††institutetext:  Center for Theoretical Physics of the Universe, Institute
for Basic Science, Daejeon 34126,
South Korea

# Precision axion physics with running axion couplings

Kiwoon Choi111Electronic address<EMAIL_ADDRESS>Sang Hui Im222Electronic
address<EMAIL_ADDRESS>Hee Jung Kim333Electronic address:
<EMAIL_ADDRESS>and Hyeonseok Seong444Electronic address:
<EMAIL_ADDRESS>
###### Abstract

We study the renormalization group running of axion couplings while taking
into account that the Standard Model can be extended to its supersymmetric
extension at a certain energy scale below the axion decay constant. We then
apply our results to three different classes of axion models, i.e. KSVZ-like,
DFSZ-like, and string-theoretic axions, and examine if string-theoretic axions
can be distinguished from others by having a different pattern of low energy
couplings to the photon, nucleons and electron. We find that the low energy
couplings of string-theoretic axions have a similar pattern as those of KSVZ-
like axions but yet reveal a sizable difference which might be testable in
future axion search experiments. We also note that the coupling of KSVZ-like
QCD axions to the electron is dominated by a three-loop contribution involving
the exotic heavy quark, gluons, top quark and Higgs field.

###### Keywords:

axions, axion couplings, effective theories

††preprint: CTPU-PTC-21-26

##  1 Introduction

Axions are one of the most compelling candidates for physics beyond the
Standard Model (SM) of particle physics 0807.3125 ; 1510.07633 ; 2003.01100 .
Axions were originally postulated to solve the strong CP problem Peccei:1977hh
; Weinberg:1977ma ; Wilczek:1977pj  and later turned out to be a natural
candidate for dark matter Preskill:1982cy ; Abbott:1982af ; Dine:1982ah .
Generically there can also be axions not necessarily associated with the
strong CP problem, which are often dubbed axion-like particle (ALP). Such ALPs
can be much lighter than the QCD axion solving the strong CP problem and may
constitute dark matter with interesting astrophysical signatures 2101.11735 .

A key feature of axions is that the field variable is periodic as

$\displaystyle a(x)\,\cong\,a(x)+2\pi f_{a},$ (1)

where the scale parameter $f_{a}$ is termed the axion decay constant. Axions
can be naturally light as their couplings are constrained by an approximate
non-linear $U(1)_{\rm PQ}$ symmetry which involves a shift of the axion field:

$\displaystyle U(1)_{\rm PQ}:\quad a(x)\,\,\,\rightarrow\,\,\,a(x)+{\rm
constant}.$ (2)

Axions may originate from the phase of complex scalar field Peccei:1977hh ;
Weinberg:1977ma ; Wilczek:1977pj ; Kim:1979if ; Shifman:1979if ; Dine:1981rt ;
Zhitnitsky:1980tq  as

$\displaystyle\sigma(x)=\rho(x)e^{ia(x)/f_{a}},$ (3)

and then $f_{a}$ is determined by the vacuum expectation value of $\rho$ as
$f_{a}=\sqrt{2}\langle\rho\rangle$. In the following discussion, such axions
will be termed “field-theoretic axions”. It is obvious that the field region
around $f_{a}=0$ for field-theoretic axions is well described by an effective
field theory in which $U(1)_{\rm PQ}$ is linearly realized. There are two
types of field-theoretic axion models which are phenomenologically viable and
have been widely discussed in the literature555 Here we consider only
relatively simple models for field-theoretic axions. For a discussion of more
involved models, e.g. those with hierarchical axion couplings, see for
instance 2012.05029 . : the Kim-Shifman-Vainshtein-Zakharov (KSVZ) axion
models Kim:1979if ; Shifman:1979if  in which all SM fields are neutral under
the linearly-realized $U(1)_{\rm PQ}$ and the Dine-Fischler-Srednicki-
Zhitnitsky (DFSZ) axion models Dine:1981rt ; Zhitnitsky:1980tq  in which the
SM fermions carry nonzero $U(1)_{\rm PQ}$-charges.

Axions can also arise from the zero modes of a higher-dimensional $p$-form
gauge field $C^{(p)}$ which has a quantized coupling to $(p-1)$-brane in the
underlying UV theory hep-th/0605206 ; 0905.4720 :

$\displaystyle
a(x)=\int_{\Sigma_{p}}C^{(p)}_{[m_{1},..,m_{p}]}(x,y)\,dy^{m_{1}}...dy^{m_{p}},$
(4)

where $\Sigma_{p}$ is a $p$-dimensional cycle in the compact internal space
with the coordinates $y^{m}$. We will term such axions originated from
$C^{(p)}$ “string-theoretic axions”. It is widely believed that the limit with
$f_{a}=0$ for string-theoretic axions is in swampland at an infinite field
distance with infinite towers of light particles 1903.06239 .

It is well known that KSVZ axions and DFSZ axions have a different pattern of
low energy couplings, which might be testable in future axion search
experiments 0807.3125 ; 2003.01100 . For instance, at tree level, the
couplings of KSVZ axions to the SM fermions are all vanishing, while those of
DFSZ axions are of order unity in the unit of $1/f_{a}$, which results in
different ratios among the low energy axion couplings $g_{aX}$
($X=\gamma,p,n,e$) to the photon, nucleons and electron. Similarly, generic
DFSZ-like axions with ${\cal O}(1)$ tree-level couplings (again in the unit of
$1/f_{a}$) to the light quarks and electron have a clearly distinguishable
pattern of $g_{aX}$ ($X=\gamma,p,n,e$) compared to those of generic KSVZ-like
axions with negligible tree-level couplings to the light quarks and electron.
On the other hand, it has not been examined yet if string-theoretic axions can
also be distinguished from other types of axions by their low energy
couplings. As we will see, the couplings of string-theoretic axions to the SM
gauge and matter fields have a similar pattern as those of KSVZ-like axions.
Therefore it requires a careful analysis including radiative corrections
Srednicki:1985xd ; hep-ph/9306216 ; 1708.00021 ; 2002.04623 ; 2011.08205 ;
2012.05029 ; 2012.09017 ; 2012.12272  to axion couplings to see if string-
theoretic axions can be discriminated from KSVZ-like axions.

With this motivation, in this paper, we study the renormalization group (RG)
running of axion couplings in a generic context and apply the results to the
models of our interest. To examine if string-theoretic axions and KSVZ-like
axions can have a distinguishable pattern of $g_{aX}$ ($X=\gamma,p,n,e$), we
first perform a generic renormalization group analysis for the axion couplings
to matter fermions and Higgs fields while taking into account that the
Standard Model can be extended to its supersymmetric extension at a scale
below $f_{a}$. We then find that string-theoretic QCD axion and KSVZ-like QCD
axion have a significantly different value of the coupling ratio
$g_{ae}/g_{a\gamma}$, which would make it feasible to distinguish one from the
other by an experimental measurement of $g_{ae}/g_{a\gamma}$. For an axion-
like particle (ALP) which does not couple to the gluons but has a non-zero
coupling to the photon, the prospect for discrimination is much more
promising: all of the three coupling ratios $g_{aY}/g_{a\gamma}$ ($Y=p,n,e$)
for string-theoretic ALP and KSVZ-like ALP differ by about one order of
magnitude. We also find that the coupling of KSVZ-like QCD axion to the
electron is dominated by a three-loop contribution involving the heavy exotic
quark, gluons, top quarks and Higgs doublet, which was not noticed in the
previous studies Srednicki:1985xd ; hep-ph/9306216 ; 1708.00021 ; 2002.04623 ;
2011.08205 ; 2012.09017 ; 2012.12272 .

The organization of this paper is as follows. In section 2, we discuss the
relevant features of the axion couplings to gauge and matter fields at the UV
boundary scale $\mu\sim f_{a}$ for DFSZ-like, KSVZ-like and string-theoretic
axions. In section 3, we present the RG equations for axion couplings in the
context of generic effective lagrangian and compute the resulting radiative
corrections to the low energy axion couplings to the light quarks and
electron. In this analysis, we take into account that the SM can be extended
to its supersymmetric extension at an energy scale below $f_{a}$. In section
4, we apply our result to the three different classes of axions, i.e. DFSZ-
like, KSVZ-like, and string-theoretic axions, to see if they can be
distinguished from each other by their low energy couplings to the photon,
nucleons and electron. Section 5 is the conclusion.

##  2 Axion couplings to matter and gauge fields at UV scales

In this section, we discuss the axion couplings to gauge and matter fields at
the UV boundary scale $\mu\sim f_{a}$ in models of KSVZ-like, DFSZ-like, and
string-theoretic axions. We will be brief on field-theoretic KSVZ-like and
DFSZ-like axions and pay more attention to the characteristic features of
string-theoretic axions. To proceed, we adopt the Georgi-Kaplan-Randall (GKR)
field basis Georgi:1986df  which can be defined at a scale around or below
$f_{a}$, for which only the axion field transforms under $U(1)_{\rm PQ}$ as

$\displaystyle U(1)_{\rm PQ}:\,\,\,\,a(x)\rightarrow a(x)+{\rm constant},$ (5)

and all other fields in the model are invariant under $U(1)_{\rm PQ}$. Then
the relevant axion couplings at $\mu\sim f_{a}$ take the form

$\displaystyle\frac{1}{2}\partial_{\mu}a\partial^{\mu}a+\frac{\partial_{\mu}a}{f_{a}}\Big{[}ic_{\phi}(\phi^{*}D^{\mu}\phi-\phi
D^{\mu}\phi^{*})+c_{\psi}\bar{\psi}\bar{\sigma}^{\mu}\psi\Big{]}+c_{A}\frac{g_{A}^{2}}{32\pi^{2}}\frac{a(x)}{f_{a}}F^{A\mu\nu}\tilde{F}^{A}_{\mu\nu},$
(6)

where $\phi$ and $\psi$ denote canonically normalized gauge-charged scalar
fields and chiral fermions in the model, and
$F^{A}_{\mu\nu}=(G^{\alpha}_{\mu\nu},W^{i}_{\mu\nu},B_{\mu\nu})$ are the
canonically normalized gauge field strength of the SM gauge group
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$. In this description, the axion
couplings $c_{A}$ ($A=G,W,B$) to gauge fields are quantized, e.g. integer-
valued for an appropriate normalization of the gauge couplings $g_{A}^{2}$,
and invariant under the subsequent RG evolution, while the couplings
$c_{\phi}$ and $c_{\psi}$ to matter fields are real-valued and generically
have nontrivial RG evolution induced by the gauge and Yukawa interactions in
the model.

As will be discussed in detail in section 3, to examine the low energy axion
couplings in the model, one can first scale the effective lagrangian (6) down
to a low energy scale $\mu={\cal O}(1)$ GeV. The relevant axion couplings at
this scale are given by

$\displaystyle\frac{1}{32\pi^{2}}\frac{a(x)}{f_{a}}\left(c_{\gamma}e^{2}F^{\mu\nu}\tilde{F}_{\mu\nu}+c_{G}g_{s}^{2}G^{\alpha\mu\nu}\tilde{G}^{\alpha}_{\mu\nu}\right)+\sum_{\Psi=u,d,e}\frac{\partial_{\mu}a}{2f_{a}}C_{\Psi}\bar{\Psi}\gamma^{\mu}\gamma_{5}\Psi,$
(7)

where $e$ and $g_{s}$ are the electromagnetic and color gauge couplings,
respectively, and

$\displaystyle c_{\gamma}=c_{W}+c_{B},\quad C_{\Psi}=C_{\Psi}^{0}+\Delta
C_{\Psi},$ (8)

with

$\displaystyle C^{0}_{u}$ $\displaystyle=$ $\displaystyle
c_{Q_{1}}(f_{a})+c_{u^{c}_{1}}(f_{a})+c_{H}(f_{a}),$ $\displaystyle C^{0}_{d}$
$\displaystyle=$ $\displaystyle
c_{Q_{1}}(f_{a})+c_{d^{c}_{1}}(f_{a})-c_{H}(f_{a}),$ $\displaystyle C^{0}_{e}$
$\displaystyle=$ $\displaystyle
c_{L_{1}}(f_{a})+c_{e^{c}_{1}}(f_{a})-c_{H}(f_{a}).$ (9)

Here $C_{\Psi}^{0}$ ($\Psi=u,d,e$) denote the axion couplings to the light
quarks and electron evaluated at tree-level, which can be interpreted as the
couplings at the UV boundary scale $\mu=f_{a}$ in our approximation, $\Delta
C_{\Psi}$ are radiative corrections to $C_{\Psi}$ received over the scales
from $f_{a}$ to $\mu={\cal O}(1)$ GeV, which will be extensively discussed in
section 3, $c_{\phi}(f_{a})$ and $c_{\psi}(f_{a})$ are the axion couplings at
$\mu=f_{a}$ including the couplings to the Higgs doublet $H$ and the three
generations of chiral quarks and leptons
$\psi=\\{Q_{i},u^{c}_{i},d^{c}_{i},L_{i},e^{c}_{i}\\}$ ($i=1,2,3$), which
would be determined by the underlying UV-completed axion model, and the
effects of flavor mixing are ignored for simplicity. The axion-Higgs coupling
$c_{H}$ gives rise to the axion couplings to the SM fermions via the axion-$Z$
boson kinetic mixing $\partial_{\mu}aZ^{\mu}$ below the weak scale.
Alternatively, one can rotate away the axion-Higgs coupling $c_{H}$ to the
axion-fermion couplings by performing axion-dependent field redefinition
proportional to $U(1)_{Y}$ hypercharge:

$\begin{split}H\rightarrow He^{ic_{H}a/f_{a}},\quad\psi\rightarrow\psi
e^{i(Y_{\psi}/Y_{H})c_{H}a/f_{a}},\end{split}$ (10)

which results in Eq. (2).

In regard to the experimental verification of axions, the most relevant
couplings are those to the photon, nucleons and electron at scales well below
GeV:

$\displaystyle\frac{1}{4}g_{a\gamma}a\vec{E}\cdot\vec{B}+\partial_{\mu}a\left[\frac{g_{ae}}{2m_{e}}\bar{e}\gamma^{\mu}\gamma_{5}e+\frac{g_{an}}{2m_{n}}\bar{n}\gamma^{\mu}\gamma_{5}n+\frac{g_{ap}}{2m_{p}}\bar{p}\gamma^{\mu}\gamma_{5}p\right]$
(11)

which are determined by the couplings in (7) as follows:

$\displaystyle g_{a\gamma}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha_{\rm
em}}{2\pi}\frac{1}{f_{a}}\Big{(}c_{\gamma}-\frac{2}{3}\frac{m_{u}+4m_{d}}{m_{u}+m_{d}}c_{G}\Big{)}\,\simeq\,\frac{\alpha_{\rm
em}}{2\pi}\frac{1}{f_{a}}\Big{(}c_{\gamma}-1.92c_{G}\Big{)},$ $\displaystyle
g_{ap}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{p}}{f_{a}}\left(C_{u}\Delta u+C_{d}\Delta
d-\Big{(}\frac{m_{d}}{m_{u}+m_{d}}\Delta u+\frac{m_{u}}{m_{u}+m_{d}}\Delta
d\Big{)}c_{G}\right),$ $\displaystyle\simeq$
$\displaystyle\frac{m_{p}}{f_{a}}\Big{(}0.90\,C_{u}(2\,{\rm
GeV})-0.38\,C_{d}(2\,{\rm GeV})-0.48\,c_{G}\Big{)},$ $\displaystyle g_{an}$
$\displaystyle\simeq$ $\displaystyle\frac{m_{n}}{f_{a}}\left(C_{d}\Delta
u+C_{u}\Delta d-\Big{(}\frac{m_{u}}{m_{u}+m_{d}}\Delta
u+\frac{m_{d}}{m_{u}+m_{d}}\Delta d\Big{)}c_{G}\right),$ $\displaystyle\simeq$
$\displaystyle\frac{m_{n}}{f_{a}}\Big{(}0.90\,C_{d}(2\,{\rm
GeV})-0.38\,C_{u}(2\,{\rm GeV})-0.04\,c_{G}\Big{)},$ $\displaystyle g_{ae}$
$\displaystyle\simeq$ $\displaystyle\frac{m_{e}}{f_{a}}C_{e}(m_{e}),$ (12)

with $\Delta u=0.897(27)$ and $\Delta d=-0.376(27)$ at $\mu=2$ GeV in
$\overline{\rm MS}$, and $m_{u}/m_{d}=0.48(3)$ 1511.02867 ; 2003.01100 . One
of our primary concerns is if string-theoretic axions can be discriminated
from field-theoretic axions by having a distinguishable pattern of $g_{aX}$
($X=\gamma,p,n,e$).

###  2.1 Field-theoretic axions: KSVZ-like and DFSZ-like axions

Field-theoretic axion models have a UV completion with a linearly realized
Peccei-Quinn symmetry:

$\displaystyle U(1)_{\rm PQ}:\quad\sigma\rightarrow
e^{i\alpha}\sigma,\quad\phi\rightarrow
e^{iq_{\phi}\alpha}\phi,\quad\psi\rightarrow e^{iq_{\psi}\alpha}\psi,$ (13)

where $\sigma$ denotes a $U(1)_{\rm PQ}$-charged complex scalar field whose
phase field is identified as the axion as in (3). For simplicity, here we
assume that $U(1)_{\rm PQ}$ is spontaneously broken predominantly by the
vacuum expectation value of a single gauge-singlet complex scalar field
$\sigma$.

To move to the GKR field basis Georgi:1986df , one can parameterize $\sigma$
as

$\displaystyle\sigma=\frac{1}{\sqrt{2}}(f_{a}+\hat{\rho})e^{ia/f_{a}},$ (14)

where $\langle\hat{\rho}\rangle=0$, and make an axion-dependent field
redefinition:

$\displaystyle\phi\rightarrow e^{-iq_{\phi}a/f_{a}}\phi,\quad\psi\rightarrow
e^{-iq_{\psi}a/f_{a}}\psi.$ (15)

Then, $U(1)_{\rm PQ}$ is realized as the GKR form in (5), and the axion
couplings to gauge and matter fields arise as a consequence of the field
redefinition (15). The resulting couplings at $\mu\sim f_{a}$ are determined
simply by the PQ-charges of the linear $U(1)_{\rm PQ}$ symmetry (13):

$\displaystyle c_{\phi}(f_{a})=q_{\phi},\quad
c_{\psi}(f_{a})=q_{\psi},\quad{c_{A}=\sum_{\psi}2q_{\psi}{\rm
Tr}(T_{A}^{2}(\psi))},$ (16)

where $T_{A}(\psi)$ $(A=G,W,B)$ denotes the gauge charge of $\psi$ for the SM
gauge group $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$.

There are two different types of field-theoretic axion models that have been
widely discussed in the literature, DFSZ models in which the SM fermions carry
non-zero $U(1)_{\rm PQ}$ charges and KSVZ models in which all SM fermions are
neutral under $U(1)_{\rm PQ}$. A minimal non-supersymmetric (non-SUSY) DFSZ
model Dine:1981rt ; Zhitnitsky:1980tq  involves the two Higgs doublets $H_{u}$
and $H_{d}$ which have the Yukawa couplings for the up-type quark masses and
the down-type quark and lepton masses, respectively, as well as the coupling
$\kappa\sigma^{2}H_{u}H_{d}$ in the potential with $\kappa$ small enough to
allow the correct electroweak symmetry breaking. The resulting axion couplings
at $\mu\sim f_{a}$ are given by

$\displaystyle\mbox{non-SUSY DFSZ}:\quad c_{H_{u,d}}(f_{a})=-1,\quad
c_{\psi}(f_{a})=\frac{1}{2},\quad{c_{G}=c_{W}=\frac{3}{5}c_{B}=6},$ (17)

where $\psi=\\{Q_{i},u^{c}_{i},d^{c}_{i},L_{i},e^{c}_{i}\\}$ are the chiral
quarks and leptons in the SM. The corresponding tree-level axion couplings to
the light quarks and electron, which can be identified as the couplings at
$\mu\sim f_{a}$, are

$\displaystyle\mbox{DFSZ}:\quad C_{u}^{0}=2\cos^{2}\beta,\quad
C_{d}^{0}=C_{e}^{0}=2\sin^{2}\beta,$ (18)

where $\tan\beta=\langle H_{u}\rangle/\langle H_{d}\rangle$ and we used
$c_{H}=c_{H_{u}}\sin^{2}\beta-c_{H_{d}}\cos^{2}\beta$ for the coupling to the
SM Higgs doublet $H$. It is straightforward to supersymmetrize this model, for
instance with a superpotential involving a term
$\sim\sigma^{2}H_{u}H_{d}/M_{P}$ Kim:1983dt , where $M_{P}=2.4\times 10^{18}$
GeV is the reduced Planck scale. The axion couplings of this SUSY DFSZ model
at $\mu\sim f_{a}$ are

$\displaystyle\mbox{SUSY DFSZ}:\quad c_{H_{u,d}}(f_{a})=-1,\quad
c_{\psi}(f_{a})=\frac{1}{2},\quad c_{G}=6,{\quad c_{W}=4,\quad c_{B}=8},$ (19)

where now $H_{u,d}$ and $\psi$ denote the full supermultiplets including
superpartners, and $c_{W,B}$ includes the Higgsino contribution to the
$U(1)_{\rm PQ}$ anomaly. Obviously, the tree-level axion couplings to the
light quarks and electron in this SUSY DFSZ model are the same as those of its
non-SUSY counterpart which is given by (18).

In KSVZ models Kim:1979if ; Shifman:1979if , all SM fields are neutral under
$U(1)_{\rm PQ}$. However, there exist exotic $U(1)_{\rm PQ}$-charged left-
handed quark and anti-quark $({\cal Q},{\cal Q}^{c})$ which get a heavy mass
as a consequence of the spontaneous breakdown of $U(1)_{\rm PQ}$ by the vacuum
expectation value $\langle\sigma\rangle=f_{a}/\sqrt{2}$, e.g. get a mass by
the Yukawa coupling

$\displaystyle{\cal L}_{\rm Yukawa}=y_{\cal Q}\sigma{\cal Q}{\cal Q}^{c}.$
(20)

One then finds

$\displaystyle{\rm KSVZ}:\quad c_{{\cal Q}}(f_{a})=c_{{\cal
Q}^{c}}(f_{a})=\frac{1}{2},\quad
c_{H}(f_{a})=c_{\psi}(f_{a})=0,\quad{c_{A}=2{\rm Tr}(T_{A}^{2}({\cal Q}))}$
(21)

for $\psi=\\{Q_{i},u^{c}_{i},d^{c}_{i},L_{i},e^{c}_{i}\\}$, which result in

$\displaystyle{\rm KSVZ}:\quad C^{0}_{\Psi}=0\quad(\Psi=u,d,e).$ (22)

In fact, there can be a contribution to $c_{\psi}(f_{a})$ from Planck-scale-
suppressed higher-dimensional operator such as
$\sigma^{*}\partial_{\mu}\sigma\bar{\psi}\sigma^{\mu}\psi/M_{P}^{2}$, which
would give $c_{\psi}(f_{a})={\cal O}(f_{a}^{2}/M_{P}^{2})$, and therefore

$\displaystyle C_{\Psi}^{0}={\cal
O}\Big{(}\frac{f_{a}^{2}}{M_{P}^{2}}\Big{)}\lesssim{\cal
O}\left(\Big{(}\frac{g_{\rm GUT}^{2}}{8\pi^{2}}\Big{)}^{2}\right),$ (23)

where $g_{\rm GUT}$ denotes the SM gauge coupling around the GUT scale or
$f_{a}$, and we use the axion weak gravity conjecture bound
$f_{a}/M_{P}\lesssim{\cal O}(g_{\rm GUT}^{2}/8\pi^{2})$ hep-th/0601001  for
the last inequality. As in the case of DFSZ model, it is straightforward to
supersymmetrize the above KSVZ model and the resulting SUSY model also has
vanishing tree-level couplings to the SM Higgs and fermion fields.

One can consider more generic models of field-theoretic axions other than
those presented above, e.g. models with different Higgs sector for the Yukawa
couplings of the SM fermions, or additional $U(1)_{\rm PQ}$-charged fields, or
even flavor-dependent $U(1)_{\rm PQ}$-charges 1708.02111 ; 1610.01639 ;
1610.07593 ; 1709.07039 ; 2101.03173 ; 1612.05492 ; 1612.08040 ; 1712.04940 .
In the following, we use the term “DFSZ-like axions” for generic field-
theoretic axions with $C_{\Psi}^{0}={\cal O}(1)$ ($\Psi=u,d,e$) and the term
“KSVZ-like axions” for those with $C_{\Psi}^{0}=0$ (or more generically
$C_{\Psi}^{0}\lesssim{\cal O}\big{(}(g_{\rm GUT}^{2}/8\pi^{2})^{2}\big{)}$. As
is well known (e.g. see Table 1 in 1801.08127 ), DFSZ-like and KSVZ-like
axions have a clearly different pattern of the observable low energy couplings
$g_{aX}$ ($X=\gamma,p,n,e$), so can be easily distinguished from each other.

###  2.2 String-theoretic axions

String theory involves a variety of extended objects ($(p-1)$-branes) which
have quantized couplings to a $p$-form gauge field in the theory
Polchinski:1998rr . As a consequence, string models accommodating the SM
generically contain a string-theoretic axion that couples to the SM gauge and
matter fields in 4-dimensional effective theory hep-th/0605206 ; 0905.4720 .
Such axions are accompanied by their scalar partners, i.e. saxions or moduli
$\tau$, whose vacuum expectation value can be identified as the Euclidean
action of the brane instanton which has a quantized coupling to axion.

Generically there can be multiple string-theoretic axions, sometimes many
0905.4720 . Yet the low energy consequence of those axions depends crucially
on the axion masses which have a close connection with the mechanism to
stabilize their moduli partners. For instance, many of string-theoretic axions
may get a heavy mass together with their moduli partners and therefore are
decoupled from the low energy world. On the other hand, in this paper we are
mostly interested in light axions whose couplings to the SM can be probed by
precision low energy experiments, which include a QCD axion and an ultralight
ALP coupled to the photon. Such light axions are required to be much lighter
than their moduli partners, so we need a mechanism to stabilize moduli while
keeping some axions to be nearly massless. In the following, we simply assume
that underlying string model provides such a mechanism as was discussed for
instance in hep-th/0602233 ; hep-th/0611279 ; 1003.1982 ; 1004.5138 ;
1206.0819 ; 1404.3880 .

Usually, string-theoretic axion and its modulus partner $\tau$ form a complex
scalar field which corresponds to the scalar component of a chiral superfield
in 4-dimensional effective supergravity of the underlying string
compactification:

$\displaystyle T=\tau(x)+i\frac{a(x)}{f_{a}}.$ (24)

Then the axion couplings to gauge and matter fields are determined by the
moduli Kahler potential $K_{0}$, the matter Kahler metric $Z_{I}$, and the
holomorphic gauge kinetic functions ${\cal F}_{A}$ of the model, which are
given by

$\displaystyle K$ $\displaystyle=$ $\displaystyle
K_{0}(T+T^{*})+Z_{I}(T+T^{*})\Phi^{*}_{I}\Phi_{I}+...,$ $\displaystyle{\cal
F}_{A}$ $\displaystyle=$ $\displaystyle\frac{c_{A}}{8\pi^{2}}T+...,$ (25)

where $\Phi_{I}$ denote gauge-charged chiral matter superfields in the model,
and the ellipses stand for the terms higher order in $\Phi_{I}$, as well as
the non-perturbative $U(1)_{\rm PQ}$-breaking terms which would be
exponentially suppressed by $e^{-T}$. Here, for simplicity, we assume a
diagonal form of the matter Kahler metric $Z_{I}$ and ignore moduli
superfields other than $T$. Note that the factor $1/8\pi^{2}$ in ${\cal
F_{A}}$ appears due to the normalization convention of $T$ for which ${\rm
Im}(T)=a(x)/f_{a}\cong{\rm Im}(T)+2\pi$, and $c_{A}$ are rational numbers (or
integers) in this convention. It is then straightforward to find that the
decay constant and couplings of the axion $a(x)$ are given by

$\displaystyle\frac{1}{2}f^{2}_{a}=M_{P}^{2}\frac{\partial^{2}K_{0}}{\partial
T\partial T^{*}}$ (26)

and

$\displaystyle c_{\phi_{I}}=\frac{\partial\ln Z_{I}}{\partial T},\quad
c_{\psi_{I}}=\frac{\partial\ln(e^{-K_{0}/2}Z_{I})}{\partial T},\quad
c_{A}=8\pi^{2}\frac{\partial}{\partial T}{\cal F}_{A},$ (27)

where $\phi_{I}$ and $\psi_{I}$ denote the scalar component and the fermion
component, respectively, of the chiral superfield $\Phi_{I}$.

The vacuum value of $\tau$ can be constrained in several ways. As noticed
above, $\tau$ corresponds to the Euclidean action of a brane instanton which
couples to $a(x)$, and then the axion weak gravity conjecture (WGC) hep-
th/0601001  implies

$\displaystyle\tau\,\lesssim\,{\cal O}\Big{(}\frac{M_{P}}{f_{a}}\Big{)}.$ (28)

Generically this brane instanton contributes to the axion potential, which can
be parameterized as

$\displaystyle\delta V=e^{-\tau}\Lambda^{2}_{\rm
ins}M_{P}^{2}\cos\Big{(}\frac{a}{f_{a}}\Big{)},$ (29)

where $\Lambda_{\rm ins}$ is a model-dependent mass parameter determined by a
more detailed feature of the model, in particular by the fermion zero modes of
the brane instanton and the scale of supersymmetry breaking. In fact, explicit
examples in string theory suggest that those brane instantons induce a
correction of ${\cal O}(e^{-T})$ to the superpotential or to the Kahler
potential of the 4-dimensional effective supergravity Dine:1986zy ; 0902.3251
, which results in666For models with a string scale $M_{\rm st}\ll M_{P}$,
there can be an additional suppression hep-th/0602233  which is not crucial
for our subsequent discussion.

$\displaystyle\Lambda_{\rm ins}^{2}={\cal O}(m_{3/2}M_{P})~{}\,\,{\rm
or}~{}\,\,{\cal O}(m_{3/2}^{2}).$ (30)

At any rate, the brane-instanton-induced potential (29) provides a lower bound
on the axion mass:

$\displaystyle m_{a}^{2}\gtrsim e^{-\tau}\Lambda_{\rm
ins}^{2}M_{P}^{2}/f_{a}^{2},$ (31)

which implies that the brane instanton action is bounded from below as

$\displaystyle\tau\,\gtrsim\,2\ln(\Lambda_{\rm ins}/m_{a}).$ (32)

For a string-theoretic QCD axion, the brane instanton-induced axion potential
(29) should be further suppressed not to spoil the axion solution to the
strong CP problem. Specifically one needs

$\displaystyle\delta V\,\lesssim\,10^{-10}m_{\pi}^{2}f_{\pi}^{2},$ (33)

implying

$\displaystyle\tau\,\gtrsim\,\ln(10^{10}\Lambda_{\rm ins}^{2}/f_{\pi}m_{\pi})$
(34)

for a QCD axion, where $m_{\pi}$ and $f_{\pi}$ are the pion mass and the pion
decay constant, respectively.

On the other hand, for $c_{A}$ of order unity, $\tau$ is also bounded from
above:

$\displaystyle\tau\,\lesssim\,{\cal O}\Big{(}\frac{8\pi^{2}}{g^{2}_{\rm
GUT}}\Big{)}$ (35)

in order to avoid a too small unified gauge coupling $g^{2}_{\rm GUT}\simeq
1/{\rm Re}({\cal F}_{A})$. For $\Lambda_{\rm ins}\gtrsim m_{3/2}$, the lower
bound (32) or (34) and the upper bound (35) are numerically comparable to each
other. This implies

$\displaystyle\tau={\cal O}\Big{(}\frac{8\pi^{2}}{g_{\rm GUT}^{2}}\Big{)}$
(36)

at least for light string-theoretic axions which can be identified as a QCD
axion or an ALP constituting dark matter. Then, with such a large value of
$\tau$, the following order of magnitude estimates are expected to be held:

$\displaystyle\frac{\partial
K_{0}}{\partial\tau}\,\sim\,\tau\frac{\partial^{2}K_{0}}{\partial\tau^{2}}\,\sim\,\tau\frac{f_{a}^{2}}{M_{P}^{2}}\,\lesssim\,{\cal
O}\Big{(}\frac{g_{\rm GUT}^{2}}{8\pi^{2}}\Big{)},$
$\displaystyle\frac{\partial\ln
Z_{I}}{\partial\tau}\,\sim\,\frac{1}{\tau}\,=\,{\cal O}\Big{(}\frac{g_{\rm
GUT}^{2}}{8\pi^{2}}\Big{)},$ (37)

which suggest

$\displaystyle c_{\phi}(f_{a})=\omega_{\phi}\frac{g_{\rm
GUT}^{2}}{16\pi^{2}},\quad c_{\psi}(f_{a})=\omega_{\psi}\frac{g_{\rm
GUT}^{2}}{16\pi^{2}},$ (38)

with

$\displaystyle\omega_{\phi}\sim\omega_{\psi}={\cal O}(1).$ (39)

Accordingly, the tree-level couplings $C_{\Psi}^{0}$ ($\Psi=u,d,e$) of string-
theoretic axions to the light quarks and electron, which are defined in (2),
are estimated as

$\displaystyle\mbox{String-theoretic}:\quad C_{\Psi}^{0}=\,{\cal
O}\Big{(}\frac{g_{\rm GUT}^{2}}{16\pi^{2}}\Big{)}\quad(\Psi=u,d,e).$ (40)

Note that this estimate applies for string-theoretic axions which have a non-
zero coupling to at least one of the SM gauge fields and also light enough to
be identified as a QCD axion or an ALP dark matter.

The above estimate indicates that the couplings of string-theoretic axions
have a similar pattern as those of KSVZ-like axions in the sense that both
axions have $|C_{\Psi}^{0}/c_{A}|\ll 1$. Yet one might be able to distinguish
one from the other if the radiative corrections to $C_{\Psi}$ are
significantly smaller than the tree-level couplings $C_{\Psi}^{0}$ for string-
theoretic axions. In such case, the low energy couplings $C_{\Psi}$ of KSVZ-
like axions will be significantly smaller than those of string-theoretic
axions, which would leave a notable imprint in the observable axion couplings
$g_{aX}$ ($X=\gamma,p,n,e$) in (11). To see if KSVZ-like axions and string-
theoretic axions indeed have a distinguishable pattern of $g_{aX}$, we will
conduct in section 3 an RG analysis for the axion couplings to matter fields.

In fact, the estimate (38) can be elaborated with the scaling argument made in
hep-th/0609180 . For gauge fields propagating over a $d_{\rm YM}$-dimensional
cycle $\Sigma_{\rm YM}$ in 6-dimensional internal space777In heterotic string
theory, gauge fields propagate over the entire 6-dimensional internal space,
so $d_{\rm YM}=6$. In other string theories with $D$-branes, gauge fields are
confined on $D_{p}$-branes wrapping a $(p-3)$-dimensional cycle, so $d_{\rm
YM}=p-3$., the corresponding gauge couplings scale as

$\displaystyle{\rm Re}({\cal F}_{A})=\frac{1}{g_{A}^{2}}\propto{\cal
V}(\Sigma_{\rm YM})$ (41)

at leading order in $1/{\cal V}(\Sigma_{\rm YM})$, which ignores for instance
the flux densities in $\Sigma_{\rm YM}$, where ${\cal V}(\Sigma_{\rm YM})$ is
the volume (in the string frame) of $\Sigma_{\rm YM}$. On the other hand, the
physical Yukawa couplings $y_{IJK}$ are determined as

$\displaystyle
y_{IJK}=\int_{\Sigma_{IJK}}\phi_{I}^{(0)}\phi_{J}^{(0)}\phi_{K}^{(0)}\,d{\cal
V}(\Sigma_{IJK}),$ (42)

where $\phi_{I}^{(0)}$ is the zero mode wavefunction of $\Phi^{I}$ which can
propagate over a $d_{I}$-dimensional subcycle $\Sigma_{I}$ of $\Sigma_{\rm
YM}$, and $\Sigma_{IJK}$ is a $d_{IJK}$-dimensional subcycle over which the
three zero mode wavefunctions $\phi_{I}^{(0)},\phi_{J}^{(0)}$ and
$\phi_{K}^{(0)}$ can have a nonzero overlap. Here $\phi_{I}^{(0)}$ is is
normalized as

$\displaystyle\int_{\Sigma_{I}}|\phi_{I}^{(0)}|^{2}\,d{\cal V}(\Sigma_{I})=1,$
(43)

Note that

$\displaystyle\Sigma_{\rm YM}\supseteq\Sigma_{I}\supseteq\Sigma_{IJK},$ (44)

and therefore

$\displaystyle d_{\rm YM}\geq d_{I}\geq d_{IJK}.$ (45)

It has been argued in hep-th/0609180  that in the limit where ${\cal
V}(\Sigma_{\rm YM})$ is large enough and therefore the flux densities in
$\Sigma_{\rm YM}$ can be ignored, the normalized zero mode wavefunction
$\phi_{I}^{(0)}$ is simply rescaled as

$\displaystyle\phi_{I}^{(0)}\rightarrow\lambda^{-d_{I}/2d_{\rm
YM}}\phi_{I}^{(0)}$ (46)

under the rescaling

$\displaystyle{\cal V}(\Sigma_{\rm YM})\rightarrow\lambda{\cal V}(\Sigma_{\rm
YM}).$ (47)

Then the associated Yukawa couplings are rescaled as

$\displaystyle y_{IJK}\rightarrow\lambda^{(2d_{IJK}-d_{I}-d_{J}-d_{K})/2d_{\rm
YM}}y_{IJK}.$ (48)

On the other hand, the physical Yukawa couplings in 4-dimensional supergravity
are given by

$\displaystyle y_{IJK}=\frac{\lambda_{IJK}}{\sqrt{Y_{I}Y_{J}Y_{K}}},$ (49)

where $\lambda_{IJK}$ are the holomorphic Yukawa couplings in the
superpotential:

$\displaystyle W_{\rm
Yukawa}=\frac{1}{6}\lambda_{IJK}\Phi^{I}\Phi^{J}\Phi^{K}$ (50)

and

$\displaystyle Y_{I}=e^{-K_{0}/3}Z_{I}.$ (51)

Due to the holomorphicity and the $U(1)_{\rm PQ}$ symmetry, $\lambda_{IJK}$
are independent of ${\cal V}(\Sigma_{\rm YM})$. Since ${\rm Re}({\cal
F}_{A})=1/g_{A}^{2}\propto{\cal V}(\Sigma_{\rm YM})$, (48) and (49) suggest

$\displaystyle Y_{I}=e^{-K_{0}/3}Z_{I}\propto\Big{(}{\rm Re}({\cal
F}_{A})\Big{)}^{\omega_{I}},$ (52)

where the scaling weights $\omega_{I}$ are rational numbers constrained as

$\displaystyle\quad\omega_{I}+\omega_{J}+\omega_{K}=\frac{1}{d_{\rm
YM}}(d_{I}+d_{J}+d_{K}-2d_{IJK})$ (53)

for the combinations having a nonzero Yukawa coupling. This implies

$\displaystyle\frac{\partial\ln Y_{I}}{\partial
T}\simeq{\omega_{I}}\frac{\partial\ln{\rm Re}({\cal F}_{A})}{\partial
T}=\omega_{I}\frac{c_{A}g_{A}^{2}}{16\pi^{2}}$ (54)

which confirms the estimates (2.2) and (38). Although the above scaling
argument is valid only at leading order in $1/{\cal V}(\Sigma_{\rm YM})$, it
is likely to give a correct order of magnitude estimate even when higher order
corrections become non-negligible.

An interesting string model which can incorporate a QCD axion is the Large
Volume Scenario (LVS) hep-th/0502058  of Type IIB string compactification on a
Calabi-Yau (CY) space which involves a 4-cycle wrapped by $D7$ branes
supporting the SM gage fields. It has been argued that in this scenario all
moduli including the saxion partner of the QCD axion can be successfully
stabilized with $f_{a}$ exponentially lower than the Planck scale hep-
th/0602233 ; 1206.0819 . At leading order approximation to ignore the effects
of magnetic fluxes on $D7$ branes, the Kahler potential and the holomorphic
gauge kinetic function of the model are given by hep-th/0609180 ; hep-
th/0602233

$\displaystyle
K=-3\ln(T_{b}+T_{b}^{*})+\frac{(T+T^{*})^{3/2}}{(T_{b}+T_{b}^{*})^{3/2}}+\frac{(T+T^{*})^{\omega_{I}}}{(T_{b}+T_{b}^{*})}\Phi_{I}^{*}\Phi_{I},\quad{\cal
F}_{A}=\frac{1}{8\pi^{2}}T,$ (55)

where

$\displaystyle T_{b}=\tau_{b}+i\theta_{b},\quad T=\tau+i\frac{a}{f_{a}}$ (56)

and the scaling weights $\omega_{I}$ are rational numbers between 0 and 1 as
suggested by (53). Note that the factor $1/8\pi^{2}$ in ${\cal F}_{A}$ is due
to the normalization convention of ${\rm Im}(T)=a(x)/f_{a}\cong{\rm
Im}(T)+2\pi$, where $a(x)$ is the QCD axion in the model. Here the Kahler
modulus $\tau_{b}$ corresponds to the volume of an exponentially big 4-cycle
in CY space, with which the 4-dimensional Planck scale $M_{P}$ can be
exponentially bigger than the string scale $M_{\rm st}$ as

$\displaystyle\tau_{b}\sim{\cal V}_{\rm CY}^{2/3}\sim\left(\frac{M_{P}}{M_{\rm
st}}\right)^{4/3}\gg\frac{8\pi^{2}}{g_{\rm GUT}^{2}},$ (57)

where ${\cal V}_{\rm CY}$ is the CY volume, while $\tau$ describes the volume
of a moderately large 4-cycle wrapped by $D7$ branes supporting the SM gauge
fields, which determines the SM gauge couplings at the string scale as

$\displaystyle\tau=\frac{8\pi^{2}}{g_{\rm GUT}^{2}}.$ (58)

For simplicity, in (55) we ignored the part depending on other moduli while
keeping only the part relevant for the couplings and the decay constant of the
QCD axion $a(x)$. We then find

$\displaystyle
f_{a}={\frac{\sqrt{3}}{2\sqrt{2}}}\frac{M_{P}}{\tau^{1/4}\tau_{b}^{3/4}}$ (59)

together with the following couplings to gauge and matter fields:

$\displaystyle c_{A}=1,\quad c_{\phi_{I}}(f_{a})=\omega_{I}\frac{g_{\rm
GUT}^{2}}{16\pi^{2}}+{\cal O}\Big{(}\frac{1}{\tau_{b}}\Big{)},\quad
c_{\psi_{I}}=\omega_{I}\frac{g_{\rm GUT}^{2}}{16\pi^{2}}+{\cal
O}\Big{(}\frac{1}{\tau_{b}}\Big{)}.$ (60)

This model of string-theoretic axion is particularly interesting as it can
give rise to an axion decay constant anywhere between $f_{a}\sim 10^{16}$ GeV
(for $\tau_{b}\sim\tau$) and the observational lower bound $f_{a}\sim 10^{9}$
GeV (for an exponentially large $\tau_{b}\sim 10^{11}\gg\tau$). Therefore, in
section 4, we use this model as a benchmark model for string-theoretic axion
and compare its low energy couplings $g_{aX}$ ($X=\gamma,p,n,e$) with those of
KSVZ-like axions. More specifically, we will consider a model with chiral
matters on the intersections of $D7$ branes, for which $d_{\rm YM}=4$,
$d_{I}=2$ and $d_{IJK}=0$, and therefore hep-th/0609180

$\displaystyle\omega_{I}=\frac{1}{2}.$ (61)

In order to see that the order of magnitude estimate (38) is valid even when
the flux densities in $\Sigma_{\rm YM}$ are not negligible, let us consider
another example of string-theoretic axions, the model-independent axion and
the model-dependent Kahler axion in heterotic string or $M$-theory in which
the underlying YM and gravitational fluxes play a key role for the model to
accommodate a QCD axion Witten:1984dg ; Choi:1985bz ; hep-th/9605136 ; hep-
th/9706171 . For simplicity, we consider a compactification on CY space with a
single Kahler axion, which involves

$\displaystyle S=\tau_{1}+i\frac{a_{1}}{f_{1}},\quad
T=\tau_{2}+i\frac{a_{2}}{f_{2}},$ (62)

where $a_{1}\cong a_{1}+2\pi f_{1}$ and $a_{2}\cong a_{2}+2\pi f_{2}$ are the
model-independent axion and the Kahler axion, respectively, that originate
from the NS 2-form field $B$ as

$\displaystyle\partial_{\mu}a_{1}\propto\epsilon_{\mu\nu\rho\sigma}\partial^{\nu}B^{\rho\sigma},\quad
a_{2}\propto\int_{\Sigma}B_{i\bar{j}}dz^{i}\wedge dz^{*j},$ (63)

where $\Sigma$ is the 2-cycle dual to the Kahler form. In the heterotic string
regime, $\tau_{1}$ correspond to the Euclidean action of the NS5 brane
instanton wrapping the entire CY space Polchinski:1998rr  , and $\tau_{2}$ is
that of the world-sheet instanton wrapping $\Sigma$ Dine:1986zy :

$\displaystyle\tau_{1}\propto\frac{{\cal V}_{\rm CY}}{g_{\rm
st}^{2}},\quad\tau_{2}\propto{\cal V}({\Sigma}),$ (64)

where $g_{\rm st}$ is the string coupling. The moduli $\tau_{1}$ and
$\tau_{2}$ can be extrapolated to the heterotic $M$-theory regime hep-
th/9603142  where they correspond to the Euclidean actions of the $M5$ brane
instanton wrapping the CY space and the $M2$ brane instanton wrapping $\Sigma$
stretched along the 11-th dimension.

The moduli Kahler potential, matter Kahler metric, and holomorphic gauge
kinetic functions of the above model are given by hep-th/9710208

$\displaystyle K_{0}$ $\displaystyle=$
$\displaystyle-\ln(S+S^{*})-3\ln(T+T^{*}),$ $\displaystyle Z_{I}$
$\displaystyle=$
$\displaystyle\frac{1}{T+T^{*}}-\frac{\epsilon}{3}\frac{1}{S+S^{*}},$
$\displaystyle{\cal F}_{E6}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}(S+\epsilon T),\quad{\cal
F}_{E8}\,=\,\frac{1}{8\pi^{2}}(S-\epsilon T)$ (65)

where ${\cal F}_{E6}$ and ${\cal F}_{E8}$ are the gauge kinetic functions of
the visible $E_{6}$ and the hidden $E_{8}$, respectively, and

$\displaystyle{\epsilon={\frac{1}{8\pi^{2}}}\int J\wedge\left[{\rm tr}(F\wedge
F)-\frac{1}{2}{\rm tr}(R\wedge R)\right]}$ (66)

for the Kahler form $J$. Obviously the pieces proportional to $\epsilon$ are
the corrections induced by flux density, which can be interpreted as string
loop corrections in the heterotic string regime Choi:1985bz ;
Derendinger:1985cv ; Ibanez:1986xy , or as a flux-induced warping of CY
geometry along the 11-th dimension in the heterotic $M$-theory regime hep-
th/9602070 ; hep-th/9710208 . At leading order approximation which ignores
these corrections, which would correspond to the weakly coupled heterotic
string theory limit with $\epsilon\tau_{2}\ll\tau_{1}$, one finds

$\displaystyle Y_{I}=e^{-K_{0}/3}Z_{I}\simeq(S+S^{*})^{1/3}\propto\Big{(}{\rm
Re}({\cal F}_{A})\Big{)}^{1/3}$ (67)

as suggested by the scaling relation (52) with $\omega_{I}=1/3$ which is
implied by (53) for gauge and matter fields propagating over the full CY
space: $d_{\rm YM}=d_{I}=d_{IJK}=6$. However, due to the hidden $E_{8}$ gauge
group confining at high scales as well as the world sheet instanton effects, a
viable QCD axion can be obtained only in the heterotic $M$-theory limit with
$\epsilon\tau_{2}\sim\tau_{1}={\cal O}(8\pi^{2}/g_{\rm GUT}^{2})$ hep-
th/9605136 ; hep-th/9706171 . In such case, the QCD axion $a(x)$ corresponds
to a combination of $a_{1}$ and $a_{2}$ which avoids the coupling to the
hidden $E_{8}$ gauge fields. To find the decay constant and couplings of this
QCD axion, one may parameterize the model-independent axion and the Kahler
axion in terms of the QCD axion as

$\displaystyle\frac{a_{1}}{f_{1}}\propto{\epsilon\frac{a}{f_{a}}},\quad\frac{a_{2}}{f_{2}}\propto\frac{a}{f_{a}},$
(68)

where the heavy axion component is frozen at its vacuum value by the potential
induced by the hidden $E_{8}$ gauge fields. For the convenience of discussion,
here we define $f_{a}$ as a scale for which the couplings of $a(x)$ to the
$E_{6}$ gauge fields are given by $c_{A}=1$, not as a scale to define the
periodicity of $a(x)$. It is then straightforward to find

$\displaystyle f_{a}=\sqrt{2}M_{P}\frac{g_{\rm GUT}^{2}}{8\pi^{2}}\frac{g_{\rm
hid}^{2}\sqrt{(g_{\rm hid}^{2}+g_{\rm GUT}^{2})^{2}-g_{\rm hid}^{2}g_{\rm
GUT}^{2}}}{|g_{\rm hid}^{4}-g_{\rm GUT}^{4}|}$ (69)

with the couplings

$\displaystyle c_{A}=1,\quad c_{\phi_{I}}(f_{a})=\omega_{\phi}\frac{g_{\rm
GUT}^{2}}{16\pi^{2}},\quad c_{\psi_{I}}=\omega_{\psi}\frac{g_{\rm
GUT}^{2}}{16\pi^{2}},$ (70)

where

$\displaystyle\omega_{\phi}$ $\displaystyle=$ $\displaystyle-\frac{g_{\rm
hid}^{2}\left[(g_{\rm hid}^{2}+g_{\rm GUT}^{2})^{2}+2g_{\rm hid}^{2}g_{\rm
GUT}^{2}\right]}{\left(g_{\rm hid}^{2}+2g_{\rm GUT}^{2}\right)\left(g_{\rm
hid}^{4}-g_{\rm GUT}^{4}\right)},$ $\displaystyle\omega_{\psi}$
$\displaystyle=$ $\displaystyle\frac{g_{\rm hid}^{2}\left[(g_{\rm
hid}^{2}+g_{\rm GUT}^{2})^{2}-g_{\rm hid}^{2}g_{\rm
GUT}^{2}\right]}{\left(g_{\rm hid}^{2}+2g_{\rm GUT}^{2}\right)\left(g_{\rm
hid}^{4}-g_{\rm GUT}^{4}\right)},$ (71)

and $g_{\rm GUT}$ and $g_{\rm hid}$ are the gauge couplings around the string
scale for the visible $E_{6}$ and the hidden $E_{8}$, respectively, which are
determined by

$\displaystyle{\rm Re}({\cal F}_{E_{6}})=\frac{1}{g_{\rm GUT}^{2}},\quad{\rm
Re}({\cal F}_{E_{8}})=\frac{1}{g_{\rm hid}^{2}}.$ (72)

This result shows that the order of magnitude estimate (38) is valid even when
the effects of flux densities are properly taken into account.

##  3 Running of the axion couplings

The low energy axion couplings can be substantially different from the UV
boundary values discussed in the previous section because of the subsequent
renormalization group evolution. In this section we will present the RG
equations of axion couplings to the SM particles at leading order in
dimensionless couplings (i.e. Yukawa and gauge couplings) in a generic way
that can be applied to the SM, the minimal supersymmetric standard model
(MSSM), and the two Higgs doublet models (2HDMs). We will then provide a semi-
analytic formula for the solution of the RG equations as well as the numerical
results.

###  3.1 RG equations

In the Georgi-Kaplan-Randall (GKR) field basis Georgi:1986df , the axion
couplings to the SM fields and an additional Higgs doublet field in
supersymmetric models or 2HDMs can be generally written

$\begin{split}{\cal
L}_{a}=\frac{\partial_{\mu}a}{f_{a}}\left[\,\sum_{\psi}(\mathbf{c}_{\psi})_{ij}\psi_{i}^{\dagger}\bar{\sigma}^{\mu}\psi_{j}+\sum_{\alpha=1,2}c_{H_{\alpha}}H_{\alpha}^{\dagger}\overset{\leftrightarrow}{iD^{\mu}}H_{\alpha}\,\right]+\frac{a}{f_{a}}\sum_{A}c_{A}\frac{g_{A}^{2}}{32\pi^{2}}F^{A\mu\nu}\widetilde{F}^{A}_{\mu\nu},\end{split}$
(73)

where $\psi_{i}=\\{Q_{i},u_{i}^{c},d_{i}^{c},L_{i},e_{i}^{c}\\}\,(i=1,2,3)$
are the left-handed quarks and leptons in the SM and $H_{\alpha}$
($\alpha=1,2$) denote the two Higgs doublets in the model.

Figure 1: Diagrams for the Yukawa-induced renormalization group running of
$\mathbf{c}_{\psi}$.

The axion couplings to $\psi_{i}$ and $H_{\alpha}$ get renormalized by the
Yukawa interactions involving $H_{\alpha}$ and $\psi_{i}$, which is determined
at leading order by the diagrams in Fig. 1. For general 2HDMs, the Yukawa
couplings can be written as

$\begin{split}{\cal L}_{\rm
Yukawa}=\sum_{f,F,\alpha,i,j}(\mathbf{y}_{fF\alpha})_{ij}f_{i}F_{j}H_{\alpha}+\textrm{h.c.},\end{split}$
(74)

where $f_{i}=\\{u^{c}_{i},d^{c}_{i},e^{c}_{i}\\}$ are $SU(2)_{L}$-singlet
fermions, and $F_{i}=\\{Q_{i},L_{i}\\}$ are $SU(2)_{L}$-doublet fermions. The
Yukawa-induced RG running of axion couplings at leading order are previously
given in 1708.00021 ; 2002.04623 ; 2011.08205 ; 2012.09017 ; 2012.12272  for
the SM or its supersymmetric extensions. Here we provide a general expression
which can be applied to general 2HDMs as well:

$\displaystyle\left.\frac{d\mathbf{c}_{F}}{d\ln\mu}\right|_{\rm Yukawa}$
$\displaystyle=$
$\displaystyle\frac{\xi_{y}}{16\pi^{2}}\sum_{f,\alpha}\left(\frac{1}{2}\\{\mathbf{c}_{F},\mathbf{y}_{fF\alpha}^{\dagger}\mathbf{y}_{fF\alpha}\\}+\mathbf{y}_{fF\alpha}^{\dagger}\mathbf{c}_{f}^{T}\mathbf{y}_{fF\alpha}+c_{H_{\alpha}}\mathbf{y}_{fF\alpha}^{\dagger}\mathbf{y}_{fF\alpha}\right),$
$\displaystyle\left.\frac{d\mathbf{c}_{f}^{T}}{d\ln\mu}\right|_{\rm Yukawa}$
$\displaystyle=$
$\displaystyle\frac{\xi_{y}}{16\pi^{2}}\sum_{F,\alpha}\left(\frac{1}{2}\\{\mathbf{c}_{f}^{T},\mathbf{y}_{fF\alpha}\mathbf{y}_{fF\alpha}^{\dagger}\\}+\mathbf{y}_{fF\alpha}\mathbf{c}_{F}\mathbf{y}_{fF\alpha}^{\dagger}+c_{H_{\alpha}}\mathbf{y}_{fF\alpha}\mathbf{y}_{fF\alpha}^{\dagger}\right),$
$\displaystyle\left.\frac{dc_{H_{\alpha}}}{d\ln\mu}\right|_{\rm Yukawa}$
$\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\sum_{f,F}\left(c_{H_{\alpha}}{\rm
tr}(\mathbf{y}_{fF\alpha}^{\dagger}\mathbf{y}_{fF\alpha})+{\rm
tr}(\mathbf{y}_{fF\alpha}\mathbf{c}_{F}\mathbf{y}_{fF\alpha}^{\dagger})+{\rm
tr}(\mathbf{y}_{fF\alpha}^{\dagger}\mathbf{c}_{f}^{T}\mathbf{y}_{fF\alpha})\right),$

where

$\begin{split}\xi_{y}=\begin{cases}1&\textrm{for non-SUSY models}\\\
2&\textrm{for SUSY models}\end{cases}\end{split}$ (76)

Figure 2: Diagrams for the gauge-induced renormalization group running of
$\mathbf{c}_{\psi}$.

Gauge interactions also give rise to a running of axion couplings which is
determined at leading order by the diagrams in Fig. 2. A general formula for
such gauge-induced RG running of axion couplings is given in 2012.05029 ;
2012.09017 ; 2012.12272  for the SM. Here we generalize the formula for SUSY
models using a connection between the beta function of an axion coupling and
the anomalous dimension of a chiral superfield as described in 1708.00021 ;
ArkaniHamed:1998kj :

$\displaystyle\left.\frac{d\mathbf{c}_{\psi}}{d\ln\mu}\right|_{\rm gauge}$
$\displaystyle=$
$\displaystyle-\xi_{g}\sum_{A}\frac{3}{2}\left(\frac{g_{A}^{2}}{8\pi^{2}}\right)^{2}\mathbb{C}_{A}(\psi)\Big{(}c_{A}-2\sum_{\psi^{\prime}}{\rm
tr}(\mathbf{c}_{\psi^{\prime}})\mathbb{T}_{A}(\psi^{\prime})\Big{)}\mathbb{1},$
$\displaystyle\left.\frac{dc_{H_{\alpha}}}{d\ln\mu}\right|_{\rm gauge}$
$\displaystyle=$
$\displaystyle-\xi_{H}\sum_{A}\frac{3}{2}\left(\frac{g_{A}^{2}}{8\pi^{2}}\right)^{2}\mathbb{C}_{A}(H_{\alpha})\Big{(}c_{A}-2\sum_{\psi^{\prime}}{\rm
tr}(\mathbf{c}_{\psi^{\prime}})\mathbb{T}_{A}(\psi^{\prime})\Big{)},$ (77)

where

$\begin{split}\xi_{g}=\begin{cases}1&\textrm{for non-SUSY models}\\\
2/3&\textrm{for SUSY
models}\end{cases},\quad\xi_{H}=\begin{cases}0&\textrm{for non-SUSY models}\\\
2/3&\textrm{for SUSY models}\end{cases}\end{split}$ (78)

The difference between non-SUSY models and SUSY modes is due to the presence
of the axion-gaugino couplings in SUSY models. Here $\mathbb{C}_{A}(\Phi)$ and
$\mathbb{T}_{A}(\Phi)$ are the quadratic Casimir and Dynkin index of the field
$\Phi$ charged under the gauge group $A$.

In fact, Eq. (3.1) and Eq. (3.1) can be determined up to overall coefficients
by the covariance of the RG equations under two spurious symmetries of the
lagrangian. One of the symmetries is associated with the $SU(3)$ flavor
rotations of $\psi_{i}$:

$\begin{split}\psi_{i}\,\rightarrow\,(U_{\psi})_{ij}\psi_{j},\quad\mathbf{c}_{\psi}\,\rightarrow\,U_{\psi}\mathbf{c}_{\psi}U_{\psi}^{\dagger},\quad\mathbf{y}_{fF\alpha}\,\rightarrow\,U_{f}^{*}\mathbf{y}_{fF\alpha}U_{F}^{\dagger}.\end{split}$
(79)

The other involves the following axion-dependent field redefinition which is
allowed within the GKR field basis:

$\begin{split}\psi_{i}\rightarrow\psi_{i}e^{ix_{\psi_{i}}a/f_{a}},\quad
H_{\alpha}\rightarrow H_{\alpha}e^{ix_{H_{\alpha}}a/f_{a}}\end{split}$ (80)

for $x_{\psi_{i}}$ and $x_{H_{\alpha}}$ satisfying

$\begin{split}(\mathbf{y}_{fF\alpha})_{ij}\left(x_{f_{i}}+x_{F_{j}}+x_{H_{\alpha}}\right)=0,\end{split}$

with redefinition of parameters

$\begin{split}(\mathbf{c}_{\psi})_{ij}\,\rightarrow\,(\mathbf{c}_{\psi})_{ij}+x_{\psi_{i}}\delta_{ij},\quad
c_{H_{\alpha}}\,\rightarrow\,c_{H_{\alpha}}+x_{H_{\alpha}},\quad
c_{A}\,\rightarrow\,c_{A}+2\sum_{\psi_{i}}x_{\psi_{i}}\mathbb{T}_{A}(\psi_{i}).\\\
\end{split}$ (81)

From the RG equations (3.1) and (3.1), we find that the condition for a non-
trivial RG running of the couplings of field-theoretic axions can be expressed
in terms of the linearly realized $U(1)_{\rm PQ}$ symmetry. For field-
theoretic axions, the axion couplings to matter fields in the GKR field basis
at $\mu\sim f_{a}$ are identified as the PQ charges of those fields in the
field basis where $U(1)_{\rm PQ}$ is linearly realized as Eq. (13), while the
axion couplings to gauge fields in the GKR field basis is identified as the
$U(1)_{\rm PQ}$ anomaly for the gauge group. Then, from Eq. (3.1), one can see
that the RG running of axion couplings induced by a Yukawa coupling occurs
when the sum of the PQ charges of the fields involved in the corresponding
Yukawa coupling is non-vanishing. On the other hand, Eq. (3.1) tells us that
the RG running of axion couplings induced by gauge interaction starts from the
scale where the heaviest PQ- and gauge-charged fermion is integrated out.

In the following analysis we will focus on the RG evolution of axion couplings
in the MSSM and the SM. We write the Yukawa couplings in the MSSM as

$\begin{split}{\cal L}_{\rm
Yukawa}=(\tilde{\mathbf{y}}_{u})_{ij}{u^{c}_{i}}Q_{j}H_{u}+(\tilde{\mathbf{y}}_{d})_{ij}{d^{c}_{i}}Q_{j}H_{d}+(\tilde{\mathbf{y}}_{e})_{ij}{e^{c}_{i}}L_{j}H_{d}+\textrm{h.c.}\end{split}$
(82)

From the general formula Eq. (3.1) and Eq. (3.1), the full axion RG equations
at leading order in the Yukawa and gauge couplings in the MSSM come out as

$\displaystyle\frac{d\mathbf{c}_{Q}}{d\ln\mu}$ $\displaystyle=$
$\displaystyle\frac{\xi_{y}}{16\pi^{2}}\left(\frac{1}{2}\\{\mathbf{c}_{Q},\tilde{\mathbf{y}}_{u}^{\dagger}\tilde{\mathbf{y}}_{u}+\tilde{\mathbf{y}}_{d}^{\dagger}\tilde{\mathbf{y}}_{d}\\}+\tilde{\mathbf{y}}_{u}^{\dagger}\mathbf{c}_{u^{c}}^{T}\tilde{\mathbf{y}}_{u}+\tilde{\mathbf{y}}_{d}^{\dagger}\mathbf{c}_{d^{c}}^{T}\tilde{\mathbf{y}}_{d}+c_{H_{u}}\tilde{\mathbf{y}}_{u}^{\dagger}\tilde{\mathbf{y}}_{u}+c_{H_{d}}\tilde{\mathbf{y}}_{d}^{\dagger}\tilde{\mathbf{y}}_{d}\right)$
$\displaystyle-\xi_{g}\left(\frac{\alpha_{s}^{2}}{2\pi^{2}}\tilde{c}_{G}+\frac{9\alpha_{2}^{2}}{32\pi^{2}}\tilde{c}_{W}+\frac{\alpha_{1}^{2}}{96\pi^{2}}\tilde{c}_{B}\right)\mathbb{1}\,,$
$\displaystyle\frac{d\mathbf{c}_{u^{c}}^{T}}{d\ln\mu}$ $\displaystyle=$
$\displaystyle\frac{\xi_{y}}{8\pi^{2}}\left(\frac{1}{2}\\{\mathbf{c}_{u^{c}}^{T},\tilde{\mathbf{y}}_{u}\tilde{\mathbf{y}}_{u}^{\dagger}\\}+\tilde{\mathbf{y}}_{u}\mathbf{c}_{Q}\tilde{\mathbf{y}}_{u}^{\dagger}+c_{H_{u}}\tilde{\mathbf{y}}_{u}\tilde{\mathbf{y}}_{u}^{\dagger}\right)-\xi_{g}\left(\frac{\alpha_{s}^{2}}{2\pi^{2}}\tilde{c}_{G}+\frac{\alpha_{1}^{2}}{6\pi^{2}}\tilde{c}_{B}\right)\mathbb{1}\,,$
$\displaystyle\frac{d\mathbf{c}_{d^{c}}^{T}}{d\ln\mu}$ $\displaystyle=$
$\displaystyle\frac{\xi_{y}}{8\pi^{2}}\left(\frac{1}{2}\\{\mathbf{c}_{d^{c}}^{T},\tilde{\mathbf{y}}_{d}\tilde{\mathbf{y}}_{d}^{\dagger}\\}+\tilde{\mathbf{y}}_{d}\mathbf{c}_{Q}\tilde{\mathbf{y}}_{d}^{\dagger}+c_{H_{d}}\tilde{\mathbf{y}}_{d}\tilde{\mathbf{y}}_{d}^{\dagger}\right)-\xi_{g}\left(\frac{\alpha_{s}^{2}}{2\pi^{2}}\tilde{c}_{G}+\frac{\alpha_{1}^{2}}{24\pi^{2}}\tilde{c}_{B}\right)\mathbb{1}\,,$
$\displaystyle\frac{d\mathbf{c}_{L}}{d\ln\mu}$ $\displaystyle=$
$\displaystyle\frac{\xi_{y}}{16\pi^{2}}\left(\frac{1}{2}\\{\mathbf{c}_{L},\tilde{\mathbf{y}}_{e}^{\dagger}\tilde{\mathbf{y}}_{e}\\}+\tilde{\mathbf{y}}_{e}^{\dagger}\mathbf{c}_{e^{c}}^{T}\tilde{\mathbf{y}}_{e}+c_{H_{d}}\tilde{\mathbf{y}}_{e}^{\dagger}\tilde{\mathbf{y}}_{e}\right)-\xi_{g}\left(\frac{9\alpha_{2}^{2}}{32\pi^{2}}\tilde{c}_{W}+\frac{3\alpha_{1}^{2}}{32\pi^{2}}\tilde{c}_{B}\right)\mathbb{1}\,,$
$\displaystyle\frac{d\mathbf{c}_{e^{c}}^{T}}{d\ln\mu}$ $\displaystyle=$
$\displaystyle\frac{\xi_{y}}{8\pi^{2}}\left(\frac{1}{2}\\{\mathbf{c}_{e^{c}}^{T},\tilde{\mathbf{y}}_{e}\tilde{\mathbf{y}}_{e}^{\dagger}\\}+\tilde{\mathbf{y}}_{e}\mathbf{c}_{L}\tilde{\mathbf{y}}_{e}^{\dagger}+c_{H_{d}}\tilde{\mathbf{y}}_{e}\tilde{\mathbf{y}}_{e}^{\dagger}\right)-\xi_{g}\frac{3\alpha_{1}^{2}}{8\pi^{2}}\tilde{c}_{B}\mathbb{1}\,,$
$\displaystyle\frac{dc_{H_{u}}}{d\ln\mu}$ $\displaystyle=$
$\displaystyle\frac{3}{8\pi^{2}}\left(c_{H_{u}}{\rm
tr}(\tilde{\mathbf{y}}_{u}^{\dagger}\tilde{\mathbf{y}}_{u})+{\rm
tr}(\tilde{\mathbf{y}}_{u}\mathbf{c}_{Q}\tilde{\mathbf{y}}_{u}^{\dagger})+{\rm
tr}(\tilde{\mathbf{y}}_{u}^{\dagger}\mathbf{c}_{u^{c}}^{T}\tilde{\mathbf{y}}_{u})\right)-\xi_{H}\left(\frac{9\alpha_{2}^{2}}{32\pi^{2}}\tilde{c}_{W}+\frac{3\alpha_{1}^{2}}{32\pi^{2}}\tilde{c}_{B}\right),$
$\displaystyle\frac{dc_{H_{d}}}{d\ln\mu}$ $\displaystyle=$
$\displaystyle\frac{3}{8\pi^{2}}\left(c_{H_{d}}{\rm
tr}(\tilde{\mathbf{y}}_{d}^{\dagger}\tilde{\mathbf{y}}_{d})+{\rm
tr}(\tilde{\mathbf{y}}_{d}\mathbf{c}_{Q}\tilde{\mathbf{y}}_{d}^{\dagger})+{\rm
tr}(\tilde{\mathbf{y}}_{d}^{\dagger}\mathbf{c}_{d^{c}}^{T}\tilde{\mathbf{y}}_{d})\right)$
$\displaystyle+\frac{1}{8\pi^{2}}\left(c_{H_{d}}{\rm
tr}(\tilde{\mathbf{y}}_{e}^{\dagger}\tilde{\mathbf{y}}_{e})+{\rm
tr}(\tilde{\mathbf{y}}_{e}\mathbf{c}_{L}\tilde{\mathbf{y}}_{e}^{\dagger})+{\rm
tr}(\tilde{\mathbf{y}}_{e}^{\dagger}\mathbf{c}_{e^{c}}^{T}\tilde{\mathbf{y}}_{e})\right)-\xi_{H}\left(\frac{9\alpha_{2}^{2}}{32\pi^{2}}\tilde{c}_{W}+\frac{3\alpha_{1}^{2}}{32\pi^{2}}\tilde{c}_{B}\right),$

with

$\begin{split}\tilde{c}_{G}&=c_{G}-{\rm
tr\left(2\mathbf{c}_{Q}+\mathbf{c}_{u^{c}}+\mathbf{c}_{d^{c}}\right)}\,,\\\
\tilde{c}_{W}&=c_{W}-{\rm
tr\left(3\mathbf{c}_{Q}+\mathbf{c}_{L}\right)}-\frac{3}{2}\xi_{H}(c_{\widetilde{H}_{u}}+c_{\widetilde{H}_{d}})\,,\\\
\tilde{c}_{B}&=c_{B}-{\rm
tr\left(\frac{1}{3}(\mathbf{c}_{Q}+8\,\mathbf{c}_{u^{c}}+2\,\mathbf{c}_{d^{c}})+\mathbf{c}_{L}+2\mathbf{c}_{e^{c}}\right)}-\frac{3}{2}\xi_{H}(c_{\widetilde{H}_{u}}+c_{\widetilde{H}_{d}})\,.\end{split}$
(84)

The parameters $\tilde{c}_{G,W,B}$ in Eq. (84) are invariant under the field
redefinition (80). Here we note that $c_{\widetilde{H}_{u}}(=c_{H_{u}})$ and
$c_{\widetilde{H}_{d}}(=c_{H_{d}})$ contributions in $\tilde{c}_{W}$ and
$\tilde{c}_{B}$ are from the Higgsino loops.

In this work we are concerned with only the flavor diagonal components of the
axion couplings. Therefore in the following we assume that
$(\mathbf{c}_{\psi})_{ij}$ are flavor diagonal in the mass eigenstate basis:

$\displaystyle(\mathbf{c}_{\psi})_{ij}=c_{\psi_{i}}\delta_{ij},$ (85)

and approximate the Yukawa couplings in the RG equations by keeping only the
third generation components:

$\begin{split}(\tilde{\mathbf{y}}_{u})_{ij}\approx\tilde{y}_{t}\delta_{i3}\delta_{j3}\,,\quad(\tilde{\mathbf{y}}_{d})_{ij}\approx\tilde{y}_{b}\delta_{i3}\delta_{j3}\,,\quad(\tilde{\mathbf{y}}_{e})_{ij}\approx\tilde{y}_{\tau}\delta_{i3}\delta_{j3}\,.\end{split}$
(86)

In this approximation, the RG equations in Eq. (3.1) can be rewritten as

$\displaystyle\frac{dn_{u_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{16\pi^{2}}\left(3(2+\xi_{y}\delta_{i3})\tilde{y}_{t}^{2}n_{t}+\xi_{y}\tilde{y}_{b}^{2}n_{b}\delta_{i3}\right)$
$\displaystyle-\left(\frac{\xi_{g}}{\pi^{2}}\alpha_{s}^{2}\tilde{c}_{G}+\frac{9(\xi_{g}+\xi_{H})}{32\pi^{2}}\alpha_{2}^{2}\tilde{c}_{W}+\frac{(17\xi_{g}+9\xi_{H})}{96\pi^{2}}\alpha_{1}^{2}\tilde{c}_{B}\right),$
$\displaystyle\frac{dn_{d_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{16\pi^{2}}\left(\xi_{y}\tilde{y}_{t}^{2}n_{t}\delta_{i3}+3(2+\xi_{y}\delta_{i3})\tilde{y}_{b}^{2}n_{b}+2\tilde{y}_{\tau}^{2}n_{\tau}\right)$
$\displaystyle-\left(\frac{\xi_{g}}{\pi^{2}}\alpha_{s}^{2}\tilde{c}_{G}+\frac{9(\xi_{g}+\xi_{H})}{32\pi^{2}}\alpha_{2}^{2}\tilde{c}_{W}+\frac{(5\xi_{g}+9\xi_{H})}{96\pi^{2}}\alpha_{1}^{2}\tilde{c}_{B}\right),$
$\displaystyle\frac{dn_{e_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{16\pi^{2}}\left(6\tilde{y}_{b}^{2}n_{b}+(2+3\xi_{y}\delta_{i3})\tilde{y}_{\tau}^{2}n_{\tau}\right)-\left(\frac{9(\xi_{g}+\xi_{H})}{32\pi^{2}}\alpha_{2}^{2}\tilde{c}_{W}+\frac{3(5\xi_{g}+\xi_{H})}{32\pi^{2}}\alpha_{1}^{2}\tilde{c}_{B}\right),$
$\displaystyle\frac{dn_{H}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{8\pi^{2}}\left(3\tilde{y}_{t}^{2}n_{t}+3\tilde{y}_{b}^{2}n_{b}+\tilde{y}_{\tau}^{2}n_{\tau}\right)-\xi_{H}\left(\frac{9}{16\pi^{2}}\alpha_{2}^{2}\tilde{c}_{W}+\frac{3}{16\pi^{2}}\alpha_{1}^{2}\tilde{c}_{B}\right).$
(87)

where $u_{i}=(u,c,t)$, $d_{i}=(d,s,b)$, $e_{i}=(e,\mu,\tau)$, and

$\begin{split}n_{u_{i}}&\equiv c_{Q_{i}}+c_{u^{c}_{i}}+c_{H_{u}},\quad
n_{d_{i}}\,\equiv c_{Q_{i}}+c_{d^{c}_{i}}+c_{H_{d}},\\\ n_{e_{i}}&\equiv
c_{L_{i}}+c_{e^{c}_{i}}+c_{H_{d}},\quad n_{H}\,\equiv
c_{H_{u}}+c_{H_{d}}.\end{split}$ (88)

Note that the parameters defined in Eq. (88) are invariant under the field
redefinition (80), so they may be directly related to physically observable
quantities as we will see below. In Eq. (87), the RG coefficients
$\tilde{c}_{G},\tilde{c}_{W}$ and $\tilde{c}_{B}$ are scale-dependent because
the axion-fermion couplings in Eq. (84) are running, while

$\begin{split}\frac{dc_{A}}{d\ln\mu}=0\quad(A=G,W,B).\end{split}$ (89)

From Eq. (89) and Eq. (3.1), one can derive

$\displaystyle\frac{d\tilde{c}_{G}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{\xi_{y}}{4\pi^{2}}\left(\tilde{y}_{t}^{2}n_{t}+\tilde{y}_{b}^{2}n_{b}\right)+\xi_{g}\left(\frac{6\alpha_{s}^{2}}{\pi^{2}}\tilde{c}_{G}+\frac{27\alpha_{2}^{2}}{16\pi^{2}}\tilde{c}_{W}+\frac{11\alpha_{1}^{2}}{16\pi^{2}}\tilde{c}_{B}\right),$
$\displaystyle\frac{d\tilde{c}_{W}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{(\xi_{y}+3\xi_{H})}{16\pi^{2}}\left(3\tilde{y}_{t}^{2}n_{t}+3\tilde{y}_{b}^{2}n_{b}+\tilde{y}_{\tau}^{2}n_{\tau}\right)$
$\displaystyle+\frac{9\xi_{g}}{2\pi^{2}}\alpha_{s}^{2}\tilde{c}_{G}+\frac{(108\xi_{g}+27\xi_{H}^{2})}{32\pi^{2}}\alpha_{2}^{2}\tilde{c}_{W}+\frac{(12\xi_{g}+9\xi_{H}^{2})}{32\pi^{2}}\alpha_{1}^{2}\tilde{c}_{B}\,,$
$\displaystyle\frac{d\tilde{c}_{B}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{(17\xi_{y}+27\xi_{H})}{48\pi^{2}}\tilde{y}_{t}^{2}n_{t}-\frac{(5\xi_{y}+27\xi_{H})}{48\pi^{2}}\tilde{y}_{b}^{2}n_{b}-\frac{(5\xi_{y}+3\xi_{H})}{16\pi^{2}}\tilde{y}_{\tau}^{2}n_{\tau}$
(90)
$\displaystyle+\frac{11\xi_{g}}{2\pi^{2}}\alpha_{s}^{2}\tilde{c}_{G}+\frac{(36\xi_{g}+27\xi_{H}^{2})}{32\pi^{2}}\alpha_{2}^{2}\tilde{c}_{W}+\frac{(380\xi_{g}+27\xi_{H}^{2})}{96\pi^{2}}\alpha_{1}^{2}\tilde{c}_{B}\,.$

Eq. (87) and Eq. (3.1) constitute a complete set of RG equations given in
terms of the RG parameters invariant under the field redefinition (80).

The above RG equations are valid when the RG scale $\mu$ is higher than mass
scales of new particles beyond the standard model (BSM). For simplicity, let
us assume that the masses of the superparticles and charged Higgs boson mass
in 2HDMs are given by a common mass scale $m_{\rm BSM}$,

$\begin{split}m_{\rm BSM}\sim m_{\rm SUSY}\sim m_{H^{\pm}}.\end{split}$ (91)

Below the BSM scale ($\mu<m_{\rm BSM}$), the BSM particles are integrated out
and no longer contribute to the running of the axion couplings. We will assume
that $m_{\rm BSM}$ is well above the weak scale so that the light Higgs
doublet field is approximated as the SM Higgs field with negligible mixing
effect. Thus the RG equations are governed by the SM degrees of freedom
including the SM Higgs field $H$. As the BSM Higgs doublet field is integrated
out, the SM Yukawa couplings $y_{t},y_{b}$ and $y_{\tau}$ satisfy the matching
condition

$\begin{split}y_{t}=\tilde{y}_{t}\sin\beta,\quad
y_{b}=\tilde{y}_{b}\cos\beta,\quad
y_{\tau}=\tilde{y}_{\tau}\cos\beta\quad\textrm{at}~{}\mu=m_{\rm
BSM},\end{split}$ (92)

where $\tan\beta\equiv\langle H_{u}\rangle/\langle H_{d}\rangle$, while the
coupling between the axion and the SM Higgs field $c_{H}$ satisfies

$\begin{split}c_{H}=c_{H_{u}}\sin^{2}\beta-
c_{H_{d}}\cos^{2}\beta\quad\textrm{at}~{}\mu=m_{\rm BSM}.\end{split}$ (93)

Then the RG equations for the axion couplings below the BSM scale (but above
the weak scale) turn out to be

$\displaystyle\frac{dc_{Q_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{16\pi^{2}}y_{t}^{2}\big{(}c_{Q_{3}}+c_{u^{c}_{3}}+c_{H}\big{)}\delta_{i3}-\left(\frac{\alpha_{s}^{2}}{2\pi^{2}}\tilde{c}_{G}+\frac{9\alpha_{2}^{2}}{32\pi^{2}}\tilde{c}_{W}+\frac{\alpha_{1}^{2}}{96\pi^{2}}\tilde{c}_{B}\right),$
$\displaystyle\frac{dc_{u^{c}_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{8\pi^{2}}y_{t}^{2}\big{(}c_{Q_{3}}+c_{u^{c}_{3}}+c_{H}\big{)}\delta_{i3}-\left(\frac{\alpha_{s}^{2}}{2\pi^{2}}\tilde{c}_{G}+\frac{\alpha_{1}^{2}}{6\pi^{2}}\tilde{c}_{B}\right),$
$\displaystyle\frac{dc_{d^{c}_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\left(\frac{\alpha_{s}^{2}}{2\pi^{2}}\tilde{c}_{G}+\frac{\alpha_{1}^{2}}{24\pi^{2}}\tilde{c}_{B}\right),$
$\displaystyle\frac{dc_{L_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\left(\frac{9\alpha_{2}^{2}}{32\pi^{2}}\tilde{c}_{W}+\frac{3\alpha_{1}^{2}}{32\pi^{2}}\tilde{c}_{B}\right),$
$\displaystyle\frac{dc_{e^{c}_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{3\alpha_{1}^{2}}{8\pi^{2}}\tilde{c}_{B}\,,$
$\displaystyle\frac{dc_{H}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle\frac{3}{8\pi^{2}}y_{t}^{2}\big{(}c_{Q_{3}}+c_{u^{c}_{3}}+c_{H}\big{)},$
(94)

where we ignore the contributions from the bottom and tau Yukawa couplings
$y_{b},y_{\tau}$, and

$\begin{split}\tilde{c}_{G}&=c_{G}-\sum_{i}\big{(}2c_{Q_{i}}+c_{u^{c}_{i}}+c_{d^{c}_{i}}\big{)}\,,\\\
\tilde{c}_{W}&=c_{W}-\sum_{i}\big{(}3c_{Q_{i}}+c_{L_{i}}\big{)}\,,\\\
\tilde{c}_{B}&=c_{B}-\sum_{i}\left(\frac{1}{3}\big{(}c_{Q_{i}}+8c_{u^{c}_{i}}+2c_{d^{c}_{i}}\big{)}+c_{L_{i}}+2c_{e^{c}_{i}}\right)\,.\end{split}$
(95)

Note that there are no longer Higgsino contributions in $\tilde{c}_{W,B}$ of
Eq. (95) compared to Eq. (84). In fact the RG equations (3.1) can be recast
into a simpler form:

$\displaystyle\frac{dC_{u_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle\frac{3}{16\pi^{2}}y_{t}^{2}C_{t}(2+\delta_{i3})-\left(\frac{\alpha_{s}^{2}}{\pi^{2}}\tilde{c}_{G}+\frac{9\alpha_{2}^{2}}{32\pi^{2}}\tilde{c}_{W}+\frac{17\alpha_{1}^{2}}{96\pi^{2}}\tilde{c}_{B}\right),$
$\displaystyle\frac{dC_{d_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{16\pi^{2}}y_{t}^{2}C_{t}(6-\delta_{i3})-\left(\frac{\alpha_{s}^{2}}{\pi^{2}}\tilde{c}_{G}+\frac{9\alpha_{2}^{2}}{32\pi^{2}}\tilde{c}_{W}+\frac{5\alpha_{1}^{2}}{96\pi^{2}}\tilde{c}_{B}\right),$
$\displaystyle\frac{dC_{e_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{3}{8\pi^{2}}y_{t}^{2}C_{t}-\left(\frac{9\alpha_{2}^{2}}{32\pi^{2}}\tilde{c}_{W}+\frac{15\alpha_{1}^{2}}{32\pi^{2}}\tilde{c}_{B}\right).$
(96)

where

$\begin{split}C_{u_{i}}&\equiv
c_{Q_{i}}+c_{u^{c}_{i}}+c_{H}=n_{u_{i}}-n_{H}\cos^{2}\beta,\\\
C_{d_{i}}&\equiv
c_{Q_{i}}+c_{d^{c}_{i}}-c_{H}=n_{d_{i}}-n_{H}\sin^{2}\beta,\\\
C_{e_{i}}&\equiv
c_{L_{i}}+c_{e^{c}_{i}}-c_{H}=n_{e_{i}}-n_{H}\sin^{2}\beta.\\\ \end{split}$
(97)

for $u_{i}=(u,c,t),d_{i}=(d,s,b)$ and $e_{i}=(e,\mu,\tau)$. We remark that the
parameters defined in Eq. (97) are invariant under the SM version of the field
redefinition (80). Having Eq. (3.1), a complete set of RG equations below the
BSM scale includes the following equations for the running of
$\tilde{c}_{A}\,(A=G,W,B)$.

$\displaystyle\frac{d\tilde{c}_{G}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{4\pi^{2}}y_{t}^{2}C_{t}+\frac{6\alpha_{s}^{2}}{\pi^{2}}\tilde{c}_{G}+\frac{27\alpha_{2}^{2}}{16\pi^{2}}\tilde{c}_{W}+\frac{11\alpha_{1}^{2}}{16\pi^{2}}\tilde{c}_{B}\,,$
$\displaystyle\frac{d\tilde{c}_{W}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{3}{16\pi^{2}}y_{t}^{2}C_{t}+\frac{9}{2\pi^{2}}\alpha_{s}^{2}\tilde{c}_{G}+\frac{27}{8\pi^{2}}\alpha_{2}^{2}\tilde{c}_{W}+\frac{3}{8\pi^{2}}\alpha_{1}^{2}\tilde{c}_{B}\,,$
$\displaystyle\frac{d\tilde{c}_{B}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{17}{48\pi^{2}}y_{t}^{2}C_{t}+\frac{11}{2\pi^{2}}\alpha_{s}^{2}\tilde{c}_{G}+\frac{9}{8\pi^{2}}\alpha_{2}^{2}\tilde{c}_{W}+\frac{95}{24\pi^{2}}\alpha_{1}^{2}\tilde{c}_{B}\,.$
(98)

Below the weak scale ($\mu<m_{W}$), the top quark, Higgs boson, and $W/Z$
gauge bosons are integrated out. We ignore the corresponding threshold
corrections calculated in 2012.12272  because they are of subleading order
giving a correction to our calculation less than 10%. At this scale, the
relevant axion couplings take the form

$\begin{split}c_{\gamma}\frac{e^{2}}{32\pi^{2}}\frac{a}{f_{a}}F^{\mu\nu}\tilde{F}_{\mu\nu}+\frac{\partial_{\mu}a}{2f_{a}}\Big{[}C_{u_{i}}\bar{u}_{i}\gamma^{\mu}\gamma_{5}u_{i}+C_{d_{i}}\bar{d}_{i}\gamma^{\mu}\gamma_{5}d_{i}+C_{e_{i}}\bar{e}_{i}\gamma^{\mu}\gamma_{5}e_{i}\Big{]},\end{split}$
(99)

where $F^{\mu\nu}$ is the electromagnetic field strength, and $u_{i},d_{i}$
and $e_{i}$ denote the Dirac quarks and leptons with masses lighter than the
RG point $\mu$. Here

$\begin{split}c_{\gamma}=c_{W}+c_{B},\end{split}$ (100)

and the matching conditions for $C_{u_{i}},C_{d_{i}}$ and $C_{e_{i}}$ at the
weak scale are given in Eq. (97). The RG equations for the above axion
couplings are given by

$\displaystyle\frac{dC_{u_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{\alpha_{s}^{2}}{\pi^{2}}\tilde{c}_{G}-\frac{\alpha_{\rm
em}^{2}}{6\pi^{2}}\tilde{c}_{\gamma}\,,$
$\displaystyle\frac{dC_{d_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{\alpha_{s}^{2}}{\pi^{2}}\tilde{c}_{G}-\frac{\alpha_{\rm
em}^{2}}{24\pi^{2}}\tilde{c}_{\gamma}\,,$
$\displaystyle\frac{dC_{e_{i}}}{d\ln\mu}$ $\displaystyle\simeq$
$\displaystyle-\frac{3\alpha_{\rm em}^{2}}{8\pi^{2}}\tilde{c}_{\gamma}\,,$
(101)

where

$\begin{split}\tilde{c}_{G}(\mu)&=c_{G}-\sum_{q}C_{q}(\mu)\Theta(\mu-
m_{q}),\\\
\tilde{c}_{\gamma}(\mu)&=c_{\gamma}-2\sum_{f}N_{c}^{f}Q_{f}^{2}C_{f}(\mu)\Theta(\mu-
m_{f}),\end{split}$ (102)

with the unit step function $\Theta(x)=1$ for $x>0$ and $\Theta(x)=0$ for
$x<0$. Here $q$ stands for the quarks $u,d,s,c,b$, and $N_{c}^{f}$ and $Q_{f}$
are the number of colors and the electromagnetic charge of the SM fermion $f$
lighter than the top quark, respectively.

Finally below the QCD scale ($\mu\lesssim 1$ GeV), the relevant axion
couplings can be written as

$\displaystyle\frac{1}{4}g_{a\gamma}a\vec{E}\cdot\vec{B}+\partial_{\mu}a\left[\frac{g_{ae}}{2m_{e}}\bar{e}\gamma^{\mu}\gamma_{5}e+\frac{g_{an}}{2m_{n}}\bar{n}\gamma^{\mu}\gamma_{5}n+\frac{g_{ap}}{2m_{p}}\bar{p}\gamma^{\mu}\gamma_{5}p\right],$
(103)

where the axion-photon coupling $g_{a\gamma}$ includes the threshold
correction from the axion-pion mixing:

$\displaystyle g_{a\gamma}\simeq\frac{\alpha_{\rm
em}}{2\pi}\frac{1}{f_{a}}\Big{(}c_{\gamma}-\frac{2}{3}\frac{m_{u}+4m_{d}}{m_{u}+m_{d}}c_{G}\Big{)}\,\simeq\,\frac{\alpha_{\rm
em}}{2\pi}\frac{1}{f_{a}}\Big{(}c_{\gamma}-1.92c_{G}\Big{)},$ (104)

the axion-electron coupling $g_{ae}$ evolves down to $\mu=m_{e}$ as

$\begin{split}\frac{dg_{ae}}{d\ln\mu}\simeq-\frac{3\alpha_{\rm
em}m_{e}}{4\pi}g_{a\gamma}\quad\Big{(}g_{ae}=\frac{m_{e}C_{e}}{f_{a}}\Big{)},\end{split}$
(105)

and the axion-nucleon couplings can be computed from the couplings
$c_{G},C_{u}(\mu)$ and $C_{d}(\mu)$ at the matching scale $\mu=2$ GeV using
the lattice data 1511.02867 :

$\begin{split}g_{ap}&\simeq\frac{m_{p}}{f_{a}}\left(C_{u}\Delta u+C_{d}\Delta
d-\Big{(}\frac{m_{d}}{m_{u}+m_{d}}\Delta u+\frac{m_{u}}{m_{u}+m_{d}}\Delta
d\Big{)}c_{G}\right),\\\ &\simeq\frac{m_{p}}{f_{a}}\Big{(}0.90\,C_{u}(2\,{\rm
GeV})-0.38\,C_{d}(2\,{\rm GeV})-0.48\,c_{G}\Big{)},\end{split}$ (106)
$\begin{split}g_{an}&\simeq\frac{m_{n}}{f_{a}}\left(C_{d}\Delta u+C_{u}\Delta
d-\Big{(}\frac{m_{u}}{m_{u}+m_{d}}\Delta u+\frac{m_{d}}{m_{u}+m_{d}}\Delta
d\Big{)}c_{G}\right),\\\ &\simeq\frac{m_{n}}{f_{a}}\Big{(}0.90\,C_{d}(2\,{\rm
GeV})-0.38\,C_{u}(2\,{\rm GeV})-0.04\,c_{G}\Big{)},\end{split}$ (107)

with $\Delta u=0.897(27)$, $\Delta d=-0.376(27)$, and $m_{u}/m_{d}=0.48(3)$ at
$\mu=2$ GeV in $\overline{\rm MS}$ 1511.02867 ; 2003.01100 .888Note that the
numbers in Eq. (106) and Eq. (107) are a bit different from those in Eq. 50 of
Ref. 1511.02867 . This is because here our $C_{u}$ and $C_{d}$ are not UV
boundary parameters around $\mu=f_{a}$ but the renormalized parameters at
$\mu=2$ GeV. If we express $C_{u}(2\,\textrm{GeV})$ and
$C_{d}(2\,\textrm{GeV})$ in terms of the UV parameters, for instance, using
Eq. (120) in the following analysis, we get better agreement with those
numbers in Ref. 1511.02867  except the coefficient of $C_{t}(f_{a})$. Our
coefficient of $C_{t}(f_{a})$ from Eq. (120) is significantly different from
the one in Ref. 1511.02867 , because Ref. 1511.02867  does not take into
account the renormalization by Yukawa interactions.

###  3.2 Semi-analytic solution and numerical results

We now discuss the solution to the RG equations of the previous subsection. We
are concerned with the RG corrections to the low energy couplings
$C_{u},C_{d}$ and $C_{e}$. We will provide semi-analytic formulas for them as
well as showing fully numerical results.

For a small $\tan\beta(\lesssim 50)$, as a good approximation, one can keep
only the top Yukawa coupling $\tilde{y}_{t}$ among the three Yukawa couplings
$\tilde{y}_{t,b,\tau}$ in the RG equations (87). By integrating the RG
equations Eq. (87), Eq. (3.1), and Eq. (3.1) using this approximation, we get
for a renormalization scale $\mu<m_{W}$

$\begin{split}C_{u}&(\mu)\simeq C_{u}(f_{a})+I_{t}(m_{\rm
BSM},f_{a})\sin^{2}\beta+I_{t}^{\rm SM}(m_{t},m_{\rm BSM})\\\
&+\frac{2}{3\pi^{2}}\int_{m_{\rm
BSM}}^{f_{a}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(\alpha_{s}^{2}(\mu^{\prime})\tilde{c}_{G}(\mu^{\prime})+\frac{9\sin^{2}\beta}{16}\alpha_{2}^{2}(\mu^{\prime})\tilde{c}_{W}(\mu^{\prime})+\frac{(4+9\sin^{2}\beta)}{48}\alpha_{1}^{2}(\mu^{\prime})\tilde{c}_{B}(\mu^{\prime})\right),\\\
&+\frac{1}{\pi^{2}}\int_{m_{W}}^{m_{\rm
BSM}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(\alpha_{s}^{2}(\mu^{\prime})\tilde{c}_{G}(\mu^{\prime})+\frac{9}{32}\alpha_{2}^{2}(\mu^{\prime})\tilde{c}_{W}(\mu^{\prime})+\frac{17}{96}\alpha_{1}^{2}(\mu^{\prime})\tilde{c}_{B}(\mu^{\prime})\right),\\\
&+\frac{1}{\pi^{2}}\int_{\mu}^{m_{W}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(\alpha_{s}^{2}(\mu^{\prime})\tilde{c}_{G}(\mu^{\prime})+\frac{1}{6}\alpha_{\rm
em}^{2}(\mu^{\prime})\tilde{c}_{\gamma}(\mu^{\prime})\right),\end{split}$
(108) $\begin{split}C_{d}&(\mu)\simeq C_{d}(f_{a})-I_{t}(m_{\rm
BSM},f_{a})\sin^{2}\beta-I_{t}^{\rm SM}(m_{t},m_{\rm BSM})\\\
&+\frac{2}{3\pi^{2}}\int_{m_{\rm
BSM}}^{f_{a}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(\alpha_{s}^{2}(\mu^{\prime})\tilde{c}_{G}(\mu^{\prime})+\frac{9\cos^{2}\beta}{16}\alpha_{2}^{2}(\mu^{\prime})\tilde{c}_{W}(\mu^{\prime})+\frac{(-2+9\cos^{2}\beta)}{48}\alpha_{1}^{2}(\mu^{\prime})\tilde{c}_{B}(\mu^{\prime})\right),\\\
&+\frac{1}{\pi^{2}}\int_{m_{W}}^{m_{\rm
BSM}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(\alpha_{s}^{2}(\mu^{\prime})\tilde{c}_{G}(\mu^{\prime})+\frac{9}{32}\alpha_{2}^{2}(\mu^{\prime})\tilde{c}_{W}(\mu^{\prime})+\frac{5}{96}\alpha_{1}^{2}(\mu^{\prime})\tilde{c}_{B}(\mu^{\prime})\right),\\\
&+\frac{1}{\pi^{2}}\int_{\mu}^{m_{W}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(\alpha_{s}^{2}(\mu^{\prime})\tilde{c}_{G}(\mu^{\prime})+\frac{1}{24}\alpha_{\rm
em}^{2}(\mu^{\prime})\tilde{c}_{\gamma}(\mu^{\prime})\right),\end{split}$
(109) $\begin{split}C_{e}(\mu)\simeq&~{}C_{e}(f_{a})-I_{t}(m_{\rm
BSM},f_{a})\sin^{2}\beta-I_{t}^{\rm SM}(m_{t},m_{\rm BSM})\\\
&+\frac{2}{3\pi^{2}}\int_{m_{\rm
BSM}}^{f_{a}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(\frac{9\cos^{2}\beta}{16}\alpha_{2}^{2}(\mu^{\prime})\tilde{c}_{W}(\mu^{\prime})+\frac{3(2+\cos^{2}\beta)}{16}\alpha_{1}^{2}(\mu^{\prime})\tilde{c}_{B}(\mu^{\prime})\right),\\\
&+\frac{1}{\pi^{2}}\int_{m_{W}}^{m_{\rm
BSM}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(\frac{9}{32}\alpha_{2}^{2}(\mu^{\prime})\tilde{c}_{W}(\mu^{\prime})+\frac{15}{32}\alpha_{1}^{2}(\mu^{\prime})\tilde{c}_{B}(\mu^{\prime})\right),\\\
&+\frac{1}{\pi^{2}}\int_{\mu}^{m_{W}}\frac{d\mu^{\prime}}{\mu^{\prime}}\frac{3}{8}\alpha_{\rm
em}^{2}(\mu^{\prime})\tilde{c}_{\gamma}(\mu^{\prime}),\end{split}$ (110)

where $C_{\Psi}(f_{a})\equiv C_{\Psi}^{0}$ as defined in Eq. (2), and
$I_{t}(m_{\rm BSM},f_{a})$ and $I_{t}^{\rm SM}(m_{t},m_{\rm BSM})$ are defined
as

$\displaystyle I_{t}(m_{\rm BSM},f_{a})$ $\displaystyle\equiv$
$\displaystyle-\frac{3}{8\pi^{2}}\int_{m_{\rm
BSM}}^{f_{a}}\frac{d\mu^{\prime}}{\mu^{\prime}}\tilde{y}_{t}^{2}(\mu^{\prime})n_{t}(\mu^{\prime}),$
(111) $\displaystyle I_{t}^{\rm SM}(m_{t},m_{\rm BSM})$ $\displaystyle\equiv$
$\displaystyle-\frac{3}{8\pi^{2}}\int_{m_{t}}^{m_{\rm
BSM}}\frac{d\mu^{\prime}}{\mu^{\prime}}y_{t}^{2}(\mu^{\prime})C_{t}(\mu^{\prime}).$
(112)

In the above integrations one can ignore the running of the parameters
$\tilde{c}_{G}(\mu),\tilde{c}_{W}(\mu)$ and $\tilde{c}_{B}(\mu)$ as this
effect is of subleading order, while one has to take into account the heavy
particle threshold corrections as in Eq. (102).

The above formulas show that the RG corrections to $C_{u,d,e}$ are mainly
determined by the running parameters $n_{t},C_{t},\tilde{c}_{G,W,B}$. To get a
more instructive form for $I_{t}$ and $I_{t}^{SM}$ from the top Yukawa
interaction, one may integrate the one-loop RG equations for $\alpha_{s}(\mu)$
and the top Yukawa coupling while solving the first-order linear differential
equations for $n_{t}(\mu)$ and $C_{t}(\mu)$ in Eq. (87) and Eq. (3.1). We then
find

$\begin{split}I_{t}&(m_{\rm BSM},f_{a})\simeq-\frac{1}{2}\left(1-R_{t}(m_{\rm
BSM},f_{a})\right)n_{t}(f_{a})\\\ &-\frac{1}{3\pi^{2}}\int_{m_{\rm
BSM}}^{f_{a}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(1-R_{t}(m_{\rm
BSM},\mu^{\prime})\right)\left(\alpha_{s}^{2}(\mu^{\prime})\tilde{c}_{G}(\mu^{\prime})+\frac{9}{16}\alpha_{2}^{2}(\mu^{\prime})\tilde{c}_{W}(\mu^{\prime})+\frac{13}{48}\alpha_{1}^{2}(\mu^{\prime})\tilde{c}_{B}(\mu^{\prime})\right),\end{split}$
(113) $\begin{split}I_{t}^{\rm SM}&(m_{t},m_{\rm
BSM})\simeq-\frac{2}{3}\left(1-R_{t}^{\rm SM}(m_{t},m_{\rm
BSM})\right)C_{t}(m_{\rm BSM})\\\ &-\frac{2}{3\pi^{2}}\int_{m_{t}}^{m_{\rm
BSM}}\frac{d\mu^{\prime}}{\mu^{\prime}}\left(1-R_{t}^{\rm
SM}(m_{t},\mu^{\prime})\right)\left(\alpha_{s}^{2}(\mu^{\prime})\tilde{c}_{G}(\mu^{\prime})+\frac{9}{32}\alpha_{2}^{2}(\mu^{\prime})\tilde{c}_{W}(\mu^{\prime})+\frac{17}{96}\alpha_{1}^{2}(\mu^{\prime})\tilde{c}_{B}(\mu^{\prime})\right),\end{split}$
(114)

with

$\begin{split}R_{t}(\mu,\mu^{\prime})&\equiv\exp\left[-24\int_{\mu}^{\mu^{\prime}}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}\frac{\tilde{y}_{t}^{2}(\mu^{\prime\prime})}{32\pi^{2}}\right]\simeq\left[1-\frac{18}{7}\frac{\alpha_{t}(\mu^{\prime})}{\alpha_{s}(\mu^{\prime})}\left(1-\left(\frac{\alpha_{s}(\mu)}{\alpha_{s}(\mu^{\prime})}\right)^{7/9}\right)\right]^{-1},\end{split}$
(115) $\begin{split}R_{t}^{\rm
SM}(\mu,\mu^{\prime})&\equiv\exp\left[-18\int_{\mu}^{\mu^{\prime}}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}\frac{y_{t}^{2}(\mu^{\prime\prime})}{32\pi^{2}}\right]\simeq\left[1-\frac{9}{2}\frac{\alpha_{t}(\mu^{\prime})}{\alpha_{s}(\mu^{\prime})}\left(1-\left(\frac{\alpha_{s}(\mu)}{\alpha_{s}(\mu^{\prime})}\right)^{1/7}\right)\right]^{-1},\end{split}$
(116)

where $\alpha_{t}(\mu)\equiv\tilde{y}_{t}^{2}(\mu)/4\pi$ for $\mu>m_{\rm BSM}$
and $\alpha_{t}(\mu)\equiv y_{t}^{2}(\mu)/4\pi$ for $\mu<m_{\rm BSM}$. Eq.
(113) and Eq. (114) explicitly show potentially important contributions from
$\tilde{c}_{G,W,B}$ through the top Yukawa interaction even when
$n_{t}(f_{a})$ and $C_{t}(m_{\rm BSM})$ vanish. As we will see below, they
give rise to an important radiative correction to $C_{e}$ for KSVZ-like
axions.

Eqs. (108), (109), (110) and (111) show that the radiative corrections to
$C_{\Psi}$ $(\Psi=u,d,e)$ induced by the top Yukawa interaction and the SM
gauge interactions originate mainly from the four UV parameters
$\\{n_{t}(f_{a}),\tilde{c}_{G}(f_{a}),\tilde{c}_{W}(f_{a}),\tilde{c}_{B}(f_{a})\\}$
in the MSSM case or from
$\\{C_{t}(f_{a}),\tilde{c}_{G}(f_{a}),\tilde{c}_{W}(f_{a}),\tilde{c}_{B}(f_{a})\\}$
in the SM case. Then one can parameterize the low energy axion couplings to
the light quarks and electron as

$\displaystyle C_{u}(2\,{\rm GeV})$ $\displaystyle=$ $\displaystyle
C_{u}(f_{a})+\Delta C_{u},$ $\displaystyle C_{d}(2\,{\rm GeV})$
$\displaystyle=$ $\displaystyle C_{d}(f_{a})+\Delta C_{d},$ $\displaystyle
C_{e}(m_{e})$ $\displaystyle=$ $\displaystyle C_{e}(f_{a})+\Delta C_{e},$
(117)

where $C_{\Psi}(f_{a})$ ($\Psi=u,d,e$) are identified as the tree-level values
$C^{0}_{\Psi}$ and

$\displaystyle\mbox{SM}:$ $\displaystyle\quad\Delta
C_{\Psi}=r_{\Psi}^{t}C_{t}(f_{a})+r_{\Psi}^{G}\tilde{c}_{G}(f_{a})+r_{\Psi}^{W}\tilde{c}_{W}(f_{a})+r_{\Psi}^{B}\tilde{c}_{B}(f_{a}),$
(118) $\displaystyle\mbox{MSSM}:$ $\displaystyle\quad\Delta
C_{\Psi}=r_{\Psi}^{t}n_{t}(f_{a})+r_{\Psi}^{G}\tilde{c}_{G}(f_{a})+r_{\Psi}^{W}\tilde{c}_{W}(f_{a})+r_{\Psi}^{B}\tilde{c}_{B}(f_{a}).$

To compute the coefficients $r_{\Psi}^{X}\,(X=t,G,W,B)$ in this
parameterization, we solve the RG equations in a fully numerical way with the
running SM gauge couplings and top Yukawa coupling at two-loop order. The
resulting $r_{\Psi}^{X}$ are depicted in Fig. 4 for the SM case (left panel)
and the MSSM case (right panel). For the MSSM, we choose $\tan\beta=10$ and
the SUSY particle masses $m_{\rm SUSY}=10$ TeV. Our results show that

$\displaystyle r^{t}_{u,d,e}\sim\,{\rm few}\times 10^{-1},\quad
r^{G}_{u,d}\sim 10^{-2},\quad r^{G}_{e}\sim\,{\rm few}\times 10^{-4}-10^{-3},$
$\displaystyle r^{W}_{u,d,e}\sim\,{\rm few}\times 10^{-4}-10^{-3},\quad
r^{B}_{u,d}\sim\,{\rm few}\times 10^{-5}-10^{-4},\quad r^{B}_{e}\sim 10^{-4}$
(119)

for $10^{7}\,{\rm GeV}\lesssim f_{a}\lesssim 10^{16}\,{\rm GeV}$. For example,
for $f_{a}=10^{10}$ GeV in the SM case, we find

$\begin{split}C_{u}(2\,{\rm GeV})&\simeq
C_{u}(f_{a})-0.28\,C_{t}(f_{a})+\left[17.8\,\tilde{c}_{G}(f_{a})+0.33\,\tilde{c}_{W}(f_{a})+0.032\,\tilde{c}_{B}(f_{a})\right]\times
10^{-3}\,,\\\ C_{d}(2\,{\rm GeV})&\simeq
C_{d}(f_{a})+0.30\,C_{t}(f_{a})+\left[19.5\,\tilde{c}_{G}(f_{a})+0.48\,\tilde{c}_{W}(f_{a})+0.017\,\tilde{c}_{B}(f_{a})\right]\times
10^{-3}\,,\\\ C_{e}(m_{e})&\simeq
C_{e}(f_{a})+0.29\,C_{t}(f_{a})+\left[0.80\,\tilde{c}_{G}(f_{a})+0.54\,\tilde{c}_{W}(f_{a})+0.13\,\tilde{c}_{B}(f_{a})\right]\times
10^{-3}\,,\end{split}$ (120)

and for $f_{a}=10^{10}$ GeV, $m_{\rm SUSY}=10$ TeV and $\tan\beta=10$ in the
MSSM case,

$\begin{split}C_{u}(2\,{\rm GeV})&\simeq
C_{u}(f_{a})-0.28\,n_{t}(f_{a})+\left[17.7\,\tilde{c}_{G}(f_{a})+0.52\,\tilde{c}_{W}(f_{a})+0.036\,\tilde{c}_{B}(f_{a})\right]\times
10^{-3}\,,\\\ C_{d}(2\,{\rm GeV})&\simeq
C_{d}(f_{a})+0.31\,n_{t}(f_{a})+\left[19.4\,\tilde{c}_{G}(f_{a})+0.23\,\tilde{c}_{W}(f_{a})+0.0047\,\tilde{c}_{B}(f_{a})\right]\times
10^{-3}\,,\\\ C_{e}(m_{e})&\simeq
C_{e}(f_{a})+0.29\,n_{t}(f_{a})+\left[0.81\,\tilde{c}_{G}(f_{a})+0.28\,\tilde{c}_{W}(f_{a})+0.10\,\tilde{c}_{B}(f_{a})\right]\times
10^{-3}\,.\end{split}$ (121)

Here the difference in the value of $r^{W}_{\Psi}$ between the SM case and the
MSSM case is mainly due to a different running of the gauge couplings and top
Yukawa coupling.

Figure 3: Diagrams for the radiative corrections to the axion-electron
coupling $C_{e}$ in KSVZ-like models.

A particularly interesting aspect of our results is that the axion-electron
coupling $C_{e}$ receives a large radiative correction from the UV parameter
$\tilde{c}_{G}(f_{a})$, which was first shown in 2012.12272  for the SM case
to our knowledge. This has an important implication for KSVZ-like QCD axions.
For KSVZ-like axions, $n_{t}(f_{a})=C_{t}(f_{a})=0$ so that there is no one-
loop level radiative correction from the Yukawa interactions. In the previous
literature, the leading contribution to $C_{e}$ in KSVZ axion model was
thought to be from non-zero $c_{W}$ or $c_{B}$ as in the second diagram of
Fig. 3, or from the axion-pion mixing below the QCD scale as in the third
diagram of Fig. 3 for a minimal KSVZ model with $c_{W}=c_{B}=0$
Srednicki:1985xd ; hep-ph/9306216 . But the results Eq. (120) and Eq. (121)
indicate that there is a larger contribution from the axion-gluon coupling
$c_{G}$. This radiative correction is originated from the first diagram of
Fig. 3 involving the exotic heavy quark (${\cal Q},{\cal Q}^{c}$), gluons, top
quark and Higgs doublet at three-loop level. This process is encoded in
$I_{t}$ and $I_{t}^{\rm SM}$ (Eq. (113) and Eq. (114)) of Eq. (110). Although
it is a three-loop process, it beats the contributions from the second and
third diagrams in Fig. 3 because $\alpha_{s}^{2}\gg\alpha_{2,1}^{2}$ and
$y_{t}\sim{\cal O}(1)$. For instance, for KSVZ-like axions with $m_{\cal
Q}=10^{10}$ GeV, we find

$\displaystyle C_{e}(m_{e})$ $\displaystyle\simeq$
$\displaystyle\Big{[}0.84\,c_{G}-0.03\,c_{G}+0.28\,c_{W}+0.10\,c_{B}\Big{]}\times
10^{-3}\,~{}(\textrm{KSVZ with MSSM})$ $\displaystyle C_{e}(m_{e})$
$\displaystyle\simeq$
$\displaystyle\Big{[}0.83\,c_{G}-0.03\,c_{G}+0.54\,c_{W}+0.13\,c_{B}\Big{]}\times
10^{-3}\,~{}(\textrm{KSVZ with SM}),$ (122)

where the first term in the square bracket denotes the contribution from the
first diagram in Fig. 3, the second is the contribution from the axion-pion
mixing represented by the last diagram in Fig. 3, and the last two terms are
the contributions from the two-loop diagrams involving the electroweak gauge
bosons.

Figure 4: Radiative corrections to the axion couplings to the light quarks and
electron from
$\left\\{C_{t}(f_{a}),\tilde{c}_{G}(f_{a}),\tilde{c}_{W}(f_{a}),\tilde{c}_{B}(f_{a})\right\\}$
for the SM or from
$\left\\{n_{t}(f_{a}),\tilde{c}_{G}(f_{a}),\tilde{c}_{W}(f_{a}),\tilde{c}_{B}(f_{a})\right\\}$
for the MSSM : $\Delta
C_{\Psi}=r_{\Psi}^{t}\left[n_{t}(f_{a})~{}\textrm{or}~{}C_{t}(f_{a})\right]+\sum_{A=G,W,B}r_{\Psi}^{A}\tilde{c}_{A}(f_{a})$.
For the MSSM, we take the parameters $\tan\beta=10$ and $m_{\rm SUSY}=10$ TeV.

##  4 Distinguishing the axions by low energy precision physics

Based on the results of the previous sections, we are going to discuss the
possibility to distinguish the different classes of axion models
experimentally. As discussed in section 2, one can consider three classes of
axion models: KSVZ-like, DFSZ-like, and string-theoretic axions. They may show
different patterns of the low energy axion couplings to the nucleons and
electron after properly taking into account the relevant radiative
corrections. To include string-theoretic axions in a consistent manner, in the
following we discuss the coupling patterns in the MSSM framework which assumes
$N=1$ SUSY at scales around $f_{a}$.

Let us first consider the approximate expression of the low energy axion
couplings defined in Eq. (103) for a specific parameter point with
$f_{a}=10^{10}$ GeV, $m_{\rm SUSY}=10$ TeV and $\tan\beta=10$. Applying the
numerical result in Eq. (121) to Eq. (106) and Eq. (107), we find

$\displaystyle g_{ap}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{p}}{f_{a}}\Big{[}-0.48\,c_{G}+0.90\,C_{u}^{0}-0.38\,C_{d}^{0}\Big{.}$
$\displaystyle\Big{.}\quad\quad~{}-0.37\,n_{t}(f_{a})+\Big{(}8.6\tilde{c}_{G}(f_{a})+0.38\tilde{c}_{W}(f_{a})+0.031\tilde{c}_{B}(f_{a})\Big{)}\times
10^{-3}\Big{]},$ $\displaystyle g_{an}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{n}}{f_{a}}\Big{[}-0.037\,c_{G}-0.38\,C_{u}^{0}+0.90\,C_{d}^{0}\Big{.}$
$\displaystyle\Big{.}\quad\quad~{}+0.38\,n_{t}(f_{a})+\Big{(}11\tilde{c}_{G}(f_{a})+0.0078\tilde{c}_{W}(f_{a})-0.0094\tilde{c}_{B}(f_{a})\Big{)}\times
10^{-3}\Big{]},$ $\displaystyle g_{ae}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{e}}{f_{a}}\Big{[}\,C_{e}^{0}+0.29\,n_{t}(f_{a})+\Big{(}0.81\tilde{c}_{G}(f_{a})+0.28\tilde{c}_{W}(f_{a})+0.10\tilde{c}_{B}(f_{a})\Big{)}\times
10^{-3}\Big{]},$ (123)

where $C_{u}^{0},C_{d}^{0}$ and $C_{e}^{0}$ are the tree level couplings as
defined in Eq. (2), and the contributions from $n_{t}(f_{a})$ and
$\tilde{c}_{A}(f_{a})\,(A=G,W,B)$ represent the radiative corrections. On the
other hand, the axion-photon coupling $g_{a\gamma}$ at low energies below the
QCD scale is given by Eq. (104),

$\begin{split}g_{a\gamma}\simeq\frac{\alpha_{\rm
em}}{2\pi}\frac{1}{f_{a}}\Big{(}c_{W}+c_{B}-1.92c_{G}\Big{)}.\end{split}$
(124)

As we have defined in section 2, we refer to DFSZ-like axions as generic
field-theoretic axions with $C_{\Psi}^{0}={\cal O}(1)~{}(\Psi=u,d,e)$. Thus
for SUSY DFSZ-like axions with $f_{a}=10^{10}$ GeV, $m_{\rm SUSY}=10$ TeV and
$\tan\beta=10$,

$\displaystyle g_{ap}^{\rm DFSZ}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{p}}{f_{a}}\left[-0.48\,c_{G}+0.90\,C_{u}^{0}-0.38\,C_{d}^{0}-0.37\,n_{t}(f_{a})\right]\sim\frac{m_{p}}{f_{a}}{\cal
O}(1),$ $\displaystyle g_{an}^{\rm DFSZ}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{n}}{f_{a}}\left[-0.037\,c_{G}-0.38\,C_{u}^{0}+0.90\,C_{d}^{0}+0.38\,n_{t}(f_{a})\right]\sim\frac{m_{n}}{f_{a}}{\cal
O}(1),$ $\displaystyle g_{ae}^{\rm DFSZ}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{e}}{f_{a}}\left[\,C_{e}^{0}+0.29\,n_{t}(f_{a})\right]\sim\frac{m_{e}}{f_{a}}{\cal
O}(1).$ (125)

Here we note $n_{t}(f_{a})=0$ for the standard DFSZ models of Eq. (17) and Eq.
(19), while it can be non-vanishing for more generic DFSZ-like axions
2101.03173 . In fact, for the standard DFSZ axions,
$n_{t}(f_{a})=\tilde{c}_{A}(f_{a})=0\,(A=G,W,B)$, and the RG running is
present only below the BSM scale after the BSM Higgs doublet or Higgsinos are
integrated out giving rise to a non-zero $C_{t}(m_{\rm BSM})$ or
$\tilde{c}_{W,B}(m_{\rm BSM})$, while the running by $\tilde{c}_{G}$ is
present only after the top quark is integrated out. Thus the RG running effect
is mostly negligible for the standard DFSZ models, while there could be a
sizable RG correction from the top Yukawa coupling for more generic DFSZ-like
axions having a non-zero $n_{t}(f_{a})$.

On the other hand, KSVZ-like axions are defined as generic field-theoretic
axions having vanishing tree level couplings to the SM particles and Higgs
doublets, i.e. $C_{u}^{0}=C_{d}^{0}=n_{t}(f_{a})=0$ (or more generically
$\lesssim{\cal O}\big{(}(g_{\rm GUT}^{2}/8\pi^{2})^{2}\big{)}$ and
$\tilde{c}_{A}(f_{a})=c_{A}~{}(A=G,W,B)$. Thus for SUSY KSVZ-like axions with
$f_{a}=10^{10}$ GeV, $m_{\rm SUSY}=10$ TeV and $\tan\beta=10$,

$\displaystyle g_{ap}^{\rm KSVZ}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{p}}{f_{a}}\left[-0.48c_{G}+(0.38c_{W}+0.031c_{B})\times
10^{-3}\right],$ $\displaystyle g_{an}^{\rm KSVZ}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{n}}{f_{a}}\left[-0.026c_{G}+(0.78c_{W}-0.94c_{B})\times
10^{-5}\right],$ $\displaystyle g_{ae}^{\rm KSVZ}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{e}}{f_{a}}\,(0.81c_{G}+0.28c_{W}+0.10c_{B})\times
10^{-3}.$ (126)

In fact here it is assumed that the mass of the exotic heavy quark ${\cal
Q}+{\cal Q}^{c}$ in the model (20), i.e. $m_{\cal Q}=y_{\cal
Q}f_{a}/\sqrt{2}$, is around $f_{a}$. If $m_{\cal Q}\ll f_{a}$ with $y_{\cal
Q}\ll 1$, $\tilde{c}_{A}(f_{a})=c_{A}-c_{\cal Q}-c_{{\cal Q}^{c}}=0$, and the
RG running of axion couplings starts from the scale $\mu=m_{\cal Q}(\ll
f_{a})$ at which ${\cal Q}+{\cal Q}^{c}$ is integrated out, yielding
$\tilde{c}_{A}(m_{\cal Q})=c_{A}$. For this case, the coefficients in Eq. (4)
depends on $m_{\cal Q}$ rather than $f_{a}$. In the following analysis, we
will also consider the case $m_{\cal Q}\ll f_{a}$.

String-theoretic axions have a distinctive feature that their couplings to
matter fields are suppressed by $1/\tau$ where $\tau$ denotes the moduli
partner whose vacuum expectation value can be identified as the Euclidean
action of the brane instanton coupled to string-theoretic axion. As a
consequence, $C_{\Psi}^{0}$ ($\Psi=u,d,e$) are numerically of ${\cal O}(g_{\rm
GUT}^{2}/8\pi^{2})$ for light string-theoretic axions which couple to the
gluons or the photon, including a QCD axion and an ultralight ALP coupled to
the photon (See Eqs. (38) and (39)). As discussed in section 2, one can
parameterize the string-theoretic axion couplings to matter fields at $\mu\sim
f_{a}$ as

$\displaystyle c_{\phi}(f_{a})=\omega_{\phi}\frac{g_{\rm
GUT}^{2}}{16\pi^{2}},\quad c_{\psi}=\omega_{\psi}\frac{g_{\rm
GUT}^{2}}{16\pi^{2}},$ (127)

where the model-dependent coefficients $\omega_{\phi}$ and $\omega_{\psi}$ are
expected to be of order unity. We then find for string-theoretic axions with
$f_{a}=10^{10}$ GeV, $m_{\rm SUSY}=10$ TeV and $\tan\beta=10$,

$\displaystyle g_{ap}^{\rm string}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{p}}{f_{a}}\Big{[}g_{\rm
GUT}^{2}\Big{(}0.56(\omega_{u^{c}_{1}}+\omega_{H_{u}})+0.32\omega_{Q_{1}}-0.24(\omega_{Q_{3}}+\omega_{u^{c}_{3}}+\omega_{d^{c}_{1}})\Big{)}\times
10^{-2}\Big{.}$
$\displaystyle\Big{.}\quad\quad~{}-0.48c_{G}+(0.38c_{W}+0.031c_{B})\times
10^{-3}\Big{]},$ $\displaystyle g_{an}^{\rm string}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{n}}{f_{a}}\Big{[}g_{\rm
GUT}^{2}\Big{(}0.56(\omega_{d^{c}_{1}}-\omega_{H_{u}})+0.32\omega_{Q_{1}}+0.24(\omega_{Q_{3}}+\omega_{u^{c}_{3}}-\omega_{u^{c}_{1}})\Big{)}\times
10^{-2}\Big{.}$
$\displaystyle\Big{.}\quad\quad~{}-0.026c_{G}+(0.78c_{W}-0.94c_{B})\times
10^{-5}\Big{]},$ $\displaystyle g_{ae}^{\rm string}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{e}}{f_{a}}\Big{[}g_{\rm
GUT}^{2}\Big{(}0.63(\omega_{e^{c}_{1}}+\omega_{L_{1}})-0.44\omega_{H_{u}}+0.19(\omega_{Q_{3}}+\omega_{u^{c}_{3}})\Big{)}\times
10^{-2}\Big{.}$ (128)
$\displaystyle\Big{.}\quad\quad~{}+(0.81c_{G}+0.28c_{W}+0.10c_{B})\times
10^{-3}\Big{]}.$

---
Figure 5:  The predicted ratios of the axion couplings from the minimal QCD
axion models within the MSSM framework (green: DFSZ-like axions, black: KSVZ-
like axions, red: string-theoretic axions). For KSVZ-like axions, the heavy
exotic quark is assumed to have a mass between $10^{-3}f_{a}$ (dashed black)
and $f_{a}$ (solid black). For the MSSM parameters, we take $\tan\beta=10$ and
$m_{\rm SUSY}=10$ TeV.

From the above results, one can see that the couplings show qualitatively
different patterns depending on whether $c_{G}\neq 0$ (as for the QCD axion)
or $c_{G}=0$. Therefore let us consider the two cases separately.

For the case with $c_{G}\neq 0$ that includes the QCD axion, all of the three
classes of axions have a similar value of the coupling ratio
$g_{ap}/g_{a\gamma}$. On the other hand, the coupling ratio
$g_{an}/g_{a\gamma}$ for KSVZ-like QCD axions and string-theoretic QCD axions
is an order of magnitude smaller than that of DFSZ-like QCD axions. So one can
discriminate between those two classes of models with an experimental
measurement of $g_{an}/g_{a\gamma}$. To completely distinguish among the three
classes of QCD axions, one needs to measure another coupling ratio
$g_{ae}/g_{a\gamma}$ as it is predicted to be of different orders of magnitude
depending on the classes of axions. In Fig. 5, we numerically plot the
predicted coupling ratios $g_{an}/g_{a\gamma}$ and $g_{ae}/g_{a\gamma}$ for
each class of QCD axions over $f_{a}$ between $10^{7}$ GeV and $10^{16}$ GeV.
For DFSZ-like QCD axion, we use the model specified in Eq. (18) and Eq. (19)
as a benchmark model. For KSVZ-like QCD axion, we use the model in Eq. (20)
with an $SU(2)_{L}$-singlet exotic heavy quark ${\cal Q}$ carrying a
$U(1)_{Y}$ hypercharge $Y({\cal Q})=-1/3$, which implies
$c_{G}=1,\,c_{W}=0,\,c_{B}=2/3$. Moreover, we assume the exotic heavy quark
has a mass $m_{\cal Q}$ between $10^{-3}f_{a}$ and $f_{a}$, which corresponds
to the scale at which the axion coupling starts to run for this class of
models. For a benchmark model of string-theoretic QCD axions, we use the large
volume scenario model in Eq. (55) with a universal scaling weight
$\omega_{I}=1/2$ as in Eq. (61), and also $g_{\rm GUT}^{2}\simeq 1/2$ with the
GUT relation $c_{G}=c_{W}=(3/5)c_{B}$.

---
Figure 6:  The predicted ratios of the ALP couplings from the minimal axion
models with $c_{W}=1\,(c_{G}=c_{B}=0)$ (left) and $c_{B}=1\,(c_{G}=c_{W}=0)$
(right) within the MSSM framework (green: DFSZ-like axions, black: KSVZ-like
axions, red: string-theoretic axions). For KSVZ-like axions, the heavy color-
singlet exotic fermion charged under $SU(2)_{L}$ (left) or $U(1)_{Y}$ (right)
is assumed to have a mass between $10^{-3}f_{a}$ (dashed black) and $f_{a}$
(solid black). For the MSSM parameters, we take $\tan\beta=10$ and $m_{\rm
SUSY}=10$ TeV.

On the other hand, for axions with $c_{G}=0$, but $c_{W}\neq 0$ and/or
$c_{B}\neq 0$, e.g. an ALP coupled to the photon without a coupling to the
gluons, Eqs. (4)-(4) tell us that all the three low energy axion couplings may
distinguish among the different classes of ALP models. In Fig. 6, we
numerically plot the predicted coupling ratios $g_{aY}/g_{a\gamma}\,(Y=p,n,e)$
for each class of ALP. As an illustrative example, we use the same parameter
values as in Fig. 5 but with different $c_{A}\,(A=G,W,B)$, i.e.
$c_{G}=c_{B}=0$ and $c_{W}=1$ for the left panel, while $c_{G}=c_{W}=0$ and
$c_{B}=1$ for the right panel. For DFSZ-like ALP, a specific model for such
parameter values may include additional heavy exotic quarks that cancel the
gluon anomaly of the model in Eq. (18) and Eq. (19). For KSVZ-like ALP, we use
a similar model as in Eq. (20) with a heavy color-singlet exotic fermion
charged under $SU(2)_{L}$ only (for the left plot) or $U(1)_{Y}$ only (for the
right plot), which is assumed to have a mass between $10^{-3}f_{a}$ and
$f_{a}$. Fig. 6 shows that the axion-proton coupling may be yet hard to
discriminate string-theoretic ALPs from KSVZ-like ALPs for $c_{W}\neq 0$ since
the radiative correction becomes sizable if $f_{a}$ is near the GUT scale.

##  5 Conclusions

Axions are compact scalar fields postulated to solve various issues in
particle physics and cosmology including the strong CP problem and dark
matter. Axions may originate from the phase of complex scalar fields (field-
theoretic axions) or from the zero modes of an antisymmetric tensor ($p$-form)
gauge field (string-theoretic axions) which couples to a $(p-1)$-brane in the
underlying UV theory. Axion couplings which are most relevant for an
experimental detection of axions are those to the photon, nucleons and
electron at scale well below the QCD scale, i.e. $g_{aX}$ ($X=\gamma,p,n,e$)
which are determined mostly by the quantized couplings to the photon and
gluons, as well as the couplings to the light quarks and electron around the
QCD scale. Then, depending on the UV origin of the field variable and the
pattern of low energy couplings, axions can be categorized into three classes:
i) field-theoretic DFSZ-like axions with $R_{0}={\cal O}(1)$, ii) field-
theoretic KSVZ-like axions with $R_{0}\lesssim{\cal O}((g_{\rm
GUT}^{2}/16\pi^{2})^{2})$, and iii) string-theoretic axions with $R_{0}={\cal
O}(g_{\rm GUT}^{2}/16\pi^{2})$, where $g_{\rm GUT}$ is the gauge coupling at
scales around the axion decay constant $f_{a}$, and $R_{0}$ denotes the ratio
(in an appropriate unit) between the coupling to the photon (or gluons) and
the tree-level couplings to the light quarks and electron. Compared to the
other two, DFSZ-like axions have a clearly distinguishable pattern of $g_{aX}$
($X=\gamma,p,n,e$). However it requires a careful analysis of radiative
corrections to the axion couplings to the light quarks and electron to
discriminate string-theoretic axions from KSVZ-like axions by experimental
measurements of $g_{aX}$ or their ratios.

With this motivation, we first performed a generic renormalization group
analysis for the axion couplings to matter fermions and Higgs fields while
taking into account that the Standard Model can be extended to its
supersymmetric extension at a scale below $f_{a}$. We then applied our results
to string-theoretic axions and KSVZ-like axions to examine if they can have a
distinguishable pattern of $g_{aX}$ ($X=\gamma,p,n,e$). We find that
discriminating string-theoretic QCD axion from KSVZ-like QCD axion appears to
be quite challenging, but yet it looks feasible with a significantly different
value of the coupling ratio $g_{ae}/g_{a\gamma}$. For an axion-like particle
(ALP) which does not couple to the gluons, but has a non-zero coupling to the
photon, the prospect for discrimination is much more promising: all of the
three couplings ratios $g_{aY}/g_{a\gamma}$ ($Y=p,n,e$) for string-theoretic
ALP and KSVZ-like ALP differ by about one order of magnitude. We also find
that the coupling of KSVZ-like QCD axion to the electron is dominated by a
three-loop contribution involving the heavy exotic quark, gluons, top quarks
and Higgs doublet, which was not noticed in the previous studies.

## Acknowledgments

This work was supported by IBS under the project code, IBS-R018-D1.

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