M. Kirchbach. Germany. L. Tiator. Abstract. The pseudoscalar and pseudovector N coupling constants are calculated

Transcription

1 ON THE COUPLING OF THE MESON TO THE NUCLEON M. Kirchbach Institut fur Kernphysik, TH Darstadt, D{64289 Darstadt, Gerany L. Tiator Institut fur Kernphysik, Universitat Mainz, D{5599 Mainz, Gerany Abstract The pseudoscalar and pseudovector N coupling constants are calculated fro an eective vertex associated with the a(98)n triangle diagra. The predicted values are in agreeent with the ones concluded fro tting photoproduction aplitudes. In this context we stress the iportance of the properties of the scalar eson octet for eson physics. Introduction In contrast to the N-interaction, little is known about the N-interaction and, consequently, about the NN vertex. In the case of pion scattering and pion photoproduction the NN coupling is preferred to be pseudovector (PV), in accord with current algebra results and chiral syetry. However, because the eta ass is so uch larger than the pion ass - leading to large SU(3) x SU(3) syetry breaking - and because of the ixing there is no copelling reason to select the PV rather than the PS for for the NN vertex. The uncertainty regarding the structure of the NN vertex extends to the agnitude of the coupling constant. This coupling constant gnn 2 =4 varies between and 7 with the large couplings arising fro ts of one boson exchange potentials. Typical values obtained in ts with OBEP potentials [] can lie anywhere between 3-7. However, including the yields only sall eects in tting the NN phase shifts and, furtherore, provides an insignicant contribution to nuclear binding at noral nuclear densities. Furtherore such OBEP potentials use the eta as an eective eson to describe eects of ore elaborate two-eson correlations. This can be seen in the full Bonn potential, where the eta coupling is below and can be neglected in the calculations [2, 3]. Fro SU(3) avor syetry all coupling constants between the eson octet and the baryon octet are deterined by one free paraeter, giving g 2 NN 4 = 3 (3 4)2 g2 NN 4 : () The resulting values for the coupling constant lie between :8 and :9 for coonly used values of between :6 :65 and depend on the F and D

2 strengths chosen as the two types of SU(3) octet eson-baryon couplings. Other deterinations of the NN coupling eploy reactions involving the eta, such as p!n, and range fro gnn 2 =4 =:6 :7 [4]. Saller values are supported by NN forward dispersion relations [5] with gnn 2 =4+ g 2 NN =4 :. There is soe rather indirect evidence that also favors a sall value for g NN. In Ref. [6], Piekarewicz calculated the { ixing aplitude in the hadronic odel where the ixing was generated by NN loops and thus driven by the proton{neutron ass dierence. To bein agreeent with results fro chiral perturbation theory the NN coupling had to be constrained to the range gnn 2 =4 = :32 :53. In a very dierent approach, Hatsuda [7] evaluated the proton atrix eleent of the avor singlet axial current in the large N C chiral dynaics with an eective Lagrangian that included the U A () anoaly. In this fraework, the EMC data on the polarized proton structure function (which have been used to deterine the "strangeness content" of the proton) can be related to the NN and the NN coupling constants. Again, his analysis prefers sall values for both coupling constants. Nevertheless, fro the above discussion it sees clear that the NN coupling constant isuch saller copared to the corresponding NN value of around 4. In a recent analysis of photoproduction on the proton [8] both nature and agnitude could be deterined in a coparison of a dynaical odel with new high accuracy data fro Mainz [9]. In this calculation the resonance sector includes the S (535), P (44) and D 3 (52) states whose couplings are xed by independent electroagnetic and hadronic reactions like (;), (; ), (; ) and (; ). The nonresonant background is described by vector eson exchange contributions and s- and u-channel Born ters, where the NN coupling constant enters. By coparison with the data on total and dierential cross sections the couping constant was deterined as gnn 2 =4 :4 with a clear preference for a pseudoscalar type. The ai of this paper is to calculate the pseudovector as well as the pseudoscalar coupling constant of the eson to the nucleon. In section 2we analyze the dierent structures of the isosinglet and isotriplet axial vector nucleon currents on the quark level to otivate the sallness of the pseudovector N coupling (subsequently denoted by f NN ) relative tothe corresponding pseudovector N coupling (denoted by f NN ). In fact, in contrast to the isotriplet axial current the isosinglet axial vector current of the nucleon does not contain any coponent fored of the isodoublet u and d quarks of the rst quark generation but is deterined exclusively by the isosinglet c and s quarks belonging to the second quark generation. The (pointlike) pseudovector N coupling will be therefore exclusively deterined by the presence of strange/chared quarkoniu coponent both in the eson wave function and the nucleon current and expected to be rather sall. In this context vertex corrections can acquire iportance. We here advocate the idea to treat the coupling of strange quarkoniu to the nucleon by eans of triangular vertices involving appropriate nonstrange esons. 2

3 In section 3 we consider the long range! (a N) triangle diagra as a odel for the ixture of the pseudoscalar and pseudovector NN vertices, derive analytical expressions for g NN and f NN and x in a natural way their relative sign. The special role of the a (98)N triangle diagra as the doinant one{loop echanis for the N coupling is singled out by the circustance that the a (98) eson is the lightest eson with a two particle decay channel containing the particle []. The contributions of heavier esons such as the isotriplet a 2 (32) tensor eson with an decay channel and the isoscalar f (4), f2(525) and f 2 (72) tensor esons with decay channels will be left out of consideration because of the short range character of the corresponding triangle diagras on the one side, and because of the coparatively sall couplings of the tensor esons to the nucleon [2, ] on the other side. In section 4 we show that the sall value of gnn 2 =4 :4, as obtained fro ts of the photoproduction aplitude [8] iswell reproduced in ters of the a (98)N triangular N coupling if coplete doinance of the full a decay width by the a (98)! + decay channel is assued and use is ade by the version of the Bonn potential with the lowest value for the a N coupling constant [2]. The paper ends with a short suary. 2 The couplings of the eson to the isoscalar axial vector current of the nucleon Within the SU(3) avor syetry schee the neutral weak axial current of the nucleon J ;5 following fro the Glashow{Weinberg{Sala electroweak gauge theory is given by the atrix eleent of the corresponding quark currents as This ay be reduced to J ;5 = = J ;5 = 4 hn j u 5 u d 5 d s 5 s j Ni : (2) 2 hn j u d 3 5 2! u j Ni + d 4 <Njs 5 s j N> g A 2 u(~p ) u(~p)+ 2 Gs u(~p ) 5 u(~p) 2 ( J ;5 (I =)+J ;5 (I =)): (3) Here g A is the weak isovector axial coupling constant, G s denotes the weak isoscalar axial coupling (g A =.25, G s = :3 :4 [2]), u(~p ) stands for the Dirac bi{spinor of the nucleon, whereas J ;5 (I =)andj ;5 (I =) in turn denote the isotriplet and isosinglet axial vector nucleon currents. The atrix eleent of the isoscalar axial vector current between the = = pseudoscalar esons and a N N state is dened in the standard way as The sae arguent applies to the neglect of the f (59)N triangular vertex. 3

4 hn N j J;5 (I =)ji = if q ; hn N j J ;5 (I =)j i = if q ; hn N j J ;5 (I =)ji = if q : (4) Here n and f n denote in turn the ass and the diensionless coupling constant of the respective eson (n = ; ; ) to the hadronic vacuu. In the three avor quark odel, the wave functions of the low lying pseudoscalar esons are described as linear cobinations of quark{antiquark (qq) pairs. The physical singlet and scalar states within the pseudoscalar eson octet corresponding to the and the esons are oreover predicted to be ixed according to j i = cos P uu + dd 2ss p 6 sin P uu + dd +ss p 3 ; j (958)i = sin P uu + dd 2ss p 6 + cos P uu + dd +ss p 3 ; (5) with the ixing angle P = : [] being deterined fro ass forulae. The presence of a strange quarkoniu coponent in the pseudoscalar isoscalar esons is equivalent to a violation of the Okubo{Zweig{Iizuka rule predicting the suppression of ss! uu=dd transitions. The evaluation of the atrix eleent of the isoscalar axial vector current between the eson state and the hadronic vacuu is based on the QCD suggestion [3] that a quark q i of avor i in the pseudoscalar esons couples only to the current q i 5 qof the sae avor and that the coupling strength is avor independent 2 h NN j qi 5 q i j ni = i j n nq ij : (6) Here, j n denotes the weight of the (q jq j ) quarkoniu in the wave function of the pseudoscalar eson (n = ; ;) considered. Eq. (3) shows that in contrast to the isoscalar vector current, the isoscalar axial vector current of the nucleon contains no non{strange coponent. Because of that the pointlike coupling of the isosinglet pseudoscalar esons to the corresponding nucleon current is realized only via their strange quarkoniu coponents, in which case one has s = while for the pion one has p cos P 6 2 p 3 sin P ; (7) s = p6 sin P + 2 p 3 cos P ; (8) = p 2 : (9) 4

5 Insertion of Eqs. (7-9) into (6) and a subsequent coparison with (4) lead on one side to f = = 2 3 ; () where we ade use of the epirical value for the (diensionless) pion decay coupling constant f = 92MeV=. On the other side, with that the coupling strength is calculated as =:9428 and the values of f and f are copletely deterined by f = s ; () f = s ; (2) respectively. To get a rough understanding of the origin of the pseudovector N, N and N couplings introduced via the corresponding Lagrangians as L = (x) = f (= )NN = (x) 5 (x)@ = (x) ; (3) L (x) = f NN (x) 5 ~ ~ (x) ; (4) it is quite instructive to consider a "toy" odel in which universality of the axial currents of the pseudoscalar eson is assued for the oent (Fig. ). This would allow one to obtain the following paraetrizations g toy NN 2 N g toy NN 2 N g toy NN 2 N = f toy NN = = f toy NN = = f toy NN = G s = :962 ; (5) 2f G s 2f = :94 ; (6) g A = :9375 ; (7) 2f where use has been ade of the on{shell equivalence between the pseudoscalar and pseudovector eson nucleon couplings leading to the relation f n = n = g nn N =2 N. The usefulness of the "universality" ansatz is best deonstrated for the case of the pion where the epirical value of f NN = :26 as deduced with a good accuracy fro chiral syetry constraints is only few percent larger than the one concluded fro the "universality " arguents as f toy NN = g A=2f =:9375f NN.For the case of the charged axial vector current "universality" is equivalent to the Goldberger{ Treian (GT) relation and thus to current conservation in the chiral liit ofavanishing pion ass. For the case of the isoscalar axial vector current, however, the "toy" odel is less useful as it would suggest a GT{like relation between G s;f ;f NN and, which is unrealistic in view of the axial anoaly proble. Nontheless, the considerations given above are instructive in a sense that they clearly illustrate the fundaental dierence between the couplings of isovector and isoscalar pseudoscalar esons to the 5

6 axial nucleon current. Whereas the pseudovector = N coupling relies on the strange coponent of the axial vector current, its purely non strange coponent is relevant for the pseudovector N coupling. Eqs. (-2) together with Eqs. (5-7) lead to the following relations r = f toy NN f toy NN f f = = p 2( p cos P 6 2 p 3 sin P ) ; (8) = g NN g NN = Gs f g A f = :527 : (9) Eq. (9) shows that the N vertex appears suppressed relative to the N vertex by at least one order of agnitude. For this reason we expect the uch larger experientally observed N couplings (r :2) to be governed ainly by the eective a N triangular vertex rather than by the contact eson{current couplings considered in the "toy" odel above. In the following section we consider an eective NN vertex associated with the a N triangle diagra (Fig. 2), which is the doinant long range one loop echanis for the isoscalar axial nucleon coupling, and calculate both the values of f NN and g NN associated with this vertex. 3 The a (98)N triangular vertex for g N N and f N N The a N triangle diagra is calculated using the following eective Lagrangians for the a! + decay, the N and the a N couplings: 2 a L a (x) = f 2 a y (x) ~ (x) ~ a (x) (2) L NN (x) = f NN (x) 5 ~ (x)@ ~ (x); (2) L a NN(x) = g a NN (x)~ (x) ~ a (x): (22) Here f NN and g a NN in turn denote the pseudovector N and the scalar a N coupling constants, for which we adopt the values fnn 2 =4 =:8 and ga 2 NN =4 =:77, respectively. These values are iplied by the relativistic Bonn one boson exchange potential (OBEPQ) for the nucleon-nucleon interaction [2]. To regularize the integral in the triangle diagras in Fig. 2 we introduce the sae onopole for factors at the NN and a NN vertices as established by the Bonn potential odel. The following contribution to the NN vertex is then obtained: g NN (q 2 ) = 3 2 a 8 2 Z 2 2 f NN f a g a NN dx ln Z ( ; a ;x;q 2 )Z ( a ; ;x;q 2 ) Z ( ; a ;x;q 2 )Z ( ; a ;x;q 2 ) 6

7 + + 2 Z Z Z Z xc(x; y; y; q 2 ) dydx Z 2 ( ; a ;x;y; y; q 2 ) Z dydxx 2 ( ; a ;x; y; q 2 ) ln Z 2 ( ; a ;x; y; q 2 ) + ln Z 2( a ; ;x; y; q 2 ) Z 2 ( ; a ;x; y; q 2 ) : (23) Here and a are the cut-o paraeters in the onopole vertex factors, for which we use the values.5 GeV and 2. GeV, respectively. The functions Z ( ; 2 ;x;q 2 ) and Z 2 ( ; 2 ;x;y; q 2 ) are dened as c(x; y; y; q 2 ) = xy( + xy) 2 N + xy(x(y y) 2 )q2 ; Z ( ; 2 ;x;q 2 ) = x 2 +( 2 2 q 2 )( x)+( x) 2 q 2 ; Z 2 ( ; 2 ;x;y;y; q) = 2 N x2 y ( x)+( )xy + x 2 y(y y)q 2 : (24) The N coupling constant is obtained by setting q 2 = 2 in Eq. (23). The corresponding expression for the pseudovector coupling reads: f NN (q 2 ) = 3 2 a 8 2 Z Z 2 2 f NN f a g a NN N dydxx 2 yz 2 ( ; a ;x;y; y; q 2 ) + Z 2 ( ; a ;x;y;y ;q 2 ) : (25) 4 Results and discussion Using for f a the value of.44 extracted fro the experiental decay width [] when ascribing the total a decay width to the a! + decay channel, we obtain g NN = 2:3 ; f NN = :58 ; g 2 NN 4 =:33 ; (26) fnn 2 4 =:27 : (27) These are the quantities which we shall interpret as the values for the pseudoscalar and pseudovector coupling constants, respectively. The ain sources of uncertainty in the paraetrization of the NN coupling constants by eans of the triangular a (98)N diagra are associated with the a (98)N coupling constant and the ()= tot a fraction. The coupling constant g a NN varies between 3: and 5:79 depending on the NN potential odel version [2, ]. In view of the KK esoniu structure of the a eson [] the a N coupling will be ainly governed by the short range KKinterediate conguration and therefore expected to be sall. For 7

8 this reason we favor in the present investigation the versions of the Bonn potential with the lowest g a NN values reported. It should further be pointed out that an increase of g NN and f NN iplied by a larger g a NN value can be copensated to a large aount by the reduction of the a (98)! + partial width fro the % used by us to a lower and ore realistic value. The size of the coupling constants obtained in the present study can therefore be viewed as realistic. Note that the pseudovector N eective coupling constant associated with the a N triangle is about three ties larger as copared to the corresponding toy odel value in Eq. (). This observation underlines the iportance of eective vertices for the coupling of the strange quarkoniu to the nucleon (copare [4] for previous work). Our result can be reforulated in ters of an eective NN Lagrangian with PS-PV ixing [5] that was also discussed in eta photoproduction before [6] L NN (x) = ig (x)[ 5 +( ) i ] (x) (28) 2 N with being a ixing paraeter between the two liiting cases of PV coupling ( = ) and PS coupling ( = ). In cobination with Eqs. (26-27) we obtain = + 2 N g = g NN g 2 4 f NN g NN! =:54 (29) = f NN =4:3 (3) = :29 : (3) In Fig. 3 we show a calculation of eta photoproduction using our coupling constants in coparison with the experiental data. We also copare with the results of Ref. [8] obtained in PS coupling with their best-t coupling constantofgnn 2 =4 =:4. The average result over the angular distribution and, consequently, the total cross section is about the sae in both calculations, however, in the forward-backward asyetry our present calculation provides an even better description due to the sall PV adixture. Considering the dash-dotted lines, calculated using a large value for the g a NN coupling constant, it becoes clear that such large values for the a coupling and consequently for the coupling are ruled out by the experient. Our considerations show that towards a better understanding of the N coupling precise easureents of the a (98) decay properties as well as a better knowledge on the a NN coupling constant are needed. We arrive at the conclusion that both the pseudoscalar and pseudovector coupling constants of the eson to the nucleon see to be exhausted by the eective a N triangular vertex. Consequently, the eson cloud odel predicts realistic results for reactions involving the coupling of the ss syste to nucleons. 8

9 To suarize, we wish to stress that in calculating the NN coupling it is necessary to account for the principal dierence between the isosinglet and isotriplet axial vector currents of the nucleon on the quark level, a fact ignored by the quark odel. For this reason the three avor constituent quark odel is unable to predict the correct size for g NN. A siilar situation is observed for the case of the KN- and KNcouplings which are concluded fro photoproduction data on the nucleon to be about an order of agnitude saller than the quark odel predictions [7]. The sall value for g KN is well understood in accounting for the principal dierence between the strangeness preserving and strangeness changing axial vector currents of the nucleon on the quark level [8]. Acknowledgeents This work was partly supported by the Deutsche Forschungsgeeinschaft (SFB 2). 9

10 References [] R. Brockann and R. Machleidt, Phys. Rev. C42 (99) 965. [2] R. Machleidt, Adv. Nucl.Phys. 9 (989) 89. [3] K. Holinde, Nucl. Phys. A543 (992) 43c. [4] J.C. Peng, Proc. of the LAMPF Workshop on Photon and Neutral Meson Physics at Interediate Energies-LA-77-C, edt. by H.W. Baer et al., 987. [5] W. Grein and P. Kroll, Nucl. Phys. A338 (98) 332. [6] J. Piekarewicz, Phys. Rev. C48 (993) 555. [7] T. Hatsuda, Nucl. Phys. B329 (99) 376. [8] L. Tiator, C. Bennhold and S.S. Kaalov, Nucl. Phys. A58, 455 (994); L. Tiator et al, Proc. of the International Conference on Mesons and Light Nuclei, Straz p. Ralske, Czech Republic, 995, to be published in Few-Body Systes. [9] B. Krusche et al, Phys. Rev. Lett. 74, 3736 (995). [] Review of Particle Properties, Phys. Rev. D5 (994) 73. [] R. Machleidt, K. Holinde, and C. Elster, Phys.Rept. 49 (987) 49. [2] J. Ellis and M. Karliner, Phys. Lett. B33 (993) 3. [3] R. L. Jae, Phys.Lett. B229 (989) 275. [4] M. Kirchbach and D. O. Riska, Nucl.Phys. A594 (995) 49. [5] F. Gross, J.W. Van Orden and K. Holinde, Phys. Rev. C4 (99) R99. [6] M. Benerrouche, N.C. Mukhopadhyay, and J.F. Zhang, Phys. Rev. D5, 3237 (995). [7] T. Mart, C. Bennhold and C.E. Hyde-Wright, Phys. Rev. C5 (995) R74. [8] M. Kirchbach, L. Tiator and C. Bennhold, in preparation.

11 Figure captions Fig. Axial current doinance ("toy") odel for the PV coupling of pseudoscalar non strange esons to the nucleon. Here l is the external axial current, J (;s) M = iq (;s) M denotes the respective isovector (upper index ) or isosinglet (upper index s) axial current of the M = ; ; eson whereas A = g A u u and 5 2 As = G s u 5 u in turn stand for the isovector and isosinglet axial vector currents of the nucleon. Fig. 2 The eective a N triangular NN vertex. The full fat line denotes the eson while dashed and double lines have been used for the and a esons, respectively. Fig. 3 Dierential cross section for eta photoproduction on the proton at dierent photon lab. energies calculated with the odel of Tiator, Bennhold and Kaalov [8]. The full lines are calculated with the coupling constants of Eqs. (26-27) and the dash-dotted lines use coupling constants that were scaled up by the larger value of 5.79 for g a NN, resulting in g NN =3:78 and f NN =:8. The dotted lines show the results of Ref. [8] in a pure PS odel with gnn 2 =4 =:4. The experiental data are fro Krusche et al [9].