Tadpole diagram

A tadpole diagram contains an internal line that begins and ends at the same interaction vertex. In a scalar theory it commonly gives a momentum-independent correction to a mass or lower-order coupling.

In a theory with interaction $-g{\phi }^{†}\phi \varphi$, the neutral correlator $⟨T{\phi }^{†}(x)\phi (y)\varphi (z)⟩$ contains a connected three-leg contraction and, unless subtracted, the disconnected product of a free complex-field propagator with the induced one-point function of $\varphi$. At first order the latter is proportional to $-ig{D}_0(x-y)D_0(0)\int d^{4}w{D}_m(z-w)$. Vacuum-bubble normalization alone does not remove it. Normal ordering, a vanishing-tadpole condition, or taking the connected correlator does.

For interaction ${\lambda }_{2k}{\varphi }^{2k}/(2k)!$, one vertex with two labelled external legs has $k-1$ independently paired tadpole diagrams. Counting contractions gives symmetry factor $S=2^{k-1}(k-1)!$. In a normalized Euclidean path integral, its amputated contribution is $-{\lambda }_{2k}{T}_D(m{)}^{k-1}/S$, where $T_D$ is the tadpole integral. It is linear in the coupling but has $k-1$ loops and a highest four-dimensional ultraviolet pole of order $k-1$ for nonzero mass.

Use $-g{\varphi }^3/3!$ and $D=6-2ϵ$ with $m>0$. In an unshifted vacuum, the one-loop mean field has pole $v_{\mathrm{div}}=-g{m}^2/[4(4\pi {)}^3ϵ]$. Attaching it to a cubic vertex gives the connected but one-particle-reducible two-point insertion $(-ig)v_{\mathrm{div}}=i{g}^2{m}^2/[4(4\pi {)}^3ϵ]$. It supplements the proper bubble unless a tadpole subtraction sets the renormalized mean field to zero. A disconnected product of two one-point functions is a separate two-loop term and is not a propagator insertion.

// Target: relativistic-quantum-field.bigb

A linear counterterm can be chosen so that the renormalized one-point function, or equivalently the chosen vacuum expectation value, is zero. This cancels attached tadpole diagrams in the perturbative expansion about that vacuum. In a massive six-dimensional cubic scalar field theory, the one-loop proper tadpole is $t=g{I}_1/2$; without subtraction the stationary background shifts by $v=-t/m^2+O(g^3)$ at one-loop order. One must specify this convention when comparing reducible exchange diagrams or mass corrections.