1 Introduction: Composite Operators and BRST Cohomology

SUSY Jumps Out of Superspace

in the

Supersymmetric Standard Model

[0.5in]


John A. Dixon111cybersusy@gmail.com

Toronto, Canada

[2in] Abstract

The supersymmetric standard model (SSM) appears to be firmly grounded in superspace. For example, it would be natural to assume that all the physically important composite operators can be made by combining superfields and superspace derivatives. But even for the simplest possible, free, massless and unbroken SUSY theory in 3+1 dimensions, this is not true.

This paper shows that there is a large set of physically important composite operators in the SSM that require explicit factors of the Grassmann odd ‘’ parameters of superspace. These explicitly break superspace invariance. These composite operators will be called ‘Outfields’ here, because they are intrinsically ‘outside’ of superspace. It is not possible to write the Outfields using only superfields and superspace derivatives.

These Outfields are present, and physically important, in all chiral SUSY theories in 3+1 dimensions. However they are very well hidden. They arise from a tricky mechanism involving the field equations. The superspace violating part of the SUSY variation of an Outfield is proportional to the field equations. The field equations are ‘equivalent to zero’, but they are not equal to zero, and that is why the Outfields have gone unnoticed for a long time.

This ‘field equation’ property of the Outfields means that the Outfields can be found by computing the local BRST cohomology of chiral SUSY in 3+1 dimensions. An Outfield then (typically) consists of the sum of two terms. The first term is a field part which violates the symmetry. But the violation of the symmetry is very special: it is proportional to the field equations. The second term contains a Zinn source times a ghost. The variation of the second term then cancels the variation of the first term, so that the combination is invariant under the BRST operator.

This explicit breaking of the initial symmetry, linked to a dependence on the Zinn sources through the field equations, is a feature that is quite rare in the BRST cohomology of non-SUSY theories such as gauge theories and gravity. However, for the rigid chiral SUSY theory in 3+1 dimensions, it is an essential ingredient of the cohomology.

In this paper, the masses are assumed to arise from the spontaneous breaking of internal symmetry with a Vacuum Expectation Value (VEV) for some scalar field. In accord with the usual case for most chiral actions, it is assumed that SUSY itself is not spontaneously broken by this VEV. So the present results can be utilized, for example, for the supersymmetric standard model (SSM), where internal symmetry is spontaneously broken from down to U(1), although SUSY itself is not spontaneously broken.

The calculation of the BRST cohomology space for these theories is performed in this paper using a spectral sequence analysis, starting with the free massless theory, and then adding interactions, and then masses. There are nine nested differentials, with ten nested cohomology spaces. The first three differentials relate to the free massless theory, and they establish the basic Outfields. The next two differentials come from the coupling terms. They give rise to many constraint equations, and the result is that SUSY picks out various physical composite operators in remarkable ways that depend crucially on the details of the particle content and the couplings. The last four differentials come from the mass terms, and these impose further constraints, but they also give rise to new terms because the mass parameter is now present.

The constraints give rise to a remarkable contrast between the cohomology of SUSY and the cohomology of gauge theories. The cohomology of gauge theories and SUSY both start with the cohomology of the free massless theories. Because the gauge theory cohomology does not involve the Zinn sources, the constraints that arise for the interacting or massive theories relate to the gauged Lie algebra that one starts with. However, for the SUSY Outfields, the constraints come from the symmetries of the superpotential alone, not the whole action. The most interesting solutions arise when the relevant symmetries do not extend to the whole action. These have nothing to do with the usual gauge-type symmetries of the action. These constraints give rise to a new kind of mingling between the interactions and SUSY itself.

These general cohomology results are illustrated with some examples from a special version of the SSM, which we call the CSSM. The CSSM requires right handed neutrinos and a Higgs singlet in addition to the usual SSM. One can see that the CSSM appears to have a that is related to these SUSY constraints. For the Leptons, there is one SU(2) doublet and two SU(2) singlets. For the Higgs, the field content is reversed. There are two Higgs SU(2) doublets and one Higgs singlet. This structure gives rise to simple Outfields in the cohomology space for each of the Leptons. The Quarks work the same way.

The symmetry that creates the Outfields is softly broken by the development of the VEV when spontaneous gauge symmetry breaking occurs. This causes the Quark and Lepton Outfields to mix with the corresponding elementary Quark and Lepton superfields. This means that these Outfields leave the cohomology space when the VEV turns on.

1 Introduction: Composite Operators and BRST Cohomology

The composite operators of a quantum field theory contain a great deal of information about the theory. A list of the physically important composite operators would be expected to satisfy the following guidelines:

  1. Physically important composite operators should be invariant, or covariant, under the symmetries of the action, and

  2. Two physically important composite operators that differ by the field equations should be equivalent, because the field equations should be equivalent to zero, in some sense.

At the start, it is not obvious how to put these two properties together in a sensible way. Fortunately, it has been discovered [1, 30, 25] that the set of physically important composite operators can be naturally organized by the construction of a nilpotent BRST operator. Then the BRST cohomology of the theory yields a complete and unique list of the local operators which incorporate the invariance, or the covariance, modulo the field equations. All quantum field theories have invariances, including invariance under the transformations of special relativity. For any given choice of such invariances, one can construct the related nilpotent BRST operator. The BRST operator always has two parts, which match the guidelines above:

  1. A symmetry variation part, and

  2. A field equation part.

1.1 A Minimal BRST operator can point the way to New Directions in a Theory

When a theory has a set of symmetries, it is frequently possible to write down a number of different BRST operators for it. Some have more, and some fewer, of the total set of symmetries. For example, one might decide not to include the Lorentz transformations in a given BRST operator, even though the theory has Lorentz invariance. Such a BRST operator for a theory would typically yield a cohomology space which contains Lorentz covariant operators, whereas inclusion of Lorentz transformations in the BRST operator would be expected to restrict the cohomology space to operators which are Lorentz invariant.

In this way, a minimal BRST operator which has a minimal subset of the invariances of the starting action, is likely to result in a cohomology space with operators which are covariant, rather than invariant, under the symmetries that are not included in the BRST operator. The resulting cohomology space is likely to be larger, and more interesting, than the more restricted cohomology space that would result from a BRST operator which incorporates all possible symmetries.

For this reason, the BRST operator examined in the present paper is quite minimal. Although the symmetries of interest do include invariance under some gauge group, and under Lorentz transformations, we do not include the gauge group, or the Lorentz transformations, in the BRST operator whose cohomology we will examine here. The result is that we find a host of ‘covariant’ composite operators in the cohomology space. If we had imposed invariance under Lorentz transformations, or an internal Lie algebra, these covariant objects in the cohomology space would have been excluded.

So, in this paper, we work out the BRST cohomology for the simplest possible BRST operator for chiral rigid SUSY in 3+1 dimensions. The result is fairly complicated, and we find objects which are covariant under the Lorentz group (they have spinor indices) and under the internal Lie algebra (they have internal indices).

The calculation of the cohomology is accomplished using the mathematical machinery of spectral sequences. This very detailed task constitutes the bulk of the technical part of this paper, and it is mostly contained in the Appendices. The main body of the paper explains the results with as little technical detail as possible. The results do suggest possible further developments, but in this paper, which is already very long, we restrict the analysis to the cohomology, leaving interpretation and related issues for other papers.

1.2 Plan of this Paper

This is a long paper, and it is hard to make it much shorter, because there are ten spaces in the spectral sequence, and it makes no sense to cut related matters up into different papers. Even at this length, the paper only touches on many subjects that require a fuller treatment. The hope is that this paper introduces techniques of general applicability. The paper is divided into eight sections and nine Appendices.

Section 1 introduces the paper with remarks about the relation between composite operators and BRST cohomology, and then introduces the Sections and Appendices.

Section 2 introduces BRST cohomology in a simple general way, and explains how the field equations, through the Zinn sources, can lead to a violation of the initial symmetry in certain cases.

In Section 3, we write down the BRST operator and action for the SUSY theory. There are two ways to do this (integrating the auxiliary or not), and we use them both as a check on each other.

In Section 4, we write down the Outfield solutions for chiral SUSY, and compare them with results known from past work. These Outfields are defined by equation (58), and they constitute one of the major results of this paper. To understand the composite Outfields, we need to use the fundamental Outfields in subsection 4.2.

Section 5 summarizes the result of the spectral sequence calculation in a summary way. This leads to the Appendices which contain the details of the spectral sequence machinery.

Section 6 contains a discussion of the mapping from the spectral sequence to the Cohomology space, and also has a summary of the space and a detailed summary of the spectral sequence space for the free massless case.

Section 7 contains some specific examples of solutions of the constraint equations for a specific version of the SSM, which we call the CSSM.

Section 8 is the conclusion. It comments on the examples from the SSM which are worked out in sections 6 and 7. The paper concludes with a discussion of possible extensions of the results.

Appendix A reviews some preliminary matters, including Counting operators.

Appendix B is a treatment of the Differentials and the Space for the Massless Free Chiral SUSY Theory.

Appendix C is a treatment of the Differentials and the Space for the Massless Free Chiral SUSY Theory.

Appendix D is a treatment of the Differentials and the Space for the Massless Free Chiral SUSY Theory.

Appendix E discusses the generation and solution of the separated irregular equations for Chiral SUSY Theory.

Appendix F summarizes what is known and not known about the cohomology space of the free massless theory.

Appendix G is a treatment of the Differentials and Spaces for the Interacting Massless Chiral SUSY Theory

Appendix H is a treatment of the Differentials and Spaces for the Massive Interacting Chiral SUSY Theory.

Appendix I summarizes the situation relating to the collapse of the three spectral sequences for the free massless, interacting and massive cases considered here.

Finally, there is a Table of Contents.

2 A Simple Introduction to BRST Cohomology with Emphasis on Three Kinds of Cohomology Terms

In this section we shall review the BRST formalism [1, 25] in a simple general way. Our purpose here is to indicate how the equations of motion fit into the BRST formalism through the Zinn sources, and how this can result in a violation of the original symmetry. This explanation is very important for the SUSY theory that we will look at next.

2.1 A Simple Introduction to the BRST Operator

Any quantum field theory that possesses some kind of invariance can be analyzed using BRST cohomology. Here is how this works in the simplest case. Suppose we have an action depending on some bosonic fields :

(1)

where is a local Lagrangian. Suppose that the action is invariant under some transformation

(2)

where acts on the fields locally:

(3)

We can always arrange for the parameter to be Grassmann odd, and for the transformation to be Grassmann odd and nilpotent[1, 25]:

(4)

The procedure now is to add the following new terms to the action:

(5)

Here is a Grassmann odd ‘Zinn Justin source’ coupled [30] to the variation in (3). So now we have a new action:

(6)

Then the following identity follows from (2) for this new action:

(7)

It can be shown using the Feynman path integral formulation of the field theory that (7) is the lowest term of the following identity:

(8)

In the above, is the one particle irreducible generating functional for the full quantum field theory. It has a loop expansion in powers of :

(9)

The one loop222Higher loops are related to also through ‘canonical transformations’ [5]. functional is governed by the cohomology333The cohomology of is defined below in equation (20) of the BRST operator , which is defined by the ‘square root’ of (7):

(10)

As a result of (7), the BRS operator in (10) is nilpotent

(11)

and this also carries through to more complicated444 If there are Grassmann odd fermions as well as the bosons , as happens in SUSY, the above carries through with appropriate changes. All the formulae get to be twice as big. Frequently one also needs some Zinn sources for variation of the ghosts to complete the nilpotence, which again increases the formulae in size. For this introductory discussion we shall imagine that we are dealing with the simplest case where the identity is simply (7). It is easy, but cumbersome, to give this explanation for the full SUSY theory below, but that interferes with the simplicity of the exposition of this part. examples.

2.2 Three Kinds of Terms in and the definition of the BRST cohomology space

Given our assumption that has the simple form (10), we can write

(12)

where

(13)
(14)
(15)

The two invariance transformations can be put together to define:

(16)

and then we have

(17)

The above division of in (17) is quite general, and it applies to all kinds of , including the one for SUSY555In general we might have the following instead of the form (12): (18) together with (19) instead of (16). For example, there might be ghosts which transform, and these could form part of . For our purposes here we do not need to use this more general form..

The local BRST cohomology space of is defined by

(20)

in the space of local integrated polynomials in the fields , sources , the ‘ghost’ parameters in and the derivative operator. In the above we define

(21)
(22)

The space is the space of terms which are of the form

(23)

where is a local polynomial in the fields, Zinns, ghosts and their derivatives. So the Lagrangian in (1) is an example of and the Actions in (1) and (5) are examples of .

The space is a factor space, which means that for each class of elements which satisfy , we choose one representative to be in . Two elements and are defined to be in the same class if there exists a such that:

(24)

2.3 Three Kinds of Possible Terms in the Cohomology

There are three kinds of terms that can arise in the cohomology. These three kinds of terms depend on the division of the equations into the three parts in (12). This division is the same when one includes the complications of fermions properly.

  1. Invariant Terms of Type that do not use the Zinn sources: The simplest situation occurs when we have invariants which satisfy:

    (25)

    Since is assumed to be in the cohomology space, it also satisfies:

    (26)

    Hence, because of (12), it must also satisfy

    (27)

    This equation can be satisfied in a trivial way by having independent of and any other quantities than the fields, and then we have:

    (28)

    and

    (29)
  2. Non-invariant Terms of Type that use the field equations:

    Suppose that we have a term which is not invariant under the field variation operator:

    (30)

    Since is assumed to be in the cohomology space, it satisfies:

    (31)

    This cannot be independent of . Using (12), we see that it must also satisfy

    (32)

    The most common way for this to happen is

    (33)
  3. Non-invariant Terms of Type that do not use the field equations:

    It is also possible for an non-invariant term in the cohomology space that satisfies (30) and (32), to satisfy:

    (34)

    instead of (33).

It might appear that the above distinctions are not real, because the cohomology is defined only up to classes, as mentioned above. One can often add a boundary term to an invariant term that makes it look like a non-invariant term . However the distinction is real, because the inverse statement is not always true:

  1. Invariant Terms of Type that do not use the Zinn sources: If it is possible to find a boundary term that can be added to a given non-invariant term to make it into an invariant , then the relevant term in the cohomology is a genuine invariant term . This is the usual case for gauge theories or gravity where SUSY is not present.

  2. Non-invariant Terms of Type that use the field equations: If it is impossible to find any boundary that can be added to a given non-invariant term

    1. to make it into an invariant ,

    2. or a non-invariant term satisfying equation (34),

    then the relevant term in the cohomology is a genuine non-invariant term of the type of the type that uses the field equations. These satisfy (33). SUSY has many examples of this.

  3. Non-invariant Terms of Type that do not use the field equations: If it is

    1. impossible to find a boundary term that can be added to a given non-invariant term to make it into an invariant , but

    2. possible to find a boundary term that can be added to a given non-invariant term to make it into a non-invariant term satisfying the equation (34),

    then it is a genuine non-invariant term of the type that does not use the field equations. This happens in several examples for SUSY, for operators that have ghost charge .

We will see that SUSY has all three of these kinds of terms. The second type of term, the Non-invariant Terms of Type that use the field equations, are the ones that generate the Outfields. These non-invariant terms have explicit factors of the parameters of superspace in them.

3 The Action and the operators for Chiral SUSY with spontaneous breaking of internal symmetry

There are two possible operators for chiral SUSY in 3+1 dimensions, depending on whether one functionally integrates the auxiliary, or not.

The formulation of where the auxiliary is not integrated we will call (short for ). This formulation where the auxiliary is not integrated can be written in the usual superfield formulation, but we will need to write it out in components to solve the cohomology. Then we can write the result again in terms of superfields, except that we will find there is some tricky business in doing that, so that manifest supersymmetry is not present for the cohomology.

The formulation of where the auxiliary is integrated we will call (short for ). This formulation uses only the physical scalar and spinor particles. However, the superfields are lost at the beginning, because they require auxiliary fields. We will see that the cohomology of generates its own set of objects that behave much like superfields. But again manifest supersymmetry is not present for the cohomology.

We will find that the two versions and have isomophic cohomology spaces:

(35)

This is easy to prove in fact, because they yield exactly the same spectral sequence , and so, as we shall explain later,

(36)

Now we shall set out the two operators and the corresponding actions.

3.1 The Superfield Formulation

3.1.1 Superfield Version of the Superfield Formulation

We start with the superfield approach [26, 14, 28], which has the following action:

(37)

The Matter superfields have the component forms:

(38)

and the Zinn-Justin superfields have the component forms:

(39)

Here the supersymmety variation . Using standard methods666We drop a term in this operator, and its analogues. This causes no problems. we can derive the BRST operator in superspace form:

(40)

The action and the BRST operator satisfy:

(41)

In this paper, it will be assumed [23] that there is a vacuum expectation value

(42)

which satisfies:

(43)

Then the shift:

(44)

serves to remove the terms from the action and the operator .

3.1.2 Component Version of the Superfield Formulation

After this shift, the action can be written in components as:

(45)

The operator in components, using the language above in subsection (2.3) is:

(46)

where

(47)
(48)

and

(49)
(50)

where we use the abbreviation:

(51)

It would be natural to add rigid internal symmetry transformations, or local internal symmetry transformations coupled to Yang-Mills supersymmetry, to the above. Those additions will not be considered in this paper because we already have enough complexity for the time being. Also it should be noted that one can miss interesting developments if one is too restrictive in choosing the that one is looking at.

3.2 The Physical Formulation

3.2.1 Integration of the auxiliary fields

The auxiliary fields in a supersymmetric theory are not physical. They appear in linear and quadratic terms in the action. They do not propagate, because there are no derivatives in the quadratic terms containing them in the action. As a result, it is possible to integrate them out of the theory in an exact non-perturbative way, by simply completing the square, performing a shift in the variable, and then performing the Gaussian integration of in the Feynman path integral. The dependent terms disappear into a multiplicative constant and we are left with a new quadratic term made of the propagating fields (and sources). In the above case this new quadratic term is

(52)

where is defined by (55). This new term is the square of the term that multiplied the linear term in the auxiliary. When this integration is done, supersymmetry ceases to be manifest. For example, superfields like (38) cease to be applicable, because they use the auxiliary field . The supersymmetry is still in the theory however, and one way to understand it is by constructing the nilpotent BRST operator below in (54), and then solving for its local BRST cohomology.

Since for this case we do not have the auxiliary fields in the generating functionals, we do not include the source for the variation of the auxiliary either.

3.2.2 Component Version of the Physical Formulation

This results in the following action

(53)

and the derived operator is

(54)

The composite field is the same as the value of the equation of motion above in (51), with complex conjugate:

(55)

and we have

(56)

4 Quick Summary of the Old and New Results for the Cohomology Space for Chiral SUSY in 3+1 Dimensions

4.1 Some Old Results and Some New Results

In other papers [6, 11, 12], it was shown that the BRS cohomology of the chiral superfield, without the Zinn terms, contained terms777There are also terms with derivatives described in those papers, and the spin must be maximized, as shown in those papers. In addition there are non-chiral terms that we will discuss briefly later. that look like

(57)

The antichiral superspace integral in (57) just picks out supersymmetric invariant type terms. In this paper, we find that the generalization888Again there are also terms with derivatives, and the spin must be maximized, as proved below. of the result (57) to the case where the Zinn sources are included takes the very similar form:

(58)

The only difference from (57) is the addition of the terms in the middle of (58). These are the terms which make (58) into an Outfield, because they contain explicit factors of . These are examples of the Non-invariant Terms of Type that use the field equations referred to in section (2.3). This is one of the fundamental results of the present paper. It is established by using spectral sequences, as will be explained below.

Our first task will be to explain what is meant by the in (58). These are antichiral pseudosuperfields, and sometimes we call them dotspinors or fundamental Outfields too. They will be explained in subsection 4.2. The expression inside the integral in (58) transforms under the appropriate like an antichiral superfield:

(59)
(60)

The superspace integral in (58) just picks out pseudosupersymmetric invariant type terms. The highest component transforms as a total derivative, and the integral of that highest component generates a class of the integrated cohomology space .

(61)

The ghost number of these terms in the action ranges from zero to positive infinity. The complex conjugate terms are also in for both (57) and (58).

The general form (58) is only true for the free massless theory. For the case where there are interactions, there are constraints which remove some of the terms in (58), or else combine them together. When masses are added in addition, there are new terms here proportional to masses, and there are new constraints too. The details of the constraints will be derived using the spectral sequence.

4.2 The Fundamental Pseudosuperfields

In the next subsubsections we explain what is meant by these expressions when used in (58). It should be mentioned that this is one of the tricky steps when using the spectral sequence. The forms given in the next subsections are not easy to find, and the best way to find them seems to be to simply guess what they have to be, given the information that one has about them from the spectral sequence, which will be explained below.

4.2.1 Fundamental Expressions in the superfield approach, with

In the superfield approach, the fundamental expressions to be used in (58), for construction of are made from the superfields that we start with:

(62)
(63)

Both of these are chiral:

(64)
(65)

The first expression is the usual superfield. The second pseudosuperfield (63) is new. It is constructed out of the Zinn field and a rather strange combination of the chiral projector on with an explicit factor of the superspace coordinate . It is natural to wonder why an explicit factor of arises here. This happens because the combination of the variation of , which brings in the equation of motion term is accompanied by an explicit factor of the ghost , and this gets compensated by the factor of , as we shall see in detail below.

It is remarkable however that the cohomology is giving rise to these explicit factors of , because they do remove the manifest superspace invariance of the theory, even in this approach that strives to keep the superspace invariance by keeping the superfields, and not integrating the auxiliary field. Of course, this explicit superspace breaking does not affect the action that we started with, because these objects do not occur in the action. That raises a question of course–can we put them into a new action of some kind? That question will be dealt with in another paper.

The transformations induced by are summarized999The equation (67) is valid only for the free massless theory. The more general case is a little more complicated. See equation (83). by the following equations:

(66)
(67)

The latter equation is easy to verify explicitly, and this is important. One gets