SPIN-CHARGE SEP ARA TION FOR THE SU(3) GAUGE THEOR Y

Vol. 21, No. 31 (2006) 6457–6463 c

World Scientific Publishing Company

SPIN-CHARGE SEPARATION FOR THE SU(3) GAUGE THEORY

VLADIMIR DZHUNUSHALIEV^∗

Department of Physics and Microelectronics Engineering, Kyrgyz-Russian Slavic University,

Bishkek, Kievskaya Str. 44, 720021, Kyrgyz Republic and

Institut f¨ur Physik, Humboldt – Universit¨at zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany

dzhun@krsu.edu.kg

Received 1 August 2006

The idea of a spin-charge separation of the SU(2) gauge potential is extended to the SU(3) case. It is shown that in this case there exist different nonperturbative ground states characterized by different gauge condensateA^B

µA^B

µ 6= 0.

Keywords: Spin-charge separation; ground states.

PACS numbers: 12.38.-t, 12.38.Aw

1. Introduction

One of the main problems in quantum field theory is the quantization of strongly interacting felds. In quantum chromodynamics this problem leads to the fact that up to now we do not completely understand the confinement of quarks. Mathemat- ically the problem is connected with quartic term g²(f^ABCA^B_µA^C_ν)² in the SU(3) Lagrangian: we have no exact mathematical tools for the nonperturbative path integration of such nonquadratic Lagrangian.

In this case it is useful to have any analogy with other area of physics. In Ref. 1 the authors consider the similarity between High-Tc cuprate superconductivity in condensed matter physics and the problem of a mass gap in the Yang–Mills theory.

The authors suggest that in both cases the basic theoretical problems is the absense of a natural condensate to describe the symmetry breaking. The method which is applied in this investigation is a slave-boson decomposition.^2–5

In Ref. 6 the idea presented that is an analogy may exist between the SU(2) Yang–Mills theory in the low-temperature phase and a nematic liquid crystal. The idea is based on a spin-charge separation of the gluon field in the Landau gauge.

∗Senior Associate of the Abdus Salam ICTP.

6457

In Ref. 7 the idea proposed is that in High-Tcsuperconductivity there may exist an analog of a hypothesized flux tube between quarks in quantum chromodynamics where such flux tube essentially increases the interaction energy of two interacting quarks in comparison with the interaction energy for two electrons.

In this paper we would like to investigate such spin-charge separation for the SU(3) gauge field theory and additionally to show that the SU(2) gauge field theory may have another spin-charge separation.

2. Spin-Charge Separation

In the matrix theory⁸there exists the theorem that any real (m×n),m > nmatrix Acan be decomposed as

A=QR , (1)

where Q is an (m×n) orthogonal matrix (Q^TQ = 1) and R is (n×n) upper triangular matrix. If A is (m×n), m < n then Q is an (m×m) orthogonal matrix andR is (m×n) upper triangular matrix. Following to this theorem, we can decompose any SU(2) gauge componentA^a_µ as

A_µ^a= ˜e_µⁱΦ˜^ia, (2) wherea= 1,2,3 is the SU(2) color index and enumerates the columns;µ= 1,2,3,4 (we consider the Euclidean version of the theory) and enumerates the rows; i = 1,2,3 is an inner index which enumerates the columns. Let us introduce the unity

1 = ΛΛ⁻¹, (3)

where Λ is an SO(3) orthogonal matrix. The unity can be inserted in Eq. (2) by such a way that

A_µ^a= (˜e_µⁱΛ^ij)(Λ^kjΦ˜^ka) =e_µⁱΦ^ia, (4) wheree_µⁱ= ˜e_µ^jΛ^ji, Φ^ia= Λ^jiΦ˜^ja. This decomposition is the subject of the inves- tigation in Ref. 6. The matrixA_µ^a is a (4×3) matrix, ˜e_µⁱ is a (4×3) matrix and Φ˜^ka is a (3×3) matrix.

In Ref. 6 the idea presented is that the SU(2) Yang–Mills theory can be asso- ciated with a nematic crystal in which the “molecules” are directed in the internal SO(3) space. The adjoint “matter” field

χ^ij=X

a

Φ^aiΦ^aj (5)

can be associated with the dielectric susceptibility

˜

χαβ= ∆ ˜χX

s

n^(s)_α n^(s)_β , (6) where n^(s)α is the direction of the axis of the sth molecule; ∆ ˜χ = ˜χ_k −χ˜⊥ is the anisotropy in the diamagnetic susceptibility along and perpendicular to the molecule axis.

3. Another Decomposition of SU(2) Gauge Fields

One can present the potentialA^a_µ also as a (3×4) matrix whereaenumerates the rows andµenumerates the columns. Then the corresponding decomposition will be A^a_µ= ˜Φ^aie˜ⁱ_µ, (7) where ˜Φ^ai is the orthogonal matrix ˜Φ^aiΦ˜^aj = δ^ij and ˜eⁱ_µ is an upper triangular matrix. Again we can insert the unity 1 = ΛΛ⁻¹between ˜Φ and ˜eon the right-hand side of Eq. (7). Finally we have

A^a_µ= Φ^aieⁱ_µ, (8)

where Φ = ˜ΦΛ and e= Λ⁻¹e. Now we would like to rewrite the SU(2) Lagrangian˜ in terms of the fields Φ^ai andeⁱ_µ similar to Ref. 6. The field strengthF^a_µν is

F^a_µν =∂µA^a_ν−∂νA^a_µ+g^abcA^b_µA^c_ν

= Φ^ai(∂µeⁱ_ν−∂νeⁱ_µ) + (eⁱ_ν∂µΦ^ai−eⁱ_µ∂νΦ^ai)

+g^abcΦ^biΦ^cjeⁱ_µe^j_ν, (9)

where^abcare the SU(2) structural constants. The terms without coupling constant g can be rewritten as

Φ^ai(∂µeⁱ_ν−∂νeⁱ_µ) + (eⁱ_ν∂µΦ^ai−eⁱ_µ∂νΦ^ai)

= Φ^bi

δ^ab∂µeⁱ_ν+1

2(eⁱ_ν∂µΦ^aj−e^j_µ∂νΦ^ai)Φ^bj

−Φ^bi

δ^ab∂νeⁱ_µ+1

2(eⁱ_µ∂νΦ^aj−e^j_ν∂µΦ^ai)Φ^bj

= Φ^bi

D^ab_µν(Γ)eⁱ_ν−D^ab_νµ(Γ)eⁱ_µ

, (10)

whereD^ab_µν(Γ) is an analog of the covariant derivative with the “connection” Γ:

Γ^ab,ij_µν(eⁱ_ν) = 1

2 eⁱ_ν∂_µΦ^aj−eⁱ_µ∂_νΦ^ai

. (11)

Then the SU(2) Lagrangian can be written as

L=L0+L1+L2 (12)

with

L0= 1 4

Φ^bi[D^ab_µν(Γ)eⁱ_ν−D^ab_νµ(Γ)eⁱ_µ] , (13) L1= g

2Φ^bi

D^ab_µν(Γ)eⁱ_ν−D^ab_νµ(Γ)eⁱ_µ

^abcΦ^bkΦ^cle^k_µe^l_ν, (14) L2= g²

4

(Trχ)²−Tr(χ)²

, (15)

where

χ^ij =eⁱ_µe^j_µ. (16)

Similar to Ref. 6 the quantityχ^ij can be associated with the nematic crystal with one difference: the “molecules” are directed in the Euclidean space–time with the coordinates x^µ. Absolutely by the same way as in Ref. 6 one can calculate the ground state of the nematic associated with the Yang–Mills theory (13)–(15). If we introduce the eigenvalues of the matrixχ= diag{χ1, χ2, χ3}, the ground state χ=χ0 is defined as

4

X

i,j=1

χ⁽⁰⁾_i χ⁽⁰⁾_j = 0 (17)

with the constraints

4

X

i=1

χ⁽⁰⁾_i ≥0,

4

Y

i=1

χ⁽⁰⁾_i ≥0. (18)

The solutions of (17) and (18) are given as

χ⁽⁰⁾₁ =χ⁽⁰⁾₂ = 0, χ⁽⁰⁾₃ ≥0. (19) The most interesting in this consideration is a nonperturbative vacuum which cor- responds toχ⁽⁰⁾₃ 6= 0. Clearly, this vacuum, state is aA²-condensate

A^a_µA^a_µ

=χ⁽⁰⁾₃ 6= 0. (20)

4. SU(3) Spin-Charge Separation

In this section we would like to repeat the SU(2) matrix decomposition of the previous section for the SU(3) case.

4.1. Aµ^B gauge potential as a (4×8) matrix

This case is similar to the spin-charge separation used in Ref. 6:

A_µ^B=e_µⁱΦ^iB. (21)

The matrix e_µⁱ is orthogonal one e_µⁱe_µ^j =δ^ij; i, j = 1,2,3,4. The matrix e_µⁱ is similar to the 4-bein but with one essential difference. Generally speaking, one has

e_µⁱe_νⁱ6=δµν. (22)

Using this decomposition, one can write L_SU(3)= 1

4(F_µν^a )²=L0+L1+L2 (23)

with

L0= 1

2(Dµφ^iB)²+1

8φ^lBΦ^jB(∂µe_ν^j−∂νe_µ^j)

×[(∂µe_ν^l−∂_νe_µ^l)−e_νⁱe_αⁱ(∂µe_α^l−∂_αe_µ^l)]

−1 2

e_νⁱ∂µΦ^iB+1

2(∂µe_νⁱ−∂νe_µⁱ)

×

e_µ^j∂_νΦ^jB+1

2(∂νe_µ^j−∂_µe_ν^j)

, (24)

L1=g 2f^BCD

e_νⁱ∂_µΦ^iB+1

2(∂µe_νⁱ−∂_νe_µⁱ)Φ^iB

−[µ↔ν]

×Φ^jCΦ^kDe_µ^je_ν^k, (25) L2=g²

4f^BCDf^{BM N}A^C_µA^D_νA^M_µ A^N_ν

=g²

4χ^CMf^BCDf^{BM N}χ^DN

=−g²

4 Tr(Φf^BΦ^T)², (26)

wheref^B is the matrix (f^B)^{M N} and

χ^AB= Φ^iAΦ^iB. (27)

The covariant derivativeD_µφ^iB is defined in the following way:

Dµφ^iB =∂µΦ^iB+ Γ(e)^ij_µΦ^jB (28) and the connection Γ(e) as

Γ^ij_µ(e) =e_νⁱ(∂µe_ν^j−∂νe_µ^j). (29) In order to find possible vacuum state we should find the values of the condensate A^B_µA^B_µ = Trχ for which the potential term L2 is zero. Let the matrix χ^AB is diagonalized

χ^AB= diag{χ1, . . . , χ8}. (30) In this case

L2= g² 4

2(χ1χ2+χ1χ3+χ2χ3) +1

2(χ1χ4+χ1χ5+χ1χ6+χ1χ7+χ2χ4

+χ2χ5+χ2χ6+χ2χ7+χ3χ4+χ3χ5+χ3χ6+χ3χ7+ 4χ4χ5+χ4χ6

+χ4χ7+ 3χ4χ8+χ5χ6+χ5χ7+ 3χ5χ8+ 4χ6χ7+χ6χ8+ 3χ7χ8)

. (31)

The first term in Eq. (31) corresponds to the SU(2) subgroup (15). For the pertur- bative vacuum the solution is

χ_i= 0, i= 1, . . . ,8. (32)

The possible nonperturbative vacuum is more complicated then in the SU(2) case.

One can exist different vacuum states. The first vacuum state is similar to the SU(2) case and it is defined by the relation

χi= 0, χj 6= 0, (33)

j is a fixed number. In this case the vacuum condensate is given in the following manner:

Trχ= A^B_µA^B_µ

=χj. (34)

From Eq. (32) we see that not allχiare equivalent that means that the correspond- ing vacuum states may be nonequivalent in the contrast with the SU(2) case.

The second possibility is the case when

χi= 0 (35)

but, for example, three χ_6,7,8 6= 0. In this case we have the following relation betweenχ6,7,8:

4

3χ6χ7+χ6χ8+χ7χ8= 0 (36) butχ6,7,8are independent degrees of freedom and they may not satisfy the relation (36). Thus in this case it will be a special vacuum state and the vacuum special condensate is

A^B_µA^B_µ

=χ^AA=X

χⁱ=χ6+χ7−4 3

χ6χ7

χ6+χ7

. (37)

Other cases with four and more nonzeroχi can be considered analogously.

4.2. Aµ^B gauge potential as a (8×4) matrix In this case

A^B_µ= Φ^Bieⁱ_µ, (38)

where Φ^BiΦ^Bj =δ^ij. The same calculations as in Sec. 3 gives us F^B_µν =∂µA^B_ν−∂νA^B_µ+gf^BCDA^C_µA^D_ν

= Φ^Bi(∂µeⁱ_ν−∂νeⁱ_µ) + (eⁱ_ν∂µΦ^Bi−eⁱ_µ∂νΦ^Bi)

+gf^BCDΦ^CiΦ^Djeⁱ_µe^j_ν (39)

and the SU(3) Lagrangian

L_SU(3)=L0+L1+L2 (40)

can be written as L0 =1

4

Φ^Bi[D^AB_µν(Γ)eⁱ_ν−D^AB_νµ(Γ)eⁱ_µ , (41) L1 =g

2Φ^Bi

D^AB_µν(Γ)eⁱ_ν−D^AB_νµ(Γ)eⁱ_µ

f^ABCΦ^BkΦ^Cle^k_µe^l_ν, (42) L2 =g²

4(f^BCDf^{BM N})(Φ^CiΦ^DjΦ^{M k}Φ^{N l})(eⁱ_µe^j_νe^k_µe^l_ν). (43) Unfortunately in this case it is impossible to simplify the quartic term in the con- sequence of the specific form of the SU(3) structural constant f^ABC.

5. Summary

In this paper we have applied the spin-charge separation for the SU(3) gauge field and have shown that the SU(2) gauge field may have two different spin-charge separations. We have shown that ground states in the SU(3) case can be divided into two branches: the first one is similar to the SU(2) case, but the second branch contains special vacuum states as there exist relations between eigenvalues of the matrix A^B_µA^B_µ. The existence of these vacuum states shows that the perturbative vacuum state of the SU(3) gauge theory can be broken down to different vacuum states characterized by different gauge condensatesA^B_µA^B_µ.

Acknowledgments

The author thanks D. Ebert, M. Mueller-Preussker and the other colleagues of the Particle Theory Group of the Humboldt University for kind hospitality. This work has been supported by DAAD.

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