Fierz-Pauli Massive Gravity

Consider building a simple quadratic theory of massive gravity. In terms of the metric perturbationhµν =gµν −ηµν, quadratic general relativity takes on the form

S = Z d4x1 2hµνE µν,αβ_h αβ +hµνTµν ≡ Z d4xL_GR(hµν) +hµνTµν (1.60)

where Eµν,αβ is the quadratic differential operator which stems from the expanding out the Ricci scalar whose exact form we will not need. If we were to attempt to add a mass term to (1.60), we would add it in some combination ah2_+bh

µνhµν whereh=hµµ and all

indices are raised and lowered withηµν. However, a priori there is no obvious reasoning for

choosing particular values ofaor b.

Fierz and Pauli first demonstrated that the only stable combination combination is

∝h2_−h2

µν [58] and [6] presented a wonderfully clear explanation for why this is the proper

choice and we reproduce the argument here. The GR lagrangian (1.60) enjoys linearized diffeomorphism symmetry, hµν → hµν +∂(µξν), but any mass term Lm(hµν) we add will

generically ruin this. A useful trick is to reintroduce diffeomorphism invariance to the massive theory by introducing St¨uckelbeg fields which restore gauge invariance at the cost of introducing more fields into the theory. Though more fields are added, the total number of degrees of freedom are unchanged as the restored gauge invariance simultaneously removes degrees of freedom.

One patterns the introduction of St¨uckelberg fields after the gauge transformation. That is, everywhere inL_GR(hµν) +Lm(hµν) we make the replacementhµν →hµν+∂(µAν). Since L_GR descends from a curvature invariant, the factors of Aµ in LGR(hµν +∂(µAν)) will

disappear, but L_m(hµν +∂(µAν)) will depend on Aµ. The theory will now contain two

different fields hµν and Aµ, but also gains the gauge symmetry

hµν → hµν+∂(µξν), Aµ→Aµ−ξµ. (1.61)

Similarly, it is fruitful to make the replacementAµ→Aµ+∂µπ. A new fieldπ(x) is added,

but again we gain a U(1) gauge invariance

hµν →hµν+∂µ∂νφ , Aµ→Aµ−∂µφ , π→π+φ . (1.62)

We now explore the physics contained withinL_m(hµν+∂(µAν)+∂µ∂νπ). The physics in the

St¨uckelberg language is equivalent to the physics we started with and so any pathologies of the St¨uckelberged theory also afflict the original theoryL_GR(hµν) +Lm(hµν).

In particular, consider our generic mass term from before, L_{m ≡} ah2_+bh2

µν. Making

the St¨uckelberg replacement, we find that L_m contains

L_m⊃ahπ+bhµν∂µ∂νπ+a(π)2+b∂µ∂νπ∂µ∂νπ . (1.63)

Integrating by parts, we find thatL_{m ⊃}(a+b)(π)2 which corresponds to a ghost in the theory of massm2

ghost∼ (a+1b)2. Removing the ghost entirely corresponds to takinga=−b

and we see that this enforces the mass term to be of the Fierz-Pauli form L_{m ∝}h2₋h2_µν. In order to determine the overall sign, we need to look at the vectors. Focusing on the quadratic vector terms that arise from the St¨uckelberg replacement in L_m =a(h2_−h2

µν), we get L_{m ⊃}a (∂µAµ)2−(∂µAν)2=− 1 2aF 2 µν (1.64)

where Fµν = ∂µAν −∂νAµ is the usual field strength tensor. In order to avoid ghost

instabilities for the vectors we then must have a > 0. Finally, the dispersion relation for

hµν determines that we must normalize to a= 1₂Mpl2m2 if the graviton is to have mass m.

Therefore, the basic requirement that our theory be free from ghosts determines that the mass term take on the Fierz-Pauli form L_m = 1₂M_pl2m2(h2₋hµν)2.

We’ve figured out the correct mass term, but additionally (1.63) exhibits kinetic mix- ing, which ought be removed. Performing integrations by parts we have that L_{m ⊃}

M2 plm2

2 (h−∂µ∂νhµν)π and we can diagonalize by defining hµν =h′µν+ m

2

4 πηµν (a Weyl

rescaling). In particular, we have the relation [71]

L_GR(hµν)∼=LGR(h′µν)− M2 plm2 2 π h ′_−∂ µ∂νh′µν + 3 16M 2 plm4(∂π)2 . (1.65)

and so in terms ofh′_µν andπ we end up with

L_⊃L_GR(h′_µν)₋ 3 16M 2 plm4(∂π)2+ m2 4 πT µ µ . (1.66)

Importantly, the field redefinition has generated a coupling between the scalar and the trace of matter. The relative powers ofmappearing in the kinetic term forπ and inπ’s coupling to matter are what underly the vDVZ discontinuity in which the limitm_→0 fails to recover general relativity. To be precise, when we canonically normalize by sendingπ _∼π/Mˆ plm2

we see that m drops out entirely, i.e. schematically L _{∼ −}(∂πˆ)2 _{+ ˆπT /M}

pl and so the

coupling of the scalar to matter persists even as m_→0, which is the root of the problem. Finally we can consider higher order interactions that arise from the Fierz-Pauli mass term. Before we were working with linearized diffeomorphisms to make the St¨uckelberg replacement, but in order to find the interesting interactions arising from the mass term we need to use the full diffeomorphisms. That is, we insert the vector St¨uckelbergs via

gµν → ∂µ(xα+Aα/2)∂ν(xβ+Aβ/2)gµν and then insert the scalar via Aµ→ Aµ+∂µπ, as

these, the procedure amounts to making the replacement hµν →hµν+∂µ∂νπ+ 1 4∂µ∂απ∂ α_∂ νπ (1.67)

everywhere in the Fierz-Pauli mass term.

Given the discussions in Sec. 1.5.2, the problems now become clear. Schematically, the Fierz-Pauli term gives rise toπ self couplings of the form

L_m∼M_pl2m2(∂2π)3+ (∂2π)4 (1.68) which when canonically normalized as ˆπ _∼Mplm2π read

L_m∼ 1

Λ5₅(∂

2_π)3₊ 1

Λ8₄(∂

2_π)4 _(1.69)

where we define the set of scales Λn ≡ Mplmn−1

1/n

which are monotonically decreasing withn. For applications to our current universe, the graviton mass should be of order the Hubble scale,m_∼H_∼10−42_{GeV, which would make Λ}

5 ∼10−30Gev and Λ3 ∼10−22GeV.

Note that Λ3 is the same order as the strong coupling scale that arises in DGP (1.32).

We then recognize that we are back to the situation found in Sec. 1.5.2 where we saw that classically interactions of the form (1.69) appear to provide a realization of the Vain- shtein mechanism, which in this context would provide a potential solution for the vDVZ discontinuity, but in reality the calculation is swamped by unknown quantum corrections and we cannot actually calculate within the non-linear regime with any control.