The Gravitational Origin of the Weak Interaction s Chirality. Stephon H. S. Alexander Dartmouth College

Transcription

1 The Gravitational Origin of the Weak Interaction s Chirality Stephon H. S. Alexander Dartmouth College

2 In Colloration with A. Marciano L. Smolin

3 S. Alexandrov S.A, A. Marciano,. Altair A. Ashtekar K. Krasnov J. Plebanski Jacobson and Smolin S. Speziale Capovilla and Smolin G. Lisi L. Friedel eferences

4 Some Questions Why is the Weak Interaction Maximally parity violating? Why is the standard model Chiral? Like gravity the weak force interacts universally with all fermions

5 Chirality Gauge Theory and Parity A Lesson on Chiral Gauge Theory L = 1 4 Fa µ F aµ ig 1 A a L µj µ L ig A a µj µ ut Parity Violating if couplings are not the same. Maximal Parity violation if right handed gauge field is missing.

6 eta Decay

7 The Lesson Parity violation can emerge from a Chiral and Parity Symmetric Mother Theory. ut what could it be...

8 General elativity and F Theory Consider the Following Action I F = i F i Λ i i. F i = Λ i, D i = ( Ansatz: A = e A A e A I JSS = ɛ cd ( e a e b F cd Λ ea e b e c e d

9 Homework: Using g µν = e a µe b νη Show that e e e e ' dete ( I JSS = ɛ cd ( e a e b F cd Λ ea e b e c e d Is equivalent to Einstein-Hilbert with a Cosmological Constant

10 Isogravity Nesti, Percacci J.Phys (7, Alexander hep-th Idea: View Gravity as a gauge theory with a connection valued in SO(3, 1; C =SL(, C L SL(, C This is realized by the fact that the Complex Ashtekar-Sen varile is Chiral.

11 Mechanism Treat one SL(,C as the connection for gravity (Jacobson, Smolin 84 The other SL(,C connection as the weak interaction Mother Theory is parity invariant Q: Is there a parity violating sector (solutions and what are they?

12 Mother Theory S = Z 1 8 G 16 G Mother Theory: Extended Plebanski " cd F cd 1 g Krasnov, Smolin cd cd cd " efgh ef gh " cd F F cd,

13 The Theory is Parity Symmetric, which was shown to be a i-metric Theory with a ghost (instility We found a new Parity violating sector that is stle and has a self-consistent perturbative expansion.

14 ewrite Action with Spinorial Indices S = fied reality c A, =, 1 ndices. to two compon This A, =, 1 a Z eft and right ha A = A AA = " A A A A A " A imilarly decompose. The scalar fields ı A F A A F A 4 G 6G ( A A A A 1 ACD (A CD 1 A C D(A C D A A A A ıg ( ACD A C D Left Connection ight Connection AA ( A A A A,

15 Equations of Motion F A = ACD CD AA A 3G 4 Gg A, F A = D A C D C A A A 3G 4 Gg A ACD = 1 8 Gg W (A CD A C D = 1 8 Gg W (A C D AA = 1 4 Gg W A A t to the connection compone D A = D A = variant derivative with respe

16 Symmetric Solution Speziale showed that this theory is a bimetric theory with 8 DOF: two spin- fields, one ghost scalar (8=51 Can be obtained by expanding in g A = ( A g b A, A = ( A g b A motion (1 and (11 o And solving EOM order by order in g

17 Parity reaking Solution This phase instead has one graviton and triplet of SU( gauge fields When coupling constant g is small. The ight Handed Field Strength is dominated by the Cosmological Constant g =, Do we get a self consistent solution?

18 F A CD (A CD 4 Gg W 3G A O(g W =4ıe O(g an expression for A = Gg ( 1? F A g 6 b A = and =

19 The Action II S = Z ı A F A A F A 4 G 6G ( A A A A 81 ı ( 18 G g A CD ( A W C D 4( A A. A = A g b A S = S ( (e AA,A A,A A S (1 (b A,b A,e AA,A A,A A ading order action Z is

20 We arrive at the leading order action S ( = Z ı 4 G A F A e 1 G Z e F A µ F A g µ g ı F A F A 4g YM 9G (16 e (F (A F C D. while the angle is 1 4g YM = g apple apple = g apple(

21 Leptonic Coupling Spin index Isospin index SU( ψ L SU( a b = ( ν 1 ψ L = L e 1 L νl e L ν L a = ψ L a1, e L a = ψ L a,

22 Matter Coupling S Dirac L = Z A A (D AC (A C Variation w.r.t Lagrange Multiplier (A C = A = e AA A S Dirac L = Z Ae A A A (D Z

23 S Dirac = Z Matter Coupling II y Symmetry ight Handed Fermions: Z A A (D A C (A C Unlike the unprimed-leptons (A C = The primed fermions obey F (A C =, Which has no simple general solution

24 eality Conditions Initially we regard all fields as complex (for the lorentzian case, and then specify reality conditions which are to be imposed on the solutions of the equations of motion. Step 1 is the following way: We differentiate the right and left two forms as A = Li A i, A i A = i Step II We then use these to define the left and right Urbantke metrics g = i ac j bd k ef " ijk bdef, g L = Li ac Lj bd Lk ef " ijk bdef. In either case the correct reality conditions are g L =( g L, g =( g. In the symmetric case this tells us that both left and righ

25 In either case the correct reality conditions are (46 L L g = (g, In either case the correct reality conditions are In either case the correct reality conditions are= L g L (g g = (g,. (47 (46 L L left and right handed metrics are real, whereas In the symmetric case this tells us that both g = (g,are (46 In either case the correct reality conditions In either case the correct reality conditions are g = (g. (47 i in the asymmetric solution we learn that g is real and the Yang-Mills connection! is L L a L L., g = (g (47 g = (g (46 g = (g, (46 In the symmetric and right handed metrics are real, whereas left real and hence incase SUthis (.tells us that both i real the asymmetric symmetric case this tellswe us learn that both and handed are real, whereas ininthe solution that g = is themetrics Yang-Mills connection right. and g left (g (47!a is (g = g. (47 can be implemented these reality conditions to the inthese the asymmetric solution we learnby thatadding g is real and the Yang-Mills connection!ai is action so they real and hence in SU (. In the symmetric case this tells us that both left and righthanded handedmetrics metricsare arereal, real,whereas whereas In the symmetric case this tellswhich us that both left and right become equations of motion arise by varying new Lagrange multipliers real and hence in SU (. i L, : These can be implemented by adding these reality conditions to the action so they in the asymmetric solution we learn that g is real and the Yang-Mills connection! i a is in thecan asymmetric solution we learn that g isreality real and the Yang-Mills connection!a is These be implemented by adding these conditions to the action so they motion Z hence become equations of which arise by varying new Lagrange multipliers real and in SU (. L, : realequations and hence in motion SU (. which arise by varying ı become of new Lagrange multipliers : A A A A wrc L, These canbe be implemented by adding these reality conditions to the action so they implemented Z can F F ( S =These by adding these reality conditions to the action so they A A A A ı A Z equations 4 G 6G of motion which by varying new Lagrange : Aarise A multipliers L, A wrc become new Lagrange ı of motion FAwhich arise by FAvarying (A multipliers A A S wrcbecome = equations L, : A A A F F ( S = Z 4 G 6G A A A A Z 14 G 6G ı (A CD A 1 (A A C D A A A A wrc ı F F ( S = A A A A wrc A A A A ACD A C D A A 1 1 FA S =1 (A CD 1 (A A 4 G(A (A 6G A F AC D(A CD AAA A A CD CD A ACD 4 G 6G ACD AC D A A (A CD (A C D A A ıg CD (A (A C D A AA ACD A C D A A A ACD D ( A A A ıg A C A ıg ( ACD C D A A A A A A C D AA ( A ( (( ıg ACD A AA A C D AA A A A A ıgacd ( ( A ACD C D AA L A AA A g ( ACD L g AA L( LA AA L L C D (g. (48 (g (g. g g (g (g L Lg g L (g (48 (48. L L (g L. (48 (g L L g g g g (g. (48 (g L The equation equation of motion (8 modified by by by The equation ofof motion (8is(8 is modified The motion is modified The equation of motion (8 is modified by The equation of motion (8 is modified by g ef g ef ef CD A ef CD A g ef = ACD CDAA ( ( 4 Gg A 4 ı G4 ı Gef Ag ef. (49 A A = ACD FFA 4 Gg. (49 ef CD A AA A g 4 ı = ACD ( ( 4 Gg G.A (49 ef G FA =FFAAACD CD AA 3G(3G 4 Gg A 4 ı AA A A. = ACD 4 Gg A 4 ıgaef (49 A 3G3G A AA 3G of motion for A yields utthe the new new term term vanishes because the equation A A ut vanishes because the equation of motion for yields ut the new term vanishes because the equation of motion for A yieldsa ut the new term vanishes because thethe equation of motion for yields ut the new term vanishes because yields equation of motion for g efg ef g ef = =, (5 ef efefefa g (5 ef, = A A g =, (5 (5 ef ef A =, variation whichwhich implies that vanishes. Meanwhile, thethe reality of of g ef.. A of of enforces vanishes. Meanwhile, implies that variation enforces reality enforces the realityg ef whichwhich implies that vanishes. Meanwhile, variation of implies that enforces the reality of g efof. g ef. vanishes. Meanwhile, variation of EOM for is modified which implies that vanishes. Meanwhile, variation of. (49 (5 enforces the reality of g ef.

26 New Prediction Left handed spinor transforms like a weak doublet scalar, what entity carries this quantum number? ight handed spinor transforms like a weak singlet spin 1/ particle, what is this? These two particles transform into each other under parity

27 New Interactions Dark Hypercharge S U(1 C ( = e 4g YM (f µ f f µ f g µ g (f f f f 9 g G ( 56 e f f ( f f ( f f 4( f F A 4( f F A 4( f F A 4( f F A,

28 Novel Features the two U(1 factors have the same Yang-Mills coupling constant as the SU( L factor, so there is coupling constant unification; however they will couple differently to matter as we will see; there is a universal four point coupling of vector potentials of the form (F F which has a universal coupling f

29 Conclusion Chirality and Parity Violation in EW Theory arises from its other gravitational hand Quantum Numbers of Higgs and Sterile neutrino arise naturally. We expect new predictions for upcoming LHC (or existing LHC experiments The Cosmological constant plays an important role in parity violation. What are the cosmological consequences of this modified gravity theory? What is the indentity of the Auxillary fields? (S.A, E. Livine, A. Marciano

30

31