Continuity of Quark and Hadron Matter

Home , Hadron

Novemb er 1998 IASSNS-HEP 98/100

hep-ph/9811473

Continuity of Quark and

Hadron Matter

Thomas Schafer

and

Frank Wilczek

School of Natural Sciences

Institute for Advanced Study

Princeton, NJ 08540

Abstract

We review, clarify, and extend the notion of color- avor lo cking.

We present evidence that for three degenerate avors the qualitative

features of the color- avor lo cked state, reliably predicted for high

density,match the exp ected features of hadronic matter at low density.

This provides, in particular, a controlled, weak-coupling realization of

con nement and chiral symmetry breaking in this (slight) idealization

of QCD.

Research supp orted in part by NSF PHY-9513835.

e-mail: [email protected]

Research supp orted in part by DOE grant DE-FG02-90ER40542.

e-mail: [email protected] 1

In a recent study [1] of QCD with three degenerate avors at high density,

a new form of ordering was predicted, wherein the color and avor degrees of

freedom b ecome rigidly correlated in the groundstate: color- avor lo cking.

This prediction is based on a weak coupling analysis using a four-fermion

interaction with quantum numb ers abstracted from one gluon exchange. One

exp ects that sucha weak coupling analysis is appropriate at high density,for

the following reason [2,3,4].Tentatively assuming that the quarks start out

in a state close their free quark state, i.e. with large Fermi surfaces, one

nds that the relevantinteractions, which are scatterings the states near the

Fermi surface, for the most part involve large momentum transfers. Thus,

by asymptotic freedom, the e ective coupling governing them is small, and

the starting assumption is con rmed.

Of course, as one learns from the theory of sup erconductivity [5], even

weak couplings near the Fermi surface can have dramatic qualitative e ects,

fundamentally b ecause there are manylow-energy states, and therefore one

is inevitably doing highly degenerate p erturbation theory. Indeed, the au-

thors of [1] already p ointed out that their color- avor lo cked state, which

is constructed by adapting the metho ds of sup erconductivity theory to the

problem of high-density quark matter, displays a gap in all channels ex-

cept for those asso ciated with derivatively coupled spin zero excitations, i.e.

Nambu-Goldstone mo des. This is con nement. For massless quarks, they

also demonstrated sp ontaneous chiral symmetry breaking.

In very recentwork we [6], and others [7], have reinforced this circle

of ideas by analyzing renormalization of the e ectiveinteractions as one

integrates out mo des far from the Fermi surface. A fully rigorous treatment

will have to deal with the extremely near-forward scatterings, which are

singular due to the absence of magnetic mass for the gluons, at least in

straightforward p erturbation theory. This problem, which is presumably

technical, is any case ameliorated self-consistently for states of the color- avor

lo cking typ e, wherein all the gluons acquire mass through the Anderson-Higgs

mechanism.

In the earlier work [1], several striking analogies b etween the calculated

prop erties of the color- avor lo cked state and the exp ected prop erties of

hadronic matter at low or zero density, based on standard lore and observed

phenomenology,were noted. In addition to con nementandchiral symmetry

breaking, the authors observed that the dressed elementary excitations in the

color- avor lo cked state have the spin quantum numb ers of low-lying hadron 2

states and for the most part carry the exp ected avor quantum numb ers,

including integral electric charge (in units of the electron charge). Thus, as

we shall sp ell out immediately b elow, the gluons match the o ctet of vector

mesons, the quark o ctet matches the baryon o ctet, and an o ctet of collective

mo des asso ciated with chiral symmetry breaking matches the pseudoscalar

o ctet. However there are also a few apparent discrepancies: there is an extra

massless singlet scalar, asso ciated with the sp ontaneous breaking of baryon

numb er (sup er uidity);thereareeight rather than nine vector mesons (no

singlet); and there are nine rather than eightbaryons (extra singlet). We

will argue that these \discrepancies" are sup er cial { or rather that they are

features, not bugs.

Let us rst brie y recall the fundamental concepts of color- avor lo cking.

The case of three massless avors is the richest due to its chiral symme-

try (and adding a common mass do es not change anything essential) so we

shall concentrate on it. The primary condensate, which one calculates using

the metho ds of sup erconductivity theory near the Fermi surface, involves

diquarks. It takes the form [1]

_ _

k l

(1) + i = q i = hq q hq

2 1 ij

a b b a Lb La

Rka

Rlb

Here L, R lab el the helicity, i; j; k ; l are two-comp onent spinor indices, a; b

are avor indices, and ; are color indices. A common space-time argu-

ment is suppressed. ; are parameters (dep ending on chemical p otential,

1 2

coupling, :::) whose non-zero values emerge from a dynamical calculation.

This equation must b e interpreted carefully. The value of any lo cal quan-

tity which is not gauge invariant, taken literally, is meaningless, since lo cal

gauge invariance parametrizes the redundantvariables in the theory,and

cannot b e broken [8]. But as weknow from the usual treatment of the elec-

troweak sector in the Standard Mo del, it can b e very convenient to use such

quantities. The p oint is that we are allowed to x a gauge during interme-

diate stages in the calculation of meaningful, gauge invariantquantities {

indeed, in the context of weak coupling p erturbation theory,wemust do so.

For our present purp oses however it is imp ortant to extract non-p erturbative

results, esp ecially symmetry breaking order parameters, that we can match

to our exp ectations for the hadronic side. To do this, we can take suitable

pro ducts of the memb ers of (1) and their complex conjugates, and contract

the color indices. In this waywe can pro duce the square of the standard chiral 3

symmetry breaking order parameter of typ e hq q i and a baryon numb er vio-

L R

lating order parameter of typ e h(qqq) i, b oth scalars and singlets under color

and avor. Atthislevel only the square of the usual chiral order parameter

app ears, fundamentally b ecause our condensates preserve left-handed quark

number modulo two. This conservation law is violated by the six-quark

vertex asso ciated with instantons, and byconvolving that vertex with our

four-quark condensate we can obtain the usual two-quark chiral symmetry

breaking order parameter [9].

By demanding invariance of the diquark condensate directly,we infer the

c L R

symmetry breaking pattern SU (3) SU (3) SU (3) U (1) ! SU (3) .

Here among the initial microscopic symmetries SU (3) is lo cal color symme-

try, while the remaining factors are chiral family and baryon numb er symme-

tries. The nal residual unbroken symmetry is a global diagonal symmetry.

Indeed, the Kronecker deltas in the nal term of (1) are invariantonlyunder

simultaneous color and avor rotations, so the color and avor are \lo cked".

This lo cking o ccurs separately for the left and right handed quarks, but

since color symmetry itself is vectorial, the e ect is also to lo ckleftand

righthanded avor rotations, breaking chiral symmetry. The global baryon

numb er symmetry is, of course, manifestly broken, but quark number is con-

served mo dulo two. Pro jecting onto the gauge invariant, color singlet, sector

this implies that baryon numb er is violated only mo dulo two. The same

conclusions would emerge from analysis of the gauge invariant symmetry

generators only, up on consideration of the gauge invariant order parameters

we constructed ab ove.

Ordinary electromagnetic gauge invariance, like color symmetry, is vio-

lated by (1), but a linear combination of hyp ercharge (diagonal matrix -2/3,

1/3, 1/3) and electromagnetic charge (diagonal matrix 2/3, -1/3, -1/3) anni-

hilates the combinations correlated by color- avor lo cking, and generates a

true symmetry.Thephysical result is that there is a massless gauge degree

of freedom, representing the photon as mo di ed byitsinteraction with the

condensate. As seen by this mo di ed photon, all the elementary excitations

have appropriate charges to match the corresp onding hadronic degrees of

freedom. In particular, their charges are all integral multiples of the electron

charge [1]. This is, of course, another classic asp ect of con nement.

It was essential, in this construction, that the charges of the quarks add

up to zero. If that were not so, wewould not have b een able to nd a color

generator capable of comp ensating the violation of naive electromagnetic 4

gauge invariance. Yet it seems somewhat accidental that these charges do

add up to zero, and one would b e quite worried if any qualitative asp ect of

con nement dep ended on this accident. This worry touches the form rather

than the substance of our argument. If the quark charges did not up to zero,

it would not b e valid to ignore Coulomb repulsion. One would havetoadda

comp ensating charge background as a mathematical device, or contemplate

inhomogeneous states. Insofar as wewant to use external gauge elds as

a prob e of pure QCD, wemust restrict ourselves to those which preserve

the overall neutrality of the QCD groundstate. Fortunately,inourslightly

idealized version of QCD no awkwardness arises for the physically imp ortant

gauge eld, i.e. the physical photon.

The elementary excitations are of three typ es. The color gluons b e-

come massivevector mesons through the Anderson-Higgs mechanism. Due

to color- avor lo cking, they acquire avor quantum numb ers, which makes

them an o ctet under the residual SU (3) . The quark elds give single-

particle spin 1/2 excitations whose stability is guaranteed by the residual Z

quark (or baryon) numb er symmetry. These excitations are massive, due to

the color- avor sup erconducting gap. They form an o ctet with the quantum

numb ers of the nucleon o ctet, plus a singlet. It might seem p eculiar on rst

hearing that a single quark can b ehaveasa baryon, but rememb er that there

is a condensate of diquarks p ervading this phase. In addition there are col-

lective Nambu-Goldstone mo des, asso ciated with the sp ontaneously broken

global symmetries. These are a massless pseudoscalar o ctet asso ciated with

chiral symmetry breaking, and a scalar singlet asso ciated with baryon num-

b er violation. A common quark mass lifts the pseudoscalar o ctet, but not

the singlet, b ecause it sp oils microscopic chiral symmetry but not microscopic

baryon number.

Clearly, there are striking resemblances b etween the elementary excita-

tions of color- avor lo cked quark matter and the low-energy hadron sp ec-

trum. One is tempted to ask whether they mightbeidenti ed. More pre-

cisely, one might ask whether strongly coupled hadronic matter at low density

go es over into the calculable, weak-coupling form of quark matter just de-

scrib ed without a phase transition. If so, then the con nement and chiral

symmetry breaking calculated for the weak coupling phase not only resemble

these central prop erties of low-density QCD, but are rigorously indistinguish-

able from them. This sort of p ossibility, that Higgs and con ned phases are

rigorously indistinguishable, has long b een known to o ccur in simple abstract 5

mo dels [10 ].

As mentioned ab ove, however, at rst sight there app ear to b e several

diculties with this identi cation. Wenow debunk them in turn.

The most profound of the apparent diculties is the existence of an ex-

tra scalar Nambu-Goldstone mo de, and the related phenomenon that baryon

number is spontaneously violated (indicating, as in liquid He , sup er uid-

ity). The answer to this comes through prop er recognition of an imp ortant

though somewhat exotic phenomenon for three degenerate avorsonthe

hadron side. Several years ago R. Ja e discovered [11], in the context of

the MIT bag mo del, that a particular dibaryon state, the H, a spin 0 SU(3)

singlet with quark content (udsuds), is surprisingly light. This arose, in his

calculations, b ecause of a particularly favorable contribution from color mag-

netism. Roughly sp eaking, in the H con guration the color elds asso ciated

with the quark sources are minimized, together with the energy they would

otherwise store, by arranging b oth the colors and spins to cancel pairwise

to the greatest extent p ossible. It has b een debated, for QCD with realistic

quark masses, whether H mightbeonlyslightly ab ove the nn or n thresh-

olds. Though at this level the outcome for realistic QCD is unclear, b oth

theoretically [12] and exp erimentally [13 ], it has come to seem quite likely

that in QCD with three degenerate quarks the H will b e the particle with

smallest energy p er unit baryon numb er. Thus at any nite baryon number

density,however small, at zero temp erature one should exp ect, in this con-

text, to nd a Bose condensate of H dibaryons. This condensate gives us

precisely { i.e. with the appropriate quantum numb ers { the sup er uid we

were led to exp ect from our sup er cially very di erent considerations on the

quark matter side.

If the H is ab ove dibaryon threshold, one will have a narrow range of

chemical p otentials where baryon numb er is built up by single baryons. Based

on the same calculations [11, 12 ], it is extremely plausible that in this case

there will b e attraction in the H channel at the Fermi surface, and hence

sup er uidity of the required typ e, now through a BCS mechanism.

This sup er uidity, whatever its source, supplies us with the key to the

riddle of the missing vector meson. For once there is a massless singlet

scalar, the putative singlet vector b ecomes radically unstable, and should

not app ear in the e ective theory.Itmight b e ob jected here that the o ctet

of vector mesons is also unstable { for massless quarks { against decayinto

massless scalar and pseudoscalar mesons. A quick answer is that this is not 6

really an ob jection at all, b ecause there is no harm in having redundant states

(whose instability will app ear immediately up on more accurate calculation).

There is a much prettier and more satisfying answer, however. If weturnon

non-zero masses for the quarks the pseudoscalar o ctet (but not the singlet)

will b ecome massive. Eventually the decayofthevector o ctet (but not the

singlet) will b e blo cked, and then we will b e grateful for the prescience of the

theory in providing the appropriate degrees of freedom.

Finally, there is the question of the \extra" singlet baryon. This is the

most straightforward. In the original calculations [1], it was found that the

singlet gap is much larger than the o ctet gap. Thus the singlet baryon is

predicted to b e considerably heavier than the o ctet. This is not problematic:

a particle of this sort is exp ected in the quark mo del, it could well exist

in reality, and in any case it is radically unstable against decayinto o ctet

baryon and o ctet pseudoscalar, at least for massless or light quarks.

So all the ob jections havebeenanswered. Continuity of quark and hadron

matter, far from b eing paradoxical, now app ears as the default option.

Clearly, sup er uidity of quark/hadron matter has b een essential for the

argument. There is considerable evidence for pairing in nuclei [14]. Its full

realization is limited by the nite size of nuclei, which in turn arises from the

non-negligible strange quark mass and the Coulomb energy that arises in the

most favorable (for QCD), symmetric arrangement of neutrons and protons.

These limitations might b e relieved to some extentinheavy ion collisions ac-

companied by creation of many strange-antistrange pairs, followed bycharge

segregation. An imp ortant signature for this, emphasized by the consider-

ations ab ove, is broadening of vector mesons, esp ecially the singlet. This

e ect might b e observable in the dimuon sp ectrum.

Our considerations here are clearly relevanttoany attempt to mo del the

deep interior of neutron stars, or conditions during sup ernova and hyp ernova

explosions. To do justice to these questions, it will b e very imp ortantto

include the e ects of unequal quark masses and of electromagnetism.That

is an imp ortant task for the future.

In the remainder of this pap er we shall consider a related but simpler prob-

lem, that of extending the analysis to larger numb ers of degenerate quarks.

An imp ortant foundational result, which emerges clearly from this analysis,

is that the color- avor lo cked state for three avors, whichwas rst guessed

to b e favorable b ecause of its large residual symmetry and by analogy to the

B phase of sup er uid He , is in fact the global minimum for three avors. It 7

also reapp ears as a building blo ck for larger numb ers of avors.

The renormalization group analysis in [6, 7 ] allows one to classify p ossible

instabilities, and to assess their relative imp ortance, for small but otherwise

arbitrary couplings near the Fermi surface. It was found that the dominant

instability corresp onds to scalar diquark condensation. The analysis do es

not x the color and avor channel of this instability uniquely, indep endent

of initial conditions for the couplings, since there are two equally enhanced

marginal interactions. One gluon exchange, which dominates for weak cou-

pling, is attractive in the color anti-symmetric 3channel, and favors one of

these interactions. During the evolution this interaction will grow, while the

repulsiveinteraction in the color symmetric 6 channel is suppressed. Thus

the instability is driven by a leading interaction of the form

ac bd ad bc

L = K ( )

o n

C C (C $ C ) ; (2)

5 5 5

b c d a

where as b efore ; ;::: are color indices and a;b;::: are avor indices. The

Dirac structure of the interaction b ecomes more transparent when written in

achiral basis. Wehave

l k i

+(L $ R): (3)

ij kl

L L L L

The renormalization group analysis only provides the form of the domi-

nantinteraction, not the structure of the order parameter. In particular, it

do es not tell us whether color- avor lo cking is the preferred state in three

avor QCD. In order to answer this question, wehave to p erform a varia-

tional analysis. Since the interaction is attractiveins-wave states, it seems

clear that the dominant order parameter is an s-wave, to o. We then only

have to study the color- avor structure of the primary condensate. For this

purp ose, we calculate the e ective p otential for the order parameter

ij i

: (4) i =

ab Lb La

is a N N matrix in avor space and a N N matrix in color

f f c c

space. Overall symmetry requires that is symmetric under the combined

exchange (a ) $ (b ). Also, since the interaction only involves color and

avor anti-symmetric terms, the e ectivepotential do es not dep end on color

and avor symmetric comp onents of . This means that the e ective

ab 8

p otential has at least N (N +1)N (N +1)=4 at directions. These trivial

c c f f

at directions will b e lifted by subleading interactions not included in our

analysis. We will comment on the imp ortance of subleading terms b elow.

We calculate the e ectivepotential in the mean eld approximation. This

approximation corresp onds to resumming all \cactus" diagrams. These di-

agrams are exp ected to b e dominant b oth in the limit of large chemical

p otential and in the large N ;N limit. In the mean eld approximation, the

c f

quadratic part of the action b ecomes

a b a b

: (5) + = K M

ba L L ba ab ab ab L L

Integrating over the fermion elds we obtain the familiar tr log term in the

e ective p otential. In order to evaluate the logarithm, wehave to diagonalize

the mass matrix M. Let us denote the corresp onding eigenvalues by ( =

1;:::;N N ). These are the physical gaps for the N N fermion sp ecies.

c f f c

Adding the mean eld part of the e ective p otential, we nally obtain

V () = ( )+M : (6)

ef f

Here, ( ) is the kinetic term in the e ective action for one fermion sp ecies,

q q

d p

2 2 2 2

( )= (p ) + + (p + ) + : (7)

(2 )

This integral has an ultra-violet divergence. This divergence can b e removed

by expressing in terms of the renormalized interaction [15]. In this work

we are not really interested in the exact numerical value of the gap, but

only in the symmetries of the order parameter. For simplicity,we therefore

regularize the integral byintro ducing a sharp three-momentum cuto .

The e ectivepotential (6) dep ends on N (N 1)N (N 1)=4 parame-

c c f f

ters. We minimize this function numerically. In order to make sure that the

minimization routine do es not b ecome trapp ed in a lo cal minimum we start

the minimization from several di erent initial conditions. For the numerical

analysis wealsohaveto xthevalue of the chemical p otential , the coupling

constant K , and the cuto . Wehavechecked that the symmetry breaking

pattern do es not dep end on the values of these parameters. Wehaveused

=0:5 GeV, = 0:6GeVandK =3:33= , similar to what was considered

in [3, 4]. 9

After we determine the matrix that minimizes the e ectivepotential

we study the corresp onding symmetry breaking. Initially, there are N 1

global avor symmetries for b oth left and right handed fermions, as well

as N 1 lo cal gauge symmetries. Sup er uidity reduces the am-mountof

symmetry. In order to nd the unbroken generators we study the second

2 2 2

variation of the order parameter =( ), where (i =1;:::;N + N

i j i

f c

2) parameterizes the avor and color transformations. Zero eigenvalues of this

matrix corresp ond to unbroken color- avor symmetries. The corresp onding

eigenvectors indicate whether the unbroken symmetry is a pure color, a pure

avor, or a coupled color- avor symmetry.

Our results are summarized in Table 1. The two avor case is sp ecial. In

this case, the dominant order parameter do es not break the color symmetry

completely, and the avor symmetry is completely unbroken. This is the sce-

nario discussed in [3, 4 ]. Sub dominantinteractions can break the remaining

color symmetry, either with or without [3] avor symmetry breaking.

The main result is that, for three avors, weverify that color- avor lo ck-

ing is indeed the preferred order parameter. We nd that all quark sp ecies

acquire a mass gap, and b oth color and avor symmetry are completely

broken. There are eightcombinations of color and avor symmetries that

generate unbroken global symmetries. These are the generators of the diag-

c+L+R

onal SU (3) . Also, the quark mass gaps fall into representations (8+1)

of the unbroken symmetry.And,asmentioned ab ove, the singlet state is

twice heavier than the o ctet.

Note that in the present analysis, which only takes into account the lead-

ing interaction, the order parameter is completely anti-symmetric in b oth

color and avor. We nd . If subleading interactions are

abI

taken into account, the order parameter will have the more general form

= + . This order parameter leaves the same residual

1 2

ab a b b a

symmetry.

The main qualitative results we found for three avors extend to N >

3. Color symmetry is always completely broken, and all quarks acquire a

mass gap. The only remaining symmetries are global coupled color- avor

symmetries. For massless quarks, chiral symmetry is sp ontaneously broken.

For an o dd number of avors, there are subleading instanton op erators that,

after the dominant gap is formed, can give an exp ectation value to op erators

of the form .For even numb ers of avors, generating a non zero h i

L R L R

is more subtle. Instantons can only give an exp ectation value to op erators 10

N N N gaps (deg) =(N N ) N

c f par c f sy m

3 2 3 (4) 3 ( ) + 3 (col)

0 0

3 3 9 (8), 2 (1) 0:80 1:27 8

0 0

3 4 18 (8), 2 (4) 0:63 1:21 6

0 0

3 5 30 (5), 2 (7), 3 (3) 0:43 1:18 3

0 0

3 6 45 (16), 2 (2) 0:80 1:27 9

0 0

Table 1: Groundstate prop erties of the s-wave sup er uid state in QCD with

N = 3 colors and N avors. N = N (N 1)N (N 1)=4 is the number

c f par c c f f

of totally anti-symmetric gap parameters. The column lab eled \gaps (deg)"

gives the relative magnitude of the gaps in the fermion sp ectrum, together

with their degeneracy.Thenumerical values of the gap and the condensation

energy p er sp ecies are given in units of = 36 MeV and =0:73 MeV =fm ,

0 0

resp ectively. N is the number of unbroken color- avor symmetries.

sy m

of the form ( ) .

L R

N = N (or a multiple thereof ) is the most favorable case, in the precise

f c

sense that in this case the condensation energy p er sp ecies is maximal. If N

is a multiple of N , the dominant gap corresp onds to multiple emb eddings

of the N = N order parameter. Wehave not studied the case N 6=3

f c c

systematically, but since color and avor are interchangeable in (2), the case

N =3forvarious numb ers of avors is covered implicitly. Also, wehave

not studied the interesting case N = N !1.

f c

Acknowledgement: We wish to thank S. Treiman for some helpful ques-

tions. 11

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