Transcript Renormalization Group Running Cosmologies – from a Scale

Renormalization Group
Running Cosmologies –
from a Scale Setting to
Holographic Dark Energy
Branko Guberina
Rudjer Boskovic Institute
Zagreb, Croatia
IRGAC 2006
• Ch. I. Cosmological constant – an
introitus
• Ch. II. Cosmological constant
renormalization and decoupling – a
running scale
• Ch. III. RG running cosmologies - a scale
setting procedure
• Ch. IV. Holographic dark energy – a
setup
• Ch. V. Conclusions
Based on collaboration with
Raul Horvat
Hrvoje Stefancic
Hrvoje Nikolic
Ana Babic
Ch. I. Cosmological constant - an intro
• The Einstein equations
Rμν - ½gμνR + gμν Λ = -8πGTμν
• The notation follows Ландау-Лифшиц, Теория поля, except for the
sign of the Einstein tensor Gμν = Rμν - ½gμνR (Einstein-Eddington
convention used).
The signature is ημν = + - - - .
The covariant conservation
implies the conservation of the sum
Dμ (8πTμν(m) /M2P+ gμνΛ ) = 0.
• The energy momentum tensor Tμν is given as a functional derivative of
the total action S with respect to the metric gμν , δA/δ gμν .
• Tμν must be conserved (invariance of the theory with respect to an
arbitrary change of coordinates:
• Since the metric tensor is covariantly constant
this gives
Λ = const.
However, a number of models with Λ = Λ(t) are around today!
The first proposal: M. Bronstein, Physikalische Zeitschrift der
Sowietunion, 3 (1933) 73.
Strongly critisized by Landau. The same story for G = G(t)!
“Such models are not innocent” (A. D. Dolgov, hep-ph/0606230).
• An argument is the following: one has to derive the energymomentum tensor Tμν by functional differentiation with respect to gμν
from the total action S. This would contain the second time
derivatives of G.
• The consistent theory should start from an action S.
• For example, cf. M. Reuter, Brans - Dicke like theory, PRD 69
(2004)104022
Cosmological constant ≠ 0 – unavoidable?
• Different sources of CC:
• Zeldovich (1968) – a particle sitting at the bottom of the harmonic
oscillator (HO) potential with a minimum energy ½ω should have a
nonzero momentum due to the Heisenberg uncertainty principle.
• Quantum mechanics gives the minimum energy ½ω, indeed.
• In QFT a quantum field is a collection of infinite number of HOs
Other contributions to CC
• The natural value of the vacuum energy in broken SuGra
ρΛ ~ MP4 ≥ 1078 GeV4 .
compared to the observed value
ρΛexp ≈ 10-47 GeV4 .
QCD condensates – nonvanishing and well established
experimentally
< ¯qq > ≠ 0, < GμνGμν > ≠ 0,
with huge values
(ρΛ)QCD ≈ - 1045 ρc .
N.B. Inside the proton the gluon condensate gives a mass to the
proton
mproton = 2mu + md - ρG² l ³proton , l proton ~ 1/mπ .
Theory versus reality
Λtheory /Λreal =
= 1 000 000 000 000 000 000 000
000 000 000 000 000 000 000 000 000
000 000 000 000 000 000 000 000 000
000 000 000 000 000 000 000 000 000
000 000 000 000 000 000 000
Ch. II. Cosmological constant renormalization and
decoupling
• It is possible to study the cosmological constant without explaining its
present value.
• If treated as parameters in an action S, both G and Λ would become
running quantities in any QFT in curved space, cf.
•
N. D. Birell, P.C. W. Davies, “Quantum Fields in Curved Space”,
•
I. L. Buchbinder, S. D. Odintsov, I. L. Shapiro, “Effective Action in
Quantum Gravity”
•
Ilya Shapiro, IRGAC 2006 talk.
• A problem with the identification of the running scale.
• Normally, in QFT the renormalization scale μ is traded for some
physical observable, e. g., a typical energy/momentum of particles in
interaction.
• What is a typical graviton momentum?
Quantum field theory in curved space-time as an
effective field theory
•
•
•
S. Weinberg, Physica (Amsterdam) 96A (1979)327
A very successful Chiral perturbation theory (ChPT) as a low-energy
nonlinear realization of QCD (J. Gasser, H. Leutwyler, NPB 250(1985)465).
Loop effects make sense in ChPT, since some processes go through
loops only.
• Gravity, with quantum corrections being small up to MP, appears to be,
prima facie, even better EFT – expected to govern the effects of
Quantum gravity at low energy scale.
• However (BAD NEWS), the possible existence of the singularity in the
future, corresponding to the wavelength probed of the order of the size
of the universe – deeply in the infrared region – may break the validity
of the EFT in the IR limit.
• In fact QG may lead to a very strong renormalization effects at large
distances induced by IR divergences, cf. I. Antoniadis, E. Mottola
(1992), N. C. Tsamis, R. P. Woodard (1993, 1995, 1996).
• E. g. correction to the classical gravity, cf. Donoghue, gr-qc/9607039.
• Compare to the corrections in
nonperturbative quantum Einstein gravity, cf.
Reuter, Tbilisi talk, hep-th/0012069
The scale μ
• The meaning of the RGE scales – in the MS scheme the μ
dependence in the effective action is compensated by the running of
the parameter Λ (as in QED where the μ dependence is
compensated by the running charge e(μ). The overall action S which
contains a running Λ(μ) is scale independent.
• The physical interpretation of the RGE scale can be achieved
calculating the polarization operator of gravitons arising from particle
loops in the linearized gravity, cf. Gorbar-Shapiro, JHEP 02(2003)21
• In the physical mass-dependent renormalization scheme an arbitrary
parameter μ is usually traded for a Euclidean momentum p.
• In QCD, for example, one writes an RG equation with respect to the
momentum scaling parameter λ, p→ λp, and eliminates the
derivatives with respect to μ.
• The trade is performed by identifying the new scale with the typical
average energy (momentum) of the physical process.
¯
Einstein-Hilbert Vacuum Action
• The present age Universe should be well described by the action S
• The formulation of the theory – rather simple, cf. Shapiro, Sola, PLB
475(2000)236, JHEP 02(2002)006, see also Peccei, Sola, Wetterich,
PLB 195(1987)183.
- construct a renormalizable gauge theory (e.g., the gauged Higgs
lagrangian) in an external gravitation field.
- start with the matter action in flat space-time and replace the partial
derivatives by the covariant ones,
- replace the Minkowski metric ημν by the general one gμν,
- replace d4x by d4x(-g)½ .
The vacuum action necessary to ensure the renormalizability of the
gauged scalar Lagrangian should contain terms
R2μνρσ , R2μν , R2, □R,
with coefficients ai which are the bare parameters.
• All divergences can be removed by renormalization of the matter fields,
their masses and couplings, the bare parameters ai, Gbare, Λbare, and the
non-minimal parameter ξbare which enters the action via a term
(presently negligible)
ξφ†φR,
where φ is a scalar field.
• Per definitionem, a vacuum energy density ρΛ is
¯.
ρΛ = Λ/8πG = Λ
• The scalar field φ with the potential energy V(φ) contributes to S
• If φvac is the value of the field φ(x) which minimizes the potential V(φ),
then the lowest state has Tμν = gμν V(φvac ).
• This is a classical scalar field contribution to the vacuum energy,
e.g., for the Higgs field with potential
V(φ) = -m² φ†φ + λ(φ†φ)²,
one has the Higgs condensate contribution (at the classical level) for
the cosmological constant
ρΛHiggs = ΛHiggs/8πG = -m4/4λ .
Zero-point energy – a renormalization
• Using the dimensional regularization in d = 4 + ε and MS scheme
• Bare vs. renormalized Λ
+ the counterterm
→
¯
• Correct behavior of β(μ) = μ (∂/∂μ)Λ for μ >> M.
Decoupling failure
• Decoupling theorem – Appelquist-Carazzone valid?
• One expects corrections of the type μ²/M² to be insufficient to suppress
the quartic power of the mass.
• Assume there exist 2 particles, a heavy one with mass M, and a light
one with mass m.
• For μ >> M, m the RGE becomes (Λ¯ = Λ/8πG = ρΛ)
¯
• For m << μ << M one expects the decoupling of a heavy particle with
the suppression factor μ²/M²
¯
• The suppression factor μ²/M² is not sufficient to suppress the
contribution of the heavy scalar, since
μ² M² >> m4
• The zero point energy of a scalar field is (cf. J. Kapusta,
Finite-temperature field theory )
and the renormalized and bare CCs are (Λ¯ = Λ/8πG)
•
¯
¯
The real scalar field ZPE contribution to the cosmological
constant
• In the μ >> m limit
¯
In the opposite μ << m limit
¯
In the Standard Model
¯
A question of the running – a scale?
• Focus on the effects of
running at scales bellow
the electron mass (the
lightest particle here):
¯
¯
• The above eq. – the large masses in the μ² term drive the
numerical value far out of the range consistent with
observations, unless the expression in the brackets
vanishes.
• In a toy model – the SM – one obtains
MH² = 4 ∑I Ni mi2 – 3mZ2 – 6mW2 .
This leads to a prediction, MH2 ≈ 550 GeV.
Nucleosynthesis
• The running of the zero-point energy is ruled by the μ4 term.
• Using the expression for the energy density of the radiation during
nucleosynthesis ρR = (π²/30) g* T4, and making a natural choice μ = T,
one obtains
[ Λ (μ) – Λ (0) ] (8πGρR) -1 = 555/(32 π4 g*)
With g* = 3.36, the above expression acquires the value
0.053. This value does not disturb nucleosynthesis.
Taking μ = 0,02 eV one obtains
(8πG)-1 [ Λ (μ) – Λ (0) ] ≈ 10-48 GeV4 .
Experiments vs. theory – an example
Cf. Shapiro et al. JCAP 0402 (2004) 006
Ch. III. RG running cosmologies – A scalesetting procedure
• Based on
Ana Babic, B. G., Raul Horvat, Hrvoje Stefancic, PRD
71(2005)124041
A class of RGE-based cosmologies – the only running quantities are
ρΛ and G.
The RGE improved Einstein equation for these cosmologies
Rμν - ½gμνR = -8πG(μ)(Tμνm + gμν ρΛ(μ)) . (1)
The only physical requirement: Eq. (1) should maintain its general
covariance with the μ-dependent (and implicitly t-dependent) G(μ)
and ρΛ(μ). This leads to the following equation
• Assuming the nonvanishing  throughout the evaluation of the
Universe, one gets
• This is a master formula.
The LHS is a function of the scale factor a, and the RHS is a function
of the scale μ, i. e. one has ρm = f(μ). The inversion gives μ = f-1(ρm).
• Once the scale setting is completed, one has G(μ) and ρΛ(μ) as
functions of ρm .
• The matter energy density – the scale behavior
and the Friedmann equation
Comments on scale setting procedure
•
Our procedure lacks the first principle connection to quantum gravity.
•
We assumed that ρm is intrinsically independent of μ.
(*)
• The above equation means: as long as ρm retains its canonical
behavior, the scale μ is univocally fixed and does not even implicitly
depend on μ in the above equation.
• If one allows, e.g., for an interaction between matter and or CC and/or
G, then eq. (*) generalizes to
• Any deviation of ρm from the canonical form depends on μ. Therefore
one is not able to fix a scale without specifying the interactions a priori.
Nonperturbative quantum gravity
•
•
•
•
•
•
The exact renormalization group approach applied to quantum gravity, cf.
M. Reuter, PRD 57(1998)971, IRGAC 2006 talk.
The effective average action Γk [gμν] – basically a Wilsonian coarse-grained
free energy (M. Reuter, C. Wetterich, NPB 417(1994)181; 427((1994)291;
506(1997)483; Berges, Tetradis, Wetterich, Phys. Rep. 363(2002)223.)
The momentum scale k is interpreted as an infrared cutoff – for a
physical system with a size L, the parameter k ~ 1/L defines an infrared
cutoff.
The path integral which defines the effective average action Γk [gμν]
integrates only the quantum fluctuations with the momenta p² > k², thus
describing the dynamics of the metrics averaged over the volume (k-1)3.
For any scale k there is a Γk which is an EFT at that scale.
Large distance metric fluctuations, p² < k², are not included.
Quantum gravity model with an IR fixed point
• M. Reuter, PRD 57(1998)971, A. Bonano, M. Reuter, PRD
65(2002)043508, PLB 527(2002)9.
• In this formalism, it is possible to set up the RGE for G(μ) and Λ(μ).
• The infrared cutoff k plays the role of the general RGE scale μ.
• In the infrared limit, Λ and G run as follows:
λ* and g* are constants and k is an IR cutoff.
• Inserting the above expressions into the master formula gives
• The result does not depend explicitly on the topology K of the Universe.
• A certain level of implicit dependence exists since the expansion of the
Universe (and the dependence of ρm on time) depends on K.
• The scale-setting procedure unambiguously identifies the scale k in the
quantum gravity model with the IR fixed point.
• The above expression for k leads to
, → ΩΛ = Ωm
• Per definitionem,
ρc = 3H2/(8πG)
ΩΛ = Λ/(8πGρc)
Ωm = ρm/ρc
Ωc = -K/(H2/a2)
ΩΛ + Ωm + Ωc = 0 .
•
Using the preceding definitions one obtains an implicit expression for the
scale factor of the Universe:
• Obviously, the expansion of the universe depends on two
parameters, w and ΩK0 .
• Solutions - special choices in the limit t’→ 0 (a→ 0) (cf. BonReuter)
• (i) ΩK0 = 0, w arbitrary, one obtains
and
in agreement with Bonano-Reuter, PLB 527(2002)9.
(ii) w = 1/3, ΩK arbitrary.
•
A simple law for the scale factor
a/a0 = H0 t ,
and
This shows why the Ansatz k = ξ / t functions correctly for
cosmologies of any curvature including the radiation only
(agreement with Bonano-Reuter PLB 527(2002)9.
(iii) w = 0, ΩK ≥ 0.
•
The universe with NR matter and
arbitrary curvature -
• In this case, the scale k can no longer be expressed in the form of ansatz k =
ξ/t.
• The scale k is uniquely defined by choosing the only real solution of the
above cubic equation.
• The product t (a/a0)-3/4 (since k ~ (a/a0) -3/4 ) as a function of t
→
•
The product of the cosmic time t and (a/a0)-3/4 versus t for the flat and
the open universe with ΩK0 = 0.02. For the curved universe, the choice
of the scale differs from k ~ 1/t.
RGE cosmological model from QFT on
curved space-time
•
When the RGE scale μ is less
than ALL masses in the theory,
one may write
• The application of the scale-setting procedure yields
Results
• The scale μ
• From the study of cosmologies with the running ρΛ in QFT in curved
space-time, we know that generally
C1 ~ m2max, C2 ~ Nb – Nf ~1, C3 ~ 1/m2min , etc., and
D1 ~ 1,
D2 ~ 1/m2min , etc.
• The value of C0 can hardly surpass ρΛtoday .
• γ1 and γ2 are constants of order 1.
- a very slow running in the vicinity of μ = mmin .
- by inserting this scale into the expansion of ρΛ one arrives at
ρΛ ~ m2max m2min .
In the extreme case we can set m max ~ MPlanck , and saturate (ρΛ )today
with mmin ~ mquintessence ~ 10-33 GeV.
Ch. IV. Holographic Dark Energy
Holographic principle: A field theory overcounts the true dynamical
degrees of freedom - therefore extra nonlocal constraints on an effective
field theory are necessary.
The entropy S scales extensively in an effective quantum field theory for a
system of size L with the UV cutoff Λ:
Bekenstein: maximum entropy in a box of volume L3 grows only as the area A of
the box.
‘t Hooft, Susskind: 3+1 field theories overcount degrees of freedom.
Also, a local quantum field theory cannot correctly describe quantum gravity – too
many degrees of freedom in UV.
Exit: limit the volume of the system according to
L = size of the box = radius of a black hole,SBH = the entropy of a black hole,
The length scale L provides an IR cutoff which is determined by the UV cutoff Λ.
A. G. Cohen, D. B. Kaplan, A. E. Nelson, PRL 82 (1999) 4971
An effective field theory satisfying the constraint
unavoidably includes many states with the Schwarzschild radius larger
than the box size L.
Solution: an additional constraint on the IR cutoff 1/L which excludes all
states that lie within their Schwarzschild radius.
The maximal energy density in the effective QFT is Λ4, therefore the energy
in a given volume ~L3 should not exceed the energy of a black hole of the
same size L, i. e.,
where MP is the Planck mass.
Box of size L
•
Schwarzschild radius RS < L
Black hole of size L
• The IR cutoff L scales as Λ-2 – this bound is more restrictive
by far.
• Near the bound saturation the entropy is given by
• Cf. a review: R. Bousso, Rev. Mod. Phys. 74(2002)825.
• Subtle questions concerning black holes and the horizons, cf.
T. Padmanabhan, Class. Quantum Grav. 19(2002)5387,
T. M. Davis,et al., Class. Quantum Grav. 20(2003)2753,
P. C. W. Davies, “The implications of a holographic universe and the
nature of physical law” , to appear (priv. comm.).
Cosmological constant and holography
•
•
•
•
•
•
The usual quantum corrections to the vacuum energy density with no
IR cutoff in QFT obviously gives the wrong prediction.
Should the fields at the present energy scales fluctuate independently
over the entire horizon, or even, over the whole Universe?
Cohen et al. proposal: taking the size L to be approximately the size of
the present horizon, L ~ H-1, the vacuum energy density Λ4 is
constrained to be of the order of (meV)4 – the right value of the
experimental vacuum energy density.
The cosmological constant problem – solved?
Not really – one can always add a constant to the quantum
corrections.
However, if correct, the Cohen et al. bound eliminates the need for
fine-tuning.
Holographic Dark Energy - A Cosmological Setup
• A generalized holographic dark energy model – both the cosmological
constant (CC) and Newton’s constant GN are scale dependent.
• A cosmological setup – the renormalization group (RG) evolution of
both CC and GN within quantum field theory in curved space.
• Cohen et al. relation between the UV and IR cutoff results in an upper
bound on the zero-point energy density ρΛ.
• The largest CC
This relation is what we call a generalized dark energy model.
• A scale μ ~ 1/L represents the IR cutoff.
• Again, choosing L = H0-1 = 1028 cm, one arrives at the present
observed value for the dark energy density ρΛ = 10-47 GeV4.
Problems with the choice of the IR cutof:
•
If ρΛ is considered the energy density of a noninteracting perfect
fluid – then the choice μ = L-1 fails to recover the equation of state
(EOS) for a dark-energy dominated universe, cf. S. D. Hsu, PLB 594
(2004)13.
• Even more, choosing L = H-1 always leads to ρΛ = ρm for flat space,
thus hindering a decelerating era of the universe for redshifts z >
0.5.
• However, a correct EOS is obtained if one chooses a future event
horizon for an IR cutoff, cf. M. Li, PLB 603 (2004) 1.
• Since the equation
(1)
was derived using ZPEs, the natural interpretation of dark energy in the
above equation is through the variable, or interacting, CC with w = -1.
• The energy transfer between various components in the universe (where G
can also vary with time) is described by a generalized equation of continuity
(2)
• Overdots denote the time derivative; matter is pressureless, w = 0.
• N. B. ρΛ is affected not only by matter, but also by a time dependent G.
• The quantity GN Tμνtotal in (2) is conserved. For GN static, Tμνtotal is
conserved.
• Sourced Friedmann eqs. with holographic dark energy studied by Y.S.
Myung, hep-th/0502128.
•
•
•
•
•
•
•
Using the holographic restriction (1) and the generalized equation of continuity (2), to
constrain the parameters of the RG evolution in QFT in curved space background, R.
Horvat, H. Stefancic, B.G. JCAP 05 (2005) 001.
The variation of the CC and G in this model arises solely from particle field
fluctuations (no quintessence-like scalar fields).
For the RG running scale μ below the lowest mass in the theory, the RG laws read
The scale μ cannot be set from first principles. Assuming the convergence of tboth
series, and using the studies of cosmologies with running ρΛ and G, in the formalism
of QFT in curved space-time, one generally gets
C1 ~ m2max, C2 ~ Nb – Nf ~1, C3 ~ 1/m2min etc., and
D1 ~ 1,
D2 ~ 1/m2min , etc.
mmax and mmin are the largest and the smallest mass in the theory, respectively.
Nb and Nf are the number of bosonic and fermionic massive degrees of freedom.
C0 = the vacuum ground state, coincides here with the IR limit of CC.
•
•
The context is set by fixing the matter density law to be a canonical one, i. e.
ρm ~ a-3 – no energy transfer between ρm and other components.
Eq. (2) reduces to
•
•
N.B. The prime denotes differentiation with respect to the scale μ.
Inserting the holographic expession (1) into the above equation
(3)
•
•
•
•
Once GN (μ) is known, the IR cutoff μ is fixed.
For μ > 0, G’N < 0, then d/dt GN > 0, i.e., GN (t) increases with increasing
cosmic time t.
This implies that GN is asymptotically free – the property seen in quantum
gravity models at the 1-loop level, cf. Fradkin, Tseytlin, NPB 201 (1982) 469.
Asymptotic freedom – of some interest for galaxy dynamics and rotation
curves, cf. Shapiro, Sola, Stefancic, JCAP 01 (2005) 012
• Applying the requrement (1) of the generalized holographic dark energy
model to the RG laws for ρΛ and GN relates C and D :
•
•
•
*****************************************************************************************
Let us first discuss the choice GN = const.
In this, rather simple case, one obtains
•
•
with ρΛ ~ m2max μ2.
However, the observational data suggest that μ0 ~ H0 today.
N.B. The above relation does not imply μ = H. The continuity equation, for GN =
const., gives
•
The scale μ cannot be univocally fixed.
•
The equation
means a continuous transfer from matter to CC and vice versa (depending on the
sign of the interaction term).
The dilution of the energy density of matter causes a deviation in the ρm ~ a-3
behavior – which depends crucially on the choice for μ.
The choice μ = H was employed in Shapiro, Sola, Espana-Bonet, Ruiz-Lapuente,
PLB 574 (2003)149, JCAP 02 (2004)006.
Note: mmax ~ MP may be an effective mass describing particles with masses just
below the Planck mass.
A case ρΛ = ρΛ(t), GN = GN(t), ρm ~ a-3
• Inserting the expansions in μ for ρΛ and GN into the scaling-fixing
relation
leads to the expression for the scale μ
•
Using the estimates for Dn, one obtains
Comments on eq (***)
1. The value of μ is marginally acceptable as far as the convergence of
the ρΛ (μ) and G (μ) series is concerned.
2. In addition, from G’N < 0 one obtains
D1 ≈ C2 > 0.
3. Equation (***) shows a very slow variation of the scale μ with the scale
factor a or the cosmic time t.
4. Once the RG scale crosses below the smallest mass in the theory, it
effectively freezes at mmin ~ H0 ~ 10-33 eV.
5. This is the main result of holography – one finds a hint for possible
quintessence-like particles in the spectrum.
• What holography basically does – it expands the
particle spectrum from either side to the extremum –
the largest possible particle masses approach the
Planck mass, and the smallest possible particle
masses are around the lowest mass scale in the
universe, H0.
• The present value of ρΛ appears as the product of squared masses
of the particles lying on the top and bottom of the spectrum - a
natural solution to the coincidence problem.
Further studies
• B. G., R. Horvat, H. Nikolic, PRD 72(2005)
• The generalized holography dark energy model with scale
dependent Λ and G, has been used to set constraints on the RG
evolution of both Λ and G.
• Assuming ρm scales canonically, it was shown that the continuity
equation fixes an IR cutoff, provided a low of variation for either Λ or
G is known.
• Using the RG running CC model, with low energy scaling given by
the QFT of particles with masses near the Planck mass, in
accordance with holography, amounts to having an IR cutoff scaling
as H½.
• Such a setup, in which the only undetermined input is the true
ground state of the vacuum, yields a transient acceleration.
• **********************************
• B. G., R. Horvat, H. Nikolic, PLB 636(2006)80 – a study of
dynamical dark energy (scale dependent G) with a constant ρΛ.
• An interesting idea of comprising both the running CC and dark energy
coming from a scalar field with a temporary phantom phase obtained
with a nonphantom scalar field, which has EOS > -1 by Elizalde, Nojri,
Odintsov, Wang, PRD 71 (2005) 103504.
• Interaction dark matter with dark energy – coincidence problem
solution, cf. D. Pavon, W. Zimdahl, PLB 628(2005)206;
• Spatial curvature & holographic dark energy, cf. W. Zimdahl, D. Pavon,
astro-ph/0606555.
Conclusions
• A model with a running CC based on the RG effects
in QFT in curved space merged with the concept of
holographic dark energy density.
• Remarkable consequences for the particle spectrum.
• Consistency with holographic predictions calls for
the appearance of the (otherwise redundant)
quintessence-like scalar fields.
• Although the holographic dark energy density in this
approach is rather a ‘toy’ model, our order-ofmagnitude estimates may indicate a preference for
the ‘combined’ dark energy nature.
Thanks!
Finis enim mundi et omnis creationis homo est.
•
The scale μ?
•
Example 1. Quantum chromodynmics
[ μ (∂/∂μ) + β(α)α ∂/∂α … ] Γren (p, …, μ) = 0,
&
[ λ (∂/∂λ) + μ (∂/∂μ) + … ] Γren (λp, …, μ) = 0,
[- (∂/∂τ) + μ (∂/∂μ) + … ] Γren (λp, …, μ) = 0,
where τ = ℓn λ, and p→ λp is a momentum scaling. Eliminating μ(∂/∂μ)
one obtains the RGE in λ, with the running α (τ).
The scale μ is traded for the momentum.
*********************************************
• Example 2. QFT in curved space-time
[ μ (∂/∂μ) + … ] Γren (p, …, μ) = 0,
&
[ τ (∂/∂τ) + μ (∂/∂μ) + … ] Γren (λ² gμν , …, μ) = 0,
with
gμν → λ² gμν , τ = ℓn λ.