DOE/ER/40561-57-INT99

TRI-PP-99-24

KRL MAP-248

Effective Theory of the Triton

P.F. Bedaque^{1}^{1}1,
H.-W. Hammer^{2}^{2}2^{3}^{3}3
Present address: Department of Physics, The Ohio State University,
Columbus, OH 43210, USA,
and U. van Kolck^{4}^{4}4

Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA

TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3

Kellogg Radiation Laboratory, 106-38, California Institute of Technology,

Pasadena, CA 91125, USA

Department of Physics, University of Washington, Seattle, WA 98195, USA

We apply the effective field theory approach to the three-nucleon system. In particular, we consider neutron-deuteron scattering and the triton. We show that in this channel a unique nonperturbative renormalization takes place which requires the introduction of a single three-body force at leading order. With one fitted parameter we find a good description of low-energy data. Invariance under the renormalization group explains some universal features of the three-nucleon system —such as the Thomas and Efimov effects and the Phillips line— and the origin of symmetry in nuclei.

## 1 Introduction

Effective field theories (EFT’s) are a powerful concept designed to explore a separation of scales in physical systems [1]. For example if the momenta of two particles are much smaller than the inverse range of their interaction , observables can be expanded in powers of . Following the early work of Weinberg and others [2], EFT’s have become quite popular in nuclear physics [3, 4]. Their application, however, is complicated by the presence of shallow (quasi) bound states, which create a large scattering length . In this fine-tuned case, a simple perturbative expansion in already breaks down at rather small momenta . In order to describe bound states with typical momenta , the range of the EFT has to be extended. A certain class of diagrams has to be resummed which generates a new expansion in where powers of are kept to all orders. For the two-nucleon system a power counting that incorporates this resummation has been found recently [5, 6, 7]. A number of two-nucleon observables have been studied in the low-energy theory where pions have been integrated out [8]. Furthermore, pions can be included explicitly and treated perturbatively in this scheme [6]. This approach has successfully been applied to scattering and deuteron physics [9].

The three-nucleon system is a natural test-ground for the understanding of nuclear forces that has been reached in the two-nucleon system. Furthermore, it shows some remarkable universal features. In the context of potential models, it has been found that different models of the two-nucleon interaction that are fitted to the same low-energy two-nucleon () data predict different but correlated values of the triton binding energy and the -wave nucleon-deuteron () scattering length in the spin channel; all models fall on a line in the plane, the Phillips line [10]. Other universal features of three-body systems are the existence of a logarithmic spectrum of bound states that accumulates at zero energy as the two-particle scattering length increases (the Efimov effect [11]), and the collapse of the deepest bound state when the range of the two-body interaction goes to zero (the Thomas effect [12]). These universal features have no obvious explanation in the framework of conventional potential models. (Alternative explanations of these features that are likely equivalent to ours can be found in Ref. [13].)

The critical issue in the extension of any theory from the two- to the three-body sector is the relative size of three-body forces. A naive dimensional analysis argument that does not incorporate the fine tuning in the two-body sector suggests that three-body forces appear only in higher orders in the expansion. We have shown that this is supported by data in the channel [14]. However, for a system of three bosons a non-trivial renormalization enhances the size of three-body forces [15]. (For a different approach, see Ref. [16].) Here we will show that the renormalization of the three-nucleon system in the channel is nonperturbative and requires a single three-body force at leading order. With such a force, the EFT allows to understand the above mentioned universal features in a unified way. Being symmetric, this force also supports the view of spin-flavor as an approximate symmetry of nuclei [17]. Moreover, if its single parameter is fitted to , we find a good description of the energy dependence of scattering and of triton binding.

## 2 Preliminaries

In order to avoid the difficulties due to the long-range Coulomb force, we concentrate here on the neutron-deuteron () system. For simplicity, we also restrict ourselves to scattering below the deuteron breakup threshold where -waves are dominant. There is only one low-energy scale, MeV, where is the binding energy of the deuteron and the nucleon mass. Since is small compared to the pion mass, the pions can be integrated out, only nucleons remaining in the EFT as explicit degrees of freedom. The leading order of this pionless theory [5, 6, 7] is equivalent to the leading order in the KSW counting scheme [6] where pions are included perturbatively. As a consequence, the extension of our results to higher energies is well defined.

There are two -wave channels for neutron-deuteron scattering, corresponding to total spin and . For scattering in the channel all spins are aligned and the two-nucleon interactions are only in the partial wave. The two-body interaction is attractive but the Pauli principle forbids the three nucleons to be at the same point in space. As a consequence, this channel is insensitive to short-distance physics and precise predictions are obtained in a straightforward way [14]. There is also no three-body bound state in this channel. The channel is more complicated. The two-nucleon interaction can take place either in the or in the partial waves. This leads to an attractive interaction which sustains a three-body bound state, the triton. The channel also shows a strong sensitivity to short-distance physics as the Pauli principle does not apply. We will see that the generic features of this channel are thus very similar to the system of three spinless bosons. There is in the latter a strong cutoff dependence even though all Feynman diagrams are finite, and renormalization requires a leading-order three-body force counterterm [15].

Let us start from the assumption that the three-body force is of natural size. The lowest-order effective Lagrangian is then given by

where the dots represent higher-order terms suppressed by derivatives and more nucleon fields. are Pauli matrices operating in spin (isospin) space, respectively. The contact terms proportional to () correspond to two-nucleon interactions in the () channels. Their renormalized values are related to the corresponding two-body scattering lengths and by . Since no derivative interactions appear at this order, this Lagrangian generates only two-nucleon interactions of zero range. For practical purposes, it is convenient to rewrite this theory by introducing “dibaryon” fields with the quantum numbers of two nucleons [18]. In our case, we need two dibaryon fields: (i) a field with spin (isospin) 1 (0) representing two nucleons interacting in the channel (the deuteron) and (ii) a field with spin (isospin) 0 (1) representing two nucleons interacting in the channel. Using a Gaussian path integration, it is straightforward to show that the Lagrangian (2) is equivalent to

At first it may look like the Lagrangian (2) contains more parameters than the original one, Eq. (2). However, the scales and are arbitrary and included in Eq. (2) only to give the dibaryon fields the usual mass dimension of a heavy field. They can easily be removed by rescaling the dibaryon fields. All observables depend only on the ratios . The introduction of these dibaryon fields makes explicit the formal similarity to the Amado model [19]. We stress, however, that the introduction of “quasi-particle” fields and carries here no dynamical assumptions [5].

Since the theory is nonrelativistic, all particles propagate forward in time, the nucleon tadpoles vanish, and the propagator for the nucleon fields is

(3) |

The dibaryon propagators are more complicated because of the coupling to two-nucleon states. The bare dibaryon propagator is simply a constant, , but the full propagator gets dressed by nucleon loops to all orders as illustrated in Fig. 1.

The nucleon-loop integral has a linear ultraviolet (UV) divergence which can be absorbed in , a finite piece determined by the unitarity cut, and subleading terms that have already been omitted in Eq. (2). Summing the resulting geometric series leads to

(4) |

where and now denote the renormalized parameters. The scattering amplitude in the respective channel is obtained by attaching external nucleon lines to the dressed propagator. In the center-of-mass frame, the -wave amplitude in the , channels for energy is

(5) |

and the renormalized parameters and can be determined from

(6) |

Note that in this zero-range approximation,

(7) |

which holds only approximately, the discrepancy coming from range corrections.

## 3 Scattering and the Triton

We now use the Lagrangian, Eq. (2), to describe -scattering in the channel and the triton. The two-nucleon interactions can take place both in the and partial waves. There are two coupled amplitudes, and , as the triton can be built by adding a neutron to a proton and a neutron which are in either of the two partial waves. The amplitude which has both an incoming and outgoing dibaryon field gives the phase shifts for scattering; is coupled to the amplitude which has an incoming dibaryon and an outgoing dibaryon . Although only corresponds to elastic scattering, both amplitudes have the quantum numbers of the triton. The diagrams for the two amplitudes are obtained from the Lagrangian (2). However, the leading piece of all diagrams for is of order , where stands for either the external momentum or . Therefore the diagrams have to be summed to all orders, which is conveniently done by solving the coupled integral equations shown in Fig. 2.

After the integration over the time component of the loop-momentum and the projection onto the -waves has been carried out, we have

(8) | |||

(9) | |||

Here () denote the incoming (outgoing) momenta in the center-of-mass frame, is the total energy, and the kernel is given by

(10) |

The amplitude is normalized such that with the elastic scattering phase shift. Furthermore, we have introduced a momentum cutoff in the integral equations. Eqs. (8, 9) have previously been derived using a different method [20]. In the limit these equations do not have a unique solution because the phase of the asymptotic solution is undetermined [21]. This is the same feature exhibited by the three-boson systems [15]. For a finite this phase is fixed and the solution is unique. However, the equations with a cutoff display a strong cutoff dependence that does not appear in any order in perturbation theory. The amplitude shows a strongly oscillating behavior. Varying the cutoff slightly changes the asymptotic phase by a number of and results in large changes of the amplitude at the on-shell point . This cutoff dependence is not created by divergent Feynman diagrams. It is a nonperturbative effect and appears although all individual diagrams are superficially UV finite.

In order to understand this strong dependence, it is useful to note that the integral equations (8, 9) are symmetric in the UV. Therefore it is sufficient to consider the limit (), since the cutoff dependence is a problem rooted in the UV behavior of the amplitudes. Furthermore, we note that the equations for and decouple in the limit. The equations for and are

(11) | |||

(12) | |||

The equation for is exactly the same equation as in the case of spinless bosons while the equation for is the same equation as in the channel. The equation is well behaved and its solution is very insensitive to the cutoff [14]. Consequently, the cutoff dependence in Eqs. (8, 9) stems solely from the equation for . As an example, the cutoff dependence of is shown by the solid, dashed, and dash-dotted curves in Fig. 3 for three different cutoffs, .

Dependence on the cutoff reflects incorrect renormalization, and thus intolerable dependence on higher-energy modes. We need to ensure that only low-energy modes contribute explicitly, otherwise the power counting used to order two-body interactions —and in particular to produce Eq. (5) in lowest order regardless of the regularization procedure— would not hold. Now, since the equation for is the same as in the boson case, we already know the solution to this problem: a one-parameter three-body force counterterm that runs with the cutoff [15]. We simply replace

(13) |

in Eq. (11). The dotted, short-dash-dotted, and short-dashed curves in Fig. 3 show the effect of the three-body force on for and . It is clearly seen that the variation of for a constant has the same effect on the amplitude as varying the cutoff. Consequently, we can compensate the changes in the asymptotic phase when is varied by adjusting the three-body force term appropriately. (A more rigorous discussion of the renormalization procedure can be found in Ref. [15].)

We can obtain an approximate expression for the running of from invariance under the renormalization group. Requiring that the equation for does not change its form when the high momentum modes are integrated out, we find

(14) |

where and is a dimensionful parameter that determines the asymptotic phase of the off-shell amplitude [15]. The running of the three-body force according to Eq. (14) is shown by the solid line in Fig. 4.

The dots are obtained by adjusting such that the low-energy solution of Eq. (11) remains unchanged when is varied. The observed agreement provides a numerical justification for our procedure. The three-body force is periodic with for . In particular, the bare three-body force vanishes for a discrete set of cutoffs. Note, however, that invariance under continuous changes in the cutoff does require a non-vanishing bare three-body force.

An important point should be stressed here. We have first renormalized the two-body subamplitude (the dibaryon propagator) and then inserted the result in the three-body equation. The loop appearing in this equation was then regulated by a cutoff and renormalized by the introduction of a three-body force. This is equivalent to using separate cutoffs (for the three-body equation) and (for the dibaryon propagator) and taking first (changing the two-body parameters as a function of to keep the two-body observables fixed) and then performing the renormalization of the three-body equation discussed above. A more standard approach would be to use the same cutoff for the loops appearing in both the dibaryon propagator and the three-body equation. The fact that the approach used in this paper is legitimate becomes evident by noting that the details of the renormalization are changed from one approach to the other, but not the physical results. To see that, let us consider the changes due to the use of throughout. The two-body subamplitude would then be changed by terms suppressed by powers of , where represents the typical momentum flowing through the two-body subamplitude. When inserted in the three-body equation, these terms are important only for loop momenta ( in Eqs. (8, 9)) of the order of . In this region of integration other arbitrary choices like, e.g., the precise shape of the cutoff used (sharp or smooth) are also important. All these differences occurring at the cutoff scale are absorbed in the three-body force, which would result in a different dependence than given in Eq. (14). The physical results for small momenta (), however, are unchanged up to terms suppressed by powers of . Those terms depend on our particular choice of regulators and also break symmetries like Galilean invariance. There are many other terms of the same order in that were already discarded, since we have performed an expansion in powers of . The error introduced by this omission, however, is of higher order than we are working at.

Formally, the three-body force term is obtained by adding

to the Lagrangian (2). Eq. (3) represents a contact three-body force written in terms of dibaryon and nucleon fields. Via a Gaussian path integration it is equivalent to a true three-nucleon force,

By performing a Fierz rearrangement, it can then be shown that the three terms in Eq. (3) are equivalent. As a consequence, there is only one three-body force which is also symmetric. Therefore the choice of the three-body force, Eq. (3), is by no means arbitrary. In fact, it is the only -wave three-body force with no derivatives that can be written down. This can be seen by remembering that three fermions (antifermions) in a completely symmetric spatial wave need to be in a completely antisymmetric spin-isospin state which is a of . Consequently, the three-body force transforms as , but only the singlet is separately invariant under the spin and isospin subgroups. The two-body parameters and appear in Eq. (3) because the effect of the three-body force cannot be separated from the effect of the two-body interaction. Both are linked by the renormalization procedure. Naive power counting would suggest that the three-nucleon force scales with . The three-body force from Eqs. (3, 3), however, is enhanced by the renormalization group flow by two powers of . Using Eq. (6), it is found to scale as which makes it leading order. contains one new dimensionful parameter, , which must be calculated from QCD or determined from experiment.

Recently Mehen, Stewart, and Wise [22] remarked that the two-nucleon amplitude (5) is approximately symmetric for . They also noticed that the only -wave four-nucleon force that can be written down is symmetric. Furthermore, there are no contact interactions with more than four nucleons without derivatives because of the Pauli principle. It is also reasonable to assume that the low-energy dynamics of other nuclei is dominated by -wave interactions, as is the case for the deuteron and the triton. Therefore our symmetric three-body force together with the findings of Mehen et al. [22] gives an explanation for the approximate symmetry [17] in nuclei.

Now we are in position to solve the full equations for the broken case. Including the three-body force from above into Eqs. (8, 9), we have

(17) | |||

(18) | |||

We need one three-body datum to fix the three-body force parameter . We choose the experimental value for the scattering length, [23], and find . (For the special cutoffs with vanishing , we recover the results of Ref. [24].) Although one three-body datum is needed as input, the EFT has not lost its predictive power. We can still predict (i) the energy dependence of scattering and (ii) the binding energy of the triton.

The resulting energy dependence of scattering for three different cutoffs is shown in Fig. 5.

It is clearly seen that the introduction of the three-body force renders the low-energy amplitude cutoff independent. The scattering length is reproduced exactly because it was used to fix . The agreement for finite momentum is at least encouraging. Our experience from the channel is that the range corrections improve the agreement considerably [14]. The dashed curve in Fig. 5 gives a crude estimate of these corrections. In our calculations we take the deuteron binding energy from experiment and determine from Eq. (7). The dashed curve in Fig. 5 is obtained by taking the experimental value of as input and leaving all other parameters unchanged. From the estimated size of the range corrections, we anticipate an improved agreement once these corrections are included. (One should also keep in mind that the experimental phase-shift analysis [25] does not give any error estimate.)

The triton binding energy is obtained from the solution of the homogeneous equations corresponding to Eqs. (17, 18) for . In Fig. 6 we show the bound state spectrum as a function of the cutoff for a particular .

The shallowest bound state is the triton. Its binding energy is cutoff independent as long as . However, as is increased new deeper bound states appear whenever goes through a pole. These new bound states appear with infinite binding energy directly at the pole. As the cutoff is further increased, their binding energy decreases and becomes cutoff independent as well. The poles of can be parametrized as

(19) |

One counts bound states for . However, only the states between threshold and are within the range of the EFT. By solving Eq. (19) for the number of bound states , we then obtain

(20) |

and recover the well-known Efimov effect [11]. In the limit , an infinite number of shallow three-body bound states accumulates at threshold. That these bound states are shallow follows from the fact that the binding energy is naturally given in units of which vanishes as . Furthermore, we also recover the Thomas effect [12]. In a hypothetical world where , the range of the EFT increases and deeper and deeper physical bound states appear. Consequently, there is an infinitely deep bound state for . However, for as in the real world the deep bound states are outside the range of the EFT and their presence does not influence the physics of the shallow ones.

Moreover, the variation of the parameter gives a natural explanation for the Phillips line [10]. The three-nucleon system has been studied with many phenomenological two-nucleon potentials where particles can have momenta as high as 1 GeV. Different phase-equivalent two-body potential models have different off-shell amplitudes, which are equivalent to different three-body forces. In some cases explicit three-body potentials have also been added. These models can be viewed as particular examples of high-energy dynamics. At low energies, they must all be equivalent to the EFT with particular values for its parameters. What we have shown is that in leading order they can only differ in the value of the single three-body parameter . Varying then generates a line. In Fig. 7 we show the Phillips line obtained in the EFT compared with results from various potential model calculations [26] and the experimental values for and . Our Phillips line is slightly below the one from the potential models. We expect this discrepancy to be reduced once range corrections are taken into account.

The dynamics of QCD chooses a particular value of , which up to higher order corrections is . For this value we find MeV for the triton binding energy, to be compared with the experimental result MeV, which is known to very high precision. The theory without pions seems to work for triton physics. However, to draw definite conclusions one has to include the range corrections for both the energy dependence and the triton binding energy.

The three-body force (3) involves three nucleons at the same point (within the resolution of the EFT) in a relative wave. It cannot contribute to the channel where the three nucleons spin in the same direction. Indeed, we have found that the quartet can be well predicted with two-nucleon input alone [14]. We stress that the three-body force that arises at low energies is not the same that appears at higher energies. As momentum increases, part of the three-body contact operator (3) may be resolved into iterations of the two-body potential. The strong renormalization of this operator at low energies is not necessarily in contradiction with the potential-model-based phenomenology where three-body forces are found to be small.

## 4 Conclusions

We have studied the three-nucleon system using EFT methods. While for scattering in the channel precise predictions are obtained in a straightforward way [14], the channel is more complicated. It displays a strong cutoff dependence even though all individual diagrams are UV finite. In this channel a nonperturbative renormalization takes place similar to the case of spinless bosons. This renormalization group argument requires an symmetric three-body force which is enhanced to leading order. Together with the recent observations of Mehen et al. [22] this gives an explanation for the approximate symmetry in nuclei. Furthermore, we find that the Phillips line is a consequence of variations in the new dimensionful parameter which is introduced by the three-body force. is not determined by low-energy two-nucleon data alone and has to be determined from a three-body datum.

This is to be contrasted to other approaches to the problem that model the two-body interaction at distances of the order of the effective range as, for instance, the Amado model [19]. There a form factor is introduced in the interaction between two nucleons whose parameters are chosen to reproduce not only the two-body scattering length but also the effective range. Since the effective range expansion is not to be trusted at momenta of the order of the inverse effective range, this corresponds, in our language, to taking the three-body force equal to zero and picking a particular form for the regulator (the effective range term in the two-body subamplitude acts as a cutoff).

The leading order of the EFT gives a quantitative description of the triton binding energy and the energy dependence for scattering. In the appropriate limits for the two-body parameters and , we also recover the well known Thomas and Efimov effects. The theory shows the potential for a realistic description of the triton once range corrections are included. Immediate applications of the EFT include polarization observables in scattering and triton properties such as its charge form factor. The incorporation of the long-range Coulomb force would widen the possible applications considerably as scattering and the physics of He becomes accessible. Work on these extensions is in progress.

Finally, since in leading order the pionless theory is equivalent to a theory with explicit (perturbative) pions [6], the success in the three-nucleon system opens the possibility of applying the EFT method to a large class of systems with three or more nucleons.

## Acknowledgements

We would like to thank Jim Friar for the potential model data. This research was supported in part by the U.S. Department of Energy grants DOE-ER-40561 and DE-FG03-97ER41014, the Natural Sciences and Engineering Research Council of Canada, and the U.S. National Science Foundation grant PHY94-20470.

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