无关运算符稳定性的量子效应研究：P(X)与Galileons理论

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本文主要探讨了名为P(X)和Galileons的一类导数理论在量子效应下产生的稳定性的关键问题。P(X)理论是一类涉及动力学项的理论，在高能量物理中被广泛应用，特别是在引力理论中，如Dvali-Gabadadze-Porrati(DGP)模型和Horndeski理论中，它们与Vainshtein机制和依赖于大动力学项的宇宙学模型密切相关，例如在某些类型的 inflationary scenarios中。文章的核心内容使用了精确的重整化群方法（Renormalization Group Approach），这是一种强大的工具，它允许研究在强相互作用区域（即动力学项远大于耦合常数的区域）内，理论在各种量子修正下的行为。作者发现，即使在量子效应下，这类理论仍然保持稳定，所有的回路修正都能在动力学主导的范围内被控制，这对于理论的可行性至关重要。接着，文章强调了对称性在这些理论中的角色。尽管对称性保护了理论的量子修正形式，但理论的稳定性并不单纯取决于对称性的数量。作者通过具体的例子，如Dirac-Born-Infeld(DBI)模型，来证明即使具有额外对称性的理论，其有效性不一定能超越P(X)理论的一般特性。DBI模型在经典解和特定背景场配置中的表现证明了这一点，即使在非标准情况下，其有效性仍然类似于一般P(X)理论。作者进一步讨论了他们的研究结果对不同宇宙阶段的影响。在静态和球对称的screening机制中，研究显示通常的幂律P(X)理论相较于DBI模型，其有效性和适用范围可能更为广泛。这暗示了在某些特定环境下，选择哪种理论可能是更为合适的选择。这篇论文深入探讨了理论物理学中的核心问题，不仅揭示了P(X)和Galileon理论的量子稳定性，还强调了对称性和理论选择在不同宇宙条件下的实际应用。这对于理解这些理论在实际模型中的表现以及预测未来实验结果具有重要的理论指导意义。

JCAP11(2014)016

an operator which is suppressed from the traditional EFT interpretation, but not necessarily

from the perspective of the re-organised EFT, based on the hierarchy between derivatives. If

an operator is important in the re-organised EFT we refer to it as “technically important.”

2.1 Cut-oﬀ versus strong coupling scale

Before we proceed with the computation of the quantum corrections, it is instructive to

recapitulate the concept of regime of validity of the classical ﬁeld theory. In the literature

the diﬀerence between the concepts of cut-oﬀ, Λ

, and that of strong coupling scale, Λ, has

sometimes appeared blurred, and so we will deﬁne them here. We will also need to introduce

the notion of regularisation scale, Λ

, and infrared regulator, κ, which are independent from

both the cut-oﬀ and the strong coupling scale. The only requirement is that Λ

, κ < Λ

and Λ ≤ Λ

By deﬁnition the strong-coupling scale of a theory, Λ, is the scale at which the dominant

interactions arise and it signals the break-down of perturbative tree-level unitarity. In a

standard EFT approach, at this scale the classical solutions are no longer a good description

for the physical system at hand, and quantum corrections (i.e., loops) have to be taken into

account.

However, the breakdown of perturbative unitarity does not necessarily imply the break-

down of unitarity and hence new physics. The later scale is the cutoﬀ of the theory, the

highest scale at which the EFT can be utilised without introducing new heavy physics. The

reason the strong coupling scale and the cut-oﬀ are not necessarily the same is that the

breakdown of perturbative unitarity only indicates the breakdown of perturbation theory. In

a theory with a hermitian Hamiltonian, strongly coupled loop eﬀects may restore unitarity

postponing the true breakdown of the EFT to a higher scale.

The concept of strong coupling scale is thus very distinct from that of cut-oﬀ which

deﬁnes the onset of new physics. The practical implications of identifying the scale Λ depend

on the theory at hand, but the following statements are generically true:

1. In many cases, the strong coupling scale, Λ, coincides with the onset of new physics, in

which case Λ ∼ Λ

2. However, there can also be a hierarchy between Λ and Λ

. At the strong coupling scale,

Λ, diﬀerent scenarios may occur and we highlight that in some of them the theory

may still provide a correct description of the physics at that scale Λ, if Λ  Λ

. In

particular:

(a) In certain cases it is suﬃcient to include a ﬁnite number of loops to restore a good

description of the microphysical processes at that scale (see, for instance, ref. [47]

for an instructive ‘self-healing’ example).

(b) In most cases an inﬁnite number of diagrams contributing at the scale Λ should be

taken into account in order to provide a good description of the physical processes

at that scale. However, this does not mean that the theory necessarily loses

predictivity at the scale Λ. It only signiﬁes that, at that energy, accurate estimates

can only be obtained by applying some resummation technique.

Physical systems where an inﬁnite number of classes of loop diagrams may be

resumed to give ﬁnite results (and sometimes even close to classical results) are well

known and include Bremsstrahlung scattering (vacuum version of the Cherenkov

radiation process) [48]. See also ref. [49] for an example in a nonlinar chiral theory.

– 7 –

JCAP11(2014)016

can prove that there is no possible converging resummation, then the theory loses

predictivity at the scale Λ, at least from a standard EFT viewpoint.

Any theory which relies on irrelevant operators to make classical predictions and exhibits

a Vainshtein or screening mechanism must lie within the second set of possibilities, namely

Λ  Λ

. In the past decade, there has been a large interest in models where the strong

coupling scale, Λ, gets redressed by a large background ﬁeld conﬁguration. If this redressing

is to make sense, it is crucial to diﬀerentiate between Λ and Λ

We conclude this small detour by noting that whilst the estimate of the cut-oﬀ energy

scale of the theory can be sometimes ambiguous (since it may be diﬃcult to determine the

scale at which other ﬁelds ought to be included in the action without knowing the details of

the UV completion of the theory), the strong coupling scale is somewhat easier to assess. It

may indeed vary from the usual method in which one identiﬁes the energy scale contributing

in the perturbative expansion of scattering amplitudes in terms of Feynman diagrams. As we

mentioned before, this happens in cases where a strongly self-interacting background implies

a redressing of the interactions, which sometimes has the eﬀect of raising the naive strong

coupling scale [50]. Given these possible ambiguities, our principal goal is to obtain results

which are explicitly independent of the cut-oﬀ of the theory, Λ

, which should render them

physically trustworthy.

2.2 Cut-oﬀ dependence and the Wilson action

Divergencies in loops appear in the form of power-laws and logarithms. The central reason

for why power-law divergences should not necessarily be trusted as an indication of loop

corrections from UV physics, is that the eﬀective action, which controls the physically renor-

malised quantities, is by deﬁnition independent of power-law divergences (see, for example,

ref. [51]). To understand this we brieﬂy review the Wilsonian picture to renormalisation.

Given a ﬁeld theory for φ we deﬁne the Wilsonian action S

(φ) by integrating out all

modes in the path integral whose momenta are larger than some Λ

, which is the regulator

scale. This can be accomplished by splitting the ﬁelds into light and heavy modes, and then

the Wilsonian action, S

(φ), only depends on the modes lighter than Λ

. We must perform

this computation in Euclidean signature, which we will keep throughout the remaining of

this manuscript.

Universal prediction from the logarithmic term. The Wilson action is given by

−S

(φ)

k≥Λ

D[φ] e

−S(φ)

. (2.3)

By construction this action is strongly dependent on the chosen regulator scale Λ

. In

particular, at one-loop we expect contributions to S

(φ) which are quartic and quadratic in

. This scale may be chosen arbitrarily and need not be related with the strong coupling

scale, Λ, nor the cutoﬀ, Λ

. However, on the basis of the discussion in section 2.1, we do

require that Λ

≤ Λ

so that the integral on the right hand side is meaningful.

We can then deﬁne the Wilson action at another arbitrarily chosen scale Λ

< Λ

via

the ﬁnite integral

−S

(φ)

≤k≤Λ

D[φ] e

−S

(φ)

. (2.4)

– 8 –

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无关运算符稳定性的量子效应研究：P(X)与Galileons理论

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