Chiral parametrization of qcd vector field. Gluon sector Текст научной статьи по специальности «Физика»

2014

ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА

Сер. 4. Том 1 (59). Вып. 4

ФИЗИКА

УДК 539.1.01 V. Yu. Novozhilov

CHIRAL PARAMETRIZATION OF QCD VECTOR FIELD. GLUONSECTOR

St. Petersburg State University, 7—9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

The chiral parametrization of QCD gauge field is considered in details in an approach developed earlier for SU(2) and SU(3) cases. A color chiral field is introduced, gluons are chirally rotated, and vector component of rotated gluons is defined on condition that no new color variables appeared with the chiral field. This condition for SU(3) case associates such a vector component with SU(3)/U(2) coset plus an U(2) field. The QCD vector field in CP2 and U(2) sectors is studied in new variables of chiral parametrization. The singlet gluonium can acquire mass due to formation of dimension two vacuum condensate of induced axial-vector field. Refs 34.

Keywords: gluons, quantum chromodynamics, Cho—Faddeev—Niemi—Shabanov (CFNS) decomposition.

В. Ю. Новожилов

КИРАЛЬНАЯ ПАРАМЕТРИЗАЦИЯ ВЕКТОРНОГО ПОЛЯ КХД. СЕКТОР ГЛЮОНОВ

Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7—9

Киральная параметризация калибровочного поля КХД подробно рассмотрена в подходе, предложенном ранее для SU(2) и SU(3) случаев. Введено цветное киральное поле, глю-онные поля кирально повёрнуты, и векторная компонента повёрнутых глюонов определена условием отсутствия излишних цветных степеней свободы, вносимых киральным полем. Для случая SU(3) это условие ассоциирует векторную компоненту с комплексным проективным пространством SU(3)/U(2) плюс U(2) поле. Векторное КХД поле рассмотрено в секторах CP2 и U(2) в новых переменных киральной параметризации. Синглетный глюоний может получить массу вследствие образования вакуумного конденсата размерности 2 наведённых аксиально-векторных полей. Библиогр. 34 назв.

Ключевые слова: глюоны, квантовая хромодинамика, разложение Чо—Фаддеева—Ни-еми—Шабанова.

Dedicated to Yu. V. Novozhilov on his 90th birthday

Introduction. The monopole condensation scenario is considered as the most probable way to confinement [1-4]. Therefore, much efforts were undertaken last decade, in order to find a proper parametrization of the gauge field in QCD, which could contain necessary

topological properties. Long ago, the Faddeev soliton picture [5] of QCD excitations was the first model, where a unit three-dimensional vector n(x), n2 = 1, was introduced to describe knot solitons in four-dimensional space-time. In order to consider magnetic properties of the QCD Cho [6, 7] and independently Duan and Ge [8] used the n(x) field, as a vector in the isospin space, to build the connection [n, d^n] in the SU(2) Yang—Mills theory. The Faddeev soliton picture was revived in the Faddeev—Niemi knot model [9], and in the n-dependent parametrization of the QCD gauge field [10, 11]. Shabanov [12, 13] used the Cho connectionis to reformulate a Yang—Mills theory as a Abelian gauge theory via a nonlocal change of variables in the space of connections, rather than via a gauge fixing, and stressed an importance of coset variables with related U (1) symmetry.

The Cho—Faddeev—Niemi—Shabanov (CFNS) decomposition for two-color QCD is defined by

Atl = Cn+g-1[d^n, n] + XnX^ = 0,

so that the behaviour of all fields is fixed with respect to n. The CFNS decomposition was extended to SU(3) [13-16] and it was shown to describe monopoles.

The question of the n(x) field origin was discussed several times [14, 17-20] to the result that n(x) can be treated as a collective dynamic field, which does not introduce additional degrees of freedom. As far as physical implications are concerned, it was shown that for two colors at low energy the CFNS procedure corresponds to a spin-charge separation [11]. This is a new step in the "abelianization" program compared with maximal Abelian gauge approach [21, 22]. In CFNS picture the abelian directions are defined in a gauge invariant way.

The CFNS decomposition as a reformulation of QCD based on change of variables has been recently investigated in detail in series of papers [14, 23, 24] where, in particularly, it was found that the Wilson loop operator and the Polyakov loop operator are constructed out of first two terms in A^ alone. Studies of coset spaces appearing in the CFNS picture revealed an existence of vortex solutions [25]. The CFNS approach has been extended to the treatment of defects [26].

However, restriction to the QCD without quarks raises the question, whether such approximation is good. Due to the chiral anomaly [27, 28], at the quantum level the color gauge sector and the color chiral anomalous sector are parts of the total color space. Chiral transformations U of quarks in the Dirac Lagrangian do not change the Yang—Mills field in the gauge sector of the theory, but introduce different decompositions of the Yang—Mills field in terms of arising vector and axial vector components. Among these decompositions we are interested in those, which preserve number of the dynamical degrees of freedom (DOF) and may reveal the topological degrees. It is at the classical level, where such decompositions are equivalent, because the Dirac Lagrangian is chiral invariant. At the quantum level, the quark path integral is changed because of the anomaly, therefore, in this case, an impact of the chiral parametrization on the quark path integral requires a special investigation.

The aim of this paper to consider in details a chiral parametrization (ChP) of the QCD gauge field proposed in papers [29, 30]. We assume that the basic color field m (x) has its origin in the quark chiral field U, as a remnant of U after conditions of consistency are imposed. We introduce the decomposition of the QCD gauge field G^ via a chiral rotation: the field G^ in the Dirac Lagrangian is submited to a chiral transformation U initiated by quarks

v — vu = U ¥l + vr,

g\i - G (VU ,AU) = V^ - AU,

where yL,R denote left (right) quarks, VUand AU are correspondingly vector and axial vector components of chirally rotated G,. Then the second line would represent the chiral decomposition, if it is possible to find such a special chiral field U0, when VU is invariant under U0 and VU can be calculated as a function V^ (G), i. e. the condition det(dVU/dG)= 0 is satisfied. As a result, this defines U0 = m (x), m2 = 1, tr m = 1 up to a constant phase. In the case SU(2) we have = mCM + ^md^rn, while the axial part is orthogonal to m in color space and may be identified with —X,. Thus, this parametrization is the same as the CFNS one in SU (2). In the SU(3) case, the chiral decomposition selects the SU (3) /U (2) ~ CP2 color space (minimal symmetry), while the CFNS decomposition admits also the SU (3) /U (1)2 coset space (maximal symmetry). The conclusion of [29, 30] is that the chiral parametrization coincides with the CFNS in the gluon sector, if the unit chiral color vector m (x) is identified with the basic color vector n(x) of CFNS.

It is important that in ChP all properties of decomposition fields follow from chiral transformation rules.

The relation G, = V^ — A^ in (1) is invariant under any U-transformations. Among all matrices U there is a chiral U (1) group of matrices Ux = exp im% with x = const, which do not change DOF of VU and describe transformations of coset variables. These transformations look dangerous for the chiral decomposition, because in the quark quantum sector they generate the chiral anomaly and the Wess—Zumino—Witten topological action Wwzw. In this case Wwzw represents quantization condition similar in spirit to the Dirac quantization of monopoles.

Left-Right group SU(3)L X SU(3)R and chiral field. In this section we establish notations for the Left-Right group SU(3)L x SU(3)R and the chiral field. Consider massless fermions in external vector and axial vector fields V,, A, and the Dirac operator

D (V, A) = iY1 d + v, + Y5A„) (1)

with antihermitean fields in algebra SU(3), so that V, = —V^ita, A, = —Aaita, where ta, a = 1, 2,... 8 are generators. The chiral transformation of fermions is given by

Vl = 'L¥L, VR = 'R¥R, ¥ = ¥L + ¥r, (2)

where 'L (x) and 'R (x) are local chiral phase factors of left and right quarks yL and ¥R, represented by unitary matrices in defining representations of left SU (3)L and right SU {3)R subgroups of the chiral group Glr = SU (3)L x SU {3)R. For \|/£ = ^ (1 + Ys) V, \|Ir = ^ (1 — Ys) V) generators and tRa of left and right subgroups of Glr can be written as tLa = |(1+ Y5)K,t-Ra = 2 (1 - 75) K, [t-Lat-Rb} = 0, where Xa, a = 1, 2,..., 8 are the Gell-Mann matrices. Then quark left and right chiral phase factors 'L, 'R arise from application of operators = exp—itLawLa), |r = exp—itRawRa) to left and right quarks ¥L and ¥R. Vector gauge transformations g are associated with ta = tLa + tRa = Xa/2, i. e. g (x) has properties of the product 'L (x) |r (x) of identical left and right rotations, wL = wR = a. The generator of purely chiral transformations g5 is t5a = y5Xa/2 = tLa — tRa, thus, g5 has properties of 'L|+ for wL = wR = ©. Infinitesimally, the Dirac operator is transformed according to

6 0 = [i^aaK,m + &aK,P>}- (3)

Commutation relations for ta,t5a are given by

[ta,tfe] = ifabctc, [ta,t5b] = ifabct5c, \p5a,t5b] = ifabctc, (4)

where fabc are antisymmetrical structure constants of SU (3), we use the normalization trtatb = 1/2Sa6.

Instead of phases 'L and 'R one can work with the chiral field U = '+'L, which describes rotation of only left quark leaving right quark in peace vL ^ vL = UVL,V ^ v'r = Vr. The same result can be obtained by the chiral transformation yL ^ 'LVL, VR ^ 'RVR, followed by a vector gauge transformation with a gauge function '+. The usual chiral gauge choice is 'R = '+, then the chiral field is taken as squared left chiral phase:

U = 'L = exp ffl, n = na K, (5)

where we used the flavor notation n. This chiral gauge corresponds to the following form of chiral transformation of the Dirac field

VU = 95 (U) V = (PlU + Pr) v, VU = V?5 (U) = V (PrU + + Pl) , (6)

which is used in this paper.

To describe U in the case involving monopoles and other defects one can use the basis with the step up/down operators E±i

E± 1 = i (Xi ± i\2) ,E±2 = ^ (U ± iXs) ,E±3 = ^ (X6 ± ±ar) (7)

and diagonal generators

Hi = [E+i,E-i] ,i = 1, 2,3 (8)

with commutation relations

[[E+i, E-j], E+k] E+k + bjkE+i [[E+i, E-j], E-k] = E-k — bjkE-i (9) and the U (1) generator of SU (3) ® U (1)

Introducing notations for Z C C3, normalized SU (3) spinor ^ and the CP2 coordinates

Ua, u+, a, p =1, 2,

Z = (zuz2,z3f c c3;

Z = v/Z+Z(cpiicp2,cp3)T = —===== (wi,w2,1)T

V1 + u+u

one can parametrize the chiral field U in a form of the SU (3) group element as follows

1 A A

ft \ 1

where u = (u1,u2 is the 2 x 2 matrix

uju+

A,-o = 06:,--—

1 + §

For applications related to SO(3) monopoles and SU(3)/SO(3) coset, it is convenient to consider SU(3) in the SO(3) basis [31]. SO(3) is the maximal subgroup of SU(3). In this

case it is convenient to use two hermitian combinations: NN (x) and NN2 (x), where NN (x) is built on antisymmetric X-matrices N = nkOk, Ok = (X7, —X5, X2), nknk = 1, N3 = N, while N2 contains only symmetric X's. Then general SU(3)-chiral field is

U (a, (3) = expill (a, (3) = expi (nci. + (^{N', N"} - ^tr N'N")fij , (11)

where NN ',N" depend on SO(3) unitvectors n'k ,n'¡'. Three unit vectors plus a, p, give altogether 8 parameters of SU(3).

The Dirac Lagrangian remains invariant under chiral transformation of fermions if external fields transform correspondingly

V D (V,A) V = VUD (VU,AU) vU, (12)

where

Vu = Uj[dU+ + {V., U+} + [A, u+]i

Au = ^U[dU+ + [1/, U+] + {A, U+}}. (13)

These expressions will be used as the starting point for an introduction of chiral variables into the QCD vector field.

Chiral field as a direction in the color space. Consider the Dirac action V D (V, 0) v containing only the QCD gauge field, V^. There is no axial vector color field, A = 0 initially. After the chiral rotation of fermions we get chirally rotated gauge field

vu = [8U+ + {1/, £/+}] , Au = [8U+ + [1/, £/+]] . (14)

This is the definition of the (once) chirally rotated field. For subsequent chiral rotations one should use transformation rules (13). Now, VU is a gauge field

VU ^ gVU9+ + 9dg+,AU ^ gAUg+, (15)

while AU transforms homogeneously under gauge transformation g (x) C SU (3)L+R. Two simple relations exist for VU ^ AU, namely

VU — AU = V,,

VU + AU = U d + V)U+, (16)

which mean that if the chiral field is considered as a regular gauge transformation, then (VU ± AU),-combinations should have the same field strengths (VU ± AU) . With topological U these combinations can describe different situations. The first of these relations may be considered as a chiral parametrization for V, = V^ — AU, i. e. as a way to introduce explicitly topological variables of U into the gauge field V . The price of such a parametriza-tion would be the requirement of V,-invariance under chiral rotations U ^ U'U eliminating superfluous variables. In order to avoid this situation one should restrict the choice of the special chiral field U (x) = U0 (x), to be used for parametrization by the following conditions:

(a) the gauge part V U of rotated gluon field is invariant under chiral rotation, and

(b) V U is defined uniquely in terms of V .

Indeed, by introducing the quark color chiral field U we arrive at a system with too many degrees of freedom. To eliminate superfluous variables, one should consider a relation between gluonic fields V^ and chirally transformed field V^ and find how different parts of these fields can be made from the same material, so that chiral field variables are either fixed, or fully incorporated into gauge field variables. The invariance condition (a) means that

(Oc/' = \ iu'+ (Kr+O u' + K1 - < + U'+W] = V". (17)

In terms of the Dirac operator this condition reads

D VU, 0) = u + d VU, 0) u, (18)

that implies that the axial field does not appear. This condition defines a special chiral field

Uo = U0, U = exp iZ, dZ = 0, Uo = m exp iZ/2, (19)

where m is a hermitian SU (3) matrix with m2 = I describing a direction in the color space and Z is a real number. With the chiral field U0 an axial vector field AU° (VU°) calculated with VU instead of Vi will disappear

(C) = \ I^M + )U0 - Vf°] = 0. (20)

Note that Eqs following from the condition (a) are independent of the chiral color group.

Consider the condition (b) This can be done by studying the (V^ ^ V)-determinant. We have the following relation between 8x8 matrices of gluonic field V and chirally rotated field VU in adjoint representation

1 i 1 1

W = 2 (! + r(u))abv,b + -^(Ud.U+U, Rab (U) = - tr (K.UhU+), (21)

where the chiral field U = ^L = expi©, © = Xa©a, is defined in the chiral gauge = Here Xa, a = 1, 2..8, are the Gell-Mann matrices.

In order to calculate the determinant det ^ (1 + R (U)), it is sufficient to consider the parametrization (11) for U in the case N' = N" = N. We get

U ->■ U (N, a, |3) = exp 1 + ^^ sin a + N2 (ei|3 cos a - l) . (22)

When eigenvalues of N are placed as diag (1, —1,0), we see that a = ©3, (3 = ©s\/3 and

det^(l + R(U)) = i(l+cos2a)i(l+cos(a + (3))i(l+cos(a-(3)). (23)

This result can be easily checked in (isospin I3, hypercharge Y) basis, where the diagonal elements of R(U) = expi (I32a + Y2^), as of the adjoint representation of U, are nothing, but those of octet: pions (I = 1,Y = 0), K-mesons (I = 1/2,Y = 1/2), K-mesons (I = 1/2, Y = -1/2), o-meson (I = 0,Y = 0).

This determinant is invariant under reflection (a, P) ^ (—a, — P). It disappears for values (a, P) equal to (n/2,0), (n/2, ±n/2) and (0, n) characterizing singularity surfaces in chiral color space (i. e. 75©). The first of these sets, (n/2, 0), corresponds to SO (3) subgroup with one zero factor in det, when U (N, n/2, 0) = exp iNa = 1 + iN — N2.

For SU (3) we are interested in two coinciding zero factors of det related to two simple roots of SU (3). Together with a singularity set related to a complex root, we can define three color chiral fields U(N, a, P) for SU (3). For the pair (a, P) = (0, n) we get the chiral field

U (n, O.jt) = (1 - 2N2) exp(-i-Jt) = mi exp(-i-Jt). (24)

For (a, P) = (n/2, ±n/2) we have

U (N, 3t/2, Jt/2) = (1 ~N -N2) exp

= m2 exp U (N, 3t/2, -it/2) = {l + N - N2) exp • (25)

In all these cases mk are normalized hermitian 3 x 3 matrices in color space: m2 = I. A product of two m's is equal to the third m up to a constant phase. With the diagonal N = diag (1, —1,0), the diagonal forms of mk are given by

mj = diag ( — 1, —1,1), m2 = diag ( — 1,1,1), m0 = diag (1, —1,1). (26)

These matrrices have simple meaning: they are related to 2n rotation of diagonal operators in SU(2) subgroups of SU(3)

mj = — exp iI32n, m° = — exp iU32n, m° = exp iV32n, (27)

where /3 = ^ X3 for isospin subgroup, U3 = — ^ X3 + -^Xg, for L^-spin subgroup, V3 = | + -^Xg for y-spin subgroup. R'om the view point of the Cartan—Weyl basis these matrices correspond to rotation around simple roots (mj and m°) and around the composite root (m°). Matrices m°k are unitary equivalent.

Thus, behavior of the (V^1 —► V^) determinant det i (1 + R) shows, how to restrict chiral color space in defining the chiral field U0. Chiral invariant regions Q, are given by sets (a, P) = (0, n), (n/2, ±n/2), where the chiral field U = mk is represented correspondingly by one of 3x3 unit matrices mk, k =1, 2, 3 up to a constant phase. Our choice is

U0 = —Sm0S+ = Sm0S+

m,

m° = diag (1, l, _l) = A Xg + I = exp in (y - ^ , (28)

where S is a unitary SU(3) transformation and Y is the color hypercharge. Independence of m0 under global group U (1) £ SU (3) is essential. It is introduced as a chiral field U0 up to a phase. Also, it is a group element of SU (3) up to a phase and in this quality participates in the CP2 involution operation (see para 4). It is the generator of U (1) invariance group of CFNS decomposition (para 5). All field quantities constructed from the color field m do not depend on this global phase.

The chiral field Uq = m is an orbit of SU (3) through the hypercharge Y = Xg/i/3. Then S is defined up to right multiplication S ^ Sh, where h G U (2), i. e. S is in the coset

SU(3)/U (2) = CP2.

Such manifold is known as the complex projective space CP2. Under the gauge transformation g e SU (3)R+L we have m ^ gmg+. It means that the direction m is changed by gauge transformations defined in terms of CP2 variables. By a special choice of the gauge ("unitary gauge") this direction may be made coinciding with m0 = 2Y + 1/3. Thus, CP2 variables describe orientational degrees of freedom contained in the unit color vector m. As we shall see below, this quantity brings quark chiral topological defects into the gluon field.

Structure of m rotated fields: m-even and m-odd fields. The rules (13) of chiral transformation for the gauge field GM (in absence of an axial vector field in initial state) by the special chiral field

U0 = m, m2 = 1, tr m = 1 define essential features of rotated fields structure. Let us rewrite the rules (13) in the form

G™ = \ (m {GM, m} + m<9Mm) ,

AT = \ (m [Gn> ™] + md^m) = im^ (G) m = I [m, D, (G) m] , (29)

where DM (G) m = dMm + \GM, m] and we used identity DM (G) m2 = 0. It follows from (11), that with respect to m the gauge field G may be subdivided into commuting G|| and anticommuting G± parts

G = Gii + G±,

[G|hm]=0, {G±,m} = 0. (30)

Only Gii contributes to Gm and only G± contributes to Am

GZ = Gil, + - m<9wm

||M HM' 2 M

A? = G±tl+-mdtlm. (31)

From m2 = 1 follows that d^m anticommutes with m, hence we have the relations

{G^,m} = {G^m} ; (32a)

[G™ m] = -dxm; (326)

{A,m} =0; (32c)

[Am,m]=2Amm; (32d)

[m,Dil (Gm) m] = 0; (32e)

[m, [dm, dm]] =0. (32.f )

Within the part G|| we distinguish an abelian field mC^ and an U (2) field Q^ both commuting with m

Gii = mC, + Q^, Q, m] = 0, C = tr mG(33)

so that

G™=G||M + im9Mm = l/p + gM, (34)

where we introduced a special notation

= mCM + \mc\m, (35)

for the CP2 field satisfying the invariance condition m D (VP; 0) m =D (VP; 0) • The matrix m is covariantly constant

D (Vf) m = 0. (36)

To complete the set of relations for m-rotated fields we remind two identities

Gm Am _ yo

n - A = G

Gm + Am = m G m + md^m.

Transformations of the Dirac operator D (G; 0) due to consequent chiral rotations by

Uo = m

v = (PiTO + v y™ = y(PRm + PL), PL<R = i (1 ± Ys) (37)

are given schematically by

D (G; 0) ^ \j/m D (G; 0) vm =D (Gm; Am) ^ \j/m D (Gm; Am) Vm =D (G; 0) =D (Gm - Am; 0). (38)

In the simplest case when the gauge field is absent, G^ = 0; we get

'1 „ 1

P (0, 0) \|/m =ip QmdMm, im^m) =Jfi (r°,r°) ,

(39)

where

rii = onA'm = 1 [m> <Vn] (40)

is a m-invariant connection.

This connection (the Cho connection) rj° is an important element of the decomposition of Yang—Mills field proposed by [8, 9, 14] or the CFN-decomposition.

Chiral paramertrization. The chiral parametrization starts when we introduce into

m

tm

this line of identities the field Gm including the connection r0 and represent the gauge field

G = r0 + mC, + Q, + X, (41)

and consider the parametrisation of X, in accordance with its properties as the axial vector field (y—Am). According to (30) X, should be equal to G^: while Q, belongs to G||

{X;,m} =0, [Q,,m] =0.

It is easy to verify that the Dirac operator has a property

m D (Vf + Q; + X;, 0) m =D (Vf + Q,, -X,), (42)

and we can use for (—X,) all relations derived for Am. 444

as

Let us construct X» = G^» = —A^ as a field anticommuting with U0 = m. Note that this property does not depend on a particular color group. Within the algebra SU (3) the matrix m0 = diag (1,1, —1) anticommutes with Xt,t = 4, 5, 6, 7. Hence, a gauge transformed matrix m = Sm0S+ anticommutes with SXtS +. Then, X» can be represented as

X» = SXtxt,»S+,t = 4, 5, 6, 7, (43)

with color invariant fields xt». Another example of matrices anticommuting with m provide derivatives d»m and md»m, which we use to construct a contribution Y» to AU

Y» = %>d»m + ixmd»m, (44)

where ^ and x are colorless functions. Analogous terms with derivatives exist in the two-color decomposition [9]. However, anticommutativity with the matrix m is not sufficient to define G± = X» + Y». We should count number of degrees of freedom contained in the chiral parametrization and consider gauge fixing. Thus, the relation G± = X» + Y» gives us only a choice of eligible terms which can be different for different applications.

Thus, as it follows from expressions for the QCD vector field ('gluons') V» there are two distinct sectors in the chiral decomposition of V» = V»1 + X»:

(a) the CP2-sector with the dynamical abelian field C». The space CP2 enters the scene with the chiral field U = m = Sm0S + , as the SU(3) orbit through m0. The matrix m is the main element, which defines the vector field in the sector (V^f2) 2 = mCM + ^md^ni, including the direction of an abelian field C» in the SU(3) space. The axial component A» anticommutes with m.

(b) an U (2) sector with the field (V») U(2) = Q»; the chiral field U = m commutes with Q.

Transformation properties of all field operators follow directly from those (13) of the Left-

Right group. Let us recapitulate them in detail. After chiral field is fixed at U = U0 = m, it is only the standard gauge transformation g (x) G U (3) common for gluons and quarks (i. e. for G» and m) which is left for all variables. With g (x) ~ 1 + to we get

SG» = —D» (G) to, 6m = [to, m], SC» = —md»to, SA? = [to, A^] ,

6rM = [to, rM] - ^ (mc^cum - <9Mto) , (45)

6QM = [to, QM] + i (mc^cum + <9M to) .

However, when we go over from matrix m = Sm0S + to its building blocks in terms of m. e. of S, we find a hidden U (1) group related to local transformation Sa3 ^ Sa3 exp ix leaving invariant the projector P = ^ (1 — m) = 5'diag (0,0,1) 5'+. This U (1) group is not a subgroup of the gauge SU (3) L+R of QCD. It comes as a hidden signature of quarks. Its existence has been shown by Shabanov [13].

CP2 sector in the chiral parametrization. The space CP2 in QCD emerged in the parametrization of the vector field, when the chiral color direction as a remnant of the quark chiral field, was introduced into the gauge field. Being by origin a quark variable the field m (x) can be considered also as a gluonic variable.

m is a 3x3 hermitean matrix m = maXa; a = 0,1, 2,... 8, with the properties m2 = 1, tr m = 1. The matrix m can be expressed as

2 1 1

m = SmoS+, m0 = diag (1,1, -1) = -== X8 + - = 2Y +

S e SU (3) ; SS+ = S+S = 1;

m is an orbit of SU(3) through hypercharge Y = -^Xg and describes the complex projective space CP2 in terms of matrix elements of S. Due to unitarity of S there are different ways of description of CP2:

(a) Representation of S in terms of the tangent bundle T of Xg, when

S = exp iT; T = am; a = Xtat; t = 4; 5; 6; 7; T + = T; (46)

a is given by a3a = aa3 = = 1; aap = 0; so that ^ m represent four indepen-

dent variables of CP2 .

0 0 \

0 0 ^ ; (47)

0 J

where a =1; 2; is normalized according to = 1, and ^ m represent four independent variables of CP2. Due to relation a3 = a we have the closed expression

S = 1 + iasinm + a2 (cosm - 1); a^ = a|3 = 1; aa3 = 0. (48)

The chiral matrix m anticommutes with a; {am} = 0; as a result

m = S2 m0. (49)

The hidden U (1) symmetry is described by transformation ^ eiX^a which leaves a2 invariant.

SU(3) matrices Q commuting with m can be built on S-transforms of Q0 = XqQ, q = 1; 2; 3; 8, namely Q = SQ0S+.

(b) Representation of m in terms of projection operators.

One can consider CP2 as a space of hermitean 3x3 projection operators

P = Pa Xa; a = 0; 1; 2... 8; P 2 = P; trP = 1; X0 = I = diag^; 1; 1) (50)

To arrive at CP2 space, we can start with P° = ^ (1 — mo) = diag(0, 0,1) and construct P = SP°S+ = m), where S is a general unitary SU(3) matrix (which always can be

rewritten in a form containing only the tangent bundle T). If we consider a given expression for a projection operator, it is necessary to check defining relations tr P = 1;P2 = P, which lead to the following conditions on Pa

1 1 P

P0 = -, PaPa = -;dabcPaPb = y, (51)

where dabc are symmetric structure constants d,abc = i tr (X0{X&, Xc}) with a,b,c ^ 0. This definition describes CP2 as a submanifold of the 8-dimensional Euclidean space Rg. or of R9 with fixed P0.

Projection operators P can be realized by means of normalized SU(3) spinors ty = (tyi; ty2; ty3) considered as unit vectors | ty y in C3 modulo the phase

Pij = tyity+; mij = -2Pij + bij; (52)

assuming ty = 0. SU(3) spinors ty and ty+ represent matrix elements Sk3 and S+- of unitary S in definition m = Sm0S +. This representation should be remembered in understanding expressions P = tyty+ and [P; ty] = ty.

In CP2 space points q and q' = qeja connected by an U (1) transformation are equiva-A point of CP2 in terms of q is

Xa (q) = q+Xaq, (53)

SU (3) spinors q and q+ anticommute with m

{m, q} = {1 — 2P, q} = 2q — 2 {qq+, q} = 0.

In terms of unitary matrix S introduced by the relation m = Sm0S + we have qj = Sj3, j= a, 3, so that qT = (i^i sin to, i^2 sin to, cos to).

Normalized SU(3) spinors are not convenient in treatment of nontrivial topological configurations. We can parametrize the CP2 space by a set of coordinates z = (z1,z2,z3), which is invariant to multiplication by a complex scalar z ^ Xz. Then a projection operator can be written as

z^zt

P, = -T^-. (54)

z+zk

Coordinate z can be parametrized by four angles

z = (sin 8 cos , sin 8 sin , cos 0) , (55)

where 0 < 8, ^ < n/2; 0 < |3, y < 2n.

QCD vector field and Effective gluonic lagrangian. Let us consider the Lagrangian for the gluon field

1 2

L = ~ 7^2 tr GMV

and express it in CP2 variables.

QCD vector field in CP2 sector. We consider the gauge field Vjf2 = mCM + ^m<9Mm (35) (the CNFS connection) and corresponding field strength

= mCp + i [¿>Mm, dym] . (56)

We use the representation of m in terms of projection operators (52). After straitforward calculations we get for the CNFS connection in the CP2 variables

J [m, dltm] = IP, dltP] = 2 (5MP - BM) (57)

where the magnetic (classical) vector 6» and magnetic matrix B» are given by

6M = \ !'(: ''"<( " ''M ; <( > , (58)

(Bm )ij = \ (¿V<Pi<P/ " VAWj) ■ (59)

When the relation q+qj = 1 is used, we have 6» = qtd»qi.

To avoid overloading the formula, we omit the Latin indices in the further presentation of the material.

Finally, the gauge field in CP2 variables is

= (1 — 2W+) C + 2 (b,W+ — B,) = C + (D— DbC<p • <p+) = C + V*, (60) where ( ) ( )

DbC = d — b, + C,),D bC = d + b, — C„).

We introduce notation

V* = (^C<P+ — DlC? • q>+) . (61)

Note the relations

DbCq>+ • ? = —Cn<P+«P, DbC? = C<p+<p.

The GFN field strength j [d^m, dym] in Eq. (56) in CP2 variables looks complicated

1

4

- [<9Mm, dvm] = Pbf„ + 2BMV, (62)

where

b,v = rç+d^ — dv^J+d, ^ (63)

BMV = - (£>£<p£vV - . (64)

1

We use notations D£ = d. - b. D£ = d. + b..

It is convenient to introduce new matrixes B.v and R.v

Bf iv — £>Mv — 7T Riv • (65)

The matrixes B.v and R.v are given by expressions

= \ (3Mcpavcp+ - <9vcpdMcp+) (66)

Rv = (b¡dv (tyty+) - bvd. (tyty+)) (67)

with properties

tr B2V = 52,, [BM, Bv] = I (6MV + BMV), {cpcp+, Bpy} = 0, { i?MV, cpcp+ } = 0. (68)

The abelian part

Cv = d.Cv - dvC (69)

can be picked out in the gauge field strength (56)

C = Cv + V*; (70)

where one can express V.^ in terms of matrixes B.v and Rt

^ Hi UCliiiO U1 mOUIAΠanu

V* = OT+ (—2C,v + b, v) + 2B ,v — Rv. (71)

QCD vector field in U (2)-sector. The gauge field in U (2) sector is Q (x). In order to deal with Q and q, we replace Q ^ Q^ = PqQ^Pq, introducing the projector Pq = (1 — onto the subgroup U (2). It follows

Q^q = 0, q+Qn = 0, Qv^q = — d^Qvq)

[a^ m,Qy] = + = — [w+^Qv]

The field QM (x) does not commute with the connection ^md^m

md^m, Qv

= —-rm [m, d^Qy] = ^ {cpcp+, <9MQV} .

22

Field strength V^2 in the U(2) sector consists out of Q^v = — dvQ^ and mixed term

+ [Q^V?]

VU2 = Q,v + [Vf, Qv] + [Q„ = Q,v + s^

where

5mv = \ {<P<P+,Qi*v} •

(72)

Axial vector AJ arising after chiral rotation U = m from GConsider the axial vector part A^ = A^ in the decomposition G^ = V^1 — A^. The field A^ anticommutes with m, hence, taking account of {m, q} = 0 one can write the decomposition of A^ as

A = K^q+ + qK+, tr A = 0, q+K = 0, K+q = 0.

(73)

We remind that q = 0. These terms correspond to tangential structures for m in SU(3) 5'r|i5'+ and t = 1,2; in combination riiiv^1) + r|21\(2\ where r|i = ^ (X4 + ¿X5),

r\2 = 5 (^6 + ih)

A = E (SntK(t)S+ + Sn+KWs+) . (74)

t

Indeed, considering this expression in detail we have with S+k = q+, Sk3 = qk (SntK(t) S +) kk/ = Sk'tK^S+k = Kk q+,

(S K(t)S+) kk, = Sk3K(t)S+k, = qk K+ — decomposition (73). Contribution of A^ to V^ is given by

A ^, Av

(K,Kv+ - K,K+) - qq+tr KK+ - KK+)

(75)

This term commutes with m

AH, Av

Consider A^v = D^Av — DvA^, where

DHAv — ( d^Av +

0.

+ Q„ A

(76) 449

We get directly

= - DCqK+ + KvDbCq+)

Qu, Ay = (Qu Kvq+ - qK+Qu.

Finally, D.Av does not contain derivatives of ty and ty+

d av = Vky} q+ + qVuK+ ,

where

Thus, the term Auv in Guv anticommutes with m

A^, mj

0.

The field strength The decomposition of the field strength is

where we remind

Guv = Vf + V^v2 +

V''

v

A u, Av

- ( DuAv - DvAn

Cuv + Vuv

V* = (-2Cuv + + DjqDV - D'bqDbq+

= Qmv + SMV = <3Mv + ^ {cpcp+, QMV} , QMV = dMQv - dvQM

Au, Av

(KK+ - K+Kv) - qq+tr (KK+ - K+Kv)

Du^4v - DvAu

VuK4v - VvKu) q+ - q ( VK+ - VvK+

(77)

(78)

(79)

VuKv = (öu + Qu + &u - 2Cu) KKv ; ^uK+ = - Qu - bu + K+. (80)

(81)

The Yang—Mills lagrangian. The Yang—Mills lagrangian can be written as

1

2g1

L = ~ tr + V^ + [Ap, Av] y2 + i2v)

1

V

tr {C2v - 4qq+Cv + (V*)2 + (V^)2 + 2V^V^2 + AA2

+ 2[Au,Av](V^ + VU2) + , (82)

where

tr V*

2

(-2C^ + b») + tr(2B

tr b2

tr r2

= - [(Dl^D^) (%+A-cp) - (%+A.cp) (D^+D^)} = 5; = 2 (bu<9vq+ - bvduq+) (b^q - bvdltq)

tr BliyRliy = 4&2<9vq+<9vq - 2b* bv (d^+dvq + dvq+ d^)

liV^liV — — ^^^v vy^T "V^

tr — V* = (2Cv - b*v) tr Kv

1

tr [A*, Av] Q*v = tr K^vQ^v - T+Q*vT tr K

+ (K+Ky - K+KJ -(Dfa+D?cp - D^+Dfa)

tr Av] S*v — 0

tr (A,„) 2 = 2 tr (v*K+ - VvK+^ (V& - VvKp)

tr [A„, Av]2 = tr (KK+ - KvK++)2 + (tr (KK+ - KvK++))2

Chiral U (1) invariance of CFNS expression. The chiral decomposition G* in the CFNS form is invariant under unitary chiral transformations U by definition G* = V^ - AU, because G* is an U-invariant combination. When we consider only the Yang—Mills sector, this invariance has no consequences. However, when the quark sector is also involved, we should transform chirally quarks and take into account the quantum anomaly of the quark path integral. If the chiral field U does not contain new degrees of fredom compared with the CFNS form, the corresponding quantum anomaly will reflect properties of m, as a common variable of quarks and gluons.

Consider the chiral field

Ux = exp imx, d*x = 0, (83)

which leaves invariant the CFNS connection VP = mCM + ^md^ni in the following sense

(UV%+ + md*m) = VP. (84)

Then the field G* = Vp + Q* - A* after the chiral transformation Ux will become G* = Vx + Q* - A* with a gauge part V? and an axial vector part A*

\ (U*. - A) Ui + ^ - A + U&U+) (U,AU+

A = \ (A - UXAJJ+) . (85)

In the q-parametrization of A*, where m =1 -

Vx = - cos x (e-ixyK+ + K^+e^

A* = i sin x ('e-l'xT>K+ - Kpy+e"11^ .

It is the interval n/2 ^ x ^ 0 which is essential for CD: at x = 0 there is no axial part (Vf-Al) = (Vp - A*, 0) , at x = n/2 the gauge part Vx is the standard CFN field VP, while A* ^ A*.

The hidden U (1) invariance of CFNS decomposition, which does not belong to the gauge group, was found for the SU (2) QCD [13]. This hidden symmetry of CFNS is of chiral origin.

In order to eliminate the superfluous variable, an additional condition is introduced in CFNS [13, 20] ( )

D* (Vp A* = 0 (86).

This condition is covariant under gauge transformations, but breaks Ux invariance. Indeed, calculating variation of this condition we get for 6Ux = im&x

6 {D, (Vx) A,} = iSx {(V,K,ty+ - tyV,K+) + 2K+K, (tyty+ - 1)} , (87)

where the first term acts in the CP2 space, while second one belongs to U (2) space with Q,, as a gauge field. At low energies, when dynamics of Q, is neglected, the additional condition is equivalent to

V,K, = (a, + Q, + b, - 2C,) K, = 0. (88)

Note, that at the end point x = n/2 the condition (86) is satisfied.

Singlet gluonium and bilinear condensate. We consider gluonium within the framework of the Chiral parametrisation (CP) of the gauge field in QCD SU (3). Acccording to CP, there are two main sectors in QCD: CP2 sector and U(2) sector. We are interested in gluonium arising in the CP2 sector containing an abelian field of SU (3). Similar problem has been discussed within the CFNS decomposition in SU(N)/U(1)wsetting, and the gluonium under consideration was referred as FAG (free abelian gluon). We shall use this name to underline similitude of problems. However, all gluoniums are free and abelian, because they do not interact among themselves. In general, gluoniums are composite particles and they should be described by some kind of collective field. In the case of FAGs, interaction gives only the mass to the free abelian field. In CP the fundamental unit matrix responsible for topology (central in the CFN decomposition) is provided by the quark chiral field. This restricts the form of such matrix and involves quarks in its calculation.

Consider from equation (82) the lagrangian of an Abelian field C,. It is convenient to write it in variables P = tyty+, P, = d,P

L(C) = -^(C2y- 4q„trP [PM, Pv] + trA%y) (89)

with

A, = (1 - P) K,P + PK+ (1 - P),

where K, = K,ty+, K+ = tyK+. Note, that there is no cross term C,v [A,,AV] in L (C). Second term in L is multiplied by the magnetic field strength trP [P,, Pv] = b,v. Retaining only C, as a gauge field in A,v = D,Av - DvA, and writing D,Av = d,Av + [-2PC,, Av] we get

trA2v = 2trQ (V,Kv+ - VvK+) (V,Kv - VvK,) = -16C,C,Kv+Kv. (90)

It follows from this relation that if there is a space-like vacuum condensate (all fields are anti-hermitian)

CK = (K+K) = ±( trA2), (91)

then the gluonium C can acquire the mass

m2C = -2CK. (92)

An existence of dimension d =2 condensate was conjectured from different considerations [32-34]. An obvious candidate for a bilinear condensate is the vacuum value (G^) of the

QCD gauge field G* = V** - A*. However, this quantity is not gauge invariant. Therefore, instead of quantity (G2} one considers usually ((G2)min), a condensate of its minimal value.

In our case, the condensate (K+K} is a gauge invariant quantity, because A* is in an adjoint representation. But we do not know how CK enters in experimental and lattice data.

Conclusions. We have considered the classical pure gluonic part of QCD and shown that the CFNS decomposition arises quite naturally from the assumption that the basic color unit n-vector of CFNS is a remnant of the quark chiral field and represents a common variable of gluons and quarks. However, the presence of quark chiral DOF's in the CFNS decomposition raises the consistency problem, because the Yang—Mills gauge field does not depend on quarks kinematically. It is this basic assumption that enables usually to consider a decomposition of the QCD gauge field separately, without invoking quarks. Thus, we should show that there are no chiral transformations involving the basic color vector m, which leave the gauge field invariant, but produce change in the quark sector.

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Статья поступила в редакцию 30 июня 2014 г.

Контактная информация

Novozhilov Victor Yurievich — Dr. Sci., Professor; e-mail: [email protected]