Calculation of zero-norm states and reduction of stringy scattering amplitudes

(1)

Progress of Theoretical Physics, Vol. 114, No. 1, July 2005

Calculation of Zero-Norm States and Reduction of Stringy Scattering Amplitudes

Jen-ChiLee∗)

Department of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwan, R. O. C.

(Received December 22, 2004)

We give a simplified method to generate two types of zero-norm states in the old covari-ant first qucovari-antized (OCFQ) spectrum of open bosonic string. Zero-norm states up to the fourth massive level and general formulas of some zero-norm tensor states at arbitrary mass levels are calculated. On-shell Ward identities generated by zero-norm states and the

factor-ization property of stringy vertex operators can then be used to argue that the string-tree

scattering amplitudes of the degenerate lower spin propagating states are fixed by those of higher spin propagating states at each fixed mass level. This decoupling phenomenon is, in contrast to Gross’s high-energy symmetries, valid to all energies. As examples, we explic-itly demonstrate this stringy phenomenon up to the fourth massive level (spin-five), which justifies the calculation of two other previous approaches based on the massive worldsheet sigma-model and Witten’s string field theory (WSFT).

§1. Introduction

The theory of string, as a consistent quantum theory, has no free parameter and an infinite number of states. It is thus conceivable that there exists huge hidden symmetry group which is responsible for the ultraviolet finiteness of the theory. In fact, it was conjectured by Gross1) more than a decade ago that an infinite broken gauge symmetries get restored at energy much higher than the Planck energy. More-over, he conjectured that, for the closed string, there existed an infinite number of linear relations among the scattering amplitudes of different string states that are valid order by order and are of the identical form in string perturbation theory as

α goes to inﬁnity. As a result, the scattering amplitudes of all string states can be expressed in terms of, say, the dilaton amplitudes. A similar result was presented in Ref. 2) for the open string case.

Soon after, it was discovered that3) the equations of motion for massive back-ground fields of the degenerate positive-norm propagating states can be expressed in terms of those of higher spin propagating states at each fixed mass level. This decoupling phenomenon was argued to be arisen from the existence of two types of zero-norm states with the same Young representations as those of the degenerate positive-norm states in the OCFQ spectrum. This was demonstrated by using mas-sive worldsheet sigma-model approach in the lowest order weak field approximation but valid to all orders in α, and thus was, in contrast to Gross’s result, valid to

all energies. To compare with the usual sigma-model loop (α) approximation, this

result was argued to be a sigma-model n+1 loop result for the n-th massive level

∗)_{E-mail: [email protected]}

at National Chiao Tung University Library on April 26, 2014

http://ptp.oxfordjournals.org/

(2)

(spin-n+1).3)–5) This calculation applies to both open and closed string cases. In a recent paper,6) the same decoupling phenomenon was demonstrated by using WSFT for the open string case up to the spin-five level. It was shown that the background fields of these degenerate positive-norm states can be gauged to the higher rank fields at the same mass level.

In this paper, we will derive this interesting stringy decoupling phenomenon from the third and a more direct method, namely, the S-matrix approach. The key was to explicitly calculate both types of zero-norm states7) in the OCFQ spectrum. An infinite number of nonlinear relations between string scattering amplitudes of different string states with the same momenta at each fixed mass level can then be written down.8) By nonlinearity, one means that the coefficients among scattering amplitudes of different string states depend on the center of mass scattering angle

φ_CM through the dependence of momentum k.9) These relations, or stringy on-shell Ward identities are, as in Gross’s case, valid order by order and are of the identi-cal form in string perturbation theory since zero-norm states should be decoupled from the string amplitudes at each order of string perturbation theory. These Ward identities, together with the factorization property of stringy vertex operators, will be used in this paper to express the scattering amplitudes of the degenerate lower spin propagating states in terms of those of higher spin propagating states, and thus reduce the number of independent scattering amplitudes at each fixed mass level. These Ward identities and the resulting decoupling phenomenon are, in contrast to Gross’s high-energy symmetries, valid to all energies. However, these nonlinear Ward identities, which are valid to all energies, are difficult to solve. The high-energy limit of these stringy Ward identities are recently9) used to explicitly prove Gross’s conjecture on linear relations among high-energy scattering amplitudes of different string states with the same momenta. It was shown that these stringy Ward identi-ties get simplied as α → ∞, and the number of independent scattering amplitudes reduces further. As a result, there is only one independent component of high energy scattering amplitude at each fixed mass level. All other components of high energy scattering amplitudes are proportional to it. Moreover, the proportionality constants between scattering amplitudes of different string states are calculated. These pro-portionality constants were found to be independent of the scattering angle φ_CMand the loop order χ of string perturbation theory as conjectured by Gross.1), 2) For the case of string-tree amplitudes, a general formula can even be given9)to determine all high energy stringy scattering amplitudes for arbitrary mass levels in terms of those of tachyons – another conjecture by Gross.1)

It is now clear that zero-norm states are of crucial importance to uncover the fundamental symmetries of string theory.9) The power of zero-norm states and their direct relation to spacetime w∞ symmetry and Ward identities10) of toy 2D string model were stressed in Ref. 11). A general formula of 2D zero-norm states at an arbitrary mass levels with Polyakov’s momentum was given in terms of Schur Poly-nomials. These zero-norm states were shown to carry the charges of w∞ symmetry, which was used to determine the tachyon scattering amplitudes without any integra-tion. In §2 of this paper, with the help of a simpliﬁed method to construct D=26 stringy positive-norm vertex operators,12) we will ﬁrst tabulate Young diagrams of

(3)

D=26 zero-norm states at each mass level given Young diagrams of positive-norm

states at the same mass level. A consistent check of counting of number of zero-norm states by using the background ghost fields in WSFT was given in Ref. 6). Here we go one step further and invent a simplified method to explicitly construct D=26 stringy zero-norm states. As examples, we calculate all relevant zero-norm states up to the spin-five level. General formulas of some zero-norm tensor states at an arbi-trary mass levels will also be given. In §3, we then use these zero-norm states and their corresponding stringy Ward identities, together with the factorization property of stringy vertex operators, to explicitly show the reduction of string-tree scattering amplitudes of degenerate positive-norm propagating states up to the spin-five level. This calculation justifies two previous independent calculations based on the massive worldsheet sigma-model approach3) and WSFT approach.6)

§2. Calculation of zero-norm states

The vertex operator of a physical state of open bosonic string

|Ψ =Cµ1...µmα µ1 −n1...α µm −nm|0; k , [αµm, ανn] = mηµνδm+n (1) is given by13) Ψ (z) =Cµ1...µmNm : (∂znjxµj_)eik·X(z)_:, ₍₂₎

where Nm = im{(nj− 1)!}−1. In the OCFQ spectrum, physical states in Eq. (1) are subject to the following Virasoro conditions

(L0− 1) |Ψ = 0, L1|Ψ = L2|Ψ = 0, (3a,b) where Lm = 1₂ ∞ −∞ : αm−n· αn: (4)

and α0 ≡ k. The solutions of Eqs. (3a,b) include positive-norm propagating states

and two types of zero-norm states. The latter are14)

Type I : L−1|x , where L1|x = L2|x = 0, L0|x = 0; (5) Type II : L−2+3 2L 2 −1 |x , where L1|x = L2|x = 0, (L0+ 1)|x = 0. (6)

Equations (5) and (6) can be derived from Kac determinant in conformal ﬁeld the-ory. While type I states have zero-norm at any spacetime dimension, type II states have zero-norm only at D=26. The existence of type II zero-norm states signals the importance of zero-norm states in the structure of the theory of string. It is straightforward to solve positive-norm state solutions of Eq. (3a, b) for some low-lying states, but soon becomes practically unmanageable. The authors of Ref. 12)

(4)

gave a simple prescription to solve the positive-norm state solutions of Eqs. (3a, b). The strategy is to apply the Virasoro conditions only to purely transverse states, so that the zero-norm states will be got rid of at the very beginning. This prescription simpliﬁed a lot of computation although some complexities remained for low spin states at higher levels. Our aim here, on the contrary, is to generate zero-norm states in Eqs. (5) and (6), so that all physical state solutions of Eq. (3) will be completed. Let us ﬁrst assume we are given positive-norm state solutions of some mass level

n. The number of positive-norm degree of freedom at mass level n (M2 = 2(n − 1))

is given by N24(n), where15) ND(n) = _2πi1 dx xn+1 ∞ k=1 1 1− xk _D . (7)

On the other hand, the number of physical state degree of freedom is given by N25(n)

in view of the constraints in Eq. (3a,b). The discrepancy is of course due to physical zero-norm states given by solutions of Eqs. (5) and (6). That is, among 25 chains of αµm oscillators one chain forms zero-norm states. Thus we can easily tabulate Young diagrams of zero-norm states at each mass level given Young diagrams of positive-norm states at the same mass level calculated by the simpliﬁed prescription in.12) For example, positive-norm state at mass level n = 4 gives zero-norm states + + +• , posive-norm state gives zero-norm states + + and positive-norm state gives zero-norm states +•. This completes the zero-norm states at mass level n = 4. Young diagrams of zero-zero-norm states up to mass level M2 = 10, together with positive-norm states calculated in Ref. 12), are listed in Appendix A. A consistent check of counting of zero-norm states by using background ghost ﬁelds in WSFT was given in Ref. 6).

To explicitly calculate zero-norm states is another issue. Suppose we are given some low-lying positive-norm state solutions. It is interesting to see the similarity between Eqs. (3a, b) and Eqs. (5) and (6) for |x and |x. The only diﬀerence is the “mass shift” of L0 equations. As is well-known, the L1and L2 equations give

the transverse and traceless conditions on the spin polarization. It turns out that, in many cases, the L1and L2 equations will not refer to the L0 equation or on-mass-shell condition. In these cases, a positive-norm state solution for |Ψ at mass level n will give a zero-norm state solution L−1|x at mass level n + 1 simply by

taking |x = |Ψ and shifting k2 by one unit. Similarly, one can easily get a type II zero-norm state (L−2+ 3₂L2₋₁)|x at mass level n + 2 simply by taking |x = |Ψ and shifting k2 by two units. For those cases where L1 and L2 equations do refer to L0 equation, our prescription needs to be modiﬁed. We will give some examples

to illustrate this method. Note that once we generate a zero-norm state, it soon becomes a candidate of physical state |Ψ to generate two new zero-norm states at even higher levels.

1. The ﬁrst zero-norm state begin at k2 = 0. This state is suggested from the positive-norm tachyon state |0, k with k2 = 2. Taking |x = |0, k and shifting k2 by one unit to k2= 0, we get a type I zero-norm state.

(5)

L−1 |x = k · α−1|0, k ; |x = |0, k , −k2= M2 = 0. (8)

2. At the ﬁrst massive level k2 =−2, tachyon suggests a type II zero-norm state L−2+3 2L 2 −1 |x =1 2α−1·α−1+ 5 2k ·α−2+ 3 2(k ·α−1) 2 _{|0, k ; |x = |0, k , −k}2 _{= 2.} (9) Positive-norm massless vector state suggests a type I zero-norm state

L−1|x = [θ·α−2+(k·α−1)(θ·α−1)]|0, k ; |x = θ·α−1|0, k , −k2= 2, θ·k = 0. (10)

However, massless singlet zero-norm state (8) does not give a type I zero-norm state at the ﬁrst massive level k2=−2 since L₁ equation on state (8) refers to L0equation, k2 = 0. This means that L1 will not annihilate state (8) if one shifts the mass to

k2 =−2.

3. At the second massive level k2 = −4, positive-norm massless vector state suggests a type II zero-norm state

L−2+3 2L 2 −1 |x =4θ · α−3+1 2(α−1· α−1)(θ · α−1) + 5 2(k · α−2)(θ · α−1) +3 2(k · α−1) 2_{(θ · α} −1) + 3(k · α−1)(θ · α−2) |0, k ; |x = θ · α−1|0, k , −k2 = 4, k · θ = 0. (11)

However, massless singlet zero-norm state (8) does not give a type II zero-norm state at mass level k2=−4 for the same reason stated after Eq. (10). Positive-norm spin-two state at k2 =−2 suggests a type I zero-norm state

L−1|x = [2θµναµ−1αν−2+ kλθµναλµν−1 ]|0, k ; |x = θµναµν−1|0, k , −k2= 4,

k · θ = ηµνθµν = 0, θµν = θνµ, (12)

where αλµν₋₁ ≡ αλ₋₁αµ₋₁αν₋₁. Similar notations will be used in the rest of this paper.

Vector zero-norm state with k2 =−2 in Eq. (10) does not give a type I zero-norm state for the same reason stated after Eq. (10). In this case, however, one can modify

|x to be

Ansatz: |x = [aθ · α₋₂+ b(k · α−1)(θ · α−1)]|0, k ; −k2= 4, θ · k = 0, (13) where a, b are undetermined constants. L0 equation is then trivially satisﬁed and L1, L2 equations give a : b = 2 : 1. This gives a type I zero-norm state

L−1|x = ₁ 2(k · α−1) 2_{(θ · α} −1) + 2θ · α−3+3₂(k · α−1)(θ · α−2) +1 2(k · α−2)(θ · α−1) |0, k ; −k2= 4, θ · k = 0. (14)

(6)

Similarly, we modify the singlet zero-norm state with k2 =−2 in Eq. (9) to be Ansatz: |x = ₅ 2ak · α−2+ 1 2bα−1· α−1+ 3 2c(k · α−1) 2 _{|0, k ; −k}2 _{= 4,} ₍₁₅₎

where a, b and c are undetermined constants. L1 and L2 equations give 5a + b + 3k2c = 0, 5k2a + 13b + 3

2k

2_{c = 0.} ₍₁₆₎

For k2=−4, we have a : b : c = 5 : 9 : 17₆ . This gives a type I zero-norm state

L−1|x = ₁₇ 4 (k · α−1) 3₊9 2(k · α−1)(α−1· α−1) + 9(α−1· α−2) +21(k · α−1)(k · α−2) + 25(k · α−3) |0, k ; (17) −k2= 4.

This completes the four zero-norm states at the second massive level. Note that state (17) was calculated in Ref. 7) without modiﬁcation. The coeﬃcients there thus need to be corrected although the main results remain valid. It is interesting to note that the Young tableau of zero-norm states at level M2 = 4 are the sum of those of all physical states at two lower levels, M2= 2 and M2= 0, except the singlet zero-norm state due to the dependence of L1 and L2 equations on L0 condition in state (8). For

those cases that L1 and L2 equations not referring to L0 condition, our construction

gives us a very simple way to calculate zero-norm states at any mass level n given those of positive-norm states at lower levels constructed by the simpliﬁed method in Ref. 12). When the modiﬁed method was needed to calculate a higher mass level zero-norm state from a lower mass level physical state like Eq. (8), an inconsistency may result and one gets no zero-norm state. This explains the discrepancy of singlet zero-norm states at levels M2= 2, 4,8 and a vector zero-norm state at level M2 = 10. 4. Similar method can be used to calculate zero-norm states at level M2 = 6. We will just list those which are relevant for the discussion in section III. They are (from now on, unless otherwise stated, each spin polarization is assumed to be transverse, traceless and is symmetric with respect to each group of indices as in Ref. 12))

L−1|x = θµνλ(kβαµνλβ₋₁ + 3α₋₁µναλ₋₂)|0, k ; |x = θ_µνλαµνλ₋₁ |0, k , (18) L−1|x = [kλθµναµ−1λαν−2+ 2θµναµ−1αν−3|0, k ; |x] = θµναµ−1αν−2|0, k , where θµν =−θ_νµ, (19) L−1|x = 2θµναµν₋₂+ 4θµναµ₋₁αν₋₃+ 2(kλθµν+ k(λθµν))αλµ−1αν−2 +2 3kλkβθµνα µνλβ −1 |0, k ; |x =2θµναµ₋₁αν₋₂+2 3kλθµνα µνλ −1 |0, k (20)

(7)

and L−2+3 2L 2 −1 |x =3θµναµν₋₂+ 8θµναµ₋₁αν₋₃+ kλθµν+ 15 2 k(λθµν) αλµ₋₁αν₋₂ + ₁ 2ηλβθµν+ 3 2kλkβθµν αµνλβ₋₁ |0, k ; |x = θµναµν−1|0, k . (21)

Note that |x in Eq. (20) has been modiﬁed as we did for Eq. (13). To further illustrate our method, we calculate the type I singlet zero-norm state from Eq. (17) as following

Ansatz : |x = [a(k · α₋₁)3+ b(k · α−1)(α−1· α−1) + c(k · α−1)(k · α−2)

+d(α−1· α−2) + f (k · α−3)]|0, k ;

−k2_{= 6.} ₍₂₂₎

The L1 and L2 equations can be easily used to determine a : b : c : d : f = 37 : 72 : 261 : 216 : 450. This gives the type I singlet zero-norm state

L−1|x = [a(k · α−1)4+ b(k · α−1)2(α−1· α−1) + (2b + d)(k · α−1)(α−1· α−2)

+(c + 3a)(k · α−1)2(k · α−2) + c(k · α−2)2

+d(α−2· α−2) + b(k · α−2)(α−1· α−1)

+(2c + f )(k · α−3)(k · α−1) + 2d(α−1· α−3) + 3f (k · α−4)]|0, k ,

−k2 _{= 6.} ₍₂₃₎

5. We list relevant zero-norm states at level M2 = 8 from the known positive-norm states and zero-positive-norm states at level M2 = 4, 6. They are

L−1|x = (kβθµνλγαµνλγβ−1 + 4θµνλγαµνλ−1 αγ−2)|0, k ; |x = θµνλγαµνλγ−1 |0, k , (24) L−1|x = θµνλ ₃ 4kβkγα µνλγβ −1 + 3kβαµνβ−1 α−2λ + 3kβα(µνλ−1 αβ)−2 +6α(µ₋₁ανλ)₋₂ + 6α(µν₋₁αλ)₋₃ |0, k ; |x = θµνλ ₃ 4kβα µνλβ −1 + 3αµν−1αλ−2 |0, k , (25) L−2+3₂L2−1 |x = θµνλ ₃ 2kβkγ+ 1 2ηγβ αµνλβγ₋₁ + kγ ₁ 2α µνλ −1 αγ−2+ 8α(µνλ−1 αγ)−2 +3α(µ₋₁ανλ)₋₂ + 6α(µν₋₁αλ)₋₃ |0, k ; |x = θµνλαµνλ−1 |0, k , (26)

(8)

L−1|x = θµν,λ(kγαγµν₋₁ αλ₋₂+ 2αµ₋₁ανλ₋₂+ 2αµν₋₁αλ₋₃)|0, k ;

|x = θµν,λαµν₋₁αλ₋₂|0, k , where θµν,λ is mixed symmetric, (27)

Note that the modiﬁed method was used in Eqs. (25) and (28).

6. Finally, we calculate general formulas of some zero-norm tensor states at arbitrary mass levels by making use of general formulas of some positive-norm states listed in Ref. 12). a. L−1θµ1...µmα µ1...µm −1 |0, k = θµ1...µm(kλα λµ1...µm −1 + mαµ−21αµ−12...µm)|0, k , (30)

where −k2 = M2 = 2m, m = 0, 1, 2, 3 · · · . For example, m = 0, 1 give Eqs. (8) and (10). b. L−2+3₂L2−1 θµ1...µmα µ1...µm −1 |0, k = θµ1...µm ₃ 2kνkλ+ 1 2ηνλ ανλµ₋₁ 1...µm+3 2m(m − 1)α µ1µ2 −2 αµ−13...µm +(1 + 3m)αµ₋₁1...µm−1αµ₋₃m + ₃ 2(m + 1)k(λθµ1...µm)+ 3 2mkµmθµ1...µm−1λ αµ₋₁1...µmαλ₋₂ |0, k , (31)

where−k2 = M2= 2m + 2, m = 0, 1, 2 · · · . For example, m = 0, 1 give Eqs. (9) and (11). c. L−1θµ1...µm−2,µm−1α µ1...µm−2 −1 αµ−2m−1|0, k = θµ1...µm−2,µm−1[kλα λµ1...µm−2 −1 αµ−2m−1+ (m − 2)α−1µ1...µm−3αµ−2m−2µm +2αµ₋₁1...µm−2αµ₋₂m−1]|0, k , ... (32)

(9)

where −k2 = M2 = 2m, m = 3, 4, 5 · · · . For example, m = 3, 4 give Eqs. (19) and (27). d. L−2+3 2L 2 −1 θµ1...µm−2,µm−1α µ1...µm−2 −1 αµ−2m−1|0, k = θµ1...µm−2,µm−1 ₃ 2kλkν + 1 2ηλν αµ₋₁1...µm−2λναµ₋₂m−1+ 6kλαµ−11...µm−2λαµ−3m−1 + ₃ 2m − 2 kλαµ−11...µm−2αµ−2m−1λ+ 2(m − 2)α−1µ1...µm−3αµ−2m−2αµ−3m−1 +11αµ₋₁1...µm−2αµ₋₄m−1+ kλαµ₋₁1...µm−3λαµ₋₂m−2µm−1 +(m − 3)αµ₋₁1...µm−4αµ₋₂m−3µm−2µm−1 |0, k , ... ₍₃₃₎

where −k2 = M2 = 2m + 2, m = 3, 4, 5.... For example, m = 3 gives Eq. (29). e. L−1θµ1...µm−4,µm−3µm−2 αµ₋₁1...µm−4αµ₋₂m−3µm−2−4 3α µ1...µm−3 −1 αµ−3m−2 = θµ1...µm−4,µm−3µm−2 kλαλµ−11...µm−4αµ−2m−3µm−2+ (m − 4)αµ−11...µm−3αµ−2m−4µm−3µm−2 +16 3 α µ1...µm−4 −1 αµ−3m−3αµ−2m−2+ 4 3kλα λµ1...µm−3 −1 αµ−3m−2 + 4αµ−11...µm−3αµ−4m−4 , ... ₍₃₄₎ where −k2 = M2 = 2m, m = 5, 6 · · · .

f. The zero-norm states of Eq. (30) can be used to generate new type I zero-norm states by the modiﬁed method as following

L−1θµ1...µm _m m + 1kλα λµ1...µm −1 + αµ−21αµ−12...µm |0, k = _m m + 1kνkλθµ1...µmα νλµ1...µm −1 + m(k(λθµ1...µm)+ kλθµ1...µm)α µ1 −2αλµ−12...µm +m(m − 1)θµ1...µmα µ1µ2 −2 αµ−13...µm+ 2mθµ1...µmα µ1 −3αµ−12...µm |0, k , (35)

where −k2 = M2 = 2m + 2, m = 1, 2, 3 · · · . For example, m = 1, 2 and 3 give Eqs. (14), (20) and (25). Note that the coefficient of the first term in Eq. (35) has been modified to _m+1m . Similarly, new type II zero-norm states can also be

con-structed.

These are examples of some higher spin zero-norm states at arbitrary mass levels. As in the case of positive-norm states, the complexity of the calculation increases when calculating lower spin zero-norm states for higher levels. Fortunately, for our purpose in this paper, it is usually good enough to calculate higher spin zero-norm states as it will become clear in the next section. For those formulas with transverse trace12)

ηT_µν = η_µν− kµkν/k2, (36)

(10)

the modiﬁed method should be used, and we have no general formulas for them. Each zero-norm state calculated in this section corresponds to an on-shell Ward identity, which can be easily written down. As an interesting example8) to illustrate the importance of zero-norm state, the inter-particle Ward identity for two propa-gating states at the second massive level (M2 = 4) was calculated to be (k · θ = 0)

₁

2kµkνθλ+ 2ηµνθλ

T_2,χ(µνλ)+ 9kµθνT2,χ[µν]− 6θµT2,χµ = 0, (37)

where we have chosen, say, v1(k1) to be the vertex operator constructed from D2

vector zero-norm state obtained by antisymmetrizing those terms which contain

αµ₋₁αν₋₂ in the original type I, Eq. (14), and type II, Eq. (11), vector zero-norm states and kµ ≡ k1µ. Note that v2, v3 and v4 can be any string states (including zero-norm states), and we have omitted their tensor index for the cases of excited string states in Eq. (37). T_2,χ s in Eq. (37) are the second massive level, χ-th order

string-loop amplitudes. At this point, {T_2,χ(µνλ), T_2,χ(µν), T_2,χµ } is identiﬁed to be the

amplitude triplet of the spin-three state and T[µν]is identiﬁed to be the amplitude of

the antisymmetric spin-two state.8) Eq. (37) thus relates the scattering amplitudes of two different string states at the second massive level. It is important to note that Eq. (37) is, in contrast to the high-energy α → ∞ result of Gross, valid to all string-loop and all energy α, and its coefficients do depend on the center of mass scattering angle φ_CM, which is defined to be the angle between−→k1 and−→k3, through the dependence of momentum k . This angular dependence disappears in the high-energy limit of Eq. (37),9) which is consistent with Gross’s result. The inter-particle gauge symmetry corresponding to Eq. (37) can be calculated to be7)

δC(µνλ)= 1 2∂(µ∂νθλ)− 2η(µνθλ) , δC[µν]= 9∂[µθν], (38)

where ∂νθν = 0, (∂2−4)θν = 0 are the on-shell conditions of the D2vector zero-norm

state. C_(µνλ) and C_[µν] are the background fields of the symmetric spin-three and antisymmetric spin-two states respectively at the second mass level. Equation (38) is the result of the first order weak field approximation but valid to all energy α in the generalized σ-model approach. It is important to note that the decoupling of D2 vector zero-norm state implies simultaneous change of both C(µνλ) and C[µν],

thus they form a gauge multiplet. This important stringy phenomenon can also be justiﬁed in WSFT.6), 8) A second order weak ﬁeld calculation implies an even more interesting spontaneously broken inter-mass level symmetry in string theory.16)

§3. Reduction of degenerate state’s amplitude

The decoupling of degenerate positive-norm states was first discovered in Ref. 3) by using generalized sigma-model approach. It was recently justified by using WSFT for the open string case up to the spin-five level.6) This stringy phenomenon begins to show up at spin-four level of open bosonic string. The explicit form of four positive-norm states at spin-four level can be found in Ref. 13). According to the

(11)

decoupling conjecture, the spin-two and the scalar positive-norm states should be decoupled. That is, their amplitudes are determined from those of two other higher spin states. Let us begin the discussion by ﬁrst making an important observa-tion. According to Eq. (2), the vertex operator corresponding to αµνλγ₋₁ is Aµνλγ = :∂xµ∂xν∂xλ∂xγeik·x:. Due to the factorization structure of this tensor vertex, which

results from the strong constraint of 2D worldsheet conformal symmetry, the

ampli-tude corresponding to Aµνλγ is fixed by its traceless, transverse spin part µνλγ. In

particular, the longitudinal parts of Aµνλγ are determined by µνλγ through the Lorentz extension; and the trace parts of Aµνλγare ﬁxed by the conformal extension. This means that given the on-shell amplitude of µνλγ, the amplitude Tµνλγ of Aµνλγ

is ﬁxed. HereTµνλγ is deﬁned to be the four-point function containing the rank -four tensor : ∂xµ∂xν∂xλ∂xγeik·x: and three tachyons. Due to the factorization structure of stringy vertex operator, the string-tree scattering amplitude are factorized in the momentum kµ carried by the vertex vertex. Let us use a simpler rank-two tensor to illustrate the trace fixing or conformal extension. Given a factorized symmetric rank-two tensor constructed from a D-vector kµ (kµ will correspond to momentum for the scattering amplitudes in the later discussion)

Aµν = kµkν+ cηµν = kµkν−k 2 Dη µν₊k2 D + c ηµν, (39)

where we have decomposed Aµν into a traceless spin part and a trace part containing a scalar c independent of the spin part, the trace part of Aµν is not ﬁxed by the spin part of Aµν. Now for the homogeneous factorized tensor, c = 0 in Eq. (39). The

traceless spin part of Eq. (39) gives us D(D+1)₂ components which is of order D2, while the factorized symmetric rank-two tensor Aµν contains only D independent components which are components of kµ. It is thus easy to see that the trace part of

Aµνis ﬁxed by the spin part of the tensor. Thus, knowing the spin part of Aµνmeans knowing the whole tensor. This result can be easily generalized to the decomposition of a homogeneous factorized tensor Aµν = kµ₁k₂ν, which contains only 2D independent components in contrast to the number of components of the spin part, which is of the order D2. Similar results can be obtained for homogeneous factorized higher rank

tensors. Note that this factorized property can only be seen in the ﬁrst order weak ﬁeld approximation3) (or vertex operator consideration), and does not show up in the zeroth order spectrum.

With the observation discussed above in mind, we can now discuss the decoupling phenomenon at level four. It was pointed out3)that the positive-norm spin-two state can be gauged to a gauge which contains only αµνλγ₋₁ and αµν₋₁αλ₋₂ terms by making use of the gauge transformations induced by the type I and the type II spin-two zero-norm states, Eqs. (20) and (21), to be

₁ 3kλµν+ 1 2k(λµν) αλµ₋₁αν₋₂+ ₁₃ 174kαkβµν+ 3 58ηαβµν αµναβ₋₁ |0, k , (40)

(12)

where µν is a symmetric traceless and transverse spin-two tensor. Since the

rank-four amplitude T_3,χµναβ is fixed by the spin-four amplitude and the mixed-symmetric rank-three amplitude T_3,χλµν is fixed by the mixed-symmetric spin-three amplitude, the amplitude of the two state in Eq. (40) is determined by those of the spin-four and the mixed-symmetric spin-three states. (Note that T_3,χ(λµν) is fixed by the spin-four amplitude T_3,χµναβ due to the existence of a totally symmetric spin-three zero-norm state Eq. (18) at this level.) In fact, T_3,χµναβ with χ = 1 can be explicitly calculated to be8) T_3,1µνλγ = Γ (− s 2 − 1 Γ (−₂t− 1) Γ (u₂ + 2) _s2 4 − s _s2 4 − 1 k₃µkν₃k₃λkγ₃ −tt2 4 − 1 (s + 2)k(µ₁ kν₁kλ₁kγ)₃ +3st 2 _s 2 + 1 _t 2+ 1 k₁(µk₁νk₃λkγ)₃ −ss2 4 − 1 (t + 2)k₁(µkν₃k₃λkγ)₃ + _t2 4 − t _t2 4 − 1 k₁µk₁νk₁λk₁γ , (41)

where s = −(k1 + k2)2, t = −(k2 + k3)2, and u =−(k1 + k3)2are the Mandelstam

variables. We have chosen the second state to be the tensor and have done the

SL(2, R) gauge ﬁxing and restricted to the s − t channel by setting x1 = 0, 0 ≤

x2 ≤ 1, x3 = 1, x4 = ∞. One easily sees from Eq. (41) that there are no terms

containing ηµνon the right hand side of T_3,1µνλγ. This is due to the normal ordering of the tensor vertex operator :∂xµ∂xν∂xλ∂xγeik·x :, and there is no contribution of terms resulting from contraction within the tensor vertex when doing the amplitude calculation. Thus the trace part of the rank-four amplitude is ﬁxed by the spin-four amplitude by the conformal extension mentioned in the beginning of this section. That is, the rank-four amplitude T_3,1µνλγ is ﬁxed by the spin-four amplitude. This result can be easily generalized to N-point amplitudes containing more than one tensor state.

Take a representative of the positive-norm scalar state at this mass level to be13) η_µν+13 3 kµkν αµν₋₂+ ₂₀ 9 kµkνkρ+ 2 3kµηνρ+ 13 3 kρηµν αµν₋₁αρ₋₂ + ₂₃ 81kµkνkρkσ+ 32 27kµkνηρσ+ 19 18ηµνηρσ αµνρσ₋₁ |0, k . (42)

It turns out that one can not gauge away the first term in Eq. (42) by using the gauge transformations induced by the two singlet zero-norm states as in the case of positive-norm spin-two state. However, since the amplitude corresponding to αµν₋₂ has been fixed by those of two higher spin states, we conclude that the positive-norm scalar state amplitude is again fixed by those of two higher spin states. This concludes the justification of decoupling conjecture for spin-four level. We stress

(13)

here that the mechanisms that is responsible for this decoupling is the existence of two-types of zero-norm states and the factorization of stringy vertex, which are both due to 2D inﬁnite dimensional worldsheet conformal symmetry.

The positive-norm states at level ﬁve were calculated in Ref. 12) to be

µνλβγαµνλβγ−1 |0, k , (43) µνλ,βαµνλ−1 αβ−2|0, k , (44) µ,νλ αµ₋₁ανλ₋₂−4 3α µν −1αλ−3 |0, k , (45) ₄ 5!(D + 5)µνλη T βγαµνλβγ−1 + µνλ αµ₋₁ανλ₋₂−4 3α µν −1αλ−3 |0, k , (46) ₅ 6(D + 1)η T (µνλ)βαµνλ−1 αβ−2+ µν αµ₋₂αν₋₃−1 2α µ −1αν−4 |0, k , (47) and _{D − 2} 80(D + 3)η T (µνηTλβγ)αµνλβγ−1 + ηT_µνλ−1₂(D − 1)(µηTν)λ αµν₋₁αλ₋₃ +3 4(Dµη T νλ− ηTµ(νλ))αµ−1ανλ−2) |0, k . (48)

According to our decoupling conjecture, states (46), (47) and (48) should be de-coupled. Note that states (27) and (45) are diﬀerent in the α_is operator content

although they share the same Young diagram. One corresponds to αµ₋₁ανλ₋₂ and the other αµν₋₁αλ₋₃ or vice versa. With the explicit form of zero-norm states calculated

in§3, we can now justify the decoupling conjecture at level ﬁve. The terms α(µ₋₁ανλ)₋₂

and α(µν₋₁αλ)₋₃ in Eq. (46) can be gauged away by zero-norm states in Eqs. (25) and (26), and the amplitude corresponding to α(µνλ₋₁ αβ)₋₂ is ﬁxed by that of αµνλβγ₋₁ through zero-norm state in Eq. (24) and our observation discussed in the beginning of this section. Thus the amplitude of state (46) is ﬁxed by those of states (43) and (44). Now turn to state (47). The terms α[µ₋₂αν]₋₃ and α[µ₋₁αν]₋₄ can be gauged away by zero-norm states in Eqs. (28) and (29), the amplitudes corresponding to α(µ₋₁ανλ)₋₂

and α(µν₋₁αλ)₋₃ are ﬁxed by those of states in Eqs. (43) and (44) through zero-norm states in Eqs. (25) and (26). Finally the amplitude of mixed-symmetric αµ₋₁ανλ₋₂ (or

αµν₋₁αλ₋₃) is fixed by those of states (43), (44) and (45). Thus the amplitude of state (47) is fixed by those of states (43), (44) and (45). Similar analysis shows that the amplitude of state (48) is again fixed by those of states (43), (44) and (45). This completes the justification of our decoupling conjecture at level five.

The decoupling calculation presented in this paper by the S-matrix approach can be easily generalized to the closed string theory by making use of the simple

(14)

relation between closed and open string amplitudes in Ref. 17). A similar gener-alization to the closed string theory can also be done for the massive worldsheet sigma-model approach. Our calculation in this section justiﬁes two previous inde-pendent calculations based on the massive worldsheet sigma-model approach3) and WSFT approach.6)

Acknowledgements

I would like to thank Physics Departments of National Taiwan University and Simon-Fraser University, where part of this work was completed during my sabbatical visits. I thank Chuan-Tsung Chan and Pei-Ming Ho for many valuable discussions. This work is supported in part by a grant of National Science Council and a travelling fund of government of Taiwan.

Appendix

The Young tabulations of all physical states solutions of Eq. (3) up to level six, including two types of zero-norm state solutions of Eqs. (5) and (6), are listed in the following table:

massive level positive-norm states zero-norm states

M2= −2 • M2= 0 • (singlet) M2= 2 , • M2= 4 , , 2 × , • M2= 6 , , , • , , 2 × , 3 × , 2 × • M2= 8 , , , , , , , 2 × , 2 × , 4 × , 5 × , 3 × • M2= 10 , , , , , , ,2 × , , • , 2 × , , 3 × , 4 × , 4 × , 7 × , 8 × , 6 × •

Note that the Young tabulations of zero-norm states at level n are subset of the sum of all physical states at levels n-1 and n-2.

References 1) D. J. Gross, Phys. Rev. Lett.60 (1988), 1229.

D. J. Gross and P. Mende, Phys. Lett. B197 (1987), 129; Nucl. Phys. B 303 (1988), 407. 2) D. J. Gross and J. L. Manes, Nucl. Phys. B326 (1989), 73.

3) J. C. Lee, Phys. Rev. Lett.64 (1990), 1636.

4) J. M. F. Labastida and M. A. H. Vozmediano, Nucl. Phys. B312 (1989), 308.

(15)

5) I. L. Buchbinder, V. A. Krykhtin and V. D. Pershin, Phys. Lett. B348 (1995), 63. I. L. Buchbinder, E. S. Fradkin, S. L. Lyakhovich and V. D. Pershin, Phys. Lett. B304 (1993), 239.

6) H. C. Kao and J. C. Lee, hep-th/0212196; Phys. Rev. D67 (2003), 086003. 7) J. C. Lee, Phys. Lett. B241 (1990), 336.

J. C. Lee and B. Ovrut, Nucl. Phys. B336 (1990), 222.

8) J. C. Lee, Z. Phys. C63 (1994), 351; Prog. Theor. Phys. 91 (1994), 353.

9) C. T. Chan and J. C. Lee, hep-th/0312226; Phys. Lett. B611 (2005), 193; Nucl. Phys. B 690 (2004), 3.

C. T. Chan, P. M. Ho and J. C. Lee, Nucl. Phys. B708 (2005), 99.

10) For a review see I. R. Klebanov and A. Pasquinucci, hep-th/9210105, and references therein.

11) T. D. Chung and J. C. Lee, Phys. Lett. B350 (1995), 22; Z. Phys. C 75 (1997), 555. J. C. Lee, Eur. Phys. J. C1 (1998), 739.

12) J. L. Manes and M. A. H. Vozmediano, Nucl. Phys. B326 (1989), 271. 13) R. Sasaki and I. Yamanaka, Phys. Lett. B165 (1985), 283.

S. Weinberg, Phys. Lett. B156 (1985), 309.

14) M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vol. I (Cambridge Uni-versity Press).

15) A. M. Polyakov, Gauge Fields and Strings (Harwood ac. pub. 1987), see section 9.10. 16) J. C. Lee, Phys. Lett. B326 (1994), 79.

M. Evans and B. A. Ovrut, Phys. Rev. D39 (1989), 3016.

17) H. Kawai, D. C. Lewellen and S.-H. H. Tye, Nucl. Phys. B269(1986), 1.