T E Cocolios

A N Andreyev

S Antalic

A Barzakh

B Bastin

J Büscher

I G Darby

W Dexters

D V Fedorov

V N Fedosseev

K T Flanagan

S Franchoo

G Huber

M Huyse

M Keupers

U Köster

Yu Kudryavtsev

E Mané

B A Marsh

P Molkanov

R D Page

M.D Seliverstov

A M Sjoedin

I Stefan

J Van De Walle

P Van Duppen

M Venhart

S Zemlyanoy

Structure of 191 Pb from αand β-decay spectroscopy

Keywords: numbers: 23.20.Nx Internal conversion, 23.60.+e α decay, 27.80.+w 190 ≤ A ≤ 219, 29.38.-c Radioactive beams, 32.10.Fn Fine and hyperfine structure

Complementary studies of 191 Pb have been made in the β decay of 191 Bi at LISOL (CRC) and in the α decay of 195 Po at ISOLDE (CERN). Fine structures in the α decay of the low-spin and high-spin isomers of 195 Po have been fully resolved. Identification of the parent state is made possible via isomer selection based on narrowband laser frequency scanning. The α-particle and γ-ray energies have been determined with greater precision. New α-particle and γ-ray energies are identified. Branching ratios in the decay of 195 Po and 191 Pb have been examined.

Introduction

Shape coexistence is an important phenomenon in the region of neutron-deficient lead isotopes and extensive experimental and theoretical studies have been performed [START_REF] Heyde | Coexistence in odd-mass nuclei[END_REF][START_REF] Wood | Coexistence in even-mass nuclei[END_REF][START_REF] Oros | Shape coexistence in the light Po isotopes[END_REF][START_REF] Andreyev | A triplet of differently shaped spin-zero states in the atomic nucleus 186 Pb[END_REF][START_REF] Van Duppen | Shape coexistence around the Z = 82 closed shell probed by α-decay[END_REF][START_REF] Julin | Intruder states in very neutron-deficient Hg, Pb and Po nuclei[END_REF][START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF][START_REF] Andreyev | In-beam α-decay spectroscopy of 191 Po and evidence for triple shape coexistence at low energy in the daughter nucleus 187 Pb[END_REF][START_REF] Bender | The Z = 82 shell closire in neutron-deficient Pb isotopes[END_REF][START_REF] Niksić | Shape coexistence in the relativistic Hartree-Bogoliubov approach[END_REF][START_REF] Van De Vel | Fine structure in the α decay of 188,192 Po[END_REF][START_REF] Smirnova | Self-consistent approach to deformation of intruder states in neutron-deficient Pb and Po[END_REF][START_REF] Frank | Evolving shape coexistence in the lead isotopes: the geometry of configuration mixing in nuclei[END_REF][START_REF] Hellemans | Configuration mixing in the neutron-deficient 186-196 Pb isotopes[END_REF][START_REF] Grahn | Lifetimes of intruder states in 186 Pb, 188 Pb and 194 Po[END_REF]. With 109 neutrons, 191 Pb is located in the heart of this region. In the present study, we report on the α decay of 195 Po and on the β decay of 191 Bi, both parents of 191 Pb. The fine structure in the α decay of the low-spin isomer 195 Po ls and high-spin isomer 195 Po hs has already been studied in an earlier experiment using the gas-filled separator RITU [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF]. An intruder state was identified at 597 keV, suggested as possibly originating from shape coexistence. However, limitations in the production mechanism and measuring conditions prevented determining the conversion coefficient of the decay from that level. The observation of an E0 component in this decay would indicate that the two connected states are of the same spin and parity.

The ground-state and isomer properties of the polonium isotopes have been studied in a campaign of experiments at CERN ISOLDE (Run I in 2007 for 193-200,202,204 Po and Run II in 2009 for 191-192,195,196,201,203,206-211,216,218 Po) using in-source laser spectroscopy, in an approach similar to that followed for the neutron-deficient lead isotopes [START_REF] Witte | Nuclear charge radii of neutron deficient lead isotopes beyond N =104 mid-shell investigated by in-source laser spectroscopy[END_REF][START_REF] Seliverstov | Charge radii and magnetic moments of ofdd-A 183-189 Pb isotopes[END_REF]. This study allowed the fine structure in the α decay of 195 Po to be revisited. The relevant parts of the decay chain at mass 195 are shown in Fig. 1.

In a set of complementary experiments, the β decay of the neutron-deficient 191,193 Bi isotopes has been studied at the LISOL facility (CRC, Louvain-La-Neuve, Belgium) as part of a wider survey of the neutron-deficient bismuth isotopes [START_REF] Van Duppen | Observation of low-lying J π = 0 + states in the single-closed-shell nuclei 192-198 Pb[END_REF][START_REF] Coenen | α decay of neutron-deficient odd Bi nuclei: shell-model intruder states in Tl and Bi isotopes[END_REF][START_REF] Van Duppen | β + /electron-capture decay of 192,194,196,198,200 Bi: experimental evidence for low lying 0 + states[END_REF][START_REF] Griffin | Decay of mass-separated 3.0 min 195g Bi to levels in 195 Pb and shape coexistence in the neutron-deficient odd-mass Pb isotopes[END_REF][START_REF] Vanhorenbeeck | Highly converted transitions in 197 Pb: evidence for shape coexistence[END_REF]. Comparing the results from the two decay studies gives insight in the determination of the spin of several energy levels in 191 Pb. 195 Po at ISOLDE and evidence for shape coexistence in 191 Pb

Fine structure α decay of

Experimental procedure

The proton beam from the CERN PS-Booster (1.4 GeV, 1.4 µA on average) impinged upon a UC x target (50 g•cm -2 ). The proton pulses were separated by an integer number of periods of 1.2 s in a sequence referred to as the supercycle. Nuclei produced in the spallation reaction diffused out of the target matrix and effused to the high-temperature (≈ 2300 K) RILIS ion source cavity. The atoms were then irradiated with three different laser beams to excite the outer electron resonantly from the polonium atom beyond its ionisation potential and thus create a Po + ion [START_REF] Cocolios | Resonant laser ionization of polonium at rilis-isolde for the study of grounden isomer-state properties[END_REF]. The ions were then extracted from the ion source cavity, accelerated by DC electrical fields to an energy of 50 keV and mass separated in the dipole magnet of the ISOLDE General Purpose Separator. Note that elements with a low enough ionisation potential, such as thallium, may also be surface ionised. Isobaric contaminants may therefore be present in the mass-separated beam. Yields of a few ten thousands ions per second were delivered for both Note that the order and energy difference x and y between the low-and high-spin isomers in respectively 195 Po and 191 Pb are not known.

195 Po hs isomers, consistent with those published in Ref. [START_REF] Cocolios | Resonant laser ionization of polonium at rilis-isolde for the study of grounden isomer-state properties[END_REF]. The ions were implanted in one of ten carbon foils (20 µg•cm -2 ) mounted on a rotating wheel. In Run I, a single Si detector (active area 150 mm 2 , thickness 300 µm) was mounted behind the foil at the implantation position. During Run II, that same position was surrounded by two Si detectors, a circular detector behind the foil (back detector, active area 300 mm 2 , thickness 300 µm) and an annular detector in front of the foil (front detector, active area 450 mm 2 , thickness 300 µm) that let the ion beam pass through. The total solid angle coverage was improved from 33% to 66% of 4π. The Full Width at Half Maximum (FWHM) energy resolution of those detectors for α particles with E α = 5.5 MeV was 25 keV, 20 keV and 30 keV, respectively. A coaxial HPGe detector was placed behind the back Si detector outside the vacuum chamber for Run II. Its energy resolution (FWHM) was 4.3 keV at E γ = 1.3 MeV. The wheel was rotated synchronously with every second PS-Booster supercycle to remove the relatively long-lived bismuth and lead daughter activities.

The α-particle energy spectra are shown in Fig. 2, in which only α particles emitted in the decay of 195 Po and of the daughters 191 Pb (after α decay) and 195m Bi (after β + /EC decay) can be seen. The 194 Po peak in the spectrum originates from mass contamination of A = 194 in A = 195 due to the tail of the mass line of that isotope. No 195 Tl contribution can be directly observed in this spectrum as it is a pure β + /EC-decaying isotope. A very intense γ-ray transition at 384 keV, coming from the internal decay of 195m Tl [START_REF] Diamond | Isomeric levels in the light thallium isotopes[END_REF], is however observed in the γ-ray energy spectrum shown in Fig. 3.

Features of the α spectra

Several features are observed in the α-particle energy spectra in Fig. 2, in addition to the main peaks. Let's consider first the low-energy tails observed on the α peaks, which are most visible on the two main transitions at E α = 6606 keV and E α = 6699 keV. These two tails are labelled a and b in the second spectrum and can be explained by the energy loss of the α particles emitted through the carbon foil. Calculations indicate that at a beam energy of 50 keV, the polonium isotopes are implanted at a depth of 25 nm [START_REF] Ziegler | SRIM-2003[END_REF] compared to the 90 nm thickness of the foil. This tail has a cut-off due to the limited angular range covered by the detectors and allowed by the foil mount geometry, and is more prononced in the back detector.

A high-energy tail is also present and is attributed to the random summing of the energy of an α particle from the decay of 195 Po with that of a positron from the β decay of 195 Tl. The maximum β summing for the two main peaks in Fig. 2, corresponding to the energy loss of a β particle in a Si detector, is indicated by d and e.

Another feature of the α-particle energy spectrum, labelled c in Fig. 2, is a broad shoulder on top of the high-energy tail from β summing. This extra shoulder has been identified through α-γ coincidences. Fig. 4 shows the energy of the α particles as a function of the α-γ time difference ∆t for events in coincidence with the 384 keV γ-ray transition from the internal decay of 195m Tl (see inset in Fig. 3). The two broad bands at an α energy of 6606 keV and 6699 keV are due to random coincidences with the 384 keV transition. Both bands exhibit a side structure that seems to vanish at ∆t ≈ 1200 ns. This is attributable to the true coincidence between the 384 keV γ ray and the strongly-converted (α tot = 161 [START_REF] Zh | Nuclear data sheets for A = 195[END_REF]) transition at 99 keV in the decay of 195m Tl. The energy of the electron emitted by the latter sums with the energy of the α particles randomly. The observed time dependence of the side structure is due to the integration time of the amplifier. For this setup, complete integration is achieved in 1000 ns, beyond which partial summing occurs.

Fine-structure α decay and branching ratio determination

The fine-structure decays of 195 Po have been investigated using the α-γ coincidence data. The γ-ray energy spectra in prompt coincidence with the α-decay peaks of 195 Po ls,hs identified around E α = 6000 keV and E α = 6400 keV are shown in Fig. 5. Two previously observed transitions at 597.2(5) keV and 669.6(5) keV [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF] can be seen. Additional transitions at 143(1), 214.8(5), 383(1), 427(1) and 483 [START_REF] Heyde | Coexistence in odd-mass nuclei[END_REF] keV are identified for the first time. The energy of the transitions at 214.8 and 383 keV add up to 597.8 [START_REF] Smirnova | Self-consistent approach to deformation of intruder states in neutron-deficient Pb and Po[END_REF] keV and are therefore consistent with a cascade that decays from the 597.2 keV level. The energy of the α particles populating each state can be determined by gating on the γ-ray transitions separately to produce γ-gated α-particle energy spectra, shown in Fig. 6. The two known α decays around 6050 keV can be cleanly identified as [START_REF] Heyde | Coexistence in odd-mass nuclei[END_REF] an α decay with energy E α = 6047 [START_REF] Van Duppen | Shape coexistence around the Z = 82 closed shell probed by α-decay[END_REF] transition and (2) an α decay with energy E α = 6027(5) keV in coincidence with the 597.2 keV γ-ray transition. This 6027 keV α decay is also observed in coincidence with the 383 keV and 214.8 keV γ-ray transitions. Also in coincidence with the 214.8 keV γ-ray transition are α decays at energies 5985 [START_REF] Niksić | Shape coexistence in the relativistic Hartree-Bogoliubov approach[END_REF] keV, 6047 keV and 6399 [START_REF] Niksić | Shape coexistence in the relativistic Hartree-Bogoliubov approach[END_REF] keV. The α particles at energy 5985 and 6399 keV are observed for the first time in the α decay of 195 Po. In the spectrum of the 427 keV γ transition, the new α-decay peak at an energy of 5985 keV is also observed and suggests therefore a second cascade with the transitions at 214.8 and 427 keV. The new energy level populated by this α decay is then at 642(1) keV. The presence of the 6047 keV α-decay peak in coincidence with the 214.8 keV transition shows that there is a possible connection between the highspin and the low-spin structures. The two γ rays with energy 143 keV and 483 keV, showing a coincidence with the fine-structure α-decay at 6047 keV from the decay of the high-spin isomer 195 Po hs , are possible candidates for this connection. However, since no γ -γ coincidences are available, those transitions remain unplaced in the level scheme of 191 Pb. The assignment of the different components to the decay of the low-or high-spin isomer of 195 Po has been discussed in Ref. [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF] in terms of the Q α value, from which it was concluded that the 6027 -597.2 keV decay comes from the low-spin 195 Po ls isomer while the 6047 -669.6 keV decay comes from the high-spin 195 Po hs isomer. The same technique confirms that the new decays 5985 -427 -214.8 keV and 6399 -214.8 keV arise from the decay of the low-spin 195 Po ls isomer.

Using the different atomic hyperfine profiles of the two nuclear states in 195 Po, it is also possible to enhance the production of one of the isomers over the other. The use of such isomeric beams has already been demonstrated for other elements [START_REF] Andreyev | Nuclear spins, magnetic moments and α-decay spectroscopy of long-lived isomeric states in 185 Pb[END_REF]. The hyperfine spectra for both nuclear states using the second transition of the ionisation scheme at 843.38 nm, as described in Ref. [START_REF] Cocolios | Resonant laser ionization of polonium at rilis-isolde for the study of grounden isomer-state properties[END_REF], is shown in Fig. 7. The regions enhancing the production of each isomer with respect to the other are highlighted. The backgroundsubstracted α-particle energy spectra for each laser frequency range are selectively combined to create isomerically-pure α-particle energy spectra, shown in Fig. 8. The fine-structure α particles can then be unambiguously assigned to the decay of one or the other isomer of 195 Po. Note however that other isotopes of polonium can be identified in those spectra due to the poorer mass resolution in Run I (as compared to Run II) and to their dependence on the laser frequency. The transitions, their assignments, energies and relative intensities are summarised in Table 1.

Combining the relative intensities with the partial information available in Ref. [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF], we can extract the conversion coefficient of the transition at 597.2 keV in two ways. First, the relative intensity of the 6027 keV α-decay of 195 Po ls , I α = 0.34(3)%, measured in this study, can be compared with the known contribution from the α-γ chain, I αγ = 0.17(5)%, given in Ref. [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF], yielding a conversion coefficient α = 0.9 [START_REF] Julin | Intruder states in very neutron-deficient Hg, Pb and Po nuclei[END_REF]. Alternatively, we can rescale the known value of α = 0.8(3) for the 669.6 keV transition in 191 Pb hs from Ref. [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF] by comparing the α branching ratios and γ intensities in the decay of 195 Po ls,hs , yielding a value of α = 0.5 [START_REF] Oros | Shape coexistence in the light Po isotopes[END_REF]. In both cases, the influence of the 382.8 keV transition from that level is also taken into account. Note that the coincidence between the α decays at 6027 keV and 6047 keV with the lead x rays 4) 63 [START_REF] Ziegler | SRIM-2003[END_REF] [28] 191 Pb ls 0.051(5) 0.013(5) [START_REF] Hornshøj | Widths for sand d-wave α-decay of neutron-deficient isotopes with Z ≤ 82[END_REF] (Fig. 6 viii) confirms that the transitions at 597.2 and 669.6 keV have a significant conversion coefficient. From the weighted average between those two independent estimates, a conversion coefficient of 0.6(3) is found compared to the calculated values α tot (E2) = 0.019, α tot (M1) = 0.068 and α tot (E1) = 0.007. From this we conclude that the 597.2 keV transition has a large E0 contribution, confirming the matching spin and parity assignments for the low-spin isomer and the 597.2 keV level in 191 Pb. Based on the spin assignment, this level is a good candidate for a state with a different shape than the low-spin isomer, as suggested in Ref. [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF]. Moreover, such a large E0 component to its decay is only possible if the two connected levels have similar configurations and indicates that the two shape-coexisting states are also considerably mixed [START_REF] Wood | Coexistence in even-mass nuclei[END_REF]. A quantitative determination of the amount of mixing cannot be based on this result alone and requires additional experimental information (e.g. the excited state lifetime).

We now consider the α-decay branching ratios of 195 Po ls and 191 Pb ls . Accounting for the different lifetimes and the fraction of the implants that recoil out of the carbon foil after emitting an α particle [START_REF] Wauters | The influence of recoil losses in subsequent α decay on the determination of α-branching ratios[END_REF], accurate branching ratios for the low-spin states can be extracted. The α branching ratio for 191 Pb ls is found to be b α = 0.051(5)%.The beta-decay branch of 195 Po ls can be estimated by looking at the α decay of its daughter nucleus 195m Bi. Although the precision on its branching ratios is limited [START_REF] Coenen | α decay of neutron-deficient odd Bi nuclei: shell-model intruder states in Tl and Bi isotopes[END_REF], the fraction of 195 Po ls that β-decays is small enough that a good accuracy for the b α ( 195 Po) may still be achieved. A value of b α = 94(4)% is found, assuming that the β decay of 195 Po ls only populates the I = 1 2 + state in 195 Bi. Those values are consistent with the previous literature values [START_REF] Wauters | Alpha decay properties of neutron-deficient polonium and radon nuclei[END_REF][START_REF] Hornshøj | Widths for sand d-wave α-decay of neutron-deficient isotopes with Z ≤ 82[END_REF], as shown in Table 2, but are more precise thanks to larger statistics. No new branching ratios could be determined in the decay of 195 Po hs or 191 Pb hs as the α decays of the high-spin 195 Bi and 191 Pb hs are not observed.

Using the formalism of Rasmussen [START_REF] Rasmussen | Alpha-decay barrier penetrabilities with an exponential nuclear potential: eveneven nuclei[END_REF], Hindrance Factors (HF) in the decay of 195 Po with respect to 194,196 Po [START_REF] Wauters | Alpha decay properties of neutron-deficient polonium and radon nuclei[END_REF], assuming no change in angular momentum (∆l = 0), are calculated and given in Table 1. Most of the HF values in the main component and in the fine structure decay of 195 Po ls and 195 Po hs are low (1 -3). This means that those α decays are unhindered and that the spin and parity of the mother and daughter states are the same. The conclusions presented in Ref. [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF] are thus all confirmed. The high HF value (465(114)) of the 6399 keV α decay, however, indicates a change in spin or configuration between 195 Po ls and the 214.8 keV excited level in 191 Pb ls . In the case of the 5985 keV α decay, the slightly higher HF value (12(2)) points to a different nature of the 642 keV level in 191 Pb ls compared to the 195 Po ls isomer but nonetheless with a likely spin assignment ( 3 2 -). Finally, the HF in the decay of 191 Pb ls with respect to 190,192 Pb [START_REF] Wauters | The alpha-branching ratios of the 188,190,192 Pb isotopes[END_REF] shows also a small value of 0.46 [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF], consistent with an unhindered decay to the I π = 3 2 (-) isomer 187m Hg [START_REF] Bonn | Optical pumping of neutron deficient 187 Hg[END_REF]. This offers an experimental confirmation, according to the ∆l = 0 α-decay strong rule, that the spin assignment of the 191 Pb ls isomer is indeed I π = 3 2 (-) , and thus similarly for the excited state at 597.2 keV and for the 195 Po ls isomer. Similarly, by combining the spin assignment 13 2 (+) in 191 Pb hs [START_REF] Dutta | Measurement of the isotope shift and hyperfine splitting of 190,191,193,197 Pb isotopes by collinear laser spectroscopy[END_REF] with the identification of an E0 component in the decay of the 669.6 keV level [START_REF] Van De Vel | Identification of low-lying proton-based intruder states in 189-193 Pb[END_REF], a spin assignment 13 2 (+) is confirmed for that excited state. The spin assignment 13 2 (+) of the high-spin 195 Po hs isomer is also confirmed through the low HF (1.6(1)/2.1(1)). 191,193 Bi in LISOL and migration of the ν2f 5/2 single-particle energy level in neutron-deficient lead isotopes

β decay of

The α and β decays of the neutron-deficient 192-196 Bi isotopes were studied at the LISOL facility (CRC, Louvain-La-Neuve, Belgium) in the 1980s [START_REF] Van Duppen | Observation of low-lying J π = 0 + states in the single-closed-shell nuclei 192-198 Pb[END_REF][START_REF] Coenen | α decay of neutron-deficient odd Bi nuclei: shell-model intruder states in Tl and Bi isotopes[END_REF][START_REF] Van Duppen | β + /electron-capture decay of 192,194,196,198,200 Bi: experimental evidence for low lying 0 + states[END_REF][START_REF] Griffin | Decay of mass-separated 3.0 min 195g Bi to levels in 195 Pb and shape coexistence in the neutron-deficient odd-mass Pb isotopes[END_REF][START_REF] Vanhorenbeeck | Highly converted transitions in 197 Pb: evidence for shape coexistence[END_REF]. The radioactive nuclei were produced in fusion-evaporation reactions using 14 N, 16 O and 20 Ne beams on natural Ir (37.3% 191 Ir, 62.7% 193 Ir), natural Re (37.4% 185 Re, 62.6% 187 Re) and 181 Ta targets, respectively. The radioactive recoils were subsequently ionised in a plasma ion source, mass separated and implanted in an aluminised mylar tape. Single γ-ray energy spectra were recorded with two coaxial HPGe detectors.

The lists of observed γ-ray energies in the β decay of 191,193 Bi are given in Tables 3 and4. For 193 Bi, relative efficiencies I γ are also given. By matching the summed energies of several γ-ray transitions with observed γ-ray energies, possible cross-over transitions in the decay of some excited levels are proposed. Note that true summing in the γ detectors alone cannot explain the observed relative intensities and that those transitions are therefore real.

From the study of the decay of the isotopes 195,197 Bi [START_REF] Griffin | Decay of mass-separated 3.0 min 195g Bi to levels in 195 Pb and shape coexistence in the neutron-deficient odd-mass Pb isotopes[END_REF][START_REF] Vanhorenbeeck | Highly converted transitions in 197 Pb: evidence for shape coexistence[END_REF], it has been observed that the most intense transition leading to the low-spin isomer in the lead daughter isotope is the decay of the 5 2 -excited state to the 3 2 -state and was interpreted as the transition between the ν2f 5/2 and ν3p 3/2 quasiparticle states. By comparing the relative intensities of the different transitions presented in Table 4, the 174.5 keV level in 193 Pb is therefore a good candidate for a similar configuration, dominated by a quasiparticle excitation in the ν2f 5/2 level. In the case of 197 Pb [START_REF] Vanhorenbeeck | Highly converted transitions in 197 Pb: evidence for shape coexistence[END_REF], a calculation using a surface delta interaction [START_REF] Arvieu | Analysis of the properties of the odd-mass single closed shell nuclei with a surface delta interaction[END_REF] concluded that these levels had actually almost 100% one-quasiparticle character. A similar behaviour might be expected for 193 Pb.

In 191 Pb, the 214.8 keV level is populated both by the β decay of the high-spin I π = ( β can yet assert for the purity of the wavefunctions of these levels far from stability, the single-particle energy levels cannot be directly deduced from these systematics. Note however that they mostly arise from a single neutron particle or hole in the ν3p 1/2 , ν3p 3/2 , ν2f 5/2 and ν1i 13/2 orbitals.

Conclusion

In conclusion, using resonant laser ionisation, a new study of the α decay of 195 Po was made. α-particle and γ-ray energies in the α decay of the low-spin and highspin isomers, branching ratios and a conversion coefficient have been extracted with better precision. Hindrance Factor values for the α decay confirm the spin assignments + in neutron-deficient odd-A Pb isotopes. These levels are dominated by one-quasiparticle excitations of the ν3p 1/2 , ν3p 3/2 , ν2f 5/2 , and ν1i 13/2 orbitals, respectively. This extends the previous systematics presented in Ref. [START_REF] Griffin | Decay of mass-separated 3.0 min 195g Bi to levels in 195 Pb and shape coexistence in the neutron-deficient odd-mass Pb isotopes[END_REF]. Note that the order of the lowest energy states (

I π = 3 2 (-)

Figure 2 .

 2 Figure 2. α-particle energy spectra measured at mass number 195 while ionising polonium (Run II) using the front detector (top) and the back detector (bottom) over a period of 4 hours. The peaks are labelled in the top spectrum with their assignments. The low-and high-energy tails of the two main α lines, indicated by doted arrows in the bottom spectrum, are labelled for the discussion in the text.

Figure 3 .

 3 Figure 3. γ-ray energy spectrum measured at mass number 195 while ionising polonium (Run II), in coincidence with any decay event in the Si detectors. The γ rays that are not labelled are all attributed to the β decay of 195g Tl. Inset: decay scheme of 195m Tl [24].

Figure 4 .

 4 Figure 4. (Color online) Energy of the α particles as a function of the α-γ time difference ∆t for events in coincidence with the γ-ray transition at 384 keV from the internal decay of 195m Tl (Run II). The y axis shows the energy recorded in the Si detector while the x axis shows the time difference between the Si detector and the Ge detector.

Figure 5 .

 5 Figure 5. From top to bottom: background-and random-subtracted γ-ray energy spectra in coincidence with the α particles with energy around 6399 keV (top), 6047 keV, 6027 keV or 5985 keV (bottom) from the fine structures in the decay of 195 Po (Run II). The γ rays are labelled according to their energy in keV.

Figure 6 .

 6 Figure6. (i-viii) Background-and random-subtracted γ-gated α-particle energy spectra (Run II). The α-particle energies are labelled in keV. The intensity fluctuations around 6.6-6.7 MeV arise from the very intense α decays at 6606 and 6699 keV and do not represent true α-γ coincidences. (v) The summed α-electron energies are observed due to the proximity of the 384 keV γ-ray from the internal decay of 195m Tl. (viii) Similarly, the highly-converted 99 keV transition from the internal decay of 195m Tl results in coincidences between x rays and the summed α-electron events.

2 Figure 7 .

 27 Figure 7. (Color online) Hyperfine structure of 195 Po ls (full triangles, black line) and 195 Po hs (open triangles, grey line) using the atomic transition at 843.38 nm from the ionisation scheme of polonium (Run I). The range used to study 195 Po ls is limited by the red dot-dashed lines while those for 195 Po hs are limited by the blue dashed lines. The background (off-resonance) range is taken on the far right (Frequency > 4.5 GHz) of the spectrum.

Figure 8 .

 8 Figure 8. (Color online) Background-substracted isomerically-purified α-particle energy spectra of 195 Po ls (red-bottom) and 195 Po hs (blue-top) using the ranges indicated in Fig. 7 (Run I). The α-decay peaks from 195 Po are labelled according to their energy in keV while the peaks not directly associated with the decay of 195 Po are labelled according to the decaying isotope. The α particle with energy 6399 keV cannot be seen in this spectrum because it lies under the low-energy tail of the more intense 6606 keV α-particle energy peak.

9 2- 2 ( 2 - 2 -, 3 2 -, 5 2 -, and 13 2 +

 9222222 ) state in191 Bi isotope and in the α decay of the low-spin I π = 3 -) state in 195 Po ls isotope. Considering the high HF in the α decay, only spin assignments of I = 5 2 or 7 2 are possible. Based on the systematics of the low-excitation energy levels in the neighbouring odd-A Pb isotopes, a spin assignment (5) is proposed, corresponding similarly as before to a mostly one-quasiparticle ν2f 5/2 level. The systematics of the lowest energy levels with spin 1 are shown in Fig.9. Since no data

Figure 9 .

 9 Figure 9. Systematics of the lowest-lying energy levels with spin and parity 1 2 -,

3

 3

2 (-) , 13 2 (

 22 +) ) in191 Pb and their relative excitation energy y are not known.

  195 Po ls and

								3 2	(-)	0
								195 Po ls 4.64 s
	13 2	(+)	y + 669.7		( 3 2	-)	642	5985 keV, HF=12
					3	(-)	597.2	6027 keV, HF=2.0
					2		427
								383
		669.7		( 5 2	-)	597.2 214.8	6399 keV,
								HF=465
	13 2	(+)	y		3 2	(-)	0 214.8	6606 keV,
	2.18 min 191 Pb hs			191 Pb ls 1.33 min	HF=1.6
			3 2	(-)	0		5310 keV,
				187 Hg			HF=0.46
				2.4 min

Table 1 .

 1 Properties of the fine-structure decays of 195 Po ls and 195 Po hs : α-particle energy E α and relative intensity I α , Hindrance Factor with respect to the ground-stateto-ground-state α decay of the neighbouring even-A nuclei HF, energy of the excited state in the daughter nucleus E * and conversion coefficient α of the direct decay of that excited level to the isomer in 195 Po.

	Isotope E α [keV]	I α [%]	HF	E * [keV]	α
	195 Po ls 6606(5) [28] 99.56(2)	1.6(1)	0
	6399(10)	0.054(12) 465(114) 214.8(5)
	6027(5)	0.34(3)	2.0(2)	597.2(5)	0.6(3)
	5985(10)	0.036(3)	12(2)	642(1)
	195 Po hs 6699(5) [28] 99.83(1)	1.6(1)	0
	6047(5)	0.17(1)	2.1(2)	669.6(5) 0.8(3) [7]

Table 2 .

 2 Branching ratios in the decay of 195 Po ls and 191 Pb ls from this work and the literature.

	Isotope b α [%]	b lit α [%]
	195 Po ls	94(

Table 3 .

 3 List of γ-ray energies E γ and relative intensities I ls,hs γ in the α decay of 195 Po ls,hs , respectively, observed in the excited structure of 191 Pb from the α decay of 195 Po and the β decay of 191 Bi.

	Isotope E γ [keV] I ls γ [%] I hs γ [%] Origin Coincident γ-ray
	191 Pb 143(1)	3.5(10)	α, β
	214.8(5)		α, β	383, 427
	383(1)	9(2)	α	214.8
	427(1)	7(2)	α	214.8
	483(1)	13(3)	α
	597.2(5)	100	α
	669.6(5)	100	α, β
	708.26		β
	820.2		β
	954.7		β
	1082.3		β
	1117.71

  for the low-spin isomers 191 Pb ls and 195 Po ls . Using the fine structure decay of 195 Po ls , the conversion coefficient for the decay of the 597.2 keV energy level to the lowest energy state in 191 Pb ls is measured for the first time, confirming the spin assignment I π = 3 level a good candidate for shape-coexisting state as observed in the neighbouring Pb isotopes. A new level is found at 214.8 keV and is also observed in the β decay of 191 Bi, allowing a spin assignment of I π = ( 5 2 -). It is a good candidate for the onequasiparticle excitations ν2f 5/2 level and extends the systematics of low-lying energy levels in the neutron-deficient lead isotopes down to 191 Pb. Finally, a new level is also found at 642 keV. A tentative spin assignment I

	2	(-) . The large value of the conversion coefficient confirms the presence
	2 of a strong E0 component in the 597.2 keV transition and makes the 597.2 keV 3	(-)

π = ( 3 2 -) is proposed based on the evaluated HF.

Table 4 .

 4 List of γ-ray energies E γ and relative intensities I γ with respect to the 174.5 keV transition observed in the excited structure of 193 Pb from the β decay of 193 Bi. The proposed cross-over transitions are based on summed γ-ray energies matching an observed γ-ray energy with too large relative intensity to be attributed to summing effects in the detector. 191 Pb 193 Pb 195 Pb 197 Pb 199 Pb 201 Pb 203 Pb 205 Pb 207 Pb

	Isotope E γ [keV] I γ [%] Possible cross-over
	193 Pb	174.5	100
		196.8	5.4
		290.6	7.8
		320.1	7.7
		354	8.7
		505.9	5.2
		554.2	38
		621.2	9.2
		681.1	48
		687.2	12.4
		711.1	48.8
		739.1	13.5
		750.1	6.3	196.8 + 554.2
		818.5	14.2	196.8 + 621.2
		861.8	20	174.5 + 687.6
		873.9	29.4	320.1 + 554.2
		995.7	23.8
		1022.3	12.8
		1049.1	9.9	174.5 + 873.9
		1116.1	8.4
		1124.7	5.2
		1171.6	10.1	174.5 + 995.7
				354 + 818.5
		1630.6	0.4	505.9 + 1124.7

Acknowledgments

We would like to thank the ISOLDE collaboration and the CRC team for providing excellent beams, and the GSI Target group for making the carbon foils. This work was supported by FWO-Vlaanderen (Belgium), GOA/2004/03 (BOF-K.U.Leuven), by the IUAP -Belgian State Belgian Science Policy -(BriX network P6/23), by the European Commission within the Sixth Framework Programme through I3-EURONS (Contract RII3-CT-2004-506065), by the U.K. Science and Technology Facilities Council, and by the Slovak grant agency VEGA (Contract No. 1/0091/10).


Néstor Armesto
email: nestor.armesto@usc.es

Matteo Cacciari
email: cacciari@lpthe.jussieu.fr

Tetsufumi Hirano
email: hirano@phys.s.u-tokyo.ac.jp

James L Nagle
email: jamie.nagle@colorado.edu

Carlos A Salgado
email: carlos.salgado@usc.es

Constraint fitting of experimental data with a jet quenching model embedded in a hydrodynamical bulk medium

Keywords:

We present a global fit to single-and double-inclusive suppression data of high-p T particles in central Au+Au collisions at top RHIC energy. We also include in this analysis data on heavy quarks via their D and B meson semi-leptonic decays (i.e. non-photonic electrons). The analysis is based on the parton quenching weights for medium-induced gluon radiation computed in the BDMPS approximation then embedded in a hydrodynamical description of the bulk medium. Our results indicate that values of the transport coefficient q more than four times larger than perturbative estimates are preferred by experimental data. This confirms previous calculations based on simpler implementations of the medium geometry or only the single-inclusive suppression. We also comment on the statistical compatibility of the heavy quark data within a radiative only energy loss scenario, and on the sensitivity of the results to nuclear modification of the parton distribution functions (PDFs) and to assumptions on the energy loss during times in the collision prior to the hydrodynamical behavior.

Introduction

The experimental program at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) has established the jet quenching phenomenon as a fundamental tool in the study of hot matter in heavy-ion collisions [1] -see [2] for reports on the subject. A variety of observables in the large transverse momentum region of the produced particle spectrum is contained under this generic name: the suppression of the single-inclusive yields of both light hadrons and nonphotonic electrons; the suppression of the double-inclusive azimuthal correlations; direct photonhadron correlations; the effects on identified jets, jet shapes; etc. A subset of these observables are also available, though with different expected magnitude, at the Super Proton Synchrotron (SPS) at CERN [3], and all of these observables will play a major role in the upcoming heavy ion program at the Large Hadron Collider (LHC) at CERN.

Another essential discovery at RHIC is the good description of the data, at small and moderate transverse momentum, achieved by a hydrodynamical modeling of the matter produced in heavy-ion collisions [1]. In particular, azimuthal asymmetries compatible with a hydrodynamical flow starting at very early times and with negligible viscosity have been measured [START_REF] Huovinen | [END_REF].

Although there is no general consensus, several jet quenching studies [5,6,7,8,9] require an interaction of partons with the medium which exceeds that estimated from the asymptoticaly large temperature behavior of perturbative Quantum Chromodynamics (pQCD) [10,11,12,13].

Corrections to these estimates (arising from higher orders in perturbation theory) are not known for quantities as e.g. q but indications exist that they could be sizable [14]. It is also worth recalling here that the relevant scale in the perturbative expansion, both in the medium and in the vacuum, is the transverse momentum of the radiated gluon. For an opaque medium case, this quantity acquires additive contributions from multiple soft scatterings which can be described in terms of light-like Wilson lines [2,15]. The averages over medium configurations of these non-perturbative quantities are usually encoded in the single parameter q. Subleading terms in the high-energy approximation involved in calculations of radiative energy loss [2] are not yet known. Thus, the conclusion of the large strength remains, but caution is warranted on the precise interpretation of the actual magnitude obtained.

These findings about the large partonic interaction strength in jet quenching and the hydrodynamical behavior of the bulk matter have led to the claim that the produced medium is strongly coupled [16]. Such a conclusion is of critical importance for our understanding of QCD. Therefore it must be substantiated by detailed studies which should consider: i) As many experimental observables as available; ii) A detailed modeling of the medium compatible with experimental data on soft particle production; iii) A statistical analysis of the uncertainties in the constraints on the medium coming from both the experimental data and the theoretical implementation of energy loss and medium modeling.

The goal of the present paper is to constrain the properties of the medium which can be studied through the description of its effects on high-p T particles. The medium is described by a hydrodynamical model adjusted to fit experimental data on the soft part of the spectrum. The modeling of the energy loss, considered as purely due to medium-induced gluon radiation, is that proposed some years ago [17,18,19,20] and used in several phenomenological studies of RHIC data [5,6,9,21]. The coupling of a hydrodynamical medium description to an energy loss calculation has already several antecedents [22,23,24,25,7,8,9]. Fits to experimental data using χ 2 -minimization of statistical uncertainties for single-and double-inclusive data were presented in [26], and including statistical and systematic uncertainties in [27,21].

In this work, we extend previous studies by both including a hydrodynamical modeling of the medium and considering the case of inclusive pion, dihadron and heavy quark suppression within the same analysis. The energy loss distributions will be taken from [20] for light quarks and gluons and from [28] for heavy quarks. The observables related to direct photons-hadron correlations and identified jets have not been accessible at present energies until recently [29,30]. However, additional statistics and refinements in the experimental analysis, and also on the theoretical implementation -in particular, the development of a Monte-Carlo approach to jet medium-modifications [31], are required before physics conclusions can be drawn. For this reason they will not be considered here.

The plan of the paper is as follows: In Section 2 the hydrodynamical model of the medium will be briefly described (with full details provided in [32,33]). In Section 3 the pQCD formalism used to compute single-and double-inclusive particle production will be outlined. In Section 4 the implementation of energy loss will be reviewed. Finally, our results will be shown and discussed in Section 5, and the conclusions presented in Section 6.

Hydrodynamical medium modeling

The energy loss of an energetic parton is determined by the properties of the medium along the parton trajectory [2]. In relativistic heavy-ion collisions, the medium itself evolves dynamically together with the propagation of high-p T partons. Full three-dimensional (3D) ideal hydrodynamics [32,33] is employed in our calculations to describe the space-time evolution of the hot matter created in Au+Au collisions at top RHIC energy ( √ s NN = 200 GeV).

We solve the equations of energy-momentum conservation

∂ µ T µν = 0 (2.1)
in full 3D space (τ, x, y, η s ) under the assumption that local thermal equilibrium is reached at an initial time τ 0 =0.6 fm/c and maintained thereafter until freeze-out. Here τ , η s , x and y are proper time, space-time rapidity, and two transverse coordinates perpendicular to the beam axis, respectively. Ideal hydrodynamics is characterized by the following energy-momentum tensor:

T µν = ( + P )u µ u ν -P g µν , ( 2.2)
where , P and u µ are energy density, pressure and local four velocity, respectively. We neglect the finite net-baryon density which is small near mid-rapidity at RHIC where all calculations will be performed. For the high temperature (T > T c = 170 MeV) Quark-Gluon Plasma (QGP) phase we use the equation of state (EOS) of an ultra-relativistic massless parton gas (u, d, s quarks and gluons) with a bag pressure B:

p = 1 3 ( -4B). (2.3)
The bag constant is tuned to be B 1 4 = 247 MeV to match the pressure of the QGP phase to that of a hadron resonance gas at critical temperature T c = 170 MeV. On the other hand, a hadron resonance gas model including all hadrons up to the mass of the ∆(1232) is employed for the low temperature phase (T < T c ). The hadron resonance gas EOS employed in this study implements chemical freeze-out at T ch = 170 MeV [33] for simultaneous reproduction of particle ratios and spectra at low p T . We assume that the hadronic matter behaves like a perfect fluid all the way down to the kinetic decoupling at T dec = 100 MeV.

The initial entropy distribution in the transverse plane is proportional to a linear combination of the number density of participants, ρ part , and that of binary collisions, ρ coll [34] as given below:

s ∝ (1 -x)ρ part + xρ coll .
(2.4)

The proportionality constant and the fraction of soft and hard components (specified by x = 0.15) are so chosen that the centrality dependence of charged particle multiplicity observed by PHOBOS [35] is reproduced [34].

For the initial condition of the flow velocity, Bjorken's scaling solution [36], namely u x (τ 0 ) = u y (τ 0 ) = u ηs (τ 0 ) = 0, is employed. The hydrodynamic model has been tested against experimental data at RHIC energies such as centrality and p T dependencies of spectra for pions, kaons and protons up to 1.5 GeV/c [23,25,34]. We will use these hydrodynamic solutions, in particular the local energy density along the trajectory of partons, to fit various high-p T data and to reliably constrain the transport coefficient q. The hydrodynamic results such as temperature and energy density as functions of the space-time position are publicly available [37].

Perturbative cross sections

The production of a hadron h at transverse momentum p T and rapidity y can be described by the factorized expression

dσ AA→h dp T dy = dx 2 x 2 dz z i,j x 1 f A i (x 1 , Q 2 )x 2 f A j (x 2 , Q 2 ) dσ ij→kl d t D k→h (z, µ 2 F ), (3.1)
where f A i (x, Q 2 ) are the parton distribution functions (PDFs) for a colliding object, either a proton or a nucleus A, x is the momentum fraction of the hadron carried by the parton, and D k→h (z, µ 2 F ) are the fragmentation functions for a parton k into a hadron h, with z the momentum fraction of the parton carried by the hadron. We utilize CTEQ6L and CTEQ6M [38] free proton parton densities depending on whether the calculation is at leading order (LO) or next-to-leading order (NLO). We take the factorization and renormalization scales as Q 2 = (p T /z) 2 and the fragmentation scale as µ F = p T for light meson production, see e.g. [39]. For the case of heavy quarks, we set all scales equal to

Q 2 = µ 2 F = M 2 + p 2
T for consistency with the comparison to experimental data in the proton-proton case [40]. When the fragmentation takes place in a medium, the corresponding medium-modified fragmentation function is modeled as [41]

D med k→h (z, µ 2 F ) = dζ 1 -ζ P (ζ)D vac k→h z 1 -ζ , µ 2 F , ( 3.2)
where P (ζ) is the probability distribution of a fractional energy loss, ζ = ∆E/E, of the fast parton in the medium [42] and the vacuum fragmentation function, D vac k→h , is taken from a known set, in this case from de Florian, Sassot and Stratmann [43]. Eq. (3.2) assumes that no modification of the virtuality evolution takes place in the medium, the only effect being the energy loss of the leading parton when traveling through the medium. This assumption has been studied recently [44], mainly motivated by the need of a more sophisticated approach for more exclusive observables. However, the fact that the medium-induced gluon radiation does not present collinear or soft divergences, due to formation time effects, indicates that the corrections to (3.2) are O(1/p T ).

In the case of proton-proton collisions, the description of the data using Eq. (3.1) is good even at lowest order (LO, with appropriate K-factors) for light hadrons produced at central rapidities [39].

For particles containing heavy quarks, the Fixed-Order plus Next-to-Leading-Log (FONLL) approach [45] is used to resum the potentially large log(M Q /p T ) terms, with M Q the mass of the heavy quark. This scheme leads to a good description of the Tevatron data [46,47,48]. At RHIC, both the PHENIX and the STAR Collaboration have measured the yield of non-photonic electrons -4 - [49,50], which are expected to be dominantly produced by the semileptonic decay of the heavy quarks inside the hadrons. In this case, the comparison of data with FONLL is reasonably good for PHENIX, while the STAR data appear underestimated by a factor ∼ 5. The discrepancy between the two experimental data set should obviously be better understood before addressing in more detail the agreement with the theoretical prediction.

The relative contribution of charm and beauty to the final yield of electrons is not known precisely, as it suffers from large theoretical uncertainties due to the variation of the actual value of the mass of the quarks and unknown higher orders [40], see [51] for recent experimental efforts. This uncertainty is particularly relevant for the medium effects as we will discuss below (also [52]).

Besides single-particle inclusive cross sections, we are interested in the case of two-particle correlations. The corresponding expression is similar to Eq. (3.1), but now two partons created in the hard scattering fragment independently. The fragmentation of each parton is again described by D vac k→h or D med k→h for the vacuum and the medium respectively. Notice that here we will be interested in two-particle correlations which are well separated in azimuthal angle (back-to-back correlations), so that correlations from the fragmentation of a single parton into two final hadrons are negligible [53,54].

Energy loss implementation

The medium-modified fragmentation functions are computed using the Salgado-Wiedemann quenching weights P (∆E/ω c , R) [20]. They depend on two quantities, ω c and R, which for a static medium of length L and transport coefficient q, are given by

ω c = 1 2 qL 2 , R = ω c L. ( 4.1)
In a dynamical medium we make use of the following scaling relations [55,5,7,8,9]

ω ef f c (x 0 , y 0 , τ prod , φ) = dξ ξ q(ξ), (4.2)
[qL] ef f (x 0 , y 0 , τ prod , φ) = dξ q(ξ), (4.3)

R ef f (x 0 , y 0 , τ prod , φ) = 2[ω ef f c (x 0 , y 0 , τ prod , φ)] 2 (qL) ef f (x 0 , y 0 , τ prod , φ) , ( 4.4)
corresponding to an equivalent static scenario to write the effective quantities. These quantities are for a parton produced at mid-rapidity where the measurements under consideration have been performed, and at transverse position (x 0 , y 0 ) and time τ prod . Each parton traverses the medium with a straight-line trajectory given by azimuthal angle φ and with local transport coefficient q(ξ). As such, the problem is reduced to specifying the relation between the local value of the transport coefficient at a given point, ξ, on the trajectory and the medium properties as given by our hydrodynamical model. Based on the estimates in [10], we define

q(ξ) = K qQGP (ξ) K • 2 3/4 (ξ). (4.5)
The energy density along the parton trajectory (ξ) is then taken from the hydrodynamical model of the medium and K is a fit parameter when comparing to experimental data. This parameter then can indicate any departure with respect to the ideal gas qQGP estimated in [10].

The production point of the hard scattering is characterized by a production weight w(x 0 , y 0 ), as computed through the product of the profile functions of the two nuclei,

w(x 0 , y 0 ) = T Au (x 0 , y 0 )T Au ( b -(x 0 , y 0 )), (4.6)
with the profile function given by a 3-parameter Fermi distribution at a given impact parameter b taken from Ref. [56]. The trajectory is then defined by an angle φ and the average of a given quantity O is computed as

O = 1 N dφdx 0 dy 0 w(x 0 , y 0 )O(x 0 , y 0 , φ), ( 4.7)
with N = 2π dx 0 dy 0 w(x 0 , y 0 ). For example, we may compute the average value of q as follows:

q = 1 N dφdx 0 dy 0 w(x 0 , y 0 ) [qL] ef f (x 0 , y 0 , φ) 2 2ω ef f c (x 0 , y 0 , φ) . (4.8)
The medium-modified fragmentation functions are computed in the same manner:

D med k→h (z, µ 2 F ) = 1 N dφdx 0 dy 0 w(x 0 , y 0 ) dζ 1 -ζ P (x 0 , y 0 , φ, ζ)D vac k→h z 1 -ζ , µ 2 F , ( 4.9)
where P (x 0 , y 0 , φ, ) is the corresponding quenching weight for the effective parameters (4.2) and (4.4). In the case of two-particle correlations, two fragmentation functions are computed for particles traveling in opposite directions -i.e. measuring different medium lengths and density profiles:

D med k→h (x 0 , y 0 , φ, z, µ 2 F ) = dζ 1 -ζ P (x 0 , y 0 , φ, ζ)D vac k→h z 1 -ζ , µ 2 F , ( 4.10)
with the average given by

D med k→h 1 (z 1 , µ 2 F,1 )D med l→h 2 (z 2 , µ 2 F,2 ) = (4.11) 1 N dφdx 0 dy 0 w(x 0 , y 0 )D med k→h 1 (x 0 , y 0 , φ, z 1 , µ 2 F,1 )D med l→h 2 (x 0 , y 0 , π -φ, z 2 , µ 2 F,2
). (4.12)

Energy loss for times prior to hydrodynamic behavior

In a dynamical medium like the one considered here, there is an ambiguity on the value of the transport coefficient, defined by Eq. (4.5), for values smaller than the thermalization time τ 0 . One extreme case is to take q(ξ) = 0 for ξ < τ 0 . The absence of any energy-loss effect for these very early times is a strong assumption since thermalization is not necessary in the approach in which the quenching weights have been computed [2]. To quantify this uncertainty, we consider three different extrapolations for the time from the hard parton production to the thermalization time (τ 0 = 0.6 fm/c in the present calculation):

• Case i): q(ξ) = 0 for ξ < τ 0 ;

• Case ii): q(ξ) = q(τ 0 ) for ξ < τ 0 ; and

• Case iii): q(ξ) = q(τ 0 )/ξ 3/4 for ξ < τ 0 .

These extrapolations range from the most extreme assumption of no effect at all before the thermalization time (Case i)) to a continuous interaction from the production time (taken to be τ prod 0.04 fm/c) with a free-streaming medium with energy density dropping as (ξ) ∼ 1/ξ (Case iii)). As it turns out, the extracted value of the parameter K in Eq. (4.5) is only modestly sensitive to different assumptions about the very early time.

Both the quenching weights and the hydrodynamical profiles are taken from the publicly available codes [57] and [37], respectively.

Results and discussion

The experimental data used in our analysis are given in terms of the nuclear modification factors

R AA = dN AA /dp 2 T dy N coll dN pp /dp 2 T dy , I AA = D AA (z T , p trig T ) D pp (z T , p trig T ) (5.1)
for single-and double-inclusive measurements, respectively. They are computed using the corresponding medium-modified fragmentation functions, (3.2), in the cross sections, (3.1), for the AA case and the corresponding unmodified (vacuum) ones for the proton-proton case appearing in the denominators of (5.1). In (5.1), the hadron-triggered fragmentation function is defined as [58] D AA (z T , p trig T ) ≡ p trig T dσ h 1 h 2 AA /dy trig dp trig T dy assoc dp assoc T dσ h 1 AA /dy trig dp trig T .

(5.2)

Here, z T = p assoc T /p trig T , and we use as factorization scales the p T of the hadrons. We highlight that the quantities defined in this manner are, in general, very different from the fragmentation functions measured, for example, in e + e -annihilation at similar virtualities due to the strong bias produced by triggering on a steeply falling perturbative particle spectrum.

The data sets included in our analysis are all for Au+Au collisions at top RHIC energy, √ s NN = 200 GeV and are listed below.

1. 'Light': PHENIX data on R AA for π 0 with p π 0 T > 5 GeV in the 0 -5 % centrality class [21], 14 data points. 1 For computational time reasons, we make the calculations with fixed p trig T = 10 GeV. We have checked that the results are not sensitive to changes in this value. With this choice of p trig T and p assoc T , the D AA (z T , p trig T ) measured [59] in d+Au collisions at top RHIC energy is well described.

-7 -3. 'Heavy': R AA for non-photonic electrons with p e T > 3 GeV, for 0 -10 % from PHENIX [60], 11 data points; and for 0 -5 % from STAR [50], 9 data points.

In order to obtain the best value of the K-parameter (or equivalently q) we perform a modified χ 2 -analysis that treats statistical and systematic uncertainties in a consistent manner, as detailed in [21]. For any set of data with central values {y i }, three types of uncertainties σ t i , t = A (statistical plus uncorrelated systematics added in quadrature), B (correlated systematics) and C (normalization uncertainties in %), are considered and a modified χ2 for each set of theoretical values {y th i } (characterized by one given value of parameter K in the present case) is obtained by minimizing χ2 = exp. points

[y i + ε B σ B i + ε C y i σ C -y th i ] 2 σi 2 + ε 2 B + ε 2 C , (5.3) σi = σ A i y i + ε B σ B i + ε C y i σ C y i , ( 5.4)
with respect to the two parameters ε B , ε C . Note that our calculations currently do not explicitly include theoretical systematic uncertainties in the above formulation. Instead we consider various modifications of the theoretical calculations and their impact of the experimentally constrained results. In the case of the the 'heavy' set, there is a theoretical uncertainty of ±8 % coming from the variation of the renormalization and factorization scales and the heavy quark mass, see [52]. Since as will be shown, the theory tends to predict less suppression than the experimental data, we take the lowest values inside this band as the theoretical input {y th i } for the fits. It is notable that in the 'light' case, we follow the treatment of systematic uncertainties in [21] where the type B uncertainties allow a tilt of the data points (considering cases of correlation or anticorrelation as a function of p T ). In the 'dihadron' case, we include the statistical uncertainties quoted in [59] and an additional 7% normalization uncertainty as used in the constraint analysis [27]. In the 'heavy' case, both experiments quote systematic uncertainties which are included in the analysis here. Note that we consider the type B uncertainties to move the data points up and down together (and do not consider the anti-correlation case here). We discuss these uncertainties for the 'heavy' case in more detail later.

We present the results for different values of the proportionality factor between the transport coefficient and the energy density, Eq. (4.5), K = K /1.46, where K = 0.5, 1, 2, 3, . . . , 20. For the 'heavy' case we extended the range up to K = 40. This grid in K is then used to obtain the best χ2 values.

Light hadrons

Our first study corresponds to a global fit of R AA and I AA within the model and assuming a frozen value of q for the very early times -Case ii) previously discussed. Additionally, the EKS98 set [61] is taken for the nuclear corrections to the PDFs. This fixes our canonical choices for the calculation.

In Fig. 1 we plot the curves for different K-factors for both R AA and I AA and compare with the neutral pion data suppression from PHENIX and the corresponding suppression of the back-toback hadron-triggered fragmentation functions from STAR. Shown in Fig. 2 as solid lines are the -8 - calculated χ2 results for both R AA and I AA comparisons. For our canonical choice, the constraint on K agrees within one standard deviation uncertainties. We then sum the two χ2 values as a function of K and obtain a common global fit result of K = 4.1 ± 0.6.

Comparison of different prescriptions for τ < τ 0

We now repeat the analysis for the two other assumptions for q for times τ < τ 0 = 0.6 fm/c. The result of this analysis leads to the computed values of the χ2 for light hadrons R AA and I AA shown in Fig. 2 (dotted and dotted-dashed lines) and to the corresponding determination of the central values and the uncertainty band for ∆ χ2 = 1. The results are relatively insensitive to the choice of very early time treatment, except for R AA in Case i). This can be understood by the following example contribution: the parton travels for the first 0.6 fm/c without any effect and the strong dependence of jet quenching on the in-medium path length forces the value of K (and q) to increase.

It is notable that in the way the averaged transport coefficient is computed in our approach, Eq. (4.8), the actual value of q is different for a given value of K depending on the assumption for the initial times in the evolution -this is simply because the average value of the energy density is different in the three cases. For example, the values at the minimum K = 4.1 are q 6.2, 8.7, 138.5 GeV 2 /fm for the Cases i), ii) and iii) respectively. This also shows the large ambiguity in the determination of an average value of q for an evolving medium: unrealistically large values of q are possible due to the averaging procedure, in particular when the extrapolation is divergent, as in Case iii). This is the reason why in this paper, we have made the choice of quoting only values of the K-factor instead of actual values of the averaged transport coefficients. The fact that our results are very solid under the two assumptions ii) and iii) provides strong support to the way in which the hydrodynamical medium and the jet quenching calculations are merged.

] 3/4 ε /[2 q ≡ K 0 2 4
τ < τ )=0 for τ ( q Case i): 0 τ < τ ) for 0 τ ( q )= τ ( q Case ii): 0 τ < τ for 3/4 τ ) 0 τ ( q )= τ ( q Case iii):

The effect of the nuclear modification of parton densities

We now explore the uncertainty associated with the choice of nuclear parton distribution functions. We consider three situations: i) nPDFs as given by the EKS98 analysis [61]; ii) nPDFs as given by the EPS08 analysis [62]; iii) proton PDFs, i.e. without any nuclear modification. These three cases comprise two extreme situations2 . In this study, the impact parameter dependence of the nuclear PDF corrections is not taken into account explicitly. This may impact the surface bias from parton energy loss, and is in part, why we also consider the no shadowing case as an extreme.

In Fig. 3 we plot again the value of χ2 for each choice of the nuclear PDFs and the corresponding central values and uncertainty bands at the ∆ χ2 = 1 level. The differences are, in this case, smaller than the uncertainties from the modeling of the initial times of medium evolution. Interestingly, the results for I AA are again somewhat less sensitive to these uncertainties than the ones for R AA -the reason being the cancellation in the double ratio for the former.

Non-photonic electrons and heavy quarks suppression

We now study the 'heavy' quark case of non-photonic electron data within our canonical set-up, i.e. Case ii) for extrapolations at small times and EKS98 for nuclear PDFs. Notice that within the radiative energy loss formalism, the energy loss of heavy quarks receives corrections due to the mass terms in the propagators whose net effect is a reduction of the amount of radiated energy. It is a generic prediction of the formalism that the energy loss of a heavy quark is smaller than that of a light quark. The computation of this feature, first pointed-out in Ref. [64], is performed in the approximation M/E 1 in Refs. [65,66,67]. The quenching weights for the multiple-soft scattering approximation used here are found in [57].

] 3/4 ε /[2 q ≡ K 0 2 4
It was previously mentioned that the calculation has been performed in the M/E 1 limit, when M and E are the mass and the energy of the heavy quark. This limit is reasonable for the charm quark case in the region under study. The case of the bottom quark is, however, less clear, as its mass and energy are comparable for a non-negligible region of the phase space under study. Further mass term corrections have not as yet been computed in this formalism.

We now proceed as in the case of the light quarks and compute, for different values of the K-factor, the R AA of the non-photonic electrons from heavy-meson decays computed within the FONLL approach [45]. As expected, the suppression is reduced due to the finite mass of the heavy quarks, which is in qualitative contradiction with the experimental data as shown in Fig. 4. As noted in Ref. [52], the unknown relative contribution from charm and bottom decays to the final electrons introduces an additional uncertainty in this comparison. Here, an 8% theoretical uncertainty is used, as obtained by varying the heavy quark masses and the renormalization and factorization scales in the calculation [52]. The theoretical uncertainty band is likely larger, especially for larger values of q. This uncertainty is not taken for the case in which only the charm quark contributes to the final electron yield as the uncertainty band is much smaller in this case.

We compute again the χ2 values for different values of the K-factor. In Fig. 4 we do that for the central theoretical value of R AA diminished by the 8% theoretical uncertainty as stated above. The first observation is that the values of χ2 never reach a minimum in the studied region of K (thus favoring even larger values beyond the current calculation range). Thus, the 'heavy' data sets appear incompatible with the 'light' and 'dihadron' data sets within this theoretical framework. However, it is striking that the minimum (within our K =factor range) is χ2 min ≈ 37 for the PHENIX data (with 11 data points) and χ2 min ≈ 1.9 for the STAR data (with 9 data points). For the STAR data sample [50], around p T ≈ 5 GeV, the typical statistical uncertainties are of order 32%, the box type systematic uncertainties drawn on the points are of order 29%, and the normalization uncertainty is 26%. In the publication, the box type uncertainties are stated as uncorrelated [50]. If they are point-to-point uncorrelated, we should add them in quadrature with the statistical uncertainties (type A). As an exercise, we use the full statistical constraint procedure with the theory comparison as a simple second-order polynomial (R AA = a + b × p T + c × p 2 T ). One finds a best fit with a = +0.70, b = -0.13 GeV -1 , and c = +0.01 GeV -2 with a total χ2 = 0.5. This represents an extreme in goodness-of-fit with a p-value = 99.8%. In this paper, we have thus treated the STAR uncertainties drawn as boxes around the points as standard type B (assuming the points are in fact correlated). Even in this case, the total χ2 = 1.1 remains an extremely good match (i.e. reflecting much smaller-than-expected fluctuations of the individual data points). This results in a rather improbable situation in that even a value K = 4.1 as determined by the 'light' and 'dihadron' data has a reasonable p-value for agreement with the 'heavy' data set discussed here. Note that a χ2 test is more stringent in that in order for K = 4.1 to be the true answer, it means that K = 30 (for example) is not. Thus, since the χ2 for K = 30 is so small (and much smaller than for K = 4.1), it rules out K = 4.1 by the ∆ χ2 test, but not by a simple p-value check.

For the PHENIX data sample [60], around p T ≈ 5 GeV, the typical statistical uncertainties are of order 20%, the type B uncertainties of order 19%, and the normalization uncertainty is 7%. In this analysis, we consider only the correlated case for the type B uncertainties. We again compare the data with a simple second-order polynomial and find the best values as a = +1.6, b = -0.4 GeV -1 , and c = +0.03 GeV -2 and a total χ2 = 15.2. This is considerably better than the best fit with the parton energy loss calculation, but still only has a p-value ≈ 5%. We note that of order 30% of the total χ2 comes from the single data point at p T = 7 GeV. The details discussed above regarding the PHENIX and STAR experimental uncertainties may be indicative of not yet fully understood systematic uncertainties and/or correlations amongst these uncertainties. Future data sets, including with upgraded detectors to specifically tag the displaced vertex from the D and B meson decays, will be crucial to resolving these experimental issues.

In order to further study these data, we make two new computations of the χ2 in Fig. 5 and6. First we compute the R AA without shadowing corrections. Now the effects are much more visible than for light mesons as anti-shadowing is a 20 -25% effect here. Second, we compute the suppression when only the charm quark contributes to the non-photonic electrons yield, obtaining a much better fit as expected -which would be, in fact, compatible with the determination of K from light mesons. This second observation provides additional support for the need of an experimental identification of the D and B meson suppression.

Summary and conclusions

We have performed a fit to several heavy-ion observables in the large transverse momentum region of the spectrum. We combine a realistic hydrodynamical medium profile, obtained from fits to the soft part of the spectrum, with a jet quenching calculation within the radiative energy loss formalism in the BDMPS approximation. The analysis of the light meson data provides the best fit to R AA and I AA with parameter K = 4.1 ± 0.6, for the K-factor of the fitted transport coefficient with respect to a perturbative estimate from Ref. [10]. This large value is in qualitative agreement with previous estimates, using different medium profiles. This analysis reveals that a common treatment of both one-and two-particle inclusive observables leads to statistically compatible results. An initial investigation of a sub-set of the theoretical uncertainties reveal the robustness of this fit to reasonable changes of these particular assumptions.

A similar procedure has been applied to study the case of heavy quarks, as measured via nonphotonic electrons. The baseline spectrum for the heavy quarks has been computed in the FONLL approach. Here, the first observation is that the smaller suppression of the formalism, when FONLL contributions from charm and bottom are taken into account, leads to a contradiction with the qualitative trend of experimental data. The second observation is the large difference of the χ2 values of the formalism when comparing the theory to the STAR or PHENIX data sets. Values of the computed minimum χ2 are very different with the two experimental data sets, which make it challenging to make firm statements about the statistical agreement between theory and data. We have also studied the influence of the shadowing corrections in PDFs, and found them to play a more relevant role for heavy quarks compared with the case of light mesons. Finally, we have studied the case in which all non-photonic electrons come from decays of charmed mesons. In this latter case, the comparison gives improved agreement with experimental data due to the smaller mass effect of the charm quark with respect to the bottom quark on the radiative energy loss.

A common fit of all these data ('light', 'dihadron', 'heavy') within a single approach and the control over the theoretical and experimental uncertainties remains an open issue in jet quenching phenomenology.

[GeV] . The upper panels show the comparison with PHENIX data [60] and the lower panels the comparison to the STAR data [50]. The small difference in centrality, 0-10% in PHENIX and 0-5% in STAR is taken into account. The FONLL scheme is used in both calculations.

-18 -

2 .= 8 -

 28 'Dihadron': STAR data on I AA for Au+Au/d+Au for charged particles in the 0-5 % centrality class [59], for p trig T 15 GeV and p trig T > p assoc T > 3.0 GeV, 6 data points 1 .

Figure 1 :

 1 Figure 1: Nuclear modification factors R AA for single-inclusive (left) and I AA for hadron-triggered fragmentation functions (right) for different values of K = K /1.46, where K = 0.5, 1, 2, 3, . . . , 20. The theoretical curves are for the canonical calculation (i.e. EKS98 for the nPDF and Case ii) for the treatment of very early times). The curve corresponding to the minimum of the common fit to R AA and I AA data: K = 4.1, is shown by a thick solid line.

Figure 2 :

 2 Figure 2: Left: χ2 -values for different values of K for light hadrons and for the three different extrapolations for ξ < τ 0 considered here: Case i), Case ii) and Case iii) are plotted with dotted, solid and dotted-dashed lines respectively. Thin lines correspond to single-inclusive π 0 data from PHENIX (R AA ) and thick ones to the double-inclusive measurements by STAR (I AA ). Right: the corresponding central values (minima of the χ2 ) and the uncertainties computed by considering ∆ χ2 = 1 are shown.

Figure 3 :

 3 Figure 3: Left: χ2 -values for different values of K for light hadrons and for three assumptions on the nuclear effects in PDFs: EKS98 (solid lines) [61], EPS08 [62] (dotted lines) and no shadowing corrections (dashed-dotted lines). As in the previous Fig., thin lines correspond to single-inclusive π 0 data from PHENIX (R AA ) and thick ones to the double-inclusive measurements by STAR (I AA ). Right: the corresponding central values (minima of the χ2 ) and the uncertainties computed by considering ∆ χ2 = 1 are shown.

Figure 4 :

 4 Figure4: Nuclear modification factor R AA and the computed values of χ2 for non-photonic electrons from heavy quark decays for the same values of the parameter K as in Fig.1-the thick solid line corresponds to the same value of K as the one if Fig.1. The upper panels show the comparison with PHENIX data[60] and the lower panels the comparison to the STAR data[50]. The small difference in centrality, 0-10% in PHENIX and 0-5% in STAR is taken into account. The FONLL scheme is used in both calculations.

Figure 5 :Figure 6 :

 56 Figure 5: Same as Fig. 4 but without nuclear corrections to the PDFs. The horizontal dashed lines in the right panels indicate the minimum of χ2 and the corresponding values for one standard deviation, ∆ χ2 = 1, and two standard deviations, ∆ χ2 = 4. Notice that if no local minimum is found in the range of K under study (lower panels corresponding to STAR data) these lines are not shown.

Notice that for a more formal analysis, the use of the uncertainties computed using the Hessian method as done in the recent EPS09 set[63] may be used.

Acknowledgments

The work of N.A. and C.A.S. has been supported by Ministerio de Ciencia e Innovación of Spain under projects FPA2005-01963, FPA2008-01177 and contracts Ramón y Cajal (NA and CAS), by Xunta de Galicia (Consellería de Educación) (NA and CAS) and through grant PGIDIT07PXIB206-126PR (NA), and by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042) (NA and CAS). The work of T.H. was partly supported by Grant-in-Aid for Scientific Research No. 19740130 and by Sumitomo Foundation No. 080734. J.L.N. acknowledges funding from the Division of Nuclear Physics of the U.S. Department of Energy under Grant No. DE-FG02-00ER41152.