strong cp and the up quark mass...

Patrick DraperUCSB

SCIPP Reunion Apr 2015

Strong CP and the Up Quark Mass Revisited

I arrived at UCSC in 2011 for a 3-year postdoc, and liked it so much they still haven’t totally managed to get rid of me.

My Time at SCIPP

For that, and for much of my education,many thanks to

Michael, Howie, Stefano, and Tom, and the excellent students and postdocs

I also owe a considerable debt to the Los Gatos/UCSC carpool.

As many of you know, SCIPP is an outstanding place to work.

Other members of the carpool made sure my education was well-rounded

“There aren’t enough small numbers to meet the many demands made of them.” R.K. Guy 1988

Why is GN/GF << 1? (electroweak hierarchy problem)

Why is Λ/Mp4 << 1? (cosmological constant problem)

Why is θQCD << 1? (strong CP problem)

Why are ε,η << 1? (fine-tuning of inflaton potential)

The Era of Fine-Tuning Problems

Outline

• The Strong CP Problem

• Brief Review of Solutions & Theoretical Shortcomings

• Two Lattice Tests of mu≠0

Neutron EDM: Theory

Crewther et al 1979for , mu ⇠ md/2

|dn| < 2.9⇥ 10�26 e cm ✓ . 10�10=>

dn ' g⇡NNg⇡NN

4⇡2mNlog(mN/m⇡)

' 10�16✓ e cm

CPV QCD Lagrangian: LCP = ✓g2

32⇡2Gµ⌫Gµ⌫ � iq(Im Mq)�5q

✓ = ✓ +Arg Det MqU(1)A anomaly ⇒ physics depends on

The Strong CP Problem

CP must be violated to give the baryon asymmetry.

CP is explicitly broken in the SM by the CKM phase.

So why is dn so small?

What’s so special about ?✓

Perhaps the worst fine-tuning problem with no especially obvious anthropic crutch.

Standard Solutions

Nelson-Barr. spontaneous CPV, =0, arrange s. th. ImDetM=0 and δ13~1

mu=0. explicit CPV, removed with high quality U(1)A

PQ. explicit CPV, removed with high quality U(1)PQ

✓

✓

In each case there are serious model-building challenges.

high-quality ≡ to a part in 1010, only broken by QCD anomaly

Radiative StabilityQuality/Genericity of SymmetriesCosmology…

U(1)PQ must be nonlinear ⇒ axion

✓

Spontaneous CPV (Nelson-Barr)

Dine Leigh Kagan 1993Hiller Schmaltz 2001Vecchi 2014Dine PD

Low-energy predictions?

✓ = 0

Bento Branco Parada 1991Barr 1984, Nelson 1984

Challenges:

(1) Higher dim ops violating Im Det M=0

(2) Radiative stability of SCPV potential

(3) Stability of

—better with lower scales mCP/Mp << 1

—better with SUSY or strong dynamics (“vev for a pion”)

—Very difficult to maintain with soft SUSY or in composite models

Models require many moving parts to explain one fairly unimportant number.

Peccei-Quinn Mechanism

V ⇠✓

faMp

◆n

f2aa

2�4+n/Mnp

vs

⇒

VQCD ⇠✓m2

⇡f2⇡

f4a

◆f2a (a+ ✓)2

Peccei Quinn 1977

Kamionkowski March-Russell 1992Primary challenge: quality of the U(1)PQ

Cosmology suggests fa ~1011 GeV, need to suppress up to dim 12?

Compelling mechanism in string theory Witten 1984

imaginary parts of some fields have discrete shift symmetry(a/fa)FFpermits . fa ≳1015 GeV QCD anomaly might be dominant

But axion/saxion cosmology typically requires high susy breaking scale + some tuning of initial conditions

Banks Dine Graesser 2006Banks Kaplan Nelson 1993

Vanishing Up-Quark Mass

Theory: similar to axion. High-quality PQ

unbroken string theory PQ?discrete symm/alignment?what makes mu special?

Banks Nir Seiberg 1994

IR, large N: ~ axion⌘0

Witten 1984

Primary challenge: Excluded

Testing mu=0

First-order ChPT:

But this is in the IR. Chiral anomaly + SU(3)xSU(3) permits nonpert. additive mass renormalization

renormalization does not generate if bare mu~0✓

@tmu = �mu + C(g)am⇤dm

⇤s@tmu = �mu + C(g)am⇤

dm⇤s

(UV: mu/md ~ 10-10)

Gell-Mann Oakes Renner 1968, Weinberg 1977

Georgi McArthur 1981

So we need to know mu in the UV. Lattice QCD

Over the last decade, multiple groups have done very impressive calculations to fit the light quark spectrum.

Best simulations: pion masses < 200 MeV, lattice spacings < (2 GeV)-1, box sizes 4x pion wavelength.

Most simulations work in isospin limit mu=md and use extra information to fix mu-md

e.g. from η→3π (insensitive to EM corrections)

mu on the Lattice

Flavor Lattice Averaging GroupFLAG 2014

PQ is perhaps the most compelling of the known solutions.

Yet no solution so far is “so beautiful it must be right.” Challenges arise when embedding in “more natural” frameworks

It would be interesting to have “high-level” cross-checks of mu≠0, looking at closely-related physics on the lattice.

Worth revisiting the excluded theories, particularly when the calculations are complicated

Lots of effort dedicated to axion searches!

Quark Masses and ChPT

Fit the UV mu/md with the meson spectrum & 2nd order ChPT?

� = 2BM , U = exp(i⇡a�a/F )

bare quark masseschiral condensate

low energy const

pseudoscalar mesons

Additive renormalization of mu is 2nd order in masses (md*ms*)

Gasser Leutwyler 1985

KM Ambiguity

L � r1�Tr(�†U�†U)� Tr(�†U)2

�+ h.c.

If 𝛘 (i.e. M) is not known, there is a reparametrization redundancy:

Banks Nir Seiberg 1994

� ! �+8a

F 2(det�†)/�† , r1 ! r1 + a

r1 =1

2(L8 � L6 � L7)

⇒ a shift in M ∝ (additive renormalization) can be absorbed in r1

No. Ambiguity in 2nd order chiral lagrangian. Kaplan Manohar 1986

Extra information needed (QCD).

Lattice

If nonperturbative physics did generate most or all of mu, would correspond to a large r1.

The lattice can measure quantities sensitive to r1 ⇒ alternate test of mu

Example: the pion mass dependence on the strange quark mass

The minimum amount of information the lattice needs to provide is to fix the KM ambiguity.

⇒ fit quark masses, or fit the LECs. Cohen Kaplan Nelson 1999

Dine PD Festuccia

dashen correction. So I could just say we take this splitting from lattice and it is insensitive to mu.

�2

�1' 1

ms

m2⇡ � (m2

K0 �m2K±)QCD

m2⇡ + (m2

K0 �m2K±)QCD

(mu)e↵mdms

' 1

ms

m2⇡ � (m2

K0 �m2K±)QCD

m2⇡ + (m2

K0 �m2K±)QCD

so we get a constraint on this dependence:

' 5 GeV�1

Take a simplifying approximation where r1 is the dominant LEC. Set mu to zero and move r1mdms into mu,eff with a KM transformation. Then the LO mu/md formula applies to mu,eff :

mu,eff controls the ms dependence of the pion mass in these limits:

Straightforward to include corrections from other LECs

m2⇡ = �1 [(mu)e↵ +md] ⌘ �1md + �2mdms

use simulations at two values of ms to extract the ratio:

�2

�1⇡

m2⇡1

�m2⇡2

m2⇡2ms1 �m2

⇡1ms2

SU(3) LECs have been looked at on the lattice literature. Can use to infer β2/β1

From MILC ’09, very roughly we find

Consistent with mu>0 (and validity of 2nd order ChPT)

✓�2

�1

◆

obs

' (1± 1) GeV�1

It would be interesting to see this sharpened and interpreted as a consistency check of mu.

So far LEC computations haven’t been done by most groups (mainly just MILC) and results and uncertainty estimates differ

“it seems fair to say that there are open issues regarding the convergence of (extended versions of) SU(3) ChPT on Nf = 2+1 ensembles” Dürr 2013

“the situation remains unsatisfactory in the sense that for each Nf only a single determination of high standing is available” FLAG 2014

It would also be interesting to test the physics of the additive mass renormalization on the lattice.

In the UV, small instantons contribute to 2nd term Georgi McArthur 1981Choi Kim Sze 1988

@tmu = �mu + C(g)am⇤dm

⇤s

Instanton contributions to correlation functions often ill-defined / incalculable due to IR divergent size integral

On the lattice, is there an independent test of small instantons?

There are certain correlators that vanish in PT in the chiral limit and at short distances their leading nonperturbative corrections are calculable

Finite Green Functions in QCD

Examples:

Field strengths make single instanton contribution IR finite at leading order

hu�µ⌫Fµ⌫d(x) d�⇢⇡F

⇢⇡s(y) s�↵�F

↵�u(z)i


⇢⇡s(y) su(z)i

hudsF (x) udsF (y)i

Fermion bilinears require 3 units of axial charge violation ⇒ instanton


⇢⇡uss(0)i

…

Dine Festuccia Pack WuDine PD Festuccia


⇢⇡uss(0)i = � 8

7⇡4C(g)⇤9|x|�4

Example:

C(g) = 1.15⇥ (2⇡4)↵�6

Finite mass perturbative correction: Gm ' 1

(4⇡2)4mumdms

x

10

subdominant for physical values of m and x > 10-2 Λ-1

Corrected at order by fluctuations around the instanton: ↵/⇡

Can push off to with (↵/⇡)2 FF ! FF

|x|�4h:uuddss:i (~10% for x~mτ-1)

Short-distance correlators of high-dimension operators are challenging.

But of interest: (1) test the lattice instanton density (2) probe the QCD IR cutoff on ρ

Possible to choose correlators carefully to avoid disconnected contributions, large corrections, etc.

Lattice

Summary

* No known solution of strong CP is completely compelling

* The variation of the pion mass with ms is a QCD prediction that can be interpreted as a consistency check of mu>0

* Small instantons additively renormalize mu, and on the lattice they can be tested directly with mostly-IR-finite Green functions

Thanks!

strong cp and the up quark mass...

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