Scattering Amplitudes in
Quantum Field Theory
Lance Dixon (CERN & SLAC)
EPS HEP 2011
25 July 2011
The S matrix reloaded
•
Almost everything we know experimentally about gauge theory is based
on scattering processes with asymptotic, on-shell states, evaluated in
perturbation theory.
•
Nonperturbative, off-shell information very useful, but often more
qualitative (except for lattice gauge theory).
All perturbative scattering amplitudes can be computed with Feynman
diagrams – but that is not necessarily the best way, especially if there is
hidden simplicity.
On-shell methods can be much more efficient, and provide new insights.
Use analytic properties of the S matrix directly, not Feynman diagrams.
•
•
•
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
2
Three Applications of On-Shell Methods
• QCD: a very practical application, needed for
quantifying LHC backgrounds to new physics
talk by G. Zanderighi
• N=4 super-Yang-Mills theory: lots of simplicity,
both manifest and hidden. A particularly beautiful
application of on-shell and related methods –
see also talk by N. Beisert
• N=8 supergravity: amazingly good UV behavior,
beyond expectations, unveiled by on-shell methods
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
3
The Analytic S-Matrix
Bootstrap program for strong interactions: Reconstruct scattering
amplitudes directly from analytic properties: “on-shell” information
• Poles
• Branch cuts
Landau; Cutkosky;
Chew, Mandelstam;
Eden, Landshoff,
Olive, Polkinghorne;
Veneziano;
Virasoro, Shapiro;
… (1960s)
Analyticity fell out of favor in 1970s with the rise of QCD & Feynman rules
Now resurrected for computing amplitudes in perturbative QCD as
alternative to Feynman diagrams! Perturbative information now
assists analyticity. Works for many other theories too.
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
4
Perturbative unitarity bootstrap
• S-matrix a unitary operator between in and out states
S† S = 1
 unitarity relations (cutting rules) for amplitudes
• Reconstruct (multi-)loop amplitudes from cuts efficiently, due to
simple structure of tree and lower-loop helicity amplitudes
• Generalized unitarity reduces everything to (simpler) trees
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
5
Granularity vs. Plasticity
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
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Generalized Unitarity (One-loop Plasticity)
Ordinary unitarity:
put 2 particles on shell
Generalized unitarity:
put 3 or 4 particles on shell
Trees recycled into loops!
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
7
Black Holes
• “The most perfect macroscopic objects there are
in the universe: the only elements in their
construction are our concepts of space and time”
S. Chandrasekhar
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
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Scattering Amplitudes
• The most perfect microscopic objects there are
in the universe?
• When gravitons scatter, the only elements in the
construction are again our concepts of space
and time
L. Dixon
Scattering Amplitudes in QFT
h
h
h
h
EPS HEP11 Grenoble
25 July
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g
g
Gauge Theory Amplitudes
g
• Seem less universal at first sight:
• Have to specify gauge group G, representation R
for matter fields, etc.
• However, full amplitudes An can be assembled
from universal, G,R-independent “color-stripped”
or “color-ordered” amplitudes An :
color
L. Dixon
Scattering Amplitudes in QFT
universal
EPS HEP11 Grenoble
25 July
10
g
Through the clouds of gas and dust
• Obscure astrophysical black holes, but also
make them detectable, their physics much richer
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
11
Through the clouds of soft interactions
• Obscure hard QCD amplitudes at their core,
but also make the
physics much richer
F. Krauss
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
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Relativistic Gauge Amplitudes
• In relativistic limit, particle masses
mb ,mW, mt , … become unimportant, making
gauge amplitudes still more universal.
• In QCD, if we could go to extremely high
energies, then asymptotic freedom, as  0,
 we would need only leading order cross
sections, i.e. tree amplitudes
=
L. Dixon
Scattering Amplitudes in QFT
+…
EPS HEP11 Grenoble
25 July
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Universal Tree Amplitudes
• For pure-glue trees, matter particles (fermions, scalars)
cannot enter, because they are always pair-produced
• Pure-glue trees are exactly the same in QCD as in
maximally supersymmetric gauge theory,
N=4 super-Yang-Mills theory:
QCD
L. Dixon
=
Scattering Amplitudes in QFT
+… =
EPS HEP11 Grenoble
N=4
SYM
25 July
14
Tree-Level Simplicity
• When very simple QCD tree amplitudes were found,
first in the 1980’s
Parke-Taylor formula (1986)
… the simplicity was secretly due to N=4 SYM
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
15
All N=4 SYM trees in closed form
Drummond, Henn, 0808.2475
For example, for 3 (-) gluon helicities and
[n-3] (+) gluon helicities, extract from:
5 (-) gluon helicities and [n-5] (+) gluon helicities:
These formulas can immediately be used for QCD
(with external quarks too) LD, Henn, Plefka, Schuster, 1010.3991
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
16
Beyond trees
• For precise Standard Model predictions
at colliders [talk by Zanderighi]
• For investigating of formal properties of
gauge theories and gravity
 need loop amplitudes
Where the fun really starts
– textbook methods
quickly fail, even with
very powerful computers
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
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Different theories differ at loop level
N=4 SYM
• QCD at one loop:
coefficients are all rational functions – determine algebraically
from products of trees using (generalized) unitarity
well-known scalar one-loop integrals,
master integrals, same for all amplitudes
rational part
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
18
Gauge Hierarchy
of Amplitude Simplicity
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
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N=4 super-Yang-Mills theory
all states in adjoint representation, all linked by N=4 supersymmetry
• Interactions uniquely specified by gauge
group, say SU(Nc), 1 coupling g
• Exactly scale-invariant (conformal) field theory:
b(g) = 0 for all g
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
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Planar N=4 SYM and AdS/CFT
• In the ’t Hooft limit,
fixed, planar diagrams dominate
• AdS/CFT duality
suggests that weak-coupling perturbation
series in l for large-Nc (planar) N=4 SYM
should have hidden structure, because
large l limit  weakly-coupled gravity/string theory
on large-radius AdS5 x S5
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
Maldacena
25 July
21
AdS/CFT in one picture
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
22
Four Remarkable, Related
Structures Unveiled Recently in
Planar N=4 SYM Scattering
• Exact exponentiation of 4 & 5 gluon amplitudes
• Dual (super)conformal invariance
• Strong coupling “soap bubbles”
• Equivalence between (MHV) amplitudes & Wilson loops
Properties all related in some way to AdS/CFT.
To be explored in more detail tomorrow in talk by N. Beisert
Outstanding question:
Can these structures be used to solve exactly for
all planar N=4 SYM amplitudes?
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
23
Dual Conformal Invariance
Broadhurst (1993); Lipatov (1999); Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160
Conformal symmetry acting in momentum space,
on dual or sector variables xi
First seen in N=4 SYM planar amplitudes in the loop integrals
x1
x4
k
x2
x5
invariant under inversion:
x3
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
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Dual conformal invariance at higher loops
• Simple graphical rules:
4 (net) lines into inner xi
1 (net) line into outer xi
• Dotted lines are for
numerator factors
All integrals entering planar 4-point amplitude at 2, 3, 4,
and 5 loops are of this form!
Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248
Bern, Carrasco, Johansson, Kosower, 0705.1864
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
25
Constrained Amplitudes
• After all loop integrations are performed, amplitude
depends only on external momentum invariants,
• Dual conformal invariance (xi inversion) fixes form of
amplitude, up to functions of invariant cross ratios:
• Since
, no such variables for n=4,5
 4,5 gluon amplitudes totally fixed, to
6
1
exact-exponentiated form (BDS ansatz)
• For n=6, precisely 3 ratios:
5
+ 2 cyclic perm’s
L. Dixon
Scattering Amplitudes in QFT
4
EPS HEP11 Grenoble
2
3
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Constrained Integrands
• Before loop integrations performed, even 4-gluon amplitude has
invariant integrands. But only a limited number of possibilities.
• Use generalized unitarity to determine correct linear combination,
by matching to a general ansatz for the integrand.
• Convenient to chop loop amplitudes all the way down to trees.
• For example, at 3 loops, one encounters the product of a
5-point tree and a 5-point one-loop amplitude:
Cut 5-point loop amplitude further,
into (4-point tree) x (5-point tree),
in all inequivalent ways:
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
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Strategy works through at least 5 loops
Bern, Carrasco, Johansson, Kosower, 0705.1864
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
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Beyond the planar approximation
• Beyond the large-Nc limit, or in theories other than
N=4 SYM, dual conformal invariance doesn’t hold.
• However, another recently discovered set of
relations for integrands can be used to get many
contributions from a handful of planar ones:
• “Color-kinematics duality”, or BCJ relations.
• Old history (at 4-point tree level) dating back to
discovery of radiation zeroes.
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
29
Radiation Zeroes
• Mikaelian, Samuel, Sahdev (1979) computed
g
• Found “radiation zero” at
• Held independent of (W,g) helicities
• Implies a connection between
– “color” (here electric charge Qd)
– kinematics (cosq)
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
30
Color-Kinematic Duality
• Extend to other 4-point non-Abelian gauge amplitudes
Zhu (1980), Goebel, Halzen, Leveille (1981)
• Massless all-adjoint gauge theory:
• Group theory  3 terms not independent (color Jacobi
identity):
• In suitable “gauge”, find “kinematic Jacobi identity”:
•
Structure extends also to arbitrary number of legs
Bern, Carrasco, Johansson, 0805.3993
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
31
Color-Kinematic Duality at loop level
• Consider any 3 graphs connected by a Jacobi identity
• Color factors obey
Cs
•
=
Ct
–
Cu
=
nt
–
nu
Duality requires
ns
• Powerful constraint on structure of integrands
• Can always check afterwards using generalized unitarity
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
32
Simple 3 loop example
Using
we can relate non-planar topologies to planar ones
=
2
3
1
4
-
In fact all N=4 SYM 3 loop topologies related to (e)
Carrasco, Johansson, 1103.3298
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
33
UV Finiteness of N=8 Supergravity?
• Quantum gravity is nonrenormalizable by power counting:
the coupling, Newton’s constant, GN = 1/MPl2 is dimensionful
• String theory cures divergences of quantum gravity by introducing
a new length scale at which particles are no longer pointlike.
• Is this necessary? Or could enough (super)symmetry, allow a
point particle theory of quantum gravity to be perturbatively
ultraviolet finite?
• A positive answer would have profound implications, even if it is
just in a “toy model”.
• Investigate by computing multi-loop amplitudes in N=8 supergravity
DeWit, Freedman (1977); Cremmer, Julia, Scherk (1978); Cremmer, Julia (1978-9)
and then examining their ultraviolet behavior.
Bern, Carrasco, LD, Johansson, Kosower, Roiban, hep-th/0702112;
BCDJR, 0808.4112, 0905.2326, 1008.3327, 1108.nnnn
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
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28 = 256 massless states, ~ expansion of (x+y)8
SUSY
24 = 16 states
~ expansion
of (x+y)4
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
35
Gravity from gauge theory (tree level)
Kawai, Lewellen, Tye (1986) derived relations
between open & closed string amplitudes
Low-energy limit gives N=8 supergravity amplitudes
as quadratic combinations of N=4 SYM amplitudes
respecting factorization of Fock space,
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
,
25 July
36
Gravity from color-kinematics duality
Given a color-kinematics satisfying gauge amplitude,
e.g.
with
, then gravity amplitude
is simply given by erasing color factors and squaring
numerators, e.g.
Bern, Carrasco, Johansson, 0805.3993, 1004.0476;
Bern, Dennen, Huang, Kiermaier, 1004.0693
Like KLT relations,
gravity = (gauge theory)2,
amplitudes respect factorization of Fock space,
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
37
AdS/CFT
vs. KLT
AdS
=
CFT
gravity
weak
=
gauge theory
strong
KLT
gravity
weak
L. Dixon
=
Scattering Amplitudes in QFT
(gauge theory)2
weak
EPS HEP11 Grenoble
25 July
38
KLT Copying
N=4 SYM amplitude (full color, not just planar), plus KLT
relations & generalized unitarity, gives all information
needed to construct N=8 amplitude at same loop order.
For example, at 3 loops:
N=8 SUGRA
N=4 SYM
N=4 SYM
rational function of Lorentz products
of external and cut momenta;
all state sums already performed
With a color-kinematics-duality respecting gauge amplitude,
passing to the gravity amplitude is trivial!
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
39
“Color-kinematics duality” at 3 loops
Bern, Carrasco, Johansson, 0805.3993, 1004.0476
N=4
N=8 SYM
SUGRA
[
]2
[
]2
Linear in
[
4 loop amplitude
has similar dual
representation!
[
]2
]2
[
]2
BCDJR, to appear
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
40
N=8 SUGRA as good as N=4 SYM in UV
Amplitude representations have been found through
4 loops with same UV behavior as N=4 SYM
 same critical dimension Dc for UV divergences:
• But N=4 SYM is known to be finite for all L.
• Therefore, either this pattern has to break at some
point (L=5???) or else N=8 supergravity would be a
perturbatively finite point-like theory of quantum
gravity, against all conventional wisdom!
• L=5 results critical (several bottles of wine at stake)
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
41
Conclusions
• Scattering amplitudes have a very rich structure,
not only in gauge theories of Standard Model, but
especially in highly supersymmetric gauge theories
and supergravity.
• Much of this structure is impossible to see using
Feynman diagrams, but has been unveiled with the
help of unitarity-based methods
• Among other applications, these methods have
• had practical payoffs in the “NLO QCD revolution”
• provided clues that planar N=4 SYM amplitudes
might be solvable in closed form
• shown that N=8 supergravity has amazingly good
ultraviolet properties
• More surprises are surely in store in the future!
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
42
From “science” to “technology”
N=4 SYM
QCD
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
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Extra Slides
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
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Many more QCD trees from N=4 SYM
Gluinos are adjoint, quarks are fundamental
Color not a problem because it’s easily
manipulated – common to work with
“color stripped” amplitudes anyway
No unwanted scalars can enter
amplitude with only one fermion line:
Impossible to destroy them once
created, until you reach two fermion
lines:
Can use flavor of gluinos to pick out desired QCD
amplitudes, through (at least) three fermion lines
(all color/flavor orderings), and including V+jets trees
LD, Henn, Plefka, Schuster, 1010.3991
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
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Infinities
• “The infinite can be appreciated only
by the finite.”
- J. Brodsky, Watermark
• “The finite can be appreciated only
after removing the infinite.”
- Quantum Field Theory
Two types of infinities:
• Ultraviolet  renormalization
• Infrared  factorization
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
46
Dimensional Regulation in the IR
One-loop IR divergences are of two types:
Soft
Collinear (with respect to massless emitting line)
Overlapping soft + collinear divergences
imply leading pole is
at 1 loop
at L loops
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
47
Infrared Factorization
Loop amplitudes afflicted by IR (soft & collinear) divergences
jet function (spin-dependent)
hard function (finite as e  0)
soft function (color-dependent)
In any theory, including QCD, S and J exponentiate.
Surprise: for planar N=4 SYM, in some cases,
full amplitude does too.
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
48
Large Nc Planar Simplification
coefficient of
• Planar limit color-trivial: absorb S into Ji
• Each “wedge” is the square root of the well-studied process
“gg  1” (the exponentiating Sudakov form factor):
L. Dixon
Scattering Amplitudes in QFT
EPS HEP11 Grenoble
25 July
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