Transcript LDEPSHEP11x

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.
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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
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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.
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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
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Granularity vs. Plasticity
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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!
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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
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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
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h
h
h
h
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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
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universal
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g
Through the clouds of gas and dust
• Obscure astrophysical black holes, but also
make them detectable, their physics much richer
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Through the clouds of soft interactions
• Obscure hard QCD amplitudes at their core,
but also make the
physics much richer
F. Krauss
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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
=
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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
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+… =
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N=4
SYM
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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
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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
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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
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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
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Gauge Hierarchy
of Amplitude Simplicity
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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
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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
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Maldacena
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AdS/CFT in one picture
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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?
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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
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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
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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
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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:
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Strategy works through at least 5 loops
Bern, Carrasco, Johansson, Kosower, 0705.1864
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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.
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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)
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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
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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
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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
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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
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28 = 256 massless states, ~ expansion of (x+y)8
SUSY
24 = 16 states
~ expansion
of (x+y)4
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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,
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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,
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AdS/CFT
vs. KLT
AdS
=
CFT
gravity
weak
=
gauge theory
strong
KLT
gravity
weak
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(gauge theory)2
weak
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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!
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“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
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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)
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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!
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From “science” to “technology”
N=4 SYM
QCD
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Extra Slides
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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
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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
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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
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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.
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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):
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