The Born series for S -wave quartet nd scattering at small cutoff values
aa r X i v : . [ nu c l - t h ] J a n Archive of Applied Mechanics manuscript No. (will be inserted by the editor)
Shung-Ichi Ando
The Born series for S -wave quartet nd scattering atsmall cutoff values Received: date / Accepted: date
Abstract
Perturbative expansions, the Born series, of the scattering length and the amplitude of S -wave neutron-deuteron scattering for spin quartet channel below deuteron breakup threshold arestudied in pionless effective field theory at small cutoff values. A three-body contact interaction isintroduced when the integral equation is solved using the small cutoffs. After renormalizing the three-body interaction by using the scattering length, we expand the integral equation as the ordinary andinverted Born series. We find that the scattering length and the phase shift are considerably wellreproduced with a few terms of the inverted Born series at relatively large cutoff values, Λ ≃
100 MeV.
Keywords
Inverted Born series · Small cutoffs · S -wave quartet nd scattering After Weinberg suggested the application of chiral perturbation theory, a low energy effective fieldtheory of QCD, to nuclear force [1], a lot of works have been done during last two decades. Forreviews, see, e.g., Refs. [2; 3; 4; 5]. Meanwhile, new facilities for the rare isotope beams, in whichexotic nuclei near the neutron (proton) drip line can be created, provide us a new opportunity forinvestigation [6; 7; 8]. Such a limit close to the drip line where the single neutron (proton) separationenergy vanishes can be theoretically recognized as unitary limit (to be discussed below), and thus theunitary limit would be a good theoretical starting point to study the exotic nuclei near the nucleardrip line. In this work, we discuss a perturbative expansion of an integral equation, as the invertedBorn series which can be related to the unitary limit, by sending the cutoff parameter to small values(along with the ordinary Born series). We study it in S -wave neutron-deuteron ( nd ) scattering for spinquartet channel in pionless effective field theory (EFT).Pionless EFT [9; 10] is one of low energy EFTs for few nucleon systems, in which one chooses atypical momentum scale Q smaller than the pion mass, m π , and thus the pions are regarded as heavydegrees of freedom and integrated out of effective Lagrangian. Thus the large scale of the pionlessEFT is Λ ≃ m π , and the theory provides us a systematic expansion scheme in terms of Q/Λ . Inapplications of the pionless EFT, much more attention has been paid to S -wave nd scattering for spindoublet ( S = 1 /
2) channel because the one-nucleon-exchange interaction becomes “singular” [11; 12].To control the singularity one promotes a three-body contact interaction to leading order (LO), and thecoupling constant of the contact interaction exhibits so called limit-cycle when the scale parameter Λ issent to the asymptotic limit. This behavior is understood to be associated with “Efimov effect”. On theother hand, an application of pionless EFT to S -wave nd scattering for spin quartet ( S = 3 /
2) channel
Shung-Ichi AndoDepartment of Physics Education, Daegu University, Gyeongsan 712-714, Republic of KoreaTel.: +82-53-8506975Fax: +82-53-8506979E-mail: [email protected] is regarded well known because the phase shift δ of the quartet nd scattering is almost perfectlydescribed by two effective range parameters in two-body deuteron channel, and no cutoff dependenceis reported [13; 14].In this work, we revisit the S -wave spin quartet nd scattering below deuteron breakup thresholdin pionless EFT and reduce the cutoff parameter Λ smaller than m π . The small scale limit might beinteresting for studying, e.g., a relation between the pionless EFT and a Halo/Cluster EFT [15] inwhich one may choose a typical scale smaller than the deuteron breakup momentum, and the deuteronis never broken into two nucleons and could be regarded as a cluster or an elementary field [16].When the value of Λ is sent to significantly smaller than m π , the cutoff dependence emerges evenin the quartet channel. Thus we introduce a three-body contact interaction when we solve the integralequation using the small cutoff values. After renormalizing the strength of the three-body interaction byusing the scattering length a of the S -wave spin quartet nd scattering we expand the integral equationfor the scattering length a and the scattering amplitude in terms of the Born series up to next-to-next-to leading order (NNLO) . We expand the Born series around so called trivial and non-trivialfixed point studied in the renormalization group analysis by Birse, McGovern, and Richardson [17].The trivial fixed point corresponds to weak coupling limit where all interactions vanish, whereas thenon-trivial fixed point does to unitary limit where the scattering length becomes infinity or the bindingenergy vanishes. Around the unitary limit, the Born series is realized as the inverted Born series andexpanded around the inverse of the amplitude [18]. We find that, by reducing the cutoff value, both a and δ are considerably well reproduced by a few terms of the inverted Born series at relatively largecutoff values, Λ ≃
100 MeV.This work is organized as the following. In Sec. 2, effective Lagrangian is displayed. In Sec. 3,integral equations for the scattering length and the amplitude are given, and the coupling of the three-body interaction is renormalized by the scattering length a . In Sec. 4, the scattering length and theamplitude are expanded in terms of the ordinary and inverted Born series, and the numerical resultsare obtained. Finally, in Sec. 5, the discussion and conclusions are presented. The effective Lagrangian for the S -wave spin quartet nd scattering in pionless EFT reads [10; 12] L = L N + L t + L , (1)where L N is the standard one-nucleon Lagrangian in the heavy-baryon formalism, L N = N † (cid:26) iv · D + 12 m N (cid:2) ( v · D ) − D (cid:3) + · · · (cid:27) N , (2)where v µ is a velocity vector satisfying a condition v = 1, D µ is the covariant derivative, and m N isthe nucleon mass. The dots denote higher order terms which are not relevant in the present work. L t is dibaryon effective Lagrangian for S state, L t = σ t t † i (cid:26) iv · D + 14 m N (cid:2) ( v · D ) − D (cid:3) + ∆ t (cid:27) t i − y t n t † i h N T P ( S ) i N i + h.c. o + · · · , (3)where σ t is a sign factor, σ t = ± t i is a dibaryon field for spin triplet ( S ) state. D µ is the covariantderivative for the dibaryon field. ∆ t is the mass difference between the dibaryon and two nucleons, m t = 2 m N + ∆ t . y t is a coupling constant for dibaryon-nucleon-nucleon ( dN N ) vertex. P ( S ) i is aprojection operator for two nucleons in the S state, P ( S ) i = 1 √ τ σ σ i , (4)where τ a and σ i are Pauli matrices for isospin and spin, respectively. Those two constants, ∆ t and y t , and the sign factor σ t (= −
1) are determined by two effective range parameters, deuteron bindingmomentum γ and effective range ρ d , in N N scattering for S channel. We note that this is not an expansion scheme in EFT, but in terms of the Born series. We do not have aclear expansion parameter as
Q/Λ . = + + + Fig. 1
Diagrams for S -wave nd scattering in spin quartet ( S = 3 /
2) channel. A shaded blob denotes scatteringamplitude a line (a double-line) nucleon (dibaryon), and a double line with a filled circle dressed dibaryonpropagator (see Fig. 2). = + + + ...
Fig. 2
Diagrams for a dressed dibaryon propagator. See the caption of Fig. 1 as well.
We obtain L , an effective Lagrangian of three-body contact interaction in terms of a nucleon anda dibaryon field for the S -wave spin quartet nd state, as L = − m N g ( Λ ) y t N † (cid:2) ( σ · t ) † ( σ · t ) + 3( σ · t )( σ · t ) † (cid:3) N + · · · . (5)The expression of the interaction is the same as that of the spin projection operator of spin-1/2and spin-1 field into spin-3/2 state. See, e.g., Eq. (A4.37) in Ref. [19]. We note that the three-bodyinteraction is a higher order term in the S -wave spin quartet nd scattering in pionless EFT, and itis not necessary to be introduced when the value of the cutoff Λ is about m π or larger, Λ ≥ m π , atleading order (LO). In addition, as we will see below, it becomes ineffective when we solve the integralequation using the ordinary cutoff value. We introduce it, however, so as to reproduce the scatteringlength a in the quartet channel (and the strength of the coupling g ( Λ ) is adjusted) when the Λ valueis reduced significantly smaller than m π , Λ < m π . S -wave nd scattering amplitude for quartet channelDiagrams for S -wave nd scattering for the spin quartet ( S = 3 /
2) channel are given in Fig. 1, whereasthose for the two-body part in Fig. 1, the dressed dibaryon propagator, are given in Fig. 2. The integralequation of the S -wave spin quartet nd scattering without the three-body contact interaction is knownas [14] t ( p, k ) = − πmρ d K ( a ) ( p, k ; E ) − π Z Λ dqK ( a ) ( p, q ; E ) q t ( q, k ) − γ + ρ d (cid:0) γ − q + mE (cid:1) + q q − mE , (6)where t ( p, k ) is the half-off-shell scattering amplitude and p ( k ) is the magnitude of off-shell final state(on-shell initial state) relative three momentum in CM frame. K ( a ) ( p, q ; E ) is the one-nucleon-exchangeinteraction, K ( a ) ( p, q ; E ) = 12 pq ln (cid:18) p + q + pq − mEp + q − pq − mE (cid:19) , (7) There are some variations of the spin 3/2 projection operator for the spin 1 and 1/2 fields: P ij / = 16 ( σ i σ j + 3 σ j σ i ) = 23 δ ij − iǫ ijk σ k = δ ij − σ i σ j . where E = − γ m N + m N k : γ and ρ d are the deuteron binding momentum and the effective range,respectively. A sharp cutoff Λ is introduced in the loop integral.We note that because a singularity at q ≃
197 MeV which represents an unphysical deeply boundstate appears in the propagator, the dressed dibaryon propagator is usually expanded in terms of theeffective range, ρ d , to avoid an effect from the unphysical bound state. However, we are interested inreducing the cutoff value, less than Λ = 197 MeV, so we do not expand the dressed dibaryon propagatorin the integral equation in Eq. (6).Thus the on-shell T-matrix is given by T ( k, k ) = p Z d t ( k, k ) p Z d , (8)where Z d is the deuteron wavefunction normalization factor, Z d = γρ d / (1 − γρ d ). In terms of thehalf-off-shell scattering amplitude T ( p, k ), we have the integral equation as T ( p, k ) = − πm N ρ d Z d K ( a ) ( p, k ; E ) − π Z Λ dqK ( a ) ( p, q ; E ) γ + q γ + ( q − k )1 − ρ d (cid:16) γ + q γ + ( q − k ) (cid:17) q T ( q, k ) q − k − iǫ . (9)3.2 Integral equation for scattering lengthThe scattering length, a , of the spin quartet nd scattering is obtained by the on-shell scatteringamplitude with zero momentum as T (0 ,
0) = − πm N a . (10)Now we introduce a half off-shell amplitude (or a half off-shell scattering length) as T ( p,
0) = − πm N a ( p, , (11)where a (0 ,
0) = a . Thus we have an integral equation in terms of a ( p,
0) as a ( p, Λ ) = 8 Z d ρ d K ( a ) ( p, − B ) − π Z Λ dqK ( a ) ( p, q ; − B ) γ + q γ + q − ρ d (cid:16) γ + q γ + q (cid:17) a ( q, Λ ) , (12)where we have included the Λ dependence in the scattering length a ( p, Λ ) in the above expressionthough a ( p, Λ ) is insensitive to Λ when Λ ≃ m π or larger. B is the deuteron binding energy, B = γ /m N .Now we choose Λ = 197 MeV, and solve Eq. (12) numerically. We have two input parameters, γ and ρ d , whose values are known as γ = 45 . ρ d = 1 .
76 fm, and thus have a th ≡ a (0 , Λ = 197 MeV) = 6 .
34 fm . (13)We reproduce results obtained by Bedaque, Hammer, and van Kolck [13; 14], a = 6 . ± .
10 fm, andGriesshammer [20], 6 . ± .
020 fm, up to NNLO in pionless EFT. It agrees well with those obtained byBedaque and Griesshammer [21], 6 . ± . et al. [22], 6.321 ... 6.347 fm, from various combinations of modern N N potentials and three-bodyforces, and the experimental datum [23], a exp. = 6 . ± .
02 fm . (14) -1 0 1 2 3 4 5 6 7 0 50 100 150 200 a ( p , ; Λ ) ( f m ) p (MeV) 6 8 10 12 14 16 18 20 0 50 100 150 200 a ( , ; Λ ) ( f m ) Λ (MeV) Fig. 3
Left panel: Half off-shell a ( p, Λ ) (fm) with Λ = 197 MeV as a function of off-shell momentum p (MeV). Right panel: Cutoff dependence of scattering length a (0 , Λ ) (fm) without the three-body interaction. In the left panel of Fig. 3, we numerically calculate and plot the half off-shell scattering length, a ( p, Λ ), at Λ = 197 MeV as a function of the off-shell momentum p . We find that the off-diagonalpart of the scattering length quickly decreases and becomes almost negligible at p > m π . This factmay indicate the insensitivity of the amplitude for the quartet channel to the value of Λ when Λ = m π or larger, Λ ≥ m π .We now reduce the value of Λ . In the limit where Λ is sent to zero, we have the on-shell scatteringlength a (0 , Λ ) as lim Λ → a (0 , Λ ) = 83 1 γ (1 − γρ d ) ≃ .
19 fm . (15)In the right panel of Fig. 3, we plot the cutoff Λ dependence of the scattering length a (0 , Λ ). One cansee that there is almost no Λ dependence at Λ >
100 MeV, whereas the significant cutoff dependenceappears in the calculated scattering length a (0 , Λ ) when the value of the cutoff is reduced less than γ (= 45 . γ , the scattering length a (0 , Λ ) has thesignificant cutoff dependence. To make the result cutoff independent, as discussed before, we introducethe three-body contact interaction g ( Λ ) which is supposed to take account of physics integrated outdue to the small cutoff, and thus have the integral equation as a ( p,
0) = 8 Z d ρ d (cid:20) γ + p + g ( Λ ) (cid:21) − π Z Λ dq h K ( a ) ( p, q ; − B ) + g ( Λ ) i γ + q γ + q − ρ d (cid:16) γ + q γ + q (cid:17) a ( q, , (16)where we have removed the Λ dependence from the half off-shell scattering length a ( p,
0) because the Λ dependence from the integral should be cancelled by g ( Λ ).In the limit that the cutoff Λ becomes large, at Λ = 197 MeV, the coupling constant g ( Λ ) shouldvanish, g ( Λ ) →
0. On the other hand, in the limit that the cutoff Λ vanishes, one has a th = 83 Z d ρ d (cid:18) γ + g (0) (cid:19) , (17)where g (0) = − . · · · fm to reproduce the value of a (0 , Λ = 197 MeV) in Eq. (13). Thus weconsider a range of g ( Λ ), − . ≤ g ( Λ ) ≤ ), in this work. a ( , ) ( f m ) g( Λ ) (fm ) Λ = 2 γγ γ a -12-10-8-6-4-2 0 0 20 40 60 80 100 120 140 160 180 200 g ( Λ ) ( f m ) Λ (MeV) a B = 0 Fig. 4
Left panel: a (0 ,
0) (fm) in the cases of three fixed cutoff values Λ = 2 γ , γ , and 0 . γ as functions of g ( Λ )(fm ). A dashed horizontal line for a th is also included. Right panel: g ( Λ ) (fm ) which reproduce a th (curve)and three-body bound state with B = 0 (dashed curve) as functions of Λ (MeV). In the left panel of Fig. 4, we numerically calculate a (0 ,
0) from Eq. (16) and plot curves of a (0 , Λ , Λ = 2 γ , γ , and 0.5 γ as functions of g ( Λ ). A dashed horizontal line for a th is also included in the figure. One can notice that the curve for Λ = 2 γ is insensitive to g ( Λ ), except fora singular point which corresponds to a three-body bound state with zero binding energy, B = 0, eventhough the value of g ( Λ ) is significantly changed. We note that the S -wave three-body force for thespin quartet channel may be suppressed due to the Pauli principle (by applying the antisymmetrizationoperator) in the conventional potential model calculation. We can reproduce the same effect (except forthe appearance of the bound state) as that of the Pauli principle when we solve the integral equationusing the normal cutoff value. The curves with Λ = γ and 0 . γ , on the other hand, show a sensitivityto g ( Λ ) and vary widely and smoothly.In the right panel of Fig. 4, we numerically calculate g ( Λ ) from Eq. (16) and plot curves of g ( Λ )which reproduce a th (curve) and the three-body bound state with B = 0 (dashed curve) as functionsof Λ . One can see that when the cutoff value is large, Λ >
160 MeV, the value of g ( Λ ) almost vanishes.This may indicate the effect of the Pauli principle for the spin quartet channel. As the cutoff valueis further reduced smaller than Λ = 160 MeV, we need nonzero value of g ( Λ ) where the short rangelength scale of the theory becomes r (= Λ − ) > .
24 fm. This length scale might be regarded longenough to be out of the range of the Pauli principle. We note that the non-vanishing three-bodycontact interaction we obtained here may not be a genuine one, but correspond to the one induced bythe exchanging nucleon propagator of the larger momentum than the value of Λ , which connects twotwo-body interactions and makes the effective three-body one at the small cutoff values. The similarobservation that a three-body force is generated from two-body forces at small cutoffs in the SRGanalysis is reported in Ref. [24]. One can also see that the three-body bound state with B = 0 appearswhen the strength of g ( Λ ) becomes stronger than that of g ( Λ ) which reproduces a th and Λ is largerthan about 60 MeV in the figure. We find that the curve of g ( Λ ) for the three-body bound state with B = 0 varies smoothly, whereas that of g ( Λ ) which reproduces a th has a plateau like shape at themiddle of the range of Λ , Λ ≃ ∼
100 MeV. We use the curve of g ( Λ ) which reproduces a th whenstudying the perturbation expansions, the ordinary and inverted Born series, in the following. Now we expand the scattering length and the amplitude in terms of the ordinary and inverted Bornseries, as discussed in the introduction. -2 0 2 4 6 8 10 12 14 0 10 20 30 40 50 a ( f m ) Λ (MeV)LONLONNLOa a ( f m ) Λ (MeV)LONLONNLOa Fig. 5
Left panel: Scattering length a obtained from the ordinary Born expansion as functions of Λ (MeV).Curves labeled by “LO” are results at leading order, “NLO” up to next-to-leading order, and “NNLO” upto next-to-next-to leading order in the both panels. A horizontal line of a th is also included. Right panel:Scattering length a obtained from the inverted Born expansion as functions of Λ (MeV). a . Thus the scattering length a (0 ,
0) in Eq. (16) is expanded as a = a (0 ,
0) = 8 Z d ρ d [ b + b + b + · · · ] , (18)where b = 1 γ + g ( Λ ) , (19) b = Z Λ dq (cid:20) γ + q + g ( Λ ) (cid:21) F ( q ) , (20) b = Z Λ dq (cid:20) γ + q + g ( Λ ) (cid:21) F ( q ) × Z Λ dq ′ h K ( a ) ( q, q ′ , − B ) + g ( Λ ) i F ( q ′ ) (cid:20) γ + q ′ + g ( Λ ) (cid:21) , (21)with F ( q ) = − π γ + q γ + q − ρ d (cid:16) γ + q γ + q (cid:17) . (22)This expansion corresponds to that around the weak coupling limit. On the other hand, we consideranother expansion, the inverted Born series, of the scattering length as1 a = 3 ρ d Z d (cid:20) b − b b − b b + b b + · · · (cid:21) . (23)In the left and right panel of Fig. 5, we numerically calculate and plot curves of the scatteringlength a obtained from the ordinary and inverted Born series, respectively, as functions of Λ . Curveslabeled by “LO” are results which include only the leading order term, b , those by “NLO” are resultsup to next-to-leading order (NLO) which include first two terms in the brackets in Eq. (18) and (23),and those by “NNLO” are results up to next-to-next-to leading order (NNLO) which include all termsin the brackets. In the left panel of Fig. 5, one can see that a region of the cutoff Λ , where the curves of a obtainedfrom the terms up to NLO and NNLO in the ordinary Born expansion agree with a th , is quite small,up to about 10 MeV. It would be a natural consequence of the perturbation around the weak couplinglimit because the perturbation would converge when the scale Λ becomes smaller than a typical scaleof the process. In the present case, it may be a th (1 /a th ≃ . Λ becomes much smaller than 1 /a th . In the right panel of Fig. 5, on the other hand, we findthat the region where a th is reproduced is remarkably broaden, up to about 100 MeV due to the twoor three terms (up to NLO and NNLO, respectively) of the inverted Born series.4.2 The ordinary and inverted Born series for the scattering amplitudeBefore expanding the integral equation for the scattering amplitude in terms of the ordinary andinverted Born series, we check how the three-body interaction g ( Λ ) can reproduce the phase shift δ atthe small cutoff values. In Fig. 6, we numerically calculate and plot the phase shift δ below deuteron -70-60-50-40-30-20-10 0 0 10 20 30 40 50 δ ( deg . ) k (MeV) Λ = 197 MeV140 MeV100 MeV60 MeVAv18 Fig. 6
Phase shift δ (deg.) of the S -wave nd scattering for spin quartet channel below deuteron breakupthreshold as functions of k (MeV). Curves are obtained by solving the integral equation at cutoff values, Λ = 197,140, 100, and 60 MeV with the three-body interaction g ( Λ ) renormalized by a th . Plus signs ”+” denote resultsfrom a modern potential model (Av18) [25]. The deuteron breakup momentum is k br ≃ . breakup threshold as functions of the on-shell momentum k by solving the integral equation at smallcutoff values, Λ = 197, 140, 100, and 60 MeV. The deuteron breakup momentum is k br ≃ . T ( k, k ) = − πZ d m N ρ d [ B + B + B + · · · ] , (24)with B = V ( k, k ) = K ( a ) ( k, k ; E ) + g ( Λ ) , (25) B = Z Λ dq V ( k, q ) G ( q, k ) V ( q, k ) , (26) B = Z Λ dqV ( k, q ) G ( q, k ) Z Λ dq ′ V ( q, q ′ ) G ( q ′ , k ) V ( q ′ , k ) , (27)where G ( q, k ) = − π γ + q γ + ( q − k )1 − ρ d h γ + q γ + ( q − k ) i q q − k − iǫ . (28)This corresponds to the expansion around the weak coupling limit. We also have an expansion aroundthe inverse of the T-matrix as1 T ( k, k ) = − m N ρ d πZ d (cid:20) B − B B − B B + B B + · · · (cid:21) . (29)To calculate the phase shift δ from the ordinary Born series in Eq. (24) we employ the relation δ = 12 i ln (cid:20) i km N π T ( k, k ) (cid:21) , (30)We note that when the amplitude expanded in the ordinary Born series is truncated, it breaks unitarycondition and the phase shift becomes a complex number [26]. On the other hand, to calculate thephase shift from the inverted Born expansion in Eq. (29) we use the formula k cot δ = 3 πm N Re T ( k, k ) , (31)with Re T NLO ( k, k ) = − m N ρ d πZ d (cid:20) B − ReB B (cid:21) , (32) Re T NNLO ( k, k ) = − m N ρ d πZ d (cid:20) B − ReB B − ReB B + ( ReB ) − ( ImB ) B (cid:21) , (33)where B has real part only. This expansion preserves the unitary condition, at least up to NNLO.In Fig. 7, we numerically calculate and plot real and imaginary part of the phase shift δ of S -wavespin quartet nd scattering obtained from the truncated ordinary Born series in Eq. (24). Curves up toNLO are obtained by including first two terms, B and B , and those up to NNLO by including allthree terms, B , B , and B , in Eq. (24). We have fixed Λ at Λ = 13 MeV because we found in theleft panel of Fig. 5 that the scattering length a th are fairly well reproduced when the cutoff value isabout up to 10 MeV. A line labeled by “Full” obtained from the calculation without the truncationat Λ = 197 MeV is also included in the figure. One can see that the real parts of δ agree with the fullresult up to about 6 MeV for NLO and 9 MeV for NNLO. In addition, the imaginary parts sharplydecrease around the edge of the cutoff value Λ = 13 MeV, and it indicates that the unitary conditionis broken. The broken unitary condition up to NLO is also partly cured by including the higher orderterm, B , at NNLO.In Figs. 8 and 9, we numerically calculate and plot curves of the phase shift δ obtained from theinverted Born series up to NLO and NNLO in Eqs. (32) and (33), respectively, where the cutoff valuesare chosen Λ = 140, 100, and 60 MeV. In addition, plus signs “+” labeled by “Full” in the figuresdenote the results from the calculation without the truncation at Λ = 197 MeV. We find in Fig. 8 thatthe curves considerably well converge to that of the full calculation as the cutoff value decreases, andthe truncated inverted Born series up to NLO fairly well reproduces the result of the full calculationat Λ = 60 MeV. In Fig. 9, we can see that the convergence to the full result, as reducing the cutoffvalue, becomes faster due to the inclusion of the higher order terms. This line is the same as that at Λ = 197 MeV in Fig. 6.0 -50-45-40-35-30-25-20-15-10-5 0 5 0 2 4 6 8 10 12 14 δ ( deg . ) k (MeV)NLO (real)NLO (imag)NNLO (real)NNLO (imag)Full Fig. 7
Real and imaginary part of phase shift δ (deg.) of S -wave spin quartet nd scattering, as functions of k (MeV), obtained from the truncated ordinary Born series up to NLO and NNLO where the cutoff is fixedat Λ = 13 MeV. A dashed line labeled by “Full” obtained from the calculation without the truncation and Λ = 197 MeV is also included. -70-60-50-40-30-20-10 0 0 10 20 30 40 50 δ ( deg . ) k (MeV) Λ = 140 MeV100 MeV60 MeVFull Fig. 8
Phase shift δ (deg.) of S -wave spin quartet nd scattering, as functions of k (MeV), obtained from theinverted Born series up to NLO with Λ = 140, 100, and 60 MeV. Plus signs “+” labeled by “Full” are obtainedfrom the calculation without the truncation at Λ = 197 MeV. In this work, we studied the perturbative expansions, as the ordinary and inverted Born series, ofthe integral equation for the S -wave nd scattering for the spin quartet channel below the deuteronbreakup threshold in pionless EFT at the small cutoff values. The three-body contact interaction isintroduced when the integral equation is solved by using the small cutoff values. After the strength ofthe three-body interaction is renormalized by using the scattering length a , we expand the integralequation for the scattering length and the amplitude, as the ordinary and inverted Born series, upto NNLO. We find that the scattering length (the phase shift) is considerably well reproduced by afew terms of the ordinary and inverted Born series as we reduce the cutoff values to about 10 MeV -70-60-50-40-30-20-10 0 0 10 20 30 40 50 δ ( deg . ) k (MeV) Λ = 140 MeV100 MeV60 MeVFull Fig. 9
Phase shift δ (deg.) obtained from the inverted Born series up to NNLO with Λ = 140, 100, and60 MeV. See the caption of Fig. 8 as well. (10 MeV) and to about 100 MeV (60 MeV), respectively. Therefore, the inverted Born expansion inthe present process can be a relevant approximation with a significantly larger valid momentum thanthe ordinary Born expansion.In the present particular process, the S -wave spin quartet nd scattering, one may regard that thereis no advantage by sending Λ to small values because the physical observables, the scattering length a and the phase shift δ , are well described (without fitting any unknown parameters) by the twoeffective range parameters in the deuteron channel at the usual large scale of the pionless theory, Λ ≃ m π . However, it could be a useful limit when one studies, e.g., a relation between pionless EFTand a Halo/Cluster EFT whose large scale smaller than m π , such as a deuteron cluster theory for areaction whose typical scale Q is smaller than the deuteron breakup momentum [16].In addition, it may be the interesting observation that the physical observables can be well describedby a few terms of the inverted Born series, being closely related to the unitary limit, at the relativelylarge cutoff values. If this property were common in some class of the interactions and/or in the unitarylimit, it could provide us a useful method to make a non-perturbative interaction perturbative and/orto be used in studies of the exotic nuclei near the drip line. Now we are studying the property ofthe inverted Born expansion by employing a renormalization group analysis, and it is to be reportedseparately. Acknowledgements
The author would like to thank K. Kubodera for reading the manuscript and Y.-H. Songfor discussion. This work is supported by the Basic Science Research Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0023661) and(2012R1A1A2009430).
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