Hadronic Violation Theory Mike Snow Indiana University I am not and never have been a member of the Communist Party theorist, And if you ask me I’ll take the 5th Amendment

But since there is no theory for hadronic parity violation, the damage that an experimentalist can do to this subject is limited

The reason why there is no theory is because such a theory would need to understand -quark correlation effects in a two system. We are still trying to understand single quark effects in a one-nucleon system From all of our previous experience with many-body systems, we know that it is very important to understand the ground state of the theory, and if the ground state is not boring it is typically highly correlated. The QCD ground state is not boring: in fact it has two condensates ( and quark-antiquark pairs) which are difficult to probe directly The NN weak silently probes quark-quark in a way which does not excite the ground state of QCD.

What, you ask, is QCD? U(1) Local Gauge Invariance ⇒ QED If we demand that the electromagnetic Lagrangian be invariant under local (space-time-dependent) U(1) gauge transformations: 1 ! (x) " ei# (x)! (x), A " A + $ #(x) µ µ e µ It is possible if we add to the free Dirac L an interaction term

µ µ –V = –j emAµ = +e ψ γ ψ Aµ of a vector current with a new gauge field so that: " = i!" µ # ! $ m!! + e!" µ! A %". ! µ µ Adding new term is equivalent to defining a covariant derivative: i" ( x) %µ # Dµ & %µ $ ieAµ such that Dµ! # e Dµ! under local gauge transformation Also must add kinetic term quadratic in new vector () field. To maintain local gauge invariance, use : 1 1 N.B. A term F A A -A (x)* -A (x)* F µ! / &µ ! % &! µ . &µ + ! + &!$ ( %&! + µ + &µ$ ( = µ! 2 µ , e ) , e ) ~m AµA would violate local gauge µ µ 1 µ! ' LQED =" (i# &µ % m)" + e"# " Aµ % Fµ! F 4 invariance ⇒ massless photon!

µ! ! Euler-Lagrange eq’n with respect to Aµ ⇒ Maxwell Eq’ns: "µ F = j . Hence, U(1) local gauge invariance ⇒ QED, a covariant field theory of charged interacting with a massless vector field satisfying Maxwell Eq’ns and originating from a conserved vector current ! QCD Interactions Arise from are -1/2 Dirac particles that come in 3 colors: R, G, B Antiquarks carry anti-color = “color propagating backwards in time”: R, G, B Quarks and antiquarks interact by exchanging bi-colored gluons that form an SU(3) color octet:

λ8 ∑ λ8 λ8 |B〉 GB +3 RB

|G〉 |R〉 [RR − GG]/√2 GR RG λ3 λ3 ∑ λ3 -2 -1 +1 +2 |R〉 |G〉 [RR+GG−2BB]/√6 -3 |B〉 BR BG

For example, from the point of view of The color singlet combination: color flow: {RR + GG + BB}/√ 3 is not q available to gluons. q q q − GB SU(3)-Color Local Gauge Invariance ⇒ QCD For quarks of a given flavor, the general local gauge transformation allows color changes, in addition to space-time-dependent phase rotations: i ( x)Tˆ quark field q(x)$Uˆq(x) # e "%a a q(x) with q(x) a 3 ! component color state ˆ ˆ ˆ [Ta ,Tb ]=ifabcTc with real fabc antisymmetric under index pairwise swap Non-commutivity of generators ⇒ non-Abelian group ⇒ surprising consequences of insisting on local gauge invariance. Introducing interaction with vector gauge fields (here, corresponding to 8 gluons) must be done with the transformation which mixes gluon colors! 1 Ga Ga (x) f (x)Gc . µ $ µ " #µ%a " ! abc %b µ g b,c Last term embodies coupling among gluon fields, and demands consequent term in gauge-invariant field strength tensor:

a a a b c Gµ! " #µG! $ #!Gµ $ g% fabc Gµ G! & SU(3) locally gauge $ invariant b, c ( + 1 " µ µ ˆ a a µ! QCD = q (i' #µ $ m )q $ g * q' %Taq- Gµ $ Gµ! Ga . ! ) a , 4 This non-Abelian (Yang-Mills) gauge field theory involves not only QED- like qqg vector coupling vertices, via 2nd term, but also ggg and gggg vertices via 3rd term ⇒ QCD ! So, If mγ = mgluon = 0, Why is the Between Hadrons Short (~1 fm) Range? QCD vs. QED In QCD, gluons carry color, while do not carry ! q q 8 independent gluons represent − possible color-anticolor combos, GB excluding singlet (RR + GG + gluon BB) ⇒ 3- and 4-gluon q q interactions allowed!

Gluon-gluon interactions:  strongly affect polarization of vacuum ⇒ many gluons and qq pairs excited near a color charge  replace charge screening of QED (bare charge larger than seen from a distance) with color anti-screening  make color very strong at long distance (low momentum xfer) ⇒ “infrared slavery”, weak at short distance (high mom. xfer) ⇒ “asymptotic freedom”  confine color: stretch color “flux tube” btwn quarks too far and it breaks, leading to formation  imply colorless hadrons interact via residual inter’n that

can be viewed as meson exchange, of range λπ ≈ 1.4 fm QCD ⇒ QED-like Vector Currents, With Important Differences Introduced by Gluon Self-Coupling The gluon loops change the sign of the correction from QED. For SU(3) color, Gross, Wilczek + … & Politzer (2004 Nobel Prize) found:

2 2 )s (µ ) )s (Q ) = . ) (µ 2 ) & Q2 # 1 s (33 2n )ln + ' flavors $ 2 ! The + sign in the denominator ⇒ color 12( % µ " force weakens with increasing Q2 or decreasing distance ⇒ asymptotic freedom. Confirmed by measurements of QCD processes vs. energy. Perturbation approach works only for 2 hard (high Q ) collisions where αs small compared to 1. Even then, can’t expect QED-like precision from first order ! α ≳ 1 for Q2 < 1 GeV 2, blows up for momentum transfers ~ s αs ∴ for hadrons of size ~ 1 fm ΛQCD ≈ 200-300 MeV ⇒ infrared slavery. Quarks and gluons confined within colorless hadrons. Complexity of QCD Vacuum ⇒ Rich Hadron Substructure As seen by a high-energy probe, a hadron shares its momentum among many partons, not just 3 valence quarks (or a valence qq pair), but a multitude of gluons and sea qq pairs as well.

Reactions that transfer large momentum, like the EM processes pictured below, excite the nucleon with small wavelength. The larger the momentum transfer, the more gluons and sea quarks we see.

xf(x) in from fits to world PDF database Phase Diagram for QCD Ordinary, zero-temperature, Fermi liquid nuclear matter represents just one point in a more general view of the possible states of strongly interacting matter!

As one raises T or in- creases density, quarks & gluons should re-emerge as the funda- mental degrees of freedom.

Increasing net baryon density → At densities several times higher than normal nuclear matter saturation point, as obtainable in the cores of stars, speculation focuses on “color superconducting” states: bi-colored pairs of quarks condense in the lowest-E state, giving gluons an effective mass via coupling to the pairs. At low baryon density but high temperature, a transition to a plasma of quarks & gluons is anticipated. Collisions of heavy nuclei at RHIC attempt to form matter retracing the path of the early . Constituent vs. Current Quarks and Hadron Mass

For hadrons built from quarks: Effective Quark Mass in Matter m m"current" (millions of electron volts) hadron >> ! u,d 1000000 where mcurrent are inferred from QCD 100000 QCD Mass Higgs Mass Lagrangian. Hadron structure models 10000 invoke effective “dressed” (constituent) 1000 quark inside hadronic matter: 100

constituent current 10 mu,d " 300 MeV vs. mu,d ! few MeV; constituent current 1 ms " 500 MeV vs. ms ! 100 MeV; u d s c b t constituent current mc " 1.65 GeV vs. mc ! 1.3 GeV. Quark Flavor

Similar phenomena occur in condensed matter physics: for example the effective mass for the motion of an electron in matter is not the same as its “free” mass. Once again, we see that QCD exhibits the features of a condensed, many-body medium. Not only that, for light quarks u and d, the size of the effect is HUGE! To understand this requires an explanation of , the chiral of QCD, and its apparent “spontaneous” breakdown Helicity and Chirality Energy eigenstates of free are also states of well-defined helicity λ ≡ ½ (σ⋅ p)/ | p | = ± ½ . Operator σ⋅ p in HDirac ⇒ no other spin component commutes with H and can be sharp in an energy eigenstate. ! Choosing zˆ|| p, it is easy to show that : ! ! * pz / | p | ' * pz / | p | ' ! (1) (1) ! ( % ( % (1) ! *, + pˆ 0 '*u ' *(E + m)ulower / | p |' 0 0 #+ u ( p) for p > 0 upper ( % ( % ( % z ( ! %( % = (1) ! = ! = ! = " ( 0 pˆ %( (1) % ((E m)u / | p |% ((E m) / | p |% (| p | /(E m)% (1) ! ) , + &)ulower & ) $ upper & $ + !$ u ( p) for pz < 0 ( % ( % ) 0 & ) 0 & States of good helicity cannot be states of good parity in general, ∧ because parity and σ⋅p operators do not commute!

5 γ is called the “chirality” ' 0 1ˆ $ operator. The operators ) 5 ( i) 0) 1) 2) 3 = % 2 " = ) 5† ! () 5 )2 =1ˆ ; ) 5,) µ = 0. % ˆ " 4 { } ½(1 + γ 5) and ½(1 − γ 5) are &12 0 # QM projection operators u( p) 1 (1 5 )u( p) 1 (1 5 )u( p) which project out right- = 2 + " + 2 ! "

and left-handed chiral = uR ( p) + uL ( p) components, respectively, 1 5 1 5 of u(p). The Dirac energy But : 2 (1+ " )v( p) = vL ( p), 2 (1! " )v( p) = vR ( p) and helicity eigenspinors Helicity and chirality are therefore not identical are not γ 5 eigenstates concepts if m>0. Helicity is conserved, but is unless m=0. frame-dependent: γ 5 commutes with Lorentz trans. QED and QCD Vector Currents: Parity Conservation, Chiral Symmetry, and (Approximate) Helicity Conservation Under parity:ψψ (scalar) is unchanged, ψ γ 5ψ (pseudoscalar) changes sign, and # () 0( J %µ % µ % 0 µ 0 J vector ; vector &( ) ( =() ) ) ( = + µ = " µ'0 !$() ( #$() 5) 0( J %µ % 5 µ % 0 5 µ 0 J axial axial &( ) ) ( =() ) ) ) ( = $ µ = " 5 µ'0 !() ) ( Both QED and QCD interactions involve the conserved Dirac (4-vector) current ψ γ µ ψ . The product of vector currents is even under parity, so both QED and QCD conserve parity. Furthermore, there is no vector coupling between L- and R spinors, leading to a property known as chiral symmetry: µ † 0 µ † 1 5† 0 µ 1 5 uL! uR = uL! ! uR = u 2 (1" ! )! ! 2 (1+ ! )u = 1 u†! 0 (1+ ! 5 )! µ (1+ ! 5 )u = 1 u! µ (1" ! 5 )(1+ ! 5 )u = 0. 4 4 e− λ = −½ λ = +½ λ′ = +½ λ′ = +½ For m=0 or E >>m, e− λ′ = −½ this means that λ = +½ e−(−E,−p) λ = −½ helicity is conserved at the vertex λ′ = −½ e−(−E,−p) Chiral Symmetry of QCD in Massless Quark Limit QCD symmetries important for light quarks can be seen by rewriting L in terms of chiral [(1±γ5)/2] projections of the quark spinors: % ( 1 " µ µ ˆ a a µ+ QCD = q (i! "µ # m )q # g ' q! $Taq* Gµ # Gµ+ Ga ; & a ) 4 % ( % ( " flavor j L µ ˆ a L R µ ˆ a R QCD = q j ! ' i"µ # g$TaGµ * q j + q j ! ' i"µ # g$TaGµ * q j & a ) & a )

1 a µ+ L R R L # Gµ+ Ga # m j ( q j q j + q j q j ) ! 4 For flavors where mj << ΛQCD (certainly u and d, perhaps s) we can neglect the mass terms, and also mu~ md. Then two simplifications occur: 1) L is flavor-independent, ∴ invariant under flavor transformations ⇒ SU(2) (good) or SU(3) flavor symmetry (not so good). 2) L-handed and R-handed quarks interact separately, without mutual coupling ⇒ L is invariant under independent flavor

transformations for L- and R-handed quark sectors. This is SU(2)L × SU(2)R [or SU(3)L × SU(3)R ] chiral symmetry. Chiral symmetry ⇒ separately conserved L & R flavor-changing currents:

µ, L L µ ˆ L µ, R R µ ˆ R ˆ J! = q " t! q , J! = q " t! q with t! = 3 SU(2) or 8 SU(3) flavor generators; µ, L µ, R µ,V µ, A µ,V µ ˆ µ, A µ 5ˆ $µ J! = $µ J! =0 % $µ J! = $µ J! = 0 where J! # q" t! q; J! # q" " t! q. Isospin Symmetry in QCD Isospin: Introduced (1930’s) by Heisenberg, Wigner to distinguish n and p as two “substates” of the same particle (“nucleon”, N), distinguished by their isospin projection, in analogy to a spin-1/2 particle.

Thus: | p! =| I = 1/ 2, I3 = +1/ 2!; | n! =| I = 1/ 2, I3 = "1/ 2! Where subscript “3” denotes analog of z-projection in abstract 3-dimensional “isospin space”, where isospin operators generate rotations, just as normal angular momentum operators generate rotations in ordinary 3-dimensional physical space. In NN system, in analogy to two coupled spin-1/2 particles, we have: | I 1, I 1 | I (1) 1/ 2 | I (2) 1/ 2 | p(1) p(2) ; = 3 = + ! = 3 = + ! 3 = + ! = ! Triplet of 1 I=1 states, | I = 1, I3 = 0! = {| I3(1) = +1/ 2!| I3(2) = "1/ 2! + | I3(1) = "1/ 2!| I3(2) = +1/ 2!} 2 Symm. 1 under {| p(1)n(2)! +| n(1) p(2)!}; 2 particle interchange | I = 1, I3 = "1! =| I3(1) = "1/ 2!| I3(2) = "1/ 2! = | n(1)n(2)!;

1 | I = 0, I3 = 0! = {| I3(1) = +1/ 2!| I3(2) = "1/ 2! " | I3(1) = "1/ 2!| I3(2) = +1/ 2!} Singlet 2 Antisym. 1 {| p(1)n(2)! " | n(1) p(2)!} I=0 state 2 Other Examples of Isospin Multiplets:

+ 0 − : π , π , π , all with mπ ≈ 140 MeV, spin-0 ⇒

+ 0 " |# ! =| I = 1, I3 = +1!; |# ! =| I = 1, I3 = 0!; |# ! =| I = 1, I3 = "1!

++ + 0 − Deltas: Δ , Δ , Δ , Δ , all with mΔ ≈ 1233 MeV, spin-3/2, excited states of the nucleon first observed as resonances in ++ -nucleon scattering I = 3/2 Δ ⇒ Δ

Quark level: u and d are the fundamental isospin doublet – isospin is a subset of “flavor” – the small bare mass difference m m is (mainly) 0 u ≠ d Δ responsible for small violations of isospin conservation in strong interaction processes. Isospin Symmetry in Pseudoscalar and Vector

K0(498) K+(494) ds Y us Y Pseudoscalars have +1 +1 S=0 and q and q in L=0 η0(547) η′0(958) state ⇒ negative parity − du -ud π (140) I π+(140) I -1/2 +1/2 3 -1/2 +1/2 3 -1 π0(135) +1 -1 +1

-1 -1 us -ds The vector mesons − 0 K (494) K (498) have L=0, S=1 ⇒ JP=1−. They are Flavor octet Flavor singlet typically several K*0(896) K*+(892) hundred MeV higher ds Y us Y in mass than the +1 +1 light pseudoscalars.

− du -ud ρ (770) I ρ+(770) I -1/2 +1/2 3 -1/2 +1/2 3 -1 ρ0(770) +1 -1 +1 In both cases, isospin -1 -1 us -ds symmetry leads to K*−(892) K*0(896) near-degenerate masses in a multiplet Flavor octet Flavor singlet Isospin Symmetry in the JP=3/2 + and 1/2 + (L=0)

Y ddd duu ddu uuu Δ(1232) I=3/2

dds uds uus *(1385) I=1 I3 Σ

dss uss Ξ*(1530) I=1/2

sss Ω−(1672) I=0

udd Y uud n p +1 Both sets of L=0 Σ0(1193) baryons exhibit − uds + Σ (1189) dds uus Σ (1197) isospin symmetry I -1 -1/2 uds +1/2 +1 3 Λ0(1116)

-1 Ξ−(1321) dss uss Ξ0(1315) QCD Ground State Breaks Chiral Symmetry, Gives Large “Effective Mass” To Light Quarks

Light hadrons form SU(2), but not SU(2)L × SU(2)R flavor multiplets. Why?

LQCD invariant under parity (L ↔ R) and, for mq=0, under chiral trans- formations. But parity and chiral (L vs. R) operators do not mutually commute! Could accommodate by mass degeneracy of opposite parity hadrons, but this is not observed, e.g., no 0+ to match 0– pions.

Chiral symmetry is explicitly broken by the quark masses, but these effects are very small for u,d. Most of the breaking is attributed to spontaneous breaking of chiral symmetry in the QCD vacuum:

L R R L † 0 ! q! q 0 " 0 # 0 ! q ! q 0 = 0 ! q ! q 0 " 0, Effective Quark Mass in Matter L L R R while 0 ! q ! q 0 = 0 ! q ! q 0 = 0. (millions of electron volts) 1000000 SSB of a continuous symmetry ⇒ 100000 QCD Mass Higgs Mass massless scalar Goldstone . 10000 – Here need pseudoscalar (0 ) bosons 1000 ± 0 (triplet for SU(2) χSB ⇒ π , π ; octet for 100

SU(3) ⇒ π, K, η) to enable quark helicity- 10 flip: R qL q 1 u d s c b t L=1 0– Quark Flavor QCD Summary: Confinement and Chiral Symmetry Breaking

Renormalizable gauge field theory, a generalization of QED to an interaction with 3 types of (R,G,B).

Vector interaction of quarks with gluons ->LQCD conserves parity (like EM)

In mq=0 limit and for u and d quarks, a SU(2)L × SU(2)R symmetry in L. This full symmetry does not appear in the QCD spectrum: spontaneous breaking of chiral symmetry in the QCD vacuum occurs. The dynamical mechanism for how this happens is not understood. χSB reduces the remaining symmetry to SU(2) transformations among the u and d flavors (isospin). Due to small current quark masses, this symmetry is approximate (good to ~1%) but still very useful

QCD gauge coupling becomes large at large distance according to perturbation theory. Quarks “must” be permanently confined since we have never isolated one.The mechanism by which quarks are permanently confined is not completely understood.

m q 0 ! q ! q 0 ~ 50 M e V only, so where does the proton mass come from? energy from the 1 a µ! QCD “gluon condensate”! < 0 | Gµ! Ga | 0 > 4 Reminder: Basic Features of the Weak Interactions Charge-changing weak currents are mediated by W ± exchange; neutral weak currents mediated by Z 0 exchange. µ− ν µ ν ν − W 0 e− Z

νe q q Heavy exchange ⇒ very short range, often approximated by contact interaction at low and moderate momentum transfers.

!c 0.197 GeV # fm !18 $W = 2 = = 0.0025 fm = 2.5"10 m MW c 80.4 GeV

Charged weak currents change quark flavor, but neutral currents don’t:

d d u u Λ → nπ 0 n u u p Λ d d n d u s d does 0 W− Z proceed e− u OK π0 via W u νe exchange Parity Violation and the Structure of the Weak Current Parity violation in weak interactions was first predicted by Lee & Yang (1956) to (νe )R account for observed K+ decay to both even (ππ ) and odd (πππ ) parity final + states. Subsequently confirmed in nuclear β -decay by C.S. Wu at NBS via (e− ) 60Co 60Ni* L 〈 s ⋅ p 〉 ≠ 0 pseudoscalar correlation, Co e J=5 J=4 consistent with emission of left-handed – (negative helicity) e and right-handed νe . Weak currents can therefore not be pure vector u ! µ u or pure axial vector u ! µ ! 5 u in form. Observation of only L-handed suggests equal mixture of the two: µ µ 1 5 e.g., J = u"! 2 (1# ! )ue .

In the , parity violation is included by allowing the () weak interaction to act only on L chiral component of u(p) particle spinors (“maximal” parity violation). Why Nature chooses to do this is not yet fully understood, but it is pretty enough to be interesting. Leptonic Weak Process ( Decay)

k νµ & g - 1 5 # p M ( ( = u+ (k). (1( . )uµ ( p) ) µ *e + µ+ e $ µ ! – 2 2 µ % " – q p′ – W e q q & g - , # $( -, + 2 ! MW & g , 1 5 # % " u ( p'). (1( . )u ((k') –k′ νe 2 2 $ e + e ! q ( MW % 2 2 "

– – using the Feynmann rules for the tree-level amplitude. In µ → e νµνe , 2 2 momentum transfer q << MW , so to excellent approximation:

) g , 1 5 & 1 ) g 1 5 & M # # * u+ (k)- (1# - )uµ ( p) 2 ue ( p")- , (1# - )u+ (#k") µ *e + µ+ e ' µ $ ' e $ ( 2 2 % MW ( 2 2 %

GF , 5 5 ! u+ (k)- (1# - )uµ ( p) [ue ( p")- , (1# - )v+ (k")] with the weak coupling [ µ ] e 2 g 2 2 constant first introduced by Fermi : GF ! 2 . 8MW Semileptonic Weak Process: Neutron and Nuclear β-Decay In weak interactions of quarks bound within baryons (spin ≠ 0), the charged current need not take the simple V – A form found for . In general, allow for various bilinear coupling forms: * ' µ 2 µ 2 µ$ 2 µ 5 2 5 µ J baryon (x) =! f ( f1(q )" + if2 (q )# q$ + g1(q )" " + g3(q )" q %! i ($!#!" $!!#!!" $!#!" $!#!"% ) vector tensor axial vector pseudoscalar & Nuclear β-decay proceeds via: e – e +

V ν V ν ud W – e or ud W + e d u u d n d d p p d d n u u u u

– For free , only n → pe νe (τ = ? s) energetically allowed. Inside nucleus, energy release depends on of initial and final-state nucleons. In any case, q2 → 0 and no evidence seen of need for tensor or pseudoscalar terms in nucleon decay:

2 2 2 q ! 0, f1(q ) ! gV , g1(q ) ! g A " µ µ 5 + + J n! p (x) =$ p (x)% (gV + g A% )#ˆ $ n (x), with #ˆ an isospin raising operator (Selected) Feynman Rules in Electroweak Theory f f1 2 γ EW " ieQ ! µ # ieQ " ! " iM ! f1 f2 gµ! – (e ) f1 " i 2 f d′ (u ) νe q 2 + – g 1 5 W (W ) g 1 5 " i ! µ (1" ! ) # i ! " (1# ! ) 2 2 2 2 u (d ) 2 e+ ( ) ′ g ! q q / M νe ! i µ" µ " W q2 ! M 2 W f f1 2

0 g 1 f f 5 g µ 1 f1 f1 5 Z # 2 2 # i ! c # c ! $ i ! (cV $ cA ! ) ( V A ) cos 2 cos"W 2 "W 2 f f1 g q q / M 2 Standard Model Predictions: µ" ! µ " Z ! i 2 2 Q c f = T 3 c f = T 3 – q ! M Z f A f V f 1 2Q sin2θ $ip'x "=±1 f W e.g., Zµ = # µe , with # µ = " {0;1,± i ,0 }; 1 2 νe ,νµ ,ντ 0 1/2 2 On-shell W or 1 ! ! – – – 1 2 Z: "=0 e ,µ ,τ –1 –1/2 " (1" 4sin !W ) # = {| p |;0,0, E } for zˆ|| p 2 µ M 1 8 2 → free particle u,c,t 2/3 1/2 2 (1" 3 sin !W ) state µ (" ) (" ) pµ p! 1 4 2 % p # µ = 0, # µ *#! = $gµ! + 2 . d,s,b -1/3 –1/2 " 2 (1" 3 sin !W ) & " M Nonleptonic Weak Processes: NN Weak Interaction Although the quark-quark weak interactions at tree level can be written down in the same way as for leptonic and semileptonic processes, now BOTH the quarks in the initial AND final states are bound within hadrons by QCD. For the weak interaction between nucleons, some (of the many) possible processes are (not showing the gluons!):

u u u u n p p d d d d p u d u u W – Z d d d u n n d d p d d n u u u u

u u u u p d u p p d d p u d u u W – Z d d d d n d d n d d n u u n u u

Many possible amplitudes can contribute, but nonperturbative QCD makes direct calculation of the amplitudes difficult. Is there anything simple we can say? qq Weak Interaction: Isospin Dependence The quark-quark weak interaction at below the W and Z mass can also be written in a “current-current” form, with contributions from charged currents and neutral currents 2 2 g † µ g † µ M CC = 2 Jµ,CC JCC ; M NC = 2 2 Jµ,NC JNC 2MW cos !W M Z

µ 1 µ 5 $ cos! sin! ' $ d ' µ 1 µ q q 5 JCC = u " (1# " )& ) & ) ; JNC = * q " (cV # cA" )q 2 % #sin! cos! ( % s ( q=u,d 2

The weak interaction does NOT conserve isospin, but QCD conserves isospin to a good approximation. Therefore we can classify what are the possible isospin changes from qq weak interactions as follows:

2 2 Charged current: ΔI=0,2 (~ V ud), ΔI=1 (~ V us) : ΔI=0,1,2.

The ΔI=1 terms comes only from the quark-quark neutral currents in the absence of strange quarks (due to small size of Vus, we expect this term to 2 be suppressed anyway by a factor of V us~0.05 These terms are about the same size, so any large differences in different channels can only come from the ground state dynamics of QCD. This is one of the few ways to probe the nonperturbative ground state of QCD NN Weak Interaction: Independent Amplitudes Consider elastic NN scattering at low enough energies that the nucleons are nonrelativistic. In this case, we can describe the scattering using phase shifts in a partial wave expansion. For spinless particles, you remember the usual form of the partial wave expansion # sin kr & ! ' + ( e ikr b P cos ( 2 ! ) e ikz f k 2 , ! $r$" #$" ) ! ! ( %) = + ( %) * ! = 0 kr r # 1 f (k 2 ,%)= (2! + 1) e 2i(! & 1 P (cos%) with real phase shifts ( ) 2ik ( ) ! ! !=0 In our case, since the nucleons are spin 1/2, we need the spin- dependent generalization of the partial wave expansion: e ikr & """#e ik z S , m + S , m S , m M (k 2 ,% ) S , m r #$ i Si ! f S f f S f i Si r S , m f S f And we can measure the elements of M. Recall that the parity operator inverts space: ! ! Pˆf (r) = f ($r); under Pˆ, (x, y, z) % ($x,$ y,$z), (r,#,!) % (r," $#," +!) ∧ ∧ P 2 = 1 ⇒ parity eigenfcns are either even (eigenvalue P = +1) or odd (P = -1) ˆ L PYLM (!,") = YLM (# $ !,# + ") = ($1) YLM (!,") for spherical harmonics So, states of good orbital angular momentum quantum # L have well-defined spatial parity (−1)L. NN Weak Interaction: Independent Amplitudes

If we consider low energy NN scattering, the parity violating amplitudes are those that connect L=0 and L=1 two-nucleon states. We can classify them.

Total NN (two identical fermion) wave function NN ! ! "tot (1,2) =# space (r1,r2 )| Stot , Sz !| Itot , I3! must be A under interchange, restricting Pauli-allowed L,S,I combinations:

Itot = 1 (isospin-S ):

1 1 1 Space-S (even L) ⊗ spin-A (Stot = 0) ⇒ S0 , D2 , G4 , … 3 3 or Space-A (odd L) ⊗ spin-S (Stot = 1) ⇒ P0,1,2 , F2,3,4 , … (2S+1) LJ notation, with L=0,1,2,3,4,… I = 0 (isospin-A ): tot denoted as S,P,D, 1 1 Space-A (odd L) ⊗ spin-A (Stot = 0) ⇒ P1 , F3 , … F,G,… 3 3 3 Space-S (even L) ⊗ spin-S (Stot = 1) ⇒ S1 , D1,2,3 , G3,4,5 , … Only upper 2 rows available for pp or nn. For np system, all combinations available. We therefore have 5 independent NN parity-violating amplitudes:

3 1 3 3 1 3 S1⇔ P1(ΔI=0, np); S1⇔ P1(ΔI=1, np); S0⇔ P0(ΔI=0,1,2; nn,pp,np) Some Approaches to Non-Perturbative QCD Lattice QCD: solve QCD numerically for quarks and gluons confined to finite space-time lattice; extrapolate to continuum limit (lattice spacing → 0) and to light quark masses (so-called chiral limit) to predict properties of hadrons and of quark-gluon matter. Great recent progress & success! Concerns: provides limited insight; CPU limitations ⇒ limits to applications, extrapolation uncertainties. : impose QCD symmetry constraints on L written in terms of effective degrees of freedom (not quarks and gluons) most relevant at low energies; adjust numerous coupling strength parameters to reproduce experiment (à la Standard Model). Used for low-E πN and NN interactions, and to extrapolate lattice

QCD results to mq → 0. Rigor at given expansion order, but applicability range limited to low momentum transfer processes. QCD-inspired models: e.g., constituent-quark potential models of hadrons; one-boson-exchange models of NN interaction, chiral models, instanton models, etc. Wide applicability and predictive power, once parameters set for simple systems, but (except for chiral EFT!) lose guarantee of controlled expansion. Fortunately for us, strong NN interaction is well-measured and understood. Weak NN can be added as a perturbation Symmetry Constraints on the Strong NN Potential At low E, the NN system can be treated by solving the Schrödinger eq’n with a Hermitian potential that may depend on the vector separation and on the nucleon spins, isospins and momenta. Okubo and Marshak (1958) wrote down the most general potential form that maintains invariance under: space & time translations and rotations; parity and time reversal; isospin rotations and particle interchange. ! ! ! ! ! ! ! ! VNN = V0 (r) +V" (r)"1 #" 2 +V! (r)!1 #! 2 +V"! ("1 #" 2 )(!1 #! 2 ) central ! ! ! ! ! ! ! ! ! Spin-orbit, with 1 + VLS (r) L # S + VLS! (r)(L # S )(!1 #! 2 ) S " 2 (!1 + ! 2 ) ! ! ! ! ! ! + VT (r) S12 + VT! (r) S12 !1 #! 2 Tensor, with S12 $ 3(!1 " rˆ)(! 2 " rˆ)#!1 "! 2 ! ! ! ! V (r)Q V (r)Q 1 ! ! + Q 12 + Q! 12 !1 #! 2 Quadratic spin-orbit, with Q12 " {(#1 ! L)(# 2 ! L) 2 ! ! ! ! ! ! ! ! ! ! ! ! + VPP (r)("1 # p)(" 2 # p) + VPP! (r)("1 # p)(" 2 # p)(!1 #! 2 ) ! ! + (# 2 ! L)(#1 ! L)} Momentum-dependent tensor Can give rise to Can give rise to preferential L-R asym. orientation of separation vs. spins in scattering of vertically polar- Some “forbidden” terms: ized beam. ! ! rˆ S ! p violates parity rˆ ! ! ! ($1 #$ 2 )! L violates time reversal S12 < 0 ! ! ! $ "$ ! L violates isospin S not conserved ( 1 2 ) ( ) S12 > 0 Experiment ⇒ Phase Shifts ⇐ Potential Model

Real δl ⇒ angular redistribution, but no loss of incident beam flux from elastic channel. Phase shifts for each contributing space-spin state and 3 3 3 3 tensor mixing parameters εJ (e.g., S1- D1 , P2- F2 ) are adjusted to fit elastic data. Potential models are tuned to fit phase shifts as a fcn. of E. E.g., some low-L phase shifts from np scattering exp’ts + a potential model fit to them (solid curves), up to pion production threshold: Major Inferred Features of the Strong NN Potential

1) VNN has finite range R, because δlsj remains small for l > kR. S-waves

dominate for Elab ≲ 10 MeV. 2) Longest-range part of central potential is attractive, with range characteristic of one-π exchange 3) Short-range (~0.5 fm) repulsive 1 core needed to account for δ ( S0 ) sign change near Elab ≅ 250 MeV. 4) Polarization measurements indicate L⋅ S term, including short-range

part affecting P-waves at Elab ≳ 100 MeV. 3 5) Phase-shift differences among P0,1,2 indicate need for tensor force.

6) Backward cross section peak for np at Elab ~ few hundred MeV ⇒ important charge-exchange (τ1⋅τ2 ) contribution. 1 7) Low-energy behavior of δ ( S0 ) ⇒ appreciable isospin violations: a << 0 ⇒ C (r ! a) i- , sin-0 ) r R e 0 cos nearly . !=0 ( > )%k%&%0& = * -0 + ' 1 r + kr ( bound S0 state # -0 %k%&%0& ! ka , a $ scattering length; nuclear nuclear exp' t # anp = !23.74 ± 0.02 fm " a pp = !17.3 ± 0.4 fm " ann = !18.9 ± 0.4 fm Do NN Potentials Have Predictive Power for Light Nuclei? Other models employing more ad hoc short-range potentials do comparably well for NN with similar # parameters. How do these all do for A=3 systems? Two bound states: 3H (triton) ≅ 3He ≅ with tighter binding, smaller size than deuteron: n n p p p n N.B. Mn > Mp ⇒ L = 0 L = 0 M(3H) – M(3He) = 3 magnetic moment : µ / µN = 2.979 ! µ p / µN = "2.127 ! µn / µN 530 keV ⇒ H β binding energy: = – 8.482 MeV = – 7.718 MeV -decays to 3He max with Ee = 18.6 Last N separation energy: Sn= 6.26 MeV Sp = 5.49 MeV 1/ 2 keV. Endpoint r 2 = 1.70 fm = 1.87 fm ch used to test mν . Faddeev eq’ns ⇒ exact Schrödinger sol’n for 3 bodies subject to mutual 2- body finite-range inter’ns in terms of: (allows off-shellnucleons) Jacobi coordinates All NN potential models under- ! ! ! ! bind A=3 nuclei and predict size R = 1 r + r + r too large by ~5-10%. 3 ( i j k ) ! 2 ! ! ! y = r ! 1 r + r i 3( i 2 ( j k )) ! ! ! r = 1 r ! r i 2 ( j k ) Triton binding energy (MeV)

Cutoff parameter for 3N force

Three-Nucleon Needed to Understand Light Nuclei

Understanding A=3 and 4 binding requires introduction N N N of NNN interactions that cannot be treated as sequential π NN interactions. There are many possible diagrams -- Δ ones treated most often are of following type: π Inclusion of a couple of selected diagrams of this type, with parameters adjusted to fit A=3 and A=4 (28.3 MeV) N N N binding energies, yields excellent predictions for energy levels of all other light nuclei up to A~12. These are computer-intensive variational calculations, exploiting Monte-Carlo sampling of many complex multi- parameter trial wave functions, e.g., evaluating: ˆ &"Hˆ ˆ %trial Oe %trial !exact O !exact = lim ˆ " #$ &"H %trial e %trial NNN inter’ns should have at least as complex a spin & iso- spin structure as NN, but are far

less constrained by scattering data. How to proceed? Chiral Effective Field Theory ⇒ Common Framework for NN, NNN and NNNN Interactions One can include NN terms in L and order chiral L = L + L + L with terms (as in χPT) in !! !N NN (2) (4) powers of pion mass/ L!! = L!! + L!! +...; momentum over Λ ≅ 1 χSB (1) (2) (3) ...; GeV. Consistent L!N = L!N + L!N + L!N + treatment through given (0) (2) (4) L NN = L NN + L NN + L NN +... order then automatically includes 3N, 4N inter’ns: Parameterize NN contact terms at each order in terms of spin-momentum operators that preserve desired symmetries, e.g.:

(0) ! ! ! ! V ( p#, p) =CS +C T ! 1 "! 2 adds 2 parameters.

(2) ! ! 2 2 V ( p$, p) =C q +C k + (4) ! ! 1 2 adds 7 V ( p ! , p ) adds 15 params. 2 2 ! ! ! ! ! (C 3q +C 4k )% 1 !% 2 + C 5 [# iS ! (q " k )] params. ! ! ! ! ! ! ! ! N2LO 3N force adds 2 params., etc. + C 6 (% 1 ! q)(% 2 ! q)+ C 7 (% 1 ! k )(% 2 ! k ) Potentials from Chiral EFT Account Well for NN Data

~ same quality of fit as OBE models, with similar # or fewer params.

If EFT gives consistent basis for under- standing 3N, 4N inter’ns, it will represent considerable advance. Theory still under construction Different Theoretical Approaches to weak NN interaction

N N Meson exchange

STRONG (PC) Lweak WEAK (PNC)

N N • Kinematic: 5 S->P transition amplitudes in elastic NN scattering [Danilov]

• QCD effective field theory: χ perturbation theory [Liu, Holstein, Musolf, et al, incorporates chiral symmetry of QCD]

• Dynamical model: meson exchange model for weak NN [effect of qq weak interactions parametrized by ~6 couplings, Desplanques, Donoghue, Holstein,…]+ QCD model calculations

• Standard Model [need QCD in strong interaction regime, lattice+EFT extrapolation (Beane&Savage)] Meson Exchange Model (DDH) and other QCD Models

N assumes π, ρ, and ω exchange PV dominate the low energy PNC NN potential as they do for !, ",# strong NN

PC Weak meson-nucleon couplings N 0 1 2 0 1 fπ , hρ , hρ , hρ , hω , hω to be determined by experiment

fπ now calculated to be ~3E-7 with QCD sum rules (Hwang+Henley, Lobov) and SU(3) soliton model (Meissner+Weigel), calculation in chiral in progress (Lee et al). ∆I = 0, 1 Weak NN Constraints in DDH Model

Present goal: perform measurements in few body systems interpretable in terms of weak NN, at low E to apply χPT

Nonzero P-odd effects seen experimentally in p-p and p-α

Next step: see P-odd effects in low energy neutron reactions NN χPT coefficients and quantum numbers (Liu07)

Partial wave transition I ↔ I’ ΔI n-n n-p p-p EFT coupling

3 3 1 mρt S1 ↔ P1 0 ↔ 1 √

3 1 0 mλt S1 ↔ P1 0 ↔ 0 √ nn 1 3 0 mλs S0 ↔ P0 1 ↔ 1 √ √ √ np 1 3 1 mλs S0 ↔ P0 1 ↔ 1 √ √ pp 1 3 2 mλs S0 ↔ P0 1 ↔ 1 √ √ √

3 3 1 Cπ [~ f S1 ↔ P1 0 ↔ 1 √ π]

First 5 couplings are allowed s->p transition amplitudes in NN elastic scattering in “pionless” EFT limit (same as Danilov parameters)

Last coupling is long-range part of weak pion exchange [in DDH~ fπ] NN χPT coefficients and observables (Liu07)

EFT coupling (partial wave np Aγ np Pγ nD Aγ nα φ np φ pp Az pα Az mixing)

3 3 mρt ( S1- P1) -0.09 0 1.4 -2.7 1.4 0 -1.07

3 1 mλt ( S1- P1) 0 0.7 1.2 1.3 -0.6 0 -0.54

nn 1 3 mλs ( S0- P0) 0 0 0.6 1.2 0 0 -0.48

np 1 3 mλs ( S0- P0) 0 -0.16 0.5 0.6 2.5 0 -0.24

pp 1 3 mλs ( S0- P0) 0 0 0 0 0 -0.45, 0 -0.78

π 3 3 -0.3 0 0 0 0.3 0 0 C ( S1- P1)[~ fπ] experiment 0.6 1.8 42 8 -0.93, -3.3 (10-7 ) ±2.1 ±1.8 ±38 ±14 -1.57 ±0.9 ±0.2

Column gives relation between PV observable and weak couplings in EFT with pion Needs calculations of PV in few body systems (NN done, others in progress) “Conclusion”: the Weak NN Interaction on one slide

~1 fm NN repulsive core (from Pauli principle applied to quarks)->1 fm range for strong NN

|N>=|qqq>+|qqqqq>+…=valence+sea quarks+gluons+… interacts through strong NN force, mediated by mesons |m>=|qq>+… Interactions have long (~1 fm) range, QCD conserves parity

Both W and Z exchange possess much smaller range [~1/100 fm] weak If the quarks are close, the weak interaction can act, which violates parity at a length scale small compared to that set by ΛQCD 2 2 2 2 -6 Relative weak/strong amplitudes: ~[e /m W]/[g /m π]~10 Quark-quark weak interaction induces NN weak interaction Visible using parity violation q-q weak interaction: an “inside-out” probe of strong QCD