UV/IR Poles and the Renormalized-Amplitude Interface

A regulated loop amplitude is an intermediate mathematical object, not yet a physical prediction. Its Laurent poles can have ultraviolet or infrared origin; its finite coefficients depend on normalization, regulator, spin-state prescription, and analytic continuation. Off-shell Green functions also depend on gauge fixing, whereas a consistently regulated on-shell amplitude between physical states should pass gauge-parameter or Ward-identity checks. A reliable handoff to renormalization therefore carries the complete definition of the object and labels how each pole was diagnosed.

Required background. Anatomy of a Loop Integral supplies the singular-region diagnosis; Dimensional Regularization as an Amplitude Tool supplies the common $d=4-2\epsilon$ regulator; and LSZ Reduction: Poles, Residues, and Stable States explains which stable external-leg residues are part of the scattering-amplitude normalization.

Helpful background. Regulators, Cutoffs, and Continuum Limits distinguishes a regulator from a physical cutoff, while The 1PI Effective Action and Mean-Field Equations identifies the vertex functions from which counterterm conditions are imposed.

The regulated amplitude before subtraction

Fix an external-state basis. For a single-coupling expansion in which each additional loop raises the coupling degree by two, let $p_0$ be the leading coupling power and write

$$\mathcal A_{\mathrm{bare}}(\epsilon,\mu) =\sum_{L\ge0}g_0^{\,p_0+2L}\, \mathcal A_{\mathrm{bare}}^{(L)}(\epsilon,\mu),$$

with the loop measure and any factors of $4\pi$ and $e^{\gamma_E}$ stated, including which scale is called $\mu$. A theory with several independent couplings instead requires a multi-index expansion rather than this shorthand. The dimensionally continued measure, scale insertion, and gamma-function normalization are developed in Weinzierl 2022, §2.4.2, pp. 28–39. At fixed loop order,

$$\mathcal A_{\mathrm{bare}}^{(L)} =\sum_{k=-K_L}^{\infty}\epsilon^k\mathcal A_k^{(L)}.$$

These coefficients may be vectors or matrices in color and spin space. They still depend on bare masses and couplings. Calling this expression “renormalized” before the bare parameters and external field normalizations have been related to specified renormalized ones hides the main operation that remains to be done.

Dimensional regularization simultaneously controls ordinary UV and soft or collinear IR behavior but permits different continuations of internal and external spin states. Beyond the leading poles, that regularization-scheme choice can affect singular and finite coefficients; Catani gives an explicit QCD discussion in Catani 1998, §2.1, preprint pp. 2–3, PDF.

Distinguishing ultraviolet from infrared poles

There is no intrinsic typographical difference between $1/\epsilon_{\mathrm{UV}}$ and $1/\epsilon_{\mathrm{IR}}$: both labels describe the origin of the same analytic regulator pole. The distinction comes from an independent diagnostic.

• Evaluate the relevant Green function at a nonexceptional Euclidean off-shell point. This removes on-shell soft and collinear singularities; remaining local poles are UV candidates.
• Introduce an auxiliary small mass or off-shellness. A pole that becomes a logarithm of that long-distance scale is IR.
• Analyze homogeneous momentum regions. Large loop momenta identify UV contributions; soft or collinear scalings identify IR contributions.
• Compare with known locality or factorization structure. UV counterterms are local polynomials in external momenta and masses, whereas on-shell IR operators correlate external directions and charges.

For UV-renormalized on-shell massless QCD amplitudes, the double and single IR poles factorize as operators on lower-order amplitudes; the one-loop structure and its color correlations are displayed in Catani 1998, §3, preprint pp. 5–7, PDF. That is a theory-specific factorization result, not permission to relabel every double pole “IR” without checking the external kinematics.

A scaleless integral is the sharp warning. It vanishes in dimensional regularization even though an auxiliary separation can expose canceling UV and IR poles. Zero therefore does not mean “no singular regions.”

The massive off-shell bubble makes the UV side explicit:

$$B(p^2)=\frac{i}{(4\pi)^2} \left[ \frac1\epsilon-\gamma_E+\log4\pi +\int_0^1\mathrm dx\, \log\frac{\mu^2}{\Delta(x)-i0} \right]+O(\epsilon).$$

Its pole coefficient is independent of $p^2$ and the masses and is therefore local. Putting a massless bubble on shell instead makes it scaleless: its zero combines UV and IR endpoint contributions rather than identifying either one. The same written $1/\epsilon$ acquires its label only from this region and kinematic information.

What must cross into renormalization

The receiving calculation needs more than the coefficients of negative powers of $\epsilon$. Preserve:

• the Lagrangian parameters and whether they are bare or already partially redefined;
• perturbative order and coupling normalization;
• external-state, color, helicity, and LSZ-residue conventions;
• gauge fixing and dimensional spin-state prescription;
• $d$, $\mu$, loop-measure prefactors, and the exact Laurent order retained;
• masses, invariants, momentum orientation, $i0$ prescription, and branch choices;
• a pole-by-pole UV/IR diagnosis and the diagnostic used;
• finite terms needed at the target order, including $O(\epsilon)$ terms that can multiply later poles;
• Ward-identity, crossing, symmetry, and independent numerical checks already performed.

Counterterm definitions, subtraction scheme, running parameters, anomalous dimensions, operator mixing, and matching are deliberately not chosen here. Those choices act on this preserved input. At observable level, the reusable validation matrix tracks virtual, real-radiation, scale, scheme, and numerical cross-checks; the present page supplies its amplitude-level inputs rather than a duplicate matrix.

A local-versus-long-distance check

Compare two scalar bubbles. A massive off-shell bubble has a logarithmic UV pole whose coefficient is independent of the external invariant; it can be canceled by a local counterterm. A massless on-shell bubble with $p^2=0$ is scaleless and vanishes, but splitting its endpoints reveals UV and IR contributions. The same bare symbol $1/\epsilon$ is therefore classified differently only after the kinematics and regions are stated.

For a genuine on-shell gauge amplitude the IR poles should be kept until they are combined with the appropriate real-emission or factorization terms. Removing them with UV counterterms would violate the long-distance physics. Conversely, leaving UV poles to “cancel against real radiation” confuses two distinct mechanisms.

Common pitfalls

Reporting only the finite remainder. A finite remainder is defined relative to a subtraction operator, scale, and scheme. Without those definitions it cannot reconstruct the regulated amplitude or be compared safely.

Dropping positive powers of epsilon too early. An $O(\epsilon)$ one-loop term can multiply a $1/\epsilon$ counterterm or subtraction operator at the next order and contribute to a finite result.

Treating external residues as optional normalization. For stable external particles the LSZ residue convention is part of the amplitude definition. Resonances and infraparticles require a different observable-level treatment rather than an ordinary external-leg factor.

Exercises

1. If an off-shell Euclidean version of a graph retains a momentum-independent $1/\epsilon$ pole, what evidence supports a UV label? The IR on-shell regions have been removed and the pole has the local form required of a counterterm.
2. Why retain $\mathcal A_1\epsilon$ when the present finite coefficient is $\mathcal A_0$? A downstream $1/\epsilon$ renormalization or factorization operator can turn it into a finite contribution.

Where the regulated amplitude goes next

• Renormalized Perturbation Theory and Counterterm Rules converts the preserved bare data into a finite, scheme-specified perturbative expansion.
• Infrared Cancellation, Regulators, and Matching Consistency uses the explicit UV/IR separation when full- and effective-theory amplitudes are compared.

References

• Catani, Stefano. “The Singular Behaviour of QCD Amplitudes at Two-Loop Order.” Physics Letters B 427 (1998): 161–171. doi:10.1016/S0370-2693(98)00332-3. Open PDF.
• Weinzierl, Stefan. Feynman Integrals. Cham: Springer, 2022. doi:10.1007/978-3-030-99558-4. Open PDF.