Add fundamental matter: masses, flavors, and decoupling

Before this lesson: BPS masses and the stability of particles; Fixed points, scaling, and conformal symmetry.

How can a massive quark disappear from the low-energy theory while leaving a measurable imprint on its strong-coupling scale? We will follow one hypermultiplet through three descriptions: its mass matrix, the central charge, and a degenerating curve. Their agreement fixes factors that dimensional analysis alone cannot determine.

The preparation is BPS masses and stability and fixed points and scaling. We work in flat four-dimensional spacetime with gauge group SU(2), not a quotient of it. The theory has N=2 supersymmetry, eight real supercharges; four are manifest in N=1 notation. The gauge rank is one. Complex hypermultiplet masses preserve all eight supercharges of the action and of a generic Coulomb vacuum.

Start with the actual matter mass

A full fundamental hypermultiplet consists of two N=1 chirals, a column $Q^i$ in the doublet and a row $\widetilde Q_i$ in its dual. Here $i=1,\ldots,N_f$ labels flavors, not colors. With canonical matter kinetic terms, the relevant superpotential is

W=\sum_i\left(\sqrt2\,\widetilde Q_i\Phi_{\rm SW}Q^i +m_i\widetilde Q_iQ^i\right). \tag{SW14.1}

The adjoint scalar includes the gauge coupling relative to a canonical adjoint field: $\Phi_{\rm SW}=g\Phi_{\rm can}$ . The masses $m_i$ are fixed complex parameters of dimension one. They are not the adjoint deformation that reduced the action to N=1 in the confinement example. Equation (SW14.1) is the matter interaction in Seiberg–Witten (3.1).

Take one flavor and the classical Coulomb configuration $\phi_{\rm SW}=a_{\rm cl}\operatorname{diag}(1,-1)/2$ . The two color components have superpotential coefficients

\mathcal M_\pm=m\pm\frac{a_{\rm cl}}{\sqrt2}, \qquad M_\pm=|\mathcal M_\pm|, \qquad u_{\rm cl}=\operatorname{Tr}\phi_{\rm SW}^2=\frac{a_{\rm cl}^2}{2}. \tag{SW14.2}

Eliminating the matter auxiliaries gives $|\mathcal M_\pm|^2(|Q_\pm|^2+|\widetilde Q_\pm|^2)$ ; the two Weyl fermions have the same Dirac mass. This derives the absolute values rather than identifying a complex coefficient with a positive mass. One component becomes massless at $a_{\rm cl}=\sqrt2m$ , hence $u_{\rm cl}=m^2$ . Quantum mechanically the special coordinate $a(u)$ replaces the classical coordinate in the exact central charge; it is not equal to $\sqrt{2u}$ at arbitrary coupling.

Put the flavor charge in the central charge

For each flavor, choose the phase symmetry $Q^i\mapsto e^{i\alpha_i}Q^i$ , $\widetilde Q_i\mapsto e^{-i\alpha_i}\widetilde Q_i$ . Denote its charge by $S_i$ . At zero masses the connected flavor Lie algebra is $\mathfrak{so}(2N_f)$ : the doublet is pseudoreal, so the dual doublets can be converted to doublets with the invariant antisymmetric tensor. The resulting $2N_f$ chirals have a symmetric flavor pairing in the adjoint Yukawa interaction. The $S_i$ select commuting rotations in this algebra. This statement does not fix the faithful global flavor group on every charged sector.

The first-paper convention used so far assigns a W boson electric charge $n_e=1$ , so a fundamental has $n_e=\pm1/2$ . The second paper instead uses integer fundamental charges. Define its coordinates explicitly:

A=\frac a2,\qquad B=a_D,\qquad q=2n_e,\qquad p=n_m. \tag{SW14.3}

The functions $a(u),a_D(u)$ now belong to the theory with matter; only the earlier charge units carry over. Thus a W has $q=\pm2$ , a fundamental has $q=\pm1$ , and $dB/dA=2\,da_D/da$ . This is a change of normalization, not an integral determinant-one change of the original period basis. The physical Dirac pairing becomes $qp'-pq'=2(n_en_m'-n_mn_e')$ . Seiberg–Witten, convention note and (11.3)–(11.7)

In these coordinates the flavored charge is

\boxed{\quad Z=pB+qA+\sum_i\frac{S_i m_i}{\sqrt2}, \qquad M\geq\sqrt2\,|Z| . \quad} \tag{SW14.4}

For the elementary states with $p=0,q=\pm1,S=1$ , this gives $\sqrt2|Z|=|m\pm\sqrt2 A|$ , reproducing (SW14.2) at weak coupling. CPT reverses all charges, including $S$ , and preserves the mass. The flavor term and its coefficient agree with Seiberg–Witten (5.3).

Flavor weights need not all be integer in this elementary normalization; magnetically charged sectors can carry spinor weights of the flavor algebra. Neither (SW14.4) nor an allowed weight constructs a stable particle. Representation constraints and the chamber-dependent spectrum remain additional information.

A curve that remembers the mass

For one flavor use the exact source curve

y^2=F(x,u) =x^2(x-u)+\frac14m\Lambda_1^3x-\frac1{64}\Lambda_1^6 . \tag{SW14.5}

Here $[x]=[u]=2$ , $[y]=3$ , and $[\Lambda_1]=1$ . This is input from the quantum solution, Seiberg–Witten (12.8); the checks below do not derive its nonzero quantum coefficients from symmetry alone.

A useful differential is

\lambda_2=-\frac{\sqrt2}{4\pi}\frac{y\,dx}{x^2}, \qquad \frac{\partial\lambda_2}{\partial u} =\frac{\sqrt2}{8\pi}\frac{dx}{y}. \tag{SW14.6}

The second equality follows directly from $2y\,\partial_u y=-x^2$ , at fixed $x,m,\Lambda_1$ . It has precisely the source normalization (15.2). Choose period cycles and their continuation so that $A\sim\sqrt{2u}/2$ on the weak electric branch. As usual, this asymptotic choice and the charge dictionary are part of the specification.

Unlike the pure-theory differential, $\lambda_2$ has mass-dependent residues. At the two points over $x=0$ , put $y_0=\pm i\Lambda_1^3/8$ and expand $y=y_0+y_1x+\cdots$ . The coefficient of $x$ in $y^2=F$ gives

y_1=\frac{m\Lambda_1^3}{8y_0}=\mp im, \qquad 2\pi i\,\operatorname{Res}_{x=0}\lambda_2 =\mp\frac m{\sqrt2}. \tag{SW14.7}

A small positively oriented loop around a pole therefore changes a period by a flavor-mass contribution. This explains why transporting cycles past poles can shift $A,B$ by constants proportional to $m$ . A compensating change of flavor-charge coordinates preserves $Z$ ; Exercise 1 makes that statement explicit. Absolute periods require this bookkeeping even though their $u$ -derivatives do not.

For comparison, Tachikawa’s variables $X,z,\Lambda_T$ obey $2\Lambda_T(X-m)/z+\Lambda_T^2z=X^2-u$ . The explicit substitution

x=-\Lambda_T^2z,\qquad y=-i\Lambda_T^2(Xz-\Lambda_T),\qquad \Lambda_1^3=-8\Lambda_T^3 \tag{SW14.8}

gives (SW14.5). Moreover $X\,dz/z$ , divided by $2\pi i\sqrt2$ , equals $\lambda_2$ plus an exact differential. Consequently $a_T=\sqrt2 A$ , $a_{D,T}=\sqrt2 B$ , and Tachikawa’s bound has no outer $\sqrt2$ . The physical mass $m$ and modulus $u$ agree. The phase and factor in the curve scale are essential; the symbol $\Lambda$ alone does not specify them. Tachikawa (5.1.4), (5.2.1)–(5.2.6)

Integrate out the heavy flavor

First derive the expected scale matching. In the ordinary fundamental trace normalization $T(\mathbf2)=1/2$ , the N=1 vector and adjoint chiral contribute $3T(\mathrm{adj})-T(\mathrm{adj})=4$ . Each full hyper has two fundamental chirals and subtracts one. Thus the N=2 holomorphic coefficient is

b_{N_f}=4-N_f,\qquad \Lambda_{N_f}^{\,b_{N_f}} =E^{\,b_{N_f}}\exp[2\pi i\tau_{\rm UV}(E)]. \tag{SW14.9}

Here $\tau_{\rm UV}=\theta/(2\pi)+4\pi i/g^2$ is the microscopic coupling; it is not the integer-charge Abelian period ratio. Holomorphy makes this running perturbatively one-loop exact. Instanton corrections to low-energy periods remain. Tachikawa (3.1.3)–(3.1.10)

Matching the coupling at a heavy threshold $m$ , with compatible finite scale definitions, gives $\Lambda_{\rm low}^{b+1}=m\Lambda_{\rm high}^b$ . The complex mass, including its phase, occurs here; replacing it by $|m|$ would lose the theta-angle information. For the curve definitions in (SW14.5), the calibrated relation is exactly

\Lambda_0^4=m\Lambda_1^3. \tag{SW14.10}

Now take $m\to\infty$ , $\Lambda_1^3\to0$ , holding $\Lambda_0^4$ fixed and keeping $u$ finite. Substitution, rather than dropping every term containing $\Lambda_1$ , gives

F=x\left(x^2-ux+\frac{\Lambda_0^4}{4}\right) -\frac{\Lambda_0^8}{64m^2} \ \longrightarrow\ x\left(x^2-ux+\frac{\Lambda_0^4}{4}\right). \tag{SW14.11}

The surviving curve degenerates at $u=\pm\Lambda_0^2$ . It is the pure curve in the second-paper package, (11.7). Its relation to the earlier $(x^2-\Lambda_0^4)(x-u)$ curve is a degree-two covering relation, called an isogeny. It changes the period lattice in the way (SW14.3) requires; merely renaming $x$ would not perform this change.

The controlled low-energy regime is $E,|a|\ll |m|$ , away from additional light thresholds. At fixed nonzero $\Lambda_1$ , sending $m$ to infinity also changes $\Lambda_0$ ; it does not compare one fixed pure theory. The simultaneous limit above does.

Locate all three degenerations

A double branch point obeys $F=\partial_xF=0$ . Eliminate $u$ using $u=(3x^2+m\Lambda_1^3/4)/(2x)$ . The remaining equation is $32x^3-8m\Lambda_1^3x+\Lambda_1^6=0$ . Equivalently, the cubic discriminant is

\operatorname{disc}_xF=-\frac{\Lambda_1^6}{16} \left[ u^3-m^2u^2-\frac98m\Lambda_1^3u +m^3\Lambda_1^3+\frac{27}{256}\Lambda_1^6 \right]. \tag{SW14.12}

To expand its finite roots, substitute (SW14.10), divide the bracket by $-m^2$ , and solve near $u=\pm\Lambda_0^2$ . For positive $m,\Lambda_0$ ,

u_\pm=\pm\Lambda_0^2-\frac{\Lambda_0^4}{16m^2} +O\!\left(\frac{\Lambda_0^6}{m^4}\right), \qquad u_{\rm q}=m^2+\frac{\Lambda_0^4}{8m^2} +O\!\left(\frac{\Lambda_0^6}{m^4}\right). \tag{SW14.13}

For example, the finite-root equation is $u^2-\Lambda_0^4-(u^3-9\Lambda_0^4u/8)/m^2+O(m^{-4})=0$ . Writing $u=\pm\Lambda_0^2+c/m^2$ yields $c=-\Lambda_0^4/16$ . The sum of the three roots is $m^2$ , which fixes the displayed heavy-root correction.

In units $\Lambda_0=1$ , direct double-root solutions give:

$m$	$u_-$	$u_+$	$u_{\rm q}$
4	−1.00387620	0.99606275	16.00781345
8	−1.00097466	0.99902152	64.00195314
16	−1.00024402	0.99975574	256.00048828

The distant singularity approaches the elementary condition $u=m^2$ ; the two finite ones recover the pure theory. This identification is controlled at large mass. Continuing labels to small mass requires periods and paths, not a permanent declaration that one root is always “the quark.”

A separate period check is more demanding than checking the polynomial limit. At $u=3$ , let $e_1<e_2<e_3$ be the three real roots of $F$ . Orient the electric cycle to go from $e_1$ to $e_2$ on the sheet with positive $y$ , then return on the other sheet. Substituting $x=e_1+(e_2-e_1)\sin^2\theta$ removes the endpoint square roots and gives

\frac{dA}{du}=\frac{\sqrt2}{2\pi} \int_0^{\pi/2} \frac{d\theta}{\sqrt{e_3-e_1-(e_2-e_1)\sin^2\theta}}.

This gives $dA/du=0.2086486387$ for $m=4$ and $0.2086562059$ for $m=16$ , approaching $0.2086567104$ . The limit equals one half of the earlier first-paper $da/du$ . Doubling $m$ reduces the difference by approximately four, as the $m^{-2}$ curve correction predicts.

What changes at four flavors?

For $N_f=4$ , $b=0$ . At zero masses the theory has an exactly marginal gauge coupling: it is a family of N=2 superconformal theories, rather than an asymptotically free theory specified by a generated scale. A generic nonzero $u$ is still a Coulomb vacuum that introduces a scale; it is not the conformally invariant vacuum at the origin. The flavor Lie algebra is $\mathfrak{so}(8)$ . Tachikawa, section 9.1

One must not insert $b=0$ into (SW14.9) and read $\Lambda^0$ as a generated scale. A mass restores dimensional running below its threshold. The numerical matching coefficient depends on the chosen coupling coordinate and scale calibration. For example, the second paper’s weak-coupling limit uses $\Lambda_3=64q_{\rm SW}^{1/2}m_4$ , with $q_{\rm SW}=e^{2\pi i\tau_{\rm SW}}$ . Here $\tau_{\rm SW}$ is the conformal curve’s period-ratio parameter in the second-paper normalization. Hold the product $64q_{\rm SW}^{1/2}m_4=\Lambda_3$ fixed as $m_4\to\infty$ , $q_{\rm SW}\to0$ . This is its specific (16.39), not a license to set every threshold coefficient to one in every source.

Four fundamental hypers are not an adjoint hyper. Their gauge representation and interactions differ, so a vanishing beta function does not establish N=4 supersymmetry. The adjoint-matter comparison is a separate calculation.

Practice

1. Continue around an elementary threshold

Near $A_0=m/\sqrt2$ , take the sourced local behavior $B=c-i(A-A_0)\log[(A-A_0)/\mu_{\rm ref}]/(2\pi)+\text{regular}$ , where $\mu_{\rm ref}\ne0$ makes the logarithm dimensionless. Continue once counterclockwise. Find the transformation of $(m/\sqrt2,B,A)^T$ , and the compensating transformation of the charge row $(S,p,q)$ . Check an arbitrary central charge and identify the locally vanishing electric-flavor combination.

Hint 1

The logarithm acquires $2\pi i$ . The mass is a fixed parameter and does not acquire this shift.

Hint 2

Write the affine transformation as a three-by-three matrix. Transform the charge row by its inverse.

Solution

The shift is $B'=B+A-m/\sqrt2$ , while $A'=A$ and $m'=m$ . Thus

\Pi_f'= \begin{pmatrix}1&0&0\\-1&1&1\\0&0&1\end{pmatrix}\Pi_f, \qquad (S',p',q')=(S+p,p,q-p).

Indeed,

(S+p)\frac m{\sqrt2}+p\left(B+A-\frac m{\sqrt2}\right)+(q-p)A =S\frac m{\sqrt2}+pB+qA.

The state $p=0,q=-1,S=1$ , or its CPT conjugate, has $Z=-A+m/\sqrt2$ and vanishes at $A_0$ . The matrix records analytic continuation and charge-coordinate compensation; it does not prove that every transformed charge is a stable state. The local logarithm and its mass shift are Seiberg–Witten (6.1)–(6.4).

2. Decouple two flavors in either order

Use the two-flavor source curve

y^2=\left(x^2-\frac{\Lambda_2^4}{64}\right)(x-u) +\frac{m_1m_2\Lambda_2^2}{4}x -\frac{(m_1^2+m_2^2)\Lambda_2^4}{64}.

First take $m_2\to\infty$ at fixed $\Lambda_1^3=m_2\Lambda_2^2$ . Then remove $m_1$ . Compare with reversing the order and with taking both large while fixing $\Lambda_0^4=m_1m_2\Lambda_2^2$ . What information would be lost by replacing either complex mass by its modulus?

Hint 1

In the first limit, the term $m_2^2\Lambda_2^4$ stays finite, even though $\Lambda_2$ tends to zero.

Hint 2

In the simultaneous limit express every correction using $1/m_1$ and $1/m_2$ . No assumption that the two masses are equal is needed.

Solution

The first limit gives $y^2=x^2(x-u)+m_1\Lambda_1^3x/4-\Lambda_1^6/64$ , exactly (SW14.5) with mass $m_1$ . The next limit fixes $\Lambda_0^4=m_1\Lambda_1^3=m_1m_2\Lambda_2^2$ . Interchanging the labels yields the same relation.

In the simultaneous limit the right side is

x\left(x^2-ux+\frac{\Lambda_0^4}{4}\right) -\frac{\Lambda_0^8(x-u)}{64m_1^2m_2^2} -\frac{\Lambda_0^8}{64}\left(\frac1{m_1^2}+\frac1{m_2^2}\right).

Both corrections vanish whenever both masses become large at fixed low-energy scale. Each term has mass dimension six. Their phases, and the phase of the surviving scale, depend on $m_1m_2$ ; absolute-value replacement would erase that holomorphic threshold information. The given curve and calibration are Seiberg–Witten (13.4)–(13.6).

3. Separate a conformal theory from its vacuum

Consider the massless four-flavor theory. Does $b=0$ mean its generic Coulomb vacuum is conformally invariant, that it has sixteen supercharges, or that adding one nonzero mass preserves conformal invariance? Explain each answer and identify what the mass preserves.

Hint 1

Distinguish a dimensionless coupling, a dimensionful expectation value, and a dimensionful parameter in the action.

Hint 2

A fundamental hypermultiplet and an adjoint hypermultiplet are different representations even when their contributions can cancel the same beta coefficient.

Solution

The massless theory has a dimensionless exactly marginal coupling. At $u=0$ its vacuum can preserve conformal symmetry. Choosing $u\ne0$ , with $[u]=2$ , introduces a scale through the state and spontaneously breaks that symmetry; this does not create a running coupling in the underlying massless action.

The theory has eight real supercharges. Four doublet hypers are not one adjoint hyper, and cancellation of a beta coefficient does not supply extra supersymmetry transformations.

A fixed nonzero mass is a relevant dimensionful deformation of the action. It breaks conformal invariance explicitly while preserving N=2 supersymmetry. Below a heavy mass the remaining three-flavor theory has $b=1$ , so its matched scale can be proportional to that mass times a dimensionless function of the original coupling. Generic Coulomb vacua continue to preserve all eight Poincaré supercharges.

The useful checkpoint is to reproduce one quark mass, one flavor residue and the fixed- $\Lambda_0$ decoupling limit in a single convention. The adjoint-hypermultiplet lesson introduces the new field content and mass analysis needed to reach N=4 and its N=2-star deformation.

Use the second-paper companion to return to the original convention note and reconstruct the degree-two curve map, including the differential and its cycle normalization.