A Theory of Weak Interaction Dynamics

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1 Open Access Lbrary Journal 2016, Volume 3, e3264 ISSN Onlne: ISSN Prnt: A Theory of Weak Interacton Dynamcs Elahu Comay Charactell Ltd., Tel-Avv, Israel How to cte ths paper: Comay, E. (2016) A Theory of Weak Interacton Dynamcs. Open Access Lbrary Journal, 3: e Receved: November 29, 2016 Accepted: December 12, 2016 Publshed: December 1, 2016 Copyrght 2016 by author and Open Access Lbrary Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.0). Open Access Abstract Problems wth the electroweak theory ndcate the need for a consstent weak nteractons theory. The analyss presented n ths work s restrcted to the relatvely smple case of elastc scatterng of a neutrno on a Drac partcle. The theory presented heren assumes that the neutrno s a massve partcle. Furthermore, the d- 2 menson L of the Ferm constant G F as well as ts unversal property are used as elements of the theory. On ths bass, t s assumed that weak nteractons are a dpole-dpole nteracton medated by a weak feld. An nteracton term that represents weak nteractons s added to the Drac Lagrangan densty. The dentty 0 ψ ψ γ s used n an analyss whch proves that the nteracton volates party because t conssts of two terms-a vector and an axal vector. Ths outcome s n accordance wth the expermentally confrmed V-A property of weak nteractons. Subject Areas Theoretcal Physcs Keywords Weak Interacton Dynamcs, Lagrangan Densty, Party Nonconservaton, V-A 1. Introducton Weak nteractons have some unque propertes that cannot be found n other knds of nteractons. The followng ponts llustrate ths clam. Weak nteractons do not conserve party. Weak nteractons do not conserve flavor. The tme duraton of weak processes span many orders of magntude. For example, the neutron s mean lfe s 880 sec whereas that of the top quark s about sec [1]. (Here examples of a very long mean lfe of about 10 9 years, lke that of the 40 K nucleus, are omtted because ths effect s due to the dfference n the quantum DOI: /oalb December 1, 2016

2 mechancal angular momentum of the nucle nvolved n the process.) The weak nteractons couplng constant s wrtten n unts that have the dmenson of energy 2 2, G F = GeV (see [2], pp. 19, 212). These weak nteractons propertes ndcate that ts theory should have a specfc structure. Another ssue s the exstence of unsettled problems wth the electroweak theory. Some of these problems are mentoned n the second secton. Consderng ths state of affars, the present work descrbes elements of a consstent weak nteractons theory. As a frst step, the dscusson s restrcted to the smplest case of an elastc neutrno scatterng (see [3]). A fundamental property of Quantum Feld Theory (QFT) s the descrpton of a par- tcle by means of a functon the takes the form ψ ( x) (see e.g [4], p. 299). Here x denotes a sngle set of four space-tme coordnates. It means that such a functon des- crbes a pontlke partcle. Indeed, x can descrbe the poston of a partcle at a gven tme but not ts dstrbuton around ths pont. Expermental data support the pontlke property of elementary partcles (see e.g. []). Evdently, two ponts cannot collde. Therefore, a medatng feld s requred for a descrpton of a scatterng process of pontlke partcles. The need for a medatng feld n weak nteractons s analogous to a correspondng property of quantum electrodynamcs (QED), where Maxwellan felds nteract wth a pontlke charge of an elementary partcle. Electrodynamcs s certanly the best physcal theory because t has many expermental supports as well as a tremendous number of specfc applcatons n contemporary technology. The present work ams to construct a weak nteracton theory that has a certan smlarty wth QED. In partcular, t follows the structure of QED and uses a weak nteracton term that s added to the Lagrangan densty of the system. Specfc aspects of ths smlarty are descrbed below n approprate places. Unts where = c = 1 are used. Greek ndces run from 0 to 3 and Latn ndces run from 1 to 3. The metrc s dag. ( 1, 1, 1, 1). Square brackets [] denote the dmenson of the enclosed expresson. In a system of unts where = c = 1 there s just one d- menson, and the dmenson of length, denoted by [ L ], s used. In partcular, energy and momentum take the dmenson L 1 2. Problems wth the Electroweak Theory and the dmenson of a dpole s [ L ]. Several theoretcal problems of the electroweak theory are brefly presented n ths secton. A fundamental prncple used heren s the correspondence between QFT and quantum mechancs. S. Wenberg has used the followng words for descrbng ths prncple: Frst, some good news: quantum feld theory s based on the same quantum mechancs that was nvented by Schroednger, Hesenberg, Paul, Born, and others n , and has been used ever snce n atomc, molecular, nuclear and condensed matter physcs (see [4], p. 49). Ths prncple can also be found n pp. 1-6 of [6]. Hereafter, ths relatonshp s called Wenberg correspondence prncple. The followng revew artcle states that t s now recognzed that neutrnos can no 2/10

3 longer be consdered as massless partcles (see [3], p. 1307). It means that the neutrno s an ordnary massve Drac partcle whch s descrbed by a 4-component spnor. (The argument also apples to a Majorana neutrno.) Ths expermental evdence does not ft the orgnal structure of the Standard Model where the neutrno s treated as a 2-component massless partcle [7]. In the followng lnes t s proved that ths property of the neutrnos s nconsstent wth expressons that have the factor ( 1± γ ). The factor ( 1 ± γ ) has been proposed for a two-component massless Wel neutrno (see [2], p. 219, 367). For example, t s used n a descrpton of an electron-neutrno nteracton (see [2], pp ) ( ± ) ψ O e 1 γ ψ ν. Here O represents an approprate operator whch operates on ψ ν. It turns out that ths expresson does not hold for a massve Drac neutrno. Indeed, operatng wth ( 1 ± γ ) on a motonless spn-up Drac spnor, one obtans 1 0 ± ± =. (2) ± ± 1 0 ± Here the γ matrces notaton s that of [8], p. 17. The rght hand sde of (2) s a Drac spnor that has an nfnte energy-momentum (see [8], p. 30). It means that the operator ( 1± γ ) casts a motonless Drac partcle nto an unphyscal state. Furthermore, a product of two γ matrces s used for a boost of a Drac partcle (see [8], p. 21). Hence, ( 1± γ ) commutes wth the boost operator. For ths reason, the operator ( 1± γ ) casts any Drac spnor nto the unphyscal state of an nfnte energy-momentum. Evdently, every bass of the Hlbert space of a Drac partcle s made of physcally acceptable states of ths quantum partcle. Therefore, a state that has an nfnte energy-momentum s not ncluded n a Hlbert space. Hence, the requred matrx elements cannot be calculated. Ths s an example showng that the factor ( 1± γ ) s nconsstent wth the Wenberg correspondence prncple. The factor ( 1± γ ) s used for adaptng the Standard Model to the V-A property of the weak nteractons (see [2], pp ). The contradcton obtaned above ndcates that the Standard Model s nconsstent wth the V-A property of the weak nteractons. Let us examne other electroweak contradctons. The W ± and the Z partcles are fundamental elements of the electroweak theory. It turns out that the theoretcal structure of these partcles s nconsstent wth fundamental physcal requrements. Let us begn wth the W ±, whch are a partcle/antpartcle that carry a postve/negatve charge, respectvely. These partcles must perform two dfferent physcal tasks: In a weak process they play the role of an nteracton medator that carres the nteracton between two fermons. On the other hand, n an electromagnetc nteracton they act as a charge carrer and the electromagnetc nteracton s medated by Maxwellan felds. It turns out that these two dfferent tasks cannot be accomplshed smultaneously. For example, the electromagnetc nteracton term s j µ Aµ, where j µ denotes a (1) 3/10

4 conserved 4-current that satsfes the contnuty equaton j µ =, µ 0. It s well known that ths requrement s satsfed by a Drac partcle (see [8], p. 24). By contrast, the electro- weak theory s more than 40 years old but there s stll no self-consstent expresson for the 4-current of the W ± (see [9]). The lack of a consstent expresson for the 4-current of the W ± s mplctly admtted by the general communty. Indeed, theoretcal groups workng n reputable research centers apply unusual procedures n a calculaton of the W ± electromagnetc nteractons. Thus, the theoretcal group of the D0 faclty at Fermlab and that of the LHC faclty at CERN use an effectve expresson for ths purpose. (see Equaton (1) n [10] and Equaton (3) n [11], respectvely). By contrast, calculatons of electromagnetc nteractons of a Drac partcle are based on a theoretcally vald expresson. The orgn of ths contradcton can be brefly explaned. Due to a wdely acceptable rule, the followng substtuton s ntroduced n order to account for the electromagnetc nteractons µ µ eaµ (see [12] p. 84). The W ± Lagrangan densty contans a product of two functons that contan µ eaµ. Hence, the Ws Noether 4-current j µ depends lnearly on A µ. It follows that the electromagnetc nteracton term j µ Aµ s quadratc n A µ and n the electrc charge e. Ths s nconsstent wth Maxwellan electrodynamcs, where Maxwell equatons are derved from a Lagrangan functon whch depends lnearly on the 4-potental A µ (see [13], p. 7). Concluson: the W ± has no 4-current. The lack of a consstent expresson for the W's conserved 4-current also volates the Wenberg correspondence prncple, because the Schroednger equaton has a consstent expresson for a conserved densty and current (see [14], pp. 3-). It can be shown that the electroweak Z boson suffers from an analogous contradcton where densty of a massve partcle cannot be defned. Thus, ths partcle s descrbed by a real functon (see [1], p. 307). For ths reason, the Z boson has no self-consstent expresson for densty [16]. It should be ponted out that densty s the 0-component of the 4-current. And ndeed, QFT textbooks do not show a consstent expresson for the 4-current of the Z boson. Thus, the electroweak Z boson suffers from a contradcton whch s analogous to the above mentoned contradcton of the electroweak W ± bosons. In partcular, a Hlbert space cannot be constructed wthout a self-consstent expresson for densty. Hence, the electroweak Z boson s nconsstent wth the Wenberg correspondence prncple, because a Hlbert space s an ndspensable element of quantum mechancs (see [4], p. 49). The contradctons of the electroweak theory ndcate that a consstent theory of weak nteractons s needed. Evdently, expermental data provde clues for a constructon of such a theory. These ssues are dscussed n the rest of ths work. 3. Fundamental Elements of a Theory of Weak Interacton Dynamcs The weak nteracton theory constructed below ams to follow the theoretcal structure of QED. Hence, the man problem s how to construct an expresson that represents weak nteractons n the form of a term of the Lagrangan densty of a Drac partcle. In 4/10

5 µ the case of electromagnetc nteracton, the 4-current of a Drac partcle s ψγ ψ (see [8], pp ), and the correspondng nteracton term of the Lagrangan densty s (see [12], p. 86) L = e A (3) EM nt ψγ µ µ ψ. It means that the electromagnetc nteracton term s the contracton of the 4-vector of µ the γ matrces wth the 4-vector A µ of the external electromagnetc potental A µ, where the latter depends on the four space-tme coordnates. Here the strength of the nteracton s 2 α e 1/137. (4) Ths quantty s a dmensonless Lorentz scalar. The weak nteracton theory descrbed heren abdes by the expermental evdence where, n the unts = c = 1, the dmenson of the Ferm constant s (see [2], pp. 19, 212) 2 [ G ] L F =. () Hence, the requred weak nteracton term dffers from ts electromagnetc counterpart (3), where the electrc charge s a dmensonless Lorentz scalar. Followng () and the dpole s dmenson [ L ], t s assumed here that the weak nteractons s a dpole-dpole nteracton. Thus, the theory must resolve two problems: 1) What s the structure of the weak feld that medate the nteracton between the weak dpoles? 2) What s the form of the weak nteracton term of the system s Lagrangan densty? A resoluton of the frst problem s qute smple. The weak feld of a weak dpole takes the Maxwellan-lke form of an axal dpole. Ths dpole s carred by every elementary spn-1/2 partcle. Hereafter, the strength of ths elementary weak dpole s denoted by d. Ths symbol dffers from the mathematcal symbol d whch s used n ntegrals, etc. d s a Lorentz scalar that has the dmenson [ L ]. The weak dpole strength d takes the same value for all elementary spn-1 2 partcles,.e., t s nde- pendent of the partcle s mass. Ths property s consstent wth the unversal feature of the Ferm constant G F. Evdently, an ant-partcle carres a weak dpole of the opposte sgn. It means that approprate formulas of dpole felds can be taken from electrodynamcs. Furthermore, ths scheme provdes unque and well known relatonshps between the weak source and the assocated weak felds. It s explaned below that the Maxwellan-lke form of the weak feld satsfes the requred dmenson of every term of the Lagrangan densty. The tensoral form of ths feld has magnetc-lke components and electrc-lke components. Its explct structure s (see [13], p. 6) 0 x y z x 0 z y =. y z 0 x z y x 0 (6) /10

6 The callgraphc letters, and are used n order to dstngush between weak felds and ther correspondng electromagnetc felds. Let us turn to the second problem. Lke all other terms of the Lagrangan densty, the electromagnetc nteracton term (3) s a dmensonless Lorentz scalar. It holds for the nteractons of the Drac partcle s electrc charge, whch s a dmensonless Lorentz scalar. Hence, the problem s to fnd the form of an analogous expresson for the elementary weak dpole, whch has the nherent dmenson [ L ]. The non-covarant formula for the energy of a magnetc dpole m nteractng wth an external magnetc feld s (see [17], p. 186) U = m B. (7) Ths formula ndcates how to construct the requred expresson. It must depend lnearly on the Maxwellan-lke weak feld of a weak dpole. The very small lmt of the neutrno mass [1] means that n actual experments the neutrno s an ultrarelatvstc partcle where v 1. Therefore, a Lorentz transformaton of the tensor (6) proves that the absolute value of the vector components and of the axal vector components of ths tensor are practcally the same (see [13] p. 66). (8) In order to construct a Lorentz scalar term for the Lagrangan densty of the weak nteractons, one must contract the tensor (6) wth another tensor whch depends on the Drac γ matrces. Evdently, a second rank antsymmetrc tensor s requred for ths purpose. Ths tensor can be readly taken from the lterature (see [8] p. 21) σ ( γγ µ ν γγ ν µ ). (9) 2 Let us wrte down the explct form of (9) as a 4 4 matrx whose entres are the approprate products of two γ µ matrces. 0 γγ 0 1 γγ 0 2 γγ 0 3 γγ γγ 1 2 γγ 1 3 σ. (10) γγ 0 2 γγ γγ 2 3 γγ 0 3 γγ 1 3 γγ Here the ant-commutaton of two dfferent γ matrces s used and the denomnator 2 of (9) s removed. The weak nteracton term of the Lagrangan densty s obtaned from a contracton of (10) and (6), tmes the scalar factor d, whch represents the weak dpole strength. Hence, the term whch s analogous to the electromagnetc nteracton term (3) s ( ) dψσ ψ= 2d ψ γγ + γγ γγ + γγ ψ. (11) Note that the pure magnary factor should not be confused wth the ndces. In order to ft to the numercal notaton of the γ s ndex, the ndces 1, 2, 3 denote the felds' components xyz,,, respectvely. Hereafter, (11) s called the prmary weak nteracton term. The form of the left hand sde of (11) s analogous to the electro- 6/10

7 magnetc nteracton term of the Lagrangan densty (3). In both cases a γ -dependent quantty s contracted wth an external feld and the factors e,d respectvely denote the ntensty of the nteracton. The structure of the prmary weak nteracton term (11) satsfes the Lagrangan densty requrements. Due to fundamental laws of tensor algebra, the full contracton of the two tensors proves that ths term s a Lorentz scalar. The dpole s feld decreases lke 3 r (see [17], p. 182) and t s multpled by the dpole s strength of the source, whose dmenson s [ L ]. Hence, the dmenson of the weak feld s L 2. The factor d of the dpole of the partcle that nteracts wth the weak feld adds the dmenson [ L ]. It follows that the dmenson of the operator of (11) s L 1, whch s the same dmenson as that of the electromagnetc 4-potental A µ of (3). Ths concluson proves that the defnton sayng that the weak feld takes a Maxwellan-lke form s not arbtrary. As stated n the ntroducton, the purpose of ths work s to fnd a theory of weak nteractons processes where flavor s conserved. It means, a descrpton of an elastc neutrno scatterng on a Drac partcle. Ths process s analogous to the elastc scatterng of an electron. Thus, the problem s to fnd the form of the nteracton of the weak dpole of a spn-1 2 partcle wth the Maxwellan-lke weak feld of another spn-1/2 partcle. It means, fndng the matrx element of the transton of a Drac spnor from ts ntal state ψ to ts fnal state ψ f due to ts nteracton wth the feld Here O ( ) f ( ) M = ψ O r d r (12) 3 w ψ. w r s the weak nteracton operator and (12) s analogous to the expresson of an elastc scatterng of an electron on a charged target (see equaton (6.3) n [2], p. 186). The prmary weak nteracton term (11) s used for obtanng an expresson for the ntegrand of the scatterng formula (12). For ths end, the dentty ψ 0 ψ γ (13) s used (see [8], p. 24). The followng calculaton proves that γ 0 of (13) changes dramatcally the form of the prmary nteracton term (11). Indeed, substtutng (13) 0 nto (11) and usng γ = γ 0, one fnds d ψγσ ψ=2d ψ γγγ + γγγ γγγ + γγγ ψ ( ) ( ) ( ) ( ) = 2d ψ γ γγγγγ + γγγγγ γγγγγ ψ = 2dψ γ γγ γγ γγ ψ = 2dψ γ γγ ψ k k In the second lne of (14) three terms are multpled by 1 = γ γ, k, k j. In the thrd lne, the γ s are reordered and the the ant-commutaton relaton γ µ γ ν + γ ν γ µ = 2g s used. The pseudoscalar γ γγγγ s substtuted. The three γ matrces are anthermtan. Therefore, the frst term of (14) s Her- (14) 7/10

8 mtan. It s analogous to the three Drac α matrces, whch are used n the Drac Hamltonan. The followng calculaton shows that also the product of the γ matrces of the second term of (14) are Hermtan. Indeed, ( ) γγ = γ γ = γγ = γγ. Hence, the two terms of (14) are Hermtan operators whch correspond to the vector V and the axal vector A parts of the weak nteractons, respectvely. Relaton (8) means that these terms are contracted wth 3-vectors that practcally have the same absolute value. It means that the weak nteracton theory whch s derved above proves that weak processes do not conserve party. Ths result s also consstent wth the equal weght of V and A n the well known V-A form of weak nteractons [18] [19]. It means that the dpole structure of the weak nteracton theory developed heren proves that an nteracton of a neutrno wth a Drac partcle s n accordance wth the party volatng V-A form of weak nteractons. 4. Concludng Remarks Ths work ams to make the frst step towards the constructon of a consstent weak nteracton theory. As such, t examnes the relatvely smple process of an elastc neutrno scatterng. Lke the case of other theores, t must take some knds of expermentally related nformaton that s used as a bass for the mathematcal structure of the theory. Ths work uses just one specfc knd of expermental nformaton whch s 2 the Ferm constant. The theory uses the dmenson of the Ferm constant [ GF ] = L and ts unversal feature. Ths nformaton s used for constructng a weak nteracton term whch s added to the Lagrangan densty of a Drac partcle. The general structure of the theory s smlar to that of QED. It comprses two knds of physcal objects, a weak axal dpole whch s assocated wth a massve Drac partcle and a weak feld that medates the nteracton between two weak dpoles. The need for such a feld s deduced from the pontlke attrbute of an elementary quantum partcle. The mathematcal structure of the theory s bult on these ssues and the result proves that weak nteractons do not conserve party. Ths theoretcal result s n accordance wth a well known property of weak nteractons. Ths success encourages a further research n ths drecton. The Lagrangan densty obtaned above descrbes weak nteractons of two spn-1/2 Drac partcles whch s medated by a weak feld. In actual cases of scatterng experments, one should remove other, much stronger nteractons. Therefore, one of the nteractng partcles must be a neutrno (or an ant-neutrno). Other aspects of weak nteractons should be analyzed. Here are some ponts: 1) The weak dpole depends on spn orentaton. Takng nto account that n a neutrno scatterng the target s a macroscopc body, one must calculate how a neutrno nteracts wth an electron whose wave functon s not an egenfuncton of s z. The same problem arses n a collson of a neutrno wth a baryon, where the proton (1) 8/10

9 spn crss ndcates that the spn orentaton of a baryonc quark s practcally undetermned. 2) Flavor changng processes should be calculated. 3) The CKM matrx as well as the neutrno oscllaton ndcate that there are three knds (called generatons) of weak dpoles whch nteract wth each other. In hadrons, weak nteractons cause transton wthn generatons and n hadrons and leptons they also cause transton between generatons. Ths evdence certanly complcates the structure of a comprehensve weak nteracton theory. References [1] Patrgnan, C., et al. (Partcle Data Group) (2016) Revew of Partcle Physcs. Chnese Physcs C, 40, Artcle ID: [2] Perkns, D.H. (1987) Introducton to Hgh Energy Physcs. Addson-Wesley, Menlo Park. [3] Formaggo, J.A. and Zeller, G.P. (2012) From ev to EeV: Neutrno Cross Sectons across Energy Scales. Revews of Modern Physcs, 84, [4] Wenberg, S. (199) The Quantum Theory of Felds. Vol. I, Cambrdge Unversty Press, Cambrdge. [] Dehmelt, H. (1988) A Sngle Atomc Partcle Forever Floatng at Rest n Free Space: New Value for Electron Radus. Physca Scrpta, 1988, T22. [6] Rohrlch, F. (2007) Classcal Charged Partcle. World Scentfc, New Jersey. [7] Blenky, S.M. (201) Neutrno n Standard Model and beyond. Physcs of Partcles and Nucle, 46, [8] Bjorken, J.D. and Drell, S.D. (1964) Relatvstc Quantum Mechancs. McGraw-Hll, New York. [9] Comay, E. (2013) Further Problems wth Integral Spn Charged Partcles. Progress n Physcs, 3, 144. [10] Abazov, V.M., et al. (D0 Collaboraton) (2012) Lmts on Anomalous Trlnear Gauge Boson Couplngs from WW, WZ and Wγ Producton n pp collsons at s = 1.96 TeV. Physcs Letters B, 718, [11] Aad, G., et al. (2012) Measurement of the WW Cross Secton n s=7 TeV pp Collsons wth the ATLAS Detector and Lmts on Anomalous Gauge Couplngs. Physcs Letters B, 712, [12] Bjorken, J.D. and Drell, S.D. (196) Relatvstc Quantum Felds. McGraw-Hll, New York. [13] Landau, L.D. and Lfshtz, E.M. (200) The Classcal Theory of Felds. Elsever, Amsterdam. [14] Landau, L.D. and Lfshtz, E.M. (199) Quantum Mechancs. Pergamon, London. [1] Wenberg, S. (1996) The Quantum Theory of Felds. Vol. 2, Cambrdge Unversty Press, Cambrdge. [16] Comay, E. (2016) Problems wth Mathematcally Real Quantum Wave Functons. Open Access Lbrary Journal, 3, e [17] Jackson, J.D. (197) Classcal Electrodynamcs. John Wley, New York. 9/10

10 [18] Glashow, S. (2009) Message for Sudarshan Symposum. Journal of Physcs: Conference Seres, 196, Artcle ID: [19] Feynman, R.P. and Gell-Mann, M. (198) Theory of the Ferm Interacton. Physcal Revew, 109, Submt or recommend next manuscrpt to and we wll provde best servce for you: Publcaton frequency: Monthly 9 subject areas of scence, technology and medcne Far and rgorous peer-revew system Fast publcaton process Artcle promoton n varous socal networkng stes (LnkedIn, Facebook, Twtter, etc.) Maxmum dssemnaton of your research work Submt Your Paper Onlne: Clck Here to Submt Or Contact servce@oalb.com 10/10