Reevalua'on of Neutron Electric Dipole Moment with QCD Sum Rules

Transcription

1 Reevalua'on of Neutron Electric Dipole Moment with QCD Sum Rules Natsumi Nagata Nagoya University Na1onal Taiwan University 5 November, 2012 J. Hisano, J. Y. Lee, N. Nagata, and Y. Shimizu, Phys. Rev. D85 (2012)

2 Outline 1. Introduc1on 2. QCD Sum Rules (review) 3. NEDM with QCD Sum Rules 4. Conclusion & Prospects

3 1. Introduc'on

4 Neutron Electric Dipole Moment (nedm) Hamiltonian (non- rela1vis1c) Lagrangian This interac1on violates parity (P) and 1me- reversal (T) invariance. CP (CPT theorem)

5 Neutron Electric Dipole Moment (nedm) d P E T d E d E

6 nedm from CKM matrix in the Standard Model CKM matrix source of CP- viola1on in the SM Its contribu1on to the nedm is quite suppressed because it is flavor- changing interac1on. I. B. Khriplovich and A. R. Zhitnitsky (1982), T. Mannel and N. Uraltsev (2012). Experimental limit (Ins1tut Laue- Langevin) Phys.Rev.Le`. 97, (2006). The nedm provides a clean, background- free probe of the CP- viola1ng interac1ons in physics beyond the SM.

7 Effec1ve Lagrangian (QCD scale; up to dimension 5) θ term quark EDMs quark CEDMs CP- viola1ng parameters (physical parameter)

8 Goal Experimental limit on nedm Limits on parton- level We use the method of QCD Sum Rules Some QCD parameters are extracted from the results obtained with the lahce simula1ons Results We derive a conserva1ve limit for the contribu1ons of the CP viola1ng operators compared with the previous calcula1ons.

9 2. QCD Sum Rules

10 QCD Sum Rules M. Shifman, A. Vainshtein, V. Zakharov, Nucl. Phys. B147, 385 (1979) M. Shifman, A. Vainshtein, V. Zakharov, Nucl. Phys. B147, 448 (1979) The correlator of the hadron fields is evaluated in terms of Operator Product Expansion (OPE) Short- distance contribu1on Perturba1ve QCD Long- distance contribu1on VEVs of quark/gluon operators By using the dispersion rela1ons, it is related with the sum of the contribu1ons of the hadronic states.

11 Correla1on func1on Spectral representa1on j(x): a hadron field Spectral func1on

12 Dispersion rela1on Let us connect the region of q 2 << 0 (short- distance, perturba1ve) with that of q 2 > 0 (long- distance, physical) The Cauchy formula Choosing the contour in the right figure, q 2 follows. Here, we use

13 Dispersion rela1on Since the above expression has ultraviolet divergence, we subtract from Π(q 2 ) the first few terms of its Taylor q 2 = 0. In the case of q 2 << 0, one can calculate it perturba1vely (quark picture) It encodes informa1on of Hadron spectrum However, this expression is inconvenient due to the presence of unknown subtrac1on terms. Moreover, li`le is known about the spectral func1on.

14 Borel transforma1on M: Borel mass Borel transforma1on eliminates the subtrac1on terms in the dispersion rela1on exponen1ally suppresses the contribu1ons from excited resonances and the con1nuum states

15 QCD sum rules By applying the Borel transforma1on to the dispersion rela1on, We obtain, This equa1on is so- called QCD sum rules

16 Operator product expansion (OPE) Π(q 2 ) (q 2 < 0) is evaluated in terms of the operator product expansion (OPE). Here we deal with the long- & short- distance contribu1on separately. Short- distance contribu1on included in the Wilson coefficients C i (Q 2 ) Long- distance contribu1on included in the VEVs (condensates) <0 O i 0>

17 Hadron contribu1on Im Π(s) Contribu1on of one- par1cle state (pole) + excited/con1nuum states (branch cut) The former has informa1on we want to extract The la`er is suppressed by the Borel transforma1on, but oven causes theore1cal uncertainty. Excited/con1nuum states Quark- hadron duality Appropriate model

18 3. NEDM with QCD Sum Rules

19 Effec1ve Lagrangian (QCD scale; up to dimension 5) quark EDMs quark CEDMs CP- viola1ng parameters (physical parameter)

20 Calcula1on The correlator of neutron currents is evaluated in two ways: (i) Operator product expansion Short- distance Long- distance (ii) Phenomenological model nedm d n Previous work M. Pospelov and A. Ritz (1999, 2001) Connect aver Borel transforma1on

21 Correla1on func1on of the neutron fields Neutron field: η n (x) Correla1on func1on We extract the nedm from the correlator. Extra phase factor α n mixes the nedm with the magne1c dipole moments. We focus on the chiral invariant term

22 Neutron interpola1ng field In order to evaluate the correlator, we need to express the neutron field as a composite operator of quark fields with the same quantum numbers as those of neutron. We find that β = 1 is an op1mal choice because it suppresses the higher- order contribu1on removes the mixing effects of currents

23 Phenomenological calcula1on Double pole Single pole No pole We assume A: const. and B 0 in the following calcula1on

24 OPE calcula1on F µν, θ,d q, d q η n (0) η n (x) F µν, θ,d q, d q F µν, θ,d q, d q η n (0) η n (x) η n (0) η n (x) d q d q We carry out the calcula1on up to the N.L.O.

25 OPE result Λ: an arbitrary parameter with mass- dimension 1. Θ is a linear combina1on of the CP- viola1ng parameters. χ, κ, are the QCD parameters determined elswhere.

26 QCD Sum Rule Now we connect the two results aver the Borel transforma1on Borel mass dependence of the r.h.s. M: Borel mass 3 M 4 exp(m n 2 /M 2 ) (GeV 4 ) One can pick out nedm from the tangent line to the func1on shown in the lev figure. Borel Mass M 2 (GeV 2 )

27 Error es1mate Single / Double Borel Mass M 2 (GeV 2 ) Ra1o of Single and Double pole contribu1ons The sum rule gives the values of d n and A at a given Borel Mass M. Central value of d n is determined where the Double pole contribu1on is dominant. We es1mate the error of the calcula1on by requiring that the Single pole contribu1on is less than 30% of the Double pole contribu1on.

28 λ n (Lahce results) We extract the value of λ n from the lahce simula1ons. Y. Aoki et. al. (2008) In previous work, λ n is also evaluated by using the QCD sum rules. The lahce QCD value is several 1mes larger than that evaluated based on the QCD sum rules. D. B. Leinweber (1997) The resultant nedm value is smaller than those in the previous literature.

29 Results (phen) (OPE) (Lahce) By subs1tu1ng the QCD parameters, we obtain J. Hisano, J. Y. Lee, N.N., Y. Shimizu (2012) This result is about 70% smaller than previous results. It gives a conserva1ve limit for the contribu1ons of the CP- viola1ng operators.

30 4. Conclusion & Prospects

31 Chiral perturba1on theory π γ K γ n p n n n Σ R.J. Crewther, P. Di Vecchia, G. Veneziano, E. Wi`en (1979) A. Pich, E. de Rafael (1991) J. Hisano, Y. Shimizu (2004) Strange content of nucleon Unknown matrix elements

32 Future prospects π γ K γ n p n n n Σ Hadron- loop calcula1on is carried out in the chiral perturba1on theory. CP- odd meson- baryon couplings are evaluated with QCD sum rules. Lahce results are used for the QCD parameters K. Fuyuto, J. Hisano, N.N., in prepara1on.

33 Conclusion We have evaluated the nedm based on the method of the QCD sum rules. By using input parameters obtained from the lahce simula1on, we have derived a conserva1ve limit for the contribu1ons of the CP- viola1ng operators.

34 Backup

35 Chiral rota1on By using the ciral transforma1ons, we move to a convenient basis. U(1) A transforma1on Θ- term is rotated into γ 5 - mass term. SU(3) A transforma1on The vacuum is aligned in a ``good direc1on. Tadpoles for pseudo- scalar mesons should vanish.

36 Effec1ve Lagrangian

37 Peccei- Quinn mechanism Experimental limits on the nedm lead to strong CP problem Peccei- Quinn (PQ) mechanism Θ a(x): axion field R. D. Peccei and H. R. Quinn (1977) Θ vanishes dynamically

38 Axion poten1al where CP- viola1ng interac1ons generate the linear term non- zero Θ is induced.

39 nedm with PQ symmetry c.f.) M. Pospelov and A. Ritz (2001)

40 Higher- dimensional operators 4- Fermi operators Their contribu1on is suppressed by light quark masses. Weinberg operator comparable to the CEDM contribu1on generated by integra1ng out the 4- Fermi operators of heavy quarks J. Hisano, K. Tsumura, M. J. S. Yang, Phys. Le`. B713 (2012) 473.

41 Parameters for Condensates