light composite scalars from lattice gauge theory beyond qcd · light composite scalars ethan neil...
TRANSCRIPT
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
Motivation: lattice beyond QCD
3
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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.
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars12
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.)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
Simulation details:SU(3) with 8 ‘flavors’
13
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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.)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
List of spectroscopy targets
17
(Using QCD nomenclature for states, but remember everything is Nf=8.)
main focus
used to test lattice artifacts
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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)
⌘
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
• 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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
• Fit-range scan to check for systematic error from cutting on t:
20
Ae↵ ⌘ C(t)e+meff t
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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.
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars22
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
Spectroscopy results
23
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
Spectrum overview
24
(LSD preliminary)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
Spectrum overview
25
(LSD preliminary)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
Vector meson
26
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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:
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars28
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.)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars29
• 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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars30
• Another test of KSRF1 passes in a totally different theory, SU(4) with Nf=2 in fundamental and AS2 irreps
(Tel Aviv/Colorado preliminary)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars31
(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)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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:
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
Scalar 0++ meson
33
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars34
(LSD preliminary)
Scalar (σ, 0++) meson
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
• 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++
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars36
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!
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars38
• 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)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars39
• 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)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
What is the low-energy effective theory?
41
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
◆
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars44
M2⇡ = 2Bm(1 + ↵mm+
zmt0
+ �mmpt0)
F⇡ = F (1 + ↵Fm+zFt0
+ �Fmpt0)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars45
• Note that FP changes by nearly 100% over our mass range…chiPT shouldn’t work self-consistently!
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars47
↵(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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars48
OX
= CX
m1/(1+�
?) +DX
m+zxp
t01/(1+!
?)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars50
M2P =
Cm
F 2�yP
+zMpt0
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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.)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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
ᬩ
§�
13
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
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)
Ethan Neil (CU Boulder/RIKEN BNL)Light Composite Scalars
Backup
54