Connected Tree Diagrams and Amputated Amplitudes

A connected tree graph becomes an invariant amplitude by removing the external propagators through LSZ, placing the external momenta on shell, and multiplying the remaining vertices, internal propagators, and external wave functions. The overall momentum-conserving delta function is stripped off; all connected tree topologies compatible with the labeled external states must then be summed. A tree has no undetermined loop momentum, so its result is assembled algebraically from external momenta rather than integrated over a free momentum. For scalar external states with polynomial local vertices it is rational in the invariants; explicit spinor or polarization wave functions can carry additional basis-dependent kinematic structure.

Required background. Momentum-Space Feynman Rules supplies vertex and propagator factors. LSZ Reduction: Poles, Residues, and Stable External States supplies external-leg amputation, residue factors, and the stable-state hypotheses.

From the connected correlator to the scattering amplitude

Use four-dimensional Lorentzian kinematics and

$$\langle f|iT|i\rangle_c =i(2\pi)^4\delta^{(4)}(P_f-P_i)\mathcal M_{fi}.$$

For scalar interpolating fields, define the connected raw time-ordered correlator and its Fourier transform by

$$\begin{aligned} D_{F,c}^{(n)}(x_1,\ldots,x_n) &=\langle\Omega|\mathrm T\{\phi_1(x_1)\cdots\phi_n(x_n)\}|\Omega\rangle_c,\\ \widetilde D_{F,c}^{(n)}(q_1,\ldots,q_n) &=\int\prod_a\mathrm d^4x_a\, e^{+i\sum_a q_a\cdot x_a}D_{F,c}^{(n)}(x_1,\ldots,x_n). \end{aligned}$$

Let $p_a$ denote positive-energy physical momenta and set $\eta_a=+1$ for incoming and $-1$ for outgoing particles. Reduction evaluates this forward Fourier transform at $q_a=-\eta_a p_a$: the incoming phase is $e^{-ip_a x_a}$ and the outgoing phase is $e^{+ip_a x_a}$. Choose one-particle state phases so that each field overlap is $\sqrt{Z_a}>0$. Near the simultaneous isolated stable scalar poles, the correlator factorizes as

$$\widetilde D_{F,c}^{(n)}(\{-\eta_a p_a\}) \sim \left[\prod_{a=1}^{n}\frac{iZ_a^{1/2}} {p_a^2-m_a^2+i0}\right] i(2\pi)^4\delta^{(4)}\!\left(\sum_a\eta_a p_a\right) \mathcal M,$$

The delta is unchanged under simultaneous reversal of all its arguments. LSZ multiplies the displayed correlator by exactly

$$\prod_{a=1}^{n}\frac{p_a^2-m_a^2}{i\sqrt{Z_a}}$$

and takes every external leg on shell, as a residue extraction with the stated boundary prescription. This operation already includes residue normalization; no further $Z_a^{-1/2}$ is applied. At tree level for canonically normalized elementary fields, $Z_a=1$ at the order being used. Internal propagators remain because they describe propagation between interaction vertices.

For example, replacing every field in a four-point correlator by $2\phi$ multiplies its pole part by $2^4$. Each new overlap is $2\sqrt Z$, so the four amputators supply $2^{-4}$ and the physical amplitude is unchanged. If a field is rescaled by a negative real number while state phases remain fixed, use its signed overlap in place of $\sqrt Z$; taking a new positive root without the corresponding state rephasing would lose that sign.

The direct diagrammatic object is $i\mathcal M$. Schwartz derives the momentum-space rules and the separation of the single overall delta function in Schwartz 2014, § 7.3, printed pp. 93–96; the external-line rule is precisely the LSZ cancellation, not a license to erase an internal line. An operator-based derivation keeps the one-particle overlap explicit in Weinberg 1995, § 10.3, Eqs. (10.3.1)–(10.3.7), pp. 437–438, using the pole theorem of § 10.2. Its opposite metric and Fourier phases and its delta-normalized states must be translated before comparing factors with the displayed formula; the canonical LSZ derivation performs the reduction with this site’s state and Fourier conventions.

Enumerate before evaluating

For fixed labeled external states, proceed in this order:

1. list interaction vertices and the fields incident on each vertex;
2. draw every connected acyclic graph with the required external half-edges;
3. discard graphs forbidden by charge, species, or other exact quantum numbers;
4. assign a momentum to every internal line and impose conservation at each vertex;
5. multiply all factors, including external spinors or polarizations; and
6. use the interaction factorials to define each vertex factor, and divide by a residual graph automorphism only if it leaves the labeled external legs and all incidence data fixed. Standard labeled tree-scattering graphs normally have no residual factor.

For a connected graph with $V$ vertices, $I$ internal lines, and $L$ independent loop momenta,

$$L=I-V+1.$$

Thus a tree satisfies $I=V-1$ and $L=0$. This is a decisive topology check: after the vertex delta functions are used, no unconstrained momentum integral may remain. Conversely, a graph with a cycle is not tree level even if it contains only a few vertices.

The counting can be seen directly in momentum space. There is one integration for each internal momentum and one delta function for each vertex. In a connected graph, $V-1$ independent vertex delta functions fix internal momenta, while the last becomes the overall conservation delta function. The number of integrations left is therefore

$$I-(V-1)=I-V+1=L.$$

This argument also explains why a disconnected graph would leave more than one conservation delta function and is not part of the connected invariant amplitude.

A two-vertex exchange check

Take positive free masses and the formal perturbative interaction

$$\mathcal L_{\mathrm{int}}=-\frac{g}{2}\phi^2\chi,$$

The cubic potential by itself is unbounded below, so this calculation does not assert an exact stable vacuum for the bare model. It isolates the exchange contribution; a stable completion may also add contact terms to the total amplitude. The $\phi\phi\chi$ vertex is $-ig$. For a labeled exchange momentum $P=p_1+p_2$, the amputated connected graph gives

$$i\mathcal M_s =(-ig)\frac{i}{P^2-m_\chi^2+i0}(-ig) =-\frac{i g^2}{s-m_\chi^2+i0}.$$

There are two vertices and one internal line, hence $I-V+1=0$. The four external $\phi$ propagators that appeared in the correlator are absent. The expression has no residual integration, and in four dimensions $[g]=1$, so $g^2/(s-m_\chi^2)$ is dimensionless as a four-point $\mathcal M$ must be. The pole and its residue are checked in Physical Poles and Tree-Level Factorization.

If the external scalars are identical, the $t$- and $u$-channel attachments are distinct contractions and must also be included. A symmetry factor in a rate is a separate matter: indistinguishable final-state phase space carries $1/n!$, whereas the amplitude is first built from all attachments. Srednicki 2006 draft, § 10, pp. 88–91, and § 11, p. 98, PDF keeps these amplitude and rate combinatorics separate.

A six-point $\phi^4$ tree is a useful enumeration test. Tree topology requires $V=2$ and $I=1$, so each quartic vertex receives three of the six labeled external legs. Choosing a three-element subset and identifying it with its complement gives

$$\frac12\binom63=10$$

distinct trees. In an all-incoming convention, each contributes an internal propagator with momentum $P_A=\sum_{i\in A}p_i$, where the unordered partition $A|A^c$ has $|A|=3$. The count is ten, not a graph symmetry factor of ten: the ten terms are different momentum channels and all belong in the amplitude.

Checks that catch different mistakes

• Topology: verify $I=V-1$ for every connected tree and no free loop momentum remains.
• Momentum routing: reverse an internal momentum. A scalar result must be unchanged; a fermion numerator changes form but the consistently ordered chain does not.
• Dimensions: in four dimensions an $n$-point amplitude has mass dimension $4-n$ in the chapter’s relativistic state normalization. External spinors and polarizations are part of $\mathcal M$ and must be included in this count.
• Permutation symmetry: exchange identical bosonic labels or antisymmetrize identical external fermions as required.
• Analytic structure: a local contact graph is polynomial; every displayed tree pole must be traceable to an internal propagator.
• Gauge consistency: only the complete gauge-related graph set is tested by a Ward replacement.

Failure of the stable-pole hypothesis is not repaired by diagram trimming. Resonances, infraparticles, and confined fields are not ordinary LSZ external states. Anatomy of a Loop Integral begins when $L>0$, while Cross Sections and Decay Rates supplies the flux and phase-space integration needed after $\mathcal M$ is known. Automated graph generation is not developed on this page.

Check your understanding

For two $\phi\phi\chi$ vertices, redraw the same exchange with the internal momentum reversed and show that the stripped scalar amplitude is unchanged. Then count vertex delta functions and prove that exactly one overall delta function survives.

Solution

The scalar propagator depends on $P^2$, so reversing $P\to-P$ leaves it unchanged. Starting with one integration over $P$, the two vertex deltas may be written as $\delta^{(4)}(p_1+p_2-P)$ and $\delta^{(4)}(P-p_3-p_4)$. The first fixes $P=p_1+p_2$; the second becomes the single overall $\delta^{(4)}(p_1+p_2-p_3-p_4)$. After LSZ removes the four external poles, the stripped result is $\mathcal M_s=-g^2/(s-m_\chi^2+i0)$, with no remaining integral.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, § 7.3, printed pp. 93–96. doi:10.1017/9781139540940.
• Srednicki, Mark. Quantum Field Theory. Author’s prepublication draft, 2006, § 10, pp. 88–91, and § 11, p. 98; pagination of this draft. Open PDF.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995; 2005 paperback, 2012 printing consulted, § 10.3, pp. 437–438. doi:10.1017/CBO9781139644167.