Weak Structure of the Nucleon

Weak Structure of the Nucleon

Doron GazitInstitute for Nuclear Theory

University of Washington, Seattle.

From a few body physicist’s perspective

Doron Gazit - The weak structure of the nucleon

2

Motivation. Interaction of weak probes with nuclei. Part I: Weak structure of the nucleon. Part II: Some studies of weak interaction in light

nuclei. Application to astrophysics. Summary and outlook.

August 2009

Outline

See S. Bacca’s talk


4August 2009

Why is the weak structure of the nucleon interesting?

The response of a nucleon to an external weak probe at low energy◦ Only the probe is perturbative.◦ One would like to constrain the non-

perturbative response: To study the fundamental theory To acquire a predictive quality.


5August 2009

A few body physicist’s perspective:A precision era:

◦ Available accurate ab-initio methods.

◦ Consistent currents and potentials from cPT.

◦Allow parameter free calculations with sub-percentage accuracy, with nucleonic dof.


6

Nuclear weak processes

( )iP,EP iì =i

( )ìP E , Pf f f=

€

k1μ = mμ ,0( )

( )2k,kk 2ì2 =

€

q0μ = ω,q( )

€

W±, Z 0

€

ˆ H W ~ − d3 r x ˆ j μa∫ r x ( ) ˆ J μ ,−a r x ( )

€

W ±,Z 0 propagator =gμν +

qμqν

MW ,Z2

q2 + MW ,Z2 →

q<<M W ,Z

gμν

MW ,Z2

Nuclear current

Lepton current

August 2009


7

The standard model dictates the quark currents:

When sandwiched between nucleonic/nuclear states, the strong interaction induces form factors.

cPT offers a venue to characterize these form factors, at low energies.

August 2009

Electroweak currents in the standard model

€

V aμ = q γ μ τ a

2q Aa

μ = q γ μγ 5τ a

2q I μ = 1

2q γ μq

€

J±μ = V±

μ + A±μ

J0μ = (1− 2sin2 θW )V0

μ − 2sin2 θWI μ + A0μ

See J. S. Real’s talk

Doron Gazit - JLab Theory seminar

8

cPT approach for low-energy EW nuclear reactions:

Weak current

Chiral Lagrangian

QCDLow energy EFT

Nöther current

Wave function

s

Nuclear Hamiltonian

Nuclear Matrix

Element


9

Forces in D-less cPT The leading order NNN

forces are at N2LO. They include 2 new

contact parameters. No new parameters at

N3LO.

August 2009Weinberg, van Kolck, Ordonez, Meissner, Epelbaum, Nogga, Bernard, Kaiser, Krebs, Machleidt, Entem…

See H. Krebs’ talk


10August 2009

D-less cPT based weak currents to fourth order

Contact term1 pion exchange

Nucleon-pion interaction, NO new parameters

T.-S. Park et al, Phys. Rev. C 67, 055206 (2003); DG PhD thesis arXiv: 0807.0216

Contact term

€

ˆ d R

Single nucleon current

Gårdestig, Phillips, Phys. Rev. Lett. 98, 232301 (2006); DG, Quaglioni, Navratil, Phys. Rev. Lett. 103, 102502 (2009).

See T.-S. Park’s talk


11

The Weak Structure of the nucleon

€

N p( )Vaμ N ' p'( ) = u p'( ) FV q2( )γ μ +

iFM q2( )2MN

σ μν qν + FS q2( )qμ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥τ a

2u p( )

N p( ) Aaμ N ' p'( ) = u p'( ) GA q2( )γ μ +

GP q2( )2MN

qμ +iGT q2( )

2MN

σ μν qν

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥γ 5

τ a

2u p( )

Vector

Axial

Weak magnetism

Induced Pseudoscalar

Second class currents

Weinberg Phys. Rev., 112, 1375 (1958)

p'

pInduced scalar

Induced Pseudotensor

€

u Pμ( ) =pn ⎛ ⎝ ⎜

⎞ ⎠ ⎟€

qμ = p'μ −pμ

€

γμ, γ 5

August 2009

Single nucleon current


12

The quark currents have a specific behavior under G-parity Cexp(-iπT2) .

Since isospin is not a symmetry of the strong force, induced second class currents are allowed in nuclear reactions.

They are expected to be suppressed by a factor:

No experimental evidence for second class currents!

August 2009

Second class currents

€

md − mu

mn + mp


13

Vector structure of the nucleon.

August 2009


14

At zero momentum transfer:

The fact that FV is not renormalized at low energies, led to the Conserved Vector Current hypothesis.

August 2009

Determination of Vector couplings

€

1 =normalizationof nucleonic w. f .

pV+0 q = 0( ) n = FV 0( )


15

CVC hypothesizes:◦ The vector parts of the charge changing current and the isovector

piece of the electromagnetic curret are three components of a vector in isospace.

◦ All 3 components are conserved.

The vector and induced-weak-magnetic form factors are equal to their electromagnetic counter parts, including the momentum dependence.

The induced scalar form factor vanishes. CVC implies that Siegert theorem holds for weak reactions. An excellent approximation in the nuclear sector.

◦ According to cPT, CVC holds to 2×10-4.

August 2009

Conserved Vector Current (CVC) hypothesis

Gerstein, Zeldovich, Sov. Phys. JETP 2, 576 (1956)Feynman, Gell Mann, Phys. Rev. 109, 193 (1958)

€

V γ , V±

cosθC

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Kaiser, Phys. Rev. C, 64, 028201 (2001)


16

Tests of CVC – Nuclear b decay

Superallowed transitions

0+0+

Only vector current contributes.

The nuclear matrix element:

Towner & Hardy define “nucleus independent” half-life:

August 2009

ene

€

ft ~ 1FV

2 MF2

€

Ft = ft 1−δC( ) 1+ δR( )

€

MF2 = ψ i τ ± ψ f

2=

= T T +1( ) − TZ Tz +1( )[ ] 1−δC( )


17August 2009

Latest survey of superallowed b decays

Hardy & Towner, Phys. Rev. C 79, 055502 (2009)

€

Ft = 3071.8 ± 0.83 sec

€

Vud = 0.97424(22)

Vud2 + Vus

2 + Vub2 = 0.99995(61)

meFs FV = −(0.0011± 0.0013)

Miller & Schwenk, Phys. Rev. C 78, 035501 (2008).


18

Superallowed decay of 10C – an outstanding problem for few-body physics “Needs” a nuclear

correction of 0.72%. T&H suggest 0.52±0.04%. An existing NCSM

calculation:

August 2009

10C

10BEB=12.0507(4) MeV

EB=15.6988(4) MeV

ft=3041.7±4.3

(98.53(2)%)

(1.4645(19)%)

Caurier et al., Phys. Rev. C 66, 024314(2002)

(0+,1)

(0+,1) E*=1.7415

(1+,0) E*=0.71835

(3+,0)


19

Axial structure of the nucleon

August 2009


20

Assuming no second class current.

The axial current is not conserved, even in the chiral limit. ◦ Partial conservation of the axial current (PCAC):

The axial constant is renormalized, in a relativistic manner:

August 2009

Axial structure of the nucleon

Bernard, Elouadrhiri, Meissner, J. Phys. G: Nucl. Part. Phys. 28, R1 (2002).

€

N p( ) Aaμ N ' p'( ) = u p'( ) GA q2( )γ μ +

GP q2( )2MN

qμ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥γ 5

τ a

2u p( )

€

2MNGA q2( ) + q2

2MN

GP q2( ) = 2 mπ2 Fπ

mπ2 − q2 GπN q2( )

€

gA =1.2695(29)

€

GA q2( ) = gA 1+rA

2

6q2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+ O q4( )

See O. Zimmer and S. Ando’s talks


21August 2009

Lattice calculations of the axial constant


22

One can asses the axial constant through AdS/QCD correspondence – using a conformal “cousin” theory of QCD which has a gravitational analogue in 5 dimensions.

A systematic way of including weak interactions into the AdS/QCD dictionary was recently proposed.

Using Sakai-Sugimoto model one gets a parameter free prediction: gA≅1.3.

Calculations of other weak form factors as well as nucleon forces are underway.

August 2009

Axial constant from AdS/QCD correspondence

DG, Yee, Phys. Lett. B670, 154 (2008).


23

The axial current is not conserved! Thus, its extension to nuclei is not trivial.

August 2009

Axial current in nuclei


24

A calculation of 3H b decay using cPT interaction and

currents

DG, Quaglioni, Navratil, Phys. Rev. Lett. 103, 102502 (2009)

August 2009

The calculation uses Idaho N3LO NN potential,Combined with N2LONNN force.


25August 2009

Step 1: use the trinuclei binding energies to find a cD-cE relation

Navratil et al., Phys. Rev. Lett. 99, 042501 (2007).


26August 2009

Step 2: calibrate cD according to the triton half life.

€

−0.3 ≤ cD ≤ −0.1 ⇓cE ∈ −0.220,−0.189[ ]

Doron Gazit - JLab Theory seminar

27

A prediction of 4He


29August 2009

A closer look into the weak axial correlations in 3H

NNN are not important???•The NNN force has a negligible effect.•Specific character of the NN force has minor effect, as long as it is “state of the art”•Caliration of cD is robust – depends weakly on the force.• Is this the origin for the success of EFT*?


30August 2009

Other checks

PANIC08 31

EFT* approach for low-energy nuclear reactions:

Weak curren

t

Chiral Lagrangian

QCDLow energy EFT

Nöther current

Wave functions

€

i ˆ J μa f

2

Phenom. Hamiltonian

Solution of Schrödinger equation

T.-S. Park et al, Phys. Rev. C 67, 055206 (2003), M. Rho nucl-th/061003; DG, Nir Barnea, Phys. Rev. Lett. 98, 192501 (2007); O’Connor, DG et al. Phys. Rev. C (2008).


32

Consistent calculations of weak and strong effects are possible.

The weak sensitivity of the weak decay to the NNN force make it an ideal candidate to constrain the NNN parameters.

The calibration of cD looks robust, whereas the value of cE will probably change when including 3NF N3LO potential.

Now we’re ready to look at the axial constant evolution in nuclei.

August 2009

Conclusions


33

Quenching of gA in nuclear matter?

b decay of 6He

August 2009

Vaintraub, Barnea, DG, Phys. Rev. C, 79 065501 (2009).€

ft ~ 1gA

2 MA2

€

MA2 = ψ i GTψ f

2

Surveys of β-decay rates of nuclei suggest that gAis gradually suppressed from ~1.27 to 1 (fully utilized A≅40).

0+1+: gA(q0)=1 in the quark

level. gA(q0)=1.27 in the

nucleon level. gA(q0)1 inside nuclei???


34

This is not surprising:◦ Axial current is not conserved.◦ Nucleons interact in nuclei.

However: ◦ A VMC calculation of the β decay 6He(0+)6Li(1+) used

AV18/UIX with phenomenological MEC and found: Single nucleon GT strength overestimates by 4% the

experimental strength. Adding MEC worsens the discrepancy to 5.4%.

Are the VMC wave functions to blame?

Are the MEC to blame? An exotic effect?

August 2009

gA Quenching in nuclear matter

Schiavilla and Wiringa, Phys. Rev. C 65, 054302 (2002).

Pervin et al., Phys. Rev. C 76, 064319 (2007).


35

We use the HH method to solve the 6 body problem, with JISP16 NN potential.

We use fourth order axial MEC calibrated in the triton.

Very rapid convergence:

August 2009

6He b decay

See A. Shirokov’s talk

E∞(6He)=28.70(13) MeVEexp(6He)=29.269 MeVE∞(6Li)=31.46(5) MeVEexp(6Li)=31.995 MeVGT|LO=2.225(2)GT=2.198(2)


36August 2009

6He b decay The contact interaction

that does not exist in pheno. MEC, has a opposite sign with respect to the long range one.

The final GT is just 1.7% away from the experimental one!

MEC brings the theory closer to experiment!

No dependence on the cutoff!

|GT|JISP16(6He)=2.198(7)|GT|exp(6He)=2.161(5)

ContactOPEC


37

The inclusion of cPT based MEC is helpful, even when one uses phen. interaction.

The conclusion is that the weak correlations inside the nucleus can lead to the observed suppression.

RPA surveys of μ capture showed that suppression is needed only in GT channel – consistent with MEC.

cPT estimation for the suppression of gA in infinite nuclear matter: ◦ dgA/gA~+8% - +13% due to long range MEC.◦ dgA/gA~-28% due to contact interaction.◦ dgA/gA~-15% - -20% total.

August 2009

6He b decay and a hint to heavier nuclei

Zinner, Langanke, Vogel, Phys. Rev. C 74, 024326 (2006).

Park, Jung, Min, Phys. Lett. B409, 26 (1997).


38

Weak structure of the nucleon from μ capture

μ capture on 3He

August 2009

DG, Phys. Lett. B666, 472 (2008).


39

In QCD, the induced pseudoscalar form factor gP depends on the axial form factor.

Adler, Dothan and Wolfenstein: HBcPT verified this result and connected it to

QCD, as well as allowed corrections to the formula.

A comparison to experiment needs higher momentum than b decays – μ capture.

August 2009

Induced Pseudoscalar

€

GP q2( ) = 4MN gπN Fπ

mπ2 − q2 − 2

3MN

2 rA2

Adler, Dothan, Phys. Rev. 151, 1257 (1966).Bernard, Kaiser, Meissner, Phys. Rev. D 50, 6899 (1994), Kaiser, Phys. Rev. C 67, 027002 (2003).€

gP q2( ) =mμ

2MN

GP q2( )


40

Since μ is close to the atom so the capture probability is bigger: .

The rates become comparable for Z~10. In proton, 0.16% branching ratio of OMC.

August 2009

Ordinary muon captureeμ

€

n e€

ne

€

τ μfree = 2.197019(21) ×10−6 sec

€

aBμ = h

Zmμcα= me

mμ

~1/ 207{

aBe

€

Z ⋅ψ1S 0( )2

~ mμ me( )3Z 4

nμ


41

The branching ration is very small (10-8 in hydrogen).

August 2009

Radiative Muon Capture

nμ

γ


42

Due to the huge effects of the nuclear structure, studying the weak structure of the nucleon in muon capture processes has reduced to the proton.

Studies of OMC and RMC on hydrogen are hard:◦ Depend on the transition rate between ortho- and

para-hydrogen.◦ Have small branching ratios.

August 2009

Muon capture on the proton


43August 2009

Induced pseudo scalar from μ-p The MuCap result:

is consistent with cPT prediction:

but with far bigger uncertainty. The RMC result clearly deserves more work,

though probably in the atomic side.

More information is needed from other nuclei.

€

gPexpr = 7.3 ±1.2

€

gP = 8.26 ± 0.23

€

gPexpr =12.8 ±1.1

RMC: G. Jonkmans et al., Phys. Rev. Lett. 77, 4512 (1996) OMC: V. A. Andreev et al., Phys. Rev. Lett. 99, 0322002 (2007).


44

For the (exclusive) process 3He(μ-,nμ) 3Han incredible measurement (0.3%) exists:

ab-initio calculations, based on phenomenological MEC or D excitation:◦ Congleton and Truhlik [PRC, 53, 956 (1996)]:

150232 Hz.◦ Marcucci et. al. [PRC, 66, 054003(2002)]: 14844

Hz. The main critique – too much freedom, without

microscopic origin. ◦ Did not include radiative corrections increase the cross

section by 3.00.4%.August 2009

OMC by 3He: 3He(μ-,nμ) 3H

Ackerbauer et al, Phys. Lett. B417, 224 (1998).

€

Γ μ−+3He → ν μ +t( )stat=1496 ± 4 Hz

Czarnecki, Marciano, Sirlin, Phys. Rev. Lett 99, 032003 (2007)


45

Using the EIHH method to solve for the wave functions, with AV18/UIX potential:

Only free parameter calibrated using triton half-life.

To be compared with: The dependence on the nuclear model is

negligible. The role of MEC ~ 12%! (compare to the 2% in

triton b decay where it was calibrated). This allows to constrain the weak structure of

the nucleon.August 2009

OMC on 3He: 3He(μ-,nμ) 3H

€

Γ =1499(2)Λ (3)NM (5)t (6)RC =1499 ±16 Hz

€

ΓEXP =1496 ± 4 Hz


46August 2009

Resulting form factors:Form Factor This work Theoretical

estimationExperimental

Pseudo-scalar gP(q2=-0.954mμ

2)

8.13±0.5 7.99±0.2(HBcPT)

gP(q2=-0.88mμ2)=

7.3±1.1

Induced scalar meFS/FV

(0.5±2)×10-4 - -0.0011±0.0013 (Towner & Hardy)

Induced pseudo-tensor GT

QCD sum rules

€

gt

gA

= −0.1(0.68)

€

gt

gA

= −0.0152(53)

€

gt

gA

< 0.36 at 90%

H. Shiomi, J. Korean Phys. Soc. 29 (1996) S378.


47

Few body nuclear physics acts as a pivot between QCD and heavy nuclei.

The current precision era in few-body nuclear physics provides an opportunity to study the weak structure of the nucleon:◦ Using precision measurements of weak interactions in

nuclei one can constrain the bare form factors, as well as their “evolution” inside nuclei, without free parameters.

◦ Constraints on strong properties are possible.◦ In particular, the upcoming MuSun measurement of μ

capture on the deutron will enable:to calibrate the 3NF at the 2-body level!

August 2009

Conclusions and Outlook


48

◦ Going to heavier nuclei, mainly A=6-8 and A=10, within cPT, should be a holy grail, as it will open the door to new constraints of CVC and second class currents.

Microscopic understanding of weak reactions validates cross-sections predicted for astrophysics, which are out of reach experimentally.

Using AdS/QCD for the calculation of weak couplings of the nucleon seems like a good approximation!

Open questions:◦ Role of D excitations in weak reactions within cPT?◦ Role of a1?◦ How far can we go in momentum transfer within cPT?

August 2009

Conclusions and Outlook


49August 2009


50

Precision era in few-body nuclear physics

ab-initio calculations HBcPT

Available methods for solving exactly the Schrödinger equation for few body systems, from their nucleonic degrees of freedom.

HH NCSM GFMC FY

High precision nuclear interaction, phenomenological or cPT based.

Consistent microscopic approach for the construction of (meson exchange) currents in the nucleus.

August 2009

Allows parameter free calculations of nuclear wave functions and low-energy reaction rates, with sub-percentage accuracy.

Allows extraction of the weak structure of the nucleon from the strongly-correlated nuclear wave function.

Offers a hint on the in-medium evolution of the weak structure.


51

Contrary to the vector coupling, the axial constant is renormalized.◦ Had the quarks were non-relativistic:

◦ The deviation is a reflection of the relativistic dynamics of the u and d quarks in the nucleon.

Thus, its numerical calculation is a test for our understanding of QCD.

Still, experiment provides the most accurate result.

August 2009

The axial constant

€

gA = r σ r τ = 5 /3

€

gA =1.2695(29)

€

GA q2( ) = gA 1+rA

2

6q2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+ O q4( )

See O. Zimmer and S. Ando’s talks


52

At finite momentum: From neutrino scattering:

From pion-electroproduction:

This axial radius discrepancy was solved in Baryon cPT, which allowed including finite pion mass in the pion-electroproduction.

The “radius” measured in pion-electroproduction:

August 2009

Axial radius – cPT success I

€

rA2 = 0.666 ± 0.014 fm2

€

rA2 = 0.639 ± 0.010 fm2

€

rA2

π −elec.= rA

2 + 364Fπ

2 1− 12π 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Bernard, Kaiser, Meissner, Phys. Rev. Lett. 691, 877 (1992).

€

GA q2( ) = gA 1+rA

2

6q2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+ O q4( )


53

Schindler et al. have included a1 in their manifestly Lorentz invariant cPT.

They showed that it has an effect only at higher energy.

August 2009

Role of a1 at higher energies:

Schindler et al, Phys. Rev. C, 75, 025202 (2007)


54

gA Quenching in nuclear matter

Renormalization of gV(q0) Renormalization of gA(q0)

FV(q0)=1 in the quark level.

FV(q0)=1 in the nucleon level.

FV(q0)=1 inside nuclei.

gA(q0)=1 in the quark level.

gA(q0)=1.27 in the nucleon level.

gA(q0)1 inside nuclei???

August 2009

“Restoration of axial symmetry”. The implications are immense, e.g., weak

reaction rates in supernovae.

Documents

Weak Structure of the Nucleon