Charged rho superconductor in the presence of magnetic field and rotation
aa r X i v : . [ nu c l - t h ] A ug Charged rho superconductor in the presence of magnetic field and rotation
Gaoqing Cao
School of Physics and Astronomy, Sun Yat-sen University, Guangzhou 510275. (Dated: August 20, 2020)In this work, we mainly explore the possibility of charged rho superconductor (CRS) in thepresence of parallel magnetic field and rotation within three-flavor Nambu–Jona-Lasino model. Byfollowing similar schemes as in the previous studies of charged pion superfluid (CPS), the CRS isfound to be favored for both choices of Schwinger phase in Minkovski and curved spaces. Due to thestability of the internal spin structure, charged rho begins to condensate at a smaller threshold ofangular velocity than charged pion for the given large magnetic fields. Even the axial vector mesoncondensation is checked – the conclusion is that CRS is the robust ground state at strong magneticfield and fast rotation, which actually sustains to very large angular velocity.
I. INTRODUCTION
Nowadays, several extraordinary conditions can be re-alized in the terrestrial relativistic heavy ion collisions(HICs), such as strong electromagnetic (EM) field [1–4] and fast rotation [5–7]. Under such circumstances,the properties of quantum chromodynamics (QCD) sys-tem are quite interesting and attractive topics. Actually,magnetic field and rotation share some similar effect, thusthe proposal of chiral magnetic effect is right followed bythat of chiral vortical effect around 2008. These anoma-lous transport phenomena were intensively studied sincethen [8–10], and recently a very important breakthroughhas been acheived in the BES II experiment of STARgroup [11]. Nevertheless, along with the discoveries ofmagnetic catalysis effect at zero temperature [12, 13] andglobal polarization of Λ hyperon in peripheral HICs [14–16], some unexpected features emerge and still requireproper explanations: the inverse magnetic catalysis ef-fect [18, 19] and the ”sign puzzles” of the local polariza-tions [16, 20–23]. In some sense, the extreme conditionsopen a wide realm for the searching of new phases, such asCRS in pure magnetic field [24, 25], neutral pseudoscalarsuperfluid in parallel EM field [26–29] and CPS in parallelmagnetic field and rotation (PMR) [30]. Among others,the possibility of CRS was under fierce debate since itsproposal [24, 25, 31–36], mainly concerning the internalquark-antiquark effect on rho mesons.Recently, the existence of CPS in PMR became alsocontroversial according to the studies in Nambu–Jona-Lasinio (NJL) model, where one is immersed in the am-biguity of the definition for Schwinger phase [37, 38]. Thebreaking effect of rotation on the internal spin structureof charged pion was checked in these works for choices ofSchwinger phase in Minkovski (SPM) and curved (SPC)spaces, respectively. It turned out that CPS is neverfavored for SPC and only favored in the intermediateregime of angular velocity for SPM [38]. As mentionedin the conclusion of Ref. [37], the spins of valence quarkand antiquark are along the same direction in rho vec-tor mesons; thus, the spin-up ρ + meson is stable in thepresence of either strong magnetic field B or large ro-tating angular velocity Ω along z direction. Due to the mass reduction in B and effective isospin chemical poten-tial generated by Ω, it is quite probable that CRS wouldoccur in PMR and keep robust to very fast rotation. Sim-ilar to the electric field discussed in Ref.[26], the rotationterm breaks the semi-positivity of fermion determinantein the partition function. Therefore, the study in suchsetup is free from the constraint of Vafa-Witten (VW)theorem [33, 39], which was previously adopted as themain point against CRS in pure magnetic field. To showthe importance of rotation effect on CRS, we’d like tomention the interesting results found in Ref.[40]: At fi-nite isospin chemical potential µ I , CPS is always favoredover CRS; but CRS would finally manage to overwhelmCPS with Ω increasing.After getting some intuitions from the Weinberg modelin Sec.II, the paper keeps a similar structure as our pre-vious work Ref. [37]. In Sec.III, we present the formalismfor SU (3) NJL model in rotating frame with a parallelmagnetic field, where the simplest forms of vector inter-actions are introduced to explore rho meson physics [35].Then, the quadratic coefficient in Ginzburg-Landau ex-pansion will be evaluated analytically in Sec.IV with thechoice of SPM in Sec.IV A and of SPC in Sec.IV B, re-spectively. Eventually, the numerical results will be illu-minated in Sec.V to check the stability of QCD systemagainst CRS and we give a simple conclusion in Sec.VI.The natural units c = ~ = k B = 1 are used throughout. II. INTUITIONS FROM WEINBERG MODEL
From the chiral effective Weinberg model [41] with pionand rho mesons the fundamental degrees of freedom, theLagrangian density can be extended to the case withPMR ias L = 12 ( D µ π ) † · D µ π (1 + π † · π f π ) − m π π † · π π † · π f π ! − ρ † µν · ρ µν + m ρ " ρ µ + g ρ π × D µ π m ρ (1+ π † · π f π ) † · " ρ µ + g ρ π × D µ π m ρ (1+ π † · π f π ) ± i e F µν ρ ∓ µ ρ ± ν − B , (1)where chiral symmetry is nonlinearly realized through theterm 1 + π † · π f π in the denominators, and magnetic and ro-tation effects are encoded in the covariant derivative D µ .Neglecting all the self-interactions of pions for simplicity,the Lagrangian density is then reduced to L = − B (cid:2) ( D µ π ) † · D µ π − m π π † · π (cid:3) − ρ † µν · ρ µν + m ρ ρ † µ · ρ µ ± i e F µν ρ ∓ µ ρ ± ν + g ρ (cid:2) ρ † µ · ( π × D µ π )+( π × D µ π ) † · ρ µ (cid:3) . (2)Here, the isovectors are defined in the electric chargeeigenstates: π = ( π , π − , π + ) and ρ = ( ρ , ρ − , ρ + ), andwe assume for convenience that the rho vector mesonsare in the spin eigenstates: ρ µ = ( ρ t , ρ ↓ , ρ , ρ ↑ ). Take ρ mesons for example, the charge eigenstates are related tothe isospin ones ρ i ( i = 1 , ,
3) as ρ = ρ , ρ ± = ρ ∓ iρ √ , and the spin eigenstates are defined by the Lorentz com-ponents ρ µ ( µ = t, x, y, z ) as ρ = ρ z , ρ ↑ / ↓ = ρ x ∓ i ρ y √ . In the vacuum, the strength tensors of ρ mesons aredefined in a similar way as those of gauge fields in the SU (2) Yang-Mills theory: ρ aµν ≡ ∂ µ ρ aν − ∂ ν ρ aµ + g ρ ǫ abc ρ bµ ρ cν with the coupling constant given by g ρ = √ m ρ /f π [41].These tensors can be rearranged in the charge eigenstatesso that the magnetic effect can be introduced directly bychanging ∂ µ to covariant derivative D µ ≡ ∂ µ + i qA µ with q the particle charge. Then, we get the strength tensorsof charge-definite ρ as ρ µν = ∂ µ ρ ν − ∂ ν ρ µ + i g ρ ( ρ − µ ρ + ν − ρ − ν ρ + µ ) , (3) ρ ± µν = D ± µ ρ ± ν − D ± ν ρ ± µ , D ± µ ≡ ∂ µ ± ieA µ ∓ i g ρ ρ µ , (4)where we find that ρ ± µν can be simply present in Abelianforms with the redefinition of the gauge field as A µ − g ρ e ρ µ .In accordance with the spin eigenstates, the correspond-ing covariant derivatives are related to the Lorentz com-ponents as: D ( ± )0 = D ( ± ) z , D ( ± ) ↑ / ↓ = i D ( ± ) x ∓ i D ( ± ) y √ ρ mesons. Consideringa constant magnetic field along z direction, we choosethe symmetric gauge for the vector potential: A µ =(0 , By/ , − Bx/ , ± i e F µν ρ ∓ µ ρ ± ν = ± eB (cid:16) ρ ∓↓ ρ ±↑ − ρ ∓↑ ρ ±↓ (cid:17) . Furthermore, according to the discussions in Ref.[43, 44],the effect of rotation along z direction can be simply in-troduced through the modification of temporal derivative ∂ t to D t = ∂ t − i Ω (cid:16) ˆ L z + ˆ S z (cid:17) , where ˆ L z ≡ − i ( x∂ y − y∂ x ) and ˆ S z are the orbital angularmomentum (OAM) and spin operators, respectively.Now, without applying any boundary condition to acylindrical system with radius R , the diagonal kineticparts of the Lagrangian can be expressed explicitly onthe basis of energy k , momentum k , Landau level n and OAM quantum number l as L k = − π − l (cid:2) ( k − i Ω l ) + k l + k + m π (cid:3) π l − π ± n, ± l (cid:2) ( k ± i Ω l ) + (2 n + 1) | eB | + k + m π (cid:3) π ∓ n, ∓ l − ρ − s, − l (cid:2) ( k − i Ω( l + s )) + k l + k + m ρ (cid:3) ρ s,l + 12 ρ t, − l (cid:2) ( k − i Ω( l + s )) + k l + k + m π (cid:3) ρ t,l − ρ ±− s,n, ± l (cid:2) ( k − i Ω( ∓ l + s )) + (2 n + 1 ± s ) | eB | + k + m ρ (cid:3) ρ ∓ s,n, ∓ l + 12 ρ ± t,n, ± l (cid:2) ( k ± i Ω l ) + (2 n + 1) | eB | + k + m ρ (cid:3) ρ ∓ t,n, ∓ l , (5)where particularly the summations over n, l, s should be understood with s = − , , n ≥ l ∈ ( −∞ , ∞ ) for neutralparticles and l ∈ ( − n, [ N ] − n ) for charged ones [43]. Note that the OAM is given by − l for negative charged particleand [ N ] ≡ h | qB | R i is the number of magnetic flux quantization. Especially, we choose the simplest Lorentz gauge D µ ρ µ = 0 for ρ mesons, then the commutations [ D µ , D ν ] from ρ † µν · ρ µν give rise to extra kinetic terms the sameas the strength tensor couplings. The left particle coupling parts of the Lagrangian involve the self-interactions of ρ mesons, which are quite the same as those of W/Z bosons in the electroweak theory, and the ρππ interactions whoseexplicit forms can be illuminated as L ρππ = g ρ nh(cid:16) ρ −↓↑ ,n, − l π + n ′ ,l ′ − ρ + ↓↑ ,n,l π − n ′ , − l ′ (cid:17) (cid:0) ± k ′′ l ′′ π l ′′ ∓ (cid:1) / √ − (cid:16) ρ − ,n, − l π + n ′ ,l ′ − ρ +0 ,n,l π − n ′ , − l ′ (cid:17) k ′′ π l ′′ − i (cid:16) ρ − t,n, − l π + n ′ ,l ′ − ρ + t,n,l π − n ′ , − l ′ (cid:17) ( k − i Ω l ′′ ) π l ′′ i − h(cid:16) ρ −↓ ,n, − l p ( n ′′ + 1) | eB | π + n ′′ +1 ,l ′′ − + ρ + ↓ ,n,l p n ′′ | eB | π − n ′′ − , − l ′′ − (cid:17) π l ′′ − − (cid:16) ρ −↑ ,n, − l p n ′′ | eB | π + n ′′ − ,l ′′ +1 + ρ + ↑ ,n,l p ( n ′′ + 1) | eB | π − n ′′ +1 , − l ′′ +1 (cid:17) π l ′′ +1 − (cid:16) ρ − ,n, − l k ′ π + n ′ ,l ′ − ρ +0 ,n,l k ′ π − n ′ , − l ′ (cid:17) π l ′′ − i (cid:16) ρ − t,n, − l ( k − i Ω l ′ ) π + n ′ ,l ′ − ρ + t,n,l ( k + i Ω l ′ ) π − n ′ , − l ′ (cid:17) π l ′′ i + h ρ ↓ ,l (cid:16) π − n ′ , − l ′ p ( n ′′ + 1) | eB | π + n ′′ +1 ,l ′′ − + π + n ′ ,l ′ p n ′′ | eB | π − n ′′ − , − l ′′ − (cid:17) − ρ ↑ ,l (cid:16) π − n ′ , − l ′ p n ′′ | eB | π + n ′′ − ,l ′′ +1 + π + n ′ ,l ′ p ( n ′′ + 1) | eB | π − n ′′ +1 , − l ′′ +1 (cid:17) − ρ ,l (cid:16) π − n ′ , − l ′ k ′′ π + n ′′ ,l ′′ − π + n ′ ,l ′ k ′′ π − n ′′ , − l ′′ (cid:17) − iρ t,l (cid:16) π − n ′ , − l ′ ( k − i Ω l ′′ ) π + n ′′ ,l ′′ − π + n ′ ,l ′ ( k + i Ω l ′′ ) π − n ′′ , − l ′′ (cid:17)io . (6)Here, as before, the summations over the quantum num-bers of all the relevant particles should be understood.We note that the coordinate integrations haven’t beencarried out yet in Eq.(6), that is why there seem no con-nections among the quantum numbers of the interactingparticles.In the following, we skip the complicated interactionparts and only focus on the kinetic parts of the La-grangian to get some physical intuitions. From Eq.(5),we surprisingly notice that rotation even induces effectivechemical potentials for neutral pion and rho, thus the π ( ρ ) condensation is expected when | Ω l | > m π ( | Ω( l + s ) | > m ρ ). Moreover, in the presence of a magnetic field,the π seems much easier to condense than π ± with thecondition for the latter: | Ω l | > p m π + | eB | . All thesepuzzles can be consistently solved when we combine therestriction of causality together with boundary condi-tions [43, 45, 46]. As a matter of fact, causality constrainsthe angular velocity to Ω ≤ /R and the Dirichlet bound-ary conditions, requiring the wave functions to vanish atthe boundary R , discretizes the transverse momentumsuch that | k l | > ( | l | + 2) /R for each l . Thus, the excita-tion energy E = ( k l + k + m ) / of the neutral particlesatisfies E > | k l | > | Ω( l + s ) | , which eventually prevents any accumulation of π or ρ meson.Next, we discuss a bit more about the effect of bound-ary condition on π ± in a background magnetic field. For π + with l >
0, the excitation energy for transverse dy-namics is [30] E nl ( eB ) ≡ q m π + | eB | (2 λ nl + 1)with the boundary condition F ( − λ nl , l + 1 , N ) = 0and the quasi Landau levels 0 ≤ λ l < λ l < λ l < . . . .We’ve checked numerically that there is always a win-dow of l satisfying l > λ l + 1) N when N &
7, which means the unstable condition Ω l > E l ( eB ) can be real-ized for large enough Ω and eB . This is consistent withCPS found in Ref. [30]. However, one problem is still left:if we don’t artificially set l ≤ [ N ] as in Ref. [30], it seemsthat the lowest total particle energy E Ω0 l = E l ( eB ) − Ω l is not bound from below, which would cause a disaster ofinfinite condensate density according to Ref. [30]. Actu-ally, the answer is that λ l becomes quite large for large l , which then makes sure that E Ω0 l > −∞ in the limit l → ∞ .For the purpose of intuition, we show the scaled di-mensionless energy˜ E Ω0 l ≡ p | eB | (2 λ l + 1) − lR p | eB | = q λ l + 1 − l √ N for different values of N in Fig.1, where positive featurescan always be identified at large l . And with the increas-ing of N , that is, the enhancement of eB for a given R , more π + can be condensed as more are trapped inthe system. It should be pointed out that finite lowerboundaries exist for the total energies of any chargedparticles. In pure magnetic field, it was found that the = = = - - l E ˜ l Ω FIG. 1. The dimensionless energy ˜ E Ω0 l as a function of theorbital angular momentum l for different values of N . degeneracy of l is automatically restricted to ≤ [ N ] forthe lowest Landau level when the boundary condition isapplied [43]. We even check in advance that the degen-eracy decreases with the Landau level n , see the plainsin Fig.2. Nevertheless, for the case N = 15 in Fig.1,the much wider unstable window of l disfavors the use ofthe artificial upper bound [ N ] for l when large angularvelocity (Ω . /R ) is involved. = n = = =
40 60
80 100 - l λ l n FIG. 2. The three lowest quasi Landau levels λ nl ( n = 0 , , l for N = 100. Finally, turn to the charged ρ vector mesons, the storyis quite different because of their non-vanishing spins.In the presence of PMR, the effective mass of ρ + ↑ or ρ −↓ decreases as q m ρ + | eB | (2 λ l −
1) on one hand [24], theeffective isospin chemical potential increases as Ω( l + 1)on the other hand. Then, it seems that ρ + ↑ condensationwould overwhelm the π + condensation to be the trueground state when Ω is large enough thatΩ( l + 1) − q m ρ + | eB | (2 λ l − > Ω l − E l ( eB ) > . As the VW theorem might forbid the decreasing of com-posite ρ + ↑ mass to zero in pure magnetic field [33–36], theestimation of the ρ + ↑ mass is not correct at all for large B in the point particle picture. However, as mentionedin the introduction, Ω invalids the proof of the theoremthus the isospin chemical potential effect of Ω( l + 1) canstill be qualitatively correct in the point particle picture.As a strong support of this point, we’d like to mentionthat the isospin effect was first discussed in chiral per-turbation theory with pions the fundamental degrees offreedom [47], and the proposed CPS was well verified bythe effective NJL model [48, 49] and lattice QCD simu-lations [50, 51] with quarks the fundamental degrees offreedom. III. NAMBU–JONA-LASINIO MODEL INROTATING FRAME
In order to explore the possibility of charged rho con-densation more realistically, we adopt the SU (3) NJL model with u, d and s quarks the fundamental degreesof freedom [52]. In the rotating frame, the action of thesystem can be conveniently given in curved spacetime by S = Z d x q − det( g µν ) L ( ¯ ψ, ψ ) , (7)where the Lagrangian density can be extended from theusual one [52, 53] to L NJL = ¯ ψ ( i /D − m ) ψ + G S X a =0 [( ¯ ψλ a ψ ) + ( ¯ ψiγ λ a ψ ) ]+ L − G V h(cid:0) ¯ ψγ µ τ a ψ (cid:1) + (cid:0) ¯ ψγ µ γ τ a ψ (cid:1) i L = − K X s = ± Det ¯ ψ Γ s ψ (8)by further adopting the four fermion vector interactionchannels with coupling constant G V . Compared to thetwo-flavor NJL model, the advantage of three-flavor NJLmodel is that there the vacuum superconductivity or CRScannot happen in pure magnetic field [35] which is con-sistent with lattice QCD simulations [33, 34, 36].In the Lagrangian, ψ = ( u, d, s ) T represents the three-flavor quark field and m = diag( m , m , m ) is thecurrent quark mass matrix. The longitudinal and trans-verse covariant derivatives with PMR effect are defined,for symmetric gauge, respectively as D = ∂ t − i Ω (cid:16) ˆ L z + ˆ S z (cid:17) , D = ∂ z and D = ∂ x + iQ By/ , D = ∂ y − iQ Bx/ Q = diag( q u , q d , q s ). For the four-fermion interaction terms, λ = q I and Gell-Mann ma-trices λ i ( i = 1 , . . . ,
8) are defined in three-flavor space,so the extra diagonal terms ( ¯ ψλ ψ ) and ( ¯ ψλ ψ ) allowmass splitting among all the flavors in contrary to thetwo-flavor case [35]. The U A (1) symmetry violating term L [54] only involves scalar-pseudoscalar channels withthe determinant defined in flavor space, Γ ± = 1 ± γ and K the coupling constant. Now, we only consider nonzerochiral condensations σ i ≡ h ¯ ψ i ψ i i , where the correspon-dence between the Arabian denotations i = 1 , , , d , s should be under-stood for the flavors. The six fermion interactions in L can be reduced to effective four fermion ones in Hartreeapproximation [52], then the Lagrangian density only in-volves four fermion effective interactions: L = ¯ ψ ( i /D − m ) ψ + X a,b =0 (cid:2) G − ab ( ¯ ψλ a ψ )( ¯ ψλ b ψ )+ G + ab ( ¯ ψiγ λ a ψ )( ¯ ψiγ λ b ψ ) (cid:3) − G V h(cid:0) ¯ ψγ µ τ a ψ (cid:1) + (cid:0) ¯ ψγ µ γ τ a ψ (cid:1) i , (9)where the non-vanishing elements of the symmetric coupling matrices G ± are given by [52] G ∓ = G S ∓ K X f=u , d , s σ f , G ∓ = G ∓ = G ∓ = G S ± K σ s , G ∓ = G ∓ = G S ± K σ d , G ∓ = G ∓ = G S ± K σ u ,G ∓ = G S ∓ K σ s − σ u − σ d ) , G ∓ = ∓ √ K
12 (2 σ s − σ u − σ d ) , G ∓ = −√ G ∓ = ∓ √ K σ u − σ d ) . (10)In the case h σ i i 6 = 0, the inverse quark propagators of different flavors are given by the introductions of the dynamicalmasses and covariant derivatives as S − i ( x, x ′ ) = i /D − m i , m i = m i − G S σ i + K X jk ǫ ijk σ j σ k . (11)By using the eigenfunction reconstruction method, we have given the propagator of a fermion with positive or negativecharge [37]; so the u and d/s quark propagators are respectively S u ( x, x ′ ) = ∞ X n =0 X l Z Z dp dp z (2 π ) i e − ip ( t − t ′ )+ ip z ( z − z ′ ) (cid:0) p l +0 (cid:1) − ( ε un ) + iǫ ( h P ↑ χ + n,l ( θ, r ) χ + ∗ n,l ( θ ′ , r ′ ) + P ↓ χ + n − ,l +1 ( θ, r ) χ + ∗ n − ,l +1 ( θ ′ , r ′ ) i(cid:0) γ p l +0 − γ p z + m u (cid:1) − h P ↑ χ + n,l ( θ, r ) χ + ∗ n − ,l +1 ( θ ′ , r ′ ) + P ↓ χ + n − ,l +1 ( θ, r ) χ + ∗ n,l ( θ ′ , r ′ ) i p n | qB | γ ) q → q u , (12) S d / s ( x, x ′ ) = ∞ X n =0 X l Z ∞−∞ dp dp z (2 π ) i e − ip ( t − t ′ )+ ip z ( z − z ′ ) (cid:0) p l − (cid:1) − ( ε d/sn ) + iǫ ( h P ↑ χ − n − ,l − ( θ, r ) χ −∗ n − ,l − ( θ ′ , r ′ ) + P ↓ χ − n,l ( θ, r ) χ −∗ n,l ( θ ′ , r ′ ) i(cid:0) γ p l − − γ p z + m d / s (cid:1) + h P ↑ χ − n − ,l − ( θ, r ) χ −∗ n,l ( θ ′ , r ′ ) + P ↓ χ − n,l ( θ, r ) χ −∗ n − ,l − ( θ ′ , r ′ ) i p n | qB | γ ) q → q d , (13)where p ls = p +Ω (cid:0) l + s (cid:1) , the dispersion relations ε in =( p z + 2 n | q i B | + m i ) / and P ↑ / ↓ = (1 ± σ ) are thespin projectors. Here, the normalized auxiliary functionsare defined for positive and negative charged particlesrespectively as χ + n,l ( θ, r ) = (cid:20) | qB | π n !( n + l )! (cid:21) e i lθ ˜ r l e − ˜ r / L ln (cid:0) ˜ r (cid:1) , (14) χ − n,l ( θ, r ) = (cid:20) | qB | π n !( n − l )! (cid:21) e i lθ ˜ r − l e − ˜ r / L − ln (cid:0) ˜ r (cid:1) , (15)where the dimensionless radius ˜ r = | qB | r / L ln ( x ) is nonvanishing only for n ≥
0. Actually, in a rotating system, a boundary must be ap-plied due to causality, then the propagators would nolonger keeps the forms of Eqs.(12) and (13). But for con-venience, we still adopt these forms and constrain theOAM as l ∈ [ − n, N − n ] [38]. Armed with that, thequark masses can be evaluated through the gap equationsgiven by the self-consistent definitions of chiral conden-sations as: σ i ≡ h ¯ ψ i ψ i i = − iV Tr S i . (16)By adopting vacuum regularization, the explicit form ofthe gap equations are − σ f = N c m f 3 π h ˜Λ f (cid:16) (cid:17) − ln (cid:16) ˜Λ f + (cid:16) (cid:17) (cid:17)i + N c m f π Z ∞ dss e − m f2 s (cid:18) q f Bs tanh( q f Bs ) − (cid:19) − N c m f n max X n =0 S N f X l =0 Z ∞−∞ dp z π α n ε f n h f ( ε f n + Ω nl ) + f ( ε f n − Ω nl ) i (17)with the reduced cutoff ˜Λ f = Λ /m f . Compared to that given in Ref. [37] but implicitly implied, the Landau lev-els are cut off by n max ( ≪ N f ) here, which was checkedto be a good approximation for large B and Ω = 0.For a continuous transition, the effective potential canbe expressed as the form in Ginzburg-Landau (GL) the-ory V eff ( σ f , ∆) = V eff ( σ f ,
0) + A ∆ + B ∆ + . . . , (18)where V eff ( σ f ,
0) is the corresponding thermodynamic po-tential giving rise to the gap equations in Eq.(17). Wenote that ∆ can be an order parameter for any kindof mesonic superfluid or superconductor with the rele-vant coefficients A and B determined by their interac-tions with quarks. To avoid too much complexity, we as-sume the transition is solely determined by the quadraticone A [37]: If A <
0, the meson condensation is fa-vored; and if A a < A b <
0, we would assume meson a is more preferred to condensate than meson b . We’ve discovered previously that A is almost consistent withthe inverse mesonic propagator in random phase approx-imation (RPA) except for some subtle discussions on theSchwinger phases of charged mesons [37, 38, 55]. As il-luminated in Ref. [35], the bare form of the coefficient isgiven by A = 14 G + Πwith the polarization function defined through thefermion loop asΠ = iV Tr (cid:2) S ( x, y )Γ M ∗ S ( y, x )Γ M e − i Φ M (cid:3) . (19)Here, V is the space-time volume, the trace should betaken over the internal and coordinate spaces, and e − i Φ M is the compensated Schwinger phase. IV. CALCULATIONS OF THE QUADRATIC COEFFICIENT
This section is mainly devoted to calculating the quadratic efficient explicitly. The interaction vertices betweenquarks and (pseudo-)scalar and vector mesons have been listed in Ref. [35] as:Γ σ/σ ∗ = − , Γ π /π ∗ = − iγ τ , Γ π ± = − iγ τ ± , Γ ¯ ω µ / ¯ ω ∗ µ = ¯ γ ± µ , Γ ¯ ρ µ / ¯ ρ ∗ µ = ¯ γ ± µ τ , Γ ¯ ρ ± µ = ¯ γ ± µ τ ± , (20)where ¯ γ ± µ = ( γ , γ ± iγ √ , γ ∓ iγ √ , γ ). In the following, we mainly focus on the ¯ ρ +1 mode, that is, the rho meson withspin long the magnetic field. The insertion of the fermion propagators from Eq.(17) into the polarization functionEq.(19) is explicitlyΠ ¯ ρ +1 = − iS n max X n =0 X l n max X n ′ =0 X l ′ Z ∞−∞ dp π Z ∞−∞ dp z π Tr ( h P ↑ χ + n,l ( θ, r ) χ + ∗ n,l ( θ ′ , r ′ ) + P ↓ χ + n − ,l +1 ( θ, r ) χ + ∗ n − ,l +1 ( θ ′ , r ′ ) i × (cid:0) γ p l +0 − γ p z + m u (cid:1) − h P ↑ χ + n,l ( θ, r ) χ + ∗ n − ,l +1 ( θ ′ , r ′ ) + P ↓ χ + n − ,l +1 ( θ, r ) χ + ∗ n,l ( θ ′ , r ′ ) i γ p nq u B ) × ( − h P ↓ χ − n ′ − ,l ′ − ( θ ′ , r ′ ) χ −∗ n ′ − ,l ′ − ( θ, r ) + P ↑ χ − n ′ ,l ′ ( θ ′ , r ′ ) χ −∗ n ′ ,l ′ ( θ, r ) i (cid:16) γ p l ′ − − γ p z − m d (cid:17) γ + iγ √ − h P ↓ χ − n ′ − ,l ′ − ( θ ′ , r ′ ) χ −∗ n ′ ,l ′ ( θ, r ) + P ↑ χ − n ′ ,l ′ ( θ ′ , r ′ ) χ −∗ n ′ − ,l ′ − ( θ, r ) i γ p n ′ | q d B | γ − iγ √ ) γ − iγ √ × e − i Φ h(cid:0) p l +0 (cid:1) − ( ε un ) i h(cid:0) p l ′ − (cid:1) − ( ε dn ′ ) i , (21)where the trace is over the Dirax and the coordinate spaces. By completing the trace over the Dirac space, theexpression becomes quite simple:Π ¯ ρ +1 = − N c iS n max X n =0 X l n max X n ′ =0 X l ′ X r,r ′ X θ,θ ′ Z ∞−∞ dp π Z ∞−∞ dp z π e − i Φ h(cid:0) p l +0 (cid:1) − ( ε un ) i h(cid:0) p l ′ − (cid:1) − ( ε dn ′ ) i (cid:16) p l +0 p l ′ − − p z − m u m d (cid:17) χ + n,l ( θ, r ) χ + ∗ n,l ( θ ′ , r ′ ) χ − n ′ ,l ′ ( θ ′ , r ′ ) χ −∗ n ′ ,l ′ ( θ, r ) , (22)where we define P r,r ′ = R ∞ rdr R ∞ r ′ dr ′ and P θ,θ ′ = R π dθ R π dθ ′ . Consistent with the form in pure magneticfield [35] but different from that of charged pion [37], only the term with numerator independent of Landau levelssurvives and there is only one kind of combination of χ + χ + ∗ and χ − χ −∗ . By the way, for the corresponding axialvector ¯ a +1 with interaction indices Γ ¯ a +1 = iγ Γ ¯ ρ +1 , the polarization function is the same as that of ¯ ρ +1 except that thesign of the mass term is changed, that is, − m u m d → + m u m d . We’ve shown in Ref. [35] that the contribution of themass term is negative to Π ¯ ρ +1 , hence m ¯ a +1 > m ¯ ρ +1 in the chiral symmetry breaking phase. Then it turns out that the¯ a +1 superconductor is neither favored in magnetic field, but it still needs to be checked in PMR where this term canbe positive for ¯ ρ +1 . A. For Schwinger phase in Minkovski space
To calculate Eq.(22) further, we choose the Schwinger phase of the form in Minkovski space, that is, Φ M = e R yx A µ ( z ) dz µ = eB sin( θ − θ ′ ) rr ′ . Then, the integrals over the polar angles can be completed to giveΠ ¯ ρ +1 = − N c iS n max X n,n ′ =0 N u X l =0 N d X l ′ =0 X r,r ′ Z ∞−∞ dp π Z ∞−∞ dp z π e − eB ( r + r ′ ) (cid:16) p ( l − n )+0 p ( n ′ − l ′ ) − − p z − m u m d (cid:17)(cid:20)(cid:16) p ( l − n )+0 (cid:17) − ( ε un ) (cid:21) (cid:20)(cid:16) p ( n ′ − l ′ ) − (cid:17) − ( ε dn ′ ) (cid:21) n ! n ′ ! l ! l ′ ! (cid:18) q u B (cid:19) l − n +1 (cid:18) | q d B | (cid:19) l ′ − n ′ +1 J l + l ′ − n − n ′ (cid:18) eB rr ′ (cid:19) ( rr ′ ) l + l ′ − n − n ′ F nl,n ′ l ′ ( q u B, | q d B | ; r, r ′ ) , (23)where the auxiliary function F is defined as F nl,n ′ l ′ ( q u B, | q d B | ; r, r ′ ) ≡ Y x = r,r ′ L l − nn (cid:18) q u B x (cid:19) L l ′ − n ′ n ′ (cid:18) | q d B | x (cid:19) . (24)Here, we find that the LLL combination of u and d quarks contribute to the term F nl,n ′ l ′ ( q u B, | q d B | ; r, r ′ ) with n = n ′ = 0, due to the special structure of χ + χ + ∗ and χ − χ −∗ in Eq.(22). For charged pion, this kind of combinationis absent [37] due to the fact that the LLLs of of u and d quarks cannot form spin singlet at all.For convenience, we redefine the radii to dimensionless ones ¯ r = ( eB/ / r and ¯ r ′ = ( eB/ / r ′ , then Eq.(23)becomesΠ ¯ ρ +1 = − N c iS n max X n,n ′ =0 N u X l =0 N d X l ′ =0 Z ∞−∞ dp π Z ∞−∞ dp z π (cid:16) p ( l − n )+0 p ( n ′ − l ′ ) − − p z − m u m d (cid:17) F nl,n ′ l ′ (˜ q u , | ˜ q d | ) (cid:20)(cid:16) p ( l − n )+0 (cid:17) − ( ε un ) (cid:21) (cid:20)(cid:16) p ( n ′ − l ′ ) − (cid:17) − ( ε dn ′ ) (cid:21) , (25) F nl,n ′ l ′ (˜ q u , | ˜ q d | ) = n ! n ′ ! l ! l ′ ! ˜ q l − n +1 u | ˜ q d | l ′ − n ′ +1 X ¯ r, ¯ r ′ e − ¯ r r ′ )22 J l + l ′ − n − n ′ (¯ r ¯ r ′ ) (¯ r ¯ r ′ ) l + l ′ − n − n ′ F nl,n ′ l ′ (2˜ q u , | ˜ q d | ; ¯ r, ¯ r ′ ) (26)with ˜ q = q/e . It is useful to transform the numerator p ( l − n )+0 p ( n ′ − l ′ ) − − p z − m u m d of the integrand in Eq.(25) to thefollowing form:12 (cid:20)(cid:16) p ( l − n )+0 (cid:17) − ( ε un ) + (cid:16) p ( n ′ − l ′ ) − (cid:17) − ( ε dn ′ ) − Ω nl,n ′ l ′ + ( m u − m d ) + 2 n q u B + 2 n ′ | q d B | (cid:21) , (27)where Ω nl,n ′ l ′ = ( l + l ′ − n − n ′ + 1)Ω. Then the polarization function becomesΠ ¯ ρ +1 = − N c iS n max X n,n ′ =0 N u X l =0 N d X l ′ =0 F nl,n ′ l ′ (˜ q u , | ˜ q d | ) Z ∞−∞ dp π Z ∞−∞ dp z π (cid:16) p ( l − n )+0 (cid:17) − ( ε un ) + 1 (cid:16) p ( n ′ − l ′ ) − (cid:17) − ( ε dn ′ ) + − Ω nl,n ′ l ′ + ( m u − m d ) + 2 n q u B + 2 n ′ | q d B | (cid:20)(cid:16) p ( l − n )+0 (cid:17) − ( ε un ) (cid:21) (cid:20)(cid:16) p ( n ′ − l ′ ) − (cid:17) − ( ε dn ′ ) (cid:21) . (28)Shifting to Euclidean space through the transformations: p → iω m and − i R ∞−∞ dp π → T P ∞ m = −∞ and completingthe summation over the fermion Matsubara frequency ω m = (2 m + 1) πT , we haveΠ ¯ ρ +1 = − N c S n max X n,n ′ =0 N u X l =0 N d X l ′ =0 F nl,n ′ l ′ (˜ q u , | ˜ q d | ) X s = ± Z ∞−∞ dp z (2 π ) ( tanh (cid:18) ε un − s Ω nl, T (cid:19) ε un " − Ω nl,n ′ l ′ + ( m u − m d ) + 2 n q u B + 2 n ′ | q d B | ( ε un − s Ω nl,n ′ l ′ ) − ( ε dn ′ ) + (cid:0) ε dn ′ ↔ ε un , nl ↔ n ′ l ′ , q u ↔ | q d | (cid:1)) . (29)The temperature and rotation dependent part can be separated out as Π ¯ ρ +1 − Π ¯ ρ +1 (cid:12)(cid:12) Ω → ,T → , which should be convergentsimilar to that of charged pion [37]. Here, the subtracting term is just the polarization function in pure magneticfield, which has been regularized in Ref. [35] as h Π B ¯ ρ +1 − Π o ( B )¯ ρ +1 i + Π Λ¯ ρ +1 . We close this section by listing the relevantterms: Π B ¯ ρ +1 = − N c π Z d ss Z − d u e − s [ m u + + m u − ] (cid:18) m u m d + 1 s (cid:19) [1+tanh ( q u Bsu + )] [1 − tanh ( q d Bsu − )] tanh( q u Bsu + ) q u Bs + tanh( q d Bsu − ) q d Bs (30)with u ± = (1 ± u ) /
2, Π o ( B )¯ ρ +1 is the weak B expansion of Π B ¯ ρ +1 to order o ( B ) and the term with three-momentumcutoff Λ isΠ Λ¯ ρ +1 = − N c Z Λ0 k dkπ ( E u E d + m u m d + k ) E u E d ( E u + E d ) − N c Z Λ0 k dk π (cid:26) q u B ( E u + E d ) (cid:20)(cid:18) E u E d + m u m d + k E + 1 E u + 1 E d (cid:19) − ( m u − m d ) + k E E d (cid:21) − ( u ↔ d ) (cid:27) . (31) B. For Schwinger phase in curved space
Next, we choose the Schwinger phase of the form in curved space, that is, Φ M = eB sin[ θ − θ ′ + Ω( t − t ′ )] rr ′ . Bytaking variable transformation of the angle: θ − θ ′ → θ − θ ′ − Ω( t − t ′ ), we find that the corresponding polarizationfunction can be modified from Eq.(25) by changing p ( l − n )+0 to p ( n ′ − l ′ )+0 , that is,Π ¯ ρ +1 = − N c iS n max X n,n ′ =0 N u X l =0 N d X l ′ =0 Z ∞−∞ dp π Z ∞−∞ dp z π (cid:16) p ( n ′ − l ′ )+0 p ( n ′ − l ′ ) − − p z − m u m d (cid:17) F nl,n ′ l ′ (˜ q u , | ˜ q d | ) (cid:20)(cid:16) p ( n ′ − l ′ )+0 (cid:17) − ( ε un ) (cid:21) (cid:20)(cid:16) p ( n ′ − l ′ ) − (cid:17) − ( ε dn ′ ) (cid:21) . (32)Then, in a similar process as the previous section, the summation over the Fermion Matsubara frequency givesΠ ¯ ρ +1 = − N c S n max X n,n ′ =0 N u X l =0 N d X l ′ =0 F nl,n ′ l ′ (˜ q u , | ˜ q d | ) X s = ± Z ∞−∞ dp z (2 π ) ( tanh (cid:18) ε un + s Ω ,n ′ l ′ T (cid:19) ε un + tanh ε dn ′ − s Ω ,n ′ l ′ T ! ε dn ′ + (cid:2) − Ω +( m u − m d ) +2 n q u B +2 n ′ | q d B | (cid:3) ε un tanh (cid:18) ε un + s Ω
32 0 ,n ′ l ′ T (cid:19) ( ε un − s Ω) − ( ε dn ′ ) + 1 ε dn ′ tanh (cid:18) ε dn ′ − s Ω
12 0 ,n ′ l ′ T (cid:19)(cid:0) ε dn ′ − s Ω (cid:1) − ( ε un ) ) . (33)In contrary to that of charged pion [38], the angular velocity Ω can still plays a role of effective isospin chemicalpotential to ¯ ρ +1 in this case, see the denominators. Finally, one should keep in mind that the regularization to Eq.(32)is performed in the same way as that of SPM. V. POSSIBILITY OF CHARGED RHOSUPERCONDUCTOR
In order to carry out numerical calculations, we choosethe following parameters for the scalar-pseudoscalar sec-tor: m u = m d = 5 . , m s = 140 . , Λ = 602 . , G S Λ = 1 .
835 and K Λ = 12 .
36 [56]. Toavoid artifacts, the vector coupling constant is fixed to G V Λ = 2 .
527 by fitting to the vacuum mass of ρ meson: m vρ = 0 . R = 20 / √ eB and constrain the rotation by Ω R ≤ m f ( G e V ) ± ( M ) ± ( C ) ± ( M ) ± ( C ) - - - - ( GeV ) (cid:4) ( G e V ) m u m d m s eB = FIG. 3. The evolutions of the dynamical quark masses m f (upper panel) and the quadratic GL expansion coefficients A (lower panel) with the angular velocity Ω at the magneticfield eB = 0 . . In the lower panel, the coefficients ofcharged rho condensation are compared to those of chargedpion for both choices of SPM and SPC, denoted by ”M” and”C”, respectively. We choose two strong enough magnetic fields for illu-mination: eB = 0 . and eB = 1 . , whichare on different sides of the minimum point of the ¯ ρ +1 mass found in our previous work [35]. The numerical re-sults are shown in Figs.3 and 4, respectively. As can beseen in the upper panels, the quark masses all decreasewith Ω in both cases, but the chiral symmetry restora-tion ( χ SR) shows a crossover feature for eB = 0 . and a first-order one for eB = 1 . . Along with the χ SR, the quadratic GL expansion coefficients are eval-uated for charged rho meson with both choices of SPMand SPC, see the lower panels in comparison with thoseof charged pions. For either choice of Schwinger phase,the CRS can always happen and is favored over the CPSwith large enough Ω.We’ve checked for eB = 0 . that the CPS is in-deed disfavored with SPM at large Ω ( ≥ .
026 GeV) thusqualitatively consistent with that found in Ref. [38]. Butfor charged rho meson, the quadratic coefficient keepsdecreasing to an order of −
100 GeV at Ω = 0 .
026 GeVwithout any signature of turning up. The discontinuity inFig.4 seems to contradict with the continuous transitionassumption in the GL approach, but it surely demon-strates an instability to the χ SR phase. The situationmight be similar to that of diquark condensation at thecritical baryon chemical potential, so here can probably m f ( G e V ) ρ ± ( M ) ρ ± ( C ) π ± ( M ) π ± ( C ) - - - Ω ( GeV ) ( G e V ) m s m u m d eB = FIG. 4. The evolutions of the dynamical quark masses m f (upper panel) and the quadratic GL expansion coefficients A (lower panel) with the angular velocity Ω at the magnetic field eB = 1 . . The conventions are the same as in Fig.3. be a first-order transition to CRS. In the lower panelof Fig.4, one note that the coefficients A increase forcharged pion but decrease for charged rho at the criticalΩ ( ∼ .
25 MeV). At last, though not illuminated in theplots, it has been checked that the charged axial vectorcondensation might be favored over χ SR or CPS phaseat large enough Ω but never over CRS phase.
VI. CONCLUSIONS AND DISCUSSIONS
In this work, the possibility of charged rho supercon-ductor in the presence of parallel magnetic field and rota-tion was intuitively studied in Weinberg model and exten-sively explored within SU (3) Nambu–Jona-Lasino model.The charged π and ρ condensations were both well con-vinced in the point particle picture. By following simi-lar schemes as the previous works on CPS [37, 38], theCRS was found to be favored over chiral symmetry break-ing, χ SR and CPS phases at large Ω, for both choices ofSchwinger phase in Minkovski and curved spaces. As thechiral partner of ρ mesons, the charged axial vector me-son was even checked in advance; and it turned out thatCRS is still robust against that at large Ω. Indeed, theNJL model study qualitatively supports the intuitionsabout the rotation effect on mesons in the point particleapproximation.In the future, more realistic but complicated studywill be performed to looking for the true ground state ofQCD system in PMR by taking into account the bound-0ary condition and inhomogeneous forms of condensatesconsistently [43, 57]. As discussed in Sec.II, the effectiveregime of l should be determined by the total energy self-consistently and can be much greater than N for large Ω.In this case, the CPS or CRS is expected to emerge at asmaller threshold of Ω compared to what we found herein NJL model. One should notice that: Though CRS ismore favored for the chosen magnetic fields, there is stilla window of Ω for CPC phase when B is relatively weak. Eventually, as the PMR is relevant to the circumstancein peripheral heavy ion collisions, it will be interestingto explore the possible signatures for the competitionsamong χ SR, CPS and CRS in experiments.
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