UV/IR Relations in QFT and Gravity

• UV/IR relations are correlations between short-distance and long-distance physics that reveal deep structural links in quantum gravity, QFT, and string theory.
• They arise through mechanisms such as forward-limit graviton exchanges, holographic entropy bounds, modular invariance, and celestial amplitudes to constrain analytic scattering behavior.
• This phenomenon challenges traditional Wilsonian decoupling and provides insights into gravitational scattering, inflation models, and nonlocal field theories.

UV/IR relations are correlations between ultraviolet and infrared data in quantum field theory, gravity, and string theory that persist despite the expectation of Wilsonian decoupling. In the literature considered here, they arise when forward-limit graviton singularities constrain Regge asymptotics, when holography and the species bound correlate $\Lambda_{\rm UV}$ and $\Lambda_{\rm IR}$, when modular invariance exchanges the $\tau_2\to0$ and $\tau_2\to\infty$ regions of the string worldsheet, when celestial Mellin transforms convert UV softness into analytic constraints in boost space, and when black-hole or nonlocal wave phenomena make late-time or long-distance observables depend on UV structure (Herrero-Valea et al., 2022, Castellano et al., 2021, Aoufia et al., 11 Mar 2026, Arkani-Hamed et al., 2020, Ho et al., 2022).

1. Taxonomy of UV/IR relations

The literature does not present a single universal UV/IR formula. Rather, it exhibits several technically distinct mechanisms that all correlate short-distance and long-distance physics. In some cases the relation links cutoffs; in others it links asymptotic behavior of amplitudes, analytic structure, or the validity of EFT descriptions. A useful organizing principle is whether the relation is imposed by analyticity, by holography and entropy, by modular invariance, or by the causal and spectral structure of the theory.

Setting	UV/IR mechanism	Representative consequence
Gravitational scattering	IR graviton singularities constrain UV asymptotics	Forward-limit positivity becomes UV-sensitive
Species/holography	$\Lambda_{\rm UV}$ correlated with $\Lambda_{\rm IR}$	Towers become light as $\Lambda_{\rm IR}\to0$
String worldsheet	Modular invariance exchanges UV and IR on $\mathcal{F}$	Universal scaling of $\Lambda$, $g$, and $\Lambda_{\rm IR}$0
Celestial amplitudes	Mellin transform ties UV softness to pole structure	Meromorphy with poles on negative even integers
Black-hole evaporation	UV cutoff or higher-derivative terms affect late-time IR flux	Hawking radiation becomes cutoff-sensitive
Nonlocal or instantaneous models	Dispersion or noncommutativity mixes high momentum and low energy	Emergent IR poles, turbulence, or superluminal fronts

A recurring misconception is that UV/IR relations are relevant only in quantum gravity. The papers considered here show broader scope: the phenomenon also appears in celestial amplitude theory, noncommutative field theory, wave propagation with nontrivial dispersion, and Lorentz-invariant but instantaneous two-dimensional models (Arkani-Hamed et al., 2020, Craig et al., 2019, Ito et al., 2022, Dubovsky et al., 2011).

2. Forward-limit graviton exchange and dispersive UV/IR matching

In gravitational $\Lambda_{\rm IR}$1 scattering, the central obstruction is infrared singular behavior in the forward limit. For elastic scattering of identical states with Mandelstam variables satisfying $\Lambda_{\rm IR}$2, the amplitude behaves in the IR as

$\Lambda_{\rm IR}$3

so the usual forward-limit positivity analysis based on twice-subtracted dispersion relations becomes ill-defined as $\Lambda_{\rm IR}$4 (Herrero-Valea et al., 2022).

The key result is that cancellation of the $\Lambda_{\rm IR}$5 pole and $\Lambda_{\rm IR}$6 terms requires a nontrivial relation between UV and IR behavior. Writing the high-energy imaginary part as $\Lambda_{\rm IR}$7 and taking the proper double-scaling limit $\Lambda_{\rm IR}$8, the paper derives the central Laplace-transform constraint

$\Lambda_{\rm IR}$9

with $\tau_2\to0$0 and $\tau_2\to0$1 fixed by the IR amplitude. In this regime the asymptotic solution takes the meromorphic form

$\tau_2\to0$2

This identifies a precise UV completion requirement imposed by IR graviton exchange.

A second major consequence is that the infinite arc at $\tau_2\to0$3, عادة neglected in nongravitational positivity arguments, need not vanish. For the UV asymptotic amplitude considered in the paper, the arc contribution in the forward-limit functional is

$\tau_2\to0$4

where the pole term is fixed by the asymptotic imaginary part while the finite constant $\tau_2\to0$5 depends on the UV real part and is not constrained by unitarity. The twice-subtracted relation therefore becomes

$\tau_2\to0$6

so gravitational positivity bounds for the $\tau_2\to0$7 coefficient are not trustworthy unless $\tau_2\to0$8 is estimated or bounded. By contrast, bounds for higher powers such as $\tau_2\to0$9 and for $\tau_2\to\infty$0-derivatives remain robust because the graviton pole carries only $\tau_2\to\infty$1 dependence.

The paper’s QED-plus-gravity example makes the point explicit. In minimally coupled QED, assuming $\tau_2\to\infty$2 produces an apparent mismatch at order $\tau_2\to\infty$3 and would suggest new physics at $\tau_2\to\infty$4. A nonzero finite arc term instead cancels the offending contribution without introducing low-scale new physics. This suggests a “reverse bootstrap” in which IR data fixes UV real-part contributions.

3. Entropy bounds, species scales, and cosmological field-range bounds

A distinct class of UV/IR relations comes from holography. Applying the Covariant Entropy Bound to a $\tau_2\to\infty$5-dimensional EFT in a box of size $\tau_2\to\infty$6 gives the correlation

$\tau_2\to\infty$7

while quantum gravity further identifies the physical UV cutoff with the species scale

$\tau_2\to\infty$8

For a tower with masses $\tau_2\to\infty$9, this yields

$\Lambda_{\rm UV}$0

A single KK tower gives $\Lambda_{\rm UV}$1 in any dimension, whereas multiple towers or string-like towers allow $\Lambda_{\rm UV}$2; the paper also derives the lower bound $\Lambda_{\rm UV}$3 for the lightest tower from the Cohen–Kaplan–Nelson collapse constraint (Castellano et al., 2021).

Because realistic compactifications often contain several towers, the species scale must be computed iteratively. For contributing towers with density parameters $\Lambda_{\rm UV}$4 and characteristic scales $\Lambda_{\rm UV}$5, the effective tower is

$\Lambda_{\rm UV}$6

This algorithm is central in the analysis of AdS and dS limits, where identifying $\Lambda_{\rm UV}$7 with curvature reproduces AdS Distance Conjecture scaling and yields explicit lower bounds on the exponent.

In cosmology with a horizon, the same logic becomes a bound on field range. Taking $\Lambda_{\rm UV}$8 and using the UV/IR mixing ansatz

$\Lambda_{\rm UV}$9

together with $\Lambda_{\rm IR}$0 and $\Lambda_{\rm IR}$1, the EFT is valid only on a finite interval

$\Lambda_{\rm IR}$2

with

$\Lambda_{\rm IR}$3

Hence

$\Lambda_{\rm IR}$4

This challenges several inflationary models, including $\Lambda_{\rm IR}$5-attractors and modular-invariant inflation, because slow roll often persists all the way to the EFT boundary rather than ending within the allowed window (Cribiori et al., 3 Jul 2025).

The entropy-based UV/IR relation also underlies holographic dark energy. In the entropic-force construction, the CKN relation implies $\Lambda_{\rm IR}$6, and choosing the IR scale as the future event horizon gives

$\Lambda_{\rm IR}$7

so the Friedmann equation acquires the additional term associated with holographic dark energy (Li et al., 2010).

4. Worldsheet modular invariance, decompactification, and non-renormalization

On the string worldsheet, UV/IR relations are enforced by modular invariance itself. At one loop, modular and conformal invariance imply universal parametric scalings for vacuum energy, gauge couplings, and higher-derivative Wilson coefficients in a species or decompactification limit. For operators $\Lambda_{\rm IR}$8 in $\Lambda_{\rm IR}$9 spacetime dimensions, the one-loop string-frame coefficients scale as

$\Lambda_{\rm IR}\to0$0

where $\Lambda_{\rm IR}\to0$1 controls the infinite-distance limit and $\Lambda_{\rm IR}\to0$2 is the volume power of the emergent higher-dimensional EFT. Correspondingly, the vacuum energy and gauge couplings obey the parametric inequalities

$\Lambda_{\rm IR}\to0$3

which reproduce the form of holographic bounds and support the magnetic weak-gravity conjecture and the dark-dimension scenario (Aoufia et al., 11 Mar 2026).

The underlying mechanism is the modular $\Lambda_{\rm IR}\to0$4-transformation $\Lambda_{\rm IR}\to0$5, which exchanges the small-$\Lambda_{\rm IR}\to0$6 and large-$\Lambda_{\rm IR}\to0$7 regions of the torus integral. In the species limit, modular invariance forces the decompactifying internal sector to be non-chiral, implies vanishing spectral gap, and therefore enforces the emergence of an infinite tower of states. This is a worldsheet realization of UV/IR mixing: the UV end of moduli space constrains IR EFT data, and vice versa.

A complementary result is the non-renormalization theorem derived from the hidden clash between modular invariance and smooth decompactification:

Any four-dimensional closed string theory which can be realized as a geometric compactification from a higher-dimensional string theory will inherit the precise stricter internal cancellations of the higher-dimensional theory from which it is obtained despite the compactification.

The associated corollary states that in a modular-invariant theory with $\Lambda_{\rm IR}\to0$8 large extra dimensions, misaligned supersymmetry and UV/IR mixing eliminate all running for $\Lambda_{\rm IR}\to0$9 regardless of $\mathcal{F}$0; for $\mathcal{F}$1 they eliminate all running for $\mathcal{F}$2 and leave at most logarithmic running for $\mathcal{F}$3 (Abel et al., 2024).

The concrete gauge-threshold example on $\mathcal{F}$4 shows how this works. Field-theoretic KK intuition would predict power-law running above the compactification scale, but modular invariance pairs KK and winding sectors so that the putative $\mathcal{F}$5 term cancels. Above the first KK threshold one gets plateaux and localized “stringy pulses,” not sustained power-law running. At $\mathcal{F}$6, scale duality $\mathcal{F}$7 forces a vanishing derivative, so the effective running enters a fixed-point-like regime. The same logic applies to the Higgs mass and other quantities sensitive to quadratic or logarithmic divergences.

5. Celestial amplitudes and boost-space analyticity

Celestial amplitudes provide another UV/IR realization because the Mellin transform to boost eigenstates is sensitive to both high-energy and soft behavior. For four-particle scattering, the celestial amplitude reduces to a single Mellin integral,

$\mathcal{F}$8

where $\mathcal{F}$9 is the total boost weight. If the momentum-space amplitude admits the IR expansion

$\Lambda$0

then the celestial amplitude is meromorphic with poles only at negative even integers,

$\Lambda$1

and the residues are precisely the IR EFT coefficients (Arkani-Hamed et al., 2020).

The UV side is equally restrictive. Would-be poles on the positive real axis, familiar from power-law UV behavior in field theory, are erased in quantum gravity because exclusive amplitudes are exponentially suppressed by black-hole production,

$\Lambda$2

This yields analyticity for $\Lambda$3 and asymptotic growth

$\Lambda$4

so black-hole thermodynamics governs the large-positive-$\Lambda$5 behavior. In this sense, UV softness becomes a statement about meromorphy and pole placement in boost space.

Infrared divergences are isolated by hard/soft factorization. In both QED and gravity, the exclusive celestial amplitude factorizes into a conformally soft factor and a hard factor, with the soft piece given by a current-algebra correlator of Goldstone modes associated with asymptotic symmetries. Conformal Faddeev–Kulish dressing absorbs the IR divergences so that the dressed celestial amplitude equals the hard factor. A common misconception is that celestial amplitudes merely repackage standard amplitudes; the celestial construction instead makes UV completion, infrared structure, black-hole asymptotics, and EFT residues coexist in a single analytic object.

6. Horizons, nonlocal propagation, and toy models of UV/IR mixing

In black-hole evaporation, UV/IR relations appear as late-time sensitivity of Hawking radiation to UV physics. For a Schwarzschild black hole in the Unruh vacuum, the standard Planckian spectrum arises from Bogoliubov coefficients with exponentially blue-shifted precursors, $\Lambda$6. Imposing a UV cutoff on the Kruskal frequency therefore suppresses late-time radiation after the scrambling time

$\Lambda$7

and more generally

$\Lambda$8

In the absence of an explicit UV cutoff, higher-derivative couplings to the background generate exponentially large time delays and particle-creation amplitudes, so an IR time cutoff feeds back into an effective UV bound on near-horizon frequencies. The paper’s conclusion is explicit: Hawking radiation is not robust against certain UV and IR effects (Ho et al., 2022).

Wave propagation provides a different cautionary example. A common expectation is that the front velocity equals the UV limit of the phase velocity, $\Lambda$9. For parity-conserving higher-spatial-derivative theories this remains true, but explicit counterexamples show that front velocities need not coincide with the UV limit of phase velocity in general dispersion relations. Superluminal fronts can arise from IR or intermediate-scale structure even when the UV limit is luminal (Ito et al., 2022). This is a direct warning against treating superluminality as a purely UV question.

The two-dimensional “Lineland” models make UV/IR mixing exact at the level of the dispersion relation,

$g$0

Energies therefore become arbitrarily small at high spatial momentum. The consequences are striking: thermal equilibrium is impossible, every excitation is metastable toward decay into shorter-wavelength modes, and the system exhibits an indefinite turbulent cascade. Yet approximate decoupling can still hold in many low-energy observables, which shows that UV/IR mixing does not automatically eliminate all EFT usefulness (Dubovsky et al., 2011).

Noncommutative field theory supplies a final toy model. There UV divergences are transmogrified into infrared scales, Lorentzian Yukawa theory develops a new infrared pole accessible in the $g$1-channel, and the softly-broken noncommutative Wess–Zumino model illustrates how UV-finiteness and UV/IR mixing compete as soft terms are varied relative to the cutoff (Craig et al., 2019). This suggests that UV/IR mixing can sometimes act as an alternative to purely Wilsonian naturalness, although the associated Lorentz violation sharply limits direct phenomenological transfer.

Across these settings, UV/IR relations do not abolish short-distance/long-distance distinctions; rather, they reorganize them. Gravitational scattering shows that IR singularities can determine UV asymptotics, entropy bounds show that UV and IR cutoffs are jointly constrained, modular invariance shows that UV and IR are exchanged on the worldsheet, celestial amplitudes encode EFT residues and black-hole softness in one meromorphic structure, and horizon or nonlocal systems show that late-time or front behavior may remain UV-sensitive. The general lesson is that in gravity, string theory, and several nonlocal QFT analogues, ultraviolet and infrared physics are often coupled by exact structural principles rather than by phenomenological accident.