Nucleon Short-Range Correlations (SRCs)

• Nucleon Short-Range Correlations (SRCs) are high-density, high-momentum nucleon fluctuations occurring at sub-femtometer separations, driven by repulsive cores and tensor forces.
• Experiments using electron scattering and triple-coincidence measurements reveal universal k⁻⁴ momentum scalings and clear proton-neutron dominance in 2N-SRCs.
• SRC insights refine nuclear models by explaining increased nucleon kinetic energies, altered binding properties, and connections to the EMC effect and QCD dynamics.

Nucleon Short-Range Correlations (SRCs) are high-density, high-momentum fluctuations within the nuclear many-body system, in which two or three nucleons reside in close spatial proximity (r ≲ 1 fm), resulting in large relative momenta far above the Fermi momentum and significant modification of nuclear momentum distributions and response functions. SRCs arise fundamentally due to the short-range repulsive core and tensor components in the nucleon–nucleon (NN) interaction, which fundamentally disrupt the mean-field paradigm at sub-femtometer length scales and drive a universal high-momentum tail in nuclear wavefunctions across the chart of nuclides.

1. Formal Definition, Scale Separation, and Contact Structure

At low and moderate relative momenta (k ≲ k_F), nuclear structure is governed by the mean-field shell model, where nucleons propagate independently in an effective potential. However, at short distances and higher momenta (k ≫ k_F), the strong repulsive core and tensor forces in the NN interaction induce non-perturbative two-body correlations. The nuclear ground-state wavefunction can thus be decomposed as: $$|\Psi\rangle = |\Phi_{\text{MF}}\rangle + \sum_{i<j} f_{ij}|\Phi_{\text{MF}}\rangle + \sum_{i<j<k} g_{ijk}|\Phi_{\text{MF}}\rangle + ...$$ where $f_{ij}$ encodes two-body (2N) short-range correlations and $g_{ijk}$ captures three-body (3N) SRCs. Modern theory places these notions on a rigorous footing through the generalization of Tan’s contact formalism: $$H_{\mathrm{eff}} = H_{\rm MF} + \sum_{\alpha} C_{\alpha} \hat{O}_{\alpha},$$ where $\alpha$ runs over all relevant two-nucleon spin-isospin channels, $\hat{O}_{\alpha}$ is a local contact operator, and $C_\alpha$ is the channel-dependent nuclear contact (Weiss et al., 2016).

In momentum space, this formalism yields universal high-momentum scaling: $n_A(k)\xrightarrow{k \gg k_F} \frac{C_A}{k^4},$ where $C_A$ is proportional to the number of SRC pairs in nucleus $A$ (the “contact”). Experimental extractions and theory confirm this universal tail, extending up to $k \sim$ 4–5 fm$^{-1}$ in all nuclei (Weiss et al., 2016, Alvioli et al., 2013).

2. Two-Nucleon SRCs: Structure, Isospin, and Momentum Distributions

2N-SRCs manifest as tightly correlated nucleon pairs with:

• Large relative momenta ($k_{\rm rel} \gtrsim 2$ fm$^{-1}$)
• Small center-of-mass momentum ($K_{\rm cm} \lesssim k_F$)
• Spatial separation $r_{ij} \lesssim 1$ fm.

Ab initio and cluster calculations reveal a pronounced correlation "hole" in the coordinate-space two-body density at short distances, which is nearly identical to the deuteron for all $A$ (universality). At large $k_{\rm rel}$ and small $K_{\rm cm}$, the 2N momentum density factorizes: $$n_{NN}(k_{\rm rel}, K_{\rm cm}) \approx n_D(k_{\rm rel})\, n_{\mathrm{cm}}^{A}(K_{\rm cm}),$$ where $n_D$ is the deuteron distribution (dominant for the S=1,T=0 pn channel), and $n_{\mathrm{cm}}^{A}$ is an $A$-dependent Gaussian with width $\sigma_{\rm cm} \sim 140$–$170$ MeV/c (Cohen et al., 2018, Alvioli et al., 2013, Alvioli et al., 2011).

The isospin-spin dependence is dominated by pn pairs in the $S=1, T=0$ channel, driven by the tensor force. In light and medium nuclei, the fraction of pp or nn-SRC pairs is $<10\%$ that of the pn-SRC pairs. For medium nuclei, exclusive (e,e′pp) to (e,e′pn) cross section ratios are $\sim$1:20 (Sargsian, 2018). However, in $A=3$ systems ($^3$He, $^3$H), this pn-dominance is less extreme, with $R_{np/pp} \sim 4.3$ (Li et al., 2022, Meng et al., 2023).

3. Experimental Extraction and High-Precision Observables

SRCs are probed predominantly by inclusive and exclusive high-energy electron and proton scattering:

• Inclusive Cross-section Ratios: At Bjorken $x_B = Q^2/(2m_N \omega) > 1.4$ and $Q^2 \gtrsim 1.4$ GeV$^2$, the per-nucleon cross-section plateaus at

$a_2(A) = 2 \frac{\sigma^A}{A \sigma^D},$

reflecting the relative probability of 2N-SRCs in $A$ compared to deuteron (Fomin, 2012, Dai et al., 2016).

• Triple-Coincidence Knockout (e,e′pN): Detects the struck nucleon and its recoil partner in back-to-back kinematics, specifying the SRC pair’s relative and CM momentum; uses kinematic cuts ($|p_{\text{miss}}|\gtrsim 300$ MeV/c, $|K_{\mathrm{cm}}| \lesssim$ 1 fm$^{-1}$) (Cohen et al., 2018, Higinbotham, 2010).
• Bremsstrahlung $\gamma$-Ray Probes: Measurement of high-energy $\gamma$ yield from $^{124}$Sn+$^{124}$Sn at 25 MeV/u directly determines the SRC fraction via spectral hardening, with $f_{\mathrm{SRC}} = 20\pm3\%$ (Xu et al., 14 Apr 2025).

Tables summarizing SRC scale parameters:

Nucleus	$\sigma_{\rm cm}$ [MeV/c]	$f_{\rm SRC}$ [%]	$a_2$
$^{12}$C	140	20	4.65
$^{56}$Fe	160	--	4.75
$^{208}$Pb	170	--	5.13
$^{124}$Sn	--	20	--

These parameters confirm the weak $A$-dependence of $K_{\rm cm}$ and the near-universality of SRC scaling in heavy nuclei (Cohen et al., 2018, Xu et al., 14 Apr 2025, Dai et al., 2016).

4. Three-Nucleon SRCs: Scaling, Extraction, and Evidence

3N-SRCs correspond to configurations with three nucleons at short relative distances sharing large momenta. Theoretically, their probability $a_3(A)$ scales quadratically with $a_2(A)$ if 3N-SRCs arise via two successive NN-SRC interactions (sequential pn-induced mechanism): $a_3(A) \approx [a_2(A)]^2$ (Sargsian et al., 2019, Sargsian, 2018). Observation of a scaling plateau in inclusive A(e,e′)X ratios at $x_B > 2$, where

$$R_3(A/Z) = \frac{\sigma_A/(A)}{\sigma_{^{3}\mathrm{He}}/3} \approx [a_2(A)/a_2(^{3}\mathrm{He})]^2,$$

is taken as evidence for 3N-SRCs. Confirming this scaling provides insight into three-body nuclear dynamics and genuine 3N forces.

Recent measurements in $^3$H and $^3$He find that the highest-momentum nucleons ($\alpha_{3N} > 1.4$) yield cross-section ratios consistent with near-isospin symmetry, supporting the universality of SRC-driven dynamics even in few-body systems (Li et al., 24 Apr 2024).

5. Nuclear Structure Impact: Momentum, Binding, and Bulk Properties

The high-momentum tail induced by SRCs leads to several universal effects:

• The kinetic energy per nucleon in LCA or ab initio wavefunctions nearly doubles compared to mean-field, with most of the increase localized at $r \lesssim 2$ fm (Cosyn et al., 2021, Ryckebusch et al., 2014).
• The fraction of nucleons in the SRC-dominated ($k > k_F$) tail is 15–25% in medium/heavy nuclei (Weiss et al., 2016, Xu et al., 14 Apr 2025).
• SRC pair probability correlates linearly with the binding energy per nucleon (pairing term removed), which reflects average local density or nucleon virtuality (Dai et al., 2016).
• In asymmetric systems, correlations invert the mean-field expectation for nucleon kinetic energies: the minority nucleon type becomes preferentially more energetic (Ryckebusch et al., 2014, Cosyn et al., 2021).

SRC-induced modifications also generate a decrease in rms nuclear radii by a few percent and reduce neutron skins in neutron-rich systems (Cosyn et al., 2021).

6. QCD and EMC Effect Connections

There is a demonstrated correlation between the magnitude of SRCs (through $a_2(A)$ or $R_{2N}$) and the slope of the nuclear EMC effect (suppression in $F_2^A/F_2^D$ for $0.3 < x_B < 0.7$), suggesting that local high-density fluctuations responsible for SRCs are also the sites of in-medium quark modification (Fomin, 2012, West, 2020). Models posit that QCD-scale dynamics, such as the formation of color-antitriplet $[ud]$ diquarks across SRC pairs, directly perturb quark distributions and structure functions in nuclei.

From a hadronic perspective, effective mass extractions yield $m_{\mathrm{SRC}}=852\pm18$ MeV per nucleon (vs. 940 MeV free nucleon mass), attributed to QCD trace anomaly or bag-model vacuum energy loss in highly overlapping SRC pairs (Wang et al., 2020).

7. Theoretical and Experimental Outlook

Ongoing and future experimental efforts focus on:

• Precision mapping of 3N-SRCs across isotopic chains using both inclusive and exclusive final states and light-cone kinematics ($\alpha > 1.6$) (Sargsian et al., 2019, Fomin et al., 2023, Li et al., 24 Apr 2024).
• Disentanglement of tensor versus central correlations and their isospin structure in few-body and heavier nuclei, especially using light-mirror systems ($^3$H/$^3$He) (Li et al., 2022, Meng et al., 2023).
• Validation and extension of contact formalism and universal scaling laws via ab initio (GFMC, VMC, NCSM) methods and extension to three-body contact operators (Weiss et al., 2016).
• Incorporation of SRC effects in global models for neutrino–nucleus scattering, nuclear symmetry energy, neutron-star equation of state, and nuclear partonic distributions (Cuyck et al., 2016, West, 2020).

Theoretical understanding of SRCs is critical for next-generation nuclear structure models, high-energy astrophysics, neutrino physics, and the broader connection between nucleonic and partonic degrees of freedom.

---

References:

(Cohen et al., 2018) (Li et al., 2022) (Wang et al., 2020) (Sargsian et al., 2019) (Dai et al., 2016, Xu et al., 14 Apr 2025) (Sargsian, 2018) (Fomin et al., 2023) (Weiss et al., 2016) (Higinbotham, 2010, Meng et al., 2023) (Cosyn et al., 2021) (Alvioli et al., 2013) (Li et al., 24 Apr 2024) (Ryckebusch et al., 2014, West, 2020) (Fomin, 2012) (Alvioli et al., 2011)