Fourth dimension of the nucleon structure: Spacetime analysis of the timelike electromagnetic proton form factors

As is well known, spacelike proton form factors expressed in the Breit frame may be interpreted as the Fourier transform of static space distributions of electric charge and current. In particular, the electric form factor is simply the Fourier transform of the charge distribution $F\left(q\right)=\int e^{i\vec{q}·\vec{r}}\rho \left(r\right)d^{3}r$. We do not have an intuitive interpretation of the same level of simplicity for the proton timelike form factor appearing in the reactions $e^{+}{e}^{-}\leftrightarrow \overline{p}p$. However, one may suggest that, in the center-of-mass frame, where $q_{\mu }{x}^{\mu }=qt$, a timelike electric form factor is the Fourier transform $F\left(q\right)=\int e^{iqt}R\left(t\right)dt$ of a function $R\left(t\right)$ expressing how the electric properties of the forming (or annihilating) proton-antiproton pair evolve in time. Here we analyze in depth this idea and show that the functions $\rho \left(r\right)$ and $R\left(t\right)$ can be formally written as the time and space integrals of a unique correlation function depending on both time and space coordinates.

Figure 1

Most recent data on the TL proton generalized FF as a function of $q^2$, from Refs. [5, 6] (black open circles), Ref. [8] (red triangles), together with the calculation from Eq. (2) (blue dash-dotted line), Eq. (3) (red dashed line), Eq. (4) (green long-dashed line), and Eq. (5) (black solid line).

Figure 2

Feynman diagrams for reactions (8) labeled as SL in the figure, (9) ${\mathrm{TL}}_{+}$, and (10) ${\mathrm{TL}}_{-}$. In the one-photon exchange approximation, electromagnetic FFs are functions characterizing the vertex coupling the virtual photon to the hadron current (thick dashed line in the figure).

Figure 3

Subdiagram participating in the three reactions (14, 15, 16). Since they may be considered physical channels of the same reaction, all of them may be described by the same diagram and by the same amplitude by changing the values of the components of the four-momenta $P_A,P_B,q$, exploiting crossing symmetry. Formally, we consider these momenta as all entering. So, $q$ coincides with the physical four-momentum of the virtual photon in the channel $T{L}_{+}$ where a $\overline{p}p$ pair is created, while $q=-q^{\prime }$, where $q^{\prime }$ is the physical four-momentum of the virtual photon, in the reverse channel $T{L}_{-}$ where a virtual photon is produced by $\overline{p}p$ annihilation. Similar considerations apply to $P_A$ and $P_B$ [see Eqs. (14, 15, 16) for the correspondence between the formal arguments of the amplitude and the physical momenta].

Figure 4

Left image shows one of the possible chains of events that at resolved level lead to proton-antiproton formation from a virtual photon. In this figure the proton is schematized in a simplified form, as composed by a charged quark plus a neutral compact diquark. So $\gamma \to \overline{p}p$ requires at least two pair-creation vertices, in the four-points $X_1$ and $X_2$. Right image shows the same process at an unresolved level of analysis. Only one vertex is present in the four-point $X$, where the $\overline{p}p$ pair is directly created by the photon. The relation between $X,X_1$, and $X_2$ is determined by Eq. (33), which in this simple case will be of the form $X=w_1{X}_1+w_2{X}_2$. The corresponding geometry is represented in Fig. 5.

Figure 5

Absolute (left image) and relative (right image) coordinates for the chain of events leading to $\overline{p}p$ formation from a virtual photon as shown in Fig. 4. (left) According to Eqs. (33), $X_1,X$, and $X_2$ lie along a straight line, which is represented as a thick dashed line in the figure. This line does not correspond to any physical particle; we just use it to highlight the relative position of the three points. The continuous thin straight lines at ${45}^{\circ }$ and ${135}^{\circ }$ represent the light cone of $X$. (right) The same geometry as the left image, but using the relative four-coordinates $x_1$ and $x_2$ introduced in Eqs. (34). With this transformation, $X$ becomes the origin. The four-coordinate $x$ that is argument of the spacetime form factor $F\left(x\right)$ coincides with $x_1$, the four-point where the photon creates the first quark-antiquark pair.

Figure 6

$R\left(t\right)=cos\left(Mt\right)e^{-a\left|t\right|}$, with $M=1$ GeV, $a=0.4$ GeV, as in Eq. (51). The retarded ($\overline{p}p$ creation) and advanced (annihilation) contributions of Eq. (52) are distinguished.

Figure 7

Dotted line shows $R_1\left(t\right)=cos\left(M_{1}t\right)e^{-a\left|t\right|}$, with $M=1$ GeV, $a=0.4$ GeV. Thick-soft double-dotted line shows $R_2\left(t\right)=cos\left(M_{2}t\right)e^{-b\left|t\right|}$, with $M_2=0.6$ GeV, $b=0.1$ GeV. Continuous line shows convolution $[R_1\left(t\right)*R_2\left(t\right)]=\int d\tau R_1\left(\tau \right)R_2(t-\tau )$ according to Eq. (54).

Figure 8

Sequence of oscillators corresponding to the response function of Fig. 7. In equilibrium both the black masses overlap with the gray circle. They can move horizontally under the action of elastic forces $F_1$ and $F_2$, graphically represented as springs, or of external forces. When any of these masses is subject to an instantaneous external impulse $\propto \delta (t-t_0)$ at the time $t_0$, its later displacement from the equilibrium position is described by the Green response function $R_i(t-t_0)$. A short impulse by an external force $\delta \left(t\right)$ at $t=0$ causes the displacement $R_1\left(\tau \right)$ of the first mass at the later time $\tau$. The displacement of the first mass acts as an external force on the second mass and may be decomposed into short impulses: $R_1\left(\tau \right)=\int d{\tau }^{\prime }{R}_1\left({\tau }^{\prime }\right)\delta (\tau -{\tau }^{\prime })$. Since each short impulse at the time $\tau$ produces a response $R_2(t-\tau )$ of the second mass, the resulting displacement of the second mass is $\int d\tau R_2(t-\tau )R_1\left(\tau \right)$.