Humboldt-Universität zu Berlin
Mathematisch-Naturwissenschaftliche Fakultät I
Institut für Physik
The Spectrum of Static-light Baryons
in Twisted Mass Lattice QCD
BACHELORARBEIT
eingereicht von
CHRISTIAN WIESE
geboren am 04. Februar 1987 in Hagen
Aufgabensteller: Prof. Michael Müller-Preussker
Zweitgutachter: PD Dr. Karl Jansen
Abgabedatum: 21.07.10
Abstract
I compute the static-light baryon spectrum with Nf = 2 flavors of sea quarks
using Wilson twisted mass lattice QCD. As valence quarks I consider light quarks
with three different masses with corresponding pion masses of 336 MeV, 417 MeV
and 517 MeV. I extract masses of states with isospin I = 0 and I = 1, with light
cloud angular momentum j = 0 and j = 1, and with parity P = + and P = −. I
present a preliminary extrapolation in the light u/d quark mass and compare my
results with available experimental results and results of other lattice groups.
Zusammenfassung
In dieser Arbeit wird das Spektrum von statisch-leichten Baryonen mit Wilson
Twisted Mass Lattice QCD und Nf = 2 Quarkflavors berechnet. Als Valenzquarks werden leichte Quarks mit mit drei verschiedenen Massen verwendet (mit
zugehörigen Pionmassen von 336 MeV, 417 MeV und 517 MeV). Es werden die
Massen von Zuständen mit Isospin I = 0 und I = 1, Gesamtdrehimpuls der leichten Freiheitsgrade j = 0 und j = 1, und Parität P = + and P = − berechnet. Es
wird eine erste grobe Extrapolation in der leichten Quarkmasse durchgeführt und
die Ergebnisse mit verfügbaren experimentellen Ergebnissen und den Ergebnissen
von anderen Gittergruppen verglichen.
1
CONTENTS
Contents
1 Introduction
2
2 Basic Principles
2.1 QCD in the continuum . . . . . . . . . .
2.2 Computing masses with the path integral
2.3 Wilson lattice QCD . . . . . . . . . . . .
2.4 The Wilson twisted mass formalism . . .
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approach
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3
3
3
4
5
3 Static-light Baryons in QCD
3.1 Static-light baryon creation operators
3.1.1 Gauge invariance . . . . . . .
3.1.2 Parity . . . . . . . . . . . . .
3.1.3 Spin . . . . . . . . . . . . . .
3.1.4 Isospin . . . . . . . . . . . . .
3.2 Experimental results . . . . . . . . .
3.3 List of operators . . . . . . . . . . .
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7
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9
10
4 Static-light Baryons in Twisted Mass Lattice QCD
4.1 The twisted mass symmetries . . . . . . . . . . . . . .
4.2 The correlation matrices . . . . . . . . . . . . . . . . .
4.3 Relation of the matrix elements . . . . . . . . . . . . .
4.3.1 γ5 hermiticity, time reversal, charge conjugation
4.3.2 Cubic rotations . . . . . . . . . . . . . . . . . .
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6 Results and Interpretation
6.1 Effective masses . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Extrapolation to the physical point . . . . . . . . . . . . . . . . .
21
21
25
7 Summary, Conclusion and Outlook
7.1 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . .
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
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8 Bibliography
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5 Simulation Setup and Analysis Details
5.1 Setup of the simulation . . . . . . . . . . . . . . . . . . . . .
5.2 Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Smearing techniques . . . . . . . . . . . . . . . . . . . . . .
5.4 Solving and interpreting the generalized eigenvalue problem
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1 INTRODUCTION
1
2
Introduction
A bottom baryon is a bound state of a bottom quark and two light quarks. Bottom baryons are among the heaviest bound states that can be measured. They
are very hard to measure experimentally due to the fact that the baryon number
has to be conserved, i.e. two baryons have to be created at a time.
In 1991, the first bottom baryon, Λb , was measured from data of the SFM detector at CERN. The mass value had a rather large error, because in this experiment
and the following electron positron collisions, the cross section for producing a
bottom baryon was very small.
Only when hadron colliders were able to achieve higher energies, more bottom
baryons could be measured. The second bottom baryon, Σb , was measured in
2007 with data of the CDF detector at the Tevatron experiment. Up to now,
only five states of bottom baryons have been measured experimentally.
In this thesis I am going to use QCD to compute the spectrum of these bottom
baryons. I will try to reproduce the experimental results and to make predictions
for states that have not been measured experimentally yet.
A useful device for performing numerical QCD is lattice QCD, which is a nonperturbative approach that reduces the continuous spacetime to a discrete lattice.
Nevertheless, when treating fermions in lattice QCD naively, a doubling problem
occurs. There are several formalisms avoiding this problem; here twisted mass
Wilson fermions will be used. Furthermore, it is very hard to perform computation with a dynamic bottom quark in lattice QCD, because a very fine lattice
spacing has to be used in order to resolve the UV oscillation of this heavy particle.
Therefore, I am going to use a static approach, i.e. restrict the heavy quark on
one single point in space. This means that in the continuum limit, the self-energy
of this particle is infinite. In lattice computations it is possible to obtain a finite
result which, however, is physically not meaningful because of its dependence on
the lattice spacing. Nevertheless, when computing differences of these masses,
the self-energy cancels, thus the result becomes independent of the lattice spacing
and physically meaningful. This baryon is called a static-light baryon.
In QCD, a static-light baryon is characterized by its isospin, the parity and angular momentum of the light cloud and the quark flavors. With regard to these
quantum numbers, a baryon creation operator can be built, from which the mass
of the baryon can be extracted when using the path integral approach.
After extracting the masses, the results have to be adjusted because of the used
approximations, such as unphysical heavy up and down quark masses, the static
approximation and a finite lattice spacing.
3
2 BASIC PRINCIPLES
2
2.1
Basic Principles
QCD in the continuum
When trying to describe QCD one needs a Lagrangian that depends on the
fermionic fields ψ, ψ̄, which represent the fermions and antifermions and a gauge
field Aµ , which represents the gluon field.
There is a term describing the dynamics of the fermions and their interaction
with the gauge field, and one term describing the dynamics of the gauge field.
The QCD lagrangian is:
LQCD [ψ, ψ̄, Aµ ] = LF [ψ, ψ̄, Aµ ] + LG [Aµ ]
(2.1)
c
Note that ψ = ψD,f
(x), where D is the Dirac, c is the color and f is the flavor
a a
index and Aµ = T Aµ where T a are the generators of SU(3), i.e. the eight GellMann matrices.
In the QCD Lagrangian LF is the fermionic action, which depends on the fermionic
fields and the gauge fields, and which describes the dynamics of the fermions. One
has to write down a lagrangian for every quark flavor which is taken into account.
X
ψ̄f (iγ µ Dµ + m) ψf
(2.2)
LF [ψ, ψ̄, Aµ ] =
f
Dµ is the covariant derivative and defined as:
Dµ ψ = ∂µ ψ − igAµ ψ
(2.3)
LG is the gauge field action. It describes the motion of the gauge field and is
given as:
1 a µν,a
LG [Aµ ] = − Fµν
F
4
(2.4)
a
Here Fµν
is the QCD field-strength tensor defined as:
a
Fµν
= ∂µ Aaν − ∂ν Aaµ − ig[Aaµ , Aaν ]
2.2
(2.5)
Computing masses with the path integral approach
QCD can be quantized using the path integral approach. Observables which
can be computed are ground state expectation values of time ordered products of
operators denoted by O = T {O1 , O2 , ...}. Here I will use the Euclidean Formalism
in order to obtain numerical results.
Z
1
DψDψ̄DAµ O[ψ, ψ̄, Aµ ]e−SE [ψ,ψ̄,Aµ ]
(2.6)
hΩ|Ô|Ωi =
N
4
2 BASIC PRINCIPLES
R
Here N = DψDψ̄DAµ e−SE [ψ,ψ̄,Aµ ] and SE [ψ, ψ̄, Aµ ] is the action of the system,
depending on the spinor fields ψ, ψ̄ and the gauge field Aµ .
In this thesis, I need a method to compute the energy differences of excitations
with well-defined quantum numbers.
For this purpose one can derive the following two-point function:
X
hΩ|O(t)O(0)|Ωi =
hΩ|O|nihn|O|Ωie−(En −EΩ )t
(2.7)
n
Note that hΩ|O|nihn|O|Ωi = hΩ|O|ni2 = C will vanish when the energy eigenstate n has not the same quantum numbers as O|Ωi.
When taking the limit t → ∞ only the first term of the sum will maintain, due
to growing energy differences En − EΩ .
lim hΩ|O(t)O(0)|Ωi = Ce−(E0 −EΩ )t
(2.8)
⇒ lim − lnhΩ|O(t)O(0)|Ωi = ∆Et − ln(C)
(2.9)
hΩ|O(t + 1)O(0)|Ωi
= ∆E
hΩ|O(t)O(0)|Ωi
(2.10)
t→∞
t→∞
⇒ lim ln
t→∞
This means that if a suitable operator O(t) can be found, i.e. a operator which
has requested quantum numbers, one is able to compute the energy difference
∆E between the vacuum energy EΩ and the energy of the ground state E0 with
these quantum numbers.
2.3
Wilson lattice QCD
Up to now it is impossible to solve QCD analytically. Therefore, I will use the
numerical approach of lattice QCD.
In lattice QCD, the Euclidean spacetime is replaced by a four-dimensional lattice. The fermionic fields ψ(x), ψ̄(x) are represented by discrete vectors ψi , ψ̄i ,
i ǫ {x, D, c, f }, which are located on the lattice sites.
The gauge field Aµ (x) is now represented by the so-called links or parallel transporters Ux,aµ̂ , which are located between neighboring lattice sites. In the continuum they are connected in the following way:
 x+aµ̂

Z
Ux,aµ̂ = P̂ exp ig
Aµ (y)dyµ
(2.11)
x
One possible lattice discretization of the gauge field action can be written down
as follows using the tree-level Symanzik gauge action:
!
4
4
X
X
βX
1×1
1×2
b0
{1 − Re Tr(Ux,µ,ν
)} + b1
{1 − Re Tr(Ux,µ,ν
)} (2.12)
SG [U] =
3 x
µ,ν=1
µ,ν=1
5
2 BASIC PRINCIPLES
1×1
1×2
where b1 = −1/12 and b0 = 1 − 8b1 . Ux,µ,ν
is the plaquette term, Ux,µ,ν
a
rectangular Wilson loop. When choosing b1 = 0, one obtains the well-known
plaquette action.
When doing computations with a naive action S = SF + SG , where the fermionic
action is written as
X γµ
∗
4
ψ̄
(∇µ + ∇µ ) + m ψ
(2.13)
SF [ψ, ψ̄, U] = a
2
x,f
with
1 †
Uµ (x)ψ(x + aµ̂) − ψ(x)
a
1
∇∗µ ψ(x) = − [Uµ (x − aµ̂)ψ(x − aµ̂) − ψ(x)]
a
the so-called fermionic doubling problem will occur.
A way to circumvent the problem is the Wilson fermionic action:
X
SF [ψ, ψ̄, U] = a4
ψ̄ (DW + m) ψ
∇µ ψ(x) =
(2.14)
(2.15)
(2.16)
x
γµ
ar
(∇µ + ∇∗µ ) + ∇µ ∇∗µ
(2.17)
2
2
The additional term is the twisted mass term, which will vanish in the continuum
limit.
DW =
2.4
The Wilson twisted mass formalism
One of the main reasons for using Wilson twisted mass lattice QCD is that it
allows to achieve an O(a) improvement [1]. Consequently, the fermionic part of
the well-known QCD action changes to
X
SF [χ, χ̄, U] = a4
χ̄ (DW + m + iµγ5 τ3 ) χ
(2.18)
x
χd . τ3 is the Pauli matrix in flavor space and µ is the so-called
where χ = χu
twisted mass.
The physical spinors can be obtained by applying the following twist rotation to
the twisted mass spinors in the continuum:
ψ = exp (iωγ5 τ3 /2) χ, ψ̄ = χ̄ exp (iωγ5τ3 /2)
(2.19)
Here χ and χ̄ are the spinors in the twisted basis, while ψ and ψ̄ are spinors in
the physical basis.
The twist angle ω satisfies the relation:
tan ω =
µR
mR
(2.20)
2 BASIC PRINCIPLES
6
where µR and mR are renormalized masses. At maximal twist, the twist angle is
ω = π/2.
Now the action of the physical field is given as:
SF [ψ, ψ̄, U] =
ar
γ
X
µ
a4
ψ̄(x)
(∇µ + ∇∗µ ) − iγ5 τ 3 − ∇µ ∇∗µ + mcr + µ ψ(x) (2.21)
2
2
x
7
3 STATIC-LIGHT BARYONS IN QCD
3
3.1
Static-light Baryons in QCD
Static-light baryon creation operators
As seen in chapter 2.2, a creation operator O is needed in order to compute the
masses of excited states of a sector. I want to find an operator which creates a
static-light baryon at position x. Therefore, one needs three QCD spinors, where
one represents a static quark, the other two the light quarks. From the operator
I require gauge invariance as well as well-defined parity, spin and isospin.
As an ansatz I want to write down the following operator and show that it satisfies
the requirements.
O = ǫabc Qa (ψ1b )T CΓ(ψ2c )
(3.1)
In this expression Q is the heavy quark spinor, ψ are the light quark spinors in
the physical basis, C = γ0 γ2 is the charge conjugation matrix, and Γ is a suitable
γ combination.
3.1.1
Gauge invariance
In order to create a physical state, the creation operator has to be invariant under
gauge transformations. In QCD, a spinor transforms under gauge transformation
as
ψ a → ψ a′ = Gab ψ b
(3.2)
where Gab ǫ SU(3), which implies det(G) = 1, GG† = 1 and G is 3 × 3 matrix.
Transforming the baryon creation operator gives us the following result:
O → O′ = ǫabc Gad Qd Gbe (ψ1b )T CΓGcf (ψ2f )
d b T
f
abc
ad be cf
= ǫ
G G G Q (ψ1 ) CΓ(ψ2 )
= ǫdef det(G)Qd (ψ1b )T CΓ(ψ2f )
f
e T
def d
(3.3)
= ǫ Q (ψ1 ) CΓ(ψ2 )
Thus, the operator is invariant under gauge transformation, as required.
3.1.2
Parity
In QCD, the action is invariant under the following parity transformation:
T
ψ → ψ ′ = γ0 ψ
′T
(3.4)
T
⇒ ψ C → ψ C = −ψ Cγ0
(3.5)
3 STATIC-LIGHT BARYONS IN QCD
8
One can easily see that the parity of the light cloud, which consists of the light
quarks and gluons is only defined by γ0 Γγ0 i.e. by the choice of Γ.
When transforming the static quark one gets:
O ′ = γ0 O
(3.6)
This means that a well-defined parity can be reached by multiplying the operator
0
with p = 1±γ
.
2
0
, i.e. the parity I will refer to later only
In my analysis, I will choose p = 1+γ
2
depends on the light quark parity. As one will see in section 4.2 the correlator
0
would vanish when choosing p = 1−γ
.
2
3.1.3
Spin
Under rotation a spinor transforms as:
ψ → ψ′ = e
αa
γ γ γ
2 0 5 a
ψ
T
αa
αa
⇒ ψ T C → (ψ ′ )T C = e 2 γ0 γ5 γa ψ C = ψ T Ce− 2 γ0 γ5 γa
(3.7)
This means that the spin of the light quarks depends on the expression
e−
αa
γ γ γ
2 0 5 a
Γe
αa
γ γ γ
2 0 5 a
(3.8)
Hence, a γ5 γ0 combination for Γ gives a light cloud angular momentum of 0,
because one observes invariance under rotation, while putting Γ = γj leads to a
light cloud angular momentum of 1.
Note, that the light quarks have a relative angular momentum of 0, i.e. are in a
S wave and their total angular momentum does only depend on their spin.
The heavy quark has a spin of 12 , i.e. with a light angular momentum of 0 one gets
a total angular momentum of 12 while a light angular momentum of 1 provides
two states, J = 21 and J = 32 . Yet, because of my treatment of the heavy quark
in the static approximation, there is no spin splitting, i.e. states are labelled by
the integer light quark spin.
3.1.4
Isospin
The two light quarks form a isospin dublet, which transforms like a spin 21 particle
with the up quark corresponding to spin up and the down quark corresponding
to spin down.
It is known from quantum mechanics that a symmetric combination of two spin
1/2 particles gives a spin 1 state, while an antisymmetric combination equals a
spin 0 state.
3 STATIC-LIGHT BARYONS IN QCD
9
This means that I can use the following expressions in order to generate a welldefined isospin:
O(ψ1 = ψu , ψ2 = ψd ) + O(ψ1 = ψd , ψ2 = ψu ) → I 2 = 1, Iz = 0
O(ψ1 = ψu , ψ2 = ψd ) − O(ψ1 = ψd , ψ2 = ψu ) → I 2 = 0, Iz = 0
O(ψ1 = ψu , ψ2 = ψu ) → I 2 = 1, Iz = 1
O(ψ1 = ψd , ψ2 = ψd ) → I 2 = 1, Iz = −1
(3.9)
(3.10)
(3.11)
(3.12)
where ψu is a spinor of an up and ψd the spinor of a down quark.
Nevertheless, this symmetrization/antisymmetrization is not necessary for the
operators, because they show the following behavior under flavor exchange:
O(ψ1 = ψd , ψ2 = ψu ) = ±O(ψ1 = ψu , ψ2 = ψd )
(3.13)
Therefore, operators which are invariant under flavor exchange create an isospin 1
state, while operators which change sign create an isospin 0 state or are identical
0.
Note that in this thesis I will focus on operators which have Iz = 0, i.e. contain
an up and a down quark.
3.2
Experimental results
Because my calculations of the static-light baryons are expected to be a good approximation of bottom baryons it is interesting to compare them to experimental
results [2].
I will only state experimental measured baryons containing up and down quarks,
because my thesis focuses on these states. Note, however, that it is part of a
bigger project which also considers states containing strange quarks [13].
• Λb
In the quark model this is a udb state with I = 0 and J P = (1/2)+ .
The measured mass is m = 5619.7(12) MeV, the difference to the B meson
is m − m(B) = 341 MeV
• Σb
In the quark model this is a udb, uub, ddb triplet with I = 1 and J P =
(1/2)+ .
The measured mass of the Σ+
b is m = 5807.8(27) MeV, the difference to the
B meson is m − m(B) = 528 MeV
The measured mass of the Σ−
b is m = 5815.2(20) MeV, the difference to the
B meson is m − m(B) = 536 Me
The mass of the Σ0b has not been measured.
• Σ∗b
In the quark model this is a udb state with I = 1 and J P = (3/2)+ .
10
3 STATIC-LIGHT BARYONS IN QCD
The measured mass of the Σ∗+
is m = 5829.0(34) MeV, the difference to
b
the B meson is m − m(B) = 550 MeV
The measured mass of the Σ∗−
is m = 5836.4(28) MeV, the difference to
b
the B meson is m − m(B) = 557 MeV
The mass of the Σ∗0
b has not been measured.
Note that in the static approach Σb and Σ∗b are a single state and consequently
have the same mass.
The mass of the B meson is m(B) = 5279.5(5) MeV.
3.3
List of operators
Considering all possible operators of the form
O = ǫabc Qa (ψub )T CΓ(ψdc )
(3.14)
which create states with well-defined quantum numbers with light quarks in a
relative S-wave, one finds the following operators stated in table 1. In the last
column of the table one can find the name of the state if it has been experimentally
observed.
creation operator
jP
J
I
exp. name
Γ = γ5
0+
1/2
0
Λb
Γ = γ0 γ5
0+
1/2
0
Λb
Γ=1
0−
1/2
0
Γ = γ0
0−
1/2
1
Γ = γj
1+ 1/2, 3/2 1
Σb
Γ = γ0 γj
1+ 1/2, 3/2 1
Σb
Γ = γj γ5
1− 1/2, 3/2 0
Γ = γ0 γj γ5
1− 1/2, 3/2 1
Table 1: List of baryon creation operators
4 STATIC-LIGHT BARYONS IN TWISTED MASS LATTICE QCD
4
11
Static-light Baryons in Twisted Mass Lattice
QCD
In the previous chapter I illustrated how QCD spinors form a baryon creation
operator and I explained which operators can be used to create states with welldefined quantum numbers.
Nevertheless, these operators have to be changed slightly, because I use gauge
configurations which were created using the twisted mass action.
In the twisted mass formalism there are different symmetries due to the explicit
O(a) breaking of parity and isospin of the Wilson term. After the computation
one can rotate the operators in the twisted basis back into the physical basis and
one can use table 1 to interpret the results.
4.1
The twisted mass symmetries
It is very important that the symmetries of twisted mass lattice QCD (tm lQCD)
differ from the well known QCD symmetries listed above.
Due to the flavor breaking in the twisted mass formalism the absolute value of the
isospin I 2 is not a symmetry anymore. Only the third component of the isospin
Iz is still a conserved quantity1 .
Because of the parity breaking of the Wilson term, the QCD parity is also not a
symmetry. Hence, the twisted mass parity, which is parity and flavor exchange,
replaces the the QCD parity as quantum number.
χ → χ′ = γ0 τ1 χ
(4.1)
In general one can find for the twisted mass parity and the third component of
the isospin [P (tm) , Iz ] 6= 0, i.e. they are not conserved independently and can not
be used as quantum numbers for our states. Yet, if Iz = 0, the commutator is
[P (tm) , Iz ] = 0. This means that in my computations, where only Iz = 0 states
are considered, I can use both P (tm) and Iz as quantum numbers.
The twisted mass action is still invariant under the quark field rotation, i.e. the
angular momentum is still a conserved quantity as well. For the computations it
is important to group the operators into sectors corresponding to their twisted
mass quantum numbers (P (tm) , j, Iz ) as done in table 2.
One should note that the j = 1 sector, is split into three sectors (γ1 , γ2 , γ3 ),
which have to be computed independently, but can be averaged at the end due
to rotational symmetry.
1
In principle, the infinite number of angular momentum representations has to be replaced
by the five discrete representations of the cubic rotation group Oh . As I am only considering
angular momenta 0 and 1, no severe difficulties are expected and I stick to the continuum
notation.
4 STATIC-LIGHT BARYONS IN TWISTED MASS LATTICE QCD
O = ǫabc Qa (χbu )T CΓ(χcd )
twisted basis
P (tm) j
physical basis
P j I
Γ = γ5
−
0
Γ = γ5
+ 0 0
Γ = γ0 γ5
−
0
Γ = γ0
− 0 1
Γ = γ0
−
0
Γ = γ0 γ5
+ 0 0
+
0
Γ=1
− 0 0
Γ = γj
+
1
Γ = γj γ5
− 1 0
Γ = γ0 γj
+
1
Γ = γ0 γj
+ 1 1
Γ = γj γ5
+
1
Γ = γj
+ 1 1
Γ = γ0 γj γ5
−
1
Γ = γ0 γj γ5
− 1 1
Γ=1
12
Table 2: twisted mass quantum number sections
Consequently, there are four independent correlation matrices from which masses
can be extracted.
There are further twisted mass symmetries, such as γ5 hermiticity, time reversal
and charge conjugation. We will use these symmetries in order to find a relation
between the following correlation matrix elements.
4.2
The correlation matrices
As shown in section 2.2, one gets important information computing the correlation function:
C̃(t) = hΩ|O(t)O(0)† |Ωi
(4.2)
Due to the twisted mass formalism a mixing of the quantum numbers appears,
e.g. in the twisted mass sector P (tm) = −, there are states with P = + and
states with P = − in the physical basis. For this reason and in order to get
better statistics I would like to consider all creation operators of one sector for
the spectrum. This means that the correlation function becomes a correlation
matrix:
C̃Γj ,Γk (t) = hΩ|OΓj (t)OΓk (0)† |Ωi
(4.3)
I will discuss later how this matrix can be used in order to extract low-lying
baryon masses.
4 STATIC-LIGHT BARYONS IN TWISTED MASS LATTICE QCD
13
Using the baryon creation operator (3.1), one can compute the correlation matrix, using the following relation:
= ǫabc
†
OΓ (x)† = ǫabc Qa (χbu )T CΓ(χcd )
(χ̄bd )γ0 (CΓ)† γ0 (χ̄cu )T Q̄a γ0 = ±ǫabc (χ̄bd )CΓ(χ̄cu )T Q̄a γ0
(4.4)
The correlation then is:
Z
abc def
DχD χ̄DU
C̃Γj ,Γk (t) = ǫ ǫ
Qa (t)Q̄d (0)Trspin CΓ1 χcd (t)χ¯d f (0) CΓ2 (χbu )T (t)(χ¯u e )T (0) γ0 (4.5)
The integration over χχ̄ can be performed by using their Grassmann properties.
This gives us the inverse of the Dirac matrix, which is the propagator of the
quark:
ab
χa (x1 )χ̄b (x2 ) = (D χ )−1 (x1 , x2 )
(4.6)
The static approximation for the bottom quark is carried out by choosing an
infinite heavy mass and by using the heavy quark effective theory (HQET)[5].
This implies that one can analytically simplify the heavy quark propagator to:
−1 ab
DQ
(x, y) = δ (3) (x − y)U ab (x, x0 ; y, y0)
AB
(1 ∓ γ0 )AB
exp (∓M(y0 − x0 ))
(4.7)
2
Here, U(x0 ; y0 ) is the Wilson line from time x0 to y0 and M is the heavy quark
mass. When a quark is propagating in positive time direction, ∓ is −. In negative
time direction, ∓ is +. An antiquark behaves in the opposite way.
Inserting these relations one can write down the correlation function
as following,
R
using that h...i is the path integral over gauge configurations DU:
(1 − γ0 )γ0
C̃Γj ,Γk (t) = ǫabc ǫdef
exp (−M · t)
2
E
D
cf
be
(4.8)
U ad (t, 0)Trspin CΓ1 (D χu )−1 (t, 0) CΓ2 (D χd )−1 (t, 0)
The exp (−M · t) term can be omitted, because I am only considering mass differences between bottom particles.
The spin structure of this expression is given by (1 − γ0 )γ0 /2 and is therefore
trivial and can easily be included in the analysis. Thus, I will omit it. The result
for the correlation matrix is as follows:
C̃Γj ,Γk (t) = ǫabc ǫdef
E
D
χu −1 cf
χ −1 be
ad
d
(t, 0) CΓ2 (D )
(t, 0)
(4.9)
U (t, 0)Trspin CΓ1 (D )
This matrix can be directly computed on the lattice.
4 STATIC-LIGHT BARYONS IN TWISTED MASS LATTICE QCD
4.3
14
Relation of the matrix elements
When using symmetries of the twisted mass action, it can be shown that there
are certain relations between the elements of the correlation matrix C̃Γj ,Γk (t).
We can use the relations to average over the matrix elements in order to get
better statistics.
4.3.1
γ5 hermiticity, time reversal, charge conjugation
The twisted mass propagator behaves in the following way under certain transformations:
γ5 hermiticity:
time reversal
†
−1
−1
D (χ) (x1 , x2 ) = τ1 γ5 D (χ) (x2 , x1 ) γ5 τ1
−1
−1
D (χ) (t1 , t2 ) = τ1 γ0 γ5 D (χ) (−t1 , −t2 )γ5 γ0 τ1
(4.10)
(4.11)
charge conjugation
T
−1
−1
D (χ) (x1 , x2 ) = γ0 γ2 D (χ) (x2 , x1 ) γ2 γ0
(4.12)
Using these symmetries a number of statements can be made about the relation
of the matrix elements:
γ5 hermiticity, for example, allows us to make the following statement about the
correlation matrix:
†
C̃Γ1 ,Γ2 (−t) = HΓ1 HΓ2 C̃Γ2 ,Γ1 (t)
(4.13)
Where HΓj is ±1 depending on the γ matrices in Γ.
Time reversal helps us to find the relation between the correlation matrix in
positive and in negative time direction:
C̃Γ1 ,Γ2 (−t) = TΓ1 TΓ2 C̃Γ1 ,Γ2 (t)
(4.14)
Where TΓj is ±1.
With charge conjugation I am able to make a statement about the behavior
of the correlation matrix when exchanging the quarks:
C̃Γ1 ,Γ2 (−t) = CΓ1 CΓ2 C̃Γ2 ,Γ1 (t)
(4.15)
4 STATIC-LIGHT BARYONS IN TWISTED MASS LATTICE QCD
O = ǫabc Qa (χbu )T CΓ(χcd )
15
creation operator TΓj CΓj HΓj
Γ = γ5
+
+
−
Γ = γ0 γ5
−
−
+
Γ = γ0
−
+
+
Γ = γj
−
+
+
Γ = γ0 γj
+
−
−
Γ = γj γ5
−
−
+
Table 3: Signs for symmetry transformations
Where CΓj is ±1.
The factors HΓj , TΓj , CΓj can be found in Table 3.
An exemplary calculation for these relations can be found in the appendix.
Combining these three symmetries, one will get relations between some of the
matrix elements. Moreover, it turns out that matrix elements are either real or
purely imaginary. As an example I would like to take Γ1 = γ0 and Γ2 = γ5 . With
this choice I get the following relation:
†
C̃γ0 ,γ5 (−t) = − C̃γ5 ,γ0 (t)
(4.16)
C̃γ0 ,γ5 (−t) = −C̃γ0 ,γ5 (t)
C̃γ0 ,γ5 (−t) = +C̃γ5 ,γ0 (t)
(4.17)
(4.18)
From these relations one can conclude for example, that the correlation function
is purely imaginary and that an exchange of the Γ matrices will cause a change
in sign.
If one wants to check these symmetry relations numerically, one has to compute
the correlation functions of the related elements, plot them as in figure 1 and
check if the symmetry is fullfilled.
In this example, one can see that time reversal is fullfilled. One can also see that
the correlation function is purely imaginary.
If one wants to check all these symmetries, including the cubic rotation, there are
over thousand pairs of correlation functions that need to be plotted, because different smearing levels have to be combined. I will discuss the details on smearing
later in this thesis. I plotted all possible correlation functions and checked the
symmetry relations:
16
4 STATIC-LIGHT BARYONS IN TWISTED MASS LATTICE QCD
Re(C̃γ0 ,γ5 (t))
Im(C̃γ0 ,γ5 (t))
-Re(C̃γ0 ,γ5 (−t))
-Im(C̃γ0 ,γ5 (−t))
Correlation function C̃
1.5e-05
1e-05
5e-06
0
0
1
2
3
4
5
Time separation t
6
7
8
Figure 1: Time reversed correlation functions for Γ1 = γ0 and Γ2 = γ5 , obtained with
15 gauge configurations
1.2e-06
Re(C̃γ1 γ5 ,γ1 γ5 (t))
Im(C̃γ1 γ5 ,γ1 γ5 (t))
Re(C̃γ2 γ5 ,γ2 γ5 (t))
Im(C̃γ2 γ5 ,γ2 γ5 (t))
Re(C̃γ3 γ5 ,γ3 γ5 (t))
Im(C̃γ3 γ5 ,γ3 γ5 (t))
Correlation function C̃
1e-06
8e-07
6e-07
4e-07
2e-07
0
-2e-07
0
1
2
3
4
5
Time separation t
6
7
8
Figure 2: Cubic rotated correlation functions for Γ1 = Γ2 = γj γ5 , obtained with 15
gauge configurations
4 STATIC-LIGHT BARYONS IN TWISTED MASS LATTICE QCD
4.3.2
17
Cubic rotations
The last symmetry I will use is the behavior of the propagator under 90 degree
cubic rotations. The propagator behaves in the following way under rotation
around the j axis:
−1
Q(χ) (x1 , x2 ) =
(1 − γ5 γj γ0 ) (χ) −1
(1 − γ0 γj γ5 )
√
√
Q
(Rx1 , Rx2 )
2
2
(4.19)
One can use this relation to connect the matrices of the vector baryon sectors,
where Γ contains a γj .
C̃Γ1 (γ1 ),Γ2 (γ1 ) = C̃Γ1 (γ2 ),Γ2 (γ2 ) = C̃Γ1 (γ3 ),Γ2 (γ3 )
(4.20)
In figure 2 one can see that the symmetry is fullfilled, because the correlation
function for different γj are identical within statistical errors.
5 SIMULATION SETUP AND ANALYSIS DETAILS
5
5.1
18
Simulation Setup and Analysis Details
Setup of the simulation
For my simulations I used (L/a)3 ×T /a = 243 ×48 gauge configurations produced
by the European Twisted Mass Collaboration (ETMC). The gauge action is the
tree-level Symanzik action. The fermionic action is the Wilson twisted mass
action at maximal twist (κ = 0.160856) with Nf = 2 degenerated flavors. The
gauge configurations were computed at β = 3.9, corresponding to a current lattice
spacing of a = 0.079(2) fm [3].
I considered twisted mass values of µ = 0.0040, µ = 0.0064 and µ = 0.010, which
correspond to pion masses of mπ = 336 MeV, mπ = 417 MeV and mπ = 517 MeV.
The following preliminary results were obtained with 270 gauge configurations:
200 with µ = 0.0040, 40 with µ = 0.0064 and 30 with µ = 0.010.
There are more calculations in progress, which will be included in a planned
publication [13].
5.2
Inversions
In order to be able to use all-to-all propagators I use stochastic sources (cf.[11]
and references therein). Here the following equation has to be solved. Note that
the indices j, k replace x, c, D.
Qkj φj = ξk
(5.1)
Where ξ is a stochastic source which has to satisfy the relation
hξj ξk† i = δj,k
(5.2)
Note that h...i denotes the average over many sources. It can be implemented
√
by randomly
√ the following elements of the complex unit circle (1/ 2,
√
√ inserting
−1/ 2, i/ 2, −i/ 2) as source elements.
This relation needs to be fullfilled because if one wants to compute the propagator
one has to compute:
φj = D −1 jk ξk
(5.3)
⇒ φj ξl∗ = D −1 jk ξk ξl†
(5.4)
⇒ hφj ξl∗ i = D −1 jk hξk ξl† i
(5.5)
⇒ hφj ξl∗ i = D −1 jk δk,l
(5.6)
⇒ hφj ξl∗ i = D −1 jl
(5.7)
For my computation, I used 12 up and 12 down inversions in order to reduce the
noise. For the baryon correlation function these are 12 × 12 samples, i.e. it is
5 SIMULATION SETUP AND ANALYSIS DETAILS
19
good to use more sources of one gauge configuration.
In order to reduce the noise further I will use timeslice dilution. This amounts
to sources having only elements on one timesslice Ts , and one can only compute
a propagator which starts at a defined time Ts .
5.3
Smearing techniques
In my simulations I used different smearing techniques in order to obtain a good
overlap of my signal with the ground state (cf. [12] and references therein).
The fermionic fields were smeared using the Gaussian smearing. I used three
different smearing levels NGauss = {10, 40, 90}, κGauss = 0.5 in order to build a
correlation matrix. Thus, I might have a chance to also determine excited states.
The spatial links were smeared using APE smearing NAPE = 40, αAPE = 0.5.
This smearing was optimized by choosing a number of smearing steps where the
effective mass is the smallest for a small number of time separations, because
excited states have a strong contribution there.
The temporal links were smeared according to the HYP2 static action in order to
reduce strong ultraviolet fluctuations and therefore improving the signal to noise
ratio.
5.4
Solving and interpreting the generalized eigenvalue
problem
As discussed in the previous chapters in all sectors there is a correlations matrix
containing information about the low-lying states.
This means that computing the effective mass, in order to obtain a mass plateau,
is not as trivial, as sketched in section 2.2.
Instead, I use the generalized eigenvalue problem (GEP).
Starting with a correlations matrix:
C̃ij (t) = hΩ|Oi (t)Oj (0)† |Ωi
(5.8)
one can solve the following eigenvalue problem [4]
C̃(t)vn (t, t0 ) = λn (t, t0 )C̃(t0 )vn (t, t0 ), n = 1, ..., N t > t0
(5.9)
and compute the effective mass as:
meff
n =
1
λn (t, t0 )
ln
a λn (t + a, t0 )
(5.10)
As one can see, one gets n effective masses, where n is the dimension of the correlation matrix. In the limit t → ∞, these effective masses approach the n lowest
masses in the corresponding sector.
5 SIMULATION SETUP AND ANALYSIS DETAILS
20
In order to interpret these states it is necessary to study the eigenvectors vn .
When plotting the squared absolute value of the components |vj |2 of effective
mass n, one can identify the dominating creation operator Oj by the largest
value |vj |2 .
The operator has to be rotated back into the pseudo physical basis. Then its
quantum numbers can be identified, by using table 2. An example for this will
be given in the next chapter.
21
6 RESULTS AND INTERPRETATION
6
Results and Interpretation
6.1
Effective masses
I extracted the following effective masses from the four correlation matrices I took
into account.
At the present level of statistics, corresponding to 200 gauge configurations, I am
only able to present preliminary results from the effective masses of the ground
state. I was able to observe excited states, but the signal was not good enough
to extract reliable masses.
One can find my computed effective masses for µ = 0.0040 and the plateau fits,
from which mass values can be determined, in Figure 3, 5, 6 and 7. The fitting
range was chosen so that χ2 /dof is O(1), and it has been checked that the results
were equal within errors when fitting at larger t values.
In figure 4 it can be seen how quantum numbers of states can be extracted from
the correlation matrices. Here, the operator with Γ = γ5 has the largest value
of |vj |2 for the ground state. After rotating the operator back into the physical
basis, as done in table 2, the operator still has Γ = γ5 , and consequently, the
quantum numbers I = 0, j P = 0+ .
The operator with Γ = γ0 γ5 , which has quantum numbers I = 1, j P = 0− after
rotation, has also a value of |vj |2 , which is significantly different from 0. This
is consistent, because only an approximative rotation is performed. Thus, one
should only consider the dominant value of |vj |2 .
All results can be found in table 4.
m − m(B) [MeV] m − m(B) [MeV] m − m(B) [MeV]
jP I
µ = 0.0040
µ = 0.0064
µ = 0.0100
state
0+ 0
461(24)
474(45)
515(29)
Λb
1+ 1
689(23)
708(40)
734(44)
Σb , Σ∗b
0− 0
1358(56)
1343(69)
1338(63)
?
1− 1
1091(49)
1302(56)
1249(34)
?
Table 4: Results from the ground state effective masses of the 4 correlation matrices
for different pion masses
22
6 RESULTS AND INTERPRETATION
1.2
Effective mass meff
1
0.8
0.6
0.4
0.2
0
0
m = 0.5847 ± 0.0099 (χ2 /dof = 0.85)
2
4
6
8
Time separation t
10
12
Figure 3: Effective mass of 3 x 3 correlation matrix. OΓ = ǫabc Qa (χbu )T CΓχcd with
Γǫ{γ5 , γ0 , γ0 γ5 }. Ground state → QCD quantum numbers I = 0, j P = 0+
1.2
j = Oγ0
j = Oγ0 γ5
j = Oγ5
Eigenvector |vj |2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
Time separation t
8
10
Figure 4: Absolute value of Eigenvectors
for ground state of 3 x 3 correlation matrix.
OΓ = ǫabc Qa (χbu )T CΓχcd with Γǫ{γ5 , γ0 , γ0 γ5 }.
23
6 RESULTS AND INTERPRETATION
1.2
m = 0.6773± 0.0067 (χ2 /dof = 1.63)
Effective mass meff
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
Time separation t
10
12
Figure 5: Effective mass of 3 x 3 correlation matrix. OΓ = ǫabc Qa (χbu )T CΓχcd with
Γǫ{γj , γj γ5 , γ0 γj }. Ground state → QCD quantum numbers I = 1, j P = 1+
1.2
Effective mass meff
1
0.8
0.6
0.4
0.2
0
0
1
m = 0.9617 ± 0.0327 (χ2 /dof = 0.03)
2
3
4
Time separation t
5
6
7
Figure 6: Effective mass of 3 x 3 correlation matrix. OΓ = ǫabc Qa (χbu )T CΓχcd with
Γǫ{1}. Ground state → QCD quantum numbers I = 0, j P = 0−
24
6 RESULTS AND INTERPRETATION
1.2
m = 0.8402± 0.0198 (χ2 /dof = 0.11)
Effective mass meff
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
Time separation t
7
8
9
Figure 7: Effective mass of 3 x 3 correlation matrix. OΓ = ǫabc Qa (χbu )T CΓχcd with
Γǫ{γ0 γj γ5 }. Ground state → QCD quantum numbers I = 0, j P = 0−
mass difference m − m(B) in MeV
900
800
700
600
500
400
300
200
Mass of Λb
Mass of Σb
Experiment
100
0
0 1352
3362
4172
Pi mass squared m2π in MeV2
5172
Figure 8: Extrapolation in the light quark mass to the physical point for the Λb and
Σb state
25
6 RESULTS AND INTERPRETATION
6.2
Extrapolation to the physical point
As mentioned above, µ = 0.0040 equals a pion mass of 336 MeV. Compared to
the physical quark mass of mπ ≈ 135 MeV, the quark mass I use is too heavy.
At present, there are no simulations available with smaller mπ at same lattice
size. Therefore, I did further computations with higher quark masses and tried
an extrapolation to the physical point.
Due to the fact that I only have three different masses with preliminary statistics,
I attempt a linear extrapolation in the pion mass squared.
As given in figure 7, a linear fit seems to be consistent with the current statistics. One can also see that my results are still too high in comparison to the
available experimental results. All the results from extrapolation can be seen in
table 5. In this table one will also find my result times the Sommer parameter
r0 = 0.420 fm[3] in order to obtain a dimensionless result which makes it easier
to compare with the results of other groups, which might use different methods
of scale setting.
Note that the experimental results are the differences between the baryon masses
and the mass of the B meson m(B) = 5279.5(5) MeV. I could equally state the
differences between the baryonmass and the mass of the B∗ meson m(B ∗ ) =
5325.1(5) MeV, which is even smaller. This is because the lattice value for the B
meson I used to compare my results with was also obtained using the static approximation [12]. This means that without the interpolation in the heavy quark
mass, this value can not clearly be identified with the B or the B∗ meson mass.
This might result in a systematic error of the order of m(B ∗ )−m(B) = 45.6 MeV.
jP I
m − m(B) [MeV] Experiment [MeV]
r0 (m − m(B)) exp. name
0+ 0
428(42)
339.2(14)
0.91(9)
Λb
1+ 1
661(37)
525...560
1.41(8)
Σb , Σ∗b
0− 0
1370(98)
−
2.9(2)
?
1− 1
999(68)
−
2.1(1)
?
Table 5: Results from extrapolation in the light quark mass to the physical point in
comparison to experimental results and as dimensionless quantity
26
7 SUMMARY, CONCLUSION AND OUTLOOK
7
Summary, Conclusion and Outlook
7.1
Summary and conclusion
At the beginning I showed how a baryon can be created in QCD by writing
down the baryon creation operator. I calculated the quantum numbers of the
operator and explained how they are modified when using twisted mass lattice
QCD. I pointed out that due to the twisted mass parity and isospin breaking, it
is necessary to compute correlation matrices instead of correlation functions. I
also explained how to handle the correlation matrices and how to interpret their
results.
I extracted the masses of four QCD states for different light quark masses and
performed an extrapolation in the light quark mass to the physical point.
When comparing my results for the Λb and the Σb with the available experimental
results I came to the conclusion that the masses I extracted are around 100 MeV
higher. This might be due to the following reasons:
• It is not clear whether the linear fit performed in the light quark mass is appropriate for the physical light quark region. Therefore, chiral perturbation
theory for static-light baryons might be more appropriate.
• One also has to check to what extent the static limit or lattice artifacts
influence the results.
• Another point which can cause the strong discrepancy to the experiment
is the scale setting. The scale of the gauge configurations I used was set
by the ETMC using the mass and the decay constant of the pion. Nevertheless, this scale corresponds to r0 = 0.420 fm, while the scale setting of
other groups, who use e.g. Ω− baryons to set their scale, corresponds to
r0 = 0.47 − 0.50 fm. When comparing my dimensionless results to results
of another lattice group whose scale setting correspond to r0 = 0.49 fm, as
done in table 6, one can see that the results agree within errors.
State r0 (m − m(B)) my results r0 (m − m(B)) [Burch et al.] [9]
Λb
0.91(9)
0.89(14)
Σb
1.41(8)
1.38(14)
Table 6: Comparison of my dimensionless result with the results of the lattice QCD
group in Regensburg
7 SUMMARY, CONCLUSION AND OUTLOOK
7.2
27
Outlook
With better statistics, which will be available in the near future, excited states
can be identified and their mass can be measured. Furthermore, better statistics
and more values of µ could tell us, whether a linear fit in m2π is able to suitably
describe the chiral extrapolation in the light quark mass. Here, it could also be
useful to develop a suitable chiral perturbation theory for static-light baryons.
The ETMC already provides an ensemble of gauge configurations with lighter
quark masses on a 323 × 64 lattice. Yet, running contractions on this lattice is
beyond the scope of this thesis.
Moreover, a continuum limit has to be performed by choosing gauge configuration
with smaller lattice spacings.
In order to check for the systematic errors when choosing an infinitely heavy quark
and to observe the spin splitting, which is caused by the bottom quark and can
be observed in the experiment, one can perform an interpolation in the heavy
quark mass. For this purpose, one could use the experimental masses for the
corresponding charmed baryons in a first step and perform a linear interpolation
in 1/mQ .
Instead of only considering up and down quarks, it is possible to use a partially
quenched action in order to compute a propagator of a strange quark. Thus
one can extract masses of the Ωb , the Ξb and other states. This is currently in
progress [13], but not part of this thesis.
8 BIBLIOGRAPHY
8
28
Bibliography
References
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[arXiv:0707.4093 [hep-lat]].
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[3] R. Baron et al. [ETM Collaboration], “Light Meson Physics from Maximally
Twisted Mass Lattice QCD,” arXiv:0911.5061 [hep-lat].
[4] B. Blossier, G. von Hippel, T. Mendes, R. Sommer and M. Della Morte,
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[9] T. Burch, C. Hagen, C. B. Lang, M. Limmer and A. Schafer, “Excitations
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REFERENCES
29
[13] M. Wagner, C. Wiese, “The spectrum of static-light baryons in twisted mass
lattice QCD,” to be published.
REFERENCES
30
Appendix
Examplary calculation for relation of the matrix elements of the correlation matrix using the symmetries of the twisted mass action
Here: γ5 -hermiticity for C̃(t)γ0 γ5 ,γ0
=
=
=
=
=
=
⇒
C̃γ0 γ5 ,γ0 (t)
D
E
χu −1 be
ǫabc ǫdef U ad (t, 0)(γ2 γ5 )BC [(D χd )−1 ]cf
(t,
0)(γ
)
[(D
)
]
(t,
0)
2 FE
BE
CF
D
E
cf
ad
χu −1 †
χd −1 †
be
abc def
U (t, 0)(γ2 γ5 )BC (γ5[(D ) ] γ5 )CF (0, t)(γ2)F E [γ5 ((D ) ] γ5 )BE (0, t)
ǫ ǫ
D
E
χd −1 † be
−ǫabc ǫdef U ad (t, 0)(γ5γ2 )BC ([(D χu )−1 ]† )cf
(0,
t)(γ
)
([(D
)
]
)
(0,
t)
2 FE
BE
CF
D
E†
(0,
t)
ǫabc ǫdef U ad (t, 0)(γ2 γ5 )BC [(D χu )−1 ]fFcC (0, t)(γ2 )F E [(D χd )−1 ]eb
EB
D
E†
χu −1 f c
(0,
t)(γ
γ
)
[(D
)
]
(0,
t)
ǫabc ǫdef U da (0, t)(γ2 )F E [(D χd )−1 ]eb
2 5 BC
EB
FC
D
E†
χu −1 be
ǫabc ǫdef U ad (−t, 0)(γ2 )BC [(D χd )−1 ]cf
(−t,
0)(γ
γ
)
[(D
)
]
(−t,
0)
2
5
F
E
BE
CF
†
C̃γ0 γ5 ,γ0 (t) = C̃γ0 ,γ0 γ5 (−t)
From line 2 to line 3 γ5 -hermiticity has been used.
REFERENCES
31
Selbstständigkeitserklärung
Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig verfasst und
keine anderen als die angegebenden Quellen und Hilfsmittel verwendet habe.
Berlin, den 21.07.2010
Christian Wiese