Grassmannian Integral for General Gauge Invariant Off-shell Amplitudes in N=4 SYM

In this paper we consider tree-level gauge invariant off-shell amplitudes (Wilson line form factors) in $\mathcal{N}=4$ SYM with arbitrary number of off-shell gluons or equivalently Wilson line operator insertions. We make a conjecture for the Grassmannian integral representation for such objects and verify our conjecture on several examples. It is remarkable that in our formulation one can consider situation when on-shell particles are not present at all, i.e. we have Grassmannian integral representation for purely off-shell object. In addition we show that off-shell amplitude with arbitrary number of off-shell gluons could be also obtained using quantum inverse scattering method for auxiliary $\mathfrak{gl}(4|4)$ super spin chain.


Introduction
Following Witten's twistor string theory [1] we have witnessed an enormous progress in understanding the structure of S-matrix of N = 4 SYM as well as other gauge theories (see [2,3] for a review). The central role in these achievements was played by newly developed computational methods, such as BCFW recursion [4,5] for tree amplitudes and generalized unitarity (see [2] and references therein) for loop amplitudes. The power of the mentioned on-shell methods is largely due to the introduction of new variables, such as helicity spinors, momentum twistors [6], link variables [7] together with an extensive use of superspace methods [8,9]. The use of on-shell methods resulted in the explicit answers for amplitudes both at high orders of perturbation theory and/or with large number of external legs (see [2,3] for a review and reference therein). The latter have led to several important all-loop conjectures as well as to the discovery of underlying integrable structure of N = 4 SYM S-matrix [10][11][12][13][14][15][16][17][18][19]. We should mention an important research direction originated with the connected RSV prescription [20,21]. Within the latter N = 4 SYM amplitudes are expressed in terms of the localized integrals over the moduli space of n-punched Riemann spheres. The next progress along this direction was due to the introduction of scattering equations [22][23][24][25][26], which were later generalized to loop level [27][28][29][30] and derived from ambitwistor string theory [31].
The on-shell methods were also successfully applied to partially off-shell objects, such as form factors in N = 4 SYM theory . The latter may be viewed as the amplitudes of the processes, in which classical field coupled through gauge invariant operator O produces an on-shell quantum state. Grassmannian representation is no exception and can be applied to form factors as well [76][77][78][79].
Another interesting off-shell objects, which will be the subject of the present paper, are gauge invariant off-shell amplitudes [80][81][82][83][84][85][86][87][88] (also known as reggeon amplitudes within the context of Lipatov's effective lagrangian), which one encounters within k T -or highenergy factorization approach [89][90][91][92] as well as in the study of processes at multi-regge kinematics. For other studies of off-shell currents and amplitudes see [93][94][95][96][97]. In the study of form factors we are dealing with local gauge invariant color singlet operators, for example operators from stress-tensor operator supermultiplet [47][48][49][50][51][99][100][101][102]. However, we may consider also gauge invariant non-local operators, for example Wilson loops (lines) or their products. This way we may consider gauge invariant off-shell amplitudes as form factors of Wilson line operators or their products [80][81][82][83][84][85][86][87][88]. An insertion of Wilson line operator corresponds to the off-shell or reggeized gluon in such formulation. In our previous paper [103] we presented a conjecture for Grassmannian integral representation of gauge invariant off-shell amplitudes with one leg off-shell and shown how they could be described in a language of auxiliary gl(4|4) super spin chain. The purpose of this paper is to extend the results of [103] to the case of amplitudes with multiple off-shell gluons.
This paper is organized as follows. In section 2 after recalling necessary definitions we formulate a conjecture for Grassmannian integral representation of gauge invariant amplitudes with multiple off-shell gluons. Here we also formulate a hypothesis for the structure of on-shell diagrams for off-shell amplitudes. The appendix A contains a check of the latter in the case of 3-point amplitude with two off-shell gluons. In section 3 we use explicit examples with known BCFW answers [80] to perform checks of our conjecture. The explicit calculations are performed using both spinor helicity and momentum twistor representations. Section 4 is devoted to the auxiliary spin chain description of the off-shell amplitudes and finally we come with our conclusion.

Gauge invariant off-shell amplitudes and regulated integral over Grassmannian
One way to define gauge invariant amplitudes containing off-shell gluons is in terms of form factors of Wilson line operators [86]: Here t c are SU(N c ) generators 1 and we also assumed so called k T -decomposition of the off-shell gluon momentum k, k 2 = 0: where p is the gluon direction (also known as the off-shell gluon polarization vector), such that p 2 = 0, p · k = 0 and x ∈ [0, 1]. There is a freedom in such decomposition, which could be parametrized by an auxiliary light-like four-vector q µ , so that Using the fact, that now k µ T is transverse both with respect to p µ and q µ vectors, the offshell gluon transverse momentum k µ T could be expanded in the basis of two "polarization" vectors 2 as [80]: It is easy to see, that k 2 = −κκ * and using Schouten identities it could be shown, that both κ and κ * are independent of auxiliary four-vector q µ [80]. Another useful relation, which follows directly from k T decomposition and will be used often in what follows, is given by Note, that Wilson line operator we use to describe off-shell gluon is colored. It is invariant δW c p (k) = 0 under local infinitesimal gauge transformations δA µ = [D µ , χ] with χ vanishing at infinity x → ∞. At the same time it transform in the adjoint representation under global SU(N c ) transformations with constant χ as [82,83]: (2.6) 1 The color generators are normalized as Tr(t a t b ) = δ ab 2 Here we used helicity spinor decomposition of light-like four-vectors p and q.
Using Wilson line operator defined above the gauge invariant amplitude with one off-shell and n on-shell gluons can be written as [86]: Here asterisk denotes an off-shell gluon and p, k, c are its direction, momentum and color index correspondingly. Next i or ε + i and color index c i . Amplitudes with multiple off-shell gluons can be represented in a similar fashion: where p i is the direction of the i'th (i = 1, ..., n) off-shell gluon and k i is its momentum. As a function of kinematical variables this function is given by where λ p,j ,λ p,j are spinors coming from helicity spinor decomposition of of j'th Wilson line direction vector p j . Note, that we can consider a situation where only off-shell gluons are present (correlation function of Wilson line operators): In practical calculations it is more convenient to deal with the color ordered versions of the above amplitudes. The original amplitudes are then recovered through the color decomposition 3 : A * n+m (1 ± , . . . , m ± , g * m+1 , . . . , g * m+n ) = g n−2 σ∈S n+m /Z n+m tr (t a σ(1) · · · t a σ(n) ) × × A * n+m σ(1 ± ), . . . , σ(g * n+m )) .

(2.11)
Here S n+m is the set of all permutations of n + m objects and Z n+m is the subset of cyclic permutations. We will also sometimes write g ± i instead of i ± to denote on-shell gluon to make some formulas more transparent.
In the case of N = 4 SYM one can also consider other then gluons on-shell states from N = 4 supermultiplet. The way to do it is to combine all sixteen on-shell states of N = 4 SYM into one on-shell chiral superfield [8]: 3 See for example [98,103].
where g + , g − denote creation/annihilation operators of gluons with +1 and −1 hecilities, ψ A are creation/annihilation operators of four Weyl spinors with negative helicity −1/2, ψ A are creation/annihilation operators of four Weyl spinors with positive helicity and φ AB stand for creation/annihilation operators of six scalars (anti-symmetric in the SU(4) R Rsymmetry indices AB). In what follows we will also need superstates defined by the action of superfield creation/annihilation operators on vacuum. For n-particle superstate we have Ω 1 Ω 2 . . . Ω n | ≡ n i=1 0|Ω i and corresponding N = 4 SYM superamplitudes could be written as where the explicit dependence of A * m+n Ω 1 , . . . , g * m+n amplitude on kinematical variables is given by The color ordered versions of all types of off-shell gauge invariant amplitudes considered before could be efficiently computed using an off-shell generalization [80,81] of the original on-shell BCFW recursion [4,5]. Later in [103] we presented a conjecture for Grassmannian integral representation of tree level color ordered version of (2.7) amplitude A * k,n+1 (here 4 k is related to overall helicity λ Σ of on-shell particles given by λ Σ = n + 2 − 2k). In our consideration in [103] n on-shell particles were treated in manifestly supersymmetric way, while the off-shell gluon remained unsupersymmetrized. To be more explicit 5 , let us consider the following Grassmannian integral over Gr(n + 2, k): , and external kinematical variables are defined as Here k is the off-shell gluon momentum. It is also assumed, that the off-shell momentum k is k T decomposed (2.2), so that p = λ pλp and q = ξξ. We claim, that making an appropriate choice of integration contour Γ, which is likely to be given by "tree contour" Γ tree [32] of n + 2 point N k−2 MHV on-shell amplitude, the gauge invariant amplitude with one gluon off-shell and n on-shell particles in N = 4 SYM could be written as [103]: This relation was successfully verified for arbitrary n and k = 2, 3. The factor Reg. in (2.15) can be considered as a deformation or IR regulator of the Grassmannian integral for on-shell amplitudes. Namely, the Grassmannian integral representation of on-shell amplitude A k n is given by (Γ = Γ tree ): .
As it is known, this integral is singular in the holomorphic soft limit (we take limit p → 0 such that for p = λλ, λ → ǫλ andλ →λ and ǫ → 0) with respect to any momentum of external on-shell particle p i . The behavior of the integral in such limit is controlled by soft theorems [104][105][106][107][108][109][110][111][112][113][114][115][116][117][118][119][120]. The same behavior also holds for the holomorphic soft limit with respect to the on-shell momenta of the Ω k n+2 Grassmannian integral. The soft behavior of the amplitude with one gluon off-shell A * k,n+1 with respect to the holomorphic soft limits of the on-shell variables p and q parameterizing off-shell gluon is however different. In this limit it must be regular and the following relation should hold 6 (q = ξξ, ǫ → 0) with the helicity of the on-shell gluon with momentum p n+1 = p equal to −1. The behavior of the Ω k n+2 with respect to such limit is similar [103]: where contours Γ tree and Γ ′ tree are identical except that Γ tree include poles at the zeros of the minors 7 (n − k + 4 · · · 1) up to (n + 1 · · · k − 3). The insertion of Reg. function is exactly what gives the desired behavior of Ω k n+2 and in fact the form of Reg. function can be fixed by this requirement. We are expecting the same property of regularity in the holomorphic soft limits with respect to variables parameterizing off-shell gluons also in the case of gauge invariant amplitudes with several off-shell gluons (Wilson line insertions) A * k,m+n (which is color ordered version of (2.8) and k is related to the total helicity λ Σ of on-shell particles as λ Σ = m − 2k + 2n). Namely, the amplitude should be regular in the limit ξ j → ǫξ j , ǫ → 0, j = 1, . . . , n where ξ j is the spinor associated with the auxiliary vector q i from k T decomposition of i'th off-shell gluon momentum k i . If the Grassmannian integral description is possible for such objects, then the Grassmannian integral should be regular in corresponding limits. Here we want to conjecture a representation for such integral using this regularity requirement. Moreover, we want to describe the external kinematical data in a way similar to the case of one off-shell gluon. The last requirement suggests that we should consider integral over Gr(k, m + 2n) Grassmannian, where m is the number of on-shell momenta, and n is the number of off-shell onces.
In the case of amplitudes with one off-shell gluon A * k,n+1 integrands of corresponding Grassmannian integrals (2.15) contained products of delta functions of linear combinations of kinematical data and Grassmannian local coordinates (given by elements of C matrix) together with the product of consecutive minors of C matrix identical to the case of the Grassmannian integral L k n+2 [Γ]: We want to use similar ingredients also in the case of n off-shell gluons. In addition we need an insertion of some regulating function Reg(m + 1, . . . , m + n). which should regulate holomorphic soft limit behavior with respect to ξ j variables. Thus, we are going to consider Grassmannian integral of the form , where external kinematical variables are chosen as Reg(j + m), Direct evaluation of this Grassmannian integral for some explicit examples, presented in next section, show that indeed it is likely to be the correct representation for A * k,m+n . That is, we want to show that given an appropriate choice of integration contour Γ = Γ tree the following identity holds Here, as before, Ω i is an i-th on-shell N = 4 chiral superfield and g * j are off-shell gluons (Wilson line operator insertions). In addition, it is likely that Γ tree can be chosen identical to the case of A k,m+2n on-shell amplitude at least in some cases.
In [103] we have also shown 9 , that on-shell diagrams for scattering amplitudes with one leg off-shell are given by corresponding on-shell diagrams for on-shell scattering amplitudes with one of the vertexes exchanged for off-shell vertex, see Fig. 1. We expect that in the case of several off-shell legs the corresponding on-shell diagrams could be also obtained by similar procedure, that is exchanging on-shell vertexes containing off-shell legs with offshell vertexes. Recalling our cutting and gluing procedure from [103] it is easy to convince yourself that it is true in a case when off-shell legs are separated by on-shell ones. In a case when off-shell legs stay next to each other it is not as obvious. To see that it is actually the case, let us consider as example 3-point amplitude with two legs off-shell. In this case the corresponding on-shell diagram is the first diagram in Fig. 3 times the corresponding Reg. function (product of Reg. functions from two off-shell vertexes). The latter could be taken out of integration sign 10 , while the original off-shell diagram is reduced (see Fig. 3) to the on-shell diagram corresponding to our Grassmannian representation using equivalence moves from Fig. 2.
Finally, to end this section, we would like to stress the following feature of our conjecture. Namely, we may consider the situation when there are no external on-shell degrees of freedom at all (m = 0). In this case we obtain the Grassmannian integral representation of color ordered correlation function of Wilson line operators (2.10):

Examples and checks
Now we are going to reproduce results of BCFW recursion [80] for different off-shell amplitudes containing multiple off-shell gluons. We will start with calculations using spinor helicity representation and later see how similar computations could be performed using momentum twistor representation.

spinor helicity representation
Let us first consider the simplest cases when Grassmannian integral fully localizes on delta functions. In the case of A * k,1+2 (g + 1 , g * 2 , g * 3 ) amplitude k = 2 and the corresponding Grassmannian integral is given by Here we integrate over Gr(2, 5) Grassmannian and the integral is fully localized on delta functions, so that the choice of integration contour prescription could be skipped. The Reg. functions are given by (m = 1) 10 See appendix A for details.
Now, we should comment on the conventions used for spinor products. First, we label all products of ǫ αβ λ α i λ β j and ǫαβλα iλβ j spinors as ij and [ij]. Next, to obtain final expressions we need to use spinor redefinitions from (2.24). In the present case with m = 1 and n = 2: λ 1 = λ 1 and spinors λ i with i = 2, . . . , 5 are expressed in terms of ξ 1+j , λ p 1+j , j = 1, 2. We will also use bra and ket notation for spinors λ i ≡ |i ,λ i ≡ |i] sometime when it will make formulas more clear. Taking into account k T decomposition of off-shell momenta k 2 and k 3 the above expression for Ω 2 1+4 is rewritten as (here and below p i = λ iλi , p 2 i = 0, which is exactly the result of BCFW recursion from [80]: As we already said before, the off-shell gluons could be actually arbitrary distributed among on-shell ones. The general formula for such configurations will look rather complicated, but particular examples are not. As an example, let us reproduce known answer for A * k,n+2 (g * 1 , g + 2 , . . . , g + i−2 , g * i−1 , g + i , . . . , g + n ) amplitude. As before k = 2 and we are integrating over Gr(2, n + 4) Grassmannian. In this case the integral is also localized on delta functions (note that in this particular example we use different labels for Reg. functions compared to other examples since the positions of off-shell gluons are different) and is given by

33) with
Evaluating the above integral we get (for saving space we skip momentum conservation delta function δ 4 (k 1 + p 2 + . . .
which after relabeling spinor variables can be rewritten as The latter result is in agreement with the result of BCFW recursion from [80] together with replacement of spinors λ l ,λ l ,η l , l = i, i + 1 with Here k j is the momentum for the j-th off-shell gluon and the label j should be chosen in such a way that the consecutive numeration of particle momenta is restored. All other spinor labels should be relabeled accordingly. To obtain n off-shell gluon insertions this operation should be repeated n − 1 times. This way (2.23) shows the result of these steps when off-shell gluons are inserted "one after another". Next, let us consider A * k,1+2 (g − 1 , g * 2 , g * 3 ) amplitude. In this case k = 3 and the integral over Gr (3,5) Grassmannian is again localized on delta functions:  [51] . (3.43) Expressing spinors in terms of external kinematical data using (2.24) with m = 1 and n = 2 together with k T decomposition of off-shell momenta k i (q i = ξ iξi , i = 2, 3): and similar expressions forλ 2 ,λ 4 we get (here we again dropped momentum conservation delta function δ 4 (p 1 + k 2 + k 3 )): This is exactly the result of BCFW recursion [80] for A * 3,1+2 (g − 1 , g * 2 , g * 3 ) amplitude: It is interesting to consider amplitudes with only off-shell states present (m = 0). The simplest example of this kind is given by A * k,0+3 (g * 1 , g * 2 , g * 3 ) amplitude. In this case k = 3 and integration goes over Gr(3, 6) Grassmannian. The Grassmannian integral in this case is no longer trivial and does not localizes on delta functions. It can be however reduced to the integral over one complex parameter τ , which in its turn could be evaluated by taking residues (see [32,103]). The result of BCFW recursion for this amplitude is given by [80]: where P ′ is the permutation operator shifting all spinor and momenta labels by +1 mod(3). It should be stressed that this object can be considered as a correlation function of three gauge invariant operators given by Wilson lines. Now, let us proceed with the Grassmannian integral itself. The conjectured Grassmannian integral representation in this case is given by the following integral where As we already mentioned this integral can be reduced to the integral over single complex parameter τ . Next, we fix GL(3) gauge as in [3] in the case of NMHV 6 amplitude. The minors (123), (345), (561) in this case will became linear functions of parameter τ and we choose integration contour identical to the case of NMHV 6 amplitude. This choice means, that we are interested in residues at the zeros of minors (123) where we used the definition of external kinematical variables from (2.24) with m = 0 in the argument of momentum conservation delta function. The other residues can be obtained by the action of permutation operator P shifting labels of λ i andλ i spinors by +1 mod(6) . (3.55) The other residues are then given by So, we see that indeed the following relation holds 3 j=1,2,3 , (3.57) and our conjectured Grassmannian integral correctly reproduces 3-point amplitude with three off-shell gluons or equivalently color ordered correlation function of three Wilson line operators: Now let us consider two 4-point amplitudes with two off-shell gluons, which we will need when discussing the vacuums for amplitudes with two off-shell gluons in the context of the auxiliary gl(4|4) spin chain in the next section. The Grassmannian integral representation for A * 2,2+2 (1 * , 2 * , 3 + , 4 + ) is given by and solving delta functions constraints we get Performing required spinor substitutions the considered off-shell amplitude is given by . (3.64) in total agreement with the result of BCFW recursion [80]. The consideration of A * (1 * , 2 * , 3 − , 4 − ) amplitude is similar. The Grassmannian integral representation in this case is given by   and performing required spinor substitutions with c i ≡ p i ξ i and k 2 i = κ i κ * i , we get (dropping obvious momentum conservation delta function): This result is again in agreement with BCFW recursion [80].

momentum twistor representation
Using the results 11 of [103] and the discussion in previous section it is easy to see, that in the momentum twistor space the Grassmannian integral representation for amplitudes with two off-shell gluons takes the following form 12 : .
where the numeration of columns goes up to mod(m + 4)) and Reg. functions are given by the following expressions
(3.73) 11 We are referring the reader to [103] for the momentum twistor notation used here. 12 The generalization to the cases with more off-shell legs is straightforward, also note that Ω 2 m+4 = A * 2,m+2 .
In the case of k = 2 the matrix D is zero dimensional, all its consecutive minors equal to one and nonconsecutive to zero. So, the integral in (3.71) is zero dimensional, integrand equal to 1 and the result is given by A * 2,m+4 . For the k = 3 case we have where Reg.(m + 2) = 1 . (3.75) and the amplitude is given by

(3.76)
Here Γ tree is the [1, 2 contour for A 3,m+4 on-shell amplitude and for c ij coefficients we get Now let us proceed with the particular examples of 4-point amplitudes with two offshell gluons, which will be required in the next section when considering auxiliary spin chain. The A * 2,2+2 (1 + , 2 + , 3 * , 4 * ) amplitude is given 13 by leg relabeling in equation (3.64): 13 See the discussion of k = 2 case above.
In the case of A * 3,2+2 (Ω 1 , Ω 2 , g * 3 , g * 4 ) amplitude the results of the general k = 3 case considered above give:  . (3.84) We have checked, that expressed in terms of helicity spinors this expression reproduces the result of BCFW recursion [80]. For A * 4,2+2 (1 − , 2 − , g * 3 , g * 4 ) amplitude k = 4 and gauge fixed D matrix takes the form The delta functions constraints completely fix its entries in this case and we get (i = 1, 2): Then the Grassmannian integral for A * 4,2+2 (1 − , 2 − , g * 3 , g * 4 ) evaluates to We have checked that this expression is consistent with (3.70) up to spinor and momenta relabeling. An interesting question is the spurious pole cancellation. As an example, let us consider m = 3 case, which is an analog of n = 7 k = 3 on-shell amplitude. In the case of the on-shell amplitude we have a term . (3.89) The general statement that (3.76) is free of spurious poles is not (very) easy to formulate, but the discussed examples together with our results for amplitudes with one gluon off-shell make us believe in the self-consistency of the presented conjecture. Still, the question of the correct choice of integration contour Γ tree for the general m + 2n, k case is open. In all examples discussed above the choice Γ tree = Γ [1,2 n+2m gives correct results, but it will be really surprising that such a choice will indeed work for example for obtaining A * n,0+2n amplitude from Ω n 0+2n .

Off-shell amplitudes and auxiliary gl(4|4) spin chain
In [103] we have shown how (deformed) gauge invariant amplitudes with one leg off-shell could be described using quantum inverse scattering method (QISM) and auxiliary gl(4|4) spin chain. The purpose of this section is to show how this description extends to the case of amplitudes with multiple off-shell gluons. Originally auxiliary spin chain description of the on-shell tree-level amplitudes appeared as a result of investigations of their symmetry properties. First, the Yangian symmetry, combining invariance under superconformal and dual superconformal transformations [9] was proven for on-shell tree-level amplitudes in [10]. Next, it was claimed [38,39] that the Grassmannian integral representation for on-shell amplitudes (2.19) is the most general form of rational Yangian invariant. The study of tree-level scattering amplitudes within the context of QISM was started in [13,14], with the introduction of the notion of spectral parameter, which was later interpreted as a deformed particle helicity. Next, the authors of [15,16] proposed to study certain auxiliary spin chain monodromies. The introduced monodromies depended on an extra auxiliary spectral parameter, while the spectral parameters of [13,14] played the role of spin chain inhomogeneities. Yangian invariants and thus on-shell amplitudes are then found as the eigenstates of these monodromies. Further, [17,18] provided a systematic classification of Yangian invariants obtained within QISM. Later QISM description was extended to form factors 14 [77] and amplitudes with one leg off-shell [103]. It should be noted, that in the last two cases the Yangian invariance is explicitly broken by the corresponding vacuum states. Still, the machinery of QISM could be applied in those cases also. The auxiliary gl(4|4) spin chain in the case of on-shell amplitudes arises by writing Yangian invariance condition as a system of eigenvalue equations for the elements of a suitable monodromy matrix M(u) [15][16][17]: M ab (u)|Ψ = C ab |Ψ . (4.90) Here u is the auxiliary spectral parameter and C ab are monodromy eigenvalues. The monodromy eigenvectors |Ψ are the elements of the Hilbert space V = V 1 ⊗ . . . ⊗ V n with V i being a particular gl(4|4) non-compact representation built using a single family of Jordan-Schwinger harmonic superoscillators w A , w B , A, B = 1 . . . 8. The latter could be conveniently written in terms of Heisenberg pairs Here [·, ·} denotes graded commutator and | · | -grading. A vacuum state required in the construction of Yangian invariants |Ψ n,k corresponding to the on-shell N k−2 MHV n-point tree-level amplitudes A n,k is given by where δ + i ≡ δ 2 (λ i ) is the vacuum for the positive helicity state at position i and δ − i ≡ δ 2 (λ i )δ 4 (η i ) is the corresponding vacuum for negative helicity state. In the following we will also need a graphical notation for the above vacuum states introduced in [77]: The monodromy matrix of the auxiliary spin chain expressed in terms of Lax operators reads where v i are spin chain inhomogeneities and Lax operators L i (u, v) are given by Here, the matrix e ab acting in the auxiliary space is given by (e ab ) cd = δ ac δ bd and the action of Lax operators on vacuum states is given by The solution of the eigenvalue equation (4.90) provides us with the expressions for Yangian invariants labeled by the permutations σ with minimal 15 decomposition σ = (i 1 , j 1 ) . . . (i P , j P ) [15,17,18]: where [15] (see also [16]) Here Γ is the Euler gamma function and To describe amplitudes with one leg off-shell in [103] we required one additional ingredient -the vacuum state corresponding to minimal off-shell amplitude:  where (k is the off-shell gluon momentum and p is its direction) (4.101) Then, for example the deformed 16 off-shell amplitude A * 2,3+1 could be written as [103]: where λ i ,η i are defined in (4.101) andū Using definition of R operators (4.98) we get [103]: The off-shell amplitude A * 2,3+1 is recovered by setting deformation parameters to zero To see what should be done to apply QISM machinery to amplitudes with multiple off-shell gluons let us recall that we are actually have a systematic procedure to construct Figure 7: On-shell diagram construction via BCFW bridges for A * 2+3 amplitude a given on-shell diagram starting from its corresponding permutation [33]. The latter procedure is known as a BCFW bridge addition construction. First, the permutation is decomposed into a chain of consequent transpositions. Then each transposition (i, j) is interpreted as a BCFW bridge. Finally, the obtained BCFW bridges are applied to a corresponding empty vacuum diagram with the prescribed values of k and n 17 . The BCFW bridge addition operation is given by It is precisely the steps we are going through within QISM approach, the only difference is that BCFW bridges or R operators are deformed now. So, to get QISM description of a given on-shell diagram corresponding to some factorization channel of the off-shell amplitude under consideration we divide it into a vacuum state and a sequence of bridge additions or R operators acting on it. The examples of this division are shown in Figs. 4-7. The large black vertexes in the above figures correspond to minimal off-shell vertexes (see Fig. 1), while small black and white vertexes to usual 3-point MHV and MHV vertexes correspondingly. The easiest way to get explicit expressions of vacuum states, corresponding to the parts of on-shell diagrams above the dashed line (see Figs. 4-7), is to either use off-shell BCFW recursion [81,87] or our Grassmannian representation. This way in the case of amplitudes with two off-shell gluons vacuum states are given (restoring the dependence on Grassmann variables for on-shell states when needed) by the sum of A * 2,2+2 (1 + , 2 + , g * 3 , g * 4 ) (3.82), A * 3,2+2 (Ω 1 , Ω 2 , g * 3 , g * 4 ) (3.83) and A * 4,2+2 (1 − , 2 − , g * 3 , g * 4 ) (3.87) multiplied by the required number of vacuum states for extra on-shell states (4.93).
Finally, in out previous paper [103] we have shown, that gauge invariant amplitudes with one leg off-shell are no longer eigenvectors of monodromy matrix of the auxiliary gl(4|4) spin chain. The latter, however, turn out to be eigenvectors of corresponding transfer matrix. The last property was the consequence of multiplicative renormalizability of amplitudes with one leg off-shell. Thus, in the case of multiple off-shell gluons we are again expecting amplitudes to have the same properties, in particular they should be eigenvectors of transfer matrix.

Conclusion
In this paper we presented a conjecture for Grassmannian integral representation of N = 4 SYM tree level gauge invariant off-shell amplitudes containing arbitrary number of offshell gluons or equivalently Wilson line form factors with an arbitrary number of Wilson line operator insertions. The conjecture was successfully verified on multiple examples known in the literature [80]. We have also derived some new closed formulas for offshell amplitudes with arbitrary number of on-shell particles and fixed number of off-shell gluons. In addition we discuss the relation of our Grassmannian representation with the integrability approach to the amplitudes of N = 4 SYM.
It is remarkable that within our approach we can obtain Grassmannian integral representation for amplitudes without on-shell particles et all, i.e. for correlation functions of Wilson line operators (pure off-shell objects). This observation leads us to conjecture that in fact all gauge invariant observables in N = 4 SYM (scattering amplitudes, form factors and correlation functions of various gauge invariant operators, not necessary local) can be uniformly represented in one way or another in terms of integrals over Grassmannian or its subsets.
There are some open questions and possible further developments along the lines considered in this article. First, it would be interesting to consider supersymmetrized version of Wilson line operators. So far we have treated all on-shell states in manifestly supersymmetric way, while the off-shell states were restricted to gluons only. It is tempting to claim that the fully supersymmetric version of off-shell amplitude, where both on-shell states and Wilson line operators are treated in supersymmetric way, will be given just by Ω k m+2n [Γ] without any constraints on the Grassmann counterparts of 2n helicity spinor variables parameterizing n off-shell momenta k i . However, we think that more accurate consideration is necessary.
Next, it would be very interesting to fully uncover geometrical picture behind conjectured here Grassmannian representation for off-shell amplitudes as well as similar representations for form factors [71,72,77,79]. It is interesting to see if the "Amplituhedron" picture can be extended to all possible gauge invariant observables in N = 4 SYM. In addition the combinatorial nature of modifications to on-shell diagram formalism used here as well as in [77,79,103] remains mostly unexplored.
Despite the relation between integrable systems and Grassmannian representation of off-shell amplitudes discussed here and in [103] some important questions remain unan-swered. Namely, the present relation allows us to use auxiliary spin chain to add on-shell legs to the amplitudes and it would be interesting if there is a similar approach based on quantum inverse scattering method which would allow us to add additional off-shell legs.
In conclusion, we would like to note that it would be interesting to extended scattering equations and ambitwistor string approaches for the case of gauge invariant off-shell amplitudes considered here. An important topic is the calculation of loop corrections to gauge invariant off-shell amplitudes. Also, it is extremely interesting to see how the ideas presented here and in [103] work in other theories, for example in gravity and supergravity, where we have also a well developed approach based on high-energy effective lagrangian [121][122][123][124][125], see also [126][127][128][129][130] for similar research along this direction. 1 Ω is assumed): Ω = p 1 ξ 2 ξ 2 p 2 κ * 2 p 1 p 2 p 1 ξ 3 ξ 3 p 3 κ * 3 p 1 p 3 L 2 5 = p 1 ξ 2 ξ 2 p 2 κ * 2 p 1 p 2