Crossing Symmetric Dispersion Relations in QFTs

For 2-2 scattering in quantum field theories, the usual fixed t dispersion relation exhibits only twochannel symmetry. This paper considers a crossing symmetric dispersion relation, reviving certain old ideas in the 1970s. Rather than the fixed t dispersion relation, this needs a dispersion relation in a different variable z, which is related to the Mandelstam invariants s, t, u via a parametric cubic relation making the crossing symmetry in the complex z plane a geometric rotation. The resulting dispersion is manifestly three-channel crossing symmetric. We give simple derivations of certain known positivity conditions for effective field theories, including the null constraints, which lead to two sided bounds and derive a general set of new non-perturbative inequalities. We show how these inequalities enable us to locate the first massive string state from a low energy expansion of the four dilaton amplitude in type II string theory. We also show how a generalized (numerical) Froissart bound, valid for all energies, is obtained from this approach.


Introduction
Dispersion relations provide non-perturbative representations for scattering amplitudes in quantum field theories [1,2]. The usual way to write dispersion relations in the context of 2-2 scattering of identical particles is to keep one of the Mandelstam invariants, usually t, fixed and write a complex integral in the variable s. This approach naturally leads to an s − u symmetric representation of the amplitude. Then, one imposes crossing symmetry as an additional condition. A similar approach can also be developed for Mellin amplitudes for conformal field theories. Recent developments in this direction include [3], [4], [5].
The amplitude's resulting representation not having manifest three-channel crossing symmetry may appear to be a drawback. For instance, in perturbative quantum field theories, when we compute Feynman diagrams, the amplitude's resulting expansion has manifest crossing symmetry. In the worldsheet formulation of string theory, the tree level sphere diagram, for instance, is manifestly crossing symmetric without having the explicit three-channel split. Hence, it would be somewhat disappointing if a non-perturbative approach failed to have this elegance inbuilt into its starting point.
The question naturally arises: Is there a manifestly crossing symmetric version of these dispersion relations? In the 1970s, this question was briefly considered in a few papers, for example, in 1972 by Auberson and Khuri in [6] and in 1974 by Mahoux, Roy, and Wanders in [7]. Unfortunately, due to the technical complications involved, barring for a smattering of a few papers (e.g., [8]), this approach has not been well explored in the literature. We will follow [6] and revive this line of questioning again.
In the CFT context, Polyakov's work in [9] proposed a fully crossing symmetric bootstrap, which was developed in [10,11]. However, this approach currently lacks a non-perturbative derivation for d ≥ 2.
Our methods in this paper will enable us to address this important question in the near future [12].
Dispersion relations give a window to understanding how analyticity and unitarity assumptions for the high energy behaviour of amplitudes constrain low energy physics contained in effective field theories [13]. Our manifestly crossing symmetric approach not only leads to a simpler and unifying derivation of recently considered positivity constraints in effective field theories [14,15,16,17], but also enables us to write down a completely general set of positivity constraints on the Wilson coefficients. In particular, we will provide straightforward derivations of many of the upper bounds on the ratios of Wilson coefficients, as well as the null constraints listed in [16,17], leading to the lower bounds. Our formalism will enable us to write down general formulae for the upper bounds and the independent null constraints.
We will consider two novel applications of our constraints. First, we will use them to find the location of the first massive string pole from the low energy expansion for the tree-level four-dilaton scattering in type II string theory (eg. [18]). Second, our approach enables us to derive a numerical upper bound on the total scattering cross-section of identical particles valid at all energies. This is a generalization of the famous Froissart bound [19]. We will conclude with some future directions.

Crossing symmetric dispersion relation
We begin by considering cubic hypersurfaces [6] in the variables where s, t, u are usual Mandelstam variables. Explicitly, these hypersurfaces will be given by (s 1 (z) − a) (s 2 (z) − a) (s 3 (z) − a) = −a 3 , with a being a real parameter 1 . The s i 's can be parametrized via where z k are cube roots of unity and we will restrict − µ 3 ≤ a < 2µ 3 [6]. Importantly, a = s 1 s 2 s 3 s 1 s 2 +s 2 s 3 +s 3 s 1 . The amplitude M(s 1 , s 2 ) can be written as an analytic function of (z, a) i.e., M(z, a) = M (s 1 (z), s 2 (z)).
M (s 1 (z), s 2 (z)) has physical cuts for s k ≥ 2µ 3 , k = 1, 2, 3. The image of the three physical cuts in the z-plane is defined via V (a) = V 1 (a) ∪ V 2 (a) ∪ V 3 (a). For our region of interest − 2µ 9 < a < 0, it is  Figure 1: The image of the three physical cuts V (a) in the z-plane for − 2µ 9 < a < 0, µ = 4.
where A (z , a) is the discontinuity of the amplitude across V (a), defined in eq (A.4).
For the completely crossing symmetric case (eg. π 0 π 0 → π 0 π 0 ) the above dispersion relation simplifies dramatically 2 , in terms of the s 1 , s 2 , s 3 variables: where A (s 1 ; s 2 ) is the s-channel discontinuity and

An inversion formula
For the situation where we have no massless poles, or where we subtract them out, the crossing symmetric amplitude has an expansion which can be seen by writing s k 's in terms of (z, a). Now identifying z 3 (z 3 −1) 2 = −x 27a 2 , we can series expand in powers of x. We obtain We can now extract the coefficient of a m . Notice that a = y/x, which means that the coefficient of a m will be W n−m,m in eq.(2.5). The s-channel discontinuity has a partial wave expansion over even spins , we find the coefficient of a m leading to the inversion formula (2.8) Note that W 0,0 = α 0 , which is the subtraction constant.
Eq.(2.8) allows two lines of investigation. (1). For n ≥ m with n ≥ 1, we get the coefficients in terms of a (s 1 ). Since partial wave unitarity implies 0 ≤ a (s 1 ) ≤ 1, we can find positivity constraints on W n−m,m . (2). For n < m with n ≥ 1, the coefficients should vanish (as needed by eq.(2.5)), which give rise to non-trivial constraints on a (s 1 ). These sum rules or "null constraints" are instrumental in getting the lower bounds on W n−m,m in EFTs. We will use these sum rules to put a bound on the total cross section at any s 1 , generalizing the famous Froissart bound.

Positivity constraints
Now from eq (2.8), we can derive inequalities involving W p,q . Note that p (j) 's involve derivatives of . Specifically we find the useful expression for r ≥ 1 We use the unitarity constraints 0 ≤ a (s 1 ) ≤ 1 as well as properties of C Since the range of s 1 in eq.(2.8) starts at 2µ/3, we have introduced δ as a convenient variable.
Non-perturbative constraints on W p,q By using eq.(3.1), we find that is positive for n ≥ m. Note that in the sum, (−1) k+m spoils the definite sign of B ( ) n,m (s 1 ). We can search for χ A solution is easily found using the following recursion relation: . (3.7) Explicitly, for the first few cases, we have which we will refer to as non-perturbative constraints to differentiate from the effective field theory constraints to be derived next.

Constraints on W p,q in EFTs
In order to derive EFT bounds, we start with eq.(2.8) and write (3.13) The n = 2 case of eq. . The derivations of [14], [16], [17] are based on fixed-t dispersion relation, while our derivation is manifestly crossing symmetric and directly involve W n−m,m .
A straightforward application of our general formulae is the examination of the n m limit. We for n m. We have checked that tree level type II string theory, to be discussed below, respects this.

Null constraints
To derive lower bounds, we make use of the n < m vanishing conditions arising from eq.(2.8). In the large δ limit, we have 4 for m > n, n ≥ 1. For example m = 2, n = 1 gives D (1,2) = 2 ( +2α)(−11−10α+2 ( +2α)) (2α+1)(2α+3) , which was first derived in [16] from fixed-t dispersion relations 6 . Once these null constraints are in place, a judicious use of Cauchy-Schwarz inequality as used in [16], or a more constraining numerical argument used in [17] can be pursued to derive lower bounds. The existence of such bounds was originally emphasized in [21].
Our approach gives completely general expressions for the independent null constraints.
Then, can we say where the first massive string pole would occur? More precisely in eq. (3.10), what is the maximum δ 0 we can use? Using the methods in this paper, we can address this question.

Bound on total scattering cross-section
In this section, will exploit the null constraints arising from m > n in (2.8) to bound total scattering cross-sections. We will be in d = 4 and we will use the standard notation s = s 1 + 4/3 with µ = 4. The null constraints read for m > n, n ≥ 1. For m + n ≤ , m > n, n ≥ 1 we can verify that g for m + n ≤ L max , m > n, n ≥ 1. We have placed the contributions arising from ≥ L max + 2 on the right which gives the inequality. From unitarity, we know that 0 ≤ a (s) ≤ 1. The inequalities (4.4) impose further conditions on the a (s). We convert the integral over s in (4.4) as a sum by defining Nmax . Using the constraints (4.4), we want to bound the total scattering cross section σ(s) 8 (2 + 1)a (s) .  the spin sum is suggested by the figure but appears to be slow for higher values of s. Also it is clear that theσ found using a typical S-matrix living on the boundary of the so-called river arising from the S-matrix bootstrap [23] or even the lake boundary in [24], is far below the numerical bound presented.
Note that Froissart bound 10 i.e.,σ s 16 log 2 (s/s 0 ) (which is not valid for lower value of s we are considering) is below the numerical bound. The main utility of our numerical bound is that it is valid for any s ≥ 4 unlike the Froissart bound which is valid for s 4. It will be fascinating to derive analytic bounds using the present method and see if a stronger than Froissart bound is possible at higher energies.
In existing derivations of the Froissart bound, the role of crossing symmetry has not been explored 11 .

Discussion
The crossing symmetric dispersion relation approach, presented in this paper, promises to open up a new and efficient way to constrain field theories. It will be worth exploring these ideas further, as clearly, at least for the cases considered here, the crossing symmetric approach provides a more direct derivation of certain constraints. It will be very interesting to connect the ideas and techniques in this paper with the "EFT-hedron" picture in [26]. Another place where we expect these crossing symmetric dispersion relations to play an important role [27] is the formulation of the dual S-matrix bootstrap in higher dimensions. So far, an explicit attempt has only been made in 2 dimensions [28].
On the CFT side, we will show in [12] how the manifestly crossing symmetric method extended to CFT Mellin amplitudes leads to the sum rule constraints arising from the two-channel dispersion relation presented in [3], [4]. Furthermore, the CFT generalized null constraints admit a straightforward derivation and are needed to show the equivalence. This suggests that the manifestly crossing symmetric dispersion relation will not only be more systematic but will have more constraints than what is easily derivable in the two-channel symmetric approach.