Covariant and helicity formalisms - Indico · 2018. 11. 21. · Crossing symmetry in tensor...

Covariant and helicity formalisms

Alessandro Pilloni

Joint Physics Analysis Center

Salamanca – September 25, 2017

A. Pilloni Covariant and helicity formalisms 1/14

Recipes to build an amplitude

The literature abounds with discussions on the optimal approachto construct the amplitudes for the hadronic reactions

I Helicity formalismJacob, Wick, Annals Phys. 7, 404 (1959)

I Covariant tensor formalismsChung, PRD48, 1225 (1993)Chung, Friedrich, PRD78, 074027 (2008)Filippini, Fontana, Rotondi, PRD51, 2247 (1995)Anisovich, Sarantsev, EPJA30, 427 (2006)

The common lore is that the former one is nonrelativistic,especially when expressed in terms of LS couplingsand the latter takes into account the proper relativistic corrections


How helicity formalism works

I Helicity formalism enforces the constraints about rotationalinvariance

I It allows us to fix the angular dependence of the amplitudeI What about energy dependence?

Example: Λb → ψΛ∗ → pK

ΛΛ

μ

μ

μ

μ

ψ

ppKKθ θ

φ

θ

*

+

−

+

−K − −ψ Λ

Λ

b

*ψ *

φ = 0

Λ

Λ

φ μψ*

Λ

b

lab frame

rest frame0

0

rest frame

∗

x

z

b

Λ

rest frame

LHCb, PRL115, 072001 (2015)

Each set of angles is defined in a different reference frameA. Pilloni Covariant and helicity formalisms 3/14

How helicity formalism works

I Helicity formalism enforces the constraints about rotationalinvariance

I It allows us to fix the angular dependence of the amplitudeI What about energy dependence?

Example: Λb → ψΛ∗ → pK

MΛ∗λΛ0b, λp ,∆λµ ≡

∑n

∑λΛ∗

∑λψ

HΛ0b→Λ

∗n ψ

λΛ∗ , λψD

12λΛ0b, λΛ∗−λψ(0, θΛ0b

, 0)∗

HΛ∗n→Kpλp , 0

DJΛ∗nλΛ∗ , λp

(φK , θΛ∗ , 0)∗

RΛ∗n (mKp)D1λψ ,∆λµ

(φµ, θψ, 0)∗,

LHCb, PRL115, 072001 (2015)

Each set of angles is defined in a different reference frameA. Pilloni Covariant and helicity formalisms 3/14

How tensor formalisms work

The method is based on the construction of explicitly covariantexpressions.

I To describe the decay a→ bc, we first consider thepolarization tensor of each particle, εiµ1...µji

(pi )

I We combine the polarizations of b and c into a “total spin”tensor Sµ1...µS (εb, εc)

I Using the decay momentum, we build a tensor Lµ1...µL(pbc) torepresent the orbital angular momentum of the bc system,orthogonal to the total momentum of pa

I We contract S and L with the polarization of a

Tensor ×RX (m) which contain resonances and form factors


What do we know?

I Energy dependence is not constrained by symmetry

I Still, there are some known properties one can enforce

RX (m) =B′LXΛ0b

(p, p0, d)

(p

MΛ0b

)LXΛ0b

BW(m|M0X , Γ0X )B ′LX (q, q0, d)(

q

M0X

)LXI Kinematical singularities: e.g. barrier factors (known)

I Left hand singularities (need model, e.g. Blatt-Weisskopf)

I Right hand singularities = resonant content (Breit Wigner,K-matrix...)


Kinematical singularities

I Kinematical singularities appear because of the spin of theexternal particle involved

I We can write the most general covariant parametrization ofthe amplitude astensor of external polarizations ⊗ scalar amplitudes

I Scalar amplitudes must be kinematical singularities free

I They can be matched to the helicity amplitudes

I We can get the minimal energy dependent factor

I Any other additional energy factor would be model-dependent


Crossing symmetry in tensor formalisms

I The process B → D̄ππ is composed of scalar particles onlyLHCb, PRD92, 032002 (2015)

I One defines the helicity angle in the isobar rest frame, thenthe amplitude is Lorentz Invariant

I Let’s consider the ρ intermediate state, B → D̄ρ(→ ππ)

A =m2B + s −m2D

2m2Bcos θ × qp = E

(B)ρ

mBcos θ × qp

I The factors p and q are the L = 1 expected barrier factors.The additional factor is analytical in s, not a kinematicalsingularity. Why is it there?


Crossing symmetry in tensor formalisms

B

D̄ π2

π1

L = 1ρ

L = 1

The tensor amplitude is given by

p(B)D · p

(ρ)π , where p

(B)D is the

breakup momentum in the B

frame, and p(ρ)π the decay

momentum in the isobar frame

A =m2B0 + s −m

2h3

2m2B0

pq cos θ

B

D π2

π1

L = 1ρ

L = 1

However, one can consider thescattering process just in theisobar rest frame.

A = pq cos θ

By crossing symmetry theamplitudes must be the same.

The usual implementation fails crossing symmetry


Example: B → ψπK

To consider the effect of spin, let’s consider B → ψπKWe focus on the parity violating amplitude for the K ∗ isobars,scattering kinematics

B(0+)

ψ(1−)

K(0−)

π(0−)

K∗(0+, 1−...)

ψ

π

B

Kp2

p1p3

p4

θs

p =λ

1/212

2√s, q =

λ1/234

2√s, zs =

s(t − u) + (m21 −m22)(m23 −m24)λ

1/212 λ

1/234


Helicity amplitude

Aλ =1

4π

∞∑j=|λ|

(2j + 1)Ajλ(s) djλ0(zs)

d jλ0(zs) = d̂jλ0(zs)ξλ0(zs), ξλ0(zs) =

(√1− z2s

)λd̂ jλ0(zs) is a polynomial of order j − |λ|


Helicity amplitude

Aλ =1

4π

∞∑j=|λ|

(2j + 1)Ajλ(s) djλ0(zs)

d jλ0(zs) = d̂jλ0(zs)ξλ0(zs), ξλ0(zs) =

(√1− z2s

)λd̂ jλ0(zs) is a polynomial of order j − |λ|The kinematical singularities of Ajλ(s) can be isolated by writing

Aj0 =m1p√s

(pq)j Âj0 for j ≥ 1,

Aj± = q (pq)j−1 Âj± for j ≥ 1,

A00 =p√s

m1Â00 for j = 0,


Identify covariants

Two helicity couplings → two independent covariant structuresImportant: we are not imposing any intermediate isobar

Aλ(s, t) = εµ(λ, p1)

[(p3 − p4)µ −

m23 −m24s

(p3 + p4)µ

]C (s, t)

+ εµ(λ, p1)(p3 + p4)µB(s, t)


Identify covariants

Two helicity couplings → two independent covariant structuresImportant: we are not imposing any intermediate isobar

C (s, t) =1

4π√

2

∑j>0

(2j + 1)(pq)j−1Âj+(s) d̂j10(zs)

B(s, t) =1

4πÂ00 +

1

4π

4m21λ12

∑j>0

(2j + 1)(pq)j

×

[Âj0(s)d̂

j00(zs) +

s + m21 −m22√2m21

Âj+(s) zs d̂j10(zs)

].

Everything looks fine but the λ12 in the denominatorThe brackets must vanish at λ12 = 0⇒ s = s± = (m1 ±m2)2,Âj+ and Â

j0 cannot be independent


General form and comparisons

Âj+ = 〈j − 1, 0; 1, 1|j , 1〉gj(s) + fj(s)

Âj0 = 〈j − 1, 0; 1, 0|j , 0〉s + m21 −m22

2m21g ′j (s) + f

′j (s)

gj(s±) = g′j (s±), and fj(s), f

′j (s) ∼ O(s − s±)

All these four functions are free of kinematic singularity.



Âj+ = 〈j − 1, 0; 1, 1|j , 1〉gj(s) + fj(s)

Âj0 = 〈j − 1, 0; 1, 0|j , 0〉s + m21 −m22

2m21g ′j (s) + f

′j (s)


′j (s) ∼ O(s − s±)


Comparison with ordinary LS

gj(s) = g′j (s)

s + m21 −m222m1√s

=

(2j − 12j + 1

)1/2Ĝ jj−1(s),

fj(s) =m1√sf ′j (s) = p

2

(2j + 3

2j + 1

)1/2Ĝ jj+1(s).

If the Ĝ jL are the usual Breit-Wigner, the g′, f ′ have singularities



Âj+ = 〈j − 1, 0; 1, 1|j , 1〉gj(s) + fj(s)

Âj0 = 〈j − 1, 0; 1, 0|j , 0〉s + m21 −m22

2m21g ′j (s) + f

′j (s)


′j (s) ∼ O(s − s±)


Comparison with tensor formalisms (j = 1)

Aλ = εµ(λ, p1)

(−gµν + P

µPν

s

)Xν(q)gS(s)

+ ερ(λ, p1)Xρµ(p)

(−gµν + P

µPν

s

)Xν(q)gD(s)



Âj+ = 〈j − 1, 0; 1, 1|j , 1〉gj(s) + fj(s)

Âj0 = 〈j − 1, 0; 1, 0|j , 0〉s + m21 −m22

2m21g ′j (s) + f

′j (s)


′j (s) ∼ O(s − s±)


Comparison with tensor formalisms (j = 1)

g1 = g′1 =

4π

3gS , f1 =

2πλ123s

gD , f′

1 = −4πλ12

3s

s + m21 −m22m21

gD .

If the gS , gD are the usual Breit-Wigner, the g′, f ′ are fine

but for singularities at s = 0


Few words on form factors and Blatt-Weisskopf

I Left hand singularities are due to dynamics →model-dependent!

I They originate from the cross-channel exchanges in scatteringprocesses

I These singularities are far from the physical region, but modifythe high-energy behaviour (e.g. suppress the barrier factors)

I In nonrelativistic potential theory, they can be univocallycalculated depending on the potential shape, e.g.Blatt-Weisskopf for square well+centrifugal potential

I Best practice: explore different form factor models,check the stability of the resonant parameters


Conclusions

I There is no God-given recipe to build the right amplitude, butone can ensure the right singularities to be respected

I The LS couplings are perfectly relativistic, although the naiveimplementation fails to respect the analyticity requirements

I The tensor formalisms have some of the right features, butexplicitly violates crossing symmetry

Misha Mikhasenko, Tomasz Skwarnicki, Adam Szczepaniak


Covariant and helicity formalisms - Indico · 2018. 11. 21. · Crossing symmetry in tensor...

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