Axions - Theory

Roberto Peccei

UCLA

Helen Quinn Symposium

SLAC April 16, 2010

Axions - Theory• 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- Σfqf (-iD + mf) qf

in the limit mf → 0 has a large global symmetry: U (N)Vx U (N)A

qf → [e iaTa/2]ff’ qf’ ; qf → [e iaTa5/2]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

form and break U(2)A down spontaneously and no mixed parity multiplets

0dduu

• 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 [aa +] 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

Aa

J5

Ab

with fermions going around the loop

• This anomaly gives a non-zero divergence

where , even in symmetry limit

• Hence, in the mf → 0 limit, although formally QCD is invariant under a U(1)A transformation

qf ->ei/25qf

the chiral anomaly affects the action

• However, matters are not that simple!

μνaμνa

μμ FxFd

π

NgαJxdαWδ

~

324

2

2

54

aa FFNg

J~

32 2

2

5

FFa 2/1

~

• This is because the pseudoscalar density entering in the anomaly is, in fact, a total divergence [Bardeen]:

where

K= Aaa [Fa -g/3 fabc Ab Ac]

• This makes W a pure surface integral

W= g2N/322 dK

Hence, using the naïve boundary condition

Aa =0 at dK = 0

U(1)A appears to be a symmetry again!

μμμνa

μνa KFF ~

• 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

dK 0

and thus U(1)A is not a symmetry of QCD

• This is most easily understood for SU(2) QCD and in Ao

a=0 gauge [Callan Dashen Gross]. In this case one has only spatial gauge fields Ai

a

• Under a gauge transformation the Aia gauge

fields transform as:

½aAia ≡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 i2n as r [n=0, 1, 2,…]

• The winding number n is related to the Jacobian of an S3 S3 map and is given by

kn

jn

inijk

32

3

AAArTrεd24π

ign

• This expression is closely related to the Bardeen current K. Indeed, in the Ao

a=0 gauge only K0≠0 and one finds for pure gauge fields:

K0=-g/3ijkabc Aia Aj

b Akc =4/3ig ijkTr Ai

Aj Ak

• The true vacuum is superposition of these, so-called, n-vacua and is called the -vacuum:

|> = e -in |n>

• Easy to see that in vacuum to vacuum transitions there are transitions with dK 0

n|t= + - n|t= - = g2/322 dK |t=+ t= -

= g2/322 d3r K0 |t=+ t= -

• Pictorially, one has

_-3 _--2 _ -1 _ 0 _1 _ 2 _3 _4 _ t =+

g2/322 dK =0 g2/322 dK =2

_-4 _-3 _--2 _ -1 _ 0 _1 _ 2 _3 _4 _ t = -

• In detail one can write for the vacuum to vacuum transition amplitude

+<|>- =eim e -in+<m|n>- = ei n +<n+|n>-

• Here the difference in winding numbers is given by

• Using the usual path integral representation for +<|>- one sees that

which allows one to re-interpret the term as an addition to the usual QCD action

)~

32(| 4

2

2][

aa

AiS FxFdg

Ae eff

aaQCDeff FxFd

gSS

~

324

2

2

aa

tt FxFd

gKd

g ~3232

42

2

2

2

• Because one cannot neglect the UA(1) anomaly in ≠ 0 sectors, the topological charge density

Q = ; = d4x Q

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 M2

o] Q2 +…

aa FFg ~

32 2

2

• 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 M2

o] Tr [ln -ln † ] = ½ [F M2o] +...

• Hence the last two terms in Leff reduce, effectively, to:

½ i Q Tr [ln -ln † ] + [1/ F2 M2

o] 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 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

• This term violates P and T, but conserves C, and thus can produce a neutron electric dipole moment of order dn

e mq/Mn2

μνaμνaθ FF

π

gθL

~

32 2

2

The Strong CP Problem

• 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 ]:

eiQ5 | > = | + >

• 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/322 K

• The proof is a bit involved but worth going thru

• As a result, the associated time independent chiral charge Qc5 = d3x Jc5

o shifts under gauge transformations which change the n-vacua

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 mu0

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:

• Last term needed to give chiral anomaly of JPQ

and acts as an effective potential for axion field

• Minimum of potential occurs at <a>=-fa/ total

μνaμνa

μ

μμ

μνaμνa

FFπ

gξ

aψaL

aaFFπ

gθLL

~

32f];f/[

~

32

2

2

aaint

21

2

2

totalSM

μνaμνa

μμ FF

π

gξ

~

32J

2

2

PQ

0~

32f 2

2

a

eff

aμνa

μνa FF

π

gξ

a

V

• 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]

• 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

aμνaμνaa FF

aπ

gξ

a

Vm

~

32f 2

2

a2eff

22

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

• Defining x=v2/v1 and vF= √(v12 + v2

2), the axion is the common phase field in 1 and 2 which is orthogonal to the weak hypercharge

• See that LYukawa is invariant under the U(1)PQ transformation

a a vF ; uRi e-ix uRi ; dRi lRi e-i/x dRi lRi

..221 chLdQuQL RjLiijRjLidijRjLi

uijYukawa

1

0v

2

1;

0

1v

2

1FF v/a

22v/a

11xixi ee

• 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 =-F2/4 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)†] where

and under PQ-transformations

x/

x

F0

0;v

i

i

e

eaa

du

d

du

u

xv/

v/x

mm

m0

0mm

m

;0

0F

F

Me

eA

ia

ai

• 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 m2/ m

2

m2/ 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>> m2

• Coefficient in front of a in Lanomaly details the relative strength of the couplings of and a to

.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

aa FF~

• 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 = mf /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 ma

st [vF / fa]

a = 3 [f / fa] ; a = 0 [f / fa]μν

μνγγaγγa FFa

Kπ

αL

~

f4 a

phys

• 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

and one finds

K a=Ng(x +1/x)[mu/(mu+md)]

FFJPQ

~4

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= ; 0 = (1-Ng)/2Ng

K a=

where X1=2v22/vF

2 , X2=2v12/vF

2

)mm(3

mm4

3

4

ud

ud

ud

ud

g

21

mm

mm

2N

XX

2

1

• Note that since in both the KSVZ model and the DFSZ model m=1 the axion mass is given by the formula:

ma = ma

st [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

• This is difficult to understand for WIMP CDM since gravitational forces always produce irrotational velocity fields

• 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

0 v

dt

vd

• 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

Q =

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

aa FFg ~

32 2

2

• 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!