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Amputation removes the external propagator factors from a momentum-space correlation function. For a connected four-point function, its diagrammatic external lines each carry a full two-point function; multiplying by the corresponding inverse two-point kernels leaves an interaction kernel.

This operation helps separate propagation to and from the interaction from the interaction itself. It is part of the connection between Green functions and scattering amplitudes, although physical external states also require on-shell limits and the appropriate pole-residue normalization. An amputated connected function need not be one-particle irreducible: it may still contain internal lines whose removal disconnects it. External-line removal and internal irreducibility are distinct operations.

A Feynman integral is the momentum-space integral associated with a Feynman diagram: each independent loop momentum is integrated with measure $d^{d}k/(2\pi {)}^d$, and propagators, vertex numerators and momentum conservation determine the integrand. Its definition also requires a pole prescription and, when divergent, a regularization in quantum field theory. Feynman parameters combine propagator denominators and permit shifts of loop momenta. A massless two-propagator example is $\int d^{d}k/[(2\pi {)}^d{k}^2(p-k{)}^2]$, with its causal prescriptions understood. Subtracting local counterterms turns a regulated diagram into a finite contribution to a renormalized amplitude.

The Feynman propagator is a vacuum time-ordered two-point function, with the conventional overall factors stated. For a free real scalar field, $⟨0|T\varphi (x)\varphi (y)|0⟩$ has momentum-space form $i/(p^2-m^2+i0)$ in the usual Minkowski convention.

Time ordering places the later field to the left, and the pole prescription selects the vacuum boundary condition. The propagator is the Green function used for internal lines in perturbative amplitudes; it is not a directly measured probability for a particle to follow a particular path. It can be nonzero at spacelike separation without enabling superluminal signalling. Causal response is instead organized by the retarded Green function and local commutators, so these different two-point functions must not be interchanged.

Feynman rules translate a Feynman diagram into a contribution to a scattering amplitude or correlation function in perturbation theory. They specify factors for external states, interaction vertices, internal propagators, momentum-conservation constraints and loop integrals, with statistics-dependent signs and symmetry factors. For quantum electrodynamics, a photon-fermion vertex contains the coupling $e$, while a photon propagator supplies an inverse squared four-momentum. The complete amplitude also contains currents, so propagator powers alone do not determine the energy dependence of a scattering cross-section.

Completing the square in the integral over an intermediate position shows that two free-particle propagators compose to the propagator for the sum of their positive time intervals. For equal slices, the quadratic identity isThe Fresnel integral contributes a factor proportional to $\sqrt{k/(k+1)}$; multiplying successive factors gives $1/\sqrt{N}$. A consistent convergence regulator fixes the phases of these square roots.

Gauge fixing imposes a condition on a gauge field to control the many field representatives belonging to the same gauge orbit. Without this step, pure-gauge variations give zero modes of the quadratic field operator, preventing its inverse from defining a propagator.

For example, a covariant gauge adds a term depending on ${\partial }_{\mu }{A}^{\mu }$ to make perturbative inversion possible. The Faddeev-Popov determinant, often represented using Faddeev-Popov ghost fields, accounts for the change of variables along the gauge orbit. Consistent physical predictions must be independent of the chosen representative. The procedure is locally useful in perturbation theory; the Gribov ambiguity warns that a simple local condition need not select exactly one representative on every gauge orbit globally.

Consider one massive gauge field with $m>0$ after Higgs mechanism. For this comparison perform a Wick rotation and take a real nonzero momentum in Euclidean space $k_E$. The transverse and longitudinal orthogonal projectors areHere “longitudinal” means parallel to the off-shell four-momentum, not the physical longitudinal polarization of an on-shell massive gauge boson. The R-xi gauge quadratic vector operator and its inverse areMultiplication verifies $K_{\xi }{D}_{\xi }=P^T+P^L=I$, since $P^T{P}^L=0$ and $(P^{T,L}{)}^2=P^{T,L}$. The unitary gauge vector operator lacks the gauge fixing derivative term in the longitudinal sector, givingThus every eigenvalue of $D_{\xi }$ falls as $1/k_E^2$ at fixed finite positive $\xi$, whereas $D_U$ has the constant longitudinal eigenvalue $1/m^2$. Unitary gauge consequently obscures ordinary power counting; this is not a failure of the physical theory.

For a worked comparison choose $|k_E|=10m$. The dimensionless longitudinal eigenvalue $m^2{D}_L$ is $1$ in unitary gauge, $1/101$ at $\xi =1$, and $1/11$ at $\xi =10$. Its crossover is $|k_E|\sim \sqrt{\xi }m$: taking $\xi \to \infty$ first gives a constant, whereas taking $|k_E|\to \infty$ first gives zero. The two limits do not commute. Contracting the propagator between conserved external conserved currents projects away $P^L$, but one cannot impose that simplification separately on every internal line of an interacting Non-Abelian gauge theory.

With $W=-i\log Z$, $\phi =\delta W/\delta J$ and $\Gamma =W-\int J\phi$, variation gives $\delta \Gamma /\delta \phi =-J$. Differentiating through the source-to-field map givesThe inverse is taken on the regulated nonsingular fluctuation space. Since $W^{(2)}=i{C}_2$, the full connected propagator is determined by the inverse effective-action quadratic kernel. Factors of $i$ and the minus sign depend on the explicitly chosen Minkowski Legendre transform convention.

// Target: perturbative-quantum-field-theory.bigb

For the polar parametrization of the Abelian Higgs field with $D=\partial +ieA$, adding $-H^2/(2\xi )$ cancels quadratic vector-phase mixing. The cross terms reduce, modulo a total derivative, to $-2e(1+h/f)\pi A^{\mu }{\partial }_{\mu }h$; interactions remain. The Faddeev-Popov operator is proportional to $\square +\xi e^2(f+h{)}^2$, so the Faddeev-Popov ghosts couple to $h$. The vector propagator falls as $k^{-2}$ at fixed finite $\xi$, but the polar kinetic and nonlinear gauge fixing vertices include terms of mass dimension above four. Manifest polynomial power counting instead uses Cartesian scalar variables with conventional R-xi gauge.

For positive kinetic coefficient $\zeta$ in ${\mathcal{L}}_0=\zeta ({\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -m^2{\varphi }^{\ast }\varphi )$, the canonical momentum is ${\pi }_{\varphi }=\zeta {\dot{\varphi }}^{\ast }$. Positive-energy oscillator modes with unit oscillator commutators have amplitude $1/\sqrt{2\zeta E_{\mathbf{p}}}$. The mixed time-ordered propagator is $C={\zeta }^{-1}{D}_F$, with $D_F$ the ordinary scalar propagator. This also follows by inverting the quadratic form in the path integral. Rescaling $\varphi$ changes interaction coefficients as well as external correlation-function factors.

The two Bhabha scattering tree-level Feynman diagrams are the two panels of Solution with a photon mediator: $s$-channel annihilation with internal momentum $p_1+p_2$, and $t$-channel exchange with internal momentum $p_1-p_3$. Each interaction vertex preserves the electron species. There is no independent $u$-channel Feynman diagram, since the outgoing electron and Positron are distinguishable.

Removing the color charge factor from Solution and replacing $g$ by $e$ givesCrossing symmetry interchanges $s$ and $t$ for the exchange Feynman diagram, givingSquaring the sum of the two scattering amplitudes also produces the interference term $2\mathrm{Re}\overline{{\mathcal{M}}_s{\mathcal{M}}_t^{\ast }}$. The given third term must be this cross term: it has both propagator denominators $st$, whereas a squared individual Feynman diagram has $s^2$ or $t^2$. With the correct relative fermionic sign, its value is $4e^4{u}^2/(st)$. ThusThe coefficient of the last displayed fraction is positive, but $s>0$ and $t<0$ in the physical scattering region, so this interference is destructive. The relative sign originates in the two possible external fermion contractions; changing an external-state phase changes both scattering amplitudes together.^[ref]

In unitary gauge the quadratic action is that of a Proca field. Use Minkowski metric $(+---)$ and add a source $J_{\mu }{A}^{\mu }$. Integration by parts, dropping the boundary term, givesWith Fourier transform convention $A(x)=\int d^{4}k{e}^{-ik\cdot x}A(k)/(2\pi {)}^4$, its kernel isFor $k^2\ne 0$, introduce the transverse and longitudinal momentum projectorsThey obey $P_T{P}_L=0$ and $P_T+P_L=1$ as mixed-index linear operators. HenceThe final expression is regular at $k^2=0$; the apparent singularities of the individual projectors have cancelled.

The normalized Gaussian functional integral is evaluated by completing the square with $A\mapsto A+K^{-1}J$:The source-independent functional determinant cancels in this ratio. Two source derivatives insert two fields, so the covariant Feynman propagator is $i{K}^{-1}$, with the Feynman i-epsilon prescription at the physical pole:This is the Proca propagator. Its pole residue has the three physical particle polarizations of a massive spin-one gauge boson. No additional gauge-fixing term is needed to invert this already gauge-fixed massive quadratic action. In a full interacting Higgs mechanism calculation, any field-dependent unitary gauge measure belongs to the interacting theory; the free quadratic gauge boson propagator above is unaffected.

Set the external four-momenta to zero after identifying the proper diagrams. In each box or triangle, at least one endpoint of the internal photon is a cubic vertex adjacent to an external scalar of zero momentum. That vertex is proportional to $l_{\rho }$, whereas the supplied numerator obeys $l^{\rho }{P}_{\rho \sigma }(l)=0$. All seven such graphs vanish algebraically. This use of the transverse numerator is stronger than merely saying that one has chosen a generic Lorenz gauge.

The two surviving seagull vertex bubbles have vertex factors $2i{e}_D^2{g}^{\mu \rho }$ and $2i{e}_D^2{g}^{\sigma \nu }$. Combining their two propagators givesThis scalar-QED four-point vertex pole in transverse gauge calculation determines the proportionality constant as well as the required tensor structure. Keep $m>0$ while extracting the ultraviolet pole. Lorentz covariance inside the regulated integral replaces $l^{\mu }{l}^{\nu }$ by $g^{\mu \nu }{l}^2/D$, giving the factor $(D-1)g^{\mu \nu }/D$.

For its scalar integral, Feynman parameter givesThere is no infrared divergence at $l=0$ with nonzero $m$. At pole order the two photon assignments and the factor $3/4$ therefore yieldThe common engineering scale factor is included in $e_D^2$; finite logarithms are immaterial to the pole. Adding $2i{e}_D^2{\delta }_4{g}^{\mu \nu }$ cancels it whenThis is minimal subtraction scheme, up to optional finite terms. It agrees with the transverse-gauge scalar QED wave-function pole and $Z_4=Z_2$. Setting $m=0$ at the outset would instead make the zero-momentum integral scaleless and obscure its separate ultraviolet divergence.

The tree-level Feynman diagrams are the two ordered quark-exchange graphs and the $s$-channel gluon-exchange graph with a three-gluon vertex. There is no two-gluon-two-quark contact term in the ordinary QCD Lagrangian density. Let $q=k_1+k_2=p_1+p_2$, $r_t=p_1-k_1$, $r_u=p_1-k_2$, and $s=q^2$, $t=r_t^2$, $u=r_u^2$. Define $i\mathcal{M}$ as the product of the stated Feynman rules.

Suppressing external colour indices, the quark graphs areThe massless Dirac equations give $\bar{u}/p_1=0$ and $/p_{2}v=0$. ConsequentlyThe last equality uses $\bar{u}/qv=0$. The quark-exchange sum is therefore $-g^2\bar{u}/k_1[T^a,T^b]v$.

For the gluon graph, use all incoming momenta $(k_1,k_2,-q)$ in the three-gluon tensor:Its double contraction is $k_{1\mu }{k}_{2\nu }{V}^{\mu \nu \rho }=s(k_2-k_1{)}^{\rho }/2$. The vertex and gluon propagator giveCombining the graphs yieldsThus the requested spinor proportionality factor is $-g^2\bar{u}(p_1)/k_{1}v(p_2)$ when the colour matrix is displayed separately; retaining it inside the spinor bilinear makes its external colour indices explicit. For a convention calling the Feynman-rule product itself $\mathcal{M}$, rather than $i\mathcal{M}$, the entire result gains one factor of $i$.

For two distinct commuting real scalar fields ${\chi }_1,{\chi }_2$ in place of the labelled fermion pair, with interaction $g\varphi {\chi }_1{\chi }_2$, there is no anticommutation sign on closing the loop and no spinor trace. The two propagators are simply multiplied. In terms of physical time-ordered scalar contractions $C_1,C_2$, the connected contribution isThe statistics and symmetry factors of a two-propagator loop determine the replacement. Relative to the fermionic formula in ii, the loop-statistics sign reverses. Scalar source kernels normalized like the stated Gaussian give the same overall minus sign after the four contraction factors are converted. There is no spin multiplicity.

If both replacement fields instead mean the same real field $\chi$, the coupling normalization and the identical-line symmetry factor must be specified. With interaction $(g/2)\varphi {\chi }^2$, the cubic vertex is $ig$ and its two-identical-line loop carries $1/2$, giving coefficient $-g^2/2$. With the literal coefficient $g\varphi {\chi }^2$, its vertex is $2ig$ and the coefficient is $-2g^2$. These cases are distinct from two separately labelled real fields; suppressing their combinatorics would hide a factor-of-two ambiguity in the replacement.

Contract the numerator of the quoted light cone gauge propagator with $n_{\mu }$:Thus every internal gluon emerging from a ghost-gluon vertex contributes a factor $n_{\mu }{\tilde{\Delta }}^{\mu \nu }=0$. Every ghost-dependent contribution to a physical gluon or matter amplitude vanishes. Equivalently, the strict condition $n\cdot A=0$ removes the Faddeev-Popov ghost interaction in i, leaving the field-independent determinant $\det (n\cdot \partial )$, which cancels from normalized correlators.

This is ghost decoupling in strict axial gauge, not a statement that the free Faddeev-Popov ghost propagator itself is zero. The algebra assumes a consistent axial-gauge pole prescription for $n\cdot p=0$ and treatment of residual gauge zero modes. The overall sign assigned to the displayed vector propagator does not enter the contraction proof. With the mostly-minus convention and the usual $-F^2/4$ action, its conventional prefactor is $-i/(p^2+i0)$; the source prints $+i$. Either prefactor gives the same zero contraction, so no sign correction is needed to obtain the requested ghost-decoupling result.

The quadratic part of the covariant-derivative term contains $g^2{v}^2{A}_{\mu }{A}^{\mu }/2-gv{A}_{\mu }{\partial }^{\mu }b$. Its mixing term cancels the $+gv{A}_{\mu }{\partial }^{\mu }b$ in the stated gauge fixing term. Thus the vector mass and diagonal quadratic action areFor nonzero $p^2$ introduce the transverse and longitudinal transverse and longitudinal momentum projectors $P_T^{\mu \nu }=g^{\mu \nu }-p^{\mu }{p}^{\nu }/p^2$ and $P_L^{\mu \nu }=p^{\mu }{p}^{\nu }/p^2$. After integration by parts, the inverse quadratic kernel isTheir orthogonality makes the inversion immediate. The gauge-boson propagator isEquivalently,The projector formula explains the physical massive pole and the gauge-dependent longitudinal pole. The apparent $1/p^2$ in the projectors cancels in the equivalent expression; it is not an extra massless physical state. At $\xi =1$ the numerator reduces to $g^{\mu \nu }$. At fixed momentum the $\xi \to \infty$ limit gives the unitary gauge propagator $-i(g^{\mu \nu }-p^{\mu }{p}^{\nu }/M_A^2)/(p^2-M_A^2+i0)$. The relation between this limit and ultraviolet power counting is not uniform in momentum.^[ref]

First fix a convention that is implicit in the source. For the stated free generating functionalthe time-ordered Wick contractions are $S_F=⟨T\psi \bar{\psi }{⟩}_0=-iS$ and ${\Delta }_F=⟨TAA{⟩}_0=-i\Delta$. These follow directly by the source differentiation rules in part (i). Thus the kernels called propagators in the question are distinguished here from the contractions, which already include their factors of $i$.

At first order in the QED interaction, write $a$ for its integration point. The normalized three-point correlation function isThe zeroth-order term vanishes because the free photon one-point correlation function is zero. There are two nonzero Wick contractions of the four fields:The first term pairs the two external fermions directly and closes the two fields at the interaction into a fermion loop; its minus sign is the closed fermion loop sign. The second term joins the external fermions through the interaction, giving the connected contribution. This is also obtained by differentiating the quartic term $(i\bar{\eta }S\eta {)}^2/2$ in the free generating functional; the two choices of bilinear cancel its factor $1/2$.

Insert $S_F=-iS$ and ${\Delta }_F=-i\Delta$. Since $(-ie)(-i{)}^3=e$, the result is exactlyThe trace term is a disconnected product of a photon tadpole diagram with the free external fermion propagator; the other term is the connected three-point correlation function. In this QED calculation the disconnected term vanishes, as established next. With a convention using $S_F,{\Delta }_F$ themselves as the symbols for the propagators, the equivalent formula retains the prefactor $-ie$ and the connected-minus-trace ordering above; mixing these two conventions would give an erroneous overall phase.

For commuting real scalar fields, remove the closed fermion loop sign and replace the Dirac propagators by scalar propagators. There are no spinor indices, hence no matrix trace. With two independent real scalar fields ${\chi }_1,{\chi }_2$ and interaction $g\varphi {\chi }_1{\chi }_2$, the same normalization givesThe overall sign has reversed relative to part (i) because $(ig{)}^2$ is now unaccompanied by a closed fermion loop sign. The same formula holds in the source's kernel convention, since there are again four propagators.

There is an additional convention to specify if the two symbols are intended to denote the same real scalar field. For the literal interaction $g\varphi {\chi }^2$, there are two identical connected Wick contractions between ${\chi }^2(x)$ and ${\chi }^2(y)$, so the coefficient becomes $-2g^2$. If the conventional interaction is instead $(g/2)\varphi {\chi }^2$, each vertex is $ig$ and the loop has symmetry factor $1/2$, giving coefficient $-g^2/2$. In both interpretations the closed fermion loop sign and spinor matrix trace disappear; the differing numerical factor comes from the interaction normalization and field identity.

Source PDF: mmathphys.physics.ox.ac.uk/sites/g/files/vzsiac2991/files/mmathphys/documents/media/aqft_for_particle_physics_2024.pdf

Solutions to all seventeen lettered parts. Use natural units, the Minkowski metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$, Feynman slash notation $/p={\gamma }^{\mu }{p}_{\mu }$, and the Feynman i-epsilon prescription in internal propagators. Define the scattering amplitude by $S_{fi}={\delta }_{fi}+i(2\pi {)}^4{\delta }^{(4)}(p_f-p_i){\mathcal{M}}_{fi}$. The gauge covariant derivative convention in question 3 is specified there to match the printed three-boson Feynman rule.