JHEP02(2013)028
Published for SISSA by Springer
Received: December 6, 2012 Accepted: January 10, 2013 Published: February 6, 2013
A note on on-shell recursion relation of string
amplitudes
Yung-Yeh Chang,a Bo Feng,b,c Chih-Hao Fu,b,a Jen-Chi Lee,a Yihong Wangd and Yi Yanga
aDepartment of Electrophysics, National Chiao Tung University,
1001 University Street, Hsinchu, Taiwan, R.O.C.
bCenter of Mathematical Science, Zhejiang University,
38 Zheda Road Hangzhou, 310027 P.R China
cZhejiang Institute of Modern Physics, Zhejiang University,
38 Zheda Road Hangzhou, 310027 P.R China
dDepartment of Physics and Astronomy, Stony Brook University,
Stony Brook, NY 11794-3800, U.S.A.
E-mail: [email protected],[email protected],
[email protected],[email protected],
[email protected],[email protected]
Abstract:In the application of on-shell recursion relation to string amplitudes, one chal-lenge is the sum over infinite intermediate on-shell string states. In this note, we show how to sum these infinite states explicitly by including unphysical states to make complete Fock space.
Keywords: Scattering Amplitudes, Bosonic Strings
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Contents1 Introduction 1
2 A brief review of BCFW on-shell recursion relation 3
3 Example I: BCFW of 4-tachyon amplitude in bosonic open string theory 3
3.1 Pole structure extraction 4
3.2 Interpreting pole expansion formula from BCFW perspective 4
3.3 Summing over physical states 5
3.3.1 Explicit calculation 10
4 Example II: BCFW of 5-tachyon amplitude in bosonic open string theory 11
4.1 Pole expansion 11
4.2 Four point scattering amplitude 13
4.3 Calculation of residue SM 14
5 The general proof 17
5.1 String theory calculation 17
5.2 The proof 18
5.3 Practical method for summing over physical states 19
6 Scattering with higher spin particles 19
7 Conclusions 24
A Mathematical identity 25
B Decoupling of Ghosts in string amplitude 26
B.1 DDF states 26
B.2 Decoupling of ghosts in string amplitude 27
B.3 Decoupling of ghosts in BCFW on-shell recursion relation 28
1 Introduction
Whilst its application requires merely the knowledge of analytic structure of the scattering amplitude of interest, the on-shell recursion relation (BCFW) [1,2] has achieved tremen-dous success in calculations of scattering amplitudes, a task would very often seem prac-tically impossible using conventional methods even when there are only a few of external
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particles involving gluons or gravitons.1 In contrast to perturbative off-shell formulation, the on-shell recursion relation uses fewer-point physical amplitude as building blocks,
A(123 . . . n) = X
poles
AL(ˆ12 . . . , ˆPh)
1
P2 AR(− ˆP−h, . . . n), (1.1)
thereby avoiding large amount of unnecessary cancelation in intermediate step of compu-tations. An important point of eq. (1.1) is the sum over all possible physical poles and allowed helicity configurations. Generalization of on-shell relation to string amplitudes was pioneered in [4, 5] and [6] and further elaborated in [7–9]. Recent applications at 4-point and to eikonal Regge limit can be found in [10] and [11] respectively. The validity of on-shell recursion relation in string theory context was argued both from the better convergent UV behavior generically observed in string amplitudes and from analyzing explicit expressions of string amplitudes.
However, when applying on-shell recursion relation to string amplitudes, we are facing the problem of summing over infinite number of physical states in (1.1). Although it could be done in principle, there is no efficient algorithm doing so. For scattering amplitudes of tachyons, based on known analytic expressions, it has been conjectured in [7] that amplitudes can be effectively reduced to factorization of two lower-point tachyon-like sub-amplitudes.
In this paper, we provide an algorithm to do the sum over infinity number of phys-ical states in (1.1). Applying our algorithm to tachyon amplitudes, we see that the sum over physical states at each mass level predicted by open string theory does produce the conjectured scalar-behaved residue observed in [6]. In contrast with the experiences with amplitude calculations in field theory, the key of our algorithm is to enlarge the sum over intermediate physical states to over intermediate complete Fock space states. The zero con-tributions of extra states are guaranteed by no-ghost theorem (i.e., the Ward-like identity in string theory).2
The structure of this paper is organized as the following: in section 2, we present a very brief review of BCFW on-shell recursion relation of generic field theory amplitudes. In section 3 we start with the familiar 4-point Veneziano amplitude as an example and demonstrate how the tachyonic recursion relation can be understood from carrying out sum directly. Section4consists of analysis on 5-point string amplitudes, in which case the pole structure becomes much more complicated. A discussion on pole structure of generic n-point amplitude is presented in section5. In section6we consider higher-spin scatterings and demonstrate that generically the mathematical connection between BCFW and tachy-onic recursion descriptions can be found in the generating function for Stirling number of the first kind outlined in appendix A, while the relation between on-shell condition and decoupling of unphysical states is discussed in appendix B.
1A review of the principles of BCFW on-shell recursion relation as well as its some applications can be found in [3].
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2 A brief review of BCFW on-shell recursion relationIn this section we provide a short review of on-shell recursion relation [1,2]. Derivation of BCFW on-shell recursion relation starts from taking analytic continuation of amplitudes. An amplitude can be regarded as function of complex momenta defined by standard Feyn-man rules. When the momenta of a pair of particle lines Feyn-manually chosen are shifted in a complex q-direction,
bk1(z) = k1+ zq, bkn(z) = kn− zq, (2.1)
with q2= q · k
a= q · kn= 0, the shifted amplitude A(z) defines a complex function. While
the explicit analytic structure of amplitude is determined by individual theory and does not concern us here, A(z) thus defined will contain simple poles produced by propagators, which is the consequence of local interaction and the null condition of q. From Cauchy’s Theorem, integrating over a contour large enough to enclose all finite poles yields
I dz A(z) z = A(0) + X poles α Resz=zα, (2.2)
where an unshifted amplitude A(0) contributes as residue at z = 0 and residues from other finite poles assume the form as cut-amplitudes, Reszα = −A(zα)
1
P2AR(zα). In
var-ious theories shifted amplitudes posses convergent large-z asymptotic behavior and the integral (2.2) vanish, we are then entitled to write down the BCFW recursion relation3
An= X poles X physical states AL(. . . , P (zα)) 2 P2+ M2AR(−P (zα), . . .), (2.3)
where the first sum is over all finite simple poles zα of z, and the second sum is over all
physical states at the given simple pole za.
3 Example I: BCFW of 4-tachyon amplitude in bosonic open string theory
As was demonstrated in the previous section, a key feature making BCFW on-shell re-cursion relation possible is that in perturbative field theory, at tree-level amplitude can often be determined entirely from its poles and related residues. The locations of poles are determined by propagators while the residues, by factorization properties. Same analytic structure holds for string theory, with one complication: there is an infinite number of poles and related residues. As an consequence, there are several expressions for amplitudes, for example, the Veneziano formula assumes the form of a worldsheet integral, making the pole structure obscured. In [6] through binomial expansions of these integral formulas, the pole structure can be made manifest. In this section, we will use four-point tachyon amplitude as an example to demonstrate our idea and method.
3We have assumed the boundary contribution to be zero. If it is no zero, we need to modify recursion relation, see [12].
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3.1 Pole structure extraction
Consider the four tachyon scattering amplitude in bosonic open string theory, given by Koba-Nielson formula as
A(1234) = Z 1
0
dz2(1 − z2)k3·k2z2k2·k1, (3.1)
where we have used the conformal symmetry to fix z1 = 0, z3 = 1 and z4 = +∞. For
arbitrary complex power w we have following binomial expansion (x − y)w= ∞ X a=0 w a ! xw−aya (3.2) where coefficient w a ! is defined as w a ! = w(w − 1)(w − 2) . . . (w − a + 1) a! (3.3)
Applying (3.2) to (1 − z2)k3·k2 and collecting relative terms we have
A(1234) = ∞ X a=0 k3· k2 a ! (−)a Z dz2z 1 2(k1+k2)2+a−2 2 (3.4)
where we have used the mass-shell condition for tachyon that k12 = k22 = −M2 = +2.4 The worldsheet integration can be explicitly carried out, producing an s-channel propagator.5 Inserting it back, we obtain
A(1234) = ∞ X a=0 k3· k2 a ! (−)a 2 (k1+ k2)2+ 2(a − 1) (3.5)
3.2 Interpreting pole expansion formula from BCFW perspective
Having derived an explicit analytic expression (3.5) for tree-level four tachyon scattering amplitude, it is then interesting to see if the result can be understood in the language of BCFW on-shell recursion relation. We choose the shifted pair to be (1, 4) to be consistent with the manifest s-channel expansion. Assuming there is no boundary contribution for on-shell recursion relation, equation (3.5) should be given by on-shell recursion relation (2.3): An= X poles X physical AL(. . . , P (zα)) 2 P2+ M2AR(−P (zα), . . .) (3.6) 4We have used the convention α′
= 1/2, so the mass of bosonic open string state is M2 = −2 + 2P∞
n=1α−n·αn
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In denominator we see infinitely many single poles occurs at za=
(k1+ k2)2+ 2(a − 1)
−2q · (k1+ k2)
, a = 0, 1, . . . (3.7)
where P = k1+ k2 and the mass square Ma2 = 2(a − 1) for every integer a is precisely the
mass spectrum prescribed by bosonic open string theory. In addition, matching residues of (3.5) with (3.6) indicates that, at each level a, there should be a number of physical states, collectively yielding X states h AL(1, 2, Pah(za))AR(−Paeh(za), 3, 4) = (−1)a k3· k2 a ! . (3.8)
Thus to understand (3.5) from BCFW recursion relation (2.3), we need to be able to interpret the scalar-behaved residue (3.8) as sum over physical states at each fixed level a. 3.3 Summing over physical states
Before undertaking a state-by-state calculation of residues over bosonic string spectrum, let us make a slight detour and consider how the analytic structure featuring intermediate states fits into the picture of BCFW on-shell recursion relation in quantum field theory. Although in Feynman rules scalar, fermion and gauge boson each are assigned with a propagator in distinct representations, we note that the propagator appearing in BCFW recursion relation (3.6) is always scalar-like. The reason is following. For example, if the intermediate particles are massless fermions, BCFW recursion relation reads
A ∼ X
h=±
AL(σL, Ph)AR(−P−h, σR). (3.9)
We can rewrite the on-shell sub-amplitude AL(σL, Ph) =Pa=1,2AeL(σL, Ph)auh(P )a, i.e.,
we have decomposed the on-shell amplitude into two parts: wave function for external on-shell particle P and the rest. Similar decomposition can be done for AR(−P−h, σR).
Thus the sum over physical states becomes A ∼ eAL(σL, Ph) X h us(P )us(p) ! e AR(−P−h, σR) ∼ eAL(σL, Ph) (γ · P ) eAR(−P−h, σR) (3.10) where in the middle, γ · P is exactly the factor needed to translate scalar propagator into the familiar fermion propagator.
A similar mechanism supports the translation from scalar propagator into gauge boson propagator when summed over physical states, but with some subtleties. The sum over two transverse physical states for gauge boson is (ǫ+µǫ−ν + ǫ−µǫ+ν) while the familiar Feynman
gauge uses gµν. In fact, in 4-dimensions we need four polarization vectors, and
gµν = ǫ+µǫ−ν + ǫ−µǫν++ ǫLµǫTν + ǫTµǫLν (3.11)
where ǫL
µ and ǫTµ are longitude and time-like polarization vector [13]. The reason that these
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give same answer depends crucially on Ward Identity of gauge theory, i.e., if all (n − 1) particles are physical polarized while the n-th particle is longitude (i.e., proportional to kµ), the amplitude is zero. Thus we have
X all states AL(σL, Ph)AR(−Peh, σR) ∼ eAµL(σL, P )gµνAeνR(−P, σR) ∼ eAµL(σL, P ) ǫ+µǫ−ν + ǫ−µǫ+ν + ǫLµǫTν + ǫTµǫLν eAνR(−P, σR) ∼ eAµL(σL, P ) ǫ+µǫ−ν + ǫ−µǫ+ν eAνR(−P, σR) = X physical states AL(σL, Ph)AR(−P−h, σR) (3.12)
Having understood the effect of summing over physical states from quantum field theory, let us return to the problem of interpreting scalar-behaved residue (3.5) as sum over physical states. In old covariant quantization framework, the Fock space in bosonic open string theory is constructed by linear combinations of states obtained from acting creation modes successively on ground state
αµ1 −n1α µ2 −n2. . . α µn −nn|0; ki . (3.13)
Generically, a Fock state can carry Nµ,1-multiple of αµ−1mode operators6and Nµ,2-multiple
of αµ−2 mode and so on. In the following discussions we use the set of numbers {Nµ,n} as
label of normalized Fock state |{Nµ,n}, ki = QD−1 µ=0 Q∞ n=1 (αµ−n)Nµ,n √ nNµ,nNµ,n! |0, ki . (3.14)
Physical states however, in addition must satisfy Virasoro constraints (L0 − 1) |φi = 0,
Lm>0|φi = 0 and constitute only a subset in Fock space. An immediate consequence is
that physical states are automatically on the mass-shell, −k2= M2 = 2(N − 1), where N
is the level N = DX−1 µ=0 ∞ X n=1 nNµ,n . (3.15)
Note however, for a generic Fock state its center-of-mass momentum kµ and modes {Nµ,n}
are considered as independent degrees of freedom and does not a priori satisfy mass-shell condition, and yet in a BCFW on-shell recursion relation, Fock states that happen to be the on mass-shell are picked out because as we have seen from (3.7) that only these states contribute to residues.
Now we come to our central point. The prescription given by BCFW on-shell recursion relation is to sum over physical states satisfying on-shell condition plus remaining Virasoro constraints Lm>0|φi = 0. However, a rather technical difficulty carrying out above
pre-scription in string theory is that it requires the knowledge of physical polarization tensor at arbitrarily high mass level N , which is very hard to write down explicitly. To bypass the
6It should be emphasized that αµ −1 and α
ν
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problem, inspired by the observation given in [13] for gauge theory (3.11), we can enlarge the sum over physical states to all states in Fock space satisfying on-shell condition. The fact that these two sums are same is guaranteed by the famous “No-Ghost Theorem”.7 With this understanding, we can write
An = X poles X physical AL(. . . , P (zα)) 2 P2+ M2AR(−P (zα), . . .) = X poles X Fock AL(. . . , P (zα)) 2 P2+ M2AR(−P (zα), . . .) = X poles X Fock ( eAL(. . . , P (zα)) · ξP 2 P2+ M2AeR(−P (zα), . . .) · ξ∗(P ) (3.16)
where at the last step we have stripped away the polarization tensor of intermediate state P from on-shell amplitude. Since the sum is taken over whole Fock space, we are free to choose any convenient basis, for example, the one given in (3.14), to perform the sum. Thus if we take pair (1, n) to conduct BCFW-deformation and sum over the polarization tensor of intermediate state, BCFW on-shell relation of a string amplitude reads
An = nX−2 i=2 +∞ X N=0 X {Nµ,n} D φ1(bk1)|V2(k2) . . . Vi(ki)|{Nµ,n}, bP E 2T{N µ,n} (Pit=1ki)2+ 2(N − 1) D {Nµ,n}, bP |Vi+1(ki+1) . . . Vn−1(kn−1)|φn(bkn) E (3.17) In this formula, the first sum is over the splitting of particles into left and right handed sides while the second sum is over poles fixed by the mass level N . The third sum is over all allowed choice of the set {Nµ,n} as long as they satisfy (3.15). The tensor structure
T{Nµ,n} is determined by the set {Nµ,n}. To demonstrate the rule for the tensor structure,
we list the tensor structure for first three levels:
• Level N = 0: For the first level, all Nµ,n= 0 so we have T = 1.
• Level N = 1: The choice is Nµ,1 = 1 for µ = 0, 1, . . . , D − 1, thus we have T = gµν,
i.e., we have φ1| . . . Vi αµ−1|0; P 2gµν P2+ 2(N − 1) 0; P |αν+1 Vi+1. . . |φn (3.18)
where when we conjugate αµ−1|0; P we get0; P |αν +1
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• Level N = 2: There are several choices and the structure is given by
DX−1 µ,ν=0 φ1| . . . Vi αµ−2 √ 2|0; P 2gµν P2+ 2(N − 1) 0; P |α ν +2 √ 2 Vi+1. . . |φn + X 0≤µ1<µ2≤D−1 X 0≤ν1<ν2≤D−1 φ1| . . . Vi α−1µ1α−1µ2|0; P 2gµ1ν1 gµ2ν2 P2+ 2(N − 1) 0; P |αν2 +1α ν1 +1 Vi+1. . . |φn + DX−1 µ,ν=0 * φ1| . . . Vi (αµ−1)2 √ 2 |0; P + 2(gµν)2 P2+ 2(N − 1) 0; P |(α ν +1)2 √ 2 Vi+1. . . |φn (3.19)
where at the second line, to avoid repetition, we must have the ordering 0 ≤ µ1 <
µ2≤ D − 1.
• Level N = 3: There are several choices which are given respectively by T1 = DX−1 µ,ν=0 φ1| . . . Vi αµ−3 √ 3|0; P 2gµν P2+ 2(N − 1) 0; P |α ν +3 √ 3 Vi+1. . . |φn T2 = DX−1 µ1,µ2,ν1,ν2=0 φ1| . . . Vi αµ1 −2 √ 2α µ2 −1|0; P 2gµ1ν1gµ2ν2 P2+ 2(N − 1) 0; P |αν2 +1 αν1 +2 √ 2 Vi+1. . . |φn T3 = X 0≤µ1<µ2<µ3≤D−1 X 0≤ν1<ν2<ν3≤D−1 φ1| . . . Vi αµ−11α−1µ2αµ−13|0; P 2gµ1ν1gµ2ν2gµ3ν3 P2+ 2(N − 1) 0; P |αν3 +1α ν2 +1α ν1 +1 Vi+1. . . |φn T4 = DX−1 µ1,µ2,ν1,ν2=0 * φ1| . . . Vi (αµ1 −1)2 √ 2 (a−1) µ2|0; P + 2(gµ1ν1)2gµ2ν2 P2+ 2(N − 1) * 0; P |(α+1)ν2 (αν1 +1)2 √ 2 Vi+1. . . |φn + T5 = DX−1 µ,ν=0 * φ1| . . . Vi (αµ−1)3 √ 3! |0; P + 2(gµν)3 P2+ 2(N − 1) 0; P |(α ν +1)3 √ 3! Vi+1. . . |φn So we have N = 3 : T1+ T2+ T3+ T4+ T5 (3.20)
These examples demonstrate the general pattern of tensor structures. However, be-cause when we have several oscillators with same n, there are freedoms with the choice of µ, we need to distinguish if these µ are same or different from each other. This makes the tensor structure a little bit of complicated. This complication can be simplified further.
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For example, at the level N = 2, we have X 0≤µ1≤µ2≤D−1 X 0≤ν1≤ν2≤D−1 αµ1 −1α µ2 −1gµ1ν1gµ2ν2α ν2 +1αν+11 = 1 2 DX−1 µ16=µ2=0 DX−1 ν16=ν2=0 αµ1 −1α µ2 −1gµ1ν1gµ2ν2α ν2 +1α ν1 +1 = DX−1 µ16=µ2=0 DX−1 ν16=ν2=0 αµ1 −1α µ2 −1 √ 2 gµ1ν1gµ2ν2 αν2 +1αν+11 √ 2 (3.21)
With this rewriting, the second and third line of (3.19) can be combined to
DX−1 µ1,µ2,ν1,ν2=0 φ1| . . . Vi αµ1 −1α µ2 −1 √ 2 |0; P 2gµ1ν1gµ2ν2 P2+ 2(N − 1) 0; P |α ν1 +1α ν2 +1 √ 2 Vi+1. . . |φn (3.22)
Similar argument can show that the sum T3, T4, T5 of (3.20) gives
T3+ T4+ T5 = DX−1 µi,νi=0 φ1| . . . Vi αµ1 −1α µ2 −1α µ3 −1 √ 3! |0; P 2gµ1ν1gµ2ν2gµ3ν3 P2+ 2(N − 1) 0; P |α ν3 +1α ν2 +1α ν1 +1 √ 3! Vi+1. . . |φn
It is easy to see that when multiple operators of the same mode n are present in the Fock state, each may or may not be carrying the same Lorentz index 0 , or 1 , or . . . , or D − 1, the general pattern is given by the expansion (a0+ a1+ . . . + aD−1)Nn/Nn! where
ai = α−ni giiα+ni . The coefficient of term (α0−n)n0(α−n1 )n1. . . (αD−n−1)nD−1 in the Fock state
is given by the coefficient of term an0 0 a
n1 1 . . . a
nD−1
D−1 with Nn = PDi=0−1ni in the expansion,
which reads 1 N !C N n0C N−n0 n1 C N−n0−n1 n2 . . . C nD−1 nD−1 = 1 N ! N ! QD−1 i=0 (ni)! (3.23) thus we can drop the µ1 < µ2 < . . . arrangement and rewrite the sum in (3.17) as
X {Nµ,n} {Nµ,n}; bP E T{Nµ,n} D {Nµ,n}; bP = X P nnNn=N ( ∞ Y n=1 (αµNn,1 −n α µNn,2 −n . . . α µNn,Nn −n ) p Nn!nNn ) 0; bPE ∞ Y n=1 (gµNn,1νNn,1gµNn,2νNn,2. . . gµNn,NnνNn,Nn) D 0; bP (∞ Y n=1 (ανNn,1 +n α νNn,2 +n . . . α νNn,Nn +n ) p Nn!nNn ) (3.24) Having the simplified version (3.24), we can give following explicit calculations.
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3.3.1 Explicit calculation Recalling the vertex of tachyon
V0(k, z) =: eik·X(z):= Z0W0, (3.25) where Z0 = eik·x+k·p ln z = eikxzk·p+1 = zk·p−1eik·x (3.26) and W0 = e P∞ n=1znnk·α−ne− P∞ n=1z −n n k·αn, (3.27)
it is easy to calculate the left three-point amplitude
h0; −k1|V0(k2, z)|{Nµ,n}; P i = δ(k1+ k2+ P ) DY−1 µ=0 ∞ Y m=1 (−kµ2)Nµ,m p mNµ,mN µ,m! (3.28)
where N is the level defined in (3.15) and the right three-point amplitude h{Nµ,n}; P |V0(k3, z)|0; k4i = δ(P − k3− k4) DY−1 µ=0 ∞ Y m=1 (kµ3)Nµ,m p Nµ,m!mNµ,m (3.29)
Using (3.28) and (3.29) it is easy to calculate first few mass levels. In fact, the same calculation has been done in our simplification leading to the simplified tensor structure (3.24). Thus we have when N = 0, it is 1, while when N = 1 it is (−k2· k3). Finally when
N = 2 it is (k2·k3)(k2·k3−1)
2 . They do satisfy (3.8) for N = 0, 1, 2.
For general level N , from (3.24), (3.28) and (3.29) we find IN = X P nNn=N Y (−k2· k3)Nn Nn!nNn (3.30) Let us define N = ∞ X n=1 nNn, J = ∞ X n=1 Nn (3.31)
with obviously that J ≤ a, then using the definition (A.2) of Stirling number of the first kind, IN can be rewritten as
IN = (−)N N X J=1 S(N, J) N ! (k2· k3) J = (−)N k2· k3 N ! (3.32)
where we have used the formula (A.1).8 This is exactly the result (3.8) we try to prove.
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4 Example II: BCFW of 5-tachyon amplitude in bosonic open stringtheory
Having shown that a 4-point Veneziano amplitude can be indeed described by BCFW on-shell recursion relation, let us consider the 5-tachyon scattering amplitude, which contains slightly richer analytic structure because unlike 4-point amplitude with only pole s12, there
are two types of poles from s12, s123 for deformation (1,5). Multiple pole structure is seen
for general amplitudes, we need to study this simplest nontrivial example.
4.1 Pole expansion
The Koba-Nielson formula for 5-point tachyon amplitude is given by A(12345) = Z 1 0 dz3 Z z3 0 dz2(1 − z3)k4·k3(1 − z2)k4·k2(z3− z2)k3·k2z2k2·k1z3k3·k1. (4.1)
where we have fixed z1 = 0, z4 = 1, z5 = ∞. Unlike in quantum field theory, where
ana-lytic behavior of an amplitude is transparent from Feynman rules, kinematic dependence in Koba-Nielson’s formulation were implicitly introduced through exponents of worldsheet integration variables, making it less easier to locate poles. However as we have seen in the previous section, worldsheet integrals can be explicitly carried out after binomial ex-pansions. Expanding (z3− z2)k3·k2 with respect to z2, which is the variable that assumes
smaller value (than z3), and expand similarly (1 − z2)k4·k2 and (1 − z3)k4·k3 we have
(1 − z2)k4·k2 = ∞ X a=0 k4· k2 a ! (−)az2a, (z3− z2)k3·k2 = ∞ X b=0 k3· k2 b ! (−)bzks·k2−b 3 z2b, (1 − z3)k4·k3 = ∞ X c=0 k4· k3 c ! (−)czc3, (4.2)
Grouping z2 and z3 dependence in equation (4.1) together we arrive
A(12345) = ∞ X a,b,c=0 k4· k2 a ! k3· k2 b ! k4· k3 c ! (−)a+b+c × Z 1 0 dz3 Z z3 0 dz2zk33·(k1+k2)−b+cz k1·k2+a+b 2 (4.3)
Carrying out the integration in order, i.e., Rdz2 first and then Rdz3 we obtain
A(12345) = ∞ X a,b,c=0 k4· k2 a ! k3· k2 b ! k4· k3 c ! (−)a+b+c × 2 s12+ 2(a + b − 1) 2 s123+ 2(a + c − 1) , (4.4)
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where we have used s12= (k1+ k2)2, s123 = (k1+ k2+ k3)2, and the mass-shell conditions
for tachyons, k12 = k22 = k32 = 2.
Now we consider the pole structure under the deformation (2.1) with pair (1, 5). For s12, the poles are located at
zN =
(k1+ k2)2+ 2(N − 1)
−q · (k1+ k2)
, N = a + b = 0, 1, . . . (4.5)
while for s123 the poles are located at
wM =
(k1+ k2+ k3)2+ 2(M − 1)
−q · (k1+ k2+ k3)
, M = a + c = 0, 1, 2, . . . (4.6) Using the BCFW recursion relation, we have
A(1, 2, 3, 4, 5) =X zN 2 s12+ 2(N − 1)RN +X wM 2 s123+ 2(M − 1)SM (4.7)
where RN and SM are corresponding residues of poles.
Residue RN: from (4.4) we can read out the residue RN as
RN = ∞ X a, b = 0 a + b = N ∞ X c=0 k4· k2 a ! k3· k2 b ! k4· k3 c ! (−)a+b+c 2 bs123(zN) + 2(a + c − 1) (4.8) Noticing that bs12(zN) + k23+ 2k3· bk12(zN) + 2(a + c − 1) = 2k3· bk12(zN) + 2(c − b + 1) we can rewrite k4· k3 c ! (−)c 2 bs123(zN) + 2(a + c − 1) = k4· k3 c ! (−)c " 1 k3· bk12(zN) + (c − b + 1) # = ∞ X c=0 Z 1 0 dz3z3k3·(ˆk1+k2)−b+c k4· k3 c ! (−)c = Z 1 0 dz3zk3·(ˆk1+k2)−b(1 − z3)k4·k3, (4.9)
The reason we write the sum over c as the integration is clear: the subamplitude at the right handed side should be A( bP , 3, 4, b5). With this rewriting we have
RN = ∞ X a, b = 0 a + b = N k4· k2 a ! k3· k2 b ! (−)N Z 1 0 dz3zk3·(ˆk1+k2)−b(1 − z3)k4·k3 (4.10)
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Residue SM: from (4.4) we can read out the residue SM as
SM = ∞ X a, c = 0 a + c = M ∞ X b=0 k4· k2 a ! k3· k2 b ! k4· k3 c ! (−)a+b+c 2 bs12(wN) + 2(a + b − 1) (4.11) Using ∞ X b=0 k3· k2 b ! (−)b ˆ k1· k2+ (a + b) + 1 z=wM = ∞ X b=0 Z 1 0 dz2z ˆ k1·k2+a+b 2 k3· k2 b ! (−)b = Z 1 0 dz2z2kˆ1·k2+a(1 − z2)k3·k2, (4.12)
which remind us the subamplitude A(b1, 2, 3, bP ), we get another form
SM = ∞ X a, c = 0 a + c = M k4· k2 a ! k4· k3 c ! (−)M Z 1 0 dz2z ˆ k1·k2+a 2 (1 − z2)k3·k2 (4.13)
4.2 Four point scattering amplitude
Now we try to reproduce the same residue from the BCFW recursion relation. To do this, we need to calculate the three point and four point amplitudes with one general Fock state. The three point case has been given in section3. Now we give the four point result.
First let us consider a simple example
0, k4|V0(k3, z3) V0(k2, z2)α−mµ |0, k1 (4.14)
where V0(k, z) stands for tachyon vertex operator (B.1) inserted at z, and the initial state
αµ−m|0, k1 is raised from the ground state by a −m mode operator. Following the standard
treatment moving this mode operator to the left until it finally annihilate the final state we obtain
(−k2µzm2 − k3µzm3 ) h0, k4|V0(k3) V0(k2)|0, k1i . (4.15)
In addition to all-tachyon amplitude we receive factors (−kµ2z2m− kµ3z3m) picked up from
the commutator
[: eik·X(z) : , αµ−m] = −kµzm : eik·X(z): . (4.16) For a generic normalized Fock state (3.14) we repeat the same manipulation, moving mode operators αµ−m one by one to the left, picking up a factor (−kµzm) when passing a
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tachyon vertex V (k, z). Putting all together we finally have
h0, p|V0(k3) V0(k2)| {Nµ,m} , k1i = * 0, p|V0(k3) V0(k2) DY−1 µ=0 ∞ Y m=1 (α−mµ )Nµ,m p Nµ,m!mNµ,m|0, k 1 + = DY−1 µ=0 ∞ Y m=1 (−k2µzm2 − k µ 3z3m)Nµ,m p Nµ,m!mNµ,m h0, p|V0(k3) V0(k2)|0, k1i (4.17) where h0, p|V0(k3) V0(k2)|0, k1i is known.
Similarly, if the Fock state defines the final state instead of the initial state of an amplitude we move mode operator αµm to the right hand side, yielding
h{Nµ,m} , k5|V0(k4) V0(k3)|0, pi = * 0, k5| DY−1 µ=0 ∞ Y m=1 (αmµ)Nµ,m p Nµ,m!mNµ,m V0(k4) V0(k3)|0, p + . = DY−1 µ=0 ∞ Y m=1 (k4µzm 4 + k µ 3zm3 )Nµ,m p Nµ,m!mNµ,m h0, k5|V0(k4) V0(k3)|0, pi . (4.18)
It is worth to notice that the factors picked up by modes have different signs from (4.17) due to the fact that opposite signs were assigned to positive and negative modes in a tachyon vertex operator, W0= e P∞ n=1znn k·α−ne−P ∞ n=1znn k·αn (4.19) so that [αµm, : eik·X(z) :] = kµzm : eik·X(z) : (4.20) 4.3 Calculation of residue SM
Having above preparation, we can calculate residue by summing over immediate Fock states at given mass level M . In other words, at level M , we should have
SM = Z dz2 X {Nµ,m} D 0, ˆk5|V0(k4)|{Nµ,m}, ˆp E D {Nµ,m}, ˆp|V0(k3) V0(k2)|0, ˆk1E z4=z3=1 , (4.21) where the summation is over modes{Nµ,m} at fixed mass level N =Pµ,m (m × Nµ,m), so
b
p, bk5, bk1 are all fixed by M . Before giving the general discussion, let us see a few examples:
• Level N = 0: at N = 0, Nµ,m must be all zero, so that equation (4.21) simply yields
S0= Z dz2 D 0, ˆk5|V0(k4)|0, ˆp E D 0, ˆp|V0(k3) V0(k2)|0, ˆk1E z4=z3=1 = 1 × Z 1 0 dz2zk2·ˆk1(1 − z2)k3·k2, (4.22)
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• Level N = 1: the N = 1 state can only arise from states having a single Nµ,m = 1
for µ = 0, . . . , D − 1, while powers of other modes remain zero S1 = X µ,ν Z dz2 D 0, ˆk5|V0(k4)|Nµ,1, ˆp E gµνDNν,1, ˆp|V0(k3) V0(k2)|0, ˆk1E z4=z3=1 = Z 1 0 dz2(−k4) · (k3z3+ k2z2) zk2·ˆk1(1 − z2)k3·k2 z 3=1 (4.23) In addition to the usual tachyonic Koba-Nielson formula we obtain a factor − (k4· k3) z3 − (k4· k2) z2|z3=1. These two terms correspond to (a, c) = (0, 1) and
(1, 0) respectively.
• Level N = 2: the first non-trivial case happens at N = 2. As in the previous mass level we receive an additional term to the tachyonic formula. For Nµ,2 states
this factor is −12 k4 · k3z23+ k2z22
, while for states with Nµ1,1 = Nµ2,1 = 1 and
0 ≤ µ1 < µ2 ≤ D − 1 the factor is 12[k4· (k3z3+ k2z2)]2− 12Pµ[k µ
4(k3z3+ k2z2)µ]2,
and for states with Nµ,1 = 2 we obtain Pµ[k µ
4(k3z3+ k2z2)] 2
. Adding all these contribution gives −1 2 k4· k3z 2 3+ k2z22 +1 2[k4· (k3z3+ k2z2)] 2 (4.24) = (k4· k3)(k4· k3− 1) 2 z 2 3+ (k4· k2)(k4· k2− 1) 2 z 2 2+ (k4· k3)(k4· k2)z3z2
Explicit expansion into series shows again agreement with
k4· k2 a ! k4· k3 c !
(−)a+cz2a, with the first, second, third terms corresponding to (a, c) = (0, 2), (2, 0) and (1, 1) respectively.
For general level N =P∞n=1n Nn in addition to the all-tachyon formula we have9
X partitions of N into {Nn} ∞ Y n=1 [−k4· (k3z3n+ k2z2n)] Nn Nn! nNn = X partitions of N into {Nn} Y n ∞ X Nn(2)=0 Nn Nn(2) ! Nn! nNn (k4· k3)Nn−N (2) n zn(Nn−Nn(2)) 3 (k4· k3)N (2) n zn N (2) n 2 . (4.25) where in the second line above we expanded the numerator with respect to power of z2,
which we denote as Nn(2). Introducing the notation Nn(3) = Nn− Nn(2), the combinatorial 9Note that at every step these factors are produced in the same pattern observed in the 4-point case, as was discussed in appendix B, except with k3 now replaced by k3z3n+ k2z2n.
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factor can be written as Nn Nn(2) ! 1 Nn! nNn = 1 Nn(2)! (Nn− Nn(2))! nNn = 1 Nn(2)! N(3)! nN (2) n nNn(3)
Now we notice that in equation (4.25), summing over partitions of fixed Nn into Nn(2) and
Nn(3) first and then summing over partitions of N into {Nn} secondly can be replaced by
summing over partitions of N directly into {Nn(2)} and {Nn(3)}, so (4.25) can be written as
X partitions into Nn(2),Nn(3) Y n 1 Nn(2)! N(3)! nN (2) n nNn(3) (k4· k2)N (3) n (k 4· k3)N (2) n zn Nn(2) 2 z n Nn(3) 3 . (4.26) Defining K =X n Nn(2), J ≡X n Nn(3), a =X n n Nn(2), c =X n n Nn(3), (4.27) sum in equation (4.26) can be divided into summations over partitions of {Nn(2)} and {Nn(3)}
with fixed J, K, a, c at first, and then summing over J, K, and a,10 i.e., equation (4.26) is equal to X a X J,K S(c, J) c! S(a, K) a! (k4· k2) J (k4· k3)K z2az3c, (4.28)
where Striling numbers of the first kind are given by
S(a, K) = X partitions Nn(2) a! Nn(2)! nN (2) n , S(c, J) = X partitions Nn(3) c! Nn(3)! nN (3) n , (4.29)
Now we are almost done. Summing equation (4.28) over J and K yields k4· k2 a ! k4· k3 c ! (−)a+cza2z3c. (4.30)
Inserting the result back into (4.21) we see that SM Z dz2 D 0, ˆk5|V0(k4)|{Nµ,m}, ˆp E D {Nµ,m}, ˆp|V0(k3) V0(k2)|0, ˆk1E z4=z3=1 =X a Z dz2 D 0, ˆk5|V0(k4)|0, ˆp E D 0, ˆp|V0(k3) V0(k2)|0, ˆk1 E × k4· k2 a ! k4· k3 c ! (−)a+cz2az3c z 4=z3=1 = M X a=0,a+c=M k4· k2 a ! k4· k3 c ! (−)a+c Z 1 0 dz2z ˆ k1·k2+a 2 (1 − z2)k3·k2, (4.31)
which is the form (4.13) we want to prove.
The other residue RN can be derived from BCFW prescription following similar
pro-cedures.
10However note that c should not be summed over here because the mass level (a + c) =P nn(N
(2) n + Nn(3)) = N is understood as a fixed number at every pole.
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5 The general proofHaving done above two examples, we would like to have a general understanding. The method we will use in this section will be a little different although it is easy to translate languages between these two approaches.
5.1 String theory calculation
In open string theory, the ordered tree-level amplitude is given by AM = gM−2
Z
δ(yA− y0A)δ(yB− y0B)δ(yc− yc0)(yA− yB)(yA− yC)(yB− yC) M Y i=2 θ(yi−1− yi) M Y j=1 dyj 0; 0 V (ky11, y1). . .V (kMyM, yM) 0; 0 (5.1) Using three delta-function, we can take yM = 0, y2 = 1, y1 = ∞, so the amplitude can be
written as AM = gM−2 Z 1 0 dy3 Z y3 0 dy4. . . Z yM −2 0 dyM−1 φ1(k1) V (k2, 1) V (k3, y3) y3 . . .V (kM−1, yM−1) yM−1 φM(kM) (5.2) where we have used the definition of initial state and final state
|Λ; ki = limy
→0
VΛ(k, y)
y |0; 0i , hΛ; k| = limy→∞yVΛ(k, y) |0; 0i (5.3)
Next we define yi = z3z4. . . zi with i = 3, . . . , M − 1, from which we can solve
z3 = y3, zi =
yi
yi−1
, i = 4, . . . , M − 1 (5.4)
Now let us fix all yi except transform yM−1 = zM−1yM−2, then using
VΛ(k, z) = zL0VΛ(k, z = 1)z−L0 (5.5) we get . . . Z 1 0 dzM−1yM−2yML0−1−2V (kM−1, 1)yM−L−10+1|φM(kM)i = . . . Z 1 0 dzM−1yML0−2−1zML0−1−2 V (kM−1, 1) |φM(kM)i
where we have used the physical condition (L0 − 1) |φMi = 0. Now we change yM−2 =
zM−2yM−3, then we have . . . Z 1 0 dzM−2yM−3yLM0−2 V (kM−2, 1) yM−2 y−L0 M−2 Z 1 0 dzM−1yML0−2−1zML0−1−2V (kM−1, 1) |φM(kM)i = . . . Z 1 0 dzM−2yM−3yLM0−2−2V (kM−2, 1) Z 1 0 dzM−1zML0−1−2V (kM−1, 1) |φM(kM)i = . . . Z 1 0 dzM−2yML0−3−1z L0−2 M−2 V (kM−2, 1) 1 L0− 1 V (kM−1, 1) |φM(kM)i
where we have used R01dzzL0−2 = 1
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Comparing expressions from last two steps, we see that we can iterate this procedure to AM = gM−2 φ1 V2(k2) 1 L0− 1 V3(k3) . . . 1 L0− 1 VM−1(kM−1) φM (5.6) Form (5.6) is the convenient one to compare with BCFW recursion relation, because lo-cations of poles are clearly indicated by propagator L1
0−1. For example, for 1
L0−1 between
vertex operators Vi and Vi+1, pole locations are given by
1
2(k1+ . . . + ki)
2
+ N − 1 = 0, N = 0, 1, 2, . . . (5.7)
Now let us consider the (1, M )-deformation given in (2.1) and use ziN to indicate the
solution obtained from equation (5.7) with k1 → k1+ zq. Because it has been proved that
boundary contribution is zero under the deformation at least for some kinematic region, we have immediately AM = gM−2 MX−2 i=2 ∞ X N=0 2Ri,N (k1+ k2+ . . . + ki)2+ 2(N − 1) (5.8) where
Ri,N = hΦi,N|Ψi,Ni
hΦi,N| = φ1(k1+ zi,Nq)| V2(k2) 1 L0− 1 V3(k3) . . . 1 L0− 1 Vi(ki) |Ψi,Ni = Vi+1(ki+1) 1 L0− 1 . . . VM−1(kM−1) φM(kM − zi,Nq) (5.9) What we want to prove is that residue Ri,N can be obtained from summing over
interme-diate physical states prescribed by BCFW on-shell recursion relation.
5.2 The proof
Now we give our proof. First, we notice that both states hΦi,N| , |Ψi,Ni are physical states,11
thus in the frame work of DDF-state construction, both physical states can be written as |sphyi + |fi, where |fi is the DDF-state while |sphyi is physical spurious states. Using the
property of spurious state, we have
hΦi,N|Ψi,Ni = sLi,N + fi,NL |sRi,N+ fi,NR
=fi,NL |fi,NR (5.10) Having established (5.10) we insert identity operator in the Fock space with given momentum Pi,N = k1+ zi,Nq + k2+ . . . + ki and annihilated by (L0− 1), so
fi,NL |fi,NR = X
i
D
fi,NL |ψi†(Pi,N)E ψi(Pi,N)|fi,NR
(5.11) where set {|ψi(Pi,N)i} can be any normalized orthogonal basis. In DDF-frame work, a
general state can be written as the linear combination of |ki , |si , |fi, i.e., a choice of the
11The proof can be found in a standard text, for example in Superstring Theory by Green, Schwarz and Witten [16] (chapter 7, vol. 1.).
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basis is |ki , |si , |fi. Using the definition of states, we see immediately that hs|fi = 0 and hk|fi = 0, thus
fi,NL |fi,NR = X
i
D
fi,NL |fi†(Pi,N)E fi(Pi,N)|fi,NR
= X
i
D
sLi,N+ fi,NL |fi†(Pi,N)E fi(Pi,N)|sRi,N + fi,NR
= X i D Φi,N|fi†(Pi,N) E hfi(Pi,N)|Ψi,Ni (5.12)
Using (5.10) and (5.12) we see immediately Ri,N = X i D Φi,N|fi†(Pi,N) E hfi(Pi,N)|Ψi,Ni (5.13)
which is the prescription given by BCFW recursion relation. Thus we have given our proof. 5.3 Practical method for summing over physical states
Having shown that BCFW recursion relation gives the right string amplitude, we need to explain how to sum over physical states. The difficulty of the sum is that the physical state is hard to describe in general, i.e., we do not know how to write down polarization vector for a given physical state. However, from the equivalent between (5.11) and (5.12) we see that we can replace the sum over all physical states to the sum over whole Fock space with given momentum and annihilated by (L0− 1). For the Fock space, there is a freedom with
the choice of basis and the one convenient for real calculation is oscillation basis defined in (3.14). Thus the residue can be calculated by
Ri,N = X {Nµ,n} D Φi,N|{Nµ,n}; bP E T{Nµ,n} D {Nµ,n}; bP |Ψi,N E = X P nnNn=N * Φi,N ( ∞ Y n=1 (αµNn,1 −n α µNn,2 −n . . . α µNn,Nn −n ) p Nn!nNn ) 0; bP + ∞ Y n=1 (gµNn,1νNn,1gµNn,2νNn,2. . . gµNn,NnνNn,Nn) * 0; bP (∞ Y n=1 (ανNn,1 +n α νNn,2 +n . . . α νNn,Nn +n ) p Nn!nNn ) Ψi.N + (5.14)
6 Scattering with higher spin particles
Having established the general method given in (5.14), let us consider scatterings when higher spin particles are present. However, before doing this, let us recall some results coming from scattering amplitudes of pure tachyons. By checking with (3.32) and (4.28), we see that residues are given as series of Lorentz invariants ki· kj with coefficients given
by Stirling number of the first kind s(N, J) =P{Nn}Q∞n=1N 1
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of ki · kj reproduces the residue in combinatorial form observed in [6]. This relation is
established by writing generating function of Stirling number into two different forms eX ln(1−z)= e−X (z+z22 + z3 3 +... )= e−X ze−X z2 2 e−X z3 3 . . . = 1 + (−)Xz + (−) 2 2! X 2z2+ . . . 1 + (−)Xz 2 2 + (−) 2X z2 2 2 + . . . ! . . . (6.1) and (1 − z)X = ∞ X a=1 (−)as(a, J)a! XJza=X a (−)a Xa ! za (6.2)
by matching power of z and setting X = k2· k3. In fact, it is straightforward to see that
residues in an arbitrary n-point pure tachyon scattering amplitude can be read off from products of generating functions
eX23ln(1−z23)eX24ln(1−z24). . . eXn−2,n−1ln(1−zn−2,n−1) (6.3)
with Xij = ki · kj, zij = zj/zi, and residues in tachyonic recursion relation can be found
through binomial expansion of
(1 − z23)X23(1 − z24)X24. . . (1 − zn−2,n−1)Xn−2,n−1. (6.4)
Having recalled the experience from tachyon amplitude, now we discuss the scattering amplitude of 3-tachyon and 1-vector, which is given by
A(1234) = Z 1 0 dz2 z2 D 0, k1 ǫ2· ˙X : eik2·X(z2): : eik3·X(z3): 0, k 4E z3=1 (6.5) = Z 1 0 dz2 −ǫ2· k1(1 − z2)k3·k2zk21·k2−1+ ǫ2· k3(1 − z2)k2·k3−1zk21·k2 (6.6) where 2 means that the second particle is a vector. As in the case of pure tachyon scattering we binomially expanding (1 − z2)k3·k2 in (6.5) and integrating over z2, yielding
A(1234) = − ∞ X a=0 (−)aǫ2· k1 k3· k2 a ! 2 (k2+ k1)2+ 2(a − 1) + ∞ X a=1 (−)a−1ǫ2· k3 k3· k2− 1 a − 1 ! 2 (k2+ k1)2+ 2(a − 1) . (6.7)
We are interested in relating residue in (6.7) with residue given by BCFW prescription D 0, k1 ǫ2· ˙X : eik2·X(z2): |{Nµ,m}, pi T{Nµ,m}h{Nµ,m}, p| : e ik3·X(z3) : 0, k4E z2=z3=1 . (6.8) It is straightforward to see at the first few levels, residues in (6.7) agree with those pre-scribed by (6.8) table 1.
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intermediate state |{Nµ,m}i T{Nµ,m}h{Nµ,m}| contribution ∼ ǫ2· k3
N = 0 |0i h0| absent N = 1 α µ −1 √ 1 |0i ηµνh0| αν 1 √ 1 (−) (ǫ2· k3) N = 2 αµ−2 √ 2 |0i ηµνh0| αν 2 √ 2 P µ1<µ2 αµ1−1 √ 1 αµ2−1 √ 1 |0i ηµ1ν1ηµ2ν2h0| αν 1 √ 1 αν21 √ 1 1 √ 2! α√µ−1 1 α√µ−1 1 |0i (ηµν) 2 h0|√1 2! αν 1 √ 1 αν 1 √ 1 (−) (ǫ2· k3) (ǫ2· k3) (k3· k2)
Table 1. Residues of 3-tachyon, 1-vector scattering for first three levels
Note that algebraically, the first term proportional to ǫ2 · k1 in (6.7) was obtained
from moving an operator ǫ2· α0 in ǫ2· ˙X(z2) = ǫ2· (α−1z21+ · · · + α0z20+ α1z−12 + . . . ) to
the left, acting upon final state |0, k1i in the standard process of normal ordering, which
simply reproduces the pure tachyon residue since rest of its kinematic dependence was contributed from 0, k1 : eik2·X(z2) : : eik3·X(z3): 0, k4 . It is therefore straightforward to
show that, following the same expansion as in the case of pure tachyon scattering, at each mass level residue contributed from this term is connected to BCFW prescription by generating function for Stirling number of the first kind. New structure however, is found in the second term proportional to ǫ2· k3 in (6.7), which was produced by moving positive
mode operators α1z2−1+ α2z2−2+ . . . in ǫ2· ˙X(z2) = ǫ2· (α−1z21+ · · · + α0z20+ α1z−12 + . . . )
to the right and contracting with intermediate states. For example when we have a Fock state α
µ1 −qαµ2−r √q√r √1
2!|0, pi as intermediate state, equation (6.8) reads
* 0, k1 (ǫ2· ∞ X n=1 αnz2−n)e− 1 nk2·αnz −n 2 α µ1 −qα µ2 −r √q√ r 1 √ 2! 0, p + ηµ1µ2ην1ν2 0, p αν1 q ανr2 √q√ r 1 √ 2!e −1 nk3·αnz n 3 0, k4 z2=z3=1 . (6.9)
Contribution proportional to ǫ2· k3 is produced by contracting an αq or αr in ǫ2· ˙X(z2)
with Fock state, yielding (ǫ2· k3) × q q z3 z2 q (k3· k2) r z3 z2 r + (k3· k2) q z3 z2 q (ǫ2· k3) × r r z3 z2 r z 2=z3=1 . (6.10) Therefore generically residue (6.8) proportional to ǫ2· k3 at level N = a is given by zaterm
expansion coefficient of the derivative of generating function (ǫ2· k3) (k2· k3) z d dze (k2·k3) ln(1−z) (6.11) = (ǫ2· k3) (k2· k3) z d dz e−X ze−Xz22 e−X z3 3 . . . .
Note that we may as well express the generating function (6.11) above as (ǫ2· k3) z
d
dz [ln(1 − z)] e
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from which it is obvious that BCFW prescription yields the same residue as tachyonic recursion relation of 1-vector 3-tachyon amplitude, since the tachyonic recursion relation was derived from binomial expansion of standard worldsheet integral formula that takes the same form as (6.12).
Explicit recursion relation. Here we present an explicit calculation of the term pro-portional to ǫ2· k3 in eq. (6.7). By using eq. (6.8), the term proportional to ǫ2· k3 with
mass level N can be calculated by gluing two 3-point functions IN = X {PmNm=N } D k1; 0 ∞ X n=1 ǫ2· αn V0(k2) {Nm} ; P E T{Nm} D {Nm} ; P V0(k3) k4; 0 E z 2=1 .
For convenience, let us denote the two 3-point functions as AL= AL(k1, k2, P ) = D k1; 0 ∞ X n=1 ǫ2· αn V0(k2) {Nm} ; PE z2=1 , (6.13) AR= AR(P, k3, k4) = D {Nm} ; P V0(k3) k4; 0 E z2=1 . (6.14)
The term AR was obtained in eq. (3.29) previously, while AL can be calculated to be (we
ignore the momentum dependent part) AL= ∞ X n=1 D 0 ǫ2· αn Y∞ m=1 e−k2·αmm α µ −m Nm p mNmNm! 0E (6.15) = ∞ X n=1 D 0 ǫ2· αn " e−k2·αnn α µ −n Nn p nNnN m! # ∞ Y m=1,m6=n e−k2·αmm α µ −m Nm p mNmN m! 0E. (6.16) In the presence of ǫ2· αn term, one notes that only term of order (Nn− 1) in the
Tay-lor expansion of exp − k2· αn/n
inside the square bracket will contribute. By using [ αµm, ανn] = mδm+nηµν , we get AL= ∞ X n=1 " (−)Nn−1n N nǫµ2(kµ2)Nn−1 p nNnNn! # ∞ Y m=1,m6=n − k2µ Nm p mNmNm! . (6.17)
Combining AR and AL and summing over all states with PmmNm = N yields
IN = X {N=PmmNm} (−) N kǫ2· k3 2· k3 ∞ Y m=1 − k2· k3 Nm mNmNm! . (6.18)
We can now use the definition of Stirling number of the first kind to get IN = ǫ2· k3 N X J=1 s(N, J) N ! (−) N−1N (k 2· k3)J−1. (6.19)
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Finally the expression can be further reduced to IN = ǫ2· k3 (−)N−1 k2· k3− 1 N − 1 . (6.20)
In the following, instead of the operator method adopted previously, we will use path-integral approach [17] to calculate the generating function for the rank-two tensor, three tachyons amplitude. As a warm up exercise, we first use this method to rederive eq. (6.12) for the vector, three tachyons amplitude. We first note that the amplitude can be written as
A = Z Y1 i=1 dzi< eik1X(z1)ǫ2· ∂X(z2)eik2X(z3)eik3X(z3)eik4X(z4)> (6.21) = Z Y4 i=1 dzi< eik1X(z1)eik2X(z2)+iǫ2·∂X(z2)eik3X(z3)eik4X(z4)>|linear in ǫ2 (6.22) = Z Y4 i=1 dziexp −X l<j klµkjν< Xµ(zl)Xν(zj) > − X j6=2 ǫ2µkjν < ∂Xµ(zl)Xν(zj) > |linear in ǫ2 (6.23) = Z 1 0 dz(1 − z) k2·k3zk1·k2 ǫ2· k1 z − ǫ2· k3 1 − z . (6.24)
In the last equality, we have used the worldsheet SL(2, R) to set the positions of the four vertex at 0, z, 1 and ∞, and the propagator < Xµ(z
l)Xν(zj) >= −ηµνln(zl− zj). Note
that the term proportional to ǫ2· k1 has been considered previously for the calculation of
four tachyons amplitude. One can now see from eq. (6.23) that the generating function for amplitude proportional to the term ǫ2· k3 is
G1 = exp{−k3·k2[− ln(1−z)]}exp{−ǫ2·k3z d dz[− ln(1−z)]} |linear in ǫ 2 (6.25) = (ǫ2· k3)z d dz[ln(1 − z)] exp{k 3·k2[ln(1−z)]} (6.26)
which is the same with eq. (6.12). Therefore the derivative of generating function in eq. (6.11) can be traced back to the derivative part ∂Xµ of the vector vertex. We now
generalize the calculation to the higher spin cases. For example, for the spin two case
A = Z Y4 i=1 dzi< eik1X(z1)ǫ2µν· ∂Xµ(z2)∂Xν(z2)eik2X(z2)eik3X(z3)eik4X(z4)> (6.27) = Z Y4 i=1 dzi< eik1X(z1)eik2X(z2)+iǫ (1) 2 ·∂X(z2)+iǫ(2)2 ·∂X(z2)eik3X(z3)eik4X(z4)>| multilinear in ǫ(1)2 ,ǫ (2) 2 (6.28) = Z 1 0 dz(1 − z) k2·k3zk1·k2 " ǫ(1)2 · k1 z − ǫ(1)2 · k3 1 − z # " ǫ(2)2 · k1 z − ǫ(3)2 · k3 1 − z # (6.29)
where ǫ(l)3µǫ(j)3ν is to be identified with ǫ3µν. Note that the terms proportional to kµ1k1ν and
kµ1kν
3 have been considered previously for the calculation of four tachyons and one vector,
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which can be expressed as A4 = ∞ X a=2 k2· k3− 2 a − 2 (−1)a−2(k 2 1+ k2)2+ 2(a − 1) ǫ2µνk3µkν3. (6.30)
The generating function for this term can be seen from eq. (6.28) as
G2 = exp{−k3·k2[− ln(1−z)]}exp n −ǫ(1)2 ·k3zdzd[− ln(1−z)] o exp n −ǫ(2)2 ·k3zdzd[− ln(1−z)] o |multilinear in ǫ(1) 2 ,ǫ (2) 2 (6.31) = ǫ(1)2 · k3 z d dz[ln(1 − z)] exp{ k3·k2 2 [ln(1−z)]} ǫ(1)2 · k3 z d dz[ln(1 − z)] exp{ k3·k2 2 [ln(1−z)]} (6.32) = ∞ X a=2 k2· k3− 2 a − 2 (−1)a−2ǫ2µνk3µkν3za. (6.33)
Eq. (6.32) contains product of two derivative terms which again can be traced back to ∂Xµ∂Xν part of the spin two vertex. After setting z = 1 in eq. (6.33) above, one can
match with the correct result in eq. (6.30).
The calculation above can be generalized to arbitrary higher spin vertex. We thus conclude that generically generating function for Stirling number of the first kind connects BCFW precription with scalar-like recursion relation to arbitrary high spin level scatter-ing, provided that the corresponding derivatives in its worldsheet integral expression are included.
7 Conclusions
Starting from the familiar 4-point Veneziano formula we have demonstrated that the scalar-like recursion relation observed by Cheung, O’Connell and Wecht in [6] and by Fotopoulos in [7] can indeed be understood from BCFW on-shell recursion relation of string ampli-tudes. We showed that explanation to the absence of higher-spin modes was very much like a similar mechanism observed in BCFW on-shell recursion relation of gauge theory amplitudes: while in gauge theory Ward identity guarantees that two unphysical degrees of freedom necessary to make up for the completeness relation [13]
gµν = ǫ+µǫ−ν + ǫ−µǫν++ ǫLµǫTν + ǫTµǫLν (7.1)
decouple, in bosonic string amplitude the No-Ghost Theorem does the same thing to de-couple necessary unphysical degrees of freedom that make up for the whole Fock space completeness relation, which makes the translation between covariant and scalar-behaved on-shell relations of string amplitudes. The freedom to translate on-shell recursion relation between Fock state and physical state is especially of practical interests since writing down polarization tensors for generic physical high-spin modes can be quite complicated in string theory context.
Although our method can be used to calculate string scattering amplitudes using the on-shell recursion relation, it may be not the best way to do so. However, it could provide
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another point of view to discuss some analytic properties of string theory along, for example, the work of Benincasa and Cachazo [18], and the work of Fotopoulous and Tsulaia [9], based on consistency using different BCFW-deformations to calculate amplitudes. It can also be used to discuss possible loop amplitudes using unitarity cut method [14,15].
Acknowledgments
We would like to thank R. Boels, F. Cachazo, D. Skinner for valuable discussions. B.F would like to thank the hospitality of Perimeter Institute where this work was presented. This work is supported, in part, by fund from Qiu-Shi and Chinese NSF funding under contract No.11031005, No.11135006, No. 11125523. CF is supported by National Science Council, 50 billions project of Ministry of Education and National Center for Theoretical Science, Taiwan, Republic of China. We would also like to acknowledge the support of S.T. Yau center of NCTU.
A Mathematical identity
Stirling Number of the first kind: the Stirling numbers of the first kind is defined from the generation function
(x)n≡ x(x − 1) . . . (x − n + 1) = n
X
k=0
s(n, k)xk (A.1)
where (x)n is the Pochhammer symbol for the falling factorial and when n = 0, (x)0 ≡ 1.
Using this, we can see that s(0, 0) = 1 but s(n, 0) = 0 if n 6= 0.
The signed Stirling numbers of the first kind are defined such that the number of permutations of n elements which contain exactly m permutation cycles is the nonnegative number |s(n, m)| = (−)n−ms(n, m) = n!X {Nt} ∞ Y t=1 1 Nt!tNt , XtNt= n, m = X Nt (A.2)
There are other ways to see above identities. Considering following Taylor expansion I1= (1 − z)X = ∞ X a=0 X a ! (−)aza (A.3)
which can be expanded by following alternative way expX ln(1 − z)= exp " (−X) z +1 2z 2+1 3z 3 + · · · + n1zn+ · · · # = ∞ X N=0 N X J=0 s(N, J) N ! (−X) JzN = ∞ X N=0 N X J=0 s(N, J) N ! (−) NXJzN (A.4)
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B Decoupling of Ghosts in string amplitudeThe content in this section can be found in [16]. In bosonic string theory, physical states are required to satisfy Virasoro constraints (L0− 1) |φi = 0 and Lm>0|φi = 0. As we have
seen in section3.2, the first of these two types of constraints was implemented as on-shell condition (3.7) so that it is satisfied by intermediate states that appear in BCFW recursion relation. In this appendix we prove that ghosts decouples from BCFW recursion relation. As a consequence we are allowed to introduce freely the physical states, for which the remaining Virasoro constraint Lm>0|φi = 0 applies, or generic Fock states as intermediate
states in the recursion relation. For the purpose of argument needed in this proof we first divide Fock space into three subspaces according to DDF construction.
B.1 DDF states
A standard DDF state is defined by acting a string of transverse Ai
−noperators on tachyonic vacuum |fi = Ai1 −n1A i2 −n2. . . A im −nm|0; p0i , (B.1) where DDF operator Ai
n is prescribed as the Fourier zero mode of vector vertex operator
Vj(nk0, τ ) = ˙Xj(τ )einX +(τ ) , Ain= 1 2π Z 2π 0 ˙ Xi(τ )einX+(τ )dτ, i = 1, . . . , D − 2 , (B.2) and p0 = (p+0, p−0, pi0) = (1, −1, 0). It is easy to show that Lm>0|fi = 0 since Lm
com-mutates with all Ai
−n while Lm>0|0; p0i = 0. For L0, using that L0 0; p20
= α′p2 0 = 1
we get (L0 − 1) |0; p0i = 0. The DDF states thus defined are positive definite, as can be
easily checked using the commutation relation [Ai m, A
j
n] = mδijδm+n. We shall denote in
the following a generic DDF state as |fi. Note however, that in the standard construction these DDF states are automatically on the N -mass-shell,
b p |fi =p0+ k0 X ni |fi (B.3)
so that (p0 + N k0)2 = p20 + 2N = 2 + 2N , where we introduced k0 = (k+0, k−0, ki0) =
(0, −1, 0), and here N =Pni. In order to describe Fock states in DDF language, where
center-of-mass momentum kµ and mode number N are considered independent, let us
define generalized off-shell DDF-like state, starting again from tachyonic vacuum but with momentum q + N k0, |fioff−shell= Ai1 −n1A i2 −n2. . . A im −nm|0; q + N k0i . (B.4)
Note that we shift ground state momentum by equal and opposite of the amount that is going to be shifted by DDF operators so that subsequent operations produces an off-shell state with arbitrary momentum q and mode eigenvalue N = Pi ni. In addition to DDF
operators we introduce operators Km, defined as
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and consider states constructed by operating a string of Virasoro generator L−n and K−m on DDF-like state |fioff−shell carrying off-shell momentum q in the following order
|{λ, µ}, fi = Lλ1 −1L λ2 −2. . . L λn −nK−1µ1 . . . K µm −m|fioff−shell . (B.6)
The set of states |{λ, µ} , fi withPrλr+Psµs+Pni = N are linearly independent and
constitutes a basis that spans level-N subspace at fixed momentum q. In the following discussions for convenience we drop the lower script that distinguishes DDF state |fi and DDF-like state |fioff−shell, while it is understood that the center-of-mass momentum is considered as a independent degree of freedom, on-shell or not, whenever a DDF basis is referred to.
B.2 Decoupling of ghosts in string amplitude
States (B.6) can be divided into two types. The first type is with L−n in front, so it is spurious state |si. The second one is without L−nand we denote it as |ki. Thus any state
in the Fock space can be uniquely decomposed as
|φi = |si + |ki (B.7)
where |si is the spurious state and |ki is the form in (B.6) without any L−n in front of the expression. Since |si , |ki are linear independently, if |φi is the eigenstate of L0, so are
|si , |ki. This means that if
(L0− 1) |φi = 0, =⇒ (L0− 1) |si = (L0− 1) |ki = 0 (B.8)
Next we show that if the state |φi is physical state, the decomposed states |si and |ki are also physical states.
Because |si is spurious and physical when |φi is physical, we have hs|si = hs|ki = 0, so hφ|φi = hk|ki. We can decompose |ki = |fi + ekEwhere |fi is DDF state and |ki is the form of (B.6) without string of L but at least one of K−m. By the property of K−m, it is easy to shown that Dek|ekE=Dek|fE= 0, so finally we have hφ|φi = hk|ki = hf|fi. This is the familiar result known as the “No-ghost Theorem” for string amplitude, which can also be characterized as the absence of negative norm among general physical state |φi.
In fact, there is a stronger statement. Using [Lm, Kn] − nKm+n and Lm>0|fi = 0, it
can show that if |ki is physical, then ekE= 0 in the expansion of |ki = |fi + ekE. Thus we see that the general physical state |φi can be written
|φi = |fi + |si (B.9)
where |fi is a DDF state and |si is a spurious physical state. The appearance of spurious physical state |si, i.e., the transformation |fi → |fi + |si is the string-theoretic analog of a gauge transformation.
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B.3 Decoupling of ghosts in BCFW on-shell recursion relation
In section 5 we saw that pole structure in a bosonic string amplitude is manifest when expressed in algebraic form
AM = hφ1| V2∆V3. . . Vi∆Vi+1. . . VM−1∆ |φMi . (B.10)
Residue at the (i − 1)-th pole at mass level N is therefore given by the sum of products X
level-N states
hφ1(k1+zi,Nq)| V2∆V3. . . Vi|{Nµ,m}, ˆpih{Nµ,m}, ˆp| Vi+1. . . ∆VM−1|φM(kM−zi,Nq)i ,
(B.11) where the above sum is taken only over intermediate Fock states that happen to be on the level-N mass-shell. Note that in BCFW recursion relation the mode eigenvalues {Nµ,m}
and center-of-mass momentum ˆp of intermediate states were originally considered as inde-pendent. It is because {Nµ,m} and ˆp assume the valuesPNµ,m = N and 12pˆ2(z)+N −1 = 0
that a pole was created atz = zi,N in the first place, so that at pole the mass-shell condition
is automatically satisfied. Consider the state
|φRi = Vi+1∆Vi+2. . . ∆VM−1|φMi (B.12)
that appears on the right side of equation (B.11). Since we are only interested in its product with on-shell states, let us operate on it a projection operator P1. For the purpose
of proving decoupling of ghosts, first we would like to show that
Lm>0P1|φRi = 0, (B.13)
where we defined Pkas a projection operator which projects states to subspace with L0 = k.
Using [L0, Lm] = −mLm, we find L0LmP1|αi = (1−m)LmP1|αi, so LmP1 = P1−mLmP1=
P1−mLm, thus we need to prove
P1−mLm|φRi = 0, m > 0 (B.14)
Using P1−m(−L0− m + 1) = 0, we get
P1−m(Lm− L0− m + 1) |φRi = 0, m > 0 (B.15)
Finally we arrive at the identity
(Lm− L0− m + 1)VN∆VN+1. . . ∆VM−1|φMi = 0, m > 0 (B.16)
Note that a vertex V has conformal dimension one, therefore satisfies [Lm, V (k, z)] = zm+1 d dz + mz m V (k, z). (B.17)
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Now using the (B.17) and set z = 1 (since we have τ = 0 which is crucial) we have
[Lm− L0, V ] = mV, or (Lm− L0− m + 1)V = V (Lm− L0+ 1) (B.18)
where dzd has been canceled. Using Virasoro algebra it is straightforward to show that (Lm− L0+ 1) 1 L0− 1 = 1 L0+ m − 1 (Lm− L0− m + 1) (B.19)
Thus (B.18) and (B.19) give (Lm− L0− m + 1)V 1 L0− 1 = V 1 L0+ m − 1 (Lm− L0− m + 1) (B.20)
so (Lm− L0− m + 1) can be pushed step by step all the way to the right until it meets
|φMi, and we obtain (Lm− L0+ 1) |φMi = 0 because |φMi is physical. From the argument
above we see that when on-shell, |φMi satisfies Virasoro constraints and is therefore a
physical state. It is straightforward to see that the same argument applies to state |φLi =
Vi∆Vi−1. . . ∆V2|φ1i.
Proof: having done all the preparations we are now finally ready to derive our proof. We note that in the algebraic expression (B.11) for residue at mass level N , the summation of outer products of Fock states |{Nµ,m}, ˆpi T{Nµ,m}h{Nµ,m}, ˆp| over level-N subspace works
as a projection operator that maps |φRi and |φLi into the level-N subspace, so that if we
decompose in this sector |φRi and |φLi according to DDF basis into |si +
˜kE+ |fi, the residue (B.11) reads X level-N states hφL|{Nµ,m}, ˆpi T{Nµ,m}h{Nµ,m}, ˆp|φRi = D sL+ ˜kL+ fL|sR+ ˜kR+ fR E = hfL|fRi . (B.21)
As argued in the decoupling of ghosts in amplitudes, spurious state |si drop out from (B.21) because both |φRi and |φLi are physical, and we remove subsequently
˜kE states since D ˜ k|˜kE=Dk|f˜ E= 0.
Inserting complete states again, but this time in DDF basis, into the product hfL|fRi,
hfL|fRi = X i hfL|si+ ki+ fii hsi+ ki+ fi|fRi =X i hfL|fii hfi|fRi = X i hfL+ sL|fii hfi|fR+ sRi =X i hφL|fii hfi|φRi (B.22)
and we see that spurious and ˜kE intermediate states drop out for the same reason, thus summing over the whole intermediate Fock space is equivalent to summing over the physical subspace.
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