#  Gravitational Contact Interactions —
— Does the Jordan Frame Exist?
In Honor of Graham G. Ross

Christopher T. Hill<EMAIL_ADDRESS>Fermi National Accelerator Laboratory
P.O. Box 500, Batavia, Illinois 60510, USA

###### Abstract

Scalar–tensor theories, with Einstein Hilbert and non-minimal interactions,
$\sim M^{2}R/2-\alpha\phi^{2}R/12$, have graviton exchange induced contact
interactions. These always pull the theory back into a formal Einstein frame
in which $\alpha$ does not exist. In the Einstein frame, contact terms are
induced, that are then absorbed back into other couplings to define the
effective potential (or equivalently, the renormalization group). We show how
these effects manifest themselves in a simple model by computing the effective
potential in both frames displaying the discrepancy.

††preprint: FERMILAB-CONF-22-699

##  I Introduction

Today we are here to honor the life and career of our colleague, Graham
Ross.111 Invited Talk, ”Workshop on the Standard Model and Beyond,
Graham G. Ross Memorial Session,” Corfu, Sept. 1, 2022.  I’ve known Graham for
nearly 50 years, beginning at Caltech when I was a graduate student in the
mid-1970’s and he was a Research Scientist. He became a mentor and good
friend.

Back then, we worked on the new theory, QCD, exploring the photon structure
functions, photon , and nonleptonic weak interactions, nonleptonic , with
renormalization group (RG) methods. In the latter we followed the pioneering
work of Shifman , where I was first introduced to “contact interactions” via
the so-called “penguin diagrams” (so named in ref.penguins ). Contact
interactions will play a central role in my talk today.

In 1981, I visited Oxford where Graham had joined the faculty. There we
discussed the possibility that the top quark would be very heavy. He had some
ideas based upon the RG equation for the top quark Yukawa coupling, $g_{top}$.
He proposed that $g_{top}/g_{QCD}$ would flow into the infra-red to a fixed
ratio (a tracker solution), which predicted a top quark mass of about 110 GeV
(with Brian Pendleton PR ). Note that at this time the experimental limit on
the top quark mass was of order $\sim 10$ GeV, and most people believed it
would appear at about $\sim 3\times m_{b}\sim 15$ GeV.

Inspired by the Pendleton-Ross idea, I studied the RG equations and found the
predicted top quark mass would most probably be near an “infrared quasi-fixed
point,” or “focus point,” $\sim 220$ GeV fixed  assuming standard model
running to the high scales. People were quite skeptical, however, about a
heavy top quark (some people promised to “eat their hats” if the top mass
exceeded $\sim 40-90$ GeV but I don’t recall beholding any such feasts). In
the late 1980’s Graham and I overlapped at CERN for a year and we worked on
various generalizations of axions schiz , which, particularly in the neutrino
sector, proved to have potentially interesting cosmological implications late
.

In the 1990’s it was becoming clear that the top quark was surprisingly heavy.
Bardeen, Lindner and I considered a composite Higgs boson as a boundstate of
top and anti-top quarks BHL . We found that the composite solution implied
that $g_{top}$ should be near the RG quasi–fixed point value, inspired from
the work of Pendleton and Ross a decade earlier. The top quark finally weighed
in at the Tevatron in 1995 at $175$ GeV, shy of fixed point prediction by
about 20%. I remain of the opinion that this is suggesting an intimate
dynamical relationship between the top quark and Higgs boson, though we may
not yet have the full theory (see CTH3  for a sketch of some new ideas).

Graham Ross was a universal physicist, able to move from phenomenology to
theory comfortably. Above all, he paid close attention to experimental
results. He was a key player in developing many ideas surrounding the MSSM in
the ‘80’s and ‘90’s. However, circa 2014 he became increasingly interested in
alternatives, such as the Weyl invariant gravity theories which had become an
active arena of research Weyl ; authors ; HillRoss ; staro . We then renewed
our collaboration, and that carried us to his untimely passing.

I feel a great loss, of a dear friend and colleague, an amiable person of the
highest integrity, an energetic researcher with a deep passion for real world
physics and its essential interface with experiment, and someone I’ve known
for much of my life. We leave a half dozen unfinished papers on our laptops.

I would like to talk today about one of my more recent works with Graham Ross,
because I feel it is foundational to quantum field theory and model
independent. This work came out of the Covid pandemic, and we had daily
multiple email exhanges, sometime skype calls, during this time.

The subject of my talk is the field theoretic structure of non-minimal
gravitationally coupled scalars in the regime in which the Planck mass is
present (as opposed to any putative pre-Planckian era). The question that
emerges from this is: “Does the Jordan frame really exist?” You’ll see what I
think as the story unfolds. All of this is described in the paper HillRoss ,
and some of the implications are tackled in staro . A follow-on paper with one
of Graham’s former students, Dumitru Ghilencea, is nearing completion Ghil .

##  II Non-Minimal Couplings of Scalars to Curvature

Over the years there has been considerable interest in Brans-Dicke, scalar-
tensor, and scale or Weyl invariant theories. These have in common fundamental
scalar fields, $\phi_{i}$, that couple to gravity through non-minimal
interactions, $\sim M^{2}R+F(\phi_{i})R$. The Planck mass, $M$ can co-exist
with these non-minimal couplings, or the scalars may acquire VEV’s that
dynamically generate it authors . When the theory is proscribed with
non–minimal interactions we say it is given in a “Jordan frame.”

A key tool in the analysis of these models is the Weyl transformation, Weyl .
This involves a redefinition of the metric, $g^{\prime}=\Omega(\phi_{i})g$, in
which $g$ comingles with the scalars. $\Omega$ can be chosen to lead to a new
effective theory, typically one that has a pure Einstein-Hilbert action, $\sim
M^{2}R$, in which the non-mimimal interactions have been removed. This is
called the “Einstein frame.” Alternatively, one might use a Weyl
transformation to partially remove a subset of scalars from the non-minimal
interactions $\sim M^{2}R+F^{\prime}(\phi_{i})R$, where $F^{\prime}$ is
optimized for some particular application.

It is a priori unclear, however, how or whether the original Jordan frame
theory can be physically equivalent to the Einstein frame form and how the
Weyl transformation is compatible with a full quantum theory Duff Ruf .
Nonetheless, many authors consider this to be a valid transformation and a
symmetry of Weyl invariant theories, and many loop calculations permeate the
literature which attempt to exploit apparent simplifications offered by the
Jordan frame.

Graham Ross and I showed that any theory with non-minimal couplings contains
contact terms HillRoss . These are generated by the graviton exchange
amplitudes in tree approximation and they are therefore ${\cal{O}}(\hbar^{0})$
and therefore classical. The contact terms occur because emission vertices
from the non-minimal interaction are proportional to $q^{2}$ of the graviton,
while the Feynman propagator is proportional to $1/q^{2}$. The cancellation of
$q^{2}\times 1/q^{2}$ therefore leads to point-like interactions that must be
included into the effective action of the theory at any given order of
perturbation theory. The result is that the non-minimal interactions disappear
from the theory and Planck suppressed higher dimension operators appear with
modified couplings.

The structural form of the theory when the contact term interactions are
included corresponds formally to a Weyl transformation of the metric that
takes the theory to the pure Einstein frame with higher dimension, Planck
suppressed, interactions. In the pure Einstein-Hilbert action there are no
classical contact terms, but at loop level they can be generated, and must be
removed as part of the renormalization group. However, by virtue of the
contact terms, nowhere is a metric redefinition performed (hence the issue of
a Jacobian in the measure of the gravitational path integral in going to the
Einstein frame becomes moot).

This means that, provided we are interested in the theory on mass scales below
$M$, the Jordan frame is an illusion and doesn’t really exist physically. In
the Jordan frame the contact terms are hidden, but they are always present,
ergo, even though the action superficially appears to be Jordan, it isn’t, and
remains always in an Einstein frame.

Efforts to compute quantities, such as effective potentials (or equivalently,
$\beta$-functions), in the Jordan frame, while ignoring the contact terms,
will yield incorrect results. Nonetheless, though the non-minimal interactions
are not present in the classical Einstein frame, they are regenerated by loops
and the potentials and RG equations are modified by this effect.

In a simple theory in the Jordan frame, where the nonminimal coupling is
$-\alpha\phi^{2}R/12$, this raises the question of how to understand the fate
of $\alpha$? With a single scalar with quartic and other interactions,
$\lambda_{i}$, the usual “naive” calculation of a $\beta$-function in the
Jordan frame (“naive” means ignoring contact terms) yields the form:

$\displaystyle\frac{\partial\alpha(\mu)}{\partial\ln(\mu)}=\beta_{\alpha}(\lambda_{i})\equiv(1-\alpha)\gamma_{\alpha}(\lambda_{i})$
(1)

where the factor $(1-\alpha)$ reflects the fact that when $M^{2}=0$ and
$\alpha=1$ (conformal limit) the kinetic term of $\phi$ disappers and $\phi$
becomes static parameter).

However, the contact terms (or a Weyl tranformation) remove $\alpha$ and leave
an Einstein frame with only the Einstein-Hilbert term, $M^{2}R$ and the $\phi$
couplings $\lambda_{i}^{\prime}$ (in what follows primed couplings refer the
Einstein frame and un-primed to Jordan frame). This means that $N$ couplings,
$(\alpha,\lambda_{i})$, in the Jordan frame have become $N-1$ couplings,
$(\lambda_{i}^{\prime})$, in the Einstein frame. Therefore any physical
meaning ascribed to $\alpha$ or the $\beta_{\alpha}$ function is apparantly
lost.

Three Feynman diagrams (shown below as D1, D2, D3, in Figs.(2,3,4)) contribute
to $\beta_{\alpha}$ in the Jordan frame. One of them multiplies $\alpha$ in
the Jordan frame (D3) and yields the $(1-\alpha)$ factor in eq.(1). But even
with $\alpha=0$ in the Einstein frame, two diagrams (D1 and D2) exist and
reintroduce a perturbative $\delta\alpha$, for a small step in scale
$\delta\mu/\mu$. This is then removed by the contact terms, but leads to
correction terms in the renormalization of the $\lambda_{i}^{\prime}$. We are
therefore sensitive to the same scale breaking information in the Einstein
frame that one has in the Jordan frame, which is encoded into
$\gamma_{\alpha}$. This does not, however, imply that the resulting
calculations in the Jordan and Einstein frames are then consistent! I will
explicitly demonstrate the inconsistency through calculation pf effective
potentials (the RG equations of the couplings can always be read off from the
effective potentials).

If we stayed in the Jordan frame, with nonzero $\alpha$, and naively computed
the same effective potential (“naively” means ignoring contact terms), into an
Einstein frame, we would obtain a different result. The difference is a term
proportional to $\alpha$ in the Jordan frame. Equivalently, going initially to
the Einstein frame and running with the RG, does not commute with running
initially in the Jordan frame and subsequently going to the Einstein frame!

We turn presently to a brief discussion of contact terms in general and review
a simple toy model from HillRoss  that is structurally similar to the
gravitational case. We then summarize gravitational contact terms (and refer
the reader to HillRoss  for details). We then exemplify the Einstein frame
renormalization group compared to the naive Jordan frame result, which ignores
contact terms, and illustrate the discrepancy.

##  III Contact Interactions

Generally speaking “contact interactions” are point-like operators that are
generated in the effective action of the theory in perturbation theory. They
may arise in the UV from ultra-heavy fields that are integrated out, such as
the Fermi weak interaction that arises from integrating out the heavy
$W$-boson. They may also arise in the IR when a vertex in the theory is
proportional to $q^{2}$ and cancels against a $1/q^{2}$ propagator. The
gravitational contact term we discuss presently is of the IR form.

###  III.1 Contact Terms in Non-gravitational Physics

Contact terms arise in a number of phenomena. Diagrammatically they can arise
in the IR when a vertex for the emission of, e.g., a massless quantum, of
momentum $q_{\mu}$, is proportional to $q^{2}$. This vertex then cancels the
$1/q^{2}$ from a massless propagator when the quantum is exchanged. This
$q^{2}/q^{2}$ cancellation leads to an effective pointlike operator from an
otherwise single-particle reducible diagram.

For example, in electroweak physics a vertex correction by a $W$-boson to a
massless gluon emission induces a quark flavor changing operator, e.g.,
describing $s\rightarrow d$+gluon, where $s(d)$ is a strange (down) quark.
This has the form of a local operator:

$\displaystyle g\kappa\overline{s}\gamma_{\mu}T^{A}d_{L}D_{\nu}G^{A\mu\nu}$
(2)

where $G^{A\mu\nu}$ is the color octet gluon field strength and $\kappa\propto
G_{Fermi}$.

This implies a vertex for an emitted gluon of 4-momentum $q$ and polarization
and color, $\epsilon^{A\mu}$, of the form
$g\kappa\overline{s}\gamma_{\mu}T^{A}d_{L}\epsilon^{A\mu}\times q^{2}+...$.
However, the gluon propagates, $\sim 1/q^{2}$, and couples to a quark current
$\sim g\epsilon^{A\mu}\overline{q}\gamma_{\mu}T^{A}q$. This results in a
contact term:

$\displaystyle
g^{2}\kappa\left(\frac{q^{2}}{q^{2}}\right)\overline{s}\gamma^{\mu}T^{A}d_{L}\overline{q}\gamma_{\mu}T^{A}q\;=\;g^{2}\kappa\overline{s}\gamma^{\mu}T^{A}d_{L}\overline{q}\gamma_{\mu}T^{A}q$

The result is a 4-body local operator which mediates electroweak transitions
between, e.g., kaons and pions Shifman , also known as “penguin diagrams”
penguins . Note the we can rigorously obtain the contact term result by use of
the gluon field equation within the operator of eq.(2),

$\displaystyle D_{\nu}G^{A\mu\nu}=g\overline{q}\gamma^{\mu}T^{A}q.$ (4)

This is justified as operators that vanish by equations of motion, known as
“null operators,” will generally have gauge noninvariant anomalous dimensions
and are unphysical Deans .

Another example occurs in the case of a cosmic axion, described by an
oscillating classical field, $\theta(t)=\theta_{0}\cos(m_{a}t)$, interacting
with a magnetic moment, $\vec{\mu}(x)\cdot\vec{B}$, through the
electromagnetic anomaly $\kappa\theta(t)\vec{E}\cdot\vec{B}$. A static
magnetic moment emits a virtual spacelike photon of momentum $(0,\vec{q})$.
The anomaly absorbs the virtual photon and emits an on-shell photon of
polarization $\vec{\epsilon}$, inheriting energy $\sim m_{a}$ from the cosmic
axion. The Feynman diagram, with the exchanged virtual photon, yields an
amplitude,
$\propto(\theta_{0}\mu^{i}\epsilon_{ijk}q^{j})(1/\vec{q}^{\;2})(\kappa\epsilon^{k\ell
h}q_{\ell}m_{a}\epsilon_{h})\sim(\kappa\theta_{0}m_{a}\vec{q}^{\;2}/\vec{q}^{\;2})\vec{\mu}\cdot\vec{\epsilon}$.
The $\vec{q}^{\;2}$ factor then cancels the $1/\vec{q}^{\;2}$ in the photon
propagator, resulting in a contact term which is an induced, parity violating,
oscillating electric dipole interaction:
$\sim\kappa\theta(t)\vec{\mu}\cdot\vec{E}$. This results in cosmic axion
induced electric dipole radiation from any magnet, including an electron CTHa
.

###  III.2 Illustrative Toy Model of Contact Terms

To illustrate the general IR contact term phenomenon, consider a single
massless real scalar field $\phi$ and operators $A$ and $B,$ which can be
functions of other fields, with the action given by:

$\displaystyle
S=\int\frac{1}{2}\partial\phi\partial\phi-A\partial^{2}\phi-B\phi$ (5)

Here $\phi$ has a propagator ${i}/{q^{2}}$, but the vertex of a diagram
involving $A$ has a factor of $\partial^{2}\sim-q^{2}$. This yields a
pointlike interaction, $\sim q^{2}\times({i}/q^{2})$, in a single particle
exchange of $\phi$, and therefore implies contact terms.

At lowest order in perturbation theory consider the diagram with $\phi$
exchange in Fig.(1). This involves two time-ordered products of interaction
operators:

$\displaystyle{T}\;\;i\\!\\!\int\\!A\partial^{2}\phi\times
i\\!\\!\int\\!B\phi\rightarrow\frac{iq^{2}}{q^{2}}\ AB\;\;\;=\;i\int\\!AB$
$\displaystyle\frac{1}{2}{T}\;\;i\\!\\!\int\\!A\partial^{2}\phi\times
i\\!\\!\int\\!A\partial^{2}\phi\rightarrow\frac{i(q^{2})^{2}}{2q^{2}}A^{2}\\!=\\!\frac{i}{2}\int\\!A\partial^{2}A.$

where $d^{4}x$ is understood in the integrals amd the $\frac{1}{2}$ factor in
the $A\partial^{2}A$ term comes from the second order in the expansion of the
path integral $\exp(i\int A\partial^{2}\phi)$. Note that we also produce a
nonlocal interaction $-{i}B^{2}/{2q^{2}}.$

Figure 1: Contact terms in the toy model are generated by diagrams with
exchange of $\phi$ (dashed). In gravity, with non-minimal term
$\sim\int\\!\sqrt{-g}F\left(\phi\right)R$ and matter field Lagrangian
$\sim\int\\!\sqrt{-g}L\left(\phi\right)$ then $A$ is replaced by $F(\phi)$ and
$B$ is replaced by $L(\phi)$, and the dashed line is a graviton propagator.

We thus see that we have diagrammatically obtained a local effective action:

$\displaystyle
S=\int\frac{1}{2}\partial\phi\partial\phi+\frac{1}{2}A\partial^{2}A+AB+\makebox{
{long distance}}$ (7)

Of course, we can see this straightforwardly by “solving the theory,” by
defining a shifted field:

$\displaystyle\phi=\phi^{\prime}-\frac{1}{\partial^{2}}\left(\partial^{2}A+B\right)$
(8)

Substituting this into the action $S$ and integrating by parts yields:

$\displaystyle
S=\int\frac{1}{2}\partial\phi^{\prime}\partial\phi^{\prime}+\frac{1}{2}A\partial^{2}A+AB+\frac{1}{2}B\frac{1}{\partial^{2}}B$
(9)

An equivalent effective local action that describes both short and large
distance is then,

$\displaystyle
S=\int\frac{1}{2}\partial\phi\partial\phi+\frac{1}{2}A\partial^{2}A+AB-{B\phi}$
(10)

The contact terms have become pointlike components of the effective action,
while the remaining long distance effects are produced by the usual massless
$\phi$ exchange. Note that the derivatively coupled operator $A$ has no long
distance interactions due to $\phi$ exchange. Moreover, in the effective
action of eq.(10) we have implicitly “integrated out” the $A\partial^{2}\phi$,
which is no longer part of the action and is replaced by new operators
$\frac{1}{2}A\partial^{2}A+AB$. We will see that this is exactly what happens
with gravity, where the $A\partial^{2}\phi$ term is schematically the
nonminimal $F(\phi)R(g)\sim F(\phi)\partial^{2}h$ term in a weak field
expansion of gravity $g=\eta+h$.

One can also adapt the use of equations of motion to obtain eq.(10) from the
action eq.(5) but this requires care. For example, the insertion of the $\phi$
equation of motion, into $A\partial^{2}\phi$ correctly gives the $AB$ term but
misses the factor of $1/2$ in the $A\partial^{2}A$ term. We can therefore do a
trick of defining a “modified equation of motion,” where we supply a factor of
$1/2$ on the term $A\partial^{2}A$, e.g. substitute:

$\displaystyle\partial^{2}\phi=-\partial^{2}A-B\rightarrow-\frac{1}{2}\partial^{2}A-B$
(11)

in place of the $\partial^{2}\phi$ in the second term of eq.((5)) to obtain
eq.(10).

##  IV Gravitational Contact Terms

Consider a general theory involving scalar fields $\phi_{i},$ an Einstein-
Hilbert term and a non-minimal interaction:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;\;S=\int\sqrt{-g}\left(\frac{1}{2}M^{2}R\left(g_{\mu\nu}\right)+\frac{1}{2}F\left(\phi_{i}\right)R\left(g_{\mu\nu}\right)+L\left(\phi_{i}\right)\right)$
$\displaystyle=S_{1}+S_{2}+S_{3}$ (12)

where we use the metric signature and curvature tensor conventions of CCJ .
$S_{1}$ is the kinetic term of gravitons

$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}M^{2}\int\sqrt{-g}R$ (13)

and becomes the Fierz-Pauli action in a weak field expansion.

$S_{2}$ is the non-minimal interaction, and takes the form:

$\displaystyle
S_{2}=\frac{1}{2}\int\sqrt{-g}F\left(\phi_{i}\right)R\left(g_{\mu\nu}\right)$
(14)

$S_{3}$ is the matter action with couplings to the gravitational weak field:

$\displaystyle S_{3}=\int\sqrt{-g}\;L\left(\phi_{i}\right)$ (15)

The Lagrangian takes the form

$\displaystyle
L\left(\phi_{i}\right)=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi_{i}\partial_{\nu}\phi_{i}-W(\phi_{i})$
(16)

with potential $W(\phi_{i})$. The matter lagrangian has stress tensor and
stress tensor trace:

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\phi_{i}\partial_{\nu}\phi_{i}-g_{\mu\nu}\left(\frac{1}{2}g^{\rho\sigma}\partial_{\rho}\phi_{i}\partial_{\sigma}\phi_{i}-W(\phi_{i})\right).$
$\displaystyle T$ $\displaystyle=$
$\displaystyle-\partial^{\sigma}\phi_{i}\partial_{\sigma}\phi_{i}+4W(\phi_{i})$
(17)

There are then three ways to obtain the contact term:

(1) GRAVITON EXCHANGE CONTACT TERM

In reference HillRoss  the corresponding Feynman diagrams of Figure 1 with are
evaluated arising from single graviton exchange between the interaction terms.

We treat the theory perturbatively, expanding around flat space. Hence we
linearize gravity with a weak field $h_{\mu\upsilon}$:

$\displaystyle
g_{\mu\upsilon}\approx\eta_{\mu\upsilon}+\frac{h_{\mu\upsilon}}{M}.$ (18)

The scalar curvature is then:

$\displaystyle R$ $\displaystyle=$ $\displaystyle R_{1}+R_{2}$ $\displaystyle
MR_{1}$ $\displaystyle=$ $\displaystyle\biggl{(}\partial^{2}h\
-\partial^{\mu}\partial^{\nu}h_{\mu\nu}\biggr{)}$ $\displaystyle M^{2}R_{2}$
$\displaystyle=$
$\displaystyle-\frac{3}{4}\partial^{\rho}h^{\mu\nu}\partial_{\rho}h_{\mu\nu}-\frac{1}{2}h^{\mu\nu}\partial^{2}h_{\mu\nu}+...$
(19)

(see HillRoss  for the complete expression for $R_{2}$). $S_{1}$ then becomes:

$\displaystyle
S_{1}=\frac{1}{2}M^{2}\int\sqrt{-g}R=\frac{1}{2}M^{2}\int\\!\\!\left(R_{1}+R_{2}+\frac{1}{2}\frac{h}{M}R_{1}\right)$
$\displaystyle=\frac{1}{2}\int
h^{\mu\nu}\biggl{(}\frac{1}{4}\partial^{2}\eta_{\mu\nu}\eta_{\rho\sigma}-\frac{1}{4}\partial^{2}\eta_{\mu\rho}\eta_{\nu\sigma}$
$\displaystyle\qquad\qquad-\frac{1}{2}\partial_{\rho}\partial_{\sigma}\eta_{\mu\nu}+\frac{1}{2}\partial_{\mu}\partial_{\rho}\eta_{\nu\sigma}\biggr{)}h^{\rho\sigma}.$
(20)

Note that the leading term, $\int R_{1}$, is a total divergence and is
therefore zero in the Einstein-Hilbert action, and what remains of eq.(IV) is
the Fierz-Pauli action. This is key to the origin of the contact terms. The
non-minimal interaction, $S_{2}$, then takes the leading form:

$\displaystyle
S_{2}=\frac{1}{2}\int\\!\\!\sqrt{-g}\;F\left(\phi\right)R\left(g\right)\rightarrow\frac{1}{2}\int\\!\\!F\left(\phi\right)R_{1}\left(g\right)$
$\displaystyle\qquad=\int\frac{1}{2M}F\left(\phi\right)\Pi^{\mu\nu}h_{\mu\nu}$
(21)

where it is useful to introduce the transverse derivative,

$\displaystyle\Pi^{\mu\nu}=\partial^{2}\eta^{\mu\nu}\
-\partial^{\mu}\partial^{\nu}.$ (22)

$S_{2}$ involves derivatives, and is the analogue of the $A\partial^{2}\phi$
term in eq.(5). It will therefore generate contact terms in the gravitational
potential due to single graviton exchange. $S_{3}$ is the analogue of the
$B\phi$ term in eq.(5), and this situation will closely parallel the toy
model.

In HillRoss  we developed the graviton propagator, following the nice lecture
notes of Donoghue et. al., Donoghue . We remark that we found a particularly
useful gauge choice,

$\displaystyle\partial_{\mu}h^{\mu\nu}=w\partial^{\nu}h$ (23)

where $w$ defines a single parameter family of gauges. The familiar De Donder
gauge corresponds to $w=\frac{1}{2}$, while the choice $w=\frac{1}{4}$ is
particularly natural in this application, and the gauge invariance of the
result is verified by the $w$-independence (we verify the Newtonian potential
from graviton exchange between static masses in $w$ gauge; see HillRoss ).

We can the compute single graviton exchange between the interaction terms of
the theory. A diagram with a single $S_{2}$ vertex and single $S_{3}$ vertex
is the analogue of $AB$ in the toy model and yields:

$\displaystyle-i\left\langle T\;S_{2}S_{3}\right\rangle=\int
d^{4}x\;\frac{F\left(\phi_{i}\right)}{2M^{2}}T(\phi_{i})$ (24)

Also we have the pair $\left\langle S_{2}S_{2}\right\rangle$ which corresponds
to $\frac{1}{2}A\partial^{2}A$ in the toy model and yields:

$\displaystyle-i{\left\langle T\;S_{2}S_{2}\right\rangle}$ $\displaystyle=$
$\displaystyle-\int
d^{4}x\;\frac{3}{4M^{2}}\;F\left(\phi_{i}\right)\\!\partial^{2}F\left(\phi_{i}\right)$

Hence, the action becomes

$\displaystyle S=S_{1}+S_{3}+S_{CT}$ (26)

where

$\displaystyle S_{CT}=\int
d^{4}x\biggl{(}-\frac{3}{4M^{2}}F\partial^{2}F+\frac{1}{2M^{2}}FT\biggr{)}$
(27)

Note the sign of the $F\partial^{2}F$ is opposite (repulsive) to that of the
toy model $A\partial^{2}A$. Since this is a tree diagram effect, it is
classical.

(2) WEYL TRANSFORMATION

In eq.(IV) we define:

$\displaystyle\Omega^{2}=\left(1+\frac{F\left(\phi_{i}\right)}{M^{2}}\right)$
(28)

and perform a Weyl transformation on the metric:

$\displaystyle g_{\mu\nu}(x)\rightarrow\Omega^{-2}g_{\mu\nu}(x)\qquad
g^{\mu\nu}(x)\rightarrow\Omega^{2}g^{\mu\nu}(x)$
$\displaystyle\sqrt{-g}\rightarrow\sqrt{-g}\Omega^{-4}$ $\displaystyle
R(g)\rightarrow\Omega^{2}R(g^{\prime})+6\Omega^{3}D\partial\Omega^{-1}$ (29)

and the action of eq.(V) becomes:

$\displaystyle
S\rightarrow\int\sqrt{-g}\biggl{(}\frac{1}{2}M^{2}R\left(g\right)$
$\displaystyle-3M^{2}\partial_{\mu}\left(1+\frac{F}{M^{2}}\right)^{1/2}\\!\\!\partial^{\mu}\left(1+\frac{F}{M^{2}}\right)^{-1/2}$
$\displaystyle+\frac{1}{2}\left(1+\frac{F}{M^{2}}\right)^{-1}\\!\\!\\!\partial_{\mu}\phi_{i}\partial^{\nu}\phi^{i}-\left(1+\frac{F}{M^{2}}\right)^{-2}\\!\\!\\!W(\phi_{i})\biggr{)}$

Keeping terms to $O({1\over M^{2}})$ and integrating by parts we have:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S=S_{1}+S_{3}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\int
d^{4}x\bigg{(}-\frac{3F\left(\phi_{i}\right)\partial^{2}F\left(\phi_{i}\right)}{4M^{2}}+\frac{F\left(\phi_{i}\right)T\left(\phi_{i}\right)}{2M^{2}}\bigg{)}$
(31)

The Weyl transformed action is identically consistent with the contact terms
of eq.(27) above, to first order in $1/M^{2}$.

Hence, contact terms arise in gravity with non-minimal couplings to scalar
fields due to graviton exchange and their form is equivalent to a Weyl
redefinition of the theory to the Einstein frame. However we emphasie that the
contact terms are not a Weyl tranformation since they do not involve a
rescaling of the metric, and there would therefore be no Jacobian ghost terms
introduced into the path integral. Hence any theory with a non-minimal
interaction $\sim F(\phi)R$ will lead to contact terms at order $1/M^{2}$.
What we say presently only applies on scales below $M$, hence the Jordan frame
would, at best, apply only in a UV completion of the theory to pre-Planckian
scales.

(3) USE OF $R$ (MODIFIED) EQUATION OF MOTION

The Einstein equation with the nonminimal term is:

$\displaystyle
M^{2}G_{\alpha\beta}=-T_{\alpha\beta}-D_{\mu}(D_{\nu}F(\phi_{i}))+g_{\mu\nu}D^{2}F(\phi_{i})$
$\displaystyle M^{2}R=T-3D^{2}F(\phi_{i})$ (32)

We can use a simple trick to obtain the contact term via a modified the
equation of motion for $R$. We supply a factor of $1/2$ in the last term which
is the analogue of the $\partial^{2}A$ term as in eq.(11):

$\displaystyle
R^{\prime}=\frac{1}{M^{2}}\biggl{(}T-\frac{3}{2}D^{2}F(\phi_{i})\biggr{)}$
(33)

Then substitute $R^{\prime}$ for $R$ in the non-minimal term $FR$ of the
action of eq.(V)

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;\;S\rightarrow\int\sqrt{-g}\left(\frac{1}{2}M^{2}R\left(g_{\mu\nu}\right)+\frac{1}{2}F\left(\phi_{i}\right)R^{\prime}\left(g_{\mu\nu}\right)+L\left(\phi_{i}\right)\right)$
$\displaystyle\qquad=S_{1}+S_{3}$ $\displaystyle\qquad+\int
d^{4}x\bigg{(}-\frac{3F\left(\phi_{i}\right)\partial^{2}F\left(\phi_{i}\right)}{4M^{2}}+\frac{F\left(\phi_{i}\right)T\left(\phi_{i}\right)}{2M^{2}}\bigg{)}$

In the RG calculation we will only need the exact equation of motion for $R$
in the Einstein frame (without the pseudo $-3D^{2}F/2$ term), so this
ambiguity does not arise.

##  V  Effective Potetial in a Simple Model

Consider the following action:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{Jordan}=\int\sqrt{-g}\biggl{(}\frac{1}{2}\partial\phi\partial\phi-\frac{\lambda_{1}}{4}\phi^{4}-\frac{\lambda_{2}\phi^{6}}{12M^{2}}$
$\displaystyle-\frac{\gamma}{12}\frac{\phi^{2}}{M^{2}}\partial\phi\partial\phi-\frac{\alpha}{12}\phi^{2}R+\frac{1}{2}M^{2}R\biggr{)}$
(35)

This is the most general action for a real scalar field with a $Z_{2}$
symmetry $\phi\rightarrow-\phi$ valid to O($M^{-2}$) with Einstein gravity and
assuming $\phi$ is massless, $m^{2}=0$. Here we do not include a term
$\phi^{4}R/M^{2}$ since, after use of equations of motion, $R\sim M^{-2}$ such
a term would enter the physics at O($M^{-4}$).

We can go to the Einstein frame by implementing the CT. We find that the
effect of single graviton exchange to eq.(V) yields a new action to order
$M^{-2}$:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{Einstein}=\\!\int\\!\sqrt{-g}\biggl{(}\frac{1}{2}\partial\phi\partial\phi-\frac{\lambda^{\prime}_{1}}{4}\phi^{4}-\frac{\lambda^{\prime}_{2}\phi^{6}}{12M^{2}}$
$\displaystyle-\frac{\gamma^{\prime}\phi^{2}}{12M^{2}}\partial\phi\partial\phi+\frac{1}{2}M^{2}R\biggr{)}$
(36)

where:

$\displaystyle\gamma^{\prime}=\gamma-\alpha-\alpha^{2}$
$\displaystyle\lambda_{1}^{\prime}=\lambda_{1}$
$\displaystyle\lambda_{2}^{\prime}=\lambda_{2}+\alpha\lambda_{1}$ (37)

We see that to first order in $M^{-2}$ in $S_{Einstein}$ we have three
interaction terms, though the original action $S_{Jordan}$ displayed four
interaction terms. In the Einstein frame action we see that $\alpha$ has
disappeared having been absorbed into redefining the other coupling constants.
This indicates that the nonminimal term in $S_{Jordan}$ with coupling $\alpha$
is unphysical. Moreover, if we are careful to incorporate the effects of the
contact term, then $S_{Einstein}$ will “close” under renormalization.

We define,

$\displaystyle Z=1-\frac{\gamma^{\prime}\phi^{2}}{6M^{2}}$ (38)

and the Einstein action becomes:

$\displaystyle
S_{Einstein}=\\!\int\\!\sqrt{-g}\biggl{(}\frac{1}{2}Z\partial\phi\partial\phi-\frac{\lambda^{\prime}_{1}}{4}\phi^{4}-\frac{\lambda^{\prime}_{2}\phi^{6}}{12M^{2}}$
$\displaystyle+\frac{1}{2}M^{2}R\biggr{)}$ (39)

We do a background field calculation where we shift
$\phi\rightarrow\phi_{0}+\sqrt{\hbar}Z_{0}^{-1/2}\hat{\phi}$, where we define

$\displaystyle Z_{0}=1-\frac{\gamma^{\prime}\phi_{0}^{2}}{6M^{2}}$ (40)

We will treat $\phi\rightarrow\phi_{0}$ as a constant and compute the one
loop, $\cal{O(\hbar)}$ effective potential in $\phi_{0}$ integrating out the
quantum fluctuations $\hat{\phi}$ .222 In a forthcoming paper with Ghilenca
Ghil  we will give the general effective action where the backgound field is
treated as non-constant and dynamical. The present analysis parallels a
Coleman-Weinberg potential CW . Expand the action with the shifted field to
O$(\hat{\phi}^{2})$, dropping terms odd in $\hat{\phi}$ for constant
$\phi_{0}$.

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{Einstein}=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\sqrt{-g}\biggl{(}\frac{1}{2}\partial\hat{\phi}\partial\hat{\phi}-\frac{1}{2}B\hat{\phi}^{2}-V_{0}(\phi_{0})+\frac{1}{2}M^{2}R\biggr{)}$

where,

$\displaystyle Z_{0}=1-\frac{\gamma^{\prime}\phi_{0}^{2}}{6M^{2}}$
$\displaystyle
B=Z_{0}^{-1}\biggl{(}{3\lambda^{\prime}_{1}}\phi_{0}^{2}+\frac{5\lambda^{\prime}_{2}}{2M^{2}}\phi_{0}^{4}\biggr{)}$
$\displaystyle={3\lambda^{\prime}_{1}}\phi_{0}^{2}+\frac{5\lambda^{\prime}_{2}+\gamma^{\prime}\lambda^{\prime}_{1}}{2M^{2}}\phi_{0}^{4}$
$\displaystyle
V_{0}(\phi_{0})=\frac{\lambda^{\prime}_{1}}{4}\phi_{0}^{4}+\frac{\lambda^{\prime}_{2}}{12}\frac{\phi_{0}^{6}}{M^{2}}$

First we consider the non-curvature terms, with flat Minkowsky space metric
$g_{\mu\nu}=\eta_{\mu\nu}$. We obtain the effective potential from the log of
the path integral described briefly in the Appendix, eq.(71), obtained by
integrating out $\hat{\phi}^{2}$. Presently we plug $m^{2}\rightarrow B$ into
the path integral expression of eq.(71),

$\displaystyle\Gamma_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}B^{2}L+O\biggl{(}\frac{B^{3}}{\Lambda^{2}}\biggr{)}$
(43)

and we define the log as:

$\displaystyle L=\frac{1}{16\pi^{2}}\ln\frac{\Lambda}{\mu}$ (44)

with a generic infrared cut-off mass scale $\mu$. Hence, squaring $B$ to
O($\hbar$), the resulting potential is to ${\cal{O}}(\phi_{0}^{6})$:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{0}=-\biggl{(}\frac{9\lambda^{\prime
2}_{1}}{2}\phi_{0}^{4}+\frac{(15\lambda^{\prime}_{2}\lambda^{\prime}_{1}+3\gamma^{\prime}\lambda^{\prime}_{1}{}^{2})}{2}\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}L$
(45)

Note that $\mu\rightarrow\phi_{0}$ in the logarithm this is essentialy a
Coleman-Weinberg potential CW .

###  V.1  Inclusion of Gravitational Effects

Consider the weak field approximation to gravity. We choose $w$-gauge of
eq.(23) and obtain

$\displaystyle\partial_{\alpha}h^{\alpha\beta}=w\partial^{\beta}h,\qquad
R=(1-w)\partial^{2}h/M$ (46)

Up to linear terms in $h_{\mu\nu}\hat{\phi}^{2}/M$ we have,

$\displaystyle
S_{Einstein}\rightarrow\int\Biggl{(}\frac{1}{2}\partial_{\mu}\hat{\phi}\partial^{\mu}\hat{\phi}+\frac{1}{4}\frac{h}{M}\partial_{\mu}\hat{\phi}\partial^{\mu}\hat{\phi}+\frac{1}{2}M^{2}R$
$\displaystyle-\frac{h^{\mu\upsilon}}{2M}\partial_{\mu}\hat{\phi}\partial_{\nu}\hat{\phi}-\frac{1}{2}\biggl{(}1+\frac{1}{2}\frac{h}{M}\biggr{)}B\hat{\phi}^{2}-V_{0}(\phi_{0})\Biggr{)}$
(47)

The contributions to the potential from the Feynman diagrams of Figs.(1,2) are
then,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{D1}=-\frac{\bigl{(}1+2w\bigr{)}}{12M}B\partial^{2}hL$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{D2}=\frac{B}{4M}\partial^{2}hL$
(48)

Full details will be given in Ghil . Noting that,

$\displaystyle\frac{1}{4M}\partial^{2}h-\frac{(1+2w)}{12M}\partial^{2}h=\frac{(1-w)}{6M}\partial^{2}h=\frac{1}{6M}R$

Hence we have the net contribution to the potential,

$\displaystyle\Gamma_{\alpha}\equiv\Gamma_{D1}+\Gamma_{D2}=\frac{1}{6}BR\;L=\frac{\lambda_{1}\phi_{0}^{2}}{2}RL$
(50)

We thus see that there is a $\delta\alpha\phi_{0}^{2}R/12$ term generated from
the $3\lambda^{\prime}_{1}\phi_{0}^{2}$ term in B:

$\displaystyle\delta\alpha=6\lambda^{\prime}_{1}L+O(1/M^{2})$ (51)

Note that there are other terms proportional to $R$, but we only keep leading
terms in $1/M^{2}$ in $\delta\alpha$, since $R\sim 1/M^{2}$.

To implement the contact term we use, in eq.(50), the leading order $R$
equation of motion in the Einstein frame,

$\displaystyle
R=\frac{1}{M^{2}}T=\frac{1}{M^{2}}4\times\biggl{(}\frac{\lambda_{1}}{4}\phi_{0}^{4}-\frac{9\lambda^{2}_{1}}{2}\phi_{0}^{4}L\biggr{)}+{\cal{O}}\biggl{(}\frac{1}{M^{4}}\biggr{)}$
$\displaystyle\rightarrow\lambda_{1}\frac{\phi_{0}^{4}}{M^{2}}+{\cal{O}}(\hbar)$
(52)

We then have

$\displaystyle\Gamma_{\alpha}=\frac{1}{2}\lambda^{\prime}_{1}\phi^{2}_{0}R\;L\rightarrow\frac{\lambda^{\prime
2}_{1}\phi_{0}^{6}}{2M^{2}}L+{\cal{O}}\frac{1}{M^{4}}$ (53)

Therefore, combining all effects

$\displaystyle\Gamma_{Einstein}\equiv V_{0}+\Gamma_{0}+\Gamma_{\alpha}=$
$\displaystyle\frac{\lambda_{1}}{4}\phi_{0}^{4}+\frac{\lambda_{2}}{12}\frac{\phi_{0}^{6}}{M^{2}}$
$\displaystyle-\frac{1}{2}\biggl{(}9\lambda^{2}_{1}\phi_{0}^{4}+(15{\lambda_{1}\lambda_{2}}+3\lambda^{2}_{1}\gamma-[\lambda_{1}^{2}])\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}L$
(54)

where the term in $[..]$ comes from the gravitational effects of D1 and D2.

Figure 2:  Diagram D1.  Figure 3:  Diagram D2.

From eq.(V.1) we can extract the $\beta$-functions for $\lambda_{1}$ and
$\lambda_{2}$. Define $D=16\pi^{2}\partial/\partial\ln\mu$, and we have:

$\displaystyle D\lambda_{1}=18\lambda_{1}^{2}$ $\displaystyle
D\lambda_{2}=90\lambda_{1}\lambda_{2}-6\lambda_{1}^{2}+18\lambda^{2}\gamma^{\prime}$
(55)

In Ghil  we obtain the RG equation for $\gamma^{\prime}$, but this requires an
effective action for non-constant $\phi_{0}$.

###  V.2 Comparison to a Conventional Calculation of $\beta_{\alpha}$
in Jordan Frame Neglecting Contact Terms

We now compute the potential in the Jordan frame where we neglect the contact
term, as is conventionally done. Here we again use the background field
method.

Return to $S_{Jordan}$, and begin by shifting
$\phi\rightarrow\phi_{0}+\sqrt{\hbar}\hat{\phi}$ and expanding to
O$\hat{\phi}^{2}$ where $\phi_{0}$ is a constant classical background field
and $\sqrt{\hbar}\hat{\phi}$ is a quantum fluctuation. Henceforth we will
suppress the factor of $\sqrt{\hbar}$ but we will compute loops to order
$\hbar$. The shifted action where we drop the linear terms in $\hat{\phi}$
becomes to order $\hbar$:

$\displaystyle
S_{Jordan}=\int\sqrt{-g}\biggl{(}\frac{1}{2}g^{\mu\upsilon}\partial_{\mu}\hat{\phi}\partial_{\nu}\hat{\phi}$
$\displaystyle-\frac{1}{2}Z_{0}^{-1}A\hat{\phi}^{2}R-\frac{1}{2}B\hat{\phi}^{2}-V_{0}(\phi_{0})+\frac{1}{2}M^{2}R\biggr{)}$
(56)

where, we have the relations of eq.(V), but with unprimed couplings replacing
the primed ones, and

$\displaystyle A=\frac{\alpha}{6}$ (57)

Expanding to linear terms in $h_{\mu\nu}/M$ in weak field gravity the action
eq.(V.2) becomes

$\displaystyle
S\rightarrow\int\Biggl{(}\frac{1}{2}\partial_{\mu}\hat{\phi}\partial^{\mu}\hat{\phi}+\frac{h}{4M}\eta^{\mu\nu}\partial_{\mu}\hat{\phi}\partial_{\nu}\hat{\phi}-\frac{h^{\mu\upsilon}}{2M}\partial_{\mu}\hat{\phi}\partial_{\nu}\hat{\phi}$
$\displaystyle-\frac{1}{2}Z_{0}^{-1}AR{\hat{\phi}^{2}}-\frac{1}{2}B\hat{\phi}^{2}-B\frac{h}{4M}\hat{\phi}^{2}-V_{0}(\phi_{0})+\frac{1}{2}M^{2}R\Biggr{)}$

Figure 4:  Diagram D2.

Neglecting the contact terms we see, in addition to the non-curvature
potential obtained previously in eq.(45), the action eq.(V.2) now (naively)
generates three diagrams linear in the curvature, D1, D2, and D3 of
Figs(2,3,4). The additional D3 diagram (which is absent in the Einsteain
frame) is:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{D3}=-Z_{0}^{-1}ABRL=-\frac{\alpha}{6}BRL+O\frac{1}{M^{2}}R$
(59)

Hence we have from eqs.(50,59) for (D1+D2+D3):

$\displaystyle\Gamma_{D3}+\Gamma_{\alpha}=-\frac{1}{6}(\alpha-1)BR\;L$
$\displaystyle=-\frac{1}{6}(\alpha-1)\biggl{(}{3\lambda_{1}}\phi_{0}^{2}+\frac{5\lambda_{2}+\gamma\lambda_{1}}{2M^{2}}\phi_{0}^{4}\biggr{)}RL$
(60)

Therefore, combining all effects:

$\displaystyle\Gamma_{Jordan}=$
$\displaystyle\frac{\lambda_{1}}{4}\phi_{0}^{4}+\frac{\lambda_{2}}{12}\frac{\phi_{0}^{6}}{M^{2}}+\frac{\alpha}{12}\phi_{0}^{2}R+\frac{\gamma}{12M^{2}}\phi_{0}^{2}R$
$\displaystyle-\frac{1}{2}L\biggl{(}9\lambda^{2}_{1}\phi_{0}^{4}+(15{\lambda_{1}\lambda_{2}}+3\lambda^{2}_{1}\gamma)\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}$
$\displaystyle-\frac{1}{6}(\alpha-1)\biggl{(}{3\lambda_{1}}\phi_{0}^{2}+\frac{5\lambda_{2}+\gamma\lambda_{1}}{2M^{2}}\phi_{0}^{4}\biggr{)}RL$
(61)

This contains the conventional radiative correction and renormaliztion group
running of $\alpha$

$\displaystyle\alpha\rightarrow\alpha+6\lambda_{1}(1-\alpha)L$ (62)

We again use the leading order $R$ equation of motion in the Einstein frame as
in eq.(V.1) to implement the contact term, but must now keep the
$\lambda^{2}_{1}L$ term,

$\displaystyle
R\rightarrow(\lambda_{1}-18\lambda^{2}_{1}L)\frac{\phi_{0}^{4}}{M^{2}}+{\cal{O}}(\hbar)$
(63)

hence

$\displaystyle\Gamma_{Jordan}\equiv
V_{0}+\Gamma_{0}+\Gamma_{\alpha}+\Gamma_{D3}$
$\displaystyle=\frac{\lambda_{1}}{4}\phi_{0}^{4}+\frac{\lambda_{2}+\alpha\lambda_{1}}{12}\frac{\phi_{0}^{6}}{M^{2}}-\frac{2\lambda^{2}_{1}\alpha\phi_{0}^{6}}{M^{2}}L$
$\displaystyle-\frac{1}{2}L\biggl{(}9\lambda^{2}_{1}\phi_{0}^{4}+(15{\lambda_{1}\lambda_{2}}+3\lambda^{2}_{1}\gamma-[\lambda_{1}^{2}])\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}$
$\displaystyle=$
$\displaystyle\frac{\lambda^{\prime}_{1}}{4}\phi_{0}^{4}+\frac{\lambda^{\prime}_{2}}{12}\frac{\phi_{0}^{6}}{M^{2}}+\frac{\lambda^{\prime
2}_{1}\alpha\phi_{0}^{6}}{2M^{2}}L\biggl{(}8-{3}\alpha\biggr{)}$
$\displaystyle-\frac{1}{2}L\biggl{(}9\lambda^{\prime
2}_{1}\phi_{0}^{4}+(15{\lambda^{\prime}_{1}\lambda^{\prime}_{2}}+3\lambda^{\prime
2}_{1}\gamma^{\prime}-[\lambda^{\prime}_{1}{}^{2}])\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}$

where we’ve converted to the Einstein frame primed variables via eqs.(V).
Comparing to eq.(V.1) we therefore see an inconsistency between the potentials
$\Gamma_{Einstein}$ and $\Gamma_{Jordan}$ at O($\hbar$):

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{Einstein}-\Gamma_{Jordan}=\frac{(8-3\alpha)\alpha\lambda^{\prime
2}_{1}\phi_{0}^{6}}{2M^{2}}\hbar L$ (65)

The fault lies in the Jordan frame calculation which does not implement the
contact term.

One might attempt to interpret this as a quantum “frame anomaly.” We see some
vague parallels with the axial anomaly here, but we think the issue is more of
an error that one is making in computing naively. Such an anomaly
interpretation would require identifying an operator that plays a special roll
in the Weyl tranformation and that has matrix elements given by the rhs of
eq,(65). Perhaps this can be related to the Weyl current, but we have not yet
explored this possibility.

##  VI CONCLUSIONS

The Weyl transformation acting on the Jordan frame, to remove non-minimal
interactions leading to the minimal Einstein frame, is identical to
implementing the contact terms HillRoss . If one didn’t know about the Weyl
transformation one would discover it in the induced contact terms in the
single graviton exchange potential involving non-minimal couplings. The Weyl
transformation is powerful as it is fully non-perturbative. Technically it can
provide a useful check on the normalization and implementation of the graviton
propagators in various gauges. But the contact term stipulates that the
mapping to the Einstein frame is dynamical and inevitable, and does not
involve field redefinitions.

In a model with non-minimal coupling $-\alpha\phi^{2}R/12$ this implies that
the parameter $\alpha$ doesn’t really exist physically. Computing
$\beta$-functions in a Jordan frame without implementing the contact term will
yield incorrect results. Implementing the contact term yields the Einstein
frame and results computed there will have no contact term ambiguities.

However, the Einstein frame has a loop induced contact term which is then
absorbed back into the potential terms by the contact terms (equivalently, a
mini-Weyl transformation, or use of the $R$ equation of motion). The use of
the $R$ equation of motion on the non-minimal term is analogous to the use of
the gluon field equation for the electroweak penguin. It is likely the Deans
and Dixon Deans  constraints on null operators applies to gravity as well.
This implies generally that there are pitfalls in directly interpreting any
physics in the Jordan frame.

We emphasize that our analysis applies strictly to a theory with a Planck mass
term. A Weyl invariant theory, where $M=0$, is nonperturbative and our
analysis is then inapplicable, and the Jordan frame may then be physically
relevant. Indeed, there is no conventional gravity in this limit since the
usual $M^{2}R$ (Fierz-Pauli) graviton kinetic term does not then exist. Hence
in this limit one would have to appeal to a UV completion, e.g., string theory
or $R^{2}$ gravity, etc.

In the case of an $R^{2}$ UV completion theory we view the formation of the
Planck mass by, e.g., inertial symmetry breaking, i.e., as a dynamical phase
transition, similar to a disorder-order phase transition in a material medium
FHR . An intriguing point to note is that if we have an $R^{2}$ UV completion,
then the graviton propagator is $\propto 1/q^{4}$. But then the $AR\sim
Ahq^{2}$ vertex implies a $1/q^{2}$ graviton exchange amplitude, which means
an inverse square law “pseudo-gravitational force” exists even above the
Planck scale to fields that couple non-minimally. This is remarkable to us and
one of many issues to develop further in this context.

Acknowledgements

Part of this work was done at Fermilab, operated by Fermi Research Alliance,
LLC under Contract No. DE-AC02-07CH11359 with the United States Department of
Energy.

##  Appendix A Projection-Regulated Feynman Loops

The loop induced effective potential for $\phi_{0}$ provides a useful way to
extract all of the $\beta$-functions of the various coupling constants. The
potential $\Gamma(\phi_{0})$ is the log of the path integral: $\Gamma=i\ln P$.
In the case of a real scalar field with mass term we consider the free action,

$\displaystyle\frac{1}{2}\int
d^{4}x\biggl{(}\partial\phi\partial\phi-m^{2}\phi^{2}\biggr{)}$ (66)

we have for the path integral:

$\displaystyle
P=\underset{k}{\prod}\biggl{(}k^{2}-m^{2}\biggr{)}^{-1/2}=\det\biggl{(}k^{2}-m^{2}\biggr{)}^{-1/2}$
(67)

whre $k=(k_{0},\vec{k})$ is the $4$-momentum hence, we have:

$\displaystyle\Gamma=i\ln
P=-\frac{i}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\ln\biggl{(}k^{2}-m^{2}+i\epsilon\biggr{)}$
(68)

This can be evaluated with a Wick rotation to a Euclidean momentum,
$k\rightarrow k_{E}=(ik_{0},\vec{k})$, and a Euclidean momentum space cut off
$\Lambda$:

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{\Lambda}\frac{d^{4}k_{E}}{(2\pi)^{4}}\ln\biggl{(}\frac{k_{E}^{2}+m^{2}}{\Lambda^{2}}\biggr{)}$
(69)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{1}{64\pi^{2}}\biggl{(}\Lambda^{4}\ln\frac{\Lambda^{2}+m^{2}}{\Lambda^{2}}-m^{4}\ln\frac{\Lambda^{2}+m^{2}}{m^{2}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\qquad-\frac{1}{2}\Lambda^{4}+\Lambda^{2}m^{2}\biggr{)}+(\makebox{irrelevant
constants})$

The cutoff can be viewed is a spurious parameter, introduced to make the
integral finite and arguments of logs dimensionless, but and not part of the
defining action. The only physically meaningful dependence upon $\Lambda$ is
contained in the logarithm, where it reflects scale symmetry breaking by the
quantum trace anomaly. Powers of $\Lambda$, e.g.,
$\Lambda^{4},\Lambda^{2}m^{2}$, spuriously break scale symmetry and are not
part of the classical action Bardeen .

It is therefore conceptually useful to have a definition of the loops in which
the spurious powers of $\Lambda$ do not arise. This can be done by defining
the loops applying projection operators on the integrals. The projection
operator

$\displaystyle
P_{n}=\biggl{(}1-\frac{\Lambda}{n}\frac{\partial}{\partial\Lambda}\biggr{)}$
(70)

removes any terms proportional to $\Lambda^{n}.$ Since the defining classical
Lagrangian has mass dimension 4 and involves no terms with $\Lambda^{2}m^{2}$
or $\Lambda^{4},$ we define the regularized loop integrals as:

$\displaystyle\Gamma$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{2}P_{2}P_{4}\int_{0}^{\Lambda}\frac{d^{4}k_{E}}{(2\pi)^{4}}\ln\biggl{(}\frac{k_{E}^{2}+m^{2}}{\Lambda^{2}}\biggr{)}$
(71) $\displaystyle=$
$\displaystyle-\frac{1}{64\pi^{2}}m^{4}\biggl{(}\ln\frac{\Lambda^{2}}{m^{2}}\biggr{)}+O\biggl{(}\frac{m^{6}}{\Lambda^{2}}\biggr{)}$

where we take the limit $\Lambda>>m$ to suppress $O(m^{6}/{\Lambda^{2}})$
terms and we are interested only in the log term (not additive constants) This
means that the additive, non-log terms, e.g. $c^{\prime}m^{2}$, are
undetermined, and the only physically meaningful result is the
$\ln(\Lambda^{2}/m^{2})$ term. $\Lambda$ can be swapped for a renormalization
scale $\mu$. Interestingly, if we define the integral as $\int\rightarrow
P_{1}P_{2}P_{3}...P_{\infty}\int$, the action on the logs will lead to the
Euler constant that arises in dimensional regularization, hinting at a mapping
to the dimensionally regularized result.

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# Sensitivity of YAC to measure the light-component spectrum of primary cosmic
rays at the “knee” energies

L M Zhai1, J Huang1, D Chen2, M Shibata3, Y Katayose3, Ying Zhang1, J S Liu1,
Xu Chen1, X B Hu1,4 and Y H Lin1 1 Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  2 National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100012, China  3 Faculty of Engineering, Yokohama National University,
Yokohama 240-8501, Japan  4 Department of Physics, Shandong University, Jinan
250100, China<EMAIL_ADDRESS>
###### Abstract

A new air-shower core-detector array (YAC : Yangbajing Air-shower Core-
detector array) has been developed to measure the primary cosmic-ray
composition at the “knee” energies in Tibet, China, focusing mainly on the
light components. The prototype experiment (YAC-I) consisting of 16 detectors
has been constructed and operated at Yangbajing (4300 m a.s.l.) in Tibet since
May 2009. YAC-I is installed in the Tibet-III AS array and operates together.
In this paper, we performed a Monte Carlo simulation to check the sensitivity
of YAC-I+Tibet-III array to the cosmic-ray light component of cosmic rays
around the knee energies, taking account of the observation conditions of
actual YAC-I+Tibet-III array. The selection of light component from others was
made by use of an artificial neural network (ANN). The simulation shows that
the light-component spectrum estimated by our methods can well reproduce the
input ones within 10% error, and there will be about 30% systematic errors
mostly induced by the primary and interaction models used. It is found that
the full-scale YAC and the Tibet-III array is powerful to study the cosmic-ray
composition, in particular, to obtain the energy spectra of protons and helium
nuclei around the knee energies.

###### pacs:

98.70.Sa, 96.50.sb, 96.50.sd

††: J. Phys. G: Nucl. Part. Phys.

Keywords: cosmic rays, hadronic interaction, knee, composition, energy
spectrum

##  1 Introduction

It is well known that the all-particle spectrum of primary cosmic rays follows
a power law of dJ/dE $\propto$ E-γ, but steepens at energies around
4$\times$1015 eV where the power index $\gamma$ changes sharply from $\sim$2.7
to $\sim$3.1 [1, 2]. Such structure of the all-particle energy spectrum is
called the “knee”, which is considered to be closely related to the origin,
acceleration and propagation mechanism of cosmic rays. In order to explain the
existence of the knee, many hypotheses and mechanisms [3, 4] have been
proposed. Although all these approaches can well describe the knee structure,
there are much discrepancies in the prediction of the individual components at
the knee region. Thus, it is critical to measure the primary chemical
composition or mass group at energies 50-10 000 TeV, especially, to measure
the primary energy spectra of individual component and determine a break
energy of the spectral index for individual component. Direct cosmic-ray
measurements on board balloons or satellites are the best way to study the
chemical composition, while the maximum energy they can cover is up to 1014
eV/nucleon at most due to limited detection area or exposure time. We may have
no choice but to rely on ground-based air-shower (AS) measurements to study
the primary chemical composition around the knee.

The study of cosmic-ray composition around the knee was done by a hybrid
experiment of the emulsion chambers (ECs), the burst detectors (BDs) and the
AS array (Tibet-II), where ECs and BDs of total area 80 m2 were set up near
the center of the AS array and operated for three years [5, 6, 7]. The
threshold energy of ECs capable of analyzing the fine structure of AS cores is
about 1 TeV, so that it is not difficult to separate the AS events induced by
light-component of protons and helium nuclei, while the energy range of
primary particles is limited to be above $\sim$200 TeV for protons and
$\sim$400 TeV for helium nuclei [6]. This experiment suggests that the flux of
light component is less than $\sim$30% of the total, resulting in that the
knee is dominated by nuclei heavier than helium [7]. A demerit of this
experiment is that there are few statistics of the high-energy core events due
to the high detection threshold energy of the experiment as mentioned above.
To improve this problem, a new air-shower core detector named YAC (Yangbajing
Air shower Core detector) has been developed and improved so as to meet our
requirements.

One important improvement is to lower the detection threshold energy of
primary particles to several times 10 TeV, about one order of magnitude
smaller than the previous experiment. With this improvement, the energy
spectra of individual components measured by YAC will overlap with those of
direct measurements, which may help us to examine the knee of light component,
such as “proton knee” or “helium knee”. Another important improvement of YAC
is its ability to count the number of shower particles passing through each
detector in a wide dynamic ranging from 1 to 106 particles, making it possible
to observe the primary cosmic rays in the energy range from $\sim$10 TeV to
$\sim$10 PeV.

Until now, we have constructed and operated YAC-I as a prototype of full-scale
YAC comprising 400 core detectors. YAC-I is a small array consisting of 16
core detectors which were placed near the center of the Tibet-III AS array as
shown in Fig. 1, while being able to observe a lot of AS-core events in the
energy around $10^{14}$ eV by the operation of a few months. As the primary
composition around this energy region is fairly well known by the direct
observations [8, 9, 10], the data from YAC-I may be used to test the
interaction models such as SIBYLL2.1, QGSJETII-04 and EPOS-LHC being currently
used in the Monte Carlo (MC) simulations. In this paper, we discuss the
performance and sensitivity of YAC for observing light-component spectrum of
primary particles through MC simulations based on the YAC-I experiment.

##  2 YAC-I Experiment

Figure 1: Schematic view of YAC-I and Tibet-III AS array. Open squares :
scintillation detectors of the Tibet-III array. For details of the Tibet-III,
see the paper [1]. Filled red squares : core detectors of the YAC-I array.
YAC-I consists of 16 detector units of 0.2 m2 each, which cover the area of
about 10 m2. Each detector of YAC-I consists of a lead plate with the
thickness of 3.5 cm (6.3 radiation lengths), a supporting iron plate with the
thickness of 0.9 cm (0.5 radiation lengths) and a plastic scintillator with
the thickness of 1 cm.

The YAC-I array consists of 16 core detectors which are placed on a 4$\times
4$ square grid covering the area of about 10 m2 as shown in Fig. 1 and has
been operating since May 2009, together with the Tibet-III AS array. Each core
detector of YAC-I comprises a plastic scintillator with the size of 40 cm
$\times$ 50 cm and a lead plate with the thickness of 3.5 cm (6.3 radiation
lengths) being put on the scintillator. The lead plate is used to select AS
particles and cores capable of having sufficient energy to create cascade
showers in the lead plate and pass through the scintillator. The plastic
scintillator in the core detector is divided into 10 pieces of the width of 4
cm and the scintillation lights are collected through wavelength shifting
(WLS) fibers as shown in Fig. 2. Such design ensures the geometrical
uniformity of detector response within 5 %. The details about the hardware of
YAC detector is described in [11, 12]. Two photomultiplier tubes (PMTs) of
high-gain (HAMAMATSU: R4125) and low-gain (HAMAMATSU: R5325) are equipped to
cover a wide dynamic range from 1 MIP(Minimum Ionization Particle) to 106 MIPs
as seen in Fig. 2. The corresponding linearity and saturation effect of PMT
and scintillator were examined by use of cosmic-ray muons and electron beams
provided by the beam facility of BEPCII (Beijing Electron Positron Collider,
IHEP, China) [13]. The stability of the PMT gain was checked and corrected
using cosmic-ray muons.

Figure 2:  Schematic view of the scintillation detector, the top view (a) and
the back view (b). The scintillator is divided into 10 pieces of 4 cm width
and scintillation lights from each piece are collected through the wavelength
shifting (WLS) fibers.

In this experiment, the YAC-I array works to observe air-shower cores and
their accompanying ASs are observed simultaneously with the Tibet-III AS
array. The Tibet-III provides information on the arrival time and direction of
each air shower and its AS size corresponding to the energy of primary
particle [1]. When a YAC-I event is triggered, its accompanying AS is
simultaneously recorded by the Tibet-III array and the matching between YAC-I
and Tibet-III events is made by their arrival time stamps recorded by a GPS
clock.

##  3 Monte Carlo Simulation

We have carried out a full Monte Carlo (MC) simulation on the development of
air showers in the atmosphere using the simulation code Corsika [14] (version
7.3500). Three hadronic interaction models, including SIBYLL2.1 [15], EPOS-LHC
(v3400) [16] and QGSJETII-04 [17], are used to generate the air-shower events
in the atmosphere. For the primary cosmic rays, we examined three composition
models, namely, “He-poor”, “He-rich” and “Gaisser-fit” models, in order to
evaluate the systematic errors attributable to primary composition models. The
“He-poor” model is based on the HD (Heavy Dominant) model mentioned in the
paper [1], and it is slightly revised to match with the new all-particle
energy spectrum [1], to be the new “He-poor” model. The “He-rich” model is the
“Model B (a lightly harder spectrum than the previous on by taking account of
the nonlinear effects)” mentioned in the paper [4]. The “Gaisser-fit” model is
the “three-population” model mentioned in the paper [18]. The proton spectra
of the former two models are fitted to the direct measurements at the low
energy and consistent with the spectrum obtained from the Tibet-EC experiment
at the high energy. The He spectrum of He-poor model coincides with the
results from RUNJOB, but the He spectrum of He-rich model coincides with the
results from JACEE, ATIC2 and CREAM. The Gaisser-fit model fits to a higher He
model (almost same as our He-rich model) at the low energy range and to the
KASCADE-QGSJET data at high energy range in which light components (P and He)
dominate in the chemical composition. In all models mentioned above, each
component is summed up so as to match with the all-particle spectrum with a
sharp knee, which was obtained with the Tibet-III AS array [1].

Table 1 is a summary of the fractions of the components (P, He, Medium and Fe)
of the three composition models in given energy regions for three primary
models. The energy spectra of individual components (or mass groups) for three
primary models are shown in Fig. 3. It is seen that all the individual
components of the three models in the low energy range (less than 100 TeV) are
in good agreement with direct measurements while differ significantly at
higher energy. The all-particle spectra of three models, however, coincide
with each other and reproduce the sharp knee structure as well.

Table 1: The fractions of individual components in the assumed primary cosmic-
ray spectra of He-poor, He-rich and Gaisser-fit models.

  * Composition | Components |  $10^{13}-10^{14}$ eV |  $10^{14}-10^{15}$ eV |  $10^{15}-10^{16}$ eV
---|---|---|---|---
Models | (%) | (%) | (%)
He-poor | P | 31.5 | 22.7 | 9.6
He | 22.4 | 18.8 | 9.7
Medium | 26.6 | 26.6 | 26.1
Fe | 19.5 | 31.9 | 54.6
He-rich | P | 31.1 | 26.3 | 10.0
He | 25.1 | 28.7 | 17.5
Medium | 32.4 | 34.4 | 50.3
Fe | 11.3 | 10.6 | 22.2
Gaisser-fit | P | 32.8 | 29.0 | 19.6
He | 34.4 | 37.4 | 37.4
Medium | 20.1 | 20.4 | 25.3
Fe | 12.7 | 13.2 | 17.7

In this simulation, primary cosmic rays at the top of the atmosphere within
the zenith angles smaller than 60 degrees are thrown into the atmosphere
isotropically and the minimum energy of primary cosmic rays is set to 40 TeV.
All shower particles in the atmosphere are traced down to the minimum energy
of 1 MeV. The AS events generated are randomly dropped onto the area of 32.84
m $\times$ 32.14 m, which is a 15 m wider in each side of the YAC-I array.
This dropping area is large enough to collect the AS events more than 99.5%
under our core-event selection conditions (see below in the text). Observation
of the MC events is made with the same method as that of the experiment.

The detector responses to shower particles falling on the detectors of
(YAC-I+Tibet-III) array are calculated using the Geant4 [21] (version 9.5),
where the detector performance, trigger efficiency and effective area are
adequately taken into account based on the experimental conditions. The number
of charged particles passing through the scintillator is defined as the PMT
output (charge) divided by that of the single-particle peak. The single-
particle peak is determined by a probe calibration [1, 11] using cosmic rays,
typically muons. The value of single-particle peak is measured as 1.98 MeV for
YAC-I detectors and 6.28 MeV for the Tibet-III detectors. These values are
used in this MC simulation.

Figure 3: Primary cosmic-ray composition for He-poor, He-rich and Gaisser-fit
models compared with those of direct measurements (ATIC2 [8], JACEE [19],
RUNJOB [20], CREAM [9, 10]) and the sum of all components (all-particle
spectrum) compared with the results obtained by the Tibet-III experiment [1] .

The main purpose of this work is to check the sensitivity of YAC to observe
the light component of primary cosmic rays as well as to evaluate the
systematic errors by adopting different primary composition models and
interaction models mentioned above. For this, we selected five combinations of
interaction models and primary composition models. Four combinations of
SIBYLL2.1+He-rich, SIBYLL2.1+He-poor, EPOS-LHC+He-poor and EPOS-LHC+Gaisser-
fit are to check the sensitivity of YAC to the light component and
uncertainties due to the adoption of the different composition models. Other
three combinations of SIBYLL2.1+He-poor, QGSJETII-04+He-poor and EPOS-LHC+He-
poor are to check the interaction models and also uncertainties under the same
primary composition model. It has value to point out here that there is no
serious difference among the current interaction models on the particle
production in the forward region and proton-air inelastic cross sections in
our concerned energy region from 10 TeV to $10^{4}$ TeV since all the models
are well tuned using recent accelerator data including LHC, while there are
big differences among primary composition models because of a lack of direct
observation data at energies above $\sim$200 TeV.

The number of air-shower events generated for each model is 7.40$\times$107,
6.57$\times$107, 4.67$\times$107, 6.25$\times$107 and 5.18$\times$107,
respectively, as shown in Table 2. The analysis of these MC events was made by
the same method used in the experiment.

##  4 Analysis

Information on the size $N_{e}$ and arrival direction of each air shower event
hitting both YAC-I and Tibet-III arrays can be easily obtained from the MC
events observed with the Tibet-III AS array simultaneously. Details of its
analysis are found in the paper [1].

From the YAC-I array, we can obtain the following five quantities reflecting
the characteristic of AS cores : (1) ${N_{hit}}$, the number of “fired”
detectors with ${N_{b}}$$\geq$ 200, where $N_{b}$ is the number of particles
(burst size) observed by each core-detector ; (2) $\sum$${N_{b}}$, the total
sum of $N_{b}$ of fired detector ; (3) ${N_{b}}$top, the maximum ${N_{b}}$
among the fired detectors ; (4) $\langle R\rangle$, the mean lateral spread
defined as $\langle R\rangle$ = $\sum$${r_{i}}$/($N_{hit}$-1) ; (5) $\langle
N_{b}R\rangle$, the mean energy-flow spread defined as $\langle
N_{b}R\rangle=\sum({N_{b}^{i}}\times{r_{i}})/N_{hit}$, where $N_{b}^{i}$
denotes the number of particles observed in $i$-th fired detector and $r_{i}$
represents the lateral distance from the burst center ($X_{c},Y_{c}$), where
($X_{c},Y_{c}$) = $\left(\frac{\sum N_{b}^{i}x_{i}}{\sum N_{b}^{i}},\frac{\sum
N_{b}^{i}y_{i}}{\sum N_{b}^{i}}\right)$. It is confirmed that the five
quantities mentioned above are basic and enough to separate the light
component (P+He) from others. A use of ANN-method [22] may further improve the
quality of separation.

Table 2: Statistics of the data sets selected in MC simulation.

  * Models | Primaries | Core events
---|---|---
(E $\geq$ 40 TeV) | (Mode energy: $\sim$200 TeV)
SIBYLL2.1+He-rich | 7.40$\times$107 | 64 331
SIBYLL2.1+He-poor | 6.57$\times$107 | 47 580
QGSJETII-04+He-poor | 4.67$\times$107 | 31 928
EPOS-LHC+He-poor | 6.25$\times$107 | 42 137
EPOS-LHC+Gaisser-fit | 5.18$\times$107 | 49 390

In order to obtain the light-component spectrum using the data from both
arrays of YAC-I and Tibet-III, we select the high-energy core events by
imposing the conditions of ${N_{b}}\geq 200$, $N_{hit}\geq 4$,
${N_{b}}^{top}\geq 1500$ and ${N_{e}}\geq 80\,000$. The mode energy of primary
particles producing such high energy core events is then estimated to be about
200 TeV. The statistics of the data-sets selected based on the five models are
listed in Table 2. Shown in Fig. 4 is the effective $S\Omega$ of YAC-I array
to observe the AS-core events satisfying the event select conditions, where
$S$ denotes the detection area and $\Omega$ the solid angle. The effective
$S\Omega$ depends weakly on the model used, but its difference is found to be
smaller than 25% in our concerned energy range.

Figure 4:  The effective $S\Omega$ of YAC-I array to observe the light-
component for various models used in MC.

##  5 Results and Discussion

We check the sensitivity of YAC array to the interaction models and primary
cosmic-ray models using the high-energy core events selected under the
conditions discussed in the previous section.

###  5.1 Total burst-size spectrum and mean lateral spread of AS-cores

It is well known that the absolute intensity of the total burst sizes depends
sensitively on the increase of cross sections, inelasticity, and also on the
primary cosmic-ray composition. Shown in Fig. 5-(a) is the integral total
burst-size spectrum ($\sum N_{b}$ (SumNb)) obtained by the respective MC model
for comparison. The $\sum N_{b}$ spectra obtained by five MC models are
compared each other by taking the flux ratio to that by the SIBYLL2.1+He-rich
model in Fig. 5-(b). It is seen that the EPOS-LHC+Gaisser-fit model gives the
highest flux in all $\sum N_{b}$ region. According to our MC simulation, the
observed AS cores in the size region of $\sum N_{b}$ = 2$\times 10^{3}$ \-
4$\times 10^{5}$ are produced mostly by the light component (P+He) with its
primary energies of several times 1014 \- 1015 eV. The fraction of light
component in the primary of this energy region is about 66% for Gaisser-fit
model while about 55% for the He-rich model and 42% for the He-poor model as
seen in Table 1. It should be, however, noted that about 70% of the observed
high-energy core events are originated by the light component, that is, the
contribution from other nuclei is fairly small.

Figure 5: (a) Integral $\sum N_{b}$ (SumNb) spectra obtained from five MC
models, (b) The intensity ratios of $\sum N_{b}$ to that obtained by the
SIBYLL2.1+He-rich model, (c) The mean energy-flow lateral spreads $\langle
N_{b}R\rangle$ in the respective energy interval for five MC models.

Here, we discuss the uncertainties due to different interaction models in
reference to the spectrum of high-energy core events. Before entering in
discussion, we should first remind that the production rate of high-energy
core events is most sensitive to the energy per nucleon of primary particles.
Thus, the mean energy per nucleon of helium nuclei is 1/4 of protons when
compared at the same primary energy and also the interaction mean free path of
helium nuclei in the air is about a half of that of protons. For the He-poor
model, it is seen that the flux of protons is slightly higher than that of
helium nuclei or almost same in the energy region over about 300 TeV and also
their power indices are almost same as learned from Fig. 3 and Table 1. If we
combine this with above discussion, it may be allowed to ignore the
contribution from helium nuclei to the core events observed in this primary
model, that is, it is regarded as those produced by protons. Under this
assumption, it may be noticed from Fig. 5-(b) that the flux values by
SIBYLL2.1, QGSJETII-04 and EPOS-LHC using the same primary model of He-poor
match well with an error of smaller than 10%, which may be attributed to
uncertainties due to the interaction models used. A deviation of QGSJETII-04
in the core-size region above about 2$\times 10^{5}$ may be due to a low
statistics of the events, that is, within a statistical fluctuation (at most
2$\sigma$ level).

On the other hand, when we arrange the light component flux of protons and
helium nuclei in descending order in the $10^{2}$ -$10^{4}$ TeV region, it
becomes as Gaisser-fit $>$ He-rich $>$ He-poor. It is then confirmed that the
flux of core events is in the same order as the primary light-component flux
and also the shape of each primary spectrum is well reflected in the
corresponding core-event spectrum, as seen in Fig. 5-(b). A typical example is
seen in the case of Gaisser-fit primary model. This means that high energy
core-event observation with YAC is very sensitive to the primary light-
component spectrum.

A correlation between $\langle N_{b}R\rangle$ and $\sum N_{b}$ is shown in
Fig. 5-(c). This figure tells us that the Gaisser-fit primary model gives
smaller lateral spread than others, while the QGSJETII-04+He-poor model gives
larger spread than others. As protons with the long interaction mean free path
can penetrate deep in the atmosphere and produce AS cores near the observation
level, resulting in giving smaller lateral spread. The Gaisser-fit primary is
light-component dominant as seen in Fig. 3, so that the core spread by this
model should be smaller than others. When the primary model is fixed as He-
poor, the mean spread of QGSJETII-04 is slightly larger than that of
SIBYLL2.1. This may slightly depend on the interaction model since the energy
spectrum of secondary particles in the very forward region (Feynman $x\sim$
0.1 - 0.3) produced at collision in the SIBYLL2.1 model is harder than that of
QGSJETII-04, that is, the former contains slightly larger number of very high-
energy secondaries than the latter. The difference of the intensity and
lateral spread between both models could be attributed to the number of very
high energy particles as those penetrate deep in the atmosphere. It should be
noted that the lateral spread of AS-cores is mostly caused by Coulomb
scattering of shower electrons and positrons in the atmosphere, not by
transverse momentum of secondaries produced at collisions except within the
depth of 1-2 radiation lengths from the interaction point in the atmosphere.
In connection with this, the energy loss of AS cores generated by QGSJETII-04
may be faster than those by SIBYLL2.1, resulting in giving lower AS-core
intensity. Hence, a precise AS experiment like the (YAC + Tibet-III AS) array
will be able to examine the interaction models to some extent.

###  5.2 Sensitivity of YAC-I to observe the light-component spectrum around
the knee

In this analysis, we use the ANN technique to separate the light-component
from others. This method is shown to be quite effective for such purpose as
confirmed by our previous works [6, 7]. In this ANN analysis, we use the
following seven quantities : (1) $N_{hit}$, (2) $\sum$$N_{b}$, (3)
$N_{b}^{top}$, (4) $\langle R\rangle$, (5) $\langle N_{b}R\rangle$, (6)
$N_{e}$ and (7) $\theta$ (zenith angle). These are input to the ANN with 35
hidden nodes and 1 output unit. To train the ANN in separating light-component
(P+He) from other nuclei, the input patterns for light-component and others
are set to 0 and 1, respectively. We then define a critical value of $T_{c}$
to calculate the corresponding purity and selection efficiency of the selected
(P+He)-like events.

Figure 6:  ANN output pattern value (T) distribution of training (P+He) events
based on the EPOS-LHC+He-poor model. The average selection purity and
efficiency over whole energy range of (P+He)-like events are 95%, 76% at
${T_{c}}$ = 0.4. Table 3: The ratios of (P+He)/All at three phases of
analysis, before and after ANN training based on the MC models. In this table
the second column represents the ratio of primary (P+He) flux to the all-
particle flux in the energy region above 40 TeV. The third column represents
the ratio of true (P+He) events contained in the observed high-energy core
events selected by the condition mentioned in the text. The fourth column
represents the ratio of true (P+He) events contained in the ANN trained core
events with $T\leq T_{c}=0.4$. The fifth column represents the ratio of the
number of ANN trained core events with the output $T\leq 0.4$ to that of all
ANN trained core events ($0\leq T\leq 1$).

  * Models | Primaries | Core events | After ANN training
---|---|---|---
(P+He)/all(%) | (P+He)/all(%) | Purity (%) | Efficiency (%)
SIBYLL2.1+He-rich | 56.2 | $71.1\pm 0.2$ | $92.0\pm 0.2$ | $70.1\pm 0.3$
SIBYLL2.1+He-poor | 46.8 | $69.7\pm 0.2$ | $94.0\pm 0.2$ | $71.1\pm 0.3$
QGSJETII-04+He-poor | 46.8 | $69.5\pm 0.3$ | $94.6\pm 0.2$ | $77.1\pm 0.4$
EPOS-LHC+He-poor | 46.8 | $69.3\pm 0.2$ | $94.6\pm 0.2$ | $75.8\pm 0.3$
EPOS-LHC+Gaisser-fit | 67.0 | $87.4\pm 0.1$ | $96.1\pm 0.2$ | $67.8\pm 0.3$

Figure 6 shows the ANN output distribution trained using the EPOS-LHC+He-poor
model. As seen in this figure, the events with ${T_{c}}\leq 0.4$ could be
regarded as the (P+He)-like events, and the average selection purity and
efficiency over whole energy range of (P+He)-like events are 95% and 76%,
respectively. Table 3 is a summary of the ratios of the (P+He)/All before and
after ANN training analysis based on the five MC models over the whole energy
range. Thanks to the performance of the YAC-I array, the ratio of the
(P+He)/All before ANN training (core events) has already reached $\sim$70% by
the core-event selection conditions ($\sim$87% for Gaisser-fit model, as seen
in Table 3). With the ANN training, the purity of selected (P+He)-like events
is further increased up to $\sim$95% as learned from Table 3. This high
quality data set is used for reconstructing the light-component primary
spectrum. Table 3 teaches us that after ANN training the difference of the
selection purity and efficiency is within $\sim$6% among three hadronic
interaction models, and $\sim$8% among three primary composition models.
Overall uncertainties due to ANN training for the MC models are then estimated
to be about 10%.

###  5.3 Expected light-component spectrum

Using the ANN trained by the EPOS-LHC+He-poor events, we select the (P+He)
like events from all the observed events and also obtain the selection purity
and efficiency. The primary energy $E_{0}$ of each selected events is then
estimated using the AS size $N_{e}$ obtained by the Tibet-III. The relation
between air shower size $N_{e}$ and primary energy $E_{0}$ is expressed as

$E_{0}=\alpha\times N_{e}^{\beta},$

where the parameters of $\alpha$ and $\beta$ are estimated from AS events
generated by the MC for the Tibet-III AS array, while $\alpha$ and $\beta$
depending on the zenith angle of air showers. Details of this procedure is
described in the paper [1]. Shown in Fig. 7 is the correlation between $E_{0}$
and $N_{e}$ of the (P+He)-like events selected by the ANN method. The solid
line denotes the best fit curve for this correlation, and the parameter values
of $\alpha=0.794$ and $\beta=1.005$ are then obtained for air showers with
$\sec\theta\leq 1.1$. The energy resolution is also estimated as about 25% at
energies around 200 TeV.

Figure 7:  Scatter plots of the primary energy ($E_{0}$) and the estimated
shower size ($N_{e}$) of (Proton+Helium)-like events based on EPOS-LHC+He-poor
model with $\sec\theta\leq 1.1$. Solid line shows the fitting result of
$E_{0}$ = 0.794$\times$ ${N_{e}}^{1.005}$ GeV.

We also checked a dependence of the correlation of $N_{e}$ and $E_{0}$ on the
interaction and primary composition model, and obtained the similar
relationship in the other models. We then found that there is less than 10%
difference for the determination of the primary energy by use of different
interaction and primary composition models.

Figure 8 shows the estimated primary energy spectrum of the light-component
(P+He) in comparison with the assumed primary spectrum. It is seen that the
estimated energy spectrum well reproduces the assumed one within about 10%
errors. Almost same results are obtained for other MC models.

Figure 8:  The estimated energy spectrum of (Proton+Helium) compared with the
assumed (input) one based on SIBYLL2.1+He-rich model.

In this work, we discuss the systematic errors coming from the models used in
the MC simulation for deriving the primary (P+He) spectrum using the
(YAC-I+Tibet-III) array. The systematic errors caused by each step of the
analysis procedures are investigated including the dependence of the MC data
on the interaction models, the primary composition models and the algorithms
for the primary mass identification. We summarize these systematic errors as
follows: 1) errors due to the observation efficiency ($S\Omega$ of (P+He))
depending on the interaction and primary composition models are found to be
smaller than 25%; 2) errors due to the selection of (P+He)-like events with
ANN training are about 10%, in which the model dependence on the purity and
efficiency is totally included; 3) errors due to the estimation of primary
energy with use of the conversion from $N_{e}$ to $E_{0}$ are 10%, in which
the dependence on both interaction and primary composition models are taken
into account; 4) errors due to the reconstruction procedures of the primary
light-component spectrum from the observed core events are estimated to be
smaller than 10%. The total systematic errors are then estimated to be about
30% as the square root of quadratic sum of those four systematic errors, which
may be somewhat overestimated because of a little correlation among four error
estimation parts.

##  6 Summary

In this paper, we have carried out a full MC simulation to examine the
capability of measuring the energy spectrum of primary light-component (P+He)
of cosmic rays at the knee energies using the (YAC-I+Tibet-III) array. The
models used in this MC simulation are SIBYLL2.1, EPOS-LHC (v3400) and
QGSJETII-04 for the interaction and He-poor, He-rich and Gaisser-fit for the
primary cosmic-ray composition. The Corsika code was used to generate AS
events in the atmosphere and the Geant4 code was used to treat the shower
particles entering in the detectors. The air-shower core events observed with
the YAC-I array were analyzed to select those induced by the light component
using the ANN technique. In this paper, we focused on the sensitivity of the
YAC-I array to observe the light component of cosmic rays around the knee and
discussed the systematic errors coming from the the models used in the MC
indispensable for obtaining the result which should be independent on the
model as possible. It is shown that the YAC+Tibet AS array is powerful to
study the primary cosmic-ray chemical composition, in particular, to obtain
the energy spectrum of light-component (P+He) of cosmic rays at the knee
energies. A full-scale YAC consisting of 400 core detectors covering more than
5000 m2 could be operated together with the Tibet-III array in the very near
future.

The authors would like to express their thanks to the members of the Tibet
AS$\gamma$ collaboration for the fruitful discussion. This work is supported
by the Grants from the National Natural Science Foundation of China
(Y11122005B, Y31136005C and Y0293900TF) and the Chinese Academy of Sciences
(H9291450S3) and the Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, CAS. The Knowledge Innovation Fund (H95451D0U2 and
H8515530U1) of IHEP, China also provide support to this study.

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