Institute for Mathematical Physics Some Properties of the Alday–maldacena Minimum Some Properties of the Alday-maldacena Minimum

The Alday-Maldacena solution, relevant to the n = 4 gluon amplitude in N = 4 SYM at strong coupling, was recently identified as a minimum of the regularized action in the moduli space of solutions of the AdS5 σ-model equations of motion. Analogous solutions of the Nambu-Goto equations for the n = 4 case are presented and shown to form (modulo the reparametrization group) an equally large but different moduli space, with the Alday-Maldacena solution at the intersection of the σ-model and Nambu-Goto moduli spaces. We comment upon the possible form of the regularized action for n = 5. A function of moduli parameters za is written, whose minimum reproduces the BDDK one-loop five-gluon amplitude. This function may thus be considered as some kind of Legendre transform of the BDDK formula and has its own value independently of the Alday-Maldacena approach.


Introduction and conclusions
An ǫ-regularized minimal action in AdS 5 σ-model was defined recently [1], and shown to reproduce the external momentum dependence of the BDS formula [2] for the n = 4-gluon amplitude in N = 4 Super Yang-Mills (SYM) theory. In [3]- [20] one may find generalizations and discussions of this important result. In a previous paper [12] it was demonstrated that the Alday-Maldacena solution is just one member of a large family of solutions; a rather distinct one, though, since it corresponds to a minimum of the classical σ-model action in the moduli space M σ n of all solutions in d = 4 dimensions (i.e. for ǫ = 0). Throughout this paper we shall use the notation and results of [12], to which we refer the reader. We shall keep the parameter n explicit in various formulas and symbols, even though, as it will be clear in the text, many of the statements will refer specifically to the cases n = 4 or 5.
Let us recall that in d = 4 dimensions and for n = 4 the moduli space of solutions constructed in [12] was parametrized by {z a , v 1 , φ} with a = 1, . . . , n enumerating the sides of the auxiliary polygon Π, Figure 1, formed by the null 4-momenta p a of the external gluons and lying at the boundary of AdS 5 at z = ∞.
It is possible, that these are all the solutions with the particular boundary conditions corresponding to the above process. In [12] they were obtained under the assumption (ansatz) that the lagrangian L σ = constant = 2. In any case, in what follows we shall use M σ n to denote this part of the moduli space. The SO(4, 2) symmetry of AdS 5 relates some of these solutions, but it does not act transitively on M σ n . Specifically, only the z a moduli are affected by this group. In addition, v 1 is an inessential modulus, since no a E-mail: mironov@itep.ru; mironov@lpi.ru b E-mail: morozov@itep.ru c E-mail: tomaras@physics.uoc.gr physical quantity depends on it. Essential moduli are the ratio z 1 z 3 /z 2 z 4 and the angle φ. The latter is not affected by SO(4, 2), but only by some larger hidden group, related presumably to the integrability of the σmodel. It is important to point out that by definition the Lagrangian density is constant, namely L σ = 2, on the entire M σ n . Thus, the corresponding action integral diverges and needs regularization. The ǫ-regularization used in [1] breaks not only the integrability, but also the SO(4, 2) symmetry, so that the regularized action becomes a non-trivial (z and φ-dependent) function on the moduli space. As shown in [12], the Alday-Maldacena solution is exactly at the minimum of this function. Incidentally, the regularization leaves unbroken the Lorentz subgroup of SO(4, 2) (which, however, is partly broken by the boundary conditions) and the two rescalings of z a that preserve the products z 1 z 3 and z 2 z 4 .
The present note is a little further development along the lines of [12]. Our purpose is on the one hand to clarify the difference between the σ-model and Nambu-Goto actions in connection with the above approach, and on the other to attempt a generalization to the five-gluon amplitude.
Specifically, in Section 2, we consider what happens if the σ-model action is replaced by the Nambu-Goto (NG) one -a question raised but left unanswered in Section 4.7 of [12]. We conclude that for n = 4 the two moduli spaces are equally large, i.e. dim M σ not expected a priori, because of the classical inequivalence of the two actions. We would like to recall here, that the σ-model is being considered without the Virasoro constraints, which would render the two models classically equivalent. As shown, the relevant solutions are parametrized by the same parameters, but they are different in the two models and the corresponding moduli spaces do not coincide. The essential moduli in the σ-model case are the ratio z 1 z 3 /z 2 z 4 and φ, while in the NG case no essential moduli are made from z a . Instead, the angle φ between the two k-vectors gets complemented by the ratio of their lengths. This simple description, however, requires careful definition of the manifold M N G 4 . The reason is that, in contrast to the σ-model case, the NG action is invariant under arbitrary reparametrizations of the world sheet and, therefore, the entire space of solutions is infinite dimensional, incomparably larger than that of the σ-model. In such a situation, it is natural to define the moduli space by factoring out the reparametrization group with coordinate transformations vanishing at infinity. Then the moduli space of solutions with a given asymptotic behaviour at infinity is finitedimensional and is actually obtained by linear transformations of the world-sheet coordinates. Similarly, it is natural to eliminate the 2d rotations and displacements, since the 2d Poincare invariance is common to the σ-model and NG actions. Next, the ǫ-regularization preserves the 2d reparametrization invariance of the NG action, therefore, again in contrast to the σ-model case, the regularized NG action is constant on the entire M N G In Section 3, guided by the pictorial representation given in [12] and the results for n = 4, we make an attempt to guess the form of the n > 4 regularized action A n (z 1 , . . . , z n ; ǫ) on the moduli space M σ n , parametrized by a conjectured set of parameters z a with a = 1, 2, ..., n. In addition, we present an ansatz for the constraint, generalization of its n = 4 counterpart, which is argued to be reasonable for n = 5. The action is minimized under the constraint and reproduces the BDDK formula [22] for the one-loop 5-gluon amplitude F (1) 5 , which eventually exponentiates to the BDS formula [2] for the full strong coupling n = 5 scattering amplitude. Hopefully, this action will eventually be derived, as in the n = 4 case, from exact solutions of the σ-model with subtle growing asymptotics. At this point however, it may just serve as a useful guide through the tedious and not particularly transparent calculations of the regularized integrals in [1] and [12].
It is important to emphasize here, that the finite partÃ n of the n-point action integral will be defined independently of the Alday-Maldacena regularization. Consequently, it may be thought of as a kind of Legendre transform of the BDDK formula and in this sense has its own value and significance. Such a function for n > 5 would have an advantage, because the {z a } are independent variables, while there are many relations between the n(n − 3)/2 parameters t ab , of which only 3n − 10 are independent. Construction ofÃ n for n > 5 is a challenging problem beyond the scope of the present paper.
2 Moduli space of NG solutions for n = 4 In this section and in order to make the comparison easier, we shall make a parallel presentation of the solutions of interest in the Nambu-Goto (NG) and the σ-model field equations.

Solving the NG equations of motion for n = 4
As explained in [12] the most relevant variables for the description of the Alday-Maldacena result are (z, v), which are actually five of the six flat coordinates (Y − , Y, Y + ), describing the embedding of AdS 5 into R 6 ++++−− . In these variables the equations of motion acquire the simple form and the difference between the σ-model and NG cases lies in the expression for H, namely, we have while In the above formulas i, j = 1, 2 label 2d coordinates on the world sheet, is the AdS-induced metric on the world sheet, is made from algebraic complements, and the two lagrangian densities are and respectively. In [12] it was suggested to make the anzatz G ij = constant in the differential equations (2) and (3), solve them with appropriate boundary conditions and finally consider the self-consistency of this anzatz as an algebraic equation (6) where u are the world sheet coordinates. The 2d vectors k a are constrained in different ways for different actions: G ij k a i k a j = 2 det G in the NG case (12) Correspondingly, the parameters v a are fixed by the boundary conditions [12], which express them in terms of the external momenta p a and {z a }. These boundary conditions restrict the number of exponentials in (10) to the number n of sides in the polygon Π: a = 1, . . . , n. One of the v-vectors, say v 1 , remains undefined; we called it inessential modulus in the introduction. The essential moduli are {z a } modulo u transformations and { k a } modulo (11) or (12).
One is left with a set of non-trivial algebraic equations, namely that G ij obtained by substitution of (10) into (6) is constant, i.e. independent of u: n a,b=1 Here E a = e ka u , E a+b = E a E b , k ab = k a − k b , P ab = z a z b (p a + . . . + p b−1 ), while further details about notation can be found in [12]. In what follows we concentrate on the case of n = 4, where this simple ansatz indeed works. Equation (14) is actually a system of relations for coefficients in front of various exponentials. Many coefficients can be cancelled if we choose k 3 = − k 1 = k −1 and k 4 = − k 2 = k −2 so that the four k-vectors form diagonals of a parallelogram. Next, comparison of the coefficients in front of z 2 1 E 1+1 on both sides of (14) gives: where η 2 = z 2 z 4 t 24 . Similarly, from the coefficient of z 2 2 E 2+2 one obtains with ξ 2 = z 1 z 3 t 13 . Together these two equations imply the consistency relation on the parameters z a ξ 2 + η 2 = z 1 z 3 t 13 + z 2 z 4 t 24 = 1 (17) already familiar from [12], and the explicit expression for G ij , All other relations, following from (14), are then automatically satisfied. For example, the coefficient of z 1 z 2 E 1+2 on the right hand side of (14) receives contributions from a + b + c + d = 1 + 1 + 2 + 3 and 1 + 2 + 2 + 4, and using the above relations one has the last expression being equal to the coefficient of the same term on the left hand side. Similarly for the coefficients of E 0 = 1.
In the σ-model case (11) leads to As soon as the two vectors k 1 and k 2 have equal lengths, the corresponding parallelogram has to be a rectangle. The remaining essential modulus is the angle φ between the two vectors, their common length being an inessential modulus (scaling of the lagrangian). Another essential modulus is ξ 2 or η 2 = 1 − ξ 2 . Rescalings of parameters z a , which leave ξ 2 and η 2 intact, are induced by constant shifts of the coordinate vectors u. Analogously, in the NG case, one obtains from (12) If we parametrize the two NG k-vectors through k 1 = (α, β) and k 2 = (γ, δ), then det G = (αδ − βγ) 2 ξ 2 η 2 , and (21) is equivalent to the system of equationš It follows that ξ 2 N G = η 2 N G = 1 2 (24) with no restriction on the vectors k 1 and k 2 . The corresponding parallelogram in this case can be arbitrary and the two essential moduli are the angle φ between k 1 and k 2 and the ratio of their lengths, | k 1 |/| k 2 | = (α 2 + β 2 )/(γ 2 + δ 2 ). It is clear that arbitrary vectors k 1 and k 2 give rise to NG solutions, because they can be made arbitrary by linear transformations of u i , which are part of the 2d reparametrization invariance of the NG equations i.e. part of the symmetry group of M N G 4 .

On the relation between NG and σ-model solutions
In the previous section we determined the moduli spaces of both the σ-model and the NG equations for n = 4 in the framework of the ansatz (10). Actually, the σ-model case corresponds to the trace of equation (14) with respect to the indices i, j [12], while in the NG case one should contract withǦ ij instead of δ ij . Our result is that the moduli spaces, while both two-dimensional, are essentially different. Moreover, there is no one-to-one correspondence between the solutions. Does this contradict the widespread belief that the NG and the σ-model are equivalent? It does not, as we shall argue next, because of the Virasoro constraints. Notice that the σ-model dealt with in [1] and [12] does not take into account the Virasoro constraints, which are crucial in the proof of the above equivalence. So, the two models are actually different and, not surprisingly, lead to different answers.
More specifically, recall that the idea behind the equivalence of the NG and σ-model formalisms is based on the consideration of the more general Polyakov action [23]: where G ab is the target space metric, made from dynamical fields like in (6), X a ≡ (r, v) and g ij is the auxiliary field of 2d-metric. The equations of motion for the dynamical fields then read while variation with respect to the 2d-metric gives Inserting (27) into (26), one reproduces the NG equations, while, taking advantage of the local symmetries of the Polyakov action to choose g ij = δ ij , one obtains the σ-model equations. This choice is a gauge-fixing and can always be achieved by a proper transformation of the world sheet variables u i 2 . Based on this, one may argue that any solution of the NG equations can be converted into a solution of the σ-model: once a G N G ij is found, it can always be diagonalized by a coordinate transformation. In our context, with G ij constant, this transformation u N G → u σ is linear, and is given simply by or equivalently, using the explicit form of k N G a and k σ a , It is always possible to find such a transformation with non-unit Jacobian, in order to convert the two NG k-vectors with different lengths into two σ-model k-vectors with equal lengths. Clearly, the above N G → σ mapping has a non-trivial kernel. It has enough parameters to map different NG solutions into the same σ-model solution; it is not an isomorphism of the two moduli spaces. The converse, however, is not true: one cannot convert an arbitrary σ-model solution into a NG one. For this, one would have in addition to satisfy the gauge condition G ij ∼ δ ij . For instance, linear transformations of coordinates u cannot change parameters ξ 2 and η 2 . The parameters z a of a particular solution (10) are rescaled by shifts of u, u → u + a, but z 1 and z 3 = z −1 or z 2 and z 4 = z −2 are rescaled in opposite directions (since k −a = − k a ), so that ξ 2 and η 2 = 1 − ξ 2 remain intact. This implies that it is not possible to use the gauge freedom of the Polyakov equations to convert σ-model solutions with generic ξ 2 = 1/2 into NG solutions, which all have ξ 2 = η 2 = 1/2. Generic coordinate-u reparametrizations (linear or otherwise) change the two tensors g ij and G ij simultaneously, and the desired transformation g σ ij , G σ ij ? −→ g N G ij , G N G ij is generically in contradiction with the other two properties, namely and (27) These relations are all compatible if and only if G σ ij ∼ δ ij , which is not true for a generic σ-model solution, but only for those with ξ 2 σ = η 2 σ = 1/2. This, as stated in the begining of this section, is a concrete manifestation of the well known fact [24] that Polyakov's σ-model, which is classically equivalent to the NG theory, reproduces the ordinary σ-model, but together with the Virasoro constraints.
A consequence of the above discussion is that the regularized σ-model and NG actions of even a common solution do not coincide. Naively, since substitution of the 2d-metric (27) into the Polyakov action (25) reproduces the NG action, one would expect that the σ-model and NG actions coincide, provided the Virasoro constraint is satisfied. This is true, but ambiguous, since both actions are infinite. The regularization proposed in [1] does not change only the target space metric G ab in both actions, which would leave them equal. Instead, it spoils the Virasoro constraint and should lead a priori to different actions! Indeed, it was explicitly checked [20] that, even in the n = 4 case, the two actions are different. However, they differ by an inessential additive constant. It would be instructive to examine their difference for higher n.
3 Guess of the action integral for n = 5 As explained in [12], it is not straightforward to generalize to n > 4 our solutions with exponential behavior at infinity. So, it is not obvious how to extend our approach to these cases and have so far been unable to find relevant solutions. Nevertheless, one can still try to guess the form of the regularized action integral A n (z 1 , ..., z n ; ǫ) for n ≥ 5, whose minimum will lead to the BDDK formula for the one-loop amplitude F (1) n of n-gluon scattering. For that, let us assume that we have an n-parameter set of solutions of the σ-model with the appropriate asymptotics, parametrized by z a , with a = 1, 2, ..., n. In addition, we must assume a regularization scheme [1] with parameter ǫ, as well as a constraint analogous to (17).
We split the action integral into the infinite A (n) ǫ and finiteÃ n pieces and guided by the pictorial representation of the BDDK formula [12] and by our n = 4 results, we write (up to an additive inessential constant 1/ǫ 2 term) with a + n ≡ a for all values of a. We neglected any additional angular variables like φ of the n = 4 case, assuming that such parameters enter in an especially simple way, like it happened in the case of n = 4 in [12], where | sin φ| −1 was a common factor in front of the entire A 4 . Notice that for n = 4 (32) reproduces the expression derived in [12].
To guess a reasonable generalization of the constraint is more difficult. For general n one has to worry about the presence of terms with higher powers of t ab in the expression for the constraint. For instance, the first non-trivial such term would be a<b<c<d z a z b z c z d t ab t cd . However, for n = 4, 5 such a term, as well as all analogous with higher powers of t vanish identically. In what follows, we shall consider the constraint n a<b z a z b t ab = 1 (33) but, one should remember, that this particular form may be oversimplified and irrelevant for n ≥ 6. Our goal is to minimize (32) under the constraint (33). Let us start with the simpler problem of minimizing A (n) ǫ under the above constraint, which is introduced with a lagrange multiplier λ. The position z For n = 4, the solution is, up to the invariance z 4 , given by [12] z (0) a = 1 2t a,a+2 Similarly, for n = 5 we obtain instead  38) and (41). The polygon is nothing but Π from Figure 1, whose edges are associated with external momenta. τ -parameters τ ab = log t ab are associated with diagonals and za with the vertices of the polygon. The marked diagonals correspond to differences of τ -parameters in the equations.