Abstract:
Frank Dodd (Tony) Smith, Jr., November 2007 and January 2008 - Apppendix re E8 geometry etc
(Click here for pdf version) (Click here to see about F4, E6, and E8 and a root vector movie.)
Garrett Lisi in hep-th/0711.0770 said ".... The building blocks of the standard model and gravity are fields over a four dimensional base manifold. ... Relying on the algebraic structure of the exceptional Lie groups, the fermions may also be recast as Lie algebra elements and included naturally ... the entire ensemble corresponds to a uniquely beautiful Lie group - the largest simple exceptional group, E8. ... The weights of ... 222 elements - corresponding to the quantum numbers of all gravitational and standard model fields - exactly match 222 roots out of the 240 of E8. ... The action for everything, [is] chosen by hand to be in agreement with the standard model with gravity included via the MacDowell-Mansouri technique....".Jacques Distler in November 2007 on his web blog Musings said "... I ...[Jacques Distler am]... not going to talk about spin-statistics, or the Coleman-Mandula Theorem ... that could render Garrett's idea a non-starter ... Instead, I will confine myself to a narrow question in group representation theory. ...
There are two ... noncompact real form[s] of E8 ...
- E8(8) [with] Spin(16) as a maximal compact subgroup ..[where]... the 248 [dimensions of E8]... decompose... as 248 = 120 + 128 ...
- E8(-24) [with] SU(2) x E7 as a maximal compact subgroup ..[where]... the 248 [dimensions of E8]... decompose... as 248 = (3,1) + (1,133) + (2,56)
... I ...[Jacques Distler am here]... deliberately not being careful about such factors of Z2 ... for ease of presentation ...
you [Garrett Lisi] state that G is embedded in E8(-24). Now you say it's embedded in Spin(7,1) x Spin(8) ... That's not a subgroup of E8(-24). It is a subgroup of E8(8). ...".
Garrett Lisi replied "... I [Garrett Lisi] made a mistake in thinking that so(7,1) + so(8) is in the Lie algebra of E IX [ = E8(-24) ] , when in fact it's in ... [ EVIII = E8(8) ]... Thanks. ...".
Bee (Sabine Hossenfelder) said in November on her blog "... I ...[ Bee have ]... complained ..
- about the absence of coupling constants throughout the paper [ hep-th/0711.0770 ] ...
- there is no base manifold present whatsoever ...[in]... the elements of the [E8 Lie] algebra ...
- he [ Garrett Lisi ] has to choose the action by hand to reproduce the SM ...
- for his [ Garrett Lisi's ] theory to work he needs ... the cosmological constant ... to be the size of about the Higgs vev, i.e. roughly 12 orders of magnitude too large
Jacques Distler in November 2007 on his web blog Musings said "... If you want to include the MacDowell-Mansouri Spin(3,1)o, along with the Standard Model gauge group, in E8, then there is not enough "room" to also include 3 generations of quarks and leptons in the 248. That was what Lisi was aiming for. And I think we are all agreed that it doesn't work. ...".
So, this paper is written based on Garrett Lisi's ideas in hep-th/0711.0770 with some modifications to satisfy the objections of Jacques Distler and Bee (Sabine Hossenfelder):
- the structure is based on EVII = E8(8) with 248 = 120 + 128
- there are calculations of coupling constants (force strengths) as well as particle masses and K-M parameters
- the base manifold spacetime is part of E8(8) itself
- the Lagrangian for Gravity plus the Standard Model is based on natural structural relations among various parts of E8(8)
- the Dark Energy (cosmological constant) : Dark Matter : Ordinary Matter ratio is calculated, with results consistent with WMAP
- the second and third generations of fermions are composites of some of the 248 elements of E8 and are not directly related to triality
- triality is useful in establishing relations among fermions, the base manifold, and gauge bosons, which relations indicate that the model satisfies Coleman-Mandula and spin-statistics
For "ease of presentation", sometimes I will be sloppy about such things as signature, distinguishing between Pinors and Spinors, precise group structure distinctions such as between SU(3)xSU(2)xU(1) and S(U(2)xU(3)) = U(1) x SU(2) x SU(3) / I(2) x I(3), etc. I hope that the real meanings will be clear from context.
Any errors in this paper are not Garrett Lisi's fault.
Spin(16) is the bivector Lie algebra of the real Clifford algebra Cl(16)
As Ramon Llull showed about 700 years ago in his Wheel A, the 16 basis vectors of Cl(16) (vertices/letters) combine to form 120 bivectors (vertex pair lines) of Cl(16) which act as the 120 generators of the Lie algebra Spin(16).
The real Clifford algebra 8-periodicity tensor product factorization
In terms of the 28 bivectors of the first Cl(8) factor and the 28 bivectors of the second Cl(8) factor and the 64 product-of-vectors, the 112 are:
Note that in the above image some of the 240 E8(8) vertices are projected to the same point:
The 128 can be seen as the sum 64 + 64 of two 8x8 square-matrices each being 64-dim (colored red and green on the following diagram).
Note that in the above image some of the 240 E8(8) vertices are projected to the same point:
Note that in the above image some of the 240 E8(8) vertices are projected to the same point:
In the above, the black underlined 4+4 = 8 correspond to the 8 E8 Cartan subalgebra elements that are not represented by root vectors, and the black non-underlined 1+3+3+1 = 8 correspond to the 8 elements of 256-dim Cl(8) that do not directly correspond elements of 248-dim E8.
A stereo view of a 24-cell (the 4th dimension color-coded red-green-blue with green in the middle)
shows that the 4-dim 24-cell has a 3-dim central polytope that is a cuboctahedron
the 12 vertices of which form the root vector polytope of the 16-dim U(2,,2) = U(1) x SU(2,2) , where 15-dim rank 3 SU(2,2) = Conformal Group Spin(2,4) produces Gravity by the MacDowell-Mansouri mechanism (see Rabindra Mohapatra, Unification and Supersymmetry (2nd edition, Springer-Verlag 1992), particularly section 14.6).
Since this group structure acts directly on the 8-dim Kaluza-Klein M4 x CP2, it acts on the associative part given by the associative 3-vector PSI of the dimensional reduction Quaternionic structure
(such as occurs due to dimensional reduction of physical spacetime from 8-dim Octonionic to 4-dim Quaternionic by freezing out (at energies lower than the Planck/GUT region) a Quaternionic substructure of 8-dim Octonionic vector space)
which is the spatial part of the M4, so that the M4 on which it acts has signature -+++
The U(1) of U(2,2) provides the complex phase of propagators.
The 28 6-vectors of Cl(8) correspond to a 28-dim rank 4 Spin(8) Lie algebra after introduction of Quaternionic structure into the E8 physics model
(such as occurs due to dimensional reduction of physical spacetime from 8-dim Octonionic to 4-dim Quaternionic by freezing out (at energies lower than the Planck/GUT region) a Quaternionic substructure of 8-dim Octonionic vector space )
by using the co-associative 4-vector PHI of the chosen Quaternionic structure to map any 6-vector A to a bivector A /\ PHI,
and so mapping the 28 6-vectors onto 28 bivectors that form a 28-dim Lie algebra.
The process is somewhat analagous to using a co-associative 4-vector PHI' in Cl(7) to define a cross-product in 7-dim vector space for vectors a, b (see F. Reese Harvey, Spinors and Calibrations (Academic 1990)) by a x b = *((a /\ b) /\ PSI)
A stereo view of a 24-cell (the 4th dimension color-coded red-green-blue with green in the middle)
shows that the 4-dim 24-cell has a 3-dim central polytope that is a cuboctahedron
that is the root vector polytope of 15-dim rank 3 Spin(6) = SU(4) that includes 8+1 = 9-dim SU(3)xU(1) = U(3) in the Twistor construction of 6-dim CP3 = SU(4) / U(3)
Projection into a 2-dim space for the root vectors of the rank 2 group SU(3) gives
where the 6 purple vertices form the hexagonal root vector polygon of 8-dim rank 2 SU(3) and the 6 gold vertices correspond to the 6 dimensions of the CP3 Twistor space.
Introduction of a Quaternionic CP3 Twistor space "... induces a mapping of projective spaces CP3 -> QP1 ...[with]... fibres ... CP1 ..." (see R. O. Wells, Complex Geometry in Mathematical Physics (Les Presses de l'Universite de Montreal 1982), particularly section 2.6).
Since CP1 = SU(2) / U(1) an introduction of Quaternionic structure into the E8 physics model
(such as occurs due to dimensional reduction of physical spacetime from 8-dim Octonionic to 4-dim Quaternionic by freezing out (at energies lower than the Planck/GUT region) a Quaternionic substructure of 8-dim Octonionic vector space )
gives weak force SU(2) through QP1 = Sp(2)/ Sp(1)xSp(1) = Spin(5) / SU(2)xSU(2) or, equivalently, through CP3 containing CP2 = SU(3) / U(2) .
Since the U(1) of U(3) = SU(3) x U(1) is Abelian, it does not correspond to a root vector vertex and therefore does not appear in the root vector diagrams.
Since this group structure is produced by a co-associative 4-vector PHI, it acts on the co-associative part of 8-dim Kaluza-Klein M4 x CP2, which is the CP2 4-dim Internal Symmetry Space of signature ++++
As described by N. A. Batakis in Class. Quantum Grav. 3 (1986) L99-L105, the U(2) = SU(2) x U(1) acts on the CP2 as little group, or local isotropy group, while the SU(3) acts globally on the CP2 = SU(3) / U(2) = SU(3) / SU(2) x U(1)
With respect to the Cl(8) grading, the first 8 of the 8x8 = 64 is the vector space, and therefore is a natural 8-dim spacetime that after introduction of a preferred Quaternionic substructure
(such as occurs due to dimensional reduction of physical spacetime from 8-dim Octonionic to 4-dim Quaternionic by freezing out (at energies lower than the Planck/GUT region) a Quaternionic substructure of 8-dim Octonionic vector space)
becomes a 4-dim plus 4-dim Kaluza-Klein space of the form M4 x CP2 as described by N. A. Batakis in Class. Quantum Grav. 3 (1986) L99-L105,
The M4 of signature -+++ contains an associative 3-dim spatial structure, while the CP2 of signature ++++ has a co-associative 4-dim structure.
So, the first 8 of the 8x8 = 64, denoted by 8_v , represents 4+4 = 8-dim M4 x CP2 Kaluza-Klein space, where the compact CP2 is small.
As to the second 8 of the 8_v x 8,
it lives in the 7-vectors of the Cl(8) grading,
and it should represent the 8 Dirac Gammas of the Cl(8) Clifford algebra, so denote it by 8_G so that
The 128 is the 128-dim rank 8 symmetric space E8 / Spin(16) of type EVIII known as Rosenfeld's octo-octonionic projective plane (OxO)P2 (see Arthur L. Besse, Einstein Manifolds (Springer 1987) and Boris Rosenfeld, Geometry of Lie Groups (Kluwer 1997)).
Since it is a plane (of 2 8x8 octo-octonionic dimensions), it has structure 128 = 64 + 64 = 8x8 + 8x8.
Since it is a half-spinor space (of Spin(16)) its elements are fundamentally fermionic, so
As to the second 8 in the 8_f+ x 8 = 64 and the 8_f- x 8 = 64
it should represent the 8 Dirac Gammas of the Cl(8) Clifford algebra, so denote it by 8_G so that :
Note also that the fermion particles are fundamentally all left-handed, and the fermion antiparticles are fundamentally all right-handed. The other handednesses are not different fundamental states, but arise dynamically due to special relativity transformations that can switch handedness of particles that travel at less than light-speed (i.e., that have more than zero rest mass).
At energies below the Planck/GUT level, the Octonionic structure of the model changes, by freezing out of a preferred Quaternionic substructure, from Real/Octonionic 8-dim spacetime to Quaternionic -+++ associative 4-dim M4 Physical Spacetime plus Quaternionic +++ co-associative 4-dim CP2 = SU(3) / SU(2) x U(1) Internal Symmetry Space.
After Quaternionic structure freezes out,
transform from 8x8 real matrices to 4x4 Quaternionic matrices
As can be seen in this chart (from F. Reese Harvey, Spinors and Calibrations (Academic 1990))
is transformed into
the 8x8x4 = 256-dim Cl(2,6) = M(8,Q) = 8x8 Quaternionic Matrix Algebra
and
is transformed into
the 4x4x4 = 64-dim Cl(2,4) = M(4,Q) = 4x4 Quaternionic Matrix Algebra
and
become
the 2-Quaternionic-dim (8-Real-dim) column vectors 2_Q_v , 2_Q_f+ , 2_Q_f-
and
become
the 2-Quaternionic-dim (8-Real-dim) row vectors 2_Q_G
so that
There is a Spin(8)-type Triality among the three 64 things
The model has:
From the point of view of high-energy 8-dim space, in which gauge boson terms have dimension 1 in the Lagrangian and fermion terms have dimension 7/2 in the Lagrangian, the Triality gives a Subtle Supersymmetry
The natural Lagrangian for the model is
The objective is to reduce the integral over the 8-dim Kaluza-Klein M4 x CP2 to an integral over the 4-dim M4.
Since the U(2,2) acts on the M4, there is no problem with it.
Since the CP2 = SU(3) / U(2) has global SU(3) action, the SU(3) can be considered as a local gauge group acting on the M4, so there is no problem with it.
However, the U(2) acts on the CP2 = SU(3) / U(2) as little group, and so has local action on CP2 and then on M4, so the local action of U(2) on CP2 must be integrated out to get the desired U(2) local action directly on M4.
Since the U(1) part of U(2) = U(1) x SU(2) is Abelian, its local action on CP2 and then M4 can be composed to produce a single U(1) local action on M4, wo there is no problem with it.
That leaves non-Abelian SU(2) with local action on CP2 and then on M4, and the necessity to integrate out the local CP2 action to get something acting locally directly on M4. This is done by a mechanism due to Meinhard Mayer, The Geometry of Symmetry Breaking in Gauge Theories, Acta Physica Austriaca, Suppl. XXIII (1981) 477-490 where he says:
"... We start out from ... four-dimensional M [ M4 ] ...[and]... R ...[that is]... obtained from ... G/H [ CP2 = SU(3) / U(2) ] ... the physical surviving components of A and F, which we will denote by A and F, respectively, are a one-form and two form on M [M4] with values in H [SU(2)] ...the remaining components will be subjected to symmetry and gauge transformations, thus reducing the Yang-Mills action ...[on M4 x CP2]... to a Yang-Mills-Ginzburg-Landau action on M [M4] ... Consider the Yang-Mills action on R ...S_YM = Integral Tr ( F /\ *F ) ... We can ... split the curvature F into components along M [M4] (spacetime) and those along directions tangent to G/H [CP2] .
We denote the former components by F_!! and the latter by F_?? , whereas the mixed components (one along M, the other along G/H) will be denoted by F_!? ... Then the integrand ... becomes
Tr( F_!! F^!! + 2 F_!? F^!? + F_?? F^?? ) ... The first term .. becomes the [SU(2)] Yang-Mills action for the reduced [SU(2)] Yang-Mills theory ...
the middle term .. becomes, symbolically, Tr Sum D_! PHI(?) D^! PHI(?) where PHI(?) is the Lie-algebra-valued 0-form corresponding to the invariance of A with respect tothe vector field ? , in the G/H [CP2] direction ...
the third term ... involves the contraction F_?? of F with two vector fields lying along G/H [CP2] ... we make use of the equation [from Mayer-Trautman, Acta Physica Austriaca, Suppl. XXIII (1981) 433-476, equation 6.18]
2 F_?? = [ PHI(?) , PHI(?) ] - PHI([?,?]) ... Thus, the third term ... reduces to what is essentially a Ginzburg-Landau potential in the components of PHI:
Tr F_?? F^?? = (1/4) Tr ( [ PHI , PHI ] - PHI )^2 ... special cases which were considered show that ...[the equation immediately above]... has indeed the properties required of a Ginzburg_Landau-Higgs potential, and moreover the relative signs of the quartic and quadratic terms are correct, and only one overall normalization constant ... is needed. ...".
(see also S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I, Wiley (1963), especially section II.11)
So,
As to
At low (where we do experiments) energies a Quaternionic structure freezes out, splitting the 8-dim spacetime into a 4-dim physical spacetime M4 and a 4-dim internal symmetry space CP2.
First generation fermion particles are represented by octonions as follows:
Octonion Fermion
Basis Element Particle
1 e-neutrino
i red up quark
j green up quark
k blue up quark
e electron
ie red down quark
je green down quark
ke blue down quark
First generation fermion antiparticles are represented by octonions in a similiar way.
Second generation fermion particles and antiparticles are represented by pairs of octonions.
Third generation fermion particles and antiparticles are represented by triples of octonions.
There are no higher generations of fermions than the Third. This can be seen geometrically as a consequence of the fact that if you reduce the original 8-dimensional spacetime into associative 4-dime M4 physical spacetime and coassociative 4-dim CP2 Internal Symmetry Space then if you look in the original 8-dimensional spacetime at a fermion (First-generation represented by a single octonion) propagating from one vertex to another there are only 4 possibilities for the same propagation after dimensional reduction:
1 - the origin o and target x vertices are both in the associative 4-dimensional physical spacetime
4-dim Internal Symmetry Space -------------- 4-dim Physical SpaceTime ---o------x---
in which case the propagation is unchanged, and the fermion remains a FIRST generation fermion represented by a single octonion o
2 - the origin vertex o is in the associative spacetime and the target vertex * in in the Internal Symmetry Space
4-dim Internal Symmetry Space ----------*---
4-dim Physical SpaceTime ---o----------
in which case there must be a new link from the original target vertex * in the Internal Symmetry Space to a new target vertex x in the associative spacetime
4-dim Internal Symmetry Space ----------*--- 4-dim Physical SpaceTime ---o------x---
and a second octonion can be introduced at the original target vertex in connection with the new link so that the fermion can be regarded after dimensional reduction as a pair of octonions o and * and therefore as a SECOND generation fermion
3 - the target vertex x is in the associative spacetime and the origin vertex o in in the Internal Symmetry Space
4-dim Internal Symmetry Space ---o---------- 4-dim Physical SpaceTime ----------x---
in which case there must be a new link to the original origin vertex o in the Internal Symmetry Space from a new origin vertex * in the associative spacetime
4-dim Internal Symmetry Space ---o---------- 4-dim Physical SpaceTime ---O------x---
so that a second octonion can be introduced at the new origin vertex O in connection with the new link so that the fermion can be regarded after dimensional reduction as a pair of octonions o and o and therefore as a SECOND generation fermion
4 - both the origin vertex o and the target vertex * are in the Internal Symmetry Space,
4-dim Internal Symmetry Space ---o------*--- 4-dim Physical SpaceTime --------------
in which case there must be a new link to the original origin vertex o in the Internal Symmetry Space from a new origin vertex O in the associative spacetime, and a second new link from the original target vertex * in the Internal Symmetry Space to a new target vertex x in the associative spacetime
4-dim Internal Symmetry Space ---o------*--- 4-dim Physical SpaceTime ---O------x---
so that a second octonion can be introduced at the new origin vertex O in connection with the first new link, and a third octonion can be introduced at the original target vertex * in connection with the second new link, so that the fermion can be regarded after dimensional reduction as a triple of octonions O and o and * and therefore as a THIRD generation fermion.
As there are no more possibilities, there are no more generations, and we have:
and
we now have a Lagrangian for the model
Path integrals give a Quantum theory via the classical Lagrangian set out above.
The Lagrangian set out above is only valid in a (possibly small) neighborhood of spacetime. To get a more global theory, the local Lagrangians must be patched together. To do that, look at it from a Cl(8) point of view, and consider that, using 8-periodicity of real Clifford algebras, taking tensor products of factors of Cl(8)
allows construction of arbitrarily large real Clifford algebras as composites of lots of local Cl(8) factors.
As to how to combine local Lagrangians in terms of E8, note that there are 7 independent Root Vector Polytopes / Lattices of type E8, denoted E8_1, E8_2, E8_3, E8_4, E8_5, E8_6, E8_7. Some of them have vertices in commmon, but they are all distinct.
All of the 7 independent Root Vector Polytope Lie algebras E8_i correspond to E8 Lattices consistent with Octonion Multiplication, and the the 7 Lie algebras / Lattices / Root Vector Polytopes E8_i are related to each other as the 7 Octonion imaginaries i,j,k,e,ie,je,ke , so the copies of E8 might combine according to the rules of octonion multiplication, globally arranging themselves like integral octonions.
Such a Spin Foam model might be related to the 26-dim Bosonic String model described in CERN preprint CERN-CDS-EXT-2004-031 in which fermions come from orbifolding and the 7 independent E8_i are used in constructing D8 branes.
Michio Hashimoto, Masaharu Tanabashi, and Koichi Yamawaki in their paper at hep-ph/0311165 describe models with T-quark condensate for Higgs in 8-dimensional Kaluza-Klein spacetime with 4 compact dimensions, like M4 x CP2 of the E8 model, and calculate that
Renormalization running up and down from that point on a plot of Higgs mass v. Tquark mass
shows that the point ( M_H = 176-188 , M_T = 172-175 ) is right on the Triviality Bound curve for as Standard Model with high-energy cut-off at the Planck energy 10^19 GeV (see hep-ph/0307138 ) and
There is not much data for a T-quark-Higgs state around ( M_H = 239 , M_T = 220 ), but perhaps the LHC might shed light on that.
As to a T-quark-Higgs state around ( M_H = 143-160 , M_T = 130-145 ) , it is not conventionally accepted that there is any evidence for such a state, but my opinion about data analysis is that there is such evidence. For example, the initial CDF and D0 histograms for semileptonic events
both independently show a tall narrow peak (green) in the 130-145 GeV range for the Tquark mass. Since mass calculations used in this E8 model had been done prior to those histograms, and had predicted a tree-level (about 10% or so accuracy) value of the Tquark mass of about 130 GeV, those independent CDF and D0 results indicate a probability around 4 sigma for M_T = 130-145 (see an entry on Tommaso Dorigo's blog around 5 September 2007).
In my opinion, recent results from CDF and D0 are still consistent with the existence of a Tquark-Higgs state around ( M_H = 143-160 , M_T = 130-145 ), but the consensus view is otherwise. However, I disagree with that consensus, based on how I see exeperimental data, such as:
Dilepton data described by Erich Ward Varnes in Chapter 8 of his 1997 UC Berkeley PhD thesis about D0 data at Fermilab:
"… there are six t-tbar candidate events in the dilepton final states … Three of the events contain three jets, and in these cases the results of the fits using only the leading two jets and using all combinations of three jets are given …".
There being only 6 dilepton events in Figure 8.1 of Varnes's PhD thesis
it is reasonable to discuss each of them, so (mass is roughly estimated by me looking at the histograms) here they are:
In terms of 3 Truth Quark mass states - high around 220 GeV or so - medium around 170 GeV or so - low around 130-145 GeV or so - those look like:
This, and other more recent experimental subtleties ( see for example www.tony5m17h.net/ and other pages on my web site there ), reinforce my view:
In my opinion, recent results from CDF and D0 are still consistent with the existence of a Tquark-Higgs state around ( M_H = 143-160 , M_T = 130-145 ), but the consensus view is otherwise.
The model Lagrangian (just looking at spacetime and gauge bosons and ignoring spinor fermions etc) is the integral over spacetime of gauge boson terms, so THE FORCE STRENGTH IS MADE UP OF TWO PARTS:
Ignoring for this exposition details about the 4-dim internal symmetry space, and ignoring conformal stuff (Higgs etc), the 4-dim spacetime Lagrangian gauge boson term is the integral over spacetime as seen by gauge boson acting globally of the gauge force term of the gauge boson acting locally for the gauge bosons of each of the four forces:
In the conventional Lagrangian picture, for each gauge force the gauge boson force term contains the force strength, which in Feynman's picture is the probability to emit a gauge boson, in either an explicit ( like g |F|^2 ) or an implicit ( incorporated into the |F|^2 ) form. Either way, the conventional picture is that the force strength g is an ad hoc inclusion.
What I am doing is to construct the integral such that the force strength emerges naturally from the geometry of each gauge force.
To do that, for each gauge force:
1 - make the spacetime over which the integral is taken be spacetime AS IT IS SEEN BY THAT GAUGE BOSON, that is, in terms of the symmetric space with GLOBAL symmetry of the gauge boson:
2 - make the gauge boson force term have the volume of the Shilov boundary corresponding to the symmetric space with LOCAL symmetry of the gauge boson. The nontrivial Shilov boundaries are:
The result is (ignoring technicalities for exposition) the geometric factor for force strength calculation.
GLOBAL: Each gauge group is the global symmetry of a symmetric space
LOCAL: Each gauge group is the local symmetry of a symmetric space
The nontrivial local symmetry symmetric spaces correspond to bounded complex domains
The nontrivial bounded complex domains have Shilov boundaries
GLOBAL AND LOCAL TOGETHER: Very roughly think of the force strength as
That is (again very roughly and intuitively): the geometric strength of the force is given by the product of
When you calculate the product volumes (using some normalizations etc that are described in more detail here below ), you see that roughly:
so since (as Feynman says) force strength = probability to emit a gauge boson means that the highest force strength or probability should be 1
I normalize the gravity Volume product to be 1, and get results roughly ( for example, the fine structrure constant calculation gives 1/137.03608 but is rounded off here as 1/137 ):
There are two further main components of a force strength:
CONSIDER MASSIVE GAUGE BOSONS: I consider gravity to be carried by virtual Planck-mass black holes, so that the geometric strength of gravity should be reduced by 1/Mp^2 and I consider the weak force to be carried by weak bosons, so that the geometric strength of gravity should be reduced by 1/MW^2 That gives the result:
FINALLY, CONSIDER RENORMALIZATION RUNNING FOR THE COLOR FORCE: That gives the result:
The use of compact volumes is itself a calculational device, because it would be more nearly correct, instead of
to use
However, since the strongest (gravitation) geometric force strength is to be normalized to 1, the only thing that matters is RATIOS, and the compact volumes (finite and easy to look up in the book by Hua) have the same ratios as the noncompact invariant measures.
In fact, I should go on to say that continuous spacetime and gauge force geometric objects are themselves also calculational devices, and
that it would be even more nearly correct to do the calculations with respect to a discrete generalized hyperdiamond Feynman checkerboard.
Some of this material was written in connection with email discussion with Ark Jadczyk. More details can be found on my web site at www.valdostamuseum.org/hamsmith/
The force strength of a given force is
alphaforce = (1 / Mforce^2 \)
( Vol(MISforce))
( Vol(Qforce) / Vol(Dforce)^( 1 / mforce ))
where:
alphaforce represents the force strength;
Mforce represents the effective mass;
MISforce represents the part of the target
Internal Symmetry Space that is available for the gauge
boson to go to;
Vol(MISforce) stands for volume of MISforce,
and is sometimes also denoted by the shorter notation Vol(M);
Qforce represents the link from the origin
to the target that is available for the gauge
boson to go through;
Vol(Qforce) stands for volume of Qforce;
Dforce represents the complex bounded homogeneous domain
of which Qforce is the Shilov boundary;
mforce is the dimensionality of Qforce,
which is 4 for Gravity and the Color force,
2 for the Weak force (which therefore is considered to
have two copies of QW for each spacetime HyperDiamond link),
and 1 for Electromagnetism (which therefore is considered to
have four copies of QE for each spacetime HyperDiamond link)
Vol(Dforce)^( 1 / mforce ) stands for
a dimensional normalization factor (to reconcile the dimensionality
of the Internal Symmetry Space of the target vertex
with the dimensionality of the link from the origin to the
target vertex).
The Qforce, Hermitian symmetric space,
and Dforce manifolds for the four forces are:
Gauge Hermitian Type mforce Qforce
Group Symmetric of
Space Dforce
Spin(5) Spin(7) / Spin(5)xU(1) IV5 4 RP^1xS^4
SU(3) SU(4) / SU(3)xU(1) B^6(ball) 4 S^5
SU(2) Spin(5) / SU(2)xU(1) IV3 2 RP^1xS^2
U(1) - - 1 -
The geometric volumes needed for the calculations are mostly taken from the book Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains (AMS 1963, Moskva 1959, Science Press Peking 1958) by L. K. Hua [with unit radius scale].
Note ( thanks to Carlos Castro for noticing this ) that the volume lisrted for S5 is for a squashed S5, a Shilov boundary of the complex domain corresponding to the symmetric space SU(4) / SU(3) x U(1).
Note ( thanks again to Carlos Castro for noticing this ) also that the volume listed for CP2 is unconventional, but physically justified by noting that S4 and CP2 can be seen as having the same physical volume, with the only difference being structure at infinity.
Note also that
Force M Vol(M)
gravity S^4 8pi^2/3 - S^4 is 4-dimensional
color CP^2 8pi^2/3 - CP^2 is 4-dimensional
weak S^2 x S^2 2 x 4pi - S^2 is a 2-dim boundary of 3-dim ball
4-dim S^2 x S^2 =
= topological boundary of 6-dim 2-polyball
Shilov Boundary of 6-dim 2-polyball = S^2 + S^2 =
= 2-dim surface frame of 4-dim S^2 x S^
e-mag T^4 4 x 2pi - S^1 is 1-dim boundary of 2-dim disk
4-dim T^4 = S^1 x S^1 x S^1 x S^1 =
= topological boundary of 8-dim 4-polydisk
Shilov Boundary of 8-dim 4-polydisk =
= S^1 + S^1 + S^1 + S^1 =
= 1-dim wire frame of 4-dim T^4
Also note that for U(1) electromagnetism, whose photon carries no charge, the factors Vol(Q) and Vol(D) do not apply and are set equal to 1, and from another point of view, the link manifold to the target vertex is trivial for the abelian neutral U(1) photons of Electromagnetism, so we take QE and DE to be equal to unity.
Force M Vol(M) Q Vol(Q) D Vol(D)
gravity S^4 8pi^2/3 RP^1xS^4 8pi^3/3 IV5 pi^5/2^4 5!
color CP^2 8pi^2/3 S^5 4pi^3 B^6(ball) pi^3/6
weak S^2xS^2 2x4pi RP^1xS^2 4pi^2 IV3 pi^3/24
e-mag T^4 4x2pi - - - -
Using these numbers, the results of the
calculations are the relative force strengths
at the characteristic energy level of the
generalized Bohr radius of each force:
Gauge Force Characteristic Geometric Total
Group Energy Force Force
Strength Strength
Spin(5) gravity approx 10^19 GeV 1 GGmproton^2
approx 5 x 10^-39
SU(3) color approx 245 MeV 0.6286 0.6286
SU(2) weak approx 100 GeV 0.2535 GWmproton^2
approx 1.05 x 10^-5
U(1) e-mag approx 4 KeV 1/137.03608 1/137.03608
The force strengths are given at the characteristic
energy levels of their forces, because the force
strengths run with changing energy levels.
The effect is particularly pronounced with the color
force.
The color force strength was calculated using a simple
perturbative QCD renormalization group equation
at various energies, with the following results:
Energy Level Color Force Strength
245 MeV 0.6286
5.3 GeV 0.166
34 GeV 0.121
91 GeV 0.106
Taking other effects, such as Nonperturbative QCD,
into account, should give
a Color Force Strength of about 0.125 at about 91 GeV
The E8 model Lagrangian (for this message just looking at spacetime and spinor fermions and ignoring gauge bosons etc) has
an Integral over 8-dim spacetime of a spinor fermion particle and antiparticle term,
in which first-generation fermion particles correspond to octonion basis elements
and first-generation fermion antiparticles correspond to octonion basis elements
At low (where we do experiments) energies a specific quaternionic submanifold freezes out, splitting the 8-dim spacetime into a 4-dim M4 physical spacetime plus a 4-dim CP2 internal symmetry space and creating second and third generation fermions that live (at least in part) in the 4-dim CP2 internal symmetry space and correspond respectively to pairs and triples of octonion basis elements.
Ignoring for this exposition details about the 4-dim CP2 internal symmetry space, and ignoring conformal stuff (Higgs etc), and considering for now only first generation fermions, the 4-dim spacetime Lagrangian spinor fermion part is:
In the conventional picture, the spinor fermion term is of the form m S S* where m is the fermion mass and S and S* represent the given fermion. Although the mass m is derived from the Higgs mechanism, the Higgs coupling constants are, in the conventional picture, ad hoc parameters, so that effectively the mass term is, in the conventional picuture, an ad hoc inclusion.
What I am doing is to NOT put in the mass m as an ad hoc Higgs coupling value,
but to construct the integral such that the mass m emerges naturally from the geometry of the spinor fermions.
To do that, make the spinor fermion mass term have the volume of the Shilov boundary corresponding to the symmetric space with LOCAL symmetry of the Spin(8) gauge group with respect to which the first generation spinor fermions can be seen as +half-spinor and -half-spinor spaces.
Then, take the the spinor fermion volume to be the Shilov boundary corresponding to the same symmetric space on which Spin(8) acts as a local gauge group that is used to construct 8-dimensional vector spacetime:
the symmetric space Spin(10) / Spin(8)xU(1) corresponds to a bounded domain of type IV8 whose Shilov boundary is RP^1 x S^7
Since all the first generation fermions see the spacetime over which the integral is taken in the same way ( unlike what happens for the force strength calculation ), the only geometric volume factor relevant for calculating first generation fermion mass ratios is in the spinor fermion volume term.
Since the physcally observed fermions in this model correspond to Kerr-Newman Black Holes, the quark mass in this model is a constituent mass.
Consider a first-generation massive lepton (or antilepton, i.e., electron or positron). For definiteness, consider an electron E (a similar line of reasoning applies to the positron).
Since the electron cannot be related to any other massive Dirac fermion, its volume V(electron) is taken to be 1.
Consider a first-generation quark (or antiquark). For definiteness, consider a red down quark I (a similar line of reasoning applies to the others of the first generation).
Therefore first-generation quarks or antiquarks can by gluons, weak bosons, or decay occupy the entire volume of the Shilov boundary RP1 x S7, which volume is pi^5 / 3, so its volume V(quark) is taken to be pi^5 / 3.
Consider graviton interactions with first-generation fermions.
MacDowell-Mansouri gravitation comes from 10 Spin(5) gauge bosons, 8 of which are charged (carrying color or electric charge).
2 of the charged Spin(5) gravitons carry electric charge. However, even though the electron carries electric charge, the electric charge carrying Spin(5) gravitons can only change the electron into a ( tree-level ) massless neutrino, so the Spin(5) gravitons do not enhance the electron volume factor, which remains
6 of the charged Spin(5) gravitons carry color charge, and their action on quarks (which carry color charge) multiplies the quark volume V(quark) by 6, giving
The 2 Spin(5) gravitons carrying electric charge only cannot change quarks into leptons, so they do not enhance the quark volume factor, so we have (where md is down quark mass, mu is up quark mass, and me is electron mass)
The proton mass is calculated as the sum of the constituent masses of its constituent quarks
which is close to the experimental value of 938.27 MeV.
In the first generation, each quark corresponds to a single octonion basis element and the up and down quark constituent masses are the same:
First Generation - 8 singletons - mu / md = 1
Second and third generation calculations are generally more complicated ( some details are given here below ) with combinatorics indicating that in higher generations the up-type quarks are heavier than the down-type quarks. The third generation case, in which the fermions correspond to triples of octonions, is simple enough to be used in this expository overview as an illustration of the combinatoric effect:
8^3 = 512 triples
mt / mb = 483 / 21 = 161 / 7 = 23
Here is a summary of the results of calculations of tree-level fermion masses (quark masses are constituent masses):
The use of compact volumes is itself a calculational device, because it would be more nearly correct, instead of
to use
However, since the strongest (gravitation) geometric force strength is to be normalized to 1, the only thing that matters is RATIOS, and the compact volumes (finite and easy to look up in the book by Hua) have the same ratios as the noncompact invariant measures.
In fact, I should go on to say that continuous spacetime and gauge force geometric objects are themselves also calculational devices, and
that it would be even more nearly correct to do the calculations with respect to a discrete generalized hyperdiamond Feynman checkerboard.
Fermion masses are calculated as a product of four factors: V(Qfermion) x N(Graviton) x N(octonion) x Sym
The ratio of the down quark constituent mass to the electron mass is then calculated as follows:
Consider the electron, e. By photon, weak boson, and gluon interactions, e can only be taken into 1, the massless neutrino. The electron and neutrino, or their antiparticles, cannot be combined to produce any of the massive up or down quarks. The neutrino, being massless at tree level, does not add anything to the mass formula for the electron. Since the electron cannot be related to any other massive Dirac fermion, its volume V(Qelectron) is taken to be 1.Next consider a red down quark ie. By gluon interactions, ie can be taken into je and ke, the blue and green down quarks. By also using weak boson interactions, it can be taken into i, j, and k, the red, blue, and green up quarks.
Given the up and down quarks, pions can be formed from quark-antiquark pairs, and the pions can decay to produce electrons and neutrinos.
Therefore the red down quark (similarly, any down quark) is related to any part of S^7 x RP^1, the compact manifold corresponding to { 1, i, j, k, ie, ie, ke, e } and therefore a down quark should have a spinor manifold volume factor V(Qdown quark) of the volume of S^7 x RP^1.
The ratio of the down quark spinor manifold volume factor tothe electron spinor manifold volume factor is just
V(Qdown quark) / V(Qelectron) = V(S^7x RP^1)/1 = pi^5 / 3. Since the first generation graviton factor is 6,
md/me = 6V(S^7 x RP^1) = 2 pi^5 = 612.03937
As the up quarks correspond to i, j, and k, which are the octonion transforms under e of ie, je, and ke of the down quarks, the up quarks and down quarks have the same constituent mass
Antiparticles have the same mass as the corresponding particles.
Since the model only gives ratios of massses, the mass scale is fixed so that the electron mass me = 0.5110 MeV.
and the constituent mass for the up quark is mu = 312.75 MeV.
These results when added up give a total mass of first generation fermion particles:
As the proton mass is taken to be the sum of the constituent masses of its constituent quarks
The theoretical calculation is close to the experimental value of 938.27 MeV.
The third generation fermion particles correspond to triples of octonions. There are 8^3 = 512 such triples.
The triple { 1,1,1 } corresponds to the tau-neutrino.
The other 7 triples involving only 1 and e correspond
to the tauon:
The symmetry of the 7 tauon triples is the same as the symmetry of the 3 down quarks, the 3 up quarks, and the electron, so the tauon mass should be the same as the sum of the masses of the first generation massive fermion particles. Therefore the tauon mass is calculated at tree level as 1.877 GeV.
The calculated Tauon mass of 1.88 GeV is a sum of first generation fermion masses, all of which are valid at the energy level of about 1 GeV.
However, as the Tauon mass is about 2 GeV, the effective Tauon mass should be renormalized from the energy level of 1 GeV (where the mass is 1.88 GeV) to the energy level of 2 GeV. Such a renormalization should reduce the mass. If the renormalization reduction were about 5 percent,
The 1996 Particle Data Group Review of Particle Physics gives a Tauon mass of 1.777 GeV.
Note that all triples corresponding to the tau and the tau-neutrino are colorless.
The beauty quark corresponds to 21 triples.
They are triples of the same form as the 7 tauon triples, but for 1 and ie, 1 and je, and 1 and ke, which correspond to the red, green, and blue beauty quarks, respectively.
The seven triples of the red beauty quark correspond to the seven triples of the tauon, except that the beauty quark interacts with 6 Spin(0,5) gravitons while the tauon interacts with only two.
The beauty quark constituent mass should be the tauon mass times the third generation graviton factor 6/2 = 3, so the B-quark mass is
The calculated Beauty Quark mass of 5.63 GeV is a consitituent mass, that is, it corresponds to the conventional pole mass plus 312.8 MeV.
Therefore, the calculated Beauty Quark mass of 5.63 GeV corresponds to a conventional pole mass of 5.32 GeV.
The 1996 Particle Data Group Review of Particle Physics gives a lattice gauge theory Beauty Quark pole mass as 5.0 GeV.
The pole mass can be converted to an MSbar mass if the color force strength constant alpha_s is known. The conventional value of alpha_s at about 5 GeV is about 0.22. Using alpha_s (5 GeV) = 0.22, a pole mass of 5.0 GeV gives an MSbar 1-loop Beauty Quark mass of 4.6 GeV, and
If the MSbar mass is run from 5 GeV up to 90 GeV, the MSbar mass decreases by about 1.3 GeV, giving an expected MSbar mass of about 3.0 GeV at 90 GeV.
DELPHI at LEP has observed the Beauty Quark and found a 90 GeV MSbar Beauty Quark mass of about 2.67 GeV, with error bars +/- 0.25 (stat) +/- 0.34 (frag) +/- 0.27 (theo).
Note that the theoretical model calculated mass of 5.63 GeV corresponds to a pole mass of 5.32 GeV, which is somewhat higher than the conventional value of 5.0 GeV. However, the theoretical model calculated value of the color force strength constant alpha_s at about 5 GeV is about 0.166, while the conventional value of the color force strength constant alpha_s at about 5 GeV is about 0.216, and the theoretical model calculated value of the color force strength constant alpha_s at about 90 GeV is about 0.106, while the conventional value of the color force strength constant alpha_s at about 90 GeV is about 0.118.
The theoretical model calculations gives a Beauty Quark pole mass (5.3 GeV) that is about 6 percent higher than the conventional Beauty Quark pole mass (5.0 GeV), and a color force strength alpha_s at 5 GeV (0.166) such that 1 + alpha_s = 1.166 is about 4 percent lower than the conventional value of 1 + alpha_s = 1.216 at 5 GeV.
Note particularly that triples of the type { 1, ie, je } , { ie, je, ke }, etc., do not correspond to the beauty quark, but to the truth quark.
The truth quark corresponds to the remaining 483 triples, so the constituent mass of the red truth quark is 161/7 = 23 times the red beauty quark mass, and the red T-quark mass is
The blue and green truth quarks are defined similarly.
All other masses than the electron mass (which is the basis of the assumption of the value of the Higgs scalar field vacuum expectation value v = 252.514 GeV), including the Higgs scalar mass and Truth quark mass, are calculated (not assumed) masses in the E8 model.
These results when added up give a total mass of third generation fermion particles:
The second generation fermion particles correspond to pairs of octonions.
There are 8^2 = 64 such pairs. The pair { 1,1 } corresponds to the mu-neutrino. The pairs { 1, e }, { e, 1 }, and { e, e } correspond to the muon.
Compare the symmetries of the muon pairs to the symmetries of the first generation fermion particles.
The pair { e, e } should correspond to the e electron.
The other two muon pairs have a symmetry group S2, which is 1/3 the size of the color symmetry group S3 which gives the up and down quarks their mass of 312.75 MeV.
Therefore the mass of the muon should be the sum of
According to the 1998 Review of Particle Physics of the Particle Data Group, the experimental muon mass is about 105.66 MeV.
Note that all pairs corresponding to the muon and the mu-neutrino are colorless.
The red, blue and green strange quark each corresponds to the 3 pairs involving 1 and ie, je, or ke.
The red strange quark is defined as the three pairs 1 and i, because i is the red down quark.Its mass should be the sum of two parts:
Unlike the first generation situation, massive second and third generation leptons can be taken, by both of the colorless gravitons that may carry electric charge, into massive particles. Therefore the graviton factor for the second and third generations is 6/2 = 3.
Therefore the symmetry part of the muon mass times the graviton factor 3 is 312.75 MeV, and the red strange quark constituent mass is
The blue strange quarks correspond to the three pairs involving j, the green strange quarks correspond to the three pairs involving k, and their masses are determined similarly.
The charm quark corresponds to the other 51 pairs. Therefore, the mass of the red charm quark should be the sum of two parts:
Therefore the red charm quark constituent mass is
The blue and green charm quarks are defined similarly, and their masses are calculated similarly.
The calculated Charm Quark mass of 2.09 GeV is a consitituent mass, that is, it corresponds to the conventional pole mass plus 312.8 MeV.
Therefore, the calculated Charm Quark mass of 2.09 GeV corresponds to a conventional pole mass of 1.78 GeV.
The 1996 Particle Data Group Review of Particle Physics gives a range for the Charm Quark pole mass from 1.2 to 1.9 GeV.
The pole mass can be converted to an MSbar mass if the color force strength constant alpha_s is known. The conventional value of alpha_s at about 2 GeV is about 0.39, which is somewhat lower than the teoretical model value. Using alpha_s (2 GeV) = 0.39, a pole mass of 1.9 GeV gives an MSbar 1-loop mass of 1.6 GeV, evaluated at about 2 GeV.
These results when added up give a total mass of second generation fermion particles:
As with forces strengths, the calculations produce ratios of masses, so that only one mass need be chosen to set the mass scale.
In the E8 model, the value of the fundamental mass scale vacuum expectation value v = <PHI> of the Higgs scalar field is set to be the sum of the physical masses of the weak bosons, W+, W-, and Z0,
whose tree-level masses will then be shown by ratio calculations to be 80.326 GeV, 80.326 GeV, and 91.862 GeV, respectively,
and so that the electron mass will then be 0.5110 MeV.
The relationship between the Higgs mass and v is given by the Ginzburg-Landau term from the Mayer Mechanism as
or, in the notation of hep-ph/9806009 by Guang-jiong Ni
where the Higgs mass M_H = sqrt( 2 sigma )
Ni says: "... the invariant meaning of the constant lambda in the Lagrangian is not the coupling constant, the latter will change after quantization ... The invariant meaning of lambda is nothing but the ratio of two mass scales:
which remains unchanged irrespective of the order ...".
Since <PHI>^2 = v^2, and assuming at tree-level that lambda = 1 ( a value consistent with the Higgs Tquark condensate model of Michio Hashimoto, Masaharu Tanabashi, and Koichi Yamawaki in their paper at hep-ph/0311165 ), we have, at tree-level
In the E8 model, the fundamental mass scale vacuum expectation value v of the Higgs scalar field is the fundamental mass parameter that is to be set to define all other masses by the mass ratio formulas of the model and
so that
To get W-boson masses, denote the 3 SU(2) high-energy weak bosons (massless at energies higher than the electroweak unification) by W+, W-, and W0, corresponding to the massive physical weak bosons W+, W-, and Z0.
The triplet { W+, W-, W0 } couples directly with the T - Tbar quark-antiquark pair, so that the total mass of the triplet { W+, W-, W0 } at the electroweak unification is equal to the total mass of a T - Tbar pair, 259.031 GeV.
The triplet { W+, W-, Z0 } couples directly with the Higgs scalar, which carries the Higgs mechanism by which the W0 becomes the physical Z0, so that the total mass of the triplet { W+, W-, Z0 } is equal to the vacuum expectation value v of the Higgs scalar field, v = 252.514 GeV.
First, look at the triplet { W+, W-, W0 } which can be represented by the 3-sphere S^3. The Hopf fibration of S^3 as
gives a decomposition of the W bosons into the neutral W0 corresponding to S^1 and the charged pair W+ and W- corresponding to S^2.
The mass ratio of the sum of the masses of W+ and W- to the mass of W0 should be the volume ratio of the S^2 in S^3 to the S^1 in S3.
The ratio of the sum of the W+ and W- masses to the W0 mass should then be
Since the total mass of the triplet { W+, W-, W0 } is 259.031 GeV, the total mass of a T - Tbar pair, and the charged weak bosons have equal mass, we have
The charged W+/- neutrino-electron interchange must be symmetric with the electron-neutrino interchange, so that the absence of right-handed neutrino particles requires that the charged W+/- SU(2) weak bosons act only on left-handed electrons.
Each gauge boson must act consistently on the entire Dirac fermion particle sector, so that the charged W+/- SU(2) weak bosons act only on left-handed fermion particles of all types.
The neutral W0 weak boson does not interchange Weyl neutrinos with Dirac fermions, and so is not restricted to left-handed fermions, but also has a component that acts on both types of fermions, both left-handed and right-handed, conserving parity.
However, the neutral W0 weak bosons are related to the charged W+/- weak bosons by custodial SU(2) symmetry, so that the left-handed component of the neutral W0 must be equal to the left-handed (entire) component of the charged W+/-.
Since the mass of the W0 is greater than the mass of the W+/-, there remains for the W0 a component acting on both types of fermions.
Therefore the full W0 neutral weak boson interaction is proportional to (M_W+/-^2 / M_W0^2) acting on left-handed fermions and
(1 - (M_W+/-^2 / M_W0^2)) acting on both types of fermions.
If (1 - (M_W+/-2 / M_W0^2)) is defined to be sin( theta_w )^2 and denoted by K,
and if the strength of the W+/- charged weak force (and of the custodial SU(2) symmetry) is denoted by T,
then the W0 neutral weak interaction can be written as W0L = T + K and W0LR = K.
Since the W0 acts as W0L with respect to the parity violating SU(2) weak force
and as W0LR with respect to the parity conserving U(1) electromagnetic force of the U(1) subgroup of SU(2), the W0 mass mW0 has two components:
the parity violating SU(2) part mW0L that is equal to M_W+/-
the parity conserving part M_W0LR that acts like a heavy photon.
As M_W0 = 98.379 GeV = M_W0L + M_W0LR, and as M_W0L = M_W+/- = 80.326 GeV, we have M_W0LR = 18.053 GeV.
Denote by *alphaE = *e^2 the force strength of the weak parity conserving U(1) electromagnetic type force that acts through the U(1) subgroup of SU(2).
The electromagnetic force strength alphaE = e^2 = 1 / 137.03608 was calculated above using the volume V(S^1) of an S^1 in R^2, normalized by 1 / sqrt( 2 ).
The *alphaE force is part of the SU(2) weak force whose strength alphaW = w^2 was calculated above using the volume V(S^2) of an S^2 \subset R^3, normalized by 1 / sqrt( 3 ).
Also, the electromagnetic force strength alphaE = e^2 was calculated above using a 4-dimensional spacetime with global structure of the 4-torus T^4 made up of four S^1 1-spheres,
while the SU(2) weak force strength alphaW = w^2 was calculated above using two 2-spheres S^2 x S^2, each of which contains one 1-sphere of the *alphaE force.
Therefore
and the mass mW0LR must be reduced to an effective value M_W0LReff = M_W0LR / 1.565 = 18.053/1.565 = 11.536 GeV for the *alphaE force to act like an electromagnetic force in the E8 model:
where the physical effective neutral weak boson is denoted by Z0.
Therefore, the correct E8 model values for weak boson masses and the Weinberg angle theta_w are:
M_W+ = M_W- = 80.326 GeV;
M_Z0 = 80.326 + 11.536 = 91.862 GeV;
Sin(theta_w)^2 = 1 - (M_W+/- / M_Z0)^2 = 1 - ( 6452.2663 / 8438.6270 ) = 0.235.
Radiative corrections are not taken into account here, and may change these tree-level values somewhat.
The Kobayashi-Maskawa parameters are determined in terms of the sum of the masses of the 30 first-generation fermion particles and antiparticles, denoted by Smf1 = 7.508 GeV,
and the similar sums for second-generation and third-generation fermions, denoted by Smf2 = 32.94504 GeV and Smf3 = 1,629.2675 GeV.
The reason for using sums of all fermion masses (rather than sums of quark masses only) is that all fermions are in the same spinor representation of Spin(8), and the Spin(8) representations are considered to be fundamental.
The following formulas use the above masses to calculate Kobayashi-Maskawa parameters:
The factor sqrt( Smf2 /Smf1 ) appears in s23 because an s23 transition is to the second generation and not all the way to the first generation, so that the end product of an s23 transition has a greater available energy than s12 or s13 transitions by a factor of Smf2 / Smf1 .
Since the width of a transition is proportional to the square of the modulus of the relevant KM entry and the width of an s23 transition has greater available energy than the s12 or s13 transitions by a factor of Smf2 / Smf1
the effective magnitude of the s23 terms in the KM entries is increased by the factor sqrt( Smf2 /Smf1 ) .
The Chau-Keung parameterization is used, as it allows the K-M matrix to be represented as the product of the following three 3x3 matrices:
1 0 0
0 cos(gamma) sin(gamma)
0 -sin(gamma) cos(gamma)
cos(beta) 0 sin(beta)exp(-i d13)
0 1 0
-sin(beta)exp(i d13) 0 cos(beta)
cos(alpha) sin(alpha) 0
-sin(alpha) cos(alpha) 0
0 0 1
The resulting Kobayashi-Maskawa parameters for W+ and W- charged weak boson processes, are:
d s b u 0.975 0.222 0.00249 -0.00388i c -0.222 -0.000161i 0.974 -0.0000365i 0.0423 t 0.00698 -0.00378i -0.0418 -0.00086i 0.999
The matrix is labelled by either (u c t) input and (d s b) output, or, as above, (d s b) input and (u c t) output.
For Z0 neutral weak boson processes, which are suppressed by the GIM mechanism of cancellation of virtual subprocesses, the matrix is labelled by either (u c t) input and (u'c't') output, or, as below, (d s b) input and (d's'b') output:
d s b d' 0.975 0.222 0.00249 -0.00388i s' -0.222 -0.000161i 0.974 -0.0000365i 0.0423 b' 0.00698 -0.00378i -0.0418 -0.00086i 0.999
Since neutrinos of all three generations are massless at tree level, the lepton sector has no tree-level K-M mixing.
According to a Review on the KM mixing matrix by Gilman, Kleinknecht, and Renk in the 2002 Review of Particle Physics:
"... Using the eight tree-level constraints discussed below together with unitarity, and assuming only three generations, the 90% confidence limits on the magnitude of the elements of the complete matrix are
d s b u 0.9741 to 0.9756 0.219 to 0.226 0.00425 to 0.0048 c 0.219 to 0.226 0.9732 to 0.9748 0.038 to 0.044 t 0.004 to 0.014 0.037 to 0.044 0.9990 to 0.9993
... The constraints of unitarity connect different elements, so choosing a specific value for one element restricts the range of others. ... The phase d13 lies in the range 0 < d13 < 2 pi, with non-zero values generally breaking CP invariance for the weak interactions. ... Using tree-level processes as constraints only, the matrix elements ...[ of the 90% confidence limit shown above ]... correspond to values of the sines of the angles of s12 = 0.2229 +/- 0.0022, s23 = 0.0412 +/- 0.0020, and s13 = 0.0036 +/- 0.0007. If we use the loop-level processes discussed below as additional constraints, the sines of the angles remain unaffected, and the CKM phase, sometimes referred to as the angle gamma = phi3 of the unitarity triangle ... is restricted to d13 = ( 1.02 +/- 0.22 ) radians = 59 +/- 13 degrees. ... CP-violating amplitudes or differences of rates are all proportional to the product of CKM factors ... s12 s13 s23 c12 c13^2 c23 sind13. This is just twice the area of the unitarity triangle. ... All processes can be quantitatively understood by one value of the CKM phase d13 = 59 +/- 13 degrees. The value of beta = 24 +/- 4 degrees from the overall fit is consistent with the value from the CP-asymmetry measurements of 26 +/- 4 degrees. The invariant measure of CP violation is J = ( 3.0 +/- 0.3) x 10^(-5). ... From a combined fit using the direct measurements, B mixing, epsilon, and sin2beta, we obtain: Re Vtd = 0.0071 +/- 0.0008 , Im Vtd = -0.0032 +/- 0.0004 ... Constraints... on the position of the apex of the unitarity triangle following from | Vub | , B mixing, epsilon, and sin2beta. ...".
In hep-ph/0208080, Yosef Nir says: "... Within the Standard Model, the only source of CP violation is the Kobayashi-Maskawa (KM) phase ... The study of CP violation is, at last, experiment driven. ... The CKM matrix provides a consistent picture of all the measured flavor and CP violating processes. ... There is no signal of new flavor physics. ... Very likely, the KM mechanism is the dominant source of CP violation in flavor changing processes. ... The result is consistent with the SM predictions. ...".
Consider the three generations of neutrinos:
and three neutrino mass states: nu_1 ; nu_2 : nu_3
and the division of 8-dimensional spacetime into
The lightest mass state nu_1 corresponds to a neutrino whose propagation begins and ends in physical Minkowski spacetime, lying entirely therein. According to the E8 model, the mass of nu_1 is zero at tree-level and it picks up no first-order correction while propagating entirely through physical Minkowski spacetime, so the first-order corrected mass of nu_1 is zero.
Since only two of the three neutrinos have first-order mass, and since in the E8 model theneutrinos are not Majorana particles, there is no neutrino CP-violation or phase at first order.
Consider the neutrino mixing matrix
nu_1 nu_2 nu_3 nu_e Ue1 Ue2 Ue3 nu_m Um1 Um2 Um3 nu_t Ut1 Ut2 Ut3
Assume the simplest mixing scheme with a massless nu_1 andnu_3 with no nu_e component so that Ue3 = 0
or, in conventional notation, mixing angle theta_13 = 0 = sin(theta_13) and cos(theta_13) = 1.
Then we have (as described in the 2004 Particle Data Book):
nu_1 nu_2 nu_3 nu_e cos(theta_12) sin(theta_12) 0 nu_m -sin(theta_12)cos(theta_23) cos(theta_12)cos(theta_23) sin(theta_23) nu_t sin(theta_12)sin(theta_23) -cos(theta_12)sin(theta_23) cos(theta_23)
Assume that nu_3 has equal components of nu_m and nu_t so that Um3 = Ut3 = 1/sqrt(2)
or, in conventional notation, mixing angle theta_23 = pi/4.
Then we have:
nu_1 nu_2 nu_3 nu_e cos(theta_12) sin(theta_12) 0 nu_m -sin(theta_12)/sqrt(2) cos(theta_12)/sqrt(2) 1/sqrt(2) nu_t sin(theta_12)/sqrt(2) -cos(theta_12)/sqrt(2) 1/sqrt(2)
The heaviest mass state nu_3 corresponds to a neutrino whose propagation begins and ends in CP2 internal symmetry space, lying entirely therein.
According to the E8 model the mass of nu_3 is zero at tree-level but it picks up a first-order correction propagating entirely through internal symmetry space by merging with an electron through the weak and electromagnetic forces, effectively acting not merely as a point
but as a point plus an electron loop at both beginning and ending points
so the first-order corrected mass of nu_3 is given by
where the factor (1/sqrt(2)) comes from the Ut3 component of the neutrino mixing matrix so that
= 1.4 x 5 x 10^5 x 1.05 x 10^(-5) x (1/137) eV =
= 7.35 / 137 = 5.4 x 10^(-2) eV.
Note that the neutrino-plus-electron loop can be anchored by weak force action through any of the 6 first-generation quarks at each of the beginning and ending points, and that the anchor quark at the beginning point can be different from the anchor quark at the ending point, so that there are 6x6 = 36 different possible anchorings.
The intermediate mass state nu_2 corresponds to a neutrino whose propagation begins or ends in CP2 internal symmetry space and ends or begins in physical Minkowski spacetime, thus having only one point (either beginning or ending) lying in CP2 internal symmetry space where it can act not merely as a point but as a point plus an electron loop.
According to the E8 model the mass of nu_2 is zero at tree-level but it picks up a first-order correction at only one (but not both) of the beginning or ending points
so that so that there are 6 different possible anchorings for nu_2 first-order corrections, as opposed to the 36 different possible anchorings for nu_3 first-order corrections,
so that the first-order corrected mass of nu_2 is less than the first-order corrected mass of nu_3 by a factor of 6,
so the first-order corrected mass of nu_2 is
= 9 x 10^(-3)eV.
Therefore: the mass-squared difference D(M23^2) is
= ( 2916 - 81 ) x 10^(-6) eV^2 =
= 2.8 x 10^(-3) eV^2
and
the mass-squared difference D(M12^2) is
= ( 81 - 0 ) x 10^(-6) eV^2 =
= 8.1 x 10^(-5) eV^2
Set theta_12 = pi/6= 0.866 so that cos(theta_12) = 0.866 = sqrt(3)/2 and sin(theta_12) = 0.5 = 1/2 = Ue2 = fraction of nu_2 begin/end points that are in the physical spacetime where massless nu_e lives. Then we have for the neutrino mixing matrix:
nu_1 nu_2 nu_3 nu_e 0.87 0.50 0 nu_m -0.35 0.61 0.71 nu_t 0.35 -0.61 0.71
Gravity and the Cosmological Constant come from the MacDowell-Mansouri Mechanism and the 15-dimensional Spin(2,4) = SU(2,2) Conformal Group, which is made up of:
According to gr-qc/9809061 by R. Aldrovandi and J. G. Peireira:
"... If the fundamental spacetime symmetry of the laws of Physics is that given by the de Sitter instead of the Poincare group, the P-symmetry of the weak cosmological-constant limit and the Q-symmetry of the strong cosmological-constant limit can be considered as limiting cases of the fundamental symmetry. ...... N ...[ is the space ]... whose geometry is gravitationally related to an infinite cosmological constant ...[and]... is a 4-dimensional cone-space in which ds = 0, and whose group of motion is Q. Analogously to the Minkowski case, N is also a homogeneous space, but now under the kinematical group Q, that is, N = Q/L [ where L is the Lorentz Group of Rotations and Boosts ]. In other words, the point-set of N is the point-set of the special conformal transformations.
Furthermore, the manifold of Q is a principal bundle P(Q/L,L), with Q/L = N as base space and L as the typical fiber. The kinematical group Q, like the Poincare group, has the Lorentz group L as the subgroup accounting for both the isotropy and the equivalence of inertial frames in this space. However, the special conformal transformations introduce a new kind of homogeneity. Instead of ordinary translations, all the points of N are equivalent through special conformal transformations. ...
... Minkowski and the cone-space can be considered as dual to each other, in the sense that their geometries are determined respectively by a vanishing and an infinite cosmological constants. The same can be said of their kinematical group of motions: P is associated to a vanishing cosmological constant and Q to an infinite cosmological constant.
The dual transformation connecting these two geometries is the spacetime inversion x^u -> x^u / sigma^2 . Under such a transformation, the Poincare group P is transformed into the group Q, and the Minkowski space M becomes the cone-space N. The points at infinity of M are concentrated in the vertex of the cone-space N, and those on the light-cone of M becomes the infinity of N. It is interesting to notice that, despite presenting an infinite scalar curvature, the concepts of space isotropy and equivalence between inertial frames in the cone-space N are those of special relativity. The difference lies in the concept of uniformity as it is the special conformal transformations, and not ordinary translations, which act transitively on N. ..."
Therefore, our Flat Expanding Universe should, according to the cosmology of the model, have (without taking into account any evolutionary changes with time) roughly:
As Dennnis Marks pointed out to me, since density rho is proportional to (1+z)^3(1+w) for red-shift factor z and a constant equation of state w:
so that the ratio of their overall average densities must vary with time, or scale factor R of our Universe, as it expands.
Therefore, the above calculated ratio 0.67 : 0.27 : 0.06 is valid only for a particular time, or scale factor, of our Universe.
Since WMAP observes Ordinary Matter at 4% NOW, the time WHEN Ordinary Matter was 6% would be at redshift z such that 1 / (1+z)^3 = 0.04 / 0.06 = 2/3 , or (1+z)^3 = 1.5 , or 1+z = 1.145 , or z = 0.145. To translate redshift into time, in billions of years before present, or Gy BP, use this chart
from a www.supernova.lbl.gov file SNAPoverview.pdf. to see that the time WHEN Ordinary Matter was 6% would have been a bit over 2 billion years ago, or 2 Gy BP.
In the diagram, there are four Special Times in the history of our Universe:
Those four Special Times define four Special Epochs:
As to how the Dark Energy /\ and Cold Dark Matter terms have evolved during the past 2 Gy, a rough estimate analysis would be:
The Ordinary Matter excess 0.06 - 0.04 = 0.02 plus the first-order CDM excess 0.27 - 0.18 = 0.09 should be summed to get a total first-order excess of 0.11, which in turn should be distributed to the /\ and CDM factors in their natural ratio 67 : 27, producing, for NOW after 2 Gy of expansion:
for a total calculated Dark Energy : Dark Matter : Ordinary Matter ratio for NOW of 0.75 : 0.21 : 0.04
The quark content of a charged pion is a quark - antiquark pair: either Up plus antiDown or Down plus antiUp. Experimentally, its mass is about 139.57 MeV.
The quark is a Naked Singularity Kerr-Newman Black Hole, with electromagnetic charge e and spin angular momentum J and constituent mass M 312 MeV, such that e^2 + a^2 is greater than M^2 (where a = J / M).
The antiquark is a also Naked Singularity Kerr-Newman Black Hole, with electromagnetic charge e and spin angular momentum J and constituent mass M 312 MeV, such that e^2 + a^2 is greater than M^2 (where a = J / M).
According to General Relativity, by Robert M. Wald (Chicago 1984) page 338 [Problems] ... 4. ...:
"... Suppose two widely separated Kerr black holes with parameters ( M1 , J1 ) and ( M2 , J2 ) initially are at rest in an axisymmetric configuration, i.e., their rotation axes are aligned along the direction of their separation.Assume that these black holes fall together and coalesce into a single black hole.
Since angular momentum cannot be radiated away in an axisymmetric spacetime, the final black hole will have momentum J = J1 + J2. ...".
The neutral pion produced by the quark - antiquark pair would have zero angular momentum, thus reducing the value of e^2 + a^2 to e^2 .
For fermion electrons with spin 1/2, 1 / 2 = e / M (see for example Misner, Thorne, and Wheeler, Gravitation (Freeman 1972), page 883) so that M^2 = 4 e^2 is greater than e^2 for the electron. In other words, the angular momentum term a^2 is necessary to make e^2 + a^2 greater than M^2 so that the electron can be seen as a Kerr-Newman naked singularity.
Since the magnitude of electromagnetic charge of each quarks or antiquarks less than that of an electron, and since the mass of each quark or antiquark (as well as the pion mass) is greater than that of an electron, and since the quark - antiquark pair (as well as the pion) has angular momentum zero, the quark - antiquark pion has M^2 greater than e^2 + a^2 = e^2.
( Note that color charge, which is nonzero for the quark and the antiquark and is involved in the relation M^2 less than sum of spin-squared and charges-squared by which quarks and antiquarks can be see as Kerr-Newman naked singularities, is not relevant for the color-neutral pion. )
Therefore, the pion itself is a normal Kerr-Newman Black Hole with Outer Event Horizon = Ergosphere at r = 2M ( the Inner Event Horizon is only the origin at r = 0 ) as shown in this image
from Black Holes - A Traveller's Guide, by Clifford Pickover (Wiley 1996) in which the Ergosphere is white, the Outer Event Horizon is red, the Inner Event Horizon is green, and the Ring Singularity is purple. In the case of the pion, the white and red surfaces coincide, and the green surface is only a point at the origin.
According to section 3.6 of Jeffrey Winicour's 2001 Living Review of the Development of Numerical Evolution Codes for General Relativity (see also a 2005 update):
"... The black hole event horizon associated with ... slightly broken ... degeneracy [ of the axisymmetric configuration ]... reveals new features not seen in the degenerate case of the head-on collision ... If the degeneracy is slightly broken, the individual black holes form with spherical topology but as they approach, tidal distortion produces two sharp pincers on each black hole just prior to merger.... Tidal distortion of approaching black holes ...
... Formation of sharp pincers just prior to merger ..
... toroidal stage just after merger ...
At merger, the two pincers join to form a single ... toroidal black hole.
The inner hole of the torus subsequently [ begins to] close... up (superluminally) ... [ If the closing proceeds to completion, it ]... produce[s] first a peanut shaped black hole and finally a spherical black hole. ...".
In the physical case of quark and antiquark forming a pion, the toroidal black hole remains a torus. The torus is an event horizon and therefore is not a 2-spacelike dimensional torus, but is a (1+1)-dimensional torus with a timelike dimension.
The effect is described in detail in Robert Wald's book General Relativity (Chicago 1984). It can be said to be due to extreme frame dragging, or to timelike translations becoming spacelike as though they had been Wick rotated in Complex SpaceTime.
As Hawking and Ellis say in The LargeScale Structure of Space-Time (Cambridge 1973):
"... The surface r = r+ is ... the event horizon ... and is a null surface ...... On the surface r = r+ .... the wavefront corresponding to a point on this surface lies entirely within the surface. ...".
A (1+1)-dimensional torus with a timelike dimension can carry a Sine-Gordon Breather, and the soliton and antisoliton of a Sine-Gordon Breather correspond to the quark and antiquark that make up the pion.
Sine-Gordon Breathers are described by Sidney Coleman in his Erica lecture paper Classical Lumps and their Quantum Descendants (1975), reprinted in his book Aspects of Symmetry (Cambridge 1985), where Coleman writes the Lagrangian for the Sine-Gordon equation as ( Coleman's eq. 4.3 ):
L = (1 / B^2 ) ( (1/2) (df)^2 + A ( cos( f ) - 1 ) )
and Coleman says:
"... We see that, in classical physics, B is an irrelevant parameter: if we can solve the sine-Gordon equation for any non-zero B, we can solve it for any other B. The only effect of changing B is the trivial one of changing the energy and momentum assigned to a given soluition of the equation. This is not true in quantum physics, becasue the relevant object for quantum physics is not L but [ eq. 4.4 ]L / hbar = (1 / ( B^2 hbar ) ) ( (1/2) (df)^2 + A ( cos( f ) - 1 ) )
An other way of saying the same thing is to say that in quantum physics we have one more dimensional constant of nature, Planck's constant, than in classical physics. ... the classical limit, vanishingf hbar, is exactly the same as the small-coupling limit, vanishing B ... from now on I will ... set hbar equal to one. ...
... the sine-Gordon equation ...[ has ]... an exact periodic solution ...[ eq. 4.59 ]...
f( x, t ) = ( 4 / B ) arctan( ( n sin( w t ) / cosh( n w x ))
where [ eq. 4.60 ] n = sqrt( A - w^2 ) / w and w ranges from 0 to A. This solution has a simple physical interpretation ... a soliton far to the left ...[ and ]... an antisoliton far to the right. As sin( w t ) increases, the soliton and antisoliton mover farther apart from each other. When sin( w t ) passes thrpough one, they turn around and begin to approach one another. As sin( w t ) comes down to zero ... the soliton and antisoliton are on top of each other ... when sin( w t ) becomes negative .. the soliton and antisoliton have passed each other. ...[
This stereo image of a Sine-Gordon Breather was generated by the program 3D-Filmstrip for Macintosh by Richard Palais. You can see the stereo with red-green or red-cyan 3D glasses. The program is on the WWW at http://rsp.math.brandeis.edu/3D-Filmstrip. The Sine-Gordon Breather is confined in space (y-axis) but periodic in time (x-axis), and therefore naturally lives on the (1+1)-dimensional torus with a timelike dimension of the Event Horizon of the pion. ...]
... Thus, Eq. (4.59) can be thought of as a soliton and an antisoliton oscillation about their common center-of-mass. For this reason, it is called 'the doublet [ or Breather ] solution'. ... the energy of the doublet ...[ eq. 4.64 ]
E = 2 M sqrt( 1 - ( w^2 / A ) )
where [ eq. 4.65 ] M = 8 sqrt( A ) / B^2 is the soliton mass. Note that the mass of the doublet is always less than twice the soliton mass, as we would expect from a soltion-antisoliton pair. ... Dashen, Hasslacher, and Neveu ... Phys. Rev. D10, 4114; 4130; 4138 (1974). A pedagogical review of these methods has been written by R. Rajaraman ( Phys. Reports 21, 227 (1975 ... Phys. Rev. D11, 3424 (1975) ...[ Dashen, Hasslacher, and Neveu found that ]... there is only a single series of bound states, labeled by the integer N ... The energies ... are ... [ eq. 4.82 ]
E_N = 2 M sin( B'^2 N / 16 )
where N = 0, 1, 2 ... < 8 pi / B'^2 , [ eq. 4.83 ]
B'^2 = B^2 / ( 1 - ( B^2 / 8 pi ))
and M is the soliton mass. M is not given by Eq. ( 4.675 ), but is the soliton mass corrected by the DHN formula, or, equivalently, by the first-order weak coupling expansion. ... I have written the equation in this form .. to eliminate A, and thus avoid worries about renormalization conventions. Note that the DHN formula is identical to the Bohr-Sommerfeld formula, except that B is replaced by B'. ... Bohr and Sommerfeld['s] ... quantization formula says that if we have a one-parameter family of periodic motions, labeled by the period, T, then an energy eigenstate occurs whenever [ eq. 4.66 ]
[ Integral from 0 to T ]( dt p qdot = 2 pi N,
where N is an integer. ... Eq.( 4.66 ) is cruder than the WKB formula, but it is much more general; it is always the leading approximation for any dynamical system ... Dashen et al speculate that Eq. ( 4.82 ) is exact. ...
the sine-Gordon equation is equivalent ... to the massive Thirring model. This is surprising, because the massive Thirring model is a canonical field theory whose Hamiltonian is expressedin terms of fundamental Fermi fields only. Even more surprising, when B^2 = 4 pi , that sine-Gordon equation is equivalent to a free massive Dirac theory, in one spatial dimension. ... Furthermore, we can identify the mass term in the Thirring model with the sine-Gordon interaction, [ eq. 5.13 ]
M = - ( A / B^2 ) N_m cos( B f )
.. to do this consistently ... we must say [ eq. 5.14 ]
B^2 / ( 4 pi ) = 1 / ( 1 + g / pi )
....[where]... g is a free parameter, the coupling constant [ for the Thirring model ]... Note that if B^2 = 4 pi , g = 0 , and the sine-Gordon equation is the theory of a free massive Dirac field. ... It is a bit surprising to see a fermion appearing as a coherent state of a Bose field. Certainly this could not happen in three dimensions, where it would be forbidden by the spin-statistics theorem. However, there is no spin-statistics theorem in one dimension, for the excellent reason that there is no spin. ... the lowest fermion-antifermion bound state of the massive Thirring model is an obvious candidate for the fundamental meson of sine-Gordon theory. ... equation ( 4.82 ) predicts that all the doublet bound states disappear when B^2 exceeds 4 pi . This is precisely the point where the Thirring model interaction switches from attractive to repulsive. ... these two theories ... the massive Thirring model .. and ... the sine-Gordon equation ... define identical physics. ... I have computed the predictions of ...[various]... approximation methods for the ration of the soliton mass to the meson mass for three values of B^2 : 4 pi (where the qualitative picture of the soliton as a lump totally breaks down), 2 pi, and pi . At 4 pi we know the exact answer ... I happen to know the exact answer for 2 pi, so I have included this in the table. ...
Method B^2 = pi B^2 = 2 pi B^2 = 4 pi
Zeroth-order weak coupling
expansion eq2.13b 2.55 1.27 0.64
Coherent-state variation 2.55 1.27