††thanks: A. G. and D. K. R. contributed equally to this work.††thanks: A. G.
and D. K. R. contributed equally to this work.

# Observation of Rydberg blockade due to the charge-dipole interaction between
an atom and a polar molecule

Alexander Guttridge  Department of Physics, Durham University, South Road,
Durham, DH1 3LE, United Kingdom  Joint Quantum Centre Durham-Newcastle, Durham
University, South Road, Durham, DH1 3LE, United Kingdom  Daniel K. Ruttley
Department of Physics, Durham University, South Road, Durham, DH1 3LE, United
Kingdom  Joint Quantum Centre Durham-Newcastle, Durham University, South Road,
Durham, DH1 3LE, United Kingdom  Rosario González-Férez  Instituto Carlos I de
Física Teórica y Computacional, and Departamento de Física Atómica, Molecular
y Nuclear, Universidad de Granada, 18071 Granada, Spain  H. R. Sadeghpour
ITAMP, Center for Astrophysics $|$ Harvard $\&$ Smithsonian, Cambridge, MA
02138 USA  C. S. Adams<EMAIL_ADDRESS>Department of Physics, Durham
University, South Road, Durham, DH1 3LE, United Kingdom  Joint Quantum Centre
Durham-Newcastle, Durham University, South Road, Durham, DH1 3LE, United
Kingdom  Simon L. Cornish<EMAIL_ADDRESS>Department of Physics,
Durham University, South Road, Durham, DH1 3LE, United Kingdom  Joint Quantum
Centre Durham-Newcastle, Durham University, South Road, Durham, DH1 3LE,
United Kingdom

###### Abstract

We demonstrate Rydberg blockade due to the charge-dipole interaction between a
single Rb atom and a single RbCs molecule confined in optical tweezers. The
molecule is formed by magnetoassociation of a Rb+Cs atom pair and subsequently
transferred to the rovibrational ground state with an efficiency of 91(1)%.
Species-specific tweezers are used to control the separation between the atom
and molecule. The charge-dipole interaction causes blockade of the transition
to the Rb(52s) Rydberg state, when the atom-molecule separation is set to
$310(40)$ nm. The observed excitation dynamics are in good agreement with
simulations using calculated interaction potentials. Our results open up the
prospect of a hybrid platform where quantum information is transferred between
individually trapped molecules using Rydberg atoms.

Ultracold dipolar systems, such as atoms optically excited to Rydberg states
and polar molecules, have emerged as promising platforms for quantum
simulation and computation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].
Rydberg atoms exhibit strong, long-range interactions that can be exploited to
engineer quantum entanglement and multi-qubit gates [15, 16, 17, 18, 19, 20,
12]. At the heart of this approach is the Rydberg blockade mechanism, where
strong van der Waals interactions between neighbouring Rydberg atoms prevent
simultaneous excitation of multiple atoms within a certain radius. Ultracold
polar molecules also exhibit long-range interactions and possess a rich
manifold of long-lived rotational states which can be coupled using microwave
fields [21, 22, 23] to realise high-fidelity quantum operations [24, 25, 26,
27, 28, 29]. Recent advances in optical tweezer arrays of Rydberg atoms [10,
11, 12, 13] and ultracold molecules [30, 31, 32, 33, 34, 35] create the
prospect of a new frontier research direction into hybrid atom-molecule
systems.

A hybrid system composed of polar molecules and Rydberg atoms trapped in
optical tweezer arrays offers a way to combine the advantages of both
platforms. For example, quantum information can be encoded in the long-lived
internal states of the molecule, and gates can be performed utilising the
strong, long-range interactions of Rydberg atoms [36, 37, 38, 39]. This
combines the fast high-fidelity interactions and readout possible with Rydberg
atoms [20, 40] with the long coherence times and radiative lifetimes of polar
molecules [21, 22, 23]. In addition, this hybrid system offers new
capabilities, such as nondestructive readout of the molecular state [41, 42],
cooling of molecules using Rydberg atoms [43, 44], and photoassociation of
giant polyatomic Rydberg molecules [45, 46, 47].

Realising controlled interactions between molecules and Rydberg atoms remains
an outstanding challenge. These interactions extend beyond the van der Waals
and dipole-dipole interactions which have been widely used in single-species
Rydberg systems [11] and the dipole-dipole interactions recently observed
between polar molecules [48, 49, 50, 51]. The long-range interaction of the
Rydberg electron with the permanent dipole, d, of the polar molecule takes, in
first order, the form of a charge-dipole interaction [45, 52]. The interaction
arises when the internal field due to the Rydberg electron and atomic core
polarizes the molecular dipole, $V_{\mathrm{cd}}({\bf r},{\bf
R}_{\mathrm{am}})={B{\bf N}^{2}-{\bf d}\cdot{\bf F({\bf r,{\bf
R}_{\mathrm{am}})}}}$. Here r is the electron position, ${\bf
R}_{\mathrm{am}}$ is the dipole position with respect to the atomic core, and
${\bf N}$ and $B$ are the quantum operators for molecular rotation and the
associated rotational constant respectively. The internal electric field is
${\bf F}=\frac{e({\bf r}-{\bf R}_{\mathrm{am}})}{|{\bf r}-{\bf
R}_{\mathrm{am}}|^{3}}+\frac{e\mathbf{R}_{\mathrm{am}}}{\mathrm{R}_{\mathrm{am}}^{3}}$,
giving rise to an anisotropic $1/\mathrm{R}_{\mathrm{am}}^{2}$ interaction.
For micron-scale separations, achievable in optical lattices and optical
tweezers, these interactions are predicted to be strong enough to preclude the
excitation of an atom to a Rydberg state in the presence of a molecule [36,
37].

In this Letter, we demonstrate Rydberg blockade due to the charge-dipole
interaction in a hybrid platform composed of a single 87Rb atom and a single
87Rb133Cs molecule confined in separate optical tweezers. The molecule is
prepared in the rovibrational ground state using a combination of
magnetoassociation and coherent optical transfer. We use species-specific
optical tweezers to control the atom-molecule separation down to $\sim 300$ nm
without significant collisional loss. At this separation, we find that
excitation of the Rb atom to a Rydberg state is suppressed.

Figure 1: (a) Energy shift of the pair state $\ket{r}+\ket{G}$ as a function
of the separation $\mathrm{R}_{\mathrm{am}}$ between the Rb Rydberg atom and
the RbCs molecule. The shaded region is shown inset to highlight the region
relevant to the blockade measurements. (b) A surface plot of the radial
electron density of the Rydberg electron for $\mathrm{R}_{\mathrm{am}}$ $=230$
nm. (c) Schematic of the experiment showing the atom and molecule trapped in
species-specific optical tweezers separated along the $y$-axis, which is also
the direction of the applied magnetic field $B$.

The hybrid atom-molecule system is described within the Born-Oppenheimer
approximation. The adiabatic Hamiltonian contains the charge induced-dipole
interaction [45], which holds for diatomic molecules with dipole moments below
the critical Fermi-Teller dipole moment $d_{\textrm{cr}}=1.639$ Debye [53] and
for s-wave scattering of the slow electron from the molecule [54]; see [55]
for more details. The trapping potentials due to the optical tweezers and the
effect of the magnetic field are neglected in the theoretical description. In
Fig. 1(a) we show the resulting interaction shift as a function of the
separation $\mathrm{R}_{\mathrm{am}}$ between a Rb atom in Rydberg state
$\ket{r}=\ket{52\mathrm{s}}$ and a RbCs molecule in the rovibrational ground
state $\ket{G}=\ket{\mathrm{X}^{1}\mathrm{\Sigma}^{+},v=0,N=0}$, with a
molecular-frame dipole moment of ${d}=1.225$ Debye [56] and a rotational
constant of $0.490$ GHz [57]. Here the molecular state is labelled by the
vibrational quantum number $v$ and the rotational quantum number $N$. At a
distance $\sim 300$ nm we see the onset of a large shift arising from the
charge-dipole interaction. The modulations in the energy arise from the
oscillatory nature of Rydberg electron wavefunction. Figure 1(b) shows the
Rydberg electron density for our system with $\mathrm{R}_{\mathrm{am}}$ $=230$
nm. At this distance, the symmetric distribution we expect for a Rydberg
electron in an s state is perturbed by the presence of the polar molecule. The
outermost minimum of the Rydberg electron wavefunction sets the range of
interactions; this occurs at 220 nm for the state $\ket{r}$.

The experimental geometry is shown in Fig. 1(c). The atom and molecule are
prepared in species-specific tweezers. Both particles predominantly occupy the
motional ground state of their respective traps. The tweezer light propagates
parallel to the $z$-axis and the magnetic field points along the $y$-axis. The
tweezer separation, $\mathrm{R}_{\rm t}$, is set by controlling the relative
tweezer alignment in all three spatial dimensions.

The atom-molecule separation, $\mathrm{R}_{\mathrm{am}}$, is determined from
the difference in the resulting potential minima. When the tweezers overlap,
the atom-molecule separation is reduced compared to the tweezer separation
($\mathrm{R}_{\mathrm{am}}<\mathrm{R}_{\rm t}$) due to the effect of each
potential on the other species. For each measurement, we repeat an
experimental sequence many times. Fluctuations in the relative alignment of
the tweezers occur from shot to shot with an estimated standard deviation of
50 nm in each coordinate. Atomic fluorescence images are taken at the start
and end of each sequence to determine the occupancy of each tweezer; molecules
are detected by reversing the association procedure and imaging the resulting
atom pair in separate tweezers. We apply various post-selection criteria on
the tweezer occupancies to obtain statistics for different experimental
scenarios [55].

Figure 2: Formation of rovibrational ground state RbCs molecules in optical
tweezers. (a) Electronic potential curves for RbCs highlighting the pump and
Stokes laser transitions that couple states $\ket{F}$, $\ket{E}$ and
$\ket{G}$. (b) Formation of weakly bound molecules by magnetoassociation of
atom pairs using a Feshbach resonance at 197 G. The top panel shows the
magnetic field ramps used to form molecules and then navigate the near-
threshold bound states shown in the middle panel. STIRAP is performed at 181.6
G when the molecule occupies the state $\ket{F}$ (indicated point). The lower
panel shows the pump-induced loss of weakly bound molecules. The atom-pair
survival probability $P_{11}$ is measured after reversing the field ramp to
dissociate any remaining molecules. (c) Atom-pair survival probability as a
function of the one-photon detuning $\Delta_{\mathrm{1p}}$ of the STIRAP
lasers when the other laser is on single-photon resonance for a round trip
$\ket{F}$ $\rightarrow$ $\ket{G}$ $\rightarrow$ $\ket{F}$. (d) Repeated STIRAP
transfers between states $\ket{F}$ and $\ket{G}$ with a one-way efficiency of
91(1)%. The dashed lines show the experimental contrast; the shaded regions
are the uncertainties. The inset shows the pulse profiles for a round-trip
transfer.

Our experiments begin by loading single 87Rb and 133Cs atoms into species-
specific optical tweezers from magneto-optical traps [58]. After determining
the trap occupation and performing rearrangement, the atoms are further cooled
using Raman sideband cooling [59, 60, 61] and transferred to the hyperfine
states
${\ket{f\mathrm{=}1,m_{f}=1}}_{\mathrm{Rb}}+{\ket{f\mathrm{=}3,m_{f}=3}}_{\mathrm{Cs}}$.
To produce a molecule, we must prepare a Rb+Cs atom pair in the relative
motional ground state of a single tweezer. We achieve this by merging a single
817 nm tweezer containing a Rb atom into a 1065 nm tweezer containing a Cs
atom.

The Rb+Cs atom pair is converted into a weakly bound RbCs molecule by
magnetoassociation using an interspecies Feshbach resonance at 197 G [62, 63,
64, 35]. The electronic potential energy curves for RbCs are shown in Fig.
2(a). The magnetic field ramps used to associate and later dissociate the atom
pair are shown in the upper panel of Fig. 2(b); the central panel shows the
energy levels that these ramps navigate to access the state $\ket{F}$ at 181.6
G, shown by the point. The formation of weakly bound RbCs molecules is
detected using pump-induced loss to other molecular states [55, 65, 66, 67]
which precludes atom-pair recovery when the association and merging steps are
reversed (Fig. 2(b) lower panel).

We transfer the weakly bound molecule to the rovibrational ground state
$\ket{G}$ using two-photon stimulated Raman adiabatic passage (STIRAP) [68,
69], as previously demonstrated for bulk gases of RbCs molecules [67, 56, 65,
70]. In Fig. 2(c) we show the probability of recovering the atom pair after a
round trip $\ket{F}$ $\rightarrow$ $\ket{G}$ $\rightarrow$ $\ket{F}$ as a
function of the single-photon detuning of either the pump or Stokes lasers,
with the other laser held on single-photon resonance. When the Stokes laser is
not resonant, the pump laser causes loss to other molecular states via
$\ket{E}$. When the pump laser is far from resonant, the molecules remain in
state $\ket{F}$ throughout the transfer sequence.

In Fig. 2(d), we characterise the STIRAP efficiency using repeated transfers
back and forth between states $\ket{F}$ and $\ket{G}$. An odd number of
successful one-way transfers results in the molecule occupying state
$\ket{G}$, whereas an even number of transfers returns it to state $\ket{F}$.
Only molecules that occupy state $\ket{F}$ at the end of the sequence are
dissociated back into atom pairs for detection. The offset of the odd points
indicates the combined efficiency of the cooling, merging, and
magnetoassociation stages of the experiment; in 50(1)% of runs we do not form
a molecule, and thus reimage the atom pair independent of the STIRAP pulses.
The maximum contrast between the odd and even points is limited primarily by
the 35(5) ms lifetime of molecules in state $\ket{F}$ in the trap and the need
to allow the magnetic field to stabilise before STIRAP [55]. We measure a one-
way transfer efficiency of 91(1)%, consistent with the best reported
efficiencies for RbCs in bulk gases [67, 70].

To observe blockade, the charge-dipole interaction between the Rydberg atom
and the molecule must be greater than the transition linewidth. For our
system, this requires interactions shifts $\gtrsim 1$ MHz. Our calculations in
Fig. 1 predict the atom-molecule distance must be $\lesssim 300$ nm to observe
this effect, a distance smaller than the beam radii of the individual tweezers
($\sim 1$ µm). We cannot acheive sub-micron separations by loading both
species into the same tweezer, as the expected lifetime due to collisional
loss is $<1$ ms [71]. Instead we utilise species-specific tweezers at
wavelengths of 1065 nm for the molecule and 817 nm for the atom. For the Rb
atom, the ratio of polarisabilities for these wavelengths is
$\alpha_{817}^{\mathrm{Rb}}/\alpha_{1065}^{\mathrm{Rb}}\sim 6.3$ [72], so that
it is confined predominantly in the 817 nm tweezer (the “atom tweezer”).
Conversely, for the RbCs molecule
$\alpha_{1065}^{\mathrm{RbCs}}/\alpha_{817}^{\mathrm{RbCs}}\sim 4.5$ [73] so
that it is confined predominantly in the 1065 nm tweezer (the “molecule
tweezer”). Typical trap potentials are illustrated in the insets of Fig. 3.

Figure 3: Collisions between ground state RbCs molecules and Rb atoms held in
separate species-specific optical tweezers. Particle survival probabilities
are plotted as a function of the tweezer separation,
$\mathrm{R}_{\mathrm{t}}$. For the molecule we report the atom-pair survival
probability, post selected on cases where a weakly bound molecule was formed
[55]. Upper panel: experimental runs where either a single Rb atom (blue
squares) or a single RbCs molecule (red circles) is present. Lower panel: runs
where both the atom and the molecule are present. The dashed lines (and shaded
regions) correspond to the mean values (and errors) from the 1-body cases.
Insets: The potential energy of the atom (blue) and molecule (red) resulting
from their own tweezer (dashed lines) and both tweezers (solid lines) for
$\mathrm{R}_{\mathrm{t}}=2000$ nm (left) and $\mathrm{R}_{\mathrm{t}}=410$ nm
(right).

We investigate loss due to collisions between a Rb atom and a RbCs molecule by
sweeping the position of the atom tweezer to a variable distance
$\mathrm{R}_{\mathrm{t}}$ from the molecule tweezer. The particles are held at
this separation for 9.5 ms before the sweep of the atom tweezer is reversed
and the particle survival probabilities are measured. The results are
presented in Fig. 3. For the molecule, we report the atom-pair survival
probability, post-selected on cases where a weakly bound molecule was formed
using a protocol that exploits the species-specific nature of the tweezers
[55]. The upper panel shows the one-body survival probabilities from
experimental runs, when _either_ the Rb atom _or_ the RbCs molecule is
present. The atomic survival probability is 97.2(4)%. For the molecule signal,
we observe a survival probability of 48(2)%, primarily caused by loss prior to
STIRAP due to the short lifetime of the state $\ket{F}$ in the trap. By
compensating for the return STIRAP efficiency, we predict that a molecule in
$\ket{G}$ is present in 53(3)% of runs in which a weakly bound molecule is
created. The lower panel in Fig. 3 shows the two-body survival probabilities
for runs in which _both_ an atom and a weakly bound molecule are initially
prepared. When the tweezers are brought together, the wavefunctions of the
particles begin to overlap and collisions cause loss of both the molecule and
atom. We observe a reduction in the atom survival probability by 58(6)%,
commensurate with the probability a molecule in state $\ket{G}$ is present.
From a Gaussian fit to the loss feature we find the loss falls to $1/e^{2}$ of
its maximum value at $\mathrm{R}_{\mathrm{t}}=250(20)$ nm.

Figure 4:  (a) Survival probability of the Rb atom as a function of the
Rydberg pulse duration for $\mathrm{R}_{\mathrm{am}}$ $=310(40)$ nm. Atoms
excited to $\ket{r}$ are ejected from the trap and lost. Events are post-
selected on the detection of a molecule in $\ket{G}$ (purple squares) or
unsuccessful formation of a molecule (green circles). The solid lines show the
results of simulations using the Lindblad master equation [55] using our
estimated atom-molecule separation. (b) Rb atom survival probability as a
function of the two-photon detuning, $\Delta$, using a 1 µs pulse for
$\mathrm{R}_{\mathrm{am}}$ $=700(40)$ nm (upper panel) and
$\mathrm{R}_{\mathrm{am}}$ $=310(40)$ nm (lower panel). For each separation,
the detuning is defined relative to the transition centre in the absence of a
molecule. Symbols are as in (a) and solid lines show the results of
simulations using the estimated atom-molecule separations.

To demonstrate blockade, we use an experimental routine similar to that used
in Fig. 3 with a hold time of 3 ms when the tweezers are close together. Two-
photon excitation of the Rb atom to the Rydberg state $\ket{r}$ is performed
during the hold time with the trapping light still present [55]. Atoms excited
to state $\ket{r}$ are anti-trapped and ejected from the tweezers, mapping
Rydberg excitation onto atom loss. To suppress collisional loss we hold the
tweezers at a separation $\mathrm{R}_{\rm t}=420(40)$ nm, significantly
greater than the width of the feature reported in Fig. 3. Here the error
represents the systematic uncertainty from the alignment calibrations. By
calculating the combined potential of the tweezers, we find this equates to an
atom-molecule separation of $\mathrm{R}_{\mathrm{am}}$ $=310(40)$ nm [55].

In Fig. 4, we demonstrate the blockade of the Rydberg transition of the Rb
atom when a RbCs molecule in state $\ket{G}$ is present. Figure 4(a) shows the
survival probability of the Rb atom as the Rydberg pulse duration is varied.
For experimental runs where the molecule tweezer is empty (green circles), we
observe clear Rabi oscillations between states $\ket{g}$ and $\ket{r}$. In
contrast, for runs where a molecule in state $\ket{G}$ is present (purple
squares), we observe a suppression of the excitation to state $\ket{r}$. Here,
the presence of the molecule shifts the energy of state $\ket{r}$ through the
charge-dipole interaction and thus blockades excitation during the Rydberg
pulse. The frequency of the residual Rabi oscillations is almost identical to
that for the unblockaded case. This is due to the sharp onset of the
interaction shown in Fig. 1(a) combined with shot-to-shot variations in the
relative alignment of the tweezers. For runs with the largest separations, the
energy shift is smaller than the linewidth of the Rydberg transition leading
to a signal at the unshifted Rabi frequency.

To simulate the expected excitation dynamics, we solve the Lindblad master
equation [55]. We use the pair-state energy shifts shown in Fig. 1(a) to
include a distance-dependent energy shift. We account for the fact that the
atom and molecule are prepared in the motional ground state of their
respective tweezers by averaging the interaction over the ground-state
wavefunction of relative motion. We also include experimental imperfections
such as dephasing from laser frequency noise and shot-to-shot fluctuations in
the relative alignment of the tweezers. Using our best estimates of the
separation, we find good agreement between the results of the simulation and
the experiment, as shown by the solid lines in Fig. 4(a).

The results shown in Fig. 4(b) demonstrate the effect of changing the atom-
molecule separation on the Rydberg blockade. In this experiment, we fix the
pulse duration to approximate a $\pi$-pulse and scan the two-photon detuning
of the light driving the Rydberg transition. For $\mathrm{R}_{\mathrm{am}}$
$=700(40)$ nm shown in the upper panel, the molecule is far beyond the
outermost minimum of the Rydberg electron wavefunction and the charge-dipole
interaction is negligible. Here, the dominant interaction is van der Waals
leading to a shift of $\sim 0.1$ kHz [74]. Consequently, the presence of a
molecule does not affect the the Rydberg excitation. However, for
$\mathrm{R}_{\mathrm{am}}$ $=310(40)$ nm shown in the lower panel, the
presence of a molecule leads to an observed shift of the Rydberg transition to
lower energy, as expected. The transition is significantly broadened due to
the sensitivity of the charge-dipole interaction to the atom-molecule
separation. The broadening causes a concomitant reduction in the signal
amplitude. Both these effects are reproduced by simulations using the same
parameters as in Fig. 4(a).

In conclusion, we have demonstrated blockade of the transition to the Rb(52s)
Rydberg state due to the charge-dipole interaction with a RbCs molecule in the
rovibrational ground state. This represents the first observation of a charge-
dipole induced shift in an ultracold atomic setting and opens up many new
research directions. The blockade we have observed provides a mechanism for
conditional and non-destructive state readout of the molecule [36, 37]. A
single Rydberg atom can also mediate effective spin-spin interactions between
a pair of molecular dipoles, allowing the simulation of models of
magnetization [75]. For molecules prepared in the $N=2$ rotational state, our
calculations for the Rb(52s) Rydberg state predict that resolvable, deeply
bound states exist for separations of $\sim 220$ nm. This offers the
possibility to photoassociate giant polyatomic Rydberg molecules [45, 46, 76,
47]. By selecting Rydberg and molecular states which interact via resonant
dipole-dipole interactions, the Rydberg blockade radius can be increased to
several microns, enabling high-fidelity entangling gates between molecules
mediated by strong interactions with neighbouring Rydberg atoms [39, 38]. This
presents the tantalising prospect of a hybrid platform where quantum
information is transferred between individually trapped molecules using
Rydberg atoms.

###### Acknowledgements.

We thank S. Spence for earlier experimental work, A. L. Tao for assistance in
the setup of the STIRAP laser system, A. C. Baldock and X. Yang for
informative simulations of the atom and molecule system, and H. J. Williams
and S. A. Gardiner for helpful discussions. We acknowledge support from the UK
Engineering and Physical Sciences Research Council (EPSRC) Grants
EP/P01058X/1, EP/V047302/1, and EP/W00299X/1, UK Research and Innovation
(UKRI) Frontier Research Grant EP/X023354/1, the Royal Society, and Durham
University. R.G.F. gratefully acknowledges financial support by the Spanish
projects PID2020-113390GB-I00 (MICIN), PY20-00082 (Junta de Andalucía) and
A-FQM-52-UGR20 (ERDF-University of Granada), and the Andalusian Research Group
FQM-207. H.R.S. acknowledges support from the NSF through a grant for ITAMP at
Harvard University. The data presented in this paper are available from
http://doi.org/10.15128/r2fj236215f.

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# Discontinuities of banana integrals in dispersion relation representation

Xu-Liang Chen1 Peng-Fei Yang1 Wei Chen1, 2<EMAIL_ADDRESS>1School
of Physics, Sun Yat-sen University, Guangzhou 510275, China
2Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou 516000, Guangdong Province,
China

###### Abstract

We derive the discontinuities of banana integrals using the dispersion
relation iteratively. We find a series of identities between the parameterized
discontinuities of banana integrals (p-DOBIs). Similar to elliptic integrals,
these identities enable the reduction of various p-DOBIs to be a linear
combination of some fundamental ones. We present a practical application of
p-DOBIs for the massive three-loop case and establish the expression of
generalized dispersion relation, which enables us to obtain the dispersion
relation representation of arbitrary banana integrals. Moreover, we propose a
hypothesis for generalized dispersion relation and p-DOBIs, which provides a
simple way to calculate the discontinuities and transform dispersion relation
representation to p-DOBIs.

Banana integrals, Discontinuity, Dispersion relation, p-DOBIs

Introduction. — Multi-loop Feynman integrals play a crucial role in the
perturbative quantum field theory. Many of these integrals can be well
understood by expressing in terms of Multiple Polylogarithms (MPLs) Weinzierl
(2022); Duhr (2015); Bourjaily et al. (2021). However, there are some physics
processes involving complex elliptic Feynman integrals, which will go beyond
the space of MPLs, for instance, the massive banana (or sunrise) integrals
Bourjaily et al. (2022). They have already given important contributions to
the physics of Higgs decays, $\gamma\gamma$ production, $t\bar{t}$ production
and so on. Many progresses have been made on computing such integrals
analytically in recent years Adams and Weinzierl (2019); Adams et al. (2018,
2016, 2014, 2015); Adams and Weinzierl (2018); Bloch and Vanhove (2015); Bloch
et al. (2015, 2017); Bönisch et al. (2021, 2022); Bourjaily et al. (2019);
Broedel et al. (2019a); Broedel et al. (2015, 2018); Broedel et al. (2019b);
de la Cruz and Vanhove (2024); Duhr et al. (2023); Forum and von Hippel
(2023); Frellesvig et al. (2021); Klemm et al. (2020); Lairez and Vanhove
(2023); Pögel et al. (2023a); Pögel et al. (2024); Pögel et al. (2023b); Pögel
et al. (2022); Primo and Tancredi (2017a, b); Remiddi and Tancredi (2016);
Remiddi and Tancredi (2017); Vanhove (2021, 2019, 2014).

It is well-known that all elliptic integrals can be expressed linearly in
terms of three fundamental integrals. Furthermore, the maximal cut of two-loop
banana integrals is associated with elliptic integrals. In Ref. Remiddi and
Tancredi (2017), a new class of functions called E-polylogarithm are defined
as a generalization of MPLs and successfully applied to two-loop banana
integrals. These functions exhibit similar structures to elliptic integrals.
This inspires us to the intriguing question of whether similar properties hold
for higher-loop banana integrals. In this letter, we provide a positive answer
at the massive three-loop level.

We define a novel class of functions as a natural parameterization of the
discontinuities of banana integrals (p-DOBIs). Utilizing the technique of
elliptic integrals, we obtain a series of identities between p-DOBIs. At
three-loop level, we express all p-DOBIs as the linear combination of several
fundamental ones. Furthermore, we demonstrate the application of p-DOBIs in
deriving the Picard-Fuchs operators and reveal that the homogenous solutions
of Picard-Fuchs equation can be given by slightly altering the definition of
p-DOBIs.

To express an arbitrary banana integral with p-DOBIs, we establish the
expression of generalized dispersion relation (GDR), which is crucial to deal
with integrals involving high-power propagators. Furthermore, we propose a
hypothesis to simplify the dispersion relation (DR) representation to p-DOBIs.

The banana as iteration of the bubble. — An $L$-loop banana integral is
defined as

$\begin{gathered}B_{n_{0}n_{1}\cdots
n_{L}}(q^{2};\bm{m})=\int\prod_{i=1}^{L}\frac{d^{D}k_{i}}{\text{i}\pi^{\frac{D}{2}}}\frac{1}{\mathcal{D}_{0}^{n_{0}}\mathcal{D}_{i}^{n_{i}}}\,,\end{gathered}$
(1)

where $\mathcal{D}_{i}=k_{i}^{2}-m_{i}^{2}$ and
$\mathcal{D}_{0}=\left(q-\sum_{i=1}^{L}k_{i}\right)^{2}-m_{0}^{2}$ are the
inverse propagators, $\bm{m}=\\{m_{0},m_{1},\cdots,m_{L}\\}$ is the set of
masses, and $q$ is the external momentum. In this work, we shall use the
following notations for convenience: $m_{ij}\equiv m_{i}+m_{j}$, $m_{0\cdots
L}^{2}=(m_{0}+\cdots+m_{L})^{2}$, $s_{ij}\equiv\sqrt{s_{i}}+m_{j}$ ,
$s_{i\bar{j}}\equiv\sqrt{s_{i}}-m_{j}$,
$\lambda_{n}\equiv\lambda(s_{n},s_{n-1},m_{n}^{2})$, where
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2xz$, $s_{0}\equiv m_{0}^{2}$ ,
$\rho_{n_{0}n_{1}\cdots
n_{L}}(q^{2};\bm{m})\equiv\pi^{-1}\text{Im}B_{n_{0}n_{1}\cdots
n_{L}}(q^{2};\bm{m})$. The discontinuity of the one-loop
$B_{11}(s;m_{0},m_{1})$ can be described by the spectral function

$\rho_{11}(s;m_{0},m_{1})\equiv\frac{1}{\pi}\text{Im}B_{11}(s;m_{0},m_{1})=2\lambda_{1}^{-\frac{1}{2}}.$
(2)

We will limit our discussion to the $D=2-2\epsilon$ dimensions and
$\epsilon^{0}$ order.

A two-loop banana integral can be represented as iteration of one-loop bubbles
Remiddi and Tancredi (2016)

$\begin{gathered}\begin{aligned} &B_{1^{3}}(q^{2};\bm{m})\equiv
B_{111}(q^{2};\bm{m})\\\
=&\int\frac{d^{D}k_{2}}{\text{i}\pi^{\frac{D}{2}}}\int_{m_{01}^{2}}^{\infty}ds_{1}\frac{-\rho_{11}(s_{1};m_{0},m_{1})}{(q-k_{2})^{2}-s_{1}}\frac{1}{\mathcal{D}_{2}}\\\
=&\int_{m_{01}^{2}}^{\infty}ds_{1}\int_{s_{12}^{2}}^{\infty}ds_{2}\frac{\rho_{11}(s_{1};m_{0},m_{1})\rho_{11}(s_{2};\sqrt{s_{1}},m_{2})}{q^{2}-s_{2}}\\\
=&4\int_{m_{012}^{2}}^{\infty}ds_{2}\int_{m_{01}^{2}}^{s_{2\bar{2}}^{2}}ds_{1}\frac{\lambda_{1}^{-\frac{1}{2}}\lambda_{2}^{-\frac{1}{2}}}{q^{2}-s_{2}},\,\end{aligned}\end{gathered}$
(3)

where the discontinuity of the propagator $(q^{2}-s_{2}+i0^{+})^{-1}$ is
$-\delta(s_{2}-q^{2})$ for $q^{2}\geq m_{012}^{2}$. Thus the two-loop DOBI is

$\begin{gathered}\begin{aligned}
\rho_{1^{3}}(s_{2};\bm{m})=-4\int_{m_{01}^{2}}^{s_{2\bar{2}}^{2}}ds_{1}\lambda_{1}^{-\frac{1}{2}}\lambda_{2}^{-\frac{1}{2}}.\end{aligned}\end{gathered}$
(4)

Similarly, the $L$-loop ($L\geq 2$) DOBI can be written as

$\begin{gathered}\begin{aligned}
\rho_{1^{L}}(s_{L};\bm{m})=(-1)^{L-1}2^{L}\prod_{i=2}^{L}\left(\int_{m_{0\cdots(i-1)}^{2}}^{s_{i\overline{i}}^{2}}\frac{ds_{i-1}}{\sqrt{\lambda_{i}}}\right)\frac{1}{\sqrt{\lambda_{1}}}.\end{aligned}\end{gathered}$
(5)

Meanwhile, the real part is given by the Cauchy principal value of the
integral

$\begin{gathered}\begin{aligned}
\text{Re}B_{1^{L}}(q^{2};\bm{m})=\mathcal{P}\int_{m_{0\cdots
i}^{2}}^{\infty}ds_{L}\frac{\rho_{1^{L}}(s_{L};\bm{m})}{s_{L}-q^{2}}.\end{aligned}\end{gathered}$
(6)

One can prove that the results of Eq. (6) numerically agrees with the Bessel
representation Groote et al. (2007, 2012); Groote and Körner (2019) for
$q^{2}<m_{0\cdots L}^{2}$. For $q^{2}>m_{0\cdots L}^{2}$, the results in Eqs.
(5)-(6) numerically agree with the analytical continuation expression of
Bessel representation in Appendix A.

Identities between discontinuities of banana integrals. — Since the above
results, we define the p-DOBIs as the following form

$\displaystyle\mathcal{I}_{k_{1}}^{a_{1}}(s_{2};\bm{m})=\int^{s_{2\bar{2}}^{2}}_{m_{01}^{2}}ds_{1}(\lambda_{1}\lambda_{2})^{-\frac{1}{2}}(s_{1}-a_{1})^{k_{1}}\,,$
(7)
$\displaystyle\mathcal{I}_{k_{1},\cdots,k_{n}}^{a_{1},\cdots,a_{n}}(s_{n+1};\bm{m})$
$\displaystyle=$ $\displaystyle\int_{m_{0\cdots
n}^{2}}^{s_{(n+1)\overline{(n+1)}}^{2}}ds_{n}\frac{(s_{n}-a_{n})^{k_{n}}}{\sqrt{\lambda_{n+1}}}\mathcal{I}_{k_{1},\cdots,k_{n-1}}^{a_{1},\cdots,a_{n-1}}(s_{n};\bm{m})\,,$
(8)

where $a_{1},\cdots,a_{n}$ are some useful parameters and we shall discuss
them later. Comparing to Eq. (5),
$\mathcal{I}_{0,\cdots,0}^{a_{1},\cdots,a_{L-1}}(s_{L};\bm{m})$ is
proportional to $\rho_{1^{L}}(s_{L};\bm{m})$ so that it can be used to
describe DOBIs. In Ref. Remiddi and Tancredi (2017),
$\mathcal{I}_{k_{1}}^{0}(s_{2})$ has been adopted to calculate the two-loop
equal mass banana integrals. Since $\mathcal{I}_{k_{1}}^{a_{1}}(s_{2})$ is an
elliptic integral, all these integrals can be represented by three fundamental
p-DOBIs. We will show that this can be generalized to the two-fold p-DOBIs
$\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})$ corresponding to the three-
loop integrals.

We start from the following identity of $\mathcal{I}_{k_{1}}^{a_{1}}(s_{2})$

$\displaystyle\int^{s_{2\bar{2}}^{2}}_{m_{01}^{2}}ds_{1}\frac{d}{ds_{1}}\left[\sqrt{\lambda_{1}\lambda_{2}}(s_{1}-a_{1})^{k_{1}}\right]=0\,,$
(9)

where $a_{1}\neq s_{2\bar{2}}^{2},m_{01}^{2}$. Noting that the upper and lower
integral limits are the zero points of $\lambda_{2}$ and $\lambda_{1}$,
respectively. Replacing $\lambda_{n}^{\frac{1}{2}}$ to
$\lambda_{n}^{-\frac{1}{2}}\lambda_{n}$ in Eq. (9) and using Eq. (7), we find
a series of identities between p-DOBIs and all these
$\mathcal{I}_{k_{1}}^{a_{1}}(s_{2})$ can be represented by three fundamental
p-DOBIs

$\begin{gathered}\\{\mathcal{I}_{-1}^{a_{1}},\,\mathcal{I}_{0}^{a_{1}},\,\mathcal{I}_{1}^{a_{1}}\\}.\end{gathered}$
(10)

For $\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})$, we consider the similar
equation as

$\displaystyle\int^{s_{3\bar{3}}^{2}}_{m_{012}^{2}}ds_{2}$
$\displaystyle\frac{d}{ds_{2}}\left[\int^{s_{2\bar{2}}^{2}}_{m_{01}^{2}}ds_{1}\frac{\sqrt{\lambda_{2}\lambda_{3}}}{\sqrt{\lambda_{1}}}(s_{1}-a_{1})^{k_{1}}(s_{2}-a_{2})^{k_{2}}\right]=0\,.$
(11)

We can further find the identities for
$\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})$ by using the identities of
one-fold p-DOBIs obtained above. Setting
$a_{1}=(m_{0}-m_{1})^{2}=0,a_{2}=(m_{0}-m_{1}+m_{2})^{2}=1$ for the equal mass
case $m_{0}=m_{1}=m_{2}=1$, the
$\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})$ with
$\\{k_{1}\geq-1,k_{2}\geq 0\\}$ can be represented by five fundamental p-DOBIs

$\begin{gathered}\vec{I}=\\{\mathcal{I}_{-1,1}^{0,1},\mathcal{I}_{-1,2}^{0,1},\mathcal{I}_{0,0}^{0,1},\mathcal{I}_{1,0}^{0,1},\mathcal{I}_{1,1}^{0,1}\\}\,,\end{gathered}$
(12)

in which the following well-known identity Remiddi and Tancredi (2016);
Remiddi and Tancredi (2017)

$\begin{gathered}\mathcal{I}_{1}^{0}(s_{2})=(1+\frac{1}{3}s_{2})\mathcal{I}_{0}^{0}(s_{2})\end{gathered}$
(13)

is utilized to achieve Eq. (12). As an example,

$\begin{gathered}\begin{aligned}
\mathcal{I}_{-1,3}^{0,1}=&\frac{1}{2}(4s_{3}-s_{3}^{2})\mathcal{I}_{-1,1}^{0,1}+\frac{1}{2}(s_{3}^{2}+8s_{3}+32)\mathcal{I}_{0,0}^{0,1}\\\
-&\frac{1}{2}(11s_{3}+24)\mathcal{I}_{1,0}^{0,1}+4\mathcal{I}_{1,1}^{0,1}+\frac{3}{2}s_{3}\mathcal{I}_{-1,2}^{0,1}\,.\end{aligned}\end{gathered}$
(14)

One can easily check this result numerically.

To derive the differential equations for two-fold p-DOBIs, we study the
following integral $\frac{d}{ds_{3}}\vec{I}$ by replacing
$\lambda_{3}^{\frac{1}{2}}$ to $\lambda_{3}^{-\frac{1}{2}}\lambda_{3}$

$\begin{gathered}\frac{d}{ds_{3}}\int^{s_{3\bar{3}}^{2}}_{m_{012}^{2}}ds_{2}\int^{s_{2\bar{2}}^{2}}_{m_{01}^{2}}ds_{1}\frac{\sqrt{\lambda_{3}}}{\sqrt{\lambda_{1}\lambda_{2}}}(s_{1}-a_{1})^{k_{1}}(s_{2}-a_{2})^{k_{2}}\,,\end{gathered}$
(15)

which then can be represented with
$\frac{d}{ds_{3}}\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})$. On the other
hand, Eq. (15) can be expressed directly by
$\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})$ after finishing the
derivatives. The differential equations
$\frac{d}{ds_{3}}\vec{I}=\mathcal{A}^{(1)}\vec{I}$ can be established with
different $\\{k_{1},k_{2}\\}$, in which $\mathcal{A}^{(1)}$ is a $5\times 5$
matrix. There are only simple poles in the matrix $\mathcal{A}^{(1)}$, as
shown in Appendix B. The $n$-th differential equations
$\frac{d^{n}}{ds_{3}^{n}}\vec{I}=\mathcal{A}^{(n)}\vec{I}$ can be easily
derived from the first order ones.

These identities can be used to calculate the Picard-Fuchs operator for
$\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})$

$\hat{L}(s_{3})\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})\equiv\left(\sum_{i=0}^{n_{max}}P_{i}(s_{3})\frac{d^{i}}{ds_{3}^{i}}\right)\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})=0,$
(16)

where $\hat{L}(s_{3})$ is the Picard-Fuchs operator and $P_{i}(s_{3})$ is a
polynomial of $s_{3}$. The polynomial coefficients in $P_{i}(s_{3})$ can be
determined by letting the coefficient of each fundamental integral be zero for
different values of $s_{3}$. Assuming $\hat{L}(s_{3})$ be the third-order
differential operator and the order of $P_{i}(s_{3})$ be less than four, we
obtain the well-known Picard-Fuchs operator for
$\mathcal{I}_{0,0}^{a_{1},a_{2}}(s_{3})$ Müller-Stach et al. (2014); Vanhove
(2019); Lairez and Vanhove (2023); Broedel et al. (2019a); Pögel et al.
(2022); Primo and Tancredi (2017a); de la Cruz and Vanhove (2024):

$\begin{gathered}\begin{aligned}
\hat{L}(s_{3})&=\left(s_{3}^{4}-20s_{3}^{3}+64s_{3}^{2}\right)\frac{d^{3}}{ds_{3}^{3}}\\\
&+\left(6s_{3}^{3}-90s_{3}^{2}+192s_{3}\right)\frac{d^{2}}{ds_{3}^{2}}\\\
&+\left(7s_{3}^{2}-68s_{3}+64\right)\frac{d}{ds_{3}}+(s_{3}-4)\,.\end{aligned}\end{gathered}$
(17)

In principle, all $\mathcal{I}_{k_{1},k_{2}}^{a_{1},a_{2}}(s_{3})$ can be
expressed as series with the method of Frobenius.

We choose different integral intervals in Eq. (8) to define functions similar
to p-DOBIs. If these functions still satisfy Eq. (Discontinuities of banana
integrals in dispersion relation representation) and Eq. (15), the identities
and differential equations remain valid. Therefore, we can choose these
functions as another two homogeneous solutions for Eq. (16), e.g.

$\begin{gathered}\begin{aligned}
\int^{s_{33}^{2}}_{s_{3\bar{3}}^{2}}ds_{2}\int^{s_{2\bar{2}}^{2}}_{m_{01}^{2}}ds_{1}\frac{1}{\sqrt{\lambda_{1}\lambda_{2}\lambda_{3}}}\,,\\\
\int^{s_{33}^{2}}_{s_{3\bar{3}}^{2}}ds_{2}\int^{s_{22}^{2}}_{s_{2\bar{2}}^{2}}ds_{1}\frac{1}{\sqrt{\lambda_{1}\lambda_{2}\lambda_{3}}}\,.\end{aligned}\end{gathered}$
(18)

We need eight fundamental p-DOBIs for the four-mass case, which are chosen as
$\vec{I}_{4m}=\\{\mathcal{I}_{-1,0}^{a_{1},a_{2}},\mathcal{I}_{-1,1}^{a_{1},a_{2}},\mathcal{I}_{-1,2}^{a_{1},a_{2}},\mathcal{I}_{0,0}^{a_{1},a_{2}},\mathcal{I}_{0,1}^{a_{1},a_{2}},\mathcal{I}_{0,2}^{a_{1},a_{2}},\mathcal{I}_{1,0}^{a_{1},a_{2}},\mathcal{I}_{1,1}^{a_{1},a_{2}}\\}$.
The parameters are assigned as
$a_{1}=(m_{0}-m_{1})^{2},\,a_{2}=(m_{0}-m_{1}+m_{2})^{2}$. The differential
equations $\vec{I}_{4m}$ and Picard-Fuchs operators can be derived following
the same method as above. We obtain the same results for
$\mathcal{I}_{0,0}^{a_{1},a_{2}}$ as those in Refs. de la Cruz and Vanhove
(2024); Lairez and Vanhove (2023). The parameters $a_{1},a_{2}$ can also be
other values, however, the pseudo-thresholds are recommended in our analyses.

Generalized dispersion relation. — Integrals with high-power propagator are
unavoidable issues to express arbitrary banana integral with p-DOBIs. However,
it may lead to incorrect results by applying DR directly to integrals with
high-power propagator.

Considering the one-loop bubble with equal mass $m_{0}=m_{1}=1$ :

$B_{2,1}(q^{2})=\int\frac{d^{2}k}{\text{i}\pi}\frac{1}{\left[(q-k)^{2}-1\right]^{2}(k^{2}-1)}\,,$
(19)

the imaginary part is $\text{Im}B_{2,1}(s)=2\pi
s^{-\frac{1}{2}}(s-4)^{-\frac{3}{2}}$. Substituting $\text{Im}B_{2,1}(s)$ into
DR

$\Pi(q^{2})=\int_{s_{N}}^{\infty}\frac{ds}{\pi}\frac{\text{Im}\Pi(s)}{s-q^{2}}\,,$
(20)

one will achieve the expression as

$B_{2,1}(q^{2})=\int_{4}^{\infty}ds\frac{2}{(s-q^{2})s^{\frac{1}{2}}(s-4)^{\frac{3}{2}}}\,,$
(21)

which is obviously divergent and thus incorrect. Such diverging problems were
discussed very early in Ref. Toll (1956). However, the explicit calculation
was not given. In this work, we establish the generalized dispersion relation
to deal with such divergent behavior at finite points and give their complete
expressions.

In general, a holomorphic function $\Pi(q^{2})$ in the closed contour $C$ is
expressed as

$\Pi(q^{2})=\left(\int_{\Gamma}+\int_{\gamma}\right)\frac{ds}{2\pi\text{i}}\frac{\Pi(s)}{s-q^{2}}+\int_{s_{N}}^{\infty}\frac{ds}{\pi}\frac{\text{Im}\Pi(s)}{s-q^{2}}\,,$
(22)

in which there is a branch cut starting at $s_{N}$ on the real axis, as shown
in Fig. 1. The contribution of the small circle $\gamma$ is usually neglected
in the expression of DR, since it appears only in integrals with high-power
propagators. This contribution can be considered directly by the cumbersome
calculations of the integral in the small circle Palameta et al. (2018).

Figure 1: Contour for integration of DR.

Similar to the subtractions on the large circle $\Gamma$, one can also do
subtractions on the small circle by letting
$\Pi_{\gamma}(q^{2})=(q^{2}-s_{N})^{n_{\gamma}}\Pi(q^{2})$ to avoid the
divergences at $s_{N}$. Substituting $\Pi_{\gamma}(q^{2})$ into DR with
subtractions, we find the GDR at the subtraction point $a$:

$\begin{gathered}\begin{aligned}
\Pi(q^{2})=\sum_{i=0}^{n_{\Gamma}-1}\frac{(q^{2}-a)^{i}\left[(q^{2}-s_{N})^{n_{\gamma}}\Pi(q^{2})\right]^{(i)}|_{q^{2}=a}}{\Gamma(i+1)(q^{2}-s_{N})^{n_{\gamma}}}\\\
+\int_{s_{N}}^{\infty}\frac{ds}{\pi}\frac{\text{Im}\Pi(s)}{s-q^{2}}\left(\frac{q^{2}-a}{s-a}\right)^{n_{\Gamma}}\left(\frac{s-s_{N}}{q^{2}-s_{N}}\right)^{n_{\gamma}},\end{aligned}\end{gathered}$
(23)

where $n_{\Gamma},n_{\gamma}$ are the subtraction times on the large circle
and small circle, respectively. Since $n_{\Gamma}$ is determined by
$\Pi_{\gamma}(q^{2})$ rather than $\Pi(q^{2})$, one should firstly find
$n_{\gamma}$ in GDR similar with the way to find $n_{\Gamma}$ in DR. The
expression of GDR in Eq. (23) can be naturally degenerated to DR for
$n_{\gamma}=0$.

For the equal mass one-loop bubble, the complete expression of
$B_{2,1}(q^{2})$ can be obtained from GDR in Eq. (23) for
$n_{\gamma}=1,n_{\Gamma}=1$ and subtraction point at $a=0$

$B_{2,1}(q^{2})=\frac{2}{q^{2}-4}+\int_{4}^{\infty}\frac{ds}{\pi}\frac{\text{Im}B_{2,1}(s)}{s-q^{2}}\frac{q^{2}}{s}\frac{s-4}{q^{2}-4}.$
(24)

This expression numerically agrees with the result in Mellin-Barnes
representation Boos and Davydychev (1991).

The result of GDR is very useful to deal with the divergent banana integrals
with high-power propagators. Using the same method in Eq. (24), we can write
the discontinuity of the two-loop $B_{211}(s_{2};1,1,1)$ as

$\displaystyle\rho_{211}(s_{2};1,1,1)$ $\displaystyle=$ $\displaystyle
2\rho_{11}(s_{2};2,1)+\int_{4}^{\infty}ds_{1}\frac{4}{s_{1}}\rho_{21}(s_{1};1,1)\rho_{11}(s_{2};2,1)$
$\displaystyle-$
$\displaystyle\int_{4}^{s_{2\bar{2}}^{2}}ds_{1}\rho_{21}(s_{1};1,1)\rho_{11}(s_{2};\sqrt{s_{1}},1)\,,$
(25)

in which the GDR of $B_{2,1}(q^{2})$ in Eq. (24) is adopted. In Eq.
(Discontinuities of banana integrals in dispersion relation representation),
the third term, which is divergent, comes from only the DR representation. The
first two terms come from the GDR, in which the second term is also divergent.
However, the divergences in the second and third terms cancel out exactly to
each other so that $\rho_{211}(s_{2};1,1,1)$ is finite. The GDR representation
of $B_{211}(q^{2};1,1,1)$ is obtained by substituting Eq. (Discontinuities of
banana integrals in dispersion relation representation) back into Eq. (23)
with $n_{\Gamma}=n_{\gamma}=1$ and $a=0$:

$\displaystyle B_{211}(q^{2};1,1,1$
$\displaystyle)=-\frac{9B_{211}(0;1,1,1)}{q^{2}-9}$
$\displaystyle+\int_{9}^{\infty}ds_{2}\frac{\rho_{211}(s_{2};1,1,1)}{s_{2}-q^{2}}\frac{q^{2}}{s_{2}}\frac{s_{2}-9}{q^{2}-9},$
(26)

which is convenient for numerical integration. However, one should note that
the real part contains the Cauchy principal value integral for $q^{2}>9$,
which may reduce numerical accuracy.

Using the result of GDR, an arbitrary banana integral can be represented as an
iteration of bubbles. Meanwhile, it is convenient to find the discontinuities.

Hypothesis about GDR and p-DOBIs.  — In this section, we try to discuss the
integrals arising in GDR. Let’s discuss the divergence of the third term from
DR in Eq. (Discontinuities of banana integrals in dispersion relation
representation)

$\begin{gathered}\begin{aligned}
&-\int_{4}^{s_{2\bar{2}}^{2}}ds_{1}\rho_{21}(s_{1};1,1)\rho_{11}(s_{2};\sqrt{s_{1}},1)\\\
=&-4\int_{4}^{s_{2\bar{2}}^{2}}ds_{1}(\lambda_{1}\lambda_{2})^{-\frac{1}{2}}(s_{1}-4)^{-1}=-4\mathcal{I}_{-1}^{4}(s_{2})\,.\end{aligned}\end{gathered}$
(27)

We use the same trick of Eq. (9) to deal with $\mathcal{I}_{-1}^{4}(s_{2})$.
The equation should be modified as the following for the parameter
$a_{1}=m_{01}^{2}=4$ at the normal threshold:

$\begin{gathered}\begin{aligned}
&\left[\sqrt{\lambda_{1}\lambda_{2}}(s_{1}-a_{1})^{k_{1}}\right]\bigg{|}_{s_{1}=m_{01}^{2}}^{s_{1}=s_{2\bar{2}}^{2}}\\\
=&\int^{s_{2\bar{2}}^{2}}_{m_{01}^{2}}ds_{1}\frac{d}{ds_{1}}\left[\sqrt{\lambda_{1}\lambda_{2}}(s_{1}-a_{1})^{k_{1}}\right].\end{aligned}\end{gathered}$
(28)

For $k_{1}=-1$,

$\displaystyle\frac{\sqrt{\lambda_{1}\lambda_{2}}}{(s_{1}-4)}\bigg{|}_{s_{1}=4}^{s_{1}=s_{2\bar{2}}^{2}}=$
$\displaystyle\mathcal{I}_{2}^{4}(s_{2})-2(s_{2}-1)(s_{2}-9)\mathcal{I}_{-1}^{4}(s_{2})$
$\displaystyle-(s_{2}-5)\mathcal{I}_{1}^{4}(s_{2})\,.$ (29)

The boundary term at the left hand side (LHS) shows how
$\mathcal{I}_{-1}^{4}(s_{2})$ diverges. In general, one can solve Eq.
(Discontinuities of banana integrals in dispersion relation representation)
and substitute $-4\mathcal{I}_{-1}^{4}(s_{2})$ back into Eq. (Discontinuities
of banana integrals in dispersion relation representation).

On the other hand, we find an alternative way to determine the DOBI
($s_{2}>9$):

$\displaystyle\rho_{211}(s_{2};1,1,1)$ $\displaystyle=$
$\displaystyle-4\left[\mathcal{I}_{-1}^{4}(s_{2})+\frac{1}{2(s_{2}-1)(s_{2}-9)}\frac{\sqrt{\lambda_{1}\lambda_{2}}}{(s_{1}-4)}\bigg{|}_{s_{1}=4}^{s_{1}=s_{2\bar{2}}^{2}}\right]$
$\displaystyle=$
$\displaystyle\frac{2(s_{2}-5)\mathcal{I}_{1}^{4}(s_{2})-2\mathcal{I}_{2}^{4}(s_{2})}{(s_{2}-1)(s_{2}-9)}\,,$
(30)

in which we have used Eq. (Discontinuities of banana integrals in dispersion
relation representation) to obtain the finite result in the last step.

It shows that we can simply calculate the term from DR iterations and reduce
it to p-DOBIs. Using the identities between p-DOBIs, we obtain the final
results containing only finite p-DOBIs. The boundary term with divergence can
be directly removed. This is our hypothesis about GDR and p-DOBIs. If this
hypothesis is true, one can in principle express any banana integral with
p-DOBIs.

Using this hypothesis, we have calculated the DOBIs $\rho_{211}(s_{2};1,2,3)$,
$\rho_{311}(s_{2};1,1,1)$ , $\rho_{2111}(s_{3};1,1,1,1)$, and our calculations
are in good agreement with the results of GDR.

According to Eq. (13), only two independent one-fold p-DOBIs are needed in the
equal mass case. It is well-known that $B_{1^{3}}(q^{2};1,1,1)$ and
$B_{211}(q^{2};1,1,1)$ are two master integrals for equal mass sunrise
integral at top sector Tarasov (1997); Laporta and Remiddi (2005), which is
actually suggested in Eq. (Discontinuities of banana integrals in dispersion
relation representation). It is possible to extend this idea to high-loop or
multi-mass cases.

Summary. — In this letter, we define the p-DOBIs by parameterizing the DOBIs
obtained in DR representation. The identities among p-DOBIs enable a variety
of p-DOBIs to be expressed linearly in terms of some fundamental p-DOBIs. The
p-DOBIs identities are useful tools for elliptic Feynman integrals. Similar to
the IBP identities revealing the structure of Feynman integrals, the p-DOBIs
identities can unveil the underlying structure of the maximal cut. We have
demonstrated several potential applications of this approach, which is capable
of handling multi-scale integrals and has the potential for extension to
higher loops. The properties of p-DOBIs make it possible to generalize the
definition of E-polylogarithm functions to three loops, which is a powerful
tool for exploring higher-order $\epsilon$ expressions of banana integrals
Remiddi and Tancredi (2017).

The DR iteration will give incorrect divergent result for integrals involving
high-power propagators. We establish the GDR to solve such problems by
considering the subtractions on both small and large circles of the
integration contour. With the knowledge of sub-loop integrals, the GDR
iterations may have potential for studying arbitrary Feynman integral. We
further find that one only needs to consider the term from DR iterations.
Assuming the hypothesis about GDR and p-DOBIs, we can determine the multi-loop
DOBIs in terms of finite p-DOBIs by ignoring the divergent boundary term.

Acknowledgments. — Xu-Liang Chen thanks Zhi-Zhong Chen for useful discussions.
This work is supported by the National Natural Science Foundation of China
under Grant No. 12175318, the Natural Science Foundation of Guangdong Province
of China under Grant No. 2022A1515011922.

##  Appendix A Appendix A: Analytical continuation of Bessel representation

Inspired by Ref. Zhang et al. (2010), we find the analytical continuation
expression of Bessel representation for the L-loop banana integrals
($q^{2}>0$):

$\begin{gathered}B_{1^{L}}(q^{2};\bm{m})=-\int_{0}^{\infty}dr\frac{r}{2}J_{0}(r\sqrt{q^{2}})\prod_{i=0}^{L}\left(i\pi
H_{0}^{(2)}(m_{i}r)\right)\,,\end{gathered}$ (31)

where $J_{0}(x)$ is the first kind Bessel function and $H_{0}^{(2)}(x)$ is the
second kind Hankel function. For $0<q^{2}<m_{0\cdots L}^{2}$, the Hankel
representation gives the same result with the Bessel representation:

$\begin{gathered}B_{1^{L}}(q^{2};\bm{m})=\int_{0}^{\infty}dr\frac{r}{2}J_{0}(r\sqrt{-q^{2}})\prod_{i=0}^{L}\left(-2K_{0}(m_{i}r)\right),\end{gathered}$
(32)

where $K_{0}(x)$ is the second kind modified Bessel function.

##  Appendix B Appendix B: Matrix $\mathcal{A}^{(1)}$

The explicit form of matrix $\mathcal{A}^{(1)}$ is

$\begin{gathered}\mathcal{A}^{(1)}=\left(\begin{matrix}0&0&0&0&0\\\
\frac{1}{2}&0&-\frac{1}{2}&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
\frac{1}{2}&0&-\frac{3}{2}&\frac{1}{2}&0\end{matrix}\right)\frac{1}{s_{3}}+\left(\begin{matrix}-\frac{1}{2}&\frac{1}{8}&\frac{1}{2}&-\frac{3}{8}&0\\\
0&0&-4&3&-\frac{1}{2}\\\
0&-\frac{3}{16}&-\frac{9}{4}&\frac{27}{16}&-\frac{3}{16}\\\
0&-\frac{1}{4}&-3&\frac{9}{4}&-\frac{1}{4}\\\
0&0&0&0&0\end{matrix}\right)\frac{1}{s_{3}-1}+\left(\begin{matrix}\frac{1}{2}&-\frac{1}{6}&-\frac{13}{6}&\frac{5}{3}&-\frac{1}{6}\\\
-\frac{1}{2}&\frac{1}{6}&\frac{13}{6}&-\frac{5}{3}&\frac{1}{6}\\\
-\frac{1}{2}&\frac{1}{6}&\frac{13}{6}&-\frac{5}{3}&\frac{1}{6}\\\
-\frac{1}{2}&\frac{1}{6}&\frac{13}{6}&-\frac{5}{3}&\frac{1}{6}\\\
\frac{1}{2}&-\frac{1}{6}&-\frac{13}{6}&\frac{5}{3}&-\frac{1}{6}\end{matrix}\right)\frac{1}{s_{3}-4}\\\
+\left(\begin{matrix}-\frac{1}{2}&\frac{1}{12}&\frac{10}{3}&-\frac{7}{3}&\frac{5}{24}\\\
0&\frac{2}{3}&\frac{32}{3}&-\frac{26}{3}&\frac{2}{3}\\\
0&\frac{1}{24}&\frac{2}{3}&-\frac{13}{24}&\frac{1}{24}\\\
0&\frac{1}{6}&\frac{8}{3}&-\frac{13}{6}&\frac{1}{6}\\\
0&\frac{4}{3}&\frac{64}{3}&-\frac{52}{3}&\frac{4}{3}\end{matrix}\right)\frac{1}{s_{3}-16}+\left(\begin{matrix}1&-\frac{1}{24}&-\frac{19}{6}&\frac{25}{24}&-\frac{1}{24}\\\
8&-\frac{1}{3}&-\frac{76}{3}&\frac{25}{3}&-\frac{1}{3}\\\
\frac{1}{2}&-\frac{1}{48}&-\frac{19}{12}&\frac{25}{48}&-\frac{1}{48}\\\
2&-\frac{1}{12}&-\frac{19}{3}&\frac{25}{12}&-\frac{1}{12}\\\
16&-\frac{2}{3}&-\frac{152}{3}&\frac{50}{3}&-\frac{2}{3}\end{matrix}\right)\,.\end{gathered}$
(33)

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