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Use the classically equivalent Polyakov action, fix worldsheet diffeomorphisms and Weyl invariance, and perform a Wick rotation of the string worldsheet. In complex coordinates, the worldsheet ghost action for the closed string isThe worldsheet ghost fields are anticommuting. Their conformal weights are $b:(2,0)$, $c:(-1,0)$ and $\bar{b}:(0,2)$, $\bar{c}:(0,-1)$. The two chiral sectors are counted separately.

In either chiral sector, each embedding coordinate is a free boson conformal field theory with central charge one. For the bc system, putting $J=2$ in $c_{bc}=-2(6J^2-6J+1)$ gives $c_{bc}=-26$. Cancellation of the worldsheet Weyl anomaly requiresThe antiholomorphic sector gives the same equation, rather than doubling the required critical dimension of string theory. This dimension count assumes flat target spacetime with just the embedding fields; more general internal conformal field theories change the matter accounting.

For the spinning string, the worldsheet Majorana fermions have conformal weight $1/2$. At $J=1/2$, the complex anticommuting bc system has $c=1$; a real Majorana spinor contributes half of this. Hence $D$ real worldsheet Majorana fermions contribute $D/2$ per chiral sector. Ramond sector versus Neveu–Schwarz sector changes their mode numbers and vacuum energies, not this local central charge.

The additional local worldsheet supersymmetry has a fermionic gauge parameter, so its superconformal ghosts are commuting fields forming a beta-gamma system, with conformal weights $\beta :3/2$ and $\gamma :-1/2$ in the holomorphic sector. Commuting statistics reverse the sign of the bc system formula, givingTogether with the reparameterization worldsheet ghost fields, their total central charge is $-26+11=-15$. ThusThese cancellations make the string worldsheet gauge symmetries compatible with quantization; in the BRST symmetry formulation they enter the nilpotence condition for the BRST operator.

After Wick rotation, the zero-momentum bubble integral with cutoff regularization isThe three channels contribute $3{\lambda }^{2}I(0)/2$. For $\Lambda \gg m$, this is $\frac{3{\lambda }^2}{32{\pi }^2}[\log ({\Lambda }^2/m^2)-1]$, so the zero-momentum renormalization condition is enforced byand hence $F=e{m}^2$. Keeping the exact cutoff expression merely replaces $F$ by the finite cutoff-dependent quantity defined by $\log ({\Lambda }^2/F)=16{\pi }^{2}I(0)$.

For $H={\hat{p}}^2/(2m)+V(\hat{x})$, divide elapsed time $t$ into $J$ slices $\Delta t=t/J$ and apply the Trotter product formula. Inserting complete position and momentum bases gives the regulated phase-space path integralThe endpoint values are $x_0=x_i$, $x_J=x_f$. Each Gaussian momentum integration in a phase-space path integral can be performed explicitly, leaving the configuration-space path integralIts measure is the time-sliced limit of $(m/(2\pi i\hslash \Delta t){)}^{J/2}{\prod }_{j=1}^{J-1}d{x}_j$, with the usual infinitesimal convergence prescription. The integral sums amplitudes, not positive classical probabilities. Operator ordering remains part of the time-slice definition. Expanding a short-time kernel recovers the Schrödinger equation; composing kernels reproduces the time-evolution operator.

In the classical limit, rapid phase cancellation suppresses paths away from stationary action. Varying $S$ with fixed endpoints gives $m\ddot{x}=-V^{\prime }(x)$. The semiclassical result sums the classical-path phases multiplied by fluctuation functional determinants; for a quadratic Hamiltonian the fluctuation integral is Gaussian and this expansion is exact.

For quantum statistical mechanics, continue $t=-i\tau$ by a Wick rotation. The Euclidean action and thermal trace giveThe trace closes position-space paths. In a many-particle trace one also sums over endpoint permutations with the appropriate exchange signs. In a coherent state field formulation, bosonic fields are periodic and fermionic coherent states instead give antiperiodic Grassmann fields, hence bosonic and fermionic Matsubara frequencies $2\pi n/(\beta \hslash )$ and $(2n+1)\pi /(\beta \hslash )$. ^[ref]

The quantum-to-classical imaginary-time mapping views a local $d$-dimensional quantum problem as a classical-looking field theory in $d+1$ dimensions, with imaginary-time extent $\beta \hslash$. A genuine classical probability interpretation additionally requires a nonnegative weight; fermion signs and Berry terms can obstruct it.

For a smooth potential in the high-temperature or small-$\hslash$ limit, imaginary-time paths fluctuate only over the thermal de Broglie wavelength ${\lambda }_T=\sqrt{2\pi \beta {\hslash }^2/m}$. If this is short compared with the potential's variation scale, the diagonal free kernel givesThis classical phase-space limit of the thermal kernel is the classical phase space partition function for one degree of freedom. A dilute identical-particle gas also needs its $1/N!$ factor; when exchange is appreciable, classical statistics fails. As a check, a single quantum harmonic oscillator gives $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}\to 1/(\beta \hslash \omega )$, precisely the classical phase-space integral.

A Wick rotation $t=-i{\tau }_E$ changes real-time evolution into $e^{-H{\tau }_E/\hslash }$. At ${\tau }_E=\beta \hslash$, taking the trace integrates equal endpoints:Thus temperature sets the imaginary-time circumference, and bosonic coordinate paths are periodic. The measure is fixed by the short-time propagator. The trace sums paths, not just a single stationary trajectory; a quadratic action is special because the saddle plus Gaussian fluctuations is exact.

The real-time path integral sums $e^{iS/\hslash }$ over paths. In a classically forbidden region, the dominant semiclassical contribution is clearer after Wick rotation: the weight becomes $e^{-S_E/\hslash }$ and stationary paths obey $m\ddot{q}=V^{\prime }(q)$, equivalent to motion in the inverted potential. A finite-action path joining degenerate minima is an instanton.

For $V(q)=\lambda (q^2-a^2{)}^2$, $\lambda >0$, zero Euclidean mechanical energy gives $m{\dot{q}}^2/2=V$. ThusThe reversed trajectory is an anti-instanton.^[ref] Expanding $q=q_I+\eta$ produces the fluctuation operator $-{\partial }_{{\tau }_E}^2+V^{\prime \prime }(q_I)/m$. Its translation zero mode is handled by integrating the collective centre ${\tau }_0$, not by dividing by a zero determinant. The remaining determinant and Jacobian form an attempt-frequency prefactor $A$.

When $S_0/\hslash \gg 1$, well-separated events form a dilute gas with rate scale $\kappa =A{e}^{-S_0/\hslash }$. Paths returning to the same well have an even number of events, while paths ending in the other well have an odd number. The sums yield $\cosh (\kappa \mathcal{T})$ and $\sinh (\kappa \mathcal{T})$, givingThe ground state is the lower symmetric combination of localized well states: tunnelling lifts the apparent classical degeneracy. This splitting is nonperturbative in $\hslash$, beyond a power series about either minimum. In a metastable well a bounce, with a negative fluctuation mode, describes decay rather than the stable doublet splitting. Finite temperature, overlapping events or a low barrier require improvements beyond the dilute approximation.

For one coordinate with Hamiltonian $H={\hat{p}}^2/(2m)+V(\hat{x})$, the propagator is $K(x_f,t_f;x_i,t_i)=⟨x_f|e^{-iHT/\hslash }|x_i⟩$, where $T=t_f-t_i>0$. Divide $T$ into $M$ intervals $\delta t$, use the Trotter product formula to separate kinetic and potential evolution, and insert complete position and momentum bases at each slice. The momentum integral is Gaussian:Multiplying the short-time kernels gives the time-sliced configuration-space path integral,with $x_0=x_i$, $x_M=x_f$. Continuum notation abbreviates this limit as ${\int }_{x_i}^{x_f}\mathcal{D}x{e}^{iS[x]/\hslash }$, where $S=\int dt[m{\dot{x}}^2/2-V(x)]$. The measure includes the displayed normalization; it is not an unweighted count of paths. A small damping or analytic-continuation prescription defines the oscillatory integrals. The paths contribute complex amplitudes, so alternative histories interfere before probabilities are calculated. This is the central difference from a classical statistical mixture of trajectories.^[ref]

When actions vary rapidly relative to $\hslash$, stationary phase selects classical paths. Put $x=x_{\mathrm{cl}}+\eta$, with $\eta$ zero at the endpoints. Vanishing first variation gives $m{\ddot{x}}_{\mathrm{cl}}+V^{\prime }(x_{\mathrm{cl}})=0$. The quadratic term isThe resulting Gaussian fluctuation determinant is the semiclassical propagator prefactor. In one dimension, away from caustics,The branch convention makes $\nu =0$ for a direct free-particle path before any caustic. The Maslov index tracks the phase changes as fluctuations pass through a focal singularity. In several dimensions replace the endpoint derivative by the Van Vleck determinant. The approximation sums all relevant classical paths; real-time action need only be stationary, not a minimum. It fails in this simple form at a caustic or coalescing saddles, where a uniform approximation is needed. Quadratic actions have no higher fluctuation terms, making their Gaussian evaluation exact. For example $S_{\mathrm{cl}}=m(x_f-x_i{)}^2/(2T)$ gives the exact free-particle kernel $\sqrt{m/(2\pi i\hslash T)}{e}^{i{S}_{\mathrm{cl}}/\hslash }$. Tunnelling involves classically forbidden regions and can be described by imaginary-time saddles rather than real classical trajectories.

Quantum statistical mechanics follows by Wick rotation $t=-i\tau$. The imaginary-time evolution operator is $e^{-\tau H/\hslash }$, with Euclidean actionTaking a thermal trace identifies and integrates the endpoint positions:This thermal trace as a closed Euclidean path integral explains the periodic imaginary-time direction. A density-matrix element instead retains open endpoints. For many identical particles the trace also sums over endpoint permutations, with a plus sign for bosons and permutation parity for fermions. In coherent-state language this becomes periodic bosonic fields and antiperiodic Grassmann fields, with corresponding Matsubara frequencies.

The Euclidean weight resembles a classical Boltzmann weight. A time-sliced local quantum field in $d$ spatial dimensions becomes a classical-looking field model in $d+1$ dimensions, with anisotropic couplings and extra extent $\beta \hslash$. Thus zero-temperature quantum criticality can be treated using an infinitely extended imaginary-time direction, while nonzero temperature imposes a finite thickness. The dynamical scaling of time need not equal the spatial scaling: the dynamical critical exponent specifies their relationship. This quantum-to-classical imaginary-time mapping requires qualification when weights have signs or imaginary terms; such a functional integral is not automatically an ordinary positive classical probability distribution. Even with a positive weight, the mapping concerns equilibrium statistics, not literal dissipative real-time trajectories. For a free particle the Euclidean kernel is a diffusion kernel with diffusion coefficient $\hslash /(2m)$, illustrating the mathematical correspondence.

The classical high-temperature limit is a further approximation, rather than simply another name for imaginary time. If quantum level spacings and exchange effects are negligible, a one-particle partition function in $d$ dimensions becomesFor indistinguishable particles there is a factor $1/N!$ in the dilute classical gas, and exchange is negligible when the number density times the thermal de Broglie wavelength to power $d$ is small. For a harmonic oscillator, the exact quantum result $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}$ tends to $1/(\beta \hslash \omega )$ as $\beta \hslash \omega \to 0$, recovering the classical phase-space integral. stationary phase explains classical mechanics; the short imaginary-time, weak-exchange limit explains classical statistical mechanics. These are related but distinct limits of the same quantum formulation.

The trace integrates the diagonal thermal kernel over a single fundamental interval, $Z={\int }_0^{2\pi }d{\varphi }_0⟨{\varphi }_0|e^{-\beta \hat{H}}|{\varphi }_0⟩$. On the circle a closed path can wind an integer number $m$ of times. Lift it to the real covering coordinate so its endpoints are $\varphi (0)={\varphi }_0$ and $\varphi (\mathcal{T})={\varphi }_0+2\pi m$, with $\mathcal{T}=\hslash \beta$.

Under Wick rotation $t=-i\tau$, $\dot{\varphi }=i{\partial }_{\tau }\varphi$, so the real-time phase $iS/\hslash$ becomes minus the Euclidean action divided by $\hslash$, withThus sum the path integral $e^{-S_E/\hslash }$ over these lifted endpoint conditions and integrate ${\varphi }_0$ from zero to $2\pi$. This gives the requested winding-sector representation of the trace. The flux term is a total derivative, so it leaves the local classical equation unchanged, but its weight is $e^{i2\pi fm}$ and distinguishes different winding sectors. This is the Aharonov-Bohm phase in a thermal path integral, rather than a force from a magnetic field on the ring itself.

The positive denominators in this part are Euclidean. Starting with the Minkowski spacetime Feynman rules, first perform a Wick rotation; a bare Minkowski metric integral without a pole prescription would not equal the displayed positive expression. Work initially with $m^2>0$ and $\Re n>D/2$, where the Euclidean massive loop integral converges.

Use Schwinger parameterization followed by the Gaussian integral:The remaining Gamma function integral givesThe factorial form is for positive integral $n$. Dimensional regularization defines the result elsewhere by analytic continuation.

For the scalar bubble integral, combine the two denominators with a Feynman parameter $x$ and shift the integration variable to $\ell =k-(1-x)p_E$. ThenFor $D=4-ϵ$ this isThe Gamma function recurrence gives $\Gamma (ϵ/2)=2/ϵ+O(1)$, and the parameter integral is $1+O(ϵ)$. ConsequentlyThis ultraviolet divergence is independent of the external momentum. The assumption $m^2>0$ avoids an infrared divergence. In particular the completely massless zero-momentum integral is a scaleless integral and vanishes in dimensional regularization; its ultraviolet and infrared poles must be separated before assigning it the displayed ultraviolet residue.

Source correction: equation 11 on the original page has $\Gamma (\alpha -1/2)$. The dimensionally regulated integral instead contains $\Gamma (\alpha -d/2)$. This is not an interchangeable convention: for $\alpha =2$ and $d=4-ϵ$, the correct factor $\Gamma (ϵ/2)$ has a logarithmic ultraviolet pole, whereas the printed $\Gamma (3/2)$ is finite.

To obtain the correct expression, shift the loop momentum and perform a Wick rotation. For positive Euclidean ${\mathcal{M}}^2$, introduce a Schwinger parameterization:The derivation first holds in the convergent range and defines the result elsewhere by analytic continuation. For integer $\alpha$, rotating back giveswith the branch inherited from $i0$. This is the Lorentzian continuation of the Euclidean massive loop integral.

Set $p=0$, $\alpha =1$, $d=4-ϵ$. Since $\Gamma (-1+ϵ/2)=-2/ϵ+O(1)$,Consequently the directed tadpole in part d givesThe counterterm is $-i{\delta }_{m^2}$, so cancelling only the pole in the minimal subtraction scheme requiresThis is the quartic complex-scalar mass renormalization result. The scale factors and logarithms contribute finite terms at this order and do not change the displayed pole coefficient. A real-field check gives a tadpole coefficient $\frac{1}{2}(3g/2+g/2)=g$, in agreement with the oriented complex-field computation.

There are three independent nonfactorized momentum integrals, namely a tadpole integral, the three-line sunset diagram integral, and the five-line banana integral. The six displayed Feynman diagrams involve six complete integral expressions:The distinction between independent building blocks and their products removes an ambiguity in counting “different momentum integrals”. For the strict divergent-wave-function renormalization interpretation of ii, the graphs in c reduce to $T,T^2,B_5$, comprising three full integral expressions and two independent nonfactorized building blocks.

Make a Wick rotation and use $m>0$ to separate ultraviolet divergences from infrared divergences. Write $D_E(k)=1/(k^2+m^2)$, and let $T,B_r(P)$ be the Euclidean versions of the integrals above, with their harmless overall factors of $i$ removed. A spherical ultraviolet cutoff givesThus the one-vertex quartic graph is linearly divergent, and the one-vertex sextic graph is quadratically divergent as $T_{\Lambda }^2$.

For the other two building blocks, the spatial Fourier transform is especially transparent. The three-dimensional Euclidean scalar propagator isThe convolution theorem converts the $r$ parallel lines into $G(x{)}^r$. Angular integration gives, with a short-distance cutoff $a={\Lambda }^{-1}$,This is a different convenient cutoff regularization for the connected block; power-divergent coefficients depend on the cutoff convention, whereas the conclusions about powers and the presence of a kinetic logarithm do not. The sinc function is understood to be $1$ at $P=0$.

For $r=3$, the integrand starts as ${\rho }^{-1}+O(1)$, soIts divergent part is independent of external momentum. In particular,The finite difference and its finite derivative show why the pure three-line sunset diagram does not need a divergent wave-function renormalization in three dimensions.

For $r=5$, the small-distance Taylor expansion isConsequentlyIn particular, the nonzero $P^2\log \Lambda$ term requires a genuine divergent wave-function renormalization, in addition to the mass counterterm. The coefficients here refer to the displayed position-space cutoff and have not included Feynman-diagram symmetry factors or bare couplings.

The comparison with the superficial degree of divergence isThus the actual leading power agrees with $\omega$ in every case. The additional logarithms in products come from ultraviolet subdivergences, which uniform power counting alone does not resolve.

For $T{B}_3$ and $T^2{B}_3$, differentiation with respect to $P^2$ removes the logarithmic divergence of $B_3$, but leaves divergent factors $T$ or $T^2$ multiplying a finite, nonpolynomial function of $P^2$. These are subdivergent contributions, not new nonlocal counterterms: subtracting each tadpole diagram subdivergence and the momentum-independent $B_3$ subdivergence removes them. For example, in a factorized subtraction scheme $T{B}_3\mapsto (T-T_{\mathrm{div}})(B_3-B_{3,\mathrm{div}})$, and likewise for $T^2{B}_3$. The remaining $B_5$ kinetic logarithm explains the strict interpretation in ii. A dimensional regularization scheme can suppress power divergences; it does not invalidate this cutoff-based comparison with $\omega$.

Continue with the cubic scalar field theory of c, and use precisely its counterterm signs. Take $m^2>0$ to separate ultraviolet divergences from infrared divergences. Make a Wick rotation and impose the rotationally invariant cutoff regularization $|q_E|<\Lambda$. PutFor example, spherical coordinates give $I_1=C{\int }_0^{{\Lambda }^2}{t}^2/(t+m^2)dt$. Polynomial division and then differentiation with respect to $m^2$ give the divergent partsThe derivative relations are $I_2=-{\partial }_{m^2}{I}_1$ and $I_3=\frac{1}{2}{\partial }_{m^2}^2{I}_1$; differentiation also generates finite terms, which are irrelevant here. The first supplied integral identity is consistent with these divergent parts after Wick rotation.

The second supplied expansion is incorrect. Write $A=p^2-m^2$. Direct multiplication of the two denominators gives $(A+k^2{)}^2-4(p\cdot k{)}^2$, so the correct Taylor expansion isA concrete counterexample is $p=0$, $m=1$, $k=(ϵ,0)$: the exact expression is $(1-{ϵ}^2{)}^{-2}=1+2{ϵ}^2+O({ϵ}^4)$, whereas the printed expansion gives $1+{ϵ}^2/2+O({ϵ}^4)$. We therefore compute the self-energy without that identity.

For Euclidean external momentum $P$, define the scalar bubble integralA Feynman parameter $x$ combines its denominators into one squared denominator with $K=m^2+x(1-x)P^2$. Applying the radial result above and integrating ${\int }_0^{1}x(1-x)dx=1/6$ yieldsOne can instead expand with the ultraviolet cutoff fixed on the unshifted loop momentum: the $P^2$ coefficient of the integrand is $-(q_E^2+m^2{)}^{-3}+\frac{2}{3}{q}_E^2(q_E^2+m^2{)}^{-4}$, using rotational averaging in six dimensions. Its leading radial term is $-1/(3q_E^6)$, giving the same logarithm. Shifting a sharp integration boundary changes finite terms in this scalar bubble integral, and does not change the displayed divergent coefficients.

To fix all signs and symmetry factors together, expand the one-loop Euclidean effective action around a background scalar field:These terms are the tadpole diagram, the two-point bubble diagram, and the triangle Feynman diagram. The tadpole diagram has factor $1/2$; the two-point bubble diagram also has factor $1/2$ after two differentiations with respect to the background scalar field. The local three-point interaction vertex is $g^3{I}_3$. Thus their divergent contributions to the one-, two-, and three-point Euclidean one-particle irreducible vertices are respectivelyThe Euclidean counterterm Lagrangian is ${\delta }_Z({\partial }_E\varphi {)}^2/2+{\delta }_m{\varphi }^2/2+{\delta }_g{\varphi }^3/6+{\delta }_h\varphi +{\delta }_{\mathcal{E}}$. Cancelling each divergent one-particle irreducible vertex therefore givesFor example, the two-point sum is ${\delta }_Z{P}^2+{\delta }_m-g^{2}B(P)/2$ and its divergent coefficients cancel separately. Finite counterterms depend on the chosen renormalization conditions. The zero-mass limit can use a positive subtraction scale in the logarithms after infrared regularization.

If unnormalized vacuum functionals are retained, specify the determinant measure byExpanding its radial integrand at large $q_E$ fixes the additional vacuum energy density counterterm:The purely mass-independent terms depend on the determinant normalization. Normalized Green functions eliminate this entire constant.

Four external $\Phi$ lines require four cubic interaction vertices, since each such interaction vertex contains only one heavy field. Their other eight endpoints must form four light internal lines. A connected Feynman diagram therefore has $V=I=4$ and $L=1$: the leading contribution is a one-loop box diagram, of order $g^4$, with four $\varphi$ scalar propagators. Charge-two incoming and outgoing heavy lines attach alternately around the box, as shown in panel (d) of b's figure. Each incoming heavy line supplies the two outgoing light-charge arrows at its corner, and each outgoing heavy line absorbs them. With external labels fixed there is one such box and its symmetry factor is one.

For $|\mathbf{p}|\ll M\ll m$, the external energies are also small compared with $m$, so expand the loop integrand in all external four-momenta. Its leading term isA Wick rotation and the Euclidean massive loop integral giveHence, more precisely than just its requested scaling,The mass dimension $[g]=1$ makes $g^4/m^4$ dimensionless, as a four-dimensional scattering amplitude should be. The leading amplitude has no extra identical-particle multiplicity.

Set the renormalized $g_3$ to zero. For a six-scalar vertex, b gives $\omega =-n_2$, so a new overall logarithmic divergence can only occur in graphs with $n_2=0$. With $b=6,f=0$, the line counts become $I_f=n_1$ and $I_b=n_1-3$, hence $L=n_1-2$.

At one loop this forces $n_1=3$ and $I_b=0$: the only candidate is a closed triangle Feynman diagram with two external scalar legs at each vertex. Its numerator at large loop momentum is proportional toThis vanishes for the specified two-component gamma matrices, each of which is traceless. Replacing one leading $/k$ by the fermion mass or an external momentum lowers the superficial degree of divergence from zero to at most $-1$. Thus the apparent logarithm cancels. Graphs with a $g_2$ vertex already have negative overall degree; the pure-scalar one-loop triangle, for instance, has $\omega =-3$. There is no one-loop ultraviolet subdivergence inside these proper one-loop vertices. ThereforeThe six-point correlation function need not vanish: finite terms are allowed. Nor is it correct to say that all odd gamma-matrix traces vanish in $2+1$ dimensions. In this representation,The cancellation above follows from repeating the same leading loop momentum, not from a false four-dimensional trace rule.

At two loops, $n_1=4$, $I_f=4$ and $I_b=1$ are possible. A closed fermion box with a scalar line joining two of its vertices leaves six external scalar legs. It has $L=2$, $\omega =0$, and four leading gamma matrices; the one-loop odd-numerator argument no longer applies.

A background-field check demonstrates that its logarithm is actually nonzero. Put $\varphi =v+\eta$ with constant $v$. Define the background fermion mass $\mu =m+g_1{v}^2/2$, the background scalar squared mass ${\mathcal{B}}^2=M^2+g_2{v}^2/2$, and the induced three-leg Yukawa coupling $y=g_{1}v$. The two-loop quantum effective potential contains the sunset diagram with two fermion lines and one scalar line. After a Wick rotation, its relevant integral, apart from a nonzero overall combinatorial factor, isHere the dots and gamma-matrix traces are Euclidean. With $A=k^2+{\mu }^2$, $B=l^2+{\mu }^2$, $C=(k-l{)}^2+{\mathcal{B}}^2$, its numerator is exactlyThe first three terms reduce to products of one-loop tadpole integrals, which have no logarithmic pole near three dimensions in dimensional regularization. The remaining term isFor positive regulator masses, the large-momentum region of $H$ has a positive, nonzero logarithmic divergence. One can see this directly by restricting the relative angles and the ratio $|k|/|l|$ to compact intervals away from zero: the remaining common radial integral is proportional to ${\int }^{\Lambda }dr/r$. Its proper one-loop subgraphs are ultraviolet convergent.