scattering amplitudes in quantum field theory lance dixon (cern & slac) eps hep 2011 25 july...

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


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


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


Granularity vs. Plasticity


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


• 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


h

hh

h


• 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


g

gg

g

color universal


Through the clouds of gas and dust

• Obscure astrophysical black holes, but also make them detectable, their physics much richer



Through the clouds of soft interactions

• Obscure hard QCD amplitudes at their core, but also make the

physics much richer


F. Krauss


• 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


= + …


• 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


= + … =QCD N=4 SYM


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


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


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 methodsquickly fail, even withvery powerful computers


Different theories differ at loop level

• QCD at one loop:


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


Gauge Hierarchyof Amplitude Simplicity



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


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


AdS/CFT in one picture


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


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:


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


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


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:


Strategy works through at least 5 loopsBern, Carrasco, Johansson, Kosower, 0705.1864


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.


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


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


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


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


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


28 = 256 massless states, ~ expansion of (x+y)8

SUSY

24 = 16 states ~ expansion of (x+y)4


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,


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,


AdS/CFT vs. KLT

AdS = CFT

gravity = gauge theoryweak strong

KLT

gravity = (gauge theory)2

weak weak


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!


“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


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)


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!


From “science” to “technology”

N=4 SYM

QCD


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


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


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


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)


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

scattering amplitudes in quantum field theory lance dixon (cern & slac) eps hep 2011 25 july...

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