I.1 Crossing, Unitarity and Feynman Diagrams

demystifying Strong Interactions I.1 crossing, unitarity and Feynman diagrams

Yuri Dokshitzer

LPTHE, Jussieu basic properties of strong interactions

ﬁnite interaction radius 13 interaction probability is O(1) at the distances r r = 10− cm ∼ 0 hadrons are intrinsically relativistic objects

Indeed, to probe the distances r0 = 1/μ ~ 1 fm, momenta k ∼ μ ~ 140 Mev are necessary, which correspond to the proton velocity v ~μ/mp ~ 1/6. (Treated as a small parameter, it is this 1/ 6 to which the nuclear physics owes its existence.)

At the same time, if we substitute for the proton a π -meson (whose mass is mπ =μ ) we get v~1 and the very possibility of a non-relativistic approach disappears.

Symmetries of the hadron spectrum and of the strong interactions. Quarks Flavour symmetry SU(3) (u-d-s quarks) Chiral symmetry SU(2) (“massless” u and d quarks) “ Old Theory ”

Current algebra Additional symmetries appear at very high energies

Effective “Chiral Theory” Theory of Complex Angular Momenta ( small energy pion-nucleon interactions )

QCD momentum sumrules String theory why bother?.. We believe QCD to be the theory of hadrons and their interactions; actually, we are pretty sure it is, indeed. ( still in a strange position, between knowing and believing... )

Which features of strong dynamics are more general than QCD itself? How general? What are they driven by?

Nowadays many a notion of the “old theory” reappear in the QCD context : Reggeons and “Reggeon Field Theory”, Pomeron, Froissart regime, shadowing, ...

Lacking the knowledge of what is speciﬁc for QCD, and what is not, we are prone to making mistakes. Just one but rather dramatic example : now and then, there appears a theoretical papers predicting existence of long range (Van der Waals) forces between hadrons ...

This is not a prediction. This is but a repeating demonstration of our ignorance of how QCD - built on the massless gluon ﬁeld - really applies to the world of hadrons ...

... Letʼs forget about QCD, for the time being causality and analyticity Consider two-particle scattering in the coordinate space Free particle propagators

reduce to the product of the free wave function, depending on the time ordering of x and y points:

And for the outgoing lines, vice versa,

The interaction amplitude, in the momentum space, takes the form

For the forward scattering case ( )

For the causality to hold, we must have where f0 should not contribute:

for a physical momentum causality and analyticity

We arrive at the integral representation for the amplitude:

The phase:

The theta-functions ensure that the phase of the exponent is positively deﬁnite:

As a consequence, M(E1)≡M(s) is a regular analytic function in the upper half-plane of complex energies E1.

Reversing the logic:

We need to have

Otherwise, the response would not vanish at small but ﬁnite negative times

To be on the safe side, one usually imposes a stronger restriction on the scattering amplitude,

(polynomial boundary) crossing

Let’s calculate the number of independent variables characterizing an n-leg amplitude: 4 ( n -1) -n -6 = 3n-10 For example, 2->2 = 2 invariants, 2->3 = 5 invariants, etc. momentum components overall energy-momentum conservation on-mass-shell conditions subtract 3 rotations and 3 boosts Mandelstam variables for the 2->2 scattering

One diagram describes 3 related processes

Crossing reactions on the Mandelstam plane unitarity S-matrix

Introducing the relativistically invariant scattering amplitude,

the scattering cross section reads

Unitarity condition:

(in the T-invariant theory, Tab=Tba )

Optical theorem: crossing Crossing is speciﬁc for the relativistic theory.

It is the information coming from cross-channels that makes a major difference between the relativistic theory and the usual non-relativistic quantum mechanics

In non-relativistic quantum mechanics, interaction is described by means of a potential which can be chosen practically arbitrarily.

In the relativistic theory, unitarity in cross-channels (t, u) imposes severe restrictions on the ``interaction potential” in the s-channel.

Causality ensures that scattering amplitudes are analytic functions of momenta. An analytic function is characterized by its singularities.

Let us look into the structure of singularities of interaction amplitudes.

Interesting as it is for its own sake, this exercise will also teach a lesson of major general importance singularities What is the physical meaning of singularities of the scattering amplitude? If we were to draw diagrams, as if in a QFT, in the Born approximation we’d have 3 graphs:

pole in energy Scattering amplitude in NQM is given by the expression

with the exact w.f. which we can express via the Green function:

Discrete energy level (bound state) pole in momentum transfer

the Born scattering amplitude no Van-der-Waals forces, develops singularity when the integral diverges for some complex q since all the hadrons 2 are massive power tail -q0 =t = 0

Yukawa potential Singularities of arbitrary Feynman diagrams: Landau rules

The Feynman integral corresponding to a diagram with n internal lines has the structure

with ! the number of independent integrations - loops.

We need to ﬁnd the conditions under which this integral becomes singular in external variables sik = (pi + pk)2 – Lorentz invariants formed by four-momenta of the external particles.

Use the Feynman identity:

to obtain for the amplitude the multiple integral

with the quadratic form in the particle momenta

Since the integrand depends analytically on the integration variables, is the only potential source of singularity.

This inhomogeneous quadratic form form can be diagonalized by an orthogonal transformation:

with Landau rules Take the external momenta such that all the invariants s ik are negative. Then the energy integrations can be transformed,

and the denominator becomes positively deﬁnite:

Consequently, the integral yields a regular, real amplitude A n ! :

n 1 − We are left with n−1 integrations over parameters 0 ≤ αi ≤ 1 restricted by the condition 1 α = α 0 . − i n ≥ i=1 An equation Δ(α, sik) = 0 determines a surface in the n-dimensional space of αi. ! A singularity appears when this surface touches for the ﬁrst time the integration domain when we change sik. For each variable αi this can happen in two ways: 1. either a zero of Δ collides with the endpoint of the integration interval, αi = 0, 2. or two zeroes simultaneously arrive from the complex plane and pinch the integration interval :

The same condition applies to the last variable, i=n. This follows from the two facts:

is a homogeneous linear function in α i ,

and is the minimum of in all loop momenta: Indeed, at the extremum point in qk, for i=1,2,... n-1 independent variables we have Landau rules

Then,

Together with ∑ α i =1, these relations impose 4 ! + n +1 conditions on 4 ! + n variables. This means that a solution may exist only for speciﬁc values of external momenta.

The corresponding equation f(sik) = 0 determines the ʻLandau surfaceʼ for the position of a singularity of the amplitude in the space of invariants sik.

This equation may be resolved, e.g. to determine the position of a singularity in the invariant energy s for ﬁxed momentum transfer variables: s = s0(t, u, . . .), or vice versa. n 2 2 Now recall the original deﬁnition of the form (α ,q,p) = α (m k ) i i − i i=1 ! Landau rules Derivative over the loop momentum gives the equations

resembling Kirchhoff current law equations for electric circuits, with momentum ki playing the role of the current, and αi that of resistance.

The derivative over αi produces the second condition :

showing that each internal line either has to have an on-mass-shell momentum or should be dropped from consideration (short-circuited). position of singularities threshold singularity

Solution corresponding to positive

- two-particle threshold position of singularities anomalous singularity (deuteron)

Letʼs look for singularity in the momentum transfer Q2

Kinematical relations

The so-called anomalous singularity is positioned at When ( ? ) position of singularities anomalous singularity (continued)

Physical example - electromagnetic scattering off a deuteron

Form factor - electric charge (proton) distribution inside the deuteron : Proton wave function

1 q2 − F (q) 1+ ! !m ! "

A fast falloff of the form factor of loosely bound deuteron is due to the presence of the anomalous singularity close to the physical region of the scattering reaction

If not for the anomalous singularity, the scattering amplitude would have only vary at a much larger scale of the two-proton threshold, 4m2 >> Q02 position of singularities box singularity

Reduction examples:

Take for simplicity particle masses all equal, pi2 = ki2 = m2 Using the symmetry, the system reduces to

Equation for the Landau surface follows:

“Karplus curve” character of a singularity

Consider the characteristic function near its extremum in all integration variables

Rescaling the integration variables,

near the singularity we get

giving

A simple power counting shows that the integral for N converges when

(differentiate over s) character of singularities

n = 3, L = 1, E = 4*1 − 3 − 1 = 0; A ~ ln (s0 − s)

1 n = 4, L = 1, E = 4*1 − 4 − 1 = -1; A ~ √s s − 0

1 n = 4, L = 1, E = 4*1 − 5 − 1 = -2; A ~ s s − 0

The singularity seems to get stronger and stronger with the increase of the number of lines in the loop...

The equation for the Landau surface contains invariants.

Only among them are independent.

The two numbers coincide for the pentagram.

A plausible conclusion: a simple pole is the strongest possible singularity. Landau rules

the pattern of singularities of the interaction amplitudes:

the position of each singularity is determined by masses of real hadrons

the character of the singularity derives from the topology of the interaction process

the magnitude of a singularity is expressed in terms of the physical on-mass-shell amplitudes

These conclusions go beyond the perturbation theory which we have employed to derive them.

The reason for that lies in the unitarity property:

the series of Feynman diagrams (though having little sense in a theory of strongly interacting objects) formally solve the unitarity conditions. ﬁnite interaction range A + Cr + m =0 ! backward scattering peak A + Cr resonances A + s-channel U dispersion relations A + Cr + s-channel U Pomeranchuk theorem Ca + Cr Froissart theorem Ca + m =0 ! interaction radius grows with energy A + t-channel U