Hadron properties in the nuclear medium

The QCD vacuum shows the dynamical breaking of chiral symmetry. In the hot and/or dense QCD medium, the chiral order parameter such as $⟨\overline{q}q⟩$ is expected to change as a function of temperature $T$ and density $\rho$ of the medium, and its experimental detection is one of the main challenges in modern hadron physics. In this paper, the theoretical expectations for the in-medium hadron spectra associated with partial restoration of chiral symmetry are discussed as well as the current status of experiments with an emphasis on the measurements of properties of mesons produced in near-ground-state nuclei.

Figure 1

The running coupling constant determined from $\tau$ decay, ${\rm Y}$ decay, deep inelastic scattering, $e^{+}{e}^{-}$ annihilation, and $Z$-boson resonance shape and width (14).

Figure 2

The masses of $u$, $d$, and $s$ quarks at the scale $\kappa =2\mathrm{GeV}$ constrained by various hadron masses using QCD sum rules and lattice QCD simulations (14).

Figure 3

(Color online) Light hadron spectrum obtained from lattice QCD simulations with dynamical $u,d,s$ quarks in the Wilson fermion formalism. The hadron masses are extrapolated to the physical quark masses (determined by $m_{\pi }$, $m_K$, and $m_{\Omega }$) using the data in the interval $\frac{1}{2}(m_u+m_d)(\kappa =2\mathrm{GeV})=3.5–67\mathrm{MeV}$. The spatial lattice volume $V$ and the lattice spacing $a$ are ${\left(2.9\mathrm{fm}\right)}^3$ and $0.09\mathrm{fm}$, respectively. Horizontal bars denote the experimental values (17).

Figure 4

(Color online) Effective quark mass $M$ as a function of the Euclidean momentum $q$ obtained from lattice QCD simulations without dynamical quarks (quenched) and with three dynamical quarks (unquenched) (41).

Figure 5

Light scalar $\left(S\right)$, pseudoscalar $\left(P\right)$, vector $\left(V\right)$, and axial-vector $\left(A\right)$ mesons. Their spin and parity are denoted as $J^p$. Thresholds of the $\pi \pi$, $\pi K$, and $KK$ decays are also shown (259).

Figure 6

(Color online) Normalized chiral condensate $\Delta =[{⟨\overline{u}u⟩}_T-(\widehat{m}∕m_s){⟨\overline{s}s⟩}_T]∕[{⟨\overline{u}u⟩}_0-(\widehat{m}∕m_s){⟨\overline{s}s⟩}_0]$ as a function of $T$ for two different lattice spacings, $a=0.24\mathrm{fm}$ $(N_{\tau }=4)$ and $a=0.17\mathrm{fm}$ $(N_{\tau }=6)$ calculated by the lattice QCD simulations with dynamical $u,d,s$ quarks in the staggered fermion formalism (60).

Figure 7

Comparison of the linear density approximation of the chiral condensate (dashed lines) and the results with higher order terms originating from nucleon-nucleon correlations calculated by the in-medium chiral perturbation theory (solid lines) (142).

Figure 8

Thermal susceptibilities in three different channels ($\pi$, $\sigma$, and $a_0$) for two-flavor QCD with staggered fermion on the $8^3\times 4$ lattice with $m_{u,d}a=0.02$. The vertical axis denotes $\sqrt{1∕{\chi }_{\mathcal{O}}}$ (145). The horizontal axis denotes the bare lattice coupling ${\beta }_{\mathrm{lat}}=6∕g^2$ which is a decreasing function of the lattice spacing $a$ and hence a increasing function of temperature $T=1∕N_{t}a$ with $N_t$ being a number of temporal lattice sites.

Figure 9

Spectral function in the isovector channel in the vacuum [${\rho }_V^{\mathrm{vac}}\left(s\right)$ with $s={\omega }^2$] obtained from the $e^{+}{e}^{-}$ annihilation into even numbers of pions (162).

Figure 10

The binding energies with finite-size Coulomb potential only, $B_{\mathrm{Coul}}$, and Coulomb plus optical potential, $B_{\text{full}}$, are calculated. The energy shifts $B_{\mathrm{Coul}}-B_{\text{full}}$ are shown as the solid bars for pionic $1s$, $2p$, and $3d$ states for ${}^{115}\mathrm{Sn}$. The shifts due to the real local terms in the potential are shown by dashed bars. Dotted bars are the results with all real terms (local plus nonlocal) in the optical potential (244).

Figure 11

A schematic drawing of the PSI pionic-hydrogen setup (107).

Figure 12

The $s$-wave parameters $b_0$ vs $\mathrm{Re}{B}_0$ obtained by fitting the x-ray data; T: (228), B1-3: (27, 28, 29), SM1-2: (215), K1-2: (157), and E: (84). The line is $b_0+0.215\mathrm{Re}{B}_0=-0.028$.

Figure 13

(Color online) The potential and wave functions of the pionic lead atom. (a) The pion optical potential for ${}^{208}\mathrm{Pb}$. The finite-size Coulomb potential is expressed by the dotted curve and the one with the optical potential by the solid curve. The imaginary part is depicted by the dashed curve. (b) The pionic wave functions of the $1s$, $2s$, and $2p$ states in coordinate space. The dashed and solid curves are obtained with the finite-size Coulomb potential and the optical potential. The half-density radius $R_0$ of ${}^{208}\mathrm{Pb}$ is indicated by the broken line (237).

Figure 14

A schematic of pionic-atom states with x-ray transitions down to the last orbital and deeply bound inner orbits with large widths which cannot be populated following the x-ray cascade. They have large widths due to nuclear absorption but are still discrete states with ${\Gamma }_n<E_n-E_{n-1}$ (261).

Figure 15

Strong-interaction shift (top) and width (bottom) of pionic atoms. The continuous lines join points calculated with the best-fit optical potential (29).

Figure 16

Energy levels of pionic atoms of ${}^{90}\mathrm{Zr}$ and ${}^{208}\mathrm{Pb}$. The results with the finite-size Coulomb potential (i.e., taking account of the nuclear charge distribution) are shown by dashed bars, while those with the optical potential by solid bars with hatched area indicating the level widths (239).

Figure 17

Diagrams for proton-pick-up pion-transfer (a) $(n,d)$ and (b) $(d,{}^3\mathrm{He})$ reactions to form pionic bound states on a neutron-hole state (132).

Figure 18

Momentum transfers in the $(n,d)$ and $(d,{}^3\mathrm{He})$ reactions on Pb for $Q=-130$ and $-140\mathrm{MeV}$ as a functions of the incident energy per nucleon (132).

Figure 19

The GSI S160 data compared with theory. (a) The ${}^{208}\mathrm{Pb}(d,{}^3\mathrm{He})$ spectrum. The abscissa is the reaction $Q$ value and the ${\pi }^{-}$ emission threshold $(Q=-140.14\mathrm{MeV})$ is shown by a dotted line. The $p(d,{}^3\mathrm{He}){\pi }^0$ reaction peak used as a calibration is also shown in the $Q$ value scale of the ${}^{208}\mathrm{Pb}(d,{}^3\mathrm{He})$ reaction kinematics. (b) Theoretical prediction by 132. A FWHM resolution of $0.5\mathrm{MeV}$ is assumed (262).

Figure 20

The Sn isotope dependence of the total $(d,{}^3\mathrm{He})$ spectrum for the pionic-atom formation at $T_d=500\mathrm{MeV}$ with $300\mathrm{keV}$ experimental resolution. The target nucleus is indicated (244).

Figure 21

The fragment separator of GSI used for the $(d,{}^3\mathrm{He})$ spectroscopy.

Figure 22

Double differential cross sections vs the ${}^3\mathrm{He}$ kinetic energy of the ${}^{124,120,116}\mathrm{Sn}(d,{}^3\mathrm{He})$ reactions measured at 0° at the incident deuteron energy of $503.388\mathrm{MeV}$. The scales of the ${\pi }^{-}$ binding energies in ${}^{123,119,115}\mathrm{Sn}$ are also indicated. From 226.

Figure 23

Likelihood contours in the $\{b_1,\mathrm{Im}{B}_0\}$ plane from the simultaneous fitting of $\{B_{1s},{\Gamma }_{1s}\}$ of the $1s$ pionic states in the three Sn isotopes and three light symmetric nuclei (226).

Figure 24

Difference $\Delta r_{np}$ between the rms radii of the neutron and proton distributions as deduced from the antiprotonic atom x-ray data as a function of $\delta =(N-Z)∕A$. The proton distributions were obtained from electron-scattering data (69) (Sn nuclei) or from muonic atom data (88) (other nuclei). The full line represents the linear relationship between $\delta$ and $\Delta r_{np}$ as obtained from a fit to the experimental data. From 243.

Figure 25

Difference $\Delta r_{np}$ between the rms radii of the neutron and proton distributions for various tin isotopes deduced from proton elastic scattering at $295\mathrm{MeV}$ (229) (squares) compared with the values adopted in the analysis of 226 (filled circles), which used the relation $\Delta r_{np}=(1.01\pm 0.15)(N-Z)∕A+(-0.04\pm 0.03)$ obtained by 243.

Figure 26

(Color online) In-medium isovector $b_1\left(\rho \right)$ compared to the vacuum isovector term $b_1$. The gray band, the experimental result from 226, is compared with chiral calculations by 73 (dark band), 170 (dashed line), and 92 (dashed-dotted line). The point is a phenomenological fit by 188. From 73.

Figure 27

(Color online) Imaginary part of the $\pi \text{−}\pi$ amplitude $T_{00}\left(E\right)$ in the scalar-isoscalar channel calculated in the chiral-unitary model (49).

Figure 28

A typical reconstructed event in CHAOS for the ${\pi }_i^{+}\to {\pi }^{+}{\pi }^{-}p$ reaction on ${}^{12}\mathrm{C}$. WCs are the four wire chambers and CFTs are the CHAOS first-level trigger counters, which are also used for particle identification. Of the 20 CFTs, two are removed to free the pion beam path. From 40.

Figure 29

Invariant-mass distributions (diamonds) for the ${\pi }^{+}\to {\pi }^{+}{\pi }^{-}$ and ${\pi }^{+}\to {\pi }^{+}{\pi }^{+}$ reactions on ${}^2\mathrm{H}$, ${}^{12}\mathrm{C}$, ${}^{40}\mathrm{Ca}$, and ${}^{208}\mathrm{Pb}$. The shaded regions represent the results of phase-space simulations for the pion-production reaction $\pi A\to \pi \pi N[A-1]$ (40).

Figure 30

The composite ratios for the CHAOS data. (Left) $C_{\pi \pi }^A$, the bin-by-bin ratio (see text) of $\pi \pi$ invariant-mass distributions for the two reactions ${\pi }^{+}A\to {\pi }^{+}{\pi }^{+}{A}^{\prime }$ and ${\pi }^{+}p\to {\pi }^{+}{\pi }^{+}n$ as a function of the $M_{\pi \pi }$ energy. The nuclei $\left(A\right)$ examined are ${}^2\mathrm{H}$ (which plays the role of a proton, $p$), ${}^{12}\mathrm{C}$ (open diamonds), ${}^{40}\mathrm{Ca}$ (full triangles), and ${}^{208}\mathrm{Pb}$ (open squares). (Right) Same as the left panel but for the ${\pi }^{+}\to {\pi }^{+}{\pi }^{-}$ reaction channel (39).

Figure 31

Comparison of the CB and CHAOS composite ratios. (Left) Experimental results for the $2{\pi }^0$ invariant-mass distributions obtained for the H, D, C, Al, and Cu targets corrected for Crystal Ball acceptance. The vertical scale is in arbitrary units. The solid lines show the results of calculations made for the TAPS group by Rapp, and the dashed line is the prediction by Vicente Vacas (224). (Right) The composite ratios $C_{\pi \pi }^A$ as a function of the $\pi \pi$ invariant mass for the ${}^{12}\mathrm{C}$ target. Full diamonds, the $C_{{\pi }^0{\pi }^0}^C$ distribution deduced from the CB data of 224; open diamonds, the CHAOS $C_{{\pi }^{+}{\pi }^{-}}^C$ distribution taken from 39 (53).

Figure 32

Setup of the TAPS detector at the Mainz MAMI accelerator. The beam entered the target chamber from the lower right edge (148; 35).

Figure 33

The $\pi \pi$ invariant mass spectra measured by the TAPS Collaboration. (Left) Mass-number-normalized differential cross sections of the reaction $A(\gamma ,{\pi }^0{\pi }^0)$ with $A={}^1\mathrm{H},{}^{12}\mathrm{C},{}^{\mathrm{nat}}\mathrm{Pb}$. (Right) The same for the reaction $A(\gamma ,{\pi }^0{\pi }^{+})$. Both panels are for incident photons in the energy range of $400–460\mathrm{MeV}$ (solid circles). Error bars denote statistical uncertainties. The dotted curves indicated phase-space distributions determined by the Monte Carlo model, and the solid curves are predictions by 205. From 171.

Figure 34

(Color online) Pion-pion invariant-mass distributions compared to results of the BUU model (48). The bars at the bottom represent the systematic uncertainty of the data; the dashed lines represent the error band for the BUU calculation (35).

Figure 35

Top view of the DLS at Bevalac (263).

Figure 36

The dilepton spectrum for $\mathrm{Ca}+\mathrm{Ca}$ at $1.0A\mathrm{GeV}$ measured by the DLS Collaboration (circles) (198) compared with the hadronic cocktail assuming the free $\rho$ spectral function (43).

Figure 37

A side view of HADES. The RICH detector, consisting of a gaseous radiator, a carbon fiber mirror, and a tilted photon detector, is used for electron identification. Two sets of multiwire drift chambers are placed in front and behind the magnetic field to measure particle momenta. A time-of-flight (TOF/TOFINO) wall accompanied by a preshower detector at forward angles is used for an additional electron identification and trigger purposes. For a reaction time measurement, a start detector is located near the target (212).

Figure 38

(Color online) Direct comparison of the dilepton pair-mass distributions measured in $\mathrm{C}+\mathrm{C}$ at $1A\mathrm{GeV}$ by HADES (within the DLS acceptance) and at $1.04A\mathrm{GeV}$ by DLS (198). Statistical and systematic errors are shown. Overall normalization errors (not shown) are 20% for the HADES and 30% for the DLS data points (11).

Figure 39

HELIOS/3 setup (15).

Figure 40

(Color online) $M_{\mu \mu }$ distribution normalized to the charged-particle multiplicity obtained by the HELIOS/3 Collaboration for S-W (filled squares) and $p\text{−}\mathrm{W}$ (open squares) collisions at $200A\mathrm{GeV}$ (15).

Figure 41

(Color online) Schematic view of the CERES spectrometer with a radial drift TPC. The latter was added in NA45/2 for the $\mathrm{Pb}+\mathrm{Au}$ runs (168).

Figure 42

(Color online) CERES inclusive $e^{+}{e}^{-}$ mass spectra of $450\mathrm{GeV}$ $p\text{−}\mathrm{Be}$, $p\text{−}\mathrm{Au}$, and $200A\mathrm{GeV}$ S-Au collisions (8, 9). Plotted is the number of electron pairs per charged particle both in the acceptance and per event. Contributions from various hadron decays as expected from $p\text{−}p$ collisions are shown together with their sum (thick line), and the systematic error on the latter is indicated by the shaded area (2).

Figure 43

(Color online) Layout of the NA60 detectors in the vertex region (245).

Figure 44

Isolation of an excess above the hadron decay cocktail (see text). Total data (open circles), individual cocktail sources (solid), difference data (thick triangles), sum of cocktail sources, and difference data (dashed) (19; 65).

Figure 45

(Color online) Comparison of the excess mass spectrum for the semicentral bin to model predictions made for In-In at ${\phantom{|}d{N}_{ch}∕d\eta |}_{\eta =0}=140$. Cocktail $\rho$ (thin solid line), unmodified $\rho$ (dash-dotted line), in-medium broadening $\rho$ (57; 202; 203) (thick solid line), and in-medium moving $\rho$ related to 44, 46 and 165 (dashed line). The errors are purely statistical. The systematic errors of the continuum are about 25%. From 19 and 66.

Figure 46

(Color online) CERES spectra. (Left) CERES unlike-sign pair yield (histogram) and combinatorial background (dashed curve). (Right) Invariant $e^{+}{e}^{-}$ mass spectrum compared to the expectation from hadronic decays (2).

Figure 47

(Color online) CERES $e^{+}{e}^{-}$ pair yield after subtraction of the hadronic cocktail. In addition to the statistical error bars, systematic errors of the data (horizontal ticks) and the systematic uncertainty of the subtracted cocktail (shaded boxes) are indicated. The broadening scenario (long-dashed line: 203; 247) is compared to a calculation assuming a density-dependent dropping $\rho$ mass [dotted line in (a): 44, 45, 46] and to a broadening scenario excluding baryon effects [dotted line in (b)]. From 2.

Figure 48

The PHENIX experiment at RHIC. From 6.

Figure 49

(Color online) Invariant $e^{+}{e}^{-}$ pair yield observed by PHENIX in $\mathrm{Au}+\mathrm{Au}$ minimum bias and in $p+p$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$. The curve is the hadronic cocktail for $p+p$. The $p+p$ spectrum and the cocktail curve were normalized to the $\mathrm{Au}+\mathrm{Au}$ yield in the $m_{e^{+}{e}^{-}}<100\mathrm{MeV}∕c^2$ region (7; 235; 3).

Figure 50

A plan view of the TAGX spectrometer. From 133.

Figure 51

Schematic view of the experimental setup of the E325 spectrometer: (a) the top view and (b) the side view. The side view shows the cross section along the center of the kaon arm (192; 216).

Figure 52

Kinematical distributions of $e^{+}{e}^{-}$ pairs in the $\varphi$ mass region $0.95\to M_{e^{+}{e}^{-}}<1.05\mathrm{GeV}∕c^2$ detected in the E325 spectrometer (points with error bars) together with the simulation result using the JAM nuclear cascade code (histogram; 186). (Left) $\beta \gamma$ distribution. (Right) Rapidity $y$ vs pair transverse momentum $p_T$ (180).

Figure 53

(Color online) Invariant-mass spectra of $e^{+}{e}^{-}$ for the (a) C and (b) Cu targets. The solid lines are the best-fit results, which is the sum of the known hadronic decays, $\omega \to e^{+}{e}^{-}$ (dashed line), $\varphi \to e^{+}{e}^{-}$ (thick dash-dotted line), $\eta \to e^{+}{e}^{-}\gamma$ (dash-dotted line), and $\omega \to e^{+}{e}^{-}{\pi }^0$ (dotted line) together with the combinatorial background (long-dashed line). $\rho \to e^{+}{e}^{-}$ is not visible (187).

Figure 54

(Color online) Invariant-mass spectra of $e^{+}{e}^{-}$. The combinatorial background and the shapes of $\eta \to e^{+}{e}^{-}\gamma$ and $\omega \to e^{+}{e}^{-}{\pi }^0$ were subtracted. The result of the model calculation considering the in-medium modification for the (a) C and (b) Cu targets. The solid lines show the best-fit results. In (a) and (b), the shapes of $\omega \to e^{+}{e}^{-}$ (dotted line) and $\omega \to e^{+}{e}^{-}$ (dash-dotted line) were modified according to the model using the formula $m_V\left(\rho \right)∕m_V\left(0\right)=1-k(\rho ∕{\rho }_0)$ with $k=0.092$ (187).

Figure 55

Obtained $e^{+}{e}^{-}$ distributions with the fit results. The target and $\beta \gamma$ region are shown in each panel. The points with error bars represent the data. The solid lines represent the fit results with an expected $\varphi \to e^{+}{e}^{-}$ shape and a quadratic background. The dashed lines represent the background (180).

Figure 56

Confidence ellipsoids for the modification parameters $k_1$ and $k_2^{\mathrm{tot}}$ in cases (i) in (a) and (ii) in (b). The values of $\Delta {\chi }^2$’s in both panels are the differences from the ${\chi }_{\min }^2(=316.4)$ at the best-fit point in case (i) which is shown by the cross in (a). The best-fit point in case (ii) is shown by the closed circle in (b) and also in (a) since the ordinates are common to both cases in parameter space (180).

Figure 57

The CLAS detector. (Top) Side view of the CLAS detector in Hall B of J-Laboratory, with the photon-tagging system. (Bottom) CLAS cutaway view (169).

Figure 58

(Color online) The CLAS $e^{+}{e}^{-}$ invariant mass spectra. (Left) Normalized combinatorial background for individual targets compared with the data. (Right) Result of the fits to the $e^{+}{e}^{-}$ invariant-mass spectrum obtained for the ${}^2\mathrm{H}$ (top), C (middle), and Fe-Ti (bottom) data (257). The curves on the right panels are Monte Carlo calculations by the BUU model for various vector-meson decay channels (76; 77).

Figure 59

Individual Breit-Wigner/${\mu }^3$ fits to the $\rho$ mass spectra (background and $\omega ,\varphi$ contributions subtracted) (257).

Figure 60

(Color online) Side view of the Crystal Barrel and TAPS detector combination.

Figure 61

(Color online) The effect of pion rescattering in the CBELSA/TAPS experimenet. (Left) The ${\pi }^0\gamma$ mass distribution obtained from a Monte Carlo simulation of the process $\gamma +\mathrm{Nb}\to {\pi }^0\gamma +X$ at $E_{\gamma }=1.2\mathrm{GeV}$. The spectrum is decomposed into different contributions corresponding to (a) the fraction of $\omega$ mesons decaying outside the nucleus, (b) the fraction of $\omega$ mesons decaying inside for which the ${\pi }^0$ does not rescatter, and (c) the fraction of $\omega$ mesons decaying inside the nucleus for which ${\pi }^0$ rescatters. In the simulation a drop of the $\omega$ mass by 16% at normal nuclear density was assumed. (Right) The same as the left panel, with the additional condition of $T_{{\pi }^0}>150\mathrm{MeV}$ (172).

Figure 62

(Color online) The ${\pi }^0\gamma$ invariant mass spectra measured by the CBELSA/TAPS experiment. (a) Inclusive ${\pi }^0\gamma$ invariant-mass spectra for momenta $<500\mathrm{MeV}∕c$. Upper histogram: Nb data; lower histogram: $\mathrm{L}{\mathrm{H}}_2$ target reference measurement. The dashed lines indicate fits for the respective background. (b) ${\pi }^0\gamma$ invariant mass for the Nb data (solid histogram) and $\mathrm{L}{\mathrm{H}}_2$ data (dashed histogram) after background subtraction. The error bars show statistical uncertainties only. The solid curve represents the simulated line shape for the $\mathrm{L}{\mathrm{H}}_2$ target (242).

Figure 63

Mass-number $\left(A\right)$ dependence of the transparency ratio $T$. The dotted line corresponds to ${\sigma }_A\propto A^{0.72}$. The solid and dashed curves show the theoretical calculation (51) without (solid curve) and with (dashed curve) Pauli-blocking correction for the $\varphi$ meson scattering angle in the laboratory frame of 0° (135).

Figure 64

(Color online) Experimentally determined transparency ratio normalized to the carbon data in comparison with a theoretical Monte Carlo simulation (147) (left) and a BUU calculation (177) (right) varying the width at $1.1\mathrm{GeV}∕c$ momentum. The width is given in the nuclear rest frame. Only statistical errors are shown (158).

Figure 65

(Color online) Two-pion invariant-mass distributions for ${\pi }^0{\pi }^0$ photoproduction of ${}^{12}\mathrm{C}$ and ${}^{208}\mathrm{Pb}$ for $E_{\gamma }=0.4–0.46\mathrm{GeV}$ calculated by 48. Results without final-state interactions (black dash-dotted lines) and with FSI (black solid lines). These are compared with the results obtained by 205, with in-medium final $\pi \pi$ interaction (solid lines) and with the free-space $\pi \pi$ interaction (dashed lines).

Figure 66

Comparison of the dielectron invariant-mass spectrum (carbon target) of E325 (circles) and CLAS-g7 (triangles).