Candidate: Baum Izhak (XXIII ciclo)

Supervisor: Prof. Guido Martinelli

Co-Supervisors: Prof. Lubicz Vittorio (Roma3)

Prof. Simula Silvano (Roma3)

Hadronic Matrix Elements for Weak Transitions

Beyond the Standard Model

PhD Research Project

Dip. di Fisica, Università di Roma “La Sapienza”

June 4, 2009

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Outline

● Motivation: Why calculate Weak matrix elements?

● Introduction to lattice theory and methodology● A few details of our calculations

● Our results so far: the K→ ElectroMagnetic and ChromoMagnetic matrix elements

● Conclusions

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The Standard Model (SM) describes extremely well all physical phenomena at current energies [Mele 2008; Martin 2008]

but there are still signs of new physics.

● Hierarchy problem –divergence of higgs mass

● Dark matter: no candidate !

● Baryogenesis ↔ CP violation: too small !

● Flavor problem: mt=170 GeV, m

u=5 MeV

and others...

Why calculate Weak matrix elements?The Standard Model – notable problems, solutions (SUSY)

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Why calculate Weak matrix elements?The Standard Model – notable problems, solutions (SUSY)

The Standard Model (SM) describes extremely well all physical phenomena at current energies

but there are still signs of new physics.

● Large hierarchy problem – divergence of higgs mass

● Dark matter: no candidate!

● Baryogenesis ↔ CP violation: too small!● Small hierarchy = Flavor problem: m

t=170 GeV, m

u=5 MeV

and others...

Most studied theory beyond the SM: SuperSymmetry (SUSY), solves many of the problems of the SM, but not all

Among the most popular theories beyond the SM is SUperSYmmetry (SUSY), which solves many of the problems of the SM, but not all

✔ Hierarchy problem – Higgs mass not diverging with cutoff

✔ Dark matter: neutralino 0

✔ Baryogenesis ↔ CP violation: easily accomodated

✗ Flavor problem: remains unexplained in SUSY

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Why calculate Weak matrix elements?

BR( KL

0 → ℓ∓ℓ ) = 67.55 0.17%

BR( K → ℓℓ ) = 8.30 0.09%

These BR's in the SM depend on the vector form factor:

(K→ ℓℓ) f

+(0)2

Which is calculated from the lattice matrix element:*t->q^2

Kaon Kℓ3

semileptonic decays as precision tests of the SM

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Why calculate Weak matrix elements?

● Rare Kaon decays have not been detected yet:

BR( KL

0 → ℓℓ∓ )exp < 6.6·10-10

● In the SM are estimated to be:

BR( K→e e )SM ~1.5·10-12

BR( K→)SM ~3·10-10

● New physics can be the leading contribution,

with Electromagnetic and ChromoMagnetic operators:

Kaon rare semileptonic decays as new physics probes

My work so far has been concentrated on these operators.

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Lattice theory [Montvay 94 ; Martinelli 88]

Explicitly calculate the path integral in discrete Euclidean space-time:

The lattice acts as a regulator:

● UV regulator – lattice step 1/a

● IR regulator – finite size 1/L

finite a and L contribute to systematic errors

aL

T

L

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Lattice theoryIn early days: ● cost of dynamic fermions too high → quenched approximation

● Light fermion masses: ● expensive ● Mesons too large with respect to L

Today:● Unquenched calculations are feasible

● light (u,d) fermions are still out of reach (but getting closer).

→ calculate at higher masses and extrapolate to physical mass → another systematical error

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Lattice methodologyexample: 2-point pion propagator

Calculate correlation functions of the operator

at large times at p=0, we can calculate the pion mass:

0 5 10 15 20 251E-3

1E-2

1E-1

1E+0 C_2ptsfitm

= (306 2) MeV

Z = (2.33 0.03)·10- 2

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Details of our calculations

● Lattice step: a~0.09 fm

● Lattice size: L=24, T=48

● Dynamical flavors: nf=2

● Our pion mass range: 300 < mπ < 600 MeV

● (mπphysical = 140 MeV)

● Twisted boundary conditions: Ψ (L)=ei2πθ Ψ (0)

→ smallest momentum: pmin

=θ ·2π /L

● Improved Symanzik gauge action

● Maximally twisted mass fermion action

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<EM>plateuax = 0.01465(35)

Results so farThe EM operator matrix element

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Results so farEM operator tensor form factor

fT

q2a2

fT (0) = 0.393(8)

= 0.59(6)pole fit

The EM operator definition:

Parametrize to get tensor form factor:

Pole fit in momentum:

m = 306(2) GeV, mK = 536(2) GeV

And overall renormalization constant, calculated non-perturbatively:Z

T=0.75

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Results so farEM operator, extrapolation to physical mass

EM operator extrapolated: m → mphys

ffT T (0) = 0.345(11)(0) = 0.345(11)Previous result [becirevic 2001]

fT (0) = 0.77(6)

Our result is smaller by a factor~2 (but with better statistical accuracy), work in progress

m2a2

fT (0)

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Results so farCM operator matrix element

<C

M>

q2a2

<CM(0)> = 0.0168(27) = 2.41.1

pole fit

The CM operator definition:

Requires an additive renormalization,because of mixing with scalar current:

m = 377(2) GeV, mK = 558(2) GeV

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Results so far

MK

2a2 = m2a2

<CM(0)>bare

<QS(0)>

c/a = -0.020(4)

CM operator renormalization

Impose renormalization condition: in the chiral limit for zero momentum,

m=mK=0, p=0

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Results so far

CM operator extrapolated: m → mphys

<CM(0)><CM(0)>renormrenorm = 0.0017 = 0.0017 0.00160.0016

● Up to an overall renormalization constant, to be calculated

● Imposes an upper bound on the CM operator matrix element

● No previous lattice calculations, this constitutes a good result !

m2a2

<C

M(0

)>ren

orm

CM operator renormalized

Extrapolation to physical mass

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Conclusions● The lattice approach enables calculation of new

theoretical results based on first principles.

● For <|QEM

|K> Better accuracy than previously achieved can be obtained for phenomenologically interesting quantities, with the ETMC results.

● Our calculation of <|QCM

|K> provides the first lattice estimate of this matrix element. In this case, even an upper bound on the matrix element would represents an information of relevant phenomological intrerest.

● More work is needed in checking the results and estimating errors.

I wish to thank my supervisors for a lot of help and

many enlightening discussions

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[1] B. Mele, PhD course lecture notes (2008) ; S. P. Martin,”A Supersymmetry Primer,'' arXiv:hep-ph/9709356 (v5 2008)

[2] I. Montvay and G. Munster, ”Quantum fields on a lattice”, Cambridge, UK: Univ. Pr. (1994) ; G. Martinelli, ”Phenomenology From Lattice QCD,'' GIFT Seminar 1988:146-198

[3] D. Becirevic, V. Lubicz, G. Martinelli and F. Mescia, Phys. Lett. B501, 98 (2001) [arXiv:hep-ph/0010349]

References