Heavylifting of gauge theories by cosmic inflation
HJE
Heavylifting of gauge theories by cosmic in ation
Soubhik Kumar 0 1
Raman Sundrum 0 1
0 College Park, MD 20742 , U.S.A
1 Maryland Center for Fundamental Physics, Department of Physics, University of Maryland , USA
Future measurements of primordial nonGaussianity can reveal cosmologically produced particles with masses of order the in ationary Hubble scale and their interactions with the in aton, giving us crucial insights into the structure of fundamental physics at extremely high energies. We study gaugeHiggs theories that may be accessible in this regime, carefully imposing the constraints of gauge symmetry and its (partial) Higgsing. We distinguish two types of Higgs mechanisms: (i) a standard one in which the Higgs scale is constant before and after in ation, where the particles observable in nonGaussianities are far heavier than can be accessed by laboratory experiments, perhaps associated with gauge uni cation, and (ii) a \heavylifting" mechanism in which couplings to curvature can result in Higgs scales of order the Hubble scale during in ation while reducing to far lower scales in the current era, where they may now be accessible to collider and other laboratory experiments. In the heavylifting option, renormalizationgroup running of terrestrial measurements yield predictions for cosmological nonGaussianities. If the heavylifted gauge theory su ers a hierarchy problem, such as does the Standard Model, con rming such predictions would demonstrate a striking violation of the Naturalness Principle. While observing gaugeHiggs sectors in nonGaussianities will be challenging given the constraints of cosmic variance, we show that it may be possible with reasonable precision given favorable couplings to the in ationary dynamics.
Cosmology of Theories beyond the SM; Gauge Symmetry

3
Squeezed limit of cosmological correlators
The inin formalism for cosmological correlators
Useful gauges for general coordinate invariance
Observables
NG from single eld in ation in the squeezed limit
NG from multi eld in ation in the squeezed limit
NG from Hubblescale masses in the squeezed limit
4
GaugeHiggs theory and cosmological collider physics
The central plot and its connections to the literature
High energy physics at the Hubble scale
Heavylifting of gaugeHiggs theory
5
eld slow roll in ation
and coupling strengths of e ective theory
Visibility of a Higgs scalar
Visibility of a massive gauge boson
Gauge theory with a heavy Higgs scalar
1 Introduction 2 Preliminaries 8
Concluding remarks and future directions
6.1.1
6.1.2
6.2.1
6.2.2
Visibility of a Higgs scalar
Visibility of a massive gauge boson
7
Detailed form of NG mediated by h
6
NG in the E ective Goldstone description of in ationary dynamics
6.1
Minimal Goldstone in ationary dynamics
Leading terms in the e ective theory and power spectrum
Higher order terms
6.2 Incorporating gaugeHiggs theory into the Goldstone E ective description
{ i {
A.1
Massive elds
A.2 In aton mode functions
B
HJEP05(218)
1
Introduction
Cosmic In ation (see [1] for a review), originally invoked to help explain the homogeneity
and
atness of the universe on large scales, provides an attractive framework for
understanding inhomogeneities on smaller scales, such as the spectrum of temperature
uctuations in the Cosmic Microwave Background (CMB) radiation. These
uctuations are
consistent with an almost scaleinvariant, adiabatic and Gaussian spectrum of primordial
curvature perturbations R [2]. The approximate scale invariance of these uctuations can
be naturally modeled as quantum oscillations of the in aton eld in a quaside Sitter (dS)
spacetime. The adiabaticity property implies that among the elds driving in ation, there
is a single \clock", the in aton, which governs the duration of in ation and the subsequent
reheating process. Finally, Gaussianity of the present data [3] re ects very weak couplings
among in ationary and gravitational elds. While these features point to successes of the
in ationary paradigm, few details of the fundamental physics at play during in ation have
emerged. Observing small nonGaussianity (NG) of the uctuations could change this
situation radically, giving critical insights not only into the in ationary dynamics itself but
also into the particle physics structure of that era.
Interactions of the in aton with itself or other elds during or immediately after in
ation can lead to a nonGaussian spectrum of R. However, NG can also be developed after
uctuation modes reenter the horizon at the end of in ation. This can happen for various
reasons, including, nonlinear growth of perturbations under gravity during structure
formation (see [4, 5] for reviews in the context of CMB and LargeScale Structure). Therefore
{ 1 {
it is crucial to understand and distinguish this latter type of NG which can \contaminate"
the invaluable primordial NG. In this paper we will assume that this separation can be
achieved in future experiments involving LargeScale Structure (LSS) surveys [6] and
21cm cosmology [7, 8], to reach close to a cosmicvarianceonly limited precision. With this
quali er, a future measurement of NG can reveal important clues as to the underlying
in ationary dynamics. For example, for the case single eld slowroll in ation, there is a
minimal amount of NG mediated by gravitational interactions [9, 10], while lying several
orders of magnitude below the current limit on NG, can be achievable in the future.
There also exist a variety of models which predict a larger than minimal NG (see [
11, 12
]
for reviews and references to original papers). A common feature among some of these
models is the presence of additional elds beyond the in aton itself. Such nonminimal
structure can be motivated by the need to capture in ationary dynamics within a fully
theoretically controlled and natural framework. If those additional elds are light with
mass, m
H (where H denotes the Hubble scale during in ation), they can oscillate and
coevolve along with the in aton during in ation. These elds can generate signi cant NG
after in ation, with a functional form approximated by the \local" shape [3, 13, 14].
On the other hand, the additional elds can be heavy with masses m & H. Such elds
can be part of \quasisingle eld in ation" which was introduced in [15] and further
developed in [16{22]. In the presence of these massive elds, the threepoint correlation function
of the curvature perturbation R has a distinctive nonanalytic dependence on momenta,
hR(~k1)R(~k2)R(~k3)i / k33 k13
in the \squeezed" limit where one of 3momenta becomes smaller than the other two. In
the above,
(m) =
+ i
3
2
r m2
H2
9
4
;
where m is the mass of the new particle. The nonanalyticity re ects the fact that the
massive particles are not merely virtual within these correlators, but rather are physically
present \onshell" due to cosmological particle production, driven by the in ationary
background timedependence. Such production is naturally suppressed for m
H, which is
re ected by a \Boltzmannlike suppression" factor in the proportionality constant in (1.1).
The only e ect of m
H particles is then virtualmediation of interactions among the
remaining light elds [23]. At the other extreme, for m
H the distinctive nonanalyticity
is lost. Hence, we are led to a window of opportunity around H, where the nonanalytic
dependence of the threepoint function is both nontrivial and observable, and can be used to
do spectroscopy of masses. Furthermore, if a massive particle has nontrivial spin [21, 24],
there will be an angledependent prefactor in (1.1), which can enable us to determine
the spin as well [25]. These observations point to a program of \Cosmological Collider
Physics" [21], which has an unprecedented reach into the structure of fundamental physics
measurements is ultimately constrained by cosmic variance, very roughly in the ball park of
hRRRi
hRRi 2
3
1
pN21cm
where we have assumed the number of modes accessible by a cosmic variance limited 21cm
experiment is N21cm
the full potential of the program.
1016 [7]. Achieving such a precision is very important for realizing
In this paper, we couple gaugeHiggs theories with m
H to in ationary
dynamics and ask to what extent the associated states can be seen via the cosmological
collider physics approach. The contributions of massive particle to the three point function
hR(~k1)R(~k2)R(~k3)i can be represented via \inin" diagrams in (quasi)dS space such as in
gure 1. From
gure 1(a), we see that since the in aton has to have the internal quantum
numbers of the vacuum,1 the same has to be true for the massive particles. The particles
must therefore be gauge singlets. Keeping this fact in mind, let us analyze the two scenarios
that can arise during in ation.
2HHyH with
H
The gauge theory may be unbroken during in ation. Gauge singlet 1particle states
then can mediate NG via tree diagrams as shown in
gure 1(a). This is also the case
that has been analyzed extensively in the literature. On the other hand, gauge charged
states can contribute via loops, as shown in
gure 1(b), but are expected to be small.
Alternatively, the gauge theory may be (partially) Higgsed during in ation. Then the
massive particle in
gure 1(a) need only be a gauge singlet of a residual gauge symmetry,
but may be charged under the full gauge group. This possibility, which has received less
attention in the literature (however, see [27, 28] for a related scenario), will be our primary
focus. There are two ways in which such a Higgsing can happen, as we discuss now.
First, such a breaking can be due to a
xed tachyonic mass term for the Higgs H,
H. In this case, the gaugeHiggs theory remains Higgsed after in ation
ends and its massive states can annihilate away as universe cools giving rise to standard
cosmology. Grand uni ed theories are examples of gauge extensions of Standard Model
(SM) containing very massive new particles and which are strongly motivated by existing
lower energy experimental data. For example, nonsupersymmetric uni cation is suggested
1In the context of Higgs in ation [26] however, in aton is the physical charge neutral Higgs eld.
{ 3 {
by the near renormalizationgroup convergence of SM gauge couplings in the 1013{1014 GeV
range, right in the highscale in ation window [2] of opportunity for cosmological collider
physics!2 NG detection of some subset of these massive states could give invaluable clues
to the structure and reality of our most ambitious theories. It is also possible that Hmass
states revealed in NG are not connected to speci c preconceived theories, but even this
might provide us with valuable clues about the far UV.
Another very interesting and testable option is a tachyonic \mass" term of the form
L
cRHyH, where R is the Ricci scalar and c > 0 parametrizes the nonminimal coupling
of Higgs to gravity. The e ects of nontachyonic terms for this form with c < 0 have been
considered before (see e.g. [31, 32]). Note, spontaneous breaking triggered by c > 0 is
completely negligible at low temperatures, say below 100 GeV. Whereas in the scenario
above we needed the gaugeHiggs theory to fortuitously have states with m
H, here
we naturally get the Higgs particle at H for c
O(1). Furthermore, if (gauge coupling
Higgs VEV)
H, we also get massive gauge bosons at H. In this way such a nonminimal
coupling can lift up a gauge theory with a relatively low Higgs scale today, which we can
access via collider or other probes, to the window of opportunity of cosmological collider
physics during the in ationary era. We will call this the \heavylifting" mechanism. To
make this idea concrete, we consider the example of heavy lifting the SM.
During the in ationary era the SM weak scale v can be lifted to be very high, but we
do not know where precisely because of the unknown parameter c (even if we knew H).
However, this uncertainty drops out in mass ratios,
mh =
mZ
mh =
mW
mh =
mt
p
2 2 h
While the top t and W boson can only appear in loops gure 1(b), the physical Higgs h
and the Z can appear in
gure 1(a) giving us one prediction in this case. However, an
important subtlety of the couplings on the r.h.s. of the ratios above is that they are not
those measured at the weak scale but rather are the results of running to
H. But it is
well known that the SM e ective potential develops an instability around 1010{1012 GeV
because of the Higgs quartic coupling running negative (see [33] and references therein
for older works). Since the in ationary H can be higher, the Higgs
eld can sample
values in its potential beyond the instability scale. Whether this is potentially dangerous
for our universe has been considered before (see e.g. [31{37]). But it is possible that this
instability is straightforwardly cured once dark matter (DM) is coupled to the SM. A simple
example [38{41] would be if future experiments determine that DM is a SM gauge singlet
2Uni cation at such scales is disfavored in minimal uni cation schemes by proton decay constraints, but
viable in nonminimal schemes such as that of refs. [29, 30].
{ 4 {
scalar S stabilized by a Z2 symmetry, S !
couplings are given by the Higgs portal coupling and scalar selfinteraction,
S. Then the most general renormalizable new
theory up to even high scale in ation energies.
Imagine a discovery of such a DM (S) is made in the coming years, along with a
measurement of k and its mass mS (and possibly a measurement of or at least a bound
on,
S).
Also, imagine a measurement of H is obtained via detecting the primordial
tensor power spectrum. Then we can use the Renormalization Group (RG) to run all the
measured couplings to the high scale H. These would then allow us to compute the runup
couplings needed to make a cosmological veri cation of (1.4). Such a veri cation of this
NexttoMinimal SM (NMSM) would give strong evidence that no new physics intervenes
between TeV and H. Since this NMSM clearly su ers from a hierarchy problem (worse
than the SM), the precision NG measurements would therefore provide us with a test of
\unnaturalness " in Nature, perhaps explained by the anthropic principle [42, 43]. Whether
the naturalness principle is undercut by the anthropic principle or by other considerations
is one of the most burning questions in fundamental physics.
Of course, the heavylifting mechanism may also apply to nonSM \dark" gaugeHiggs
sectors, which we may uncover by lower energy experiments and observations in the coming
years, or to gaugeHiggs extensions of the SM which may emerge from collider experiments.
In this way, there may be more than one mass ratio of spin0 and spin1 particles that
might appear in NG which we will be able to predict. As we will show, such new gauge
structure may be more easily detectable in NG than the (NM)SM, depending on details of
its couplings. It is important to note that di erent gauge theory sectors in the current era,
with perhaps very di erent Higgsing scales, can be heavylifted to the same rough scale H
during in ation, with their contributions to NG being superposed.
The heavylifting mechanism may not be con ned to unnatural gaugeHiggs theories.
For example, if low energy supersymmetry (SUSY) plays a role in stabilizing the
electroweak hierarchy, a suitable structure of SUSY breaking may permit the heavylifting
mechanism to work. Heavylifting can then provide us with a new test of naturalness! Possibly
nontachyonic squarks and sleptons in the current era were tachyonic during in ation,
higgsing QCD or electromagnetism back then. We leave a study of the requisite
SUSYbreaking structure for future work. Cosmological collider physics studies incorporating
SUSY but restricted to gauge singlet elds have appeared in [17, 44].
NG potentially provide us with the boon of an ultrahigh energy \cosmological
collider", but cosmic variance implies it operates at frustratingly low \luminosity"! We will
see that this constrains how much we can hope to measure, even under the best
experimental/observational circumstances. For example, a pair of spin1 particles appearing
in the NG will be more di cult to decipher than only one of them appearing, due to
{ 5 {
the more complicated functional form of the pair that must be captured in the limited
squeezed regime under cosmic variance. And yet, we would ideally like to see a rich
spectrum of particles at H. The key to visibility of new physics under these harsh conditions is
then determined by the strength of couplings to the in aton. This is the central technical
consideration of this paper, taking into account the signi cant suppressions imposed by
(spontaneously broken) gauge invariance. We study this within two e ective eld theory
frameworks, one more conservative but less optimistic than the other. Single eld slowroll
in ation gives the most explicit known construction of in ationary dynamics, but we will
see that minimal models under e ective eld theory control give relatively weak NG
signals, although still potentially observable. We also consider the more agnostic approach in
which the dynamics of in ation itself is parametrized as a given background process [45],
but in which the interactions of the gaugeHiggs sector and in aton uctuations are
explicitly described. This will allow for larger NG signals, capable in principle of allowing even
multiple particles to be discerned.
This paper is organized as follows. We start in section 2 by reviewing the inin
formalism and its use in calculation of the relevant nonGaussian observables. We also include a
discussion of di erent gauges and conventions used for characterizing NG. Then in section 3
we review the signi cance of the squeezed limit of cosmological correlators, both in the
absence and presence of new elds beyond the in aton. In particular, we review the derivation
of (1.1). In section 4 we discuss some general aspects of gaugeHiggs theory dynamics
during in ation and elaborate upon the two alternatives for Higgs mechanism discussed above.
We then specialize in section 5 to slowroll in ation where we study the couplings of
Higgstype and Ztype bosons to the in aton in an e ective eld theory (EFT) framework. In
section 6 we describe parallel considerations in the more agnostic EFT approach mentioned
above. The two levels of e ective descriptions are then used in sections 7 and 8
(supplemented by technical appendices A{D) to derive some of the detailed forms of NG due to
Higgstype and Ztype exchanges respectively. We conclude in section 9.
Hubble units. In this paper, the Hubble scale during in ation is denoted by H. To
reduce clutter, from now on we will set H
1 in most of the numbered equations, with
a few exceptions where we explicitly write it for the sake of clarity. Factors of H can
be restored via dimensional analysis. However, we will refer explicitly to H in the text
throughout, again for ease of reading, and in the unnumbered equations within the text.
2
2.1
Preliminaries
The inin formalism for cosmological correlators
Primordial NG induced by in aton
uctuations are calculated as \inin" expectation
values [46] of certain gaugeinvariant (products of) operators at a xed instant of time towards
the end of in ation, denoted by tf . The expectation needs a speci cation of the quantum
state. The notion of \vacuum" is illde ned because spacetime expansion gives a
timedependent Hamiltonian, H(t). However, for very short distance modes/physics at some
very early time ti, the expansion is negligible and we can consider the state to be the
{ 6 {
Minkowski vacuum, j i. As such modes redshift to larger wavelengths at tf , the state at
tf can then be taken to be given by U (tf ; ti)j i, where
t
f
i R dtH(t)
U (tf ; ti) = T e t
i
In order to capture arbitrarily large wavelengths at tf in this manner, we formally take
ti !
1. (For free elds, the state de ned in this way at nite times, is the BunchDavies
\vacuum".) Then the desired latetime expectation value of a gauge invariant operator Q
is given in the Schroedinger picture by, h jU (tf ; ti =
1)yQU (tf ; ti =
1)j i.
Now the calculation of the expectation value becomes standard. First, we go over to
the interaction picture, and second we employ the standard trick of continuing the early
evolution slightly into complex time to project the free vacuum j0i onto the interacting
vacuum j i. Thus we arrive at the inin master formula,
h jU (tf ; ti)yQU (tf ; ti)j i = h0jT e
+i
t
R
f
1(1+i )
dt2HiInt(t2)
QI (tf )T e
i
t
R
f
1(1 i )
dt1HiInt(t1)
j0i: (2.2)
In the above, the subscript I denotes that the corresponding operator is to be evaluated
in the interaction picture. Finally, Hint(t) is the interaction part of the Hamiltonian of
the
uctuations, i.e. H = H0 + Hint with H0 being quadratic in
uctuations.
We note
that the antitime ordered product also appears in (2.2). The perturbative expansion of
cosmological correlators of the above general type is facilitated as usual by expanding in
products of Wick contractions, given by inin propagators. This leads to a diagrammatic
form, illustrated in gure 2.
2.2
Useful gauges for general coordinate invariance
Metric and in aton
uctuations are not gauge invariant under di eomorphisms. Hence
we now review two useful gauges and a gauge invariant quantity characterizing the scalar
perturbations during in ation. Our discussion will be brief and for more details the reader
is referred to [9, 47]. For simplicity, we will specialize here to single eld slowroll in ation,
but the considerations are more general.
The metric of dS space is given by,
ds2 =
dt2 + a2(t)d~x2;
with a(t) = eHt being the scale factor in terms of Hubble scale H. To discuss the gauge
choices, it is useful to decompose the spatial metric hij dxidxj in presence of in ationary
backreaction as follows [47],
hij = a2(t) (1 + A) ij +
where, A; B; Ci; ij are two scalars, a divergenceless vector, and a transverse traceless tensor
perturbation respectively. The in aton eld can also be decomposed into a classical part
0(t) and a quantum
uctuation (t; ~x),
(2.3)
(2.4)
(2.5)
(t; ~x) = 0(t) + (t; ~x):
{ 7 {
Using the transformation rules of the metric and scalar eld, it can be shown that the
quantity [48],
R
A
2
1
_
0
;
is gauge invariant. This is the quantity that is conserved on superhorizon scales for single
eld in ation [13, 49{52]. Although R seems to depend on more than one scalar uctuation,
there is only one physical scalar
uctuation which is captured by it. This is because
among the ve scalar uctuations in the metric plus in aton system, two are nondynamical
constraints and two more can be gauged away by appropriate di eomorphisms, leaving only
one uctuation. To make this manifest, we can do gauge transformations which set either
to zero in (2.6) to go to spatially at and comoving gauge respectively. The rst of
these will be most useful for simplifying inin calculations involving Hubblescale massive
particles external to the in ation dynamics, while the second one is useful for constraining
the squeezed limit of the NG due to in ationary dynamics itself.
Spatially
at gauge [9]. In this gauge the spatial metric (2.4) becomes
Gauge invariant answers can be obtained by writing
becomes in this gauge,
in terms of R using (2.6), which
Comoving gauge [9]. In this gauge the spatial metric (2.4) looks like
hij = a2(t) ((1 + A) ij + ij ) ;
= 0. This means the gauge invariant quantity R evaluated
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
hij = a2(t) ( ij + ij ) :
R =
1
_
0
:
R =
A
2
;
Pk
hR(~k)R( ~k)i0;
{ 8 {
with quantum in aton eld
in the new gauge becomes,
which lets us rewrite the spatial metric (2.4) as
hij = a2(t) ((1 + 2R) ij + ij ) ;
with R being conserved after horizon exit (in single eld in ation).
2.3
Observables
Having discussed the gauge choices, we now move on to discussing the observables. The
power spectrum for the density perturbations is given by,
where the 0 denotes the notation that momentum conserving delta functions are taken
away i.e.
hR(~k1)
R(~kn)i = (2 )3 3(~k1 +
+ ~kn)hR(~k1)
R(~kn)i0 :
The power spectrum can be evaluated to be
where the r.h.s. is to be evaluated at the moment of horizon exit k = aH for a given
kmode. Since di erent kmodes exit the horizon at di erent times and H_24 has a slow time
dependence, the combination k3Pk is not exactly kindependent, and we can write
0
1 1
The crude estimate of cosmic variance (1.3) translates to F
10 4{10 3. It is often
conventional in the literature to typify the size of NG by the value of F at the equilateral
point,
Since we are mostly interested here in the squeezed limit for future signals, k3
we will explicitly compute F in that limit, referring to fNL only in the context of current
NG limits (see subsection 6.1). In terms of the quantum in aton eld , the function F
can be rewritten as,
data [2] we get, ns
power spectrum takes the form,
0:96 and H_24 = 8:7
0
where 1
ns is the tilt of the power spectrum and k is a \pivot" scale. From Planck
10 8 at k = 0:05 Mpc 1. In position space, the
hR(~x1)R(~x2)i
jx1
1
x2jns 1
:
To calculate the bispectrum we will be interested in evaluating hR(~k1)R(~k3)R(~k3)i. By
translational invariance the three momenta form a triangle, and by rotational invariance
we are only interested in the shape and size of the triangle, not in the orientation of
the triangle. Furthermore since we also have approximate scale invariance, we do not
care about the overall size of the triangle, so e ectively the momentum dependence of
bispectrum is governed only by the ratios kk31 and kk21 . We denote the bispectrum by the
function B(k1; k2; k3),
B(k1; k2; k3) = hR(~k1)R(~k3)R(~k3)i0:
It is convenient to de ne a dimensionless version of this,
F (k1; k2; k3) =
_
0
h (~k1) (~k2) (~k3)i0
h (~k1) ( ~k1)i0h (~k3) ( ~k3)i0 jk3 k1;k2 ;
and where the r.h.s. is evaluated at the point of horizon exit for each mode.
{ 9 {
(2.15)
Squeezed limit of cosmological correlators
NG from single eld in ation in the squeezed limit
In single eld in ation, NG in the squeezed limit is proportional to the tilt of the in aton
power spectrum [9, 53, 54], i.e.
F Single Field(k1; k2; k3)jk3 k1;k2 = (1
ns) + O
(3.1)
k3
k1
2
:
Let us go to comoving gauge (2.11) to demonstrate this. We are interested in
computing hRh(~k1)Rh(~k2)Rs(~k3)i0, where the subscript h(s) means the associated momentum is
hard(soft). We de ne position space coordinates ~xi to be conjugate to momentum ~ki. In
the limit k3
j~x1 ~x2j
k1; k2 we are interested in an \Operator Product Expansion (OPE)" regime,
j~x1
~x3j. Consider just the leading treelevel structure of the associated diagram
in gure 2, and rst focus on just the boxed subdiagram. We see that for this subdiagram
the soft line is just a slowlyvarying background eld in which we are computing a hard
2point correlator. Thus,
hRh(x1)Rh(x2)Rs(x3)i
hhRh(x1)Rh(x2)iRs( x1+2x2 )Rs(x3)i:
(3.2)
The e ect of the soft mode Rs is just to do the transform ~x ! (1 + Rs)~x of (2.11) within
the leading 2point function of (2.16):
hRh(x1)Rh(x2)iRs
1
(jx1
x2j(1 + Rs))ns 1
(jx1
1
x2j)ns 1 (1
ns)Rs
x1 + x2 :
2
(3.3)
To get the middle expression, we have taken Rs to be approximately constant over distances
of order j~x1
~x2j, a good approximation since k3 ! 0. The last expression follows by
expanding in (small) Rs, evaluated at the midpoint (x~1 + x~2)=2. We have also dropped a
Rsindependent piece since that goes away when we consider the three point function.
Thus the three point function becomes,
hRh(x1)Rh(x2)Rs(x3)i
hhRh(x1)Rh(x2)iRs( x1 +2x2 )Rs(x3)i
(1
ns)
(jx1
1
x2j)ns 1 (jx1
1
x3j)ns 1
:
(3.4)
Fourier transforming to momentum space,
hRh(~k1)Rh(~k2)Rs(~k3)i0
(1
(1
1
1
ns) k4 ns k4 ns
3
1
ns)hRh(~k1)Rh( ~k1)i0hRs(~k3)Rs( ~k3)i0;
(3.5)
leading to (3.1). Subleading corrections proportional to
are absent by rotational
k3
k1
invariance, so the leading corrections are order
The importance of the above expression lies in the fact that in the squeezed limit any
value of F Single Field bigger than O(1
ns) will signal the presence of new physics beyond
single eld in ationary dynamics. In particular, next we comment on what can happen
to the squeezed limit if we have multiple light (m
H) elds (\multi eld in ation") or
m
3.2
H
elds (\quasi single eld in ation" [16]) during in ation.
NG from multi eld in ation in the squeezed limit
If we have light elds with m
H, other than the in aton, then during in ation those
elds can lead to larger NG in the squeezed limit than (3.1), see [55] and references therein.
This can be understood again via similar inin diagrammatics to gure 2. In this case it is
again true that we have to evaluate the hard two point function in the background of some
soft mode, and correlate the result with a R soft mode. However, since Rs is no longer the
only soft mode in the theory,
hRh(x1)Rh(x2)isoft mode 6= hRh(x1)Rh(x2)iRs :
Thus the derivation in the previous subsection does not go through.
Consequently
F Multi Field in the squeezed limit is no longer constrained to be order (1
ns), but rather
it becomes model dependent.
3.3
NG from Hubblescale masses in the squeezed limit
The situation changes quite a lot if we have particles with m
H. Such particles can
modify the bispectrum in a way that in the squeezed limit F contains a nonanalytic part,
F nonanalytic
/ f ( )
94 and f ( ) is a calculable function of the mass of the particle, which is
k3
k1
than H.
of the order 1 when
1 but is \Boltzmann suppressed"
e
for large . We have a
proportionality sign in (3.7) because there are model dependent prefactors which can take
either large or small values, thus from (3.7) itself we can not get a complete estimate of
NG. We will spell out the model dependent prefactors later.
The crucial aspect of (3.7) is that F now contains a nonanalytic dependence on
along with other analytic terms. Importantly this nonanalytic behavior can not be
captured by any single or multi eld in ation models where all the masses are much smaller
(3.6)
(3.7)
(c) Triple Exchange Diagram. Note that all these diagrams rely on mixing between the in aton
uctuation and massive scalar in the (implicit) nontrivial background of rolling 0(t).
as a two point function. The 0 background causing mixing is not explicitly shown, as in
gure 3.
(b) The same \OPE" approximation expressed an in atonh three point function with one in aton
leg set to zero momentum to now explicitly represent the background 0
.
This dependence also encodes the information about the mass of the Hubble scale
particle, via the exponent , [16{21]. If the massive particle has a nonzero spin(s), then
F nonanalytic has an additional factor dependent on Legendre polynomials, Ps(cos ), where
k1 k^3 = cos . If we can measure this angular dependence precisely enough then we can
^
get the information about spin as well [21, 24]. Furthermore, such angular dependence
is absent in purely single eld and some of the multi eld in ation models. This can in
principle help us in distinguishing the \signal" of m
H particles from the \background"
of m
H particles.
We see as we go to the region, m
H, the leading behavior reverts to being analytic
k1
k3 3=2 3=2
, and indistinguishable from purely single eld or multi eld in ation. This
means it is observationally challenging to reach the region m
H and still distinguish and
measure m accurately. Also, at the other extreme, for m
H cosmological production is
strongly Boltzmann suppressed, so observation will again be di cult. Therefore we are led
to a window around H for doing spectroscopy of masses and spins.
Let us brie y explain the form of (3.7), rst concentrating on just the soft
k3dependence. In presence of new particles with m
H there are additional contributions
to the bispectrum beyond those in gure 2. At tree level we can have three diagrammatic
forms, as shown in
gure 3. These are called single, double and triple exchange diagram
based on the number of massive propagators [24]. In the squeezed limit, we are once again
interested in calculating hRh(~k1)Rh(~k2)Rs(~k3)i0 i.e. correlation of two hard modes with a
soft mode. In position space, this again corresponds to an \OPE" limit, jx12j
jx13j, where
the hard subdiagram is given by an e ective local vertex, depicted in gure 4 by the round
black blob. The strength of this e ective vertex is then given by the hard twopoint function
in the background of the massive but k3soft eld, which is predominantly k3independent.
Tracking only the k3dependence is then given by the twopoint correlator shown in gure 4.
The leading k3 dependence can be worked out by the scaling properties of the elds
involved, which can be read o from their classical late time asymptotics. For a general
scalar eld,
9 . This means, O1(2)(~x) can be thought of as an operator with
4
! 0, dS isometry generators in 4D acts as generators of the conformal group in 3D
space. However, the leading e ect of the in ationary background is to break this conformal
invariance of latetime correlators, but only weakly for slow roll. Using the simple scaling
symmetry subgroup of the 3D conformal invariance, we nd3;4
hO (~x2)Rs(~x3)iinf / jx23j
:
should have the factor k323 i
However, it is well known that 2point correlators of di ering scale dimension vanish if
conformal invariance is exact, therefore the implicit proportionality \constant" is suppressed
by slowroll parameters here. Fourier transforming and writing
. We can now put back the k1 dependence, which again by
the above scale invariance can only enter into the expression for F as shown in (3.7).
4
4.1
GaugeHiggs theory and cosmological collider physics
The central plot and its connections to the literature
Having commented on NG and the squeezed limit in general, we focus on what kind of
signature a gauge theory coupled to the in aton will have on NG. In particular we study
signatures of Higgs scalars and gauge bosons. Nontrivial spin of heavy particles in the
context of slowroll in ation was rst considered in ref. [21], primarily for even spin. In
ref. [24] both even and odd spin were considered. In both [21] and [24] no assumptions were
made on the origins of the heavy masses. Here, we will impose the stringent constraints
following from assuming that the heavy masses arise via the Higgs mechanism of
weaklycoupled gauge
eld theory, in particular for spins 0 and 1. In particular, the relevant
nonlinear terms coupling the gaugeHiggs sector and the in aton will be more suppressed
by requiring gauge invariance than would be the case for massive elds unconnected to a
Higgs mechanism.
For example, consider the interaction of a pair of massive spin1 particles, Z , with a
pair of in aton elds. Without considering a gauge theoretic origin for Z, a low dimension
interaction respecting (approximate) in aton shift symmetry has the form,
1
(3.8)
(3.9)
(4.1)
3There can be subleading slowroll
(1
ns) corrections to the exponent which are neglected here.
4Note that 4D dimensionful parameters, such as the Planck scale, do not break this 3D conformal or
scale invariance.
where
is of order the cuto
of EFT. However, if Z is a Higgsed gauge boson, the
analogous interaction must arise from
1
(4.2)
Crucially the interaction between Z and in aton has to happen in the presence of the Higgs
eld H since we are assuming in aton to be a gauge singlet. Assuming the gauge theory
is spontaneously broken, we see that the gauge theory interaction has a suppression of the
order m2Z = 2 compared to the nongauge theoretic case.5 This argument can be generalized
to all the gauge boson interaction terms that we consider below. This also makes a general
point that it can be harder to see NG due to gauge sector particles compared to nongauge
theoretic states. This is especially true for spin1 particles as we saw above.
In [27, 28] the signature of gauge theory was considered, focusing on unbroken
electroweak symmetry during the in ationary phase as well as the scenario of Higgs in ation
(in which the in aton is identi ed with the physical Higgs eld). In a general gauge theory
with unbroken gauge symmetry, the gauge bosons will be massless up to (small) loop
corrections [56, 57]. Nontrivial spectroscopy must then proceed via gaugecharged matter, which
can only appear in loops by charge conservation and the singlet nature of the in aton. Such
loops are di cult (but depending on speci c models, may not be impossible6) to observe
for several reasons. First, at one loop [21], F loop
f~( ) kk31 3+2i
+ g~( ) kk31 3 2i
+
so the fallo is faster compared to (3.7) as one goes to smaller k3. Second, for heavier
masses the Boltzmann suppression goes as e 2
because there is now a pair of massive
particles involved. Thirdly, we will obviously have the loop factor (
161 2 ) suppression.
This gives us motivation to look for bigger tree level e ects which will be present if
gauge symmetry is broken spontaneously during in ation. In [27, 28] such a scenario was
mentioned although the primary focus was on Higgsin ationlike scenarios in which the
Higgs VEV is very large compared to H and consequently the massive gauge bosons are too
heavy to be seen via NG due to Boltzmann suppression. The situation is much better if one
keeps the gauge theory and in aton sectors distinct, with gauge symmetry spontaneously
broken and Higgs VEV not too much larger than H. This is the case we focus on, and we will
see that such scenarios can give rise to observable NG for both spin0 and spin1 particles.
Since the Hubble scale during in ation can be very high (H . 5
1013GeV), in ation
and the study of NG provides an exciting arena to hunt for new particles. In this regard
two distinct possibilities arise. We discuss them next.
5The nongauge theoretic case can be viewed as the limit of the gauge case where mZ
. For example,
a QCD
meson or a spinone superstring excitation cannot be housed in pointparticle EFT, except in the
marginal sense where the e ective cuto is
mZ, where the constraints of gauge invariance disappear.
6For example one can imagine working in an e ective theory of in ation with its cuto
& H, however
if we have a cuto
very close to Hubble then the gauge theory spectrum is no longer separated from the
states coming from some UV completion of the eld theory, and measurements of NG cannot be translated
robustly into information about the gauge theory alone. Such a scenario, of course, is still interesting, but
we do not focus on that in this paper.
HJEP05(218)
1
2
c1
+
We could imagine a scenario in which there exists some new spontaneously broken gauge
theory at H. Then some of the gaugecharged matter and gauge elds may become singlets
under the residual unbroken gauge symmetry. Bosons of this type, spin0 and spin1, can
therefore have Hubble scale masses, couple to the in aton, and leave their signatures on
NG at treelevel. For simplicity here, we focus on spontaneously broken U(1) gauge theory
with no residual gauge symmetry, but is straightforward to generalize to the nonabelian
case. For example, we can imagine a scalar in the fundamental representation of SU(N )
breaking the symmetry to SU(N
1). Then the gauge boson associated with the broken
diagonal generator plays the role of the massive U(1) gauge boson that we consider now.
Let us focus on the case of single eld slowroll in ation. We write an e ective theory
with cuto
. Since we are interested in e ects of gauge theory on NG, we will write down
higher derivative interaction terms between the gauge sector and in aton. But we will not
be explicit about higher derivative terms containing gauge sector elds alone or the in aton
alone, although we will ensure that such terms are within EFT control.
The lagrangian containing the in aton
(with an assumed shift symmetry), the Higgs
(H) and gauge bosons (not necessarily the SM Higgs and gauge bosons) has the form
1
2
L =
Mp2lR + LGauge Theory
2
V ( ) + Liinntf + Lint
infgauge;
(4.3)
where LGauge Theory contains all the terms (including higher derivative terms) containing
gauge theory
elds alone. V ( ) is a generic slow roll potential.
Liinntf contains higher
derivative terms containing in aton alone. For our purpose the interesting interaction
terms between gauge theory and the in aton are contained in Lint
below assuming an UV cuto
and a set of dimensionless EFT coe cients ci,
infgauge, which we write
(HyD H) +
(HyD H) +
:
c22 (@ ) H H +
2 y
c44 (@ )2Z2
(4.4)
In Liinntf, the rst term gives a quadratic mixing between Higgs and Z0. It also couples Higgs,
Z and the in aton. But it does not contain any quadratic mixing between the in aton and
Z; and also none between the in aton and Higgs. But we do see, from
gure 3, that we need
one or more quadratic mixings between the in aton and the massive particle of interest.
Such quadratic mixing does arise from the second and the fth term, which give quadratic
mixing of the in aton with Higgs and Z respectively. The third term gives, among other
interactions, the interaction between an in aton and a pair of Zs. We have not written
operators coming from the expansion in
terms we have already considered.
Hy2H
since these will be subdominant to the
To unpack (4.4) we can go to the unitary gauge for U(1) gauge theory and write down
some of the relevant terms,
h
v
;
Lint
infgauge = 1;Z Z0h + 1;hh h_ +
2v
1;Z Z0h2 +
+
y
H H
+ 5;h _ h_
+ 1
2h
v
+
_
0
_
0
Z
1;Z =
5;Z =
Im(c1) _0mZ
Im(c5) _20mZ v ;
5
1;h =
5;h =
Re(c1) _0
2
Re(c5) _20v :
5
;
2 _h2 +
2v
Z h
(hp+v) T
2
2 =
2c2 _0v ;
2
where we have expanded the Higgs eld in unitary gauge H = 0
and the in aton
eld
term via the parameter
=
c2 2_02 . We also have several quadratic mixing parameters, i,
4.3
Heavylifting of gaugeHiggs theory
Until now, we have been discussing theories with Higgs physics intrinsically of order H.
Now although a future detection of m
H particles via NG will be very interesting in its
own right, given that H may well be orders of magnitude beyond the energies of foreseeable
particle colliders, we would not have valuable complementary access to this physics in the
lab. But as discussed in the introduction, the alternative is the \heavylifted" scenario,
in which m
H during in ation and again yields observable NG, and yet m
H in the
current postin ationary era and therefore conceivably is accessible to collider and other
\lowenergy" probes.
Given a gauge theory at low energy, we can consider adding a nonminimal coupling
of the Higgs to gravity, cRHy
H to the lagrangian (4.3), where we will consider c of order
one. This gives a Higgs e ective potential of the form,
Ve (H) = hjHj
4
2
hjHj
2
cRHyH;
While the curvature is negligible in the current era, during in ation we have R
so that for c > 0, the symmetry breaking scale setting gaugeHiggs physical masses is
naturally of order H.
We can also see how this \heavylifting" mechanism appears in
Einstein frame in which the in aton and Higgs potential get modi ed to
(V ( ) + V (H)) ! (V ( ) + V (H))= 4
V ( ) 1
+ V (H);
(4.8)
4cHyH
Mp2l
!
(4.5)
(4.6)
(4.7)
12H2,
2 = 1 + 2cHyH is the Weyl scaling factor used to get to Einstein frame and we
have kept the leading correction in cHyH=Mp2l.7 For NG, the discussion in the previous
subsection then carries over from this point.
As we elaborated in the introduction, one interesting fact about the heavylifting
mechanism is that it is testable. This requires a knowledge of the couplings of the gauge theory
sector in the current era, where they may be accessible at collider energies, and a
measurement of H during in ation, as for example via the primordial tensor power spectrum.
We can then use the renormalization group to run those couplings up to H, and thereby
predict the mass ratios of spin0 and spin1 h and Z type particles (bosons charged under
the full gauge symmetry which are singlets of the unbroken gauge symmetry) as they were
in the in ationary epoch when they contributed to NG. Here the richer the set of h and Z
type particles, and hence the larger the set of mass ratios, the less precision we would need
to measure each ratio in NG in order to be convinced that we are seeing the same gauge
theory in both regimes.
5
eld slow roll in ation
We saw in the previous section that the leading interaction between in aton and gauge
theory is captured by (4.4) and (4.5). These can be used to estimate the magnitudes
of NG induced by h and Z. However, the parameters appearing in those two lagrangians
have to satisfy several consistency requirements. We rst discuss such restrictions and then
proceed with the estimation of NG. Our discussion in this section will be in the context of
slow roll in ation.
and coupling strengths of e ective theory
We start with the restriction on
, which we saw in the previous section sets the most
optimistic suppression scale for higherdimensional interactions relevant to NG. We imagine
that
roughly represents the mass scale of heavy particles that have been integrated out
to give the e ective nonrenormalizable couplings we need between the gauge sector and
in aton. We can therefore think of them as mass \mediators" of the requisite e ective
interactions. But in general, if such mediators couple substantively to both the in aton and
to the gauge sector, they will also mediate in aton (nonrenormalizable) selfinteractions,
roughly powers of
. In order for the e ective expansion in these powers to be
controlled, we should require
to exceed the in ationary kinetic energy [23],
7There may in addition be direct Higgsin aton couplings even before the Weylrescaling to Einstein
frame, in which case the Einstein frame couplings may be modi ed from that above. However, even this
modi cation would have to share similar features, namely that during in ation the Higgs mass parameter
is e ectively raised to the H2scale and in the current postin ationary era the Higgs mass parameter is
much smaller in order to t the current electroweak data. Therefore, we will not pursue this more general
modi ed lagrangian, for simplicity.
>
q
_0:
(5.1)
In our ensuing discussion of single eld in ation, we will take this bound to hold. We will
assume an approximate in aton shift symmetry during in ation, allowing the 4 to be only
as big as the slowlyrolling kinetic energy rather than a larger scale.
The potential energy of the in aton eld V ( ) gives rise to an even higher energy scale
1
V 4 , which is bigger than
q _0. Approximate shift symmetry during in ation keeps this
scale from spoiling the EFT expansion in higherdimension operators, but after in ation
this symmetry may be signi cantly broken and the higher scale can then a ect dynamics
signi cantly. In particular, EFT with
< V 4 can break down at reheating, signaling that
1
the scale mediators can be reheated and subsequently decay. However, the NG produced
and described by the controlled e ective theory during in ation are already locked in on
superhorizon scales and are insensitive to the subsequent postin ationary breakdown of
the EFT.
Furthermore,
in
theories
involving
large
\vacuum"
expectation
values,
nonrenormalizable operators in the UV theory can become superrenormalizable
(or marginal) in the IR, once some elds are set to their expectation values. There is
then the danger of such e ective superrenormalizable couplings becoming strong in the
IR, and outside perturbative control, or becoming e ective mass terms which are too
large phenomenologically and have to be
netuned to be smaller. This general concern
is realized in the present context, because of the large classical expectation given by
H2, as well as large hHi > H within some of the interesting parameter space. We
nd that these issues are avoided for su ciently small ci in (4.4) with,
c
i
O
H=
q
_
0 ;
which we take to hold from now on. We go into more detail on such restrictions in the
next subsection.
To concretely illustrate the above considerations, consider the following set up. We
imagine a theory, with a cuto
interact directly. Thus a term like 102 (@ ) H H is absent in the lagrangian. However, we
2 y
0 & V 14 >
q _0, in which the in aton and Higgs do not
assume the presence of a \mediator" gaugesinglet particle
with mass m
talks to both the in aton and Higgs separately via the terms,
q _0, which
will lead to a controlled e ective theory expansion. These
parameters are reproduced naturally if we take,
H; 0
1
V 4
1=4q _0; m
q _0 in
i
O
H=
q _
0
1
0
Then below m , we can integrate
out to write an e ective coupling between the in aton
Now in the previous paragraphs we have stated that the choice of
&
q _
0 and
(5.2)
(5.3)
(5.4)
HJEP05(218)
e
l
e
l
g
n
isFh0.005
0.010
5
5
10
20
50
HJEP05(218)
Slowroll description (B.17) with c2 = p _
m=1.9H
m=2.2H
m=1.6H
Double exchange diagram
As derived in appendix B in (B.31),
i 2
32
where, A( ) and s( ) are mass dependent coe cients: A( ) =
sin( 4 + i
2 ); and s( ) can be represented by the integral, s( ) = R01 dxx2 e ixJ+(x)x3=2+i
where, J+(x) is a somewhat complicated function given in (B.23).
We exemplify the
strength of NG in table 3 for the benchmark values, 2 = 0:2; h = 0:5. We illustrate the
momentum dependence of F double in
h
gure 9 and 10.
In the special case of single eld
mass
j
0.2
0.0
3
2
1
1
2
e
l
b
u
o
dFh 0
k1
k3
k1
k3
m=1.6H
m=1.9H
mass
mass
j
j h
slowroll in ation, using lagrangian (5.6), F double takes an identical form to (B.31) except
h
the coupling constants are now di erent (B.29),
k3
k1
3=2
i
k3
2k1
+ (
!
) :
(7.7)
The strength of the NG then, for the same set of benchmark values, c2 = p _
q _0, is shown in table 4. The shape dependence is identical to gures 9 and 10, so
H , h = 2H_20 ,
0
not shown explicitly.
7.3
Triple exchange diagram
The triple exchange diagram has been calculated in [61], but we include it here for
completeness and comparison to the other diagrams. As derived in appendix B in (B.35),
h
where, A( ) is the same coe cient as introduced above and t( ) = R01 dxx4 J+(x)2x 23 +i .
We exemplify the strength of NG below for the benchmark values, 2 = 0:2 and h = 0:5
in table 5. We illustrate the momentum dependence of F triple in
gure 11 and 12.
In
the special case of single eld slowroll in ation, using lagrangian (5.6), F triple takes an
identical form except the coupling constants are now di erent (B.34),
h
The strength of the NG then, for the same set of benchmark values, c2 = p _
q _0, is shown in table 6. The shape dependence is identical to gures 11 and 12, so
not shown explicitly.
k3 2
3
k1
)
(7.8)
h
!
) : (7.9)
H , h = 2H_20 ,
0
5
10
20
50
lep 2
itr h
F
0
1
3
4
5
k1
k3
h
k1
k3
mass
1.6 H
1.9 H
2.2 H
j h
f triple
j
mass
f single
j
Detailed form of NG mediated by Z
To discuss the form of F for NG mediated by Z, we again rst focus on the Goldstone
effective description as before, and specialize to the single eld slowroll description following
that. Since the triple exchange diagram is too small to make any observable contribution
we will restrict ourselves to single and double exchange diagrams.
The Goldstone e ective lagrangian needed for this case is given by (6.32) which we
rewrite,
k^3 and k^1,
2
Im(d1)mZ v _ cZ0
2 f 2
cZ
(8.1)
In this case in the squeezed limit, F (k1; k2; k3) is a function of kk31 and also the angle between
F =
f ( )
+ f ( )
k1
k1
sin2
= k^3 k^1. We also see that F falls faster with kk31 . The angle dependence, in
principle, gives an important handle to determine the spin1 nature of Z. Recently in [25]
it was analyzed to what extent future galaxy surveys can constrain mass and spin. A
forecast using 21cm cosmology would also be important and possibly more constraining.
Now we give the expressions for f ( ) for the single exchange diagram, leaving the
details for the appendix D. The computation of double exchange diagram will not be
performed in this paper, however using the mixed propagator formalism [61] it can be
done. Here, we will only give some reasonable estimates.
8.1
Single exchange diagram
As derived in appendix D in (D.6),
Z
v
2
( 2 2i )(1 i sinh(
i
cosh(
)
k1
+(
!
We illustrate the strength of NG, for the parameter choices,
= 8H in table 7. We see the strengths are quite weak, hence 21cm
cosmology is critical if we are to see NG due to the single exchange diagram. Note that even
an imprecise measurement should be readily distinguishable from scalarmediated NG and
(8.2)
(8.3)
!
) ;
NG purely due to the in ationary dynamics (analytic in kk31 ), due to the nontrivial angular
dependence.
We now discuss single eld slowroll in ation. The relevant lagrangian for a
nonnegligible Zmediated signal arises when the associated Higgs scalar h is heavy enough
that its onshell propagation is Boltzmann suppressed, but can be integrated out to yield
new Z vertices, as in (5.21). It has an identical structure to the Goldstone lagrangian (8.1)
above, as shown in D. Hence F can be obtained just by the replacement,
vmZ
2
!
1;Z 2
m2 :
We see for 1;Z = 1; 2 = 1; mh = 3H we have roughly the same strength of NG as the
e ective Goldstone theory. However, we get parametrically bigger NG in both e ective
theories from the double exchange diagram in gure 3, which we now discuss.
Double exchange diagram
As we mentioned above, in this paper we will give only an estimate of the double exchange
diagram. As we have explained in section 3, in the squeezed limit diagrams factorize into
contributions from hard and soft processes. This means in gure 3(b), the Z propagator
having hard momenta k2 is expected to be a function of O(1) (in Hubble units). In that
approximation the diagram then has the same topology as the single exchange diagram.
However, as can be seen from the lagrangian (8.1), the parametric strength of the diagram
goes like
2
f 2
2
;
(8.5)
(8.4)
which has the enhancement by f2 . Thus, while we saw that the singleexchange
contri2
bution was at best marginally detectable in the future, the doubleexchange contribution
should be much more promising in magnitude for
5{10H, v
2 3H, with fZ
0:1{1.
We leave a precise calculation of this for later work, to hopefully con rm this expectation.
Moving to the case of single eld slowroll in ation, from (5.21) arising from integrating
out the associated heavy h, we see that the doubleexchange diagram is parametrically
This should yield a weak but detectable signal.
enhanced over singleexchange by a factor of Hvm_2h0 , so that fZ
9
Concluding remarks and future directions
0:01 for v
q _0; mh
3H.
Cosmological Collider Physics builds on the distinctive nonanalytic momentum
dependence of primordial NG mediated by particles with masses m
H, in contrast to the
analytic dependence of NG due purely to the in ationary dynamics, driven by elds with
m
H. In this paper, we focused on the question of whether gaugetheories with such
ultrahigh
H mass scales could be detected by this means, since such theories are
obviously very highly motivated. If the gauge symmetry is unbroken during in ation,
gaugecharged states can only a ect primordial NG via very small looplevel e ects, di cult to
observe. However, we showed that when the gaugesymmetry is (partially) Higgsed, the
Z
Goldstone EFT
Slowroll Models
with
5H
with
10H
with
60H
1{10
0:1{1
0:1{1
0:01{0:1
0:01{0:1
0:001{0:01
Higgstype spin0 and Ztype spin1 bosons can contribute at tree level to potentially
observable NG. The simplest e ective vertices one can write connecting the gaugeHiggs states
q _
0
to the in aton so as to mediate NG are nonrenormalizable, suppressed at least by powers
of the cuto of the in ationary EFT, , representing the threshold of even heavier physics
that has been integrated out. The largest NG will then come by considering the lowest
consistent . We studied these NG within two e ective descriptions of the in ationary
dynamics: a) generic slowroll in ation models, and b) the e ective Goldstone description
of in aton quantum
uctuations. In slowroll, the minimal cuto
is given by the scale of
kinetic energy of the rolling in aton eld,
60H. The e ective Goldstone description
is more agnostic about in ationary dynamics, treating this as a given classical background
process, in which case
can be as low as a few H. Of course, the detailed strengths of
NG, F , that we get in the two cases are modelparameter dependent, but we can brie y
summarize the results in sections 7 and 8 in table 8. The dimensionless bispectrum F
(see (2.17), (2.18)) given above is the maximum value taken in the squeezed regime. Based
on the above table, several remarks are in order. While the above choices for EFT cuto s
lead to an observable strength of NG, we cannot make the cuto s much bigger, since the
NG falls rapidly as a function of squeezing and the observable precision is limited by cosmic
variance, F
10 4{10 3, (1.3). The scale of Higgsing, v, is also relevant to our
theoretical control. Higgsing obviously relaxes the tight constraints of gauge invariance, allowing
treelevel NG. But there are nontrivial constraints of the gauge structure following from
having to expand observables in powers of v= . In the UV limit v
, the constraints of
gaugeinvariance disappear altogether. To stay in theoretic control, we have chosen v . 31
in our studies.
We have used e ective nonrenormalizable vertices for this paper, but it is obviously of
great interest and importance to seek a more UVcomplete level of theoretical description
to have greater con dence in the opportunity to detect gauge theory states in NG. We
see that the strength of NG is bigger when it is mediated by h's compared to mediation
by Z's. Furthermore, if cosmological collider physics turns out to be in a purely
gaugetheoretic domain, then we would not see any states with spin > 1, and their associated
angular dependences. Spin > 2 mediated NG would signal a breakdown of pointparticle
eld theories, perhaps signaling the onset of string theoretic structure. On the other hand,
observing spins 0; 1 only, with stronger spin0 signals, would give strong evidence for the
structure studied above. While the (NM)SM gives only one h and one Z, extensions of it
(for example, even just some colored scalars) or whole new gauge sectors are capable of
giving multiple h=Ztype states to observe.
We have argued that a strong possibility for mgauge theory
H is that they arise
via a \heavylifting" mechanism from much lowerscale gauge theories in the current era.
If these gauge theories are already seen at lowerscale terrestrial experiments, then the
renormalization group allows us to predict expected mass ratios in NG. In principle, such
corroboration would provide spectacular evidence for the large range of validity of such
gauge theories, and the absence of intervening (coupled) states. However, we cannot hope
to get a very precise measurement of such mass ratios, given cosmic variance. But if we are
ever in the position to predict even a few such ratios, modestly precise measurements in
NG would still be compelling. Alternatively, of course, we may discover wholly unexpected
gaugestructure within the NG, at least dimly seen.
There are multiple future directions which remain to pursue. There is obviously the
need for an explicit calculation of the doubleexchange diagram involving Ztype particles
which would provide a check for our estimates. Cosmological correlations derived from
in ationary expansion are famously nearly spatially scaleinvariant. But in large regimes
of slowroll in ation or in the Goldstone description, the correlators are actually nearly
spatially conformally invariant, that is they are close to the isometries of dS spacetime.
In this paper, we have assumed this regime of in ationary dynamics. But it is possible
to relax this assumption of approximate conformal invariance, and just keep approximate
scale invariance, for example allowing a small speed of in aton
uctuations, cs
1, which
can give rise to larger NG [45, 63, 64], even allowing us to probe loop e ects of charged
states. This remains to be explored. There is also the generic question of how e ciently
we can use NG templates to look for simultaneous presence of spin0 and spin1 particles,
with a \background" of in ationary NG as well as latetime e ects. Recent preliminary
studies in these directions appear in [25, 62] which suggest that some of the stronger signals
we describe above would be visible with reasonable precision.
We can view the heavylifting mechanism as leveraging unnaturalness, by noting
that the lowdimension Higgs mass term of elementary Higgs
elds is very \unstable"
to curvaturerelated corrections. In that sense, con rming heavylifting of an unnatural
gaugeHiggs theory, such as the (NM)SM, would be a strong sign that naturalness is
massively violated in Nature. Of course, the validity of naturalness is one of the burning
debates and concerns within fundamental physics. But it is also possible that terrestrial
experiments show us a natural theory, such as a supersymmetric gauge theory. One can
then consider the possibility of heavylifting of such a natural theory. Depending on the
nature of supersymmetrybreaking it is possible that the lifted gauge theory exhibits a
different pattern of supersymmetry soft breaking and associated Higgsing than the unlifted
theory in the current era. We leave an investigation of supersymmetric gaugeHiggs theory
for the future.
We have seen that invaluable information on the gaugetheoretic structure of the laws
of nature can be imprinted on cosmological NG, but we have also seen that these signals
are extremely weak given cosmic variance. To have any chance of seeing and deciphering
such exciting physics will require pushing experimental precision and understanding of
systematic uncertainties to the their limits. Heavylifting indeed!
2
2
= 0:
This can be solved in terms of Hankel (or equivalently, Bessel) functions. After Fourier
transforming to ~kspace, we can write a general classical solution as,
( ; ~k) = c1(
) 23 Hi(1)( k ) + c2(
.
As usual, to canonically quantize the theory, we elevate the coe cients c1; c2 to linear
The quantum eld thereby has the form,
combinations of creation and destruction operators, a~y ; a~k, on the BunchDavies vacuum.
k
Acknowledgments
The authors would like to thank Nima ArkaniHamed, Julian Munoz, Marc Kamionkowski,
David E. Kaplan for helpful discussions. This research was supported in part by the NSF
under Grant No. PHY1620074 and by the Maryland Center for Fundamental Physics
(MCFP).
A
Scalar elds in dS space
The metric for the Poincare patch of dS spacetime in Hubble units can be written as
! 1
lim fk( ) = (
r 1 eik :
2k
d 2 + d~x2
2
:
A.1
Massive elds
We want to get the mode functions for a quantum eld in dS. This can be obtained by rst
solving the classical equation of motion and then by canonically quantizing the theory. Let
us start by writing the equation of motion,
( ; ~k) = fk( )a~y + fk( )a ~k;
k
where the mode functions, fk( ) and fk( ) (or equivalently, the linear combinations referred
to above), are determined as follows. We rst nd the conjugate momentum
demand, [ ( ; ~x); ( ; ~y)] = i 3(~x
Wronskian condition on the mode functions at
~y) and [a~k; ay ] = (2 )3 3(~k
~k0
1,
~k0). This gives the
fk( )fk0 ( ) fk0 ( )fk( ) = i 2:
To impose the Bunch Davies vacuum we demand fk( ) / eik , and using the Wronskian
condition (A.6) we can also x the normalization of fk( ) up to a phase. In summary,
we demand
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
p
2
p
2
fk( ) = (+ie i =4)
e =2(
) 23 Hi(1)( k ):
This can be satis ed by choosing
and
Here, we have introduced some phase factors which are just conventions, which drop out
HJEP05(218)
when we calculate propagators.
We note that in (2.2) we have both time and antitime ordered expressions appearing.
Since a propagator involves two mode functions, we can have a total of four types of
propagators depending on the mode functions coming from either the time or antitime ordered
part. We denote time(antitime) ordering by a +( ) sign. As an example, a propagator
G+ (k; ; 0) means the mode function with argument ( 0) is coming from time(antitime)
ordering. Similarly, G++(k; ; 0) means both the mode functions are coming from time
ordering. Thus we can write
G++(k; ; 0) = fk( 0)fk( ) ( 0
) + fk( )fk( 0) (
G+ (k; ; 0) = fk( 0)fk( ) ;
G +(k; ; 0) = fk( )fk( 0) ;
G
(k; ; 0) = fk( 0)fk( ) (
0) + fk( )fk( 0) ( 0
0) ;
) :
Among these four, G+ ; G++ are conjugates of G +; G
respectively, and so we only
have two independent propagators.
A.2
In aton mode functions
Mode functions for massless elds, in particular the in aton, follow by using
= 3i=2
in (A.8) and (A.9), which gives
fk( ) =
fk( ) =
(1
ik )eik
p2k3
(1 + ik )e ik
p2k3
;
:
A.3
Some useful relations for diagrammatic calculations
For later use we also note a few relations involving Hankel and hypergeometric functions
that arise upon evaluating the Feynman diagrams for the NG correlators of interest. We
can write the following integral involving Hankel functions in terms of a hypergeometric
(A.8)
(A.9)
(A.10)
(A.11)
(n+1+i ) (n+1 i )2F1(n+1+i ; n+1 i ; n+3=2; (1 p)=2);
(n+1 i ) (n+1+i )2F1(n+1 i ; n+1+i ; n+3=2; (1 p)=2):
It is useful later to approximate these expressions for large p by using limiting forms of
Hankel and hypergeometric functions,
e =2Hi(2)(z) =
=2Hi(1)(z) =
i
(i )(z=2) i e =2 + ( i )(z=2)+i e
=2 ;
( i )(z=2)+i e =2 + (+i )(z=2) i e
=2 :
We also need large negative argument expansion of hypergeometric function,
2F1(a; b; c; z) =
(b) (c
a) (c)
( z) a +
(c) (a
(a) (c
( z) b:
function (valid for real
and
i < 12 ),
e =2 Z 1
= p
= p
0
( i=2)n
2 (n+3=2)
=2 Z 1
0
dxxne ipxHi(2)(x)
dxxne+ipxHi(1)(x)
B.1
NG due to h exchange
Calculation of single exchange diagram
We will use the inin formula (2.2) to calculate NG due to the single exchange diagram
which is depicted in
gure 3(a). We begin by reviewing this calculation in the context of
single eld slowroll in ation, as originally performed in [21]. The relevant terms in the
lagrangian (5.6) for such a diagram are
L
2 _0
:
Note in (2.2) we have both time and antitime ordering. Thus each vertex can contribute
either from time or antitime ordering. So an inin diagram with n vertices gives rise to
2n subdiagrams. These subdiagrams di er in the type of propagators used for the massive
particle and in atons. For example if both the vertices are coming from time ordering, we
should use G++ for the massive propagator as de ned in (A.10). We call the subdiagram
containing G++ to be I++. Thus for the single exchange diagram we have four subdiagrams
which we denote by I++; I+ ; I +; I
depending on which kind of massive propagator has
been used. However, to compute the entire three point function due to single exchange
diagram, we have to consider only two subdiagrams, since the other two are related by
complex conjugation. For example, we will calculate only I
and I+
which are related
to I++ and I + respectively by complex conjugation. We sum all four contributions to
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(B.1)
get the nal answer. To clarify the above comments, we write the expressions for four
subdiagrams schematically,
I
/ ( i)( i)
Z d Z d 0
4
0
4
g (k3; 0)g~ (k1; k2; )G
(k3; ; 0):
(B.2)
The prefactors i arise depending on whether we use e i R Hdt for time ordering or e+i R Hdt
for antitime ordering. g ; g~
are in aton bulkboundary propagators (which we de ne
below); and G
are bulkbulk propagators (A.10) for h.
B.1.1
Calculation of I+
Let us start with I+
diagram. This diagram factorizes into a product of two integrals
with one coming from time ordering and another from anti time ordering.
Antitime ordered contribution.
We rst calculate in aton contribution using in
aton mode function (A.11),
+ i
1
2
:
g (k; )
h _( ; ~k) ( 0 ! 0; ~k)i =
to write the anti time ordered contribution as
Antitime Ordered Contribution = (+i)
Using the mode functions (A.9) and relation (A.13) we get
Antitime Ordered Contribution =
Time ordered contribution. Let us rst calculate the in aton contribution again.
Now we have to do a little more work since based on (B.1) we see that we have to nd the
contraction which can be schematically written as h j(@ )2i. Writing g~+(k1; k2; ) as,
g~+(k1; k2; )
we get, using in aton mode function (A.11),
g~+(k1; k2; ) =
1
2k13
+ 2
( ik1i)( ik2i)
1
2k23
(1
eik12 k k
2 2 4
1 2
ik1 )(1
2k132k23
ik2 ) eik12 ;
where we have de ned, k12 = k1 + k2. We can simplify this by removing some dependent
factors by writing the above as an operator D acting on eik12 , where
D
k k
1 2@k12 + ( ~k1 ~k2)(1
2 2 2
Then,
g~+(k1; k2; ) =
2
4k13k23 Deik12 :
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
Thus the time ordered contribution looks like
Time Ordered Contribution = ( i)
1
4k13k23 D
1
Z 0 d eik12 fk3( ):
2
The integral can be evaluated using (A.12) and (A.8) to get a hypergeometric function.
Since we will be interested in squeezed limit, k3
k1; k2, we can expand the answer
using (A.16). We then get
Time Ordered Contribution
1
= 4p2k13k23pk3 D
The action of D simpli es in the squeezed limit,
( 2i ) (1=2+i ) p
1=2 i
(2i ) (1=2 i ) p
1=2+i
(1=2 i )
2
(1=2+i )
2
HJEP05(218)
8
Dk12 = (
2)k122+ ;
(B.10)
(B.11)
:
(B.12)
using which,
1
8
2
Thus we have
I
+ I+
2
= _20 64k12k22k32
1
+ ( !
):
Time Ordered Contribution = 4p2k13k23pk3
(3=2+i )(5=2+i )
( 2i ) (1=2+i ) 2
(1=2 i )
k12 2k3
k12
Now we are ready to put together both the contributions:
1
I+ = _20 64k12k22k32 (1=2 + i )2 ( 2i )(3=2 + i )(5=2 + i )
1=2 i
k1
k3
+( !
):
(B.13)
1=2 i
+ ( !
):
(B.14)
B.1.2
Calculation of I
I
diagram is in general complicated since it does not factorize into
and 0 integrals. But
we can still calculate the nonanalytic terms in k3 in the squeezed limit. This is because in
the squeezed limit, 0 integral contributes when
0
integral is dominant when
O( k112 ). Thus when k3
(A.10) drops out and the integral approximately factorizes. The 0 integral then is
identical to what we had for I+ ; whereas the only change for integral is that k12 !
k12.
O( k13 ), whereas the contribution of
k12, one of the step functions
1
2
+ i
2
+ i
+ i
k1
k3
12 i
1
ei ( 12 i )
(B.15)
h (~k1) (~k2) (~k3)i = _0 16k12k22k32
2
2
1
The full three point function can be written as a sum of I++; I+ ; I +; I
. This gives
(1+i sinh(
This gives F as a function of k3
as de ned in (2.18) to be
1
4
1
2
2
2
2
2
k1
3
2
3
2
1
2
5
2
5
2
3
2
h
(1+i sinh(
For the case of the Goldstone e ective description of in ation, with
f , the
relevant terms in the lagrangian are given by (6.28). The calculation of F single follows
identical steps as above with the di erence that the operator D = k k
1 2 k12
h
52 + i
! 12 12 + i
32 + i . Hence the nal answer
into, the replacement
reads (taking d2 = 1),
h
(1+i sinh(
Calculation of double exchange diagram
We see from the gure 3(a) that there is a quadratic mixing between in aton and Higgs
eld h. For numerical simpli cation one can de ne a mixed propagator which captures
this mixing [61]. Using this mixed propagator, we then calculate double exchange diagram
numerically. We rst focus on single eld slow roll in ation.
B.2.1
Mixed propagators
A mixed propagator is characterized by the 3 momentum (say k) owing through the line
and bulk time coordinate . We have to sum over all time instants 0 where the mixing
occurs. The bulk time coordinate
can be part of either time or antitime ordering. Let us
rst consider the case where
comes from time ordering, and denote that mixed propagator
by, D+( ; k). Then we have two possibilities, (a) 0 comes also from time ordering, in which
case the we can write the contribution of the mixing part of the entire diagram as,
k3 12 +i
k1
k3 32 +i
k1
k3 32 +i
k1
(B.16)
(B.17)
(B.18)
) :
I+ =
Z 0 d 0
1
0
4 g+(k; 0) (
0)f ( )f ( 0) + ( 0
)f ( 0)f ( ) ;
(B.19)
and, (b) 0 comes from antitime ordering, for which we get,
I
= +
g (k; 0)f ( 0)f ( ):
(B.20)
The overall signs are due to again e i R dtH and we have omitted a factor of +i for simplicity
which we will restore in the nal expression for the mixed propagator. We can then rewrite
I+ as
f ( )
0
4 g+(k; 0)f ( 0):
(B.21)
So the whole contribution of the mixed propagator when
comes from time ordering is
D+( ; k) = (+i)(
2)(I+ +I )
= (+i)(
2) 2iIm(I )+f ( )
0
4 g+(k; 0)f ( 0) f ( )
0
4 g+(k; 0)f ( 0) :
We have restored the factor of +i and also put the mixing vertex
Using the mode functions for in atons (A.11) and massive scalars (A.8) and (A.9), and
also the relations, (A.13), (A.12), we can evaluate the mixed propagator analytically. For
convenience, we de ne the function J+(
k) = 8k3 D+( ; k):
J+(x) = x 2
2 sech(
3 hp
)e =2
Hi(2)(x)ei =4 + H(1i) (x)e i =4
2pxHi(1)(x) (csch(
2pxHi(2)(x) (csch(
)F( ; x) + (1
coth(
)F( ; x)
(1 + coth(
; x))
where x =
k , and F( ; x) is given in terms of hypergeometric function 2F2,
F( ; x) =
1
1
x
i ) 2 + i 2
2F2
1
2
1
2
3
2
i ;
i ;
i ; 1
2i ; 2ix : (B.24)
We will be using the small argument limit of J+(x),
3
J+(x) = A( )x 2 (x=2)i + A(
3
)x 2 (x=2) i ;
where A( ) =
) ( i ) sin( 4 + i 2 ).
B.2.2
Threepoint function
After we incorporate appropriate internal lines into mixed propagators, the double exchange
diagram e ectively contains a single vertex. Thus we only have two diagrams to calculate,
with one of them being the conjugate of the other. Let us start with the time ordered
contribution. Since the lagrangian (5.6) contains a term of the form _
0
_h2 then we get
+ i
0
Z d
4
k
2 2
1
2k13
eik1 D+( ; k2)D+( ; k3):
(B.26)
(B.22)
(B.23)
(B.25)
Once again we will be interested in the limit k3
k1, so using the small argument
Z d2 eik1 D+( ; k1)D+( ; k3)
expansion of J+,
+i _
0
1
2k1
2 2
2 2
2
64k13k33
2 _0
2
2 16
k3
k1
k3
k1
3=2
k3
2k1
3=2
k3
2k1
k1
2
2 _0 64k13k33 A( )s( )
k3
k1
3=2 k3
2k1
k1
3=2 k3
2k1
i !
; (B.27)
where we have de ned s( ) = R01 dxx2 e ixJ+(x)x3=2+i . The full three point function then
becomes, after adding the antitime ordered contribution,
)s (
+ ( !
):
(B.28)
This gives F as a function of kk31 after summing over permutation k1 $ k2,
A (
)s (
+ ( !
): (B.29)
terms of the function, s( ) = R01 dxx2 e ixI+(x)x3=2
we have,
For m < 32H we get an appropriately modi ed version of the above expression. In
2 2
2 8
k1
k3 3=2+
.
F double =
h
B( )Im(s( ))
(B.30)
where B( )
2 +1 ( )p2= sec(
kept a contribution of the form
For the Goldstone description we see from (6.28) that the functional form for F double
h
is identical to above. For the overall coe cient we change, 22 ! 21 2( 2v)2f 2, and hence,
) sin( =4
=2). In deriving the above we have not
i 2
32
F double = 2( 2vf )2
h
A (
)s (
k3
k1
3=2
k3
2k1
+ ( !
):
(B.31)
B.3
Calculation of triple exchange diagram
Using the mixed propagator, the triple exchange diagram can also be calculated in an
identical manner [61]. Using the cubic Higgs vertex, h2v h3 we can write the time ordered
diagram as,
( i) hv Z d
2
4 D+( ; k1)D+( ; k2)D+( ; k3):
(B.32)
Again in the limit k3
k1 we can use the small argument expansion of J+(x),
( i) hv Z d
2
= ( i) hv
2
4 D+( ; k1)D+( ; k2)D+( ; k3)
3 3
2
83k13k33 A( )t( )
3
k3 2 k3
k1
2k1
3
k3 2 k3
k1
2k1
i !
; (B.33)
) ( i ) sin( 4 + i 2 ) and t( ) = R01 dxx4 J+(x)2x 32 +i . After
adding the antitime ordered contribution and permutation k1 $ k2 we get
h
A (
)t (
k3
2k1
+ (
For the Goldstone description we see from (6.28) that the functional form for F triple is
identical to above. For the overall coe cient we change, 32 ! ( 2v)3, and hence,
h
A (
)t (
k1
k3
2k1
+ (
128
d x
4 p
g
4
1
2
m2Z2 ;
r F
= m2Z :
r Z
= 0:
C
Massive vector elds in dS space
Here we will derive mode functions for massive spin1 elds [24], which will be useful in
the next appendix in computing NG mediated by Ztype particles.
C.1
Mode functions in momentum space
We start with the lagrangian,
where F
= r Z
r Z . Variation of the action yields the equation of motion,
Taking the divergence of both sides and also using the fact that r r F
/ R
F
= 0,
we get
Mode functions for Z are then obtained by solving (C.2) and (C.3). The NG correlators
involve mixing the in aton with the Z, which is constrained by the spatial rotation and
translation invariance. Therefore only the longitudinal Z polarization, which is a spatial
scalar, can appear since the in aton is obviously scalar. We are interested in the mode
functions for this degree of freedom. It is shared between the timelike component Z and
the longitudinal spatial component Zlong = Z~ k^ with k^ being a unit vector pointing in the
direction of propagation.
Fourier transforming from ( ; ~x) to ( ; ~k) coordinates, the constraint equation reads,
2
From (C.2) and (C.4) we get the equation of motion for the component Z ,
2
(m2 + 2)
2
Z = 0:
i
):
(B.34)
h
):
(B.35)
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
This is almost identical to the equation of motion for the scalar (A.3). The solutions
are again given in terms of Hankel functions, but with
After we obtain the mode function for Z , that for Zlong is simply obtained from the
constraint equation (C.4). In parallel with the case of scalars, the quantum
eld is obtained
by elevating the free superposition coe cients in the general classical solution to linear
combinations of creation and destruction operators on the BunchDavies vacuum,
2 = m2 + 2
94 = m2
1
4
Z ( ; ~k) = hk;0( )b~y + hk;0( )b ~k;
k
(C.6)
(D.1)
where the mode functions are
hk;0( ) = N e 2 (
hk;l( ) = N e 2
2k
) 23 Hi(1)( k )
1
) 2
For single eld slowroll in ation, the lagrangian involving the in aton and spin1 Z
relevant for singleexchange (diagram (a) in gure 3) is given by (5.21) to be
Hi(1)( k )+k Hi(1+)1( k ) k H(1) ( k ) ; (C.8)
i 1
i p k . In the above b and by are annihilation and creation operators for the
ik1i
2k13
3
2
longitudinal degree of freedom for spin1.
D
NG due to Z exchange
L =
_Z + _
0
where
contribution to future work, although we have estimated its strength and it seems readily
= 1m;Z2h 2 . As discussed earlier, we defer the computation of the doubleexchange
detectable in future measurements. We have also shown earlier that the tripleexchange
contribution is suppressed and can be neglected.
We now parallel the steps taken in the calculation of NG for the case of singleexchange
of a scalar h. We start with I + diagram. In this case the time ordered and antitime
ordered contribution factorize, and we can evaluate them separately.
3
2
f1
; 2p +ipf1
; 2p +f2
; 2p
ipf2
; 2p
f3
; 2p +ipf3
; 2p
2
2
Time ordered contribution.
(+i )
k32 02
2k33
Antitime ordered contribution.
i
0
= _0 16k2k12k323 mZ
4
p
2
k
2 2
2
2k23
e i4 (k^1 k^3)
Z 0 d
1
1
1
1
2
1
2
eik3 0 hk;0( ) =
1
p
3
4 2 mZ k32
De ning p = kk13 ,
3
2
+ i
3
2
: (D.2)
e ik2
e ik1 (1+ik1 )k^3ihk;l( )
(D.3)
We multiply the above two contributions and also sum over I++; I
; I+
diagrams.
Finally we sum over permutations ~k1 $ ~k2 to get,
Z
2
3
2
2
cosh(
( 2 2i )(1 i sinh(
k3 52 +i
k1
= k^1 k^3 and we have used large negative argument expansion of hypergeometric
For the Goldstone description, we see from (6.32) that the functional form of F single is
identical to the above. The overall coe cient is changed to
! 2
vmZ (taking Im(d1) = 1).
where
function.
Hence we get
Z
(D.4)
(D.5)
(D.6)
HJEP05(218)
where di erent integrals involving Hankel functions have been evaluated using (A.13),
f1(n; p) =
f2(n; p) = (+i)
f3(n; p) = ( i)
(n + 3=2) p
1
(+i=2)n
(n + 3=2) p
(+i=2)n
(n + 3=2) p
1
1
(n + 1
i ) (n + 1 + i )
(n + 2
i ) (n + i )
(n + 2 + i ) (n
i )
2F1(n + i ; n + 2
i ; n + 3=2; (1
p)=2) ;
2F1(n + 2 + i ; n
i ; n + 3=2; (1
p)=2) :
2F1(n + 1 + i ; n + 1
i ; n + 3=2; (1
p)=2) ;
v
2
2 1
2
3
2
2
2
cosh(
( 2 2i )(1 i sinh(
k1
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