Ghost anomalous dimension in asymptotically safe quantum gravity

0.29 MB·English

by  Astrid Eichhorn

Ghost anomalous dimension in asymptotically safe quantum gravity
Astrid Eichhorn and Holger Gies
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at Jena,
Max-Wien-Platz 1, D-07743 Jena, Germany E-mail: [email protected], [email protected]
(Dated: January 27, 2010)
We compute the ghost anomalous dimension within the asymptotic-safety scenario for quantum
gravity. For a class of covariant gauge fixings and using a functional RG scheme, the anomalous
dimension ηc is negative, implying an improved UV behavior of ghost fluctuations. At the non-
Gaußian UVfixedpoint, weobservea maximum valueofηc ≃−0.78 fortheLandau-deWittgauge
withinthegiven schemeandtruncation. Most importantly,thebackreaction oftheghost flowonto
the Einstein-Hilbert sector preserves the non-Gaußian fixed point with only mild modifications of
the fixed-point values for the gravitational coupling and cosmological constant and the associated
0 critical exponents; also their gauge dependence is slightly reduced. Our results provide further
1 evidencefor theasymptotic-safety scenario of quantumgravity.
0
2
I. INTRODUCTION metry group, a continuum formulation based on the
n
gauge-variant degrees of freedom requires gauge fixing
a
J The asymptotic-safety scenario, introduced by Wein- conventionally accompanied by Faddeev-Popov ghosts.
Whereas most studies have concentrated on gauge-
7 berg in 1979 [1–3], provides for a promising route to a
2 fully consistent and predictive theory of quantum grav- invariant building blocks of the theory, the investiga-
ity. Purelywithinthe frameworkofstandardlocalquan- tion of the renormalization flow of the ghost sector has
] started only recently [41]. Even though the ghosts do
h tum field theory, gravity can become renormalizable at
not belong to the physical state space, their fluctuations
t an interacting non-Gaußian fixed point in the ultravio-
- can, in principle, contribute in an important manner to
p let (UV). A verification of this scenario requires a non-
gauge-invariant observables, depending on the details of
e perturbativeanalysisoftherenormalizationflowinduced
h bythefluctuationsofthegravitationaldegreesoffreedom the gauge. In fact, in the Landau gauge in Yang-Mills
[ theory, the ghost sector can even carry crucial physical
of the theory.

informationinthestronglyinteractingregimeofthethe-
1 Convincing evidence for this scenario has been col-
ory, as is, for instance, expressed in the Kugo-Ojima or
v lected in the past years, using the metric field itself as
3 the microscopic degree of freedom. Based on the pio- Gribov-Zwanzigerscenariosforconfinement: here,signa-
3 tures for the physical property of confined color charges
neering work of Reuter [4], the renormalization flow of
0 occur in the form of a strongly divergentghost propaga-
generic gravitational theories including Einstein-Hilbert
5 torintheinfrared,whereasthegluonpropagatorisfinite
terms and higher-order curvature invariants has been
.
1 studied within the functional RG [5–27] and other field- or even zero [42–46]. These properties can be related to
0 the existence of Gribov copies, i.e., the non-uniqueness
theoretical methods [28, 29]. These results do not only
0 ofthe gauge-fixingconditionandthe relatedzerosofthe
agree on the existence of a non-Gaußianfixed point, but
1
Faddeev-Popov operator [43, 44]. This problem also ex-
also on the classification of renormalization group (RG)
:
v relevantandirrelevantdirections. Inparticular,thenum- ists in gravity [47, 48], where it is even more challenging
i due to the highly non-trivial topology of the diffeomor-
X ber of RGrelevantdirections correspondsto the number
phism group.
of physical parameters to be fixed for the theory to be
r
a fully predictive; there are strong indications from high- As an indicator for the relevance of ghost fluctuations
ordercalculationsthatthisnumberisfinite[23–25],lend- at a UV fixed point of asymptotically safe Quantum
ing substantial support to the scenario that Quantum Gravity, we compute the ghost anomalous dimension in
Einstein Gravity (QEG) is asymptotically safe and ap- this work. On the one hand this measures how these
plicable to diverse physical scenarios. For instance, the gauge-variant degrees of freedom propagate in a given
implications of QEG for cosmology as well as black hole gauge, on the other hand the ghost backreaction on the
physics have been studied in [30–32]. Some features of renormalization flow of the graviton sector provides for
theasymptotic-safetyscenarioarealsoreminiscenttore- another nontrivial test of the existence of the UV fixed
sults obtained within other nonperturbative approaches point.
to a quantum field theory of gravity [33–36]. Such investigations require a non-perturbative tool
Theconceptofasymptoticsafety,ofcourse,extendsto which supports an approach to quantum gravity within
generalquantumfieldtheories: for instance,correspond- systematic and consistent approximation schemes. Such
ing scenarios curing the triviality problem of the Higgs an advanced framework is provided by the functional
sector of the particle-physics standard model have been renormalization group which facilitates studies not only
developed in the pure matter sector [37, 38] and also in- of a given action but of the renormalization flow of
cluding gravity [39, 40]. generic effective actions in theory space (for reviews for
As for any quantum field theory with a local sym- gauge theories, see [49]). The functional RG can be for-
2
mulatedintermsofaflowequationforascale-dependent and focus on the single wave function renormalization
effectiveaverageactionΓ withk denotingamomentum Z .
k c
scale. TheWetterichflowequation[50]relatestheflowof During the course of the flow, we also identify the full
Γ to the trace of the full non-perturbative propagator, metric γ with the background metric g¯ , once the
k µν µν
propagator of the fluctuation field is determined. This
1 1
∂ Γ = STr ∂ R , (1) approximation appears justified for many aspects of the
t k 2 Γ(2)+R t k Einstein-Hilbert sector [53]. More generally, we expect
k k
bimetric truncationsto be mandatoryfor a computation
where t = lnk and Γ(2) denotes the second functional of(momentum-dependent)propagatorsandvertices. For
k
derivative with respect to the fields. The regulatorfunc- instance in Yang-Mills theories, the distinction between
tionR actsasamomentum-andk-dependentmassterm fluctuation and background is crucial for the computa-
k
and cuts off momentum modes with p2 . k2. Accord- tion of gauge-invariantinfrared observables [54].
ingly, the limit k is related to the classical action Thepaper isorganizedasfollows: Sect.II summarizes
→ ∞
S [51], whereas the full quantum effective action Γ is our method to extract the ghost anomalous dimension
cl
obtained for k 0. andpresentsgeneralresultsforaclassofcovariantgauges
→
As Γ is a functional in the infinite dimensional space and a spectrally adjusted regulator. In Sect. III, explicit
k
of all operators compatible with the symmetry con- results for the Landau-deWitt gauge and the deDonder
straints, an exact solution of the Wetterich equation is gauge are given. We discuss the implications of our re-
difficult. Approximate solutions can be found by trun- sults and conclude in Sect. IV. Many technical details
catingasystematicexpansionoftheeffectiveaction. The and explicit representations may be found in the appen-
reliability of an approximation can be studied not only dices.
by checking its internal consistency but also by actively
monitoring the convergence of physical quantities at in-
creasing order in this expansion. II. GHOST ANOMALOUS DIMENSION
In order to extract gauge-invariantparts of the action
functional, the background-field formalism [52] is most In order to extract the anomalous dimension of the
convenient. Here the full metric γµν is split into a back- ghost, we project the flow equation onto the running of
ground g¯µν and a fluctuation metric hµν: the ghost wave function renormalization. We use a de-
composition of Γ(2)+R = + into an inverse prop-
γµν =g¯µν +hµν. (2) agator matrix k= Γ(2)k[c¯=P0 =Fc]+R , including the
P k k
Accordingly, we denote covariant derivatives compatible regulator but no external ghost fields, and a fluctuation
with the full or background metric by Dµ or D¯µ, re- matrix = Γ(2)[c¯,c] Γ(2)[c¯= 0 = c] containing exter-
spectively. In the present work, we study the following nalghosFtfieldsk. Thec−ompkonentsof areeitherlinearor
truncation of the effective action in four dimensions: bilinear in the ghostfields, as higherFordersdo not occur
in our truncation. We may now expand the right-hand
Γ =Γ +Γ +Γ , (3)
k kEH kgf kgh side of the flow equation as follows:
where 1
∂ Γ = STr [Γ(2)+R ]−1(∂ R ) (8)
t k 2 { k k t k }
ΓkEH = 2κ¯2ZN(k)Z d4x√γ(−R+2λ¯(k)), (4) = 1STr∂˜tln + 1 ∞ (−1)n−1STr∂˜t( −1 )n,
Z (k) 2 P 2 n P F
Γkgf = N d4x√g¯g¯µνFµ[g¯,h]Fν[g¯,h], (5) nX=1
2α
Z where the derivative ∂˜ in the second line by definition
t
with acts only on the k dependence of the regulator, ∂˜ =
t
∂ R δ . Sinceeachfactorof containsacouplingto
F [g¯,h]=√2κ¯ D¯νh 1+ρD¯ hν . (6) extternkaδlRfikelds, this expansion siFmply corresponds to an
µ µν µ ν
− 4 R
(cid:18) (cid:19) expansion in the number of vertices.
To bilinear order in the external ghost and antighost,
Herein, κ¯ = (32πGN)−12 is related to the bare Newton we may directly neglect all contributions beyond
constantGN. Theghosttermwithawavefunctionrenor- ( −1 )2. Diagrammatically, the remaining terms
malization Zc is given by cPorresFpond to a tadpole and a self-energy diagram.
Irrespectiveofourchoiceofgaugeintheghostsector,i.e.,
Γ = √2 d4x√g¯Z c¯ D¯ρg¯µκγ D
kgh c µ κν ρ for all values of the gauge parameter ρ, we find that the
−
Z (cid:16) tadpole does not contribute to the ghost anomalous di-
1
+ D¯ρg¯µκγ D (1+ρ)D¯µg¯ρσγ D cν.(7) mensionasthecorrespondingvertexiszero;thisisshown
ρν κ ρν σ
− 2
in appendix A.
(cid:17)
In this work we neglect the possibility of a different flow We take advantage of the possibility to specify a con-
for the two tensor structures in the kinetic ghost term crete backgroundforcomputationalpurposesin orderto
3
and scalar (σ and h) degrees of freedom:
p p
h (p) = hT (p)+i µ v (p)+i ν v (p)
µν µν ν µ
p2 p2
p p 1 1
+ µ νσ(p) p δ σ(p)+pδ h(p), (11)
p2 − 4 µν 4 µν
where h(p)=γµνh is the conformal mode. In this de-
µν
composition of the graviton, the quantities −1 and
are6 6matricesinfieldspaceφ=(hT ,v ,Pσ,h,c¯ ,c F).
× µν µ µ µ
Inthefollowing,weconfineourselvestothegaugechoice
ρ α for which the propagator matrix is diagonal in
FIG. 1: Tadpole diagram with the two-graviton-ghost- →
the graviton modes and off-diagonal in the ghosts. Ac-
antighost vertex denoted by a dot and self-energy diagrams
with graviton-ghost-antighost vertex denoted by a diamond. cordingly, the product −1 2 decomposes into four
P F
The crossed circle represents the regulator insertion ∂tRk termsfromthefourdifferentgravitonmodes,whichmay
(cid:0) (cid:1)
which occurs on both of the two propagators for the self- be considered separately. The corresponding matrix ele-
energy diagram. ments of −1 and which build up the self-energy di-
P F
agram can be decomposed into elementary propagators
andvertices for these modes, listed indetail in appendix
project unambiguously onto the running couplings. For B. Incidentally, the gauge choice ρ α becomes prob-
the running of the ghost wave function renormalization, lematic for ρ = α = 3: here, both t→he inverse ghost as
it suffices to choose a flat background, i.e. g¯µν = δµν, well as the inverse scalar gravitonpropagatorsdevelop a
as this allows for a clean separation between the ki- zeromode. Theinversepropagatoristhusnotinvertible,
neticghostoperatorandhigher-orderoperatorscoupling signaling that the gauge fixing is no longer complete for
ghostbilinearstocurvatureinvariants. Itisimportantto this choice. Nevertheless, the most important standard
stress that this choice of background has nothing to do gauge choices α = 0 or α = 1 are unaffected by this
with aperturbative reasoning,where one wouldconsider problem.
small metric fluctuations around a flat background. For
The matrix multiplication in Eq. (8) to second order
the projectionofthe Wetterich equation,any convenient
intheverticescanbe performedstraightforwardly. Also,
background that helps distinguishing different operators
the trace over Lorentz indices and momentum variables
unambiguouslycanbeusedinordertoextractthecorre-
canefficiently be performedbystandardmeansandsim-
sponding β functions. In fact, we are allowed to choose
plifiesbyusingtheprojectionruleasspecifiedbyEq.(9).
different backgrounds for extracting the flow of different Finally evaluating the ∂˜ derivative necessitates a more
operators. Thenotionofabackgroundinthesenseofthe t
explicit form of the regulator. We choose a spectrally
truequantumexpectationvalueofthefieldcanfinallybe
and RG-adjusted regulator [55, 56],
computed, once the full effective action is known.
On a flat background, we project the flow equation in
Fourier space onto the running wave function renormal- R (p2)=Γ(2)(p2)r Γ(k2)(p2) . (12)
ization by k k Zk2 !
η = ∂ lnZ (9)
c − t c Herein,the wavefunctionrenormalizationZ intheregu-
→ ←
latorisadaptedtoeachmode,suchthatthecutoffscales
1 1 ∂ d4q˜ δ δ
= δαγ ∂ Γ . foreachmodeareequal;thisimpliesthatthefactorofZ
−√2Z 4 ∂p˜2 (2π)4 δc¯α(p˜) t kδcγ(q˜)
c Z also contains the numerical prefactors from Eq. (B7) for
  the transverse, vector, scalar and conformal mode. This
OurconventionsforthefunctionalGrassmannianderiva- adjustmentisuseful,asdifferenteffectivecutoffscalesfor
tives are such that different modes could artificially alter the results within
→ ← a truncation.
δ d4p δ Wetherebyarriveatthefollowingflowequation,where
c¯µ(p)M (p)cν(p) =δ(p˜,q˜)M (p˜),
δc¯α(p˜) (2π)4 µν δcγ(q˜) αγ thefirstlineisthetransversetracelesscontribution. The
Z
(10) secondline isduetothe transversevectormode,andthe
where δ(p,q) = (2π)4δ(4)(p q). In the following, we last two lines result from the two scalar modes, respec-
− ′
decompose the fluctuation metric into irreducible repre- tively. Theterms r areduetotheexternalmomentum
∼
sentations of the Poincar´e group in momentum space, p˜flowingthroughtheinternalghostlineandbeingacted
i.e., transverse traceless (hT ), transverse vector (v ), upon by the ∂ -derivative in Eq. (9).
µν µ p˜2
4
dp2 5(α 7)
η = √2Z ∂˜ p2 −
c c t 16π2 "18(α 3)κ¯2Z p2 2λ¯(k) 1+r p2−2λ¯(k) √2Z 1+r p2
Z − N − k2 c k2
α (cid:0) (cid:1)(cid:16) (cid:16) 1 (cid:17)(cid:17)1p2 r(cid:16)′ kp22 (cid:16) (cid:17)(cid:17)
+ (13)
κ¯2Z p2 2λ¯(k) 1+r p2−2λ¯(k) √2Z 1+r p2 3 − 4k21+(cid:16)r p(cid:17)2 
N − k2 c k2 k2
(cid:0) (cid:1)(cid:16) (cid:16) (cid:17)(cid:17)α (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) p2 r′ kp22
2(α 7)+3
−18(α−3)κ¯2ZN (α−3)p2+4αλ¯(k) 1+r p2+k4α2αλ¯−(3k) √2Zc 1+r kp22  − k21+(cid:16)r kp(cid:17)22 
(cid:0) (cid:1)1(cid:18) (cid:18) (cid:19)(cid:19) (cid:16) (cid:16) (cid:17)(cid:17) p2 r′ kp22 (cid:16) (cid:17)
+ (3α 4) α
6(α−3)κ¯2ZN (α−3)p2+4λ¯(k) 1+r p2+k42αλ¯−(k3) √2Zc 1+r kp22  − − k21+(cid:16)r kp(cid:17)22 #
(cid:0) (cid:1)(cid:18) (cid:18) (cid:19)(cid:19) (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17)
The Landau-deWitt gauge α = 0 clearly plays a dis- we arrive at an explicit form for the ghost anomalous
tinguished role as only the transverse traceless and the dimension,whichisgiveninappendixDfortheLandau-
conformal mode propagate. This feature appears gener- deWitt (α=0) and the deDonder (α=1) gauge.
ically in all diagrams with more than one vertex as the
Endowingtheghostpropagatorwithanon-trivialwave
propagatoris α. ThisfavorstheLandau-deWittgauge
∼ functionrenormalizationηcfeedsbackintotheflowequa-
from a computational point of view. As it is moreover
tionsintheEinstein-Hilbertsectorbyvirtueoftheghost
a fixed point of the renormalization group flow [57, 58],
loop. We evaluate the gravitonanomalous dimension η
N
we consider the Landau-deWitt gauge as our preferred
andtheflowofthecosmologicalconstant∂ λbyexpand-
t
gauge choice.
ingthepropagatorsinabasisoftensor,vectorandscalar
hyperspherical harmonics on a d sphere and find the β
functions given in Appendix C.
III. RESULTS
This flow of the Einstein-Hilbert sector has been ana-
lyzed in a variety of computations in the literature, e.g.
For explicit computations, we use an exponential reg-
[4,6,9,13,18,20]. Fromatechnicalviewpoint,ourcom-
ulator shape function r(y),
putationalmethoddiffersfromtheliteratureaswedonot
relyonheat-kernelresults. Apartfromthisminordetail,
1
r(y)= . (14) our results are in perfect agreement with the literature;
ey 1
inparticular,ourchoiceofaspectrallyadjustedregulator
−
corresponds to type III in [18]. The new modifications
Such a smooth shape function is advantageous here as
arising from the flow of the ghost sector correspond to
the evaluation of η = ∂ lnZ and ∂ λ require up to
N t N t
− the additional terms η .
second-order derivatives. c
∝
Introducing the dimensionless Newton coupling and cos- FortheUVfixed-pointsearch,weinserttheexpression
mological constant, for η into Eq. (C2) and Eq. (C1). Imposing the fixed-
c
point condition η = 2 and ∂ λ = 0, we then find the
N t
1 −
G= 32πκ¯2ZNk2−d ⇒ ∂tG=(d−2+ηN)G fimxaetdrixp,owinhticvhalmueasyabnedcoemigpeanrveadluteostθh1e,2irocfoutnhteersptaarbtislitiny
λ=λ¯k−2 ∂ λ= 2λ+k−2∂ λ¯,(15) a truncation where η =0:
t t c
⇒ −
gauge G∗ λ∗ G∗λ∗ θ1,2 ηc
Landau-deWittwith ηc =0 0.270068 0.378519 0.102226 2.10152 ±i1.68512 0
Landau-deWittwith ηc 6=0 0.28706 0.316824 0.0909475 2.03409±i 1.49895 -0.778944
deDonderwith ηc =0 0.181179 0.480729 0.0870979 1.40864 ±i1.6715 0
deDonderwith ηc 6=0 0.207738 0.348335 0.0723625 1.38757±i1.283 -1.31245
5
The gaugedependences forthe fixed-pointvaluesλ∗ and GΛ
0.10
G∗ agree with those found e.g. in [8] for different regu-
0.09
lators. The value of the ghost anomalous dimension also
exhibitsasizablegaugedependence. Ourresultsprovide 0.08
evidence that ηc < 0 is valid for all admissible choices 0.07
of α with a maximum value of η = 0.778944 be-
c,max 0.06
−
ing acquired in the Landau-deWitt gauge in the present
0.05
truncation. This is due to the fact that the conformal
0.04
mode contributeswith apositive signto ηc (forG∗ >0),
whereastheothermodestypicallycontributewithaneg- Ηc
-3 -2 -1 0 1 2 3
ative sign. As the scalar and the vector contribution are
both proportional to α, η decreases with increasing α.
c
Another important observation is that the inclusion FIG.3: Fixed-pointvalueof theproductGλ asafunction of
of the ghost anomalous dimension on average leads to a ηc treated as a free parameter in thedeDondergauge.
weaker gauge dependence in the Einstein-Hilbert sector,
as mightbe readofffromtable III. This canbe takenas
a signature for the convergence of the expansion of the Re@ΘD
effective action. 1.55
We observe a rather strong dependence of the
cosmological-constantfixedpointλ∗ontheghostanoma- 1.50
lous dimension, whereas G∗ shows only a mild variation.
In order to test the stability of the non-Gaußian fixed 1.45
point in the Einstein-Hilbert sector against further vari-
ationsoftheghostsector,letusstudytheη dependence 1.40
c
of the fixed point from a more general viewpoint. For
this, we consider η as a free parameter for the moment, Ηc
c -3 -2 -1 0 1 2 3
and then solve the fixed-point equations for G and λ as
wellasthecorrespondingcriticalexponentsasafunction Im@ΘD
ofη . This canmimicthe influence ofhigher-orderghost 2.5
c
operators on the true value of η .
c
2.0
G 1.5
0.24
1.0
0.22
Ηc
0.20
-3 -2 -1 0 1 2 3
0.18
0.16 FIG. 4: Real part and positive imaginary part of the critical
exponents(eigenvaluesofthestabilitymatrix)intheEinstein-
Ηc
-3 -2 -1 0 1 2 3 Hilbert sector as a function of ηc treated as a free parameter
inthedeDondergauge. Thecorrespondingself-consistent so-
Λ lution for theghost anomalous dimension is ηc ≃−1.31245.
0.6
Forany η [ 3,3],whichwe expect to coverthe physi-
c
∈ −
0.5 callyrelevantrange,wefindaphysicallyreasonablefixed
point which providesfurther evidence for the stability of
0.4
the asymptotic-safety scenario. Quantitatively, the cos-
0.3 mologicalconstantgetssignificantlyreducedfordecreas-
Ηc ing ghost anomalous dimension. The values of G∗ show
-3 -2 -1 1 2 3 the opposite behavior, but much less pronounced. The
universal product G∗λ∗ becomes less sensitive to ηc for
positive η .
c
FIG.2: Fixed-pointvaluesofG(upperpanel),λ(lowerpanel) Furthermore, let us investigate the “TT approxima-
asafunctionofηctreatedasafreeparameterinthedeDonder tion”,whereallgravitonmodesexceptforthetransverse
gauge with ρ = 1 = α. The corresponding self-consistent traceless mode are not allowed to propagate. This ap-
solutionfortheghostanomalousdimensionisηc ≃−1.31245. proximationhas effectively been adopted in the study of
6
the ghost-curvaturecouplingin[41]. As onthe onehand Eventhoughtheghostanomalousdimensionturnsout
the transversetracelessmode is the genuine spin-2mode to be large, i.e., of order one, its influence on the fixed-
characteristic for gravity,and on the other hand the TT point properties in the Einstein-Hilbert sector remains
approximation is technically much easier to deal with, rather moderate, confirming earlier approximations that
it is interesting to see whether this ”transverse traceless have neglected η . On the other hand, the values that
c
approximation” is able to capture the essential physics, η can acquire in standard gauges indicate that ghost
c
such as the existence of the fixed point and the number contributions can become important in future analyses.
of relevant directions. In the TT approximation and in The anomalous dimension η parameterizes the scale
c
the Landau-deWitt gauge, we find a fixed point with a dependence of the wavefunction renormalizationZ . As
c
negative value of λ and a positive G. As there are RG the functional RG flow is local in momentum space, we
trajectories where the negative cosmological constant is canconsiderthescaledependenceofZ asanestimatefor
c
acquiredonlyintheUV,thisfixedpointisphysicallyad- the momentum dependence of the fully dressed inverse
missible. Thecriticalexponentsinthisapproximationno ghost propagator:
longerformacomplexconjugatepairofeigenvaluesinthe
Einstein-Hilbert sector,but two positive realeigenvalues Gc(p2)−1 Zc(k2 =p2)p2 (p2)1−η2c. (17)
∼ ∼
θ =2.808, θ = 1.075. The ghost anomalous dimension
1 2
isgivenbyη = 0.5515. Weconcludethattheexistence The corresponding correlator in position space reads
c
−
of the non-Gaußian fixed point as well as the number of
1
′
relevantdirections is correctlyreproducedin the TT ap- G (x,x) . (18)
proximation. It is an interesting open question whether c ∼ |x−x′|2+ηc
this feature of the transverse traceless approximation is
As η < 0 for all cases studied in this work, the propa-
c
generic and can be confirmed also in other truncations.
gationofghostmodesis suppressedinthe UVcompared
As another approximation of our full result, we can
to the perturbative propagator. For those gauges (e.g.,
taketheperturbativelimit,i.e.,considertheleadingcon-
α = 0 and α = 1) where 2 < η < 0, the dressed cor-
c
tributioninGtoη . Forvanishingcosmologicalconstant, −
c relator still develops a singularity at short distances. In
but arbitrary gauge parameter α (=ρ), we find
general, the UV behavior of higher-order diagrams in-
volving ghosts is improved for η < 0 compared to a
5α3 26α2+5α+124 G c
η = − + (G2). (16) perturbative framework.
c − 4π(α 3)2 O Itisinterestingtocomparethistothegravitonanoma-
(cid:0) − (cid:1)
lous dimension which is η = 2 at the fixed point,
N
Again, we observe that ηc is negative for all admissible implying an even weaker UV pr−opagation, Gh(x,x′)
gauge parameters α 0, with a singularity at α = 3 ln x x′ [15]. At first sight, one might expect tha∼t
due to incomplete gau≥ge fixing as discussed above. The a c|om−plet|e calculation of the ghost anomalous dimen-
Lbeainndgauin-dethWeitsta-mgaeugreanligmeitasisaηtct=he−n93oπ1nG-G≃au−ßi1a.n09fi6x4Ged, sainotneeshaoucaldncaellslaotiroensuoltf ginauηgce =mo−de2s ianndordgherosttomgoudaers-
point. Of course, this result is non-universal but scheme near the fixed point. However, this cancellation could
dependentinfourdimensions,asthepower-countingRG also come about for differing anomalous dimensions if
critical dimension is d=2. this difference is balanced by a corresponding running
Let us finally compare our findings to very recent re- of the vertices with ghost legs. A mechanism of this
sults by Saueressig and Groh, where ηc = 1.85 was ob- type has for instance been observed at a non-Gaußian
−
tained in the deDonder gauge applying a cutoff scheme IR fixed point in Landau-gauge Yang Mills theory [60].
withoutspectraladjustment[59]. Thequantitativevaria- On the other hand, if this difference in the propagation
tionisintherangefamiliarfromtypicalregulatordepen- of ghost and graviton modes persisted (say, if the ver-
dencies already observed in earlier calculations, see e.g. tices did not differ much from a trivial scaling), η >η
c N
[18]. Mostimportantly,thequalitativepicturesmutually would indicate a dominance of ghost modes for the RG
confirm each other. running. This could be a signature of the dominance
of field configurations near the Gribov horizonalong the
lines of the Gribov-Zwanziger scenario in gauge theories
IV. CONCLUSIONS [43]. We emphasize that a complete discussionof this is-
sue also requiresto distinguish betweenthe background-
We have computed the ghost anomalous dimension in field anomalous dimension ηN and the corresponding
quantumgravityinvariouscovariantgauges. Withinthe fluctuation-field anomalous dimension.
asymptotic-safety scenarioof quantum gravity,our work Another consequence of ηc < 0 can be read off from
represents a further extension of existing approximation theinducedscalingoftheverticesinvolvingghostmodes.
schemes for the construction of renormalizable RG tra- Consideravertexwithmghostandmantighostlegspa-
jectories. Our results provide further evidence for the rameterized by a coupling g¯(m). After renormalizing the
1/2
existence of a non-Gaußian fixed point which facilitates ghost fields with Z , the β function for the renormal-
c
a UV-complete construction of quantum gravity. ized coupling g(m) g¯(m)/Zm receives a contribution of
∼ c
7
the type where φ(p) = hT (p),v (p),σ(p),h(p),c (p),c¯ ( p)
µν µ µ µ −
andi,j labelthefieldcomponents. Here,wehavechosen
∂tg(m) =mηc+.... (19) the momentum-s(cid:0)pace conventions for the anti-ghost op(cid:1)-
For ηc < 0, this increases the RG relevance of this ver- posite to those of the ghost, i.e., if cµ(p) denotes a ghost
tex in comparison to standard power-counting. A sim- with incoming momentum p then c¯µ(q) denotes an anti-
ple example is provided by the ghost-curvature coupling ghostwithoutgoingmomentumq. Theghostpropagator
ζ¯c¯µRcµ considered in [41] (m = 1 in this case). In a is an off-diagonalmatrix,
∼
TTapproximation,therenormalizedcouplinghasturned
out to be RG relevant for η = 0. This conclusion now
remains unchanged, as our vcalues ηc < 0 even enhance −1 = 0 Γ(k2,c)c¯(p,q)+Rk −1 (B2)
the RG relevance of this operator. Pgh Γ(k2,c¯)c(p,q))+Rk 0 !
−1
0 Γ(2) (p,q)+R
Acknowledgments =  Γ(2) (p,q)+R −1 (cid:16) k,c¯c 0 k(cid:17) 
k,cc¯ k
The authors would like to thank F. Saueressig,  
K. Groh, M. Reuter and J.M. Pawlowski for helpful dis- (cid:16)0 −1 (cid:17) 
cussions. This work was supported by the DFG under = Pc¯−c1 P0cc¯ !, (B3)
contract No. Gi 328/5-1 (Heisenberg program) and GK
1523/1.
where
Appendix A: Vanishing of the tadpole diagram → ←
δ δ
Γ(2) (p,q) = Γ = Γ(2) (p,q)).
k,cc¯,µν δc ( p) kghδc¯ ( q) − k,c¯c,µν
The vanishing of the graviton tadpole contribution to µ − ν −
(B4)
the running of Z can be shown by making use of the
c
second variation of the Christoffel symbol,
δ2Γλ = δ δΓλ Within our truncation, the ghost propagator reads ex-
σν σν
plicitly
1
= δ(cid:0) γλτ(cid:1)(D h +D h D )
2 σ τν ν στ − τhσν
(cid:18) (cid:19) −1
(2)
= −hλτ(Dσhτν +Dνhστ −Dτhσν). (A1) Γk,c¯c+Rk µν (B5)
(cid:16) (cid:17)
Varying the ghost kinetic term twice with respect to the 1 ρ 1p p 1
= δ + − µ ν δ4(p q).
metric produces the following type of terms: √2Z p2 µν 3 ρ p2 (1+r(y)) −
c (cid:18) − (cid:19)
δ2(γ D cν) = δ2(γ Γν )cλ
κν ρ κν ρλ
= (2h δΓν +γ δ2Γν )cλ. (A2) The graviton propagators are obtained from the second
κν ρλ κν ρλ
variationoftheEinstein-Hilbertandthegauge-fixingac-
Inserting the firstand second variationof the Christoffel
tion. Setting γ =g¯ =δ after the functional varia-
µν µν µν
symbolfromEq.(A1)leadstoacancellationbetweenthe
tion yields the following expression in Fourier space:
twoterms. Accordingly,thesecondvariationoftheghost
kinetic term with respect to the metric vanishes (for all
choicesofρ). Hence,thereisnogaugeinwhichagraviton δ2ΓkEH+gf
tadpole can contribute to the running of the ghost wave d4p 1
= κ¯2 hαβ( p) (δ δ +δ δ )p2
functionrenormalization. (Clearlyamoregeneralkinetic (2π)4 − 4 αµ βν αν βµ
ghosttermwith the volumeelementd4x√γ wouldresult Z h
1 1
inatadpolecontributionfromthesecondvariationofthe δ p2δ +δ p p (p p δ +p p δ )
αβ µν αβ µ ν β µ να α ν µβ
− 2 − 2
volume element.)
1 1
+ λ δ δ (δ δ +δ δ )
k αβ µν αµ βν αν βµ
2 − 2
(cid:18) (cid:19)
Appendix B: Vertices and propagators 1 1 1+ρ
+ (p p δ +p p δ ) p p δ
α µ βν β µ βµ α β µν
α 2 − 2
In this appendix, we derive the building blocks for the (cid:16)(1+ρ)2
+ δ p2δ h (p). (B6)
expansion (8) of the flow equation in terms of the quan- αβ µν µν
16
tities and . In the following, we always aim at a (cid:17)i
P F
Euclidean flat background. Our conventions for 2-point
InsertingtheYorkdecomposition(11)intoEq.(B6)then
functions are given by
resultsinthefollowingexpression,fromwhichtheinverse
→ ←
propagatorsfollow directly by functional derivatives:
δ δ
(2)
Γ (p,q)= Γ , (B1)
k,ij δφ ( p) kδφ (q)
i j
−
8
d4p 1 p2
δ2Γ = Z κ¯2 hTαβ( p) p2 2λ hT ( p)+vβ( p) 2λ v (p)
kEH+gf N (2π)4 − 2 − k αβ − − α − k β
Z3 3h α (cid:0) (cid:1) ρ α (cid:18) 1 (cid:19) ρ2 3α
+ σ( p) p2 − 4λ σ(p)+h( p)3 − p2σ( p)+ h( p) p2 − +4λ h(p) .(B7)
k
− 16 α − − 8α − 16 − α
(cid:18) (cid:19) (cid:18) (cid:19) i
In this work, we confine ourselves to the gauge choice where the last term in the transverse traceless projector
ρ αwherethe propagatormatrix becomesdiagonalin P removes the trace part.
TT
→
the graviton modes. The vector and transverse traceless
tensor propagators go along with transverse and trans- The resulting propagators together with the regulator
verse traceless projectors, respectively. In d-dimensional R constitute the term in the expansion of the flow
k
Fourier space, these projectors read equation (8). P
p p
µ ν
P (p) = δ ,
Tµν µν − p2 The termcarriesthe dependence onthe ghostfields
F
thatcouplevia verticestothe fluctuationmodes. To ob-
1
PTTµνκλ(p) = (PTµκPTνλ+PTµλPTνκ) tain these vertices, we vary the ghost action once with
2
respect to the metric and then proceed to a flat back-
1
P P , (B8) ground, yielding:
Tµν Tκλ
−d 1
−
δΓ
kgh
1 1
= √2Z d4x√g¯c¯µ D¯ρh D¯ +D¯ρ D¯ h +D¯ρh D¯ (1+ρ)D¯ hρD¯ (1+ρ)D¯ [D¯ hλ] cν.
− c µν ρ ν µρ ρν µ− 2 µ ν ρ− 4 µ ν λ
Z d4p d4(cid:16)q (cid:2) (cid:3) 1 p (cid:17)
√2Z c¯µ(p+q)h (p)cκ(q) q (p+q) δρδσ+δσδρ κ (pρ+qρ)δσ +(pσ+qσ)δρ
→ − c (2π)4(2π)4 ρσ − · 2 µ κ µ κ − 2 µ µ
q Z 11+ρ(cid:16) (cid:0) 1+(cid:1) ρ (cid:0) (cid:1)
µ ((pσ+qσ)δρ+(pρ+qρ)δσ)+ (p +q )(δρqσ+δσqρ)+ p δρσ(p +q )
− 2 κ κ 2 2 µ µ κ κ 4 κ µ µ
d4p d4q (cid:17)
= √2Z c¯µ(p+q)cκ(q) V(T)ρσ(p,q)hT (p)+V(v)ρ(p,q)v (p)++V(σ)(p,q)σ(p)+V(h)(p,q)h(p) .
− c (2π)4(2π)4 κµ ρσ κµ ρ κµ κµ
Z (cid:16) (cid:17)
Here,weintroducedthe Yorkdecomposition(11)forthe sponding vertices connecting ghost and anti-ghost with
gravitonfluctuation, suchthat wecanreadoffthe corre- the gravitoncomponents:
1 p q 11+ρ
V(T)ρσ(p,q) = q (p+q) δρδσ+δσδρ κ qρδσ+qσδρ µ (qσδρ+qρδσ)+ (p +q )(δρqσ+δσqρ),
κµ − · 2 µ κ µ κ − 2 µ µ − 2 κ κ 2 2 µ µ κ κ
V(v)ρ(p,q) = 2i p q ((cid:0)p+q)1 δρδ(cid:1)σ+δσδ(cid:0)ρ pκ qρδ(cid:1)σ+(pσ+qσ)δρ qµ ((pσ+qσ)δρ+qρδσ)
κµ p2 σ − · 2 µ κ µ κ − 2 µ µ − 2 κ κ
(cid:16)
11+ρ (cid:0) (cid:1) (cid:0) (cid:1)
+p (p +q )(δρqσ+δσqρ) (B9)
2 2 µ µ κ κ
1 3 3 ρ (cid:17) 1 ρ 1 1
V(σ)(p,q) = p p p2+q2+ − q p +q q p2 − p2δ q2+q p
κµ −p2 κ µ 4 2 · µ κ4 2 − 4 µκ ·
h 1 (cid:18) 1 ρ (cid:19)1+ρ (cid:18) (cid:19) (cid:0) (cid:1)
+p q p2+ − q p +p q p2
κ µ µ κ
2 2 · 8
(cid:18) (cid:19) i
1 1 ρ 1+ρ
V(h)(p,q) = q (p+q)δ +p (p +q )(ρ 1)+p p q q − +p q .
κµ 4 − · µκ κ µ µ − µ κ− µ κ 2 µ κ 2
(cid:16) (cid:17)
From this, the four possible fluctuation matrix entries be evaluated:
contributing to the quantity in the expansion (8) can
F
9
and similarly for the other graviton modes.
→ ←
δ δ
Γ(2) (q,p) = Γ
hTc δhT ( q) kghδc (p)
µν − κ
= √2Z c¯τ(p q)V(T) ( q,p)
− c − κτµν −
→ ←
δ δ
Γ(2) (q,p) = Γ
hTc¯ δhT ( q) kghδc¯ ( p)
µν − κ −
= √2Z cτ(q p)V(T) ( q,q p)
c − τκµν − −
→ ←
δ δ
Γ(2) (q,p) = Γ Appendix C: β functions in the Einstein-Hilbert
chT δc ( q) kghδhT (p)
κ − µν sector
= √2Z c¯λ(p q)V(T) (p, q)
c − κλµν −
→ ←
δ δ
(2)
Γc¯hT(q,p) = δc¯ (q)ΓkghδhT (p) In the Einstein-Hilbert sector, we find the following β
κ µν functions, obtained with a spectrally and RG adjusted
= √2Z cλ(q p)V(T) (p,q p),(B10) regulator with exponential shape function in d=4:
− c − λκµν −
∂ λ+2λ
t
12(∂ λ+2λ)Ei 4λ
= 361π Gλ(−150(∂tλ+2λ)Ei(2λ)− t α 3(cid:16)−α−3(cid:17) −150Li2 e2λ +6Li2 e−α4−λ3 +36Li2 e2αλ
− (cid:0) (cid:1) (cid:16) (cid:17) (cid:0) (cid:1)
6
+ (α 3)2 −3α(3α−2)(∂tλ+2λ)Ei(2αλ)(α−3)2−9αLi2 e2αλ (α−3)2+(7α−9) Li2 e−α4α−λ3 (α−3)
− (cid:0) (cid:1) h (cid:16) (cid:17)
4αλ
2α(∂ λ+2λ)Ei 2iπλ α(39α 166)+219 +∂ λ 25ln 1+e2λ (α 3)2
t t
− (cid:18)−α−3(cid:19)i− h − i " (cid:0)− (cid:1) −
+ 2ln 1+e−α4−λ3 (α 3)+α 3(3α 2)ln 1+e2αλ (α 3)2+2(7α 9)ln 1+e−α4α−λ3
− − − − − − − #!
(cid:16) (cid:17) h (cid:0) (cid:1) (cid:16) (cid:17)i
+ 1 e−4(αα+−13)λ 2e4(αα+−13)λ 18iπ(α 3)α(3α 2)λ+(7α 27) π2 3ηc 3eα4α−λ3(α 3)ηN
α 3 − − − − − − −
− (cid:16) (cid:0) (cid:1)(cid:17)
2(3α−1)λ 2((α−1)α+2)λ 4λ
+ 75e α−3 (α 3)ηN+9e α−3 (α 3)(3α 2)ηN 3eα−3(7α 9)ηN
− − − − − !)
8αλLi2 e−α4α−λ3 4αe−α4−α3λ 4αλ 4αλ −4αλ
+ 9G" α(cid:16)−3 (cid:17) + (α−3)2(∂tλ+2λ)(α+eα−3 4αλ(cid:18)Ei(cid:18)−α−3(cid:19)+iπ(cid:19)−(α−3)Li2he α−3 i!−3)
+ 6 2λ(∂tλ+2λ)(Ei(2αλ)+iπ)α2+∂tλLi2 e2αλ α+2Li3 e2αλ +4Li3 e−α4α−λ3 e−α4α−λ3ηN
" # −
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
3e2αλ(2∂ λα+4λα+η )+5 4λ(∂ λ+2λ)(Ei(2λ)+iπ)+2∂ λLi e2λ +4Li e2λ e2λ(2∂ λ+4λ+η )
t N t t 2 3 t N
− " − #
(cid:0) (cid:1) (cid:0) (cid:1)
+ (α 43)2 Li3 e−α4−λ3 (α−3)2−4∂tλLi2 e−α4−λ3 (α−3)+e−α4−λ3 (4∂tλ+8λ−(α−3)ηN)(α−3)
− (cid:16) (cid:17) (cid:16) (cid:17)
4λ
+ 16iπλ(∂ λ+2λ)+16λ(∂ λ+2λ)Ei +8(η 4ζ(3)) , for λ>0. (C1)
t t c
(cid:18)−α−3(cid:19)! − #!
10
G 144λln 1 e−α4α−λ3 α2
η = 54e2αλ(∂ λ+2λ)α2 − 36e2αλ(∂ λ+2λ)α 18(3α 2)(∂ λ+2λ)Ei(2αλ)α
N 36π" t − (cid:16)(α 3)2 (cid:17) − t − − t
−
−4αλ
24λln 1 e α−3 α 144λ
− 18(3α−2) 2iπλ+e2αλ(∂tλ+2λ)−∂tλln −1+e2αλ α− (cid:16)α− 3 (cid:17) + (α 3)2 ln 1−e−α4α−λ3 α
(cid:0) (cid:0) (cid:1)(cid:1) − − (cid:16) (cid:17)
12(∂ λ+2λ)Ei 4λ
− t α 3(cid:16)−α−3(cid:17) + 12(α(7α−3)29)(∂tλ+2λ) ln −1+e−α4α−λ3 −Ei −α4αλ3 α+27e2αληNα
− − (cid:18) (cid:16) (cid:17) (cid:18) − (cid:19)(cid:19)
− 4λ
3e α−3ηNα 150(∂ λ+2λ)Ei(2λ) 300λln 1 e2λ +150∂ λln 1+e2λ +300λln 1+e2λ
t t
− α 3 − − − − −
−
− 4λ − 4λ (cid:0) (cid:1) − 4λ(cid:0) (cid:1) (cid:0) (cid:1)
24λln 1 e α−3 24λln 1+e α−3 12∂tλln 1+e α−3
− (cid:16)α −3 (cid:17) + (cid:16)−α 3 (cid:17) + (cid:16)α− 3 (cid:17) −150Li2 e2λ +6Li2 e−α4−λ3
− − −4αλ − (cid:0) (cid:1) (cid:16) (cid:17)
6(7α 9)Li2 e α−3 2(α 9) π2 3η
+ 18(2 3α)Li e2αλ + − − − c 12 π2 3η +75e2λη 18e2αλη
− 2 α 3(cid:16) (cid:17) − α 3 − − c N− N
− (cid:0)− (cid:1)
− 4λ (cid:0)−4αλ(cid:1) (cid:0) (cid:1)
9e α−3ηN 3e α−3(9 7α)ηN
+ + −
α 3 α 3
#
− −
for λ>0. (C2)
Appendix D: Ghost anomalous dimension transverse traceless contribution and the last lines are
due to the conformal mode:
In the Landau-deWitt gauge (α = 0), we find the fol-
lowing expression for η , where the three lines are the
c
35
η = e−4λG e6λ 6∂ λ+η 3η +6 +e8λη +3e4λ η +8λ+4 +12 e2λ ∂ λ(6λ 1)
cL t c N N c t
−162π − − −
(cid:16)
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2)
+ λ( η +12λ 2) η λ E ( 6λ)+12e2λ ∂ λe2λ(4λ 1)+λ η +4λ+e2λ( η +8λ 2) E ( 2λ)
N c 1 t c N 1
− − − − − − − − −
(cid:3) (cid:17) (cid:16) h i(cid:17)
+ 12 η λ+e2λ 6λ∂ λ+∂ λ+(η +η 16λ+2)λ +e4λ( 4λ∂ λ+∂ λ+η 8λ+2)λ Γ(0, 4λ)
c t t c N t t N
− − − − − !
(cid:16) (cid:16) (cid:17) (cid:17)
2 4λ
+ e−8λ/3G 8e4λ/3 λ(8λ 3η )+e4λ/3 ∂ λ(8λ 3)+λ( 3η +16λ 6) Ei
c t N
243π − (cid:16) − (cid:2) − − − (cid:3)(cid:17) (cid:18) 3 (cid:19)
8λ
+ 8 3η λ+e8λ/3 ∂ λ(8λ 3)+λ( 3η +16λ 6) +e4λ/3 3∂ λ(4λ 1)+λ(32λ 3(η +η +2)) ) Ei
c t N t c N
− − − − − − 3
h (cid:2) (cid:3) (cid:2) (cid:3) i (cid:18) (cid:19)
+ 3 e8λ/3 3η +e4λ/3 4∂ λ+η + 3+e4λ/3 η +6 +16λ+12
c t c N
" − −
(cid:16) (cid:16) (cid:16) (cid:17) (cid:17) (cid:17)
+ 8 η λ+e4λ/3( 4λ∂ λ+∂ λ+(η 8λ+2)λ) Ei(4λ) . (D1)
c t t N
− − #!
(cid:16) (cid:17)
The exponential factors result from the spectral adjust- In the deDonder or harmonic gauge (α = 1), we have
ment of the regulator. The expression is linear in G, as to take contributions from all modes into account. Ac-
each contributing diagram contains exactly one graviton cordingly, we arrive at the following expression, which
propagator. decomposes into transverse traceless (D2), vector (D3),