Harmony of Scattering Amplitudes: From to

BNL Physics Colloquium Nov 12, 2013 Zvi Bern, UCLA

1 Scattering amplitudes Scattering of elementary particles is fundamental to our ability to unravel microscopic laws of nature.

Smooth running of the at CERN emphasizes importance of scattering amplitudes. Vast subject. Here we give some examples of advances of past few years in understanding and calculating 4 month program at Simons Center scattering in . 2 Outline Will outline some new developments in understanding scattering amplitudes. Surprising harmony. 1. Scattering using Feynman diagrams. Obscures harmony. 2. Modern on-shell methods for scattering. 3. Sample applications: — LHC Physics — Reexamination of compatibility of quantum mechanics and . W q e º q g0 g g g g 3 Feynman Diagrams jets

p quark

p jets of Higgs boson event in CMS detector hadrons Note jets of particles

Quantum chromodynamics (QCD) describes the strong nuclear force binding quarks and into protons and neutrons.

QCD describes in great detail, scattering of protons at particle colliders. 4 Scattering Amplitudes Every graduate student in particle physics learns how to calculate scattering amplitudes via Feynman diagrams. gluon scattering Feynman propagator

Coupling constant

“color” factor In principle this is a complete solution for small coupling

Tree – Leading Order (LO) Loop – Next-To-Leading Order (NLO) Quantum correction Feynman diagrams give us a perturbative expansion in Planck’s constant ¹h 5 Scattering Amplitudes

In principle this is a complete solution for scattering with small coupling constants. Systematic!

In practice not so easy: • Proper way to calculate? Many scales, large logs, infrared issues, strong coupling, non-perturbative contributions, physical states, etc. In this talk we do not address these issues.

• A more practical problem is just dealing with the Feynman diagrams. In this talk we discuss this aspect. 6 State-of-the-Art Calculations

In 1948 Schwinger computed anomalous magnetic moment of the electron.

In 2010 typical 1-loop modern example:

g : gluon -- carrier of strong force q: quark -- matter W, Z : Weak boson carrier of weak interaction

60 years later at 1 loop only 2 or 3 legs more than

Schwinger if we use traditional Feynman diagrams. 7 Tree-level example: Five gluons

Force carriers in QCD are gluons. Similar to photons of QED except they self interact. Consider the five-gluon amplitude:

Used in calculation of pp 3 jets at CERN If you evaluate this following textbook Feynman rules you find… 8 Result of evaluation (actually only a small part of it):

Messy combination of momenta and gluon polarization vectors. 9 Reconsider Five-Gluon Tree

Xu, Zhang and Chang With a little Chinese magic, i.e. helicity states: and many others

Use a better organization of color charges:

Motivated by the color organization of open string amplitudes. 10 Mangano and Parke Example of loop difficulty

Consider a tensor integral:

Note: this is trivial on modern computer. Non-trivial for larger numbers of external particles.

Evaluate this integral via Passarino-Veltman reduction. Result is …

11 Result of performing the integration

Calculations explode for larger numbers of particles or loops. Clearly, there should be a better way! 12 W + n Jet NLO Calculations for the LHC

Suppose you said “I’m going not shave until I finish calculating”

W + 1 jet W + 2 jets W + 3 jets Still no Feynman diagram calculation

13 Why are Feynman diagrams difficult for high-loop or high-multiplicity processes?

• Vertices and propagators involve unphysical gauge-dependent off-shell states. An important origin of the complexity. Individual Feynman diagrams unphysical

Einstein’s relation between momentum and energy violated in the loops. Unphysical states! Not gauge invariant. • All steps should be in terms of gauge invariant on-shell physical states. On-shell formalism.

Don’t violate Einstein’s relation! ZB, Dixon, Dunbar, Kosower (1998) 14 Heisenberg Feynman diagram loops violate on shellness because they encode the uncertainty principle. Loops: You can create new particles even with insufficient energy as long as you destroy them quickly enough. Theorem: Off-shellness or energy conservation violation is essential for getting the correct answer because of uncertainty relation.

It looks like an on-shell formalism will fail to capture everything.

15 Sneaking Past Heisenberg

Favorite Theorem: Theorems in physics are bound to be misleading or have major loopholes.

What’s the loophole here?

• Want to reconstruct the complete amplitude using only on-shell physical information. • Keep particles on shell in intermediate steps of calculation, not in final results.

16 Modern Unitarity Method

on-shell ZB, Dixon, Dunbar and Kosower Two-particle cut: Systematic assembly of complete amplitudes from cuts for any number of Three-particle cut: particles or loops.

ZB, Dixon and Kosower; Generalized ZB, Morgan; Britto, Cachazo, Feng; unitarity as a Ossala,Pittau,Papadopoulos; practical tool. Ellis, Kunszt, Melnikov; Forde; Badger on-shell and many others

Reproduces Feynman diagrams except intermediate steps of calculation based on physical quantities not unphysical ones. 17 On-Shell Recursion for Tree Amplitudes

Britto, Cachazo, Feng and Witten (BCFW) Physical scattering amplitudes constructed from on-shell physical scattering amplitudes with fewer particles! on-shell amplitude

Works by making clever use of Cauchy’s residue theorem Same construction works in Brandhuber, Travaglini, Spence; Cachazo, Svrcek; Benincasa, Boucher-Veronneau, Cachazo; Arkani-Hamed and Kaplan, Hall

In , extended to loops. Arkani-Hamed et al. 18 G. Salam, ICHEP 2010

19 20 G. Salam, La Thuile 2012

21 2013: NLO W+5j [BlackHat+Sherpa: Bern et al] [unitarity] BlackHat: C++ implementation of on-shell methods for one-loop amplitudes

ZB, Dixon, Febres Cordero, Hoeche, Ita, Kosower, Maitre, Ozeren BlackHat is a C++ package for numerically computing one-loop matrix elements with 6 or more external particles. • Input is on-shell tree-level amplitudes. • Output is numerical on-shell one-loop amplitudes. On-shell methods used to achieve the speed and stability required for LHC phenomenology at NLO. Other (semi) on-shell packages: — Helac-NLO: Bevilacqua, Czakon, Ossola, Papadopoulos, Pittau, Worek — Rocket: Ellis, Giele, Kunszt, Melnikov, Zanderighi — SAMURAI: Mastrolia, Ossola, Reiter, Tramontano — MadLoop: Hirchi, Maltoni, Frixione, Frederix, Garzelli, Pittau — GoSam: Cullen, Greiner, Heinrich, Luisoni, Mastrolia, Ossola, Reiter, Tramontano

— NGluon: Badger, Biedermann, Uwer 22

ATLAS Comparison against NLO

ATLAS compared data against NLO theoretical predictions

Powerful experimental confirmation of NLO approach.

Validate in regions where no new physics is expected and look for discrepancies on tails of distributions where new physics may be hiding.

number of jets Theory predictions from BlackHat used to aid search for . W +1, 2, 3, 4 jets inclusive 23 W + 5 Jets in NLO QCD Hadronic energy spectrum W q e º q g0 g g g g A triumph for on-shell methods! Recently completed.

arXiv:1304.1253 LO/NLO ratio LO/NLO

Transverse hadronic energy • A new level for “state-of-the-art” LHC theory: First NLO QCD 2 6 process. Examples of Harmony in Scattering Amplitudes

25 Gravity vs Gauge Theory Consider the Einstein gravity Lagrangian curvature Flat-space metric field metric Infinite number of complicated interactions + … terrible mess

Compare to gauge-theory Lagrangian on which QCD is based

Only three and four point interactions Gravity seems so much more complicated than gauge theory.

Does not look harmonious! 26 Three Vertices Standard Feynman diagram approach. Three-gluon vertex:

Three-graviton vertex:

About 100 terms in three vertex Naïve conclusion: Gravity is a nasty mess. Where is the harmony? 27 Simplicity of Gravity Amplitudes

People were looking at gravity the wrong way. On-shell viewpoint much more powerful.

On-shell three vertices contains all information: gauge theory:

“square” of gravity: Yang-Mills vertex.

Using modern methods, any gravity scattering amplitude constructible solely from on-shell 3 vertex. Higher-point vertices irrelevant!

Harmony is revealed! 28 Gravity vs Gauge Theory Consider the Einstein gravity Lagrangian flat metric graviton field

metric Infinite number of irrelevant interactions!

+ … Simple relation to gauge theory

Compare to gauge-theory Lagrangian on which QCD is based.

Only three-point interactions

no Gravity seems so much more complicated than gauge theory.

Does not look harmonious! 29 Duality Between Color and Kinematics

momentum dependent coupling color factor kinematic factor constant

Color factors based on a Lie algebra:

Jacobi Identity

Use 1 = s/s = t/t = u/u to assign 4-point diagram to others. n c n c n c tree = g2 s s + t t + u u A4 s t u ³ ´ Color factors satisfy Jacobi identity: Numerator factors satisfy similar identity: Color and kinematics sing the same tune! 30 Duality Between Color and Kinematics Consider five-point tree amplitude: ZB, Carrasco, Johansson (BCJ) color factor kinematic numerator factor

Feynman propagators

a3a4b ba5c ca1a2 a3a4b ba2c ca1a5 a3a4b ba1c ca2a5 c1 f f f ; c2 f f f ; c3 f f f ´ ´ ´

c1 c2 + c3 = 0  n1 n2 + n3 = 0 ¡ ¡ Claim: We can always find a rearrangement so color and kinematics satisfy the same algebraic constraint equations.

Recent progress on unraveling mysterious relations. BCJ, Bjerrum-Bohr, Feng, Damgaard, Vanhove, ; Mafra, Stieberger, Schlotterer; Tye and Zhang; Feng, Huang, Jia; Chen, Du, Feng; Du, Feng, Fu; Naculich, Nastase, Schnitzer O’Connell and Montiero; Bjerrum-Bohr, Damgaard, O’Connell and Montiero 31 Higher-Point Gravity and Gauge Theory

ZB, Carrasco, Johansson color factor kinematic numerator factor QCD: Feynman propagators

Einstein Gravity: sum over diagrams with only 3 vertices Same kinematic numerators appear in gravity and gauge theory!

Gravity and QCD kinematic numerators sing same tune!

Cries out for a unified description of the sort given by .

32 Loop-Level Generalization ZB, Carrasco, Johansson (2010) Each loop represents power of ¹h sum is over kinematic diagrams numerator color factor

gauge theory

propagators 2 gravity

symmetry factor Loop-level conjecture is identical to tree-level one except for symmetry factors and loop integration. Double copy works if numerator satisfies duality. 33 BCJ Gravity integrands are free! Ideas generalize to loops: color factor

kinematic (k) (i) (j) numerator If you have a set of duality satisfying numerators. To get:

gauge theory gravity theory simply take color factor kinematic numerator

ck nk

Gravity loop integrands follow from gauge theory! 34 Applications of On-shell Methods to

35 Quantum Gravity

Often repeated statement: “Einstein’s theory of General Relativity is incompatible with quantum mechanics.”

To a large extent this is based on another often repeated statement: “All point-like quantum theories of gravity are ultraviolet divergent and non-renormalizable.”

Where do these statements come from and are they true?

36 Compatibility of Gravity with Quantum Mechanics

Dimensionful coupling

Gravity:

Gauge theory:

Extra powers of loop momenta in numerator means integrals are badly behaved in the UV. This is the origin (but not only issue) of the simplistic statement that quantum mechanics is not compatible with General Relativity.

Is this correct? 37 Test case: N = 8 Supergravity The best theories to look at are supersymmetric theories. Not viable theories of nature but an excellent test case. Supersymmetry is a symmetry between bosons (forces) and fermions (matter) We consider the N = 8 supergravity theory of Cremmer and Julia. 1 graviton + 254 other physical states The states are carefully arranged to have a higher degree of symmetry

Reasons to focus on N = 8 supergravity: • With more supersymmetry suspect better UV properties. • High symmetry implies technical simplicity.

In the late 70’s and early 80’s supergravity was seen as the primary means for unifying gravity with other forces. 38 Ferrara, Freedman, van Nieuwenhuizen Opinions from the 80’s

If certain patterns that emerge should persist in the higher orders of perturbation theory, then … N = 8 supergravity in four dimensions would have ultraviolet divergences starting at three loops. Green, Schwarz, Brink, (1982)

It is therefore very likely that all supergravity theories will diverge at three loops in four dimensions. … The final word on these issues may have to await further explicit calculations. Marcus, Sagnotti (1985)

The idea that all supergravity theories diverge at 3 loops has been widely accepted wisdom for over 30 years, with a only handful of contrarian voices.

39 Feynman Diagrams for Gravity

SUPPOSE WE WANT TO CHECK IF CONSENSUS OPINION IS TRUE

~1020 No surprise it has 3 loops never been TERMS calculated via − Calculations to settle Feynman diagrams. this seemed utterly hopeless! − Seemed destined for 4 loops ~1026 dustbin of undecidable TERMS questions.

31 5 loops ~10 More terms than TERMS atoms in your brain! Unitarity Method + Color/Kinematics Duality

Z B, Carrasco, Dixon, Johansson, Roiban For N = 8 supergravity.

3 loops ~10 TERMS

4 loops ~102 TERMS Much more manageable!

5 loops ~103 TERMS (Not yet done)

We now have the ability to settle the 35 year debate and determine the true UV behavior gravity theories. Complete Three-Loop Result ZB, Carrasco, Dixon, Johansson, Kosower, Roiban; hep-th/0702112 ZB, Carrasco, Dixon, Johansson, Roiban arXiv:0808.4112 [hep-th] Obtained via on-shell unitarity method:

Three-loop is not only ultraviolet finite it is “superfinite”—cancellations beyond those needed for finiteness in four dimensions.

Finite for D < 6

42 More Recent Opinion In 2009 Bossard, Howe and Stelle had a careful look at the question of how much supersymmetry can tame UV divergences.

In particular … suggest that maximal supergravity is likely to diverge at four loops in D = 5 and at five loops in D = 4 …

Bossard, Howe, Stelle (2009)

Bottles of wine were at stake!

We had the tools to collect the wine.

43 N = 8 Supergravity Four-Loop Calculation ZB, Carrasco, Dixon, Johansson, Roiban (2010)

50 distinct planar and non-planar diagrammatic topologies

“I’m not shaving until we finish the calculation” — John Joseph Carrasco John Joseph shaved!

UV finite for D = 4 and 5 actually finite for D < 11/2 Very very finite. 44 New Challenges Recent papers argue that trouble starts at 5 loops and by 7 loops we have valid potential UV divergence in D = 4 accounting for known symmetries (suggesting it diverges). Bossard, Howe, Stelle; Elvang, Freedman, Kiermaier; Green, Russo, Vanhove ; Green and Bjornsson ; Bossard , Hillmann and Nicolai; Ramond and Kallosh; Broedel and Dixon; Elvang and Kiermaier; Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger

Based on this a reasonable person would conclude that N = 8 supergravity almost certainly diverges at 7 loops in D = 4.

45 N = 8 Sugra 5 Loop Calculation ZB, Carrasco, Johannson, Roiban

~500 such diagrams with ~1000s terms each

Being reasonable and being right are not the same.

Place your bets: • At 5 loops in D = 24/5 does 5 loops N = 8 supergravity diverge? • At 7 loops in D = 4 does N = 8 supergravity diverge? Kelly Stelle: Zvi Bern: English wine California wine D8R4 counterterms “It will diverge” “It won’t diverge” 46 N = 8 Sugra 5 Loop Calculation ZB, Carrasco, Johannson, Roiban

~500 such diagrams with ~1000s terms each

Being reasonable and being right are not the same

Place your bets: • At 5 loops in D = 24/5 does 7 loops N = 8 supergravity diverge? • At 7 loops in D = 4 does N = 8 supergravity diverge? David Gross: Zvi Bern: California wine California wine 8 4 D R counterterms “It will diverge” “It won’t diverge” 47 Scene from “The Theory”

48 N = 4 supergravity

N = 4 supergravity at 3 loops ideal test case. Similar to N = 8 supergravity at 7 loops. Cremmer, Ferrara and Scherk Consensus had it that it would almost certainly diverge in UV at 3 loops.

Bossard, Howe, Stelle; Bossard, Howe, Stelle, Vanhove We completed computation last year. ZB, Davies, Dennen, Huang Three-loop UV divergences cancel completely!

• Duality between color and kinematic shown to be source of “magical cancellations” in a simpler D = 5 two-loop case. ZB, Davies, Dennen, Huang

• Four loop computation completed recently: It diverges but it is tied to quantum anomaly which is not present in N > 5 supergravity. ZB, Davies, Dennen, Smirnov, Smirnov

49 Outlook

• UV finiteness of N = 8 supergravity, given up for dead twice, is back in business, with new surprises. I hope to be able to collect at least one bottle of wine in the next year (even if English).

• I don’t know if we will be able to find a completely satisfactory description of Nature via supergravity.

• But at least people are looking again at this possibility and we uncovered some interesting things along the way. 50 Summary • Reformulated perturbative quantum field theory into a form allowing us to attack difficult fundamental problems. • Non-trivial state-of-the-art calculations for LHC physics. Good impact on the experiments.

• Remarkable harmony in scattering amplitudes. Two examples: — color kinematics. — gravity ~ (gauge theory)2 • Perturbative gravity and gauge theories not so different! Surprisingly tame in the UV.

With the new “on-shell” theoretical tools we can expect many more exciting developments in the coming years. 51 Further Reading

If you wish to read more see following non-technical descriptions.

Hermann Nicolai, PRL Physics Viewpoint, “Vanquishing Infinity” http://physics.aps.org/articles/v2/70

Z. Bern, L. Dixon, D. Kosower, May 2012 Scientific American, “Loops, Trees and the Search for New Physics”

Anthony Zee, Quantum Field Theory in a Nutshell, 2nd Edition is first textbook to contain modern formulation of scattering and commentary on new developments. 4 new chapters.

52 Some amusement

YouTube: Search “Big Bang DMV”, first hit. 20 seconds into the video.

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