Transcript Constructing QCD One

Darren Forde (SLAC & UCLA)
CONSTRUCTING QCD ONE-LOOP
AMPLITUDES
arXiv:0704.1835 (To appear this evening)
OVERVIEW
Motivations for precision calculations NLO
and one-loop amplitudes already given.
Unitarity bootstrap technique combining,
• Unitarity cuts in D=4 for cut-constructible pieces.
• On-shell recurrence relations for rational pieces.
Focus on the cut-constructable terms here,
•A new method for extracting scalar bubble
and triangle coefficients.
Coefficients from the behaviour of the free
integral parameters at infinity.
THE UNITARITY BOOTSTRAP
Focus on these terms
Unitarity bootstrap technique

Cut-constructible from gluing together trees in D=4,
 i.e. unitarity techniques in D=4.
  missing rational pieces in QCD. [Bern, Dixon, Dunbar,
Kosower]

Rational from one-loop on-shell recurrence relation.

Alternatively work in D=4-2ε, [Bern, Morgan], [Anastasiou, Britto, Feng,
[Berger, Bern, Dixon, DF, Kosower]
Kunszt, Mastrolia]

Gives both terms but requires trees in D=4-2ε.
ONE-LOOP INTEGRAL BASIS

A one-loop amplitude decomposes into
Rational terms
 
2
Rn  r

 4 

2

  bi
 i
  cij
ij
  dijk
ijk



Quadruple cuts freeze the integral  boxes [Britto, Cachazo, Feng]
l
l1
l3
l2
1 2 2
2
2
dijk l 
)
A
(
l
)
(l0
0, lA1 1 (lijk0,
l

0,
l
l4 (lijklijk
; a 2 2 ijk ; a 3A3
ijk ;
a)A
;a )
2 a 1
2
TWO-PARTICLE AND TRIPLE CUTS


What about bubble and triangle terms? Additional coefficients
Triple cut  Scalar triangle coefficients?
  dijk
cij
k

Two-particle cut  Scalar bubble coefficients?
bi
  cij
j

Disentangle these coefficients.
  dijk
jk
Isolates a single triangle
DISENTANGELING COEFFICIENTS



Approaches,
 Unitarity technique, [Bern, Dixon, Dunbar, Kosower]
 MHV one-loop cut-constructible by joining MHV vertices
at two points, [Bedford, Brandhuber, Spence, Traviglini], [Quigley, Rozali]
 Integration of spinors, [Britto,Cachazo,Feng] + [Mastrolia] + [Anastasiou,
Kunst],
 Solving for coefficients, [Ossola, Papadopoulos, Pittau]
 Recursion relations, [Bern, Bjerrum-Bohr, Dunbar, Ita]
Large numbers of processes required for the LHC,
 Automatable and efficient techniques desirable.
Can we do better?
TRIANGLE COEFFICIENTS

Coefficients, cij, of the triangle integral, C0(Ki,Kj), given by
cij    Inf A1 A2 A3   t 
t 0
Triple cut of the triangle C0(Ki,Kj)
K3
Single free integral parameter in l
S (  S2 ) ˆ 
l   t Kˆ 1  12
K2
  S1S2
A3
A2
K1
A1
l 
K2
Series expansion in t at infinity
a0  a1t1  a2t 2 
 amaxt max
1 S2 (  S1 ) ˆ 
ˆ
K

K
1
2
t  2  S1S2
  Kˆ 1 Kˆ 2 Kˆ 1
S1 ˆ
S2 ˆ
ˆ
ˆ
K

K

K
,
K

K

K1
Masslessly Projected momentum 1
1
2
2
1


SIX PHOTONS
[Nagy, Soper], [Binoth, Heinrich, Gehrmann, Mastrolia],
Papadopoulos,
6 λ‘s [Ossola,
top and
bottom Pittau]

3-mass triangle of A6(-+-+-+)  the triple cut integrand
2
2
2
l1h2 l2 3  l1h51
16 A(l ,1 ,232,il ) A(l2 ,3 , 4 , l1 ) A(l1h1 ,5 ,6 , l h )
l 2 l2 2 l1 4 l2 4 l 6 l1 6
Extra propagator
h


h2
2
 Box terms
No propagator  Triangle
2
2
2
Propagator ↔ pole in t,
Rest t A1 A2 A3
ˆ 1 Kˆ 3 Kˆ 5
max
K
a111tA2 A3 (ta) max
dtaInf
 t    dl D0
S1 (  S2 ) ˆ 32  dtA11A22 A33 1dt
0 A
ˆ
l 2  t K1 2  2
K2i2 
t {l}
tj
boxes
poles {j}
2
2
2
  S1S2
2   Kˆ 2 Kˆ 4 Kˆ 6
1
1
1
The scalar triangle coefficient
j
2 solutions to γ  divide by 2

The complete coefficient.
VANISHING INTEGRALS

In general higher powers of t appear in [Inf A1A2A3](t).
a0  dt  a1  dtt  a2  dtt 2 

 amax  dtt max
Integrals over t vanish for chosen parameterisation, e.g.
(Similar argument to [Ossola, Papadopoulos, Pittau])
 dtt

4
d
 l
Kˆ 1 l Kˆ 2
l 2l12l22
Kˆ 1 K1 Kˆ 2 C1  Kˆ 1 K 2 Kˆ 2 C2  0
In general whole coefficient given by
cij    Inf A1 A2 A3   t 
t 0
ANOTHER TRIANGLE COEFFICIENT

3-mass triangle coefficient of e e   q q  g  g  in the
14:23:56 channel. [Bern, Dixon, Kosower]
2 λ‘s top and bottom
i
l2 5
2
ll2
2
23
2
14 56 4l2 2l ll1 l1l2
Independent of t
Series expand in t around infinity

i

2 
 Kˆ 1 5
2
23
2

S 
S1 1  1  14 56 4 Kˆ 1
 

2 Kˆ 1
WHAT ABOUT BUBBLES?

The bubble coefficient bj of the scalar bubble integral
B0(Kj)
1
b j  i  Inf  Inf A1 A2  ( y )  (t )
 Inf A1 A2 A3  (t )
1 

t 0, y m 
2 triangles
m 1
t T ( i )
Two-particle cut of the bubble B0(Ki)
Two free integral parameter in l
A2
K1
A1
max y≤4
amax y 0 max y
a10 1 a20 2
a00  a10 y  a20y   amax y 0 y
2
3
max y  1
1
2
t  a01  a11 y  a21 y   amax1 y max y 
+

 t max t a0max t  a1max t y1  a2max t y 2 
S
l   t Kˆ 1  1 (1  y )  

l 
y ˆ
K1   
t
S
    Kˆ 1   , Kˆ 1  K1  1 

 amax y max t y max

NON-VANISHING INTEGRALS

Similar to triangle coefficients, but depends upon t.


1
 Inf A1 A2 A3  (t )

Box and triangle coeff’s
2 triangles
t T ( i )
Two free parameters implies
 Rest t j A1 A2 A3 
Res y  yl 


t

t
  Res y  yl A1 A2 A3  
j




Inf
Inf
A
A
(
t
)
(
y
)

Inf
(
t
)







{l} 

1 2


 poles

y  yl
y  yl
poles {j},{l}
 
 
Two-particle cut contrib
One extra Pole in y,
looks like a triangle
Contains bubbles
 dtt
lP  0  y0  t
y fixed at pole
 xP
[ Kˆ1P]
Kˆ 1 K1   C1  Kˆ 1 K 2   C2  0
S
l   t Kˆ 1  1 (1  y0 )  

l 
y0 ˆ 
K1   
t
TRIPLE-CUT CONTRIBUTIONS

Example: Extract bubble of three-mass linear triangle,
d

4
l
a  l b
l 2 (l  K1 ) 2 (l  K 2 ) 2
Cut l2 and (l-K1)2 propagators, gives integrand
a l b
Single pole
  Kˆ 2  K 2 
S2 ˆ
K1

Series expand y and
(l  K 2 )
then t around ∞, 1
m
No “triangle” terms as  dtt  0
set t  0, y 
m 1
 a  Kˆ 1 b  S1 a  Kˆ 2 b 

2
  S1S2
 2  S1S2
2

Complete coefficient.
TRIPLE CUT CONTRIBUTIONS CONT.


Multiple poles  Can’t choose χ so that all integrals in t
vanish.
Sum over all triangles containing the bubble,
 Renormalisable theories, max of t3.
 Integrals over t known, Cij a constant, e.g. C11=1/2
 S1  
i
 dtt    
i


i
K 2 Kˆ 1
 K1.K 2 
 K1.K 2   S1S2
i 1
i
 Cij
S 2j 1
 K1.K 2 
Gives equivalent, χ independent result
2
j 1
2
B
(
K
1 )
j 1 0






(
K
.
K
)
a
K
b
S
a
K
b
1
1
2
1
1
2



2
2
2  ( K1.K 2 )  S1S2
( K1.K 2 )  S1S2 


OTHER APPLICATIONS

Comparisons against the literature
 Two minus all gluon bubble coefficients for up to 7 legs.
[Bern, Dixon, Dunbar, Kosower], [Bedford, Brandhuber, Spence, Travigini]
N=1 SUSY gluonic three-mass triangles for A6(+-+-+-),
A6(+-++--). [Britto, Cachazo, Feng]
 Various bubble and triangle coefficients for processes
of the type e e   q q  g  g. [Bern, Dixon, Kosower]


Bubble and three-mass triangle coefficients for six photon
A6(+-+-+-) amplitude. [Nagy, Soper], [Binoth, Heinrich, Gehrmann, Mastrolia],
[Ossola, Papadopoulos, Pittau]
CONCLUSION
“Compact”
coefficients from
2 steps,
Efficient and
easy to
implement.
• Momentum
parameterisation.
• Series expansion in
free parameters at
infinity.
• Scalar bubble and
triangle coefficients.
Use with unitarity
bootstrap for
complete
amplitude.
ON-SHELL RECURSION RELATIONS

Recursion using on-shell amplitudes with fewer legs,
[Britto, Cachazo, Feng] + [Witten]
Two reference legs “shifted”,
P2
P2





i i  
i  j , j j  
i   j


i P j
i P j


1
P2
Intermediate momentum leg is on-shell.



Final result independent of the of choice shift.
Complete amplitude at tree level.
At one loop need the cut pieces [Berger, Bern, Dixon, DF, Kosower]

Combining both involves overlap terms.