One-loop corrections in the SU(2) sector via unitarity

In this section, we demonstrate in detail how to calculate one-loop corrections to minimal form factors via the on-shell method of unitarity.⁷ We focus on composite operators from the SU(2) sector for explicitness.

Operators in the SU(2) sector are formed from two kinds of scalar fields with one common SU(4) index, say X =φ₁₄ and Y =φ₂₄. According to (2.13), the colour-ordered minimal tree-level super form factor of such operators is given by a polynomial in ˜η_i¹η˜_i⁴ and

˜

η_i²η˜⁴_i multiplied by a momentum-conserving delta function; e.g. for O = tr(XXY X . . .), we have

FˆO,L⁽⁰⁾ (1, . . . , L;q) = (2π)⁴δ⁴(q− XL

i=1

λ_iλ˜_i) ˜η¹₁η˜⁴₁η˜₂¹η˜₂⁴η˜²₃η˜⁴₃η˜¹₄η˜₄⁴· · ·+ cyclic permutations . (3.14) We encode the one-loop corrections in the interaction operator I⁽¹⁾ defined in (3.10).

At this loop order, only two fields in the composite operator can interact at a time, and those have to be neighbouring in order to produce a single-trace structure. Hence, we can write I⁽¹⁾ in terms of its interaction density I_{i i+1}⁽¹⁾ as

I⁽¹⁾= XL i=1

I_{i i+1}⁽¹⁾ , (3.15)

whereI_{i i+1}⁽¹⁾ acts on the fieldsiand i+ 1 and cyclic identification i+L∼iis understood.

We depictI_{i i+1}⁽¹⁾ as

I_{i i+1}⁽¹⁾ = ^Ii⁽¹⁾ , (3.16)

where we in general only specify the first field the density acts on in the case that this determines all fields it acts on unambiguously.

In the SU(2) sector, we can write I_{i i+1}⁽¹⁾ explicitly as a differential operator in the fermionic variables:

I_{i i+1}⁽¹⁾ = X2 A,B,C,D=1

(I_i⁽¹⁾)^{ZC ZD}_ZAZBη˜_i^C ∂

∂η˜_i^Aη˜^D_i+1 ∂

∂η˜^B_i+1 , (3.17) where Z₁ =X and Z₂ =Y. The matrix element (I_i⁽¹⁾)^{ZC ZD}_ZAZB encodes the contributions of all interactions that transform the fieldsZ_A,Z_B in the operator to external fieldsZ_C,Z_D.

As the global charges are conserved, the only non-vanishing matrix elements are (I_i⁽¹⁾)^XX_XX, (I_i⁽¹⁾)^XY_XY , (I_i⁽¹⁾)^{Y X}_XY , (I_i⁽¹⁾)^{Y X}_{Y X}, (I_i⁽¹⁾)^XY_{Y X} and (I_i⁽¹⁾)^{Y Y}_{Y Y} . (3.18) Moreover, matrix elements that are connected by a simple relabelling ofXandY coincide, i.e.

(I_i⁽¹⁾)^XX_XX = (I_i⁽¹⁾)^{Y Y}_{Y Y}, (I_i⁽¹⁾)^XY_XY = (I_i⁽¹⁾)^{Y X}_{Y X}, (I_i⁽¹⁾)^{Y X}_XY = (I_i⁽¹⁾)^XY_{Y X}. (3.19) Hence, we only have to calculate (I_i⁽¹⁾)^XX_XX, (I_i⁽¹⁾)^XY_XY and (I_i⁽¹⁾)^{Y X}_XY. As already mentioned, this can be achieved via the on-shell method of unitarity.

7For reviews of unitarity for scattering amplitudes, see e.g. [52, 198, 199].

3.2 One-loop corrections in the SU(2) sector via unitarity 45 The general idea behind unitarity [56,57] and generalised unitarity [58] is to reconstruct processes, such as scattering amplitudes, form factors or correlation functions, at loop order by applying cuts. Here, a cut denotes replacing one or more propagators according to

i

l²_i →2πδ⁺(l_i²) = 2πδ(l²_i)Θ(l⁰_i), (3.20) whereδ⁺(l²_i) denotes the delta function picking the positive-energy branch of the on-shell condition l²_i = 0. This delta function can be explicitly written using the Heaviside step function Θ, as shown on the right hand side of (3.20). On such a cut, the process factorises into the product of one or more tree-level or lower-loop processes. In the case of unitarity, the cut has to result in exactly two factors and corresponds to a discontinuity in one of the kinematic variables. For generalised unitarity, more general cuts are possible.⁸ Through-out this work, we are using four-dimensional (generalised) unitarity, i.e. we evaluate the expressions on the cut in D = 4 dimensions.⁹ This allows us to use the simpler building blocks in four dimensions but requires that the results are lifted to D= 4−2ε, leading to some subtleties as discussed in the next chapter.

In this section, we apply unitarity at the level of the integrand, i.e. we aim to reconstruct the integrand of the minimal one-loop form factors via cuts. As each interaction density I_{i i+1}⁽¹⁾ depends only on a single scale, namelys_{i i+1} = (p_i+p_i+1)², it is sufficient to consider the double cut corresponding to the discontinuity ins_{i i+1}. For convenience, we set i= 1.

On this cut, the minimal one-loop form factor ˆF_O,L⁽¹⁾ factorises into the product of the minimal tree-level form factor ˆF_O,L⁽⁰⁾ and the four-point tree-level amplitude ˆA⁽⁰⁾4 , as shown in figure 3.1:¹⁰

(2π)^D is reduced to a phase-space integral, which for a general number of particlesnand general Dis given by

Z

Alternatively, we can write the integration as Z which is not integrated over; cf. the discussion below (2.4). In our conventions, the super-spinor-helicity variables corresponding to −l_i are related to those corresponding to l_i as λ_−li =−λ_li, ˜λ_−li = ˜λ_li, ˜η_−li = ˜η_li.

8See [200] for a discussion of the relation between cuts and discontinuities across the corresponding branch cuts in generalised unitarity.

9For a review ofD-dimensional unitarity, see [201].

10Here and throughout the first part of this work, we are splitting off the dependence on the gauge group generators T^a to work with colour-ordered objects as defined in (2.8) and (2.9). The contractions of the generators T^a in the trace factors of (2.8) and (2.9) can be performed via (1.6), which is trivially done in the case of planar cuts.

q p1

Figure 3.1: The double cut of the minimal one-loop form factor ˆFO,L⁽¹⁾ in the channel (p₁+p₂)².

where we denote by the superscript and the vertical bar the specified component defined in (2.7) dressed with the corresponding ˜η factors. For instance,

Aˆ⁽⁰⁾₄ (−l₂,−l₁,1|^X,2|^X) = ˆA⁽⁰⁾₄ (−l₂,−l₁,1^X,2^X)˜η₁¹η˜₁⁴η˜₂¹η˜⁴₂. (3.25) Inserting the expression (B.1) for the super amplitude and performing the integration over the fermionic variables, we find Here, we have defined dLIPS as dLIPS multiplied by the momentum-conserving delta] function from the amplitude. In general,

dLIPS]_n,{l} = dLIPS_n,{l}(2π)^Dδ^D

k=ip_k is the total external momentum traversing the cut, which is p_1,2 = p₁ +p₂ in the case under consideration. The Schouten identity (2.5) and momentum conservation yield

h12ihl₁l₂i

h1l₁ih2l₂i =−(p1+p₂)²

(p₁−l₁)² . (3.28)

Thus, the cut of this one-loop form factor is proportional to the cut of the triangle integral:¹² FˆO,L⁽¹⁾ (1,2,3, . . . , L)

11This case was already treated in [129].

12Here, the depicted cut integral denotes the double cut of (A.13) but with measureR _d^D_l

(2π)^D instead of

3.2 One-loop corrections in the SU(2) sector via unitarity 47 (I_i⁽¹⁾) ^XX_XX ^XY_XY ^{Y X}_XY

i

i+1

s_{i i+1} -1 -1 0

i

i+1

0 -1 +1

Table 3.1: Linear combinations of Feynman integrals forming the matrix elements for the minimal one-loop form factors in the SU(2) sector.

Now, we lift the result to the D-dimensional uncut expressions, i.e. we conclude that the same proportionality exists between the uncut one-loop form factor and the uncut triangle integral. The precise rules for this lifting procedure, which in particular removes the factor iin (3.29), are explained in appendix A.1. We find that

(I₁⁽¹⁾)^XX_XX=−s₁₂

p1

p2

. (3.30)

A similar calculation for (I₁⁽¹⁾)^{Y X}_XY shows that Fˆ_O,L⁽¹⁾(1,2,3, . . . , L)

(I₁⁽¹⁾)^{Y X}_XY,s12

= ˜η₁²η˜₁⁴η˜₂¹η˜₂⁴Fˆ_O,L⁽⁰⁾ (1^X,2^Y,3, . . . , L)i





p1

p2

l1

l2



 , (3.31) and hence

(I₁⁽¹⁾)^{Y X}_XY =

p1

p2

. (3.32)

This is an explicit example where the one-loop form factor is not proportional to the tree-level form factor of the same composite operator, making it necessary to promote I^(ℓ) to operators as done in (3.10).

The case (I₁⁽¹⁾)^XY_XY can be calculated in complete analogy to the previous cases. We have summarised the results of the three calculations in table 3.1. Note that the different matrix elements satisfy

(I_i⁽¹⁾)^XY_XY + (I_i⁽¹⁾)^{Y X}_XY = (I_i⁽¹⁾)^XX_XX. (3.33) This identity is in fact a consequence of the Ward identity (2.18) for the generators

J¹_i = ˜η¹_i ∂

∂η˜_i² + ˜η²_i ∂

∂η˜¹_i , J²_i =−i˜η_i¹ ∂

∂˜η_i² +i˜η_i² ∂

∂η˜¹_i , J³_i = ˜η¹_i ∂

∂η˜¹_i −η˜²_i ∂

∂η˜²_i (3.34) of SU(2). Applying (2.18) once for the tree-level form factor in (3.10) and once for its one-loop correction, we find

[J^A,I⁽¹⁾] = 0, (3.35)

which implies (3.33).

Defining the identity operator

we can recast the one-loop corrections into the following form:

I_{i i+1}⁽¹⁾ =−s_{i i+1} Explicit expressions for the one-mass triangle integral and the bubble integral are given in (A.13) and (A.12), respectively. The divergence of the triangle integral is

−s_{i i+1} and the one of the bubble integral is

i

i+1

= 1

ε+O(ε⁰), (3.40)

where γ_cusp⁽¹⁾ and G₀⁽¹⁾ were given in (3.4) and (3.5), respectively. Comparing (3.38) to the general form (3.13), we can immediately read off the one-loop dilatation operator in the SU(2) sector as

This is exactly the Hamiltonian density of the integrable Heisenberg XXX spin chain and perfectly agrees with the result first obtained in [25].

After this warm-up exercise, let us now derive the complete one-loop dilatation operator via generalised unitarity.

3.3 One-loop corrections for all operators via generalised

Im Dokument Form factors and the dilatation operator in N = 4 super Yang-Mills theory and its deformations (Seite 44-48)