Testing a new method for scattering in finite volume in the $\phi^4$ theory

We test an alternative proposal by Bruno and Hansen [1] to extract the scattering length from lattice simulations in a finite volume. For this, we use a scalar $\phi^4$ theory with two mass nondegenerate particles and explore various strategies to implement this new method. We find that the results are comparable to those obtained from the L\"uscher method, with somewhat smaller statistical uncertainties at larger volumes.


Introduction
Lattice QCD has been shown to be a powerful tool to determine scattering quantities from first principles. The standard approach is the Lüscher method [2], which relates the finite-volume spectrum obtained from the lattice to the infinite-volume scattering amplitude. It has been applied to many physical systems, including results at the physical point-see Ref. [3] for a review. The formalism has also been recently extended to three particles with three different but conceptually equivalent formulations available in the literature at present [4][5][6][7][8], see Refs. [9,10] for recent reviews.
In Ref. [1], the authors propose a new strategy to extract scattering quantities. Henceforth, this will be referred to as the BH method. This approach is based on the usage of four-point functions rather than energy levels. The hope is that this approach can be generalised more easily to multi-hadron processes.
As pointed out by the authors, the case of threshold kinematics is particularly favourable as it allows for a direct extraction of the scattering length, with the πN channel being one concrete example. a e-mail: garofalo@hiskp.uni-bonn.de In this letter we test this novel approach in a scalar φ 4 theory. Using this theory has proven to be an excellent test bed for novel scattering studies, as shown in Refs. [11][12][13]. In order to mimic the πN case, we consider two mass nondegenerate real scalar particles. We explore the necessary techniques, and the optimal approach to use the BH method at threshold. Moreover, we compare to the standard Lüscher approach and find good agreement.

Description of the Model
The Euclidean model used here is composed by two real scalar fields φ i , i = 0, 1 with the Lagrangian with nondegenerate (bare) masses m 0 < m 1 . The Lagrangian has a Z 2 ⊗ Z 2 symmetry φ 0 → −φ 0 ⊗ φ 1 → −φ 1 , which prevents sectors with even and odd number of particles to mix.
To study the problem numerically, we define the theory on a finite hypercubic lattice with lattice spacing a and a volume T · L 3 , where T denotes the Euclidean time length and L the spatial length. We define the derivatives of the Lagrangian (eq. (1)) on a finite lattice as the finite differences ∂ µ φ(x) = 1 a (φ(x + aµ) − φ(x)). In addition, periodic boundary conditions are assumed in all directions. The discrete action is given in Ref. [12] for the complex scalar theory, but it is trivial to adapt to this case. We set a = 1 in the following for convenience.

arXiv:2107.04853v1 [hep-lat] 10 Jul 2021
In Ref. [1], Bruno and Hansen derived a relation between the scattering length a 0 and the following combination of Euclidean four-point and two-point correlation functions at the two-particle threshold: with the time ordering t f > t > t i > 0. The relation of C BH 4 to the scattering length reads where µ 01 = (M 0 M 1 )/(M 0 +M 1 ) is the reduced mass. It is defined in terms of the renormalized masses M 0 and M 1 of the two particles. These masses can be extracted as usual from an exponential fit at large time distances of the two-point correlation functions In the following we discuss three different strategies to extract the scattering length: 1. We attempt a direct fit of eq. (3) the the data. 2. We include an overall constant in the fit to account for the O (t − t i ) 0 effect. 3. We make use of a shifted function at fixed t i and t f , , which cancels the constant term. We then determine a 0 by fitting to t With the second strategy-the red band in fig. 1-one is able to start fitting at significantly smaller t-values. The data are well described with a χ 2 /dof ∼ 0.2 For the third approach, we study ∆ t C BH 4 (t). This is shown in fig. 1 as blue circles, and the blue band represents the best fit result with error. The main advantage of the last strategy is that it allows us to extract the physical information at smaller t without introducing extra parameters in the fit. Indeed, the data looks almost constant over the complete t-range available. Only very close to t i the square root term might become visible.
For this third strategy, which looks most promising from a systematic point of view, we also investigate the dependence on the choice for t i and t f . This is shown in fig. 2 for the same ensemble as in fig. 1. We do not observe any significant systematic effect stemming -40 -20 0 5 10 from excited state contributions when changing t i or t f . However, we clearly see significantly smaller statistical uncertainties with smaller t i and t f values.

Comparison to the Lüscher method
In this section we compare the BH method described above with the Lüscher threshold expansion [14]. The latter relates the two-particle energy shift, defined as ∆E 2 = E 2 − M 0 − M 1 , to the scattering length a 0 via with c 1 = −2.837297, c 2 = 6.375183 and E 2 being the interacting two-particle energy. E 2 can be extracted from C 2 (t) = φ 1 (t)φ 0 (t)φ 1 (0)φ 0 (0) , whose large-t behaviour is Note that the last term is a known thermal pollution due to finite T in the presence of periodic boundary conditions. Using M 0 and M 1 as input determined from the corresponding two-point functions, the only additional parameter is B 2 .
Alternatively, it is possible to eliminate the second term defining Lüscher   Fig. 3 Comparison of a 0 computed with BH method eq. (4) with t i = 2 and t f = 10 (blue circles), with t i = 3 and t f = 16 (red triangles) and Lüscher method eq. (5) (black squares). The horizontal bands correspond to the weighted average of each method. and then taking the finite derivative The two-particle energies obtained from eq. (6) are compatible to those from eq. (8). The results are reported in table 1, along with the values for the scattering length a 0 computed from E 2 using eq. (5).
The comparison between the BH and the Lüscher method is depicted in fig. 3 for all our ensembles. The values are compatible with each other, however the BH method gives systematically larger values for a 0 .

Conclusion
In this letter, we have investigated the BH method, proposed in Ref. [1], using a scalar theory on the lattice. We have indeed verified that it is a viable method to obtain the scattering length, and that it produces results that are compatible with those of the Lüscher method [14]. The most reliable strategy to analyse the four-point function is found to be the use of finite differences in time to remove an overall constant term.
We observe a systematic difference between the Lüscher and BH method. Interestingly, for each ensemble separately both determinations appear compatible. The systematic trend becomes evident only after averaging over all runs, as shown in the bands of fig. 3. This might be attributed to different lattice artefacts, since both methods represent different estimators for a 0 . We are not able to check this hypothesis here, because we cannot take the continuum limit. However, the different  Table 1 Values of a 0 , M 0 , M 1 and E 2 measured. The column ∆tC BH corresponds to the value of a 0 fitted with eq. (4) fixing t i = 3 and t f = 16 or t i = 2 and t f = 10, the column C BH + c is the result of the fit eq. (3) adding a constant term. The two-particle energy E 2 is computed form C 2 with the fit of eq. (6) and from ∆C 2 with eq. (8). The corresponding value of a 0 computed with the Lüscher method is reported in the corresponding columns.
systematics of the two methods offer in general a useful opportunity for cross checks.
The statistical error is similar in both approaches. Also the scaling in L appears to be similar, with maybe a slight advantage for the BH method. However, any advantage of one method compared to another one will in general depend on the theory considered. We conclude that it seems promising to use the BH method in lattice QCD for instance for πN scattering, where also the lattice spacing dependence could be investigated.