Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 48 (2015)

Is cosmological constant screened in Liouville gravity with matter?
T. Inami,∗1,∗2 Y. Koyama,∗3 Y. Nakayama,∗4 and M. Suzuki∗5,∗6
Recent observation of dark energy in our universe
has led to the conviction that the cosmological constant Λ has an inﬁnitesimal positive value. It meansthat our space-time is de √Sitter (dS) space with theHubble constant H being Λ. It has been proposedthat the strong infrared (IR) divergence property ofquantum corrections on dS space may explain thesmallness of Λ in our current universe (so called cosmological constant problem). The Einstein equation describes the relation between the space-time Ricci tensor Rµν and the energy momentum (EM) tensor Tµν
due to the presence of matter. In vacuum, where Tµν is
proportional to the metric, the Einstein equation takesthe formκ
1Rµν − gµν R + gµν Λeﬀ = 0, Λeﬀ = Λ − Tρ ρ , (1)2Dwhere R is the scalar curvature, κ = 8πG with G being
Newton’s constant. The vacuum contribution of Tµν
is now combined with Λ to deﬁne the eﬀective cosmological constant Λeﬀ .
In view of this expression, we may wonder if a large
value of Tµν cancels the large value of Λ, yielding a very
1 We study
small value of Λeﬀ that we observe today.?)
this question in 2-dimensional (2D) Liouville gravity.
We are interested in the dS solution of Liouville gravity, which can describe the interaction between scaler
ﬁeld and gravity. The 2D cosmological constant has
two components: the coupling of the Liouville potential and the trace of the EM tensor. By using Weyltransformation to 2D metric gµν (gµν = e2φ ĝµν ), weobtain the equation,SL+mat[∫ [Φ, φ]√1 µνQĝ ∂µ Φ∂ν Φ += − d2 x −ĝR̂Φ
4πb24πb]1Λ(2)+ e2Φ + ĝ µν ∂µ φ∂ν φ + e2Φ V (φ) .
κ2The ﬁrst three terms are Liouville gravity. At thispoint, it is important to note that the negative valueof Λ corresponds to dS space in Liouville gravity.
As a concrete matter Lagrangian, we have studied amassless scalar ﬁeld theory with λφ4 interaction minimally coupled to Liouville gravity.

∗1
∗2
∗3
∗4
∗5∗6√√11L = − g µν ∂µ φ∂ν φ −g − λφ4 −g + ∆L,
24!

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RIKEN Nishina Center
Department of Physics, National Taiwan University
Department of Physics, National Tsing-Hua University
Walter Burke Institute for Theoretical Physics, California
Institute of Technology
Department of Physics, Shizuoka University
Department of Information Science and Technology, Graduate School of Science and Technology, Shizuoka University

λλx

λλxz

zb1b2Fig. 1. Part of λ2 -order corrections to the EM tensor Tµν.

where ∆L consists of counter-terms. In dS space, themassless scalar propagator contains IR divergence inthe long wavelength limit and the IR logarithm appears because the cutoﬀ regularization of the IR di2-4 we havevergence. Based on the in-in formalism,?,?,?)
computed the VEV of the EM tensor to the order ofλ2 .
Λeﬀ ∼ Λ + (Weyl anomaly) ++κλln2 a(η)
32π 2

κλ2
1ln4 a(η),
8π (4π)2 H 2(4)11 −Htis the scale factor and η = − Hewhere a = − Hηis conformal time. The resulting VEV has time dependence through the IR logarithms, and as a consequence, the eﬀective cosmological constant shows thescreening eﬀect at late time such that the absolutevalue decreases with time. This should be in contrast
with the situations where D > 2, in which the cosmo5
logical constant is anti-screened in the λφ4 theory.?)
To claim that the observed dS breaking eﬀects arephysical, we need to ﬁnd whether they may be eliminated from the local counter-terms. Here, we discussrather unfamiliar time-dependent IR counter-terms.
This possibility plays a crucial role to understand the(in)equivalence between the Sine-Gordon model andthe massive Thirring model in dS space. We are indeed able to recover the dS invariance by adding timedependent IR counter-terms to the naive perturbative computations using the dS breaking propagator.
Within the perturbation theory we have studied, however, a similar mechanism does not seem to be workable in λφ4 theory. This fact supports the claim thatthe observed screening mechanism of the cosmologicalconstant should be physical. This sensitive issue will
be further discussed in our future publication.

References
1) J. Polchinski: hep-th/0603249.
2) J. S. Schwinger: J. Math. Phys. 2, 407 (1961).
3) L. V. Keldysh: Zh. Eksp. Teor. Fiz. 47, 1515 (1964)[Sov. Phys. JETP 20, 1018 (1965)].
4) S. Weinberg: Phys. Rev. D 72, 043514 (2005).
5) V. K. Onemli and R. P. Woodard: Class. Quant. Grav.
19, 4607 (2002).

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