S-matrix Bootstrap

There is an old idea that it might be possible to bootstrap the S-matrix of an interacting quantum field theory just by imposing a set of consistency conditions. In full generality, this approach is too hard because the analytic structure of the scattering amplitudes is extremely complex due to the multi-particle production channels. However, some theories admit a limit that turns off multi-particle production and the bootstrap approach becomes feasible. One example is the weak coupling limit of a string theory: the tree-level scattering amplitude, as a function of the Mandelstam invariants, has a simple analytic structure with only simple poles associated to the production of single-string states. The planar limit of a confining gauge theory is another important example, with mesons and glueballs playing the role of string states.

We shall focus on the 4 particle scattering amplitude \(A(s,t,u)\) of this class of theories, where

\begin{equation*} s=-(p_1+p_2)^2\ ,\ \ \ \ \ \ t=-(p_1+p_3)^2\ ,\ \ \ \ \ \ u=-(p_1+p_4)^2\ , \end{equation*}

are the usual Mandelstam invariants. In addition,

\begin{equation*} s+t+u=4m_0^2 \end{equation*}

where \(m_0\) is the mass of the external particles (all the same for simplicity).

We shall assume the following:

Analyticity: the amplitude is analytic everywhere, except at a discrete set of single poles along the positive real axis of \(s\), \(t\) and \(u\).
Crossing: \(A(s,t, u)= A(t,s,u)=A(s,u,t)\)
Unitarity: the residues of the poles are polynomials that can be expanded in partial waves with positive coefficients.
Regge behaviour: \(A \approx f(t) s^{\alpha(t)}\) at fixed \(t\) and large \(s\) away from the real axis.

Let us write

\begin{equation*} A(s,t,u) = \oint_s \frac{ds'}{2\pi i} \frac{A(s',t,u')}{s'-s} \end{equation*}

where \(u=4m^2-s-t\) and \(u'=4m^2-s'-t\). We write \(A(s,t,u)\) to make crossing symmetry explicit but the amplitude is really just a function of 2 variables, which we take to be \(s\) and \(t\). We can now deform the contour to pick up the contributions from the poles at \(s'=m^2\) and \(u'= m^2\). This gives

\begin{equation*} A(s,t,u) =\sum_{m^2} \frac{Res(A, s=m^2)}{s-m^2} +\sum_i \frac{Res(A, u=m^2)}{u-m^2} \end{equation*}

if \(\alpha(t)<0\) so that we can drop the contribution from infinity in the \(s'\) complex plane. We can now use unitarity to write

\begin{equation*} Res(A, s=m^2) = \sum_{l} c(m^2,l) P_l\left( \frac{m^2-4m_0^2+2t}{m^2-4m_0^2}\right) \end{equation*}

where \(c(m^2,l) \ge 0\), \(P_l\) is a Gegenbauer polynomial in \(d\) space dimensions (we are dropping normalization constants)

\begin{equation*} P_l(x)= \ _2F_1\left( -l, d + l - 2, \frac{d - 1}{2}, \frac{1-x}{2}\right) \end{equation*}

and we used

\begin{equation*} \cos \theta = \frac{u-t}{u+t}=\frac{m^2-4m_0^2+2t}{m^2-4m_0^2}\ . \end{equation*}

Finally, we conclude that

\begin{equation*} A(s,t,u) = \sum_{m^2,l} c(m^2,l) P_l\left( \frac{m^2-4m_0^2+2t}{m^2-4m_0^2}\right)\left(\frac{1}{s-m^2} + \frac{1}{u-m^2} \right)\ ,\ \ \ \ \ {\rm for}\ \alpha(t)<0\ ,\ \end{equation*}

which is explicitly symmetric under \(s \leftrightarrow u\). The bootstrap equation follows from requiring invariance under \(s \leftrightarrow t\),

\begin{equation*} 0=A(s,t,u)-A(t,s,u) = \sum_{m^2,l} c(m^2,l) \left( G_{m^2, l} \left(s,t\right) - G_{m^2, l} \left(t,s\right) \right) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) \end{equation*}

where

\begin{equation*} G_{m^2, l} \left(s,t\right)=P_l\left( \frac{m^2-4m_0^2+2t}{m^2-4m_0^2}\right) \left(\frac{1}{s-m^2} + \frac{1}{4m_0^2-s-t-m^2} \right) \end{equation*}

The sum in equation \((1)\) converges if \(\alpha(t)<0\) and \(\alpha(s)<0\).

Assuming that the external scalar particle also appears as a pole in the amplitude we can rewrite \((1)\) as follows

\begin{equation*} 1 = \sum_{m^2,l}\tilde{c}(m^2,l) \,F_{m^2, l} \left(s,t\right) \end{equation*}

where

\begin{equation*} \tilde{c}(m^2,l) = \frac{c(m^2,l)}{c(m^2,0)}\ge 0 \end{equation*}

and

\begin{equation*} F_{m^2, l} \left(s,t\right) = \frac{G_{m^2, l} \left(s,t\right) - G_{m^2, l} \left(t,s\right)}{G_{m_0^2, 0} \left(t,s\right) - G_{m_0^2, 0} \left(s,t\right)} = \frac{(t-m_0^2)(s-m_0^2)}{s-t} \left( G_{m^2, l} \left(s,t\right) - G_{m^2, l} \left(t,s\right) \right) \end{equation*}

The sum rule above is very similar to the one used in arXiv:0807.0004 in the context of the conformal bootstrap. The goal of this project is to apply the techniques recently develop in that context to the S-matrix bootstrap.

Wiki

Open Bootstrap Collaboration / S-matrix Bootstrap

S-matrix Bootstrap