Explore hadronic matrix elements in the timelike region: from HVPs to rare kaon decays

Transcription

1 Explore hadronic matrix elements in the timelike region: from HVPs to rare kaon decays Xu Feng (ØÌ) Workshop on Parton Distributions in Modern Era, Beijing, 07/14/2017

2 Introduction Main topic of this workshop is DIS and PDF Lepton-hadron scattering process `1 + N(P) `2 + X (P X ) Hadronic current amplitude unkown Jµ(P, q) = P X J µp Object of study: Probe the parton structure of the nucleon Study QCD dynamics at the confinement scale Hadronic tensor from cross section W µ = 1 4 (2 ) 4 X 4 (P + q P X )PJ µ P X P X J P 2/35

3 Start with a simpler case Lepton-hadron scattering e + e scattering Hadronic tensor for DIS Hadronic vacuum polarization (HVP) 3/35

4 Hadronic vacuum polarization function 4/35

5 HVP function in Euclidean spacetime Hadronic vacuum polarization function µ (Q) = d 4 xe iqx 0T {J µ (x)j (0)}0 On lattice Discrete Fourier transform assigns the discrete momentum Q i = n i (2 L) q 2 = i Q 2 i < 0 always spacelike Π(Q 2 ) HVP function at spacelike region Q 2 [GeV 2 ] 5/35

6 Analytic continuation method Non-QCD state: photon is a superposition of a complete set of hadron states Analytic continuation method proposed by Ji & Jung, PRL 86 (2001) 208 dt e!t d 3 xe i qx 0T {J µ (x)j (0)}0 = dt e!t C µ (q, t)! is photon energy, input by hand, thus can be tuned continuously Worry about e!t divergent for large t? 0T {J µ (t)j (0)}0 exponentially decreases as e E t Subtract (0) without Q 2 extrapolation (! 2 ) (0) = dt e!t 1! 2 t2 2 C zz(0, t) 6/35

7 Analytic continuation method: results Tuning photon s energy! Cover both spacelike and timelike Determine (Q 2 ) at Q 2 = Π(Q 2 ) n 2 =0, {µ,ν}={x,x} conventional n 2 =1, {µ,ν}={x,x} -0.2 Π(Q 2 ) n 2 =2, {µ,ν}={x,x} n 2 =3, {µ,ν}={x,x} Q 2 [GeV 2 ] Q 2 [GeV 2 ] XF, Hashimoto, Hotzel, Jansen, et al, PRD 88 (2013) /35

8 Demonstration: Minkowski spacetime In Minkowski spacetime, in the framework of QCD+QED Using the LSZ reduction formula (k, )J M (0) (k, )J M (0) = i lim k k " µ(k, ) k 2 d 4 xe ik x T{A M µ (x)j M (0)} where A M µ (x) is a photon field defined in Minkowski spacetime Treat QED part perturbatively: exp(is int ) = 1 + is int +, S int = e d 4 xa M µ (x)j M µ (x) At O(e) we have (k, )J M (0) = ie" µ(k, ) d 4 ye iky T{J M µ (y)j M (0)} 8/35

9 Demonstration: Euclidean spacetime In Euclidean spacetime, look at M µ (t 0, k) = d 3 xe i k x A E µ(x, t 0 )J E (0) t 0 A E µ( k) (k, )e!t0 (k, )J E (0)! < EV : neglect hadronic states, suppressed by e (E V!)t 0 Neglect three-photon states, suppressed by powers of QED coupling Employ pqed M µ (t 0, k) = e dt d 3 ye i k y D µ ( k, t 0 t) J E (y, t)j E (0) Photon propagator dk 0 2 D µ (k)e ik0t = dk 0 2 We can write µ k k 2 e ik0t = e!t 2! µ M µ (t 0, k) e dt e!t0 t d 3 ye i k y J E µ (y, t)j E (0) 9/35

10 Demonstration of the analytic continuation We get in Euclidean spacetime (k, )J E (0) = e " µ( ) dt e!t d 3 ye i k y Jµ E (y, t)j E (0) Remember that in Minkowski spacetime (k, )J M (0) = ie" µ(k, ) d 4 ye iky T{Jµ M (y)j M (0)} Thus the validity of analytic continuation method is demonstrated This method is originally proposed by [Ji & Jung, PRL 86 (2001) 208, PRD 64 (2001) ] Besides for HVPs, analytic continuation has also been used for lat. calc. of charmonium radiative decay and 0 Dudek & Edwards, PRL 97 (2006) Cohen, Lin, Dudek, Edwards, PoS LAT (2008) 159 XF, Aoki, Fukaya, Hashimoto et.al. PRL 109 (2012) CLQCD, Chen et. al, EPJC 76 (2016) / 35

11 0 Chiral anomaly at m u = m d = 0 Lattice QCD at m u = m d 0 non-trivial test of chiral anomaly Non-QCD final state: (p 1, 1) (p 2, 2) 0 (q) d 4 xe ip1x 0T {J µ (x)j (0)} 0 (q) = µ p1 p 2 F 0 (m, 2 p1, 2 p2) 2 Analytic continuation: dt e!t d 3 xe i p1 x 0T {J µ (x)j (0)} 0 (q) Lattice result for F 0, using overlap chiral fermion F(m π 2,0,0) FS corrected m π [GeV ] XF, Aoki, Fukaya, Hashimoto et.al. PRL 109 (2012) / 35

12 Explore the timelike region, above hadron production threshold 12 / 35

13 Highly non-trivial above hadron production threshold BES III Provide rich experimental data for low energy hadron physics R = (e+ e hadrons) (e + e µ + µ ) 13 / 35

14 Require nonperturbative lattice QCD In low energy region QCD asympototic freedom pqcd fails in the low energy region QCD color confinment hard to explain hadron spectrum e.g. resonance, bound state We need nonperturbative treatment lattice QCD Solve QCD from first-principals, don t rely on model assumption 14 / 35

15 e + e + - (e + e + ) at GeV, from [BES III, Phys. Lett. B753 (2016) 629] 1400 )) [nb] FSR π + π - (γ e + σ bare (e s' [GeV] s < 0.9 GeV, hadronic final state dominated by + resonant peak at 0.77 GeV No essential di erence for e + e + at 0.6 GeV and 0.77 GeV resonance is a dynamical property for scattering at 0.77 GeV 15 / 35

16 Timelike pion form factor From (e + e + ), timelike pion form factor can be extracted (e + e + ) = 0 (e + e + )F (s) 2 0 : tree-level cross section by assuming ± as a point-like particle F (s) 2 describes the internal EM structure for scattering, resonance, timelike pion form factor three in one Spacelike form factor from J µ γ π + π + single hadron in i/f state, F (q 2 ) is a real function, LQCD works well Timelike form factor from J µ 0 γ multi-hadron in final state, F (q 2 ) is complex, di π + π cult for LQCD 16 / 35

17 correlation function Euclidean correlation functions are the primary objects in Lattice QCD Euclidean time t it, operator O(t) = e Ht O(0)e Ht Single pion correlation function: (t) = x ū 5 d(x, t) C (t) = 0 (t) (0)0 = e E n t n 0 2 n At large t, only ground state n = saturates Two-pion correlation function C (t) = 0( ) (t)( )(0)0 = e E n t,n 0 2 n Lattice size is finite afewfm discrete spectrum 17 / 35

18 scattering phase What information can we obtain from discrete E? If no interaction between, E is simply determined by lattice size L E free = m 2 + p 21 + m 2 + p 2 2, p 1,2 = 2 L n 1,2 With interaction, E di ers from E free E = E n E free 0 contains the information of QCD interaction ππ scattering in finite volume ππ scattering in infinite volume π π E ππ π π ππ, out ππ, in = e 2iδ π π π π Lüscher establish a relation between E and (k) = n (q), n Z, q = (E) k 2 L, E = 2 m 2 + k 2 18 / 35

19 Lattice results for the I = 1, P-wave scattering phase use 3 Lorentz frames; at each frame, calculate lowest 2 E 6 scattering phase to map out the resonance region 1 sin 2 (δ) 0.5 CMF MF1 MF2 sin 2 (δ)=1=>am R ae CM XF, K. Jansen, D. Renner, PRD83 (2011) (one of the early work on ) 19 / 35

20 Recent lattice results Sophisticated calculation of I = 1 scattering at m = 236 MeV [Hadron Spectrum Collaboration, ] 1) distillation smearing 2) variational method 3) anisotropic lattice a t = a s 3.5 4) multi-channel approach / 35

21 Matrix element with multi-hadron state ( p) ( p)j µ, final state involves multi-hadrons Maiani-Testa no-go theorem 91: LQCD can t explore the transition involving multi-hadrons Lattice results physical ones, requiring lattice size L For a large, quasi-infinite volume, p = 2 L n 0 Ground state dominance at large t leads to (p) ( p) (0) (0) A E =2m Dense discrete energy level hard to extract excited state Only information of A at E = 2m Solution We must work in the finite volume [Lellouch, Lüscher, 00] Finite-size correction must be treated properly 21 / 35

22 Finite volume approach (I) d 3 x 0J µ (x, t)j µ (0)0 V =,nj µ 0 V 2 e E n t + V n From the correlation function of vector current operator J µ, we can get E n : canbeusedtodetermine,nj µ 0 V 2 : What information can we obtain? Recall Lüscher s formula (q) + (k) = n V, finite volume sums infinite volume integrals [Lin, Martinelli, Sachrajda, Testa, 01] de V (E), n V (E) = dn de = q (q) + k (k) E 4 k 2 V (E) can be viewed as the density of states at energy E 22 / 35

23 Finite volume approach (II) Finite volume V d 3 x 0J µ (x, t)j µ (0)0 V V de V (E),EJ µ 0 V 2 e Et Infinite volume: spectral representation for vector-current correlator Final relation C (t) = d 3 x 0J µ (x, t)j µ (0)0 = de f (E)e Et 2m f (E) = k 3 E F (s) 2, s = E 2 = 4(m 2 + k 2 ) F (s) 2 = 3 (q) q + µ0 V 2 Result is correct, but demonstration is not very satisfactory Equivalent integral does not mean equivalent integrand Integral 2m runs over inelastic region New demonstration [Christ, XF, Martinelli, Sachrajda, 15] 23 / 35

24 Timelike pion form factor 180 F (s) 2 = 3 (q) q m π =380 MeV + µ0 V m π =290 MeV δ(e) GS form for δ(e) GS form for δ(e) F π (E) GS+polynomials GS form for F π 15 GS+polynomials GS form for F π E [GeV] E [GeV] XF, Aoki, Hashimoto, Kaneko, PRD91 (2015) / 35

25 Rare Kaon decays ν ν Z ν ν e, µ, τ ū, c, t W W s d s d K + W π + K + ū, c, t π + u u u u H W (x) H W (0)K Rare Kaon decay vs N(P)J µ (x) J (0)N(P) Deep inelastic scattering 25 / 35

26 K + + : Experiment vs Standard model ν ν Z ν ū, c, t s d K + W π + u u Z-exchange ν e, µ, τ W W s d K + ū, c, t π + u u W -W K + + : largest contribution from top quark loop, thus theoretically clean EM H e G F 2 2 sin 2 tx t (x t ) ( sd) V A ( ) V A W N Probe the new physics at scales of N 1 2 MW = O(10 TeV) Past experimental measurement is 2 times larger than SM prediction Br(K + + ) exp = [BNL E949, 08] Br(K + + ) SM = 9.11 ± [Buras et. al., 15] but still consistent with > 60% exp. error 26 / 35

27 New experiment New generation of experiment: NA62 at CERN aims at observation of O(100) events [ ] 10%-precision measurement of Br(K + + ) Latest results reported at FPCP 2017 Detector installation completed in % of 2016 data no event yet Full 2016 data O(1) events 27 / 35

28 2 nd -order weak interaction and bilocal matrix element Hadronic matrix element for the 2 nd -order weak interaction T T dt + T [Q A (t)q B (0)] K + = + Q A nnq B K + n M K E n + + Q B nnq A K + 1 e (M K E n)t M K E n For E n > M K, the exponential terms exponentially vanish at large T For E n < M K, the exponentially growing terms must be removed n : principal part of the integral replaced by finite-volume summation possible large finite volume correction when En M K [Christ, XF, Martinelli, Sachrajda, PRD 91 (2015) ] 28 / 35

29 Low lying intermediate states O S=1 O S=0 ν ν s d K + u e, µ, τ π + u ν e, µ, τ ν s ū, c d K + O S=1 O S=0 π + u u O Z ν ν s d K + Q 1,2 π + u u O Z ν ū, c s d K + Q 1,2 π + u u ν 29 / 35

30 Short-distance divergence [Christ, XF, Portelli, Sachrajda, PRD 93 (2016) ] SD divergence appears in Q A (x)q B (0) when x 0 Introduce a counter term X Q 0 to remove the SD divergence p 1 p p 2 loop {Q A Q B } RI = X(µ p 2 0,a) = 0 i =µ 2 0 The coe p 4 Q RI A Q RI B cient X is determined in the RI/(S)MOM scheme The bilocal operator in the MS scheme can be written as d 4 xt[q MS A (x)qb MS MS (0)] p 3 p 1 p 4 Q RI 0 = Z A Z B d 4 xt[qa lat QB lat lat ] + X lat RI + Y RI MS Q 0 (0) p 2 p 3 X lat RI is calculated using NPR and Y RI MS calculated using PT 30 / 35

31 Results for charm quark contribution Charm quark contribution P c P c = Pc SD + P c,u NNLO QCD [Buras, Gorbahn, Haisch, Nierste, JHEP 0611 (2006) 002]: P SD c = 0.365(12) Phenomenological ansatz [Isidori, Mescia, Smith, NPB 718 (2005) 319] P c,u = 0.040(20) Lattice = 420 MeV, m c = 860 MeV [Bai, Christ, XF, Lawson, Portelli, Sachrajda, PRL 118 (2017) ] P c = (±13) stat (±32) scale ( 45) FV P c Pc SD = (±13) stat (±32) scale ( 45) FV As a smaller m c is used, P c is also smaller Cancellation in W -W and Z-exchange diag. leads to small P c P SD c Important to perform the calculation at physical m and m c 31 / 35

32 K `+` CP conserving decay: K + +`+` and K S 0`+` Involve both - and Z-exchange diagram, but -exchange is much larger l + s K u γ,z W c u l d π u Unlike Z-exchange, the -exchange diagram is LD dominated By power counting, loop integral is quadratically UV divergent EM gauge invariance reduces divergence to logarithmic c u GIM cancellation further reduces log divergence to be UV finite 32 / 35

33 Lattice calculation strategy Focus on -exchange Hadronic part of decay amplitude is described by a form factor T µ +,S (p K, p ) = d 4 xe iqx (p )T {J em(x)h µ S=1 (0)}K + K S (p K ) = G F M 2 K (4 ) 2 V +,S(z) z(k + p) µ (1 r 2 )q µ with q = p K p, z = q 2 M 2 K, r = M M K The target for lattice QCD is to calculate the form factor V +,S (z) Lattice calculation strategy: [Christ, XF, Portelli, Sachrajda, PRD 92 (2015) ] Use conserved vector current to protect the EM gauge invariance Use charm as an active quark flavor to maintain GIM cancellation 33 / 35

34 First exploratory calculation on K + +`+` Use ensemble, N conf = 128 [Christ, XF, Jütter, Lawson, Portelli, Sachrajda, PRD 94 (2016) ] a 1 = 1.78 GeV, m = 430 MeV m K = 625 MeV, m c = 530 MeV Momentum dependence of V + (z) V + (z) = a + + b + z V (z) a + = 1.6(7), b + = 0.7(8) z = q 2 /MK 2 K + + e + e data + phenomenological analysis: a + = 0.58(2), b + = 0.78(7) [Cirigliano, et. al., Rev. Mod. Phys. 84 (2012) 399] V j (z) = a j + b j z + jr 2 + j (z z 0 ) G F M 2 K r 4 K V (z) =a + bz p = 2 L (1, 0, 0) p = 2 L (1, 1, 0) p = 2 L (1, 1, 1) 1 + z (zr ) , j = +, S r 2 V F V (z) loop Experimental data only provide d dz square of form factor V +(z) 2 Need phenomenological knowledge to determine the sign for a +, b + 34 / 35

35 Summary HVPs at spacelike momenta standard lattice QCD calculation Ji & Jung s analytic continuation method allows to probe the region below hadron production threshold Lüscher s finite-volume method allows to probe the region above hadron production threshold Timelike pion form factor Rare Kaon decays For Deep inelastic scattering a challenging task as highly above hadron production threshold Keh-Fei s talk on hadronic tensor Huey-Wen s talk on quasi-pdf 35 / 35