druk na prawach rękopis* It A P 0 It T NR 1336/PH -
TOVA&DS A COMBINATORIAL DESCRIPTION OF SPACE AND STRONG INTERACTIONS
P, ŻENCZYKOWSKI *
Deportment of Theoretical Physics, Institute of Nuclear Physic*,
ul. E. Redzikownkiego 152, 31-JU2 Kraków, Poland
September 1986
Abstract-;
We propose a reinterprets tion of a. auccessful phenomenological approach to hadron o elf-energy effects known ma the unitarised quark model. General arguments are given that the proper descrip¬ tion of strong interactions may require abandoning the assigne- nent of primary role to continuous concepts such ma position and momentum in favour of disoret* ones such as spin or V-apin. The reinterpretation exploits an analogy between th« V-spin diagrams occuring in the calculations of hadronic loop effects and the spin network idea of Penrose. A connection between the S—acatrix ap¬ proach to hadron masses and the purely algebraic approach charac¬ teristic of the quark model is indicated. Several hadron mass re¬ lations generated by a resulting SU(6)„-group-theor«tic expres¬ sion ire presented and discussed. Results of an attempt to gene¬ ralise the scheme to th« description of hadron vertices are re¬ ported.
* address after September 15, 19861 Department of Hiysics, College of Physical Science , University of Guelph, Guelpb, Ontario, Canada. 1. Introduction
Despite Important auco»»««t of local field theory the funda¬
mental questions concerning the precise connection between the
classioal maorcsoopio spaoe-time satisfying Elnstelnian looality
and the non-looal properties of quantum theory are still unre¬
solved. Tinas - although quantum ohromodynamios offers the pos¬
sibility of describing the -world of hadrons in terms of a local
quantum field theory of quarks and gluons - one nay still argue
that in a siore fundamental quantun approach to strong interactions
one should not assume the olassioal apncv-tlas oontlnuusi as one
of the primary oonoepts of the theory.
In accordance with the latter idea the present paper proposes
a different Interpretation of the suooesful phenomenologioal ap¬
proach to the question of hadronio self-energy effeots studied
recently [1,2]. I5ae suocess of the mase formulas derived within
this approaoh [1] is here attributed to a suggested tight oon-
nection between strong interactions and the supposed quantum
origin of macrosoopio space. In addition the paper contain* also more oomplete meson mass formulas, discussions on the generality
of mass relations obtained previously and results of an attempt
to apply the scheme to the description of hadron vertices.
General arguments supporting the view that the problem of strong interactions is perhaps closely related to the presupposed quan¬
tum origin of olassioal macroscopic spaoe are presented in Seo- tion 2. In this section the spin network idea of Penrose is briefly recalled. The point of view is adopted that it consti¬ tutes a part of a more complicated disorete network - from whioh macroecopic spaoe and strong interactions properties should be derived.
In Section 3 a simple algorithm allowing th» calculation of
the combined spin-flavour dependence of meson and boryon mass
differences is given. This algorithm - suggested by S-matrix
considerations - is interpreted in the spirit of the discrete
quantum network idea.
An attempt to apply the scheme to the description of vertex
symmetry breaking is briefly reported in Section h.
Flnallyf the standard interpretation of th» phenomenological approach of [1,2J and the relnterpretetlon of this paper are
Juxtaposed in the last Section. 2. General
In quantum fleld-theoretlc approaches apace-tine provides • classical background, Intuitively thought of aa a medium in which tho particlss propagate according; to the rules of quantum thwory. A prevalent opinion is that the concept of classical space starts to be in conflict vlth quantum theory belov ~10 •*•* en only * where quantum gravitation effects should be considered. This la aruch belov «< 1O* cai - the characteristic distance of strong interactions and -v10 cm - the distance at which Q. E.D. baa been experimentally verified. It may aeem therefore that the idea of linking strong interactions with the quantum origin of space is not tenable. It is however known that quantum theory possesses non-local properties [3-5]* Furthermore the experiment of Aspect [6] has confirmed that physical systems may exhibit strongly non¬ local featuree (over macroscopic distances of order 10 m). The ex¬ perimental results are both In excellent agreement with quantum theory and In violent disagreement with conventional - aa drawn from special relativity - ldeaa about the propagation of cau •" Influences* Consequently, the opinion Is more and more often ex¬
pressed C5l that our vlev on the nature of space-time requires a thorough revision. Since the problem ooours already for flat space-time It seems that this conflict between quantum theory and our concept of space should be resolved before one attempts to take gravitation Into acoount. Ihus, Planck length need not cor¬ respond to any critical distance at which one wight expect the elucidation of the origin of conflict between quantum rules and our understanding of space-time. Similarly, the distance 10 cm obtained from the experiment- ally verified application of local field theory doaa not eet up an upper limit for ouch a critical distance. In fact both the algorithm of quantum theory and th« Aspect experiment suggest that no ouch critical diatanoe exists.
In the framework of local field theories both classical (e.g. con¬ tinuous apace-time) and quantum (e.g. quantisation prescription, rulea of calculation) concepts are built in as primary conoepts. Within such theories it ia therefore impossible to study th» con¬ ceivable possibility that they ara actually related. It has bean proposed in various contexts (7-1°] that continuous claasical
space-tine should be considered secondary and that the primary concepts should be quantum-theoretical and irwt likely discrete [1OJ. The points of space-tine vould then be derived concepts, fne moat obvious physical concept whioh has well-known discrete (as opposed to continuous) quantum properties and which ia tightly connected with the notion of space-time itself Is angular momen¬ tu*. The idea of Penrose [10] wea therefore to atart from an¬ gular momentum and build from It the concept of apace in aome way. ma baaic idea begins with the problem of defining the di¬ rection of apin projection of* say, spin i/2t particle* Such a particle has only tvo "directions" to choose freai (forget about the dlraotion provided by the continuous (l) woman turn). Penrose writes then [10] i "whether these possibilities are up* and 'down or 'right* and 'left" depends on how things ere connected with the macroscopic world. Since we do not want to *h
1 ture (which thua could be thought of as quaai-claesical) belong¬ ing to the discrete system under consideration* Ihls orientation, can b« determined through an "experiment" in. which the perticie and th* larger structure examining it combina and/or ax oh an ga spin. Tnia ia a typical aiiuaiion »aoiffifc«r«d la qusatuxs systess:
th* outcoma of an experiment dependa both on th* examined obj*ct and on th* whole experimental set-up* In thia way on* is l*d to th* atudy of spin networks and th* closely r*lat*d 3n-J symbols which should b* among th* basic structur*a of tha combinatorial approach to macroscopic apace*
In th* aiinplcat version of hla approaoh Fmnroam oonaid*r*d a unlv*ra* built out of auoh. spin natvorka and att*mpt*d to d*fin* *ngl*9 and b*nc* rotations in t*rms of tb,*a* apin atructur*s> Later h* vas l*d to th* study of oonforaal group in which rota¬ tions and translation* ar* tr*at*d on *qual footing [11] . XL though th* resulting- seh*a* was originally lnt*nd*d to approach in a diffarant way th* problem of th* quantieation of gravity, attempta to apply it to elementary particle physics and in par¬ ticular to strong interaction* have also been nade [12,133* 3be
propoaed partlole olasslfleation [.14-13] doea not seem, however, to correspond in a direct way to th* standard pattern In which elementary "particles" are grouped into generations built froai leptona and three-coloured quarks, both species coming in weak- -iaoapln doublets* Such a correspondence, with quarks as eon- formal seml-splnors, can be obtained [16] at th* expense of enlarging the original conformal group only - an indication that the ray of combining rotations and translations In the original tvlator scheme may not be wholly correct nor sufficient. Putting aalde thle acheae as a whole it nay still be argued that the spin network Idea should be very well suited to the description of strong interactions in which spin seems to play a v The sin of this paper is to propose a combinatorial expression for the spin dependence of hadron masses. The association ot the problem of maas with the problem of strong Interactions may aoetn unjustified at present. However, the clarity of the concept of mass is somewhat blurred in strong interactions. In standard ap¬ proaches one first assumes the existence of polntlilce quarks with¬ in hadrons and assigns them current masa aa if they were free ordinary partlclea like leptons. Then, through a confinement dogma these "particles" become unobaervable and the "long-distance" "constituent• quark masses* are believed to energe as more ap¬ propriate. In the description of many low-energy properties of hadrons. Setting apart any apeoific implementation of this general * the best available at present paraateter-free amtsl Tor baryon •egnetio atoaenta {tram which constituent quark Masses were origi¬ nally determined) la tbs amdsl of Sctawinger [18] In which the "oeostltuent • quark swss Is by definition half the aass ot the correspoadlag Tseter ateson* schema. It la obvious that any such aah»m» is compoeed of -two logical stapa of which, tae aacond (confinement) is contrived to partially eancal the assumption (aaslgnement of masa to a quark) mada in tha first stop. If the oonoapt of maaa cannot be aesig- n*d to a olfflplc quark (not avan s»ro mass) - tha problem of con- finamani doea aot exist since Is all qu»rk secrębea AM naturally looks for an objaot with, conventional particla attributes: Although we do not want to assign right from tha bag-inning continuous concepts such aa position or momentum to a quark they might •merge in an appropriate large structure limit, In accor¬ dance with the general aplrit of the combinatorial approach. In tha reals of strong interactions the only objects to which tha concept of mass could be applied'ahould be hadrona taemaelvea* Since several quark model auccesses reat on the assumption that quark ie an orthodox particle vith all its attributes - to sub¬ stantiate the above considerations it ia neceasary to reproduce succeaaful quark model relations using tba concept of hadron mass alone. In the following section a number of auch relations (and a few additional onea) are actually derived in this -way* The ba- aic ingredient of the approach ia a combinatorial prescription for the apln dependence of hadron masses. This prescription is abatracted from the phenoatanologlcal approach of £1 ,2} (based on the S-roatrlx ideas) •• tha leading tarn of hadron eelf-energy differences and In this paper is considered fundamental. S' > 3. Hadron Masses The concept of quark mass will be not used below in the deriva tion of hadron mass relationships, the aucuess of the quark model approach [19-21] foroea us to assume however that hadron* should be described ss quantum statss composed of quark-antiquark pairs (for mesona) end of three quarks (for baryone). Ihe terra "quark" describes here a spin-flavour index only. la accordance with the ideas of the preceding section we do not assign momentum to a aln» gle quark (not to a hadron constructed in this van only through its correlation with the macroscopic world tha concept of conti nuous momentum is thought to be eventually feaeignable to a hadron). No insight is proposed on the origin of flavour quantum number» "Ground-state" mesons transform as 1 @>35, "ground-stats" baryona aa 56 (i.e. symmetric) representation of SU(6)_. To comply with the Pauli exclusion principle the colour quantum number should be assigned to quarks aa well. It is however not necessarily related to QCO colourt in thia paper gluoas ere considered nonexistent. The only role of colour is to provide an explanation for why one should work with the 56- (and not 20-) dimensional representation of su(6) . from now on wa shall therefore forget about it. d In phenomenological studies performed in [1,2,22-25] the cru cial ingredient was the consideration of the leading contribution to hadron self-energy coming from the symmetry-related aet of two-hadron intermediate states. To estimate this contribution an assumption concerning three-point hadron vertices was needed. Ibis is the assumption of oU(6) -symmetry [ Z€ 3 (actually the assum ption is a little »trong»r since the couplings 1-35-35, 35-35-35 or 1-56-56f 35-56-56 sre assumed to bs related by the quark mo del). Since ft» г the vertices involving sxclted hadrons the use of э a phenosienological nodal in at present necessary 'we restrict our attention to "ground-state" mesons and baryons only. The estimate of the self-energy contribution to the mass of had- ron A due to its coupling to hadrona B and C (Flg»1) involves the calculation of the square w. of the relevant SU(6) Clebsch-Gor- don coefficient, called henceforth a weight. As shown in [1,23] the leading contribution from the loop shown In Fig.1 is vhere C0)C< are unknown constants. Ca may contain an additi¬ ve dependence on flavour, but the spin dependence comes from the second term in Eq.(i) only. Formula (i) can be derived by expan¬ ding any specific expression for the loop of flg.1 to first order in mass differences /**. — Let the mesons A,B,C belong to 1- or 35-dljnenaional represen¬ tation of SU(6)S> the weight needed in Eq.(1) (we allow fcr the 10 possibility of mixing among two states of I a O, Y t= 0, hence In general A. if A~) is then with the SU(6)¥ Clebsch-Gordłłn coefficient given by vA-i m 'i'a© conventional S-spin values and its projections (^A» Ai ^ S_,mo) uniquely determine the corresponding values of W-epin '•Wc^ needed in Eq.(2b). The flavour coupling is F(A;6,C) » Tr(A6C-CBA) , = A , ) = A ^> ic and A.3j».. are standard 3x3 meson 1 1 matrices (App.C of £i}). 1Sio divleion Into SU(2)„ singlat and triplat depends of coura« on the choice of the auxiliary spin projection axis. Expression (2a) ia however independent of this choice - aa it should be if such an axia were to have no physical meaning' at tliia o tag*, Assume for the moment that n>_ and m„ are all identical. lhe summation over B,C in Eq.(2a) can then be porformed and if A is not a flavour-singlet vector particle the result is independent of A. Thus (up to a possible additive flavour1 dependence of C.) all particles but the vector flavour-singlet are degenerate. For the latter case the sum over B,C ofw, weights is smaller since the couplings of SU(6)„ sing.\et vanish by Eq.(2c). Then, the con¬ stant C_ takes on a different value in tuo original derivation of Eq,, (1) and. In general, the porticlo is not degenerate with other members of 1 (£)35. Ve discuss possible remedies for this vector flavour-singlet problem a little further* Below it is accoptod that tha I a 0, i s O vector sector' (in which mixing with flavour singlet occurs) cannot bo described through i^qa (1,2) without aotoe modifications. Thus no formulas resulting from the applica¬ tion of £qe (1,2) to this sector shall be discussed. Let us now get rid of tho unknown constant CQ in "ją, (1) (to¬ gether with Its possible additive flavour dependence). By forming appropriate combinations of meson mass differences the following flvt* equations are obtained (two additional equations with para¬ meters describing the sector mixed with the vector flavour sin¬ glet hove been dropped): 12 (3b) Dc) 5sl2 siw 2ST (3d) 18) o ot m partlol* standa tor it» «• fonaatloa er states froa «h« SUC3)»ay»»»tric to th« idtjrałoal sis ta affaotad by \W> «md ainllarly In tha paaudoacalar aaotor with Iu5> \ Jto In Bq«(3) tha alxlne ia daacrlbad tqr § - tha angle of deviation fro* ldaal mixingt f Cv) " DKV) KV) Va aaataaa in tna Taotor aaotor for B and C aaaonat = O W . (5b) * ^ (5c) ona obtalna froai Eq. (3a)t and aftar tna dlaconallsatton of tna raaalnlaff aquatlona -3«)) ona gatat IV * • fe = 4 C, Sine* C. flhouXd b« peslttT* (it corresponds [23] •.go to tl» first d»riveitiT» of th* a»satlr* and riming anlft function) It follows from Eqo(6atcfa) that 15 •AI i i i «- (70) froa Eqa(6b,d) w* hav« CL - -g- (provided Q-X or KHfC+O). If 1? >^ «Łd VC >K it follow* froa Bq.(7o) that O£<0 In agr**a*nt with th* axparlmantal sign, łor Op • -'ł3° ("p«r- f«ot" wiring) v« got for tlw loft- and right-band aldoa of £q« (7b, c) tho mnbiri gathorod in Tabl* 1* In* e*a* vlth partlol* aynbol* standlnc for raaa* aquara* la gi.r«n tnor* aa v*ll« tbm mljcinę or th* light paoudoaoalar atat*a (oompoaod of u»dta quarka) vltn tli* haavy onaa (aada of ot***) la *xp*ot*d to inoroaa* tha r«a.a« In Eqa(7b,e) tnua iaqwoTine tfr* agraawant with axparfant. Conrvraaly, if Op la fnroad to to* aqual to s*ro tn* partlcl* of alddan *tranc*n*aa d*eoupl*a froai 0 and X what abould dimini ah tha r.h.a. of fiq.(7b) « It la lnt*r*atlne to look at a almpllflod aituatlon In wbloh Intamal pa*udoaoal«r aaaona ocua- poaad of naw quark tjrp*a (q) ar* aaauwd to b* unan\x*d with "Otf. thm wai^ata ot tha contributions trom aaaona containing quark q to th* light aaaon awaaaa ar* then given in mol* Z» Froa labl* e it follow* that haavy flaroura do not oontrlbut* to th* •*•* dlff*r*no** of light aaaonat tha q-qoark contributions to th* laft- and right-hand aldaa of Bq*(7) •«• idaatloally aare* A d*tallad atudr of th* Mixing problaa for »f > 3 1* byroad th* •o*p* of this ••par da* to tb* rasidlr iaopaaaing (vita Mf) 16 bar oX » It should be noted that the vector flavour-singlet problem depends on the number of flavours Hf i the SU(N,)-slnglet stete oontalns smaller and smaller admixtu'e of the SU(2)-{SU(3)-) singlet when N_ —*• °° . This suggests a possible way of dealing with the singlet problem for any finite number'of flavours. For the state • (uu + d3) to be an approximate eigenstate of the infinite mass matrix it is sufficient that the ratio of sums ovr k> 2(utd) of the off-diagonal and of the diagonal contributions from the internal (UQL.) + (uqk) pairs is close to zero* lnie can be achieved through an appropriate breaking u" vertex symmetry* .Another possibility of dealing with the vector flavour-singlet problem vats mentioned in £273 t it requires the consideration of the "excited" states (conjugated by O~parlty to the "ground" sta¬ tes), in addition to the "ground" states themselves. At present only a phenomenological treatment of the involved vertices Is pos¬ sible, however• Since the vector flavour-singlet problem is absent from the baryon sector it is interesting to examine the *toeavy" quark contribution in this sector, for the "light" quarks the SV(6)V Clebach-Gordan coefficient for baryon B going into meson M and baryon A (meson first convention) is given by L&^A] (8.) 17 vi tit th« flavour coupling* t ,M, A\ - -Tr(M)TrteA) HV(BMA) (8b) [g , H and tha atandard aaaignamant for maaon (M) and baryon (B,A) «a- tricaa (App.C of f|] ). Tha vai^ita naadad in Eq. (1) ara cec) "VI la Ł1] it haa baan ahovn that tha contribution froai tha "ground" -atata intaraadiata hadrona eonpoaad of a,d,a quarka praaarraa tha Gall-Mann-Gfcubo formula for oetat and tha •ąuml apaclng rula 18 for the decuplet. Th* signs of £&- N and Q—X splittings vere shown to be related and identical. Eq.(1) and Eq.(8c) in addition to STT(3) mass formulas lead to Z- 3A- 2 A +2N +2^*= o with Bq.(9a) being the SU(6) relation of £28-29} and ^l*(9b) - th« de Rujule-Georgi-Glaahow formula £30] for the (S~AVC^""^^ ratio of moaa differences. It may be checked after lengthy but atraightforvard calculations (uaing suitably modified Eq.(8b) or Table 1 ot [1] ) that a comple¬ te set of ground-state intermediate hadrons containing an additio¬ nal quark q doaa not contribute to the naaa combinations (9a,b) of external baryona. Tnla cancellation occurs independently of what are the mass relations batw*«n the q-containing hadrons th«si- selvea. Gn the other hand the Gall-Mann—dcubo and the equal spa¬ cing SU(3) formulae ara fulfilled for external lines provided the following equalities hold for the internal Unas as vailt 0 where "Wxta ( "OWH^ ) denotes the mass of spin 1- •*•*• with up and/or down quarka syanMtriaed (the nass of the fully sym¬ metric in flavour spin - -2_ atate). If Eqa(9a,b,c) and the 19 SU(3)'maaa formula• ara fulfilled for tha internal lines than Eqe(9c) for the axtarnal baryona ara alao fulfilled. Apart from a degenerate caaa (a specific single value of C.. (baryon sector)) Eqa(9c) raeuit alao by solving Eq(i) for Nf * k (and the SU(8) vertox symme tr y). Eq.(i) can alao ba applied to tha question of iaoapln violating met a (i differences of hadrons. Such calculations hove racantly b«en dona for tha ground-state baryona [25] where there ara ten li¬ nn arly lndepended masa diffarancaa. It has baan ahovn that Eq.(i) laada tot 1) six sun rulaa for baryon maaaaa - which follow from tha aa- sumad SU(6) symmatry and mr» satisfied by other modala aa vail ;i) additional prescriptions for four combinations of baryon mas- Mi which, togathar with a phanomenological astiaata of con¬ tributions from othar posalbla aourcas, describe tha obaarvad pattern of iaoapin violating mass diffarancea vary well. A shortened version of Table 1 of £25] obtained after neglecting possible dependence of CQ in Eq.(1) on the third component of isospin is given here as table 3. Table 3 reveals that the contributions of Eq,(1) alone suffice to predict In agreement with experiment (after subtracting the electromagnetic contribution) tvo out of three ratios of mass combinations of point 2) above. One still lacks s part of the necesaary experimental input required by Eq.(1) to compute the numerator of the trird ratio. The agreement seen in Table 3 is very interestingt three mass differences which customarily are thought to be of different dynamical origins are apparently cor¬ rectly related by a single group-theoretical prescription. (One of mass combinations considered, namely p- /VL •+• .±^$t„fV 2(S—2* )) measures what ±a usually thought to aria* from the quark masa difference a - m.) It should ba reminded that In the phenomenological study of £251 leading to Tabla 3 all th» dependence on momantum variabla has baan totally ignored. Yet tha resulting nvunbara ara in a surprisingly good agreement with experiment*, tha success achieved under the aa- sumption of neglecting the momentum altogether may b* Interpreted as corroborating the "discrete" spirit of the combinatorial ap¬ proach o and tha need to introduce a discrete "predecessor" of momentum in such a vay which vould not Tiolate succesful vasa predictions of this section^ the predicted ratio of [ p-y is J0% larger than experiaMntally, This Bight Indicate tha ne¬ cessity of considering flavour aynaatry braaldng In the -rartioas 21 k. Vertices Data on baryon magnetic moments (supplemented with the assump¬ tion of vector meson dominance [18]) and other data on hedronic couplings [31] indicate that SO(6) and flavour SU(3) symmetries are broken In a peculiar nonadditive way [32,33]. This experimen¬ tally observed pattern of vertex rymmetry breaking haa not been explained in any scheme as yet. TJiua, an Interesting question is what are the predictions of the group-theoretical approach model¬ led upon the treatment of masses of the previous oectlon. Be low the results of such a calculation [31*] are shortly sketched. Consider therefore the diagrams of Fig*2 -which directly descri¬ be the group—theoret leal structure of the baryon-meson-baryon (BMA.) vertex and are analogous to Bqa(2a) or (8c). Each vertex component in Fig.2a corresponds to a STJ(6)„ Glabsch-Gordan coef¬ ficient of Eq.(2b) or Eq.(8a). Direction of arrows (incident * a» outgoing) and vertex orientation correspond to the bra-ket des¬ cription and meson first convention of the SU(6)V. Clebsch-Gordan coefficients respectively. Relative normalisation and phases in £qa(2b,8) have been chosen consistently so that a simple product of OG coefficients possesses the required SU(6)„ symmetry pro- pert lea after the summation over Internal states Is carried out. One contributions from Flg.2a and 2b are therefore, respectively: 22 To calculate (10) apart from Eqa(2b,8a) on* needs alao Ck* aA; M for sny M S 35 of SU(6)„ (bu4 not for tb.« sixtglat) on* ha« L - 6 56 iłiu», if tha DMMI eorraaponrtlng to Internal llnaa ara d«gan»ra«= ta ao that such sumaationa can actually ba parformad - tha raaul- ting BMA rartax Is SU(6)v-aynBiatrical. Introducing tha breaking of tha SU(6)„-v*rtax ajnaiatry through maaa diffarancaa of internal linea only and eatimating it (as before in th* maaa aactor itaelf) to flrat order in maaa differancea otam gata expressions of the following structure (M £ 35 of STT(6)„)t (loop contribution) - i_.6 -^бИМА^ +- -i- (possible 2weig~rule violating term) where ŁQ, U are constants and WA... etc. are weights which have been calculated on a computer. It turns out that Zweig-rule violating terms vanish if К —1£ = ^v^-^./ what Is not far from experlmont. Such terms were therefore neglected. From £qe(l2), upon aseumlng the dominance of vector mesons [18] one can derive a set of formulas for barren magnetic moments and magnetic transitions. There are two free parameters in thee* for» mules: the size IX. (adjusted to fitM,») of the SU(6) symmetric term (from freedom In L) which may be argued to be approxima tely determined by -2WN/Wp ( [1&3) «nd the sis* С of the correction (from Ц)« Using for simplicity perfect mixing for pseudosealar and ideal mixing for vector mesons and inserting physical hadron masses into Eqa(12) one gete finally [3*0 (Eqs(l3) will all the weights and thus with all mass dependence shown explicitly are complicated and their presentation here is redundant) = /W/* |0+*O + f O+wc) {,3) №lf = -|0+^C) + §(/1+9.3 C) 2k c) -^ It turns out that no choice parameter C in Eq,(i3) can explain the observed pattern of nonadditivities In boryon magnetic mo¬ menta and magnetic tranaitio&s [32,18]. Thus, the combinatorial scheme as considered so far ia able to describe properly the pat" tern of hadron masses but fails when one attempts to apply it in the most na.i've way to the description of vertex symmetry breaking. Success in the description of masses and failure in the descrip¬ tion, of vertices is a feature of all other "succesful" contempo- rary schemes as.veil. It should be noted that the attempt of this section is based on the simplest possible guess concerning the algebraic structure of vertices, justified by the success of Eq.(1) only. Within the S-matrix approach the W-spin projection axis oannot be identical for all three subvertices of the diagrams corresponding to Fig.2, as opposed to the situation encountered in Fig.1, Thus, the in¬ troduction of momentum of intermediate particles into the des¬ cription of vertex symmetry breaking may seem necessary. Clearly, if the general idea of this paper is a sound on*t momentum should be introduced in a way compatible with the discrete spirit of the approach, Tte twistor approach does not lead, however, to the conventional classification of leptons and quarks [1^,15]. Thus the question how momentum should be introduced is, from tho author's point of view, still an open problem. 5. Concluding Remarks la standard approaches quarka are thought to ba polntllka partiolea confined Into extended hadrona. Apart from the Inter action of quarka vlthin hadrona (which - among other effecta - lead to quark confinement) there are alao lnteractiona of hadrona themselves leading to hadronie ееIf-energy ahifts and other had- ron-level effects. All thia phyaica takoв place In ordinary apace and Eq.(1) conatitutea the leading term of the extremely compli cated contribution from the "hadronie cloud11. Thia vaa the point of view adopted in paper 11] . In thia paper group-theoretic discrete concepta are considered fundamental and it la thought that concepta auch aa continuous apace, momentum and, conaequently, the field-theoretic descrip tion of quark degrees of freedom In hadrona should emerge in an appropriate large structure limit only. Eq,(1) is then thought to provide a connection between the S-matrix approach to hadron mas ses through self-energies (with classical momentum constituting one of the primary concepta of the approach) and the purely al gebraic approach to hadron ттаашв from which the naa ve quark model has originated. The question of a possible correspondence between other contributions of the S-matrix approach and the al gebraic approach remains of course open. Whether the interpretation advocated for in this paper should be considered aa more acceptable than the orthodox point of view a- dopted in [l} can be decided only by further attempts to develop algorithms baaed solely on discrete concepts, and capable of lin king together varioua experimental facts. 26 Figur* and Tabla Captions Fig.1. The A •—•> BC —^ A loop. One signs -(+) denote the clockwise (antidoelcirise) ordering of particles in a vertex (O-G coefficient Fig.2. The group-theoretic structure of the B > MA vertext a) meson M emitted from internal meson line, b) meson M emitted from internal baryon line, The signs -(+) near vertices denote the clockwise (anticlockwise) ordering of particles in a vertex* Tabl© 1, Comparison ot £q»(7b#c) vith experiment. 2o Weights of contributions from Internal masons containing "heavy" quark q. Pseudosoalar (vector) loesons are de¬ noted by (uq) ((uq)*) if the "light" quark Is non- strange and by (aq) ((śq)H) if it Is strange} T»bl* 3, Comparison with experiment of predicted ratios of the combinations of isospin violating mass diffarencas which are asnsltlva to the dynamical input (point 2) in the text). M M, 8 ^ 28 Table 1. f masses maes squares < l.h.a. r.h.s. l.h.s. r.h.a. ag.(7b) .627 .355 .566 .3'n «g. (7c) .396 .167 .550 .252 Xabl* 3. ratio of mass combination* prediction (Bq.(l)) experiment (electro¬ magnetic contribution subtracted) -.25 ± .0* -.19 ± .06 +.23 - .09 +.51* t .18 Tabl* 2. *^^^_ B,C 1 A ^-^^^ (uq ) (.uq)* (uq)"t5q)" total T 2 2 (uq)(uq) (u5)CSq)" (uq)"C5q)" total k k i 3 •f (uq)(q«)" (uq)"(q5) (uq)*(qi)" total K 1 1 2 (uq)(ql) (uq)(ql)" (oq)*(qS) total L 2 2 ? h J J J M [uq)(uq)* («q)liq)" uq)"(uq)" (8q-) (Sq)* total 2 0* t«tt («» 0 0 2. 0 0 0 ~k 0 • 0 0 C 2 0 0 0 2 0 k 30 References [1] P. Żenczyko*?*k:lt Ann,, of Pbys.LN.Y.) 162.(1986) &53. [2] N.A. Tdfrnqviat, Ann. of Fhya. (N. Y. ) 123, (1979) 1. [3.] J.S, Bell, Physics (N. Y.) 1., (1964) 195. [4] H.P. Stapp, Phys. Rev. |>3_, 09?l) J303. [5J J.F. Clauser, A. Shimony, Rep. Progr. Phya. jłj_, (19?8) 1882, [6] A. Aspect, J. Delibard, G. Roger, Phya. Rev. Lett. jł°_, 1,1982) 1804. f}] CM. Patton, J.A. Wheeler (,19?5) Is Physios Legislated by Cosmogony in "Quantum Gravity, an Oxford Symposium", (iaham, C.J., R. 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