Glueball Masses from the Lattice - ectstar.eu · Glueballs Biagio Lucini Lattice setup Conﬁging theories (Near-) conformal theories Conclusions and outlook Glueball Masses from

Glueballs

Biagio Lucini

Lattice setup

Configingtheories

(Near-)conformaltheories

Conclusionsand outlook

Glueball Masses from the Lattice

Biagio LuciniSwansea University

QCD-TNT-III, Trento, September 2013

Biagio Lucini Glueballs

Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Motivations

The existence of gauge-invariant bound states of gluons implied byconfinement

However, glueball states have not yet been convincingly identified inexperiments

Glueball-like states can play a key role also in strongly interacting dynamicsbeyond the standard model

A calculation from first principles using lattice techniques can serve as aguidance to theoretical models and experimental searches


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Outline

1 Lattice setup

2 Glueballs in QCD-like theories

3 Glueballs and (near-)conformal dynamics

4 Conclusions and outlook


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Outline

1 Lattice setup





Glueballs

Biagio Lucini

Lattice setup

Configingtheories



The Lattice


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Lattice action for full QCD

Path integral

Z =

∫(DUµ(i)) (det M(Uµ))Nf e−Sg(Uµν(i))

with

Uµ(i) = Pexp(

ig∫ i+aµ̂

iAµ(x)dx

)and

Uµν(i) = Uµ(i)Uν(i + µ̂)U†µ(i + ν̂)U†ν(i)

Gauge part

Sg = β∑i,µ

(1−

1NRe Tr(Uµν(i))

), with β = 2N/g2

0


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Masses from correlators

Trial operators Φ1(t), . . . ,Φn(t) with the quantum numbers of the state of interest

Cij(t) = 〈0| (Φi(0))† Φj(t)|0〉

= 〈0| (Φi(0))† e−HtΦj(0)eHt|0〉

=∑

n

〈0| (Φi(0))† |n〉〈n|e−HtΦj(0)eHt|0〉

=∑

n

e−∆Ent〈0| (Φi(0))† |n〉〈n|Φj(0)|0〉

=∑

n

c∗incjne−∆Ent = δij∑

n

|cin|2e−amnt →t→∞

δij|ci1|2e−am1t


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Variational principle

1 Find the eigenvector v that minimises

am1(td) = −1td

logv∗i Cij(td)vj

v∗i Cij(0)vj

for some td2 Fit v(t) with the law Ae−m1t to extract m1

3 Find the complement to the space generated by v(t)4 Repeat 1-3 to extract m2, . . . ,mn

Sources of systematics

Need a good overlap of the eigenvectors with the state of interest

Need a large variational basis including all possible states overlapping withthe required one

Need to keep under control finite size and lattice artefacts

Care should be taken in assigning the spin


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Outline

1 Lattice setup





Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Glueballs in the quenched approximation

++ −+ +− −−PC

0

2

4

6

8

10

12

r 0mG

2++

0++

3++

0−+

2−+

0*−+

1+−

3+−

2+−

0+−

1−−2−−3−−

2*−+

0*++

0

1

2

3

4

mG (G

eV)

(From Morningstar and Peardon, hep-lat/9901004 )


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



QCD at large N

Generalisation of QCD: SU(N) gauge theory (possibly enlarged with Nf fermions inthe fundamental representation)

Taking the limit g2 → 0, N →∞, λ = g2N fixed simplifies the theory and one cansee that:

Quark loop effects ∝ 1/N ⇒ The N =∞ limit is quenched

Mixing glueballs-mesons ∝ 1/√

N ⇒ No mixing between glueballs andmesons at N =∞Meson decay widths ∝ 1/N ⇒ mesons do not decay at N =∞OZI rule ∝ 1/N ⇒ OZI rule exact at N =∞

↪→ The simpler large N phenomenology can explain features of QCD

phenomenology in a quenched setup that removes most of the practical

computational difficulties for QCD (and SU(3))


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Large N limit on the lattice

The lattice approach allows us to go beyond perturbative and diagrammaticarguments. For a given observable

1 Continuum extrapolationDetermine its value at fixed a and NExtrapolate to the continuum limitExtrapolate to N →∞ using a power series in 1/N2

2 Fixed lattice spacingChoose a in such a way that its value in physical units is common tothe various NDetermine the value of the observable for that a at any NExtrapolate to N →∞ using a power series in 1/N2

Study performed for various observables both at zero and finite temperature for

2 ≤ N ≤ 8


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Glueball masses at large N

[BL, Teper and Wenger, JHEP 0406 (2004) 012]


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Masses at N =∞

0++ m√σ= 3.28(8) +

2.1(1.1)N2

0++∗ m√σ= 5.93(17)− 2.7(2.0)

N2

2++ m√σ= 4.78(14) +

0.3(1.7)N2

Accurate N =∞ value, normal O(1/N2) correction


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Glueball spectrum at aTc = 1/6

0

0.5

1

1.5

2

2.5

3

am

Ground StatesExcitations[Lucini,Teper,Wenger 2004]

A1++ A1

-+ A2+- E++ E+- E-+ T1

+- T1-+ T2

++ T2--T2

-+T1++E-- T2

+-

Figure 20: The spectrum at N = !. The yellow boxes represent the large N extrapolation of masses

obtained in ref. [38].

0 2 4 6 8 10 12 14 16 18

!!M2

2"#

0

1

2

3

J

PC = + +PC = + -PC = - +PC = - -

Figure 21: Chew-Frautschi plot of the glueball spectrum

– 36 –

[BL, Rago and Rinaldi, JHEP 1008 (2010) 119]


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Meson spectrum (at fixed a)

^Fπ

^fρ

ρ a0

a1

b1

π∗ ρ∗a

0

*a

1

*b

1

*

0

1

2

3

4

5

m /

√σ

SU(∞)SU(2)

SU(3)

SU(4)

SU(5)

SU(6)

SU(7)

SU(17)

[Bali et al., JHEP 06 (2013) 071]


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Large N vs. experiments

0

1

2

3

4

5

m /

√σ

Ground statesExcited states

Experiment, √σ = 395 MeV

Experiment, √σ = 444 MeV

mq = m

ud

^Fπ

^fρ

π ρ a0

a1

b1

0

4.6

9.2

13.8

18.4

23.0

m /

^ F

∞

[Bali et al., JHEP 06 (2013) 071]


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Back to QCD

0+ + 0+− 0−+ 0−− 1−− 1+− 1+ + 2+ + 2−+ 2+− 2−− 3−− 3+− 3−+ 3+ +1

2

3

4

5

6

7

M G

eV

PDG exptAnisovich et al.Glueball, this work

[Gregory et al., JHEP 1210 (2012) 170]


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Outline

1 Lattice setup





Glueballs

Biagio Lucini

Lattice setup

Configingtheories



The spectrum for a QCD-like theory

At high fermion masses the theory is nearly-quenched

At low fermion masses the relevant degrees of freedoms are thepseudoscalar mesons


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Locking

All spectral mass ratios depend very mildly on m below the locking scale


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Locking near the Banks-Zaks point

If this scenario is valid beyond BZ, at all scales� Λ

mV/mPS ' 1 + ε

mPS � σ1/2

mG/σ1/2 '

[mG/σ

1/2](YM)


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Spectrum in SU(2) Nf = 2 Adj

0.0 0.2 0.4 0.6 0.8 1.0a mPCAC

0.5

1.0

1.5

2.0

2.5

3.0

a M

valu

es a

t mP

CA

C =

∞

PS meson mass0

++ glueball mass

2++

glueball mass

σ1/2 (σ = string tension)

[Del Debbio, BL, Patella, Pica and Rago, Phys. Rev. D80 (2009) 074507]


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Spectrum in SU(2) Nf = 2 Adj

0.0 0.2 0.4 0.6 0.8 1.0a mPCAC

0.5

1.0

1.5

2.0

2.5

3.0a

M

valu

es a

t mP

CA

C =

∞

PS meson mass0

++ glueball mass

2++

glueball mass

σ1/2 (σ = string tension)

The pseudoscalar is always higher in mass than the 0++ glueball, as predicted bythe locking scenario at high fermion mass

These theories naturally provide a light scalar⇒ is this a route to understanding

the mechanism of electroweak symmetry breaking?


Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Outline

1 Lattice setup





Glueballs

Biagio Lucini

Lattice setup

Configingtheories



Conclusions and outlook

Lattice calculations are an (increasingly more) useful tool to understand thefate of glueballs in QCD↪→ Need to control better mixing with scattering and meson states

Valuable information can be provided by lattice calculations in the large Nlimit↪→ Need to take the continuum limit

The dynamics of glueball is heavily influenced by the proximity to theconformal window↪→ New classes of theories need to be explored in order to see if stronglyinteracting BSM dynamics is a viable mechanism of electroweek symmetrybreaking


Documents

Glueball Masses from the Lattice - ectstar.eu · Glueballs Biagio Lucini Lattice setup Conﬁging theories (Near-) conformal theories Conclusions and outlook Glueball Masses from