Symmetries and Mass Splittings in QCD$_2$ Coupled to Adjoint Fermions

20 Pages·0.16 MB·English

by  Joshua Boorstein

EFI-93-69
hep-th/9401044
Symmetries and Mass Splittings
4
9 in QCD Coupled to Adjoint Fermions
2
9
1

n
a
J

1 Joshua Boorstein and David Kutasov
1

Enrico Fermi Institute
1
v and Department of Physics
4
4 University of Chicago
0
1 Chicago, IL 60637, USA
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4
9
Two dimensional QCD coupled to fermions in the adjoint representation of the gauge
/
h
t group SU(N), a useful toy model of QCD strings, is supersymmetric for a certain ratio
-
p
e of quark mass and gauge coupling constant. Here we study the theory in the vicinity of
h
: the supersymmetric point; in particular we exhibit the algebraic structure of the model
v
i and show that the mass splittings as one moves away from the supersymmetric point
X
r obey a universal relation of the form M 2(B) M 2(F) = M δm+O(δm3). We discuss the
a i − i i
connection ofthis relationto string and quark model expectations and verify it numerically
for large N. At least for low lying states the O(δm3) corrections are extremely small. We
also discuss a natural generalization of QCD with an infinite number of couplings, which
2
preserves SUSY. This leads to a Landau – Ginzburg description of the theory, and may be
useful for defining a scaling limit in which smooth worldsheets appear.
12/93
1. Introduction.
Two dimensional Yang – Mills theory coupled to adjoint matter has been argued to
be an interesting toy model for studying QCD strings [1], [2], being probably the simplest
confining gauge theory which undergoes a deconfining transition in the leading order in the
1/N expansion [2]. In string theory there is a qualitative difference between models with
an exponential growth in the density of states with mass (and consequently a Hagedorn
transition), and ones where such growth is absent [3] – the two are separated by the famous
c = 1 barrier. It is natural to expect a similar “transition” in gauge theory as well. Since
large N gauge theories with finite densities of states are known to be described by strings
[4], [5], it seems important to “cross the c = 1 barrier” in gauge theory and understand the
nature of the relation to strings in that regime. QCD coupled to adjoint matter provides
2
a (hopefully) simple case in which this regime can be quantitatively studied.
In a recent paper [2] it has been shown that some aspects of the model hint at a
stringy structure. In particular, the masses of certain winding modes around compact
(Euclidean) time exhibit an interesting dependence on the parity of the winding number,
which is difficult to understand in gauge theory but is very natural in string theory; the
model with adjoint fermions exhibits an in general softly broken supersymmetry (SUSY),
reminiscent of similar results in string theory [3]; and the spectrum of the appropriate
2d Coulomb potential (ignoring pair production) contains an infinite number of “Regge
trajectories” with an exponential density of (bosonic and fermionic) bound states, at high
mass.
In this paper we are going to focus on the supersymmetry found in [2] for QCD
2
coupled to adjoint fermions, and its (explicit) breaking by the quark mass term. The
main questions we wish to address here are the algebraic structure of the supersymmetric
theory and possible generalizations thereof, with the hope that the supersymmetric theory
may be exactly solvable (perhaps at large N), and the structure of the theory near the
supersymmetric point. This should shed light on questions like the applicability of stringy
constraints on the spectrum found in [3] to (this) gauge theory.
In Section 2 we describe QCD coupled to quarks in the adjoint representation of
SU(N), the light – cone quantization of the model, and the SUSY which arises for a
specific ratio of the quark mass and the gauge coupling. In Section 3 we examine the
theory perturbed away from the supersymmetric point by a change of the mass of the
constituent quarks. We show that the splitting in the mass squared of the super – partner
1
bound states is universal, at least to second order in the deviation of the quark mass from
the supersymmetric mass. This allows us to examine the behavior of:
Z(β) = Tr( )Fe−βM2 (1.1)
−
near the supersymmetric point, which plays an important role in string theory [3]. Section
4 contains some numerical results; we verify the universal mass splitting by numerically
calculating the mass of the lowest lying excitations in the bosonic and fermionic sectors
as a function of the constituent quark mass, and studying the appropriate differences. In
addition we study numerically the distribution of certain signs needed for the evaluation
of (1.1). In section 5 we embed QCD in an infinite dimensional space of theories all of
2
which are supersymmetric. These theories are parametrized by a superpotential W(Φ)
and exhibit an intriguing connection to (super –) Landau – Ginzburg theories, a fact that
may be useful to understand them better. We conclude in section 6.
2. QCD Coupled to Adjoint Fermions.
2
Consider the theory of real (Majorana) fermions in the adjoint representation of the
gauge group SU(N), described by the Lagrangian,
1
= F2 +ψ¯γµD ψ +mψ¯ψ (2.1)
L 8g2 µν µ
where F = ∂ A +[A ,A ], D ψ = ∂ ψ +i[A ,ψ], ψ is a traceless hermitian anti-
µν [µ ν] µ ν µ µ µ ab
commuting matrix, m is the (bare) fermion mass and g the gauge coupling. We follow the
conventions of [2] which we will review next to establish the notation.
Two dimensional gauge theories look especially simple in light – cone quantization [6],
[7], which has been recently applied to this model in [1], [2]. We denote by ψ the right
ab
moving fermions and by ψ¯ the left moving ones. The SU(N) currents, J+ = ψ ψ ,
ab ab ac cb
J− = ψ¯ ψ¯ form right and left moving level N affine Lie algebras, respectively. In the
ab ac cb
gauge Aab = 0 the Lagrangian (2.1) takes the form:
−
1
= (∂ A )2 +iψ∂ ψ +iψ¯∂ ψ¯ 2imψ¯ψ +A J+. (2.2)
L 4g2 − + + − − +
The equations of motion for A , ψ¯ do not involve derivatives with respect to x+, the light
+
– cone “time”; it is easy to integrate them out to obtain an action solely in terms of the
right moving fermions ψ:
1 1
= iψ∂ ψ +g2J+ J+ +im2ψ ψ. (2.3)
Lψ + ∂2 ∂
− −
2
Quantization on constant x+ surfaces gives rise to the momentum operator:
P+ =i dx−ψ∂ ψ
−
Z
(2.4)
1 1
P− = dx− im2ψ ψ g2J+ J+ .
Z (cid:18)− ∂ − ∂2 (cid:19)
− −
Expanding ψ(x+ = 0) in modes:
ψ (x−) = 1 ∞ dkψ (k)e−ikx− (2.5)
ab ab
2√π Z
−∞
and imposing the canonical anticommutation relation, ψ (x−),ψ (0) = 1δ(x−)δ δ
{ ab cd } 2 a,d c,b
we find the mode anticommutation relations:
ψ (k),ψ (k′) = δ(k +k′)δ δ . (2.6)
ab cd a,d c,b
{ }
ψ (k) with k 0 are creation operators, whereas the ones with k 0 are annihilation
ab
≤ ≥
operators. The light – cone vacuum is chosen such that:
ψ (k) 0 = 0 k > 0. (2.7)
ab
| i ∀
The momentum operators (2.4) are normal ordered in the standard fashion, and take the
form:
∞
P+ = dkkψ ( k)ψ (k)
ab ba
Z −
0 (2.8)
∞ dk ∞ dk
P− =m2 ψ ( k)ψ (k)+g2 J+( k)J+(k)
Z k ab − ba Z k2 ab − ba
0 0
where J+(k) is given by (for k = 0):
ab 6
∞
J+(k) = dpψ (p)ψ (k p). (2.9)
ab Z ac cb −
−∞
J+(0) = ∞dpψ ( p)ψ (p) must annihilate all physical states (due to confinement).
ab 0 ac − cb
R
Physical states are obtained by acting with raising operators ψ ( k) on the vacuum
ab
−
(2.7). P+ is diagonal in this basis; therefore, eigenmodes of P− are also eigenmodes of
the mass operator, M2 2P+P−. Since P+ commutes with all the operators mentioned
≡
below, we will usually set it to 1 from now on to simplify some formulae.
It waspointedout in[2]that thedynamicsdescribed by (2.6), (2.8)issupersymmetric,
at least for m2 = g2N. In particular, the operator:
1 1 ∞
Q+ = dx−ψ ψ ψ = dpψ ( p)J+(p) (2.10)
3√N Z ab bc ca 3√N Z ab − ba
−∞
3
was shown in [2] to commute with the light – cone Hamiltonian:
[Q+,P−] = 0, for m2 = g2N. (2.11)
In addition, a simple calculation shows that:
(Q+)2 = P+. (2.12)
Thus Q+ is a conserved charge. The supersymmetry transformation:
ǫ
δψ (p) =ǫ Q+,ψ (p) = J+(p)
ab { ab } √N ab (2.13)
δJ+(p) =ǫ[Q+,J+(p)] = ǫ√Npψ (p)
ab ab − ab
acts non-linearly on ψ.
Clearly (2.10) can not be the full symmetry of the light – cone Hamiltonian. The
theory we are discussing is left – right symmetric, and although the chiral gauge chosen
(A = 0) makes the symmetry non – manifest, physics must be symmetric; hence, there
−
must exist another supercharge Q− with the properties (for m2 = g2N):
[Q−,P±] = 0, (Q−)2 = P− (2.14)
−
(the minus sign in the last of equations (2.14) will be convenient later.) Indeed, we shall
show shortly that the appropriate conserved charge is:
g 1 1 1
Q− = dp dp dp δ(p +p +p )( + + )ψ (p )ψ (p )ψ (p )
1 2 3 1 2 3 ab 1 bc 2 ca 3
− 3 Z p p p
1 2 3
(2.15)
∞ dp
=g ψ ( p)J+(p).
Z p ab − ba
−∞
One can of course verify that Q− (2.15) indeed satisfies (2.14) by explicit calculation,
but that involves careful normal ordering and is somewhat tedious. A simpler algebraic
derivation which is also useful for the next sections is the following. Consider the light –
cone Hamiltonian of the theory P− (2.8), written in the form:
P− = P− +αH (2.16)
susy 0
where P− is the supersymmetric Hamiltonian ( (2.8) with m2 = g2N), α = m2 g2N,
susy −
and:
∞ dk
H = ψ ( k)ψ (k). (2.17)
0 ab ba
Z k −
0
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Now define an operator F(γ) by
F(γ) = exp(γH )Q+exp( γH ). (2.18)
0 0
−
Expanding F in powers of γ we have:
∞ γn
F(γ) = Q (2.19)
n
n!
nX=0
where Q is given by the standard Baker-Hausdorff formula, i.e.
n
Q = [H ,[H ,[H ,...[H ,Q+]]]...]. (2.20)
n 0 0 0 0
Writing out Q explicitly we have
n
( )n 1 1 1
Q = − dp dp dp δ(p +p +p )( + + )nψ (p )ψ (p )ψ (p ) (2.21)
n 1 2 3 1 2 3 ab 1 bc 2 ca 3
3√N Z p1 p2 p3
where the integration extends over the complete three dimensional space. Note that:
Q− = g√NQ . (2.22)
1
NowwenotethatF2(γ) = P+ isindependent ofγ. RequiringthatF2(γ)(2.19)should
not depend on γ gives rise to an infinite tower of anticommutation relations satisfied by
the Q ’s. The order γ relation is Q+,Q = Q+,Q− = 0. At order γ2 we find:
n 1
{ } { }
Q+,Q = Q ,Q . (2.23)
2 1 1
{ } −{ }
After some elementary algebra it is possible to write Q in the form:
2
1 ∞ dp
Q = ψ ( p)J+(p). (2.24)
2 √N Z p2 ab − ba
−∞
Anti-commuting thisoperatorwithQ (= Q+) withthehelp of(2.13)and using thederived
0
relationship (2.23) we find that (Q−)2 = P− as expected.
−
To summarize, we find that QCD coupled to adjoint matter at its supersymmetric
2
point, m2 = g2N, has a standard (1,1) SUSY algebra,
(Q±)2 = P±; Q+,Q− = 0. (2.25)
± { }
Note the hermiticity properties of the supercharges (2.10), (2.15): (Q±)† = Q±. This is
±
the origin of the peculiar signs in eqs (2.14), (2.25).
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3. The vicinity of the supersymmetric point.
The light – cone Hamiltonian P− (2.8) can be parametrized as:
P−(α) = (Q−)2 +αH (3.1)
0
−
with α as in (2.16) the deviation of the constituent quark mass squared from its super-
symmetric value g2N, and H given by (2.17). At α = 0 the spectrum is supersymmetric;
0
bosonic states B are paired with fermionic ones, F = Q+ B :
| i | i | i
P− B = M2 B ; P− Q+ B = M2Q+ B . (3.2)
susy| i | i susy | i | i
Actually, one can do slightly better and diagonalize the “mass” operator Q+Q−,
M ≡
which satisfies 2 = P− . Eigenstates of satisfy:
M susy M
Q− B = MQ+ B . (3.3)
| i | i
Here M can be either positive or negative, and the relative signs of the M’s corresponding
to different bound states B will actually play a role later.
| i
At anyrate, asweturnonα in(3.1), themassesofthedegenerate bosonsandfermions
(3.2) are expected to change in a complicated way. However, as we shall now show, the
mass splittings exhibit a simple universal behavior, at least to second order in α. Indeed,
consider a bosonic eigenstate of the light – cone Hamiltonian P−(α):
P−(α) B = M2(α) B . (3.4)
α B α
| i | i
In general, there is no reason for F Q+ B to be an eigenstate of P−(α) (of course,
α α
| i ≡ | i
this is the case at α = 0 due to supersymmetry (3.2)). Nevertheless, it turns out that F
α
| i
is actually an eigenstate of P−(α) to first order in α. To verify that, we compute (unless
stated otherwise, we put g√N = 1 from now on):
P−(α) M2(α) F = P−(α),Q+ B = α H ,Q+ B = αQ− B (3.5)
B α α 0 α α
− | i | i | i | i
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
where in the last step we have used a result from section 2:
Q− = H ,Q+ . (3.6)
0
(cid:2) (cid:3)
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Now, to first order in α we can replace αQ− B αQ− B or, by (3.3):
α α=0
| i → | i
αQ− B = αM Q+ B +O(α2) = αM F +O(α2).
α B α B α
| i | i | i
Substituting this in (3.5), we find that F is indeed an eigenstate to this order, and the
α
| i
mass splitting is:
M2(α) M2(α) = αM +O(α2). (3.7)
F B B
−
This can also be written in the form: M (α) M (α) = α + O(α2), where we remind
F − B 2
the reader that the masses M can be positive or negative depending on the sign in (3.3).
Thus, at least to first order in α the mass splittings in this model are highly universal.
What happens at higher orders in α? One can easily show using standard techniques
that if B and F are eigenstates of P−(α), (3.1) with eigenvalues M2(α), M2(α)
| iα | iα B F
respectively, then
F Q− B
M2(α) M2(α) = ααh | | iα (3.8)
F − B F Q+ B
α α
h | | i
To find the mass splitting to second order in α we need to keep terms up to first order in
the ratio of inner products in (3.8). To that order, we can substitute F = Q+ B (see
α α
| i | i
(3.5), (3.6)), so:
B Q+Q− B
M2(α) M2(α) = ααh | | iα +O(α3) (3.9)
F − B B B
α α
h | i
Using (3.3) it is easy to see that (3.9) implies that the order α2 term in δM2 vanishes and,
finally,
M2(α) M2(α) = αM +O(α3). (3.10)
F B B
−
As mentioned in the introduction, and explained in [2], it would be very interesting to
calculate the partition sum Z(β) (1.1) (at least at large N). In conventional string theory
this would have the property that lim Z(β) = finite, despite the fact that the separate
β→0
contributions of bosons and fermions to (1.1) diverge as Z (β),Z (β) βaexp(b/β) as
B F
≃
β 0. It is far from clear in QCD that Z(β 0) is indeed (or should be) finite. To first
→ →
order in α, the calculation of this quantity reduces to:
Z(β) = Tr ( )Fe−βM2(α) = Tr Q+,e−βP−(α) Q+ = αβTr Q+Q−eβ(Q−)2 (3.11)
B,F B B
− −
h i
where, again, (3.6) has been used in the last step. In terms of the eigenstates M of
i
equation (3.3) (which we recall can be positive or negative) we find:
Z(β) = αβ M exp( βM2)+O(α2) (3.12)
i i
−
Xi∈B
7
where the sum over i runs only over bosonic bound states. Clearly, if all (or most) of the
M had the same sign, Z(β 0) would diverge as exp(c/β). A finite limit would imply
i
→
almost complete cancellations between the different terms in the sum, and would require
large numbers of positive and negative M at high M .
i
| |
In the next section we shall study the signs of M numerically. We shall also numeri-
i
cally verify (3.10). This is necessary because in the derivation above we have used certain
properties of the Hilbert space = B , F which are not a priori guaranteed. In
α α α
H {| i | i }
particular, it is not obvious that operators like H ,Q+Q− act well on ; in the ‘t Hooft
0 α
H
model [8] which is analogous to adjoint QCD in some respects, similar operators are actu-
ally singular in certain regions of parameter space. While we do not expect this to be the
case here, it seems useful to check (3.10), at least in simple examples.
4. Numerical Results
The system of equations involved in solving (3.4) form an infinite number of multi-
variable integral equations which have so far resisted all attempts at an exact solution. It
is however possible to reduce this problem to the tractable problem of diagonalizing finite
matrices. This is done by discretizing light – cone momentum; we will not describe the
details of the discretization, which appear in [1], [9].
Following the convention of [1] we write P+P− in the form
P+P− = xH +H (4.1)
0 1
where x = m2/g2N, H is as in (2.17), and H = ∞ dkJ ( k)J (k). Note that x is
0 1 0 k2 ab − ba
R
related to α in (2.16) by x = α+1, i.e. x = 1 corresponds to the supersymmetric point.
Below we shall investigate some aspects of the mass spectrum as a function of x at large
N. We also look at the spectrum of Q+Q−, in particular the distribution of the signs of
M in (3.3).
There are two main difficulties with obtaining reliable numerical results for the spec-
trum of QCD coupled to adjoint matter:
2
1) The size of the matrices one needs to diagonalize (even at large N) increases rapidly
with the cutoff. This is especially problematic when one is studying highly excited states,
whose wavefunctions are (generically) rapidly varying with momentum, or contain many
quarks (or both).
8
2) It is clear that the condition of normalizability of the light – cone wavefunction should
play a crucial role in selecting physical states and making the spectrum discrete. In large
N adjoint QCD there is much more room than in the ‘t Hooft model [8] for states with
2
finite norm at finite cutoff (of course all states have finite norm then) to become non –
normalizableasthecutoffisremoved. Thisisdue to thepresence ofsectors witharbitrarily
many adjoint quarks. It is very difficult in practice to follow the states while increasing
the cutoff and check whether they survive in the continuum limit.
Nevertheless, it was shown in [1], [9] that for a few low lying states these effects are
numericallysmall. Inparticular, thelowest lyingexcitationinthefermionic sectorcontains
to a high precision three adjoint quarks, while the bosonic one has significant components
only in the 2,4,6 quark sectors. Thus, to verify (3.10) we diagonalized (4.1) truncating to
the above mentioned sectors and continuing the results for the lowest lying state from finite
cutoff (K = 24 for bosons and K = 25 for fermions in the notation of [9]) to infinite one
using a certain Pade approximation. The results are exhibited in Figs 1,2. The uncertainty
in the masses squared due to the various truncations and extrapolations is estimated to
be 2 3%.
−
In Fig. 1 we plot the masses squared of the lowest lying excitations in the bosonic
and fermionic sectors as a function of the ratio x = m2/g2N in the region 0 < x < 1.5.
The mass of the fermionic bound state at zero constituent quark mass is calculated to be
5.72. That of the boson is 10.77. For x = 1 (the supersymmetric point) we obtain values
of 25.73 and 25.82 respectively for the bosonic and fermionic states. All of the above are in
good agreement with [9]. We see that to a good approximation the numerical calculation
reproduces the qualitative features of the theory at least for the lowest lying state. Note
also that as expected the individual masses show non – linear behavior away from the
supersymmetric point.
In Fig. 2 we plot M2(x) M2(x). According to (3.10) this plot should be a straight
F − B
line near x = 1; surprisingly, we find a straight line over the full range of the graph. The
deviation of the points from the straight line fit of Fig. 2 is significantly less than the
uncertainties mentioned above. The slope of the straight line of Fig. 2 is 5.14, whereas
(3.10) predicts [9]: √25.8 = 5.08. The two agree to within the accuracy of our numerical
analysis.
The sign of the slope can also be determined by diagonalizing Q+Q−. We find numer-
ically that Q+Q− is positive on the bosonic lowest lying state, which means (using (3.10))
that the fermion is heavier than the boson for m2 > g2N and vice versa. This again
9