Amplitude-bootstrap assumptions and open problems

At tree level one might choose rational functions with specified physical poles, helicity weights and maximum momentum degree. A finite basis then turns constraint matching into linear or polynomial equations for coefficients. Choosing the degree cutoff is physical power-counting input, not an innocent computational convenience. Excluding an allowed local operator from the ansatz can manufacture apparent uniqueness by removing the very coefficient one should have determined from matching.

Crossing must also be implemented with the intended state and phase conventions. For identical scalars it becomes permutation symmetry of the invariant function, while spinning or charged particles require the corresponding crossed helicities and species. Dimensional analysis constrains coefficients but does not usually determine their values. A bootstrap result should therefore name its spectrum, spacetime dimension, perturbative order, analytic class, symmetries and normalization before claiming that an amplitude has been fixed.

Multiplying the function by a vanishing channel invariant and taking its pole limit fixes r but removes both contact terms in the example. Leading soft information can leave additional regular functions invisible as well. More restrictive soft behavior can eliminate some coefficients, but such a requirement must be derived for the intended theory or explicitly imposed. It is not legitimate to borrow an enhanced soft theorem from a different effective field theory merely because it makes the equations solvable.

The remaining coefficient space can be characterized as the null space of the linear constraints on a fixed ansatz. A nonzero null vector represents a deformation invisible to those tests. Independent normalization measurements, matching calculations, higher-order data or an appropriately justified asymptotic condition may remove it. Checking a few generic numerical points can expose mistakes, but only a complete functional analysis within the declared space proves the absence of further invisible directions.

At loop level the function space includes logarithmic branch cuts and sometimes functions beyond ordinary polylogarithms. Branch choices, constants not detected by discontinuities, rational terms and regulator poles matter. Solving a restricted cut or symbol problem is valuable without necessarily fixing the complete integrated amplitude. Infrared subtraction and ultraviolet renormalization define which finite object is being bootstrapped, and changing those conventions can change that object while preserving a physical observable after all ingredients are combined.

Research questions include finding complete yet economical function spaces, obtaining useful numerator representations with stated locality and power counting, and proving that chosen cuts and boundary data are sufficient in a specified setting. Progress on one multiplicity or helicity sector does not settle these questions for arbitrary quantum field theories. Conversely, a failed ansatz need not refute the underlying theory; its excluded contact structures or analytic functions may be the problem. Transparent assumptions make both success and failure scientifically interpretable.