Nucleon-deuteron scattering from an effective field theory

4 June 1998

Physics Letters B 428 Ž1998. 221–226

Nucleon-deuteron scattering from an effective field theory P.F. Bedaque a

a,1

, U. van Kolck

b,2,3

Institute for Nuclear Theory, UniÕersity of Washington, Seattle, WA 98195-1560, USA b Department of Physics, UniÕersity of Washington, Seattle, WA 98195-1560, USA Received 3 November 1997; revised 6 March 1998 Editor: J.-P. Blaizot

Abstract We use an effective field theory to compute low-energy nucleon-deuteron scattering. We obtain the quartet scattering length using low energy constants entirely determined from low-energy nucleon-nucleon scattering. We find a th s 6.33 fm, to be compared to a e x p s 6.35 " 0.02 fm. q 1998 Published by Elsevier Science B.V. All rights reserved.

There has been considerable interest lately in a description of nuclear forces from the low-energy effective field theory ŽEFT. of QCD. ŽFor a review, see Ref. w1x.. Following a program suggested by Weinberg w2x, the leading components of the nuclear potential have been derived w3x and a reasonable fit to two-nucleon properties has been achieved w4x. The correct formulation of the nuclear force problem within the EFT method is important because it will allow a systematic calculation of nuclear properties consistently with QCD. One would like, for example, to devise a theory of nuclear matter rooted in a hadronic theory that treats chiral symmetry correctly and yields the well-known few-nucleon phenomenology. One hopes that after a number of parameters of the EFT are either calculated from first principles or fitted to a set of few-nucleon data, the theory can be used to predict other reactions involving light nuclei and features of heavier nuclei. However, some issues concerning renormalization in this non-perturbative context and fine-tuning in the two-nucleon S-waves have been raised in Refs. w5,6x and are still not fully resolved w7,8x. The fine-tuning necessary to bring a Žreal or virtual. bound state very close to threshold generates a scattering length a much larger than other scales in the problem. At momenta of O Ž1ra., mesons can be integrated out and the characteristic mass scale m of the underlying theory controls the size of the other effective range parameters; for example, the effective range r 0 ; 2rm. Once the leading order contributions, which give rise to a, are included to all orders, the EFT at momenta O Ž1ra. becomes an expansion in powers of 1rŽ a m .. Kaplan w6x has noticed that the interactions that generate a non-zero r 0 can also be resummed by the introduction of a baryon number

1

E-mail: [email protected]. E-mail: [email protected] 3 Address after Jan 1 1998: Kellogg Radiation Lab 106-38, California Institute of Technology, Pasadena, CA 91125; E-mail: [email protected]. 2

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 4 3 0 - 4

P.F. Bedaque, U. Õan Kolck r Physics Letters B 428 (1998) 221–226

222

two state of mass D s 2rMa < r 0 <, which in lowest order in a derivative expansion couples to two nucleons with a strength g 2r4p s 1rM 2 < r 0 <. In Ref. w6x it was shown how this works in the two-nucleon 1 S0 channel. Analogous considerations hold for the 3 S1 channel, where they are similar to the old quasi-particle approach of Weinberg w9x. In this paper we consider the application of these ideas to the three-nucleon system. Our goal here is to calculate some of the three-nucleon parameters that are dominated by the leading interactions in the EFT without pions. We show in particular that the quartet scattering length in neutron-deuteron scattering can be predicted once the EFT is constrained by low-energy two-nucleon data. Such an attempt to a model-independent or ‘‘universal’’ approach is not a new idea; it permeates for example the work of Efimov Žsee, e.g., w10x. and Amado Žsee, e.g., w11x.. However, as we will show, the EFT formulation is much easier to implement, from both conceptual and practical standpoints. For momenta of order 1ra Žthe momentum scale relevant for zero-energy Nd scattering., we can integrate out mesons and consider an EFT with only nucleons N. Interactions are then described by a tower of nucleon contact operators with an increasing number of derivatives. Amplitudes in leading order are given by a zero-range four-nucleon interaction iterated to all orders. Corrections come in powers of 1rŽ amp . w8x. The next two orders in this expansion, 1rŽ amp . ; r 0r2 a and 1rŽ amp . 2 ; Ž r 0r2 a. 2 , stem from one and two insertions of a two-derivative four-nucleon operator giving rise to a non-zero r 0 . It is advantageous to sum all the contributions coming from this operator, which can be easily done because they appear in a geometric series. The resulting interaction is equivalent to the s-channel propagation of a dibaryon, and therefore can be obtained more directly by the introduction of a dibaryon field. Since in both I s 0 and I s 1 S-wave two-nucleon channels we observe Žone real, one virtual. bound states near threshold, we consider two dibaryon fields, T Ž D . of spin zero Žone. and isospin one Žzero.. The most general Lagrangian invariant under parity, time-reversal, and small Lorentz boosts is

ž

L s N † iE 0 q y

gT 2

=

2

2M

/

ž

q . . . N q T † P yi E 0 y

Ž T † P Ns 2 tt 2 N q h.c. . y

gD 2

=

2

4M

/

ž

q DT q . . . T q D† P yi E 0 y

Ž D† P Nt 2 ss 2 N q h.c. . q . . .

=

2

4M

/

q DD q . . . D

Ž 1.

Here the DT , D and g T , D are undetermined parameters and ‘‘ . . . ’’ stands for higher order terms. ŽNote that the effects of non-derivative and two-derivative four-nucleon terms can be absorbed into a redefinition of DT , D and g T , D and higher order four-nucleon terms.. In this non-relativistic theory all particles propagate forward in time, nucleon tadpoles vanish and, as a consequence, there is no dressing of the nucleon propagator, which is simply i

SN Ž p . s 0

p y

p2 2M

.

Ž 2.

qie

The propagators for dibaryons are more complicated, because of the coupling to two-nucleon states. The dressed propagators consist of the bubble sum in Fig. 1, which amounts to a self-energy contribution proportional to the

Fig. 1. Dressed dibaryon propagator.

P.F. Bedaque, U. Õan Kolck r Physics Letters B 428 (1998) 221–226

223

Fig. 2. NN amplitude.

bubble integral. This integral is proportional to the Žlarge. mass M, and it is this enhancement that gives rise to non-perturbative phenomena and leads eventually to the existence of bound states. The integral is also ultraviolet divergent and requires regularization. Introducing a cut-off L we find a linear divergence A L, a cut-off independent piece which is non-analytic in the energy, a term that goes as Ly1 and terms that are higher order in Ly1 . The first and third terms can be absorbed in renormalization of the parameters of the Lagrangian Ž1.; in what follows we omit a label R that should be attached to these parameters, i.e., DT , D and g T , D stand for the renormalized parameters. Higher order terms are neglected because they are of the same order as interactions in the ‘‘ . . . ’’ of the Lagrangian Ž1.. A dibaryon propagator has therefore the form 1

iSD Ž p . s p0 y

p

2

4M

y DD q

Mg D2

ž /( 2p

y Mp 0 q

p2 4

.

Ž 3.

yie qie

Note that such a dressed propagator has two poles at p 0 s p 2r4M y B, p 0 s p 2r4M y Bd e e p and a cut along the positive real axis starting at p 0 s p 2r4M. The NN amplitude can now be obtained directly from SD Ž p . as in Fig. 2. In the center-of-mass, the on-shell I s 0, J s 1 S-wave amplitude at an energy E s k 2rM is 3

TN N Ž k . s

4p M

1 y

2pDD

Ž Mg D2 .

2p q

Ž M 2 g D2 .

,

Ž 4.

k 2 y ik

which is exactly equivalent to the effective range expansion. An analogous result holds for the I s 1 S-wave. The four parameters DT , D and g T , D can then be fixed from the experimentally known scattering lengths and effective ranges. The NN amplitude has shallow poles at B ; 1rMa2 which are associated with the deuteron in the 3 S1 channel and with the virtual bound state in the 1S0 channel. The effective theory has also an additional deep bound state in each channel at Bd e e p ; 4rMr 02 , which is outside the range of validity of the EFT. From the triplet parameters 3a s 5.42 fm and 3 r 0 s 1.75 fm w12x we find DD s 8.7 MeV and g D2 s 1.6 P 10y3 MeVy1 . The resulting deuteron binding energy is B s 2.28 MeV. From the singlet parameters 1 a p p s y17.3 fm, 1a n p s y23.75 fm, 1a n n s y18.8 fm, 1 r 0 p p s 2.85 fm, 1 r 0 n p s 2.75 fm, and 1 r 0 n n s 2.75 fm w13x we find the averages DT s y1.5 MeV and g T2 s 1.0 P 10y3 MeVy1 . With the parameters so determined, we turn now to possible predictions in low-energy nucleon-deuteron scattering. For simplicity we restrict ourselves to scattering below the deuteron break-up threshold 4 , where the S-wave is dominant. There are two S-wave channels, corresponding to total spin J s 3r2 and J s 1r2. In the quartet only D contributes while in the doublet T also appears. The Nd scattering amplitude TNd from the same interactions is given by the diagrams in Fig. 3, which can be summed up by solving an integral equation in the quartet and a pair of coupled integral equations in the doublet channel.

4

In contrast to an EFT with an ‘‘elementary’’ deuteron field, we can in principle extend our results above the break-up threshold Žas long as the typical momentum remains much smaller than the pion mass. at the expense only of a greater numerical effort.

P.F. Bedaque, U. Õan Kolck r Physics Letters B 428 (1998) 221–226

224

Fig. 3. Nd amplitude.

The first diagram in the right-hand-side of Fig. 3 gives a contribution of order ; Mg 2rp 2 ; a 2rMr 0 . The remaining graphs mix different orders in r 0ra, since they involve the dibaryon propagator; they contribute at order ; g 4 M 2rp D ; Ž a 2rMr 0 .Ž1 q O Ž r 0ra. q . . . .. Other two-body contributions not included in Fig. 3 are suppressed by at least three powers of r 0r2 a. For instance, P-wave interactions arise from a term in the Lagrangian with two derivatives and a coefficient of O Ž1rMmp3 .; substitution of one of the dibaryon propagators by a P-wave interaction vertex would thus be supressed by ; Ž r 0ra. 3 in comparison to the leading order. Likewise, the effect of the higher derivative term Žnot written explicitly in Ž1.. responsible for the shape parameter Ž; k 4 . in the effective range expansion of the nucleon-nucleon interaction is ; k 4 r 03, and is thus also suppressed by Ž r 0ra. 3 compared to the leading piece ; 1ra. The diagrams in Fig. 3 are power-counting finite, but this does not preclude the existence of relevant contact interactions between the nucleon and the dibaryons. Pion exchange that would generate such interactions can be expected to be larger for the I s 1 dibaryon T, and therefore predominantly affect the J s 1r2 channel. Moreover, in the J s 3r2 channel, all spins are aligned and the three nucleons cannot be at the same position, so a six-nucleon vertex should contain at least two derivatives and have a coefficient ; 1rMmp6 ; its contribution is thus expected to be suppressed in relation to the leading order graph by six powers of r 0ra. We will return to the doublet case in a future publication. Here we study the quartet channel, expected to be much less sensitive to the details of the physics of momenta of O Ž mp .. An enormous simplification comes about because the s-channel interaction due to the dibaryon is both local and separable. This allows us to write a simple integral equation that sums all the graphs in Fig. 3. Performing the integration over the time-component of the loop 4-momentum, we find that the conveniently normalized on-shell amplitude as a function of the initial Žfinal. center-of-mass 3-momentum k Ž p . satisfies

y

3Ž p 2 y k 2 . 8M

2

g D2

1 q 4p

ž(

4

Ž p 2 y k 2 . q MB y 'MB d3l

y1 s

3

2

Ž p q kr2. q MB

y

H Ž 2p .

/

t Ž p, k . 2

p yk 2 yie

1 3

l2qlPpqp 2y

tŽ l,k. 3 4

2

k 2 q MB

l yk 2 yie

.

Ž 5.

Note that all terms in a perturbative expansion of t in Ž5. are of the same order Ž; 1r 'MB .. It is straightforward but tedious to show that the wave function Ž2p . 3d Ž p y k . q t Ž p, k .rŽ p 2 y k 2 y i e . corresponding to a scattering solution indeed satisfies the Schrodinger equation derived from the Lagrangian Ž1.. ¨

P.F. Bedaque, U. Õan Kolck r Physics Letters B 428 (1998) 221–226

225

Fig. 4. Function aŽ x . for h s 0.40 without Žsolid line. and with Ždashed line. cut-off.

At zero energy Ž k ™ 0. the calculation simplifies 5 : only the S-wave, dependent on the magnitudes of momenta, contributes to the scattering, and we can perform the angular integration directly. It is also convenient to normalize all quantities to 'MB . Defining x s pr 'MB , aŽ x . s

'MB

t

4p

p

ž' / MB

,0 ,

Ž 6.

and introducing FŽ x, z. s

1 xz

ln

x 2 q z 2 q 1 q xz

ž

x 2 q z 2 q 1 y xz

/

,

Ž 7.

Eq. Ž5. becomes 3 4

1

yh q 1q

(

1q

3 4

aŽ x . s y x

2

1 2

x q1

1

y

`

H dzF Ž x , z . aŽ z . . p 0

Ž 8.

Note that there is only one parameter h s 2p'MB rM 2 g D2 s3 r 0'MB r2 s 0.40 in this equation. The value of the function aŽ x . at x s 0 gives the Nd scattering length in units of 1r 'MB . The same equation was previously obtained and solved in the zero-range limit Žh ™ 0. w15x. We have solved Eq. Ž8. numerically for h s 0.40 by the Nystrom method w16x. The solution aŽ x . is plotted as the solid line in Fig. 4. The pole in aŽ x . around x ; 4.4 is associated with the spurious deep two-body pole. Its presence allows intermediate states where two nucleons fall into this deep state while the other has extra energy. The interesting point is that even though the effective theory makes nonsensical predictions outside its domain of validity, like the existence of this new state, the low-x part of the curve is insensitive to the large-x behavior, and the prediction for the scattering length is sensible 6 . In order to demonstrate this more explicitly 5

Because we account for the range of the NN interaction, we are able to calculate the energy-dependence of Nd scattering close to threshold as well w14x. 6 Since this deep pole is a ghost and does not appear in initial or final states, it should not cause problems in any process where typical momenta are within the range of validity of the EFT.

226

P.F. Bedaque, U. Õan Kolck r Physics Letters B 428 (1998) 221–226

we have also solved Eq. Ž8. with a cut-off two-nucleon amplitude without the deep pole. For a cut-off of 150 MeV we obtain the broken line in Fig. 4. The quartet scattering length is 4 a s yaŽ0.r 'MB . For h s 0 Žand B fixed., we reproduce the result 4 a s 5.09 fm of Ref. w15x. Taking into account the finite range Žh s 0.40. we obtain ŽFig. 4. 4 a s 6.33 fm with an uncertainty from higher orders of ; "0.10 fm. This result obtained with no free parameters is in very good agreement with the experimental value of 4 a s 6.35 " 0.02 fm w17x. We thank David Kaplan for extensive discussions. Discussions with Ben Bakker, Joe Carlson, Vitaly Efimov, Jim Friar, Walter Glockle, and Martin Savage are also acknowledged. UvK is grateful to Justus Koch for ¨ hospitality at NIKHEF where part of this work was carried out. This research was supported in part by the DOE grants DOE-ER-40561 ŽPFB. and DE-FG03-97ER41014 ŽUvK..

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