Julie Cass SULI Program 2011 SLAC National Accelerator Center Advisor Josef Frisch
Scattering Amplitudes in Quantum Field Theory Lance Dixon (CERN & SLAC) EPS HEP 2011 25 July 2011.
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Transcript of Scattering Amplitudes in Quantum Field Theory Lance Dixon (CERN & SLAC) EPS HEP 2011 25 July 2011.
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
EPS HEP11 Grenoble 25 July 4L. Dixon Scattering Amplitudes in QFT
The Analytic S-MatrixBootstrap program for strong interactions: Reconstruct scattering amplitudes directly from analytic properties: “on-shell” information
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.
• Poles
• Branch cuts
EPS HEP11 Grenoble 25 July 5L. Dixon Scattering Amplitudes in QFT
Perturbative unitarity bootstrap
• Reconstruct (multi-)loop amplitudes from cuts efficiently, due to simple structure of tree and lower-loop helicity amplitudes
• S-matrix a unitary operator between in and out statesS† S = 1 unitarity relations (cutting rules) for amplitudes
• Generalized unitarity reduces everything to (simpler) trees
EPS HEP11 Grenoble 25 July 6L. Dixon Scattering Amplitudes in QFT
Granularity vs. Plasticity
EPS HEP11 Grenoble 25 July 7L. Dixon Scattering Amplitudes in QFT
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!
EPS HEP11 Grenoble 25 July 8
• “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
Black Holes
L. Dixon Scattering Amplitudes in QFT
EPS HEP11 Grenoble 25 July 9
• 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
Scattering Amplitudes
L. Dixon Scattering Amplitudes in QFT
h
hh
h
EPS HEP11 Grenoble 25 July 10
• 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 :
Gauge Theory Amplitudes
L. Dixon Scattering Amplitudes in QFT
g
gg
g
color universal
EPS HEP11 Grenoble 25 July 11
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 12
Through the clouds of soft interactions
• Obscure hard QCD amplitudes at their core, but also make the
physics much richer
L. Dixon Scattering Amplitudes in QFT
F. Krauss
EPS HEP11 Grenoble 25 July 13
• 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
Relativistic Gauge Amplitudes
L. Dixon Scattering Amplitudes in QFT
= + …
EPS HEP11 Grenoble 25 July 14
• 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:
Universal Tree Amplitudes
L. Dixon Scattering Amplitudes in QFT
= + … =QCD N=4 SYM
EPS HEP11 Grenoble 25 July 15
Parke-Taylor formula (1986)
• When very simple QCD tree amplitudes were found, first in the 1980’s
… the simplicity was secretly due to N=4 SYM
Tree-Level Simplicity
L. Dixon Scattering Amplitudes in QFT
All N=4 SYM trees in closed formDrummond, Henn, 0808.2475
For example, for 3 (-) gluon helicities and [n-3] (+) gluon helicities, extract from:
5 (-) gluon helicities and [n-5] (+) gluon helicities:
L. Dixon Scattering Amplitudes in QFT 16EPS HEP11 Grenoble 25 July
These formulas can immediately be used for QCD (with external quarks too) LD, Henn, Plefka, Schuster, 1010.3991
EPS HEP11 Grenoble 25 July 17
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
L. Dixon Scattering Amplitudes in QFT
Where the fun really starts – textbook methodsquickly fail, even withvery powerful computers
EPS HEP11 Grenoble 25 July 18
Different theories differ at loop level
• QCD at one loop:
L. Dixon Scattering Amplitudes in QFT
rational part
well-known scalar one-loop integrals,master integrals, same for all amplitudes
coefficients are all rational functions – determine algebraicallyfrom products of trees using (generalized) unitarity
N=4 SYM
EPS HEP11 Grenoble 25 July 19
Gauge Hierarchyof Amplitude Simplicity
L. Dixon Scattering Amplitudes in QFT
EPS HEP11 Grenoble 25 July 20L. Dixon Scattering Amplitudes in QFT
N=4 super-Yang-Mills theory
• Interactions uniquely specified by gauge group, say SU(Nc), 1 coupling g
• Exactly scale-invariant (conformal) field theory: b(g) = 0 for all g
all states in adjoint representation, all linked by N=4 supersymmetry
EPS HEP11 Grenoble 25 July 21L. Dixon Scattering Amplitudes in QFT
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
Maldacena
EPS HEP11 Grenoble 25 July 22L. Dixon Scattering Amplitudes in QFT
AdS/CFT in one picture
EPS HEP11 Grenoble 25 July 23L. Dixon Scattering Amplitudes in QFT
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
Outstanding question: Can these structures be used to solve exactly forall planar N=4 SYM amplitudes?
Properties all related in some way to AdS/CFT.To be explored in more detail tomorrow in talk by N. Beisert
EPS HEP11 Grenoble 25 July 24L. Dixon Scattering Amplitudes in QFT
Dual Conformal Invariance
Conformal symmetry acting in momentum space,on dual or sector variables xi
First seen in N=4 SYM planar amplitudes in the loop integrals
Broadhurst (1993); Lipatov (1999); Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160
x5
x1
x2
x3
x4
kinvariant under inversion:
EPS HEP11 Grenoble 25 July 25L. Dixon Scattering Amplitudes in QFT
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/0610248Bern, Carrasco, Johansson, Kosower, 0705.1864
EPS HEP11 Grenoble 25 July 26L. Dixon Scattering Amplitudes in QFT
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 exact-exponentiated form (BDS ansatz) • For n=6, precisely 3 ratios:
+ 2 cyclic perm’s
1
2
34
5
6
EPS HEP11 Grenoble 25 July 27L. Dixon Scattering Amplitudes in QFT
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 a5-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:
EPS HEP11 Grenoble 25 July 28L. Dixon Scattering Amplitudes in QFT
Strategy works through at least 5 loopsBern, Carrasco, Johansson, Kosower, 0705.1864
EPS HEP11 Grenoble 25 July 29L. Dixon Scattering Amplitudes in QFT
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.
EPS HEP11 Grenoble 25 July 30L. Dixon Scattering Amplitudes in QFT
Radiation Zeroes• Mikaelian, Samuel, Sahdev (1979) computed
• Found “radiation zero” at
• Held independent of (W,g) helicities• Implies a connection between
– “color” (here electric charge Qd) – kinematics (cosq)
g
EPS HEP11 Grenoble 25 July 31L. Dixon Scattering Amplitudes in QFT
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
EPS HEP11 Grenoble 25 July 32L. Dixon Scattering Amplitudes in QFT
Color-Kinematic Duality at loop level
• Consider any 3 graphs connected by a Jacobi identity
• Color factors obey
Cs = Ct – Cu
• Duality requires
ns = nt – nu • Powerful constraint on structure of integrands
• Can always check afterwards using generalized unitarity
EPS HEP11 Grenoble 25 July 33L. Dixon Scattering Amplitudes in QFT
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
EPS HEP11 Grenoble 25 July 34L. Dixon Scattering Amplitudes in QFT
UV Finiteness of N=8 Supergravity?
• Quantum gravity is nonrenormalizable by power counting: the coupling, Newton’s constant, GN = 1/MPl
2 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
EPS HEP11 Grenoble 25 July 35L. Dixon Scattering Amplitudes in QFT
28 = 256 massless states, ~ expansion of (x+y)8
SUSY
24 = 16 states ~ expansion of (x+y)4
EPS HEP11 Grenoble 25 July 36L. Dixon Scattering Amplitudes in QFT
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,
EPS HEP11 Grenoble 25 July 37L. Dixon Scattering Amplitudes in QFT
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, 1004.0693Like KLT relations, gravity = (gauge theory)2,amplitudes respect factorization of Fock space,
EPS HEP11 Grenoble 25 July 38L. Dixon Scattering Amplitudes in QFT
AdS/CFT vs. KLT
AdS = CFT
gravity = gauge theoryweak strong
KLT
gravity = (gauge theory)2
weak weak
EPS HEP11 Grenoble 25 July 39L. Dixon Scattering Amplitudes in QFT
KLT CopyingN=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 productsof external and cut momenta; all state sums already performed
With a color-kinematics-duality respecting gauge amplitude,passing to the gravity amplitude is trivial!
EPS HEP11 Grenoble 25 July 40L. Dixon Scattering Amplitudes in QFT
“Color-kinematics duality” at 3 loopsBern, Carrasco, Johansson, 0805.3993, 1004.0476
N=4 SYMN=8 SUGRA[ ]2
[ ]2 [ ]2
[ ]2
[ ]2
Linear in
4 loop amplitudehas similar dualrepresentation!BCDJR, to appear
EPS HEP11 Grenoble 25 July 41L. Dixon Scattering Amplitudes in QFT
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)
EPS HEP11 Grenoble 25 July 42L. Dixon Scattering Amplitudes in QFT
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 43
From “science” to “technology”
N=4 SYM
QCD
EPS HEP11 Grenoble 25 July 44L. Dixon Scattering Amplitudes in QFT
Extra Slides
Many more QCD trees from N=4 SYM
Gluinos are adjoint, quarks are fundamental
Color not a problem because it’s easilymanipulated – 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 fermionlines: 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 45EPS HEP11 Grenoble 25 July
EPS HEP11 Grenoble 25 July 46
Infinities
• “The infinite can be appreciated only by the finite.” - J. Brodsky, Watermark
L. Dixon Scattering Amplitudes in QFT
• “The finite can be appreciated only after removing the infinite.” - Quantum Field Theory
Two types of infinities:• Ultraviolet renormalization• Infrared factorization
EPS HEP11 Grenoble 25 July 47L. Dixon Scattering Amplitudes in QFT
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
EPS HEP11 Grenoble 25 July 48L. Dixon Scattering Amplitudes in QFT
Infrared Factorization
In any theory, including QCD, S and J exponentiate. Surprise: for planar N=4 SYM, in some cases, full amplitude does too.
Loop amplitudes afflicted by IR (soft & collinear) divergences
soft function (color-dependent)
jet function (spin-dependent)
hard function (finite as e 0)
EPS HEP11 Grenoble 25 July 49L. Dixon Scattering Amplitudes in QFT
Large Nc Planar Simplification
• 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):
coefficient of