Results from Lattice QCDcdavies/talks/lepton-photon-2013.pdf · 2015. 6. 3. · Hadron spectrum Hadron structure QCD parameters CKM elements ... Quantum chromodynamics 0.11 0.12 0.13!

Results from Lattice QCD

Christine DaviesUniversity of Glasgow

Lepton-Photon 2013 San Francisco, June 2013

Tuesday, 25 June 2013

QCD is a key part of the Standard Model but quark confinement is a complication/interesting feature.

ATLAS@LHC

vs

Properties of hadrons calculable from QCD if fully nonperturbative calculation is done - can test QCD/search for new physics and determine parameters (to 1%).

q and annihilate

q�Vqq�CKM element

Run: 202742Event: 98595

Ks

Ks + + Tag

1630804-076 CLEO-cDs → µν

Ds → Kπππ


Nuclear physics

Particle physics

Astrophysics

Applications of Lattice QCD/Lattice field theory

Hadron spectrum Hadron structure

QCD parameters

CKM elementsQCD at high temperatures and densitiesTheories beyond the

Standard Model

Quantum gravity

Nuclear massesand properties

Glueballs and exotica

Annual proceedings of lattice conference:http://pos.sissa.it/

condensed matter physicscomputational physicscomputer science ...

Giedt, Panero: Lat2012

Doi: Lat2012

Young: Lat2012

Lombardo, Aarts: Lat2012; Satz: Saturday

Lin: Lat2012Dudek, Mohler: Lat2012

Tarantino: Lat2012

Shintani: Lat2011

Laiho: Lat2011

Boyle: Lat2012


http://pos.sissa.it

http://pos.sissa.it

Lattice QCD = fully nonperturbative QCD calculation RECIPE• Generate sets of gluon fields for Monte Carlo integrn of Path Integral(inc effect of u, d, s (+ c) sea quarks)• Calculate averaged “hadron correlators” from valence q props.

• Determine and fix to get results in physical units.

a mq

• Fit as a function of time to obtain masses and simple matrix elements

a• extrapolate to for real world. *now have phys mu,d*

a = 0, mu,d = phys


Example parameters for gluon configurations being made using two different formalisms for handling quarks.

mass of u,d quarks

real world 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.005 0.01 0.015 0.02 0.025 0.03

m2 /

GeV

2

a2 / fm2

MILC HISQ, 2+1+1BMW clover, 2+1

Need volume:

mu,d ≈ ms/10

mu,d ≈ ms/27

mπL > 3

u,d,s,c in seaHighly Improved Staggered Quarks

u,d,s in sea

and >500 independentconfigurations

MILC:1212.4768BMW: 1011.2711

* NEW! physical u/d now possible *

RBC/UKQCD: dw quarks underway


Results for the masses of mesons that are long-lived and so can be well-characterised in experiment

0

2

4

6

8

10

12

MES

ON

MA

SS (G

eV/c

2 )

c J/c'

'

hc c0c1c2

bb' '

''

b0b1(1P)b2b0b1(2P)b2

(1D)hb(1P)

hb(2P)

Bc

Bc'

BsB

Bs*

B*

Bc*

Bc*'

Bc0*

DsD

K

exptfix params

postdcnspredcns

2008

2011 2012

2005

Agreement very good - errors typically a few MeV, need to worry about em, mu-md ..

b quark

c quark

u, d, s quarks

HPQCD: 1207.5149


4

1000

1500

2000

2500

3000

FIG. 1: Spectrum of isovector mesons extracted in [13] with a pion mass of ∼ 700MeV. Masses (in MeV) are averages over twocomputed volumes, ∼ (2.0 fm)3, (2.5 fm)3. Box size indicates the statistical uncertainty on the extracted mass. Ellipses indicatethat there may be heavier states in that JPC but we did not determine them reliably in [13]. States are color coded by assignedL-wave supermultiplet as described in the text. States colored grey could not be assigned to an L-wave supermultiplet.

qq̄ JPC

L = 0(S)S = 0 0−+

S = 1 1−−

L = 1(P )S = 0 1+−

S = 1 (0, 1, 2)++

L = 2(D)S = 0 2−+

S = 1 (1, 2, 3)−−

L = 3(F )S = 0 3+−

S = 1 (2, 3, 4)++

L = 4(G)S = 0 4−+

S = 1 (3, 4, 5)−−

TABLE I: qq̄ 2S+1LJ , JPC supermultiplets.

nificantly lighter than positive parity states, 0+−, (2+−)2.As well as more 1−+ states, much higher in the spectrum

are 0−− and 3−+ states. We will argue that these exotic

states can be explained as being hybrid mesons.

Under examination, the spectrum of non-exotic JPC

mesons shown in Figure 1 displays JPC patterns con-

sistent with qq̄ 2S+1LJ structure (as presented in Table

I). These patterns are identified in Figure 1, where we

show apparently complete S, P,D, F -wave supermulti-

plets3 , and a partial G-wave supermultiplet indicated

3 The terminology “supermultiplet” is a historical one, where fla-vor multiplets of a given JPC were first identified and later dif-

by the presence of 4−−, 4−+ states. There appear to

be higher mass recurrences of several of these supermul-

tiplets, which in analysis to be described later we will

suggest could be radial excitations of qq̄.

Once these assignments based upon degeneracy pat-

terns are made, we can see a near-degenerate set of states

with JPC = 0−+, 1−−, 2−+ remains unassigned (colored

blue and indicated with a star (�) in Figure 1). No super-

multiplet in Table I contains such a set, and we notice

that the mass-scale of these states is very close to the

lightest exotic state, the 1−+. We hypothesize that these

states could be members of the lightest hybrid meson

supermultiplet.

To explore the hypothesis that we have a supermulti-

plet of hybrid mesons embedded within a spectrum of qq̄states, we take advantage of the “overlap” information

provided by the two-point correlator analysis described

in the previous section. In the same variational analysis

that determines the mass spectrum, mn, we also deter-

mine the matrix-elements, Zni =

�n��Oi

��0�, that encode

the degree to which operator Oi “overlaps with” state n.To the extent that we understand the structure of our

operators, we can infer some information about the in-

ternal structure of the extracted states from the relative

size of these overlaps.

ferent JPC having a common proposed bound-state structurewere related.

Mapping excited states is harder .. Hadron Spectrum:1004.4930Dudek: Lattice2012BGR: 1112.1601Light isovectors:

exotics

“normal” JPC

“hybrids” ?

1−−, (0, 1, 2)−+lightest hybrid multiplet = = Swave qq(0−+, 1−−)× g(1+−)

π, ρ, . . .(huge basis of single-hadron operators, but a=0.12 fm (anisotropic), mass π = 700 MeV)

strong overlap with “gluey” operators π1(1600),π(1800),π2(1880) + 1−−?see also charmonium:Hadron Spectrum:1204.5425


80 90 100 110 120

ms (2 GeV) (MeV)

RBC-UKQCD 11

BMW 11

Blum 10

HPQCD 10

Dominiguez 08

Chetyrkin 06

9. Quantum chromodynamics 29

overall χ2 to the central value is determined. If this initial χ2 is larger than the numberof degrees of freedom, i.e. larger than the number of individual inputs minus one, thenall individual errors are enlarged by a common factor such that χ2/d.o.f. equals unity.If the initial value of χ2 is smaller than the number of degrees of freedom, an overall,a-priori unknown correlation coefficient is introduced and determined by requiring thatthe total χ2/d.o.f. of the combination equals unity. In both cases, the resulting finaloverall uncertainty of the central value of αs is larger than the initial estimate of aGaussian error.

This procedure is only meaningful if the individual measurements are known not tobe correlated to large degrees, i.e. if they are not - for instance - based on the sameinput data, and if the input values are largely compatible with each other and with theresulting central value, within their assigned uncertainties. The list of selected individualmeasurements discussed above, however, violates both these requirements: there areseveral measurements based on (partly or fully) identical data sets, and there are resultswhich apparently do not agree with others and/or with the resulting central value, withintheir assigned individual uncertainty. Examples for the first case are results from thehadronic width of the τ lepton, from DIS processes and from jets and event shapes ine+e− final states. An example of the second case is the apparent disagreement betweenresults from the τ width and those from DIS [264] or from Thrust distributions in e+e−

annihilation [278].

0.11 0.12 0.13!! (("" ))s ##

LatticeDIS e+e- annihilation

$-decays

Z pole fits

Figure 9.3: Summary of values of αs(M2Z) obtained for various sub-classes

of measurements (see Fig. 9.2 (a) to (d)). The new world average value ofαs(M2

Z) = 0.1184 ± 0.0007 is indicated by the dashed line and the shaded band.

Due to these obstacles, we have chosen to determine pre-averages for each class ofmeasurements, and then to combine those to the final world average value of αs(MZ),using the methods of error treatment as just described. The five pre-averages aresummarized in Fig. 9.3; we recall that these are exclusively obtained from extractionswhich are based on (at least) full NNLO QCD predictions, and are published inpeer-reviewed journals at the time of completing this Review. From these, we determinethe new world average value of

αs(M2Z) = 0.1184 ± 0.0007 , (9.23)

June 29, 2012 14:54

28 9. Quantum chromodynamics

0.11 0.12 0.13!! (("" ))s ##

HPQCD (Wilson loops)

Maltman (Wilson loops)

JLQCD (Adler functions)

PACS-CS (vac. pol. fctns.)

0.11 0.12 0.13!! (("" ))s ##

ALEPH (j&s)

Abbate et al. (T)

JADE (j&s)OPAL (j&s)

Dissertori et al. (3j)

0.30 0.32 0.34 0.36!! (("" ))s $$

Narison

Davier

BaikovBeneke

Maltman

Caprini

0.11 0.12 0.13!! (("" ))s ##

ABKMBBGJRMSTW

!"# !$#

!%#!&#

HPQCD (c-c correlators)

NNPDF

SM review

Figure 9.2: Summary of determinations of αs from hadronic τ -decays (a), fromlattice calculations (b), from DIS structure functions (c) and from event shapes andjet production in e+e−-annihilation (d). The shaded bands indicate the averagevalues chosen to be included in the determination of the new world average of αs.

model and constraints on new physics from data at the Z-pole, αs(M2Z) = 0.1197± 0.0028

will be used instead, as it is based on a more constrained data set where QCD correctionsdirectly enter through the hadronic decay width of the Z. We note that all theseresults from electroweak precision data, however, strongly depend on the strict validityof Standard Model predictions and the existence of the minimal Higgs mechanism toimplement electroweak symmetry breaking. Any - even small - deviation of nature fromthis model could strongly influence this extraction of αs.

Determination of the world average value of αs(M2Z)

A non-trivial exercise consists in the evaluation of a world-average value for αs(M2Z).

A certain arbitrariness and subjective component is inevitable because of the choice ofmeasurements to be included in the average, the treatment of (non-Gaussian) systematicuncertainties of mostly theoretical nature, as well as the treatment of correlations amongthe various inputs, of theoretical as well as experimental origin. In earlier reviews[243–245] an attempt was made to take account of such correlations, using methods asproposed, e.g., in Ref. 281, and - likewise - to treat cases of apparent incompatibilitiesor possibly underestimated systematic uncertainties in a meaningful and well definedmanner:

The central value is determined as the weighted average of the different input values.An initial error of the central value is determined treating the uncertainties of allindividual measurements as being uncorrelated and being of Gaussian nature, and the

July 9, 2012 19:53

Lattice QCD sets world averages for quark masses and Direct access to parameters in QCD Lagrangian means systematic errors smaller

a variety of lattice methods agree

non-lattice methods have larger errors

PDG

PDG

αs

Lattice calcs now adding QED for accurate mu/md Izubuchi:Lattice2012; RM123: 1303.4896

av:94.3(1.2) MeV

av:0.1184(7)


Meson decay constants

2012

B → τν

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

DEC

AY

CO

NST

AN

T (G

EV)

UNFLAVORED FLAVORED

+

J/c

’

b

’

’’Bc

BsB+

Ds

D+

K+

Lattice QCD predictionsLattice QCD postdictions

experiment

HPQCD1302.2644

Parameterises hadronic information needed for annihilation rate to W or photon: Γ ∝ f2

HPQCD1208.2855


Constraining new physics with lattice QCD

6

come from experiment, but there are two complicationsthat result from simplifications in the simulations. Thefirst is that the simulation does not include electromag-netism. The second is that mu = md in the simulation,while in reality mu = 0.48(10)md [1].

The most appropriate pion mass for fπ+ is theneutral-pion mass (134.9766(6) MeV [1]). All π mesonswould have this mass in a world without electromag-netism—our simulations, for example—up to very small(quadratic) corrections from the u−d mass difference.These corrections are estimated at 0.32(20)MeV for Mπ+

in [27]. For our purposes, it is sufficient to take 0.32MeVas the uncertainty in the pion mass, and ignore the dis-tinction between charged and neutral pions:

Mphysπ = 134.98(32)MeV (15)

This pion mass corresponds in our simulation to alight-quark mass of m� = (mu + md)/2. The corre-sponding kaon mass is one for an s� meson. This is theroot-mean-square average of the K+ and K0 masses withadditional small corrections for electromagnetism:

(MphysK )2 ≡ 1

2

�(M2

K+ +M2K0)

−(1 +∆E)(M2π+ −M2

π0)�. (16)

∆E would be zero if electromagnetic effects in the K sys-tem mirrored those of the π. In fact it is closer to 1.Recent lattice calculations [28–30] that include electro-magnetic effects give values in the region 0.6-0.7. Wetake ∆E = 0.65(50) to conservatively encompass theseresults and this gives

MphysK = 494.6(3)MeV. (17)

Tuning the pion mass to Mphysπ and the kaon mass to

MphysK in our fits sets the strange-quark mass to its phys-

ical value, and the light-quark mass to the average m� ofthe u and d masses. This light-quark mass is correct, towithin our errors, for the valence quarks in the pion, andfor sea quarks in all three mesons.

This tuning is not correct, however, for theK+’s valence light-quark, which is a u quark, withmass 0.65(9)m�. This difference produces a small butsignificant downward shift in fK+ . To compute the cor-rected K+ decay constant, we evaluate our fit formulaswith a pion mass given by

�0.65(9)Mphys

π , while adjust-ing the kaon mass so that 2M2

K − m2π is unchanged (to

leave the s-quark mass unchanged). These adjustmentsare made only for the valence-quark masses in the K+;the valence-quark masses in the pion and ηs, as speci-fied by Mphys

π and MphysK , are left unchanged, as are the

sea-quark masses in each of the mesons.

D. Fit Results

We fit w0 times each of the decay constants and eachηs mass in Table III to the formulas above, as functions

0 1805

0.156

0.158

0.160

0.162

0.164

f K(G

eV)

0.05 0.10 0.15 0.20m2

π/(2m2K −m2

π)

0.130

0.135

0.140

0.145

0.150

f π(G

eV)

FIG. 2: Fit results for the π, K, and ηs decay constantsas functions of the light-quark mass for three different lat-tice spacings: 0.15 fm (top/blue), 0.12 fm (middle/green), and0.09 fm (bottom/red). The data shown are from Table III,with corrections for errors in the s masses, and for finite-volume errors. The lines show our fit with the best-fit valuesof the fit parameters. The dashed line is the a = 0 extrap-olation, and the gray band shows our continuum results atthe physical light quark mass point with m� = (mu +md)/2.The current experimental result for fπ+ is also shown (blackpoint). Note that the three plots are against very differ-ent scales in the vertical direction: the range covered in thefπ plot is 10 times larger than that covered in the fηs plot.

of the pion and kaon masses and w0. We also fit theexperimental value for fπ+ = 130.4(2)MeV to our for-mula evaluated at the physical pion and kaon masses,Eqs. (15, 17)). These fits are all done simultaneouslyusing the same parameters for the fit functions in eachcase, and including the correlations between π, K and ηsresults discussed in Section II.The results for the decay constants, as a function of the

light-quark mass, are shown in Figure 2. For each decay

Vus/Vud

� νl

Annihilation of to W allows CKM element determination given decay constants from lattice QCD

K/π

* results at physical u/d quark masses* fK/fπ

Γ(K+ → �ν)

Γ(π+ → �ν)

|Vus|fK+

|Vud|fπ+

= 0.27598(35)Br(K+)(25)EM

expt for

fK+

fπ+from lattice gives CKM

HPQCD:1301.1670

HISQ on MILC configs = mu,d/ms


fK+

fπ+

= 1.1916(21)

|Vus||Vud|

= 0.23160(29)expt(21)EM (41)latt

Vud from nuclear decay now needs improvement for unitarity test!β

Comparison of results (note: ) fK+ < fK

* results at physical u/d quark masses*

(28)Br(20)EM (40)latt(5)Vud

|Vus| = 0.22564

1− |Vud|2 − |Vus|2 − |Vub|2

= −0.00009(51)

clover

HISQ

HISQ

HISQ

domain-wall

domain-wall

asqtad

good agreement from different formalisms

RM123:1303.4896 gives by 0.40(4)%

1.15 1.17 1.19 1.21 1.23 1.25

HPQCD, 1303.1670

MILC, 1301.5855

ETMC, Lattice2013

BMW, 1001.4692

HPQCD, 0706.1726

LvW, 1112.4861

MILC, 1012.0868

RBC/UKQCD1011.0892

fK/fnf=2+1

fK+/f +

nf=2+1+1

twisted mass


Pseudoscalar meson physics with four dynamical quarks D. Toussaint,

Figure 1: fD and fDs on the different ensembles. Fits that are quadratic and linear in a2, or use only physicalquark mass ensembles are superimposed on the data. More details are given in the text.

3. Interpolate or extrapolate to the charm valence mass where MDs is correct. This fixes mc.4. Find md −mu from E&M adjusted K0 −K+ mass difference [8]5. Find, by interpolation or extrapolation, fK at the adjusted up-quark mass.6. Find fD and MD (a check) at the adjusted down and charm masses.7. Find fDs at the adjusted strange and charm masses.8. Find Mηc (check) at the adjusted charm-quark mass.

We have also used a tuning procedure where we replaced fπ and Mπ by the decay constant,fp4s, and mass, Mp4s, of a fictitious meson with degenerate valence quark masses at 0.4 timesthe strange-quark mass, using the values for fp4s and Mp4s in MeV determined from three flavorasqtad data. Finally, we varied the procedure by first fixing the lattice spacing from the static-quarkpotential, and then tuning the valence quark masses as described above.

In Eq. 2.1 we list results from the above procedure for the most influential ensemble (a ≈ 0.09fm with physical masses). These results are adjusted for finite size and and some electromagneticeffects, but are not adjusted for mistuned sea-quark masses or extrapolated to the continuum limit.The errors here are statistical only but, as discussed above, errors from scale setting and valence-quark mass tuning are included in the statistical error here. The D+, D0 and ηc masses are not usedin the tuning, so they can be compared to their experimental values, shown in parentheses.

a = 0.08792(10) fm aml = 0.001333(5) ams = 0.03648(11) amc = 0.4323(7)mu/md = 0.480(6) ms/ml = 27.36(3) mc/ms = 11.851(17)fK = 155.12(22) MeVMD0 = 1867.8(1.5) MeV (cf 1864.8) MD+ = 1870.2(1.2) MeV (cf 1869.6)Mηc = 2980.2(3) MeV (cf 2980.3(1.2))fD = 208.75(1.03) MeV fDs = 246.60(20) MeV fDs/ fD = 1.181(6)

(2.1)

3. Continuum extrapolationThe results for fD and fDs on each ensemble are shown in Fig. 1. In this figure the physical

quark mass ensembles, labelled as “ml = 0.04ms”, have the smallest statistical errors. This is

3

Constraining new physics with lattice QCD: fDs , fD


MILC:1210.8431

MILC (Lattice2013):

agree well with previous HPQCD:

new results using HISQ quarks on MILC 2+1+1 configs

fDs = 248.0(2.5)MeV

fD = 208.3(3.4)MeVHPQCD:1008.4018

HPQCD:1206.4936fDs Comparison

Average of CLEO-c [PRD80,112004(2009)], BaBar [PRD82,091103(2010)] and Belle Preliminary.

[MeV

]s

Df

220

240

260

280

300

4.5 MeV±Experimental W.A.: 257.2 2.5 MeV±Lattice (HPQCD): 248.0

µ!+µ " +

sD #!+# " +

sD

CLEO-c BaBar Belle Preliminary

Average of experimental determinations is consistent within 1.8σ with most

precise lattice QCD calculation by HPQCD.

Need further lattice QCD results with comparable precision to confirm the

calculation by HPQCD.

A. Zupanc (KIT) Ds → �ν and fDs CHARM2012, May 2012 29 / 30

experimental update: new Belle resultsfDs = 257.2(4.5)MeV

Zupanc:charm2012

World av:

fDs = 247.2(2.2)MeV

fD = 211.4(1.6)MeV

2σ above theoryTuesday, 25 June 2013

Constraining new physics with lattice QCD: fBs , fB3

B and Bs are fit separately; priors used in the fit are

described in [11]. The amplitudes and energies from the

fits are given in Tables IV and V. a3/2Φ(0)q is the matrix

element of the leading current J (0)0 and a3/2Φ(1)

q that of

J (1)0 and J (2)

0 , whose matrix elements are equal at zero

meson momentum. Notice that the statistical errors in

Φ do not increase on the physical point lattices, because

they have such large volumes.

We take two approaches to the analysis. The first is

to perform a simultaneous chiral fit to all our results for

Φ,Φs,Φs/Φ and MBs − MB using SU(2) chiral pertur-

bation theory. The second is to study only the physical

u/d mass results as a function of lattice spacing.

For the chiral analysis we use the same formula and

priors for MBs − MB as in [11]. Pion masses used in

the fits are listed in Table V and the chiral logarithms,

l(M2π), include the finite volume corrections computed

in [18] which have negligible effect on the fit. For the

decay constants the chiral formulas, including analytic

terms up to M2π and the leading logarithmic behaviour,

are (see e.g. [19]):

Φs = Φs0(1.0 + bsM2π/Λ

2χ) (5)

Φ = Φ0

�1.0 + bl

M2π

Λ2χ

+1 + 3g2

2Λ2χ

�−3

2l(M2

π)

��(6)

The coefficients of the analytic terms bs, bl are given

priors 0.0(1.0) and Φ0,Φs0 have 0.5(5). To allow for

discretisation errors each fit formula is multiplied by

(1.0 + d1(Λa)2 + d2(Λa)4), with Λ = 0.4 GeV. We ex-

pect discretisation effects to be very similar for Φ and Φs

and so we take the di to be the same, but differing from

the di used in the MBs −MB fit. Since all actions used

here are accurate through a2 at tree-level, the prior on

d1 is taken to be 0.0(3) whereas d2 is 0.0(1.0). The di areallowed to have mild mb dependence as in [11]. The ratio

Φs/Φ is allowed additional light quark mass dependent

discretisation errors that could arise, for example, from

staggered taste-splittings.

Error % ΦBs/ΦB MBs −MB ΦBs ΦB

EM: 0.0 1.2 0.0 0.0a dependence: 0.01 0.9 0.7 0.7chiral: 0.01 0.2 0.05 0.05g: 0.01 0.1 0.0 0.0stat/scale: 0.30 1.2 1.1 1.1operator: 0.0 0.0 1.4 1.4relativistic: 0.5 0.5 1.0 1.0total: 0.6 2.0 2.0 2.1

TABLE VI: Full error budget from the chiral fit as a per-centage of the final answer.

The results of the decay constant chiral fits are plot-

ted in Figs. 1 and 2. Extrapolating to the physical

point appropriate to ml = (mu + md)/2 in the absence

of electromagnetism, i.e. Mπ = Mπ0 , we find ΦBs =

0.00 0.05 0.10 0.15 0.20 0.25M2

π/M2ηs

1.14

1.16

1.18

1.20

1.22

1.24

1.26

(f B

s�MB s)/(f B√MB)

Physical point

Set 1Set 2Set 3Set 4


FIG. 1: Fit to the decay constant ratio ΦBs/ΦB . The fitresult is shown in grey and errors include statistics, and chi-ral/continuum fitting.

0.00 0.05 0.10 0.15 0.20 0.25M2

π/M2ηs

0.35

0.40

0.45

0.50

0.55

f Bq�

MB q

(GeV

)3/2

fB√MB

fBs�MBs

Physical point



FIG. 2: Fit to the decay constants ΦBs and ΦB . Errors on thedata points include statistics/scale only. The fit error, in grey,includes chiral/continuum fitting and perturbative errors.

0.520(11) GeV3/2, ΦB = 0.428(9) GeV3/2, ΦBs/ΦB =

1.215(7). For MBs −MB we obtain 86(1) MeV, in agree-

ment with the result of [11].

Figs 3 and 4 show the results of fitting MBs − MB

and decay constants from the physical point ensembles

only, and allowing only the mass dependent discretisation

terms above. The results are ΦBs = 0.515(8) GeV3/2,

ΦB = 0.424(7) GeV3/2, ΦBs/ΦB = 1.216(7) and MBs −MB = 87(1) MeV. Results and errors agree well between

the two methods and we take the central values from the

chiral fit as this allows us to interpolate to the correct

pion mass.

Our error budget is given in Table VI. The errors that

are estimated directly from the chiral/continuum fit are

those from statistics, the lattice spacing and g and other

chiral fit parameters. The two remaining sources of error

in the decay constant are missing higher order corrections

in the operator matching and relativistic corrections to

the current. We estimate the operator matching error by

allowing in our fits for an amb-dependent α2s correction to

the renormalisation in Eq. 4 with prior on the coefficient

= ml/ms* results at physical u/d quark masses*

HPQCD: 1302.2644.

Uses improved NRQCD for b quark and HISQ u/d and s quarks on HISQ 2+1+1 gluon configsfBs = 224(5)MeV

!"#! µµ" !"#$% %&%$ '$(&) *+ !, -*./ -$00" !"#$%1%&%$1'$(&)1*+1!,1-*./1-$00

#%$'*(.$'1234251! 66718149:;;<=:997>?=@A

" !$+5*.*B$1.C1DE1*+1,!!,

231∝ .&+Fβ G1,HI

" 2$5.1#%$5$+.10*J*.1*51K%CJ1LMNO

2342 ! 7 H ?= P Q A=R LS

$%&

234251! 6671T1H:U>?=@P1 Q1A=R1LS1

" NC%1./$1!,1#%$'*(.*C+

SVLW $X#$(.5 P 5*Y+&0 &+' ?ZSVLW1$X#$(.51P15*Y+&01&+'1?Z

W&([Y%C"+'1$B$+.51*+1./$1JC5.

5$+5*.*B$1W*+1*+1Z1KW@?11 :12&([Y%C"+'1*5

'CJ*+&.$' W) 5$J*0$#.C+*( '$(&)5'CJ*+&.$'1W)15$J*0$#.C+*(1'$(&)5

CK1'*KK$%$+.1W1\"&%[5

" 9σ $B*'$+($1-*./1Z1KW@?1;σ CW5$%B&.*C+1-*./1FKW@?1

11BaBar Symposium April 2009

Standard Model processes:

Br(Bs → µ+µ−) =3.47(19)× 10−9

Nov. 2012Tuesday, 25 June 2013

B, Bs decay constant update 2013

NOTE: fBs < fDs now quite clear

(but Belle 1208.4678 = 211(28) MeV)

185(3) 225(3) averages


150 175 200 225 250 275 300fBx

/ MeV

PDG av BR(B-> ) + PDG av Vub

HPQCD NRQCD1302.2644

HPQCD NRQCD1202.4914HPQCD HISQ 1110.4510FNAL/MILC 1112.3051

ETMC Lattice2013ALPHA 1210.7932

fB fBs fB,expt-av

fB+

fB0

u, d seau, d, s sea

u, d, s, c sea


0 0.5

aml/(ams)physical

0.97

0.98

0.99

1

f 0 (q2 =0

)

HISQ on asqtad result (total error)chiral extrapolation in the continuumchiral extrapolation for a=0.15fm a = 0.12 fm (Nf = 2+1+1 HISQ)a = 0.15 fm (Nf = 2+1+1 HISQ)chiral extrapolation for a=0.12fmchiral extrapolation for a=0.09fma = 0.09 fm (Nf = 2+1+1 HISQ)a = 0.06 fm (Nf = 2+1+1 HISQ)

chi2/dof [dof] = 0.28 [8] p = 0.99

f+(0) = 0.9712(24)

Constraining new physics with lattice QCD: form factorsK → π�ν K

π

*MILC in progress: HISQ results with physical u/d quarks - expect ~0.3% error*

NEW - now results with full continuum and chiral extrapolation. Different formalisms agree

0.955 0.96 0.965 0.97 0.975

MILC, 1212.4993HISQ

RBC/UKQCD1305.7217domain wall

fK (q2=0), nf=2+1

with experiment:|Vus|fKπ(0) = 0.2163(5)

1005.2323

Vus

|Vus| = 0.2238(11)

fKπ(0) = 0.9667(40)

fKπ(0) = 0.9670

�+27

−50

�

f+(q2), f0(q

2)

f+(0) = f0(0)


Form factor shapes and dependence on spectator quark:

TABLE II. Input parameters for the chiral extrapolation [22, 53].

rsχPT Ensemble

quantity C020 C007 F0062 F0124

r1/a 2.821123 2.738591 3.857729 3.788732

µ0 6.234000 6.234000 6.381592 6.381592

r21a2∆P 0 0 0 0

r21a2∆A 0.2052872 0.2052872 0.0706188 0.0706188

r21a2∆T 0.3268607 0.3268607 0.1153820 0.1153820

r21a2∆V 0.4391099 0.4391099 0.1523710 0.1523710

r21a2∆I 0.5369975 0.5369975 0.2062070 0.2062070

r21a2δ′V −0.05 −0.05 −0.03 −0.03

r21a2δ′A −0.28 −0.30 −0.15 −0.16

We treat the chiral extrapolations of the B → D and Bs → Ds data slightly differently.For B → D, we analyze only full QCD data points, i.e., mx = ml. Then, since the strangesea quark in all ensembles is tuned within several per cent of its physical mass, mh ≈ ms, thedependence of the form factors on the sea and valence quark masses cannot be disentangled.Therefore, we drop the parameter c1,+ when fitting the B → D data. For our Bs → Ds

data, on the other hand, the strange valence quark is tuned close to its physical mass for allthe ensembles we analyze, mx = ms. As a result, we cannot disentangle the valence quarkdependence. Therefore, we discard the parameter c0,± when fitting the Bs → Ds data, andestimate the tuning error of ms a posteriori in Sec. VID.

The rSχPT expression for logs1-loop(Λχ, w) contains several low-energy constants used inrSχPT to describe the masses and decay constant of light pseudoscalar mesons. The valueswe use for these parameters are taken from Refs. [22, 53] and given in Table II. The χPTexpressions also require the D(s)-D∗

(s) splitting ∆(c) and the pion decay constant fπ; we take

0.75

0.8

0.85

0.9

0.95

1

1.05

1 1.05 1.1 1.15 1.2

h+

w

B->D: !2/dof=5/20, p-value=1.0

physical007/050020/050

0062/0310124/031

0.75

0.8

0.85

0.9

0.95

1

1.05

1 1.05 1.1 1.15 1.2

h+

w

Bs->Ds: !2/dof=7.6/20, p-value=0.99

physical007/050020/050

0062/0310124/031

FIG. 6. Chiral-continuum extrapolation of h+(w) for B → D (left) and Bs → Ds (right) decaysbased on the four ensembles listed in Table I. The blue bands show only the statistical errors and

the red curves are the chiral and continuum limits.

13

TABLE II. Input parameters for the chiral extrapolation [22, 53].

rsχPT Ensemble

quantity C020 C007 F0062 F0124

r1/a 2.821123 2.738591 3.857729 3.788732

µ0 6.234000 6.234000 6.381592 6.381592

r21a2∆P 0 0 0 0

r21a2∆A 0.2052872 0.2052872 0.0706188 0.0706188

r21a2∆T 0.3268607 0.3268607 0.1153820 0.1153820

r21a2∆V 0.4391099 0.4391099 0.1523710 0.1523710

r21a2∆I 0.5369975 0.5369975 0.2062070 0.2062070

r21a2δ′V −0.05 −0.05 −0.03 −0.03

r21a2δ′A −0.28 −0.30 −0.15 −0.16

We treat the chiral extrapolations of the B → D and Bs → Ds data slightly differently.For B → D, we analyze only full QCD data points, i.e., mx = ml. Then, since the strangesea quark in all ensembles is tuned within several per cent of its physical mass, mh ≈ ms, thedependence of the form factors on the sea and valence quark masses cannot be disentangled.Therefore, we drop the parameter c1,+ when fitting the B → D data. For our Bs → Ds

data, on the other hand, the strange valence quark is tuned close to its physical mass for allthe ensembles we analyze, mx = ms. As a result, we cannot disentangle the valence quarkdependence. Therefore, we discard the parameter c0,± when fitting the Bs → Ds data, andestimate the tuning error of ms a posteriori in Sec. VID.

The rSχPT expression for logs1-loop(Λχ, w) contains several low-energy constants used inrSχPT to describe the masses and decay constant of light pseudoscalar mesons. The valueswe use for these parameters are taken from Refs. [22, 53] and given in Table II. The χPTexpressions also require the D(s)-D∗

(s) splitting ∆(c) and the pion decay constant fπ; we take

0.75

0.8

0.85

0.9

0.95

1

1.05

1 1.05 1.1 1.15 1.2

h+

w

B->D: !2/dof=5/20, p-value=1.0

physical007/050020/050

0062/0310124/031

0.75

0.8

0.85

0.9

0.95

1

1.05

1 1.05 1.1 1.15 1.2

h+

w

Bs->Ds: !2/dof=7.6/20, p-value=0.99

physical007/050020/050

0062/0310124/031

FIG. 6. Chiral-continuum extrapolation of h+(w) for B → D (left) and Bs → Ds (right) decaysbased on the four ensembles listed in Table I. The blue bands show only the statistical errors and

the red curves are the chiral and continuum limits.

13

Fermilab/MILC:1202.6346.

0 0.5 1 1.5 2q2 (GeV2)

0.8

1

1.2

1.4

f 0(q2 ) o

r f+(q

2 )

f0

f+

b → cc → s

HPQCD: 1305.1462.

determined ff ratio to use in normln of Bs → µ+µ−

form factors same for light or strange spectator

compare f+ shape parameters: lattice and expt.

D → KDs → ηs

D → K�ν

spectator


B semi-leptonic decays and constraints on new physics B decays with in final state are sensitive to form factors suppressed by lepton masses for so may see new physics.

τe, µ

3

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1 1.1 1.2 1.3 1.4 1.5 1.6

w

z-parameterization f0z-parameterization f+

BaBar ’10

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1 1.1 1.2 1.3 1.4 1.5 1.6

w

simulated extrapolated

FIG. 1. The form factors f+ (top, red) and f0 (bottom, blue)from lattice QCD. The range of simulated recoil values is tothe left of the vertical line. The filled colored bands show theinterpolation/extrapolation of the numerical lattice data overthe full kinematic range using the z parameterization. Forcomparison, the experimental measurement from BaBar [26]is shown as solid filled circles (using |Vcb| = 41.4 × 10−3 [17,26]).

and the results are given in the upper panel of Table Vof the same work. As seen in Fig. 1, our result for f+(q2)agrees very well with experimental measurements [26]over the full kinematic range. This nontrivial check givesconfidence in the extrapolation of f0(q2), which cannotbe obtained experimentally and for which lattice-QCDinput is crucial. In particular, lattice-QCD uncertaintiesare smallest near q2 = (MB−MD)2, so the discussion be-low hinges principally on our calculation of f0(q2) nearthis point, the validated f0(0) = f+(0), and a smoothconnection between the two limits.

We calculate the Standard-Model B → D�ν partial de-cay rates into the three generations of leptons using theseform factors and Eqs. (1) and (2) with GS = GT = 0,GV = GFV ∗

cb. The resulting distributions are plotted inFig. 2. To illustrate the role of the scalar form factorf0(q2), we also show the rates with only the contribu-tions from f+(q2). Due to the significant helicity sup-pression, the differential decay rates into light leptonsare well-approximated by a single contribution from theform factor f+(q2). For B → Dτν, however, the contri-bution from the scalar form factor f0(q2) comprises halfof the Standard-Model rate.

Given the lattice-QCD determinations of f+(q2) andf0(q2) we can obtain the Standard-Model values for R(D)and PL(D). These are the primary results of this let-ter, and we now discuss the sources of systematic uncer-tainty. In Ref. [15], many statistical and several system-atic errors cancelled approximately or exactly in the ratiofBs→Ds�ν0 /fB→D�ν

0 studied there. Some of these do notcancel (as well) in R(D) and PL(D), however, because

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

dΓ

/dq

2 (

|Vcb

|210

12s-1

GeV

-2)

q2 (GeV

2)

B→Deν

B→Dµν

B→Dτν

FIG. 2. Differential decay rates for B → Deν (green), B →Dµν (blue), and B → Dτν (red) in the Standard Model.The black dot-dashed curves show the rates calculated withf0(q

2) = 0.

they affect f+(q2) and f0(q2) differently.

Table I shows the error budgets for R(D) and PL(D).The statistical error in R(D) is significant (3.7%) dueto the different phase-space integrations in the numera-tor and denominator, whereas for PL(D) the correlatedstatistical fluctuations largely cancel. For the same rea-son, the errors in R(D) arising from the extrapolation tothe physical light-quark masses and the continuum limit(1.4%) and to the full q2 range (1.5%), are much largerthan for PL(D). We estimate the error from the chiral-continuum extrapolation by comparing the results for fitswith and without next-to-next-to-leading order analyticterms in the chiral expansion. We estimate the error fromthe z extrapolation by varying the range of synthetic dataused in the z fit, including an additional pole in the fitfunction, and including higher powers of z. The specificchiral and z-fit variations considered are enumerated inTable VI of Ref. [15] and discussed in detail in the sur-rounding text. The remaining sources of uncertainty inTable I do not contribute significantly to the quantitiesstudied in Ref. [15], so we describe them in greater detailbelow.

We determine the bare heavy-quark masses in our sim-ulations by tuning the parameters κb and κc in the heavy-quark action such that the kinetic masses of the pseu-doscalar Bs and Ds mesons match the experimentally-measured values [20]. In practice, it is easier to work withthe form factors h±(w) on the lattice, which are linearcombinations of f+,0(q2) [15]. We study how the formfactors h±(w) depend on κb,c by re-computing the formfactors on some ensembles at values of κb,c slightly aboveand below the default ones, and extracting the slopeswith respect to κb,c. We use these slopes to correct ourresults for R(D) and PL(D) slightly from the simulatedκ values to the physical ones, and conservatively take the

B → D�ν Fermilab/MILC: 1206.4992.

Br(B → Dτν)

Br(B → D�ν)= 0.316(14)

� = e, µBaBar:1205.5442;1303.0571 = 0.440(72)

f+, f0

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0 4 8 12 16 20 24

dBl /

dq

2 [10-7

/ GeV

2 ]

q2 [GeV2]

J/! ! (2S) BelleBABARCDFLHCb [16]LHCb [17]

(a) Belle [14], BABAR [13], CDF [15], and LHCb [16, 17] dataand the Standard Model contribution to dB!/dq

2, ! = e, µ.

0

5e-09

1e-08

1.5e-08

2e-08

2.5e-08

0 4 8 12 16 20 24q2 [GeV2]

f+f0fTinputsCi

(b) Error components for dB!/dq2.

0

0.1

0.2

0.3

12 15 18 21 24

dB" /

dq

2 [10-7

/ GeV

2 ]

q2 [GeV2]

! (2S)

(c) Predicted Standard Model contribution to dBτ /dq2.

0

2e-09

4e-09

6e-09

8e-09

1e-08

12 15 18 21 24q2 [GeV2]

f+f0fTinputsCi

(d) Error components for dBτ/dq2.

FIG. 3: (left) Standard Model differential branching fractions and experiment. (right) Form factor, input parameter,and Wilson coefficient (Ci) contributions to the error. The total error is the sum in quadrature of the components.

where θ! is the angle between the B and "− as mea-sured in the dilepton rest frame. The “flat term” F !

H ,introduced by Bobeth et al. [41], is suppressed by m2

! inthe Standard Model and is potentially sensitive to newphysics [1, 5]. The “forward-backward asymmetry” A!

FB

is zero in the Standard Model (up to negligible QEDcontributions [41, 42]) so is also a sensitive probe of newphysics. The flat term [41]

F !H(q2low, q

2high) =

∫ q2highq2low

dq2 (a! + c!)

∫ q2high

q2lowdq2 (a! + 1/3 c!)

(10)

is constructed as a ratio to reduce uncertainties. Eval-uated in experimentally motivated q2 bins, values forF e,µ,τH are given in Tables II and III.

SUMMARY AND OUTLOOK

Employing the first unquenched lattice QCD formfactors [12], we calculate: Standard Model differentialbranching fractions; branching fractions integrated overexperimentally motivated q2 bins; ratios of branchingfractions potentially sensitive to new physics; and the flatterm in the angular distribution of the differential decayrate. Where available, we compare with experiment andprevious calculations. For q2 >

∼ 10 GeV2 our results aremore precise than previous SM predictions. For all q2

our results are consistent with previous calculations andexperiment.

Predictions for observables involving the ditau finalstate are particularly precise and potentially sensitive tonew physics. Given this combination, measurements ofBτ , Rτ

! , or FτH by experimentalists would be particularly

interesting and welcome.

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0 4 8 12 16 20 24

dBl /

dq

2 [10-7

/ GeV

2 ]

q2 [GeV2]

J/! ! (2S) BelleBABARCDFLHCb [16]LHCb [17]

(a) Belle [14], BABAR [13], CDF [15], and LHCb [16, 17] dataand the Standard Model contribution to dB!/dq

2, ! = e, µ.

0

5e-09

1e-08

1.5e-08

2e-08

2.5e-08

0 4 8 12 16 20 24q2 [GeV2]

f+f0fTinputsCi

(b) Error components for dB!/dq2.

0

0.1

0.2

0.3

12 15 18 21 24

dB" /

dq

2 [10-7

/ GeV

2 ]

q2 [GeV2]

! (2S)

(c) Predicted Standard Model contribution to dBτ /dq2.

0

2e-09

4e-09

6e-09

8e-09

1e-08

12 15 18 21 24q2 [GeV2]

f+f0fTinputsCi

(d) Error components for dBτ/dq2.

FIG. 3: (left) Standard Model differential branching fractions and experiment. (right) Form factor, input parameter,and Wilson coefficient (Ci) contributions to the error. The total error is the sum in quadrature of the components.

where θ! is the angle between the B and "− as mea-sured in the dilepton rest frame. The “flat term” F !

H ,introduced by Bobeth et al. [41], is suppressed by m2

! inthe Standard Model and is potentially sensitive to newphysics [1, 5]. The “forward-backward asymmetry” A!

FB

is zero in the Standard Model (up to negligible QEDcontributions [41, 42]) so is also a sensitive probe of newphysics. The flat term [41]

F !H(q2low, q

2high) =

∫ q2highq2low

dq2 (a! + c!)

∫ q2high

q2lowdq2 (a! + 1/3 c!)

(10)

is constructed as a ratio to reduce uncertainties. Eval-uated in experimentally motivated q2 bins, values forF e,µ,τH are given in Tables II and III.

SUMMARY AND OUTLOOK

Employing the first unquenched lattice QCD formfactors [12], we calculate: Standard Model differentialbranching fractions; branching fractions integrated overexperimentally motivated q2 bins; ratios of branchingfractions potentially sensitive to new physics; and the flatterm in the angular distribution of the differential decayrate. Where available, we compare with experiment andprevious calculations. For q2 >

∼ 10 GeV2 our results aremore precise than previous SM predictions. For all q2

our results are consistent with previous calculations andexperiment.

Predictions for observables involving the ditau finalstate are particularly precise and potentially sensitive tonew physics. Given this combination, measurements ofBτ , Rτ

! , or FτH by experimentalists would be particularly

interesting and welcome.

f+, f0, fT

B → K�+�−HPQCD: 1306.0434; 1306.2384.

� = e, µ

� = τ


-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0q2 in GeV2

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

f +(q2 )

exptvcoarse f+coarse f+fine f+expt fitlattice fit

Electromagnetic transitions provide important tests

Test lattice QCD vs direct experiment at low spacelike q2 ; <r2> sensitive to light quark mass

γ

*HPQCD preliminary at physical u/d masses*

NA7

pion electromagneticform factor

ETMC: 0812.4042; JLQCD:0810.2590; Mainz:1306.2916

1.2

1.4

1.6

1.8

2

2.2

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

V(0

)

a2 / fm2

HISQtwisted mass

expt

J/ψ → γηc

1206.14451208.2855

CLEO-c

Different quark formalisms agree. Lattice more accurate than experiment.


Muon anomalous magnetic moment �µ = ge

2m�S

aµ =g − 2

2= O(10−3)anomaly

Hadronic corrections to the muon g−2 from lattice QCD T. Blum

Table 1: Standard Model contributions to the muon anomaly. The QED contribution is through α5, EWα2, and QCD α3. The two QED values correspond to different values of α , and QCD to lowest order (LO)contributions from the hadronic vacuum polarization (HVP) using e

+e− → hadrons and τ → hadrons, higher

order (HO) from HVP and an additional photon, and hadronic light-by-light (HLbL) scattering.

QED 11658471.8845(9)(19)(7)(30)×10−10 [2]11658471.8951(9)(19)(7)(77)×10−10 [2]

EW 15.4(2)×10−10 [5]QCD LO (e+e

−) 692.3(4.2)×10−10, 694.91(3.72)(2.10)×10−10 [3, 4]LO (τ) 701.5(4.7)×10−10 [3]HO HVP −9.79(9)×10−10 [6]HLbL 10.5(2.6)×10−10 [9]

The HVP contribution to the muon anomaly has been computed using the experimentallymeasured cross-section for the reaction e

+e− → hadrons and a dispersion relation to relate the real

and imaginary parts of Π(Q2). The current quoted precision on such calculations is a bit more thanone-half of one percent [3, 4]. The HVP contributions can also be calculated from first principlesin lattice QCD [8]. While the current precision is significantly higher for the dispersive method,lattice calculations are poised to reduce errors significantly in next one or two years. These willprovide important checks of the dispersive method before the new Fermilab experiment. Unlikethe case for aµ(HVP), aµ(HLbL) can not be computed from experimental data and a dispersionrelation (there are many off-shell form factors that enter which can not be measured). While modelcalculations exist (see [9] for a summary), they are not systematically improvable. A determinationusing lattice QCD where all errors are controlled is therefore desirable.

In Sec. 2 we review the status of lattice calculations of aµ(HVP). Section 3 is a presentationof our results for aµ(HLbL) computed in the framework of lattice QCD+QED. Section 4 gives ourconclusions and outlook for future calculations.

Z

W

Z...

Figure 1: Representative diagrams, up to order α3, in the Standard Model that contribute to the muonanomaly. The rows, from to top to bottom, correspond to QED, EW, and QCD. Horizontal solid linesrepresent the muon, wiggly lines denote photons unless otherwise labeled, other solid lines are leptons,filled loops denote quarks (hadrons), and the dashed line represents the higgs boson.

3

e.g. from QED: µγ

Blum lattice2012:1301.2607

BNL, to be improved at FNAL and J-PARC= 3σ mismatch

QCD contribn from hadronic vac. pol and ‘light-by-light’.

e+e−from

a = 0.086 fm, L = 2.8 fma = 0.078 fm, L = 3.7 fma = 0.078 fm, L = 2.5 fma = 0.078 fm, L = 1.9 fma = 0.061 fm, L = 2.9 fma = 0.061 fm, L = 1.9 fm

dispersive analysisa → 0 result

Preliminary

m2PS

[

GeV2]

aN

f=2+1+1

µ0.250.20.150.10.050

7e-08

6e-08

5e-08

4e-08

3e-08

2e-08

1e-08

0

ETM underway : nf=2+1+1

IntroductionThe hadronic vacuum polarization (HVP) contribution (O(α2))

The hadronic light-by-light (HLbL) contribution (O(α3))aµ Implications for new physicsaµ(HLbL) Summary/Outlook

Hadronic vacuum polarization (HVP)

+The blobs, which represent all possible intermediate hadronicstates, are not calculable in perturbation theory, but can becalculated from

� dispersion relation + experimental cross-section for

e+e−(and τ) → hadrons ahad(2)µ = 14π2

�∞4m2

πds K (s)σtotal(s)

� first principles using lattice QCD,

a(2)hadµ =�απ

�2 �∞0 dQ2 f (Q2)Π(Q2) [Lautrup and de Rafael 1969, Blum 2002]

Tom Blum (UConn / RIKEN BNL Research Center)Masashi Hayakawa (Nagoya University)Taku Izubuchi (BNL / RIKEN BNL Research Center)The muon anomalous magnetic moment

IntroductionThe hadronic vacuum polarization (HVP) contribution (O(α2))

The hadronic light-by-light (HLbL) contribution (O(α3))aµ Implications for new physicsaµ(HLbL) Summary/Outlook

HLbL

+ + ...Blobs: all possible hadronic states

Model estimates put this O(α3) contribution atabout (10−12)×10−10 with a 25-40% uncertainty

No dispersion relation a’la vacuum polarization

Lattice regulator: model independent, approxima-tions systematically improvable

Tom Blum (UConn / RIKEN BNL Research Center)Masashi Hayakawa (Nagoya University)Taku Izubuchi (BNL / RIKEN BNL Research Center)The muon anomalous magnetic moment

500 600 700 800aµ(HVP) x 1010

ETM 1103.4818

Aubin,Blumhep-lat/0608011

RBC/UKQCD1107.1497

Mainz 1112.2894

nf=2+1

nf=2

aexptµ − aSMµ ≈ 29(8)× 10−10

Progress on HVP HLBL needs QCD+QED

~10x10-10

clover-quenched s

domain wall

asqtad

twisted mass


K → ππ ∆I = 1/2and the rule

Christ: Lattice2012; RBC/UKQCD: 1212.1474; 1111.1699

450 times more likely than K → ππI=0K → ππI=2

3

Li

i

s j j

L

π

πK

Contraction 1�.

Lij

s j iL

π

πK

Contraction 2�.

FIG. 1: The two contractions contributing to ReA2.

They are distinguished by the color summation (i, jdenote color). s denotes the strange quark and L that

the currents are left-handed.

we find ReA2 = (1.381±0.046±0.258) 10−8 GeV to which

the EWP operators contribute −0.0171 10−8 GeV [4, 5]

(the physical value is ReA2 = 1.479(4)10−8 GeV). We

therefore neglect the EWP operators in the following dis-

cussion. Chiral symmetry implies that Q3/2(27,1) does not

mix with the EWP operators so that ReA2 is propor-

tional to its lattice matrix element; the constant of pro-

portionality is the product of the Wilson coefficient, the

renormalization constant, finite-volume effects and kine-

matical factors (see [5] for a detailed discussion, including

an explicit demonstration that the mixing is indeed neg-

ligible in the DWF simulation).

Fierz transformations allow the K → ππ correlation

function of Q3/2(27,1) to be reduced to the sum of the two

contractions illustrated in Fig. 1, labeled by 1� and 2�.

The two contractions are identical except for the way

that the color indices are summed. A2 is proportional to

the matrix element extracted from the sum 1�+ 2�. The

main message of this letter is our observation from all

three simulations that 1� and 2� have opposite signs and

are comparable in size. This is illustrated in Fig. 2 for the

results at physical kinematics from [4, 5], where we plot

1�, - 2� and 1� + 2� as functions of t. We extract A2 by

fitting 1�+ 2� in the interval t ∈ [5, 19] where there is a

significant cancellation between the two terms. A similar,

although not quite so pronounced cancellation occurs at

threshold for physical masses and for the heavier masses

studied in [3, 10], see Fig. 3 for example.

We stress that it is only the correlation function 1�+ 2�which has a time behaviour corresponding to E(ππ)2 .

Because the calculation is performed in a finite-volume

E(ππ)2 �= E(ππ)0 and 1� and 2� individually have an

isospin 0 component. If E(ππ)2 = mK then 1�+ 2� is in-

dependent of t away from the kaon and two-pion sources,

and this is what we observe, particularly in Fig. 2 where

the energies are matched most precisely.

It has been argued that the factorisation hypothe-

sis [14] works reasonably well in reproducing the experi-

mental value of A2 (see e.g. Sec.VIII-4 in [15]). In this

approach, the gluonic interactions between the quarks

combining into different pions are neglected and A2 is

related to the decay constant fπ and the K�3 form factor

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 4 8 12 16 20 24

C2,2(∆

,t)� 10

9�

t

1�− 2�

1�+ 2�

FIG. 2: Contractions 1�, - 2� and 1�+ 2� as functions of

t from the simulation at physical kinematics and with

∆ = 24 .

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 2 4 6 8 10 12 14 16 18 20

C2,2(∆

,t)� 10

8�

t

1�− 2�

1�+ 2�


t from the simulation at threshold with mπ � 330MeV

and ∆ = 20.

close to zero momentum transfer. On the basis of color

counting, one might therefore expect that 2� � 1/3 1�,

whereas, for physical kinematics, we find 2� � −0.7 1�and that nevertheless 1�+ 2� leads to the correct result

for A2. Thus the expectation based on the factorisation

hypothesis proves to be unreliable here.

Following the discovery that 1� and 2� have opposite

signs we examined separately the two contributions to

the matrix element �K̄0|(s̄d)L(s̄d)L|K0� which contains

the non-perturbative QCD effects in neutral kaon mix-

ing [11]. The two contributions correspond to Wick con-

tractions in which the two quark fields in the K0 interpo-

lating operator are contracted i) with fields from the same

current in (s̄d)L(s̄d)L and ii) with one field from each of

the two currents. Color counting and the vacuum inser-

3

Li

i

s j j

L

π

πK

Contraction 1�.

Lij

s j iL

π

πK

Contraction 2�.













































0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 4 8 12 16 20 24

C2,2(∆

,t)� 10

9�

t

1�− 2�

1�+ 2�



∆ = 24 .

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 2 4 6 8 10 12 14 16 18 20

C2,2(∆

,t)� 10

8�

t

1�− 2�

1�+ 2�



and ∆ = 20.















nonperturbative enhancement of matrix elementsof 4-quark operators from weak Hamiltonian

3

Li

i

s j j

L

π

πK

Contraction 1�.

Lij

s j iL

π

πK

Contraction 2�.













































0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 4 8 12 16 20 24

C2,2(∆

,t)� 10

9�

t

1�− 2�

1�+ 2�



∆ = 24 .

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 2 4 6 8 10 12 14 16 18 20

C2,2(∆

,t)� 10

8�

t

1�− 2�

1�+ 2�



and ∆ = 20.















I=2 amplitude now calculated directly in lattice QCD; I=0 underway

Find cancellation in A2 between different color contractions of key operator (same effect will enhance A0) .

HW

HW

naive expectn: (ME2)=(ME1)/3Tuesday, 25 June 2013

Ken Wilson

1936-2013

Originator of Lattice QCD


Future• sets of ‘2nd generation’ gluon configs now have at physical value (so no extrapoln) or down to 0.05fm (so b quarks are ‘light’) also can include charm in the sea now.

mu,da

Conclusion• Lattice QCD results for gold-plated meson masses, decay constants and form factors provide stringent tests of QCD/Standard Model. • Gives QCD parameters and some CKM elements to 1-2% and constrains Beyond the Standard Model physics.

• v. high statistics/large volumes needed for harder calculations (precision baryon physics, flavor singlet /glueball spectroscopy, excited states, nuclear physics) will become available with increased computer power...Lattice 2013: Mainz, July 29th www.lattice2013.uni-mainz.de


http://www.lattice2013.uni-mainz.de

http://www.lattice2013.uni-mainz.de

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Results from Lattice QCDcdavies/talks/lepton-photon-2013.pdf · 2015. 6. 3. · Hadron spectrum Hadron structure QCD parameters CKM elements ... Quantum chromodynamics 0.11 0.12 0.13!