Two-Loop Gravity amplitudes from four dimensional Unitarity

0.19 MB·

by  David C. Dunbar

January 25, 2017
Two-Loop Gravity amplitudes from four dimensional Unitarity
David C. Dunbar, Guy R. Jehu and Warren B. Perkins
College of Science,
Swansea University,
Swansea, SA2 8PP, UK
January 25, 2017
Abstract
We compute the polylogarithmic parts of the two-loop four and five-graviton amplitudes where
the external helicities are positive and express these in a simple compact analytic form. We use
7 these to extract the ln(µ2) terms from both the four and five-point amplitudes and show that these
1
match the same R3 counterterm.
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PACS numbers: 04.65.+e
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I. INTRODUCTION
Computing the scattering amplitudes of a quantum theory using its singular and analytic
structure has a long history with many notable sucesses [1]. Recently the five-point all-
plus helicity amplitude has been computed in QCD: first the integrands were determined
using the method of maximal cuts [2] and D-dimensional unitarity and then the integrals
were evaluated yielding a compact analytic form [3]. In D-dimensional unitarity the cuts
are computed in D = 4 2ǫ dimensions where, typically, the components of the cuts are
−
considerably more complicated than in four dimensions. In [4] it was demonstrated that four-
dimensional unitarity techniques [5, 6] blended with a knowledge of the singular structure
of the amplitude could reproduce this form in a straightforward way. The four-dimensional
approach was also used to calculate the six-point amplitude [7, 8] which was subsequently
verified [9].
Here we apply these techniques to gravity amplitudes with a particular emphasis on their
Ultra-Violet (UV) behaviour. Understanding the UV structure of quantum gravity necessi-
tates studying the two-loop amplitude. ’t Hooft and Veltman [10] demonstrated the one-loop
finiteness of on-shell amplitudes in quantum gravity by showing the available counterterms
in four dimensions made no contribution to perturbative amplitudes. This cancellation does
not persist to two loop and Goroff and Sagnotti [11, 12] in a landmark calculation were able
to compute a UV infinity of the form
209
AR (1.1)
∼ 80ǫ ×
where AR is the functional form generated by an R3 counterterm. A feature of the com-
putation was the appearance of sub-divergences which were cancelled diagram by diagram.
In [13] this computation was revisited using evanescent operators which arise at one-loop
to remove the sub-divergences in the four-point two-loop amplitude. For gravity coupled to
various matter multiplets they found UV terms
a
1 +a ln(µ2) AR (1.2)
2
ǫ ×
(cid:16) (cid:17)
and noted that the coefficient of ln(µ2), a , was more robust and simpler than a . Specif-
2 1
ically, when coupled to n (non-propagating) three form fields a obtained a contribution
3 1
proportional to n but a did not. Additionally when coupled to scalars and vectors a was
3 2 2
simply proportional to the the difference between the number of bosonic and fermionic de-
grees of freedom. As was argued in ref [13], the coefficient a has physical content since after
2
renormalisation the amplitude depends upon a but not a .
2 1
In this article we explore and compute, up to rational terms, the four and five-point two-
loop amplitude in quantum gravity where all the external gravitons have positive helicity.
We use the techniques which have proven successful for the gluon amplitude: the amplitude
is organised using a knowledge of its singularity structure, then four-dimensional unitarity is
used to determine the logarithmic and dilogarithmic parts. Since ln(µ2) only appears in the
combination ln(K2/µ2) we can extract the coefficient of ln(µ2) using unitarity. We obtain
the same coefficient of ln(µ2) for the four-point amplitude as in ref. [13]. We also obtain the
coefficient of ln(µ2) for the five-point amplitude and show that this matches to the same R3
counterterm.
2
II. STRUCTURE OF THE AMPLITUDE
As a convention we remove the coupling constant factors from the full n-point L-loop
(L)
amplitude, using
n
M
i(κ/2)n−2+2L(r )L
(L)(1, ,n) = Γ M(L)(1, ,n) (2.1)
Mn ··· (4π)L(2−ǫ) n ···
where r = Γ2(1 ǫ)Γ(1+ǫ)/Γ(1 2ǫ).
Γ
− −
We then organise the amplitude according to its singularity structure. The amplitude has
both Infra-Red (IR) and UV singularities in the dimensional regulation parameter ǫ. The
all-plus amplitudes, which are finite at one-loop, can be divided as:
M(2) =M(1)I(1) + G(2) + F(2) + R(2) + (ǫ), (2.2)
n n n n n n O
(1)
wherethefirsttermcontainstheIRsingularitiesoftheamplitude. ThefunctionI is[14–17]
n
n 1 µ2 ǫ
I(1) = s (2.3)
n − ǫ2 ij s
" i<j (cid:18)− ij(cid:19) #
X
and M(1) is the all-ǫ form of the one-loop amplitude (with s = (k + k )2). The leading
n ij i j
singularity of I(1) is only ǫ−1 since
n
n
1 1
s = 0 (2.4)
ǫ2 ij ǫ2 ×
!
i<j
X
by momentum conservation and the leading singularity is then
n
1
s ln( s /µ2) M(1). (2.5)
ǫ × ij − ij × n
!
i<j
X
The amplitude also has finite logarithmic terms. In our four dimensional formulation these
arise in two-particle cuts and have the form of one-loop bubble integral functions,
n 1 µ2 ǫ
G(2) = c . (2.6)
n ijǫ s
i<j (cid:18)− ij(cid:19)
X
where c are rational functions of λ and λ¯ .1
ij k k
The all-plus two-loop amplitude in QCD does not contain this term [19]. This give rise to
1 As usual, a null momentum is represented as a pair of two component spinors pµ = σµ λαλ¯α˙. For real
αα˙
momenta λ= λ¯∗ but for complex momenta λ and λ¯ are independent [18]. We are using a spinor helicity
formalism with±the usual spinor products habi=ǫαβλαaλβb and [ab]=−ǫα˙β˙λ¯αa˙λ¯βb˙.
3
ǫ−1 and ln(µ2) terms in the combination
1
c +ln(µ2) . (2.7)
ij
× ǫ
!
i<j (cid:18) (cid:19)
X
There may be other sources of ǫ−1 terms not directly determined by unitarity. The function
(2) (2)
F contains the remaining polylogarithms of the amplitude and R contains the remaining
n n
rational terms. In dimensional regularisation the internal momenta lie in D = 4 2ǫ and it
−
is really D-dimensional unitarity which should be used to reconstruct the amplitude. Conse-
quently, four dimensional unitarity is not sensitive to the rational terms, however it does give
considerable simplification. The rational terms may be computed by complementary meth-
ods such as recursion. For the case of QCD, the rational terms for n = 5,6 were computed
(2)
by recursion starting from the four point amplitude. We will not compute R here: it not
5
(2)
being necessary for our analysis and R not being available.
4
The all-plus two-loop amplitude is a particularly simple amplitude. The all-plus helicity
tree amplitude vanishes,
M(0)(1+,2+, ,n+) = 0 (2.8)
n ···
This can be seen as a consequence of supersymmetric Ward identities [20]. These imply that
thisamplitude vanishestoallordersinperturbationtheoryinsupersymmetric theories. Since
the n-graviton amplitudes for pure gravity coincide with those for supersymmetric theories
at tree level then the gravity tree amplitude also vanishes.
The one-loop four-point amplitude for pure gravity is [21]2
st 2 (s2 +t2 +u2)
M(1)(1+,2+,3+,4+) = +O(ǫ) (2.9)
4 − 12 23 34 41 120
(cid:18)h ih ih ih i(cid:19)
and the n-point amplitude can be expressed as [22]
( 1)n
M(1)(1+,2+,3+, ,n+) = − h(a,M,b)h(b,N,a)tr3(aMbN)+O(ǫ) (2.10)
n ··· 960
a,b,M,N
X
where h(a,M,b) are the “half-soft” functions of ref. [22]. The summation is over pairs of legs
(a,b) and partitions (M,N) of the remaining legs where both the sets M and N have at least
one element. The half-soft functions we need for the five-point amplitude are
1 1 [cd] 1
h(a, c ,b) = , h(a, c,d ,b) = . (2.11)
{ } ac 2 cb 2 { } ac ad cd cb db
h i h i h ih ih ih ih i
When coupled minimally to additional bosons and fermions these amplitudes are multiplied
by a factor (N N )/2 where N is the total number of bosonic/fermionic degrees of
B F B/F
−
freedom. A key feature of the one-loop amplitudes is that they are, to order ǫ0, rational
functions and as such have no cuts in four dimensions. Thus if computing amplitudes using
cuts in four dimensions they are indivisible and can be treated as a vertex. The only four
dimensional cuts of the n-point all-plus amplitude are shown in fig. 1. For this helicity ampli-
tude the only tree amplitudes necessary to compute the cuts are the three point amplitudes
2 We use for four point kinematics s s12, t s14 and u s13.
≡ ≡ ≡
4
and the Maximally-Helicity-Violating (MHV) tree amplitudes [23].
a) + b) c) d)
+ +
+
− +
+
− +
+ +
+ + + + + +
− − − − − −
+ +
+ − − +
+ +
− + − + − +
+ + + +
+ +
FIG. 1: Four dimensional cuts of the two-loop all-plus amplitude involving an all-plus one-loop
vertex (indicated by )
•
(1) (1)
The four-dimensional calculation gives the coefficient of I to be M to leading order in
n n
ǫ. As in the QCD case, we promote this to the all-ǫ form. The all-plus one-loop amplitudes
in eq. (2.2) for four and five-points are known to all orders in ǫ [22, 24]
[12]2[34]2
M(1)(1+,2+,3+,4+) = 2 I1234[µ8]+I1243[µ8]+I1423[µ8]
4 12 2 34 2 4 0 4 0 4 0
h i h i
(cid:0) [12][23][34][45][51](cid:1)
M(1)(1+,2+,3+,4+,5+) = β I123(45)[µ8] 2 I12345[µ10]+Perms
5 123(45) 4 0 − 12 23 34 45 51 5 0
h ih ih ih ih i
(2.12)
where
β = [12]2[23]2h(1, 4,5 ,3) (2.13)
123(45)
− { }
and I [µ2m] are the n-point integral functions with µ2m inserted where µ are the 2ǫ co-
n 0 0 0 −
ordinates in dimensional regularisation, dDx = d4xd−2ǫµ . The superscripts denote the
0
ordering and clustering of the external legs. These are related to scalar integrals in higher
R R
dimensions [24, 25],
I [µ2r] = ǫ(1 ǫ) (r 1 ǫ)(4π)rID=4+2r−2ǫ . (2.14)
m 0 − − ··· − − m
III. THE FOUR-POINT ALL-PLUS HELICITY AMPLITUDE
The four point all-plus helicity amplitude has some significant simplifications. Specifically,
the quadruple cuts vanish3 and there are only one-mass triangle and bubble contributions.
In fact this amplitude is sufficiently simple that using the one loop amplitude as a vertex
both the triangle and bubble functions can be obtained simply from the two-particle cuts [21]
3 In computing the quadruple cuts for a four-point amplitude the only non-vanishing product of cut am-
plitudes has alternating MHV and MHV three-point vertices at the corners. This precludes any box
functions for the four-point all-plus amplitude.
5
with the result,
2( s)1−ǫ +2( t)1−ǫ +2( u)1−ǫ
M(2)(1+,2+,3+,4+) = M(1)(1+,2+,3+,4+) − − −
4 4 × ǫ2
(cid:18)
2s(3u2+3t2 2s2)( s/µ2)−ǫ 2t(3s2 +3u2 2t2)( t/µ2)−ǫ
− − − −
− (s2 +t2 +u2) ǫ − (s2 +t2 +u2) ǫ
2u(3t2 +3s2 2u2)( u/µ2)−ǫ
− − +rational terms . (3.1)
− (s2 +t2 +u2) ǫ
(cid:19)
This expression contains ǫ−1 and the ln(µ2) terms in the combination,
30stu 1
M(1)(1+,2+,3+,4+) +ln(µ2)
4 × (s2 +t2 +u2) × ǫ
(cid:18) (cid:19)
2
1 st 1
= stu +ln(µ2) . (3.2)
− 4 12 23 34 41 × × ǫ
(cid:18)h ih ih ih i(cid:19) (cid:18) (cid:19)
If gravity is coupled to matter with N 2 additional bosonic degrees of freedom and
B
−
N fermionic degrees of freedom, the one-loop all-plus amplitude is multiplied by a factor of
F
(N N )/2 and the calculation follows through as above. The all-plus amplitude in this
B F
−
theory is thus the expression above with the replacement
1 N N
B F
− (3.3)
− 4 −→ − 8
which matches the result of [13]. Consequently, we note that four dimensional unitarity gives
the correct renormalised amplitude up to rational terms although, as was known in ref [21],
the coefficient of ǫ−1 does not match the field theory calculation.
IV. THE FIVE POINT ALL-PLUS AMPLITUDE
This amplitude contains functions, particularly dilogarithms, that are not present in the
four-point amplitude. These are contained in the box contributions shown in fig. (2). The
box contribution is readily evaluated using a quadruple cut [26].
a+
e+ ℓ
− 2 + b+
+ •+
ℓ ℓ
1 3
− −
− +
d+ ℓ4 c+
FIG. 2: The labelling and internal helicities of the quadruple cut.
6
With the labelling of fig. 2 the cut momenta are
cd c P e P ed
¯ de cd ¯
ℓ = h iλ λ , ℓ = h | |λ , ℓ = h | |λ , ℓ = h iλ λ , (4.1)
1 d e 2 e 3 c 4 d c
ec ec ec ec
h i h i h i h i
giving the coefficient of the box function
1
= M(1)(a+,b+,ℓ+, ℓ+) M(0)( l−,c+,l+) M(0)( l−,d+,l−) M(0)( l+,e+,l−)
C{a,b},c,d,e 2 4 3 − 2 × 3 − 3 4 × 3 − 4 1 × 3 − 1 2
1 [ab]6[cd]2[ed]2
= ab 2 ec 2 + ae 2 bc 2 + ac 2 be 2 (4.2)
240 ec 4 ! h i h i h i h i h i h i
h i
(cid:0) (cid:1)
This is the coefficient of the integral function I1m(s ,s ,s ) where [27]
4 cd de ab
2 (µ2ǫ)
I1m(S,T,M2) = ( S)−ǫ +( T)−ǫ ( M2)−ǫ
4 − ST − ǫ2 − − − −
"
h i
M2 M2 1 S π2
+Li 1 + Li 1 + ln2 + + (ǫ) (4.3)
2 2
− S − T 2 T 6 O
#
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
and overall factors have been removed according to the normalisation of eq. (2.1).
This integral function splits into singular terms plus a remainder I1m = I1m:IR + I1m:F
4 4 4
where
2 (µ2ǫ)
I1m:IR(S,T,M2) ( S)−ǫ +( T)−ǫ ( M2)−ǫ . (4.4)
4 ≡ −ST − ǫ2 − − − −
" #
h i
The one-mass integral function is
(µ2ǫ)
I1m(K2) = ( K2)−1−ǫ. (4.5)
3 ǫ2 −
and the two-mass triangle function is,
(µ2ǫ)( K2)−ǫ ( K2)−ǫ
I2m K2,K2 = − 1 − − 2 . (4.6)
3 1 2 ǫ2 ( K2) ( K2)
− 1 − − 2
(cid:0) (cid:1)
The boxes, one-mass and two-mass triangles all have IR infinite terms of the form
(µ2ǫ)
( K2)−ǫ (4.7)
ǫ2 −
The coefficients of the triangle contributions can be evaluated using triple cuts [28–31] and
a canonical basis [32]. Summation over the box and triangle contributions gives an overall
7
coefficient of M(1),ǫ0(a+,b+,c+,d+,e+), i.e
5
I1m + I1m + I2m
C{a,b},c,d,e 4 C{a,b,c},d,e 3 C{a,b},c,{d,e} 3 IR
=M(cid:16)X(1),ǫ0(a+,b+,c+,d+,Xe+) n 1s Xµ2 ǫ (cid:17) (4.8)
5 × −ǫ2 ij s
i<j (cid:18)− ij(cid:19)
X
where M(1),ǫ0(a+,b+,c+,d+,e+) is the order ǫ0 truncation of the one-loop amplitude. A key
5
step is to promote the coefficient of these terms to the all-ǫ form of the one-loop amplitude
which then gives the correct singular structure of the amplitude. We have confirmed relation-
ship (4.8) for the n-point amplitude by computing the triple and quadruple cuts at specific
kinematic points up to n = 10.
Consequently, we can obtain the compact explicit analytic form for the dilogarithmic
remainder part of the amplitude
1 [ab]6[cd]2[ed]2
F(2) = ab 2 ec 2 + ae 2 bc 2 + ac 2 be 2
5 240 ec 4 ! h i h i h i h i h i h i
XP30 h i (cid:18) (cid:19)
2 s s 1 s π2
Li 1 ab +Li 1 ab + ln2 cd + , (4.9)
2 2
× −s s − s − s 2 s 6
cd de!" (cid:18) cd(cid:19) (cid:18) de(cid:19) (cid:18) de(cid:19) #
where thepermutation sum is over the30 independent permutations of thelegs ( a,b ,c,d,e)
{ }
after factoring out for the symmetries ( a,b ,c,d,e) ( b,a ,c,d,e) ( a,b ,e,d,c).
{ } ≡ { } ≡ { }
Note that the coefficients in the F(2) term contain ec −4 singularities. On this singularity
5 h i
the integral function vanishes and F(2) has ec −3 singularities. These are spurious and not
5 h i
(2)
present in the full amplitude. They cancel against the G terms as we will discuss at the
5
end of the next section.
V. COEFFICIENT OF ln(s/µ2)
We determine the presence of the ln(s /µ2) functions using two-particle cuts. The coef-
ab
ficient has two contributions as shown in fig.3.
− + + −
a+ + a+ +
A: + B : +
b+ + b+ +
− + + −
FIG. 3: Contributions to the two-loop amplitudes involving an all-plus loop (indicated by )
•
We determine these using canonical forms. The canonical basis approach [32] is a sys-
tematic method to determine the coefficients of triangle and bubble integral functions in a
8
one-loop amplitude from the three and two-particle cuts. A two particle cut is of the form
C i d4ℓ δ(ℓ2)δ(ℓ2)Atree( ℓ ,a, ,b,ℓ ) Atree( ℓ , ,ℓ ). (5.1)
2 ≡ 1 1 2 1 − 1 ··· 2 × 2 − 2 ··· 1
Z
The product of tree amplitudes appearing in the two-particle cut canbe decomposed in terms
of canonical forms ,
i
H
Atree( ℓ , ,ℓ ) Atree( ℓ , ,ℓ ) = e (ρ ,ℓ ), (5.2)
1 − 1 ··· 2 × 2 − 2 ··· 1 iHi k j
X
where the e are coefficients independent of ℓ . We then use substitution rules to replace
i j
the (ρ ,ℓ ) by rational functions H [ρ ,P] to obtain the coefficient of the bubble integral
i k j i k
H
function as
e H [ρ ,P]. (5.3)
i i k
i
X
Here we use this technique treating the one-loop all-plus vertex as a tree amplitude.
The canonical forms we need and their substitutions are
aℓ cℓ Cc [C P a Ac [A P a
0 (A,C;a,c,ℓ ,ℓ ) h 1i h 2i H0 [A,C;a,c;P] = h i | | i + h i | | i
H1,1 1 2 ≡ Aℓ Cℓ −→ 1,1 CA [C P C AC [A P A
1 2
h ih i h i | | i h i | | i
B ℓ C ℓ [A P B [A P C
0 (A;B ;C ;ℓ ,ℓ ) h 1 1ih 1 2i H0 [A;B ;C ;P] = | | 1i | | 1i. (5.4)
H2x 1 1 1 2 ≡ Aℓ Aℓ −→ 2x 1 1 [A P A 2
1 2
h ih i | | i
where P is the cut-momentum P = k + k . We take the product of amplitudes in the
a b
···
cut and split them into a sum of terms of type in eqn.(5.4) using partial fractioning
n−1 ℓX n n−1 Y X 1 n 1
j=1 h ji = j=1 h i ji = α (5.5)
n ℓY Y Y × ℓY i ℓY
Q i=1h ii i=1 Q l6=ih l ii ! h ii i=1 h ii
X X
Q Q
Forthefive-point amplitude thecutshave P = k +k . Workingspecifically withs = s ,
a b ab 45
the two-particle cut has two contributions
A :M(0)(1+,2+,3+,l−,l−) M(1)(4+,5+,l+,l+)
5 2 1 × 4 1 2
B :M(1)(1+,2+,3+,l+,l+) M(0)(4+,5+,l−,l−) (5.6)
5 2 1 × 4 1 2
From the term A we obtain a contribution to the coefficient of
3 2 2
CA = c αrβrH0 [Ar,ar;Br,br;P]+ c αrβrH0 [Ar;Br,br;P] (5.7)
45 r i j 1,1 i j 3 3 r i i 2x i 3 3
r=1,2 i=1 j=1,j6=i r=1,2 i=1
X X X X X
with
2 BrAr brar
αr = l=1h l ii βr = h 1 ii (5.8)
i ArAr i arar
Ql6=ih l ii l6=ih l ii
where Q Q
[45]6[23] [45]6[12]
c = c = (5.9)
1 2
−120 23 13 120 12 13
h ih i h ih i
9
and
A1 = 1 , 2 , 3 , A2 = 2 , 3 , 1
{| ii} {| i | i | i} {| ii} {| i | i | i}
a1 = 1 , 2 , a2 = 2 , 3
{| ii} {| i | i} { i} {| i | i}
B1 = 5 , 5 , K 1] B2 = 5 , 5 , K 3]
{| ii} {| i | i − 45| } {| ii} {| i | i − 45| }
b1 = b2 = 4 , 4 (5.10)
{| ii} {| i} {| i | i}
Contributions fromthe second configurationaremorecomplicated, but using relationships
between the different terms to simplify the final expression we obtain
CB = TB +TB +TB (5.11)
45 1,2,3,4,5 2,1,3,4,5 3,2,1,4,5
with
5 5 5
[23][45]
TB = α β H0 [i,j;A ,B , 4,5 ]+ α β H0 [i;A ,B , 4,5 ]
1,2,3,4,5 120 23 45 i j 1,1 4 4 { } i i 2x 4 4 { }
!
h ih i i=2 j=2,j6=i i=2
X X X
(5.12)
where
3 iA 3 jB
α = k=1h ki β = k=1h ki (5.13)
i j
il jl
l∈Q{2,3,4,5}−{i}h i l∈Q{2,3,4,5}−{j}h i
Q Q
and
A = B = 1 ,K 1],K 1],K 1] (5.14)
k k
{| i} {| i} {| i | | | }
Thus the amplitude contains
1
(CA +CB) ln(s /µ2) (5.15)
45 45 ǫ − 45
(cid:18) (cid:19)
The full amplitude thus has ǫ−1 and ln(µ2) terms of
1
(CA +CB)ln(s )+ (CA +CB) +ln(µ2) (5.16)
ij ij ij ij ij ǫ
i<j i<j (cid:18) (cid:19)
X X
We have determined the above expression using four-dimensional unitarity which isolates the
coefficient of ln(s ). The value of ln(µ2) then follows. The coefficient of ǫ−1 in this is tied
ij
to the ln(µ2) but is, presumably, not the value which would be obtained from a field theory
calculation.
Thebubblecoefficients containspurious ec −3 singularitieswhichmustcancel [33]against
h i
(2)
the singularities in the F term. Specifically the singularities are of the form
5
1
ln(s ),ln(s ),ln(s )+permutations (5.17)
ec 3 ×{ ab dc ed }
h i
(2)
where the permutations are of of a,b,d. The F term contains dilogarithms but near the
5
10