BEC-BCS crossover driven by the axial anomaly in the NJL model
aa r X i v : . [ h e p - ph ] S e p TKYNT-10-13, INT-PUB-10-044
BEC-BCS crossover driven by the axial anomalyin the NJL model
Hiroaki Abuki ∗ , Gordon Baym † , Tetsuo Hatsuda ∗∗ and Naoki Yamamoto ‡ ∗ Department of Physics, Tokyo University of Science, Tokyo 162-8601, Japan † Department of Physics, University of Illinois, 1110 W. Green St., Urbana, Illinois 61801, USA ∗∗ Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan ‡ Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA
Abstract.
We study the QCD phase structure in the three-flavor Nambu–Jona-Lasinio model,incorporating the chiral-diquark interplay due to the axial anomaly. We demonstrate that for acertain range of model parameters, the low temperature critical point predicted by a Ginzburg-Landau analysis appears in the phase diagram. In addition, we show that the axial anomaly presentsa new scenario for a possible BEC-BCS crossover in the color-flavor locked phase of QCD.
Keywords:
QCD, quark matter, superconducting, Bose-Einstein condensate
PACS:
INTRODUCTION
The phases of QCD at finite temperature T and quark chemical potential m are beingactively studied. In particular, at low T and high m a color superconducting (CSC)phase [1, 2] characterized by a diquark condensate h qq i is expected to appear owingto the attractive interaction between quarks, provided either by one-gluon exchange orby instantons. On the other hand, when the system at finite m is heated, a transition to aquark-gluon plasma (QGP) takes place at a (pseudo-)critical temperature.Our understanding of the phase structure based on the lattice simulations is stillimmature due to the severe sign problem at finite m . Therefore the analyses so far haverelied mainly on specific models of QCD, such as the Nambu–Jona-Lasinio model [3, 4],the Polyakov–Nambu–Jona-Lasinio (PNJL) model [5], and etc. Some of these modelindicated the possible existence of a critical point located at high T [6].The interesting possibility of a second critical point at rather low temperature in thecolor-flavor locked (CFL) phase was recently predicted on the basis of general Ginzburg-Landau (GL) analysis [7]. Moreover, this critical point has proven to make a quark-hadron continuity possible [8, 9, 10]. In two-flavor QCD, similar critical points havebeen found in the NJL model [11], although their origin is different from axial anomaly.This work is aimed at locating this critical point in the ( m , T ) -phase diagram usingthe phenomenological NJL model [12]. Starting with the three-flavor NJL model incor-porating the axial anomaly induced chiral-diquark interplay, we study the location ofthe new critical point and its dependence on the strength of the anomaly. We also ob-serve that the axial anomaly triggers a crossover between a Bose-Einstein condensedstate (BEC) of diquark pairing and Bardeen-Cooper-Schrieffer (BCS) diquark pairing[13, 14, 15, 16, 17] in the CFL phase. NCORPORATING THE AXIAL ANOMALY IN NJL MODEL
The Lagrangian of the NJL model with three-flavors consists of three terms: L = ¯ q ( i g m ¶ m − m q + mg ) q + L ( ) + L ( ) , (1)where q = ( u , d , s ) is the flavor triplet quark field, m q is a flavor symmetric quark mass( m u = m d = m s ), and m is the chemical potential for conserved quark number. L ( ) and L ( ) are the four-fermion and six-fermion interactions, respectively. As usual we set L ( ) = L ( ) c + L ( ) d with the standard choice [3, 4] L ( ) c = G tr ( f † f ) , L ( ) d = H tr [ d † L d L + d † R d R ] , (2)where f i j ≡ ( ¯ q R ) ja ( q L ) ia , ( d L ) ai ≡ e abc e i jk ( q L ) jb C ( q L ) kc , and ( d R ) ai ≡ e abc e i jk ( q R ) jb C ( q R ) kc ,with a , b , c and i , j , k the color and flavor indices, and C the charge conjugation operator.The flavor U ( ) generators t a ( a = , · · · ,
8) are normalized so that tr [ t a t b ] = d ab , and t A and l A ′ with A , A ′ = , , ( ) color,respectively. L ( ) is invariant under SU ( ) L × SU ( ) R × U ( ) A × U ( ) B symmetry. Theinteraction L ( ) c produces attraction of q ¯ q pairs, leading to the formation of a chiralcondensate. Similarly L ( ) d leads to attraction of qq pairs in the color-anti-triplet andspin-parity 0 ± channel, inducing a color-flavor locked (CFL) condensate [1]. We treatthe two couplings G and H as independent parameters.The six-fermion interaction in our model consists of two parts, L ( ) = L ( ) c + L ( ) c d . L ( ) c is the standard Kobayashi-Maskawa-’t Hooft (KMT) interaction [18], L ( ) c = − K ( det f + h . c . ) . (3)This interaction is not invariant under U ( ) A symmetry, which accounts for the axialanomaly in QCD due to instantons. Consequently the mass of the h ′ meson becomeslarger than that of the other pseudoscalar octet Nambu-Goldstone (NG) bosons ( p , h , K )for positive value of K . On the other hand, the term (3) makes the chiral phase transitionfirst-order as a function of T at m = L ( ) c d = K ′ (cid:16) tr [( d † R d L ) f ] + h . c . (cid:17) . (4)It is this term that is responsible for the aforementioned low temperature critical point.We assume K ′ >
0, so that qq pairs in the positive parity channel, h d L i = −h d R i , areenergetically favored. We keep K and K ′ as independent parameters [12].The condensates favored by the interaction L ( ) + L ( ) are the flavor-symmetricchiral and diquark condensates in the spin-parity 0 + channel, defined by h f i j i = ( c / ) d i j , h d Lai i = −h d Rai i = ( s / ) d ai . (5) ABLE 1.
Two sets of parameters in the present three-flavor NJL model:The momentum cutoff is fixed at L = . M and the chiral condensate c at vacuum are also given. m q [MeV] G L H L K L M [MeV] c / [MeV]Set I 0 1.926 1.74 12.36 355.2 − − Here the condensate order parameters are c and s .It is straightforward to derive the thermodynamic potential at the mean-field level [12] W ( c , s ; m , T ) = U ( c , s ) − Z | p |≤ L d p ( p ) (cid:229) ± (cid:2) w ± + w ± (cid:3) − T Z d p ( p ) (cid:229) ± h ( + e − w ± / T ) + ln ( + e − w ± / T ) i , (6)where L is a momentum cutoff to regulate the vacuum energy, U ( c , s ) = G c + H | s | − K c − K ′ | s | c , (7)is a constant term which is needed to cancel double counting of the interactions, and w ± = q ( p M + p ± m ) + ( D ) , w ± = q ( p M + p ± m ) + D (8)are the dispersion relations for the quasi-quarks in the octet and singlet representations,with M and D the dynamical Dirac and Majorana masses, defined as M = m q − (cid:18) G − K c (cid:19) c + K ′ | s | , D = − (cid:18) H − K ′ c (cid:19) | s | . (9)These equations imply that c < m q , while s isgenerally complex; the thermodynamic potential is a function of | s | . PHASE STRUCTURE AND DISCUSSION
The phase structures can be determined numerically by looking for the values of c and s that minimize the thermodynamic potential in Eq. (6). We follow the parameter choiceof [4]. We show in Table 1 two sets of parameters we adopt, Set I and Set II respectively.We vary the strength of the chiral-diquark coupling (the K ′ term) by hand. We work inthe flavor SU ( ) limit, assuming m u = m d = m s ≡ m q for simplicity.We show in Fig. 1 the phase structures for Set I (massless case) in the upper panel,and those for Set II (massive case) in the lower panel. Panels (a) and (b) show the resultswithout and with the K ′ -term; in (b) we have taken K ′ = . K with K = . / L as FIGURE 1.
The phase structure in the ( m , T )-plane in the three-flavor NJL model without (a) and with(b) the K ′ term. The Upper and lower panels present the results in the massless case I and the massivecase II respectively. Phase boundaries with a second-order transition are denoted by a single line and afirst-order transition by a double line. The dashed-dot line at high T in case II shows the chiral crossoverline, while the dotted line in (b) denotes the BEC-BCS crossover. See [12] for further details. a representative value. The phase diagrams contain a CFL phase with s = ( ) baryon number broken, and other two phases both characterized with s =
0, a Nambu-Goldstone (NG) phase with c = c = m q =
0) or c ∼ m q = m q since it acts as an external symmetry breaking source of chiralsymmetry breaking and thus smears the strength of the phase transition.The effect of nonvanishing K ′ can be seen by comparing (a) and (b). We indeed seethat the low temperature critical point shows up at the other end of the line of the firstorder chiral phase transition. This is , as discussed in [7], because the K ′ -term acts as anexternal field for c , which turns the first-order chiral phase transition into a crossoverin the CFL phase where s =
0. Note that the CFL phase in the panel (b) accompanies anonzero chiral condensate c = L ( ) c d .The axial anomaly, for sufficiently large chiral-diquark coupling K ′ , not only triggersthe low T critical point, but also drives a BEC-BCS crossover in the CFL phase, asdiscussed in [17]. Within an NJL-type model such a BEC regime appears for sufficiently FIGURE 2.
The phase diagram in the ( m , K ′ ) -plane at T = large pairing attraction, H , in the qq -channel [14, 15, 16]. The novel feature here isthat the axial anomaly helps to realize the BEC regime through its contribution to theeffective qq coupling. This can be easily seen by extracting from Eq. (4) the dominantzero mode ( f i j ∼ d i j c /
2) contribution to the quark-quark interaction L ( ) c d ∼ K ′ | c | tr (cid:2) ( d R − d L ) † ( d R − d L ) (cid:3) − K ′ | c | tr (cid:2) ( d R + d L ) † ( d R + d L ) (cid:3) . (10)The first term increases the effective attraction between quarks in the 0 + channel,while the second term is repulsive and suppresses the 0 − pairing. Thus when the chiralcondensate is nonvanishing, as in the NG phase, the axial anomaly helps the formationof a diquark BEC condensate.In fact it is possible to show that at sufficiently large K ′ there are nine diquark boundstates with mass M D ( m , T ) ≤ M ( m , T ) where M ( m , T ) is the dynamical Dirac mass ofquarks at m and T . Each diquark complex scalar has quark number ± m . Thus when 2 m hits M D ( m , T ) from below a BEC condensate muststart to form. Then the condition for the onset of a BEC approaching from the NG phase(NG-BEC boundary) is given by the condition [20, 14]2 m = M D ( m , T ) . (11)In order to see how the BEC domain in the CFL phase grows as a function of K ′ , weshow in Fig. 2 the phase diagram in the ( m , K ′ )-plane for massless quarks. The first-order line separating the CFL and NG phases for small K ′ eventually terminates at thecritical point P. On the other hand, for K ′ sufficiently large, a BEC regime of bounddiquarks appears across a second-order phase transition at a critical chemical potential m = M D ( m , ) / + channel. As a result, a BEC-BCS crossover or even the first orderBEC-BCS transition can be realized in the CFL phase.Finally we note that very recently the extension of our analysis incorporating theeffect of heavy strange quark mass was reported [21]. It still remains an important taskto extend the analyses imposing the charge neutrality and b -equilibrium conditions.The numerical calculations were carried out on Altix3700 at YITP in Kyoto University. REFERENCES
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