BCFW 遞迴關係式與玻色開弦中樹圖的散射振幅

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國

國立

立

交

交通

通

大

大學

學

電

電子

子

子物

物

物理

理

理研

研

研究

究

究所

所

碩

士

論

文

BCFW 遞

遞

迴

迴關

關

關係

係

係式

式

式與

_與

_與玻

_玻

_玻色

_色

_色開

開

開弦

弦

弦中

中

中樹

_樹

_樹圖

_圖

_圖的

的

的散

散

_射

_射振

_振

_振幅

幅

BCFW recursion relation and tree level amplitudes

for bosonic open string

研究

生：張永業

指導教授：李仁吉

教

_授

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BCFW 遞

遞

迴

迴關

關

關係

係

係式

式

式與

_與

_與玻

_玻

_玻色

_色

_色開

開

開弦

弦

弦中

中

中樹

_樹

_樹圖

_圖

_圖的

的

的散

散

_射

_射振

_振

_振幅

幅

BCFW recursion relation and tree level amplitudes

for bosonic open string

研究

生：張永業

_{Student: Yung-Yeh Chang}

指導教授：李仁吉

教

_授

_{Advisor: Prof. Jen-Chi Lee}

國立交通大學

電子物理研究所

碩士論文

A Thesis

Submitted to Department of ElectroPhysics

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in

ElectroPhysics

January 2013

Hsinchu, Taiwan, Republic of China

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BCFW 遞

遞

迴

迴關

關

關係

係式

係

式

式與

_與

_與玻

_玻

_玻色

_色

_色開

開

開弦

弦

弦中

中

中樹

_樹

_樹圖

_圖

_圖的

的

的散

散

_射

_射振

_振

_振幅

幅

學生：張永業

指導教授：李仁吉

教

_授

國立交通大學電子物理研究所

摘

要

在本篇論文中，我們將BCFW遞迴關係式的應用從場論散射振幅推廣到弦論。計算弦論的散_{射振幅困難之處在於需要對無限多個中間物理態 (Intermediate physical state) 做} 加總。我們將加總的範圍從物理態擴大到全部的 Fock states 解決了這個問題，而且成功地利用此方法計算出四個快子(tachyon)的散射振幅；並更進一步計算出一個任意物理態與三個快子的散射振幅。除此之外，我們了解到利用上述的方法計算散射振幅須要生成函數(generating function)的輔助，且利用路徑積分得到此生成函數的一般結構_。

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BCFW recursion relation and tree level amplitudes for bosonic

open string

Student: Yung-Yeh Chang

Advisor: Prof. Jen-Chi Lee

Department of ElectroPhysics

National Chiao Tung University

ABSTRACT

In this thesis, we extend the application of BCFW recursion relation to string tree-level amplitudes. In contrast to the field theory calculation, we encounter the difficulty of summing over all intermediate physical states with infinite tower of mass levels. We develop a method to resolve this difficulty by enlarging the sum over all intermediate physical states to an easier sum over the entire Fock space of string spectrum. The calculation is successfully applied to the 4-tachyon amplitude and then to the cases of one arbitrary higher spin state and 3-tachyon amplitudes. We also figure out a generating function for summing the infinite poles of string spectrum in the BCFW string amplitude calculation. The generic structure of this generating function for higher spin scattering amplitude can be obtained from the standard path integral calculation of string scattering amplitude.

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Acknowledgement

First and foremost, I would like to express my gratitude to Prof. Jen–Chi Lee, my supervi-sor, for his patient guidance throughout my master studies. I am thankful for the valuable comments and suggestions of the thesis committee Prof. Pei–Ming Ho, Prof. Chong–Sun Chu and Prof. Yi Yang, which made this master thesis more readable. Furthermore, I would also like to acknowledge Dr. Chih–Hao Fu for his help and collaboration in this project. I appreciate Dr. George Moutsopoulos, Dr. Yoshihiro Matsuka, Dr. Shang–Yu Wu and Pei–Hung Yuan for helpful discussions.

I thank my friends and fellow graduate students, Yi–Shuan Lin, Sheng–Hong Lai and Yao–Yuan Shih for stimulating discussions in physics, for working together on the homework, and for all the fun we have had. I would also like to thank a special person, my girl friend Wei–Jyun Yu, for her unselfish support and dedication.

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Chapter 1 Introduction

The calculation of scattering amplitudes has been a central issue in quantum field theory in which Feynman’s rule provides a clear picture in the calculation of scattering processes. However, with the increasing number of external particles, the efficiency of this method was restricted since the number of Feynman diagrams increase tremendously. As a result, the previous statement may need to be changed from calculating scattering processes to how to calculate scattering amplitudes more efficiently. To do so, many novel theories such as spinor method, color–ordered technique and BCFW on-shell recursion relation popped out one after another during the past few decades.

The BCFW method was initially proposed for gauge field theory. It merely relies on the general complex analytic structures of scattering amplitudes. The original higher point scattering amplitude can then be expressed as sum of products of lower point on-shell scattering amplitudes. As a result, one can recycle the calculation for lower point functions to simplify the calculation for higher point functions.

The success of BCFW calculation of scattering amplitudes in quantum field theory motivates us to extend the calculation to string theory. In this thesis, we extend the application of BCFW recursion relation to string tree-level amplitudes. In contrast to the field theory calculation, we encounter the difficulty of summing over all intermediate physical states with infinite tower of mass levels. We develop a method to resolve this difficulty by enlarging the sum over all intermediate physical states to an easier sum over the entire Fock space of string spectrum. The calculation is successfully applied to the

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4-1.1. LITERATURE REVIEWS

tachyon amplitude and then to the cases of one arbitrary higher spin state and 3-tachyon amplitudes.

This thesis is organized as following. In Chapter 2, we give a brief introduction to BCFW recursion relation in quantum field theory, and some basics of string theory. Chapter 3 is divided into two parts: the first part contains spinor semiology. Then, in the second part, we adopt the BCFW recursion relation to compute a concrete example, namely, 4-gluon color-ordered scattering amplitude. In Chapter 4, we begin with the familiar four-point Veneziano formula, and demonstrate that how one can extend BCFW method to four-tachyon string scattering amplitude. In Chapter 5, we extend BCFW method to string scattering processes with a higher spin vertex and 3 tachyons. Finally, we give a brief conclusion of this thesis. The last three chapters are mainly based on our paper [1], which has been accepted in January, 2013.

1.1 Literature Reviews

In [2,3], Britto, Cachazo, Feng and Witten (BCFW) proposed a recursion relation for scattering amplitudes of Yang–Mills theory based on deforming the momenta and taking the analytic continuation over the complex plane. After doing so, the amplitude can then be characterized by its poles and the corresponding residues. This feature allowed one to express the higher point scattering amplitude in terms of sum of products of two lower-point on-shell scattering amplitudes.

The extension of BCFW recursion relation from field theory to string theory was initiated by Rutger Boels, Kasper Jens Larsen, Niels A. Obers and Marcel Vonk in [4]. They showed that BCFW technique is applicable for all open 4–point amplitudes in flat space. They also conjectured that BCFW method could be extended to higher point amplitudes and to closed string cases.

In 2010, Clifford Cheung, Donal O’Connell, and Brian Wecht [5] demonstrated that all tree-level amplitudes possessed convergent asymptotic behavior and thus allowed ap-plication of BCFW recursion relation. Furthermore, in this paper pole structures were made manifest through binomially expanding the Koba–Nielson formula for tachyon

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am-1.1. LITERATURE REVIEWS

plitudes.

In [6], Angelos Fotopoulos proposed how to construct the Veneziano amplitude via BCFW procedure by applying conjectured 3-point function with two tachyons and an arbitrary intermediate massive state. Namely, for four-point function, the product of two 3-point tachyon-liked amplitudes can be produced by summing over all massive interme-diate states.

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Chapter 2 Preliminaries

In this chapter, we provide a very concise introduction to BCFW recursion relation as well as some background knowledge of string theory.

2.1 Review of BCFW recursion relation

BCFW on–shell recursion relation [2,3] allows us to express on–shell amplitudes as sums of products of relatively lower–point on–shell amplitudes. It is known that, from the Feynman’s rules, the essential ingredients for tree–level amplitudes are propagators and vertices, which implies that scattering amplitudes are rational functions in terms of kine-matic variables. If we shift two of the external momenta into complex plane by

ˆ

k1(z) = k1+ zq, kˆn(z) = kn− zq. (2.1)

Energy-momentum conservation is manifestly preserved, ˆk1+ ˆkn= k1+ kn. We also need

to impose these constraints q2 _{= q · k}

1 = q · kn = 0 so as to preserve the on–shell conditions

ˆ k2

1 = k12 and ˆk22 = k22 of the deformed pair. Amplitudes are now complex functions with

simple poles, which is the consequence of the light-like condition of the shifted momentum q, locating at the propagators . An complex amplitude with simple poles can be expresses as A(z) =X a Ra z − za . (2.2)

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2.2. STRING THEORY

Taking the analytic continuation over the whole complex plane yields 1 2πi I _dz z A(z) = A(z = 0) + X poles zα6=0 Res A(z) z z=zα . (2.3) A simple pole manifestly exists at z = 0, which reproduces the un–shifted amplitude A(z = 0) while other residues from other finites poles form the products of two relatively lower–point on–shell amplitudes ResA(z)_z

z=zα

= −AL(za)_P12AR(za). If we assume no

boundary contributions as z → ∞ in the left–handed side of equation (2.3), we can then write down the BCFW on–shell recursion relation for n external particles

An(1, 2, · · · , n) = X poles za X physcal h AL(ˆ1, 2, · · · , ˆPah) 1 P2 a AR( ˆPah, a + 1 · · · , ˆn). (2.4)

The first summation indicates that we have to sum over all finite poles zawhile the second

summation is over all physical intermediate states at a given simple pole za.

2.2 String theory

2.2.1 The classical version

Nambu–Goto action

Assume the universe is composed by one dimensional objects, so-called strings, instead of point particles, then we have the corresponding relativistic action

SN G = − T Z dA = − T Z dσdτ −det∂X µ ∂σα ∂Xν ∂σβ ηµν 1₂ = − T Z dσdτ q ( ˙X · X0₎2_{− ˙}_X2_X02_. _(2.5)

This is the well–known Nambu-Goto action. X(σ, τ ) is the t rack swept by the one dimensional object, called w orldsheet, which is parametrized by the coordinate σ and by the evolution τ . While A is the area of X(σ, τ ) bounded by σ and τ . The notations ˙X and X0 mean

˙

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2.2. STRING THEORY

From the principle of least action δS = 0, we have the equations of motion by calculating the Euler–Lagrange equation

∂ ∂τ ∂L ∂ ˙Xµ + ∂ ∂σ ∂L ∂X0µ = 0. (2.7)

This method is the same with classical mechanics : we vary the path and fix the initial point and the end point. The canonical momentum conjugated to Xµ _{can be obtained by}

definition Πµ= ∂L ∂ ˙Xµ = −T ( ˙X · X 0_)X0µ_{− (X}0₎2_X_˙µ h (X0_{· ˙}_X)2_{− ( ˙}_X)2_(X0₎2i 1 2 . (2.8)

Clearly to see that due to the square root appears in the denominator, quantizing this theory is rather complicated.

Polyakov action

In order to avoid the difficulty in quantizing the Nambu–Goto action, the Polyakov action SP is therefore proposed, which is given by

SP = − T 2 Z dσdτ √ hhαβ∂αXµ∂βXνηµν, (2.9)

where h ≡ −dethαβ. From δS = 0, It is easy to get the equations of motion with

respect to the variation of δ(Xµ) and δhαβ, w.r.t. δXµ⇒ ∂α √ hhαβ∂βXµ = 0, (2.10) w.r.t. δhαβ ⇒ ∂αXµ∂βXµ− 1 2hαβh γδ_∂ γXµ∂δXµ= 0. (2.11)

Equation (2.11) comes from the variantion with respect to the induced metric hαβ, i.e.

∂L/∂hαβ = 0. Thus, [7] hαβ = 2∂αXµ∂βXµ hγδ_∂ γXµ∂δXµ . (2.12)

Substitute the metric tensor (2.12) above back to the Polyakov action, we are able to re–derive the Nambu–Goto action.

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2.2. STRING THEORY

The action (2.9) is left invariant with 3 symmetries: Poincare, reparametrization and Weyl rescaling. Using these degrees of freedom, the induced metric hαβ _{can be simplified}

to the two–dimensional Minkowski metric ηαβ

hαβ = ηαβ = diag (−1, 1). (2.13) Replace hαβ _{with η}αβ _{in equation (2.10), the E.O.M. becomes a wave equation,}

namely ∂α∂αXµ= ∂τ2− ∂σ2 Xµ = 4∂+∂−Xµ= 0, (2.14) where ∂+ = ∂ ∂σ+ , ∂− = ∂ ∂σ− (2.15) with σ± ≡ τ ± σ.

To solve for Xµ _{for open strings, boundary conditions have to be imposed at the}

endpoints, i.e. Xµ(τ, σ = 0) = Xµ(τ, σ = π) = 0 and X0µ(τ, σ = 0) = X0µ(τ, σ = π) = 0 . The equation of motion (2.14) is the two–dimensional wave equation with d’Alembert’s solution of wave equation, i.e.

Xµ(σ+, σ−) = X_Rµ(σ−) + X_Lµ(σ+) = xµ− ipµlnz + iX n6=0 1 nα µ nz −n . (2.16)

Note that we have set the Regge slope α0 = 1/2, σ = 0 1 _{[8] and change the variable}

z ≡ exp(iτ ) in the second line of the above equation. The requirement for the reality of Xµ _implies

αµ_−n= (α_nµ)†. (2.17) The Poisson brackets of Xµ _{and the canonical momentum Π}µ _{≡ ∂L/∂ ˙}_X

µ = T ˙Xµ 2

1_{This assumption is often used for vertex operators. It means that the emission of a state at the end}

of the string σ = 0 at the proper time τ . See GSW book, chapter 7.

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2.2. STRING THEORY

are defined in that way similar to classical mechanics

[Xµ(σ), Πν(σ0)]_P.B.= −ηµνδ(σ − σ0). (2.18) The subscripts ”P.B.” denotes the Poisson bracket. This definition quickly leads to the Poisson brackets for the position xµ _{and momentum p}ν _{in the C.M. frame, and for}

the Fourier modes αµ

n of Xµ, i.e.

[αµ_m, αν_n]_P.B. = imδm+nηµν,

[xµ, pν]P.B. = −ηµν. (2.19)

Now let us turn to the constraint equations (2.11) of αµ

m’s. Equation (2.11) demands that

all of the Fourier modes of world sheet Xµ(τ, σ) have to obey T++= T−− = 0 in classical

level. For open string, we are able to do the Fourier transformation to express T++ and

T−− in terms of the ladder operators, yielding

Lm = T Z π 0 eimσT+++ e−imσT−− dσ = T 4 Z π −π eimσ ˙X + X0 2 dσ = 1 2 ∞ X n=−∞ αm−n· αn = 0. (2.20)

2.2.2 The quantum version

The first quantization of open bosonic strings is presented in this subsection. It is known that one standard method of getting into the quantum physics from classical is to promote the physical quantities and Fourier modes to operators. This is equivalent by the substi-tution: replace the classical Poisson brackets with commutator, i.e. [· · · ]P.B. → −i[· · · ].

Thus, equation (2.18) and (2.19) need to be rewritten as follows

[Xµ(σ), Πν(σ0)] = iηµνδ(σ − σ0),

[αµ_m, αν_n] = mδm+nηµν, (2.21)

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2.2. STRING THEORY

Similar analogy are able to be made with simple harmonic oscillators in quantum mechanics. If we normalize αµ m’s such that aµm ≡ αµm/ √ m, then [aµ m, aνn †_{] = η}µν_δ m−n.

The physical interpretation of aµ_m’s is also very much similar to that in simple harmonic oscillators. For αµ

m with m > 0, it lowers a physical state and as a result a µ

m>0|0i = 0.

In contrast, an operator aµ−m with m > 0 rises the level of a physical state. Since the

world sheet contains momentum which does not share the same Hilbert space with the oscillation operators. Therefore, a completely ground state for an bosonic open string can be denoted as |0; pi satisfies

αµ_m>0|0; pi = 0, (2.22) ˆ

pµ_{|0; pi = p}µ_{|0; pi.} _(2.23)

The constraints of the classical theory correspond to the vanishing of the energy momen-tum tensors T++ and T−− as shown in equation (2.20). In quantum level, the vanishing

of Lm in classical theory should be replaced by the positive frequency modes annihilate a

physical state |ψi, that is

Lm>0|ψi = 0. (2.24)

This is much like the Gupta–Bleuler treatment in quantizing the E.M. theory. But L0

should be discussed independently since there exists an ordering ambiguity due to normal ordering. The normal–ordered expression of L0 is

L0 = 1 2α 2 0+ ∞ X n=1 α−n· αn (2.25)

up to a to–be–determined constant say a. We include a and demand that a physical state |ψi must satisfy

(L0− a)|ψi = 0. (2.26)

Choose a = 1 to avoid ghosts. In addition, equation (2.26) carries the information of mass M of open strings. From M2 = −p2 and the number operator N ≡P

kα−k· αk, we have M2 = − 2 + 2 ∞ X k=1 α−k · αk = 2(N − 1). (2.27)

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2.2. STRING THEORY

For example, the scalar ground state (N = 0) with M2 = −2 is tachyon. In contrast, the first excited state N = 1 is given by · α−1|0; pi , which has M2 = 0 and thus is a

massless vector particle with polarization .

Equation (2.24) and (2.26) form the essential conditions for physical states. Lm and

L0are the so–called Virasoro generators of bosonic open strings satisfying the the Virasoro

algebra

[Lm, Ln] = (m − n)Lm+n+

D(m3− m)

12 δm+n, (2.28) where D means the dimension of the space–time, which is 26 if we choose a = 1.

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Chapter 3 Spinor semiology and the application

of BCFW rercursion relation

In this Chapter, a brief introduction to the spinor notations [9–11] is given by solving Dirac equation of a massless particle. After these kind of notations have been introduced, some Lorentz invariant quantities are created in terms of these notations. Having the above preparations, those quantities will be used to build up the 4–gluon scattering amplitudes by employing the BCFW technique. This calculation is provided in the last section.

3.1 Spinor notations

In this section, we would like to introduce the spinor notations. Throughout the whole section, the Lorentz signature ηµν = diag(−1, 1, 1, 1) is used. At first, consider a spin-1/2

particle with momentum p. Its behaviors can be understood by solving the Dirac equation

γ · p ψ(p) = 0 (3.1)

with p2 = 0. The gamma matrices in the Dirac representation (or standard representation) are γ0_D =   I 0 0 −I  , (γi)D =   0 σi −σi 0   (3.2)

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3.1. SPINOR NOTATIONS with σi, i = 1, 2, 3 σ1 =   0 1 1 0  , σ2 =   0 −i i 0  , σ3 =   1 0 0 −1  . (3.3) Those are the well–known Pauli matrices.

In order to distinguish the Dirac solutions from the Weyl solutions, we add “W ” to denote the case in Weyl representation and “D” for Dirac. The solution of Dirac equation (3.1) are often written as

ψ(p) =   ψA ψB  . (3.4)

Expanding equation (3.1) yields

−p0ψA+ ~σ · ~pψB = 0, −~σ · ~pψA+ p0ψB = 0. (3.5)

Above equations (3.5) give us two choices for the solutions of ψA and ψB. They are

respectively ψA= ψB, ψA= −ψB. For positive energy p0 > 0, we have

ψA= ψB⇒ ψ+(p) = 1 √ 2         √ p+ √ p−eiφp √ p+ √ p−eiφp         , (3.6)

where the subscript ” + ” on ψ(p) means positive helicity and p+ = p0+ p3, p− =

p0− p3 and eiφp ₌ p1 + ip2 pp2 1 + p22 = p√1+ ip2 p+p− . (3.7)

For the case of ψA= −ψB, the corresponding wave function should be

ψA = −ψB⇒ ψ−(p) = 1 √ 2         √ p−e−iφp −√p+ −√p−e−iφp √ p+         . (3.8)

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3.1. SPINOR NOTATIONS

Besides the Dirac representation, the so–called Weyl representation is also common to see.

In Weyl representation, the gamma matrices are

γµ=   0 − (¯σµ)αβ˙ − (σµ)_{α ˙}_β 0  . (3.9)

σµ and ¯σµ are defined in the following

σµ = (I, ~σ) , σ¯µ= (I, −~σ) . (3.10)

~σ are still the three Pauli matrices.

The off-diagonal gamma matrices in (3.9) imply ψ(p) could be divided into the com-bination of two 2-component spinors obey different kinds of transformation, i.e.

ψ(p) =   ξβ˙ ηα  . (3.11)

The lower undotted index α and the upper dotted one ˙β label the components of spinors η and ξ with both of the indices running from 1 to 2. The transformations of η and ξ are assigned in the following: If we denote ˜η = η†, ξ = ξ˜ †, and the SL(2, C) transformation matrix A with det(A) = 1, then we have

η0 = Aη → ηα, (3.12) ˜ η0 = (η0)† = ˜ηA†→ ˜ηα˙, (3.13) ξ0 = ξA−1 → ξα_, _(3.14) ˜ ξ0 = (ξ0)†= A−1†ξ → ˜˜ ξα˙. (3.15) The consistency of the index structure implies following indices assignments for the transformation matrices: A → Aαβ, A†→ A† β˙ ˙ α, A−1 → A−1 β α , A−1† → A−1†α˙ ˙ β (3.16)

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For the reason that the Lagrangian of Dirac equation have to be Lorentz invari-ant, σµ, ¯σµ, A and the Lorentz transformation Lµν are required to satisfy the following

relations:

AσµA†= Lµνσν (3.17)

A†−1σ¯µA−1 = Lµνσ¯ν. (3.18)

Consistency requires the following indices assignments for σµ and ¯σµ, i.e.

σµ→ (σµ)_{α ˙}_β, σ¯µ→ ( ¯σµ)αβ˙ . (3.19)

If we define a 2 × 2 matrix as the following

≡ iσ2 =   0 1 −1 0  = − −1 . (3.20)

where σ2 is the Pauli matrix. We can soon find out, by direct calculation, that

(σµ) T

−1 = ¯σµ. (3.21)

If we add the indices into the above equation (3.21), it leads to α ˙˙γ(σµ)_γδ_˙ −1 δβ = (¯σµ) ˙ αβ ⇒ (σµ)_βα˙ = −1 ˙ β ˙γ(¯σµ) ˙ γδ δα. (3.22)

From equation (3.22), we can immediately see that changes a upper undotted (a lower dotted) index into an lower undotted (an upper dotted). In contrast to , −1 changes an upper dotted (a lower undotted ) index into a lower dotted (an upper undotted) . Therefore, the index structures for and − are

αβ = α ˙˙β, −1

αβ

= −1_{α ˙}_˙_β (3.23) such that

ηα = αβηβ, ηα = (−1)αβηβ. (3.24)

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Expanding the Dirac equation (3.1), η and ξ satisfy

~σ · ~p

|~p| ξ = −ξ, ~ σ · ~p

|~p| η = η. (3.25)

For positive energy |~p| = p0 > 0, the equation for η means the solution has positive

helicity or right–handed while negative helicity or left–handed for ξ. The Weyl represen-tation is related to Dirac represenrepresen-tation by a similarity transformation

(γµ)W = S (γµ)DS−1,

where S is the transformation matrix given by

S = √1 2   I −I I I  . (3.26)

This transformation also implies that the solutions of Dirac equation could be transformed to each other by

ψ(W )(p) = Sψ(D)(p). (3.27) Thus, the Weyl solutions could be obtained through (3.27)

ψ₊(W )(p) = Sψ(D)₊ (p) =         0 0 η1 η2         =         0 0 √ p+ √ p−eiφp         , ψ₋(W )(p) = Sψ(D)₋ (p) =         ξ˙1 ξ˙2 0 0         =         √ p−e−iφp −√p+ 0 0         . (3.28)

Lorentz scalars could be represented as ¯ψψ , where ¯ψ ≡ ψ†γ0_{. The two Weyl basis η}

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3.1. SPINOR NOTATIONS the way hpipji ≡ ¯ψ (D) − (pi) ψ (D) + (pj) = η(pi)βη(pj)β = −hpjpii, (3.29) [pipj] ≡ ¯ψ (D) + (pi) ψ (D) − (pj) = ξ(pi)_β˙ξ(pj) ˙ β _{= −[p} jpi]. (3.30)

They are simply called the angle bracket for h· · · i and square bracket for [· · · ]. In the above, we have taken

|pii = η(pi)β, hpj| = η(pj)β, (3.31)

|pi] = ξ(pi) ˙ β

, [pj| = ξ(pj)_β˙. (3.32)

The spinors are also related to its null momentum by the identities

Pb ˙a ≡ (σ · p)˙ba = |pi[p|, (3.33)

P˙ab ≡ (¯σ · p)˙ab = −|p]hp|. (3.34) Furthermore, some properties of these brackets could be derived by directly calculation:

[pipj] = hpjpii∗, (3.35)

hpipji[pjpi] = −2pi· pj. (3.36)

So far, some useful Lorentz scalars have been made by contracting the an upper dotted index with a lower dotted index spinor, or an upper undotted index with a lower undotted index such as in (3.29) and (3.30). As mentioned at the beginning of this section, we would like to use these Lorentz invariances as building blocks to build up scattering amplitudes. If we consider amplitudes include massless vector bosons, polarizations also require to be rewritten in the language of spinors. Polarization vectors with definite helicities for bosons can be represented as [12,13]

(k, q)µ₊= hq|σ µ_|k] √ 2hqki, (k, q) µ −= − [q|¯σµ_|ki √ 2[qk] . (3.37) Here q is some chosen light–like momentum, called reference momentum.

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3.2. CALCULATING COLOR–ORDERED 4–GLUON AMPLITUDE THROUGH BCFW RECURSION RELATION TECHNIQUE

3.2 Calculating color–ordered 4–gluon amplitude through

BCFW recursion relation technique

Before starting calculating the color–ordered amplitude, we would like to introduce how does ”color–ordered” amplitude come from [11].

The gauge transformation for QCD for a spin-1/2 particles is

ψ0 = eiPaTaλa_ψ. _(3.38)

The generators Ta obey the commutative relation

[Ta, Tb] = i

√

2fabcTc. (3.39)

If we are primarily interested in the SU (Nc), Ta can be used as basis for any Nc× Nc

matrices. Ta’s also satisfy the relation N2 c−1 X a=1 (Ta) j1 i1(Ta) j2 i2 = δ j2 i1δ j1 i2 − 1 Nc δj1 i1δ j2 i2. (3.40)

The structure constants (color factors) fabc can be obtained from

fabc= −

i √

2T r ([Ta, Tb] Tc) . (3.41) Let us now consider the amplitude for four incoming gluons. With the coupling constant suppressed, the scattering amplitude M is composed of s, t and u channels

M = Ms+ Mt+ Mu (3.42)

with

Ms = fa1a2bfa3a4bAs, Mu = fa1a3bfa2a4bAu, Mt = fa1a4bfa3a2bAt. (3.43)

From equation (3.41) and (3.40), product of two color factors, for example fa1a2bfa3a4b,

can be decomposed as sum of traces of product of Ta’s, i.e.

fa1a2bfa3a4b = −

1

2[T r(Ta1Ta2Ta3Ta4) − T r(Ta1Ta2Ta4Ta3) − T r(Ta1Ta3Ta4Ta2) + T r(Ta1Ta4Ta3Ta2)] .

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This color factor is indeed belonged to the s-channel. Similar results can be obtained for both the t and u channels, they are respectively

fa1a3bfa2a4b = − 1 2[T r(Ta1Ta3Ta2Ta4) − T r(Ta1Ta3Ta4Ta2) − T r(Ta1Ta2Ta4Ta3) + T r(Ta1Ta4Ta2Ta3)] , (3.45) fa1a4bfa3a2b = − 1 2[T r(Ta1Ta4Ta3Ta2) − T r(Ta1Ta4Ta2Ta3) − T r(Ta1Ta3Ta2Ta4) + T r(Ta1Ta2Ta3Ta4)] . (3.46) With the color decomposition, we can write amplitude M in (3.42) as

M =

2,3,4

X

j6=k6=l

M (1jkl) T r(Ta1TajTakTal). (3.47)

M (1jkl) inside the summation in the above equation are called the “color-striped” am-plitudes since the color factors fabc has been striped away. We find

M (1432) = M (1234) = −1 2(As+ At) , M (1243) = M (1342) = 1 2(As+ Au) , M (1324) = M (1432) = −1 2(−Au+ At) . (3.48) We can immediately find out that the sum of the right hand sides of the above three equations add up to zero, which means that there are only two independent color–striped amplitudes.

Now that most of the preliminaries have been developed, we shall start calculating the color–ordered amplitude M (1−2−3+₄+_{) = −}1

2(As+ At). By choosing the reference

momenta which satisfy q1 = q2 = p4, q3 = q4 = p1 , the four-gluon vertex amplitude and

the t-channel Atgive zero contribution, only As survives. One can carry out this example

through standard Feynman’s rule, and it turns out the answer is M 1−2−3+4+ = h12i

4

h12ih23ih34ih41i. (3.49) We next directly borrow the above result and use BCFW recursion relation to re–construct this amplitude. We can use contour integral to write

1 2πi I _dz z M ˆ1 − 2−3+ˆ4+ = M 1− 2−3+4+ |z=0+ X poles, za Res " M ˆ1−2−3+_ˆ₄+ z # |z=za. (3.50)

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If the boundary terms vanish as z → 0, we have

M 1−2−3+4+ |za=0 = − X poles, za6=0 Res " M ˆ1−2−3+_ˆ₄+ z # z=za . (3.51)

The momenta deformation in equation (2.1) is equivalent to deform the spinors in the way that

| ˆp1i = |p1i, [ ˆp1| = [p1| + z[p4|,

| ˆp4i = |p4i − z|p1i, [ ˆp4| = [p4|. (3.52)

It is easy to verify that momentum is conserved by

4 X i=1 |pii[pi| = 4 X i=1 σ · pi = σ · (ˆp1+ p2 + p3+ ˆp4) = σ · (p1+ p2+ p3+ p4). Thus, ˆ p1+ p2+ p3+ ˆp4 = p1+ p2+ p3+ p4 = 0.

Obviously, a simple pole arises from the propagator when the intermediate vector ˆ

p2 _{= (ˆ}_p

1+ p2)2 goes on-shell, that is

−(ˆp1+ p2)2 = 0 = hˆp1p2i[p2pˆ1] = hp1p2i[p2p1] + zhp1p2i[p2p4], (3.53)

and the pole occurs at

z = za= −

[p2p1]

[p2p4]

. (3.54)

It is convenient to express the propagator as 1 (ˆp1+ p2)2 = Ra z − za (3.55) in which Ra= −1/hp1p2i[p2p4].

After doing the analytic continuation, the residue (3.52) gives rise from this pole is

Res A(z) z z=za = M (ˆ1−2−pˆ+)Ra za M (ˆp−3+ˆ4+). (3.56)

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If momentum p’s are allowed to be complex, three–point functions can be defined, they are M 1−2−3+ = h12i 4 h12ih23ih31i, M 1 +₂+₃−₌ [12]4 [12][23][31]. (3.57) Explicit calculation of equation (3.57) are offered in the Appendix A. Then from (3.57), we have M (ˆ1−2−pˆ+)M (ˆp−3+ˆ4+) = hp1p2i 3_[p 3p4]3 hp2pi[ˆˆ pp3] [p4p]hˆˆ pp1i . (3.58)

Put the shifted momenta into (3.58), its denominator goes like

hp2pi[ˆˆ pp3] [p4p]hˆˆ pp1i

= {hp2| ( |p1i ( [p1| + z[p4| ) + |p2i[p2| ) |p4]} · {hp1| (|p1i ([p| + z[p4|) + |p2i[p2|) |p3i}

= hp2p1i[p1p4]hp1p2i[p2p3]. (3.59)

From the fact that hpipii = [pipi] = 0, some of the terms in the denominator of

equation (3.58) will vanish. Finally, M (ˆ1−2−pˆ+_{)M (ˆ}_p−₃+_ˆ₄+_{) could be found out to be}

independent of z, i.e. M (ˆ1−2−pˆ+)M (ˆp−3+ˆ4+) = hp1p2i 3_[p 3p4]3 hp2p1i[p1p4]hp1p2i[p2p3] . (3.60)

Use the result of Raas well as equation (3.54) and (3.60), the residue (3.56) becomes

[p2p4] [p2p1] · 1 hp1p2i[p2p4] · hp1p2i 3_[p 3p4]3 hp2p1i[p1p4]hp1p2i[p2p3] . (3.61)

After applying the following formulae,

hp2p1i[p1p4] = −hp2p3i[p3p4],

hp1p2i[p2p3] = −hp1p4i[p4p3],

hpipji = −hpjpii,

[pipj] = −[pjpi].

, then equation (3.56) becomes

− h12i

4

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Put a minus sign in the above equation, we can soon recover the result of equation (3.49).

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Chapter 4 Four-tachyon scattering amplitude

When applying BCFW on-shell recursion relation to string amplitudes, pole structures become obscure if we formulate the amplitudes by say Koba–Nielson formulas. We begin with the familiar four–point Koba–Nielson formula, and review how the pole structures are made manifest through binomially expanding this integral formula in [5]. Later, we use our algorithm to solve for the difficulty of summing over infinite number of physical states by enlarging the sum over all physical states to over the completely Fock states [1]. This algorithm comes from the inspiration from the Ward identity in field theory. Finally, we find a mathematical connection between the residue prescribed from BCFW and the generating function for Stirling number of the first kind. This connection is quite useful for the further evaluation when we extend the application of our algorithm to the amplitudes containing higher spin particles.

4.1 Poles extraction

The Koba–Nielson formula for four-tachyon scattering amplitude is given by A(1234) =

Z 1

0

dz2(1 − z2)k2·k3z2k1·k2, (4.1)

where we use gauge fixing to set z1 = 0, z3 = 1 and z4 = +∞. For the purpose of

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4.1. POLES EXTRACTION (x − y)a = ∞ X N =0 a N xa−N(−y)N, (4.2) where the combinatorial factor a

N is defined to be a N ≡ a(a − 1)(a − 2) · · · (a − N + 1) N ! . (4.3)

Inserting the result in equation (4.2) to (4.1), the Koba–Nielson formula becomes

A(1234) = ∞ X N =0 k₂· k3 N (−1)N Z 1 0 zk1·k2+N 2 dz2. (4.4)

Carry out the world sheet integral over z2 and use the mass-shell conditions for

tachyons k2

1 = k22 = −M2 = +2, we have k1· k2 = (k1+ k2)2/2 − 2. After doing so, the

s–channel propagator emerges [5] A(1234) = ∞ X N =0 k₂· k3 N (−1)N 2 (k1+ k2)2 + 2(N − 1) . (4.5)

Having extracted the propagator 2/ [(k1+ k2)2+ 2(N − 1)] from the tree–level tachyon

amplitude, we next would like to re–construct (4.5) by BCFW technique. Manually choose the pair of deformation to be k1 and k4, i.e.

ˆ

k1(z) = k1+ zq, kˆ4(z) = k4− zq (4.6)

with q2 = k1·q = k4·q = 0. Assume there is no boundary contributions when z approaches

to infinity, (4.5) could be expressed by the BCFW recursion relation A (1, 2, 3, 4) = X poles zN X physical h AL ˆ 1, 2, ˆPh 2 (k1+ k2)2+ 2(N − 1) AR ˆPh, 3, ˆ4 . (4.7)

Obviously, there are infinite number of poles exist in the denominator due to infinite tower of mass levels of the intermediate states. For an arbitrary mass level N , they are

zN = −

(k1+ k2)2+ 2 (N − 1)

2q · k2

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4.2. SUMMING OVER ALL PHYSICAL STATES

Compare equation (4.5) and (4.7) at each level N and we have X physical h AL(ˆ1, 2, ˆPh) AR( ˆPh, 3, ˆ4) = (−)N   k2· k3 N  . (4.9) Thus, in order to verify the validity of BCFW in the tree level string amplitudes, we have to be able to handle the scalar residue at the left–handed side of equation (4.9) as sum over all intermediate physical states at a given mass level N and prove the equality of (4.9).

4.2 Summing over all physical states

Before straightly doing the summation over all intermediate states in equation (4.9), let us first take a look at a scattering process such as e+_e− _{→ e}+_e− _{in field theory . The}

propagator of this process is gauge boson. If we shift the first and the fourth particles, the BCFW recursion relation reads

A ∼ X

state h

Aµ_L(ˆ1, 2, ˆPh)ηµνAνR( ˆP −h

, 3, ˆ4). (4.10) The intermediate states are composed of massless bosons. In (3+1)D flat space–time, we have [14] ηµν = +µ − ν + − µ + ν + T µ L ν + L µ T ν, (4.11) where + µ and −

µ are the two transverse polarizations with definite helicities while Tµ

and L_µ are respectively the time—like and longitudinal 4–vector. We can replace ηµν in

equation (4.10) with that in (4.11). But since the Ward identity of gauge theory governs that, for a scattering of n particles, if all (n − 1) particles are physical polarized while the n-th particle carries unphysical polarization, the amplitude vanishes. Thus, we have

A ∼ Aµ_L(ˆ1, 2, ˆPh) _µ+−_ν + −_µ+_ν + T_µL_ν + L_µT_ν Aν_R( ˆP−h, 3, ˆ4)

= Aµ_L(ˆ1, 2, ˆPh) +_µ−_ν + −_µ+_ν Aν_R( ˆP−h, 3, ˆ4). (4.12) Originally in (4.10), we sum over all intermediate states. It turns out that it is equivalent to summing only physical states since the time–like and longitudinal polarization are clearly unphysical.

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4.2. SUMMING OVER ALL PHYSICAL STATES

Having had a glimpse at the scattering amplitude in field theory, we then come to the main task of converting the scalar residue (4.9) as a summation over physical states. Firstly, we have to understand how to construct an arbitrary state from ladder operators with general mass level N in the Fock space [15]. In general, a Fock state can be built up by successively acting creation operators αµ_−m with m > 0 on a ground state |0; P i, that is | {Nm} ; P i = " _∞ Y m=1 (αµ−m) Nm √ Nm! mNm # |0; P i. (4.13) The above Fock state | {Nm} ; P i carries N1–multiple of αµ−1 mode operators and N2–

multiple of αµ₋₂ mode operators and so on. We use {Nm} to label the normalized Fock

state and, for simplicity, denote αµ1

−mα µ2 −m· · · α µNm −m = α µ −m Nm . The number Nm of

the m-th mode operators must satisfy N =

∞

X

m=1

m Nm. (4.14)

It should be emphasized that different tensor indices of α−m are treated as different

oper-ators although they have the same mode. However, generic Fock states include ”ghosts” such as α0†_m|0i since it has negative norm, i.e. h0α_m0α0†_m0i < 0. Thus, there comes a problem, we need to get rid of these kind of unphysical states from Fock space while summing over all intermediate states. It turns out to be a difficult objective since we do not know so far how to construct the polarization tensor for a general mass level N . With the implication from the previous discussion of the electron–positron (e+_e− _{→ e}+_e−₎

scattering process at the beginning of this subsection, we are able to avoid this difficulty by enlarging the range of summation from the physical states to the entire Fock space of string spectrum. This fact is guaranteed by the so–called ”No-Ghost theorem”. Hence, equation (4.5) can be written as

An(1, 2, 3, 4) = X poles zN X F ock AL ˆ 1, 2, ˆP 1 P2_{+ M}2AR ˆP , 3, ˆ4 . (4.15)

With this understanding, we are able to write the left–handed side of the residue equation (4.9) as IN ≡ X {P mNm=N } D − ˆk1; 0 V0(k2) {Nm} ; ˆP E T{Nm} D {Nm} ; ˆP V0(k3) ˆ k4; 0 E z2=1 (4.16)

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4.3. LEVEL MATCHING

for a given mass level N . T{Nm} = ηµ1ν1ηµ2ν2· · · ηµNmνNm is the polarization tensor for

a set of intermediate states {Nm} with definite mass level N , which satisfies equation

(4.14). Thus, we can see a subscript {Nm} at the bottom of T{Nm}. Equation (4.16) is

expected to equal

k₂ · k3

N

(−1)N. (4.17)

This equality will be explicitly proved in the following subsection 4.4. We can similarly generalize equation (4.7) to an arbitrary n-point function. If we choose k1 and kn to be

the deformation pair, then the BCFW recursion relation for this scattering amplitude can be calculated by An(1, · · · , n) = X i X N X {N =P mmNm} h−ˆk1|V2(k2) · · · Vi−1(ki−1) | {Nm} ; ˆPii 2T{Nm} P2 i + 2 (N − 1) h{Nm} ; ˆPi| Vi(ki) · · · Vn−1(kn−1)| ˆkni, (4.18) where Pi = P

i ki is the momenta flow from the adjacent external particles.

4.3 Level matching

In order to quickly match the residue (4.17) getting from the Koba–Nielson formula with that obtaining from BCFW prescription (4.16), one can do level matching to see whether our result is correct or not. For future reference, the explicit expressions of the first 4 levels from equation (4.17) are provided in the following

N = 0 ⇒ I0 = 1, N = 1 ⇒ I1 = −k2· k3, N = 2 ⇒ I2 = k2· k3(k2· k3− 1) 2! , N = 3 ⇒ I3 = − k2· k3(k2· k3− 1) (k2· k3− 2) 3! .

• Level N = 0: For N = 0, all Nm = 0. We have T = 1. Thus, the contribution

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4.3. LEVEL MATCHING I0 = X h −ˆk1; 0 | V0(k2) |0; ˆpi × 1 × h0; ˆp | V0(k3) | ˆk4; 0i z2=1 = 1. (4.19) As was expected!

• Level N = 1: In this case, there is only one intermediate state: { α−1}.

T = ηµν. Put αµ−1 into (4.16), residue I1 of this level could be easily worked out

I1 = h −ˆk1; 0 | V0(k2) (αµ−1) |0; ˆpi ηµνh0; ˆp | (αν1) V0(k3) | ˆk4; 0i _z 2=1 = − k2· k3. (4.20)

Agrees with our argument!

• Level N = 2: In this case, the intermediate states for N = 2 are shown as follows:

{α√−2 2, α2 −1 √ 2!}. I2 = T1+ T2. (4.21)

I2 is respectively constituted of 2 components from contracting the right-handed

and left-handed 3–amplitudes with 2 distinct intermediate states. For T1, the tensor

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4.3. LEVEL MATCHING

sets of tensor indices are required. Thus, its contribution reads T2 = h −ˆk1; 0 | e P∞ n=1n1k2·α−ne− P∞ n=11nk2·αn α µ1 −1α µ2 −1 √ 2 |0; ˆpi ηµ1ν1ηµ2ν2 h0; ˆp | α ν1 1 α ν2 1 √ 2 eP∞n=1n1k3·α−n| ˆk 4; 0 i = k µ1 2 k µ2 2 √ 2 ηµ1ν1ηµ2ν2 kν1 3 k ν2 3 √ 2 = (k2· k3) 2 2 . (4.23)

Frankly, the sum of the two components reveals our prediction! I2 = T1 + T2 = − k2· k3 2 + (k2 · k3)2 2 = k2· k3(k2· k3− 1) 2! . (4.24)

• Level N = 3: Three intermediate states are needed to be taken into account: {α√−3 3, α−2α−1 √ 2 , α3 −1 √ 3!}. I3 = T1+ T2+ T3. (4.25)

T1 comes from the contribution of α−3/

√

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4.4. EXPLICIT CALCULATION OF RESIDUE

As for T2, the tensor structure is T = ηµ1ν1ηµ2ν2. So its contribution reads

Furthermore, T3 could be also computed, and its tensor structure is obviously T =

ηµ1ν1ηµ2ν2ηµ3ν3. T3 = D − ˆk1; 0 e P∞ n=1 1 nk2·α−ne− P∞ n=1 1 nk2·αn α µ1 −1α µ2 −1α µ3 −1 √ 3! 0; ˆp E ηµ1ν1ηµ2ν2ηµ3ν3 D 0; ˆp αν1 1 α ν2 1 α ν3 1 √ 3! eP∞n=1 1 nk3·α−n ˆ k4; 0 E = D − ˆk1; 0 (−k2· α1)3 3! αµ1 −1α µ2 −1α µ3 −1 √ 3! 0; ˆp E ηµ1ν1ηµ2ν2ηµ3ν3 D 0; ˆp αν1 1 α ν2 1 α ν3 1 √ 3! (k2 · α−1)3 3! ˆ k4; 0 E = − (k2· k3) 3 3! . (4.28)

Thus, sum T1, T2 and T3 together

I3 = T1+ T2+ T3 = − k2· k3 3 + (k2· k3) 2 2 − (k2· k3) 3 3! = − k2· k3(k2· k3− 1) (k2· k3− 2) 3! . (4.29)

Same as we have claimed before.

4.4 Explicit calculation of residue

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calcu-4.4. EXPLICIT CALCULATION OF RESIDUE

higher although it seems to be straightforward. What we are going to do in this section is to demonstrate the explicit calculation of equation (4.16) for an arbitrary mass level N , and prove the equality of (4.16) and equation (4.17).

At first, recall the tachyon vertex operator

V0(k, z) =: eik·X(Z) := Z0W0, (4.30)

where

Z0 = eik·x+k· ˆplnz = zk· ˆp−1eik·x. (4.31)

ˆ

p in equation (4.31) is the momentum operator. W0 is the pure oscillation part of V0,

which is W0 = e P∞ n=1 zn nk2·α−ne− P∞ n=1 z−n n k2·αn. (4.32)

Those conventions and definitions could be found in Green, Schwarz and Witten’s book [8].

4.4.1 Explicit calculation of 3–point amplitude

Considering the on-shell amplitude, z2 can be set to be 1. Thus, we just have to take care

of the oscillation part W0. We use AR and AL to denoted the right and left 3–amplitudes

AL = AL(ˆk1, k2, ˆP ) = D − ˆk1; 0 V0(k2) {Nm} ; ˆP E z2=1 , (4.33) AR = AR(− ˆP , k3, ˆk4) = D {Nm} ; ˆP V0(k3) ˆ k4; 0 E z2=1 . (4.34) Calculating AR

Put the definition of | {Nm} ; P i and V0 into AR, we have

AR= D 0; ˆP ∞ Y m=1 (αν m) Nm √ mNmN m! ! eP∞n=11nk3·α−n ˆ k4; 0 E =D0; ˆP ∞ Y m=1 (αν m) Nm √ mNmN m! em1k3·α−m ˆ k4; 0 E =D0; ˆP ∞ Y m=1 (αν m) Nm √ mNmN m! · 1 Nm! k3· α−m m Nm ˆ k4; 0 E . (4.35)

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Using the commutative relation [αµ

m, αnν] = mδm+nηµν and the identity h0|αanαa−n|0i =

a!na, where a denotes the power of αn not a tensor index, yields

AR= ∞ Y m=1 kν 3 Nm √ mNmN m! . (4.36) Calculating AL

AL can also be able to carry out by following the same manners

AL= D − ˆk1; 0 e −P∞ n=1 1 nk2·αn ∞ Y m=1 (α_−mµ )Nm √ mNmN m! ! 0; ˆP E = D − ˆk1; 0 ∞ Y m=1 e−m1k2·αm (α µ −m) Nm √ mNmN m! 0; ˆP E . (4.37)

For the reason that the number of creation and annihilation operators inside the Dirac bracket must be the same otherwise it will vanish. Use the Taylor expansion to expand the exponential part and only the Nm-th order term survives. Then,

AL= D − ˆk1; 0 1 Nm! −1 mk2· αm Nm (αµ−m) Nm √ mNmN m! 0; ˆP E = ∞ Y m=1 − kµ₂Nm √ mNmN m! . (4.38)

4.4.2 Contracting A

R

and A

L

Using equation (4.16), (4.36) and (4.38), it is easy to calculate IN for a general mass level

N . IN = X {P mNm=N } ∞ Y m=0 − k₂µNm √ mNmN m! ηµ1ν1ηµ2ν2· · · ηµNmνNm kν 3 Nm √ mNmN m! = X {P mmNm=N} ∞ Y m=1 (−k2· k3)Nm Nm! mNm . (4.39)

Notice that Nm and m satisfy the following two relations:

X m mNm = N, X m Nm = J. (4.40)

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Using equation (B.9) and the definition (B.8) of Stirling number of the first kind in the Appendix as a generating function , IN can be rewritten as

IN = N X J =0 |s(N, J)| N ! (−k2· k3) J ₌k2· k3 N (−)N. (4.41) This is the result we have expected!

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Chapter 5 Scattering with higher spin particles

Having a great success in the full–tachyon amplitude, now let us generalize our algorithm to the scattering amplitude contains an arbitrary spin state and three tachyons. As a warmup exercise, we first demonstrate how to explicitly calculate the residue from BCFW prescription of the scattering amplitude of one vector and 3 tachyons by using our algorithm. The corresponding generating function can be found as well but which is slightly different from the four–tachyon case. Then, through path integral approach, the generic structure of generating function of scattering amplitude of an arbitrary spin vertex and 3 tachyons can be systematically worked out [1].

5.1 Scattering amplitude of one vector and 3 tachyons

5.1.1 Algebraic calculation

The vertex operator of a massless vector is

V (k, z) =: · ˙Xeik·X(z) : . (5.1) The 1–vector 3–tachyon scattering amplitude is given by the following integration

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5.1. SCATTERING AMPLITUDE OF ONE VECTOR AND 3 TACHYONS A(1¯234) = Z 1 0 h−k1; 0| 2· ˙X(z2)V0(k2, z2) V0(k3, z3) |k4; 0idz2 = (2· k3) Z 1 0 dz2|1 − z2|k2·k3−1|z2|k1·k2 − (2· k1) Z 1 0 dz2|1 − z2|k2·k3|z2|k1·k2−1, (5.2)

where ¯2 means the second particle is chosen to be a vector. 2 here is the polarization

of the incoming vector particle carrying momentum k2 which obey k22 = k2 · 2 = 0. By

applying gauge fixing, we set z1 = 0, z3 = 1 and z4 = ∞. As the same with the pure

tachyon scattering case, we expand ( 1 − z2)k2

·k3−1 _{and ( 1 − z}

2)k2

·k3 _{by using equation}

(4.2), and integrate over z2, which yields

A(1¯234) = (2· k3) ∞ X N =1 k₂· k3− 1 N − 1 (−)N −1 2 (k1+ k2)2+ 2(N − 1) − (2· k1) ∞ X N =0 k2· k3 N (−)N 2 (k1+ k2)2+ 2(N − 1) , (5.3)

where N is the mass level of the intermediate states. The propagator 2/[ (k1+k2)2+2(N −

1) ] has now been extracted. It is obvious that the residue of equation (5.3) is consisted of two terms, one of the term is proportional to 2· k1 and the other one is proportional

to 2· k3. The term proportional to 2· k1, i.e. k2_N·k3(−1), is simply corresponding to the

4-tachyon amplitude. But, a new term comes from the other one proportional to 2· k3.

The residue of 1–vector 3–tachyon amplitude from BCFW prescription can be written as X {P mmNm=N} h 0; −k1| (2· ˙X) V (k2, z2) | {Nm} ; piT{Nm}h{Nm} ; p| V (k3, z3) |0; k4i|z2=z3=1. (5.4) Differ from the 4–tachyon amplitude, an extra term 2· ˙X locates at the second particle.

Algebraically, the term proportional to 2· k1 in equation (5.3) can be obtained by acting

the operator 2· ˆp in 2· ˙X = 2·(ˆp +

P

n=1αne−inτ)1 to the bra h −k1; 0|, which reproduces

1P

n=1α−neinτ gives no contribution since h 0 | α−n= 0, for n > 0. Thus, 2· ˙X could be equivalently

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5.1. SCATTERING AMPLITUDE OF ONE VECTOR AND 3 TACHYONS

−2·k1. The rest of its kinematic dependence could be easily shown to be the pure tachyon

residue as in equation (4.16), that is X

{P mNm}

h−k1; 0| V (k2, z2) | {Nm} ; pi T{Nm}h {Nm} ; p| V (k3, z3) |0; k4i,

which confirms our prediction. And yet, the term proportional to 2 · k3 is

IN = X {P mNm=N } D − k1; 0 _X∞ n=1 2· αnz2−n V0(k2) {Nm} ; p E T{Nm} D {Nm} ; p V0(k3) k4; 0 E z2=1 . (5.5)

Residue (5.5) has to be equal to (2· k3) ∞ X N =1 k₂· k3− 1 N − 1 (−)N −1 (5.6)

in equation (5.3). The next step is to show the equality between equation (5.5) and (5.6). For this example, the level matching and evaluation for the generating function are left in the appendix C.

5.1.2 Explicit derivation for the related residue of interest

Having done the basic algebraic calculation in the previous subsection, we would like to explicitly deduce the equality between equation (5.5) and (5.6). For a general mass level N , this kinematic dependence IN can be derived from gluing two 3-point amplitudes, that

is IN = X {P mNm=N } D − k1; 0 _X∞ n=1 2· αn V0(k2) {Nm} ; p E T{Nm} D {Nm} ; p V0(k3) k4; 0 E z2=1 .

We denote the two 3-point on–shell amplitudes (the left part and right one) as

AL = AL(k1, k2, P ) = D − k1; 0 _X∞ n=1 2· αn V0(k2) {Nm} ; p E z2=1 , AR = AR(−P, k3, k4) = D {Nm} ; p V0(k3) k4; 0 E z2=1 .

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The explicit calculation of 3-point amplitudes

AR has been obtained before, which is

AR= ∞ Y m=1 kν 3 Nm √ mNmN m! . (5.7)

On the other hand, the algebraic structure of AL is new to us, which is given by

AL= D 0 _X∞ n=1 2· αn V0(k2) {Nm} E z2=1 = ∞ X n=1 D 0 2· αn V0(k2) ∞ Y m=1 αµ_−mNm √ mNm_N m! 0 E z2=1 . (5.8)

After setting z2 = 1, the tachyonic vertex V0(k2) becomes

V0(k2) = exp h_X∞ p=1 k2· α−p p i exph− ∞ X p=1 k2· αp p i .

When moving 2· αnto the right, it will first encounter the term exp P ∞

p=1 (k2· α−p)/p

. But remember that the polarization 2 and the momentum k2 are orthogonal; therefore,

only the identity of this term survives after αn contracting with α−p. Thus

AL= ∞ X n=1 D 0 2· αn _Y∞ m=1 e−k2·αmm αµ−m Nm √ mNm_N m! 0 E = ∞ X n=1 D 0 2· αn " e−k2·αnn αµ_−nNn √ nNnN n! # _∞ Y m=1,m6=n e−k2·αmm αµ_−mNm √ mNmN m! 0 E . (5.9)

Using Taylor expansion to expand the exponential part inside the square bracket in the equation (5.9) and moving those things to the left, we see that only the (Nn− 1)-th order

term do not vanish. Thus,

AL= ∞ X n=1 D 0 2· αn " 1 (Nn− 1)! − k2· αn n Nn−1 αµ_−n Nn √ nNn_N n! # ∞ Y m=1,m6=n e−k2·αmm αµ−m Nm √ mNmN m! 0 E . (5.10)

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Using the commutative relation [ α_mµ , αν_n] = mδm+nηµν, we will reach at

AL = ∞ X n=1 (" (−)Nn−1_{n N} nµ2(k µ 2)Nn−1 √ nNn_N n! # _∞ Y m=1,m6=n − kµ₂Nm √ mNm_N m! ) . (5.11) Gluing AL and AR

Combine the left 3-point amplitude AL in equation (5.11) with the right one AR in (5.7),

we obtain D − k1; 0 _X∞ n=1 2· αn V0(k2) {Nm} ; p E T{Nm} D {Nm} ; p V0(k3) k4; 0 E = ∞ X n=1 (" (−)Nn−1_nN n(2· k3) (k2· k3)Nn−1 nNnN n! # _∞ Y m=1,m6=n − k2· k3 Nm mNmN m! ) = ∞ X n=1 " − nNn(2 · k3) k2· k3 # _∞ Y m=1 − k2· k3 Nm mNmN m! . (5.12)

For any given mass level N , we have N =

∞

X

n=1

nNn.

Thus, the result (5.12) we just obtain above can be written as

− N 2· k3 k2· k3 ∞ Y m=1 − k2· k3 Nm mNm_N m! . (5.13)

Summing over all physical states, we have the residue IN:

IN = X {N =P mmNm} (−) N 2· k3 k2· k3 ∞ Y m=1 − k2· k3 Nm mNmN m! .

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5.2. GENERATING FUNCTION FROM PATH INTEGRAL APPROACH

5.1.3 Recover the result

The generating function for the residue proportional to 2 · k3 is

₂· k3 k2· k3 z d dz e k2·k3ln(1−z) = (2· k3) z d dz[ln(1 − z) ] e k2·k3ln(1−z)_. _(5.14)

The quick evaluation of this generating function is left in Appendix C.2. Use the relations (B.3) and (B.4) in the appendix and IN can be rewritten as

IN = (−)N −1N 2· k3 k2· k3 N X J =0 s(N, J ) N ! (k2· k3) J _(5.15) = 2· k3 N X J =1 s(N, J ) N ! (−) N −1_{N (k} 2· k3)J −1. (5.16)

Refering to (B.10) in the appendix and setting X = k2· k3 in that relation, we soon

recover the result:

IN = 2· k3 (−)N −1

k2· k3− 1

N − 1

.

5.2 Generating function from path integral approach

As we shall see in the previous examples of full–tahcyon and 1–vector 3–tachyon am-plitude, we can always find a generating function related to the residue prescribed by BCFW. In the following, instead of the operator method used previously, we adopt path integral approach [16] to calculate the generating functions. We first take the 1–vector 3–tachyon amplitude as an example to illustrate how to re–derive its generating function (5.14) from path integral formalism.

Note that the amplitude can be written as A(1¯234) =

Z 4 Y

i=1

dzih: eik1·X(z1) : : 2· ˙Xeik2·X(z2):: eik3·X(z3) : : eik4·X(z4) :i, (5.17)

where h· · · i is short for h0| · · · |0i. For convenience, one can exponentiate 2· k3 up to the

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5.2. GENERATING FUNCTION FROM PATH INTEGRAL APPROACH identity [8] h: eA1 _{:: e}A2 _{: · · · : e}AM _{:i = exp} h P i<jhAiAji i

. Use the world sheet SL(2, R) to set z1 = 0, z3 = 1 and z4 = ∞. We have [16]

A(1¯234) = Z 4 Y i=1 dzih: eik1·X(z1) : : eik2·X(z2)+2· ˙X :: eik3·X(z3) : : eik4·X(z4):i|linear in 2 (5.18) = Z 4 Y i=1 dzie− P j<lkjµklνhXµ(zj)Xν(zl)+i P j6=22µkjνh ˙X(z2)X(zj)i_| linear in 2 (5.19) = Z 1 0 dz2(1 − z2)k2 ·k3 zk1·k2 2 2· k1 z2 − 2· k3 1 − z2 . (5.20)

The propagator in the above equation is hXµ_(x)Xν_{(y)i = −η}µν_{ln(x − y). The term}

proportional to 2· k1 has been considered before, which is the same as in the case of pure

tachyon amplitude. Thus, we just neglect the analysis on it. The term of our interest is the one proportional to 2 · k3 in equation (5.20). Please note that, in the previous

discussion we binomially expand (1 − z)k2·k3 _{and take the world sheet integral to obtain}

the propagator. Obviously, zk1·k2 _{has been integrated away, hence does not involve in the}

generating function. This implies that the generating function should be G1 = e{k2·k3ln(1−z2)}e n 2·k3z2_dz2d ln(1−z2) o |linear in 2 = (2· k3) z2 d dz2 [ln(1 − z2)] e[k2·k3ln(1−z2)], (5.21)

which is exactly the same with (5.14). Please note that there is an extra term z_dzdln(1 − z) in equation (5.21) if we compare this equation with the generating function for the pure tachyon case. This result can be viewed from the term 2· ˙X in the vertex operator, which

results in h ˙Xµ(x)Xν(y)i in equation (5.19), h ˙Xµ(x)Xν(y)i = ix d

dxhX

µ_(x)Xν_{(y)i = −iη}µν_x d

dxln(x − y). (5.22) Following the same spirit, one can generalize this method to a more complicated case such as a vertex containing arbitrarily n-multiple of · ˙X’s, that is

V (k2, z2) =:

(1)₂ · ˙X (2)₂ · ˙X× · · · ×(n)₂ · ˙Xeik2·X(z2)_{: .} _(5.23)

This higher spin particle has mass level n, therefore, k2

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5.2. GENERATING FUNCTION FROM PATH INTEGRAL APPROACH

path integral formalism, the amplitude should be

A(1¯234) = Z 4 Y i=1 dzih: eik1·X(z1) : : (1)₂ · ˙X (2)₂ · ˙X× · · · ×(n)₂ · ˙Xeik2·X(z2) _: : eik3·X(z3) _{: : e}ik4·X(z4) _:i _(5.24) = Z 4 Y i=1 dziheik1·X(z1) : : eik2·X(z2)+ (1) 2 · ˙X+ (2) 2 · ˙X+···+ (n) 2 · ˙X : : eik3·X(z3) _{: : e}ik4·X(z4)_:i| linear in (1)₂ ,(2)₂ ,··· ,(n)₂ (5.25) = Z 4 Y i=1 dziexp{− X j<l kjµklνhXµ(zj)Xν(zl)i + i X j6=2 (1)_2µkjνh ˙Xµ(z2)Xν(zj)i+ iX j6=2 (2)_2µkjνh ˙Xµ(z2)Xν(zj)i + · · · + i X j6=2 (n)_2µkjνh ˙Xµ(z2)Xν(zj)i}|_{multilinear in}(1) 2 , (2) 2 ,··· , (n) 2 (5.26) = Z 1 0 dz (1 − z)k2·k3 zk1·k2 " (1)₂ · k1 z − (1)₂ · k3 1 − z # " (2)₂ · k1 z − (2)₂ · k3 1 − z # × · · · × " (n)₂ · k1 z − (n)₂ · k3 1 − z # . (5.27)

Having the above experience, some other terms can be obtained from amplitudes containing relative lower spin vertex. Thus, we know the new term is the one proportional to(1)₂ · k3 (2)₂ · k3 × · · · ×(n)₂ · k3 , and thus it is Z 1 0 dz (1 − z)k2·k3_zk1·k2    (−)n (1)₂ · k3 (2)₂ · k3 × · · · ×(n)₂ · k3 (1 − z)n    = (−)n (1)₂ · k3 (2)₂ · k3 × · · · ×(n)₂ · k3 Z 1 0 dz2(1 − z2)k2 ·k3−n zk1·k2 2 . (5.28) Binomially expand (1 − z2)k2 ·k3−n

at the end of equation (5.28), we reach

∞ X a=0 Z 1 0 dz2   k2· k3− n a  (−)az k1·k2+a 2 = ∞ X a=n   k2· k3− n a − n  (−)a−n 2 (k1+ k2)2+ 2 (a − 1) . (5.29)

From the previous experience, throw the propagator in the above equation (5.29) away, and the rest should be the residue from BCFW prescription. The generating

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func-5.2. GENERATING FUNCTION FROM PATH INTEGRAL APPROACH

tion Gn for this new term can be seen from equation (5.26)

Gn = e n (1)₂ ·k3z_dzdln(1−z) o e n (2)₂ ·k3z_dzdln(1−z) o × · · · × e n (n)₂ ·k3z_dzdln(1−z) o e{k·k3ln(1−z)}_| multilinear in (1)₂ , (2)₂ , ··· ,(n)₂ (5.30) = (1)₂ · k3 z d dz ln (1 − z) (2)₂ · k3 z d dz ln (1 − z) × · · · × (n)₂ · k3 z d dzln (1 − z) ek2·k3ln(1−z) _(5.31) = (1)₂ · k3 (2)₂ · k3 × · · · ×(n)₂ · k3 X∞ a=n   k2· k3− n a − n  (−) a za. (5.32) After setting z = 1 in equation (5.32), this is exactly the residue from BCFW prescrip-tion of the new term in equaprescrip-tion (5.29). Equaprescrip-tion (5.31) contains n derivative terms z(d/dz)ln(1 − z). They can be traced back to the n–multiple of · ˙X’s in the vertex operator.

From the above arguments, we can conclude that the derivative term z (d/dz) ln(1−z) appears in the generating function can be traced back to the term · ˙X of the vertex operator. This term · ˙X will later results in h ˙Xµ_(x)Xν_{(y)i ∝ x (d/dx) ln(x − y) and thus}

exists in the generating function. The key feature connecting the generating function with vertex operator has been made manifest now, generalization to more complicated configuration of vertex operators can be established. For example, consider this emission vertex for the second excited state

V (k2, z2) =: 2· ¨X(z2)eik2·X(z2) : . (5.33)

Having the previous experience, to obtain the generating function of the vertex (5.33), we have to calculate h ¨X(x)X(y)i

h ¨Xµ(x)Xν(y)i = −ηµνx d dxx

d

dxln(x − y). (5.34) Multiplying z(d/dz)z(d/dz)ln(1 − z) by 2· k3 and the factor exp [k2· k3ln(1 − z)] leads

to the generating function of the vertex (5.33) G2 = (2· k3) z d dzz d dz [ln(1 − z)] e k2·k3ln(1−z)_. _(5.35)

The above calculation can then be generalized to the vertex operator having arbitrarily n-th order derivative of Xµ_{(τ ), namely}

V (k2, z2) =: 2· ∂τ(n)X eik2

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5.3. BRIEF SUMMARY

Following the same procedures, we first evaluate h∂ n_Xµ_(x) ∂τn x Xν(y)i = ηµν x d dx n ln(x − y), (5.37) and the generating function of the scattering amplitude with vertex (5.36) and three tachyons can be easily calculated by

Gn= (2· k3) z d dz n ln(1 − z)ek2·k3ln(1−z)_. _(5.38)

Combining the algebraic structures of equation (5.31) and (5.38), we are able to obtain the generating function of a scattering amplitude of 3 tachyons and one arbitrary spin state, for example

1 · ˙X 2· ¨X 3· ¨X · · · : eik·X _:, _(5.39) is given by G = Y1z d dzln(1 − z) Y2z d dzz d dzln(1 − z) Y3z d dzz d dzln(1 − z) · · · eX ln(1−z)_, (5.40) where Y1 = 1· k, Y2 = 2· k and Y3 = 3· k.

5.3 Brief summary

In the first section of this chapter, we first generalize our algorithm from four–tachyon amplitude to an amplitude of one vector and three tachyons. Generating function for this case still exists. In section 5.2, we show how to adopt path integral approach to systematically find out the generating function of a scattering amplitude of one arbitrary string state and 3 tachyons.

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Chapter 6 Conclusion

Let us summarize what we have achieved in the thesis. Motivated by the success of BCFW recursion relation in field theory, we are interested in applying this method to string theory. In field theory, Feynman’s rule helps us to write down scattering amplitude systematically. In BCFW calculation of field theory, the BCFW poles are determined by Feynman propagators. However, when applying BCFW method to string amplitudes, the BCFW poles for string amplitude such as Veneziano amplitude are not manifest. In [5], Clifford Cheung et at binomially expanded the integrand of Veneziano amplitude to extract the pole structure of string amplitude.

In this thesis, we extend the application of BCFW recursion relation to string tree-level amplitudes. In contrast to the field theory calculation, we encounter the difficulty of summing over all intermediate physical states with infinite tower of mass levels. We develop a method to resolve this difficulty by enlarging the sum over all intermediate physical states to an easier sum over the entire Fock space of string spectrum. The zero contributions of extra states are guaranteed by the no-ghost theorem in the open bosonic string theory. In this calculation, we do produce the conjectured scalar-behaved residue observed in [5]. The calculation is successfully applied to the 4-tachyon amplitude, and then to the cases of one arbitrary higher spin state and 3-tachyon amplitudes. For the cases of higher spin scatterings, we figure out a generating function for summing the infinite poles of string spectrum in the BCFW string amplitude calculation. We also find out that we can use identities of the Stirling number of the first kind to sum over the string

BCFW 遞迴關係式與玻色開弦中樹圖的散射振幅

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研究

生：張永業

指導教授：李仁吉

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研究

生：張永業

_{Student: Yung-Yeh Chang}

指導教授：李仁吉

_授

_{Advisor: Prof. Jen-Chi Lee}

國立交通大學

電子物理研究所

碩士論文