• Nie Znaleziono Wyników

Four-nucleon force contribution to the binding energy of ^{4}He

N/A
N/A
Protected

Academic year: 2022

Share "Four-nucleon force contribution to the binding energy of ^{4}He"

Copied!
8
0
0

Pełen tekst

(1)

DOI:10.1051/epjconf/20100305006

© Owned by the authors, published by EDP Sciences, 2010

Four-nucleon force contribution to the binding energy of

4

He

A. N o g g a 1,2’3, D. R ozpędzik 3, E. E p elb au m 1,4, W. G lö ckle 5, J. G olak 3, H. K am ada 6, R. S k ibiński 3, and H. W itała 3 1 F orschungszentrum Jülich, Institut fUr K ernphysik (Theorie) and Jü lich C enter for H adron Physics, D -52425 Julich,

G erm any

2 F orschungszentrum Julich, Institute for A dvanced S im ulation, D -52425 Julich, G erm any

3 M. S m oluchow ski Institute o f Physics, Jag iello n ian U niversity, PL-30059 Krakow , Poland

4 H elm holtz-Institut für Strahlen- und K ernphysik and B ethe C enter for T heoretical Physics, U niversität B onn, D -53115 B onn, G erm any

5 In stitut fur theoretische Physik II, R uhr-U niversitat B ochum , D -44780 B ochum , G erm any

6 D epartm ent o f Physics, Faculty o f E ngineering, K yushu Institute o f Technology, 1 -1 S ensuicho, Tobata, K itakyushu 8 04-8550, Japan

Abstract. We study the four-nucleon force contribution to the binding energy of 4He in the framework of chiral nuclear interactions. The four-nucleon forces start to contribute in the next-to-next-to-next-to leading order. We discuss our power counting expectations for the size of the 4N contribution and then explicitly calculate it in first order perturbation theory. Our expectations agree with the results. Quantitatively, the contribution might be significant. This motivates further studies in more complex nuclei.

1 Introduction

O ne o f the m ain goals o f nuclear physics is the und erstan d ­ ing o f the properties o f nuclei based on nuclear in terac­

tions. It is g enerally accepted that the H am iltonian for a nuclear system is driven by nucleon-nucleon (NN) pair in­

teractions, for w hich highly accurate m odels have been de­

veloped [1-3]. B ut the application o f these m odels to light nuclei [4-8] has show n that N N interactions alone are not able to provide a sufficiently accurate description o f the data. T his led to the conclusion that three-nucleon forces (3N F ’s) are required to describe nuclei based on m icro ­ scopic interactions 1. M odels for th ree-nucleon (3N) inter­

actions exist starting w ith the venerable F uijita-M iyazaw a force [9]. S uch m odels have b een refined by repulsive short distance pieces [1 0] or im plem enting constraints by n-nu- cleon (nN) scattering [11,12]. U nfortunately, due to the phenom enological character o f N N interactions, none o f these m odels is based on a com m on footing w ith any o f the m odern accurate N N interactions 2. T his is how ever a basic requirem ent o f any com bination o f N N and 3N forces since both cannot be defined independently o f each other[15]. N evertheless, brute-force com binations o f such N N and 3N force m odels, that are tuned to at least describe the 3H binding energy, give quite reasonable results for 3N scattering observables [16] and binding energies o f light

a e-mail: a .n o g g a @ f z - ju e lic h .d e

1 The term microscopic interactions refers to interactions among nucleons as basic constituents of nuclei.

2 For an attempt to derive NN and 3N forces from a unified approach see [13,14].

nuclei [4 -6 ,1 7 ,8 ]. B ut at the sam e tim e, such results show deficiencies that indicate that part o f the nuclear H am ilto­

nian is not understood sufficiently w ell.

A system atic schem e to derive the nuclear H am iltonian is based on chiral perturbation theory (ChPT). H ere the ap ­ proxim ate but spontaneously broken chiral sym m etry o f the QCD L agrangian is im plem ented in an effective field theory in term s o f nucleon and pion fields. Chiral sy m m e­

try constrains the possible couplings o f these fields, espe­

cially for the pions being the pseudo-G oldstone bosons re ­ lated to the spontaneous sym m etry breaking. D ue to these constraints, the L agrangian and all diagram s can then be expanded in term s o f j - , w here Q is a typical m om en­

tum o f the considered process or the pion m ass and Ax is the chiral sym m etry b reaking scale o f the order o f the p m eson or nucleon mass. For low m om enta and system s w ith nucleon num ber A = 0 or 1, this leads to a pertur- bative expansion o f the relevant am plitudes. For A > 2, this expansion cannot be perturbative, since bound states (the nuclei) exist. W einberg recognized that diagram s w ith purely nucleonic interm ediate states, so called reducible diagram s, are responsible for this non-perturbativeness. He therefore suggested to expand a potential (the sum o f all irreducible diagram s) using the standard pow er counting o f ChPT. R educible diagram s can then by sum m ed up to infinite order by solving the S chrödinger equation based on such a potential [18,19]. This explains naturally w hy N N forces are driving the nucleon H am iltonian and m ore- and-m ore-nucleon interactions becom e less-and-less im ­ portant. It also enables us to derive N N and m ore-nucleon interactions w ithin the sam e fram ew ork.

This is an Open Access article distributed under the terms o f the Creative Commons Attribution-Noncommercial License 3.0, which

(2)

Fig. 1. Non-zero contributions to class I of 4NF diagrams k g6A.

Fig. 2. Non-zero contributions to class II of 4NF diagrams k g4A.

It turns out that leading order (LO ,Q °), next-to-leading order ( N L O ,^ 2), next-to-next-to-leading order (N2LO,Q®) and next-to-next-to-next-to-leading order (N3LO,Q^) term s o f the chiral expansions are required to obtain N N inter­

actions that have an accuracy com parable to the m odern phenom enological ones [20,21]. The leading 3 N F ’s appear in N 2LO [22,23] , som e parts o f the subleading term s have been form ulated [24] but not applied yet. T he leading four- nucleon force (4NF) is o f order Q4 and has been derived in [25,26]. In these proceedings w e report on the application o f this 4NF, nam ely a calculation o f its contribution to the binding energy o f 4He. T his w ork goes beyond our first estim ate o f this 4N F contribution [27], since w e now take the 4He w ave function in its full com plexity into account.

T his is especially im portant for a reliable estim ate o f short range contributions o f the 4NF.

W e start introducing the chiral 4N F in Sec. 2. T hen w e turn to the m ore technical aspects and define the in­

gredients o f the actual calculations in Sec. 3. In Sec. 4, w e discuss our pow er counting expectations for the size o f the 4NF. The num erical results have been obtained using a M onte-C arlo approach that is introduced in Sec. 5. T he re­

sults are given in Sec. 6, w hich leads us to the conclusions and the outlook in the final section.

2 Chiral interactions and 4NF's

g \ C r .

Fig. 4. Non-zero contributions to class V of 4NF diagrams k g2A c t .

F our-nucleon interactions have already been discussed in the 19 8 0 ’s [28,29]. A t the tim e, the conclusion w as that the contribution is probably sm all enough to be neglected.

G iven that 3 N F ’s w ere know n to be m uch m ore im portant but m uch less understood at the tim e than today, it w as re a ­ sonable to neglect the 4 N F ’s based on the results obtained.

B ut it is tim ely to reconsider this part o f the interactions now for tw o reasons. Firstly, w e are now in position that m uch m ore accurate nuclear structure calculations are p os­

sible, the aim being to predict the m asses even o f drip line nuclei. F or such an endeavor, the accuracy o f the u n d erly ­ ing forces needs to be m uch higher and 4 N F ’s m ight b e­

com e quantitatively im portant. Secondly, w e have now a system atic schem e to derive consistent NN , 3N and 4N in­

teractions based on chiral perturbation theory. T herefore,

w e are now in the position to derive the com plete leading contribution o f 4 N F ’s consistently to the chiral N N and 3N interactions.

H ere, w e restrict ourselves to the effective theory w ith ­ out explicit A isobar degrees o f freedom . We stress that due to the strong coupling o f the n N system to the A and the sm all difference o f the nucleon and A m ass, the in clu ­ sion o f A m ight be advisable. For the N N and for 3 N F ’s, this has been done already (see e.g. [3 0 -33]). T he results confirm the im portance o f A ’s in nuclear interactions. For the 4NF, the contribution due to A ’s w as estim ated in [34]

based on a phenom enological approach. T his study indi­

cated that the A contribution to 4 N F ’s is sm all. It w ill be interesting to look at this estim ate again based on chiral

(3)

Fig. 5. Non-zero contributions to class VII of 4NF diagrams k g2A Cr

interactions and to confirm that this conclusion persists in­

dependent o f the choice for cutoffs, but this is beyond the scope o f this study.

N eglecting A ’s and based on the pow er counting o f W einberg, one only gets N N interactions in LO (Q0) and N LO (Q2) and additional N N interactions and first 3 N F ’s in N 2LO (Q 3). Up to this o rder the com plete nuclear H am il­

tonian has been derived. The Q4 term s (N3LO) have been com pletely form ulated for the N N force and they proved to be quantitatively im portant for an accurate description o f N N data [20,21] up to the pion production threshold.

A t this point, parts o f the subleading 3 N F ’s have been fo r­

m ulated [24], but they have not been applied yet. T he ap­

proach has been review ed in [35,36]. For the explicit ex­

pressions o f the N N and 3N forces, w e refer to Ref. [37]

w here all results have been derived in the sam e schem e o f unitary transform ations that w as used for the 4 N F ’s ap­

plied here. T he approach is w ell suited to end up w ith standard nuclear potentials that can be directly applied to few -nucleon system s since all interactions are m anifestly energy-independent. Since the sam e approach w as used for the N N and 4N force, it is insured that both are consistent to each other.

A d d itionally to the Q4 N N and 3 N F ’s, there are also first 4 N F ’s in this order. T he derivation o f the com plete set o f these term s has been done in Refs. [25,26] and show ed that the leading 4N F does not only consist o f pion ex­

change pieces, but also o f short range pieces that are di­

rectly linked to corresponding short range pieces o f the N N interaction. It is useful to classify the contributions ac­

cording to their dependence on the axial-vector coupling constant gA and the low energy constants (L E C ’s) Ct. In Refs. [25,26] eight classes have been identified. Som e o f the contributions are zero, therefore only class I (k gA), class II (k gA), class IV (k gA Ct) , class V (k gA Ct) , and class V II (k gA C 2T) term s have to be considered. In Figs. 1 to 5, w e sum m arize the topologies o f the diagram s contributing to the 4NF. T he diagram s show n v isualize ex­

pressions that have been derived algebraically. N ote that som e o f the diagram s look as if they are reducible itera-

200

100

0 -

CS LO CT LO CT NLO CT NNLO CS NLO CS NNLO

A [fm-1]

Fig. 6. Cutoff dependence of the LO LEC’s CS and CT for various orders of the chiral interaction.

tions o f N N or 3N interactions. We how ever only consider the irreducible parts here, w hich naturally separate in the expressions derived. To arrive at the final expressions, it is also m andatory to study 3N forces consistently. It turns out that the requirem ent that 3 N F ’s are renorm alizable further constrains the expressions for the 4 N F ’s. For details, w e refer to Ref. [26] w here also the final expressions o f the 4N F can be found.

A s m entioned earlier, there is a relation o f the short range part o f the leading 4N and LO N N interactions. The LO N N interaction consists o f the 1n-exchange and two contact interactions

Vlo = - 9A 2 F n

o~i q o~2 q

q'2 + in2n T 1■ T2+ Cs + Ct( T \ ■ O 2

Here, q is the m om entum transfer from one nucleon to the other and o , ( t ) are P auli m atrices acting in spin (isospin) space o f nucleon i. T he strength o f the 1n exchange is d e­

term ined by the axial-vector coupling constant gA and the pion decay constant Fn = 92.4 MeV. The strength o f the contact term s is param eterized by the L E C ’s Cs and Ct, w hich are determ ined by a fit to N N scattering data and/or the deuteron properties. Interestingly, parts o f the 4N F d e­

pend on Ct stressing the strong relation o f N N and 4 N F ’s and the need for consistent com binations o f N N and m ore- nucleon interactions.

Fortunately, the strength o f the 4N F is com pletely d e­

term ined by L E C ’s that also appear in the leading N N in ­ teraction. O ur estim ate below w ill therefore be com pletely param eter independent.

T he nuclear interaction needs to be regularized in or­

der to obtain a w ell-defined S chrodinger equation. T his is usually done by m ultiplying the potential m atrix elem ents w ith a cutoff functions, e.g. exponentials

^ (p '.p ) ^ exp ( “ ( ^ ) ) k(p'>p) e x p ( _ (a) ) ' The cutoff functions dependent on the relative m om enta o f the nucleons (here p and p ') and a cutoff A. T he pow er n is usually chosen betw een 4 and 8. B elow w e w ill show leading order results for A = 2 to 7 fm- 1 and higher order results for A = 500 to 600 MeV.

A s can be seen from Fig. 6, the Cs and Ct are strongly cutoff dependent. F rom naturalness, one w ould expect that

2

(4)

Table 1. BE’s £ (3He) and £ (4He) for 3He and 4He for selected phenomenological models and LO,NLO and N2LO chiral inter­

actions compared to experiment. For chiral interactions, the cut­

off dependence is indicated given the minimal and maximal bind­

ing energy obtained in our calculations. All energies und the cut­

offs are given in MeV.

interaction £ (3He) £ (4He)

AV18+Urbana-IX -7.72 28.5

CD-Bonn+TM99 -7.74 -28.4

LO -5.4 ... -11.0 -15.1 ...-39.9

NLO -6.99 ...-7.70 -24.4 . . . -28.8 N2LO -7.72 ...-7.81 -27.7 ...-28.6

Expt. -7.72 -28.3

CS and C T are o f the order o f 100 G eV -2 . For CS , this nat­

uralness estim ate holds for m ost cutoffs and orders w ith a few exceptions. S u ch exceptions can be linked to the ap­

pearance o f spurious bound states in the N N system [38].

G enerally, the Cs/t for such A can be large, but their co n ­ tribution to interactions are nevertheless natural, since the short distance w ave function is suppressed for such A . C o n ­ trarily, Ct is m uch sm aller than the naturalness estim ate.

T his has been observed already in [39] and can be traced back to the approxim ate W igner sym m etry o f nuclear in­

teractions. It w ill be interesting to study the im portance o f term s o f the 4N F proportional to Ct below.

3 4He wave functions

We are going to estim ate the 4N F contribution in first or­

der perturbation theory. For such an estim ate the expecta­

tion value o f the relevant operators w ith respect to the 4H e w ave function has to be calculated. T herefore, w e w ould like to sum m arize briefly w hich w ave functions enter our calculations.

A lth o u g h consistent results can only be obtained based on chiral nuclear interactions, w e have also perform ed cal­

culations based on the m odern phenom enological in terac­

tions AV18 [1] and C D -B onn [2]. In both cases, w e au g ­ m ent the N N interaction by phenom enological 3 N F ’s based on 2n exchange, w hich have been adjusted to the 3H b in d ­ ing energy and, for U rbana, also to nuclear m atter density.

For details on this adjustem ent see Refs. [5,40].

For studying the cutoff dependence o f the expectation values, it is also useful to study the 4N F for the leading order w ave functions. H ere, w e follow the schem e o f Ref. [38]. H ow ever, w e only consider s-w ave interactions, so that only tw o L E C ’s need to be adjusted, w hich w e fit to the deuteron binding energy and the 4S0 phase shift at

1 M eV laboratory energy.

O u r m ost consistent calculations are based on chiral nuclear interactions o f order N LO and N2LO. H ere, w e ap­

ply the interactions o f Ref. [21] w hich have been derived in the sam e fram ew ork as the 4NF. T his guarantees consis­

tency o f both parts o f the interaction. In order N 2LO also the leading 3 N F ’s are included. T he relevant L E C ’s have

been adjusted to the 3H binding energy and the nd doublet scattering length as outlined in [23].

In order to obtain the w ave functions, w e solve Yaku- bovsky equations in m om entum space in a partial w ave basis [5]. For the representation o f the w ave function, w e take angular m om enta up to l = 6 into account. T his re ­ quires a large num ber o f partial w ave channels o f the or­

der o f 1 0 0 0 for the representation o f the w ave functions.

For the M onte C arlo schem e described below, w e need to transform the w ave function from this partial w ave b a ­ sis to a basis depending on three-m om enta and individual nucleon spin/isospin projections. T his transform ation has been im plem ented quite efficiently, still it takes the bulk o f the com putational resources.

W e sum m arize the binding energy results in Table 1.

O ne can see that the binding energies are w ell described for the phenom enological and chiral N 2LO interactions. Due to the correlation o f the 3H, 3H e and 4He binding energies, this is not surprising. A t LO and N L O , the binding energy o f 3H cannot be adjusted, so that the dependence is still rather strong. The chiral expansion o f binding energies is g enerally slow ly converging, since the cancelation o f k i­

netic and potential energy enhances sm all contributions to the interaction.

4 Power counting estimate

Before actually doing an explicit calculations for the 4NF, w e w ould like to estim ate its contribution based on g e n ­ eral pow er counting argum ents and previous experience.

A lthough potential energies are no observables, it is useful to estim ate higher order contributions to the binding en ­ ergy based on the expectation values o f the N N potential, since the chiral expansion is perform ed for this potential.

Since typical m om enta in nuclei are o f the order o f the pion m ass, the sm all scale o f the expansion is usually assum ed to be o f this order. T here are som e discussions on the large chiral sym m etry b reaking scale. W hereas for purely pio- nic processes, loop contributions can be w ell estim ated as­

sum ing Ax ~ 4n fn « 1 GeV, this is probably not a valid choice for processes involving nucleons. Pion production is not explictly included into the chiral interactions and the m om entum scale associated w ith such processes is o f the order o f 400 MeV. A t the sam e tim e, the m om entum cutoff o f higher order chiral interactions is in the sam e order o f m agnitude. T herefore, w e use this value for our estim ate o f higher order contributions. The expansion factor then becom es « 0.35.

A X

W ith this choice, w e can estim ate the 3NF contribution to the binding energy. Since it is o f order Q3, w e expect that the 3NF contributes 4 % o f the potential energy to the binding energy. In Table 2, w e present the expectation v al­

ues o f the N N and 3N interactions for four different calcu­

lations o f the 4H e binding energy. The N 2LO calculations coincide w ith the ones w e w ill use for the evaluation o f the 4NF. A dditionally, w e show results for N 3LO calculations based on the chiral interactions o f Ref. [20]. N ote that also here the 3NF is only up to order N2LO.

(5)

Table 2. BE’s E and expectation values of the NN ((Vnn)) and 3N ((V3NF)) interactions for 4He. All energies and the cutoffs are given in MeV. “DR” indicates that loops for the chiral interaction of Ref. [20] are regularized using dimensional regularization. The experimental BE is -28.30 MeV.

interaction A / A E (Vnn) < V3 mf) < V3 n f) /( Vnn)

N2LO 4 5 0/700 -27.65 -84.56 -1 . 1 1 1.3%

N2LO 600 / 700 -28.57 -93.73 -6.83 7.3 %

N3LO-3NF-A 500/D R -28.27 -99.45 -4.06 4.1 %

N3LO-3NF-B 500/D R -28.24 -98.92 -7.10 7.2 %

LO-LAM=7 w f Class IV contribution

Fig. 7. 4NF contribution to the binding energy of 4He. Ten in­

dependent MC results are shown for the class IV contribution based on the LO wave function with A = 7 fm-1. Error bars are estimates for the single run standard deviations. The line is the average of all ten runs. The band indicates the standard deviation of the combined runs.

W e find that the expectation values o f the 3NF is indeed o f the order o f 4 %. T his seem s to back our very conser­

vative choice for Ax . We note that the inclusion o f .d’s into the effective theory shifts part o f the 3NF to N LO . H ere, w e strictly stick to ^ -le s s ChPT and estim ate the higher order contributions based on Ax = 400 MeV.

The 4N F is order Q 4. B ased on the expectation values o f N N potential given in the table, w e can estim ate that the 4N F contributes approxim ately 1 M eV to the binding en­

ergy. Such a contribution is not negligible in nuclear stru c­

ture calculations. T his estim ate is also in line w ith the o b ­ servation that an accurate description o f N N data requires potentials up to order N3LO. It is therefore necessary to m ake an explicit calculation to get a m ore reliable estim ate o f its contribution.

5 Numerical technique

In this section, w e w ant to introduce briefly the num erical m ethod used for the evaluation o f the pertinent integrals.

Since w e base our estim ate o f the 4N F contribution on first order perturbation theory, w e need to calculate the expec­

tation value o f the 4N F w ith respect to 4He w ave functions.

T his leads to integrals o f the form

W = ^ f d p u d p 3 d q 4 d p \2d p '3d q4

a a '

<^|P12P3q4«> < ... | V4|. . . ) <p 12 'P3 'q 4 ' a | W)

Fig. 8. Same as Fig. 7 for a calculation of the class I contri­

bution for the NNLO wave function with A = 550 MeV and A = 600 MeV.

S f

d p u d p3 d q4 d p[ 2 d p3 d q4

w(P12, P3, q4; P l2, p ,3, q4)

< W |p12P3q4 a ) < ... | V4I. . . ) <p 12 'p3 'q4 ' a ' | W) w( p1 2, p3, q4; p12, p3, q4)

(1) H ere, p i2 p3 q4 (p i2 ' p3 ' q 4 ' ) are incom ing and outgoing Ja co b i m om enta in the 4N system . The 4N F m atrix ele­

m ent is <. . . |V4| . . .) and depends on these m om enta and the incom ing and outgoing spin/isospin channels a and a (labeling all possible com binations o f spin/isospin p ro­

je c tio n s o f the four nucleons). The 4He w ave functions

<p12 ' p3 ' q4 ' a ' | W) are also given in term s o f the Ja co b i m o ­ m enta and a and w is a w eig h t function to be discussed below.

We have not perform ed a partial w ave decom position.

T herefore, the dim ensionality o f the integral is m uch to high to be calculated w ith standard techniques. A M onte Carlo (MC) schem e is m uch better suited for this purpose.

We found that an im portance sam pling sim ilar to the M e tro ­ polis algorithm [41] is required to keep the com putational needs sm all and increase the accuracy.

U sually such an im portance sam pling is guided by the square o f the w ave function. In configuration space, this quantity is perfectly suited as a w eig h t function since it is then autom atically norm alized to one at least as long as the operators are local. For m om entum space, the stru c­

ture is m ore com plicated, since the integrals require w eight functions w ith higher dim ensionality as in configuration space. T his im plies that a sim ple square o f the w ave fu n c­

tion is not useful for the im portance sam pling anym ore.

T his problem could be solved by perform ing part o f the

0

0 6 0 2 8

run run

(6)

integrals using standard m ethods as has been successfully done in [42]. We found this approach less practical in our case, since the three- and four-nucleon o perators w ould re­

quire to perform high dim ensional integrations using sta n ­ dard integration m ethods.

O ur solution w as to give up w eig h t functions based on the w ave functions o f the system , but choose a ratio ­ nal ansatz instead. T he param eters o f the ansatz w ere then adjusted so that the standard deviation in test cases w as m inim ized. In this way, w e w ere able to im prove the ac­

curacy sufficiently. A t the sam e tim e, the w eig h t function could be analytically norm alized to one so that the calcu­

lations becam e feasible.

E.g. w e choose for the im portance sam pling for inte­

grals o f the form o f Eq. (1) a w eig h t function depending on the six integration variables p 2- = p 12, p 3, q 4, p [ 2, p3

and q4

w ( P t2> P3> q4> P1 2, P3, q ^ = w [ p'1 2, p'3, q4> P1 2, P3, Pi)

_

I I ( r - 3) ( r - 2) ( r - 1)

Cp, ^

. .

= M f a + c „ y ( )

For sim plicity, the ansatz only depends on the m agnitude o f the m om enta. W ith the param eters Cpi and r the shape o f the w eig h t functions can be influenced. T he ansatz g u ar­

antees (for large enough r) th at the w eig h t function is nor­

m alized to one.

In practice, w e used a M ath em a tica script to generate the num erical expressions o f the potential m atrix elem ents for each a / a reliably. T he resulting code lines could be directly included in a FO R TR A N code evaluating the high dim ensional integrals given above.

In order to check the statistical character o f our M C re­

sults, w e perform ed for each quantity 1 0 independent cal­

culations. Figs. 7 and 8 sum m arize the results o f tw o re p ­ resentative sets o f calculations. T he error bars are standard deviations o f the single runs. T he line is the average o f all the ten runs and the shaded region indicates the standard deviation o f the average. It is reassuring that the app ro x i­

m ately 2/3 o f the single runs overlap w ith the average in b oth cases w ithin one standard deviation. T his is a n o n ­ trivial confirm ation o f the statistics o f the runs.

W ith this technique, w e w ere able to estim ate the 4N F contribution for the different interactions reliably. We are now in the position to discuss the results in the next section.

6 Results

We start the discussion o f the results based on LO w ave functions for w hich w e w ere able to investigate a large range o f cutoffs. Som e exem plary results are show n in Figs. 9 and 10. The actual contribution o f the tw o show n parts o f the 4N F are strongly cutoff dependent. T his is not surprising and reflects the fact that the potential is not o b ­ servable. E ven for the large cutoff the 4N F contribution is stable and rem ains below or around 1 MeV. T his is not only true for these tw o exam ples, but also for all the other classes o f diagram s.

Fig. 9. Cutoff dependence of the class I contribution to the 4NF for the LO wave functions. Error bars are the statistical errors of the MC evaluation.

Fig. 10. Cutoff dependence of the class IV contribution to the 4NF for the LO wave functions. Error bars are the statistical er­

rors of the MC evaluation.

For the larger cutoffs, it is not necessarily true that the 4N F is perturbative. For 3 N F ’s, w e know that first order perturbation theory is insufficient for som e higher cutoffs [43] or for som e phenom enological m odels [44]. T h ere­

fore, the large cutoff results have to be taken w ith som e care, although w e have no indication that perturbation th e­

ory is not appropriate for these estim ates.

Surprisingly, w e find a rather large contribution from class IV. T his class depends on Ct, w hich generally is sm aller than natural due to W igner sym m etry. B ut our re ­ sults for the expectation values in LO for class IV is not u nnaturally sm all. In fact, as can be seen in the figures, it is larger than class I contributions for m ost cutoffs. A t this point, w e do not fully understand this enhancem ent o f the LO results.

T he com plete final results are depicted in Fig. 11. We show results for the different classes separately. The bars indicate the range o f the results for different cutoffs (or for AV18 and C D -B onn for the phenom enological m od­

els). For the chiral interactions, the param eters o f the 4NF are com pletely fixed by the N N interaction. For the p h e­

n om enological ones, the strength o f the contact pieces w as fixed arbitrarily to Ct = 10 G eV -2 . T his choice is below the naturalness estim ate and is m eant to take W igner sy m ­ m etry into account. For com pleteness, w e also show the sum o f all contributions.

T he phenom enological and LO estim ates tend to be larger than the ones for N LO and N 2LO interactions. In b oth cases, w e observe that class IV contributions are larger

(7)

Fig. 11. Expectation values of the 4NF for various chiral and phe­

nomenological interactions. Contributions of different classes are shown separately. The width of the bars indicates the dependence on the cutoff for the chiral interactions and the band spanned by AV18 and CD-Bonn for the phenomenological interactions.

than expected by the size o f Ct. We also note that the CD- B onn and AV18 results are close to each other, although the LO results are strongly cutoff dependent. T his is an unusually behavior, since g enerally cutoff dependence for LO results show s up as a strong m odel dependence for p h e­

nom enological calculations. It has to been seen in future, w h eth er this dependence can be traced back to the cutoff dependence o f the binding energy.

N L O and N 2LO results are m ore interesting since for these the N N interactions are strictly consistent w ith the 4N F and the binding energy is already described reaso n ­ ably. T he results for these interactions are sm aller. A s ex­

pected, the class IV, V and V II contributions are suppressed because o f W igner sym m etry. Individually, the class I and class II contributions are o f the order o f 500 keV. The sum o f both is sm aller (around 300 keV) since b oth parts can­

cel each other partly. We also note that for som e cutoffs the 4N F acts attractively or repulsively. We again stress that the potential is not observable. T herefore, w e cannot expect results to be independent o f the cutoff.

7 Conclusions and outlook

In sum m ary, w e have studied the 4N F contribution to the binding energy o f 4H e in the fram ew ork o f chiral p erturba­

tion theory. To this aim, w e m ade use o f a M C technique in m om entum space that enabled us to calculated the high dim ensional integrals required for the evaluation o f the ex­

pectation values. The schem e allow s one to generate the m ost com plex parts o f the code using M athem atica. This w ay, the expressions can be reliably transferred into our FO R TR A N codes.

By now it is clear that 3 N F ’s give im portant co n tri­

butions to the bin d in g energies o f nuclei. Based on the pow er counting, 4N F contributions m ight still be signifi­

cant. W e found by explicit calculation that the 4N F co n ­ tribution is som ew hat sm aller than the pow er counting es­

tim ate o f 1 M eV at least w h en the higher order chiral in­

teractions are used. T he individual contributions o f class IV to V II are suppressed due to W igner sym m etry, so that only class I and II contributions are non-negligible. For

4He these tw o classes cancel each other in parts so that the com plete contribution for this case is below or up to approxim ately 300 keV in m agntitude.

A lthough this w ould probably be considered as a neg­

ligible contribution, som e care has to be taken before final conclusion on 4 N F ’s can be m ade. Firstly, the p henom eno­

logical interactions tend to lead to larger 4NF. B ut m ost im portantly, the 4N F contribution could be larger for nu ­ clei w ith a different spin/isospin structure than 4He. In this case the class I and II contributions m ight add construc­

tively im plying a visible contribution o f 4 N F ’s. T his has to be studied in m ore detail in future.

This work was supported by the Polish Ministry of Science and Higher Education under Grants No. N N202 104536 and No. N N202 077435. It was also partially supported by the Helmholtz Association through funds provided to the virtual institute “Spin and strong QCD”(VH-VI-231) and and to the young investigator group “Few-Nucleon Systems in Chiral Effective Field Theory”

(grant VH-NG-222) and by the European Community-Research Infrastructure Integrating Activity “Study of Strongly Interacting Matter” (acronym HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of EU. The numerical calculations have been performed on the supercomputer cluster of the JSC, Jülich, Germany.

References

1. R.B. W iringa, V.G.J. Stoks, R. Schiavilla, Phys. Rev.

C 51, 38 (1 9 9 5 ),n u c l - t h / 9 4 0 8 0 1 6

2. R. M achleidt, Phys. Rev. C 63, 024001 (2001), n u c l - t h / 0 0 0 6 0 1 4

3. V.G.J. Stoks, R.A .M . K lom p, C.P.F. Terheggen, J.J. de Swart, Phys. Rev. C 49, 2950 (1994), n u c l - t h / 9 4 0 6 0 3 9

4. A. N ogga, H. K am ada, W. G lockle, Phys. Rev. Lett.

8 5 ,9 4 4 (2 0 0 0 ),n u c l - t h / 0 0 0 4 0 2 3

5. A. N ogga, H. K am ada, W. G lockle, B.R. B arrett, Phys.

Rev. C 65, 054003 (2 0 0 2 ),n u c l - t h / 0 1 1 2 0 2 6

6. M. Viviani, A. Kievsky, S. Rosati, Phys. Rev. C 71, 024006 (2005), n u c l - t h / 0 4 0 8 0 1 9

7. P. N avratil, B.R. B arrett, Phys. Rev. C 57, 3119 (1998), n u c l - t h / 9 8 0 4 0 1 4

8. S.C. Pieper, R.B. W iringa, A nn. Rev. N ucl. Part. Sci.

51, 53 (2001), n u c l - t h / 0 1 0 3 0 0 5

9. J. Fujita, H. M iyazaw a, Prog. Theor. Phys. 17, 360 (1957)

10. J. C arlson, V.R. P andharipande, R.B. W iringa, Nucl.

Phys. A 401, 59 (1983)

11. H.T. C oelho, T.K. Das, M .R. R obilotta, Phys. Rev.

C 28, 1812 (1983)

12. S.A. C oon et al., N ucl. Phys. A 317, 242 (1979) 13. D. P luem per, J. Flender, M.F. Gari, Phys. Rev. C 49,

2370 (1994)

14. J.A . Eden, M.F. Gari, Phys. Rev. C 53, 1510 (1996), n u c l - t h / 9 6 0 1 0 2 5

15. W.N. Polyzou, W. G lockle, Few B ody Syst. 9(2-3), 97 (1990), ISSN 0177-7963

(8)

16. H. W itała et al., Phys. Rev. C 63, 024007 (2001), n u c l - t h / 0 0 1 0 0 1 3

17. P. N avratil, W.E. O rm and, Phys. Rev. C6 8, 034305 (2003), n u c l - t h / 0 3 0 5 0 9 0

18. S. W einberg, Phys. Lett. B 251, 288 (1990) 19. S. W einberg, N ucl. Phys. B 363, 3 (1991)

20. D.R. Entem , R. M achleidt, Phys. Rev. C6 8, 041001 (2003), n u c l - t h / 0 3 0 4 0 1 8

2 1. E. E pelbaum , W. G lockle, U .G. M eißner, N ucl. Phys.

A 747, 362 (2 0 0 5 ),n u c l - t h / 0 4 0 5 0 4 8 22. U. van K olck, Phys. Rev. C 49, 2932 (1994)

23. E. E pelbaum , A. N ogga, W. G lockle, H. K am ada, U.G. M eißner, H. W itała, Phys. Rev. C6 6, 064001 (2002), n u c l - t h / 0 2 0 8 0 2 3

24. V. B ernard, E. E pelbaum , H. K rebs, U.G. M eißner, Phys. Rev. C 77, 064004 (2008), a r X i v : 0 7 1 2 . 1 9 6 7

[ n u c l - t h ]

25. E. E pelbaum , Phys. Lett. B 639, 456 (2006)

26. E. E pelbaum , Eur. Phys. J. A 34, 197 (2007), a r X i v : 0 7 1 0 . 4 2 5 0 [ n u c l - t h ]

27. D. R ozpedzik et al., A cta Phys. Polon. B 37, 2889 (2006), n u c l - t h / 0 6 0 6 0 1 7

28. H. M cM anus, D.O. Riska, Phys. Lett. B 92, 29 (1980) 29. M .R. R obilotta, Phys. Rev. C 31, 974 (1985)

30. C. O rdonez, L. Ray, U. van K olck, Phys. Rev. C 53, 2086 (1 9 9 6 ),h e p - p h /9 5 1 1 3 8 0

31. N. K aiser, S. G erstendorfer, W. W eise, N ucl. Phys.

A 637, 395 (1 9 9 8 ),n u c l - t h / 9 8 0 2 0 7 1

32. H. K rebs, E. E pelbaum , U.G. M eißner, Eur. Phys. J.

A 32, 127 (2 0 0 7 ),n u c l - t h / 0 7 0 3 0 8 7

33. E. E pelbaum , H. K rebs, U.G. M eißner, N ucl. Phys.

A 806, 65 (2008), a r X i v : 0 7 1 2 . 1 9 6 9 [ n u c l - t h ] 34. A. D eltuva, A.C. Fonseca, P.U. Sauer, Phys. Lett.

B 660, 471 ( 2 0 0 8 ),a r X i v : 0 8 0 1 . 2 7 4 3 [ n u c l - t h ] 35. P.F. B edaque, U. van K olck, Ann. Rev. N ucl. Part. Sci.

52, 339 (2002), n u c l - t h / 0 2 0 3 0 5 5

36. E. E pelbaum , H.W. H am m er, U.G. M eißner (2008), (accepted for publication in Rev. M od. Phys.), a r X i v : 0 8 1 1 . 1 3 3 8 [ n u c l - t h ]

37. E. E pelbaum , Prog. Part. N ucl. Phys. 57, 654 (2006), n u c l - t h / 0 5 0 9 0 3 2

38. A. N ogga, R.G .E. T im m erm ans, U. van Kolck, Phys.

Rev. C 72, 054006 (2005), n u c l - t h / 0 5 0 6 0 0 5 39. E. E pelbaum , U .G. M eißner, W. G lockle, C. Elster,

Phys. Rev. C 65, 044001 (2002), n u c l - t h / 0 1 0 6 0 0 7 40. B.S. Pudliner, V.R. P andharipande, J. C arlson, S.C.

Pieper, R.B. W iringa, Phys. Rev. C 56, 1720 (1997), n u c l - t h / 9 7 0 5 0 0 9

41. N. M etropolis, A.W. R osenbluth, M .N. R osenbluth, A .H . Teller, E. Teller, J. Chem . Phys. 21, 1087 (1953) 42. R.B. W iringa, R. Schiavilla, S.C. Pieper, J. C arlson,

Phys. Rev. C 78, 021001 (2008), a r X i v : 0 8 0 6 . 1 7 1 8 [ n u c l - t h ]

43. A. N ogga, S.K. Bogner, A. S chw enk, Phys. Rev. C 70, 061002 (2 0 0 4 ),n u c l - t h / 0 4 0 5 0 1 6

44. A. B om elburg, Phys. Rev. C 34, 14 (1986)

Cytaty

Powiązane dokumenty

Ex- plosive mixtures of dust and air may form during transport (e.g. in bucket elevators) and during the storage of raw mate- rials such as cereals, sugar and flour. An explosion

After a jump, the potential function forces the price to return rapidly toward its average level with the higher reversal rate (a so-called fast relaxation process) than the

Do podstawowych narzędzi polityk prowadzowanych przez uczelnie wyż- sze w celu zwiększenia zatrudnialności swych studentów oraz ,,wygładzania” procesu przejścia

Wyraźny wzrost wielkości dochodów gminnych jednostek samorządu terytorialnego w Polsce był przejawem pozytywnych zmian zachodzących w gospodarce i społeczeństwie,

W związku z deklaracjami Prezesa UOKiK, wpro- wadzeniem programu dla sygnalistów i wejściem w życie ustawy o roszczeniach związanych z na- ruszeniem prawa konkurencji

I Ogólnopolskie Forum Dyrekto- rów Archiwów Diecezjalnych w Polsce odbyło się w Poznaniu 20-21 paździer- nika 2014 roku, II Ogólnopolskie Forum Dyrektorów Archiwów Diecezjalnych

The ninth volume of Logopedia Silesiana presents a new and unified structure of the journal, now including both theoretical articles (be it reviews or original contributions) and

Wspomnienia zaproszonych księży zostały wcześniej tak ukierunkowane, aby z nich wyłoniła się całość dziejów Wydziału Teologicznego i Seminarium Duchow­ nego w