light composite scalars from lattice gauge theory beyond qcd · light composite scalars ethan neil...

Light Composite Scalars from Lattice Gauge Theory

Beyond QCDEthan T. Neil (Colorado/RIKEN BNL)

Fermilab HEP Theory Seminar - September 7, 2017

(arXiv:1601.04027, and upcoming)

Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars

Outline

• Motivation: lattice beyond QCD

• Simulation details

• Spectroscopy results (preliminary)

• Probing the low-energy effective field theory (even more preliminary)

2


Motivation: lattice beyond QCD

3


Composite BSM physics?• The Standard Model has many

theoretical puzzles, including Higgs naturalness and dark matter.

• Lots of theory proposals exist to resolve both of these problems by extending the SM. In fact, QCD has examples of mechanisms that can be useful (cosmically stable particles, massive scalars w/o naturalness.)

• The idea of composite BSM (from non-Abelian gauge dynamics) is to generalize these ideas from QCD and put them to use.

http://2.bp.blogspot.com/-xmz69lhESTU/VKlVuUPzV7I/AAAAAAAAAyU/ZMXIhIfo1II/s1600/higgs.jpg

DM

http://2.bp.blogspot.com/-xmz69lhESTU/VKlVuUPzV7I/AAAAAAAAAyU/ZMXIhIfo1II/s1600/higgs.jpg


Composite Higgs• The mass of the pi+ meson in QCD is subject to radiative corrections

from the photon:

5

• Quadratically divergent (3α/4π) Λ2 contribution to the mass! Fine-tuning problem if Λ >> 140 MeV. But we know this is an EFT: for Λ ~ 800 MeV, we see other resonances, and eventually quarks and gluons.

• If Higgs is composite with new resonances ~ TeV scale, then that naturalness problem is solved in the same way! (Potential little hierarchy between mh and Λ, but this could be due to symmetry e.g. h is a pseudo-Goldstone boson.)

on , ~ -

th- + . .

ft. .

.

IT IT IT IT

F

H c + 4- Fermi + en,f- ^

etc ( > > TED

µ , a. , ⇒ ¥.( few ten ,

smf- ✓ →so eev


Composite Higgs, in detail1

6

• Fundamental Higgs terms removed

L = LSM � Lh + LHC + Lint

(credit to C. Pica for this nice approach to describing the “anatomy” of composite Higgs)

• EW breaking, and SM mass terms. No fundamental scalar —> four-fermion operators:

LHC = �1

4F aµ⌫F

µ⌫,a +

NfX

i=1

i�µDµ i

1

⇤2UV

ffOB1

⇤2UV

fOF

(extended technicolor) (partial compositeness)

• New strong “hypercolor” gauge+fermion interactions:

or

1 more info: TASI lectures by R. Contino, arXiv:1005.4269

Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars7

on , ~ -

th- + . .

ft. .

.

IT IT IT IT

F

H c + 4- Fermi + en,f- ^

etc ( > > TED

µ , a. , ⇒ ¥.( few ten ,

smf- ✓ →so eev

with lattice, can work non-perturbatively here!

most pheno studies done here


Lattice and Composite BSM• Lattice gauge theory is numerical and non-perturbative - a

great tool for working with QCD (I don’t need to tell you that here!)

• Moving beyond QCD, we can “turn the dials” to study more general theories with applications to BSM models:

8

• (Multiple reps are interesting for partial compositeness, new lattice results recently - but that’s another talk.)

(Nc,Nf,R): SU(Nc) gauge theory, Nf fermions in irrep R


Philosophy of lattice BSM approach

• In the future: lattice can play a similar role as in QCD if a composite theory is discovered (precision), once we understand what the underlying field content is.

• Present: lattice acts more like a “parameter scan” of the large theory space. We can’t study every possible strongly-coupled model, but we can study several and look for regularities (or surprises.)

• Gives the most likely places to look, until we know more. (i.e., lattice studies can find new lampposts to look under.)

9


Gauge theory and the β-function

• Interesting phase structure as Nc and Nf are varied - easiest to understand in terms of β-function.

10

β(g)

g∗

UV IR

g

g(μ)

g∗

μ

�(↵) ⌘ @↵

@(logµ2)

= ��0↵2 � �1↵

3 � ...

�0 =1

4⇡

✓11

3Nc �

2

3Nf

◆

�1 =1

16⇡2

✓34

3N2

c �13

3Nc �

1

Nc

�Nf

◆

(R=fundamental)

• With Nf small (QCD), coupling grows in IR and confines

• Large enough Nf creates an infrared fixed point. Scale invariance in IR, no confinement or chiral symmetry breaking.


Theory space

11

CBZ

“CBZ” = “Conformal Banks-Zaks”: expansion where conformal fixed point coupling is deeply perturbative

Conformal transition is an active area of lattice research


C. Pica, plenary talk at Lattice 2016 - BSM presentations at that year’s conferenceTheory space, from the lattice

(note: bands are analytic estimates of the edge of the conformal transition, probably not reliable.)


Simulation details:SU(3) with 8 ‘flavors’

13


SU(3) with Nf=8• Why this theory?

• “QCD-like”: highly optimized code available for SU(3) gauge group (adding more duplicate fermions much easier)

• “Near-conformal”: lattice results so far hint this theory is very near the conformal transition

• No particular pheno model connection, although some attempts have been made to connect with “walking technicolor”1,2

14

1 S. Matsuzaki and K. Yamawaki, arXiv:1201.4722 2 LatKMI collaboration, arXiv:1302.6859

(I’ll touch on some other simulations too, but this is the main focus.)

Lattice Strong Dynamics CollaborationXiao-Yong Jin James Osborn

Rich Brower Claudio Rebbi Evan Weinberg

Anna Hasenfratz Ethan Neil

Oliver Witzel

Pavlos Vranas

Joe Kiskis

Tom Appelquist George Fleming Andy Gasbarro

15

Ethan Neil Enrico Rinaldi

David Schaich

Graham Kribs


Lattice setup and ensembles• nHYP staggered fermions

with fundamental-adjoint plaquette action

• High statistics for determination of 0++ scalar

• MπL~5 or larger for all ensembles used in main analysis

• Wilson flow scale t0 used for scale setting. (Note: strong dependence on am!)

16

(4-flavor ensembles for cross-check.)


List of spectroscopy targets

17

(Using QCD nomenclature for states, but remember everything is Nf=8.)

main focus

used to test lattice artifacts


Extracting the pion mass (a warm-up)

• Pion mass comes from connected two-point function:

18

Gktoj

~at )~ Ox or¥a

CIAASHINE##0¥Glop ) G Ctt ) C- ( ort )

=ZDH ) - C Ct )

C(t) ⇠NXX

i=0

Ai

⇣e�Mit + e�Mi(Nt�t)

⌘


• Fit to standard sum of exponentials to determine mass:

19

C(t) ⇠NXX

i=0

Ai

⇣e�Mit + e�Mi(Nt�t)

⌘A

e↵⌘

C(t)e

+m

efft


• Fit-range scan to check for systematic error from cutting on t:

20

Ae↵ ⌘ C(t)e+meff t


Extracting the sigma mass• For flavor-singlet states (like σ), propagator from source/sink to

itself does not vanish —> double-trace (disconnected) diagrams:

21

Gktoj

~at )~ Ox or¥a

CIAASHINE##0¥Glop ) G Ctt ) C- ( ort )

=ZDH ) - C Ct )

• Correlation between two different traces over entire lattice volume has significantly worse signal-to-noise

• We use stochastic estimation (6 U(1) wall sources/config) with dilution (suppresses off-diagonal noise correlations). Fermion operator only.


10

rameterize S(t) and �C(t) by

S(t) = a

d

cosh [M0++(Nt

/2� t)]

+ b

d

(�1)t cosh [M⌘sc(Nt

/2� t)]

�C(t) = a

c

cosh [Ma0(Nt

/2� t)]

+ b

c

(�1)t cosh [M⇡sc(Nt

/2� t)] .

The staggered parity partner of the 0++ meson is an594

isosinglet pseudoscalar, and thus is technically an ⌘-like595

state. However, it is a taste nonsinglet: its full quan-596

tum numbers are �4�5⇠4⇠5. It is therefore not a↵ected597

by the U(1) axial anomaly. For this reason, it is unin-598

teresting, and we will not discuss it further. TODO:599

Add cite: justify why it is not interesting, can’t600

(meaningfully? simply?) take continuum limit to601

learn about the 8 flavor ⌘

0/⇡0 like state. There’s602

a reason staggered eta-prime measurements go603

through the pain of using the right spin-taste604

state.605

TODO: Add discussion of vacuum subtraction:606

vacuum is noisier than signal of correlator. Easy607

way to avoid it: finite di↵erences. Show plot.608

In practice, even with the finite di↵erence, it is di�cult609

to extract the ground state from both S(t) and �C(t):610

both channels su↵er from strong excited-state contami-611

nation, especially because we use both point sources and612

point sinks. (Above we say stochastic sources for613

�C(t)...)614

D(t) is not a well-defined correlator in the sense that itdoes not uniquely couple to one set of quantum numbers.Numerically, we can still parameterize it as a linear com-bination of S(t) and �C(t). Ignoring the wrap-around ofperiodic boundary conditions TODO: which I shouldreally just get rid of everywhere except the ex-plicit example, we can write

D(t) =4

N

f

{S(t)� (�C(t))}

= a

d

e

�M0++ t � a

c

e

�Ma0 t

+ (�1)t�b

d

e

�M⌘sc t � b

c

e

�M⇡sc t . (15)

For M0++< M

a0 , we can extract the mass of the 0++615

from D(t) alone for asymptotically large t [8]. This corre-616

lator appears to have smaller excited-state contamination617

than S(t), leading to longer plateaus in e↵ective-mass618

plots. TODO: Add a plot, discuss features. Re-619

mark that D(t) curves down—negative amplitude620

states manifest in parameterization of the corre-621

lator.622

• Fit to just D(t) is complicated: negative-623

amplitude a0 state is di�cult to resolve. Ad-624

dressed with Bayesian constraints (Ethan’s625

method). Need discussion from Ethan on626

the tests he did.627

• End of the day: quote multiple errors. Sta-628

tistical error from fit to just D, 2D � C.629

FIG. 7. Small test to check the di↵erences between the 2D(t)correlator and the 2D(t) � C(t) correlator masses. Switchto range plot...?

FIG. 8. The spectrum of the 8-flavor theory.

Systematic error from varying t

min

, t

max

.630

Can also combine all in quadrature, naively.631

We’ll need multiple plots to highlight this.632

In Fig. 7 we report M⇡

and M

⇢

along with the scalar633

mass obtained from the non-linear fit of the 2D(t) cor-634

relator and the 2D(t)�C(t) correlator. TODO: Prob-635

ably will cite the following at some point in this636

section... [74]637

V. SPECTRUM RESULTS AND DISCUSSION638

At some point in this section we’ll discuss the639

KSRF relations [77, 78]...640

Compare connected spectrum with Ref. [79]...641

Compare spectrum with Ref. [? ]...642

Ref. [43] argues that our growing M

V

/M

P

is due643

to finite-volume e↵ects...644

Linear vs. power-law chiral extrapolations as645

in arXiv:1405.4752??? These may not be worth-646

• Joint fit to S=2D-C and D, both of which contain M0++ but have different systematics:

S(t)

D(t)

10

rameterize S(t) and �C(t) by

S(t) = a

d

cosh [M0++(Nt

/2� t)]

+ b

d

(�1)t cosh [M⌘sc(Nt

/2� t)]

�C(t) = a

c

cosh [Ma0(Nt

/2� t)]

+ b

c

(�1)t cosh [M⇡sc(Nt

/2� t)] .

The staggered parity partner of the 0++ meson is an594

isosinglet pseudoscalar, and thus is technically an ⌘-like595

state. However, it is a taste nonsinglet: its full quan-596

tum numbers are �4�5⇠4⇠5. It is therefore not a↵ected597

by the U(1) axial anomaly. For this reason, it is unin-598

teresting, and we will not discuss it further. TODO:599

Add cite: justify why it is not interesting, can’t600

(meaningfully? simply?) take continuum limit to601

learn about the 8 flavor ⌘

0/⇡0 like state. There’s602

a reason staggered eta-prime measurements go603

through the pain of using the right spin-taste604

state.605

TODO: Add discussion of vacuum subtraction:606

vacuum is noisier than signal of correlator. Easy607

way to avoid it: finite di↵erences. Show plot.608

In practice, even with the finite di↵erence, it is di�cult609

to extract the ground state from both S(t) and �C(t):610

both channels su↵er from strong excited-state contami-611

nation, especially because we use both point sources and612

point sinks. (Above we say stochastic sources for613

�C(t)...)614

D(t) is not a well-defined correlator in the sense that itdoes not uniquely couple to one set of quantum numbers.Numerically, we can still parameterize it as a linear com-bination of S(t) and �C(t). Ignoring the wrap-around ofperiodic boundary conditions TODO: which I shouldreally just get rid of everywhere except the ex-plicit example, we can write

D(t) =4

N

f

{S(t)� (�C(t))}

= a

d

e

�M0++ t � a

c

e

�Ma0 t

+ (�1)t�b

d

e

�M⌘sc t � b

c

e

�M⇡sc t . (15)

For M0++< M

a0 , we can extract the mass of the 0++615

from D(t) alone for asymptotically large t [8]. This corre-616

lator appears to have smaller excited-state contamination617

than S(t), leading to longer plateaus in e↵ective-mass618

plots. TODO: Add a plot, discuss features. Re-619

mark that D(t) curves down—negative amplitude620

states manifest in parameterization of the corre-621

lator.622

• Fit to just D(t) is complicated: negative-623

amplitude a0 state is di�cult to resolve. Ad-624

dressed with Bayesian constraints (Ethan’s625

method). Need discussion from Ethan on626

the tests he did.627

• End of the day: quote multiple errors. Sta-628

tistical error from fit to just D, 2D � C.629

FIG. 7. Small test to check the di↵erences between the 2D(t)correlator and the 2D(t) � C(t) correlator masses. Switchto range plot...?

FIG. 8. The spectrum of the 8-flavor theory.

Systematic error from varying t

min

, t

max

.630

Can also combine all in quadrature, naively.631

We’ll need multiple plots to highlight this.632

In Fig. 7 we report M⇡

and M

⇢

along with the scalar633

mass obtained from the non-linear fit of the 2D(t) cor-634

relator and the 2D(t)�C(t) correlator. TODO: Prob-635

ably will cite the following at some point in this636

section... [74]637

V. SPECTRUM RESULTS AND DISCUSSION638

At some point in this section we’ll discuss the639

KSRF relations [77, 78]...640

Compare connected spectrum with Ref. [79]...641

Compare spectrum with Ref. [? ]...642

Ref. [43] argues that our growing M

V

/M

P

is due643

to finite-volume e↵ects...644

Linear vs. power-law chiral extrapolations as645

in arXiv:1405.4752??? These may not be worth-646


Spectroscopy results

23


Spectrum overview

24

(LSD preliminary)


Spectrum overview

25

(LSD preliminary)


Vector meson

26


Vector Meson Saturation• Saturation of vector channel by a single resonance

(ρ) gives a phenomenological model of low-energy quantities, based on rho mass and width.

• VMS works well in QCD (~10%) for some things, e.g. KSRF1 relations:

• What happens in other strongly-coupled models?

27

1K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett.16, 255 (1966); Riazuddin and Fayyazuddin, Phys. Rev. 147,1071 (1966).

electroweak decay

HC-strong decayM⇢

F⇡⇠ 1p

Nc

large-Nc:


LSD Collaboration, arXiv:1601.04027

• Test of one KSRF relation (Fρ/Fπ), nice agreement, little mass dependence

• Other relation gives gρππ from Mρ/Fπ (convention: Fπ~93 MeV in QCD.)


• Another test of VMS: pion vector form factor. Works very well for light “pions” in SU(2) with Nf=2 fundamental (above).

• More directly, the vector meson should give a resonant contribution to the timelike pion form factor. Harder calculation, but in progress.

A. Hietanen, R. Lewis, C. Pica and F. Sannino, arXiv:1308.4130


• Another test of KSRF1 passes in a totally different theory, SU(4) with Nf=2 in fundamental and AS2 irreps

(Tel Aviv/Colorado preliminary)


(Tel Aviv/Colorado preliminary)(Tel Aviv/Colorado preliminary)

• Results for rho-pi coupling different, but seem to roughly match large-N expectations (~1/N0.5 or 1/N)


Vectors in composite Higgs

• Bounds from direct searches (blue), indirect bounds on ξ (dashed lines.) With likely gρ from lattice, LHC direct searches may not have enough reach, but future colliders can probe directly.

• Large coupling gives width Γ/M over 10%. Focused searches for large-width objects might help.

32

(Thamm, Torre and Wulzer, 1502.01701)

large Nc

M⇢ = g⇢fnote different convention:


Scalar 0++ meson

33


(LSD preliminary)

Scalar (σ, 0++) meson


• σ degenerate with π, much lighter than ρ! Seen by two groups in this theory (LSD + LatKMI), over a wide range in mass…

35

(LSD preliminary)

Mass dependence of 0++


σπ

LatHC Collaboration, talk at “Lattice for BSM Physics 2016”

• Similar outcome, different theory - SU(3), 2 fermions in AS2 representation.

C.)NONAKA)(SCALAR)Collabora/on)) SCGT2015)

κ&Dependence&of&EffecAve&Masses�

(m�a)2

with)disconnected)diagram�

)connected)diagram�

tetra�

ρ�2m�a

•  Molecule)))))R))Small)effect)of)disconnected))))))))diagram))))R πRπ)scaXering)state?)

•  Tetra)(plateau)?)))))R)Disconnected)diagram)is)))))))important.)))))R))small)overlap)to))))))))ground)state)of)molecule)

arXiv1412.3909[hep[lat]�

Very different from QCD, where mass dependence of σ is strong!(slide from C. Nonaka)• QCD looks very different at heavy mass!


• Our own test of Nf dependence (and of systematics in our σ extraction procedure): calculate at Nf=4 as well

Nf=8 Nf=4

(matched on t0)


• Our own test of Nf dependence (and of systematics in our σ extraction procedure): calculate at Nf=4 as well

Nf=8Nf=4

(matched on mρ)


An aside: the eta-prime

• Pseudoscalar counterpart of sigma is the flavor-singlet η’

• Expected to be heavy due to chiral anomaly; should get even heavier proportional to Nf1

40Preliminary

Compare using a common

reference scalePseudoscalar flavor-singlet becomes heavier with

increasing number of flavors

~2x

~3x

LatKMI preliminary, from Lattice 2017 (E. Rinaldi) - matches expected scaling!

1 R. Kaiser and G. Leutwyler, arXiv:hep-ph/0007101


What is the low-energy effective theory?

41


In search of an EFT

• Low-energy EFT is used extensively in lattice QCD: chiral perturbation theory (chPT) -> modeling quark-mass dependence. (Lattice chPT -> include lattice artifacts ~ an too.)

• We need to know the appropriate low-energy EFT to model our Nf=8 data and answer questions about the chiral limit, etc. Suspicious of chPT due to light σ.

• If the right EFT here is not chPT, then we have something novel which might be very useful for model-building!

42


Model 1: chiral perturbation theory

• Tried and tested in QCD and beyond, particularly with lattice results at unphysical masses.

• Contains no light, narrow 0++ state! QCD σ is a two-pion scattering state, very broad.

43

L =F 2

4h(DµU)†DµUi

�F 2

4h�†U + U†�i Nonlinear:

Symmetry:

U ! LUR†

SU(N)L ⇥ SU(N)R ! SU(N)V

U = exp

✓i

f⇡⇡aT a

◆


M2⇡ = 2Bm(1 + ↵mm+

zmt0

+ �mmpt0)

F⇡ = F (1 + ↵Fm+zFt0

+ �Fmpt0)


• Note that FP changes by nearly 100% over our mass range…chiPT shouldn’t work self-consistently!


Model 2: mass-deformed CFT

• We see a non-conformal system, but we are explicitly breaking scale symmetry at finite mass. Maybe Nf=8 is IR conformal?

• If so, can see remnant of CFT behavior, restored as mass sent to zero. Seen in other theories on lattice, such as SU(3) Nf=12.

• Test hypothesis by fitting. (This isn’t really a complete EFT, more of a framework, but we can treat it similarly.)

46


↵(µ)

µ

↵?

⇤ = a�1M

m(µ)

µ

⇤ = a�1

m(⇤)

M

M

m(µ) = m(⇤)

✓⇤

µ

◆�?

M ⇠ m1/(1+�?)

• “Seed mass” at high scale runs into the IR

• Fermions screen out, theory confines at M

• Confinement scale depends on m, governed by anomalous mass dimension at fixed point

Mass deformation of IR CFT

All hadron masses scale ~ M


OX

= CX

m1/(1+�

?) +DX

m+zxp

t01/(1+!

?)


Model 3: dilaton EFT• Explicit attempt to expand chiral EFT by adding a

light scalar DOF, identified as a “dilaton” (i.e. Goldstone of scale symmetry breaking.)

• Several recent papers in this direction1,2,3

• Focus on framework of Appelquist, Ingoldby, Piai3.

49

L =f2⇡

4

✓�

fd

◆2

h@µU†@µUi

+m2

⇡f2⇡

4

✓�

fd

◆y

hU + U†i

• Predict relations between F and M for π and σ. Fσ needed for many of them, so we test one specific equation.

1Matsuzaki and Yamawaki, arXiv:1311.3784 2Golterman and Shamir, arXiv:1603.04575 3Appelquist, Ingoldby and Piai, arXiv:1702.04410


M2P =

Cm

F 2�yP

+zMpt0


Model 4*: linear sigma model• Work in progress (A. Gasbarro). σ and flavored scalars

transform under flavor symmetry like η’ and π’s.

51

L =1

2h@µM†@µMi+ m2

�

4hM†Mi

�m2� �m2

a

8f2hM†Mi2 � m2

a

8f2h(M†M)2i2

Symmetry:SU(N)L ⇥ SU(N)R ! SU(N)V

M ! LMR†

Linear scalars:M ⌘ US S =

�pNf

+ aaT a

M2� � 3M2

⇡

• However, simplest version of this EFT predicts at tree level:

• Can’t describe our data! Work in progress on generalization (more than one way to organize power counting with scalars included.)


Other observables• Looking at more observables should give

better discrimination between models

• Pi-pi scattering (preliminary, right) in particular should be sensitive to σ-π interactions. Lattice calc + EFT work are in progress.

• Dilaton models give concrete predictions for Fd, the “dilaton decay constant”. However, extraction requires full dilatation current D, which includes gluonic operator - need gluon extraction of 0++ state as well.

52

More Information from Pion Scattering

• Scattering length agrees well with LO XPT when plotted against physical (computed) values of Mπ/Fπ

• Plotted against bare quark mass, very poor agreement with LO XPT

• Scattering data in process of being analyzed with various EFT options

Andrew Gasbarro Walking Gauge Theory EFT

Ref [1]

Ref [1]M

π A

Mπ

A /m

q

� �� 22

2

2

2161

161

FBm

FMAM q

LO SS S

SS

� ¸

¹

·¨©

§�

13


Summary• Composite BSM is interesting, but strong

coupling is hard. Lattice can help (with UV completion.)

• Vector-meson saturation seems to work well beyond QCD, and gρππ~6 (in my convention) is fairly insensitive to fermion mass/number

• Hints of a light 0++ scalar - what is the EFT for this + “pions”? Adding more mass points, observables to discriminate. Dilaton EFT promising at first glance.

• Suggestions for EFT tests are welcome…as are efforts to connect to realistic composite Higgs scenarios —> pheno!

53

(LSD preliminary)


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54

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