# Probing gluon GTMDs of the proton in deep inelastic diffractive dijet
production at HERA

Barbara Linek<EMAIL_ADDRESS>University of Rzeszów, ul. Pigonia 1,
PL-35-959 Rzeszów, Poland  Marta Łuszczak<EMAIL_ADDRESS>University of
Rzeszów, ul. Pigonia 1, PL-35-959 Rzeszów, Poland  Wolfgang Schäfer
<EMAIL_ADDRESS>Institute of Nuclear Physics, Polish Academy of
Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland  Antoni Szczurek
<EMAIL_ADDRESS>Institute of Nuclear Physics, Polish Academy of
Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland

###### Abstract

We calculate several differential distributions for diffractive dijets
production in $ep\to e^{\prime}{\rm jet\,jet}p$ in the pQCD dipole approach
using off diagonal unintegrated gluon distributions (GTMDs). Different models
from the literature are used. We concentrate on the contribution from
exclusive $q\bar{q}$ dijets.

Results of our calculations are compared to H1 and ZEUS data, including
specific experimental cuts in our calculations. In general, except of one
GTMD, our results are below the HERA data. The considered mechanism is
expected to gives a sizeable contribution to the ZEUS data, while it is
negligible in the kinematics of the H1 measurement.

This is in contrast to recent results from the literature where the
normalization was adjusted to some selected distributions of H1 collaboration
and no agreement with other observables was checked.

The ZEUS data provide stricter limitations on the GTMDs than the H1 data. We
conclude, based on comparison to different observables, that the calculated
cross sections are only a small fraction of the measured ones which contain
probably also processes with pomeron remnant. Alternatively the experimental
data could be explained by inclusion of $q\bar{q}g$ component.

We present also azimuthal correlations between the sum and the difference of
dijet transverse momenta. The cuts on transverse momenta of jets generate
corresponding azimuthal correlations which can be misidentified as due to
elliptic gluon distributions.

##  I Introduction

The $ep$ scattering at the HERA collider was proven to be an ideal tool for
studying proton structure and in particular gluon distributions in the proton
in the region of very small Bjorken-$x$ or small gluon momentum fraction
Newman and Wing (2014). The dijet production is a particularly valuable final
state in this context. Here we concentrate on the exclusive, or diffractive,
dijet production in the $ep\to ejjp$ reaction, with the final state proton in
its ground state as measured at HERA by the H1 Aaron  _et al._ (2012) and ZEUS
Abramowicz  _et al._ (2016) collaborations. Experimentally, the exclusivity
requirement on the final state is very demanding. Often rather a semiexclusive
reaction $ep\to jjxp$ or even $ep\to jjxp^{*}$ is measured, where $x$ denotes
additional hadronic activity in the diffractive system, besides the dijet,
whereas $p^{*}$ stands for a possible excitation of the proton with particle
production in its fragmentation region.

Diffractive processes, which fulfill the additional requirement of a large
rapidity gap between proton and the diffractive final state, are a testbed on
ideas for the QCD Pomeron. These processes are conveniently described in the
color dipole approach, where the leading order process for diffractive masses
$M^{2}\sim Q^{2}$ is the diffractive excitation of the $q\bar{q}$ state of the
virtual photon Nikolaev and Zakharov (1992). There are several related
approaches to that problem, see for example the reviews Wüsthoff and Martin
(1999); Ashery (2006); Frankfurt  _et al._ (2022). One of them relies on the
Pomeron having partonic structure, and is based on collinear factorization
involving diffractive parton distributions Watt  _et al._ (2007). This
approach strictly applies to inclusive diffraction. Calculations have been
performed to high orders (NNLO) in the perturbative expansion Britzger  _et
al._ (2018). The photoproduction limit, which has applications to
ultraperipheral heavy ion collisions has been addressed in this framework in
Guzey and Klasen (2019).

In this work, we adopt the formalism which starts from the color dipole
approach, but is formulated in momentum space. Here the information contained
in the impact parameter space dipole amplitude is transferred to off-forward
transverse momentum dependent gluon distributions, commonly referred to as
GTMDs (generalized transverse momentum dependent distributions) in the
literature. For reviews which explore the relation to the gluon Wigner
function, see Refs. Pasechnik and Taševský (2023); Boussarie  _et al._ (2023).
At large jet momenta the forward diffractive amplitude probes directly the
unintegrated gluon distribution of the target Nikolaev and Zakharov (1994);
Nikolaev  _et al._ (2001). This approach is appropriate in the small-$x$ or
high-energy limit, and takes into account only the transverse momentum
transfer. The longitudinal momentum transfer, and the so-called skewedness is
taken into account in a collinear factorization formalism based on generalized
parton distributions used in Ref. Braun and Ivanov (2005). This work also
includes $q\bar{q}$ exchanges in the $t$-channel, which are relevant at not
too large rapidity gap. In Ref. Linek  _et al._ (2023) we discussed different
models of GTMDs in application to the $pA\to c\bar{c}pA$ process. In this case
no data is available so far and the measurement is challenging. Here we will
use essentially the same formalism to the case in which both H1 and ZEUS data
are available. We intend to compare the results of the formalism and different
GTMDs to the existing data. Other recent calculations of diffractive dijet
production in the color dipole or GTMD approaches are in Refs. Altinoluk  _et
al._ (2016); Salazar and Schenke (2019); Mäntysaari  _et al._ (2021);
Boussarie  _et al._ (2019); Hagiwara  _et al._ (2016, 2017); Reinke Pelicer
_et al._ (2019); Boer and Setyadi (2021, 2023). Some of these papers
concentrate on photoproduction or heavy quarks only. We have some overlap with
the work Boussarie  _et al._ (2019), who use the Golec-Biernat–Wüsthoff
parametrization Golec-Biernat and Wüsthoff (1998) of the dipole amplitude. For
the corresponding gluon distribution, our results and conclusions agree with
theirs. We also use the GTMDs proposed/fitted in Refs.Boer and Setyadi (2021,
2023), however we differ in our conclusions from these works.

##  II Formalism

In Fig.1 we show the four pQCD diagrams contributing to the exclusive
diffractive production of dijets.

Figure 1: Feynman diagrams for the diffractive production of dijets in
electron-proton collisions, discussed in the present paper.

###  II.1 Kinematic variables

In this paper we will use the standard kinematic variables: for incoming and
outgoing lepton four-momenta $k,k^{\prime}$, we have the photon four-momentum
$q=k-k^{\prime}$, with $q^{2}=-Q^{2}<0$. Let $P$ be the four-momentum of the
incoming proton, then the $ep$ cm-energy squared is $s=(k+P)^{2}\sim 2k\cdot
P$ and the so-called inelasticity $y$ is defined as

$\displaystyle y={q\cdot P\over k\cdot P}\,,$ (1)

so that the square of cms energy of the $\gamma^{*}p$ system $W^{2}_{\gamma
p}=(q+P)^{2}$ is calculated from

$\displaystyle W_{\gamma p}^{2}=ys-Q^{2},$ (2)

where we neglected the proton mass. The standard Bjorken variable $x_{Bj}$

$\displaystyle x_{Bj}={Q^{2}\over 2P\cdot q}={Q^{2}\over Q^{2}+W_{\gamma
p}^{2}-m_{p}^{2}}\,.$ (3)

The diffractive jets carry momentum fractions $z,1-z$ of the photon and
momenta $\vec{p}_{\perp 1},\vec{p}_{\perp 2}$ transverse to the photon-proton
collision axis. The square of the dijet invariant mass is

$\displaystyle M_{jj}^{2}={p_{\perp 1}^{2}+m_{q}^{2}\over z}+{p_{\perp
2}^{2}+m_{q}^{2}\over 1-z}-\Delta_{\perp}^{2},$ (4)

where $\vec{\Delta}_{\perp}=\vec{p}_{\perp 1}+\vec{p}_{\perp 2}$ is the
transverse momentum of the dijet system, which for the exclusive limit of
interest here, equals (negative of) the momentum transfer to the proton. We
also use the standard diffractive variables

$\displaystyle\beta={Q^{2}\over Q^{2}+M_{jj}^{2}},$ (5)

which is the fraction of the Pomeron momentum carried by the struck quark. The
fraction of the proton momentum carried by the pomeron is

$\displaystyle x_{\mathrm{I\\!P}}={x_{Bj}\over\beta}.$ (6)

It is also interesting to calculate the azimuthal angle correlations between
the sum and the difference of the jets transverse momenta

$\displaystyle\cos\phi_{\vec{P}_{\perp}\vec{\Delta}_{\perp}}={\vec{P}_{\perp}\cdot\vec{\Delta}_{\perp}\over
P_{\perp}\Delta_{\perp}}\,,$ (7)

where

$\displaystyle\vec{P}_{\perp}={1\over 2}(\vec{p}_{\perp 1}-\vec{p}_{\perp
2})\,,\qquad\vec{\Delta}_{\perp}=\vec{p}_{\perp 1}+\vec{p}_{\perp 2}\,.$ (8)

We will also use jet rapidities $\eta_{1}$ and $\eta_{2}$ which are important
only when experimental cuts are imposed (see Table 1).

###  II.2 Cross section

In order to compare to HERA data with appropriate experimental cuts, we want
to perform our calculations at the $ep$ level. The cross section can be
written in terms of photon fluxes and $\gamma^{*}p$ cross sections as

$\displaystyle{d\sigma^{ep}\over dydQ^{2}d\xi}$ $\displaystyle=$
$\displaystyle{\alpha_{em}\over\pi yQ^{2}}\Big{[}\Big{(}1-y+{y^{2}\over
2}\Big{)}{d\sigma_{T}^{\gamma^{*}p}\over
d\xi}+(1-y){d\sigma_{L}^{\gamma^{*}p}\over d\xi}\Big{]}.$ (9)

Here $d\xi=dzd^{2}\vec{P}_{\perp}d^{2}\vec{\Delta}_{\perp}$, and $T,L$ stand
for the transverse and longitudinal photons. We have neglected interferences
between photon polarizations, which vanish if one averages over the angle
between electron scattering plane and hadronic plane. The $\gamma^{*}p\to
q\bar{q}p$ cross sections read

$\displaystyle{d\sigma^{\gamma^{*}p}_{T}\over
dzd^{2}\vec{P}_{\perp}d^{2}\vec{\Delta}_{\perp}}$ $\displaystyle=$
$\displaystyle 2N_{c}\alpha_{em}\sum_{f}e_{f}^{2}\int d^{2}\vec{k}_{\perp}\int
d^{2}\vec{k}^{~{}^{\prime}}_{\perp}T\big{(}Y,\vec{k}_{\perp},\vec{\Delta}_{\perp}\big{)}T\big{(}Y,\vec{k}^{~{}^{\prime}}_{\perp},\vec{\Delta}_{\perp}\big{)}$
(10) $\displaystyle\times$
$\displaystyle\Bigg{\\{}\Big{(}z^{2}+(1-z)^{2}\Big{)}\left[{(\vec{P}_{\perp}-\vec{k}_{\perp})\over(\vec{P}_{\perp}-\vec{k}_{\perp})^{2}+\epsilon^{2}}-{\vec{P}_{\perp}\over
P^{2}_{\perp}+\epsilon^{2}}\right]\cdot\left[{(\vec{P}_{\perp}-\vec{k}^{~{}^{\prime}}_{\perp})\over(\vec{P}_{\perp}-\vec{k}^{~{}^{\prime}}_{\perp})^{2}+\epsilon^{2}}-{\vec{P}_{\perp}\over
P^{2}_{\perp}+\epsilon^{2}}\right]$ $\displaystyle+$ $\displaystyle
m_{f}^{2}\left[{1\over(\vec{P}_{\perp}-\vec{k}_{\perp})^{2}+\epsilon^{2}}-{1\over
P^{2}_{\perp}+\epsilon^{2}}\right]\cdot\left[{1\over(\vec{P}_{\perp}-\vec{k}^{~{}^{\prime}}_{\perp})^{2}+\epsilon^{2}}-{1\over
P^{2}_{\perp}+\epsilon^{2}}\right]\Bigg{\\}},$
$\displaystyle{d\sigma^{\gamma^{*}p}_{L}\over
dzd^{2}\vec{P}_{\perp}d^{2}\vec{\Delta}_{\perp}}$ $\displaystyle=$
$\displaystyle
2N_{c}\alpha_{em}4Q^{2}z^{2}(1-z)^{2}\times\sum_{f}e_{f}^{2}\int
d^{2}\vec{k}_{\perp}\int
d^{2}\vec{k}^{~{}^{\prime}}_{\perp}T\big{(}Y,\vec{k}_{\perp},\vec{\Delta}_{\perp}\big{)}T\big{(}Y,\vec{k}^{~{}^{\prime}}_{\perp},\vec{\Delta}_{\perp}\big{)}$
(11) $\displaystyle\times$
$\displaystyle\left[{1\over(\vec{P}_{\perp}-\vec{k}_{\perp})^{2}+\epsilon^{2}}-{1\over
P^{2}_{\perp}+\epsilon^{2}}\right]\cdot\left[{1\over(\vec{P}_{\perp}-\vec{k}^{~{}^{\prime}}_{\perp})^{2}+\epsilon^{2}}-{1\over
P^{2}_{\perp}+\epsilon^{2}}\right],$

with

$\displaystyle\epsilon^{2}=z(1-z)Q^{2}+m_{f}^{2}.$ (12)

Notice, that as e.g. in Refs. Boer and Setyadi (2021, 2023), the transverse
momentum $\vec{\Delta}_{\perp}$ enters only into the GTMD, and the so-called
impact factor that describes the $\gamma^{*}\to q\bar{q}$ transition and
coupling of quarks to gluons, is taken in the forward direction. We do not
discuss the subtleties related to this approximation (see for example Ref.
Linek  _et al._ (2023)), which would become crucial only in the discussion of
more subtle angular correlations. Our formulae here differ e.g. from those in
Boer and Setyadi (2021), where quark masses were neglected. The effect of mass
is important for the $c\bar{c}$ dijets, as will be discussed in the results
section.

The unintegrated gluon density matrix $f$ encodes the same information as the
generalized transverse momentum distribution (GTMD) of gluons in the proton
target. It was proven that in practical numerical applications $f$ and $T$
agree up to the factor $1/2$ (for details see Ref. Linek  _et al._ (2023))

$\displaystyle f\Big{(}Y,{\vec{\Delta}_{\perp}\over
2}+\vec{k}_{\perp},{\vec{\Delta}_{\perp}\over
2}-\vec{k}_{\perp}\Big{)}\to-{1\over
2}\,T(Y,\vec{k}_{\perp},\vec{\Delta}_{\perp})\,.$ (13)

###  II.3 GTMDs

The mentioned GTMD is a representation of the diffraction amplitude in
momentum space often used in the literature (see e.g. Refs. Hagiwara  _et al._
(2016, 2017); Reinke Pelicer  _et al._ (2019); Pasechnik and Taševský (2023))
which is the Fourier transform of the dipole amplitude

$\displaystyle
T(Y,\vec{k}_{\perp},\vec{\Delta}_{\perp})=\int{d^{2}\vec{b}_{\perp}\over(2\pi)^{2}}{d^{2}\vec{r}_{\perp}\over(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\,e^{-i\vec{k}_{\perp}\cdot\vec{r}_{\perp}}\,N(Y,\vec{r}_{\perp},\vec{b}_{\perp})\,.$
(14)

We decided to choose normalization that is consistent with Ref. Reinke Pelicer
_et al._ (2019). However, such a Fourier transform does not converge to zero
and requires regularization, which is often a Gaussian cutoff function

$\displaystyle
T(Y,\vec{k}_{\perp},\vec{\Delta}_{\perp})=\int{d^{2}\vec{b}_{\perp}\over(2\pi)^{2}}{d^{2}\vec{r}_{\perp}\over(2\pi)^{2}}e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}e^{-i\vec{k}_{\perp}\cdot\vec{r}_{\perp}}\,N(Y,\vec{r}_{\perp},\vec{b}_{\perp})\,e^{-\varepsilon
r_{\perp}^{2}}\,.$ (15)

The value of the $\varepsilon$ parameter has a significant impact on the
distribution of the obtained cross-sections, which was shown in Ref. Linek
_et al._ (2023). For the purposes of this analysis, we decided to assume
$\varepsilon=(0.5\,\rm{fm})^{-2}$, similarly to Refs. Reinke Pelicer  _et al._
(2019); Boer and Setyadi (2021); Linek  _et al._ (2023). The Fourier expansion
of the dipole amplitude allows to distinguish an isotropic contribution and an
elliptical term depending on the orientation of the dipole,

$\displaystyle
N(Y,\vec{r}_{\perp},\vec{b}_{\perp})=N_{0}(Y,r_{\perp},b_{\perp})+2\,\cos(2\phi_{br})\,N_{\epsilon}(Y,r_{\perp},b_{\perp})+\dots\,,$
(16)

however we only introduce the isotropic part of the dipole amplitude in the
present analysis.

We consider five different GTMD models. Two of them are parameterizations of
off-forward gluon density matrices for diagonal unintegrated gluon
distribution compatible with the Golec-Biernat-Wüsthoff (GBW model) Golec-
Biernat and Wüsthoff (1998) and Moriggi-Paccini-Machado (MPM model) Moriggi
_et al._ (2020) for which we use the diffractive slope $B=4$
$\textrm{GeV}^{-2}$,

$\displaystyle f\Big{(}Y,{\vec{\Delta}_{\perp}\over
2}+\vec{k}_{\perp},{\vec{\Delta}_{\perp}\over
2}-\vec{k}_{\perp}\Big{)}={\alpha_{s}\over 4\pi N_{c}}\,{{\cal
F}(x_{\mathrm{I\\!P}},\vec{k}_{\perp},-\vec{k}_{\perp})\over
k_{\perp}^{4}}\,\exp\Big{[}-{1\over 2}B\vec{\Delta}^{2}\Big{]}\,.$ (17)

We also included models that are the Fourier transforms of the dipole
amplitude described by equation (15). We use also the bSat model of Kowalski
and Teaney Kowalski and Teaney (2003) (KT model) with parameters from Table I
fit 3 of Kowalski and Teaney (2003) and two models based on the McLerran-
Venugopalan approach McLerran and Venugopalan (1994) \- proposed by Iancu-
Rezaeian Iancu and Rezaeian (2017) (MV-IR model) and the Boer-Setyadi Boer and
Setyadi (2021) (MV-BS model) fitted to the H1 experimental data. Both of these
models are independent of the $x_{\mathrm{I\\!P}}$, therefore we modified the
MV-IR model using $\lambda=0.277$ as

$\displaystyle T_{\rm MV-IR}^{\rm
mod}(Y,\vec{k}_{\perp},\vec{\Delta}_{\perp})=T_{\rm MV-
IR}(\vec{k}_{\perp},\vec{\Delta}_{\perp})\,e^{\lambda
Y},\quad\,Y=\ln\Big{[}{0.01\over x_{\mathrm{I\\!P}}}\Big{]}\,.$ (18)

It should be noted that more realistic extensions of the MV-IR model have been
proposed in the literature, see e.g. Ref. Mäntysaari  _et al._ (2021).
Adapting the MV-BS model according to Boer and Setyadi (2021) we introduce
$\chi=1.25$ (see below),

$\displaystyle N_{0}(r_{\perp},b_{\perp})=-{1\over 4}r_{\perp}^{2}\chi
Q_{s}^{2}(b_{\perp})\ln\Big{[}{1\over r_{\perp}^{2}\lambda^{2}}+e\Big{]},$
(19)

with

$\displaystyle Q_{s}^{2}(b_{\perp})={4\pi\alpha_{s}C_{F}\over
N_{c}}\exp\Big{[}{-b_{\perp}^{2}\over 2R_{p}^{2}}\Big{]}\,.$ (20)

In principle, $\chi$ can be treated as a free parameter to be adjusting to
experimental data.

##  III Numerical results

Here we will present our results for both H1 and ZEUS cuts. The details of the
cuts are described in Aaron  _et al._ (2012) and Abramowicz  _et al._ (2016)
papers and summarized in Table 1.

H1 cuts | ZEUS cuts
---|---
$4<Q^{2}<110\,\rm GeV^{2}$ | $Q^{2}>25\,\rm GeV^{2}\,$
$x_{\mathrm{I\\!P}}<0.1$ | $x_{\mathrm{I\\!P}}<0.01$
$0.05<y<0.7$ | $0.1<y<0.64$
$-1<\eta_{1,2}<2.5$ | $\eta_{1,2}<2$
$p_{\perp 1}>5\,\rm GeV$ | $p_{\perp 1,2}>2\,\rm GeV$
$p_{\perp 2}>4\,\rm GeV\,$ | $M_{jj}>5\,\rm GeV\,$
$\left|t\right|<1\,\rm GeV^{2}$ |  $90<W_{\gamma p}<250\,\rm GeV\,$
Table 1: Cuts used by the H1 and ZEUS collaborations.

In Table 2 we present our phase space integrated cross sections for five
different models of GTMDs. Quite different results are obtained for the
different models. Results for light $q\bar{q}$ and $c\bar{c}$ dijets
production are presented separately. For each model, the $c\bar{c}$
contribution is between 50 and 70$\%$ of the cross section of the light quark
dijets cross section. We placed in this table also experimental cross sections
as measured by the H1 and ZEUS collaborations. All GTMDs except the MV-BS
model underpredict experimental data, while the latter model is in the
ballpark of H1 data but dramatically overpredicts the ZEUS results. Not taking
into account cuts on jet momenta $p_{\perp 1,2}$ results in a several-fold
increase in the total cross-sections. Here the
$x_{\mathrm{I\\!P}}$-independent GTMD Boer and Setyadi (2021) was used.

GTMD | H1, | H1, | H1, | ZEUS, | ZEUS, | ZEUS,
---|---|---|---|---|---|---
approach |  light $q\bar{q}$ |  $ep\to c\bar{c}$ | no $p_{\perp 1,2}$ cuts |  light $q\bar{q}$ |  $ep\to c\bar{c}$ | no $p_{\perp 1,2}$ cuts
|  $\sigma$ $(pb)$ |  $\sigma$ $(pb)$ |  $\sigma$ $(pb)$ |  $\sigma$ $(pb)$ |  $\sigma$ $(pb)$ |  $\sigma$ $(pb)$
GBW | 26.35 | 19.91 | 10900.86 | 13.57 | 6.67 | 337.11
MPM | 147.94 | 108.26 | 10151.00 | 43.61 | 20.47 | 313.17
MV-BS | 404.06 | 269.75 | 10999.73 | 1346.11 | 624.55 | 3117.95
KT | 21.29 | 15.20 | 5957.65 | 12.57 | 5.67 | 52.60
MV-IR | 243.20 | 155.21 | 11784.75 | 37.83 | 17.62 | 91.18
DATA | 254 | - | 72 | -
Table 2: Total cross section for H1 and ZEUS conditions and different
approaches.

Figure 2: $Q^{2}$ dependence of the cross section for H1 (left) and ZEUS
(right) kinematics for different GTMDs.

Figure 3: Jet transverse momentum dependence of the cross section for H1
(left) and ZEUS (right) kinematics for different GTMDs.

Figure 4: Mandelstam $t$ dependence of of the cross section for H1 (left) and
ZEUS (right) kinematics for different GTMDs.

Figure 5: Inelasticity dependence of the cross section for H1 (left) and ZEUS
(right) kinematics for different GTMDs.

Figure 6: $x_{\mathrm{I\\!P}}$ distribution dependence of the cross section
for H1 (left) and ZEUS (right) kinematics for different GTMDs.

Figure 7: $\beta$ dependence of the cross section for H1 (left) and ZEUS
(right) kinematics for different GTMDs.

We now turn to the discussion of distributions in various kinematical
variables. We show distributions in the $Q^{2}$, jet transverse momentum,
Mandelstam $t$, inelasticity $y$, pomeron $x$ and standard diffractive
variable $\beta$ both for H1 and ZEUS kinematics along with the corresponding
experimental data. In Fig. 2, we show the cross section differential in
$Q^{2}$. At large $Q^{2}>20\,\rm{GeV}^{2}$, for all GTMDs, the shape of the
cross section reasonably well agrees with the H1 data. However there are large
differences in the normalization of the cross section, as is evident already
from Table 2. The largest contribution is obtained for the MV-BS model. For
the kinematics of the ZEUS data, the MV-BS result differs by almost two orders
of magnitude from other GTMDs. A similar good agreement of the shapes of
distributions is observed for the jet transverse momentum, see Fig.3. However
in the $t$ distribution shown in Fig. 4 the slope of MV-BS model is slightly
different from the other GTMDs.The Boer-Setyadi GTMD gives results that are
close to the H1 data. So at first look the distributions seems to be supported
by the data. The distribution in inelasticity (or electron energy loss) $y$ is
shown in Fig.5. The shape is reasonably well reproduced for all distributions,
although the falloff towards large $y$ is a bit too strong.

The situation changes for the distribution in $\log_{10}x_{\mathrm{I\\!P}}$,
shown in Fig. 6. notice, that $\Delta Y\equiv\ln(1/x_{\mathrm{I\\!P}})$ is
essentially the rapidity gap between proton and diffractive system. Here the
MV-BS distribution is evidently disfavoured by the H1 data, similarly to the
MV-IR and MPM models whereas the GBW and KT GTMDs give cross section which is
not in contradiction with the data. It is worth noting that in the H1
kinematics we have rather large $x_{\mathrm{I\\!P}}>0.01$. Here one does not
expect the Pomeron exchange to dominate, but rather the contribution of
$q\bar{q}$ exchanges described by the secondary Reggeons will take over. See
for example the case of $pp$ scattering in Ref. Łuszczak  _et al._ (2015),
where secondary Reggeons give a sizeable contribution. Therefore, it should
not be expected that all experimental data will be correctly reproduced by the
dipole approach which is appropriately used for, say, small
$x_{\mathrm{I\\!P}}<0.01$. In fact is the ZEUS data for which one can expect
the Pomeron or gluon GTMD contribution to be appropriate.

The ZEUS collaboration has measured the distribution in $\beta$, which we show
for the kinematic cuts of both H1 and ZEUS in Fig.7. Here we see, that the MV-
BS GTMD dramatically overpredict the ZEUS data while MPM and MV-IR do not
exceed the values of experimental data within their measurement uncertainties.
It appears that the Boer-Setyadi GTMD distribution is artificially large in
order to describe the H1 data, where gluons arguably do not dominate. The
distributions in $x_{\mathrm{I\\!P}}$ and $\beta$ show that the agreement of
the MV-BS model above is rather artificial. We caution the reader, that even
for the ZEUS data it would likely be inappropriate to fit the GTMD to data
using the present approach. There are many indications Abramowicz  _et al._
(2016), that the $q\bar{q}g$ three-parton contribution is important in this
context (see e.g. Ref. Bartels  _et al._ (1999)). It seems that the KT and GBW
models are the most realistic ones.

For completeness in Fig. 8 we show distribution in azimuthal angle between the
sum and the difference of the jet transverse momenta. Such studies were
performed before for $c\bar{c}$ correlations in $pA$ collisions Linek  _et
al._ (2023). We predict interesting azimuthal correlations for dijets at HERA.
The straight horizontal line corresponds to the case without cuts on jet
transverse momenta. The figures show that the correlation is mainly due to the
cuts. In principle, such azimuthal correlation can be also due to elliptical
gluon distributions Linek  _et al._ (2023), not taken into account in the
present paper. The observed effect can be therefore easily misinterpreted as
observation of an elliptical glue. Of course, the elliptical glue can generate
further effect on top of azimuthal correlations generated by the cut effect.

In addition we show distributions in $M_{jj}$, $z$, $\phi_{jj}$ and $W_{\gamma
p}$ in Fig. 9, 10, 11, 12, respectively.

The distributions in $M_{jj}$ are characterized by almost identical shapes for
all GTMD models and both kinematic ranges, in contrast to the distribution in
$z$, see Fig. 10. The maximum is visible for H1 cuts in this example at
$z=0.5$ for all models except MV-BS, the distribution of which is almost flat
in the range of $0.1-0.9$. The remaining models differ as to the value of the
mentioned maximum. The situation is the opposite in the ZUES kinematics, where
for $z=0.5$ there is a minimum of the distributions of all models.
The high peak at $\phi_{jj}=180\deg$ in the distribution in the azimuthal
angle between produced jets for the H1 kinematics from Fig. 11 shows that the
jets are almost back-to-back, while in the case of ZEUS cuts that effect is
less pronounced.

Taking into account the distributions discussed above, we conclude that the H1
Aaron  _et al._ (2012) and ZEUS Abramowicz  _et al._ (2016) data contain
mechanisms other than those shown in Fig. 1, which will be the subject of
further investigations.

Figure 8: Distributions in the azimuthal angle $\phi$ between
$\vec{P}_{\perp}$ and $\vec{\Delta}_{\perp}$ normalised to the total cross
section for the H1 and ZEUS kinematics for different GTMDs.

Figure 9: Invariant mass dependence of the cross section for H1 (left) and
ZEUS (right) kinematics for different GTMDs.

Figure 10: $z$ dependence of the cross section for H1 (left) and ZEUS (right)
kinematics for different GTMDs. Please note the normalization.

Figure 11: Azimuthal angle between produced jets dependence of the cross
section for H1 (left) and ZEUS (right) kinematics for different GTMDs. Please
note the normalization.

Figure 12: $W_{\gamma p}$ dependence of the cross section for H1 (left) and
ZEUS (right) kinematics for different GTMDs.

Figure 13: $Q^{2}$ distributions of transverse and longitudinal components of
the cross section for the MPM and the KT GTMDs for H1 (left) and ZEUS (right)
kinematics without $Q^{2}$ cuts.

Figure 14: $z$ distributions of transverse and longitudinal components of the
cross section for the MPM and the KT GTMDs for H1 (left) and ZEUS (right)
kinematics.

Finally in Fig. 13 and Fig. 14 we wish to show separately the transverse and
longitudinal components to the cross section as a function of photon
virtuality and light-front momentum fraction. At high $Q^{2}$ the longitudinal
component dominates only for the H1 kinematics, which is associated with their
behavior in the $z$ distributions. The relative relation between contributions
depends on the model of GTMD.

Up to now we include only light-quark jets similarly to ref. Boer and Setyadi
(2021). In Fig. 15, 16, 17 we show what happens when we add in addition
$c\bar{c}$ dijets. Including $c\bar{c}$ may cause increase of the cross
section by as much as $50\%$ for ZEUS and $70\%$ for H1 kinematics. This goes
beyond the scope of the present analysis.

Another topic which requires further studies is inclusion of Sudakov effects
(see e.g. Hatta  _et al._ (2020)), but this goes beyond the scope of the
present paper.

As shown above, the MV-BS model overtimes some distributions, therefore we
also show distributions in $\beta$ and Mandelstam $t$ of the MV-BS model
taking into account four different values of the $\chi$ factor in Fig. 18. It
is shown that its value have no impact to the shapes of distributions but
changes their normalization of the cross section.

Figure 15: Contributions of light and $c\bar{c}$ dijets as a function of
$\beta$ for the KT GTMD for H1 (left) and ZEUS (right) kinematics.

Figure 16: Contributions of light and $c\bar{c}$ dijets as a function of
$M_{jj}$ for the KT GTMD for H1 (left) and ZEUS (right) kinematics.

Figure 17: Contributions of light and $c\bar{c}$ dijets as a function of
$Q^{2}$ for the KT GTMD for H1 (left) and ZEUS (right) kinematics.

Figure 18: $\beta$ and $t$ distributions for ZEUS kinematic for the MV-BS
model with different $\chi$ values.

##  IV Conclusions

In the present paper we have discussed dijet correlations in the $ep\to epjj$
process. The corresponding differential distributions have been calculated
using gluon GTMDs (generalized transverse momentum dependent gluon
distributions) of the proton. Different GTMDs from the literature have been
employed. We have calculated distributions in many kinematical variables and
compared our results to existing H1 and ZEUS data. The Boer-Setyadi MPM and
MV-IR GTMDs reasonably well describe many observables for the H1 kinematics,
but fail to describe distributions in $x_{\mathrm{I\\!P}}$ distribution and
strongly overshoots the cross section differential in $\beta$ for the ZEUS
kinematics. Here it needs to be pointed out that it is the ZEUS kinematics,
where gluon GTMDs are expected to contribute. Some of the other GTMD
distributions are consistent with the H1 and ZEUS data. The most realistic
gluon distributions appear to be the KT and GBW GTMDs, which give a very small
contribution for H1, as it should be, and give a sizeable contribution at
$\beta>0.5$ for the ZEUS cuts. This means that the considered mechanism is not
sufficient. In view of the analysis presented in Abramowicz  _et al._ (2016),
one should include also the diffractive excitation of the $q\bar{q}g$
component. Alternatively, one could try to explain the data within the
resolved pomeron picture. This will be done elsewhere.

We have also calculated azimuthal angle correlations between the sum and
difference of jet transverse momenta. As we use forward impact factors and our
GTMDs have no elliptic part, these correlations are purely an effect of
experimental cuts. As we have demonstrated, the cuts on jet transverse momenta
generate modulation in $\phi_{\Delta_{\perp}P_{\perp}}$ which could be
misidentified as the presence of elliptic gluon distribution.

In Ref. Boussarie  _et al._ (2019) only the GBW distribution was used for the
dijet production, and the results shown in this paper are similar to our
calculations for the same model of GTMD. Our analysis includes also a
comparison of several different GTMD models and presents the distributions in
many more kinematic variables.

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# Detection of hidden photon dark matter using the direct excitation of
transmon qubits

Shion Chen(a), Hajime Fukuda(b), Toshiaki Inada(a), Takeo Moroi(b,c)†, Tatsumi
Nitta(a), Thanaporn Sichanugrist(b) (a)International Center for Elementary
Particle Physics (ICEPP), The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-0033, Japan
(b)Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-0033, Japan
(c)QUP (WPI), KEK, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan

###### Abstract

We propose a novel dark matter detection method utilizing the excitation of
superconducting transmon qubits. Assuming the hidden photon dark matter of a
mass of $O(10)\ \mu{\rm eV}$, the classical wave-matter oscillation induces an
effective ac electric field via the small kinetic mixing with the ordinary
photon. This serves as a coherent drive field for a qubit when it is resonant,
evolving it from the ground state towards the first-excited state. We evaluate
the rate of such evolution and observable excitations in the measurements, as
well as the search sensitivity to the hidden photon dark matter. For a
selected mass, one can reach $\epsilon\sim 10^{-12}-10^{-14}$ (where
$\epsilon$ is the kinetic mixing parameter of the hidden photon) with a single
standard transmon qubit. A simple extension to the frequency-tunable SQUID-
based transmon enables the mass scan to cover the whole $4-40\ \mu{\rm eV}$
($1-10\ {\rm GHz}$) range within a reasonable length of run time. The
sensitivity scalability along the number of the qubits also makes it a
promising platform in accord to the rapid evolution of the superconducting
quantum computer technology.

†Corresponding author.

Introduction: After the discovery of the anomalous velocity behavior of the
galaxies in the Coma cluster reported by F. Zwicky [1, 2], the hypothetical
mass source – dark matter (DM) – has been an outstanding mystery in physics.
Even though various strong indirect evidences have been established through
astrophysical and cosmological observations, the absence of the direct
detection leaves its particle-physics properties still largely unknown.

Numerous efforts have been implemented for the direct DM detection.
Experiments using nuclear (electron) recoil processes provide excellent
sensitivities to DM of a mass around GeV to TeV (GeV to sub-GeV) scale (for a
dedicated review, see, e.g., Ref [3]). The recoil technique becomes less
effective for DM lighter than $\sim 10\ {\rm MeV}$ due to the recoil energy
below the detection threshold that is typically $O(1)\ {\rm keV}$.

Different detection regimes are clearly needed for exploring lighter DM. For
DM weakly coupled to electromagnetic interaction (e.g., the hidden photon or
axion), haloscope experiments [4, 5] using microwave cavities have provided
the leading sensitivity below $O(1)\ {\rm meV}$ [6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 18].
Since such light DM can be treated as a classical matter-wave due to its high
number density within their de Broglie wavelength, the DM-converted photons
can be accumulated in the matched resonant modes. This results in a detectably
sizable electric signal read out from an antenna.

Alternatively, it has also been pointed out that the excitation processes in
condensed-matter systems can be used to probe very light DM (see, e.g., Ref
[32]). Since the energy gaps in condensed-matter systems are generally much
smaller than $O(1)\ {\rm keV}$, the DM absorption may exhibit distinctive
excitation signatures. The goal of this letter is to extend this idea to the
superconducting quantum bits (qubits), known as artificial atoms, by
exploiting their favorable features: a strong coupling to electric fields,
tunable energy gaps, and the precise readout/control functionality enabling
easy detection and manipulation of the excitation. We focus on the two-level
system of the lowest two energy states of the qubits, namely, the ground state
($|g\rangle$) and the first excited state ($|e\rangle$). The energy gap is
typically $O(1-10)\ \mu{\rm eV}$, corresponding to the frequency of $O(1-10)\
{\rm GHz}$. The strong coupling allows the efficient DM absorption of a mass
corresponding to the energy gap, driving the qubits from $|g\rangle$ to
$|e\rangle$.

In this study, we target the oscillating hidden photon field as the DM. While
being a well-motivated DM candidate, it is also a natural constituent arising
from a large class of the string-inspired model in particle physics [33].
Through the kinetic mixing with the photon, the hidden photon oscillation
yields a weak coherent effective electromagnetic field which causes the
$|g\rangle\leftrightarrow|e\rangle$ transition. We estimate the transition
rate from $|g\rangle$ to $|e\rangle$ driven by this feeble coherent field
assuming a standard transmon qubit architecture [34]. We find that, even with
a single qubit in a typical specification of today’s transmon, one can
reasonably probe the unexplored parameter region of the hidden photon. A wide
frequency tunability ($\sim 1-10\ {\rm GHz}$) is easily acquired by extending
to a transmon based on Superconducting Quantum Interference Device (SQUID).
This is particularly remarkable given the much less required engineering
effort compared to the typical haloscope experiments targeting a similar
frequency range. The sensitivity scalability along the number of qubits is
also discussed for future prospects.

Transmon qubit and hidden photon: Let us start with introducing the
interaction between the transmon qubit and hidden photon. The natural unit
(i.e., $c=\hbar=1$) is used throughout this letter.

A transmon qubit is modeled as a closed circuit loop consisting of a capacitor
element and a non-linear inductance realized by a Josephson junction or a
SQUID. The Hamiltonian for the system is described by

$\displaystyle{\cal H}_{0}=\frac{1}{2}CV^{2}-J\cos\hat{\theta},$ (1)

where $\hat{\theta}$ is the phase difference across the Josephson junction,
and $C$ is the capacitance. $J$ is positive-valued in our convention. This is
constant when a Josephson junction is considered as the inductance element,
while it is tunable for a SQUID through the magnetic flux bias applied. The
voltage difference $V$ between the Josephson junction is related to
$\hat{\theta}$ as

$\displaystyle V=(2e)^{-1}\dot{\hat{\theta}},$ (2)

where $e$ is the electric charge of the electron. The conjugate momentum of
$\hat{\theta}$, denoted as $\hat{n}$, is introduced to proceed with the
canonical quantization:

$\displaystyle\hat{n}\equiv Z\dot{\hat{\theta}},$ (3)

with

$\displaystyle Z\equiv(2e)^{-2}C.$ (4)

Notice that $\hat{n}=CV/2e$ can be regarded as the total charge in units of
$2e$, and $\hat{\theta}$ and $\hat{n}$ satisfy the commutation relation
$[\hat{\theta},\hat{n}]=i$. The Hamiltonian is then written as

$\displaystyle{\cal
H}_{0}=\frac{1}{2Z}\hat{n}^{2}-J\cos\hat{\theta}=\frac{1}{2C}(2e\hat{n})^{2}-J\cos\hat{\theta}.$
(5)

The energy levels of this system are unequally spaced; the ground and the
first excited states, denoted as $|g\rangle$ and $|e\rangle$, respectively,
are used for the transmon qubit. The excitation energy from $|g\rangle$ to
$|e\rangle$ is denoted as $\omega$. Then, the Hamiltonian of the system is
reduced to approximately

$\displaystyle{\cal H}_{0}=\omega|e\rangle\langle e|.$ (6)

For convenience of the later discussion, we also define

$\displaystyle\hat{a}\equiv\frac{1}{\sqrt{2\omega Z}}\left(\hat{n}-i\omega
Z\hat{\theta}\right),~{}~{}~{}\hat{a}^{\dagger}\equiv\frac{1}{\sqrt{2\omega
Z}}\left(\hat{n}+i\omega Z\hat{\theta}\right),$ (7)

which satisfy $[\hat{a},\hat{a}^{\dagger}]=1$. These correspond to the
annihilation and creation operators when approximating the potential as a
parabolic one, where $J$ and $\omega$ are related as $J\simeq Z\omega^{2}$.

If there exists the hidden photon DM, an effective electric field is induced
by the hidden photon oscillation. In the mass-eigenstate basis, the
interaction terms of the electron field $\Psi_{e}$ with the electromagnetic
(EM) photon $A_{\mu}$ and hidden photon $X_{\mu}$ are given by

$\displaystyle{\cal L}_{\rm
int}=e\bar{\Psi}_{e}\gamma^{\mu}\left(A_{\mu}+\epsilon
X_{\mu}\right)\Psi_{e},$ (8)

where $\epsilon$ is the kinetic-mixing parameter. Assuming that the
oscillating hidden photon is the DM, we denote the hidden photon field around
the earth as

$\displaystyle\vec{X}=\bar{X}\vec{n}_{X}\cos m_{X}t,$ (9)

where $m_{X}$ is the hidden photon mass, $\bar{X}$ is the amplitude of the
oscillation and $\vec{n}_{X}$ is the unit vector pointing to the direction of
$\vec{X}$. The amplitude is related to the local density of the DM as

$\displaystyle\rho_{\rm DM}=\frac{1}{2}m_{X}^{2}\bar{X}^{2}.$ (10)

The effective electric field which a qubit would sense is given by

$\displaystyle\vec{E}^{\rm(eff)}=\vec{E}^{\rm(EM)}+\vec{E}^{(X)},$ (11)

where $\vec{E}^{(X)}$ is the field induced by the DM, and $\vec{E}^{\rm(EM)}$
is the peripheral ordinary electric field considered in the later discussion.
Using Eqs. (9) and (10) one obtains:

$\displaystyle\vec{E}^{(X)}=-\epsilon\dot{\vec{X}}=\bar{E}^{(X)}\vec{n}_{X}\sin
m_{X}t,$ (12)

with

$\displaystyle\bar{E}^{(X)}\equiv\epsilon
m_{X}\bar{X}=\epsilon\sqrt{2\rho_{\rm DM}}.$ (13)

Hereafter, we consider the case that $\vec{E}^{\rm(EM)}$ and $\vec{E}^{(X)}$
have the same time dependence; this is the case in particular when considering
a qubit located in a cavity-like metallic package as discussed later. We
parameterize the total effective electric field as

$\displaystyle\vec{E}^{\rm(eff)}=\bar{E}^{\rm(eff)}\vec{n}_{E}\sin m_{X}t,$
(14)

where $\vec{n}_{E}$ is the unit vector pointing to the direction of
$\vec{E}^{\rm(eff)}$. We also define a coefficient:

$\displaystyle\kappa\equiv\frac{\bar{E}^{\rm(eff)}}{\bar{E}^{(X)}}.$ (15)

With the effective electric field, the voltage difference of the capacitor
becomes $V+d\bar{E}^{\rm(eff)}\cos\Theta\sin m_{X}t$, with $\Theta$ being the
angle between $\vec{n}_{E}$ and the normal vector of the conductor plate.
Concentrating on the terms up to the linear order in $\epsilon$, the
Hamiltonian is modified as ${\cal H}={\cal H}_{0}+\Delta{\cal H}$ such that

$\displaystyle\Delta{\cal H}=CVd\bar{E}^{\rm(eff)}\cos\Theta\sin
m_{X}t=2\eta\sin m_{X}t(\hat{a}+\hat{a}^{\dagger}),$ (16)

where

$\displaystyle\eta\equiv\frac{1}{2\sqrt{2}}d\bar{E}^{\rm(eff)}\sqrt{C\omega}\cos\Theta=\frac{1}{2}\epsilon\kappa
d\sqrt{C\omega\rho_{\rm DM}}\cos\Theta.$ (17)

Calculating the matrix elements $\langle g|\Delta{\cal H}|e\rangle$ and
$\langle e|\Delta{\cal H}|g\rangle$ presuming that the excited state is well
approximated as $|e\rangle\simeq\hat{a}^{\dagger}|g\rangle$, the following
effective Hamiltonian expression is obtained:

$\displaystyle{\cal H}=\omega|e\rangle\langle e|+2\eta\sin
m_{X}t(|e\rangle\langle g|+|g\rangle\langle e|),$ (18)

describing the interaction between the transmon qubit and the hidden photon
DM.

Time evolution of the qubit: Now we show that the DM-induced field
$\bar{E}^{\rm(eff)}$ causes the Rabi oscillation of the qubit, a coherent
drive between $|g\rangle$ and $|e\rangle$. For a qubit state

$\displaystyle|\Psi(t)\rangle=\psi_{g}(t)|g\rangle+e^{-i\omega
t}\psi_{e}(t)|e\rangle,$ (19)

the time evolution is given by

$\displaystyle i\frac{d}{dt}|\Psi(t)\rangle={\cal H}|\Psi(t)\rangle,$ (20)

namely,

$\displaystyle i\frac{d}{dt}\left(\begin{array}[]{c}\psi_{g}\\\
\psi_{e}\end{array}\right)=\left(\begin{array}[]{cc}0&-i\eta(e^{-i(\omega-
m_{X})t}+e^{-i(\omega+m_{X})t})\\\ i\eta(e^{i(\omega-
m_{X})t}+e^{i(\omega+m_{X})t})&0\end{array}\right)\left(\begin{array}[]{c}\psi_{g}\\\
\psi_{e}\end{array}\right).$ (27)

Suppose that the qubit frequency is tuned to be equal to the hidden photon
mass, i.e., $\omega=m_{X}$. Neglecting the fast oscillating component
(rotation wave approximation), the evolution equation reduces to

$\displaystyle i\frac{d}{dt}\left(\begin{array}[]{c}\psi_{g}\\\
\psi_{e}\end{array}\right)\simeq\left(\begin{array}[]{cc}0&-i\eta\\\
i\eta&0\end{array}\right)\left(\begin{array}[]{c}\psi_{g}\\\
\psi_{e}\end{array}\right).$ (34)

Assuming that the qubit is initially at the ground state, i.e.,
$\psi_{g}(0)=1$ and $\psi_{e}(0)=0$, we obtain

$\displaystyle\psi_{g}(t)\simeq\cos\eta t,~{}~{}~{}\psi_{e}(t)\simeq\sin\eta
t.$ (35)

The transition probability from the ground state to the excited state is
$p_{ge}(t)=|\psi_{e}(t)|^{2}\simeq\sin^{2}\eta t$, corresponding to a Rabi
oscillation at a frequency of $\eta$.

Note that the discussion above is valid only within the coherence time of the
system $\tau$, i.e., $t<\tau$. The coherence time can be defined for the DM
and the qubit individually ($\tau_{X}$ and $\tau_{\rm q}$, respectively). The
former is estimated to be $\tau_{X}\sim 2\pi/m_{X}v_{X}^{2}$ with $v_{X}\sim
10^{-3}$ being the hidden photon velocity. For the latter, the longitudinal
coherent time $T_{1}$ is relevant here since the dephasing is highly
suppressed in the transmon limit ($ZJ\ll 1$) [34]. A $T_{1}$ of $\sim
100~{}\mu{\rm s}$ is commonly achieved in the recent experiments [35, 36, 37].
The coherence time for the system is dictated by the shorter one, i.e.,
$\tau\simeq\mbox{min}(\tau_{X},\tau_{\rm q})$, which is usually given by that
of the qubit. Hereafter, we parameterize $\tau\equiv 2\pi Q/\omega$, with $Q$
being the quality factor.

Assuming that $\tau\ll\eta^{-1}$, the transition probability from $|g\rangle$
to $|e\rangle$ within the coherence time is evaluated as

$\displaystyle p_{*}\equiv p_{ge}(\tau)\simeq(\eta\tau)^{2}.$ (36)

Numerically, the transition probability is given by

$\displaystyle p_{ge}(\tau)\simeq$
$\displaystyle\,0.12\times\kappa^{2}\cos^{2}\Theta\left(\frac{\epsilon}{10^{-10}}\right)^{2}\left(\frac{f}{1\
{\rm GHz}}\right)$ $\displaystyle\,\times\left(\frac{\tau}{100\ \mu{\rm
s}}\right)^{2}\left(\frac{C}{0.1\ {\rm pF}}\right)\left(\frac{d}{10\ \mu{\rm
m}}\right)^{2}$ $\displaystyle\,\times\left(\frac{\rho_{\rm DM}}{0.45\ {\rm
GeV/cm}^{3}}\right),$ (37)

where $f\equiv\omega/2\pi$, which is related to the hidden photon mass as

$\displaystyle f\simeq 0.24\ {\rm GHz}\times\left(\frac{m_{X}}{1\ \mu{\rm
eV}}\right)$ (38)

when $\omega=m_{X}$.

For a typical transmon qubit, the circuit parameters are $C\sim O(0.1)\ {\rm
pF}$, $d\sim O(10-100)\ \mu{\rm m}$, and $f\sim 1-10\ {\rm GHz}$. One finds
that the excitation probability after a $100~{}{\rm\mu s}$ of evolution can be
as high as 10% for $\epsilon=10^{-11}$, which is about the current general
limit set by the cosmological and astrophysical constraints for $m_{X}\sim
1-10\ \mu{\rm eV}$ [38].

Experimental setup and the measurement protocol: A typical setup for transmon
measurements [39] is employed in the search, which is briefly described as
follows. A transmon and the readout resonator are fabricated on a silicon or
sapphire substrate chip, which are capacitively coupled through a Coplanar
Waveguide (CPW) transmission line on the same chip. The chip is mounted on a
printed circuit board to interface the CPW and the coaxial cables via the wire
bonding. These are packaged in a metallic shield thermally attached to the
mixing chamber plate of the dilution refrigerator and cooled down to $\sim 10$
mK. The standard dispersive readout [40] is used to determine the qubit state,
i.e., a microwave pulse is sent to the readout resonator on the qubit chip and
the state-dependent amplitude/phase of the transmitted signal is measured. The
transmitted signal undergoes the staged amplification by first the quantum
amplifiers (e.g., Josephson parametric amplifier [41] or Josephson traveling
wave parametric amplifier [42]), heterostructure field effect transistor
amplifiers, and the ordinary room temperature amplifiers to achieve a good
single-shot readout fidelity.

For the detection of the DM signal, we consider the following protocol. A
single or multiple SQUID-based transmon qubit(s) are used for the search,
where the number of qubits is denoted as $n_{\rm q}$.

  * 1.

All the qubits are set to the ground state at $t=0$, and are left for the
evolution until $t=\tau$.

  * 2.

The qubit states are then read out. A single-shot readout is assumed with a
$O(100)~{}{\rm ns}$ of readout time.

  * 3.

For a given $\omega$, the process 1–2 are repeated $n_{\rm rep}$ times. The
total number of the qubit measurements is $N_{\rm try}\equiv n_{\rm q}n_{\rm
rep}$. The number of expected excitations is $N_{\rm sig}=p_{*}N_{\rm try}$,
ignoring the inter-qubit interference.

  * 4.

The process 3 is repeated in different qubit frequency $\omega$ by detuning
the SQUID-based transmon.

The search is a resonant-finding where the DM signal exhibits a sharp peak on
the continuum baseline in the excitation rate, at a particular qubit frequency
corresponding to the DM mass.

Observation of the false positive excitations (“background events”) needs to
be taken into account. In the regime of $T>1$ mK (with $T$ being the
temperature of the qubits) the dominant source of the background events is
considered to be the thermal excitation. The expected number of such
background events is

$\displaystyle N_{\rm bkg}=e^{-\omega/T}N_{\rm try}.$ (39)

In a typical dilution refrigerator, $T=10$ mK can be solidly achieved. One may
even expect $T\sim 1$ mK or lower by introducing Continuous Nuclear
Demagnetization Refrigerator (CNDR) [43]. However, it has been argued that the
actual temperature of the qubits may be different due to the quasi-particle
creation processes inside the qubit chip (e.g., by cosmic ray radiation).
These cause a significant level of excitation reportedly equivalent to a
thermal residual of $\sim 30~{}{\rm mK}$ [44]. Hereafter, we assume two
temperature scenarios, i.e., 30 mK as the conservative estimate based on the
currently available setup, and 1 mK assuming the usage of CNDR and the
circumvention of the quasi-particle problem in the future.

The readout error can be also a potential source of the background given that
the typically achieved single-shot fidelity is about 99 % [45]. However, since
this is often limited by the longitudinal decoherence of the $|e\rangle$
during the measurement, it can be drastically improved by utilizing the higher
excited states [46] when one only needs to distinguish between the ground
state and the others. It also worths pointing out that the background level
can be measured and validated using the “sideband” given the nature of
resonant-finding searches where signals can only peak at a specific
frequencies.

One unique feature about this search is that the signal excitation is a
coherent Rabi oscillation. Therefore, the qubit frequencies with suspiciously
high excitation rates can be further re-examined by explicitly performing the
Rabi oscillation measurements with various time interval. Particularly for
relatively large $\epsilon$ ($>10^{-12}$) yielding high $p_{*}$ ($>10\ \%$),
the signal would reveal a characteristic sinusoidal variation in the
$|e\rangle$ population as the function of time lapse before decohering. It
would be more distinct in case of multiple qubits as all of them experience
the same coherent excitation by signals, helping discerning from spurious
backgrounds and claiming the discovery.

The packaging effect on the field: The effect of the qubit packaging on the
electric field requires a careful consideration. As it is often a metallic
container electrically covering the qubit chip, it effectively becomes a
microwave cavity. The effective electric field $\vec{E}^{\rm(eff)}$ projected
onto the cavity wall would vanish at the cavity wall, since the secondary
field ($\vec{E}^{\rm(EM)}$) is provoked by the electrons in the metal reacting
to the DM-included field $\vec{E}^{\rm(X)}$. Importantly, however, since the
phases of $\vec{E}^{\rm(EM)}$ and $\vec{E}^{(X)}$ have different spatial
evolution, the field cancellation becomes imperfect off the wall. While
$\vec{E}^{(X)}$ can be regarded as spatially homogeneous since the interaction
of hidden photon is extremely weak and the effects of the cavity on the
dynamics of hidden photon can be safely ignored, $\vec{E}^{\rm(EM)}$ is
dependent on the position $\vec{x}=(x,y,z)$. The field configuration of
$\vec{E}^{\rm(EM)}$ inside the cavity is obtained by solving
$\Box\vec{E}^{\rm(EM)}=0$ and $\vec{\nabla}\vec{E}^{\rm(EM)}=0$
simultaneously, with the boundary condition at the cavity wall,
$[\vec{E}^{\rm(EM)}_{\parallel}+\vec{E}^{(X)}_{\parallel}]_{\rm wall}=0$,
where the subscript “$\parallel$” indicates vectors projected onto the cavity
wall. Consider, e.g., a case where a transmon is placed inside a cylinder-
shaped cavity with its conductor plate perpendicular to the cylinder axis
(defined as $z$-axis). As only the $z$-component of the fields are relevant,
we find

$\displaystyle{E}^{\rm(EM)}_{z}(\vec{x})=-\frac{J_{0}(m_{X}r)}{J_{0}(m_{X}R)}{E}^{(X)}_{z},$
(40)

where $r\equiv\sqrt{x^{2}+y^{2}}$ is the radial distance from the cylinder
axis, $R$ is the radius of the cylinder, and $J_{0}$ is the Bessel function of
the first kind. Note that $m_{X}$ is assumed not equal to any of the cavity
mode frequencies here, which ensures $J_{0}(m_{X}R)\neq 0$. Then, for the
qubit located at $r=r_{\rm q}$, we obtain a $\kappa$ factor (defined in Eq.
(15)):

$\displaystyle\kappa^{\rm(cylinder)}=1-\frac{J_{0}(m_{X}r_{\rm
q})}{J_{0}(m_{X}R)},$ (41)

representing the enhancement or reduction on the total field due to the cavity
effect. Notably, when $m_{X}$ is close to some of the cavity mode frequencies
$\omega_{\rm c}$ satisfying $J_{0}(\omega_{\rm c}R)=0$, $|\kappa|$ can be much
greater than unity and the excitation rate is enhanced. In general, $\kappa$
depends on the geometry of the cavity as well as on the location of the qubit.
A detailed analysis is beyond the scope of this letter. We just comment here
that, if the size of the cavity is of the order of $\omega^{-1}$, $|\kappa|$
is expected to be $\sim O(1)$ or larger.

Sensitivity estimation: The signal rate is calculated from Eq. (37) with
$C=0.5\ {\rm pF}$, $d=100\ \mu{\rm m}$, $Q=10^{6}$, $\kappa=1$, $\rho_{\rm
DM}=0.45\ {\rm GeV}/{\rm cm}^{3}$, and with the angular average
$\cos^{2}\Theta\rightarrow\frac{1}{3}$ being taken. The number of qubits are
assumed to be either $n_{\rm q}=1$ or $100$. For the background, only thermal
excitation contribution is considered assuming $T=30\ {\rm mK}$ or $1\ {\rm
mK}$. The search sensitivity is derived by comparing $N_{\rm sig}$ and $N_{\rm
bkg}$. Here, we apply a simple form of $N_{\rm sig}/\sqrt{N_{\rm bkg}}$ as the
proxy to the significance in the unit of Gaussian-equivalent standard
deviation. The following criterion is used for the DM detection in the study:

$\displaystyle N_{\rm sig}>\mbox{max}(3,5\sqrt{N_{\rm bkg}}),$ (42)

which requires either $5\sigma$ where there is substantial amount of
backgrounds, or minimum 3 signal events in a highly background-free regime.

The qubit frequency scan is considered in the range of $1\leq f\leq 10\ {\rm
GHz}$, corresponding to the DM mass of $4-40\ \mu{\rm eV}$. The step width is
$\Delta\omega=\omega/Q$. Taking $Q=10^{6}$, the number of the scan points is
$\sim 2\times 10^{6}$, and the measurement time for each scan point is taken
to be $\sim 14\ {\rm sec}$ which is chosen so as the total time for the scan
fits within one year. The readout time ($O(100~{}{\rm ns})$) and the interval
between the readout ($10~{}\mu{\rm s}$, based on Ref [10]) is neglected in the
evaluation as they are short enough compared with the coherence time $\tau$.

Figure 1: Contours of constant $p_{*}\equiv p_{ge}(\tau)$ on $m_{X}$ vs.
$\epsilon$ plane ($10^{-1}$, $10^{-3}$, $10^{-5}$, and $10^{-7}$, from the
top). Parameters of $C=0.5\ {\rm pF}$, $d=100\ \mu{\rm m}$, $Q=10^{6}$,
$\kappa=1$ and $\rho_{\rm DM}=0.45\ {\rm GeV}/{\rm cm}^{3}$ are assumed. The
gray-shaded region is excluded by the cosmological and astrophysical
constraints [38] (dark gray) and the existing hidden-photon search experiments
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 18] (light gray) based on the summary in Ref. [47].
The blue shaded regions indicate the sensitivity with the 1-year scan over the
frequency range for $n_{\rm q}=1$ (dark blue) and $100$ (light blue) (more
details in the main text) assuming the thermal noise of $T=1\ {\rm mK}$. The
dashed lines show the sensitivity with $T=30\ {\rm mK}$ with the top (bottom)
line corresponding to $n_{\rm q}=1$ ($100$) respectively.

Fig. 1 shows the projected sensitivity of our proposed experiment. The
contours of constant $p_{*}$ on $m_{X}$ vs. $\epsilon$ plane is also overlaid.
The gray-shaded region is excluded by cosmological and astrophysical
constraints [38] (dark gray) and the existing hidden-photon search experiments
in this frequency range [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 18] (light gray). The upper
dark blue (lower light blue) shaded regions indicate the regions fulfilling
the discovery criteria defined in Eq. (42) with $n_{\rm q}=1$ $(100)$ assuming
$T=1\ {\rm mK}$. The dashed lines show the sensitivity with $T=30\ {\rm mK}$
with the top and bottom line corresponding to $n_{\rm q}=1$ and $100$,
respectively. Notice that the discovery reach is insensitive to the value of
$Q$ in this evaluation ignoring the measurement time or interval, since
$p_{*}\propto Q^{2}$ while $N_{\rm try}\propto Q^{-2}$. The unexplored
frequencies in the $1-10$ GHz range can be fully covered by the mass scan with
one year of the measurement time. The sensitivity of the hidden photon DM
search of our proposal is comparable to or better than those of other
proposals with condensed-matter excitations (e.g., electric excitations [48,
49, 50, 51, 52], phonon [53, 54], magnon [55], and condensed-matter axion [56,
57, 58]).

Conclusions and discussion: In this letter, we have proposed a new detection
scheme for the hidden photon DM using transmon qubits. Due to the small
kinetic mixing with the ordinary EM photon, an effective ac electric field is
induced that coherently drives a transmon qubit from the ground state toward
the first-excited state when it is resonant. We have calculated the rate of
such excitation (see Eq. (37)), and evaluated the hidden DM search sensitivity
assuming the thermal excitation ($1-30$ mK) as the sole source of background.
Using a standard SQUID-based transmon, the sensitivity can reach $\epsilon\sim
10^{-12}-10^{-14}$ with a $\sim 14\ {\rm sec}$ of measurement for a single
frequency, and with a one year to complete the scan over the $4-40\ \mu{\rm
eV}$ ($1-10$ GHz) range.

There are a few considerations left for the future studies that can further
boost the sensitivity. (1) Qubit design optimization maximizing the electric
dipole moment, where more aggressive transmon parameters and complex circuit
design can be sought. (2) Coherent multi-qubit excitation, in an analogy to
Dicke’s superradiance [59, 60], can be also explored. While dismissed in this
letter, the interference effect can in principle yield $\propto n_{q}^{2}$
enhancement as opposed to $\propto n_{q}$ in the excitation rate, which is
particularly relevant when $n_{q}$ becomes larger. (3) The packaging effect
can be further investigated. So far, we focus on a relatively simple setup: a
cylinder-shaped cavity with the off-resonant frequency of the hidden photon.
More detailed understanding on the dependencies of $\kappa$ may allow the use
of a high-$Q$ cavity package resonating to both the DM and the qubits.

The search scheme can be also directly benefited from the exponential
advancement of the large-scale NISQ computers led by, e.g., IBM [61] or Google
[62]. Since the requirements and the experimental setup are almost identical,
the improved qubit multiplicity and coherence in the NISQ machines will scale
the typical sensitivity of this experiment as well. Technically, it might be
even possible to perform the experiment with the existing NISQ machines in a
parasitic manner by using their idle or calibration time during the operation.

Finally, we point out that the physics cases of the search can be widely
extended beyond the hidden photon DM, such as the axion DM or other non-DM
transient energy density such as dark radiation.

Acknowledgments: We thank to Atsushi Noguchi and Shotaro Shirai for discussing
and providing the key feedback on the experimental setup and the possible
background contributions. TM was supported by JSPS KAKENHI Grant Numbers
18K03608 and 22H01215.

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