The -quantized systematics of quark-dominated elementary particle lifetimes

Transcription

1 The -quantized systematics f quark-dminated elementary particle lifetimes Malclm H. Mac Gregr* Fellw, American Physical Sciety 130 Handley St., Santa Cruz, CA URL: thepwerfalpha.cm mhmacgreg@al.cm April 23, 2012 Abstract The experimental lifetime systematics f the 36 lng-lived quark and particle metastable grund states is displayed graphically n a lgarithmic glbal -grid spaced in pwers f the fine structure cnstant ~ 1/137 and centered n the ± lifetime. The hadrn lifetimes separate int fur nn-verlapping lifetime grups, each dminated by a single quark flavr. These in turn divide int slw flavr-breaking electrweak decays and fast flavr-cnserving paired-quark and radiative decays, separated by 4 lifetime gaps. The lng-lived electrweak subgrups feature cntral lifetime (CL) bands that map nt the -grid lines, plus particles that are displaced by factrs f 2, 3 r 4 frm the CL. The quark lifetime dminance rule is c > b > s. The neutrn and mun lifetimes lie n the glbal -grid. The taun lifetime fits int the c-quark lifetime grup. PACS numbers: Kt, Fr, b *Frmerly f Lawrence Livermre Natinal Labratry 1

2 1. Intrductin and summary Tw imprtant experimental prperties f an elementary particle are its mass m and its mean life r lifetime. Experimental measurements f these quantities have becme very accurate in the past few decades. The updated 2011 editin f Review f Particle Physics (RPP) [1] lists apprximately 171 particle states 3 leptns, 167 hadrns, and the tp quark t that have welldetermined mass and lifetime values. The hadrn states are accunted-fr as cmbinatins f the Standard Mdel u, d, s, c, b quark flavrs. The glbal systematics f particle and quark masses has been well-studied [1], but the glbal systematics f particle and quark lifetimes has received less attentin. The hadrn lifetimes span 28 rders f magnitude (plus the stable electrn and prtn), and they separate int lng-lived shrt-lived 21 ( 10 sec) quark grund-state cnfiguratins and 21 ( 10 sec) particle excited states. Our analysis here is centered n the lng-lived quark grund states, which cntain unique infrmatin abut the prperties f the particles and quarks. Their lifetime regularities reveal the s, c, b flavr structure f the particles, and lead t the cnclusin that particle lifetimes are dictated by their dminant quark flavrs. In the present article, we display plts f the experimental lifetime data that illustrate and delineate imprtant features f the quark and particle lifetime systematics. As a guide t this discussin, we summarize here the main cnclusins frm this glbal lifetime analysis. (1) The particle mean lives [1] can be separated int metastable unpaired-quark grund state decays with lifetimes > sec, paired quark-antiquark grund-state decays with lifetimes frm t sec, and excited-state decays with lifetimes frm t sec (Fig. 1). (2) The unpaired-quark and paired-quark grund states are each divided int fur discrete lifetime grups (PS), (s), (b), (c) in the rder f decreasing lifetimes, with each grup dminated by a single quark flavr, as fllws: (PS), the pseudscalar (u, d, s) up and dwn quark mesns and strange quark kans; (s), the strange hyperns; (b), the bttm mesns and baryns; (c), the charm mesns and baryns. These grups fllw the quark dminance rule c > b > s (Fig. 2). (3) The > sec grund-state lifetimes i are quantized in pwers f e 2 / c1/137 i with respect t an -spaced lgarithmic lifetime grid i / that is anchred n the lifetime (Figs. 3, 4, 6-9), and is dented here as the lifetime "glbal -grid". x 2

3 (4) The flavr-cnserving paired-quark grund state lifetimes are shrter than the matching flavr-breaking unpaired-quark grund state lifetimes by factrs f apprximately 4 (Fig. 3). (5) Each f the fur unpaired-quark (PS), (s), (b), (c) lifetime flavr grups has a "central lifetime" (CL) band f particles plus particles that are separated frm the CL by integer lifetime ratis f 2, 3, r 4 (Figs. 4 and 5). (6) The (PS), (s), (b) CL central lifetimes crrespnd t the glbal -grid integers xi 0, 1, 2, respectively, but the (c)-grup CL (and the ther charmed-particle grund states and the leptn) are shifted ff the x i 2 grid psitin by a cmmn factr f 3 (dented here as the lifetime "charm crrectin factr") in the directin f shrter lifetimes (Figs. 4 and 11). (7) The nn-strange (PS),,, ' mesn lifetimes crrespnd t the glbal -grid integer psitins xi 0, 4, 5, 6, respectively (Figs. 6-9). (8) The strange (PS) L K, K, K S kans are -quantized n a lifetime lcal -grid that is anchred n the K lifetime and is shifted frm the glbal -grid by a factr f 2 twards shrter lifetimes (Fig. 10). (9) The neutrn and the leptns are -quantized n anther lcal grid that is anchred n the and lifetime and is shifted frm the glbal -grid by a factr f ~1.6 twards shrter lifetimes (Figs ), and they exhibit the fllwing experimental lifetime ratis: (a) 4 /, where the mun and neutrn lifetimes represent flavr-cnserving and n flavr-breaking decays, respectively (see cnclusin (4) abve and Figs ). (b) 3 / /3 (t 2% accuracy!), where the hadrnic charm crrectin factr f 3 is applied here t the leptnic lifetime (see (6) abve and Fig. 12). (10) A plt f particle lifetimes versus particle masses (Fig. 14) illustrates the lifetime and mass similarities between the leptn and the charmed (c)-grup mesns. This suggests that the weakly-interacting taun is cnnected in a generatinal sense with the c quark family, and nt with the b and t quark generatin family, as is generally prtrayed. This cnclusin is reinfrced by the accurate 3 / /3 lifetime rati displayed in (9b) abve, which uses the hadrnic charm crrectin factr f 3 fr the taun lifetime. 3

4 2. The glbal structure f particle lifetimes Fig. 1 shws the 170 measured particle mean lifetimes [1], tgether with the tp quark lifetime, which are pltted n a lgarithmic scale that extends frm the neutrn ( ~ 10 3 sec) t the W and Z gauge bsns ( ~ sec). These lifetimes divide naturally int fur znes: Zne 1. The regin with lifetimes lnger than 10 7 sec, which cntains the stable prtn and electrn and metastable neutrn and mun (and their crrespnding antiparticles). These are the fur lwest-mass spin ½ fermin states. The nuclens (p and n) carry the baryn quantum number, and the leptns (e and ) carry the electrn and mun leptn quantum numbers, respectively. Zne 2. The lifetime regin frm 10 7 t sec, which cntains 24 hadrns and the tau leptn. The 24 hadrn states are unpaired-quark grund states, whse decays require flavrbreaking electrweak gauge bsn transfrmatins that slw dwn the decay rates. The taun () carries the tau leptn quantum number. Zne 3. The lifetime regin frm t sec, which cntains 10 hadrn states that are mainly paired-quark grund states whse decays d nt require flavr-breaking transfrmatins. Zne 4. The 133 particles with lifetimes shrter than sec (1 zeptsec), which are excited states that decay back dwn t the metastable grund states. The experimental lifetime regularities displayed in the present article ccur mainly in the metastable grund-state particles (including leptns), which have lifetimes > 1 zeptsecnd. The very shrt Zne 4 mean lifetime f the tp quark t is deduced frm its resnance width /. A crucial feature f the t quark lifetime is that it is shrter than the time theretically required t cmbine with anther quark and frm a particle [2]. Thus this represents the direct determinatin f an intrinsic bare quark lifetime. 3. The c > b > s quark dminance rule and the fur quark lifetime families Fig. 2 displays the experimental flavr regularities in the lifetimes f 32 quark grund-state particles, all f which have metastable lifetimes > sec. The 24 unpaired-quark grund state lifetimes in Zne 2 (tp) clump int fur flavr-separated grups, which are labeled (PS), (s), (b), (c), and the 10 paired-quark and/r radiative decay grund state lifetimes in Zne 3 (bttm) clump int a matching set f flavr-separated grups. These lifetime flavr grups reflect 4

5 the lifetimes f individual quarks within the particle, which are the same fr bth mesns and baryns. The lifetime flavr grups are created in accrdance with the quark dminance rule c > b > s, where the shrtest quark lifetime in a particle dictates its decay. The wrkings f this quark dminance rule are spelled ut in the captin t Fig. 2. The c quark has the shrtest "intrinsic" lifetime, fllwed by b quark and then the s quark. This isn't the rder we expect t find, since high-mass excitatins such as the b quark usually are mre unstable than much-lwer-mass excitatins such as the c quark. The nn-strange and pseudscalar (PS) mesn lifetimes ccur in the (PS) lifetime grupings, as d the strange K L and K pseudscalar kan lifetimes, but the strange K S kan lifetime appears in the (s) flavr grup. The wrkings f the c > b > s quark dminance rule are illustrated mst clearly by the B c ( cb, bc ) and B s ( bs, bs) mesn lifetimes, which are labeled in Fig. 2. The B c mesn is cmpsed f a b and c quark-antiquark pair. The B c mass is intermediate between the bb and cc masses, but the B c lifetime is nt intermediate between the b and c lifetimes: it falls squarely in the middle f the (c) flavr grup. Similarly, the B s mesn is cmpsed f a b and s quarkantiquark pair. The B s mass is intermediate between the ss and bb mesn masses, but the lifetime falls directly in the narrw (b) flavr grup. This same quark dminance rule carries ver t the baryn lifetimes, where the c B s csd lifetime appears in the (c) flavr grup, and the b bsd lifetime appears in the (b) flavr grup. These tw baryn states are labeled in Fig. 2. The u and d quarks als ccur in particle states f the (s), (b) and (c) lifetime flavr grups, but any effect they have n lifetimes is superseded by the c > b > s quark dminance rule As can be bserved in Fig. 2, the separatin distances between the matching unpaired-quark and paired-quark (PS), (s), (b), (c) lifetime flavr grups are smewhat larger than the factr f 10 7 shift that is emplyed between the Zne 2 and Zne 3 crdinate ranges in Figs. 1 and 2. These very large unpaired-quark t paired-quark separatin distances are clearly f theretical interest. As a clue t their rigin, we nte that the K and K S mesns displayed in Fig. 2, which bth feature K decay mdes, and which ccupy similar psitins near the shrt-lifetime bundaries f the (PS) and (s) flavr grups, have the lifetime rati This is K K S 5

6 within 1% f the numerical value 1 / , where e 2 / c is the fine structure cnstant. The ccurrence f the cnstant in the K t KS lifetime rati, and in the (PS) t (s) unpairedquark flavr grup separatin distance, suggests that may als play a rle in the spacings f the matching unpaired-quark t paired-quark (PS), (s), (b), (c) flavr-grup pairs f Fig. 2. This suggestin is reinfrced by the experimental 4 lifetime gaps that are bserved in these matching flavr-grup pairs, as discussed in the next sectin and displayed in Fig The "glbal -grid" scaling and 4 gaps f the (PS), (s), (b), (c) lifetime flavr grups The lifetimes in the elementary particle displays f Figs. 1 and 2 were pltted in units f secnds. In rder t investigate the quantitative nature f the elementary particle flavr grups delineated in Fig. 2, we cmbine the 32 hadrn particle states f Fig. 2 int a single 10-7 t sec lifetime dmain, and we add in 5 clsely-related hadrns. Then, in rder t ascertain if the fine structure cnstant e 2 / c1/137 enters int the lifetime systematics, we select the mesn lifetime ratis t that f the as the reference lifetime, and we express the lifetimes i f the ther particles as by means f the equatin x which defines the lifetime glbal i i /, -grid. If the lgarithms x i t the base have integer values, then the cnstant is relevant t the systematics. The results f this lifetime representatin are displayed in Figure 3, which cntains 37 metastable hadrn states. Evidence f an -quantizatin f particle lifetimes appears mst clearly in the unpaired-quark flavr grups. The,,, ' nn-strange pseudscalar mesns in the (PS) lifetime regin at the tp f Fig. 3 prvide strng evidence fr a lifetime scaling in pwers f These lifetimes span 6 pwers f, r 13 rders f magnitude, and they clsely match the grid lines f the superimpsed lifetime -grid, which are spaced by factrs f 137. This result is reinfrced by the unpaired-quark (s) flavr grup lifetimes, which are a factr f 137 are shrter than the reference lifetime, and the (b) flavr grup lifetimes, which are a factr f (137) 2 shrter. These tw flavr grups ccupy the x i = 1 and x i = 2 grid lines, respectively. The unpaired-grup (c) flavr grup lifetimes d nt fall n an -grid line, but are shifted by a factr f 3 past the x i = 2 grid 6

7 line tward shrter lifetimes. Evidence f a discrete flavr grup substructure is visible in the (PS), (s), and (c) unpaired-quark lifetimes. This substructure is displayed in Fig. 4 and analyzed in detail in Sec. 4. Figs. 2 and 3 demnstrate the manner in which the metastable ( > 1 zsec) elementary particle grund-state lifetimes srt int fur flavr grups that each cntain slw flavr-breaking unpaired-quark decays and fast flavr-cnserving paired-quark and radiative decays. It is f interest t list the experimentally measured particles in these fur flavr grups: (PS) The pseudscalar mesns,k,k (unpaired); L,, ' (paired). (s) The s-quark flavr grups K, S,,,,, (unpaired); (radiative decay). (b) The b-quark flavr grups B,B,B,,, (unpaired); 1S, 2S, (paired). 3S s b b b (c) The c-quark flavr grups D,D,D,B,,,, (unpaired); 1S s c c c c c J/, J/ (paired), 2S * c c c D,,, (excited states). These lifetime flavr grups cntain mesns and baryns cmbined tgether in the same lifetime patterns. This indicates that an individual flavred quark acts the same way with respect t lifetimes whether it is in a mesn r a baryn. These quarks dictate the (nn-verlapping ) flavr grup lifetimes, which are separated by factrs f 1/137. Anther salient feature in each f these flavr grups is that the rati f the paired t unpaired lifetimes is a factr f rughly 4, with n rgue lifetimes appearing in the intervening 4 lifetime gaps. This is a huge leap mre than 8 rders f magnitude and it reflects the small cupling cnstants f the unpaired-quark electrweak decays as cmpared t the much larger cupling cnstants f the paired-quark and radiative strng decays. These apprximate 4 lifetime gaps invlve bth mesns and baryns. It is instructive t calculate the numerical values f the 4 gaps displayed in Fig. 3, which are as fllws: (PS) (133) 4 ; (s) (214) 4 ; (b) (105) 4 ; (c) mesn (90) 4 ; (c) baryn (182) 4. These ~ 4 gap sizes straddle the value (137) 4 that represents strict scaling. An imprtant pint t cnsider is that the 37 metastable hadrns represented here include all f the quark grund states, and there are n vilatins f the scaling systematics n rgue particles. Histrically, the verall framewrk fr this lifetime systematics was apparent in the early experimental data, and was published 7

8 prir t the discveries f the c quark [3] and the b quark [4], which have served t flesh ut the glbal lifetime -grid. The experimental lifetimes displays Figs. 2 and 3 feature an -quantized lifetime grid that emplys the renrmalized cupling cnstant 2 e/ c1/137, and nt the running cupling cnstant 2 ( q ) that is used fr theretical calculatins. The running cnstant matches the renrmalized value 1/137 at lw-energy scales 2 ( q 0), but increases t 2 ( q ) 1/128 at high-energy scales 2 2 mw ( q ) [5]. The accuracy f the fit f the experimental lifetimes t the renrmalized -grid indicates that it is the renrmalized cupling cnstant which is relevant t the decay triggering prcess. This cnclusin will be reinfrced when we extend the -grid t encmpass the lng-lived mun and neutrn (Sec. 7) 5. Quark flavr-grup central lifetimes (CL) and factr-f lifetime substructure Figure 4 is an expanded glbal -grid plt f the (PS), (s), (b), (c) unpaired-quark lifetime flavr grups displayed in Figs 2 and 3. It illustrates the fact that f these grups cntains a clear-cut "central lifetime" (CL), tgether with a grup substructure that brackets the CL, and which appears in the frm f apprximately integer lifetime ratis f 2, 3, r 4 between related particles. The vertical slid lines in Fig. 4 dente the average CL value fr each grup. These fur unpaired-quark flavr grups cntain the fllwing CL particles, respectively: (PS) The ( ) pseudscalar pin, which is bracketed between the reference lifetime in the equatin i / (s) The, hyperns, which are bracketed between the hyperns and K S kan. The, line in the directin f shrter lifetimes. The K x K L and K kans, and is the i that defines the lifetime glbal -grid., hyperns and the, average CL is displaced slightly frm the x i = 1 -grid and K s mesns bth have decay mdes, and their lifetime rati f clsely matches the -grid spacing f (b) The ( B,B,B,,, s b b b ) bttm mesns and baryns. All f the b-quark particles are In the CL grup, which is centered n the x i = 2 -grid line. s c c (c) The ( D,D,B, ) charm mesns and baryn, which are bracketed between the D ± 8

9 charm mesn and the c, c, c charm baryns. The c-quark CL is displaced frm the b-quark CL at x i = 2 by a factr f 3 t the -grid value x i = These CL grups demnstrate the wrkings f the quark dminance rule c > b > s. The mesn is squarely in the c-quark CL grup, and the Bs Bc bc bs mesn is squarely in the b-quark CL grup. The ne cmmn feature in the b-quark set f CL particles is the b quark itself, and the ne cmmn feature in the c-quark set f CL particles is the c quark. Thus it seems empirically clear that each particle lifetime is dictated by the stability f its single dminant quark. The particle grupings displayed in Fig. 4 cntain eight apprximate factr-f-2 lifetime ratis amng related particles, tgether with tw factr-f-3 ratis and tw factr-f-4 ratis. This brings up the questin as t whether these experimental lifetime ratis represent intrinsically integer values, r are just cincidental results. This questin is addressed in Fig. 5, where the numerical values f these ratis are arrayed s as t btain average values fr the factr-f-2, factr-f-3 and factr-f-4 lifetime intervals. As shwn in Fig. 5, the average lifetime rati fr each grup is within a few percent f being an exact integer. This suggests that there is a structural cmpnent f sme kind which perates as a decay trigger in these decays. This decay trigger is superimpsed n the verall stability f the dminant quark itself, which exhibits a dependence n the renrmalized fine structure cnstant. The Standard Mdel has wrked very well in the calculatin f partial decay channels fr these particles, and it has been suggested that the ttal lifetime r decay width f an elementary particle is merely the sum ver the available decay channels. Hwever the bserved glbal structure f these lifetimes indicates that the decay prcess is initiated in a manner which is t sme extent independent f the decay channels and f the available phase space. 6. The -quantizatin f the pseudscalar (PS) mesn lifetimes The lw-mass pseudscalar mesn family cntains five nnstrange mesns (,,,, ') (K,K,K,K) and fur strange kans L S. Their lifetimes are displayed in Figs They prvide sme f the mst accurate examples f glbal lifetime -scaling (Fig. 3), the 4 gap between unpaired-quark and paired-quark decays (Fig. 3), and a factr-f-2 lifetime substructure 9

10 (Figs. 4 and 5). They als exhibit a unique sequential spacing f the,, ' lifetimes by apprximate factrs f (Fig. 3), and a precise -spacing f the K and K S lifetimes (Figs. 2 and 4). In Sec. 6 we evaluate the accuracy f the PS quantizatins fr the,,, ' K,K,K mesns, and in Sec. 7 we discuss the ramificatins f the L S kans. Figure 6 shws the,,, ' lifetime experimental data, pltted in the frm f lifetime ratis n the glbal -quantized lifetime grid f Figs. 3 and 4. The accuracy f the -scaling is visually apparent. The numerical values f the lifetime lgarithms x i are shwn under the data pints. Their deviatins frm integer 4, 5 and 6 values are all less than 0.7%. This figure demnstrates that the,, ' PS mesn lifetimes accurately scale with respect t the reference lifetime by factrs f S 4, S 5, S 6, respectively, where the scaling factr is S ~ 137. Figure 7 displays calculated numerical values f the scaling factr S fr the experimental lifetime ratis shwn in Fig. 6. The ratis f the / and / ' lifetimes are nly qualitatively cmparable t the value S = 137, but the glbal scaling factrs S that emply the rati t the lifetime and extend ver several pwers f S clsely bracket this value. The large span f values encmpassed by the,,, ' PS mesn lifetimes, and the accuracy with which they fall n the lifetime -grid, can be demnstrated by writing dwn their experimental values, and then cmparing them t the values btained under the assumptin f a precise scaling in pwers f. This cmparisn is as fllws, where the lifetimes are quted in zeptsecnds (10 21 sec): Mesn Experimental lifetime (zsec) Calculated -scaled lifetime (zsec) Accuracy = 26,033,000,000, = 84, = % ' = % The calculated -grid lifetimes reprduce the experimental values with a lifetime accuracy level f rughly 15% ver thirteen rders f magnitude. This makes it pssible t deduce the lifetime 10

11 systematics in ne lcal regin f values, and then link this systematics t ther lcal lifetime regins f the glbal lifetime netwrk, as demnstrated fr example in Fig. 3. Anther way t assess the accuracy f the lifetime scaling in pwers f S 1 is t calculate the least-squares sum 2 ( S) f the calculated lifetimes i (S) t the experimental lifetimes (exper) i, weighted by the accuracies f the experimental values, as a functin f S. The equatin fr 2 ( S) is [6 where 2 i( S) i(exper) ( S), i i (exper) (exper) i is the experimental errr f the ith measurement. The calculated lifetimes 2 x i have the frm i ( S) S, x i 4,5,6 (see Fig. 7). The value f S that minimizes the chisquared sum is the mst accurate scaling factr fr this data set. This methd nly wrks well in a statistical sense if all f the data have rughly cmparable errr limits. Otherwise, the 2 sum is dminated by the fits t the data that have very small errr bars.[6] The results f this calculatin fr the,,, ' data set are displayed in Figure 8. As can be seen, the 2 (S) minimum ccurs at the value S min = 139.1, which clsely agrees with the lifetime -grid value This 2 analysis reinfrces the cnclusin frm Figs. 6 and 7 that these lifetimes exhibit a dependence n the renrmalized fine structure cnstant. The 2 ( S) analysis is expanded in Fig. 9 t include the CL central-grup s-quark and b-quark lifetimes f Fig. 4. The excluded unpaired-quark lifetimes in Fig. 4 invlve either factr f displacements frm the lifetime -grid, r, in the case f the c-quark lifetimes, an verall factr f 3 displacement. But the eleven particles that enter int the chi-square sum f Fig. 9 require n crrectins t the experimental data pints. As can be seen, the 2 ( S ) minimum fr these eleven particles is at S min = , which matches the lifetime glbal -grid spacing t an accuracy f 0.6%. This result indicates that the identificatin f the unpaired-quark central lifetimes CL in Fig. 4 as the lifetimes which are naturally spaced by factrs f relative t the unpairedquark lifetime is an experimentally justified assumptin. 11

12 7. The kan "lcal -grid" centered n the K ± mesn The pseudscalar mesn and kan experimental lifetimes were pltted as the (PS) unpairedquark and paired-quark lifetime grups in Fig. 3, and the factr-f-2 spacings f the kans were displayed in Fig. 4. In Sec. 6 we analyzed the lifetime spacings f the nnstrange,,, ' mesns (Figs. 6-9). We nw similarly analyze the strange K,K,K L S kans. The kan lifetimes are shifted by factrs f 2 ff the -anchred glbal lifetime -grid. Thus it seems apprpriate t plt them n a "lcal -grid" that is anchred n the K lifetime. This plt is shwn in Fig. 10. The K and K kans are related by their S hadrnic decay mdes, and their lifetimes fall accurately n the shifted kan lcal -grid. Their experimental lifetime rati f is within 0.9% f the value , which indicates that the fine structure cnstant is relevant t bth the kan and pin lifetimes. The is a factr f 4 lnger than that f the K. The K kan has a hadrnic decay mde, and its lifetime L lifetime is the gemetric mean f the K lifetimes, and is the CL central lifetime f the (PS) unpaired-quark lifetime grup (Fig. 2). In Sec. 6 we demnstrated that the nnstrange fall accurately n the glbal -grid that is centered n the the strange K,KS K and L,,, ' pseudscalar mesn lifetimes mesn. In Sec. 7 we shwed that pseudscalar kan lifetimes fall accurately n a lcal -grid that is centered n the K kan. In Sec. 8 we extend these results t include a mixed set f particles whse lifetimes fall n a lcal -grid that is centered n the ± mun. 8. The lifetime -quantizatin f the mun, neutrn and "charm-crrected" taun One reasn fr studying the experimental systematics f elementary particle lifetimes is t ascertain the requirements these results impse n theretical lifetime frmalisms. A secnd reasn is t find ut if the particle lifetimes reveal smething abut the nature f the particles themselves. And, as displayed here, they d. Histrically, the symmetries f particle masses, spins, istpic spins and hypercharges were used t deduce the existence f quark substates within the particles. We can see frm the lifetimes displayed in Fig. 2 that the existence f quark-like b- 12

13 jects r quark flavrs can be inferred frm the manner in which these particle lifetimes grup tgether in nn-verlapping flavr-grup regins. This lifetime infrmatin serves t crrbrate the reality f quarks as cmpnents f hadrns. The range f particle lifetimes in each f these individual flavr grups is evidently dictated by a single dminant quark flavr, which perates in the same manner in mesns as it des in baryns (Figs. 3-5). A third reasn fr studying lifetime systematics is t see if they reveal unsuspected relatinships amng the particles. Our studies up t the present pint have been mainly centered n the systematics f the 24 unpaired-quark hadrnic grund states that are displayed in Fig. 4, which ccupy the Zne (2) lifetime regin that extends frm 10 7 t sec (Fig. 1). In the present sectin we expand this systematics by adding in the tw lng-lived Zne (1) particles the hadrnic neutrn and leptnic mun and als the Zne (2) leptnic taun. The weaklyinteracting and leptns are cmmnly regarded as being unrelated t the strnglyinteracting hadrns. Hwever, we will nw demnstrate that they, and als the metastable neutrn, have lifetimes that fit int and extend the anchred glbal -grid systematics. Furthermre, these lifetimes furnish sme infrmatin abut the generatinal matching f the three families f quark pairs t the three families f leptn-neutrin pairs. The neutrn, mun and taun lifetimes were displayed and labeled in Fig. 1, but have nt subsequently been included. Their additin extends the bserved lifetime regularities in an interesting and unique manner. Figure 11 displays the lifetimes f the 24 unpaired-quark grund-state particles f Fig. 4, tgether with the neutrn, and lifetimes, which are all pltted n the glbal -grid. The neutrn and mun lifetimes ccupy new psitins n the -grid, which are bth clse t -quantized integer values f the lifetime lgarithms x i. Thus they extend the accurate lifetime scaling in pwers f. The taun lifetime, hwever, des nt match the glbal - grid. Instead, it appears as a member f the c-quark flavr cluster, which is shifted away frm the x i = 2 -grid psitin by a factr f 3 t x i = The x 2 -grid itself is ccupied by the b-quark flavr cluster. It turns ut, as we will demnstrate, that the precise lifetime psitining f the taun leads t unexpected and phenmenlgically significant results. A subtle but imprtant feature in Fig. 11 is the fact that the neutrn and i lifetimes are shifted ff the glbal -grid psitins x i = 5 and 1, respectively, and tward shrter lifetimes, 13

14 by factrs f 1.42 and 1.62, and the lifetime is shifted ff the factr-f-3-displaced c-quark - grid psitin at x i = by a factr f (Nte the analgus factr-f-2 shift ff the glbal -grid and tward shrter lifetimes displayed by the pseudscalar kan lcal -grid plt in Fig. 10.) This cmmn lifetime shift suggests that the neutrn, mun and taun are in sme manner related. It als indicates that the leptnic taun shares the characteristic c-quark factrf-3 lifetime shift (c-quark crrectin factr) f Dx i = that applies t all eight hadrnic unpaired c-quark grund states in Figs. 4 and 11. Thus we have the hadrnic c-quark lifetime prperties being reflected in the weakly-interacting and featureless (quarkless) taun. T investigate these implicatins in mre detail, we plt these lifetimes n a lcal -grid that is centered n the lifetime, as shwn in Fig. 12. Figure 12 shws the standard -spaced lgarithmic lifetime grid, but with the leptnic instead f the hadrnic as the reference lifetime, and it displays just the experimental neutrn and lifetimes and the "charm-crrected" crr 3 lifetime. As can be seen, these lifetimes are spaced by accurate pwers f The 3 gap that separates the and crr lifetimes is unique in the particle lifetime systematics. Using this systematics as a guide, we deduce the fllwing leptn lifetime equatin: 3 3 ( ). This equatin is remarkable fr bth its accuracy and its implicatins. The equatin cnnects tw lifetimes, which are six rders f magnitude apart, t an verall accuracy f 2%. It is infrmative t shw the numerical values frm this equatin, using lifetimes in micrsecnds (sec): (exp erimental lifetime) sec (experimental lifetime) sec (calculated lifetime) sec (2.1% accuracy) We can draw three cnclusins frm these results: (1) the leptnic taun accurately shares the factr-f-3 "c-quark lifetime shift" (with respect t the b-quark particles n the glbal -grid) that ccurs fr the hadrnic c-quark mesns and baryns; (2) 3 is a relevant scaling factr fr the separatin f the leptnic and lifetimes, in the same sense that is the relevant scaling factr fr the separatin f the hadrnic unpaired-quark and paired-quark flavr-grup life- 14

15 times displayed in Fig. 3; (3) the renrmalized fine structure cnstant accurately applies t these metastable leptn lifetimes. The ther significant result we can btain frm the lifetimes displayed in Fig. 12 is the relatinship it suggests between the neutrn and mun lifetimes. These are separated by the same 4 lifetime gap that is bserved in each f the (PS), (s), (b), (c) flavr grups f Fig. 3. This gap is between the slw flavr-changing unpaired-quark decays and the fast flavr-cnserving pairedquark r radiative decays, and it divides each flavr grup int tw well-separated subgrups. The free neutrn decay int a prtn, electrn and electrn antineutrin (which has the neutrn lifetime displayed in Fig. 2) is a classic textbk example [7] f a weak-interactin flavrtransfrming decay that cnverts a d int a u quark, as mediated by a virtual W gauge bsn. Leptn number is cnserved in this decay. The mun decay int an electrn, mun neutrin and electrn antineutrin cnserves quark flavr (since leptns d nt have quarks), and it als cnserves leptn number. Thus mun decay frmally serves as a flavr-cnserving prcess. Hence the experimental 4 gap between the neutrn and mun lifetimes is in line with the 4 gaps shwn in Fig. 3, but nly if the neutrn and mun are related particles, since the 4 gaps displayed in Fig. 3 are all within flavr-related particle grups. Thus the mun-t-neutrn lifetime rati f 4 displayed in Fig. 12 indicates that these particles are in sme sense tied tgether, which is als indicated by their similar lifetime displacements n the -grid f Fig centered lifetime glbal We can demnstrate the accuracy f the lw-mass neutrn-t-mun 4 lifetime rati in Fig. 12 by cmparing it t the ther bserved lw-mass flavr-breaking-t-flavr-cnserving 4 lifetime rati, which is the t lifetime rati displayed in Figs. 3, 6 and 7. These tw sets f lifetimes are pltted tgether in Fig. 13, where the lng-lived flavr-changing particle decay in each case is used t anchr the lifetime a-grid, and the shrt-lived flavr-cnserving decay lifetime appears at a separatin distance f 4 (eight rders f magnitude). These are the nly examples bserved belw 1 GeV, and hence are lgically f fundatinal imprtance. They represent the lwest-mass bsn decays (the pins) and the lwest-mass fermin decays (the neutrn and mun). The lifetime relatinship between the and lifetime relatinship between the hadrnic neutrn and leptnic mun is nt. mesns is a familiar result, but the 15

16 The bservatins we have made here with respect t the neutrn, mun and taun are based n lifetime systematics. Additinal infrmatin can be btained frm mass systematics. Figure 14 displays a mass vs. lifetime plt fr the unpaired-quark flavr grups f Fig. 4, with the mun and taun added in. In Fig. 4 the particle x-axis lgarithmic lifetimes were arbitrarily spread ut alng the y axis fr cmparisn purpses. In Fig. 14 we use the same lifetime x axis as in Fig. 4, but we nw emply the y axis t represent the particle mass. This mass-lifetime plt shws that the tau leptn, which has a lifetime that is in the (c) flavr grup, als has a mass that is right in the mass regin f the charmed D mesns. Thus bth the lifetime and mass f the tau place it as a member f the cnventinal (s, c) quark generatin f particle states, and nt as a member f the (b, t) quark generatin (where it is custmarily prtrayed [8]). This mass-lifetime plt als cntains an example that illustrates the wrkings f the c > b > s quark-flavr dminance rule (Fig. 2). The B c mesn, which cntains bth a b quark and a c quark, has the largest mass f the unpaired b-quark particles, and is psitined far abve the ther c-quark masses, and yet its lifetime falls squarely n the c-quark CL central-lifetime vertical axis. Hence the c > b > s quark dminance rule is independent f the mass values f the particles. As the final result here, we cite sme particle mass values that reinfrce the neutrn-mun relatinship suggested by their lifetimes systematics. In Fig. 13 we cmpared the similar 4 lifetime ratis f the (n, ) and (, ) pairs f particles. We can extend these lifetime results t include masses by cmparing the linear mass ratis f the (, neutrn, ) fermin lifetime triad f Fig. 12 and the (,, ') bsn mass triad f Fig. 6, where ( )/2 is the average pin mass. These tw triads shw interesting similarities, as displayed in Fig. 15: (1) The pin is the lightest metastable bsn, and the mun is the lightest metastable fermin. (2) The heavier masses in each triad are accurate multiples f the mass f the lightest particle. (3) The bsn and fermin triads bth have linear mass ratis: bsn: (,, ') = 1::4::7; fermin: (, n, ) = 1::9::17. (4) The experimental accuracies f these mass ratis are: 4 (0.2%); ' 7 (0.3%); n 9 (1.2%); 17 (1.1%). The (,, ') triad cntains three clsely-related hadrn states, whereas the (, n, ) triad has a hadrn interspersed between tw leptns, and yet their numerical accuracies are remarkably similar. The mass systematics f Fig. 15 reinfrces the lifetime systematics f Figures 6 and 12. In 16

17 particular, the clse interrelatinship between the mun and the neutrn emerges frm bth their mass and their lifetime systematics. It des nt appear in the current frmulatins f Standard Mdel theries. This suggests that in assessing theries and mdels fr elementary particles, bth masses and lifetimes (mass stability) need t be taken int cnsideratin. References 1. Particle Data Grup, J. Phys. G 37, (2010) 2. T. Aaltnen et al., arxiv: v1 [hep-ex] 23 Aug M. H. Mac Gregr, Nuv Cim. 8A, 235 (1972), pp ; Phys. Rev. D9, 1259 (1974), pp M. H. Mac Gregr, Phys. Rev. D13, 574 (1976). 5. Ref. 2, p R. A. Arndt and M. H. Mac Gregr, Methds in Cmputatinal Physics 6, 253 (1966). 7. A. Seiden, Particle Physics (Addisn-Wesley, San Francisc, 2005), pp. 12 and M. Veltman, Facts and Mysteries in Elementary Particle Physics (Wrld Scientific, Singapre, 2003), frnt cver and pp

18 Figure 1. The measured lifetimes f 171 particle states, as listed in RPP 2011 [1]. The lifetimes in secnds are pltted n a lgarithmic scale that spans 28 rders f magnitude, and that separates int fur distinct lifetime znes. The Zne 4 hadrn excited states decay dwn t the grund states via strng decays that cnserve flavr, and they have shrt lifetimes f less that sec (1 zsec). The hadrn grund states divide int tw factr-f-10 7 lifetime znes, which reflect the flavr-breaking that is invlved in the decay prcess. The paired-quark mesn grund states, and the hypern state, all f which cnserve flavr and/r have radiative decays, are in the t sec Zne 3 lifetime range. The unpaired-quark hadrn grund states, which require flavr-changing electrweak decays, are all in the 10 7 t sec Zne 2 lifetime range. The nly massive elementary particles with lifetimes lnger than 10 7 sec are the Zne 1 metastable mun and neutrn and the stable electrn and prtn. The shrted-lived excitatins are the very massive Zne 4 W and Z gauge bsns and tp quark t. 18

19 Figure 2. A lifetime plt f 32 metastable hadrnic quark flavr grund states, with each hadrn labeled by its dminant quark substate. The Zne 2 tp sectin displays the unpaired-quark lifetimes in the 10 7 t sec range, and the bttm sectin displaying the crrespnding pairedquark lifetimes in the t sec range. The lifetimes in each sectin separate int fur nnverlapping quark flavr grups, labeled as (PS) black, (s) blue, (b) red, (c) green, with mesns and baryns gruped tgether. The flavred quarks each have a characteristic range f lifetime values, and the shrtest flavr range is dminant. The quark dminance rules are as fllws: (1) All particles that cntain a c quark are in the (c) flavr grup. (2) All particles that cntain a b quark but n c quark are in the (b) flavr grup. (3) The hyperns, which cntain s quarks but n b r c quark, are in the (s) flavr grup. (4) The nn-strange pseudscalar (PS) mesns are in the (PS) flavr grup, whereas the strange PS kans appear in bth the (PS) grup (K,K L) and the (s) grup (K S ). These results establish the empirical quark flavr dminance rule c > b > s fr hadrn lifetimes, with the pseudscalar mesns frming a slightly mre cmplex subset f particles. The K /KS 1 lifetime rati f matches the fine structure cnstant value

20 Figure 3. The lifetimes f 37 metastable quark grund states, shwing the -quantizatin f the unpaired-quark (PS) black, (s) blue, (b) red, and (c) green flavr grups, and the 4 lifetime gaps between the unpaired-quark and crrespnding paired-quark and radiative flavr grups. The 32 metastable hadrn grund states f Fig. 2 are displayed here, tgether with the and ' pseudscalar mesns and three c-quark baryn excited states that have lifetimes slightly shrter than 1 zsec. The lifetimes are pltted as ratis t the reference lifetime, using a lgarithmic representatin with the base 1 /137. The nnstrange,,, ' pseudscalar mesn lifetimes match x i = 0, 4, 5, 6 -grid lines, re- spectively. The unpaired-quark (s) and (b) flavr grups match the x i = 1 and 2 -grid lines, and the unpaired-quark (c) grup is displaced by a factr f 3 frm x i = 2 in the directin f shrter lifetimes (see Fig. 4). Anther prminent feature in Fig. 3 is the divisin f the (PS), (s), (b), (c) flavr-grup lifetimes int lng-lived and shrt-lived cmpnents. The lifetimes in the paired quark flavr grups are shrter than the lifetimes in the crrespnding unpaired-quark grups by factrs f apprximately 4. The intervening 4 gaps frm a lifetime desert regin where n particle states have been bserved. The results displayed in Fig. 3 demnstrate that an -spaced lifetime grid anchred n the ± lifetime is an apprpriate framewrk (a glbal -grid) n which t cmbine the 37 grund-state hadrns int a unified glbal lifetime pattern. They als shw that mesns and baryns ccur tgether in the same pattern flavr-grupings, which are dictated by individual flavred quarks within the particles. 20

21 Fig. 3 21

22 Figure 4. Unpaired-quark (PS) black, (s) blue, (b) red, (c) green flavr-grup CL central lifetimes and factr f grup substructures. This figure displays the 24 unpaired-quark grund-state lifetimes f Figs. 2 and 3, shwn pltted n a glbal lifetime -grid centered n the mesn. The vertical lines dente the CL central lifetime in each quark flavr grup The (PS) grup CL (the ) defines the x i = 0 -grid line The (s) grup CL is slightly ff the x i = 1 -grid line. The (b) grup CL is accurately placed n the x i = 2 -grid line. The (c) grup CL is displaced tward shrter lifetimes by a factr f 3 with respect t the (b) grup CL. The B c mesn is a member f the (c) grup CL particles, and the B s mesn is a member f the (b) grup CL particles, in accrdance with the c > b > s quark dminance rule. The lifetimes that d nt fall n the central lifetimes exhibit a lifetime substructure in which the lifetime ratis between related particles ccur with apprximately integer values f 2, 3 r 4. There are eight factr-f-2 ratis, tw factr-f-3 ratis, and tw ± factr-f-4 ratis (including K / K ) displayed in Fig. 4. These lifetime ratis are L als displayed graphically in Fig. 5, where they are gruped tgether and averaged. 22

23 Figure 5. The factr f integer lifetime ratis that were displayed and labeled in Fig. 4 are shwn here in Fig. 5, where they are gruped tgether by integer value and averaged. The eight factr-f-2 lifetime ratis are between related particles except fr the /(, ) rati, which emplys the (s) flavr grup CL central lifetime, and the D/( C B/3) / rati, which emplys the (c) flavr grup CL central lifetime. The tw factr-f-3 lifetime ratis are between particles (which feature ne flavred quark) and particles (which feature three flavred quarks). Althugh the experimental lifetime ratis 2, 3, and 4 are nly apprximately equal t integers, the average ver all f the measured ratis fr a given integer type is within a few percent f being an exact integer. This suggests an underlying particle integer decay mechanism that has smaller effects as perturbatins. In additin t the integer lifetime ratis displayed here, the / vectr mesn lifetime rati is The and have lifetimes slightly shrter than 1 zsec, and hence are nt included in the present metastable particle cmpilatin. 23

24 Figure 6. The accurate glbal quantizatin f the nnstrange pseudscalar mesns, whse lifetimes extend ver 6 pwers f, r 13 rders f magnitude. The ± mesn, the lwest-mass and the lngest-lived (except fr K ) hadrn state, serves as the reference lifetime fr this glbal -grid. The numerical values f the lifetime lgarithms x i are displayed belw the experimental data pints, and are clse t the integer values that dente precise -scaling. In detail, the deviatins f the lgarithms x i frm the exact integers 4, 5 and 6 are 0.65%, 0.26% and 0.50%, respectively. The 4 lifetime gap between the unpaired-quark ± and paired-quark mesns is als displayed in Fig. 3, where it is cmpared t similar 4 gaps in the ther lifetime flavr grups. The factr f ~ lifetime spacings in the,, ' excitatin sequence are unique in the elementary particle lifetime data set, as can be seen in Fig. 3. L 24

25 Figure 7. Experimental values f scaling factrs S fr the,,, ' pseudscalar mesn lifetimes, pltted against the glbal -grid framewrk f Fig. 6. The left equatin defines the x i data pints n the lifetime -grid, and the right equatin defines the scaling factr S under the assumptin that the x i are exact integers. As can be seen, the lcal lifetime ratis / 1 and / ' are nly qualitatively cmparable t the -quantized value 137, but the S values fr the,, ' lifetimes when they are expressed as ratis t the lifetime clsely match the -grid scaling. This agreement demnstrates the usefulness f the lifetime glbal -grid fr crrelating the systematics f the metastable grund-state hadrn lifetimes ver many rders f magnitude. 25

26 Figure 8. The least-squares sum 2 (S) fr the nnstrange,, ' pseud-scalar mesns, pltted as a functin f the scaling factr S (see Fig. 7). The chi-squared minimum at S = gives the best fit f the theretical S-scaled lifetimes t the experimental lifetimes, which are weighted by the accuracy f the experimental data. [6] This scaling factr clsely matches factr-f-137 grid lines f the glbal - grid that is emplyed in Figs. 3, 4 and 6. 26

27 Figure 9. The 2 ( S ) sum fr eleven metastable particles, pltted as a functin f S. These eleven particles are the,, ' mesns f Fig. 8, the CL central-grup, s-quark hyperns f Fig. 4, and the CL centralgrup B,B,B, s b, b, b b-quark mesns and baryns f Fig. 4. Nne f these particles invlve factr f integer displacements frm the lifetime -grid. As can be seen, the shape f the 2 ( S ) curve is similar t that f Fig. 8, but the minimum has been shifted t the value S min = This matches the -grid spacing f t an accuracy f 0.6%. Hence the cncept f unpaired-quark lifetime quantizatin in pwers f, which is visually apparent in the glbal lifetime plts f Figs. 3, 4 and 6, is quantitatively brne ut by the least-squares plts f Figs. 8 and 9. This 2 minimum als substantiates the use f the ± mesn t anchr the glbal lifetime -grid, and it verifies that this grid is based n the renrmalized 1 fine structure cnstant 137, and nt the running cnstant (q 2 ) that 2 increases frm ( q ) 1/137 at q 2 2 = 0 t 1/128 at q 2 m [5]. W 27

28 Figure 10. The strange K, K L, K S pseudscalar kan lifetimes, shwn pltted n an -quantized lcal -grid that is anchred n the K lifetime. Bth the K and K lifetimes match -grid lines, which demnstrates the relevance f lcal -grids fr shifted as well as central lifetimes. The nnstrange lifetime is als displayed here t illustrate hw it serves as the central lifetime f the K L,, K S lifetime triad, which frms the unpaired-quark (PS) subgrup in Fig. 2. This mixed kan and mesn triad cntains tw accurate factr-f-2 lifetime ratis, as well as a factr-f-4 rati between the tw kans (see Figs. 4 and 5). S 28

29 Figure 11. The hadrnic unpaired-quark lifetime plt f Fig. 4 with the hadrnic neutrn and leptnic mun and taun added in. The vertical slid lines thrugh the (PS), (s), (b) and (c) flavr-grup data pints dente the central lifetimes CL fr each grup. The (PS) and (b) CL's lie n the x i = 0 and x i = 2 -grid lines, respectively, but the (s) CL is slighter shrter than the x i = 1 -grid line. The (c) flavr grup has fur particles n the CL at x i = , and fur particles with integer-displaced lifetimes (see Fig. 4), all f which are shifted tward shrter lifetimes by a cmmn factr f 3 (the c-quark crrectin factr Dx i = ) with respect t the x i =2 -grid line. The neutrn and mun lifetimes ccupy the x i = 5 and x i = 1 glbal -grid lines, but are shifted tward shrter lifetimes by factrs f 1.42 and 1.62, respectively. The taun is similarly shifted with respect t the c- quark CL by a factr f This suggests that these three lifetime shifts may be related. In Fig. 12 we replt these three lifetimes n a lcal -grid that is anchred n the lifetime instead f the lifetime. 29

30 Figure 12. The neutrn and lifetimes, shwn pltted tgether with the crr 3 lifetime (using the c-quark charm crrectin factr 3 shwn in Fig.4), and displayed n an - quantized lifetime grid that is anchred n the lifetime. The leptnic t crr lifetime rati is 3 (t 2% accuracy ver a span f 6 rders f magnitude). This suggests that the tau leptn, which des nt cntain a c quark, nevertheless carries the c-quark lifetime flavr systematics. This surprising link between hadrnic and leptnic lifetime behavir can als be inferred, althugh in a smewhat different frm, frm the neutrn t lifetime rati f 4. This is the lifetime rati that ccurs between the slw flavr-breaking and fast flavrcnserving hadrnic decays displayed in Fig. 3. The slw neutrn-t-prtn decay invlves a flavr-breaking d t u quark cnversin. The fast t e decay cnserves quark flavrs (it has nne). Thus the neutrn t lifetime rati f 4 frmally fllws the flavrbreaking t flavr-cnserving rule f Fig. 3, but here ne particle is a hadrn and the ther is a leptn. Hence the bserved lifetime factr f 4 cnceptually links the hadrnic neutrn t the leptnic mun. Similarly, the bserved lifetime factr f 3 3 links the leptnic taun t the hadrnic charm family (if we assume that the lifetime factr f 3 represents an apprpriate leptnic lifetime interval). Hence the experimental lifetime systematics f Figs. 11 and 12 suggests a cmmnality f sme hadrn and leptn lifetime prperties. 30

31 Figure 13. A cmparisn f the flavr-changing neutrn and charged pin lifetimes with the flavr-cnserving mun and neutral pin lifetimes, which are each a factr f 4 shrter. As can be bserved, the paired neutrn-mun and / lifetime ratis, which each span eight rders f magnitude, are accurately quantized in pwers f. These tw sets f lifetimes are displaced frm each ther by a factr f 5 (Fig. 1). The surprising result here is that whereas the and are clearly-related particles, the n and are cmmnly assumed t be unrelated. In additin t the neutrn-mun lifetime relatinship displayed here, there is als a mass relatinship that invlves the neutrn, mun, and taun, as described in the text and displayed in Fig

32 Figure 14. The 24 unpaired-quark hadrn grund states f Fig. 4 plus the and ± leptns, shwn n a mass-lifetime plt that has the lgarithmic lifetime -grid as the abscissa and the linear masses in MeV as the rdinate. The vertical slid lines indicate the central lifetime in each flavr grup regin (Fig. 4). The clse assciatin f the ± leptn mass with the mass grup f c-quark D mesns and charm baryns is cnsistent with the fact that its lifetime carries the factr-f-3 charm crrectin factr. Hwever, the clse assciatin f the B c bc mesn mass with the mass grup f b-quark B mesns and bttm baryns is nt cnsistent with its lifetime, which falls squarely n the central lifetime (vertical line) f the charm flavr family. This shws that the c > b > s quark dminance rule (Fig. 2) fr dictating particle lifetimes supersedes the effects f the particle mass values. Particle mass grupings d nt necessarily lead t crrespnding lifetime flavr grupings. 32