QFT for particle and nuclear physics

Choose this pathway if your question begins with quarks, leptons, gauge bosons, hadrons, or nuclei and must end with a quantitative prediction. Before reading farther, name the calculation:

I will calculate [observable] for [process or system] in [kinematic regime], to [stated accuracy], using [inputs], and compare it with [data or another calculation].

“Study QCD” is not yet a target. “Compute the fiducial dilepton invariant-mass spectrum at next-to-leading order,” “determine a semileptonic hadron form factor in a stated kinematic range,” and “predict a low-energy neutrino–deuteron breakup rate in pionless EFT” are targets. This pathway organizes the interfaces needed to reach such an answer; it is not a claim that one sequence completes the Standard Model or nuclear physics.

Required background. QED and Yang–Mills theory supplies gauge and color conventions; LSZ and tree amplitudes supplies external-state normalization; renormalization and the renormalization group connects scales; effective field theory and matching controls changes of degrees of freedom; and infrared-safe observables and synthesis connects amplitudes to measurable quantities. Enter at the first result in that chain that you cannot reproduce for your chosen process.

Begin with a named prediction

Take these three actions before choosing a specialist chapter.

1. Write the final mathematical object. Is it an amplitude, decay rate, partonic cross section, hadronic form factor, nuclear response, fiducial distribution, or likelihood? Include external states, kinematics, units, and desired perturbative or EFT order.
2. Mark every change of description. Write a chain such as “gauge theory → hard amplitude → Wilson coefficients → hadron or nuclear matrix element → detector bins.” Beside each arrow, name the theorem, matching calculation, fitted quantity, or response model that justifies it.
3. Test the earliest vulnerable step. Check a Ward or Slavnov–Taylor identity, an Abelian or decoupling limit, dimensions and group factors, and cancellation of the renormalization or factorization scale to the claimed order. The first failed check identifies where to begin.

A useful shared spine has five stages:

Stage	Object carried forward	Decisive question
Consistent theory	fields, representations, parameters, gauge fixing, and physical-state conditions	Is the action gauge consistent, and do unphysical polarizations cancel?
Short-distance prediction	renormalized amplitude or hard coefficient at a stated order	Are normalization, color, analytic continuation, and infrared conditions correct?
Scale separation	matched coefficients and their running	Which modes were removed, what is the expansion parameter, and does scale dependence cancel?
Long-distance physics	PDFs, hadron matrix elements, nuclear interactions, or response functions	Is the nonperturbative input defined in the same scheme and within a controlled domain?
Reported result	fiducial observable, response-folded expectation, and statistical model	Do the theory and measured objects have the same definition, bins, correlations, and validity range?

Use Standard Model Assembly and Consistency when the task requires the full field and parameter map rather than one gauge sector. Assembly is a consistency step, not a substitute for the long-distance or measurement layers.

Follow one interaction through four physical layers

A heavy neutral vector provides a running example. Let

$$\begin{aligned} \mathcal L_V={}&-\frac14 V_{\mu\nu}V^{\mu\nu} +\frac12 M^2V_\mu V^\mu\\ &+V_\mu\left(g_q\,\bar q\gamma^\mu q +g_\ell\,\bar\ell\gamma^\mu\ell\right). \end{aligned}$$

This is a schematic resolved-mediator interaction. If $V_\mu$ is a gauge boson, its charges, mass generation, anomalies, and additional states must be supplied by a consistent theory. At momentum transfer $Q\ll M$, its classical equation of motion gives

$$V_\mu =-\frac{g_qJ^q_\mu+g_\ell J^\ell_\mu}{M^2} +\mathcal O\!\left(\frac{\partial^2}{M^4}\right),$$

where $J_q^\mu=\bar q\gamma^\mu q$ and $J_\ell^\mu=\bar\ell\gamma^\mu\ell$. Substitution yields

$$\mathcal L_{\mathrm{EFT}} \supset -\frac{g_qg_\ell}{M^2} \left(\bar q\gamma^\mu q\right) \left(\bar\ell\gamma_\mu\ell\right).$$

The operator has dimension six, its coefficient has mass dimension $-2$, and the next local term is suppressed by $Q^2/M^2$. Classical elimination, operator completeness, and power counting are developed in Georgi 1993, pp. 209–219.

Matching is only the first interface. A general operator basis runs and mixes:

$$C_i(\mu)=U_{ij}(\mu,M)C_j(M), \qquad \mathcal A =\sum_i C_i(\mu)\, \langle f\lvert O_i(\mu)\rvert i\rangle.$$

The scale dependence of the coefficients must cancel that of the matrix elements through the calculated order. Conserved currents can protect particular combinations, but that must be established rather than assumed.

Layer	Degrees of freedom and object	New information required
Partonic	quarks, gluons, leptons; a hard amplitude or partonic cross section	renormalized parameters, color sums, perturbative order, and an infrared-safe definition
Hadronic	color-singlet external states or incoming hadrons	PDFs, fragmentation functions, form factors, spectral information, or other nonperturbative matrix elements
Nuclear	nucleons, pions when resolved, and nuclear states	nuclear interactions and currents, low-energy constants, regulator tests, and many-body truncation
Detector and inference	reconstructed objects, bins, and event counts	acceptance and migration, backgrounds, nuisance parameters, covariance or likelihood, and calibration

For a hard collision of hadrons $A$ and $B$, a typical factorized prediction is

$$\begin{aligned} \frac{d\sigma_{AB}}{dO} ={}&\sum_{a,b} f_{a/A}(\mu_F)\otimes f_{b/B}(\mu_F) \otimes \frac{d\hat\sigma_{ab}}{dO}(Q;\mu_R,\mu_F)\\ &+\mathcal O\!\left[ \left(\frac{\Lambda_{\mathrm{QCD}}}{Q}\right)^p \right]. \end{aligned}$$

The partonic coefficient is not the hadronic cross section: PDFs, scheme, and factorization scale are part of the leading-power result. The formula also does not prove factorization for every measurement; validity depends on the process, observable, inclusiveness, and soft and Glauber regions Collins, Soper, and Sterman 1989, §§2–5, pp. 1–37.

For an exclusive hadron transition the interface is instead

$$\mathcal A_{H_i\to H_f} = \sum_k C_k(\mu) \langle H_f\lvert O_k(\mu)\rvert H_i\rangle.$$

For a low-energy nuclear transition it may become

$$\mathcal A_{A\to A'} = \sum_k C_k(\mu) \left\langle\Psi_{A'}\left\lvert O_k^{(1\mathrm b)}+O_k^{(2\mathrm b)}+\cdots \right\rvert\Psi_A\right\rangle,$$

where one- and many-body currents must use the same power counting and regulator treatment as the nuclear states. Finally, a typical forward-folded expectation in reconstructed bin $r$ is

$$\lambda_r(\theta,\eta) =\mathcal L_{\mathrm{int}} \sum_t R_{rt}(\eta)\,\sigma_t(\theta)+b_r(\eta).$$

Here $\sigma_t$ is the truth- or fiducial-bin prediction, $R_{rt}$ is the response matrix, $b_r$ is the background, and $\eta$ denotes nuisance parameters. A likelihood built from $\lambda_r$ belongs to the inference layer; it cannot repair an undefined cross section or a missing nuclear matrix element.

Choose the branch by the unresolved physics

The branches are not mutually exclusive. Begin with the first one containing the missing interface, and combine them only when the named observable actually crosses them.

QCD, hadrons, and nuclear EFT

Choose this branch when the obstacle is color dynamics or the change from quarks and gluons to hadrons or nuclei.

• For $Q\gg\Lambda_{\mathrm{QCD}}$, begin with Perturbative QCD and Partons. Produce a hard coefficient or partonic observable together with the relevant collinear, transverse-momentum, threshold, or jet factorization statement.
• For a spectrum, resonance, form factor, parton distribution, or heavy-quark expansion, continue to Hadrons and Heavy Quarks. Identify whether each input comes from lattice QCD, dispersion theory, a fit, a sum rule, or an EFT, and preserve its scheme, correlations, and domain.
• For nuclei or very low-energy nucleons, continue to Nuclear and Few-Body EFT. Use pionless EFT when all resolved momenta are well below the pion mass; use chiral EFT when pion exchange is resolved. State the breakdown scale, regulator window, fitted low-energy constants, many-body truncation, and current operators. This organization is reviewed by Epelbaum, Hammer, and Meißner 2009, §§II–IV, pp. 1776–1817.

Use Confinement, Screening, and Mass Gaps in Controlled Regimes when the question itself concerns a confinement diagnostic or controlled mechanism. It is not a mandatory detour before every hadronic calculation, and a fitted matrix element is not a proof of confinement.

Electroweak, flavor, and neutrinos

Choose this branch when symmetry breaking, chiral currents, mixing, or lepton-sector propagation defines the observable.

• Electroweak Theory and the Higgs supplies mass eigenstates, weak currents, the Fermi limit, and input schemes. State whether the calculation uses a resolved $W$ or $Z$, a low-energy four-fermion interaction, or a pole or fiducial Higgs observable.
• Quark Flavor and CP supplies CKM factors, weak effective Hamiltonians, meson mixing, and CP-sensitive amplitudes. A decay combines $\sum_i C_i(\mu)\langle Q_i(\mu)\rangle$; neither factor is separately scheme independent. The matching–running–matrix-element separation is developed in Buchalla, Buras, and Lautenbacher 1996, §§2–3, pp. 1129–1150.
• Neutrino and Lepton Physics supplies neutrino masses, mixing and propagation, charged-lepton flavor violation, and Majorana probes. Keep production, propagation, interaction cross section, nuclear response, and detector reconstruction separate. An oscillation probability is not a neutrino–nucleus event prediction.

This branch can determine the short-distance current while the QCD or nuclear branch determines its matrix element. Rare hadron decays, beta decay, and neutrino scattering commonly require both.

Precision tests and portals

Choose this branch for a controlled comparison between a baseline prediction and a possible deformation.

1. Use Precision Standard Model: Observables and Inference to decide whether the compared quantity is a pole parameter, pseudo-observable, fiducial cross section, released likelihood, or combined fit. Fix the input scheme, overlap, correlations, and nuisance model first.
2. Use Consistent Extensions and Portals to test fields, charges, anomalies, vacuum structure, widths, interference, and decoupling. Use an EFT only when every relevant invariant transfer is below the heavy scale; resolve the mediator near a pole or threshold.
3. Compare theories at the same observable layer. A partonic new-physics amplitude cannot be compared directly with reconstructed counts, and a fitted Wilson coefficient needs its basis, scale, truncation, and likelihood.

Likelihood-ratio tests and nuisance profiling have asymptotic and boundary conditions; the standard formulae and hypotheses are given in Cowan, Cranmer, Gross, and Vitells 2011, §§2–3. This branch teaches a comparison method, not current bounds or a ranking of extensions.

Know where the prediction can fail

Perturbation theory. A small coupling is insufficient when logarithms of widely separated scales compensate it. Fixed order must then be reorganized or resummed. Near $\Lambda_{\mathrm{QCD}}$, perturbative running does not calculate the spectrum or hadron matrix elements.

Factorization. A theorem is specific to a process, observable, and hierarchy. Endpoint regions, non-global measurements, Glauber exchange, thresholds, or power corrections can require a new theorem or invalidate the chosen one. Scale variation estimates missing terms inside an assumed factorization; it does not prove factorization.

Effective theories. All relevant momenta and masses must remain below the breakdown scale. Near the heavy-vector pole the local interaction fails even if its coefficient is fitted precisely. Matching, running, and matrix elements must use compatible bases and schemes.

Hadronic and nuclear inputs. Carry forward regulator and scheme dependence, fitted data, finite-volume or discretization effects, and correlations. Nuclear EFT adds a many-body expansion and can require interactions to be iterated; naive relativistic loop counting need not survive that reorganization.

States and measurements. LSZ applies directly to stable asymptotic particles, not to resonances, confined partons, or detector objects. Hadronization, detector response, backgrounds, and statistical calibration are distinct approximations. Precision at the final layer cannot compensate for an uncontrolled earlier interface.

Worked checks

Match the heavy vector

Starting from the heavy-vector Lagrangian, derive the semileptonic coefficient, its mass dimension, and the leading low-energy correction.

Solution

At leading order in derivatives,

$$J_\mu=g_qJ^q_\mu+g_\ell J^\ell_\mu, \qquad \mathcal L_V=\frac12M^2V_\mu V^\mu+V_\mu J^\mu+\cdots.$$

The equation $M^2V_\mu+J_\mu=0$ gives $V_\mu=-J_\mu/M^2$. Therefore

$$\frac12M^2V^2+V\cdot J =-\frac{J^2}{2M^2}.$$

The cross term in $J^2$ is $2g_qg_\ell J_q\cdot J_\ell$, so

$$C_{q\ell}=-\frac{g_qg_\ell}{M^2}.$$

Each current has dimension $3$, the operator has dimension $6$, and $C_{q\ell}$ has dimension $-2$. The propagator expansion

$$\frac{1}{q^2-M^2} =-\frac1{M^2} \left(1+\frac{q^2}{M^2}+\cdots\right)$$

shows a relative correction of order $q^2/M^2$. The sign agrees with classical elimination for the interaction convention used here.

Carry a partonic result to a detector bin

An infrared-finite $d\hat\sigma_{q\bar q\to\ell^+\ell^-}/dm_{\ell\ell}$ is called a prediction for reconstructed dilepton counts. What is missing?

Solution

First convolve with PDFs in a declared factorization scheme:

$$\frac{d\sigma_{AB}}{dm_{\ell\ell}} =\sum_{a,b} f_{a/A}\otimes f_{b/B}\otimes \frac{d\hat\sigma_{ab}}{dm_{\ell\ell}}.$$

Fix the perturbative order, $\mu_R$, $\mu_F$, active flavors, electroweak inputs, and fiducial lepton and radiation definitions. Bin the particle-level prediction with the published measurement function, then forward-fold it:

$$\lambda_r =\mathcal L_{\mathrm{int}}\sum_tR_{rt}\sigma_t+b_r.$$

The response and background model add nuisance parameters and correlations; a stated likelihood then defines the inference. Partonic calculation, PDF convolution, fiducial definition, detector response, and likelihood are noninterchangeable inputs.

Combine branches for a neutrino–nucleus test

For a heavy-vector constraint from neutrino–deuteron breakup at momentum transfer well below both $M$ and $m_\pi$, give the minimal chain and its two primary validity tests.

Solution

The portal branch supplies a consistent mediator theory and $C(M)=-g_qg_\nu/M^2$. EFT running carries a complete operator basis to the hadronic scale. The electroweak and neutrino branch fixes the Standard Model current, neutrino state, flux, and interference.

Because $Q\ll m_\pi$, the nuclear branch uses pionless EFT for the deuteron states and one- and two-body currents. Its low-energy constants and truncation uncertainty enter the nuclear matrix element. The breakup distribution is then folded with flux, efficiency, resolution, backgrounds, and likelihood.

The leading tests are

$$\frac{Q^2}{M^2}\ll1, \qquad \frac{Q}{m_\pi}\ll1.$$

The first controls removal of the mediator; the second controls pionless EFT. The scale and scheme must also cancel between coefficients and nuclear matrix elements. No one branch supplies this complete prediction.

Leave with a usable calculation plan

Begin specialist work when you can state:

• the final amplitude, rate, distribution, response, or likelihood and its kinematic domain;
• fields, external states, gauge and normalization conventions, and order;
• every hierarchy, matching scale, running interval, and first omitted power;
• the factorization theorem or hadron/nuclear matrix element connecting short and long distances;
• the origin, scheme, correlations, and domain of each nonperturbative input; and
• the map to measured bins, including response, backgrounds, nuisance parameters, and uncertainty checks.

If the missing result is an amplitude or infrared cancellation, move to scattering calculations for phenomenology. For matching, mixing, or multiscale running, use renormalization and EFT for working researchers. For lattice, Monte Carlo, or reproducibility work, use computational field theory onboarding. Or return to the pathway guide.

References

• Buchalla, Gerhard, Andrzej J. Buras, and Markus E. Lautenbacher. “Weak Decays Beyond Leading Logarithms.” Reviews of Modern Physics 68 (1996): 1125–1244. DOI.
• Collins, John C., Davison E. Soper, and George Sterman. “Factorization of Hard Processes in QCD.” In Perturbative Quantum Chromodynamics, edited by A. H. Mueller, 1–91. Singapore: World Scientific, 1989. DOI.
• Cowan, Glen, Kyle Cranmer, Eilam Gross, and Ofer Vitells. “Asymptotic Formulae for Likelihood-Based Tests of New Physics.” European Physical Journal C 71 (2011): 1554. DOI.
• Epelbaum, Evgeny, Hans-Werner Hammer, and Ulf-G. Meißner. “Modern Theory of Nuclear Forces.” Reviews of Modern Physics 81 (2009): 1773–1825. DOI.
• Georgi, Howard. “Effective Field Theory.” Annual Review of Nuclear and Particle Science 43 (1993): 209–252. DOI.