March 25, 2022

# Three-loop corrections to the quark and gluon decomposition of the QCD trace
anomaly and their applications

Kazuhiro Tanaka Department of PhysicsDepartment of Physics Juntendo University
Juntendo University Inzai Inzai Chiba 270-1695 Chiba 270-1695 Japan Japan
<EMAIL_ADDRESS>
###### Abstract

In the QCD energy-momentum tensor $T^{\mu\nu}$, the terms that contribute to
physical matrix elements are expressed as the sum of the gauge-invariant quark
part and gluon part. Each part undergoes the renormalization due to the
interactions among quarks and gluons, although the total tensor $T^{\mu\nu}$
is not renormalized thanks to conservation of energy and momentum. We show
that, through the renormalization, each of the quark and gluon parts of
$T^{\mu\nu}$ receives a definite amount of anomalous trace contribution, such
that their sum reproduces the well-known QCD trace anomaly. We provide a
procedure to derive such anomalous trace contribution for each quark/gluon
part to all orders in perturbation theory, and obtain the corresponding
explicit formulas up to three-loop order in the $\overline{\rm MS}$ scheme in
the dimensional regularization. We apply our three-loop formulas of the
quark/gluon decomposition of the trace anomaly to calculate the anomaly-
induced mass structure of nucleons as well as pions. Another application of
our three-loop formulas is a quantitative analysis for the constraints on the
twist-four gravitational form factors of the nucleon, $\bar{C}_{q,g}$.

trace anomaly, energy-momentum tensor, QCD, gravitational form factors

##  1 Introduction

The QCD energy-momentum tensor $T^{\mu\nu}$ is known to receive the trace
anomaly [1] as

$T^{\mu}_{\mu}=\eta_{\mu\nu}T^{\mu\nu}=\frac{\beta(g)}{2g}F^{2}+\left(1+\gamma_{m}(g)\right)m\bar{\psi}\psi\
,$ (1)

representing the broken scale invariance due to the quantum loop effects, with
the beta-function $\beta$ for the QCD coupling constant $g$ and the anomalous
dimension $\gamma_{m}$ for the quark mass $m$. Here, $\eta_{\mu\nu}$ is the
the metric tensor, and $F^{2}$ ($=F_{a}^{\mu\nu}{F_{a}}_{\mu\nu}$) and
$\bar{\psi}\psi$ denote the renormalized composite operators dependent on a
renormalization scale. The symmetric energy-momentum tensor $T^{\mu\nu}$ is
expressed as the sum of the gauge-invariant quark part $T^{\mu\nu}_{q}$ and
gluon part $T^{\mu\nu}_{g}$, as ($D^{\mu}=\partial^{\mu}+igA^{\mu}$,
$R^{(\mu}S^{\nu)}\equiv\left(R^{\mu}S^{\nu}+R^{\nu}S^{\mu}\right)/2$)

$T^{\mu\nu}=\frac{1}{2}\bar{\psi}\gamma^{(\mu}i\overleftrightarrow{D}^{\nu)}\psi+\left(F_{a}^{\mu\rho}{{F_{a}}_{\rho}}^{\nu}+\frac{\eta^{\mu\nu}}{4}F_{a}^{\lambda\rho}{F_{a}}_{\lambda\rho}\right)\equiv
T^{\mu\nu}_{q}+T^{\mu\nu}_{g}\ ,$ (2)

using the QCD equations of motion (EOM), up to the ghost and gauge-fixing
terms that are not relevant for the following discussion. Classically, we
have, ${\eta_{\mu\nu}}T_{q}^{\mu\nu}=m\bar{\psi}\psi$ and
${\eta_{\mu\nu}}T_{g}^{\mu\nu}=0$, up to the terms that vanish by the EOM, but
(1) does not coincide with the quantum corrections to the $m\bar{\psi}\psi$
operator, reflecting that renormalizing the quantum loops and taking the trace
do not commute. We note that the total tensor $T^{\mu\nu}$ of (2) is not
renormalized; it is a finite, scale-independent operator, because of the
energy-momentum conservation, $\partial_{\nu}T^{\mu\nu}=0$, while
$T_{q}^{\mu\nu}$ and $T_{g}^{\mu\nu}$ are not conserved separately and
$T_{q}^{\mu\nu}$ as well as $T_{g}^{\mu\nu}$ is subject to regularization and
renormalization. This fact suggests that each of $T_{q}^{\mu\nu}$ and
$T_{g}^{\mu\nu}$ should receive a definite amount of anomalous trace
contribution, such that their sum reproduces (1). The corresponding trace
anomaly for each quark/gluon part is derived up to two-loop order in [2]. The
extension to the three-loop order is worked out in [3], demonstrating that the
logic to determine the quark/gluon decomposition of the trace anomaly holds to
all orders in perturbation theory. In the MS-like (MS, $\overline{\rm MS}$)
schemes in the dimensional regularization, we obtain

$\eta_{\mu\nu}T_{q}^{\mu\nu}=m\bar{\psi}\psi+\frac{{{\alpha_{s}}}}{{4\pi}}\left({\frac{{{n_{f}}}}{3}{F^{2}}+\frac{{4{C_{F}}}}{3}m\bar{\psi}\psi}\right)+\cdots\
,\;\;\;\;\;\eta_{\mu\nu}T_{g}^{\mu\nu}=\frac{{{\alpha_{s}}}}{{4\pi}}\left({-\frac{{11{C_{A}}}}{6}{F^{2}}+\frac{{14{C_{F}}}}{3}m\bar{\psi}\psi}\right)+\cdots\
,$ (3)

for $n_{f}$ flavor and $N_{c}$ color with $C_{F}=(N_{c}^{2}-1)/(2N_{c})$ and
$C_{A}=N_{c}$, where the ellipses stand for the two-loop (${\cal
O}(\alpha_{s}^{2})$) as well as three-loop (${\cal O}(\alpha_{s}^{3})$)
corrections, whose explicit formulas are presented in [2, 3]. The sum of the
two formulas of (3) coincides with (1) at every order in $\alpha_{s}$
($=g^{2}/(4\pi)$).

##  2 Renormalization mixing at three loop

We sketch how the formulas (3) are obtained. First of all, the renormalization
of $T_{q}^{\mu\nu}$, $T_{g}^{\mu\nu}$ of (2) is not straightforward. Indeed,
$T_{q}^{\mu\nu}$, $T_{g}^{\mu\nu}$ are composed of the twist-two (traceless
part) and twist-four (trace part) operators and the renormalization mixing
between the quark part and gluon part also arises. To treat them, we define a
basis of independent gauge-invariant operators up to twist four,

$\displaystyle
O_{q}=i\bar{\psi}\gamma^{(\mu}\overleftrightarrow{D}^{\nu)}\psi\
,\;\;\;\;\;\;\;\;\;\;\;\;O_{q(4)}=\eta^{\mu\nu}m\bar{\psi}\psi\
,\;\;\;\;\;\;\;\;\;\;\;\;O_{g}=-F^{\mu\lambda}F^{\nu}_{\ \lambda}\
,\;\;\;\;\;\;\;\;\;\;\;\;O_{g(4)}=\eta^{\mu\nu}F^{2}\ ,$ (4)

and the corresponding bare operators, $O_{k}^{B}$. The renormalization
constants are introduced as

$\displaystyle
O_{g}=Z_{T}O_{g}^{B}+Z_{M}O_{g(4)}^{B}+Z_{L}O_{q}^{B}+Z_{S}O_{q(4)}^{B}\
,\;\;\;\;\;\;O_{q}=Z_{\psi}O_{q}^{B}+Z_{K}O_{q(4)}^{B}+Z_{Q}O_{g}^{B}+Z_{B}O_{g(4)}^{B}\
,$ (5) $\displaystyle O_{g(4)}=Z_{F}O_{g(4)}^{B}+Z_{C}O_{q(4)}^{B}\
,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;O_{q(4)}=O_{q(4)}^{B}\
,$ (6)

where, for simplicity, the mixing with the EOM operators as well as the BRST-
exact operators is not shown, as their physical matrix elements vanish and
they do not affect our final result [4]. Here, $O_{g}$, as well as $O_{q}$, is
a mixture of the twist-two and -four operators, and the corresponding twist-
four components receive the contributions of the twist-four operators
$O_{g(4)}$ and $O_{q(4)}$. The two formulas of (6) reflect, respectively, that
the twist-four operator $O_{g(4)}$ mixes with itself and another twist-four
operator $O_{q(4)}$, and that $O_{q(4)}$ is renormalization group
(RG)-invariant (see [5, 2, 3]).

Subtracting the traces from both sides of the equations (5), $O_{k}$ and
$O_{k}^{B}$ are, respectively, replaced by the corresponding twist-two parts,
$O_{k(2)}$ and $O^{B}_{k(2)}$, such that the twist-four contributions drop
out. The renormalization constants $Z_{T},Z_{L},Z_{\psi}$ and $Z_{Q}$ remain
in the resulting equations that represent the flavor-singlet mixing of the
twist-two spin-2 operators, and thus can be determined by the second moments
of the DGLAP splitting functions which are known up to the three-loop accuracy
[6].

For the renormalization mixing (6) at twist four, the Feynman diagram
calculation of $Z_{F}$ and $Z_{C}$ is available to the two-loop order [5].
Moreover, it is shown [3] that the constraints imposed by the RG invariance of
(1) allow to determine the form of $Z_{F}$ as well as $Z_{C}$ in the MS-like
schemes, completely from $\beta(g)$ and $\gamma_{m}(g)$, which are known to
five- and four-loop order in the literature, respectively.

Therefore, six renormalization constants $Z_{T},Z_{L},Z_{\psi},Z_{Q},Z_{F}$
and $Z_{C}$ among ten constants arising in (5), (6) are available to a certain
accuracy in the MS-like schemes, and they take the form,

$Z_{X}=\left(\delta_{X,T}+\delta_{X,\psi}+\delta_{X,F}\right)+\frac{a_{X}}{\epsilon}+\frac{b_{X}}{\epsilon^{2}}+\frac{c_{X}}{\epsilon^{3}}+\cdots\
,$ (7)

in the $d=4-2\epsilon$ spacetime dimensions with $X=T,L,\psi,Q,F$, and $C$;
here, $a_{X},b_{X},c_{X},\ldots,$ are the constants given as power series in
$\alpha_{s}$, and $\delta_{X,X^{\prime}}$ denotes the Kronecker symbol.
However, $Z_{M}$, $Z_{S}$, $Z_{K}$ and $Z_{B}$ still remain unknown. It is
shown [3] that these four renormalization constants can be determined to the
accuracy same as the renormalization constants (7), by invoking that they
should also obey (7) with $X=M,S,K,B$, and that the RHS of the formulas of (5)
are, in total, UV-finite. Thus, all the renormalization constants in (5), (6)
are determined up to the three-loop accuracy, and this result allows us to
derive the three-loop formulas [3] for (3), by calculating the trace part of
(5).

##  3 Anomaly-induced mass structure of hadrons

The QCD trace anomaly (1) signals the generation of a nonperturbative mass
scale, say, the nucleon mass $m_{N}$: Taking the matrix element of (1) in
terms of a hadron state $|h(p)\rangle$ with the 4-momentum $p^{\mu}$ as
$p^{2}=m_{h}^{2}$, and using the fact that $\langle
h(p)|T^{\mu\nu}|h(p)\rangle=2p^{\mu}p^{\nu}$, we obtain

$2m_{h}^{2}=\langle h(p)|T^{\mu}_{\mu}|h(p)\rangle=\langle
h(p)|\left(\frac{\beta(g)}{2g}F^{2}+\left(1+\gamma_{m}(g)\right)m\bar{\psi}\psi\
\right)|h(p)\rangle\ ,$ (8)

so that almost all of the hadron mass $m_{h}$ could be attributed to the
quantum loop effects in QCD responsible for the trace anomaly. Based on (8)
for the nucleon ($h=N$), it is frequently argued that the entire mass $m_{N}$
comes from gluons in the chiral limit. However, the partition of QCD loop
effects as (3) shows that the latter statement would not be suitable: Indeed,
(3) allows us to separate (8) as

$2m_{h}^{2}=\langle
h(p)|\eta_{\lambda\nu}T^{\lambda\nu}_{g}(\mu)|h(p)\rangle+\langle
h(p)|\eta_{\lambda\nu}T^{\lambda\nu}_{q}(\mu)|h(p)\rangle\ ,$ (9)

and, evaluating (3) with $N_{c}=3$, $n_{f}=3$ at the renormalization scale
$\mu$, one finds [3]

$\displaystyle\eta_{\lambda\nu}T^{\lambda\nu}_{g}(\mu)=\left(-0.437676\alpha_{s}(\mu)-0.261512\alpha_{s}^{2}(\mu)-0.183827\alpha_{s}^{3}(\mu)\right)F^{2}(\mu)+\cdots\
,$ (10)
$\displaystyle\eta_{\lambda\nu}T^{\lambda\nu}_{q}(\mu)=\left(0.0795775\alpha_{s}(\mu)+0.0588695\alpha_{s}^{2}(\mu)+0.0216037\alpha_{s}^{3}(\mu)\right)F^{2}(\mu)+\cdots\
,$ (11)

with the ellipses associated with the operator $m\bar{\psi}\psi$. The nucleon
($h=N$) in the chiral limit gives

$\frac{\langle
N(p)|\eta_{\lambda\nu}T^{\lambda\nu}_{q}(\mu)|N(p)\rangle}{\langle
N(p)|\eta_{\lambda\nu}T^{\lambda\nu}_{g}(\mu)|N(p)\rangle}=-0.181818-0.0258682\alpha_{s}(\mu)+0.0424613\alpha_{s}^{2}(\mu)\
,$ (12)

where $\frac{n_{f}}{3}/(-\frac{{11{C_{A}}}}{6})\cong-0.181818$. Eqs. (9)-(12),
combined with $\langle N(p)|F^{2}|N(p)\rangle<0$, show that the gluon- and
quark-loop effects make the nucleon mass heavy and light, respectively, with
the magnitude of the former being five times larger than that of the latter.
From (12), the $\mu$-dependence of this result for the relative size of the
gluon/quark loop effects in the chiral limit is rather weak. It is also worth
noting that the total sum (9) of (10) and (11) allows us to constrain the
matrix element of $F^{2}$ as $\langle N(p)|F^{2}(\mu=1~{}{\rm
GeV})|N(p)\rangle\simeq-8.61m_{N}^{2}$, using
$\alpha_{s}(1~{}\rm{GeV})=0.47358\ldots$, as the three-loop running coupling
constant in the $\overline{\rm MS}$ scheme with $\alpha_{s}(M_{Z})=0.1181$. We
note that the neglected four-loop contributions are expected to produce
corrections less than ten percent because $\alpha^{3}_{s}(1~{}\rm{GeV})\simeq
0.1$.

Next, we consider the pion case, for which the PCAC relation,
$-\left(m_{u}+m_{d}\right)\langle
0|\left(\bar{u}u+\bar{d}d\right)|0\rangle=2f_{\pi}^{2}m_{\pi}^{2}$, with
$f_{\pi}$ the pion decay constant, indicates $m_{\pi}^{2}\sim m$ as $m\to 0$.
Eq. (8) for the pion ($h=\pi$) implies,
$\left.\bigl{\langle}\pi(p)\bigl{|}F^{2}\bigr{|}\pi(p)\bigr{\rangle}\right|_{m\to
0}=0$, in the chiral limit $m\to 0$. Eq. (8) also gives the relation among the
${\cal O}(m)$ terms: When the substitution,
$\bigr{|}\pi(p)\bigr{\rangle}\to\bigr{|}\pi(p)\bigr{\rangle}_{0}+\bigr{|}\pi(p)\bigr{\rangle}_{1}+\ldots$,
is made, where
$\bigr{|}\pi(p)\bigr{\rangle}_{0}\equiv\left.\bigr{|}\pi(p)\bigr{\rangle}\right|_{m=0}$
and $\bigr{|}\pi(p)\bigr{\rangle}_{1}$ is the ${\cal O}(m^{1})$-term, we have,
$\bigl{\langle}\pi(p)\bigl{|}m\bar{\psi}\psi\bigr{|}\pi(p)\bigr{\rangle}\to\
_{0}\bigl{\langle}\pi(p)\bigl{|}m\bar{\psi}\psi\bigr{|}\pi(p)\bigr{\rangle}_{0}$
and $\bigl{\langle}\pi(p)\bigl{|}F^{2}\bigr{|}\pi(p)\bigr{\rangle}\to\
_{0}\bigl{\langle}\pi(p)\bigl{|}F^{2}\bigr{|}\pi(p)\bigr{\rangle}_{1}+\
_{1}\bigl{\langle}\pi(p)\bigl{|}F^{2}\bigr{|}\pi(p)\bigr{\rangle}_{0}$, up to
the corrections of ${\cal O}(m^{2})$. The pion mass can also be calculated as
the mass shift due to the ordinary first-order perturbation theory in the
quark mass term in the QCD Hamiltonian, as [7]

$m_{\pi}^{2}=\
_{0}\bigl{\langle}\pi(p)\bigl{|}m\bar{\psi}\psi\bigr{|}\pi(p)\bigr{\rangle}_{0}\
,$ (13)

and, combining this with the above results for the ${\cal O}(m)$ terms in (8),
we obtain

$\bigl{(}1-\gamma_{m}(g)\bigr{)}m_{\pi}^{2}=\bigl{\langle}\pi(p)\bigl{|}\frac{\beta(g)}{2g}F^{2}\bigr{|}\pi(p)\bigr{\rangle}\
,$ (14)

to the ${\cal O}(m)$ accuracy. Therefore, up to the corrections of ${\cal
O}(m^{2})$, the terms associated with the $F^{2}$ operator and the
$m\bar{\psi}\psi$ operator in the RHS of (8) contribute to $m_{\pi}^{2}$
according to the relative weights, $\left(1-\gamma_{m}(g)\right)$ and
$\left(1+\gamma_{m}(g)\right)$, respectively; here,
$\gamma_{m}(g)=0.63662\alpha_{s}+0.768352\alpha_{s}^{2}+0.801141\alpha_{s}^{3}\simeq
0.559$, at the three-loop accuracy. Substituting (13) and (14) into (9) with
(3), we find

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{1}{2m_{\pi}^{2}}\bigl{\langle}\pi(p)\bigl{|}\eta_{\lambda\nu}T^{\lambda\nu}_{g}(\mu)\bigr{|}\pi(p)\bigr{\rangle}=0.611111-0.12215\alpha_{s}(\mu)-0.124659\alpha_{s}^{2}(\mu)-0.0430357\alpha_{s}^{3}(\mu)\
,$ (15)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{1}{2m_{\pi}^{2}}\bigl{\langle}\pi(p)\bigl{|}\eta_{\lambda\nu}T^{\lambda\nu}_{q}(\mu)\bigr{|}\pi(p)\bigr{\rangle}=0.388889+0.12215\alpha_{s}(\mu)+0.124659\alpha_{s}^{2}(\mu)+0.0430357\alpha_{s}^{3}(\mu)\
,$ (16)

and, using $\alpha_{s}(1~{}\rm{GeV})=0.47358\ldots$, we obtain

$\frac{1}{2m_{\pi}^{2}}\bigl{\langle}\pi(p)\bigl{|}\eta_{\lambda\nu}T^{\lambda\nu}_{g}(\mu=1~{}{\rm
GeV})\bigr{|}\pi(p)\bigr{\rangle}=0.521\
,\;\;\;\frac{1}{2m_{\pi}^{2}}\bigl{\langle}\pi(p)\bigl{|}\eta_{\lambda\nu}T^{\lambda\nu}_{q}(\mu=1~{}{\rm
GeV})\bigr{|}\pi(p)\bigr{\rangle}=0.479\ ,$ (17)

which hold to the ${\cal O}(m)$ accuracy. Again, the $\mu$-dependence of the
result is rather weak, but (17) shows the structure different completely from
the nucleon case: Both the gluon- and quark-loop effects produce roughly half
of the pion mass. This may be a particular nature as a Nambu-Goldstone boson.

##  4 Anomaly constraints on the nucleon’s twist-four gravitational form
factor

The nucleon matrix element of each term in (2) is parameterized as
($|p\rangle\equiv|N(p)\rangle$)

$\langle
p^{\prime}|T_{q,g}^{\mu\nu}|p\rangle=\bar{u}(p^{\prime})\Bigl{[}A_{q,g}(t)\gamma^{(\mu}\bar{P}^{\nu)}+B_{q,g}(t)\frac{\bar{P}^{(\mu}i\sigma^{\nu)\alpha}\Delta_{\alpha}}{2m_{N}}+D_{q,g}(t)\frac{\Delta^{\mu}\Delta^{\nu}-\eta^{\mu\nu}t}{4m_{N}}+\bar{C}_{q,g}(t)m_{N}\eta^{\mu\nu}\Bigr{]}u(p)\
,$ (18)

in terms of the gravitational form factors $A_{q,g}(t),B_{q,g}(t),D_{q,g}(t)$,
and $\bar{C}_{q,g}(t)$ [9, 10], where $\Delta=p^{\prime}-p$,
$\bar{P}=\left(p+p^{\prime}\right)/2$, $t=\Delta^{2}$,
$\bar{P}^{2}=m_{N}^{2}-t/4$, and $u(p)$ is the nucleon spinor. $A_{q,g}(t)$
and $B_{q,g}(t)$ are familiar twist-two form factor, obeying the forward
($t\to 0$) sum rules, $A_{q}(0)+A_{g}(0)=1$,
$[A_{q}(0)+B_{q}(0)+A_{g}(0)+B_{g}(0)]/2=1/2$, representing a sharing of the
total momentum and total angular momentum by the quarks/gluons. $D_{q,g}(t)$,
$\bar{C}_{q,g}(t)$ have also received considerable attention recently [8, 9,
10, 11]: $D_{q,g}(t)$ are related to the so-called D term, $D\equiv
D_{q}(0)+D_{g}(0)$ [9]. For $\bar{C}_{q,g}(t)$, exact manipulations for the
divergence of (2) yield the operator identities,
$\partial_{\nu}T_{q}^{\mu\nu}=\bar{\psi}g{F^{\mu\nu}}{\gamma_{\nu}}\psi$,
$\partial_{\nu}T_{g}^{\mu\nu}=-F_{a}^{\mu\nu}D_{ab}^{\rho}F_{\rho\nu}^{b}$, up
to the terms that vanish by the EOM, so that $\bar{C}_{q}(t)=-\bar{C}_{g}(t)$,
$\bar{C}_{q}(t)\propto\langle
p^{\prime}|\bar{\psi}igF^{\mu\nu}\gamma_{\nu}\psi|p\rangle$, and
$\bar{C}_{g}(t)\propto\langle
p^{\prime}|F_{a}^{\mu\nu}iD^{\rho}_{ab}F^{b}_{\rho\nu}|p\rangle$ hold [10],
showing that $\bar{C}_{q,g}(t)$ are of twist four. We note that
$\bar{C}_{q,g}(t)$ are relevant to the force distribution inside the nucleon
[9, 12] and the nucleon’s transverse spin sum rule [13].

To see the consequence of (3) on $\bar{C}_{q,g}(t)$, we take the trace of the
forward limit, $\Delta^{\mu}\to 0$, of (18):

${\bar{C}}_{q}(t=0)=-{\bar{C}}_{g}(t=0)=-\frac{1}{4}A_{q}(t=0)+\frac{1}{8m_{N}^{2}}\langle
p|\eta_{\lambda\nu}T_{q}^{\lambda\nu}|p\rangle\ ,$ (19)

and substitute (3) into the second term, and the three-loop DGLAP evolution
into $A_{q}(\mu)\equiv A_{q}\left(t=0,\mu\right)$ as a flavor-singlet spin-2
operator renormalized at the scale $\mu$.
$\bar{C}_{q}(\mu)\equiv\bar{C}_{q}(t=0,\mu))$ of (19) reads [2]

$\bar{C}_{q}(\mu)=-\frac{1}{4}\left(\frac{n_{f}}{4C_{F}+n_{f}}+\frac{2n_{f}}{3\beta_{0}}\right)+\left(\frac{2n_{f}}{3\beta_{0}}+1\right)\frac{\left\langle
p\right|m\bar{\psi}\psi\left|p\right\rangle}{8m_{N}^{2}}-\frac{4C_{F}A_{q0}+n_{f}\left(A_{q0}-1\right)}{4\left(4C_{F}+n_{f}\right)}\left(\frac{\alpha_{s}\left(\mu\right)}{\alpha_{s}(\mu_{0})}\right)^{\frac{8C_{F}+2n_{f}}{3\beta_{0}}}+\cdots\
,$ (20)

with $\beta_{0}\equiv(11C_{A}-2n_{f})/3$, where $\mu_{0}$ is a certain
starting scale, $A_{q0}\equiv A_{q}(\mu_{0})$, and $\left\langle
p\right|F^{2}\left|p\right\rangle$ has been eliminated in favor of the nucleon
mass $m_{N}$ using (8). Here, explicitly shown are the leading order (LO)
terms that are derived from the contributions at the one-loop accuracy in
(19); the ellipses denote the NLO and NNLO terms derived from the two- and
three-loop contributions in (19), respectively. The first few terms
independent of $\mu$ in (20) represents the asymptotic value which is
completely determined by the values of $N_{c}$ and $n_{f}$ in the chiral
limit. Substituting $N_{c}=3$ and $n_{f}=3$, we obtain

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\left.{\bar{C}_{q}(\mu)}\right|_{{n_{f}}=3}}=\left.-0.146+0.306\frac{{\langle
p|m\bar{q}q\left|p\right\rangle}}{{2{m_{N}^{2}}}}+\left(0.09-0.25A_{q0}\right){\left({\frac{{{\alpha_{s}}\left(\mu\right)}}{{{\alpha_{s}}({\mu_{0}})}}}\right)^{\frac{{50}}{{81}}}}+{\alpha_{s}}(\mu)\
\right(0.006$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\left.0.08\frac{{\langle
p|m\bar{q}q\left|p\right\rangle}}{{2{m_{N}^{2}}}}{\rm{}}+\left(0.013-0.035A_{q0}\right){{\left({\frac{{{\alpha_{s}}\left(\mu\right)}}{{{\alpha_{s}}({\mu_{0}})}}}\right)}^{\frac{{50}}{{81}}}}+\left(0.035A_{q0}-0.028\right){{\left({\frac{{{\alpha_{s}}\left(\mu\right)}}{{{\alpha_{s}}({\mu_{0}})}}}\right)}^{-\frac{{31}}{{81}}}}\right)+\cdots\
,$ (21)

where the NLO as well as LO terms are explicitly shown, and the ellipses stand
for the NNLO terms. For illustration, we plot (21) as a function of $\mu$ in
the chiral limit: Fig. 1(a) shows the results up to the LO, NLO, and NNLO
accuracy; the NLO as well as NNLO corrections give a few percent level
effects, reflecting the small numerical coefficients arising in (21), and the
NLO and NNLO corrections tend to cancel; the apprach to the asymptotic value,
$-0.146$, is quite slow. The NNLO result is separated, in Fig. 1(b), into the
individual contributions of each term in (19), the first (twist-2) term and
the second (anomaly) term; both twist-2 and anomaly terms produce the
important effects.

To summarize, the quark/gluon decomposition of the QCD trace anomaly allows us
to study the hadrons from new aspects, revealing e.g., quite different pattern
between the nucleon and the pion.

Figure 1: The nucleon’s gravitational form factor ${\bar{C}}_{q}(t=0,\mu)$ at
the NNLO (3-loop) accuracy: (a) the LO, NLO, and NNLO calculations; (b) the
contributions of the first (twist-2) and second (anomaly) terms of (19).

Acknowledgments: This work was supported by JSPS KAKENHI Grant Number
JP19K03830.

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