The HBT Effect

Before going through the methodology of measuring and interpreting femtoscopic correlations in high-energy physics, let us mention that the origins of this field go back to R. Hanbury Brown and R. Q. Twiss, who investigated stars with optical and radio telescopes, at Jordell Bank and later at the Narrabri Observatory (Hanbury Brown et al. Reference Hanbury Brown and Twiss1956). They discovered an unexplained correlation between signals of two telescopes, interpreted as coincidental photon detections happening ‘too frequently’ at the two telescopes, as compared with the expected frequency of random coincidences. They furthermore observed that this correlation (excess number of coincidences) decreases when increasing the distance between the two telescopes. Hanbury Brown later wrote in his autobiography:

Indeed, while these experiments were repeated in tabletop experiments as well, it took several decades to unravel the physics and mathematics behind this phenomenon, now called the HBT effect. There are two simple ways of understanding it. The first is (based on Baym Reference Baym1998) to consider the optical or radio signal as spherical waves being emitted from a point $a$ within the source, with the amplitude

$${A_a}\left( r \right) = {1 \over {\left| {r - {r_a}} \right|}}\alpha {e^{ik\left| {r - {r_a}} \right| + i{\Phi _a}}}$$

where $r$ is the spatial point at which we investigate the wave, ${r_a}$ is the origin of the wave, $\alpha $ is its strength, $k$ its wavenumber, and ${{\rm{\Phi }}_a}$ is its phase, assumed to be a random variable with a uniform distribution in case of thermal emission. A similar amplitude can be expected from another point of the source; let us denote that one with a similar expression, but with index $b$ and strength $\;\beta $ . Let us assume that the source comprises only these two points, denoted by $a$ and $b$ . Then at any point in space, the wave amplitude from the source can be written as

$$A\left( r \right) = {A_a}\left( r \right) + {A_b}\left( r \right)$$

At the detectors, a sum of the waves emitted from both points of the source is detected. The observed intensity in detector A is then the square of the amplitude:

$${I_A} = {\left| {A\left( {{r_A}} \right)} \right|^2} = {1 \over {{L^2}}}\left( {{{\left| \alpha \right|}^2} + {{\left| \beta \right|}^2} + {\alpha ^{\rm{*}}}\beta {e^{ik\left( {{r_{bA}} - {r_{aA}}} \right) + i\left( {{\Phi _b} - {\Phi _a}} \right)}} + c.c.} \right)$$

where ${r_A}$ is the location of detector A, $L$ is the average distance between the source points and the detectors, while ${r_{bA}}$ is the distance of source point b and detector A, similarly for ${r_{aA}}$ , and c.c. denotes the complex conjugate of the preceding term. The observed signal is then this intensity, averaged over a sufficiently large time interval, where phase information cancels out if the source is chaotic:

$${\langle I_A\rangle } = {\langle I_B\rangle } = {1 \over {{L^2}}}\left( {{{\left| \alpha \right|}^2} + {{\left| \beta \right|}^2}} \right)$$

The joint (or coincident) intensity is, however, the product of intensities, where the phase information is retained even after averaging, resulting in an excess term as compared with the product of time-averaged intensities:

$$\langle {I_A}{I_B}\rangle = \langle {I_A}\rangle \langle {I_B}\rangle \left( {1 + {1 \over 2}\cos k\left( {{r_{aA}} - {r_{bA}} + {r_{aB}} - {r_{bB}}} \right)} \right)$$

and

$${C_{AB}} = {{\langle{ I_A}{I_B}\rangle} \over {\langle{I_A}\rangle\langle{I_B}\rangle}} \approx 1 + {1 \over 2}\cos {{kRd} \over L}$$

where the last approximation can be confirmed by expanding the sum of the distances in terms of d/L and R/L, where $d$ is the distance of the detectors and $R$ is the distance of the investigated source points. The factor 1/2 in front of the cosine is specific to this configuration of two source points; the general result is the excess term, decreasing with increasing detector distance or source size. In a general source, an origin-dependent source strength $\alpha \left( {{r_s}} \right)$ would appear, and the sum would go over all possible ${r_s}$ values; in fact, it would an integral instead of the ${A_a}\left( r \right) + {A_b}\left( r \right)$ sum written above.

A second way of understanding the HBT effect is when one considers the detection of two quantum-mechanical particles stemming from sources a and b, in detectors A and B. If we consider the two detection events separately, the corresponding wave functions are (assuming plane waves here for simplicity, but the calculation would work equivalently for spherical waves):

$${\rm{\Psi }}_B^b = {e^{ik{r_{bB}} + i{{\rm{\Phi }}_{b}}}} {\ \rm and\ } {\rm{\Psi }}_A^a = {e^{ik{r_{aA}} + i{{\rm{\Phi }}_a}}}$$

If these particles are identical bosons (particles with integer intrinsic angular momentum, obeying Bose-Einstein statistics, e.g., photons), then the two-particle wave function must be symmetric with respect to particle exchange (cf. Bose–Einstein statistics), and the detection events are also entangled, expressible with the wave function

$${\Psi _{AB}} = {1 \over {\sqrt 2 }}\left( {\Psi _A^a\Psi _B^b + \Psi _B^a\Psi _A^b} \right)$$

Then the single and joint detection probabilities are the modulus squares of the above wave functions for a single particle and the pair, respectively. In case of a setting similar to the one discussed in the classical derivation, the result in this case is:

$${C_{AB}} = {{P\left( {A,B} \right)} \over {P\left( A \right)P\left( B \right)}} = \;1 + \cos k\left( {{r_{aA}} - {r_{bA}} + {r_{aB}} - {r_{bB}}} \right) \approx 1 + \cos k{{Rd} \over L}$$

This second derivation can be extended to fermions (particles with half-odd-integer intrinsic angular momentum, obeying Fermi-Dirac statistics) as well, in which case a similar anti-correlation is observed. Furthermore, for an extended source, a source weight could appear as a prefactor in ${\rm{\Psi }}$ and an integral over $r$ would have to be performed. Figure 2 shows example data from the measurement of Sirius by Hanbury Brown and Twiss, along with a calculation of such an extended source.

Figure 2. Correlation data from Sirius and a theoretical curve based on the calculation of a star of angular size 0.0063’, reproduced from Hanbury Brown et al. (Reference Hanbury Brown and Twiss1956).

Femtoscopy

Independent of the HBT-discovery, Goldhaber and collaborators observed a similar correlation for pairs (Goldhaber et al. Reference Goldhaber, Fowler, Goldhaber and Hoang1959) of pions (hadrons made up of one quark and one antiquark of the lightest kind), detected in high-energy proton–proton collisions. A clear understanding of the background of the two related phenomena is owed to articles by Glauber, Baym and others (Glauber Reference Glauber1963; Baym Reference Baym1998). The main result, applicable today in high-energy physics, is that the two-particle momentum correlation function is connected to the particle creation probability density. This can be understood based on the connection of the single-particle momentum distribution ${N_1}\left( p \right)$ expressed as

$${N_1}\left( p \right) = \smallint S\left( {r,p} \right){\left| {{{\rm{\Psi }}_{\rm{p}}}\left( r \right)} \right|^2}{\rm{d}}r$$

where $S\left( {r,p} \right)$ is the particle creation probability density (also called single-particle source, or simply source) at the space-time point $r$ and momentum $p$ (in other words, the infinitesimal probability of creating a particle with momentum $p$ in point $r$ , which would be a sum of two Dirac delta distributions in the above two-source example), ${{\rm{\Psi }}_{\rm{p}}}\left( r \right)$ is the single-particle wave function of a particle with momentum $p$ , and the integral runs over the entire space-time. Similarly, the two-particle momentum distribution ${N_2}\left( {{p_1},{p_2}} \right)$ can be expressed as

$${N_2}\left( {{p_1},{p_2}} \right) = \smallint S\left( {{r_1},{p_1}} \right)S\left( {{r_2},{p_2}} \right){\left| {{{\rm{\Psi }}_{{{\rm{p}}_1},{p_2}}}\left( {{r_1},{r_2}} \right)} \right|^2}{\rm{d}}{r_1}{\rm{d}}{r_2}$$

where

$${\Psi _2}\left( {{r_1},{r_2}} \right) = {1 \over {\sqrt 2 }}\left( {{\Psi _{{{\rm{p}}_1}}}\left( {{r_1}} \right){\Psi _{{{\rm{p}}_2}}}\left( {{r_2}} \right) + {\Psi _{{{\rm{p}}_2}}}\left( {{r_1}} \right){\Psi _{{{\rm{p}}_1}}}\left( {{r_2}} \right)} \right)$$

is the two-particle symmetrized wave-function (for identical bosons). Assuming plane waves (i.e., neglecting interactions), defining $q = {p_1} - {p_2},K = \left( {{p_1} + {p_2}} \right)/2$ and using the approximation that $q \ll K$ , one obtains for the correlation function of identical bosons

$$C\left( {q,K} \right) = {{{N_2}\left( {q + K,q - K} \right)} \over {{N_1}\left( {q + K} \right){N_1}\left( {q - K} \right)}} \approx 1 + {{{{\left| {\smallint {\rm{d}}rS\left( {r,K} \right){e^{iqr}}} \right|}^2}} \over {{{\left| {\smallint {\rm{d}}rS\left( {r,K} \right)} \right|}^2}}} = 1 + {{{{\left| {\tilde S\left( {q,K} \right)} \right|}^2}} \over {{{\left| {\tilde S\left( {0,K} \right)} \right|}^2}}}$$

where the tilde denotes the Fourier transform in the first variable. If one introduces the spatial correlation function as

$$D\left( {r,K} \right) = \smallint S\left( {\rho + {r \over 2},K} \right)\ S\left( {\rho - {r \over 2},K} \right){\rm{d}}\rho $$

and assumes it to be normalized to unity, then one obtains

$$C\left( {q,K} \right) = 1 + \smallint {\rm{d}}rD\left( {r,K} \right){e^{iqr}} = 1 + {\tilde D}\left( {q,K} \right)$$

The main result here is that the momentum correlation function is directly and simply connected to the spatial (pair) distribution. In femtoscopy, one often assumes a parametric shape for the $r$ -dependence of $S$ or $D$ , and then the momentum dependence is expressible through the dependence of the parameters on $K$ . For example, if the source is a Gaussian:

$$S\left( {r,K} \right)\sim\exp \left[ { - {{{r^2}} \over {2R{{\left( K \right)}^2}}}} \right]$$

or equivalently

$$D\left( {r,K} \right)\sim\exp \left[ { - {{{r^2}} \over {R{{\left( K \right)}^2}}}} \right]$$

then

$$C\left( {q,K} \right) = 1 + \exp \left[ { - {q^2}R{{\left( K \right)}^2}} \right]$$

hence a source of some given width $R$ creates correlations of inverse width $1/R$ . This is illustrated by Figure 3; and this width is then also called the HBT radius (after the HBT effect).

Figure 3. A source of width $R$ creating correlations of width $1/R$ .

This simple picture neglects several phenomena, such as final-state interactions (FSI), secondary particle production and multiparticle correlations. For details about these, see for example Adare et al. (Reference Adare, Aidala, Ajitanand, Akiba, Akimoto and Alexander2018) and Nagy et al. (Reference Nagy, Purzsa, Csanád and Kincses2023). We only note here that in case of secondary particle production from decays of long-lived resonances, the correlation strength decreases (Adare et al. Reference Adare, Aidala, Ajitanand, Akiba, Akimoto and Alexander2018), and the correlation function (for the same Gaussian source as above) becomes

$$C\left( {q,K} \right) = 1 + \lambda \left( K \right) \cdot \exp \left[ { - {q^2}R{{\left( K \right)}^2}} \right]$$

where $\lambda \left( K \right)$ is the (momentum-dependent) correlation strength, which can be interpreted as the squared fraction of the primordially produced particles (Adare et al. Reference Adare, Aidala, Ajitanand, Akiba, Akimoto and Alexander2018). Let us note at this point that, in experimental analyses, usually the dependence on ${m_T} = \sqrt {{m^2} + {K^2}} $ (where $m$ is the identified particle’s mass) is investigated instead of the dependence on $K$ .

Lévy-stable Sources

Due to the central limit theorem and thermodynamics leading to Gaussians, the natural assumption for the source shape is that of a Gaussian, and this assumption has been explored in a multitude of experimental and phenomenological investigations (Lisa et al. Reference Lisa, Pratt, Soltz and Wiedemann2005). In this article we refrain from discussing those and instead focus on a different avenue of research. As mentioned in several references (Csörgő et al. Reference Csörgő, Hegyi and Zajc2004; Csanád et al. Reference Csanád, Csörgő and Nagy2007 Adare et al. Reference Adare, Aidala, Ajitanand, Akiba, Akimoto and Alexander2018), measurements (see, for example, Adler et al. Reference Adler, Afanasiev, Aidala, Ajitanand, Akiba and Alexander2007 or references in Adare et al. Reference Adare, Aidala, Ajitanand, Akiba, Akimoto and Alexander2018) suggest phenomena beyond the Gaussian distribution. A straightforward generalization of the central-limit theorem for random variables without a finite variance or mean leads to a general class of distributions. Out of these we focus here on symmetric Lévy-stable distributions (Csörgő et al. Reference Csörgő, Hegyi and Zajc2004), defined as

$${\cal L}\left( {r;\alpha, R} \right) = {\left( {2\pi } \right)^{ - 3}}\smallint {\rm{d}}q{e^{iqr}}{e^{ - {1 \over 2}{{\left| {qR} \right|}^\alpha }}}$$

where $r$ is the (three-dimensional) variable of the distribution, Lévy-index $\alpha $ and Lévy-scale $R$ are its parameters, $q$ is an integration variable, and $qr$ is the scalar product of $q$ and $r$ . The distributions are stable for $0 \lt \alpha \le 2$ , meaning that a sum of random variables with such distribution converges to a random variable with the same distribution. In case of $\alpha \lt 2$ and three dimensions, these distributions exhibit a power law tail with exponent $ - \left( {1 + \alpha } \right)$ . For $\alpha = 2$ , the Gaussian distribution is retained, while for $\alpha = 1$ one obtains the Cauchy (Lorentz) distribution. Note that the Lévy-scale $R$ can also be called the HBT radius, but one has to keep in mind that it is conceptually a different observable from the one extracted from Gaussian measurements (except if $\alpha $ is close to 2).

If the single-particle source is of the Lévy-stable shape discussed above, then the pair source is also Lévy-stable with the same exponent, but modified width:

$$S\left( r \right) = {\cal L}\left( {r;\alpha, R} \right) \Rightarrow D\left( r \right) = {\cal L}\left( {r;\alpha, {2^{1/\alpha }}R} \right)$$

where we dropped the dependence on mean momentum $K$ for simplicity.

There may be several reasons for the appearance of these distributions, see a short review in Csanád et al. (Reference Csanád and Kincses2024). One example is anomalous diffusion (Csanád et al. Reference Csanád, Csörgő and Nagy2007), as illustrated in Figure 4, where it is also apparent that in case of anomalous diffusion (also called Lévy flight) the appearance of large step sizes is much more frequent than in normal diffusion, creating a heavy tail in the resulting distribution.

Figure 4. Examples for normal and anomalous diffusion in two dimensions. (Images modified from Wikipedia.)

Another example for the reasons of appearance of Lévy-stable distributions may be a second-order phase transition in the vicinity of the CEP, as detailed in Csörgö et al. (Reference Csörgő, Hegyi, Novák and Zajc2006). It has been conjectured that at the critical point, the Lévy-exponent $\alpha $ introduced above is identical to the critical correlation exponent $\eta $ , and hence drops to a much smaller value than 2, even down to values near 0.5 (Csörgö et al. Reference Csörgő, Hegyi, Novák and Zajc2006; Csanád et al. Reference Csanád and Kincses2024).

Experimental Results and Interpretations

As detailed above, femtoscopy can be used to explore the probability density distribution of particle creation from the sQGP. The most abundant particles are pions; hence, their correlations can be measured most precisely (in terms of statistical uncertainty). The considerations in the previous section motivate the accurate measurement of the Lévy exponent $\alpha $ , but also measuring $R$ with the correct source assumption is of crucial importance (Csanád et al. Reference Csanád and Kincses2024). In this section we discuss several recent experimental results on these parameters.

The detailed centrality and average transverse mass dependence of these two-pion Lévy HBT parameters have been recently measured in 5.02 TeV Pb+Pb collisions with the Compact Muon Solenoid (CMS) experiment at the LHC (Tumasyan et al. Reference Tumasyan2024). Here no particle identification was applied; instead, all detected particles were assumed to be pions. This reduces the correlation strength but does not change its shape, as non-pions do not contribute to the measured correlations. This analysis found qualitatively similar trends of the ${m_T}$ -dependence of Lévy parameters as the first measurement of Lévy parameters in heavy-ion collisions, measured in 200 GeV Au+Au collisions by the Pioneering High Energy Nuclear Interaction eXperiment (PHENIX) at RHIC (Adare et al. Reference Adare, Aidala, Ajitanand, Akiba, Akimoto and Alexander2018):

• • a nearly constant $\alpha \left( {{m_T}} \right)$ ,

• • a nearly constant $\lambda \left( {{m_T}} \right)$ at large ${m_T}$ (when rescaled by the pion fraction),

• • and an $R$ decreasing with ${m_T}$ as $R \propto \sqrt {{m_T}} $ .

However, measured $\alpha $ values obtained by CMS were found to be around 1.6–2.0, significantly larger than those of PHENIX (where a mean value of $\alpha \approx 1.2$ was found in 0–30% Au+Au collisions). The $\alpha $ and $R$ values measured by CMS are shown in Figures 5 and 6.

Figure 5. Lévy-index $\left\langle \alpha \right\rangle $ averaged over pair transverse mass, as a function of the number of participant nucleons (number of protons and nucleons involved in the collision, ${N_{part}}$ ), measured by CMS at LHC (figure reproduced from Tumasyan et al. Reference Tumasyan2024).

Figure 6. Lévy scale $R$ as a function of pair transverse mass ${m_T}$ , measured by CMS at LHC. Solid lines show linear fits based on hydrodynamic predictions (figure reproduced from Tumasyan Reference Tumasyan2024).

A similar analysis in the Solenoidal Tracker at RHIC (STAR) experiment was also performed in 200 GeV Au+Au collisions (Kincses Reference Kincses2024). Similar values, ${m_T}$ -dependence, and centrality dependence for the Lévy index $\alpha $ , Lévy scale $R$ and correlation strength $\lambda $ were found as in Adare et al. (Reference Adare, Aidala, Ajitanand, Akiba, Akimoto and Alexander2018). Let us note in particular the decrease of $\lambda \left( {{m_T}} \right)$ at small ${m_T}$ , attributed to a possible indirect signal of in-medium mass modification of the $\eta '$ meson (Adare et al. Reference Adare, Aidala, Ajitanand, Akiba, Akimoto and Alexander2018), a special particle made up of a quark and an antiquark from one of the three lightest kinds, with equal probability. The trends of the parameters versus ${m_T}$ are also similar to the abovementioned CMS analysis (noting the difference in the accessible ${m_T}$ interval); while the largest difference is that in the STAR measurement, $\alpha $ decreases for central collisions. Results by STAR on $\alpha $ and $R$ are shown in Figures 7 and 8.

Figure 7. Lévy index $\alpha $ as a function of pair transverse mass ${m_T}$ , measured by STAR at RHIC. Solid lines show constant fits (figure reproduced from Kincses Reference Kincses2024).

Figure 8. Lévy scale $R$ as a function of pair transverse mass ${m_T}$ , measured by STAR at RHIC. Solid lines show fits with an empirical curve (figure reproduced from Kincses Reference Kincses2024).

Measurements by the NA61/SHINE (SPS Heavy Ion and Neutrino Experiment) at the Super Proton Synchrotron (SPS) were reported for Be+Be collisions in Adhikary et al. (Reference Adhikary, Adrich, Allison, Amin, Andronov and Anticic2023) and for Ar+Sc collisions in Pórfy (Reference Pórfy2024). These collision systems are initially significantly smaller than those in Au+Au collisions at RHIC and in Pb+Pb collisions at the LHC. The $\alpha $ values measured in Be+Be collisions were found to be around 1.6–2.0, similar to those obtained at the LHC. The ones measured in Ar+Sc collisions were found to be around 1.0–1.4, similar to those obtained at RHIC. The origin of this behaviour is as yet unexplained. The Lévy scale $R$ shows a similar trend to the one at RHIC or LHC, but with smaller values, corresponding to the much smaller initial system size. Results from NA61/SHINE on $\alpha $ and $R$ are shown in Figures 9 and 10. Note furthermore that the correlation strength $\lambda $ does not show the characteristic decrease at small ${m_T}$ observed at RHIC (and mentioned above). This hints at a difference in the physical processes occurring at SPS and at RHIC.

Figure 9. Lévy-index $\alpha $ as a function of pair transverse mass ${m_T}$ , measured by NA61/SHINE at SPS. Also shown are lines for special values of $\alpha $ (figure reproduced from Pórfy Reference Pórfy2024).

Figure 10. Lévy scale $R$ as a function of pair transverse mass ${m_T}$ , measured by NA61/SHINE at SPS (figure reproduced from Pórfy Reference Pórfy2024).

Summary

Several phenomenological and experimental results related to femtoscopy have been presented. It was found that Lévy sources are observed at a wide range of collision energies, at SPS, at RHIC, and at the LHC. Multiple physical phenomena may influence the appearance of these sources: two examples discussed above are anomalous diffusion and critical phenomena. At LHC energies, as well as at higher RHIC energies, only the first (anomalous diffusion) may influence experimental correlation functions, as no critical phenomena are expected in this collision energy range. At lower RHIC energies or at the SPS, these may play a role as well. Observations, on the other hand, show varying values of Lévy exponent $\alpha $ : they are around 1.6–2.0 in SPS Be+Be collisions and LHC Pb+Pb collisions, while $\alpha $ values around 1.0–1.4 are found in SPS Ar+Sc collisions and RHIC Au+Au collisions. Furthermore, the opposite centrality dependence of $\alpha $ was found at RHIC and at LHC. Hence, it is clear that not only the collision energy but also the system size may influence these trends, and further explorations are needed to unravel the physical phenomena determining the shape of the particle-emitting source. Altogether, the measured Lévy exponents indicate a source that is clearly not Gaussian (i.e., $\alpha \lt 2$ ), thus underlining the importance of using Lévy-distributed source shapes.

The Lévy scale $R$ was also measured in Be+Be and Ar+Sc collisions at SPS, in Au+Au collisions at RHIC and in Pb+Pb collisions at the LHC. A characteristic dependence on the transverse mass ${m_T}$ was found at every energy and centrality: $R$ decreases with ${m_T}$ , as predicted based on the expanding nature of the fireball from which hadrons are created at the freeze-out. This dependence calls for detailed calculations and simulations to understand its origin: traditional hydrodynamic calculations used in deriving this relationship were based on a Gaussian assumption for the source shape.

Finally, the correlation strength $\lambda $ was also measured in the abovementioned systems. At RHIC, a decrease at small ${m_T}$ was found, as well as a saturation at larger ${m_T}$ values. The accessible range at LHC allowed only for the confirmation of the second trend. On the other hand, at SPS the mentioned decrease seemed to be absent, a difference that is expected to be explored in greater detail in higher precision SPS data as well.

The findings outlined above emphasize the importance of incorporating Lévy-shaped sources in femtoscopic analyses of momentum correlations measured in ultra-relativistic collisions of nuclei. To unravel the physical processes influencing the qualitative and quantitative details behind these observations, further phenomenological studies are required.