lhc-theorie – heavy exotics at lhc´ -...

Introduction Colleagues Multiquarks (t6 t6) (QQqq) Conclusions

LHC-Theorie – Heavy exotics at LHC

available at http://lpsc.in2p3.fr/theorie/Richard/SemConf/Talks.html

Jean-Marc Richard

Laboratoire de Physique Subatomique et CosmologieUniversite Joseph Fourier–IN2P3-CNRS–INPG

Grenoble, France

West-Cost LHC Theory, November 21, 2008

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Table of contents

1 Introduction

2 Recent work by my colleagues

3 Stable or narrow multiquark states?

4 Dodecatoplet (t6 t6)First speculationsRevised estimates

5 Double charm exoticsFavourable symmetry breakingImproved four-body calculation

6 Conclusions

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Introduction

LHC-TheorieMany activities in France related to LHC (Orsay, Paris, Annecy,etc.)Small network including Lyon and Grenoble to coordinate studiesrelated to LHCWe : Aldo Deandrea, Sacha Davidson et al. (Lyon), MichaelKlasen, Sabine Kraml, Ingo Schienbein, JMR (Grenoble) +postdocs + thesis students + visitorsAim at developing collaboration West-Cost–Lyon–GrenoblemeetingSome recent work by my colleaguesRemark on the dodecatoplet t6 t6

New calculation of (QQqq) with better treatment of confiment.

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Recent or ongoing work by my colleagues

Sabine Kraml, several recent papers on NMSSM, unification, andparticipation in several working groups,Carole Weydert (thesis with Klasen) is studying the production ofcharged Higgs in minimal extensions of SM, and beyond.Aldo Deandrea, MSSM, ILC working group, flavor physics atLHC, hadron physics (η → 3π0, . . .)Sacha Davidson, Flavor physics at LHC plus many other things,Ingo Schienbein et al. studied the hadroproduction of D and Bmesons, for which a serious disagreement existed between data(too high) and predictions (too low). While data↘, theoreticalestimates↗. In these processes involving heavy quarks, thedifficulty was to keep the mass terms and resum the Logs.Another work by Ingo dealt with NuTeV results. See next slide,his results reproducing quite well the results, with less shadowingand less antishadowing than in most previous works. These PDFenter a number of processes at LHC. See next slide.

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Hadroproduction of D and B mesons [see arXiv:0807.2215]

dσ/dpT (nb/GeV) p p– → D*+ X

GM-VFNS

√S = 1.96 TeV

-1 ≤ y ≤ 1

pT (GeV)

1

10

10 2

10 3

10 4

5 7.5 10 12.5 15 17.5 20 22.5 25

p p– → B+ X

√S = 1.96 TeV

-1 ≤ y ≤ 1

GM-VFNZM-VFN

FFN (no FF)

FFN (old Input)

µi=mT, mb=4.5 GeV

pT (GeV)

dσ/dpT (nb/GeV)

1

10

10 2

10 3

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25

[GeV]T

p10 20 30 40 50 60 70 80 90 100

X)

[nb/

GeV

]+

B→ p

(p

T/d

pσ

d

-310

-210

-110

1

10

210

310 =1.96 TeVS

-0.6 < y < 0.6

GM-VFNZM-VFNFFN (no FF)

=4.5 GeVb

, mT=mi

µ

PDFs and nuclear corrections for the LHC [see arXiv:0807.2215]

x-110 1

]F

eν 2

R[F

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.202=5 GeV2QA=56, Z=26

fit A2

KP

SLAC/NMC HKN07 (NLO)

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Production of γ,Z ,Z ′

Work by Klasen, Ledroit, Morel, Fuks, etc.Theoretical estimates using fixed order and resummation vs. moreconventional approaches using fixed order, MC+ parton shower, etc.

1

I - Du Z aux Z’ II - Les Z’ au LHC III - Les études Z’

Resommatio

n

+6%

-3%

Les corrections QCD d’ordre supérieurLes corrections QCD d’ordre supérieur

Corrections au vertex

Emission de particules réelles Calculs théoriques :

Ordre fixe + resommation

Approche MC :

Ordre fixe (MC@NLO)

+ Parton shower (Herwig)

Facteurs K

PDF !

5%

Incertitudes théoriquesNLO + Resommation " !10%

Variation d’échelle

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Comparison PYTHIA, MC@NLO, joint resummation

M [GeV]900 1000 1100 1200

LO

σ/dσd

1.15

1.2

1.25

1.3

1.35

PYTHIAResummationMC at NLO

(LHC)- l+ l→, Z, Z’ γ →p p

[GeV]T

q0 10 20 30 40 50 60 70

]-1

[fb

GeV

T/d

qσ

d

0

2

4

6

8PYTHIAResummation MC at NLO

(LHC)- l+ l→, Z, Z’ γ →p p

[Fuks, Klasen, Ledroit, Li, Morel (2008)]1 TeV Z ′; PYTHIA (LO/LL+), MC@NLO (NLO/LL), resummation (NLO/NLL).Mass-spectrum normalized to leading order:

* PYTHIA (power shower ): mass-spectrum multiplied by a K -factor of 1.26.* Good agreement between MC@NLO and resummation.

Transverse-momentum distribution:* PYTHIA spectrum much too soft, peak not well predicted.* Good agreement between MC@NLO and resummation.

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Stable or narrow multiquark states?

Many scenarios for multiquark (meta) stability,Coherences in the chromomagnetic operator σi .σj λi .λjH(uuddss) < (uds) + (uds) [Jaffe] not found, orP(Qqqqq) < (Qq) + (qqq) [Lipkin, Gignoux et al.] not seen!Chiral dynamics Late light pentaquark not confirmed!Nuclear forces X (3872) predicted as DD∗ “molecule”. Perhapsother examples.Other speculations will become accessible to experimental tests,in particular at LHC.Below: brief discussion of tn tm toplets and(QQqq) with charm or beauty = 2.

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Dodecatoplet (t6t6)

Higgs exchange

−αHexp(−µH r)

r, with αH = g2

t /(4π) and gt ∼ 1.

Does it bind tn tm? (Frogatt, H.B. Nielsen)Up to n ≤ 6 and m ≤ 6, behave as bosons.Optimistic estimate by Nielsen and Frogatt, who neglected theDebye factor!Corrected by a Hartree (self consistent effective one-particlepotential) by Shuryak et al. → upper variational bound on groundstate energyIn fact, the calculation of the self-Yukawian boson system alreadyavailable in the literature (Pacheco et al.) and a lower bound isalso possible.

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Dodecatoplet (t6t6)

If Higgs exchange alone, by scaling, the only parameter isG = mtαH/µH

for 2-body, no binding for G ≤ 1.68 (Blatt and Jackson, 1949),i.e., µH ≤ 8.2 GeV for αH = 1/(4π).For (t6 t6), estimate µH . 29 GeV By Shuriak, and µH . 31 GeVfrom Pacheco et al.Perhaps slightly heavier with a better variational calculation,From the lower-bound on the ground-state energy, the criticalmass [again for αH = 1/(4π)], cannot exceed µ(c)

H = 49 GeV.

If αH bigger, µ(c)H inversely proportional.

Higher tn tm not excluded (with Fermi statistics). Rememberclusters of 3He atoms: (3He)n bound for n & 35!

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Limits on the t6t6 energy

NG/2

!N/(N ! 1)

!!2.5

!!2.0

!!1.5

!!1.0

!!0.5

|2

|2.5

|3.0

|3.5

|4.0

|4.5

••

•

•

•

•

FIG. 1: !N/(N ! 1), where !N is the N -body energy problem as a function of GN/2, where G is the coupling. Solid curve: exact 2-body

energy !2 which is also a lower bound to !N/(N ! 1) for N > 2. Dashed curve: Gaussian variational approximation !N/(N ! 1), valid forany N . Dots: variational hyperscalar approximation !12/11 for the 12-body case.

C. Simple variational upper bound for theN -body system

If one describes the relative motion using (N ! 1) Jacobi variables

x1 = r2 ! r1 , x2 =2r3 ! r1 ! r2"

3, · · · (6)

from the rescaled positions, and removes the centre-of-mass motion, the N -body Hamiltonian reduces to

!N!1!

i=1

!i ! G!

i<j

v(rij) . (7)

If the above Gaussian is generalised as"N!1

i=1"a(xi) and taken as trial function, the variational energy reads

!N = mina

[(N ! 1)t(a) ! N(N ! 1)Gp(a)/2] , (8)

which is easily estimated, and becomes a better and better approximation as N increases.

Note in (8) the obvious relation

!N (G) = (N ! 1) !2(NG/2) , (9)

which, as we shall see below, becomes a constraining inequality if the approximate energies are replaced by the exact ones.

It also indicates that if the number N of bosons is large enough, binding is achieved however small is the coupling G. Thispossibility of binding systems whose subsystems are unbound, first pointed out by Thomas [6], is nowadays refereed to as

“Borromean binding” . For references, see, e.g., [7].

Note also that the variational energies obey the same scaling laws, virial theorem, etc., as the exact ones. This is stressed

in [4], and can be traced back up to very early papers dealing with quantum systems [8, 9].

D. Hyperscalar upper bound forN -body systems

The above trial wave function"

i "a(xi) is just one particular function, namely Gaussian, of the hyperradius r > 0 definedas

r2 = x21 + · · ·x2

N!1 . (10)

3

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Double charm exotics

Favourable symmetry breakingStarts from (QQqq) = (M,M,m,m) with M = mincrease M and decrease m, while M−1 + m−1 = ConstantThen E(QQqq) decrease while E(threshold) = constantThus for M/m large enough, stability [Ader et al, Heller et al.,Brink et al., Lipkin, Nussinov, Rosina et al., Semay et al., etc.But the “critical” M/m likely to require more that charm = 2 in”current” models

Better treatment of confinementCoulomb (one-gluon exchange) has a colour factor λi .λj

The same factor λi .λj also applied to confinement in earlymodels!Hence σ r12 (quarkonium) becomes σ[r12 + r23 + r31] for baryonsBut there is no justification!

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Steiner-tree model of confinement-1

The linear q − q potential of mesonsinterpreted as minimising the gluon energy in the flux tube limit

Artru, Dosch, Merkuriev, etc., proposed a bet-ter ansatz, often verified and rediscovered(strong coupling, adiabatic bag model (Kuti etal.), flux tube (Kogut et al.), lattice QCD, etc.)

The q − q − q potential of baryons is with thejunction optimised, i.e., fulfilling the conditionsof the well-known Fermat-Torricelli problem.

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Steiner-tree model of confinement-2

Generalisation to tetraquarks [e.g., Sugunama et al., Lattice QCD]

V4 = min(Vf ,VS)

combination offlip-flop Vf (already used in its quadratic version by Lenz et al.)

Vf = λmin(r13 + r24, r23 + r14)

and Steiner-tree VS

VS = λmink,`(r1k + r2k + rk` + r`3 + r`4) .

This QCD-inspired potential is more favourable and shouldreasonably lead to stable (ccqq) states, to found together withdouble-charm baryons (QQq)

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The Steiner tree model of tetraquarks-3

As an illustration, we consider two variants of a purely linearpotential

1 the additive model

H1 =X

i

p2

2mi− 3

16

Xi<j

λ(c)i .λ

(c)j rij

2 The Steiner-tree model

H2 =X

i

p2

2mi+ V4

H1 does not bind for masses (m,m,m,m) but for masses(M,M,m,m), if M/m & 5J. Carlson and V.R. Pandharipande concluded that H2 does notbind, butthey used too simple trial wave functions for the 4-body problem,and did not consider unequal masses.

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Tetraquarks in the minimal-path model-1

Numerical study

Vijande, Valcarce and R. revisited the calculation of Carlson at al.with a basis of correlated Gaussians (matrix elements painfullycalculated numerically), and obtained stability for (QQqq) even forM/m = 1, and better stability for M/m� 1.Rigorous study

Hyam Rubinstein (Melbourne) and JMR demonstrated binding in thelimit of very large M/mProof

H2 ≤

[p2

x

M+

√3

2|x |

]+

[p2

y

m+

√3

2|y |

]+

[p2

z

2µ+ |z|

]which is exactly solvable.

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The inequality comes from a standard reduction pattern of Steinertrees (Melznak’s circles or torroıdal domain associated to subsets ofpoints)

A flavour of the proof. In the3-body case, Steiner treelinked to Napoleon’stheorem.JA + JB + JC = CC′ whereC′ makes an externalequilateral triangleassociated to the side AB.

Well-known property of theFermat-Torricelli problem.(C′ belongs to the torroıdaldomain associated to AB)

A

B

C

J

B

A C

D

I J

Figure 1: Generalisation of the linear quark–antiquark potential of mesons to baryons (left) andtetraquarks (right).

JA

B

C

JA

B

C

A!

B!

C !

JA + JB + JC = CC !

Figure 2: Left: the junction is at the intersection of the arcs from which each side is seen under120". Right: the junction as a side product of Napoleon’s theorem

1

A

B

C

D

I

J

x yz

Figure 3: First inequality.

A

B

E

E !

C

D

F

F !

I

J

P

Q

HK

Figure 4: Construction of the minimal string in the planar case.

2

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The analogue for the planar tetraquark is

A

B

E

E !

C

D

F

F !

I

J

P

Q

HK

Figure 5: Construction of the minimal string in the planar case.

3

VS = JA + JB + JK + KC + KD = EF

The minimal network linking (A,B,C,D) is the maximal distancebeween {E ,E ′} and {F ,F ′}, which are the torroıdal domainsassociated to (A,B) and (C,D) (= points completing an equilateral tr.)

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In space, still

VS = JA + JB + JK + KC + KD = EF

whereE ∈ CAB= torroıdal domain ofquarks AB, (equilateral circle)F ∈ CCD= torroıdal domain ofantiquarks CD,VS is the maximal distancebetween the circles CAB andCCD, which is less than thedistance between the centresand the sum of radii.

AB

CAB

CD

CCD

I

J

E

F

Figure 5: The confining potential for the tetraquark sys-tem (ABCD) is the minimal length of the tree IA+IB+IJ+JB+JC when I and J are varied. It is also the max-imal distances between the circles CAB and CCD, i.e., thedistance EF . The circle CAB is centered at the middle ofA and B, has AB as axis and a radius |AB|

!3/2, and

CCD has analogous properties in the antiquark sector.

hadrons does not require the mass of baryons to be computed within less than 1MeV of accuracy.What matter are the pattern of ordering and the properties beyond the mass spectrum. On theother hand, it is crucial to treat the four-body problem very precisely, even in simple models,to compare within the same dynamical scheme, the mass of the tetraquark states to that of twomesons, i.e., to indicate whether the model predicts he existence of stable multiquarks. In otherwords, it is required to solved the four-body problem much beyond he simplest approximation.This means that the potential has to be estimated a very large number of times.We are convincedthat the geometric properties of the Steiner tree will be of great help, to reduce considerably theamount of numerical minimisation when estimating the value of the potential.

It is our intend to extend this investigation to the case the pentaquark (one antiquark andfour quarks) and hexaquark configurations (six quarks), which have been much debated in recentyears.

7

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Conclusions

Intense activity of my colleagues forLHC,(t6 t6) probably unbound,Better models of confinement beyondnaive additive models, plus Fermat,Torricelli, Steiner, Melznak and evenNapoleon, suggest double-flavourexotics to be found at LHCThe hadron family already rich, butlikely to welcome new members,

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