Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity

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by  Temple He

Loop-Corrected Virasoro Symmetry of 4D Quantum
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Gravity
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n
Temple He, Daniel Kapec, Ana-Maria Raclariu and Andrew Strominger
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J

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] Center for the Fundamental Laws of Nature, Harvard University,
h
t Cambridge, MA 02138, USA
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Abstract
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v
6 Recentlyaboundaryenergy-momentumtensorTzz hasbeenconstructedfromthesoftgravi-
9 ton operator for any 4D quantum theory of gravity in asymptotically flat space. Up to an
4
0 “anomaly” which is one-loop exact, Tzz generates a Virasoro action on the 2D celestial sphere
0 at null infinity. Here we show by explicit construction that the effects of the IR divergent part
.
1 of the anomaly can be eliminated by a one-loop renormalization that shifts Tzz.
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Contents
1. Introduction 1
2. Tree-Level Energy-Momentum Tensor 2
3. One-Loop Correction to the Subleading Soft Graviton Theorem 5
4. One-Loop Correction to the Energy-Momentum Tensor 6
A. IR Divergence of One-Loop T Correction 8
zz
1 Introduction
Recently it has been demonstrated [1] that any theory of gravity in four asymptotically flat
dimensions has, at tree-level, a Virasoro or “superrotational” symmetry that acts on the celestial
sphere at null infinity. This verified conjectures in [2–6] and follows from the newly discovered
subleading soft graviton theorem [7]. Except for an anomaly arising from one-loop exact infrared
(IR) divergences [8–12], this subleading soft theorem extends to the full quantum theory. However,
the implications of this anomaly for the Virasoro symmetry of the full quantum theory are not
understood, and the exploration of such implications comprises the subject of the present paper.
Thereareseveralopenpossibilities. OneisthattheVirasoroactionisdefinedintheclassicalbut
not in the quantum theory. If so, anomalous symmetries still have important quantum constraints
thatwouldbeinterestingtounderstand. AsecondpossibilityisthattheVirasoroaction actsonthe
full quantum theory, but that the generators and symmetry action are renormalized at one-loop.
This is suggested by the discussion in [13], where it is pointed out that the very definition of a
scattering problem in asymptotically flat gravity requires an infinite number of exactly conserved
charges and associated symmetries, as well as by [14], which found that the anomaly vanishes with
an alternate order of soft limits. A third possibility is that the implications can only be properly
formulated in a Faddeev-Kulish [15–20] basis of states (constructed for gravity in [21]), in which
case all IR divergences are absent. After all, IR divergences preclude a Fock-basis -matrix for
S
quantum gravity and, although we have become accustomed to ignoring this point, it is hard to
discuss symmetries of an object which exists only formally!
In this paper we give evidence which is consistent with, but does not prove, the second hy-
pothesis, which states that the Virasoro action persists to the full quantum theory but requires the
generators tohave aone-loopcorrection. Weusetherecentconstructionofa2Denergy-momentum
tensor T found in [22,23], wherez is a coordinate on the celestial sphere, in terms of soft graviton
zz
modes. The tree-level subleading soft theorem [7] implies that insertions of T in the tree-level
zz
-matrix infinitesimally generate a Virasoro action on the celestial sphere. At one-loop order, the
S
1
subleading soft theorem has an IR divergent term with a known universal form. This spoils the
Virasoro action generated by T insertions. However, we show explicitly that the effects of the
zz
IR divergent term can be removed by a certain shift in T that is quadratic in the soft graviton
zz
modes. Thepossibility thatthis couldbeachieved byasimpleshiftisfarfrom obviousandrequires
a number of nontrivial cancellations.
This does not demonstrate that there is a Virasoro action on the full quantum theory generated
by a renormalized energy-momentum tensor, as there may also be an IR finite one-loop correction
to the subleading soft theorem. At present, little is known about such finite corrections. In all
cases which have been analyzed [8,9], the finite part of the correction vanishes. Yet, there is no
known argument that this should always be the case, and this remains an open issue for us.
Theoutlineof this paperis as follows. In section two we fixconventions and recall the construc-
tion of the tree-level soft graviton energy-momentum tensor. We then reproduce the derivation of
theone-loop exactIRdivergentcorrections tothesubleadingsoftgraviton theoreminsectionthree.
Finally, this divergence is rewritten in section four in terms of the formalmatrix element of another
quadratic soft graviton operator, effectively renormalizing the tree-level energy-momentum tensor.
2 Tree-Level Energy-Momentum Tensor
In this section, we review the derivation of the 2D tree-level energy-momentum tensor living
on the celestial sphere at null infinity [22,23]. Asymptotic one-particle states are denoted by p,s ,
| i
where p is the 4-momentum and s is the helicity, and such states are normalized so that
p′,s′ p,s = (2π)3 2p0 δ ′δ(3)(p~ p~′) . (2.1)
h | i ss −
(cid:0) (cid:1)
An n-particle -matrix element is denoted by
S
out in , (2.2)
n
M ≡ h |S| i
where in p ,s ; ;p ,s and out p ,s ; ;p ,s . Consider the amplitude
1 1 m m m+1 m+1 n n
| i≡ | ··· i h | ≡ h ··· |
± (q) out;q, 2 in , (2.3)
n+1
M ≡ h ± |S| i
consisting of n external hard particles along with an additional external graviton that has momen-
tum p q, energy p0 ω, and polarization ε± . Denoting the same amplitude without the
n+1 n+1 µν
≡ ≡
extra external graviton as , the tree-level soft graviton theorem states that
n
M
lim ± (q) = S(0)± +S(1)± + (q) , (2.4)
n+1 n n n
ω→0M O M
h i
where the leading and subleading soft factors are given by
κ n pµpνε± (q) iκ n ε± (q)pµq
S(0)± = k k µν , S(1)± = µν k λ λν , (2.5)
n 2 p q n − 2 p q Jk
k k
k=1 · k=1 ·
X X
2
respectively. Here, κ √32πG is the gravitational coupling constant, and λν is the total angular
k
≡ J
momentum operator for the kth particle.
Asymptotically flat metrics in Bondi gauge take the form
ds2 = du2 2dudr+2r2γ dzdz¯
zz¯
− −
(2.6)
2m
+ Bdu2+rC dz2+rC dz¯2+DzC dudz+Dz¯C dudz¯+ .
r zz z¯z¯ zz z¯z¯ ···
Here, γ 2 istheroundmetricontheS2 andD istheassociated covariant derivative. The
zz¯ ≡ (1+zz¯)2 z
coordinates (u,r,z,z¯) are asymptotically related to the standard Cartesian coordinates according
to
1
x0 = u+r , xi = rxˆi(z,z¯) , xˆi(z,z¯) = (z+z¯, i(z z¯),1 zz¯) . (2.7)
1+zz¯ − − −
A massless particle with momentum p crosses the celestial sphere at a point (z ,z¯ ). In the
k k k
helicity basis, the particle momentum and polarization can be parameterized by an energy ω and
k
this crossing point. It follows from (2.7) that
z +z¯ i(z z¯ ) 1 z z¯
pµ = ω 1, k k , − k − k , − k k , k = 1, ,n ,
k k 1+z z¯ 1+z z¯ 1+z z¯ ···
(cid:18) k k k k k k(cid:19)
1 1
ε+(p ) = ( z¯ ,1, i, z¯ ) , ε−(p )= ( z ,1,i, z ) ,
µ k k k µ k k k
√2 − − − √2 − −
(2.8)
z+z¯ i(z z¯) 1 zz¯
qµ(z) = ω 1, ,− − , − ωqˆµ(z) ,
1+zz¯ 1+zz¯ 1+zz¯ ≡
(cid:18) (cid:19)
1 1
ε+(q) = ( z¯,1, i, z¯) , ε−(q)= ( z,1,i, z) ,
µ µ
√2 − − − √2 − −
± ± ±
where the soft graviton polarization tensor is taken to be ε = ε ε .
µν µ ν
Now, the perturbative fluctuations of the gravitational field have a mode expansion given by
d3q 1
hout x0,~x = ε¯α (q)aout(q)eiq·x+εα (q)aout(q)†e−iq·x , (2.9)
µν (cid:0) (cid:1) αX=±Z (2π)3 2ωq h µν α µν α i
where aout(q)† and aout(q) are the standard creation and annihilation operators for gravitons obey-
α α
ing the commutation relations
aout(p),aout(q)† = (2π)3(2ω)δ δ(3)(p~ ~q) . (2.10)
α β αβ
−
(cid:2) (cid:3)
The transverse components of the metric fluctuations near I+are given by
1
C (u,z,z¯) κ lim ∂ xµ∂ xνhout u+r,rxˆ(z,z¯) . (2.11)
z¯z¯ ≡ r→∞ r z¯ z¯ µν
(cid:0) (cid:1)
The large-r saddle-point approximation yields
iκ ∞
C (u,z,z¯) = εˆ dω aout ω xˆ e−iωqu aout(ω xˆ †eiωqu , (2.12)
z¯z¯ −8π2 z¯z¯Z0 qh − q − + q i
(cid:0) (cid:1) (cid:1)
3
where
1 2
εˆz¯z¯ = r2∂z¯xµ∂z¯xνε+µν(q)= (1+zz¯)2 . (2.13)
Note that (2.12) is an intuitively plausible result since it states that the graviton field operator at
a point (z,z¯) on the celestial sphere has an expansion in plane wave modes whose momenta are
aimed towards that point.
The Bondi news tensor N = ∂ C has Fourier components
zz u zz
ω iωu ω iωu
N due N , N due N . (2.14)
zz zz z¯z¯ z¯z¯
≡ ≡
Z Z
The zero mode of the news tensor is defined by
1 κ
N(0) lim Nω +N−ω = εˆ lim ωaout ωxˆ +ωaout ωxˆ † . (2.15)
z¯z¯ ≡ 2 ω→0 z¯z¯ z¯z¯ −8π z¯z¯ω→0 − +
(cid:16) (cid:17) h i
(cid:0) (cid:1) (cid:0) (cid:1)
Similarly, the first zero energy moment of the news is defined by
i iκ
N(1) lim ∂ Nω N−ω = εˆ lim 1+ω∂ aout ωxˆ aout ωxˆ † . (2.16)
z¯z¯ ≡ −2ω→0 ω z¯z¯− z¯z¯ 8π z¯z¯ω→0 ω − − +
(cid:16) (cid:17) h i
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(0)
All of these quantities have nonvanishing -matrix insertions even as ω 0. The operator N
S → z¯z¯
projects onto the leading Weinberg pole in the soft graviton theorem [24], so its matrix elements
are tree-level exact and are given by
κ
out N(0) in = εˆ lim ωS(0)− out in , (2.17)
h | z¯z¯ S| i −8π z¯z¯ω→0 n h |S| i
where
n
κ ω (z z )
S(0)− = (1+zz¯) k − k . (2.18)
n −2ω (z¯ z¯ )(1+z z¯ )
k k k
k=1 −
X
(1)
In a similar fashion, N projects onto the subleading (1) term in the soft graviton theorem. At
z¯z¯
O
tree-level, its matrix elements are given by
iκ
out N(1) in = εˆ S(1)− out in , (2.19)
h | z¯z¯ S| i 8π z¯z¯ n h |S| i
where
κ n (z z )2 2h
S(1)− = − k k Γzk h ∂ + s Ω . (2.20)
n 2 z¯ z¯ z z − zkzk k − zk | k| zk
k=1 − k (cid:20) − k (cid:21)
X
In this expression, Γz is the connection on the asymptotic S2, h and h¯ are the conformal weights
zz k k
given by
1 1
h s ω ∂ , h¯ s ω ∂ , (2.21)
k k k ω k k k ω
≡ 2 − k ≡ 2 − − k
(cid:0) (cid:1) (cid:0) (cid:1)
4
1
and Ω is the corresponding spin connection. As was demonstrated in [22], (2.19) implies that
z
insertions of the operator
4i γww¯
T d2w D3N(1) (2.22)
zz ≡ κ2 z w w w¯w¯
Z −
into the tree-level -matrix reproduce the Ward identity for a 2D conformal field theory:
S
n
h h 1
out T in = k + k Γzk + ∂ s Ω out in . (2.23)
h | zzS| i Xk=1"(z−zk)2 z−zk zkzk z−zk (cid:0) zk −| k| zk(cid:1)#h |S| i
3 One-Loop Correction to the Subleading Soft Graviton Theorem
The matrix element (2.17) is exact because the leading Weinberg pole in the soft graviton
expansion is uncorrected. On the other hand, the subleading theorem which governs the (1)
O
termsinthesoftgraviton expansiondoeshavequantumcorrections thatmodifythematrixelement
(2.19) [8]. These corrections are known to be one-loop exact, and arise from IR divergences in soft
exchanges between external lines. Indeed, they must be present in order to cancel (within suitable
inclusive cross-sections) IR divergences that arise from the Weinberg pole. The divergent part of
this one-loop correction was derived in [8], which we will we now review.
The loop expansion of the n-particle scattering amplitude is
∞
= (ℓ)κ2ℓ , (3.1)
n n
M M
ℓ=0
X
wherewefactored outtheκ2 term thatcomes along witheach additionalloop.2 Indimensionalreg-
ularization withd = 4 ǫ,thedivergentpartof theone-loop n-pointgraviton scattering amplitudes
−
is universally related to the tree amplitude according to [25,26]
σ
(1) = n (0) , (3.2)
Mn div ǫ Mn
(cid:12)
(cid:12)
with (cid:12)
1 n µ2
σ (p p )log . (3.3)
n ≡ −4(4π)2 i · j 2pi pj
i,j=1 − ·
X
The ǫ−1 singularity is due exclusively to IR divergences because pure gravity is on-shell one-
O
loop finite in the UV and has no collinear divergences. Using (3.2) and applying the tree-level soft
(cid:0) (cid:1)
theorem involving a negative-helicity soft graviton, we obtain
(1)−(q) q→0 σn+1 S(0)− +S(1)− (0) . (3.4)
Mn+1 div −−−→ ǫ n n Mn
(cid:12) (cid:16) (cid:17)
1The zweibein is (cid:0)e+,e−(cid:1)=√2γzz¯(cid:0)dz(cid:12)(cid:12),dz¯(cid:1) and Ω±± =±21(cid:0)Γzzzdz−Γz¯z¯z¯dz¯(cid:1).
2In addition, there is a factor of κn−2 in each (ℓ) dueto then external lines.
Mn
5
Wewouldliketoexpandtheaboveequationinpowersofthesoftenergyω. Toproceed,weseparate
σ into two terms, one with the soft graviton momentum q and one without:
n+1
1 n µ2
′ ′
σ = σ +σ , σ (p q)log . (3.5)
n+1 n n+1 n+1 ≡ −2(4π)2 i· 2pi q
i=1 − ·
X
Note that σ′ = (ω) as the logω term vanishes by momentum conservation, while σ = ω0 .
n+1 n
O O
We then find, up to ω0 ,
(cid:0) (cid:1)
O
(1)− q→0 (cid:0)S(0(cid:1))− +S(1)− (1) + σn′+1S(0)− (0) 1 S(1)−σ (0) , (3.6)
Mn+1 div −−−→ n n Mn div ǫ n Mn − ǫ n n Mn
(cid:12) (cid:16) (cid:17) (cid:12) (cid:16) (cid:17)
where Sn(1)− (cid:12)(cid:12)in the last term acts only on th(cid:12)(cid:12)e scalar σn. The anomalous term consists of the last
two terms on the right-hand-side of the above equation and is ω0 . It is a universal correction
O
to the subleading soft theorem from IR divergences.
(cid:0) (cid:1)
Thus far, we have been focusing on the IR divergent part of the one-loop amplitude. However,
the one-loop amplitude also has a finite piece:
(1) = (1) + (1) . (3.7)
n n n
M M div M fin
(cid:12) (cid:12)
It is expected from [8] that (cid:12) (cid:12)
(cid:12) (cid:12)
(1)− q→0 S(0)− +S(1)− (1) +∆ S(1)− (0) , (3.8)
Mn+1 fin −−−→ n n Mn fin fin n Mn
(cid:12) (cid:16) (cid:17) (cid:12)
where∆finSn(1)− istheone-l(cid:12)(cid:12)oopfinitecorrectiontothenega(cid:12)(cid:12)tive-helicity subleadingsoftfactorSn(1)−.
Given that the subleading soft graviton theorem is one-loop exact [8], it follows that the all-loop
soft graviton theorem is
− q→0 S(0)− +S(1)− +κ2 σn′+1S(0)− 1 S(1)−σ +∆ S(1)− , (3.9)
Mn+1 −−−→ n n ǫ n − ǫ n n fin n Mn
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:16) (cid:17)
where the terms in square brackets proportional to κ2 are the IR divergent and finite parts of the
anomaly.
Little appears to be currently known about ∆ S(1)−. In all explicitly checked cases, including
fin n
all identical helicity amplitudes and certain low-point single negative-helicity amplitudes, it was
demonstrated that there are no IR finite corrections to the subleading soft graviton theorem [8,9],
implying ∆ S(1)− = 0 for these cases. Nevertheless, we are unaware of any argument indicating
fin n
that this term always vanishes, or on the contrary that its form is universal. In the absence of such
information, we will restrict our consideration to the universal divergent correction given in (3.6).
4 One-Loop Correction to the Energy-Momentum Tensor
The one-loop corrections (3.9) to the subleading soft factor are expected to result in corrections
to thetree-level Virasoro-Ward identity (2.23). Inthissection, weshow thatthis is indeed thecase.
6
Moreover, wefindthat theeffects of theuniversaldivergent correction in (3.6)can beeliminated by
a corresponding one-loop correction to the energy-momentum tensor. That is, whenever we have
∆ S(1)− = 0, the shifted energy-momentum tensor obeys the unshifted Virasoro-Ward identity
fin n
(2.23).
(1)
The tree-level matrix elements of the operator N are given by (2.19). At one-loop level, the
z¯z¯
matrix elements acquire a divergent correction of the form
iκ3 σ′ 1
out N(1) in = εˆ lim 1+ω∂ n+1S(0)− S(1)−σ out in . (4.1)
h | w¯w¯S| i div 8π z¯z¯ω→0 ω ǫ n − ǫ n n h |S| i
(cid:12) (cid:18) (cid:16) (cid:17)(cid:19)
(cid:12) (cid:0) (cid:1)
It immediately follows from (2.22) that the IR divergent one-loop correction to the T Ward
(cid:12) zz
identity is given by
out ∆T in
zz
h | S| i
κ γww¯ (4.2)
= d2w D3 εˆ lim(1+ω∂ ) σ′ S(0)− (S(1)−σ ) out in ,
−2πǫ z w w w¯w¯ω→0 ω n+1 n − n n h |S| i
Z − h (cid:16) (cid:17)i
where
out T in out∆T in . (4.3)
zz div zz
h | S| i| ≡ h | S| i
It is far from obvious, but nevertheless possible, to rewrite this in terms of the zero modes of the
Bondi news. This computation is done explicitly in appendix A, and we find that ∆T can be
zz
expressed as
2 γww¯
∆T = d2w 2N(0)D N(0) +D N(0)N(0) . (4.4)
zz −πκ2ǫ z w ww w w¯w¯ w ww w¯w¯
Z − (cid:16) (cid:16) (cid:17)(cid:17)
Hence, the shifted energy-momentum tensor, given by
T˜ = T ∆T , (4.5)
zz zz zz
−
obeys the unshifted Ward identity
n
h h 1
out T˜ in = k + k Γzk + ∂ s Ω out in (4.6)
h | zzS| i Xk=1"(z−zk)2 z−zk zkzk z−zk (cid:0) zk −| k| zk(cid:1)#h |S| i
to all orders, whenever ∆ S(1)− = 0.
fin n
This result seems interesting for a number of reasons. First of all, note that while the renor-
malized soft factor contains logarithms and explicit dependence on the renormalization scale, such
terms do not appear in the anomalous contribution to the energy-momentum tensor. Furthermore,
the fact that the divergence takes the form of a matrix element involving only the local operators
N(0)(w) and N(0)(w) allows us to perform an “IR renormalization” of the operator T by sub-
ww w¯w¯ zz
tracting away the divergent operator. The form of the divergence, when rewritten in terms of the
soft graviton operators, is reminiscent of the forward limit of a scattering amplitude. However, it
remains to be seen whether or not there are finite corrections to the Ward identity (4.6) and, if so,
whether or not they can be eliminated by a further finite shift of the energy-momentum tensor.
7
Acknowledgements
We are grateful to D. Harlow, Z. Komargodski, P. Mitra, B. Schwab, and A. Zhiboedov for
useful conversations. This work was supported in part by DOE grant DE-FG02-91ER40654.
A IR Divergence of One-Loop T Correction
zz
In this appendix, we explicitly compute the matrix elements of ∆T given in (4.2) by
zz
out∆T in
zz
h | S| i
κ γww¯ (A.1)
= d2w D3 εˆ lim(1+ω∂ ) σ′ S(0)− (S(1)−σ ) out in .
−2πǫ z w w w¯w¯ω→0 ω n+1 n − n n h |S| i
Z − h (cid:16) (cid:17)i
For completeness, we recall the expressions for the leading and subleading tree-level soft factors:
n µ ν ± n ± µ
κ p p ε (q) iκ ε (q)p q
S(0)± = k k µν , S(1)± = µν k λ λν . (A.2)
n 2 p q n − 2 p q Jk
k k
k=1 · k=1 ·
X X
Since S(1)− acts on a scalar in (A.1), the action of is given by
n kµν
J
∂ ∂
σ = i p p σ . (A.3)
Jkµν n − kµ∂pν − kν∂pµ n
(cid:20) k k(cid:21)
Using (A.2), (A.3), and momentum conservation, it follows that
1
∆ S(1)− σ′ S(0)− S(1)−σ
div n ≡ ǫ n+1 n − n n
hκ n (p(cid:16) ε−)2 (cid:17)i p q µ2 (A.4)
= i · (p q)log j · (p ε−)(p ε−)log .
4(4π)2ǫ " pi q j · pi pj − i· j · 2pi pj#
i,j=1 · · − ·
X
Momentum conservation implies that (A.4) is independent of both the soft energy ω and the
renormalization scale µ. It follows that (A.1) becomes
κ γww¯
out ∆T in = d2w D3 εˆ ∆ S(1)− out in . (A.5)
h | zzS| i −2π z w w w¯w¯ div n h |S| i
Z − (cid:16) (cid:17)
Before proceeding, it is useful to define the quantity
√2
εˆ ∂ xˆi(w)ε+(q(w)) = , (A.6)
w¯ ≡ w¯ i 1+ww¯
so that εˆ = εˆ εˆ . It is then straightforward to show that
w¯w¯ w¯ w¯
D2 εˆ ε− = 0 ,
w w¯
(cid:16) D2(cid:17)q = 0 ,
w (A.7)
(p ε−)2
Dw¯D εˆ i · = 2πω δ(2)(w z ) ,
w w¯w¯ p qˆ − i − i
i !
·
8
where qµ = ωqˆµ. Using the first of the above identities, we have
µ2
D3 εˆ (p ε−)(p ε−)log = 0 , (A.8)
w w¯w¯ i · j · 2p p
i j!
− ·
which implies
κ n (p ε−)2 p q
D3 εˆ ∆ S(1)− = D3 εˆ i· (p q)log j · . (A.9)
w w¯w¯ div n 4(4π)2ǫ w w¯w¯ pi q j · pi pj!
(cid:16) (cid:17) Xi,j · ·
To evaluate this, we distribute the covariant derivatives via the product rule and first compute the
term
n (p ε−)2 p q
D3 εˆ i· (p q) log j · =
w w¯w¯ p q j · p p
i ! i j
i,j=1 · ·
X (A.10)
n
p qˆ
2πγ ω (p qˆ)D δ(2)(w z )+3(p ∂ qˆ)δ(2)(w z ) log j · ,
− ww¯ i j · w − i j · w − i p pˆ
j i
iX,j=1 h i ·
where we have used the last two identities of (A.7) along with momentum conservation. Similarly,
using (A.7) and momentum conservation, we find
n (p ε−)2 p q n (p ε−)2(p ∂ q)2
3 D D εˆ i· (p q) D log j · = 3 D εˆ i · j · w ,
w w w¯w¯ p q j · w p p w w¯w¯ p q p q
i,j=1 " i· ! (cid:18) i· j(cid:19)# i,j=1 " i· j · #
X X
n (p ε−)2 p q n (p ε−)2 2
εˆ i· (p q)D3 log j · = εˆ i · (p ∂ q)3 .
iX,j=1 w¯w¯ pi·q j · w(cid:18) pi·pj(cid:19) iX,j=1 w¯w¯ pi·q (pj ·q)2 j · w
(A.11)
Finally, using momentum conservation and the relationship between the soft momenta and polar-
ization vectors [23]
1
ε+ = ∂ qˆ , (A.12)
µ w √γ µ
(cid:18) ww¯ (cid:19)
we find
n p ∂ qˆ 2 n (p ε+)2
j w j
· = εˆ · . (A.13)
p qˆ ww p qˆ
j=1 (cid:0) j · (cid:1) j=1 j ·
X X
Substituting (A.10) and (A.11) into (A.9), and then using (2.17) along with (A.13), we find
2 γww¯
out ∆T in = d2w out 2N(0)D N(0) +3D N(0)N(0) in
h | zzS| i −πκ2ǫ z w − w¯w¯ w ww w ww w¯w¯ S
2 Z γ−ww¯ D (cid:12)(cid:12)h (cid:16) (cid:17)i (cid:12)(cid:12) E (A.14)
= d2w out(cid:12) 2N(0)D N(0) +D N(0)N(0) in(cid:12) ,
−πκ2ǫ z w ww w w¯w¯ w ww w¯w¯ S
Z − D (cid:12)h (cid:16) (cid:17)i (cid:12) E
(cid:12) (cid:12)
which is precisely (4.4). (cid:12) (cid:12)
9