Published for SISSA by Springer
Received: August 17, 2016
Revised: October 17, 2016 Accepted: November 7, 2016
Published: November 18, 2016
Aspects of Galileon non-renormalization
Garrett Goon,a Kurt Hinterbichler,b Austin Joycec and Mark Troddend
aDepartment of Applied Mathematics and Theoretical Physics, Cambridge University, Wilberforce Road, Cambridge, CB3 0WA, U.K.
bPerimeter Institute for Theoretical Physics,
31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
cEnrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago,S. Ellis Avenue, Chicago, IL 60637, U.S.A.
dCenter for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania,S. 33rd Street, Philadelphia, PA 19104, U.S.A.
E-mail: mailto:[email protected]
Web End [email protected] , [email protected], mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected]
Abstract: We discuss non-renormalization theorems applying to galileon eld theories and their generalizations. Galileon theories are similar in many respects to other derivatively coupled e ective eld theories, including general relativity and P (X) theories. In particular, these other theories also enjoy versions of non-renormalization theorems that protect certain operators against corrections from self-loops. However, we argue that the galileons are distinguished by the fact that they are not renormalized even by loops of other heavy elds whose couplings respect the galileon symmetry.
Keywords: E ective eld theories, Global Symmetries, Scattering Amplitudes
ArXiv ePrint: 1606.02295
JHEP11(2016)100
Open Access, c
The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2016)100
Web End =10.1007/JHEP11(2016)100
Contents
1 Introduction 1
2 Review of Galileons and their non-renormalization 32.1 The non-renormalization theorem 4
3 Non-renormalization and power counting 53.1 General power counting 53.2 Galileons 73.3 General relativity 83.4 P (X) theories 93.5 Conformal dilaton 10
4 Coupling to heavy elds 114.1 Integrating out elds via functional determinants 134.2 Galileons 154.3 General relativity 164.4 P (X) theories 174.4.1 T coupling 174.4.2 DBI 184.5 Conformal dilaton 20
5 Conclusions 21
A Evaluating functional determinants 22
1 Introduction
The galileons [1] are a fascinating class of higher-derivative scalar e ective eld theories which display rich and varied structure and phenomenology. They have elegant geometrical origins as the description of brane uctuations in the DGP model [2, 3] (further elaborated upon in [47]), describe the helicity zero mode of a ghost-free interacting massive spin-2 eld [8, 9], are the key players in interesting IR modications of GR which display Vainshtein screening [10, 11] near heavy objects [1, 12], and possess an S-matrix with many special properties [1317].1
1The galileons also possess unusual features: for solutions around heavy sources, perturbations can propagate superluminally [1, 18] (though this can be alleviated in other examples [19, 20]), and, treated in isolation, there are arguments that galileons have no local, Lorentz invariant UV completion [21, 22] (however, when incorporated into full massive gravity, these obstructions are lifted in some cases [23]). See [2429] for interpretations of these features in terms of non-standard UV completions.
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In this paper we focus on another property of galileons: their non-renormalization theorem. Certain galileon operators are not renormalized by galileon loops [3, 30]. We are interested in understanding both the importance of this fact and how it compares to other, supercially similar, non-renormalization theorems obeyed by other e ective eld theories.
The simplest example of a galileon is a single scalar eld, (x), which obeys a shift symmetry linear in coordinates,
(x) 7(x) + c + bx , (1.1)
with c, b constant. Any term built out of , and its derivatives, will be strictly invariant under (1.1). However, there also exist special operators with fewer than two derivatives per , which are not strictly invariant, but rather are invariant up to a total derivative. The cubic galileon interaction is the canonical term of this type
Scubic =Z d4x
1 3 ()2 , (1.2)
with some strong coupling scale. These special operators are reviewed in section 2.1.
The statement of the non-renormalization theorem is that loops of galileon elds only serve to renormalize the higher derivative operators built from . For example,
in (1.2) doesnt run. This is in accord with general folklore stating that terms invariant only up to a total derivative are typically protected in some way against quantum corrections. Examples are the Wess-Zumino-Witten (WZW) term [31] in the chiral Lagrangian and Chern-Simons terms in three dimensional Yang-Mills theory [32] whose coe cients dont run and, further, are quantized [31, 33]. The special galileon operators to which the nonrenormalization theorem applies are, in fact, an analogue of the WZW term [7], albeit there is no argument to suggest their coe cients should be quantized.
One puzzle is that the theorem is simultaneously non-trivial and trivial, in some sense. It is non-trivial in that there exists a diagrammatic proof of the theorem [30] which heavily relies on the detailed structure of the special galileon operators. It is trivial in that the same conclusions also essentially follow from dimensional analysis arguments applied to self-loop graphs in dimensional regularization, with no reference to the detailed form of the galileon operators. These dimensional analysis arguments can be made for many other massless, derivatively-coupled theories including General Relativity (GR), P (X) theories and the conformal dilaton eld (alternatively known as the conformal galileon [1, 34]). Certain low-dimension operators in these other theories are also not renormalized by self loops.
What, then, is the non-trivial content of the galileon renormalization theorem? Are the renormalization properties of galileons qualitatively di erent from those of GR, P (X) and conformal dilaton, or do they follow similarly from the derivative expansion of e ective eld theory?
In this paper, we argue that the essential di erence comes once we consider loops of heavy elds which couple to the galileon, (x). The galileon renormalization theorem implies that the special galileon operators are not renormalized even by heavy elds provided that they couple in a way which respects the galileon symmetry. This e ect is not visible by considering only self-loops in dimensional regularization, which captures
2
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1
2()2
only logarithmic corrections. It is, loosely speaking, captured by power divergences in graphs. The detailed diagrammatic proof of the galileon renormalization theorem tells us that the entire quantum contribution to the galileon vanishes, including power corrections. This suggests that coupling heavy elds to the galileon should not renormalize the galileon operators, and indeed this is what we will nd in explicit examples. In contrast, coupling heavy elds to GR, P (X) and the conformal dilaton do a ect the operators in these theories which are not renormalized by the logarithmic part of self loops. In this precise sense the galileon non-renormalization theorem is stronger.
In section 2 we review galileon theories in more detail and discuss the detailed version of the non-renormalization theorem. In section 3 we review how the non-renormalization theorem follows from dimensional analysis, apply the same arguments to other theories, and discuss the motivations behind our above statements. In section 4 we illustrate how heavy physics a ects the operators in these theories by coupling in a massive scalar eld and integrating it out. The methods used for integrating out the heavy eld are summarized in appendix A, and we conclude in section 5.
Conventions. Throughout we use mostly plus metric signature. We denote the at-space dAlembert operator by 2 .
2 Review of Galileons and their non-renormalization
In this section we briey review some of the basic properties of the galileon and the nonrenormalization theorem. A galileon scalar eld, (x), has an action which is invariant under the extended shift symmetry (1.1). In order for the action to be invariant under this symmetry, the interaction terms must involve derivatives. Many of the operators invariant under (1.1), powers of 2 for example, will lead to higher order equations of motion (EOM) and hence will generically run afoul of Ostrogradskis theorem [35], leading to instabilities (see [36, 37] for nice reviews). These instabilities are not problematic as long as the theory is treated as an e ective eld theory (EFT) [3840].
Interestingly, not all operators invariant under the galileon symmetry are of this type; there exist a nite number of operators which have fewer than two derivatives per eld and thus are not constructed from the invariant building block . In addition, they yield strictly second order equations of motion. The existence of such operators opens up a regime in which we can reliably study the non-linear, classical phenomena dictated by these terms, while consistently ignoring the e ects of the higher derivative operators discussed in the preceding paragraph [12, 41]. This can be thought of in analogy with Einstein gravity: there is a regime where classical non-linearities are important, for example in the vicinity of the event horizon of a black hole, while quantum mechanical phenomena are (at least from the point of view of local e ective eld theory) expected to be unimportant.
In d spacetime dimensions, there are d + 1 of these special operators, all of which change by a total derivative under (1.1). In four dimensions, they take the form [1]
L1 =
L2 = ()2
L3 = 2()2
3
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L4 = ()2 (2)2 ()2 L5 = ()2 (2)3 + 2()3 32()2
. (2.1)
These operators can be compactly written using the Levi-Civita symbol, which makes many of their properties manifest,
Ln 1n1n41n1n411 n1n1 . (2.2)
The anti-symmetric structure of the epsilons guarantees that having two derivatives with either a or index acting on a vanishes, making it easy to see that the galileon terms have second order equations of motion and shift under the symmetry (1.1) by a total derivative.
For the remainder of the paper, we will follow common conventions and refer to the special terms in (2.1) alone as galileons. All other terms compatible with the galileon symmetry will simply be called higher order operators.
2.1 The non-renormalization theorem
Loops of elds dont renormalize the galileon interactions (2.2) at any order in perturbation theory. This was rst noted for the cubic galileon theory in [3] and then extended to the fully general case in [30]. The argument of [30] is phrased in terms of the 1PI action and is diagrammatic. We present a simple path integral version of the same argument here. The detailed form of the galileon interactions is crucial to each of these arguments.
Consider calculating the 1PI e ective action via the background eld method [42] for a galileon Lagrangian, L() = P5
i=1 ciLi with Li as in (2.1) (higher order operators with
more derivatives can also be added to the action without altering the conclusions of the following argument). The e ective action, [
], for an arbitrary eld prole
(x) can be
derived by taking the bare Lagrangian L(), expanding the eld about the background
=
+ and path integrating over the uctuation keeping only bubble diagrams which are 1PI with respect to uctuation lines:
exp i [
] =
Z1PI D exp iS[
, (2.4)
we can then integrate the 1 derivative by parts, to turn this operator into
L3( + ) 12341234 11 22
. (2.5)
It is clear that this argument will generalize to all of the galileon operators; note that this relies crucially on the particular structure of these terms.
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+ ] . (2.3)
We would like to verify that there are no corrections to the coe cients of the galileon operators (2.1). We perform the replacement =
+ in the galileon operators (2.1).
When written in terms of tensors, it is immediately clear that we can always integrate the result by parts so that every
factor has exactly two derivatives acting upon it. For instance, making the replacement in the cubic operator we nd that L3 generates terms of
the form
L3( + ) 12341234 1 122
In the resulting path integral (2.3), S[
+ ] thus contains terms involving with either zero, one or two derivatives acting upon it, but it depends on
strictly through the
] will be build from propagators
and derivatives thereof. All generated terms must therefore have at least two derivatives per
, while the galileons (2.1) have fewer than this, hence the galileons are not renormalized by self loops.2 Similar arguments hold in scalar-tensor generalizations of the galileon [43], and underlie the technical naturalness of ghost-free massive gravity [44].
3 Non-renormalization and power counting
We now argue that the non-renormalization theorem as previously stated also follows as a simple statement about power counting in e ective eld theory, and that essentially all derivatively-coupled theories enjoy a similar non-renormalization for their leading operators.
3.1 General power counting
To make invariant statements about non-renormalization in EFTs, we will want to be able to estimate the way that various diagrams scale with the external momenta of particles. These estimates will allow us to quickly check whether an operator can be renormalized by a loop diagram.
We will want to make estimates of the scaling of observables in theories of the form
Le = 4
Xjcj fj+dj Oj (, ) . (3.1)
Here Oj (, ) stands for any operator built out of and derivatives thereof, cj are order
one dimensionless coe cients, and fj and dj count the number of elds and derivatives appearing in Oj, respectively. In (3.1), we have assumed that only one scale enters
the Lagrangian, for simplicity. The extension to multiple scales is straightforward, but unnecessary for our interests. Throughout the body of this paper, we will only discuss massless theories (apart from a short discussion in the conclusions of how the following estimates and results change when the theory is massive).
In the theory (3.1), the momentum dependence of an N-point scattering amplitude3 (or o -shell amputated correlator) M(N), can easily be estimated, following [45]. The overall
mass dimension of the amplitude is 4 N, every loop integral leads to an integration R
1k2 (we only consider bosonic theories). The only other way factors of momenta can appear is through derivatives in the interaction terms, Oj (, ). Denoting the number of loops in the diagram by L, the number of
internal lines by I, and the number of vertices with i lines and n derivatives by V(i,n), we
2Because the galileon is massless, is expected to have non-local terms involving objects like log @2 and @2, so one might worry that powers of inverse @2s could somehow cancel out the derivatives acting on
s. The dimensional analysis arguments of the next section ensure that this doesnt happen.
3The special case of N = 2 can be thought of as the amputated vacuum polarization diagram.
5
combination 2
. Path integrating over , the result [
and vertices which depend on
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d4k, and every internal line contributes
[summationtext]i,n nV(i,n). Dividing by powers of to ensure the correct dimension, we obtain the estimate
M(N) 4N
k
nd that the amplitude scales as k4L2I+
4L2I+[summationtext]i,n nV(i,n), (3.2)
where k represents some combination of the external momenta. The result (3.2) can simplied somewhat with the use of simple graph-theoretic identities [45]. First, the number of internal and external lines are related via
N + 2I =Xi,niV(i,n). (3.3)
Similarly, the number of internal lines is related to the number of loops in the graph via
L = 1 + I X
i,n
V(i,n) . (3.4)
It is convenient to use (3.4) to eliminate I from (3.2), which leads to the nal power-counting estimate for M(N),
M(N)(k) 4N
k
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2L+2+[summationtext]in(n2)V(i,n), (3.5)
which depends only on the number of vertices (and the number of derivatives contained therein) and the number of loops. Using (3.5), it is easy to check what types of diagrams can renormalize coe cients in the Lagrangian, as we will see in the following sections.
The formula (3.5) requires two important comments:
In writing (3.2), we have implicitly assumed that we are using dimensional regular
ization or some other mass-independent regularization scheme. This ensures that the only scales which can emerge from loop integrals correspond to factors of external momenta. Had we instead used a mass-dependent regularization such as Pauli-Villars or a cuto , then factors of the regularization mass scale UV would also appear in (3.2).
Physical results are of course regulator independent, so nothing essential is lost by using a mass-independent scheme. Thus, unless stated otherwise, we use dimensional regularization for all calculations.
Logarithmic factors log(k2/2), with the regularization scale, are not captured by
the power counting estimate. Therefore, one should think of (3.5) as also potentially containing logarithmic factors when the diagrams involve loops. Dependence on the regularization scale is only through these logarithmic factors.
We now use this power-counting estimate (3.5) to explore the behavior of various derivatively-coupled EFTs.
6
k6 k12 k12 k12 k12
Figure 1. The various 2 2 diagrams built solely from galileon operators (2.1), up to one loop,
and their scaling with external momenta. It is clear that the loop diagrams contribute at higher orders in momenta than the tree amplitude.
3.2 Galileons
First, we can re-derive the non-renormalization theorem for the galileon in this language. Consider a galileon scattering process with N external legs. Both the galileon operators and higher derivative terms m(2)n are included in the action.
Start by examining tree diagrams built solely from vertices drawn from the special galileon terms (2.1). The operator with i elds has 2i2 derivatives, meaning that P
2)V(i,n) = 2(N 2), as follows from the topological relations (3.3) and (3.4) evaluated at
L = 0. Therefore,
M(N)gal. tree 4N
Now, we apply the estimate (3.5) to all possible loop diagrams with N external legs and attempt to build anything with the scaling (3.6), corresponding to a renormalization of the galileon operators. We will nd that this is not possible. It is easy to check that loop diagrams built only from galileon vertices cannot renormalize the original operators. Using (3.3) and (3.4) for a diagram with L loops, we now have
Pi,n(n 2)V(i,n) = 2(N +
2(N1)+6L. (3.7)
An L-loop diagram thus carries 6L more powers of k than the original tree diagrams, and therefore none of these graphs can renormalize the original interactions. Further, evaluating (3.7) at L = 1, it can be deduced that the higher derivative operators of the form (2)n or (2)n are not renormalized by loops of , either. This is in agreement with explicit computations of the 1PI e ective action [3, 12].
If we also use higher derivative vertices, then a vertex with i external legs has at least 2i 2 derivatives. This changes the relevant sum to a lower bound P
i,n(n 2)V(i,n)
2(N +2L2). Such diagrams therefore have at least as many powers of k as (3.7), meaning
that they also will not renormalize the galileon operators.
The above analysis is merely a formalization of the intuition that quickly becomes obvious when one draws diagrams. Consider 2 2 scattering to one-loop using only
galileon operators, as shown in gure 1. The tree diagrams built from L4 ()2(2)2
or two insertions of L3 ()22 clearly scale as k6, while the loop diagrams can
be easily estimated to scale as k12. Thus, the loops cannot renormalize the tree-level
contribution. (See also [13] for galileon power counting.)
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i,n(n
k
2(N1). (3.6)
2L 2), yielding the estimate
M(N)gal. loop 4N
k
Note that in contrast to section 2.1 nowhere in the preceding argument did we have to make any use of the detailed structure of the galileon interactions. Instead, the result just follows from the fact that there are certain numbers of derivatives per and that the galileon is massless.4 In fact, as we will see next, the galileon is not even the unique theory which has a non-renormalization theorem of this type, it is a generic property of derivatively coupled theories.
3.3 General relativity
As a rst example, consider calculating graviton scattering diagrams in pure Einstein gravity with the cosmological constant tuned to zero,
S = M2Pl 2
Z d4xgR + , (3.8)
where contains higher order operators mRn. Perturbing the metric about at space
as g = + 1
MP h, the Einstein-Hilbert term is of the schematic form
S Z d4x
Each interaction vertex now has exactly two derivatives, so that a tree diagram with N external legs built from Einstein-Hilbert vertices has the scaling
M(N) M4NPl
MPl now plays the role of . In comparison, an N-point, L-loop diagram built from the Einstein-Hilbert terms scales as5
M(N) M4NPl
Building loops using vertices drawn from the higher order operators contained in the
in (3.8) only increases the scaling with k.
Therefore, we see that loops cannot correct the Einstein-Hilbert vertices: the Planck mass is not renormalized in pure at-space GR. Additionally, graviton loops will not cause the cosmological constant to be renormalized and so there is no cosmological constant problem in pure GR. These statements have long been known6 [46, 4850].
4If the eld were massive then some of the ks in (3.5) could correspond to factors of mass m instead of external momenta and the above analysis would not be guaranteed to work. This is further discussed in the conclusions.
5Precisely the same estimate appears in DeWitts early paper on quantum gravity [46].
6At one-loop, the only counterterms needed are those proportional to R2 and to R2 (the other dimension-4 counterterm, proportional to R2 , is degenerate with the others by the Gauss-Bonnet theorem). This is the origin of the statement that pure GR is one-loop nite in four dimensions: the divergences all correspond to redundant operators which can be eld redened away and hence do not contribute to the S-matrix. The two-loop calculation was performed in [47], where it was found that a non-redundant counterterm R R R is required. The scaling of all of these results agrees with the estimate (3.11).
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1
Xn=0
h
MPl
n(h)2. (3.9)
k
MPl
2. (3.10)
k
MPl
2+2L. (3.11)
3.4 P (X) theories
As the next example, consider the e ective eld theory of a scalar with derivative self-interactions which have at most one derivative per eld. Specically we consider Lagrangians of the form L = 4P (X), where X 12()2/ 4 and P is an arbitrary
function. Theories of this type can be considered the leading terms in a derivative expansion of theories which possess a shift symmetry 7 + c. Consequently, they arise
in myriad places, perhaps most famously as the EFT of the Nambu-Goldstone mode for a complex scalar with a symmetry breaking Mexican hat potential. P (X) models have been used extensively in theoretical cosmology, both in K-ination models [51, 52], and to drive late-time acceleration in K-essence models [5355]. The ghost condensate [56] is another example in this class. In string theory, the scalar part of the action for D-branes is a special case of a P (X) theory the Dirac-Born-Infeld (DBI) model [57]. The DBI model enjoys an enhanced symmetry, = x +, and has been used in various cosmologies [5861]. Shift-symmetric models have also found application in condensed matter settings, the pure P (X) theories we consider describe the e ective action of superuids [62, 63] and suitable multi-eld generalizations can describe general uids [64, 65].
All operators in a P (X) Lagrangian will have at least one derivative per and, after su cient integrating by parts, can be made strictly invariant under the shift symmetry; there are no analogues of the galileon operators (2.1) for P (X) theories (apart from the trivial tadpole term L ).
A generic P (X) action can be written as a Taylor series7
S =
Z d4x 4
N+4L, (3.13)
where we have used the fact that all the operators of interest have one derivative per eld, so
Pi,n(n 2)V(i,n) = Pi,n(i 2)V(i,n) = N + 2L 2, after application of (3.3) and (3.4).
We therefore see that the contribution from loops to a given N-point amplitude is suppressed by positive powers of k/ , so the leading momentum contribution to a given amplitude is not corrected by loops. This is another example of non-renormalization: the cn coe cients in (3.12) cannot be changed by loops, because this would require correcting the tree level amplitudes, and loop contributions have too many powers of k to do so.
Note that no part of this argument relies upon the precise form of P (X), or equivalently the precise relations between the various cn. We therefore see that an arbitrary function
P is radiatively stable in this sense. In fact, because each loop adds a factor of k4 to the amplitude, we can actually further deduce that if we added operators of the schematic form
7Assuming that P (X) is an analytic function of X, which will be the case if Minkowski is a sensible vacuum of the theory. There are interesting examples where this assumption fails, e.g., [66, 67].
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1
Xn=1cnXn + , (3.12)
where contains terms with more derivatives per . Using (3.5), we nd that a diagram
with N external legs built from the operators in (3.12) scales as
M(N) 4N
k
L Xn to the action, these would not be renormalized by loops of , either. This is in
accord with the results of [68], who argued that an arbitrary functional form for P (X) is radiatively stable when considering self-loops and tracking only logarithmic divergences by explicitly computing the 1-loop e ective action, and that the leading corrections come with 4 additional derivatives.
3.5 Conformal dilaton
Finally, we consider the theory of the conformal dilaton the Goldstone of the spontaneous breaking of conformal symmetry down to Poincar. In addition to the normal linear action of the Poincar group, this theory is invariant under the following symmetries
= c(1 + x) , = b 2x + 2xx x2
, (3.14)
for constant c and b, which nonlinearly realize the conformal group, SO(4, 2). The scalar also has a geometric interpretation as the small-eld limit of the brane-bending mode of a Minkowski brane embedded in an Anti-de Sitter bulk [4, 5].8
This theory arises in various contexts. It was proposed as a type of IR completion of the galileon [1], and for this reason it often goes by the name conformal galileon. It was also studied long ago by Volkov as a prototypical example of a spontaneously broken spacetime symmetry [72]. In [73] it was argued that the conformal dilaton shares many properties with gravity (including a version of the CC problem). It has been used to construct alternative scenarios to cosmological ination [34, 7476] and also plays a prominent role in the proof of the a-theorem in four dimensions [77, 78].
Despite the complicated appearance of the non-linear symmetries in (3.14), it is easy to construct invariant actions for , by simply building di eomorphism invariant actions using the e ective metric g = e2. The kinetic term for is just given by the Einstein-
Hilbert term (with the wrong overall sign):
Skin =
2 . (3.16)
8The worldvolume theory of a at co-dimension one brane in an AdS space nonlinearly realizes the conformal group even away from the small-eld limit. This nonlinear realization is actually equivalent to the parameterization we consider, with the two theories related by a complicated eld redenition [6971].
9Note that (3.15) alone is a free theory in disguise, as can be seen via the eld redenition
= e.
When other terms are added to the action, this is no longer true, of course.
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Z d4xgR[g] = Z d4x
2 e2()2 , (3.15)
after integrations by parts. The free kinetic term ()2 is accompanied by an innite set
of specic interactions9 n()2. The cosmological constant term yields an exponential
potential g = e4 and higher order operators are built from higher order curvature
invariants made from the Riemann tensor and its covariant derivatives.
The O(R2) operators in this theory are particularly interesting and require a brief
discussion. The three operators {R2, R2, R2} are degenerate with each other and,
after integrations by parts, only yield a single independent operator
2
12
2
LR2 2+ ()2
This follows the expected counting: the Gauss-Bonnet theorem removes one combination and the vanishing of the Weyl tensor removes another, resulting in the above redundancy.
However, there also exists another four derivative operator which cannot be written in terms of four dimensional curvature invariants, is di erent from (3.16), and is symmetric under (3.14), up to a total derivative:
Lwz ()4 + 22()2. (3.17)
The operator (3.17) is the only operator in the EFT without a four-dimensional geometric description. It has a natural interpretation as a Wess-Zumino term, hence the notation, and can be derived by coset methods applied to the breaking pattern SO(4, 2) SO(3, 1) [7].10
It is a direct analogue of the special galileon operators (2.1) and the Wess-Zumino-Witten term of the chiral Lagrangian [7]. Finally, the operator also appears in the at space limit of the Wess-Zumino anomaly functional (for the a anomaly of a 4D CFT) [77, 78].
We now apply the power counting formula (3.5). As in the General Relativity case, we will need to tune the cosmological constant term, g = e4, to zero in order to have a
Poincar invariant solution to expand about. After canonically normalizing, 7/ , we
start by considering arbitrary diagrams built from only the kinetic term and its associated interactions (3.15) (as these have the fewest derivatives). We estimate that the L-loop diagram scales as
M(N)e2()2 4N
From this estimate, it appears that that one-loop diagrams constructed from (3.15) alone will renormalize the 4-derivative operators (3.17) and (3.16). However, as noted in footnote 9, (3.15) is really a free kinetic term in disguise and hence any S-matrix element constructed solely from vertices taken from this operator will vanish, after all diagrams are summed up [79, 80], so the expression (3.18) actually vanishes.
Our loop diagrams must therefore use at least one insertion of a 4-derivative (3.16) (3.17), or higher, vertex. Using a single four-derivative vertex and arbitrarily many vertices from the kinetic operator, we nd
M(N) 4N
so the rst operators that can possibly be renormalized are the 6-derivative R3 and R R terms. Replacing the 4-derivative vertex with a higher order operator or going to
higher loops only increases the scaling with k. Therefore, loops of only renormalize 6-derivative, and higher order, operators. Thus, the potential, kinetic terms and 4-derivative terms do not run in the pure conformal dilaton theory.
4 Coupling to heavy elds
Given the apparent ubiquity of non-renormalization statements for derivatively coupled theories, we are led to ask: what is special about the galileon non-renormalization theorem?
10A di erent way of deriving (3.17) is by constructing curvature invariants for g = e2 in arbitrary d, where the Gauss-Bonnet term no longer vanishes, before taking a limit to d 4 [1].
11
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k
2L+2. (3.18)
k
2L+4, (3.19)
Specically, the detailed argument of section 2.1 would seem to be superuous given that the same results can be derived by dimensional analysis. Does the detailed proof of nonrenormalization in some way distinguish the galileon from GR, P (X) and the dilaton?
We argue that indeed it does; the galileon non-renormalization theorem ensures that even coupling additional heavy elds to will not cause the galileon operators to be renormalized. As we will see, this is in stark contrast to the other theories we have considered, whose leading operators will generically be corrected by heavy elds.
A heuristic argument for the above claim is the following. All of the estimates we have performed so far have assumed a mass-independent regulator. Consider, instead, using a cuto , Pauli-Villars or some other mass-dependent scheme. A new scale UV will now arise from loop diagrams and complicate the estimates. For instance, the 4-point loop diagram coming from two insertions of the X2 ()4/ 4 vertex in the P (X) theory will now
have the schematic form
M(4)X2,X2
whereas only the rst term appears in dimensional regularization. This does not change the conclusions of section 3.4, but just complicates the expressions. In particular, only the rst term in (4.1) has a logarithmic divergence and we would again conclude that P (X) loops only cause 8-derivative and higher order operators to run. Still, we generically nd power law divergences11 proportional to k6 and k4, corresponding to X2 and X2
operators which, we saw, receive no running from self-loops.
In contrast, if we used a cuto to calculate the loops contributing to 2 2 scattering
in the purely galileon theory, we would nd an expression of the form
M(4)gal. loops
There are power divergences corresponding to (2)4 and (2)4 operators (which,
we found, do not run from loops), but no power divergence corresponding to the galileon operators (2.1). This is assured by the detailed non-renormalization theorem of section 2.1: all contributions to the galileon operators, logarithmic and power law, are vanishing, regardless of the regulator.
Power laws capture, in some rough sense, the e ect of coupling heavy elds to the theory. For instance, consider pure 4 theory in d = 4,
L =
12
JHEP11(2016)100
k
8+
6+
uv
uv
2 k
4 k
4, (4.1)
k
12+
10+
uv
uv
2 k
4 k
8. (4.2)
4!4 . (4.3) Calculating the one-loop vacuum polarization diagram using a cuto , one nds that there is a quadratic divergence which needs to be removed by a counterterm, m2 2UV. Essentially, this quadratic divergence12 indicates the generic result of coupling heavy elds to .
11Taking power law divergences too seriously can yield misleading conclusions, see, e.g. [81]. Logarithms provide the sharpest information, and these power law corrections should really be thought of as an estimate of the logarithmic part of the result of integrating out a heavy particle.
12There is also a logarithmic divergence, indicating that m2 runs with (m2) m2, but this is not crucial for the following discussion, so it is omitted.
1
2()2
m2
2 2
For instance, imagine that the action (4.3) represented the leading terms in the EFT that arose from integrating out a heavy eld of mass M which couples to in the following way,
LUV =
1
2()2
m2
g22 2 + . . . . (4.4)
The one-loop vacuum polarization diagram for , with running in the loop, now generates logarithmic running with (m2) gM2. This expression for the beta function is valid only
for energies larger than M. At energies below s mass, this heavy eld quickly decouples
and its contribution to beta functions is driven to zero; see [82] for a relevant review. So, if m2(EUV) is the value of s mass squared parameter at a scale EUV M, then its value
at very low energies EIR M (denoted by m2(EIR)) is given by an expression of the form m2(EIR) m2(EUV) + gM2 log
EUV
M
2 2
1
2( )2
M2
2 2
, (4.5) up to O(1) factors. Above, weve ignored all other sources of running and, again, used the
fact that the heavy matter decouples at energies below M in order to only run m2 between the scales EUV and M (as opposed to between EUV and EIR, which would be incorrect).
The above line of reasoning is an example of the sense in which power divergences demonstrate the existence of hierarchy problems. The quadratic divergence we found in the low energy theory 2UV mimics the gM2 log EUV/M correction13 above with the rough
correspondence UV M. Though only logarithmic divergences are unambiguous [81],
power laws serve as acceptable proxies for how heavy physics can a ect parameters in the action and the conclusions reached through either analysis are typically in agreement.
Since GR, P (X) and the conformal dilaton have no non-renormalization argument which is regulator independent, we expect their amplitudes to generically have power divergences corresponding to all possible operators, hence we expect that the terms which are not renormalized by self-loops will still be sensitive to the e ects of heavy elds. We expect that the same will not be true of galileons: coupling them to heavy elds will not renormalize the special operators (2.1).
The remainder of the paper is devoted to examining this claim in detail. We couple a heavy matter eld to all of the theories we discussed previously and integrate out the heavy eld. For simplicity, we couple in a non-self-interacting scalar of mass M, being careful to use couplings which respect the symmetries of the e ective theory. We rst show that the lowest-derivative galileon operators are not a ected by this procedure. We then go on to Einstein gravity, P (X) theories, and the conformal dilaton in turn and demonstrate that heavy elds can a ect the coe cients of all the operators of interest.
4.1 Integrating out elds via functional determinants
In this section, we briey introduce the formalism we employ to integrate out the heavy eld, . For simplicity, we only consider a non-self-interacting heavy scalar coupled
13For to be active at low energies, we must have m(EIR) M. This is very unnatural, as it is only true if m2(EUV) lies in a very narrow window in which it can cancel o most of the gM2 log EUV/M term, otherwise m2(EIR) O(M2). A priori, there is no good reason why this should be the case, which is the usual hierarchy problem.
13
JHEP11(2016)100
to the elds of interest in a manner preserving the symmetry of the EFT so that the action is quadratic in . In this case, the path integral over can be done exactly, and the e ective action for is given by a functional determinant
exp iSe [] =Z D exp iS, = det
2S,
1/2exp iS, =0 . (4.6)
Much machinery has been built to evaluate determinants of this type, for instance using heat kernels [83, 84]. We choose to evaluate the functional determinant perturbatively, in powers of the eld . To facilitate this, it is often best to write the determinant as a contribution to the e ective action:
exp i Se = det
2S,
JHEP11(2016)100
1/2= exp 12 Tr log 2S,
. (4.7)
As a simple example, consider the two scalar action from the previous section (4.4),
L, =
1
2()2
m2
2 2
1
2( )2
M2
2 2
g22 2. (4.8)
The contribution to the action from integrating out is then
Se = i2 Tr log
2S,
=
i2 Tr log 2
M2 g2
. (4.9)
Dividing through by a factor of the propagator this shift can be absorbed into the normalization of the path integral, and corresponds to canceling the vacuum bubbles of the eld we can cast this as
Se = i2 Tr log
1
(4.10)
We now want to evaluate this expression perturbatively in by expanding the logarithm
Se =
i2 Tr
12 M2
g2
+ . (4.11)
Each term in this expression can be mapped to a particular Feynman diagram contribution. We rst evaluate the O(2) contribution to the action; in order to do this, we introduce sets
of position and momentum eigenstates as outlined in appendix A and trace over momentum eigenstates14
=
i 2
12 M2
g2
i4 Tr
12 M2
g2 1
2 M2
g2
Z d4p hp|
12 M2
g2|pi =
ig
2
Z d4x 2(x) Z
d4p
(2)4
1p2 + M2 . (4.12)
The (divergent) integral can then be evaluated in dimensional regularization to yield the contribution to the e ective action
Se Z d4x
gM2 (4)2
1 log M/ (x)2, (4.13)
14The mapping between this situation and the notation in equation (A.18) is SJ(n)[ip]J = g(x)2, S1 I(0)[ip]I = (p2 + M2)1.
14
where we absorbed all nite terms into the denition of . The 1/ divergence is canceled by a counterterm for the mass, after which we can smoothly take 0, and the logarithm
combines with the mass term in the full action to form
Se Z d4x
m2 2gM2(4)2 log M/ (x)2 . (4.14)
Demanding that the action be independent of the arbitrary mass scale15 yields the beta function (m2) = 2gM
2
(4)2 , in accord with our previous expression (4.5). This result agrees with the standard diagrammatic analysis.
For emphasis, the beta function we just derived is only valid at energies larger than the mass of the heavy particle. Its behavior is approximately piecewise:
(m2)
, (4.15)
though the exact form of the low energy behavior is scheme dependent [82]. The decoupling at energies below M is entirely general and simply represents the fact that short distance physics has little e ect on long distance physics. In the following sections, we will derive many more beta functions and they should all be understood in the above manner, being only valid at energies larger than the heavy mass scale (which we always write as M).
4.2 Galileons
First, consider the galileon. We couple a heavy scalar to the galileon in a galileon invariant way and integrate it out. None of the galileons are ever a ected; this is for the same reason that underlies the proof of the non-renormalization theorem in section 2.1: for the coupling between and to be invariant, should only appear through 2 (or with more derivatives) and path integrating over therefore only generates terms built strictly from 2.
For example, consider the case where there is a linear in coupling, so that there will also be tree-level contributions to the e ective action which comes from eliminating via its equation of motion. In order to see that this will not lead to galileon terms, note that there is essentially only one way to write an invariant linear coupling: L f(n2) , with f(n2) an arbitrary scalar function of and derivatives thereof. It can be reasoned that any other galileon invariant coupling can be put in this form after integrations by parts. Therefore, the classical EOM will take the schematic form
(2 + M2) = f(n2) + , (4.16) where contains self-interaction terms. Then, in order to integrate out at tree-level,
one would merely take the original Lagrangian L(, ) and replace by
7
1
M2
1
2
JHEP11(2016)100
2
(4)2 E [greaterorsimilar] M
0 E [lessorsimilar] M
2gM
1 2M2 + . . . f(n2) . (4.17)
15More generally, wavefunction renormalization factors also have to be taken into account when determining the beta function, but simply demanding independence from will be su cient for all the examples we consider. The general procedure is given in, for example, [85], where it is phrased in terms of the 1PI action. The functional determinants we consider can be thought of as (part of) the one-loop contribution to the 1PI action, as calculated using the background eld method [42].
15
This will never lead to galileons, since all elds have too many derivatives acting upon them.
As an example of an explicit loop computation, take the galileon invariant coupling to to be
Sint =Z d4x
1
2( )2
2 22 . (4.18)
Integrating out , the e ective action then contains the terms
Se = i2 Tr log 2 M2 +
M2
2 2 +
2 . (4.19)
Notice that inside the functional trace the eld already has two derivatives acting on it. Hence, it is already clear that no operators with fewer than two derivatives will be generated from integrating out . We can check this explicitly by computing the lowest order corrections in derivatives, at order 2 we nd
Se Z d4x
2 322 2
JHEP11(2016)100
1 log M/ (2)2 +1 1922M2
3(2)3 + , (4.20)
where indicates terms which are both higher order in elds with 2 derivatives per eld
and terms with more derivatives per eld. We see from this cubic action that in particular the cubic galileon is not generated. This continues to be true at higher order only terms with at least two derivates per eld are generated.
Again, these results can be seen to follow from the fact that the interaction always involves a power of 2 which always results in e ective actions built from 2. A more interesting case would be if there were a Wess-Zumino like coupling between and which was invariant under (1.1) up to a total derivative but could not be integrated by parts into a form where there are at least two derivatives per , however we are not aware of any such couplings. Note that if the coupling is not invariant, for example the standard
T matter coupling often considered in studies of the Vainshtein mechanism, then these
non-renormalization statements do not strictly hold [86].
Weve been somewhat agnostic about the role of the heavy eld, but one might have hoped that it could play some signicant part in the UV completion of the nonrenormalizable galileon theory. The above calculations then provide explicit evidence against the possibility of UV completing galileons in a standard, local manner, in accordance with the general arguments of [21] against any such completion.
4.3 General relativity
Now we will see that the non-renormalization of the Planck mass and cosmological constant of section 3.3 does not hold upon integrating out a heavy eld. Consider coupling minimally a heavy scalar, , to GR,
S =
Z d4xg
M2PlR
2 +
1
2 ( M2) . (4.21)
16
The result of integrating out is well known (see e.g. [87], or [88, 89]). There is a contribution to the cosmological constant which can be computed from the zero momentum part of the diagram
. (4.22)
The contribution to the e ective action is
Se Z d4x g +
M4 322
1 log M/ . (4.23)
We introduce the renormalized CC as = R + with = 1
M4322 , take 0 safely
and then demand that Se be independent of to yield the beta function for ,
( ) = M4
322 . (4.24)
There is a contribution to the Planck mass which can be computed from the order 2 part of the diagram(4.25)
which leads to a contribution to the e ective action of the form
Se Z d4xg M2Pl
M2 482
JHEP11(2016)100
1 log M/
R2 + . (4.26)
In order to cancel o the pole, we introduce a renormalized Planck mass and associated counterterm. Then, demanding that the answer is independent of the renormalization scale, , yields the beta function for the Planck mass
(M2Pl) = M2
482 . (4.27)
4.4 P (X) theories
In this section we couple to P (X) theories, and integrate it out to see to the renormalization of P (X). We do this in several di erent ways.
4.4.1 @@ T coupling
The couplings we consider must respect the 7 + c symmetry of P (X) theories. One
coupling which satises this criterion is coupling the stress tensor of the eld to derivatives of , as might arise from certain brane-world constructions,
L = 4P (X)
1
2( )2
M2
2 2 +
4 T ( ). (4.28)
Here the stress tensor for is
T( ) =
1
2( )2 +
M2
2 2 . (4.29)
17
Integrating out , the e ective action then contains the generated terms
Se = i2 Tr log M2
M2
4 ()2
2 4 +
4 ()2
2 4 .
(4.30)
The simplest possible computation we can do is to check if this coupling to the heavy eld induces a wavefunction renormalization of the kinetic term 12()2. The only
operator that can contribute to this is the M2 4 ()2 term in (4.30), so we just have to
compute the single insertion trace involving this operator:
=
i2 Tr
1 log M/ .
(4.31)
We see that the P (X) kinetic term does get renormalized by the heavy eld, i.e., an anomalous dimension16 is acquired, when we couple to as in (4.28). In comparison, loops of in a pure P (X) theory do not induce such an anomalous dimension.
4.4.2 DBI
Next, we consider a special case of P (X) theories: the DBI Lagrangian. This theory describes the dynamics of a Minkowskian 3-brane embedded in a ve dimensional spacetime. The action is built through the induced metric on the brane,
g = + 1 4 , (4.32)
and its associated curvature invariants, including the extrinsic curvature tensor K =
. In particular, the DBI kinetic term comes from the volume element:
SDBI = 4 Z d4xg = 4 Z d4xr1 +
()2
JHEP11(2016)100
12 M2
M2
4 ()2 = Z d4x
1
2()2
M4 (4)2 4
4 . (4.33)
In addition to the 7 + c symmetry of all P (X) theories, DBI is further symmetric
under 7 + b(x + ). In this context, the P (X) symmetry is the worldvolume
consequence of higher-dimensional translation invariance of the brane along the transverse direction, while the second symmetry is a consequence of higher-dimensional boosts mixing brane directions with the transverse direction.
Couplings to a heavy scalar which are DBI-invariant are easy to engineer by utilizing the induced metric g in (4.32). The simplest such coupling takes the form:
Scoupling =
2 2 . (4.34)
We would like to understand how the presence of this heavy scalar renormalizes the tension . To calculate this, we will determine the contributions to the ()2 and ()4 terms
in the e ective action, and verify that they match the expansion in (4.33).
16The anomalous dimension is = M
2
Z d4xg
1
2 g
M2
2(4)2 2 , as can be derived using eq. (1a.1.39) of [85].
18
The functional determinant we want to compute is (again, we drop the contribution from g)
Se = i2 log det
M2
= i
2 Tr log
M2
(4.35)
built from covariant derivatives with respect to g. The easiest way of computing
is to use
= 1
g
with
= g
gg
= 2
1 4 2
1 4
+ 1
8 2()2 +
JHEP11(2016)100
1
2 8 ()2 +
1 8 ()2 , (4.36)
where we have used g =
4 , 1/q1 + ()
2
2
4 , and indices are raised and lowered with . We are only looking for terms which can generate terms ()2 or ()4, and the pieces in (4.36) which have two derivatives acting on a cannot give rise
to these. The relevant trace is then reduced to
Se = i
2 Tr log 2 M2
1 4 +
1 8 ()2 . (4.37)
The contribution to the ()2 term is given by a single insertion trace over the operator:
=
i2 Tr
1 2M2
1 4 =Z d4x
1
2()2
M4 322 4
1 log M/ .
(4.38)
and, summing up both a single and double insertion trace, the ()4 terms are given by
+ =
Z d4x
18 4 ()4
1 log M/ . (4.39)
The relative coe cient matches precisely what we obtain from the expansion of the volume element:
4g 1 +
1
2()2
M4 322 4
18 4 ()4 + . . . (4.40)
Comparing to (4.33) the renormalized tension is seen to run as
( 4) = M4
322 . (4.41)
Therefore, we explicitly see that a heavy scalar renormalizes the innite tower of Xn
operators which appear in the DBI action.
Note that we have essentially just repeated the calculation of the CC running of section 4.3: couples to via minimal coupling to g, which is just a specic choice of metric. It is therefore not surprising that the two beta functions (4.41) and (4.24) agree.
19
4.5 Conformal dilaton
Finally, we couple to the conformal dilaton eld, denoted by . We take to transform as a primary eld of weight so that under the conformal symmetries transforms as
= c ( + x) , = b 2 x + 2xx x2
, (4.42)
with constant c, b, while the dilaton transforms non-linearly as in equation (3.14).
After xing the mass and scaling dimension of and canonically normalizing, there exists a one-parameter class of two derivative interactions which are quadratic in and symmetric under (4.42) and (3.14):
Sint =Z d4x
e2(1 )
JHEP11(2016)100
2 2
+ 2 2
M2e2(2 )
e2(1 )
2 2()2
e2(1 ) . (4.43)
The = 1, = 0 case was considered in [77], as a check of their arguments in the proof of the a-theorem.
Working to fourth order in derivatives, we integrate out and nd, after a lengthy calculation:
Se =Z d4x h
dV e4 dRe2()2 + dR
2 ( )2
2 + ()2
i
2 1 log M/ (4.44)
+
Z d4x h
fV e4 fRe2()2 + fR
2 ( + ()2)2 + fWZ + 2 ()2
i
,
where the coe cients of the divergent and nite terms are:
dV dR dR2
fV fR fR2
fWZ
= 1
(4)2
M4 2
M2( + 1),
1
2 ( + 1)2
3M4 8
M23
160 + ( +)( +1)3
1 180
. (4.45)
In order to consistently calculate (4.45) using dimensional regularization, one must also work with the invariant operators g , gR, etc. (where, again, g = e2)
as constructed in arbitrary dimensions. This induces additional factors of s, as in g = ed = e(4), which contribute non-trivially to the nite parts of the functional
determinant. Such terms are necessary in obtaining a conformally invariant answer.
We see that the Wess-Zumino term receives a nite renormalization. The coe cient of the Wess-Zumino term, fWZ, is notably independent of both and , as it should be, since fWZ is directly related to the a-anomaly which characterizes a fundamental property of the free scalar eld which shouldnt depend on how one couples the dilaton to . Its numerical value is in full agreement with the result in [77].
20
5 Conclusions
Many massless, derivatively coupled e ective theories have non-renormalization theorems that follow simply from dimensional analysis, without reference to the detailed structure of the interactions. Essentially, their interactions have so many derivatives that the self-loops can only a ect operators of a certain minimum dimension. Among the theories in this class are General Relativity, P (X) theories, the conformal dilaton (sometimes called the conformal compensator, or conformal galileon), and galileons.
Galileons, however, possess a stronger, diagrammatic non-renormalization theorem that depends on the detailed structure of the galileon interaction terms [30]. We have interpreted the extra strength of the galileon non-renormalization theorem as the statement that the galileon operators are not renormalized even by loops of other heavy elds, as long as they are coupled in a galileon invariant way. In the other theories like GR or P (X), the leading operators are renormalized by these heavy loops.
We have tested this interpretation by coupling a heavy scalar eld to each of these theories in ways that respect the symmetries of each theory. None of the ve special galileon operators are a ected by heavy elds. In comparison, the operators in the GR, P (X) and conformal dilaton theories which were not a ected by self-loops are renormalized by loops of the heavy eld.
Finally, we note that even if the galileon symmetry is not an exact symmetry but is weakly broken, the non-renormalization theorem still controls corrections to the galileon terms, rendering them proportional to the small breaking, a fact which can be useful, for example in constructing technically natural cosmological models [90, 91].
As an example of this, we note another di erence in the non-renormalization theorem of galileons compared with other theories that becomes apparent when we consider deforming the theory with a mass. In the power counting formula (3.5), we were able to nd the scaling of arbitrary diagrams with powers of the scale and the external momenta of the diagram k. When the theory is massless, k truly refers to an external momenta, but if there are massive particles running in the amputated diagram, then factors of k can also represent mass scales. For instance, consider adding a mass term to the leading operators in a P (X) theory:
L =
1
2()2
JHEP11(2016)100
4 ()4 + . . . . (5.1)
The power counting formula (3.5) still tells us that the tree and one-loop 2 2 diagrams
scale as
M(4)()4
k
m2
2 2
4, M(4)()4,()4 k
8, (5.2)
respectively. However, the meaning is di erent now, since some of the ks can correspond to mass scales and so in addition to terms like k
8 we can also have terms like m
4k
4.
This would correspond to a running of with () 2
m
4.
Therefore, we see that generically adding mass scales ruins the analyses in section 3. However, there is an exception with the galileons. Though it would seem that powers of k2 could turn into factors of m2 in the power counting formulae and result in running
21
galileon couplings, the detailed version of the non-renormalization theorem ensures that this does not occur. This can be seen from the path integral proof in section 2.1; adding a mass term changes nothing about the argument. This is yet another way that the galileons are di erent from GR, P (X) and the conformal dilaton: the non-renormalization theorem survives when the theory is deformed by a mass [92].
Acknowledgments
We thank Brando Bellazzini for helpful correspondence. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. This work was made possible in part through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reect the views of the John Templeton Foundation (KH). This work was supported in part by the Kavli Institute for Cosmological Physics at the University of Chicago through grant NSF PHY-1125897, an endowment from the Kavli Foundation and its founder Fred Kavli, and by the Robert R. McCormick Postdoctoral Fellowship (AJ). The work of M.T. was supported in part by US Department of Energy (HEP) Award DE-SC0013528. GG gratefully acknowledges support from a Starting Grant of the European Research Council (ERC StG grant 279617).
A Evaluating functional determinants
In this appendix, we review the procedure for evaluating the traces in the functional determinants. For another recent application of this general technique, see [93]. The general object we are interested in computing is the 1-loop correction to the action
(1) = i
2 log det
2S
ij
JHEP11(2016)100
=
i2 Tr log
2S
ij
, (A.1)
where the i are the elds in the theory. Dening 2S
ij
Sij we can expand in powers of
i as
Sij = S(0)ij + S(1)ij + S(2)ij , (A.2)
after factoring out S0ij (which is eld independent), we can expand the logarithm to obtain
(1) = i
2 Tr log S(0)
1 + S1(0)S(1) + S1(0)S(2) +
(A.3)
= i
2 Tr log S(0)+
i2 Tr
S1(0)S(1)
+ i
2 Tr
S1(0)S(2)
+ i
2 Tr
S1(0)S(1)S1(0)S(1)
+ ,
where we have employed matrix notation S(n) S(n)ij. The piece Tr log S(0) is indepen
dent of the elds in the action, and just represents a constant shift of the vacuum energy of the theory, so we will often discard it.
In order to evaluate the traces in (A.3), it is useful to make a quantum-mechanical analogy, the operators whose trace we want to evaluate are local functions of coordinates
22
and derivatives: S(n)ij(x, ). We therefore promote the coordinates and derivatives to operators acting on a Hilbert space as
x 7x , 7ip. (A.4)
We then introduce sets of position eigenstates, |xi and momentum eigenstates |pi, which
satisfy the orthogonality and completeness relations
hx|yi = (x y) hp|ki = (p k) (A.5)
Z ddx |xihx| =
Z ddp |pihp| =
1(2)d/2 eipx. (A.7)
The action of the x and p operators on these eigenstates is the obvious one
x|xi = x|xi (A.8) p|pi = p|pi. (A.9)
Any local operator O(x, ) is of the form
O(x, ) = O(x) + O1(x)1 + O12(x)12 + OI(x)I, (A.10)
where I is a multi-index. The coe cients O12(x) in this expression are built out of the
elds i and their derivatives. To evaluate the traces, we make the replacement (A.4)
O(x, ) 7 OI(x)[ip]I. (A.11)
We will evaluate the traces in momentum space, so we want to evaluate the matrix element of this operator between momentum eigenstates
hk|OI(x)[ip]I|pi = Z ddx hk|OI(x)|xihx|[ip]I|pi = Z ddx OI(x)[ip]Ihk|xihx|pi
= Z ddx ei(kp)x(2)dOI(x)[ip]I = (2)d
ddk
(2)d eikx
JHEP11(2016)100
1
1. (A.6)
The inner product between these two bases is given by
hx|pi =
I(k p)[ip]I, (A.12)
where tilde refers to the Fourier transform.17 with this rule we can evaluate any trace that we encounter.
17We employ the following convention for Fourier transformation
f(x) =
[integraldisplay]
~f(k) ~f(k) =
[integraldisplay] ddx eikxf(x). (A.13)
23
Propagator. The matrix element of the operator S(0)() = SI(0)ij[ip]I, with all the SI(0)ij = const. can be evaluated as
hk|SI(0)ij[ip]I|pi = SI(0)ij[ip]I(p k). (A.14)
The matrix elements for the propagator S1(0) can be evaluated similarly:
hk|S1I(0)ij[ip]I|pi = S1I(0)ij[ip]I(p k). (A.15)
Single insertion trace. Frequently, we will want to compute the trace involving an insertion of a single operator of order n in the background elds. This takes the form
i2 Tr
S1(0)S(n)
= i
Z ddp hp|S1I ij(0)[ip]I SJ(n)ij(x)[ip]J|pi (A.16)
= i
2
2 J(n)ij(0) Z
ddp
(2)d [ip]JS1I ij(0)[ip]I, (A.17)
where to get to the second line we have inserted a complete set of momentum eigenstates and used the formulae (A.12) and (A.15). The factorJ(n)jk(0) seems at rst sight somewhat strange, but we can rewrite the Fourier transform at zero momentum as an integral over all of position space to obtain the form
i2 Tr
S1(0)S(n)
= i
2
Z ddx SJ(n)ij(x) Z
JHEP11(2016)100
ddp
(2)d [ip]JS1I ij(0)[ip]I, (A.18)
which is of the form that we evaluate in the text.
Double insertion trace. We will also want to compute the trace involving two insertions of the operator S1(0)S(1), which can be evaluated as
i4 Tr (S1(0)S(1)S1(0)S(1))=
i 4
Z ddphp|S1I ij(0)[ip]ISJ(1)jk(x)[ip]JS1K kl(0)[ip]KSL(1)li(x)[ip]L|pi
=
i 4
Z
ddp
(2)d
ddq
(2)d S1I ij(0)[ip]I
J(1)jk(p q)[iq]JS1K kl(0)[iq]K
L(1)li(q p)[ip]L. (A.19)
This can be simplied by shifting q 7q + p so that we have
i4 Tr h
(S1(0)S(1))2i
=
i 4
Z
ddq
(2)d
J(1)jk(q)
L(1)li(q) (A.20)
Z
ddp
(2)d S1I ij(0)[ip]I[i(q + p)]JS1K kl(0)[i(q + p)]K[ip]L
The integral over q can now be thought of as a convolution in Fourier space at zero momentum, so transforming back to position space, we obtain:
i4 Tr h
(S1(0)S(1))2i
=
i 4
Z ddxSI(1)li(x) (A.21)
Z
(2)d [ + ip]IS1J ij(0)[ip]JS1K kl(0)[ + ip]K[ip]L SL(1)jk(x),
where the derivatives should be understood as acting on SJ(1)(x).
24
ddp
Higher insertions. This pattern generalizes to higher numbers of insertions. At nth
order for an operator at th order in the elds we have
i(1)n1n Tr h
(S1(0)S())ni
(2)d [ + ip]IS1J ij(0)[ip]JS1K kl(0)[ + ip]KSL()jk(x)[ + ip]L
S1M lm(0)[ + ip]MSN()mn(x)[ip]N ,
where in the above expression all derivatives should be thought of as acting on everything to their right.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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Abstract
Abstract
We discuss non-renormalization theorems applying to galileon field theories and their generalizations. Galileon theories are similar in many respects to other derivatively coupled effective field theories, including general relativity and P (X) theories. In particular, these other theories also enjoy versions of non-renormalization theorems that protect certain operators against corrections from self-loops. However, we argue that the galileons are distinguished by the fact that they are not renormalized even by loops of other heavy fields whose couplings respect the galileon symmetry.
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