The Mainz (g 2) project: Status and Perspec8ves

Transcription

1 : Status and Perspec8ves Hartmut Wi=g PRISMA Cluster of Excellence, Ins6tute for Nuclear Physics and Helmholtz Ins6tute Mainz MITP Workshop Determina)on of the Fundamental Parameters in QCD 9 March 2016

2 The Standard Model ager the Higgs discovery Standard Model fully established but cannot account for: Mass and scale hierarchies: m top /m e > m Higgs m Planck Dark maeer and dark energy Amount of CP viola6on to sustain maeer/an6maeer asymmetry 2

3 The Standard Model ager the Higgs discovery Standard Model fully established but cannot account for: Mass and scale hierarchies: m top /m e > m Higgs m Planck Dark maeer and dark energy Amount of CP viola6on to sustain maeer/an6maeer asymmetry Explore the limits of the Standard Model Search for new par6cles and phenomena at higher energy Search for enhancement of rare phenomena Compare precision measurements to SM predic6ons 3

4 The Standard Model ager the Higgs discovery Standard Model fully established but cannot account for: Mass and scale hierarchies: m top /m e > m Higgs m Planck Dark maeer and dark energy Amount of CP viola6on to sustain maeer/an6maeer asymmetry Explore the limits of the Standard Model Search for new par6cles and phenomena at higher energy Search for enhancement of rare phenomena Compare precision measurements to SM predic6ons Control over hadronic uncertain8es 3

5 The Standard Model ager the Higgs discovery Standard Model fully established but cannot account for: Mass and scale hierarchies: m top /m e > m Higgs m Planck Dark maeer and dark energy Amount of CP viola6on to sustain maeer/an6maeer asymmetry Explore the limits of the Standard Model Search for new par6cles and phenomena at higher energy Search for enhancement of rare phenomena Compare precision measurements to SM predic6ons Control over hadronic uncertain8es 3

6 Precision Tests of the Standard Model Anomalous magne8c moment of the muon: a µ 1 2 (g 2) µ Dispersion theory: Model es6mates: a HVP µ = (692.3 ± 4.2 ± 0.3) a HLbL µ = based on Rexp(e + e hadrons) ( (105 ± 26) (116 ± 39)

7 Precision Tests of the Standard Model Running of electroweak mixing angle Running of sin 2 θw at low energies discriminates between different scenarios for New Physics Challenge for theory: hadronic contribu6ons 5

8 The Mainz (g 2)μ project Collaborators: N. Asmussen, A. Gérardin, J. Green, O. Gryniuk, G. von Hippel, H. Horch, H. Meyer, A. Nyffeler, V. Pascalutsa, A. Risch, HW M. Della Morte, A. Francis, B. Jäger, V. Gülpers, G. Herdoíza Topics: Hadronic vacuum polarisa6on Light-by-light scaeering Running of αem and sin 2 θw Determina6on of αs from vacuum polarisa6on func6on 6

9 The muon (g 2) in La=ce QCD 7

10 La=ce QCD approach to HVP Convolu6on integral over Euclidean momenta: [Lautrup & de Rafael; Blum] µ (Q) = a HVP µ =4 2 Z 1 Z 0 dq 2 f(q 2 ) (Q 2 ) (0) d 4 x e iq (x y) hj µ (x)j (y)i (Q µ Q µ Q 2 ) (Q 2 ) J µ = 2 3 u µu 1 3 d µd 1 3 s µs +... La^ce momenta are quan6sed: Q µ = 2 L µ Determine VPF Π(Q 2 ) and addi6ve renormalisa6on Π(0) Sta6s6cal accuracy of Π(Q 2 ) deteriorates as Q 0 8

11 La=ce QCD approach to HVP Convolu6on integral over Euclidean momenta: [Lautrup & de Rafael; Blum] a HVP µ =4 2 Z 1 0 dq 2 f(q 2 ) (Q 2 ) (0) Integrand peaked near Q 2 ( p 5 2)m 2 µ m 2 µ Q 2 f(q 2 ) Q 2 [GeV 2 ] 9

12 La=ce QCD approach to HVP Convolu6on integral over Euclidean momenta: [Lautrup & de Rafael; Blum] a HVP µ =4 2 Z 1 0 dq 2 f(q 2 ) (Q 2 ) (0) Integrand peaked near Q 2 ( p 5 2)m 2 µ L = 20 fm m 2 µ 10 fm 5 fm Q f(q 2 ) Accurate determina6on requires large sta6s6cs on large volumes! Q 2 [GeV 2 ] 9

13 La=ce QCD approach to HVP Hybrid method: [Golterman, Maltman & Peris, Phys Rev D90 (2014) ] Π(Q 2 ) Padé approx. numerical interpola6ons P.T. 0.1 GeV 2 Q 2 10

14 La=ce QCD approach to HVP Hybrid method: [Golterman, Maltman & Peris, Phys Rev D90 (2014) ] Π(Q 2 ) Padé approx. numerical interpola6ons P.T. 0.1 GeV 2 Q 2 Determine Π(0) from Padé approxima6on in small-momentum region 10

15 La=ce QCD approach to HVP Hybrid method: [Golterman, Maltman & Peris, Phys Rev D90 (2014) ] Π(Q 2 ) Padé approx. numerical interpola6ons P.T. 0.1 GeV 2 Q 2 Determine Π(0) from Padé approxima6on in small-momentum region Requires sub-percent accuracy in u,d-part for Q 2 = O(0.1 GeV 2 ) 10

16 La=ce QCD approach to HVP Main issues: Sta6s6cal accuracy at the sub-percent level required Reduce systema6c uncertainty associated with region of small Q 2 accurate determina6on of Π(0) Perform comprehensive study of finite-volume effects Include quark-disconnected diagrams Include isospin breaking: mu md, QED correc6ons 11

17 Low-momentum region of Π(Q 2 ) Apply twisted boundary condi6ons to access low-q 2 regime: (x + Le µ )=e i µ (x) ) Q µ = 2 L + µ L Twist Fourier 0.25 Π(q 2 ) (aq) 2 [Della&Morte,&Jäger,&Jü0ner,&H.W.,&JHEP&1203&(2012)&055;&PoS&&LATTICE2012&(2012)&175]& 12

18 Low-momentum region of Π(Q 2 ) Apply twisted boundary condi6ons to access low-q 2 regime: 0.28 (x + Le µ )=e i µ (x) ) Q µ = 2 L + µ L Twist Fourier 0.25 Π(q 2 ) (aq) 2 m = 190 MeV [Della&Morte,&Jäger,&Jü0ner,&H.W.,&JHEP&1203&(2012)&055;&PoS&&LATTICE2012&(2012)&175]& 12

19 Low-momentum region: Time moments Expansion of VPF at low-q 2 : (Q 2 )= 0 + 1X Q 2j j j=1 Vacuum polarisa6on for Q =(!,~0) : kk (!) =a 4 X x 0 e i!x 0 X ~x hj k (x)j k (0)i Spa6ally summed vector correlator: G(x 0 )= a 3 X ~x hj k (x)j k (0)i 13

20 Low-momentum region: Time moments Expansion of VPF at low-q 2 : Vacuum polarisa6on for Q =(!,~0) : (Q 2 )= 0 + 1X j=1 Q 2j j kk (!) =a 4 X x 0 e i!x 0 X ~x hj k (x)j k (0)i Spa6ally summed vector correlator: G(x 0 )= a 3 X ~x hj k (x)j k (0)i Time moments: G 2n a X x 0 [Chakraborty et al., Phys Rev D89 (2014) ] n o x 2n 0 G(x 0 )=( 2n! 2 ˆ (! 2 )! 2 =0 13

21 Low-momentum region: Time moments Expansion of VPF at low-q 2 : Vacuum polarisa6on for Q =(!,~0) : (Q 2 )= 0 + 1X j=1 Q 2j j kk (!) =a 4 X x 0 e i!x 0 X ~x hj k (x)j k (0)i Spa6ally summed vector correlator: G(x 0 )= a 3 X ~x hj k (x)j k (0)i Time moments: G 2n a X x 0 [Chakraborty et al., Phys Rev D89 (2014) ] n o x 2n 0 G(x 0 )=( 2n! 2 ˆ (! 2 )! 2 =0 Expansion coefficients: (0) 0 = 1 2 G 2, j =( 1) j+1 G 2j+2 (2j + 2)! 13

22 Time-Momentum Representa8on Integral representa6on of subtracted VPF ˆ (Q 2 ) (Q 2 ) (0) (Q 2 ) (0) = 1 Z 1 Q Z 2 dx 0 G(x 0 ) Q 2 x sin Qx 0 0 G(x 0 )= d 3 x hj k (x)j k (0)i [Bernecker & Meyer, Eur Phys J A47 (2011) 148] [Francis et al. 2013; Feng et al. 2013; Lehner & Izubuchi 2014, Del Debbio & Portelli 2015, ] Q 2 is a tuneable parameter No extrapola6on to Q 2 = 0 required; related to 6me-moments Must determine I = 1 vector correlator G(x0) for x0 Include two-pion states to capture long-distance behaviour 14

23 Time-Momentum Representa8on (Q 2 ) (0) = 1 Z 1 Q Z 2 dx 0 G(x 0 ) Q 2 x sin Qx 0 0 G(x 0 )= d 3 x hj k (x)j k (0)i [Gülpers et al., arxiv: ; Francis et al., arxiv: ] 15

24 Current data sets and sta8s8cs Nf = 2 flavours of O(a) improved Wilson fermions Three values of the la^ce spacing: a = 0.076, 0.066, fm Pion masses and volumes: m min = 185 MeV, m L> measurements per ensemble To be processed: Nf = 2+1 flavours of O(a) improved Wilson fermions; tree-level Symanzik gauge ac6on; open boundary condi6ons Five values of the la^ce spacing; physical pion mass 16

25 Comparison: Fits versus Time moments Construct Padé approximants either from fits or 6me moments Padé [1,1]: Fit VPF data 0 Fit Padé [1,1] for < 0.5 GeV 2 Q 2 low (Q 2 ) m = 185 MeV Q 2 [GeV 2 ] 17

26 Comparison: Fits versus Time moments Construct Padé approximants either from fits or 6me moments Padé [1,1]: Fit VPF data 0 Fit Padé [1,1] for < 0.5 GeV 2 Q 2 low (Q 2 ) m = 185 MeV Q 2 [GeV 2 ] 17

27 Comparison: Fits versus Time moments Construct Padé approximants either from fits or 6me moments Padé [1,1]: Fit Padé [1,1]: Time-moments VPF data Fit Padé [1,1] for < 0.5 GeV 2 Q 2 low (Q 2 ) m = 185 MeV Q 2 [GeV 2 ] 17

28 Comparison: Fits versus Time moments Construct Padé approximants either from fits or 6me moments Padé [1,1]: Fit Padé [1,1]: Time-moments VPF data Fit Padé [1,1] for < 0.5 GeV 2 Q 2 low (Q 2 ) m = 185 MeV Q 2 [GeV 2 ] Low-order Padé approximants consistent for Q 2 < 0.5 GeV 2 Apply trapezoidal rule to evaluate convolu6on integral for Q GeV 2 17

29 Comparison: TMR versus fits Interpolate G(x0) by cubic spline for x0 < x0,cut 18

30 Comparison: TMR versus fits Interpolate G(x0) by cubic spline for x0 < x0,cut x 0,cut G(x 0 )=A e e m Vx 0 18

31 Comparison: TMR versus fits Interpolate G(x0) by cubic spline for x0 < x0,cut x 0,cut G(x 0 )=A e e m Vx 0 Mild dependence on x0,cut 18

32 Comparison: TMR versus fits Interpolate G(x0) by cubic spline for x0 < x0,cut Comparison with determined from Padé fits a (ud)hvp µ a (ud)hvp µ TMR, a =0.076 fm TMR, a =0.066 fm TMR, a =0.049 fm P[1,1], a =0.076 fm P[1,1], a =0.066 fm P[1,1], a =0.049 fm x 0,cut G(x 0 )=A e e m Vx m 2 [GeV 2 ] Mild dependence on x0,cut 18

33 Comparison: TMR versus fits Interpolate G(x0) by cubic spline for x0 < x0,cut Comparison with determined from Padé fits a (ud)hvp µ a (ud)hvp µ TMR, a =0.076 fm TMR, a =0.066 fm TMR, a =0.049 fm P[1,1], a =0.076 fm P[1,1], a =0.066 fm P[1,1], a =0.049 fm x 0,cut G(x 0 )=A e e m Vx 0 Mild dependence on x0,cut m 2 [GeV 2 ] Consistent results with similar accuracy 18

34 Chiral and con8nuum extrapola8ons Use collec6on of different func6onal forms, e.g. a (ud)hvp µ up, down a=0.0 fm a= fm a= fm a= fm a = fm a = fm a = fm Fit A: b 0 + b 1 m 2 + b 2 m 2 ln(m 2 )+b 3 a Fit B: 300 b 0 + b 1 m 2 + b 2 m 4 + b 3 a m 2 [GeV 2 ].. a (s)hvp µ strange a = fm a = fm a = fm Perform cuts in pion mass and la^ce spacing La^ce spacing effects clearly resolved for larger quark masses m 2 [GeV 2 ] 19

35 Chiral and con8nuum extrapola8ons Use collec6on of different func6onal forms, e.g. a (ud)hvp µ up, down a=0.0 fm a= fm a= fm a= fm a = fm a = fm a = fm Fit A: b 0 + b 1 m 2 + b 2 m 2 ln(m 2 )+b 3 a Fit B: 300 b 0 + b 1 m 2 + b 2 m 4 + b 3 a m 2 [GeV 2 ].. a (c)hvp µ charm a = fm a = fm a = fm Perform cuts in pion mass and la^ce spacing La^ce spacing effects clearly resolved for larger quark masses m 2 [GeV 2 ] 19

36 Summary on a µ hvp Final errors es6mated via extended frequen6st method a (ud)hvp µ a (s)hvp µ a (c)hvp µ Pade fit Time moments TMR Pade fit Time moments TMR Preliminary Pade fit Time moments TMR Different methods consistent at the level of 1σ Time moments sta6s6cally most precise Strange quark contribu6on consistent with other recent results Overall accuracy dominated by u, d contribu6on 20

37 Disconnected Contribu8ons Electromagne6c current correlator with u, d, s quarks: G`s (x 0 ):= Z d 3 x J `s k (x)j `s k (0), Iden6fy connected and disconnected contribu6ons: G`s (x 0 )= 5 9 G`con(x 0 )+ 1 9 Gs 1 con(x 0 ) Z d 3 x Tr S`(x, x) k G`s disc(x 0 )= 9 G`s J`s µ = 2 3 u ku 1 3 d kd 1 3 s ks disc (x 0) Tr [S s (x, x) k ] {x! 0} 21

38 Disconnected Contribu8ons Electromagne6c current correlator with u, d, s quarks: G`s (x 0 ):= Z d 3 x J `s k (x)j `s k (0), Iden6fy connected and disconnected contribu6ons: G`s (x 0 )= 5 9 G`con(x 0 )+ 1 9 Gs 1 con(x 0 ) Z d 3 x Tr S`(x, x) k G`s disc(x 0 )= 9 G`s J`s µ = 2 3 u ku 1 3 d kd 1 3 s ks disc (x 0) Tr [S s (x, x) k ] {x! 0} [Gülpers et al., arxiv: ; V. Gülpers, PhD Thesis 2015] 21

39 Disconnected Contribu8ons Electromagne6c current correlator with u, d, s quarks: G`s (x 0 ):= Z d 3 x J `s k (x)j `s k (0), Iden6fy connected and disconnected contribu6ons: G`s (x 0 )= 5 9 G`con(x 0 )+ 1 9 Gs 1 con(x 0 ) Z d 3 x Tr S`(x, x) k G`s disc(x 0 )= 9 G`s J`s µ = 2 3 u ku 1 3 d kd 1 3 s ks disc (x 0) Tr [S s (x, x) k ] {x! 0} [Gülpers et al., arxiv: ; V. Gülpers, PhD Thesis 2015] 21

40 Disconnected Contribu8ons Non-zero disconnected contribu6on can be resolved: Disconnected contribu6on for x0 : 1 9 G`s disc G`s G = G 1 G 9 1 2G s con G`con x 0!1! 1 9 Failure to resolve non-zero signal provides systema6c error es6mate [Gülpers et al., arxiv: ; V. Gülpers, PhD Thesis 2015] 22

41 Running of electroweak couplings 23

42 Running of α phenomenological approach Fine structure constant: (Q 2 )= 1 (Q 2 ) Hadronic contribu6ons phenomenological approach: had (Q 2 )= Q2 3 Z 1 4m 2 ds R had(s) s(s Q 2 ) c.f. a HVP µ = mµ 3 2 Z 1 4m 2 ds R had(s) ˆK(s) s 2 (5) had (M 2 Z) = ( ± 1.38) 10 4 Error on Δαhad limits accuracy of Standard Model tests 24

43 Running of α Euclidean approach Vacuum polarisa6on func6on: had (Q 2 )=4 (Q 2 ) (0) Adler func6on: D(Q 2 )= 12 2 d (Q2 ) d ln Q 2 = 3 d d ln Q 2 had (Q 2 ) [H. Horch, G. La_ce 2015] La^ce QCD: similar accuracy as phenomenological approach 25

44 Running of sin 2 θw Defini6on: sin 2 W (Q 2 )=sin 2 (0) {z } sin 2 W (Q 2 ) SM published sin 2 θ W (µ) ongoing planned Q W (Cs) Q W (e) SLAC NuTeV screening antiscreening Q W (p) Mainz LEP 1 Tevatron Q W (Ra) Q W (p) JLab Q W (e) JLab edis SLD CMS µ [GeV] [Erler & Su, PPNP 71 (2013) 119] Dispersive approach requires separa6on of contribu6ons from up/down-type quarks 26

45 Running of sin 2 θw Euclidean approach: Z µ (Q) = Z d x e iq x V Z µ (x)j (0) V Z µ = V 3 µ sin 2 W J µ V 3 µ = 1 4 u µu d µ d s µ s + c µ c +... Z (Q 2 )= 3 (Q 2 ) sin 2 W (Q 2 ) had sin 2 W (Q 2 )= e2 sin 2 Z (Q 2 ) Z (0) 0 Spin-off of calcula6on of running of Δαhad 27

46 Running of sin 2 θw Preliminary results: Connected contribu6ons: Mainz ETMC had sin 2 θw ( ˆQ 2 ) ˆQ 2 [GeV 2 ] [H. Horch, G. Herdoíza, V. La_ce 2015] 28

47 Running of sin 2 θw Preliminary results: Connected contribu6ons: Disconnected contribu6ons: Mainz ETMC had sin 2 θw ( ˆQ 2 ) ˆQ 2 [GeV 2 ] [H. Horch, G. Herdoíza, V. La_ce 2015] 28

48 Running of sin 2 θw Preliminary results: Connected contribu6ons: Disconnected contribu6ons: Mainz ETMC had sin 2 θw ( ˆQ 2 ) ˆQ 2 [GeV 2 ] Long-distance behaviour of total correlator limited by accuracy of disconnected contribu6on systema6c error es6mate [H. Horch, G. Herdoíza, V. La_ce 2015] 28

49 Hadronic Light-by-Light ScaYering 29

50 La=ce approaches to HLbL Several different methods, e.g. QCD+QED simula6ons [T. Blum et al., 2008 ] Here: compute HLbL amplitude from Euclidean four-point func6on of the vector current E µ 1 µ 2 µ 3 µ 4 (P 4 ; P 1,P 2 )= Z d 4 x 1 d 4 x 2 d 4 x 4 e i P a P a x a hj µ1 (x 1 )J µ2 (x 2 )J µ3 (0)J µ4 (x 4 )i Pilot study: compute forward scaeering amplitude of transversely polarised virtual photons: M TT ( Q 2 1, Q 2 2, ) = e4 4 R µ 1 µ 2 R µ3 µ 4 E µ 1 µ 2 µ 3 µ 4 ( Q 2 ; Q 1,Q 1 ) [J. Green, O. Gryniuk, G. von Hippel, H.B. Meyer, V. Pascalutsa, PRL 115 (2015) ] 30

51 La=ce approaches to HLbL Relate forward amplitude to cross sec6on for! hadrons M TT ( Q 2 1, Q 2 2,v) M TT ( Q 2 1, Q 2 2, 0) = Z 1 0 d 0...[ 0( 0 )+ 2 ( 0 )] [J. Green, O. Gryniuk, G. von Hippel, H.B. Meyer, V. Pascalutsa, PRL 115 (2015) ] 31

52 La=ce approaches to HLbL Relate forward amplitude to cross sec6on for! hadrons M TT ( Q 2 1, Q 2 2,v) M TT ( Q 2 1, Q 2 2, 0) = Z 1 0 d 0...[ 0( 0 )+ 2 ( 0 )] [J. Green, O. Gryniuk, G. von Hippel, H.B. Meyer, V. Pascalutsa, PRL 115 (2015) ] 31

53 La=ce approaches to HLbL Relate forward amplitude to cross sec6on for! hadrons MTT( Q 2 1, Q 2 2, ) MTT( Q 2 1, Q 2 2, 0) M TT ( Q 2 1, Q 2 2,v) M TT ( Q 2 1, Q 2 2, 0) 10 5 m = 324 MeV, Q 2 1 =0.377 GeV 2 (GeV 2 ) = Z 1 0 d 0...[ 0( 0 )+ 2 ( 0 )] Q 2 2 (GeV 2 ) [J. Green, O. Gryniuk, G. von Hippel, H.B. Meyer, V. Pascalutsa, PRL 115 (2015) ] 31

54 La=ce approaches to HLbL Relate forward amplitude to cross sec6on for! hadrons MTT( Q 2 1, Q 2 2, ) MTT( Q 2 1, Q 2 2, 0) M TT ( Q 2 1, Q 2 2,v) M TT ( Q 2 1, Q 2 2, 0) 10 5 m = 324 MeV, Q 2 1 =0.377 GeV 2 (GeV 2 ) = Z 1 0 d 0...[ 0( 0 )+ 2 ( 0 )] Comparison with phenomenological model for σ0 and σ Q 2 2 (GeV 2 ) [J. Green, O. Gryniuk, G. von Hippel, H.B. Meyer, V. Pascalutsa, PRL 115 (2015) ] 31

55 32

56 Prospects 32

57 Summary Hadronic vacuum polarisa8on Major effort within La^ce QCD community; focus on reducing overall uncertainty to the sub-percent level Several complementary formalisms TMR, 6me moments readily carry over to La^ce QCD with open boundary condi6ons Good progress in compu6ng quark-disconnected contribu6ons Hadronic light-by-light scayering Several concepts are being explored Aim for 20 30% overall accuracy 33