For $\underline{\lambda} \in \GT(\lambda)$, let $\lambda_b^{(i)}$ be the lowest weight of the $\Uc_q(\Glie_k)$-module $L(\lambda^{(i)};\Glie_k)$ defined in <ref>. Then $\lambda^{(i)}_b = \mu^{(i)}_b$ if and only if $\lambda^{(i)} = \mu^{(i)}$. In the following, we show that $\lambda^{(i)}_b = \mu^{(i)}_b$ under the condition $f_{i,\underline{\lambda}} = f_{i,\underline{\mu}}$.

Assume first  $1 \leq i \leq M$. Write $\lambda_b^{(i)} = \sum_{s=1}^i a_{i+1-s} \epsilon_s$ and $\mu_b^{(i)} = \sum_{s=1}^i a_{i+1-s}' \epsilon_s$. Then

a_1 ≥a_2 ≥⋯≥a_i,  a_1' ≥a_2' ≥⋯≥a_i'.

Since $v_{\underline{\lambda}}^*$ is contained in a sub-$\Uc_q(\Glie_i)$-module of $L(\lambda)^*$ isomorphic to $L(-\lambda_b^{(i)};\Glie_i) \otimes \BC_{|\lambda - \lambda^{(i)}|}$, we get an explicit formula for $f_{i,\underline{\lambda}}$:

f_i,λ = ∏_s=1^i (q^-a_i+1-s - z a q^2(s-1)+ a_i+1-s),  f_i,μ = ∏_s=1^i (q^-a_i+1-s' - z a q^2(s-1) + a_i+1-s').

Note that $s-1+a_{i+1-s} < t-1+a_{i+1-t}$ for $1 \leq s < t \leq i$. Hence $a_s = a_s'$ for $1 \leq s \leq i$.

Next assume $i = M+k$ with $1 \leq k \leq N$. Let us write

λ^(M+k)_b = ∑_l=1^M x_M+1-l ϵ_l +  ∑_j=1^k y_k+1-j ϵ_M+j,  μ^(M+k)_b = ∑_l=1^M x_M+1-l' ϵ_l + ∑_j=1^k y_k+1-j' ϵ_M+j.

Then the $x_l,y_j$ (similar for the $x_l',y_j'$) verify the following conditions:

(A)  $x_1 \geq x_2 \geq \cdots \geq x_M$ and $y_1 \geq y_2 \geq \cdots \geq y_k$;

(B)  if $y_j \leq r$, then $x_{r+1} = 0$ (with the convention $x_{M+1} = x_{M+2} = \cdots = 0$).

Our aim is to show that $x_l = x_l'$ and $y_j = y_j'$ for $1 \leq l \leq M$ and $1 \leq j \leq k$. Note that $f_{M+k,\underline{\lambda}} = f_{M+k,\underline{\mu}}$ implies $\frac{N(z)}{D(z)} = \frac{N'(z)}{D'(z)}$

N(z) = ∏_l=1^M (1 - za q^2(l-1) + 2 x_M+1-l ),   D(z) = ∏_j=1^k (1 - z a q^2(M-k-1+ j)- 2 y_j  )

and $N'(z),D'(z)$ are similarly defined using $x_l',y_j'$. We shall prove $N(z) = N'(z)$. If one of the fractions $\frac{N(z)}{D(z)}, \frac{N'(z)}{D'(z)}$ is in reduced form, then by counting the zeros and the poles we get $N(z) = N'(z)$. Suppose therefore neither $\frac{N(z)}{D(z)}$ nor $\frac{N'(z)}{D'(z)}$ is reduced.

Let  $P$ (resp. $Z$) the set of poles (resp. zeros) not including $\infty$ of the rational function $\frac{N(z^{-1})}{D(z^{-1})}$. Let $P_0$ (resp. $Z_0$) be the set of zeros of $D(z^{-1})$ (resp. $N(z^{-1})$). Define similarly $P',Z',P_0',Z_0'$. Then

P = P',  Z = Z',   P ⊊P_0,  Z ⊊Z_0.

Moreover, there exists $1 \leq j_1 < j_2 < \cdots < j_s \leq k$ such that

P_0 ∖P = {aq^2(M-k-1+j_t - y_j_t)  | 1 ≤t ≤s }.

One can  find a unique $1 \leq i_s \leq M$ such that $aq^{2(i_s-1+x_{M+1-i_s})} = aq^{2(M-k-1+j_s - y_{j_s})}$. So

y_j_s = M-k+j_s-i_s-x_M+1-i_s ≤M-k+j_s-i_s.

Now Condition (B) above says that $x_{M-k+j_s-i_s+1} = 0$. In view of Condition (A) ($j_s \leq k$)

x_M+1-i_s = 0,  y_j_s = M-k+j_s-i_s.

Again Condition (A) says that $x_{M+1-l} = 0$ for $1\leq l \leq i_s$. It follows that

{aq^2(l-1)+x_M+1-l | i_s < l ≤M } ⊆Z,  N(z) = ∏_l≤i_s(1-zaq^2(l-1))∏_l>i_s(1-zaq^2(l-1)+x_M+1-l) .

Similar analysis of the rational function $\frac{N'(z)}{D'(z)}$ leads to $i_s'$ with the above properties. We are therefore led to show that $i_s = i_s'$. Assume the contrary: say $i_s > i_s'$. Then observe first from $Z = Z'$ that

x_M+1-l = x_M+1-l'   for l > i_s.

Next, we have $a q^{2(i_s-1+x_{M+1-i_s}')} \in Z' = Z$. Hence there must be $1 \leq j \leq M$ such that

j-1+x_M+1-j = i_s - 1 + x_M+1-i_s'.

Since $aq^{2(i_s-1+x_{M+1-i_s})} \notin Z$ by assumption, $j \neq i_s$.
When $j < i_s$, $x_{M+1-j} = 0$ and the above equation can not hold. So $j > i_s$. It follows that $x_{M+1-j} = x_{M+1-j}'$ and $i_s - 1 + x_{M+1-i_s}' = j-1 + x_{M+1-j}'$, which is impossible as $x_{M+1-i_s}' \leq x_{M+1-j}'$. This completes the proof.

As in Proposition <ref>, all the $\ell$-weight spaces of $\ev_a^* L(\lambda)^*$ are one-dimensional of  the form $\BC v_{\underline{\lambda}}$ with $\underline{\lambda} \in \GT(\lambda)$.

$\tilde{\chi}_q (\ev_a^* L(\lambda)^*)  \in \BZ[[A_{i,b}]^{-1} : i \in I_0, b \in aq^{2\BZ+1}]$ for $\lambda \in \CP$ and $a \in \BC^{\times}$.

First observe from Definition <ref> that

(A_i,a)_i = q 1-zaq^-1/1-zaq,  (A_i,a)_j = 1   for j ≠i.

As in the proof of Corollary <ref>, let $\lambda \in \CP$. Let $Y^{\lambda}$ be the associated $\Glie$-Young diagram. Let $f^- \in \CB(Y^{\lambda})$ be the $\Glie$-Young tableau such that the associated Gelfand-Tsetlin vector $v_{\GT(f^-)} \in L(\lambda)$ is a lowest weight vector. For simplicity, let us denote $\underline{\mu} := \GT(f^-)$. In view of Proposition <ref>, it is enough to show that for all $i \in I$ and for all $\underline{\lambda} \in \GT(\lambda)$, the power series $f_{i,\underline{\mu}}(z) f_{i,\underline{\lambda}}(z)^{-1}$ can be written as a product of $q \frac{1-zbq}{1-zbq^{-1}}$ with $b \in \BC^{\times}$. We consider the case $i = M+k$ with $1 \leq k \leq N$. (The case $i \leq M$ is similar and simpler.)

Let $g := GT_{\lambda}^{-1}(\underline{\lambda})$. Then as in the proof of Corollary <ref> we have

g(x,y) ≤f^-(x,y)   for (x,y) ∈Y^λ.

As before, for $\theta \in \CP_{M_i,N_i}$, let $\theta_b$ be the lowest weight of the $\Uc_q(\Glie_i)$-module $L(\theta; \Glie_k)$. Write

μ^(M+k)_b = ∑_j=1^M x_M+1-j ϵ_j + ∑_l=1^k y_k+1-j ϵ_M+j,   λ^(M+k)_b = ∑_j=1^M x_M+1-j' ϵ_j + ∑_l=1^k y_k+1-j' ϵ_M+j.

Then as in the proof of Proposition <ref> we have

\begin{align*}
& f_{i,\underline{\mu}} = \prod_{j=1}^M (q^{-x_{M+1-j}} - zaq^{2(j-1) + x_{M+1-j}})  \times \prod_{l=1}^k (q^{y_{k+1-l}} - zaq^{2(M-l) - y_{k+1-l}})^{-1},  \\
& f_{i,\underline{\lambda}} = \prod_{j=1}^M (q^{-x'_{M+1-j}} - zaq^{2(j-1) + x'_{M+1-j}})  \times \prod_{l=1}^k (q^{y'_{k+1-l}} - zaq^{2(M-l) - y'_{k+1-l}})^{-1}.
\end{align*}

It is therefore enough to show that $x_j \leq x_j'$ and $y_l \leq y_l'$ for $1 \leq j \leq M$ and $1 \leq l \leq k$.

Let us verify $y_l \leq y_l'$ for $1 \leq l \leq k$. (The case for $x$ is parallel.) For this, we use the description in <ref> on the relationship between Young diagrams and lowest weight vectors. Namely, we have (Definition <ref>)

\begin{eqnarray*}
y_{k+1-j} &=& \sharp \{ x \in \BZ_{>0}  \ |\ (x, N+1-j) \in Y^{\lambda} \} = \sharp (f^-)^{-1}(M+k),  \\
y'_{k+1-j} &=& \sharp \{ x \in \BZ_{>0} \ |\  (x, k+1-j) \in g^{-1}(I_{M_i,N_i}) \}.
\end{eqnarray*}

Suppose that $(x,N+1-j) \in Y^{\lambda}$. Then $f^-(x,N+1-j) = M+k$. It follows that $g(x,N+1-j) \leq M+k = i$. Hence $g(x,k+1-j) \leq g(x,N+1-j) \leq i$. In other words,

{ x ∈_>0   | (x, N+1-j) ∈Y^λ }  ⊆{ x ∈_>0  |  (x, k+1-j) ∈g^-1(I_M_i,N_i) }.

This says that $y_l \leq y_l'$, as desired.

Let $(M,N) = (1,2)$ and $\Glie = \mathfrak{gl}(1,2)$. Consider the $U_q(\Gaff)$-module $W_{k,a}^{(2)}$. Let us compute its normalized $q$-character by using dual Gelfand-Tsetlin basis. The $\Glie$-Young diagram in this case is

Y = { (i,1) ∈^2 | 1 ≤i ≤k }.

There are $2k+1$ $\Glie$-Young tableaux of shape $Y$, namely $f_i,g_j$ with $0 \leq i \leq k, 0 \leq j \leq k-1$:

f_i(l,1) =
2     (1 ≤l ≤i),

3     (i < l ≤k),
g_j(l,1) =
1     (l = 1),

2     (1<l ≤j+1),

3     (j+1 < l ≤k).

Let $v_i,w_j \in W_{k,a}^{(2)}$ be dual Gelfand-Tsetlin vectors associated with $f_i,g_j$. Then $v_0$ is a highest $\ell$-weight vector. The action of $C_q(\Glie)$ on these vectors becomes:

\begin{align*}
& C_1(z) v_i = (1-za) v_i, \quad C_1(z) w_j = (q^{-1} - z a q) w_j, \\
& C_2(z) v_i = (1-za) (q^i - z a q^{-i})^{-1} v_i, \quad C_2(z) w_j = (1-za)(q^{j+1}-za q^{-j-1})^{-1} w_j.
\end{align*}

It follows that

\begin{equation*}
\widetilde{\chi}_q (W_{k,a}^{(2)}) = 1 + (1 + [A_{1,aq}]^{-1}) \sum_{l=1}^k \prod_{j=1}^l [A_{2,aq^{-2j+1}}]^{-1}.
\end{equation*}

§ GELFAND-TSETLIN BASIS OF ASYMPTOTIC REPRESENTATIONS

In this section, we study the normalized $q$-character for the asymptotic modules $L_{r,a}^+$ and $L_{r,a}(b)$ constructed in previous sections <ref>-<ref> and we establish Gelfand-Tsetlin basis for these modules (provided that $b$ is generic).

§.§ Gelfand-Tsetlin basis.

For $k \in I$, let $Y_q(\Glie_k)$ be the subalgebra of $Y_q(\Glie)$ generated by the $s_{ij}^{(n)},(s_{ii}^{(0)})^{-1}$ with $i,j \leq k$. Then there exists  canonical isomorphisms of superalgebras

Y_q(𝔤𝔩(M_k,N_k)) ≅Y_q(_k) ⊂U_q(_k) ≅U_q(𝔤𝔩(M_k,N_k)).

We have therefore a chain of sub-superalgebras of $Y_q(\Glie)$:

Y_q(_1) ⊂Y_q(_2) ⊂⋯⊂Y_q(_M+N-1) ⊂Y_q(_M+N) = Y_q().

Let $V$ be a $Y_q(\Glie)$-module.  A basis  $(v_j)_{j \in \Lambda}$ of $V$ is called a Gelfand-Tsetlin basis if

(B1) $V$ is semi-simple as a $Y_q(\Glie_k)$-module for all $k \in I_0$ and simple as a $Y_q(\Glie)$-module;

(B2) for all $j \in \Lambda$ there exists a sequence $(S_s)_{s\in I}$ such that: $S_s$ is a simple sub-$Y_q(\Glie_s)$-module; $S_s \subset S_{s+1}$ for $1 \leq k < M+N$; $v_j \in S_1$;

(B3) for all $1 \leq k \leq M+N$ and for $S$ a simple sub-$Y_q(\Glie_s)$-module, the decomposition of $\Res_{Y_q(\Glie_{s-1})}^{Y_q(\Glie_s)} S$ into simple sub-$Y_q(\Glie_{s-1})$-modules is multiplicity-free.

Now we can state the central result of this section.

Let $r \in I_0, a,b \in \BC^{\times}$  be such that $b \notin \pm q^{\BZ}$. Then the $Y_q(\Glie)$-modules $L_{r,a}^+$ and $L_{r,a}(b)$ admit Gelfand-Tsetlin bases.

The idea of proof goes as follows. Fix $r \in I_0$. There exists an inductive system $(V_k, F_{k,l})$ constructed in <ref>-<ref> where $V_k$ are properly defined Kirillov-Reshetikhin modules. The representations of $Y_q(\Glie)$ on $L_{r,a}^+$ and on $L_{r,a}(b)$ are built upon the same underlying space, namely the inductive limit $(V_{\infty}, F_k)$. For all $k \in \BZ_{>0}$, the $Y_q(\Glie)$-module $V_k$ admits a Gelfand-Tsetlin basis. Furthermore the structural maps $F_{k,l}: V_l \longrightarrow V_k$ respect Gelfand-Tsetlin bases on both sides.  From this we construct directly a basis of $V_{\infty}$ which serves at the same time as a Gelfand-Tsetlin basis for $L_{r,a}^+$ and for $L_{r,a}(b)$.

The next two subsections are devoted to the detailed proof of this theorem in cases $r \leq M$ and $r > M$. Along the proof, we shall find the simple sub-$Y_q(\Glie_k)$-modules in (B2)  in a combinatorial way (by using semi-infinite Young tableaux).

Before the proof, let us make the following convention.
For $1 \leq s \leq M+N$, related to the $q$-Yangian $Y_q(\Glie_s) \cong Y_q(\mathfrak{gl}(M_s,N_s))$ we can define in exactly the same way the category $\BGG_s$, the group $\ADP_{M_s,N_s}$, the generalized simple roots $A_{i,a;s} \in \ADP_{M_s,N_s}$ for $1 \leq i < s$, the ring $(\CEl)_s$, the $q$-character $\chi_q^s$ and the normalized $q$-character $\tilde{\chi}_q^s$ . We identify $\ADP_{M_s,N_s}$ (resp. $(\CEl)_s$) with a subset of $\ADP$ (resp. $\CEl$) in the following natural way:

f = (f_i: 1 ≤i ≤k) ↦f' = (f_i': 1 ≤i ≤M+N),  f_i' =
f_i     (i ≤k),

1     (i > k).

Then under this identification, $A_{i,a;s} = A_{i,a}$ for $1 \leq i < k$ as seen from Definition <ref>.

§.§ Proof of Theorem <ref> when $r \leq M$.

Fix $a \in \BC^{\times}$. Let us be in the situation of the first paragraph of <ref>, so that we have an inductive system of vector superspaces $(V_k, F_{k,l})$ with inductive limit $(V_{\infty}, F_k)$. Note that we have fixed a highest $\ell$-weight vector $v_k \in V_k$ for all $k \in \BZ_{>0}$ such that $F_{k,l} v_l = v_k$ whenever $k > l$. Moreover, we take $v_{\infty} = F_1(v_1)$. Denote $(\rho^k,V_k), (\rho^+,V_{\infty}), (\rho^b, V_{\infty})$ the representations of $Y_q(\Glie)$-modules $W_{k,aq^{2k}}^{(r)}, L_{r,a}^+, L_{r,a}(b)$. Let $t: \BZ_{>0} \longrightarrow \BZ_{>0}$ be a strictly increasing function such that the $F_{k,t(l)} \rho^k(s_{ij}^{(n)}) F_{k,l}$ for $k > t(l)$ are well-defined. (See Proposition <ref>.)

§.§.§ Compatibility of Gelfand-Tsetlin bases and structural maps.
Fix $l,k \in \BZ_{>0}$ such that $l < k$. Choose for all $l > 0$ a Gelfand-Tsetlin basis $(v_{\underline{\lambda}}: \underline{\lambda} \in \GT(l\varpi_r))$ of $V_l$ satisfying the properties of Corollary <ref>. (Note that $V_k \cong L(k\varpi_r)$ as $\Uc_q(\Glie)$-modules.)

In the above situation, we have: for all $\underline{\lambda} = (\lambda^{(i)}) \in \GT(l\varpi_r)$,

F_k,l v_λ ∈^× v_ μ

with $\underline{\mu} = (\mu^{(i)}) \in \GT(k\varpi_r)$ defined by $\mu^{(i)} = \lambda^{(i)} + (k-l) \begin{cases}
\varpi_i & (i \leq r), \\
\varpi_r & (i > r).
\end{cases}$

This comes from Remark <ref> combined with (the proof of) Proposition <ref> on relationship between Gelfand-Tsetlin vectors and $\ell$-weight spaces and on multiplicity-free property of $q$-character, in view of the definition of $F_{k,l}$ in <ref>.

§.§.§ Semi-infinite Young tableau and Gelfand-Tsetlin bases.
Based on Lemmas <ref> and <ref>, we introduce another index set of Gelfand-Tsetlin basis of $V_k$. This will be convenient for the statement of results.

Let $Y^{(r)}$ be the subset of $\BZ^2$ consisting of $(i,j)$ such that $1 \leq i \leq r$ and $j < 0$. Let $\CB^{(r)}$ be the set of functions $f: Y^{(r)} \longrightarrow I$ satisfying (T1)-(T3) in Definition <ref> and: for all $1 \leq i \leq r$, there exists $j < 0$ such that $f(i,j) = i$. Such an $f$ is also called a semi-infinite Young tableau. For $k > 0$, let $\CB^{(r)}_k$ be the subset of $\CB^{(r)}$ consisting of such $f$ that $f(i,j) = i$ for $j < -k$. We have therefore a chain of subsets of $\CB^{(r)}$:

_1^(r) ⊆_2^(r) ⊆_3^(r) ⊆⋯,   ^(r) = ⋃_k=1^∞ _k^(r).

There is a canonical bijective map $\pi_k:\CB(Y^{k\varpi_r}) \longrightarrow \CB_k^{(r)}$ sending $g$, a $\Glie$-Young tableau of shape $Y^{k\varpi_r}$, to a semi-infinite Young tableau $f$ where $f(i,j) = \begin{cases}
i & (j < -k), \\
g(i,j+k+1) & (j \geq -k).
\end{cases}$
Let $f_0 \in \CB^{(r)}$ be such that $f_0(i,j) = i$. By using Lemma <ref>, Corollary <ref> and Lemma <ref>, one can construct by induction on $k > 0$, a Gelfand-Tsetlin basis $\{ v[k,f] | f \in \CB_k^{(r)} \}$ of $V_k$ verifying the following properties.

(a) $v[k,f_0] = v_k$ for all $k > 0$.

(b) Let $l < k$ and $f \in \CB_l^{(r)}$. Then $F_{k,l} v[l,f] = v[k,f]$.

(c) $v[k,f] \in \BC^{\times} v_{\underline{\mu}}$ with $\underline{\mu} = GT_{k\varpi_r} (\pi_k^{-1}f) \in \GT(k\varpi_r)$.

For $f \in \CB^{(r)}$, define $v[f] := F_k v[k,f]$ whenever $f \in \CB_k^{(r)}$. Note that $v[f] \in V_{\infty}$ is well-defined thanks to (b). Moreover $v[f_0] = v_{\infty}$, and $\{v[f]\ |\ f \in \CB^{(r)} \}$ is a basis of $V_{\infty}$.

For $f \in \CB^{(r)}$ and $s \in I$, let $S_s(f)$ be the sub-vector-superspace of $V_{\infty}$ spanned by the $v[g]$ where $g \in \CB^{(r)}$  satisfies $g(i,j) = f(i,j)$ whenever $f(i,j) > s$ or $g(i,j) > s$. Clearly

v[f] ∈S_1(f) ⊆S_2(f) ⊆⋯⊆S_M+N-1(f) ⊆S_M+N(f) = V_∞.

For $1\leq i \leq \min(r,s)$, define $l_i := \sharp \{ j<0 \ |\ f(i,j) > s \}$. Set $\lambda_s(l,f) := \sum_{i=1}^{\min(r,s)} (l-l_i) \epsilon_i$. Let $f_{\downarrow s} \in \CB^{(r)}$ be the semi-infinite Young tableau obtained from $f$ by replacing any $f(i,j)\leq  s$ with $i$. At last,
let $\Phi^l,\Phi^+,\Phi^b \in \ADP$ such that

ρ^l(C_i(z)) v_l = (Φ^l)_i(z) v_l,  ρ^+(C_i(z)) v_∞ = (Φ^+)_i(z) v_∞,  ρ^b(C_i(z)) v_∞ = (Φ^b)_i(z) v_∞.

The precise formulas for $\Phi^l,\Phi^+,\Phi^b$ will be given in the proof of Lemma <ref>.

Let $f \in \CB_l^{(r)}$ and $s \in I$. The sub-vector-superspace $F_l^{-1} S_s(f) \subseteq V_l$ is a simple sub-$Y_q(\Glie_s)$-module isomorphic to $\ev_{aq^{2l}}^* L(\lambda_s(l,f); \Glie_s) \otimes D$ where $D$ is a sign module. Moreover, $v[l,f_{\downarrow s}]$ is a highest $\ell$-weight vector of the $Y_q(\Glie_s)$-module $F_l^{-1}S_s(f)$.

Indeed, $F_{l}^{-1} S_s(f)$ is the sub-vector-superspace of $V_l$ spanned by the $v[l,g]$ where $g \in \CB_l^{(r)}$ and $g(i,j) = f(i,j)$ whenever $g(i,j) > s$ or $f(i,j) > s$. In view of (c), the map $\pi_l$, and the correspondence between Young tableaux and Gelfand-Tsetlin patterns in Lemma <ref>, the lemma is evident.

For $s \in I$ and $f \in \CB^{(r)}$, the sub-vector-superspace $S_s(f)$ of $V_{\infty}$ is stable by $\rho^x(Y_q(\Glie_s))$ whenever $x \in \{+,b\}$.

By definition, $S_s(f)$ is stable by the $\rho^x(s_{ii}^{(0)})$. (See also Lemma <ref> below for more general statements.) It is enough to show that $S_s(f)$ is stable by the $\rho^x(\widehat{s}_{ij}^{(n)})$ where $n \in \BZ_{\geq 0}$ and $1 \leq i,j \leq s$. Let us assume $f \in \CB_{v}^{(r)}$.

For $k > v$, the preceding lemma says that  $F_{k}^{-1} S_s(f) \subseteq V_k$ is stable by the $\rho^k(\widehat{s}_{ij}^{(n)})$ with $1 \leq i,j \leq s$. It follows that ($v < k < t(k) < u$)

\begin{eqnarray*}
&\ & F_{u,t(k)}^{-1} \rho^u(\widehat{s}_{ij}^{(n)}) \underline{F_{u,k}  F_k^{-1}} S_s(f)  \subseteq F_{u,t(k)}^{-1} \underline{ \rho^u(\widehat{s}_{ij}^{(n)}) F_u^{-1} S_s(f)}  \\
&\subseteq & \underline{F_{u,t(k)}^{-1}  F_{u}^{-1}} S_s(f) \subseteq F_{t(k)}^{-1} S_s(f).
\end{eqnarray*}

On the other hand, we know from <ref> the following asymptotic formula

F_u,t(k)^-1  ρ^u(s_ij^(n)) F_u,k  = A_k + q^2u B_k    for u > t(k)

where $A_k,B_k: V_k \longrightarrow V_{t(k)}$ are linear operators depending only on $i,j,n$ and inductively on $k$. As $u$ can be arbitrarily large, we must have

A_k F_k^-1 S_s(f) ⊆F_t(k)^-1 S_s(f),  B_k F_k^-1S_s(f) ⊆F_t(k)^-1 S_s(f)

for all $k > v$. But this says exactly that $\rho^+(\widehat{s}_{ij}^{(n)})$ (resp. $\rho^b(\widehat{s}_{ij}^{(n)})$), being defined as inductive limit of the $A_k$ (resp. $A_k + b^2 B_k$), stabilizes $S_s(f)$, as desired.

Based on the proof of the above lemma, the following lemma is clear.

Let $f \in \CB^{(r)}$ and let $s \in I$. Then $v[f_{\downarrow s}] \in S_s(f)$ is a highest $\ell$-weight vector for the $Y_q(\Glie_s)$-modules $(\rho^x,S_{f})$ whenever $x \in \{+,b\}$. Moreover

\begin{align*}
& \rho^b(s_{ii}(z)) v[f_{\downarrow s}] = v[f_{\downarrow s}] \begin{cases}
b(q^{-l_i} - zaq^{l_i}) & (1 \leq i \leq \min(r,s)), \\
1 - z a b^2 & (\min(r,s) < i \leq s),
\end{cases} \\
& \rho^+(s_{ii}(z)) v[f_{\downarrow s}] = v[f_{\downarrow s}] \begin{cases}
q^{-l_i} - z a q^{l_i} & (1 \leq i \leq \min(r,s)), \\
1 & (\min(r,s) < i \leq s).
\end{cases}
\end{align*}

Here as usual $l_i := \sharp \{j< 0 \ |\ f(i,j) > s \}$ for $1 \leq i \leq \min(r,s)$.

Lemmas <ref>-<ref> can be rephrased as follows. The original inductive system $(V_k, F_{k,l})$ admits a sub-inductive-system of vector superspaces of the form:

( _aq^2k^* L(λ_s(k,f);_s), F_k,l ),  λ_s(k,f) = ∑_i=1^min(r,s) (k - l_i) ϵ_i.

The two asymptotic constructions in <ref> and in <ref> can be restricted to this sub-inductive-system, resulting in asymptotic modules $(\rho^+,S_s(f))$ and $(\rho^b,S_s(f))$ over $Y_q(\Glie_s)$.

For $f \in \CB^{(r)}$, there exists uniquely  $\Phi_f \in \ADP$ such that whenever $f \in \CB^{(r)}_l$:

ρ^l(C_i(z)) v[l,f] = (Φ^lΦ_f)_i(z) v[l,f],  ρ^x(C_i(z)) v[f] = (Φ^xΦ_f)_i(z) v[f],  x ∈{+,b}, i ∈I.

Moreover, $\Phi_f$ is a product of the $A_{i,x}^{-1}$ where $i \in I_0$ and $x \in a q^{2\BZ + 1}$.

For such $l$ and $f$, $v[l,f]$ is a Gelfand-Tsetlin vector. In view of Proposition <ref>, $\BC v[l,f]$ is an $\ell$-weight space of $(\rho^l,V_l)$. Hence by Corollary <ref>  there exists a unique $\Phi_f$ as a product of the $A_{i,x}^{-1}$ with $i \in I_0, x \in a q^{2\BZ +1}$ such that

ρ^l (C_i(z)) v[l,f] = (Φ^lΦ_f)_i(z) v[l,f].

Now Lemma <ref> says that $\rho^k(C_i(z)) v[k,f] = (\Phi^k\Phi_f)_i(z) v[k,f]$ for all $k > l$.

To compute $\rho^b(C_i(z)) v[f]$, remark that for all $k > l$, by assumption $F_{k,l} v[l,f] = v[k,f]$. Hence for all $k > t(l)$

F_k,t(l)^-1 ρ^k(C_i(z))F_k,l v[l,f] = (Φ^kΦ_f)_i(z) v[t(l),f]

On the other hand, in view of the following explicit formulas for $\Phi^k,\Phi^+,\Phi^b$:

\begin{eqnarray*}
(\Phi^k)_i(z) &=& \begin{cases}
q^{ki} \prod_{j=1}^i (1-za\theta_j^{-1}) & (i \leq r), \\
q^{kr} \prod_{j=1}^r (1-za\theta_j^{-1}) \times \prod_{j=r+1}^i(1-za\theta_j^{-1}q^{2k})^{d_j} & (i > r),
\end{cases} \\
(\Phi^+)_i(z) &=&  \begin{cases}
\prod_{j=1}^i (1-za\theta_j^{-1}) & (i \leq r), \\
\prod_{j=1}^r (1-za\theta_j^{-1}) & (i > r),
\end{cases}  \\
(\Phi^b)_i(z) &=&  \begin{cases}
b^i \prod_{j=1}^i (1-za\theta_j^{-1}) & (i \leq r), \\
b^r \prod_{j=1}^r (1-za\theta_j^{-1}) \times \prod_{j=r+1}^i(1-za\theta_j^{-1}b^2)^{d_j} & (i > r),
\end{cases}
\end{eqnarray*}

we see from the construction of $\rho^b$ in <ref> that (by replacing $q^k$ with $b$ everywhere)

ρ^b (C_i(z)) v[f] = (Φ^k Φ_f)_i(z)|_q^k →b v[f] = (Φ^bΦ_f)_i(z) v[f].

Finally, for $\rho^+(C_i(z)) v[f]$, from the above formulas of $\Phi^k,\Phi^+$ we deduce that there exists a formal power series $h_i(w) \in 1 + w\BC[[w]]$ for $i \in I$ such that for all $k > t(l)$,

\begin{eqnarray*}
&&(\Phi^k)_i(z) (\Phi^k)_i(0)^{-1} = (\Phi^+)_i(z) h_i(zq^k), \quad (\Phi^+)_i(0) = 1  \\
&&F_{k,t(l)}^{-1} \rho^k(C_i(z)C_i(0)^{-1}) F_{k,l} v[l,f] = h_i(zq^k) (\Phi^+ \Phi_f)_i(z) (\Phi^+\Phi_f)_i(0)^{-1}  v[t(l),f].
\end{eqnarray*}

From Gauss decomposition observe  that $C_i(z)C_i(0)^{-1} \in \widehat{Y}_q(\Glie)[[z]]$. Now from the construction of $\rho^+$ in <ref> we obtain (by taking $q^k = 0$)

ρ^+(C_i(z)C_i(0)^-1) v[f] = (Φ^+Φ_f)_i(z) (Φ^+Φ_f)_i(0)^-1 v[f].

It is therefore enough to compute $\rho^+(C_i(0)) v[f]$. For this, note that in $(\rho^l,V_l)$,

|v[l,f]|_ = lϖ_r + ∑_i=1^r∑_j=1^l (ϵ_f(i,j-l-1) - ϵ_i).

Hence in $(\rho^+,V_{\infty})$, the $\BQ$-degree of $v[f]$ is given by the second term of the RHS. From the proof of Proposition <ref> we get an explicit expression of the $(\Phi_f)_i(0)$:

ρ^+(C_i(0)) v[f] = (Φ_f)_i(0) v[f] = (Φ^+Φ_f)_i(0) v[f].

In consequence, $\rho^+(C_i(z)) v[f] = (\Phi^+\Phi_f)_i(z) v[f]$, as desired.

We arrive at the following important consequence of the preceding lemmas: simplicity of $S_{M+N}(f)$ for $\rho^+$ and $\rho^b$.

Let $1 \leq r \leq M$ and $a,b \in \BC^{\times}$. Then

χ_q (L_r,a^+) = χ_q (L_r,a(b)) = lim_k →∞ χ_q (W_k,aq^2k^(r))

as formal power series in the $[A_{i,x}]^{-1}$ where $i \in I_0$ and $x \in a q^{2\BZ+1}$. Furthermore, $L_{r,a}^+$ is a simple $Y_q(\Glie)$-module. $L_{r,a}(b)$ is simple if $b \notin \pm q^{\BZ_{>-r}}$.

Let us explain firstly that the limit of normal $q$-characters above makes sense. For $V$ a module in category $\BGG$ with well-defined normalized character (Remark <ref>), set

χ_q (V) = ∑_Φ∈ d(Φ, V) [Φ] ∈.

Then according to Proposition <ref>, for $k \in \BZ_{>0}$ and $\Phi \in \ADP$, either $d(\Phi, V_k) = 0$, or $d(\Phi, V_k) = 1$, in which case $\Phi$ is a product of the $A_{i,x}^{-1}$ with $i \in I_0, x \in a q^{2\BZ+1}$. Furthermore, Lemma <ref> indicates that for $l < k$

d(Φ, V_l) ≤d(Φ, V_k).

Combining with the facts that $\widetilde{\chi}_q(V)$ is a refinement of $\widetilde{\chi}(V)$ and that $\lim\limits_{k\rightarrow \infty} \widetilde{\chi}(V_k)$ exists (Equation (<ref>)), we conclude that so does $\lim\limits_{k\rightarrow\infty} \widetilde{\chi}_q (V_k)$, and it is a formal power series in the $[A_{i,x}]^{-1}$ (where $i \in I_0, x \in a q^{2\BZ+1}$) with coefficients  $0$ or $1$.

Next the character formulas are immediate consequences of Lemma <ref>:

χ_q (L_r,a^+) = χ_q (L_r,a(b)) = ∑_f ∈^(r) [Φ_f] =  lim_k →∞ χ_q (W_k,aq^2k^(r)).

Thirdly we show that $L_{r,a}^+$ is a simple $Y_q(\Glie)$-module, imitating the proof of <cit.>. According to Proposition <ref>, $L_{r,a}^+$ has a simple sub-quotient, denoted by $S_{r,a}^+$, which contains a highest $\ell$-weight vector $w$ such that $s_{ii}(z) w = w \begin{cases}
1 - z a & (i \leq r), \\
1 & (i > r).
\end{cases}$ Furthermore, for $x \in \BC^{\times}$, there exists a simple $Y_q(\Glie)$-module $S_{r,x}^-$ in category $\BGG$ containing a highest $\ell$-weight vector $u_x$ such that $s_{ii}(z) u_x = u_x  \begin{cases}
1 & (i \leq r), \\
1 - z x & (i > r).
\end{cases}$
Let us show that $\widetilde{\chi}_q (L_{r,a}^+) = \widetilde{\chi}_q(S_{r,a}^+)$. In other words, if $l > 0$ and $d(\Phi, V_l) = 1$, then $\Phi$ must appear in $\widetilde{\chi}_q (S_{r,a}^+)$. Indeed, for all $k > l$, we can find a one-dimensional $Y_q(\Glie)$-module $D$ such that:

(F) $D \otimes V_k$ is a  simple sub-quotient of $S_{r,a}^+ \otimes S_{r,aq^{2k}}^-$.

It follows from Remark <ref> that $\Phi$ must appear in $\widetilde{\chi}_q (S_{r,a}^+) \widetilde{\chi}_q (S_{r,aq^{2k}}^-)$. In other words, there exists $\Phi_k^+$ (resp. $\Phi_k^-$) in $\widetilde{\chi}_q (S_{r,a}^+)$ (resp. in $\widetilde{\chi}_q (S_{r,aq^{2k}}^-)$) such that

Φ= Φ_k^+ Φ_k^-.

Since  $\varpi(\Phi),\varpi(\Phi_{\pm}^k) \in - \BQ_{\geq 0}$, the set $\{\Phi_k^+ : k > l \}$ is finite. The pigeonhole principle indicates that there exists an infinite sequence $(k_n)_{n > 0}$ of integers such that

l < k_1 < k_2 < k_3 < ⋯,   Φ_k_1^- = Φ_k_2^- = Φ_k_3^- = ⋯.

Lemma <ref> below says that $\Phi^{-}_{k_1} \in \sigma \DP$, so does $\Phi_{k_1}^+ \Phi^{-1}$. The later, as a product of the $A_{i,x}^{\pm 1}$, must be $1$. So $\Phi = \Phi^+_{k_1}$ appears in $\widetilde{\chi}_q (S_{r,a}^+)$, as desired.

At last we show that $L_{r,a}(b)$ is simple provided that $b \notin \pm q^{\BZ_{-r}}$. For this, let $S(r,a,b)$ be a simple sub-quotient of $L_{r,a}(b)$ containing a highest $\ell$-weight vector $u$ such that $s_{ii}(z) u = u \begin{cases}
b - z a b & (i \leq r), \\
1 - z a b^2 & (i > r).
\end{cases}$ Let $l > 0$. There exists a one-dimensional $Y_q(\Glie)$-module $D$ such that:

(F1) $D \otimes V_l$ is a simple sub-quotient of $S(r,a,b) \otimes S(r,ab^2,b^{-1}q^l)$.

Assume $d(\Phi, V_l) = 1$. There exist a monomial $\Phi_1$ in $\widetilde{\chi}_q (S(r,a,b))$ and a monomial $\Phi_2$ in $\widetilde{\chi}_q (S(r,ab^2, b^{-1}q^l))$ such that $\Phi = \Phi_1 \Phi_2$. On the other hand, from the definition of $S(r,a,b)$ and from the normalized $q$-character of $L_{r,a}(b)$ we deduce that there exists $k > l$ such that

d(Φ_1, W_k,aq^2k^(r)) = 1 = d(Φ_2, W_k,ab^2q^2k^(r)).

Note that $d(\Phi, W_{k,aq^{2k}}^{(r)}) = 1$ from Lemma <ref>. Now Lemma <ref> forces $\Phi_2 = 1$. Hence $\Phi = \Phi_1$ appears in $\widetilde{\chi}_q (S(r,a,b))$. It follows that $L_{r,a}(b) \cong S(r,a,b)$ is simple.

Let $W$ be a $Y_q(\Glie)$-module in category $\BGG$ admitting normalized character. Let $(k_n)_{n \in \BZ_{>0}}$ be a strictly increasing sequence of integers. For $n \in \BZ_{>0}$ take $W_n := \Phi_{q^{2k_n}}^* W$. If $\Phi \in \ADP$ is such that $d(\Phi, W_n) > 0$ for all $n$, then $\Phi \in \sigma \DP$.

Let $\theta: \ADP \longrightarrow \ADP$ be the group isomorphism $(f_i(z)) \mapsto (f_i(zq))$. Then for $n \in \BZ_{>0}$,

χ_q (W_n) = ∑_Ψ∈ d(Ψ, W) [θ^2k_n Ψ].

One can therefore find $\Phi_n \in \ADP$ such that $\Phi = \theta^{2k_n} \Phi_n$ and $d(\Phi_n, W) > 0$. As $W$ is in category $\BGG$ and the $\varpi(\Phi_n)$ are the same, $\{\Phi_n\ | \ n > 0\}$ is a finite set. By pigeonhole principle, there exists a sub-sequence $(l_n)_{n\in \BZ_{>0}}$ of strictly increasing integers such that $\Phi_{l_1} = \Phi_{l_n}$ for all $n > 0$. It follows that $\Phi_{l_1} = \theta^{2l_2-2l_1} \Phi_{l_1}$. This forces $\Phi_{l_1} \in \sigma (\DP)$, so does $\Phi$.

Corollary <ref> says that $L_{r,a}(b)$ is simple if $b \in \pm q^{\BZ_{>-r}}$. In the case $r = M$, it is possible to determine the $b \in \pm q^{\BZ_{-r}}$ making $L_{M,a}(b)$ non-simple as $L_{M,a}(b)$ is of dimension $2^{MN}$. In the case $r < M$, if $b \in \pm q^k$ with $k \geq 0$, then $L_{r,a}(b)$ has a finite-dimensional simple sub-quotient isomorphic to $W_{k,aq^{2k}}^{(r)}$ (up to tensor product by one-dimensional modules, with the convention that $W_{0,a}^{(r)}$ is the trivial module). Since $L_{r,a}(b)$ is infinite-dimensional, it is not simple. It remains to find integers $1 \leq s < r$ such that $L_{r,a}(\pm q^{-s})$ is not simple. Unfortunately, limited by Lemma <ref>, we still do not know the answer.

The proof of Corollary <ref> applied perfectly if we replace $S_{M+N}(f)$ with more general $S_s(f)$, we obtain the following corollary, whose proof is omitted, on normalized $q$-character formulas and on simplicity of asymptotic modules.

Let $f \in \CB_l^{(r)}, s \in I$ and let $b \in \CB^{\times}$.

(1) The $Y_q(\Glie_s)$-modules $(\rho^+, S_s(f))$ and $(\rho^b, S_s(f))$ are in category $\BGG_s$ and

χ_q^s (ρ^+, S_s(f)) = χ_q^s (ρ^b, S_s(f)) = lim_k→∞ χ_q^s (_aq^2k^* L(λ_s(k,f); _s))

as formal power series in the $[A_{i,x}]^{-1}$ with $1 \leq i < s$ and $x \in a q^{2\BZ+1}$.

(2) The $Y_q(\Glie_s)$-module $(\rho^+, S_s(f))$ is simple, while the $Y_q(\Glie_s)$-module $(\rho^b,S_s(f))$ is simple if $b \notin \pm q^{\BZ}$.

§.§.§ End of proof of Theorem <ref> when $r \leq M$.

Assume $b \notin \pm q^{\BZ}$. Let $x \in \{+,b\}$. Consider the $Y_q(\Glie)$-module $(\rho^x, V_{\infty})$. According to Corollary <ref>, its basis $\{v[f]\ |\ f \in \CB^{(r)} \}$ together with the chains (one for each $f \in \CB^{(r)}$)

v[f] ∈S_1(f) ⊆S_2(f) ⊆S_3(f) ⊆⋯⊆S_M+N-1(f) ⊆S_M+N(f) = V_∞

satisfies (B1)-(B2) in the definition of a Gelfand-Tsetlin basis in <ref>. Observe that for $f,g \in \CB^{(r)}$ and $s \in I$, by definition of $S_s(f)$, Lemma <ref> and Corollary <ref> the following statements are equivalent:

(1) $v[f] \in S_s(g)$;

(2) $\{(i,j) \in Y^{(r)}\ |\ f(i,j) > s \} = \{(i,j) \in Y^{(r)}\ |\ g(i,j)  > s \}$;

(3) $(\rho^x,S_s(f))$ and $(\rho^x,S_s(g))$ are isomorphic as $Y_q(\Glie_s)$-modules;

(4) $S_s(f) = S_s(g)$.

Now (B3) on multiplicity-free property of the decomposition of $\Res_{Y_q(\Glie_{s-1})}^{Y_q(\Glie_s)}(\rho^x, S_s(f))$ into simple sub-$Y_q(\Glie_{s-1})$-modules, which are the $S_{s-1}(g)$, is obvious.  $\Box$

§.§ Proof of Theorem <ref> when $r > M$.
The proof is almost identical to the case $r \leq M$. We explain the main steps.

Take $V_l := W_{l,a}^{(r)}$ which is a tensor product of $\ev_a^*L(l\varpi_r)$ with a sign module. We borrow the notations from <ref>, so that $v_k \in V_l, v_{\infty} \in V_{\infty}, F_{k,l}, F_k, \rho^k, \rho^+, \rho^b, \Phi^k,\Phi^+,\Phi^b$ and the strictly increasing function $t$ will be used freely. In the present case,

\begin{eqnarray*}
(\Phi^k)_i(z) &=& \begin{cases}
\prod_{j=1}^i (1-za\theta_j^{-1})^{d_j} & (i \leq r), \\
\prod_{j=1}^r(1-za\theta_j^{-1})^{d_j} \times \prod_{j=r+1}^i (q^k - z a \theta_j^{-1} q^{-k})^{-1} & (i > r),
\end{cases}  \\
(\Phi^b)_i(z) &=& \begin{cases}
\prod_{j=1}^i (1-za\theta_j^{-1})^{d_j} & (i \leq r), \\
\prod_{j=1}^r(1-za\theta_j^{-1})^{d_j} \times \prod_{j=r+1}^i (b - z a \theta_j^{-1} b^{-1})^{-1} & (i > r),
\end{cases}  \\
(\Phi^+)_i(z) &=& \begin{cases}
\prod_{j=1}^i (1-za\theta_j^{-1})^{d_j} & (i \leq r), \\
\prod_{j=1}^r(1-za\theta_j^{-1})^{d_j}  & (i > r).
\end{cases}
\end{eqnarray*}

Set $Y^{(r)}$ to be the subset of $\BZ^2$ consisting of $(i,j)$ such that $1 \leq j \leq M+N-r$ and $i > 0$. Define $\CB^{(r)}$ to be the set of functions $f: Y^{(r)} \longrightarrow I$ satisfying (T1)-(T3) in Definition <ref> and: for all $1\leq j \leq M+N-r$, there exists $i>0$ such that $f(i,j) = r+j$. Let $\CB^{(r)}_k$ be the subset of $\CB^{(r)}$ consisting of $f$ such that $f(k+1,j) = r+j$ for all $1 \leq j \leq M+N-r$.

For $s \in I$ and $f \in \CB^{(r)}_l$, the subset $Y_s(l,f)$ of $Y^{(r)}$ consisting of $(i,j)$ such that $i\leq l$ and $f(i,j) \leq s$ is easily seen to be a $\Glie_s$-Young diagram. Set $\mu_s(l,f) := \Ym_{M_s,N_s}(Y_s(l,f))$. (See Remark <ref>.) Let $\lambda_s(l,f) \in \BP_{M_s,N_s}$ be the highest weight of the simple $\Uc_q(\Glie_s)$-module $L(\mu_s(f);\Glie_s)^*$. Then we have the following observation.

(1) If $s \leq r$, then $\lambda_s(l,f) = \lambda_s(k,f)$ whenever $l < k$ and $f \in \CB^{(r)}_{l}$.

(2) If $s > r$, then $\lambda_s(k,f) = \lambda_s(l,f) - (k-l)\sum_{j=r+1}^s \epsilon_j$ whenever $l < k$ and $f \in \CB^{(r)}_{l}$.

Set $f_0 \in \CB^{(r)}$ to be such that $f(i,j) = j+r$ for all $1 \leq j \leq M+N-r$.

According to <ref>, for all $l > 0$, there is a dual Gelfand-Tsetlin basis $\{ v_{l,f}\ |\ f \in \CB^{(r)}_l \}$ of $V_l$ such that for all $s \in I$ and $f \in \CB_l^{(r)}$ the vector $v_{l,f}$ is contained in a simple sub-$Y_q(\Glie_s)$-module of $V_l$ of the form:

_a^* L(λ_s(l,f);_s) ⊗D ≅_a^* L(μ_s(l,f);_s)^* ⊗D

where $D$ is a sign module (irrelevant to us). Now from Proposition <ref>, Corollary <ref>, their proofs, and the construction of $F_{k,l}$ in <ref> we conclude that

For $l < k, f \in \CB_l^{(r)}$ we have $F_{k,l} v_{l,f} \in \BC^{\times} v_{k,f}$.

Now we can choose by induction on $k$ a basis $\{v[k,f]\ |\ f \in \CB_k^{(r)} \}$ of $V_k$ such that

(a) $F_{k,l} v[l,f] = v[k,f]$ whenever $k > l$ and $f \in \CB_l^{(r)}$;

(b) $v[l,f_0] = v_l$ for all $l > 0$;

(c) $v[l,f] \in \BC^{\times} v_{l,f}$ for $f \in \CB_l^{(r)}$.

Define $v[f] := F_k v[k,f] \in V_{\infty}$ whenever $f \in \CB^{(r)}_k$. As before, for $f \in \CB^{(r)}$ and $s \in I$, let $S_s(f)$ be the sub-vector-superspace of $V_{\infty}$ spanned by the $v[g]$ where $g \in \CB^{(r)}$ verifies $g(i,j) = f(i,j)$ whenever one of them is bigger than $s$.
Now Corollary <ref> stays the same in our situation, as the proof of which requires only the multiplicity-free property of normalized $q$-character of evaluation modules. As in <ref>, we conclude the proof of Theorem <ref> in the case $r > M$.  $\Box$

Remark <ref> remains true in our situation. Let $s \in I$ and let $f \in \CB^{(r)}_l$. Write

λ_s(l,f) = ∑_i=1^s x_i ϵ_i.

Let $v[f_{\downarrow s}] \in S_s(f)$ be a highest $\ell$-weight vector for both $(\rho^+,S_s(f))$ and $(\rho^b,S_s(f))$. If $s \leq r$, then $S_s(f)$ is finite-dimensional and for $1 \leq i \leq s$

ρ^+(s_ii(z)) v[f_↓s] =  ρ^b(s_ii(z)) v[f_↓s] = (q_i^x_i - za q_i^-x_i) v[f_↓s].

If $s > r$, then $S_s(f)$ is infinite-dimensional and

\begin{eqnarray*}
\rho^+(s_{ii}(z)) v[f_{\downarrow s}] &=& v[f_{\downarrow s}] \begin{cases}
q_i^{x_i} - z a q_i^{-x_i} & (1 \leq i \leq r),  \\
q^{-x_i-l} & (r < i \leq s),
\end{cases} \\
\rho^b(s_{ii}(z)) v[f_{\downarrow s}] &=& v[f_{\downarrow s}] \begin{cases}
q_i^{x_i} - z a q_i^{-x_i} & (1 \leq i \leq r),  \\
bq^{-x_i-l} - z a b^{-1} q^{l+x_i} & (r < i \leq s).
\end{cases}
\end{eqnarray*}

In this case, $(\rho^+,S_s(f))$ and $(\rho^b,S_s(f))$ can be seen as the asymptotic modules obtained from the following sub-inductive-system

( _a^* L(λ_s(k,f);_s) ⊗D_k, F_k,l )

of the original one $(V_k,F_{k,l})$ by carrying out the asymptotic constructions in <ref> and <ref>.
Here the $D_k$ are some properly defined sign modules.

§.§ Consequences on normalized $q$-character.
Let us go back to <ref>. The following corollary is now direct in view of Corollary <ref> (which holds for all $r \in I_0$) applied to $S_{M+N}(f)$ and Examples <ref>-<ref>.

Let $f = (f_i:  i \in I_0) \in \CR$. Assume that  $f_i(\infty) \neq 0$ for all $i \in I_0$. Then $\widetilde{\chi}_q (V(f))$ is a formal power series in the $[A_{i,x}]^{-1}$ where $i \in I_0$ and $x \in \BC^{\times}$. In particular, if $S$ is a finite-dimensional simple $U_q(\Gaff)$-module, then the normalized $q$-character $\widetilde{\chi}_q (S)$ is a polynomial in the $[A_{i,x}]^{-1}$.

Let $r \in I_0$ be such that $M< r < M+N$. Consider the $U_q(\Gaff)$-module $L_{r,a}(b)$. By pulling it back with respect to $f_{J,I}: U_q(\Gafft) \longrightarrow U_q(\Gaff)$, we get a $U_q(\Gafft)$-module $f_{J,I}^* L_{r,a}(b)$ such that

s_ii;J(z) f_J,I^* v = f_J,I^* v
b-zab^-1     (i ≤M+N-r),

1-za     (i > M+N-r)

where $v \in L_{r,a}(b)$ is a highest $\ell$-weight vector of $L_{r,a}(b)$. By comparing the normalized characters and highest $\ell$-weights of $f_{J,I}^* L_{r,a}(b)$ and $L_{M+N-r,ab^{-2};J}(b)$ we conclude that

Let $M< r < M+N$ and $a,b \in \BC^{\times}$. Assume that $b \notin \pm q^{\BZ_{>-M-N+r}}$. Then the $U_q(\Gaff)$-module $L_{r,a}(b)$ is simple.

Let us rephrase Lemma <ref> and Corollary <ref> in another way. Fix $i \in I_0$ and $a \in \BC^{\times}$. For $b \in \BC \setminus \{a\}$,  set $\theta_{i,a,b} \in \CR$ to be

(θ_i,a,b)_j = 1 (j ≠i),  (θ_i,a,b)_i = 1-za/1-zb.

Then whenever $b$ is generic (for example $b \notin q^{2\BZ}$), we have

χ_q (V(θ_i,a,b)) = χ_q (V(θ_i,a,0)).

In other words, when $b$ is generic, $\widetilde{\chi}_q (V(\theta_{i,a,b}))$ is independent of $b$. Note that a similar phenomenon happens for more general rational functions than the $\frac{1-za}{1-zb}$, as we have observed in Equation (<ref>) concerning the $q$-Yangian $Y_q(\mathfrak{gl}(1,1))$.

[1]R. Baxter,
Partition function of the eight-vertex lattice model,
Ann. Phys. 70 (1972), 193-228.

[2]N. Beisert, W. Galleas and T. Matsumoto,
A quantum affine algebra for the deformed Hubbard chain,
J. Phys. A: Math. Theor. 45 (2012), 365206.

[3]V. Bazhanov, A. Hibberd and A. Khoroshkin,
Integrable structure of $W_3$ conformal field theory, quantum Boussinesq theory and boundary affine Toda theory,
Nucl. Phys. B 622 (2002), 475-547.

[4]G. Benkart, S. Kang and M. Kashiwara,
Crystal bases for the quantum superalgebra $U_q(\mathfrak{gl}(m,n))$,
J. Amer. Math. Soc. 13 (2000), 295-331.

[5]V. Bazhanov and S. Lukyanov,
Integrable structure of quantum field theory: classical flat connections versus quantum stationary states,
Preprint arXiv: 1310.4390.

[6]V. Bazhanov, S. Lukyanov and A. Zamolodchikov,
Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz,
Commun. Math. Phys. 177 (1996), 381-398.

[7]V. Bazhanov, S. Lukyanov and A. Zamolodchikov,
Integrable structure of conformal field theory II. Q-operator and DDV equation,
Commun. Math. Phys. 190 (1997), 247-278.

[8]V. Bazhanov, S. Lukyanov and A. Zamolodchikov,
Integrable structure of conformal field theory III. The Yang-Baxter relation,
Commun. Math. Phys. 200 (1999), 297-324.

[9]V. Bazhanov and Z. Tsuboi,
Baxter's $Q$-operators for supersymmetric spin chains,
Nuclear Phys. B 805 (2008), 451-516.

[10]I. Damiani,
La $\mathcal{R}$-matrice pour les algèbres quantiques de type affine non tordu,
Ann. Sci. Ecole Norm. Sup. 31 (1998), 493-523.

[11]E. Frenkel and D. Hernandez,
Baxter's relations and spectra of quantum integrable models,
Preprint arXiv:1308.3444.

[12]E. Frenkel and E. Mukhin,
The Hopf algebra $\textrm{Rep} (U_q\widehat{\mathfrak{gl}_{\infty}})$,
Sel. Math., New Ser. 8 (2002), no. 4, 537-635.

[13]E. Frenkel and N. Reshetikhin,
The $q$-character of representations of quantum affine algebras and deformations of $\mathcal{W}$-algebras,
Recent Developments in Quantum Affine Algebras and related topics, Cont. Math. 248 (1999), 163-205.

[14] L. Faddeev, N. Reshetikhin and L. Takhtajan,
Quantization of Lie groups and Lie algebras,
Leningrad Math. J. 1 (1990), 193-225.

[15]L. Faddeev, N. Reshetikhin and L. Takhtajan,
Quantization of Lie groups and Lie algebras,
Yang-Baxter equation in Integrable Systems, Advanced Series in Mathematical Physics, Vol 10, Singapore (1989), 299-309.

[16]R. Gade,
Universal $R$-matrix and graded Hopf algebra structure of $U_q(\widehat{gl}(2|2))$,
J. Phys. A: Math. Gen. 31 (1998), 4909-4925.

[17] D. Hernandez,
Representations of quantum affinizations and fusion product,
Transform. Groups 10 (2005), no. 2, 163-200.

[18]D. Hernandez,
Simple tensor products,
Invent. Math. 181 (2010), 649-675.

[19]D. Hernandez,
Spectre des systèmes intégrables quantiques et représentations linéaires,
Gazette des Mathématiciens 141 (2014), 34-47.

[20] D. Hernandez and M. Jimbo,
Asymptotic representations and Drinfeld rational fractions,
Compos. Math. 148 (2012), no. 5, 1593-1623.

[21]I. Heckenberger, F. Spill, A. Torrielli and H. Yamane,
Drinfeld second realization of the quantum affine superalgebras of $D^{(1)}(2,1;x)$ via the Weyl groupoid,
Publ. RIMS. Kyoto B 8 (2008), 171-206.

[22]I. Ip and A. Zeitlin,
Q-operator and fusion relations for $C_q^{(2)}(2)$,
Lett. Math. Phys. 104 (2014), 1019-1043.

[23]P. Kulish and A. Zeitlin,
Superconformal field theory and SUSY $N=1$ KdV hierarchy II: the $Q$-operator,
Nuclear Phys. B 709 (2005), 578-591.

[24]E. Mukhin and C. Yang,
Affinization of category $\mathcal{O}$ for quantum groups,
Trans. Amer. Math. Soc. 2014, arXiv:1204.2769.

[25] A. Molev,
Gelfand-Tsetlin bases for classical Lie algebras,
Handbook of Algebra, Vol 4, ed. M. Hazewinkel, Elsevier 2006, pp. 109-170.

[26]A. Molev,
Combinatorial bases for covariant representations of the Lie superalgebra $\mathfrak{gl}_{m|n}$,
Bulletin of the Institute of Mathematics, Academia Sinica 6 (2011), 415-462.

[27]T. Palev, N. Stoilova and J. Van der Jeugt,
Finite-dimensional representations of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$ and related $q$-identities,
Commun. Math. Phys. 166 (1994), 367-378.

[28]V. Serganova,
Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra $\mathfrak{gl}(m|n)$,
Sel. Math., New Series 2 (1996), 607-651.

[29]Z. Tsuboi,
Asymptotic representations and $q$-oscillator solutions of the graded Yang-Baxter equation related to Baxter $Q$-operators,
Preprint arXiv:1205.1471.

[30] H. Zhang,
Representations of quantum affine superalgebras,
to appear in Math. Z., Preprint arXiv:1309.5250.

[31]H. Zhang,
RTT realization of quantum affine superalgebras and tensor products,
Preprint arXiv:1407.7001.


# Definition of gravitational local spatial densities for spin-0 and spin-1/2
systems

J. Yu. Panteleeva  Institut für Theoretische Physik II, Ruhr-Universität
Bochum, D-44780 Bochum, Germany  E. Epelbaum  Institut für Theoretische Physik
II, Ruhr-Universität Bochum, D-44780 Bochum, Germany  J. Gegelia  Institut für
Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany  High
Energy Physics Institute, Tbilisi State University, 0186 Tbilisi, Georgia
U.-G. Meißner  Helmholtz Institut für Strahlen- und Kernphysik and Bethe
Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany
Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center
for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany  Tbilisi
State University, 0186 Tbilisi, Georgia

(10 July 2023)

###### Abstract

We work out details of defining the spatial densities corresponding to the
gravitational form factors of spin-0 and spin-1/2 systems using spherically
symmetric sharply localized wave packets. The expressions for the spatial
densities are provided in the frames with both zero and non-zero expectation
values of the momentum operator.

##  I Introduction

In close analogy with the electric charge density of hadrons Hofstadter:1958 ;
Ernst:1960zza ; Sachs:1962zzc , the interpretation of Fourier transforms of
gravitational form factors in the Breit frame as local densities of various
physical quantities characterizing the system was suggested in Refs.
Polyakov:2002wz ; Polyakov:2002yz ; Polyakov:2018zvc . For systems whose
intrinsic size is comparable with their Compton wavelength, this
identification of spatial density distributions with the Fourier transforms of
the electromagnetic and gravitational form factors was criticized in Refs.
Burkardt:2000za ; Miller:2007uy ; Miller:2009qu ; Miller:2010nz ;
Jaffe:2020ebz ; Miller:2018ybm ; Freese:2021czn . This issue has raised much
interest recently Lorce:2020onh ; Lorce:2022cle ; Guo:2021aik ; Lorce:2018egm
; Freese:2021mzg ; Chen:2022smg ; Freese:2022fat ; Carlson:2022eps . The
light-front approach has been used to define purely intrinsic electromagnetic
spatial densities as two-dimensional distributions in the impact parameter
space Burkardt:2000za ; Miller:2007uy ; Miller:2009qu ; Miller:2010nz ;
Guo:2021aik . The relationship between these densities and the non-
relativistic three-dimensional distributions in the Breit frame in terms of
the Abel transform was studied in Refs. Panteleeva:2021iip ; Freese:2021mzg .
The phase-space approach of Refs. Lorce:2020onh ; Lorce:2022cle ;
Lorce:2018egm  allows one to define fully relativistic three-dimensional
spatial densities. However, these densities do not have a strict probabilistic
interpretation due to their dependence on both coordinates and momenta. In
Refs. Freese:2021czn ; Lorce:2018egm , the two-dimensional light-front Fourier
transforms of the energy-momentum tensor form factors have been interpreted as
the spatial densities of the energy-momentum and the internal forces in
hadrons. In Ref. Epelbaum:2022fjc , a definition of spatial densities of local
operators for systems with arbitrary Compton wavelengths has been suggested by
considering an example of the charge density of a spin-$0$ system, see also
Ref. Fleming:1974af  for an early study along this line. Recently this novel
definition was also applied to the electromagnetic spatial densities for
spin-1/2 systems Panteleeva:2022khw . For illuminating comments and extensions
on the approach using sharply localized packets see Ref. Carlson:2022eps .

In this work we consider the one-particle matrix elements of the energy-
momentum tensor (EMT) in localized states of spin-0 and spin-1/2 systems.
Specifying the one-particle state by a spherically symmetric wave packet and
sharply localizing it in space we define spatial distributions corresponding
to gravitational form factors in the zero average momentum frame (ZAMF). Next,
we generalize the new definition to Lorentz-boosted frames. Analogously to
Ref. Jaffe:2020ebz , which dealt with the electromagnetic case, we also
consider the static approximation and discuss the limitations of the results
obtained in this limit.

Recently in Ref. Freese:2022fat  it has been claimed that the idea of
obtaining internal densities via wave packet localization is fallacious,
because it does not reproduce the “true internal densities”. The authors first
clarify what they understand under true internal densities of hadrons and how
these are related to the physical densities obtained from expectation values
of local operators. Next, using the formulated criteria they observe that the
wave packet localization does not result in meaningful internal densities for
instant form coordinates. We believe that the parameterization of the physical
densities in terms of internal densities and smearing functions which is
suggested in Ref. Freese:2022fat  as the basis for defining the true internal
densities has an oversimplified form and is inadequate for a complicated
hadronic system in a state described by a wave packet.

Our work is organized as follows. In Sec. II we define spatial densities of
the EMT in the ZAMFs of a spin-0 and a spin-1/2 systems and consider the
static approximation. Spatial densities in moving reference frames are
considered in Sec. III. Section IV contains our interpretation of the obtained
results and we summarize in Sec. V.

##  II Gravitational densities in the zero average momentum frame

Spatial densities of the EMT defined via localized states differ significantly
from the ones of the electromagnetic current. A superposition of eigenstates
of the charge operator, which makes the localized packet, remains an
eigenstate of the charge operator with the same eigenvalue. However this is
not the case for the energy-momentum operator. That is, a packet which is a
superposition of one-particle eigenstates of the four-momentum operator with
different four-momenta is not an eigenstate of the same operator. On the other
hand, such a state does not contain admixtures of states with particle-
antiparticle pairs and, therefore, can be used to define spatial distributions
for one-particle states.

Below we work out details of the spatial densities corresponding to the EMT of
spin-0 and spin-1/2 systems in the ZAMF and in moving frames. Throughout this
work we closely follow the notations of Ref. Jaffe:2020ebz . We choose the
four-momentum eigenstates $|p,s\rangle$ characterizing our system to be
normalized as

$\langle
p^{\prime},s^{\prime}|p,s\rangle=2E(2\pi)^{3}\delta_{s^{\prime}s}\delta^{(3)}({\bf
p^{\prime}}-{\bf p})\,,$ (1)

where $(p,s)$ and $(p^{\prime},s^{\prime})$ are momentum and polarization of
the initial and final states of a spin-1/2 system, respectively. For spin-0
systems we have analogous expressions without polarizations $s$ and
$s^{\prime}$ (this is also the case for the expressions below, which are
applicable to both spin-0 and spin-1/2 systems). Further, we have $p=(E,{\bf
p})$, with $E=\sqrt{m^{2}+{\bf p}^{2}}$, where $m$ is the particle mass.

When calculating matrix elements of the EMT we use normalizable Heisenberg-
picture states written in terms of wave packets as follows:

$|\Phi,{\bf X},s\rangle=\int\frac{d^{3}{p}}{\sqrt{2E(2\pi)^{3}}}\,\phi(s,{\bf
p})\,e^{-i{\bf p}\cdot{\bf X}}|p,s\rangle,$ (2)

where ${\bf X}$ is the spatial translation vector whose interpretation will be
discussed below and $\phi(s,{\bf p})$ is the profile function satisfying the
normalization condition

$\int d^{3}{p}\,|\phi(s,{\bf p})|^{2}=1\,.$ (3)

To define the density distributions in the ZAMF of the system we use
spherically symmetric wave packets.111The average momentum of the system
vanishes in states corresponding to such packets. In the case of spin-1/2
systems, the profile functions are also chosen to be spin-independent, i.e.
$\phi(s,{\bf p})=\phi({\bf p})=\phi(|{\bf p}|)$. It is convenient to define
dimensionless profile functions

$\phi({\bf p})=R^{3/2}\,\tilde{\phi}(R{\bf p})\,,$ (4)

where $R$ specifies the size of the wave packet. A sharp localization of the
system is achieved by taking small values of $R$.

###  II.1 Spin-0 particles

To define spatial densities associated with the EMT of a scalar particle we
consider its matrix element in a state specified by Eq. (2) for a spin-0 state
and take its limit when $R\to 0$. Using the parametrization of matrix elements
of the EMT in single-particle momentum eigenstates in terms of the form
factors $\Theta_{1}(q^{2})$ and $\Theta_{2}(q^{2})$ Donoghue:1991qv ;
Kubis:1999db  we obtain:

$\displaystyle t^{\mu\nu}_{\phi}({\bf x}-{\bf X})$
$\displaystyle=\langle\Phi,{\bf X}|\hat{T}^{\mu\nu}({\bf x},0)|\Phi,{\bf
X}\rangle=\int\frac{d^{3}{p^{\prime}}d^{3}{p}}{(2\pi)^{3}\sqrt{4E^{\prime}E}}\,\phi^{\star}({\bf
p^{\prime}})\,\phi({\bf p})\langle p^{\prime}|\hat{T}^{\mu\nu}({\bf
x},0)|p\rangle$
$\displaystyle=\int\frac{d^{3}{P}\,d^{3}q}{(2\pi)^{3}\sqrt{4EE^{\prime}}}\,\bigl{[}\left(q^{2}g^{\mu\nu}-q^{\mu}q^{\nu}\right)\Theta_{1}\left(q^{2}\right)+2P^{\mu}P^{\nu}\Theta_{2}\left(q^{2}\right)\bigr{]}\phi\bigg{(}{\bf
P}-\frac{\bf q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf
q}{2}\bigg{)}\,e^{-i{\bf q}\cdot({\bf x}-{\bf X})}\,,$ (5)

where we have introduced new variables $q=p^{\prime}-p$ and
$P=(p+p^{\prime})/2$. In terms of these variables the energies are given as
$E=(m^{2}+{\bf P}^{2}-{\bf P}\cdot{\bf q}+{\bf q}^{2}/4)^{1/2}$ and
$E^{\prime}=(m^{2}+{\bf P}^{2}+{\bf P}\cdot{\bf q}+{\bf q}^{2}/4)^{1/2}$.

By applying the method of dimensional counting of Ref. Gegelia:1994zz , the
leading contribution in Eq. (5) for $R\to 0$ can be obtained without
performing the integration over momenta and without specifying the expressions
for the form-factors and the spherically symmetric profile function. For the
form factors $\Theta_{1}\left(q^{2}\right)$ and $\Theta_{2}\left(q^{2}\right)$
decaying for large $q^{2}$ as $1/q^{4}$ and $1/q^{2}$, or faster,
respectively,222 In Ref. Tanaka:2018wea  using perturbative QCD the
large-$q^{2}$ behavior of the gravitational form factors of the (pseudo)scalar
hadrons has been found to be $\Theta_{1}\left(q^{2}\right)\sim 1/q^{4}$ and
$\Theta_{2}\left(q^{2}\right)\sim 1/q^{2}$. However, according to Ref.
Tong:2021ctu  in case of pions perturbative QCD leads to
$\Theta_{1}\left(q^{2}\right)\sim 1/q^{2}$, $\Theta_{2}\left(q^{2}\right)\sim
1/q^{2}$ for large $q^{2}$. Notice that our derivation would not be applicable
for the latter case. the only non-vanishing contribution for $R\to 0$
originates from the large ${\bf P}$ region. This contribution can be obtained
by substituting ${\bf P}=\tilde{\bf P}/R$ and expanding the resulting
integrand in Eq. (5) in powers of $R$ around $R=0$. Doing so and introducing a
new variable ${\bf r}={\bf x}-{\bf X}$ we obtain

$\displaystyle t^{\mu\nu}_{\phi}({\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}\tilde{P}d^{3}{q}}{(2\pi)^{3}}\left(\frac{\tilde{P}^{\mu}\tilde{P}^{\nu}}{\tilde{P}R}\,\Theta_{2}\left[\tilde{q}^{2}\right]+\frac{R}{2\tilde{P}}\left(\tilde{q}^{2}g^{\mu\nu}-\tilde{q}^{\mu}\tilde{q}^{\nu}\right)\Theta_{1}\left[\tilde{q}^{2}\right]\right)|\tilde{\phi}(\tilde{\bf
P})|^{2}\,e^{-i{\bf q}\cdot{\bf r}}+\text{Rest},$ (6)

where we introduced four-vectors
$\tilde{q}^{\mu}=({\bf\hat{\tilde{P}}}\cdot{\bf q},{\bf q})$ and
$\tilde{P}^{\mu}=(\tilde{P},{\bf\tilde{P}})$ with $\tilde{P}=|{\bf\tilde{P}}|$
and ${\bf\hat{\tilde{P}}}={\bf{\tilde{P}}}/\tilde{P}$. Notice that we wrote
explicitly only terms that contribute to the final expressions for the
densities, see the discussion in Sec. IV, and ”Rest” stands for all other
terms of the expansion. Using Eq. (3) and spherical symmetry we arrive at the
final form of the local density distributions in the ZAMF of the system

$\displaystyle t^{\mu\nu}({\bf r})$
$\displaystyle=N_{\phi,R}\,\int\frac{d^{2}\hat{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\hat{\tilde{P}}^{\mu}\hat{\tilde{P}}^{\nu}\,\Theta_{2}\left[-{\bf
q}_{\perp}^{2}\right]e^{-i{\bf q}\cdot{\bf
r}}-N_{\phi,R,2}\,\int\frac{d^{2}\hat{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\left(\tilde{q}^{\mu}\tilde{q}^{\nu}+{\bf
q}_{\perp}^{2}g^{\mu\nu}\right)\,\Theta_{1}\left[-{\bf
q}_{\perp}^{2}\right]\,e^{-i{\bf q}\cdot{\bf r}}+\text{Rest}\,,$ (7)

where $\hat{\tilde{P}}^{\mu}=\left(1,{\bf\hat{\tilde{P}}}\right)$, ${\bf
q}_{\perp}={\bf\hat{\tilde{P}}}\times\left({\bf
q}\times{\bf\hat{\tilde{P}}}\right)$, ${\bf
q}_{\perp}^{2}\equiv-\tilde{q}^{2}$ and

$\displaystyle N_{\phi,R}$ $\displaystyle=$
$\displaystyle\frac{1}{R}\int\,d\tilde{P}\tilde{P}^{3}|\tilde{\phi}({|{\bf\tilde{P}}|})|^{2}\,,$
$\displaystyle N_{\phi,R,2}$ $\displaystyle=$
$\displaystyle\frac{R}{2}\int\,d\tilde{P}\tilde{P}|\tilde{\phi}({|{\bf\tilde{P}}|})|^{2}\,.$
(8)

Unlike the electromagnetic case (see Refs. Epelbaum:2022fjc  and
Panteleeva:2022khw ), the dependence on the form of the profile function and
the size of the packet remains in the normalization factors $N_{\phi,R}$ and
$N_{\phi,R,2}$ of the local density distributions. Notice that for $R\to 0$
the first term in Eq. (7) goes to infinity while the second term vanishes.

Postponing the interpretation of the expression in Eq. (7) to Sec. IV we now
consider the “static” approximation. The local densities of Refs.
Polyakov:2002wz ; Polyakov:2002yz ; Polyakov:2018zvc  in terms of the Fourier
transforms of the form factors in the Breit frame emerge by expanding the
integrand in Eq. (5) in powers of $1/m$ up to leading-order terms before
performing the integration Jaffe:2020ebz  and localizing the wave packet by
taking $R\to 0$ limit.333 Notice that the $R\to 0$ and $m\to\infty$ limits do
not commute as pointed out in Ref. Epelbaum:2022fjc , and therefore the static
expression does not emerge from Eq. (7) by expanding it in powers of $1/m$.
The resulting expressions after the $1/m$-expansion of the integrand have the
form:

$\displaystyle t^{00}_{\phi,{\rm naive}}({\bf r})$
$\displaystyle=m\int\frac{d^{3}{P}\,d^{3}q}{(2\pi)^{3}}\,\Theta_{2}\left(-{\bf
q}^{2}\right)\phi\bigg{(}{\bf P}-\frac{\bf
q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf q}{2}\bigg{)}\,e^{-i{\bf
q}\cdot{\bf r}},$ $\displaystyle t^{0i}_{\phi,{\rm naive}}({\bf r})$
$\displaystyle=\int\frac{d^{3}{P}\,d^{3}q}{(2\pi)^{3}}\,P^{i}\,\Theta_{2}\left(-{\bf
q}^{2}\right)\phi\bigg{(}{\bf P}-\frac{\bf
q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf q}{2}\bigg{)}\,e^{-i{\bf
q}\cdot{\bf r}},$ $\displaystyle t^{ij}_{\phi,{\rm naive}}({\bf r})$
$\displaystyle=\frac{1}{2m}\int\frac{d^{3}{P}\,d^{3}q}{(2\pi)^{3}}\,\bigl{[}\left({\bf
q}^{2}\delta^{ij}-q^{i}q^{j}\right)\,\Theta_{1}\left(-{\bf
q}^{2}\right)+2P^{i}P^{j}\,\Theta_{2}\left(-{\bf
q}^{2}\right)\bigr{]}\phi\bigg{(}{\bf P}-\frac{\bf
q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf q}{2}\bigg{)}\,e^{-i{\bf
q}\cdot{\bf r}}.$ (9)

To localize the wave packet we take the $R\to 0$ limit again by using the
method of dimensional counting. The desired result is obtained by substituting
${\bf P}={\bf\tilde{P}}/R$, expanding the integrands in Eq. (9) in powers of
$R$ around $R=0$ and keeping explicitly only terms which are relevant, we
obtain

$\displaystyle t^{00}_{{\rm naive}}({\bf r})$ $\displaystyle=$ $\displaystyle
m\int\frac{d^{3}{\tilde{P}}\,d^{3}q}{(2\pi)^{3}}\Theta_{2}\left(-{\bf
q}^{2}\right)\,|\tilde{\phi}({\bf{\tilde{P}}})|^{2}e^{-i{\bf q}\cdot{\bf
r}}+\text{Rest}=\int\frac{d^{3}q}{(2\pi)^{3}}\,m\,\Theta_{2}\left(-{\bf
q}^{2}\right)\,e^{-i{\bf q}\cdot{\bf r}}+\text{Rest}\,,$ $\displaystyle
t^{0i}_{{\rm naive}}({\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\tilde{P}}\,d^{3}q}{(2\pi)^{3}}\,\frac{\tilde{P}^{i}}{R}\,\Theta_{2}\left(-{\bf
q}^{2}\right)\,|\tilde{\phi}({\bf{\tilde{P}}})|^{2}\,e^{-i{\bf q}\cdot{\bf
r}}+\text{Rest}=0+\text{Rest}\,,$ $\displaystyle t^{ij}_{{\rm naive}}({\bf
r})$ $\displaystyle=$ $\displaystyle\frac{4\pi\,\delta^{ij}}{3mR^{2}}\int
d\tilde{P}\,\tilde{P}^{4}\,|\tilde{\phi}({\bf{\tilde{P}}})|^{2}\int\frac{d^{3}q}{(2\pi)^{3}}\,\Theta_{2}\left(-{\bf
q}^{2}\right)e^{-i{\bf q}\cdot{\bf r}}$ (10) $\displaystyle+$
$\displaystyle\frac{1}{2m}\int\frac{d^{3}q}{(2\pi)^{3}}\,\left({\bf
q}^{2}\delta^{ij}-q^{i}q^{j}\right)\Theta_{1}\left(-{\bf
q}^{2}\right)e^{-i{\bf q}\cdot{\bf r}}+\text{Rest}\,,$

where we used Eq. (3) and the fact that for spherically symmetric packets
integral over an odd function of ${\bf\tilde{P}}$ vanishes. The $t^{00}_{{\rm
naive}}$ and the second term of $t^{ij}_{{\rm naive}}$ in Eq. (10) coincide
with the corresponding expressions of spatial densities obtained as the
Fourier transforms of the gravitational form factors in the Breit frame.
Notice that both of these terms do not depend on the packet profile function
and the size of the packet.

###  II.2 Spin-1/2 particles

The matrix elements of the EMT of a spin-1/2 system in one-particle
eigenstates of the four-momentum operator are parametrized in terms of three
form factors as follows Polyakov:2018zvc :

$\langle p^{\prime},s^{\prime}|\hat{T}^{\mu\nu}({\bf
x},0)|p,s\rangle=e^{-i{\bf q}\cdot{\bf
x}}\bar{u}(p^{\prime},s^{\prime})\left[A(q^{2})\frac{P^{\mu}P^{\nu}}{m}+iJ(q^{2})\frac{P^{\mu}\sigma^{\nu\alpha}q_{\alpha}+P^{\nu}\sigma^{\mu\alpha}q_{\alpha}}{2m}+D(q^{2})\frac{q^{\mu}q^{\nu}-g^{\mu\nu}q^{2}}{4m}\right]u(p,s)\,,$
(11)

where the momenta $q$ and $P$ are defined as in the previous section and the
Dirac spinors are normalized as
$\bar{u}(p,s^{\prime})u(p,s)=2m\,\delta_{s^{\prime}s}$. The matrix element of
the EMT in localized states specified by Eq. (2) is written as

$\displaystyle t^{\mu\nu}_{\phi}(s^{\prime},s,{\bf x}-{\bf X})$
$\displaystyle\equiv\langle\Phi,{\bf X},s^{\prime}|\hat{T}^{\mu\nu}({\bf
x},0)|\Phi,{\bf X},s\rangle$
$\displaystyle=\int\frac{d^{3}{P}\,d^{3}{q}}{(2\pi)^{3}\sqrt{4EE^{\prime}}}\,\,\bar{u}(p^{\prime},s^{\prime})\left[A\left((E-E^{\prime})^{2}-{\bf
q}^{2}\right)\frac{P^{\mu}P^{\nu}}{m}+iJ\left((E-E^{\prime})^{2}-{\bf
q}^{2}\right)\frac{P^{\mu}\sigma^{\nu\alpha}q_{\alpha}+P^{\nu}\sigma^{\mu\alpha}q_{\alpha}}{2m}\right.$
$\displaystyle\left.+D\left((E-E^{\prime})^{2}-{\bf
q}^{2}\right)\frac{q^{\mu}q^{\nu}-g^{\mu\nu}q^{2}}{4m}\right]u(p,s)\,\phi\bigg{(}{\bf
P}-\frac{\bf q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf
q}{2}\bigg{)}\,e^{-i{\bf q}\cdot({\bf x}-{\bf X})}.$ (12)

Taking the $R\to 0$ limit in Eq. (12) by using the method of dimensional
counting in a complete analogy to the case of spin-0 systems, and assuming
that form factors $A(q^{2})$, $J(q^{2})$ and $D(q^{2})$ decay for large
$q^{2}$ as $1/q^{2}$, $1/(q^{2})^{3/2}$ and $1/q^{4}$, or faster,
respectively,444This condition is actually satisfied by the large-$q^{2}$
behavior of the gravitational form factors of the proton obtained in
perturbative QCD in Refs. Tanaka:2018wea ; Tong:2021ctu ; Tong:2022zax , in
our notation corresponding to $A(q^{2})\sim J(q^{2})\sim 1/q^{4}$ and
$D(q^{2})\sim 1/q^{6}$, modulo logarithms. we obtain for the operators in spin
space

$\displaystyle t^{\mu\nu}_{\phi}({\bf r})$
$\displaystyle=N_{\phi,R}\int\frac{d^{2}\hat{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\left[\frac{i}{2m}\left(\hat{\tilde{P}}^{\mu}({\bm{\sigma}}_{\perp}\times{\bf
q})^{\nu}+\hat{\tilde{P}}^{\nu}({\bm{\sigma}}_{\perp}\times{\bf
q})^{\mu}+{\bf\hat{\tilde{P}}}\cdot({\bm{\sigma}}_{\perp}\times{\bf
q})(\delta^{\mu 0}\hat{\tilde{P}}^{\nu}+\delta^{\nu
0}\hat{\tilde{P}}^{\mu})\right)J\left(-{\bf q}_{\perp}^{2}\right)\right.$
$\displaystyle\left.+\hat{\tilde{P}}^{\mu}\hat{\tilde{P}}^{\nu}A\left(-{\bf
q}_{\perp}^{2}\right)\right]e^{-i{\bf q}\cdot{\bf
r}}+\frac{1}{2}\,N_{\phi,R,2}\,\int\frac{d^{2}\hat{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\,\left(\tilde{q}^{\mu}\tilde{q}^{\nu}+g^{\mu\nu}{\bf
q}_{\perp}^{2}\right)\,D\left(-{\bf q}_{\perp}^{2}\right)\,e^{-i{\bf
q}\cdot{\bf r}}+\text{Rest}\,,$ (13)

where ${\bf r}={\bf x}-{\bf X}$ and $({\bm{\sigma}}_{\perp}\times{\bf
q})^{0}=0$. Again, we wrote explicitly only those terms which contribute to
the final expressions for the densities in Sec. IV.

Postponing again the discussion of the obtained result to the Sec. IV, we
consider next the static approximation. Expanding the integrand in Eq. (12) in
powers of $1/m$ up to leading-order terms we obtain for the operators in spin
space

$\displaystyle t^{00}_{\phi,{\rm naive}}({\bf r})$
$\displaystyle=m\int\frac{d^{3}{P}\,d^{3}{q}}{(2\pi)^{3}}\,\,A\left(-{\bf
q}^{2}\right)\phi\bigg{(}{\bf P}-\frac{\bf
q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf q}{2}\bigg{)}\,e^{-i{\bf
q}\cdot{\bf r}},$ $\displaystyle t^{0i}_{\phi,{\rm naive}}({\bf r})$
$\displaystyle=\int\frac{d^{3}{P}\,d^{3}{q}}{(2\pi)^{3}}\,\left[A\left(-{\bf
q}^{2}\right)P^{i}-\frac{i}{2}\,\epsilon^{ijk}q^{j}\sigma^{k}\,J\left(-{\bf
q}^{2}\right)\right]\phi\bigg{(}{\bf P}-\frac{\bf
q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf q}{2}\bigg{)}\,e^{-i{\bf
q}\cdot{\bf r}},$ $\displaystyle t^{ij}_{\phi,{\rm naive}}({\bf r})$
$\displaystyle=\frac{1}{m}\int\frac{d^{3}{P}\,d^{3}{q}}{(2\pi)^{3}}\,\left[A\left(-{\bf
q}^{2}\right)\,P^{i}P^{j}+\frac{1}{4}\,D\left(-{\bf q}^{2}\right)\,\left(-{\bf
q}^{2}\delta^{ij}+q^{i}q^{j}\right)\right.$
$\displaystyle\left.-\frac{i}{2}\,\left(\epsilon^{ilk}q^{l}P^{j}+\epsilon^{jlk}q^{l}P^{i}\right)\sigma^{k}\,J\left(-{\bf
q}^{2}\right)\right]\phi\bigg{(}{\bf P}-\frac{\bf
q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf q}{2}\bigg{)}\,e^{-i{\bf
q}\cdot{\bf r}}.$ (14)

To localize the wave packet by taking the limit $R\to 0$ we use the method of
dimensional counting and obtain the following expressions for the “naive”
spatial densities:

$\displaystyle t^{00}_{{\rm naive}}({\bf r})$ $\displaystyle=$ $\displaystyle
m\int\frac{d^{3}{q}}{(2\pi)^{3}}\,A\left(-{\bf q}^{2}\right)e^{-i{\bf
q}\cdot{\bf r}}+\text{Rest},$ $\displaystyle t^{0i}_{{\rm naive}}({\bf r})$
$\displaystyle=$
$\displaystyle-\frac{i}{2}\epsilon^{ijk}\sigma^{k}\int\frac{d^{3}{q}}{(2\pi)^{3}}q^{j}J\left(-{\bf
q}^{2}\right)e^{-i{\bf q}\cdot{\bf r}}+\text{Rest},$ $\displaystyle
t^{ij}_{\phi,{\rm naive}}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}\int\,d\tilde{P}\tilde{P}^{4}|\tilde{\phi}({\bf{\tilde{P}}})|^{2}\,\frac{4\pi\,\delta^{ij}}{3m}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,\,A\left(-{\bf
q}^{2}\right)\,e^{-i{\bf q}\cdot{\bf r}}$ (15) $\displaystyle+$
$\displaystyle\frac{1}{4m}\int\frac{d^{3}{q}}{(2\pi)^{3}}D\left(-{\bf
q}^{2}\right)\,\left(-{\bf q}^{2}\delta^{ij}+q^{i}q^{j}\right)\,e^{-i{\bf
q}\cdot{\bf r}}+\text{Rest}\,,$

where we made use of Eq. (3) and the fact that the integral over
${\bf\tilde{P}}$ of an odd function of this variable vanishes. Analogously to
the case of a spin-0 system, the $t^{00}_{{\rm naive}}$, $t^{0i}_{{\rm
naive}}$ and the second term of $t^{ij}_{\phi,{\rm naive}}$ in Eq. (15)
coincide with the corresponding expressions of spatial densities obtained as
the Fourier transforms of the gravitational form factors in the Breit frame.

Defining the mean squared energy radius of a system via

$\langle r^{2}\rangle=\frac{\int d^{3}r\,t^{00}({\bf r})r^{2}}{\int
d^{3}r\,t^{00}({\bf r})}\,,$ (16)

for a spin-1/2 system in the static approximation of Eq. (15) we obtain
$\langle r^{2}\rangle_{{\rm naive}}=6A^{\prime}(0)$, while Eq. (13) leads to
$\langle r^{2}\rangle=4A^{\prime}(0)$. Analogous result holds also for spin-0
systems. The same smaller mean squared radius, compared to the Breit-frame
result, has been obtained previously in Ref. Freese:2021czn  using the two-
dimensional formalism utilizing light-front coordinates.

##  III Gravitational densities in moving frames

###  III.1 Spin-0

To generalize the results of the previous section to moving frames we replace
$\phi({\bf p})$ with its boosted expression $\phi_{\bf v}({\bf p})$, given in
terms of the spherically symmetric ZAMF quantity as follows Hoffmann:2018edo

$\phi_{\bf v}({\bf p})=\sqrt{\gamma\Big{(}1-\frac{{\bf v}\cdot{\bf
p}}{E}\Big{)}}\,\phi\big{[}{\bf p}_{\perp}+\gamma({\bf p}_{\parallel}-{\bf
v}E)\big{]}\,,$ (17)

where ${\bf v}$ is the boost velocity, $\gamma=(1-v^{2})^{-1/2}$, ${\bf
p}_{\parallel}=({\bf p}\cdot{\bf\hat{v}}){\bf\hat{v}}$, ${\bf p}_{\perp}={\bf
p}-{\bf p}_{\parallel}$ and $E=\sqrt{m^{2}+{\bf p}^{2}}$.

In exact analogy to the case of the ZAMF, we obtain for a spin-0 system in the
$R\to 0$ limit using the method of dimensional counting:

$t^{\mu\nu}_{\phi,{\bf v}}({\bf
r})\\!\\!=\\!\\!\int\\!\frac{d^{3}\tilde{P}\,d^{3}{q}}{(2\pi)^{3}}\,\gamma\left(\\!1\\!-\\!{\bf
v}\cdot{\bf\hat{\tilde{P}}}\\!\right)\\!\Biggl{\\{}\\!\frac{\tilde{P}\,\hat{\tilde{P}}^{\mu}\hat{\tilde{P}}^{\nu}}{R}\Theta_{2}\left[\tilde{q}^{2}\right]\\!+\\!\frac{R}{2\tilde{P}}\left[\tilde{q}^{2}g^{\mu\nu}\\!-\\!\tilde{q}^{\mu}\tilde{q}^{\nu}\right]\\!\Theta_{1}\\!\left[\tilde{q}^{2}\right]\\!\Biggl{\\}}\\!\,\Big{|}\tilde{\phi}\left[{\bf\tilde{P}}_{\perp}\\!+\\!\gamma(\bf{\tilde{P}}_{\parallel}\\!-\\!{\bf
v}\tilde{P})\right]\\!\Big{|}^{2}\\!e^{-i{\bf q}\cdot{\bf r}}+\text{Rest},$
(18)

where ${\bf\hat{\tilde{P}}}$, $\hat{\tilde{P}}^{\mu}$ and $\tilde{q}^{\mu}$
are defined as in the previous section. Next, we change the integration
variable
${\bf\tilde{P}}\to{\bf\tilde{P}}^{\prime}={{\bf\hat{v}}}\times\big{(}{\bf\tilde{P}}\times{\bf\hat{v}}\big{)}+\gamma\big{(}{\bf\tilde{P}}\cdot{\bf\hat{v}}-v\tilde{P}\big{)}{\bf\hat{v}}$,
denote ${\bf\hat{m}}\equiv{\bf\hat{\tilde{P}}^{\prime}}$ and introduce a
vector-valued function

${\bf n}\big{(}{\bf
v},{\bf\hat{m}}\big{)}={\bf\hat{v}}\times\big{(}{\bf\hat{m}}\times{\bf\hat{v}}\big{)}+\gamma\big{(}{\bf\hat{m}}\cdot{\bf\hat{v}}+v){\bf\hat{v}}\,.$
(19)

Taking into account that $\tilde{\bf
P}={\bf\hat{v}}\times\big{(}{\bf\tilde{P}}^{\prime}\times{\bf\hat{v}}\big{)}+\gamma\big{(}{\bf\tilde{P}}^{\prime}\cdot{\bf\hat{v}}+v\tilde{P}^{\prime}\big{)}{\bf\hat{v}}$,
it follows that ${\bf\hat{n}}={\bf\hat{\tilde{P}}}$. The Jacobian of the
change of variables ${\bf\tilde{P}}\to{\bf\tilde{P}}^{\prime}$ cancels the
first factor in the integrand in Eq. (18), and we finally obtain

$\displaystyle t^{00}_{\phi,{\bf v}}({\bf r})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\int\frac{d{\bf\hat{m}}\,d\tilde{P}^{\prime}\tilde{P}^{\prime
2}\,d^{3}{q}}{(2\pi)^{3}}\;\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,e^{-i{\bf
q}\cdot{\bf
r}}\Biggl{\\{}\frac{\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})}{R}\,\Theta_{2}\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]-\frac{{\bf
q}^{2}R}{2\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})}\,\Theta_{1}\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]\Biggr{\\}}+\text{Rest}\,,$ $\displaystyle
t^{0i}_{\phi,{\bf v}}({\bf r})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\int\frac{d{\bf\hat{m}}\,d\tilde{P}^{\prime}\tilde{P}^{\prime
2}\,d^{3}{q}}{(2\pi)^{3}}\;\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,e^{-i{\bf
q}\cdot{\bf
r}}\Biggl{\\{}\frac{\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime}){\hat{n}}^{i}}{R}\,\Theta_{2}\left[({\bf\hat{n}\cdot
q})^{2}\\!-\\!{\bf q}^{2}\right]\\!\\!-\\!\\!\frac{R}{2}\frac{{\bf\hat{n}\cdot
q}}{\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})}\,q^{i}\Theta_{1}\left[({\bf\hat{n}\cdot
q})^{2}\\!-\\!{\bf q}^{2}\right]\\!\\!\Biggr{\\}}+\text{Rest}\,,$
$\displaystyle t^{ij}_{\phi,{\bf v}}({\bf r})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\int\frac{d{\bf\hat{m}}\,d\tilde{P}^{\prime}\tilde{P}^{\prime
2}\,d^{3}{q}}{(2\pi)^{3}}\;\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}e^{-i{\bf
q}\cdot{\bf
r}}\Biggl{\\{}\frac{\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})}{R}\,{\hat{n}}^{i}{\hat{n}}^{j}\Theta_{2}\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]$ (20) $\displaystyle-$
$\displaystyle\frac{R}{2}\;\frac{1}{\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})}\,\left[\left(({\bf\hat{n}\cdot
q})^{2}-{\bf
q}^{2}\right)\delta^{ij}+q^{i}q^{j}\right]\Theta_{1}\left({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]\Biggr{\\}}+\text{Rest}\,.$

Using the spherical symmetry of
$\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}$, the integration over
$\tilde{P}^{\prime}$ factorizes out in Eq. (III.1). To carry out the remaining
angular integration over ${\bf\hat{m}}$ in spherical coordinates we align the
$z$\- and $x$-axes along the ${\bf v}$ and ${\bf q}_{\perp}$ directions,
respectively, denote $\eta=\cos\theta$ and obtain

$t^{\mu\nu}_{\bf v}({\bf
r})=N_{\phi,R}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,\bar{t}^{\,\mu\nu}\left(q_{\parallel},{q}_{\perp}\right)\,e^{-i{\bf
q}\cdot{\bf
r}}+N_{\phi,R,2}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,\bar{t}_{2}^{\,\mu\nu}\left(q_{\parallel},{q}_{\perp}\right)\,e^{-i{\bf
q}\cdot{\bf r}}+\text{Rest}\,,$ (21)

with $q_{\parallel}\equiv{\bf\hat{v}}\cdot{\bf q}$, $q_{\perp}\equiv|{\bf
q}_{\perp}|$ and

$\displaystyle\bar{t}^{\,\mu\nu}(q_{\parallel},{q}_{\perp})$
$\displaystyle=\int_{-1}^{+1}d\eta\int_{0}^{2\pi}d\phi\,\frac{\Omega^{\mu}\Omega^{\nu}}{\gamma(1+v\eta)}\Theta_{2}\left[{\bar{q}}^{2}\right]\,,$
$\displaystyle\bar{t}_{2}^{\,\mu\nu}(q_{\parallel},{q}_{\perp})$
$\displaystyle=-\int_{-1}^{+1}d\eta\int_{0}^{2\pi}d\phi\,\frac{1}{\gamma(1+{v}\eta)}\,\left[\bar{q}^{\mu}\bar{q}^{\nu}-{\bar{q}}^{2}g^{\mu\nu}\right]\Theta_{1}\left[{\bar{q}}^{2}\right]\,,$
(22)

where
$\Omega^{\mu}=(\gamma(1+v\eta),{\bf\hat{\omega}_{\perp}+\gamma(\hat{\omega}_{\parallel}+{\bf
v})})$,
$\bar{q}^{\mu}=(\big{[}\sqrt{1-\eta^{2}}\cos\phi\,q_{\perp}+\gamma(\eta+v)q_{\parallel}\big{]}/(\gamma(1+v\eta)),{\bf
q})$ and
${\hat{\omega}}=(\sqrt{1-\eta^{2}}\cos\phi,\sqrt{1-\eta^{2}}\sin\phi,\eta)$.

In the infinite momentum frame (IMF) with $v\to 1$, $\gamma\to\infty$, we
obtain

$\displaystyle\bar{t}^{\,\mu\nu}(q_{\parallel},{q}_{\perp})$
$\displaystyle=4\pi\,\gamma\,\hat{v}^{\mu}\hat{v}^{\nu}\,\Theta_{2}\left[-{\bf
q}_{\perp}^{2}\right]\,,$
$\displaystyle\bar{t}_{2}^{\,\mu\nu}(q_{\parallel},{q}_{\perp})$
$\displaystyle=-\frac{2\pi}{\gamma}\,\alpha\,\left[{\bf
q}_{\perp}^{2}g^{\mu\nu}+q_{v}^{\mu}q_{v}^{\nu}\right]\Theta_{1}\left[-{\bf
q}_{\perp}^{2}\right]\,,$ (23)

where $\hat{v}^{\mu}=(1,{\bf\hat{v}})$, $q_{v}^{\mu}=(q_{\parallel},{\bf q})$
and

$\alpha=\lim_{v\to 1}\int_{-1}^{+1}\frac{d\eta}{1+v\eta}\,.$ (24)

Notice that $\alpha/\gamma\sim\sqrt{1-v}\ln(1-v)$ when the $v\to 1$ limit is
taken. Substituting these expressions into Eq. (21) for the IMF we obtain:

$\displaystyle t^{00}_{\rm IMF}({\bf r})$ $\displaystyle=$ $\displaystyle
N_{\infty}\,\delta(r_{\parallel})\tilde{\Theta}_{2}\left[{\bf
r}_{\perp}\right]+N_{0}\left[\delta(r_{\parallel})\,\frac{\partial^{2}\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]}{\partial{\bf r}_{\perp}^{k}\partial{\bf
r}_{\perp}^{k}}+\frac{\partial^{2}\delta(r_{\parallel})}{\partial{\bf
r}_{\parallel}^{k}\partial{\bf r}_{\parallel}^{k}}\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]\right]+\text{Rest}\,,$ $\displaystyle t^{0i}_{\rm IMF}({\bf
r})$ $\displaystyle=$ $\displaystyle
N_{\infty}\,\delta(r_{\parallel})\hat{v}^{i}\tilde{\Theta}_{2}\left[{\bf
r}_{\perp}\right]+N_{0}\,\hat{v}^{k}\left(\frac{\partial^{2}\delta(r_{\parallel})}{\partial{\bf
r}_{\parallel}^{k}\partial{\bf r}_{\parallel}^{i}}\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]+\frac{\partial\delta(r_{\parallel})}{\partial{\bf
r}_{\parallel}^{k}}\frac{\partial\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]}{\partial{\bf r}_{\perp}^{i}}\right)+\text{Rest}\,,$
$\displaystyle t^{ij}_{\rm IMF}({\bf r})$ $\displaystyle=$ $\displaystyle
N_{\infty}\,\delta(r_{\parallel})\hat{v}^{i}\hat{v}^{j}\tilde{\Theta}_{2}\left[{\bf
r}_{\perp}\right]+N_{0}\biggl{(}-\delta(r_{\parallel})\,\frac{\partial^{2}\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]}{\partial{\bf r}_{\perp}^{k}\partial{\bf
r}_{\perp}^{k}}\,\delta^{ij}+\frac{\partial^{2}\delta(r_{\parallel})}{\partial{\bf
r}_{\parallel}^{i}\partial{\bf r}_{\parallel}^{j}}\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]+\frac{\partial\delta(r_{\parallel})}{\partial{\bf
r}_{\parallel}^{i}}\frac{\partial\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]}{\partial{\bf r}_{\perp}^{j}}$ (25) $\displaystyle+$
$\displaystyle\frac{\partial\delta(r_{\parallel})}{\partial{\bf
r}_{\parallel}^{j}}\frac{\partial\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]}{\partial{\bf
r}_{\perp}^{i}}+\delta(r_{\parallel})\frac{\partial\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]}{\partial{\bf r}_{\perp}^{i}\partial{\bf
r}_{\perp}^{j}}\biggr{)}+\text{Rest}\,,$

where $N_{0}=\frac{2\pi\,\alpha}{\gamma}N_{\phi,R,2}$, $N_{\infty}=4\pi\gamma
N_{\phi,R}$. Furthermore the Fourier transforms of the form factors are given
by

$\displaystyle\tilde{\Theta}_{i}\left[{\bf
r}_{\perp}\right]=\int\frac{d^{2}{q}_{\perp}}{(2\pi)^{2}}\,\Theta_{i}\left[-{\bf
q}_{\perp}^{2}\right]e^{-i{\bf q}_{\perp}\cdot{\bf r}_{\perp}}.$ (26)

###  III.2 Spin-1/2

For a spin-1/2 system in a moving frame, we consider the state specified by
the following wave packet

$\displaystyle|\Phi,{\bf X},s\rangle_{\bf v}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{p}}{\sqrt{2E(2\pi)^{3}}}\,\sqrt{\gamma\Big{(}1-\frac{{\bf
v}\cdot{\bf p}}{E}\Big{)}}\,\phi\big{[}\Lambda_{\bf v}^{-1}{\bf
p}\big{]}\,e^{-i{\bf p}\cdot{\bf
X}}\sum_{s_{1}}D_{s_{1}s}\Big{[}W\Big{(}\Lambda_{\bf v},\frac{\Lambda_{\bf
v}^{-1}{\bf p}}{m}\Big{)}\Big{]}|p,s_{1}\rangle,$ (27)

where $\gamma=(1-v^{2})^{-1/2}$, $E=\sqrt{m^{2}+{\bf p}^{2}}$, $\Lambda_{\bf
v}^{-1}{\bf p}={\bf\hat{v}}\times\big{(}{\bf
p}\times{\bf\hat{v}}\big{)}+\gamma\big{(}{\bf
p}\cdot{\bf\hat{v}}-vE\big{)}{\bf\hat{v}}$ with $\Lambda_{\bf v}$ denoting the
Lorentz boost from the ZAMF to the moving frame. The
$D_{s_{1}s}\left[W\right]$ matrices build the spin-1/2 representation of the
Wigner rotations Weinberg:1995mt .

For the matrix elements of the EMT of a spin-1/2 system in a moving frame we
obtain the following expressions in the $R\to 0$ limit using the method of
dimensional counting:

$\displaystyle t^{00}_{\phi,{\bf v}}({\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\gamma\left(1-{\bf
v}\cdot{\bf\hat{{\tilde{P}}}}\right)\left\\{\frac{\tilde{P}}{R}\left(1-\frac{i}{2m}\,\epsilon^{ijk}\sigma^{k}q^{i}{\hat{\tilde{P}}}^{j}\right)A\left[\tilde{q}^{2}\right]+\frac{\tilde{P}q^{i}}{2Rm}\,\left[\Sigma^{i}_{\bf
v,\hat{m}},{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\right]_{-}J\left[\tilde{q}^{2}\right]\right.$
$\displaystyle+$ $\displaystyle\left.\frac{R\,{\bf
q}^{2}}{4\tilde{P}}\,\left(1-\frac{i}{2m}\,\epsilon^{lnk}\sigma^{k}q^{l}{\hat{\tilde{P}}}^{n}\right)D\left[\tilde{q}^{2}\right]\right\\}\Big{|}\tilde{\phi}\bigl{(}{\bf\tilde{P}}^{\prime}\bigr{)}\Big{|}^{2}\,e^{-i{\bf
q}\cdot{\bf r}}+\text{Rest}\,,$ $\displaystyle t^{0i}_{\phi,{\bf v}}({\bf r})$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\gamma\left(1-{\bf
v}\cdot{\bf\hat{{\tilde{P}}}}\right)\Biggl{\\{}\tilde{P}\,\frac{{\hat{\tilde{P}}}^{i}}{R}\left(1-\frac{i}{2m}\,\epsilon^{ljk}\sigma^{k}q^{l}{\hat{\tilde{P}}}^{j}\right)A\left[\tilde{q}^{2}\right]$
$\displaystyle-$
$\displaystyle\frac{i\tilde{P}}{4R\,m}\biggl{[}i\,{\hat{\tilde{P}}}^{i}q^{j}\left[\Sigma^{j}_{\bf
v,\hat{m}},{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\right]_{-}+i\,{\bf\hat{\tilde{P}}}\cdot{\bf
q}\left[\Sigma^{i}_{\bf
v,\hat{m}},{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\right]_{-}-\epsilon^{ijk}q^{j}{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\left[\Sigma^{k}_{\bf
v,\hat{m}},{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\right]_{-}\biggr{]}J\left[\tilde{q}^{2}\right]$
$\displaystyle+$ $\displaystyle\frac{R\,{\bf\hat{\tilde{P}}}\cdot{\bf
q}\,q^{i}}{4\tilde{P}}\,\left(1-\frac{i}{2m}\,\epsilon^{lnk}\sigma^{k}q^{l}{\hat{\tilde{P}}}^{n}\right)\,D\left[\tilde{q}^{2}\right]\Biggr{\\}}\,\Big{|}\tilde{\phi}\bigl{(}{\bf\tilde{P}}^{\prime}\bigr{)}\Big{|}^{2}\,e^{-i{\bf
q}\cdot{\bf r}}+\text{Rest}\,,$ $\displaystyle t^{ij}_{\phi,{\bf v}}({\bf r})$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\gamma\left(1-{\bf
v}\cdot{\bf\hat{{\tilde{P}}}}\right)\Biggl{\\{}\frac{\tilde{P}\,{\hat{\tilde{P}}}^{i}{\hat{\tilde{P}}}^{j}}{R}\,\left(1-\frac{i}{2m}\,\epsilon^{lnk}\sigma^{k}q^{l}{\hat{\tilde{P}}}^{n}\right)A\left[\tilde{q}^{2}\right]$
(28) $\displaystyle-$
$\displaystyle\frac{i\tilde{P}}{4R\,m}\biggl{[}i\,{\bf\hat{\tilde{P}}}\cdot{\bf
q}\left({\hat{\tilde{P}}}^{i}\left[\Sigma^{j}_{\bf
v,\hat{m}},{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\right]_{-}+{\hat{\tilde{P}}}^{j}\left[\Sigma^{i}_{\bf
v,\hat{m}},{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\right]_{-}\right)$
$\displaystyle-$
$\displaystyle\left(\epsilon^{jkl}{\hat{\tilde{P}}}^{i}q^{k}+\epsilon^{ikl}{\hat{\tilde{P}}}^{j}q^{k}\right){\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\left[\Sigma^{l}_{\bf
v,\hat{m}},{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\right]_{-}\biggr{]}J\left[\tilde{q}^{2}\right]$
$\displaystyle+$
$\displaystyle\frac{R(\tilde{q}^{i}\tilde{q}^{j}+\delta^{ij}\tilde{q}^{2})}{4\tilde{P}}\left(1-\frac{i}{2m}\,\epsilon^{lnk}\sigma^{k}q^{l}{\hat{\tilde{P}}}^{n}\right)\,D\left[\tilde{q}^{2}\right]\Biggr{\\}}\,\Big{|}\tilde{\phi}\bigl{(}{\bf\tilde{P}}^{\prime}\bigr{)}\Big{|}^{2}\,e^{-i{\bf
q}\cdot{\bf r}}+\text{Rest}\,,$

where $[A,B]_{-}=AB-BA$,
${\bf\tilde{P}}^{\prime}={\bf\hat{v}}\times\big{(}{\bf\tilde{P}}\times{\bf\hat{v}}\big{)}+\gamma\big{(}{\bf\tilde{P}}\cdot{\bf\hat{v}}-v\tilde{P}\big{)}{\bf\hat{v}}$
and the unit vector ${\bf\hat{m}}$ is defined as
${\bf\hat{m}}\equiv{\bf\hat{\tilde{P}}}^{\prime}$. Further, the
$\Sigma^{i}_{\bf v,\hat{m}}$ refers to the Wigner-rotated spin operator

$\Sigma^{i}_{\bf v,\hat{m}}=D^{\dagger}\left[W(\Lambda_{\bf
v,\hat{m}})\right]\sigma^{i}D\left[W(\Lambda_{\bf v,\hat{m}})\right],$ (29)

with

$D\left[W(\Lambda_{\bf v,\hat{m}})\right]=\lim_{R\to
0}D\left[W\Big{(}\Lambda_{\bf v},\frac{\Lambda^{-1}_{\bf
v}\big{(}{\bf\tilde{P}}/R\big{)}}{m}\Big{)}\right].$ (30)

Next, we change the integration variables
${\bf\tilde{P}}\to{\bf\tilde{P}}^{\prime}$ to obtain

$\displaystyle t^{00}_{\phi,{\bf v}}({\bf r})\\!\\!\\!$
$\displaystyle=\\!\\!\\!$
$\displaystyle\frac{1}{R}\int\frac{d{\bf\hat{m}}\,d{\tilde{P}^{\prime}}\tilde{P}^{\prime
2}\,d^{3}{q}}{(2\pi)^{3}}\;\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})\left\\{\left(1-\frac{i}{2m}\,\epsilon^{ijk}\sigma^{k}q^{i}{\hat{n}}^{j}\right)A\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]\right.$ $\displaystyle+$
$\displaystyle\left.\frac{1}{2m}\,q^{i}\left[\Sigma^{i}_{\bf
v,\hat{m}},{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{-}J\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]\right\\}\,e^{-i{\bf q}\cdot{\bf r}}$
$\displaystyle+$
$\displaystyle\frac{R}{2}\int\frac{d{\bf\hat{m}}\,d{\tilde{P}^{\prime}}\tilde{P}^{\prime
2}\,d^{3}{q}}{(2\pi)^{3}}\;\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,\frac{{\bf
q}^{2}}{2\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})}\left(1-\frac{i\,\epsilon^{lnk}\sigma^{k}q^{l}{\hat{n}}^{n}}{2\,m}\right)D\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]e^{-i{\bf q}\cdot{\bf r}}+\text{Rest}\,,$
$\displaystyle t^{0i}_{\phi,{\bf v}}({\bf r})\\!\\!\\!$
$\displaystyle=\\!\\!\\!$
$\displaystyle\frac{1}{R}\int\frac{d{\bf\hat{m}}\,d{\tilde{P}^{\prime}}\tilde{P}^{\prime
2}\,d^{3}{q}}{(2\pi)^{3}}\,\;\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,\,\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})\,\biggl{\\{}{\hat{n}}^{i}\,\left(1-\frac{i\,\epsilon^{ljk}\sigma^{k}q^{l}{\hat{n}}^{j}}{2\,m}\right)A\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]$ $\displaystyle-$
$\displaystyle\frac{i}{4m}\biggl{[}i\,{\hat{n}^{i}}q^{j}\left[\Sigma^{i}_{\bf
v,\hat{m}},{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{-}+i\,{\bf\hat{n}}\cdot{\bf
q}\left[\Sigma^{i}_{\bf
v,\hat{m}},{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{-}-\epsilon^{ijk}q^{j}{\bf\hat{n}}\cdot{\bm{\sigma}}\left[\Sigma^{i}_{\bf
v,\hat{m}},{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{-}\biggr{]}J\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]\biggr{\\}}\,e^{-i{\bf q}\cdot{\bf r}}$
$\displaystyle+$
$\displaystyle\frac{R}{2}\int\frac{d{\bf\hat{m}}\,d{\tilde{P}^{\prime}}\tilde{P}^{\prime
2}\,d^{3}{q}}{(2\pi)^{3}}\;\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,\frac{{\bf\hat{n}\cdot
q}\,{q}^{i}}{2\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})}\left(1-\frac{i\,\epsilon^{lnk}\sigma^{k}q^{l}{\hat{n}}^{n}}{2\,m}\right)D\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]\,e^{-i{\bf q}\cdot{\bf r}}+\text{Rest},$
$\displaystyle t^{ij}_{\phi,{\bf v}}({\bf r})\\!\\!\\!$
$\displaystyle=\\!\\!\\!$
$\displaystyle\frac{1}{R}\int\frac{d{\bf\hat{m}}\,d{\tilde{P}^{\prime}\tilde{P}^{\prime
2}}\,d^{3}{q}}{(2\pi)^{3}}\,\;\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,\gamma\left(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime}\right)\biggl{\\{}{\hat{n}}^{i}{\hat{n}}^{j}\left(1-\frac{i}{2m}\,\epsilon^{lnk}\sigma^{k}q^{l}{\hat{n}}^{n}\right)A\left[({\bf\hat{n}\cdot
q})^{2}-{\bf q}^{2}\right]$ (31) $\displaystyle-$
$\displaystyle\frac{i}{4m}\biggl{[}i\,{\bf\hat{n}}\cdot{\bf
q}\left({\hat{n}}^{i}\left[\Sigma^{j}_{\bf
v,\hat{m}},{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{-}+{\hat{n}}^{j}\left[\Sigma^{i}_{\bf
v,\hat{m}},{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{-}\right)-\left(\epsilon^{jkl}{\hat{n}}^{i}q^{k}+\epsilon^{ikl}{\hat{n}}^{j}q^{k}\right){\bf\hat{n}}\cdot{\bm{\sigma}}\left[\Sigma^{l}_{\bf
v,\hat{m}},{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{-}\biggr{]}$
$\displaystyle\times$ $\displaystyle J\left[({\bf\hat{n}\cdot q})^{2}-{\bf
q}^{2}\right]\biggr{\\}}\,e^{-i{\bf q}\cdot{\bf r}}$ $\displaystyle+$
$\displaystyle\\!\\!\frac{R}{2}\\!\\!\int\\!\\!\\!\frac{d{\bf\hat{m}}\,d{\tilde{P}^{\prime}}\tilde{P}^{\prime
2}d^{3}{q}}{(2\pi)^{3}}\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,\frac{q^{j}{q}^{i}+\delta^{ij}\left(({\bf\hat{n}\cdot
q})^{2}\\!-\\!{\bf
q}^{2}\right)}{2\gamma(\tilde{P}^{\prime}+{v}\tilde{P}_{\parallel}^{\prime})}\left(\\!1\\!-\\!\frac{i}{2m}\epsilon^{lnk}\sigma^{k}q^{l}\,{\hat{n}}^{n}\\!\right)\\!D\\!\left[({\bf\hat{n}\cdot
q})^{2}\\!-\\!{\bf q}^{2}\right]\\!e^{-i{\bf q}\cdot{\bf
r}}\\!+\\!\text{Rest}.$

The expressions in Eq. (31) simplify considerably in two extreme cases. First,
in the particle’s ZAMF, with $v=0$, $\gamma=1$ and $D\left[W\left(\Lambda_{\bf
0},{\bf\hat{m}}\right)\right]={\rm identity}$, we have ${\bf n}\big{(}{\bf
v},{\bf\hat{m}}\big{)}={\bf\hat{m}}$, so that we can replace the integration
measure $d{\bf\hat{m}}$ by $d{\bf\hat{n}}$.

Next, in the infinite-momentum frame (IMF) with $v\to 1$ and
$\gamma\to\infty$, the vector-valued function ${\bf\hat{n}}$ turns to
$\bf{\hat{v}}$, so that the integrands in Eq. (31) depend on ${\bf\hat{m}}$
only through the Wigner rotation matrices. We use the spherical symmetry of
$\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}$ and factor out the
integration over $\tilde{P}^{\prime}$ in Eq. (31). To carry out the remaining
angular integration over ${\bf\hat{m}}$ in spherical coordinates we align the
$z$\- and $x$-axes along the ${\bf v}$ and ${\bf q}_{\perp}$ directions,
respectively, and denote $\eta=\cos\theta$. Finally we obtain the following
expressions for spatial densities in the IMF:

$t^{\mu\nu}_{\bf v}({\bf
r})=N_{\phi,R}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,\bar{t}^{\mu\nu}\left(q_{\parallel},{q}_{\perp}\right)\,e^{-i{\bf
q}\cdot{\bf
r}}+N_{\phi,R,2}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,\bar{t}_{2}^{\mu\nu}\left(q_{\parallel},{q}_{\perp}\right)\,e^{-i{\bf
q}\cdot{\bf r}}+\text{Rest}\,,$ (32)

with

$\displaystyle\bar{t}^{\mu\nu}\left(q_{\parallel},{q}_{\perp}\right)$
$\displaystyle=4\pi\gamma\,\left\\{\hat{v}^{\mu}\hat{v}^{\nu}\left(1+\frac{i}{2m}\,\epsilon^{ilk}\hat{v}^{i}\sigma^{k}q^{l}\right)A\left[-{\bf
q}_{\perp}^{2}\right]+\frac{i}{4m}\biggl{(}\hat{v}^{\mu}({\bm{\sigma}}_{\perp}\times{\bf
q})^{\nu}+\hat{v}^{\nu}({\bm{\sigma}}_{\perp}\times{\bf q})^{\mu}\right.$
$\displaystyle\left.{}+(\delta^{\mu 0}\hat{v}^{\nu}+\delta^{\nu
0}\hat{v}^{\mu}){\bf\hat{v}}\cdot({\bm{\sigma}}_{\perp}\times{\bf
q})\biggr{)}J\left[-{\bf q}_{\perp}^{2}\right]\right\\}\,,$
$\displaystyle\bar{t}_{2}^{\mu\nu}\left(q_{\parallel},{q}_{\perp}\right)$
$\displaystyle=\frac{\pi}{\gamma}\,\alpha\left({q}_{v}^{\mu}{q}_{v}^{\nu}+g^{\mu\nu}{\bf
q}_{\perp}^{2}\right)\left(1+\frac{i}{2m}\,\epsilon^{ilk}\hat{v}^{i}\sigma^{k}q^{l}\right)D\left[-{\bf
q}_{\perp}^{2}\right]\,,$ (33)

where $({\bm{\sigma}}_{\perp}\times{\bf q})^{0}=0$ and $\alpha$ is defined in
Eq. (24). Substituting Eq. (33) in Eq. (32) we obtain, analogously to Eq.
(26), expressions for the two-dimensional spatial distributions in the IMF.

##  IV Interpretation of the results

We are now in the position to discuss the physical interpretation of the new
expressions obtained in the current work. It is clear from Eq. (III.1) that in
the IMF, the matrix elements of the EMT in localized states represent two-
dimensional distributions in the plane perpendicular to the velocity of the
reference frame. Analyzing Eqs. (21) and (23) we conclude that there are two
types of contributions: those depending on the velocity of the reference frame
and thus characterizing the movement of the system as a whole and the other
contributions which are related to internal properties of the system under
consideration. The contribution generated by $\bar{t}^{\mu\nu}$ corresponds to
the motion of the system as a whole, while the term generated by
$\bar{t}_{2}^{\mu\nu}$ is not related to this motion and, therefore, should be
interpreted as related to internal characteristics. Notice that essentially
the same argument supporting separate interpretation of different
contributions to the matrix elements of EMT has been given previously in Refs.
Freese:2021mzg ; Freese:2022fat .

Integrating Eq. (21) for the IMF over all possible directions of the frame
velocity, it is easily seen that the resulting two terms are proportional to
the corresponding terms in Eq. (7), i.e. to the terms which involve the same
form factors.555Notice that while the coefficients of proportionality are
different for different terms, the behavior for $R\to 0$ remains the same,
i.e. the first term diverges while the second term vanishes. Thus, the spatial
distributions in the ZAMF, given by Eq. (7), can be understood as an integral
over all directions of the IMF velocity. As evidenced by Eq. (III.1), the
spatial distributions are given by two dimensional images of the system in the
IMF. The same can be concluded for the spin-1/2 system using equations (13)
and (32). Thus the three-dimensional “picture” of the system emerges as a
“composition” of its all possible two-dimensional cuts.

As mentioned above for the IMF, the term corresponding to
$\bar{t}_{2}^{\mu\nu}$ characterizes the internal structure. Therefore also in
the ZAMF the term

$t^{ij}_{2}({\bf
r})=N_{\phi,R,2}\,\int\frac{d^{2}\hat{n}\,d^{3}{q}}{(2\pi)^{3}}\,\left(-q^{i}q^{j}+{\bf
q}_{\perp}^{2}\delta^{ij}\right)\,\Theta_{1}\left[-{\bf
q}_{\perp}^{2}\right]\,e^{-i{\bf q}\cdot{\bf r}}$ (34)

characterizes the distribution of internal forces. Notice that while the
normalization of this quantity depends on the wave packet profile function its
spatial distribution is uniquely determined by the EMT form factor. We
identify the traceless and the trace parts via

$t^{ij}_{2}({\bf
r})=\left(\frac{r^{i}r^{j}}{r^{2}}-\frac{1}{3}\,\delta^{ij}\right)s(r)+\delta^{ij}p(r)\,,$
(35)

where

$\displaystyle s({\bf r})$ $\displaystyle=$ $\displaystyle
N_{\phi,R,2}\int\frac{d^{2}\hat{n}d^{3}{q}}{(2\pi)^{3}}\left(\frac{{\bf
r}^{2}{\bf q}^{2}-3({\bf r}\cdot{\bf q})^{2}}{2\,{\bf
r}^{2}}\right)\Theta_{1}\left[-{\bf q}_{\perp}^{2}\right]e^{-i{\bf q}\cdot{\bf
r}}$ $\displaystyle=$ $\displaystyle N_{\phi,R,2}\,\int
d^{2}\hat{n}\,r\frac{d}{d{r}}\frac{1}{r}\frac{d}{d{r}}\left(\delta(r_{\parallel})\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]\right),$ $\displaystyle p({\bf r})$ $\displaystyle=$
$\displaystyle
N_{\phi,R,2}\int\frac{d^{2}\hat{n}\,d^{3}{q}}{(2\pi)^{3}}\left({\bf
q}_{\perp}^{2}-\frac{{\bf q}^{2}}{3}\right)\Theta_{1}\left[-{\bf
q}_{\perp}^{2}\right]\,e^{-i{\bf q}\cdot{\bf r}}$ (36) $\displaystyle=$
$\displaystyle N_{\phi,R,2}\int
d^{2}\hat{n}\left(\frac{1}{3}\frac{1}{r^{2}}\frac{d}{dr}r^{2}\frac{d}{dr}-\frac{1}{r_{\perp}^{2}}\frac{d}{dr_{\perp}}r_{\perp}^{2}\frac{d}{dr_{\perp}}\right)\left(\delta(r_{\parallel})\tilde{\Theta}_{1}\left[{\bf
r}_{\perp}\right]\right)\,.$

The quantities $s(r)$ and $p(r)$ have been interpreted in Refs.
Polyakov:2002yz ; Polyakov:2018zvc  as the shear force and the pressure,
respectively.666Notice that this interpretation has been questioned recently
in Ref. Ji:2021mfb . It is evident from Eq. (IV) that the overall
normalizations of these quantities depend on the packet while their functional
form is uniquely determined by the form factor $\Theta_{1}$. It is not
surprising that the normalization factors of the pressure and shear force
distributions vanish in the limit of sharply localized states as these
functions are related to the variation of the action with respect to the
spatial metric $g_{ik}({\bf r})$. This variation corresponds to a change of
the location of the system in three-dimensional space, which vanishes for
sharply localized states. Notice that the shape of these distributions does
not depend on the localization of the system and is uniquely determined by the
corresponding form factor.

In exact analogy to the case of the spin-0 system, by integrating the IMF
density in Eq. (32) over all possible directions of the frame velocity, we
obtain terms involving different form factors of a spin-1/2 system, which are
proportional to the corresponding terms in Eq. (13). Thus, analogously to Eq.
(35), the term

$\displaystyle t^{ij}_{2}(s^{\prime},s,{\bf
r})=\frac{1}{2}\,N_{\phi,R,2}\,\delta_{s^{\prime}s}\,\int\frac{d^{2}\hat{n}\,d^{3}{q}}{(2\pi)^{3}}\,\,\left(q^{i}q^{j}-\delta^{ij}{\bf
q}_{\perp}^{2}\right)\,D\left(-{\bf q}_{\perp}^{2}\right)\,e^{-i{\bf
q}\cdot{\bf r}}$ (37)

can be interpreted as the distribution of the shear force and the internal
pressure.

Figure 1: Normalized energy, pressure and shear force density distributions in
the static approximation (dashed lines) and in the ZAMF (solid lines).
Normalizations of densities are chosen by dividing the corresponding matrix
elements of the EMT by their integrals over the three dimensional space.

As mentioned above, a superposition of eigenstates of the energy-momentum
vector with different eigenvalues is not an eigenstate of this operator.
Therefore, in general, the interpretation of the $t^{00}({\bf r})$ and
$t^{0i}({\bf r})$ in terms of the energy and momentum distributions seems to
be problematic. However, in sharply localized states, where the wave packets
are dominated by large momenta, these quantities can be interpreted as energy
and momentum spatial distributions, respectively.777 Notice here that for
these terms we do not offer separate interpretation of the $\bar{t}$ and
$t_{2}$ contributions and therefore the second are dropped as compared to the
first. For a spin-0 system in the ZAMF we have:

$t^{00}({\bf
r})=N_{\phi,R}\,\int\frac{d^{2}\hat{n}\,d^{3}{q}}{(2\pi)^{3}}\,\Theta_{2}\left[-{\bf
q}_{\perp}^{2}\right]\,e^{-i{\bf q}\cdot{\bf r}},$ (38)

and for the spin-1/2 system in the ZAMF we obtain the following expressions

$\displaystyle t^{00}({\bf r})$
$\displaystyle=N_{\phi,R}\,\int\frac{d^{2}\hat{n}\,d^{3}{q}}{(2\pi)^{3}}\,A\left[-{\bf
q}_{\perp}^{2}\right]\,e^{-i{\bf q}\cdot{\bf r}},$ (39) $\displaystyle
t^{0i}({\bf r})$
$\displaystyle=N_{\phi,R}\dfrac{i\varepsilon^{jkl}}{2m}\,\int\frac{d^{2}\hat{n}\,d^{3}{q}}{(2\pi)^{3}}\,\sigma_{\perp}^{k}q^{l}\left(\delta^{ij}+\hat{n}^{i}\hat{n}^{j}\right)\,J\left[-{\bf
q}_{\perp}^{2}\right]\,e^{-i{\bf q}\cdot{\bf r}}.$ (40)

On the other hand, in the static approximation the states are defined by
packets with sizes much larger than the Compton wavelength of the system. For
this case, our expressions of the spatial distributions in the static
approximation for a spin-0 system, Eq. (10), and for a spin-1/2 system, Eq.
(15), coincide with the corresponding Breit-frame expressions for the $t^{00}$
and $t^{0i}$ components, while for the $t^{ij}$ components the Breit-frame
expression reproduces the part describing the internal structure only. Thus,
the Breit-frame expressions indeed correspond to the system in the ZAMF in a
state described by a packet with the characteristic scale much larger than the
Compton wavelength of the system. For such a packet the integral is governed
by momenta much smaller than the mass of the system and, therefore, replacing
the corresponding energies by the first term in the expansion
$E=\sqrt{m^{2}+{\bf p}^{2}}=m+{\bf p}^{2}/(2m)+\cdots$ is a good
approximation. That is, the packet is dominated by eigenstates of the energy
with the same eigenvalue $m$, and therefore it is also an eigenstate of the
energy operator with the eigenvalue $m$. Thus, $t^{00}({\bf r})$ can be
interpreted in this case as the spatial distribution of the mass, which is the
same as the full energy of the system in the ZAMF in the static approximation.
More details of interpretation of the Fourier transforms of the gravitational
form factors in the Breit frame in terms of various spatial distributions can
be found in Ref. Polyakov:2018zvc . Taking into account that sharp
localization of the system requires a huge amount of energy, it is not
surprising that the normalization factor for the energy and the momentum
distributions $N_{\phi,R}$ explodes. Again, the functional form of these
densities is uniquely determined by the corresponding form factors. To
illustrate the comparison of the static approximation and the ZAMF
expressions, in Fig. 1 we plot the normalized energy, pressure and shear force
density distributions for a hypothetical spin-0 system with gravitational form
factors $\Theta_{1}(q^{2})=1/(1-q^{2}/\Lambda^{2})^{3}$ and
$\Theta_{2}(q^{2})=1/(1-q^{2}/\Lambda^{2})^{2}$, where we take
$\Lambda=1\,{\rm GeV}$.

Finally, as the spatial densities are given in terms of ${\bf r}={\bf x}-{\bf
X}$, ${\bf X}$ should be interpreted as the position of the center-of-gravity
of the system.

##  V Summary and conclusions

To summarize, in the current work we considered the one-particle matrix
elements of the EMT in localized states of spin-0 and spin-1/2 systems.
Specifying the one-particle states by spherically symmetric wave packets,
independent of the spin polarization for the case of spin-1/2 systems, and
sharply localizing system we obtained the definition of spatial distributions
which are independent of the specific radial form of the packet. This
definition is applicable for an arbitrary relative size of the Compton
wavelength and other internal characteristics of the system. We obtained the
expressions for the newly defined quantities in the ZAMF as well as in moving
reference frames. Writing the spatial distributions in the ZAMF as integrals
of the corresponding expressions in the IMF over all possible directions, we
offered a physical interpretation of the obtained results.

Finally, using the results of Ref. Jaffe:2020ebz , we also considered the
static approximation and obtained the spatial distributions, which coincide
with the Fourier transforms of the form factors in the Breit frame as
originally suggested as spatial distributions of various quantities
Polyakov:2018zvc .

###### Acknowledgements.

We thank H. Alharazin for helpful discussions. This work was supported in part
by BMBF (Grant No. 05P21PCFP1), by DFG and NSFC through funds provided to the
Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD” (NSFC
Grant No. 11621131001, DFG Project-ID 196253076 - TRR 110), by ERC
NuclearTheory (grant No. 885150), and ERC EXOTIC (grant No. 101018170), by CAS
through a President’s International Fellowship Initiative (PIFI) (Grant No.
2018DM0034), by the VolkswagenStiftung (Grant No. 93562), by the EU Horizon
2020 research and innovation programme (STRONG-2020, grant agreement No.
824093), and by the Heisenberg-Landau Program 2021.

## References

  * (1) R. Hofstadter, F. Bumiller, and M. R. Yearian, Rev. Mod. Phys. 30, 482 (1958).
  * (2) F. J. Ernst, R. G. Sachs and K. C. Wali, Phys. Rev. 119, 1105-1114 (1960).
  * (3) R. G. Sachs, Phys. Rev. 126, 2256-2260 (1962).
  * (4) M. V. Polyakov and A. G. Shuvaev, [arXiv:hep-ph/0207153 [hep-ph]].
  * (5) M. V. Polyakov, Phys. Lett. B 555, 57 (2003).
  * (6) M. V. Polyakov and P. Schweitzer, Int. J. Mod. Phys. A 33 (2018) no.26, 1830025.
  * (7) M. Burkardt, Phys. Rev. D 62 (2000), 071503(R), [erratum: Phys. Rev. D 66 (2002), 119903(E)].
  * (8) G. A. Miller, Phys. Rev. Lett. 99, 112001 (2007).
  * (9) G. A. Miller, Phys. Rev. C 79, 055204 (2009).
  * (10) G. A. Miller, Ann. Rev. Nucl. Part. Sci. 60 (2010), 1-25.
  * (11) R. L. Jaffe, Phys. Rev. D 103 (2021) no.1, 016017.
  * (12) G. A. Miller, Phys. Rev. C 99, no.3, 035202 (2019).
  * (13) A. Freese and G. A. Miller, Phys. Rev. D 103, 094023 (2021).
  * (14) C. Lorcé, Phys. Rev. Lett. 125, no.23, 232002 (2020), [arXiv:2007.05318 [hep-ph]].
  * (15) C. Lorcé, P. Schweitzer and K. Tezgin, [arXiv:2202.01192 [hep-ph]].
  * (16) Y. Guo, X. Ji and K. Shiells, Nucl. Phys. B 969, 115440 (2021), [arXiv:2101.05243 [hep-ph]].
  * (17) C. Lorcé, H. Moutarde and A. P. Trawiński, Eur. Phys. J. C 79, no.1, 89 (2019), [arXiv:1810.09837 [hep-ph]].
  * (18) C. E. Carlson, [arXiv:2208.00826 [hep-ph]].
  * (19) A. Freese and G. A. Miller, Phys. Rev. D 105, no.1, 014003 (2022), [arXiv:2108.03301 [hep-ph]].
  * (20) Y. Chen and C. Lorcé, [arXiv:2210.02908 [hep-ph]].
  * (21) A. Freese and G. A. Miller, [arXiv:2210.03807 [hep-ph]].
  * (22) J. Y. Panteleeva and M. V. Polyakov, Phys. Rev. D 104 (2021) no.1, 014008, [arXiv:2102.10902 [hep-ph]].
  * (23) E. Epelbaum, J. Gegelia, N. Lange, U.-G. Meißner and M. V. Polyakov, Phys. Rev. Lett. 129, no.1, 012001 (2022), [arXiv:2201.02565 [hep-ph]].
  * (24) G. N. Fleming, Physical Reality & Math. Descrip., 357 (1974).
  * (25) J. Y. Panteleeva, E. Epelbaum, J. Gegelia and U.-G. Meißner, Phys. Rev. D 106, no.5, 056019 (2022), [arXiv:2205.15061 [hep-ph]].
  * (26) J. F. Donoghue and H. Leutwyler, Z. Phys. C 52, 343-351 (1991).
  * (27) B. Kubis and U.-G. Meißner, Nucl. Phys. A 671, 332-356 (2000) [erratum: Nucl. Phys. A 692, 647-648 (2001)], [arXiv:hep-ph/9908261 [hep-ph]].
  * (28) J. Gegelia, G. S. Japaridze and K. S. Turashvili, Theor. Math. Phys. 101, 1313-1319 (1994).
  * (29) K. Tanaka, Phys. Rev. D 98, no.3, 034009 (2018), [arXiv:1806.10591 [hep-ph]].
  * (30) X. B. Tong, J. P. Ma and F. Yuan, Phys. Lett. B 823, 136751 (2021), [arXiv:2101.02395 [hep-ph]].
  * (31) X. B. Tong, J. P. Ma and F. Yuan, JHEP 10, 046 (2022), [arXiv:2203.13493 [hep-ph]].
  * (32) S. E. Hoffmann, [arXiv:1804.00548 [quant-ph]].
  * (33) S. Weinberg, “The Quantum theory of fields. Vol. 1, : Foundations,” Cambridge University Press (2005-06-02).
  * (34) X. Ji and Y. Liu, Phys. Rev. D 106, no.3, 034028 (2022), [arXiv:2110.14781 [hep-ph]].