IRGAC 2006

Renormalization Group Running Cosmologies – from a Scale Setting to Holographic Dark Energy

Branko Guberina

Rudjer Boskovic Institute Zagreb, Croatia

• 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 energy- momentum 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 , R2

μν , 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 ρΛ

Higgs = 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 – 3mZ

2 – 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 scale-setting 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

• Per definitionem,

ρc = 3H2/(8πG) ΩΛ = Λ/(8πGρc)

Ωm = ρm/ρc Ωc = -K/(H2/a2)

ΩΛ + Ωm + Ωc = 0 .

, → ΩΛ = Ωm

• 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 ΩK

0 .

• 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/m2

min , 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 m2

min .

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

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.

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

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 Black hole of size L

• Schwarzschild radius RS < 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 Tμν

total 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/m2

min etc., and D1 ~ 1, D2 ~ 1/m2

min , 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-of-magnitude 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.