CC BY
17
YflK 539.1.01
BecTHHK Cn6ry. Cep. 4. 2012. Bun. 3
V. Yu. Novozhilov
COLOR SOLITONS IN THE EXTENDED CHIRAL GROUP Ex*
Introduction. Diquarks, which were introduced in 1964 by Gell-Mann [1], became an efficient tool for studying various processes in hadron physics. The concept of diquark have been recently considered in different applications: quark-diquark systematics of baryons [2], spectroscopy and Regge trajectories of heavy baryons [3]. Diquark correlations in the quark-gluon plasma are important for understanding the dynamics of heavy-ion collisions, processes in the early Universe and possibly the cores of the neutron stars [4]. Between the new physics candidates proposed to explain the tt asymmetry measured in the Tevatron [5] there are some scalar diquarks [6].
The extended chiral group Ex [7, 8] was introduced in the bosonisation approach [10, 11] to derive the effective action of diquark fields. The corresponding extended chiral transformation depends both on pseudoscalar and diquark fields as parameters. While at the classical level the chiral symmetry is broken by quark mass, the extended chiral (Ex) symmetry is broken by quark mass and gluon fields. Ex-group is U(2N) for N internal degrees of freedom, N = NcNf. Non-anomalous (measure preserving) generators span the Lie algebra of O(2N), anomalous generators belong to the coset U(2N)/O(2N). Anomalous generators describe chiral rotations and transformations with diquark variables ("diquark" rotations), non-anomalous part consists out of gauge transformations and combined chiral "diquark" rotations. It was assumed that Ex-symmetry breaking due to quark masses and gluon fields is soft in the sense that the action for bosonised diquark fields can be obtained by integrating corresponding Ex-anomaly. Colorless chiral fields after bosonisation give rise to Goldstone particles — pseudoscalar mesons. At low energies bosonised diquark parameters of Extransformations with quantum numbers of lightest JP = 0+ wd-diquark was treated as a Goldstone-like particle. The Ex -group in this case is SU(4), non-anomalous transformations are just gauge transformations SU(3) x U(1) and the diquark Goldstone degrees of freedom belongs to CP3 = SU(4)/SU(3) x U(1). In the limit of vanishing gluon condensate and current quark masses the wd-diquark introduced a la Goldstone becomes massless.
Understanding world of color solitons is important in an non-perturbative approach to the low energy hadron physics. One of the problems is to include chiral anomalies [12, 13] in the total (gauge plus chiral, or anomalous) color space and to study changes of gauge space.
In the gauge sector of SU(2) QCD theory, color solitons were reported by Faddeev—Ni-emi [14-16] and Cho et al. [17-19]. In the quark chiral sector, the Skyrmion model for constituent quarks (qualitons) was discussed by Kaplan [20-22], and it was shown [23, 24] that isolated color solitons (i. e. solitons on the background of vacuum gluon field) can be classically stabilized by the chromomagnetic vacuum field in the cases of two colors, one flavor and three colors, one flavor, their mass can be evaluated and intersoliton potential displays confinement behavior.
The aim of this letter is to discuss a novel type of topological color solitons, existing in the extended chiral group Ex, as was proposed in the paper [25]. We consider isolated solitons which are defined as solitons in vacuum background field. We describe vacuum in a phenomenological way [26] through condensates and assume that cubic gluon condensate is zero.
* This work is supported by RFBR grant 10-02-00881-a. © V. Yu. Novozhilov, 2012
Extended chiral group Ex. The extended chiral group Ex is the group of all gauge and chiral transformations leaving quark Lagrangian invariant, if external fields are transformed accordingly [7, 8]. In absence of external fields, Ex is the group of global color and flavor transformations leaving free quark Lagrangian invariant. The basic Dirac spinor V is 8-com-
ponent, as considered first by Pauli and Gursey:
* = &
and the Dirac Lagrangian L can be rewritten as L = (1/2)VTFV with F represented in the block form
C$ -DT
F : ,
D
where D = iy^(dv, + V + y5a^) is the Dirac operator with external fields, "T" means transposition and $ = 7^(4*511 + Ys4v) contains external vector diquark fields, $ = yo^yo-
To avoid difficulties with the Majorana spinors in finding a counterpart of F in the Euclidean path integral over V, one should take special carej7, 9] keeping in mind that what is necessary for chiral actions is only to calculate det F. To this end we use the hermitian operator
g=(d * V $ Dc
Dc = C-1DTC, with det G = det F and required positivity properties [8]. Thus, having in mind the chiral anomaly and related effective action, one should study transformations of operator G induced by quark transformation 6V
6V = -QV, G ^ G' = exp(-S + y5©) Gexp(S + y5©), (1)
where antihermitian matrices Q, S, © are given by
Q = pu (a + y5x) + P22 (a* - j5X*) + P12 (£ + Y5®) C + p2i (-%* + Y5m*) C; S = pna + p22 a* + pi2^ + p2i^*; © = P11X - P22X* + P12® - P21®*-
We introduced notation: pab is a 2 x 2 block matrix with elements (pab)cd = 6acbbd. We have a+ = -a, x+ = -x, = -roT = ro. The quark baryon number is b = (P11 - p22)/3 = ps/3. ^
Transformations with s do not change the quark path integral, while transformations with © induce chiral anomalies. The Lie algebras with S + y5© and S + © are isomorphic. For Nc colors and Nf flavors, generators S + © are in algebra of U (2N), N = NcNf. Non-anomalous generators a are in algebra of SU (N) and include color SU (Nc); the left-right group GLR = SU (N)L x SU (N)R has a, x as generators, while (S + ©)sp = ^na + + p22a* + p12ro - p21ro* is in algebra of symplectic group Sp (2N), because (S + ©)sp ip2 + + ip2 (S + ©)sp = 0. In Ex non-anomalous generators S are in algebra of the orthogonal group O (2N) and include in addition to block diagonal generators a also non-diagonal generators which arise from non-commutativity of anomalous generators ©LR = p11 x -- p22X* and ©sp = p12ro - p21ro* [8]. Anomalous generators ©sp of the symplectic group Sp (2N) belong to the coset Sp (2N) /SU (N). We see that there are two distinguished
subgroups GLR and Sp (2N) of Ex with the same block diagonal non-anomalous generators S0 = pna + p22a* and different anomalous parts ©LR and ©sp, which do not require introduction of additional non-anomalous generators In the case of entire group Ex, anomalous generators © include both x and ro and belong to the coset SU (2N) /O (2N).
Effective action. The chiral field is U = exp©; calculation of det{exp(y5©)Gexp(y5©)} / det G by integration of anomaly follows the standard bosonization procedure [10, 11]. After eliminating external color axial fields, av = 0, we get the chiral action for 2N internal degrees of freedom W (U) = Weff (U) — Wwz with an effective Lagrangian Leff (U) and the Wess—Zumino term WWZ
LeS(U) = trcJ {^D^UDi'U-1 1
+
384n2 1
\ [UDJJ-1,UDVU~1} R2 - (UDyU~1UDvU~1)2
+ y^ô (\udll~u-\udyu~l]{gyv + ugypu-1) + g^g"^1) j
(2)
where coefficients contain additional factor 1/2 coming from square root of quark determinant; Dn = d^ + [Gp, *], Gp = puG^ + P22(-GT) + pi2$ + p2i$- The kinetic term depends on a phenomenological parameter /J. As in the case of GLR color solitons [23], we do not take into account the term which contains higher derivatives and reflects presence of radial excitations.
Choice of background field. Colour configurations are always associated with background colour field because of necessity to maintain colour gauge invariance. In this respect, the case of colour solitons is quite different from the case of flavor solitons, where there is no flavour gauge invariance, and the external flavour gauge field can be eliminated from the chiral action. We consider the colour gauge group SU(2) with antihermitian generators Ta = Ta/(2i), where Ta are the Pauli matrices.
The background colour field should be chosen according to the problem under consideration. Our first step is to study a single colour soliton, i. e. a soliton in the vacuum of gluonic field. The gluonic vacuum is characterized by the condensate
= (*o, é o-o-*0>) = é e^G™ ^ 0, (3)
4n2 4n2
that is by the non-zero vacuum expectation value of the Yang—Mills Lagrangian for the full quantum field O^ represented by the background vacuum field G^ in our approximation. According to phenomenological descpription Cg y 0, so that G^ is a chromomagnetic field in the real case of SU(3) gauge group. The vacuum field strength Gki in the temporal gauge G0 = 0 is constant up to a time independent gauge transformation. The effective Lagrangian Leff is invariant under gauge transformations of background fields and the chiral field.
We shall consider the simplest case of a chromomagnetic vacuum background field, when it is an Abelian-type field which is a product a coordinate vector field Vk and a SU(2) color vector na
1 1 T
G% = Vkna, Vk = --Vkixi = --eklmxlVmB, Gk = gGak(4) 2 2 2i
where na is a constant unit vector in the colour space, vm is a constant unit vector in coordinate space, vmB = (1/2)emlkVlk is the vacuum chromomagnetism and B is related to
the condensate Cg = ^ B2. In the vacuum all directions na and v; are equivalent, so that it is necessary to average over them at the end. Such a choice of vacuum field does not lead to stability troubles in QCD; although imaginary terms were detected at one-loop level [29], they disappear in all-loop treatment [30].
Solitons in the symplectic group. Let us consider the symplectic group Sp (2N) with a in SU (N), N = NcNf. The chiral field Usp is a mapping from S3 to ©sp = p+ro — p_ro* with ro = roT. For the whole group (with both a and ro) such mappings belong to n3 = Z . Excluding a we get:
(a) two or three colors, one flavor: ro = Xaroa, a =1, 3, 4, 6, 8, no spherical solitons;
(b) two colors, two flavors: ro = Abrob, Ab are nine traceless symmetrical matrices o2 x t2 and o^ x k,l = 0,1, 3; no spherical solitons;
(c) three colors, two flavors; non-anomalous part Ssp is given by 12 x 12 matrix with a in SU (6) built on SU (3) matrices Xa and flavor Pauli matrices Tk, while anomalous part ©sp together with Ssp span Sp(12) algebra. Symmetric matrix ro contains fields with diquark quantum numbers associated with both symmetrical matrices Xf x Tf and both antisymmetrical ones X-^ x t2. Antisymmetric matrices XA = (X2, —X5, X7) = (Ok); (Ok)ij = = —i^kij in color O (3) algebra we combine with unit coordinate vector rk, rkrk = 1 into r = = Okrk. We retain only those parameters ro, which enter with generators of O (3), introduce the shape function Fsp (R) and write the anomalous part Osp and the chiral field as
©sp = P+«T2Ofc- P-it2Ofc= i%2nrFsp, n = Pi cosx - p2 sinx, Jsp = exp ©Sp
Usp = exp©sp = 1 + it2nrsinF + f2 (cosF — 1), R2 = xkxk, (5)
assuming that x is constant. Isospin matrices Ik = (р3т1, т2, рзХз) commute with ©sp. Usp can describe two conjugate color solitons
u± = ^ (1 ± Т2Л) exp (±irFsp(R)) (6)
made out of fields -Э^ with diquark quantum numbers. ©sp satisfies ©3p = —i©spF2p. In this special case transformations close in a smaller group, namely O (6) ~ SU (4), and a belong to U(3) ~ SU(3) x U (1), while Goldstone diquark degrees of freedom n©sp belong to the complex projective space CP3 = SU (4) /SU (3) x U (1) [8]. Note that n anticommutes with the quark baryon number b = p3/3.
Thus, the commutator [Gk, Usp] with the background vacuum field
Gk = PiiGk + P22 (-GT), Gk = VkNa, N = XaNa, NaNa = 1
contains anticommutator {N + NT, r} in order to preserve the form of Usp we have to restrict Ssp to a common subgroup O (3) of SU (3) x SU (3)*. This is possible for vacuum background field Gak = VkNa, N = XaNa, tr N3 = 0 in the gauge G^ = 0. Then
Gk = P11 Gk + p22 ( — GT) = (P11 + p22 ) Vk NN,
where Nk = OiNi, N3 = W. Finally, the soliton subgroup Gso\ in symplectic Sp (12) is given by
(s + 75©)sp1 = iOk (ak + 75t2^rkFsp) + ip3T3aF,
where aF is a flavor parameter. For comparison, in the left-right group (S + J5©)lr with both a and x in SU (3) algebra the analogue soliton subgroup is
(S + 75©)L°R = iOk (ak + Y5p3rkflr) .
Thus,
Gsol = O (3)L x O (3)R x U (1)p ,
but in the left-right group Gsol the baryon direction p3 is replaced by another one t2 n- If we introduce the baryon SU (2)b with generators (t2pi,t2p2, p3)/(2i), then t2n can be reached from p3 by rotation-
Let us consider the complete Ex group with anomalous generators © = (x, w):
a) two or three colors, one flavor: © is built on generators (p3XA, Xs, p±XS), no spherical solitons, no constituent quarks, if generators with p± are present and Ex is not reduced to the left-right subgroup;
b) two colors, two flavors: no SU(2) and O(3) subgroups in the same sense;
c) three colors, two flavors: most general © is built on generators
(p3XAtS, p3XStA, p±XSts, p±XAtA) and among them there are two O(3) subgroups with generators
e±(0(3))~ i(i±t)(x2,-x5,xr),
where Z = p31T cos k + nT2 sin k, [S, Z] = 0 and n = p1 cos x — p2 sin x, Z2 = 1- Angles k and x are two constant parameters: x comes from the symplectic group Sp(12), while k describes mixing of left-right and symplectic contributions to Z- If k = 0, no symplectic contribution is present. If k = n/2, there is no left-right contribution. The chiral field for Ex in this case is U = exp iZrF (R), it can be obtained by SU (2)b rotation from Usp or Ulr- In this special case
S + Y5© = iOk (ak + YrxrkF),
when S is invariant under rotations of baryon SU (2)b, no additional gauge parameters ^ arise from repetition of such transformation- Thus, although we started from different subgroups of the extended chiral group for NC = 3, Np = 2, the resulting structure of solitonic chiral subgroups is the same: it is O (3)l x O (3)r , where left L and right R are defined by projectors (1 ± y5p)/2, where a block matrix p with p2 = 1, is linear in the Pauli matrices p1 t2, p2T2, p3 of the baryon SU (2)b- The reason is that the group Ex includes quarks and antiquarks-
Topological charge tx (U) for a soliton U in the left-right subgroup of Ex is related to the quark baryon number b = p3 /Nc
1 1 [
h {U) = 2 2A^NC J Sxiiijk tr{p3UDiU+UDjU+UDkU+} (7)
and coincides with the baryon number of soliton U - For three colors and Nf flavors tx starts from (2/3)Npf (flavor analogous case is skyrmion as dibaryon studied by Balachandran et al- [28]-
The topological charges for solitons in symplectic subgroup Sp (12) and complete group Ex are winding numbers in directions p^ = t2n and pE = Z within the baryon SU (2)b-space
1 1 [
tA (U) = J d3*^ tv{pAUdiU+UdjU+UdkU+}, A =±, E, (8)
and do not describe solitonic baryon numbers- These winding numbers start from ±4/3-Boundary conditions for the shape functions F (R) in both cases trivially follow from [27, 28] -
Finiteness of soliton mass follows from the static effective Lagrangian Leff and asymptotic behavior of the shape function F (R) at large R. The mass is given by the positive definite functional, as it can be easily verified; asymptotics of F (R) is defined by the kinetic term. The kinetic term averaged over directions Nk and vt of background vacuum field Gf = ViNk, Vi = — ^ £ijtrjVtR\/Gg/2 in O (3) color and coordinate spaces takes the following form
1 — f2 jo t?'2 , of 2 i ^ 2n j-,2 \ • 2 r , f 4 , 1 2^ j-,2\ p ,\2
A' = J Npfi +2 ^+ -^cgR2J sin2 F+ +-^CgR2J (cosF - 1)-J , (9)
where Cg is the gluon condensate. Thus, asymptotic behavior at large R of the shape function F (R) is governed by the similar equation as in the case of color SU (2) soliton in [23]. We use the result
F (/0i?rf exp , R oo, (10)
which guarantees that the mass M = —4^/ dRR2Lstat is finite for positive condensate Cg, i. e. for chromomagnetic vacuum field. It follows from the effective Lagrangian (2), that the soliton mass is invariant under SU (2)b rotation.
Conclusions. We have shown that classically stable, finite mass topological solitons exist in the extended chiral group Ex. In the case of three colors, two flavors their status is described by the special chiral group O (3)¿ x O (3)r with notions of left L and right R defined in terms of (1 ± y5Z)/2, where Z is a constant non-abelian charge direction in the SU (2)b and p3 is baryon number direction. In the case of isolated soliton, Z = const. Will be a covariant notion, when Z commutes with direction N of background vacuum SU (3) field, as it is for tr NV3 = 0. The vacuum background field defines also an asymptotic behavior of the shape function F (R); the field should be chromomagnetic. This pattern based on establishing mapping from S3 to anomalous part © of Ex and using properties of vacuum background field can be followed in more complicated cases. This expansion of the world of color solitons is not accompanied by the widening of pure QCD gauge space: it is still SU(3); no additional gauge degrees ^ were still required to accommodate novel solitons. However, these degrees ^ are likely to appear, if vacuum is to be described by set of condensates. Novel type of solitons may be essential in discussion of baryon asymmetry and baryon number nonconservation.
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Статья поступила в редакцию 3 апреля 2012 г.
