Rowperfect is the next best thing to being in a boat ...without the early mornings.
The closer an off-water rowing machine is to rowing on-water, the better the training will be for improving actual rowing performance on-water.
Essentially, training is providing the body with stimuli to adapt. The more accurate and sport specific the stimuli, the better the results of the training.
Rowperfect is the first rowing machine to replicate the dynamics of rowing on-water—allowing it to deliver the most sport-specific training available today.
Every other rowing machine in history is a 'stationary' ergometer, designed to replicate the dynamics of a boat bolted into the floor. This differs considerably from that of a light racing shell floating freely on the water. Instead of the main part of the movement being done by the boat, it is now the rower who has to do all the movement. The rower moves his mass relative to the environment, back and forth, over the full length of the slides, and the change in velocity at both turning points is high. Stationary rowing machines react similarly to a boat that is infinitely heavy. The heavier the boat, the more the rower has to accelerate and decelerate his body mass during the stroke/recovery cycle and the more force needed. Rowperfect was developed to eliminate the drawbacks of traditional stationary ergometers. Rowperfect is the best simulation of a rowing boat currently available. Start rowing on the Rowperfect and immediately the similarity to the real thing is obvious. A real catch, without the slack that characterises standard rowing machines, is followed by a solid, fluid stroke without a shock load on the spine. A real recovery ensues, with a very real warning when the whole is badly co-ordinated. The dynamics of a floating boat are accurately reproduced in Rowperfect. It is mass balanced in the same way as a racing shell which is floating freely on the water. The flywheel/stretcher carriage represents the boat; its weight matches the average weight of the boat per rower. Both the carriage and the rower are freely movable along the main bar (the water). Similar to the boat, the carriage is balanced dynamically between the force of the legs on the stretcher (the acceleration and deceleration forces of this boat simulating part and the rower) and the force exerted on the handle. The crucial difference between the two photos is that in the right photograph of the original Rowperfect Classic the flywheel carriage slides freely on the main bar. Seemingly a small visual difference becomes a monumental difference.
Observe the strain on the back and legs and over-compression of the knees, caused by the full body momentum.
And probably the worst part for rowers—poor technique scores well on stationary ergometers.
Ordinary rowing machines are potentially damaging to backs and knees and bad for technique. As the Rowperfect breaks the learning process into easily managed steps. The result is you can learn to row much quicker, without the bad habits, which are acquired, and then ingrained, when we learn in a boat from day one.


text:
- International Celebrity Site dedicated to dirty gossip, leaked photos & video.
Christine Taylor Nude
Christine Joan Taylor-Stiller (born July 30, 1971) is an American actress and comedian. Wikipedia
Born: 30th July, 1971, Allentown, Pennsylvania, USA
Height: 5' 6½" (169 cm)
Claim to Fame: Marcia on Brady Bunch Movie, Zoolander
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The third season of Friends, an American situation comedy created by David Crane and Marta Kauffman, premiered on NBC on September 19, 1996. The Nineteenth Street Theater building is home of Civic Theatre of Allentown, a historic community arts center that produces live theater, runs educational programs, and screens art house films. This year also eliminated the categories of "Most Desirable Female" and "Most Desirable Male." Desperate But Not Serious is a 1999 movie directed by Bill Fishman. Party Girl is a 1995 film directed by Daisy von Scherler Mayer starring Parker Posey.
Christine Taylor and Vincent Schiavelli guest starred in the pilot episode, the only episode filmed. Dedication is a 2007 American romantic comedy film starring Billy Crudup and Mandy Moore. Chappaqua is a hamlet and census-designated place in the town of New Castle, in northern Westchester County, New York, United States. Ross Eustace Geller, Ph.D. Maureen Denise McCormick is an American actress, author, and recording artist.
Benjamin Edward "Ben" Stiller is an American actor, comedian, writer, producer, and director. "The Van Buren Boys" is the 148th episode of the sitcom Seinfeld, and name of a fictional New York street gang. Calendar Girl is a 1993 film starring Jason Priestley, Gabriel Olds, and Jerry O'Connell. Room 6 is a 2006 horror film starring Christine Taylor, Shane Brolly, Jerry O'Connell, and Ellie Cornell. This is a following list of the MTV Movie Award winners and nominees for Best On-Screen Duo. The First Time is a 2012 teen romantic comedy film written and directed by Jon Kasdan and stars Britt Robertson, Dylan O'Brien, James Frecheville and Victoria Justice.
It starred Edward Herrmann, Christine Taylor and Veronica Hamel. Their sign is one hand with all the fingers 'up' and spread out. He spent many years in the comedy team Stiller and Meara with his wife, Anne Meara. Kristen Stewart & Robert Pattinson won for "The Twilight Saga films" in four consecutive years. The Wedding Singer is a 1998 romantic comedy film written by Tim Herlihy and directed by Frank Coraci. Claire Guyot is a French actress and singer, known for her prolific dubbing of foreign productions. David Scott Lascher is an American actor best known for his roles in Blossom, Sabrina, the Teenage Witch, and the Nickelodeon show Hey Dude, in which he starred with Christine Taylor. The Brady Bunch Movie is a 1995 American comedy film based on the 1969–1974 television series The Brady Bunch.
Katie Green
Jessica Sutta
Lindsey Wixson
Jessica Marais
Silvina Luna
© 2015 Desnuda.Top - All Rights Reserved

redpajama_set_name: RedPajamaCommonCrawl


text:
// Copyright 2011 Hakan Kjellerstrand hakank@gmail.com
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
package com.google.ortools.contrib;

import com.google.ortools.Loader;
import com.google.ortools.constraintsolver.DecisionBuilder;
import com.google.ortools.constraintsolver.IntVar;
import com.google.ortools.constraintsolver.Solver;
import java.io.*;
import java.text.*;
import java.util.*;

public class DivisibleBy9Through1 {
  /**
   * A simple propagator for modulo constraint.
   *
   *
This implementation is based on the ECLiPSe version mentioned in "A Modulo propagator for
   * ECLiPSE"
   * http://www.hakank.org/constraint_programming_blog/2010/05/a_modulo_propagator_for_eclips.html
   * The ECLiPSe Prolog source code: http://www.hakank.org/eclipse/modulo_propagator.ecl
   */
  public static void my_mod(Solver solver, IntVar x, IntVar y, IntVar r) {
    long lbx = x.min();
    long ubx = x.max();
    long ubx_neg = -ubx;
    long lbx_neg = -lbx;
    int min_x = (int) Math.min(lbx, ubx_neg);
    int max_x = (int) Math.max(ubx, lbx_neg);

    IntVar d = solver.makeIntVar(min_x, max_x, "d");

    // r >= 0
    solver.addConstraint(solver.makeGreaterOrEqual(r, 0));

    // x*r >= 0
    solver.addConstraint(solver.makeGreaterOrEqual(solver.makeProd(x, r).var(), 0));

    // -abs(y) < r
    solver.addConstraint(solver.makeLess(solver.makeOpposite(solver.makeAbs(y).var()).var(), r));

    // r < abs(y)
    solver.addConstraint(solver.makeLess(r, solver.makeAbs(y).var().var()));

    // min_x <= d, i.e. d > min_x
    solver.addConstraint(solver.makeGreater(d, min_x));

    // d <= max_x
    solver.addConstraint(solver.makeLessOrEqual(d, max_x));

    // x == y*d+r
    solver.addConstraint(
        solver.makeEquality(x, solver.makeSum(solver.makeProd(y, d).var(), r).var()));
  }

  /**
   * toNum(solver, a, num, base)
   *
   *
channelling between the array a and the number num
   */
  private static void toNum(Solver solver, IntVar[] a, IntVar num, int base) {
    int len = a.length;

    IntVar[] tmp = new IntVar[len];
    for (int i = 0; i < len; i++) {
      tmp[i] = solver.makeProd(a[i], (int) Math.pow(base, (len - i - 1))).var();
    }
    solver.addConstraint(solver.makeEquality(solver.makeSum(tmp).var(), num));
  }

  /**
   * Solves the divisible by 9 through 1 problem. See
   * http://www.hakank.org/google_or_tools/divisible_by_9_through_1.py
   */
  private static void solve(int base) {
    Solver solver = new Solver("DivisibleBy9Through1");

    //
    // data
    //
    int m = (int) Math.pow(base, (base - 1)) - 1;
    int n = base - 1;

    String[] digits_str = {"_", "0", "1", "2", "3", "4", "5", "6", "7", "8", "9"};

    System.out.println("base: " + base);

    //
    // variables
    //

    // digits
    IntVar[] x = solver.makeIntVarArray(n, 1, base - 1, "x");

    // the numbers. t[0] contains the answe
    IntVar[] t = solver.makeIntVarArray(n, 0, m, "t");

    //
    // constraints
    //
    solver.addConstraint(solver.makeAllDifferent(x));

    // Ensure the divisibility of base .. 1
    IntVar zero = solver.makeIntConst(0);
    for (int i = 0; i < n; i++) {
      int mm = base - i - 1;
      IntVar[] tt = new IntVar[mm];
      for (int j = 0; j < mm; j++) {
        tt[j] = x[j];
      }
      toNum(solver, tt, t[i], base);
      IntVar mm_const = solver.makeIntConst(mm);
      my_mod(solver, t[i], mm_const, zero);
    }

    //
    // search
    //
    DecisionBuilder db = solver.makePhase(x, solver.INT_VAR_DEFAULT, solver.INT_VALUE_DEFAULT);
    solver.newSearch(db);

    //
    // output
    //
    while (solver.nextSolution()) {
      System.out.print("x: ");
      for (int i = 0; i < n; i++) {
        System.out.print(x[i].value() + " ");
      }
      System.out.println("\nt: ");
      for (int i = 0; i < n; i++) {
        System.out.print(t[i].value() + " ");
      }
      System.out.println();

      if (base != 10) {
        System.out.print("Number base 10: " + t[0].value());
        System.out.print(" Base " + base + ": ");
        for (int i = 0; i < n; i++) {
          System.out.print(digits_str[(int) x[i].value() + 1]);
        }
        System.out.println("\n");
      }
    }
    solver.endSearch();

    // Statistics
    System.out.println();
    System.out.println("Solutions: " + solver.solutions());
    System.out.println("Failures: " + solver.failures());
    System.out.println("Branches: " + solver.branches());
    System.out.println("Wall time: " + solver.wallTime() + "ms");
  }

  public static void main(String[] args) throws Exception {
    Loader.loadNativeLibraries();
    int base = 10;

    if (args.length > 0) {
      int new_base = Integer.parseInt(args[0]);
      if (new_base > 10) {
        // Note: The next valid base after 10 is 14 and
        // the number 559922224824157, which is too large in this model.
        System.out.println("Sorry, max allowed base is 10. Setting base to 10.");
      } else if (new_base < 2) {
        System.out.println("Sorry, min allowed base is 2. Setting base to 2.");
        base = 2;
      } else {
        base = new_base;
      }
    }

    DivisibleBy9Through1.solve(base);
  }
}

redpajama_set_name: RedPajamaGithub


text:
//********************************************************************
//*
//* Warning: This file was generated by ecore4CPP Generator
//*
//********************************************************************

#ifndef OCLPACKAGE_HPP
#define OCLPACKAGE_HPP

// namespace macro header include
#include "ocl/ocl.hpp"

#include "ecore/EPackage.hpp"

namespace ecore
{
	class EParameter;
	class EEnumLiteral;
	class EEnum;
	class EOperation;
	class EDataType;
	class EPackage;
	class EAnnotation;
	class EClass;
	class EReference;
	class EStringToStringMapEntry;
	class EAttribute;
	class EGenericType;
}

namespace ocl
{
	namespace Evaluations
	{
		class EvaluationsPackage;
	}

	namespace Expressions
	{
		class ExpressionsPackage;
	}

	namespace Types
	{
		class TypesPackage;
	}

	namespace Values
	{
		class ValuesPackage;
	}

}

namespace ocl
{
	/*!
	The Metamodel Package for the ocl metamodel. This package is used to enable the reflection of model elements. It contains all model elements
	which were described in an ecore file.
	*/

	class OCL_API oclPackage : virtual public ecore::EPackage
	{
		private:
			oclPackage(oclPackage const&) = delete;
			oclPackage& operator=(oclPackage const&) = delete;

		protected:
			oclPackage(){}

		public:
			//static variables
			static const std::string eNAME;
			static const std::string eNS_URI;
			static const std::string eNS_PREFIX;

			virtual std::shared_ptr<ocl::Evaluations::EvaluationsPackage> getEvaluations_Package() const = 0;
			virtual std::shared_ptr<ocl::Expressions::ExpressionsPackage> getExpressions_Package() const = 0;
			virtual std::shared_ptr<ocl::Types::TypesPackage> getTypes_Package() const = 0;
			virtual std::shared_ptr<ocl::Values::ValuesPackage> getValues_Package() const = 0;

			//Singleton Instance and Getter
			private:
				static std::shared_ptr<oclPackage> instance;
			public:
				static std::shared_ptr<oclPackage> eInstance();
	};
}
#endif /* end of include guard: OCLPACKAGE_HPP */


text:
Election Johnnie To
By Matt McMillan
In Election, a sprawling tale of gangland power struggle in Hong Kong, Johnnie To reclaims the glory of Hong Kong cinema gone by. While not channelling the balletic uber-violence of '80s gangster masters like John Woo, Election goes a long way in reviving the flagging Hong Kong film industry. Every two years the Wo Shing Society, one of the oldest Triads (Chinese organised crime gangs), elects one of their younger bosses as chairman. This year the election is between level-headed Lok (the brilliantly subtle Simon Yam) and his polar opposite, the maniacally feral Big D (the versatile Tony Leung Ka Fai). When Big D's bribes fail to make him chairman, he decides to machete tradition and take power by force. But once things kick off, Lok's mild-mannered veneer is peeled back to reveal a cold, ruthless powerbroker willing to do whatever it takes to protect the Wo Shing Society. More a complex and densely plotted slow burn than an over-the-top opera of spraying bullets, Election features a huge cast of characters, each with their own motivation and role to play. The police want to avoid a messy gang war, "uncles" (big bosses) want to maintain the status quo and ambitious young gang members work their own angles trying to gain power. The film may be a little short on action and come across as slow and convoluted to some viewers but it pays off in the end with a grim resolution that lacks the heroic redemption of its Hollywood brethren. Comparisons to Infernal Affairs (remade by Scorsese as The Departed) and The Godfather Trilogy are both obvious and fair. Election is a dark, slick, intricately rewarding and instructive journey into the underbelly of Hong Kong's triads. (Alliance Films)
More Johnnie To
'The Queen of Black Magic' Ticks All the Boxes of Good Horror Directed by Kimo Stamboel
Ratu Ilmu Hitam, or The Queen of Black Magic, is a remake of the 1981 Indonesian film of the same name and the latest addition to Shudder's...
'Night Stalker' Is a Fascinating but Uncomfortably Glamorous Dive into Richard Ramirez's Crimes Directed by James Carroll and Tiller Russell
It's the summer of 1985, and the city of Los Angeles is being terrorized by the Night Stalker. A serial killer, serial rapist, kidnapper and...
'The White Tiger' Offers Incisive Commentary on India's Class Division Directed by Ramin Bahrani
It might be tempting to compare The White Tiger to Slumdog Millionaire, but you'd do a disservice to the former specifically and the genre o...
'Psycho Goreman' Is Bloody Good Fun Directed by Steven Kostanski
In her 1964 essay Notes on 'Camp', Susan Sontag says of camp that it is "the love of the exaggerated." In other words, what is campy isn't n...
'Hunted' Gets Bored with Its Own Protagonist Directed by Vincent Paronnaud
Hunted, a Belgian-French-Irish production, is the latest in a long line of survival horror films that focus on a woman fighting back against...


text:
Home / Tag Archives: stolen
Tag Archives: stolen
The records of Müller interviews received by CNN show how much Trump is pushing for stolen e-mails
November 2, 2019 US 0 3
CNN sued the Department of Justice for access to Mueller's witness interview notes. With the release of this weekend, for the first time, a glimpse behind the scenes of Müller's investigative work outside of court proceedings and the report itself is thrown. On the orders of a judge, the Ministry of Justice will continue to publish monthly new tranches of …
Texas MPs say missing New Hampshire couple was murdered, camper stolen
November 2, 2019 US 0 11
The bodies of James and Michelle Butler from Rumney, New Hampshire, were found this week in a shallow grave found on Padre Island, Texas, MPs said. (Facebook) In a shallow grave on a beach in South Texas, bodies of a missing New Hampshire couple were found whose pickup and camper van were stolen and reportedly driven to Mexico. The corpses …
The arrest occurred after a shirt by Tom Brady was stolen from the Patriots' Hall of Fame
October 26, 2019 Sports 0 3
Foxborough Massachusetts police arrested 33-year-old Zanini Cineus on Thursday, accusing him of stealing memorabilia from an exhibit in the Patriots' Hall of Fame. Among the stolen goods was a game-worn Tom Brady jersey, which Cineus wore under his jacket, police said, when the police found him. But wearing the jersey did not seem to bring Cineus any of Brady's outstanding …
Salvador Dalí etching stolen from the gallery of San Francisco in "Snatch and Run": NPR
October 15, 2019 Entertainment 0 8
The Salvador Dalí etching of 1966 Burning Giraffe worth $ 20,000 was stolen from a gallery in San Francisco on Sunday. Dennis Rae Fine Art Hide Caption Switch caption Dennis Rae Fine Art The Salvador Dalí etching of 1966 Burning Giraffe worth $ 20,000 was stolen from a gallery in San Francisco on Sunday. Dennis Rae Fine Art A thief …
In the car stolen disabled dog from Florida found: "Zorra is gone, she is dead"
October 8, 2019 US 0 4
A disabled dog, resting in a car stolen last week in Florida and requesting the desperate owner of the dog to ask for her return, was found dead, a rescue team said. Zorra, a 13-year-old Husky Shepherd mix with paralyzed hind legs, was found dead Monday. Over 100 abandoned Everglades Florida dogs have been posted online. The car in which …
Stolen truck crashes through entrance gate at Naval Station in Corpus Christi, Texas
A man was arrested at Naval Air Station Corpus Christi on Monday morning after falling through a security gate in a stolen car that was being pursued by local police. stated on his Facebook page at 8am: "This is not an exercise" and there was an "unauthorized person" on the site. A short time later, the base stated that a …
Jewelry worth $ 353,000 is said to have been stolen from homes in the Trump Tower
September 23, 2019 US 0 2
Detectives in New York City are investigating reports of two thefts in homes in Trump Tower on Sunday, stealing more than $ 350,000 worth of gems, including a $ 117,000 diamond bracelet from a 59-year-old apartment. Stock The thefts were first published by the New York Post. One woman reported that her wardrobe on the 42nd floor was missing a …
Blenheim Palace: Golden toilet stolen from Churchill's birthplace
September 14, 2019 Entertainment 0 2
Written by Emily Dixon Augusta Anthony, CNN A full 18-karat gold toilet was stolen from the Blenheim Palace in Oxfordshire, England, the mansion of former Prime Minister Winston Churchill. The fully functional toilet was installed as part of an exhibition by Italian artist Maurizio Cattelan titled "Victory Is Not an Option", which was only made public on Thursday. The palace …
Tallahassee PD may have just falsified the stolen angle of AEW & Jericho
September 4, 2019 Entertainment 0 1
We're less than 24 hours away from Chris Jericho's loss or theft of the AEW World Cup. Then All Elite releases a promo from Jericho and turns the shoot into a work for their shows. And we already have another bizarre twist in this typical pro-wrestling story … A reporter who reported the story for the Tallahassee Democrat saw the …
Mike Trout writes history with the 200th stolen base
September 1, 2019 Sports 0 3
ANAHEIM – Mike Trout wrote Saturday night's Angels' 10-4 comeback against the Red Sox at Angel Stadium Story as he recorded his 200th stolen career to become the youngest player to join the 200-homer , 200-stolen base club. Trout, who came into play with 283 careers ANAHEIM – Mike Trout wrote Saturday night's Angels' 10-4 comeback against the Red Sox …
Page 1 of 1112345 » 10...Last »


text:
\section*{\label{sec::introduction}Introduction}

With the last generation of particle detectors in space, physics of charged cosmic rays (CRs) has become a precision discipline.
During the last decade, the space-based spectrometers PAMELA and AMS-02, which is borne to the International Space Station,
have driven measurement uncertainties in the CR fluxes as low as the percent level in an energy range from 1~GeV to a few TeV.
They have measured  the CR nuclear
\cite{PAMELA_Adriani:2011cu,2014ApJ...791...93A,AMS-02_Aguilar:2015_ProtonFlux,AMS-02_Aguilar:2015_HeliumFlux,Aguilar:2016vqr}
and leptonic (positron and electron)
\cite{2009Natur.458..607A,2011PhRvL.106t1101A,2013PhRvL.110n1102A,2014PhRvL.113l1102A,2014PhRvL.113v1102A}
components, as well as CR antiprotons
\cite{2010PhRvL.105l1101A,AMS-02_Aguilar:2016_AntiprotonFlux}.
The most recent antiproton flux measurement by AMS-02 extends from 1 to 400 GeV with an uncertainty of 5\% for almost the whole energy range.
The new precise flux data have stimulated various analyses on Galactic CR propagation and particle dark matter
annihilation into antimatter (see, e.g., \cite{Kappl:2014_pbarCrossSection,2016PhRvD..94l3019K,Cuoco:2017rxb,Winkler:2017xor}).
However, to infer correct conclusions on any modeling and interpretation, an accurate description of the underling antiproton production is necessary.
It is generally established that the bulk of antiprotons in our Galaxy is produced by the interaction of CRs
on the interstellar medium (ISM)~\cite{Donato:2008jk}, conventionally called secondary antiprotons.
In practice, the dominant contribution is provided by the proton-proton ($pp$) channel, namely CR proton on ISM hydrogen,
and either the CR projectile or the ISM target replaced by helium (He$p$, $p$He, and HeHe). Heavier channels can contribute at the few percent level.

In a first paper \cite{Donato:2017ywo} (hereafter DKD17), we discussed the requirement on cross section measurements
to determine the antiproton source term at the uncertainty level of AMS-02 flux data.
Here we invert our perspective and seek to determine the source term and its uncertainty from existing cross section measurements.
In general, there are two different strategies to parametrize the energy-differential cross section which enters the source term calculation.
The first possibility is to find an analytic description of the fully-differential and Lorentz invariant cross section performing a fit to cross section data. Then, a Lorentz transformation and angular integration are applied to find the energy-differential cross section.
This strategy was first pursued by~\cite{TanNg:1983_pbarCrossSectionParametrization}.
The other option is to use Monte Carlo predictions to extract the required cross section.
This was, for example, done with DTUNUC \cite{Simon_Antiproton_CS_Scaling_1998, Donato:2001_DTUNUC}
and more recently using EPOS-LHC and QGSJET-II-04 by Kachelriess {\it et al. }~\cite{Kachelriess:2015_pbarCrossSection}  (hereafter KMO).
These Monte Carlos, EPOS and QGSJET, were originally developed for very high-energy interactions as for example occurring in CR air showers,
but a special training to low-energy data allows to apply them to Galactic CR antiprotons.
However, this approach only gives robust results above an antiproton energy of 10~GeV,
which is a drawback considering that AMS-02 antiproton data reach down to 1~GeV.
Analytic parameterizations overcome this limit,  since they are directly based on the relevant low-energy data.
We pursue an analytic parametrization in the following.
With respect to previous analysis \cite{diMauro:2014_pbarCrossSectionParametrization, Kappl:2014_pbarCrossSection, Winkler:2017xor},
we exploit new data by the NA61 experiment~\cite{Aduszkiewicz:2017sei} in the $pp$ channel and the first ever determination of $p$He data by the LHCb experiment~\cite{Graziani:2017}.

\section{\label{sec::current_status} State of the art  on the antiproton source term }

We first shortly review the basic equations and procedure to calculate the cosmic antiproton source term.
The source term $q_{ij}$ originating from the interaction of the CR component $i$ on the ISM component $j$ is given by a convolution integral
of the energy-differential antiproton production cross section $d\sigma_{ij}/dT_{\bar{p}}$ with the incoming CR flux $\phi_i$
and the ISM target density $n_{\mathrm{ISM},j}$ over the CR kinetic energy per nucleon $T_i$:
\begin{eqnarray}
  \label{eqn::sourceTerm_1}
  q_{ij}(T_{\bar{p}}) &=& \int\limits_{T_{\rm th}}^\infty dT_i \,\,
                                    4\pi \,n_{\mathrm{ISM},j} \, \phi_i  (T_i) \, \frac{d\sigma_{ij}}{d T_{\bar{p}}}(T_i , T_{\bar{p}}).
\end{eqnarray}
Here $T_{\rm th}$ is the energy threshold for antiproton production and the factor $4\pi$ corresponds the angular integration of an isotropic CR flux.
However, experiments do not directly measure the energy-differential cross section
but rather the fully-differential cross section, usually expressed in a Lorentz invariant form
\begin{eqnarray}
  \label{eqn::inv_XS}
  \sigma_{\mathrm{inv}} = E \frac{d^3\sigma}{dp^3} (\sqrt{s},x_\mathrm{R},p_{\mathrm{T}}),
\end{eqnarray}
where $E$ is the total antiproton energy and $p$ its momentum.
It is typically a function of the kinematic variables
$\sqrt{s}$, $x_\mathrm{R}=E_{\bar{p}}^*/E_{\bar{p}}^\mathrm{max\,*}$, $p_{\mathrm{T}}$,
which are the center-of-mass (CM) energy, the antiproton energy divided by the maximal antiproton energy in the CM frame,
and the transverse antiproton momentum, respectively\footnote{The superscript * denotes that a quantity is taken in the CM frame.}.
Notice that the kinematic variables always refer
to the nucleon-nucleon CM frame. Sometimes it is convenient to replace $x_\mathrm{R}$ by the so-called Feynman variable
$x_f=2p_{\mathrm{L}}^*/\sqrt{s}$
which is twice the longitudinal antiproton momentum divided by the
CM energy.
To obtain the energy-differential cross section in \eqnref{sourceTerm_1},
the kinetic variables are transferred into the LAB frame, {\it i.e.} the frame where the target particle is at rest, by means of a Lorentz transformation.
Suitable variables in the LAB frame are the kinetic energies of the CR $T_i$ and of
the antiproton $T_{\bar{p}}$, and the angle $\theta$ of the produced antiproton with respect to the incident CR.
Finally, the integral over the solid angle $\Omega$ relates to the energy-differential cross sections:
\begin{eqnarray}
\label{eqn::energyDifferentialToInvCS}
\frac{d\sigma_{ij}}{d T_{\bar{p}}}(T , T_{\bar{p}})
  &=&  p_{\bar{p}}  \int d \Omega \,\, \sigma_\mathrm{inv}^{(ij)}(T_i, T_{\bar{p}}, \theta).
\end{eqnarray}
More details and explicit calculations are presented in DKD17.

The dominant channels for the production of
secondary antiprotons are $pp$ at roughly 50-60\% of the total spectrum, and $p$He and He$p$ at 10-20\% each, while the channels
involving heavier incoming CRs contribute only up to a few percent (see below).
Until very recently, no measurements of the helium channels were available, rendering the $pp$ channel the baseline
for any scaling to proton-nucleus ($pA$) or nucleus-nucleus ($AA$) channels.
Often this was done in a very simplified  assumption, where each nucleon of the target interacts with each nucleon of the projectile, such that
the nucleon-nucleon interaction scales according to the $pp$ one. To first order, one expects a re-scaling of $pp$ by a factor $A^D$.
The value of the parameter $D$ is typically chosen between $2/3$,
corresponding to an approximation in which the target area is estimated from a classical sphere, and $1$, if all nucleons interact completely independently.
A parameterization for the cross sections involving helium (as projectile and/or target) should be derived phenomenologically
from data in this specific channel. In order to be reliable, the data should cover a wide portion of the kinematical parameter space
relevant for antiproton energies of interest. As we will discuss below, the first data on $p$He scattering do not fulfil this condition.
Therefore, in the following of this paper we  rely on re-scaling from $pp$ cross sections,
which implies that uncertainties from this channel almost directly translate into the helium and heavier channels.
Given the importance of the $p+p\rightarrow\bar{p} + X$ channel, we review it below.

\subsection{The proton-proton channel}

The latest analyses of the antiproton production cross section were done by
Di Mauro {\it et al. }~\cite{diMauro:2014_pbarCrossSectionParametrization} and
Kappl {\it et al. }~\cite{Kappl:2014_pbarCrossSection}, which were both triggered by newly available precise data
from the NA49 experiment \cite{NA49_Anticic:2010_ppCollision}. NA49 measures antiprotons in a fix-target $pp$ collision at $\sqrt{s}=17.3$~GeV.
The two analyses follow slightly different strategies.
On the one hand,  \cite{diMauro:2014_pbarCrossSectionParametrization} combined the NA49 data with a series of old data sets
spanning a range in CM energies from $\sqrt{s}=6.1$~GeV to $200$~GeV.
They perform a global fit to extract a parametrization of the fully-differential Lorentz invariant cross section.
On the other hand, \cite{Kappl:2014_pbarCrossSection} relies on NA49 data and exploits the scaling invariance of the cross section above $\sqrt{s} =$10~GeV,
namely the fact that the cross section does not depend on $\sqrt{s}$: $\sigma_{\mathrm{inv}} (\sqrt{s}, x_\mathrm{R}, p_{\mathrm{T}}) \approx \sigma_{\mathrm{inv}}(x_\mathrm{R}, p_{\mathrm{T}})$.
Below 10~GeV, the scaling invariance is supposed to be violated.
The two analyses pointed out  two issues not  considered in previous parameterizations: isospin violation and hyperon induced production.
In order to calculate the total amount of antiprotons produced in our Galaxy, one has to include all the particles which decay into antiprotons,
namely antineutrons and antihyperons.
Traditionally,  it has been assumed that antiproton and antineutron production in $pp$ collisions is equal, and the
antiproton source term has simply been multiplied by a factor 2 to account for the contribution from antineutron decays.
Indeed, NA49 data \cite{Fischer:2003xh} indicate an enhanced production of antineutrons with respect to the antiproton one.
Following \cite{Winkler:2017xor}, we consider a $\sqrt{s}$ dependent  isospin violation, which is estimated not to exceed  20\%.
The second issue has a similar origin. A fraction of the total antiproton yield originates from an intermediate antihyperon,
which subsequently decays to an antiproton. The NA49 collaboration
explicitly corrects and subtracts antiprotons originating from hyperons. However,
the hyperon correction in older experiments is not always clearly taken into account, and data are not easily comparable.
The usual assumption is that those experiments  were not able to distinguish between primary (prompt) antiprotons and intermediate hyperon states,  and contain a hyperon contamination  which is of the order of 30\%-60\%.
In an update of  \cite{Kappl:2014_pbarCrossSection}, Winkler \cite{Winkler:2017xor} discusses the energy dependence of isospin violation and hyperon production.
Furthermore, he points out  that the scaling invariance of the cross section is broken above $\sqrt{s}=50$~GeV
such that the $p_{\mathrm{T}}$-shape and normalization of the cross section require to be adjusted. High-energy collider data
are used to specify and parametrize the scaling violation.
Finally, above $\sqrt{s}$=10~GeV the analytic result in \cite{Winkler:2017xor} agrees with the Monte Carlo approach
by KMO, hinting that towards high energies the descriptions become robust, which is expected since the cross sections are constrained by precise NA49 and LHC data.
Below 10~GeV the situation is different, because the relevant data taken in the 70's or 80's incorporate large (systematic) uncertainties.

\begin{figure}[t!]
	\includegraphics[width=1.\linewidth]
	    {Fig_1.pdf}
  \caption{ Fraction of the $pp$ source term originating from the kinematic parameter space of the cross section
            which is experimentally determined by NA49 and NA61.
            The contribution is normalized to the total $pp$ source term. The NA61 data are taken for $\sqrt{s}=7.7$~GeV to $17.3$~GeV (blue dot-dashed line),
            while the NA49 is taken at $\sqrt{s}=17.3$~GeV and here assumed to be valid
            in the range 15-20~GeV (solid red line). The red dashed line is obtained assuming that the NA49 data are valid
            in the $\sqrt{s}$ range from 10 to 50~GeV, while the dotted blue one is obtained extending the validity of NA61 data
            up to $\sqrt{s}=50$~GeV. More explicitly, we assume a range from $\sqrt{s}=7.7$~GeV to $50$~GeV.}
	\label{fig::contribution_pp}
\end{figure}

Very recently the NA61 experiment published antiproton cross section measurements at four different CM energies
$\sqrt{s}$=7.7, 8.8, 12.3 and 17.3~GeV, corresponding to beam proton energies $T_p$=31, 40, 80, and 158 GeV, respectively \cite{Aduszkiewicz:2017sei}.
The data are corrected for hyperons and, compared to NA49, extend to lower $\sqrt{s}$.
To see how much the NA61 data improve our knowledge about the $pp$ antiproton source term, we conduct the following exercise.
We calculate the fraction of the $pp$ source term originating from the kinematic parameter space of the cross section
which is experimentally determined by NA49 and NA61, respectively.
\figref{contribution_pp} shows this fraction normalized to the total $pp$ source term, $i.e.$ integrated on the whole kinematic parameter space.
In more detail, the source term in \eqnref{sourceTerm_1} contains an integral over $T_p$, or equivalently $\sqrt{s}$,
while NA49 data are taken for one fixed value of $\sqrt{s}$.
In order to extract meaningful results we have to know the cross section over a non-zero range in $\sqrt{s}$.
A conservative assumption is that the NA49 cross section is known in a small range around 17.3 GeV, we choose $\sqrt{s}=15$ to 20~GeV.
From \figref{contribution_pp} we draw the conclusion
that the experimental data of NA49 (narrow $\sqrt{s}$ range) contribute 20\% to the antiproton source spectrum,
peak around $T_{\bar{p}}=30$~GeV, and quickly decrease towards smaller or larger energies.
The information contained in this data gets totally negligible for $T_{\bar{p}} < 15$ GeV and $T_{\bar{p}} > 70$ GeV.
In contrast to NA49, the NA61 experiment performed runs also at lower $\sqrt{s}$, which significantly
improves the coverage of the contribution to the source spectrum.
The  experimental data of NA61 account for up to 70\% and peak at $T_{\bar{p}}$ around 8~GeV.
As a matter of fact, the contribution of the true experimental data to the total source spectrum
covers a relatively small range in$T_{\bar{p}}$. One might wonder how this
can lead to an accurate determination of the source term spectrum. The reason is the
theoretical assumption of scaling invariance, according to which  the cross section is independent of $\sqrt{s}$ in a range from 10 to 50~GeV~\cite{Winkler:2017xor}.
In other words, we can pretend to know the cross section from $\sqrt{s}=10$ to 50~GeV from a single measurement within the range.
We therefore extend the validity of both the experiments accordingly. The results in \figref{contribution_pp}  show that
the NA49 parameter space can contribute between 70\% and 80\% from $T_{\bar{p}}\sim10$ to 100~GeV.
Above this energy, the determination of the source spectrum requires further data at large $\sqrt{s}$ describing the scaling violation.
The extended NA61 data coincide with NA49 above $T_{\bar{p}}\sim20$~GeV, while significantly
improving the coverage of the source spectrum at lower energies down to 5~GeV.
The baseline for our calculation in \figref{contribution_pp}  is the cross section parametrization
derived later in this paper (Param.~II-B).
However, the results are expected to be robust against changing the actual parametrization.

The conclusion of this exercise is that, in order to constrain the $pp$ source term for $T_{\bar{p}}\lsim 5$~GeV,
it is necessary to have additional low-energy data available.
Indeed, the currently available cross section measurements below $\sqrt{s}\approx7$~GeV  contain large systematic uncertainties,
such that a good determination is hard to obtain.
We notice that it would be useful to collect precise data at low $\sqrt{s}$ to fix the antiproton spectrum in all the energy range
where CR data are now provided with an extremely high accuracy \cite{AMS-02_Aguilar:2016_AntiprotonFlux}.
Especially, progress could be made by a $p+p\rightarrow \bar{p} +X$
experiment at energies below  $\sqrt{s}=7$~GeV.
In \appref{app_source_term_fraction} we show how data from NA61 at $\sqrt{s}=6.3$~GeV could improve the cross section coverage of the $pp$ source term.
A detailed study of the complete relevant parameter space is discussed in DKD17.

\subsection{The nuclear channels}

In addition to the production of antiprotons from $pp$ scatterings,
the $p$He and He$p$ channels contribute a large fraction of  the total source term.
\begin{figure}[b!]
	\includegraphics[width=1.\linewidth]
	    {Fig_2.pdf}
  \caption{ Relative contribution of the various production channels to the total secondary antiproton source spectrum. The four dominant channels
            $pp$, $p$He, He$p$, and HeHe are given individually. We group heavy CR nuclei scattering off hydrogen and helium in the ISM:
            CNO, NeMgSi, Fe, and LiBeB. By heavy ISM we denote CR proton and helium scattering off the rare ISM components CNONeMgSiFe.
           }
	\label{fig::contribution_sourceTerm}
\end{figure}
This information may be inferred from \figref{contribution_sourceTerm}, where we plot the relative contribution of each
production channel obtained by changing the incoming CR nuclei and the ISM components.
The production cross sections are taken from the results we present in \secref{pA_fit} (Param. II-B).
In the figure, $pp$, $p$He, He$p$, HeHe label the CR-ISM nucleus. For heavier CR nuclei, we group the
reactions of LiBeB, CNO, Fe and NeMgSi CR nuclei over the ISM ($p$ and He). We also consider the contribution
from CR $p$ and He scattering off  the subdominant heavy ISM components accounted for the  CNONeMgSiFe nuclei.
The CR fluxes have been taken as follows:
$p$ from \cite{AMS-02_Aguilar:2015_ProtonFlux}, He from \cite{AMS-02_Aguilar:2015_HeliumFlux},
Li, Be and B from \cite{Aguilar:2018njt}, C and O from~\cite{Aguilar:2017hno}, N from~\cite{Yan:2017_TalkXSCRC}, while for all the heavier nuclei we apply the rescaling to oxygen flux as in \cite{1990AandA...233...96E}.
For the ISM composition, we assume $n_{\rm H} = 1 \; {\rm cm}^{-3}$, $n_{\rm He} = 0.1\, n_{\rm H} $, while the abundance for heavier nuclei
is taken from \cite{Przybilla:2008ph}.
It is clear from the figure that the channel involving He, both projectile and target, constitute 30-40\%  of the total
spectrum depending on the antiproton energy.
The heavier primary CNO nuclei contribute a non negligible few percent at the AMS energies.
All the other contributions considered in this study turn out to be negligible.

Until very recently the  cross sections involving He nuclei were not experimentally determined, and
all calculations rely on re-scaling and extrapolation from $pp$ and $pA$ measurements,
where $A$ is typically carbon, but sometimes  heavier nuclei up to lead.
The strategy for re-scaling was either based on Monte Carlo simulations, as performed with DTUNUC at low energies
\cite{Donato:2001_DTUNUC} or KMO at high energies,
or on fitting parameterizations to the scarce $pA$ data, as performed by Duperray {\it et al. }
\cite{Duperray:2003_pbarCrossSectionParametrizationForPA}.
The LHCb collaboration provides now the first ever measurement of  $p+\mathrm{He} \rightarrow {\bar{p}} + X$~\cite{Graziani:2017}, where the incident
LHC protons of 6.5~TeV momentum scatter off a fixed-target helium (corresponding to $\sqrt{s} = 110$~GeV).
The LHCb detector can measure antiprotons with a momentum between 10 and 100 GeV and transverse momentum varying between 0.5 and 3.4 GeV.
In~\cite{Reinert:2017aga} these data are compared to the parametirization of~\cite{Winkler:2017xor} showing reasonable agreement.
\begin{figure}[b!]
	\includegraphics[width=1.\linewidth]
	    {Fig_3.pdf}
  \caption{ Similar to \figref{contribution_pp}, but for the nuclear channel.
            It shows the fraction of the antiproton source term which is
            covered by the kinematic parameter space of the cross section
            measurements by NA49 $p$C and LHCb $p$He.
            Specifically, we assume a range of $\sqrt{s}=15$-20 GeV for NA49 and 100-120 GeV for LHCb.
            Each contribution is normalized to the total source term of the specific  channel. }
	\label{fig::contribution_pHe}
\end{figure}
\figref{contribution_pHe} shows the fraction of the LHCb parameter space to the $p$He and He$p$ source terms.
We make the conservative assumption that the cross section is only known in a small (roughly 10\%) range around the measured $\sqrt{s}$.
In this case, the contribution to the $p$He channel is at the permille level, peaking at an energy between between 10 and 100 GeV,
while the contribution to the He$p$ channel is significantly larger at the 4\% level.
The different coverage of the $p$He source spectrum in the inverse He$p$ kinematic configuration depends on the fact that  in the CM frame all but one
LHCb data point correspond to backwards scattering in the $p$He system, or equivalently forwards scattering in the He$p$ system.
The source term integral in \eqnref{sourceTerm_1} enhances the high-energy forward scattering due to the convolution with the steeply falling CR flux.
Since in any case the contribution of the LHCb data to the source terms is very small,  it is impossible to base
the calculation of the $p + {\rm He} \rightarrow \bar{p} +X$ production solely on LHCb data.
In the  parameterization of  the $p$He cross section,  we will therefore rely on a re-scaling of the  $pp$ ruled by
the $p$C data from NA49 \cite{Baatar:2012fua}, taken at $\sqrt{s}=17.3$~GeV.
Their contribution to the source term, as visible in \figref{contribution_pHe},  is comparable in energy and amount to the $pp$ contribution from NA49.

The important conclusion from \figref{contribution_pHe} is that the current LHCb data are not yet sufficient to give a full picture of the
the antiproton production spectrum in the helium channels and its uncertainties.
The contribution of the incoming $p$ or He at the highest energy contribute only a small fraction to the
produced antiprotons, in particular, referring to AMS-02 energies.
This result is due to the fact that during the computation of the source spectrum  the cross section is folded with an incident beam, namely the CR flux, which follows an energy power law with index of about $-2.7$.
Nonetheless, the LHCb data contain valuable information:
It shows for the first time how well the rescaling from the $pp$ channel applies to a helium target and how the cross section extrapolation to
high energies works.
Moreover, finding an agreement between LHCb data and predictions based on $pp$ and $p$C,
 increases trust in our current approaches and models.
The way to improve the contribution of LHCb and the significance of its data is to increase the antiproton detection threshold above 100~GeV and/or lowering the incident proton energy below 1~TeV.
In  \appref{app_source_term_fraction} we present predictions for the contribution with LHCb data at lower CM energies. Furthermore,
we give an update of the results from DKD17  in \appref{app_par_space} to determine the
whole relevant parameter space of $pA$ cross sections to interpret AMS-02 data. The update takes into account the asymmetry of the
cross section, namely it is given in terms of $x_f$ instead of $x_\mathrm{R}$.

\section{\label{sec::pp_fit} Fitting the proton-proton channel}

The proton-proton channel is relevant since it contributes about 40\% of the total and, furthermore, it is
the baseline for re-scaling to heavier nuclei, and for treating the contribution from antineuterons and hyperons.
Its accurate determination is of central importance, since any uncertainty in $pp$ directly translates into all the other channels.
In the following we test and update the most recent analytic parametrizations
by Di Mauro {\it et al. } \cite{diMauro:2014_pbarCrossSectionParametrization} and
Winkler \cite{Winkler:2017xor}, employing the  NA49 \cite{NA49_Anticic:2010_ppCollision} and
the newly available NA61 data \cite{Aduszkiewicz:2017sei}. To reduce systematic biases we will try to discard most of the old data sets.
Before turning to the fit results, we devote separate discussions to hyperons and isospin violation,
the cross section parameterizations, the cross section data sets, and the fitting procedure.

\subsection{Isospin violation and hyperons}
The fits that we are going to perform are on the prompt antiproton production, so that  antineutrons or antihyperons
which subsequently decay into antiprotons are excluded from the fit.
The estimate of the antiproton source term in the Galaxy requires the addition of these contributions by re-scaling from the prompt production
\begin{eqnarray}
  \label{eqn::XS_galaxy}
  \sigma_{\mathrm{inv}}^\mathrm{Galaxy} = \sigma_{\mathrm{inv}} (2 + \Delta_{\mathrm{IS}} + 2 \Delta_{\Lambda}),
\end{eqnarray}
where $\Delta_{\mathrm{IS}}$ is the enhancement factor of antineutron with respect to antiproton production
 and $\Delta_\Lambda$  is the hyperon factor\footnote{We assume that the antiproton and antineutron production from hyperons is equal.}.
The investigations in \cite{Winkler:2017xor} indicate that the factors $\Delta_{\mathrm{IS}}$ and $\Delta_\Lambda$ are energy dependent.
We adopt these results and shortly repeat the analytic formulas for completeness:
\begin{eqnarray}
  \label{eqn::isospin}
  \Delta_\mathrm{IS} &=& \frac{c_1^\mathrm{IS}}{1+(s/c_2^\mathrm{IS})^{c_3^\mathrm{IS}}},
\end{eqnarray}
with
$c_1^\mathrm{IS} = 0.114$, $c_2^\mathrm{IS} = (144\,\mathrm{GeV})^2$, and $c_3^\mathrm{IS} = 0.51$
and
\begin{eqnarray}
  \label{eqn::hyperon}
  \Delta_\Lambda &=& 0.81 \left(c_1^\Lambda + \frac{c_2^\Lambda}{1+(c_3^\Lambda /s)^{c_4^\Lambda}} \right),
\end{eqnarray}
with $c_1^\Lambda = 0.31$, $c_2^\Lambda = 0.30$, $c_3^\Lambda = (146\,\mathrm{GeV})^2$, and $c_4^\Lambda = 0.9$.
The uncertainties of these parameters have been determined in  \cite{Winkler:2017xor}.
Their impact on the antiproton spectrum is discussed later in this paper.

\subsection{Cross section parametrization}

We use two parameterizations in the fit: Eq.~(12) by Di Mauro  {\it et al. } \cite{diMauro:2014_pbarCrossSectionParametrization} (hereafter Param. I)
and Winkler~\cite{Winkler:2017xor} (Param. II).
Both formulae are given for the Lorentz invariant cross section in the CM frame as a function of
the kinetic variables $\sqrt{s}$, $x_\mathrm{R}$, and $p_{\mathrm{T}}$.
Param.~I depends on 8 fit parameters $\mathcal{C}=\lbrace C_1...C_8 \rbrace$
\begin{eqnarray}
  \label{eqn::param_diMauro}
  \sigma_{\mathrm{inv}} (\sqrt{s}, x_\mathrm{R}, p_{\mathrm{T}}) &=&\sigma_{\mathrm{in}} (1-x_\mathrm{R})^{C_1} \exp(-C_2 x_\mathrm{R})           \\ \nonumber
              &&\times \left[ C_3\left(\sqrt{s}\right)^{C4} \exp(-C_5 p_{\mathrm{T}}) \right.  \\ \nonumber
            && \left.+ C_6\left(\sqrt{s}\right)^{C7} \exp\left(-C_8 p_{\mathrm{T}}^2\right) \right].
\end{eqnarray}
The pre-factor $\sigma_{\mathrm{in}}$ is the total inelastic $pp$ cross section and its energy-dependent form is given
in~\cite{diMauro:2014_pbarCrossSectionParametrization} (Appendix~B).
We note that  this parametrization allows freedom for the scaling with $\sqrt{s}$ and $p_{\mathrm{T}}$.
Especially, it includes an increasing normalization $\sigma_{\mathrm{in}}(s)$ which is determined by a separate fit to data.

Param.~II depends only on 6 parameters $\mathcal{C}=\lbrace C_1...C_6\rbrace$ and is given by
\begin{eqnarray}
  \label{eqn::param_Winkler}
  \sigma_{\mathrm{inv}} (\sqrt{s}, x_\mathrm{R}, p_{\mathrm{T}}) &=& \sigma_{\mathrm{in}} R \, C_1  (1-x_\mathrm{R})^{C_2} \\ \nonumber
  &&\times \left[ 1 + \frac{X}{\mathrm{GeV}}(m_T - m_p) \right]^{\frac{-1}{C_3 X}},
\end{eqnarray}
where $m_T=\sqrt{p_T^2 + m_p^2}$.
The factor
\begin{eqnarray}
  \label{eqn::param_Winkler_III}
   R  &=&
   \begin{cases}
     1 & \sqrt{s} \geq 10\,\mathrm{GeV} \\
     \left[ 1 + C_5 \left(10-\frac{\sqrt{s}}{\mathrm{GeV}}\right)^5 \right]     &  \text{elsewhere}        \\
     \quad \times \exp \left[ C_{6} \left( 10 - \frac{\sqrt{s}}{\mathrm{GeV}} \right)^2 \right. \\
     \quad \times \left. (x_\mathrm{R} - x_{R,\mathrm{min}} )^2  \right]
   \end{cases}
\end{eqnarray}
describes the scaling violation of the cross section at low $\sqrt{s}$, and $x_{R,\mathrm{min}}=m_p/E_{\bar{p}}^{\mathrm{max}*}$.
As before, $\sigma_{\mathrm{in}}$ is the total inelastic cross section, whose form is determined to be
\begin{eqnarray}
  \label{eqn::param_Winkler_II}
  \sigma_{\mathrm{in}} &=& c_{\mathrm{in},1} + c_{\mathrm{in},2} \log\left( \sqrt{s} \right) + c_{\mathrm{in},3} \log^2\left( \sqrt{s} \right), \end{eqnarray}
with $c_{\mathrm{in},1} = 30.9$~mb, $c_{\mathrm{in},2} = -1.74$~mb, and $c_{\mathrm{in},3} = 0.71$~mb.
Finally, the last factor of \eqnref{param_Winkler} describes the scaling violations at large $\sqrt{s}$. This factor contains the parameter
\begin{eqnarray}
  \label{eqn::param_Winkler_IV}
  X &=& C_4 \log^2 \left( \frac{\sqrt{s}}{4 m_p} \right).
\end{eqnarray}
The scaling violation at large energies affects the cross section parametrization in two ways.
Firstly, the total inelastic $pp$ cross section rises according to \eqnref{param_Winkler_II} and, secondly,
the $p_{\mathrm{T}}$ shape is changed as described by the last factor of \eqnref{param_Winkler}.
Scaling violations were intensively studied in by Winkler~\cite{Winkler:2017xor} and found not to affect the behavior
of the cross section below $\sqrt{s}=50$~GeV.
In this analysis we are interested in low-energy part, where NA61 adds new data. A closer look at \eqnref{param_Winkler} reveals that the parameter $C_3$ determines the $p_{\mathrm{T}}$ shape at low energies,
while $C_4$ regulates the strength of alteration towards high energies.
So, we  fix the parameter $C_4=0.038$ \cite{Winkler:2017xor}, while allowing the other 5 parameters to vary freely.

\subsection{Data}

The main data sets to constrain the fit on $\sigma_{p+p\rightarrow \bar{p}+X}$ are the NA49  \cite{NA49_Anticic:2010_ppCollision} and
 NA61 \cite{Aduszkiewicz:2017sei} ones.
However, the discussion about \figref{contribution_pp}  revealed the necessity of a further
data set at low energies to fix the antiproton source term below $T_{\bar{p}} = 5$~GeV.
There are only two available data sets at these energies: Dekkers {\it et al. } \cite{Dekkers:1965zz} taken at $\sqrt{s}=6.1$ and 6.7~GeV and
Allaby {\it et al. }~\cite{Allaby:1970jt}  at $\sqrt{s}=6.15$~GeV.
We use  the measurements by Dekkers {\it et al. }, while
the data set by Allaby {\it et al. }~\cite{Allaby:1970jt} is not
taken into account because it contains very small statistical errors in combination with large systematic and normalization uncertainties.
When fitting Param.~I, we add data from the BRAHMS experiment, which is taken in $pp$ collisions at $\sqrt{s}=200$~GeV \cite{Arsene:2007jd},
in order to fix the freedom of the high-energy behavior in this parameterization.
In the case of Param.~II, we fixed the high-energy behavior (see discussion above) and, thus,
the additional data set is not necessary. A summary of all $pp$ data is given in \tabref{pp_data}.
\begin{table}
\caption{ Summary of all $pp$ data sets, their available CM energies, and references.
          Moreover, we declare which parametirzation (I~or~II) is used and which scale uncertainty
          $\sigma_{\mathrm{scale}}$ is adopted in the fits (see \eqnref{chiSquare_scale}). }
\label{tab::pp_data}
\begin{tabular}{l c c c c c}
  \\ \hline \hline
  Experiment & $\sqrt{s}\quad \mathrm{[GeV]}$ & $\sigma_{\mathrm{scale}}$ & I & II & Ref.  \\ \hline
  NA49 & 17.3 & 6.5\% & $\times$ & $\times$ & \cite{NA49_Anticic:2010_ppCollision} \\
  NA61 & 7.7, 8.8, 12.3, 17.3 & 5\% & $\times$ & $\times$ & \cite{Aduszkiewicz:2017sei} \\
  Dekkers {\it et al.}  & 6.1, 6.7 & 10\% & $\times$ & $\times$ & \cite{Dekkers:1965zz} \\
  BRAHMS & 200  & 10\% & $\times$ &  & \cite{Arsene:2007jd} \\ \hline \hline
\end{tabular}
\end{table}
The NA49 and NA61 collaborations  explicitly determine the prompt antiproton flux, namely, hyperon-induced antiprotons are subtracted
from the original data.
However, for older experiments the situation is not completely clear. Since hyperons have a very short life-time,
they usually decay inside the detector and can contribute to the measurement. Modern detectors, such as NA49, NA61 and LHCb,
can reconstruct a primary vertex and discard hyperon-induced antiprotons.
The usual assumption for older experiments is that they did not distinguish between hyperon-induced and prompt antiprotons.
Thus, to use their data, in our case Dekkers {\it et al. } and BRAHMS, we subtract the hyperon contribution according to \eqnref{hyperon}.
Since antineutrons have a far longer lifetime compared to hyperons, they never decay inside the detector and do not require a similar correction.

\subsection{Fit Procedure}

We perform a $\chi^2$-fit using the \textsc{Minuit} package from \textsc{Root}\footnote{\url{https://root.cern.ch}} software framework in order to minimize the $\chi^2$,
which is divided into two terms:
\begin{eqnarray}
  \label{eqn::chiSquare}
  \chi^2 (\mathcal{C}, \omega )               &=& \chi_\mathrm{stat}^2 (\mathcal{C}, \mathcal{\omega})   +   \chi_\mathrm{scale}^2(\mathcal{\omega}).
\end{eqnarray}
The first term accounts the statistical information and contains a sum over all the data points $i_k$ of all the experiments $k$ from 1 to $L$:
\begin{eqnarray}
  \label{eqn::chiSquare_stat}
  \chi_\mathrm{stat}^2 (\mathcal{C}, \omega) &=& \sum \limits_{k=1}^{L} \sum\limits_{i_k}
                                         \frac{\left( \omega_k {\sigma_{\mathrm{inv}}}_{,i_k} -
                                         \sigma_{\mathrm{inv}}(  \mathcal{C},  \mathcal{T} )_{,i_k}     \right)^2}{\omega_k^2 \sigma_{i_k}^2}. \quad
\end{eqnarray}
Here ${\sigma_{\mathrm{inv}}}_{,i_k}$ is the $i_k$  data point  for invariant cross section having total uncertainty $\sigma_{i_k}$, which is taken as
 the quadratic sum of  statistical and systematical uncertainties of each data point if both are stated separately.
The cross section parametrization $\sigma_{\mathrm{inv}}( \mathcal{C},\mathcal{T} )$ is evaluated (either for Param.~I or II) at the parameter set $\mathcal{C}$
and the kinematic variables
$ \mathcal{T} = {\sqrt{s}}^{(i_k)}, {x_\mathrm{R}}^{(i_k)}, {p_{\mathrm{T}}}^{(i_k)}$.
For each data set we allow a re-scaling  by a constant factor $\omega_k$, which penalizes the $\chi^2$
 by the second term in \eqnref{chiSquare}, explicitly  given by
\begin{eqnarray}
  \label{eqn::chiSquare_scale}
  \chi_\mathrm{scale}^2(\omega ) &=& \sum \limits_{k=1}^{L}
                                           \frac{\left( \omega_k - 1 \right)^2}{ \sigma_{\mathrm{scale},k}^2},
\end{eqnarray}
where $\sigma_{\mathrm{scale},k}$ is the scale uncertainty for each data set (see \tabref{pp_data}). Note that the sum in \eqnref{chiSquare_stat} runs over every single data point, while the sum in \eqnref{chiSquare_scale} only runs over all data sets. Indeed, a scale uncertainty requires that all points are correlated. Moving up or down all the data points of each set  by the same factor is only penalized once, not for each data point. This is in contrast to the treatment in \cite{diMauro:2014_pbarCrossSectionParametrization} (although stated differently in the paper).  NA49  explicitly states scale uncertainties of 6.5\%.
For the other experiments we estimate the scale uncertainty according to the average size of the systematic uncertainties to be
5\% for NA61, and 10\% for Dekkers {\it et al. } and BRAHMS.
During the fit we simultaneously adjust $\mathcal{C}$ and the nuisance parameters $\omega=\omega_1 ... \omega_L$, leaving in total 12 free parameters for Param.~I and 8 for Param.~II. In practice, we use the \textsc{Root} algorithms in two steps to minimize our $\chi^2$. First, the MIGRAD algorithm determines a good parameter estimate, then the HESSE algorithm confirms  these parameters and gives a more reliable covariance matrix.

\subsection{Results}

\begin{table}[b!]
\caption{  Fit results to the $p+p\rightarrow\bar{p} + X $ cross section.
           The full correlation matrices are given in \tabref{Correlation_pp_diMauro} and \tabref{Correlation_pp_Winkler}. \\
           $^{(*)}$ The parameter $C_4$ is fixed, {\it i.e.} not included in the fit, in Param.~II (see text for details). }
\label{tab::Fit_results_pp}
\begin{tabular}{ l  c  c }
 \hline \hline
                         &   with Param.~I                   &   with Param.~II                   \\ \hline
$C_1$                     &   $3.50\pm 0.64$                  & $(5.02\pm 0.22)\times 10^{-2}$     \\
$C_2$                     &   $5.59\pm 0.85$                  & $7.790\pm0.077$                    \\
$C_3$                     &   $(4.00\pm 0.73)\times 10^{-2}$  & $(1.649\pm 0.012)\times 10^{-1}$   \\
$C_4$                     &   $-0.251\pm0.071$                & ${\it (3.800\pm 0.057)\times 10^{-2} \,}^{(*)}$   \\
$C_5$                     &   $2.651\pm0.097$                 & $(4.74\pm 2.59)\times 10^{-4}$     \\
$C_6$                     &   $(3.78\pm 0.53)\times 10^{-2}$  & $3.70\pm0.64$                      \\
$C_7$                     &   $(4.3\pm 4.3)\times 10^{-2}$    & - \\
$C_8$                     &   $2.695\pm0.047$                 & - \\
$\omega_\mathrm{BRAHMS}$  &   $1.115\pm0.079$                 & - \\
$\omega_\mathrm{Dekkers}$ &   $1.051\pm0.068$                 & $1.090\pm0.090$                 \\
$\omega_\mathrm{NA49}$    &   $1.059\pm0.039$                 & $1.061\pm0.044$                 \\
 $\omega_\mathrm{NA61}$   &   $0.936\pm0.036$                 & $0.932\pm0.038$                 \\  \hline \hline
\end{tabular}
\end{table}

We find that both parametrizations \eqnref{param_diMauro} and~\eqref{eqn::param_Winkler}, result in a good fit to the $pp$ cross section data, giving a
$\chi^2$/ndf of 1.30 and 1.18 for Param.~I and Param.~II, respectively. The best fit parameters with the relevant
$1\sigma$ uncertainties are summarized in~\tabref{Fit_results_pp}. We present the full correlation matrix in the Appendix.
Furthermore, we demonstrate in \tabref{Fit_properties_pp} that
reduced $\chi^2$s for all data sets are close to one, namely,
all data sets in the fit are consistent with
each other, if we allow for a re-scaling by the nuisance parameters $\omega_k$.
There seems to be a general tendency to scale down the NA61 data by 6-7\%, while increasing the NA49 data by about 6\% and
the Dekkers data by 5-9\% in both parametrizations.
 Especially, we note that NA49 and NA61 have an overlap in the kinetic parameter space at $\sqrt{s}=17.3$~GeV. After the slight re-scaling the two data sets are in agreement with each other.

\begin{table}[t!]
\caption{Fit quality of the $pp$ channel. The first row reports the global fit, while the other ones show the contribution of the single data sets to the $\chi^2$. }
\label{tab::Fit_properties_pp}
\begin{tabular}{ l c c }
\hline \hline
                                          &   with Param.~I  &   with Param.~II   \\ \hline
$\chi^2$/ndf                            &   534.7/411      &   464.7/394    \\
$\chi^2_\mathrm{BRAHMS}$ (data points)    &   27.6   (21)    &       -        \\
$\chi^2_\mathrm{Dekkers}$(data points)    &    9.8   (10)    &   8.3    (10)  \\
$\chi^2_\mathrm{NA49}$   (data points)    &   211.4 (143)    &   179.0 (143)  \\
$\chi^2_\mathrm{NA61}$   (data points)    &   286.0 (249)    &   277.4 (249)  \\  \hline \hline
\end{tabular}
\end{table}

\begin{figure}[b!]
	\includegraphics[width=1.\linewidth]{Fig_4.pdf}
  \caption{ The differential cross section $d\sigma/dT_{\bar{p}} (p+p\rightarrow \bar{p} + X)$ for prompt antiprotons,
  			 at the representative proton energies $T_p$ = 20 GeV, 450 GeV and 6.5 TeV.
  			 The dashed (solid) line and the red (blue) band are the result of our analysis for Param. I and Param. II.
  			 The uncertainty band corresponds to the $2\sigma$ confidence interval.
  	         We report for comparison some literature estimations (see text for details).
  			 Tables with the full cross section results are provided in the Supplemental Material to this paper. }
\label{fig::comparison_XS_pp}
\end{figure}

\begin{figure}[b!]
	\includegraphics[width=1.\linewidth, trim={0  4.8cm 0 0},clip]{Fig_5a.pdf}
	\includegraphics[width=1.\linewidth, trim={0 0 0 18.2cm},clip]{Fig_5b.pdf}
  \caption{ Source term of prompt antiprotons originating from $pp$ collisions and its uncertainty induced by
            the cross section fits of Param.~I (red dashed) and Param.~II (blue solid), respectively.
            For comparison, we show the antiproton source term from previous parametrizations Di Mauro {\it et al.} , Winkler and KMO.
            The central panel displays the ratio $k$ to the best fit of Param.~II
            and the shaded uncertainty bands correspond to the $2\sigma$ confidence interval.
            For completeness the lower panel contains the $1\sigma$ envelope of the $n$-dimensional $\chi^2$ distribution
            (see discussion in the text for details).
          }
	\label{fig::comparison_pp}
\end{figure}

In \figref{comparison_XS_pp} we report our results for the differential cross section
$d\sigma/dT_{\bar{p}} (p+p\rightarrow \bar{p} + X)$ for the production of prompt antiprotons, at the representative proton energies   $T_p = 20$ GeV, 450 GeV and 6.5 TeV.
The uncertainty bands are derived from our fits (see discussion about \figref{comparison_pp}), and go from 20\% at $T_p = 20$ GeV to 10\% for  $T_p =450$ GeV and 6.5 TeV.
For the lower $T_p$ value the to predictions are mildly compatible, while they
almost overlap for higher energies. We also report some estimates from recent literature, showing some discrepancy with our
findings mostly at low $T_{\bar{p}}$ for the Di Mauro and the Winkler parameterizations. The Monte Carlo based KMO parameterization has been divided by 2.3 since
it accounts all the antiprotons produced in the interaction, {\it i.e.} not only the prompt ones.
We provide tables with the total (antiprotons from prompt production, from antineutron and antihyperon dacay) cross section for a full $T_p$ scan - as well as for a number of other incoming particles on $p$ and He -  in the Supplemental Material.

We calculate the $\bar{p}$ source term from the two cross section parameterizations, and compare them two previous predictions.
We remind again that the fit is performed to the prompt antiproton production, and consequently the source term
calculated according to \eqnref{sourceTerm_1} and displayed in \figref{comparison_pp} does not include antiprotons from neutron and hyperon decay.
To calculate the  fit uncertainty we sample random points in the parameter space $(\mathcal{C},\omega)$ from the full correlation matrix and verify each point against the total $\chi^2$ from \eqnref{chiSquare}.
Then we compute the $\chi^2$ profile as function of the source term, separately at each energy. The uncertainty band at 1$\sigma$ ($n\sigma$) is given by $\Delta \chi^2 = 1$ ($\Delta \chi^2 = n^2$). The interpretation of this $1\sigma$ region is that in 68.3\% of all cases the source term falls within the band.  We checked that the size of the uncertainty band grows approximately linearly with the $\sigma$-interval. Therefore, we show only the $2\sigma$ band in our plots, a different confidence level may be obtained by rescaling.
An alternative useful quantity is the envelope of the $n$-dimensional $\chi^2$ distribution at $1\sigma$ level, where $n$ is the number of free parameters in the fit (Param.~I: $n=12$, Param.~II: $n=8$). In other words, this envelope is built from the set which contains of 68.3\% of the source term realizations and has the lowest $\chi^2$ values. We show this envelope in \figref{comparison_pp} for comparison and note that it almost coincides with the $3\sigma$ band.
Finally, we obtain an uncertainty - solely from the cross section fit - of $\pm8$\%. With Param.~II it increases to about $\pm15$\% below 5~GeV.
The source terms from Param.~I and Param.~II are consistent within the fit uncertainties.
In particular, from $T_{\bar{p}}=1$~GeV to a few hundred GeV, the agreement between the two models
is very good. Above 500~GeV, Param.~II provides an antiproton spectrum systematically higher than Param.~I.
In~\cite{diMauro:2014_pbarCrossSectionParametrization}, it was already pointed out that Param.~I - due to the employed data sets -
gives  reliable results  up to a few hundred GeV.
Param.~II, which employs different data sets at the highest $\sqrt{s}$,  is likely to give a more trustworthy result at high energies.
The comparison of previous analysis by Di~Mauro, Winkler and KMO reveals several interesting features.
The direct (because involving the same parametrization, just different data sets) comparison between Param.~I and Di Mauro shows that the source term predictions
are very close between 10 and 100~GeV, while Param.~I source spectrum stands systematically lower below 10~GeV and above 100~GeV.
This is probably the effect of the hyperon subtraction for Dekkers and BRAHMS data, which was considered here but not
in~\cite{diMauro:2014_pbarCrossSectionParametrization}. Param.~II and Winkler are consistent within uncertainties. Especially, above 30~GeV
the two predictions also coincide with KMO. Below 10~GeV KMO clearly deviates and overpredicts the antiproton source spectrum.

Concluding, both parametrizations for the $\sigma_{p+p\rightarrow \bar{p} + X}$  result in compatible $\bar{p}$ source terms for the energy range measured by AMS-02. The difference in the shape
of the two parametrizations is only few percent in the range of $T_{\bar{p}}= 5$ to 100~GeV, however, at 1~GeV and 1~TeV it grows to 10\%, where Param.~I is slightly softer at high energies.

\section{\label{sec::pA_fit} Fitting the proton-nucleus channel}

The proton-nucleus channels contribute between 40 and 50\% of the total secondary antiproton production.
However, the currently available data on  antiproton production measurement in $pA$ collisions are not sufficient to
allow an individual description of each relevant channel, especially $p$He (see discussion in \secref{current_status}).
We use $p$C data by NA49 and $p$He data by LHCb to determine a re-scaling factor for the $pA$ and, specifically,
$p$He cross sections from the $pp$ cross section.

\subsection{Cross section parametrization}
Antiproton production in $pp$ collisions is by definition symmetric under a reflection along the beam axis in the CM frame,
while this is not necessarily the case in $pA$ collisions (in the nucleon-nucleon CM frame).
Actually, NA49 $p$C data~\cite{Baatar:2012fua} reveals that the cross section is not symmetric between forward and backward production.
It is plausible that the binding of the nucleons in the nucleus has an effect on the antiproton production and breaks the symmetry.
Since a description of the cross section in terms of $x_\mathrm{R}$ which intrinsically expects symmetry is inconsistent, we will use $x_f$ instead in the following whenever we discuss $pA$ channels  .
Following the description by NA49, \cite{Kappl:2014_pbarCrossSection} exploits a re-scaling of $pp$ cross section in terms of overlap functions.
The idea is to split the antiproton production into two components produced by projectile and target,
where the antiprotons from each component are produced mainly forward directed. Separately adjusting the overlap functions allows to accommodate the asymmetry.
\begin{figure*}[t!]
	\includegraphics[width=.5\linewidth]{Fig_6a.pdf}\includegraphics[width=.5\linewidth]{Fig_6b.pdf}
  \caption{ Comparison of LHCb data to the fit with Param.~I-B (left) and Param.~II-B (right).
            The grey band corresponds to 2$\sigma$ uncertainty in the fit. The LHCb data agree better with Param.~II and, therefore,  they
            select this model for the high-energy behavior of the Lorentz invariant cross section.
          }
	\label{fig::LHCb_data}
\end{figure*}
\begin{table}[b!]
\caption{Projectile overlap function $F_\mathrm{pro}(x_f)$. The definition is taken from \cite{Baatar:2012fua}. }
\label{tab::Overlapfunction}
\begin{tabular}{ c l | c l}
\hline \hline
 $x_f$     &  $F_\mathrm{pro}$&  $x_f$     &  $F_\mathrm{pro}$   \\ \hline
-0.250     &   0.0000  &  0.250     &   1.0000  \\
-0.225     &   0.0003  &  0.225     &   0.9997  \\
-0.200     &   0.0008  &  0.200     &   0.9992  \\
-0.175     &   0.0027  &  0.175     &   0.9973  \\
-0.150     &   0.010   &  0.150     &   0.990   \\
-0.125     &   0.035   &  0.125     &   0.965   \\
-0.100     &   0.110   &  0.100     &   0.890   \\
-0.075     &   0.197   &  0.075     &   0.803   \\
-0.050     &   0.295   &  0.050     &   0.705   \\
-0.025     &   0.4     &  0.025     &   0.6     \\
           &           &  0.000     &   0.5     \\ \hline \hline
\end{tabular}
\end{table}
The inclusive Lorentz invariant cross section of $p + A \rightarrow \bar{p} + X$ scattering is given by
\begin{eqnarray}
  \label{eqn::param_pA}
  \sigma_{\mathrm{inv}}^{pA} (\sqrt{s}, x_f, p_{\mathrm{T}}) &=& f^{pA}(A, x_f, \mathcal{D}) \,\,  \sigma_{\mathrm{inv}}^{pp}(\sqrt{s}, x_\mathrm{R}, p_{\mathrm{T}}), \quad
\end{eqnarray}
where $A$ is the mass number of the nucleus and $\mathcal{D}=(D_1,D_2)$ are the two fit parameters.
Explicitly, the factor $f^{pA}$ is defined by:
\begin{eqnarray}
  \label{eqn::param_pA_II}
  f^{pA} &=&  A^{D_1} \left[ A^{D_2} \left(1+\frac{N}{A}\Delta_\mathrm{IS} \right) F_\mathrm{pro}(x_f) \right. \\ \nonumber
         && \left. \qquad  \qquad +   F_\mathrm{tar}(x_f) \vphantom{\frac{N}{A}}  \right].
\end{eqnarray}
$F_\mathrm{pro}(x_f)$ and $F_\mathrm{tar}(x_f)$ are the projectile and target overlap functions.
They fulfil $F_\mathrm{tar}(x_f) = F_\mathrm{pro}(-x_f)$ and
$F_\mathrm{tar}(x_f) + F_\mathrm{pro}(x_f) = 1$ and are defined in \tabref{Overlapfunction}. $N$ is the number of neutrons in the nucleus.
The form factor $f^{pA}$ is motivated by~\cite{Baatar:2012fua,Kappl:2014_pbarCrossSection}. Its $A$ dependence is chosen such that in the case of
$A=1$ we retain proton-proton scattering.
We remind that the kinetic variables $x_f$ and $\sqrt{s}$ refer to the nucleon-nucleon CM frame,
where proton and nucleus have the same velocity, not the same momentum.
Consequently, the CM energy $\sqrt{s}$ is the colliding energy of the nucleon with the proton.

The fit procedure is analogous to the $pp$ case discussed in the previous section.
However, here the parameters $\mathcal{C}$ from \eqnref{param_pA} are fixed, in other words the $pp$ scattering is unaltered,
and only the new parameters $\mathcal{D}$ are varied in the fit.
The definition of our $\chi^2$ is equivalent to \eqnref{chiSquare}, when $\mathcal{C}$ is replaced by $\mathcal{D}$ and $k$ runs over the experiments with $pA$ data. As before we allow for nuisance parameters $\omega$ of each data set.

\subsection{Data}

\begin{table}[t!]
\caption{ Main properties of the NA49 $p$C and LHCb $pHe$ data sets: available CM energies,
scale uncertainty $\sigma_{\mathrm{scale}}$ adopted in the fits, the parameterization (I~or~II) used in the fit,
 references. Labels A and B refer to use of NA49 $p$C data alone or NA49 $p$C and LHCb $pHe$, respectively. }
\label{tab::pA_data}
\begin{tabular}{l c c c c c c c }
  \\ \hline \hline
       & $\sqrt{s}\quad \mathrm{[GeV]}$ & $\sigma_{\mathrm{scale}}$ & I-A & I-B & II-A & II-B & Ref.  \\ \hline
  NA49 & 17.3 & 6.5\% & $\times$   & $\times$ & $\times$ & $\times$ & \cite{Baatar:2012fua} \\
  LHCb & 110  & 6.0\% &            & $\times$ &          & $\times$ & \cite{Graziani:2017} \\ \hline \hline
\end{tabular}
\end{table}

We exploit the data from NA49 and LHCb on $p$C and $p$He scattering, respectively.
Both experiments use a fixed target nucleus  while the incident proton is accelerated to
158~GeV in NA49 and 6.5~TeV in LHCb.
Their data are very precise, and both experiments correct for antihyperons, \textit{i.e.} they remove antiprotons originating from the decay of intermediate antihyperon states.
We summarize the main experimental information in~\tabref{pA_data}.

\subsection{Results}

\begin{table}[b!]
\caption{Fit quality of $f^{pA}$ for the different $pp$ Param.~I and II, and for the different data sets A (NA49 $p$C) and B (NA49 $p$C, LHCb $p$He).
The first row shows the result of the fit, while the second and third rows report the split contribution from the $p$C NA49 and $p$He LHCb data sets.
In brackets are the numbers of data points entering in the fit. The italic numbers are not the result of a minimization, but the  $\chi^2$ on LHCb data with the parameters fixed by NA49 $p$C data.}
\label{tab::Fit_properties_pA}
\begin{tabular}{ l c c  c c}
\hline \hline
                             &   \multicolumn{2}{c}{Param.~I}        &    \multicolumn{2}{c}{Param.~II}      \\
                             &    A                &    B            &    A                &    B            \\ \hline
$\chi^2$/ndf               &   153.0/118         &   1296.3/253      &   131.2/118         &   326.3/253     \\
$\chi^2_\mathrm{NA49}$       &   153.0 (121)       &   155.3 (121)   &   131.2 (121)       &   131.8 (121)   \\
$\chi^2_\mathrm{LHCb}$       & \textit{1266 (136)} &   1141 (136)    & \textit{212.4 (136)}&   194.5 (136)   \\ \hline \hline
\end{tabular}
\end{table}

\begin{table*}[t!]
\caption{  Fit results of $f^{pA}$ for the different $pp$ Param.~I and II, and for different data sets A (NA49 $p$C) and B (NA49 $p$C, LHCb $p$He).
           }
\label{tab::Fit_results_pA}
\begin{tabular}{ l c c c c }
\hline \hline
                          &   \multicolumn{2}{c}{Param.~I}            &    \multicolumn{2}{c}{Param.~II}              \\
Parameter                 &    A                  &    B              &    A                  &    B                  \\ \hline \hline
$D_1$                     &   $0.830\pm 0.012$    & $0.825\pm 0.012$  &   $0.825\pm 0.012$    & $0.828\pm 0.012$      \\
$D_2$                     &   $0.149\pm 0.013$    & $0.167\pm 0.012$  &   $0.154\pm 0.013$    & $0.145\pm 0.012$      \\
$\omega_\mathrm{NA49}$    &   $1.000\pm 0.025$    & $1.001\pm0.024$   &   $1.000\pm 0.025$    & $0.997\pm0.024$       \\
$\omega_\mathrm{LHCb}$    &   -                   & $0.900\pm0.015$   &   -                   & $1.034\pm0.018$       \\ \hline \hline
\end{tabular}
\end{table*}

\begin{figure*}[t!]
	\includegraphics[width=0.5\textwidth]{Fig_7a.pdf}\includegraphics[width=0.5\textwidth]{Fig_7b.pdf}
  \caption{ The differential cross section $d\sigma/dT_{\bar{p}} (p+{\rm He} \rightarrow \bar{p} + X)$ (left) and
  $d\sigma/dT_{\bar{p}}({\rm He} + p\rightarrow \bar{p} + X)$ (right) for prompt antiprotons, at the representative incident energies
  $T_p$ = 20 GeV, 450 GeV and 6.5 TeV. The dashed (solid) line and the relevant red (blue) band are the result of our analysis for Param. I and Param. II.
  The uncertainty band corresponds to the $2\sigma$ confidence interval.
  We report for comparison some literature estimations (see text for details). Tables with the full cross section results are provided in the Supplemental Material to this paper.
           }
	\label{fig::comparison_XS_pHe}
\end{figure*}

\begin{figure*}[t!]
	\includegraphics[width=0.5\textwidth]{Fig_8a.pdf}\includegraphics[width=0.5\textwidth]{Fig_8b.pdf}
  \caption{ CR $p$He (left panel) and He$p$ (right panel) antiproton source term with the uncertainty on cross sections
            for the best fit of Param.~I-B and II-B, {\it i.e.} with NA49 $p$C and LHCb $p$He data.
            Uncertainties are given at the $2\sigma$ confidence interval.
           }
	\label{fig::comparison_pHe}
\end{figure*}

We perform four fits to determine, first, the goodness of the parametrizations (I~and~II) from the $pp$ fit for the
interpretation of nuclei data and, secondly, the impact of LHCb data  by excluding (case A) or including (case B) them in the fits.
\tabref{Fit_properties_pA} comprises the results of all four fits.
The fits with $p$C data alone (without LHCb data) I-A and II-A converge to a
$\chi^2/$ndf of 1.3 and 1.1, respectively, leaving the conclusion that the NA49 proton-carbon data fits very well to a rescaled $pp$ cross section.
In the second step, we use the fit results to predict the $p$He cross section and to compare it to LHCb data.
Param.~I shows a large difference between data and the prediction, measured by a $\chi^2/$ndf from LHCb alone of
9.3. On the other hand, Param.~II gives a $\chi^2/\mathrm{ndf}=1.6$, hinting already the good agreement with Param.~II rescaled by the form factor $f^{pA}$ fixed on $p$C data.
Including the LHCb data in the fit  does not change the general picture. The quality of the fit slightly improves  to 8.4 and 1.4 in both cases  I-B and II-B, respectively.
We conclude that Param.~II results in a much better description of the $p$He data by LHCb.
The best fit values for all parameters are summarized in \tabref{Fit_results_pA}.
Our results for Param.~II agree with \cite{Winkler:2017xor} ($i.e.$  $\langle\nu_\mathrm{He}\rangle=1.25$ there is comparable with
$4^{D_2}=1.22$ and $4^{1-D1}=1.27$). However, for $p$C we obtain a 10\% lower value of $12^{D_2}=1.43$ or $12^{1-D1}=1.53$ instead
of $\langle\nu_\mathrm{C}\rangle=1.6$.
\figref{LHCb_data} displays the comparison of the LHCb data to the cross section prediction. It is visible that
the $p_{\mathrm{T}}$-shape of Param.~I does not fit well the data. This shape is inherited from the $pp$ data, and it is therefore
unlikely to improve the fit by a mere refinement of the  $f^{pA}$ parametrization. We remind that Param.~II includes corrections to
the $p_{\mathrm{T}}$-shape due to scaling violation \cite{Winkler:2017xor}.
Finally, we remark that we explicitly tried a fit solely on LHCb data, but since
the data contain, apart from one data point, only points for antiprotons produced in backward direction it cannot constrain the asymmetry imposed by $D_2$ and the parameters $D_1$ and $D_2$ turn out to be degenerate.
To calculate $\sigma_{\mathrm{He} + p \rightarrow \bar{p} +X}$ we use a generalization of \eqnref{param_pA_II}:
\begin{eqnarray}
  \label{eqn::param_pA_III}
  f^{A_1A_2} &=&  A_1^{D_1} A_2^{D_1} \left[ A_1^{D_2} \left(1+\frac{N_1}{A_1}\Delta_\mathrm{IS} \right) F_\mathrm{pro}(x_f) \right. \qquad \\ \nonumber
         && \left. \qquad  \qquad +   A_2^{D_2} \left(1+\frac{N_2}{A_2}\Delta_\mathrm{IS} \right) F_\mathrm{tar}(x_f) \vphantom{\frac{N}{A}}  \right].
\end{eqnarray}
We cross-checked the validity of this approach by taking the $p$He cross section
and transforming it to the frame where  the proton is at rest. The two methods give the same result.
Similarly to \figref{comparison_XS_pp}, in \figref{comparison_XS_pHe} we report the results for the differential cross section
$d\sigma/dT_{\bar{p}}(p+{\rm He}\rightarrow \bar{p} + X)$ (left panel) for the representative proton energies $T_p$ = 20 GeV, 450
GeV and 6.5 TeV. The right panel contains the same information but for incoming He on $p$.
To determine the fit uncertainty we sample points from the correlation matrices of the $pp$ and $pA$ fits (see \appref{app_corr_matr}).
To constrain the $pA$ fit at the 2$\sigma$  confidence level
we require that the total $\chi^2$ (sum of $pp$ and $pA$ fit)
lies within a range of $\Delta\chi^2$=4 compared to our best fit values.
Additionally, we reflect the fact that the $pp$ fit is a prior of the $pA$ fit by requiring that also its
$\chi^2$ is within a range of $\Delta\chi^2$=4.
The conclusions are similar to  the
$pp$ case. We provide a full table for the total cross sections of a number of incident nuclei and their isotopes, and for both $p$ and
He fixed target in the Supplemental Material to this paper.

We use the fit results to calculate source terms for the $p$He and He$p$ production channels.
The results are shown in \figref{comparison_pHe}.
In general, the form and uncertainty of $p$He and He$p$ look similar to $pp$, since both depend on the $pp$ parametrization.
The fact that He$p$ is harder than $p$He comes from the CR flux which is harder for He compared to $p$.
The two parameterizations are compatible within uncertainties in the AMS-02 $T_{\bar{p}}$ energy range, while Param.~I implies a slightly softer $\bar{p}$ spectrum w.r.t. Param.~II.
The agreement with former parametrizations Winkler and KMO is unchanged compared to the $pp$ study. However, the re-scaled Di Mauro {\it et al. } shows large deviation in the shape at high energies. We recommend  to use the re-scaling from this paper instead.

\section{\label{sec::final_spectrum} The total antiproton source term}

\begin{figure}[b!]
	\includegraphics[width=.5\textwidth]{Fig_9.pdf}
  \caption{Source terms of CR antiprotons and separate  CR-ISM contributions, grouped  following the prescriptions in \figref{contribution_sourceTerm}.
  The shaded bands report the $2 \sigma$ uncertainty due to prompt $\bar{p}$ production cross sections as derived in this paper.
  In the bottom panel we show the relative uncertainty on the total source term. The grey band refers to the prompt $\bar{p}$'s
  only, while the outer lines quantify the additional uncertainty due to isospin violation and to hyperon decay.
   }
	\label{fig::comparison_all}
\end{figure}
The results obtained in the previous sections can be joint to compute the total antiproton source term in the Galaxy, including antineutrons and
antihyperons, and the contributions from nuclei heavier than helium. The latter, as shown in \figref{contribution_sourceTerm}, give a contribution which is not negligible when compared to errors on the $\bar{p}$ flux measured by AMS-02. The  CR CNO on $p$ or He contributes
to the source term at the few percent level each. Even the heavier CR primaries NeMgSi and Fe may contribute above 1\%.
We note that our fit is tuned to He and C data and therefore the uncertainty on cross sections is extrapolated for CR sources heavier than CNO.
The total $\bar{p}$ source term is plotted in \figref{comparison_all}, along with the contribution for every production channel.
We use the same inputs for CR fluxes and ISM components as discussed in the context of \figref{contribution_sourceTerm}.
It is visible how the measured hardening of CR nuclei fluxes with respect to protons \cite{AMS-02_Aguilar:2015_HeliumFlux,Aguilar:2017hno}
results in a corresponding hardening of the antiproton source term \cite{Donato:2010vm}.
The rescaling from the prompt $\bar{p}$ production follows \eqnref{XS_galaxy}.
We also plot the uncertainty band from the production cross sections, as determined in the fits to data on prompt antiprotons.
In order to include the production from neutron and hyperon decays, we pick the parameters
as declared in \cite{Winkler:2017xor}, and namely
$c_1^\mathrm{IS} = 0.114\pm0.1$  for the determination of $\Delta_{\rm IS}$ (see \eqnref{isospin}), and
$0.81\pm0.04, c_1^\Lambda = 0.31\pm0.0375$, $c_2^\Lambda = 0.30\pm0.0125$ for the determination of $\Delta_\Lambda$ (see \eqnref{hyperon}).

The results in \figref{comparison_all} show that the uncertainty due to prompt cross sections (bottom panel) is at the level of $\pm8$\% at 2$\sigma$ above $T_{\bar{p}}=5$~GeV.
At $T_{\bar{p}}\leq 5$~GeV it increases to $\pm15$\% at 1~GeV. Adding the uncertainties from  isospin violation in the antineutron production  and from hyperon decays, the uncertainty on the total antiproton source term ranges
$\pm12$\% from high energies down to about few GeV, and increases to $\pm20$\% below that value.
Above $T_{\bar{p}}=50$~GeV the total antiproton source spectrum can be approximated by  a power law  with an index of about $-2.5$ .

\section{\label{sec::happy} Conclusions}

The role of high-energy particle physics in the interpretation of CR data receives increasing attention,
since data from space are provided with improving precision.
AMS-02 on the International Space Station collected data of CR nuclei, leptons, and antiprotons with unprecedented accuracy,
often pushing uncertainties down to few percent in a large range of energy from the GeV to the TeV scale.
The fluxes of secondary CRs, which are produced in interactions with the ISM, depend on the inclusive production
cross sections provided by high-energy particle experiments. In particular, this applies to CR antiprotons whose origin is believed
to be dominantly secondary.
Consequently, the interpretation of the antiproton flux in terms of CR propagation or the search for a possible primary component, such as
for example dark matter annihilation or decay, relies on the accurate modeling of secondary production. The underlying cross sections
should be provided at least at  the same accuracy level as CR measurements.

In this paper,  we analyze the first-ever data on the inclusive cross section $p + \mathrm{He} \rightarrow \bar{p} +X$
collected by the LHCb collaboration at Cern, with beam protons at $T_p=6.5$~TeV and a fixed helium target.
Since the coverage of the kinematic parameter space of this data do not allow a standalone parametrization,
we apply a rescaling from $p + p \rightarrow \bar{p} +X$ cross section.
Therefore, we update the most recent parametrizations from Di Mauro {\it et al. } (Param.~I) and Winkler (Param.~II) exploiting the newly available NA61 data.
Then we determine the rescaling factor to proton-nucleus using $p$He data from LHCb and $p$C data from NA49 (taken at $\sqrt{s} = 110$ and 17.3~GeV, respectively).
The LHCb $p$He data clearly prefer Param.~II. All other data result in equally good fits for both parametrizations.
Moreover, the LHCb data show for the first time how well the rescaling from the $pp$ channel applies to helium target. By using $pp$, $p$He and $p$C data we estimate the uncertainty on the
Lorentz invariant cross section for $p + \mathrm{He} \rightarrow \bar{p} + X$.
This uncertainty is dominated by $p+p \rightarrow \bar{p} + X$ cross section, which translates into all channels
since we derive them using the $pp$ cross sections.

Finally, we use our cross sections to compute the antiproton source terms and their uncertainties for all the production channels, considering also nuclei heavier than He both in CRs and the ISM.
At intermediate energies from $T_{\bar{p}}=5$~GeV up to a few hundred GeV the prompt source terms derived from Param.~I and II are
compatible within uncertainties, which are at the level of $\pm8$\% at the 2$\sigma$ level and increase to $\pm15$\% below $T_{\bar{p}}=5$~GeV.
The uncertainty is dominated by $p+p \rightarrow \bar{p} + X$ cross section, which translates into all channels.
Antineutron- and  hyperon-induced production increases the uncertainty by an additional 5\%.
Overall the secondary antiproton source spectrum is affected by an uncertainty of up to $\pm20$\%.
Moreover, we find that
CR CNO makes up to few percent of the total source term and should always be considered.
In the Supplemental Material to this paper, we provide the energy-differential cross sections, which are required to calculate the source spectrum, for all relevant isotopes.
We quantify the necessity of new data on antiproton production cross sections, and pin down the kinematic parameter space
which should be covered by future data.

\section{\label{sec::acknowledgments}Acknowledgments}
We warmly thank G. Graziani and G. Passaleva for useful discussions, and P. Von Doetinchem for many useful insights on NA61 data.

\bibliographystyle{apsrev4-1.bst}

redpajama_set_name: RedPajamaArXiv


Jan. 12, 2016 10:00 a.m.
SURREY — Mayor Linda Hepner teared up as she was unexpectedly recognized for 30 years of civic service to the City of Surrey during a city council meeting Monday night.
City manager Vincent Lalonde put in a special request to do so at the top of the meeting.
Hepner received applause as Lalonde announced the mayor has recently reached a "significant milestone" in serving the city for 30 years through her work both an elected official and as city staff.
"Thank you for embarrassing me so early in the new year," said Hepner with a smile.
Donna Jones, who Hepner mentored, spoke during the mayor's recognition.
This experience, said Jones, gives Hepner the insight to "draw information and inspiration" from both sides in her work as mayor.
Hepner was first hired to the city as then-city manager Mike Jones' executive assistant.
Hepner would go on to hold several titles, including business development officer, manager of corporate administration and manager of economic development.
Then, in 2005, she was elected to city council and became Surrey's mayor in 2014.
"Quite a journey indeed," remarked Jones.
Former city manager Mike Jones – the man who hired Hepner – wrote a letter that Donna read to all at the meeting.
"Back then I knew you were good, but I did not know you would blossom into Wonder Woman," said Mike, praising Hepner for her "outstanding" performance in all the roles she's held.
"Now as mayor you have the leading political role with great influence in the present and for the future," said Mike.
Lalonde presented Hepner with a framed copy of Mike's letter.


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text:
Steve Lukather
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Toto and The Players Club / Mascot Label Group today reveal the title track from 'With A Little Help From My Friends' released today, June 25. Steve Lukather shares, "We are keeping the music of Toto alive with a little help from our friends." Joseph Williams offers, "In the midst of a global pandemic and an unprecedented lockdown we realized a few things. Musicians must play and people need to get together to watch them. It was then Luke, Dave and […]
TOTO Reveal Live Video For 'You Are The Flower'
TOTO and The Players Club / Mascot Label Group have revealed the second live video, 'You Are The Flower', to be taken from their upcoming live release, 'With A Little Help From My Friends' – out June 25. Talking about the song Joseph Williams says, "Among some of the deeper cuts is one of my all-time favorite songs, 'You Are The Flower' by the original – Bobby Kimball. Being a massive fan (and a good friend), I dedicate the performance to him. " The performance captures one special night on November 21, 2020 when Steve Lukather, Joseph Williams and David Paich appeared with the new line-up of TOTO for a global event originating from Los Angeles, CA. Joining Lukather, Williams and Paich for this next chapter in their indelible history […]
Toto To Release 'With A Little Help From My Friends'
Toto and The Players Club / Mascot Label Group have announced the global release of 'With A Little Help From My Friends' on June 25. With this they have revealed the performance of 'Till The End'. The performance captures one special night on November 21, 2020 when Steve Lukather, Joseph Williams and David Paich appeared with the new line-up of Toto for a global event originating from Los Angeles, CA. Joining Lukather, Williams and Paich for this next chapter in their indelible […]
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Steve Lukather Reveals Video For New Song 'Serpent Soul'
Steve Lukather and The Players Club / Mascot Label Group present 'Serpent Soul', the second track featured on Luke's upcoming new studio album 'I Found The Sun Again'. The video presents the photos and video content captured in the recording studio by Mini Carlsson when the album was tracked. In reference to 'Serpent Soul', Luke offers, "You can fucking figure out who the song […]