Background Fields and Gauging

This chapter is organized by one question: what is done with the symmetry data? A current source probes local operators. A connection on a fixed bundle probes the symmetry globally. Gauging is the additional operation that quotients by transformations and sums or integrates over gauge data, thereby constructing a new theory. The symmetry of that new theory must then be determined again; it cannot be read off from the parent action alone.

Enter through a current source if the goal is a Ward identity or response kernel, through a background bundle if topology and holonomy matter, or through the gauging diagnostic if the immediate question is whether a proposed construction really defines a new gauge theory. These routes meet, but they are not one compulsory sequence. In particular, the response page branches from source calculus, while the gauging route passes through global background data.

The scope is ordinary zero-form symmetry. The chapter develops external sources, background connections, spurions, local counterterms, continuous and finite gauging, and the first residual or dual symmetries that follow. Detailed anomaly cancellation, gauge fixing, boundary gauge symmetry, interacting gauge dynamics, transport, higher-form gauging, and categorical post-gauging structure are continuations rather than assumptions here.

Helpful background. The Generating Functional supplies normalized source differentiation, while Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities supplies the global geometry of a connection. Neither is required merely to use this overview.

Parent volume. Symmetry and Gauge Structure

Jump to: choose an entry route · review the chapter

Enter by the operation you need

Reader goal	Entry and route	Capability at the end
Generate current insertions or connected current kernels	Quantum Currents, Improvements, and Conservation + The Generating Functional $\to$ Current Sources and Generating Functionals	Differentiate $W[a]$ with the source sign, time ordering, seagulls, and contact terms controlled
Formulate a symmetry probe on nontrivial spacetime topology	Current Sources + Bundle Connections $\to$ Coupling to Background Gauge Fields and Bundles	Specify $W[M;P,a]$ , its patching data, holonomies, and covariant Ward identity without making $a$ dynamical
Separate structural response from transport and scheme choices	Current Sources + Localized Transformations and Ward–Takahashi Identities $\to$ Spurions, Local Counterterms, and Symmetry Response	Distinguish exact, explicitly broken, and spurion-covariant source families and classify the local ambiguity of response
Decide whether a proposed “gauging” is only a background coupling	Background Gauge Fields + Symmetry, Gauge Redundancy, and Duality $\to$ Background Fields versus Dynamical Gauging	List the quotient, measure, action, sectors, weights, counterterms, and consistency conditions still missing
Construct a gauged theory and determine what symmetry remains	Background versus Gauging + Vector, Principal, and Associated Bundles $\to$ Gauging Continuous and Finite Symmetries $\to$ Residual, Quotient, and Emergent Dual Symmetries	Build the continuous or finite sum with global sectors included, then determine the faithful inherited and emergent symmetry

Three short diagnostics identify the most useful repair:

Source calculus. Ready: explain why $\delta W/\delta a_\mu$ inserts a current and why a second derivative can contain a seagull term, then enter Current Sources and Generating Functionals. Unsure: inspect whether the action itself depends quadratically on the source before identifying the second derivative with a two-current correlator. Repair: use The Generating Functional and Quantum Currents, Improvements, and Conservation.
Global geometry. Ready: distinguish a connection from one local potential and a flat connection from a trivial background, then enter Coupling to Background Gauge Fields and Bundles. Unsure: ask whether a flat connection can still have nontrivial holonomy around a noncontractible loop. Repair: use Bundle Connections and Vector, Principal, and Associated Bundles.
Physical operation. Ready: say whether a transformation acts on physical operators or identifies descriptions, then enter Background Fields versus Dynamical Gauging. Unsure: ask whether the proposed gauge field is an argument of the functional or an integration variable. Repair: use Symmetry, Gauge Redundancy, and Duality.

Fixed, transformed, and summed data

The following table is the chapter’s conceptual map. “Held fixed” means not summed or integrated during one evaluation of $Z$ ; “transformed or varied” means comparing neighboring or gauge-related source configurations to obtain identities. Reading across any row then answers what is fixed, what is compared, what is summed, and what conclusion is justified.

Status of symmetry-coupled data at each stage
Construction	Held fixed	Transformed or varied	Summed or integrated	What it establishes
Current source	A prescribed local `a`_μ and all other external sources	`a`_μ and operator sources under a localized symmetry variation	Only the original dynamical fields	Current insertions, connected kernels, and source-space Ward identities
Background connection	A bundle `P` → `M` and a connection `a` on it	Local representatives, transition functions, and charged sections coherently	Only the original dynamical fields	Holonomy-sensitive probes and globally meaningful covariant identities
Spurion family	Backgrounds together with transforming couplings such as `h`(`x`)	The whole source family	Only the original dynamical fields	Covariance between theories at different source values, not necessarily a symmetry at fixed sources
Gauged theory	Spectator backgrounds and the chosen gauging data	Gauge transformations become identifications	Gauge connections or finite bundles, matter fields, and all admitted sectors	A new operator algebra and state space, provided the sum is consistent
Post-gauging theory	The complete gauging data	Candidate inherited and dual transformations	No additional sum until another gauging is performed	The faithful residual, quotient, extended, or emergent symmetry

In local notation, a fixed background gives

Z_{\mathcal T}[M;P,a] = \int_{\text{fields on }P}\mathcal D\Phi\, e^{iS_{\mathcal T}[\Phi;P,a]}, \qquad W_{\mathcal T}=-i\log Z_{\mathcal T}.

Here $(P,a)$ is an argument of the functional. It is not integrated over. Source differentiation therefore changes the prescribed probe. Define the renormalized source-dependent insertion by $[\mathcal J_A^\mu(x;a)]_R \equiv[\delta S_{\mathcal T}/\delta a_\mu^A(x)]_R$ ; then

\frac{\delta W_{\mathcal T}}{\delta a_\mu^A(x)} = \left\langle[\mathcal J_A^\mu(x;a)]_R\right\rangle_a,

up to the source normalization and renormalization prescription declared on the leaf. At zero background this reduces to the chosen current representative. At nonzero background it can include source-dependent terms; for the charge-one scalar, $\mathcal J^\mu=j^\mu+2a^\mu\phi^\dagger\phi$ . Higher derivatives produce connected time-ordered distributions together with the contact and seagull terms forced by the full source-dependent action. This construction is developed in Schwartz 2014, § 14.3, pp. 261–264.

Gauging a continuous subgroup $H$ instead has the schematic form

\begin{aligned} Z_{\mathcal T/H}[M] = \sum_{[P_H]} \int_{\substack{\operatorname{Conn}(P_H)\times \operatorname{Fields}(P_H)\\/\mathcal G(P_H)}} \mathcal D a\,\mathcal D\Phi\, \exp i\bigl( S_{\mathcal T}+S_{\mathrm{gauge}} +S_{\mathrm{top}}+S_{\mathrm{ct}} \bigr). \end{aligned}

The formula is intentionally schematic: the admissible bundles, boundary conditions, quotient, measure, local gauge action, topological weights, counterterms, and regulator are part of the definition. Perturbative gauge fixing is a way to represent the quotient; it is not a replacement for these global choices. The local scalar and Yang–Mills constructions underlying this schematic sum are developed in Schwartz 2014, §§ 8.3–8.6, pp. 120–132.

For a finite group $H$ , there is no Lie-algebra-valued photon to integrate. On a closed spacetime, the corresponding groupoid sum is schematically

Z_{\mathcal T/H}[M] = \sum_{[P_H]\ \mathrm{flat}} \frac{e^{iS_{\mathrm{top}}[P_H]}} {|\operatorname{Aut}(P_H)|}\, Z_{\mathcal T}[M;P_H].

Thus the temporal sum enforces the gauge projection, possibly twisted by the chosen topological weight, while spatial bundles supply twisted sectors. Fixed flat backgrounds, the sum over them, and possible topological weights are distinguished explicitly in Gaiotto et al. 2015, § 2, pp. 5–10, arXiv PDF. Local Ward invariance is not enough: the exponentiated functional must also descend under large transformations and on the admitted nontrivial bundles.

Convention bridge

This chapter inherits the site’s mostly-minus $(+,-,\ldots,-)$ metric— $(+---)$ in four dimensions—natural units, Lorentzian $e^{iS}$ weight, Hermitian generators, and $+ip\cdot x$ Fourier transform. The following additions recur often enough to state once:

A background connection absorbs the coupling: $a=gA$ and $D=\mathrm d-ia$ . For an active transformation $\psi\mapsto U\psi$ , $a^U=UaU^{-1}-i(\mathrm dU)U^{-1}$ and $f=\mathrm da-ia\wedge a$ . In the Abelian case, $\psi\mapsto e^{i\alpha}\psi$ and $a\mapsto a+\mathrm d\alpha$ .
Holonomy uses $\mathcal P\exp(+i\oint a)$ . A local potential is never a substitute for the bundle, transition functions, or admitted boundary data.
Early source calculations may normalize $\mathcal Z[a]=Z[a]/Z[0]$ . Gauging pages retain relative sector normalizations and topological phases, so an overall normalization may no longer be discarded silently.
$W$ generates connected kernels. The ordinary effective action $\Gamma$ is one-particle irreducible with respect to the elementary-field sources that were Legendre transformed, not automatically with respect to the current source $a$ .
Source and Ward calculations are Lorentzian and time ordered unless stated otherwise. Compact torus sums used for finite gauging are Euclidean. A retarded or Kubo kernel requires an additional real-time prescription.
Every gauging statement must specify the spacetime dimension, global group, faithful action or kernel, charge normalization, bundle sectors, boundary conditions, coupling placement, and allowed local counterterms whenever these affect the result.

Exact chapter guide

1. Current Sources and Generating Functionals

Current Sources and Generating Functionals asks how a prescribed source encodes current observables. This is the chapter’s source-calculus entry. It starts from a renormalized quantum current and a normalized generating functional, derives the first and second source derivatives, and keeps the scalar seagull and distributional contacts visible. It also separates $W$ from an ordinary 1PI effective action. Afterward, you can reconstruct the complete connected source kernel and state which local terms came from the source-dependent action.

Prepare with Quantum Currents, Improvements, and Conservation and The Generating Functional. The 1PI Effective Action and Mean-Field Equations is useful if the connected/1PI distinction is the main issue. Continue either to the global-background page or directly to the response branch.

2. Coupling to Background Gauge Fields and Bundles

Coupling to Background Gauge Fields and Bundles replaces the local shorthand $a_\mu$ by a connection on a fixed bundle. This is the global-geometry foundation of the gauging route. It explains patching, curvature, holonomy, charged sections, and the covariant Ward identity, while keeping the bundle and connection external. Flat does not mean trivial, and a finite-group background need not possess any globally defined one-form. Afterward, you can specify the globally meaningful argument $W[M;P,a]$ and test whether local formulas patch consistently.

Prepare with the source page and Bundle Connections; Vector, Principal, and Associated Bundles supplies a deeper geometric repair. Continue to the background-versus-gauging diagnostic; the response page is also useful but is not a hard step on that route.

3. Spurions, Local Counterterms, and Symmetry Response

Spurions, Local Counterterms, and Symmetry Response treats an enlarged family of sources. This is the chapter’s independent response branch. A transforming coupling can make that family covariant even when one fixed numerical coupling explicitly breaks the symmetry. Complete response includes current-current distributions, seagulls, and time-ordering contacts. Allowed finite background-local counterterms shift local polynomial pieces but not arbitrary separated-point or nonanalytic content. Afterward, you can separate forced distributional terms, adjustable local terms, and genuinely nonlocal response without calling a time-ordered kernel a transport coefficient.

Prepare with the current-source page and Localized Transformations and Ward–Takahashi Identities. The global-background page is recommended when nontrivial bundles matter. Continue to Sources, Linear Response, and Kubo Formulae only after the additional real-time and state assumptions have been supplied.

4. Background Fields versus Dynamical Gauging

Background Fields versus Dynamical Gauging is the conceptual pivot. Its operational test is simple: a connection that remains an argument is a background; a gauge connection that is quotiented and integrated or summed is dynamical gauge data. Gauging also changes which charged insertions are local observables and which sectors belong to the theory. Afterward, you can inspect a proposed construction and list exactly which additional data are needed before it defines a gauged theory.

For the extended-operator specialization, the higher-form gauging process map tracks the same gate, sector sum, operator attachment, residual action, and dual symmetry without duplicating the chapter’s general construction.

Prepare with the global-background page and Symmetry, Gauge Redundancy, and Duality. Continue to the construction page only after the group, global form, sectors, measure, action, counterterms, boundaries, and anomaly condition have been stated.

5. Gauging Continuous and Finite Symmetries

Gauging Continuous and Finite Symmetries implements the distinction in two cases and is the chapter’s construction step. Continuous gauging integrates connections with local fluctuations and a chosen gauge action. Finite gauging performs a weighted groupoid sum over flat bundles, including both projection and twisted sectors. Possible discrete torsion is a choice constrained by the group and dimension, not an arbitrary phase for every sector. Afterward, you can write the appropriate continuous or finite construction with its quotient, sectors, weights, and gaugeability conditions explicit.

Prepare with the diagnostic page and Vector, Principal, and Associated Bundles. What Is an Anomaly? is recommended before deciding that the symmetry is gaugeable. Continue to the post-gauging page to determine the actual symmetry of the result.

6. Residual, Quotient, and Emergent Dual Symmetries

Residual, Quotient, and Emergent Dual Symmetries begins only after the gauging data are fixed. This is the post-gauging interpretation step. Transformations of the parent theory first have to preserve those data and normalize the gauged subgroup; the resulting candidate quotient must then be reduced to its faithful action on the new theory. Finite Abelian gauging can separately produce a dual symmetry acting on twisted sectors. Afterward, you can distinguish the faithful inherited symmetry from a sector-born dual symmetry and state when the two may mix.

Prepare with the construction page. Quantum Implementations, Projective Actions, and Central Extensions helps when inherited and dual transformations combine nontrivially. Continue to higher-form symmetry when the dual operators are not ordinary local charges.

One scalar through the chapter

Take a charge-one complex scalar $\phi$ and, when needed, the deformation

\Delta\mathcal L =h\,\phi^N+h^*(\phi^\dagger)^N, \qquad N\in\mathbb Z,\quad N\ge 2.

With $D=\mathrm d-ia$ and $\phi\mapsto e^{i\alpha}\phi$ , spurion covariance assigns $h\mapsto e^{-iN\alpha}h$ . This example is representative because one faithful $U(1)$ action supports a local current, nontrivial background bundles, controlled breaking to a finite subgroup, and both continuous and finite gauging. The same example tests every change of status:

Stage	What the scalar example establishes
Current source	Differentiating with respect to $a_\mu$ inserts the $U(1)$ current; the $a^2\phi^\dagger\phi$ term supplies the scalar seagull
Global background	$\phi$ is a section of $L$ and $h$ of $L^{-N}$ ; a nowhere-zero frozen $h$ trivializes $L^N$ , and a background preserving it also obeys $(\mathrm d+iNa)h=0$ , leaving only flat $\mathbb Z_N$ holonomy
Spurion and response	At $h=0$ , with an invariant regulator and measure and no anomaly, the complete current response is transverse; at fixed $h\ne0$ , its longitudinal part is tied to mixed $h$ and $h^*$ kernels rather than being set to zero
Background versus gauging	A fixed $a$ probes the scalar theory; gauging the full $U(1)$ requires $h=0$ or additional compatible structure; gauging the exact $\mathbb Z_N$ at fixed $h$ is a different operation; adding a dynamical charge- $-N$ field defines yet another theory
Continuous and finite gauging	Four-dimensional scalar electrodynamics illustrates continuous local fluctuations; a two-dimensional $\mathbb Z_N$ example illustrates spatial twisted sectors and temporal projection
Post-gauging symmetry	At $h=0$ , for gaugeable data that preserve the parent $U(1)$ , the inherited candidate is $U(1)/\mathbb Z_N$ ; its faithful action has changed charge normalization. Gaugeable finite cyclic data in two dimensions instead produce a dual $\widehat{\mathbb Z}_N$ acting on twisted sectors

For the two-dimensional finite example, choose $\phi(x+L)=e^{2\pi i k/N}\phi(x)$ . A convention that assigns the inverse transition function to the same holonomy relabels $k\mapsto-k$ and changes no sector content. Temporal insertion projects onto invariant states, whereas the dual character acts on the spatial label $k$ ; these are different operations.

The scalar does not represent non-Abelian stabilizers, non-invertible or categorical post-gauging structure, or a general higher-form symmetry. Those cases require their own global data and are not inferred from this thread.

The ordinary dual zero-form statement is special to two dimensions. In $d$ dimensions, gauging a finite Abelian zero-form symmetry can instead produce a $(d-2)$ -form dual symmetry Gaiotto et al. 2015, § 3, p. 14, arXiv PDF. Double gauging has its cleanest elementary form for finite Abelian groups and still depends on normalization and topological weights; the two-dimensional sector construction is analyzed in Bhardwaj and Tachikawa 2018, § 2.1, pp. 4–5, eqs. (2.1)–(2.4), arXiv PDF.

Distinctions that must remain separate

Source versus connection. A local current source and a connection are successive descriptions of an external probe. The second includes global patching and holonomy; neither is dynamical.

Background covariance versus gauge redundancy. A background bundle automorphism identifies presentations of the fixed external argument, and a more general transformation can map one source value to another; the source is not integrated and no Gauss-law projection is imposed. After gauging, the corresponding transformations are quotiented in the dynamical configuration space. Even a kinetic-looking term does not make a field dynamical if that field is never integrated over.

Spurion covariance versus exact symmetry. Transforming $h$ describes a covariant family. Holding nonzero $h$ fixed explicitly breaks the charge-one $U(1)$ to transformations satisfying $e^{iN\alpha}=1$ within that $U(1)$ . The complete action can still have additional transformations not contained in the original circle group.

Contact terms, counterterms, and current improvements. Seagulls and time-ordering contacts are forced parts of a complete distribution. Allowed finite source-local counterterms shift prescribed local pieces. A current improvement is an operator redefinition and can also change separated current correlators; these are not three names for one ambiguity. On nontrivial finite backgrounds, allowed local topological terms can also shift global partition-function phases. Once gauged, distinct permitted topological weights can define inequivalent theories rather than two prescriptions for one theory.

Structural response versus transport. A time-ordered vacuum kernel organizes Ward identities and source dependence. Conductivity or another transport coefficient additionally requires a state, retarded continuation, limits, and subtraction conventions.

Projection versus finite gauging. Projecting the untwisted Hilbert space is incomplete. Spatial twisted sectors, automorphism factors, and allowed topological weights are part of the finite gauge theory.

Inherited versus emergent symmetry. After gauging $H\subset G$ , writing $G/H$ is justified only when $H$ is normal and the full gauging data are preserved. More generally one first obtains a data-preserving normalizer quotient and then takes its faithful image. A dual symmetry is new sector data; it need not form a direct product with the inherited part. Residual, Quotient, and Emergent Dual Symmetries derives this criterion and states its limitations.

Gauging-induced versus renormalization-group emergence. The dual symmetry created by a specified topological sum is not the same mechanism as an approximate or exact symmetry emerging at a long-distance fixed point.

Check a gauging proposal

A proposed gauging is specified well enough only when all of the following questions have definite answers:

Which symmetry? Give the global group or global form, its action, its kernel, and the subgroup being gauged.
Which configurations? State the admitted bundles or cocycles, boundary conditions, defects, and topological sectors.
Which identification and measure? State the gauge group acting on each configuration, the quotient or gauge-fixing representation, and the continuous measure or finite automorphism weights.
Which dynamics and weights? Give the local gauge action, couplings, counterterms, theta-like or discrete topological terms, and regulator.
Why is it consistent? Check small and large transformations, nontrivial bundles, boundaries, and any anomaly that would prevent the exponentiated functional from descending.
What changed physically? Identify the local and extended observables, state sectors, formerly charged fields that ceased to be local, and any new constraints.
What acts afterward? Compute the faithful inherited symmetry and any dual or higher-form symmetry; do not infer a direct product without checking its composition law.

If a proposal supplies only $D_\mu=\partial_\mu-igA_\mu$ but does not say whether $A_\mu$ is integrated over, it has not yet distinguished a background coupling from gauging. If it keeps only invariant states for a finite group, it has not yet included the spatial twisted sectors. These two failures locate the missing operation without requiring a model-dependent calculation.

Check your understanding

A sound response should satisfy the criterion after each prompt.

Retrieval — classify the status of a source. Name the three statuses of $a$ used in the chapter. Use: Current Sources, Background Gauge Fields, and Background versus Gauging. Success criterion: identify a local current source, a connection on a fixed bundle, and an integrated or summed gauge field, with the first two nondynamical. Repair: return to Background versus Gauging if the integration variable cannot be identified.

Explanation — detect global background data. Explain why flat does not mean trivial. Use: Coupling to Background Gauge Fields and Bundles. Success criterion: name holonomy or transition functions and a noncontractible loop on which a flat connection can be detected. Repair: review Bundle Connections if only curvature was checked.

Derivation check — reconstruct complete response. Inspect a proposed second derivative of $W[a]$ that contains only $\langle \mathrm T jj\rangle_c$ . Use: Current Sources and Spurions and Response. Success criterion: check the source-dependent action for seagulls and the time-ordered product for contact terms before accepting the formula. Repair: redo the scalar-source differentiation on Current Sources.

Convention translation — preserve connection covariance. Rewrite a formula that uses $D=\mathrm d+iA$ and $\phi\mapsto e^{-i\alpha}\phi$ in this chapter’s convention. Use: Coupling to Background Gauge Fields and Bundles. Success criterion: change the field, connection-transformation, and holonomy signs coherently rather than flipping one sign in isolation. Repair: verify $D^U(U\phi)=U(D\phi)$ using Bundle Connections.

Comparison — separate a source family from one theory. Compare a transforming spurion $h(x)$ with a fixed nonzero $h$ . Use: Spurions and Response. Success criterion: distinguish covariance of the family from the residual symmetry of one theory and include the conjugate $h^*$ contribution. Repair: return to the scalar source identity on Current Sources.

Transfer — gauge one factor of a product symmetry. A two-dimensional theory has a gaugeable $\mathbb Z_2^{(A)}\times\mathbb Z_2^{(B)}$ symmetry. Gauge only the first factor. Use: Gauging Continuous and Finite Symmetries and Residual, Quotient, and Emergent Dual Symmetries. Success criterion: sum over flat $\mathbb Z_2^{(A)}$ bundles with projection and twisted sectors, retain $\mathbb Z_2^{(B)}$ only when the gauging data preserve it, and test whether the dual $\widehat{\mathbb Z}_2^{(A)}$ combines directly or by a nontrivial action or extension. Repair: apply the finite-gauging checklist on Background versus Gauging before naming the post-gauging group.

Failure diagnosis — find an incomplete finite gauging. A derivation declares a finite group gauge theory after projecting the untwisted Hilbert space. Use: Background versus Gauging and Gauging Continuous and Finite Symmetries. Success criterion: identify the missing twisted sectors and ask whether topological weights and an anomaly have been checked. Repair: apply the finite-gauging checklist on Background versus Gauging.

Synthesis — determine post-gauging symmetry. Given $H\subset G$ , describe the result from parent theory to post-gauging symmetry. Use: Background versus Gauging, Gauging Continuous and Finite Symmetries, and Residual, Quotient, and Emergent Dual Symmetries. Success criterion: specify the background probe, gauging data, operator/state change, data-preserving normalizer, faithful image, and any separately emergent dual symmetry. Repair: restart with the seven questions in Check a gauging proposal.

Continue by the missing structure

For obstruction rather than construction, go to What Is an Anomaly? and then the rest of Anomalies, Inflow, and Matching.
For the local and global content of an ordinary gauge theory, continue to Gauge Fields, Redundancy, and Observable Content and then Dynamical Gauge Fields and Matter.
For a perturbative representation of the quotient, continue to The Faddeev–Popov Construction.
For genuine response and transport, continue to Sources, Linear Response, and Kubo Formulae or Current Vertices and Ward-Consistent Response.
For finite gauging in two-dimensional conformal field theory, continue to Cosets and Orbifolds.
For the rigorous limits of reconstruction from sectors, continue to Sector Completeness, Field Algebras, and Classification Limits.
For a dual symmetry carried by extended operators, continue to Gauging a Higher-Form Symmetry.
For boundary-sensitive gauge transformations, continue to Proper and Improper Gauge Transformations.
To choose a different symmetry problem rather than continue this branch, return to the Symmetry and Gauge Structure volume overview.

References

Bhardwaj, Lakshya, and Yuji Tachikawa. “On Finite Symmetries and Their Gauging in Two Dimensions.” Journal of High Energy Physics 03 (2018): 189. DOI. Open PDF
Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF
Schwartz, Matthew D. Quantum Field Theory and the Standard Model. 1st ed. Cambridge: Cambridge University Press, 2014. DOI