Finitevolume effects to the HVP contribution to the muon g2
The anomalous magnetic moment of the muon, $(g2)_\mu$, has become a central focus in the broader particle physics community, due to significant tension between the best experimental [1][2] and theoretical determinations [3][4][5]. The $\sim 3$ to $4$ sigma discrepancy represents a real opportunity to discover new physics beyond the Standard Model (BSM), especially given the incredibly clean determinations on both the experimental and theoretical sides, together with the quadratic sensitivity to BSM effects, $(g2)_\mu^{\text{BSM}} \sim (m_\mu / \Lambda_{\text{BSM}})^2$, where $m_\mu$ is the muon mass and $\Lambda_{\text{BSM}}$ the scale of the putative new physics.
Experiments are underway at both Fermilab [6][7][8][9] and JPARC [10][11], with an update expected some time this year, and at least a factor of 2 uncertainty reduction targeted in the coming years. The theory community is committed to maximizing the impact of this update, by providing a Standard Model determination of comparable overall uncertainty. The effort on this side is also very advanced and is summarized in detail in a forthcoming theory white paper [12].
On the theoretical side, the leading uncertainties in $a_\mu \equiv (g2)_\mu/2$, arise from hadronic contributions, generated via the coupling of QCD fields to the muonphoton vertex. These break into three categories: the leading hadronicvacuumpolarization (HVP) contribution, $a_{\text{HVP}}$, the leading hadroniclightbylight and the subleading corrections to the HVP. Outstanding progress has been made in the determination of the latter two contributions such that these are well in line to reach the overall target uncertainty [13][14][15]. Since both the hadronic lightbylight and the subleading HVP are suppressed relative to the leading HVP, the targeted relative uncertainties here are $\sim 10%$ and, despite the complicated nature of these quantities, well within reach; see again ref. [12]. In this work we restrict attention to the $a_{\text{HVP}}$ contribution, for which subpercent uncertainty is required to reach the overall $(g2)_\mu$ precision target.
Numerical lattice QCD (LQCD) provides an ideal tool in the determination of $a_{\text{HVP}}$, and many leading collaborations have already presented very advanced calculations [16][17][18][19][20][21][22][23][24][25][26][14][27][28][29][30][31]. The observable can be directly extracted from a Euclidean electromagneticcurrent twopoint function and is thus wellsuited to highprecision lattice determinations. To make progress in practice, a deep theoretical understanding of all uncertainties is crucial, with the dominant sources being discretization effects, scalesetting uncertainty, statistical uncertainty (especially for large separations of the vector currents as well as those arising from quarkdisconnected diagrams) and, finally, uncertainties arising from the effects of working in a finitevolume spacetime.
Projects

Dr. Maxwell T. Hansen (Edinburgh University) and Prof. Dr. Agostino Patella (HumboldtUniversität zu Berlin & IRIS Adlershof & DESY Zeuthen) have calculated [32][33] finitevolume corrections to $a_{\text{HVP}}$ by means of an effective theory of pions with generic local interactions in the isospinsymmetric limit. The interaction Lagrangian can be arbitrarily complicated, and in order to make sense of the Feynman integrals, an ultraviolet cutoff that preserves all relevant symmetries is assumed. However, the resulting formulae and their proof are insensitive to these details.

Sofie Martins (CP3Origins) has generalized the calculation of finitevolume corrections to $a_{\text{HVP}}$ to the case of Cparity boundary conditions in the spatial directions. She is also studying finitevolume corrections to the isospinbreaking corrections to $a_{\text{HVP}}$.
Documents and links
Bibliography
[1] 
G.W.
Bennett,
B.
Bousquet,
H.N.
Brown,
G.
Bunce,
R.M.
Carey,
P.
Cushman,
G.T.
Danby,
P.T.
Debevec,
M.
Deile,
H.
Deng,
et al.
Measurement of the negative muon anomalous magnetic moment to 0.7 ppm Phys.Rev.Lett. 92 (2004) 161802 arXiv: hepex/0401008 [hepex] 

[2] 
G.W.
Bennett,
B.
Bousquet,
H.N.
Brown,
G.
Bunce,
R.M.
Carey,
P.
Cushman,
G.T.
Danby,
P.T.
Debevec,
M.
Deile,
H.
Deng,
et al.
Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL Phys.Rev.D 73 (2006) 072003 arXiv: hepex/0602035 [hepex] 
[3] 
Fred
Jegerlehner,
Andreas
Nyffeler
The Muon g2 Phys.Rept. 477 (2009) 1 arXiv: 0902.3360 [hepph] 
[4] 
Fred
Jegerlehner
Muon g – 2 theory: The hadronic part EPJ Web Conf. 166 (2018) 00022 arXiv: 1705.00263 [hepph] 
[5] 
Michel
Davier,
Andreas
Hoecker,
Bogdan
Malaescu,
Zhiqing
Zhang
Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon $g2$ and ${\alpha (m_Z^2)}$ using newest hadronic crosssection data Eur.Phys.J.C 77 (2017) 827 arXiv: 1706.09436 [hepph] 
[6] 
R.M.
Carey,
K.R.
Lynch,
J.P.
Miller,
B.L.
Roberts,
W.M.
Morse,
Y.K.
Semertzides,
V.P.
Druzhinin,
B.I.
Khazin,
I.A.
Koop,
I.
Logashenko,
et al.
The New (g2) Experiment: A proposal to measure the muon anomalous magnetic moment to +0.14 ppm precision 
[7] 
J.
Grange,
V.
Guarino,
P.
Winter,
K.
Wood,
H.
Zhao,
R.M.
Carey,
D.
Gastler,
E.
Hazen,
N.
Kinnaird,
J.P.
Miller,
et al.
Muon (g2) Technical Design Report arXiv: 1501.06858 [physics.insdet] 
[8] 
David
Flay
Precision Magnetic Field Calibration for the Muon $g2$ Experiment at Fermilab PoS ICHEP2016 (2017) 1075 
[9] 
Ran
Hong
Experiences from the Commissioning and First Physics Run of the Fermilab Muon g2 Experiment arXiv: 1810.03729 [physics.insdet] 
[10] 
K.
Shimomura
Muonium in JPARC; from fundamental to application Hyperfine Interact. 233 (2015) 89 
[11] 
Yutaro
Sato
Muon g2/EDM experiment at JPARC PoS KMI2017 (2017) 006 
[12] 
T.
Aoyama,
N.
Asmussen,
M.
Benayoun,
J.
Bijnens,
T.
Blum,
M.
Bruno,
I.
Caprini,
C.M.
Carloni Calame,
M.
Cè,
G.
Colangelo,
et al.
The anomalous magnetic moment of the muon in the Standard Model Phys.Rept. 887 (2020) 1 arXiv: 2006.04822 [hepph] 
[13] 
Thomas
Blum,
Norman
Christ,
Masashi
Hayakawa,
Taku
Izubuchi,
Luchang
Jin,
Chulwoo
Jung,
Christoph
Lehner
Connected and Leading Disconnected Hadronic LightbyLight Contribution to the Muon Anomalous Magnetic Moment with a Physical Pion Mass Phys.Rev.Lett. 118 (2017) 022005 arXiv: 1610.04603 [heplat] 
[14] 
Davide
Giusti,
Vittorio
Lubicz,
Guido
Martinelli,
Francesco
Sanfilippo,
Silvano
Simula,
Cecilia
Tarantino
HVP contribution of the light quarks to the muon $(g  2)$ including isospinbreaking corrections with TwistedMass fermions PoS LATTICE2018 (2018) 140 arXiv: 1810.05880 [heplat] 
[15] 
Nils
Asmussen,
Antoine
Gerardin,
Jeremy
Green,
Oleksii
Gryniuk,
Georg
von Hippel,
Harvey B.
Meyer,
Andreas
Nyffeler,
Vladimir
Pascalutsa,
Hartmut
Wittig
Hadronic lightbylight scattering contribution to the muon g – 2 on the lattice EPJ Web Conf. 179 (2018) 01017 arXiv: 1801.04238 [heplat] 
[16] 
T.
Blum
Lattice calculation of the lowest order hadronic contribution to the muon anomalous magnetic moment Phys.Rev.Lett. 91 (2003) 052001 arXiv: heplat/0212018 [heplat] 
[17] 
Florian
Burger,
Xu
Feng,
Grit
Hotzel,
Karl
Jansen,
Marcus
Petschlies,
Dru B.
Renner
FourFlavour LeadingOrder Hadronic Contribution To The Muon Anomalous Magnetic Moment JHEP 02 (2014) 099 arXiv: 1308.4327 [heplat] 
[18] 
Bipasha
Chakraborty,
C.T.H.
Davies,
G.C.
Donald,
R.J.
Dowdall,
J.
Koponen,
G.P.
Lepage,
T.
Teubner
Strange and charm quark contributions to the anomalous magnetic moment of the muon Phys.Rev.D 89 (2014) 114501 arXiv: 1403.1778 [heplat] 
[19] 
Bipasha
Chakraborty,
C.T.H.
Davies,
J.
Koponen,
G.P.
Lepage,
M.J.
Peardon,
S.M.
Ryan
Estimate of the hadronic vacuum polarization disconnected contribution to the anomalous magnetic moment of the muon from lattice QCD Phys.Rev.D 93 (2016) 074509 arXiv: 1512.03270 [heplat] 
[20] 
T.
Blum,
P.A.
Boyle,
T.
Izubuchi,
L.
Jin,
A.
Jüttner,
C.
Lehner,
K.
Maltman,
M.
Marinkovic,
A.
Portelli,
M.
Spraggs
Calculation of the hadronic vacuum polarization disconnected contribution to the muon anomalous magnetic moment Phys.Rev.Lett. 116 (2016) 232002 arXiv: 1512.09054 [heplat] 
[21] 
Bipasha
Chakraborty,
C.T.H.
Davies,
P.G.
de Oliviera,
J.
Koponen,
G.P.
Lepage,
R.S.
Van de Water
The hadronic vacuum polarization contribution to $a_{\mu}$ from full lattice QCD Phys.Rev.D 96 (2017) 034516 arXiv: 1601.03071 [heplat] 
[22] 
T.
Blum,
P.A.
Boyle,
L.
Del Debbio,
R.J.
Hudspith,
T.
Izubuchi,
A.
Jüttner,
C.
Lehner,
R.
Lewis,
K.
Maltman,
M.
Krstić Marinković,
et al.
Lattice calculation of the leading strange quarkconnected contribution to the muon $g − 2$ JHEP 04 (2016) 063 , JHEP 05 (2017) 034 arXiv: 1602.01767 [heplat] 
[23] 
M.
Della Morte,
A.
Francis,
V.
Gülpers,
G.
Herdoíza,
G.
von Hippel,
H.
Horch,
B.
Jäger,
H.B.
Meyer,
A.
Nyffeler,
H.
Wittig
The hadronic vacuum polarization contribution to the muon $g2$ from lattice QCD JHEP 10 (2017) 020 arXiv: 1705.01775 [heplat] 
[24] 
D.
Giusti,
V.
Lubicz,
G.
Martinelli,
F.
Sanfilippo,
S.
Simula
Strange and charm HVP contributions to the muon ($g  2)$ including QED corrections with twistedmass fermions JHEP 10 (2017) 157 arXiv: 1707.03019 [heplat] 
[25] 
Sz.
Borsanyi,
Z.
Fodor,
C.
Hoelbling,
T.
Kawanai,
S.
Krieg,
L.
Lellouch,
R.
Malak,
K.
Miura,
K.K.
Szabo,
C.
Torrero,
et al.
Hadronic vacuum polarization contribution to the anomalous magnetic moments of leptons from first principles Phys.Rev.Lett. 121 (2018) 022002 arXiv: 1711.04980 [heplat] 
[26] 
D.
Giusti,
F.
Sanfilippo,
S.
Simula
Lightquark contribution to the leading hadronic vacuum polarization term of the muon $g2$ from twistedmass fermions Phys.Rev.D 98 (2018) 114504 arXiv: 1808.00887 [heplat] 
[27] 
T.
Blum,
P.A.
Boyle,
V.
Gülpers,
T.
Izubuchi,
L.
Jin,
C.
Jung,
A.
Jüttner,
C.
Lehner,
A.
Portelli,
J.T.
Tsang
Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment Phys.Rev.Lett. 121 (2018) 022003 arXiv: 1801.07224 [heplat] 
[28] 
C.T.H.
Davies,
C.
DeTar,
A.X.
ElKhadra,
E.
Gámiz,
Steven
Gottlieb,
D.
Hatton,
A.S.
Kronfeld,
J.
Laiho,
G.P.
Lepage,
Yuzhi
Liu,
et al.
Hadronicvacuumpolarization contribution to the muon’s anomalous magnetic moment from fourflavor lattice QCD Phys.Rev.D 101 (2020) 034512 arXiv: 1902.04223 [heplat] 
[29] 
Antoine
Gérardin,
Marco
Cè,
Georg
von Hippel,
Ben
Hörz,
Harvey B.
Meyer,
Daniel
Mohler,
Konstantin
Ottnad,
Jonas
Wilhelm,
Hartmut
Wittig
The leading hadronic contribution to $(g2)_\mu$ from lattice QCD with $N_{\rm f}=2+1$ flavours of O($a$) improved Wilson quarks Phys.Rev.D 100 (2019) 014510 arXiv: 1904.03120 [heplat] 
[30] 
Sz.
Borsanyi,
Z.
Fodor,
J.N.
Guenther,
C.
Hoelbling,
S.D.
Katz,
L.
Lellouch,
T.
Lippert,
K.
Miura,
L.
Parato,
K.K.
Szabo,
et al.
Leading hadronic contribution to the muon magnetic moment from lattice QCD Nature 593 (2021) 51 arXiv: 2002.12347 [heplat] 
[31] 
Vera
Gülpers
Recent Developments of Muon g2 from Lattice QCD PoS LATTICE2019 (2020) 224 arXiv: 2001.11898 [heplat] 
[32] 
Maxwell T.
Hansen,
Agostino
Patella
Finitevolume effects in $(g2)^{\text{HVP,LO}}_\mu$ Phys.Rev.Lett. 123 (2019) 172001 arXiv: 1904.10010 [heplat] 
[33] 
Maxwell T.
Hansen,
Agostino
Patella
Finitevolume and thermal effects in the leadingHVP contribution to muonic ($g − 2$) JHEP 10 (2020) 029 arXiv: 2004.03935 [heplat] 