MAKING SENSE OF THE NAMBU-JONA-LASINIO MODEL VIA SCALE INVARIANCE
Philip D. Mannheim
University of Connecticut
Seminar at Cern Scaling Workshop
January 2019
Living Without Supersymmetry – the Conformal Alternative and a Dynamical Higgs Boson, J. Phys. G 44, 115003 (2017). (arXiv:1506.01399 [hep-ph]) Mass Generation, the Cosmological Constant Problem, Conformal Symmetry, and the Higgs Boson, Prog. Part. Nucl. Phys. 94, 125 (2017). (arXiv:1610:08907 [hep-ph])
1 The status of the chiral-invariant Nambu-Jona-Lasinio (NJL) four-fermi model is quite equivocal. It serves as the paradigm for dynamical symmetry breaking and yet it is not renormalizable. So one looks to obtain dynamical symmetry breaking in a gauge theory instead. An early attempt was Maskawa and Nakajima (1974). They studied the quenched (i.e. bare) photon, planar graph approximation to an Abelian gauge theory, 1 L = − F F µν + ψγ¯ µ(i∂ − e A )ψ − m ψψ,¯ (1) QED−FF 4 µν µ 0 µ 0 with the first two graphs and their iterations but not the non-planar third:
Figure 1: The first few graphs in the fermion self-energy Schwinger-Dyson equation
With S−1(p) = p/ − B(p2) they obtained
Z p2 2 2 Z ∞ 2 2 3α 2 q B(q ) 2 B(q ) B(p ) = m0 + dq 2 2 2 2 + dq 2 2 2 . (2) 4π 0 p [q + B (q )] p2 [q + B (q )] 2 Solutions to this equation depend on whether α = α0 = e0/4π is less than or greater than π/3.
2 On using a cutoff Λ2, for α ≤ π/3 they obtained
p2 (ν−1)/2 3α1/2 3αm Λν−1 Σ(p2) = m , ν = ± 1 − , m = , (3) m2 π 0 2π(1 − ν) mν−1 while for α > π/3 they obtained m cos[((3α/π − 1)1/2/2)ln(p2/m2) + σ] 3mα cos[(3α/π − 1)1/2ln(Λ/m) + τ] B(p2) = , m = − , (4) (p2/m2)1/2 0 2π(µ2 + 1)1/2(Λ/m) where σ is an integration constant, and τ = σ + arctan(3α/π − 1)1/2.
For α > π/3 the bare mass m0 will vanish identically if we set 3α 1/2 Λ π − 1 ln + τ = . (5) π m 2 This corresponds to dynamical chiral symmetry breaking, and one finds a massless pseudoscalar boson.
For α < π/3 (viz. ν < 1) it initially again appears that the bare mass is zero. However this time m0 only vanishes in the limit of infinite cutoff. As noted by Baker and Johnson (1971) at the same time the multiplicative −1/2 ¯ 1−ν ¯ renormalization constant Zθ that renormalizes ψψ diverges as Λ , so that m0ψψ is non-zero, and the chiral symmetry is broken in the Lagrangian. Despite the fact that the Schwinger-Dyson equation now becomes homogeneous and despite the fact that one is looking at its non-trivial self-consistent solution, there is then no Goldstone boson and this is known as the Baker-Johnson evasion of the Goldstone theorem.
Conventional wisdom: One can get dynamical symmetry breaking in a gauge theory if the coupling is big enough.
3 Now we already know of a counter-example to this wisdom. In BCS one obtains dynamical symmetry breaking no matter how weak the induced four-fermi interaction between the electrons is, provided it is attractive. This is not a two-body effect (which would require strong coupling) but a many-body effect due to the filled Fermi surface.
However, the quenched planar graph approximation is not valid for strong coupling, since non-planar graphs are of the same order as planar graphs. The all-order quenched planar plus non-planar graph solution was found by Baker, Johnson and Wiley (1961), and it is of the form 2 γθ(α)/2 2 γθ(α)/2 ˜−1 −p − i ˜ −p − i S (p, m) = p/ − m + i, ΓS(p, p, 0, m) = , (6) m2 m2 for both the inverse fermion propagator and the insertion of ψψ¯ into it. This solution holds for any value of α weak or strong, and for all values corresponds to a theory with a ¯ fundamental m0ψψ, and no dynamical symmetry breaking. The Maskawa-Nakajima phase transition at α = π/3 is just an artifact of using a perturbative result outside of its domain of validity.
4 Further insight into the JBW solution is given by the renormalization group equation ∂ ∂ m + β(α) S˜−1(p, m) = −m[1 − γ (α)]Γ˜ (p, p, 0, m). (7) m ∂α θ S So if β(α) = 0 (which it is in the quenched approximation for any value of α, or for non- quenched if β(α) = 0 at a fixed point) one gets scaling with anomalous dimensions, with the dimension of θ = ψψ¯ being given by
dθ(α) = 3 + γθ(α), γθ(α) = ν − 1. (8) There is no Maskawa-Nakajima broken symmetry solution since that would require ν = 1/2 (1 − 3α/π) to become complex [γθ(α) = −1 + i(3α/π − 1) ] and anomalous dimensions are real.
So how can we get dynamical symmetry breaking?
To this end we note that if one has γθ(α) = −1 then dθ(α) = 2 and the four-fermion interaction becomes renormalizable (suggested in Mannheim (1975) and proven to all orders in the four-fermi coupling in Mannheim(2017)). Thus if we couple a scale invariant QED with γθ(α) = −1 to a four-fermion interaction we can then get a renormalizable NJL model and dynamical symmetry breaking. All that is required is to dress the point four-fermi 2 2 −1/2 vertices so that instead of Γ˜S(p, p, 0, m) = 1 one has Γ˜S(p, p, 0, m) = (−p /m ) , with quadratic divergences becoming log divergences.
5 1 THE NAMBU-JONA-LASINIO (NJL) CHIRAL FOUR-FERMION MODEL
1.1 Quick Review of the NJL Model as a Mean-Field Theory in Hartree-Fock Approximation
Introduce mass term with m as a trial parameter and note m2/2g term Z h g g i I = d4x iψγ¯ µ∂ ψ − [ψψ¯ ]2 − [ψiγ¯ ψ]2 NJL µ 2 2 5 " # Z m2 Z g m2 g = d4x iψγ¯ µ∂ ψ − mψψ¯ + + d4x − ψψ¯ − − ψiγ¯ ψ2 µ 2g 2 g 2 5
INJL = IMF + IRI, mean field plus residual interaction (9) Hartree-Fock approximation m2 m hΩ | ψψ¯ − |Ω i = hΩ | ψψ¯ − |Ω i2 = 0, (10) m g m m g m Z 4 ¯ d k 1 m hΩm|ψψ|Ωmi = −i Tr = , (11) (2π)4 p/ − m + i g Satisfied by self-consistent M, and defines g−1 MΛ2 M 3 Λ2 M − + ln = . (12) 4π2 4π2 M 2 g
6 1.2 Vacuum Energy
Z 4 Z 4 d p d p (m) = i Tr ln p/ − m + i − i Tr ln p/ + i (2π)4 (2π)4 m2Λ2 m4 Λ2 m4 = − + ln + (13) 8π2 16π2 m2 32π2 is quadratically divergent. m2 ˜(m) = (m) − g m4 Λ2 m2M 2 Λ2 m4 = ln − ln + . (14) 16π2 m2 8π2 M 2 32π2 is only log divergent, with double-well potential emerging, but still cutoff dependent.
7 1.3 Higgs-Like Lagrangian Vacuum to vacuum functional due to m(x)ψψ¯ : hΩ(t = −∞)|Ω(t = +∞)i = eiW (m(x)) X 1 Z W (m(x)) = d4x ...d4x Gn(x , ..., x )m(x )...m(x ), n! 1 n 0 1 n 1 n Z 1 W (m(x)) = d4x −(m(x)) + Z(m(x))∂ m(x)∂µm(x) + ...... 2 µ
Figure 2: Vacuum energy density (m) via an infinite summation of massless graphs with zero-momentum point mψψ¯ insertions.
2 ¯ Figure 3: ΠS(q , m(x)) developed as an infinite summation of massless graphs, each with two point mψψ insertions carrying momentum qµ (shown as external lines), with all other point mψψ¯ insertions carrying zero momentum. Eguchi and Sugawara (1974), Mannheim (1976): Z d4x Λ2 1 1 I = ln ∂ m(x)∂µm(x) + m2(x)M 2 − m4(x) . (15) EFF 8π2 M 2 2 µ 2 ¯ 5 ¯ µ 5 Set φ = hΩm|ψ(1 + γ )ψ|Ωmi. Couple to an axial gauge field via ψgAγ γ Aµ5ψ. Get effective Higgs: Z d4x Λ2 1 1 g2 I = ln |(∂ − 2ig A )φ(x)|2 + |φ(x)|2M 2 − |φ(x)|4 − A F F µν5 . (16) EFF 8π2 M 2 2 µ A µ5 2 6 µν5
8 1.4 The Collective Tachyon Modes when the Fermion is Massless
Z Z 4 2 4 ip·x ¯ ¯ d p 1 1 ΠS(q ) = d xe hΩ|T (ψ(x)ψ(x)ψ(0)ψ(0))|Ωi = −i Tr , (2π)4 p/ + i p/ + /q + i Z Z 4 2 4 ip·x ¯ ¯ d p 1 1 ΠP(q ) = d xe hΩ|T (ψ(x)iγ5ψ(x)ψ(0)iγ5ψ(0))|Ωi = −i Tr iγ5 iγ5 . (17) (2π)4 p/ + i p/ + /q + i
Λ2 q2 Λ2 q2 Π (q2) = Π (q2) = − − ln − . (18) S P 4π2 8π2 −q2 8π2
2 2 2 2 g 1 TS(q ) = g + gΠS(q )g + gΠS(q )gΠS(q )g + ... = 2 = −1 2 , 1 − gΠS(q ) g − ΠS(q )
2 2 2 2 g 1 TP(q ) = g + gΠP(q )g + gΠP (q )gΠP (q )g + ... = 2 = −1 2 . 1 − gΠP(q ) g − ΠP(q ) (19)
Z−1 1 Λ2 T (q2) = T (q2) = ,Z = ln (20) S P (q2 + 2M 2) 8π2 M 2
Tachyonic poles, but at cutoff independent masses. Normal vacuum is unstable. g−1 takes care of the quadratic divergence.
9 1.5 The Collective Goldstone and Higgs Modes when the Fermion is Massive
Z 4 2 d p 1 1 ΠS(q ,M) = −i Tr (2π)4 p/ − M + i p/ + /q − M + i Λ2 M 2 Λ2 (4M 2 − q2) (4M 2 − q2) Λ2 = − + ln + + ln 4π2 4π2 M 2 8π2 8π2 M 2 1 (4M 2 − q2)3/2 (4M 2 − q2)1/2 + (−q2)1/2 − ln . (21) 8π2 (−q2)1/2 (4M 2 − q2)1/2 − (−q2)1/2
Z 4 2 d p 1 1 ΠP(q ,M) = −i Tr iγ5 iγ5 (2π)4 p/ − M + i p/ + /q − M + i Λ2 M 2 Λ2 q2 Λ2 (4M 2 − q2) = − + ln − ln + 4π2 4π2 M 2 8π2 M 2 8π2 (8M 4 − 8M 2q2 + q4) (4M 2 − q2)1/2 + (−q2)1/2 + ln . (22) 8π2(−q2)1/2(4M 2 − q2)1/2 (4M 2 − q2)1/2 − (−q2)1/2
R−1 R−1 T (q2) = S ,T (q2) = P , (23) S (q2 − 4M 2) P q2
1 Λ2 R = R = ln . (24) S P 8π2 M 2
The scalar Higgs mass is finite and of order the dynamical fermion mass, and residue is determined.
10 2 SCALE INVARIANT QED COUPLED TO FOUR FERMI THEORY AT γθ(α) = −1
1 g g L = − F F µν + ψγ¯ µ(i∂ − eA )ψ − [ψψ¯ ]2 − [ψiγ¯ ψ]2 QED−FF 4 µν µ µ 2 2 5 1 m2 g m2 g = − F F µν + ψγ¯ µ(i∂ − eA )ψ − mψψ¯ + − ψψ¯ − − ψiγ¯ ψ2 4 µν µ µ 2g 2 g 2 5 = LQED−MF + LQED−RI. (25)
2 −1/2 2 −1/2 ˜−1 −p − i ˜ −p − i S (p) = p/ − m + i, ΓS(p, p, 0) = (26) µ2 µ2
2 ¯ 2 as renormalized at µ . With dimension of (ψψ) dropping from 6 to 4 when γθ = −1, quadratic divergences become logarithmic, and four-fermion interaction becomes renormalizable to all orders in g (Mannheim 2017).
mµ2 Λ2 m hΩ |ψψ¯ |Ω i = − ln = . (27) m m 4π2 mµ g
µ2 Λ2 1 Λ2 4π2 − ln = ,M = exp . (28) 4π2 Mµ g µ µ2g Gap equation gives −g ∼ 1/lnΛ2. Thus g is negative, i.e. attractive, and becomes very small as Λ → ∞, with BCS-type essential singularity in gap equation at g = 0. Hence dynamical symmetry breaking with weak coupling.
11 2.1 Vacuum Energy
" # i Z d4p m2 −p2 − i−1 m2µ2 Λ2 1 (m) = Tr ln 1 − = − ln + (29) 2 (2π)4 p2 + i µ2 8π2 mµ 2 and is only log divergent. Due to presence of m2/2g term we obtain the completely finite m2 m2µ2 m2 ˜(m) = (m) − = ln − 1 , (30) 2g 16π2 M 2 which we recognize as a double-well potential, dynamically induced.
We thus see the power of dynamical symmetry breaking. It reduces divergences. Moreover, since m2/2g is a cosmological term, dynamical symmetry breaking has a control over the cosmological constant problem that an elementary Higgs field potential does not. When coupled to conformal gravity, the cosmological constant problem is completely solved.
12 2.2 Higgs-Like Lagrangian
Figure 4: Dynamically generated double-well potential for the renormalized vacuum energy density when γθ(α) = −1.
2 ¯ Figure 5: ΠS(q , m(x)) developed as an infinite summation of massless graphs, each with two dressed mψψ insertions carrying momentum qµ (shown as external lines), with all other dressed mψψ¯ insertions carrying zero momentum.
Mannheim (1978): 1 L = −˜(m(x)) − m(x)[Π (−∂ ∂µ, m(x)) − Π (0, m(x))]m(x) + ... EFF 2 S µ S m2(x)µ2 m2(x) 3µ = − ln − 1 + ∂ m(x)∂µm(x) + .... (31) 16π2 M 2 256πm(x) µ
13 2.3 The Collective Tachyon Modes when the Fermion is Massless
µ2 Λ2 Π (q2, m = 0) = Π (q2, m = 0) = − ln − 3 + 4 ln2 . (32) S P 4π2 (−q2)
2 g 1 TS(q ) = 2 = −1 2 , 1 − gΠS(q ) g − ΠS(q ) 2 g 1 TP(q ) = 2 = −1 2 , (33) 1 − gΠP(q ) g − ΠP(q )
q2 = −Mµe4ln2−3 = −0.797Mµ, (34)
31.448Mµ T (q2) = T (q2) = (35) S P (q2 + 0.797Mµ)
14 2.4 The Collective Goldstone Mode when the Fermion is Massive
Z d4p (p2)(−p2) − m2µ2 Π (q2 = 0, m) = −4iµ2 . P (2π)4 ((p2 + i)2 + m2µ2)2 Z d4p 1 1 = 4iµ2 = . (36) (2π)4 (p2 + i)2 + m2µ2 g
128πM 57.446M T (q2) = = . (37) P 7µq2 µq2
2.5 The Collective Higgs Mode when the Fermion is Massive
1/2 2 q0(Higgs) = (1.480 − 0.017i)(Mµ) , q (Higgs) = (2.189 − 0.051i)Mµ. (38)
2 2 q0(Higgs) = (1.480 − 0.017i)M, q (Higgs) = (2.189 − 0.051i)M , (39)
Higgs mass is close to dynamical fermion mass, but above threshold, and thus has a width. In a double well elementary Higgs field theory Higgs mass is real. Width can be used to distinguish an elementary Higgs from a dynamical one.
15 3 WE GET LOCAL CONFORMAL SYMMETRY FOR FREE
µν To implement local Lorentz invariance for a Dirac spinor (i.e. ψ → exp(ωµν(x)M )ψ where ωµν(x) depends µ a ab on x ), one introduces a set of vierbeins Vµ and a spin connection ωµ , where 1 −ωab = V b∂ V aν + V bΓλ V aν, Γλ = gλα(∂ g + ∂ g − ∂ g ). (40) µ ν µ λ νµ µν 2 µ να ν µα α νµ In a standard curved Riemannian space the massless Dirac action is given by Z 1 I = d4x(−g)1/2iψγ¯ aV µ(∂ + Γ )ψ, Γ = (γ γ − γ γ )ωab. (41) D a µ µ µ 8 a b b a µ
The Dirac action ID possesses four local translation invariances and six local Lorentz invariances. However, ID also possesses one local conformal invariance as well, since it is left invariant under
2β(x) a β(x) a −3β(x)/2 gµν(x) → e gµν(x),Vµ (x) → e Vµ (x), ψ(x) → e ψ(x), (42) with arbitrary spacetime-dependent β(x).
bc Local Lorentz invariance implies local conformal invariance, with Γµ = Σbcωµ acting as a gauge field for local conformal transformations, just as Aµ is the gauge field for local gauge transformations. In ψ → eβ+iαψ, gauging β is gravity, gauging α is Yang-Mills.
Thus unify fundamental forces with gravity through gauging real and imaginary parts of fermion phase. No longer need to unify them using a Lie group, So no need to evade Coleman-Mandula theorem, and thus no need for supersymmetry.
16 4 CONFORMAL GRAVITY
The Weyl tensor Cλµνκ 1 1 C = R − (g R − g R − g R + g R ) + Rα (g g − g g ) (43) λµνκ λµνκ 2 λν µκ λκ µν µν λκ µκ λν 6 α λν µκ λκ µν λν obeys g Cλµνκ = 0, and has the remarkable property that under a local conformal transformation gµν → 2β(x) λ λ e gµν(x), all derivatives of β(x) drop out identically and C µνκ → C µνκ. µν Analog to Maxwell tensor Fµν → Fµν under a gauge transformation and to g Fµν = 0.
In terms of a dimensionless coupling constant αg the action IW Z Z 1 I = −α d4x(−g)1/2C Cλµνκ = −α d4x(−g)1/2 R Rλµνκ − 2R Rµκ + (Rα )2 (44) W g λµνκ g λµνκ µκ 3 α µν is the unique locally conformal invariant action for the gravitational sector (compare FµνF ).
17 5 WE GET LOCAL CONFORMAL GRAVITY FOR FREE Z Z 1 I = −α d4x(−g)1/2C Cλµνκ = −2α d4x(−g)1/2 R Rµκ − (Rα )2 . (45) W g λµνκ g µκ 3 α
For Dirac action ID path integration over the fermion fields is direct, and leads (’t Hooft) to Z ¯ [Dψ][Dψ] exp(ID) = exp(iIEFF), Z 1 I = d4x(−g)1/2C R Rµν − (Rα )2 (46) EFF µν 3 α with leading term with log divergent constant C. (Path integration is same as a one loop Feynman diagram.) Conformal gravity is unavoidable. Standard Dirac action is ghost free, so conformal gravity must be ghost free also. Confirmed by Bender and Mannheim (2008) with conformal gravity being a non-Hermitian PT theory. When we set Z 4 Z 4 d p d p (m) = i Tr ln p/ − m + i − i Tr ln p/ + i (47) (2π)4 (2π)4 we were only looking at energy difference. But gravity sees full Z 4 d p (m) = i Tr ln p/ − m + i (48) (2π)4 and it is quartically divergent. Cancelled by graviton loop quartic divergence.
18 5.1 Conformal Gravity and the Cosmological Constant Problem
M 4 M 2 L = − Rα . (49) EFF 16π2 512π α
M 2 gµν M 4 T µν = i ψγ¯ µ∂νψ − Rµν − Rα − gµν . (50) M ~ 256π 2 α 16π2
2 1/2! c (1 + z) q0 dL = − 1 − 1 + q0 − 2 , (51) H0 q0 (1 + z)
60 Structure of the theory is such that q0 cannot be of order 10 but is required to obey: −1 ≤ q0 ≤ 0. This is consistent with the data with fitted q0 = −0.37. Predict departures from ΩM (t0) = 0.3, ΩΛ(t0) = 0.7 standard gravity at high redshift.
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Figure 6: Hubble plot expectations for q0 = −0.37 (highest curve) and q0 = 0 (middle curve) conformal gravity and for ΩM (t0) = 0.3, ΩΛ(t0) = 0.7 standard gravity (lowest curve). 20 5.2 Conformal Gravity and the Dark Matter Problem
4 3 0 r ∇ B(r) = (T 0 − T r) = f(r). (52) 4αgB(r)
1 Z r r04 B(r) = − dr0f(r0) 3r02r + 6 0 r Z ∞ 1 0 0 03 0 2 − dr f(r )(3r + r r ) + B0(r), (53) 6 r
∗ ∗ 2 2 2 N β c R R R vLOC = 3 I0 K0 2R0 2R0 2R0 R R − I1 K1 2R0 2R0 N ∗γ∗c2R2 R R + I1 K1 , (54) 2R0 2R0 2R0
γ c2R N ∗β∗c2 N ∗γ∗c2R γ c2R v2 = v2 + 0 − κc2R2 → − + + 0 − κc2R2. (55) TOT LOC 2 R 2 2
β∗ = 1.48 × 105cm, γ∗ = 5.42 × 10−41cm−1, −30 −1 −54 −2 γ0 = 3.06 × 10 cm , κ = 9.54 × 10 cm , (56) Fit 138 galaxies with VISIBLE N ∗ of each galaxy (i.e. galactic mass to light ratio M/L) as only variable, ∗ ∗ ∗ with β , γ , γ0 and κ all universal, and with NO DARK MATTER, and with 276 fewer free parameters than 2 2 −30 −1 in dark matter calculations. Works since (v /c R)last ∼ 10 cm for every galaxy.
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UGC 11819 0 0 2 4 6 8 10 12 FIG. 5: Fitting to the rotational velocities26 of the LSB 21 galaxy sample 80 80 120 100
100 60 80 60 80 60 40 40 60 40 40 20 20 20 20 DDO 168 DDO 170 M 33 NGC 55 0 0 0 0 0 1 2 3 4 0 2 4 6 8 10 12 0 2 4 6 8 0 2 4 6 8 10 12 120 140 120 250 100 120 100 200 100 80 80 80 150 60 60 60 100 40 40 40
50 20 20 20 NGC 247 NGC 300 NGC 801 NGC 1003 0 0 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 0 10 20 30 40 0 5 10 15 20 25 30 100 250 250 80 80 200 200 60 60 150 150 40 40 100 100
20 20 50 50
NGC 1560 NGC 2683 NGC 2998 NGC 3109 0 0 0 0 0 2 4 6 8 10 0 5 10 15 20 25 30 35 0 10 20 30 40 0 1 2 3 4 5 6 7
300 350 100 250 250 300 80 200 250 200 60 150 200 150 150 40 100 100 100
50 50 20 50 NGC 5033 NGC 5371 NGC 5533 NGC 5585 0 0 0 0 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 50 0 2 4 6 8 10 12 14
300 350 140 100
250 120 300 80 250 200 100
80 200 60 150 60 150 40 100 40 100
50 20 20 50 NGC 5907 NGC 6503 NGC 6674 UGC 2259 0 0 0 0 0 10 20 30 40 0 5 10 15 20 0 10 20 30 40 50 60 0 2 4 6
350 300 300 250 250 200 200 150 150
100 100
50 50 UGC 2885 Malin 1 0 0 0 10 20 30 40 50 60 70 0 20 40 60 80 100 FIG. 6: Fitting to the rotational velocities27 of the Miscellaneous 22 galaxy sample 140 120 80 120 100
100 80 60 80 60 40 60 40 40 20 20 20 F 568-V1 F 574-1 UGC 731 0 0 0 0 5 10 15 0 2 4 6 8 10 12 14 0 2 4 6 8 10 120 100 80 100 80 60 80
60 60 40 40 40
20 20 20 UGC 3371 UGC 4173 UGC 4325 0 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7
80 100 80
80 60 60 60
40 40 40
20 20 20
UGC 4499 UGC 5414 UGC 5721 0 0 0 0 2 4 6 8 0 1 2 3 4 0 2 4 6 8
100 60 100 50 80 80 40 60 60 30 40 40 20
20 10 20 UGC 5750 UGC 7232 UGC 7323 0 0 0 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 0 1 2 3 4 5
FIG. 7: Fitting to the rotational velocities (in km28 sec−1) of the 24 dwarf galaxy sample – Part 1 100 50 120
100 80 40
80 60 30
60 40 20 40
20 10 20 UGC 7399 UGC 7524 UGC 7559 0 0 0 0 5 10 15 20 25 30 0 2 4 6 8 0.0 0.5 1.0 1.5 2.0 2.5 40 80 100
80 30 60
60
20 40 40
10 20 20
UGC 7577 UGC 7603 UGC 8490 0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 2 4 6 8 0 2 4 6 8 10
120 80 100 150
60 80 100 60 40
40 50 20 20 UGC 9211 UGC 11707 UGC 11861 0 0 0 0 2 4 6 8 0 5 10 15 20 0 2 4 6 8 10 12
100 120 80 100 80
60 80 60 60 40 40 40
20 20 20 UGC 12060 UGC 12632 UGC 12732 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 12 14
FIG. 8: Fitting to the rotational velocities29 of the 24 dwarf galaxy sample – Part 2 60 50 80 50 40
60 40 30 30 40 20 20
20 10 10 UGC 3851 UGC 4305 UGC 4459 0 0 0 0 2 4 6 8 10 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 60 60 100
50 50 80
40 40 60
30 30 40 20 20
20 10 10 UGC 5139 UGC 5423 UGC 5666 0 0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10
FIG. 9 Fitting to the rotational velocities of the 6 dwarf galaxy sample
100 140 300 80 120 250 100 60 200 80 150 40 60 100 40 20 20 50 DDO 154 UGC 128 Malin 1 0 0 0 0 10 20 30 40 50 0 20 40 60 80 0 50 100 150 FIG. 10: Extended distance predictions for DDO 154, UGC 128, and Malin 1.
Predict that velocities should fall since v2 → −κc2R2. But v2 cannot go negative, so galaxies should end and have maximum size of no more than 150 Kpc or so.
30 6 SUPERSYMMETRY VERSUS CONFORMAL SYMMETRY – THE SCORECARD
(1) In flat space physics an interplay between bosons and fermions can render logarithmically divergent Feynman diagrams finite. Conformal symmetry takes care of quartic and quadratic as well.
(2) An interplay between bosons and fermions can cancel the perturbative quadratic diver- gence that an elementary Higgs scalar field would possess (the hierarchy problem). Problem does not exist if Higgs is composite.
(3) The existence of fermionic supersymmetry generators allows one to evade the Coleman- Mandula theorem that forbids the combining of spacetime and bosonic internal symmetry generators in a common Lie algebra. No need to put spacetime in a Lie group
(4) With the inclusion of supersymmetry one can potentially achieve a unification of the cou- pling constants of SU(3)×SU(2)L ×U(1) at a grand unified energy scale. Not addressed by conformal symmetry, but would need something beyond standard model.
31 (5) In the presence of gravity an interplay between bosons and fermions can cancel the quartic divergence in the vacuum energy. Ditto conformal gravity since conformal gravity graviton is a boson. All contributions to vacuum energy under control and cosmological constant problem solved.
(6) Cancellation of perturbative infinities can also be found in supergravity, the local version of supersymmetry. Local conformal symmetry also cancels infinities, including those associated with mass generation.
(7) With supersymmetry one can construct a consistent candidate quantum theory of gravity, string theory, which permits a possible unification of all of the fundamental forces and a metrication (geometrization) of them. Conformal gravity is a consistent quantum gravity theory. Only requires four dimensions and no new particles. Admits of a metrication of the fundamental forces.
(8) Finally, with supersymmetry one has a prime candidate for dark matter. With con- formal gravity, dark matter not needed, nor dark energy either.
32 7 THE MORAL OF THE STORY
With the vacuum of quantum field theory being a dynamical one, in a sense Einstein’s ether has reemerged. Only it has reemerged not as the mechanical ether of classical physics that was excluded by the Michelson-Morley experiment, but as a dynamical, quantum-field- theoretic one full of Dirac’s negative energy particles, an infinite number of such particles whose dynamics can spontaneously break symmetries. The type of physics that would be taking place in this vacuum depends on how symmetries are broken, i.e. on whether the breaking is by elementary Higgs fields or by dynamical composites. If the symmetry is broken by an elementary Higgs field, then the Higgs boson gives mass to fundamental gauge bosons and fermions alike. However, if the breaking is done dynamically, then it is the structure of an ordered vacuum itself that generates masses, with the mass generation mechanism in turn then producing the Higgs boson.
In the dynamical case then mass produces Higgs rather than Higgs produces mass. In the dynamical case we should not be thinking of the Higgs boson as being the “god particle”. Rather, if anything, we should be thinking of the vacuum as being the “god vacuum”.
33 Properties of the THINGS 18 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI DDO 0154 LSB 4.2 0.007 0.8 8.1 0.03 0.003 0.45 1.12 [?][?][?][?] IC 2574 LSB 4.5 0.345 4.2 13.1 0.19 0.098 0.28 1.69 [?][?][?][?] NGC 0925 LSB 8.7 1.444 3.9 12.4 0.41 1.372 0.95 4.17 [?][?][?][?] NGC 2403 HSB 4.3 1.647 2.7 23.9 0.46 2.370 1.44 2.89 [?][?][?][?] NGC 2841 HSB 14.1 4.742 3.5 51.6 0.86 19.552 4.12 5.83 [?][?][?][?] NGC 2903 HSB 9.4 4.088 3.0 30.9 0.49 7.155 1.75 3.75 [?][?][?][?] NGC 2976 LSB 3.6 0.201 1.2 2.6 0.01 0.322 1.60 10.43 [?][?][?][?] NGC 3031 HSB 3.7 3.187 2.6 15.0 0.38 8.662 2.72 9.31 [?][?][?][?] NGC 3198 HSB 14.1 3.241 4.0 38.6 1.06 3.644 1.12 2.09 [?][?][?][?] NGC 3521 HSB 12.2 4.769 3.3 35.3 1.03 9.245 1.94 4.21 [?][?][?][?] NGC 3621 HSB 7.4 2.048 2.9 28.7 0.89 2.891 1.41 3.18 [?][?][?][?] NGC 3627 HSB 10.2 3.700 3.1 8.2 0.10 6.622 1.79 15.64 [?][?][?][?] NGC 4736 HSB 5.0 1.460 2.1 10.3 0.05 1.630 1.60 4.66 [?][?][?][?] NGC 4826 HSB 5.4 1.441 2.6 15.8 0.03 3.640 2.53 5.46 [?][?][?][?] NGC 5055 HSB 9.2 3.622 2.9 44.4 0.76 6.035 1.87 2.36 [?][?][?][?] NGC 6946 HSB 6.9 3.732 2.9 22.4 0.57 6.272 1.68 6.39 [?][?][?][?] NGC 7331 HSB 14.2 6.773 3.2 24.4 0.85 12.086 1.78 9.61 [?][?][?][?] NGC 7793 HSB 5.2 0.910 1.7 10.3 0.16 0.793 0.87 3.61 [?][?][?][?]
34 Properties of the Ursa Major 30 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI NGC 3726 HSB 17.4 3.340 3.2 31.5 0.60 3.82 1.15 3.19 [?][?][?][?] NGC 3769 HSB 15.5 0.684 1.5 32.2 0.41 1.36 1.99 1.43 [?][?][?][?] NGC 3877 HSB 15.5 1.948 2.4 9.8 0.11 3.44 1.76 10.51 [?][?][?][?] NGC 3893 HSB 18.1 2.928 2.4 20.5 0.59 5.00 1.71 3.85 [?][?][?][?] NGC 3917 LSB 16.9 1.334 2.8 13.9 0.17 2.23 1.67 4.85 [?][?][?][?] NGC 3949 HSB 18.4 2.327 1.7 7.2 0.35 2.37 1.02 14.23 [?][?][?][?] NGC 3953 HSB 18.7 4.236 3.9 16.3 0.31 9.79 2.31 10.20 [?][?][?][?] NGC 3972 HSB 18.6 0.978 2.0 9.0 0.13 1.49 1.53 7.18 [?][?][?][?] NGC 3992 HSB 25.6 8.456 5.7 49.6 1.94 13.94 1.65 4.08 [?][?][?][?] NGC 4010 LSB 18.4 0.883 3.4 10.6 0.29 2.03 2.30 5.03 [?][?][?][?] NGC 4013 HSB 18.6 2.088 2.1 33.1 0.32 5.58 2.67 3.14 [?][?][?][?] NGC 4051 HSB 14.6 2.281 2.3 9.9 0.18 3.17 1.39 8.52 [?][?][?][?] NGC 4085 HSB 19.0 1.212 1.6 6.5 0.15 1.34 1.11 10.21 [?][?][?][?] NGC 4088 HSB 15.8 2.957 2.8 18.8 0.64 4.67 1.58 5.79 [?][?][?][?] NGC 4100 HSB 21.4 3.388 2.9 27.1 0.44 5.74 1.69 3.35 [?][?][?][?] NGC 4138 LSB 15.6 0.827 1.2 16.1 0.11 2.97 3.59 5.04 [?][?][?][?] NGC 4157 HSB 18.7 2.901 2.6 30.9 0.88 5.83 2.01 3.99 [?][?][?][?] NGC 4183 HSB 16.7 1.042 2.9 19.5 0.30 1.43 1.38 2.36 [?][?][?][?] NGC 4217 HSB 19.6 3.031 3.1 18.2 0.30 5.53 1.83 6.28 [?][?][?][?] NGC 4389 HSB 15.5 0.610 1.2 4.6 0.04 0.42 0.68 9.49 [?][?][?][?] UGC 6399 LSB 18.7 0.291 2.4 8.1 0.07 0.59 2.04 3.42 [?][?][?][?] UGC 6446 LSB 15.9 0.263 1.9 13.6 0.24 0.36 1.36 1.70 [?][?][?][?] UGC 6667 LSB 19.8 0.422 3.1 8.6 0.10 0.71 1.67 3.09 [?][?][?][?] UGC 6818 LSB 21.7 0.352 2.1 8.4 0.16 0.11 0.33 2.35 [?][?][?][?] UGC 6917 LSB 18.9 0.563 2.9 10.9 0.22 1.24 2.20 4.05 [?][?][?][?] UGC 6923 LSB 18.0 0.297 1.5 5.3 0.08 0.35 1.18 4.43 [?][?][?][?] UGC 6930 LSB 17.0 0.601 2.2 15.7 0.29 1.02 1.69 2.68 [?][?][?][?] UGC 6973 HSB 25.3 1.647 2.2 11.0 0.35 3.99 2.42 10.58 [?][?][?][?] UGC 6983 LSB 20.2 0.577 2.9 17.6 0.37 1.28 2.22 2.43 [?][?][?][?] UGC 7089 LSB 13.9 0.352 2.3 7.1 0.07 0.35 0.98 3.18 [?][?][?][?]
35 Properties of the LSB 20 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI DDO 0064 LSB 6.8 0.015 1.3 2.1 0.02 0.04 2.87 6.05 [?][?][?][?] F563-1 LSB 46.8 0.140 2.9 18.2 0.29 1.35 9.65 2.44 [?][?][?][?] F563-V2 LSB 57.8 0.266 2.0 6.3 0.20 0.60 2.26 6.15 [?][?][?][?] F568-3 LSB 80.0 0.351 4.2 11.6 0.30 1.20 3.43 3.16 [?][?][?][?] F583-1 LSB 32.4 0.064 1.6 14.1 0.18 0.15 2.32 1.92 [?][?][?][?] F583-4 LSB 50.8 0.096 2.8 7.0 0.06 0.31 3.25 2.52 [?][?][?][?] NGC 0959 LSB 13.5 0.333 1.3 2.9 0.05 0.37 1.11 7.43 [?][?][?][?] NGC 4395 LSB 4.1 0.374 2.7 0.9 0.13 0.83 2.21 2.29 [?][?][?][?] NGC 7137 LSB 25.0 0.959 1.7 3.6 0.10 0.27 0.28 3.91 [?][?] ES [?] UGC 0128 LSB 64.6 0.597 6.9 54.8 0.73 2.75 4.60 1.03 [?][?][?][?] UGC 0191 LSB 15.9 0.129 1.7 2.2 0.26 0.49 3.81 15.48 [?][?][?][?] UGC 0477 LSB 35.8 0.871 3.5 10.2 1.02 1.00 1.14 4.42 [?][?] ES [?] UGC 1230 LSB 54.1 0.366 4.7 37.1 0.65 0.67 1.82 0.97 [?][?][?][?] UGC 1281 LSB 5.1 0.017 1.6 1.7 0.03 0.01 0.53 3.02 [?][?][?][?] UGC 1551 LSB 35.6 0.780 4.2 6.6 0.44 0.16 0.20 3.69 [?][?][?][?] UGC 4325 LSB 11.9 0.373 1.9 3.4 0.10 0.40 1.08 7.39 [?][?][?][?] UGC 5005 LSB 51.4 0.200 4.6 27.7 0.28 1.02 5.11 1.30 [?][?][?][?] UGC 5750 LSB 56.1 0.472 3.3 8.6 0.10 0.10 0.21 1.58 [?][?][?][?] UGC 5999 LSB 44.9 0.170 4.4 15.0 0.18 3.36 19.81 5.79 [?][?][?][?] UGC 11820 LSB 17.1 0.169 3.6 3.7 0.40 1.68 9.95 8.44 [?][?][?][?]
36 Properties of the LSB 21 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI ESO 0140040 LSB 217.8 7.169 10.1 30.0 20.70 3.38 8.29 [?][?][?] NA ESO 0840411 LSB 82.4 0.287 3.5 9.1 0.06 0.21 1.49 [?][?] ES NA ESO 1200211 LSB 15.2 0.028 2.0 3.5 0.01 0.20 0.66 [?][?] ES NA ESO 1870510 LSB 16.8 0.054 2.1 2.8 0.09 1.62 2.02 [?][?][?] NA ESO 2060140 LSB 59.6 0.735 5.1 11.6 3.51 4.78 4.34 [?][?][?] NA ESO 3020120 LSB 70.9 0.717 3.4 11.2 0.77 1.07 2.37 [?][?] ES NA ESO 3050090 LSB 13.2 0.186 1.3 5.6 0.06 0.32 1.87 [?][?] ES NA ESO 4250180 LSB 88.3 2.600 7.3 14.6 4.79 1.84 5.17 [?][?][?] NA ESO 4880490 LSB 28.7 0.139 1.6 7.8 0.43 3.07 4.34 [?][?] ES NA F571-8 LSB 50.3 0.191 5.4 14.6 0.16 4.48 23.49 5.10 [?][?][?][?] F579-V1 LSB 86.9 0.557 5.2 14.7 0.21 3.33 5.98 3.18 [?][?][?][?] F730-V1 LSB 148.3 0.756 5.8 12.2 5.95 7.87 6.22 [?][?][?] NA UGC 04115 LSB 5.5 0.004 0.3 1.7 0.01 0.97 3.42 [?][?][?] NA UGC 06614 LSB 86.2 2.109 8.2 62.7 2.07 9.70 4.60 2.39 [?][?][?][?] UGC 11454 LSB 93.9 0.456 3.4 12.3 3.15 6.90 6.79 [?][?][?] NA UGC 11557 LSB 23.7 1.806 3.0 6.7 0.25 0.37 0.20 3.49 [?][?][?][?] UGC 11583 LSB 7.1 0.012 0.7 2.1 0.01 0.96 2.15 [?][?][?] NA UGC 11616 LSB 74.9 2.159 3.1 9.8 2.43 1.13 7.49 [?][?][?] NA UGC 11648 LSB 49.0 4.073 4.0 13.0 2.57 0.63 5.79 [?][?][?] NA UGC 11748 LSB 75.3 23.930 2.6 21.6 9.67 0.40 1.01 [?][?][?] NA UGC 11819 LSB 61.5 2.155 4.7 11.9 4.83 2.24 7.03 [?][?][?] NA
37 Properties of the Miscellaneous 22 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI DDO 0168 LSB 4.5 0.032 1.2 4.4 0.03 0.06 2.03 2.22 [?][?][?][?] DDO 0170 LSB 16.6 0.023 1.9 13.3 0.09 0.05 1.97 1.18 [?][?][?][?] M 0033 HSB 0.9 0.850 2.5 8.9 0.11 1.13 1.33 4.62 [?][?][?][?] NGC 0055 LSB 1.9 0.588 1.9 12.2 0.13 0.30 0.50 2.22 [?][?][?][?] NGC 0247 LSB 3.6 0.512 4.2 14.3 0.16 1.25 2.43 2.94 [?][?][?][?] NGC 0300 LSB 2.0 0.271 2.1 11.7 0.08 0.65 2.41 2.69 [?][?][?][?] NGC 0801 HSB 63.0 4.746 9.5 46.7 1.39 6.93 2.37 3.59 [?][?][?][?] NGC 1003 LSB 11.8 1.480 1.9 31.2 0.63 0.66 0.45 1.53 [?][?][?][?] NGC 1560 LSB 3.7 0.053 1.6 10.3 0.12 0.17 3.16 2.16 [?][?][?][?] NGC 2683 HSB 10.2 1.882 2.4 36.0 0.15 6.03 3.20 2.28 [?][?][?][?] NGC 2998 HSB 59.3 5.186 4.8 41.1 1.78 7.16 1.75 3.43 [?][?][?][?] NGC 3109 LSB 1.5 0.064 1.3 7.1 0.06 0.02 0.35 2.29 [?][?][?][?] NGC 5033 HSB 15.3 3.058 7.5 45.6 1.07 0.27 3.28 3.16 [?][?][?][?] NGC 5371 HSB 35.3 7.593 4.4 41.0 0.89 8.52 1.44 3.98 [?][?][?][?] NGC 5533 HSB 42.0 3.173 7.4 56.0 1.39 2.00 4.14 3.31 [?][?][?][?] NGC 5585 HSB 9.0 0.333 2.0 14.0 0.28 0.36 1.09 2.06 [?][?][?][?] NGC 5907 HSB 16.5 5.400 5.5 48.0 1.90 2.49 1.89 3.44 [?][?][?][?] NGC 6503 HSB 5.5 0.417 1.6 20.7 0.14 1.53 3.66 2.30 [?][?][?][?] NGC 6674 HSB 42.0 4.935 7.1 59.1 2.18 2.00 2.52 3.57 [?][?][?][?] UGC 2259 LSB 10.0 0.110 1.4 7.8 0.04 0.47 4.23 3.76 [?][?][?][?] UGC 2885 HSB 80.4 23.955 13.3 74.1 3.98 8.47 0.72 4.31 [?][?][?][?] Malin 1 LSB 338.5 7.912 84.2 98.0 5.40 1.00 1.32 1.77 [?][?][?][?]
38 Properties of the 30 Dwarf Galaxy Sample 2 2 Galaxy Distance LB i (R0)disk Rlast MHI Mdisk (M/LB)disk (v /c R)last 9 B ◦ 9 9 B −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) F568-V1 78.20 2.15 40 3.11 17.07 2.32 16.00 7.45 2.95 F574-1 94.10 3.42 65 4.20 13.69 3.31 14.90 4.35 2.77 UGC 731 11.80 0.69 57 2.43 10.30 1.61 3.21 4.63 1.91 UGC 3371 18.75 1.54 49 4.53 15.00 2.62 4.49 2.91 1.78 UGC 4173 16.70 0.33 –5 4.43 12.14 2.24 0.07 0.20 1.21 UGC 4325 11.87 1.71 41 1.92 6.91 1.04 6.51 3.82 4.37 UGC 4499 12.80 1.01 50 1.46 8.38 1.15 1.80 1.79 2.37 UGC 5414 9.40 0.49 55 1.40 4.10 0.57 1.13 2.29 3.31 UGC 5721 7.60 0.48 +10 0.76 8.41 0.57 1.90 3.96 2.27 UGC 5750 56.10 4.72 64 5.60 21.77 1.00 3.68 0.78 1.03 UGC 7232 3.14 0.08 59 0.30 0.91 0.06 0.14 1.76 7.64 UGC 7323 7.90 2.39 47 2.13 5.75 0.70 4.19 1.75 4.59 UGC 7399 24.66 4.61 2.32 32.30 6.38 5.42 1.18 1.32 UGC 7524 4.12 1.37 46 3.02 9.29 1.34 5.29 3.86 2.67 UGC 7559 4.20 0.04 61 0.87 2.75 0.12 0.05 1.32 1.43 UGC 7577 2.13 0.05 –15 0.51 1.39 0.04 0.01 0.20 1.18 UGC 7603 9.45 0.80 78 1.24 8.24 1.04 0.41 0.52 1.81 UGC 8490 5.28 0.95 +10 0.71 11.51 0.72 1.53 1.61 1.47 UGC 9211 14.70 0.33 44 1.54 9.62 1.43 1.23 3.69 1.55 UGC 11707 21.46 1.13 68 5.82 20.30 6.78 9.89 8.76 1.77 UGC 11861 19.55 9.44 50 4.69 12.80 4.33 45.84 4.86 6.55 UGC 12060 15.10 0.39 40 1.70 9.89 1.67 3.45 8.94 2.00 UGC 12632 9.20 0.86 46 3.43 11.38 1.55 4.26 4.97 1.82 UGC 12732 12.40 0.71 39 2.10 14.43 3.23 4.11 5.76 2.40 UGC 3851 4.85 2.33 2.07 11.70 1.32 0.47 0.20 1.37 UGC 4305 2.34 0.41 –5 0.68 4.75 0.31 0.08 0.20 1.18 UGC 4459 3.06 0.03 27 0.60 2.47 0.04 0.01 0.20 0.97 UGC 5139 4.69 0.20 14 0.96 3.58 0.21 0.05 0.24 1.19 UGC 5423 7.14 0.14 45 0.61 1.97 0.05 0.28 2.01 1.82 UGC 5666 3.85 2.53 56 3.56 11.25 1.37 1.96 0.77 1.88
39 ¯ 7.1 Non-Zero Vacuum Expectation Value for ψψ and the condition γθ(α) = −1
Even if m0 vanishes in the limit of infinite cut-off, physical mass could still be zero since it satisfies a homogeneous equation. However (Mannheim 1974, 1975), if
γθ(α) = −1, (57) then ˜(m) < ˜(m = 0), with the energy density of the vacuum having a double-well potential form and minimum at non-zero m: m2µ2 m2 ˜(m) = ln − 1 . (58) 16π2 M 2
Compatibility of short-distance Wilson expansion with massive Johnson-Baker-Willey propagator S−1(p) ∼ p/−(−p2)γθ(α)/2 also gives γθ(α) = −1 (Mannheim 1975). Specifically, in a scale invariant theory the Wilson expansion is of the form
¯ ¯ 2 2 γθ(α)/2 ¯ T (ψ(x)ψ(0)) = hΩ0|T (ψ(x)ψ(0))|Ω0i + (µ x ) : ψ(0)ψ(0) : (59) where the normal ordering is done with respect to the unbroken massless vacuum |Ω0i. Now take matrix element in the spontaneously broken vacuum |Ωmi, to obtain 1 S˜(p) = + (−p2)(−γθ(α)/2−2), S˜−1(p) = p/ − (−p2)(−γθ(α)−2)/2. (60) p/
−1 2 γ (α)/2 Compatibility with S (p) ∼ p/ − (−p ) θ then gives γθ(α) = −γθ(α) − 2, i. e. γθ(α) = −1. Also shows that vacuum of Jonhnson-Baker-Willey electrodynamics is a broken vacuum even though no Goldstone boson. We expain this conundrum below. ¯ Thus in this talk we explore symmetry breaking with γθ(α) = −1, i. e. with D[ψψ] = 2, which makes the four-fermion interaction [ψψ¯ ]2 non-perturbatively renormalizable (Mannheim 1975).
40 7.2 Distinguishing a Dynamical Higgs Boson from a Fundamental One Z Z g Z(η,¯ η) = D[ψ,¯ ψ] exp i d4x ψγ¯ µ(i∂ − eA )ψ − (ψψ¯ )2 + ηψ¯ + ψη¯ (61) µ µ 2 Z Z(η,¯ η) = D[ψ,¯ ψ, σ] " !# Z g g σ 2 × exp i d4x ψγ¯ µ(i∂ − eA )ψ − (ψψ¯ )2 + − ψψ¯ + ηψ¯ + ψη¯ (62) µ µ 2 2 g
Z Z σ2 Z(η,¯ η) = D[ψ,¯ ψ, σ] exp i d4x ψγ¯ µ(i∂ − eA )ψ − σψψ¯ + + ηψ¯ + ψη¯ (63) µ µ 2g Z Z 4 µ Z(0, 0) = D[σ] exp i d x (−˜(σ) + Z(σ)∂µσ∂ σ/2 + ...) Z Z σ2(x)µ2 σ2(x) 3µ = D[σ] exp d4x − ln − 1 + ∂ σ(x)∂µσ(x) + .... 16π2 M 2 256πσ(x) µ (64)
Elementary and dynamical scalar Feynman diagrams are identical when Z scalar is off-shell. Since there is no d4xJ(x)σ(x) source term for σ(x), on- shell there are differences, such as the Higgs width. newpage
41 8 Conformal Symmetry Challenges Supersymmetry
8.1 Cancellation of Infinities
Supersymmetry cancellations are perturbative. Conformal invariance cancellations are non-perturbative.
In QED key infinities are Z3 and m0 = m − δm, with Z1 and Z2 being gauge dependent. If β(α) = 0 then Z3 is finite. And if γθ(α) < 0 then δm is finite.
γ (α) Λ2 γ2(α) Λ2 Λ2 γθ(α)/2 m = m 1 + θ ln + θ ln2 + .... = m → 0. (65) 0 2 m2 8 m2 m2
2 2 2 If γθ = −1 then ˜ = − m /2g is finite, as are the scalar and pseudoscalar TS(q ), TP (q ) channels in the fermion-antifermion scattering amplitude.
42 8.2 Supersymmetry Treatment of the Vacuum Energy Density
µν ¯ µ ν TM = i~ψγ ∂ ψ (66)
Z ∞ µ ν µν 2~ 3 k k hΩ0|TM |Ω0i = − 3 d k . (67) (2π) −∞ ωk
µν µ ν µν hΩ0|TM |Ω0i = (ρM + pM)U U + pη , (68)
Z ∞ 00 2~ 3 ρM = hΩ0|TM |Ω0i = − 3 d kωk, (69) (2π) −∞
Z ∞ 2 11 22 33 2~ 3 k pM = hΩ0|TM |Ω0i = hΩ0|TM |Ω0i = hΩ0|TM |Ω0i = − 3 d k . (70) 3(2π) −∞ ωk
µν ηµνhΩ0|TM |Ω0i = 3pM − ρM = 0 (71)
K4 K4 ρ = −~ , p = − ~ . (72) M 4π2 M 12π2
4 2 2 4 2 2 4 ~K m K m 4~ K m ρM = − − + ln − , 4π2 4π2~ 16π2~3 m2 32π2~3 4 2 2 4 2 2 4 ~K m K m 4~ K 7m pM = − + − ln + , (73) 12π2 12π2~ 16π2~3 m2 96π2~3 Supersymmetry (i. e. add in boson loop with opposite overall sign) cancels the quartic divergence, but not the quadratic one unless superpartners are degenerate in mass, which they are known not to be.
43 8.3 Conformal Gravity Treatment of the Vacuum Energy Density Gravity couples to energy density and not to energy density difference. Hence in the presence of gravity one cannot normal order away any infinities such as zero point infinities. Must cancel them by counterterms (provided they can naturally be induced) or by gravity itself, provided it is consistent quantum-mechanically as it has zero-point energy density too. Z Z 1 I = −α d4x(−g)1/2C Cλµνκ ≡ −2α d4x(−g)1/2 R Rµκ − (Rα )2 , (74) W g λµνκ g µκ 3 α
1 1 2 W µν = gµν(Rα );β + Rµν;β − Rµβ;ν − Rνβ;µ − 2RµβRν + gµνR Rαβ − gµν(Rα );β 2 α ;β ;β ;β ;β β 2 αβ 3 α ;β 2 2 1 + (Rα );µ;ν + Rα Rµν − gµν(Rα )2, (75) 3 α 3 α 6 α
µν µν µν µν µν −4αgW + TM = 0,TUNIV = TGRAV + TM = 0, (76)
Z ∞ µ ν µν 2~ 3 Z(k)k k hΩ0|TGRAV|Ω0i = 3 d k , (77) (2π) −∞ ωk Conformal gravity cancellation when fermion m = 0 if graviton Z(k) = 1. Comparison with standard gravity.
1 Z Z I = − d4x(−g)1/2Rα ,I = − d4x(−g)1/2Λ (78) EH 16πG α Λ 1 1 − Rµν − gµνRα = T µν. quantum = quantum? or classical = classical? (79) 8πG 2 α M 1 µν 1 µν α µν − R − g R α = hΩ|TM |Ωi. classical = matrix element of quantum (80) 8πG 2 CL
44 1 µν 1 µν α µν − R − g R α = hΩ|TM |ΩiFIN. classical = finite matrix element of quantum (81) 8πG 2 CL 4 4 00 ~K M hΩM |TGRAV|ΩM i − − = 0. (82) 4π2 16π2~3 Z(k) readjusts if fermion is massive – gravity is quantized by its source and not quantized independently.
2 2 2 2 2 1/2 iM 2 2 2 1/2 iM kZ(k) = (k + iM /~ ) − + (k − iM /~ ) + . (83) 4~2(k2 + iM 2/~2)1/2 4~2(k2 − iM 2/~2)1/2
µν µν µν µν (TGRAV)DIV + (TM )DIV = 0, (TGRAV)FIN + (TM )FIN = 0. (84)
45 8.4 Conformal Invariance and the Metrication and Unification of the Fundamental Forces 1 Λλ = gλα(∂ g + ∂ g − ∂ g ), (85) µν 2 µ να ν µα α νµ
ab ab b λ aν −ω˜µ = −ωµ + Vλ δΓ νµV , (86)
ab b aν b λ aν ba −ωµ = Vν ∂µV + Vλ Λ µνV = ωµ . (87)
1 Z I = d4x(−g)1/2iψγ¯ aV µ(∂ + Σ ω˜bc)ψ + H. c., (88) D 2 a µ bc µ
2i 1 δΓλ = − gλα (g A + g A − g A ) + gλα(Q + Q − Q ). (89) µν 3 να µ µα ν νµ α 2 µνα νµα ανµ Z 4 1/2 ¯ a µ bc 5 ID = d x(−g) iψγ Va (∂µ + Σbcωµ − iAµ − iγ Sµ)ψ, (90)
1 Sµ = (−g)−1/2µαβγQ . (91) 8 αβγ Z 4 1/2 ¯ a µ bc i i 5 i i ID = d x(−g) iψγ Va (∂µ + Σbcωµ − igV T Aµ − igAγ T Sµ)ψ. (92)
Z 1 1 1 1 I = d4x(−g)1/2C R Rµν − (Rα )2 + Gi Gµν + Si Sµν , (93) EFF 20 µν 3 α 3 µν i 3 µν i
Z 1 1 1 I + I = d4x(−g)1/2 − 2α R Rµν − (Rα )2 − Gi Gµν − Si Sµν . (94) W YM g µν 3 α 4 µν i 4 µν i
Z g I = − d4x(−g)1/2 FF ψT¯ iψψT¯ iψ + ψiγ¯ T iψψiγ¯ T iψ , (95) FF 2 5 5
IUNIV = ID + IW + IYM + IFF. (96)
Z 1 I = d4x(−g)1/2C − M 4(x) + M 2(x)Rα − (∂ + iA )M(x)(∂µ − iAµ)M(x) . (97) MF 6 α µ µ Potential theory of everything – string theory with extra dimensions and supersymmetry is a theory of more than everything.
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