la˛ice qcd for hadron and nuclear physics: sca˛ering and ... · they respect conservation laws:...
TRANSCRIPT
![Page 1: La˛ice QCD for hadron and nuclear physics: sca˛ering and ... · They respect conservation laws: if isospin of the colliding particles is 3/2, resonance must have isospin 3/2 ( resonance)](https://reader033.vdocuments.mx/reader033/viewer/2022050212/5f5e2d21d7244c362f7583c7/html5/thumbnails/1.jpg)
La�ice QCD for hadron and nuclear physics: sca�ering and
many-particle systems
Sinéad M. Ryan
Trinity College Dublin
National Nuclear Physics Summer School, MIT July 2016
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Plan
A look at the spectrum of single-particle states
Resonances
Two particles in a box
Accessing resonance information from finite volume calculations
Toy models
Recent simulations
Spectrosopy at finite temperature
Summary
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The spectrum of light states
Using the technology we have discussed, the spectra of mesons and baryons can be
determined precisely.
On the menu:
0.5
1.0
1.5
2.0
2.5
exotics
isoscalar
isovector
YM glueball
negative parity positive parity
Is this everything?
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Resonances and scattering states
We have assumed that all particles in the spectrum are stable
Many (the majority) are not.
A resonance is a state that forms eg when colliding two particles and then decays
quickly to sca�ering states.
They respect conservation laws: if isospin of the colliding particles is 3/2, resonance
must have isospin 3/2 (∆ resonance)
Usually indicated by a sharp peak in a cross section as a function of c.o.m. energy of
the collision.
Can la�ice qcd distinguish resonances and sca�ering states?
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Mesons and Isospin Strange particles Resonances The quark model
Resonances in e+e−
→ hadrons
σee→hadrons(Eee)
Note the more or less sharpresonances on a comparably flat“continuum”, coming frome+e− → qq̄
(We will discuss this in more detail!)
They are (apart from the Z ) allrelated to qq̄-bound states.
Zoom into J/Ψ
Note: Here width around 3MeV completely determinedby detector (ΓJ/Ψ = 87keV)
F. Krauss IPPP
Introduction to particle physics Lecture 4
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Maiani-Testa no-go theorem
In
States
Out
States
Importance sampling Monte-carlo simulations rely on a path integral with positive
definite probability measure: Euclidean space
Maiani-Testa: sca�ering matrix (S-matrix) elements cannot be extracted from
infinite-volume Euclidean-space correlation functions (except at threshold)
Michael 1989 and Maiani, Testa (1990)
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Maiani-Testa (2)
Can understand this since:
Minkowski space: S-matrix elements complex functions above kinematic thresholds
Euclidean space: S-matrix elements are real for all kinematics - phase information lost
La�ice simulations with dynamical fermions admit strong decays eg for light-enough
up and down dynamical quarks ρ→ ππ
τ
τ
τΦ
Φ
Φ
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Maiani-Testa (3)
Can the la�ice get around the no-go theorem to extract the masses and widths of
such unstable particles?
Yes - use the finite volume.
Computations done in a periodic box
momenta quantised
discrete energy spectrum of stationary states→ single hadron, 2 hadron ...
sca�ering phase shi�s→ resonance masses, widths deduced from finite-box spectrum
B. DeWi�, PR 103, (1956) - sphere
M. Lüscher, NPB (1991) - cube
Two-particle states and resonances identified by examining the behaviour of energies in
finite volume
For elastic two-body resonances (Lüscher): M1M2 → R→ M1M2
−→ Volume dependence of energy spectrum
−→ Phase shi� in infinite volume
−→Mass and width of resonance - parameterising the phase shi� e.g. with Breit Wigner.
Note: This is a rapidly developing field. I will add some refs for recent work or see La�iceConf talks.
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Particles in a box
Spatial la�ice of extent L with periodic boundary conditions
Allowed momenta are quantized: p = 2πL (nx , ny , nz ) with ni ∈ {0, 1, 2, . . . L− 1}
Energy spectrum is a set of discrete levels, classified by p: Allowed energies of a
particle of mass m
E =
√
√
√
m2 +
�
2π
L
�
2
N2with N2 = n2
x + n2
y + n2
z
Can make states with zero total momentum from pairs of hadrons with momenta
p,−p.
“Density of states” increases with energy since there are more ways to make a
particular value of N2e.g. {3, 0, 0} and {2, 2, 1}→ N2 = 9
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Avoided level crossings
Consider a toy model with two states (a resonance and a two-particle decay mode) in
a box of side-length L
Write a mixing hamiltonian:
H =
�
m g
g 4πL
�
Now the energy eigenvalues of this hamiltonian are given by
E± =(m + 4π
L )±Ç
(m− 4πL )2 + 4g2
2
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Avoided level crossings
6 8 10 12 14 16 18 20
mL
0
0.5
1
1.5
2E/m
g/m=0.1
g/m=0.2
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Avoided level crossings
Ground-state smoothly changes from resonance to two-particle state
Need a large box. This example, levels cross at mL = 4π ≈ 12.6
Example: m = 1 GeV state, decaying to two massless pions - avoided level crossing is
at L = 2.5fm.
If the decay product pions have mπ = 300 MeV, this increases to L = 3.1fm
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Lüscher’s method
Relates the spectrum in a finite box to the sca�ering phase shi� (and so resonance
properties)
Lüscher’s formula
det
�
cot δ(E∗n ) + cotϕ(En, ~P, L)�
= 0
and cotϕ a known function (containing a generalised zeta function).
The idea dates from the quenched era. To use it in a dynamical simulation need
energy levels at extraordinary precision. This is why it has taken a while ....
An alternative approach called the potential method by HALQCD is also in use [PRL99
(2007), 022001] - less robust, certainly less widely used.
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Lüscher’s method
Z00 is a generalised Zeta function:
Zjs(1, q2) =∑
n∈Z 3
r jYjs(θ,ϕ)
(n2 − q2)s
[Commun.Math.Phys.105:153-188,1986.]
With the phase shi�, and for a well-defined resonance, can fit a Breit-Wigner to
extract the resonance width and mass.
δ(p) ≈ tan−1
�
4p2 + 4m2
π − m2
σ
mσΓσ
�
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Lüscher (3): considering ρ→ ππ
For non-interacting pions, the energy levels of a 2 pion system in a periodic box of
length L are
E = 2
Ç
m2
π + p2 p = 2π|~n|/L
and ~n has components ni ∈ N.
In the interacting case the energy levels are shi�ed
E = 2
Ç
m2
π + p2 p = (2π/L)q
where q is no longer constrained to orginate from a quantised momentum mode.
In the presence of the interaction, energy eigenvalues deviate from the
noninteracting case
These deviations contain the information on the underlying strong interaction -
yielding resonance information via Luscher formulism.
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Schrödinger equation
Exercise: find the phase shi� for a 1-d potential
V(x) = V0δ(x − a) + V0δ(x + a)
Now compute the spectrum in a finite box and use Lüscher’s method to compare the
two
L
En
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Test: O(4) Sigma model
8 9 10 11 12 13 14 15 16 17 18 19L
1.0
1.5
2.0
2.5
3.0
Ene
rgy
M.Peardon and P. Giudice
Spectrum of O(4) model in bro-
ken phase
Phase shi� inferred from
Lüscher’s method
0.3 0.4 0.5 0.6p
-2.0
-1.0
0.0
1.0
2.0δ(
p)
DataFit
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ρ→ ππ
Start with an “easy” system, I = 2ππ and test methods there. Interaction not strong
enough to form a resonance, but is weak and repulsive.
In the real world want to study rho→ ππ in isospin I = 1
This involves disconnected diagrams which is already a complication - although in
principle doable.
I = 1 case is now studied (distillation has helped a lot here)
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I=2 ππ scattering
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Resolve shi�s in masses away from non-interacting values
Orange boxes: possible ππ∗ sca�ering states
Dashed lines: non-interacting pion pairs
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I=2 ππ scattering - not physical pions
-70
-60
-50
-40
-30
-20
-10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
HooglandLosty
CohenDurusoyex
pt.
HadSpec - Dudek et al, 1011.6352 &
1203.6041
-50
-40
-30
-20
-10
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
-15
-10
-5
0
5 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Non-resonant sca�ering in S-wave and D-waves.
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towards physical pions in I=2 ππ.
à
à
à
à
àà
à
à
à
à
àà
ô
ô
ô
ôô
ô
æ
æ
æ
æ
æ
æ
ì
ì
ì
ì
ìì
ì
ì
ì
àà
ôô
ææ
ìì
àà
Hoogland et al '77Cohen et al '73Durosoy et al '73Losty et al '74NPLQCD '11
0.0 0.1 0.2 0.3 0.4 0.5-40
-30
-20
-10
0
k2 HGeV2L
∆Hd
egre
esL
NPLQCD, 1107.5023. Combines chiral perturbation theory and la�ice results at
mπ ∼ 396MeV to predict phase shi� at the physical pion mass.
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ππ in I=1
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The inelastic threshold
Lüscher’s method is based on elastic sca�ering.
Since mπ is small, most resonances are above this threshold
It will be crucial to ensure we have a comprehensive basis of operators that createmulti-hadron states.
Going beyond elastic. The method is generalised for: moving frames; non-identical
particles; multiple two-particle channels, particles with spin, by many authors.
det
�
t−1(E) + iρ(E)− M(E, L)�
= 0
relates the sca�ering t matrix to the discrete spectrum of states in finite volume
(coupled channels).
The precision and robustness of some numerical implementations is now very
impressive. [See e.g. talks at La�ice 2015 & 2016]
First coupled-channel resonance in a la�ice calculation
πK → ηK by D. Wilson et al 1406.4158 and 1507.02599
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Recent (and very recent) results
-30
0
30
60
90
120
150
180
1000 1200 1400 1600
0.7
0.8
0.9
1.0 1000 1200 1400 1600
-0.02
-0.01
0
0.01
0.02(a) (b) (c)
910 920 930 940 950 960
-30
0
30
60
90
120
150
180
1000 1200 1400 1600
0.7
0.8
0.9
1.0 1000 1200 1400 1600
πK → ηK by D. Wilson et al 1406.4158 and 1507.02599
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More coupled channels!
Is a0 a qq̄ state or dominated by a KK̄ molecular configuration?
-30
30
60
90
120
150
180
1000 1050 1100 1150 1200 1250 1300
0.250.500.75
1000 1050 1100 1150 1200 1250 1300
/ MeV
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1000 1050 1100 1150 1200 1250 1300 / MeV
HSC, Wilson et al 1602.05122
Phase shi�s, inelasticity and amplitudes (for mπ ∼ 400MeV)
Find an S-wave resonance in a two-coupled channel region - πη,KK̄ , includes limited
3-channel sca�ering - (πη,KK̄ , πη′). Resonance pole has large coupling to KK̄ .
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H-Dibaryon
A bound 6-quark state (udsuds) first proposed by Ja�e (1977) in MIT bag-model - at
81MeV below ΛΛ threshold.
La�ice calculations [NPLQCD, HALQCD] find H-dibaryon bound but at quark
masses larger than physical pion.
Extracting resonance parameters from ΛΛ
A linear chiral extrapolation does not discriminate between bound/unbound at the
physical pion mass. Does suggest a state in I=0, J=0, s=-2 (ΛΛ) that is just
bound/unbound.
More work to be done for good understanding
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X(3872) - a first look
Prelovsek & Leskovec 1307.5172
ground state: χc1(1P)
DD̄∗ sca�ering mx: pole just below thr.
Threshold ∼ mu,d and mc discretisation?
Padmanath, Lang, Prelovsek 1503.03257
3.4
3.55
3.7
3.85
4
4.15
4.3
4.45
En [
GeV
]
Lat. - OMM17
Lat. - OMM17 - O
-c c
Lat.(only O-c c
)
D(0) -D*(0)
J/Ψ(0) ω(0)
D(1) -D* (-1)
J/Ψ(1) ω(-1)
ηc(1) σ(-1)
X(3872) not found if cc̄ not in basis.
Also results from Lee et al 1411.1389
Within 1MeV of D0D̄0∗, 8MeV of D+D∗ thresholds: isospin breaking e�ects important?
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Z+c
An “exotic” hadron i.e. does not fit in the quark model picture.
There are a number of exploratory calculations on the la�ice.
Challenges:The Z+
c (and most of the XYZ states) lies above several thresholds and so decay to
several two-meson final states
requires a coupled-channel analysis for a rigorous treatment
on a la�ice the number of relevant coupled-channels is large for high energies.
State of the art in coupled-channel analysis:Lüscher: Kπ,Kη [HSC 2014,2015]
HALQCD: Zc [preliminary results]
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Z+c- First look on the lattice
Prelovsek, Lang, Leskovec, Mohler: 1405.7615
Lattice
D(2) D*(-2)
D*(1) D*(-1)
J/ψ(2) π(−2)ψ
3 π
D(1) D*(-1)ψ
1Dπ
D* D*η
c(1)ρ(−1)
ψ2S
π
D D*j/ψ(1) π(-1)η
c ρ
J/ψ π
Exp.
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
E[G
eV]
Lattice
a b c
without 4Q
13 expected 2-meson e’states found
(black)
no additional state below 4.2GeV
no Z+c candidate below 4.2GeV
Similar conclusion from Lee et al [1411.1389] and Chen et al [1403.1318]
Why no eigenstate for Zc? Is Z+c a coupled channel e�ect? What can other groups say?
Work needed!
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Many other states being investigated
Tetraquarks:
Double charm tetraquarks (JP = 1+, I = 0) by HALQCD [PLB712 (2012)]
a�ractive potential, no bound tetraquark state
Charm tetraquarks: variational method with DD∗,D∗D∗ and tetraquark operators
finds no candidate.
Y(4140)
Ozaki and Sasaki [1211.5512] - no resonant Y(4140) structure found
Padmanath, Lang, Prelovsek [1503.03257] considered operators:
cc̄, (c̄s)(̄sc), (c̄c)(̄ss), [c̄s̄][cs] in JP = 1+
. Expected 2-particle states found and χc1,X(3872)
not Y(4140).
.
.
.
See Prelovsek @ Charm2015 for more
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Exploratory studies of scattering/many-body systems
Characterised by:
New methods (developed/applied in last 5 years)
algorithmic: distillation allows access to all elements of propagators and construction of
sophisticated basis of operators.
theoretical: spin-identification; construction of multi-hadron operators etc
Generally high statistics, improved actions etc - results can be very precise.
Systematic errors not all controlled in exploratory studies: e.g. no continuum
extrapolation, relatively heavy pions ...
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A di�erent frontier: finite temperature and density QCD
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Spectroscopy at finite temperature
We have heard about finite temp QCD.
One avenue of investigations: States made from heavy quarks are expected to act as
a probe of dynamics of the QGP
There are interesting results coming from RHIC and CERN for the melting and
suppresion of such states.
Can la�ice say anything? It is a challenge!
(Without details here) the thermal correlator is
C(τ) ∼∫ ∞
0
dωρ(ω,T)K(ω, τ,T), (p = 0).
C(τ) sampled discretely but ρ has values for continuous ω
An ill-posed problem!
Maximum entropy methods (MEM) can be used but can be unstable and
model-dependent
New ideas needed!
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Effective masses at finite T
0 20 40 60 800
0.05
0.1
0.15
0.2
0.25
aτm
eff
3S
1 (vector)
0 20 40 60 80
τ/aτ
0.2
0.3
0.4
aτm
eff
3P
1 (axial-vector)
T=0.42Tc
T=1.05Tc
T=1.40Tc
T=2.09Tc
Anisotropic la�ices ξ = 6, Nf = 2, as = 0.15fm
Note P wave behaviour at T > Tc
Appears to rule out pure exponential decay at high T for P waves.
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Results: maximum entropy analysis
ηb MEM
ω [GeV]
0
200
400
600
ρ(ω
)/M
2T/T
c=0.42
T/Tc=1.05
T/Tc=1.05
T/Tc=1.20
T/Tc=1.20
T/Tc=1.40
9 10 11 12 13 140
200
400
600T/T
c=1.40
T/Tc=1.68
9 10 11 12 13 14
T/Tc=1.68
T/Tc=1.86
9 10 11 12 13 14 15
T/Tc=1.86
T/Tc=2.09
1S0(pseudoscalar)
ηb
1S survives to highest T examined.
excited states not discernable at 1.4 ® T /Tc ® 1.68⇒ melting or suppression?
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Finite density QCD
Would like to explore in finite μ with
la�ice qcd
μ 6= 0⇒ M not γ5 hermitian.
No longer have a (positive) probability
weight for Monte Carlo simulations -
sign problemThere are “work-arounds” but no
solutions (yet!)
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Executive Summary & Outlook
There is much that I did not cover in these lectures
I chose to focus on methods, new and old, for the “basic” building blocks of
spectroscopy
... and described their successful applications as well as some pitfalls
La�ice hadron/nuclear physics is moving rapidly at the moment as new techniques
emerge
Many challenges remain e.g. no general framework for extracting sca�ering
amplitudes involving more than two hadrons. Clever ideas needed!
There will be lots more experimental data in the near future and to keep pace will be
challenging
Thanks for listening and enjoy the rest of the school!