The nucleon-nucleon interaction

Transcription

1 TOPICAL REVIEW The ucleo-ucleo iteractio R. Machleidt ad I. Slaus Deartmet of Physics, Uiversity of Idaho, Moscow, Idaho 83844, U. S. A. Triagle Uiversities Nuclear Laboratory (TUNL), Duke Statio, Durham, North Carolia 27706, U. S. A. ad Rudjer Boskovic Istitute, Zagreb, Croatia Abstract. We review the major rogress of the ast decade cocerig our uderstadig of the ucleo-ucleo iteractio. The focus is o the low-eergy regio (below io roductio threshold), but a brief outlook towards higher eergies is also give. The items discussed iclude charge-deedece, the recise value of the πnn coulig costat, hase shift aalysis ad high-recisio NN data ad otetials. We also address the issue of a roer theory of uclear forces. Fially, we summarize the essetial oe questios that future research should be devoted to. Submitted to: J. Phys. G: Nucl. Part. Phys. 1. Itroductio The uclear force has bee at the heart of uclear hysics ever sice the field was bor i 1932 with the discovery of the eutro by Chadwick [1]. I fact, durig the first few decades of uclear hysics, the term uclear forces was ofte used as syoymous for uclear hysics as a whole [2]. There are good reasos why the uclear force lays such a outstadig role. The iteractio betwee two ucleos is basic for all of uclear hysics. The traditioal goal of uclear hysics is to uderstad the roerties of atomic uclei i terms of the bare iteractio betwee airs of ucleos. With the oset of quatumchromodyamics (QCD), it became clear that the ucleo-ucleo (NN) iteractio is ot fudametal. Nevertheless, eve today, i ay first aroach towards a uclear structure roblem, oe assumes the ucleos to be elemetary articles. The failure or success of this aroach may the teach us somethig about the relevace of subuclear degrees of freedom. The NN iteractio has bee ivestigated by a large umber of hysicists all over the world for the ast 70 years. It is the emirically best kow iece of strog iteractios; i fact, for o other samle of the strog force a comarable amout of exerimetal data has bee accumulated. The oldest attemt to exlai the ature of the uclear force is due to Yukawa [3]. Accordig to his theory, massive bosos (mesos) mediate the iteractio betwee two ucleos. This idea sawed the sister discilie of article hysics. Although, i the light of QCD, meso theory is ot erceived as fudametal aymore, the meso exchage cocet cotiues to rereset the best workig model for a quatitative ucleo-ucleo otetial.

2 Nucleo-ucleo iteractio 2 Historically, it tured out to be a formidable task to describe the uclear force just heomeologically, ad it took a quarter cetury to come u with the first semiquatitative model [4] i Ever sice, there has bee substatial rogress i exerimet ad theory of the uclear force. Most basic questios were settled i the 1960 s ad 70 s such that i recet years we could cocetrate o the subtleties of this eculiar force. I this toical review, we will reort the chief rogress of the ast decade. The focus will be o the low-eergy regio (below io roductio threshold). Summaries of earlier eriods ad a edagogical itroductio ito the field ca be foud i refereces [5, 6]. I the 1990 s, major issues cocerig the NN iteractio have bee: charge-deedece, the recise value of the πnn coulig costat, imroved hase shift aalysis, high-recisio NN data, high-recisio NN otetials, QCD ad the uclear force, NN scatterig at itermediate ad high eergies. We will ow review these toics oe by oe. 2. Charge deedece By defiitio, charge ideedece is ivariace uder ay rotatio i isosi sace. A violatio of this symmetry is referred to as charge deedece or charge ideedece breakig (CIB). Charge symmetry is ivariace uder a rotatio by about the y-axis i isosi sace if the ositive z-directio is associated with the ositive charge. The violatio of this symmetry is kow as charge symmetry breakig (CSB). Obviously, CSB is a secial case of charge deedece. CIB of the strog NN iteractio meas that, i the isosi T =1state,the roto-roto (T z = +1), eutro-roto (T z = 0), or eutro-eutro (T z = 1) iteractios are (slightly) differet, after electromagetic effects have bee removed. CSB of the NN iteractio refers to a differece betwee roto-roto () ad eutro-eutro () iteractios, oly. The charge deedece of the NN iteractio is subtle, but i the 1 S 0 state it is well established. The observatio of small chargedeedet effects i this state is ossible because the scatterig legth of a almost boud state acts like a owerful magifyig glass o the iteractio. The curret uderstadig is that o a fudametal level the charge deedece of uclear forces is due to a differece betwee the u ad dow quark masses ad electromagetic iteractios amog the quarks. A cosequece of this are mass differeces betwee hadros of the same isosi multilet ad meso mixig. Therefore, if CIB is calculated based uo hadroic models, the mass differeces betwee hadros of the same isosi multilet, meso mixig, ad irreducible mesohoto exchages are cosidered as major causes. For reviews o charge deedece, see refereces [7, 8, 9]. We will ow summarize recet develomets (that are ot cotaied i ay of these reviews).

3 Nucleo-ucleo iteractio Charge symmetry breakig Exerimet. As discussed, the scatterig legths i the 1 S 0 state for,, ad scatterig (deoted by a, a,ada, resectively) are the best evidece for the charge-deedece of uclear forces. While we have well-established values for a ad a sice may decades, the eutro-eutro scatterig legth cotiues to be a tough roblem. The basic reaso for this is that, so far, we have ot bee able to coduct ay direct measuremets of a usig free eutro-eutro collisios [10]. All curret values are extracted from multi-article reactios the aalyses of which are beset with large theoretical ucertaities. The rocesses that are believed to have the smallest ucertaities are µ + d ν µ + +, (1) π + d γ + +, (2) + d + +. (3) While, there are o data o the first reactio, the other two rocesses have bee studied reeatedly. I 1998, a very carefull study of the π iduced reactio was ublished [11] ad, i 1999, a reewed thorough ivestigatio of the eutro iduced rocess was accomlished [12], yieldig results that are i erfect agreemet, amely, D(π,γ) [11] : a = ± 0.53 fm, (4) D(, ) [12] : a = 18.7 ± 0.6 fm, (5) which ca be summarised by a = 18.6 ± 0.4fm. (6) Correctig for the eutro-eutro magetic iteractio, the ure uclear value is: a N = 18.9 ± 0.4fm. (7) This summarizes the status by the ed of Ufortuately, this is ot the hay ed of the story that everybody had hoed for. To roerly discuss the ew (ad old) roblems, we will first rovide more details cocerig the two tyes of reactios for which exerimets have bee coducted. Over the ast 20 years, there have bee three ideedet studies of the reactio π + d γ + +. I oe case [13], oly the γ sectrum was measured, while i the other two cases [14, 11], kiematically comlete exerimets were erformed measurig the γ ad a eutro i the fial state. The results are: a = ± 0.34 (stat.) ± 0.26 (syst.) ± 0.30 (theor.) fm = ± 0.52 fm [13], (8) a = ± 0.42 (stat.) ± 0.39 (syst.) ± 0.30 (theor.) fm = ± 0.65 fm [14], (9) a = ± 0.05 (stat.) ± 0.44 (syst.) ± 0.30 (theor.) fm = ± 0.53 fm [11]. (10) Owig to the high satial resolutio of the gamma ray detector i referece [11], it was ossible to assess the systematic errors due to ucertaities i the modellig of the stoed io distributio i the target ad i target vertex recostructio i the Mote Carlo simulatio. Therefore, the systematic ucertaities of the kiematically comlete studies are ow much better uderstood, ad a very high statistical accuracy

4 Nucleo-ucleo iteractio 4 i referece [11] makes the exerimetal ucertaity comarable to the theoretical oe i the extractio of a from the reactio D(π,γ). The combied result from all three studies gives the ew world average for the D(π,γ) reactio a = ± 0.27 (exer.) ± 0.30 (theor.) fm = ± 0.40 fm. (11) I summary, the reactio π + d γ + + aears to be i good shae. Ufortuately, we caot say the same about the eutro-iduced deutero breaku rocess. Util the recet ivestigatio by the TUNL grou, Gozalez-Trotter et al. [12], all studies of the reactio + d + + gave for a values that differed from that obtaied from the D(π,γ) rocess. Theoretical ucertaities i extractig a from the eutro-iduced deutero breaku are much larger, as we will exlai ow. First, i reactios with more tha two ucleos i the fial state, three ucleo forces (3NF) modify the cross sectio. It was suggested [15] that the 3NF is the reaso why a extracted from the D(, ) rocess differs from that obtaied from the D(π,γ) reactio. The 3NF are a atural cosequece of strog iteractios. Therefore, 3NF do exist, but the questio is how sigificat they are ad, i articular, do they affect a secific cofiguratio of the d breaku wherefrom oe extracts a. There are several idicatios for ossible 3NF effects i uclear hysics: 3 H bidig eergy, uclear matter bidig eergy, 4 He bidig eergy ad egative arity excited states, 3 He ad 4 He oe-body desity distributios, bidig eergies ad radii of some uclei, 17 O magetic momet form factor, d cature, A y i elastic d scatterig, ad sace star, fial state iteractio (FSI) ad quasifree scatterig (QFS) cofiguratios i the d breaku [7, 16, 17]; but oe of them rovided coclusive iformatio o 3NF. It was ossible to recocile all values of a extracted from the studies of the reactio D(, ) i the eergy domai of 10 to 50 MeV with those obtaied from the D(π,γ) rocess usig the Fujita-Miyazawa 3NF [7]. However, the reaalyses of these d breaku rocesses [18] gave values that differed cosiderably from those quoted by the origial authors. Though ay reaalysis is clouded by the lack of all relevat iformatio, the mai reaso is the fact that origial aalyses used simle S- wave searable NN otetials, while the re-aalyses were doe usig the rigorous three body theory of Glöckle et al [16]. Obviously, the claimed theoretical ucertaities i the origial aers were uderestimated. Secod, the magetic iteractio modifies the value of the 1 S 0 scatterig legth extracted from the eutro iduced deutero breaku. It was show [19] that for the eutro-icku cofiguratio i the eutro-iduced deutero breaku leadig to the FSI there is a magetic iteractio i the 1 S 0 state which is reulsive thereby decreasig the absolute value of a. Deedig o the NN otetial (hard core or soft core), imulse aroximatio estimates of the effect of the magetic iteractio i the icku cofiguratio chages a from 18.5 fmto 17.2 oreve 16.4 fm [19]. The correctio for the kockout cofiguratio has the oosite sig sice it is domiated by the magetic iteractio betwee a eutro ad a roto i the 1 S 0 state. The situatio is more comlex for the eutro-roto FSI, sice the FSI occurs i the 1 S 0 ad 3 S 1 states. The determiatio of a by Gozales-Trotter et al [12] has two characteristic features: first, it uses the rigorous theory [16] icludig, i additio to several realistic NN otetials, also the Tucso-Melboure 3NF, ad secod, it erforms a highaccuracy comariso of eutro-eutro ad eutro-roto FSI i the 1 S 0 state by

5 Nucleo-ucleo iteractio 5 measurig cross sectios of the reactio D(, ) for idetical kiematic coditios (the agle of the two emitted ucleos iteractig i the 1 S 0 fial state is 28 to 43 deg) at the icidet eutro eergy of 13 MeV. Therefore, the eutro-roto scatterig legth, a, becomes the stadard for determiig a. By comarig the extracted value for a ad its ucertaity, it was ossible to set a uer limit of 0.2 ± 0.6 fm o ay ossible effects due to 3NF ifluecig the extracted value of a. Of course, it is ossible that the effect of the magetic iteractio discussed by Slobodria et al [19] are egligible i the eergy/agular regio studied by Gozalez-Trotter et al [12]. O the other had, it should be stressed that the rigorous calculatios by Glöckle et al [16] do ot iclude the electromagetic iteractio, ad that there are ow may idicatios of the shortcomigs of the Tucso-Melboure 3NF. I the year of 2000, a ew study of the eutro-eutro FSI i the D(, ) reactio at the icidet eutro eergy of 25.3 MeV was ublished by the eutro grou at Bo [20]. The data were aalyzed usig the rigorous theory [16]. The extracted value is, a = ± 0.4 fm, (12) which is i drastic disagreemet with the result of the TUNL grou [12] ublished i 1999 ad also with those obtaied from the D(π,γ) rocess. While most of the revious kiematically comlete studies of the reactio D(, ) emloy a thick, active deuterated target measurig the eergy of the roto, ad detectig two eutros at early the same agle o the same side of the icidet eutro, this recet measuremet [20] uses a thi deuterium target ad detects a eutro at Θ = 55.5 deg ad a roto at Θ =41.15 deg. The advatage of this geometry is the reductio of the strog cross talk betwee eutro detectors ad the reductio i losses from eutro multile scatterig. This geometrical cofiguratio has the added advatage that the locus cotais QFS besides FSI ad, therefore, rovides a built-i ormalizatio. Ideed, ormalizig the data to QFS yields a very similar value: a = ± 0.35 fm. (13) Neither the use of differet NN otetials or the iclusio of the Tucso-Melboure 3NF i the rigorous calculatio roduces oticeably differet results for a. This geometry at this icidet eergy is the regio where the 3NF effect of the Tucso- Melboure otetial is very small. The relimiary result by the Bo grou usig the same icidet eergy of 25.3 MeV gave a good fit to the FSI sectrum usig a = 24 fm. The disagreemet betwee the two most recet studies, Gozalez-Trotter et al (TUNL) [12] ad Huh et al (Bo) [20], oes agai the roblem of how comletely do we uderstad the iteractios ivolved i the three ucleo roblem, secifically the 3NF. It also suggests that additioal exerimetal studies at differet icidet eergies ad at differet agles might be useful i resolvig the roblem. Whe we use for a the value obtaied from the D(π,γ) studies [equatio (11)], correct it for the magetic momet iteractio [equatio (7)], ad comare it to the corresodig value [8]: a N = 17.3 ± 0.4 fm, (14) the charge-symmetry is broke by the followig amout, a CSB a N an =1.6 ± 0.6 fm. (15)

6 Nucleo-ucleo iteractio 6 Table 1. CSB differeces of the 1 S 0 effective rage arameters caused by ucleo mass slittig. 2π deotes the sum of all 2π-cotributios ad πρ the sum of all πρ-cotributios. TBE (o-iterative two-boso-exchage) is the sum of 2π, πρ, ad (πσ + πω). ki. e. OBE 2π πρ πσ + πω TBE Total Emirical a CSB (fm) ± 0.6 fm r CSB (fm) ± 0.12 fm Recommeded values for the corresodig effective rages are [8], r N =2.75 ± 0.11 fm, (16) r N =2.85 ± 0.04 fm, (17) imlyig r CSB r N rn =0.10 ± 0.12 fm. (18) Traditioally, it was believed that the meso mixig exlais essetially all CSB effects. The largest cotributio came from ρ ω mixig, ad there was very meager kowledge of π η mixig. Recetly, CSB was studied by the comariso of two charge-symmetric rocesses: D(π +,η) ad D(π,η) i the eergy regio of the η threshold. The result for the ratio of the two rocesses i this eergy regio is R = dσ + /dσ =0.937±0.007, after a hase sace correctio is made for the differece i the threshold eergies of the two reactios [21]. The deviatio of R from 1 is a idicatio of CSB which is mostly due to π η mixig. A heomeological fully relativistic model, which is based o couled chael Nπ Nη amlitudes, takes ito accout differet ad FSI ad exlicitly icludes π η mixig, was develoed [22] ad comared to the data yieldig for the π η mixig agle the value of (1.5 ± 0.4) deg, cosistet with the mixig agle determied from article decays ad isosiforbidde rocesses as well as with several other theoretical redictios [21] Theory. The differece betwee the masses of eutro ad roto reresets the most basic cause for CSB of the uclear force. Therefore, it is imortat to have a very thorough accoutig of this effect. The most trivial cosequece of ucleo mass slittig is a differece i the kietic eergies: for the heavier eutros, the kietic eergy is smaller tha for rotos. This raises the magitude of the scatterig legth by 0.25 fm as comared to. Besides the above, ucleo mass slittig has a imact o all meso-exchage diagrams that cotribute to the uclear force. I 1998, the most comrehesive ad thorough calculatio of these CSB effects ever coducted has bee ublished [23]. The ivestigatio is based uo the Bo Full Model for the NN iteractio [24]. Here, we will summarize the results. For this we devide the total umber of meso exchage diagrams that is ivolved i the uclear force ito several classes. Below, we reort the results for each class. (i) Oe-boso-exchage (OBE, figure 1) cotributios mediated by π 0 (135), ρ 0 (770), ω(782), a 0 /δ(980), ad σ (550). I the Bo Full Model [24], the σ describes oly the correlated 2π exchage i ππ S-wave (ad ot the ucorrelated 2π exchage sice the latter is calculated exlicitly, cf. figure 2). Charge-symmetry is broke by the fact that for scatterig the roto mass is

7 Nucleo-ucleo iteractio 7 ß 0 ; ff 0 ;ρ 0 ;!; ffi ß 0 ; ff 0 ;ρ 0 ;!; ffi (a) (b) Figure 1. scatterig. Oe-boso-exchage (OBE) cotributios to (a) ad (b) used i the Dirac siors reresetig the four exteral legs [figure 1(a)], while for scatterig the eutro mass is alied [figure 1(b)]. The CSB effect from the OBE diagrams is very small (cf. table 1). (ii) 2π-exchage diagrams. This class cosists of three grous; amely the diagrams with NN, N ad itermediate states, where refers to the baryo with si ad isosi 3 2 ad mass 1232 MeV. The most imortat grou is the oe with N itermediate states which we show i figure 2. Part (a) of figure 2 alies to scatterig, while art (b) refers to scatterig. Whe chargedio exchage is ivolved, the itermediate-state ucleo differs from that of the exteral legs. This is oe of the sources for CSB from this grou of diagrams. The 2π class of diagrams causes the largest CSB effect (cf. table 1 ad dashed curve i figure 3). (iii) πρ-exchages. Grahically, the πρ diagrams ca be obtaied by relacig i each 2π diagram (e. g., i figure 2) oe io by a ρ-meso of the same charge state. The effect is tyically oosite to the oe from 2π exchage. (iv) Further 3π ad 4π cotributios (πσ+ πω). The Bo otetial also icludes some 3π-exchages that ca be aroximated i terms of πσ diagrams ad 4πexchages of πω tye. The sum of the two grous is small, idicatig covergece of the diagramatic exasio. The CSB effect from this class is essetially egligible. The total CSB differece of the siglet scatterig legth caused by ucleo mass slittig amouts to 1.58 fm (cf. table 1) which agrees well with the emirical value 1.6 ± 0.6 fm. Thus, ucleo mass slittig aloe ca exlai the etire emirical CSB of the siglet scatterig legth [25]. This is a remarkable result. The imact of the various classes of diagrams o CSB hase shift differeces are show i figure 3. The total effect is the largest i the 1 S 0 state where it is most oticable at low eergy; e. g., at 1 MeV, the hase shift differece is 1.8 deg. The differece decreases with icreasig eergy ad is about 0.15 deg at 300 MeV, i 1 S 0. The CSB effect o the hase shifts of higher artial waves is small; i P ad D waves, tyically i the order of 0.1 deg at 300 MeV ad less at lower eergies. This fact may suggest that CSB i artial waves other tha L = 0 may be of o relevace.

8 Nucleo-ucleo iteractio 8 + ß 0 ß 0 ++ ß ± ß ß @ ß ß ± (a) 0 ß 0 ß 0 ß ± ß ß @ ß ß ± (b) Figure 2. Two-io-exchage cotributios with N itermediate states to (a) ad (b) scatterig. I refereces [26] it was show that this is ot true: CSB beyod the S waves is crucial for the exlaatio of the Nole-Schiffer aomaly. Before fiishig this subsectio, a word is i lace cocerig other mechaisms that cause CSB of the uclear force. Traditioally, it was believed that ρ 0 ω mixig exlais essetially all CSB i the uclear force [8]. However, recetly some doubt has bee cast o this aradigm. Some researchers [27, 28, 29, 30] foud that ρ 0 ω exchage may have a substatial q 2 deedece such as to cause this cotributio to early vaish i NN. Our fidig that the emirically kow CSB i the uclear force ca be exlaied solely from ucleo mass slittig (leavig essetially o room for additioal CSB cotributios from ρ 0 ω mixig or other sources) fits well ito this sceario. O the other had, Miller [9] ad Coo ad coworkers [31] have advaced couter-argumets that would restore the traditioal role of ρ-ω exchage. The issue is uresolved. Good summaries of the cotroversial oits of view ca be foud i refereces [9, 32, 33]. Fially, for reasos of comleteess, we metio that irreducible diagrams of π ad γ exchage betwee two ucleos create a charge-deedet uclear force. Recetly, these cotributios have bee calculated to leadig order i chiral erturbatio theory [34]. It turs out that to this order the πγ force is chargesymmetric (but does break charge ideedece).

9 Nucleo-ucleo iteractio 9 Figure 3. CSB hase shift differeces δ δ (without electromagetic iteractios) for laboratory kietic eergies T lab below 300 MeV ad artial waves with L 1. The CSB effects due to the kietic eergy, OBE, the etire 2π model, ad πρ exchages are show by the dotted, dash-trile-dot, dashed, ad dash-dot curves, resectively. The solid curve is the sum of all CSB effects Charge ideedece breakig The emirical values for the siglet effective rage arameters are [35]: a = ± fm, r =2.77 ± 0.05 fm. (19) It is useful to defie the followig averages: ā 1 2 (an + an )= 18.1 ± 0.6 fm, (20) r 1 2 (rn + rn )=2.80 ± 0.12 fm. (21) Igorig CSB, the CIB differeces i the effective rage arameters are give by: a CIB ā a =5.64 ± 0.60 fm, (22) r CIB r r =0.03 ± 0.13 fm. (23) The major cause of CIB i the NN iteractio is io mass slittig. Based uo the Bo Full Model for the NN iteractio [24], the CIB due to io mass slittig

10 Nucleo-ucleo iteractio 10 ß 0 ß 0 + ß ± (a) (b) Figure 4. scatterig. Oe-io exchage (OPE) cotributios to (a) ad (b) has bee calculated carefully ad systematically i referece [36]. We will discuss ow the various classes of diagrams ad their cotributios to CIB. (i) Oe-io-exchage (OPE). The CIB effect is created by relacig the diagram figure 4(a) by the two diagrams figure 4(b). The effect caused by this relacemet ca be uderstood as follows. I orelativistic aroximatio ad disregardig isosi factors, OPE is give by V 1π (g π,m π )= g2 π (σ 1 k)(σ 2 k) 4M 2 m 2 π + k 2 FπNN(Λ 2 πnn, k ) (24) with M the average ucleo mass, m π the io mass, ad k the mometum trasfer. The above exressio icludes a πnn vertex form-factor, F πnn, which deeds o the cutoff mass Λ πnn ad the magitude of the mometum trasfer k. ForS =0adT =1,whereS ad T deote the total si ad isosi of the two-ucleo system, resectively, we have 01 gπ 2 k 2 V 1π (g π,m π )= m 2 π + k2 4M 2 F πnn 2 (Λ πnn, k ), (25) where the suerscrits 01 refer to ST. Ithe 1 S 0 state, this otetial exressio is reulsive. The charge-deedet OPE is the, 01 V 1π = 01 V 1π (g π 0,m π 0) (26) For edagogical reasos, we use simle, aroximate exressios to discuss the effects from io exchage. Note, however, that i the calculatios of referece [36] relativistic time-ordered erturbatio theory is alied i its full comlexity ad without aroximatios. Table 2. CIB cotributios to the 1 S 0 scatterig legth, a CIB, ad effective rage, r CIB, from various comoets of the NN iteractio. OPE 2π πρ πσ + πω Total Emirical a CIB (fm) ± 0.60 r CIB (fm) ± 0.13

11 Nucleo-ucleo iteractio 11 for scatterig, ad 01 V 1π =201 V 1π (g π ±,m π ±) 01 V 1π (g π 0,m π 0) (27) for scatterig. If we assume charge-ideedece of g π (i. e., g π 0 = g π ±), the all CIB comes from the charge slittig of the io mass, which is [37] m π 0 = MeV, (28) m π ± = MeV. (29) Sice the io mass aears i the deomiator of OPE, the smaller π 0 -mass exchaged i scatterig geerates a larger (reulsive) otetial i the 1 S 0 state as comared to where also the heavier π ± -mass is ivolved. Moreover, the π 0 -exchage i scatterig carries a egative sig, which further weakes the OPE otetial. The bottom lie is that the otetial is more reulsive tha the otetial. The quatitative effect o a CIB is such that it exlais about 60% of the emirical value (cf. table 2). This has bee kow for a log time. Due to the small mass of the io, OPE is a sizable cotributio i all artial waves icludig higher artial waves; ad due to the io s relatively large mass slittig (3.4%), OPE creates relatively large charge-deedet effects i all artial waves (see dashed curve i figure 5). (ii) 2π-exchage diagrams. We ow tur to the CIB created by the 2π exchage cotributio to the NN iteractio. There are may diagrams that cotribute (see referece [36] for a comlete overview). For our qualitative discussio here, we ick the largest of all 2π diagrams, amely, the box diagrams with N itermediate states, figure 6. Disregardig isosi factors ad usig some drastic aroximatio, the amlitude for such a diagram is V 2π (g π,m π )= g4 π 72 16M 4 25 d 3 [σ ks k] 2 (2π) 3 (m 2 π + k2 ) 2 (E + E 2E q) FπNN(Λ 2 πnn, k ) FπN (Λ 2 πn, k ), (30) where k = q with q the relative mometum i the iitial ad fial state (for simlicity, we are cosiderig a diagoal matrix elemet); E = M ad E = M with M = 1232 MeV the -isobar mass; S is the si trasitio oerator betwee ucleo ad. For the πn coulig costat, f πn, the quark-model relatioshi fπn 2 = f πnn 2 is used [24]. For small mometum trasfers k, this attractive cotributio is roughly roortioal to m 4 π. Thusforthe2π exchage, the heavier ios will rovide less attractio tha the lighter oes. Charged ad eutral io exchages occur for as well as for, ad it is imortat to take the isosi factors carried by the various diagrams ito accout. They are give i figure 6 below each diagram. For scatterig, the diagram with double π ± exchage carries the largest factor, while double π ± exchage carries oly a small relative weight i scatterig. Cosequetly, scatterig is less attractive tha scatterig which leads to a icrease of a CIB by 0.79 fm due to the diagrams of figure 6. The crossed diagrams of this tye reduce this result ad icludig all 2π exchage diagrams oe fids a total effect of 0.36 fm [36]. (iii) πρ-exchages. This grou is, i ricile, as comrehesive as the 2π-exchages discussed above. Grahically, the πρ diagrams ca be obtaied by relacig i

12 Nucleo-ucleo iteractio 12 Figure 5. CIB hase shift differeces δ δ [with δ (δ + δ )/2] for laboratory kietic eergies T lab below 300 MeV ad artial waves with orbital agular mometum L 1. The CIB effects due to OPE, the etire 2π model, πρ exchages, ad (πσ + πω) cotributios are show by the dashed, dash-dot, dash-trile-dot, ad dotted curves, resectively. The solid curve is the sum of all CIB effects. each 2π-diagram oe of the two ios by a ρ-meso of the same charge-state. This cotributio to CIB (dash-trile-dot curve i figure 5) is geerally small, ad (i most states) oosite to the oe from 2π. (iv) Further 3π ad 4π cotributios (πσ+ πω). As discussed, the Bo otetial also icludes some 3π-exchages that ca be aroximated i terms of πσ diagrams ad 4π-exchages of πω tye. These diagrams carry the same isosi factors as OPE. The CIB effect from this class is very small, excet i 1 S 0 (dotted curve i figure 5). Notice that this effect has always the same sig as the effect from OPE (dashed curve), but it is substatially smaller. The reaso for the OPE character of this cotributio is that πσ revails over πω ad, thus, determies the character of this cotributio. Sice sigma-exchage is egative ad sice, futhermore, the roagator i betwee the π ad the σ exchage is also egative, the overall sig of the πσ exchage is the same as OPE. Thus, it is like a shortraged OPE cotributio.

13 Nucleo-ucleo iteractio 13 ß 0 ß ± + ß 0 ++ ß ± (a) ß 0 ß 0 ß ± ß ± 0 ß 0 + ß ± 0 ß 0 + ß ± (b) Figure 6. 2π-exchage box diagrams with N itermediate states that cotribute to (a) ad (b) scatterig. The umbers below the diagrams are the isosi factors. Cocerig the siglet scatterig legth, the CIB cotributios discussed exlai about 80% of a CIB (cf. table 2). Ericso ad Miller [38] arrived at a very similar result usig the meso-exchage model of Partovi ad Lomo [39]. The sum of all CIB effects o hase shifts is show by the solid curve i figure 5. Notice that the differece betwee the solid curve ad the dashed curve (OPE) i that figure reresets the sum of all effects beyod OPE. Thus, it is clearly see that OPE domiates the CIB effect i all artial waves, eve though there are substatial cotributios besides OPE i some states, otably 1 S 0 ad 3 P 1. I referece [36], also the effect of rho-mass slittig o the 1 S 0 effective rage arameters was ivestigated. Ufortuately, the evidece for rho-mass slittig is very ucertai, with the Particle Data Grou [37] reortig m ρ 0 m ρ ± =0.4 ± 0.8 MeV. Cosistet with this, m ρ 0 = 769 MeV ad m ρ ± = 768 MeV, i. e., a slittig of 1 MeV was assumed, i the exloratory study of referece [36]. With this, oe fids a CIB = 0.29 fm from oe-rho-exchage, ad a CIB =0.28 fm from the o-iterative πρ diagrams with NN itermediate states. Thus, idividual effects are small ad, i additio, there are substatial cacellatios betwee the two classes of diagrams that cotribute. The et result is a vaishig effect. Thus, eve if the rho-mass slittig will be better kow oe day, it will ever be a great source of CIB. Aother CIB cotributio to the uclear force is irreducible io-hoto (πγ) exchage. Traditioally, it was believed that this cotributio would take care of the remaiig 20% of a CIB [38, 40, 41]. However, a recetly derived πγ otetial based

14 Nucleo-ucleo iteractio 14 uo chiral erturbatio theory [34] decreases a CIB by about 0.5 fm, makig the discreacy eve larger. Thus, it is a matter of fact that about 25% of the charge-deedece of the siglet scatterig legth is ot exlaied at this time. 3. The πnn coulig costat For the uclear force, the io is the most imortat meso. Therefore, it is crucial to have a accurate uderstadig of the coulig of the io to the ucleo. I the 1990 s, we have see a cotroversial discussio about the recise value for the πnn coulig costat. We will first briefly review the evets ad the discuss i which way the NN data imose costraits o this imortat coulig costat. From 1973 to 1987, there was a cosesus that the πnn coulig costat is gπ 2/4π =14.3 ± 0.2 (equivalet to f π 2 =0.079 ± ). This value was obtaied by Bugg et al. [43] from the aalysis of π ± data i 1973, ad cofirmed by Koch ad Pietarie [44] i Aroud that same time, the eutral-io coulig costat was determied by Kroll [45] from the aalysis of data by meas of forward disersio relatios; he obtaied gπ 2 /4π = ± 0.40 (equivalet to 0 fπ 2 =0.080 ± 0.002). 0 The icture chaged i 1987, whe the Nijmege grou [46] determied the eutral-io coulig costat i a artial-wave aalysis of data ad obtaied gπ 2 /4π =13.1 ± 0.1. Icludig also the magetic momet iteractio betwee rotos 0 i the aalysis, the value shifted to ± 0.13 i 1990 [47]. Triggered by these evets, Ardt et al. [48] reaalysed the π ± data to determie the charged-io coulig costat ad obtaied gπ 2 /4π =13.31 ± I subsequet work, the ± Nijmege grou also aalysed,, adπn data [49]. The status of their work as of 1993 is summarized i Ref. [50] where they claim that the most accurate values are obtaied i their combied ad aalysis yieldig gπ 2 /4π =13.47 ± (equivalet to fπ 2 = ± ) ad g 2 0 π /4π = ± 0.05 (equivalet to ± fπ 2 = ± ). The latest aalysis of all π ± data below 2.1 GeV coducted ± by the VPI grou usig fixed-t ad forward disersio relatio costraits has geerated gπ 2 /4π =13.75 ± 0.15 [51]. The VPI NN aalysis extracted g 2 ± π /4π ad gπ 2 /4π 13.9 as well as the charge-ideedet value g 2 ± π /4π 13.7 [52, 53]. Also Bugg ad coworkers have erformed ew determiatios of the πnn coulig costat. Based uo recise π ± data i the MeV rage ad alyig fixed-t disersio relatios, they obtaied the value gπ 2 /4π =13.96 ± 0.25 ± (equivalet to fπ 2 = ± ) [54]. From the aalysis of NN elastic data ± betwee 210 ad 800 MeV, Bugg ad Machleidt [55] have deduced gπ 2 /4π = ± Usig πnn Lagragias as defied i the authoritative review [42], the relevat relatioshis betwee the seudoscalar io coulig costat, g π, ad the seudovector oe, f π,are ( ) g 2 2 π 0 4π = 2M f 2 m π 0 = f 2 π 0 (31) π ± ad ( ) g 2 2 π ± 4π = M + M f 2 m π π ± ± = f 2 π ±. (32) with M = MeV the roto mass, M = MeV the eutro mass, ad m π ± = MeV the mass of the charged io.

15 Nucleo-ucleo iteractio 15 Table 3. Imortat coulig costats ad the redictios for the deutero ad some hase shifts for five models discussed i the text. A B C D E Emirical Imortat coulig costats g 2 π0/4π g 2 π ±/4π κ ρ The deutero Q (fm 2 ) (2) a η (4) b A S (fm 1/2 ) (8) c P D (%) P 0 hase shifts (deg) 10 MeV (17) d 25 MeV (53) d 50 MeV (9) d a Corrected for meso-exchage currets ad relativity. b Referece [62]. c Referece [63]. d Nijmege multi-eergy hase shift aalysis [64] ± 0.39 ad gπ 2 /4π =13.94 ± Thus, it may aear that recet determiatios show a cosistet tred towards a lower value for g π with o idicatio for substatial charge deedece. However, this is ot true ad for a comrehesive overview of recet determiatios of the πnn coulig costat, see referece [56]. I articular, there is oe determiatio that does ot follow the above tred. Usig a modified Chew extraolatio rocedure, the Usala Neutro Research Grou has deduced the charged-io coulig costat from high recisio charge-exchage data at 162 MeV [57]. Their latest result is gπ 2 /4π =14.52 ± 0.26 [58]. We ote that the ± method used by the Usala Grou is cotroversial [59, 60]. Sice the io lays a crucial role i the creatio of the uclear force, may NN observables are sesitive to the πnn coulig costat, g π. We will discuss here the most romiet cases ad their imlicatios for a accurate value of g π. We will focus o the deutero, NN aalyzig owers A y, ad the siglet scatterig legth. Other NN observables with sesitivity to g π are si trasfer coefficiets. Cocerig the latter ad their imlicatios for g π, we refer the iterested reader to refereces [55, 61] The deutero The crucial deutero observables to cosider are the quadruole momet, Q, adthe asymtotic D/S state ratio, η. The sesitivity of both quatities to g π is demostrated i table 3. The calculatios are based uo the CD-Bo otetial [65, 66]) which belogs to the ew geeratio of high-recisio NN otetials that fit the NN data below 350 MeV with a erfect χ 2 /datum of about oe. The umbers i table 3 are a udate of earlier calculatios of this kid [67, 68] i which older NN otetials were alied. However, there are o substatial differeces i the results as comared to the earlier ivestigatios.

16 Nucleo-ucleo iteractio 16 Figure 7. 3 P 0 hase shifts of roto-roto scatterig as redicted by Model A ad E (g 2 π /4π =13.6, solid lie), B 0 (κρ =3.7, dash-3dot), C (g2 π0 /4π =14.0, dashed), ad D (g 2 π0/4π = 14.4, dash-dot). The solid dots rereset the Nijmege multi-eergy hase shift aalysis [64]. For meaigful redictios, it is imortat that all deutero models cosidered are realistic. This requires that besides the deutero bidig eergy (that is accurately reroduced by all models of table 3) also other emirically well-kow quatities are correctly redicted, like the deutero radius, r d, ad the trilet effective rage arameters, a t ad r t. As it turs out, the latter quatities are closely related to the asymtotic S-state of the deutero, A S, which itself is ot a observable. However, it has bee show [63] that for realistic values of r d, a t,adr t, the asymtotic S-state of the deutero comes out to be i the rage A S = ± fm 1/2. Thus, A S lays the role of a imortat cotrol umber that tells us if a deutero model is realistic or ot. As ca be see from table 3, all our models ass the test. Model A of table 3 uses the curretly fashioable value for the πnn coulig costat g 2 π/4π =13.6 which clearly uderredicts Q while η is redicted satisfactorily. Oe could ow try to fix the roblem with Q by usig a weaker ρ-meso tesorcoulig to the ucleo, f ρ. It is customary to state the stregth of this coulig i terms of the tesor-to-vector ratio of the ρ coulig costats, κ ρ f ρ /g ρ. Model A uses the large value κ ρ = 6.1 recommeded by Hoehler ad Pietarie [69]. Alteratively, oe may try the value imlied by the vector-meso domiace model for the electromagetic form factor of the ucleo [70] which is κ ρ =3.7. This is doe i our Model B which shows the desired imrovemet of Q. However, a realistic model for the NN iteractio must ot oly describe the deutero but also NN scatterig. As discussed i detail i referece [71], the small κ ρ caot reroduce the ɛ 1 mixig arameter correctly ad, i additio, there are serious roblems with the 3 P J hase shifts, articularly, the 3 P 0 (cf. lower art of table 3 ad figure 7). Therefore, Model B is urealistic ad must be discarded. The oly arameters left to imrove Q are g π ad the πnn vertex form-factor, F πnn (cf. equatio 24, above). As for the ρ meso, F πnn is heavily costraied by NN hase arameters, articularly, ɛ 1. The accurate reroductio of ɛ 1 as determied i the Nijmege multi-eergy hase shift aalysis [64] essetially leaves o room

17 Nucleo-ucleo iteractio 17 Table 4. χ 2 /datum for the fit of the world A y data below 350 MeV (subdivided ito three eergy rages) usig differet values of the πnn coulig costat. Coulig costat g 2 π 0 /4π Eergy rage (# of data) A C D 0 17 MeV (45 data) MeV (148 data) MeV (624 data) for variatios of F πnn oce the ρ meso arameters are fixed. Thus, we are fially left with oly oe arameter to fix the Q roblem, amely g π. As it turs out, for relatively small chages of gπ 2 /4π there is a liear relatioshi, as demostrated i table 3 by the redictios of Model A, C ad D which use gπ 2 /4π =13.6, 14.0, ad 14.4, resectively. Cosistet with earlier studies [67, 68], oe fids that gπ/4π is eeded to correctly reroduce Q. However, a io coulig with gπ/4π creates roblems for the 3 P 0 hase shifts which are redicted too large at low eergy (cf. lower art of table 3 ad figure 7). Now, a oe-boso-exchage (OBE) model for the NN iteractio icludes several arameters (about oe doze i total). Oe may therefore try to imrove the 3 P 0 by readjustig some of the other model arameters. The vector mesos (ρ ad ω) have a strog imact o the 3 P 0 (ad the other P waves). However, due to their heavy masses, they are more effective at high eergies tha at low oes. Therefore, ρ ad ω may roduce large chages of the 3 P 0 hase shifts i the rage MeV, with little imrovemet at low eergies. The bottom lie is that i site of the large umber of arameters i the model, there is o way to fix the 3 P 0 hase shift at low eergies. I this articular artial wave, the io coulig costat is the oly effective arameter, at eergies below 100 MeV. The hase shifts of the Nijmege aalysis [64] as well as the hases roduced by the VPI grou [72] require gπ 2 /4π Notice that this fidig is i clear cotradictio to our coclusio from the deutero Q. There aears to be a way to resolve this roblem. Oe may assume that the eutral io, π 0, coules to the ucleo with a slightly differet stregth tha the charged ios, π ±. This assumtio of a charge-slittig of the πnn coulig costat is made i our Model E where we use gπ 2 /4π =13.6 adg 2 0 π /4π =14.4. ± This combiatio reroduces the 3 P 0 hase shifts at low eergy well ad creates a sufficietly large deutero Q Aalyzig owers I our above cosideratios, some hase shifts layed a imortat role. I ricile, hase shifts are othig else but a alterative reresetatio of data. Thus, oe may as well use the data directly. Sice the days of Gammel ad Thaler [4], it is well-kow that the trilet P -wave hase shifts are fixed essetially by the NN aalyzig owers, A y. Therefore, we will ow take a look at A y data ad comare them directly with model redictios. I figure 8, we show high-recisio A y data at 9.85 MeV from Wiscosi [73]. The theoretical curves show are obtaied with gπ 2 /4π =13.2 (dotted), 13.6 (solid), 0

18 Nucleo-ucleo iteractio 18 Figure 8. The roto-roto aalyzig ower A y at 9.85 MeV. The theoretical curves are calculated with g 2 π 0 /4π =13.2 (dotted), 13.6 (solid, Model A), ad 14.4 (dash-dot, Model D) ad fit the data with a χ 2 /datum of 0.98, 2.02, ad 9.05, resectively. The solid dots rereset the data take at Wiscosi [73]. ad 14.4 (dash-dot) ad fit the data with a χ 2 /datum of 0.98, 2.02, ad 9.05, resectively. Clearly, a small coulig costat aroud 13.2 is favored. Sice a sigle data set is ot a firm basis, we have looked ito all A y data i the eergy rage MeV. Our results are reseted i table 4 where we give the χ 2 /datum for the fit of the world A y data below 350 MeV (subdivided ito three eergy rages) for various choices of the eutral πnn coulig costat. It is see that the A y data at low eergy, articularly i the eergy rage 0 17 MeV, are very sesitive to the πnn coulig costat. A value g 2 π 0 /4π 13.6 is clearly referred, cosistet with what we extracted from the sigle data set at 9.85 MeV as well as from the 3 P 0 hase shifts. Next, we look ito the A y data. A sigle samle is show i figure 9, the A y data at 12 MeV from TUNL [74]. Predictios are show for Model A (solid lie), D (dash-dot), ad E (dash-trile-dot). The charge-slittig Model E fits the data best with a χ 2 /datum of 1.00 (cf. table 5). We have also cosidered the etire A y data measured by the TUNL grou [74] i the eergy rage MeV (31 data) as well as the world A y data i the eergy rages 0 17 MeV (120 data). It is see that there is some sesitivity to the πnn coulig costat i this eergy rage, while there is little sesitivity at eergies above 17 MeV (cf. table 5). Cosistet with the tred see i the 12 MeV data, the larger data sets below 17 MeV show a clear referece for a coulig costat aroud 14.4 if there is o charge slittig of g π. This imlies that without charge-slittig it is imossible to obtai a otimal fit of the ad A y data. To achieve this best fit, charge-slittig is eeded, like g 2 π 0 /4π =13.6 adg 2 π ± /4π =14.0, as cosidered i colum 5 of table 5. The drastic charge-slittig of Model E is ot favored by the more comrehesive A y data sets. The balace of the aalysis of the ad A y data the is: g 2 π 0 /4π 13.6 ad g 2 π ± /4π Notice that this slittig is cosistet with our coclusios from the deutero. Thus, we have ow some idicatios for charge-slittig of g π from two

19 Nucleo-ucleo iteractio 19 Figure 9. The eutro-roto aalyzig ower A y at 12 MeV. The theoretical curves are calculated with g 2 π 0/4π = g2 π ±/4π = 13.6 (solid lie, Model A), g 2 π 0/4π = g2 π ±/4π =14.4 (dash-dot, Model D), ad the charge-slittig g2 π 0/4π = 13.6, g 2 π ±/4π =14.4 (dash-3dot, Model E). The solid dots rereset the data take at TUNL [74]. very differet observables, amely the deutero quadruole momet ad aalyzig owers. Therefore, it is worthwhile to look deeer ito the issue of charge-slittig of the πnn coulig costat. Ufortuately, there are severe roblems with ay substatial charge-slittig for two reasos. First, theoretical work [77] o isosi symmetry breakig of the πnn coulig costat based uo QCD sum rules comes u with a slittig of less tha 0.5% for gπ 2 ad, thus, caot exlai the large charge slittig idicated above. Secod, a roblem occurs with the covetioal exlaatio of the charge-deedece of the siglet scatterig legth, which we will exlai ow. Table 5. χ 2 /datum for the fit of various sets of A y data usig differet values for the πnn coulig costats. Coulig costats g 2 π 0 /4π; g2 π± /4π Eergy, data set (# of data) 13.6; ; ; ; ; 14.4 A C D E 12 MeV [74] (9 data) MeV [74] (31 data) MeV world data (120) MeV [75] (85 data) MeV world data (416)

20 Nucleo-ucleo iteractio Charge-deedece of the siglet scatterig legth ad charge-deedece of the io coulig costat Here, we aregoig to showi detail how charge-slittigof the πnn coulig costat affects the charge-deedece of the 1 S 0 scatterig legth. It will tur out that the suggested charge-slittig of g π causes a disaster for our established uderstadig of the charge-deedece of the siglet scatterig legth. Our above cosideratios suggest charge-slittig of g π, like gπ 2 0/4π =13.6, (33) gπ 2 ±/4π =14.4, (34) cf. Model E of table 3. We will ow discuss how this charge-slittig of g π affects a CIB (more details ca be foud i the origial aer referece [76]). Accidetally, this slittig is i relative terms about the same as the io-mass slittig; that is gπ 0 g π ±. (35) m π 0 m π ± As discussed (cf. equatios (25) ad (30) ad text below these equatios), for zero mometum trasfer, we have roughly for oe-io exachage ( ) 2 gπ OPE (36) ad for 2π exchage TPE m π ( gπ m π ) 4, (37) which is ot uexected, ayhow. O the level of this qualitative discussio, we ca the redict that ay ioic charge-slittig satisfyig equatio (35) will create o CIB from io exchages. Cosequetly, a charge-slittig of g π as give i equatios (33) ad (34) will wie out our established exlaatio of CIB of the NN iteractio. I referece [76], accurate umerical calculatios based uo the Bo mesoexchage model for the NN iteractio [24] have bee coducted. The details of these calculatios are selled out i referece [36] where, however, o charge-slittig of g π was cosidered. Assumig the g π of equatios (33) ad (34), oe obtais the a CIB redictios give i the last colum of table 6. It is see that the results of a accurate calculatio go eve beyod what the qualitative estimate suggested: the covetioal CIB redictio is ot oly reduced, it is reversed. This is easily uderstood if oe recalls [cf. equatios (25) ad (30)] that the io mass aears i the roagator (m 2 π + k2 ) 1. Assumig a average k 2 m 2 π, the 7% charge slittig of m2 π will lead to oly about a 3% charge-deedet effect from the roagator. Thus, if a 6% charge-slittig of gπ 2 is used, this will ot oly override the io-mass effect, it will reverse it. Based uo this argumet ad o the umerical results, oe ca the estimate that a charge-slittig of gπ 2 of oly about 3% (e. g., gπ 2 /4π =13.6adg 2 0 π /4π =14.0) ± would erase all redictios of CIB i the siglet scatterig legth derived from io mass slittig. Besides io mass slittig, we do ot kow of ay other essetial mechaism to exlai the charge-deedece of the siglet scatterig legth. Therefore, it is ulikely that this mechaism is aihilated by a charge-slittig of g π. This may be take as a idicatio that there is o sigificat charge slittig of the πnn coulig costat.

21 Nucleo-ucleo iteractio 21 Table 6. Predictios for a CIB i uits of fm without ad with the assumtio of charge-deedece of g π. No charge-deedece of g π Charge-deedet g π: g 2 π 0/4π = g2 π ±/4π =14.4 g2 π0/4π =13.6 g 2 π ±/4π =14.4 1π π πρ, πσ, πω Sum Emirical 5.64 ± Coclusios Several NN observables ca be idetified that are very sesitive to the πnn coulig costat, g π. They all carry the otetial to determie g π with high recisio. I articular, we have show that the A y data below 17 MeV are very sesitive to g π ad imly a value gπ 2/4π The A y data below 17 MeV show moderate sesitivity ad the deutero quadruole momet shows great sesitivity to g π ;both observables imly gπ 2 /4π The two differet values may suggest a relatively large charge-slittig of g π. However, a charge-slittig of this kid would comletely destroy our established exlaatio of the charge-deedece of the siglet scatterig legth. Sice this is ulikely to be true, we must discard the ossibility of ay substatial charge-slittig of g π. The coclusio the is that we are faced with real ad substatial discreacies betwee the values for g π based uo differet NN observables. The reaso for this ca oly be that there are large, ukow systematic errors i the data ad/or large ucertaities i the theoretical methods. Our homework for the future is to fid these errors ad elimiate them. Aother way to summarize the curret cofused situatio is to state that, resetly, ay value betwee 13.2 ad 14.4 is ossible for gπ 2 /4π deedig o which NN observable you ick. If we wat to i dow the value more tightly, the we are faced with three ossible scearios: g π is small, gπ 2 /4π 13.6: The deutero η ad scatterig at low eergies are described well; there are moderate roblems with the A y data below 17 MeV. The most serious roblem is the deutero Q. Meso-exchage curret cotributios (MEC) ad relativistic correctios for Q of fm 2 or more would solve the roblem. Preset calculatios redict about fm 2 or less. A serious reivestigatio of this issue is called for. g π is large, gπ/4π : The deutero Q is well reroduced, but η is redicted too large as comared to the most recet measuremet by Rodig ad Kutse [62], η =0.0256(4). Note, however, that all earlier measuremets of η came u with a larger value; for examle, Borbely et al. [78] obtaied η =0.0273(5). There are o objectively verifiable reasos why the latter value should be less reliable tha the former oe.

22 Nucleo-ucleo iteractio 22 The deutero η carries the otetial of beig the best observable to determie g π (as oited out reeatedly by Ericso [63, 79] i the 1980 s); but the usettled exerimetal situatio soils it all. The A y data at low eergy are described well. The most serious roblem are the A y data below 100 MeV. g π is i the middle, 13.6 gπ/4π : we have all of the above roblems, but i moderate form. I coclusio, to arrive at a accurate value for g π, there is a lot of homework to do for theory ad exerimet. 4. Phase shift aalysis I site of the large NN database available i the 1990 s, covetioal hase shift aalyses are by o meas erfect. For examle, the hase shift solutios obtaied by Bugg [80] or the VPI/GWU grou [72] tyically have a χ 2 /datum of 1.3 or more, for the eergy rage MeV. This may be due to icosistecies i the data as well as deficiecies i the costraits alied i the aalysis. I ay case, it is a matter of fact that withi the covetioal hase shifts aalysis, i which the lower artial waves are essetially ucostraied, a better fit caot be achieved. About two decades ago, the Nijmege grou embarked o a rogram to substatially imrove NN hase shift aalysis. To achieve their goal, the Nijmege grou took two decisive measures [64]. First, they rued the database; i.e., they scaed very critically the world NN database (all data i the eergy rage MeV laboratory eergy ublished i a regular hysics joural betwee Jauary 1955 ad December 1992) ad elimiated all data that had either a imrobably high χ 2 (more tha three stadard deviatios off) or a imrobably low χ 2 ; of the 2078 world data below 350 MeV 1787 survived the sca, ad of the 3446 data 2514 survived. Secod, they itroduced sohisticated, semi-heomeological model assumtios ito the aalysis. Namely, for each of the lower artial waves (J 4) a differet eergydeedet otetial is adjusted to costrai the eergy-deedet aalysis. Phase shifts are obtaied usig these otetials i a Schroediger equatio. From these hase shifts the redictios for the observables are calculated icludig the χ 2 for the fit of the exerimetal data. This χ 2 is the miimized as a fuctio of the arameters of the artial-wave otetials. Thus, strictly seakig, the Nijmege aalysis is a otetial aalysis; the fial hase shifts are the oes redicted by the otimized artial-wave otetials. I the Nijmege aalysis, each artial-wave otetial cosists of a short- ad a log-rage art, with the searatio lie at r =1.4 fm. The log-rage otetial V L (r >1.4 fm) is made u of a electromagetic art V EM ad a uclear art V N : V L = V EM + V N (38) The electromagetic iteractio ca be writte as V EM () =V C + V VP + V MM () (39) for roto-roto scatterig ad V EM () =V MM () (40) for eutro-roto scatterig, where V C deotes a imroved Coulomb otetial (which takes ito accout the lowest-order relativistic correctios to the static