Narayana, K L

Patil, S B

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A  physical m odel from  the mass em pirics o f  tw o-particle  B aryon resonance states and a postu lation  o f  m edium  and low  strong interactionsK. L. Nauayana ai d^ S. J3. PatilPhy^ 'ik,'^  Department, tShivaji VnivemUy, Kolhapur [Ueceivod 29 October J075)Indtan J. Phys. 50, 993-1002 (1976)Empirical luads formula A-\-B\/L(L^\)-{-CL{L \-\) witli Lbomg tlio j'olal ivo orbital angular monituiliiin Ik Ibuncl to hold good for the two particle N. A, A sysUnnH of {n N) and Nk~ i‘OHoiianc(\s. The OHtimated paramott'is from exporimental inasB data arc suggested to interpret a physical model of the two particle resonant states and that each state to possi^ ss a predominantly siirg](i partich  ^ character admixed wit.h a low jiossibility of rigid rotator oi' Bohr type structui'os Utilizing the iiniYi'rsal fundamental length concepi. of AnastaKHo\  ^ Siiiha and (Sivaram further it has boon established that the different. N, A, A stati‘s involve an additive medium stroug and a subtractive lowst-rong iutci'actions giving I'lse to a resonance charge” or the usual resonance coupling constant for t‘ach resonant stat,e.1. I ntboductionAttempts at deriving empirical rolataons from known ('xperirnental data, for obtaining obsei vi^ d masses of st able particles and n'sonances aie reported earlier by a numlxir of investigators,Two ajiproaches to this end are, one those whicli make use of the (inhereni ) symmetry models and the othei Avhieh consists in fitting certain linear oi non­linear I'ldat.ions by itomputatioiial procedure,s As examiilos of t he first approaclimention may be made of the works by Oneda et at (1974), Budh Bam (196(i), Gerald Rosen (1972), Carruthers ei al (1907), Iowa (19711), Lee et al (1974«. 19746) But this approach, in the words of Carruthers et al, should nut be constructed as evidence that the later approach of conventional methods as inferior In the secoml category we mention specifically of the investigations by Sarma (1903), Regge (1900), Stornhoinier (190S), Agarwal (1971), Maglie (1900), French et al (1907), Battachaijee (1970, 1971, 1974), JNiarayaua et al (1970)111 reforoneo to the resonance systems eonsidoxed in the present woik, follow­ing Sarma (1963) it is convenient to o]as,sil'y the resonance K as belong to specific pair of particles L and N, if the dominant mode of decay of k is given byK -> L + A993... (1)0 9 4 K , L. Narayana and S. B. PatilThe (empirical regularity of Buch two particle resonance systems has been suggested by him to satisfy the relation_  V M L ± }) (i2)where q is tlui contn* ol mass momentum and L  is i Jie relative angular momentum. The mass oJ‘ j’esonanc;e state according to him is given by the relation1/* ^  \/g“+W i^“-hV(Z‘^ +’«'2“ (3)wliicJi on ditterentiation with respect ji — ~{L~\ i ) ‘“ leads to the Kegge’s formuJa vizd(iI ... (4)Hero R is interpreted by Hcggi' (I960), Chew (1962) to b(* the effccitive size of the compound syst,em. But the presumption the ]iaramoter h is constant, ■w'c find by utilising the recent mass data as not coi riHJt and hence justifying a frosli look at this hypothesis.Semi-theorotica] foimulae given by Gerald Kosen (1972) as (extensions ol Schwinger’s formulae (ScliAvinger 1963) for bar yon o(;iiiplet and baryon resonance docmplet, (which adopt the fi'actional hyper charge and iso-spin etc (juantum numberrs) are found to be of considerable success in explaining t he mass splittings, but they inalce no emphasis or reference to the luiture ol' decay modes ol th(‘ difforoiit rtjsonaiit states.French et al have classified about 12 sequences each characterising a parti- (iiilar decay mode, but they include both the stangc and non-strange particles and consider only a linear mass relation of the type M  — .sH-i?i(w-l-l), with a‘, t as constants and n an integral quantum number. However, they advocate a rigid rotatoj' or Bohr-type rnod(4, consistent with the earlier findings of Maglic (1966) for baryon masses, and do not mention thi) possibility of a dependence of M directly on nLinear mass lelations of several tjqies have also boon examined by Storn- heimer (1968) adopting the genera] formula qrm -\-rm — M  and-genei’ated by various values of p and r Agarwal (1971) who adopts this formula for just thro() s(jqueuces of particles and resonances draws an analogy with the Thomson's pluin-ijudding model of the atom to interpret the p  and r parameters. Ho points out however the pos.sible non-linear character of the graphs of mass values and fails to comment anything about the significance of different decay modes of the masses. On the other hand, though Battaoharjee (1974) adopts a poly­nomial equation to interpret the masses, ho mixes up all the elementary particles with no classification at all.E m p irica l mass form ula 995The object o f present work is tJierefoi’e to seek an inherent physical mode] of the two-paiticle baryon resonance systems and to discuss the non-linear cliarae- ter oi‘ their mass relations Wo also indicat e a possible connectivity of the work with the unitary symmetry models of baryon lesonances. Implication of our mass formula, utilizing the universal fundamental length concept of Anastassov (1974) iSinha iSr, Sivaram (1974) has been given, leading to the postulation of medium-strong and low-strong interactions involved in the formal,ion of resonant states.2. M ethod and  R esultsIn the table J listed are the various two jiartick* rosonanc(‘ masses, spin, parity assignments and predominant modes of decay.states, theirTable 1 Particle data used, in t he present, workKesonaiiee^  SystoniOrbitalAngularluonieritum,spill Paril Vretitro oJmass Mass (Mov) momentum (exp)jtN RoaouaticoH jV J 1/2 3/2- 4.66 15202 1/2 r./2^ .672 16883 1/2 7/2- 888 21904 1/2 9/2H 90.6 22206 1/2 13/2+ 11.64 2660S 1/2 n /2 ' 1366 3030nN ReBonancoH A 2 a/2 7/2+ 74 L 19604 3/2 ll/2^ 1023 2420f) 3/2 16/2+ 1260 28608 3/2 19/2+ 1475 3230A'/c- Rosonancea A 1 2/2 3/2- 429 1650o 1/2 6/2^ ' 642 1814:i 1/2 7/2- 913 23604 1/2 9/2+ - 2312*6 1/2 13/2+ - 2712*8 1/2 17/2' — 3071*Since no definite exp. values are available, we used values given by Gerald Rosen (1972).li'u'Rt v^ 'o did a study of tho dopciidence of contro of iriass momentum on^/L{L-\-\.) with L b(Mtig tlie rolatis’-o orbital angular momentum, for the following two curves.Case I . TriV -^resomince, iV-systom o f spin. /V — 1 with Z  — 1, 2, 3, 4, 0, 8.Case II ■ TriV -^resonance. A-system ('f siDins, *S^ ^  -J with L ^  2 4, ti, 8Two curves result in tJie case 1 and only one for the Case II and these ai e shown in figiiL’c^ I. SiiK^ e these graphs are not linear we are to take that' the relation given996 K. L. Narayana and S- B. Patil1. Graph of the coiitio of mass momoutum q verBua the factor -v/X(l+l) for tho N- system (odd L), A-system (ovon h) and JV-system (ovon X).by Sarina vis; eq (2) as inadequate to describe the centre o f mass-momentum dopondeuce on tho rolativi  ^ orbital angulai momentum. Our modified relation is,... (5)E m pirical m ass form ula 997whore / ,  g, h eonstfuiis are rletormiried by a stal.iHtical lit. The obtained value.K lor these eonstants are listed in table 2.Table 2. Calculated constants Ibi’ the f/-rolatioiiRolat,ioiiC n u a la n ts JS s y s te m  A -s y s to in  (e v e n  X )A r-fivs le in  (X  o d d )/ 1)3-3502 3 5 2 -0 7 1 4 1 5 0 -0 0U 2 l3 d )6 6 8 1 6 9 -5 0 7 5 2 1 0 -0 0h - 7 - 6 0 8 9 4 - 4 - 3 0 7 3 0 -0 0Fig 2. Graph nr the mass values il/versus the factor for the A^aystem (odd X),A -^system (oven L), A-systom (oven X), A-system (odd X) and A-systom (even X)Figure 2 giv’^ os the curves representing the relation for both the above cases I and II. and in addition for the experimentally observed998 K. L, Narayana and S. B. PatilmasRos tho Nk“  resonance A-systems, with both the possibilities L odd and L even Tlio evaluated values for constants A, B, C are listed in table 3.Table 3. Calculated constants of the JIf* relationABaiV-syHtom(lor ovon L) A-systom957-9939321-43603-9-133161338-6805269-8666-4-0767,^'L[L+\)AGL{L-\ ])(for odd L)A -^system A-Rystein1126-0000 1200 1380303-0120 336 350-8-66282 0 03. Discussion and ConclusionsAt first it must be pointed out that the simple relation given Sarma (1963) viz eq. (2) (with njl =  c =  1), has an analogy with DeBrogbs’s I'olation A =  hjp, for the wave length A associated with a particle of monjentum p. Th(^  analogy however fails in that the relation of Sarma involves in the numerator a multiple of the relative orbital angular momentum of the stale, though that both the relations agree as far the dimensions are c^xamined. Again from the curves ol figure 1 for tlio 7rA-Resonance, A-systcm and A-system {L fiven) states, it is clear that a simple linear relation may not be (iorrect. The L odd states whidi imply the linear relation, are inadequate as only f.Avo non-ambiguous experimental values could bo used, in each aoquonco.The physical significance of the deviations from a linear relation while in view of the analogy with tlui DeBroglie’s relation may be mentioned, that, instead of a single parameter, such as the Quantum wavelength associated with a particle, the resonance state requires a description in terms of more number of cliaracteris- tic parameters.The set o f parameters from the suggested relation in our work being (/, g, h) or [A, B, G) respectively as one uses either the centre of mass momentum or the experimentally observed mass of the any chosen resonance state.A constant term, as it occurs in the relationJIf* ^  + «h ‘  +  V .. (6)obviously refers to the rest mass energy of the resonance system. While in simple relationM *  =  A + B y / L { h - [ - \ )  -\ -G L [L + \ ) -  (7)E m pirical m ass form ula 999the Gonstaut 1/erm A though may be taken to represent the rest mass energy ol' the system, it is to be recognized as being the same for a numbei’ of distinct statics differing in their total angulai momentum. It maiy also bo tempting to identify the constant term ol the JIf, with the mass of L — 0 particles ol the system, as has been done by Gerald Rosen (1972) in hi.s formulation lollowing Hchwmgcr (1963) of a fraction hypercharge and isospin etc semi-theoroUcal mass formula for Baryon mass spectra. But. we piefer to retain the Ij 0 massifs mferred from the values of f  (or alternatively from the intercopt on the ordinate of the graphs) as a eharactoi’istic mass of tlu^  resonance system. These mass values denoted Mt„t aie ccuniiared with those of L - -  0 particle mass values denot ed^Cip hi table 4 The for the A-systmu of resonances appe^ arsTable 4 Gompari.son of the Mi^i with-K ob o jia iices  S ysL omMintInl/fiicepl.(M e v )n N  R cH u n a n co  N  AN  K~ R e s o n a n o o967-9931) 133K 080,6inaHM fo r  L  -=  0 (A le v )939-0 1230-1230(Mov)18-3939(108-0706-103 0805)to bo aiiarmousl^  ^ largo. TJiis .suggests that L --  0, A-i’iisonaiil stat(' must bo distiinguished from the L =  {), A-partielo state (Vasavada cl ol 1966, Kmi cl cil 1972). The reason for distiugui,siring between and values (or tin*particle L — 0 and the L -  0 resonant, stales) in gerrcial thejidoro is that we presume essentially different stiengths ol stiong mtemetion as may^  be resjionsihh* for the lormation of /y =  0 resonant states oi‘ the L 0  particles This would become clearly established at siibsetjuont. stage of this paperAs regards the se(!ond parameter (j1 oui* w'oik it. is convenient to considerq f-\ gy/L{L-\-\) ~\-L{L-\ 1) Here if /  and li vanish, wc note that, g — \jh giverr by Sarnia (1903) and thus simply/- corresponds to a single parariieti'r oi‘ a fmidamental length— associated Avith the state as defined by him This jiara- meter also gives the direct, dcpeiideucc on the relative orbital angular momentum o f either qor  in our relations. Rroni the values given in table 2 it is noteworthy that /  and h are small compared to g excepting in the ease of A systeiu. The g values for the diffori*ut syst/Cms considered approximatclj'^ are oJ' the same order o f magnitude, which is perhaps the reason why 8arina could obtain a charac­ter Lstic b parameterThe occurrence of L(i>4-J) in. the empirical mass formulae has been prcviouslj'  ^interpreted by French ct al (1907) and Maglic (1900) as indicative of a rigid- rotator or Bohr-type jh^^sical modal for baryon resonances of both strange and iioii‘-8trango character. But in view that the data by that procedure does not1 0 0 0 K. L. Narayana and S. B. Patiln>pi’()(liico Iho accopiod values oi angular momenta, French et al (1967) interpret in terms of a classical pictui'o of the jiioii field coupled to the nucleon source Muough a non-rolaliv'istic limiL ol the relativistic mtoraction, g{rr A i^). Thus they sugg(!st one can consider an induced moment of inertia m the nucleon to explain the ob,served baryon masses reasonably and w^ ith the accepted values of their angular momenta.Contraiy lo this pioiiosition of Bohi-type or rigid rotator model we get a negativif value for thi' coefficient h or G oi' L{L~] 1) term in our relations, excepting in the castis ol j[/odd but where these Vanisli. The negative sign we suggest to intm-pret as an o]iposiMl nature of the resonance system to exist as a rigid I'otatoi. The suhtiact,ive tendency m tJie energy cannot, be argued as the possi­bility of minimization in enmgy and hence towards <i stable rigid rotatoi model ol resonance, system because the older ol negativii energies involved (withm the mass limits considered) are very small. The positive' contribution is relatn-ely compart'd lo tJie contributions from a rigid rotator typt' structure (i.c., from the third term mvtdvmg the factor L{Ij-\-\))Mext by (ixtending the conct'pt of liindamental length introduced by Anastassov (1974), Shivaram I'v' Sinha (1974) as one ol the universal constants, to hold good also lor the resonance systems, wo derive the respt'ctive coupling con.st.ants associated witli each mass value of the sy.stt'iiis. For exainjilt', il the mass of the A-rtrsonance state of orbital angular momentum L ~  2, with the firedominant nN mode of decay is chosen thtm,f7r“tic1690^ nC~~rr... (8)Wr is the i-esonance mass obtained Irom eq. (6) /„ is the fundamental lengt.h and f/y“/fec IS coupling constantUtilising the value of/,, given by Hivarain fSmha (1974) — 2-8 X 10“ ^^ ems.we get.fie 2;5-971 ... (9)for this ca.se it is note-wrothy that, tins valuer is about t.wici' thal. for the strong interaction of (‘lementary pa.i tielt's Wo define tlierefori* (/, as some kind of a resonance charge, similar to the baiyonic or nuclear (diargc. Interesting further, is avc gotjv-U-  -  \ n  ^ ^/^L 1-1) \~C^L{L+l) (10)Al Gwith yU — - etc. and similar expression for and 6^ .^ The evaluatedEmjpirical mass form ula 1 0 0 1coupling constants associated with the dilTorent resonance masse,s aie listed in table 5 and table 0 sIiom s^ t,he values The lists o f (joupling (‘onstantsare useful for studies of the unitary syiumtcry models. Further the relation given above impbes that every few baKsic couplmg consianls, A ,^ are enoughto generate the J‘c s^onant statics of a given j esonaucu system and that these various states arise,y only as a oousoquonce of ihoii relative orbital angular momentum values. We ,state that the constants A ,^ B^ , reprosont physically the strong interactron, the medium-strong interaction and low-stioiig jiiteraclions ros- pectivoly ol a two particle resonance* system Tire low-strong ijiteractions how­ever is opposed to both the nretliurn-strong and strong mteractious. With increasing j-esonance mass this postulation results in a dominant low-strong, mteractioir contribution. Beyond the limit ol mass values considered horC; recent expenmontaJ lindirrg has l>e(‘n found to have an exponemtial irrereaso This shows t-hat for that range the low-strong interactions may become additive and lurthor involve higher order tcrm,s. l^xtension of om' relation may be attempted as and u'-hen uirainbiguous higher experimental I’osonanco mass data becomes availableTable 5. The calculatt^d interaction strengths using tlie relationHeIqC~nOrbitalAngularJVlomontmiiTrA-RoHoiiances A^ 'HyHLnm A-syatomn/c”IleBonancesA-systom1 21 55* 23-96*2 23-971 27-66141 25-74103:j 31-06* — 32-66*\ 31-3776B 35 4035 32-71427fi 37 6S3rjR 40'43G51 38-64666R 42-93/J03 46-08822 43-57005* Exporirnontal mass value IS used to calculate the xntoiaction strengthTable b. Oalculatod constants and (J^  adopting tlu^  relationsetc.KConstantsN-Bystom A-system A-system13-58416 18-98221 15-962274-56788 3-68484 4-29679-0-12950 -0-05779 -0-122691 0 0 2 K . L. Narayana and S. B. PatilAgarwal B . K. 1971 I  ml. J r h y s  45, 406 AuoE!ta;:3f3Uv A . H . 1974 N viovo O itnento 21B, 641.Bhuttacharjoe B J. 1970 Jm liau J . P hya  44, 60; 1974 Ibvd., 48, 1075. BliatLacliaijon B . J 1971 Prot., 39tli Tml S c C ong. P a rt J/ I, Se. I l l ,  81. Biidh Ham 1966 Nuot)o C im cnto, Soriti X . 43. 721 (Jariuthor I'. & N ioto M M, 1907 PJiya P ev  163, 1040 Chow C. P  1962 P ev . M od . P hys. 34, 371)F ionoh -i. 1>., L.amh, W  H & M owat J  U. 1967 P hya . licv . 163, 1754. (loralcl Hoaon 1972 P hya  Pv.v D5. 1100,H agodioii K  1970 N u cl. P hya  B24, !J3Iow a S A  h4jjoc5l.Loaot>iiy o1‘ Baryon Sl^atos (rroj>niil).Him Y  H. & VasavacU K. V . 1972 P hya  P ev . D5, 1002,Loo T . Y . & C hon-I’Hai C. T  1974a Phya, J f e v .  DIO, 3110.Loo T  Y^  & Chfm I'wai C T  1974b P hya. R eo. DIO, 2960 Ma^lio B . 1960 AWiOOO Oitnento 45A, 949.X^larayana Iv Jj <fc Sharad N avaio 1976 JSluovu (Jiubcnto 33A, 041 Onoda S. JSe Takasugi K. 1974 Phya Rev. DIO, 3113,Heggo 1900 N u ooo  O^mevto 18, 947.Hai'iiia K . V L 1903 IT  JShiovo CHmento Sert-e X ,  30, 2S ►Schwinger .P 1903 Phya. R ev. Letters 18, 797.Sivaram C^  A Siiiha K 1974 Phyaica N cica  5, 146.Sternhoimoi H. M. 196S J^hyaical Revtrm  170, 1267.Vaeavada H  V . & H im  Y. S 1 066 P hyaiea l R eview  152, 1250.

A Physical Model from the Mass Empirics of Two-Particle Baryon Resonance States and a Postulation of Medium and Low Strong Interaction

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A Physical Model from the Mass Empirics of Two-Particle Baryon Resonance States and a Postulation of Medium and Low Strong Interaction

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