QCD Meson electromagnetic form factors from Lattice - Christine Davies University of Glasgow HPQCD collaboration
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Meson electromagnetic
form factors from Lattice
QCD
Christine Davies
University of Glasgow
HPQCD collaboration
Mainz
January 2020
PH DALI Run=16449 Evt=4055
Understanding the strong force is key to testing the SM
- binds quarks into hadrons that we see in experiment
Connecting observed hadron
properties to those of quarks
requires full nonperturbative
treatment of Quantum
Chromodynamics - lattice QCD
ATLAS@
Can calculate hadron LHC
masses and rates for πe−
simple weak/em
IP +
decays and −
Bs K
transitions to test + −
Ds
Standard Model π+
K
1cm|
Electromagnetic transitions provide good tests/probes of
hadron structure free of CKM issues. Lattice QCD
calculations can be compared directly to experiment for
I
simple cases e.g. (to be discussed here)
! b]
1) Vector meson annihilation to a photon
I (lepton pair)
a)
4⇡ 2 2 fV2
(V ! e+ e ) = ↵ Q
3 MV {Se
decay constant e e
2) Scattering of a meson from a photon as
a function of Q2
formn q
d 2 2 factor
/ |F (q )|
dq 2
Lattice QCD can also provide decay constants for mesons that cannot be accessed experimentally (e.g. neutral pseudo scalars such as ηc) and for mesons that do not exist experimentally (e.g. ηs) The same is true for form factors hence Lattice QCD can provide additional “data” with which non-lattice theorists can test model relationships between form factors and decay constants. Meanwhile, a key aim of lattice QCD calculations is to map out the space-like form factor Q2F(Q2) from low to high Q2 ahead of experiment.
Driving improved theory for form factors:
Jefferson Lab 12 GeV upgrade - new expts E12-06-101 +
E12-09-11 to determine π /K form factors to 6 GeV2
erson Lab 12 GeV Upgrade
New Hall Experimental Capabilitie
m Upgrade arc magnets
e
and supplies
Add 5 cryomodules
Hall A Hall B
Existing HRS New CLAS12, large
CHL upgrade
acceptance spectrometer
20 cryomodules
magnetic focusing
spectrometers + Big à Good hadron PID
new
Add arc
Bite + new, large20 cryomodules à Simultaneous
acceptance Super Big
Add 5 cryomodules measurement of broad
ic
Bite spectrometer phase space
Enhanced capabilities
in existing Halls
being built Hall C, JLAB
Hall
2 C
HMS + new SHMS
magnetic focusing
spectrometers
à Precision cross
sections, LT Mo
separations à
Fπ(Q )status
2
Current experimental Results
for ⇡ - and Models
for K direct determn. to 0.1 GeV2
Maris and Tandy,
055204 (2000)
à relativistic trea
variety
quarks of
(Bethe-Sa
(non-
+ Dyson-Schwing
lattice)
theory and R
Nesterenko
predictns
Phys. Lett. B115,
à Green’s functio
asymptotic
used to extract fo
form
Lattice and
Brodsky QCD de T
direct pion Phys.Rev.D77,
should be 0
determin electro- à Anti-de Sitter/C
able to do
Theory approach
ation A.P. Bakulev et al, Phys. Rev. D70,
production better …
033014 (2004)
Lattice QCD: fields defined on 4-d
discrete space-(Euclidean) time.
Lagrangian parameters: ↵s , mq a
1) Generate sets of gluon fields for
Monte Carlo integrn of Path Integral
(inc effect of u, d, s, (c) sea quarks)
2) Calculate valence quark propagators
and combine for “hadron correlators” .
Fit for hadron masses and amplitudes
• Determine a to convert results in
lattice units to physical units. Fix mq
from hadron mass
*numerically extremely challenging*
• cost increases as a ! 0, mu/d ! phys
a and with statistics, volume.
Quark formalisms
Many ways to discretise Dirac Lagrangian onto lattice.
All should give same answers.
n
Issues are: Discretisation errors at power a
Numerical speed of matrix inversion
Chiral symmetry
Fermion doubling
We use Highly Improved Staggered Quarks (HISQ) for
u, d, s and c. Also (with extrapolation) for b.
Disc. errors ↵s2 a2 , a4. Numerically fast. Chiral symm.
Some complications from doublers (‘tastes’) which
appear as discretisation effects.
HPQCD: hep-lat/0610092
Example parameters for ‘2nd generation’ calculations now
being done with staggered quarks.
“2nd generation”
lattices inc. c
mu = md MILC HISQ, 2+1+1
0.14 quarks in sea
= ml
HISQ = Highly
0.12
improved
mass of u,d
staggered quarks -
quarks 0.1
very accurate
/ GeV2
discretisation
0.08
E.Follana, et al,
3 volumes HPQCD, hep-lat/
2
0.06 0610092.
m
real mu,d ⇡ ms/10
0.04
world
m⇡ 0 = 0.02 mu,d ⇡ ms/27
135 MeV
0 Volume:
0 0.005 0.01 0.015 0.02 0.025 0.03
2 2
m⇡ L > 3
a / fm
Hadron correlation functions (‘2point functions’) give
masses and decay constants.
X T
† mn T large m0 T
h0|H (T )H(0)|0i = An e ! A0 e
n masses of all
H QCD H hadrons with
quantum
|h0|H|ni|2 fn2 mn numbers of H
An = =
2mn 2
decay constant parameterises amplitude to annihilate - a
property of the meson calculable in QCD.
1% accurate experimental info.
for f and m for many mesons!
Need accurate determination
from lattice QCD to match0.1830 Light meson decay constants
0.1825 R. Dowdall et al, HPQCD, 1303.1670.
Use w0 to fix lattice
fhs (GeV)
0.1820
0.1815 spacing, with value of w0
0.1810 fixed from fπ.
0.1805 PCAC reln for HISQ means
0.164 no renormln factors needed.
0.162 Very small discretisation
fK (GeV)
0.160 effects.
0.158 ηs meson defined within
0.156 lattice QCD as ss
0.150 meson that cannot mix.
0.145 Can calculate its decay
fp (GeV)
0.140 constant very accurately
0.135 with full chiral analysis.
0.130
M⌘s = 688.5(2.2) MeV
0.05 0.10 0.15 0.20
m2p /(2m2K m2p ) = ml /ms f⌘s = 181.14(55) MeV0.55 PDG Heavy meson decay
constants
fJ/ [GeV]
0.50 D. Hatton et al, HPQCD,
in prep.
0.45 Improved analysis
of cc case nearly
PRELIMINARY
0.40
0.0 0.2 0.4 0.6 0.8 complete (inc.
(amc )2 impact of c electric
0.500
HPQCD [1008.4018]
charge). Good
0.475
agreement with
expt. for J/ψ .
f⌘c [GeV]
0.450
0.425 Allows accurate
determination of
0.400
PRELIMINARY ηc decay constant.
0.0 0.2 0.4 0.6 0.8
(amc )2Summary of meson decay constants
Parameterises hadronic information needed
for annihilation rate to W or photon:
/ f2
0.7 Experiment : weak decays
: em decays
DECAY CONSTANT [GeV]
0.6 Lattice QCD : predictions
: postdictions
0.5
0.4
0.3
0.2
0.1
π K B ∗ B D Bs∗ Bs φ Ds Ds∗ ψ ′ ηc ψ Bc∗ Bc Υ′ ηb Υ
0
HPQCD 1208.2855, 1312,5264, 1408.5768, 1503.05762, 1703.05552, FNAL/MILC 1712.09262a)
Form factors give more info. on structure. Electromagnetic
form factors provide test for those for weak decays {SeV/c)
Simplest example is that of the meson
e
electromagnetic
e
form factor at space-like q2: rt
d 2 2
/ |F (q )| n q
dq 2 !
small Q2 (=-q2): direct ⇡e scattering I
a
(NA7, 1986, up to 0.26 GeV2); determine
a) bl
rms radius of electric charge distn. Many
lattice QCD tests of this. e.g. HPQCD, 1511.07382 e e
Fig. 1. Data on the squared modulus of F~ for Itl < l
large Qtion
2 : use electroproduction on a proton.
[1]; (b) direct 7re scattering [2-4]; (c) inverse electr
n
Done up to Q = 2.5 GeV . Key
2 2
Theexpt for JLAB
horizontal bar (b) indicates t
upgrade: extend to 6 GeV2. Pert. QCD
prediction at very high Q2. Lattice QCD? a)Perturbative QCD at very high Q2: high mom. photon
accompanied by high mom. gluon. Hard scattering factorises
from ‘distribution amplitude’ for qq in meson.
Lepage, Brodsky 1979
!
x y
⇡ + ··· ⇤
⇡
1 x 1 y
Asymptotic 2
8⇡↵s fM
2
prediction: Q !1 FM =
for meson M Q2
form factor decay constant
For lower 1 2
2
8⇡↵s (Q/2)fM X
Q2: 2
FM (Q ) = 1 + a M
2 n (Q/2)
Q n=2
Higher twist
even n for coeffs of Gegenbauer
corrns: addnl
powers of m2/Q2 ‘symmetric’ meson poly. , run to 0 at high Q2rameter [23], and has a correlated 0.5% uncertainty
physical value of w0 , fixed using f⇡ [3]. Set 1 will ⇡ ⇡
Lattice QCD calculation: ’3-point functions’
d to “very coarse”, 2 as “coarse” and 3 as “fine”.
3, 4 and 5 give the sea quarkrtmasses in lattice eunits
needed for form-factors
d = ml ). Ls and Lt are the lengths in lattice units
and time directions for each lattice. The number t
q
rations that
e.g. for pion to pion
we have used in each set is given in
h column. The final column gives the values of the J
p2 p1
transition via vector
of the 3-point function, T , in lattice units.
⇡
0.831 current J 9,12,15
aml,sea ams,sea am
c,sea sL ⇥L n
t cfg T ⇡
9 0.00235 0.0647 32⇥48 1000
2 0.00184 0.0507 0.628 48⇥64 1000 12,15,18
t
p=0
Need to calculate correlators
9 0.0012 bl
0.0363 0.432 64⇥96c] 223 16,21,26 d)
p2 Jp1
for multiple ⇡ T values and ⇡
⇡
0 T
⇡
of0How to do lattice QCD ff. J. Koponen et al, HPQCD,
1701.04250 *updated*
calculation at high Q ?
2
Need a formalism with small discretisation errors that is
numerically fast. *HISQ*
Studying ηs and ηc is faster than π and K and with smaller
stat. errors, so enabling behaviour to be mapped out to
higher Q2.
Need to test relationship of form factors to decay
constants for a range of mesons.
Also need to test relationship of form factors for different
mesons containing same struck quarks e.g. K and ηs
Can also compare form factors for different currents
(e.g scalar) vs expectations from perturbative QCD helicity
rulesResults for ηs HPQCD, 1701.04250
3 values of a (0.15fm to 0.09fm) + 2 mu/dsea (ms/5 and ms/10)
Use ‘twisted boundary 1.04
fine
coarse
v. coarse
conditions’ to insert 1.02
momentum and test 1
c2
discretisation errors as a 0.98
function of (pa).
0.96
Stat. errors are key issue. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
1.1 (pa)2
fine
coarse 0.7
v. coarse
⟨0|J5 |ηs (p)⟩/⟨0|J5|ηs (0)⟩
1.05 0.65
Q2 = 2.53 GeV2
Q2 Fηs (Q2)(GeV2 ) 0.6
1 0.55
0.5
0.95
0.45
0.4
Q2 = 0.91 GeV2
0.9
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.35
0 0.005 0.01 0.015 0.02 0.025
(pa)2 2 2
a (fm )Results - Can reach Q2 of 6 GeV2 with small stat. errors
⌘s Disc. and sea quark mass effects very small
1 physical point from
F =
1 + Q2 /M 2 ‘z-expansion’ fit
1.2 pert. QCD inc.
fine
coarse PQCD 2 ⇡ (2GeV) =
1 v. coarse
superfine [x(1 x)]0.52
Q2 Fηs (Q2)(GeV2)
Braun et al,
0.8 pole 1503.03656
0.6 using f⌘s
0.4
0.2 asymp. pert.
PQCD 1
QCD
0
0 2 4 6 8 10
no sign of asymptotic
Q2 (GeV2) HPQCD, 1701.04250 Updated to
form kicking in … add superfine pointsz-expansion p p
tcut t tcut t=q 2
z(t, tcut ) = p p
tcut t + tcut 2
tcut = 4MK
or Physics and CP Violation Conference, Vancouver, 2006
Maps t region into
-1 < z < 1 t z
| throughout semileptonic
! r=1
hoice t0 = t+ (1 − 1 − t− /t+ ).
M element |z|max
Vud 2 3.5 × 10−5
Vcb
q =0!z=00.032
Vus 0.047
Fit:
Vcs 0.051
Vcd 0.17 Figure 2: Mapping (3) of the cut t plane onto the unit
X
Vub+ Q2 /M0.28
2 circle.
i The semileptonic region
i 4 by the m
is represented sea
blue
(1 )F = 1 + line.z Ai 1 + Bi (a⇤) + Ci (a⇤) + Di
i
10
investigated by standard means. 4 Let us focus on
⇤ = 1 GeV no
ors, following just from kine- z-independent disc. errors by defn.
the form factors for pseudoscalar-pseudoscalar transi-
s. Pseudoscalar-pseudoscalar tions, defined by the matrix element of the relevant
′Higher Q2 is possible for heavier quarks - try ηc
HPQCD, LATTICE2018, 1902.03808
6
HPQCD preliminary fine pert. QCD :
superfine
ultrafine
scalar ff
5
Q2Fηc (Q2)/Fηc (0)(GeV2 )
should fall
4 more rapidly
scalar than 1/Q2 -
3 not true
vector
2 vector ff
NLO
closer to
LO
1
PQCD vector Q2 → ∞ PQCD at
NLO but
0
0 5 10 15 20 25 30 agreement
Q2 (GeV2) still not
NLO asympt. form : Melic et al, hep-
2
8⇡f ↵s (Q/2)[1 + 1.18↵s + . . .] ph/9802204 good.Electromagnetic form factors of the K from lattice QCD
For K must allow for both strange and light (u/d) current
interacting with photon - they are quite different
not expected
1.2
vc5
from pert.
vc10 QCD
1 vcphys
c5 s current:
Q2 FK (Q2)(GeV2 )
c10
0.8 f5 vector pole K and ηs
s ηs fit agree -
0.6
spectator
0.4 irrelevant
l
0.2 physical (a=0,
HPQCD preliminary mu/d=phys)
0 result from fit
0 1 2 3 4 5 6
Q2 (GeV2) J. Koponen et al, HPQCD,
in progressElectromagnetic form factors of the K from lattice QCD
K+ form factor is electric charge weighted combination
2/3* s + 1/3*u - first lattice QCD calculation of this
0.8
NA7 HPQCD Preliminary F not
0.7
changing
0.6 *5% prediction* for JLAB expt
with f as
Q2FK (Q2)(GeV2)
0.5 K+ expected
0.4 by asympt.
pert. QCD
0.3
0.2 K + PQCD Q2 → ∞
K0 = s - d
0.1
K0
- nonzero
0
0 1 2 3 4 5 6
J. Koponen et al, HPQCD,
Q2(GeV2) in prep.www.physics.gla.ac.uk/HPQCD
Conclusion
• Lattice QCD gives very accurate map of meson decay
constants now. New 1% test against experiment for J/ψ.
• Lattice QCD calculation of K+ space-like electro-
magnetic form factor underway to make predictions for
JLAB expt. - can reach Q2=4 GeV2 with 5% accuracy.
• Lattice QCD can test behaviour of meson electro-
magnetic form factors - predictions from asymptotic
pert. QCD do NOT work well.
• Q2F is flat at large Q2 (25 GeV2) but > asymptotic value .
• F depends on struck quark - independent of spectator.
• F does not depend on f in the expected way.
• helicity rules do not seem to work for scalar current.
• rethink needed?Spares
e e
⇡ electromagnetic form factor at small q2
e scattering probes ⇡ electric charge distn
2 n q
E OF THE h⇡(p
PION1 )|V
FROM FULL
µ |⇡(p LATTICE
2 )i = QCD
f+ (q )(p 1 +WITH
p2 ) µ …
mass an
a) Working at
bl
expecte
physical squared
u/d
values
quark masses
Fig. 1. Data on the squared modulus ofon F~ HISQ for Itl
(where
<
2+1+1
tion [1]; (b) direct 7re scattering [2-4]; (c) inverseconfigs,
ele
lattice QCDthe for
The horizontal barraw (b)results
indicate
effect
on o
top of chiral p
experiment
theory
determines the normalisation F.(0) =chiral
1, a
stagger
J. Koponen et
al, HPQCD,
= F⇡ (q 2 ) by: 1511.07382 affect πStat errors on form factor as a function of Q2
20
fine
coarse
stat. error in Fηs (Q2) (%)
v. coarse
15
10
5
0
0 1 2 3 4 5 6 7 8
Q2(Qa)2(GeV)2 J. Koponen et
al, HPQCD,
1701.04250.Comparison of F/f2 for different mesons
Pert. QCD expects the asymptotic value
to be the same for all - 8⇡↵s (Q/2)
30
pi
etas
25
meson
20
ηc
2
Q Fmeson(Q )/f
“plateau”
2
15
value
10 12.7
2
5
0
0 0.5 1 1.5 2 2.5 3
Q2 (GeV2)You can also read
