Positivity with Long-Range Interactions

This paper introduces a new class of infrared-finite, analytic, and unitary scattering amplitudes labeled by experimental energy resolution, which enable the derivation of infrared-safe positivity bounds for effective field theories with long-range forces in four dimensions, as demonstrated through explicit bounds for pions coupled to electromagnetism and gravity.

The Gist

The Big Problem: The "Infinite Fog"

Imagine you are trying to listen to a conversation between two people (particles) in a very quiet room. You want to understand exactly what they are saying to learn about the laws of physics.

However, in our universe, there are two invisible, long-range forces: Electromagnetism (like light and radio waves) and Gravity. These forces are like a constant, low-level hum or a thick fog that never goes away.

In the past, when physicists tried to calculate the "conversation" between particles, this fog made the math break down. The calculations would result in "infinity" or "zero," effectively drowning out the actual message. It was like trying to hear a whisper in a hurricane; the signal was lost in the noise. Because of this, for decades, physicists couldn't use a powerful tool called Positivity Bounds to test theories in our 4-dimensional universe when these forces were present. They had to pretend the forces didn't exist or move to imaginary universes to do the math.

The Solution: The "Smart Filter"

The authors of this paper invented a clever new way to look at the data. They realized that the "fog" isn't random noise; it's actually a predictable pattern based on how big your "microphone" (the detector) is.

Think of it like this:

• The Old Way: You try to record the conversation with a microphone that is infinitely sensitive. It picks up every single rustle of air, every distant wind, and the recording becomes a chaotic mess of static.
• The New Way (Stripped Amplitudes): The authors say, "Let's build a microphone with a specific size." If we know exactly how big our microphone is, we can mathematically subtract the predictable background hum that it always picks up.

They call this new, cleaned-up signal a "Stripped Amplitude."

• They take the messy, infinite signal.
• They mathematically "peel off" the infinite fog (using a formula developed by Stephen Weinberg).
• What's left is a clean, finite signal that still contains all the important physics but is free of the mathematical infinities.

The "Detector" Metaphor

The paper introduces a very physical concept: Resolution.

Imagine you are at a concert.

• If you have perfect hearing (infinite resolution), you hear every single sound, including the sound of the air molecules vibrating. This is too much information; it's overwhelming.
• If you have realistic hearing (finite resolution), you hear the music clearly, but you ignore the tiny, individual air vibrations.

The authors define their "Stripped Amplitude" based on this realistic hearing. They say, "Let's assume our detector has a limit. It can't hear sounds softer than a certain volume." By defining this limit, the math stops breaking. The "fog" becomes a manageable, calculable correction rather than a disaster.

Why This Matters: The "Rule of Law" for Physics

Once they cleaned up the signal, they could apply Positivity Bounds.

Think of Positivity Bounds as a Rule of Law for the universe. Just as a judge can look at a crime scene and say, "This story doesn't add up; the suspect is lying," physicists use these bounds to look at a theory and say, "This theory violates the laws of causality or energy conservation; it's impossible."

• Before this paper: If a theory included gravity or electromagnetism, the judge (the math) would throw up its hands and say, "I can't decide, the evidence is infinite."
• After this paper: The judge puts on the "Smart Filter," clears the fog, and says, "Okay, now I can see. This theory is valid," or "This theory is broken."

The "Sweet Spot" Discovery

One of the most interesting findings is about the size of the detector.

• If the detector is too sensitive (trying to hear the tiniest whisper), the gravity/electromagnetism effects become so huge that they swamp the interesting physics, and the bounds become useless again.
• If the detector is too insensitive, you miss the long-range effects entirely.
• The Sweet Spot: There is a "Goldilocks" zone where the detector is just right. In this zone, the long-range forces (gravity/EM) act as a small, calculable correction to the rules, rather than breaking them.

Real-World Example: Pions

To prove their idea works, they applied it to Pions (particles that make up atomic nuclei).

• They calculated how pions scatter off each other when electromagnetism and gravity are present.
• They found that the "Rules of Law" (Positivity Bounds) still hold, but they are slightly tweaked by the presence of these forces.
• For example, they found a specific number (related to gravity) that must be greater than a certain negative value. This is a concrete, testable prediction that wasn't possible before.

Summary

1. The Problem: Long-range forces (gravity/EM) make physics math explode into infinity, stopping us from testing theories.
2. The Fix: Create a "Stripped Amplitude" by mathematically removing the predictable "fog" based on the size of our detector.
3. The Result: We can now apply strict "Rule of Law" tests (Positivity Bounds) to theories in our real universe (4D space) even when gravity and electricity are involved.
4. The Takeaway: The universe has strict rules, and even with the "fog" of gravity and light, we can finally see them clearly if we use the right mathematical lens.

This paper essentially gives physicists a new pair of glasses that allows them to see the fundamental laws of the universe clearly, even when the universe is trying to hide them behind a curtain of long-range forces.

Technical Summary

1. Problem Statement

The paper addresses a fundamental obstacle in the S-matrix bootstrap program and Effective Field Theory (EFT) positivity bounds within four spacetime dimensions ($D=4$): the presence of long-range forces mediated by massless particles (photons and gravitons).

• The IR Divergence Issue: In theories with long-range interactions, standard scattering amplitudes suffer from infrared (IR) divergences. As the IR regulator is removed, these amplitudes vanish (trivialize) due to the emission of infinite soft photons/gravitons.
• Breakdown of Positivity: Traditional positivity bounds rely on dispersion relations derived from analytic, unitary, and crossing-symmetric amplitudes. Because standard amplitudes vanish in $D=4$ with massless mediators, the standard derivation of positivity constraints (which link UV completability to IR Wilson coefficients) fails.
• The $1/t$ Pole: Specifically, the exchange of massless particles introduces a $1/t$ pole in the $t$-channel. In $D=4$, this pole renders the dispersive integrals (arcs) non-integrable or leads to "non-decoupling" effects where gravitational corrections appear to persist even as the coupling $G \to 0$, creating a paradox where the unperturbed theory ($g^2 \ge 0$) seems to contradict the perturbed theory.
• Previous Limitations: Previous attempts to resolve this involved working in $D>4$ or AdS space, or introducing hard IR cutoffs, but these approaches either lacked physical meaning in $D=4$ or failed to restore unitarity and analyticity simultaneously.

2. Methodology

The authors introduce a new framework based on "Stripped Amplitudes" ($M_E$) tailored for finite-size experimental detectors.

A. The Scaling Limit and Detector Resolution

The authors define an experimental energy resolution scale $E$ (inverse proper size of the detector). They consider a specific scaling limit where the coupling constants ($\alpha$ for QED, $G$ for gravity) and the resolution $E$ are taken to zero, but specific composite parameters are held fixed:

• Electromagnetism: $\alpha_E \equiv \alpha \log(M/E) = \text{fixed}$
• Gravity: $G_E \equiv G M^2 \log(M/E) = \text{fixed}$
  Here, $M$ is a hard scale (e.g., particle mass or UV cutoff). This limit isolates the universal long-distance effects common to all sufficiently large detectors.

B. Definition of Stripped Amplitudes ($M_E$)

The core innovation is the definition of the stripped amplitude:
$$M_E \equiv \lim_{\epsilon \to 0^+} \frac{M_\epsilon}{W}$$
where:

• $M_\epsilon$ is the standard regulated amplitude.
• $W$ is the Weinberg soft exponential, which contains the universal IR-divergent factors (e.g., $W_{QED} \sim \exp(\alpha \log \dots)$).
• By factoring out $W$, the authors remove the IR divergences analytically.

C. Properties of $M_E$

The authors prove that $M_E$ possesses the necessary properties for the bootstrap program:

1. IR Finiteness: By construction, $M_E$ is finite as $\epsilon \to 0$.
2. Analyticity & Crossing Symmetry: $M_E$ is an analytic function of Mandelstam variables with correct crossing symmetry.
3. Regge Behavior: $M_E$ satisfies appropriate high-energy bounds (Froissart-Martin type) required for dispersion relations.
4. Unitarity: In the scaling limit (1.1), $M_E$ satisfies a form of unitarity. The authors show that $M_E$ effectively coincides with Faddeev-Kulish amplitudes (which use dressed asymptotic states) in this limit.
5. Functional Unitarity: Even when the $1/t$ pole creates issues for the amplitude itself, the authors define functionals (dispersive integrals smeared with a function $\psi(q)$) that satisfy positivity. The $1/t$ singularity contributes only a finite, calculable amount of "negativity" that is parametrically suppressed by the scaling limit.

3. Key Contributions

1. Resolution of the $D=4$ Positivity Paradox: The paper demonstrates that positivity bounds do exist in $D=4$ with long-range forces, provided one works with IR-finite stripped amplitudes rather than divergent standard amplitudes.
2. Finite, Calculable Corrections: Unlike previous analyses where gravity seemed to induce non-decoupling negative terms (e.g., $g^2 M^4 + \text{const} \ge 0$), the authors show that the gravitational corrections are finite, calculable, and smoothly decouple as $G \to 0$. The "negativity" is controlled by the parameter $G_E$ and is subleading to the leading-order Wilson coefficients.
3. Generalized Dispersion Relations: They derive twice-subtracted dispersion relations for $M_E$ that are free of IR divergences. These relations connect the UV physics to the IR Wilson coefficients, including all orders in the effective couplings $\alpha_E$ and $G_E$.
4. Explicit Bounds in Pion EFT: The framework is applied to the scattering of pions coupled to electromagnetism and gravity, providing explicit, improved bounds on Wilson coefficients.

4. Key Results

The authors apply their formalism to two specific scenarios:

A. Pions with Electromagnetism ($\pi^+\pi^0$ and $\pi^+\pi^+$)

• $\pi^+\pi^0$ Scattering: Since $\pi^0$ is neutral, there is no $1/t$ pole. The bounds on Wilson coefficients (e.g., $c_{2,0}, c_{3,1}$) receive logarithmic corrections controlled by $\alpha_E$. The bounds smoothly interpolate between the standard short-range result and the long-range corrected result.
• $\pi^+\pi^+$ Scattering: This case involves the problematic $1/t$ pole.
  • The authors derive bounds of the form:
    $$\bar{c}_{2,0} M^4 + (59.9 \alpha_E)^2 + O(\alpha) > 0$$
  • They show that the $1/t$ pole generates a finite, positive contribution proportional to $\alpha_E^2$, which tightens the bound rather than destroying it.
  • The bounds are continuous as $\alpha_E \to 0$, recovering the standard tree-level results.

B. Pions with Gravity ($\pi^0\pi^0$)

• The authors derive bounds for massless scalars coupled to gravity.
• The bound on the leading Wilson coefficient $\hat{c}_{2,0}$ takes the form:
  $$\hat{c}_{2,0} M^4 > -65.7 \left[ 8\pi G_E - \frac{(8\pi G_E)^2}{2\pi^2} \right] + O(G)$$
• Significance: This explicitly resolves the "non-decoupling" puzzle. The negative contribution is proportional to $G_E$ (which depends on the detector resolution $E$). As $G \to 0$ (with fixed $E$), the correction vanishes, restoring the standard bound $\hat{c}_{2,0} \ge 0$. The "negativity" is not a fundamental obstruction but a calculable, finite effect of the long-range force.

5. Significance and Implications

• Restoration of the Bootstrap in $D=4$: This work removes the primary theoretical barrier preventing the application of the S-matrix bootstrap to realistic theories containing electromagnetism and gravity in four dimensions.
• Physical Interpretation of $E$: The energy resolution $E$ is not just a mathematical regulator but a physical parameter representing the detector's sensitivity. The results imply that the space of consistent EFTs depends on the experimental resolution, a concept that bridges the gap between theoretical constraints and experimental reality.
• Parametric Control: The framework provides a systematic expansion in $G/GE = 1/\log(M/E) \ll 1$. This allows for high-precision constraints on Wilson coefficients by choosing detectors with optimal resolution (balancing the need for small $E$ to suppress corrections against the need for large $E$ to avoid strong coupling regimes).
• Future Directions: The paper opens the door for numerical bootstrap analyses in $D=4$ with long-range forces, potentially leading to tighter constraints on particle physics models and quantum gravity scenarios. It also suggests that "optimal" detectors for constraining irrelevant operators are those with finite, non-zero resolution, rather than idealized infinite-resolution detectors.

In summary, the paper successfully constructs a rigorous, IR-finite framework for deriving positivity bounds in the presence of long-range forces, proving that such forces do not invalidate the connection between UV causality and IR EFT coefficients, but rather modify them in a calculable, finite, and physically meaningful way.