Remarks on the U(1) axial symmetry in QCD at zero and non-zero temperature

Abstract

This paper is organized in two parts. The first part (Sections 2-5) is dedicated to the theory at T=0 and contains a pedagogical review of some fundamental aspects related with the chiral symmetries of QCD, the U(1) problem and its solution proposed by 'tHooft, Witten and Veneziano. In the second part (Sections 6-14) we discuss the role of the U(1) axial symmetry for the phase structure of QCD at finite temperature. One expects that, above a certain critical temperature, also the U(1) axial symmetry will be restored. We will try to see if this transition has (or has not) anything to do with the usual chiral transition: various possible scenarios are discussed. In particular, we analyse a scenario in which the U(1) axial symmetry is still broken above the chiral transition. We will show that this scenario can be consistently reproduced in the full respect of the relevant QCD Ward Identities and also using an effective Lagrangian model. A new order parameter is introduced for the U(1) axial symmetry.

Full text

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arXiv:hep-ph/0206236v1 24 Jun 2002
IFUP–TH/2002–24
June 2002
Remarks on the U(1) axial symmetry in QCD
at zero and non–zero temperature
Enrico Meggiolaro
Dipartimento di Fisica,
Universit` a di Pisa,
Via Buonarroti 2,
I–56127 Pisa, Italy.
Abstract
This paper is organized in two parts. The first part (Sections 2–5) is dedicated to the
theory at T = 0 and contains a pedagogical review of some fundamental aspects related
with the chiral symmetries of QCD, the U(1) problem and its solution proposed by ’tHooft,
Witten and Veneziano. In the second part (Sections 6–14) we discuss the role of the U(1)
axial symmetry for the phase structure of QCD at finite temperature. One expects that,
above a certain critical temperature, also the U(1) axial symmetry will be restored. We
will try to see if this transition has (or has not) anything to do with the usual chiral
transition: various possible scenarios are discussed. In particular, we analyse a scenario
in which the U(1) axial symmetry is still broken above the chiral transition. We will show
that this scenario can be consistently reproduced in the full respect of the relevant QCD
Ward Identities and also using an effective Lagrangian model. A new order parameter is
introduced for the U(1) axial symmetry.

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1. Introduction
It is generally believed that a phase transition which occurs in QCD at a finite temper-
ature is the restoration of the spontaneously broken SU(L) ⊗ SU(L) chiral symmetry in
association with L massless quarks. At zero temperature the chiral symmetry is broken
spontaneously by the condensation of q¯ q pairs and the L2− 1 JP= 0−mesons are just
the Nambu–Goldstone (NG) bosons associated with this breaking [1, 2, 3, 4, 5]. At high
temperatures the thermal energy breaks up the q¯ q condensate, leading to the restoration
of chiral symmetry. We expect that this property not only holds for massless quarks but
also continues for a small mass region. The order parameter for the chiral symmetry
breaking is apparently ?¯ qq? ≡
the non–vanishing of ?¯ qq? in the chiral limit sup(mi) → 0. From lattice determinations of
the chiral order parameter ?¯ qq? one knows that the SU(L)⊗SU(L) chiral phase transition
temperature Tch, defined as the temperature at which the chiral condensate ?¯ qq? goes to
zero (in the chiral limit sup(mi) → 0), is nearly equal to the deconfining temperature Tc
(see, e.g., Ref. [6]). But this is not the whole story: QCD possesses not only an approxi-
mate SU(L) ⊗ SU(L) chiral symmetry, for L light quark flavours, but also a U(1) axial
symmetry (at least at the classical level) [7, 8]. The role of the U(1) symmetry for the
finite temperature phase structure has been so far not well studied and it is still an open
question of hadronic physics whether the fate of the U(1) chiral symmetry of QCD has
or has not something to do with the fate of the SU(L) ⊗ SU(L) chiral symmetry. In the
following Sections we will try to answer these questions:
?L
i=1?¯ qiqi?: the chiral symmetry breaking corresponds to
• At which temperature is the U(1) axial symmetry restored?
temperature does exist!)
(if such a critical
• Does this temperature coincide with the deconfinement temperature and with the
temperature at which the SU(L) chiral symmetry is restored?
In the “Witten–Veneziano mechanism” [9, 10] for the resolution of the U(1) problem, a
fundamental role is played by the so–called “topological susceptibility” in a QCD without
quarks, i.e., in a pure Yang–Mills (YM) theory, in the large–Nclimit (Ncbeing the number
of colours):
?
A = lim
k→0lim
Nc→∞
−i
?
d4xeikx?TQ(x)Q(0)?
?
, (1.1)
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where Q(x) =
enters into the expression for the squared mass of the η′: m2
of light quark flavours taken into account in the chiral limit. Therefore, in order to study
the role of the U(1) axial symmetry for the full theory at non–zero temperatures, one
should consider the YM topological susceptibility A(T) at a given temperature T, formally
defined as a large–Nclimit of a certain expectation value ?...?T in the full theory at the
temperature T [11]:
g2
64π2εµνρσFa
µνFa
ρσis the so–called “topological charge density”. This quantity
η′ =2LA
F2
π, where L is the number
A(T) = lim
k→0lim
Nc→∞
?
−i
?
d4xeikx?TQ(x)Q(0)?T
?
. (1.2)
In other words, A(T) = lim
k→0A0(k,T) = A0(0,T), where the quantity
A0(k,T) = lim
Nc→∞
?
−i
?
d4xeikx?TQ(x)Q(0)?T
?
, (1.3)
at four–momentum k and physical temperature T, is nothing but the leading–order term
in the 1/Ncexpansion of the corresponding quantity
χ(k,T) = −i
?
d4xeikx?TQ(x)Q(0)?T, (1.4)
of the full theory, at the same four–momentum k and physical temperature T. [From
the general theory of 1/Ncexpansion it is known that A0(k,T) is of order O(1), while
successive terms coming from including fermions, are of order O(1/Nc).] It is in this
sense that one can talk about the behaviour of A(T) above or below the chiral transition
temperature Tch.
The problem of studying the behaviour of A(T) as a function of the temperature T
was first addressed, in lattice QCD, in Refs. [12, 13, 14]. Recent lattice results [15]
(obtained for the SU(3) pure–gauge theory) show that the YM topological susceptibility
A(T) is approximately constant up to the critical temperature Tc≃ Tch, it has a sharp
decrease above the transition, but it remains different from zero up to ∼ 1.2 Tc. In the
Witten–Veneziano mechanism [9, 10], a (no matter how small!) value different from zero
for A is related to the breaking of the U(1) axial symmetry, since it implies the existence
of a pseudo–Goldstone particle with the same quantum numbers of the η′(see also Ref.
[16]). Therefore, the available lattice results for the topological susceptibility show that
the U(1) chiral symmetry is restored at a temperature TU(1)greater than Tch.
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Another way to address the same question is to look at the behaviour at non–zero
temperatures of the susceptibilities related to the propagators for the following meson
channels [17] (we consider for simplicity the case of L = 2 light flavours): the isoscalar
I = 0 scalar channel σ (also known as f0in the modern language of hadron spectroscopy),
interpolated by the operator Oσ= ¯ qq; the isovector I = 1 scalar channel δ (also known
as a0), interpolated by the operator?Oδ= ¯ q? τ
π, interpolated by the operator?Oπ= i¯ qγ5? τ
η′, interpolated by the operator Oη′ = i¯ qγ5q. Under SU(2)Atransformations, σ is mixed
with π: thus the restoration of this symmetry at Tch requires identical correlators for
these two channels. Another SU(2) chiral multiplet is (δ,η′). On the contrary, under
the U(1)Atransformations, π is mixed with δ: so, a “practical restoration” of the U(1)
axial symmetry should imply that these two channels become degenerate, with identical
correlators. Another U(1) chiral multiplet is (σ,η′). (Clearly, if both chiral symmetries
are restored, then all π, η′, σ and δ correlators should become the same.) In practice, one
can construct, for each meson channel f, the corresponding chiral susceptibility
2q; the isovector I = 1 pseudoscalar channel
2q; the isoscalar I = 0 pseudoscalar channel
χf=
?
d4x?Of(x)O†
f(0)?, (1.5)
and then define two order parameters: χSU(2)⊗SU(2) ≡ χσ− χπ, and χU(1) ≡ χδ− χπ.
If an order parameter is non–zero in the chiral limit, then the corresponding symmetry
is broken. Present lattice data for these quantities seem to indicate that the U(1) axial
symmetry is still broken above Tch, up to ∼ 1.2 Tch, where the δ–π splitting is small but
still different from zero [18, 19, 20]. In terms of the left–handed and right–handed quark
fields [qL,R≡1
correlators for the δ+and π+channels:
2(1 ± γ5)q], one has the following expression for the difference between the
DU(1)(x) ≡ ?Oδ+(x)O†
= 2
?¯ uRdL(x) ·¯dRuL(0)? + ?¯ uLdR(x) ·¯dLuR(0)?
δ+(0)? − ?Oπ+(x)O†
π+(0)?
??
.(1.6)
(The integral of this quantity,
χU(1)= χδ− χπ.) What happens below and above Tch? Below Tch, in the chiral limit
sup(mi) → 0, the left–handed and right–handed components of a given light quark flavour
(up or down, in our case with L = 2) can be connected through the quark condensate,
giving rise to a non–zero contribution to the quantity DU(1)(x) in Eq. (1.6) (i.e., to the
quantity χU(1)). But above Tchthe quark condensate is zero: so, how can the quantity
?d4xDU(1)(x), is just equal to the U(1) chiral parameter
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DU(1)(x) (i.e., the quantity χU(1)) be different from zero also above Tch, as indicated by
present lattice data? The only possibility in order to solve this puzzle seems to be that
of requiring the existence of a genuine four–fermion local condensate, which is an order
parameter for the U(1) axial symmetry and which remains different from zero also above
Tch. This will be discussed in Section 7.
The paper is organized in two parts. The first part (Sections 2–5) is dedicated to
the theory at T = 0 and contains a pedagogical review of some fundamental aspects
related with the chiral symmetries of QCD (Section 2), the U(1) problem and its solution
proposed by ’tHooft (Section 3), the Witten’s mechanism (Section 4) and the Veneziano’s
mechanism (Section 5). In the second part (Sections 6–14) we discuss the role of the U(1)
axial symmetry for the phase structure of QCD at finite temperature. One expects that,
above a certain critical temperature TU(1), also the U(1) axial symmetry will be restored.
We will try to see if this transition has (or has not) anything to do with the usual chiral
transition: various possible scenarios are discussed in Section 6. In particular, we analyse
a scenario in which the U(1) axial symmetry is still broken above the chiral transition
[21, 22, 23]. In Section 7 a new order parameter is introduced for the U(1) axial symmetry.
In Section 8 we make an analysis of the relevant QCD Ward Identities (WI) in the range of
temperatures between Tchand TU(1). The saturation of the relevant QCD Ward Identities
is obtained in a 1/Ncexpansion. A new 2L–fermion effective vertex is introduced to take
care of the new order parameter for the U(1) chiral symmetry alone. In Sections 8 and
9 we introduce a new meson field associated with the new U(1) chiral order parameter
and then, in Section 9, we study the form of a new effective Lagrangian including, in
addition to the usual quark–anti-quark meson fields Uij, also the new 2L–fermion field X.
In Sections 10 and 11 we study the mass spectrum of the theory respectively above and
below the SU(L) ⊗ SU(L) chiral phase transition. In Section 12 we apply the results of
Section 11 to the “real–world” case (at T < Tch), where there are L = 3 light flavours,
named u, d and s: we derive a generalized Witten–Veneziano formula for the η′mass.
In Section 13 we show that the above results are “model–independent”, i.e., independent
on additional terms in the Lagrangian. In Section 14 we analyse the consequences of our
theoretical model on the chiral condensate ?¯ qq? and on the topological susceptibility χ,
in the full theory with quarks. In particular we are interested in their dependence on the
light quark masses (something about this is also said in the Appendix). In Section 14 we
also discuss the Nc–dependence for some relevant quantities that we have found. Finally,
the conclusions and an outlook are given in Section 15.
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