IR side of bounds on Theories with Spontaneously Broken Lorentz Symmetry

This paper derives analyticity bounds for theories with spontaneously broken Lorentz symmetry by expressing them in terms of low-energy kinematical quantities, demonstrating that gapped excitations must propagate slower than gapless ones at low momenta.

The Gist

The Big Picture: Reading the Future in the Past

Imagine you are a detective trying to solve a crime. You can't see the criminal (the "UV" or high-energy physics), but you can see the footprints they left behind in the mud (the "IR" or low-energy physics).

In the world of physics, there is a golden rule: The universe is consistent. If you know the rules of the universe at the highest energies (the "source code"), you can predict what the footprints should look like at low energies. Conversely, if you see footprints that look weird, you know the source code must be broken or impossible.

For a long time, physicists have used a tool called Analyticity Bounds to check these footprints. Think of these bounds as a "speed limit sign" for the universe. If a theory predicts that particles can travel faster than light or move in impossible ways, the math breaks down, and the theory is rejected.

The Problem:
Usually, these speed limit signs are written in a very complicated language that only works if the universe is perfectly symmetrical (Lorentz invariant). But our universe isn't always perfect. In things like superfluids (super-cold liquids) or the early universe, this symmetry is "broken." The old speed limit signs don't work well here. It's like trying to use a map of a flat city to navigate a mountain range; the rules change.

The Goal:
The authors of this paper wanted to find a new way to write these speed limit signs specifically for these "broken symmetry" worlds. They wanted to translate the complex math into a simple rule about how fast things move in the low-energy world.

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The Analogy: The Noisy Dance Floor

To understand their solution, imagine a crowded dance floor (the Superfluid).

1. The Gapless Dancers (The Phonons):
   Imagine a group of dancers who can move freely across the floor without any resistance. They are "gapless." In a normal, symmetrical universe, they would move at the speed of light. But in this superfluid, they move slower, at a specific "sound speed" (let's call it 0.5).

2. The Gapped Dancers (The Heavyweights):
   Now, imagine a second group of dancers who are wearing heavy lead boots. They can't move until they get a big push. They have a "mass gap." In the old theories, it was hard to figure out how fast these heavy dancers could go without breaking the laws of physics.

3. The Kinetic Mixing (The Tangled Rope):
   The authors introduced a twist: The light dancers and the heavy dancers are tied together by a rope. If one moves, the other is dragged along. This is called kinetic mixing. Because they are tied, the heavy dancers can't just sit still; they are forced to move, but their movement is a mix of the light dancer's speed and their own heavy inertia.

The Discovery: The "Acoustic Vector"

The authors realized that looking at the speed of the light dancers (Phase Velocity) or the heavy dancers (Group Velocity) separately didn't give a clear answer. It was like trying to guess the speed of a car by looking only at the wheels or only at the engine; neither tells the whole story when they are tangled.

Instead, they invented a new tool called the Acoustic Vector.

• The Metaphor: Imagine the dance floor has a hidden grid of invisible strings (the Acoustic Metric) that guide how the dancers move. The Acoustic Vector is like a compass needle that points exactly where the energy is flowing, taking into account both the light dancers and the heavy dancers tied together.

The New Rule (The Breakthrough):
The authors found that the "speed limit sign" for these broken-symmetry theories can be written as a simple, single rule:

"The heavy dancers (gapped excitations) must never move faster than the light dancers (gapless excitations) when they are moving slowly."

In technical terms, the "Acoustic Vector" must stay inside a specific cone (a "sound cone") defined by the speed of the light dancers.

Why This Matters

1. It's Simpler: Instead of doing pages of complex math to check if a theory is valid, you just check this one speed rule. If the heavy stuff tries to outrun the light stuff at low speeds, the theory is invalid.
2. It's Surprising: For a long time, physicists thought the rule was just "nothing can go faster than light." But this paper shows that in these special systems, the rule is actually "nothing can go faster than the sound of the system."
3. It Connects the Dots: It proves that the deep, high-energy rules of the universe (which we can't see) are directly encoded in the simple, low-speed behavior of these heavy particles.

The Conclusion

The paper is like finding a new, simpler rule for a complex game.

• Old Rule: "Don't break the math." (Hard to check).
• New Rule: "Don't let the heavy stuff outrun the light stuff." (Easy to check).

By using this new "Acoustic Vector" compass, the authors showed that even in a messy, broken-symmetry world, the universe still demands a specific order: The slow, heavy things must respect the speed of the light, fast things. If they don't, the theory is just a fantasy, not a description of reality.

Technical Summary

1. Problem Statement

The paper addresses the challenge of characterizing analyticity bounds (positivity bounds) in Effective Field Theories (EFTs) where Lorentz symmetry is spontaneously broken.

• Context: In standard relativistic QFT, analyticity of scattering amplitudes (derived from UV assumptions like unitarity, locality, and microcausality) imposes constraints on the coefficients of low-energy interactions (Wilson coefficients). These bounds are often interpreted in the IR as forbidding superluminal propagation (faster-than-light signals).
• The Gap: While this UV/IR connection is well-understood for Lorentz-invariant theories, it is difficult to derive and interpret analyticity bounds for theories with broken Lorentz symmetry. Previous attempts to interpret these bounds via IR kinematics (e.g., phase or group velocities) have been inconclusive or failed to provide momentum-independent constraints for gapped systems.
• Specific Challenge: The authors focus on EFTs with a mass gap (gapped excitations). In such systems, the standard phase velocity diverges at zero momentum, and group velocity definitions become ambiguous or fail to capture the analyticity constraints in a sharp, momentum-independent way.

2. Methodology

The authors employ a combination of EFT construction, dispersion relation analysis, and geometric optics approximations using an acoustic metric formalism.

• Model System: They study a conformal superfluid in $D=3$ spacetime dimensions coupled to a $U(1)$ gauge field. This system is chosen because:
  • It spontaneously breaks Lorentz symmetry (via a finite charge density background).
  • Analyticity bounds for its coefficients were previously derived in Ref. [27] using correlators of conserved currents ($\langle JJ \rangle$).
  • The coupling to the gauge field introduces kinetic mixing between the scalar (superfluid) and gauge modes, generating a mass gap for all physical excitations.
• Lagrangian Analysis: They start with the EFT Lagrangian (Eq. 1) containing higher-dimensional (HD) operators with coefficients $c_2, c_3$ and gauge kinetic terms. They derive the quadratic Lagrangian for fluctuations ($\pi$ and $A_\mu$) around a background $\chi = \mu t$.
• Dispersion Relations: By solving the determinant of the kinetic matrix ($\det K = 0$), they derive the dispersion relations for the physical modes. Crucially, they identify a gapped mode $\omega_-^2(p)$ whose low-energy behavior depends on the HD coefficients.
• Acoustic Metric Formalism: Instead of relying solely on phase ($v_p$) or group ($v_g$) velocities, they introduce the acoustic metric $Z^{\mu\nu}(p)$.
  • They define an acoustic vector $N^\mu = Z^{\mu\nu} p_\nu$.
  • They define a specific velocity $v = N/N^0$ derived from this vector.
  • This formalism allows them to handle both the mass gap ($C^2 \neq 0$) and momentum-dependent dispersion (dispersivity) simultaneously.

3. Key Contributions

The paper makes three primary technical contributions:

1. Reformulation of Bounds via Acoustic Vector: The authors demonstrate that the known analyticity bounds on Wilson coefficients can be expressed as a momentum-independent condition on the velocity $v$ associated with the acoustic vector $N^\mu$.

   • Specifically, the bound requires that for momenta below the mass gap, the velocity $v$ must not exceed the sound speed of the gapless mode ($c_s^2 = 1/2$ in $D=3$).
   • Condition: $v^2 \leq c_s^2 = 1/2$ for low momenta.

2. Distinction from Phase/Group Velocity: A critical finding is that neither the phase velocity nor the group velocity can satisfy a sharp, momentum-independent bound that reproduces the analyticity constraints.

   • The phase velocity diverges at $p \to 0$ for gapped modes.
   • The group velocity and phase velocity differ from the acoustic velocity $v$ due to dispersivity and the gap.
   • Only the acoustic velocity $v$ (derived from $N^\mu$) provides a consistent geometric interpretation of the bounds across the entire low-energy regime.

3. Equivalence to Current Correlators: The authors explicitly show that imposing the condition $v^2 \leq 1/2$ on the acoustic vector reproduces the exact analyticity bounds derived in Ref. [27] using $\langle JJ \rangle$ correlators. This establishes a direct bridge between the "current correlator" approach (UV/analyticity) and the "propagation of gapped modes" approach (IR/kinematics).

4. Results

• The Bound: For the conformal superfluid coupled to a gauge field, the analyticity bound on coefficients $\tilde{c}_2, \tilde{c}_3$ (where $\tilde{c} = c/(b+d)$) is:
  $$\tilde{c}_2 (1 - \xi^2) - \tilde{c}_3 \geq - \left( \frac{1 - \xi^2}{2} \right)^2 \frac{1}{\xi^2}$$
  where $\xi \in [0,1]$ is a parameter related to the momentum scale.
• IR Interpretation: This inequality is equivalent to requiring that the acoustic vector $N^\mu$ lies inside the sound-cone of the gapless excitation (defined by speed $c_s = 1/\sqrt{2}$).
  • Physically, this means gapped excitations must propagate slower than gapless excitations at low momenta.
• UV/IR Connection: The velocity $v$ interpolates between $c_s^2 = 1/2$ at $p \to 0$ and $c^2 = 1$ (speed of light) at $p \to \infty$. The analyticity bound ensures the velocity does not "overshoot" the gapless sound speed in the low-momentum regime before asymptoting to the UV speed.

5. Significance

• New IR Characterization: The paper provides a robust method to characterize UV analyticity constraints purely in terms of IR kinematic quantities, even in the presence of a mass gap and broken Lorentz symmetry.
• Refining Causality: It clarifies that "causality" in these systems is not simply about avoiding superluminality relative to the speed of light ($c=1$). Instead, the constraint is relative to the gapless sound speed of the medium. Propagation outside the $c_s=1/\sqrt{2}$ soundcone is not a causality paradox but is forbidden by analyticity.
• Generalizability: The use of the acoustic metric and acoustic vector suggests a pathway to analyzing more complex systems (e.g., cosmological models, modified gravity) where Lorentz symmetry is broken. It highlights that the "speed" relevant for bounds is the one defined by the effective geometry of the excitations, not necessarily the standard phase or group velocity.
• Resolution of Ambiguity: It resolves the difficulty of applying positivity bounds to gapped theories by identifying the correct velocity variable ($v$) that remains well-defined and physically meaningful at zero momentum.

In summary, the work successfully translates abstract analyticity bounds into a concrete geometric condition: the acoustic vector of gapped modes must remain within the sound-cone of the gapless modes, offering a deeper understanding of how UV consistency manifests in the IR dynamics of non-relativistic or broken-symmetry systems.