Higgs -quantized coupling constants for quarks and metastable particles

Transcription

1 *Formrly at Lawrnc Livrmor National Laboratory Higgs -quantizd coupling constants for quarks and mtastabl particls Malcolm H. Mac Grgor* Fllow, Amrican Physical Socity 130 Handly Strt Santa Cruz, CA April 3, 01 -mail: Abstract Higgs mass-gnration particl coupling constants g ar calculatd as suggstd by Ldrman and Hill, who us th top quark t as th rfrnc Higgs mass. Accurat -quantizd coupling constants g ar obtaind for th mtastabl lptons, constitunt quarks, proton, B c mson, and W and Z gaug bosons, whr m / c, and whr th masss ar th inrtial Einstin masss E/ c of th stats involvd. An analogous st of invrs Higgs-lik coupling constants f is also prsntd in which th lctron srvs as th rfrnc mass, as suggstd by th -boostd mass gnration structur of th xprimntal f and g coupling constant valus. PACS numbrs: 1.10.Kt, 1.60.Fr, b *Formrly at Lawrnc Livrmor National Laboratory 1

2 A ky problm in lmntary particl physics has bn th xprimntal sarch for th Higgs boson, and th clarification of its rol in gnrating particl masss. A stp towards th solution of this problm is providd by th analysis of Ldrman and Hill [1], who not that th Frmi coupling cofficint G F in th four-point function for calculating wak-intraction dcays [] corrsponds to an nrgy scal of 175 GV, which matchs th mass of th top quark t. Thy idntify th scalar Higgs fild with th fild that is rquird for th Frmi wak intractions in a fild-thory contxt, and thy commnt that th top quark mass may play a significant rol in th Higgs formalism. Th Higgs fild coupls linarly to particl masss with a coupling constant g, so Ldrman and Hill writ th Higgs lctron coupling constant g in th form m gm t, which givs g = Howvr, thy mphasiz that in th absnc of a thortical framwork for th coupling constants, th numrical valu for g tlls us almost nothing about th lctron mass [1]. In th prsnt papr, w introduc an -quantizd mpirical framwork that accuratly rproducs this numrical valu, whr / c 1/137. Exprimntal support for th Ldrman and Hill viwpoint is obtaind from th mass valus of th W and Z gaug bosons and top quark t. Du to thir thortical importanc, ths masss hav bn vry accuratly masurd, as follows: [3] m W GV, m GV, Z m GV. t Th intrsting rsult hr is that th sum of th lctrowak W and Z masss is GV, which matchs th (supposdly-unrlatd) t quark mass of 17.9 GV to an accuracy of 0.8%. Th prcision of this mass quality indicats that th mass of th t quark is in som mannr rlatd to th masss of th W and Z, and hnc th t quark mass logically plays a rol in th Higgs mass formalism, as Ldrman and Hill suggst [1]. A clu to th thortical form of th Higgs coupling constants g can b obtaind from th lctrowak liftims of th mtastabl (t > 10 1 sc) particls, which display an xprimntal quantization in powrs of th fin structur constant 1/137 that xtnds ovr many ordrs of magnit [4, 5]. This liftim -quantization logically rlats to th mass structurs of ths particls, and hnc to th valus of thir Higgs coupling constants. W can invstigat this conjctur xprimntally by tracing out a mass-linkd xcitation chain btwn th lctron and th top quark t. W hav alrady stablishd th top link of this chain, which is in th rlationship

3 btwn th W, Z and t masss. Dnoting th avrag mass of th W and Z gaug bosons as th mass unit m ( m + m )/, WZ W Z w hav m m WZ, at th 1% accuracy lvl. t Th scond link in th lctron-to-top-quark mass chain has to span a rang of masss that xtnds from th low-mass (blow 1 GV) wll-populatd rgion of obsrvd particl stats to th high-mass (abov 80 GV) sparsly-populatd rgion of th W and Z bosons and t quark. Thus this mass link has to xtnd for at last an ordr of magnit in particl mass valus. W obtain this link in a spctacular mannr by stying th Tvatron collision procss that crats th W and Z particls. Countr-rotating protons and antiprotons collid togthr in th Frmilab Tvatron at TV nrgis. Th high-nrgy protons ar rlativistically flattnd into pancaks that ach contain thr ssntially fr u and d quarks, and th collisions that produc W s and Z s ar th rar vnts (1 in ) in which th collision is a dirct hit btwn a u or d quark in th proton and a u or d quark in th antiproton, with th colliding pair rciving most of th collision nrgy. Trating th u and d quarks as qual-mass constitunt-quark mass units dnotd as, w dfin thir avragd xprimntal mass valu as m (1/ 3) m. To tst th conjctur of mass quantization, w assum that high-nrgy tvatron collisions impart an " -boost" factor of 1/ 137 incras in mass or nrgy to ach colliding plus quark mass-unit pair, thus crating a pair of high-mass a combind xprimntal mass-unit valu plus gaug boson quark mass units, which hav m + m ( m m ) / 3 85,718 MV. p p Comparing this -boostd gaug boson quark pair mass to th xprimntal WZ avrag mass, m ( m + m ) / 85,793.3 MV, WZ W Z w s that ths two masss agr to an accuracy of 0.09%. This mass quality suggsts that th W and Z vctor bosons ar formd from (u, d ) and ( u, ) quark combinations. It also stablishs th scond link in th lctron-to-top-quark mass chain a link that is -quantizd. Th bottom link that is still ndd is th proton-to-lctron mass ratio. Procding as abov, w not that th xprimntal muon-to-lctron mass ratio is 3/ and th xprimntal protonto-muon mass ratio is 9, which combin togthr to giv m / m 7/, (0.75% accuracy). p d p 3

4 This bottom link thus contains a scond -quantizd " -boost" mass xcitation. Joining ths thr lctron-to-top-quark mass links togthr givs th calculatd mass [6] m (18 / ) m GV, t which matchs th xprimntal mass m t = 17.9 GV to an accuracy of 0.1%.Th individual links in this mass xcitation chain all hav xprimntal accuracis which ar at th 1% lvl. Thus th xprimntal accuracy of th xcitation chain xtnds throughout th mass rgion from th 0.5 MV lctron to th 173 GV top quark. Th currnt xprimntal accuracy [3] of th top quark mass is 1.1 GV, or 0.6%, so th calculatd accuracy of 0.1% sms somwhat fortuitous. Th lctron-to-top-quark mass xcitation chain is diagrammd in Fig. 1, whr it portrays th following mass gnration squnc: (1) lctron to muon to proton; () proton back down to proton-quark; (3) proton-quark to gaug-boson-quark to gaug boson to top quark. This xcitation squnc disrgards quantum numbrs and has in common only th total nrgis of th particl stats, which ar proportional to th inrtial masss of th stats. Th inrtial mass of a particl is dfind by th Einstin quation E = mc. W can dnot this inrtial mass as th Einstin mass of th particl. Thus this particl systmatics can b dscribd in ithr an nrgy or an Einstin-mass framwork. Fig. 1 uss th Einstin-mass framwork. In this framwork, th quark inrtial masss ar qual to th convntional Standard Modl constitunt-quark masss [7]. Th Fig. 1 xcitation chain also appars as Fig. 3 of Rf. [8], whr it is in th nrgy framwork. Th challng implicitly containd in th Ldrman and Hill discussion of th Higgs coupling constant is to obtain a thortical basis for th numrical valu g = of th lctron coupling constant. Th mpirically-stablishd mass gnration chain displayd in Fig. 1 givs g / , which corrsponds to th calculatd mass valu m t = m / g = GV for th top quark t. This mass gnration chain provids a phnomnological basis for th numrical valu of g. Th calculatd valu for m t is vry clos to th currnt xprimntal valu of 17.9 GV [3], so w mploy this calculatd valu as th rfrnc mass for th Ldrman-Hill Higgs coupling constants g, sinc it brings th L-H calculations into prcis agrmnt with a matching st of lctron-basd coupling constants f that w dfin blow. W can xtnd th validity of this -quantizd Higgs mass gnration mchanism by applying it to th u, d, s, c, b st of quarks, th B c mson, th and lptons, th proton, and th W 4

5 and Z gaug bosons and gaug-boson quarks, which ar basic xcitations that rprsnt frmion ground stats.. Th masss of ths stats ar thir inrtial masss, which ar dtrmind from th xprimntal masss of th particls involvd. Th u, d, s, c, b quarks ar assignd constitunt-quark masss [7], whos xprimntal valus ar dfind as th particl mass dividd by th numbr of quarks and antiquarks in th particl. Th u and d quarks ar tratd as qual-mass ntitis that carry th mass-unit labl. Matching quark-antiquark binding nrgis ar small -3% at nrgis blow GV, and ngligibl abov 6 GV and ar not applid hr [9]. Quark-quark binding nrgis ar ngligibl. Th calculatd and xprimntal Higgs coupling constants for this st of particl stats ar displayd in Fig., togthr with th xprimntal data usd to dtrmin inrtial masss for ths stats. Exprimntally, th Higgs particl coupling constants g occur on thr -quantizd Higgs lvls: th g lvl, which contains just th lctron and positron; th 1 g lvl, which contains th lptons and hadrons; and th 0 g lvl, which contains th W and o Z gaug bosons and top quark t. On ach of ths lvls th lowst-mass stat acts as a unit mass, with highr-mass mtastabl xcitations occurring as multipls of th unit mass. On th lvl, th MV lctron, th only mmbr on this lvl, has th Higgs constant g = /18. Th unit mass on th 1 lvl is th 105 MV muon, whos Higgs constant is a factor of (3/ ) largr than g, so that g = /1. Th highr-mass lpton,, s, c, b quarks, B c mson, and p proton hav Higgs constants that ar accurat multipls of /1, as displayd in Fig.. Th unit mass on th 0 lvl is th 4.86 GV gaug boson quark, whos Higgs constant is a factor of 3/ largr than g so that g 1/4. Th WZ avrag-mass stat and th t quark hav Higgs con- stants that ar accurat multipls of 1/4. Th graphical logarithmic display in Fig.. of th Higgs coupling constants g of ths fundamntal particl stats rvals thir xcitation mchanisms in a particularly transparnt mannr. In 1 0 particular, th thr -quantizd Higgs xcitation lvls g,, ar clarly in vidnc. Th xprimntal data usd to obtain th xprimntal Higgs constants ar listd at th lft in th figur. Th agrmnt btwn th calculatd and xprimntal Higgs valus is at an accuracy lvl of 1%, with no fr paramtrs usd xcpt for th us of th calculatd valu for th top quark rfrnc mass. Th largst dviations ar for th s and c quarks, whos masss and 5

6 coupling constants ar dducd from th and J/ msons rspctivly, and which rquir small quark-antiquark binding nrgy corrctions [9] that ar not incld hr. Th particl stats that ar displayd in Fig. rprsnt frmion -boost xcitations of (1) th lctron to th muon, and () th proton quarks to th gaug boson quarks. Thr is also a boson -boost xcitation of th lctron to a st of psoscalar PS pion quarks, which ar rquird to rproduc th,, ', K msons. Th Higgs coupling constants g for ths particls ar multipls of th unit coupling constant /18. Ths PS msons ar discussd lswhr [10]. Th complt st of unit Higgs coupling constants g rquird for gnrating all of ths basic Einstin-mass particl ground stats is numrically as follows: m MV (J 1/); g /18; m m / MV ( J 0) ; g /18; PS PS mμ 3 m / MV ( J 1/ ) ; g /1; m 9 m / 43,18 MV ( J 1/ ) ; g 1/4. Th subscripts hav bn dfind abov. Th angular momntum J-valus rprsnt th intrinsic angular momntum ach mass unit contributs to th particl xcitation. Th ( /18) psoscalar mson mass unit PS dfind hr srvs as a spinlss mass quantum that dos not contribut angular momntum. Th factor of 3/ ratio btwn th (J = 0) 70 MV ( /18) mass unit and th ( J = 1/) 105 MV ( /1) mass unit can b attributd to th mathmatical rsult that a fully rlativistic spinning sphr of mattr is half again as massiv as its nonspinning countrpart [11]. W can obtain an altrnat formulation of th Higgs coupling constants by mploying th lctron as th rfrnc mass. This is suggstd by th fact that th four basic Higgs mass units shown abov ar dfind in trms of as th lctron mass gnrator, and ths mass units rproduc th obsrvd -quantization of th Higgs coupling constants g in Fig. 1. W dfin th lctron-basd Higgs coupling constants by th quation f m / m, and w us th invrs constant X 1/ 137 in th quations for th Higgs coupling constants, with X acting as a particl mass gnrator. In this rprsntation, th quation for th top quark Higgs coupling constant is f 18X. Th four basic mass units (in MV) ar writtn in th X-quantizd notation as t, /), /). PS ( f 1), ( f X) ( f 3X 43,18 ( f 9X i i 6

7 Th -quantizd Higgs coupling constants g of Fig. 1 transform into th X-quantizd coupling constants f displayd in Fig., whr th xprimntal particl and quark mass valus, and th calculational accuracis, ar th sam as in Fig. 1. Also indicatd in Fig. ar four groupings of particl stats whos masss link togthr in a systmatic mannr. Th rsults prsntd hr lad to th following st of obsrvations: (1) Th masss of lptons, quarks, hadrons, and gaug bosons combin togthr in a unifid xcitation formalism, which dpnds only on th total nrgis E, or quivalntly th inrtial masss m E/ c, of th stats involvd, whr w can dnot ths inrtial masss as Einstin masss. Th mass qualitis btwn mixd typs of particl ar in ssnc nrgy qualitis. () Th mass/nrgy rlationships shown hr ar basd on xprimntal data, and th accuracis of thir systmatic xcitation pattrns ar obtaind without using adjustabl paramtrs. (3) Th Frmi wak-intraction formalism is now commonly intrprtd as a fild thory, and th scalar fild that it uss has an nrgy scal that is comparabl to th mass of th top quark t. Thus th t coupling to this fild is or ordr unity, and this coupling can b convnintly usd as th rfrnc mass for dfining a st of Higgs constants g i = m i /m t, whr th scalar Frmi fild srvs as th scalar Higgs fild. (4) Particl mass rlationships can b stid in th contxt of particl Higgs constants instad of particl masss or nrgis, which brings out th mass rlationships in a transparnt mannr. (5) Th Higgs constants for th basic particl and quark "ground stats" ar -quantizd, and this mpirical -quantization is an ssntial fatur of a mathmatical formalism that tis togthr ths fundamntal ground stats in a cohrnt and comprhnsiv mannr. (6) This formalism srvs as th first stp toward a thortical framwork that, as Ldrman and Hill not [1], is rquird for a maningful intrprtation of th Higgs coupling constants. 7

8 Rfrncs. [1] L. M. Ldrman and C.T. Hill, Symmtry and th bautiful univrs (Promthius, Amhrst) 004, p. 8. [] A. Sidn, Particl Physics (Addison Wsly, San Francisco) 005, Ch. 8. [3] Rviw of Particl Physics: K. Nakamura t al. (Particl Data Group), J. Phys. G 37, (010), and 011 partial updat for th 01 dition. [4] M. H. Mac Grgor, Liftims of SU(3) groups and particls as a scaling in powrs of, Phys. Rv. D 13, 1976, pp ; Int. J. Mod. Phys. 0A 005, pp (invitd rviw articl). [5] M. H. Mac Grgor, Th powr of (World Scintific, Singapor), 007, pp and Ch.. [6] M. H. Mac Grgor, arxiv: hp-ph/ (006). [7] A. Sidn, Rf. [], Ch. 6. [8] M. H. Mac Grgor, vixra: (01). [9] For discussions of quark-antiquark binding nrgis, s Rf. [5], Sc. 3.15; Rf. [8], Fig. 4. [10] For discussions of th psoscalar msons, s Rf. [5], Sc. 3.7; Rf. [8], Figs. and 6. [11] M. H. Mac Grgor, Phys. Rv. D9, 1974, pp , App. B; Th natur of th lmntary particl (Springr-Vrlag, Brlin) 1978, Ch.6; Th nigmatic lctron (Kluwr, Dordrcht, 199, Part III; Rf. [5], Ch. 4. 8

9 Fig. 1 ThEinstin-mass diagram for th gnration of th top quark t from th lctron, showing two factor-of-137 -boosts. Th rsulting Einstin mass quation, m t (18 / ) m, which faturs th rnormalizd fin structur constant, is accurat to 0.1%. Th quark stats p and gb rprsnt nrgy-avragd proton and gaug boson u and d quarks, rspctivly. 9

10 Fig.. Calculatd and xprimntal Higgs coupling constants g, shown togthr with th accuracis of th -quantizd thortical valus (in parnthss) and th data usd for th xprimntal valus. Th thr -quantizd Higgs mass lvls ar clarly in vidnc in this graphical portrayal of th Higgs coupling constants. This -quantizd coupling constant formalism contains no adjustabl paramtrs, and its accuracy lvl is bttr than 1%, xcpt for th s and c quarks, whr low-mass binding-nrgy corrctions [9] ar ndd (but ar not applid). 10

11 Fig. 3. Th particl coupling constants of Fig. convrtd into lctron-basd X 137 Einstin mass units. Thr X-quantizd mass lvls ar clarly in vidnc. Four mass-rlatd groups of particls ar dlinatd by vrtical links, as follows: (1) th and s quarks as multipls of th muon mass; () th accurat (0.59%) linar mass spacings of th muon, proton, and tau lpton; (3) th succssiv mass triplings of th s, c, b quarks, and th Bc bc mson mass; (4) th tvatron collision X-boost from to quark pairs, which thn combin to form th W and Z gaug bosons and top quark t as a mass triad on th X Higgs lvl. 11