Abstract
The mass radius is a fundamental property of the proton that so far has not been determined from experiment. Here we show that the mass radius of the proton can be rigorously defined through the formfactor of the trace of the energy-momentum tensor (EMT) of QCD in the weak gravitational field approximation, as appropriate for this problem. We then demonstrate that the scale anomaly of QCD enables the extraction of the formfactor of the trace of the EMT from the data on threshold photoproduction of $J/\psi$ and $\Upsilon$ quarkonia, and use the recent GlueX Collaboration data to extract the r.m.s. mass radius of the proton ${\rm R_m = 0.55 \pm 0.03 \ fm}$. The extracted mass radius is significantly smaller than the charge radius of the proton ${\rm R_C = 0.8409 \pm 0.0004 \ fm}$. We attribute this difference to the interplay of asymptotic freedom and spontaneous breaking of chiral symmetry in QCD, and outline future measurements needed to determine the mass radius more precisely.
Highlights
The mass distribution is a fundamental property of a physical object
R For a particle of mass M 1⁄4 μdV, the total potential is φ 1⁄4 −GM=R, and the force acting on a probe mass m is Fg 1⁄4 −m∂φ=∂R, which yields the Newton’s law of gravity, Fg. We have reviewed this textbook derivation to show that the distributions of mass and of the trace of the energy-momentum tensor (EMT) are identical in the weak gravitational field limit; see (5)
T00 and of the trace of the EMT with respect to t at t 1⁄4 0, we find that they differ by terms Gið0Þ=ð4M2Þ that have to be compared to dGi=dtjt1⁄40 ≡ Gið0Þ=m2i that depend on the compositeness scales m2i of the corresponding form factors
Summary
The mass distribution is a fundamental property of a physical object. Yet, while a lot of information is available about the charge distribution inside the proton, nothing is known at present about its mass radius. In the chiral limit of massless quarks, the scale anomaly [3,4] expresses the trace of the energy-momentum tensor (EMT) of QCD in terms of the scalar gluon operator [5,6]. It has been proposed [7,8] that the matrix elements of this operator (which is largely responsible for the mass of the proton) can be extracted from the photoproduction of heavy quarkonia near the threshold. We will use this form factor to extract the mass radius of the proton from the GlueX data
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