The Strong CP Problem and Axions

Joint ILIAS-CAST-CERN Axion Training CERN

November 30, 2005

R. D. PecceiUCLA

The Strong CP Problem and Axions

• The U(1)A Problem of QCD

• The QCD Vacuum and the Strong CP Problem

• Approaches to the Strong CP Problem

• U(1)PQ and Axions

• Axion Dynamics

• Invisible Axion Models

• Concluding Remarks

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

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

0dduu

• However, because U(2)A is spontaneously broken symmetry, expect appearance in the spectrum of approximate Nambu-Goldstone bosons, with m 0 [ m0 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

The QCD Vacuum and the Strong CP Problem

• 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 in QCD is CP not very badly broken?

• 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

• 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= -

• 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 to re-interpret term as addition to 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

• Resolution of U(1)A problem, by recognizing complicated nature of QCD’s vacuum, effectively adds and extra term to LQCD

• This term violates P and T, but conserves C. Strong bound on the neutron electric dipole moment dn<1.1 x 10-26 ecm requires the angle to be very small [dne mq/MN

2 < 10-9 - 10-10]

• Why should this be so is the strong CP problem Problem actually worse if one considers the effect of chiral transformations on -vacuum

μνaμνaθ FF

π

gθL

~

32 2

2

• Chiral transformations, because of the anomaly, change the -vacuum [Jackiw Rebbi ]:

eiQ5 | > = | + > (see Appendix)• If one includes weak interactions, the quark

mass matrix is in general complex

LMass = -qiR Mij qjL + h. c. To diagonalize it one must, among other things,

perform a chiral transformation which changes into

total = + Arg det M• Strong CP Problem: Why is this angle, coming

from the strong and weak interactions, so small?

Approaches to the Strong CP Problem• There are three possible “solutions” to Strong

CP Problem:

i. Unconventional dynamics

ii. Spontaneously broken CP

iii. An additional chiral symmetry

However, in my opinion, only iii. is viable solution

• It is, of course, also possible that, as a result of some anthropic reasons

total = + Arg det M

just turns out to be of O(10-10), but I doubt it!

• Approaches to i. unconventional dynamics are also not very believable:

• They either suggest that B.C. which gave rise to -vacuum is an artifact [but then, what is solution to U(1)A problem?], or use periodicity of vacuum energy E() ~ cos to deduce that vanishes [but, why E/=0?]

• The second possibility, ii Spontaneously broken CP, is more interesting

• If CP is a symmetry of nature, which is spontaneously broken, then can set =0 at the Lagrangian level

• However, gets induced back at the loop-level, and to get < 10-9 one needs, in general, also to insure that 1-loop=0

• Although models exist where this is so, theories with spontaneously broken CP need complex Higgs VEVs, leading to FCNC and domain walls, and introduce recondite physics [Barr Nelson] to avoid these problems

• In my view, however, the biggest drawback for this “solution” to the strong CP problem is that experimental data is in excellent agreement with the CKM Model– a model where CP is explicitly not spontaneously broken

• Introducing iii an additional chiral symmetry is a very natural solution to strong CP problem since it, effectively, rotates -vacua away

e-i Q5 | > = | 0 >• Two suggestions for this chiral symmetry:

i. The u-quark has no mass, mu = 0ii. SM has an additional global U(1) chiral

symmetry [Peccei Quinn]• mu = 0 is disfavored by current algebra

analysis [ Leutwyler]. Further, it is difficult to understand why

Arg det M = 0 What is the origin of this chiral symmetry?

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

• 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 NG a mass]

• Calculation of axion mass first done explicitly by current algebra techniques [Bardeen Tye]

• Here will give an effective Lagrangian derivation [Bardeen Peccei Yanagida], as it also gives readily axion couplings

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:

=N/2(x +1/x)=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 one introduces 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 mo )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

Leptons also contribute to and one finds

=Ng{[3(2/3)2]x+[3(-1/3)2+(-1)2]1/x}

=4/3Ng(x +1/x)

FFJPQ

~4

• As before, we must separate out the light quark contributions in the anomaly, since they are counted by the coupling of the °and to 2.

• Adding the lepton and heavy quark contributions of the axion coupling to the em anomaly

eff=4/3Ng(x +1/x)-4/3x -1/3x

to that coming from the and mixing

3 + 5/3 0

gives, finally,

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

• Will do this for the KSVZ model because it is simple and illustrates well what we just did

• The KSVZ axion does not interact with leptons and only interacts with light quarks as the result of the strong and em anomalies

• The superheavy quark Q induces the following couplings [eQ is the em charge of Q]

FFeFF

g

f

aL Qaa

aaxion 4

3~

322

2

2

• Since in the KSVZ model the ordinary Higgs do not carry PQ charge, the only interactions of the axion come from the effective anomaly mass term which here is given by

Lanomaly=-½ mo2 [+ {[f/fa] [1/2]}a ]2

• To the above one must add the standard quadratic term coming from the light quarks

L2=-½ mo2{mu/(mu+md)[+]2+md/(mu+md)[-]2}

• Diagonalizing Lanomaly plus L2 gives the axion parameters:

m=1 ; 3=-½(md-mu)/(mu+md) ; 0 =-½

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

ma = ma

st [vF / fa] 6.3 [106 GeV / fa] eV

• The calculation of K a in this model is equally straightforward. To the contribution of the superheavy quark in the em anomaly [3eQ

2], one must add that coming from the mixing of the axion with the ° and [ 3 + 5/3 0 ]

• This gives, finally,

K a= 3eQ2 – (4md +mu)/3(mu+md)

• I will not go through the analogous calculation for the DFSZ model, but just quote the results

• It proves convenient to define X1=2v2

2/vF2 , X2=2v1

2/vF2 ,

where vF= √(v12 + v2

2) and v1 and v2 are Higgs VEVs, and to rescale fa fa/2Ng to make m≡1, so that also in the DFSZ model

ma = ma

st [vF / fa] 6.3 [106 GeV / fa] eV• One then finds 3= ; 0= (1-Ng)/ 2Ng

K a=

ud

ud

g

21

mm

mm

2N

XX

2

1

)mm(3

mm4

3

4

ud

ud

• 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 through NNNNa 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. WMAP data provides bound on CDM and axions:

Ωah2 ≤ 0.12

• Quote result of a recent calculation of axion contribution to Universe’s energy density by Fox Pierce Thomas:

Ωah2 =0.5[(fa/)/1012 GeV]7/6[i2 +

2]

Here is coefficient of PQ anomaly, I is initial misallignment angle and its mean square fluctuation and is a possible dilution factor

• For =1, and using for i an average angle i

2= <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

Concluding Remarks

• After more than 25 years, preferred solution to strong CP problem remains having a U(1)PQ symmetry in the theory and its concomitant axions

• Although Fermi scale axions have been ruled out, invisible axions models are still viable and axion oscillations could account for the dark matter in the Universe

• No totally compelling invisible axion models exist, but it is encouraging that experimentalists are actively searching for axions

Appendix: -vacua and chirality• The gauge matrices n can be obtained by

compounding: n =[1]n. It follows thus that on an n-vacuum state

1|n>=|n+1>

• Hence n-vacua 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 associated time independent chiral charge Qc5 = d3x Jc5

o, as a result, 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 changes the -vacuum [Jackiw Rebbi]

e i/N Qc5|> = | + >