S-matrix Bootstrap for Effective Field Theories: Massless Pions

TL;DR

Guerrieri, Penedones, and Vieira apply the numerical S-matrix bootstrap to bound the leading chiral Lagrangian coefficients $\alpha$ and $\beta$ for massless pions in more than two dimensions. They construct a low-energy amplitude $A(s|t,u)$ and use a $\rho$-mapping to encode analytic structure, enforcing Lorentz invariance, crossing, and unitarity to carve out the allowed $(\alpha,\beta)$ region as a function of a truncation parameter $N_{\max}$. The results show a largely stable region for $N_{\max}\gtrsim 18$, with QCD values near the boundary and logarithmic corrections refining naive dispersive bounds for large $|\alpha|$, and reveal resonance patterns along the boundary. This work demonstrates a proof of concept for higher-dimensional EFT bootstrap and points to extensions to other massless sectors and to comparisons with string-theory predictions.

Abstract

We use the numerical S-matrix bootstrap method to obtain bounds on the two leading Wilson coefficients of the chiral lagrangian controlling the low-energy dynamics of massless pions thus providing a proof of concept that the numerical S-matrix bootstrap can be used to derive non-perturbative bounds on EFTs in more than two spacetime dimensions.

Figures (7)

• Figure 1: Allowed region in the $\{\alpha,\beta\}$ space for different values of $N_{\text{max}}$ ranging from 12 to 23. For each $N_{\text{max}}$ we bound all the partial waves up to spin $L_{\text{max}}=90$ such that the spin cutoff dependence is negligible. In the inset it is shown a zoom out of the $\{\alpha,\beta\}$ space. The green region is allowed. The dashed lines denote the naive bounds \ref{['naivebounds']} obtained from dispersion relations and unitarity, neglecting the effect of logarithmic branch cuts in the scattering amplitude. Our numerical bound has a non-trivial shape in the natural range $\alpha \sim \beta \sim \frac{1}{(4\pi)^2}$ expected from naive dimensional analysis. Using values from Bijnens:1995ynBijnens:1997vq, it seems that QCD lies inside but close to the boundary of the allowed region.
• Figure 2: Discrete derivative of $\beta_\text{min}(\alpha)$ with respect to $\alpha$ for different values of $N_\text{max}$ (same colour coding as in figure \ref{['fig1']}). We see that for large negative (positive) $\alpha$ we approach the expected $-1/2$ ($0$) slope from the naive bounds \ref{['naivebounds']}. This leading linear behavior is already seen clearly in figure \ref{['fig1']}. Interestingly, there are important logarithmic corrections to this as illustrated in the inset. There, we plotted the combination $\alpha (d\beta_\text{min}/d\alpha + 1/2 \theta(-\alpha))$, involving the Heaviside $\theta$-function. This combination kills the leading linear asymptotics and extracts the coefficient of any log at large $|\alpha|$. Of course, numerical errors are magnified when we multiply by $\alpha$ which is why we get the numerical oscillations in the inset. Nevertheless, we see reasonable agreement with the analytic prediction \ref{['improvednaivebounds']} shown as red dashed horizontal lines.
• Figure 3: (a) Example of a phase shift $\delta_{0}^{0} =\text{Re}( \frac{1}{2i} \log S_0^{(0)})$ at a given point at the boundary (corresponding to $\alpha =0.062$, spin $\ell=0$ and isospin $I=0$) as a function of $\hat{s}=s/f_\pi^2$ as we increase $N_\text{max}$. The phase shift seems to have converged to a nice resonance like shape. (b) The absolute value $|S_{0}^{(0)}|$ approaches $1$ more and more as we increase $N_\text{max}$. It clearly seems as if unitarity wants to be saturated in the infinite $N$ limit. We also observe that this unitarity saturation is often achieved at the price of some more erratic behaviour at high energy which we do not control well. Given the interesting interplay between unitarity and the Aks theorem aks1aks2 as recently emphasized in Correia:2020xtr, it would be very interesting to study this unitarity (non)-saturation in much more detail. (c) To reach the optimal infinite $N$ bound we can try to fit our optimal target (in this case we are minimizing $\beta$ at fixed $\alpha$) as we increase $N_\text{max}$ and try to extrapolate.
• Figure 4: Phase shifts $\delta_{l}^{I} =\text{Re}( \frac{1}{2i} \log S_l^{(I)})$ (solid lines; left y-axis) and absolute values of the corresponding amplitudes (dashed lines; right y-axis) for the lowest spins and isospins along the boundary as function of $\hat{s}=s/f^2_\pi$, with $N_\text{max}=23$. The gray dots and error bars are experimental phase shifts for real world massive pions Protopopescu:1973shLosty:1973etGrayer:1974crEstabrooks:1974vuHoogland:1977ktBatley:2010zzaGarciaMartin:2011cn. The comparison is done by plotting the experimental data also as a function of the square of the center of mass momentum in units of $f_\pi$, i.e. $(s-4m_\pi^2)/f_\pi^2$. Here we use $m_\pi\simeq 140\text{Mev}$ and $f_\pi\simeq 93 \text{MeV}$.
• Figure 5: There is a rich pattern of resonances in the optimal S-matrices as we move along the allowed S-matrix space. On the left, we have odd spin resonances turned on and no even spin resonances. On the right we have the opposite. The real world has both types of resonances so should somehow not be at the boundary. Would be very interesting to force the presence of the $\rho$ resonance as in Andrea and repeat this analysis as done in that paper for massive pions.