Going to the light front with contour deformations

We explore a new method to calculate the valence light-front wave function of a system of two interacting particles, which is based on contour deformations combined with analytic continuation methods to project the Bethe-Salpeter wave function onto the light front. In this proof-of-concept study, we solve the Bethe-Salpeter equation for a scalar model and find excellent agreement between the light-front wave functions obtained with contour deformations and those obtained with the Nakanishi method frequently employed in the literature. The contour-deformation method is also able to handle extensions of the scalar model that mimic certain features of QCD such as unequal masses and complex singularities. In principle the method is suitable for computing parton distributions on the light front such as parton distribution functions, transverse momentum distributions and generalized parton distributions in the future.

Figure 1

Pictorial representation of the hadron-to-hadron correlator in Eq. (1) (left) and the Bethe-Salpeter wave function in Eq. (2) (right).

Figure 2

Sketch of the singularities in the complex $w$ plane for $0<\alpha <1$. Part (a) is the situation in Eqs. (22)–(23) with $i\epsilon$ in the denominators, whereas part (b) corresponds to Eq. (31) with $i\epsilon$ in the integration path. For $|\alpha |<1$ the results are identical, but for general values of $\alpha$ only (b) leads to an analytic function.

Figure 3

Light-front wave function (27) for the monopole model with $t=-0.5$, $\gamma =2$ and normalized to $\psi (0,0)$.

Figure 4

Light-front wave function (27) for the monopole in the complex ${\alpha }^2$ plane. For the integration path in Fig. 2 the function would only have support for ${\alpha }^2>0$ (thick orange curves) but vanish everywhere else.

Figure 5

Different regions in the complex $x$ plane separated by the branch cuts (33). To obtain the correct light-front wave function outside ${\mathcal{R}}_{\epsilon }$, one would need to deform the Euclidean integration contour.

Figure 6

Exemplary cut configurations in the complex $\sqrt{x}$ plane for $\alpha =0.6$, $\gamma =4$ and three different values of $\sqrt{t}$. The three lines correspond to the propagator cuts ${\sqrt{x}}_{\pm }$ (blue, red) and the cut ${\sqrt{x}}_0$ from the monopole amplitude (orange) in Eq. (53). The region ${\mathcal{R}}_{\epsilon }$ shown in blue connects the origin $\sqrt{x}=0$ with $\sqrt{x}\to \infty$.

Figure 7

Bethe-Salpeter equation.

Figure 8

Singularity structure in the complex $\sqrt{t}$ plane.

Figure 9

Branch cuts arising from the Bethe-Salpeter kernel, Eq. (64), in the complex $\sqrt{x^{\prime }}$ plane. We chose two points ${\sqrt{x}}_1=(1+2i)/\sqrt{2}$ and ${\sqrt{x}}_2=(1+2i)\sqrt{2}$ and four values of $\beta =0.2$, 1, 2, 3 in each case. The blue dashed line is an integration contour that avoids all possible cuts.

Figure 10

Calculable region in the complex $\sqrt{t}$ plane when employing contour deformations, for an arbitrary singularity position in the propagator or the amplitude.

Figure 11

Largest eigenvalue of the Bethe-Salpeter equation (63) for $\beta =4$ plotted over $\mathrm{Im}\sqrt{t}=\mathrm{Re}M/(2m)$ for six values of $\mathrm{Re}\sqrt{t}$. The physical region for bound states ($0<M<2m$) corresponds to $\mathrm{Re}\sqrt{t}=0$ and $0<\mathrm{Im}\sqrt{t}<1$. The eigenvalue obtained with the Nakanishi method is shown for comparison for $\mathrm{Re}\sqrt{t}=0.20$ (red dots).

Figure 12

Absolute value of the BS amplitude for $\alpha =-0.602$ and $\sqrt{t}=0.20+0.20i$ as a function of $x$ and $\omega$ (all variables are dimensionless). The crosses mark selected values $x_1=0.94$, $x_2=11$, $x_3=101$ and ${\omega }_1=0$, ${\omega }_2=0.5$ which are referred to in Fig. 13.

Figure 13

Dependence of the BS amplitude on $\alpha$ for fixed values of $x$ and $\omega$ as indicated in Fig. 12. The curves correspond to three different values $\sqrt{t}=0.2+\lambda i$ with $\lambda \in \{0.2,0.8,1.2\}$. The results for $-\omega$ are identical to those for $+\omega$ when exchanging $\alpha \to -\alpha$.

Figure 14

Absolute value of the light-front wave function as a function of $x$ and $\alpha$, for $\sqrt{t}=0.20+0.80i$ and $\beta =4$.

Figure 15

Dependence of the light-front wave function $\psi (\alpha ,x,t)$ on $\alpha$ for three different values $x=0.039$ (top), $x=0.326$ (center) and $x=2.05$ (bottom) and different values of $\sqrt{t}$. The results obtained with contour deformations and the SPM are compared to the Nakanishi results; the latter are restricted to $\mathrm{Im}\sqrt{t}<1$ below the threshold.

Figure 16

Same as in Fig. 15, but now as a function of $x$ for fixed $\alpha =0.668$ (top) and $\alpha =0.043$ (bottom).

Figure 17

Distribution amplitude $\varphi (\alpha )$ from contour deformations (solid lines) compared to the results from the Nakanishi method (crosses); see Fig. 15 for the legend.

Figure 18

Distribution amplitude $\varphi (\alpha )$ for $\sqrt{t}=0.5i$ in the physical region. The band is the result using contour deformations and analytic continuations and compared to the result from the Nakanishi method.

Figure 19

Distribution function $u(\alpha )$ for $\sqrt{t}=0.5i$ in the physical region, normalized to $u(0)=1$.

Figure 20

Light-front wave function for unequal masses and five different values of $ϵ$, evaluated at $x=1.22$, $\sqrt{t}=0.2+0.8i$ and $\beta =4$.

Figure 21

Complex conjugate propagator poles in the complex $w$ plane. The interpretation of the $i\epsilon$ prescription like in Fig. 2 does not return the correct limit $\delta \to 0$ (left), in contrast to the one according to Fig. 2 (right).

Figure 22

Light-front wave function (top) and distribution amplitude (bottom) for a propagator with a complex conjugate pole pair. The results correspond to $\sqrt{t}=0.2+0.8i$ and $\beta =4$; the light-front wave function is evaluated at $x=0.95$.

Figure 23

Shape of the function (b1) for $A=i$ and different values of $C=0.6\exp (i\sigma \frac{\pi }{2})$; from top left to bottom right: $\sigma =0$, 0.3, 0.5, 0.7, 0.9, 1. The blue (red) curves are the positive (negative) branches. The open points are the values of $\pm AC$, the orange circles have radii $|AC|$ and the gray circles radii $|A|$, and the orange lines connect the origin with the points $A{C}^2$. The filled squares show the intersections with the real axis.

Figure 24

Shape of the function (b5) for $iu=0.3+0.6i$ and two values of $A$. Depending on $\arg (A)\lessgtr \arg (iu)$, the branch cuts twirl in different directions.