Branches, Sheets, Analytic Continuation, and Monodromy

A “multivalued analytic function” is not an ordinary function that returns several answers at once. It is a family of compatible local holomorphic branches. Analytic continuation transports one chosen local branch along a path; monodromy records whether returning along a loop changes that branch. A branch cut is then a practical way to remove obstructing loops and display one single-valued branch.

Required background. Holomorphic Functions and Cauchy Theory supplies local holomorphy, path integrals, and the simply connected domains used in continuation.

This page develops branches, continuation, sheets, and monodromy without assuming a particular scattering model. Its controlled QFT-facing example is the two-particle threshold of a logarithmic scalar-bubble integral. Physical channel domains, unitarity, and crossing belong to Scattering.

Analytic continuation transports a germ

A function element at $z_0$ is a pair $(U,f)$, where $U$ is a neighborhood of $z_0$ and $f$ is holomorphic on $U$. Two elements define the same germ at $z_0$ when they agree on some smaller neighborhood of $z_0$. The germ remembers the local analytic function without privileging a particular disk.

Let $\gamma:[0,1]\to\Omega$ be a path with $\gamma(0)=z_0$. An analytic continuation of a germ along $\gamma$ is a finite chain of function elements

$$(U_0,f_0),\ldots,(U_N,f_N)$$

covering successive pieces of the path, with neighboring functions agreeing on the relevant connected overlap. The identity theorem makes each local handoff unique: once two holomorphic representatives agree near one transition point, they cannot be changed independently later on the same connected overlap.

Continuation can nevertheless depend on the whole path. The same starting germ may arrive at two different germs at the same endpoint if the two paths wind differently around an obstruction.

The monodromy theorem

Suppose $\Omega$ is a domain and a germ at $z_0\in\Omega$ can be analytically continued along every path $\gamma:[0,1]\to\Omega$ with $\gamma(0)=z_0$. If two such paths have the same endpoint and are homotopic in $\Omega$ relative to their endpoints, their continuations give the same endpoint germ. In particular, if $\Omega$ is simply connected, continuation is path-independent and the starting germ extends to one single-valued holomorphic function on $\Omega$.

Each hypothesis matters.

• The starting element must be fixed; a differential equation or algebraic relation can admit several local solutions.
• Continuation must exist along every path used in the homotopy.
• Simple connectedness removes path ambiguity, not genuine singularities. It does not extend a function through a pole, branch point, or natural boundary.

Analytic continuation is therefore a uniqueness statement plus an existence problem. A formula valid in one region does not continue merely because the same symbols can be written elsewhere. The germ-based statement and monodromy theorem are developed in Conway 1978, Chapter IX; a worked continuation by overlapping elements appears in Orloff 2018, Topic 13, PDF.

The logarithm is the basic monodromy example

On a connected open set $U\subset\mathbb C\setminus\{0\}$, a holomorphic logarithm is a holomorphic function $L$ satisfying

$$e^{L(z)}=z.$$

Differentiating gives

$$L'(z)=\frac1z.$$

Consequently, such an $L$ exists exactly when

$$\oint_\gamma\frac{\mathrm dz}{z}=0$$

for every closed contour $\gamma$ in $U$. A simply connected $U\subset\mathbb C\setminus\{0\}$ satisfies this condition. After choosing one value $L(z_0)$ with $e^{L(z_0)}=z_0$, define

$$L(z) = L(z_0) + \int_{z_0}^{z}\frac{\mathrm d\zeta}{\zeta}.$$

Path independence follows from the vanishing closed-contour integrals.

If a branch is continued around a closed loop $\gamma$, then

$$\Delta_\gamma L = \oint_\gamma\frac{\mathrm dz}{z} = 2\pi i\,\operatorname{Ind}(\gamma,0).$$

A once-counterclockwise loop therefore sends

$$L\longmapsto L+2\pi i.$$

The derivative $1/z$ returns to itself, but the primitive does not. This additive change is the logarithm’s monodromy.

Choosing a cut and an argument interval

Fix an angle $\theta$ and remove the ray

$$\{re^{i\theta}:r\geq0\}.$$

On the remaining domain, choose

$$\theta<\operatorname{Arg}_\theta z<\theta+2\pi$$

and define

$$\operatorname{Log}_\theta z = \ln|z| + i\operatorname{Arg}_\theta z.$$

This is one branch, not a new global logarithm on $\mathbb C\setminus\{0\}$. Moving the cut changes the displayed branch but does not move the branch point at $0$.

For the principal branch,

$$-\pi<\operatorname{Arg}z<\pi,$$

the cut is the nonpositive real axis. For $x>0$, its boundary values are

$$\operatorname{Log}(-x+i0) = \ln x+i\pi, \qquad \operatorname{Log}(-x-i0) = \ln x-i\pi.$$

The two limits differ by $2\pi i$.

These principal-branch conventions agree with NIST DLMF, accessed 2026, §§ 4.2 and 4.4.

Roots and complex powers inherit the branch

Once a logarithm branch $L$ is fixed, define

$$z^\alpha = e^{\alpha L(z)}.$$

Continuing around a loop of winding number $n$ changes this value by

$$z^\alpha \longmapsto e^{2\pi i\alpha n}z^\alpha.$$

For $\alpha=p/q\in\mathbb Q$ in lowest terms, there are $q$ distinct branches. For a noninteger irrational real exponent, repeated winding gives infinitely many distinct values.

The square root is the simplest finite example. If $w^2=z-a$, one counterclockwise circuit around $a$ gives

$$w\longmapsto-w,$$

and a second circuit returns to the original value. A sign written at one point is not enough to define a square root globally; the domain and continuation path are part of the specification.

Sheets make the local branches into one surface

For the square root, consider

$$\Sigma = \{(z,w)\in\mathbb C^2:w^2=z-a\}.$$

Projection to the $z$-plane is two-to-one away from $z=a$. At $(a,0)$, the coordinate $w$ is regular and

$$z=a+w^2.$$

Thus the apparent multivaluedness belongs to the projection, not to the function $w$ on $\Sigma$.

A cut-plane picture represents $\Sigma$ by two copies of the cut $z$-plane, with opposite banks glued so that crossing the cut moves from one copy to the other. This picture is useful, but the cut itself is not an intrinsic singularity. It can be moved as long as branch points, other singularities, and the chosen normalization are respected.

For the logarithm, repeated circuits add $2\pi i$ without returning after finitely many turns, so the corresponding surface has infinitely many sheets. Sheet labels such as “first” and “second” are therefore conventions: they are meaningful only after a base branch, cuts, and continuation paths have been stated.

Boundary values and discontinuity conventions

Let $F$ be holomorphic off a real cut. When the limits exist, define

$$F_\pm(s) = \lim_{\epsilon\to0^+}F(s\pm i\epsilon)$$

and use the convention

$$\operatorname{Disc}F(s) = F_+(s)-F_-(s).$$

Some sources reverse this sign. Others define a spectral or absorptive part by dividing the discontinuity by $2\pi i$ or $2i$. A translation must state which object is being used.

If Schwarz reflection holds,

$$F(s^*)=F(s)^*,$$

then $F_-(s)=F_+(s)^*$ on the cut and

$$\operatorname{Disc}F(s) = 2i\,\operatorname{Im}F_+(s).$$

Without that reflection property, discontinuity and imaginary part are not interchangeable.

QFT application: an equal-mass two-particle threshold

The logarithmic part of a regulated equal-mass scalar bubble has the Feynman-parameter form used in Schwartz 2014, § 16.1, pp. 302–303:

$$B(s) = \int_0^1 \operatorname{Log} \left( \frac{m^2-sx(1-x)}{\mu^2} \right) \,\mathrm dx, \qquad m>0,\quad\mu>0.$$

This page isolates its analytic structure; overall coupling factors, regulator-dependent local terms, and renormalization conditions are not needed for the branch analysis. Choose the branch for which $B(s)$ is real when $s<4m^2$. For complex $s$ away from

$$[4m^2,\infty),$$

the logarithm’s argument avoids its cut for every $x\in[0,1]$, and the integral defines a holomorphic function. On compact subsets of this cut plane, the integrand and its $s$-derivative are uniformly bounded, which justifies differentiation under the finite $x$-integral.

The threshold appears when the logarithm’s argument can vanish:

$$m^2-sx(1-x)=0.$$

Because $x(1-x)\leq1/4$, a real solution first occurs at

$$s_0=4m^2, \qquad x=\frac12.$$

For real $s>4m^2$, define

$$\beta(s) = \sqrt{1-\frac{4m^2}{s}}, \qquad x_\pm = \frac{1\pm\beta(s)}2.$$

The argument is negative precisely for $x_-<x<x_+$. On the upper bank of the $s$-cut,

$$m^2-(s+i0)x(1-x) = -|m^2-sx(1-x)|-i0$$

on this interval, so the principal logarithm contributes $-i\pi$. On the lower bank it contributes $+i\pi$.

For fixed $s>4m^2$, the roots $x_\pm$ are simple. Near either root, the logarithmic singularity is bounded, uniformly for sufficiently small $\epsilon>0$, by an integrable majorant of the form $C+|\ln|x-x_\pm||$. Away from the roots, the boundary values converge uniformly. Dominated convergence therefore permits taking the $\epsilon\to0^+$ limit under the $x$-integral. Thus

$$\begin{aligned} \operatorname{Disc}B(s) &= \int_{x_-}^{x_+}(-2\pi i)\,\mathrm dx\\ &= -2\pi i\,\beta(s), \qquad s>4m^2. \end{aligned}$$

The minus sign is fixed jointly by the definitions $B=+\int_0^1\operatorname{Log}(\cdots)\,\mathrm dx$ and $\operatorname{Disc}B=B_+-B_-$. A convention in which the loop function contains $-B$ has the opposite discontinuity.

Near threshold,

$$\beta(s) \sim \frac{\sqrt{s-4m^2}}{2m},$$

so the two boundary values meet with square-root threshold behavior. Continuing around $s=4m^2$ changes the sign of this local square-root coordinate. The logarithm can also accumulate additive $2\pi i$ changes under repeated continuation, so a global sheet label still requires a declared path and branch convention.

The calculation establishes a cut and its convention-explicit jump for this model integral. It does not derive amplitude analyticity domains, positivity, unitarity, or crossing. Those physical statements belong to Analyticity and Crossing of Amplitudes.

Failure modes and stopping conditions

“Multivalued” is not a function definition. Specify a local germ, a domain, and either a branch or a continuation path.

A branch cut is not automatically a physical singularity. Its placement is often conventional. Branch points and monodromy are invariant data; a particular drawn ray is not.

Crossing a cut without a rule loses the sheet. Record the starting bank, crossing direction, and discontinuity convention.

A principal branch is not privileged by the problem. It is a useful normalization. Boundary conditions or a physical $i0$ prescription may select a different boundary value.

Continuation does not prove a larger domain exists. Poles, accumulating singularities, and natural boundaries can stop it.

$\operatorname{Disc}F=2i\,\operatorname{Im}F_+$ needs reflection. Check $F(s^*)=F(s)^*$ before making that replacement.

Sheet numbers are not portable by themselves. Translate cuts, normalizations, and continuation paths when comparing sources.

Exercises

1. Continue a logarithm branch once counterclockwise around the origin. Derive the change from an integral rather than from an argument diagram.

   Check

   Since $L'(z)=1/z$, continuation around $\gamma(t)=re^{2\pi it}$ gives

   $$\Delta_\gamma L = \oint_\gamma\frac{\mathrm dz}{z} = 2\pi i.$$

   Thus the endpoint germ is $L+2\pi i$.

2. Define a branch of $q(z)=\sqrt{z-s_0}$ on $\mathbb C\setminus[s_0,\infty)$ by taking $0<\operatorname{Arg}(z-s_0)<2\pi$. Find the two boundary values for real $s>s_0$.

   Check

   On the upper bank, the argument tends to $0$, while on the lower bank it tends to $2\pi$. Hence

   $$q(s+i0)=\sqrt{s-s_0}, \qquad q(s-i0)=-\sqrt{s-s_0}.$$

   Crossing the cut swaps the two square-root sheets.

3. In the scalar-bubble example, show that the interval on which $m^2-sx(1-x)<0$ has length $\beta(s)$.

   Check

   The two roots are

   $$x_\pm = \frac{1\pm\sqrt{1-4m^2/s}}2.$$

   The quadratic is negative between them, and

   $$x_+-x_- = \sqrt{1-\frac{4m^2}{s}} = \beta(s).$$

   Multiplying this length by the logarithm jump $-2\pi i$ gives the stated discontinuity.

References

• John B. Conway, Functions of One Complex Variable I, 2nd ed., Chapter IX, Springer, 1978. Book record. This is the structural source for germs, analytic continuation, the monodromy theorem, and Riemann surfaces.
• NIST Digital Library of Mathematical Functions, accessed August 11, 2026, §4.2, Logarithm, Exponential, and Powers and §4.4, Special Values and Limits. These sections fix principal-branch and boundary-value conventions for the logarithm and complex powers.
• Jeremy Orloff, 18.04 Complex Variables with Applications, MIT OpenCourseWare, 2018, Topic 13, Analytic Continuation and the Gamma Function, PDF. This is the teaching source for continuation from overlapping analytic formulas and the identity-theorem uniqueness check.
• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, §16.1, pp. 302–303, especially Eqs. (16.4)–(16.12), Cambridge University Press, 2014. Book record. This is the QFT source for the regulated scalar-bubble integral and its Feynman-parameter logarithm in the spacelike region. The continuation to timelike threshold kinematics and the convention-explicit discontinuity above are derived here from the stated logarithm boundary values; they are not derived on the cited pages.