Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
Conclusions
1/77
Loops and Volumes inChiral Perturbation Theory
Johan Bijnens
Lund University
http://www.thep.lu.se/∼bijnens
Advances in Effective Field Theories Julich, Germany 7-9 November 2016
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
Conclusions
2/77
Overview
1 Overview
2 Chiral Perturbation TheoryA selective historyWhat is it?A mesonic ChPT program framework
3 Some examples at two or more loops: one masspi form-factorπ+ polarizabilityBeyond QCD or BSMLeading logarithms
4 Three flavour: more massesg − 2 HVP connected, disconnected and strangeFinite Volume
Masses at two-loopsTwistingHVP TwistedMasses twistedK`3: twisted at one-loopK`3: MILC and staggeredComments FV
5 Conclusions
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
3/77
Chiral Perturbation Theory: A selective history
Goldberger-Treiman relation:
M. L. Goldberger and S. B. Treiman, “Decay of the pimeson,” Phys. Rev. 110 (1958) 1178.
PCAC:
M. Gell-Mann and M. Levy, “The Axial Vector Current InBeta Decay,” Nuovo Cim. 16 (1960) 705.Y. Nambu, “Axial Vector Current Conservation in WeakInteractions,” Phys. Rev. Lett. 4 (1960) 380.
Nambu-Goldstone bosons:
Y. Nambu, “Quasi-Particles and Gauge Invariance in theTheory of Superconductivity”, Phys. Rev. 117 (1960) 648.J. Goldstone, “Field Theories with SuperconductorSolutions”, Nuovo Cim. 19 (1961) 154.J. Goldstone, A. Salam, S. Weinberg, “BrokenSymmetries”, Phys. Rev. 127 (1962) 965.
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
4/77
Chiral Perturbation Theory: A selective history
Current algebra:
M. Gell-Mann, “The Symmetry group of vector and axialvector currents,” Physics 1 (1964) 63.S. Fubini and G. Furlan, “Renormalization effects forpartially conserved currents,” Physics 1 (1965) 229.
Many applications:
S. L. Adler, “Sum rules for the axial vector couplingconstant renormalization in beta decay,” Phys. Rev. 140(1965) B736 Erratum: [Phys. Rev. 149 (1966) 1294]Erratum: [Phys. Rev. 175 (1968) 2224].W. I. Weisberger, “Unsubtracted Dispersion Relations andthe Renormalization of the Weak Axial Vector CouplingConstants,” Phys. Rev. 143 (1966) 1302.S. Weinberg, “Pion scattering lengths,” Phys. Rev. Lett.17 (1966) 616.
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
5/77
Chiral Perturbation Theory: A selective history
Effective Lagrangians:S. Weinberg, “Nonlinear realizations of chiral symmetry,”Phys. Rev. 166 (1968) 1568.S. R. Coleman, J. Wess and B. Zumino, “Structure ofphenomenological Lagrangians. 1.,” Phys. Rev. 177(1969) 2239.C. G. Callan, Jr., S. R. Coleman, J. Wess and B. Zumino,“Structure of phenomenological Lagrangians. 2.,” Phys.Rev. 177 (1969) 2247.
Chiral logarithms and loops:L. F. Li and H. Pagels, “Perturbation theory about aGoldstone symmetry,” Phys. Rev. Lett. 26 (1971) 1204.H. Lehmann, “Chiral invariance and effective rangeexpansion for pion pion scattering,” PLB 41 (1972) 529.G. Ecker and J. Honerkamp, “Pion pion scattering from ansu(3) x su(3) invariant lagrangian,” NPB 62 (1973) 509.P. Langacker and H. Pagels, “Chiral perturbation theory,”Phys. Rev. D 8 (1973) 4595.
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
6/77
Chiral Perturbation Theory: A selective history
S. Weinberg, “Phenomenological Lagrangians,” Physica A96 (1979) 327. (2886 citations on 01 Nov 2016 Inspire)
J. Gasser and H. Leutwyler, “Chiral Perturbation Theoryto One Loop,” Annals Phys. 158 (1984) 142. (3645)
J. Gasser and H. Leutwyler, “Chiral Perturbation Theory:Expansions in the Mass of the Strange Quark,” Nucl.Phys. B 250 (1985) 465. (3454)
5931 citations to at least one of the above
184 due to Ulf (out of more than 700 inspire items)
Some others: Oset (135), Bijnens (134), Pelaez (104),Pich (101), Scherer (87), Leutwyler (79), Ecker (74),Gasser (66), Donoghue (60), Holstein (59), V.Bernard(54), G.Colangelo (52),. . .(done by checking some names)
Now sufficiently well known to no longer be cited
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
7/77
Chiral Perturbation Theory: A selective history
Early followers (i.e. p4 and loop calculations, selection):
G. D’Ambrosio and D. Espriu, “Rare Decay Modes of theK Mesons in the Chiral Lagrangian,” PLB 175 (1986) 237.
J. L. Goity, “The Decays K 0(s) → γγ and K 0(`) → γγ inthe Chiral Approach,” Z. Phys. C 34 (1987) 341.
D. B. Kaplan and A. V. Manohar, “Current Mass Ratiosof the Light Quarks,” Phys. Rev. Lett. 56 (1986) 2004.
J. Gasser, M. E. Sainio and A. Svarc, “Nucleons withChiral Loops,” Nucl. Phys. B 307 (1988) 779.
G. Ecker, A. Pich and E. de Rafael, “Radiative KaonDecays and CP Violation in ChPT,” NPB 303 (1988) 665.
J. Bijnens and F. Cornet, “Two Pion Production inPhoton-Photon Collisions,” Nucl. Phys. B 296 (1988) 557.
J. Donoghue, B. Holstein and Y. C. Lin, “The Reactionγγ → π0π0 and Chiral Loops,” PRD37(1988)2423.
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
8/77
Chiral Perturbation Theory: A selective history
And when did Ulf start (first citation and first ChPT papers)
I. Zahed, A. Wirzba and U. G. Meissner, “Soft pioncorrections to the Skyrme soliton,” PRD 33 (1986) 830.J. Gasser and U. G. Meissner, “Chiral expansion of pionform-factors beyond one loop,” NPB 357 (1991) 90.V. Bernard, N. Kaiser and U. G. Meissner, “πK scatteringin ChPT to one loop,” NPB 357 (1991) 129.V. Bernard, N. Kaiser and U. G. Meissner, “πη scatteringin QCD,” PRD 44 (1991) 3698.V. Bernard,N. Kaiser, U. G. Meissner, “Chiral expansion ofthe nucleon’s electromagnetic polarizabilities,”PRL67(1991)1515V. Bernard,N. Kaiser,J. Gasser, U. G. Meissner, “Neutralpion photoproduction at threshold,” PLB 268 (1991) 291.V. Bernard, N. Kaiser, J. Kambor and U. G. Meissner,“Chiral structure of the nucleon,” NPB 388 (1992) 315.
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
9/77
Chiral Perturbation Theory
Exploring the consequences ofthe chiral symmetry of QCDand its spontaneous breaking
using effective field theory techniques
Derivation from QCD:H. Leutwyler,On The Foundations Of Chiral Perturbation Theory,Ann. Phys. 235 (1994) 165 [hep-ph/9311274]
For references to lectures see:http://www.thep.lu.se/˜bijnens/chpt/
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
10/77
Chiral Perturbation Theory
A general Effective Field Theory:
Relevant degrees of freedom
A powercounting principle (predictivity)
Has a certain range of validity
Chiral Perturbation Theory:
Degrees of freedom: Goldstone Bosons from spontaneousbreaking of chiral symmetry
Powercounting: Dimensional counting in momenta/masses
Breakdown scale: Resonances, so about Mρ.
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
10/77
Chiral Perturbation Theory
A general Effective Field Theory:
Relevant degrees of freedom
A powercounting principle (predictivity)
Has a certain range of validity
Chiral Perturbation Theory:
Degrees of freedom: Goldstone Bosons from spontaneousbreaking of chiral symmetry
Powercounting: Dimensional counting in momenta/masses
Breakdown scale: Resonances, so about Mρ.
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
11/77
Goldstone Bosons
Spontaneous breakdown
〈qq〉 = 〈qLqR + qRqL〉 6= 0
SU(3)L × SU(3)R broken spontaneously to SU(3)V
8 generators broken =⇒ 8 massless degrees of freedomand interaction vanishes at zero momentum
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
12/77
Goldstone Bosons
Power counting in momenta: Meson loops, Weinbergpowercounting
rules one loop example
p2
1/p2
∫d4p p4
(p2)2 (1/p2)2 p4 = p4
(p2) (1/p2) p4 = p4
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
13/77
Chiral Perturbation Theories
Which chiral symmetry: SU(Nf )L × SU(Nf )R , forNf = 2, 3, . . . and extensions to (partially) quenched
Or beyond QCD
Space-time symmetry: Continuum or broken on thelattice: Wilson, staggered, mixed action
Volume: Infinite, finite in space, finite T
Which interactions to include beyond the strong one
Which particles included as non Goldstone Bosons
Ulf has done a lot in applying it to new areas
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
14/77
Lagrangians: Lagrangian structure (mesons,strong)
2 flavour 3 flavour PQChPT/Nf flavour
p2 F ,B 2 F0,B0 2 F0,B0 2
p4 l ri , hri 7+3 Lr
i ,Hri 10+2 Lr
i , Hri 11+2
p6 c ri 51+4 C r
i 90+4 K ri 112+3
p2: Weinberg 1966
p4: Gasser, Leutwyler 84,85
p6: JB, Colangelo, Ecker 99,00; Weber 2008à Li LEC = Low Energy Constants = ChPT parametersà Hi : contact terms: value depends on definition of cur-rents/densitiesà Finite volume: no new LECsà Other effects: (many) new LECs
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
15/77
Mesons: which Lagrangians are known (nf = 3)
Loops Lorder LECs effects included
Lp2 2 strong (+ external W , γ)Le2p0 1 internal γ
L = 0 L∆S=1GFp2 2 nonleptonic weak
L∆S=1G8e2p0 1 nonleptonic weak+internal γ
Loddp4 0 WZW, anomaly
Lp4 10 strong (+ external W , γ)Le2p2 13 internal γ
L∆S=1G8Fp4 22 nonleptonic weak
L ≤ 1 L∆S=1G27p4 28 nonleptonic weak
L∆S=1G8e2p2 14 nonleptonic weak+internal γ
Loddp6 23 WZW, anomaly
Lleptonse2p2 5 leptons, internal γ
L ≤ 2 Lp6 90 strong (+ external W , γ)
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
16/77
Chiral Logarithms
The main predictions of ChPT:
Relates processes with different numbers ofpseudoscalars/axial currents
Chiral logarithms
includes Isospin and the eightfold way (SU(3)V )
Unitarity included perturbatively
m2π = 2Bm +
(2Bm
F
)2 [ 1
32π2log
(2Bm)
µ2+ 2l r3(µ)
]+ · · ·
M2 = 2Bm
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
17/77
LECs and µ
l r3(µ)
li =32π2
γil ri (µ)− log
M2π
µ2.
is independent of the scale µ.
For 3 and more flavours, some of the γi = 0: Lri (µ)
Choice of µ :
mπ, mK : chiral logs vanish
pick larger scale
1 GeV then Lr5(µ) ≈ 0
what about large Nc arguments????
compromise: µ = mρ = 0.77 GeV
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
18/77
Program availability
Making the programs more accessible for others to use:
Two-loop results have very long expressions
Many not published but available fromhttp://www.thep.lu.se/∼bijnens/chpt/
Many programs available on request from the authors
Idea: make a more general framework
CHIRON:
JB,
“CHIRON: a package for ChPT numerical results
at two loops,”
Eur. Phys. J. C 75 (2015) 27 [arXiv:1412.0887]
http://www.thep.lu.se/∼bijnens/chiron/
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
A selectivehistory
What is it?
A mesonic ChPTprogramframework
Two or moreloops
Three flavour:more masses
Conclusions
19/77
Program availability: CHIRON
Present version: 0.54
Classes to deal with Li ,Ci , L(n)i ,Ki , standardized
in/output, changing the scale,. . .
Loop integrals: one-loop and sunsetintegrals
Included so far (at two-loop order):Masses, decay constants and 〈qq〉 for the three flavour caseMasses and decay constants at finite volume in the threeflavour caseMasses and decay constants in the partially quenched casefor three sea quarksMasses and decay constants in the partially quenched casefor three sea quarks at finite volume
A large number of example programs is included
Manual has already reached 94 pages
I am continually adding results from my earlier work
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
20/77
The first one (at least some parts)
J. Gasser and U. G. Meissner, “Chiral expansion of pionform-factors beyond one loop,” NPB 357 (1991) 90.Using dispersive methods to determine the nonanalyticdependence on q2 in the pion vector and scalarform-factor for two-flavoursDone to two-loop order fully: JB, G. Colangelo and P. Talavera,
“The Vector and scalar form-factors of the pion to two loops,” JHEP 9805
(1998) 014 [hep-ph/9805389]
also known for three-flavour case: JB and P. Talavera, “Pion and
kaon electromagnetic form-factors,” JHEP 0203 (2002) 046
[hep-ph/0203049]; JB and P. Dhonte, “Scalar form-factors in SU(3) chiral
perturbation theory,” JHEP 0310 (2003) 061 [hep-ph/0307044]
Back to dispersive (just one example) B. Ananthanarayan,
I. Caprini, D. Das and I. Sentitemsu Imsong, “Parametrisation-free
determination of the shape parameters for the pion electromagnetic form
factor,” Eur. Phys. J. C 73 (2013) 2520 [arXiv:1302.6373 [hep-ph]]
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
21/77
Pion form-factor: quality of fit
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.25 -0.2 -0.15 -0.1 -0.05 0
|FπV|2
t [GeV2]
full result
p4
p6
p8
data
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
22/77
Charged pion polarizabilities: experiment
An example where ChPT triumphedReview: Holstein, Scherer, Ann. Rev. Nucl. Part. Sci. 64 (2014) 51 [1401.0140]
Expand γπ± → γπ± near threshold: (z± = 1± cos θcm)
dσ
dΩ=
dσ
dΩ Born−αm3
π
((s −m2
π
)2
4s2 (sz+ + m2πz−)
(z2−(α− β) +
s2
m4π
z2+(α + β)
)Three ways to measure: (all assume α + β = 0)
πγ → πγ (Primakoff, high energy pion beam)Dubna (1985) α = (6.8± 1.4) 10−4 fm3
Compass (CERN, 2015) α = (2.0± 0.6± 0.7) 10−4 fm3
γπ → πγ (via one-pion exchange)Lebedev (1986) α = (20± 12) 10−4 fm3
Mainz (2005) α = (5.8± 0.75± 1.5± 0.25) 10−4 fm3
γγ → ππ (in e+e− → e+e−π+π−)MarkII data analyzed (1992) α = (2.2± 1.1) 10−4 fm3
Extrapolation and subtraction: difficult experiments
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
23/77
Polarizabilities: extrapolations needed
γ + γ → π+ + π
-
Wππ
(GeV)
σto
t (
|co
s(θ
ππ)|
< 0
.6 )
(
nb
)
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
0.3 0.4 0.5 0.6 0.7 0.8
from Pasquini et al. 2008
γp → πγnγ π
Off-shell π
γ
p n
πN → πγNπ π
Off-shell γγ
N N
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
24/77
Charged pion polarizabilities: theory
ChPT:
One-loop JB, Cornet, 1986, Donoghue-Holstein 1989
α + β = 0, α = (2.8± 0.2) 10−4 fm3
input π → eνγ (error only from this)Two-loop Burgi, 1996, Gasser, Ivanov, Sainio 2006
α + β = 0.16 10−4 fm3,α = (2.8± 0.5) 10−4 fm3
Dispersive analysis from γγ → ππ:
Fil’kov-Kashevarov, 2005 (α1 − β1) = (13.0+2.6−1.9) · 10−4fm3
Critized by Pasquini-Drechsel-Scherer, 2008
“Large model dependence in their extraction”“Our calculations. . . are in reasonable agreement withChPT for charged pions”(α1 − β1) = (5.7) · 10−4fm3 perfectly possible
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
25/77
QCDlike and/or technicolor theories
One can also have different symmetry breaking patternsfrom underlying fermions
Three generic cases
SU(N)× SU(N)/SU(N)SU(2N)/SO(2N) (Dirac) or SU(N)/SO(N) (Majorana)SU(2N)/Sp(2N)
Many one-loop results existed especially for the first case(several discovered only after we published our work)
Equal mass case pushed to two loops JB, Lu, 2009-11
Majorana, Finite Volume and partially quenched addedJB, Rossler, arXiv:1509.04082
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
26/77
NF fermions in a representation of the gauge group
complex (QCD):qT = (q1 q2 . . . qNF
)Global G = SU(NF )L × SU(NF )RqL → gLqL and gR → gRqR
Vacuum condensate Σij = 〈qjqi 〉 ∝ δijgL = gR then Σij → Σij =⇒ conserved H = SU(NF )V :
Real (e.g. adjoint): qT = (qR1 . . . qRNFqR1 . . . qRNF
)qRi ≡ C qT
Li goes under gauge group as qRi
some Goldstone bosons have baryonnumberGlobal G = SU(2NF ) and q → gq
〈qjqi 〉 is really 〈(qj)TC qi 〉 ∝ JSij JS =
(0 II 0
)Conserved if gJSgT = JS =⇒ H = SO(2NF )
Real with NF Majorana fermionssome Goldstone bosons have baryonnumberGlobal G = SU(2NF ) and q → gqMajorana condensate is 〈(qj)
TC qi 〉 ∝ δij = IijConserved gIgT = I
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
26/77
NF fermions in a representation of the gauge group
complex (QCD):qT = (q1 q2 . . . qNF
)Global G = SU(NF )L × SU(NF )RqL → gLqL and gR → gRqR
Vacuum condensate Σij = 〈qjqi 〉 ∝ δijgL = gR then Σij → Σij =⇒ conserved H = SU(NF )V :
Real (e.g. adjoint): qT = (qR1 . . . qRNFqR1 . . . qRNF
)qRi ≡ C qT
Li goes under gauge group as qRi
some Goldstone bosons have baryonnumberGlobal G = SU(2NF ) and q → gq
〈qjqi 〉 is really 〈(qj)TC qi 〉 ∝ JSij JS =
(0 II 0
)Conserved if gJSgT = JS =⇒ H = SO(2NF )
Real with NF Majorana fermionssome Goldstone bosons have baryonnumberGlobal G = SU(2NF ) and q → gqMajorana condensate is 〈(qj)
TC qi 〉 ∝ δij = IijConserved gIgT = I
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
27/77
NF fermions in a representation of the gauge group
complex (QCD): qT = (q1 q2 . . . qNF)
Global G = SU(NF )L × SU(NF )R qL → gLqL andgR → gRqR
Vacuum condensate Σij = 〈qjqi 〉 ∝ δijConserved H = SU(NF )V : gL = gR then Σij → Σij
Pseudoreal (e.g. two-colours):qT = (qR1 . . . qRNF
qR1 . . . qRNF)
qRαi ≡ εαβC qTLβi goes under gauge group as qRαi
some Goldstone bosons have baryonnumberGlobal G = SU(2NF ) and q → gq
〈qjqi 〉 is really εαβ〈(qαj)TC qβi 〉 ∝ JAij JA =
(0 −II 0
)Conserved if gJAgT = JA =⇒ H = Sp(2NF )
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
28/77
Lagrangians
JB, Lu, arXiv:0910.5424, JB Rossler 1509.04082:
4 cases similar with u = exp(
i√2FφaX a
)But the matrices X a are:
Complex or SU(N)× SU(N)/SU(N):all SU(N) generators
Real or SU(2N)/SO(2N):SU(2N) generators with X aJS = JSX aT
Pseudoreal or SU(2N)/Sp(2N):SU(2N) generators with X aJA = JAX aT
Real Majorana or SU(N)/SO(N):SU(N) generators with X a = X aT
SO(2N): not usual way of parametrizing SO(2N) matrices
the two are related by a U(2N) transformation:same ChPT except for anomalous sector
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
29/77
Lagrangians
u → hug †L ≡ gRuh† for complex
u → hug † for real, pseudoreal
h is in the conserved part of the group for all cases
uµ = i(u†∂µu − u∂µu†
)→ huµh†
external fields can also be included.
a generalized mass term χ± → hχ±h† can be defined withχ± = u†χu† ± uχu
LLO = F 2
4 〈uµuµ + χ+〉L4 = L0〈u
µuνuµuν〉 + L1〈uµuµ〉〈uνuν〉 + L2〈u
µuν〉〈uµuν〉 + L3〈uµuµu
νuν〉
+L4〈uµuµ〉〈χ+〉 + L5〈u
µuµχ+〉 + L6〈χ+〉2 + L7〈χ−〉2 +
1
2L8〈χ
2+ + χ
2−〉
−iL9〈f+µνuµuν〉 +
1
4L10〈f
2+ − f 2
−〉 + H1〈lµν lµν + rµν r
µν〉 + H2〈χχ†〉 .
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
30/77
The main useful formulae
Calculating for equal mass case goes through using:
Complex :⟨X aAX aB
⟩= 〈A〉 〈B〉 −
1
NF
〈AB〉 ,
⟨X aA
⟩ ⟨X aB
⟩= 〈AB〉 −
1
NF
〈A〉 〈B〉 .
Real :⟨X aAX aB
⟩=
1
2〈A〉 〈B〉 +
1
2
⟨AJSB
T JS
⟩−
1
2NF
〈AB〉 ,
⟨X aA
⟩ ⟨X aB
⟩=
1
2〈AB〉 +
1
2
⟨AJSB
T JS
⟩−
1
2NF
〈A〉 〈B〉 .
Pseudoreal :⟨X aAX aB
⟩=
1
2〈A〉 〈B〉 +
1
2
⟨AJAB
T JA
⟩−
1
2NF
〈AB〉 ,
⟨X aA
⟩ ⟨X aB
⟩=
1
2〈AB〉 −
1
2
⟨AJAB
T JA
⟩−
1
2NF
〈A〉 〈B〉
So can do the calculations for all casesFor the partially quenched case: extension needed
JB, Rossler, arXiv:1509.04082 (done with quark-flow method)
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
31/77
φφ→ φφ
ππ scattering
Amplitude in terms of A(s, t, u)Mππ(s, t, u) = δ
abδcdA(s, t, u) + δ
acδbdA(t, u, s) + δ
adδbcA(u, s, t) .
Three intermediate states I = 0, 1, 2
Our three casesTwo amplitudes needed B(s, t, u) and C (s, t, u)
M(s, t, u) =[⟨
X aXbX cXd⟩
+⟨X aXdX cXb
⟩]B(s, t, u)
+[⟨
X aX cXdXb⟩
+⟨X aXbXdX c
⟩]B(t, u, s)
+[⟨
X aXdXbX c⟩
+⟨X aX cXbXd
⟩]B(u, s, t)
+δabδcdC(s, t, u) + δacδbdC(t, u, s) + δ
adδbcC(u, s, t) .
B(s, t, u) = B(u, t, s) C(s, t, u) = C(s, u, t) .
7, 6 and 6 possible intermediate states
All formulas similar length to ππ cases but there are somany of them
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
32/77
φφ→ φφ: aI0/n
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
aI 0/n
M2phys [GeV
2]
n = 2
LO
NLO
NNLO
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
aI 0/n
M2phys [GeV
2]
n = 3
LO
NLO
NNLO
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
aI 0/n
M2phys [GeV
2]
n = 4
LO
NLO
NNLO
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
aI 0/n
M2phys [GeV
2]
n = 5
LO
NLO
NNLO
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
33/77
Weinberg’s argument for leading logarithms
The recursive argument of renromalizable theories doesnot work
Weinberg, Physica A96 (1979) 327
Two-loop leading logarithms can be calculated using onlyone-loop: Weinberg consistency conditions
ππ at 2-loop: Colangelo, hep-ph/9502285
General at 2 loop: JB, Colangelo, Ecker, hep-ph/9808421
Proof at all orders
using β-functions: Buchler, Colangelo, hep-ph/0309049
with diagrams: JB, Carloni, arXiv:0909.5086
Extension to nucleons: JB, Vladimirov, arXiv:1409.6127
Underlying reason: nonlocal divergences must cancel
JB, Vladimirov, Polyakov, Carloni, Lanz, Kampf
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
34/77
Mass to order ~6
0
(1)
0
(2)
0
(3)
0
(4)
0
(5)
0
(6)
0
0
(7)
0
0
(8)
0
0
(9)
0
0
(10)
0
0
(11)
0
0
(12)
0
0
(13)
0
0
(14)
0
0
(15)
0
0 0
(16)
0
0 0
(17)
0
0 0
(18)
0
0 0
(19)
0
0 0
(20)
0
0 0
(21)
0
0 0
(22)
0
0
0
0
(23)
0
0
0
0
(24)
0
0
0
0
(25)
0
0
0
0
(26)
0
0
0
0
(27)
0
0
0 0
0
(28)
0
0
0 0
0
(29)
0
0
0
0
0
0
(30)
1
(31)
1
(32)
1
(33)
1
(34)
1
(35)
0
1
(36)
0
1
(37)
0
1
(38)
0
1
(39)
0
1
(40)
0
1
(41)
0
1
(42)
0
1
(43)
0
1
(44)
0
1
(45)
0
1
(46)
0
1
(47)
0
1
(48)
0
1
(49)
0
1
(50)
0
0 1
(51)
0
0 1
(52)
0
0 1
(53)
0
0 1
(54)
0
0 1
(55)
0
0 1
(56)
0
0 1
(57)
0
0 1
(58)
0
0 1
(59)
0
0 1
(60)
0
0 1
(61)
0
0 1
(62)
0
0 1
(63)
0
0
0
1
(64)
0
0
0
1
(65)
0
0
0
1
(66)
0
0
0
1
(67)
0
0
0
1
(68)
0
0
0
1
(69)
0
0
0
1
(70)
0
0
0
1
(71)
0
0
0
1
(72)
0
0
0
1
(73)
0
0
0
1
(74)
0
0
0 0
1
(75)
0
0
0 0
1
(76)
0
0
0 0
1
(77)
0
0
0 0
1
(78)
0
0
0
0
0
1
(79)
2
(80)
2
(81)
2
(82)
2
(83)
0
2
(84)
0
2
(85)
0
2
(86)
0
2
(87)
0
2
(88)
0
2
(89)
0
2
(90)
0
2
(91)
0
2
(92)
0
2
(93)
1
1
(94)
1
1
(95)
1
1
(96)
1
1
(97)
1
1
(98)
1
1
(99)
1
1
(100)
1
1
(101)
0
0 2
(102)
0
0 2
(103)
0
0 2
(104)
0
0 2
(105)
0
0 2
(106)
0
0 2
(107)
0
1 1
(108)
0
1 1
(109)
0
1 1
(110)
0
1 1
(111)
0
1 1
(112)
0
1 1
(113)
0
0 2
(114)
0
1 1
(115)
0
1 1
(116)
0
1 1
(117)
0
1 1
(118)
0
1 1
(119)
0
1 1
(120)
0
1 1
(121)
0
0
0
2
(122)
0
0
0
2
(123)
0
0
0
2
(124)
0
0
1
1
(125)
0
0
1
1
(126)
0
0
1
1
(127)
0
0
1
1
(128)
0
0
0
2
(129)
0
0
1
1
(130)
0
0
1
1
(131)
0
0
1
1
(132)
0
0
1
1
(133)
0
1
0
1
(134)
0
1
0
1
(135)
0
1
0
1
(136)
0
1
0
1
(137)
0
1
0
1
(138)
0
1
0
1
(139)
0
1
0
1
(140)
0
0
1
1
(141)
0
1
0
1
(142)
0
0
0 0
2
(143)
0
0
0 1
1
(144)
0
0
0 1
1
(145)
0
0
0 1
1
(146)
0
0
1 0
1
(147)
0
0
1 0
1
(148)
0
0
1 0
1
(149)
0
0
0 1
1
(150)
0
0
1 0
1
(151)
0
0
0
0
1
1
(152)
0
0
0
1
0
1
(153)
0
0
1
0
0
1
(154)
3
(155)
3
(156)
3
(157)
0
3
(158)
0
3
(159)
0
3
(160)
0
3
(161)
0
3
(162)
0
3
(163)
1
2
(164)
1
2
(165)
1
2
(166)
1
2
(167)
1
2
(168)
1
2
(169)
1
2
(170)
1
2
(171)
1
2
(172)
0
0 3
(173)
0
0 3
(174)
0
0 3
(175)
0
1 2
(176)
0
1 2
(177)
0
1 2
(178)
0
1 2
(179)
0
1 2
(180)
0
1 2
(181)
0
1 2
(182)
0
1 2
(183)
0
1 2
(184)
0
1 2
(185)
1
1 1
(186)
1
1 1
(187)
1
1 1
(188)
1
1 1
(189)
1
1 1
(190)
1
1 1
(191)
0
0
0
3
(192)
0
0
1
2
(193)
0
0
1
2
(194)
0
0
1
2
(195)
0
0
1
2
(196)
0
1
0
2
(197)
0
1
0
2
(198)
0
1
0
2
(199)
0
1
1
1
(200)
0
1
1
1
(201)
0
1
1
1
(202)
0
1
1
1
(203)
0
1
1
1
(204)
0
1
1
1
(205)
0
1
0
2
(206)
0
1
1
1
(207)
0
0
1
2
(208)
0
1
1
1
(209)
0
1
1
1
(210)
0
1
1
1
(211)
0
1
1
1
(212)
0
0
0 1
2
(213)
0
0
1 0
2
(214)
0
0
1 1
1
(215)
0
0
1 1
1
(216)
0
0
1 1
1
(217)
0
0
1 1
1
(218)
0
1
0 1
1
(219)
0
1
0 1
1
(220)
0
1
0 1
1
(221)
0
1
0 1
1
(222)
0
0
0
1
1
1
(223)
0
0
1
0
1
1
(224)
0
1
0
1
0
1
(225)
4
(226)
4
(227)
0
4
(228)
0
4
(229)
0
4
(230)
1
3
(231)
1
3
(232)
1
3
(233)
1
3
(234)
1
3
(235)
2
2
(236)
2
2
(237)
2
2
(238)
0
0 4
(239)
0
1 3
(240)
0
1 3
(241)
0
1 3
(242)
0
2 2
(243)
0
2 2
(244)
0
1 3
(245)
0
2 2
(246)
1
1 2
(247)
1
1 2
(248)
1
1 2
(249)
1
1 2
(250)
1
1 2
(251)
1
1 2
(252)
0
0
1
3
(253)
0
0
2
2
(254)
0
1
0
3
(255)
0
1
1
2
(256)
0
1
1
2
(257)
0
1
1
2
(258)
0
1
2
1
(259)
0
1
2
1
(260)
0
1
2
1
(261)
0
1
1
2
(262)
0
2
0
2
(263)
0
1
1
2
(264)
0
1
2
1
(265)
1
1
1
1
(266)
1
1
1
1
(267)
1
1
1
1
(268)
1
1
1
1
(269)
0
0
1 1
2
(270)
0
0
1 2
1
(271)
0
1
0 1
2
(272)
0
1
1 0
2
(273)
0
1
1 1
1
(274)
0
1
1 1
1
(275)
0
1
1 1
1
(276)
0
1
1 1
1
(277)
0
0
1
1
1
1
(278)
0
1
0
1
1
1
(279)
0
1
1
0
1
1
(280)
5
(281)
0
5
(282)
1
4
(283)
1
4
(284)
2
3
(285)
2
3
(286)
0
1 4
(287)
0
2 3
(288)
1
1 3
(289)
1
1 3
(290)
1
2 2
(291)
1
2 2
(292)
0
1
1
3
(293)
0
1
2
2
(294)
0
1
3
1
(295)
0
2
1
2
(296)
1
1
1
2
(297)
1
1
1
2
(298)
1
1
1
2
(299)
0
1
1 1
2
(300)
0
1
1 2
1
(301)
1
1
1 1
1
(302)
0
1
1
1
1
1
(303)
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
35/77
General
Calculate the divergence
rewrite it in terms of a local Lagrangian
Luckily: symmetry kept: we know result will besymmetrical, hence do not need to explicitly rewrite theLagrangians in a nice formLuckily: we do not need to go to a minimal LagrangianSo everything can be computerizedThank Jos Vermaseren for FORM
We keep all terms to have all 1PI (one particle irreducible)diagrams finite
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
36/77
Result Mass O(N + 1)/O(N)
M2phys = M2(1 + a1LM + a2L2
M + a3L3M + ...)
LM = M2
16π2F 2 log µ2
M2
i ai , N = 3 ai for general N
1 − 12
1− N2
2 178
74− 7N
4+ 5 N2
8
3 − 10324
3712− 113N
24+ 15 N2
4− N3
4 243671152
839144− 1601 N
144+ 695 N2
48− 135 N3
16+ 231 N4
128
5 − 8821144
336612400
− 1151407 N43200
+ 197587 N2
4320− 12709 N3
300+ 6271 N4
320− 7 N5
2
6 19229646676220800
158393809/3888000− 182792131/2592000 N+1046805817/7776000 N2 − 17241967/103680 N3
+70046633/576000 N4 − 23775/512 N5 + 7293/1024 N6
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
37/77
Result Mass SU(n)× SU(N)/SU(N)
M2phys = M2(1 + a1LM + a2L2
M + a3L3M + ...)
LM = M2
16π2F 2 log µ2
M2
i ai for N = 2 ai for N = 3 ai for general N
1 −1/2 −1/3 −N−1
2 17/8 27/8 9/2 N−2 − 1/2 + 3/8 N2
3 −103/24 −3799/648 −89/3 N−3 + 19/3 N−1 − 37/24 N − 1/12 N3
4 24367/1152 146657/2592 2015/8 N−4 − 773/12 N−2 + 193/18 + 121/288 N2
+41/72 N4
5 −8821/144 − 27470059186624
−38684/15 N−5 + 6633/10 N−3 − 59303/1080 N−1
−5077/1440 N − 11327/4320 N3 − 8743/34560 N5
6∗ 19229646676220800
129027731639331200
7329919/240 N−6 − 1652293/240 N−4
−4910303/15552 N−2 + 205365409/972000
−69368761/7776000 N2 + 14222209/2592000 N4
+3778133/3110400 N6
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
pi form-factor
pi polarizability
Beyond QCD orBSM
Leadinglogarithms
Three flavour:more masses
Conclusions
38/77
Numerics: π-π scattering
1
1.2
1.4
1.6
1.8
2
2.2
2.4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
a0 0/a
0 0tr
ee
M2phys [GeV
2]
0
1
2
3
4
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
a2 0/a
2 0tr
ee
M2phys [GeV
2]
0
1
2
3
4
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
39/77
Reviews and general comments
General loop review: JB, Prog. Part. Nucl. Phys. 58 (2007) 521
[hep-ph/0604043]
Recent review of LECs:JB, Ecker,Ann.Rev.Nucl.Part.Sci. 64 (2014) 149 [arXiv:1405.6488]
Highlights of my recent work:
g − 2 HVP: estimates of connected vs disconnected vsstrange contributionsFinite volume and twisting
Masses to two-loopForm-factors one-loopg − 2 HVP: two-loop
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
40/77
Two-point: Why
Muon: aµ = (g − 2)/2 and aLO,HVPµ =
∫ ∞0
dQ2f(Q2)
Π(Q2)
0.00 0.05 0.10 0.15 0.200.000
0.002
0.004
0.006
0.008
0.010
0.012
plot: f(Q2)
Π(Q2)
with Q2 = −q2 in GeV2
Figure and data:Aubin, Blum, Chau, Golterman, Peris, Tu,
Phys. Rev. D93 (2016) 054508 [arXiv:1512.07555]
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
41/77
Two-point: Connected versus disconnected
Connected Disconnectedyellow=lots of quarks/gluons
Πµνab (q) ≡ i
∫d4xe iq·x
⟨T (jµa (x)jν†a (0))
⟩jµπ+ = dγµu
jµu = uγµu, jµd = dγµd , jµs = sγµs
jµe =2
3uγµu − 1
3dγµd−1
3sγµs
Study in ChPT at one-loop:Della Morte, Juttner, JHEP 1011 (2010) 154 [arXiv:1009.3783]
Two-loop JB, Relefors [arXiv:1609.01573]
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
42/77
Two-point: Connected versus disconnected
Include also singlet part of the vector current
There are new terms in the Lagrangian
p4 only one more: 〈Lµν〉 〈Lµν〉+ 〈Rµν〉 〈Rµν〉=⇒ The pure singlet vector current does not couple tomesons until p6
=⇒ Loop diagrams involving the pure singlet vectorcurrent only appear at p8 (implies relations)
p6 (no full classification, just some examples)〈DρLµν〉 〈DρLµν〉+ 〈DρRµν〉 〈DρRµν〉,〈Lµν〉
⟨Lµνχ†U
⟩+ 〈Rµν〉
⟨RµνχU†
⟩,. . .
Results at two-loop order, unquenched isospin limit
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
43/77
Two-point: Connected versus disconnected
Πµνπ+π+ : only connected
Πµνud : only disconnected
Πµνuu = Πµν
π+π+ + Πµνud
Πµνee =
5
9Πµνπ+π+ +
1
9Πµνud
Infinite volume (and the ab considered here):
Πµνab =
(qµqν − q2gµν
)Π
(1)ab
Large Nc + VMD estimate: Π(1)π+π+ =
4F 2π
M2V − q2
Plots on next pages are for Π(1)ab0(q2) = Π
(1)ab (q2)− Π
(1)ab (0)
At p4 the extra LEC cancels, at p6 there are new LECcontributions, but no new ones in the loop parts
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
44/77
Two-point: Connected versus disconnected
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
-0.1 -0.08 -0.06 -0.04 -0.02 0
^ Ππ
+
q2 [GeV
2]
VMD
p4+p
6
p4
p6 R
p6 L
• Connected• p6 is large• Due to the
Lri loops
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
45/77
Two-point: Connected versus disconnected
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
-0.1 -0.08 -0.06 -0.04 -0.02 0
^ ΠU
D
q2 [GeV
2]
p4+p
6
p4
p6 R
p6 L
• Disconnected• p6 is large• Due to the
Lri loops
• about−1
2 connected
• − 110 is from
Π(1)ee =
59 Π
(1)π+π+ + 1
9 Π(1)ud
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
46/77
Two-point: Connected versus disconnected
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
-0.1 -0.08 -0.06 -0.04 -0.02 0
^ ΠU
D/^ Π
π+
q2 [GeV
2]
p4+p
6+VMD
p4+p
6
p4
p6 R
p6 L
• p4 and p6
pion partexactly −1
2• not true for
unsubtractedat p4(LEC)• not true for
pure LEC atp6
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
47/77
Two-point: Including strange
-0.0015
-0.001
-0.0005
0
0.0005
-0.1 -0.08 -0.06 -0.04 -0.02 0
^ ΠS
q2 [GeV
2]
p4+p
6
p4
p6 R
p6 L
VMDφ
• πconnected u,d• uddisconnected u,d• ssstrange current• us mixedstrange-u,d• strange partis very small:q2 =0 subtraction(only kaon loops)
p4 and p6 cancellargely
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
48/77
Two-point: with strange, electromagnetic current
-0.0025
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
-0.1 -0.08 -0.06 -0.04 -0.02 0
q2
5/9 π
1/9 ud
1/9 ss
-2/9 us
sum
• πconnected u,d• uddisconnected u,d• ssstrange current• usmixed s–u,d• new p6 LECcancels• VMD, VMDφnot included
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
49/77
Finite volume
Lattice QCD calculates at different quark masses, volumesboundary conditions,. . .
A general result by Luscher: relate finite volume effects toscattering (1986)
Chiral Perturbation Theory is also useful for this
Start: Gasser and Leutwyler, Phys. Lett. B184 (1987) 83, Nucl. Phys. B
307 (1988) 763
Mπ, Fπ, 〈qq〉 one-loop equal mass case
I will stay with ChPT and the p regime (MπL >> 1)
1/mπ = 1.4 fmmay need to (and I will) go beyond leading e−mπL terms“around the world as often as you like”
Convergence of ChPT is given by 1/mρ ≈ 0.25 fm
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
50/77
Finite volume: selection of earlier ChPT results
masses and decay constants for π,K , η one-loopBecirevic, Villadoro, Phys. Rev. D 69 (2004) 054010
Mπ at 2-loops (2-flavour)Colangelo, Haefeli, Nucl.Phys. B744 (2006) 14 [hep-lat/0602017]
〈qq〉 at 2 loops (3-flavour)JB, Ghorbani, Phys. Lett. B636 (2006) 51 [hep-lat/0602019]
Twisted mass at one-loopColangelo, Wenger, Wu, Phys.Rev. D82 (2010) 034502 [arXiv:1003.0847]
Twisted boundary conditionsSachrajda, Villadoro, Phys. Lett. B 609 (2005) 73 [hep-lat/0411033]
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
51/77
Papers
Finite volume at two-loops (periodic)Two-loop sunset integrals at finite volume,
JB, Bostrom, Lahde, JHEP 1401(2014)019 [arXiv:1311.3531]Finite Volume at two-loops in Chiral Perturbation Theory,
JB, Rossler, JHEP 1501 (2015) 034 [arXiv:1411.6384]Finite Volume for Three-Flavour Partially Quenched ChiralPerturbation Theory through NNLO in the Meson Sector,
JB, Rossler, JHEP 1511 (2015) 097 [arXiv:1508.07238]
Finite Volume and Partially Quenched QCD-like Effective Field
Theories, JB, Rossler, JHEP 1511 (2015) 017 [arXiv:1509.04082]
Twisted boundary conditionsMasses, Decay Constants and Electromagnetic Form-factors withTwisted Boundary Conditions,
JB, Relefors, JHEP 1405 (2014) 015 [arXiv:1402.1385]The vector two-point function with twisted boundary conditions,
JB, Relefors, to be publishedK`3 wth staggered, finite volume and twisting,
Bernard, JB, Gamiz, Relefors, to be published
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
52/77
Masses at two-loop order
Sunset integrals at finite volume doneJB, Bostrom and Lahde, JHEP 01 (2014) 019 [arXiv:1311.3531]
Loop calculations:JB, Rossler, JHEP 1501 (2015) 034 [arXiv:1411.6384]
0.001
0.01
2 2.5 3 3.5 4
∆Vm
2 π/m
2 π
mπ L
p4 Nf=2
p4+p
6 Nf=2
p4 Nf=3
p4+p
6 Nf=3
1e-06
1e-05
0.0001
0.001
0.01
2 2.5 3 3.5 4
∆Vm
2 K/m
2 K
mπ L
p4
p6
p6 Li
r only
p4+p
6
Agreement for Nf = 2, 3 for pionK has no pion loop at LO
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
53/77
Decay constants at two-loop order
Sunset integrals at finite volume doneJB, Bostrom and Lahde, JHEP 01 (2014) 019 [arXiv:1311.3531]
Loop calculations:JB, Rossler, JHEP 1501 (2015) 034 [arXiv:1411.6384]
0.001
0.01
2 2.5 3 3.5 4
−∆
VF
π/F
π
mπ L
p4 Nf=2
p4+p
6 Nf=2
p4 Nf=3
p4+p
6 Nf=3
0.0001
0.001
0.01
2 2.5 3 3.5 4
−∆
VF
K/F
K
mπ L
p4
p6
p6 Li
r only
p4+p
6
Agreement for Nf = 2, 3 for pionK now has a pion loop at LO
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
54/77
Other p6
Masses and decay constants at finite volume:
Finite volume for PQ three flavour (all cases) JB, Rossler,
JHEP 1511 (2015) 097, [arXiv:1508.07238]
QCD-like theories, normal and PQ (one valence mass, onesea mass) JB, Rossler, JHEP 1511 (2015) 017, [arXiv:1509.04082]
SU(N)× SU(N)/SU(N)SU(N)/SO(N) (including Majorana case)SU(2N)/Sp(2N)
If you want more graphs: look at the papers or play withthe programs in CHIRON
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
55/77
Twisted boundary conditions
On a lattice at finite volume pi = 2πni/L: very fewmomenta directly accessible
Put a constraint on certain quark fields in some directions:q(x i + L) = e iθ
iqq(x i )
Then momenta are pi = θi/L + 2πni/L. Allows to mapout momentum space on the lattice much betterBedaque,. . .
Small note:
Beware what people call momentum:is θi/L included or not?Reason: a colour singlet gauge transformationGSµ → GS
µ − ∂µε(x), q(x)→ e iε(x)q(x), ε(x) = −θiqx i/LBoundary conditionTwisted ⇔ constant background field+periodic
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
56/77
Twisted boundary conditions: Drawbacks
Drawbacks:
Box: Rotation invariance → cubic invariance
Twisting: reduces symmetry further
Consequences:
m2(~p2) = E 2 − ~p2 is not constant
dispersion relation so what is a “mass” not clearalready present in “moving frame” without twistRenormalized momentum is quantity-dependent
There are typically more form-factors
In general: quantities depend on more (all) components ofthe momenta
Charge conjugation involves a change in momentum
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
57/77
Two-point function: twisted boundary conditions
JB, Relefors, JHEP 05 (201)4 015 [arXiv:1402.1385]∫V
ddk
(2π)dkµ
k2 −m26= 0
〈uγµu〉 6= 0
jµπ+ = dγµu
satisfies ∂µ⟨T (jµπ+(x)jν†π+(0))
⟩= δ(4)(x)
⟨dγνd − uγνu
⟩Πµνa (q) ≡ i
∫d4xe iq·x
⟨T (jµa (x)jν†a (0))
⟩Satisfies WT identity. qµΠµν
π+ =⟨uγµu − dγµd
⟩ChPT at one-loop satisfies thissee also Aubin et al, Phys.Rev. D88 (2013) 7, 074505 [arXiv:1307.4701]
two-loop in partially quenched: JB, Relefors, in preparation
satisfies the WT identity (as it should)
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
58/77
〈uγµu〉
-3e-06
-2e-06
-1e-06
0
1e-06
2e-06
3e-06
0 π/2 π 3π/2 2π
⟨− uγµ
u⟩
twis
ted
θu
p4
p6 R
p6 L
p4+p
6
-3e-06
-2e-06
-1e-06
0
1e-06
2e-06
3e-06
0 π/2 π 3π/2 2π⟨− u
γµu
⟩ part
ially
tw
iste
d
θu
p4
p6 R
p6 L
p4+p
6
Fully twisted Partially twistedθu = (0, θu, 0, 0), all others untwistedmπL = 4 (ratio at p4=2 up to kaon loops)
For comparison:〈uu〉V ≈ −2.4 10−5 GeV3
〈uu〉 ≈ −1.2 10−2 GeV3
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
59/77
Two-point: partially twisted, one-loop
-0.0001
-8e-05
-6e-05
-4e-05
-2e-05
0
2e-05
4e-05
-0.1 -0.08 -0.06 -0.04 -0.02 0
∆V
Πµ
ν p
art
tw
ist p
4
q2
♦
sinθu
µν=00
µν=11
µν=22
µν=33
q =(
0,√−q2, 0, 0
)Π22 = Π33
~θu = L qmπ0L = 4mπ0 = 0.135 GeV
−q2Π(1)VMD = −4q2F 2
π
M2V−q2
≈5e-3· q2
0.1
diamond: periodicNote: Πµν(0) 6= 0
Correction is at the % level
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
60/77
Two-point: partially twisted, with two-loop
-0.0001
-8e-05
-6e-05
-4e-05
-2e-05
0
2e-05
4e-05
-0.1 -0.08 -0.06 -0.04 -0.02 0
∆V
Πµ
ν p
art
tw
ist p
4+
p6
q2
♦
sinθu
µν=00
µν=11
µν=22
µν=33
q =(
0,√−q2, 0, 0
)Π22 = Π33
~θu = L qmπ0L = 4mπ0 = 0.135 GeV
−q2Π(1)VMD = −4q2F 2
π
M2V−q2
≈5e-3· q2
0.1
diamond: periodicNote: Πµν(0) 6= 0
Correction from two loop is reasonable (thin lines are p4)
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
61/77
Two-point: partially twisted, one-loop
-0.0001
-8e-05
-6e-05
-4e-05
-2e-05
0
2e-05
4e-05
-0.1 -0.08 -0.06 -0.04 -0.02 0
∆V
Πµ
ν p
art
tw
ist p
4
q2
♦
sinθxu
µν=00
µν=11
µν=12
µν=33
q =
(0,
√−q2√
2,
√−q2√
2, 0
)Π11 = Π22
~θu = L qmπ0L = 4mπ0 = 0.135 GeV
−q2Π(1)VMD = −4q2F 2
π
M2V−q2
≈5e-3· q2
0.1
diamond: periodicNote: Πµν(0) 6= 0
Correction is at the % level
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
62/77
Two-point: partially twisted, one-loop
-0.0001
-8e-05
-6e-05
-4e-05
-2e-05
0
2e-05
4e-05
-0.1 -0.08 -0.06 -0.04 -0.02 0
∆V
Πµ
ν p
art
tw
ist p
4+
p6
q2
♦
sinθxu
µν=00
µν=11
µν=12
µν=33
q =
(0,
√−q2√
2,
√−q2√
2, 0
)Π11 = Π22
~θu = L qmπ0L = 4mπ0 = 0.135 GeV
−q2Π(1)VMD = −4q2F 2
π
M2V−q2
≈5e-3· q2
0.1
diamond: periodicNote: Πµν(0) 6= 0
Two loop correction again reasonable (thin lines are p4)
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
63/77
Partial twisting: masses
Bernard, JB, Gamiz, Relefors, in preparation
mπL = 3, ~θu = (θ, 0, 0), ~θd = ~θs = ~θdsea = ~θssea = 0
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π
|∆V m
2 π+|/m
2 π
θ
p1
p2
p3
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π
|∆V m
2 π+|/m
2 π
θ
p1
p2
p3
~θusea = 0 ~θusea = (π/3, 0, 0)
~p1 = (θ, 0, 0) /L, ~p2 = (θ + 2π, 0, 0) /L, ~p3 = (θ − 2π, 0, 0) /L,
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
64/77
K`3: Twisting and finite volume
There are more form-factors since Lorentz-invariance andeven cubic symmetry is broken
Masses become twist and volume dependent
All these need to be remembered in the Ward identities
Masses needed when checking Ward identities
For unquenched twisted masses, decay constants andelectromagnetic form-factor (see there for earlier work):JB, Relefors, JHEP 05 (2014) 015 [arXiv:1402.1385]
Partial twisting and quenching, staggered: masses and K`3
Bernard, JB, Gamiz, Relefors, in preparation
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
65/77
K`3
q = p − p′
〈π−(p′)|sγµu(0)|K 0(p)〉 = f+(pµ + p′µ) + f−qµ + hµ .
〈π−(p′)|(ms −mu)su(0)|K 0(p)〉 = ρ .
Ward identity: (p2 − p′2)f+ + q2f− + qµhµ = ρ
ChPT:
p4 Isopin conserving and breaking Gasser, Leutwyler, 1985
p6 Isospin conserving JB, Talavera, 2003
p6 Isospin breaking JB, Ghorbani, 2007
p4 partially quenched, staggered Bernard, JB, Gamiz, 2013
p4 Finite volume Ghorbani, Ghorbani, 2013 (q2 = 0 )p4 Finite volume, twisted, partially quenched, staggeredBernard, JB, Gamiz, Relefors, in preparation
Rare decays: p4 Mescia, Smith 2007, p6 JB, Ghorbani, 2007
Split in f+, f− and hµ not unique
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
66/77
K`3
Masses: finite volume masses with twist effect included.
p =(√
m2K (~p) + ~p2, ~p
)p′ =
(√m2π(~p′) + ~p′2, ~p′
)q2 calculated with m2
K and m2π at V =∞ will also have
volume corrections (small effect)
First: Twisting and partially quenched
Second: Staggered as well
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
67/77
K`3: infinite volume, pure loop part
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
-0.04 -0.02 0 0.02 0.04
f +,f
-,ρ[G
eV
2]
q2
f+
f+ PQ
f-
f- PQ
ρ
ρ PQ
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
-0.04 -0.02 0 0.02 0.04
q2
fµ=0
A
fµ=3
A
fµ=0
B
fµ=3
B
PQ case
The components are the well defined ones at finite volume
plots: p4 (neglecting the Lr9q2 term)
Valence masses with mπ = 135 GeV and mK = 0.495 GeV
PQ case with msea = 1.1m, mssea = 1.1ms .
case A: ~p = 0, case B: ~p′ = 0
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
68/77
K`3
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
-0.04 -0.02 0 0.02 0.04
q2
V=∞ ρ PQ
V ρ PQ A
V ρ PQ B
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
-0.04 -0.02 0 0.02 0.04
q2
V=∞ fµ=0
A
V fµ=0
A
V=∞ fµ=0
B
V fµ=0
B
ρ µ = 0
ρ∞ ≈ 0.23 GeV2
mπL = 3
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
69/77
K`3
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
-0.04 -0.02 0 0.02 0.04
q2
V=∞ fµ=0
A
V fµ=0
A
V=∞ fµ=0
B
V fµ=0
B
µ = 3
Calculate the volume corrections for exactly what you did
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
70/77
What do you calculate on the lattice?
Want f+(0) at infinite volume and physical masses
WT identity: (p2 − p′2)f+ + q2f− + qµhµ = ρ
Assume calculation at physical masses
All parts in the WTI at fixed ~p, ~p′ have finite volumecorrections: p2, p′2, q2, f−, qµhµ and ρ
Can use WTI at finite volume and then extrapolate f+ orextrapolate ρ and then use WTI
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
71/77
MILC lattices and numbers Preliminary
a(fm) ml/ms L(fm) mπ(MeV) mK (MeV) mπL
0.15 0.035 4.8 134 505 3.25
0.12 0.2 2.9 309 539 4.50.1 2.9 220 516 3.20.1 3.8 220 516 4.30.1 4.8 220 516 5.4
0.035 5.7 135 504 3.9
0.09 0.2 2.9 312 539 4.50.1 4.2 222 523 4.7
0.035 5.6 129 495 3.7
0.06 0.2 2.8 319 547 4.50.035 5.5 134 491 3.7
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
72/77
Results: ~θu = (0, θ, θ, θ) (staggered)
Finite volume part of WI divided by m2K −m2
π:
∆V m2K −∆V m2
π
m2K −m2
π
+ ∆V f+(0) +qµhµ
m2K −m2
π
=∆V ρ
m2K −m2
π
mπ mπL “mass” “f+” “hµ” “ρ”
134 3.25 0.00000 −0.00042 0.00007 −0.00036
309 4.5 0.00013 −0.00003 −0.00041 −0.00031220 3.2 0.00054 −0.00048 −0.00084 −0.00077220 4.3 −0.00007 −0.00009 −0.00005 −0.00021220 5.4 −0.00005 −0.00003 0.00001 −0.00006135 3.9 −0.00006 −0.00020 0.00005 −0.00021
312 4.5 0.00047 0.00023 −0.00068 −0.00001222 4.7 −0.00000 0.00018 −0.00003 0.00014129 3.7 −0.00013 −0.00004 0.00009 −0.00007
319 4.5 0.00052 0.00037 −0.00081 0.00008134 3.7 −0.00016 0.00045 0.00013 0.00043
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
73/77
Results: ~θu = (0, θ, 0, 0) (staggered)
Finite volume part of WI divided by m2K −m2
π:
∆V m2K −∆V m2
π
m2K −m2
π
+ ∆V f+(0) +qµhµ
m2K −m2
π
=∆V ρ
m2K −m2
π
mπ mπL “mass” “f+” “hµ” “ρ”
134 3.25 −0.00003 −0.00066 0.00008 −0.00061
309 4.5 −0.00030 −0.00017 −0.00002 −0.00049220 3.2 −0.00078 −0.00105 0.00036 −0.00148220 4.3 −0.00033 −0.00034 0.00018 −0.00049220 5.4 −0.00008 −0.00010 0.00003 −0.00015135 3.9 −0.00002 −0.00032 0.00001 −0.00033
312 4.5 −0.00019 0.00002 −0.00009 −0.00026222 4.7 −0.00024 −0.00018 0.00017 −0.00025129 3.7 −0.00003 −0.00050 −0.00001 −0.00054
319 4.5 −0.00026 0.00013 −0.00012 −0.00025134 3.7 −0.00005 −0.00058 0.00001 −0.00062
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
74/77
Results: ~θu = (0, θ, 0, 0) (not staggered)
Finite volume part of WI divided by m2K −m2
π:
∆V m2K −∆V m2
π
m2K −m2
π
+ ∆V f+(0) +qµhµ
m2K −m2
π
=∆V ρ
m2K −m2
π
mπ mπL “mass” “f+” “hµ” “ρ”
134 3.25 −0.00049 −0.00124 0.00037 −0.00137
309 4.5 −0.00033 0.00014 −0.00004 0.00022220 3.2 −0.00113 0.00077 0.00067 0.00031220 4.3 −0.00062 −0.00011 0.00046 −0.00027220 5.4 −0.00014 −0.00011 0.00010 −0.00016135 3.9 0.00004 −0.00045 −0.00008 −0.00049
312 4.5 0.00031 0.00015 −0.00009 −0.00025222 4.7 −0.00037 −0.00015 0.00027 −0.00025129 3.7 −0.00000 −0.00066 −0.00005 −0.00071
319 4.5 −0.00031 0.00015 −0.00011 −0.00027134 3.7 −0.00007 −0.00064 0.00001 −0.00070
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
HVP
Finite Volume
Masses 2-loop
Twisting
HVP Twisted
Masses twisted
K`3K`3: MILC andstaggered
Comments FV
Conclusions
75/77
Comments FV
Many results available
Range from very relevant to really small at present lattices
Partial twisting can be used to check our FV correctionson the same underlying lattice
Be careful: ChPT must exactly correspond to your latticecalculation
Programs available (for published ones) via CHIRONThose for this talk: sometime later this year (I hope)
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
Conclusions
76/77
Conclusions
A very short (and biased) history
A few older results
Finite volume, twisting, staggered,. . .
Lots of stuff left from many people (especially Ulf)
One thing left:
Loops andVolumes in
ChPT
Johan Bijnens
Overview
ChPT
Two or moreloops
Three flavour:more masses
Conclusions
77/77
Conclusions
CONGRATULATIONS ULF
“for his developments and applica-tions of effective field theories inhadron and nuclear physics, that al-lowed for systematic and precise in-vestigations of the structure and dy-namics of nucleons and nuclei basedon Quantum Chromodynamics.”