Relativistic, model-independent, three-particle quantization condition

We present a generalization of Lüscher’s relation between the finite-volume spectrum and scattering amplitudes to the case of three particles. We consider a relativistic scalar field theory in which the couplings are arbitrary aside from a ${\mathbb{Z}}_2$ symmetry that removes vertices with an odd number of particles. The theory is assumed to have two-particle phase shifts that are bounded by $\pi /2$ in the regime of elastic scattering. We determine the spectrum of the finite-volume theory from the poles in the odd-particle-number finite-volume correlator, which we analyze to all orders in perturbation theory. We show that it depends on the infinite-volume two-to-two K-matrix as well as a nonstandard infinite-volume three-to-three K-matrix. A key feature of our result is the need to subtract physical singularities in the three-to-three amplitude and thus deal with a divergence-free quantity. This allows our initial, formal result to be truncated to a finite dimensional determinant equation. At present, the relation of the three-to-three K-matrix to the corresponding scattering amplitude is not known, although previous results in the nonrelativistic limit suggest that such a relation exists.

Figure 1

Example of singular contribution to the on shell three-to-three scattering amplitude. Dashed lines are on shell, amputated, external propagators, while the solid line is a fully dressed propagator, which can in general be off shell. Filled circles represent two-to-two scattering amplitudes. The internal (solid) line can become on shell for physical external momenta, corresponding to two isolated two-to-two scattering events.

Figure 2

Diagrammatic definition of the divergence-free three-to-three amplitude, ${\mathcal{M}}_{\mathrm{df},3}$. In the subtracted term, filled circles represent on shell two-to-two scattering amplitudes ${\mathcal{M}}_2$. Dashed cuts stand for simple kinematic factors that appear between adjacent ${\mathcal{M}}_2$. These factors have the requisite poles so that the subtracted terms cancel the singularities in ${\mathcal{M}}_3$. The $\mathcal{S}$ outside the square brackets indicates that the subtracted terms are symmetrized.

Figure 3

The smooth cutoff function $H(\overrightarrow{k})\equiv J(x)$ with $x=E_{2,k}^{*2}/[4m^2]$. The function varies from 0 to 1 as $E_{2,k}^{*2}\equiv (E-{\omega }_k{)}^2-(\overrightarrow{P}-\overrightarrow{k}{)}^2$ varies from 0 to $4m^2$. Using this range of variation ensures that the function has width $\mathrm{\Delta }\approx m$ in the space of spectator momentum $\overrightarrow{k}$.

Figure 4

Skeleton expansion defining the finite-volume correlator. The rightmost blob in all diagrams represents a function of momentum ${\tilde{\sigma }}^{†}$, whose specific form is determined by the interpolating fields defining the correlator. The leftmost blob represents an analogous function, $\tilde{\sigma }$. Any insertion between these with four legs represents a two-to-two Bethe-Salpeter kernel $i{B}_2$. Any insertion with six legs represents an analogous three-to-three kernel $i{B}_3$. All lines connecting kernels and $\tilde{\sigma }$-functions represent fully dressed propagators. The dashed rectangles indicate that all loop momenta are summed rather than integrated, due to the finite-volume condition. The regions bounded by these rectangles also emphasize chains of loops that have common coordinates which prevent the diagram from factorizing. This is one of the central complications faced in this work. (For example the top line, with only three-to-three insertions, does factorize and is therefore a straightforward generalization of the two-particle case.)

Figure 5

Examples of Feynman diagrams contributing to (a) $i{B}_2$, the two-to-two Bethe-Salpeter kernel and (b) $i{B}_3$, the analogous three-to-three kernel.

Figure 6

Finite-volume correlator diagram with no kernel insertions.

Figure 7

Finite-volume correlator diagrams containing only two-to-two insertions with no change in the scattered pair.

Figure 8

Diagrammatic representation of Eq. (71).

Figure 9

(a) Diagrams contributing power-law finite-volume contributions to $C_L^{(1)}$. The dashed line for the bottom propagator indicates that the $k^0$ integration has been done and only the one-particle pole kept, giving rise to the factor of $1/2{\omega }_k$. The inset shows the effect of substituting the identity of Fig. 8. (b) Result for $C_L^{(1)}$ after grouping terms according to the number of $F$ insertions. Diagrams with no insertions combine with the terms neglected in (a) to give $C_{\infty }^{(1)}$. In diagrams with at least one insertion of $F$ the factors to the left and right are ${\sigma }^{*}+A^{\prime (1,u)}$ and ${\sigma }^{†*}+A^{(1,u)}$, respectively. The factors between $F$ insertions (denoted by black circles) are two-to-two K-matrices. The final term in the curly braces must be subtracted since it is included in the first term but is not part of the definition of $C_L^{(1)}$. (c) Definition of ${A^{\prime }}^{(1,u)}$. The superscript $u$ indicates that the unscattered particle is also the particle whose momentum is singled out by the coordinate system. Dashed lines for external momenta indicate both that they are on shell and that they are amputated.

Figure 10

Finite-volume correlator diagrams containing only two-to-two insertions, with one switch in the scattered pair.

Figure 11

Diagrammatic definitions of (a) $i{\mathcal{K}}_{3\to 3}^{(2,u,u)}$, (b) $A_L^{\prime (2,u)}$, and (c) $C_{L,0F}^{(2)}$. In (b) and (c) the dotted box encloses momenta that are summed rather than integrated. The solid circle represents the two-particle K-matrix. Other notation is as above.

Figure 12

Diagrammatic version of the decomposition of ${\mathcal{K}}_3^{(2,u,u)}(\overrightarrow{p},\overrightarrow{a},\overrightarrow{k},{\overrightarrow{a}}^{\prime })$ given in Eq. (127). External dashed lines indicate on shell momenta, whereas momenta flowing along the solid external lines are, in general, off shell. All external propagators are amputated. The first term on the right-hand side is the singular term ${\mathcal{D}}^{(2,u,u)}$, with the double dashed lines representing $G^b$. The two ${\mathcal{K}}_2$ are evaluated on shell for all momenta that flow along dashed propagators. The second term represents the divergence-free amplitude ${\mathcal{K}}_{\mathrm{df},3}^{(2,u,u)}$.

Figure 13

(a) Definition of ${A^{\prime }}^{(1,s)}$, in which the momentum singled out by the coordinate system (here $\overrightarrow{k}$) is that of a particle that scatters. (b) Definition of the symmetrized quantity ${A^{\prime }}^{(1)}$.

Figure 14

Finite-volume correlator diagrams containing only two-to-two insertions and with two switches in the scattered pair.

Figure 15

(a) Definition of $A_L^{\prime (3,u)}$. The dotted rectangle contains momenta which are summed; thus only the leftmost two-particle loop is integrated. (b) Definition of $C_{L,0F}^{(3)}$, which has two integrated and three summed loop momenta. (c) Definition of ${\mathcal{K}}_{3,L}^{(3,u,u)}$, which has a single summed momentum.

Figure 16

Decomposition of ${\mathcal{K}}_{3,L}^{(3,u,u)}$. See Fig. 15 for momentum labels. Double solid lines indicate nonsingular terms. On the top line these come from the $r^0$ contour circling poles other than the single-particle pole, while for the diagonal lines in the switch state the notation is as in Fig. 12. Double dashed lines represent the singular quantity $G^b$ sandwiched between on shell amplitudes, as in Fig. 12. The single dashed line within the loop (top propagator) indicates the on shell propagator factor $1/(2{\omega }_r)$. (a) Initial decomposition. Loop momenta inside dotted boxes are summed, while those not in a box are integrated. (b) and (c): Use of the sum-minus-integral identity, as indicated by the vertical bars and factors of $F$, leaving a remainder which is integrated. The vertical bar crosses the two propagators whose momenta are projected on shell by $F$, so that the uncrossed propagator is the spectator. ${\mathcal{K}}_{\mathrm{df},3}^{(3,u,u)}$ is given by the sum of the four terms containing loop integrals [two in (a) and one each in (b) and (c)].

Figure 17

Diagrammatic definitions of (a) $A_L^{\prime (n,u)}$, (b) $C_{L,0F}^{(n)}$, and (c) $i{\mathcal{K}}_{3,L}^{(n,u,u)}$.

Figure 18

Decomposition of ${\mathcal{K}}_{3,L}^{(4,u,u)}$. All external propagators are dropped, and the notation of Figs. 12 and 16 is used. (a) ${\mathcal{K}}_{3,L}^{(4,u,u)}$ itself [see Eq. (171)]; (b) the most singular term (with three singular propagators); (c) and (d): terms with two singular propagators and their decompositions; (e), (f), and (g): terms with one singular propagator and their decompositions; (h), (i), and (j): nonsingular terms. Terms in the decompositions are always ordered from most to least singular. The treatment of loop momenta is indicated explicitly: they are either summed (dashed box), integrated (integral sign) or the sum-minus-integral identity is used (factor of $F$). Where the order of integrals matters it is shown explicitly.

Figure 19

Contours in $w$ complex plane contributing to $f_{\widetilde{\mathrm{PV}}}(z)$. (a) $z$ real and positive; (b) $z$ above the positive real axis; (c) $z$ below the positive real axis. In each case the cross indicates the location of the $z=w$ pole, and the numbers indicate the weights associated with each contour.

Figure 20

$f_{\widetilde{\mathrm{PV}}}(x)$ for $g(w,z)=\exp (z-w)$ compared to the result of using the PV prescription, $f_{\mathrm{PV}}(x)$. The former [dashed (red)] is smooth, while the latter [solid (blue)] has a cusp at $x=0$. For $x<0$, the difference between the two functions is the pole term, the second term in Eq. (b10).