Reality of Diquarks Between Nuclear and Quark Matter

Reality of Diquarks between Nuclear and Quark Matter

Kenji Fukushima

The University of Tokyo

August 8, 2014 @ JPARC 1 Our belief:

Nuclear Matter

2 Our belief: Squeeze

Quark Matter

How it happens?

3 Our belief: Squeeze

Something???

Especially Between Nuclear and Quark Matter

4 Our belief:

Something???

This is not theoretically clarified (surprisingly) So let’s go by experiment, some people say…

5 HIC (thermal fit) established: Densest Matter Created in Heavy-Ion Collision

8GeV/nucleon-nucleon 0.2 (~30GeV in lab.) ~ saturation density ]

-3 0.15 HRG estimate based on 0.1 AGS/SPS/RHIC data

160 0.05

Baryon Density [fm Chemical 120 Freeze-out line 80 0 0 40 200 400 600 800 Temperature [MeV] Baryon Chemical Potential [MeV] 6 Thermal fit works too good: / Nuclear Physics A 00 (2010) 1–4 3 stout continuum $ 5 #$ 0.6 HRG physical # HRG physical $ $ HRG distorted stout Nt!8 $ 0.5 HRG distorted Nt!8 ! ! 4 HRG distorted asqtad Nt!8 ! !# ! ! HRG distorted Nt!12 $ !" # " stout Nt!10 ! $ # $ 0.4 " asqtad Nt!8 ! ! ! 4 ! stout Nt 8 # 2 # ! # asqtad Nt!12 T 3 ! T # $ asqtad Nt!8 $% # # " ! $ s 2 0.3 ! p4 Nt!8 $ ! 3p # p4 Nt!8 $ # Χ ! $ hisq Nt!8 # " Ε# 2 ! ! # # 0.2 $ # " ! $ # " ! $ $ $ ! ! # 0.1 1 ! $ # " !! #$ " " !!!!!!!! 0.0 0 120 140 160 180 100 120 140 160 180 T MeV T MeV

Figure 2: Left panel: strange quark susceptibility as a function of the temperature. full symbols correspond to results obtained with the asqtad, p4 and hisq actions [1, 6]. Our continuum result is indicated by the gray band. The solid line is the HRG model! result" with physical masses. The dashed and dotted lines are the HRG! model" results with distorted masses corresponding to Nt = 12 and Nt = 8, which take into account the discretization e↵ects and heavier quark masses, which characterize the results of the hotQCD Collaboration. Right panel: (✏ 3p)/T 4 as a function7 of the temperature. Open symbols are our results. Full symbols are the results for the asqtad and p4 actions at Nt = 8 [1]. Solid line: HRG model with physical masses. Dashed lines: HRG model with distorted spectrums. As it can be seen, the prediction of the HRG model with a spectrum distortion corresponding to the stout action at Nt = 8 is already quite close to the physical one. The error on the recent preliminary HISQ result [6] is larger than the di↵erence between the stout and asqtad data, that is why we do not show them here. resonances. We include all known baryons and mesons up to 2.5 GeV, as listed in the latest edition of the Particle Data Book (for an improvement of the model by including an exponential mass spectrum see [13]). We will compare the results obtained with the physical hadron masses to those obtained with the distorted hadron spectrum which takes into account lattice discretization e↵ects. Each pseudoscalar meson in the staggered formulation is split into 16 mesons with di↵erent masses, which are all included. Similarly to Ref. [9], we will also take into account the pion mass- and lattice spacing- dependence of all other hadrons and resonances. Quark number susceptibilities increase during the transition, therefore they can be used to identify this region. q T @2 ln Z They are deﬁned as = 2 , (with q = u, d, s). In the left panel of Fig. 2 we show our continuum- 2 V @(µq) µi=0 extrapolated results for the strange quark number susceptibility, in comparison with the HRG results with physical spectrum. Also shown are the hotQCD collaboration data, in comparison with the HRG model results with distorted spectrum. In the right panel of Fig. 2 we show the trace anomaly (✏ 3p) divided by T 4 as a function of the temperature. Our Nt = 8 results are taken from Ref. [14]. Notice that, for this observable, we have a check-point at Nt = 10: the results are on top of each other. Also shown are the results of the hotQCD collaboration at Nt = 8 [1] and the HRG model predictions for physical and distorted resonance spectrums. On the one hand, our results are in good agreement with the “physical” HRG model ones. It is important to note, that using our mass splittings and inserting this distorted spectrum into the HRG model gives a temperature dependence which lies essentially on the physical HRG curve (at least within our accuracy). On the other hand, a distorted spectrum based on the asqtad and p4 frameworks results in a shift of about 20 MeV to the right. In order to compare our results to those of the hotQCD collaboration, we also calculate the quantity l,s = ml ml ( ¯ l,T ¯ s,T )/ ¯ l,0 ¯ s,0) (with l = u, d). We compare our results to the predictions of the HRG h i ms h i h i ms h i model and PT [15]. To this purpose, we need to know the quark mass dependence of the masses of all resonances included in the partition function. We assume that all resonances behave as their fundamental states as functions of the quark mass, and take this information from Ref. [16]. They agree with the results obtained by our collaboration in [17].

4. Conclusions

We have presented our latest results for the QCD transition temperature. The quantities that we have studied are the strange quark number susceptibility, the chiral condensate and the trace anomaly. We have given the complete temperature dependence of these quantities, which provide more information than the characteristic temperature val- Theorists may say:

Our understanding of dense matter is so poor because of the sign problem in lattice-QCD So far, no reliable lattice results at µ =0,T=0 B 6

8 Theorists may say:

Our understanding of dense matter is so poor because of the sign problem in lattice-QCD So far, no reliable lattice results at µ =0,T=0 B 6

Is this really so? Isn’t it just a lame excuse of theorist? (Lattice is not a unique theory tool)

9 BI-TP 2009/30 CERN-PH-TH-2009-229 INT-PUB-09-060 This O(g4) calculation appearedTUW-09-19 in 2009

Cold Quark Matter

Aleksi Kurkela,1 Paul Romatschke,2 and Aleksi Vuorinen3, 4, 5 1Institute for Theoretical Physics, ETH Zurich, CH-8093 Zurich, Switzerland 2Institute for Nuclear Theory, University of Washington, Box 351550, Seattle, WA, 98195 3Faculty of Physics, University of Bielefeld, D-33501 Bielefeld, Germany 4CERN, Physics Department, TH Unit, CH-1211 Geneva 23, Switzerland 5ITP, TU Vienna, Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria We p erform an (α2)perturbativecalculationoftheequationofstateofcoldbut O s dense QCD matter with two massless and one massive quark flavor, finding that perturbation theory converges reasonably well for quark chemical potentials above 1GeV.Usingarunningcouplingconstantandstrangequarkmass, and allowing for further non-perturbative effects, our results point to a narrow range where absolutely stable strange quark matter may exist. Absent stable strangequarkmatter,our findings suggest that quark matter in compact star cores becomes confined to hadrons only slightly above the density of atomic nuclei. Finally, weshowthatequations of state including quark matter lead to hybrid star masses up to M 2M ,in ∼ ⊙ agreement with current observations. For strange stars, we find maximal masses of M 2.75M and conclude that confirmed observations of compact stars with ∼ ⊙ M>2M would strongly favor the existence of stable strange quark matter. ⊙

10 arXiv:0912.1856v2 [hep-ph] 29 Jan 2010 3

I. INTRODUCTION

The properties of cold nuclear matter at densities above that of atomic nuclei, in particular its equation of state (EoS) and the location of the phase transition to deconfined quark matter, remain poorly known to this day. The difficulty in performing first principles calculations in such systems can be traced back to the complicated non-linear and non- perturbative nature of Quantum Chromodynamics (QCD). These properties have precluded an analytic solution describing confinement, while non-perturbative numerical techniques, such as lattice QCD, are inapplicable at large baryon densities and small temperatures due to the so-called sign problem. This should be contrasted with the situation at small baryon density and large temperatures, where close to the deconfinement transition region lattice QCD has provided controlled results for the EoS as well as the nature of the transition [1, 2], while at temperatures much above the transition, the system is well described by analytic results from resummed perturbation theory [3–6]. Experimentally, the high temperature / low baryon density regime of QCD can be studied in relativistic heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) [7–10] and in the future at the Large Hadron Collider (LHC) [11]. Collisions at lowerenergy,e.g. at the Alternating Gradient Synchrotron (AGS) and Super Proton Synchrotron (SPS) [12, 13], as well as those planned at the Facility for Antiproton and Ion Research (FAIR) and RHIC [14, 15], study QCD matter at somewhat higher baryon density, and may give some insight into the EoS of cold nuclear matter. However, at truly low temperatures and supra- nuclear densities, QCD matter exists only in somewhat inconveniently located ’laboratories’: Compact stars. In the cores of compact stars, nuclear matter is expected to reach densities several times 3 that of atomic nuclei nsat 0.16 fm− , so that astrophysical observations may be able to provide critical information∼ about the EoS of strongly interacting matter in a regime inac- cessible to terrestrial experiments. Theoretically, the bulk properties of nuclear matter at or close to nsat have been studied using microscopic calculations [16] as well as phenomenological mean-field theory [17]. While giving matching results for symmetric nuclear matter (equal number of protons and neutrons) [18], in neutron rich matter, relevant for compact star cores, the theoretical predictions differ amongst themselvesbymorethan100percent for basic quantities such as the pressure at n nsat [19]. Extrapolating to higher densities further increases these differences, on top of which∼ new phenomena such as pion and kaon condensation [20, 21] and the presence of hyperons (such as Λ, Σ±) only add to enlarge the uncertainties in the EoS. At some critical density, nuclear matter is expected to undergo a phase transition to deconfined quark matter, which is theoretically well understood only at asymptotically high densities, where the QCD coupling αs is small [22]. There, the stable ground state of quark matter is that of a color superconductor, but it is not known whether such a state persists to densities closer to the deconfinement transition, or whether normal unpaired quark matter, or some novel phase, becomes favored. Even the possibility of the normal quark phase being the fundamental ground state of nuclear matter (with atomic nuclei being only metastable) has beenSaying: suggested in the so-called strange quark matter hypothesis [23–25]. This opened up the possibility for entire stars being madeThis up of self-boundis unfortunately quark matter (’strange true… stars’) [26]. 4 Interestingly, despite all the advances in our understanding of QCD, most of the analysis4 of cold but dense quark matter still continues to be performed usingtheMITbagmodel dating back 35 years [27]. In this model, the interactions between quarks are absorbed into adating phenomenological back 35 years ’bag [27]. constant’, In this model, which the is not interactions calculable between within the quarks model are but absorbed effectively into generateda phenomenological by the QCD ’bag interactions constant’, (see which Ref. is [28] not for calculable a discussion withinof thethis mo issue),del but and e isffectively simply addedgenerated to the by the pressure QCD interactionsof a non-interacting (see Ref. system. [28] for a We discussion believeof tha thist a issue), refinement and is of simply this model,added to using the a pressure perturbative of a EoS non-interacting for quark matter system. evaluated We believe with a tha runningt a refinementαs and strange of this quarkmodel, mass, using should a perturbative be quite EoS superior for quark to the matter plain evaluated bag model, with and a r —unning absentαs advancesand strange in trulyquark non-perturbative mass, should be methods quite superior — should to the replace plain the bag latter model, whene andver — aiming absent advancesfor at least in semi-quantitativetruly non-perturbative results. methods — should replace the latter whenever aiming for at least semi-quantitativeTo this end, in this results. paper we consider the perturbative evaluation of the QCD pressure at To this end, in this paper we consider the perturbative evaluation of the QCD2 pressure at zero temperature, where the state-of-the-art result is still the pioneering order αs calculation 2 ofzero Freedman temperature, and McLerran where the [29, state-of-the-art 30] and Baluni result [31]. is stillThese the authors, pioneeringhowever, order onlyαs calculation included of Freedman and McLerran [29, 30] and Baluni [31]. These authors, however, only included2 effects of the strange quark mass up to order αs, dropping the mass entirely at order αs.As 2 presenteffects of day the knowledge strange quark suggests mass a up strange to order quarkαs, droppingmass of about the mass 100 entirely MeV [32], at orderwith atomicαs.As nucleipresent corresponding day knowledge to suggests a quark a chemical strange quark potential mass of of roughly about 1 30000 MeVMeV, [32], one with can expect atomic non-negligiblenuclei corresponding strange to quark a quark mass chemical effects in potential the EoS (cf. ofRef. roughly [33]). 300 InMeV, view of one the can situation, expect non-negligible strange quark mass effects in the EoS (cf. Ref. [33]). In view2 of the situation, we believe that a perturbative calculationClassic of the cold works QCD EoS in to orde1977r αs — including the 2 completewe believe strange that a quarkperturbative mass eff calculationects — is long of the overdue. cold QCD This EoS provide to ordes ther αs motivation— including for the us tocomplete take on strange this challenge quark mass in the eff presentects —Still work.is long overdue.the state-of-the-art-result? This provides the motivation for us toOur take paperon this is challenge organized in as the follows. present work. In Section II, we introduce our notation, explain 11 howOur renormalization paper is organized is performed, as follows. and Inoutline Section the II, general we introduce structure our of the notation, computation. explain Inhow Section renormalization III, we then is performed, go through and all outlinethe different the general parts of structu the carelculation, of the computation. presenting theIn Section results for III, the we individual then go through terms and all inthe the diff enderent assembling parts of the entcalculation,ire grand presenting canonical potentialthe results of for the the system. individual Section terms IV is and devoted in the to end a detailed assembling analysis the entof ourire grand result, canonical covering aspectspotential such of the as the system. choice Section of the IVrenormalization is devoted to scale a detailed and the analysis dependenceof our of result, the result covering on theaspects strange such quark as the mass. choice Having of the renormalization gained control of scale the and perturbat the depiveendence EoS, in of Section the result V we on considerthe strange various quark applications mass. Having of it, gainedstudying control the scenarios of the perturbat of stable strangeive EoS, quark in Section matter V and we aconsider phase transition various applications between ordinary of it, studying quark matterthe scenarios and the of stable hadronstrangeic phase. quark In Section matter andVI, wea phase finally transition consider between the implications ordinary of quark our work matter on and astrophysical the hadron systicems, phase. while In Section in Section VI, VIIwe finally we draw consider our conclusions. the implications Several of technical our work details, on astrophysical as well as m systostems, of the while partial in Section results ofVII our we computation, draw our conclusions. are left to Several Appendices technical A–E. details, as well as most of the partial results of our computation, are left to Appendices A–E.

II. SETUP II. SETUP The equation of state of a thermodynamic system is dictated by the functional relation betweenThe equation some fundamental of state of aquantity, thermodynamic such as thesystem pressure is dictated or energy by the density,functional and relationvarious (usuallybetween intensive) some fundamental parameters, quantity, such as such the as temperature the pressure and or di enﬀeergyrent density,chemical and potentials. various In(usually the grand intensive) canonical parameters, ensemble, such it can as be the solved temperature from the and grand diﬀe porenttential, chemical or Landau potentials. free energy,In the grand canonical ensemble, it can be solved from the grand potential, or Landau free energy, Ω = E µN = T ln Z = PV, (1) Ω = E − µN = − T ln Z = − PV, (1) where E is the (microcanonical) energy,− and Z−the partition− function of the system. In this paper,where E weis set the out (microcanonical) to perform a perturbative energy, and evaluationZ the partition of the gra functionnd potential of the of system. QCD to In order this 4 2 gpaper,=(4 weπαs set) in out the to strong perform coupling a perturbative constant, evaluation keeping the of the temperature grand potential at zero ofbut QCD assuming to order 4 2 theg =(4 quarkπαs chemical) in the strongpotentials coupling to be constant, high enough keeping so that, the temperature due to asymptotic at zero but freedom, assuming the expansionthe quark converges chemical potentials to a satisfactory to be high degree. enough Various so that, thermod dueynamic to asymptotic quantities freedom, can then the beexpansion obtained converges from the to grand a satisfactory potential, whichdegree. itself Various is determined thermodynamic by computing quantities (minus) can then the sumbe obtained of all the from one-particle-irreducible the grand potential, (1PI) which vacuum itself is graphs determined of the bytheory. computing (minus) the sum of all the one-particle-irreducible (1PI) vacuum graphs of the theory. Assume (thinking exp.):

No sign problem: you can have whatever from lattice or Ideal exp. device: you can squeeze matter arbitrarily

12 Assume (thinking exp.):

No sign problem: you can have whatever from lattice or Ideal exp. device: you can squeeze matter arbitrarily

How to characterize “quark matter”? Is there any “order parameter” or “signature” for quark matter ?

13 “Order parameters” at finite-T:

[Polyakov Loop] Wuppertal 1.0

2 f /T !s /T e q 0.8 0.6 [Strangeness Fluct.] 0.4 Renormalized Polyakov Loop 2 Order Parameters @ p 0.2 0.0 s 2 100 150 200 250 300 350 ⇠ @µs T [MeV] 14 “Order parameters” at finite-T:

[Polyakov Loop]

fq /T Always small if T~0 e Not work as an order param. [Strangeness Fluct.] at low temperature @2p s 2 ⇠ @µs 15 “Order parameters” at finite-T:

[Polyakov Loop]

fq /T e µ µ (= µ /3) s ⇠ q B [Strangeness Fluct.] Not work as an order param. at high baryon density @2p s 2 ⇠ @µs 16 Theorists (incl. me) would claim:

Diquarks are necessary (and real) ingredients to go from Nuclear Matter to Quark Matter

Color superconductivity could be relevant already near Nuclear Matter ? (Or, undistinguishable from Nuclear Matter)

This summarizes what I am addressing

17 What are Diquarks?

Very subtle and profound question

18 What are Diquarks?

Very subtle and profound question

A clear introduction : Quark Model or MIT Bag Model

As real as Constituent Quarks

Lattice-measurable (gauge invariant) object?

19 On the lattice in the Landau gauge:

40 1.6 ρ(ω) (3−03−) diquark ρ(ω) (3−16) diquark 35 (613−) diquark 1.4 (606) diquark 30 1.2

25 1

20 κ = 0.147 0.8 κ = 0.147

15 0.6

10 0.4

5 0.2 (a) ω [GeV] (b) ω [GeV] 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5

“good”/“bad”Figure 2: Spectral functions diquarks for color anti-triplet (a) andothersextet diquarks diquark states (b). mass of theSeems316¯ diquark to be has a aboutreal object the same in value “some” in the regime chiral limit, whereas the 606 diquark mass obviously is not well represented by a unique mass. This favours the interpretation of color sextet states being unbound, in accordance with the expected repulsive q-q interaction in this channel 4. 20

Diquark state (30¯ 3)¯ (613)¯ (316)¯ (606)

ma(FSC):MEMresult 0.60(2) 0.70(3) 0.74(9) — ma(FSC):2-exp.ﬁt 0.62(2) 0.73(4) 0.77(17) 0.50(15)

Table 1: Diquark masses in the chiral limit obtained with MEM and two-exponential ﬁts. Our notation for quantum numbers of the diquark states is (Flavor, Spin, Color)-representation.

5ThermalSpectralFunctions At non-zero temperature meson correlation functions are periodic and restricted to the Euclidean time interval [0, 1/T ], ∞ cosh(ω(τ − 1/2T )) D(τ)= dωA(ω) . !0 sinh(ω/2T ) In the high temperature limit the meson spectral functions are expected to approach those of freely propagating quark anti-quark pairs. To leading order Nc 2 this is described by the free spectral function, i.e. A(ω)= 4π2 ω tanh(ω/4T ) in the scalar channel. We want to test here whether this behaviour can be reproduced on lattices with ﬁnite temporal extent Nτ .Weusethecontinuum expression for D(τ)andevaluateitatadiscretesetofEuclideantimes,τk = k/Nτ ,withk =1, 2,...,Nτ − 1. This is shown in Fig. 3(a). The reconstructed spectral functions in Fig. 3(b) were obtained with MEM, adding Gaussian noise with the variance σ(τ)=bD(τ)/τ to the exact results.

5 Volume 8, number 3 PHYSICS LETTERS 1 February 1964 or, in the notation of ref. 3), (instead of purely mathematical entities as they would be in the limit of infinite mass). Since charge and baryon number are exactly conserved, one of [ 1~ + 1~ + + Icoso the quarks (presumably u3z or d-Y) would be abso- + [~4c~ + ~54(~ + i ~.: + ~5 sin 0. lutely stable *, while the other member of the doublet would go into the first member very slowly by We thus obtain all the features of Cabibbo's picture 8) H-decay or K-capture. The isotopic singlet quark of the weak current, namely the rules I AI[ = 1, would presumably decay into the doublet by weak AY = 0 and I/x/[ =~,~ AY/AQ = +1, the conserved interactions, much as A goes into N. Ordinary A Y= 0 current with coefficient cosVolume 0, the vector 8, number matter 3 near the earth's surface wouldPHYSICS be conta- LETTERS 1 February 1964 current in general as a component of the current of minated by stable quarks as a result of high energy the F-spin, and the axial vector current transform- cosmic ray events throughout the earth's history, ing under SU(3) as the same component of another but the contamination is estimated to be so small octet. Furthermore, we have 3) the equal-time that it would never have been detected. A search commutation rules for the fourth components of the for stable quarks of charge -~ or +2 and/or stable currents : di-quarks of charge -~ or +-~ or +-~ at the highest energy accelerators would help to reassure us of the Anon-existence SCHEMATIC of real quarks.MODEL OF BARYONS AND MESONS These ideas were developed during a visit to - 2fjkl [~4(x)± ~Z~(x)] 6(x-x'), ColumbiaVolume University 8, number in 3 March 1963 ; M.the GELL-author PHYSICS MANN LETTERS 1 February 1964 would like to thankCalifornia Professor Institute Robert Serberof Technology, for Pasadena, California stimulating them. i = 1,... 8, yielding the group SU(3) × SU(3). We References Received 4 January 1964 can also look at the behaviour of the energy density 1) M.Gell-Mann, California Institute of Technology ~44(x) (in the gravitational interaction) under equal- Synchrotron Laboratory Report CTSL-20 (1961). time commutation with the operators ~24(x')+~45(x'). 2) Y.Ne'eman, Nuclear Phys.A SCHEMATIC26 (1961) 222. MODEL OF BARYONS AND MESONS That part which is non-invariant under the group 3) M.Gell-Mann, Phys.Rev. 125 (1962) 1067. If we assume 4)that E.g.: the R.H.Capps, strong Phys.interactions Rev. Letters 10of (1963) bary- 312; bern t - n~ would be zero for all known baryons and will transform like particular representations of M. GELL- MANN ons and mesons areR. E.correctly Cutkosky, J.Kalckar described and P. Tar]anne,inCalifornia terms Physics Instituteof mesons.of Technology, The Pasadena, most Californiainteresting example of such a SU(3) × SU(3), for example like (3, 3) and (3, 3) if it Letters 1 (1962) 93; 1 comes just from the masses of thethe quarks. broken "eightfoldE.Abers, way" F. Zachariasen1-3) we andare A. Ctempted . Zemaeh, Phys.to model is one in which the triplet has spin ~ and All these relations can now be abstractedlook for fromsome fundamentalRev. 132 (1963) explanation 1831; of the situa- Receivedz = -1, 4 January so that 1964 the four particles d-, s-, u ° and b ° the field theory model and used in a dispersion the- S.Glashow, Phys. Rev. 130 (1963) 2132; tion. A highly promisedR. E. Cutkosky approach and P. Tarjanne, is the Phys.purely Rev. 132dy- (1963) exhibit a parallel with the leptons. ory treatment. The scattering amplitudesnamical for "bootstrap"strong- 1354. model for all the strongly in- A simpler and more elegant scheme can be ly interacting particles on the mass shell are as- 5) P.TarjanneIf we assumeand V.L. thatTeplitz, the Phys.strong Rev. interactions Letters 11 of bary- bern t - n~ would be zero for all known baryons and sumed known; there is then a systemteracting of linear particlesdis- (1963) onswithin 447.and mesonswhich areone correctlymay try described to de- in termsconstructed of mesons. if weThe allow most non-integralinteresting example values of such for a the 1 persion relations for the matrix elementsrive isotopic of the spin6) Jamesandthe brokenstrangenessJoyee, Finnegan's"eightfold conservation Wake way" (Viking 1-3) Press,we andare New tempted charges. to modelWe canis one dispense in which theentirely triplet haswith spin the ~ andbasic weak currents (and also the electromagnetic and York,look 1939)for somep.383. fundamental explanation of the situa- z = -1, so that the four particles d-, s-, u ° and b ° broken eightfold 7)symmetry M. Gell-Mann fromand M. L~vy,self-consistency Nuovo Cimente 16 (1960) baryon b if we assign to the triplet t the following gravitational interactions) to lowest order in these tion. A highly promised approach is the purely dy- exhibit a parallel with the leptons. alone 4). Of course,705. with only strong interactions, properties: spin ½, z = -~, and baryon number -~. interactions. These dispersion relations, unsub- 8) N.namical Cabibbo, "bootstrap"Phys.Rev. Letters model 10 for(1963) all 531. the strongly in- A simpler and more elegant2 schemet can be 1 tracted and supplemented by the non-linearthe orientation com- of theteracting asymmetry particles in within the unitarywhich one may try Weto de- then constructedrefer to the if we members allow non-integral u3, d-~, values and for s-3- the of mutation rules abstracted from thespace field theory, cannot be specified;rive isotopic one spin hopes and strangenessthat in some conservation the andtriplet charges. as "quarks" We can dispense6) q and entirely the memberswith the basic of the may be powerful enough to determine all the matrix way the selection ofbroken specific eightfold components symmetry fromof the self-consistency F- anti-triplet baryon as b anti-quarksif we assign to ~1. the Baryonstriplet t the can following now be elements of the weak currents, including the effec- alone 4). Of course, with only strong interactions, properties: spin ½, z = -~, and baryon number -~. tive strengths of the axial vector current matrix * There is the alternative possibility that the quarks are 2 t 1 spin by electromagnetismunstablethePrimeval orientation under and decay the of into the weak expressionbaryon asymmetry interactionsplus anti-di-quark in the unitary or constructed We thenfrom refer quarks to the membersby using u3, the d-~, combinations and s-3- of elements compared with those of the vector current. determines the choiceanti-baryonspace of cannot isotopic plus quadri-quark. be specified; spin andIn oneany hyper- ease,hopes some that par-in some (qqq), (qqqqq),the triplet asetc., "quarks" while 6) q mesonsand the membersare made of the out It is fun to speculate about the way quarks would ticleway ofof fractionalthe diquarks selection charge of would specific have componentsto be absolutely of the F- anti-triplet as anti-quarks ~1. Baryons can now be behave if they were physical particlescharge of finite directions. mass stable. of (qcl), (qq~tcl), etc. It is assuming that the lowest Even if we considerspin bythe electromagnetism scattering amplitudes and the weak of interactions baryon configurationconstructed from (qqq) quarks gives by using just the thecombinations represen- determines the choice of isotopic spin and hyper- (qqq), (qqqqq), etc., while mesons are made out strongly interactingcharge particles directions. on the mass shell only tations 1,of (qcl),8, and (qq~tcl), 18 that etc. haveIt is assumingbeen observed, that the lowest while and treat the matrix elementsEven if we ofconsider the weak, the scattering electro- amplitudes the oflowest baryon meson configuration configuration (qqq) gives (q q) just similarly the represen- gives magnetic, and gravitationalstrongly interacting interactions particles by onmeans the mass shelljust only 1 andtations 8. 1, 8, and 18 that have been observed, while of dispersion theory,and theretreat theare matrix still elements meaningful of the andweak, electro-A formalthe lowest mathematical meson configuration model based(q q) similarly on field gives 21 magnetic, and gravitational interactions by means just 1 and 8. important questionsof regarding dispersion theory,the algebraic there are proper-still meaningful theory and can Abe formal built mathematical up for the modelquarks based exactly on field as for ties of these interactionsimportant that questions have soregarding far been the algebraicdis- proper-p, n, A intheory the oldcan beSakata built up model, for the quarksfor example exactly as3) for cussed only by abstractingties of these the interactions properties that from have soa far beenwith dis- all p,strong n, A in interactions the old Sakata model,ascribed for exampleto a neutral 3) formal field theory cussedmodel only based by abstracting on fundamental the properties fromvector a mesonwith all fieldstrong interacting interactions ascribedsymmetrically to a neutral with formal field theory model based on fundamental215 vector meson field interacting symmetrically with entities 3) from whichentities the 3)baryons from which and the mesons baryons andare mesonsthe are three the particles. three particles. Within Within such such a aframework, framework, thethe built up. built up. electromagneticelectromagnetic current current (in units(in units of of e) e) isis justjust If these entities wereIf these octets, entities we were might octets, expect we might the expect the u - u - d -d - s}s} underlying symmetryunderlying group symmetryto be SU(8) group instead to be SU(8) of instead of SU(3); it is therefore tempting to try to use unitary or ~-3~ + ~8~/J3 in the notation of ref. 3). For the SU(3); it is thereforetriplets tempting as fundamental to try toobjects. use unitary A unitary tripletor ~-3~ t +weak ~8~/J3 current, in wethe can notation take over of from ref. the 3). Sakata For the triplets as fundamentalconsists objects. of an isotopic A unitary singlet triplet s of electric t chargeweak zcurrent, model thewe form can suggested take over by Gell-Mannfrom the and Sakata L4vyT), consists of an isotopic(in units singlet of e) ands of an electric isotopic doubletcharge (u, z d) withmodel thenamely form i p7~(l+Y5)(nsuggested bycos Gell-Mann0 + h sin 8), whichand L4vyT), gives (in units of e) and ancharges isotopic z+l anddoublet z respectively. (u, d) with The anti-tripletnamely iin p7~(l+Y5)(n the quark scheme cos the 0expression + h sin 8),*** which gives has, of course, the opposite signs of the charges. i u ya(1 + y5)(d cos 0 + s sin 0) charges z+l and z respectively.Complete symmetry The among anti-triplet the members of thein the quark scheme the expression *** has, of course, the tripletopposite gives signsthe exact of eightfoldthe charges. way, while a mass difference, for example, between the isotopic dou- i u ya(1 + y5)(d cos 0 + s sin 0) Complete symmetry among the members of the * Work supported in part by the U. S. Atomic Energy triplet gives the exactblet eightfoldand singlet givesway, thewhile first-order a mass violation. Commission. For any value of z and of triplet spin, we can difference, for example, between the isotopic dou- ** This is similar to the treatment in ref. 1). See also construct baryon octets from a basic neutral baryon* Work supportedref. 5). in part by the U. S. Atomic Energy blet and singlet givessinglet the bfirst-order by taking combinations violation. (btt), Cotttt), Commission.*** The parallel with i ~e Ya( 1 + ¥5) e and i ~ ¥~(1 + ¥5)~ etc. **. From (btt), we get the representations 1 is obvious. Likewise, in the model with d-, s-, u °, For any value of z and of triplet spin, we can ** This is similarand b ° discussed to the treatmentabove, we would in ref. take 1).the weakSee cur-also construct baryon octetsand 8, fromwhile froma basic (btttt) neutral we get baryon1, 8, 10, 10, andref. 5). rent to be i(b° cos e + ~o sin e) ¥~(1 + ¥5) s- 27. In a similar way, meson singlets and octets can *** The parallel+ i(u° withcos e i- ~e~o sinYa( e) 1 +ya(1 ¥5) + ¥5)e and d-. iThe ~ part¥~(1 with + ¥5)~ singlet b by taking combinationsbe made out of (tt), (btt), (tttt), Cotttt), etc. The quantum num- n(nt-n~) = 0 is just i To ¥c~(1 + 75)(d- cos e + s- sin O). etc. **. From (btt), we get the representations 1 is obvious. Likewise, in the model with d-, s-, u °, 214 and b ° discussed above, we would take the weak cur- and 8, while from (btttt) we get 1, 8, 10, 10, and rent to be i(b° cos e + ~o sin e) ¥~(1 + ¥5) s- 27. In a similar way, meson singlets and octets can + i(u° cos e - ~o sin e) ya(1 + ¥5) d-. The part with be made out of (tt), (tttt), etc. The quantum num- n(nt-n~) = 0 is just i To ¥c~(1 + 75)(d- cos e + s- sin O).

214 846 Letters to the Editor

required to be totally symmetric m accordance with SU(6). We regard SU(6) as an approximate dynamical symmetry respected by states with L=O. In the hypothetical limit of no exchange · potentials between a quark and a diquark, we would obtain an octet Regge trajectory with the 1/2+ octet as its starting resonance (see Table I). We find J=L+1/2 and P=(-1)L=(-l)J-1/2 for resonances lying on it. In reality exchange potentials cause signature splitting, which explains the ex- Downloaded from istence of the two octet trajectories, o::(l/2+, 5/2+, ... ) and r(3/2-, 7/2-.. ·). Experimen- tally the splitting does not appear to be so large as to invalidate the concept of 846Prog. Theor. Phys. Vol. 36 (1966),Letters No. 4 to the exchangeEditor degeneracy, which was originally http://ptp.oxfordjournals.org/ introduced by Arnold3) for meson reso- Baryon Resonances in a Quark Model requirednances. Thereto be cantotally be anothersymmetric octet m trajec-ac- Masakuni IDA and Reido KoBA Y ASH! cordancetory with with J=L-1/2. SU(6). WeWe regard have SU(6)no firm asexvenmental an approximate evidence dynamical in favor symmetryof its exis- lnstttute of Phystcs respectedtence, however. by states with L=O. College of General Education University of Tokyo, Komaba, Tokyo TableIn the I. hypotheticalPossible Regge limit trajectoriesof no exchange in the · potentialslimit of betweenexchange a degeneracy.quark and aThe diquark, asterisk July 12, 1966 we indicateswould obtain trajectories an octet with Regge maximum trajectory J . ∗ FIGURE 1. N spectrum, labeled by their spin and parity as JP along the abscissa from the quark by guest on August 7, 2014 FIGURE 1. N∗ spectrum,with labeled the by 1/2+ their spin octet and parity asasJP itsalong thestarting abscissa from theresonance quark model predictions of Capstick and Roberts [2]. For eachmodel stat predictionse, its π ofN Capstickbranching and Roberts fraction [2]. For each is stat shown.e, its πN branching These fraction is shown. These An interestiong classification of mesonare compared with the(see PDG judgement Table [3] of whichI). ∗ states−We have∗ b eenfind∗ identiﬁed− ∗ withJ=L+1/2 1∗ − 2∗ ...or 3 ∗ −, 4∗ and are compared with the PDG judgement [3] of which statesprovenance, have according been to the identiﬁed legend shown. with0 1 1 2 or 3 2 4 3 J resonancesprovenance, according has torecently the legend shown.been given by P=(-1)L=(-l)J-1/SU(3)L I 2 for resonances lying on it. In reality exchange potentials cause IizukaWho1l and Sinanoglu said the2l on thefirst? basis ofPerhaps a these missing states really do1/2+ not exist.3/2- If baryons5/2+were diquark–quark7/2- systems,*L+1/2 Fig. 2, as Lichtenbergsignature and8 Tassie noted splitting, more than 40 yearswhich ago [4], explains the number of statesthe ex- quark-antiquark system. Mesons are placed Downloaded from Perhaps these missing states really do not exist.would If be baryons restricted andwere in fact diquark–quark be very like that1/2- currently systems,3/2+ observed. 5/2- L-1/2 on Regge trajectories of this system.'l ButA what has thisistence do with QCD?of the There two are several octet ways totrajectories, approach that question. o::(l/2+, Fig. 2, as Lichtenberg and Tassie noted more thanThe lattice 40 provides years a ago modeling[4], of the the real number world with of a states discretisation of space-time. Whilst the lattice5/2+, clearly cannot ... ) haveand3/2+ the complete r(3/2-,5/2- rotatio 7/2+nal7/2- symmetry ..9/2- ·). of theExperimen- real world,*L+3/2 straightforwardwould be restricted extension and in fact of be verytheir like idea thatthe currently newlyto computed ob latticeserved. spectrum, as explained by Robert Edwards [5], reveals a tally the× splitting does not appear to be baryonBut whatresonances has this dowould with QCD?lead to There too aremanypattern several very like ways the SU (6 to) approaO(3) of thech quark that3/2- model: question. certainly5/2+ not7/2- that of a pointlike diquark–quark system.so Therelarge is no suggestionas to thatinvalidate the so far undiscovered the statesconcept should L+1/2 of levels,The lattice however. provides In a this modeling Letter of thewe realconsider world with a discr10 etisationPennington of space-time. (2011) WhilstProg. the Theor. lattice Phys. clearly Vol. cannot 36 have(1966), the completeNo. 4 rotatioexchangenal symmetry degeneracy, of the1/2- real 3/2+ world,which 5/2- was originallyL-1/2

a specific three-quark model of low-lying http://ptp.oxfordjournals.org/ introduced by Arnold3) for meson reso- baryonBaryonthe newly resonances, Resonances computed latticewhich in spectrum,a Quarknecessitates as Model explained a by Robert Edwards [5], reveals1/2+ a3/2- L-3/2 pattern very like the SU (6) × O(3) of the quark model:nances. certainly notThere that ofcan a pointlikebe another octet trajec- few unobserved ones. Quarks are assumed diquark–quark system. There is no suggestion that the sotory far undiscovered with J=L-1/2. states should We have no firm to obeyMasakuni para-Fermi IDA and statistics. Reido KoBA Y ASH! exvenmentalIn a similar evidence way inwe favor get of aits decuplet exis- We supposelnstttute that baryonsof Phystcs consist ofFIGURE a 2. In QCDRegge each quark is intrajectory a triplet of color. On with the left is athe three quark 3/2+ model ofdecuplet a color as singlet baryon. On thetence, right is a diquark-quarkhowever. model of a baryon, where the diquark must be in a color qq pair (orCollege a diquark) of General and Education another quarkanti-triplet, so that in thisits model starting the baryon is like amember meson as far as color (see is concerned. Table I). Reso- movingUniversity around of itTokyo, with Komaba, orbital Tokyo angular Tablenances I. lyingPossible on Reggeit have trajectories J=L+3/2 in theand limit of exchange degeneracy. The asterisk momentum L. In order that for L=O ·our P=(-1)L=(-l)J+112• Signature splitting July 12, 1966 Whoindicates invented trajectories this word?with maximum J . model can produce the 1/2+ octet and the gives rise to the well-known 0'(3/2+, 7 /2+, by guest on August 7, 2014 3/2+ decuplet, which belong to the "56" ... ) decuplet and a fJ(5/2-, 9/2-,-· ·) decuplet. An interestiong classification of meson 0 1 2 3 ... , J 3 ofresonances SU(6), the has qq recentlypair must been be givenin a byS1 SU(3)LThere Ican be three more decuplet trajec- SU(3) 22 stateIizukaFIGURE and1l 2. andformIn Sinanoglu QCD an each quark2l onsextet. is inthe a triplet basisUnwanted of color.of a On the lefttories is a three with1/2+quark J=L+l/2,3/2- model5/2+ of a color L-1/27/2- and*L+1/2 L-3/2. levelsquark-antiquarksinglet baryon.of a 1/2+ On the system.decuplet right is a diquark-quark Mesonsand a 3/2+are model placedoctet of a bar yon, whereAgain8 the diquarkwe have must beno in aexperimental color evidence anti-triplet, so that in this model the baryon is like a meson as far as color is concerned.1/2- 3/2+ 5/2- L-1/2 canon beRegge excluded trajectories if the of threethis system.'l quarks areA suggesting their existence. There seems straightforward extension of their idea to 3/2+ 5/2- 7/2+ 9/2- *L+3/2 baryon resonances would lead to too many 3/2- 5/2+ 7/2- L+1/2 levels, however. In this Letter we consider 10 1/2- a specific three-quark model of low-lying 3/2+ 5/2- L-1/2 baryon resonances, which necessitates a 1/2+ 3/2- L-3/2 few unobserved ones. Quarks are assumed to obey para-Fermi statistics. In a similar way we get a decuplet We suppose that baryons consist of a Regge trajectory with the 3/2+ decuplet as qq pair (or a diquark) and another quark its starting member (see Table I). Reso- moving around it with orbital angular nances lying on it have J=L+3/2 and momentum L. In order that for L=O ·our P=(-1)L=(-l)J+112• Signature splitting model can produce the 1/2+ octet and the gives rise to the well-known 0'(3/2+, 7 /2+, 3/2+ decuplet, which belong to the "56" ... ) decuplet and a fJ(5/2-, 9/2-,-· ·) decuplet. of SU(6), the qq pair must be in a 3S1 There can be three more decuplet trajec- state and form an SU(3) sextet. Unwanted tories with J=L+l/2, L-1/2 and L-3/2. levels of a 1/2+ decuplet and a 3/2+ octet Again we have no experimental evidence can be excluded if the three quarks are suggesting their existence. There seems Volume 60B, number 2 PHYSICS LETTERS 5 January 1976

mentum and charge conjugation: C = (-1) L +s. Field broad exotic QQQQ states and the P-wave baryons theories are intrinsically many body problems and this states overlap broad 4QQ states. In such cases one relation cannot be derived. In the cavity approxima- might expect that mixing effects will play an essential tion to the three dimensional bag ordinary hadrons role in an unravelling of partial widths. This may pro- are made by populating quark modes. It is easy to see vide a clue to an understanding of some of the elusive that this yields many states not accessible in a quark P-wave states such as the A 1 . model. Consider for example aJ = 2 P-wave meson. In Our conclusion is that the present theoretically at- the old-fashioned quark model this state must have C tractive model for particle substructure based upon -'- +1. In the bag, two states are possible: 2(S1/2P3/21 colored quarks and gluons generates a very rich spec- Volume 60B, number 2 PtfYSICS LETTERS+ P3/2gl/2)J =2 with C = 71. (The spectroscopic5 January nota- 1976 trum in the mass range just above one GeV. At present tion is borrowed from atomic physics). only a very limited amount of this has been experimen- It might seem that the appearance of these states is tally confirmed. The possible classification of the some artifact of the way we have mistreated transla- known 0 ++ mesons as QQQQ exotic quark states may UNCONVENTIONAL STATES OF CONFINEDtional invariance QUARKS in three AND dimensions. GLUONS A look ~ at one- be beginning to confirm that theory is correct and the dimension shows that this is not the case. In one di- spectrum is indeed very rich. Further experiments are R.L. JAFFE* andmension K. JOHNSON the bag theory of free fermions is exactly sol- greatly needed together with aid from theorists on get- Laboratory Jbr Nuclear Scienceuble. and DepartmentThe Hilbert of spacePhysics, consists of a set of creation op- ting the signal out from the background generated + + Massachusetts Institute of Technology,erators: Cambridge, bn, dn, Mass. n = 02139, 0, 1 ... USA ~o which act on a vacuum from the resonances which have been firmly established I Y~p) which carries the momentum label P. The first ex- If this richness is not observed, then one must seri- Received 27 Octobercited state1975 of fermion and antifermion is ½x/2 (bldo+ + ously question the cherished theoretical model we now -+ b~ d~)[ ~p)with C = ~-1. Note the contrast with the have for hadronic substructure. The spectrum of confined colored quarks and gluons is studied in the bag theory. The masses of 3-types of unconventional states are calculated: hadrons made of gluons alone;non-relativistic QQQQ exotics harmonic and QQQ oroscillator: Qt~ states inwith that unconven- system the tional quantum numbers. Many of these states are in the massHilbert range space600-1600 is the MeV. same The but known the state0 ++ mesons with themay + be sign We wish to thank the Aspen Center for Physics members of a nonet of QQ0t~ exotics rather than P-wave Q(~is spurious. states. It is an artifact of replacing a two body where this work was begun for its hospitality, and to problem by a nailed-down potential [9]. Clearly the thank our MIT colleagues, Tom deGrand, Joe Kiskis independent reality of the "bag" plays an important and Charles Thorn for many fruitful discussions on The picture of colored quarks and massless colored roleabout in this 1600 effect. MeV. Other All orconfinement our results areschemes based whichon de- these matters. gluons, coupled ala Yang-Mills and confined to the in- traptailed quarks calculations on "kinks" with in collaboratorsscalar fields suffer at MIT a similarwhich terior of hadrons has provided a qualitative understand- fate.will Indeed be published it seems elsewhere natural to [3-5]. expect such states in ing of a wide variety of hadronic phenomena. More- any theoryAlthough in which all of thethe boundcalculations state problemhave been is carriednot References over, an approximate but specific realization of this strictlyout in a thetwo MIT body bag problem, model, wethat believe is in any that relativistic our results picture - the cavity approximation to the MIT bag theory.are characteristic of any model based on confined col- [1] A. Chodos et ah, Phys. Rev. D9 (1974) 3471; A. Chodos et al., Phys. Rev. D11 (1974) 2599. [1] - Exoticahas been successful [2] in describing the spec- oredIn the quarks bag model and gluons. the energies Deep inelastic of these scatteringstates may sup- [2] T. deGrand, R.L. Jaffe, K. Johnson and J. Kiskis, MIT trum of light hadrons (1÷~- and ~3+ baryons, 0- and 1- be portsestimated. a picture They of occurrelatively first weaklyamong interactingthe P-wave quarks Volume 60B, number 2 PtfYSICS LETTERS 5 January 1976 preprint, MIT-CTP-475, to be published in Phys. Rev. mesons) in terms of a very few parameters. Here we mesonsconfined and inbaryons. hadrons The with subject a radius of ofP-wave approximately excitations [3] R.L. Jaffe, J. Kiskis and C.B. Thorn, unpublished. wish to point out that this same picture predicts the is tooone extensiveFermi. The to detaileddiscuss here.form Weof thecontent strong ourselves confining [4] R.L. Jaffe and K. Johnson, in preparation. existence of a large number of (unconventional) states, withforces a list are of not additions important to the in usualunderstanding quark model the massesmulti- [5] R.L. Jaffe and T. deGrand, in preparation. [6] K. Johnson and C.B. Thorn, to be published. many more than have so far been observed. These in- plets.or motionAmong of the the P-wave quarks mesons, since a inconstituent addition toparticle the UNCONVENTIONAL STATES OF CONFINED QUARKS AND GLUONS ~ 17] P.G.O. Freund and Y. Nambu, Phys. Rev. Lett. 34 clude several types of exotics and states consisting of 0 ++,will 1 +-, not spend1++ and much 2 ++ timenonets in thewe findregion 0 +- where , 1 +- it , is1 ++ (1975) 1646. gluons with or without quarks. The bag model pro- andturned 2 +- nonets.around Amongby these P-wave forces. baryons, This leads in naturallyaddition to R.L. JAFFE* and K. JOHNSON H. Ffitzsch and P. Minkowski, Cal. Tech. preprint CALT- vides reliable estimates of the masses of these states to athe spectrum states of which the L is = dominated 1 [70] of SU(6)by the wekinetic find ener-an L 68-492. Laboratory Jbr Nuclear Science and Department of Physics, (cf. the light hadrons) with no new parameters. It is = 1gies [56]. of theAll constituentsof these states (quarks are predicted or gluons) to andhave to a [8] J. Willemsen, MIT preprint MIT-CTP-491. Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA [9] G.W. Brandenburg et al., SLAC preprint, to be published already possible to identify some of them with experi- massesroughly similar linear to increasethe masses in massof the as known progressively P-wave more in Nuclear Physics B. Volumementally 60B, numberobserved 2 "resonances". In particularReceivedPHYSICS we are 27 LETTERS Octobermesonsconstituents 1975 and baryons. are added However, in the inlowest many 5mode Januarycases of the the1976 P- sys- [10] S.D. Drell and K. Johnson, Phys. Rev. D6 (1972) 3248. led to classify the 0 ÷+ enhancements known as the e, wavetem. meson Structure states induced share quantum by the exchange numbers ofwith colored the mentumS* and and 6 Theas charge QQQQspectrum conjugation: states. of confined If correct, C colored = (-1) this quarks L +s.assignment andField gluons is studiedbroadgluons inexotic the is bagimportant QQQQ theory. states Thein generating masses and the of 3-typesP-wave the spectra of baryons uncon- of the ventional states are calculated: hadrons made of gluons alone; QQQQ exotics and QQQ or Qt~ states with unconven- theoriesdisrupts are furtherintrinsically the already many bodyuneasy problems state of andthe P-this stateslight overlap hadrons broad [2], 4QQand continuesstates. In suchto be cases important one here. tional quantum numbers. Many of these states are in the mass range 600-1600 MeV. The known 0 ++ mesons may be relationwave cannotmesonsmembers be inof derived. thea nonet quark of In QQ0t~ model.the cavity exotics approxima- rather than P-wave Q(~might states.The expectbag model that combinesmixing effects these will ingredients play an essentialin a frame- tion toOur the conclusion three dimensional will be thatbag eitherordinary these hadrons states will rolework in an which unravelling allows ofus partialto calculate widths. and This demonstrate may pro- are bemade found by bypopulating experimentalists quark modes. or our It confined, is easy to quark-see vide204this a clueproblem to an explicitly. understanding None ofof some our conclusionsof the elusive rest P-wave states such as the A 1 . thatgluon thisThe yieldstheory picture many of of hadrons colored states not isquarks as accessible yet andlacking massless in ina quarksome colored fun- aboutupon 1600whether MeV. one All views or our the results bag as are fundamental based on de- or as Our conclusion is that the present theoretically at- model.gluons,damental, Consider coupled dynamical for ala example Yang-Mills ingredient aJ =and 2which P-waveconfined will meson. forbidto the In thein- taileda phenomenological calculations with realization collaborators of a atfield-theoretic MIT which ef- tractive model for particle substructure based upon theteriorexistence old-fashioned of hadrons of these quark has states providedmodel or elevate this a qualitativestate them must to understand-havemuch C willfect. be published elsewhere [3-5]. Can explain why1 a0(980) heaviest without strangeness -'- +1.inghigher Inof the amasses. wide bag, variety twoOur statesdiscussion of hadronic are possible: will phenomena. be, 2(S1/2P3/2of necessity, More- coloredAlthoughThe quarks states all and we of gluonswishthe calculations to generates discuss are havea veryof beenthree rich carried classes:spec- + P3/2gl/2)Jover,brief. anWe approximate will=2 with consider C = but71. only specific(The states spectroscopic realization with mass of lessnota- this than trumoutI) gluonicinin thethe MITmass hadrons: bagrange model, statesjust abovewe without believe one quarks,GeV. that ourAt where presentresults a Exotic component is to be mixedonlycolor a very singlet limited(via state amount instantonof several of thisgluons has isbeenint.) confined experimen- by the tionpicture is borrowed - the cavityfrom atomicapproximation physics). to the MIT bag are characteristic of any model based on confined col-23 It[1] Thismight - workhas seem been is supported that successful the inappearance part [2] throughin describing of funds these provided statesthe spec- is by tallyoredsame confirmed. quarks mechanisms and The gluons. whichpossible Deep confine classification inelastic quarks; scattering of2) theType sup- 1 ex- sometrumERDA artifact of lightunder of hadronstheContract way (1÷weAT(11-1)-3069.~- andhave ~3+ mistreated baryons, 0-transla- and 1- knownportsotic quarka0 ++picture mesons states: of relatively asin QQQQparticular weakly exotic QQQQ interactingquark mesons states quarks (6Qmay and tionalmesons)* A.P. invariance Sloan in termsFoundation in threeof a very dimensions.Fellow. few parameters. A look atHere one- we beconfined QQQQQbeginning in states tohadrons confirm are lesswith that prominent a radiustheory of isand approximately correct will beand dis the dimensionwish to pointshows out that that this this is not same the picture case. In predicts one di- the spectrumone Fermi. is indeed The detailed very rich. form Further of the experimentsstrong confining are greatly needed together with aid from theorists on get-201 mensionexistence the ofbag a theorylarge number of free offermions (unconventional) is exactly sol-states, forces are not important in understanding the masses uble. The Hilbert space consists of a set of creation op- ting the signal out from the background generated many more+ + than have so far been observed. These in- or motion of the quarks since a constituent particle erators:clude bn,several dn, ntypes = 0, of1 ...exotics ~o which and act states on aconsisting vacuum of fromwill thenot resonancesspend much which time havein the been region firmly where established it is I Y~p)gluons which with carries or without the momentum quarks. Thelabel bag P. modelThe first pro- ex- turnedIf this around richness by is these not observed, forces. This then leads one naturally must seri- to citedvides state reliable of fermion estimates and ofantifermion the masses is of ½ x/2these (bld stateso+ + ouslya spectrum question which the cherishedis dominated theoretical by the kineticmodel weener- now -+ b~(cf. d~)[ the ~p)with light hadrons) C = ~-1. with Note no the new contrast parameters. with theIt is havegies for of thehadronic constituents substructure. (quarks or gluons) and to a non-relativisticalready possible harmonic to identify oscillator: some inof thatthem system with experi- the roughly linear increase in mass as progressively more Hilbertmentally space observed is the same "resonances". but the state In withparticular the + wesign are constituentsWe wish to arethank added the inAspen the lowest Center mode for Physics of the sys- is spurious.led to classify It is an the artifact 0 ÷+ enhancements of replacing aknown two body as the e, wheretem. thisStructure work wasinduced begun by forthe its exchange hospitality, of colored and to problemS* and by 6 aas nailed-down QQQQ states. potential If correct, [9]. this Clearly assignment the thankgluons our is MITimportant colleagues, in generating Tom deGrand, the spectra Joe Kiskisof the independentdisrupts further reality the of thealready "bag" uneasy plays state an important of the P- andlight Charles hadrons Thorn [2], for and many continues fruitful to discussionsbe important on here. rolewave in this mesons effect. in Otherthe quark confinement model. schemes which theseThe matters.bag model combines these ingredients in a frame- trap quarksOur conclusion on "kinks" will in bescalar that fields either suffer these astates similar will work which allows us to calculate and demonstrate fate.be Indeed found byit seems experimentalists natural to expect or our suchconfined, states quark- in this problem explicitly. None of our conclusions rest anygluon theory theory in which of hadrons the bound is as state yet lackingproblem in issome not fun- Referencesupon whether one views the bag as fundamental or as strictlydamental, a two dynamicalbody problem, ingredient that is which in any will relativistic forbid the a phenomenological realization of a field-theoretic ef- theory.existence of these states or elevate them to much [1]fect. A. Chodos et ah, Phys. Rev. D9 (1974) 3471; A. Chodos et al., Phys. Rev. D11 (1974) 2599. Inhigher the bagmasses. model Our the discussion energies willof these be, of states necessity, may The states we wish to discuss are of three classes: [2] T. deGrand, R.L. Jaffe, K. Johnson and J. Kiskis, MIT be estimated.brief. We will They consider occur onlyfirst amongstates with the P-wavemass less than I) gluonicpreprint, hadrons: MIT-CTP-475, states to without be published quarks, in Phys. where Rev. a mesons and baryons. The subject of P-wave excitations [3]color R.L. singlet Jaffe, stateJ. Kiskis of severaland C.B. gluons Thorn, is unpublished. confined by the is tooThis extensive work is supportedto discuss in here. part throughWe content funds ourselvesprovided by [4]same R.L. mechanisms Jaffe and K. which Johnson, confine in preparation. quarks; 2) Type 1 ex- with ERDAa list ofunder additions Contract to AT(11-1)-3069. the usual quark model multi- [5]otic R.L. quark Jaffe states: and T. in deGrand, particular in preparation. QQQQ mesons (6Q and [6] K. Johnson and C.B. Thorn, to be published. plets.* A.P. Among Sloan theFoundation P-wave Fellow.mesons, in addition to the QQQQQ states are less prominent and will be dis 17] P.G.O. Freund and Y. Nambu, Phys. Rev. Lett. 34 0 ++, 1+-, 1++ and 2 ++ nonets we find 0 +- , 1 +- , 1++ (1975) 1646. and 2 +- nonets. Among P-wave baryons, in addition H. Ffitzsch and P. Minkowski, Cal. Tech. preprint CALT-201 to the states of the L = 1 [70] of SU(6) we find an L 68-492. = 1 [56]. All of these states are predicted to have [8] J. Willemsen, MIT preprint MIT-CTP-491. [9] G.W. Brandenburg et al., SLAC preprint, to be published masses similar to the masses of the known P-wave in Nuclear Physics B. mesons and baryons. However, in many cases the P- [10] S.D. Drell and K. Johnson, Phys. Rev. D6 (1972) 3248. wave meson states share quantum numbers with the

204 Bare vs Constituent Meson qq¯ + qqq¯ q¯ + qqq¯ qq¯ q¯ + ⇠ ··· (Vacuum Re-organized) q q¯ + (Bag Constant) ⇠ con con M qq¯ How can we be so sure about ⇠ B qqq ⇠ beyond quantum num of Quark Model ?

24 Quark Number Scaling 894 Roy A. Lacey

.1 V 2 /n q

.05

FIG. 2: (Color online)“Scaling”v2 vs. pT (left panel) tells and KE Tus(middle a panel).lot Theabout scaled results complex in the right panel systems is obtained via nq scaling of the data shown in the middle panel. Results are shown for several particle species produced in minimum bias Au+Au collisions at √sNN = 200 GeV [26]. In principle applied to confirm exotic hadrons (S.H. Lee) August 8, 2014 @ JPARC 25 of perfect fluid hydrodynamics for the scaling of the elliptic flow coefficient v2 with eccentricity ε, system size and transverse kinetic energy KET [28–30]; they also indicate the predictions of valence quark number (nq) scaling [31–33]. The result of such scaling is illustrated in Fig. 2c; it shows that, when plotted as a function of the transverse kinetic energy KET and scaled by the number of valence quarks nq of a hadron (nq = 2 for mesons and nq = 3 for baryons), v2 shows a universal dependence for a broad range of particle species [26, 27, 34]. This has been interpreted as evidence that hydro- dynamic expansion of the QGP occurs during a phase char- acterized by (i) a rather low viscosity to entropy ratio η/s [20, 22, 23, 34] and (ii) independent quasi-particles which exhibit the quantum numbers of quarks [31–36]. The scaled v2 values shown in Fig. 2c allow an estimate of cs because the magnitude of v2/ε depends on the sound speed [30]. One such estimate [26, 27] gives cs 0.35 0.05; a value which suggests an effective equation of∼ state± (EOS) which is softer than that for the high temperature QGP [3]. It FIG. 3: (Color online) η/s vs (T Tc)/Tc for several substances as however, does not reflect a strong first order phase transition indicated. The calculated values for− the meson-gas have an associ- in which cs 0 during an extended hadronization period. This ated error of 50% [38]. The lattice QCD value T = 170 MeV [4] ∼ ∼ c sound speed is also compatible with the fact that v2(pT ) is ob- is assumed for nuclear matter. The lines are drawn to guide the eye. served to saturate in Au+Au collisions for the collision energy range √s = 60 200 GeV [37]. NN − Femtoscopic measurements involving the use of the Bowler-Sinyukov 3D HBT analysis method [in Bertsch-Pratt Hints for even more complicated reaction dynamics is fur- coordinates], have been used to probe for long-range emis- ther illustrated via a comparison of η/s values for nuclear, sions from a possible long-lived source [7–10, 12–14]. The atomic and molecular substances in Fig. 3. It shows the obser- observed RMS-widths for each dimension of the emission vation that for atomic and molecular substances, the ratio η/s source Rlong,Rside and Rout, show no evidence for such emis- exhibits a minimum of comparable depth for isobars passing sions. That is, Rout/Rside 1.0. It is somewhat paradoxical in the vicinity of the liquid-gas critical point [39, 40]. When that these observations have∼ been interpreted as a femtoscopic an isobar passes through the critical point (as in Fig. 3), the puzzle [7] – “the HBT puzzle” – despite the fact that they minimum forms a cusp at Tc; when it passes below the critical clearly reflect a rich and complex set of thermodynamic tra- point, the minimum is found at a temperature below Tc (liquid jectories for which cs = 0 (or ∆T = 0) during an extended side) but is accompanied by a discontinuous change across the hadronization period. ̸ ̸ phase transition. For an isobar passing above the critical point, EPJ Web of Conferences by more than two orders of magnitude with respect to the be used in the HERAPDF approach. Therefore direct tests fixed target experiments and cover the wide x range from of the models are possible. The full statistics of the HERA 7 10 to 0.7. At the HERA collider experiments, H1 and inclusive CC and NC data are used for NLO and NNLO ZEUS, the cross sections of NC and CC DIS are measured QCD fits resulting in HERAPDF1.5 [3]. As an example, with high precision. The measurements of the two experi- the combined NC cross sections are shown in Fig. 2 to- ments are combined and are further used to determine par- gether with QCD prediction based on HERAPDF1.5NLO. ton distribution functions HERAPDF [2] . The QCD analysis HERAPDF1.5 follows the formal- ism, model and paramatrisation assumptions as reported in [2]. The QCD predictions for the structure functions 3 HERAPDF are obtained by solving the DGLAP evolution equations at NLO (or NNLO) in the MS scheme with the renormal- isation and factorisation scales chosen to be Q2. The QCD The PDFs are determined from the structure function mea- predictions for the structure functions are obtained by the surements using the corresponding coecient functions cal- convolution of the PDFs with the NLO coecient func- culated to a certain order in perturbative QCD (pQCD). tions calculated using the general mass variable flavour The structure functions, and in turn the PDFs, depend on x number RT scheme [4]. For the parametrisation of PDFs at and Q. The x-dependence of the parton distributions is not the input scale the generic form xf(x) = AxB(1 x)C(1 + yet calculable in pQCD and has to be parametrized at a cer- Ex2) is used. The parametrised PDFs are the gluon distri- tain starting scale Q . The dependence on Q is described 0 bution, the valence quark distributions and the u-type and by the DGLAP evolution equations [1]. Starting from a d-type anti-quark distributions. The normalisation parame- parameterisation of the PDFs at a starting scale, either by Parton Distribution Function ters A are constrained by the quark number and momentum making ad-hoc assumptions on their analytical form or by sum-rules. using the neural-net technology, fits to various sets of ex- Parton Distribution Function perimental data, with HERA DIS data being the backbone, Valence and Sea Quarks and Gluons are performed within the DGLAP evolution scheme. The H1 and ZEUS HERA I+II PDF Fit 1 2 2 resulting PDFs depend on the order in which the perturba- xf xg (× 0.05) Q = 10000 GeV proton tive QCD calculation is performed, the assumptions about HERAPDF1.5 NNLO (prel.) u the PDF parametrization, the treatment of heavy quarks, 0.8 exp. uncert. valence quark constituent model uncert. u d ↵ xS ( 0.05) the choice for the value of s(MZ) and the treatment of the × parametrization uncert. uncertainties. The data sets included in the PDF fit and the 0.6 xq (x) consistency of these data sets determines the experimental valence xu uncertainty of the PDFs. v 0.4 sea-quarks xdv gluons ??? 0.2 IMF manifests H1 and ZEUS quark d.o.f. ∼1/3 )

2 x + + 3 HERA I+II NC e p (prel.) HERAPDF1.5 e p 0 HERAPDF Structure Function Working Group March 2011 10 - - -4 -3 -2 -1 (x,Q HERA I+II NC e p (prel.) HERAPDF1.5 e p 10 10 10 10 x 1 ± r,NC July 15, 16, 2014 @ 東北大学 10/152 x = 0.02 (x300.0) August 8, 2014 @ JPARC 26

10 2 x = 0.032 (x170.0) x = 0.05 (x90.0) Fig. 3. The parton distribution functions from HERAPDF1.5 x = 0.08 (x50.0) NNLO. The gluon and sea distributions are scaled down by a 10 x = 0.13 (x20.0) factor of 20. The experimental, model and parametrisation un- x = 0.18 (x8.0) certainties are shown. 1 x = 0.25 (x2.4)

-1 10 x = 0.40 (x0.7) In Fig. 3 the parton distributions HERAPDF1.5NNLO 2 2 x = 0.65 at Q = 10000 GeV are shown. In addition to the ex- -2 10 perimental uncertainties, the variation of model inputs and HERA Inclusive Working Group August 2010 2 3 4 5 parametrisation in the determination of HERAPDF are per- 10 10 10 10 Q2/ GeV2 formed and provided as additional eigenvectors. The model uncertainties are evaluated by varying the input assump- Fig. 2. Inclusive DIS cross sections for NC in e± collisions at tions on minimum Q2 of the data used in the fit, the stran- HERA. The measurements of the H1 and ZEUS experiments are + geness fraction and the masses of heavy quarks. The para- combined. Open (closed) symbols represent e p (e p) scattering. metrisation uncertainty is formed by an envelope of the The shaded curves represent QCD prediction based on HERA- PDF1.5NLO. maximal deviations from the central fit varying parametrisation assumptions. HERAPDF1.5NLO and NNLO sets are the recommended HERA PDFs to be used for the pre- The parton distributions HERAPDF [2] are determined dictions of processes at the LHC. The corresponding eigen- using only combined HERA DIS data, where the corre- vectors are available [5]. lations of the systematic uncertainties are properly taken into account. This allows the usage of the conventional 2 tolerance of 2 = 1. Since this QCD analysis is solely 4 Benchmarking HERAPDF based on ep data, the PDFs do not depend on the approach for nuclear corrections needed for fixed target data. Several The PDFs are intrinsic properties of the proton and are phenomenological schemes of heavy quark treatment can therefore process-independent. Cross section predictions What limit manifests Diquarks?

27 3¯ 3 = 8 1withthesamechargesasforqq¯.Forthisreasontheyarecalledcryptoexotics. ⊗ ⊕ qqq¯q¯ can organize alternatively into two color singlet qq¯ mesons, of course, and sophisticated modeling includes both channels (with diquarks dominating at short distances, mesons at larger distances). The non-existence of low-lying dibaryons is related to the (or at least, a) foundational problemWhat of nuclear limit physics: Why manifests do protons and neutrons Diquarks? in close contact retain their integrity? Essentially the same question arises in a sharp form for the H particle studied by Jaffe[2].IthastheconfigurationDiquarks as Inspirationuuddss.Inthebagmodelitappearsthata and as Objects single bag containing these quarks supports a spin-0 state that is quite favorable energet- ically. A calculation based on quasi-freeFrank quarks Wilczek residing∗ inacommonbag,allowingfor one-gluon exchange, indicates that H mightFebruary well 2, be2008 near or even below ΛΛ threshold, and thus strongly stable; or perhaps even below Λn threshold, and therefore stable even against lowest-order weak interactions. These possibilities appear to be ruled out both experimentally and by numerical solution of QCD, thoughAbstract possibly neither case is airtight. Good Attraction between quarks is a fundamental aspect of QCD. It is plausible that diquark correlations,several together of the most with profound repulsion aspects of low-energy between QCD diq dynamuarks,ics are suggests connected a to reason why the almost-independent-particlediquark correlations, approach including: fails paucity in of exotics this (whic case.histhefoundationofthequarkNote that for this mechanism to model and of traditional nuclear physics), similarity of mesons and baryons, color su- work requires thatperconductivity essentially at nonperturbative high density, hyperfine splittings, quark∆ inteI =1raction/2rule,andsomestriking effects, beyond one gluon exchange, must befeatures in play. of structure and fragmentation functions. After a brief overview of these issues, Idiscusshowdiquarkscanbestudiedinisolation,bothphenomenologically and numerically, and present approximate mass differences for diquarks with different quantum numbers. The mass-loaded generalization of the Chew-Frautschi formula provides an 2DiquarksasObjectsessential tool.

From all this1DiquarksasInspiration it appears that diquarks may be very useful degrees of freedom to focus on in QCD. If1.1 we’re Diquarks going in to Microscopic do that, the QCD first step should be to study them in a pure and isolated form,In electrodynamics and determine the basic their interaction parameters. between like-cha Thisrged is particles not straightforward, is repulsive. In due to confinement, sinceQCD, however,the diquarks the primary are interaction colored. between But I two believe quarks th canbeattractive.Attheere are attractive ways to do most heuristic level, this comes about as follows. Each quarkisinthe3 representation, so something approachingthat the two-quark isolating color state them,3 3 bothcan be physically either the symmetric and numerically.6 or the antisymmetric 3¯. ⊗ Of course,Antisymmetry, the same problem of course, is arises not possible for with quarks. just 1 color! Our Two considera widely separatedtions quarks will each apply to them 28 in a non-trivialgenerate way, the as color well. flux associated with the fundamental representation; if they are brought together in the 3¯,theywillgeneratethefluxassociatedwithasingleanti-fundamental, In rapidlywhich spinning is just baryons half as much. centrifugal Thus by bringing forces the lead quarks to togeth a geoer wemetry lower the where gluon a field quark at one end ofarXiv:hep-ph/0409168v2 17 Sep 2004 a line ofenergy: color there flux is attraction is joined in the to3¯ twochannel. quarks We might at expect the other. this attraction The to two-quark be roughly end then half as powerful as the quark-antiquark 3 3¯ 1.Sincequark-antiquarkattractiondrives makes a little laboratory where one can compare⊗ → good and bad diquark configurations with the energy in the attractive channel below zero, triggering condensation qq¯ =0ofqq¯ ⟨ ⟩̸ each other, assesspairs andthe chiral effects symmetry of strangeness, breaking, an attraction and (comparin even half asgwithmesons)normalizethem powerful would appear to relative to singlebe potentially quarks. quite important for understanding low-energy QCD dynamics. ∗ Famously, theSolicited Chew-Frautschi contribution to the Ian formula Kogan memorial volume, ed.M.Shifman.

1 M 2 = a + σL (2.1) organizes trajectories of resonances (Chew-Frautschi formula) with the same internal quantum numbers but diﬀerent values of JP ;hereσ is a universal constant 1.1Gev2 while ∼ a depends on the quantum numbers, and L is an orbital angular momentum, quantized in

6 3¯ 3 = 8 1withthesamechargesasforqq¯.Forthisreasontheyarecalledcryptoexotics. ⊗ ⊕ qqq¯q¯ can organize alternatively into two color singlet qq¯ mesons, of course, and sophisticated modeling includes both channels (with diquarks dominating at short distances, mesons at larger distances). The non-existence of low-lying dibaryons is related to the (or at least, a) foundational problem of nuclear physics: Why do protons and neutrons in close contact retain their integrity? Essentially the same question arises in a sharp form for the H particle studied by Jaffe[2].Ithastheconfigurationuuddss.Inthebagmodelitappearsthata single bag containing these quarks supports a spin-0 state that is quite favorable energet- ically. A calculation based on quasi-free quarks residing inacommonbag,allowingfor one-gluon exchange, indicates that H might well be near or even below ΛΛ threshold, and thus strongly stable; or perhaps even below Λn threshold, and therefore stable even against lowest-order weak interactions. These possibilities appear to be ruled out both experimentally and by numerical solution of QCD, though possibly neither case is airtight. Good diquark correlations, together with repulsion between diquarks, suggests a reason why the almost-independent-particle approach fails in this case. Note that for this mechanism to work requires that essentially nonperturbative quark interaction effects, beyond one gluon Regge Trajectory for even−L Nucleons (series IA). 6 exchange, must be in play. x 10 All Nucleons of series IA 8

7 7 2DiquarksasObjects 6 6

From5 all this it appears that diquarks may be very useful degrees of freedom to focus 5 ) 2 )

2 on in QCD. If we’re going to do that, the first step should be to study them in a pure 4 4 ∗ P and (GeV isolated form, and determine their parameters. This is not straightforward, due to (GeV FIGURE 1. N spectrum, labeled by their spin and parity as J along the abscissa from the quark 2 2 E E model predictions of Capstick and Roberts [2]. For each state, its πN branching fraction is shown. These 3 ∗ ∗ ∗ ∗ 3 confinement, since the diquarks are colored. Butare compared I believe with the th PDGere judgement are [3] attractive of which states wayshave been to identified do with 1 − 2 or 3 − 4 something approaching isolating them, both physicallyprovenance, according and tonumerically. the legend shown. 2 2 OfWhat course, the same limit problem arisesmanifests for quarks. Our considera Diquarks?tions will apply to them 1 Perhaps these missing states really do not exist. If baryons were diquark–quark systems, 1 in a non-trivial way, as well.Nucleons 2 Fig. 2, as Lichtenberg and Tassie noted more than 40 years ago [4], the number of states In rapidly spinning baryonsFitted line (E centrifugal=1.07*L + .781) forceswould lead be restricted to a geo andmetry in fact be very where like that a quark currently obatserved. one 0 0 0 1 2 3 4 5 6 end of0 a line1 of color2 flux3 is joined4 to5 two6 quarksBut atwhat the has this other. do with T QCD?he two-quark There are several end ways then to approach that question. Angular momentum (L) Angular Momentum (L) The lattice provides a modeling of the real world with a discretisation of space-time. makes a little laboratory where one can compareWhilst good the lattice and clearly bad cannotdiquark have the configurations complete rotational withsymmetry of the real world, the newly computed lattice spectrum, as explained by Robert Edwards [5], reveals a (a) each other, assess the e(b)ffects of strangeness, andpattern (comparin very like thegwithmesons)normalizethemSU (6) × O(3) of the quark model: certainly not that of a pointlike relative to single quarks. diquark–quark system. There is no suggestion that the so far undiscovered states should Famously, the Chew-Frautschi formula Regge Trajectory for even−L Deltas (series IB). Regge Trajectory for Lambdas (with [ud]−−s). 9 2 7 M = a + σL (2.1)

8 6 7 organizes trajectories of resonances (Chew-Frautschi formula) with the same internal quan- P 2 tum5 numbers but diﬀerent values of J ;hereσ is a universal constant 1.1Gev while 6 ∼ FIGURE 2. In QCD each quark is in a triplet of color. On the left is a three quark model of a color ) a depends on the quantum numbers, and L is an orbital angular momentum, quantized in ) 2 2 singlet baryon. On the right is a diquark-quark modelManifested of a baryon, where the diquark must be in a color 5 4 anti-triplet, so that in this model the baryon is like a meson as far as color is concerned. (GeV (GeV

2 4 2

E 3 E 6

3 2 Chew-Frautschi relationship 2

1 1 Deltas 2 Lambdas Fitted line (E =1.18*L + 1.429) Fitted line (E2=1.08*L + 1.211) 0 0 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Angular Momentum (L) Angular Momentum (L) 29 (c) (d)

Figure 1: Various E2 vs L plots. (a) is a plot of all nucleons of series IA, showing “even-odd eﬀect”. (b-d) are plots of prominent Regge trajectories.

12 What limitDecays manifestsDecays Diquarks? ρ-mode rρ-mode-mode l-modeλ-mode λ-mode

Hosaka et al.

(qq) really localized in space Q-(qqRatio:) [q qin+ Ratio:thisqqQ limit] [[qQq qof++ heavyqqQqqq]] [QQ ?q + qqq] Seems to be a big assumption ? ρ-mode ρ-mode ! 30 ! λ-mode λ-mode

12/03,&2012 12/03,&2012 JPARC

Here Pvac[Mq] and Pµ[Mq] have a peak at Mq = M0 and Mq = 0, respectively. Assuming that the total pressure is simply P [Mq]=Pvac[Mq]+Pµ[Mq], we can see that the existence of two separate peaks in P [Mq] requires [159], 2 ν µq ν a< 2 2 ! 2 0.076, (42) 16π M0 16π ≃ for 2-ﬂavour case with ν = 12. At the ﬁrst-order critical point the peak at Mq = 0 is just as high as the second peak at Mq M0, which determines the critical chemical potential, ≃ 4 ν µc a 2 4 . (43) ≃ 24π M0

Once a satisfies (42), the chiral phase transition with increasing µq at T = 0 should be of first order at µq µc. The actual value≃ of a is model-dependent: In the NJL model with two flavours and in the linear-σ model, a is estimated respectively as 1 νΛ2 1 =0.067 (NJL model) 2M 2 8π2 − 4G 0 % S & a = ⎧ 2 2 , (44) ⎪ Mσ fπ ⎨ 4 =0.02 0.05 (linear-σ model) 8M0 ∼ ⎪ 2 where we used the standard⎩⎪ NJL parameters; Λ = 631 MeV, GSΛ =2.19, and the resultant M0 = 336.2 MeV [10]. The uncertainty in the linear-σ model comes from the choice of the σ meson mass. Then, in both cases, the estimated a satisfies the inequality (42) implying the first-order phase transition. The critical chemical potential deduced from (43) is, in the NJL model case, given by µc =1.07M0 360 MeV which is consistent with that obtained numerically in the NJL model. ≃ Let us now argue that the first-order transition obtained as above is rather sensitive to the choice of the model Lagrangian. Indeed, it has been known that 2 the repulsive contribution to the pressure of the form +GVnq induced by a quark interaction of density-density type can totally wash out the first-order transition [160, 161, 162, 163, 164, 159]. For example, in the NJL model with the density-density interaction, the condition (42) is changed to 2 ν 2νGVµ a< 1 q , (45) 16π2 − 3π2 % & which implies that the first-order phase transition does not arise for GV > 0.25GS [162, 163, 159].

6. Formation of the diquark condensate

Finding a ground state of quark matter at T 0 with extremely large value of µ What limit manifests≈ Diquarks? q is an interesting theoretical challenge. (We use µq instead of µB throughout this section, for our central interest is the quark degrees of freedom). Let us consider the Cooper’s stability-test in quark matter [60, 61]. In the perturbative regime of QCD the one-gluon exchange potentialHigh isdensity proportional limit to a product of the quark SU(Nc) charges;

a a Nc +1 Nc 1 (t )αβ(t )α′β′ = δαβδα′β′ δαβ′ δα′β + − δαβδα′β′ + δαβ′ δα′β .(46) − 4Nc − 4Nc ' ( ' ( Attractive Triplet Repulsive Sextet

Sharp Fermi surface (1D) + Attractive Force = Cooper instability (formation of “pairs”)

31 pQCD justifies itself at high T All gluons are screened by gT or g2T ! pQCD does not justify itself at high µ Magnetic gluons never screened ! ! Insufficient justification ! CSC justifies pQCD at high µ All gluons are screened by gµ

32 Deconfinement and CSC

Asymptotic Deconﬁnement in High-Density QCD

D.H. Rischke,1,4 D.T. Son,2,4 and M.A. Stephanov3,4 1Nuclear Theory Group, Brookhaven National Laboratory, Upton, New York 11973 2Physics Department, Columbia University, New York, New York10027 3Department of Physics, University of Illinois, Chicago, Illinois 60607-7059 4RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973

We discuss QCD with two light flavors at large baryon chemical potential µ.Colorsuperconductivity leads to partial breaking of the color SU(3) group. We show that the infrared physics is governed by the gluodynamics of the remaining SU(2) group with an exponentially soft confinement scale ′ ΛQCD ∆ exp( aµ/(g∆)), where ∆ µ is the superconducting gap, g is the strong coupling, and ∼ − ≪ ′ a =2√2π/11. We estimate that, at moderate baryon densities, Λ is (10 MeV) or smaller. QCD O The confinement radius increases exponentially with density, leading to “asymptotic deconfinement.” The velocity of the SU(2) gluons is small due to the large dielectric constant of the medium.

Introduction.—Soon after the discovery of asymptotic charges below ∆ implies that the medium(2011) is transparent freedom in QCD [1] a hypothesis was put forward that, to the SU(2) gluons: there is no Debye screening and at high baryon densities, quarks (which are normally con- Meissner effect for these gluons. Mathematically, the po- µν fined in hadrons by strong forces) are liberated, i.e., nu- larization tensor Πab (q)vanishesatq =0,whichcanbe clear matter transforms into deconfined quark matter [2]. checked by a direct calculation of Π at small q33,aswas In recent years, our knowledge of dense quark matter has done in Ref. [5]. The absence of Debye screening means considerably expanded. We now understand that, in re- that a static color charge inserted into the medium can- ality, dense matter shows more intricate features than in not be completely screened as it is in hot plasmas. This is the original picture of [2]. In particular, quark matter easy to understand since all quarks carrying SU(2) color at high densities exhibits the phenomenon of color su- are bound into SU(2) singlet Cooper pairs. Analogously, perconductivity [3,4], which determines the symmetry of the Meissner effect is absent because the superconducting the ground state and the infrared dynamics. currents, which are coherent motions of the condensate, The number of light quark flavors Nf turns out to play cannot screen the magnetic field, since the condensate is acrucialrole.ThesimplestcaseisNf =2,whereupand SU(2) neutral. Thus, at first sight, it might seem that the down quarks are massless and other quarks are neglected. quarks in the medium have no effect on the gluon effec- 2 2 The following picture emerges in perturbation theory, as tive Lagrangian, which must be simply L = Fµν /(4g ), well as in instanton-inspired models. The condensation of i.e., the SU(2) Yang-Mills Lagrangian with− the coupling color antitriplet up-down diquarks breaks the color SU(3) g matching the running coupling in the original theory down to an SU(2) subgroup. Thus, five of the original at the scale ∆ [6]. However, a closer look shows that eight gluons acquire “masses” by the Meissner effect [4,5], the situation is somewhat more complicated and, in fact, similar to the Higgs mechanism. The remaining three more interesting. gluons are massless (perturbatively). Because of Cooper Although a static SU(2) charge cannot be completely arXiv:hep-ph/0011379v2 28 Jul 2001 pairing, the spectrum of quark excitations carrying SU(2) Debye screened by SU(2) neutral Cooper pairs, it can color charge has a gap ∆. still be partially screened if the medium is polarizable, In order to understand the physics below the energy i.e., if it has a dielectric constant ϵ different from unity. scale ∆ we must examine the pure gluodynamics in the If ϵ > 1, then the Coulomb potential between two static remaining unbroken SU(2) sector. As we shall see, the color charges is g2/(ϵr); i.e., the gauge coupling is effec- process of high-density “deconfinement” is quite non- tively reduced by a factor of ϵ1/2.Asexplainedinmore trivial in this case: the quarks are always confined (as- detail below, this is exactly the situation in the color- suming that SU(2) Yang-Mills theory confines), how- superconducting phase. Analogously the medium can, in ever, the confinement radius grows exponentially with in- principle, have a magnetic permeability λ =1.(Wede- creasing density. We shall also see that, at scales much note the permeability by λ instead of the more̸ common shorter than the confinement radius, the dynamics of the µ,sincethelattersymbolisalreadyusedforthechemical SU(2) gluons is similar to electrodynamics in a dielectric potential.) The dynamics of gluons is thus modified by medium with large refraction index. the dielectric constant and the magnetic permeability of The effective Lagrangian.—Below the scale ∆,weex- the medium. Hence, one needs to develop the theory of pect that the heavy (gapped) degrees of freedom decouple “gluodynamics of continuous media”, which, as far as we and the remaining fields can be described by a local effec- know, has never been encountered before. This theory, tive Lagrangian. The absence of quarks carrying SU(2) in contrast to its U(1) counterpart (the electrodynamics

1 2SC (2-flavor CSC) reads:

SU(3)C broken to SU(2)C 5 gluons (out of 8) get massive (Meissner effect)

4 quarks (out of 6=2×3) get gapped with D

3 unscreened gluons + 4 gapped quarks with D

Dominant at E < D = 2-color YM in the vacuum = confining theory

34 CFL (3-flavor CSC) reads:

SU(3)C broken completely All 8 gluons get massive (Meissner effect)

No confinement remains

Can this be a “definition” of deconfinement ?

Private communications with Gordon Baym If so, quark matter is realized only through diquarks

35 A question arises:

A diquark condensate signals for quark deconfinement

Is the diquark condensate measurable (observable) ?

Like Higgs bosons in EW, diquarks can be real then BUT Condensate should be defined gauge-invariantly qq (qq)(¯qq¯) (qq)(qq)(qq) h i h i h i

36 CSC is theoretically defined by:

Tetra-quark scalar-meson condensate

Hexa-quark (H-dibaryon, demon-deuteron, LL) condensate

breaking

Chiral Symmetry and U(1)V Symmetry

Mesons in CSC are “exotic” ones

37 Hyper Nuclear Matter CFL

Chiral σ qq¯ =0 σ =0 ⟨ ⟩∼⟨ ⟩̸ ⟨ ⟩ Symmetry σ˜ q¯2q2 small σ˜ =0 ⟨ ⟩∼⟨ ⟩∼ ⟨ ⟩̸ π qq¯ +(¯qq¯)(qq)+ π˜ (¯qq¯)(qq)+ ∼ ··· ∼ ··· Z(2) Z (2) (instanton) Z (2) Z (2) −→ V ←− L × R Super- ∆ = Σ2 + Λ2 + NΞ =0 ∆ small NN ⟨ ⟩̸ NN ∼ -ﬂuidity H =0 H =0 ⟨ ⟩̸ ⟨ ⟩̸ χ 2 nucleons + 3 diquarks χ 3 diquarks H ∼ H ∼ conﬁnement–Higgs crossover

TABLE I:Crossover Summary of qualitative of changesBEC—BCS from nuclear matter to CFL quark matter. Correlation in Hyper Nuclear Matter CFL momentum space _ qq none (apart from UA(1) breaking) pion _ _ _ _ qq qq qq qq qq _ _ q q qqq small qqq phason qq q q qq q q qq qq qq qq q q µ

BEC of colorless H BEC of colored qq BCS

FIG. 1: SchematicTetra-quark picture of structure the structural is change protected and the two-step by symmetry crossover. 38 phase where colored diquarks play an essential role. From the point of view of the NG bosons this can be seen as dissociation of the hadrons into constituent diquarks. As far as H is concerned, we can say in the following way; the hadronic phase has a Bose-Einstein condensate (BEC) of the color-singlet H-dibaryon, while the dissociated colored diquarks lead to a superconducting state at higher baryon density, and yet they compensate for their color charge to be a color-singlet in the CFL phase. In this sense, the attractive force between diquarks controls the state of matter. If the interaction is strong enough, the state is BEC-like, and otherwise, it is BCS-like. This BEC-BCS crossover looks quite diﬀerent

11 Inverse meson mass ordering in color-ﬂavor-locking phase of high density QCD

D.T. Son1,3 and M.A. Stephanov2,3 1 Physics Department, Columbia University, New York, NY 10027 2 Department of Physics, University of Illinois, Chicago, IL 60607-7059 3 RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973

October-November 1999

Meson spectrum looks similar to scalar nonet u (d¯s¯) ⇠ Abstract d (¯su¯) Instantons mix them to (qq¯) ⇠We derive the effective Lagrangian (opposite for the low-energy role massivto nonet)emesonexcitations s of the(¯u color-flavor-lockingd¯) (CFL) phase of QCD with 3 flavors oflightquarks.We compute⇠ the decay constants, the maximum velocities, and themassesofthemesons at large baryon chemical potential µ.Thedecayconstantsarelinearinµ.Themeson 39 maximum velocities are close to that of sound. The meson masses in the CFL phase arXiv:hep-ph/9910491v2 6 Dec 1999 are significantly smaller than in the normal QCD vacuum and depend only on bare quark masses. The order of the meson masses is, to some extent,reversedcompared to that in the QCD vacuum. In particular, the lightest particle is η′.

1 Nuclear Matter exhibits:

Chiral symmetry still broken (could be restored) Superfluidity of nucleons (also in nuclei)

From the symmetry point of view, such superfluid nuclear matter is undistinguishable from deconfined quark matter with diquark condensate

Such an exotic mixing may exist in nuclear matter (Quark-Hadron Continuity) 40 Not surprisingly:

Quark-Diquark Model of Baryons leads to Diquarks in Normal Nuclear Matter Bleuler et al. (1983) In the Quark-Gluon Plasma also:

Many people speculated the relevance of diquarks Inconsistent with quark # scaling? Fluctuations in low-energy BES

Shuryak (2004) 41 My personal belief:

- It is time to combine -

Modern QCD wisdom of deconfinement and color superconductivity

Phenomenology of a nuclear matter superfluid

Quark-diquark model for nuclear matter

For reality of diquarks

42 From the thermodynamic point:

Quarks and Gluons

HRG

(analyzing tool of multiplicity) Quarks 43 From the thermodynamic point:

If you have “nucleon matter” and “quark matter” at the same chemical potential, which pressure greater?

44 From the thermodynamic point:

If you have “nucleon matter” and “quark matter” at the same chemical potential, which pressure greater? p (µ2 m2 )3/2 nucleon ⇠ B B 2 2 3/2 2 p N (µ m ) N p quark ⇠ c q q ⇠ c nucleon Counter intuitive but true… All quark models underestimate p at finite density

45 From the thermodynamic point:

If you have “nucleon matter” and “quark matter” at the same chemical potential, which pressure greater?

2 2 3/2 Diquarks as pnucleon (µ m ) ⇠ B B real d.o.f. in p 2 2 3/2 2 p N (µ m ) N p quark ⇠ c q q ⇠ c nucleon Counter intuitive but true… All quark models underestimate p at finite density

46 Future of HIC heads for:

Also, NICA, JPARC coming?

1.2 Au+Au Collisions at RHIC The current data provide the most relevant measure- 1.0 ments over the widest range in µB (20 to 450 MeV) to Skellam Distribution date for the CP search, and for comparison with the σ 0.8 70-80% baryon number susceptibilities computed from QCD to

Deviation S 0.6 0-5% understand the various features of the QCD phase struc- 0.4 Net-proton ture [6, 16, 17]. The deviations of Sσ and κσ2 below 0.2 0.4

1.0 the UrQMD model which does not include a CP also has been 2 shows deviations from the Skellam expectation. Hence

σ 0.8 p+p data conclusions on the existence of CP can be made only af- κ 0.6 Au+Au 70-80% ter comparison to QCD calculations with CP behavior seen at RHIC Au+Au 0-5% which include the dynamics associated with heavy-ion 0.4 Au+Au 0-5% (UrQMD) Ind. Prod. (0-5%) collisions, such as ﬁnite correlation length and freeze-out 1.05 eﬀects. 1.00 In summary, measurements of the higher moments and S 2 0.95 their products ( σ and κσ ) of the net-proton distribu-

)/Skellam tions at midrapidity ( y < 0.5) within 0.4

σ 0.90 | | GeV/c in Au+Au collisions over a wide range of √sNN (S 0.85 and µB have been presented to search for a possible 5 6 7 8 10 20 30 40 100 200 CP and signals of a phase transition in the collisions. Colliding Energy s (GeV) NN These observables show a centrality and energy dependence, which are neither reproduced by non-CP trans- FIG. 4: (Color online) Collision energy and centrality depen- 2 port model calculations, nor by a hadron resonance gas dence of the net-proton Sσ and κσ from Au+Au and p+p 2 model. For √sNN > 39 GeV, Sσ and κσ values are simi- collisions at RHIC. Crosses, open squares and filled circles are lar for central, peripheral Au+Au collisions and p+p col- Hint to diquarks?for the efficiency (I corrected guess results of p +pso), 70-80%, and 0-5% 2 S Au+Au collisions, respectively. Skellam distributions forcor- lisions. Deviations for both κσ and σ from HRG and responding collision centralities are shown in the top panel. Skellam expectations are observed for √sNN 27 GeV. ≤ More data needed!Shaded hatched bands are the results from UrQMD [22]. In The measurements are reasonably described by assuming the middle and lower panels, the shaded solid bands are the independent production of Np and Np¯, indicating that expectations assuming independent proton and anti-proton47 there are no apparent correlations between the protons production. The width of the bands represents statistical un- and anti-protons for the observable presented. However certainties. The hadron resonance gas model (HRG) values for κσ2 and Sσ/Skellam are unity. The error bars are sta- at the lower beam energies, the net-proton measurements tistical and caps are systematic errors. For clarity, p+p and are dominated by the shape of the proton distributions 70-80% Au+Au results are slightly displaced horizontally. only. The data presented here also provides information to extract freeze-out conditions in heavy-ion collisions using QCD based approaches [35, 36]. Au+Au collisions and the peripheral collisions. The re- We thank M. Asakawa, R. Gavai, S. Gupta, F. Karsch, sults are closer to unity for √sNN = 7.7 GeV. Devia- K. Rajagopal, K. Redlich and M. A. Stephanov for dis- tions of 0-5% Au+Au data from Skellam expectations, cussions related to this work. We thank the RHIC Oper- (( Data Skellam )/ err 2 +err 2)arefoundtobe ations Group and RCF at BNL, and the NERSC Center ! stat sys most| significant− for| 19.6 GeV and 27 GeV, with values of at LBNL, the KISTI Center in Korea and the Open Sci- 3.2 and 3.4 for κσ2, and 4.5 and 5.6 for Sσ, respectively. ence Grid consortium for providing resources and sup- The deviations for 5-10% Au+Au data are smaller for port. This work was supported in part by the Offices κσ2 with values of 2.0 and 0.6 and are 5.0 and 5.4 for of NP and HEP within the U.S. DOE Office of Science, Sσ, for 19.6 GeV and 27 GeV, respectively. A reason- the U.S. NSF, CNRS/IN2P3, FAPESP CNPq of Brazil, able description of the measurements is obtained from the Ministry of Ed. and Sci. of the Russian Federation, independent production approach. The data also show NNSFC, CAS, MoST, and MoE of China, the Korean deviations from the hadron resonance gas model [31, 32] Research Foundation, GA and MSMT of the Czech Re- which predict κσ2 and Sσ/Skellam to be unity. To under- public, FIAS of Germany, DAE, DST, and CSIR of the stand the effects of baryon number conservation [33] and Government of India, National Science Centre of Poland, experimental acceptance, UrQMD model calculations (a National Research Foundation (NRF-2012004024), Min- transport model which does not include a CP) [22] for istry of Sci., Ed. and Sports of the Rep. of Croatia, and 0-5% Au+Au collisions are shown in the middle and bot- RosAtom of Russia. Finally, we gratefully acknowledge tom panels of Fig. 4. The UrQMD model shows a mono- a sponsored research grant for the 2006 run period from tonic decrease with decreasing beam energy [23]. Renaissance Technologies Corporation.