Positivity and Geometric Function Theory Constraints on Pion Scattering

This paper presents the fascinating correspondence between the geometric function theory and the scattering amplitudes with $O(N)$ global symmetry. A crucial ingredient to show such correspondence is a fully crossing symmetric dispersion relation in the $z$-variable, rather than the fixed channel dispersion relation. We have written down fully crossing symmetric dispersion relation for $O(N)$ model in $z$-variable for three independent combinations of isospin amplitudes. We have presented three independent sum rules or locality constraints for the $O(N)$ model arising from the fully crossing symmetric dispersion relations. We have derived three sets of positivity conditions. We have obtained two-sided bounds on Taylor coefficients of physical Pion amplitudes around the crossing symmetric point (for example, $\pi^+\pi^-\to \pi^0\pi^0$) applying the positivity conditions and the Bieberbach-Rogosinski inequalities from geometric function theory.

In [4] the correspondence between the famous Bieberbach conjecture (de Branges' theorem) and the non-perturbative crossing symmetric scattering amplitudes was pointed out, which established a close relationship between the Bieberbach-bounds and the bounds on the Wilson coefficients. In [3] it was pointed out that crossing symmetric scattering amplitudes are typically real functions. The pivotal ingredient in demonstrating such a correlation between the geometric function theory and the scattering amplitudes is crossing symmetric dispersion relation in a new z-variable for a fixed parameter a. The z, a variables arise by parametrizing the Mandelstam invariants in the following way. The 2-2 scattering amplitudes with O(N ) global symmetry are functions of s, t, u, the Mandelstam invariants, satisfying s + t + u = 4m 2 , where m is the mass of the external scalars. For convenience, we will work with the notation [5] [6] Fully crossing symmetric dispersion relation is written down by parametrizing s i as a function of z, a.
The parametrization is given by where a is a real parameter, − µ 3 ≤ a < 2µ 3 and z i is the cube roots of unity, after we parametrize s i as a function of z, a, the amplitude is a function of z 3 =z, a,i.e. M(z, a). The parameter a is given by a = y x , where x = − (s 1 s 2 + s 2 s 3 + s 3 s 1 ), y = −s 1 s 2 s 3 . For a crossing symmetric dispersion relation in z-variable, while keeping a-fixed [5,6], the kernel is a univalent function 1 , and the absorptive part is positive for a certain range of a. The kernel's specific form and the absorptive part's positivity enable one to re-express the dispersion relation as a Robertson representation 2 . Once a function has a Robertson representation, we can say it is a typically real function. Therefore the amplitudes M(z, a) are typically real functions. We ask here if such dispersion relation can be written for theories with O(N ) global symmetry? Do these correspondences exist for theories with global O(N ) symmetry? The answer turns out to be yes! We write down three sets of fully crossing symmetric dispersion relations for three specific combinations of isospin amplitudes. We call these combinations of isospin amplitudes to be F k (z, a), k = 0, 1, 2. We find that these combinations are in the Robertson representation. Hence we can write three sets of Bieberbach-Rogosinski inequalities. If we write p (a)a 2pzp , k = 0, 1, 2 , (1.7) the Bieberbach-Rogosinski inequalities take the form: for a range of a, which is derived in the main text (see eq (4.14)).
These three sets of dispersion relation and unitarity conditions give three sets of positivity constraints 1 A function is univalent on a domain D if it is holomorphic, and one-to-one, i.e. for all z 1 , z 2 in D, f (z 1 ) = f (z 2 ) if z 1 = z 2 . 2 In |z| < 1 a regular function F (z) is typically real if and only if it has the Robertson representation: for Taylor coefficients of the amplitudes around the crossing symmetric point. These three sets of new positivity constraints are very non-trivial. Demanding the locality, we get three sets of independent sum rules or locality constraints. These novel sets of independent sum rules constrain the theories strongly.
There is a lot of recent work on constraining the quantum field theory using dispersion relation, utilizing the analyticity and unitarity assumptions [7][8][9][10][11]. Dispersion relations are the non-perturbative depiction of scattering amplitudes. Usually, people write dispersion relations in 2-2 scattering by keeping Mandelstam invariant t-fixed and write a dispersion integral in s-variable, which leads to the s ↔ u symmetric representation of the amplitude. Imposing full crossing symmetry as an additional condition, one gets null constraints. Analogous strategies also developed in the context of conformal field theories, Mellin amplitudes. See for example [12][13][14][15].
The constraints in EFT Wilson coefficients were worked out in [16][17][18][19][20][21][22][23][24][25][26][27][28], using positivity of the partial wave and null conditions 3 . In our case, we don't use the null constraints; instead, we use positivity (of the partial wave expansion) and bounds in the Taylor series coefficients of the amplitudes in z-variable that appear from the geometric function theory.
Bieberbach-Rogosinski inequalities and the positivity conditions give two-sided bounds on the Taylor coefficients of the amplitudes of any physical process around the crossing symmetric point 4 ; these processes need not be fully crossing symmetric, only two channel symmetry is sufficient. For example, This amplitude is not fully crossing symmetric. The three sets of Bieberbach-Rogosinski inequalities and three sets of positivity conditions provide two-sided (order O(1), in [3] the O(1) is shown to be |O(1)| < 5.625) bounds on the C p,q , presented in table (3).
We have organized the paper in the following way. The definitions of the fully crossing symmetric combinations, their dispersion relations, inversion formula and sum rules are presented in Section 2. Section 3 contains positivity conditions on the Taylor coefficients of three crossing symmetric combinations.
Section 4 describes how geometric function theory for the O(N ) model can be realized. Section 5 contains the application of the geometric function theory to physical pion amplitudes and bounds on Taylor coefficients of physical amplitudes around crossing symmetric point. We conclude with a summary and future directions in section 6. We have added Appendices for multiple demonstrations and verifications.

Crossing symmetric dispersion relation for O(N ) model
The 2-2 scattering amplitude with O(N ) global symmetry can be written as where A(s i | s j , s k ) = A(s i | s k , s j ). The isospin I s-channel amplitudes M (I) (s 1 , s 2 ) are given by We want to write down crossing symmetric dispersion relation in z-variable for fixed a following [5,6] for O(N ) model. We are interested in such a crossing symmetric dispersion relation to connecting with the geometric function theory. In general full 3-channel crossing symmetry is missing in isospin amplitudes M (I) (s 1 , s 2 ). Following [29,30], we will consider independent fully crossing symmetric combinations The G 0 (s 1 , s 2 ) is the π 0 π 0 → π 0 π 0 amplitude. Dispersion relation in z-variable for G 0 (s 1 , s 2 ) is discussed in [5,6], and relations to geometric function theory is given in detail in [3,4]. We will write z-variable dispersion relations for all three of the G k (s 1 , s 2 ) and demonstrate that G k (s 1 , s 2 ) (linear combinations of them) are related to geometric function theory. We expect that G k (s 1 , s 2 ) to have the same analyticity properties as the isospin amplitude M (I) (s 1 , s 2 ). Since antisymmetry of M (1) (s 1 , s 2 ) with respect to s 2 and s 3 , prevents the denominators appearing in G 1 (s 1 , s 2 ) and G 2 (s 1 , s 2 ) to introduce any additional new singularities at s 1 = s 2 , s 2 = s 3 , s 3 = s 1 . From the large s 1 , fixed s 2 behaviour of the isospin amplitudes M (I) (s 1 , s 2 ), we can write once 5 subtracted dispersion relation in z-variable for fixed a for G k (s 1 , s 2 ).
For large s 1 fixed s 2 we have The discontinuity of G k (s 1 , s 2 ) starts at s 1 = 2µ 3 . Therefore, once we know the discontinuity of G k (s 1 , s 2 ), we can write fully crossing symmetric dispersion relations, following the same logic in [5,6]. This leads to fully crossing symmetric dispersion relation 6 is s 1 , s 2 variables where α (k) 0 is the subtraction constant. The DiscG k (s 1 ; s 2 ) is the s-channel discontinuity of G k (s 1 , s 2 ) and , a = s 1 s 2 s 3 s 1 s 2 + s 2 s 3 + s 3 s 1 .
(2.9) Partial wave expansion for G k (s 1 , s 2 ) and it's convergence has been discussed in [29] (see [31] for some applications). Since the G k (s 1 , s 2 ) have the same analyticity properties as the isospin amplitude M (I) (s 1 , s 2 ) and M (I) (s 1 , s 2 ) has cuts starting at s 1 ≥ 2µ/3, hence the domain of a is same as discussed in [5]. The range of a, we will consider here is − µ 3 ≤ a < 2µ 3 . This range of a can be enlarged to −6.71µ ≤ a < 2µ 3 as discussed in [5, see appendix ], [29, see discussion below eq. 3.10]. We will use that in our calculations in the upcoming sections.
Below we have presented the the formula for s-channel discontinuity, DiscG k (s 1 ; s 2 ): , (2.11) is a real positive number. More importantly, due to unitarity, the partial wave coefficients satisfy For our calculations, we will be only utilizing the positivity of the partial wave coefficients.

Inversion formulas and sum rules
The G k (s 1 , s 2 ) have the same analyticity properties as the isospin amplitude M (I) (s 1 , s 2 ), and they don't have any additional singularities. More importantly, all three of them are fully crossing symmetric.
Therefore, we can write (2.14) with crossing symmetric variables x = − (s 1 s 2 + s 2 s 3 + s 3 s 1 ), y = −s 1 s 2 s 3 . In the z-variable the kernel takes the form H (s 1 ; s 1 , s 2 , s 3 ) = 27a 2 z 3 (3a−2s 1 ) −27a 3 z 3 +27a 2 z 3 s 1 +(z 3 −1) 2 (s 1 ) 3 , which can be seen by writing s i 's in terms of (z, a). Now identifying crossing symmetric variable x in terms of z-variable via the relation we can series expand in powers of x. We obtain Coefficient of a m (since a = y/x) will be W n−m,m . In general, one can write The DiscG k are given in terms of the partial wave coefficients of the s-channel discontinuity of the isospin amplitudes, (2.17) We use the same convention as [6].
The Gegenbauer polynomials can be expanded as . Now we plug the formulas (2.17) for DiscG k in the equation (2.15). After that we compute the the coefficient of a m . The coefficient of a m gives us formula (2.20) In the appendix, we have verified the dispersion relation and inversion formula against O(3) Lovelace-

Sum rules
In the equation (  p,q were derived, using the positivity of the partial wave coefficients and the positivity of Gegenbauer polynomials, m r=0 Where δ 0 is the scale 7 of the theory and There exist similar kinds of (but very non-trivial) positivity conditions for W (1) p,q and W (2) p,q , which take the following form n,m,m (µ, δ 0 ) = 1. The χ (1) n,r,m (µ, δ 0 ), χ (2) n,r,m (µ, δ 0 ), Υ (1) n,r,m (µ, δ 0 ), Υ (2) n,r,m (µ, δ 0 ) are known positive coefficients. Its quite hard to get a general formula, but one can work out case by case in m. For example, below, we listed them up to m = 3.
For EFTs the analysis is modified by taking the lower limit of the integral 2µ 3 → 2µ 3 + δ 0 , here δ 0 will be the EFT scale. Since for EFTs, most of the time the low energy amplitude DiscG| is computable. Therefore, we subtract the low energy part from the full DiscG and plug back in the integration , which changes the lower limit to 2µ 3 + δ 0 . In our calculations, we will use δ 0 = 0. See [6] for more discussions. and χ (2) n,0,1 (δ, µ) = 81(2n+5) A schlicht function f (z) (normalized f (0) = 0 and f (0) = 1, see [4] for a quick overview), which is univalent inside the unit disk |z| < 1 with b p ∈ R, is also typically real function. The kernel H is a function of this kind.
If f (z) is a regular typically real function in |z| < 1, then the coefficients should satisfy the bounds (see [2,3] for proof) An important representation of typically real is the Robertson representation [32].

Robertson representation:
In |z| < 1 a regular function F (z) is typically real if and only if it has the Robertson representation: where the measure f (η) is a non-decreasing function.
From the crossing symmetric dispersion relation, we get the full amplitude as an integral of discon- univalent typically real function. For some range of a, the A(s 1 , s (+) (s 1 , a)) is non-negative. We will show below from these two facts that the full amplitude can be recast as Robertson representation; hence it is a typically real function.

Typically real functions for O(N ) model
We would consider the combinations By definition then we can write (z = z 3 ) n−m,m . (4.6) In [3] it was shown that F 0 (z, a) is a typically real function from the positivity of the Disc s G 0 s 1 ; s The kernel can be written as inz variables as (4.7) We know from [4] that Kernel is a univalent function inside the unit disk. Notice that for β 1 (a, s 1 ) = 27a 2 s 3 1 (3a − 2s 1 ) < 0, we must have a < 4µ/9, since s 1 ≥ 2µ/3 (see [3]). The dispersion relations for F k (z, a) are given by In the combination Disc 1 − 2δ 2,k 3 G 0 + δ 1,k G 1 + δ 2,k G 2 , if we use (2.10) and collect the coefficients of s 1 , a)), we find that the coefficients are always positive for s 1 ≥ 2µ 3 , N ≥ 3, µ ≥ 4 for some range of a: for k = 1, 2 the range of a is − 2 (s 1 , a) for − 2µ 9 < a < 2µ 3 , s 1 ≥ 2µ 3 . Therefore we find that the combination is always non-negative for ranges discussed above Since s 1 ≥ 2µ 3 , therefore the range of a in case of k = 0 is given by − 2µ 9 < a < 4µ 9 , while in case of k = 1, 2 range of a is given by − 2µ 9 < a < 2µ 9 . See appendix (C) for more clarifications. We change the variable s 1 to η by the equation In this changed variable, we can write F k (z, a) as (see [3] for more details) .

(4.12)
We have adopted the notation DiscF k (η, a) for Disc 1 − a) , after changing the variable s 1 → η. Since the DiscF k (η, a) are positive (for s 1 ≥ 2µ 3 , N ≥ 3, µ ≥ 4 and range of a in case of k = 0 is given by − 2µ 9 < a < 4µ 9 , while in case of k = 1, 2 range of a is given by − 2µ 9 < a < 2µ 9 ) then . (4.13) are non-decreasing functions. We can conclude that functions F k (z, a) are typically real functions.

Rogosinski bounds on
Since functions F k (z, a) are typically real functions, therefore we can readily write where κ p = − sin pϕ p sin ϕ p , csc (ϕ p ) (p cos (pϕ p ) − sin (pϕ p ) cot (ϕ p )) = 0 . For p odd, π/p < ϕ p < 3π/2p, while for p even, ϕ p = π is only solution, giving κ p = p. Notice the range of a is different for different k and for k = 1, 2, one will have to restrict to the cases N ≥ 3 , µ ≥ 4 5 Geometric function theory constraints on Pion amplitudes In this section, we discuss the geometric function theory constraints on the Pion amplitudes. We will consider the case N = 3, µ = 4 in this section. For pion amplitudes, the functions F k (z, a) are indeed typically real functions. It is more natural to work with W (k) n−m,m defined in (4.6).

Bounds on coefficients
Since the imaginary part of F k (z, a) is positive for the range given in (4.14), therefore we note that This is true since here m = 0 or we are considering the coefficients of a 0 . Which enable us to compute n,0 , which is given by Therefore we can write (derivation is similar to second equation of (3.1), see [6] for details)

Rogosinski bounds on
n−m,m + δ 2,k W n−m,m . (5.6) In the above equations, we will consider m = 1, 2, 3, . . . n. The Taylor coefficients of the amplitude around the crossing symmetric point (appearing on α  p,q for n = 2 in equation (4.14). We use the normalizations W

Bounds on Taylor coefficients of physical amplitudes
The A(s 1 | s 2 , s 3 ) is symmetric under exchange of s 2 ↔ s 3 . Therefore without lose of generality, we can Once we have the bounds on W (k) p,q , using the isospin amplitudes, namely using (2.13), we can put bounds on Taylor coefficients of any physical amplitudes around the crossing symmetric point. For example, for the reaction π + + π − → π 0 + π 0 the amplitude is  p,q for n = 3.
Using the equation (2.13) and the expansion (2.14), we find that Therefore we can write C p,q as a combinations of W (k) p,q . For example 3,0 + 1 9 W (2) 1,1  Table 3: Bounds on C p,q . Note that W (k) 1,0 = 1 in our normalization. Also note that C p,q which contains the subtraction constants W (k) 0,0 can't be bounded. In order to restore the normalization and get the two sided bounds in terms of W (k) in table (2) and use in equation (5.10).
If we put W (k) 1,0 = 1, then we recover the results that are given in (3). Using the constraints in the table (3), we find the finite region of theory space as depicted in figure (2). Some of the known theories are also indicated.

Summary and future directions
In this paper, we have applied the techniques in geometric function theory, namely typically real functions to 2-2 scattering amplitudes with global O(N ) symmetry. We have used the fully crossing symmetric dispersion relation as in [5,6].
The main results of the paper are the following: • In the case of theory with O(N ) global symmetry, there exist three independent sets of fully crossing symmetric combinations ( see (2.4), (2.5), (2.6)) of isospin amplitudes, which was studied in [29].
We have written down three collections of fully crossing symmetric dispersion relations in z-variable for fixed a.
• The dispersion relations empowered us to derive three sets of independent sum rules. These new sum rules arise because of the cancellation of unphysical powers of x.
• We have written down the partial wave expansion for the dispersion relations through partial wave expansion of the isospin amplitudes (see (2.17)). These partial wave expansions allow us to derive of • Typically real-ness of the functions (4.4) enable us to formulate the Bieberbach-Rogosinski inequalities for typically real functions (4.14).
• The Bieberbach-Rogosinski inequalities and the positivity conditions permit us to write down twosided bounds on the three groups of Taylor coefficients for pion scattering (table (2)). These three assortments of bounds allow us to establish bounds on Taylor coefficients of any physical amplitudes around the crossing symmetric point (see for example table (3)). Our bounds presented in table  Here are our immediate future directions: • These three sets of dispersion relations can be employed in the matter of CFT Mellin amplitude to make connections with Polyakov Mellin Bootstrap [35][36][37][38][39]. The upcoming work of [40] will discuss the Witten block expansion. The upcoming work of [41] will show that a crossing antisymmetric correlator could be expanded in a manifestly crossing antisymmetric basis by introducing a crossing antisymmetric dispersion relation.
• An exciting application will be to see the implications of our sum rules (2.21) to the CFT correlators of charged fields discussed in [42] and the functionals therein.
• Another charming area worth exploring is CFT Mellin amplitudes in light of geometric function theory. It will be interesting to relate CFT typically real-ness with swampland conditions considered recently in [43,44] as well as the tricky correlator bounds found in [45]. It will be very interesting to relate the geometric function theory analysis to the positive geometry of CFT, discussed in [46].
• It will be appealing to see the implications of our bounds and sum rules (2.21) to S-matrix bootstrap for pion amplitudes [33,34,47] and to the dual the S-matrix bootstrap [48].

Shapiro model
In this section, we will verify our various formulas for the Lovelace-Shapiro model [33,49].
The s-channel discontinuities are given by , We put DiscG k (s 1 , s 2 ) given in the above formulae in the dispersion relation (2.8). For numerical illustration, we truncate the k-sum at k = k max . We note that, because of δ(s 1 − 2k − 1), the s 1 integral can be easily done. The comparison is given the table below (4) where cos(θ) = 1 + 2s 2 s 1 . We will work in d = 4 for illustration purpose. Now one can calculate the partial wave coefficients We will truncate -sum to L max and k-sum to k max . Knowing the a (I) (s 1 ), we can calculate the W -6.00331 -6.00331 -6.00331 -6.00331 n−m,m for m > n, the procedure is exactly same. Now since for m > n which implies (y/x) m x n i.e negative powers of x. There should not be any negative powers of x as in the expression (2.14) due to locality. The coefficients W n,r,m (µ, δ), χ (1) n,r,m (µ, δ), χ (2) n,r,m (µ, δ) First, we take the combinations n,r,m (µ, δ)W n,r,m (µ, δ).

D Verifications of bounds with known theories D.1 The 2-loop Chiral perturbation theory
In the 2-loop chiral perturbation theory, we use the experimental values for the parameters in the amplitude (we use the amplitude given in [19,50]). We can easily see that all of these coefficients satisfy the bounds listed in table (1), (2). We can compute the C p,q using the equation ( p,q . Our bounds assumed the lower limit of the s 1 integral as 8 3 (for µ = 4), while for the O(3) Lovelace-Shapiro model lower limit is s 1 = 1. Therefore, in order to compare with the bounds given in (2), we need to multiply with appropriate powers of 3 8 . In order to match with the conventions of the EFT scale, we multiply Notice that these C p,q satisfy the inequalities listed in table (3).