FEYNMAN DIAGRAMS AND THE S-MATRIX, AND OUTER SPACE (SUMMER 2020)

DIRK KREIMER (LECT. JUNE 08, 2020)

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1.1. One-loop graphs. Consider the one-loop triangle with vertices {A, B, C} and edges {(A, B),(B, C),(C, A)}, and quadrics:

P_{AB} =k^{2}_{0} −k^{2}_{1}−k_{2}^{2}−k_{3}^{2}−M_{1},
P_{BC} = (k_{0}+q_{0})^{2}−k_{1}^{2}−k^{2}_{2} −k^{2}_{3}−M_{2},
P_{CA} = (k_{0}−p_{0})^{2}−(k_{1})^{2}−(k_{2})^{2}−(k_{3}−p_{3})^{2}−M_{3}.

Here, we Lorentz transformed into the rest frame of the external Lorentz 4-vector q =
(q_{0},0,0,0)^{T}, and oriented the space like part ofp= (p_{0}, ~p)^{T} in the 3-direction: ~p= (0,0, p_{3})^{T}.

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Using q_{0} = p

q^{2}, q_{0}p_{0} = q_{µ}p^{µ} ≡ q.p, ~p· ~p = ^{q.p}^{2}^{−p.pq.q}_{q}2 , we can express everything in
covariant form whenever we want to.

We consider first the two quadrics P_{AB}, P_{BC} which intersect inC^{4}.

The real locus we want to integrate isR^{4}, and we split this asR×R^{3}, and the latter three
dimensional real space we consider in spherical variables as R^{+}×S^{1}×[−1,1], by going to
coordinates k1 =√

ssinφsinθ,k2 =√

scosφsinθ, k3 =√

scosθ,s=k_{1}^{2}+k_{2}^{2}+k_{3}^{2},z = cosθ.

We have

P_{AB} =k^{2}_{0} −s−M_{1},
P_{BC} = (k_{0}+q_{0})^{2}−s−M_{2}.
So we learn says =k_{0}^{2}−M_{1} from the first and

k_{0} =k_{r} := M_{2}−M_{1}−q_{0}^{2}
2q0

from the second, so we set

s_{r} := M_{2}^{2}+M_{1}^{2}+ (q_{0}^{2})^{2} −2(M_{1}M_{2}+q^{2}_{0}M_{1} +q_{0}^{2}M_{2})

4q^{2}_{0} .

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The integral over the real locus transforms to Z

R^{4}

d^{4}k → 1
2

Z

R

Z

R+

√sδ_{+}(P_{AB})δ_{+}(P_{BC})dk_{0}ds×
Z 2π

0

Z 1

−1

dφδ_{+}(P_{CA})dz.

We considerk_{0}, sto be base space coordinates, whileP_{CA}also depends on the fibre coordinate
z = cosθ. Nothing depends onφ (for the one-loop box it would).

Integrating in the base and integrating also φ trivially in the fibre gives 1

2

√s_{r}
2q_{0} 2π

Z 1

−1

δ_{+}(P_{CA}(s =s_{r}, k_{0} =k_{r}))dz.

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ForP_{CA} we have

(1.1) P_{CA} = (k_{r}−p_{0})^{2}−s_{r}−~p·p~−2|~p|√

s_{r}z−M_{3} =:α+βz.

Integrating the fibre gives a very simple expression (the Jacobian of theδ-function is 1/(2√
s_{r}|~p|),
and we are left with the Omn`es factor

(1.2) π

4|~p|q_{0}.
This contributes as long as the fibre variable

z = (k_{r}−p_{0})^{2}−s_{r}−~p·~p−M_{3}
2|~p|√

sr

lies in the range (−1,1). This is just the condition that the three quadrics intersect.

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An anomalous threshold below the normal theshold appears when (m_{1} − m_{2})^{2} < q^{2} <

(m_{1}+m_{2})^{2}. In that range, s_{r} is negative, hence its square root imaginary. It follows that z
can be real only for z = 0, and this delivers

sr = (kr−p0)^{2}−~p·~p−M3,
which is negative for sufficiently large M_{3}, as expected.

On the other hand, when we leave the propagator P_{CA} uncut, we have the integral
1

2

√s_{r}
2q_{0} 2π

Z 1

−1

1

P_{CA}_{(s=s}_{r}_{,k}_{0}_{=k}_{r}_{)}dz.

This delivers a result as foreseen by S-Matrix theory [?].

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The twoδ_{+}-functions constrain thek_{0}- andt-variables, so that the remaining integrals are
over the compact domainS^{2}.

As the integrand does not depend on φ, this gives a result of the form

(1.3) 2πC

Z 1

−1

1 α+βzdz

| {z }

:=JCA

= 2πC

β lnα+β α−β = 1

2Var(Φ_{R}(b_{2}))×J_{CA},

where C = √

s_{r}/2q_{0} is intimitaly related to Var(Φ_{R}(b_{2})) for b_{2} the reduced triangle graph
(the bubble), and the factor 1/2 here is Vol(S^{1})/Vol(S^{2}).

Here, α and β are given through (see Eq.(1.1)) l_{1} ≡ ~p^{2} = λ(q^{2}, p^{2},(p +q)^{2})/4q^{2} and
l_{2} :=s_{r} =λ(q^{2}, M_{1}, M_{2})/4q^{2} as

α= (kr−p0)^{2}−l2−l1−M3, β = 2p
l1l2.
Note that

C

β = 1

pλ(q^{2}, p^{2},(q+p)^{2}) = 1
2q_{0}|~p|,
in Eq.(1.3) is proportional to the Omn`es factor Eq.(1.2).

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In summary, there is a Landau singularity in the reduced graph in which we shrink P_{CA}.
It is located at

q_{0}^{2} =s_{normal} = (p

M_{1}+p
M_{2})^{2}.

It corresponds to the threshold divisor defined by the intersection (P_{AB} = 0)∩(P_{BC} = 0).

This is not a Landau singularity when we unshrink P_{CA} though. A (leading) Landau
singularity appears in the triangle when we also intersect the previous divisor with the locus
(P_{CA} = 0).

It has a location which can be computed from the parametric approach. One finds
q^{2}_{0} = s_{anom} = (p

M_{1}+p

M_{2})^{2}+
+4M_{3}(√

λ_{2}√

M_{1} −√
λ_{1}√

M_{2})^{2}− √

λ_{1}(p^{2}−M_{2}−M_{3}) +√

λ_{2}((p+q)^{2}−M_{1}−M_{3})2

4M_{3}√
λ_{1}√

λ_{2} ,

with λ_{1} =λ(p^{2}, M_{2}, M_{3}) and λ_{2} =λ((p+q)^{2}, M_{1}, M_{3}).

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Humboldt U. Berlin

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