Note on off-shell relations in nonlinear sigma model

In this note, we investigate relations between tree-level off-shell currents in nonlinear sigma model. Under Cayley parametrization, all odd-point currents vanish. We propose and prove a generalized $U(1)$ identity for even-point currents. The off-shell $U(1)$ identity given in [1] is a special case of the generalized identity studied in this note. The on-shell limit of this identity is equivalent with the on-shell KK relation. Thus this relation provides the full off-shell correspondence of tree-level KK relation in nonlinear sigma model.


Introduction
Color-kinematic duality (or BCJ duality) [2] was discovered by Bern, Carrasco and Johansson in 2008. This duality states that Yang-Mills amplitudes can be written in a so-called BCJ formula where kinematic factors share the same algebraic properties (including antisymmetry and Jacobi identity) with color factors. BCJ duality implies relations between color-ordered amplitudes at tree-level. Specifically, the antisymmetry implies Kleiss-Kuijf relation (KK relation) [3], while the Jacobi-identity implies Bern-Carrasco-Johansson (BCJ) relation [2]. With these relations, one can reduce the number of independent tree-level color-ordered amplitudes to (n−3)!. Both KK and BCJ relations have been proven in string theory [4,5] and field theory [6][7][8][9][10].
In [1], the authors proved the U (1) identity and the fundamental BCJ relation for three-level currents with one off-shell leg. The on-shell versions of these two relations were obtained by taking on-shell limit of the off-shell leg. Using the method for generating general on-shell KK and BCJ relations by the fundamental ones [43], one can obtain all the general on-shell KK and BCJ relations which have the same formulae with the corresponding relations in Yang-Mills theory.
Although all the on-shell versions of KK and BCJ relations for tree-level amplitudes in nonlinear sigma model have been proven in [1], only two special off-shell relations, namely U (1) identity and fundamental BCJ relation, have been studied. These off-shell relations do not share the same formulae with those in Yang-Mills theory [1]. Actually, in Yang-Mills theory, there have been suggested (all leg) off-shell KK relations [44] which have the same formulae with the corresponding on-shell relations. No BCJ relation for off-shell currents in Yang-Mills theory was found 1 .
A question is whether we can find the full off-shell extensions of the general on-shell KK and BCJ relations in nonlinear sigma model. There are several possible ways to think about this question. One way is to construct the BCJ formula in nonlinear sigma model and apply the algebraic properties to the kinematic factors. The main obstacles for this approach are the infinite number of vertices and the existence of off-shell leg. Another attempt is to generate all off-shell relations from the known off-shell U (1) identity and off-shell fundamental BCJ relation. However, the existence of the off-shell leg again becomes the main trouble.
In this note, we propose a generalized U (1) identity for even-point off-shell currents J(σ) in nonlinear sigma model. As already shown in the papers [45,46], under Cayley parametrization, the odd-point currents (with even numbers of on-shell legs and one off-shell leg) have to vanish [45,46]. The generalized U (1) identity for even-point currents (with odd numbers of on-shell legs and one off-shell leg) is given by A special case of the off-shell generalized U (1) identity (1.1) is r = 1 (or s = 1). In this case, R, S (or S, R) have to be 1, 2 respectively and we arrive at the U (1) identity proven in [1] (see (2.8)). When multiplying a p 2 1 → 0 to the right hand side of the relation (1.1), we just arrive at the corresponding on-shell relation for color-ordered amplitudes A 2 σ∈OP ({α 1 ,...,αr} {β 1 ,...,βs}) which has been shown to be equivalent with the on-shell KK relation [44].
To prove the off-shell identity (1.1), we first study the eight-point identity with r = 3, s = 4 by explicit calculations with Berends-Giele recursion. Because of the complexity, it seems impossible to extend the calculation directly to a general proof. Instead, we redefine the coefficients for products of subcurrents level by level. After this redefinition, all the divisions D with R D + S D < r + s have the right coefficients in (1.1). Then we only need to prove that the coefficient for (r, s) division has the right form. By combining the U (1) identity with a generalized U (1) identity with fewer α's, we obtain a set of equations which are finally used to determine the (r, s) coefficient.
The structure of this note is following. In section 2, we review the Feynman rules, Berends-Giele recursion and the U (1) identity proved in the paper [1]. In section 3, we study the generalized U (1) identity with three elements in {α} and four elements in {β} by Berends-Giele recursion directly. It will be quite hard to extend this calculation to a general proof. In section 4, we provide another approach by redefining the coefficients of divisions with R D + S D < r + s step by step. After these redefinitions, all divisions with R D + S D < r + s already have the right coefficients. We then prove that the coefficient for (r, s) division also has the right form. At last, we conclude this work in section 5.
2 Preparation: Feynman rules, Berends-Giele recursion and U(1) identity In this section, we review Feynman rules, the Berends-Giele recursion and the U (1) identity in nonlinear sigma model 3 . Most of the notations follow the recent papers [45,46].

Lagrangian
The Lagrangian of U (N ) non-linear sigma model is where F is a constant. Using Caylay parametrization as in [45,46], the U is defined by where φ = √ 2φ a t a and t a are generators of U (N ) Lie algebra.

Trace form of color decomposition
The full tree amplitudes can be given by trace form decomposition Since traces have cyclic symmetry, the color-ordered amplitudes A also satisfy cyclic symmetry Here, p j denotes the momentum of the leg j; momentum conservation has been considered.
Since there is at least one odd-point vertex for current with odd-point lines (including the off-shell line) and the odd-point vertices are zero, we always have J(2, . . . , 2m + 1) = 0, (2.7) for (2m + 1)-point amplitudes. The currents with even points in general are nonzero and are built up by only odd numbers of even-point sub-currents. Since odd-point currents have to vanish, in all following sections of this paper, we just need to discuss on the relations among even-point currents.

The off-shell versions of U (1) identity
In [1], the authors have proven the U (1) identity for off-shell currents in nonlinear sigma model. The identity is where on the left hand side, we summed over the permutations in {α 1 } {β 1 , . . . , β 2m } with keeping the relative order in the β set. On the right hand side, we summed over the divisions of {β} into two ordered subsets.
3 Direct calculation of an eight-point example (3.1) To prove this identity, we use Berends-Giele recursion (2.6) to express all the currents on the left hand side of (3.1) by six-and four-point subcurrents. Then we collect the terms with a same vertex connected to the off-shell leg 1. After summing all the possible diagrams in each collection, the left hand side of (3.1) is expressed by • diagrams with neither six-point nor four-point substructure (see Fig. 1(B)).
Remembering that the identity (1.1) is satisfied by four-and six-point currents, we apply these lower-point identities to the four-and six-point substructures in the first class of diagram. Then diagrams in the first class are rewritten in terms of products of subcurrents containing only α or β elements. Since the second class of diagram does not have any substructure, it is already expressed by products of subcurrents containing only α or β elements. After this reduction, for a given product of subcurrents (or in other words, given division of {α} set and {β} set), we collect the coefficients together. Thus the left hand side of (3.1) is written as Figure 2. We redefine the coefficients for R + S < r + s divisions such that they are the right ones as in the general form (1.1). Then we solve the R = r, S = s coefficient.
where s ij ≡ (p i + p j ) 2 . Coefficients for each division can be classified into three types (see 1). A type-2 coefficient always cancels with a propagator of a subcurrent and divides the subcurrent into new subcurrents. For example, the coefficient in type-2 term on the second line is −( . Thus this part of contribution cancels with the second term of the type-1 coefficient of (3, 4) division. Similarly, other type-2 terms also cancel with type-1 terms for divisions with larger R D + S D . All the type-1 and type-2 terms cancel out in this way. Only the type-3 terms are left and give the right hand side of the eight-point identity (3.1).

Proof of the generalized U(1) identity for off-shell currents
In the previous section, we have provided a direct approach to an eight-point example by Berends-Giele recursion. Although the coefficients in the example were shown to have a good pattern (see table 1), it will be quite hard to extend the calculation to a general proof. One reason is that we will encounter many different lower-point substructures of the identity (1.1) when the number of {α} elements grows. Thus we have to prove the general formula (1.1) in a different way. In this section, we will show a general proof of the identity (1.1). The main idea is following: • As we have done in the eight-point example, we write the left hand side of the identity (1.1) by Berends-Giele recursion and collect the diagrams with a same vertex attached to the off-shell leg 1 (See Fig. 1(A) and (B)). Reducing the substructures by lower-point identities and putting the coefficients corresponding to each product of subcurrents together, we express the left hand side of (1.1) as follows • We show that the expression obtained in the above step can be rearranged (figure 2) into the following formula where V (r,s) is the dimensionless coefficient for the (r, s) division. The first term of (4.2) is given by sum of divisions D (R D + S D < r + s) which already have the correct coefficients in (1.1). Thus we only need to prove that the coefficient V (r,s) in the second term of (4.2) also has the right expression in (1.1), i.e., V (r,s) = δ(|r − s| − 1). • The undetermined coefficient V (r,s) has the general form 1 combining an U (1) identity and a generalized U (1) identity with fewer α's, we can prove that V (r,s) = δ(|r − s| − 1). Therefore, the generalized U (1) identity (1.1) for off-shell currents is proved.
In the remainder of this section, we will show the left hand side of (1.1) can be rearranged into (4.2) and then solve V (r,s) .

Proof of the validity of (4.2) with an undetermined coefficient V (r,s)
Now we show that the left hand side of the off-shell generalized U (1) identity (1.1) can be rearranged into the form (4.2). We start from several examples. Four-point example: The four-point example is the four-point U (1) identity (see [1]). By explicit calculation, this is given by the sum of three diagrams in Fig. 3, i.e., The identity with two α's and one β can be obtained by exchanging the roles of α and β. Before giving the next example, let us have a look at an off-shell extension of the right hand side of (4.3), i.e., we replace the three on-shell legs α 1 , β 1 and β 2 in Fig. 3 by three off-shell currents J(A 1 ), J(B 1 ) and J(B 2 ) correspondingly. From Feynman rules (2.5), the coefficient of J(A 1 )J(B 1 )J(B 2 ) is written as a linear combination of 1  with Here, J (4) (σ) denote the diagrams with the four-point vertices connected to 1, J(A 1 ), J(B 1 ) and J(B 2 ) and a (1,2) 2 = 1. Thus the off-shell extension (4.5) can be expressed by Fig. 4. We will encounter (4.4), (4.5) in higher-point cases.
Six-point example: With the four-point identity in hand, let us consider the six-point example. The first six-point example is the U (1)-identity with only one α, which has been understood. Now we consider the generalized identity with two α's Step-1 To prove this identity, we start from the left hand side. We use Berends-Giele recursion to express the currents on the left hand side. Then collect the diagrams together with same substructures of generalized U (1)-identity. After reducing diagrams containing four-point substructures of U (1)-identity by   and (p i · p j ) respectively come from the off-shell propagator and vertices (as shown in (4.1)). The second term in (4.7) is the (2, 1) division which can only get contribution from diagrams with the off-shell leg 1 directly connected to four-point vertices whose other three lines are connected to J(α 1 ), J(α 2 ) and J(β 1 , β 2 , β 3 ). The sum of such contributions is noting but the off-shell extension (4.4) in the four-point example with where the on-shell conditions of α 1 and α 2 have been used.
Let us consider the off-shell extension of the first term on the right hand side of (4.9), assuming that we have already proved V (2,3) = c ij p i · p j , the off-shell extension of V (2,3) must have the form (see Fig. 6) Following a parallel discussion, we can do the same on the six-point relation with only one α and extend the coefficient of (1, 4) division to off-shell case (see Fig. 7) (4.12) Eight-point example: We now consider an eight-point example with three α's and five β's. The formula of this example is given by (3.1) in section 3.
Step-1 To prove the eight-point example, we first express the left hand side of (3.1) by Berends-Giele recursion and then collect the contributions to a substructure of generalized U (1)-identity together. After applying generalized U (1)-identity, we get Again, we start from the R + S = 3 divisions, there are two cases corresponding to the last two terms of the above equation. These cases only get contributions from diagrams with the off-shell leg 1 connected to a four-point vertex. As shown in the six-point example, the coefficients U 5 and U 6 can be given by the off-shell extension W (1,2) (Fig. 4), particularly and U (3,4) 6 Step-2 The term (p α 1 + p α 2 + p α 3 ) 2 in U reduces J(α 1 , α 2 , α 3 ) to J(α 1 )J(α 2 )J(α 3 ) with a factor 1 2F 2 s α 1 α 3 , the term ( reduces J(β 2 , β 3 , β 4 ) to J(β 2 )J(β 3 )J(β 4 ) with a factor 1 2F 2 s β 2 β 4 . As in the four-point example, we can redefine the coefficients so that (4.13) becomes get contributions from -a) the diagrams with the off-shell leg 1 connected to six-point vertices whose other legs are attached to subcurrents containing only α or β elements (as shown in Fig. 1 (B)) -b) the diagrams with the off-shell leg 1 connected to four-point vertices, which contain substructures of generalized U (1) identity (as shown in Fig. 1 (A) ).
Both cases has correspondence in the U (2,3) 1 of (4.10) (with exchanging the roles of α's and β's) and they have the same pattern with (4.10) when the off-shell subcurrents goes on-shell.
General Discussion: In general, when we consider the generalized U (1)-identity (1.1) with r α's and s β's, we can use Berends-Giele recursion to rewrite the left hand side and collect terms corresponding to a same substructure as shown in Fig. 1. Applying the lower-point identity to the substructures and summing the coefficients for any given division, we reexpress the left hand side of (1.1) by (4. We start from the divisions with R D + S D = 3, i.e., (1, 2) division and (2, 1) division. The contributing diagrams are those in the four-point example with replacing the on-shell lines by off-shell currents. Thus it has the form of the off-shell extension (4.5). Since p 2 A i J A i and p 2 B i J B i in (4.5) will further reproduce divisions of A i and B i with coefficients 1 p 2 1 c ij p i ·p j , we absorb all these contributions into the corresponding divisions with R D + S D > 3. The only left contribution for divisions with R D + S D = 3 is the first term of (4.5) which gives rise to the expected coefficients.
Then we consider divisions with R D + S D = 5, which get both contributions from its corresponding U (r,s) D in (4.21) as well as p 2 A i J A i and p 2 B i J B i in the off-shell extension (4.5) of four-point case. Since the coefficients for divisions with R D + S D = 5 are defined in the same way with the V (r ′ ,s ′ ) (r ′ + s ′ = 5) in the six-point example, they are just the off-shell extensions (4.11), and (4.12). Again, the terms containing In the remaining part of this section, we will solve the coefficients c's to show that V (r,s) has the expected form.

Solving V (r,s)
In the above discussion, we have shown that the left hand side of the generalized U (1)-identity (1.1) could be rearranged into the form (4.2). All the coefficients of divisions in (4.2) with R D + S D < r + s are those on the right hand side of the identity (1.1). Only the coefficient V (r,s) for (r, s)-division are undetermined. Now let us prove that V (r,s) has the right form, i.e., In this case, (1.1) becomes the U (1)-identity for 2m = s + 2-point currents, which has been studied in [1]. The V (1,2) for four-point relation with one α and two β's is Hence V (1,2m−2) satisfies the form (4.23).

r > 1
To solve V (r,s) for r > 1, we consider the following combination of currents (4.27) Here we summed over divisions {ρ} → {ρ L }{ρ R } on the right hand side. Substituting above expression into (4.26) and rearranging the summations, we reexpress the combination I(α 1 | α 2 , . . . , α r ; β 1 , . . . , β s ) by The delta functions impose constraints on r = r L + r R − 1 and s = s L + s R . The only nonzero contributions are the cases with r, s satisfying r = s − 1, r = s + 1, r = s + 3.
ii) For r = s + 1, both terms with r L = s L − 1, r R = s R + 1 and terms with r L = s L + 1, r R = s R − 1 contribute. Then iii) For r = s + 3, the nonvanishing terms are those with r L = s L + 1, r R = s R + 1. Thus we get ii) the second part B (r,s) is the sum of terms with (r − 2, s) divisions containing a nontrivial subcurrent J(φ ∈ OP ({α 1 } {α i , α i+1 })).

Conclusions
In this paper, we proposed and proved the generalized U (1)-identity for tree-level off-shell currents in nonlinear sigma model. When we take on-shell limit, this relation becomes the on-shell generalized U (1) identity which is equivalent with KK relation. The U (1)-identity for off-shell currents proposed in [1] is a special case of the generalized U (1)-identity. There are several possible further extensions of this work, including the generalized off-shell BCJ relation, the loop-level extensions and the BCJ duality which implies the relations in nonlinear sigma model.