PHYSICAL REVIEW D 103, 094010 (2021) Nucleon mass radii and distribution: Holographic QCD, lattice QCD, and GlueX data Kiminad A. Mamo* Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA † Ismail Zahed Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA (Received 12 March 2021; accepted 15 April 2021; published 13 May 2021) We briefly review and expand our recent analysis for all three invariant A, B, D gravitational form factors of the nucleon in holographic QCD. They compare well to the gluonic gravitational form factors recently measured using lattice QCD simulations. The holographic A-term is fixed by the tensor T ¼ 2þþ (graviton) Regge trajectory, and the D-term by the difference between the tensor T ¼ 2þþ (graviton) and scalar S ¼ 0þþ (dilaton) Regge trajectories. The B-term is null in the absence of a tensor coupling to a Dirac fermion in bulk. A first measurement of the tensor form factor A-term is already accessible using the current GlueX data, and therefore the tensor gluonic mass radius, pressure, and shear inside the proton, thanks to holography. The holographic A-term and D-term can be expressed exactly in terms of harmonic numbers. h 2 i ≈ ð0 57–0 60 Þ2 The tensor mass radius from the holographic threshold is found to be rGT . fm ,in h 2 i ≈ ð0 62 Þ2 agreement with rGT . fm as extracted from the overall numerical lattice data, and empirical h 2 i ≈ ð0 7 Þ2 GlueX data. The scalar mass radius is found to be slightly larger rGS . fm . DOI: 10.1103/PhysRevD.103.094010 I. INTRODUCTION lattice QCD simulations [5,6], empirical measurements to support this understanding is only now emerging at A persistent and fundamental question in physics is current and dedicated electron machines [7–10]. Recently, about the origin of mass in the nucleon, and therefore in all the GlueX Collaboration [8] at JLAB has reported thresh- visible hadronic mass in the Universe. Where does it come old data for photoproduction of charmonium J=Ψ that may from, and how is it distributed inside the nucleon? Unlike start to lift the lid on some of these fundamental questions. the Higgs in electroweak theory which is at the origin of the Indeed, near threshold the elastic production of a heavy leptonic and current quark masses, most of the hadronic vector meson is likely to proceed mostly through gluons or mass of the visible Universe stems from QCD, a theory more precisely tensor glueballs, as is the case way above with almost no mass [1,2] (and references therein). threshold through the strong Pomeron exchange in the The dual quantum breaking of conformal symmetry and diffractive regime [11,12]. the spontaneous breaking of chiral symmetry in QCD are at In a recent analysis of the GlueX data using a holo- the origin of this mass without mass. These two fundamental graphic construction, we have shown [13] that the threshold phenomena are tied by strong topological fluctuations in the differential cross section is only sensitive to the tensor QCD vacuum: eternal tunneling events between gauge gravitational form factor, and suggested that this tensor vacua with different winding numbers, also known as form factor or A-term is extractable from the current data instantons and anti-instantons [3,4] (and references therein). under a minimal but universal set of holographic assump- While decisive understanding of these two phenomena tions. This allows for a first extraction of the tensor mass has been achieved theoretically and numerically using radius among other things. Remarkably, the holographic construction ties the A- and D-gravitational form factors, *[email protected] thereby allowing for the extraction of the gluonic pressure † [email protected] and shear inside the proton. For completeness, we note the holographic discussion regarding the extraction of the Published by the American Physical Society under the terms of gluon condensate in the proton using the GlueX data the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to in [14]. the author(s) and the published article’s title, journal citation, In Sec. II we briefly review and expand our arguments and DOI. Funded by SCOAP3. for the holographic A, B, D invariant gravitational form 2470-0010=2021=103(9)=094010(9) 094010-1 Published by the American Physical Society KIMINAD A. MAMO and ISMAIL ZAHED PHYS. REV. D 103, 094010 (2021) factors for the proton, extract the tensor mass radius, and gMNðzÞ → gMNðzÞþhMNðx; zÞð2Þ compare the results to the most recent lattice data. In Sec. III we use the holographic relationship between the A- and 2 ¼ ð Þ M N D-term to analyze the gluonic pressure and shear inside the with line element ds gMN z dx dx in a five- proton. The pressure inside the proton results from a delicate dimensional anti–de Sitter space or AdS5, in the double balance between the repulsive tensor glueball contribution at limit of large Nc and strong gauge coupling [19] (and short distances, and the attractive scalar glueball contribu- references therein). The form factors in (1) follow from the tion at large distances. An estimate of the scalar mass radius coupling of the irreducible representations of the metric is made. In Sec. IV we show how to use the current GlueX fluctuations hμν, to a bulk Dirac fermion with chiral data to extract empirically the A-form factor. The result is in components ψ L;R. The bulk metric fluctuations can be remarkable agreement with the holographic result and lattice decomposed in terms of the 2 ⊕ 1 ⊕ 0 invariant tensors [20] data. Our conclusions are in Sec. V. TT hμνðk; zÞ¼½ϵμν hðk; zÞþkμkνHðk; zÞ II. GRAVITATIONAL FORM FACTORS ⊥ ⊥ þ½kμAν ðk; zÞþkνAμ ðk; zÞ The standard decomposition of the energy-momentum form factor in a nucleon state is [15–18] 1 þ ημνfðk; zÞ ; ð3Þ 3 ðμ νÞα μν ðμ νÞ ip σ kα hp2jT ð0Þjp1i¼u¯ðp2Þ AðkÞγ p þ BðkÞ 2m N which is the spin-2 made of the transverse-traceless part h μ ν − ημν 2 þ ð Þ k k k ð Þ ð Þ plus the longitudinal-traceful part H, the spin-1 made of the C k u p1 ; 1 ⊥ mN transverse vector Aμ , and the spin-0 traceful part f. ðμ νÞ 1 μ ν ν μ 2 2 with a b ¼ ða b þ a b Þ, k ¼ðp2 − p1Þ ¼ t, 2 A. A-term p ¼ðp1 þ p2Þ=2, and the normalization uu¯ ¼ 2mN. Equation (5) is conserved and traceful. Throughout, To determine the A-term, we contract the energy- DðkÞ¼4CðkÞ will be used interchangeably. In holography, momentum form factor (1) with a spin-2 transverse- TT Eq. (1) sources the metric fluctuations in bulk, traceless polarization tensor ϵμν , TT μν TT μ ν hp2jϵμν T ð0Þjp1i¼u¯ðp2ÞðAðkÞϵμν γ p Þuðp1Þ Z δ A˜ pffiffiffi ¼ ¯ð Þ −ϕðzÞ ðψ 2 ð Þþψ 2 ð ÞÞ ð Þ ϵTTγμ ν ð Þ u p2 dz ge z R z L z h k; z × μν p u p1 δh0 2 Z A˜ pffiffiffi ¼ ¯ð Þ −ϕðzÞ ðψ 2 ð Þþψ 2 ð ÞÞχð Þ ϵTTγμ ν ð Þ ð Þ u p2 2 dz ge z R z L z k; z × μν p u p1 : 4 The last two lines follow from a tree level Witten diagram and the holographic dictionary in the soft wall construction as detailed in [13,19,21]. They correspond to the coupling of the transverse-traceless part of the graviton hðk; zÞ¼h0χðk; zÞ with χðk; 0Þ¼1 (dual to 2þþ tensor glueballs) to a Dirac fermion in bulk. More specifically [13]1 Z A˜ pffiffiffi ð Þ¼ −ϕðzÞ ðψ 2 ð Þþψ 2 ð ÞÞχð Þ A k 2 dz ge z R z L z k; z 1 þ a a ¼ A˜ ð1 − 2a Þð1 þ a2Þþa ð1 þ a Þð1 þ 2a2Þ H k − H k ð Þ k k k k k 2 2 5 ¼ − 2 8κ2 ð Þ¼ψð1 þ Þþγ with ak k = N. Here H x x is the harmonic number or digamma function plus Euler number. The κ ϕð Þ¼κ2 2 ð0Þ scale N follows from the dilaton profile z Nz . It is dual to the string tension in QCD. A is not fixed in holography ˜ (1-point function). We have defined A ¼ Að0Þ to encode Oð1=NcÞ corrections coming from 1-loop and higher Witten diagrams for the transverse-traceless tensor part of the energy momentum tensor. We have checked that (5) is in numerical agreement with a result in [21] modulo the overall normalization. 1In [13] there is a typo in the argument of the digamma function ψðxÞ → ψð1 þ xÞ. 094010-2 NUCLEON MASS RADII AND DISTRIBUTION: HOLOGRAPHIC … PHYS. REV. D 103, 094010 (2021) B. D-term 1 To determine the C-term or D-term (D ¼ 4C), we contract the energy-momentum form factor (1) with 3 ημν, 2 2 1 μν mN k k hp2jημνT ð0Þjp1i¼u¯ðp2Þ AðkÞ þ BðkÞ − CðkÞ uðp1Þ 3 3 12m m Z N N δ C˜ pffiffiffi 1 ¼ ¯ð Þ −ϕðzÞ ðψ 2 ð Þþψ 2 ð ÞÞ ð Þ η γμ ν ð Þ u p2 dz ge z R z L z f k; z × μν p u p1 δf0 2 3 Z C˜ pffiffiffi m ¼ ¯ð Þ −ϕðzÞ ðψ 2 ð Þþψ 2 ð ÞÞχð Þ N ð Þ u p2 2 dz ge z R z L z k; z × 3 u p1 ¼ ¯ð Þ ð Þ mN ð Þ ð Þ u p2 AS k 3 u p1 ; 6 ˜ 2 where the constant ASð0Þ¼C encodes the Oð1=NcÞ 0 ΔS → 4 þ β ðαsÞ − βðαsÞ ð8Þ corrections coming from 1-loop and higher Witten dia- αs grams for the traceful part of the energy momentum tensor.