Hadronic Weak Interactions

Matthias R. Schindler

Fundamental Neutron Physics Summer School 2015

Some slides courtesy of N. Fomin

1 / 44

Weak interactions

- One of “fundamental” interactions- Component of Standard Model- SU(2)×U(1) gauge theory- Quarks and leptons interact via weak interaction- Mediated by W, Z exchange

See lectures by S. Gardner

2 / 44

Typology of weak interactions

- Leptonic: only leptons, no quarksExample:

µ+ → e+ + νe + νµ

- Semileptonic: some leptons, some quarksExample:

K 0 → π+ + e− + νe

- Hadronic: no leptons, only quarksExample:

K 0 → π+ + π−

3 / 44

Strangeness-changing hadronic weak interactions

K 0 → π+ + π−

- Simple quark model

K 0 = ds, π+ = ud , π− = ud

- Strangeness-changing decay- Mediated by W exchange

Ignore from now on

4 / 44

Strangeness-conserving hadronic weak interactions

At low energies current-current form

L∆S=0weak =

G√2

cos2 θCJ0,†W J0

W︸ ︷︷ ︸∆I=0,2

+ sin2 θCJ1,†W J1

W︸ ︷︷ ︸∆I=1

+J†Z JZ

- Isospin structure ∆I = 0,1,2- ∆I = 1 dominated by neutral current JZ (sin2 θC ∼ 0.05)- NC JZ not observed in flavor-changing hadronic decays

5 / 44

Hadronic weak interactions

Well understood in terms of SM degrees of freedom

So what is the problem?

- Free quarks not observed→ Hadrons (N, π, . . . )- Range of weak interactions ≈ 1/MW ∼ 0.002 fm- 0.002 fm� rnucleon

- Quarks confined into hadrons by strong interactions (QCD)

6 / 44

Strong interactions

Quantum chromodynamics- SU(3) gauge theory- Quarks as matter fields- Interaction mediated by gluons

LQCD =6∑

f =1

qf (i /D −mf )qf −12

Tr (GµνGµν)

with covariant derivative Dµqf ≡ (∂µ + igAµ) qf ,

Looks simple enough?

7 / 44

Strong interactions

- QCD coupling increases for lower energies

0

0.1

0.2

0.3

1 10 102

µ GeV

α s(µ)

adapted from Particle Data Group

- Perturbation theory in αS not useful at low energies- Non-perturbative regime of QCD- Construct effective interaction on hadronic level

8 / 44

Example: Nucleon structure

Desy

9 / 44

Hadronic weak interactions at low energies

- Quarks confined in hadrons by strong interactions- Size ∼ (fm)

- Quarks interact weakly- Range ∼ 0.002 fm

Interplay of weak and nonperturbative strong interactions

- Construct PV interaction between hadrons (Here: NN)- Use to calculate observables in hadronic systems- Relate to underlying SM

10 / 44

Why should I care?

- Weak neutral current in hadron sector- Probe of strong interactions

- Weak interactions short-ranged- Sensitive to quark-quark correlations inside nucleon- No need for high-energy probe- “Inside-out probe”

- Isospin dependence of interaction strengths?→ ∆I = 1/2 puzzle (strangeness-changing )?

- Can contribute to T violating observables

11 / 44

Parity-violating NN interactions

- Parity-violating component in NN interaction- Suppressed by GF m2

π/g2πNN ∼ 10−7

- Isolate in pseudoscalar observables (~σ · ~p)- Longitudinal asymmetries- Angular asymmetries- Spin rotation- γ circular polarization- Anapole moment

12 / 44

PV in complex nuclei

Enhancements in heavy nuclei- Large parity conserving matrix amplitudes- Close level spacings between states with opposite parity

〈+|VPV |−〉E+ − E−

- Up to 10% effect (~n 139La)- Many-body problem→ Difficult to interpret

13 / 44

PV in few-nucleon systems

- No enhancements

O(10−7) effects

- High-intensity sources (neutrons, photons, . . . )- Control over systematics- Understand in terms of NN interactions

14 / 44

NPDGamma

γθ

- Radiative capture of polarized neutrons on protons- Angular asymmetry Aγ

1Γ

dΓ

d cos θ= 1 + Aγ cos θ

- Correlation between neutron spin and photon momentum- Measured at SNS

15 / 44

NPDGamma data taking completed in June 2014

Spin octet

16 / 44

• In principle, experiment can be done with just one detector, reversing the neutron spin:

• Add opposite detector at same angle

(eliminates some systematic errors):

Asymmetry Extraction

↓↑

↓↑

+

−=

YYYYAraw

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

−+

+

−=

↓↓

↓↓

↑↑

↑↑

ij

ij

ji

jiraw YY

YYYYYY

A21

ii

j

θ

17 / 44

~n 3He→ p 3H

- Charge-exchange reaction- Correlation between neutron spin and proton momentum

10 Gauss solenoid

RF spin Flipper

3He target / ion chamber

FnPB cold neutron guide

3He Beam Monitor

FNPB n3He detector (“the Cave”)

Super-mirror polarizer

Aim: extract parameters characterizing hadronic PV interactions

18 / 44

Early ideas about NN interaction

NN potential- Finite range- Massive particle exchange- Yukawa: M ∼ 100 MeV

Long-range (r & 1.5 fm): One-pion exchange

- Intermediate distance (0.7 fm . r . 1.5 fm): two-pionexchange?

- Short-distance (r . 0.7 fm): ?

19 / 44

NN potentials: basics

Symmetry constraints on possible form of NN potential- Galilei invariance- C,P,T- Isospin

V =∑

ViOi

- Operators Oi : 1, ~S · ~S, S12, ~L · ~S, (~L · ~S)2

- Coefficients Vi(r , ~p 2, ~L 2): undetermined

20 / 44

A note on potentials

Potentials are not observable

- Matrix element〈f |V |i〉

- Unitary transformation:

〈f | → 〈f |U†, |i〉 → U|i〉

- Matrix element unchanged if

V → UVU†

21 / 44

Models

Phenomenological models- Make ansatz for Vi

- Long range: one-pion exchange- ≈ 40− 50 parameters- Fit to NN scattering data- Interpretation?- Connection to QCD?

One-boson-exchange models- Long range: one-pion exchange- Intermediate and short range: σ, 2π, ρ, ω, . . .- Mi < 1 GeV- Connection to QCD?

22 / 44

PV potential

Meson-exchange models- Single-meson exchange (π±, ρ, ω) between two nucleons

with one strong and one weak vertex

π±, ρ, ω

- Barton’s theorem: no π0 exchange- Weak interaction encoded in PV meson-nucleon couplings

23 / 44

DDH model

- Hadronic weak interaction observable through flavor-changing decays, but

�SM: flavor-changing neutral currents GIM suppressed, unobservableݓ to see the hadronic neutral current, must study ΔS=0 ⇒�ݓ interactions

- NN, nuclear systems the only accessible possibilities

�must use PNC to filter out the effects of strong, E&M interactionsݓ�often modeled as a series of one-boson exchangesݓ

strong vertex⇔

π±, ρ, ω

Motivation Old Paradigm New Direction Summary S-P Amplitudes Meson-Exchange Model PV M–N Couplings Current Status

Predictions for /P Meson-Nucleon Couplings

⇥107 DDH Range Best DZ FCDH KMh1

� 0.0⇤+11.4 4.6 1.1 2.7 0.2h0

⇥ -30.8⇤+11.4 -11.4 -8.4 -3.8 -3.7h1

⇥ -0.4⇤+00.0 -0.2 0.4 -0.4 -0.1h2

⇥ -11.0⇤+-7.6 -9.5 -6.8 -6.8 -3.3h0

⇤ -10.3⇤+05.7 -1.9 -3.8 -4.9 -6.2h1

⇤ -1.9⇤+-0.8 -1.1 -2.3 -2.3 -1.0h⌅1⇥ 0.0 -2.2

M

B B’

M

B B’

B B’

M

(b) Quark Model

(c) Sum Rule

(a) Factorization

Calculations by DDH, DZ, FCDH are based on quark models, KM used thechiral soliton modelh⌅1⇥ term is usually ignored, so leaving 6 /P couplings to be checked by exps.

QCD sum rule calculations of h1� give 3⇥10�7 (HHK 98, formerly 2⇥10�8)

and 3.4⇥10�7 (Lobov 02)Lattice QCD calculations of h1

� (should be similar to g� but with a shorterrange) are proposed (e.g. Beane and Savage: matching PQQCD toPQChPT)

Cheng-Pang Liu Parity Violation in Few-Nucleon Systems

Haxton

- 6 (7) PV meson-nucleon couplings (π±, ρ, ω)1

- Estimate weak couplings (quark models, symmetries)⇒ ranges and “best values/guesses”

1Desplanques, Donoghue, Holstein (1980)24 / 44

DDH potential

VDDH =ih1πgAM√

2Fπ

(~τ1 × ~τ2

2

)z

(~σ1 + ~σ2) ·[~p1 − ~p2

2M,wπ(r)

]− gρ

(h0ρ ~τ1 · ~τ2 + h1

ρ

(~τ1 + ~τ2

2

)z

+ h2ρ

3τ z1 τ

z2 − ~τ1 · ~τ2

2√

6

)×(

(~σ1 − ~σ2) ·{~p1 − ~p2

2M,wρ(r)

}+ · · ·

)+ · · ·

with- gM : strong (PC) meson-nucleon couplings- hi

M : weak (PV) meson-nucleon couplings

- wM(r) =exp(−mM r)

4πr

25 / 44

Complications

- Connection to QCD?- Error estimates?- Consistency between different parts of calculations?

- Wave functions- Operators- External currents (e.m., weak)- PC and PV potentials (couplings, particle content,. . . )- 3N, 4N,. . . forces?

- “mesons” = “real” mesons? (1/Mρ ≈ 0.3 fm)

26 / 44

Experimental constraints

-2 0 2 4 6 8 10 12 14f - 0.12 h 1 - 0.18 h 1

-5

0

5

10

15

20

25

30

-(h0+0.7h0 )

pp

p

133Cs

19F

205Tl

18F18F19F

pα

pp❊

DDH

best value

-2 0 2 4 6 8 10 12 14f - 0.12 h 1 - 0.18 h 1

-5

0

5

10

15

20

25

30-(h0+0.7h0 )

pp

p

133Cs

19F

205Tl

18F

-2 0 2 4 6 8 10 12 14f - 0.12 h 1 - 0.18 h 1

-5

0

5

10

15

20

25

30

-(h0+0.7h0 )

pp

p

133Cs

19F

205Tl

18F

Haxton, Holstein (2013)

27 / 44

Re-analysis of pp scattering at 221 MeV- Use consistent strong couplings in PC and PV parts

!

-2 0 2 4 6 8 10 12 14

f - 0.12 h1- 0.18 h

1

-5

0

5

10

15

20

25

30

-(h0+0.7h0)

pp

p

133Cs

19F

205Tl

18F

18F19F

p!

pp

DDH

best value

-2 0 2 4 6 8 10 12 14

f - 0.12 h1- 0.18 h

1

-5

0

5

10

15

20

25

30-(h0+0.7h0)

pp

p

133Cs

19F

205Tl

18F

-2 0 2 4 6 8 10 12 14

f - 0.12 h1- 0.18 h

1

-5

0

5

10

15

20

25

30

-(h0+0.7h0)

pp

p

133Cs

19F

205Tl

18F

!

!80 !60 !40 !20

!40

!35

!30

!25

!20

0.7h!pp-h"pp (!107)

h!

pp+

0.7

h"

pp (!

10

7)

Haxton, Holstein (2013)

28 / 44

Effective theories

Potential energy difference- High school:

∆U = mgh

- When does it fail? By how much?

- College:

∆U = −GMm(

1rf− 1

ri

)= −GM

Rm(

RR + h

− 1)

= mGMR2 h

RR + h

where ri = R, rf = R + h

29 / 44

Effective theories

Potential energy difference- High school:

∆U = mgh

- When does it fail? By how much?- College:

∆U = −GMm(

1rf− 1

ri

)= −GM

Rm(

RR + h

− 1)

= mGMR2 h

RR + h

where ri = R, rf = R + h

30 / 44

Effective theories

∆U = mgh(

1− hR

+h2

R2 + · · ·)

- mgh: lowest-order approximation- Valid for h� R (separation of scales)- Systematically improvable

Example of effective theory

Further examples:- Multipole expansion for confined charge distribution- Fermi theory of weak interactions2

- . . .

2Cow-cow scattering example: Gripaios [arXiv:1506.05039]31 / 44

Effective field theories

- Low-energy theory for NN interactions- Effective degrees of freedom: nucleons, pion,. . .- Symmetry constraints

So what’s new?

- Connection to QCD through symmetries- Only relevant degrees of freedom, not more- Expansion in ratio of scales Q/Λ < 1: Power counting

32 / 44

Power counting

Perturbation theory in QED- Coupling α� 1- Expand observables in α- Calculate order by order (Feynman graphs, etc)

Perturbation theory in EFT- Possibly strong coupling C ⇒ PT in C not possible- Identify ratio of scales

- Typical low-energy (long-distance) scale E- Underlying high-energy (short-distance) scale Λ

- Expand observables in ratio E/Λ

- Calculate order by order (Feynman graphs, etc)

33 / 44

Advantages

- Model independent- Unified framework

- PC and PV potentials- External currents- 2N, 3N, 4N, . . .

- Systematically improvable- Theoretical error estimates

34 / 44

Pionless EFT- At very low energy: pion-exchange not resolved:

1~q 2 + m2

π

≈ 1m2π

(1−

~q 2

m2π

+ · · ·)

- EFT with only nucleons as degrees of freedom: EFT(6π)- Contact interactions- Order by number of derivatives- Lowest-order parity-conserving Lagrangian (partial-wave

basis)

L =N†(i∂0 +~∇2

2M)N − 1

8C(1S0)

0 (NT τ2τaσ2N)†(NT τ2τaσ2N)

− 18C(3S1)

0 (NT τ2σ2σiN)†(NT τ2σ2σiN) + . . . ,

- Connect C(1S0)0 , C(3S1)

0 , etc to effective range expansion

35 / 44

Parity violation in EFT(6π)

- Parity determined by orbital angular momentum L : (−1)L3

- Simplest parity-violating interaction: L→ L± 1- Leading order: S − P wave transitions

P S

- Spin, isospin: 5 different combinations

3Danilov (1965, ’71); Zhu et al. (2005); Phillips, MRS, Springer (2009); Girlanda (2008)36 / 44

Lowest-order parity-violating LagrangianPartial wave basis4

LPV =−[C(3S1−1P1)

(NTσ2 ~στ2N

)†·(

NTσ2τ2i↔∇N

)+ C(1S0−3P0)

(∆I=0)

(NTσ2τ2~τN

)†(NTσ2 ~σ · i

↔∇τ2~τN

)+ C(1S0−3P0)

(∆I=1) ε3ab(

NTσ2τ2τaN)†(

NTσ2 ~σ ·↔∇τ2τ

bN)

+ C(1S0−3P0)(∆I=2) Iab

(NTσ2τ2τ

aN)†(

NTσ2 ~σ · i↔∇τ2τ

bN)

+ C(3S1−3P1) εijk(

NTσ2σiτ2N

)†(NTσ2σ

kτ2τ3↔∇

jN)]

+ h.c.

- Need 5 experimental results to determine LECs

4Phillips, MRS, Springer (2009)37 / 44

Example: ~np → dγ

γθ

5

1Γ

dΓ

d cos θ= 1 + Aγ cos θ

Aγ =43

√2π

M32

κ1(1− γa1S0

) g(3S1−3P1)

5Savage (2001); MRS, Springer (2009)38 / 44

Chiral EFT for NN interaction

- Degrees of freedom: nucleons and pions- Energy scales: E ≈ mπ < Λ ≈ 1 GeV- Symmetries: Galilei, C, (P), T, chiral (non-trivial)- Lowest-order PC potential:

V LOPC = − g2

A4F 2

(~σ1 · ~q)(~σ2 · ~q)

~q 2 + m2π

(~τ1 · ~τ2) + CS + CT (~σ1 · ~σ2)

- one-pion exchange- Contact terms (short-distance operators)- Low-energy constants CS,CT

- Systematic corrections possible- Higher powers of q = pf − pi- Two-pion exchange- . . .

39 / 44

Chiral expansion of nuclear forces Chiral expansion of nuclear forces

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Zwei-Nukleon-Kraft

Führender Beitrag

Korrektur 1. Ordnung

Korrektur 2. Ordnung

Korrektur 3. Ordnung

Drei-Nukleon-Kraft Vier-Nukleon-KraftTwo-nucleon force Three-nucleon force Four-nucleon force

LO (Q0)

NLO (Q2)

N2LO (Q3)

N3LO (Q4)

(numbers from Pudliner et al. PRL 74 (95) 4396)

E. Epelbaum

40 / 44

Chiral PV NN potential

Leading order:

V LOPV = −i

gAhπ2√

2F(~σ1 + ~σ2) · ~q~q 2 + m2

π

(~τ1 × ~τ2)z

- One-pion exchange ∝ hπ6

- LO contribution to ~np → dγ

Next-to-leading order- Contact terms- TPE ∝ hπ- New γπNN contact interaction

Caveat: power counting assumes that hπ is not “small”

6Savage, Springer (1998); Kaplan et al. (1999); Zhu et al. (2005); Liu (2007); Song etal. (2011), (2012)

41 / 44

Translation between different formalisms

- DDH operator structure→ EFT(6π) at low energies- Can relate DDH and EFT couplings?- Problems

- Low-energy limit of PC potentials?- Regulator dependence in potential (µP): µ2

P4πr e−µP r

- Scale dependence in EFT couplings (µ)

Relation between couplings in DDH and EFT scale-dependent

hiDDH = f (µp, µ)hi

EFT

42 / 44

PV on the lattice- Standard Model degrees of freedom (quarks, gluons,. . . )- Calculating reactions not yet realistic- Determine PV couplings- PV quark operators on lattice between 3-quark operators

(proton, neutron, etc)-

hπ =(

1.099± 0.505 (stat.) +0.058−0.064 (syst.)

)× 10−7

- mπ ∼ 389 MeV, L ∼ 2.5 fm, as ∼ 0.123 fm- Connected diagrams only7

- Consistent with most model estimates, lower end of DDH“reasonable range”

(a) (b) (c)

7Wasem (2012)43 / 44

Bibliography

B. Desplanques, J. F. Donoghue and B. R. Holstein, AnnalsPhys. 124, 449 (1980)

E. G. Adelberger and W. C. Haxton, Ann. Rev. Nucl. Part.Sci. 35, 501 (1985)

S. L. Zhu, C. M. Maekawa, B. R. Holstein,M. J. Ramsey-Musolf and U. van Kolck, Nucl. Phys. A 748,435 (2005)

M. J. Ramsey-Musolf and S. A. Page, Ann. Rev. Nucl. Part.Sci. 56, 1 (2006)

W. C. Haxton and B. R. Holstein, Prog. Part. Nucl. Phys.71, 185 (2013)

M. R. Schindler and R. P. Springer, Prog. Part. Nucl. Phys.72, 1 (2013)

44 / 44

Spin rotation

- Phase accumulated after traversing a target of thickness L

ϕ = Re(n − 1)kL

- Index of refraction

n − 1 =ρµ

k2M,

ρ: target density, µ: reduced mass,M: scattering amplitude- Phase

ϕ = ρLµ

kRe(M)

45 / 44

Spin rotation

- Beam in z direction with +x polarization

|x+〉 =1√2

(|+〉+ |−〉)

- Helicity states |±〉 evolve with phase factor

1√2

(e−iφ+ |+〉+ e−iφ− |−〉

)=

1√2

e−iφ+

(|+〉+ ei(φ+−φ−)|−〉

)- For φ+ 6= φ−:

φPV = φ+ − φ−- Spin rotation angle per unit length L

1ρ

dφPV

dL=µ

kRe (M+ −M−) ,

46 / 44

Anapole moment

Multipole expansion of charge and current operators- P and T conserving: charge, electric quadrupole, magnetic

dipole, . . .- P and T violating: electric dipole, magnetic quadrupole, . . .- P violating, T conserving: anapole moment, . . . 8

Current matrix element

〈N(p′)|Jµ|N(p)〉 = u(p′)[γµ F1(q2) + i

σµνqν2m

F2(q2)

+1

m2 (/qqµ − q2γµ)γ5 a(q2)

+ iσµνqν

2mγ5 d(q2)

]u(p)

8Zel’dovich (1958); Flambaum et al. (1980); Haxton et al. (2002)47 / 44

Chiral Symmetry

- QCD Lagrangian

LQCD = q(i /D −M

)q − 1

4Gµν, aGµνa

- Project onto right- and left-handed fields

LQCD = (qR i /DqR+qL i /DqL−qRMqL−qLMqR)−14Gµν, aGµνa

- ForM = 0:

L0QCD = (qR i /DqR + qL i /DqL)− 1

4Gµν, aGµνa

- Invariant under qR → URqR,qL → ULqL

SU(2)L × SU(2)R × U(1)V symmetry

48 / 44

Chiral Symmetry II

Spectrum:- SU(2) multiplets

- No parity doubling

- Pions very light

Spontaneous symmetry breaking to SU(2)VPions as Goldstone bosons

Explicit symmetry breaking by mu,d 6= 0- Treat mu,d as perturbation

49 / 44