"Mass Without Mass" and Nuclear Matter
MANNQUE RHO
INSTITUT DE PHYSIQUE THÉORIQUE, CEA SACLAY, 91191 GIFSURYVETTE CÉDEX, FRANCE
Unraveling the mystery of proton's "mass without mass" via nuclear matter was the fiveyear World Class III Project at Hanyang University entitled "From Dense Matter to Compact Stars." This is a brief account of its objective and accomplishments with possible impacts on a future direction for research in nuclear physics.
In 2008 the Korean Government launched what was called "World Class University Program" with the objective to bring Korean Universities to the world's top level in basic science, and I was invited to join as a guest scholar in the program proposed by Hyun Kyu Lee at Hanyang University in the category III. The program III consisted of my spending four months a year for 5 years to help generate, and participate in, original research activities at the graduateschool level with potentials for a breakthrough in the field encompassing nuclear and hadron physics and astrophysics. The project chosen was to understand baryonic matter compressed under extreme gravitational pressure but stable against collapse into black holes, namely, massive compact stars, such as the recently found
~2solar mass objects [1], and our objective was to explore what one can learn, beginning at the forthcoming RIB accelerator "RAON" of the Institute for Basic Science (Korea), then at FAIR (Germany) and also ultimately at advanced gravitywave detectors such as aLIGO, aVirgo as a precursor to the phenomenon that takes place under extreme conditions. What transpired from the WCU/Hanyang (W/H for short) project is the glimpse into a possible source of the proton mass, up to date unknown, as revealed in nuclear processes. I dub the problem "mass without mass" borrowing the terminology from Wilczek's 1999 Physics Today article [2].
The landscape of hadronic phases has been extensively explored at high temperatures thanks to lattice QCD on the theory side and to RHIC and now LHC on the experimental side. An amazingly simple and elegant structure has been discovered in the form of highly correlated, nearly perfect liquid with quarks and gluons. In stark contrast, the situation of high density, relevant to massive compact stars, is totally different. Lattice QCD, the only trustful theoretical tool available presently in highly nonperturbative regimes, cannot access the density relevant to the interior of compact stars, so the phase structure of baryonic matter beyond the wellstudied nuclear matter density remains totally barren and is in an urgent need to be explored experimentally.
The principal theme of the W/H project was that the origin of the proton mass plays a key role in the equation of state (EoS) for compact stars, and can be explored in the forthcoming terrestrial accelerators such as RIB machines (e.g., RAON in Korea, FRIB of MSU/Michigan...), FAIR of Darmstadt/Germany, NICA at Dubna/Russia etc. and the space observatories in operation and in project. We set this as one of the primary objectives of the ambitious Korean basic science institute IBS with the RAON playing the core role.
The basic problem is that we do not know where more than 90% of the proton mass, ~1 GeV (more precisely 938 MeV), come from. The mass of all the "visible" objects around us, down to molecules and even to nuclei, can be accurately accounted for, say, by more than 98%, by the total number of protons (and neutrons) included therein. When it comes to the mass of the proton it is no longer the sum of what constitutes its mass, quarks and gluons. The recent discovery of the Higgs boson is to account for the mass of the quarks in the proton, but it does not "explain" what the proton mass is made up of. The quark masses are nearly zero, so whatever makes up the proton mass cannot be "sum of something." This problem is not like the neutrino mass, the dark matter etc. that are the unknowns in particle physics. A particle physicist would simply say the proton mass comes from "backreaction of gluon fields resisting accelerated motion of the quarks and gluons" [2]. In fact in the near future, the QCD structure of the proton will be fully mapped out at the JLab in terms of quarks and gluons. But it will still remain far short of explaining why there are no "stuffs" with mass that make up the proton mass.
Let me state the problem in more physical terms. In the standard lore established by the pioneering work of Nambu, Goldstone and others, the proton mass (and the mass of "lightquark mesons," say, the ρ meson to be specific), if one ignores the tiny u (up) and d (down)quark masses, is said to be entirely "generated dynamically." Phrased in terms of symmetries, the mass is then attributed to the spontaneous breaking of chiral symmetry (SBCS for short). The SBCS is characterized by that the local orderparameter bilinear quark field has a nonvanishing vacuum expectation value (VEV), <>_{0 } ≠ 0. If the proton mass were entirely generated by SBCS, then one could tweak the quark condensate by density or temperature to vanish, thereby making the proton mass to disappear, i.e.,
m_{N}(<>) → 0 as <> → 0. But QCD does not say that this is the entire story. In fact, it is possible to have a mass term in the proton that does not vanish when the quark condensate goes to zero without violating chiral symmetry in the chiral limit. It is easy to understand this if one recalls the SU(2)×SU(2) GellMannLévy linear sigma model with the doublet nucleons, the triplet pions and the scalar s meson (we reserve the usual notation σ for the dilaton which will be introduced below). In this model, the nucleon and the scalar s acquire masses entirely by the VEV of the s field
<s>_{0} ≠ 0 while the pion remains massless by NambuGoldstone theorem. On the other hand, it has been recognized [3] that there is nothing to prevent the nucleon mass in an effective field theory from having the form

m_{N} = m_{0} + (<>)

(1) 
such that → 0 as the condensate is dialed to zero, provided one introduces parity doublet to the nucleon. Then the nucleon mass does not vanish if m_{0} does not. And m_{0} is a chiralsinglet. What is significant is that m_{0} can be big,
m_{0} ≈ (0.60.9)m_{N}.
Now what about the mesons?
It was realized that the situation for meson masses is quite different. To address this matter, we have found it powerful to resort to an effective field theory that incorporates in addition both scalar and vector degrees of freedom, the former as (pseudo)NambuGoldstone (NG) boson, dilaton (σ), of spontaneously and explicitly broken – via the quantum (trace) anomaly – scale symmetry and the latter V = (ρ,ω) as hidden local gauge fields. This approach naturally brings the energy/momentum scale to the scale of meson mass, ~700 MeV, in nuclear dynamics. The combined symmetry, dubbed scaleinvariant hidden local symmetry, though invisible in the vacuum, enables one to describe, at low orders in scalechiral counting, baryonic matter up to the density commensurate with the interior of massive compact stars.
The scalar dilaton as a NG boson is highly controversial, because the existence of an infrared (IR) fixed point in QCD for small number of flavors, say, N_{F} ~ 3, is neither confirmed nor ruled out. However for the number of flavors N_{F} ≳ 8, it is considered to be intimately connected to the issue of dilatonic Higgs model going beyond the Standard Model. We find the scale symmetry, hidden or even if absent in the vacuum, can "emerge" as density increases to what is referred to as "dilatonlimit fixed point (DLFP)" where the scalar s goes critical in the sense of LandauGinzburg paradigm, becoming degenerate with the zeromass pion in the chiral limit. This is in fact a notion associated with the dilaton condensate that has been around for a long time since the work I did with Gerry Brown, known as BrownRho scaling [4] and it reemerges in the work at the W/H but in a completely different guise.
As for the vector mesons, it is hidden gauge symmetry that controls how their masses scale as density increases beyond the normal n_{0}. Here the gauge symmetry, invisible in QCD in the vacuum, emerges from the redundancy residing in the product of the left and right (chiral) symmetries. A particularly important role is played by the isovector ρ meson which figures importantly at high density. It has been shown by a Wilsonian renormalizationgroup analysis that as the quark condensate goes towards zero at chiral restoration, the gauge coupling g_{ρ} should drop to zero as [5]

g_{ρ} ~ <> → 0. 
(2) 
and the ρ mass, m_{ρ}, should consequently go to zero proportional to g_{ρ}.^{1} This is the "vector manifestation (VM)" and the density at which the mass vanishes is the VM fixed point density.
^{1} I should mention here that certain dilepton experiments in heavyion collisions looking for this phenomenon have been wrongly interpreted. This was pointed out in [6].
Here then is the drastic difference between the proton mass and the ρ mass in the way chiral restoration is approached. As density increases such that the quark condensate tends to zero, the ratio of the ρ mass over the proton mass is to vanish as

m_{ρ}^{*} / m_{N}^{*} ∝ <>^{*} / m_{0} → 0.

(3) 
This is distinctly at odds with the naive interpretation of "Nambu scenario" based on spontaneous breaking of chiral symmetry, which would suggest the righthand side of (3) will go to a constant ∝ 2/3, the ratio of (constituent) quark numbers.
What turned out to be pivotal in the W/H project was the discovery in 2010 of the role that topology could play in supplying a key information on dense matter that is not available from elsewhere. The model that purported to unify, through topology, mesons and baryons in one Lagrangian, first formulated in 19591960 by Skyrme [7] primarily for nuclear physics, was resuscitated in 1983 by field theorists (see [8] for References) in terms of large N_{c} QCD, and quickly generated a great excitement among many particle and nuclear theorists. Unfortunately, due to daunting mathematical difficulties with the Skyrme model and with many astute mathematical physicists gone to string theory that became fashionable then, the progress was extremely slow in formulating nuclear dynamics in the model. Most of the practitioners of the model, highly frustrated with slow progress, soon dropped the field and went over to other problems, e.g., chiral effective field theories that were being rapidly developed at about the same time. A few diehard skrymion aficionados – among whom I was one – however stubbornly stuck to the model in attempting to solve one of the most challenging nuclear physics problems, dense baryonic matter. This effort was initiated in early 2000's at the newly established KIAS (Korea Institute of Advanced Study), and then was picked up at the W/H. The discovery made at the W/H resulted from this effort and led to the result that I describe below.
There was, and is even now, no analytic or even numerical skyrmion approach to dense matter with continuum Lagrangians. It is too difficult mathematically. However dense matter can be simulated by putting skyrmions on crystal lattice. How to do this is described in various reviews and books (see for instance [9, 11]). What turns out to come out of such simulations is new and crucially important for the physics of compactstar matter we were formulating. As the crystal size is reduced, corresponding to increased density, a skyrmion in medium is found to fractionalize, at a density that we denote as n_{1/2}, into two halfskyrmions. This phenomenon is a topological effect, depending on symmetry, and hence is robust. In fact this topological phenomenon resembles closely what has taken place in some strongly correlated condensed matter systems such as fractional quantum Hall effects, chiral superconductivity, deconfined quantum critical phenomenon, etc. [9].^{2} In our case with skyrmions, it depends only on the pion degree of freedom that carries topology. The heavy degrees of freedom do not affect its existence although they do influence the precise location of the density involved, n_{1/2}. Although it cannot be pinned down theoretically, phenomenology indicates that the reasonable location for n_{1/2} is about 2n_{0} where n_{0} is the normal nuclear matter density. This is the density regime the RAON may be able to probe.
^{2} Indeed in twodimensional ferromagnets, halfskyrmions (also referred to as merons) interact via dipoledipole interactions resulting in multimeron bound states and for large topological charge Q ≫ 1, the minimal energy configuration is found to be the square lattice formed by merons [10], quite similar to the halfskyrmion phase in dense baryonic matter for density n > n_{1/2}.
Now one of the most remarkable features of the halfskyrmion phase is that the quark condensate Σ ≡ <>, when averaged over the unit cell of crystal, , goes to zero. Furthermore the skyrmion representing the nucleon turns out to have a constant mass proportional to the dilaton condensate <χ> (where χ = f_{χ}e^{σ/fχ} transforms linearly while σ is the dilaton field that transforms nonlinearly like the pion field). The mass of the skyrmion has the form

m_{skyrmion} = m'_{0} + Δ()

(4) 
and as → 0, Δ goes to zero, so it reproduces Eq. (1) with m_{0 }= m'_{0} ∝ <χ>. In the halfskyrmion phase, <χ> is more or less constant with m'_{0} ~ (0.60.9)m_{N}. This precisely reproduces the structure of the paritydoublet model. Note however the fundamental difference. Here the parity doubling emerges out of the medium whereas in the paritydoublet model, it is put in by hand. This implies that even if QCD does not, intrinsically, have the paritydoublet symmetry, it can appear as an emergent symmetry. We believe that it is the scale symmetry that emerges here.
Since a fully quantummechanical calculation in the skyrmion crystal description is not presently doable, the strategy adopted at the W/H was to map whatever robust features that could be extracted from it to an effective Lagrangian that has baryons put in explicitly as in standard nuclear chiral effective field theory. This can be done straightforwardly by coupling the baryons in a scalechiral symmetric way to the given Lagrangian with both scale and chiral symmetry breaking terms suitably incorporated. We call this Lagrangian bsHLS ("b" standing for the baryons and "s" standing for the scalar). One can then set up a scalechiral counting rule in a way parallel to the highly successful (standard) chiral perturbation theory [12]. With this suitably matched to QCD at some matching scale, one can do Wilsonian renormalizationgroup (RG) analysis for quantum manybody systems. How to do this has been formulated elsewhere in various different ways. The novelty in the W/H program was to provide the slidingvacuum – due to density – in the Lagrangian with the topological inputs playing a crucial role.
Several important impacts of the topology change come from Eqs. (2) and (4) with Eq. (1). The first is on the nuclear tensor force, one of the most important component of nuclear forces in nuclear interactions, which had been extensively studied since the early days of nuclear physics going back several decades. Now the topology change gives a totally new structure to it [13]. In the framework so formulated, the tensor force is given entirely by the exchange of a pion and a ρ. The important role of the pion has been wellknown since Yukawa's discovery of the pion: It is what binds a proton and a neutron into deuteron, and figures prominently in complex nuclei, in particular in nuclei far from the stability line. Equally significantly, the tensor force dominates in the symmetry energy, an essential part of the EoS for compact stars. What's new here however is that, firstly, the ρ meson is indispensable for the tensor force – this is being confirmed in ongoing lattice QCD calculations of nuclear forces – and, secondly, its tensor force is drastically affected by the topology change at n ~ 2n_{0} with the ρNN coupling dropping rapidly as (2). Consequently, it undergoes a dramatic change at the density n_{1/2} where the halfskyrmions appear. The net tensor force first decreases, as density increases, toward n_{1/2}, and then increases afterwards due to the suppression of the hidden gauge coupling going toward the VM fixed point. In fact the decreasing tensor force as density approaches n_{0} from below has been neatly confirmed by the long lifetime of the Carbon14, i.e., the famous C14 dating [14]. What happens at higher density as it goes above after n_{1/2} is that the topology change stiffens the EoS for compact stars for density exceeding ~2n_{0}. Without this stiffening, massive stars of ~2M_{⊙} cannot be stable against gravitational collapse.
Now what does this skyrmionhalfskyrmion topology change say about the source of the proton mass, nuclei far from stability and massive compact stars?
The proton mass deduced from the skyrmion crystal, as noted above, goes like m_{N}^{*} ∝ <χ>^{*} in n ≥ n_{1/2 }~ 2n_{0}. As implied in (3), the proton mass has a source quite different from that of the ρ mass which we think is consistent with the Nambu scenario. Now to what density does the proton mass follow <χ>, staying independent of density? We will see later that this issue comes up with the sound velocity in massive stars.
There is something special and magical, hitherto unsuspected, about the tensor force with its spin and isospin dependence, independently of where it comes from and also of the change as a function of density as manifested in the C14 dating. There is a tantalizing hint, which could perhaps be explored in RAONtype experiments, that the tensor force is not renormalized by strong interactions both inside and outside of nuclear medium [15]. In Landau Fermiliquid theory of nuclear matter, a field theory way of understanding nuclear matter and also finite nuclei, the quasiparticle interactions are "fixedpoint" quantities, becoming more accurate as k_{F}, the Fermi momentum, becomes large. As far as I know, there is no proof that the tensor force is a fixedpoint interaction. But there is a strong correlation with this tensor force and LandauMigdal G_{0}' interaction – that controls the giant GamowTeller resonances in nuclei – which is at the fixed point as Gerry Brown argued (see [15]). Let me suppose it is a fixed point interaction. Then one can look at processes, such as the monopole matrix elements measured in exotic nuclei [16], that could be studied in RAONtype accelerators probing the density regime n ≳ 2n_{0}. That would reveal the "dramatic change" of the tensor force as the density goes over n_{1/2}. It will be a challenge to figure out what experiments can probe this property that zeroes in on the problem of proton mass vs. ρ mass.
Next what about the proton mass/ρ mass issue with respect to compact stars? The answer to this question is given in the most recent publications by the postW/H collaboration [17, 18]. Once the density goes over n_{1/2}, the ρ mass runs rapidly to zero, but the proton mass, with its small dynamically generated mass Δ(Σ) vanishing, goes to a chiralinvariant constant proportional to <χ>. As mentioned, the dilaton condensate there is densityindependent. Consequently the VEV of the trace of energymomentum tensor, <θ_{μ}^{μ}>, becomes constant independent of density. Therefore ∂<θ_{μ}^{μ}> / ∂_{n} = 0. Expressed in terms of the sound velocity v_{s}, this gives (∂ε(n) / ∂n)(13(v_{s}^{2} / c^{2} )) = 0 for n > n_{1/2}. Since we do not expect LeeWicktype abnormal states in the relevant density regime, we must have
v_{s} / c = . This is a tremendously surprising result. Although there is, apparently, no known way to measure the sound velocity of dense matter in terrestrial laboratories, it has a profound implication on the structure of compactstar matter. Theoretically one would expect it to be v_{s} / c < at low density, but driven by strong interactions between nonrelativistic nucleons, increase first as density increases, and then overshoot at n > n_{0}. Finally at asymptotically high density where perturbative QCD intervenes, the sound velocity will asymptote to due to the asymptotic freedom, with <θ_{μ}^{μ}> → 0. In fact this is what is found in our theory. But in our theory, n_{1/2} is merely a couple of times the normal matter density, potentially accessible at RIB accelerators and certainly in compactstar matter, so the "conformality" seems to set in extremely precociously. How this surprising feature could appear in observables measurable at densities at or exceeding ~2n_{0} is an extremely interesting question to address in the near future. The RAON, perhaps, and surely the forthcoming accelerators such as FAIR etc. will be able to offer the glimpse into this totally uncharted region of strongly interacting matter. It could also be probed in gravity waves emitted from coalescing neutron stars to be measured in the future, for instance, in tidal deformability or Love number as predicted in the postW/H collaboration [18].
We feel what we have uncovered at the W/H is a big first step forward to "seeing" what goes on at high density. But our effort has remained largely unrecognized, not to mention appreciated, in the nuclear community, even in Korea. When the discovery of the pivotal role of the topology change in compact star physics was submitted for publication to Phys. Rev. Letter, the Divisional Associate Editor, after mediating serious differences between the authors and the referees, formally rejected the paper on the ground – to paraphrase – that "all the nuclear theorists working on the skyrmion model dropped the subject and are not working on it anymore. Therefore this paper cannot be accepted in PRL."
What we have uncovered could very well be the mere tip of a giant iceberg. There must surely be a lot to be explored. There is a fascinating and intriguing hint of a web of connections encoded in the proton mass: quarkgluon bag and skyrmions – known as "CheshireCat mechanism" [9, 19], "dyonic salts" and "popcorns" being discovered in holographic QCD from string theory etc. [9]. The celebrated mathematician Atiyah, with his theoretical physics colleagues, sees, inspired by soliton models, an even bigger picture in terms of geometry [20]. This situation could be aptly captured by the 2010 editorial in NATURE (referring to what's happening in condensed matter systems) [21]: "After reemerging from the depths of obscurity several times over, the spotlight is back on skyrmions. And a reader can only wonder what other neglected gems of mathematical ideas are tucked away in the literature, awaiting a creative scientist to recognize their value to the physical world?"
Acknowldegments: I would like to thank Yongseok Oh for inviting me to write this note and Shoji Nagamiya for his advice on its structure. Part of the material covered in this note was presented in my talks at the APCTP Workshop on Frontiers of Physics: Dense Matter from Chiral Effective Theories and at the Fujiwara Seminar at Shimoda (Japan) in tribute to Toshimitsu Yamazaki's 80h birthday.
References
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[2] F. Wilczek, "Mass without mass," Physics Today 52N11, 13 (1999); "Origins of mass," Cent. Eur. Phys. 10 (5), 1021 (2012).
[3] C. E. DeTar and T. Kunihiro, "Linear sigma model with parity doubling," Phys. Rev. D 39, 2805 (1989).
[4] G. E. Brown and M. Rho, "Scaling effective Lagrangians in a dense medium," Phys. Rev. Lett. 66, 2720 (1991).
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[7] T.H.R. Skyrme, "A unified field theory of mesons and baryons," Nucl. Phys. 31, 556 (1962).
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[19] M. Rho, A. S. Goldhaber and G. E. Brown, "Topological soliton bag model for baryons," Phys. Rev. Lett. 51, 747 (1983).
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[21] "Skyrmion Makeover: Celebrating the treasures of topological twists," NATURE 465, 17 June 2010.

Mannque Rho received his PhD degree from University of California, Berkeley, California in 1964. After a year as a postdoc, he became a permanent member at IPhT (Institut de Physique Théorique), CEA Saclay in France. He worked as Visiting Scientist at the Theory Division of CERN, Visiting Professor at Stony Brook University (Stony Brook, NY) & Nagoya University (Japan), Humboldt Professor at GSI/Darmstadt & TU/Munich (Germany) and WCU Professor at Hanyang University. He got awarded Prix Paul Langevin, FrancoGerman Humboldt Prize, Korean National Academy of Science Prize and HoAm Prize. He is currently a member of "Association pour la Science" at IPhT, CEA, Saclay. 
