Fermion Signs and Closed Loops

Fermionic diagram signs come from reordering Grassmann-odd operators or sources into a fixed reference order. Three effects must be kept distinct: relative signs between contractions of identical external fermions, the order of matrices along each open fermion line, and one additional factor $-1$ for every closed fermion loop. Arrow flow organizes contractions and charge flow, but arrows do not generate the signs by themselves.

Required background. Momentum-Space Feynman Rules fixes vertex and propagator conventions, while The Fermion Propagator fixes the ordered Dirac two-point function and numerator.

Helpful background. Grassmann Functional Integrals for Free Fermions derives the same signs from ordered source derivatives.

Fix an order before contracting

For a Dirac theory, choose once and for all:

• the order of fermionic field insertions in the correlator;
• the order of external fermion states in the bra and ket;
• whether Grassmann derivatives act from the left or right; and
• the orientation assigned to every $\psi$–$\bar\psi$ contraction.

Wick’s theorem then assigns each complete contraction the parity of the graded permutation that brings its paired fields together. For example, with the ordered fields

$$\psi_1,\quad\bar\psi_2,\quad\psi_3,\quad\bar\psi_4,$$

the selected free vacuum gives

$$\langle\mathrm T(\psi_1\bar\psi_2\psi_3\bar\psi_4)\rangle_0 =S_{12}S_{34}-S_{14}S_{32}.$$

The exchange term is negative because producing the charge-oriented pair order requires an odd permutation. It is not negative because an antiparticle has negative norm or because two contraction arcs cross. Weinberg derives the relative fermionic signs from explicit operator permutations in Weinberg 1995, § 6.1, pp. 267–268.

Changing the chosen order can multiply every term by a common sign. That common phase is convention dependent; relative signs between physically interfering terms are not.

For ordinary fermion-number-conserving Dirac diagrams, once all vertices and external states have been restored to one fixed convention, the residual statistics sign can be summarized as

$$(-1)^{L_f+P_{\mathrm{ext}}},$$

where $L_f$ is the number of closed fermion loops and $P_{\mathrm{ext}}$ is the parity of the permutation needed to restore the chosen external-fermion order. This is a check, not a replacement for ordered Wick contraction: Majorana fields, anomalous propagators, or differently ordered interaction monomials require the more general graded procedure.

Open fermion lines

For

$$\mathcal L_{\mathrm{int}}=-y\phi\bar\psi\psi,$$

the vertex is $-iy$, and the internal Dirac propagator is

$$S_F(q)=\frac{i(q\!\!\!/+m)}{q^2-m^2+i0}.$$

An open line becomes an ordered matrix chain. Starting at the external spinor on the right and following the fermion arrow toward the barred spinor on the left gives, for example,

$$\bar u(p')(-iy)S_F(q_2)(-iy)S_F(q_1)(-iy)u(p).$$

The order is physical matrix multiplication: gamma matrices and propagator numerators need not commute. One may traverse the diagram in the opposite direction only after applying the appropriate transpose or charge-conjugation identities. Srednicki gives a complete open-line reading prescription for Dirac Yukawa theory in Srednicki 2007, § 45, pp. 282–291.

If two diagrams differ only by exchanging identical external fermions, reorder both external states to the same reference order before adding them. The odd exchange supplies the relative minus required by Fermi statistics. This sign is separate from any symmetry factor of the graph.

Why every closed fermion loop contributes minus one

A closed chain has no external spinor that can absorb the endpoints of a Grassmann contraction. Closing the chain requires one extra cyclic reordering of odd fields relative to the corresponding bosonic contraction pattern, leaving a factor $-1$. Matrix indices then close into a trace. Thus each independent closed Dirac loop contributes

$$(-1)\operatorname{tr}(\text{ordered propagators and vertices}).$$

For the one-loop scalar two-point function in Yukawa theory, a convention-consistent integrand is

$$\begin{aligned} i\Pi(p) &=(-1)(-iy)^2 \int\frac{\mathrm d^d k}{(2\pi)^d} \operatorname{tr}\!\left[ \frac{i(k\!\!\!/+m)}{k^2-m^2+i0} \frac{i((k+p)\!\!\!/+m)}{(k+p)^2-m^2+i0} \right]. \end{aligned}$$

The leading $(-1)$ is the closed-loop statistics sign. The trace is over spinor indices; additional internal indices must also be traced. Reversing the loop-momentum routing changes $k$ but not this sign.

The same result follows from a fermionic Gaussian integral. With inverse free kernel $M_0$ and bosonic-background insertion $V$, integrating a Dirac field produces

$$\det(M_0+V)=\det M_0\, \exp\!\left[ \operatorname{Tr}\log(1+M_0^{-1}V) \right],$$

and expanding

$$\operatorname{Tr}\log(1+X) =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\operatorname{Tr}X^n$$

generates cyclic traces. The $1/n$ removes cyclic choices of a starting point, and the Grassmann determinant has the opposite Gaussian power from a commuting field. After the factors of $i$ and the declared vertices are restored, this supplies exactly the extra minus for each closed fermion cycle. The determinant expansion and its explicit comparison with closed fermion diagrams are developed in Srednicki 2007, § 53, pp. 326–330, especially p. 330.

A sign checklist

Source of sign	When it appears	Reliable check
vertex sign	from $iS_{\mathrm{int}}$	reconstruct the written interaction term
propagator numerator and $i$	from the inverse free kernel	multiply by $q\!\!\!/-m$
external exchange	odd permutation of identical fermionic states or insertions	restore one fixed external order
closed loop	one per independent closed odd-field cycle	compare with the Grassmann determinant expansion
ghost loop	closed loop of Grassmann-odd ghost fields	same parity rule, despite scalar Lorentz character
diagram symmetry	quotient of identical graph labels	compute separately as $1/S_G$

Keeping these entries separate makes sign debugging local. In particular, a loop minus cannot repair an incorrect interaction sign, and a symmetry factor never supplies a statistics sign.

Arrow flow, charge flow, and momentum flow

A Dirac propagator contracts $\psi$ with $\bar\psi$, so an arrow can record fermion-number flow. The momentum arrow is an independent convention. For an incoming antiparticle, charge flow and the chosen incoming momentum direction may point oppositely. Label both rather than inferring one from the other.

At a fermion-number-conserving vertex, arrows form continuous open lines or closed cycles. If the interaction violates fermion number, or if anomalous Nambu propagators are present, this simple orientation rule must be extended. The sign still comes from the ordered odd variables.

Majorana and other self-conjugate fields

For a Majorana field, particle and antiparticle are not distinct, so a globally directed fermion-number arrow is unavailable. Contractions involving $\psi\psi$ or $\bar\psi\bar\psi$ can be nonzero, charge-conjugation matrices enter, and apparently reversed chains may represent the same contraction. One must use a declared Majorana flow prescription and its vertex conventions; importing Dirac arrows and then guessing extra factors is unreliable. The closed-loop minus remains a Grassmann-statistics statement, while additional $1/2$ factors can arise from the normalization of a Majorana action and from identical contractions. A complete worked convention appears in Srednicki 2007, § 49, pp. 303–307.

Common pitfalls

Counting line crossings. A planar redraw can change the number of crossings without changing the algebra. Count a graded permutation in the declared field order.

Adding one minus per fermion propagator. Propagators carry their kernel-derived factors. The extra statistics sign is one per closed fermion loop, not one per segment.

Reversing a matrix chain by inspection. Open-line numerators and vertices are ordered matrices. Reversal requires transpose and charge-conjugation identities, not visual symmetry.

Check your understanding

Write the two Wick pairings of $\psi_1\bar\psi_2\psi_3\bar\psi_4$ and restore both to charge-oriented propagators. Identify the permutation responsible for their relative sign.

Solution

The adjacent pairing gives $S_{12}S_{34}$. For the crossed pairing, bringing $\bar\psi_4$ next to $\psi_1$ moves it past the two odd fields $\bar\psi_2,\psi_3$, while restoring the remaining contraction to the orientation $\psi_3\bar\psi_2$ contributes one further odd interchange relative to the adjacent reference ordering. Equivalently, direct recursive Wick contraction gives

$$\langle\mathrm T(\psi_1\bar\psi_2\psi_3\bar\psi_4)\rangle_0 =S_{12}S_{34}-S_{14}S_{32}.$$

Only the relative minus is convention independent; reversing the declared order of all external odd variables can change a common overall sign.

Where to continue

• Combine signs with momentum-dependent vertices: Derivative Interactions and Contact Terms checks which momentum belongs to which ordered field.
• See the same parity rule for ghosts: Gauge-Fixed Perturbation Rules, Ghost Diagrams, and Identity Checks derives the ghost action and its closed-loop sign.
• Build physical tree amplitudes: Tree Amplitudes and Gauge Consistency applies open-line chains with external-state conventions.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge: Cambridge University Press, 1995. First edition; 2005 paperback, 2012 printing consulted. DOI.