Variations on Slavnov's scalar product

We consider the rational six-vertex model on an L-by-L lattice with domain wall boundary conditions and restrict N parallel-line rapidities, N<L/2, to satisfy length-L XXX spin-1/2 chain Bethe equations. We show that the partition function is an (L-2N)-parameter extension of Slavnov's scalar product of a Bethe eigenstate and a generic state, with N magnons each, on a length-L XXX spin-1/2 chain. Decoupling the extra parameters, we obtain a third determinant expression for the scalar product, where the first is due to Slavnov [1], and the second is due to Kostov and Matsuo [2]. We show that the new determinant is a discrete KP tau-function in the inhomogeneities, and consequently that tree-level N = 4 SYM structure constants that are known to be determinants, remain determinants at 1-loop level.


Introduction
Scalar products of N -magnon states on a length-L spin chain, play a central role in studies of correlation functions in integrable spin chains [3,4]. Recently, they have appeared in studies of 3-point functions in 4-dimensional N = 4 super Yang-Mills theory, SYM 4 [5,6,7,8]. Of particular interest is the scalar product of an eigenstate of the spin-chain transfer matrix and a generic state, which in the case of integrable XXX and XXZ spin-1 2 chains was evaluated by N Slavnov as an N × N determinant [1]. Recently, I Kostov and Y Matsuo obtained a second expression for the same object as a 2N ×2N determinant [2].
In this work, we start from Izergin's determinant expression for the domain wall partition function of the rational six-vertex model on an L×L lattice [9], and require that the rapidities on N parallel lattice lines, N L/2, satisfy the Bethe equations of a length-L XXX spin-1 2 chain. We show that the result is an extended version of Slavnov's scalar product that depends on (L − 2N ) extra parameters. Taking these extra parameters to infinity, so they decouple from the partition function, we obtain a third expression for Slavnov's scalar product as an L × L determinant. We show that the new determinant expression is a discrete KP τ -function in the inhomogeneities that can be written in Casoratian form (the discrete analogue of a Wronskian). This allows to use the results of N Gromov and P Vieira [10,11] to prove that SYM 4 tree-level structure constants that are known to be determinants [8], remain determinants at 1-loop level.
0.1. Outline of contents. In Section 1, we recall basic definitions to make the presentation reasonably self-contained, and review recent results to put our own results in context. In 2, we recall basic facts related to the scalar product. In 3, we require that a subset of the parameters of Izergin's domain wall partition function are Bethe roots, and identify the result as a scalar product with extra parameters. In 4, we decouple the extra parameters of Section 3, to obtain a third determinant expression of Slavnov's scalar. In 5, we prove that the third determinant expression is equal to the second determinant expression of [2]. In 6, we give a new proof that the second determinant expression of Kostov and Matsuo is equal to the first determinant expression of Slavnov. Our proof is along the lines of Izergin's proof of the determinant expression of the domain wall partition function, and can be regarded as an alternative to the proof in [2]. In 7, we show that the structure constants that were expressed in determinant form in [8], retain their determinant form when 1-loop radiative corrections are included along the lines of [10,11]. Finally, in 8, we collect a number of remarks.

Definitions and overview
1.1. Context, notation, etc. used in this work. We restrict our attention to the rational six-vertex model and XXX spin-1 2 chain, but our conclusions extend to the trigonometric sixvertex model and XXZ spin- 1 2 chain, as well as to vertex models based on higher-spin su(2) representations and spin chains. All six-vertex configurations will have L vertical lattice lines, and all spin chains will be of length L and periodic.
In six-vertex terms, {x} is a set of free rapidities that flow in horizontal lattice lines, {y} is a set of free rapidities that flow in vertical lattice lines, and {b} is a set of rapidities that flow in horizontal lattice lines and satisfy length-L spin- 1 2 chain Bethe equations. From now on, we refer to the rapidities that flow in horizontal lattice lines as 'rapidities', and to the rapidities that flow in vertical lattice lines as 'inhomogeneities'.
In spin-chain terms, {x} is a set of free auxiliary space rapidities, {y} is a set of free quantum space rapidities, or inhomogeneities, and {b} is a set of auxiliary space rapidities that satisfy Bethe equations. From now on, and similarly to the six-vertex case, we refer to the auxiliary space rapidities as 'rapidities', and to the quantum space rapidities as 'inhomogeneities'. We use |z| for the cardinality of a set {z}.
Rapidities that satisfy Bethe equations are referred to as 'Bethe-restricted'. Six-vertex model configurations and partition functions that depend on Bethe-restricted variables are also Betherestricted. Partitions functions with a subset of rapidities set equal to a subset of the inhomogeneities are 'inhomogeneity-restricted'.
1.2. Bethe eigenstates and a generic states in XXX spin-1 2 chains. Consider a length-L periodic integrable XXX spin-1 2 chain. The Hilbert space of states H is spanned by magnon states. An N -magnon state, N = 0, 1, 2, . . . , is created by the action of N Bethe raisingoperators B(x i ), where x i , i = 1, 2, . . . , N , are free rapidities, on a pseudo-vacuum state |vac . A dual Hilbert space of states H ⋆ is analogously created by the action of Bethe lowering-operators C(x i ) on a dual pseudo-vacuum state vac|. For more details using the same notation and terms used in this work, see [12].
States characterized by free rapidities {x} are not eigenstates of the spin chain transfer matrix. They are 'generic', or 'off-shell'. States characterized by Bethe-restricted rapidities {b} are eigenstates of the spin chain transfer matrix. They are 'Bethe eigenstates', or 'on-shell'.
1.3. The scalar product of a Bethe eigenstate and a generic state. Scalar products of two magnon states play an essential role in studies of integrable spin chains. If both states are off-shell, |x 1 and |x 2 , then the scalar product x 1 |x 2 = x 2 |x 1 can be expressed in Izergin-Korepin sum form [3]. If both states are on-shell, |b 1 and |b 2 , then the scalar product vanishes unless {b 1 } = {b 2 } = {b}. In that case, b|b is the Gaudin norm [13,14].
If one state is on-shell, |b , and the other off-shell, |x , then the scalar product b|x = x|b can be evaluated in determinant form. This is the case in which we are primarily interested in this work.
1.4. The first determinant expression for the scalar product. For spin chains with su(2)symmetry, such as the XXX and XXZ spin-1 2 chains, and their higher-spin analogues, b|x was evaluated by Slavnov in determinant form [1], and therefore is frequently referred to as Slavnov's scalar product. In this work, we simply say 'scalar product', and, in light of the results in [2] and in this work, we refer to Slavnov's determinant expression as 'the first determinant expression for the scalar product'. No tractable expression, such as a determinant, is known for scalar products of an off-shell state and an on-shell state in integrable models based on higher rank algebras.
1.5. The scalar product is a discrete KP τ -function. The scalar product b|x of two Nmagnon states is a function of three sets of variables: 1. A set of Bethe-restricted rapidities {b}, of cardinality N , 2. A set of free rapidities {x}, of cardinality N , 3. A set of free inhomogeneities {y}, of cardinality L, where L is the number of sites of the spin chain that the states live on. To simplify the notation, we will frequently omit to show the dependence on {y}.
In [15,16], it was shown that b|x is a discrete KP τ -function in {x}. Since b|x is symmetric in {y} as well, it was conceivable that b|x is a τ -function in {y} as well, but it was not straightforward to show that.
1.6. Inhomogeneity-restricted scalar products. In [4,17], a class of restricted scalar products, obtained by setting a subset of {x} equal to a subset of {y}, was studied. Since the scalar products are determinants, the inhomogeneity-restricted scalar products are also determinants.
In [17], the six-vertex model configurations whose partition functions are scalar products, and their step-by-step restrictions that lead to inhomogeneity-restricted scalar products were studied in detail.
1.7. Tree-level SYM 4 structure constants that can be expressed in determinant form.
In [5,6,7], a class of tree-level 3-point functions of states that live in scalar su(2) sectors 1 were formulated in XXX spin-1 2 chain terms, and Izergin-Korepin sum expressions were obtained for their structure constants.
In [8], the tree-level structure constants of [5,6,7] were identified with the inhomogeneityrestricted scalar products of [4,17]. The basic idea of [8] was to formulate the structure constants in six-vertex terms, compare the result with the analogous formulation of the inhomogeneityrestricted scalar products in six-vertex terms of [17], and to show that the two objects are equal, up to an overall factor that is easily computed.
In [7], a special case of the 3-point functions of [5,6] where one operator is a non-BPS, while the other two operators are (essentially) BPS states was studied in detail. It turns out that these objects can also be expressed in terms of determinants that are obtained from inhomogeneityrestricted Slavnov determinants by taking a set of Bethe roots to infinity.
1.8. The 'theta morphism' of Gromov and Vieira. In studies of quantum integrability in weakly-coupled N = 4 supersymmetric Yang-Mills, SYM 4 , gauge-invariant single-trace composite operators are mapped to states in closed spin chains [18]. In particular, tree-level single-trace operators, in su(2) scalar sectors, that are eigenstates of the 1-loop mixing matrix, are mapped to states of periodic XXX spin-1 2 chains. So far, these spin chains were homogeneous in the sense that the inhomogeneities were set to the same value, which can be set to zero.
In [10,11], N Gromov and P Vieira showed that 1-loop radiative corrections can be introduced into the structure constants studied in [5,6,7], and that were expressed as determinants in [8], by switching on the inhomogeneities, that is, by choosing y i = θ, i = 1, . . . , L, computing the structure constants to lowest non-trivial order in θ, which is O(θ 2 ), as there are no order O(θ) contributions, and setting θ equal to the gauge coupling constant.
1.9. 1-loop SYM 4 structure constants that can be expressed in determinant form. In [19], structure constants of the latter 3-point functions were identified with six-vertex model configurations on (N × L)-rectangular lattices, N L, with 'partial domain wall boundary conditions', and the corresponding 'partial domain wall partition functions', pDWPF's, were studied in some detail.
In [19], we obtained two expressions for these partition functions: 1. As a determinant of an L × L matrix, Z L×L and 2. As a determinant of an N × N matrix, Z N ×N . The latter was first obtained by I Kostov [20,21]. Further, we showed that Z L×L is a discrete KP τ -function in the inhomogeneities {y}, and Z N ×N is a discrete KP τ -function in the free rapidities {x}.
Using the fact that these pDWPF's are discrete KP τ -functions in the inhomogeneities {y}, together with the recent results of N Gromov and P Vieira [10,11] allowed us to show that these structure constants remain determinants in the presence of 1-loop radiative corrections. On the other hand, the fact that we were unable to show that the inhomogeneity-restricted scalar product is a τ -function in the inhomogeneities prevented us from extending the determinant result to 1-loop level for tree-level structure constants with 3 non-BPS operators.
1.10. A second determinant expression for the scalar product. In [22,23], I Kostov and F Smirnov independently suggested that the scalar product of a Bethe eigenstate and a generic state can be obtained from Izergin's (L × L) domain wall partition function, either by sending (L − 2N ) rapidities, N L/2, to infinity, and thereby decoupling them so that one ends up with a partial domain wall partition function, then setting N of the remaining rapidities to satisfy appropriate Bethe equations [22], or by re-interpreting Korepin's domain wall configuration as the scalar product of a Bethe eigenstate that is built on the lowest-weight pseudo-vacuum (all spins down, rather than up) and a generic state, by requiring an appropriate subset of rapidities to satisfy appropriate Bethe equations [23]. The latter Bethe equations should come from 'beyond the equator' [24].
In [2], I Kostov and Y Matsuo obtained a realization of the suggestion of [22], starting from 2N × L lattice configurations, 2N L, called 'partial domain wall configurations' in [19], whose partition functions can be written in determinant form in two different ways: As the determinant of an (2N × 2N ) matrix [20,21], or as the determinant of an (L × L) matrix. In [2], Kostov and Matsuo start from the (2N × 2N ) determinant, require that N rapidities satisfy an appropriate set of Bethe equations, then show that the result is equal to Slavnov's (N × N ) determinant expression for the scalar product.
1.11. Bethe-restricted domain wall partition functions as parameter-extended scalar products. In this work, we obtain a parameter-extension of [22,2], that is not identical to but partially along the lines of [23]. We start from an (L × L) Korepin domain wall configuration, and the corresponding (L × L) Izergin determinant expression for the partition function, then we set N rapidities to satisfy appropriate Bethe equations, such that Izergin's determinant now has (L − N ) free rapidities, and split the latter into two subsets {x} of cardinality N , and {t} of cardinality (L − 2N ) 0. We interpret the resulting determinant as an (L − 2N )-parameter extension of the scalar product of an N -magnon Bethe eigenstate and an N -magnon generic state, on a length-L XXX spin-1 2 chain. Everything we say applies without obstruction to six-vertex models with trigonometric weights and the corresponding XXZ spin-1 2 chains.

1.12.
A third determinant expression for the scalar product. Taking the (L − 2N )extension parameters to infinity, having first normalized properly, we obtain a determinant of an (L × L) matrix, that we interpret as a third determinant expression of the scalar product of a Bethe eigenstate and a generic state, of N magnons each, on a length-L XXX spin-1 2 chain. We prove that by showing that our determinant expression is equal to the second determinant expression in [2]. The fact that the two have different forms follows from the fact that [2] start from the (2N × 2N ) determinant expression of the (2N × L) partial domain wall partition function, while in this work, we start from the (L×L) determinant expression of the same object.
1.13. The scalar product as a discrete KP τ -function in the inhomogeneities. In previous works [15,16], we showed that the scalar product is a discrete KP τ -function in the free rapidities, but we could not obtain the same result in terms of the inhomogeneities, even though the scalar product is symmetric in them.
As an application of the third determinant form of the scalar product obtained in this paper, we show that the scalar product is a discrete KP τ -function in the inhomogeneities.
1.14. Structure constants that are determinants at tree-level remain determinants at 1-loop level. Given that the scalar product is a discrete KP τ -function in the inhomogeneity variables {y}, it can be written in Casoratian form in any of the variables y i , i ∈ {1, 2, . . . , L}. This allows us to show that the structure constants that can be expressed in determinant form [8], retain their determinant form in the presence of 1-loop radiative corrections, when the latter are included along the lines proposed in [10,11]. This is an application of the results of this work to computations in weak-coupling N = 4 supersymmetric Yang-Mills.
The assignment of weights to the vertices of the model is shown in Figure 1.
We define the scalar product S({x}, {b}|{y}) to be the partition function of the lattice shown in Figure 2. This definition is a completely consistent graphical representation of the algebraic form of the scalar product [3]. The first determinant expression for the scalar product was found by Slavnov in [1]. Slavnov's expression is given by 3 for the Vandermonde in a set of N variables {x} = {x 1 , . . . , x N }. 2 To simplify the notation, we often leave the subscripts off the sets {b} N , {x} N , . . . when the cardinality is clear from the context. 3 Here, and in all subsequent determinants, we use Greek indices to label the rows and Latin indices to label . .

A scalar product that depends on extra parameters
We consider a domain wall partition function on an L × L lattice, whose rapidities are the union of the three sets . , x N } and whose inhomogeneities are {y} L = {y 1 , . . . , y L }. We denote such a partition function by Z({x}, {b}, {t}|{y}). Its graphical version is shown in Figure 3.
Using Izergin's determinant formula for the DWPF [9] (applied to the case where the rapidities are of mixed type 4 ), we have where indices in the products range over precisely the same values as in the determinant, namely 1 α, β N , 1 γ L − 2N , 1 j L. We adopt this convention in all analogous formulae which follow. One of the purposes of this paper is to prove the following result. The proof will be given over the course of Sections 4-6.

A third determinant expression for the scalar product
Define the function This limit was studied in [19], and the resulting object was called a 'partial domain wall partition function'. This is in reference to the fact that Z({x}, {b}|{y}) is the partition function of the rectangular lattice shown in Figure 4, with 'partial domain wall boundary conditions'.
Starting from the determinant expression (6), the limits in (7) can be taken explicitly. This procedure was explained in detail in [19], so we only quote the result here, To prove Lemma 1, it is clearly sufficient to prove the following.
. .  Lemma 2 is another version of the recent result of I Kostov and Y Matsuo [2], who proved that the Slavnov scalar product is equal to a pDWPF. The result of [2] was in the context of the formula (11) for the pDWPF, which we discuss in the next section, whereas our result is in the context of equation (8) for the pDWPF. We call (11) and (8) the second and third expression for the scalar product, respectively.

The third determinant expression equals the second
An alternative determinant expression for the pDWPF was found by Kostov in [20]. Kostov's expression for the pDWPF is 5 In contrast to the determinant (8), which is L × L, the determinant in (11) is 2N × 2N . A direct proof of the equivalence of the two determinants (8) and (11) was given in [19]. We will not repeat this proof here, and from now on treat (8) and (11) as interchangeable expressions for the pDWPF.

The second and third determinant expressions equal the first
In this section we give an alternative proof of Lemma 2. The original proof was given in [2]. The basis of our proof is to define inhomogeneity-restricted versions of both the scalar product (3) and the pDWPF (8), (11). The inhomogeneity-restricted scalar products were defined and calculated previously in [4,17]. 5 Again, we remark that the usual form of Kostov's determinant is when the rapidities and inhomogeneities are labelled uniformly, as {x} = {x 1 , . . . , x N } and {y} = {y 1 , . . . , y L }, respectively. In that case, it is given by In the case N = L, the determinant (10) becomes an alternative expression for the DWPF.
6.1. Inhomogeneity-restricted scalar products. In this section, we consider variations of the size of the sets {x}, {b}, {y}. For that reason, it is necessary to restore subscripts to these sets to indicate their cardinality. For all 0 n N we define . . , y L }. We split up the dependence on the inhomogeneities deliberately, to indicate the separate symmetry in these two sets. The case n = N is the scalar product (3) itself.
Starting from the formula (3), we can explicitly evaluate the function defined in (12). The result is the following hybrid determinant Since this object comes from the first expression for the scalar product (3), we call it the first expression for the inhomogeneity-restricted scalar product.

6.2.
Properties of the inhomogeneity-restricted scalar product. Considering the inhomogeneity-restricted scalar product (13) as a function in x n , we can show that it has the following properties.
A. It is a meromorphic function in x n of the form where ) is a polynomial of degree L − N + n − 1 in x n . B. It is symmetric in the set of variables {y N −n+1 , . . . , y L }. C. By setting x n = y N −n+1 , we obtain the recursion relation In the case n = 0, we have Since it is quite straightforward to verify that these statements are true, we only comment briefly on their proof. A is proved by showing that all poles in x n in the denominator of (13) are cancelled by a zero resulting from setting two rows of the determinant equal. Similarly, one shows that the determinant itself has only poles at the points specified in the denominator of (14). Degree counting establishes the correct degree for the polynomial in the numerator.
B is easy to prove, since the only place where {y N −n+1 , . . . , y L } appear are in the factors L l=1 (x α − y l )/(x α − y l + 1), which are obviously symmetric with respect to these variables. C follows from the definition (12) of the inhomogeneity-restricted scalar products. D comes from comparing the n = 0 case of (13) with Izergin's determinant representation of the DWPF (5). Importantly, the properties A-D uniquely determine the set of scalar products (12). If another set of functions obey the same properties, they must be equal to the inhomogeneity-restricted scalar products.
6.3. Inhomogeneity-restricted partial domain wall partition functions. In analogy with Section 6.1, we define inhomogeneity-restricted versions of the pDWPF (8), (11). This is done in precisely the same way, namely, for all 0 n N we define The case n = N is just the pDWPF itself. Explicit formulae for the inhomogeneity-restricted pDWPF can be obtained by starting from the determinants (8) or (11), and specializing the variables in the way prescribed by (17). Doing this in the case of (8) gives where the indices of the products range over the same values as in the determinant, namely, 1 α n, 1 β N , N − n + 1 j L. 6 In the case of (11), we get where we again remark that the indices in the products range over the values 1 α n, 1 β N , 1 γ N − n. Ultimately, we will show that (19) and (18) are equal to the inhomogeneity-restricted scalar product (13). For that reason, hereafter we refer to them as the second and third expression for the inhomogeneity-restricted scalar product, respectively. 6.4. Properties of the inhomogeneity-restricted pDWPF. Consider the determinant representations (18) and (19) for the inhomogeneity-restricted pDWPF, as a function in x n . We claim that the inhomogeneity-restricted pDWPF has the following properties.
A. It is a meromorphic function in x n of the form ) is a polynomial of degree (L − N + n − 1) in x n . B. It is symmetric in the set of variables {y N −n+1 , . . . , y L }.
C. By setting x n = y N −n+1 , we obtain the recursion relation In the case n = 0, we have To prove these properties, it is convenient to freely change between the expressions (18) and (19). A is proved using (19). One can easily check that all poles in x n in the denominator of (19) are cancelled by a zero from setting two rows of the determinant equal. The only poles which are present are the ones described by (20), and degree counting establishes the correct degree for the polynomial in the numerator.
B is proved using (18), which is invariant under simultaneously reordering the inhomogeneities in the Vandermonde ∆{−y} N −n+1 L and those in the determinant. C follows from the definition (17) of the inhomogeneity-restricted pDWPF. D is proved by considering the n = 0 case of (18), when it is identically the pDWPF described in (22). ) satisfy the same set of properties A-C. The only apparent difference is their initial condition, property D. In the following subsection we will show that because {b} N satisfy the Bethe equations, the right hand sides of equations (16) and (22) are in fact equal up to the sign (−) N . In doing so, we will have proved that for all 0 n N , due to the fact that the properties A-D are uniquely determining. This is of course sufficient to prove Lemma 2, which corresponds to the case n = N of equation (23).
6.6. Resolving the initial condition. We begin by adjusting the right hand side of (16), which follows from Kostov's expression for the DWPF (see (10) with N = L). Hence the right hand side of (16) can be written as where the last line follows from using the Bethe equations (2) to modify every entry of the determinant. We denote the final determinant in (25) by and from now on, treat it as a linear function in free variables {κ 1 , . . . , κ N }. On the other hand, using Kostov's pDWPF formula (10), the right hand side of (22) is given by Up to the sign (−) N , which it is necessary for us to introduce at some point, we write the determinant in (27) as where again κ i ≡ L k=N +1 (bi−y k ) (bi−y k +1) , but we treat these as free variables. We prove that where we have introduced the notationb i = b i +1. Subtracting (column j+1)/b N from (column j) for all 1 j N − 1, the final row of the determinant is only non-zero in the final entry. This reduces the size of the determinant by 1, and after extraction of common factors from the surviving rows, one obtains (bN −bi) . One can also evaluate the derivative with respect to κ N , and subtracting (column j + 1)/b N from (column j) for all 1 j N − 1, in analogy with above, we find that . Since (30) and (32) are recursion relations for the linear function D 2 (N, κ) at two different values of κ N (namely, κ N = 0 and as κ N → ∞), together with the initial condition D 2 (1, κ) = κ 1 − 1, they determine it uniquely.

7.2.
Complete symmetric functions and discrete derivatives. Following [25], the complete symmetric functions h i {y} in the set of variables {y} = {y 1 , . . . , y L } are defined as coefficients in a generating series, We define a discrete derivative ∆ l which acts on the complete symmetric functions as where the subscript l is used to denote the l-th element of the set {y}, and y l denotes the omission of that variable from the set. 7.3. Casoratian determinants. We define a Casoratian matrix Ω to be one whose entries ω i,j are symmetric with respect to a set of variables {y} = {y 1 , . . . , y L }, and satisfy where ∆ l is the discrete derivative with respect to any variable y l ∈ {y}. The determinant of a Casoratian matrix is a Casoratian determinant, and is a discrete analogue of the Wronskian.
Using the definition (37) of the discrete derivative, it is easy to see that the determinant satisfies (38), and is therefore Casoratian, for arbitrary M L and coefficients c ik , 1 i L, 1 k M . Using the Jacobi-Trudi identity for Schur functions [25], we can write (39) equivalently as where the final equality is just from matrix transposition. We will take (40) as our generic form of a Casoratian determinant. Casoratian determinants are τ -functions of the discrete KP hierarchy (see, for example, [16] for a more detailed exposition). All results stated in the sequel apply to any determinant of the form (40), and all such determinants can also be viewed as discrete KP τ -functions in the {y} variables.
7.4. Action of Gromov-Vieira mapping on Casoratian determinants. In [19], we studied the action of the GV mapping on partial domain wall partition functions, which are Casoratian determinants in their inhomogeneities {y} L . However the procedure outlined in [19] applies generally to any Casoratian determinant of the form (39) (or equivalently, (40)). We briefly review these results here. Let [|Ω|] y denote the GV mapping of the determinant (40) (the transposed version, for notational convenience), with respect to the variables {y 1 , . . . , y L }. We claim that |Ω| y = c j, 1 . . . c j,L−2 c j,L−1 + g 2 Lc j,L+1 c j,L + g 2 Lc j,L+2 1 j L We will not prove this equation here, since full details can be found in [19]. Equation (41) says that, up to higher order corrections in g, the determinant structure of Casoratian determinants is preserved under the GV mapping (35). 7.5. Restricted scalar products are Casoratian determinants. Returning to equation (18) for the inhomogeneity-restricted scalar product S({x} n , {b} N |{y} N −n , {y} N −n+1 L ), we renormalize it as follows, It is then straightforward to show that where the coefficients c ik {X} are given by ( From equation (44), we see that the (renormalized) inhomogeneity-restricted scalar products are Casoratian determinants in the set of inhomogeneities {y N −n+1 , . . . , y L }. Hence their image under the GV mapping in these variables is given by (41). We conclude that the determinant structure of tree-level SYM 4 structure constants between three non-BPS states (which are equal to inhomogeneity-restricted scalar products, [8]) is preserved under 1-loop corrections. This extends our previous result [19] in the context of tree-level SYM 4 structure constants between two BPS and one non-BPS state (which are equal to pDWPF's, [7]). Previously, we were unable to obtain such a result starting from the expression (13) for the inhomogeneity-restricted scalar product, which is not manifestly a Casoratian determinant in the inhomogeneities.

1.
Our results extend to the trigonometric six-vertex model and the corresponding XXZ spin-1 2 chain without obstruction. 2. We note the different dimensions of the determinants in the three expressions for the inhomogeneity-restricted scalar product, equations (13), (19) and (18). The first expression (13) comes directly from Slavnov's determinant for the scalar product, and is N × N . The second expression (19) comes from the pDWPF expression for the scalar product in [2], which has 2N rapidities. Hence (19) is 2N × 2N . The third expression (18), which is new in this work, comes from a specialization of an L×L determinant, and is (L − N + n)×(L − N + n). Since the second and third expressions, (19) and (18), are larger in size than the first (13), it is unclear whether these will be computationally advantageous in studies of N = 4 SYM structure constants.
3. It is tempting to conjecture that Slavnov-type scalar products in spin chains based on higher-rank algebras can also be obtained from Bethe-restricted versions of the higher-rank domain wall partition functions (giving a parameter-extended scalar product), in the limit where the extension parameters are decoupled [26].
4. We have no statistical mechanical interpretation for the extra parameters in the parameterextended Slavnov scalar product. Hopefully, these parameters will act as regularization parameters in computations of physical objects, such as correlation functions, to be removed at the end of the computation. It may also be that the parameter-extended scalar product is an expectation value of an operator that is characterized by the extension parameters.
5. The parameter-extended scalar product of Section 3 is a discrete KP τ -function in the extra parameters, which play the role of an extra set of Miwa variables.
6. We used the fact that the third determinant expression for the inhomogeneity-restricted scalar product is a discrete KP τ -function, and therefore can be explicitly written as a Casoratian determinant, together with the results of N Gromov and P Vieira [10,11], to show that treelevel SYM 4 structure constants, with three non-BPS states that are known to be determinants [8], remain determinants in the presence of 1-loop corrections. The results of [10,11] actually extend to 2-loop radiative corrections, and our computations can be straightforwardly extended to 2-loops. In [27], D Serban argued that the results of [10,11] extend to all loops, at least in the limit where all three operators are represented by asymptotically long spin chain states.
7. The second determinant expression of Kostov and Matsuo is expressed as an expectation value of free charged fermions in [2]. From this expectation value, one can deduce that the second determinant expression is a discrete KP τ -function in the inhomogeneities 7 . Writing this fermionic expectation value explicitly as a Casoratian determinant, it should be possible to use it to obtain the result of Section 7 in this work. It should also be possible to obtain the result directly from the first determinant expression of Slavnov, but we expect this to be tedious. Having different expressions for the same object, we expect that each should be easier to use for different purposes.
8. While writing the results reported in this work, we became aware of [28], where the rational Gaudin model was studied, and it was shown that the scalar product between a generic state and Bethe eigenstate can be expressed as a domain wall partition function. This observation was based on replacing the original N -magnon Bethe eigenstate (coming from the action of N spinlowering operators on the highest weight state) with an equivalent (L − N )-magnon eigenstate (coming from the action of L − N spin-raising operators on the lowest weight state). This is a different approach to the one reported in this paper, where we have considered the equivalence of the Slavnov scalar product and domain wall partition function without needing to change the Bethe roots. Hence the results of [28] are essentially unrelated to the results in this work.