Scalar products and norm of Bethe vectors in $\mathfrak{o}_{2n+1}$ invariant integrable models

1) Present a breakthrough on a previously-identified and long-standing research stumbling block.
2) The results derived are original and easily formulated and reproduced $gl_n$ known results, when opportunely specialized, of particular interest is the block determinant formula for the norm.

There are no conclusions and the intro and references are too briefly addressed.

Dear Editor,
This is an important paper where the authors continue the relevant analysis of higher rank quantum integrable models. They generalize results that they have previously obtained for the models associated to the $gl_{n}$ and $gl_{n,m}$ symmetry in collaboration with fellow researchers.
Here, they derive scalar products of the off-shell Bethe vectors as sum of partitions over Bethe parameters for the class of quantum integrable models associated to $o_{2n+1}$ higher rank symmetry. They show that the building blocks of these scalar products are the so-called highest coefficients, for which they prove recurrence relations and residue theorems that completely characterize them and that they verify reproduce the known $gl_{n}$ results, once the reduction to these last models is implemented. Notably, they are able to prove a block determinant form for the Gaudin norm of on-shell Bethe vectors of the $o_{2n+1}$ model, in this way extending results known by the same research group only for n=1.
The research material is made accessible to the general public with clear notations and explanations, this make possible to follow their proofs and argue in favor of their results even if these are somewhere quite involved.
The authors have indicated for the fulfilments of journal expectations, a breakthrough on a previously-identified and long-standing research problems, I agree on it and I think that it deserves to be published once the authors will take into accounts the following clarification requests:
a) In section 5.2, before Proposition 5.4, the authors implement a specialization on the set of the partitions of the variables putting the sets of the u and v with label 0 to the empty sets. While it is possible, following their arguments, to understand how the recurrences adapt to this specialization and rederive those of the $gl_{n}$ case, what for me is not well described is why is this specialization that bring from the $o_{2n+1}$ models to the $gl_{n}$ models. Similarly, in the intro to section 7, in line 5, it is written that “BVs of the form …, since they correspond to $Y(gl_{n})$ BVs”. I think that it would be important to comment why these BVs belongs to $Y(gl_{n})$ and why the above specification allows to go from models associated to $o_{2n+1}$ to models associated to $gl_{n}$. Indeed, this is central to understand why the authors can reproduce from the $o_{2n+1}$ case the recursion relations and block determinant of the Gaudin norm for the $gl_{n}$ case.
b) The paper misses to define the main prospectives and research directions that can emerge as consequences of the results here derived. I think that the authors should write at least a conclusion. Even the current introduction is very short in defining the research background where this research is located and the use of this type of results, e.g. when the authors refer to correlation functions, they only cite the book in citation [3], where only the primordial results on these quantities are described, while the main present and fundamental results in the Algebraic Bethe Ansatz using the Slavnov’s scalar product formula [22] are missing.
c) In the integrable models associated to $gl_3$ symmetry, there are block determinant formulae for the scalar products of two Bethe vectors associated two a periodic and twisted boundary condition, respectively. These formulae are fundamental as they are extremely easier of the sum over partition ones and they have been used moreover to generate results on dynamical objects like the form factors, all these are results generated by some/all the present authors with fellow researchers. It is then natural to ask if there is the possibility to have some block determinant formula also for the scalar products of models associated to $o_{2n+1}$, for n=2 or 3 in view of the specialization to the corresponding $gl_n$ models. If this is the case, then, the next natural question is if the authors have planned or can already state the possibility to study form factors of local operators. In the current paper is only discussed the $o_3$ case due to an isomorphism to $gl_2$.
d) It seems to me that the functions $\lambda_{i}$ are not arbitrary free functions, as they have to satisfy the condition (2.8); the “formal parameter” in (3.7) seems to be z and not u as stated before this equation.

The requested changes are listed in the report.

Publish (meets expectations and criteria for this Journal)

1. The paper gives a complete solution of a complicated and essential problem.
2. Detailed presentation of a complicated proof.
3. Explanation of the relations with the known cases, namely the $Y(\mathfrak{0}_3)$ case.
4. Extremely efficient choice of notations making several steps of the proof readable

1. Lack of presentation of the context and possible future developments

It is a very interesting and important paper for every reader following the previous work of the authors. Generalisation of the results for norms of the Bethe vectors to new algebras and a complete proof of the Gaudin formula in new cases is highly non-trivial and absolutely necessary program for the field of quantum integrability. The paper is very technical but it is the feature of the problem. I appreciated the detailed presentation of the proof, good presentation of the result (including the explicit form of the Gaudin matrix) and separate section on the simplest case $Y(\mathfrak{0}_3)$ where previously known results are recovered.

1. I would appreciate a more detailed introduction with more context and explanation of the importance of the problem as well as a conclusion indicating future possible developments (scalar products of off-shell on-shell Bethe vectors? generalisation of the Gaudin formula to other series?)

Publish (easily meets expectations and criteria for this Journal; among top 50%)