Large deflection scattering, soft radiation and KMOC formalism

This paper extends the KMOC formalism beyond large impact parameter scattering by demonstrating that soft radiative fields, specifically electromagnetic and gravitational memory, can be computed as non-perturbative inclusive observables derived from on-shell amplitudes, consistent with previous saddle-point analyses.

The Gist

The Big Picture: Predicting the Aftermath of a Cosmic Crash

Imagine two massive spaceships zooming past each other in deep space. They don't crash, but they get close enough that their gravity (or electric charge) pulls on them, causing them to swerve. This is called scattering.

When they swerve, they don't just change direction; they also send out ripples. In physics, these ripples are waves of light (photons) or gravity (gravitons). The paper asks a very specific question: Can we predict the exact pattern of these ripples using the rules of quantum mechanics, even if the spaceships swerve wildly?

The Two Main Characters

To understand the paper, we need to meet two different "methods" scientists use to solve this problem.

1. The KMOC Method (The "Gentle Nudge" Approach)

The Analogy: Imagine two billiard balls rolling on a very long, smooth table. They are far apart, so they only feel a tiny, gentle tug from each other. They barely swerve.

• How it works: The KMOC formalism (named after Kosower, Maybee, and O'Connell) is a brilliant mathematical tool designed for this "gentle nudge" scenario. It uses quantum math to predict the outcome of these gentle collisions.
• The Problem: It works perfectly when the balls are far apart (large impact parameter). But what if the balls are heading straight for each other? What if they swerve violently? The KMOC tool breaks down because it assumes the interaction is weak and gentle. It's like trying to use a thermometer to measure the temperature of a nuclear explosion; the tool isn't built for that intensity.

2. The "Saddle Point" Method (The "Crowd Counting" Approach)

The Analogy: Imagine a huge concert. You want to know how many people are leaving through a specific exit. Instead of tracking every single person's path (which is hard if the crowd is chaotic), you look at the total number of people leaving and find the "most likely" number.

• How it works: This method (developed by A. Sen and others) looks at the probability of emitting waves. It asks: "If we have a chaotic crash, what is the most likely number of waves that will be emitted?" By finding this "peak" (the saddle point), they can predict the waves without needing to know the messy details of the crash itself.
• The Advantage: This works even for violent, chaotic crashes. It relies on a universal rule: Soft Theorems. These are like the "laws of conservation" for low-energy waves. They say, "No matter how crazy the crash is, the low-energy ripples always look a certain way."

The Paper's Discovery: Merging the Two Worlds

The authors of this paper wanted to see if they could take the KMOC method (which usually fails for violent crashes) and stretch it to work for violent crashes, just like the "Saddle Point" method does.

They asked: Can we use the KMOC tool to count the waves in a chaotic crash, even if we don't know exactly how the spaceships swerved?

The Answer: Yes, but with a twist.

The "Soft Bin" Trick

Imagine you have a bucket (a "bin") that only catches the very smallest, weakest ripples (soft radiation) from the crash. The authors showed that if you only care about these tiny ripples, you don't need to know the messy details of the crash.

They proved that even if the spaceships crash violently (violating the "gentle nudge" rules), the KMOC formula still works if you focus only on these tiny ripples. The chaotic details of the crash cancel out, leaving behind a clean, universal pattern of ripples.

The Metaphor:
Think of a stormy ocean.

• The KMOC method usually tries to predict the waves by looking at the gentle breeze.
• The Saddle Point method looks at the total chaos and finds the average.
• This Paper says: "Hey, if you only look at the tiny, gentle ripples on the very surface of the water, you can use the gentle breeze math (KMOC) even during a hurricane! The huge waves don't matter for the tiny ripples."

The Catch: Gravity is Weird

The paper found that this trick works perfectly for Electromagnetism (light/electricity). If you have a chaotic crash of charged particles, the tiny ripples of light are predictable and universal.

However, Gravity is trickier.

• The Analogy: In electricity, the ripples are independent. In gravity, the ripples interact with each other. A ripple can create more ripples.
• The Problem: Because gravity ripples interact, the "tiny ripples" depend on the "big, hard ripples" that happened during the crash. To predict the tiny ripples of gravity, you actually do need to know the messy details of the crash.
• The Result: The authors showed that while you can calculate the gravity ripples using their method, you can't get a simple, universal formula like you can with light. You get stuck needing to know the "hard" details of the collision.

Summary in Plain English

1. The Goal: Scientists want to predict the "afterglow" (radiation) of cosmic collisions using quantum math.
2. The Old Tool (KMOC): Great for gentle, distant collisions. Fails for violent ones.
3. The New Idea: The authors tried to force the Old Tool to work on violent collisions by focusing only on the weakest, "softest" signals.
4. The Success: They proved it works for Light (Electromagnetism). Even in a violent crash, the soft light follows a simple, universal rule that the KMOC tool can find.
5. The Limitation: It doesn't work as cleanly for Gravity. Gravity's "afterglow" is too messy and interconnected; the soft signals depend on the hard, violent parts of the crash, so the simple formula breaks down.

In short: The paper shows that even in a chaotic universe, the faintest whispers (soft radiation) often follow simple rules that we can decode, provided we are talking about light. Gravity, however, is too loud and complicated to be tamed by the same simple trick.

Technical Summary

1. Problem Statement

The paper addresses a fundamental limitation in the KMOC (Kosower, Maybee, and O'Connell) formalism, a framework used to derive classical scattering observables (such as radiation flux and impulse) from quantum $S$-matrix amplitudes.

• The Limitation: The standard KMOC formalism relies on the assumption of large impact parameter scattering. In this regime, the exchange momenta are small (soft), allowing the classical limit to be taken perturbatively order-by-order in the Post-Minkowskian (PM) or Post-Lorentzian (PL) expansion. Consequently, the formalism struggles to describe generic scattering processes, particularly those involving large deflections (small impact parameters) where exchange momenta are unbounded and the perturbative expansion in $1/|b|$ breaks down.
• The Gap: While the KMOC formalism computes inclusive observables (like radiative flux) perturbatively, it underutilizes the non-perturbative nature of quantum soft theorems. Conversely, alternative approaches (like the saddle-point analysis by Sen and one of the authors in Ref. [2]) can compute classical soft radiation for generic scattering by extremizing probabilities, but they rely on "in-out" asymptotics rather than the "in-in" expectation values central to KMOC.
• The Goal: The authors aim to determine if the KMOC paradigm can be extended to compute classical soft radiation (specifically electromagnetic and gravitational memory) for generic scattering processes, including those with small impact parameters, without relying on the large impact parameter expansion.

2. Methodology

The authors employ a hybrid approach, combining the KMOC "in-in" formalism with the universality of multi-soft factorization theorems.

• In-In Formalism with Soft Bin: They define an inclusive observable $\hat{K}^\mu_{\text{soft}}$ representing the soft flux in a specific frequency bin $B$ at future null infinity ($\mathcal{I}^+$). The observable is the expectation value $\langle \text{in} | T^\dagger \hat{K}^\mu_{\text{soft}} T | \text{in} \rangle$.
• Violation of Goldilocks Hierarchy: Unlike standard KMOC applications, the authors consider an initial state where the impact parameter $b \to 0$. This violates the "Goldilocks hierarchy" ($l_c \ll l_w \ll |b|$), meaning the exchange momenta do not scale with $\hbar$ in the classical limit, and the hard scattering amplitude cannot be expanded perturbatively.
• Soft Factorization: They utilize the exactness of the Weinberg soft theorem in the quantum theory. For $X$ soft photons (or gravitons) emitted in a bin, the amplitude factorizes as:
  $$A_{n+X} \approx (S^{(0)})^X A_n$$
  where $S^{(0)}$ is the leading soft factor depending only on asymptotic momenta and charges.
• Saddle Point Approximation: Instead of fixing the number of emitted photons $X$ order-by-order (as in standard perturbation theory), the authors sum over all possible $X$ in the soft bin. In the classical limit ($\hbar \to 0$), this sum is dominated by a saddle point $X^*$, which is extremized. This mimics the logic of Ref. [2] but is derived within the KMOC "in-in" framework.
• Semi-Inclusive Cross Sections: To handle infrared divergences and the unbounded nature of the hard scattering, the calculation involves tracing over all "hard" photons/gravitons outside the soft bin, resulting in a semi-inclusive cross section weighted by the soft factor.

3. Key Contributions

A. Extension of KMOC to Large Deflection Scattering

The paper demonstrates that the KMOC formalism is not restricted to large impact parameters when computing soft radiative observables. By leveraging the universality of soft theorems, the authors show that the soft flux can be computed even when the hard scattering amplitude is non-perturbative (i.e., for generic impact parameters).

B. Derivation of Classical Memory via In-In Formalism

The authors derive a non-perturbative formula for electromagnetic memory using the KMOC paradigm. They show that the classical soft flux is determined by:

1. The leading soft factor $S^{(0)}$.
2. The semi-inclusive cross section of the hard scattering process.
3. An extremization condition (saddle point) for the number of soft quanta.

C. Distinction Between Electromagnetic and Gravitational Memory

A critical technical contribution is the analysis of why the electromagnetic result generalizes cleanly to gravity, while gravitational memory faces an obstruction:

• Electromagnetism: The soft flux depends only on the asymptotic charges and momenta. The semi-inclusive cross section effectively reduces to a delta function fixing the final momenta, recovering the standard classical soft theorem.
• Gravity (Non-linear Memory): Gravitational soft factorization includes a term dependent on the momenta of other emitted gravitons (the "non-linear" or Christodoulou memory term). In the KMOC in-in formalism, this requires knowledge of the inelastic amplitude (scattering into hard gravitons). Unlike the EM case, the classical limit of these inelastic amplitudes is not a pure phase, making it difficult to extract a universal soft theorem solely from the semi-inclusive cross section without additional input about hard radiation.

4. Key Results

1. Electromagnetic Memory Formula:
   The authors derive that the classical soft flux $S^\mu$ in a bin $B$ is given by:
   $$S^\mu = \sum_R \lambda_R \, |B| \, |S^{(0)}_{\text{in,out}}|^2 \, k_0^\mu$$
   where $\lambda_R$ is the classical limit of the semi-inclusive cross section for the hard process, and $S^{(0)}$ is the Weinberg soft factor. This result is independent of the details of the hard scattering amplitude (as long as the soft factorization holds) and matches the result obtained via the saddle-point analysis in Ref. [2].

2. Saddle Point Equivalence:
   They explicitly show that the sum over the number of soft photons $X$ in the KMOC formalism is dominated by a saddle point $X^* \propto 1/\hbar$. This confirms that the classical limit of the quantum expectation value naturally selects the configuration that maximizes the probability of emission, consistent with the "in-out" extremization method.

3. Gravitational Obstruction:
   For gravity, the soft flux includes a contribution from the "missing" energy carried by hard gravitons ($Y$ set). The formula becomes:
   $$S^\mu_{\text{grav}} \propto \int \sigma_{\text{semi-inclusive}} \, |S^{(0)}(1,2,R,Y)|^2 \, dY$$
   Because the soft factor $S^{(0)}$ depends on the momenta of the hard gravitons $Y$, and the classical limit of the inelastic cross section is complex (not just a phase), the authors conclude that recovering the universal classical soft graviton theorem purely from the KMOC in-in formalism is obstructed without specifying the hard radiation profile.

5. Significance

• Bridging Formalisms: The paper successfully bridges the gap between the KMOC formalism (which is powerful for perturbative, large impact parameter scattering) and non-perturbative soft theorems (which apply to generic scattering). It proves that the "in-in" expectation value approach can capture non-perturbative soft physics.
• Universality of Soft Radiation: It reinforces the idea that soft radiation (memory) is a universal feature of gauge theories and gravity, determined primarily by asymptotic symmetries and soft theorems, rather than the specific dynamics of the hard interaction.
• Limitations of Perturbative Approaches: The work highlights the specific challenges in applying perturbative scattering amplitude techniques to gravitational memory in the presence of non-linear effects (Christodoulou memory), suggesting that a complete description of gravitational memory may require more than just the $S$-matrix of the hard process.
• Future Directions: The authors suggest that their framework could be used to investigate the emergence of the logarithmic soft theorem (subleading order) non-perturbatively, a topic where current KMOC and saddle-point methods have not yet provided a complete derivation.

In summary, the paper demonstrates that while the KMOC formalism is typically restricted to small-angle scattering, its core machinery can be stretched to compute classical soft radiation for large deflection scattering by exploiting the universality of soft theorems and saddle-point extremization, successfully reproducing known results for electromagnetism and identifying specific structural hurdles for gravity.