Browsing the Web of Amplitudes

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MASTERTHESIS, 30C

Browsing the Web of Amplitudes

Author: Alexander Söderberg Supervisors: Henrik Johansson Subject Reader: Joseph Minahan

Thesis Series: FYSAST Thesis Number: FYSMAS1051

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BSTRACT

We begin by studying field-theory amplitude relations such as the Kleiss-Kuijf, Bern-Carrasco-Johansson, Kawai-Lewellen-Tye and the double copy construction, which are important ingredients in this thesis. Going beyond the field-theory limit we study how the gauge-sector of the heterotic string relates to type I amplitudes through the single valued projection of multiple zeta values. At low energy and for a U(1) gauge group (a single brane) the type I amplitudes are generated by the Born-Infeld action, whereas the corresponding heterotic amplitudes vanish in this limit. As a simple exercise we study Yang-Mills theory deformed by a F4operator, which is the first correction induced by the Born-Infeld action. This exercise is then generalized by considering the four- and six-point amplitudes in Tseytlin’s proposal for a non-Abelian Born-Infeld action. Comparing these amplitudes with those found in type I and heterotic string theory we attempt to gain more insight about the non-Abelian Born-Infeld action.

MASTERPROGRAM INPHYSICS

DIVISION OFTHEORETICALPHYSICS

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AMMANFAT TNING

Spridningsamplituder beskriver växelverkningar mellan partiklar. Till exempel annihilation där en elek-tron och dess antipartikel, en posielek-tron, kan växelverka med varandra och skapar en foton, som sedan blir ett nytt elektronpar. Med hjälp av spridningsamplituder kan man hitta tvärsnittsarean som beskriver sannolikheten för en given interaktion. För att kunna förutsäga resultatet av ett experiment är det därför viktigt att studera dessa amplituder, och hur olika amplituder förhåller sig till varandra. I gaugeteorier, det vill säga teorier som är invarianta under lokala symmetri transformationer, till exempel Yang-Mills teorin, delar vi ofta upp amplituder i ordnade delamplituder, där partiklarnas ordning i dessa delampli-tuder spelar roll. Vanligtvis är det dessa delamplidelampli-tuder som sdelampli-tuderas. I första delen utav detta arbete studeras det hur amplituder, inom samma eller mellan olika teorier, förhåller sig till varandra. Rela-tioner som till exempel KK och BCJ relaterar delamplituder inom samma teori, vilket reducerar andelen delamplituder som behöver beräknas, medan relationer såsom KLT relaterar gaugeteorier med en gravi-tationsteori.

Efter det studeras amplituder i olika supersträngteorier, och hur de förhåller sig till varandra. Olika su-persträngteorier motsvarar olika gränsvärden på M-teorin, vilket har föreslagits vara den underliggande teorin som beskriver de fyra krafterna: starka- och svaga växelverkan, elektromagnetism samt graviation. Den viktigaste relationen för det här arbetet i supersträngteori är en matematisk relation som relaterar amplituder i heterotisk strängteori med amplituder från typ I strängteori. Denna relation kommer att användas för att beräkna sexpunkts amplituder i heterotisk strängteori.

Sedan jämförs de heterotiska strängamplituderna med amplituder av samma ordning från fältteorigränsen av en kandidat till den icke-Abelska Born-Infeld-teorin. Denna teori beskriver gaugefält som lever på ett

D p-bran. Ett D-bran är generaliserad yta i strängteori som strängars ändpunkter kan befinna sig på, och

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ONTENTS

1 Amplitudes in Quantum Field Theories 3

1.1 Super Yang-Mills Amplitudes . . . 3

1.2 Supergravity Amplitudes . . . 4

1.3 The Color-Kinematics Duality . . . 5

2 Amplitudes in Superstring Theories 9 2.1 Type I Amplitudes . . . 9

2.2 Multiple Zeta Values and the Single-Valued Projection . . . 10

2.3 α0-expansion of Type I Amplitudes . . . 12

2.4 Heterotic String Amplitudes . . . 13

2.5 Graviton String Amplitudes . . . 15

2.6 Web of Amlitudes . . . 16

3 F4-Deformation of YM 17 3.1 Lagrangian in a F4-Deformation of YM . . . 17

3.2 Spinor and Polarization Technology . . . 19

3.3 Four-Point Amplitude in F4-Deformation of YM . . . 21

4 Calculating Heterotic String Amplitudes 23 4.1 Heterotic String Amplitudes as SYM Amplitudes . . . 23

4.2 Propagator . . . 24

4.3 SYM Amplitudes . . . 25

4.3.1 Three-Point Vertex . . . 25

4.3.2 Four-Point Vertex . . . 28

4.4 Heterotic String Amplitudes . . . 30

4.5 Little Group Scaling . . . 31

5 Generalized Born-Infeld Action 33 5.1 Color-Ordered Amplitudes FromL1andLF4 . . . 35

5.2 Color-Ordered Amplitudes FromL2andLYM . . . 36

5.3 Color-Ordered Amplitudes FromL1andLYM . . . 38

5.4 Non-Abelian Born-Infeld Amplitudes . . . 39

6 Comparing Full Amplitudes 41

Appendices 47

A SU(N ) Algebra Convention 49

B F4Normalization 51

I

NTRODUCTION

It is important to study scattering amplitudes in quantum field theories (QFTs) and string theories since they are required for computing the probability of processes that can be studied in experiments. At the beginning of this thesis we will study how we may relate one amplitude with another, be it within the same theory or not. Naively, in a trace basis decomposition, we need to calculate n! different color-ordered subamplitudes, but with amplitude relations such as the KK and the BCJ relations we reduce the number of independent subamplitudes to (n − 3)!. The KK relations was first derived by Kleiss and Kujif back in 1989 [1], and the BCJ relations was not derived 2008 by Bern, Carrasco and Johansson [2]. The BCJ relations has reinvigorated that there are some highly non-trivial results that needs further study.

As previously stated we may relate amplitudes from different theories with each other as well. A prime example of this is the KLT relations, which was derived by Kawai, Lewellen and Tye in 1986 [3]. The KLT relations describes supergravity (SUGRA) amplitudes as a sum over a pair of super-Yang-Mills (SYM) am-plitudes at tree level, hence we often say that SUGRA is in some sense the "square" of gauge-theories. The KLT relations is not the only relation which relates amplitudes from different theories with each other. Using the BCJ relations we may relate the gravity amplitudes with two different SYM at loop level. It is referred to as the BCJ double copy relation [4]. One important thing to note is that the BCJ double copy can in general be different from the KLT relations, and that the BCJ double copy relation holds at any loop-level while the KLT relations only stands true at tree level.

Another relation that relates amplitudes from different theories, that we will study, is the single-valued projection of amplitudes from type I string theories. The single-valued projection acts on mathemati-cal objects that are mathemati-called multiple zeta values (MZVs), or more generally on polylogarithmic functions. These objects may be found inside theα0_{-expansion of type I amplitudes, whereα}0_{is the inverse string}

tension, and by the single-valued projection of those we find the amplitudes from heterotic string the-ories. The single-valued projection has been defined by the mathematician Brown [5], and the single valued projection of amplitudes from type I string theory has been introduced by Stieberger and Tay-lor [6].

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N = 1 Super-Yang-Mills (SYM) in ten dimensions is governed by the Lagrangian [8]1

LSYM= − 1 4F a1 µνFa1µν+ i 2 ¯ Λa1Γµ_D µΛa1, DµΛa1= ∂µΛa1+i g 2 f a1a2a3_Aa2 µ Λa3. (1.1.1)

Here D_µis a covariant derivative, andΛa1_{is a spinor. The gauge group is SU(N ). Dimensional reduction}

yieldsN = 4 SYM in four dimensions. We will study this Lagrangian in higher detail in section 4.1 and focus on the color ordering of amplitudes in this section.

With all fields in the adjoint representation we find the n-point tree amplitude,Atree_n , at tree-level in a SYM theory by summing over its color-ordered subamplitudes,A , and the color generators, Ta, for the SU(N ) gauge group2[2]

Atree_{n = g}n−2 X

π∈Sn/Zn

Tr (Ta1_Taπ(2)_...Ta_{π(n))A (1,π(2,...,n)) , S}

n= Perm. (1, ..., n) _(1.1.2)

Here g is the SYM coupling constant, and we sum over all cyclic permutations of (n − 1)! legs, giving us a total of (n − 1)! different A s. This way of writing an amplitude is called the color decomposition of the theory. The color-ordered amplitudes satisfy the following cyclic and reflection properties [2]

A (1,...,n) = A (2,...,n,1) ,

A (1,...,n) = (−1)n_{A (n,n − 1,...,1) .} (1.1.3)

Let us discuss how we may reduce the number ofA s inAtree_n from (n − 1)! to (n − 3)!, see [9]. The general idea is to express two of the (n −1)! subamplitudes in terms of other subamplitudes, which fixes three legs

1_{More details about our convention is in appendix A.}

2_{The number of supercharges may vary, amplitudes may still be color-decomposed in the same way. We have reduced the n!}

inAtree_n . This yields (n −3)! terms in (1.1.2). We fix one leg, say leg n −1, using the KK relation [1], which is a more general form of the photon decoupling (or dual Ward identity) relation. The photon decoupling relation states

A (1,...,n) + A (2,1,3,...,n) + ... + A (2,...,n − 1,1,n) = 0 , (1.1.4) and the more general KK relation yields

A (1,α,n,β) = (−1)n_β X

σ∈HA (1,σ,n) , H = OP(α,β T

) . (1.1.5)

Here the ordered permutations, OP, are permutations which maintains the order of the setsα and β, nβ is the amount of elements inβ and βTis the reversed order ofβ. Finally we fix a third leg, say leg n, using the fundamental BCJ relation3

s12A (2,1,3,...,n) + (s12+ s13)A (2,3,1,4,...,n) + ...+

+¡s12+ ... + s1,n−1¢A (2,...,n − 1,1,n) = 0 .

(1.1.6)

Here sj lis the Mandelstam variables4

s_{j l:= −(k}j+ kl)2. (1.1.7)

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Originally the KLT relation [3] was derived to express the graviton string amplitude at tree level in terms of color-ordered superstring subamplitudes. However, as can be seen in [10,11] we may use the KLT relation outside of string theory to express the n-point amplitude,Mtree_n , for gravitons at tree-level in supergravity (SUGRA) in terms of SYM subamplitudes,Anand ˜An

Mtree_n = (−1)n−3κn−2 X π,ρ∈Sn−3 A [1,π(2,...,n − 2),n − 1,n]S[π(2,...,n − 2)|ρ(2,...,n − 2)]× × ˜A [1,ρ(2,...,n − 2),n − 1,n] ≡ (−1)n−3κn−2AπSπρA˜ρ, κ = 8πG . (1.2.1)

Hereκ is related to Newton’s gravitational constant, G, and Sπρis the KLT kernel. This kernel is a (n −3)!× (n−3)! matrix. We will study Sπρas well as the KLT relation in more detail when we move on to amplitudes in string theory, see section 2.1. For simplicity we think of the SYM subamplitudes,A_πand ˜A_ρas vectors of length (n − 3)!. Since we sum over orderings of two SYM subamplitudes in the KLT relation, we often say a SUGRA amplitude is found through squaring SYM amplitudes. It is important to know thatA_πand

˜

Aρ does not need to be from the same SYM theory, e.g. A may be in N = 0 SYM, i.e. YM, while ˜A is

inN = 4 SYM. Of course, different choices of SYM subamplitudes yields different SUGRA amplitudes, e.g. ifA_πandA˜_ρare bothN = 4 SYM amplitudes we get a N = 8 SUGRA amplitude, and if we A_πis a YM amplitude whileA˜_ρ is aN = 4 SYM amplitude we get a N = 4 SUGRA amplitude from the KLT relation [3].

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So far we have only studied amplitudes at tree level, e.g. the KLT relations only holds for amplitudes at tree level. Let us now discuss SYM and SUGRA amplitudes at loop level.

To begin with, let us write the n-point amplitude,Atree

n , at tree level as Atree_{n = g}n−2X j njcj Q αjp 2 αj . _(1.3.1)

Here we sum over all diagrams, the product goes over all channels (internal propagators) in the jth di-agram, nj is the kinematic numerator for the jthdiagram, cj is the color factor for the jt hdiagram and

p_αjis the momentum for theαj-channel. The color factor, cj, consists of all structure constants in the

jthdiagram, i.e. cj is a bunch of structure constants, fabc, contracted together.

Note 1. In [12], we see that in SYM we may absorb the four-point vertices into cubic ones. This is easy

to see through examples, e.g. from a four-point vertex we will pick up a factor of g2feabfecd and from a three-point vertex we have a factor of g fabc. Thus from diagrams with two three-point vertices we get the same factor g2_feab_fecd_{as that in four-point vertices, which yields us that the color-structure of four-point}

vertices is the same as that of diagrams with two three-point vertices.

~

Figure 1.3.1: Four-point vertices can be rewritten as diagrams that looks like a product of two three-point vertices in SYM.

Note 2. In (1.3.1) we have writtenAtree_n using structure constants, as opposed to (1.1.2) where we haveAtree_n

in a trace basis.

In SYM, the gauge group structure constants, fabc, satisfy the Jacobi identity, hence the cj’s will satisfy it

too, i.e.

fabefecd_{+ f}acefed b_{+ f}ad efebc_{= 0 ⇒} cj+ ck+ cl= 0 , (1.3.2)

for three different diagrams, j , k and l . For two diagrams that differ only by the permutation of the indices in fabc, say diagram j and k, we may have from antisymmetry of the fabcs that

cj= −ck. (1.3.3)

The BCJ relation for numerators tells us that in some gauge we can find njs that satisfy the same relations

as cj [2], i.e.

nj+ nk+ nl= 0 ,

nj= −nk.

(1.3.4)

The BCJ relations for the numerators, nj, may in principle contain poles [13]. This analogy between nj

Note 3. This relation is gauge dependent, however, the gauge invariant content of it is equivalent (at

tree-level) to the fundamental BCJ relation at (1.1.6) we studied earlier [14].

Figure 1.3.2: In the first diagram, the circle with arrows shows that we read off the labels for the structure constants, fabc, in a clockwise ordering. The first diagrams (in the upper row) equals zero from the Jacobi identity. Thus from antisymmetry of fabcand the BCJ relations (1.3.4) we know that the first diagram (in the upper right corner) may be written in terms of two other diagrams (the lower row).

SUGRA amplitudes relates to the amplitude (1.3.1) through the BCJ double copy relation

Mtree_{n = i}³κ 2 ´n−2_X j njn˜j Q αjp 2 αj . _(1.3.5)

Here njand ˜njare numerators from two (not necessarily the same) SYM theories. At least one of the two

numerators in this relation need to satisfy the BCJ relations in (1.3.4). This relation between SUGRA and SYM amplitudes is equivalent to the KLT relation (1.2.1) at tree level if we consider a gauge theory with all particles in the adjoint representation [4]. Unlike the KLT relation, the BCJ relation is conjectured to hold at loop level [12]. At L-loop level, i.e. for loops, we have the amplitude

AL_{n= i}Lgn−2+2LX j 1 Sj Z _dd L_p (2π)d L njcj Q αjp 2 αj . (1.3.6)

amplitude,ML_n, at L-loop level ML_{n= i}L+1³κ 2 ´n−2+2L X j 1 Sj Z _dd L_p (2π)d L njn˜j Q αjp 2 αj . (1.3.7)

Unlike the tree level case, there is no proof that one can always find a representation where we can find numerators, nj, that satisfy the BCJ relation (1.3.4) at loop level. So far it seems to hold forN = 4 SYM

for at least four loops though [13].

The reader may wonder whether one can exchange nj with another color-factor ˜cj inALn and get the

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In superstring theory, i.e. supersymmetric string theory, we consider type I (open superstrings), type II (closed superstrings) and heterotic strings. Heterotic strings are hybrids of closed bosonic strings and closed superstrings. In fact there exist two different type II strings (type IIa and type IIb) as well as two different heterotic strings (SO(32) heterotic and E 8 ⊗ E8 heterotic). The five superstring theories are dif-ferent limits of M-theory [16].

We will study superstring amplitudes and express heterotic string as well as graviton string amplitudes in terms of type I subamplitudes (or disk amplitudes). From this point of view type I amplitudes play a central role in superstring theory.

In string perturbation theory we sum over Riemann surfaces1, compared to QFTs where we sum over all possible Feynman diagrams. This makes amplitudes in string theory more mathematically challenging than QFT amplitudes hence we will only study string theory amplitudes at tree level.

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We will start with the n-point, type I amplitude,AI_n, at tree level, which we want write in a similar form as (1.1.2). In order to reduce the number of color-ordered subamplitudes,AI, from (n −1)! to (n −3)!, we use the corresponding relation for KK and BCJ, see (1.1.5) and (1.1.6), in superstring theory. This relation is called the monodromy relation [17, 18]

AI n(1, ..., n) + eiπ˜s12AnI(2, 1, 3, ..., n) + eiπ(˜s12+ ˜s13)AnI(2, 3, 1, 4..., n) + ...+ + eiπ(˜s12+...+ ˜s1,n−1)AI n(2, ..., n − 1,1,n) = 0 , ˜ sj l:= α0(kj+ kl)2= 2α0kjkl. (2.1.1)

Here ˜sj l are dimensionless Mandelstam variables andα0is the inverse string tension2. As can be seen

in [19], the field-theory limit of the KK relation (1.1.5) is the real part of the monodromy relation, and

the fundamental BCJ relation (1.1.6) is the imaginary part. According to [20, 21] we may write the type I subamplitudes,A_πI,π ∈ Sn−3, in terms of SYM subamplitudes,A_πSYM, i.e.

AI

π= FπρAρSYM, π ,ρ ∈ Sn−3., (2.1.2)

Here F_πρis a (n − 3)! × (n − 3)! matrix, where its elements are hypergeometric functions. We will write it in the same way as in [6]

F_πρ= (−1)n−3Z_πσSσρ, π ,ρ ,σ ∈ S_n−3, (2.1.3) Z_πσ= Z D(π) n−2 Y j =2 d zj Y ∆(k,l)|zkl| ˜ skl 1 z1σ(2)zσ(2)σ(3)...zσ(n−3)σ(n−2) , zk j:= zk− zj, D(π) = {0 < z_ρ(2)< ... < zρ(n−2)< 1 , zj∈ R} , ∆(k,l) = {1 ≤ k < l ≤ n − 1 ,k ∈ {1,...,n − 2} ,l ∈ {2,...,n − 1}} , (2.1.4) Sσρ= n−2 Y j =2 Ã ˜ s1 j+ j −1 X k=2 θ(j,k)˜sj k ! , θ(j,k) =   

1 , if j and k are in the same order in bothρ(2,...,n − 2) andσ(2,...,n − 2).

0 , else.

(2.1.5)

Here Z_πσis a disk integral, i.e. we integrate over a disk, Sσρis the KLT kernel (we encountered this earlier in the KLT relation, see (1.2.1)). Through a SL(2,_{C)-transformation, i.e. a Möbius transformation, we} have fixed three points

z1= 0 , zn−1= 1 , zn= ∞ , (2.1.6)

thus we only need to integrate over z2, ..., zn−2in Zπσ. All together, this yields the type I subamplitude AI

π= (−1)n−3ZπρSρσA_σSYM, π ,ρ ,σ ∈ S_n−3. (2.1.7)

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Before we study heterotic string amplitudes we will discuss both multiple zeta values (MZVs) and the single-valued projection. Integrals which appears in type I amplitudes may be written as a power series expansion inα0, which may be expressed in terms of MZV’s. The single-valued projection is an operator which only acts on MZVs, even though MZVs are numbers. The MZVs are treated as motives of the Hopf algebra, which they generate3. This projection operator is useful when we study heterotic string ampli-tudes.

MZV’s,ζn1...nr, are specific values of the generalized Riemann zeta function

4 ζn1...nr:= X 0<k1<...<kr r Y m=1 k−nm m , nj∈ {2, 3, ...} . (2.2.1)

We define r as the depth of the MZV, and we define the weight, w , of the MZV as w = r X m=1 nr . (2.2.2)

In [22] we see that we may write the MZVs as polylogarithms, G, i.e.

ζn1...nr= (−1) r_{G(0, ..., 0} | {z } nr−1 , 1, ..., 1, 0, ..., 0 | {z } n1−1 , 1 : 1) , G(a1, ..., an: z) : = Z z 0 d t t − a1 G(a2, ..., an: t ) , G(z) = 1 , aj, z ∈ C . (2.2.3)

Here the G inζn1...nr has a total of n zeros and ones. Since we may write G(a1, ..., an: z) as an integral over another polylogarithm, G(a2, ..., an: t ), with one less argument we may iterate this procedure and end up

with a generalization of G(a1, ..., an: z)5

G(a0: a1, ..., an: an+1) := Z ∆n,γ d z1 z1− a1∧ ... ∧ d zn zn− an . (2.2.4)

Here ∧ denotes the antisymmetric wedge product, γ is a path in M = C\{a1, ..., an} with endpoints

γ(0) = a0∈ M , γ(1) = an+1∈ M ,

(2.2.5)

and∆n,γis a simplex which consists of all n-tuples of z1, ...zn, onγ6. Expressing the MZV’s in terms of I

yields [22] ζn1,...,nr = (−1) r_{I (0 :ρ(n} 1, ..., nr) : 1) , ρ(n1, ..., nr) = 10−r r Y j =1 10nj _{. (2.2.6)}

The set of linear combinations of MZV’s is a ring [22], e.g. the stuffle (or quasi-shuffle) relation

ζm aζmb = ζ m a,b+ ζ m b,a+ ζ m a+b. (2.2.7)

If we act with the single-valued projection, sv(•), on a MZV, we remove all of the branch cuts in the poly-logarithms (2.2.4) in the MZV, e.g. we define the usual logarithm with a branch cut over all negative real numbers, meaning the single-valued projection of a logarithm would remove this branch cut. This oper-ation projects MZVs onto a subset of the MZVs

sv : H → Hsv, ζmn1...nr 7→ ζ

m

sv(n1, ..., nr) . (2.2.8)

Here the m-index aboveζn1...nr means that we are treating them as motives rather than numbers. The subalgebra,Hsv, to the Hopf algebra,H , is generated by the single-valued multiple zeta values (SVMZV’s), ζm sv. The SVMZV’s satisfy [5] sv¡ A(ζm_{)B (ζ}m ) +C (ζm)¢ = svA(ζm_{)svB (ζ}m ) + svC (ζm) . (2.2.9) 5_{We write G(a}

2, ..., an: z1) as an integral over G(a3, ..., an: z2) e.t.c.

As can be seen in [6], some examples of relations that the SVMZVs satisfies are ζm sv(2) = 0 , ζm sv(2n + 1) = 2ζm2n+1, n ≥ 1 , ζm sv(5, 3) = 14ζm3ζm5 , ζm sv(3, 5, 3) = 2ζm3,5,3− 2ζm3ζm3,5− 10 ¡ ζm 3 ¢2 ζm 5 . . (2.2.10)

More relations obeyed by the MZVs and SVMZVs may be found in [5, 6, 22].

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In this section we will study theα0-expansion of the hypergeometric functions F_πρ, that appear in type I subamplitudes,_AπI, see (2.1.2). This expansion will be given in terms of MZVs. We will write F_πρin a different, but equivalent, as (2.1.3), namely as generalized Euler functions [22]

F_πρ_{= (−1)}n−3 Z D(π) n−2 Y j =2 d zj Y ∆(k,l)|z kl|s˜kl j −1 X m=1 ˜ sm j zm j . (2.3.1)

Here D(π) and ∆(k,l) are the same as in (2.1.4).

Before we study the general case, we will study the cases where n = 3 as well as n = 4. For the three-point case we fix all of the coordinates with the SL(2,C) transformation (2.1.6), meaning there is nothing to integrate, i.e.

F ≡ (Fπρ) = (−1)3−3|z12|s˜12= |0 − 1|s˜12= 1 ⇒ AI= ASYM. (2.3.2)

As we can see, the three-point type I amplitude is equal the SYM amplitude, thus there is noα0-correction in this case.

In the four-point case when n = 4 we get

F = (−1)4−3 Z 1 0 d z2|z12|s˜12|z13|s˜13|z23|s˜23 ˜ s12 z12= − ˜s Z 1 0 d z2 0 − z2|0 − z2| ˜ s |0 − 1|˜t|1 − z2|u˜ = ˜s Z 1 0 d z2 z2 z₂s˜(1 − z2)u˜= ˜s Z 1 0 d z2z2s−1˜ (1 − z2)u˜=Γ(1 + ˜s)Γ(1 + ˜u) Γ(1 + ˜s+ ˜u) . (2.3.3)

According to [22] we may use the following identities Γ(1 + x) Γ(1 − x)= exp µ −2 ∞ X n=1 x2n+1 2n + 1ζ2n+1 ¶ e−i γEx_, π s ˜˜u ˜ s + ˜u sin[π(˜s+ ˜u)] sin(π˜s)sin(π ˜u)= exp

µ 2 ∞ X n=1 ζ2n 2n £ ˜s 2n + ˜u2n− ( ˜s+ ˜u)2n¤ ¶ , (2.3.4)

whereγEis the Euler–Mascheroni constant, to rewrite (2.3.3) as

We define P2nand M2n+1as

P2n≡ R2n, M2n+1≡ −R2n+1. (2.3.6)

Now for a general number of legs, F_πρas well as Rnwill differ from that above

F_πρ_{= Pπσ}Qσρ: exp µ X n≥1 ζ2n+1M2n+1 ¶ : , ¡Qσρ¢ = 1 +X n≥8 Qn, ¡Pπρ¢ = 1 +X n≥1 ζn 2P2n, P2n= F |ζn 2 , M2n+1= F |ζ2n+1. (2.3.7)

Here P_πσ,Qσρ and M2n+1are all (n − 3)! × (n − 3)! matrices, and " : " denotes normal ordering. Qn is

calculated up to n = 16 in [22], e.g. Q8,Q9and Q10is given by

Q8= 1 5ζ3,5[M5, M3] , Q9= 0 , Q10= µ 3 14ζ 2 5+ 1 14ζ3,7 ¶ [M7, M3] . (2.3.8)

As we may see, is P2nassociated withζ2n-terms, and M2n+1associated withζ2n+1-terms. The

polyloga-rithms are studied in more details in [23], and the matrices P2nas well as M2n+1can be found for different

order and multiplicity of supercharges at [24]. Theα0-expansion in (F_πρ) up to the eighth order in MZVs is given by ¡F_πρ¢ = 1 + ζ2P2+ ζ3M3+ ζ22P4+ ζ2ζ3P2M3+ ζ5M5+ ζ32P6+ 1 2ζ 2 3M32+ ζ7M7+ + ζ2ζ5P2M5+ ζ22ζ3P4M3+ ζ42P8+ 1 2ζ3ζ5{M3, M5} + 1 2ζ2ζ 2 3P2M32+ +1 5ζ3,5[M5, M3] + O (ζ9,Q10) . (2.3.9)

2.4

H

ETEROTIC

S

TRING

A

MPLITUDES

Let us now study a heterotic amplitude for massless vectors, and see how it relates to type I amplitudes. Unlike type I amplitudes, where we used the monodromy relation to reduce the number of subampli-tudes, we use the KK and the fundamental BCJ relation (see (1.1.5) and (1.1.6)) to reduce the number of subamplitudes to (n − 3)!. Similar to type I subamplitudes, we may write heterotic subamplitudes, AHET

π ,π ∈ {1,...,(n − 3)!}, in terms of SYM-subamplitudes, AτSYM[6], i.e.

AHET

π = (−1)n−3JπρSρσAσSYM, π,ρ ∈ {1,...,(n − 3)!} . (2.4.1)

Here J_πρand Sρσare both (n −3)!×(n−3)! matrices, where Sστis the same as for type I amplitudes (2.1.5), and every element in J_πρis a sphere integral, i.e. an integral where we integrate over a sphere

J_πρ= n Y j =1 Z z∈C d zj Y ˜ ∆(k,l) |zkl|2 ˜skl 1 z1µ(2)zµ(2)µ(3)...zµ(n−3)µ(n−2)zµ(n−2),n−1zn−1,nzn1× ×_z¯ 1 1µ(2)z¯µ(2)µ(3)... ¯zµ(n−3)µ(n−2)z¯µ(n−2)nz¯n,n−1z¯n−1,1 . (2.4.2)

Important to note is that the first factor (with only zs) is of order 1,µ(2,...,n −2),n −1,n, while the second factor (with only ¯zs) is of order 1,µ(2,...,n − 2),n,n − 1. We can change the order of the second of these

factors to be the same as the first one through a matrix, K_πρ, that acts as a basis change of SYM amplitudes ˜

ASYM

Example 1. When we consider four or five interacting vector multiplets, i.e. n = 4, or 5, Kπρis given by [6] • n = 4 : K =s_s˜_˜14 13 , • n = 5 : ¡K_πρ¢ = "_{( ˜s} 13+ ˜s15) ˜s34 ˜ s14s˜35 − ˜ s13s˜24 ˜ s14s˜35 −s˜12s˜34 ˜ s14s˜25 ( ˜s12+ ˜s15) ˜s24 ˜ s14s˜25 # , π ,σ ∈ {1,2} . (2.4.4)

If we fix the same three points as those we fixed when finding a type I amplitude (see (2.1.6)), and we use

K_πρto rewrite the order of the ¯zs of the second factor in (2.4.2) to be the same as the order of the zs in its

first factor, we end up with

J_πσ= (K−1)_πρI_ρσ, I_ρσ= n−2 Y j =2 Z z∈C d zj Y ∆(k,l)|z kl|2 ˜skl× ×_z 1 1ρ(2)zρ(2)ρ(3)...z_{ρ(n−3)ρ(n−2)}z¯1σ(2)z¯σ(2)σ(3)... ¯z_{σ(n−3)σ(n−2)}. (2.4.5)

Here∆(k,l) is the same as in (2.1.4). In [6] we see that we may express the sphere integrals I_ρσ, as a single-valued version of the disc integrals (2.1.4), Z_τσ, that we encountered in type I amplitudes

I_{ρσ= Kρ}τsvZ_τσ. (2.4.6)

What we do here is Taylor expand the¯¯z_{k j} ¯ ¯

˜

sk j_{factor in Z}

τσin terms ofα0(which is in ˜sk j, see (2.1.4)), and

then note that I_ρσis proportional to sv(Z )_τσ. If we plug this I_ρσback intoA_πHETat (2.4.1), we get AHET

π = (−1)n−3δπτsvZ_τσSστA_τSYM= (−1)n−3svZ_πρSρσASYM_σ . (2.4.7) Comparing with (2.1.7) yields7

AHET

π = svAπI, π ∈ Sn−3. (2.4.8)

Note 4. Not only we may write the heterotic subamplitude,A_πHET, as a single-valued type I subamplitude, sv_AπI, but we may also write the integrals, I_ρσ, in_AπHETas a single-valued version of Z_τσ, see (2.4.6).

= sv

Figure 2.4.1: Heterotic string subamplitudes are given by the single-valued projection of type I subam-plitudes.

7_{Only Z}

πρinAI

2.5

G

RAVITON

S

TRING

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MPLITUDES

If we apply the KLT relation (1.2.1) on the type I subamplitudes,A_πI, we get the graviton string scattering amplitude,MI_n, in type I string theory

M_nI _{= (−1)}n−3κn−2A_πIS˜πρA˜_ρI, π,ρ ∈ S_n−3. (2.5.1) Hereκ is the graviton coupling constant, and both A_πI and ˜A_ρIare type I subamplitudes. Sπρis a sinus version of (2.1.5) [6], i.e. ˜ Sπρ₌ n−2 Y j =2 sin à ˜ s1 j+ j −1 X k=2 θ(j,k)˜sj k ! . (2.5.2)

Let us define a gravity kernel, Gπρ, s.t.

M_nI _{= κ}n−2_AπSYMGπρA_ρSYM, Gπρ:= ˜SπσI_στS˜τρ. (2.5.3) Here I_στis the disk integral from (2.4.5). In the field theory limit, i.e. in the limit whenα0→ 0, we let

Gπρ _−→ α0_→0(−1) n−3_S˜πρ 0 ⇒ M I n −→ α0_→0(−1) n−3_κn−2_ASYM π S˜πρ0 A SYM ρ . (2.5.4)

We now rewrite the KLT relation (1.2.1) using the basis change (2.4.3)

M_nI −→

α0_→0(−1)

n−3_κn−2_ASYM

π S˜πρKρσAσSYM. (2.5.5) Comparing (2.5.4) with (2.5.5) yields

˜

Sπσ_{= ˜}Sπρ₀ (K−1)_ρσ _⇒ Gπρ_{= ˜}Sπρ₀ (K−1)_ρσI_στS˜τρ. (2.5.6) Thus from (2.5.3) we get

MI_{n= κ}n−2_AπSYMS˜πρ₀ (K−1)_ρσI_στS˜τυ_AυSYM | {z } (2.4.1) = (−1)3−n_AHET ρ = (−1)n−3κn−2AπSYMS˜πρ0 svA I ρ . (2.5.7)

As we can see mayMI

2.6

W

EB OF

A

MLITUDES

So we can relate both heterotic and graviton string amplitudes to type I amplitudes. This may indicate that type I amplitudes play a central role in superstring amplitudes. This is not the only relations between amplitudes in different superstring theories though. As can be seen in [6], are there a lot of relations between various amplitudes in superstring theory and its field theory limits. This cluster of amplitudes, and the relations between them is often referred to as the web of amplitudes.

TypeBI

Heterotic

TypeBII

Sugra

SYM

KLT

Sv

Sv

Mellin

α'→0

α'→0

α'→0

U(N)BBBBB>U(1)

BI

F⁴

α'→0

Figure 2.6.1: Part of the web of amplitudes.

C

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3

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4

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YM

3.1

L

AGRANGIAN IN A

F

-D

EFORMATION OF

YM

As an exercise we will calculate the four-point amplitude in a F4-deformation of YM, and determine its Lagrangian. This exercise will be useful in section 5 where we calculate amplitudes through six-points from Tseytlin’s proposal for a non-Abelian Born-Infeld theory [7].

We only consider completely symmetric structure constants, dabcd. There is a more general F4-deformation of YM than the one we compose in this section, but we do not study it here. The structure constants dabcd is defined as the symmetrized trace over four generators from the SU(N) gauge group, i.e.

dabcd: = 1 4!STr ³ TaTbTcTd´_≡ 1 4! h Tr³TaTbTcTd´_{+ Perm.(a, b, c, d)}i = 4 4! h Tr³TaTbTcTd´+ Perm.(b, c, d)i= 1 3! h Tr³TaTbTcTd´+ Perm.(b, c, d)i (3.1.1)

Here we have used the cyclicity of the trace. We find the Lagrangian for the F4-Deformation of YM by considering all possible combination of F4-terms1

LF4= α1dabcdF_µνa FbµνF_ρσc Fdρσ+ α2dabcdF_µνa FbνρF_ρσc Fdσµ. (3.1.2)

Here theαj’s, j ∈ {1,2}, are constants. We may find a relation between the αjs by normalizing the full

amplitudes obtained from the Lagrangian above2

A_F4(1+, 2+, 3+, 4+) =A_F4(1−, 2+, 3+, 4+) = 0 , A_F4(1−, 2−, 3+, 4+) = 2i 3d abcd 〈12〉2[34]2. (3.1.3)

We will discuss the spinor and polarization formalism in section 3.2. Here s and u are Mandelstam

vari-1_{One could also imagineL}

F4 as Tr(FaFb)Tr(FcFd) and Tr(FaFbFcFd) terms, where the trace goes over all Lorentz indices.

However, the result is the same.

ables3

s ≡ s12:= −(k1+ k2)2, t ≡ s13:= −(k1− k3)2, u ≡ s14:= −(k1− k4)2.

(3.1.4)

Using the non-Abelian field strength tensor4

F_µνa _{= ∂µ}Aa_{ν− ∂ν}Aa_{µ+ i g f}abcA_µbAc_{ν= ∂µ}A_νa_{− ∂ν}Aa_{µ+ O (A}2) , (3.1.5) we get the following Lagrangian,L_F4, in F4-theory

LF4=2dabcd∂_µAa_ν∂_ρAc_σ n α1 ³ ∂µAbν_∂ρAdσ_{+ ∂}νAbµ_∂σAdρ´_{− 2α}1∂µAbν∂σAdρ+ + 2α2 ³ ∂µAbρ∂νAdσ_{− ∂}µAbρ∂σAdν_{− ∂}µAbσ∂νAdρ´₊ + α2 ³ ∂µAbσ∂ρAdν+ ∂νAbρ∂σAdµ´o+ O (A5) . (3.1.6)

We find Feynman rules from a Lagrangian by Fourier transforming the interaction terms. The fields A_µa are usually stripped off, but for our purpose it is convenient to convert them to polarization vector com-ponents that corresponds to plane waves. For example, a Fourier transform of∂µAaνyields

∂µAaν_{→ i k}µ_²ν (3.1.7)

Here²ν_j is the polarization, and kµ_j is the momentum of the vector field Aa_ν. The index j ∈ {1,...,4} denotes which leg kµ_j as well as²µ_j belong to. The four-point amplitude in F4-theory can be directly computed from the quartic interaction in the Lagrangian, giving5

A4= 2i5 X π∈S4 daπ(1)...a_π(4)n α1 h (k_π(1)k_π(2))(k_π(3)k_π(4))(²_π(1)²_π(2))(²_π(3)²_π(4))+ +(kπ(2)²π(1))(k_π(1)²_π(2))(k_π(4)²_π(3))(k_π(3)²_π(4))¤ + − 2α1(kπ(1)kπ(2))(²π(1)²π(2))(kπ(4)²π(3))(kπ(3)²π(4))+ 2α2(kπ(1)kπ(2)) h (k_π(4)²_π(1))(k_π(3)²_π(2))(²_π(3)²_π(4))− − (²π(1)²π(4))(kπ(3)²π(2))(kπ(4)²π(3)) − (kπ(4)²π(1))(²π(2)²π(3))(kπ(3)²π(4)) i + + α2 h (k_π(1)k_π(2))(k_π(3)k_π(4))(²_π(1)²_π(4))(²_π(2)²_π(3))+ + (kπ(2)²π(1))(kπ(3)²π(2))(kπ(4)²π(3))(kπ(1)²π(4))¤ª ≡ X π∈S4 daπ(1)...a_π(4)Vµπ(1)µπ(2)µπ(3)µπ(4)_²a π(1)µπ(1)²a_π(2)µπ(2)²a_π(3)µπ(3)²a_π(4)µπ(4) =¡da1a2a3a4_Vµ1µ2µ3µ4_{+ Perm.(1, ..., 4)}¢ ²a1µ1²a2µ2²a3µ3²a4µ4 =¡Vµ1µ2µ3µ4_{+ Perm.(1, ..., 4)¢ d}a1a2a3a4² a1µ1²a2µ2²a3µ3²a4µ4 ≡ Vstruc.µ1µ2µ3µ4da1a2a3a4²a1µ1²a2µ2²a3µ3²a4µ4, V µ1µ2µ3µ4 struc. = Vµ1µ2µ3µ4+ Perm.(1, ..., 4) . (3.1.8) 3_{We let k}

1and k2be ingoing momenta, while k3and k4are outgoing. 4_{More details about our convention is in appendix A.}

HereVµ1µ2µ3µ4

struc. is the vertex in the structure constant basis. The kinematic part of the Feynman vertex is Vµjµlµmµn_{= 2i}nα 1 h (kjkl)(kmkn)ηµjµlηµmµn+ kµ_ljkµ_jlknµmkµmn i + − 2α1(kjkl)ηµjµlkµnmkµmn+ + 2α2(kjkl) h k_nµjkµl mηµmµn− ηµjµnkmµlknµm− k µj n ηµlµmkµmn i + + α2 h (kjkl)(kmkn)ηµjµnηµlµm+ k_lµjkmµlkµnmkµjn io . (3.1.9)

Using (3.1.1) gives us the vertex,V_traceµjµlµmµn, in the trace basis

A_F4≡£V_traceµ1µ2µ3µ4Tr¡Ta1...Ta4¢ + Perm.(2,3,4)¤²_a1µ1²_a2µ2²_a3µ3²_a4µ4, Vµjµlµmµn trace = 1 3!V µ1µ2µ3µ4 struc. . (3.1.10)

Note 5. As we may see, are V_traceµjµlµmµnandVµ1µ2µ3µ4

struc. completely symmetric.

From (3.1.3) we see that we are interested in the four-point subamplitude where two legs have negative polarization and the other two have positive polarization. Thus we sum over all such combinations when we find the four-point amplitude.

3.2

S

PINOR AND

P

OLARIZATION

T

ECHNOLOGY

In order to have a simpler form of the amplitude, we need to write down explicitly the polarizations,²jµj. We are going to use spinors to do this. These spinors originates from massless Dirac fields,ψ, governed by the Lagrangian

Lψ= i ¯ψ
∂ψ . (3.2.1)

Which has the following solutions to its equations of motion

∂ψ = 0 , ∂ ¯ψ = 0 ⇒

⇒ ψ±∼ u±(p)ei px+ v±(p)e−i px,



p v_±(p) = 0 , u¯_±(p)

p = 0 ⇒

⇒ v₊= (|p]a, 0)T , v−= (0, |p〉a˙)T, u¯+= ([p|a, 0) , u¯−= (0, 〈p|a˙) .

(3.2.2)

We denote the product of two spinors, |p〉a˙and |p]a, where ˙a and a are SU(2)-indices, as

〈k|a˙|p〉a˙≡ 〈kp〉 ,

[k|a|p]a≡ [kp] ,

〈k|a|p]a= [k|a|p〉a= 0 ,

| j 〉 ≡ |kj〉 , | j ] ≡ |kj] .

(3.2.3)

If we knowA [1−... j−( j + 1)+...n+], then we can findA [1+... j+( j + 1)−...n−] through the shift

Which basically halfs the amount of color-ordered amplitudes we need to find in a theory.

Spinors have the following properties [13]

k ≡ kµγµ= −|k〉[k| − |k]〈k| , 〈i j 〉[i j ] = (ki+ kj)2= 2kikj≡ −si j, [k|γµ|p〉 = 〈p|γµ|k] , (3.2.5) 〈kp〉 = −〈pk〉 , [k p] = −[pk] -Antisymmetry, 〈k1|γµ|k2〉〈p1|γµ|p2〉 = 2〈k1p1〉[k2p2] -Fierz Identity, n X j =1 〈k j 〉[ j p] = 0 -Momentum Conservation, [p j ][kl ] + [pk][l j ] + [pl ][ j k] = 0 〈p j 〉〈kl 〉 + 〈pk〉〈l j 〉 + 〈pl 〉〈 j k〉 = 0 -Schouten Identity. (3.2.6)

As already mentioned, we are considering massless fields, i.e. k2_{j = 0 ∀ j . Above γ}µare the Dirac gamma matrices and n is the number of legs of the amplitude.

The polarization,²µ_±, is given by spinor expressions

²µ−(p, q) = − 〈p|γµ|q] p 2[q p] , ² µ +(p, q) = − 〈q|γµ|p] p 2〈qp〉 , q 6= p . (3.2.7)

Here q is a null reference momentum which is not parallel with the momenta p_µof Aµ. Amplitudes never depends on q, and one can always get rid of it in one way or another through massaging our amplitudes. For simplicity we denote

²−µ j ≡ ²µ−(kj, qj) , ²+µ j ≡ ² µ +(kj, qj) . (3.2.8)

We have the following Lorentz products

²− j²−l = 〈 j l 〉[qjql] [qjj ][qll ] , ²+_j²+_{l =}〈qjql〉[ j l ] 〈qjj 〉〈qll 〉 , ²−_j²+_{l =}〈 j ql〉[qjl ] [qjj ]〈qll 〉 , kj²−_l =〈l j 〉[ j ql ] p 2[qll ] , kj²+_l =〈qplj 〉[j l ] 2〈qll 〉 . (3.2.9)

When all legs are positive/negative we will chose reference momenta as

qj= q ∀ j ∈ {1, ..., 4} ⇒ ²j²l= 0 ∀ j , l ∈ {1, ..., 4} , (3.2.10)

and in the case where we have one negative polarization (let the first leg have negative polarization) we chose reference momenta as

For the MHV amplitude we chose the reference momenta to be ½ q1= q2= 3 q3= q4= 1 ¾ ⇒ ½ ²1²2= ²3²4= ²α²3= ²1²i= 0 .

k3²α= k1²i= kα²α= ki²i= 0 , (no sum over α, i .) (3.2.12)

Leaving only²2²4, kα²β, k2²i, ki²j and k4²α,α 6= β ,i 6= j ,α,β ∈ {1,2} ,i, j ∈ {3,4} terms in the amplitude. ²2²4=〈21〉[34]

[32]〈14〉. (3.2.13)

3.3

F

OUR

-P

OINT

A

MPLITUDE IN

F

-D

EFORMATION OF

YM

Using the technology in previous section we get the amplitude6

AF4 trace(1+2+3+4+) = 2i 9 n α2 ³ [12]2[34]2+ [13]2[24]2+ [14]2[23]2´+ −α1 2 ³ [12][13][34][42] + [12][14][23][34] + [13][14][23][42]´o, AF4 trace(1−2+3+4+) = 0 , AF4 trace(1−2−3+4+) = i 4α1+ 3α2 18 〈12〉 2_[34]2_. (3.3.1)

If we compare the F4MHV amplitude with the YM amplitude, see (3.1.3), we find

i 3!dabcd4α1+ 3α2 18 〈12〉 2_[34]2 =2i 3d abcd 〈12〉2[34]2, 4α1+ 3α2= 2 . (3.3.2)

The two conditions at (3.1.3) yields

α1= −α2

4 . (3.3.3)

Here we have squared the Schouten identity (3.2.6), and compared it with the F4-amplitude in (3.3.1) where all legs have positive polarizations. Solving the constraints (3.3.2) and (3.3.3) yields the Lagrangian, LF4, in F4-deformation of YM LF4= dabcd µ F_µνa FbνρF_ρσc Fdσµ−1 4F a µνFbµνFρσc Fdρσ ¶ . (3.3.4)

This is the well-known F4connection in type I string theory, often denoted "t8F4" (see [26]).

6_{These amplitudes are calculated using Mathematica. The Mathematica code used for this thesis has been uploaded together}

C

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4

C

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H

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4.1

H

ETEROTIC

S

TRING

A

MPLITUDES AS

SYM A

MPLITUDES

In this section we will find up to six-point heterotic string amplitudes (at tree level) and compare them with amplitudes for strings attached on D p-branes. A D p-brane is a hypersurface with p spatial di-mensions [27]1. We will discuss those kind of string amplitudes in section 5. In section 2.4 we saw that color-ordered heterotic string amplitudes are given by the single-valued projection of type I subampli-tudes, see (2.4.8). Hence our starting point for heterotic string amplitudes will be theα0_{expansion of}

type I subamplitudes (2.3.9)2. AHET = svAI= svF ASYM, (4.1.1) svF = 1 + ζsv(3)M3+ ζsv(5)M5+ 1 2ζ 2 sv(3)M32+ ζsv(7)M7+ 1 2ζsv(3)ζsv(5){M3, M5}+ +1 5ζsv(3, 5)[M5, M3] + O (ζsv(9),Q10) . (4.1.2)

Here we have used both (2.2.9) and (2.2.10). We have ordered the (n − 3)! subamplitudes into vectors AHET_andAI_{, and svF is a (n −3)!×(n −3)! matrix. The Lagrangian for N = 1 SYM in ten dimensions for}

massless fields is given by (1.1.1). It is seen from this Lagrangian that we have three-point interactions between three gauge fields, Aa_µ, or one A_µaand twoΛa’s as well as four-point interactions between four

A_µa’s. We are interested in the difference between pure gluon tree-level diagrams from open strings on a brane and heterotic string theory. For pure gluon amplitudes in SYM, spinors only appear inside loops.

, , e.t.c.

Figure 4.1.1: Spinors only appear at loop level for pure gluon amplitudes in SYM.

Thus we will ignore the spinor terms in the lagrangian at (1.1.1). In order to find the color-ordered am-plitudes associated withLSYM, we first write it in terms of the gauge fields

LSYM= − 1 2∂µA a1 ν ¡∂µAa1ν− ∂νAa1µ¢ − i g fa1a2a3∂µAaν1Aa2µAa3ν+ +g 2 4 f a5a1a2_fa5a3a4_Aa1 µ Aaν2Aa3µAa4ν+ fermion terms . (4.1.3)

In the upcoming subsections we discuss ordered SYM amplitudes. We are interested in color-ordered amplitudes where we fix three legs, i.e. we wish to findASYM[123],ASYM[1234],

ASYM[12345],ASYM[13245] andASYM[123456] with all different permutations of the legs 2, 3 and 4.

4.2

P

ROPAGATOR

Let us first find the propagator. We will need it when we are to calculate contributions to amplitudes with internal gauge fields. We find the propagator by fixing the gauge. In [28] we see that the Feynman propagator, Dµν_F (k), satisfies

τµνD_Fνρ(x − y) = i δµρδ(4)(x − y) ,

S ∼

Z

ddx Aµτ_µνAν. . (4.2.1)

Hereτ_µνis some differential operator. Under a Fourier transformation we get ˜

τµν(k) ˜DνρF (k) = i δµρ. (4.2.2)

From which we may find the required propagator. When we find ˜Dµν_F we study only the_O(A2)-terms in (4.1.3) S2: = Z ddx∂_µAa1 ν (x)¡∂µAa1ν(x) − ∂νAa1µ(x)¢ = Z ddx Aa1 ν (x)¡−∂2ηνµ+ ∂µ∂ν¢ Aaµ1(x) = Z _dd_k (2π)dA˜ a1µ_(k)¡−k2η µν+ kµkν¢A˜a1ν(−k) . (4.2.3)

Thus from (4.2.1) we see that ˜Dµν_F satisfies ¡−k2_η

Ansatz 1. We may try to solve this expression for ˜Dµν_F with an ansatz

˜

Dµν_{F = α(k)η}µν_{+ β(k)k}µkν. (4.2.5)

Hereα and β are some functions which depends on k.

This ansatz together with (4.2.4) yields

−k2αδµρ+ αkµkρ− k2βkµkρ+ k2βkµkρ= −k2αδµρ+ αkµkρ≡ i δµρ. (4.2.6) As we may see, by comparingδ_µρand k_µkρterms, this is unsolvable forα, meaning that (−k2η_µν+kµk_ν) has no inverse. This problem arise from the gauge invariance of the action, meaning we need to fix the gauge in order to find ˜Dµν_F . We may add a gauge fixing term,Lgf, to the Lagrangian [28]

Lgf= 1 ξ∂µA a1 ν ∂νAa1µ. (4.2.7) Which yields · −k2ηµν+ µ 1 −1 ξ ¶ k_µk_ν ¸ ˜ Dνρ_F = i δµρ. (4.2.8)

Note 6. This gauge fix will not contribute to any amplitude.

The same ansatz (4.2.5) for ˜Dµν_F yields

−k2α(k,ξ)δµρ+ µ 1 −1 ξ ¶ α(k,ξ)kµkρ−k 2 ξ β(k,ξ)kµkρ= i δµρ. (4.2.9)

Comparingδ_µρand k_µkρterms yields ( −k2α = i ³ 1 −1_ξ´α −k_ξ2β = 0 ) ⇔ ( α = −i k2 β = −i k4(ξ − 1) ) ⇒ ˜Dµν_F = − i k2 µ ηµν− (1 − ξ)k µ_kν k2 ¶ . (4.2.10)

Let us work in the Feynman gauge, i.e.

ξ = 1 ⇒ _D˜µν

F (k) = −

iηµν

k2 . (4.2.11)

4.3

SYM A

MPLITUDES

4.3.1

T

-P

V

Knowing the propagator, we are ready to calculate color-ordered SYM amplitudes. We start with diagrams only containing three-point vertices. We find the three-point amplitude,A3, from a Fourier transform of

(1.1.2), using the cyclicity of fabc A3= −i3g X π∈S3 faπ(1)a_π(2)a_π(3)(² π(1)²π(2))(k_π(1)²_π(3)) = i g£ fa1a2a3_(² 1²2)(k1²3) + fa1a3a2(²1²3)(k1²2) + fa2a1a3(²2²1)(k2²3)+ + fa2a3a1₍² 2²3)(k2²1) + fa3a1a2(²3²1)(k3²2) + fa3a2a1(²3²2)(k3²1) ¤ = i g£¡ ηµνk₁ρ_{+ η}νρk₂µ_{+ η}ρµkν₃¢ fa1a2a3₊ +¡ ηρµk₁ν+ ηµνkρ₂+ ηνρkµ₃¢ fa1a3a2¤ ²1µ²2ν²3ρ = i g£ηµν(k1− k2)ρ+ ηνρ(k2− k3)µ+ ηρµ(k3− k1)ν¤ × ×£Tr¡Ta1_Ta2_Ta3_{¢ − Tr¡T}a1_Ta3_Ta2¢¤ ²1µ²2ν²3ρ ≡ X π∈S2 Tr¡Ta1_Taπ(2)_Ta_π(3)¢ ASYM[1π(2,3)] ,

ASYM[1ab] : = Vtraceµνρ²1µ²2ν²3ρ,

Vtraceµνρ: = i g

£

ηµν(k1− k2)ρ+ ηνρ(k2− k3)µ+ ηρµ(k3− k1)ν¤ .

(4.3.1)

HereV_traceµνρis the color-ordered, three-point vertex factor in the trace basis. As can be seen, there will be a change in sign when flipping two legs at a three-point vertex. For three-point amplitudes we must be very careful when we chose reference momenta. This follows from the singular configuration, i.e. from momentum conservation (3.2.6)

〈12〉[23] = 0 , 〈13〉[32] = 0 , 〈23〉[31] = 0 . (4.3.2) Thus either |1〉 ∼ |2〉 ∼ |3〉, or |1] ∼ |2] ∼ |3]. This means that if we would set the reference momenta equal to the momenta of one of the leg we could very easily divide by zero. Therefore, we study the three-point amplitude analytically. The color-ordered three-point amplitude is given by

ASYM[1−2+3+] = = i g à 〈1q2〉[q12] [q11]〈q22〉 〈q31〉[13] − 〈q32〉[23] p 2〈q33〉 +〈q2q3〉[23] 〈q22〉〈q33〉 〈12〉[2q1] − 〈13〉[3q1] p 2[q11] + +〈1q3〉[q13] [q11]〈q33〉 〈q23〉[32] − 〈q21〉[12] p 2〈q22〉 ! . (4.3.3)

Here we see that we must chose |1〉 ∼ |2〉 ∼ |3〉 to get a non-trivial, i.e. non-zero, amplitude. Using anti-symmetry, momentum conservation and the Schouten identity, see (3.2.6), yields

ASYM[1−2+3+] = =pi g 2 1 [q11]〈q22〉〈q33〉 n (−1)〈q21〉[q12] ³ 〈q31〉[13] − (−1)〈q31〉[13] ´ + + (−1)〈q31〉[q13] ³ (−1)〈q21〉[12] − 〈q21〉[12] ´ o =2i gp 2 〈q21〉〈q31〉 [q11]〈q22〉〈q33〉 ³ (−1)[q12][31] − [q13][12] ´ = −2p2i〈q21〉〈q31〉[q11][23] [q11]〈q22〉〈q33〉 . (4.3.4)

Which yields3 ASYM[1−2+3+] = p 2i g [23] 3 [21][13]. (4.3.6)

Yet again, we see that we will get a minus sign if we change places on two legs at a three-point vertex. Let us move on to higher point diagrams which only contains three-point vertices, see figure 4.3.1.

, , 1 1 a b 1 a b c 2 3 4 5 6 4 5

Figure 4.3.1: Diagrams with only three-point vertices. The arrows shows the direction of the momenta. We align the momenta such that either all or none are directed inwards the vertex.

If all legs have negative/positive polarization, we let all of the reference momenta to be the same. In this case, the reference momenta, qj, does not equal the momentum, kj, of any leg

qj= q 6= kl∀ j , l ∈ {1, ..., n} . (4.3.7)

If we consider both negative and positive polarizations we chose reference momenta such that all legs with negative/positive polarization have their reference momenta, q₋/q₊, equal the last/first leg’s mo-mentum, i.e.

q_{−= k}n, q+= k1 ⇒ (²1²j) = (²j²n) = 0 ∀j {1,...,n} . (4.3.8)

It is useful to note that for m negative legs, (²²)m-terms are zero. Another useful property is the Lorentz gauge condition

kj²j= 0 , j ∈ {1,...,n} , (no sum over j). (4.3.9)

We will denote color-ordered amplitudes from a diagram with e.g. two three-point vertices asT33[1234].

Using the technology we just discussed yields the color-ordered amplitude associated with the first dia-gram in figure 4.3.1 T33[1234] = V_{SYM ν}µ1µ2 D˜νρ_F V_SYMρµ3µ4²1µ1²2µ2²3µ3²4µ4 =¡i g ¢2 −i [−(k1+ k2)]2 © ηµ2ν_[k 2+ (k1+ k2)]µ1+ ηνµ1[−(k1+ k2) − k1]µ2ª × ×©ηνµ3[−(k3+ k4) − k3]µ4+ ηµ4ν[k4+ (k3+ k4)]µ3 ª ²1µ1²2µ2²3µ3²4µ4 = i g 2 2(k1k2)η µ2µ3_2kµ1 2 (−2)k µ4 3 ²1µ1²2µ2²3µ3²4µ4= − 2i g2 (k1k2) (k2²1)(²2²3)(k3²4) . (4.3.10)

Since this is proportional to (²²) we get

T33[1+2+3+4+] = T33[1−2+3+4+] = 0 . (4.3.11)

The diagram with two negative polarization equals T33[1−2−3+4+] = − 4i g2 〈12〉[12] 〈12〉[24] p 2[41] 〈21〉[43] [42]〈13〉 〈13〉[34] p 2〈14〉 = −2i g 2 〈12〉2[34]2 [12][41]〈12〉〈41〉. (4.3.12) From momentum conservation we get

½ 〈12〉[12] = −s12= −s34= 〈34〉[34] 〈12〉[41] = −〈32〉[43] = −〈23〉[34] ¾ ⇒ [34] =〈12〉[12] 〈34〉 = − 〈12〉[41] 〈23〉 . (4.3.13) If we plug this back into (4.3.12) we get the Parke-Taylor amplitude

T33[1−2−3+4+] = 2i g2 〈12〉 4

〈12〉〈23〉〈34〉〈41〉. (4.3.14)

The n-point, MHV, pure-gluon amplitude where two arbitrary legs, a and b, have negative polarization, is given by the Parke-Taylor amplitude [13]

A [1+₂+_{...(a − 1)}+_a−_{(a + 1)}+_{....(b − 1)}+_b−_{(b + 1)}+_{...(n − 1)}+_n+_{] = i}³p

2g´n 〈ab〉

4

〈12〉〈23〉...〈n1〉. The only non-trivial diagrams from figure 4.3.1 are the MHV diagrams as well as the six-point diagram with three negative polarizations. These diagrams (together with the Mathematica code used to calculate them) can all be found at the same site as the thesis.

All amplitudes calculated in this project has been shown to be gauge-invariant. A gauge transformation of the gauge field, A_µ, is given by

A_µ→ Aµ+ ∂µ ⇔ ²µ→ ²µ+ kµ.

This means that if we exchange one of the polarizations,²_µ, to its momentum, k_µ, the amplitude should be zero. We have tested this for every amplitude calculated in this project.

4.3.2

F

-P

V

A4= i g2 4 f bajal_fbaman₍² j²m)(²l²n) = i g2 4 f bajal_fbamanηµjµmηµlµn² µj²µm²µl²µn ≡ X j ,l ,m,n fbajal_fbaman_Vj l mn_² µj²µm²µl²µn = h fba1a2_fba3a4_V1234_{+ Perm.(1, ..., 4)}i_² µ1²µ2²µ3²µ4 ≡ ³ fba1a2_fba3a4V1234

struc.+ fba1a3fba4a2Vstruc.1342+ fba1a4fba2a3Vstruc.1423

Here Vj l mnis the vertex operator for the full amplitude, and_Vstruc.1 j l mis the color-ordered vertex operator in the structure constant basis. We wish to find the color-ordered amplitudes in the trace basis. Hence we write the structure constants as

fba1aj_fbalam_{= Tr}³_Tb_[Ta1_{, T}aj_]´_Tr³_Tb_[Tal_{, T}am_]´_{. (4.3.16)} In order to understand this we need to use [13]

¡Ta¢ µν¡Ta ¢ ρσ= δµσδρν− 1 Nδµ ν_δ ρσ (4.3.17)

This relation yields

Tr³Ta1_...Taj_Tb_Taj +1_...Tam´_Tr³_Tam+1_...Tak_Tb_Tak+1_...Tan´₌ = Tr¡Ta1_...Taj_Tak+1_...Tan_Tam+1_...Tak_Taj +1_...Tam_{¢ +} + 1 NTr¡T a1_...Tam_{¢ Tr¡T}am+1_...Tan_{¢ .} (4.3.18)

Which together with cyclicity of the trace and antisymmetry of the commutator yields

fba1aj_fbalam_{= Tr¡[T}a1_{, T}aj_][Tal_{, T}am_{]¢ . (4.3.19)} Using this we find the color-ordered, four-point vertex,V_trace1 j l m, fromLSYMin the trace basis

V1 j l m trace = V 1 j l m struc. − V 1m j l struc. , V 1 j l m struc. = −V 1 j ml struc. . (4.3.20)

Note 7. Not all of theV_struc.µνρσ’s in the expression above appear in the structure constant basis, e.g. _V1243

struc.,

hence we introduced the antisymmetry on the two last indices.

InsertingV_struc.µνρσ, see (4.3.15), inV_traceµνρσyields Vtraceµνρσ= −i g2

¡

ηµνηρσ− 2ηµρηνσ+ ηµσηνρ¢ . (4.3.21) Since this vertex operator is proportional toη2, the four-point diagram associated with this vertex is zero, i.e.

T4[1234] = 0 . (4.3.22)

The five- and six-point diagrams we can build withV_traceµνρσandV_traceµνρare illustrated in figure 4.3.2.

, , a b 4 5 a b c 5 6 1 1 1 a b c 5 6 1 a b 5 c 6 , a b c 5 6 1

Using the same methods as in section 3 yields

T43[1ab45] = T44[1abc56] = 0 . (4.3.23)

We also find that all of the amplitudes in figure 4.3.2 are trivial, i.e. zero, if less than two legs have negative polarization. As with the previous section, does the non-trivial amplitudes contain a lot of terms. The Mathematica code used to calculate these amplitudes has been uploaded together with the thesis.

4.4

H

ETEROTIC

S

TRING

A

MPLITUDES

As previously discussed may we find heterotic string amplitudes by ordering all of the color-ordered SYM amplitudes in a vector, and then multiply that vector with the single-valued version of the (n −3)!×(n−3)! matrix, F , see (4.1.1). We may find the M3, M5and the M7matrices at [24]. The non-trivial SYM diagrams

(up to the sixth order) are illustrated in figure 4.4.1.

, , , , + + _ _ _ + + + + _ _ _ +/ _ + + _ _ _ _ _ + + + _ _ + + + + _ _{+/ _} + + + + +

Figure 4.4.1: The non-trivial SYM amplitudes (up to sixth order).

To the lowest order in theα0_{-expansion (4.1.2), up to the five-points diagrams in heterotic string theory}4

AHET[1−2+3+] = p 2i g [23] 3 [21][13] AHET[1−2−3+4+] = 2i g2 h 1 + ˜s12s˜14 ³ ˜ s12+ ˜s14 ´ ζsv(3) + O¡(α0)5 ¢i 〈12〉 4 〈12〉〈23〉〈34〉〈41〉 (4.4.1)

4_{We don’t write out the six-point amplitudes since those expressions (as well as higher orders inζ}

sv) are very big. These

AHET[1−2−3+4+5+] = = −ip2g3 〈12〉 〈13〉〈14〉〈15〉〈45〉[15] ½ 2ζsv(3)〈12〉[35] [13][25] ³ s15− s23− s34 ´ × ×³s12+ s15+ s23+ s34+ s45 ´ ³ s12+ s23− s45 ´ × × ³ 〈14〉[24][35] + 〈15〉[25][35] − 〈14〉[23][45] ´ − 1 12 ³ 2〈13〉〈14〉[34][35] + + 2〈13〉〈15〉[35]2− 〈12〉〈14〉[23][45] − 〈14〉2[34][45] − 〈14〉〈15〉[35][45]´× × h 1 −hs˜2₁₂s˜15+ ˜s12s˜215 − ˜s34 ³ ˜ s12 ³ ˜ s12+ 2 ˜s23+ ˜s34 ´ + ˜s34s˜45+ + ˜s245 ´ i × × ζsv(3) io + O¡(α0₎5¢ (4.4.2)

Here ˜sj k is the dimensionless Mandelstam variable (2.1.4), and sj kis the "usual" Mandelstam variable

(3.2.5).

4.5

L

ITTLE

G

ROUP

S

CALING

Before we move on, let us discuss the little group scaling. Our starting point for this discussion will be the spinor completeness relation

−

p = |p〉[p| + |p]〈p| . (4.5.1)

The little group scaling are unitary transformations corresponding to rotations around the three mo-menta of a massless particle. The cross section as well as the momentum needs to be invariant under this rescaling, but not the amplitudes per se, i.e. the amplitudes are covariant. Thus from the relation above, we find that the little group scaling needs to be of the form

| j 〉 → tj| j 〉 , 〈p| → ¯t−1〈p|

| j ] → t−1j | j ] , [p| → ¯t[p|

, tj¯tj= 1 , t ∈ C . (4.5.2)

These scalings are U(1) transformations [13]. Thus the n-point amplitude in four dimensions has U(1)n -covariance5. We see that every term in the heterotic amplitudes scale in the same way, which is a sign that our amplitudes are correct.