Loops and Volumes in Chiral Perturbation Theoryhome.thep.lu.se/~bijnens/talks/juelich16.pdf · Decays and CP Violation in ChPT," NPB 303 (1988) 665. J. Bijnens and F. Cornet, \Two

$Page 1: Loops and Volumes in Chiral Perturbation Theoryhome.thep.lu.se/~bijnens/talks/juelich16.pdf · Decays and CP Violation in ChPT," NPB 303 (1988) 665. J. Bijnens and F. Cornet, \Two$
Loops andVolumes in

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Johan Bijnens

Overview

ChPT

Two or moreloops

Three flavour:more masses

Conclusions

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Loops and Volumes inChiral Perturbation Theory

Johan Bijnens

Lund University

[email protected]

http://www.thep.lu.se/∼bijnens

Advances in Effective Field Theories Julich, Germany 7-9 November 2016

http://www.thep.lu.se/~bijnens/

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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


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.

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ChPT

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4/77


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

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ChPT

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Two or moreloops


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5/77


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

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ChPT

A selectivehistory

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Two or moreloops


Conclusions

6/77


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?


Two or moreloops


Conclusions

7/77


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.

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ChPT

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ChPT

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What is it?


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8/77


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?


Two or moreloops


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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/

http://www.thep.lu.se/~bijnens/chpt/

Loops andVolumes in

ChPT

Johan Bijnens

Overview

ChPT

A selectivehistory

What is it?


Two or moreloops


Conclusions

10/77


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?


Two or moreloops


Conclusions

10/77


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?


Two or moreloops


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?


Two or moreloops


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?


Two or moreloops


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?


Two or moreloops


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?


Two or moreloops


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?


Two or moreloops


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?


Two or moreloops


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?


Two or moreloops


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/

http://www.thep.lu.se/~bijnens/chpt/

http://www.thep.lu.se/~bijnens/chiron//

Loops andVolumes in

ChPT

Johan Bijnens

Overview

ChPT

A selectivehistory

What is it?


Two or moreloops


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


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


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


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


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


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


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


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


Conclusions

26/77


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


〈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


Conclusions

27/77


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


〈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


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


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


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


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


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


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


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


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


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


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


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


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


HVP

Finite Volume

Masses 2-loop

Twisting

HVP Twisted

Masses twisted


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


HVP

Finite Volume

Masses 2-loop

Twisting

HVP Twisted

Masses twisted


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


HVP

Finite Volume

Masses 2-loop

Twisting

HVP Twisted

Masses twisted


Comments FV

Conclusions

42/77


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

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43/77


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

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44/77


-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

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45/77


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

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46/77


-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

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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

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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

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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

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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]

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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

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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

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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

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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

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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

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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

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ChPT

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Twisting

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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

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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

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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

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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


Correction from two loop is reasonable (thin lines are p4)

Loops andVolumes in

ChPT

Johan Bijnens

Overview

ChPT

Two or moreloops


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61/77


-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


Correction is at the % level

Loops andVolumes in

ChPT

Johan Bijnens

Overview

ChPT

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62/77


-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


Two loop correction again reasonable (thin lines are p4)

Loops andVolumes in

ChPT

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Twisting

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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

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Overview

ChPT

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Twisting

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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

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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

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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

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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

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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

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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

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Twisting

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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


HVP

Finite Volume

Masses 2-loop

Twisting

HVP Twisted

Masses twisted


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


HVP

Finite Volume

Masses 2-loop

Twisting

HVP Twisted

Masses twisted


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


HVP

Finite Volume

Masses 2-loop

Twisting

HVP Twisted

Masses twisted


Comments FV

Conclusions

73/77

Results: ~θu = (0, θ, 0, 0) (staggered)


π:

∆V m2K −∆V m2

π

m2K −m2

π

+ ∆V f+(0) +qµhµ

m2K −m2

π

=∆V ρ

m2K −m2

π


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


HVP

Finite Volume

Masses 2-loop

Twisting

HVP Twisted

Masses twisted


Comments FV

Conclusions

74/77

Results: ~θu = (0, θ, 0, 0) (not staggered)


π:

∆V m2K −∆V m2

π

m2K −m2

π

+ ∆V f+(0) +qµhµ

m2K −m2

π

=∆V ρ

m2K −m2

π


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


HVP

Finite Volume

Masses 2-loop

Twisting

HVP Twisted

Masses twisted


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


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


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.”

Documents

Loops and Volumes in Chiral Perturbation Theoryhome.thep.lu.se/~bijnens/talks/juelich16.pdf · Decays and CP Violation in ChPT," NPB 303 (1988) 665. J. Bijnens and F. Cornet, \Two