Ordinary and exotic baryons,

strange and charmed,

in the relativistic mean field approach

Dmitri Diakonov

Petersburg Nuclear Physics Institute

Kyoto, Feb 8, 2010

D.D., Nucl. Phys. A (2009), JETP Lett. (2009), arXiv: 0912.3175

Theoretical approaches to hadron physics

Limitations:

Perturbative QCD: only processes with large momentum transferChiral perturbation theory: only small momentaLattice: chiral physics so far inaccessible, processes in Minkowski space inaccessible

Holographic QCD

Chiral physics is critical to understand hadrons, as seen from the uncertainty principle:

When one attempts to measure a quark position to an accuracybetter than the pion Compton wave length of 1.4 fm, one producesa pion, i.e. a pair. Hence, viewing baryons as “made of threequarks” contradicts the uncertainty principle [Landau and Peierls (1931)]

The statement “nucleons are made of three quarks” is kindergarten physics, and we areimmediately punished if we attempt to take it seriously. For example,

1) “spin crisis”: only of the nucleon spin is carried by three valence quarks

2) “mass crisis”: only ¼ of the nucleon term is carried by three valence quarks:

| | 67 6 MeV2

u dN

m mN uu dd N

4MeV 7 MeV( 3) 17.5MeV

2

QQ

Both paradoxes are explained by the presence of additional pairs in baryons.QQ

0.3 0.1

5

eff , , ,A Ai AL Q i M e Q K

[models with massive quarks and (confining) gluon interactions contradict chiral symmetry, as they violate invariance under chiral rotation

5 5

, , .i i i i i ie Q Q Qe e e eQ e

In general, quarks at low momenta are governed by a Dirac Hamiltonian, with all 5 Fermi terms:

The scalar (S), pseudoscalar (P), vector (V), axial (A), and tensor (T) fields have, generally,non-local couplings to quarks, and are fluctuating quantum fields. However, in the limit they can be replaced by the non-fluctuating mean field. cN

c val seanucleon mass = N (E +E ) - (no field)

its minimum determines the mean field.

equal-time Green function

0 5 5( [ ])2

ii

ii S iP V A TH

It’s helpful to consider the limit! At least, general properties are not lost!cN

How does baryon spectrum look like at ?

(imagine number of colors is not 3 but 1003)

cN

Witten (1979): Nc quarks in a baryon can be considered in a mean field(like electrons in a large-Z atom or nucleons in a large-A nucleus).

The mean field is classical

Baryons are heavy objects, with mass .

One-particle excitations in the mean field have energy

Collective excitations of a baryon as a whole have energy

(1/ )cO N

(1/ )cO N

)( cO N

(1)O

Color field fluctuates strongly and cannot serve as a mean field,but color interactions can be Fierz-transformed into quarks interacting (possibly non-locally) with mesonic fields, whose quantum fluctuations are suppressed as .

Examples: instanton-induced interactions, NJL model, …

Important Q.: if what is smaller, 1

?s

c

mor

NcN

the answer:

splitting inside SU(3) multiplets is , numerically ~140 MeV

splitting between the centers of multiplets is , numerically ~ 230 MeV.

Hence, meaning that one can first put , obtain the degenerate SU(3)multiplets, and only at the final stage account for nonzero , leading to splittinginside multiplets, and mixing of SU(3) multiplets.

~ s cm N

~ / cN2/s cm N 0sm

sm

n pp

mean octet mass 1152 MeV

mean decuplet mass 1382 MeV

splitting inside decuplet ~ 140 MeV

difference from decuplet 230 MeV

What is the symmetry of the mean field?

Variant I (maximal symmetry): the mean field is SU(3)-flavor- and SO(3)-rotation-symmetric,as in the old constituent quark model (Feynman, Isgur, Karl,…) In principle, nothing wrong about it, except that it contradicts the experiment, predicting too many excited states !!In addition, the pion field is strong , and at large Nc must be classical, but there isno way to write the classical pion field in an SU(3) symmetric way!

Variant II : the mean field for the ground state breaks spontaneously SU(3) x SO(3)symmetry down to SU(2) symmetry of simultaneous space and isospin rotations, like in the hedgehog Ansatz

breaks SU(3) and SO(3)separately but supportsSU(2) symmetry of simultaneous spin and isospin rotations !

There is no general rule but we know that most of the heavy nuclei (large A) are not spherically-symmetric. Having a dynamical theory one has to show which symmetryleads to lower ground-state energy.

Since SU(3) symmetry is broken, the mean fields for u,d quarks, and for s quark are completely different – like in large-A nuclei the mean field for Z protons is different from the mean field for A-Z neutrons.

Full symmetry is restored when one SU(3)xSO(3) rotates the ground and one-particle excitedstates there will be “rotational bands” of SU(3) multiplets with various spin and parity.

13)( NNg

4,5,6,7,80 ( ), , 1, 2,3; 0.

aa a a x

n P r n ar

A list of structures compatible with the `hedgehog’ SU(2) symmetry:

isoscalar

isovector

Mean fields acting on u,d quarks.One-particle wave functionsare characterized bywhere K = T + J, J = L + S.

Mean fields acting on s quarks.One-particle wave functionsare characterized bywhere J = L + S.

12 functions P, Q, R must be found self-consistently from a dynamical theory.However, even if they are unknown, there are interesting implications of the symmetry.

PK

PJ

0 ( )S Q r

0 1( )V Q r

0 2 ( )i iQT n r

0 ( )a an P r

1( )ai aik kn PV rò

2 3( ) ( )ai ai a iP r n nA P r

4 5( ) ( )aij aij bij a bP r n n rT P ò ò

0 ( )S R r

0 1( )V R r

0 2 ( )i i RT n r

Ground-state baryon and lowest resonances

This is how the ground-state baryon N(940,1/2+) looks like.

SU(3) and SO(3) rotational excitations of this filling scheme form the lowest baryon multiplets: 1152(8, 1/2+) and 1382(10, 3/2+)

We assume confinement (e.g. ) meaning that the u,d and s spectra are discrete. Some of the components of the mean field (e.g. ) are C or G-odd, meaning that the twospectra are not symmetric with respect to E E

One has to fill in all negative-energy levels for u,d and separately for s quarks, and the lowest positive-energy level for u,d.

~S rV

The lowest resonances beyond the rotational band [Diakonov, JETP Lett. 90, 451 (2009)]

are (1405, ½-), N(1440, ½+) and N(1535, ½-). They are one-particle excitations:

(1405, ½-) and N(1535, ½-) are two differentways to excite an s quark level. N(1535, ½-) isin fact a pentaquark [B.-S. Zou (2008)]

N(1440, ½+) (uud) and (½+) ( ) are two different excitations of the same level of u,d quarks. is an analog of the Gamov-Teller excitation in nuclei! [when a proton is excitedto the neutron’s level or vice versa.]

uudds

uudss

Sum rule:

valid up to 1/Nc corrections

Theory of rotational bands above one-quark excitations [Victor Petrov and D.D.]

SU(3)xSO(3) symmetry is broken spontaneously by the ground-state mean field, down to SU(2). The full symmetry is restored when one rotates the ground-state baryon and its one-particle excitations in flavor and ordinary spaces. [cf. Bohr and Mottelson…]

Consider a flavor rotation of the mean field by an SU(3) matrix R, and a space rotationby an SU(2) matrix S. If R,S are constant matrices all energy levels remain the same.If R,S are slowly depending on time, one obtains the rotation Lagrangian

3 72 2 81 2

rot ,1 4

( ) ( ) ( )2 2

a a A a a au d s

a A

I IL Y K J

† †, .i R R i S S are moments of inertia.1,2 ( )cO NI

1 2( of , quarks' # #) ( of quarks).

3 3u d sY

init fin K K K is the vector sum of the grand spin of u,d quarks involved

init fin J J J is the vector sum of the spin of s quarks involved.

One needs to quantize the rotational Lagrangian.

Introduce angular momenta, flavor (F) and spin (J):

rot1( ) .a a a a a a a a

a

LIJ K J F K J

1

rot2

( ), 1,2,3,

, 4,5,6,7,

, 8,

a a

A AA

I A aL

I AF

Y A

The rotational Hamiltonian rot rotA A a aFH J L

8 38 8

1,2,3 4,5,6,7 1 1rot

1 2 22 2 2

a a A A A A a a

a A A a

F F F F F F F F F F

I IH

I

82 2

21

1( , ) ( 3 3 )

3A A

A

F F C p q p q pq p q

3

1

( 1)a a

a

F F T T

Casimir operator foran SU(3) multiplet

isospin of baryons with given Y’

All one-quark excitations entail their own rotational levels.

Some rotational bands are short, some are long.

Some rotational levels are degenerate in the leading order, some are calculably split.

22

rot2 1 2

3( ) 1 14 ( 1)

2 2 2

C r YT T

I IH

I

, u d sJ T K J

Finally, the rotational energy of the SU(3) multiplet r containing baryons with a givenhypercharge Y’ (whose isospin is T’ ) is

The possible spins J of the multiplets are found from the vector addition law above.

SU(3) multiplets are degenerate in spin J. The degenaracy is lifted in the next order 2

1

cNSplitting inside SU(3) multiplets is ( )s cO m N

Mixing of baryons with identical quantum numbers belonging to different multiplets is

, leading to the mass shifts 2~ /s cm N 2 3~ /s cm N

Example: what is the rotational band about the s quark transition?1 1

2 2s s

hypercharge Y’ = 1/3 + 1/3 + 1/3 = 1.

n pp

Y Y

T T

3'

2T

1'

2T

1 1J T

2 2

Octets: (T’ = ½) J = ½, ½, 3/2 – all degenerate up to 1/Nc^2 corrections

Decuplets: (T’=3/2) J=1/2, 3/2, 3/2, 5/2 – all degenerate up to 1/Nc^2 corrections

Parity = – .

Splitting of multiplets due to rotation:

1

3,3 / 2 ,1850) ,1/ 2 ,1615) 235 Me(

2( V

I 10 8

1

3,3 / 2 ,1382) ,1/ 2 ,1152) 230 Me(

2( V

I 10 8

The moment ofinertia is the same !

(meaning the large-Nclogic works well !)

Parity-minus rotational bands

11,

2

(1405,1/ 2 )

1 3 5, , 2 , , ,2 2 2

10 10 10

1 1 3, , , , ,2 2 2

8 8 8 1615(8,1/2-), 1710(8,1/2-),

1680(8,3/2-)

1758(10,1/2-), 1850(10,3/2-), (1930,5/2-)?

1 3 5, , 2 , , ,2 2 2

8 8 8

1895(8,3/2-),1867(8,5/2-),…?

,

1

20u d

s

1 1

2 2s s

,

3

20u d

s

1 3

2 2s s

3, (1520,3 / 2 )2

1

[ no rotational band for this excitation ]

[ no rotational band for this excitation ]

(should be degenerate)

(should be degenerate)

(should be degenerate)

Parity-plus rotational bands

1 3, , ,2 2

8 10

1 3 5, , , , ,2 2 2

8 8 8

1 3 5 7, , , , , , ,2 2 2 2

10 10 10 10

1630(8,1/2+),1732(10,3/2+)

1845(8,1/2+),1865(8,3/2+),1867(8,5/2+)

2060(10,1/2+),2087(10,3/2+),2071(10,5/2+), (1950,7/2+)?

,

10

2 u ds

, ,0 0u d u d

, ,0 2u d u d

1,2

10 1750(anti-10,1/2+)?

(should be degenerate)

(should be degenerate)

2 excited levels for u,d quarks & 2 excited levels for s quarks

seem to be capable of explaining nicely all baryon multiplets < 2 GeV, and predict a couple of new ones, but not as many as the old quark model.

To summarize:

Important: exciting P = -1 levels for u,d quarks is prohibited by Fermi statistics!

Charmed baryons from the large-Nc perspective

If one of the Nc u,d quarks is replaced by c or b quark, the mean field is still the same, andall the levels are the same! Therefore, charmed baryons can be predicted from ordinary ones!

standard charmed baryonsc

c

c

c

c

c

c

c

mean masses: 2536 2603

25702

2408

The difference 2570 – 2408 = 162 MeV =1

1

I. On the other hand, can be found

from the octet-decuplet splitting 1

1151MeV.

I It is a check that the mean field and

the position of levels do not change much from light to charmed baryons!

1

1

I

degenerate up to21/ cN

E

ud s

KPPJ

... ...

c

KP PJ -

b

(15,1/ 2 )

anti-decapenta-plet

exotic 5-quark charmed baryons

Exotic 5-quark charmed baryons are light (~2420 MeV) and can decay only weakly:

0 ,..., K K p

clear signature, especially in a vertex detector.Life time

There is also a Gamov-Teller-type transition:

E=0

u,d s

KP=0+

P=1/2+J

... ...

cb

,c cB B

,cB p cB “Beta-sub-c”

NB: is anotherpentaquark, hypothetized byLipkin and Karliner; in ourapproach it must be ~350 MeV heavier!

c uuddc

1310 s

cuudscudds

c

c

c

c

cddsu cuusd

cdssu cussd

c

c

~ B B

In fact, there must be three almost degenerate multiplets, 15,1/ 2 ) 2, 15,3 2 )( ( /

Neglecting mixing between particles with identical quantum numbers, one can write formulas for the splitting inside the anti-decapenta-plet:

, ,0 20 1:

4c

mM m

1 20

5:

2 8c

m mM

, ,0, 20 1:

2c

mM m

,0 1 20

5:

4 16c

m mM

0 20 1: 2

2c

mM m

6 masses, 3 parameters 3 “Gell-Mann – Okubo” relations:

1 4 22M MM

1 2 5 35 2 92M M M M

1 2 6 34 6MM M M

,0 1 2:cB M m m 1( )M

2( )M

3( )M

4( )M

5( )M

6( )M

Six baryons at the corners of the anti-decapenta weight diagram are explicitly exotic(i.e. cannot be composed of three quarks), the rest are crypto-exotic, and mix up with

A similar anti-decapenta-plet exists,containing the b quark, the lightestbeing the doublet with mass

0,b bbuuds B bB udds ) 130MeV 575( 0MeV.bm

, .3 6

Big question: What is the production rate of

Expected production rate at LHC [Yu. Shabelsky + D.D.] :

4: ~ 10dN

ddy

3baryons : ~ 10dN

dyc

Decays:

20~ 1Br

LHCb:

7: ~10 ?c

dN

dyB

0c

p

pK K

p pK KB

0

c

p

p

pK KB

p

typical

15 9 6 ,particles/ year 10 ~ 10 / year10 cB

Somewhat less number but still a considerable amount of events expected at LHC ! Should decay mainly as anythingb cB B

, (~ 2420) , ?c cuuds ddsB cu

,0 (~ 5750) ,b buudB s budds

Conclusions

1. Hierarchy of scales: baryon mass ~ Nc one-quark excitations ~ 1 splitting between multiplets ~ 1/Nc mixing, and splitting inside multiplets ~ m_s Nc < 1/Nc

2. The key issue is the symmetry of the mean field : the number of states, degeneracies follow from it. I have argued that the mean field in baryons is not maximal but next-to-maximal symmetric, . Then the number of multiplets and their (non) degeneracy is approximately right.

(3)3) (2)(SU SO SU

3. This scheme confirms the existence of as a “Gamov – Teller” excitation, in particular,

1,2

10

4. An extension of the same idea, based on large Nc, to charmed (bottom) baryons leads to a prediction of anti-decapenta-plets of pentaquarks. The lightest and are exotic and stable under strong decays, and should be looked for!

, (~ 2420) ( )c cuu d dsB ,0 (~ 5750) ( )b buu d dsB

antiquark distribution at low virtuality (DD, Petrov, Polyakov, Pobylitsa and Weiss, 1996) [solid] vs. phenomenological GRV distribution.

If we scan nucleons with a resolution q ~ 600 MeV, only ~65% of nucleons are made of 3 quarks, ~25% are made of 5 quarks, and ~10% of more than 5 quarks.

Nevertheless, the fraction of momentum carried by antiquarks is very small:

1 20*0.65 *0.25 *0.1 8%

5 7

Appendix

Additional conclusions

1. “Baryons are made of three quarks” contradicts the uncertainty principle. In fact, already at low resolution ~65% of nucleons are made of 3 quarks, ~25% of 5 quarks, and ~10% of more than 5.

2. The 5-quark component of baryons is rather well understood, and should be measured directly

3. Pentaquarks (whose lowest Fock component has 5 quarks) are not too “exotic” – just Gamov-Teller excitations. In addition to the narrow there is a new prediction of charmed (and bottom) pentaquarks which decay only weakly.

uudds , ,0( ) , ( )c bcuu d s B uu sB b d