Models, measurements, and effective field theory: proton capture on Beryllium-7 at next-to-leading order
We employ an effective field theory (EFT) that exploits the separation of scales in the -wave halo nucleus to describe the process up to a center-of-mass energy of 500 keV. The calculation, for which we develop the lagrangian and power counting in terms of velocity scaling, is carried out up to next-to-leading order (NLO) in the EFT expansion. The Coulomb force between Be and proton plays a major role in both scattering and radiative capture at these energies. The power counting we adopt implies that Coulomb interactions must be included to all orders in . We do this via EFT Feynman diagrams computed in time-ordered perturbation theory, and so recover existing quantum-mechanical technology such as the Lippmann-Schwinger equation and the two-potential formalism for the treatment of the Coulomb-nuclear interference. Meanwhile the strong interactions and the E1 operator are dealt with via EFT expansions in powers of momenta, with a breakdown scale set by the size of the Be core, MeV. Up to NLO the relevant physics in the different channels that enter the radiative capture reaction is encoded in ten different EFT couplings. The result is a model-independent parametrization for the reaction amplitude in the energy regime of interest. To show the connection to previous results we fix the EFT couplings using results from a number of potential model and microscopic calculations in the literature. Each of these models corresponds to a particular point in the space of EFTs. The EFT structure therefore provides a very general way to quantify the model uncertainty in calculations of . We provide details of this projection of models into the EFT space and show that the resulting EFT parameters have natural size. We also demonstrate that the only NLO corrections in come from an inelasticity that is practically of NLO size in the energy range of interest, and so the truncation error in our calculation is effectively NLO. The key LO and NLO results have been presented in our earlier papers. The current paper provides further details on these studies. We also discuss the relation of our extrapolated to the previous standard evaluation.
pacs:25.20.-x, 25.40.Lw, 11.10.Ef, 21.10.Jx, 21.60.De
The nuclear reaction creates nuclei inside the Sun, where they quickly decay to produce neutrinos. These neutrinos constitute most of the solar neutrino spectrum above 2 MeV Adelberger:2010qa; Robertson:2012ib and thus nearly the entire signal in chlorine- and water-based detectors Robertson:2012ib. Constraints on neutrino properties and solar interior composition based on this signal depend on comparisons of detected and theoretical neutrino production rates, which require the cross section Robertson:2012ib. However, this cross section must be extrapolated from experimental data above 100 keV down to solar energies around 20 keV using a theoretical model, and the error associated with model selection dominates recent evaluations Davids:2003aw; Descouvemont:2004hh; Adelberger:2010qa.
The so-called Halo effective field theory (halo EFT) developed in recent years vanKolck:1998bw; Kaplan:1998tg; Kaplan:1998we; Bertulani:2002sz; Bedaque:2003wa; Hammer:2011ye; Rupak:2011nk; Canham:2008jd; Higa:2008dn; Ryberg:2013iga; Hammer:2017tjm is well suited to the study of proton capture on Be, because there is a nice separation of scales in the B system near the proton threshold. The effective size of the nucleus can be estimated by looking at its lowest break-up channel, , which has a MeV AME2012II threshold. This translates to an effective binding momentum of MeV ( is the He-He reduced mass ). In developing our EFT below, we take this as the high momentum scale , where the effective theory breaks down, corresponding to a short distance scale fm. The other scales relevant to low-energy direct capture are small compared with . The binding energy of in the breakup channel is MeV AME2012II; AME2016II, which translates to the binding momentum MeV. Here the reduced mass while , and MeV. (In our notation, denotes a valence “nucleon” and denotes a “core” from which a larger nucleus is constructed.) The static Coulomb interaction between and the proton is important near and below threshold, so the Coulomb momentum MeV (with and the particle charges) is also a key parameter. This may be written in terms of the channel momentum and the usual Sommerfeld parameter as , and it also is small compared to . Besides “elastic” channels containing the ground state, there is a low energy excited state of , with the excitation energy MeV; the corresponding momentum MeV is again small compared to . (Note that these numbers are slightly different from those in our previous work Zhang:2014zsa, because here we update the B and masses to the latest mass evaluation AME2012II; AME2016II.) All these momenta correspond to large distance scales fm. As a result, we can consider as a “core” particle with one low-energy excitation, and as a shallow bound state with both and channels. Meanwhile the -wave interaction between and in the initial state of our reaction has scattering lengths that are markedly larger than the short-distance scale in both spin channels, and fm Angulo2003a. In that sense, the -wave interactions between core and proton are abnormally strong. Dynamics associated with all of these low-momentum scales—, , , as well as and —can be accounted for in the EFT framework. Based on this, an order-by-order expansion in a parameter that is exists for the scattering and reaction amplitudes.
The dominance of large-length-scale contributions near threshold, which is a prerequisite for rapid convergence of an EFT, has always been prominent in models of the reaction. Existing models based on ordinary quantum mechanics include potential models in which is made of structureless protons and nuclei interacting through an effective potential christyduck; barker80; davidstypel03; esbensen04; huang10; navratil06, phenomenological -matrix models that avoid an explicit potential by reducing its effects to a few fitted parameters barker95; barker00, and “microscopic” calculations in which is a collection of eight interacting nucleons. Until recently, computational limits restricted microscopic models severely, but they nonetheless incorporated important effects like core excitation and wave function antisymmetry johnson92; descouvemont94; Descouvemont:2004hh. Computational limits are less severe now, and true ab initio calculations with bare nucleon-nucleon interactions and much more complete computational bases have become possible Navratil:2011sa. As ab initio models increase in completeness and complexity, they hold the promise of reducing cross section uncertainties by producing reliable constraints complementary to those provided by experiment.
In all models, the dominance of regions with large cluster separation and zero strong interaction is the most important feature of low-energy direct capture. The main difficulty lies in quantifying the influence of the short-range interaction. It is our hope that an EFT formalism can provide a convenient language for understanding and comparing features of all models, and for fitting experimental data with a minimum of tacit assumptions. EFT might also provide a check on the computational consistencies of more complex models and a convenient parameterization for disseminating their results. It should also be useful for consistently stitching together multiple types of information from different experiments and calculations.
We have studied this reaction and its isobaric analog in EFT and presented our results in a series of short reports Zhang:2013kja; Zhang:2014zsa; Zhang:2015ajn; Zhang:2015vew (see the discussion at the end of this section). This paper serves to present all the technical details that have not been shown in those short papers and to discuss the relation to other models. Note that the same reaction has been studied in EFT in Ref. Ryberg:2014exa when our earlier reports were finished. Some content of the present paper is parallel to that in Ref. Ryberg:2014exa, as will be mentioned in the main text. In Section II, a simple model is used to illustrate the EFT Lagrangian and power counting in a Feynman diagrammatic approach. Similar theories have been developed before in systems without Coulomb effects Bedaque:2003wa; Hammer:2011ye; Rupak:2011nk. A system with -wave nuclear scattering in the presence of strong Coulomb interaction was also studied in EFT Kong:1999sf; Higa:2008dn, and the related capture has been studied in Ryberg:2013iga; Ryberg:2015lea. Here our power counting is based on so-called velocity scaling Luke:1996hj, in order to handle the effective mass scale in the nonrelativistic dynamics. The power counting for the EM interaction is also made transparent. Then we identify the relevant leading order (LO) and next-to-leading order (NLO) diagrams for - and -wave scattering, as well as particular diagrams that contribute to the capture reaction. Our - and -wave scattering calculations are based on a series of time-ordered-perturbation-theory diagrams, which yields the Lippmann-Schwinger expansion (LSE). The LSE is briefly developed in Appendix A, where we demonstrate that the Feynman diagrams and the corresponding diagrams in the LSE are the same. However, the LSE calculation is more suitable for the non-relativistic case at hand and also exposes the connection between EFT and conventional quantum-mechanical calculations, especially for the capture reaction. In addition, Coulomb effects and Coulomb wave functions (discussed in Appendix B) are well developed in the space coordinate, and the LSE enables straightforward transformation into coordinate space. The developed Lagrangian, power counting, and calculational techniques are applied directly in Section III to study the system. The major challenge there is to handle the spin degrees of freedom and the low-energy excitation of the core, but the overall structure of the Lagrangian and the power counting is the same as in the previous section.
Section IV is devoted to the capture reaction. The relevant LO and NLO diagrams are identified and calculated one-by-one using time-ordered perturbation theory. Section V begins with a discussion of the EFT in the language of existing capture models and vice versa. It concludes with the results of fitting our EFT to a selection of models Davids:2003aw; esbensen96; navratil06; Descouvemont:2004hh from the literature. This fitting lacks the ambiguity of fitting experimental data, because a model can be computed exactly and provides more information than just the total -factor. The fitted parameters locate published models unambiguously in the space of EFT parameter values, and show that our power counting works for those models.
In our previous LO calculation Zhang:2014zsa, we used the measured binding energy and scattering lengths along with ab initio asymptotic normalization coefficients (ANCs) of the bound state to fix couplings and find a curve in good agreement with available data (within the uncertainty of the EFT). The results showed significant dependence of on the -wave scattering lengths when all other parameters were kept fixed. A mistake there is corrected in the current paper, but the conclusion remains intact. The same LO calculation was applied successfully to the isospin mirror of the present reaction, i.e. in Ref. Zhang:2013kja. By comparing these two calculations Zhang:2013kja; Zhang:2014zsa, we found that isospin breaking occurs at a momentum scale at or above the breakdown scale , so that the EFT parameters in the two systems are not the same when the EM interaction is switched on and off. For the NLO amplitude developed in this paper, we took a different strategy to fix parameters. We applied Bayesian methods to analyze the modern direct capture data, constrain EFT parameters, and obtain stringent constraints on the low energy even without tight constraints on individual parameters. The major results were reported in Ref. Zhang:2015ajn with some details of the Bayesian analysis in Ref. Zhang:2015vew; we summarize these in Sec. VI. We conclude with a short summary of major results.
Ii A simple model
In this section we use a simplified model to explain the power-counting rules for the EFT Lagrangian and Feynman diagrams. We then identify the LO and NLO diagrams for proton-core scattering in the - and -waves, and show that this reproduces the Coulomb-modified effective range expansions (ERE) for the two scattering phase shifts. The energy variable in the resulting -matrix operator is then continued from the positive- (scattering) to the negative-(bound state) region in order to locate and study the shallow bound-state pole. This pole is the analog, in this simple model, of the pole in the -proton scattering amplitude. This section adds details to our previous brief reports Zhang:2013kja; Zhang:2014zsa and also lays the ground work for the realistic study of the -proton system in the next section. Although the power-counting discussion relies heavily on Ref. Luke:1996hj, we reproduce it in full here, in order to make this paper self-contained. In the process we tailor the arguments of Ref. Luke:1996hj to the context of light-nuclear reactions.
The EFT Lagrangian is:
Here , , , and are the core, proton (“nucleon”), -wave dimer, and -wave dimer fields and stands for complex conjugation. Repeated indices are implicitly summed, as they are throughout the paper. In the simple model of this section the and are spin-zero particles. Their masses are and and their charges are and . Meanwhile the and spins correspond to the -wave scattering state (zero) and the -wave bound state (one). These dimer fields are introduced to simplify the EFT calculation Kaplan:1996nv; Hammer:2011ye; Higa:2008dn; Ryberg:2014exa; Ryberg:2015lea. Both have mass , and charge . and denote the dimers’ unrenormalized binding energies. Note that the extra “” for the free--field piece of the Lagrangian is introduced to reproduce a positive -wave effective range, as will become clear in later discussion—see Eq. (17b). The interaction associated with is a contact coupling which leads to -wave scattering. A similar term generates the -wave interaction, but it is proportional to the relative velocity
where operator () picks up the velocity of the () particle, and , with the reduced mass for the system. In this coupling, the relative velocity dependence is required by Galilean invariance; the photon coupling results from minimal substitution on particle momenta.
It should be pointed out that if we take the usual convention that, under time reversal, a field with spin and spin projection transforms as , then both the - and -wave interactions in Eq. (1) are even under time reversal. Thus, under this standard convention for a spin- field, the factor of in the -- coupling that was present in our previous publications Zhang:2013kja; Zhang:2014zsa should, in fact, be absent. However, this change makes no difference to any physical amplitude that was calculated in those papers.
Adding the free Lagrangian for the photon, which is just the canonical one, with , to the matter Lagrangian of Eq. (1), then specifies the dynamics—apart from some higher-order contact interactions which will be discussed below.
ii.2 Velocity scaling
Even without considering Coulomb interactions, the Lagrangian (1) naively exhibits three distinct energy/momentum scales: the high () and low () momentum scales associated with the short- and long-distance dynamics, and the reduced mass . The appearance of particle masses obscures the power-counting discussion Hammer:2011ye. The so-called velocity scaling proposed in Ref. Luke:1996hj solves this problem by guaranteeing the correct scaling of momenta and energies for a non-relativistic theory. The low-momentum and low-energy scales are rewritten as, respectively, (relative momentum) and (relative energy).
Velocity scaling proceeds by defining the scaling factors for space and time as , and , since these are the typical space and time scales of interest in our EFT. We then scale space and time with these factors, defining new, dimensionless, co-ordinates, via and . The corresponding momentum and energy variables are and , defined by and ; again and are then of order 1 in the EFT power counting. Meanwhile, matter fields are scaled by while is scaled by , so that the normalization of the free Lagrangian is the same after rescaling. Defining a rescaled Lagrange density, , from the original action , via we find that the matter and minimal-substitution part of this rescaled Lagrange density is:
Here , and for . For simplicity, the symbols for the scaled fields, space-time derivatives and , and are kept the same as before, but now the natural expectation for all free-particle terms is that they are of order , since, e.g., , is the relative velocity operator in units of the low velocity scale . For a matter field with four-momentum the propagator is now .
Essentially by construction then, the only dependence on mass in the Lagrangian (3) comes through the fraction . The velocity scaling proposed in Ref. Luke:1996hj indeed makes it explicit that the velocity is what determines the suppression or enhancement of different terms in the EFT. As with the more standard Lagrangian written in terms of momenta, explicit factors of in strong-interaction vertices will be compensated by factors of buried in couplings, e.g., the appearance of a in the denominator (numerator) of the -wave (-wave) interaction term means that, once the natural scaling of () is taken into account, that term will be enhanced by a factor of (suppressed by a factor of ).
Photon-matter interactions reveal the full benefit of velocity scaling, and in Eq. (3) we have also included the interactions with photon fields that minimal substitution produces. The factors of in the minimal couplings of and photons are simply a consequence of the (different) roles of time and space derivatives in non-relativistic dynamics: the photon coupling is proportional to , while the transverse () one is , so transverse photons are suppressed by a factor of relative to photons. Here, in contrast to the strong interactions, the suppression is by , i.e. the velocity is to be measured in units of the speed of light—since these minimal-substitution vertices are not sensitive to the breakdown velocity it is not that ratio that controls the suppression, but the (significantly smaller) . Meanwhile, free-photon propagation is also clearer in terms of velocity scaling: for an on-shell transverse photon (e.g., the photon radiated in the reaction of interest) its momentum and energy both scale as , but for an off-shell photon (e.g., a photon exchanged between the charged core and the charged proton) the two scale as and . This is made explicit by defining a rescaled free-photon Lagrangian density , through , which is:
For the piece of the photon field that generates the Coulomb potential the rescaled propagator is then , while transverse () photons have a propagator . In the last propagator, the factor in principle should be expanded in geometric series. However, if the transverse photon goes on shell (becomes a “radiation photon”) that series needs to be resummed. A detailed discussion of this distinction can be found in Ref. Luke:1996hj. Since we will, for the most part, consider only internal photon lines that obey the kinematics and we do not reproduce that discussion here.
ii.3 Power counting for strong-interaction parameters
The power-counting for the Lagrangian will then be complete if we can determine how many powers of the high scale (now ) the strong-interaction parameters , , , and carry. Naive dimensional analysis (NDA) applied to the rescaled Lagrangian yields the scalings shown in the first row of Table 1.
We now consider the Dyson series for the propagator, , shown on the first line of Fig. 1. Based on the rescaled Lagrangian in expression (3), we can estimate the size of these diagrams by counting factors of in vertices and propagators. If we adopt NDA scaling for then it is order one, and, since the rescaled particle momentum are also , the free propagator of the -wave dimer field, is . The diagrams that constitute the leading part of the self energy are defined on the second line of Fig. 1. In this paragraph we consider only the self-energy bubble without any Coulomb interaction: the first diagram on the right-hand side (RHS) of the lower line of Fig. 1. In contrast to standard EFT power counting there is no need to keep track of factors from loops, since the scaled-momentum integration in the loop calculation is always of order . However, the one-loop self energy is enhanced, due to the presence of a factor . Then, for a natural , each term in the Dyson series for the propagator is larger than the last, thus vitiating a diagrammatic expansion for the dressed propagator . Following Refs. Kaplan:1998tg; Kaplan:1998we we ensure that each term in the Dyson series is of the same EFT order by enhancing the unrenormalized mass, to as shown in the second row of table 1. With this counting each term in the first line of Fig. 1 is of the same size (). Since we have kept the scaling unchanged this constitutes a fine tuning between the NDA estimate of the self-energy bubble and the size of the dimer’s bare mass. The fact that the rescaled while the kinetic, , piece of the inverse propagator is still then also justifies dropping the kinetic piece of the propagator at leading-order in the EFT expansion 111The factor in -p system, which may suppress the kinetic term further in this context. In general, .. In other words, under the scaling in the second line of Table 1, the scaled -wave dimer propagator can be taken to be static at LO: .
For the -wave dynamics, we follow the NDA assignments. The -wave bubble is then suppressed by a factor of —in contradistinction to the -wave bubble. We will see that this is indeed the correct power-counting conclusion, except in certain special kinematic regions. Such a power counting, in which the self-energy of the -wave dimer is suppressed relative to its kinetic part, has been used in earlier EFT studies Bedaque:2003wa; Hammer:2011ye of systems sharing the same feature of a low-energy -wave resonance. Those studies, were, however, for neutron-core scattering, and so did not consider the role of the Coulomb interaction.
ii.4 Power counting with Coulomb
Turning our attention, then, to the power counting of such Coulomb interactions, we first point out that -wave scattering with strong Coulomb effects was previously studied in this EFT in Refs. Kong:1999sf; Higa:2008dn; Ryberg:2015lea. We will now reiterate these arguments, albeit in the context of velocity-scaling, for the propagator, and discuss their extension to the propagator. As already discussed, the self-energy bubble without Coulomb-photon exchange, i.e., the first diagram on the RHS of the lower line of Fig. 1, . As we move from left-to-right in that figure each diagram has an extra exchange of a Coulomb photon. That results in the diagram acquiring an additional factor associated with the product of the two -photon vertices . (Recall that the -photon propagator is in terms of rescaled momenta.) This factor, , is known as the Sommerfeld parameter. In the energy region of interest here it is . The loop integrations also generate factors , and so, as long as , resummation of the ladder of Coulomb photon exchange diagrams is mandatory. This then defines the LO -wave dimer self-energy: . Such a self-energy, which includes the sum of the exchange of zero, one, two, …Coulomb-photon exchanges will henceforth be denoted by a shaded bubble. It follows that, in the kinematic regime , the addition of Coulomb photons does not change the order of the self energy from the order computed with the free Green’s function. It is just that now the self energy must be computed using a core-proton Green’s function that includes one-Coulomb-photon exchange to all orders in . That self energy is still—as in the case—resummed in a geometric series, as per the upper line of Fig. 1, and this procedure generates the LO propagator, , in the system.
To compute the dressed propagator at NLO, , the second diagram shown in Fig. 2 should be included. The vertex depicted there as a small filled box is the -field kinetic term that got demoted to NLO when we chose to enhance over its NDA estimate. We see that the diagram with the single insertion of this vertex , which makes it NLO compared to the LO propagator, (we established that is ). Note that if we wish to recover the effective-range expansion exactly, not just order by order in the effective range, then we must resum a geometric series involving this kinetic-energy operator. This is the content of the terms in square brackets in Fig. 2. However, strictly speaking, only the second diagram on the RHS of Fig. 2 is NLO.
The situation for the -wave dimer propagator, , is different. In Fig. 3, we show the Dyson series for this case. Since , the rescaled free propagator is also of order . Meanwhile, the -wave self-energy , since the loops generate factors of order one, for an arbitrary number of Coulomb-photon exchanges, as long as , and the coupling gives suppression by a factor of . Therefore the second diagram on Fig. 3’s RHS is and hence is NLO. However, as argued in Refs. Pascalutsa:2002pi; Bedaque:2003wa; Hammer:2011ye, when the center-of-mass energy (i.e. in terms of the unscaled momentum) is close to , the leading-order propagator becomes larger than the NDA expectation. In this regime the entire series shown on the RHS of Fig. 3 must be resummed. This is the regime that is pertinent to -wave bound states in the proton-core system, and since we are interested in as a -wave proton- bound state we must use that resummation here.
ii.5 Power counting for radiative capture
Turning our attention now to electromagnetic reactions: for a general diagram and interaction vertex the power counting is similar to that we have done so far for scattering. The size of a diagram can be established by the number of factors of that it carries. Consider the diagrams and that both describe the same capture reaction. If diagram carries a factor and diagram carries a factor of then NDA states that their ratio is:
since the explicit factors of from the Lagrangian must be compensated by factors of in couplings. (Note that this assumes that the same types of photon, longitudinal or transverse, appear in both diagrams and . Otherwise factors of , not , will appear.)
First we compute the order of a graph that results from the gauged coupling in the Lagrangian, which is just one of a set of LO diagrams for the capture reaction. (The full set can be found in Fig. 7.) This first diagram in Fig. 4 is order , since it includes a factor of from the coupling, and an additional factor of from the coupling to the radiated () photon.
Since EFT includes all interactions consistent with underlying symmetries the Lagrangian (3) must be supplemented by terms that are gauge-invariant by themsleves. The leading such term that describes the E1 transition between the -wave and the -wave dimer,
contributes to the capture reaction via the second diagram in Fig. 4, where the filled box means the coupling. This is a contact term and it renormalizes loop graphs that appear at the same order in the EFT exapnsion. The factor of ensures that this coupling is invariant under time reversal. The factor was omitted in the Lagrangians for short-distance electromagnetic operators that were given in Ref. Hammer:2011ye, but this does not affect any of the results presented there.
After the velocity scaling discussed at the beginning of this section is applied, the contact term changes to
Here the overall factor of the reduced mass has been eliminated by defining . After this scaling the second diagram in Fig. 4 is . Assuming NDA for and , and inserting the power counting for the -wave propagator identified above, , produces an overall scaling of for the graph involving the E1 contact operator that is proportional to the LEC . Thus, according to NDA, the ratio of the second and first diagram should be of order , which shows that the E1 contact term contributes to the capture into a -wave bound state at NLO. (The counting is different if the reaction proceeds from a -wave scattering state, into an -wave bound state Ryberg:2015lea.) This also means that and hence .
ii.6 Toy amplitude for -wave scattering up to NLO
Having established the power counting through the use of velocity scaling we now return to expressions in terms of momenta. The discussion can be continued in terms of velocities, and this has the benefit of yielding dimensionless integrals. But the connection with previous work in halo EFT is more straightforward if amplitudes are written in terms of momenta.
The power-counting discussion of the previous subsection is based on an expansion in Feynman diagrams. However, in practice, time-ordered perturbation theory is more suited for our calculations. In particular, the use of time-independent quantum-mechanical perturbation theory allows us to employ the Lippmann-Schwinger equation (LSE) for resummations, such as the one that takes place for Coulomb interactions between the proton and the core. This, in turn, allows us to identify Coulomb wave functions—with all their well-known properties—in our calculation.
In fact, since particle-antiparticle pair production does not exist in this EFT, the intermediate states that occur in a given Feynman diagram all have fixed particle content. For proton-core scattering this diagram is the same as a particular contribution to the LSE time-ordered perturbation theory series. The only exception is transverse photon exchange between charged particles, for which an example is shown in Fig. 5. Its LHS is the one-transverse-photon exchange Feynman diagram, which in fact equals the sum of two time-ordered perturbation theory graphs on the RHS. However in our problem, radiative corrections turn out not to affect the result at the accuracy we seek. Therefore, in the following calculation, we generate Feynman diagrams but then use the corresponding time-ordered perturbation theory expression to do the matrix element computation. This in no way affects the power counting, since the time-ordered and Feynman graphs are equivalent. The LSE in the context of our EFT is developed in Appendix A, which includes a brief discussion of quantization, Fock-state definition, and calculations of various matrix elements corresponding to vertices and propagators in a Feynman diagram. Our notation is also defined there. (While Ref. Kong:1999sf used the LSE in their EFT calculation, the connection to the original field theory is not fully explained there.).
First we compute the propagator, defined in Eq. (102) as a matrix element between plane wave states. The discussion of power counting above implies the LO free propagator is static: should be defined as ; the “” is due to the negative norm of . Notice that a general matrix element always has the factor due to total three-momentum conservation in time-ordered perturbation theory.
The self energy, , is given by the sum of the series of zero to infinitely many Coulomb-photon exchanges depicted in the second line of Fig. 1. The matrix element is written as:
where we have summed the series algebraically in the third line by defining the Coulomb Hamiltonian, . Meanwhile, as discussed in Appendix A, is the strong potential in this system, and it produces an dimer transition. The first (reading from left to right) annihilates and particles from the intermediate state and creates a in the final state, while the second annihilates the in the initial state and creates a pair in the intermediate state.
To compute , we make use of the Coulomb-distorted two-body states that are the eigenstates of (see Eq. (96)). Here we omit the subscripts on momenta in the Fock-space state; dimer momenta are still indicated as such. We also introduce the intrinsic states, defined in the center-of-mass frame, for the Coulomb Hamiltonian. With the intrinsic Hamiltonian, defined as
the intrinsic states satisfy with the relative energy equal to the energy of the pair in its center-of-mass frame. The matrix element of can then be re-expressed as:
Note that since the Lagrangian is a point coupling the probability amplitude for the conversion of an pair into a dimer is the product of the coupling and the size of the wave function at .
By introducing these intrinsic wave functions, which are the solutions of quantum-mechanical one-body problems, we can use standard quantum-mechanical results to do our calculation. Inserting a complete set of eigenstates of , and using these definitions, the matrix element becomes
It follows that the self energy is only a function of the Gallilean invariant combination , i.e.
with . Similar results have been derived in, e.g., Refs. Higa:2008dn; Ryberg:2013iga. Details about and the definition of ( is the Sommerfeld parameter) can be found in Appendix B. The integration diverges, however the integrand has been split into a finite and a divergent piece, as in the second step of Eq. (II.6). The first part yields with
, and the digamma function MathHandBook1. The divergent term can be analytically continued in terms of the space-dimension variable . The following integration is involved Kong:1999sf,
with the Riemann zeta function. When , , , and with the Euler constant; at other integers , is finite. We use the power-divergence subtraction (PDS) scheme Kaplan:1998we; Higa:2008dn and subtract the pole at . We then use the MS scheme to remove the pole in that is associated with the Coulomb interaction and obtain
where is the dimensionful scale introduced to ensure the correct overall dimensions of .
As mentioned in the power-counting discussion of the previous section, summing up all the bubble insertions shown in Fig. 1 gives . The NLO contribution to , as shown in Fig. 2, comes from the insertion of the operator , i.e. the second diagram on Fig. 2’s RHS. However in order to match the conventional ERE, we can sum all of the bracketed diagrams in Fig. 2. This leads to . If we then impose the renormalization conditions:
Even though here we work only at the level of the dimer propagator we have chosen to already write things in terms of and , which are the scattering length and effective range in the -wave ERE for scattering. We make three points before moving on to discuss the scattering -matrix in the next paragraph. First, setting , Eqs. (17a) and (17b) recovers the corresponding relationships for a system without Coulomb effects Hammer:2011ye; Zhang:2013kja. Second, the overall “” sign in the ’s free lagrangian in expression (1) is responsible for generating a positive in Eq. (17b). Third, when PDS and MS are employed, is renormalized by the self-energy loop diagram, but is not.
The scattering -matrix diagrams are shown in Fig. 6. The bracketed diagrams are due to pure Coulomb scattering. They can be analytically computed (see, e.g., Goldberger1964), and won’t be dealt with here. The so-called strong-interaction -matrix, i.e., the total -matrix with pure Coulomb scattering subtracted is
Up to NLO it is due to the last two diagrams in Fig. 6. In terms of matrix elements, it is
If the Green’s function were sandwiched between states it would be . Thus, in the following we define
with the first (second) —reading from left to right—annihilates (creates) the particle in the intermediate state and creates (annihilates) an pair in the final (initial) state. We then use the dimer completeness relation:
This matrix element can again be simplified to a delta function times the matrix element between the intrinsic states:
Since the states are already defined, the intrinsic operator is then defined by its matrix elements on this basis. On shell the strong interaction operator, evaluated on this, the “intrinsic Coulomb basis”, can be expressed in terms of a phase shift Higa:2008dn; Ryberg:2015lea
This derivation clarifies how the Coulomb modified wave function appears in halo EFT: it is the co-ordinate-space representation of the intrinsic Coulomb basis . The separation of transverse photons from Coulomb photons through velocity scaling also delineates the order at which corrections due to those photons must be considered. (See Sec. VII.) As far as strong interactions are concerned, the next correction to the ERE, which must be , occurs in the EFT via an operator that appears in the Lagrangian only at LO in the EFT expansion Mehen:1998tp.
Eq. (26) justifies the use of the notation and in the renormalization conditions in Eqs. (17a) and (17b). Since and are observables they are -independent and this, in turn, determines the -dependence of . must absorb both the divergence that gives the in Eq. (17a) and the Coulomb () divergence proportional to . Thus the short-distance physics is affected by Coulomb photons, and so the separation of physics between dimer and parts of the Fock space becomes dependent on the treatment of that short-distance physics, i.e. only model-dependent statements can be made about it. This, in turn, means that one cannot define a scheme- and scale-independent strong proton-core scattering length Kong:1999sf; Gegelia:2003ta.
ii.7 Toy amplitude for -wave scattering up to NLO, and computation of shallow bound-state properties
Fig. 3 shows the diagrams for calculating the propagator . As discussed in Sec. II.3, if we are in the kinematic regime where the free dimer propagator has a singularity then we must resum the LO self energy to all orders, i.e. compute the entire series shown in Fig. 3 Pascalutsa:2002pi; Bedaque:2003wa. As will become clear by the end of this section, this is equivalent to requiring resummation in the vicinity of a value of momentum that is both within the domain of the EFT (i.e. ) and satisfies , where is the -wave effective range and the -wave scattering volume. In our case this condition is satisfied at the bound-state pole. It is not satisfied in -proton -wave scattering, and so if our calculation were concerned with that process we could terminate the series at NLO, i.e. include the self-energy only perturbatively (cf. Ref. Hammer:2011ye) and still have NLO accuracy. However, here the properties of the bound state are crucial to the calculation of