Summations of large logarithms by parton showers

We propose a method to examine how a parton shower sums large logarithms. In this method, one works with an appropriate integral transform of the distribution for the observable of interest. Then, one reformulates the parton shower so as to obtain the transformed distribution as an exponential for which one can compute the terms in the perturbative expansion of the exponent. We apply this general program to the thrust distribution in electron-positron annihilation, using several shower algorithms. Of the approaches that we use, the most generally applicable is to compute some of the perturbative coefficients in the exponent by numerical integration and to test whether they are consistent with next-to-leading-log summation of the thrust logarithms.

Figure 1

Plot of $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$, Eqs. (248) and (249), versus $\log (\nu )$ (solid red curve). For large $\log (\nu )$ the graph is approximately a straight line, corresponding to only one factor of $\log (\nu )$, indicating that the shower generates $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$ at NLL accuracy. The dashed blue curve is $d⟨{\mathcal{I}}_2^{[2]}(\nu )⟩/d\log (\nu )$. The dotted red curve shows an approximate version of $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$ described in the text. These calculations and calculations of $⟨{\mathcal{I}}_k^{[2]}(\nu )⟩$ in later figures use full QCD color.

Figure 2

Plot of $⟨{\mathcal{I}}_3^{[2]}(\nu )⟩$, Eqs. (248) and (249), versus $\log (\nu )$ (solid red curve). The dashed blue curve is $d⟨{\mathcal{I}}_3^{[2]}(\nu )⟩/d\log (\nu )$. For large $\log (\nu )$ the graph of $d⟨{\mathcal{I}}_3^{[2]}(\nu )⟩/d\log (\nu )$ is approximately a straight line, indicating that the shower generates $⟨{\mathcal{I}}_3^{[2]}(\nu )⟩$ at NLL accuracy.

Figure 3

Plot of $(\tau /{\sigma }_{\mathrm{H}})d\sigma /d\tau$ according to Deductor with $\mathrm{\Lambda }$ ordering at $Q^2=(10\mathrm{TeV}{)}^2$ compared to the NLL expectation, Eqs. (221) and (222). In Deductor, we use a cutoff for splittings: $k_{\mathrm{T}}>1\mathrm{GeV}$ and $\mathrm{\Lambda }>1\mathrm{GeV}$. The Deductor curve is higher than the NLL curve at $\tau \approx 0.01$ and lower for small $\tau$. The Deductor calculation uses the LC+ approximation for color.

Figure 4

Plots of $(\tau /{\sigma }_{\mathrm{H}})d\sigma /d\tau$ using Deductor with $\mathrm{\Lambda }$ ordering at $Q^2=(10\mathrm{TeV}{)}^2$. The black dashed curve is the Deductor curve from Fig. 3. The blue solid curve is the NLL formula from Fig. 3. The red solid curve is the Deductor result corrected with the factor $r(\tau )$ to include more exact ${\alpha }_{\mathrm{s}}$ evolution. The purple solid curve, with noticeable statistical fluctuations, is the corrected Deductor result with two units of extra color beyond the LC+ approximation.

Figure 5

Plots of $(\tau /{\sigma }_{\mathrm{H}})d\sigma /d\tau$ using Deductor with $\mathrm{\Lambda }$ ordering at $Q^2=(10\mathrm{TeV}{)}^2$. All three curves are Deductor results corrected with the factor $r(\tau )$ to include more exact ${\alpha }_{\mathrm{s}}$ evolution. The red solid curve uses the LC+ approximation and is taken from Fig. 4. In the green solid curve the color approximation is reduced from LC+ to color using U(3) as the color group. This puts a factor $C_A/2$ at $qq\mathrm{g}$ vertices. The purple dashed curve also uses U(3) as the color group but inserts a factor $C_{\mathrm{F}}$ at the first emission.

Figure 6

Plot of $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$ versus $\log (\nu )$, as in Fig. 1, for the Deductor shower algorithm with $k_{\mathrm{T}}$ ordering. The blue dashed curve is $d⟨{\mathcal{I}}_2^{[2]}(\nu )⟩/d\log (\nu )$.

Figure 7

Plot of $⟨{\mathcal{I}}_3^{[2]}(\nu )⟩$ versus $\log (\nu )$, as in Fig. 2, for the Deductor shower algorithm with $k_{\mathrm{T}}$ ordering. The blue dashed curve is $d⟨{\mathcal{I}}_3^{[2]}(\nu )⟩/d\log (\nu )$.

Figure 8

Plot of $(\tau /{\sigma }_{\mathrm{H}})d\sigma /d\tau$ with $\mathrm{\Lambda }$ ordering and $k_{\mathrm{T}}$ ordering at $Q^2=(10\mathrm{TeV}{)}^2$. Both are compared to the NLL expectation, Eqs. (221) and (222). We use a cutoff on the transverse momentum in splittings: $k_{\mathrm{T}}>1\mathrm{GeV}$.

Figure 9

Ratio of $(\tau /{\sigma }_{\mathrm{H}})d\sigma /d\tau$ with $k_{\mathrm{T}}$ ordering to the NLL expectation, $\tau g(\tau )$, Eqs. (221) and (222). The ratio is calculated at $Q^2=(1\mathrm{TeV}{)}^2$, $Q^2=(10\mathrm{TeV}{)}^2$, and $Q^2=(100\mathrm{TeV}{)}^2$. We use a cutoff on the transverse momentum in splittings: $k_{\mathrm{T}}>1\mathrm{GeV}$ in each case.

Figure 10

Plot of $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$, as in Fig. 1, for the Deductor splitting functions with the Catani-Seymour local momentum mapping [44]. $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$ is approximately quadratic in $\log (\nu )$, indicating that ${\mathcal{I}}_2^{[2]}(\nu )$ changes the NLL result.

Figure 11

Plot of $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$, as in Fig. 1, for a shower with $k_{\mathrm{T}}$ ($\beta =0.0$) ordering and the Catani-Seymour local momentum mapping [44] according to an algorithm based on the PanLocal dipole shower of Ref. [16] with exact color. For large $\log (\nu )$, $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$ is approximately linear in $\log (\nu )$, indicating that ${\mathcal{I}}_2^{[2]}(\nu )$ leaves the NLL result intact.

Figure 12

Plot of $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$, as in Fig. 1, versus $\log (\nu )$, for a shower with $\beta =0.5$ ordering and the Catani-Seymour local momentum mapping [44] according to an algorithm based on the PanLocal dipole shower of Ref. [16] with exact color. For large $\log (\nu )$, $⟨{\mathcal{I}}_2^{[2]}(\nu )⟩$ is approximately linear in $\log (\nu )$, indicating that ${\mathcal{I}}_2^{[2]}(\nu )$ leaves the NLL result intact.

Figure 13

Plot of $⟨{\mathcal{I}}_3^{[2]}(\nu )⟩$, as in Fig. 2, for a shower with $k_{\mathrm{T}}$ ($\beta =0.0$) ordering and the Catani-Seymour local momentum mapping [44] according to an algorithm based on the PanLocal dipole shower of Ref. [16] with exact color.

Figure 14

Plot of $⟨{\mathcal{I}}_3^{[2]}(\nu )⟩$, as in Fig. 2, versus $\log (\nu )$ for a shower with $\beta =0.5$ ordering and the Catani-Seymour local momentum mapping [44] according to an algorithm based on the PanLocal dipole shower of Ref. [16] with exact color.

Figure 15

Integration regions for second splitting.