In this work we extend the results obtained in [5] on mass shifting for bosonic string to the case of superstring. The modified anomaly terms of superstring superalgebras are shown and the corresponding BRST charge is used.
Communications in Physics, Vol 19, No (2009), pp 201-204 ORBITAL OSCILLATOR COMMUTATION RELATIONS AND MASS SHIFTING FOR SUPERSTRING NGUYEN THI HA LOAN Hanoi Pedagogical University No NGUYEN HONG HA Institute of Physics and Electronics VAST Abstract In this work we extend the results obtained in [5] on mass shifting for bosonic string to the case of superstring The modified anomaly terms of superstring superalgebras are shown and the corresponding BRST charge is used I INTRODUCTION Superstring theory [1-4] is considered as a prospective direction for the construction of unified theory of all fundamental interactions On this way there are, however, some difficulties to overcome, among these is the existence of particle with negative squared mass called tachyon The authors of Ref have considered a model of bosonic string, in which the mass spectrum of component fields can be shifted as compared to conventional theory, and hence the tachyon can be automatically removed The aim of our work is to extend the result in [5] to the case of superstring It is show that in this case we can also construct a superalgebra with a modified anomaly term is such a way that the theory does not contain tachyon field without the use of GSO mechanism [6] The contents of the paper are arranged as follows In Sec II we construct the superalgebra with a modified anomaly term in the spirit of Ref In Sec III the BRST charge for superstring is treated with a modified anomaly term Sec IV is devoted to the equations of motion and the mass spectrum for component fields II MODIFIED NEVEU-SCHWARZ AND RAMOND SUPERALGEBRAS In Ref on the base of modified commutation relations for orbital oscillators [αµn , ανm] = (−n.η µν + Gµν ) δn,−m , [π µ , π ν ] = Gµν , [P µ , π ν ] = 0, (1) the Virasoro algebra is derived [Ln , Lm ] = (n − m) Ln+m + A (n) δn,−m , (2) with modified anomaly term A (n) = D n n2 − + nG2 , 12 (3) 202 NGUYEN THI HA LOAN AND NGUYEN HNG HA where Gµν is some antisymmetric tensor, G2 ≡ Gµν Gµν αµ0 = pµ + π µ , (4) π µ |0 = To extend this model to the case of superstring we processed as follows Put (ψ) Ln = L(x) n + Ln (x) where Ln are Virasoro generators related to the coordinate Xµ and satisfy the commu(ψ) tation relations (2) and (3), Ln are Virasoro generators related to the supercoordinate ψ and satisfy the commutation relations (ψ) (ψ) L(ψ) = (n − m) Ln+m + A(ψ) n , Lm n δn,−m (5) (ψ) In accordance with equations (2) and (3) we consider the case when An has an additional term proportional to nG2 , namely (ψ) A(n) = D n n2 + − 3δ + gψ nG2 24 (6) where gψ is some parameter, Now we have δ= 1, N S sector 0, R sector (x) (ψ) A(n) = A(n) + A(n) = g ≡ 12 + gψ D n(n − δ) + g.nG2 (7) According to (7) we have the superalgebra of the form 2 [Ln , Lm] = (n − m) Ln+m + D n n − + g.nG δn,−m n [Ln , Gr ] = − r Gn+r δ 2 {Gr , Gs } = 2Ln+s + D S − + gG δr,−s (8) III BRST CHARGE It is known (see e.g [7]) that the expression of BRST charge for superstring is: Q= Ln C−n + n∈Z Gλ γ−λ + λ + n λ (n − m) : C−n C−m bn+m : n,m∈Z n ( − λ) : βn+λ γ−λ C−n : − (9) : bλ+ρ γ−λ γ−ρ − a0 c0 : λ,ρ where Cn , bn are ghost and antighost oscillators satisfying the commutation relations {Cn , bm} = δn,−m , [γλ , βρ] = δλ,ρ {Cn , Cm } = 0, [γλ , γρ] = 0, {bn , bm} = [βλ , βρ] = λ, ρ ∈ Z + 12 for NS superstring, and λ, ρ ∈ Z for R superstring (10) ORBITAL OSCILLATOR COMMUTATION RELATIONS AND MASS SHIFTING FOR SUPERSTRING203 From equations (8) – (10) we can derive: Q2 = n>0 D (D − 10) n2 + gG2 − + + 2a0 8 D (D − 10) λ2 + gG2 − + + 2a0 + λ>0 nC−n Cn (11) γ−λ γλ in the NS-case, and (D − 10) n2 + gG2 + 2a0 nC−n Cn Q2 = n>0 + λ>0 (D − 10) λ2 + gG2 + 2a0 γ−λ γλ + a0 γ 2 (12) in the R-case From here it is seen that Q2 = when 1 − gG2 in the NS-case, and D = 10, a0 = 0, g = in the R-case D = 10, a0 = (13) IV EQUATIONS OF MOTION Let us consider the equation (L0 − a0 ) Ψ [X, ψ] = (14) followed from the BRST equation for string field functional QΨ [X, ψ] = (15) By inserting here the explicit expression of L0 L0 = ∞ µ − µ α−K αµK − k=1 b−λ bµλ (16) λ>0 and using the commutation relations between the oscillators αµn and bµλ , we obtain the equations of motion for component fields in the expansion expression of functional Ψ, n n ,λ λ ∞ (−i)r+3 ψµ11 µrr ,υ11 υss (x) Ψ [X, ψ] = n,s=0 µ+ n1 ·α (17) r!s! + µ+ r υ1 nr λ1 α b + bυλss |0 For example, for the component fields associated to low excited states ψ [X, ψ] = µ+ ψ (x) − iAυ (x) bυ+ − iCµ (x)α1 + |0 (18) 204 NGUYEN THI HA LOAN AND NGUYEN HNG HA we have + gG2 − ψ (x) = + gG2 Aµ (x) = + gG2 + Cµ (x) = (19) ··· Hence, the mass spectrum M is shifted by an amount gG2 as compared to conventional theory In particular, the former tachyon field ψ(x) has m2 = gG2 − which is positive when gG2 > In general, the component field nr ,λ1 λs ψµn11 µ (x) r ,υ1 υs satisfies the Klein-Gordon equation nr ,λ1 λs + M (n, λ) ψµn11 µ (x) = r ,υ1 υs with M (n, λ) = r s nj + j=1 k=1 For R superstring the result remains unchanged (20) λk + gG2 − (21) REFERENCES [1] [2] [3] [4] [5] [6] [7] M B Green, J H Schwarz, E Wetten, Superstring Theory, Cambridge University Press 1987 L Brink, M Henneaux, Principles of String Theory, Plenum Press, New York 1988 M.Kaku, Introduction to Superstring Theory, World Scientific 1989 L Brink, D Friedann, A M Polyakov, Physics and Mathematics of Strings, World Scientific 1990 Dao Vong Duc, Phuong Thi Thuy Hang, Comm.in Phys 17 (2007) 193 F Gliozzi, D Olive, J Scherk, Nucl.Phys.B 122 (1997) 253 Dao Vong Duc, Basic Principles of Quantum Superstring Theory, Publishing House for Science and Technology, VAST, 2007 Received 02 August 2008 ... the commutation relations {Cn , bm} = δn,−m , [γλ , βρ] = δλ,ρ {Cn , Cm } = 0, [γλ , γρ] = 0, {bn , bm} = [βλ , βρ] = λ, ρ ∈ Z + 12 for NS superstring, and λ, ρ ∈ Z for R superstring (10) ORBITAL. .. + 12 for NS superstring, and λ, ρ ∈ Z for R superstring (10) ORBITAL OSCILLATOR COMMUTATION RELATIONS AND MASS SHIFTING FOR SUPERSTRING2 03 From equations (8) – (10) we can derive: Q2 = n>0 D (D... related to the coordinate Xµ and satisfy the commu(ψ) tation relations (2) and (3), Ln are Virasoro generators related to the supercoordinate ψ and satisfy the commutation relations (ψ) (ψ) L(ψ) =