The Belle Collaboration

# Search for the lepton flavour violating decays $B^{+}\to
K^{+}\tau^{\pm}\ell^{\mp}$ ($\ell=e,\mu$) at Belle

S. Watanuki  0000-0002-5241-6628 G. de Marino  0000-0002-6509-7793 K. Trabelsi
0000-0001-6567-3036 I. Adachi  0000-0003-2287-0173 H. Aihara
0000-0002-1907-5964 D. M. Asner  0000-0002-1586-5790 H. Atmacan
0000-0003-2435-501X V. Aulchenko  0000-0002-5394-4406 T. Aushev
0000-0002-6347-7055 R. Ayad  0000-0003-3466-9290 V. Babu  0000-0003-0419-6912
Sw. Banerjee  0000-0001-8852-2409 M. Bauer  0000-0002-0953-7387 P. Behera
0000-0002-1527-2266 K. Belous  0000-0003-0014-2589 M. Bessner
0000-0003-1776-0439 V. Bhardwaj  0000-0001-8857-8621 B. Bhuyan
0000-0001-6254-3594 D. Biswas  0000-0002-7543-3471 D. Bodrov
0000-0001-5279-4787 G. Bonvicini  0000-0003-4861-7918 J. Borah
0000-0003-2990-1913 A. Bozek  0000-0002-5915-1319 M. Bračko
0000-0002-2495-0524 P. Branchini  0000-0002-2270-9673 T. E. Browder
0000-0001-7357-9007 A. Budano  0000-0002-0856-1131 M. Campajola
0000-0003-2518-7134 L. Cao  0000-0001-8332-5668 D. Červenkov
0000-0002-1865-741X M.-C. Chang  0000-0002-8650-6058 B. G. Cheon
0000-0002-8803-4429 K. Chilikin  0000-0001-7620-2053 K. Cho
0000-0003-1705-7399 S.-J. Cho  0000-0002-1673-5664 S.-K. Choi
0000-0003-2747-8277 Y. Choi  0000-0003-3499-7948 S. Choudhury
0000-0001-9841-0216 D. Cinabro  0000-0001-7347-6585 S. Das
0000-0001-6857-966X G. De Nardo  0000-0002-2047-9675 G. De Pietro
0000-0001-8442-107X R. Dhamija  0000-0001-7052-3163 F. Di Capua
0000-0001-9076-5936 T. V. Dong  0000-0003-3043-1939 D. Epifanov
0000-0001-8656-2693 T. Ferber  0000-0002-6849-0427 D. Ferlewicz
0000-0002-4374-1234 B. G. Fulsom  0000-0002-5862-9739 R. Garg
0000-0002-7406-4707 V. Gaur  0000-0002-8880-6134 A. Garmash
0000-0003-2599-1405 A. Giri  0000-0002-8895-0128 P. Goldenzweig
0000-0001-8785-847X E. Graziani  0000-0001-8602-5652 T. Gu
0000-0002-1470-6536 Y. Guan  0000-0002-5541-2278 K. Gudkova
0000-0002-5858-3187 C. Hadjivasiliou  0000-0002-2234-0001 S. Halder
0000-0002-6280-494X X. Han  0000-0003-1656-9413 T. Hara  0000-0002-4321-0417
K. Hayasaka  0000-0002-6347-433X H. Hayashii  0000-0002-5138-5903 D. Herrmann
0000-0001-9772-9989 W.-S. Hou  0000-0002-4260-5118 C.-L. Hsu
0000-0002-1641-430X K. Inami  0000-0003-2765-7072 G. Inguglia
0000-0003-0331-8279 N. Ipsita  0000-0002-2927-3366 A. Ishikawa
0000-0002-3561-5633 R. Itoh  0000-0003-1590-0266 M. Iwasaki
0000-0002-9402-7559 W. W. Jacobs  0000-0002-9996-6336 Q. P. Ji
0000-0003-2963-2565 S. Jia  0000-0001-8176-8545 Y. Jin  0000-0002-7323-0830 K.
K. Joo  0000-0002-5515-0087 A. B. Kaliyar  0000-0002-2211-619X H. Kichimi
0000-0003-0534-4710 C. H. Kim  0000-0002-5743-7698 D. Y. Kim
0000-0001-8125-9070 K.-H. Kim  0000-0002-4659-1112 Y.-K. Kim
0000-0002-9695-8103 K. Kinoshita  0000-0001-7175-4182 P. Kodyš
0000-0002-8644-2349 A. Korobov  0000-0001-5959-8172 S. Korpar
0000-0003-0971-0968 E. Kovalenko  0000-0001-8084-1931 P. Križan
0000-0002-4967-7675 P. Krokovny  0000-0002-1236-4667 T. Kuhr
0000-0001-6251-8049 M. Kumar  0000-0002-6627-9708 K. Kumara
0000-0003-1572-5365 A. Kuzmin  0000-0002-7011-5044 Y.-J. Kwon
0000-0001-9448-5691 J. S. Lange  0000-0003-0234-0474 M. Laurenza
0000-0002-7400-6013 S. C. Lee  0000-0002-9835-1006 P. Lewis
0000-0002-5991-622X L. K. Li  0000-0002-7366-1307 Y. Li  0000-0002-4413-6247
L. Li Gioi  0000-0003-2024-5649 J. Libby  0000-0002-1219-3247 Y.-R. Lin
0000-0003-0864-6693 D. Liventsev  0000-0003-3416-0056 T. Matsuda
0000-0003-4673-570X S. K. Maurya  0000-0002-7764-5777 F. Meier
0000-0002-6088-0412 M. Merola  0000-0002-7082-8108 F. Metzner
0000-0002-0128-264X K. Miyabayashi  0000-0003-4352-734X R. Mizuk
0000-0002-2209-6969 G. B. Mohanty  0000-0001-6850-7666 M. Nakao
0000-0001-8424-7075 L. Nayak  0000-0002-7739-914X M. Nayak
0000-0002-2572-4692 N. K. Nisar  0000-0001-9562-1253 S. Nishida
0000-0001-6373-2346 H. Ono  0000-0003-4486-0064 P. Oskin  0000-0002-7524-0936
G. Pakhlova  0000-0001-7518-3022 S. Pardi  0000-0001-7994-0537 H. Park
0000-0001-6087-2052 J. Park  0000-0001-6520-0028 S.-H. Park
0000-0001-6019-6218 A. Passeri  0000-0003-4864-3411 T. K. Pedlar
0000-0001-9839-7373 R. Pestotnik  0000-0003-1804-9470 L. E. Piilonen
0000-0001-6836-0748 T. Podobnik  0000-0002-6131-819X E. Prencipe
0000-0002-9465-2493 M. T. Prim  0000-0002-1407-7450 M. Röhrken
0000-0003-0654-2866 N. Rout  0000-0002-4310-3638 G. Russo  0000-0001-5823-4393
S. Sandilya  0000-0002-4199-4369 A. Sangal  0000-0001-5853-349X L. Santelj
0000-0003-3904-2956 V. Savinov  0000-0002-9184-2830 G. Schnell
0000-0002-7336-3246 C. Schwanda  0000-0003-4844-5028 Y. Seino
0000-0002-8378-4255 K. Senyo  0000-0002-1615-9118 M. E. Sevior
0000-0002-4824-101X W. Shan  0000-0003-2811-2218 M. Shapkin
0000-0002-4098-9592 J.-G. Shiu  0000-0002-8478-5639 B. Shwartz
0000-0002-1456-1496 F. Simon  0000-0002-5978-0289 E. Solovieva
0000-0002-5735-4059 M. Starič  0000-0001-8751-5944 M. Sumihama
0000-0002-8954-0585 T. Sumiyoshi  0000-0002-0486-3896 M. Takizawa
0000-0001-8225-3973 K. Tanida  0000-0002-8255-3746 F. Tenchini
0000-0003-3469-9377 M. Uchida  0000-0003-4904-6168 T. Uglov
0000-0002-4944-1830 Y. Unno  0000-0003-3355-765X K. Uno  0000-0002-2209-8198
S. Uno  0000-0002-3401-0480 R. van Tonder  0000-0002-7448-4816 G. Varner
0000-0002-0302-8151 K. E. Varvell  0000-0003-1017-1295 D. Wang
0000-0003-1485-2143 E. Wang  0000-0001-6391-5118 M.-Z. Wang
0000-0002-0979-8341 E. Won  0000-0002-4245-7442 X. Xu  0000-0001-5096-1182 B.
D. Yabsley  0000-0002-2680-0474 W. Yan  0000-0003-0713-0871 S. B. Yang
0000-0002-9543-7971 J. Yelton  0000-0001-8840-3346 Y. Yusa
0000-0002-4001-9748 Z. P. Zhang  0000-0001-6140-2044 V. Zhilich
0000-0002-0907-5565 V. Zhukova  0000-0002-8253-641X

###### Abstract

We present a search for the lepton-flavour-violating decays $B^{+}\to
K^{+}\tau^{\pm}\ell^{\mp}$, with $\ell=(e,\mu)$, using the full data sample of
$772\times 10^{6}$ $B\overline{B}$ pairs recorded by the Belle detector at the
KEKB asymmetric-energy $e^{+}e^{-}$ collider. We use events in which one $B$
meson is fully reconstructed in a hadronic decay mode. We find no evidence for
$B^{\pm}\to K^{\pm}\tau\ell$ decays and set upper limits on their branching
fractions at the 90% confidence level in the $(1$–$3)\times 10^{-5}$ range.
The obtained limits are the world’s best results.

###### pacs:

11.30.Er, 13.20.Fc, 13.25.Ft

††preprint:  Belle Preprint 2022-30 KEK Preprint 2022-41

Recently, there has been a resurgence of interest in the study of leptoquark
fields in light of discrepancies in semi-leptonic $B$-decays nature ,
collectively known as $B$-physics anomalies, which challenge the assumed
Lepton Flavour Universality (LFU) of fundamental interactions. These
measurements have been obtained studying two different quark transitions:
$b\to c\tau\nu$ and $b\to s\ell\ell$, where $\ell=(e,\mu)$. If confirmed by
further measurements, this would be clear evidence of New Physics in which new
heavy particles couple preferentially to second and third generation leptons.
Many extensions of the Standard Model (SM) that include violation of LFU
predict Lepton Flavour Violating (LFV) processes in hadron decays with charged
leptons in the final state glashow . In particular, the vector leptoquark
$U_{1}$ with SM quantum numbers $(\mathbf{3},\mathbf{1})_{2/3}$ has been
identified as the only single-mediator solution Angelescu:2021lln . In the
minimal scenario, $U_{1}$ provides an interesting prediction, i.e. lower
bounds on the lepton flavour violating $b\to s\tau^{\mp}\mu^{\pm}$ decay
modes, for example ${\cal B}(B\to K\tau\mu)>0.7\times 10^{-7}$. The branching
fractions for the two $\ell\tau$ charge combinations are not in general the
same, as they depend on the details of the physics mechanism producing the
decay.

Upper limits on the branching fractions for $B^{+}\to
K^{+}\tau^{\pm}\ell^{\mp}$ decays have been previously set at the 90%
confidence level (C.L.) using hadronic $B$-tagging by the BaBar collaboration
between $1.5\times 10^{-5}$ and $4.5\times 10^{-5}$ babar ; the LHCb
collaboration has studied a single mode, using $B^{+}$ mesons from
$B_{s2}^{*0}\to B^{+}K^{-}$ decays, setting a limit ${\cal B}(B^{+}\to
K^{+}\tau^{+}\mu^{-})<3.9\times 10^{-5}$ at the 90% C.L. LHCb:2020khb .

In this Letter, we report a search for $B^{+}\to K^{+}\tau^{\mp}\ell^{\pm}$
decays using the full Belle data sample recorded at the $\Upsilon(4S)$
resonance. The inclusion of the charge-conjugate decay mode is implied. This
is the first such search from Belle.

The analysis is based on the full data sample of $772\times 10^{6}$
$B\overline{B}$ pairs collected with the Belle detector Belle:2000cnh  at the
KEKB asymmetric-energy $e^{+}e^{-}$ collider Kurokawa:2001nw . The Belle
detector is a large-solid-angle spectrometer, which includes a silicon vertex
detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel
threshold Cherenkov counters (ACC), time-of-flight scintillation counters
(TOF), and an electromagnetic calorimeter (ECL) comprised of CsI(Tl) crystals
located inside a superconducting solenoid coil that provides a 1.5 T magnetic
field. An iron flux return located outside the coil is instrumented to detect
$K^{0}_{L}$ mesons and identify muons.

The analysis procedure is developed using Monte Carlo (MC) simulation based on
events generated with EvtGen evtgen, which includes final-state radiation
(FSR) effects simulated by PHOTOS photos , and the detector response is
simulated by GEANT3 geant . The $B^{+}\to K^{+}\tau^{\pm}\ell^{\mp}$ decays
are generated using a uniform three-body phase space model (PHSP); we also
consider variations in the linear combinations of the relevant operators for
the $b\rightarrow s\tau\ell$ transitions: $\mathcal{O}_{i}$ and the relative
Wilson coefficients $C_{i}^{\tau\ell}$, where $i=9,10,S,P$ Becirevic:1602 .

In each event, we require a fully reconstructed hadronic $B^{\pm}$ decay,
which we refer to as the tagged $B$ meson candidate or $B_{\rm tag}$. This is
done using the Full Event Interpretation (FEI) algorithm fei , a machine-
learning algorithm developed for $B$-tagged analyses at Belle and Belle II. It
supports both hadronic and semileptonic tagging, reconstructing $B$ mesons
across more than 4000 individual decay chains. The training is performed in a
hierarchical manner: final-state particles are first reconstructed from
detector information, then unstable particles (e.g. $D$, $D^{*}$) are built up
from these particles, and then reconstruction of $B$-mesons is performed last.
For each $B_{\rm tag}$ candidate reconstructed by the FEI, a value of the
final multi-variate classifier output, $\Sigma_{\rm FEI}$, is assigned.
$\Sigma_{\rm FEI}$ is distributed between zero and one, representing
candidates identified as being background-like and signal-like, respectively.

For the hadronic FEI, the minimal number of tracks per event satisfying
certain quality criteria is set to three, as the vast majority of $B$-meson
chains include at least three charged particles, and such a criterion is
useful for suppressing background from non-$B\overline{B}$ events.
Requirements are placed on the impact parameters to ensure close proximity to
the interaction point (IP), less than $0.5$ cm in the transverse plane and
less than $2.0$ cm along the $z$ axis (parallel to the $e^{+}$ beam). ECL
clusters that are used for $\gamma$ reconstruction are required to satisfy a
region-dependent energy threshold criterion. All the intermediate states
($\pi^{0},J/\psi,K_{S}^{0}$ and $D^{(*)}$ mesons) must pass loose cuts on the
reconstructed invariant mass and only the best candidates in terms of
$\Sigma_{\rm FEI}$ are kept. The FEI results in many $B_{\rm tag}$ candidates
per event. The number of these candidates is reduced with selections on the
beam-energy-constrained mass $M_{\rm bc}=\sqrt{(E^{*}_{\rm
beam}/c^{2})^{2}-(p^{*}_{{B}_{\rm tag}}/c)^{2}}$, and the energy difference
$\Delta E=E^{*}_{B_{\rm tag}}-E^{*}_{\rm beam}$, where $E^{*}_{\rm beam}$ is
the beam energy, and $E^{*}_{B_{\rm tag}}$ and $p^{*}_{B_{\rm tag}}$ are the
energy and momentum of the $B_{\rm tag}$ candidate in the center of mass
(c.m.) rest frame, respectively. The criteria applied are $M_{\rm bc}>5.27$
GeV/$c^{2}$ and $|\Delta E|<0.1$ GeV. Finally, the candidate with the highest
$B_{\rm tag}$ classifier output, $\Sigma_{\rm FEI}$, is selected and a loose
requirement $\Sigma_{\rm FEI}$ $>0.001$, provides further background rejection
with little signal loss.

We then search for the signal $B\to K\tau\ell$ decay in the rest of the event,
which we refer to as the signal $B$ meson candidate or $B_{\rm sig}$. The
notation $B\to K\tau\ell$ refers to one of the following four final states
that we consider, where in addition to the kaon of opposite charge to $B_{\rm
tag}$ we associate the primary lepton, $\mu$ or $e$: $B^{+}\to
K^{+}\tau^{+}\mu^{-}$ and $B^{+}\to K^{+}\tau^{+}e^{-}$ defined as
$OS_{\mu,e}$ modes because the kaon and the primary lepton have opposite
charge, and $B^{+}\to K^{+}\tau^{-}\mu^{+}$ and $B^{+}\to K^{+}\tau^{-}e^{+}$,
defined as $SS_{\mu,e}$ modes. In all cases, we require that the $\tau$ decays
to $\tau\to e\nu\overline{\nu}$, $\tau\to\mu\nu\overline{\nu}$, or
$\tau\to\pi\nu$. The combined branching fraction for these decays is 46%
ParticleDataGroup:2022pth . The $\tau\to\rho\nu$ mode, despite not being
explicitly reconstructed, significantly contributes to the $\tau\to\pi\nu$
candidates – by roughly one half – because of its large branching fraction
($\sim 25\%$).

We reconstruct $B^{+}\to K\tau^{\pm}\ell^{\mp}$ decays by selecting three
charged particles that originate from the vicinity of the IP and are not
associated with the $B_{\rm tag}$. We require impact parameters less than
$1.0$ cm in the transverse plane and less than $4.0$ cm along the $z$ axis. To
reduce backgrounds from low-momentum particles, we require that tracks have a
minimum transverse momentum of 100 MeV/$c$. From the list of selected tracks,
we identify $K^{+}$ candidates using a likelihood ratio ${\cal
R}_{K/\pi}={\cal L}_{K}/({\cal L}_{K}+{\cal L}_{\pi})$, where ${\cal L}_{K}$
and ${\cal L}_{\pi}$ are the likelihoods for charged kaons and pions,
respectively, calculated based on the number of photoelectrons in the ACC, the
specific ionization in the CDC, and the time of flight as determined from the
TOF. We select kaons by requiring ${\cal R}_{K/\pi}>0.6$, which has a kaon
identification efficiency of 83% and a pion misidentification rate of 5%.
Similarly, we select pions by requiring ${\cal R}_{\pi/K}>0.6$, which has a
pion identification efficiency of 84% and a kaon misidentification rate of 6%.

Muon candidates are identified based on information from the KLM. We require
that candidates have a momentum greater than 0.8 GeV/$c$ (enabling them to
sufficiently penetrate KLM), and a penetration depth and degree of transverse
scattering consistent with those of a muon muid . The latter information is
used to calculate a normalized muon likelihood ratio ${\cal R}_{\mu}={\cal
L}_{\mu}/({\cal L}_{\mu}+{\cal L}_{K}+{\cal L}_{\pi})$, where ${\cal L}_{\mu}$
is the likelihood for muons, for which we require ${\cal R}_{\mu}>0.9$. For
this requirement, the average muon detection efficiency is 89%, with a pion
misidentification rate of 1.5% pid .

Electron candidates are required to have a momentum greater than 0.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and are identified using the ratio of
ECL cluster energy to the CDC track momentum, the shower shape in the ECL, the
matching of the track with the ECL cluster, the specific ionization in the
CDC, and the number of photoelectrons in the ACC. This information is used to
calculate a normalized electron likelihood ratio ${\cal R}_{e}={\cal
L}_{e}/({\cal L}_{e}+{\cal L}_{\rm hadrons})$, where ${\cal L}_{\rm hadrons}$
is a product of hadron likelihoods, for which we require ${\cal
R}_{e}>~{}0.9$. This requirement has an efficiency of 92% and a pion
misidentification rate below 1% eid .

After selecting one charged kaon, one prompt lepton (electron or muon) and the
$\tau$ daughter (electron, muon or pion) with the appropriate charge
combination, we require that there are no other tracks than the ones
associated to $B_{\rm tag}$ or $B_{\rm sig}$. The charged kaon and the prompt
lepton are uniquely determined to minimize $\chi^{2}$ of the $B_{\rm sig}$
vertex fit for the prompt tracks. In case there are two possibilities in
$\tau$ daughter particle identification, $\tau$ leptonic decay has priority.
Unlike other $B$ decays involving $\tau$’s (e.g. $B\to\tau\nu$, $B\to
D^{*}\tau\nu$), the $B\to K\tau\ell$ channel has the unique property of having
the one (or two) neutrino(s) coming only from the $\tau$ itself, allowing the
signal yield to be extracted using the recoil mass, $M_{\rm recoil}$, which
should peak at the mass of the $\tau$ lepton. Such variable is easily obtained
at $B$-factories, because of the known initial kinematics and the full
reconstruction of the other $B$ in the event. In fact, if we consider the
$B_{\rm sig}$, the 4-momentum of the $\tau$ can be written as:

$p_{\tau}=p_{B_{\rm sig}}-p_{K}-p_{\ell}\\\ $ (1)

where $p_{B_{\rm sig}}$ is not known a priori. In the frame where the
$\Upsilon{(4S)}$ resonance is at rest, the two $B$ mesons are back to back,
hence:

$\displaystyle\mathbf{p}^{*}_{B_{\rm tag}}$
$\displaystyle=-\mathbf{p}^{*}_{B_{\rm sig}}$ (2)

furthermore, the two $B$’s have the same energy, which is half the energy
$\sqrt{s}$ of the $\Upsilon{(4S)}$:

$\displaystyle{E}^{*}_{B_{\rm tag}}$ $\displaystyle={E}^{*}_{B_{\rm
sig}}=\frac{\sqrt{s}}{2}.$ (3)

In order to obtain the best resolution on the $B$ variables, we replace
${E}^{*}_{B_{\rm tag}}$ with $E_{\rm beam}^{*}$, but use the reconstructed
${p}^{*}_{B_{\rm tag}}$ rather than the average value $p_{\rm
beam}^{*}=\sqrt{E_{\rm beam}^{*2}/c^{2}-m_{B}^{2}c^{2}}$. Using the condition
(2) and the substitution $E_{B}^{*}=E^{*}_{\rm beam}$ in equation (1), we
obtain:

$\begin{cases}\mathbf{p}_{\tau}^{*}&=-\mathbf{p}_{B_{\rm
tag}}^{*}-\mathbf{p}_{K}^{*}-\mathbf{p}_{\ell}^{*}\\\ {E}_{\tau}&={E}_{\rm
beam}^{*}-E_{K}^{*}-E_{\ell}^{*}\\\ \end{cases}\implies M_{\rm
recoil}^{2}=m_{\tau}^{2}=m_{B}^{2}+m_{{K}\ell}^{2}-2(E^{*}_{\rm
beam}E^{*}_{{K}\ell}/c^{4}+p^{*}_{B_{\rm
tag}}p^{*}_{{K}\ell}\cos\theta/c^{2})$ (4)

where $\theta$ is the angle between $\mathbf{p}^{*}_{B_{\rm tag}}$ and
$\mathbf{p}^{*}_{{K}\ell}$.

The main source of background consists of Cabibbo-favoured transitions from
$B^{+}B^{-}$ events. For the $OS$ configurations, where the primary lepton
charge is opposite to the $B_{\rm sig}$ charge, the dominant background comes
from semileptonic $D$ decays: $B^{+}\to\overline{D}^{0}(\to
K^{+}\ell^{-}\overline{\nu}_{\ell})X^{+}$. On the other hand, for the $SS$
configurations the primary lepton and the $B_{\rm sig}$ have the same charge
and the semileptonic $B^{+}$ decays like $B^{+}\to\overline{D}^{0}(\to
K^{+}X^{-})X\ell^{+}\nu_{\ell}$ provide the three charged particles for the
$B_{\rm sig}$ candidates. Events compatible with a $B^{+}_{\rm
sig}\to\overline{D}^{0}(\to{K^{+}\pi^{-})X^{+}}$ decay are rejected by vetoing
candidates in the range $1.81$ GeV$/c^{2}<m_{K^{+}t^{-}}<1.91$ GeV$/c^{2}$,
where $t$ denotes the primary lepton or the track from the $\tau$ in the $OS$
or $SS$ case, respectively. In the first case, only the $K\tau\mu$ modes show
such a $D^{0}$ component because of the larger probability to identify a pion
as a muon rather than an electron. In the $SS$ case, the $D^{0}$ peak is much
more prominent and relates to the $\tau\to\pi$ mode and is independent of the
flavour of the primary lepton.

We further improve the signal selection using a Boosted Decision Tree (BDT)
classification. Two classifiers are trained for the background suppression.
The first one is optimised to reduce the $B\overline{B}$ events and uses as
inputs some kinematic information as well as the topology of the $B_{\rm sig}$
and information on the rest of the event (the set of ECL clusters that are not
used for the $B_{\rm sig}$ and $B_{\rm tag}$ reconstruction). In particular we
use: the invariant mass $m_{K^{+}t^{-}}$, which helps in suppressing the
combinatorial background from charm decays, the number of ECL clusters that
are not associated with the reconstructed event and the sum of their energies,
the extended Fox-Wolfram moments KFSW , the distance from the IP of the signal
vertex and the distance between the primary kaon and each of the other two
signal tracks. For each mode only the ten most important variables are kept
for the final training, the metrics being the information gain provided by
each feature in all the decision trees used for the classifier. The threshold
$t$ on the BDT response is optimized using a figure of merit punzi , defined
as

$\mathcal{F}(t)=\frac{\epsilon(t)}{\frac{3}{2}+\sqrt{N_{\rm bkg}(t)}},$ (5)

where $\epsilon(t)$ is the efficiency for the cut $t$, $N_{\rm bkg}$
represents the number of background events surviving the cut $t$ in the signal
region defined as $1.68$ GeV/$c^{2}$ $<M_{\rm recoil}<1.87$ GeV/$c^{2}$ which
contains $\sim 80\%$ of the signal events. The MC sample used to estimate the
background corresponds to a luminosity of twice that of data. After the cut on
the first BDT output, a large fraction of the surviving background is coming
from $q\overline{q}$ ($q=u,d,s,c$) events; for this reason a second BDT
classifier is trained on these events. The input variables to suppress the
continuum background are: event-shape variables such as $R_{2}$ and the CLEO
cones cleoC  and the angle $\theta_{\rm T}$ between the thrust axes calculated
from final-state particles for the $B_{\rm tag}$ and for the rest of the event
in the c.m. frame.

We use control samples in order to evaluate systematic uncertainties related
to data/MC discrepancies and to calibrate the signal shape PDF as it is fixed
from MC simulation. The first control sample consists of $B^{+}\to
D^{-}\pi^{+}\pi^{+}$ events, generated in MC as the result of two-body
$B^{+}\to\overline{D}^{**0}(\to D^{-}\pi^{+})\pi^{+}$ decays, with
$\overline{D}^{**0}=\\{\overline{D}_{0}^{*0},\overline{D}_{2}^{*0}\\}$
according to Refs. BaBar:2009pnd ; Belle:2003nsh . This channel has similar
topology to our signal as the $D$ can be treated as the $\tau$, allowing for a
comparison of the performance of the first BDT classifier between data and MC
(Fig. 1(top) for $OS_{\mu}$). For the calibration of the efficiency of
$q\overline{q}$ suppression a second control sample $B\to J/\psi K$ is used
because of the similar final state while no usage of the $B_{\rm sig}$
topology is required (Fig. 1(bottom) for $OS_{\mu}$).

Figure 1: The BDT response of the $B\overline{B}$ suppression for the
$B^{+}\to D^{-}\pi^{+}\pi^{+}$ mode (top), and the $q\overline{q}$ suppression
for the $B^{+}\to J/\psi K^{+}$ mode (bottom) for $B^{+}\to
K^{+}\tau^{+}\mu^{-}$ case.

The signal yields for $B\to K\tau\ell$ decays are obtained by performing
unbinned extended maximum-likelihood fits to the $M_{\rm recoil}$
distributions. The PDF used to model reconstructed signal decays consists of
the sum of a reversed Crystal Ball function to model the main peak and the
high-side power-law tail, and a broad Gaussian, with the same mean parameter,
to describe the candidates with worse resolution due to imperfect $B_{\rm
tag}$ reconstruction. The background events have a smooth shape in the $M_{\rm
recoil}$ signal region, and are described by a $2$nd-order Chebychev
polynomial. The yields are floated, as well as the background shape parameters
while the parameters describing the signal PDF are fixed from the MC
simulation. We apply corrections to these parameters to account for small
differences between MC simulation and data. These correction factors are
obtained from the $B^{+}\to\overline{D}{}^{(*)0}\pi^{+}$ control samples where
$M_{\rm recoil}$ is calculated from the pion from the $B^{+}$ decay and the
$B_{\rm tag}$. We validate our fitting procedure and check for fit bias using
MC simulation. We generate large ensembles of simulated experiments in which
the $M_{\rm recoil}$ distributions are generated from the PDFs used for
fitting.

The $M_{\rm recoil}$ distributions for LFV $B\to K\tau\ell$ decays along with
projections of the fit result are shown in Fig. 2. The fitted signal yields
listed in Table 1 are consistent with zero for all four modes.

Figure 2: Observed $M_{\rm recoil}$ distributions for the four $B\to
K\tau\ell$ modes, along with projections of the fit result. The black dots
show the data, the dashed blue curve shows the background component, and the
solid red curve shows the overall fit result. The dash-dotted green curve
shows the signal PDF, with a normalization corresponding to the 90% C.L. upper
limit.

We calculate the upper limit (UL) for these modes at the $90\%$ C.L. using a
frequentist method. In this method, for different numbers of signal events
$N_{\rm{sig}}(\rm{gen})$, we generate 10000 pseudo experiments with signal and
background PDFs as obtained in the nominal data fit, with each set of events
being statistically equivalent to our data sample of 711 fb-1. We fit all
these simulated data sets, and, for each value of $N_{\rm{sig}}(\rm{gen)}$, we
calculate the fraction of MC experiments that have $N_{\rm sig}\leq
N_{\rm{sig}}(\rm{data})$. The $90\%$ C.L. upper limit is taken to be the value
of $N_{\rm{sig}}(\rm{gen})$ (called here $N_{\rm sig}^{\rm{UL}}$) for which
$10\%$ of the experiments have $N_{\rm{sig}}\leq N_{\rm{sig}}(\rm{data})$. The
upper limit on the branching fraction is then derived using the formula:

$\cal B^{\rm{UL}}$ $=\dfrac{N_{\rm sig}^{\rm{UL}}}{N_{B\overline{B}}\times
2\times f^{+-}\times\varepsilon}$,

where $N_{B\overline{B}}$ is the number of $B\overline{B}$ pairs = $(772\pm
11)\times 10^{6}$, $f^{+-}$ is the branching fraction ${\cal
B}(\Upsilon(4S)\to B^{+}B^{-})$ for charged $B$ decays (using $0.514\pm 0.006$
ParticleDataGroup:2022pth ), and $\varepsilon$ is the signal reconstruction
efficiency. By default $\varepsilon$ is obtained with signal PHSP MC samples
efficiency , while we also consider a NP model with a combination of the
effective operators $\mathcal{O}_{S,P}$ by reweighting the
$q^{2}=m_{\tau\ell}^{2}$ distribution which gives the smallest efficiency. The
systematic uncertainty in $\cal B^{\rm{UL}}$ is included by smearing the
$N_{\rm sig}$ distribution obtained from the MC fits with the fractional
systematic uncertainty. The results are listed in Table 1.

Table 1: Efficiencies, fit yields, and branching fraction upper limits at the 90% C.L. for PHSP (and NP) case.  Mode |  $\varepsilon$ (%) |  $\varepsilon^{\rm NP}$ (%) | $N_{\rm sig}$ |  $\cal B^{\rm{UL}}$ $(10^{-5})$
---|---|---|---|---
$B^{+}\rightarrow K^{+}\tau^{+}\mu^{-}$ | 0.064 | 0.058 | $-2.1\pm 2.9$ | 0.59 (0.65)
$B^{+}\rightarrow K^{+}\tau^{+}e^{-}$ | 0.084 | 0.074 | $\;\;\;1.5\pm 5.5$ | 1.51 (1.71)
$B^{+}\rightarrow K^{+}\tau^{-}\mu^{+}$ | 0.046 | 0.038 | $\;\;\;2.3\pm 4.1$ | 2.45 (2.97)
$B^{+}\rightarrow K^{+}\tau^{-}e^{+}$ | 0.079 | 0.058 | $-1.1\pm 7.4$ | 1.53 (2.08)
Table 2: Contributions to the systematic uncertainties of the measurements. Source | $K^{+}\tau^{+}\mu^{-}$ | $K^{+}\tau^{+}e^{-}$ | $K^{+}\tau^{-}\mu^{+}$ | $K^{+}\tau^{-}e^{+}$
---|---|---|---|---
Additive (events) |  |  |  |
PDF shape (mean) | $0.09$ | $0.01$ | $0.08$ | $0.08$
PDF shape (width) | $0.02$ | $0.08$ | $0.04$ | $0.07$
PDF shape ($f_{\rm sig}$) | $0.28$ | $0.16$ | $0.11$ | $0.16$
Linearity | $0.03$ | $0.04$ | $0.02$ | $0.04$
Total | $0.30$ | $0.18$ | $0.14$ | $0.20$
Multiplicative (%) |  |  |  |
$B_{\rm tag}$ calibration | $5.9$ | $5.9$ | $5.9$ | $5.9$
Track reconstruction | $1.1$ | $1.1$ | $1.1$ | $1.1$
Kaon id. | $1.3$ | $1.4$ | $1.3$ | $1.3$
Lepton id. | $0.3$ | $0.4$ | $0.3$ | $0.4$
$\tau$ daughter id. | $0.7$ | $0.7$ | $0.6$ | $0.6$
MC statistics | $1.0$ | $1.5$ | $1.2$ | $1.0$
Number of $B\overline{B}$ pairs | $1.4$ | $1.4$ | $1.4$ | $1.4$
BDT $B\overline{B}$ selection | $10.6$ | $10.0$ | $12.7$ | $12.6$
BDT $q\overline{q}$ selection | $8.8$ | $8.6$ | $9.2$ | $6.6$
$f^{+-}$ | $1.2$ | $1.2$ | $1.2$ | $1.2$
Total | $15.3$ | $14.8$ | $17.0$ | $15.7$

The systematic uncertainties in our measurements are listed in Table 2, where
additive uncertainties arise from the signal yield, while multiplicative
uncertainties are from the efficiency. Uncertainties in the shape of the PDFs
used for the signal are evaluated by varying all fixed parameters by $\pm
1\sigma$, including the correction factors to the shapes obtained from the
$B^{+}\to\overline{D}^{(*)0}\pi^{+}$ control samples, and varying the fraction
of the Gaussian ($f_{\rm sig}$) by 10%. The resulting change in the signal
yield is taken as the systematic uncertainty. The reconstruction efficiency
for $B_{\rm tag}$ evaluated via MC simulation, is corrected to account for
differences between MC and data in the branching fractions and models used for
hadronic $B$ decays. This correction is evaluated by comparing the number of
events containing both a $B_{\rm tag}$ and a semileptonic $B\to D\ell\nu$ Gelb
. The resulting correction factor is $85\pm 5$% and the uncertainty in this
value is taken as a systematic uncertainty.

The systematic uncertainty due to the charged track reconstruction is
evaluated using $D^{*+}\to D^{0}\pi^{+}$ with $D^{0}\to
K_{S}^{0}\pi^{+}\pi^{-}$, resulting in an uncertainty of 0.35% per track.
Uncertainties due to $K^{+}$ and $\pi^{+}$ (for $\tau\rightarrow\pi\nu$ mode)
identification is 1.3%, as measured with a $D^{*+}\to
D^{0}(K^{-}\pi^{+})\pi^{+}$ sample. The uncertainty due to lepton
identification is evaluated using $J/\psi\to\ell^{+}\ell^{-}$ events,
resulting in an uncertainty of 0.3% for muons and 0.4% for electrons. The
systematic uncertainty arising from the number of $B\overline{B}$ pairs is
1.4%. We compare the efficiency of the BDT selection between data and MC
samples with the control channel $B^{+}\to D^{-}\pi^{+}\pi^{+}$ for
$B\overline{B}$ suppression and $B^{+}\to J/\psi K^{+}$ for continuum
suppression, the differences between data and MC simulation are assigned as a
systematic uncertainty. We use a systematic uncertainty of 1.2% in the
fraction $f^{+-}$ ParticleDataGroup:2022pth .

We have searched for the lepton-flavour-violating decays $B^{+}\to
K^{+}\tau^{\pm}\ell^{\mp}$ using the full Belle data set. We find no evidence
for these decays and set the following upper limits on the branching fractions
at the 90% C.L.:

$\displaystyle\begin{split}&{\cal B}(B^{+}\rightarrow
K^{+}\tau^{+}\mu^{-})<0.59\times 10^{-5}\\\ &{\cal B}(B^{+}\rightarrow
K^{+}\tau^{+}e^{-})<1.51\times 10^{-5}\\\ &{\cal B}(B^{+}\rightarrow
K^{+}\tau^{-}\mu^{+})<2.45\times 10^{-5}\\\ &{\cal B}(B^{+}\rightarrow
K^{+}\tau^{-}e^{+})<1.53\times 10^{-5}\end{split}$ (6)

Our results are the most stringent limits to date.

This work, based on data collected using the Belle detector, which was
operated until June 2010, was supported by the Ministry of Education, Culture,
Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the
Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of
Nagoya University; the Australian Research Council including grants
DP180102629, DP170102389, DP170102204, DE220100462, DP150103061, FT130100303;
Austrian Federal Ministry of Education, Science and Research (FWF) and FWF
Austrian Science Fund No. P 31361-N36; the National Natural Science Foundation
of China under Contracts No. 11675166, No. 11705209; No. 11975076; No.
12135005; No. 12175041; No. 12161141008; Key Research Program of Frontier
Sciences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011;
Project ZR2022JQ02 supported by Shandong Provincial Natural Science
Foundation; the Ministry of Education, Youth and Sports of the Czech Republic
under Contract No. LTT17020; the Czech Science Foundation Grant No. 22-18469S;
Horizon 2020 ERC Advanced Grant No. 884719 and ERC Starting Grant No. 947006
“InterLeptons” (European Union); the Carl Zeiss Foundation, the Deutsche
Forschungsgemeinschaft, the Excellence Cluster Universe, and the
VolkswagenStiftung; the Department of Atomic Energy (Project Identification
No. RTI 4002) and the Department of Science and Technology of India; the
Istituto Nazionale di Fisica Nucleare of Italy; National Research Foundation
(NRF) of Korea Grant Nos. 2016R1D1A1B02012900, 2018R1A2B3003643,
2018R1A6A1A06024970, RS202200197659, 2019R1I1A3A01058933, 2021R1A6A1A03043957,
2021R1F1A1060423, 2021R1F1A1064008, 2022R1A2C1003993; Radiation Science
Research Institute, Foreign Large-size Research Facility Application
Supporting project, the Global Science Experimental Data Hub Center of the
Korea Institute of Science and Technology Information and KREONET/GLORIAD; the
Polish Ministry of Science and Higher Education and the National Science
Center; the Ministry of Science and Higher Education of the Russian
Federation, Agreement 14.W03.31.0026, and the HSE University Basic Research
Program, Moscow; University of Tabuk research grants S-1440-0321, S-0256-1438,
and S-0280-1439 (Saudi Arabia); the Slovenian Research Agency Grant Nos.
J1-9124 and P1-0135; Ikerbasque, Basque Foundation for Science, Spain; the
Swiss National Science Foundation; the Ministry of Education and the Ministry
of Science and Technology of Taiwan; and the United States Department of
Energy and the National Science Foundation. These acknowledgements are not to
be interpreted as an endorsement of any statement made by any of our
institutes, funding agencies, governments, or their representatives. We thank
the KEKB group for the excellent operation of the accelerator; the KEK
cryogenics group for the efficient operation of the solenoid; and the KEK
computer group and the Pacific Northwest National Laboratory (PNNL)
Environmental Molecular Sciences Laboratory (EMSL) computing group for strong
computing support; and the National Institute of Informatics, and Science
Information NETwork 6 (SINET6) for valuable network support.

## References

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citnum

# Reevaluating the Role of Race and Ethnicity in Diabetes Screening

Madison Coots
Harvard University Correspondence may be sent to Madison Coots at
<EMAIL_ADDRESS>Soroush Saghafian
Harvard University  David Kent
Tufts University  Sharad Goel
Harvard University

Key Messages • There is active debate over whether to consider patient race
and ethnicity when estimating disease risk. By accounting for race and
ethnicity, it is possible to improve the accuracy of risk predictions, but
there is concern that their use may encourage a racialized view of medicine. •
In diabetes risk models, despite substantial gains in statistical accuracy
from using race and ethnicity, the gains in clinical utility are surprisingly
modest. • These modest clinical gains stem from two empirical patterns: first,
the vast majority of individuals receive the same screening recommendation
regardless of whether race or ethnicity are included in risk models; and
second, for those who do receive different screening recommendations, the
difference in utility between screening and not screening is relatively small.
• Our results are based on broad statistical principles, and so are likely to
generalize to many other risk-based clinical decisions.

## Introduction

Approximately 1 in 10 Americans suffers from Type 2 diabetes. Diabetes can
lead to several serious health problems, such as heart disease, kidney
disease, and vision loss. But, if detected early, patients can receive
pharmacotherapy and make corrective changes to their diet and lifestyle to
better manage their health. Although every individual could be regularly
screened for diabetes, screening comes with monetary and non-monetary costs
(e.g., taking time off from work may result in lost income). Consequently, the
medical community recommends that only those with at least a moderate risk of
developing diabetes undergo screening. Results by Aggarwal et al. suggest that
individuals typically benefit from screening if their risk of diabetes is
above 1.5% [1]. To follow this guidance, statistical algorithms can be used to
predict the diabetes risk for each individual, recommending screening for
those with predicted risk above the threshold.

These risk predictions are often produced using variables such as age and body
mass index (BMI). There is debate over whether to additionally base risk
predictions on an individual’s race and ethnicity to account for observed
disparities in diabetes incidence rates across demographic subgroups in the
United States. [1] Past work has argued that including race and ethnicity
substantially improves the accuracy of clinical prediction models, and that
their omission could exacerbate disparities in health outcomes [7, 8].
However, there is persistent concern over the use of race and ethnicity in
estimating disease risk—for diabetes and beyond—and leading journals have
published articles critical of “race-aware” predictions [11, 3, 2]. Including
race and ethnicity as inputs to predictive models may, for instance,
inadvertently reinforce pernicious attitudes of biological determinism or lead
to greater stigmatization of individuals who are already marginalized. In part
for these reasons, several hospitals have recently moved away from reporting
race-adjusted glomerular filtration rate estimates, instead reporting a race-
unaware value, both to avoid race-based predictions and to mitigate concerns
that a race-aware model may deprioritize Black patients for kidney
transplantation [9, 4, 5].

We offer a new perspective on the statistical and clinical utility of race and
ethnicity in diabetes risk estimation. We begin by showing that racial and
ethnic minorities—particularly Asian Americans—are substantially more likely
to develop diabetes than White Americans with comparable age and BMI, in line
with past analyses. Accordingly, considering race and ethnicity can
substantially improve diabetes risk predictions for both White and non-White
individuals. However, by adopting a utility-based decision framework, we then
show that these greatly improved predictions do not result in the
commensurately large clinical benefits many might expect. This finding stems
from two patterns in the data: first, while the more accurate race-aware model
changes predictions for all patients, decisions (in terms of screening
recommendations) change for only a small fraction of patients; second, among
those who do receive different screening recommendations, the net value of
screening is relatively small since their risk of diabetes is close to the
decision threshold.

Our analysis is based on publicly available data from the National Health and
Nutrition Examination Survey (NHANES) [6]. NHANES combines interview responses
with laboratory data to provide insight into the health and nutritional status
of adults and children in the U.S. The survey is conducted every two years by
the National Center for Health Statistics and is frequently used by
researchers to assess the prevalence of major diseases and their risk factors
across the U.S. population. We use the four NHANES cycles from 2011-2018.
Following Aggarwal et al., we restricted our sample to approximately 18,000
patients who were not pregnant, were 18–70 years old, and had a BMI between
18.5 and 50.0 $\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{2}$.

## Results

### Race-unaware predictions of diabetes risk are substantially miscalibrated
by race and ethnicity.

Among the subset of individuals with a “race-unaware” estimated diabetes risk
of 1%—based on age and BMI—about 1% were in reality found to have diabetes.
But when we disaggregate this group of individuals by race and ethnicity, we
see that the model systematically underestimates diabetes risk for Asian,
Black, and Hispanic patients, while overestimating diabetes risk for White
patients. In particular, Asian patients with an estimated risk of 1% under the
race-unaware risk prediction model in reality have diabetes at a rate of
approximately 2%, double the nominal estimate. That is, of the approximately
one million Asian American adults in the U.S. with a race-unaware predicted
diabetes risk of 1%, about 2% of them in reality would be found to have
diabetes. Similarly, White patients with a predicted risk of 2% under the
race-unaware model have diabetes at a rate just under 1.5%. These
miscalibrated predictions lead to misclassifications: with a screening
threshold of 1.5%—in line with current guidelines—the race-unaware model would
incorrectly fail to recommend screening for some Asian American patients who
have relatively high risk of diabetes and would incorrectly recommend
screening for some White patients who have relatively low risk of diabetes.
The upper-left plot of Figure 1 compares the empirical rate of diabetes to the
race-unaware risk predictions across the risk spectrum, illustrating the
miscalibration of predicted risks by race and ethnicity.

Figure 1: Assessing the statistical and clinical utility of race and ethnicity
in diabetes risk estimation. Upper left: Calibration plot for a race-unaware
risk model, showing that racial minorities have significantly higher empirical
risk of diabetes compared to their race-unaware risk prediction; Lower left:
Calibration plot for a race-aware risk model, showing that predicted and
empirical risk are well-aligned across groups; Upper right: Distribution of
race-unaware predicted risk, highlighting individuals for whom the race-
unaware screening decision is at odds with their empirical utility for being
screened. Lower right: Plot showing utility as a function of race-unaware risk
predictions, indicating that individuals who receive incorrect screening
recommendations incur relatively small utility loss (highlighted in grey). In
all four panels, the horizontal axis excludes those with higher than 5%
predicted risk to highlight the region near the decision threshold.

### Race-aware models improve risk predictions.

One way to correct the observed miscalibration is to incorporate race and
ethnicity into risk predictions. The lower-left plot of Figure 1 compares the
empirical rate of diabetes to race-aware predicted risks, based on age, BMI,
and race and ethnicity. The predicted race-aware risks are largely in line
with empirical diabetes rates across groups. Further, it seems hard to rectify
the miscalibration of race-unaware models without explicitly considering race
and ethnicity, as the miscalibration persists for race-unaware models that
consider a variety of additional factors beyond age and BMI, including family
history (see Figure A1 in the Appendix for further details).

### Despite improved accuracy, race-aware predictions yield limited net
benefits.

Race-aware predicted risk scores greatly improve the calibration of estimates,
improving aggregate outcomes for all race and ethnicity groups. However,
perhaps surprisingly, the overall clinical benefits are not as large as one
might expect. To quantify the added value of race-aware risk scores, we adopt
a utility-based framework where both costs and gains in health are expressed
in dollars by specifying an exchange rate. We begin by assuming a constant
cost for screening (which we normalize to equal 1) and a constant benefit $r$
for detecting diabetes, which captures the net benefit of early detection and
treatment of the disease. The screening cost encapsulates a wide range of
monetary and non-monetary considerations, such as the cost to perform the test
and the cost of taking time away from work to be screened. Based on this
framework, the optimal policy is to screen patients if and only if their
predicted risk of diabetes exceeds $t=1/r$, which represents the “point of
indifference.” In other words, the expected benefits of screening exceed the
costs if, and only if, one’s risk of diabetes is greater than $1/r$. For the
purposes of our analysis, we set the benefit $r$ for detecting diabetes to 70,
consistent with guidelines that suggest a screening threshold for diabetes of
approximately 1.5%. (In Figure A2 in the Appendix, we show that our results
are qualitatively similar across a wide range of choices for $r$.)

Across the entire patient population, we estimate the per capita utility gain
from the race-aware model is 0.015—where we recall that the cost of screening
is normalized to equal 1. If we assume the cost of screening equals $100, then
the race-aware diabetes risk model yields an average per capita gain of
approximately $1.50 over a race-unaware model.

Race and ethnicity | Per capita utility gain
---|---
Asian | 0.077
White | 0.016
Hispanic | 0.0005
Black | 0.0004
Table 1: The per capita utility gain of a race-aware risk model over a race-
unaware model, disaggregated by race and ethnicity.

In Table 1, we show the results of this utility analysis by race and
ethnicity. Asian individuals experience the greatest utility gains from a
race-aware model, with a gain of about $7.77 per person on average. White
individuals follow with a utility gain of approximately $1.60 per person (from
the avoidance of some unnecessary screening), and Black and Hispanic
individuals experience near-zero utility gains from the use of a race-aware
model over a race-unaware model. The largest gains accrue to Asian Americans
in part because, as discussed above, the race-unaware model most significantly
underestimates diabetes risk for this group.

Given that the race-unaware risk model is starkly miscalibrated, it is perhaps
surprising that the race-aware model does not yield larger utility gains. Two
factors help explain this phenomenon. First, most individuals receive the same
screening recommendation under a race-aware model as under a race-unaware
model, as recommendations only change for the relatively few individuals close
to the decision threshold. Specifically, 84% of Asian individuals, 93% of
White individuals, 96% of Hispanic individuals, and 97% of Black individuals
receive the same screening decision under both models. The vast majority of
individuals thus accrue no gains from the race-aware model. Second, the small
number of patients near the threshold who do receive different screening
recommendations under the two risk models are relatively indifferent between
being screened and not being screened—precisely because they are close to the
screening threshold—and so accrue small utility gains.

The upper-right and lower-right plots in Figure 1 illustrate these patterns.
The upper-right plot shows the distribution of race-unaware risk scores for
Asian and White individuals, highlighting the relatively small number of
patients for whom the screening recommendation is at odds with their empirical
utility for being screened. The lower-right plot shows the empirical utility
of screening as a function of race-unaware risk score, for both Asian and
White individuals. Asian individuals with nominal race-unaware diabetes risk
lower than the 1.5% screening threshold would be counselled against screening
even though they have positive utility for screening. Analogously, some White
individuals with race-unaware risk estimates above the screening threshold
would be counselled in favor of screening even though they have negative
utility for screening. But in both cases, the loss of utility for these errors
is relatively small.

### Generalizing these results to other decisions.

Because these results are based on broad principles, we believe they are
likely to generalize to many other settings in which race-aware models might
improve predictions. The key insight here is that when the decision threshold
is determined by the “point of indifference” (i.e., where the costs and
benefits of an intervention are exactly balanced), decisions are likely to
change only for patients who are close to this “point of indifference”, thus
limiting the value of improved classification. Where risk prediction is used
in a shared decision-making context, the decision threshold is generally
determined by that point of indifference.

## Discussion

The principles illustrated here might not apply under conditions of scarcity,
where rationing of resources may yield thresholds far from the point of
indifference (e.g., organ transplantation). Consider an example where severe
budgetary constraints mean that only the riskiest 50% of Americans could be
screened for diabetes, which corresponds approximately to a 6% screening
threshold. If the “point of indifference” (from the patient perspective)
remained at 1.5%, the marginal benefit of using a race-aware over a race-
unaware approach for targeting would improve utility for racial minorities by
an order of magnitude.

Further, our analysis is contingent on the specific utility function that we
use to evaluate screening decisions. For example, we implicitly assume that
the value of detecting diabetes is comparable across race and ethnicity
groups. In settings where that is not the case, the benefits of a race-aware
approach are likely larger than what we find here [10]. Second, we assume that
the benefit of detecting diabetes is independent of an individual’s age. A
more detailed analysis might explicitly consider quality-adjusted life years.
Third, to aid interpretation, we estimate that the monetary and non-monetary
costs of screening are approximately $100 (i.e., that one util equals $100).
Given the subjective nature of some of the screening costs (e.g., an
individual’s value for time), accurately estimating this quantity is
challenging. Finally, we have primarily considered per capita utility gains,
but one could alternatively consider aggregate population-level utility
benefits, which are considerably larger. Nonetheless, we believe our
simplified analysis helps illustrate that substantial gains in statistical
accuracy may still result in modest gains in clinical utility.

Race-aware risk models can improve estimates of diabetes risk, but the
clinical value of considering race and ethnicity is smaller than the
improvement in predictions might suggest. Thus, if there are important reasons
to avoid consideration of race and ethnicity in risk prediction—for example,
from decreased trust in the medical system—the clinical benefits may not
justify their use. Our results likely extend to a variety of contexts in
medicine and beyond where the explicit use of race and ethnicity in predictive
models is contested, although the specifics in each case need to be
considered. We hope that our analytic approach helps researchers and
policymakers better understand and balance these underlying trade-offs.

## Contributors and sources

Madison Coots is a Ph.D. student in public policy. Dr. Saghafian is an
associate professor of public policy with expertise in the development and
application of methods in operations research and management science to
examine societal problems in healthcare. Dr. Kent is a general internist and
researcher with two decades of experience in clinical prediction modelling.
Dr. Goel is a professor of public policy with expertise in algorithmic
fairness and computational approaches to public policy. Coots wrote the
initial draft of the manuscript and is the guarantor. All authors reviewed and
revised the manuscript.

## Data and code availability

Data and code to reproduce our analysis are available at:
https://github.com/madisoncoots/race-in-diabetes-screening.

## Acknowledgements

Dr. Kent is funded through a “Making a Difference” and Presidential Awards
from The Greenwall Foundation.

## References

  * Aggarwal et al. [2022] Rahul Aggarwal, Kirsten Bibbins-Domingo, Robert W. Yeh, Yang Song, Nicholas Chiu, Rishi K. Wadhera, Changyu Shen, and Dhruv S. Kazi.  Diabetes screening by race and ethnicity in the United States: Equivalent body mass index and age thresholds.  _Annals of Internal Medicine_ , 175(6):765–773, 2022.
  * Basu [2023] Anirban Basu.  Use of race in clinical algorithms.  _Science Advances_ , 9(21):eadd2704, 2023.
  * Cerdeña et al. [2020] Jessica P Cerdeña, Marie V Plaisime, and Jennifer Tsai.  From race-based to race-conscious medicine: how anti-racist uprisings call us to act.  _The Lancet_ , 396(10257):1125–1128, 2020.
  * Diao et al. [2021] James A Diao, Gloria J Wu, Herman A Taylor, John K Tucker, Neil R Powe, Isaac S Kohane, and Arjun K Manrai.  Clinical implications of removing race from estimates of kidney function.  _Journal of the American Medical Association_ , 325(2):184–186, 2021.
  * Eneanya et al. [2019] Nwamaka Denise Eneanya, Wei Yang, and Peter Philip Reese.  Reconsidering the consequences of using race to estimate kidney function.  _Journal of the American Medical Association_ , 322(2):113–114, 2019.
  * for Disease Control and for Health Statistics [NCHS] Centers for Disease Control and Prevention (CDC); National Center for Health Statistics (NCHS).  National Health and Nutrition Examination Survey Data, 2022.  https://www.cdc.gov/nchs/nhanes/index.html.
  * Manski [2022] Charles F Manski.  Patient-centered appraisal of race-free clinical risk assessment.  _Health Economics_ , 31(10):2109–2114, 2022.
  * Manski et al. [2022] Charles F Manski, John Mullahy, and Atheendar Venkataramani.  Using measures of race to make clinical predictions: Decision making, patient health, and fairness.  Technical report, National Bureau of Economic Research, 2022.
  * Powe [2020] Neil R Powe.  Black kidney function matters: use or misuse of race?  _Journal of the American Medical Association_ , 324(8):737–738, 2020.
  * Thomas et al. [2021] Tainayah W Thomas, Carol Golin, Carmen D Samuel-Hodge, M Sue Kirkman, Shelley D Golden, and Alexandra F Lightfoot.  Race and gender differences in abnormal blood glucose screening and clinician response to prediabetes: a mixed-methods assessment.  _Preventive Medicine_ , 148:106587, 2021.
  * Vyas et al. [2020] Darshali A Vyas, Leo G Eisenstein, and David S Jones.  Hidden in plain sight—reconsidering the use of race correction in clinical algorithms.  _New England Journal of Medicine_ , 383(9):874–882, 2020.

##  Appendix A Appendix

Calibration plot for an extended race-unaware diabetes risk model

Figure A1: Calibration plot for an extended race-unaware risk model that
includes the following covariates in addition to age and BMI: gender, weight,
height, waist circumference, self-reported greatest weight, whether a
patient’s close family members have diabetes, whether the patient is
depressed, income, health insurance status, and whether the patient feels food
secure.

Sensitivity analysis over the reward for detecting diabetes

Figure A2: When varying the reward for diabetes detection r, we observe
relatively small changes in the benefit from the use of a race-aware risk
model across groups.

Subgroup utility plots

Figure A3: Each circle represents a different patient subgroup by age, BMI,
and race, and the circles are sized according to the number of patients in
that subgroup.


# Quantum evolution with random phase scattering

K. Ziegler  Institut für Physik, Universität Augsburg
D-86135 Augsburg, Germany

###### Abstract

We consider the quantum evolution of a fermion-hole pair in a d-dimensional
gas of non-interacting fermions in the presence of random phase scattering.
This system is mapped onto an effective Ising model, which enables us to show
rigorously that the probability of recombining the fermion and the hole decays
exponentially with the distance of their initial spatial separation. In the
absence of random phase scattering the recombination probability decays like a
power law, which is reflected by an infinite mean square displacement. The
effective Ising model is studied within a saddle point approximation and
yields a finite mean square displacement that depends on the evolution time
and on the spectral properties of the deterministic part of the evolution
operator.

###### Contents

  1. I Introduction
  2. II Summary of the main results
  3. III Quantum evolution with random phase scattering
  4. IV Functional integral representation
  5. V Discussion
    1. V.1 Hopping expansion
    2. V.2 Saddle point approximation
    3. V.3 Mean square displacement
  6. VI Conclusions and outlook
  7. A Saddle point integration

##  I Introduction

The creation of an electron-hole pair and its subsequent recombination is a
fundamental process in quantum physics with many applications in different
fields. Although there exist phenomenological descriptions of this process by
classical decay models [1, 2], for a deeper understanding a quantum approach
is required. We will focus here on a fermion-hole pair in a $d$–dimensional
system of non-interacting fermions. The pair can be created either by photons
and phonons in a real material or by injection into the system. Then the
question is, whether this pair recombines after some evolution by emitting a
photon/phonon or the fermion and the hole remain localized near the place
where they were created initially (cf. Fig. 1). Both possibilities can be
studied by measuring the return probability to the initial quantum state. This
probability depends on the spatial separation of the fermion and the hole.
Assuming that the hole is created at the site ${\bf R}$ and the fermion at the
site ${\bf R}^{\prime}$, we can define the probability $P_{{\bf R}{\bf
R}^{\prime}}$ that the system returns to the initial state over the finite
time interval $\tau$. Although it is plausible that this probability decreases
with increasing distance $|{\bf R}-{\bf R}^{\prime}|$, the law of change with
the distance depends on the interaction with the environment. For instance, on
a periodic lattice this probability has a long range behavior, which depends
on the dimensionality of the underlying space. In the following we will focus
on the effect of random phase scattering on the spatial decay of this
probability. In other words, is it possible to control the spatial fermion-
hole separation to avoid their recombination?

To analyze the evolution and calculate physical quantities, the standard
procedure would be to diagonalize the Hamiltonian $H$ of the evolution
operator $e^{-iH\tau/\hbar}$. For a translational invariant system this can be
achieved through a Fourier transformation. However, in a realistic system the
Hamiltonian $H$ is not translational invariant but subject to some disorder.
In this case the corresponding random Hamiltonian cannot be diagonalized by a
Fourier transformation. To mimic the effect of disorder in the evolution of a
quantum state we “scramble” a translational invariant $e^{-iH\tau/\hbar}$ with
a random phase factor $e^{i\alpha}$ by using the evolution operator
$U=e^{i\alpha}e^{-iH\tau/\hbar}$. This choice was inspired by the random
unitary gate models that have been discussed in the context of quantum
circuits [3, 4, 5]. The following analysis is also inspired by previous
studies of the invariant measure of transport in systems with random chiral
Hamiltonians [6]. Although this seems to be an entirely different problem,
there are some striking similarities that are reflected by their graphical
representations.

##  II Summary of the main results

The central result of this work is that the random phase scattering of non-
interacting fermions is equivalent to scattering on discrete Ising spins or on
a continuous (real) Ising field. This is a consequence of a geometric
restriction in the graphical representation due to the Fermi statistics. It
enables us to deform the Ising field integration such that the poles of the
integrand of the return probability $P_{{\bf R}^{\prime}{\bf R}}$ are avoided.
This implies an exponential decay with respect to $|{\bf R}-{\bf
R}^{\prime}|$. For an explicit evaluation of the decay we employ a saddle
point approximation of the Ising field. This provides for $h=e^{-iH\tau}$

$P_{{\bf R}^{\prime}{\bf R}}=\langle|(\phi+h)^{-1}_{{\bf R}{\bf
R}^{\prime}}|^{2}\rangle_{\phi}\approx|(\phi_{0}+h)^{-1}_{{\bf R}-{\bf
R}^{\prime}}|^{2},$

where $\phi_{0}$ is determined by a saddle point equation. The corresponding
mean square displacement reads

$[R_{\nu}^{2}]=\frac{\tau^{2}}{2}\int_{\bf
k}\frac{(\partial_{k_{\nu}}\epsilon_{\bf
k})^{2}}{[1+\phi_{0}^{2}+2\phi_{0}\cos(E_{0}+\epsilon_{\bf
k}\tau)]^{2}}\Big{/}\int_{\bf
k}\frac{1}{1+\phi_{0}^{2}+2\phi_{0}\cos(E_{0}+\epsilon_{\bf k}\tau)},$

where $\epsilon_{\bf k}$ is the dispersion of the Hamiltonian $H$ and $E_{0}$
is related to the Fermi energy. In the absence of random phase scattering we
have $\phi_{0}=1$, which implies for $E_{0}+\epsilon_{\bf k}\tau\leq{\bar{E}}$

$[R_{\nu}^{2}]\sim\frac{2\pi\tau}{3d\mu}(\pi-{\bar{E}})^{-2}$

when ${\bar{E}}<\pi$, and $[R_{\nu}^{2}]$ is infinite for ${\bar{E}}\geq\pi$.

Figure 1: a) There is a recombination of a fermion created at ${\bf R}$ and a
hole created at ${\bf R}^{\prime}$ at the blue dot. b) The localization of a
fermion and a hole near their points of creation implies an exponentially
small probability for recombination. The localization radius (or decay length)
is indicated by the blue circles. In this sketch we consider $|{\bf R}-{\bf
R}^{\prime}|$ much larger than the localization radius.

##  III Quantum evolution with random phase scattering

A system of non-interacting fermions with the Hamiltonian $H$ evolves during a
fixed time step $\tau$ with the unitary evolution operator
$U_{\tau}=e^{i\alpha}e^{-iH\tau}$. Here and subsequently we have chosen the
scale of physical quantities such that $\hbar=1$. The phases $\\{\alpha_{\bf
r}\\}$ are randomly distributed on $[-\pi,\pi)$, independently on different
lattice sites ${\bf r}$. $U_{\tau}$ acts on the $2^{|\Lambda|}$ dimensional
Hilbert space, spanned by the fermionic states $|\\{n_{\bf r}\\}\rangle$ with
occupation numbers $n_{\bf r}=0$ or $n_{\bf r}=1$ on a lattice site ${\bf
r}\in\Lambda$. $|\Lambda|$ is the number of lattice sites and
$|\\{n(0)\\}\rangle\equiv|\\{n_{\bf r}(0)\\}\rangle$ is the initial state, in
which the fermionic system is prepared at the beginning. Then we consider the
situation in which a fermion-hole pair is created by the operator
$c^{\dagger}_{{\bf R}^{\prime}}c_{\bf R}$ at time $t=0$ at different sites
${\bf R}$, ${\bf R}^{\prime}$ . To determine the spatial correlation of the
fermion-hole pair after the time $\tau$, where the quantum state is the
initial state again, we write

$\frac{\langle|\langle\\{n(0)|U_{\tau}c^{\dagger}_{{\bf R}^{\prime}}c_{\bf
R}|\\{n(0)\\}\rangle|^{2}\rangle_{\alpha}}{|\langle\\{n(0)|U_{\tau}|\\{n(0)\\}\rangle|^{2}\rangle_{\alpha}},$

which is the return probability for the initial state $|\\{n(0)\\}\rangle$. To
avoid the specific definition of the initial state, we sum over the return
probabilites of all basis states to obtain

$P_{{\bf R}{\bf R}^{\prime}}:=\frac{\langle|Tr(U_{\tau}c^{\dagger}_{{\bf
R}^{\prime}}c_{\bf
R})|^{2}\rangle_{\alpha}}{\langle|TrU_{\tau}|^{2}\rangle_{\alpha}},$ (1)

where $Tr$ is the trace of $2^{|\Lambda|}\times 2^{|\Lambda|}$ matrices. For
$\tau=0$ we have

$\langle\\{n\\}|c^{\dagger}_{{\bf R}^{\prime}}c_{\bf
R}|\\{n\\}\rangle=\delta_{{\bf R}{\bf R}^{\prime}}\delta_{n_{\bf R},1},$

such that only the particle number operator $c^{\dagger}_{\bf R}c_{\bf R}$
contributes to the trace, while the spatially separated fermion-hole pair does
not. For $\tau>0$, though, the evolution $U_{\tau}c^{\dagger}_{{\bf
R}^{\prime}}c_{\bf R}|\\{n\\}\rangle$ can create some overlap with
$|\\{n\\}\rangle$, which contributes to the trace. Thus, the return
probability $P_{{\bf R}{\bf R}^{\prime}}$ (${\bf R}^{\prime}\neq{\bf R}$) is a
measure for how effective the evolution with $U_{\tau}$ can move the fermion-
hole pair to the same site. It is plausible that this is less likely the
larger the distance $|{\bf R}-{\bf R}^{\prime}|$ is and that it increases with
increasing time $\tau$. Therefore, $P_{{\bf R}{\bf R}^{\prime}}$ decays with
this distance and may increase with $\tau$.

Besides creating a fermion and a hole simultaneously, we can also create a
hole at site ${\bf R}$ and time $t=0$, let this hole evolve for the time
$\tau$ and then annihilate it. The probability for this annihilation process
reads

$P^{\prime}_{{\bf R}{\bf R}^{\prime}}:=\frac{\langle|Tr(c^{\dagger}_{{\bf
R}^{\prime}}U_{\tau}c_{\bf
R})|^{2}\rangle_{\alpha}}{\langle|TrU_{\tau}|^{2}\rangle_{\alpha}}.$ (2)

In the following we drop the index $\tau$ for simplicity and use
$U_{\tau}\equiv U$. Assuming that $|\\{{\tilde{n}}_{\bf
q}\\}\rangle\equiv|\\{{\tilde{n}}\\}\rangle$ are eigenstates of $U$ for a
special realization of the phases $\\{\alpha_{\bf r}\\}$, we obtain

${\tilde{C}}_{{\bf Q},t}:=Tr(U^{(1-t)}c^{\dagger}_{\bf Q}U^{t}c_{{\bf
Q}})=\sum_{\\{{\tilde{n}}_{\bf q}\\},\\{{\tilde{n}}^{\prime}_{\bf
q}\\}}\langle\\{{\tilde{n}}\\}|U^{(1-t)}\\{{\tilde{n}}\\}\rangle\langle\\{{\tilde{n}}\\}|c^{\dagger}_{\bf
Q}|\\{{\tilde{n}}^{\prime}\\}\rangle\langle\\{{\tilde{n}}^{\prime}\\}|U^{t}|\\{{\tilde{n}}^{\prime}\\}\rangle\langle\\{{\tilde{n}}^{\prime}\\}|c_{{\bf
Q}}|\\{{\tilde{n}}\\}\rangle$ (3)

for $t=0,1$. With $\langle\\{{\tilde{n}}\\}|c^{\dagger}_{\bf
Q}|\\{{\tilde{n}}^{\prime}\\}\rangle\langle\\{{\tilde{n}}^{\prime}\\}|c_{{\bf
Q}}|\\{{\tilde{n}}\\}\rangle=\delta_{{\tilde{n}}^{\prime}_{{\bf
Q}},0}\delta_{{\tilde{n}}_{{\bf Q}},1}\prod_{{\bf q}\neq{\bf
Q}}\delta_{{\tilde{n}}^{\prime}_{\bf q},{\tilde{n}}_{\bf q}}$ we get

${\tilde{C}}_{{\bf Q},t}=\sum_{\\{{\tilde{n}}_{\bf
q}\\},\\{{\tilde{n}}^{\prime}_{\bf
q}\\}}\langle\\{{\tilde{n}}\\}|U^{(1-t)}|\\{{\tilde{n}}\\}\rangle\langle\\{{\tilde{n}}^{\prime}\\}|U^{t}|\\{{\tilde{n}}^{\prime}\\}\rangle\delta_{{\tilde{n}}^{\prime}_{{\bf
Q}},0}\delta_{{\tilde{n}}_{{\bf Q}},1}\prod_{{\bf q}\neq{\bf
Q}}\delta_{{\tilde{n}}^{\prime}_{\bf q},{\tilde{n}}_{\bf q}}.$

Since $U$ is diagonal in this basis with
$\langle\\{{\tilde{n}}\\}|U|\\{{\tilde{n}}\\}\rangle=\prod_{\bf
q}\langle{\tilde{n}}_{\bf q}|e^{-iE_{\bf q}{\tilde{n}}_{\bf
q}}|{\tilde{n}}_{\bf q}\rangle=\prod_{\bf q}e^{-iE_{\bf q}{\tilde{n}}_{\bf
q}}$, we obtain a product of diagonal matrix elements

$\langle\\{{\tilde{n}}\\}|U^{(1-t)}|\\{{\tilde{n}}\\}\rangle\langle\\{{\tilde{n}}^{\prime}\\}|U^{t}|\\{{\tilde{n}}^{\prime}\\}\rangle=\prod_{{\bf
q}}e^{-iE_{\bf q}{\tilde{n}}_{\bf q}(1-t)}e^{-iE_{\bf
q}{\tilde{n}}^{\prime}_{\bf q}t}.$

Inserting this into Eq. (3), we get for the sum due to the Kronecker deltas

${\tilde{C}}_{{\bf Q},t}=Tr(U^{(1-t)}c^{\dagger}_{\bf Q}U^{t}c_{\bf
Q})=e^{-iE_{\bf Q}(1-t)}\prod_{{\bf q}\neq{\bf Q}}(1+e^{-iE_{\bf
q}})=\frac{e^{iE_{\bf Q}t}}{1+e^{iE_{\bf Q}}}\prod_{{\bf q}}(1+e^{-iE_{\bf
q}}).$ (4)

Finally, we return to the real-space representation to obtain $e^{-iE_{\bf
q}}\to\hat{U}_{{\bf r}{\bf r}^{\prime}}$, where $\hat{U}$ is a
$|\Lambda|\times|\Lambda|$ matrix on the lattice, and

$Tr(Uc^{\dagger}_{{\bf R}^{\prime}}c_{\bf R})=({\bf
1}+\hat{U}^{\dagger})^{-1}_{{\bf R}{\bf R}^{\prime}}\det({\bf 1}+\hat{U})\ ,\
\ Tr(c^{\dagger}_{{\bf R}^{\prime}}Uc_{\bf R})=({\bf 1}+\hat{U})^{-1}_{{\bf
R}{\bf R}^{\prime}}\det({\bf 1}+\hat{U}),$ (5)

where $det$ is the corresponding determinant. Hence the return probabilities
become

$P_{{\bf R}{\bf R}^{\prime}}=\frac{\langle|({\bf
1}+\hat{U}^{\dagger})^{-1}_{{\bf R}{\bf R}^{\prime}}\det({\bf
1}+\hat{U})|^{2}\rangle_{\alpha}}{\langle|\det({\bf
1}+\hat{U})|^{2}\rangle_{\alpha}}\ ,\ \ P^{\prime}_{{\bf R}{\bf
R}^{\prime}}=\frac{\langle|({\bf 1}+\hat{U})^{-1}_{{\bf R}{\bf
R}^{\prime}}\det({\bf 1}+\hat{U})|^{2}\rangle_{\alpha}}{\langle|\det({\bf
1}+\hat{U})|^{2}\rangle_{\alpha}}$ (6)

due to Eqs. (1), (2). The identity $|({\bf 1}+\hat{U}^{\dagger})^{-1}_{{\bf
R}{\bf R}^{\prime}}|^{2}=|({\bf 1}+\hat{U})^{-1}_{{\bf R}^{\prime}{\bf
R}}|^{2}$ implies that $P^{\prime}_{{\bf R}{\bf R}^{\prime}}=P_{{\bf
R}^{\prime}{\bf R}}$.

##  IV Functional integral representation

For the further treatment of the return probability $P^{\prime}_{{\bf R}{\bf
R}^{\prime}}=P_{{\bf R}^{\prime}{\bf R}}$ in Eq. (6) it is convenient to
separate the random phase factor and the deterministic evolution of $\hat{U}$
as $\hat{U}_{{\bf r}{\bf r}^{\prime}}=e^{i\alpha_{\bf r}}h_{{\bf r}{\bf
r}^{\prime}}$. Then we employ a Grassmann functional integral to write

$P_{{\bf R}^{\prime}{\bf R}}=\frac{1}{\cal
N}\langle\int_{\varphi}\exp\left[\pmatrix{\varphi_{1}\cr\varphi_{2}\cr}\cdot\pmatrix{{\bf
1}+e^{i\alpha}h&0\cr 0&{\bf
1}+h^{\dagger}e^{-i\alpha}\cr}\pmatrix{\varphi^{\prime}_{1}\cr\varphi^{\prime}_{2}\cr}\right]\varphi_{1{\bf
R}}\varphi^{\prime}_{1{\bf R}^{\prime}}\varphi_{2{\bf
R}^{\prime}}\varphi^{\prime}_{2{\bf R}}\rangle_{\alpha}$ (7) $=\frac{1}{\cal
N}\langle adj_{{\bf R}{\bf R}^{\prime}}({\bf 1}+e^{i\alpha}h)adj_{{\bf
R}^{\prime}{\bf R}}({\bf
1}+h^{\dagger}e^{-i\alpha})\rangle_{\alpha}=\frac{1}{\cal N}\langle|\det({\bf
1}+e^{i\alpha}h)|^{2}|({\bf 1}+e^{i\alpha}h)^{-1}_{{\bf R}{\bf
R}^{\prime}}|^{2}\rangle_{\alpha}$

with the normalization

${\cal
N}=\langle\int_{\varphi}\exp\left[\pmatrix{\varphi_{1}\cr\varphi_{2}\cr}\cdot\pmatrix{{\bf
1}+e^{i\alpha}h&0\cr 0&{\bf
1}+h^{\dagger}e^{-i\alpha}\cr}\pmatrix{\varphi^{\prime}_{1}\cr\varphi^{\prime}_{2}\cr}\right]\rangle_{\alpha}=\langle|\det({\bf
1}+e^{i\alpha}h)|^{2}\rangle_{\alpha}.$

We note that the kernel of the quadratic form has zero modes (i.e., eigenmodes
of ${\bf 1}+e^{\alpha}h$ with vanishing eigenvalue) because the eigenvalues of
the random unitary matrices $e^{i\alpha}h$ and $h^{\dagger}e^{-i\alpha}$ are
randomly distributed on the unit circle in the complex plane. These zero modes
depend on the realization of the random phase.

In the integral (7) we pull out the phase factors by rescaling the Grassmann
fields to obtain

$P_{{\bf R}^{\prime}{\bf R}}=\frac{1}{\cal
N}\langle\int_{\varphi}\exp\left[\pmatrix{\varphi_{1}\cr\varphi_{2}\cr}\cdot\pmatrix{e^{-i\alpha}+h&0\cr
0&e^{i\alpha}+h^{\dagger}\cr}\pmatrix{\varphi^{\prime}_{1}\cr\varphi^{\prime}_{2}\cr}\right]\varphi_{1{\bf
R}}\varphi^{\prime}_{1{\bf R}^{\prime}}\varphi_{2{\bf
R}^{\prime}}\varphi^{\prime}_{2{\bf R}}\rangle_{\alpha}$ (8) $=\frac{1}{\cal
N}\int_{\varphi}\prod_{\bf r}\langle(1+e^{-i\alpha_{\bf r}}\varphi_{1{\bf
r}}\varphi^{\prime}_{1{\bf r}})(1+e^{i\alpha_{\bf r}}\varphi_{2{\bf
r}}\varphi^{\prime}_{2{\bf
r}})\rangle_{\alpha}\exp\left[\pmatrix{\varphi_{1}\cr\varphi_{2}\cr}\cdot\pmatrix{h&0\cr
0&h^{\dagger}\cr}\pmatrix{\varphi^{\prime}_{1}\cr\varphi^{\prime}_{2}\cr}\right]\varphi_{1{\bf
R}}\varphi^{\prime}_{1{\bf R}^{\prime}}\varphi_{2{\bf
R}^{\prime}}\varphi^{\prime}_{2{\bf R}}\rangle_{\alpha},$

which gives after phase averaging

$=\frac{1}{\cal N}\int_{\varphi}\prod_{\bf r}(1+\varphi_{1{\bf
r}}\varphi^{\prime}_{1{\bf r}}\varphi_{2{\bf r}}\varphi^{\prime}_{2{\bf
r}})\exp\left[\pmatrix{\varphi_{1}\cr\varphi_{2}\cr}\cdot\pmatrix{h&0\cr
0&h^{\dagger}\cr}\pmatrix{\varphi^{\prime}_{1}\cr\varphi^{\prime}_{2}\cr}\right]\varphi_{1{\bf
R}}\varphi^{\prime}_{1{\bf R}^{\prime}}\varphi_{2{\bf
R}^{\prime}}\varphi^{\prime}_{2{\bf R}}.$

We get the same result when we replace the phase factor by an Ising spin
$\\{S_{\bf r}=\pm 1\\}$ or by a real Gaussian field $\phi_{\bf r}$ which will
be called Ising field in the following. For the latter we write

$P_{{\bf R}^{\prime}{\bf R}}=\frac{1}{{\cal N}_{\phi}}\int
e^{-\frac{1}{2}\sum_{\bf r}\phi_{\bf
r}^{2}}\int_{\varphi}\exp\left[\pmatrix{\varphi_{1}\cr\varphi_{2}\cr}\cdot\pmatrix{\phi+h&0\cr
0&\phi+h^{\dagger}\cr}\pmatrix{\varphi^{\prime}_{1}\cr\varphi^{\prime}_{2}\cr}\right]\varphi_{1{\bf
R}}\varphi^{\prime}_{1{\bf R}^{\prime}}\varphi_{2{\bf
R}^{\prime}}\varphi^{\prime}_{2{\bf R}}\prod_{\bf r}d\phi_{\bf r}$ (9)
$=\frac{1}{{\cal N}_{\phi}}\int e^{-\frac{1}{2}\sum_{\bf r}\phi_{\bf
r}^{2}}adj_{{\bf R}{\bf R}^{\prime}}(\phi+h)adj_{{\bf R}^{\prime}{\bf
R}}(S+h^{\dagger})\prod_{\bf r}d\phi_{\bf r}=\frac{1}{{\cal N}_{\phi}}\int
e^{-\frac{1}{2}\sum_{\bf r}\phi_{\bf r}^{2}}|adj_{{\bf R}{\bf
R}^{\prime}}(\phi+h)|^{2}\prod_{\bf r}d\phi_{\bf r}$
$=\langle|(\phi+h)^{-1}_{{\bf R}{\bf R}^{\prime}}|^{2}\rangle_{\phi}\ \ {\rm
with}\ \ \ \langle\ldots\rangle_{\phi}=\frac{1}{{\cal N}_{\phi}}\int
e^{-\frac{1}{2}\sum_{\bf r}\phi_{\bf r}^{2}}|\det(\phi+h)|^{2}\ldots\prod_{\bf
r}d\phi_{\bf r}.$

This result is reminiscent of the average two-particle Green’s function with
respect to a Gaussian distribution of $\phi_{\bf r}$, multiplied by the
determinant term $|\det(\phi+h)|^{2}$. There are two important differences in
comparison to the average two-particle Green’s function
$\langle|(V+H_{0})^{-1}_{{\bf R}{\bf R}^{\prime}}|^{2}$ of Anderson
localization, though. The first is that the determinant can be written as a
product of the eigenvalues of $\phi+h$. This cancels poles of
$|(\phi+h)^{-1}_{{\bf R}{\bf R}^{\prime}}|^{2}$, implying that the poles of
the Green’s functions are not relevant for the $\phi_{\bf r}$ integration. In
other words, the adjugate matrix $adj_{{\bf R}{\bf
R}^{\prime}}(\phi+h)=\det(\phi+h)(\phi+h)^{-1}_{{\bf R}{\bf R}^{\prime}}$ does
not have any pole for $|\phi_{\bf r}|<\infty$, and the integration with
respect to $\phi_{\bf r}$ can be deformed in any finite area of the complex
plane. This reflects an exponential decay with a finite decay length of the
return probability $P_{{\bf R}^{\prime}{\bf R}}$. The second difference is
that $h$ is unitary and its eigenvalues are located on the unit circle of the
complex plane, while the Hamiltonian $H_{0}$ in the Anderson localization
problem is a Hermitian matrix with eigenvalues on the real axis.

The deformation of the $\phi_{\bf r}$ integration provides a rigorous but only
qualitative result regarding the decay of $P_{{\bf R}^{\prime}{\bf R}}$. For a
quantitative result of the decay we must perform the integration explicitly.
We will do that approximately within a saddle-point integration in Sect. V.2
and App. A.

##  V Discussion

First, we note that the expansion of the integrand of Eq. (7) and a subsequent
Grassmann and phase integration yields graphs with 4-vertices, where two edges
$h_{{\bf r}{\bf r}^{\prime}}$ from $\varphi_{1}$ and two Hermitean conjugate
edges $h^{\dagger}_{{\bf r}{\bf r}^{\prime}}$ from $\varphi_{2}$ are
connected. This condition is enforced by the Grassmann property, which
requires a product of $\varphi_{1{\bf r}}\varphi^{\prime}_{1{\bf
r}}\varphi_{2{\bf r}}\varphi^{\prime}_{2{\bf r}}$ at each site ${\bf r}$.
Moreover, the random phase factors glue $h$ and $h^{\dagger}$ at these
products to form a 4-vertex and to prevent a 2-vertex. The geometric property
of the 4-vertex enables us either to form loops of edges or to connect the
sites ${\bf R}$ and ${\bf R}^{\prime}$ by a string of both types of edges.
Both, the $h$ edges as well as the $h^{\dagger}$ edges form loops and an ${\bf
R}$-${\bf R}^{\prime}$ string separately. This is a consequence of the
diagonal kernel of the quadratic form in Eq. (7). Moreover, each loop carries
a factor $-1$ from the Grassmann field. Two typical examples are depicted in
Fig. 2 with the same formation of the nine black edges but with different
formations of the nine red edges. In the left example a loop and a double
string are separated by a special choice of red edges, while in the right
example there is only one connected graph.

This type of graphs is known from the invariant measure of chiral random
Hamiltonians [6]. There is a crucial difference though that is related to the
zero mode: In contrast to the random phase scattering $e^{i\alpha_{\bf
r}}h_{{\bf r}{\bf r}^{\prime}}$, the scattering of the chiral model is
$e^{i\alpha_{\bf r}}h_{{\bf r}{\bf r}^{\prime}}\sum_{{\bf
r}^{\prime\prime}}h^{\dagger}_{{\bf r}^{\prime}{\bf
r}^{\prime\prime}}e^{-i\alpha_{{\bf r}^{\prime\prime}}}$. For the latter we
have a uniform zero mode

$\sum_{{\bf r}^{\prime}}[\delta_{{\bf r}{\bf r}^{\prime}}-e^{i\alpha_{\bf
r}}h_{{\bf r}{\bf r}^{\prime}}\sum_{{\bf r}^{\prime\prime}}h^{\dagger}_{{\bf
r}^{\prime}{\bf r}^{\prime\prime}}e^{-i\alpha_{{\bf r}^{\prime\prime}}}]=0$
(10)

for any realization of the random phase.

Figure 2: Two typical graphs representing contributions to the functional
integral of the return probability $P_{{\bf R}^{\prime}{\bf R}}$ in Eq. (7).
Black (red) edges represent $h$ ($h^{\dagger}$). Both edges form (i) a loop
and (ii) a string connecting the sites ${\bf R}$ and ${\bf R}^{\prime}$ of the
fermion and the hole. The strings are contributions to the inverse matrix
elements $({\bf 1}+e^{i\alpha}h)^{-1}_{{\bf R}{\bf R}^{\prime}}$ and $({\bf
1}+h^{\dagger}e^{-i\alpha})^{-1}_{{\bf R}{\bf R}^{\prime}}$, respectively,
while loops are contributions to the determinants. There are only 4-vertices,
except for the endpoints ${\bf R}$ and ${\bf R}^{\prime}$, which are
connecting to two black and two red edges. Other edge crossings are not
connected by vertices.

###  V.1 Hopping expansion

In order to get a better understanding of the behavior of the return
probability $P_{{\bf R}^{\prime}{\bf R}}$ we return to the expression of Eq.
(6) with random phases and simplify it by neglecting the determinants. This
leads to the product of the conjugate one-particle Green’s functions of only
two individual particles:

$\langle({\bf 1}+he^{i\alpha})^{-1}_{{\bf R}{\bf R}^{\prime}}({\bf
1}+e^{-i\alpha}h^{\dagger})^{-1}_{{\bf R}^{\prime}{\bf
R}}\rangle_{\alpha}=\langle(e^{-i\alpha}+h)^{-1}_{{\bf R}{\bf
R}^{\prime}}(e^{i\alpha}+h^{\dagger})^{-1}_{{\bf R}^{\prime}{\bf
R}}\rangle_{\alpha}.$

A hopping expansion of the inverse matrices in powers of the evolution
operator $e^{i\alpha}h$ and its Hermitian conjugate can be written as a
truncated geometric series

$({\bf 1}+e^{i\alpha}h)^{-1}_{{\bf R}{\bf R}^{\prime}}({\bf
1}+h^{\dagger}e^{-i\alpha})^{-1}_{{\bf R}^{\prime}{\bf
R}}=\sum_{l,m=0}^{N-1}(e^{i\alpha}h)^{l}_{{\bf R}{\bf
R}^{\prime}}(h^{\dagger}e^{-i\alpha})^{m}_{{\bf R}^{\prime}{\bf R}},$

where the truncation with $N<\infty$ is necessary because it is not clear
whether the series converges. Since after phase averaging only $l=m$ survives,
we can ignore terms with $l\neq m$ here. This gives

$\sum_{l=0}^{N-1}(he^{i\alpha})^{l}_{{\bf R}{\bf
R}^{\prime}}(e^{-i\alpha}h^{\dagger})^{l}_{{\bf R}^{\prime}{\bf
R}}=\delta_{{\bf R}{\bf R}^{\prime}}+h_{{\bf R}{\bf
R}^{\prime}}h^{\dagger}_{{\bf R}^{\prime}{\bf R}}+\sum_{{\bf r}_{1},{\bf
r}_{1}^{\prime}}h_{{\bf R}{\bf r}_{1}}h_{{\bf r}_{1}{\bf
R}^{\prime}}h^{\dagger}_{{\bf R}^{\prime}{\bf
r}_{1}^{\prime}}h^{\dagger}_{{\bf r}_{1}^{\prime}{\bf R}}e^{i\alpha_{{\bf
r}_{1}}-i\alpha_{{\bf r}_{1}^{\prime}}}$ (11) $+\ldots+\sum_{{\bf r}_{1},{\bf
r}_{1}^{\prime},{\bf r}_{2},{\bf r}_{2}^{\prime},\ldots,{\bf r}_{N-1},{\bf
r}_{N-1}^{\prime}}h_{{\bf R}{\bf r}_{1}}h_{{\bf r}_{1}{\bf r}_{2}}\cdots
h_{{\bf r}_{N-1}{\bf R}^{\prime}}h^{\dagger}_{{\bf R}^{\prime};{\bf
r}_{N-1}^{\prime}}h^{\dagger}_{{\bf r}_{N-1}^{\prime}{\bf
r}_{N-2}^{\prime}}\cdots h^{\dagger}_{{\bf r}_{1}^{\prime}{\bf
R}}\prod_{j=1}^{N-1}e^{i\alpha_{{\bf r}_{j}}-i\alpha_{{\bf r}_{j}^{\prime}}}.$

Now we can average over the random phases to obtain

$\langle({\bf 1}+he^{i\alpha})^{-1}_{{\bf R}{\bf R}^{\prime}}({\bf
1}+e^{-i\alpha}h^{\dagger})^{-1}_{{\bf R}^{\prime}{\bf
R}}\rangle_{\alpha}=\delta_{{\bf R}{\bf R}^{\prime}}+h_{{\bf R}{\bf
R}^{\prime}}h^{\dagger}_{{\bf R}^{\prime}{\bf R}}+\sum_{{\bf r}_{1}}h_{{\bf
R}{\bf r}_{1}}h_{{\bf r}_{1}{\bf R}^{\prime}}h^{\dagger}_{{\bf R}^{\prime}{\bf
r}_{1}}h^{\dagger}_{{\bf r}_{1}{\bf R}}$ (12) $+\ldots+\sum_{{\bf r}_{1},{\bf
r}_{2},\ldots,{\bf r}_{N-1}}\sum_{\pi_{N-1}}h_{{\bf R}{\bf r}_{1}}h_{{\bf
r}_{1}{\bf r}_{2}}\cdots h_{{\bf r}_{N-1}{\bf R}^{\prime}}h^{\dagger}_{{\bf
R}^{\prime};\pi({\bf r}_{N-1})}h^{\dagger}_{\pi({\bf r}_{N-1})\pi({\bf
r}_{N-2})}\cdots h^{\dagger}_{\pi({\bf r}_{1}){\bf R}},$

where we sum with respect to all permutations $\pi_{N-1}$ of all non-
degenerate sites of $\\{{\bf r}_{1},{\bf r}_{2},\ldots,{\bf r}_{N-1}\\}$.
Although this is a compact expression, it is difficult to perform the sum over
the permutations and to calculate the corresponding values. Nevertheless, as
an important special case the identity $\pi_{N-1}=id$ can be calculated. It is
a contribution of an unrestricted random walk on the lattice. This represents
a long range correlation in the form of diffusion. However, it will be
destroyed by the determinant factor in Eq. (7), as mentioned in the previous
section, where the Ising field representation leads to an exponential decay.
In other words, the coupling of many fermions to the random phase scattering
supports localization by avoiding singularities that appear in the case of two
particles. For a quantitative result of the exponential decay we study the
mean square displacement within a saddle point approximation in the next
section.

###  V.2 Saddle point approximation

The return probability in Eq. (9) is treated within the saddle point
integration of the Ising field $\phi$ (cf. App. A). This yields

$P_{{\bf R}^{\prime}{\bf R}}=\langle|(\phi+h)^{-1}_{{\bf R}{\bf
R}^{\prime}}|^{2}\rangle_{\phi}\approx|(\phi_{0}+h)^{-1}_{{\bf R}-{\bf
R}^{\prime}}|^{2},$ (13)

where we have neglected the flucuations $\delta\phi_{\bf r}$ around the saddle
point $\phi_{0}$. This approximation enables us to factorize the return
probability as

$P_{{\bf R}^{\prime}{\bf R}}\approx|C_{{\bf R}-{\bf R}^{\prime}}|^{2}\ ,\ \
C_{{\bf R}-{\bf R}^{\prime}}=(\phi_{0}+h)^{-1}_{{\bf R}-{\bf R}^{\prime}},$

where $C_{{\bf R}-{\bf R}^{\prime}}$ can be represented by its Fourier
transform

${\tilde{C}}_{\bf k}=\frac{1}{\phi_{0}+e^{-iE_{\bf k}}}$ (14)

with the eigenvalue $E_{\bf k}$ of the translational invariant matrix $H\tau$.
Thus, the effect of the random phase scattering is associated only with the
value of $\phi_{0}$, where the latter is determined by $E_{\bf k}$ via the
saddle point equation (23). Moreover, $\phi_{0}=1$ represents the absence of
random phase scattering.

The results for the Ising field $\phi_{0}$ of App. A can be interpreted in
terms of the magnetic properties of the classical Ising model [7]. The
asymmetric shift $\cos E_{0}$ plays the role of an external magnetic field and
$\phi_{0}$ corresponds to the magnetization [8]. Thus, the effective Ising
model has a unique Ising field $\phi_{0}>0$ or $\phi_{0}<0$ when $\cos
E_{0}\neq 0$, while for $\cos E_{0}=0$ there are either two degenerate
solutions with opposite signs of $\phi_{0}$ (ferromagnetic phase) or a single
solution with $\phi_{0}=0$ (paramagnetic phase). In contrast to the classical
Ising model with a continuous transition though, Fig. 3 indicates a jump of
$\phi_{0}$ for our effective Ising model.

###  V.3 Mean square displacement

The mean square displacement provides a measure for the localization length.
It is defined as

$[R_{\nu}^{2}]:=\frac{\sum_{{\bf R}^{\prime}}({\bf R}_{\nu}-{\bf
R}^{\prime}_{\nu})^{2}P_{{\bf R}^{\prime}{\bf R}}}{\sum_{{\bf R}}P_{{\bf
R}^{\prime}{\bf R}}}=\frac{-\partial_{q_{\nu}}^{2}{\tilde{P}}_{\bf
q}\Big{|}_{{\bf q}=0}}{{\tilde{P}}_{0}},$ (15)

where ${\tilde{P}}_{\bf q}$ is the Fourier transform of the translational-
invariant $P_{{\bf R}^{\prime}{\bf R}}\equiv P_{{\bf R}^{\prime}-{\bf R}}$.
Now we study $P_{{\bf R}{\bf R}^{\prime}}=|C_{{\bf R}{\bf R}^{\prime}}|^{2}$
with the help of the saddle point integration. In this case the mean square
displacement reads

$[R_{\nu}^{2}]=\frac{\sum_{\bf R}R_{\nu}^{2}|C_{\bf R}|^{2}}{\sum_{\bf
R}|C_{\bf R}|^{2}},$ (16)

where the numerator is

$\sum_{\bf R}R_{\nu}^{2}e^{i{\bf q}\cdot{\bf R}}|C_{\bf R}|^{2}\Big{|}_{{\bf
q}=0}=-\partial^{2}_{q_{\nu}}\sum_{\bf R}e^{i{\bf q}\cdot{\bf R}}|C_{\bf
R}|^{2}\Big{|}_{{\bf q}=0}=-\partial^{2}_{q_{\nu}}\sum_{\bf R}\int_{\bf
k}\int_{{\bf k}^{\prime}}e^{i({\bf q}-{\bf k}-{\bf k}^{\prime})\cdot{\bf
R}}{\tilde{C}}_{\bf k}{\tilde{C}}^{*}_{{\bf k}^{\prime}}\Big{|}_{{\bf q}=0}$

with the Fourier transform ${\tilde{C}}_{\bf k}$ of Eq. (14). Then the ${\bf
R}$ summation can be performed and leads to a Kronecker delta, which gives for
Eq. (16)

$[R_{\nu}^{2}]=-\partial^{2}_{q_{\nu}}\int_{\bf k}{\tilde{C}}_{\bf
k}{\tilde{C}}^{*}_{{\bf q}-{\bf k}}\Big{|}_{{\bf q}=0}\Big{/}\int_{\bf
k}{\tilde{C}}_{\bf k}{\tilde{C}}^{*}_{-{\bf k}}.$

Now we assume that $C_{-{\bf k}}=C_{{\bf k}}$ to obtain eventually

$[R_{\nu}^{2}]=\int_{\bf k}|\partial_{k_{\nu}}{\tilde{C}}_{\bf
k}|^{2}\Big{/}\int_{\bf k}|{\tilde{C}}_{\bf k}|^{2},$ (17)

which becomes with Eq. (14)

$[R_{\nu}^{2}]=\int_{\bf k}\frac{(\partial_{k_{\nu}}E_{\bf
k})^{2}}{(1+\phi_{0}^{2}+2\phi_{0}\cos E_{\bf k})^{2}}\Big{/}\int_{\bf
k}\frac{1}{1+\phi_{0}^{2}+2\phi_{0}\cos E_{\bf k}}.$ (18)

A special case is the one in which the energy in Eq. (22) is symmetric with
respect to $\phi_{0}\to-\phi_{0}$. Then there exists a critical value
$\tau_{c}$ of the evolution time $\tau$: If $\tau$ exceeds $\tau_{c}$ the
saddle point is always $\phi_{0;min}=0$, as indicated in Fig. 3a). This
implies for Eq. (14) that ${\tilde{C}}_{\bf k}=e^{iE_{\bf k}}$, which yields
for the corresponding mean square displacement

$[R_{\nu}^{2}]=\int_{\bf k}(\partial_{k_{\nu}}E_{\bf k})^{2}=\tau^{2}\int_{\bf
k}(\partial_{k_{\nu}}\epsilon_{\bf k})^{2}.$ (19)

$\epsilon_{\bf k}$ is the dispersion of the Hamiltonian $H$. Thus, the mean
square displacement increases with the squared evolution time $\tau$. For
$\tau<\tau_{c}$, on the other hand, or when the energy is asymmetric with
respect to $\phi_{0}\to-\phi_{0}$, we have $\phi_{0}\neq 0$.

In the absence of random phase scattering we have $\phi_{0}=1$ directly from
Eq. (7). Then for the special case $E_{\bf k}=k^{2}\tau/2\mu$ ($0\leq
k\leq\lambda$) the mean square displacement on a $d$-dimensionsional lattice
reads

$[R_{\nu}^{2}]=\frac{\tau}{d\mu}\int_{0}^{{\bar{E}}}\frac{E^{d/2}}{(1+\cos
E)^{2}}dE\Big{/}\int_{0}^{{\bar{E}}}\frac{E^{d/2-1}}{1+\cos E}dE$ (20)

with the integration cut-off ${\bar{E}}=\lambda^{2}\tau/2\mu$. This is a
finite expression for ${\bar{E}}<\pi$, which diverges with a power law as

$[R_{\nu}^{2}]\sim\frac{2\pi\tau}{3d\mu}(\pi-{\bar{E}})^{-2}$ (21)

when we approach ${\bar{E}}=\pi$ from below. For ${\bar{E}}\geq\pi$ the mean
square displacement is always infinite without random phase scattering. This
result has the form of a diffusion relation with time $\tau$ and a divergent
diffusion coefficient for ${\bar{E}}\to\pi$ when we ignore the fact that
${\bar{E}}$ also depends on $\tau$. A possible interpretation is that the
fermion-hole pair is subject to diffusion due to its interaction with the
other fermions of the system. The divergence, on the other hand, reflects a
long range correlation of the fermion and the hole that reflects the pole of
${\tilde{C}}_{\bf k}=1/(1+e^{-iE_{\bf k}})$.

A more detailed analysis, especially for the evaluation of $\tau_{c}$,
requires specific expressions of the dispersion $\epsilon_{\bf k}$. This would
exceed the goal of this work to present a generic approach for the effect of
disorder on the recombination of fermion-hole pairs.

a)  b)

Figure 3: The Ising energy ${\cal E}(\phi)$ as defined in Eq. (24) for a
constant density of states. a) ${\cal E}(\phi)$ is plotted at the symmetry
point $E_{0}=\pi/2$ for the band width $a=1.9$ (blue curve) and $a=2.1$ (red
curve). It indicates a jump of the Ising field from two degenerate nonzero
values to $\phi_{0;min}=0$. b) ${\cal E}(\phi)$ is plotted with band width
$a=7.5$ for a symmetric band $E_{0}=0$ (red curve) and a band that is shifted
by $E_{0}=\pi/2$ (blue curve).

##  VI Conclusions and outlook

The probability $P_{{\bf R}^{\prime}{\bf R}}$, which describes the probability
to return to the initial quantum state after the creation of a fermion at site
${\bf R}$ and a hole at site ${\bf R}^{\prime}$ and their evolution, decays
always exponentially with the distance $|{\bf R}-{\bf R}^{\prime}|$ in the
presence of random phase scattering. To obtain this rigorous result a mapping
of the random phase model onto an Ising-like model was essential. This was
supplemented by an approximative calculation of this decay, based on a saddle
point integration of the effective Ising model, to get some quantitative
insight into the decay. The latter calculation is instructive, since it
demonstrates how the solution of the saddle point equation avoids the
singularities of the underlying fermion model. In the absence of random phase
scattering, one of these singularities leads to a non-exponential decay for a
sufficiently long evolution of the state with the fermion-hole pair. This is
reflected by an infinite mean square displacement of the fermion and the hole.

In our approximation we have not included the Gaussian fluctuations around the
saddle point solution. It would be interesting to include them and to
determine their effect on the decay of the return probability. In this context
it would also be useful to understand the effect of these fluctuations on the
transition from $\phi_{0}\neq 0$ to $\phi_{0}=0$ at the symmetry point under
an increasing evolution time. Another extension of our approach is the
application to the return probability of a system under periodically repeated
projective measurements [9, 10] or under randomly repeated projective
measurements [11]. Then the effect of random phase scattering on the resulting
monitored evolution could also be described by the effective Ising field
model. Even more interesting but also more challenging would be the extension
of the approach to the transition probability for the monitored evolution
under randomly repeated projective measurements [12].

##  Appendix A Saddle point integration

We approximate the integral

$\langle\ldots\rangle_{\phi}=\frac{1}{{\cal N}_{\phi}}\int
e^{-\frac{1}{2}\sum_{\bf r}\phi_{\bf r}^{2}}|\det(\phi+h)|^{2}\ldots\prod_{\bf
r}d\phi_{\bf r}$

by using a saddle-point integration. Then we determine the maximal
contribution to the integrand by assuming a uniform $\phi$ and write
$\phi_{\bf r}=\phi+\delta\phi_{\bf r}$. This enables us to approximate the
integral in terms of Gaussian fluctuation around the uniform $\phi$ with
respect to $\delta\phi_{\bf r}$, where $\phi$ must be fixed as $\phi_{0}$ at
the minimum of the Ising energy

${\cal
E}(\phi)=\frac{1}{2}\phi^{2}-\int_{-\infty}^{\infty}\log(1+\phi^{2}+2\phi\cos
E)\rho(E)dE$ (22)

with the density of states $\rho(E)$. The integrand is singular for $\phi=1$,
$E=\pi$ and for $\phi=-1$, $E=0$. These singularities yield large values for
the energy. Therefore, they do not represent lowest energy contributions of
the saddle point. This is also reflected in the curves of Fig. 3b). Moreover,
the saddle-point solution $\phi_{0}$ must satisfy

${\cal E}^{\prime}(\phi)=\phi-2\int_{-\infty}^{\infty}\frac{\phi+\cos
E}{1+\phi^{2}+2\phi\cos E}\rho(E)dE=0.$ (23)

For a constant density of states $\rho(E)$ on the interval
$[-a/2+E_{0},a/2+E_{0}]$ we get

${\cal
E}(\phi)=\frac{1}{2}\phi^{2}-\frac{1}{a}\int_{-a/2+E_{0}}^{a/2+E_{0}}\log(1+\phi^{2}+2\phi\cos
E)dE.$ (24)

A special case is $E_{0}=\pi/2$, where we have

${\cal
E}(\phi)=\frac{1}{2}\phi^{2}-\frac{1}{a}\int_{-a/2}^{a/2}\log(1+\phi^{2}-2\phi\sin
E)dE$

with the symmetry relation ${\cal E}(\phi)={\cal E}(-\phi)$. This would also
hold when the density of states is symmetric with respect to $E=\pi/2$ in Eq.
(22). The Ising energy is plotted for several values of $E_{0}$ and the band
width $a$ in Fig. 3.

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