particle physics chapter 4 weak interactions - … · un 1977 iversité hadjlakhd ar batna el! pm...

1 79 7Université Hadj Lak dh ar

B A T N AEl

Particle Physicschapter 4

Weak Interactions

Yazid Delenda

Departement des Sciences de la matiereFaculte des Sciences - UHLB

http://theorique05.wordpress.com/f422

Batna, 07 May 2015

(http://theorique05.wordpress.com/f422) Particle Physics - chapter 4 1 / 58

Weak Interactions

Like QED and QCD, the force carriers of weak interactions are spin-1bosons that couple to quarks and leptons.Force carriers of weakinteractions are three intermediate vector bosons: W+ and W− with mass80 GeV/c2, and Z0 with mass 91.2 GeV/c2.Since these gauge bosons are very massive the weak interaction is a veryshort range force (recall that range ∼ 1/M) of order 2× 10−2 fm.Unlikethe strong, electromagnetic and gravitational forces, the weak interactionis not responsible for any kind of bound systems (the bound systems forthe aforementioned forces are successively: hadrons and nuclei, atoms andmolecules, astronomical objects).The first dedicated experiment to study vector bosons is the SPSproton–anti-proton collider at CERN:

p+ p→W+ +X → l+ + νl +X

p+ p→W− +X → l− + νl +X

p+ p→Z0 +X → l− + l+ +X(http://theorique05.wordpress.com/f422) Particle Physics - chapter 4 2 / 58

Weak Interactions

p+ p→W+ +X → l+ + νl +X

p+ p→W− +X → l− + νl +X

Weak Interactions

p+ p→W+ +X → l+ + νl +X

p+ p→W− +X → l− + νl +X

Weak Interactions

p+ p→W+ +X → l+ + νl +X

p+ p→W− +X → l− + νl +X

Weak Interactions

p+ p→W+ +X → l+ + νl +X

p+ p→W− +X → l− + νl +X

Weak Interactions

as shown in figure.

W±, Z0

Weak Interactions

From the quark point of view these processes are essentially quarkanti-quark annihilation processes:

u+ d→W+, d+ u→W−

u+ u→Z0, d+ d→ Z0

Hence the signature of a W boson is a lepton with large momentumemitted at wide angles with large missing transverse momentum carriedout by the neutrino.If the weak boson has zero transverse momentum thenby momentum conservation the missing transverse momentum must equalto the transverse momentum of the lepton.The signature of the Z0 boson is a pair of leptons in the final state withlarge transverse momentum. Therefore the mass of the Z0 boson is theinvariant mass of the leptons.

Weak Interactions

u+ d→W+, d+ u→W−

u+ u→Z0, d+ d→ Z0

Weak Interactions

u+ d→W+, d+ u→W−

u+ u→Z0, d+ d→ Z0

Weak Interactions

u+ d→W+, d+ u→W−

u+ u→Z0, d+ d→ Z0

Weak Interactions Classification

Classification

The weak interactions are classified in two types:

Charged current reactions: Before the electroweak theory was formulatedall observed weak processes were charged current reactions(such as β decay) mediated by W+ or W− bosons.

Neutral current reactions: The electroweak theory actually predicted theneutral current reactions caused by the Z0 boson.

In what follows we discuss the classification of the charged currentreactions, which can be subclassified into three types:

Classification

ClassificationPurely leptonic processes

Purely leptonic processes are weak processes involving purely leptons.For example the muon decay:

µ− → e− + νe + νµ

Recall that the electromagnetic interactions can be built from the basicinteraction shown in figure [see lecture notes], which give the 8 basicvertices shown in figure [see lecture notes].In a similar way, purely-leptonicweak interaction processes can be built from a certain number of reactionscorresponding to the basic vertices shown in the following figure.

µ− → e− + νe + νµ

νℓℓ−

νℓℓ+

For example, from the right-hand vertex we can derive the eight verticesshown in figure.

νℓℓ−

νℓℓ+

For example, from the right-hand vertex we can derive the eight verticesshown in figure.

ℓ−W−

ℓ−

Similarly eight other diagrams can be built by replacing particles withanti-particles.

Weak interactions always conserve lepton quantum numbers

Diagram-wise this conservation is guaranteed by the fact that at eachvertex, there is one arrow pointing in and one pointing out.The arrowpointing-in in the initial state always represents a particle (`− or ν`) andthe arrow pointing out in the initial state represents an anti-particle (`+ orν`). In the final state the situation is the opposite.Note that processes shown in the figure above are virtual, just like thosewe met in QED, so that two or more have to be combined to conserveenergy.However processes like `+ + ν` →W+ and W− → `− + ν` do notviolate energy conservation if MW > M` +Mν` .

The leptonic vertices are characterized by the corresponding strengthparameter αW independently on lepton type involved.The strength of theweak interaction is comparable with the electromagnetic one analogous toelectron-electron scattering by photon exchange.For example the muondecay is shown in figure.

νe(http://theorique05.wordpress.com/f422) Particle Physics - chapter 4 10 / 58

Since W bosons are very heavy, interactions like this can be approximatedby a zero-range interaction shown in figure.

Taking into account spin effects, the relation between αW and GF inzero-range approximation is:

GF√2

=4παW8M2

where gW is the coupling constant in W -vertices, by definition.

ClassificationSemi-leptonic and purely hadronic processes

Semi-leptonic processes are those involving both leptons and hadrons,for example the beta decay process:

n→ p+ e− + νe

Purely hadronic process are charged-current weak reactions involvingpurely hadrons, for example the decay of the Λ particle.

Λ→ π− + p

In semi-leptonic or purely hadronic processes, constituent quarks emit orabsorb W bosons. From the lepton-quark symmetry the correspondinggenerations of quarks and leptons have identical weak interactions:

(νee−

)⇔(ud

), etc

n→ p+ e− + νe

Λ→ π− + p

(νee−

)⇔(ud

), etc

n→ p+ e− + νe

Λ→ π− + p

(νee−

)⇔(ud

), etc

For example the neutron beta decay is shown in figure.

The basic vertices for weak interactions of quarks are shown in figure.

The corresponding coupling constants do not change upon exchange ofquarks/leptons:

gud = gcs = gW

For example, an allowed reaction is:

π− → µ− + νµ (du→ µ− + νµ)

However, some reactions do not comply with the lepton-quark symmetry,for example the kaeon decay

K− → µ− + νµ (su→ µ− + νµ)

Another example is dominant decay of Λ, which is shown in figure.

gud = gcs = gW

π− → µ− + νµ (du→ µ− + νµ)

K− → µ− + νµ (su→ µ− + νµ)

gud = gcs = gW

π− → µ− + νµ (du→ µ− + νµ)

K− → µ− + νµ (su→ µ− + νµ)

W−sdu

u d}π −

For this the concept of “quark mixing” was introduced by Cabibbo. Thestrength of the charged weak interaction which involves mixing of thequarks is described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix,which is a unitary matrix.

Weak Interactions Beta decay: Fermi theory

Beta decay: Fermi theory

The weak interaction is responsible for the β decay of atomic nuclei, whichinvolves the transformation of a proton to a neutron or vice versa, andwhich allows nucleus to reach most stable p/n ratio.Examples:

10C →10 B∗ + e+ + νe

14O →14 N∗ + e+ + νe

where one proton converts into a neutron from Carbon 10 nucleus toBoron 10 nucleus, or from Oxygen 14 nucleus to Nitrogen 14 nucleus (β+

decay). The atomic number remains the same but the charge of thenucleus is reduced:

p→ n+ e+ + νe

which for free proton is forbidden by particle masses1,

1mp = 938 MeV/c2 and mn = 940 MeV/c2, neutron being heavier.(http://theorique05.wordpress.com/f422) Particle Physics - chapter 4 19 / 58

10C →10 B∗ + e+ + νe

14O →14 N∗ + e+ + νe

p→ n+ e+ + νe

10C →10 B∗ + e+ + νe

14O →14 N∗ + e+ + νe

p→ n+ e+ + νe

10C →10 B∗ + e+ + νe

14O →14 N∗ + e+ + νe

p→ n+ e+ + νe

Beta decay: Fermi theoryThus energy input is needed in these reactions. However the crossedreaction:

n→ p+ e− + νe

is energetically allowed and is the reason for instability of neutrons (β−

decay).The reaction can also happen by electron capture (a.k.a. K-capture)where a proton captures an electron from the K-shell and converts into aneutron and electron neutrino:

p+ e− → n+ νe

Historically the existence of the neutrino was inferred by conservation ofmomentum, since the emitted β particle had a spectrum of energies.Problem: Using the relativistic energy-momentum relation, show that inthe decay of a particle of mass M at rest into two-particles of masses m1

and m2, the two particles must have a definite energy.(http://theorique05.wordpress.com/f422) Particle Physics - chapter 4 20 / 58

n→ p+ e− + νe

p+ e− → n+ νe

n→ p+ e− + νe

p+ e− → n+ νe

n→ p+ e− + νe

p+ e− → n+ νe

Fermi’s explanation of β-decay (1932) was inspired by the structure of theelectromagnetic interaction.Recall that the invariant amplitude forelectromagnetic electron-proton scattering, shown in the following figure,

is given by:

Fermi’s explanation of β-decay (1932) was inspired by the structure of theelectromagnetic interaction.Recall that the invariant amplitude forelectromagnetic electron-proton scattering, shown in the following figure,

is given by:

iM = (ieupγµup)−igµνq2

(−ieueγνue) = i(eupγµup)1

q2(−eueγµue)

where we have treated the proton as a structure-less Dirac particle withcharge +e and that of the electron is −e.M is the product of the electronand proton electromagnetic currents, together with the propagator of theexchanged photon.The electron and proton electromagnetic currents takethe form:

ejeµ = −eueγµue, ejpµ = +eupγµup

in which case the matrix element becomes:

M = −e2

q2jpµj

q2(−eueγµue)

M = −e2

q2jpµj

q2(−eueγµue)

M = −e2

q2jpµj

By analogy with the current-current form, Fermi proposed that theinvariant-amplitude for β-decay be given by:

Mβ = GF (unγµup)(uνeγµue)

where GF is the weak coupling constant which remains to be determinedby experiment; GF is called the Fermi constant.

By analogy with the current-current form, Fermi proposed that theinvariant-amplitude for β-decay be given by:

Mβ = GF (unγµup)(uνeγµue)

where GF is the weak coupling constant which remains to be determinedby experiment; GF is called the Fermi constant.

In the following figure we show the Feynman diagram for this process(p+ e− → νe + n) in the zero range approximation.

e−νe

Note the charge-raising or charge-lowering structure of the weak current.We speak of these as the “charged weak currents”.

In the following figure we show the Feynman diagram for this process(p+ e− → νe + n) in the zero range approximation.

e−νe

Note the charge-raising or charge-lowering structure of the weak current.We speak of these as the “charged weak currents”.

However there is one point which was not foreseen by Fermi, that theneutrinos are left-handed particles and violate parity, so the Lagrangianmust not allow right-handed neutrinos to be created, only left handed onesare allowed.To account for this we just make the replacementγµ → γµ(1− γ5)/2 (see next section)2, where γ5 = γ5 = iγ0γ1γ2γ3.Wealso add the factor 4/

√2 to GF in order to keep the original definition of

GF which did not include γ5. The matrix element for the beta-decayprocess becomes:

Mβ =GF√

2(unγµ(1− γ5)up)(uνeγµ(1− γ5)ue)

which can be written in the form:

Mβ =4GF√

2jµeνj

2γµ alone in the Lagrangian does not violate parity in QED, however the termγµ(1− γ5)/2 does.

Mβ =GF√

Mβ =4GF√

2jµeνj

Mβ =GF√

Mβ =4GF√

2jµeνj

Mβ =GF√

Mβ =4GF√

2jµeνj

with the charge-raising weak current:

jµeν = uνeγµ 1− γ5

This is charge-raising because in the sense of Feynman diagrams thetransition e− → νe involves raising the charge by one unit (from −1 to0),while the charge-lowering current:

jpnµ = unγµ1− γ5

involves lowering the charge by one unit.To estimate GF we compute the transition amplitude:

Tβ = iMβ(2π)4δ4(pp − pn − pe − pν)

which can be calculated by making the assumption that the low energyinteraction enables us to use non-relativistic spinors in which γ1, γ2, andγ3 do not contribute.

Note that we have made the assumption that the proton and neutron arestructure-less and we ignored strong interactions.This is feasible since thebeta decay is a very low energy reaction (point-like interaction) which alsomeans strong interactions are ignorable here.Summing over spin we obtain the energy spectrum of the emitted positron:

dpe=G2F

π3p2e(Ee − E0)

where E0 is the energy released to the lepton pair, E0 = Eν + Ee.Thus, iffrom the observed positron spectrum we plot

√dΓ

as a function of Ee, we should obtain a linear plot with end point E0. Thisis called the “Kurie plot”.

dpe=G2F

π3p2e(Ee − E0)

√dΓ

dpe=G2F

π3p2e(Ee − E0)

√dΓ

dpe=G2F

π3p2e(Ee − E0)

√dΓ

dpe=G2F

π3p2e(Ee − E0)

√dΓ

Beta decay: Fermi theoryIt can be used to check whether the neutrino mass is indeed zero. Anon-vanishing neutrino mass destroys the linear behavior, particularly forEe near E0, as shown in figure.

We can crudely integrate the differential rate above, putting pe = Ee, i.e.neglecting the electron mass.We also neglect corrections due to the atomicelectrons, which distort the electron wave function.

√dΓ

µν 6= 0

√dΓ

µν 6= 0

Beta decay: Fermi theoryIn this way we obtain

τ=G2FE

We can thus extract the value of GF from experiment,

GF = 10−5m−2p

This value can be obtained from the measured β-decay lifetimes of severaldifferent nuclei. Experimental evidence thus supports the existence of theneutrino and the validity of the effective Fermi interaction. Including theabove mentioned corrections, as well as radiative corrections, and usingthe so called super allowed Fermi transitions the very precise value belowis obtained:

GF (β decay) = 1.16632× 10−5GeV−2

which has dimensions of inverse-mass. Note that the value for GF frommuon decay is slightly different (1.136).

τ=G2FE

GF = 10−5m−2p

GF (β decay) = 1.16632× 10−5GeV−2

τ=G2FE

GF = 10−5m−2p

GF (β decay) = 1.16632× 10−5GeV−2

τ=G2FE

GF = 10−5m−2p

GF (β decay) = 1.16632× 10−5GeV−2

Weak Interactions Inverse-beta decay: neutrino interactions

Inverse-beta decay: neutrino interactions

The inverse-β decay process is one in which a neutrino maybe detected bybeing absorbed by an atomic nucleus:

νe + p→ n+ e+

νe + n→ p+ e−

However this reaction requires a very large detector in order to detect asignificant number of neutrinos because of the small cross-section of thisinteraction.Anti-neutrinos were first detected in the 1950s near a nuclear reactor, asshown in the following figure.

The inverse-β decay process is one in which a neutrino maybe detected bybeing absorbed by an atomic nucleus:

νe + p→ n+ e+

νe + n→ p+ e−

However this reaction requires a very large detector in order to detect asignificant number of neutrinos because of the small cross-section of thisinteraction.Anti-neutrinos were first detected in the 1950s near a nuclear reactor, asshown in the following figure.

In this experiment two scintillation detectors were put in the vicinity of acadmium target. Anti-neutrinos with an energy above the threshold limit(1.8 MeV) interact with the protons in water, which results in neutronsand positrons.

Nuclear

Anti-neutrinos withenergy above threshod 1.8 MeV

Scintillation

Cadmium target

Detector

γ ray

νe + p → e+ + n

e+ + e− → γ + γ

n → γ ray +X

Reactor

Detector

ScintillationDetector

The resulting positron annihilates with electrons in the detector materialcreating photons with an energy of about 0.5 MeV.Pairs of photons incoincidence could be detected by the two scintillation detectors above andbelow the target.The neutrons were captured by cadmium nuclei resultingin gamma rays of about 8 MeV that were detected a few microsecondsafter the photons from a positron annihilation event.Many experiments are established to detect neutrinos from varioussources, such as solar neutrinos, atmospheric neutrinos, etc. Neutrinodetectors are often built underground in order to isolate the detector fromcosmic rays and other background radiation.

Weak Interactions Neutrino helicity

Helicity operator and projection operators

Consider the Dirac equation for free neutrinos with energy and momentumpµ = (E,~p), assuming them to be massless for now.As usual we look forsolutions of the standard form:

)e−i(Et−~p.~x)

where χ and φ are 2-component spinors which only depend onmomentum. Substituting in the Dirac equation:

i∂ψ

∂t= (−i~α. ~∇ + βm)ψ

(0 ~σ~σ 0

)e−i(Et−~p.~x)

i∂ψ

∂t= (−i~α. ~∇ + βm)ψ

(0 ~σ~σ 0

)e−i(Et−~p.~x)

i∂ψ

∂t= (−i~α. ~∇ + βm)ψ

(0 ~σ~σ 0

)e−i(Et−~p.~x)

i∂ψ

∂t= (−i~α. ~∇ + βm)ψ

(0 ~σ~σ 0

in the Dirac representation, and ~σ are the pauli matrices. We find thecoupled equations:

(0 ~σ.~p~σ.~p 0

)(χφ

so we obtain

χ =~σ.~p

Eφ, φ =

~σ.~p

These equations can be decoupled by defining the 2-component spinors:

νR =1√2

(χ+ φ), νL =1√2

(χ− φ)

which leads to the decoupled equations:

(~σ.~up)νR = +νR, (~σ.~up)νL = −νLwhere ~up = ~p/E = ~p/p (since we are dealing with a massless particle) isthe unit vector along the momentum axis of the particle.

(0 ~σ.~p~σ.~p 0

)(χφ

so we obtain

χ =~σ.~p

Eφ, φ =

~σ.~p

νR =1√2

(χ+ φ), νL =1√2

(χ− φ)

(0 ~σ.~p~σ.~p 0

)(χφ

so we obtain

χ =~σ.~p

Eφ, φ =

~σ.~p

νR =1√2

(χ+ φ), νL =1√2

(χ− φ)

(0 ~σ.~p~σ.~p 0

)(χφ

so we obtain

χ =~σ.~p

Eφ, φ =

~σ.~p

νR =1√2

(χ+ φ), νL =1√2

(χ− φ)

The operator:

2~σ.~up = ~S.~up

is known as the “helicity operator”, that is, the helicity operator is theprojection of the spin of a particle onto its momentum axis.We thereforesee that νL is an eigenfunction of H with eigenvalue −1/2, so itcorresponds to a solution to the Dirac equation with negative helicity,whileνR corresponds to positive helicity. In other words νL describes a“left-handed” neutrino while νR describes a “right-handed” one.The operator H commutes with the Hamiltonian and therefore the helicityis a good quantum number.It is not, however a Lorentz-invariant quantityfor a massive particle.If a particle of given helicity moves with a velocityβ < 1 we can overtake it and find its helicity flipped around.

The operator:

2~σ.~up = ~S.~up

The operator:

2~σ.~up = ~S.~up

The operator:

2~σ.~up = ~S.~up

The operator:

2~σ.~up = ~S.~up

The operator:

2~σ.~up = ~S.~up

The operator:

2~σ.~up = ~S.~up

Note also that under the parity operator (spacial inversion), it may easilybe shown that these components transform to one another.This meansthat if the standard model involves only νL then it clearly violates parity.The operators

O± =1

2(1± γ5),

where γ5 = γ5 ≡ iγ0γ1γ2γ3, are projection operators. They satisfy thefollowing relations:

O2± = O±

O+O− = O−O+ = 0

Note that γ5 is a hermitian matrix, so O+ and O− are hermitian as well.Also note that:

γ0γ5 = −γ5γ0 ⇒ γ0O+ = O−γ0

O± =1

2(1± γ5),

O2± = O±

O+O− = O−O+ = 0

γ0γ5 = −γ5γ0 ⇒ γ0O+ = O−γ0

O± =1

2(1± γ5),

O2± = O±

O+O− = O−O+ = 0

γ0γ5 = −γ5γ0 ⇒ γ0O+ = O−γ0

O± =1

2(1± γ5),

O2± = O±

O+O− = O−O+ = 0

γ0γ5 = −γ5γ0 ⇒ γ0O+ = O−γ0

In the Dirac representation we have:

(0 11 0

where each bloc is a 2× 2 dimensional matrix. In the chiral representationwe have:

(−1 00 1

The result of application of these operators on the wave function ψ yields

O−ψ = ψL

O+ψ = ψR

where ψL has a negative helicity and ψR has a positive helicity.

(0 11 0

(−1 00 1

O−ψ = ψL

O+ψ = ψR

(0 11 0

(−1 00 1

O−ψ = ψL

O+ψ = ψR

A particle with negative helicity has the spin anti-parallel to the directionof motion and is called a left-handed particle.Similarly ψR has positivehelicity and corresponds to a right-handed particle.

Wu experiment and the helicity of neutrinos

An experiment on the weak decay of Cobalt-60 nuclei carried out byChien-Shiung Wu and collaborators in 1957 demonstrated that parity isnot a symmetry of the universe.The experiment studied r-transitions of polarized cobalt nuclei:

60Co→60 Ni∗ + e− + νe

The nuclear spins in a sample of 60Co were aligned by an externalmagnetic field, and an asymmetry in the direction of the emitted electronswas observed. The asymmetry was found to change sign upon reversal ofthe magnetic field such that electrons prefer to be emitted in a directionopposite to that of the nuclear spin. The essence of the argument issketched in figure.

60Co→60 Ni∗ + e− + νe

Jz = 5 Jz = 4 Jz = 1

60Co 60Ni∗︷︸︸︷+e− + νe

The observed correlation between the nuclear spin and the electronmomentum is explained if the required Jz = 1 is formed by a right-handedanti-electron neutrino, νR, and a left-handed electron, e−L .

The cumulative evidence of many experiments is that indeed only νR (andνL) are involved in weak interactions. The absence of the “mirror-image”states, νR and νL, is a clear violation of parity invariance.Also, chargeconjugation (C)-invariance is violated, since C transforms a νL state intoa νL state.However, the γµ(1− γ5) form leaves the weak interactioninvariant under the combined CP operation.For example:

Γ(π+ → µ+ + νL) 6= Γ(π+ → µ+ + νR) = 0, P violation

Γ(π+ → µ+ + νL) 6= Γ(π− → µ− + νL) = 0, C violation

Γ(π+ → µ+ + νL) = Γ(π− → µ− + νR), CP invariance

In this example, ν denotes a muon neutrino.(http://theorique05.wordpress.com/f422) Particle Physics - chapter 4 41 / 58

Weak Interactions V-A interaction

V-A interaction

How do we build a theory of weak interactions with parity violation?Themost general form of the matrix element we can write is:

M∝ (uψf Ouψi)(uφ,f Ouφi) (1)

where O is a combination of γ matrices. It turns out that there are only 5bilinear covariant expressions that can be formed by the gamma matrices,which are shown in the following table

V-A interaction

Name Symbol Current Number of components Effect under parity

Scalar S ψψ 1 +

Vector V ψγµψ 4 (+,−,−,−)Tensor T ψσµνψ 6

Axial-vector A ψγµγ5ψ 4 (+,+,+,+)

Pseudo-scalar P ψγ5ψ 1 −

where σµν = i(γµγν − γνγµ)/2.

V-A interaction

You can show, for example, that the vector current ψγµψ transformsunder the parity operation:

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

on the coordinates as shown in the table.

V-A interactionA series of experiments through the end of the 1950’s led to a new form ofthe effective weak interaction:

Mβ =GF√

2[unγ

µ(1− γ5)up] [uνeγµ(1− γ5)ue]

The factor 1/√

2, introduced for historical reasons, maintains the value ofthe Fermi constant GF . The uγµu and uγµγ5u transform, under Lorentztransformations of the coordinates, respectively as a vector (V) and anaxial vector (A)3:

ψγµψ → Λµνψγνψ, vector transformation

ψγµγ5ψ → det(Λ)Λµνψγνγ5ψ, Axial vector transformation

from which the name V-A.3These are termed so because a four-vector under parity gets its spacial components

reversed (such as momentum) while axial vector (or pseudovector) is unchanged, such asangular momentum.Under parity these components also change as described.

Mβ =GF√

2[unγ

The factor 1/√

Mβ =GF√

2[unγ

The factor 1/√

V-A interaction

Parity violation comes from the fact that the behaviour of the vector andaxial vector currents under a parity transformation are different.As you cansee from the table, the vector current flips sign under parity whereas theaxial vector does not. The interference between these two terms createsthe parity violation.One can see this schematically by remembering thatwhat we observe is usually the square of the amplitude. Suppose theamplitude is pure V-A. Then:

|M|2 ∼ (V −A)(V −A) = V V +AA− 2AV

If we apply a parity transformation then the sign of the V term flips, butthe sign of the A term does not: