Cosmic rays, anti-helium, and an old navy spotlight

TL;DR

This paper develops a new method to predict the secondary flux of cosmic-ray anti-nuclei, especially anti-helium, using a volume scaling law based on HBT correlations, leading to higher flux estimates relevant for AMS02 detection.

Contribution

It introduces a novel volume scaling law for anti-nuclei production based on HBT data, improving flux predictions for cosmic-ray anti-helium.

Findings

Anti-helium production cross section is 1-2 orders of magnitude higher than previous estimates.

The new flux estimate suggests anti-helium could be detectable by AMS02 within five years.

The scaling law is consistent across various collision systems, from AA to pp.

Abstract

Cosmic-ray anti-deuterium and anti-helium have long been suggested as probes of dark matter, as their secondary astrophysical production was thought extremely scarce. But how does one actually predict the secondary flux? Anti-nuclei are dominantly produced in pp collisions, where laboratory cross section data is lacking. We make a new attempt at tackling this problem by appealing to a scaling law of nuclear coalescence with the physical volume of the hadronic emission region. The same volume is probed by Hanbury Brown-Twiss (HBT) two-particle correlations. We demonstrate the consistency of the scaling law with systems ranging from central and off-axis AA collisions to pA collisions, spanning 3 orders of magnitude in coalescence yield. Extending the volume scaling to the pp system, HBT data allows us to make a new estimate of coalescence, that we test against preliminary ALICE pp data. For anti-helium the resulting cross section is 1-2 orders of magnitude higher than earlier estimates. The astrophysical secondary flux of anti-helium could be within reach of a five-year exposure of AMS02.

Tables

Table 1. Table 1: Benchmark decayed fragmentation cross sections.

reaction	benchmark cross section (mb)
12C$\to$11B	55
12C$\to$10B	14
16O$\to$11B	25
16O$\to$10B	9
14N$\to$11B	30
14N$\to$10B	9
20Ne$\to$11B	14
20Ne$\to$10B	2
24Mg$\to$11B+10B	15

Equations

$$\frac{n_a(\mathcal{R})}{n_b(\mathcal{R})}$$

$$Q_a(\mathcal{R})$$

$$n_{\bar{p}}(\mathcal{R})$$

$$X_{\mathrm{esc}}(\mathcal{R})$$

$$X_{\mathrm{esc}}$$

$$E_A\frac{d{N}_A}{d^3{p}_A}$$

$$\frac{A}{m_p^{A-1}}{\left(\frac{4\pi }{3}{p}_c^3\right)}^{A-1}$$

$$B_A\propto V^{1-A},$$

$$B_3$$

$$⟨C_3⟩$$

$$\frac{B_3}{\mathrm{Ge}{\mathrm{V}}^4}$$

$$\frac{B_2}{\mathrm{Ge}{\mathrm{V}}^2}$$

$$B_2^{(pp)}$$

$$B_3^{(pp)}$$

$${\sigma }_{pp\to \overline{{}^3\mathrm{He}}}(\mathcal{R})$$

$$\frac{d{\sigma }_{pp\to \overline{\mathrm{t}}}({ϵ}_p,{ϵ}_{\overline{\mathrm{t}}})}{d{ϵ}_{\overline{\mathrm{t}}}}$$

$$R(x)$$

$${\Phi }_N(x,m)$$

Full text

Introduction. Composite cosmic-ray (CR) anti-nuclei like anti-deuterium ($\bar{d}$) and anti-helium ($\overline{\rm{}^{3}He}$) have long been suggested as probes of dark matter Donato:1999gy ; Baer:2005tw ; Donato:2008yx ; Brauninger:2009pe ; Kadastik:2009ts ; Cui:2010ud ; Dal:2012my ; Ibarra:2012cc ; Fornengo:2013osa ; Carlson:2014ssa ; Aramaki:2015pii , as their secondary astrophysical production was thought to be negligible Chardonnet:1997dv ; Duperray:2005si ; Ibarra:2013qt ; Cirelli:2014qia ; Herms:2016vop . But how does one actually predict the secondary flux? Astrophysical anti-nuclei are dominantly produced in pp collisions, for which laboratory cross section data is scarce or altogether absent.

We make a new attempt at tackling this problem by appealing to a scaling law of nuclear coalescence with the physical volume of the hadronic emission region. The same volume is probed by Hanbury Brown-Twiss (HBT) two-particle correlation measurements Scheibl:1998tk ; Lisa:2005dd . A common tool in heavy ion collision studies Uribe:1993tr ; Malinina:2013fhp ; Boggild:1998dx ; Adam:2015vja ; Bearden:2001sy ; Chajecki:2005zw , the HBT method owes its acronym to the inventors of intensity interferometry, utilised in the 50’s for the first angular size determination of a star outside the solar system using two US navy spotlights as light buckets Brown:1956zza ; HanburyBrown:1956bqd . In this paper we redirect the HBT idea back to astrophysics, this time in connecting accelerator data to anti-nuclei CRs.

We show that the scaling law applies to systems ranging from central and off-axis AA collisions to pA and pp collisions, spanning 3 orders of magnitude in coalescence yield. Guided by HBT data we make a new estimate of the $pp\to\overline{\rm{}^{3}He}$ cross section, that we validate against preliminary ALICE pp data.

Our results for the $\bar{p},\,\bar{d},$ and $\overline{\rm{}^{3}He}$ flux are summarised in Fig. 1. The predicted $\overline{\rm{}^{3}He}$ yield is 1-2 orders of magnitude higher than most earlier estimates Chardonnet:1997dv ; Duperray:2005si ; Ibarra:2013qt ; Cirelli:2014qia ; Herms:2016vop and the flux could reach, within uncertainties, the expected 5-yr 95%CL flux upper limit estimated for AMS02 prior to its launch kounineHebar .

The outline of the paper is as follows. We begin by calculating the secondary $\bar{p}$ flux, demonstrating along the way that the astrophysical details of CR propagation are irrelevant for the calculation of stable, relativistic, secondary CR (anti-)nuclei like $\bar{p},\,\bar{d}$ and $\overline{\rm{}^{3}He}$. The challenge, instead, is in computing the production cross sections. Invoking the HBT-coalescence relation, we derive new estimates for the $\bar{d}$ and $\overline{\rm{}^{3}He}$ yield in pp collisions, forming the basis of our results in Fig. 1. Many details are reserved to the Appendices: A. accelerator data analysis; B. comparison to previous work; C. phase space calculations; D. comments on the $\bar{p}p$ secondary source; E. benchmark fragmentation cross sections.

CR $\bar{p}$: the Galaxy is a fixed-target experiment. CR $\bar{p},\,\bar{d}$ and $\overline{\rm{}^{3}He}$ are produced as secondaries in collisions of other CRs, notably protons, with interstellar matter (ISM), notably hydrogen in the Galaxy. While the details of CR propagation are unknown, the confinement in the Galaxy is magnetic and thus different CR particles that share a common distribution of sources exhibit similar propagation if sampled at the same magnetic rigidity $\mathcal{R}=p/Z$. It is therefore natural to gauge the propagation of CR anti-nuclei from that of secondary nuclei like boron (B), formed by fragmentation of heavier CRs. For such secondaries, the ratio of densities of two specie $a,b$ satisfies an approximate empirical relation, valid at relativistic energies ($\mathcal{R}\gtrsim$ few GV) Engelmann:1990zz ; 2003ApJ…599..582W ; Katz:2009yd ,

[TABLE]

Here $Q_{a}$ denotes the net production of species $a$ per unit ISM column density,

[TABLE]

where $\sigma_{a}/m$ and $\sigma_{P\to a}/m$ are the total inelastic and the partial $P\to a$ cross section per target ISM particle mass $m$, respectively. These cross sections can (and for $\bar{p}$, $\bar{d}$ and $\overline{\rm{}^{3}He}$, do) depend on energy. In Eq. (2) we define these cross sections such that the source term $Q_{a}(\mathcal{R})$ is proportional to the progenitor species density $n_{P}(\mathcal{R})$ expressed at the same rigidity.

Eq. (1) is theoretically natural, in that it is guaranteed to apply if the relative composition of the CRs (not CR intensity, nor target ISM density) in the regions that dominate the spallation is similar to that measured locally at the solar system Ginzburg:1990sk ; Katz:2009yd , and as long as no significant energy gain/loss occurs during propagation. Restricting our analysis to $\mathcal{R}\geq 5$ GV, we expect Eq. (1) to be accurate to order 10% or so, as demonstrated by nuclei data Engelmann:1990zz ; 2003ApJ…599..582W ; Katz:2009yd ; Blum:2013zsa .

Eq. (1) is useful because we can use the measured flux of B, C, O, p, He,… to predict, e.g., the $\bar{p}$ flux 1992ApJ…394..174G ; Katz:2009yd ; Blum:2013zsa :

[TABLE]

The RHS of Eq. (3) is derived from laboratory cross section data and from direct measurements of local CRs, without reference to any detail of propagation.

The quantity

[TABLE]

known as the CR grammage, measures the column density of ISM traversed by CRs. We combine AMS02 B/C Aguilar:2016vqr and C/O AMS02:C2O with heavier CR data from HEAO3 Engelmann:1990zz and with laboratory fragmentation cross section data (see e.g. Tomassetti:2015nha ), to derive $X_{\rm esc}$ directly from measurements:

[TABLE]

Our result for $X_{\rm esc}$ agrees with the power-law approximation derived in Ref. Blum:2013zsa to 20% accuracy.

Now that we have $X_{\rm esc}$, we use the $\bar{p}$ production and loss cross sections parametrised in 1983JPhG….9..227T ; 0305-4616-9-10-015 (applying the correction in Winkler:2017xor ) together with measurements of the proton and helium Aguilar:2015ooa ; Aguilar:2015ctt flux to calculate $Q_{\bar{p}}$ and apply it in Eq. (3). Solar modulation is included as in 2003ApJ…599..582W with $\Phi=450$ MV. The result is compared to data in Figs. 1-2.

Figs. 1-2 demonstrate that the $\bar{p}$ flux measured by AMS02 Aguilar:2016kjl is consistent with secondary production Blum:2013zsa . Beyond this fact, they also demonstrate that – as far as relativistic, stable, secondary nuclei and anti-nuclei CRs are considered – the Galaxy is essentially a fixed-target experiment. Having calibrated the set-up on one species (B), one can calculate the flux of other secondaries directly from particle physics cross sections. The problem of predicting the anti-nuclei CR flux is therefore decoupled from the modelling of propagation and is reduced to calculating the relevant cross sections, to which we attend next.

Calibrating coalescence with HBT correlations. We use a coalescence ansatz Butler:1963pp ; Schwarzschild:1963zz ; Gutbrod:1988gt relating the formation of composite nucleus product with mass number $A$ to the formation cross section of the nucleon constituents:

[TABLE]

where $dN_{i}=d\sigma_{i}/\sigma$ is the differential yield, $\sigma$ is the total inelastic cross section, and the constituent momenta are taken at $p_{p}=p_{A}/A$.

The factor $R(x)$, with $x=\sqrt{s+A^{2}m_{p}^{2}-2\sqrt{s}\tilde{E}_{A}}$ and $\tilde{E}_{A}$ the centre of mass product nucleus energy, is a phase space correction that we define as in Duperray:2003tv . This becomes necessary in order to extend the coalescence analysis down to near-threshold collision energies, important for the astrophysics as well as for low energy laboratory data. Details on the derivation of $R(x)$ are given in App. C.

$B_{A}$, the coalescence factor, needs to be extracted from accelerator data. However, experimental information on $\bar{d}$ and $\overline{\rm{}^{3}He}$ production is scarce and, in the most part, limited to AA or pA collisions. For pp collisions, the most relevant system for CR astrophysics, no quantitative data exists for $pp\to\overline{\rm{}^{3}He}$, and the data for $pp\to\bar{d}$ is sparse.

Faced with this problem, previous estimates Chardonnet:1997dv ; Duperray:2005si ; Ibarra:2013qt ; Cirelli:2014qia ; Herms:2016vop of the secondary CR $\bar{d}$ and $\overline{\rm{}^{3}He}$ flux made two key simplifying assumptions:

Coalescence parameters used to fit $pp\to\bar{d}$ data were translated directly to $pp\to\overline{\rm{}^{3}He}$. More precisely, the coalescence factor $B_{A}$ was converted to a coalescence momentum $p_{c}$, via

[TABLE]

The value of $p_{c}$ found from $pp\to\bar{d}$ accelerator data was then assumed to describe $pp\to\overline{\rm{}^{3}He}$. 2. 2.

The same coalescence momentum was sometimes assumed to describe both $pA\to\bar{d}$ and $pp\to\bar{d}$.

In what follows we give theoretical and empirical evidence, suggesting that both assumptions may be incorrect. To do this, we make an excursion into the physics of coalescence.

The role of the factor $B_{A}$ is to capture the probability for A nucleons produced in a collision to merge into a composite nucleus. It is natural for the merger probability to scale as Bond:1977fd ; Mekjian:1977ei ; Csernai:1986qf

[TABLE]

where $V$ is the characteristic volume of the hadronic emission region. A model of coalescence that realises the scaling of Eq. (8) was presented in Ref. Scheibl:1998tk . A key observation in Scheibl:1998tk is that the same hadronic emission volume is probed by Hanbury Brown-Twiss (HBT) two-particle correlation measurements Lisa:2005dd . Both HBT data and nuclear yield measurements are available for AA and pA systems, allowing a test of Eq. (8).

Ref. Scheibl:1998tk proposed the following formula for the coalescence factor,

[TABLE]

Here, $m_{t}$ is the transverse mass and $R_{i}$ are the $m_{t}$-dependent HBT scales characterising the collision. For concreteness we focus on $A=3$, but the treatment of $A=2$ is analogous. The quantity $\left\langle C_{3}\right\rangle$ expresses the finite support of the 3He wave function. It may be estimated via

[TABLE]

where $b_{3}\approx 1.75$ fm is the 3He nucleus size. For $p_{t}\lesssim 1$ GeV, setting $R_{i}\approx R$, we have

[TABLE]

The extension to deuterium, with nucleus size $b_{2}=3.2$ fm, is given by

[TABLE]

The coalescence factor in AA, pA, and pp collisions, presented w.r.t. HBT scale deduced for the same systems, is shown in Fig. 3. The data analysis entering into making the plot is summarised in App. A. The data is consistent with Eqs. (11) (bottom panel) and (12) (top panel), albeit with large uncertainty.

HBT data for pp collisions Uribe:1993tr ; Malinina:2013fhp ; Aamodt:2010jj suggest $R$ in the range $0.5-1.2$ fm, indicated by letters in both panels of Fig. 3. For $pp\to\bar{d}$, direct measurements from the ISR ALBROW1975189 ; 1973PhLB…46..265A ; Henning:1977mt give

[TABLE]

As seen in the top panel of Fig. 3, this result is consistent with the intersect of Eq. (12) with the specified range of $R$. (As done in Refs. Duperray:2005si ; Ibarra:2013qt ; Cirelli:2014qia , we discard here the high-$p_{t}$ data from Serpukhov Abramov:1986ti and only show it in Fig. 3 for completeness. Details can be found in App. A.)

For $pp\to\overline{\rm{}^{3}He}$ we do not have direct experimental information. We therefore extract a rough prediction of $B_{3}$, by taking the intersect of Eq. (11) with the two ends of the relevant range for $R$. This gives the following order of magnitude estimate:

[TABLE]

marked by the two horizontal dashed lines in the bottom panel of Fig. 3.

Results from the ALICE experiment allow us to make a preliminary test of Eq. (14). Ref. Sharma:2011ya reported 20 $\overline{\rm{}^{3}He}$ and 20 $\overline{\rm t}$ in the ALICE pp $\sqrt{s}=7$ TeV run, corresponding to luminosity $\mathcal{L}\approx 2.2$ nb*-1* with a pseudo-rapidity cut $|\eta|<0.9$ and with no further $p_{t}$ cut111We thank Natasha Sharma for clarifying the experimental procedure.. The $p_{t}$-dependent efficiency for $\overline{\rm{}^{3}He}$ detection was given in Adam:2015vda . In Fig. 4 we use these parameters to calculate the expected number of $\overline{\rm{}^{3}He}$ or $\overline{\rm t}$ events and compare with data. The result supports a coalescence factor $B^{(pp)}_{3}\approx(5-8)\times 10^{-4}$ GeV4, in agreement with Eq. (14). A dedicated analysis by the ALICE collaboration is highly motivated.

CR anti-helium. Two channels produce a final state $\overline{\rm{}^{3}He}$: direct $pp\to\overline{\rm{}^{3}He}$ and $pp\to\overline{\rm t}$ with subsequent decay $\overline{\rm t}\to\overline{\rm{}^{3}He}$. The first channel should suffer some Coulomb suppression with a Gamow factor that can be estimated by $f_{\rm coul}\sim e^{-\frac{\pi\alpha m_{p}}{p_{c}}}$. Eq. (14) suggests $p_{c}\sim 0.1-0.2$ GeV, leading to $f_{\rm coul}\sim 0.8-0.9$. This is supported by experimental results on the relative yield 3He/t Sharma:2011ya ; Sharma:2016vpz ; Adam:2015vda that are consistent with $f_{\rm coul}\sim 1$. (Ref. BUSSIERE19801 reported $\overline{\rm{}^{3}He}/\overline{\rm t}<1$; however, the 3He/t data from the same publication show an opposite trend, 3He/t $\geq 1$.) In what follows, for concreteness we focus on $pp\to\overline{\rm t}$ but we include a factor of 2 increased yield from the direct $pp\to\overline{\rm{}^{3}He}$ channel.

Combining Eq. (14) with the $pp\to\bar{p}$ production cross section222A 19% hyperon contribution to the $\bar{p}$ cross section Blum:2017iol is subtracted, assuming that coalescence feeds only on prompt $\bar{p}$ and neglecting the contribution from $\overline{{}^{3}_{\Lambda}{\rm H}}$ decay. 1983JPhG….9..227T ; 0305-4616-9-10-015 , we use Eq. (6) to obtain the differential cross section $E_{\overline{\rm t}}\frac{d\sigma_{pp\to\overline{\rm t}}}{d^{3}p_{\overline{\rm t}}}=\sigma_{pp}E_{\overline{\rm t}}\frac{dN_{\overline{\rm t}}}{d^{3}p_{\overline{\rm t}}}$, where $\sigma_{pp}$ is the total inelastic pp cross section Olive:2016xmw ; Fagundes:2012rr . The effective production cross section to be used in Eq. (2) is then

[TABLE]

where

[TABLE]

The final state rigidity and energy are related by $4\mathcal{R}^{2}=\epsilon^{2}-9m_{p}^{2}$. For the inelastic cross section of $\overline{\rm{}^{3}He}$, entering the loss term in Eq. (2), we use the $\bar{p}$ cross section 1983JPhG….9..227T multiplied by 3.

The resulting $\overline{\rm{}^{3}He}$ flux is plotted in Fig. 1. A pre-launch estimate of the 18-yr 95%CL $\overline{\rm{}^{3}He}$ flux upper limit accessible with AMS02 was given in Ref. kounineHebar in terms of the $\overline{\rm{}^{3}He}/$He flux ratio. In Fig. 1, in dashed line, we plot this expected upper limit sensitivity, scaled to 5-yr exposure and multiplied by the observed He flux Aguilar:2015ctt . We learn that AMS02 may indeed detect secondary $\overline{\rm{}^{3}He}$ in a 5-yr analysis. To further quantify this result, in Fig. 5 we show the Poisson probability for a 5-yr analysis of AMS02 to detect $N\geq 1,2,3,4$ $\overline{\rm{}^{3}He}$ events. The calculation assumes an $\overline{\rm{}^{3}He}$ analysis with the same exposure as the 5-yr $\bar{p}$ analysis of Aguilar:2016kjl .

CR anti-deuterium. The $\bar{d}$ analysis is analogous to that of $\overline{\rm{}^{3}He}$. The $\bar{d}$ flux is plotted in Fig. 1, for the range of $B_{2}$ given in Eq. (13). AMS02 5-yr 95%CL $\bar{d}$ flux sensitivity in the kinetic energy range 2.5-4.7 GeV/nuc, estimated in Ref. Aramaki:2015pii , is shown by solid lines.

Summary. We calculate the flux of secondary cosmic ray $\bar{p},\,\bar{d}$ and $\overline{\rm{}^{3}He}$. Propagation details are irrelevant to the calculation as long as consistent input data, notably B/C and proton flux, are used to calibrate it. The challenge is in deriving the correct production cross section in pp collisions, the dominant astrophysical source, for which accelerator data is scarce.

Using a scaling law of coalescence with HBT data we derive a novel estimate of the yield of $\overline{\rm{}^{3}He}$ in pp collisions. Our results are consistent with preliminary pp data from ALICE, motivating a dedicated analysis of $B^{(pp)}_{3}$ by the collaboration itself. Direct $\bar{d}$ data in pp collisions are also consistent with the HBT scaling.

Our prediction for the $pp\to\overline{\rm{}^{3}He}$ cross section is larger by 1-2 orders of magnitude compared to most previous estimates in the literature. The astrophysical secondary flux of $\overline{\rm{}^{3}He}$ is potentially within reach of a five-year exposure of AMS02.

Acknowledgments. We thank Ulrich W. Heinz and Yosef Nir for reading a draft version of this work, Urs Wiedemann for useful discussions and Andrei Kounine and Natasha Sharma for clarifying experimental details pertaining to AMS02 and ALICE analyses. This research is supported by the I-CORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number 1937/12). The work of MT is supported by the JSPS Research Fellowship for Young Scientists. KB is supported by grant 1507/16 from the Israel Science Foundation and is incumbent of the Dewey David Stone and Harry Levine career development chair.