Solving all 4-point correlation functions for bosonic
open string theory in the high-energy limit
Chuan-Tsung Chan
a, Pei-Ming Ho
b, Jen-Chi Lee
c,
Shunsuke Teraguchi
d, Yi Yang
caPhysics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan, ROC bDepartment of Physics, National Taiwan University, Taipei, Taiwan, ROC cDepartment of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwan, ROC
dPhysics Division, National Center for Theoretical Sciences, Taipei, Taiwan, ROC Received 25 April 2005; received in revised form 6 July 2005; accepted 15 July 2005
Available online 8 August 2005
Abstract
We study the implication of decoupling zero-norm states in the high-energy limit, for the 26-dimensional bosonic open string theory. Infinitely many linear relations among 4-point functions are derived algebraically, and their unique solution is found. Equivalent results are also obtained by taking the high-energy limit of Virasoro constraints, and as an independent check, we compute all 4-point functions of 3 tachyons and an arbitrary massive state by saddle-point approximation. 2005 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Motivation
The high-energy limit of string theory is an old subject[1–3]. Its motivation is to find
the hypothetical hidden symmetry. It is believed that the higher-spin gauge fields in string
E-mail addresses:[email protected](C.-T. Chan),[email protected](P.-M. Ho),
[email protected](J.-C. Lee),[email protected](S. Teraguchi),[email protected]
(Y. Yang).
0550-3213/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.07.018
theory receive their masses via a spontaneous symmetry breaking mechanism[2,4]. It is then natural to expect that the gauge fields become effectively massless and the symmetry is restored in the high-energy limit. The symmetry might also explain mysterious corre-spondences such as dualities and holography.
This subject has been attacked from various directions. One can take the high-energy
limit of the worldsheet theory and use it to define string theory in the high-energy limit[5].
However there is a lot of ambiguity in taking the high-energy limit of a worldsheet. It is not clear which choice of the high-energy worldsheet action is the one suitable for the purpose of studying symmetries.
Another approach is to study consistent higher spin gauge theories[6]. It turns out that
there is no consistent scale-free nontrivial interactions in the flat spacetime. Usually people study these theories in the AdS background. In general we need a characteristic length scale
to define consistent interactions. The string scale αcannot be used because by definition
α→ ∞ in the high-energy limit.
The third approach is to examine correlation functions in string theory and look for
meaningful patterns in the high-energy limit[1–3]. This is the approach that we will take
in this work. We consider the bosonic open string theory and compare correlation functions which are different from each other by a single vertex at the same mass level. We claim that their relative ratios are completely determined at the leading order for any mass level. This should be viewed as the strongest signal there ever was for a symmetry in the high-energy limit.
The main idea of our approach is that the decoupling of zero-norm states strongly
restricts the dynamics of string theory[7]. One can show[8,9]that off-shell gauge
transfor-mations of Witten’s string field theory, after imposing the no-ghost condition, are identical to the on-shell stringy gauge symmetries generated by two types of zero-norm states. In view of the dictating role of gauge symmetry in Witten’s SFT, we believe that the
exis-tence of the huge gauge symmetry represented by zero-norm states[10,11]fixes the theory
uniquely. In particular, we will show in this paper that by taking a self-consistent
high-energy limit[12–14]of arbitrary four-point functions, it will allow us to express them at
the leading order in terms of those of tachyons. We will also take the 2D string as an ex-ample to establish the connection between the high-energy limit of zero-norm states and
the hidden symmetry. In 2D string, we will show that the high-energy limit of w∞
alge-bra generated by 2D zero-norm states[15]can be identified with the w∞symmetry of 2D
string[16].
1.2. Review
We will focus on 4-point functions in this work, although our discussion can be gener-alized to higher point correlation functions. Due to Poincaré symmetry, a 4-point function is a function of merely two parameters. Viewing a 4-point function as the scattering am-plitude of a two-body scattering process, one can choose the two parameters to be E (one
half of the center of mass energy for the incoming particles, i.e., particles 1 and 2 inFig. 1,
and φ (the scattering angle between particles 1 and 3). For convenience we will take the
Fig. 1. Kinematic variables in the center of mass frame.
the momenta of particles 3 and 4 on the X1–X2plane. Readers can refer toAppendix A
for expressions of the kinematic variables in this frame. The high-energy limit under consideration is
(1)
αE2→ ∞, φ= fixed.
Based on the saddle-point approximation of Gross and Mende [1], Gross and Manes[3]
computed the high-energy limit of 4-point functions in the bosonic open string theory. To explain their result, let us first define our notations and conventions. For a particle of
mo-mentum k, we define an orthonormal basis of polarizations{eP, eL, eTi}. The momentum
polarization eP is proportional to k, the longitudinal polarization eLis the space-like unit
vector whose spatial component is proportional to that of k, and eTi are the space-like
unit-vectors transverse to the spatial momentum. As an example, for k pointing along the
X1-direction,
(2)
k=k0, k1, k2, . . . , k25= (E, p, 0, . . . , 0), p > 0,
the basis of polarization is
eP = 1 m p2+ m2, p, 0, 0, . . . , 0, eL= 1 m p, p2+ m2, 0, 0, . . . , 0, (3) eTi= (0, 0, . . . , 1, . . .),
where m is the mass of the particle. In general, eTi(for i= 3, . . . , 25) is just the unit vector
in the Xi-direction, and the definitions of eP, eLand eT2 will depend on the motion of the
particle. For eT2, which is parallel to the scattering plane, we denote it by eT (seeFig. 1
andAppendix A). The orientations of eT2 for each particle are fixed by the right-hand rule,
k × eT2= eT3, where k is the spatial momentum of 4-vector k. We will use the notation ∂nXA≡ eA· ∂nX for A= P, L, T , Ti.
Each vertex is a polynomial of{∂nXA} times the exponential factor exp(ik · X). Among
all possible choices of polarizations for the 4 vertices in a 4-point function, we now argue that only the polarizations L and T need to be considered. The polarization P can be
gauged away using zero-norm states[9]. To see why we can ignore all Ti’s except T , we
note that a prefactor ∂nXA can be contracted with the exponent ik· X of another vertex.
The contribution of this contraction to a scattering amplitude is proportional to kA∼ E.
If kA= 0 (i.e., if A = L or T ), this is much more important in the high-energy limit
than a contraction with another prefactor ∂mXB, which gives ηAB∼ E0. Therefore, if all
polarizations in all vertices are chosen to be either L or T , the resulting 4-point function will dominate over other choices of polarizations.
According to the zeroth-order saddle-point approximation [1], the leading order of a
4-point function is (4) V (k1)V (k2)V (k3)V (k4) ∼ 4 a=1 va T .
HereT is the correlation function of 4 tachyons, according to the following rules
(5) 1 (n− 1)!∂ nXT → iE sin φ (−λ)n, (6) 1 (n− 1)!∂ nXL→ −iλn−1− 1 λ− 1 E2sin2φ 2M(−λ)n, where1 (7) λ= sin2φ 2.
A peculiar feature of(6)is that it vanishes for n= 1
(8)
∂XL→ 0.
Does this mean that all 4-point functions involving ∂XLare subleading? The answer
de-pends on how we define the notion of “subleading”. What is the reference 4-point function to be compared with?
Note that it is impossible to have a universal notion of “leading order” for all 4-point functions, since 4-point functions with vertices at higher mass levels typically dominate over those with vertices at lower mass levels in the high-energy limit. Having a well defined limit for all correlation functions is also contradictory to the fact that there is no consistent interacting higher-spin gauge field theory in the flat spacetime.
We propose that the appropriate question to ask is: How will a 4-point function change if we replace a vertex by another vertex at the same mass level? For example, we can replace
a factor ∂XLby ∂XT in one of the vertices. Naively, the former with ∂XLshould dominate
over the latter in the high-energy limit since the components of eLscale as E1and those
of eT scale as E0. However, Eq.(8)tells us that ∂XLdoes not live up to this expectation.
Nevertheless, it does not imply that ∂XLis subleading compared with ∂XT. It depends on the rest of the terms in the prefactor.
The purpose of our work is to find linear relations among 4-point functions in the
high-energy limit as a signature of the hidden symmetry proposed in[1]. This is not manifest in
the saddle-point result(5),(6), which holds for the naive leading order (0th order
saddle-point approximation). One of the main reasons is that they ignored vertices involving ∂XL.
Another main reason is that they did not try to determine the precise power factor of E
[14]in a 4-point function (a higher order correction to the saddle-point approximation is
needed), which is crucial for examining linear relations among 4-point functions.
The proper notion of “leading order” in the high-energy limit such that linear
rela-tions among 4-point funcrela-tions can be established was first discovered in[12,13]. A purely
algebraic approach utilizing zero-norm states was developed there to derive the linear rela-tions explicitly for the first few mass levels. This approach was further explored to obtain stronger results, and its connection/difference from the saddle-point computation was
ex-plained in detail in[14].
The central idea behind the algebraic approach used in[12–14]was the decoupling of
zero-norm states (i.e., the requirement of gauge invariance). A crucial step in the derivation
is to replace the polarization eP by eL in the zero-norm states. It is assumed that, while
zero-norm states decouple at all energies, the replacement leads to states that are decoupled at high energies.
The new achievements of this paper include:
(1) We studied the implication of the decoupling of zero-norm states in the high-energy limit. The 4-point functions at the leading order are specified for all mass levels. (2) Infinite linear relations among 4-point functions are obtained for all mass levels. The
ratio of any two 4-point functions at the leading order is uniquely determined. (3) We developed a “dual” description of the algebraic approach using the spurious states,
which is to take high-energy limit of the Virasoro constraints.
(4) Using saddle-point approximation, we explicitly computed all 4-point functions for 3 tachyons and one arbitrary massive state, and verified the linear relations obtained via algebraic methods.
2. Main results
For brevity, we will refer to all 4-point functions different from each other by a single vertex at the same mass level as a “family”. When we compare members of a family, we only need to specify the vertex which is changed.
A 4-point function will be said to be at the leading order if it is not subleading to any of its siblings. We will ignore those that are not at the leading order. Our aim is to find the numerical ratios of all 4-point functions in the same family at the leading order. Apparently, there are more 4-point functions at the leading order at higher mass levels. Our goal may seem insurmountable at first sight.
Saving the derivation for later, we give our main results here. A 4-point function is at the leading order if and only if the vertex V under comparison is a linear combination of
vertices of the form (9) V(n,m,q)(k)=∂XTn−m−2q∂XLm∂2XLqeik·X, where (10) n m + 2q, m, q 0.
The corresponding states are of the form
(11)
αT−1n−m−2qα−1L mα−2L q|0, k.
The mass squared is 2(n− 1). All other states involving α−2T , α−3A , . . . are subleading.
Using the notation2
(12)
T(n,m,q)=V
1V(n,m,q)(k)V3V4
,
all linear relations among different choices of V(m,n,q) (obtained from the decoupling of
spurious states at high energies) can be solved by the simple expression
(13) lim E→∞ T(n,2m,q) T(n,0,0) = −1ˆm 2m+q 1 2 m+q (2m− 1)!!, (14) lim E→∞ T(n,2m+1,q) T(n,0,0) = 0,
where ˆm =√2(n− 1). This formula tells us how to trade ∂XL and ∂2XL for ∂XT, so
that all 4-point functions can be related to the one involving only ∂XT in V2. The formula
above applies equally well to all vertices.
Since we know the value of a representative 4-point function[12,14]
(15) TT1··T2··T3··T4·· n1n2n3n4 = (−1) n2+n4 2E3sin φ CM Σ niT (Σn i), where T (n) =√π (−1)n−12−nE−1−2n sinφCM 2 −3 cosφCM 2 5−2n (16) × exp −s ln s+ t ln t − (s + t) ln(s + t) 2
is the high-energy limit of (−
s
2−1)(−
t
2−1)
(u2+2) with s+ t + u = 2Σni − 8, and we have
calculated it up to the next leading order in E. In Eq.(15), niis the number of Ti of the ith
vertex operators and Ti is the transverse direction of the ith particle. We can immediately
write down the explicit expression of a 4-point function if all vertices are nontrivial at the leading order.
2 More rigorously, V
2needs to be a physical state in order for the correlation function to be well-defined. We should keep in mind that our results should be applied to suitable linear combinations of(9), possibly together with subleading states, to satisfy Virasoro constraints.
3. Linear relations among 4-point functions
Before we go on, we recall some terminology used in the old covariant quantization.
A state|ψ in the Hilbert space is physical if it satisfies the Virasoro constraints
(17)
Ln− δn0
|ψ = 0, n 0. Since L†n= L−n, states of the form
(18)
L−n|χ
are orthogonal to all physical states, and they are called spurious states. Zero-norm states are spurious states that are also physical. They correspond to gauge symmetries. In the old covariant first quantization spectrum of open bosonic string theory, the solutions of physical state conditions include positive-norm propagating states and two types of
zero-norm states. The latter are[7]
(19) Type I: L−1|x, where L1|x = L2|x = 0, L0|x = 0, (20) Type II: L−2+3 2L 2 −1 | ˜x, where L1| ˜x = L2| ˜x = 0, (L0+ 1)| ˜x = 0.
In this section we derive the linear relations among all amplitudes in the same family by taking the high-energy limit of zero-norm states (HZNS). Solutions of HZNS for some
low lying mass level are presented in Appendix B. The first step in the derivation is to
identify the class of states that are relevant, i.e., those at the leading order. As we explained
in Section1.2, we only need to consider the polarizations eT and eL.
To get a rough idea about how each vertex operator scales with E in the high-energy
limit, we associate a naive dimension to each prefactor ∂mXAaccording to the following
rule
(21)
∂mXT → 1, ∂mXL→ 2.
The reason is the following. Each factor of ∂mXµhas the possibility of contracting with
the exponent iki· X of another vertex operator so that it scales like E in the high-energy
limit. Furthermore, components of the polarization vectors eT and eLscale with E like E0
and E1, respectively.
When we compare vertex operators at the same mass level, the sum of all the integers m
in ∂mXA is fixed. Roughly speaking, it is advantageous to have many ∂XA than having
fewer number of ∂mXA with m > 1. For example, at the first massive level, the vertex
operator ∂XT∂XTeik·Xhas a larger naive dimension than ∂2XTeik·X.
The counting of the naive dimension does not take into consideration the possibility that the coefficient of the leading order term happens to vanish by cancellation. The true dimension of a vertex operator can be lower than its naive dimension, although the reverse never happens.
Through experiences accumulated from explicit computations[12–14], we find that the
highest spin vertex
(22)
is always at the leading order in its family. Since the naive dimension of this state equals its true dimension, any state with a lower naive dimension than this vertex operator can be ignored. This implies that we can immediately throw away a lot of vertex operators at each mass level, but there are still many left. The problem is that, although there are
disadvantages to have ∂mXT with m 2 or ∂mXL with m 3 compared with having
(∂XT)m, it may be possible that having extra factors of ∂XL, which has a higher naive
dimension than ∂XT, can compensate the disadvantage of these factors. However, explicit
computations at the first few massive levels showed that this never happens.
We will now argue why this is generically true, and show in this subsection that the only states that will survive the high-energy limit at level n are of the form
(23) |n, 2m, q ≡α−1T n−2m−2qαL−12mαL−2q|0; k.
Our argument is essentially based on the decoupling of zero-norm states in the high-energy limit. However, note that when we take the high-high-energy limit, that is, when we
replace eP by eL and ignore subleading terms in the 1/E2 expansion,3 the zero-norm
states become positive norm states, although we will still call them “high-energy zero-norm states”, or HZNS for short. (This is why it is possible to derive relations among
positive-norm physical states by taking the high-energy limit of Ward identities. SeeAppendix B,
for examples.) As such, it is not essential to maintain the zero-norm condition, and we can simply take the high-energy limit of spurious states. It can be shown that, as far as the final results are concerned, the decoupling of those spurious states we are going to use in this paper are equivalent to the decoupling of high-energy zero-norm states (HZNS).
Thanks to the Virasoro algebra, we only need two Virasoro operators
(24) L−1=1 2 n∈Z α−1+n· α−n= ˆmα−1P + α−2· α1+ · · · , (25) L−2=1 2 n∈Z α−2+n· α−n=1 2α−1· α−1+ ˆmα P −2+ α−3· α1+ · · ·
to generate all spurious states. Here ˆm is the mass operator, i.e., ˆm2= −k2when acting on
the state|0, k.
3.1. Irrelevance of other states
To prove that only states of the form(23)are at the leading order, we shall prove that
(i) any state which has an odd number of α−1L is irrelevant (i.e., subleading in the
high-energy limit), and (ii) any state involving a creation operator whose naive dimension is less than its mode index n, i.e., states belonging to
(26)
α−nL , n > 2; αT−m, m > 1
is also irrelevant. We proceed by mathematical induction.
3 Strictly speaking, we need to justify the replacement eP→ eL. This is not totally trivial, and will be treated
First we prove that any state which has a single factor of α−1L is irrelevant, and that any
state with two αL−1’s is irrelevant if it contains an operator of naive dimension less than its
index.
Consider the HZNS L−1χ where χ is any state without any α−1L , and it is at level
(n− 1). Note that, except α−1L , the naive dimension of an operator is always less than or
equal to its index (we exclude α−1P as mentioned above). This means that the naive
di-mension of χ is less than or equal to (n− 1). Since we know that at level n, the state
Eq.(23)has true dimension n, when computing L−1χ in the high-energy limit, we can
ig-nore everything with naive dimension less than n. This means that we need L−1to increase
the naive dimension of χ by no less than 1. In the high-energy limit of L−1
(27)
L−1→ ˆmαL−1+ α−2L αL1 + α−2T αT1 + · · · ,
only the first term will increase the naive dimension of χ by 1. All the rest do not change the naive dimension. This means that, to the leading order,
(28)
L−1χ∼ ˆmα−1L χ .
This is a state with a single factor of α−1L and it is a HZNS, so it should be decoupled in
the high-energy limit.
Now consider an arbitrary state χ at level (n− 1) which has a single factor of α−1L . If
χ involves any operator whose naive dimension is less than its index, the naive dimension
of χ is at most (n− 1). In the high-energy limit
(29)
L−1χ→ ˆmα−1L χ+ αL−2α1Lχ+ · · · ,
except the first two terms, all other terms are irrelevant because they contain a single factor
of α−1L . As the second term has a naive dimension (n− 1) and can be ignored, we conclude
that α−1L χ is irrelevant.
The next step in mathematical induction is to show that if (a) states with (2m− 1)
factors of αL−1are irrelevant, and (b) states with 2m factors of α−1L are still irrelevant if it
also contains any of the operators in(26), then we can prove that both statements are also
valid for m→ m + 1.
Suppose χ is an arbitrary state at level (n− 1) which has 2m factors of αL−1’s. The
high-energy limit of L−1χ is given by(29). The second term has (2m− 1) factors of α−1L
and is irrelevant. The rest of the terms, except the first, are irrelevant because they contains
at least one operator from the set(26). Hence the first term is a HZNS and is irrelevant. We
have proved our first claim for (m+ 1), i.e., a state with (2m + 1) factors of α−1L decouple
at high energies.
Similarly, consider the case when χ is at level (n− 1) and has (2m − 1) factors of α−1L .
Furthermore we assume that it involves operators from the set (26). Then the first term
in(29)is what we want to prove to be irrelevant. The second term is irrelevant because
we have just proved that a state with (2m+ 1) factors of α−1L is irrelevant. The rest of the
terms are irrelevant because they have (2m− 1) α−1L ’s. Thus we conclude that both claims
3.2. Linear relations
According to the previous subsection, only states of the form(23)are relevant in the
high-energy limit. The mass of the state is√2(n− 1). The 4-point function associated with
|n, m, q will be denoted T(n,m,q). The aim of this subsection is to find the ratio between a
genericT(n,m,q)and the reference 4-point function, which is taken to beT(n,0,0).
Consider the HZNS
(30)
L−1|n − 1, 2m − 1, q ˆm|n, 2m, q + (2m − 1)|n, 2m − 2, q + 1,
where many terms are omitted because they are not of the form(23). This implies that
(31)
T(n,2m,q)= −2m− 1
ˆm T(n,2m−2,q+1). Using this relation repeatedly, we get
(32)
T(n,2m,q)=(2m− 1)!!
(− ˆm)m T
(n,0,m+q),
where the double factorial is defined by (2m− 1)!! =(2m)2mm!!. Next, consider another class of HZNS
(33)
L−2|n − 2, 0, q 1
2|n, 0, q + ˆm|n, 0, q + 1.
Again, irrelevant terms are omitted here. From this we deduce that
(34) T(n,0,q+1)= − 1 2ˆmT (n,0,q), which leads to (35) T(n,0,q)= 1 (−2 ˆm)qT (n,0,0).
Our main result(13)is an immediate result of combining(32)and(35).
4. Linear relations from Virasoro constraints
In this section we will establish a “dual description” of our approach explained above. The notion dual to the decoupling of high-energy zero-norm states is Virasoro constraints. Let us briefly explain how to proceed. First write down a state at a given mass level as
linear combination of states of the form Eq. (11)with undetermined coefficients, which
are interpreted as the Fourier components of spacetime fields. Requiring that the Virasoro
generators L1and L2annihilate the state implies several linear relations on the coefficients.
The linear relations can then be solved to obtain ratios among all fields.
To compare the results of the two dual descriptions, we note that the correlation func-tions can be interpreted as source terms for the particle corresponding to a chosen vertex. Thus the ratios among sources should be the same as the ratios among the fields, since all fields of the same mass have the same propagator. However, some care is needed for the normalization of the field variables. One should use BPZ conjugates to determine the norm of a state and normalize the fields accordingly.
4.1. Examples
To illustrate how Virasoro constraints can be used to derive linear relations among scat-tering amplitudes at high energies, we give some explicit examples in this subsection. We will calculate the proportionality constants among high-energy scattering amplitudes of
different string states up to mass levels m2= 8. The results are of course consistent with
those of previous work[12,13]using high-energy zero-norm states.
4.1.1. m2= 4
The most general form of physical states at mass level m2= 4 are given by
(36)
µνλα−1µ αν−1αλ−1+ (µν)αµ−1α−2ν + [µν]α−1µ α−2ν + µαµ−3
|0, k. The Virasoro constraints are
(37) (µν)+ 3 2k λ µνλ= 0, (38) −kν [µν]+ 3 µ− 3 2k νkλ µνλ= 0, (39) 2kν [µν]+ 3 µ− 3(kνkλ− ηνλ) µνλ= 0.
By replacing P by L, and ignoring irrelevant states (we have justified this in Section3for
the high-energy limit), one easily gets
(40)
T T T: (LLT ): (LT ): [LT ]= 8 : 1 : 3 : −3.
After including the normalization factor of the field variables4and the appropriate
symme-try factors, one ends up with
TT T T:T(LLT ):T(LT ):T[LT ]
(41) = 6 T T T: 6 (LLT ):−2 (LT ):−2 [LT ]= 8 : 1 : −1 : 1.
Here the definitions ofTT T T,T(LLT ),T(LT ),T[LT ]and similar amplitudes hereafter can be
found in[12,13]and the result obtained is consistent with the previous zero-norm state
calculation in[12]or Eq.(13).
4.1.2. m2= 6
The most general form of physical states at mass level m2= 6 are given by
µνλσαµ−1α−1ν α−1λ α−1σ + (µνλ)α−1µ α−1ν αλ−2+ µν,λαµ−1αν−1α−2λ (42) + (1) (µν)α µ −1αν−3+ [µν](1) αµ−1α−3ν + (µν)(2) α µ −2α−2ν + µαµ−4 |0, k,
where µν,λrepresents the mixed symmetric spin three states, that is, one first symmetrizes
µν and then anti-symmetrizes µλ. The Virasoro constraints are calculated to be
(43) 2kσ (µνλσ )+ (µνλ)= 0,
(44) 2kλ (µνλ)+ kλ( λµ,ν+ µλ,ν)+ 3 (µν)(1) + [µν](1) + 4 (2)(µν)= 0, (45) kµ (µν)(1) + kµ [µν](1) + 4 ν= 0, (46) 6ηλσ (µνλσ )+ 2kλ (µνλ)+ 1 2k λ( µν,λ+ νµ,λ)+ 3 (µν)(1) = 0, (47) ηµν (µνλ)+ ηµν (µν,λ)+ 4kµε(2)(µν)+ 4 λ= 0.
In the high-energy limit, similar calculation as above gives
T(T T T T ):T(T T LL):T(LLLL):TT T ,L:T(T T L):T(LLL):T(LL) = 4! (T T T T ): 4! (T T LL): 4! (LLLL):−4 T T ,L:−4 (T T L):−4 (LLL): 8 (LL)(2) (48) = 16 :4 3: 1 3:− 2√6 3 :− 4√6 9 :− √ 6 9 : 2 3,
which is consistent with the previous zero-norm state calculation in[13]or Eq.(13).
4.1.3. m2= 8
The most general form of physical states at mass level m2= 8 are given by (for
sim-plicity, we neglect terms containing α−nµ with n 3)
µνλσραµ−1α−1ν α−1λ α−1σ αρ−1+ (µνλσ )α−1µ α−1ν α−1λ α−2ρ + (µνλ)α−1µ αν−2αλ−2 (49) + µνλ,σα−1µ α−1ν α−1λ αρ−2+ µ,νλα−1µ α−2ν αλ−2 |0, k,
where µνλ,σ represents the mixed symmetric spin four states, that is, first symmetrizes
µνλ and then anti-symmetrizes µσ . Similar definition for the mixed symmetric spin three
states µ,νλ. The Virasoro constraints are calculated to be
(50) 5kσ (µνλσρ)+ 2 (µνλσ )= 0, 3kλ (µνλσ )+ 1 2k λ ( µνλ,σ+ λµν,σ+ µλν,σ)+ (µ ↔ ν) (51) + 4 (µνσ )+ µ,νσ+ ν,µσ = 0, (52) kµ (µνλ)+ 1 2k µ( µ,νλ+ µ,λν)= 0, (53) 5ηρσ (µνλσρ)+ kσ (µνλσ )+ 1 3k σ( µνλ,σ + νλµ,σ+ λµν,σ)= 0, 3ηνλ (µνλσ )+ ηνλ( µνλ,σ+ λµν,σ+ νλµ,σ)+ 4kλε(µσ λ) (54) + 2kλ(ε µ,σ λ+ εµ,λσ)= 0.
In the high-energy limit, similar calculation as above gives
T(T T T T T ): T(T T T L): T(T T T LL): T(T LLL): T(T LLLL): T(T LL): TT ,LL: TT LL,L: TT T T,L = 5! (T T T T T ): 3! × 2 (T T T L): 5! (T T T LL): 3! × 2 (T LLL): 5! (T LLLL) : 8 (T LL): 8 T ,LL: 3! × 2 T LL,L: 3! × 2 T T T ,L (55) = 32 :√2 : 2 :3 √ 2 16 : 3 8: 1 3: 2 3: √ 2 16 : 3 √ 2,
which can be checked to be remarkably consistent with the results of Eq.(13)after Young tableaux decomposition.
4.2. General mass levels
In this section we calculate the ratios of string scattering amplitudes in the high-energy limit for general mass levels by imposing Virasoro constraints. The final result will, of course, be exactly the same as what we obtained by requiring the decoupling of high-energy zero-norm states. In the presentation here we use the notation of Young’s tableaux.
We consider the general mass level m2= 2(n−1). The most general state can be written
as (56) |n = mj k j=1 1 jmjm j! µj1 · · · µjmj α µj1···µjmj −j |0, k, where we defined the abbreviation
(57) αµ j 1···µ j mj −j ≡ α µj1 −j· · · α µjmj −j ,
with mj is the number of the operator α−jµ . The summation runs over all possible
combi-nations of mj’s with the constraints
(58)
k
j=1
j mj= n and 0 mj n,
so that the total mass is n. It is obvious that k is less or equal to n. Since the upper indices {µj 1· · · µ j mj} in α µj1 −j· · · α µjmj
−j are symmetric, we used the Young tableaux notation to denote
the coefficients in Eq.(56). The direct product⊗ acts on the Young tableaux in the standard
way, for example,
(59) 1 2 ⊗ 3 = 1 2 3 ⊕ 1 2
3 .
Finally, 1/(jmjm
j!) are the normalization factors. To be clear, for example n = 4, the state
can be written as |4 = 1 4! µ 1 1 µ 1 2 µ 1 3 µ 1 4 α µ1 1 −1α µ1 2 −1α µ13 −1α µ1 4 −1+ 1 2· 2! µ 1 1 µ 1 2 ⊗ µ 2 1 α µ1 1 −1α µ1 2 −1α µ2 1 −2 (60) +1 3 µ 1 1 ⊗ µ 3 1 α µ1 1 −1α µ3 1 −3+ 1 22· 2! µ 2 1 µ22 α µ2 1 −2α µ2 2 −2+ 1 4 µ 4 1 α µ4 1 −4 |0, k.
Next, we will apply the Virasoro constraints to the state Eq.(56). The only Virasoro
con-straints which need to be considered are
(61)
with Lmthe standard Virasoro operator (62) Lm= 1 2 ∞ n=−∞ αm+n· α−n.
After taking care the symmetries of the Young tableaux, the Virasoro constraints become
L1|n = mj kµ11 k j=1 µj1 · · · µjmj + m1 i=2 µ12 · · · ˆµ1i · · · µ1m 1 ⊗ µ 1 i µ21 · · · µ 2 m2 k j=1,2 µj1 · · · µjmj + k l=3 (l− 1) µ1 2 · · · µ1m1 ⊗ ml−1 i=1 µl1−1 · · · ˆµli−1 · · · µl−1 ml−1 ⊗ µl−1 i µl1 · · · µlml k j=1,l,l−1 µj1 · · · µjmj 1 (m1− 1)! αµ 1 2···µ1m1 −1 (63a) × k j=1 1 jmjm j! αµ j 1···µ j mj −j |0, k = 0, and L2|n = mj 1 2η µ1 1µ12 k j=1 µj1 · · · µjmj + µ1 3 · · · µ 1 m1 ⊗ µ 2 1 · · · µ 2 m2+1 k µ2 m2+1 k j=1,2 µj1 · · · µjmj + m1 i=3 µ13 · · · ˆµ1i · · · µ1m 1 ⊗ µ 1 i µ31 · · · µ3m3 k j=1,3 µj1 · · · µjmj + k l=4 (l− 2) µ13 · · · µ1m 1 ⊗ ml−2 i=1 µl1−2 · · · ˆµli−2 · · · µlm−2 l ⊗ k µli−2 µl1 · · · µlm l j=1,l,l−2 µj1 · · · µjmj 1 (m1− 2)! αµ 1 3···µ1m1 −1 (63b) × k j=1 αµ j 1···µ j mj −j |0, k = 0.
A hat on an index means that the index is skipped there (and it should appear somewhere else). In the above derivation we have used the identity for the Young tableaux
1 · · · p =1 p 1+ σ(21)+ σ(321)+ · · · + σ(p···1) 2 · · · p ⊗ 1 (64) =1 p p i=1 σ(i1) 2 · · · p ⊗ 1 ,
where σ(i···j)are permutation operators.
States which satisfy the Virasoro constraints are physical states. What we are going to show in the following is that, in the high-energy limit, the Virasoro constraints turn out to be strong enough to give the linear relationship among the physical states. To take the
high-energy limit in the above equations (63a)and(63b), we replace the indices (µi, νi)
by L or T , and
(65)
kµi→ ˆmeL, ηµ1µ2→ eTeT,
where ˆm is the mass operator. The Virasoro constraints at high energy are derived in
Ap-pendix C.1 as Eqs. (C.4a) and (C.4b). To solve the constraints, we need the following
lemma to further simplify them.
Lemma. (66) T · · · T L · · · L l1 ⊗ T ··· T m2−l2 L · · · L ⊗ · · · {mj,j3} ≡ 0,
except for (i) l2= m2, mj= 0 for j 3 and (ii) l1= 2m.
This lemma is equivalent to part of the results of Section3, but will also be proved in
AppendixC.2by applying Virasoro constraints. Finally, the Virasoro constraints at
high-energy reduce to ˆm T ··· T n−2q−2−2m L · · · L 2m+2 ⊗ L ··· L q (67a) + (2m + 1) T ··· T n−2q−2−2m L · · · L 2m ⊗ L ··· L q+1 = 0, ˆm T ··· T n−2q−2−2m L · · · L 2m ⊗ L ··· L q+1 (67b) +1 2 T · · · T n−2q−2m L · · · L 2m ⊗ L ··· L q = 0,
By mathematical recursion, Eq.(67a)leads to (68a) T · · · T n−2q−2m L · · · L 2m ⊗ L ··· L q =(2m− 1)!!(− ˆm)q (2m+ 2q − 1)!! T · · · T n−2q−2m L · · · L 2m+2q ,
and similarly, Eq.(67b)leads to
(68b) T · · · T n−2q−2m L · · · L 2m ⊗ L ··· L q = − 1 2ˆm q T · · · T n−2m L · · · L 2m .
Combining Eqs.(68a) and (68b), we get
(69) T · · · T n−2q−2m L · · · L 2m ⊗ L ··· L q = − 1 2ˆm q (2k− 1)!! 4m(n− 1)m T · · · T n .
This is equivalent to Eq.(13).
To get the ratio for the specific physical states, we make the Young tableaux decompo-sition T · · · T n−2q−2m L · · · L 2m ⊗ L ··· L q (70) = min{n−2q−2m,q} l=0 T · · · T L · · · L L · · · L · l!CqlCnl−2q−2m,
where Cql =l!(q−l)!q! and we have (n− 2q − 2m) T ’s and (2m + q − l) L’s in the first
column, (l) L’s in the second column in the second line of the above equation. Therefore, we obtain T · · · T L · · · L L · · · L · l!CqlCnl−2q−2m (71) = l!C l qCnl−2q−2m min{n−2q−2m,q} l=0 l!ClqCnl−2q−2m − 1 2ˆm q (2m− 1)!! 4m(n− 1)m T · · · T n ,
which is consistent with the ratios Eqs.(41), (48) and (55)for m2= 4, 6, 8, respectively.
5. Saddle-point approximation for stringy amplitudes
In previous sections, we have identified the leading high-energy amplitudes and derived the ratios among high-energy amplitudes for members of a family at given mass levels, based on decoupling principle. While deductive arguments help to clarify the underlying assumptions and solidify the validity of decoupling principle, it is instructive to compare
it with a different approach, such as the saddle-point approximation[14]. Therefore, we shall perform direct calculations to check the results obtained above and make comparisons between these two approaches.
In this section, we give a direct verification of the ratios among leading high-energy amplitudes based on the saddle-point method. The four-point amplitudes to be calculated consist of one massive tensor and three tachyons. Since we have shown that in the high-energy limit the only relevant states are those corresponding to
(72)
αT−1n−2m−2qα−1L 2mα−2L q|0, k, −k2= 2(n − 1),
we only need to calculate the following four-point amplitude
(73) T(n,2m,q)≡ 4 i=1 dxi V1V2(n,2m,q)V3V4 , where (74) V2(n,2m,q)≡∂XTn−2m−2q∂XP2m∂2XPqeik2X2, (75) Vi≡ eikiXi, i= 1, 3, 4.
Notice that here for leading high-energy amplitudes we replace the polarization L by P . Using either path-integral or operator formalism, after SL(2, R) gauge fixing, we obtain
the s− t channel contribution to the stringy amplitude at tree level
T(n,2m,q)⇒ 1 0 dx x(1,2)(1− x)(2,3) eT · k1 x − eT · k3 1− x n−2m−2q (76) × eP · k 1 x − eP · k 3 1− x 2m −eP · k1 x2 − eP· k 3 (1− x)2 q ,
where we have simplified the inner products among momenta by defining (1, 2)≡ k1· k2.
In order to apply the saddle-point method, we need to rewrite the amplitude above into the “canonical form”. That is,
(77) T(n,2m,q)(K)= 1 0 dx u(x)e−Kf (x), where (78) K≡ −(1, 2) → s 2 → 2E 2, (79) τ ≡ −(2, 3) (1, 2)→ − t s → sin 2φ 2, (80) f (x)≡ ln x − τ ln(1 − x), (81) u(x)≡ (1, 2) ˆm 2m+q (1− x)−n+2m+2q(f)2m(f)q−eT· k3 n−2m−2q .
The saddle-point for the integration of moduli, x= x0, is defined by (82) f(x0)= 0, and we have (83) x0= 1 1− τ, 1− x0= − τ 1− τ, f (x 0)= (1 − τ)3τ−1.
From the definition of u(x), it is easy to see that
(84) u(x0)= u(x0)= · · · = u(2m−1)(x0)= 0, and (85) u(2m)(x0)= (1, 2) ˆm 2m+q (1− x0)−n+2m+2q(2m)!(f0)2m+q −eT · k 3 n−2m−2q .
With these inputs, one can easily evaluate the Gaussian integral associated with the
four-point amplitudes, Eq.(77),
1 0 dx u(x)e−Kf (x) = ! 2π Kf0e −Kf0 u(2m)0 2mm! (f 0)mKm + O 1 Km+1 (86) = ! 2π Kf0e −Kf0 (−1)n−q2 n−2m−q(2m)! m! ˆm2m+q τ −n 2(1− τ) 3n 2En+ OEn−2 .
This result shows explicitly that with one tensor and three tachyons, the energy and angle dependence for the high-energy four-point amplitudes only depend on the level n, and we can solve for the ratios among high-energy amplitudes within the same family,
lim E→∞ T(n,2m,q) T(n,0,0) = (−1)q(2m)! m!(2 ˆm)2m+q (87) = −2mˆm− 1 · · · −3ˆm −1ˆm − 1 2ˆm m+q ,
which is consistent with Eq.(13).
We conclude this section with three remarks. Firstly, from the saddle-point approach,
it is easy to see why the product of αP−1oscillators induce energy suppression. Their
con-tribution to the stringy amplitude is proportional to powers of f(x0), which is zero in
the leading order calculation. Secondly, one can also understand why only even
num-bers of α−1P oscillators will survive for high-energy amplitudes based on the structure of
Gaussian integral in Eq. (77). While for a vertex operator containing (2m+ 1) α−1P ’s,
we have u(x0)= u(x0)= · · · = u(2m)(x0)= 0, and the leading contribution comes from
u(2m+1)(x0)(x− x0)2m+1, this gives zero since the odd-power moments of Gaussian
to the scattering amplitudes, the integration range for the x variable seems to devoid of a direct application of saddle-point method. Presumably, we can apply the saddle-point ap-proximation to the full amplitudes whose integration range extends over whole real line. However, it is curious to see why s–t channel alone has the same functional form as the full amplitude in the high-energy limit. To see this, one can check that the leading
con-tribution Eq. (86)is actually “form-invariant” under any monotonous change of variable
x= ξ(y), v(y) ≡ ξ(y)u(ξ(y)), g(y)≡ f (ξ(y)). If the analytic structure of the integrand
is no concern, we can justify the use of saddle-point approximation even for the case of s–t channel.
6. Remarks
6.1. Virasoro generators at high energies
In retrospect, the main result(13)can be very easily obtained from the Virasoro
gener-ators if we accept that we only need to consider states of the form
(88)
αT−1n−m−2qα−1L mα−2L q|0, k.
On the space of these states, the Virasoro generators are effectively
(89) ˜L−1= ˆmαL −1+ α−2L α1L, (90) ˜L−2= ˆmαL −2+ 1 2α T −1a−1T , (91) ˜L−n= 0 for n 3,
where we have also replaced eP by eL.
Using these deformed Virasoro generators, we create high-energy approximations of spurious states. On the back of an envelope, one can check that the decoupling of the
spu-rious states created by ˜L−1and ˜L−2implies(32)and(35), respectively (with P replaced
by L). In short, ˜L−1tells us how to trade α−1L αL−1for α−2L , and ˜L−2tells us how to trade
α−2L for α−1T αT−1.
Similarly, the Hermitian conjugates of ˜L1and ˜L2can be used to derive the same result
by demanding that they annihilate states in the high-energy limit.
6.2. 2-dimensional string
Although we have shown that there exist infinitely many linear relations among 4-point functions which uniquely fix their ratios in the high-energy limit, it is not totally clear that there is a hidden symmetry responsible for it. However, we would like to claim that these linear relations are indeed the manifestation of the long-sought hidden symmetry of string theory, and that we are on the right track of understanding the symmetry. To persuade the readers, we test our claim on a toy model of string theory—the 2-dimensional string theory. While the hidden symmetry of the 26-dimensional bosonic string theory is still at large, the symmetry of the 2-dimensional string theory is much better understood. It is known to
be associated with the discrete states
(92)
ψJ M± ∼ ∆J, M,−i√2Xexp √2iMX(0)+ (±J − 1)φ(0).
Half of them ψJ M+ generate the w∞algebra[16]
(93) dz 2π iψ + J1M1ψ + J2M2∼ (J2M1− J1M2)ψ + (J1+J2−1)(M1+M2).
Let us now check whether the w∞ symmetry is generated by the high-energy limit of
zero-norm states. In[15], explicit expression for a class of zero-norm states was given
G+J M∼ (J − M)!∆J, M,−i√2Xexp √2(iMX+ (J − 1)φ + (−1)2J J−M j=1 (J− M − 1)! dz 2π iD J, M,−i√2X(z), j (94) × exp √2i(M+ 1)X(z) + (J − 1)φ(z) − X(0).
The notation needs some explanation. Here ∆(J, M,−i√2 X) is defined by
(95) ∆J, M,−i√2X= "" "" "" " S2J−1 S2J−2 · · · SJ+M S2J−2 S2J−3 · · · SJ+M−1 · · · · SJ+M SJ+M−1 · · · S2M+1 "" "" "" " , where (96) Sk= Sk −i√ 2 k! ∂ k X(0) , and Sk= 0 if k < 0,
and Sk({ai})’s denote the Schur polynomial defined by
(97) exp ∞ k=1 akxk =∞ k=0 Sk {ai} xk.
D(J, M, −i√2X(z), j ) is defined by a similar expression as Eq. (95), but with the j th
row replaced by{(−z)j− 1 − 2J, (−z)j− 2J, . . . , (−z)j−J −M−2}. It was shown[15]that
zero-norm states in Eq.(94)generate a w∞algebra.
In the high-energy limit, the factors ∂kXA are generically proportional to a
lin-ear combination of the momenta of other vertices, so it scales with energy E. Thus
D(J, M, −i√2X, j ) is subleading to ∆(J, M,−i√2X). Ignoring the second term in
Eq. (94)for this reason, we see that these zero-norm states indeed approach to the
dis-crete states ψJ M+ above! Thus, the w∞algebra generated by Eq.(94)is identified to w∞
symmetry in Eq.(93). This result strongly suggests that the linear relations among
corre-lation functions obtained from HZNS are indeed related to the hidden symmetry also for the 26-dimensional strings. Although we still do not know what is the symmetry group, or how it acts on states, this work sheds new light on the road to finding the answers.
Acknowledgement
The authors thank Jiunn-Wei Chen, Tohru Eguchi, Koji Hashimoto, Hiroyuki Hata, Takeo Inami, Yeong-Chuan Kao, Yoichi Kazama, Yutaka Matsuo, Tamiaki Yoneya for helpful discussions. This work is supported in part by the National Science Council and the National Center for Theoretical Sciences, Taiwan, ROC.
Appendix A. Kinematic variables and notations
For the readers’ convenience, we list the expressions of the kinematic variables involved
in the evaluation of a 4-point function in this appendix. InFig. 1, we take the scattering
plane to be the X1–X2plane. The momenta of the particles are
(A.1) k1= p2+ m2 1,−p, 0 , (A.2) k2= p2+ m2 2, p, 0 , (A.3) k3= −q2+ m2 3,−q cos φ, −q sin φ , (A.4) k4= −q2+ m2 4, q cos φ, q sin φ .
They satisfy k2i = −m2i. In the high-energy limit, the Mandelstam variables are
(A.5) s≡ −(k1+ k2)2= 4E2+ O 1/E2, (A.6) t≡ −(k2+ k3)2= −4 E2− 4 i=1m2i 4 sin2φ 2 + O 1/E2, (A.7) u≡ −(k1+ k3)2= −4 E2− 4 i=1m2i 4 cos2φ 2 + O 1/E2,
where E is related to p and q as
(A.8) E2= p2+m 2 1+ m22 2 = q 2+m 2 3+ m 2 4 2 .
The polarization bases for the 4 particles are
(A.9) eL(1)= 1 m1 p,− p2+ m2 1, 0 , eT(1)= (0, 0, −1), (A.10) eL(2)= 1 m2 p, p2+ m2 2, 0 , eT(2)= (0, 0, 1), eL(3)= 1 m3 −q, −q2+ m2 3cos φ,− q2+ m2 3sin φ , (A.11) eT(3)= (0, − sin φ, cos φ),
eL(4)= 1 m4 −q,q2+ m 4cos φ, q2+ m2 4sin φ , (A.12) eT(4)= (0, sin φ, − cos φ).
Appendix B. High-energy zero-norm states
In this subsection, we explicitly calculate high-energy zero-norm states (HZNS) of some low-lying mass level. We will also show that the decoupling of these HZNS is equivalent to the decoupling of those spurious states used in the text to derive the desired linear relations. In the old covariant first quantization spectrum of open bosonic string theory, the solutions of physical state conditions include positive-norm propagating states and two types of
zero-norm states. The latter are[7]
(B.1) Type I: L−1|x, where L1|x = L2|x = 0, L0|x = 0; (B.2) Type II: L−2+3 2L 2 −1 | ˜x, where L1| ˜x = L2| ˜x = 0, (L0+ 1)| ˜x = 0.
Based on a simplified calculation of higher mass level positive-norm states in[17], some
general solutions of zero-norm states of Eqs.(B.1) and (B.2)at arbitrary mass level were
calculated in[18]. Eqs.(B.1) and (B.2)can be derived from Kac determinant in conformal
field theory. While type I states have zero-norm at any spacetime dimension, type II states
have zero-norm only at D= 26.
The solutions of Eqs.(B.1) and (B.2)up to the mass level m2= 4 are listed as follows
[18]: (1) m2= −k2= 0: (B.3) L−1|x = k · α−1|0, k, |x = |0, k, |x = |0, k. (2) m2= −k2= 2: (B.4) 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, (B.5) L−1|x = θ· α−2+ (k · α−1)(θ· α−1)|0, k, |x = θ · α−1|0, k, θ · k = 0. (3) m2= −k2= 4: 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, (B.6) | ˜x = θ · α−1|0, k, k · θ = 0,
L−1|x = 2θµνα−1µ αν−2+ kλθµνα−1λ α−1µ αν−1 |0, k, (B.7) |x = θµνα−1µν|0, k, k · θ = ηµνθµν= 0, θµν= θνµ, L−1|x = 1 2(k· α−1) 2(θ· α −1)+ 2θ · α−3+3 2(k· α−1)(θ· α−2) +1 2(k· α−2)(θ· α−1) |0, k, (B.8) |x = 2θ· α−2+ (k · α−1)(θ· α−1)|0, k, θ · k = 0, 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, (B.9) |x = 25 2 k· α−2+ 9 2α−1· α−1+ 17 4 (k· α−1) 2 |0, k.
Note that there are two degenerate vector zero-norm states, Eq.(B.6)for type II and
Eq.(B.8)for type I, at mass level m2= 4. For mass level m2= 2, the high-energy limit of Eqs.(B.5) and (B.4)are calculated to be
(B.10) L−1(θ· α−1)|0 →√2 αL−1α−1L + αL−2|0, (B.11) L−2+3 2L 2 −1 |0 → √2 αL−2+1 2α T −1α−1T |0 (B.12) +3 2 2αL−1α−1L +√2 αL−2|0.
Note that Eq.(B.12)is the high-energy limit of the second term of type II zero-norm state.
It is easy to see that the decoupling of(B.10)implies the decoupling of(B.12). So one can
neglect the effect of(B.12)even though it is of leading order in energy. It turns out that
this phenomena persists to any higher mass level as well. This justifies that the decoupling
of HZNS is equivalent to the decoupling of those spurious states used in the text to derive the desired linear relations. By solving Eqs.(B.10) and (B.11), we get the desired linear relation,TT T :TL:TLL= 4 : −
√
2 : 1. Similarly, the high-energy limit of Eqs.(B.6)–(B.9)
are calculated to be (B.13) L−2+3 2L 2 −1 |0 → 4α−1(Tα−2L)+1 2α T −1α−1T α−1T |0 (B.14) +3 2 4α(L−1α−1L α−1T )+ 4α−1(TαL)−2|0, (B.15) L−1θµναµν−1 |0 → 2α(T−1αL)−2+ 2α−1(Lα−1L α−1T )|0, (B.16) L−1 2θ· α−2+ (k · α−1)(θ· α−1)|0 →4α(L−1αL−1α−1T )+ 4α−1(Tα−2L)|0, (B.17) L−1 25 2 k· α−2+ 9 2α−1· α−1+ 17 4 (k· α−1) 2 |0 → 0.
It is easy to see that the decoupling of Eq.(B.15)or(B.16)implies the decoupling of Eq.(B.14). By solving the equations, one getsTT T T:TLLT:T(LT ):T[LT ]= 8 : 1 : −1 : 1.
Appendix C. Virasoro constraints
C.1. High-energy limit of Virasoro constraints
To take the high-energy limit for the Virasoro constraints, we replace the indices (µi, νi)
by L or T , and
(C.1)
kµi→ ˆmeL, ηµ1µ2→ eTeT.
Eqs.(63a) and (63b)become
0= ˆm L µ12 · · · µ1m 1 k j=1 µj1 · · · µjmj + m1 i=2 µ1 2 · · · ˆµ1i · · · µ1m1 ⊗ µ 1 i µ21 · · · µ2m2 k j=1,2 µj1 · · · µjmj + k l=3 (l− 1) µ12 · · · µ1m 1 ⊗ ml−1 i=1 µl1−1 · · · ˆµli−1 · · · µl−1 ml−1 (C.2a) ⊗ µl−1 i µ l 1 · · · µ l ml k j=1,l,l−1 µj1 · · · µjmj , and 0=1 2 T T µ 1 3 · · · µ 1 m1 k j=1 µj1 · · · µjmj + ˆm µ1 3 · · · µ 1 m1 ⊗ µ 2 1 · · · µ 2 m2 L k j=1,2 µj1 · · · µjmj + m1 i=3 µ13 · · · ˆµ1i · · · µ1m 1 ⊗ µ 1 i µ 3 1 · · · µ3m3 k j=1,3 µj1 · · · µjmj + k l=4 (l− 2) µ13 · · · µ1 m1 ⊗ ml−2 i=1 µl1−2 · · · ˆµli−2 · · · µlm−2 l (C.2b) ⊗ µl−2 i µ l 1 · · · µ l ml k j=1,l,l−2 µj1 · · · µjmj .
The indices and{µji} are symmetric and can be chosen to have ljof{L} which 0 lj mj
and{T } for the rest. Thus
0= ˆm µ12 T · · · T m1−2−l1 L · · · L l1+1 k j=1 µj1 T · · · T mj−1−lj L · · · L lj + T ··· T m1−2−l1 L · · · L l1 ⊗ µ1 2 µ21 T · · · T m2−1−l2 L · · · L l2 k j=1,2 µj1 T · · · T mj−1−lj L · · · L lj + (m1− 2 − l1) µ12 T · · · T m1−3−l1 L · · · L l1 ⊗ µ2 1 T · · · T m2−l2 L · · · L l2 k j=1,2 µj1 T · · · T mj−1−lj L · · · L lj + l1 µ12 T · · · T m1−2−l1 L · · · L l1−1 ⊗ µ2 1 T · · · T m2−1−l2 L · · · L l2+1 k j=1,2 µj1 T · · · T mj−1−lj L · · · L lj + k l=3 (l− 1) µ12 T · · · T m1−2−l1 L · · · L l1 ⊗ T ··· T ml−1−1−ll−1 L · · · L ll−1 ⊗ µl−1 1 µ l 1 T · · · T ml−1−ll L · · · L ll k j=1,l,l−1 µj1 T · · · T mj−1−lj L · · · L lj + k l=3 (l− 1)(ml−1− 1 − ll−1) µ12 T · · · T m1−2−l1 L · · · L l1 ⊗ µl−1 1 T · · · T ml−1−2−ll−1 L · · · L ll−1 ⊗ µl 1 T · · · T ml−ll L · · · L ll
k j=1,l,l−1 µj1 T · · · T mj−1−lj L · · · L lj + k l=3 ll−1(l− 1) µ12 T · · · T m1−2−l1 L · · · L l1 ⊗ µl−1 1 T · · · T ml−1−1−ll−1 L · · · L ll−1−1 (C.3a) ⊗ µl 1 T · · · T ml−1−ll L · · · L ll+1 k j=1,l,l−1 µj1 T · · · T mj−1−lj L · · · L lj , and 0=1 2 µ 1 3 T · · · T m1−1−l1 L · · · L l1 k j=1 µj1 T · · · T mj−1−lj L · · · L lj + ˆm µ1 3 T · · · T m1−3−l1 L · · · L l1 ⊗ µ2 1 T · · · T m2−1−l2 L · · · L l2+1 k j=1,2 µj1 T · · · T mj−1−lj L · · · L lj + T ··· T m1−3−l1 L · · · L l1 ⊗ µ1 3 µ 3 1 T · · · T m3−1−l3 L · · · L l3 k j=1,3 µj1 T · · · T mj−1−lj L · · · L lj + (m1− 3 − l1) µ13 T · · · T m1−4−l1 L · · · L l1 ⊗ µ3 1 T · · · T m3−l3 L · · · L l3 k j=1,3 µj1 T · · · T mj−1−lj L · · · L lj + l1 µ13 T · · · T m1−3−l1 L · · · L l1−1 ⊗ µ3 1 T · · · T m3−1−l3 L · · · L l3+1 k j=1,3 µj1 T · · · T mj−1−lj L · · · L lj
