Axions - Theory
Roberto Peccei
UCLA
Helen Quinn Symposium
SLAC April 16, 2010
Axions - Theory
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Recollections
The U(1)A Problem of QCD
The QCD Vacuum
The Strong CP Problem
Solutions to the Strong CP Problem
U(1)PQ and Axions
Axion Dynamics
Invisible Axion Models
Galactic Hints for Axions
Concluding Remarks
Recollections
• The 1976-77 academic year was a special
year at Stanford, enlivened by the presence
of Steve Weinberg in the Department and
Gerard ‘t Hooft at SLAC
• For me, the highlight was the presence of
Helen Quinn, as a visitor from Harvard to the
Institute of Theoretical Physics in the
Department
• We worked happily together on the then hot
topic of instantons, whose consequences we
labored hard to understand
• In this period we wrote 3 papers together at
the end of January, March, and May 1977
(Two of which are well known, and a third
which should be!)
• Our first joint paper [Some Aspects of
Instantons, Nuovo Cimento 31A, 307 (1977) ]
is how we really learned about instantons
-how these Euclidean solutions impact
gauge theories in Minkowski space
- the bearing that they have on the validity of
perturbation theory
- their relation to the violation of quantum
numbers like B+L, broken by anomalies
• Our two better known papers [CP Conservation
in the Presence of Pseudoparticles, Phys. Rev.
Lett. 38, 1440 (1977); Constraints Imposed by
CP Conservation in the Presence of
Pseudoparticles, Phys. Rev. D16, 1791 (1977)]
introduce a global U(1) symmetry to explain
why CP is conserved in the strong interactions
• Steve Weinberg’s persistence in asking
questions was a great motivator for the U(1)PQ
solution
• Remarkably, neither Helen nor I, or Steve (or
the referees of our papers) realized that the
U(1)PQ solution implied the existence of a nearly
massless particle, the axion!
• Still remember clearly the sinking feeling I got in
late summer 1977 when Helen told me that
Weinberg (by then in Texas) called her to tell us
that there was a Goldstone Boson associated to
our “solution”.
• Shortly thereafter joint papers by Weinberg and
Wilczek appeared in which they discussed how
U(1)PQ leads to axions and how the axions get a
small mass, giving also some suggestions on
how to search for such a light, very long-lived
particle
• In this talk in honor of Helen I will discuss the
theoretical basis for axions and argue that they
indeed must exist!
The U(1)A Problem of QCD
• In the 1970’s the strong interactions had a
puzzling problem, which became particularly
clear with the development of QCD.
• The QCD Lagrangian for N flavors
LQCD = -1/4Fa Fa- Σfqf (-iD + mf) qf
in the limit mf → 0 has a large global symmetry:
U (N)Vx U (N)A
qf → [e iaTa/2]ff’ qf’ ; qf → [e iaTa5/2]ff’ qf’
Vector
Axial
• Since mu, md << ΛQCD, for these quarks mf → 0
limit is sensible. Thus expect strong interactions
to be approximately U (2)Vx U (2)A invariant.
• Indeed, experimentally know that
U(2)V = SU(2)V x U(1)V≡ Isospin x Baryon #
is a good approximate symmetry of nature
 (p, n) and (, °) multiplets in spectrum
• For axial symmetries, however, things are
different. Dynamically, quark condensates
 uu  dd  0 form and break U(2)A down
spontaneously and no mixed parity multiplets
• However, because U(2)A is a spontaneously
broken symmetry, expect appearance in the
spectrum of approximate Nambu-Goldstone
bosons, with m  0 [ m 0 as mu, md  0 ]
• For U(2)A would expect 4 such bosons (, ).
Although pions are light, m  0, see no sign of
another light state in the hadronic spectrum,
since m2>> m2 .
• Weinberg dubbed this the U(1)A problem and
suggested that, somehow, there was no U(1)A
symmetry in the strong interactions
• In the language of Chiral Perturbation Theory,
the QCD dynamics for the (, )- sector needs
to be augmented by an additional term which
breaks explicitly U(1)A, beyond the breaking
term induced by the quark mass terms.
• Defining  = exp i/F [aa +] and including a
symmetry breaking pion mass m2 ~ (mu+ md)
one has:
Leff = -¼F2 Tr  † + ¼F2 m2 Tr ( + † )
- ½M2o 2
• Provided M2o >> m2 this allows m2>> m2 , but
what is the origin of this last term?
The QCD Vacuum
• The resolution of the U(1)A problem came
through the realization that the QCD vacuum
is more complicated [‘t Hooft].
• This complexity, in effect, is what makes U(1)A
not a symmetry of QCD, even though it is an
apparent symmetry of LQCD in the limit mf → 0
• However, this more complicated vacuum gives
rise to the strong CP problem. In essence, as
we shall see, the question becomes why is CP
not very badly broken in QCD ?
• A possible resolution of the U(1)A problem
seems to be provided by the chiral anomaly
for axial currents [Adler Bell Jackiw]
• The divergence of axial currents, get quantum
corrections from the triangle graph
Aa
J5
Ab
with fermions going around the loop
• This anomaly gives 2a non-zero divergence
  J 5 
g N
 ~
F
Fa
a
2
32
~ 
Fa  1 / 2   F ,
where
even in symmetry limit
• Hence, in the mf → 0 limit, although formally
QCD is invariant under a U(1)A transformation
qf ->ei/25qf
the chiral anomaly affects the action
2
g
N
4
μ
4
μν ~
δW  α  d x  μ J 5  α
d xFa Faμν
2 
32π
• However, matters are not that simple!
• This is because the pseudoscalar density
entering in the anomaly is, in fact, a total
divergence [Bardeen]:
μν
a
F
~
Faμν   μ K μ
where
K= Aa [Fa -g/3 fabc Ab Ac]
• This makes W a pure surface integral
W= g2N/322  dK
Hence, using the naïve boundary condition
Aa =0 at    dK = 0
U(1)A appears to be a symmetry again!
• What ‘t Hooft showed, however, is that the
correct boundary condition to use is that
Aa  be a pure gauge at 
i.e. either Aa =0 or gauge transformation of 0
• It turns out that, with these B. C., there are
gauge configurations for which
 dK  0
and thus U(1)A is not a symmetry of QCD
• This is most easily understood for SU(2) QCD
and in Aoa=0 gauge [Callan Dashen Gross]. In
this case one has only spatial gauge fields Aia
• Under a gauge transformation the Aia gauge
fields transform as:
½aAia ≡Ai Ai -1 + i/g i -1
Thus vacuum configurations are either 0 or
have the form i/g i -1
• In the Aoa=0 gauge can further classify vacuum
configurations by how  goes to unity as r 
n  e i2n as r   [n=0, 1, 2,…]
• The winding number n is related to the
Jacobian of an S3 S3 map and is given by
ig 3
n
24π 2
3
i
j
k
d
rT
r
ε
A
A
A
ijk
n
n
n

• This expression is closely related to the
Bardeen current K. Indeed, in the Aoa=0 gauge
only K0≠0 and one finds for pure gauge fields:
K0=-g/3ijkabc Aia Ajb Akc =4/3ig ijkTr Ai Aj Ak
• The true vacuum is superposition of these, socalled, n-vacua and is called the -vacuum:
|> =  e -in |n>
• Easy to see that in vacuum to vacuum
transitions there are transitions with  dK  0
n|t= + - n|t= - = g2/322  dK |t=+ t= -
= g2/322  d3r K0 |t=+ t= -
• Pictorially, one has
_ _ _ _ _ _ _ _ _ t =+
-3
--2
-1
0
1
2
3
4
g2/322  dK =0
g2/322 dK =2
_ _ _ _ _ _ _ _ _ _ t = -
-4
•
-3
--2
-1
0
1
2
3
4
In detail one can write for the vacuum to
vacuum transition amplitude
im e -in <m|n> =  ei 
<|>
=e
+
+

n +<n+|n>-
• Here the difference in winding numbers  is
given by
g2
 
32 2
 d  K
 t 
t 
g2

32 2
d
4

xFa
~
Fa
• Using the usual path integral representation
for +<|>- one sees that
2
g
iSeff [ A]
4
 ~
 ( 
d xFa Fa )
   |       Ae
2 
32

which allows one to re-interpret the  term as
an addition to the usual QCD action
S eff  SQCD
g2

32 2
d
4

xFa
~
Fa
• Because one cannot neglect the UA(1)
anomaly in  ≠ 0 sectors, the topological
charge density
Q = g F  F~  ;  =  d4x Q
32
in effect, acts as a dynamical parameter.
• Thus [Di Vecchia Veneziano] Q should also
be added to the Chiral Lagrangian describing
the low energy behavior of QCD:
Leff = ¼F2 Tr  † + ¼F2 m2 Tr ( + † )
½ i Q Tr [ln -ln † ] + [1/ F2 M2o] Q2 +…
2
2
a
a
• The 3rd term in Leff is included to take into
account the anomaly in the UA(1) current, while
the 4th term is the lowest order term in a
polynomial in Q2
• Q is essentially a background field and can be
eliminated through its equation of motion:
Q = -i/4 [F2 M2o] Tr [ln -ln † ] = ½ [F M2o]  +...
• Hence the last two terms in Leff reduce,
effectively, to:
½ i Q Tr [ln -ln † ] + [1/ F2 M2o] Q2 →-½M2o 2 +
which serve to provide an additional gluonic
mass term for the  meson, solving the UA(1)
problem
This emerges
more directly
from lattice
QCD
calculations,
which show
that, indeed,
m → constant
as m → 0
The Strong CP Problem
• The resolution of the U(1)A problem, by
recognizing the complicated nature of QCD’s
vacuum, however, engenders another problem:
the strong CP problem
• As we saw, effectively, the QCD vacuum
structure adds and extra term to LQCD
2
g
μν ~
Lθ  θ
Fa Faμν
2
32π
• This term violates P and T, but conserves C,
and thus can produce a neutron electric dipole
moment of order dn  e mq/Mn2 
• The strong bound on the neutron electric dipole
moment dn<1.1 x 10-26 ecm requires the angle 
to be very small  < 10-9 - 10-10 [Baluni;
Crewther Di Vecchia Veneziano Witten]
• Why  should be this small is the strong CP
problem
• Problem is actually worse if one considers the
effect of chiral transformations on the -vacuum
• One can show that chiral transformations,
because of the anomaly, change the -vacuum
[Jackiw Rebbi ]:
eiQ5 |  > = |  +  >
• The proof is a bit involved but worth going thru
• The gauge matrices n associated with the n
vacua can be obtained by compounding:
n =[1]n, where 1|1>=|2>. It follows thus that
on an n-vacuum state
1|n>=|n+1>
• Hence n-vacua, as expected, are not gauge
invariant, but the -vacuum is:
1|>=  e-in 1 |n>=  e-in |n +1>= ei | >
• In a theory with N massless quarks there is a
conserved but gauge variant chiral current
Jc5 = J5 -g2N/322 K
• As a result, the associated time independent
chiral charge Qc5 = d3x Jc5o shifts under
gauge transformations which change the nvacua
1 Qc5 1-1 = Qc5 + N
• Consider
1e i/N Qc5|> = 1e i/N Qc5 1-1 1 |>
= e i( + ) e i/N Qc5|>
which shows that a chiral rotation indeed
changes the -vacuum [Jackiw Rebbi]
e i/N Qc5|> = | +  >
• If besides QCD one includes the weak
interactions, in general the quark mass matrix is
non-diagonal and complex
LMass = -qiR Mij qjL + h. c.
To diagonalize M one must, among other
things, perform a chiral transformation by an
angle of Arg det M which, as a result of the
Jackiw Rebbi result, changes  into
total =  + Arg det M
• Thus, in full generality, the strong CP problem
can be stated as:
Why is the angle total , coming from the strong
and weak interactions, so small? [Weinberg’s
persistent question to Helen and me!]
Solutions to the Strong CP Problem
•
Thirty years later, there are only three
possible “solutions” to the strong CP problem:
i. Anthropically total is small
ii. CP is broken spontaneously and the induced
total is small
iii. A chiral symmetry drives total → 0
• In my opinion, only iii. is a viable solution and
it necessitates introducing in the Standard
Model a new global, spontaneously broken,
chiral symmetry U(1)PQ
i. Anthropic solution
• It is, of course, possible that, as a result of
some anthropic reasons
total =  + Arg det M
just turns out to be of O(10-10). There are, after
all, such small ratios in the SM [e.g. me/mt~10-6]
• What make me doubt this “explanation” is that
the physics of the QCD vacuum (and hence )
and that of the quark mass matrix (Arg det M)
seem totally unrelated. So, why should total be
a small CP violating phase?
ii Spontaneously broken CP solution
• This second possibility is more interesting. In
fact, if CP were a symmetry of nature which is
then spontaneously broken, one can set =0
at the Lagrangian level.
• However, since CP must be spontaneously
broken,  gets induced back at the loop-level.
[Beg Tsao; Georgi; Mohapatra Senjanovic ]
• But to get  < 10-9 one needs, in general, also
to insure that 1-loop=0. This is not easy and,
furthermore, there are other problems
associated with spontaneously broken CP
violation
• Theories with spontaneously broken CP need
complex Higgs VEVs and, as Zeldovich,
Kobzarev and Okun pointed out, these give rise
to different CP domains in the Universe
• These different CP domains in the Universe are
separated by walls which have substantial
energy density and, being 2-dimensional,
dissipate slowly as the Universe cools wall ~ T
• Indeed, if these domains existed, the energy
density in the walls would badly overclose the
Universe now, unless the scale of spontaneous
CP violation was greater than Tinflation~ 1010 GeV
• Theories where CP is violated at high scales
and which have 1-loop=0 exist [Nelson, Barr],
but they are recondite and difficult to reconcile
with experiment
• Typically, the CP violating phases generated at
high scales induce small phases at low energy:
eff ~ [Mw/MHS]n 
and this is not what one sees experimentally.
• All experimental data is in excellent agreement
with the CKM Model– a model where CP is
explicitly, not spontaneously, broken and where
the CP violating phases are of O(1)
iii. A chiral symmetry drives total → 0
• This is a very natural solution to the strong CP
problem since it, effectively, rotates  -vacua
away
e-i Q5 |  > = | 0 >
• Two suggestions has been put forth for this
chiral symmetry:
i. The u-quark has no mass mu = 0 [Kaplan
Manohar]
ii. The SM has an additional global U(1) chiral
symmetry [Peccei Quinn ]
• Want to argue that only PQ solution is tenable
•
•
The “ solution” mu = 0 is disfavored by current
algebra analysis [ Leutwyler]. Furthermore, it
is difficult to understand why
Arg det M = 0
What is the origin of this chiral symmetry?
The ratio of quark masses is computable in
Chiral Perturbation Theory. Correcting for
electromagnetic effects, one arrives at leading
order to the famous Weinberg formula
This result is
confirmed by
calculations
on the
Lattice
Theoretical
predictions
summarized
in the Figure
Leutwyler
mu0
MILC Collaboration rules out mu=0
at 10
U(1)PQ and Axions
• Introducing a global U(1)PQ symmetry, which is
necessarily spontaneously broken, replaces:
total =  + Arg det M  a(x) / fa
Static CP viol. Angle
Dynamical CP cons.
Axion field
• The axion is the Goldstone boson of the broken
[Weinberg Wilczek] U(1)PQ symmetry and fa is
scale of the breaking. Hence under U(1)PQ
a(x)  a(x)   fa
• Formally, for U(1)PQ invariance the Lagrangian
of SM is augmented by axion interactions:
g2
μν ~
μ
1
L  LSM  θ total
F
F


a μ a
a
aμν
2
2
32π
2
a
g
μν ~
 Lint [ μ a / f a ; ψ ] 
ξ
F
Faμν
a
2
fa
32π
• Last term needed to give chiral anomaly of JPQ
μ
 μ J PQ
g2
μν ~
ξ
F
a Faμν
2
32π
and acts as an effective potential for axion field
• Minimum of potential occurs at <a>=-fa/ total
Veff
a
ξ g2
μν ~

Fa Faμν
2
f a 32π
a
0
• Easy to understand the physics of PQ solution.
If one neglects the effects of QCD then U(1)PQ
symmetry allows any value for <a>:
0≤ <a> ≤ 2
• Including the effects of the QCD anomaly
generates a potential for the axion field which is
periodic in the effective vacuum angle
Veff ~ cos[total + <a>/fa ]
• Minimizing this potential with respect to <a>
gives the PQ solution
<a>=-fa/ total
Much simpler proof than in original PQ paper!
• Hence theory written in terms of
aphys= a- <a>
has no longer a -term [ this is the PQ solution]
• Furthermore, expanding Veff at minimum gives
the axion a mass [anomaly gives the NGB a
mass]
V
ξ g

~
m 
2
a
2
a
eff
2

2
f a 32π 2 a
Faμν Faμν
a
• Calculation of axion mass first done explicitly
by current algebra techniques [Bardeen Tye],
but an effective Lagrangian derivation
[Bardeen Peccei Yanagida] is easier and also
readily gives axion couplings to matter
Axion Dynamics
• In the original Peccei Quinn model, the U(1)PQ
symmetry breakdown coincided with that of
electroweak breaking fa = vF, with vF  250 GeV.
• However, this is not necessary. If fa >> vF then
axion is very light, very weakly coupled and
very long lived [invisible axion models]
• Useful to derive first properties of weak-scale
axions and then generalize the discussion to
invisible axions
• To make SM U(1)PQ invariant must introduce 2
Higgs fields to absorb independent chiral
transformations of u- and d-quarks (and
leptons)
• Yukawa interactions in SM involve Higgs
LYukawa  ijuQLi 1uRj  ijd QLi 2d Rj  ij LLi 2 Rj  h.c.
• Defining x=v2/v1 and vF= √(v12 + v22), the axion
is the common phase field in 1 and 2 which
is orthogonal to the weak hypercharge
1
1
iax / v F 1
ia / xv F 0
1 
v1e
v 2e
0 ;  2 
1
2
2
 
 
• See that LYukawa is invariant under the U(1)PQ
transformation
a  a   vF ; uRi e-ix uRi ; dRi lRi e-i/x dRi lRi
• Let us focus on the quark pieces. The current
JPQ=-vF ∂ a + x Σi uiR  uiR + 1/x Σi diR  diR
identifies the strong anomaly coefficient  as:
 =Ng(x +1/x)
• To compute the axion mass and mixings from
an effective chiral Lagrangian we need to
separate out light u- and d-quarks from rest.
• For these purposes introduce, as before, a 2x2
matrix of NG fields
Σ = exp[ i(. +)/F ]
and the U(2)VxU(2)A invariant eff. Lagrangian
Lchiral =-F2/4 Tr ∂ † ∂ 
• To Lchiral must add U(2)VxU(2)A breaking terms
which mimic the U(1)PQ invariant Yukawa
interactions of the u- and d-quarks.
• This is accomplished by adding
Lmass=½(F m )2 Tr[ΣAM+(ΣAM)†]
 m

where
0
e
A
 0
ixa / v F
u
m  m

u
d

;
M


e ia / xv F 

0

0


md 
m u  m d 
and under PQ-transformations
a  a  v F
eix
;   
 0
0 
i / x 
e 
• However, Lmass only gives part of the physics.
Indeed, the quadratic terms in Lmass involving
neutral fields
L2mass=-½ mo2{mu/(mu+md)[+-xf/vFa]2
+ md/(mu+md)[--f/xvFa]2}
give the wrong ratio for m2/ m2
m2/ m2= md/mu 1.6 [ the U(1)A problem!]
and the axion is still massless
• To account for the effect of the anomaly in both
U(1)A and U(1)PQ one must add a further
effective mass term which gives the  the right
mass and produces a mass for the axion
• It is easy to see that such a term has the form
[Bardeen Peccei Yanagida]
Lanomaly=-½ Mo2 [+ {[f/vF] [(Ng-1)(x +1/x)/2]}a ]2
where Mo2  m2>> m2
• Coefficient in front of a in Lanomaly details the
relative strength of the couplings of  and a to
 ~
Fa Fa .Naively, one would imagine
{} = f/vF /2 = f/vF Ng/2(x +1/x)
However, only the contribution of heavy quarks
to the PQ anomaly should be included (hence
Ng (Ng-1)) since light quark interactions of
axions are included already in Lmass
• Diagonalization of the quadratic terms in Lmass
and Lanomaly gives both the axion mass and the
parameters for a- and a – mixing for the PQ
model.
• Convenient to define
mast = mf /vF [mumd/(mu+md)]  25 KeV
• Then can characterize all axion models by 4
parameters { m; 3 ; 0 ; K a } of O(1). To wit:
ma = m mast [vF / fa]
a = 3 [f / fa] ; a = 0 [f / fa]
Laγγ
aphys μν ~
α

K aγγ
F Fμν
4π
fa
• A simple calculation for weak-scale axions,
where fa=vF, gives:
m=Ng(x +1/x)
3=½[(x -1/x)-Ng(x +1/x)(md-mu)/(mu+md)]
0 =½(1-Ng)(x+1/x)
• To compute the coupling K a one must
consider the em anomaly of the PQ current

  J PQ

~
 
F  F
4
and one finds
K a=Ng(x +1/x)[mu/(mu+md)]
Invisible Axion Models
•
•
•
Original PQ model, where fa=vF, was long ago
ruled out by experiment.
For example, one can estimate the branching
ratio [Bardeen Peccei Yanagida]
BR(K+ + +a)  3 x 10 -5 0 2
 3 x 10 -5 (x+1/x)2
which is well above the KEK bound
BR(K+ + +nothing) <3.8 x 10 -8
However, invisible axion models, where
fa>>vF, are still viable
•
•
These invisible axion models introduce fields
which carry PQ charge but are SU(2)XU(1)
singlets
Two types of models have been proposed
i) KSVZ [Kim; Shifman Vainshtein Zakharov]
Only a scalar field  with fa= <> >> vF and a
superheavy quark Q with MQ~fa carry PQ
charge
ii) DFSZ [Dine Fischler Srednicki; Zhitnisky]
Adds to PQ model a scalar field  which
carries PQ charge and fa= <> >> vF
• For these models, one can repeat the
calculations we just did to get the axion mass
and couplings, and one finds:
KSVZ model
m=1; 3=-½(md-mu)/(mu+md); 0 =-½
K a= 3eQ2 –(4md +mu)/3(mu+md)
DFSZ model
m=1 ; 3= 1  X
 X 2 md  mu 


2  2Ng
m d  m u 
1
; 0 = (1-Ng)/2Ng
4 4m d  m u

3 3(m d  m u )
K a=
where X1=2v22/vF2 , X2=2v12/vF2
• Note that since in both the KSVZ model and the
DFSZ model m=1 the axion mass is given by
the formula:
ma = mast [vF / fa] 6.3 [106 GeV / fa] eV
• Although KSVZ and DFSZ axions are very light,
very weakly coupled and very long-lived, they
are not totally invisible
• Astrophysics gives bounds on ma since axion
emission, through e ae and Primakoff
processes causes energy loss ~1/ fa affecting
stellar evolution.
• Other upper bounds on ma come from
SN1987a, since axion emission in core collapse
affects neutrino spectrum
• Typically bounds allow axions lighter than
ma ≤ 1-10-3 eV
• Remarkably, cosmology gives a lower bound on
axion mass (upper bound on fa ) [Preskill Wise
Wilczek; Abbott Sikivie; Dine Fischler]
• Physics is simple to understand. When
Universe goes through PQ phase transition at
T~ fa >>ΛQCD anomaly ineffective and <aphys> is
arbitrary. Eventually, when Universe cools to
T~ΛQCD the axion gets a mass and <aphys>  0.
• Coherent pa=0 axion oscillations towards
minimum contribute to Universe’s energy
density and act as cold dark matter
• Typical calculation of axion contribution to
Universe’s energy density [Fox Pierce Thomas]
Ωah2 =0.5[(fa/)/1012 GeV]7/6[i2 +2]
Here  is coefficient of PQ anomaly, I is initial
misalignment angle and  its mean square
fluctuation and  is a possible dilution factor
• WMAP data provides bound on axion CDM
Ωah2 ≤ 0.12
• For =1, and using for i an average angle i2=
<2>= 2/3 and neglecting fluctuations, WMAP
data gives the following cosmological bound for
the PQ scale:
fa/ < 3 x 1011 GeV or 2.1 x 10-5 eV < ma
Galactic Hints for Axions
• These considerations suggest that there is
an axion window
2x 10-5 eV < ma < 10-3 eV
to explore – which Leslie Rosenberg will
discuss in much greater detail
• Here want to mention only an exciting feature
of axionic CDM, pointed out recently by Pierre
Sikivie, [arXiv:1003.2426] which argues
indirectly for their existence.
• CDM has little velocity dispersion and its
distribution in galactic halos is thought to have
caustics, surfaces in space with large DM
Kinney and Sikivie
• Some evidence for
caustics shown in Figure
• In self-similar infall
model [Fillmore and
Goldreich] expect, for
vrot=220 km/sec, caustics
at
An~ 40kpc / n [Jmax/0.18]
with Jmax related to
angular momentum of
the infalling CDM
• This particular caustic ring structure in halos is
associated with the CDM particles falling in with
net overall rotation, due to tidal torquing from
nearby galaxies , and hence having

 v  0
• This is difficult to understand for WIMP CDM
since gravitational forces always produce
irrotational velocity
fields


dv
  
dt
• Yang and Sikivie argue situation is different for
axions, because cold dark matter axions form a
Bose- Einstein Condensate, and thus are not
collisionless, which results in velocity fields for
the axions which are not irrotational
• Physics of BEC for axions is very nice, but I can
only sketch it here
• Important point is that BEC does indeed form
near the QCD phase transition time when axion
gets a mass
• Gravitational interactions then re-thermalize the
axionic BEC continually, so that most axions
condense into a state of lowest energy, with the
rotational modes of the axion field consistent
with the total angular momentum acquired by
tidal torquing
• This picture reproduces all features of self
similar model which agrees with galactic data
Concluding Remarks
• Galactic hints for axions are very welcome,
because theoretically their existence is really
needed!
• The solution of the UA(1) problem, through the
recognition of the more complex nature of the
QCD vacuum, makes the axial gluonic density
g
~
Q = 32 F  F 
a dynamical parameter and brings into question
why CP is conserved by the strong interactions
since, effectively,
LStrong = LQCD + total Q
with total =  + Arg det M an arbitrary angle
0 ≤ total ≤ 2
2
2
a
a
• The only natural way to understand why
total < 10-9- 10-10
is to imagine, as Helen and I did, that there is
a remnant spontaneously broken global chiral
symmetry that survives to low energy U(1)PQ
• Its associated (quasi) Nambu Goldtone boson,
the axion, renders total a dynamical parameter,
so that total  a(x) / fa and CP is conserved in
the strong interactions
LStrong = LQCD + Q a(x) / fa
AXIONS SHOULD EXIST!