6.3 Gauge Flips and Groves in the Standard Model
6.3.2 Gauge Motions
In [6], such an analysis has been performed for tree level amplitudes with all external lines corresponding to fermions, ignoring the contributions of the Higgs boson.3 In the remainder of this section, we shall now generalize this analysis to the case of SM forests consisting of diagrams with a non-vanishing number of loops.
Parallel Motions
The second higher level operation we consider, is the motion of neutral bosons and gluons along threads. In this respect, we first note that H-lines can be moved freely past otherH-lines alongZ0 orW-threads. This is a simple conse-quence of the flips permuting two externalH-lines in (6.9c) and (6.9f). In the latter case, the relevant flip is given explicitely by
→ . (6.14)
We say that H-lines can be moved parallel to Z0 and W-threads and denote this asHkZ0 andH kW, respectively.
In an analogous way, according to the relevant flips in (6.9a), (6.10b), (6.10d), (6.10e), and (6.10g),Z0-lines can be moved parallel toW and fermion threads, i. e. we haveZ0kW andZ0kf, wheref represents a general SM fermion. We display two example flips:
→ (6.15)
→ (6.16)
The same motions are possible for photon lines, with the one exception that a photon may not be moved across a neutrino line. Keeping this exception in mind, we thus getγkW andγkf.
In addition, we have the parallel motionsGkqof gluons along quark threads, following from (6.10g) and (6.10f).
Perpendicular Motions
Parallel motions of neutral bosons along a thread ` will necessary terminate when ` ends. If ` happens to end at a second thread `0 (which, of course, implies that ` is a boson line), it will often be possible to move the neutral boson from` onto`0.
For instance,Z0-lines can be moved from fermion threads onto W threads andvice versa, as the following flip in (6.10b) shows:
→
,
(6.17)
We say that Z0-lines can be moved perpendicular from fermion threads onto W-threads, andvice versa, denoting this asZ0⊥f W. Again, the same applies
for photon lines, i. e. there is a perpendicular motion γ⊥f W. Note that there is no restriction for photon lines if the photon is moved from the fermion thread onto the W-thread. For if the last line of the fermion thread was a neutrino line, the photon would not have been there in the first place.
ForH-lines, there are three perpendicular motions. Instead of displaying the relevant flips explicitely, we will refer to the respective subsets. Thus, (6.10c) impliesH ⊥f W, (6.10e) impliesH ⊥f Z0, and (6.9b) impliesH⊥Z0W.
Observe that we have chosen to define the perpendicular motions ofH-lines involving fermions as motions from the fermion thread onto the gauge boson thread. The reverse motions, from a gauge boson thread onto a fermion thread, are possible only for a massive fermion thread. As we shall later see, the lack of this gauge motion for massless fermions leads to additional structure of the SM forests with massless fermions.
Finally, for gluons we have the perpendicular motionG⊥qG, which follows from (6.2). Note that the interpretation is slightly different in this case. In particular, it does not actully make sense to speak of a gluon thread, because color can flow into all three directions at a triple gluon vertex. However, it is certainly true that a gluon coupled to quark thread can always be moved onto a neighboring gluonline. We shall understandG⊥qGin this way.
Crossover Motions
There are yet more intricate motions of neutral bosons. In particular, while a Z0-line certainly cannot move parallel to a Z0-thread, it can use aZ0-line to switch from aW-thread to anotherW-thread or a fermion thread. For the case of twoW-threads, the relevant flips come from (6.9d) and (6.9a):
→ → →
(6.18) We say that theZ0-line is movedcrossover fromW-thread toW-thread, using a Z0-line as abridge, and denote this asZ0×W Z0W.
For the case of one fermion thread and oneW-thread we need (6.10a) and the perpendicular motionZ0⊥f W:
→ → →
(6.19) Thus, we haveZ0×f Z0W. Again, for photons the identical motionsγ×W Z0W andZ0×f Z0W are possible.
Finally, using (6.9f), we can show that H has the crossover motion H × Z0HZ0. That is, a H-line can use anotherH-line as a bridge fromZ0-thread
toZ0-thread:
→ → →
(6.20) Note that H ×W Z0W and H ×f Z0W as well as H ×f Z0f, Z0×f W f, and G×qGq, although valid motions with the effect of a crossover motion, are actually sequences of perpendicular motions. In contrast, the real crossover motions displayed above cannot be represented as sequences of perpendicular motions only.
Crossing Threads
The last two operations we shall need in order to discuss the structure of SM forests are concerned with the crossing of threads. Actually, we have seen such an operation at work in the discussion of crossover motions. To see this, take a look back at (6.18). In the first diagram, we recognize twoW-threads, connected by a Z0-line. The flip to the second diagram then breaks up bothW-threads and reconnects them in a different way. In particular, if we take the arrows to indicate a direction along the threads, the tail of the first thread is joined onto the head of the second thread, and vice versa. We say that the twoW-threads have beencrossed.
To appreciate the significance of thread crossing, suppose the diagrams in (6.18) were contributions to the amplitude (or, rather, connected Green’s function) for the processW+W− →W+W−Z0. In that case, beginning and end of theW threads are distinguished by the momenta of the external W bosons.
Thread crossing then realizes the two possible ways of connecting the external W-lines into threads. As we shall see below, this generalizes to the case of more than twoW-threads.
Thread crossing is also possible for twoZ0-threads connected by anH-line, which follows readily from (6.9e). In fact, since a Z0-thread has no intrinsic direction, the broken up threads can be reconnected in all possible ways.
Absorbing Closed Threads
Thread crossing can be used to absorb a closedW-thread orZ0-thread, i. e. a W-loop orZ0-loop, into another W-thread orZ0-thread, respectively. We list this case separately, because it is very important for the structure of the SM forests.
As an example, consider the case of a closedW-thread:
→ (6.21)
In an analogous way, closed Z0-threads can be absorbed into intoZ0-threads.
Note that the absorbing thread can be either an open thread or a closed thread.
The possibility of absorbing closed threads clearly shows that the number of W-loops and Z0-loops is not invariant under gauge flips. Of course, the total number of loops remains fixed.
Elimination ofH4 Vertices
This gauge motion answers the question under which conditions a H4 vertex can be expanded by a gauge flip. This is important, because the expansion of a H4vertex as afour-pointsubdiagram into a subdiagram with twoH3vertices is not a gauge flip. Consequently, diagrams with different numbers ofH4 vertices would be candidates for separate invariant subsets, unless the five-point gauge flips can be used to perform this transformation indirectly.
This is, however, the case, as is apparent from (6.11) and (6.12). For def-initeness, we consider the former five-point subdiagrams with two W bosons.
We first use a five-point flip to replace theH4vertex by aW+W−H2vertex:
→ (6.22)
The W+W−H2 vertex can then be expanded by a gauge flip of a four-point subdiagram:
→ (6.23)
In the same way, theH4vertex can be expanded if theW-thread is replaced by a Z0-thread. Thus, if aH4 vertex is coupled to a W-thread orZ0-thread, the three H-lines not connected to the thread can be absorbed onto the thread by a gauge motion.
Summary of Gauge Motions
Let us summarize the set of valid gauge motions, i. e. the possible transforma-tions of Feynman diagrams achievable by sequences of gauge flips:
1. Elimination of internal H-lines, except for H-lines connecting two Z0 -threads.
2. The parallel motions H k Z0, H k W, Z0 k W, Z0 k f, γ k W, γkf 6=ν, Gkq.
3. The perpendicular motions Z0 ⊥f W, γ ⊥f W, H ⊥ f W, H ⊥ f Z0, H⊥Z0W, G⊥qG.
4. The crossover motionsZ0×W Z0W, Z0×f Z0W, γ×W Z0W, γ×f Z0W, H×Z0HZ0.
5. Thread crossing of two W-threads or twoZ0-threads.
6. Absorption of closedW-threads andZ0-threads.
7. Elimination ofH4vertices coupled to W-threads orZ0-threads.