# Partial wave analysis for the in-hadron condensate

Pianpian Qin  Department of Physics and Chongqing Key Laboratory for Strongly
Coupled Physics, Chongqing University, Chongqing 401331, China  Zhan Bai
Insitute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China  Muyang Chen  Department of Physics, Hunan Normal University, Changsha
410081, China  Si-xue Qin<EMAIL_ADDRESS>Department of Physics and Chongqing
Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing
401331, China

###### Abstract

In-hadron condensates, defined as the scalar form factors at zero-momentum
transfer, are investigated for flavor-symmetric mesons in pseudoscalar and
vector channels under the rainbow-ladder truncation within the Dyson-Schwinger
equations framework. We confirm the efficiency of the in-hadron condensates in
describing the effects of dynamical chiral symmetry breaking from both global
and structural perspectives by comparing the meson masses, the dimensionless
in-hadron condensates, and the partial wave decompositions of in-hadron
condensates as functions of current-quark mass. From partial wave analysis, we
infer $\pi(1300)$ is a radial excitation dominated by $s$ waves and
$\rho(1450)$ is not a $p$ wave-dominated excitation. This work provides a new
insight into the studies of hadron properties with partial wave analysis for
the in-hadron condensates.

##  I Introduction

The chiral symmetry and its breaking are essential in quantum chromo-dynamics
(QCD) and play an important role in hadron physics Chang  _et al._ (2007);
Mitter  _et al._ (2015); Braun  _et al._ (2016); Aguilar and Papavassiliou
(2011). Due to the non-vanishing current-quark mass, the chiral symmetry is
broken explicitly. The current mass of $u/d$ quark is $3$-$5$ MeV. However,
the constitute mass of $u/d$ quark inside the nucleon is $300$-$500$ MeV. It
is believed that this large mass gap stems from the so-called dynamical chiral
symmetry breaking Roberts (2020); Roberts  _et al._ (2021).

Theoretical studies on dynamical chiral symmetry breaking require non-
perturbative tools, such as effective field models Buballa (2005); Ratti  _et
al._ (2006); Song  _et al._ (2021), functional renormalization group method
Pawlowski (2007); Schaefer and Wambach (2008), Dyson-Schwinger equations
(DSEs) approach Bashir  _et al._ (2012); Roberts and Schmidt (2000); Alkofer
and von Smekal (2001); Maris and Roberts (2003); Fischer (2006), and lattice
QCD Montvay and Münster (1997); Ding and et al. (2019). In particular,
starting from the first principle, DSEs approach preserves the dynamical
chiral symmetry breaking and the quark confinement of QCD simultaneously. The
DSEs approach has been successfully applied to study the hadron properties
Roberts and Williams (1994); Kekez and Klabučar (1996); Aguilar and et al.
(2019); Barabanov and et al. (2021), the hot and dense nuclear matter Fischer
(2019); Gomes (2021); Isserstedt  _et al._ (2021), and the QCD phase
transitions Qin  _et al._ (2011a); Fischer  _et al._ (2011); Gao and Liu
(2016).

With the DSEs approach, the effects of dynamical chiral symmetry breaking can
be reflected in observable quantities such as quark mass function Miransky
(1985); Fischer and Alkofer (2003), sigma-term Hannah (1999); Flambaum  _et
al._ (2006), and hadron decay constants Maris  _et al._ (1998). The vacuum
quark condensate is the order parameter of dynamical chiral symmetry breaking,
which can measure the effects more directly Maris and Roberts (1997); Zong
_et al._ (2003). However, the vacuum quark condensate is not well defined
since it badly diverges at finite current-quark mass Zong  _et al._ (2003);
Chen  _et al._ (2021). Besides, quarks are confined inside hadrons. The
confinement effect is not considered in vacuum quark condensate, which might
be related to the large cosmological constant Brodsky  _et al._ (2010, 2012).

In Refs. Maris and Roberts (1997); Maris  _et al._ (1998), the in-hadron
condensate is first proposed from the Gell-Mann-Oakes-Renner (GMOR) relation
Gell-Mann  _et al._ (1968). This in-hadron order parameter of dynamical chiral
symmetry breaking is finite at non-vanishing current-quark mass and might
reduce the cosmological constant to the observed value Brodsky and Shrock
(2011). However, this original definition is only valid in the pseudoscalar
channel and is hard to be extended to other channels. In Ref. Chang  _et al._
(2012), the scalar form factor at zero-momentum transfer is proved as an
equivalent quantity to measure the effects of dynamical chiral symmetry
breaking, which can be easily extended to all hadrons Flambaum  _et al._
(2006). Besides, it can also describe the responses of hadron masses to the
current-quark mass, which can help to place constraints on the fundamental
constants of nature by using observational data Webb  _et al._ (1999).

The scalar form factor at zero-momentum transfer, named in-hadron condensate
hereafter, can not only measure the global effects of dynamical chiral
symmetry breaking, but also provide a structural perspective for the effects
of dynamical chiral symmetry breaking. It is known that hadrons have rich and
complicated Dirac tensor structures constrained by Poincar$\acute{e}$
covariance Hilger  _et al._ (2017); Bhagwat  _et al._ (2007); Xing  _et al._
(2021). By organizing the Dirac tensors with respect to spin and orbital
angular momentum of quarks, partial wave analysis can be used to distinguish
the contributions of corresponding partial waves to the in-hadron condensates
Macfarlane (1962); Eichmann  _et al._ (2016).

In this work, we solve the gap equation for quark propagator, the homogeneous
Bethe-Salpeter (BS) equation for BS amplitude, and the inhomogeneous BS
equation for scalar vertex within the DSEs framework under the rainbow-ladder
(RL) truncation scheme. To measure the global effects of dynamical chiral
symmetry breaking, the hadron masses and the in-hadron condensates are
calculated for the ground and first-excited states of flavor-symmetric mesons
in pseudoscalar and vector channels. Afterwards, the partial wave analyses are
performed to the in-hadron condensates to provide a structural perspective for
the effects of dynamical chiral symmetry breaking.

This paper is organized as follows. In Sec. II and III , the definition of in-
hadron condensate and the theoretical framework of DSEs approach are
introduced respectively. In Sec. IV, the calculated results and discussions
are presented. Finally, a summary is given in Sec. V.

##  II  Quark Condensates: from Vacuum to in-Hadron

In the chiral limit, the vacuum quark condensate is defined as

$-\langle\bar{q}q\rangle^{\textrm{vac}}_{0}=Z_{4}~{}\text{tr}\int_{q}^{\Lambda}S_{\hat{m}=0}(q)\,,$
(1)

where $S_{\hat{m}=0}(q)$ is the quark propagator in the chiral limit with $q$
the momentum and $Z_{4}$ is the corresponding renormalization constant.
$\int_{q}^{\Lambda}$ represents a Poincar$\acute{e}$ invariant regularization
of the four-dimensional integral with $\Lambda$ the regularization mass-scale.
The trace is over color and Dirac space.

Many successes have been achieved in describing the effects of dynamical
chiral symmetry breaking by using the vacuum quark condensate. Nevertheless,
it suffers from some problems. For example, the vacuum quark condensate is
badly divergent for finite current-quark mass. Although this divergence can be
partly eliminated by using subtraction schemes, ambiguities still exist
especially at large current-quark mass Chang  _et al._ (2007); Gao and Liu
(2016); Chen  _et al._ (2021); Williams  _et al._ (2007); Braun  _et al._
(2020); Bazavov and et al. (2012); Gao  _et al._ (2021). Besides, the
cosmological constant derived from the vacuum quark condensate is about
$10^{46}$ times larger than the observed value Brodsky and Shrock (2011). It
is argued that the quark condensate can be modified by the strong interaction
inside the hadrons, where the vacuum quark condensate fails to provide a
reasonable and realistic measure Brodsky  _et al._ (2012).

To settle these problems, it is natural to generalize the definition of quark
condensate from vacuum to in-hadron. As pion is the lightest hadron and the
Goldstone boson associated with dynamical chiral symmetry breaking, the in-
pion condensate is first investigated. In Ref. Maris  _et al._ (1998), the
Gell-Mann-Oakes-Renner (GMOR) relation Gell-Mann  _et al._ (1968) is
reconsidered, which connects the pion decay constant $f_{\pi}$ and mass
$M_{\pi}$ with the vacuum quark condensate

$f_{\pi}^{2}M_{\pi}^{2}\approx-(m_{u}+m_{d})\left\langle\bar{q}q\right\rangle^{\textrm{vac}}_{0}\,,$
(2)

where $m_{u/d}$ is the current-quark mass. $f_{\pi}$ is related to $\langle
0|\bar{q}\gamma_{5}\gamma_{\mu}q|\Pi\rangle$ with
$\left.\left.\right|\Pi\right\rangle$ the pion state. Alternatively, the
quantity in the left hand side of Eq. (2) can be obtained with the decay
constant $\rho_{\pi}$ by equating pole terms of the corresponding axial and
pseudoscalar vertices in axial-vector Ward Takahashi identity (AV-WTI) Maris
_et al._ (1998),

$f_{\pi}^{2}M_{\pi}^{2}=(m_{u}+m_{d})f_{\pi}\rho_{\pi}\,,$ (3)

where $\rho_{\pi}$ is defined as $i\rho_{\pi}=-\langle
0|\bar{q}i\gamma_{5}q|\Pi\rangle$.

Comparing Eqs. (2) and (3), one can naturally define the in-pion condensate
Maris and Roberts (1997); Brodsky  _et al._ (2010); Chang  _et al._ (2012)

$-\left\langle\bar{q}q\right\rangle_{\pi}:=f_{\pi}\rho_{\pi}\,.$ (4)

This definition is valid for non-vanishing current-quark mass and can return
to the vacuum quark condensate in the chiral limit as long as the chiral limit
residue of the bound state pole in the pseudoscalar vertex is defined as Maris
_et al._ (1998)

$\rho_{\pi}^{0}=-\frac{1}{f_{\pi}^{0}}\left\langle\bar{q}q\right\rangle^{\textrm{vac}}_{0}\,,$
(5)

where $f_{\pi}^{0}$ is the value of $f_{\pi}$ in the chiral limit. The in-pion
condensate has been used to measure the effects of dynamical chiral symmetry
breaking inside the pion Maris and Roberts (1997); Maris  _et al._ (1998);
Brodsky  _et al._ (2010). This definition in Eq. (4) is valid only for
pseudoscalar mesons and is hard to be extended to other hadrons Maris and
Roberts (1997).

In Ref. Chang  _et al._ (2012), it is proved that the in-pion condensate can
also be represented through the scalar form factor at zero-momentum transfer
$Q^{2}=0$

$S_{\pi}(0)=-\left\langle\Pi\left|\frac{1}{2}(\bar{u}u+\bar{d}d)\right|\Pi\right\rangle\,.$
(6)

Besides, the scalar form factor at zero-momentum transfer can be extended to
other channels to measure the effects of dynamical chiral symmetry breaking
inside the corresponding hadrons. Generally, the scalar form factor at zero-
momentum transfer for the flavor-symmetric meson can be represented with the
expectation value of the operator $\bar{q}q$ in the meson state
$\left.\left.\right|P\right\rangle$ Chang  _et al._ (2012)

$S_{P}(0)=-\left\langle P\left|\bar{q}q\right|P\right\rangle\,,$ (7)

where $P$ is the meson momentum and is related to the meson mass $M_{P}$ via
the on-shell condition $P^{2}=-M_{P}^{2}$. Constrained by the AV-WTI, the
scalar form factor is related to the hadron mass $M_{P}$ and the current-quark
mass $m_{q}$ by Flambaum  _et al._ (2006); Ren  _et al._ (2015)

$S_{P}(0)=M_{P}\frac{\partial M_{P}}{\partial m_{q}}\,.$ (8)

This relation can be viewed as a consequence of the Feynman-Hellmann theorem
Feynman (1939). It is found that the dimensionless quantity $S_{P}(0)/M_{P}$
is equivalent to the response of hadron mass to the current-quark mass,
therefore it is a measure of the global effects of dynamical chiral symmetry
breaking.

Within the DSEs framework, the quantity $S_{P}(0)$ can also be calculated from
the loop integration

$S_{P}(0)=\text{tr}\int_{q}^{\Lambda}\bar{\Gamma}_{P}(q;-P)S(q_{+})\Gamma_{s}(q_{+};0)S(q_{+})\Gamma_{P}(q;P)S(q_{-})\,,$
(9)

where $S(q_{\pm})$ are the renormalized dressed quark propagators with
momentum $q_{\pm}=q\pm\frac{1}{2}P$. $\Gamma_{P}(q;P)$ represents the BS
amplitude of the meson and $\Gamma_{s}(q_{+};0)$ is the scalar vertex with
zero momentum. The corresponding Feynman diagram is shown in Fig. 1. It is
obvious that Eq. (9) can be used to analyze the partial wave contributions of
the BS amplitudes to the in-hadron condensates, which makes it feasible to
probe the effects of dynamical chiral symmetry breaking from a structural
perspective.

Figure 1: The Feynman diagram for the in-hadron condensate in Eq. (9). $P$ and
$q_{\pm}$ are the momenta of the mesons and quark propagators. $Q$ is the
transfer momentum with $Q^{2}=0$.

##  III Dyson-Schwinger Equation Approach

The equations of motion for Green functions of QCD fields are described by a
set of infinite coupled equations, i.e., DSEs. The accurate solutions of the
DSEs are impossible to obtain. To solve the DSEs in practice, the truncation
schemes as well as the interaction models are necessary. In this work, the RL
truncation in Ref. Munczek (1995); Bender  _et al._ (1996) and a simplified
version of the interaction in Ref. Qin  _et al._ (2011b) are employed to solve
the quark propagator $S(q_{\pm})$, the BS amplitude $\Gamma_{P}(q;P)$, and the
scalar vertex $\Gamma_{s}(q_{+};0)$ in Eq. (9).

###  III.1 The quark propagator

The Dyson-Schwinger equation for quark propagator is known as the gap
equation, which reads Roberts and Williams (1994)

$S^{-1}(q)=Z_{2}(i\not{q}+Z_{m}m_{q})+\Sigma(q)\,,$ (10)

where $\Sigma(q)$ is the self-energy

$\Sigma(q)=g^{2}Z_{1}\int_{k}^{\Lambda}D_{\mu\nu}(l)\frac{\lambda^{a}}{2}\gamma_{\mu}S(k)\frac{\lambda^{a}}{2}\Gamma_{\nu}(k,q)\,.$
(11)

The renormalization constants $Z_{1}$, $Z_{2}$, and $Z_{m}$ correspond to the
quark-gluon vertex, quark propagator, and current-quark mass, respectively.
$\frac{\lambda^{a}}{2}$ is the fundamental representation of $SU(3)$ color
symmetry. $D_{\mu\nu}(l)$ with $l=q-k$ is the renormalized dressed gluon
propagator and $\Gamma_{\nu}(k,q)$ the renormalized dressed quark-gluon
vertex. $g$ is the coupling constant. The corresponding Feynman diagram of
Eqs. (10) and (11) is shown in Fig. 2.

Figure 2: The Feynman diagram for the gap equation in Eqs. (10) and (11).

For the dressed gluon propagator and the dressed quark-gluon vertex, the
following Ans$\ddot{a}$tz in Ref. Qin  _et al._ (2011b) is used

$Z_{1}g^{2}D_{\mu\nu}(l)\Gamma_{\nu}(k,q)=l^{2}\mathcal{G}(l^{2})D_{\mu\nu}^{\text{free}}(l)\gamma_{\nu}\,,$
(12)

where the dressed quark-gluon vertex $\Gamma_{\nu}(k,q)$ in the quark self-
energy in Eq. (11) is truncated to the tree level $\gamma_{\nu}$.
$D_{\mu\nu}^{\text{free}}(l)=\left(\delta_{\mu\nu}-l_{\mu}l_{\nu}/l^{2}\right)/l^{2}$
is the free gluon propagator in Landau gauge. The non-perturbative dressing
effect is absorbed in the effective interaction function $\mathcal{G}(l^{2})$,
which is supposed to compensate for all the missing pieces in the quark-gluon
vertex Qin  _et al._ (2011b); Maris and Tandy (1999). As the non-perturbative
features are insensitive to the ultraviolet behavior of the interaction, in
this work we adopt a simplified version of the interaction in Ref. Qin  _et
al._ (2011b), where only the infrared part is kept

$\mathcal{G}(l^{2})=\frac{8\pi^{2}}{\omega^{4}}De^{-l^{2}/\omega^{2}}\,.$ (13)

In this case the renormalization procedure can be skipped. In Eq. (13) the
parameters $D$ and $\omega$ control the strength and width of the interaction
respectively. It is found that the observables of vector and pseudoscalar
mesons are insensitive to variations of $\omega\in[0.4,0.6]$ as long as
$D\omega=\text{const.}$ In this work, $\omega=0.5$ GeV and $f_{\pi}=0.093$ GeV
is obtained with $D\omega=(0.85~{}\text{GeV})^{3}$.

###  III.2 The Bethe-Salpeter amplitude of the mesons

The BS amplitude of the mesons can be solved from the renormalized homogeneous
BS equation, which reads

$[\Gamma_{P}(q;P)]_{\alpha\beta}=\int_{k}^{\Lambda}K_{\alpha\gamma,\beta\delta}(q,k;P)\left[S(k_{+})\Gamma_{P}(k;P)S(k_{-})\right]_{\gamma\delta}\,.$
(14)

$K_{\alpha\gamma,\delta\beta}(q,k;P)$ is the quark-antiquark scattering kernel
with $\alpha$, $\beta$, $\gamma$, $\delta$ the Dirac and color indices. For
this eigen equation, solutions exist only for particular, separated values of
$P^{2}$. The corresponding Feynman diagram is shown in Fig. 3.

Figure 3: The Feynman diagram for the homogeneous BS equation in Eq. (14).

To solve the homogeneous BS equation, the truncation of the quark-antiquark
scattering kernel $K$ is performed with respect to the constraints of AV-WTI
Chang  _et al._ (2021); Qin and Roberts (2021); Qin  _et al._ (2020).
Considering that the quark-gluon vertex is truncated to the tree level in Eq.
(12), here we choose the one-gluon exchange for the scattering kernel

$K_{\alpha\gamma,\beta\delta}(q,k;P)=-\mathcal{G}(l^{2})l^{2}D_{\mu\nu}^{\text{free}}(l)\left(\gamma_{\mu}\frac{\lambda^{a}}{2}\right)_{\alpha\gamma}\left(\gamma_{\nu}\frac{\lambda^{a}}{2}\right)_{\beta\delta}\,.$
(15)

The united truncation scheme combining Eq. (12) for the quark-gluon vertex and
Eq. (15) for the scattering kernel is the so-called RL truncation Cloët and
Roberts (2014). It is the first term in a non-perturbative, systematic, and
symmetry-preserving approximation. RL truncation preserves the one-loop
renormalization group properties of QCD, and has provided a uniformly accurate
description and prediction for a wide range of hadron properties Chang  _et
al._ (2011); Bashir  _et al._ (2012); Qin  _et al._ (2019); Aguilar  _et al._
(2009).

For mesons with spin-parity $J^{P}$, the BS amplitude $\Gamma_{P}(q;P)$ can be
expanded in the corresponding tensor basis $\tau_{P}^{i}(q;P)$

$\Gamma_{P}(q;P)=\sum_{i=1}^{N}\tau_{P}^{i}(q;P)\mathcal{F}_{i}(q^{2},q\cdot
P)\,,$ (16)

where $\mathcal{F}_{i}(q^{2},q\cdot P)$ is the scalar coefficient function.
$N$ is the dimension of the tensor basis determined by the meson spin $J$. In
this work, we focus on the pseudoscalar ($J^{P}=0^{-}$) and vector
($J^{P}=1^{-}$) mesons at ground states and first-excited states, where $N=4$
for $J=0$ and $N=8$ for $J=1$. The orthogonal Dirac bases are summarized in
Tab. 1 and Tab. 2 for pseudoscalar and vector mesons respectively Hilger  _et
al._ (2017). The basis elements are organized with respect to their quark spin
$S$ and orbital angular momentum $L$ in the meson’s rest frame, where
$L=0,1,2$ correspond to $s$, $p$, $d$ waves respectively Eichmann  _et al._
(2016). In the Dirac bases, $\hat{P}=P/|P|$ is the unit vector of the meson
momentum, $q^{\mu}_{\perp}=q^{\mu}-q\cdot\hat{P}\hat{P}^{\mu}$ is the relative
momentum transverse to the corresponding meson momentum $P$. The Dirac matrix
$\gamma^{\mu}_{\perp}=\gamma^{\mu}-\gamma\cdot\hat{P}\hat{P}^{\mu}$ is also
transverse to $P$.

Table 1: The orthogonal Dirac basis for pseudoscalar ($J^{P}=0^{-}$) channel with $S$ and $L$ the spin and the orbital angular momentum. i | $\tau^{i}$ | S | L
---|---|---|---
1 | $\gamma_{5}$ | 0 | 0
2 | $\gamma_{5}\gamma\cdot\hat{P}$ | 0 | 0
3 | $\gamma_{5}\gamma\cdot q_{\perp}$ | 1 | 1
4 | $\gamma_{5}[\gamma\cdot q,\gamma\cdot\hat{P}]$ | 1 | 1
Table 2: Similar as Tab. 1 but for vector ($J^{P}=1^{-}$) channel. i | $\tau^{i}$ | S | L
---|---|---|---
1 | $\gamma^{\mu}_{\perp}$ | 1 | 0
2 | $\gamma^{\mu}_{\perp}\gamma\cdot\hat{P}$ | 1 | 0
3 | $q^{\mu}_{\perp}\mathbb{1}$ | 0 | 1
4 | $q^{\mu}_{\perp}\gamma\cdot\hat{P}$ | 0 | 1
5 | $[\gamma^{\mu}_{\perp},\gamma\cdot q_{\perp}]$ | 1 | 1
6 | $\gamma^{\mu}_{\perp}[\gamma\cdot q,\gamma\cdot\hat{P}]-2q^{\mu}_{\perp}\gamma\cdot\hat{P}$ | 1 | 1
7 | $q^{\mu}_{\perp}\gamma\cdot q_{\perp}-\frac{1}{3}q_{\perp}^{2}\gamma^{\mu}_{\perp}$ | 1 | 2
8 | $q^{\mu}_{\perp}[\gamma\cdot q,\gamma\cdot\hat{P}]-\frac{1}{3}q^{2}_{\perp}[\gamma^{\mu}_{\perp},\gamma\cdot\hat{P}]$ | 1 | 2

###  III.3 The scalar vertex

The scalar vertex can be obtained by solving the in-homogeneous BS equation.
With RL truncation, it reads

$\Gamma_{s}(q;Q)=\mathbb{1}+g^{2}\int_{k}^{\Lambda}D_{\mu\nu}(l)\frac{\lambda^{a}}{2}\gamma_{\mu}S(k_{+})\Gamma_{s}(k;Q)S(k_{-})\frac{\lambda^{a}}{2}\gamma_{\nu}\,,$
(17)

where $D_{\mu\nu}$ is the dressed gluon propagator with Ans$\ddot{a}$tz given
in Eq. (12). $S(k_{\pm})$ are the quark propagator as the solutions of Eqs.
(10) and (11). The Feynman diagram is shown in Fig. 4.

Figure 4: The Feynman diagram for the scalar vertex in Eq (17).

Eq. (17) can be solved with iterative algorithm and the solution can be
expanded with the full Dirac basis in scalar channel,

$\\{\mathbb{1}\,,\gamma\cdot Q\,,\gamma\cdot q\,,[\not{q},\not{Q}]\\}\,.$ (18)

Since the in-hadron condensate is defined as the scalar form factor at zero-
momentum transfer $Q^{2}=0$ in Eqs. (7) and (9), the Dirac basis is reduced to
$\\{\mathbb{1}\,,\gamma\cdot q\\}$.

##  IV Results and discussion

Figure 5: (Color online) The masses of the mesons $M_{P}$ as functions of the
current-quark mass $m_{q}$ in different channels: pseudoscalar ground state
($\pi$-channel), pseudoscalar first-excited state ($\pi^{\prime}$-channel),
vector ground state ($\rho$-channel) and vector first-excited state
($\rho^{\prime}$-channel). The grey dashed line represents $2m_{q}$. The light
quark region $[0,1]$ GeV (left) and the heavy quark region $[3,4]$ GeV (right)
are shown.

To measure the total effects of chiral symmetry breaking including both the
dynamical and explicit parts, the masses of the pseudoscalar and vector mesons
at ground and first-excited states are investigated as shown in Fig. 5. It is
obvious that the meson masses $M_{P}$ increase monotonically with the increase
of the current-quark mass $m_{q}$. In the heavy quark region shown in the
right panel, similar linear tendencies are found for all the channels. At
$m_{q}\lesssim 4$ GeV, the relative difference between $M_{P}$ and $2m_{q}$
are much smaller than that at $m_{q}\lesssim 1$ GeV for all these channels,
which indicates the effects of dynamical chiral symmetry breaking are weaker
compared to that of explicit chiral symmetry breaking for heavy quarks. In the
chiral limit shown in the left panel, the mass of the Goldstone boson $\pi$ is
vanished. With the increase of $m_{q}$, $M_{\pi}$ increases approximately
proportional to $\sqrt{m_{q}}$ according to Eq. (2). This nonlinearity
indicates the interactions between the quarks inside $\pi$ are significant,
which clearly demonstrates the non-perturbative nature of the light quarks.
Besides, in the chiral limit, the vector meson $\rho$ and the excited states
$\pi^{\prime}$, $\rho^{\prime}$ possess non-vanishing masses, which are
generated totally from the dynamical chiral symmetry breaking.

It is expected that the vector meson is heavier than the pseudoscalar meson
for both ground and excited states. This is the case for heavy quarks as shown
in the right panel in Fig. 5. However, for light quarks near the chiral limit,
the mass of the first-excited state in vector channel is found smaller than
that in pseudoscalar channel, i.e., $M_{\rho^{\prime}}<M_{\pi^{\prime}}$. With
the increase of the current-quark mass, there exists a crossing point at about
$0.3$ GeV, after which the mass ordering of the first-excited states are
changed to be $M_{\rho^{\prime}}>M_{\pi^{\prime}}$ and remains stable till the
heavy quark limit.

Figure 6: (Color online) Similar as Fig. 5 but for dimensionless in-hadron
condensates $S_{P}(0)/M_{P}$.

To measure the effects of dynamical chiral symmetry breaking, the
dimensionless in-hadron condensates $S_{P}(0)/M_{P}$ are calculated as
functions of current-quark mass $m_{q}$ for different channels as shown in
Fig. 6. In the heavy quark region, the current-quark mass dependencies of
$S_{P}(0)/M_{P}$ are quite weak and the differences among different channels
are negligible. According to Eq. (7), this behavior of $S_{P}(0)/M_{P}$
indicates that the responses of meson masses $M_{P}$ to the current-quark mass
$m_{q}$ are linear, which is consistent with the results shown in the right
panel of Fig. 5. Near the chiral limit, the values of $S_{P}(0)/M_{P}$ are
significant and decrease rapidly as the current-quark mass increases. Since in
the chiral limit, the meson masses are generated totally from dynamical chiral
symmetry breaking, the rapid decreasing of $S_{P}(0)/M_{P}$ probably implies
that the global effects of dynamical chiral symmetry breaking is also
decreasing. Besides, with the increase of the current-quark mass, a crossing
behavior for $S_{P}(0)/M_{P}$ between the first-excited states in pseudoscalar
and vector channels is also found. After the crossing point at about $0.8$
GeV, the ordering of the $S_{P}(0)/M_{P}$ remains stable till the heavy quark
limit. By comparing the meson masses $M_{P}$ in Fig. 5 and the dimensionless
in-hadron condensates $S_{P}(0)/M_{P}$ in Fig. 6, we find that
$S_{P}(0)/M_{P}$ and $M_{P}$ are consistent in measuring the global effects of
dynamical chiral symmetry breaking.

Figure 7: (Color online) The relative partial wave contributions to the in-
hadron condensates $S^{(S,L)}_{P}(0)/S_{P}(0)$ as functions of current-quark
mass $m_{q}$ for pseudoscalar mesons at ground state (left) and first-excited
state (right).

To probe the effects of dynamical chiral symmetry breaking from a structural
perspective, the relative partial wave contributions to the in-hadron
condensates $S^{(S,L)}_{P}(0)/S_{P}(0)$ for pseudoscalar and vector mesons are
shown in Fig. 7 and Fig. 8, respectively. The cross terms are not depicted as
their contributions can be easily obtained and not of interest here. For
pseudoscalar mesons at both ground and first-excited states, it is clear that
$s$ waves with $(S,L)=(0,0)$ contribute the majority of the in-hadron
condensates. This indicates that the dynamical chiral symmetry breaking is
mainly contributed by the $s$ waves. The ordering of the partial wave
contributions of the first-excited state is the same as that of the ground
state, which indicates the first-excited state in pseudoscalar channel is a
radial excitation. At the current-quark mass $m_{u/d}=5$ MeV, the mass of the
first-excited state in pseudoscalar channel obtained in this work is $1080$
MeV, which is close to the mass of the detected $\pi(1300)$ in experiment.
This indicates that $\pi(1300)$ is probably a radial excitation dominated by
$s$ waves.

Figure 8: (Color online) Similar as Fig. 7 but for vector mesons.

For vector mesons at ground state in the left panel of Fig. 8, the
contribution of the $s$ waves with $(S,L)=(1,0)$ dominates the in-hadron
condensates and increases steadily with the increase of the current-quark
mass. The second important contribution is from $p$ waves with $(S,L)=(0,1)$,
which shows a decreasing tendency with respect to the current-quark mass.
Similar behavior is found for $p$ waves with $(S,L)=(1,1)$ with less
contribution. The contribution of $d$ waves is negligible. In the heavy quark
limit, the contributions of $p$ and $d$ waves are vanished and the $s$ waves
contribute the vast majority.

For the first-excited state of vector mesons shown in the right panel of Fig.
8, the relative partial wave contributions to the in-hadron condensates
$S^{(S,L)}_{P}(0)/S_{P}(0)$ are found similar as the ground state at heavy
quark limit. In light quark region, with the increase of the current-quark
mass, the contribution of the $s$ waves with $(S,L)=(1,0)$ increase and the
contribution of the $p$ waves with $(S,L)=(0,1)$ decrease rapidly. A crossing
point is found at about $0.7$ GeV. Besides, the contributions of $p$ waves
with $(S,L)=(1,1)$ and $d$ waves with $(S,L)=(1,2)$ are affected by this novel
behavior of the $s$ and $p$ waves. Near the chiral limit, the ordering of the
partial wave contributions of the first-excited state is different from that
of the ground state, which indicates the first-excited state in vector channel
is not a radial excitation. At the current-quark mass $m_{u/d}=5$ MeV, the
mass of the first-excited state in vector channel obtained in this work is
$970$ MeV, which is largely different from the mass of the detected
$\rho(1450)$ in experiment. This indicates that $\rho(1450)$ is not dominated
by $p$ waves to a large degree.

From the global perspective in Fig. 6, in the heavy quark limit, the ordering
of the in-hadron condensates of the ground and first-excited states in
pseudoscalar and vector channels is as expected. In the light quark region,
the crossing point of the in-hadron condensates between the first-excited
states in pseudoscalar and vector channels is about $0.8$ GeV. From the
structural perspective in Fig. 7 and Fig. 8, in the heavy quark limit, the
partial wave contributions to the in-hadron condensates of the ground and
first-excited states in pseudoscalar and vector channels are all dominated by
$s$ waves. In the light quark region, the crossing point of the contributions
of $s$ and $p$ waves for the first-excited state in vector channel is about
$0.7$ GeV. These clearly demonstrate that, on the one hand, it is consistent
between the global and the structural perspective of in-hadron condensates in
describing the dynamical chiral symmetry breaking. On the other hand, the
global effects of dynamical chiral symmetry breaking can be understood with
the partial wave contributions to in-hadron condensates. Furthermore, the
crossing point of the meson masses in Fig. 5 is smaller than that of the in-
hardon condensates in Fig. 6. With the increase of current-quark mass, the
ordering of the meson masses is as expected after $0.3$ GeV, while the
ordering of the in-hardon condensates is not as expected until $0.8$ GeV. That
is to say, in the current-quark mass region at about $0.3$-$0.8$ GeV, the
meson mass spectra seem reasonable, but are not supported by the structural
information of mesons. This indicates that mass is not a precise measure to
the dynamical chiral symmetry breaking. In comparison, the in-hardron
condensate can not only measure the global effects of dynamical chiral
symmetry breaking, but also provide a structural perspective for the effects
of dynamical chiral symmetry breaking.

It should be noticed that the meson masses, the in-hadron condensates and the
partial wave contributions are all obtained with the DSEs approach under the
RL truncation, which truncates the quark-gluon vertex and the quark-antiquark
scattering kernel to the tree level and preserves the AV-WTI. As the
pseudoscalar mesons are strongly constrained by the symmetry embodied in AV-
WTI, the calculated properties of the first-excited state in pseudoscalar
channel is reliable qualitatively. While for the first-excited state of vector
meson, the missing contribution from the higher orders might lead to non-
negligible effects on spin-orbital interaction and thus on the partial waves,
especially when the current-quark mass is close to the chiral limit. Thus the
relative partial wave contributions to in-hadron condensates might be changed
beyond the RL truncation.

Attempts have been made beyond the RL truncation Matevosyan  _et al._ (2007);
Chang and Roberts (2009, 2012); Binosi  _et al._ (2016); Qin and Roberts
(2021). In Ref. Qin and Roberts (2021), the dressed-quark anomalous
chromomagnetic moment is included and the mass ordering of the first-excited
states in pseudoscalar and vector channels has been corrected. In future work,
it is interesting to study the meson mass spectra and the relative partial
wave contributions to in-hadron condensates with a more realistic truncation
scheme.

##  V Summary

By using the DSEs approach, the scalar form factors at zero-momentum transfer
are calculated as the in-hadron condensates for the ground and first-excited
states of the flavor-symmetric mesons in pseudoscalar and vector channels. The
responses of the dimensionless in-hadron condensates and the meson masses to
the current-quark mass are consistent in describing the global effects of
dynamical chiral symmetry breaking. In-hadron condensate also provides a
structural perspective to the effects of dynamical chiral symmetry breaking by
comparing the relative partial wave contributions. In pseudoscalar channel,
for the ground and first-excited states, the in-hardon condensates are
dominated by the $s$ wave contributions. In vector channel, for the ground
state, the in-hardon condensates are dominated by the $s$ wave contributions,
while for the first-excited state, they are dominated by the $p$ wave
contributions. By comparing the theoretical results of the meson masses in
pseudoscalar and vector channels with experimental values, we infer that the
$\pi(1300)$ is a radial excitation dominated by $s$ waves, while the
$\rho(1450)$ is not a $p$ wave-dominated excitation.

Comparing the crossing behaviors of the meson masses $M_{P}$, the
dimensionless in-hadron condensates $S_{P}(0)/M_{P}$, and the partial wave
contributions $S^{(S,L)}_{P}(0)/S_{P}(0)$, we confirm the efficiency of the
in-hadron condensates in describing the effects of dynamical chiral symmetry
breaking from both global and structural perspectives. This work provides a
new insight into the studies of hadron properties with partial wave analysis
for the in-hadron condensates. In future, we plan to extend this analysis with
a more realistic scheme beyond the RL truncation.

###### Acknowledgements.

P. Q. thanks Dr. Langtian Liu for helpful discussion. This work was partly
supported by the National Natural Science Foundation of China under Grant No.
12147102 and No. 12005060, the China Postdoctoral Science Foundation under
Grant No. 2021M700610.

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[a]João Barata

# First steps towards the quantum simulation of jet quenching

Carlos Salgado

###### Abstract

The leading order $\alpha_{s}$ effect in jet quenching corresponds to the
broadening of the jet’s transverse momentum, due to the multiple interactions
with the underlying medium. A complete understanding of momentum broadening is
critical for the success of jet quenching phenomenology.

In this talk, we introduce a strategy to quantum simulate single particle
momentum broadening in a QCD background medium. We argue that it is, in
principle, possible to extract the jet quenching parameter $\hat{q}$ from such
an algorithm. More importantly, this corresponds to the first step towards
simulating full medium induced parton showers, which is far beyond the
capabilities of classical computers.

##  1 Introduction

One of the strongest evidences for the production of a quark gluon plasma in
heavy ions collisions is the suppression of high transverse momentum jets’
yield. This phenomenon was first anticipated by Bjorken [1] and is nowadays
broadly referred to as jet quenching. Such modifications to the jets
properties, when compared to a vacuum benchmark, arise from the interactions
of the jets’ constituents with the underlying QCD medium. At eikonal accuracy,
where energy suppressed terms are neglected, the jet’s partons evolve along a
common light cone direction at fixed transverse positions with respect to the
jet core. As a result, only their individual color fields can be modified by
the medium. At this order, all partons satisfy the on-shell relation and thus
there is no radiative energy loss, and the only possible phenomenological
effect that can be studied corresponds to the broadening of the jet’s
momentum. Although single particle momentum broadening can be easily computed
using classical methods, simulating it in quantum computers is a natural
starting point for the future simulation of fully embedded jets. For these,
multi-particle quantum interferences are crucial. However, tacking such
contributions into account is particularly hard to perform in classical
computers and thus one hopes that the so-called quantum advantage might become
important to achieve such a goal in the future.

We introduce a simple quantum algorithm to simulate the evolution of a single
parton in the presence of a QCD background. The strategy followed is a quantum
analog of previous classical approaches [2] and can be extended to include in-
medium radiation.

##  2 Theoretical set up

In this section, we first detail the set up to describe the evolution of a
single hard parton in a stochastic QCD background field. From this we will
extract the effective Hamiltonian that will then be used to construct the time
evolution operator to be implemented in terms of a quantum circuit. Secondly,
we give a broad overview of the quantum simulation algorithm, which is used to
mimic the evolution of the single particle in the quantum computer. For more
details see [3].

###  2.1 Parton evolution in a QCD background

We consider a single parton with momentum
$p\equiv(\omega,{\boldsymbol{p}},p^{-})=((p^{0}+p^{z})/2,{\boldsymbol{p}},p^{0}-p^{z})$
in light cone coordinates. Here we must assume that
$\omega\gg|{\boldsymbol{p}}|\gg T$, where $T$ denotes the medium temperature
and $\omega\approx p^{0}$ is assumed to be the large component of the momentum
of the right-moving particle. The quark gluon plasma background corresponds to
a highly occupied state and thus it admits a classical (stochastic)
description in terms of a field ${\cal A}^{\mu}(x)=({\cal A}^{+}(x),{\cal
A}_{\perp}(x),{\cal A}^{-}(x))$. In the light-cone gauge, where ${\cal
A}^{+}=0$, the only non-vanishing field component is ${\cal A}^{-}$ [4]. In
addition, because the initial parton is highly boosted, the spread of its
wavefunction in $p^{-}$ must be very small. As a consequence, the spacetime
dependence of the field can be simplified to be ${\cal
A}^{-}(x^{+},{\boldsymbol{x}})\equiv{\cal A}^{-}(x^{+},{\boldsymbol{x}},0)$.
Also, in the highly boosted and collinear set up here considered, one can use
that $x^{0}=t\sim x^{3}$, so that $x^{+}\sim t$ can be viewed as the time
variable of the problem, while all the dynamics of the single parton system
are restricted to the two-dimensional transverse plane. Fig. 1 summarizes the
previous discussion on the spacetime picture of the problem.

Figure 1: Spacetime picture for a highly boosted parton (jet) evolving in the
presence of a QCD background. The medium is assumed to have a finite length
$L$ along $x^{+}$ and $p^{+}=\omega$.

In the case where one neglects the production of in-medium radiation (i.e.
suppresses all $\mathcal{O}(\alpha_{s})$ terms), the full theory is captured
by the in-medium propagator $G(t,{\boldsymbol{x}};0,{\boldsymbol{y}})$, which
obeys [5]

$\left(i\partial_{t}+\frac{\partial^{2}_{\boldsymbol{x}}}{2\omega}+g{\cal
A}^{-}(t,{\boldsymbol{x}})\cdot
T\right)G(t,{\boldsymbol{x}};0,{\boldsymbol{y}})=i\delta(t)\delta({\boldsymbol{x}}-{\boldsymbol{y}})\,.$
(1)

This describes the propagation of a scalar particle between the transverse
position ${\boldsymbol{y}}$ at time $x^{+}=0$ and the position
${\boldsymbol{x}}$ at time $x^{+}=t$. In particular, it implies that the
corresponding Hamiltonian is that of a non-relativistic particle [6, 3, 7]

${\cal H}(t)=\frac{{\boldsymbol{p}}^{2}}{2\omega}+g{\cal
A}^{-}(t,{\boldsymbol{x}})\cdot T={\cal H}_{K}+{\cal H}_{\cal A}(t)\,.$ (2)

This will be the operator to be implemented in terms of quantum circuits. We
note that that the eikonal limit mentioned above corresponds to the case where
$\frac{{\boldsymbol{p}}}{\omega}\to 0$, and thus only ${\cal H}_{{\cal A}}$
survives. Indeed, this term is diagonal in a position basis and thus only
gives rise to a phase factor. Also, in what follows we will assume that the
particle is in the singlet color representation, such that $T=1$. The
generalization of the algorithm to other color representations is discussed in
[3].

###  2.2 The quantum simulation algorithm

The quantum simulation of complex quantum systems by using simpler and
controllable ones was first envisioned by Feynman [8] in the 1980s. In its
essence, the quantum simulation algorithm consists in solving the Schrodinger
equation for a given Hamiltonian by manipulating the natural dynamics of a
given quantum system such that it mimics the evolution of a system of
interest. In the case where the controllable system is a digital quantum
computer, i.e. a collection of interconnected (idealized) $1/2$-spins which
admit the application of an universal set of fundamental quantum gates, the
algorithm can be broadly summarized as follows [9]:

  1. 1.

Input: One needs to provide a Hamiltonian ${\cal H}$ and specify the
associated Hilbert space. The initial wavefunction of the system
$|\psi_{0}\rangle$ also has to be provided. In addition, for the present
problem, the background field is assumed to be a stochastic variable. Thus,
one also has to provide an ensemble of $m$ field configurations with the
associated probability distribution.

  2. 2.

Digitization: Next, it is necessary to map the degrees of freedom of the
physical system to the qubits available in the computer. This should also
include the qubit representation of the initial wavefunction.

  3. 3.

Initial state preparation: Given the qubit representation of
$|\psi_{0}\rangle$, one needs to prepare a circuit that generates this state
in the quantum computer efficiently and accurately. In this step, one assumes
that a fiducial state can always be prepared by the computer.

  4. 4.

Time evolution: Given the discretization/digitization strategy used, one has
to represent the time evolution operator, $U=\exp(-i\int dt\,{\cal H}(t))$, in
terms of basic quantum gates. Then one applies these set of gates to the
initial state, so that the final state of the system is prepared.

  5. 5.

Measurement: Due to the special character of measurement in Quantum Mechanics,
one can not efficiently measure the full final wavefunction in general.
Instead, measurement protocols have to be implemented allowing one to access
particular information about the system requiring only a reasonable number of
measurements.

##  3 Quantum simulating single parton broadening

The quantum simulation algorithm can be immediately applied to simulate the
evolution of a single parton inside a medium; the respective quantum circuit
is detailed in Fig. 2. Note that the bottom part of the diagram denotes a
classical memory where the information about the gauge field is stored to then
be used by the quantum circuit above.

Figure 2: Overview of the quantum circuit. Single lines denote quantum
channels while double lines denote classical ones. Above each line we detail
the state being store in the circuit. The $\blacksquare$ denotes that the time
evolution gates parameters are to be determined from the gauge field.

Following the steps detailed in the previous section, the input to the system
will consist on the Hamiltonian given in Eq. 2, $m$ field configurations for
the QCD background and we shall consider the initial wavefunction to be that
of a particle with light-cone energy $\omega$ and initial traverse momentum
${\boldsymbol{p}}={\boldsymbol{0}}$. The Hilbert space under consideration
corresponds that of a single non-relativistic particle in two dimensions; thus
it is natural to describe the system either in a position basis,
$|{\boldsymbol{x}}\rangle$, or in a momentum on, $|{\boldsymbol{p}}\rangle$,
which are naturally related by a Fourier transform.

To represent the system in terms of qubit degrees of freedom, we discretize
the system in a two dimensional lattice. Any position state can be written as
$|{\boldsymbol{x}}\rangle=|a_{s}{\boldsymbol{n}}\rangle$, where $a_{s}$ the
spatial lattice spacing and ${\boldsymbol{n}}=(n_{1},n_{2})$ is a
dimensionless vector with each component taking values between $0$ and
$N_{s}-1$, with $N_{s}$ the number of lattices sites per dimension. An
equivalent lattice exists in momentum state, related to this one by a discrete
Fourier Transform; see [3, 10, 11] for more details.

Rewriting the Hamiltonian, $H={\cal H}a_{s}$, in terms of position and
momentum operators acting on the dimensionless basis introduced

$H=\frac{{\boldsymbol{P}}^{2}}{2E}+gA(t,{\boldsymbol{X}})\cdot
T=H_{K}+H_{A}(t)\,,$ (3)

we obtain that evolution operator simply reads

$U(L^{\prime},0)\equiv{\cal
T}\exp\left[-i\int_{0}^{L^{\prime}}dt\,H(t)\right]\,,$ (4)

where $L^{\prime}\equiv L/a_{s}$ is the dimensionless medium length. To
implement Eq. 4 in terms of basic quantum gates we use a simple product
formula (see [12, 13, 14, 15, 16] for more discussion on this procedure for
time dependent Hamiltonians), which leads to

$U(L^{\prime},0)\approx\prod_{k_{t}=1}^{N_{t}}\left\\{\exp\left[-iH_{K}\frac{L^{\prime}}{N_{t}}\right]\exp\left[-iH_{A}\left(k_{t}\cdot\frac{L^{\prime}}{N_{t}}\right)\frac{L^{\prime}}{N_{t}}\right]\right\\}\equiv\prod_{k_{t}=1}^{N_{t}}\left\\{U_{K}(\varepsilon_{t})U_{A}(k_{t}\cdot\varepsilon_{t},\varepsilon_{t})\right\\}\,.$
(5)

This decomposition of the evolution operator essentially consists in slicing
the evolution time $L^{\prime}$ into $N_{t}$ steps, where in each step one can
use that $e^{X+Y}\approx e^{X}e^{Y}$ for $X$ and $Y$ non-commuting operators.
Thus, the error associated with this scheme is of
$\mathcal{O}(\varepsilon_{t}^{2})$ with $\varepsilon_{t}=L^{\prime}/N_{t}$. In
the particular case of the Hamiltonian given in Eq. 3, it is convenient to
split it, at each step in the time evolution, in terms of a term governed by
$H_{K}$ and a term containing the gauge field. The reason for this separation
is because the kinetic term is diagonal in momentum space, i.e. when acting on
a state $|{\boldsymbol{p}}\rangle$ it only gives a phase

$U_{K}(\varepsilon_{t})|{\boldsymbol{p}}\rangle=\exp\left(-i\frac{\varepsilon_{t}}{2E}{\boldsymbol{p}}^{2}\right)|{\boldsymbol{p}}\rangle\,,$
(6)

while $H_{A}$ is diagonal in a position basis

$U_{A}(k_{t}\cdot\varepsilon_{t},\varepsilon_{t})|{\boldsymbol{x}}\rangle=\exp(-ig\varepsilon_{t}A(k_{t}\cdot\varepsilon_{t},{\boldsymbol{x}}))|{\boldsymbol{x}}\rangle\,.$
(7)

Thus, if in between the application of the two evolution operators one
performs a (quantum) Fourier Transform [9], the circuit can be efficiently
implemented.

The decomposition of $U_{K}$ and $U_{A}$ can be done in a straightforward
manner. For $U_{K}$ one just needs to implement an algorithm which gives a
phase to the ket state as a function of the state value. A simple strategy to
realize such an operation was introduced in [17] and more details on its
application can be found in [3]. For the gauge term, and assuming that one has
transformed from the momentum to the position basis, the implementation
requires that a classical memory with the values of the field at time
$k_{t}\varepsilon_{t}$ is provided for all lattice points. Although this in
general a classically costly implementation, we notice that in general one
provides a statistical model for the field, which makes its computation quite
efficient. Given these field values, one only needs a diagonal operator where
each entry contains the field value at a different position; such an operator
can be constructed either by classically solving a system of $N_{s}^{2}$
linear equations or by multiple applications of controlled quantum gates; see
[3, 9] for some details. A single step of this algorithm is summarized in Fig.
3.

Figure 3: Implementation of the $k_{t}^{\rm th}$ time step, as indicated in
Eq. 5.

After iterating $N_{t}$ times the above single-step circuits on the initial
wave function, one generates the final state $|\psi_{L}\rangle$. As mentioned
in the previous section, given this state one needs to implement a protocol to
efficiently extract the relevant information for the wavefunction. In the
present work, we are interest in extracting the jet quenching parameter
$\hat{q}$, which is trivially related to the average momentum squared
accumulated in the medium.

In order to this, we apply the Hadamard test as detailed in Fig. 4. We
introduce an ancillary qubit which can be either in the state $|0\rangle$ or
in the state $|0\rangle+i|1\rangle$. Then we apply the Hadamard transformation
[9] followed by a controlled application of an operator $V$. By controlled we
mean that $V$ is only applied to the final state $|\psi_{L}\rangle$ if the
ancilla qubit is in the state $|1\rangle$. Finally, one reverts the Hadamard
transform and measures the ancilla. We associate the outcome of such a
measurement to a classical random variable $\chi$ which takes value $\chi=-1$
if one measures the state $|0\rangle$ and $\chi=-1$ is one observes the state
$|1\rangle$.

Figure 4: Circuit representation of the measurement strategy.

It is then easy to show that for any $V$

$\langle\chi\rangle_{\rm
QM}\equiv\bra{\psi_{L}}V+V^{\dagger}\ket{\psi_{L}}=\Re\bra{\psi_{L}}V\ket{\psi_{L}}\,.$
(8)

if the ancilla is in the $\ket{0}$ state. If the ancilla is prepared in the
state $1/\sqrt{2}(\ket{0}+i\ket{1})$, then

$\langle\chi\rangle_{\rm QM}=\Im\bra{\psi_{L}}V\ket{\psi_{L}}\,,$ (9)

which gives access to the real and imaginary parts of unitary operator $V$. In
the case when $V$ is diagonal in the momentum basis, i.e.
$V=V_{\alpha}=\exp(i\alpha{\boldsymbol{P}}^{2})$, with $\alpha$ a free
numerical parameter, we get that

$\Re\bra{\psi_{L}}V_{\alpha}\ket{\psi_{L}}=\langle\cos(\alpha{\boldsymbol{P}}^{2})\rangle_{\rm
QM}\,,$ (10)

and

$\Im\bra{\psi_{L}}V_{\alpha}\ket{\psi_{L}}=\langle\sin(\alpha{\boldsymbol{P}}^{2})\rangle_{\rm
QM}\,,$ (11)

One can imagine that $\alpha$ can be taken sufficiently small such that to
linear order

$\langle e^{i\alpha{\boldsymbol{P}}^{2}}\rangle_{\rm QM}\approx
1+i\frac{\alpha}{a_{d}^{2}}\hat{q}L\to\langle\sin(\alpha{\boldsymbol{P}}^{2})\rangle_{\rm
QM}\approx\frac{\alpha}{a_{d}^{2}}\hat{q}L\,.$ (12)

where in the right hand side we have related the expectation value of the
operator $V$ to the jet quenching parameter, while the left hand side is
directly obtained from the quantum computer. Higher order moments of the
underlying distribution can be obtained using the same method but allowing
$\alpha$ to be larger [3].

##  4 Conclusions and outlook

In this work we have introduced a simple hybrid quantum strategy to extract
the jet quenching parameter. Although the current proposal can not outmatch
classical approaches, it offers a road for the future quantum simulation of
embedded jets.

In the future, it would be important to have a first-time quantum computation
of single particle evolution inside the medium. Due to the necessity to
discretize space and the limitations of current hardware, such simulations
might still not be able to compete with their classical counterparts. However,
we note that the study of color flow in the medium, which requires far less
quantum resources, should already be able to reveal important information
about jet evolution. Another important goal to pursue would be to extend the
present approach to include the production of gluonic radiation.

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