Higher-twist fermionic operators and DIS structure functions from the $\mathrm{AdS}/\mathrm{CFT}$ duality

The role of local higher-twist ($\tau >3$) spin-$1/2$ fermionic operators of the strongly coupled $\mathcal{N}=4$ supersymmetric Yang-Mills theory on the symmetric and antisymmetric deep inelastic scattering structure functions is investigated. The calculations are carried out in terms of the duality between $\mathcal{N}=4$ SYM theory and type IIB supergravity on ${\mathrm{AdS}}_5\times S^5$. Particularly, we explicitly obtain the structure functions for single-trace spin-$1/2$ fermionic operators in the ${\mathbf{20}}^{*}$ and ${\mathbf{60}}^{*}$ irreducible representations of $SU(4{)}_R$, corresponding to twists 4 and 5, respectively. We also calculate the contributions of other single-trace spin-$1/2$ fermionic operators in the $\mathbf{4}$, $\mathbf{20}$ and $\mathbf{60}$ irreducible representations of $SU(4{)}_R$. New important effects are found in comparison with the minimal twist ($\tau =3$) case, and they are studied thoroughly.

Figure 1

Matrix elements of the electromagnetic current inside the hadron are written on the left. The minimal coupling is represented by the first diagram. The diagrams with dotted vertices correspond to different Pauli terms. Selection rules allow for these three Pauli diagrams.

Figure 2

Full structure functions $F_1$, $F_2$ and $g_2$ (blue line) as functions of the Bjorken parameter $x$, obtained from twist-4 spin-$1/2$ fermionic operators ${\mathcal{O}}_{k=1}^{I_1,(6)}$ with $l_5=1\ge l_4\ge l_3\ge l_2\ge l_1=0$ ($\mathcal{Q}=1/2$). We also display all the contributions coming from different terms, namely: from the minimal coupling ${\beta }_m^2{F}_1^m$ (orange line); the Pauli interaction where the intermediate state ${\lambda }_{\mathcal{X}}\equiv {\lambda }_{k=1}^{-}$ is the same as the incident state ${\beta }_P^2{F}_1^P$ (green line), and where the intermediate state is different ${\beta }_{Pm}^2{F}_1^P$ (violet line), both belonging to the ${\mathbf{20}}^{*}$; the contribution from cross terms ${\beta }_m{\beta }_P{F}_1^c$ (red line); and the contributions from the Pauli interactions with intermediate states in the $\mathbf{60}$: ${\beta }_{+}^2{F}_1^{P+}$, where ${\lambda }_{\mathcal{X}}\equiv {\lambda }_{k+1}^{+}$ (brown line); and in the $\mathbf{4}$: ${\beta }_{-}^2{F}_1^{P-}$, where ${\lambda }_{\mathcal{X}}\equiv {\lambda }_{k-1}^{+}$ (light blue line).

Figure 3

Full structure functions $F_1$, $F_2$ and $g_2$ (blue line) as functions of the Bjorken parameter $x$ and the detail of each contributions, obtained from the twist-4 spin-$1/2$ fermionic operators ${\mathcal{O}}_{k=1}^{I_1,(6)}$ with $l_5=l_4=l_3=l_2=l_1=1$ ($\mathcal{Q}=3/2$). The meaning of the curves is analogous as described in Fig. 2.

Figure 4

Full structure functions $F_1$, $F_2$ and $g_2$ as functions of the Bjorken parameter $x$ obtained from the twist-4 spin-$1/2$ fermionic operators ${\mathcal{O}}_{k=1}^{I_1,(6)}$ with $(1,l_4,l_3,l_2,l_1{)}_{a=1}$.

Figure 5

Full structure functions $F_1$, $F_2$ and $g_2$ as functions of the Bjorken parameter $x$ obtained from the twist-5 spin-$1/2$ fermionic operators ${\mathcal{O}}_{k=2}^{I_2,(6)}$ with $(2,l_4,l_3,l_2,l_1{)}_{a=1}$.

Figure 6

$F_2/(2x{F}_1)$ as a function of the Bjorken parameter $x$ for different values of the operator twist. The horizontal red line indicates the Callan-Gross relation, i.e., $F_2/(2x{F}_1)=1$.

Figure 7

Data of the structure function $F_2$ for the proton extracted from Refs. [59, 60], as a function of $\log q^2$. Long-dashed lines represent the best fit for $\tau =3$. Continuous lines correspond to the best fit for twist-5 operator corresponding to the Kaluza-Klein state (2,2,2,2,2). Dotted lines (very close to the long-dashed lines) correspond to a fit with a linear combination of twist 3, 4 and 5 operators as explained in the main text. Note that in this figure $F_2$ has been multiplied by $2^{i_x}$, where $i_x$ is the number of the $x$ bin, ranging from 1 ($x=0.85$) to 3 ($x=0.65$).

Figure 8

Full structure function $g_3$ (blue line) as a function of the Bjorken parameter $x$, obtained from the contribution of the twist-4 spin-$1/2$ fermionic operators ${\mathcal{O}}_{k=1}^{I_1,(6)}$ with $l_5=1\ge l_4\ge l_3\ge l_2\ge l_1$. We distinguish the contributions from the minimal coupling ${\beta }_m^2{F}_1^m$ (orange line); the Pauli interaction between states which belong to the same Kaluza-Klein tower: where the intermediate state ${\lambda }_{\mathcal{X}}\equiv {\lambda }_{k=1}^{-}$ is the same as the incident state ${\beta }_P^2{F}_1^P$ (green line) and where the intermediate state is different ${\beta }_{Pm}^2{F}_1^P$ (violet line); the contribution from crossed terms ${\beta }_m{\beta }_P{F}_1^c$ (red line); and the contributions from the Pauli interaction involving states which are dual to operators belonging to the $\mathbf{60}$ irreducible representation of $SU(4{)}_R$: ${\beta }_{+}^2{F}_1^{P+}$, where ${\lambda }_{\mathcal{X}}\equiv {\lambda }_{k+1}^{+}$ (brown line), and also the $\mathbf{4}$ irreducible representation of $SU(4{)}_R$: ${\beta }_{-}^2{F}_1^{P-}$, where ${\lambda }_{\mathcal{X}}\equiv {\lambda }_{k-1}^{+}$ (light blue line).

Figure 9

Full structure function $g_3$ as a function of the Bjorken parameter $x$ obtained from the twist-4 spin-$1/2$ fermionic operators ${\mathcal{O}}_{k=1}^{I_1,(6)}$ with $(1,l_4,l_3,l_2,l_1{)}_{a=1}$ (a) and in (b) from the twist-5 spin-$1/2$ fermionic operators ${\mathcal{O}}_{k=2}^{I_2,(6)}$ with $(2,l_4,l_3,l_2,l_1{)}_{a=1}$.