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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 906 (2016) 105–132

www.elsevier.com/locate/nuclphysb

Constraining the effective action by a method of

external sources

Björn Garbrecht

, Peter Millington

a,b,∗

Physik-Department T70, James-Franck-Straße, Technische Universität München, 85748 Garching, Germany

School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom

Received 25

January 2016; accepted 21 February 2016

Available

online 3 March 2016

Editor: Stephan

Stieberger

Abstract

propose a novel method of evaluating the effective action, wherein the physical one- and two-point

functions are obtained in the limit of non-vanishing external sources. We illustrate the self-consistency of

this method by recovering the usual 2PI effective action due to Cornwall, Jackiw and Tomboulis, differ-

ing

only by the fact that the saddle-point evaluation of the path integral is performed along the extremal

quantum, rather than classical, path. As such, this approach is of particular relevance to situations where

the dominant quantum and classical paths are non-perturbatively far away from one-another. A pertinent

example is the decay of false vacua in radiatively-generated potentials, as may occur for the electroweak

vacuum of the Standard Model. In addition, we describe how the external sources may instead be chosen

so as to yield the two-particle-point-irreducible (2PPI) effective action of Coppens and Verschelde. Finally,

in the spirit of the symmetry-improved effective action of Pilaftsis and Teresi, we give an example of how

the external sources can be used to preserve global symmetries in truncations of the 2PI effective action.

Speciﬁcally, in the context of an O(2) model with spontaneous symmetry breaking, we show that this ap-

proach

allows the Hartree–Fock approximation to be re-organized, such that the Goldstone boson remains

massless algebraically in the symmetry-broken phase and we obtain the correct second-order thermal phase

transition.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Corresponding author at: School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD,

United Kingdom.

E-mail

addresses: garbrecht@tum.de (B. Garbrecht), p.millington@nottingham.ac.uk (P. Millington).

http://dx.doi.org/10.1016/j.nuclphysb.2016.02.022

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

106 B. Garbrecht, P. Millington / Nuclear Physics B 906 (2016) 105–132

1. Introduction

The use of effective-action techniques has become ubiquitous across theoretical physics, both

in the relativistic regime of high-energy processes and the non-relativistic setting of condensed

matter systems. Such techniques play a signiﬁcant role in the study of both perturbative and

non-perturbative effects, including phase transitions, transport phenomena and renormalization

group evolution.

The functional e

valuation of the one-particle-irreducible (1PI) effective action was ﬁrst de-

scribed by Jackiw [1] and subsequently generalized to nPI by Cornwall, Jackiw and Tomboulis

(CJT) [2]. The effective action provides a systematic means for obtaining the quantum equations

of motion for n-point correlation functions, which automatically resum inﬁnite sets of diagrams.

wever, in order to make the solution of these systems of equations tractable, we must, in

reality, ﬁnd consistent truncation schemes that preserve the underlying symmetries, and much

attention has been given to this in the literature.

It is well kno

wn that truncations of the 2PI effective action do not, in general, preserve global

and local symmetries of the effective action [3,4] due to higher-order effects. The reason for

this can be understood heuristically as follows [5]: the satisfaction of symmetry identities, such

as the Ward–Takahashi identities [6,7], in the case of global and Abelian g

auge theories, or the

Slavnov–Taylor identities [8,9], in the case of non-Abelian gauge theories, requires the can-

cellation of diagrams of different topologies. However, once the 2PI effective action has been

truncated at some ﬁnite order in the loop expansion, only a subset of all topologies are re-

summed, and the required cancellation is no longer e

xact. In the case of global symmetries, such

as O(N) models with spontaneous symmetry breaking (SSB), this problem manifests in the vio-

lation of Goldstone’s theorem [10,11], with the Goldstone bosons acquiring non-zero masses in

the SSB phase [12–15]. A number of authors ha

ve proposed solutions to this problem [14–21].

These include the so-called external-propagator method [22,23], Optimized Perturbation Theory

(OPT) (see e.g. Refs. [24,25]) and the symmetry-improved CJT effective action [26] of Pilaftsis

and Teresi (PT), in which the Ward identities are imposed through the method of Lagrange mul-

tipliers. The latter variant of the ef

fective action has the advantage that, aside from ensuring the

masslessness of the Goldstone boson in the SSB phase and the correct second-order phase tran-

sition [27], it also yields the correct decay thresholds for both the Higgs and Goldstone modes.

In addition, this approach has been shown to be free of the problem of IR di

vergences [28,29]

that arise as a result of the massless Goldstone bosons. In the case of QED, truncation of the 2PI

effective action leads to violation of the Ward–Takahashi identities, and the transversality of the

photon polarization cannot be guaranteed [30].

Once embedded in the Schwinger–K

eldysh closed-time-path (CTP) formalism [31,32] of non-

equilibrium ﬁeld theory (see also Refs. [33–37]), the CJT effective action allows the derivation

of quantum transport equations by means of the Kadanoff–Baym formalism [38,39] (see also

Refs. [40–44]). In recent years, these approaches have received a wealth of interest, not least

in applications to the e

volution of number densities in the early Universe. In addition, substan-

tial progress has been made in the non-perturbative renormalization of the effective action both

at zero and ﬁnite temperature [23,45–50]. In such cases, the physical limit of the effective ac-

tion is obtained with non-vanishing external sources, where these encode information about the

statistical ensemble of the system (see e.g.

Ref. [36] and also Refs. [43,44]).

The d

iscovery of a ∼ 125.5 GeV Higgs boson [51–53] has brought into question the sta-

bility of the electroweak vacuum of the Standard Model [54–57]. At present, state-of-the-art

calculations [58–62] (for a recent overview, see Ref. [63]) suggest that the electroweak vacuum

B. Garbrecht, P. Millington / Nuclear Physics B 906 (2016) 105–132 107

is metastable, having a lifetime longer than the current age of the Universe. However, these

estimates are subject to a number of uncertainties. These include the current precision on the ex-

perimental determination of the top-quark mass [64,65] and the potential impact of new physics

at high scales [66–71]. In addition, the standard RG impro

vement of the effective potential has

recently been challenged [72]. Finally, it remains an open question as to the correct method by

which to determine the tunneling rate when the global minimum of the potential is generated

radiatively [73]. This latter issue is relevant also to the Coleman–Weinberg scenario of SSB [74],

as well as symmetry restoration at ﬁnite temperature [75–77]. In these cases, the e

xtremal quan-

tum path of the system is non-perturbatively far away from the extremal classical path,

and

it is therefore necessary to ﬁnd ways of accounting consistently for the impact of these non-

perturbative effects on the structure of the path integral itself. It is this observation that was the

original motivation for this work.

The content of this article is as follo

ws. In section 2, we outline the method of evaluating the

effective action in which the physical limit is obtained in the presence of non-vanishing external

sources in vacuum. This is in stark contrast to the standard approach, where the physical limit

instead corresponds to vanishing e

xternal sources. The external sources may then be used to

constrain the effective action, and we will consider three concrete examples:

(i) In section 3, we choose the e

xternal sources such that the theory is forced along the extremal

quantum path, recovering the 2PI CJT effective action [2]. Subsequently, we discuss the

relevance of this method to the problem of vacuum decay in potentials where the global

minimum emerges only as a result of radiati

ve corrections.

(ii) In section 4, we sho

w how the external sources may be chosen so as to recover the two-

particle-point-irreducible (2PPI) effective action of Coppens and Verschelde (CV) [78–81],

which resums only local self-energy insertions. In addition, we will see that this method

of external sources will allow us to avoid the problem of double-counting diagrams in the

resummation without the need to isolate terms in the ef

fective action artiﬁcially.

(iii) In section 5, we consider the Hartree–F

ock approximation [82,83] in the case of a glob-

ally O(2)-invariant model with SSB. Therein, we obtain results in the spirit of the PT

symmetry-improved effective action [26,28,29] and show that the Ward identity may be

used to constrain the external sources so as to ensure that the Goldstone boson remains

massless algebraically in the SSB phase. In addition, we will also sho

w that this approach

yields the correct second-order thermal phase transition, making a strong case for the utility

of this novel method.

Finally, our conclusions and potential future directions are presented in section 6.

2. Method of external sources

In this section, we describe an alternative method of evaluating the effective action in which

the physical limit is obtained for non-vanishing external sources. We will make comparison with

the standard approach, where the physical limit is instead obtained for vanishing external sources.

For the purposes of illustration, it will sufﬁce to truncate the ef

fective action at the level of

In view of potential applications to vacuum metastability, we imply here that the most probable exit paths that are

quantum in Minkowski space can be treated as classical in Euclidean space.

108 B. Garbrecht, P. Millington / Nuclear Physics B 906 (2016) 105–132

the bi-local source. Nevertheless, this approach generalizes straightforwardly to the inclusion of

higher poly-local sources. Furthermore, in order to present the diagrammatic structures explicitly,

we will work at the two-loop level, truncating the effective action at order

. The expansion at

higher-loop orders will proceed analogously as per the general arguments that apply to the 2PI

and 2PPI effective actions, respectively.

Throughout this article, we will wo

rk in four-dimensional Euclidean spacetime. The results

derived, however, apply equally in Minkowski spacetime. Finally, for deﬁniteness, we consider

the λ

theory, with the Euclidean Lagrangian density



∂



λ

, (1)

where 

≡ (x) is a real scalar ﬁeld of mass m, ∂

≡ ∂/∂x

denotes the partial derivative

with respect to the four-dimensional Euclidean spacetime coordinate x

= (x, x

), and λ is a

dimensionless coupling.

In the presence of both a local and a bi-local source, the ef

fective action [φ, ] is deﬁned as

the Legendre transform [2]

[φ,]≡max

J,K



J,K

[φ,]

=max

J,K



−

h ln Z[J,K]+J



h





, (2)

subject to the extremization

δ

J,K

[φ,]

δJ



J =J ,K=K

= 0 , (3a)

δ

J,K

[φ,]

δK



J =J ,K=K

= 0 , (3b)

where J

and K

are the extremal sources, which we will hereafter refer to as the external

sources. The generating functional Z[J, K] has the form

Z[J,K]=



[d] exp



−



S[]−J



−





, (4)

in which S[] is the classical Euclidean action. Throughout this article, we employ the DeWitt

notation, in which continuous indices are integrated over, i.e.

≡



x J (x) φ(x) . (5)

On performing the extremization in eq. (2), we obtain the following expression for the effec-

tive action:

[φ,]=−

h ln Z[J , K]+J

[φ,]φ

[φ,]



h



. (6)

The independent one- and two-point functions, φ

and 

, are given by

δ ln Z[J,K]

δJ



J =J ,K=K

, (7a)



= 2

δ ln Z[J,K]

δK



J =J ,K=K

−

. (7b)

B. Garbrecht, P. Millington / Nuclear Physics B 906 (2016) 105–132 109

Note that the sources J

and K

are both independent of φ

and 

, whereas the external

sources J

≡ J

[φ, ] and K

≡ K

[φ, ] are both functionals of φ

and 

. As a result,

and 

are both functionals of J

and K

, but in such a way that they remain independent

of one another. Finally, by means of eq. (7), we may show that

δ[φ,]

δφ

= J

[φ,]+K

[φ,]φ

, (8a)

δ[φ,]

δ

[φ,] . (8b)

In the standard approach, eqs. (8a) and (8b) deliver the quantum equations of motion for the

physical one- and two-point functions when the external sources appearing on the right-hand

sides are set to zero. It is in this point and in the way that physical limits will be taken that the

present approach will dif

fer.

Before continuing, we remark that the local source J

is a real-valued functional. On the other

hand, the bi-local source K

is, in general, a complex-valued functional. However, it must satisfy

the Hermiticity condition K

∗

= K

, ensuring that the exponent of the generating functional

is real. It follows that the effective action in eq. (6) is a real-valued function for Z[J, K] ≥

0, acquiring a non-vanishing imaginary part iff Z[J, K] < 0. The latter situation arises in the

context of false vacuum decay, as we will discuss in the follo

wing section.

In order to recast the ef

fective action [eq. (6)] in a more useful form, we wish to eliminate

the explicit appearance of the external sources J

and K

. We must therefore ﬁrst perform the

functional integral in the generating functional Z[J , K]. This can be achieved by expanding

around the saddle-point ϕ

, which satisﬁes the source-dependent stationarity condition

δS[]

δ



 =ϕ

− J

[φ,]−K

[φ,]ϕ

= 0 . (9)

Note that the external sources J

and K

, appearing in eq. (9), are evaluated at the conﬁgura-

tions φ

and 

. By writing 

=ϕ

1/2



, where the factor of

1/2

is included explicitly

for bookkeeping purposes, the exponent of the generating functional can be expanded as follows:

S[]−J

[φ,]

−

[φ,]



= S[ϕ]−J

[φ,]ϕ

−

[φ,]ϕ



−1

[φ,]



3/2

λϕ



, (10)

in which there are no terms linear in



thanks to the stationarity condition in eq. (9). In eq. (10),

we have deﬁned the inverse two-point function

−1

[φ,]=G

−1

(ϕ) − K

[φ,] , (11)

where

−1

(ϕ) ≡

S[]

δ



 =ϕ

S[ϕ]

δϕ

= δ

(4)



−∂

+ m



, (12)

and δ

(4)

≡ δ

(4)

(x − y) is the four-dimensional Dirac delta function. For now, we shall assume

that the spectrum of the operator G

−1

is positive-deﬁnite, i.e. m

> 0. We will, however, return

to this assumption later in the context of false vacuum decay in section 3, when we will instead

take m

< 0.

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