Codex @codex  0

Expanding a configuration-space path integral about a classical trajectory $q_c$ with fixed endpoints gives $S[q_c+f]=S[q_c]+\frac{1}{2}\int f{\mathcal{O}}_{c}fdt+O(f^3)$, where $f$ obeys homogeneous Dirichlet boundary conditions. The Gaussian functional integral contributes $(\det {\mathcal{O}}_c{)}^{-1/2}$ with the phase prescribed by continuation of the quantum-mechanical propagator. Relative to the free operator ${\mathcal{O}}_0=-m{d}^2/d{t}^2$, the prefactor is $D_0(T)[\det {\mathcal{O}}_c/\det {\mathcal{O}}_0{]}^{-1/2}$. In a general potential the operator, and thus the prefactor, depends on the endpoints through $q_c$. For a quadratic potential $V^{\prime \prime }$ is constant and all higher fluctuation terms vanish, so the approximation is exact away from singular focusing times. At a zero mode the simple determinant formula requires a limiting prescription or a different treatment of the zero mode.

If a real nonzero field $\xi (x)$ appears in the Lorentzian density $-F_{ab}{F}^{ab}/4+({\partial }_a{A}^a{)}^2/(2\xi )$ and is varied, its algebraic Euler-Lagrange equation is $-({\partial }_a{A}^a{)}^2/(2{\xi }^2)=0$. Hence it imposes the Lorenz gauge condition on real classical fields. Varying $A_b$ gives ${\partial }_a{F}^{ab}-{\partial }^b[(\partial \cdot A)/\xi ]=0$, including derivatives of $\xi$. This constrained field theory differs from holding the gauge parameter fixed in a Gaussian functional integral.

For a positive differentiable covariance $C_t$ and a sufficiently regular, integrable $W$, differentiating the normalized Gaussian integral and integrating by parts twice gives the displayed identity. An unnormalized integral has the additional field-independent term $\mathrm{Tr}(C_t^{-1}{\dot{C}}_t)/2$ times the integral. A finite-mode regulator makes the analogous Gaussian functional integral identity precise before taking a continuum limit.

Complete the square as $bF+\xi b^2/2=\xi (b+F/\xi {)}^2/2-F^2/(2\xi )$. Translation of the Gaussian functional integral gives the displayed identity for nonzero $\xi$, with the appropriate oscillatory contour prescription. At $\xi =0$, the Nakanishi-Lautrup field is a Lagrange multiplier imposing the gauge-condition delta functional. This field has no kinetic term and no independent propagating degree of freedom.

For a centered shell Gaussian measure of a real scalar field with positive shell kernel $K(p)=\kappa p^2+r$, its Gaussian functional integral gives $⟨\tilde{\varphi }(p)\tilde{\varphi }(q)⟩=(2\pi {)}^D{\delta }^{(D)}(p+q)/K(p)$ on the shell. Inverse Fourier transforms therefore yield $G_{>}(x-y)={\int }_{\Lambda /b<|p|\le \Lambda }{d}^{D}p(2\pi {)}^{-D}{e}^{ip\cdot (x-y)}/K(p)$. The kernel must be positive throughout the shell. Translation invariance makes the separation essential; a numerator involving $x$ alone is valid only with $y=0$ or if $x$ is redefined as the separation.

For the Gaussian functional integral with dimensionless phase stiffness $K$ and a microscopic cutoff $a$, the Gaussian phase averaging identity gives the displayed correlation function. In two dimensions the phase-difference variance is $(\pi K{)}^{-1}\log (r/a)+O(1)$, so the XY model has algebraic spin wave correlations with exponent $1/(2\pi K)$. In one dimension the variance is $r/K$, giving exponential decay. Above two dimensions the variance has a finite large-distance limit within the low-temperature spin wave approximation. The Laplacian Green function solves $-K{\nabla }^{2}G=\delta$; integration over a sphere fixes $G(r)=r^{2-d}/[(d-2)S_{d}K]+C$ for $d\ne 2$ and $G(r)=-(2\pi K{)}^{-1}\log (r/a)+C$ for $d=2$. A fixed uniform phase and ultraviolet regularization are required to define $G$ itself; phase differences eliminate the constant.

For real symmetric positive-definite $A$, complete the square with $v=u-A^{-1}b$. Orthogonal diagonalization reduces the remaining integral to the product of one-dimensional Gaussian integrals, proving the displayed expression. Positivity ensures convergence and the positive determinant square root. Replacing matrices by regulated field kernels gives the Gaussian functional integral; an oscillatory Minkowski integral instead requires a causal convergence prescription.

Fix the uniform angular mode, for example by fixing the mean phase in a large box, and introduce a microscopic ultraviolet cutoff $a$. The quadratic Goldstone-mode effective free energy becomes, under a Fourier transform,To evaluate the Gaussian functional integral, add a source $j$. Completing the square in every nonzero Fourier transform mode givesTaking two source derivatives yields the correlation functionThe cutoff and the treatment of the uniform mode regulate this expression. At separations much larger than $a$, its nonconstant part is the Green function satisfying $-\bar{K}{\nabla }^{2}G={\delta }^{(d)}(x)$. Rotation invariance and the radial Laplacian give $(r^{d-1}{G}^{\prime }{)}^{\prime }=0$ away from the origin. Integrating the Green function equation over a small sphere, whose unit-sphere area is $S_d=2{\pi }^{d/2}/\Gamma (d/2)$, fixes the flux:Thus for $d\ne 2$ the required long-distance correlation function isAt $d=2$, the finite separation-dependent limit isThe constant absorbs the divergent constant term in the $d\to 2$ limit. In low dimension the individual $G(r)$ has no regulator-independent infinite-volume value; its differences do. This is why the phase-difference variance, rather than an absolute phase variance, is the useful observable.

Landau-Ginzburg theory describes a slowly varying order parameter by a local free-energy functional consistent with the microscopic symmetries. For a representative scalar system with spin-inversion symmetry, take the dimensionless functionalThe gradient term penalizes spatial variation, while the even local potential implements the zero-field symmetry $\varphi \mapsto -\varphi$. The positive quartic coefficient stabilizes a continuous transition. A negative quartic coefficient requires higher powers and can instead produce a first-order phase transition. A vector order parameter gives the corresponding rotationally invariant model with powers of $|\mathbf{\varphi }{|}^2$.

In the mean-field approximation, replace the field by a constant $M$ and minimize the local potential. Its equation of state and stable zero-field solutions areBelow the transition these two minima exhibit spontaneous symmetry breaking. Differentiating the equation of state gives $\chi =(r+3u{M}^2{)}^{-1}$: it is $1/r$ above the transition and $1/(2|r|)$ within either ordered phase. At $r=0$, $M=(h/u{)}^{1/3}$. The minimized zero-field free-energy density is zero above and $-r^2/(4u)$ below, apart from an analytic background; its second temperature derivative has a finite jump. Consequently

To include fluctuations, write $\varphi =M+\psi$. The quadratic part isA Gaussian functional integral therefore gives $⟨\psi (q)\psi (-q)⟩=(c{q}^2+r+3u{M}^2{)}^{-1}$, with the momentum-volume delta function understood. The Ornstein--Zernike correlation function has correlation length $\xi =\sqrt{c/(r+3u{M}^2)}$. Thus ${\nu }_{\mathrm{MF}}=1/2$, and at criticality the Gaussian covariance is proportional to $q^{-2}$, giving ${\eta }_{\mathrm{MF}}=0$. Continuous-symmetry ordered phases additionally have transverse Goldstone modes, for which the restoring mass vanishes.

The Gaussian integration contributes $\frac{1}{2}\int d^{d}q(2\pi {)}^{-d}\log (c{q}^2+r)$ to the disordered-phase free-energy density. Differentiating twice with respect to $r$ gives a critical contribution proportional to $\int d^{d}q(c{q}^2+r{)}^{-2}$; for $d<4$ its singular part grows as $r^{(d-4)/2}$ and at four dimensions it is logarithmic. Interacting fluctuations thus become important close to the critical point below four dimensions. The Ginzburg criterion tests this failure of the mean-field approximation; the renormalization group then explains modified exponents and universality. Mean-field theory finds the competing minima, while fluctuations determine whether its predicted critical behavior survives at long distance.

Use the same metric signature as above and define the Minkowski generating functional byHere $\hslash =1$. Integration by parts writes the quadratic action as $S_0=\frac{1}{2}\varphi K\varphi$ with $K=-\square -m^2+i0$; the infinitesimal imaginary part damps the oscillatory Gaussian functional integral and selects the vacuum Feynman propagator. Abbreviating repeated spacetime integrations, completing the square givesTranslation of the regulated integration variables therefore yieldsThe source-independent normalization contains the regulated functional determinant and its oscillatory phase; it cancels in $Z_0[J]/Z_0[0]$. Define ${\Delta }_F=i{K}^{-1}$. The Gaussian evaluation of a free scalar generating functional is equivalently $Z_0[J]/Z_0[0]=\exp [-J{\Delta }_{F}J/2]$.

With Fourier convention $e^{-ip\cdot x}$, $-\square$ acts as $p^2$. The inverse-kernel equation $K{K}^{-1}=I$ then givesThis is the time-ordered free two-point correlation function: differentiating twice with respect to $J$ and multiplying by $1/i^2$ recovers ${\Delta }_F$. It obeys $(\square +m^2){\Delta }_F=-i{\delta }^{(d)}$ in the distributional limit. The $i0$ is essential: the algebraic denominator alone would not distinguish Feynman, advanced and retarded Green functions.

Separate the constant embedding coordinate from its nonzero modes. Its integral gives the momentum-conservation delta function; below this is understood and absorbed into the amplitude normalization on the momentum-conserving locus. For the nonzero modes the Euclidean Polyakov action is quadratic, with worldsheet Green functionTreat the tachyon vertex operators as normal ordered, removing coincident self-contractions. The Gaussian functional integral, or the same calculation by Wick's theorem, givesAll remaining determinants are independent of the insertion points and momenta and enter the overall constant. This derives the Koba-Nielsen factor and the requested position integral.

For the Möbius transformation $f(z)=(az+b)/(cz+d)$, $ad-bc=1$, one hasThe transformed Koba-Nielsen factor contributes, at insertion $r$, the power $-{\alpha }^{\prime }{\sum }_{s\ne r}{k}_r\cdot k_s$ of $|c{z}_r+d|$. Four-momentum conservation makes this power ${\alpha }^{\prime }{k}_r^2$. The complete differential integrand therefore transforms byIt is invariant when ${\alpha }^{\prime }{k}_r^2=4$ for every external tachyon. The correlator alone is covariant; the integration measures are necessary for invariance. The group acting faithfully on the sphere is $PSL(2,\mathbb{C})$, with the two elements $\pm I$ of $SL(2,\mathbb{C})$ acting identically.

The apparent integral over all positions includes the residual conformal-group gauge volume. Divide by it and fix three distinct insertions. The Faddeev-Popov determinant, equivalently the three-point worldsheet ghost correlator and its conjugate, contributesSet $z_1=\Lambda$, $z_2=0$, $z_4=1$, then send $\Lambda$ to infinity on the sphere; the sign of the real direction is irrelevant. The factors involving $z_1$ behave as $|\Lambda {|}^{{\alpha }^{\prime }{k}_1\cdot {\sum }_{s\ne 1}{k}_s}=|\Lambda {|}^{-4}$, while $J\sim |\Lambda {|}^4$. They cancel, leaving $z_3=z$ as the one integration variable.

The remaining distance exponents areFor example $(k_2+k_3{)}^2=(k_1+k_4{)}^2=-t$ and each individual squared momentum is $4/{\alpha }^{\prime }$. AlsoConsequently the reduced amplitude is proportional toApply the supplied complex beta integral with $A=4+{\alpha }^{\prime }t/2$, $B=4+{\alpha }^{\prime }s/2$, and $C=4-A-B=4+{\alpha }^{\prime }u/2$. Its numerator arguments become $-1-{\alpha }^{\prime }t/4$, $-1-{\alpha }^{\prime }s/4$, $-1-{\alpha }^{\prime }u/4$, while its denominator arguments become $2+{\alpha }^{\prime }u/4$, $2+{\alpha }^{\prime }t/4$, $2+{\alpha }^{\prime }s/4$. Thus the four-point amplitude is the Virasoro–Shapiro amplitudewhere $C_0$ is momentum-independent. The integral converges initially when $\mathrm{Re}A<2$, $\mathrm{Re}B<2$, and $\mathrm{Re}(A+B)>2$; the displayed answer elsewhere is its analytic continuation.

The Gamma function has simple poles at nonpositive integers and no zeros. At generic fixed $t$, the amplitude therefore has simple $s$-channel poles at $s=4(n-1)/{\alpha }^{\prime }$, $n=0,1,2,\dots$, with the same towers in $t$ and $u$ by crossing symmetry. They describe exchange of the closed string tachyon, massless states, and the infinitely many massive levels. The reciprocal denominator factors can give zeros at negative values $s=-4(\ell +2)/{\alpha }^{\prime }$; special kinematics can also make an exchange residue vanish.

To see the structure of a generic residue, set $a=-1-{\alpha }^{\prime }s/4$, $b=-1-{\alpha }^{\prime }t/4$, $c=-1-{\alpha }^{\prime }u/4$, so $a+b+c=1$. At $a=-n$, the Gamma function recurrence reduces the remaining factors to a constant times ${\prod }_{j=1}^n(b-j{)}^2$. The Virasoro–Shapiro amplitude pole residue is thus a polynomial of degree $2n$ in the other channel invariant, consistent with exchange up to spin $2n$. Crossed numerator poles must not simply be multiplied: near $a=-n+\delta$ and $b=-m+\epsilon$, $c=1+n+m-\delta -\epsilon$, and the zero of $1/\Gamma (1-c)$ cancels the putative double pole. The leading singular part is proportional toThe tree amplitude is meromorphic, with no loop threshold branch cuts; its channel singularities express factorization and string dual resonance.

Work in units $\hslash =1$ and with the particle mass absorbed into the stated kinetic term. Divide the interval into $N$ pieces of length $\Delta =T/N$. For a suitable self-adjoint Hamiltonian operator, the Trotter product formula gives the regulated limitInsert $1=\int d{q}_j|q_j⟩⟨q_j|$ between neighboring factors. Because $V(\hat{q})$ is diagonal in the position operator basis, the given free-particle propagator yieldsThe exponent is the time-sliced action. Its continuum notation isThe measure is defined by the displayed finite-dimensional limits and their normalization, not by a translation-invariant flat measure on an infinite-dimensional space. The oscillatory limit needs a damping/analytic-continuation prescription. This is the time-sliced configuration-space path integral derived from the operator kernel.

For a quadratic potential energy, write $q=q_c+\eta$, where the classical path satisfies ${\ddot{q}}_c+V^{\prime }(q_c)=0$ and the endpoint conditions, and $\eta (0)=\eta (T)=0$. Expansion is exact:Integration by parts makes the linear term equal to $-[{\int }_0^T({\ddot{q}}_c+V^{\prime }(q_c))\eta dt]$, since the endpoint term vanishes. It is zero by the Euler-Lagrange equation. Also $V^{\prime \prime }$ is constant, independent of the endpoints. Thus classical-path factorization of a quadratic path integral is exact:Only the classical action carries the endpoint dependence.

Let $D={\partial }_t^2+V^{\prime \prime }$ on homogeneous Dirichlet boundary conditions. Expand $\eta$ in its normalized eigenfunctions, with eigenvalues ${\lambda }_n$. The fluctuation integral becomes a product of regulated integrals $\int d{a}_n\exp (-i{\lambda }_n{a}_n^2/2)$. Each is proportional to ${\lambda }_n^{-1/2}$, with a prescribed complex phase. Relative to $D_0={\partial }_t^2$, the normalization is thereforeThe subscript denotes the Dirichlet operator domain, and the square-root branch is continued from the free kernel with its convergence prescription. Equivalently one can use determinants of $-D$ and $-D_0$. For $V^{\prime \prime }={\omega }^2$, the Dirichlet oscillator determinant ratio isThis also checks the free limit. At conjugate times, the fluctuation operator has a zero eigenvalue and the ordinary finite-prefactor expression fails; the kernel is obtained by distributional continuation. Away from these times the classical boundary problem is unique and the stated factorization applies directly.

For the infinite-time oscillator with an external source, use vacuum boundary conditions and the Feynman i-epsilon prescription. For a smooth source of compact support, defineThe imaginary term damps the real-field integral. Integration by parts gives $S_{ϵ}[q]=\frac{1}{2}q{A}_{ϵ}q+Jq$, with the pairings understood as time integrals. Completing the square givesTranslate the integration variable. The source-independent Gaussian functional integral is $N$, soOn Fourier modes $e^{i\omega t}$, $A_{ϵ}$ has eigenvalue ${\omega }^2-m^2+iϵ$, henceFor $m>0$, closing the contour above for $t>0$ and below for $t<0$ gives $G(t)=-i{e}^{-im|t|}/(2m)$. The physical time-ordered oscillator quantum field theory propagator is $iG$, not $G$ itself. The shifted stationary path is $q_c=-GJ$ and obeys ${\ddot{q}}_c+m^2{q}_c=J$ with the same vacuum prescription. A retarded inverse would define a different boundary problem. Dividing by the zero-source integral removes $N$ and gives the vacuum source functional of a quantum oscillator. At zero mass an additional infrared prescription is needed for the inverse on constant modes.

Use units $\hslash =1$. The fluctuations about a classical path satisfy $f(0)=f(T)=0$. Expanding the action functional and integrating the term $m{\dot{q}}_c\dot{f}$ by parts givesThe linear term vanishes by the Euler-Lagrange equation and the endpoint conditions. Integrating the quadratic kinetic term by parts yields the Dirichlet fluctuation determinant of a semiclassical propagator:A finite time-slicing first turns this into a product of ordinary Gaussian integrals, whose limit is proportional to $(\det {\mathcal{O}}_c{)}^{-1/2}$. A useful normalization relative to the free operator ${\mathcal{O}}_0=-m{d}^2/d{t}^2$ isThe oscillatory square root is fixed by the short-time limit and the Feynman i-epsilon prescription. Away from zero modes this gives the semiclassical propagator; if several classical paths contribute, their saddle contributions are summed with their corresponding determinant phases.

For a general potential the prefactor depends on the endpoints through $q_c$; writing only $D(T)$ suppresses that dependence. If $V(q)=a{q}^2/2+bq+c$, the Taylor expansion has no terms beyond quadratic and $V^{\prime \prime }=a$ is independent of $q_c$. The Gaussian functional integral is then exact, and its prefactor genuinely depends only on $T$ and the potential parameters. Singular focusing times are understood by continuation of the kernel, rather than by assigning a finite value to a zero-mode determinant.

For the free-particle propagator, $q_c(t)=q_0+(q_1-q_0)t/T$ and $S_c=m(q_1-q_0{)}^2/(2T)$. The free determinant supplies the factor $\sqrt{m/(2\pi iT)}$. Its normalization can also be checked by the momentum-space representationwhere completing the square evaluates the Gaussian integral and the short-time limit gives the position-space delta function.

For the gravitational potential, the original PDF contains $-mgq$; the TeX aid drops the final $q$. The Euler-Lagrange equation is ${\ddot{q}}_c=g$, and its endpoint solution isSubstitution into the kinetic and potential terms givesSince $V^{\prime \prime }=0$, the fluctuation operator is the free one. Thus the quantum-mechanical propagator in a constant force is exactly

The quantum-mechanical propagator is $⟨q_1|e^{-iHT}|q_0⟩$, with Hamiltonian operator $H=-{\partial }_q^2/(2m)+V(q)$. For imaginary time $-i\tau$, insert an orthonormal complete set of energy eigenstates to obtainHere ${\psi }_n(q)=⟨q|n⟩$. This discrete sum assumes a discrete complete spectrum; continuous spectral components require the corresponding spectral integral. The heat-semigroup expression is well defined for an appropriate self-adjoint Hamiltonian bounded below. It is this spectral condition, not an unrestricted formal rotation for every conceivable potential, that justifies the imaginary-time kernel.

A path on the circle lifts to a path on the real line ending at $q_1+2\pi w$, where the integer $w$ is its winding number. The periodic quantum theory sums all these winding sectors with equal weight, giving the real-time winding representation of a quantum rotor kernel. For imaginary time, put $x=q_1-q_0$ and $y=\tau /m$ in the supplied Gaussian summation identity:Comparison with the spectral expansion determinesThe normalization uses $0\le q<2\pi$. The ground state is nondegenerate; levels with $n\ne 0$ have the degeneracy $n\leftrightarrow -n$. Returning to real time replaces $e^{-\tau E_n}$ by $e^{-iT{E}_n}$, with the usual limiting prescription. No twisted boundary condition or flux phase is present in the stated image sum.

Without gauge fixing, gauge-equivalent configurations are integrated repeatedly, and the quadratic gauge-field operator has zero modes along infinitesimal gauge transformations. It therefore has no inverse with which to define perturbative propagators. A gauge condition removes these directions locally in the perturbative expansion. The associated Faddeev-Popov determinant can be represented by anticommuting Faddeev-Popov ghost fields and Faddeev-Popov antighost fields.

First make the normalization explicit. For the fields literally occurring in the action with overall $1/g^2$, putThe resulting kinetic terms are canonical, and the interaction terms carry powers of $g$. We use signature $(-,+,+,+)$ and incoming Fourier modes $e^{ipx}$, consistent with the displayed $p^2-i0$ denominators. The quadratic action in the canonical fields isColor indices are diagonal. Introduce the longitudinal and transverse projectors $P_{L,\mu \nu }=p_{\mu }{p}_{\nu }/p^2$ and $P_T=\eta -P_L$. The gauge kinetic matrix is $-p^2(P_T+{\xi }^{-1}{P}_L)$, with inverse $-(P_T+\xi P_L)/p^2$. The ghost kinetic operator is $-p^2$. The Gaussian functional integral and Grassmann integral therefore giveThe longitudinal formula is understood as the Feynman-prescribed inverse of the kinetic operator; away from its poles the projector inversion is ordinary algebra. The propagators for the original unrescaled fields are $g^2$ times these expressions. Using $P_{T}p=0$ and $P_{L}p=p$ giveswith either normalization. This is the longitudinal gauge propagator contraction.

The canonical ghost interaction comes directly from the covariant derivative:For incoming antighost momentum $p$, the derivative supplies $i{p}^{\mu }$; the expansion of $e^{iS}$ supplies another $i$. Thus the ghost-gluon vertex is $g{p}^{\mu }{f}_{abc}$, as requested. For unrescaled fields its coefficient is instead $p^{\mu }{f}_{abc}/g^2$. The printed order-$g$ rule presupposes canonical normalization; keeping the overall $1/g^2$ and an order-$g$ vertex for the same unrescaled variables would mix conventions. Choosing Fourier modes $e^{-ipx}$ instead reverses the vertex sign and must be done consistently for all momentum assignments.

In Feynman gauge, $\xi =1$, the one-loop ghost two-point graph has one internal ghost line and one internal gauge line joined by two such vertices. The external ghost line is open, so it does not carry a closed ghost-loop minus sign. With an orthonormal adjoint basis, the structure constants are totally antisymmetric, and the color product along this line is $f_{adc}{f}_{cdb}=-C{\delta }_{ab}$. The two internal propagators supply $(-i{)}^2=-1$, leaving a positive color-and-vertex factor for the amputated insertion. DefineThe amputated insertion is $i{\delta }_{ab}{g}^2{I}_d(p)C$. Restoring the two external canonical ghost propagators gives the one-loop contributionThe literal unrescaled correlator in the question is $g^2{G}^{(1)}$, hence of order $g^4$, whereas its tree term is $-i{\delta }_{ab}{g}^2/(p^2-i0)$. In $d\ne 4$ the dimensionless renormalized coupling convention adds ${\mu }^{4-d}$ multiplying $g^2$ in the canonical loop term.

To evaluate the Feynman integral, combine the denominators with a Feynman parameter:Shift $\ell =k-(1-\alpha )p$. The denominator becomes $[{\ell }^2+\alpha (1-\alpha )p^2-i0{]}^2$, and the numerator is $p\cdot \ell +(1-\alpha )p^2$. The odd term vanishes in translation-invariant dimensional regularization. For spacelike $p$, a Wick rotation converts the remaining integral into a Euclidean one and cancels the prefactor $1/i$. The needed radial integral isIndeed, spherical coordinates and $u={\ell }_E^2/M^2$ leave a beta integral ${\int }_0^{\infty }{u}^{d/2-1}(1+u{)}^{-2}du=\Gamma (d/2)\Gamma (2-d/2)$; the angular factor cancels $\Gamma (d/2)$. Establish this first in a convergent range and then analytically continue in $d$. Substituting $M^2=\alpha (1-\alpha )p^2$ yieldsThis is the printed result, including the factor $1/i$ in its Minkowski-space definition. Equivalently the parameter integral is $B(d/2-1,d/2)$. Symmetry under $k\leftrightarrow p-k$ also gives $I_d(p)$ as $p^2/2$ times the massless scalar bubble, independently checking the numerator factor.

Write $ϵ=4-d$. Since $\Gamma (ϵ/2)=2/ϵ+O(1)$ and the parameter integral tends to ${\int }_0^1(1-\alpha )d\alpha =1/2$, the pole isHere $p$ is off shell and nonzero, separating the ultraviolet pole from massless infrared issues. The unrescaled amplitude has an extra factor $g^2$. The divergence is proportional to the existing ghost kinetic operator, so the one-loop ghost kinetic counterterm in Feynman gauge is local:It inserts $-i\delta Z_c{p}^2$ into the inverse propagator, cancelling the divergent insertion $+i{g}^{2}C{I}_d(p)$. With the two external propagators attached, its contribution is $+i{\delta }_{ab}\delta Z_c/p^2$, cancelling the displayed connected amplitude pole. In the original variables the same action term is $-\delta Z_c\partial \bar{c}\cdot \partial c/g^2$. If instead one defines dimensional regularization by $d=4-2ϵ$, the identical counterterm is written $\delta Z_c=g^{2}C/(32{\pi }^2ϵ)$; the factor of two is purely the regulator convention.