Relativistic three-body bound state in a 3D formulation

Background: The relativistic three-body problem has a long tradition in few-nucleon physics. Calculations of the triton binding energy based on the solution of the relativistic Faddeev equation, in general, lead to a weaker binding than the corresponding nonrelativistic calculation.

Purpose: In this work we solve for the three-body binding energy as well as the wave function and its momentum distribution. The effect of the different relativistic ingredients is studied in detail.

Method: Relativistic invariance is incorporated within the framework of Poincaré invariant quantum mechanics. The relativistic momentum-space Faddeev equation is formulated and directly solved in terms of momentum vectors without employing a partial-wave decomposition.

Results: The relativistic calculation gives a three-body binding energy which is about 3% smaller than its nonrelativistic counterpart. In the wave function, relativistic effects are manifested in the Fermi motion of the spectator particle.

Conclusions: Our calculations show that though the overall relativistic effects in the three-body bound state are small, individual effects by themselves are not necessarily small and must be taken into account consistently.

Figure 1

The analytical term $F(p_{jk},p_{jk}^{\prime },k_i)$ connecting the relativistic and the nonrelativistic right-half-shell $T$ matrices, Eq. (17), as function of the momenta $p_{jk}$ and $p_{jk}^{\prime }$ in the two-body subsystem, for a fixed third-particle momentum $k_i=5{\mathrm{fm}}^{-1}$.

Figure 2

The matrix elements of Jacobian function $N(kcos(\theta ),ksin(\theta ),x)$, Eq. (25), as function of the angles $x$ and $\theta$ calculated for different values of the momentum $k$.

Figure 3

The matrix elements of the permutation coefficient $C(kcos(\theta ),ksin(\theta ),x)$, Eq. (29), as function of the angles $x$ and $\theta$ calculated for different values of the momentum $k$.

Figure 4

The magnitude of the relativistic three-body bound-state wave function $\Psi (p_{jk},k_i,x_{p_{jk}})$ as a function of the pair momentum $p_{jk}$ and the spectator momentum $k_i$ for fixed values $x_{p_{jk}}=0$ (a) and $x_{p_{jk}}=+1$ (b) obtained with the MT-V potential [5, 18]. The differences between the relativistic and nonrelativistic wave functions are shown for fixed values $x_{p_{jk}}=0$ in panel (c) and $x_{p_{jk}}=+1$ in (d).

Figure 5

The relativistic and nonrelativistic momentum distribution functions $n\left(k_i\right)$ (solid line and solid squares) and $n\left(p_{jk}\right)$ (dashed line and solid circles) obtained from the MT-V potential [5, 18]. The function $n\left(p_{jk}\right)$ is scaled with a factor 0.1.

Figure 6

The difference between the relativistic and nonrelativistic momentum distribution functions (a) for $n\left(p_{jk}\right)$ and (b) for $n\left(k_i\right)$ calculated from the MT-V potential (solid line). The dashed line shows the difference for the case when $C=1$ in the relativistic calculation, while for the dash-dotted line $N=1$. The solid circles represent a calculation in which both $C=N=1$.