Matching of the low-x evolution kernels

TL;DR

This paper shows how to reconcile different formulations of low-x evolution kernels at NLO, enabling the construction of a Mobius invariant BFKL kernel in N=4 supersymmetric Yang-Mills theory.

Contribution

It introduces a method to match the Mobius form of the BFKL kernel with the dipole model kernel at NLO, and constructs a Mobius invariant NLO BFKL kernel in N=4 SYM.

Findings

Successfully matches the kernels in different frameworks.

Constructs a Mobius invariant NLO BFKL kernel.

Provides insights into the structure of low-x evolution at NLO.

Abstract

We demonstrate that the ambiguity of the low-x evolution kernels in the next-to-leading order (NLO) permits one to match the Mobius form of the BFKL kernel and the kernel of the colour dipole model and to construct the Mobius invariant NLO BFKL kernel in N=4 supersymmetric Yang-Mills theory.

Equations

$$⟨\vec{q}|{\vec{q}}^{\prime }⟩=\delta (\vec{q}-{\vec{q}}^{\prime }),⟨\vec{r}|{\vec{r}}^{\prime }⟩=\delta (\vec{r}-{\vec{r}}^{\prime }),⟨\vec{r}|\vec{q}⟩=\frac{e^{i\vec{q}\vec{r}}}{(2\pi {)}^{(1+ϵ)}}.$$

$$-4i(2\pi {)}^{D-2}\delta ({\vec{q}}_A-{\vec{q}}_B)\text{mbox}{disc}_s{\mathcal{A}}_{AB}^{A^{\prime }{B}^{\prime }}=⟨A^{\prime }\bar{A}|{\left(\frac{s}{s_0}\right)}^{\hat{\mathcal{K}}}\frac{1}{{\hat{\vec{q}}}_1^2{\hat{\vec{q}}}_2^2}|{\bar{B}}^{\prime }B⟩.$$

$$⟨{\vec{q}}_1,{\vec{q}}_2|\hat{\mathcal{K}}|{\vec{q}}_1^{\prime },{\vec{q}}_2^{\prime }⟩=\delta ({\vec{q}}_{11^{\prime }}+{\vec{q}}_{22^{\prime }})\frac{{\mathcal{K}}_r\left(q_1,q_1^{\prime },q\right)}{{\vec{q}}_1^2{\vec{q}}_2^2}+\delta ({\vec{q}}_{22^{\prime }})\delta \left({\vec{q}}_{11^{\prime }}\right)\left(\omega \left({\vec{q}}_1^2\right)+\omega ({\vec{q}}_2^2)\right),$$

$$⟨{\vec{q}}_1,{\vec{q}}_2|{\bar{B}}^{\prime }B⟩=4p_B^{-}\delta ({\vec{q}}_B-{\vec{q}}_1-{\vec{q}}_2){\Phi }_{B^{\prime }B}({\vec{q}}_1,{\vec{q}}_2),$$

$$⟨A^{\prime }\bar{A}|{\vec{q}}_1,{\vec{q}}_2⟩=4p_A^{+}\delta ({\vec{q}}_A-{\vec{q}}_1-{\vec{q}}_2){\Phi }_{A^{\prime }A}({\vec{q}}_1,{\vec{q}}_2),$$

$$\frac{{\mathcal{K}}_r^B({\vec{q}}_1,{\vec{q}}_1^{\prime };\vec{q})}{{\vec{q}}_1^2{\vec{q}}_2^2}=\frac{{\alpha }_s({\mu }^2)N_c}{{\pi }^2}\left(\frac{1}{{\vec{q}}_{11^{\prime }}^2}+\frac{\left({\vec{q}}_2{\vec{q}}_{11^{\prime }}\right)}{{\vec{q}}_2^2{\vec{q}}_{11^{\prime }}^2}-\frac{\left({\vec{q}}_1{\vec{q}}_{11^{\prime }}\right)}{{\vec{q}}_1^2{\vec{q}}_{11^{\prime }}^2}-\frac{\left({\vec{q}}_1{\vec{q}}_2\right)}{{\vec{q}}_1^2{\vec{q}}_2^2}\right),$$

$$\omega ({\vec{q}}^2)=-\frac{{\alpha }_s({\mu }^2)N_c}{(2\pi {)}^{2+2ϵ}}\int d^{2+2ϵ}k\left(\frac{2}{{\vec{k}}^2}-\frac{2\vec{k}(\vec{k}-\vec{q})}{{\vec{k}}^2(\vec{k}-\vec{q}{)}^2}\right).$$

$$\hat{\mathcal{K}}\to {\hat{\mathcal{O}}}^{-1}\hat{\mathcal{K}}\hat{\mathcal{O}},⟨A^{\prime }\bar{A}|\to ⟨A^{\prime }\bar{A}|\hat{\mathcal{O}},\frac{1}{{\hat{\vec{q}}}_1^2{\hat{\vec{q}}}_2^2}|{\bar{B}}^{\prime }B⟩\to {\hat{\mathcal{O}}}^{-1}\frac{1}{{\hat{\vec{q}}}_1^2{\hat{\vec{q}}}_2^2}|{\bar{B}}^{\prime }B⟩.$$

$$\hat{\mathcal{K}}\to \hat{\mathcal{K}}-[{\hat{\mathcal{K}}}^B,\hat{O}].$$

$$\hat{\mathcal{K}}\to \hat{\text{K}}=\hat{\mathcal{K}}+\frac{{\alpha }_s}{8\pi }{\beta }_0[{\hat{\mathcal{K}}}^B,\ln \left({\hat{\vec{q}}}_1^2{\hat{\vec{q}}}_2^2\right)],$$

$$\langle\vec{r}_{1},\vec{r}_{2}|\hat{\emph{K}}_{M}|\vec{r}_{1}^{\;\prime},\vec{r}_{2}^{\;\prime}\rangle=\frac{\alpha_{s}(\mu^{2})N_{c}}{2\pi^{2}}\int d\vec{\rho}\frac{\vec{r}_{12}{}^{2}}{\vec{r}_{1\rho}^{\,\,2}\vec{r}_{2\rho}^{\,\,2}}\Biggl{[}\delta(\vec{r}_{11^{\prime}})\delta(\vec{r}_{2^{\prime}\rho})+\delta(\vec{r}_{1^{\prime}\rho})\delta(\vec{r}_{22^{\prime}})-\delta(\vec{r}_{11^{\prime}})\delta({r}_{22^{\prime}})~{}\Biggr{]}$$

$$+\frac{\alpha_{s}^{2}(\mu^{2})N_{c}^{2}}{4\pi^{3}}\Biggl{[}\delta(\vec{r}_{11^{\prime}})\delta(\vec{r}_{22^{\prime}})\int d\vec{\rho}\,g_{0}(\vec{r}_{1},\vec{r}_{2};\rho)+\delta(\vec{r}_{11^{\prime}})g_{1}(\vec{r}_{1},\vec{r}_{2};\vec{r}_{2}^{\;\prime})+\delta(\vec{r}_{22^{\prime}})g_{1}(\vec{r}_{2},\vec{r}_{1};\vec{r}_{1}^{\;\prime})$$

$$+\frac{1}{\pi}g_{2}(\vec{r}_{1},\vec{r}_{2};\vec{r}_{1}^{\;\prime},\vec{r}_{2}^{\;\prime})\Biggr{]}.$$

$$g_0({\vec{r}}_1,{\vec{r}}_2;\vec{\rho })=2\pi \zeta (3)\delta \left(\vec{\rho }\right)-g({\vec{r}}_1,{\vec{r}}_2;\vec{\rho }),$$

$$g_1({\vec{r}}_1,{\vec{r}}_2;{\vec{r}}_2^{\prime })=\frac{11}{6}\frac{{\vec{r}}_{12}^2}{{\vec{r}}_{22^{\prime }}^2{\vec{r}}_{12^{\prime }}^2}\ln \left(\frac{{\vec{r}}_{12}^2}{r_{\mu }^2}\right)+\frac{11}{6}\left(\frac{1}{{\vec{r}}_{22^{\prime }}^2}-\frac{1}{{\vec{r}}_{12^{\prime }}^2}\right)\ln \left(\frac{{\vec{r}}_{22^{\prime }}^2}{{\vec{r}}_{12^{\prime }}^2}\right)$$

$$+\frac{1}{2{\vec{r}}_{22^{\prime }}^2}\ln \left(\frac{{\vec{r}}_{12^{\prime }}^2}{{\vec{r}}_{22^{\prime }}^2}\right)\ln \left(\frac{{\vec{r}}_{12}^2}{{\vec{r}}_{12^{\prime }}^2}\right)-\frac{{\vec{r}}_{12}^2}{2{\vec{r}}_{22^{\prime }}^2{\vec{r}}_{12^{\prime }}^2}\ln \left(\frac{{\vec{r}}_{12}^2}{{\vec{r}}_{22^{\prime }}^2}\right)\ln \left(\frac{{\vec{r}}_{12}^2}{{\vec{r}}_{12^{\prime }}^2}\right),$$

$$\ln r_{\mu }^2=2\psi \left(1\right)-\ln \frac{{\mu }^2}{4}-\frac{3}{11}\left(\frac{67}{9}-2\zeta (2)\right),$$

$$g_2({\vec{r}}_1,{\vec{r}}_2;{\vec{r}}_1^{\prime },{\vec{r}}_2^{\prime })$$

$$+\frac{\ln \left(\frac{{\vec{r}}_{12^{\prime }}^2{\vec{r}}_{21^{\prime }}^2}{{\vec{r}}_{11^{\prime }}^2{\vec{r}}_{22^{\prime }}^2}\right)}{4\vec{r}{}_{11^{\prime }}^2\vec{r}{}_{22^{\prime }}^2}\left(\frac{\vec{r}{}_{12}^4}{d}-\frac{\vec{r}{}_{12}^2}{\vec{r}{}_{1^{\prime }{2}^{\prime }}^2}\right)+\frac{\ln \left(\frac{\vec{r}{}_{12}^2\vec{r}{}_{1^{\prime }{2}^{\prime }}^2}{\vec{r}{}_{11^{\prime }}^2\vec{r}{}_{22^{\prime }}^2}\right)}{2\vec{r}{}_{12^{\prime }}^2\vec{r}{}_{21^{\prime }}^2}(\frac{r}{2\vec{r}{}_{1^{\prime }{2}^{\prime }}^2}$$