Perturbative results for two and three particle threshold energies in finite volume
We calculate the energy of the state closest to threshold for two and three identical, spinless particles confined to a cubic spatial volume with periodic boundary conditions and with zero total momentum in the finite-volume frame. The calculation is performed in relativistic quantum field theory with particles coupled via a interaction, and we work through order . The energy shifts begin at , and we keep subleading terms proportional to , and . These terms allow a non-trivial check of the results obtained from quantization conditions that hold for arbitrary interactions, namely that of Lüscher for two particles and our recently developed formalism for three particles. We also compare to previously obtained results based on non-relativistic quantum mechanics.
We have recently derived a quantization condition that, subject to some conditions, determines the energy spectrum for three particles in a cubic box Hansen and Sharpe (2014, 2015a). The formalism is fully relativistic, and extends earlier work in non-relativistic effective field theories Polejaeva and Rusetsky (2012); Briceno and Davoudi (2013). One application of this general result is to determine how the energy of the state lying closest to the three-particle threshold depends on the box size . We have developed such a threshold expansion through in Ref. Hansen and Sharpe (2015b).
This expansion also allows a test of the formalism. Since the energy shifts are proportional to , the kinematics becomes nonrelativistic for large enough . Thus we can compare our results to those obtained using non-relativistic quantum mechanics in Refs. Beane et al. (2007); Tan (2008). This comparison is successful for the three leading terms (proportional to with ), but turns out not to be useful for the term. This is because the three-particle interaction enters at this order, and there is no a priori relation between a non-relativistic contact interaction and a relativistic three-particle amplitude at threshold.111One indication that there cannot exist a simple a priori relation between the non-relativistic and relativistic three-body interactions is that both require regularization and are scale and scheme dependent, with different regularization schemes needed for the two theories. In fact, in the relativistic context the situation is even more complicated. This is because, as has been known for a long time (see, e.g. Refs. Rubin et al. (1966); Brayshaw (1968); Potapov and Taylor (1977a, b), and our recent discussion in Ref. Hansen and Sharpe (2014)), the standard three-particle scattering amplitude diverges at threshold. To obtain a finite threshold amplitude one must perform subtractions, e.g. following the methods introduced in Refs. Hansen and Sharpe (2014, 2015b). These, however, are not unique—many definitions are possible. Thus one can only use the terms as a method for determining this relation, and not as a check on our threshold expansion.
We have thus turned to perturbation theory (PT) as an alternative tool to provide a test of the results from the threshold expansion. We pick the simplest interacting, perturbatively-renormalizable relativistic QFT—scalar theory—and determine the threshold energy shift through , keeping terms scaling as with in the large volume expansion. Cubic order is sufficient to provide a non-trivial check on the threshold expansion developed in Ref. Hansen and Sharpe (2015b). Furthermore, although the theory has no bare six-point vertex, there is an induced three-particle scattering amplitude starting at , and the subtraction methods developed in Refs. Hansen and Sharpe (2014, 2015b) are tested at one-loop order by our calculation. In particular, as part of our calculation, we have worked out the subtracted three-particle amplitude at threshold through cubic order, using the subtraction defined in Ref. Hansen and Sharpe (2015b).
We have carried out the calculation both for two and three particles, so as to provide further cross-checks. In particular, we can compare the result of our perturbative threshold expansion for two particles with that obtained using the relativistic finite-volume two-particle formalism developed in Refs. Luscher (1986, 1991). The agreement we find, as described below, gives us confidence in our methodology.
We can also compare to the results for two particles obtained using non-relativistic quantum mechanics in Ref. Beane et al. (2007). Since the three-particle amplitude does not enter into this result, the comparison is unambiguous. We find a discrepancy in terms proportional to , and make a suggestion for its source.
The remainder of this article is organized as follows. In the following section we give an overview of our methodology. Results for the two-particle threshold energy are then worked out in Sec. III. In Sec. IV we calculate the subtracted, three-particle scattering amplitude at threshold. In Sec. V we determine the three-particle threshold energy and express it in terms of this subtracted amplitude as well as two-to-two scattering observables. We make the comparisons to prior results in Sec. VI, and present some conclusions.
Three appendices contain technical details. Appendix A derives two identities for finite-volume momentum sums. Appendix B works out the s-wave scattering length and effective range in theory. Finally, appendix C develops the threshold expansion for two particles that follows from Lüscher’s quantization condition out to .
Ii Overview of calculation and methodology
We consider the relativistic QFT described by the Euclidean Lagrangian density
with a scalar field. Although this theory contains no six-particle local interaction, a scattering amplitude is induced, as discussed below. This theory has a symmetry, under which , that forbids amplitudes involving an odd number of fields. This symmetry is also assumed in the general three-body formalism of Refs. Hansen and Sharpe (2014, 2015a). We have included counterterms for wavefunction and mass renormalization. These are tuned so that is the physical, infinite-volume mass and that the residue of the infinite-volume propagator at the mass pole is unity. Note that we do not include an explicit counterterm for the coupling , preferring to work initially with the bare coupling and later describe its renormalization.
To determine the energies of states close to threshold, we calculate even and odd particle-number correlators in finite volume. Choosing convenient overall factors, these are defined by
The subscript “” indicates that the spatial volume is restricted to a cube of side , with periodic boundary conditions applied to . Thus the allowed momenta are , with a vector of integers. We work in Euclidean space, with the Euclidean time, which is taken to have infinite range. We choose to place all fields at zero spatial momentum since the threshold state in the absence of interactions consists of particles at rest. Thus our creation and annihilation operators will have large ( in PT) overlap with the actual threshold state even in the presence of interactions. This is a convenience, but is not strictly necessary since all we need is for our operators to have some overlap with the threshold state.
The general form of these two correlators is known in terms of the eigenstates of the Hamiltonian of the theory. Assuming, as we do henceforth, that , we have
Due to the symmetry, it is possible to separate states with even- and odd-particle quantum numbers into and respectively. This implies that contains a contribution from the vacuum state, which [given the inclusion of the in its definition, Eq. (2)] leads to a growing exponential of . Similarly, contains an exponentially growing contribution from a single-particle state. Such growing exponentials might be problematic in a numerical simulation, but can be readily identified in an analytic calculation.
Our aim is to pick out from the infinite sum of exponential contributions, those corresponding to the states nearest threshold. We know that there is only one such state for each correlation function, since there is only one such state in the free theory (with all particles at rest) and we are making an infinitesimal perturbation. For these threshold states, the quantities vanish as as a sum of powers of , up to possible logarithmic corrections—which in fact do not arise at the order we work—and exponentially suppressed terms. The latter, which behave as , we neglect throughout. Such corrections are also dropped in Lüscher’s general two-particle quantization condition and in our general three-particle formalism. As mentioned above, in this work we expand the threshold energy shift, , in both and , working through and .
As is well known (see, e.g., Refs. Luscher (1986, 1991); Beane et al. (2007); Tan (2008)), the leading contribution to the threshold energy shift is proportional to . As we now explain, this implies that we can unambiguously pick out from the the contribution from the near-threshold state. Consider first the behavior of from excited states. The lightest such state adds a minimal unit of relative momentum between two of the particles, so that , where with . For , the energy shift can be expanded as , and so we see that the excited states are parametrically separated from the near-threshold state, whose energy shift scales as as noted above. Thus there can be no avoided level crossings, and, in an analytic calculation, we can unambiguously identify the excited state contributions. We stress that it is crucial to discard these exponentials before expanding in , so that the contributions do not become confused with the ground-state energy shift. Excited states involving more particles (e.g. five-particle states in ) are even more obviously separated, since then , which does not vanish as . Similarly, far-subthreshold states, which contain less particles (e.g single particle states in ), have , and the exponentials can also be easily separated.
In light of the previous discussion, the method we use is as follows. We calculate and order by order in PT, and remove by hand the contributions from exponentially growing far-subthreshold states and from exponentially falling excited states. The resulting subtracted correlators we call . We know that these have the form
Thus if we expand in powers of , and keep only the constant and linear terms,
then the threshold energy shift is given by
The advantage of this method is that it allows us to keep track of powers of in a straightforward manner. An alternative approach would be to identify the infinite set of perturbative diagrams leading to the exponential behavior in Eq. (8), with its associated renormalization factor . This requires working to all orders in in a subset of diagrams. We have used this alternate method as a check on our results, though we present no details here.
where, again, is the energy of a free particle with a finite-volume momentum . For each loop there will be a sum over spatial momentum restricted to the allowed finite-volume values. In addition there will be an integral over the Euclidean time of each intermediate vertex. For given values of each of the spatial momenta, this is simply an integral over exponentials, with the integrand depending on the time ordering due to the absolute values in the propagators, Eq. (11). Thus the integrals are trivial, but keeping track of all the time orderings is less so. We have used two independent Mathematica codes to ensure that all terms are included. After the vertex integrals, one can read off which terms are exponentially falling or growing, drop these by hand, and thus obtain . Expanding in powers of leads to the results for and [see Eq. (9)] and thus to [using Eq. (10)].
The final stage of the calculation is, for each loop, to sum over the finite-volume momenta. This is done by converting sums to integrals, which can be absorbed into infinite-volume loop contributions, plus a finite-volume residue leading to the desired power-law terms. The methods for converting sums to integrals are variants and extensions of those used in deriving the general finite-volume quantization conditions Luscher (1986, 1991); Hansen and Sharpe (2014) and their threshold expansions Hansen and Sharpe (2015b). The results we need are collected in Appendix A.
As a simple illustration of these methods, consider the diagrams of Figs. 1(a) and 2(a). By construction, both diagrams lead to . In particular, the contraction factors cancel the and in Eqs. (2) and (3), respectively. There are thus no exponentially growing or falling terms to remove by hand, and at this order.
Since only the constant term in is non-vanishing at leading order, the perturbative expansions of the quantities of interest can be written:
Inserting these expansions into Eq. (10), we find
Thus a third-order calculation of requires a third order result for but only a second-order result for .
Another simple example of our methods is provided by the disconnected diagram of Fig. 1(b). This diagram clearly has no dependence on , so when we use Eq. (2) we find that . Dropping this exponentially growing term leads to , so this diagram makes no contribution to the threshold energy.
The final example we consider here is the connected diagram Fig. 1(c), which involves a single vertex. All four propagators have vanishing spatial momenta, so we only need to do the integral over the vertex position, . The resulting contribution to the correlator is
Here there are no exponentially growing or falling terms to remove, and we find the first nontrivial result for the threshold energy shift
as well as an contribution to the constant term
We now describe three classes of diagram that we do not need to calculate explicitly, although they contribute to the at the order we work. This is because they either do not contribute to , or they lead only to changes in the overall normalization of the correlators, , but not to the energy shifts . Examples of these diagrams are shown in Fig. 3 for and Fig. 4 for .
The first class involves self-energy and counterterm insertions on the diagrams described above (i.e., those in Figs. 1 and 2). All diagrams in Fig. 3 and those of Fig. 4 (a)-(c) are examples of this class. We first note that contributions involving tadpole diagrams, such as those in Figs. 3(a)-(f) and Fig. 4(a), are cancelled identically by the counterterm. This is because the loops are independent of the external momenta and thus lead only to mass renormalization. The only subtlety is that actually cancels the infinite-volume version of the tadpole diagram, in which the loop is integrated rather than summed. However, since the difference between the sum and the integral is exponentially suppressed, scaling like , no contribution remains in our expansion.222The difference between loop sums and integrals is exponentially suppressed as long as there are no cuts through the loops in which, for threshold kinematics, all particles can go on shell. Several examples where this is not the case are described in subsequent sections.
The other renormalization diagrams that contribute at the order we work are those that do not involve tadpoles. These are those shown in Figs. 3(g)-(i) and Figs. 4(b) and (c), together with related diagrams in which the renormalizations appear on different external lines, and diagrams containing the corresponding counterterms. Note that renormalizations appear only on external propagators at this order. For all these diagrams the loop sums can be replaced by integrals—there are no power-law finite-volume residues. Thus, if the propagators were evaluated on mass shell, the contributions of these diagrams would exactly cancel with those containing the appropriate mass and wavefunction counterterms (given our convention that the residue at the mass pole is unity). In fact, the external propagators, while evaluated at vanishing spatial momenta, are not Fourier transformed in time. Thus they contain excited state contributions. However, once these are removed, following our general prescription described above, we expect the general argument to hold and the contributions of these diagrams and those with the corresponding counterterms to cancel exactly. We have checked that this is the case by explicitly calculating these diagrams.
The second class of diagrams are those exemplified by Figs. 4(d), (e) and (f), as well as those obtained by horizontal reflection. The characteristic feature of this class is that there is a disconnected propagator joining two of the external fields at either the initial or final time, multiplied by a “one-to-three” correlator. This factorization is maintained as higher order corrections are included. The disconnected propagator has no dependence, and so provides only an overall factor. Thus, applying our methodology to this class of diagrams amounts to studying the three-particle threshold energy using the one-to-three correlator. As noted above, this is a legitimate approach, since one can use any interpolating fields with the correct quantum numbers. In particular, this class of diagrams alone must give the same result for as that obtained from the full . Thus we can drop these one-to-three diagrams without changing the result for the energy shift. We have checked this argument by explicitly calculating all the diagrams in this class up to order . Note that, since one vertex is needed to convert the initial single particle into three, only a second-order result for is obtained.
The third and final class of diagrams are those exemplified by Figs. 4(g), (h) and (i). Here one is effectively calculating the threshold energy shift using the “one-to-one” correlator, with the disconnected propagators at each end only changing the overall factor. Once again, this class of diagrams alone, analyzed using our method, must yield the correct result for , and so can be dropped.
Iii Two particle energy shift
In this section we calculate the threshold energy shift for two particles. We work to in PT and keep terms up to in the volume expansion. We have already obtained the contribution linear in , Eq. (20), so we start here with the quadratic term, which arises from Figs. 1(d) and (e).
For Fig. 1(d), the form of the result depends on whether the loop momentum, , vanishes or is non-vanishing. This is because one of the terms which enters the energy shift in the case becomes an excited state exponential, to be discarded, in the case of non-vanishing loop momenta. The zero-momentum contribution is
The term is the second term in the expansion of , with given in Eq. (20), and does not concern us here. For we find the contribution
The superscript indicates that ultraviolet (UV) regularization is required, although the choice of regulator is unimportant. The last term in the summand is from excited states, and is dropped in , as explained in the previous section. The remaining two parts of the summand diverge for , and are converted to integrals plus finite-volume residues using the results in Appendix A. For example, the term proportional to leads to the following contribution to :
where we have used Eq. (102) and introduced the shorthand .
Turning to Fig. 1(e), we note that does not need to be treated separately. After dropping excited-state contributions, we obtain
Here the sum can be replaced by an integral, since the summand is nonsingular and we are dropping exponentially suppressed volume dependence.
Here is the one-loop integral defined in Appendix B, Eq. (111), while is the finite integral
Two comments are in order. First, the terms are exactly those needed to convert the factor of multiplying the first-order results in Eqs. (20) and (21) into the renormalized . The latter is defined in Eqs. (112)-(113), which we reproduce here for clarity:
where is the two-particle scattering length. Second, we have truncated at , since higher order terms in lead to contributions to of . This is because multiplies when it contributes to , as can be seen from Eq. (17).
Now we move to third order. As is clear from Eq. (17), we only need to determine the term linear in , , in order to obtain . Furthermore, we can drop any contributions falling as or faster.
We begin with Fig. 1(f). If both loop momenta vanish, the diagram is proportional to and can be dropped. If only one loop momentum vanishes, we find (dropping higher order terms)
The UV divergent integral in Eq. (32) combines with that in the result from Fig. 1(i) [given in Eq. (38) below] to give a term proportional to . This turns out to be exactly the contribution needed to convert the factor of multiplying the term in Eq. (22) to .
If both momenta are non-vanishing, the result factorizes into a product of loop sums333This factorization occurs because, in order to obtain a contribution proportional to , the times of the vertices must satisfy . The set-up is then essentially the same as in the calculation of a threshold scattering amplitude, for which we know, from using Feynman diagrams, that the contributions from the two loops factorize. This is true both in finite and infinite volume..
The only product of finite-volume residues that falls slowly enough to be included is
Terms involving a single finite-volume residue multiplied by the integral give part of the contribution needed to convert the factor of multiplying the and terms in Eq. (25) into . Terms involving two integrals contribute to two-loop renormalization, generating part of the term which converts to in the contribution to .
The sums in both loops in Fig. 1(g) and (h) can all be converted into integrals, and these integrals are exactly those obtained when the same diagrams are evaluated as contributions to infinite-volume scattering. It follows that these diagrams contribute only to renormalization of lower-order terms.
This leaves Fig. 1(i), and its horizontal reflection. Here we must treat the cases and separately (see the label in the figure), since the separation between ground and excited states is different in the two cases. For , the contribution to the threshold correlator is
The UV divergent integral leads to a renormalization of lower-order terms, as described above in the discussion following Eq. (32).
For , the diagram contributes
where and . We can replace the sum over with an integral since the summand is regular, leading to
Here we have assumed that the UV regulator maintains rotational invariance, and used Eq. (102). The first term in Eq. (42) contributes to the two loop renormalization of , while the second completes the renormalization of the term in Eq. (25). The third term completes the renormalization of the term in Eq. (25), leaving a finite residue which we now calculate.
To determine we must expand for small :
where the are functions of . Performing the angular average over , and picking out the term quadratic in , we find
Carrying out the algebra, and the resulting finite integrals, we find
The last term is part of the renormalization of the term in Eq. (25), as mentioned above. The first two terms complete the finite residues at the order we work.
where the “renorm. parts” are the above-described contributions that convert to in lower-order terms.
Iv Divergence-free three-particle scattering amplitude at threshold
The threshold energy shift in the three-particle case, , depends, at , on the three-particle scattering amplitude, . As is well know, however, is singular for certain choices of external momenta and, in particular, at threshold (see, e.g., Refs. Rubin et al. (1966); Brayshaw (1968); Potapov and Taylor (1977a, b)). This is a well-understood physical singularity, described, for example in Ref. Hansen and Sharpe (2014). It implies that the term in cannot depend on itself, but rather on a subtracted version which is finite at threshold. In Ref. Hansen and Sharpe (2014) we provide one possible definition for a subtracted amplitude, which we call the divergence-free amplitude. This definition is general, working for particles of any masses and both at and away from threshold. It is, however, a cumbersome definition to implement (e.g. involving an infinite number of subtractions, all but three of which are finite for degenerate particles at threshold). Our analysis of the threshold expansion of the three-particle quantization condition suggests instead using a simpler quantity, which we call , whose definition involves only the minimum necessary subtractions Hansen and Sharpe (2015b). This quantity is motivated and discussed at length in Ref. Hansen and Sharpe (2015b).
The purpose of this section is to calculate to , so that we can express our perturbative result in terms of this infinite-volume quantity. Its definition is Hansen and Sharpe (2015b)
Here is an IR cutoff, whose definition will be explained in the context of the following calculation. There are three subtraction terms, involving , and , respectively. Only the first two terms enter at the order we work, since . We give the definitions of the relevant parts of and below.
The diagrams that contribute to at the order we work are those of Figs. 2(i)-(o), now interpreted as Feynman diagrams in infinite volume. We first consider the diagram, Fig. 2(i), which is also reproduced in Fig. 5(a) along with the momentum labels we use. This gives a contribution that diverges at threshold, since the intermediate propagator goes on shell. The prescription of Eq. (49) corresponds here simply to working away from threshold with general momenta, making the subtraction (here of ), and then sending all external momenta to zero Hansen and Sharpe (2015b). The contribution to the scattering amplitude is
where , and we have used the vanishing of the total spatial momentum. The denominator of the propagator is
where and, here, . This denominator vanishes when all the spatial momenta tend to zero, for then and both tend to . It also vanishes for non-zero momenta if (i.e. if ) and we must avoid such above-threshold divergent momentum configurations. The subtraction term corresponding to this diagram is the leading order part of Hansen and Sharpe (2015b)
By construction this has the same pole at as , so that the difference is finite (both at threshold and for above-threshold divergent momenta):
The final steps needed to obtain are to symmetrize over the external momentum assignments,444Since the particles are identical, interchanging and does not lead to a different assignment, so there are only three choices for each of the initial and final states and to take the threshold limit. Since the limiting value is independent of the choice of external momenta, symmetrization gives a factor or , and the total contribution from this diagram is
We next consider the diagrams in which the vertices in the tree diagram receive one-loop corrections. The full set of of these are those involving the s-, t- and u-channel loops shown in Figs. 5(b), (c) and (d), respectively, as well as the corresponding corrections to the right-hand vertex. The subtraction in this case is somewhat subtle so we provide a more detailed explanation.
A key point in the following is that the intermediate propagator (with momentum ) is off shell, and so the loop corrections to the vertices at either end of this propagator differ in general from those which enter the on-shell scattering amplitude. This is true despite the fact that three of the four legs are on shell (e.g. those with momenta , and for the left-hand vertex). In fact, at one-loop order, the s-channel loops [e.g. Fig. 5(b)] do not depend on the off-shellness of , while the t and u-channel loops [e.g. Figs. 5(c) and (d)] do.
This point is important because the general form of the subtraction term, , replaces the loops at either end of the central propagator with on-shell scattering amplitudes, as well as changing the form of the propagator (as described above for the tree-level diagram). This feature of the subtraction is crucial, since it means that it is given in terms of physical quantities. The general form is quite complicated because it involves scattering amplitudes in all partial waves Hansen and Sharpe (2015b). However, since we are interested in the subtracted amplitude at threshold, we need only the part of that involves s-wave scattering, and we need this only close to threshold. Specifically, we have
where the superscript “” refers to the subfigures within Fig. 2. Here is the magnitude of the spatial momentum of evaluated in the CM frame, and is the same quantity for . The result (57) is simply the threshold expansion of the two-particle scattering amplitude, obtained using Eqs. (108), (112) and (119) in Appendix B. Note that this is expressed in terms of . Note also that the leading order part of Eq. (56) is the subtraction (53) that we used for the tree-level diagram.
As noted above, only the t- and u-channel loops differ from the on-shell scattering amplitude. The part which differs is, close to threshold, proportional to , where, for example, and when the left-hand vertex is being corrected. Now , and is the same irrespective of whether is on or off shell. Thus the difference between on- and off-shell is equal to the difference between on and off-shell, which is just . The key point is that this cancels the denominator of the propagator, leaving a finite residue in the threshold limit.
We can now perform the subtraction, take the threshold limit, and multiply by the momentum permutation factor of 9, to obtain the final result from the diagrams of Fig. 5:
Note that we have written this result in terms of the renormalized coupling.
where is the UV cutoff function discussed in Appendix A. is to be integrated over with an IR cutoff , so that the integral is finite. The definition of the IR-regulated three-particle scattering amplitude, is, in general, quite involved, as it requires applying a cut-off both on loop momenta and also on the external momenta. However, the external momenta scale as , which implies that the IR cutoff they induce in the loop integral is weaker than that from the direct cutoff at . This means that we can set the external momenta to zero. Doing so, we find that
where the subscript on the integral indicates the IR cutoff. Combining with the integral of leads to an IR convergent integral in which one can set :
We stress that this is a UV convergent integral that has a finite value, but that this value depends on the choice of cutoff function .
Next we consider the “twisted bull’s head” diagram of Fig. 2(m). This diagram is convergent in the IR, so no subtraction is needed, and it can be evaluated directly at threshold. We find
The final diagrams involve intermediate single-particle states. These also have no IR divergences, require no subtractions, and can be evaluated directly at threshold. Since Fig. 2(n) is a tree diagram, it is simple to evaluate, and yields
The one-loop corrections to this diagram are given by Fig. 2(o) and the similar diagram where the right-hand vertex is corrected. Note that although this looks like an s-channel loop and might be expected to have t- and u-channel partners, in fact, no other diagrams exist. Evaluating the loop we find that the sum of the two diagrams gives
The UV divergence here is exactly that needed to convert in (65) to , up to a finite residue:
Combining results from all diagrams, we obtain