F = 2 FCNCs in general SUSY models to NLO · NPB 619 (2001) gluinos Dominant contribs, when NOT...

Diego GuadagnoliTechnische Universitaet Muenchen

∆F = 2 FCNCsin general SUSY models

to NLO

Outline⇨ Introduction: theoretical framework (OPE + RGE), the MSSM and the Mass Insertion Approximation (MIA)

⇨ Motivation and status of the calculations for the F = 2 systems

⇨ NLO Calculation of the F = 2 Hamiltonian: schemes considered, checks of the calculation

⇨ Phenomenological analysis: Bs system in the light of ∆Ms by CDFD∅

D.Guadagnoli, Ringberg 06, October 26, 2006

Renormalization scale: ~ few GeV

Coefficient Functions (CFs):encode physics beyond

Operator Matrix Elements:encode physics below

✔ Theoretical input

⟨ Bs0∣H full

SMH fullSUSY∣Bs

0⟩

SUSY contributions: Minimal SupersymmetricStandard Model (MSSM)

⟨B s0∣H full

MSSM∣Bs0⟩=∑

i

dim 6

C i ⟨B s0∣Qi ∣B s

0 ⟩ contributions fromdim 6 op 's

The full amplitude is expanded as:NEGLIGIBLE

Step Perturbative calculation of the CFs at some “high” energy scale, O(SUSY masses).Thus avoid large logs.

Step RGE evolution of the CFs(and of the matrix elem’s) down to the scale .ADM for the op. basis needed (perturbatively).

Step NONPerturbative calculation of the op. matrix elem’sbetween the physical meson states.Lattice QCD mandatory.

Experimentalobservables:

∆Ms , s mesonantimesonoscillation amplitude

⟨ Bs0∣H full

SM∣Bs0⟩depend on


mass basis flavor basis

mass eigenstatesbasis

flavor × chiralitybasis

On the Mass Insertion Approx (MIA)

☞ Redefinition of the squark fields: flavor × chirality basis vs. mass basis

uLI =Z U

† I iU i

d LI = Z D

† I iD i

“Left” squarks

uRI = ZU

† I3 iU i

d RI =Z D

† I3iDi

“Right” squarks

i = 1,...,6

Z D† M1 L

2 0 ⋯

0 M 2 L2 ⋯

⋮ ⋮ ⋱Z D = m112 12

LL ⋯

21LL m22

2 ⋯⋮ ⋮ ⋱ ≃ m2 1 0 12

LL ⋯

21LL 0 ⋯⋮ ⋮ ⋱

The ‘MIA’ Hyp #1:diag terms

~ degenerate

Hyp #2:nondiag small

I = flavor

squark propagator(flavor basis + MIA)

≃diagonal

+ ×NONdiag.: Mass Insertion

SUSY source

of FCNC's

‘rotation’


∆MsMSSM / ∆Ms

SM

Buras et al.NPB 619 (2001)

gluinos

Dominant contribs, when NOT assuming any symmetry in the squark matrix

Higgses

Dominant contribs (negative),for large tan

M sH−box∝ −tan 2

M sH−DP∝ − tan 4

M sMSSM = M s

SM

M sg M s

H−box M sH−DP M s

M s0

neutralinos:negligible

For low/moderate tan , SUSY contrib’s = gluinos

Naïve ranking of gluino contributions

One has in general

M s

M sg ≤ 10−2

u×ud×d

charginos

Ball et al.PRD 69 (2004)

SUSY contributions to ∆Ms

Looking at single ’s,bounds are hierarchical

LL (or RR) only ☞ | | 0.4

(LR , RL) ☞ b s does better

LL × RR only ☞ | | 0.02÷0.05

Assuming Mgluino / Msquark ⋲ 1, one finds

M sg

M sSM≃ 500GeVm q

2

O 1 LL2 RR2

O 10 {LR2 RL2 & LRRL }

O 100 LLRR D.Guadagnoli, Ringberg 06, October 26, 2006

‘State of the art’ of the calculations for F = 2 Hamiltonians

Step ➊ CFs(MSUSY)

SM: complete to NLOBuras et al.

1990Herrlich + Nierste

1994, ‘95 & ‘96

MSSM: complete to LOGabbiani et al.

1996

Step ➋CFs evolution:

CFs(MSUSY) CFs()

SM + MSSM: complete to NLO Ciuchini et al.

1998

Step ➌Hadronic Matrix

Elements

SM + MSSM: quenched lattice estimateBécirévic et al.

2001

calculation to 2 loops of the anomalous dimension matrix for the basisQi (dim = 6)

for the basis Qi (dim = 6) [unquenched (= full QCD) estimate highly desirable]


NLO available only for the 2HDM and for charginos

☞ NLO corrections to the CFs in the MSSM were still missing (... and important)

Motivations

M.Ciuchini et al.JHEP 10 (1998)

2) Existing phenomenological analyses are affected by a residual scheme and dependence naively corresponding to an O(S(MSUSY)) error.

D.Bécirévic et al.NPB 634 (2002)

3) They allow to reach within the MSSM the same precision as that in the SM.

1) A completetoNLO analysis requires both the NLO ADM for the operator basis (already present) and of the NLO corrections to the CFs. If one or both the ingredients are missing, the error in the analyses is a NLO one.

(LL & RR contribs.) vs. scale

2 Abs{⟨ Bd∣Heff B=2∣Bd ⟩}

Large scale dependence, but note that ☞ CFs are proportional to S

2

☞ ADM entries for the new ∆F = 2 operators (gluino exch.) are surprisingly large

NLO calculationof the ∆F=2 Hamiltonian

(M.Ciuchini et al., 2006)

same external statesas in the full theory

1 loop correctionsin the effective theory

CFs are obtained enforcing (at the scale MSUSY) the following equality (‘matching’ fulleff)

Amplitude in the full theory

LO = + + +

F.Gabbiani et al.NPB 477 (1996)

It becomes important ✔ to automatize as much of the calculation as possible✔ to be sure not to forget any diagram

external states witharbitrary kinematics2 loop amplitude

in the full theory

EffFull

⟨b d∣H fullMSSM∣b d ⟩ = ∑

i

dim 6

C i ⟨b d∣Qi∣b d ⟩R = ∑i j

dim 6

C i r ij⟨ b d∣Q j∣bd ⟩tree

CFs calculation

NLO = + another 70 1PI, “nonequivalent” diagrams

Ciuchini, Franco, D.G., Lubicz, Porretti, Silvestrini

JHEP 09 (2006)

NLO calculation: general strategy

The whole calculation was performed within the program Mathematica

Generation of all the2 loop Wick contractions

in the MSSM (strong)

Manipulations onthe single diagram.

In particular: gamma matrix reduction

Basis: package FeynArts✔ it generates all the 2 loop topologies with insertion of vertices in a given model✔ it writes the single diagram in the form of a Feynman Amplitude

Basis: package TRACER✔ it allows to calculate traces of gamma structures in arbitrary dimensions✔ it allows the use of different regularization schemes. In particular DRED e NDR

Modification of the implemented model

(MSSM) for allowing the most general squark

mixing

Integration over the loop momenta

Use of both schemes(def. of the

evanescents mandatory)

✔ adopting the Mass Insertion Approximation (MIA), only ≤ 3 loop masses (‘feasible’)✔ integrals diverge both UV and IR. Use of 2 IR regs. as a check: gluon mass and IRdimensional

Creation of theFeynman Amplitudefor the single diagrams


Some checks of the calculation

The CFs were calculated ☑ in two regularization schemes (DRED & NDR)

☑ with two IR regulators: ‘gluon mass’ & dimensional

Case A: DRED, with IRreg = gluon mass Note residual (1/IR) divergences after renormalization

Necessary to include LO contributions fromDRED evanescents

Case B: DRED, with IRreg = dim

1 st check

Case C:NDR, with

IRreg = gluon mass

2 nd check

Notes for the check A vs. C✔ Consider the Z of scheme changing (DRED – NDR) in the effective theory✔ NDR breaks SUSY: add suitable finite corrections to masses and SU(3) couplings

Martin + VaughnPLB 318, 331 (93)


Impact of NLO CFs

dependence of the amplitude: case “LL only”

= LO= NLO

scale dependence~10 % (LO) ~ 3.5% (NLO)


Impact of NLO CFs: continued

dependence of the amplitude: case “LR only”

= LO= NLO

scale dependence~6 % (LO) ~ 2% (NLO)


Phenomenologicalanalysis

(Bs system)

Constraining 's

a) Generate values for the matrix element by ✔ extracting the exp input (2) with normal distributions

✔ extracting Abs['s] and Arg['s] with flat distributions

b) Calculate observables O in (1), with the values generated in a), and give them a ‘gaussian weight’, i.e. weigh with Exp[(Oth – Oexp)

2]/(22exp)

MonteCarlo

ms = 17.33−0.210.42±0.07 ps−1

F=1 decays:b sb s l l−

Exp quantities

Exp input

(2)

ms = 2 Abs {⟨ Bs∣Heff B, S=2∣B s⟩}

(1)

Theory relations

and MSSM formulae for the decays

Phenomenological analysisWith the O(S) corrections to the CFs it becomes possible a full NLO analysis

☑ SM couplings & masses

ϱ, from treelevel processes

Evaluation of the matrix element⟨ Bs∣Heff

B , S=2∣B s⟩...and the other input in (1) bag parameters from the lattice


⇨General procedure: Our matrix element is written as follows

Constraints on Re[] and Im[]

⟨ Bs∣HeffB , S=2∣B s⟩ = Re ASMi Im ASMASUSY Re AB

232i Im AB232

SM part

⇨We switch on one 2AB at a time: LL only, RR only, LL=RR, LR, RL or LR=RL

Example 1: constraints on Re[] vs. Im[], case “LL only”

SUSY part proportional “to a given 2

AB”

Always pairs of 's appear in the

ampliude

yaxis: Im[(23)LL]xaxis: Re[(23)LL]

Constraints□ = ms

□ = b s □ = b s l+ l □ = all

tan = 10(23)LL

[-0.1,+0.05] + i [0.2,0.2]


yaxis: Im[(23)XY]xaxis: Re[(23)XY]

Constraints□ = ms

□ = b s □ = b s l+ l □ = all

LL, RR insertions: the Ms constraint is relevant / fundamental

LL only, tan =3

⇩[-0.15,+0.15] + i [0.25,0.25]

RR only, tan =3

⇩[-0.4,+0.4] + i [0.9,0.9]

LL=RR, tan =3

⇩[-0.05,+0.05] + i [0.03,0.03]

LL=RR, tan =10

⇩[-0.03,+0.03] + i [0.02,0.02]

yaxis: Im[(23)XY]xaxis: Re[(23)XY]

Constraints□ = ms

□ = b s □ = b s l+ l □ = all

LR, RL insertions: F=1 constraints rule

LR only, tan =3

⇩[-0.0025,+0.01] + i [0.015,0.015]

RL only, tan =3

⇩[-0.008,+0.008] + i [0.008,0.008]

Remarks

Constraints on LL and LL=RR mass insertions are severe

Constraints on LR and RL mass insertions are very severe

Do these constraints have a severe impact on theBs – mixing phase ?


Bs – mixing phase

✔ In the SM one has Arg M12SM ≡ Arg{⟨ Bs∣H eff , SM

B , S=2∣Bs⟩} = 22 ≃ 0.04

What is the allowed range for with the previous limits on the ’s ?Arg M 12MSSM

LL only, tan =3~ 10 × SM value are allowed

LR only, tan =3no sizable deviations

from the SM

RR only, tan =3~ 100 × SM value are easy to get

(but RR is still mildly constrained...)

LL=RR, tan =3~ 100 × SM value are again easy

(yet LL=RR is severely constrained!)

The CP asymmetry in Bs will provide a truly fantastic probe!


Bs – mixing phase

✔ PDF’s for sin 2 s in the LL, RR and LL=RR cases

LL only, tan =3 RR only, tan =3 LL=RR, tan =3


= LO= NLO

Example: constraints on Abs[] , case “LL only”

Impact of NLO CFs

Previous phenomenological analyses used LO values for the CFs. What is the effect of going to the NLO ?

✔ NLO tends to slightly relax limits on 's

✔ The effect is a percent level one. Many percent for the “LR = RL” case.


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F = 2 FCNCs in general SUSY models to NLO · NPB 619 (2001) gluinos Dominant contribs, when NOT...

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