana de San Nicolas de Hidalgo
(UMSNH), Morelia, Mexico hc Benemérita Universidad Autónoma de Puebla (BUAP),
Mexico hd Universidad Autónoma de Chiapas (UNACH), Chiapas, Mexico he Centro
Mesoamericano de Física Teórica (MCTP), Mexico ia National Centre for Nuclear
Research, Lodz, Poland ib Faculty of Physics, University of Warsaw, Poland ja
Institute of Space Science ISS, Magurele, Romania ka Joint Institute for
Nuclear Research, Dubna, Russia kb Skobeltsyn Institute of Nuclear Physics,
Lomonosov Moscow State University, Russia kc Space Regatta Consortium,
Korolev, Russia la Institute of Experimental Physics, Kosice, Slovakia lb
Technical University Kosice (TUKE), Kosice, Slovakia ma Universidad de León
(ULE), León, Spain mb Instituto Nacional de Técnica Aeroespacial (INTA),
Madrid, Spain mc Universidad de Alcalá (UAH), Madrid, Spain md Universidad
Politécnia de madrid (UPM), Madrid, Spain me Instituto de Astrofísica de
Canarias (IAC), Tenerife, Spain na KTH Royal Institute of Technology,
Stockholm, Sweden oa Swiss Center for Electronics and Microtechnology (CSEM),
Neuchâtel, Switzerland ob ISDC Data Centre for Astrophysics, Versoix,
Switzerland oc Institute for Atmospheric and Climate Science, ETH Zürich,
Switzerland pa Space Science Laboratory, University of California, Berkeley,
USA pb University of Chicago, IL, USA pc Colorado School of Mines, Golden, CO,
USA pd University of Alabama in Huntsville, Huntsville, AL, USA pe Lehman
College, City University of New York (CUNY), NY, USA pf NASA Marshall Space
Flight Center, Huntsville, AL, USA pg University of Utah, Salt Lake City, UT,
USA ph Georgia Institute of Technology, GA, USA pi University of Iowa, Iowa
City, IA, USA pj NASA Goddard Space Flight Center, Greenbelt, MD, USA pk
Center for Space Science & Technology, University of Maryland, Baltimore
County, Baltimore, MD, USA pm Fairfield University, Fairfield, CT, USA