Simple Rules for Quantum Field Theory

Wichita State University Physics Seminar

Matthew Buican

Rutgers University

May 6, 2015

1

Overview

• QFT is everywhere

• The QFT zoo

• The Wilsonian picture

• Anomalies and dualities

• Further organizing principles and rules

• Particle physics and naturalness

2

QFT is Everywhere: Condensed Matter

• 2D Ising model

H = −1T

∑〈ij〉 σi · σj

• Physics organized according to long and short distances (rela-

tive to the correlation length).

3

QFT is Everywhere: Condensed Matter (cont...)

• At short distances, power-law correlators with well-defined di-

mensions.

〈σ(r)σ(0)〉 = r−2∆σ

• At Tc correlation length diverges and above holds everywhere.

Physics scale and (conformally) invariant.

• Described by known 2D CFT with dimensions of operators

matching known analytic solutions, e.g. ∆σ = 18. Can move off

criticality, e.g., by deforming by ε ∼ σ2 (this is relevant, ∆ε = 1,

and Z2-invariant).

4

QFT is Everywhere: Condensed Matter

• Duality: S → S′, T → T ′, σi → σ′i... The σ′i are non-local

“disorder” operators in the original theory... T ′ = −2log tanhT−1. In

particular, the T →∞ limit corresponds to the T → 0 limit (and

vice versa).

• Some results also in 3D. Wilson-Fisher, bootstrap, etc.

• More exotic connections (e.g., closely related cousins appear

to describe certain aspects of four and six-dimensional supercon-

formal theories!).

5

QFT is Everywhere: Particle Physics

• QFT explains this:

• Standard Model is astonishingly successful...

6

QFT is Everywhere: Particle Physics (cont...)

• “Simple” set of rules for diverse phenomena

L ∼ (Fµν)2 + ψ̄iDψi + yijHψiψj + · · ·+m|H|2 − λ|H|4 .

• Describes all non-gravitational forces... “Explains” masses offundamental particles...

• ...But it still has its mysteries: why is the Higgs light (looksfine tuned: relation to Ising...)? Where does the flavor hierar-chy come from? Why is the particle content what it is (someconstraints from anomalies)? Doesn’t look that minimal, etc(maybe different measures of minimality?).

• Hint of deeper principles: SUSY? Hint of new modules of QFT(maybe new CFTs)?

7

QFT is Everywhere: Gauge/Gravity duality

• Surprisingly, (non-gravitational) QFT seems to know about

gravity: AdS/CFT

• Deep questions in quantum gravity (black hole physics, etc.)

may have field theoretical explanations.

8

The QFT Zoo

• QFT is diverse and complicated: Lots of different kinds of

theories, infinite dimensional Hilbert space, many-body system,

etc...

• Particle collisions are messy... Understanding them requires

tedious calculations that often break down...

9

The QFT Zoo (cont...)

• Maybe there are interesting ideas from classical algebraic ge-ometry that help (Arkani-Hamed et. al.)

• Usually these kinds of tools are associated with SUSY (I’llreturn to this symmetry later)...

10

The Vacuum is Also Complicated

• In any case, the quantum vacuum is a complicated state (but

also contains the seed of some interesting order).

• Interesting entanglement properties of the vacuum:

11

The Vacuum is Also Complicated (cont...)

• Leads to very non-trivial insights into quantum gravity (Ryu,

Takayanagi):

12

The Vacuum is Also Complicated (cont...)

• ...And vice-versa:

A

B

C

A

B

C

A

B

C

=

�

S

A+B

+ S

B+C

� S

A+B+C

+ S

B

A

B

C

A

B

C

A

B

C

=

�

S

A+B

+ S

B+C

� S

A

+ S

C

13

Attempting to Tame the Quantum Zoo

• What are the organizing principles? How do we meaningfullycount different d.o.f.’s?

• We usually start with a simple set of rules. Sometimes, theycan be encoded in a Lagrangian:

L =1

g2Tr(Fµν)2 + ψ̄Dψ + · · · (1)

• Things quickly become complicated: couplings become strong,new d.o.f.’s emerge. Need computers, new ideas, etc.

• Sometimes, if we are lucky, a new and simple set of rulesemerges in nature, e.g.

L = f2|∂π|2 + · · · (2)

• But how do we understand this systematically and generally?

14

The Wilsonian Picture

• Start with a short-distance definition.

• Integrate out (massive) degrees of freedom / “coarse grain”

the system.

∫Λ<k<Λ+dΛ dΦe−S[Φ]

• New degrees of freedom typically emerge as in QCD (at short

distances quarks and gluons while at long distances pions, etc.).

• Deep and powerful way of thinking, but need more traction

(especially for strongly coupled theories)... Would be nice to get

some concrete and simple numbers to help out...

15

Anomalies...

• Anomalies are important objects in QFT. They are describedin terms of some anomaly coefficients (essentially coefficients ofsome contact terms of symmetry currents).

• They describe violation of a symmetry at coincident points incorrelation functions.

• In free theories, they are easy to compute: kqiqjqk =∑A qiqjqk.

• Global anomalies are handles.16

Anomalies... (cont...)

• ’t Hooft provided an ingenious argument using background

gauge fields that shows that whatever anomaly we have in the

UV we must have the same anomaly in the IR.

• Argument holds as long as the symmetry is unbroken; if spon-

taneously broken, have a Goldstone term that compensates the

difference.

• They are very nice because they are non-perturbatively invari-

ant under the RG flow.

17

...and Dualities

• And we can often use these anomalies to check various dualities

/ RG hypotheses.

• Particularly powerful when combined with more symmetry, es-

pecially supersymmetry (Seiberg).

• Seiberg’s “electric” theory

SU(Nc) SU(Nf)× SU(Nf) U(1)R U(1)B

Q Nc Nf × 1 1− NcNf

1

Q̃ N̄c 1× N̄f 1− NcNf

−1

18

...and Dualities (cont...)

• Seiberg’s “magnetic” theory

SU(Nf −Nc) SU(Nf)× SU(Nf) U(1)R U(1)Bq Nf −Nc N̄f × 1 Nc

NfNc

Nf−Ncq̃ N̄f − N̄c 1× N̄f

NcNf

− NcNf−Nc

M 1 Nf ×Nf 2− 2NcNf0

• More rigorous checks when have SUSY—have moduli spaces

(additional geometrical object) and also a natural symmetry of

the SUSY algebra itself.

• Extremely rigorous checks using superconformal index

19

Further Organizing Principles: the Extreme Ends of theWilsonian RG Flow

• There is an added bonus to the Wilsonian picture: if the QFTis UV complete, deep UV (and deep IR) should be scale invariant(which may in fact also be conformally invariant).

CFTIR

CFTUV

+ M O

• If not UV complete, may need string theory, see e.g. Klebanov-Strassler.

20

Further Organizing Principles: the Extreme Ends of the

Wilsonian RG Flow (cont...)

• Generally many new symmetries emerge in the UV and the IR(including a zoo of internal ones). Because different operatorsdecay, have different UV completions of same IR theory, etc.Will return to this...

• Can say a lot about CFTs. They have a nice “Taylor expan-sion” for products of operators (OPEs) which is associative:

f12k f34k

f14k

f23k

φk

φ1

φ2 φ3

φ4

�

k

= φk

φ1

φ2 φ3

φ4

�

k

21

Further Organizing Principles: the Extreme Ends of the

Wilsonian RG Flow (cont...)

• Use crossing symmetry to get non-trivial info on operator di-

mensions and various numbers that describe collective behavior

of CFT: τij, c,... Begin to put order in space of CFTs and then

also space of QFTs..

• Can even relate theories in different dimensions... For example,

2d/4d and 2d/6d correspondences.

22

Further Organizing Principles: Conformal Anomalies

• In 2D have

T = cE2

This guy obeys an interesting rule (also decreases in an interest-

ing way with energy) (Zamolodchikov).

cUV > cIR

Can be seen by studying 〈TT 〉 correlator in 2D.

23

Further Organizing Principles: Conformal Anomalies

• Conformal anomalies a and c: contact terms in three and twopoint functions of stress tensor. In curved background:

T = aE4 + cW2

•More conceptual reason for why above holds: conformal anomalymatching.

• Idea: add dilaton to compensate UV and IR anomalies (Ko-margodski and Schwimmer).

• Also can prove that a decreases in 4D

aUV > aIR

24

Further Organizing Principles: Conformal Anomalies(cont...)

• Can compute same information from entanglement entropy—e.g., entanglement through entangling points in 2D computes c!Then use SSA (Casini and Huerta) to prove that cUV > cIR.Can use EE to isolate a in 4d as well! Some proposals on howto use this to again prove aUV > aIR.

• Now, in certain large classes of theories, we can also com-pute this information just by knowing the list of certain types ofoperators (MB, Nishinaka, Papageorgakis), (Ardehali, Liu,Szepietowski), (Di Pietro and Komargodski), · · ·

• In any case, this a and c give reasonable measures of numberof degrees of freedom (matching the Wilsonian intuition).

25

Further Organizing Principles: Additional Anomalies andSUSY

• SUSY has more currents. In particular, it has a “supercurrent”for the supercharge (that takes bosons into fermions and vice-versa)

{Qα, Q̄α̇

}∼ Pαα̇

• Emergent symmetries in the IR can mix with it. We can definethis mixing abstractly for CFTs as being measured by the normof some special reference current and check that it is boundedby a quantity computed in the UV (at least in a large class oftheories). (MB)

• Emergent symmetry currents jµ = i(φ†σµφ−φσµφ†)+ · · · relatedto |φ|2 by SUSY...

26

Particle Physics and Naturalness

• Recall Higgs mass is very small. From CM perspective, this issurprising. In terms of Feynman Diagrams (the virtual top quarkcontribution), it is also surprising

• SUSY, roughly speaking, provides a canceling loop. But SUSYis broken. So cancelation is true as long as SUSY breakingtransmitted to top partner (the so-called “stop”) in a suppressedfashion so that it is still reasonably light... Still not (completely)ruled out by LHC!

27

SUSY, Anomalies, and Naturalness

• Stop mass is just

L ⊃ m|φ|2

• Can this be related to an emergent symmetry current by (ap-proximate SUSY)? Looks like it can. Also, if it is the specialreference current discussed above, then the operator is boundedby UV data (according to the rule we discussed above), whichmeans can cook up theories that “naturally” provide light stops.

• Still no experimental confirmation of this (or anything elsebeyond the standard model) at the LHC, though...

• Realization in condensed matter?28

The Future

• Understand better the landscape of QFTs. There are almost

certainly more rules to be discovered (I didn’t yet mention odd

dimensions, etc.)!

• Major advances in QFT. Unfortunately, doesn’t yet look like

nature makes much use of these advances...

• In any case, we are moving into a new era with new “experi-

ments”: OPE and conformal bootstrap operator collisions give

new data we need to make sense of (via a new “phenomenol-

ogy”).

29