∎

11institutetext: A. Baroni 22institutetext: Theoretical Division, Los Alamos
National Laboratory, Los Alamos, NM 87545, USA
22email<EMAIL_ADDRESS>33institutetext: G. B. King 44institutetext:
Department of Physics and the McDonnell Center for the Space Sciences at
Washington University in St. Louis, MO, 63130, USA
44email<EMAIL_ADDRESS>55institutetext: S. Pastore 66institutetext:
Department of Physics and the McDonnell Center for the Space Sciences at
Washington University in St. Louis, MO, 63130, USA
66email<EMAIL_ADDRESS>
# Electroweak Currents from Chiral Effective Field Theory ††thanks: SP and GBK
are supported by the U.S. Department of Energy under contract DE-SC0021027,
and through the Neutrino Theory Network (SP) and the FRIB Theory Alliance
award DE-SC0013617 (SP). AB is supported by the U.S. Department of Energy
Early Career Award program, Office of Science, Office of Nuclear Physics.

Alessandro Baroni  Garrett B. King  Saori Pastore

(Received: date / Accepted: date)

###### Abstract

Since the pioneering work of Weinberg, Chiral Effective Field Theory
($\chi$EFT) has been widely and successfully utilized in nuclear physics to
study many-nucleon interactions and associated electroweak currents. Nuclear
$\chi$EFT has now developed into an intense field of research and is applied
to study light to medium mass nuclei. In this contribution, we focus on the
development of electroweak currents from $\chi$EFT and present applications to
selected nuclear electroweak observables.

###### Keywords:

Chiral Effective Field Theory Nuclear Structure Nuclear Reactions

A major objective of nuclear theory is to explain the structure and dynamics
of nuclei and their interaction with electroweak probes in a fully microscopic
approach. In such an approach, nucleons interact with each other via many-body
(primarily, two-and three-nucleon) interactions, and with external electroweak
probes, such as electrons, neutrinos, and photons, via many-body currents
describing the couplings of these probes to individual nucleons and many-body
clusters of correlated nucleons. Over the past sixty years, several highly
accurate phenomenological interactions Wiringa:1994wb ; Machleidt:2000ge  have
been developed and successfully applied to study nuclear properties. Despite
this success, phenomenological theories are hardly improvable; moreover, their
connection to the underlying theory ultimately governing nuclear dynamics,
that is Quantum Chromodynamics (QCD), is ambiguous. Chiral effective field
theory ($\chi$EFT), formulated by Weinberg in the Nineties Weinberg:1990rz ;
Weinberg:1991um ; Weinberg:1992yk , resolves these shortcomings.

Figure 1: Diagrams illustrating the contributions to the axial current up to
N3LO. Nucleons, deltas, pions, and external weak fields are denoted by solid,
thick-solid, dashed, and wavy lines, respectively. The square in panel b
represents relativistic corrections, while the dot in panels d denotes a
vertex induced by subleading terms in the $\pi$-nucleon chiral Lagrangian
Baroni:2016xll .

$\chi$EFT is a low-energy approximation of QCD whose degrees of freedom are
bound states of QCD (e.g., pions, protons, neutrons, etc.). It exploits the
symmetries exhibited by QCD in the low-energy regime, in particular chiral
symmetry, to constrain and determine the interactions of pions among
themselves and with other baryons. The pion couples to these particles by
powers of its momentum $Q$, and the Lagrangians describing these interactions
can be expanded in powers of $Q/\Lambda$, where $\Lambda\sim 1$ GeV represents
the chiral-symmetry breaking scale and characterizes the convergence of the
expansion. Therefore, the validity of the theory is confined to kinematic
regions where the constraint $Q\ll\Lambda$ is realized. The coefficients of
the chiral expansion, or low-energy constants (LECs), are unknown and need to
be fixed by comparison with experimental data or calculated by nonperturbative
QCD computational methods such as Lattice QCD Aoki:2016frl ; Bijnens:2014 ;
Orginos:2015aya ; Shanahan:2016 ; Detmold:2021 ; Beane:2014ora ;
Savage:2016kon ; Tiburzi:2017 ; Davoudi:2020 ; Cirigliano:2020 ; Parreno:2021
.

This extremely powerful approach provides an expansion of the Lagrangians in
powers of a small momentum as opposed to an expansion in the strong coupling
constant, restoring de facto the applicability of perturbative techniques also
in the low-energy regime. Due to the chiral expansion it is then possible, in
principle, to evaluate an observable to any degree of desired accuracy and to
know a priori the hierarchy of interactions contributing to the (low-energy)
process under study. The systematic expansion in $Q/\Lambda$ naturally
arranges the operators in increasing numbers of nucleons. For example, in the
$Q/\Lambda$ power counting, three-nucleon forces are suppressed with respect
to two-nucleon forces, and so on. Another crucial feature of $\chi$EFT is that
many-body electroweak currents can be readily and consistently constructed
within the same $\chi$EFT adopted to derive the many-nucleon interactions.

Since the pioneering work of Weinberg Weinberg:1990rz ; Weinberg:1991um ;
Weinberg:1992yk , this calculational scheme has been widely utilized in
nuclear physics Binder:2018pgl ; Epelbaum:2008ga ; Machleidt:2011zz ;
Entem:2014msa ; Entem:2003ft ; Epelbaum:2002vt ; Epelbaum:2014sza ;
Gezerlis:2014zia ; Epelbaum:1998ka ; Epelbaum:1999dj ; Kaiser:1997mw ;
Krebs:2007rh ; Krebs:2012yv ; Krebs:2018jkc ; Ordonez:1995rz ; vanKolck:1994yi
; Piarulli:2014bda ; Piarulli:2016vel ; Piarulli:2017dwd ; Gandolfi:2020pbj ;
10.3389/fphy.2019.00245 ; Ekstrom:2013kea ; Ekstrom:2017koy ; Park:1993jf ;
Park:1995pn ; Park:1998wq ; Park:2002yp ; Phillips00 ; Phillips2003 ;
Phillips05 ; Walzl2001 ; Pastore:2008ui ; Pastore:2009is ; Pastore:2011ip ;
Piarulli:2012bn ; Kolling:2009iq ; Kolling:2011mt ; Krebs:2016rqz ;
Krebs:2020rms ; Baroni:2015uza ; Baroni:2016xll ; Klos:2016omi ; Song:2007 ;
Song:2009 ; Schiavilla:2018udt  and nuclear $\chi$EFT has developed into an
intense field of research. In this contribution, we focus on the development
of electroweak currents from $\chi$EFT Park:1995pn ; Park:1998wq ; Park:2002yp
; Phillips00 ; Phillips2003 ; Phillips05 ; Walzl2001 ; Pastore:2008ui ;
Pastore:2009is ; Pastore:2011ip ; Piarulli:2012bn ; Kolling:2009iq ;
Kolling:2011mt ; Krebs:2016rqz ; Krebs:2020rms ; Baroni:2015uza ;
Baroni:2016xll ; Klos:2016omi  and present applications to selected
observables in light and medium mass nuclei Bacca:2014tla ; kolling2012 ;
Rozpedzik ; Gysbers:2019uyb ; Pastore:2012rp ; Pastore:2014oda ;
Pastore:2017uwc ; King:2020wmp .

Nuclear electroweak current (j) and charge ($\rho$) operators can be expressed
as a sum of one and many-body contributions, namely

$\displaystyle\rho({\bf q})$ $\displaystyle=$
$\displaystyle\sum_{i}\rho_{i}({\bf q})+\sum_{i<j}\rho_{ij}({\bf
q})+\cdots\,,$ $\displaystyle{\bf j}({\bf q})$ $\displaystyle=$
$\displaystyle\sum_{i}{\bf j}_{i}({\bf q})+\sum_{i<j}{\bf j}_{ij}({\bf
q})+\cdots\,,$

where q is the momentum carried from the external field and the dots denote
three-body operators and beyond. To illustrate the kind of currents emerging
from a $\chi$EFT with pions, nucleons, and delta excitations, in Fig. 1 we
show the vector axial current at tree-level up to next-to-next-to-next-to-
leading order (N3LO) in the chiral expansion at zero momentum transfer. The
major contribution is from the leading order single-nucleon operator (panel
a), this corresponds to the standard Gamow-Teller operator. Two-body
corrections enter at N2LO with a transition current where a nucleon is excited
into a delta, which decays into a pion that gets reabsorbed by a second
nucleon (panel c). At N3LO, there is another one-pion range two-body current
(panel d), along with a contact current (panel e). Because of the power
counting, kinematic effects can also be accounted for and arranged within the
power expansion. A term of this kind is illustrated in panel b, representing a
relativistic correction to the leading one-body operator. All the LECs
entering these operators are experimentally known except for that entering the
contact term in panel e, which is determined by fits to experimental data
Baroni:2018fdn .

Figure 2: Calculated deuteron magnetic form factor from Ref. Piarulli:2012bn
(magenta hatched band) compared with results from Ref. kolling2012  (purple
solid band). Experimental data are from Refs. STANFORD65 ; MAINZ81 ; SACLAY85
; BONN85 ; SLAC90 ; SICK2001

The pioneering derivations of both vector and axial vector currents up to one-
loop contributions are from Park, Min and Rho Park:1993jf ; Park:1995pn  who
worked within covariant perturbation theory with pions and nucleons as degrees
of freedom. These currents, have been used by Song and collaborators in
several hybrid calculations in $A\leq 3$ nuclei Song:2007 ; Song:2009 . The
electromagnetic charge operator was first investigated within $\chi$EFT by
Walzl et al. in Ref. Walzl2001  and Phillips in Refs. Phillips00 ;
Phillips2003 ; Phillips05 . More recently, derivations based on two different
implementations of time-ordered perturbation theory appeared in the
literature. One is from the so-called Pisa-JLab group Pastore:2008ui ;
Pastore:2009is ; Pastore:2011ip ; Piarulli:2012bn  and the other is from the
Bochum-Bonn group Kolling:2009iq ; Kolling:2011mt ; Krebs:2019 . The latter is
based on the method of the unitary transformation Okubo:1954  that decouples,
in the Hilbert space of pions and nucleons, states consisting of nucleons only
from those including both pions and nucleons. The two derivations differ in
the treatment of reducible diagrams Pastore:2009is ; Kolling:2011mt ;
Krebs:2016rqz ; Baroni:2018fdn . When calculating box diagrams entering the
electromagnetic charge and current operators Pastore:2011ip ; Krebs:2020rms ,
the two methods lead to results that are in agreement. However, as discussed
at length in Refs. Baroni:2018fdn ; Krebs:2020rms , the two groups find
different results for the box diagrams in the loop contributions to the axial
current operator. The (minor) numerical impact of this difference has been
investigated in Ref. Baroni:2018fdn . Further checks are underway to clarify
the origin of these differences Krebs:2020box . Both formulations, have been
used in calculations of electroweak observables in (primarily) light nuclei.

Figure 3: Magnetic moments in nuclear magnetons for $A\leq 9$ nuclei from Ref.
Pastore:2014oda . Black stars indicate the experimental values from Refs.
Borremans05 ; Tilley02 ; Tilley04 , while blue dots (red diamonds) represent
Green’s Function Monte Carlo calculations which include the LO one-body
currents (one-body plus two-body currents at N3LO) from $\chi$EFT.

Figure 4: Ratios of Green’s function Monte Carlo calculations to experimental
values of the Gamow-Teller reduced matrix elements in the 3H, 6He, 7Be, 8B,
8Be, 8He and 10C weak transitions from Refs. Pastore:2017uwc ; King:2020wmp .
Theory predictions correspond to the $\chi$EFT axial current at LO (empty
symbols) and up to N3LO (filled symbols).

In Fig. 2, we show calculations of the deuteron magnetic form factor based on
$\chi$EFT currents calculated by Kölling et al. in Ref. Kolling:2009iq  and
Piarulli et al. in Ref. Piarulli:2012bn . The theoretical results are in very
good agreement with each other and with the experimental data for values of
momentum transferred $q\lesssim 3$ fm-1. $\chi$EFT currents are first used for
nuclei with $A>3$ in Ref. Pastore:2012rp  where they are included when
studying magnetic moments and electromagnetic transitions in $A\leq 10$
systems. Two-body currents always improve on the agreement between
experimental data and theoretical calculations. A long standing under-
prediction Utsuno:2004wn  by less sophisticated theoretical estimations of the
measured 9C magnetic moment is in Ref. Pastore:2012rp  explained by the
presence of a 40% correction generated by two-body electromagnetic currents.
This enhancement can be appreciated in Fig. 4 by comparing blue dots
(representing calculations based on the single nucleon paradigm) and red
diamonds (representing calculations with two-body electromagnetic currents).

Axial currents are tested primarily in beta decays and electron capture
processes for which data are readily available and known for the most part
with great accuracy. The long-standing problem of the systematic
overprediction of Gamow-Teller beta decay matrix elements Chou:1993zz  in
simplified nuclear calculations, the so-called ‘$g_{A}$ problem’, has been
recently addressed by several

Figure 5: Comparison of experimental ($y$-axis) and theoretical ($x$-axis)
Gamow-Teller matrix elements for medium-mass nuclei. The theoretical results
were obtained using (i) a bare Gamow-Teller one-body operator, (ii) Gamow-
Teller one-body operator consistently evolved with the Hamiltonian
Gysbers:2019uyb , and (iii) a consistently-evolved Gamow-Teller operator that
includes both one- and two-body currents. See Ref. Gysbers:2019uyb  for
details.

groups Pastore:2017uwc ; King:2020wmp ; Gysbers:2019uyb . The authors in Refs.
Pastore:2017uwc ; King:2020wmp  calculated the Gamow-Teller matrix elements in
$A=6-10$ nuclei accounting systematically for many-body effects in nuclear
interactions and coupling to the axial current both derived in $\chi$EFT. The
agreement of the calculations with the data is excellent and the
‘$g_{A}$-problem’ can be resolved in light nuclei largely by correlation
effects in the nuclear wave functions. A summary of these calculations is
reported in Fig. 4. Similar results for these light nuclei obtained using the
No-core shell model are reported in Ref. Gysbers:2019uyb .

The $\chi$EFT approach in recent year is being implemented in studies of
medium mass nuclei Gysbers:2019uyb . As a representative of this class of
electroweak calculations we show the results of Ref. Gysbers:2019uyb  on beta
decay matrix elements represented in Fig. 5. Here, the authors demonstrate
that the quenching in the nuclear matrix elements arises primarily from
$\chi$EFT axial two-body currents and strong correlations in the nucleus.
Nuclei from $A=3$ to 100Sn are analyzed with $\chi$EFT predictions in
agreement with experimental data.

There has been exceptional progress in studying nuclear physics using
$\chi$EFT. In the last two decades, this framework rooted in the symmetries
and breaking pattern of QCD has allowed for the calculation of many low-energy
nuclear processes, such as electromagnetic reactions and $\beta$ decays in
both light and medium-mass nuclei, has reached a remarkable agreement with
experiment, and has contributed to solving long-standing anomalies in nuclear
theory. As chiral interactions and currents are being refined and pushed to
higher orders, we have entered the precision era of this powerful framework.

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Public-Key Based Authentication Architecture for IoT Devices Using PUF

Haji Akhundov1 Erik van der Sluis2 Said Hamdioui1 Mottaqiallah Taouil1

Delft University of Technology, Delft, The Netherlands

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Intrinsic ID B.V.,
Eindhoven, The Netherlands

Nowadays, IoT is a trending topic in the computing world.
Notably, IoT devices have strict design requirements and are often referred to as constrained devices.
Therefore, security techniques and primitives that are lightweight are more suitable for such devices, e.g., SRAM PUF and ECC.
SRAM PUF is an intrinsic security primitive that is seeing widespread adoption in the IoT segment.
ECC is a public-key algorithm technique that has been gaining popularity among constrained IoT devices.
The popularity is due to using significantly smaller operands when compared to other public-key techniques such as RSA (Rivest Shamir Adleman).
This paper shows the design, development, and evaluation of an application-specific secure communication architecture based on SRAM PUF technology and ECC for constrained IoT devices.
More specifically, it introduces an ECDH public-key based cryptographic protocol that utilizes PUF-derived keys as the root-of-trust for silicon authentication.
Also, it proposes a design of a modular hardware architecture that supports the protocol.
Finally, to analyze the practicality as well as the feasibility of the proposed protocol, we demonstrate the solution by prototyping and verifying a protocol variant on the commercial Xilinx Zynq-7000 APSoC device.


# Fixed target measurements at LHCb for cosmic rays physics

GIACOMO GRAZIANI  INFN, Sezione di Firenze, via Sansone 1, 50019 Sesto
Fiorentino (FI), Italy

###### Abstract

The LHCb experiment has the unique possibility, among the LHC experiments, to
be operated in fixed target mode, using its internal gas target. The energy
scale achievable at the LHC, combined with the LHCb forward geometry and
detector capabilities, allow to explore particle production in a wide
Bjorken-$x$ range at the $\sqrt{s_{\scriptscriptstyle\rm NN}}\sim 100$
$\mathrm{\,Ge\kern-1.00006ptV}$ energy scale, providing novel inputs to
nuclear and cosmic ray physics. The first measurement of antiproton production
in collisions of LHC protons on helium nuclei at rest is presented. The
knowledge of this cross-section is of great importance for the study of the
cosmic antiproton flux, and the LHCb results are expected to improve the
interpretation of the recent high-precision measurements of cosmic antiprotons
performed by the space-borne PAMELA and AMS-02 experiments.

##  1 LHCb as a fixed target detector

The LHCb detector [1] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks, which are predominantly produced at high $\eta$
in $p$ $p$ collisions at the LHC. The forward geometry and excellent
vertexing, tracking and particle identification (PID) capabilities [2], which
are key features for the reconstruction of heavy flavour decays, make it also
an ideal tool to study interactions of the LHC beams with a fixed target. Such
target is provided by the SMOG (System for Measuring Overlap with Gas) device
[3, 4], through which tiny amounts of a noble gas (He, Ne, Ar) can be injected
inside the primary LHC vacuum around the LHCb vertex detector (VELO). The
design gas pressure in the VELO region is $2\times 10^{-7}$ mbar, which is
small enough not to significantly perturb the LHC operation. The device was
originally conceived to determine the machine luminosity using a beam gas
imaging technique[4]. Since 2015, LHCb has started to exploit SMOG to perform
a set of physics runs, using special fills not devoted to $p$ $p$ physics,
with different beam and target configurations, allowing a wealth of unique
production studies. One of the main goals of this program is the study of
heavy flavour production in proton-ion collisions with different target mass
number at $\sqrt{s_{\scriptscriptstyle\rm NN}}\sim 100$
$\mathrm{\,Ge\kern-1.00006ptV}$, an intermediate energy between the existing
data collected at SPS and RHIC/LHC accelerators. These measurements can shed
light on the cold nuclear matter effects affecting the production of the most
relevant probes that are used for detecting quark-gluon plasma at higher
energy density. Another attractive feature of the fixed target configuration
is the access to the large Bjorken-$x$ region in the target nucleus. Nuclear
PDFs in this region are sensitive to antishadowing effects and to possible
contributions from intrinsic charm and beauty. The first results for charm
production, obtained from a data set of proton-argon collisions at
$\sqrt{s_{\scriptscriptstyle\rm NN}}=110$ $\mathrm{\,Ge\kern-1.00006ptV}$
corresponding to a few $\mbox{\,nb}^{-1}$, have been recently released [LHCB-
CONF-2017-001]. Though the results are still limited by the data size, the
observed differential ${D}^{0}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ yields are already expected to provide constrains on nuclear PDFs at
large $x$. Exclusive particle production studies in this kinematic range can
also provide crucial inputs to the modelling of cosmic ray showers in the
atmosphere and in the cosmos.

##  2 Cosmic collisions at LHCb

The measurement discussed in the following is motivated by the high-precision
determination of the $\overline{p}$/$p$ ratio in cosmic rays, up to the energy
of 350 $\mathrm{\,Ge\kern-1.00006ptV}$, achieved during the last years by the
space-borne PAMELA [6] and AMS-02 [7] experiments. The investigation of the
antimatter content in cosmic rays is recognized as a primary tool for the
understanding of high-energy astrophysical phenomena and the measurements of
the antiproton fraction outside of the Earth’s atmosphere provides a sensitive
indirect probe for Dark Matter. The interpretation of these measurements is
currently limited by the uncertainty on the expected amount of secondary
antiprotons produced by spallation of primary cosmic rays on the interstellar
medium. State-of-the-art calculations [8, 9, 10] show that the experimental
results are still compatible with the secondary $\overline{p}$ production,
tough data indicate a larger $\overline{p}$ flux at high energy with respect
to most predictions. The largest uncertainty on the prediction is due,
particularly in the 10–100 GeV range, to the limited knowledge of the
$\overline{p}$ production cross-section in the relevant processes. In
particular, no data for $\overline{p}$ production exist for $p\text{He}$
collisions.

##  3 Measurement of antiproton production in $p\text{He}$ collisions

LHCb performed the first measurement of $\overline{p}$ production in
$p\text{He}$ collisions by operating SMOG with helium during special fills
with limited number of proton bunches, accelerated to 6.5 TeV
($\sqrt{s_{\scriptscriptstyle\rm NN}}$ =110 $\mathrm{\,Ge\kern-1.00006ptV}$).
Most of the data were collected in a single LHC fill during about five hours
in May 2016. Events were triggered with a minimum bias requirement, fully
efficient on the collisions producing an antiproton within the detector
acceptance. The measurement is performed from collisions occurring in an 80 cm
long fiducial region, where the best reconstruction efficiency is achieved.
Antiprotons are counted in two-dimensional bins in momentum ($p$) and
transverse momentum ($p_{\mathrm{T}}$), in the range $12<\mbox{$p$}<110$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $0.4<\mbox{$p_{\mathrm{T}}$}<4$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The kinematic limits are dictated by
the acceptance of the two ring-imaging Cherenkov detectors (RICH) providing
particle identification. The first one covers the range $2<\eta<4.4$ and
allows $\overline{p}$/${K}^{-}$ separation in the momentum range 10-60
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, while the second has acceptance
$3<\eta<5$ and actively identifies antiprotons between 30 and about 110
$\mathrm{\,Ge\kern-1.00006ptV}$. The analysis described in this document
covers only the prompt $\overline{p}$ production, namely the antiprotons
produced directly in the $p\text{He}$ collision, or from resonances decaying
via the strong interaction. The component due to hyperon decays, treated here
as a background component and subtracted from the result, will be the subject
of a dedicated study.

###  3.1 Reconstruction

Candidates are selected by requiring a negative track in the kinematic range
of interest after applying quality requirements on the reconstruction of the
track and of the collision primary vertex (PV), whose position must be
compatible with the beam geometry and lie within a 80 cm long fiducial region
where the best reconstruction efficiency is achieved. These requirements are
almost fully efficient in simulation, while allowing to suppress vertices from
decays, secondary collisions, or combinatorial track association in events
produced by collisions occurring upstream of the fiducial region. The PV
reconstruction efficiency is estimated from simulation. The average value
varies with $z$ from 76% in the most upstream region to 95% around the nominal
collision point, with a mild dependence on the $p_{\mathrm{T}}$ of the
candidate $\overline{p}$. The related systematic uncertainty is evaluated by
weighting the simulation to account for discrepancies in the PV topology
description with respect to data. The reconstruction efficiency for prompt
antiprotons, $\epsilon_{\rm rec}$, including acceptance effects and the
tracking detector efficiency, is determined from simulation in three-
dimensional bins of $p$, $p_{\mathrm{T}}$ and $z$. The bins are chosen to be
small enough to minimize the dependence on the assumed spectra in simulation,
but are at least five times larger than the resolution in each variable. The
size of the kinematic bins is chosen according to the expected spectrum of
reconstructed antiprotons, while twelve bins of equal size are used for $z$.
To control systematic uncertainties, only bins where $\epsilon_{\rm rec}$
exceeds 35% are retained. The tracking efficiency predicted by the simulation,
averaged over $z$, ranges from 40 to 80% depending on the track kinematics. A
correction determined from calibration samples in $p$ $p$ data, of order 1%,
is applied to account for imperfections in the simulation of the tracking
detector response.

###  3.2 Particle identification

Figure 1:  Example of two-dimensional DLL distributions for a particular bin
($21.4<\mbox{$p$}<24.4{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},1.2<\mbox{$p_{\mathrm{T}}$}<1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$),
illustrating the performance of the RICH detectors in separating ${\pi}^{-}$,
${K}^{-}$ and $\overline{p}$ particles. The distribution in data is shown on
the left, while the templates for the four categories, obtained from
calibration samples (from simulation for the ghost component), are shown on
the right.

Antiprotons within the selected sample of negative tracks are identified
through the response of the RICH detectors, from which two variables are
built, $\mathrm{DLL(p-\pi)}$ and $\mathrm{DLL(p-K)}$, representing the
difference of the log likelihood between the proton and pion and the proton
and kaon hypothesis. The fraction of antiprotons among the negative tracks is
determined from the two-dimensional distribution of the DLL variables. Three
sets of templates for the different particles species are considered:
predicted by the simulation, obtained from calibration samples in the
$p\text{He}$ data set, and from calibration samples in the LHCb $p$ $p$ data
collected in 2016. The $p\text{He}$ calibration samples consist of selected
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$
decays for pions, ${\mathchar 28931\relax}\rightarrow{p}{{\pi}^{-}}({\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}\rightarrow{\overline{p}}{{\pi}^{+}})$ for (anti)protons and
$\phi\rightarrow{{K}^{+}}{{K}^{-}}$ for kaons. Calibration samples in $p$ $p$
data have much larger size, and, thanks to the available
${{D}^{*\pm}}\rightarrow{{D}^{0}}({{K}^{\mp}}{{\pi}^{\pm}}){{\pi}^{\pm}}$
selection, provide better coverage of the region with high $p$ and
$p_{\mathrm{T}}$ for kaons. On the other hand they are characterized by a much
larger detector occupancy, which is critical for the RICH performance, with
respect to the $p\text{He}$ events. Such difference is taken into account by
weighting the $p$ $p$ events according to the observed charged track
multiplicity. The most appropriate calibration templates are chosen for each
kinematic region, while the related systematic uncertainty is estimated from
their comparison. The fraction of antiprotons in each kinematic bin of the
selected sample is determined with a two-dimensional extended binned maximum
likelihood fit, where the ($\mathrm{DLL(p-\pi)}$, $\mathrm{DLL(p-K)}$)
distribution in data is fitted as the sum of four components: ${\pi}^{-}$,
${K}^{-}$, $\overline{p}$ and ghost tracks. The latter component, whose
template is obtained from simulation, is needed to account for candidate
tracks which can not be unambiguously matched to a particle. This occurs in
simulation in about 2% of cases. The DLL distributions for data and
calibration samples, illustrating the RICH performance, are shown in Fig. 1
for an arbitrary kinematic bin.

###  3.3 Backgrounds

The selected antiprotons that are not prompt are treated as a background and
are subtracted from the selected sample. Such background is suppressed by
requiring that the impact parameter (IP), which is reconstructed through the
VELO with a resolution of
$\left(15+29/\mbox{$p_{\mathrm{T}}$}({\mathrm{\,Ge\kern-1.00006ptV\\!/}c})\right){\,\upmu\mathrm{m}}$,
is compatible with zero. The residual nonprompt background varies in
simulation between 3% at the lowest $p_{\mathrm{T}}$ values and 1% at high
$p_{\mathrm{T}}$. In 90% of cases, this background is due to hyperon decays,
while in the remaining cases the antiprotons come from a secondary collision
in the detector material and are mistakenly assigned to the primary vertex.
The average background level is constrained from the tail of the IP
distribution in data to be $(2.6\pm 0.6)\%$, where the uncertainty is
systematic and is estimated by varying the fraction of nonprompt
$\overline{p}$ in simulation within the range where a good agreement with data
is observed.

Another background to be considered is due to the possible contamination of
the gas target. The rate of collisions on the LHC residual vacuum is evaluated
by acquiring part of the data without the injected helium gas, while using the
same vacuum-pumping configuration of the data taking with gas. The yield
measured in these special data are scaled according to the corresponding
number of protons on target, and the relative average contribution from
residual vacuum is evaluated to be $(0.7\pm 0.2)\%$, where the uncertainty is
systematic. In Fig. 2, the normalized PV multiplicity distributions for data
with and without injected gas are compared. The multiplicity is slightly lower
on average for the collisions on the residual gas, though it exhibits a longer
tail. This confirms that residual vacuum is dominated by hydrogen, with a
small contribution from elements heavier than helium, as indicated by the rest
gas analysis performed by the LHC vacuum group in the absence of beam.

Figure 2: PV track multiplicity distributions for collisions on residual
vacuum or on the helium target.

###  3.4 Normalization

The SMOG device does not presently allow a precise calibration of the injected
gas pressure. Instead, the normalization for the $\overline{p}$ production
measurement is provided by observing a process with a well-known cross-
section. Single electrons scattered off by the proton beam can be observed
within the LHCb acceptance. For a 6.5 TeV proton beam, in the corresponding
kinematic range the scattering is purely elastic. The cross-section in the
polar angle range $3<\theta<27$ mrad, outside of which the electrons can not
be reconstructed in LHCb, is 180.6 $\mu$b. Though this is three orders of
magnitude below the total nuclear inelastic cross-section, events are expected
to have a distinct signature, with a single low-momentum and
low-$p_{\mathrm{T}}$ electron track visible in the detector, with little or no
other activity. Background events which could mimic this signature are
expected from soft nuclear interactions where the candidate electron is either
the product of a photon conversion, or a light hadron from a central exclusive
production event. In both cases the background is charge symmetric. This
allows to model the background from events with a single positively charged
track (referred to as single positrons in the following). Multiplicity
distributions in data confirm that background-dominated regions close to the
signal are charge symmetric.

Figure 3: Distributions of (left) momentum and (right) $p_{\mathrm{T}}$ for
(top plot) single electron and single positron candidates; (bottom plot)
background subtracted electron candidates, compared with the distributions in
simulation, which are normalized to data.

Single electrons candidate events are selected through a loose kinematic
selection on the track and applying veto requirements on any detector activity
not compatible with the elastic scattering hypothesis. The selection yields
16569 single $e^{-}$ candidates and 9548 $e^{+}$ candidates. The signal yield
is obtained by the difference of the two components. The background subtracted
kinematic distributions are shown on Fig. 3. An excellent agreement with the
simulation is observed, confirming the validity of the charge-symmetry
hypothesis for the background.

The luminosity is determined from the background-subtracted yield of scattered
electrons $N_{e}$, the known cross-section $\sigma_{{p}{e^{-}}}$ and the
electron reconstruction efficiency $\epsilon_{\rm e}$, as
$\mathcal{L}=N_{e}/(Z_{He}\times\sigma_{{p}{e^{-}}}\times\epsilon_{\rm e})$,
where $Z_{He}=2$ is the helium atomic number. Possible effects of gas
ionization were evaluated and are expected to be negligible. The
reconstruction efficiency, evaluated from simulation, is limited by the soft
momentum and transverse momentum spectrum. Electrons lose a sizable fraction
of their energy through bremsstrahlung in the beam pipe and detector material,
and large acceptance effects are caused by the spectrometer magnetic field.
The relative systematic uncertainty on $\epsilon_{\rm e}$ is estimated to be
6% from the stability of the result when varying the main selection criteria,
and in particular from the ability of the simulation to describe the large
modulation of the efficiency with the electron azimuthal angle. The result is
$\mathcal{L}=0.443\pm 0.011\pm 0.027$ nb-1, where the first uncertainty is
statistical, and the second is systematic, dominated by the uncertainty on
$\epsilon_{\rm e}$.

Table 1: Relative uncertainties on the $\overline{p}$ production cross-section measurement. The ranges refer to the variation among kinematic bins.  Source | Statistical | Systematic | Systematic
---|---|---|---
|  | (correlated) | (uncorrelated)
Data size |  $0.7-10.8\%$ |  |
| ($<3\%$ for most bins) |  |
Normalization | $2.5\%$ | $6.0\%$ |
Event and PV requirements |  | $0.3\%$ |
PV reco |  | $0.8\%$ |
Tracking |  | $2.2\%$ | $3.2\%$
Nonprompt background |  | $0.3-0.7\%$ |
Residual vacuum background |  | $0.1\%$ |
Efficiency of IP requirement |  |  | $1.0\%$
PID |  | $1.2-5.0\%$ | $0-26\%$
|  |  | ($<10\%$ for most bins)
Simulated sample size |  |  | $0.8-15\%$
|  |  | ($<4\%$ for $\mbox{$p_{\mathrm{T}}$}<2$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$)
Figure 4: Total relative uncertainty for the cross-section measurement in each
kinematic bin, in per cent.

###  3.5 Uncertainties

Table 1 summarizes the uncertainties on the cross-section measurement. The
precision of the measurement is limited by the systematic uncertainty. The
largest uncertainty which is correlated among all kinematic bins is the
aforementioned relative 6% on the normalization. The uncorrelated uncertainty
is dominated for most bins by the error on the $\overline{p}$ fraction from
the PID analysis. Large relative uncertainties, up to 26%, affect the bins at
the borders of the detector acceptance and, for the intermediate momentum
region, in the transition region between the two RICH detectors, at $\eta\sim
4.4$. For the other regions, the accuracy is typically a few per cent. The
relative total uncertainty in each bin is illustrated in Fig. 4. It amounts to
a relative 10% or less for most of the accessible $p_{\mathrm{T}}$ regions.

A major difference between the fixed target configuration and the standard $p$
$p$ data taking in LHCb is the extension of the luminous region. Particular
care is devoted to evaluate the dependence on the $z$ of the different
experimental effects. The $\overline{p}$ yield normalized to the electron
yield is found to be independent of $z$ within the statistical uncertainty.
The stability of the result is also checked as a function of the absolute time
and of the time within the LHC orbit, excluding unexpected biases related to
the beam time structure.

###  3.6 Results

The antiprotons candidates are counted from a sample of 33.7 million selected
$p\text{He}$ collisions, from which a sample of 1.4 million antiprotons is
determined by the PID analysis. The double differential $\overline{p}$
production cross-section $d^{2}\sigma/dp\,dp_{\mathrm{T}}$ is computed in each
kinematic bin after correcting for the reconstruction efficiency and the
background. The results are compared in Fig. 5 with the predictions of the
EPOS LHC [11] model, which is used in the LHCb simulation.

Figure 5: Result for the $\overline{p}$ cross-section measurement, compared to
absolute predictions from different models. The plots show the ratio of data
over simulation as a function of $p_{\mathrm{T}}$ in the 18 momentum bins, for
(black round closed symbols) EPOS LHC, (red squared closed symbols) EPOS 1.99,
(green squared open symbols) HIJING 1.38, and (violet round open symbols)
QGSJETII-04. The error bars represent the uncorrelated uncertainty for each
measurement. The additional correlated uncertainty, shown only for EPOS LHC
but also relevant to the other cases, is indicated by the red dashed lines.

The double differential shape, notably the momentum spectrum, is found to be
in good agreement with the simulation, while the absolute production rate is
larger on average by about a factor 1.5. The data are also compared with three
other models implemented in the CRMC [12] package v1.5.6: EPOS 1.99 [13],
HIJING 1.38 [14] and QGSJET II-04 [15].

The total inelastic cross-section is also determined from the measured total
yield of recorded collisions within the fiducial region. The PV reconstruction
efficiency for inelastic collisions is predicted, assuming the EPOS LHC model,
to be $56.4\pm 2.0\%$. The result is $\sigma_{\rm inel}^{\rm
LHCb}(p\text{He},\sqrt{s_{\scriptscriptstyle\rm
NN}}=110\mathrm{\,Ge\kern-1.00006ptV})=(140\pm 10)\text{\leavevmode\nobreak\
mb}$ which is larger than the EPOS LHC prediction by a factor $1.19\pm 0.08$,
implying that the measured $\overline{p}$ multiplicity per inelastic collision
is significantly larger in data. The multiplicity predicted by the pre-LHC
version of EPOS is in better agreement with data. HIJING predicts a lower
inelastic cross-section (100 mb), while it reproduces well the measured
absolute $\overline{p}$ production cross-section values. QGSJET matches the
measured values at very low $p_{\mathrm{T}}$, while it exhibits a harder
$p_{\mathrm{T}}$ spectrum than data.

##  4 Conclusions and Outlook

The LHCb experiment has recently opened the way to the use of the LHC beams
for fixed target physics. The first measurement of antimatter production in
$p\text{He}$ collisions is one of the first results of this novel program.
Further details on this work can be found in the related LHCb conference
contribution [16]. The results are expected to contribute to reduce the
uncertainty on the prediction for the secondary antiproton flux in cosmic
rays. Further development of this study in the near future is foreseen, with
the inclusion of data collected at $\sqrt{s_{\scriptscriptstyle\rm NN}}=86.6$
GeV during November 2016, and the measurement of the contribution due to
hyperon decays. Several other measurements relevant to the modelling of cosmic
ray interactions will be possible in the near future: production of light
charged particles, including deuterons, photons and charmed particles with
different gas targets.

## Acknowledgments

We are indebted to our colleagues from the cosmic ray community, O. Adriani,
L. Bonechi, F. Donato and A. Tricomi, who suggested this measurement, and
would like to thank F. Donato and T. Pierog for their advice.

## References

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