Towards the QED beta function and renormalons at $1/N_f^2$ and $1/N_f^3$

We determine the $1/N_f^2$ and $1/N_f^3$ contributions to the QED beta function stemming from the closed set of nested diagrams. At the order of $1/N_f^2$, we discover a new logarithmic branch cut closer to the origin when compared to the $1/N_f$ results. The same singularity location appears at $1/N_f^3$, and these correspond to a UV renormalon singularity in the finite part of the photon two-point function.

Figure 1

Left: The ratio test applied to the expansion coefficients of ${\beta }_{\text{nested}}^{(2)}(K)$ reveals that the radius of convergence is $K=3$. A Richardson extrapolation is used to accelerate the convergence of the series. Right: The prefactor of the leading large-order behavior is determined to be $-\frac{1}{2n(n-1)3^n}$, which leads to a much faster convergence than $-\frac{1}{2n^2{3}^n}$. See Appendix pp2.

Figure 2

Nested QED beta function at $\mathcal{O}(1/N_f^2)$. At $K=3$ the beta function is finite, but it has a logarithmic branch cut.

Figure 3

(Conformal) Padé approximants of the nested QED beta function at $\mathcal{O}(1/N_f^2)$ with the branch cut subtracted; see Eq. (14). The Padé approximants seem to hint toward a pole or a branch cut at $K=\frac{15}{2}$.

Figure 4

(Conformal) Padé approximants of the nested QED beta function at $\mathcal{O}(1/N_f^3)$. The Padé approximants seem to hint toward a pole or a branch cut at $K=3$.

Figure 5

Real and imaginary parts of the Borel transform of the finite part of the renormalized photon two-point function at the order of $1/N_f$. This Borel transform is obtained from the analytic formula (a20) in Appendix pp1-s2. We see UV renormalon singularities at $t=3k$ with $k=1,2,\dots ,\infty$ and IR renormalon singularities at $t=-3k$ with $k=2,3,\dots ,\infty$.

Figure 6

The $1/N_f^2$ coefficients of the $1/\epsilon$ part of the nested diagrams and corresponding counterterm are each factorially divergent. As these are not separately RG independent, we choose ${\mu }^2=-p^2/4\pi$, $p^2$ being the external momentum. Their sum produces a RG-independent convergent series according to the large-order behavior given in Eq. (11).

Figure 7

Real and imaginary parts of the Borel transform of the finite part of the renormalized photon two-point function at the order of $1/N_f^2$. This Borel transform has been reconstructed from the finite number of expansion coefficients using the Padé-conformal method described in Appendix pp3. We see a UV renormalon singularity at $t=+3$, an IR renormalon singularity at $t=-6$, and a hint of a further IR renormalon singularity at $t=-9$.

Figure 8

Real and imaginary parts of the Borel transform of the finite part of the renormalized photon two-point function at the order of $1/N_f^3$. This Borel transform has been reconstructed from the finite number of expansion coefficients using the Padé-conformal method described in Appendix pp3. We see a UV renormalon singularity at $t=+3$, an IR renormalon singularity at $t=-6$, and a hint of a further IR renormalon singularity at $t=-9$.

Figure 9

Topologies contributing to the beta function at $1/N_f^2$.

Figure 10

Large-order behavior after taking two derivatives of the nested beta function. Left: The ratio test reveals that the radius of convergence is $K=3$. Right: The prefactor of the leading large-order behavior is determined to be $-\frac{1}{18·3^n}$, indicating a simple pole at $K=3$.

Figure 11

Conformal Padé approximant of the nested QED beta function at $\mathcal{O}(1/N_f^2)$ compared to the exact result and a Padé approximant of the same order.