One-loop dilatation operator - Form factors and the dilatation operator in N = 4 super Yang-Mil

notation, the ﬂow of each fundamental gauge group indexi, j, k, l, . . . = 1, . . . , N is depicted by a line. Each closed line yields a factor of δⁱ_i =N.

The perturbative expansion becomes particularly simple in the so-called ’t Hooft, large N or planar limit N → ∞, g_YM → 0 with the ’t Hooft coupling λ=g²_YMN ﬁxed [19]. In this limit, only Feynman diagrams contribute that are planar with respect to their double-line notation, i.e. with respect to their colour structure. It should be stressed that this notion of planarity is in general diﬀerent from planarity in momentum space, where all external momenta are understood to point outwards. Both notions coincide for Feynman diagrams containing only elementary interactions and hence in particular for scattering amplitudes. However, this ceases to be the case in the presence of gauge-invariant local composite operators, as these are colour singlets but have a non-vanishing momentum.

Figure 1.1 shows an example of a Feynman diagram that is planar in double-line notation (b) but leads to a Feynman integral that is non-planar in momentum space (c). Whether a diagram is planar or non-planar in the sense of the ’t Hooft limit can in general only be determined via N counting after all colour lines have been closed. Following [29], we call a diagram withnopen double lines planar if it contributes at the leading order in N after a planar contraction of these lines with ann-point vertex at inﬁnity. Of course, whether a diagram contributes at leading order to a particular process depends on the contraction of the colour lines in this process. In particular, subdiagrams of elementary interactions which appear to be suppressed in the ’t Hooft limit as they are non-planar and have a double-trace structure can contribute at the leading order when contracted with composite operators in a certain way. The reason for this is that the complete planar contraction of the composite operator with a trace factor in the double-trace interaction leads to an additional power of N compared to the contraction with a single-trace interaction. As the mechanism behind this enhancement of the N-power requires the interaction range to equal the number of ﬁelds in the single-trace operator, i.e. the length L of the spin chain, it is called ﬁnite-size eﬀect. The right factor in ﬁgure 1.1 (a) shows a non-planar double-trace diagram; the fact that it is non-planar can be seen when taking all external ﬁelds to point outwards. However, the depicted contraction of this double-trace interaction with a composite operator leads to the planar diagram in ﬁgure 1.1 (b). One source of the double-trace structure of the interaction can be a sequence of ﬁelds wrapping around the composite operator [29]. We will encounter this wrapping eﬀect in both parts of this work

— as well as a new ﬁnite-size eﬀect in the second part.

1.4 One-loop dilatation operator

In this section, we give a short introduction to the quantum corrections to the dilatation operator, in particular at one-loop order. These play a major role in both parts of this work.

Deﬁning the eﬀective planar coupling constant

g² = λ

(4π)² = g_YM² N

(4π)² , (1.18)

a d b

a b× cc dd

(a) tr(T^aT^b) tr(T^cT^d)×tr(T^cT^d)

a b

(b) N²tr(T^aT^b)

Figure 1.1: Planarity and non-planarity in double-line notation and momentum space:

(a) contraction of a non-planar double-trace interaction with a composite operator, (b) resulting planar diagram in double-line notation, (c) corresponding non-planar momentum space integral. In double-line notation the operator is depicted as a grey blob, while it is depicted by a double line in momentum space. (We trust that the reader will not confuse both kinds of double lines.)

the dilatation operator can be expanded as⁴ D=

X∞ ℓ=0

g^2ℓD^(ℓ). (1.19)

At ℓ-loop order, connected interactions can involve at most ℓ+ 1 ﬁelds of a composite operator of lengthLat a time. Moreover, in the planar limit, these have to be neighbouring ﬁelds in the same trace factor. Hence, in this limit theℓ-loop dilatation operatorD^(ℓ) can be written as sum of a density (D^(ℓ))_i...i+ℓ that acts on ℓ+ 1 neighbouring sites of the corresponding spin chain:

D^(ℓ)= XL

i=1

(D^(ℓ))_i...i+ℓ, (1.20)

where cyclic identiﬁcation i+L ∼i is understood. The study of the dilatation operator can be simpliﬁed by restricting to closed subsectors, which are deﬁned via constraints on the various quantum numbers [27].

The complete one-loop dilatation operator density (D⁽¹⁾)_{i i+1} of N = 4 SYM theory was ﬁrst calculated by Niklas Beisert via a direct Feynman diagram calculation in the SL(2) sector that was then lifted to the complete theory via symmetry [27]. It was later shown that (D⁽¹⁾)_{i i+1}is completely ﬁxed by symmetry apart from one global multiplicative constant [30]. Several diﬀerent representations of (D⁽¹⁾)i i+1 exist.

The ﬁrst kind of representation employs the following decomposition of the tensor product of two singleton representations [27]:

VS⊗ VS = M∞

i=0

Vj. (1.21)

4Beyond one-loop order and certain subsectors, also odd powers ofgcan occur in the expansion (1.19).

In this thesis, however, we restrict ourselves to cases where even powers suffice.

1.4 One-loop dilatation operator 29 Denoting the projection operator to the subspaceVj as

P_j :VS⊗ VS −→ Vj, (1.22)

the one-loop dilatation operator density can be written as (D⁽¹⁾)_{i i+1} = 2

i is the j^th harmonic number. Though quite compact, this represen-tation is not very useful in direct calculations of anomalous dimensions due to the lack of handy expressions forP_j.

For direct calculations, a second kind of representation is advantageous, which is known as harmonic action. It uses the oscillators deﬁned in section 1.2, which can be combined into one superoscillator

A^†_i = (a^†1_i ,a^†2_i ,b^†˙1_i ,b^†˙2_i ,d^†1_i ,d^†2_i ,d^†3_i ,d^†4_i ). (1.24) We specify the individual component oscillators of A^†_i by superscripts A_i as A^†A_i ⁱ, i.e.

A^†1_i = a^†1_i , . . . ,A^†8_i = d^†4_i . Using these superoscillators, the one-loop dilatation operator density can be written as a weighted sum over all their reorderings [27]:

(D⁽¹⁾)_{1 2}A^†A_s₁¹· · ·A^†A_s_nⁿ|0 i= wherendenotes the total number of oscillators at both sites,n₁₂(n21) denotes the number of oscillators changing their site from 1 to 2 (2 to 1) and the Kronecker delta ensures that the resulting states fulﬁl the central charge constraint. The coeﬃcient is given by

c(n, n12, n₂₁) =

(2h ¹₂n

if n₁₂=n₂₁= 0, 2(−1)¹⁺ⁿ¹²ⁿ²¹B ¹₂(n12+n₂₁),1 +¹₂(n−n₁₂−n₂₁)

else, (1.26) where B denotes the Euler beta function.

An integral formulation of the latter representation was found by Benjamin Zwiebel in [190].⁵ Deﬁning

(A^†_i)^~ⁿⁱ = (a^†1_i )^a¹ⁱ(a^†2_i )^a²ⁱ(b^†˙1_i )^bⁱ^˙1(b^†˙2_i )^bⁱ^˙2(d^†1_i )^d¹ⁱ(d^†2_i )^d²ⁱ(d^†3_i )^d³ⁱ(d^†4_i )^d⁴ⁱ , (1.27) the representation (1.25) can be recast into the form

(D⁽¹⁾)_{1 2}(A^†₁)^~ⁿ¹(A^†₂)^~ⁿ²|0i= 4δ_C₂_,0

5An alternative integral representation of the harmonic action can be found in [9] and an operatorial form in [101, 103].

The equivalence of (1.28) and (1.25) can be seen from the known integral representations B(x, y) = 2

Z ^π₂

dθ(sinθ)^2x−1(cosθ)^2y−1, h(y) = 2

Z ^π₂

dθcotθ 1−(cosθ)^2y .

(1.30)

Note that the ﬁrst summand in the integral on the right hand side of (1.28) acts as regu-larisation, altering the divergent integral representation of B(0, y) such that it yieldsh(y) instead. In chapter 3, we will ﬁnd that the integral representation (1.28) emerges naturally when deriving the complete one-loop dilatation operator via ﬁeld theory alone.

Beyond one-loop order, several complications arise. The range of the interaction on a composite operator of length L is naturally bounded by L. If this bound is saturated, the dilatation operator acts on the whole spin chain at once, invalidating the notion of an interaction density. At this point, ﬁnite-size eﬀects set in, which will be a main topic of the second part of this thesis.⁶ Moreover, the length of a composite operator is not a conserved quantum number beyond one-loop order. The leading length-changing contributions to the dilatation operator are completely ﬁxed by symmetry and were found in [99]. In certain subsectors of the theory, such as the SU(2) sector, the length L is connected to global charges of the theory, and hence preserved. In this thesis, we will not consider cases where length-changing occurs.

6In fact, we will find a new kind of finite-size effect in the second part of this thesis, which already sets in at one-loop order in the deformed theories.

Part I

Form factors

Chapter 2

Introduction to form factors

In this chapter, we start our investigation of form factors in N = 4 SYM theory. We review some important concepts that underlie the modern study of scattering amplitudes as well as form factors in section 2.1. This also allows us to introduce our notation and conventions. In section 2.2, we then calculate the minimal tree-level form factors of all operators via Feynman diagrams and relate them to the spin-chain picture. We discuss the general problems arising for non-minimal and loop-level form factors in section 2.3.

While section 2.1 is a review of well known results, section 2.2 is based on original work by the author ﬁrst published in [4].

2.1 Generalities

The physical quantities we are going to study are form factors of local gauge-invariant composite operators O. For such an operator, the form factor is deﬁned as the overlap between the state created by O(x) from the vacuum |0i and an n-particle on-shell state h1, . . . , n|,¹ i.e.

FO,n(1, . . . , n;x) =h1, . . . , n|O(x)|0i. (2.1) This deﬁnition reduces to the one for the scattering amplitude when settingO=1:

An(1, . . . , n) =h1, . . . , n|0i. (2.2) In both cases, the on-shell state is speciﬁed by the momentap_i, the helicitiesh_i, the ﬂavours and the gauge-group indicesa_i of the on-shell ﬁeldsi= 1, . . . , n. We take all on-shell ﬁelds to be outgoing.

Most on-shell techniques that were developed for scattering amplitudes work in mo-mentum space.² Hence, as a ﬁrst step, we Fourier transform (2.1) from position space to momentum space:

FO(1, . . . , n;q) = Z

d⁴xe^−iqxh1, . . . , n|O(x)|0i= Z

d⁴xe^−iqxh1, . . . , n|e^ixPO(0) e^−ixP|0i

= (2π)⁴δ⁴ q− Xn

i=1

p_i

h1, . . . , n|O(0)|0i,

(2.3)

1Thenexternal fields are set on-shell using Lehmann-Symanzik-Zimmermann (LSZ) reduction [191].

2For recent reviews about on-shell techniques for scattering amplitudes, see [52, 53].

where we have used (1.7) in the second line and the delta function in the third line guar-antees momentum conservation. We depict the Fourier-transformed form factor as shown in ﬁgure 2.1.

FO,n q

···

Figure 2.1: The form factor of an operator Oin momentum space. The non-shell ﬁelds with momenta p_i (p²_i = 0 for i= 1, . . . , n) are depicted as single lines while the operator with oﬀ-shell momentum q (q² 6= 0) is depicted as double line. The direction of each momentum is indicated by an arrow.

Using the matrices σ^α_µ^α^˙ = (1, σ₁, σ₂, σ₃)^α^α^˙, where σ_i are the usual Pauli matrices, the four-dimensional momenta p^µ_i can be written in terms of 2×2 matrices as p^α_i^α^˙ = p^µ_iσ^α_µ^α^˙. The on-shell condition p²_i = p^µ_ip_i,µ = 0 then translates to detp = 0. Hence, on-shell momenta p^α_i^α^˙ can be expressed as products of two two-dimensional spinors λ^α_p_i and ˜λ^α_p^˙_i, which transform in the anti-fundamental representations of SU(2) and SU(2), respectively:

p^α_i^α^˙ =λ^α_p_iλ˜^α_p^˙_i. (2.4) These are known as spinor-helicity variables. For real momenta and Minkowski signature, they are conjugate to each other, i.e. ˜λ^α_p^˙_i =±(λ^α_p_i)^∗, where the + (−) occurs for positive (negative) energy. Moreover, multiplying λ^α_p_i by a phase factor t∈C, |t|= 1, and ˜λ^α_p^˙_i by t⁻¹ leaves the momentum p^α_i^α^˙ invariant. The behaviour of amplitudes and form factors under this transformation is called little group scaling. In order to simplify notation, we will frequently abbreviate λ^α_p_i and ˜λ^α_p^˙_i as λ^α_i and ˜λ^α_i^˙, respectively. Contractions of the spinor-helicity variables are denoted ashiji=ǫ_αβλ^α_iλ^β_j and [ij] =−ǫ_α_˙_β_˙λ˜^α_i^˙˜λ^β_j^˙. They satisfy the so-called Schouten identities

hijihkli+hikihlji+hilihjki= 0, [ij][kl] + [ik][lj] + [il][jk] = 0. (2.5) Furthermore, we can use Nair’sN = 4 on-shell superﬁeld [55] to describe external ﬁelds of all diﬀerent ﬂavours and helicities in a uniﬁed way:

Φ(p_i,η˜_i) =g₊(p_i) + ˜η_i^Aψ¯_A(p_i) +η˜_i^Aη˜_i^B

2! φ_AB(p_i) +ǫ_ABCDη˜^A_i η˜_i^Bη˜_i^C

3! ψ^D(p_i) + ˜η_i¹η˜_i²η˜³_iη˜⁴_i g₋(p_i), (2.6) whereg₊ (g−) denotes the gluons of positive (negative) helicity and ˜η^A_i ,A= 1,2,3,4, are anticommuting Graßmann variables transforming in the anti-fundamental representation of SU(4). We can then combine all component amplitudes into one super amplitude and all component form factors into one super form factor. The component expressions can be extracted from the super expressions by taking suitable derivatives with respect to the ˜η_i^A

Im Dokument Form factors and the dilatation operator in N = 4 super Yang-Mills theory and its deformations (Seite 27-35)