Null polygonal Wilson loops in full \(\mathcal N=4\) superspace (Q2899449)

The one-loop expectation value NEWLINE\[NEWLINE{g^2N\over 64\pi^2} M^{(1)}[C]= {1\over N} \oint_C \oint_C {1\over 2} (\text{Tr\,}AA'),NEWLINE\]NEWLINE of a null polygonal Wilson loop \({\mathcal W}[C]\) in \(N=4\) super Yang-Mills theory in full superspace is computed. A null polygon is a sequence of points \((x_j,\theta_j,\overline\theta_j)\) which are joined by null lines. The shape of the Wilson loop is a null polygon in a superspace [\textit{N. Beisert} et al., J. Phys. A, Math. Theor. 45, No. 26, 47 p. (2012; Zbl 1258.81075)]. \(C\) is the contour of a null polygon, \(A\) and \(A'\) are one copy of the gauge connection for each of the two integrals, where NEWLINE\[NEWLINEA= A^++ A^-+d\Lambda.NEWLINE\]NEWLINE Then setting \(M^{(1)}_{\pm\pm'} C= 64\pi^2\oint_C \oint^\prime_C{1\over 2}\langle A^{\pm} A^{\prime\pm\prime}\rangle\), we have NEWLINE\[NEWLINEM^{(1)}[C]= M^{(1)}_{++}[C]+ M^{(1)}_{--}[C]+ 2M^{(1)}_{+-}[C].NEWLINE\]NEWLINE On the null line, join vertices \(j\) and \(j+1\), \(A^{\pm}\) are exact: \(A^{\pm}_j= dG^{\pm}_j\). The one-loop expectation value is given by NEWLINE\[NEWLINE\sum^n_{j,k=1} \Biggl({1\over 2}\langle 0|G^+_{j-1,j} G^+_{k-1,k}|\rangle+{1\over 2}\langle 0|G^-_{j-1,j} G^-_{k-1,k}|0\rangle+\langle 0|G^+_{j-1,j} G^-_{k-1,k}|0\rangle\Biggr).NEWLINE\]NEWLINE Then the main results of this paper are:NEWLINENEWLINE1. The chiral-chiral expectation value \(\langle 0|G^+_{j-1,j} G^+_{k-1,k}|0\rangle\) is given by NEWLINE\[NEWLINE-{1\over 4\pi^2} {\langle j-1j\rangle\langle k- 1k\rangle \delta^{0|4}(\theta_{k,j}| x^+_{k,j}|\overline\rho])\over (x^+_{k,j})^2\langle j-1| x^+_{k,j}|\overline\rho]\langle j|x^+_{k,j}|\overline\rho] \langle k-1| x^+_{k,j}|\overline\rho]\langle k| x^+_{k,j}|\overline\rho]}.NEWLINE\]NEWLINE Here \(p\) is a reference spinor and can be interpreted as an integral constant (\S4.4. (4.30)).NEWLINENEWLINE 2. The mixed-chiral expectation value \(\langle 0|G^+_{j-1,j} G^-_{k-1,k}|0\rangle\) is given by NEWLINE\[NEWLINE\begin{multlined} {1\over 64\pi^2} \Biggl(-\text{Li}_2\Biggl({\langle j-1| x^{+-}_{j,k}|k]\langle j| x^{+-}_{j,k}|k- 1]\over\langle j-1| x^{+-}_{j,k}| k-1]\langle j| x^{+-}_{j,k}|k- 1]\langle j| x^{+-}_{j,k}| k]}\Biggr)+\\ {1\over 2}\log(\langle j-1| x^{+-}_{j,k}|k- 1]\langle j| x^{+-}_{j,k}[k]) \log\Biggl({\langle j-1| x^{+-}_{j,k} |k]\langle j|x^{+-}_{j,k}|k- 1]\over\langle j-1| x^{+-}_{j,k}|k- 1]\langle j| x^{+-}_{j,k} |k]}\Biggr)\Biggr).\end{multlined}NEWLINE\]NEWLINE Here \(\text{Li}_2(z)= \sum_n{z^n\over n^2}\) is the dilogarithm function (\S4.5. (4.39)).NEWLINENEWLINE Making the ansatz \(A_{\alpha\alpha}= D_{\alpha\beta} B^\beta_\alpha+ D_{\alpha\alpha}\) (\S3.2) and using mode expansion (\S3.3. (3.17)) and the light gauge condition (\S3.5. (3.23)) of \(B\), \(G^{\pm}_j\) are computed in closed form (\S4.3. (4.15), (4.16)). To derive (4.30), a variable change \(\overline\eta=\overline\zeta[\overline\lambda \overline\rho]\) is used. Then introducing the differential operators NEWLINE\[NEWLINE{\mathcal D}_{\ell}= -i\langle\ell| \sigma^\mu|\overline\rho] {\partial\over\partial x^{+\mu}_{k,j}}NEWLINE\]NEWLINE and computing \({\mathcal D}_{j-1}{\mathcal D}_j{\mathcal D}_{k-1}{\mathcal D}_k(1/(x^+_{k,j})^2)\), (4.30) is derived.NEWLINENEWLINE (4.39) is obtained by showing the equality NEWLINE\[NEWLINE{\partial^2\over\partial a\partial d}\langle 0|G^+_{j-1,j} G^-_{k-1,k}|0\rangle= {\partial^2\over\partial b\partial c}\langle 0|G^+_{j-1,j} G^+_{k-1,k}|0\rangle= -{1\over 64\pi^2} {1\over ad-bc}.NEWLINE\]NEWLINE These formulas have convenient expression in terms of twister variables. This is explained in \S5 together with an explanation of twister variables. In \S6, the author studies regularizations of the mixed-chirality contributions and compute regularized expectation values of the axial framing and super-Poincaré forms (\S6.1. (6.9), \S6.2. (6.11)). Then by using these calculations, the boxed Wilson loop (\S6.3. (6.19)) is confirmed to be finite and superconformal and its explicit expression agrees with that of \textit{L. F. Alday} et al. [J. High Energy Phys. 2011, No. 4, 41 p. (2011; Zbl 1250.81071)]. \S7, the last section concerns Yangian symmery and anomalies. In the first two sections (\S2 and \S3), reviews of \(N=4\) super-Yang-Mills theory in superspace and gauge field propagator are given. Some technical details, such as position space calculations, are given in an appendix.